LMFDB - Embedded newform 4160.2.a.bu.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4160,2,Mod(1,4160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4160.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4160 = 2^{6} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4160.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4,0,0,0,4,0,0,0,4,0,0,0,-8,0,0,0,16,0,0,0,4,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$33.2177672409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.84776$ of defining polynomial
Character	$\chi$	$=$	4160.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61313 q^{3} +1.00000 q^{5} +3.69552 q^{7} +3.82843 q^{9} +1.08239 q^{11} +1.00000 q^{13} +2.61313 q^{15} -7.65685 q^{17} +1.08239 q^{19} +9.65685 q^{21} +5.67459 q^{23} +1.00000 q^{25} +2.16478 q^{27} +8.82843 q^{29} +8.47343 q^{31} +2.82843 q^{33} +3.69552 q^{35} -10.4853 q^{37} +2.61313 q^{39} -2.00000 q^{41} -6.94269 q^{43} +3.82843 q^{45} -11.0866 q^{47} +6.65685 q^{49} -20.0083 q^{51} -9.31371 q^{53} +1.08239 q^{55} +2.82843 q^{57} +9.37011 q^{59} +8.82843 q^{61} +14.1480 q^{63} +1.00000 q^{65} +13.2513 q^{67} +14.8284 q^{69} +11.5349 q^{71} +0.828427 q^{73} +2.61313 q^{75} +4.00000 q^{77} -8.28772 q^{79} -5.82843 q^{81} +3.69552 q^{83} -7.65685 q^{85} +23.0698 q^{87} -11.6569 q^{89} +3.69552 q^{91} +22.1421 q^{93} +1.08239 q^{95} -5.31371 q^{97} +4.14386 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{9} + 4 q^{13} - 8 q^{17} + 16 q^{21} + 4 q^{25} + 24 q^{29} - 8 q^{37} - 8 q^{41} + 4 q^{45} + 4 q^{49} + 8 q^{53} + 24 q^{61} + 4 q^{65} + 48 q^{69} - 8 q^{73} + 16 q^{77} - 12 q^{81}+ \cdots + 24 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^9 + 4 * q^13 - 8 * q^17 + 16 * q^21 + 4 * q^25 + 24 * q^29 - 8 * q^37 - 8 * q^41 + 4 * q^45 + 4 * q^49 + 8 * q^53 + 24 * q^61 + 4 * q^65 + 48 * q^69 - 8 * q^73 + 16 * q^77 - 12 * q^81 - 8 * q^85 - 24 * q^89 + 32 * q^93 + 24 * 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
$3$	2.61313	1.50869	0.754344	−	0.656479i	$-0.227955\pi$
$3$	2.61313	1.50869	0.754344	+	0.656479i	$0.227955\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.69552	1.39677	0.698387	−	0.715720i	$-0.253901\pi$
$7$	3.69552	1.39677	0.698387	+	0.715720i	$0.253901\pi$
$8$	0	0
$9$	3.82843	1.27614
$10$	0	0
$11$	1.08239	0.326354	0.163177	−	0.986597i	$-0.447826\pi$
$11$	1.08239	0.326354	0.163177	+	0.986597i	$0.447826\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.61313	0.674706
$16$	0	0
$17$	−7.65685	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$17$	−7.65685	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$18$	0	0
$19$	1.08239	0.248318	0.124159	−	0.992262i	$-0.460377\pi$
$19$	1.08239	0.248318	0.124159	+	0.992262i	$0.460377\pi$
$20$	0	0
$21$	9.65685	2.10730
$22$	0	0
$23$	5.67459	1.18323	0.591617	−	0.806219i	$-0.298490\pi$
$23$	5.67459	1.18323	0.591617	+	0.806219i	$0.298490\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.16478	0.416613
$28$	0	0
$29$	8.82843	1.63940	0.819699	−	0.572795i	$-0.194141\pi$
$29$	8.82843	1.63940	0.819699	+	0.572795i	$0.194141\pi$
$30$	0	0
$31$	8.47343	1.52187	0.760936	−	0.648827i	$-0.224740\pi$
$31$	8.47343	1.52187	0.760936	+	0.648827i	$0.224740\pi$
$32$	0	0
$33$	2.82843	0.492366
$34$	0	0
$35$	3.69552	0.624657
$36$	0	0
$37$	−10.4853	−1.72377	−0.861885	−	0.507104i	$-0.830716\pi$
$37$	−10.4853	−1.72377	−0.861885	+	0.507104i	$0.830716\pi$
$38$	0	0
$39$	2.61313	0.418435
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−6.94269	−1.05875	−0.529376	−	0.848388i	$-0.677574\pi$
$43$	−6.94269	−1.05875	−0.529376	+	0.848388i	$0.677574\pi$
$44$	0	0
$45$	3.82843	0.570708
$46$	0	0
$47$	−11.0866	−1.61714	−0.808570	−	0.588400i	$-0.799758\pi$
$47$	−11.0866	−1.61714	−0.808570	+	0.588400i	$0.799758\pi$
$48$	0	0
$49$	6.65685	0.950979
$50$	0	0
$51$	−20.0083	−2.80173
$52$	0	0
$53$	−9.31371	−1.27934	−0.639668	−	0.768651i	$-0.720928\pi$
$53$	−9.31371	−1.27934	−0.639668	+	0.768651i	$0.720928\pi$
$54$	0	0
$55$	1.08239	0.145950
$56$	0	0
$57$	2.82843	0.374634
$58$	0	0
$59$	9.37011	1.21988	0.609942	−	0.792446i	$-0.291193\pi$
$59$	9.37011	1.21988	0.609942	+	0.792446i	$0.291193\pi$
$60$	0	0
$61$	8.82843	1.13036	0.565182	−	0.824966i	$-0.308806\pi$
$61$	8.82843	1.13036	0.565182	+	0.824966i	$0.308806\pi$
$62$	0	0
$63$	14.1480	1.78248
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	13.2513	1.61891	0.809454	−	0.587183i	$-0.199763\pi$
$67$	13.2513	1.61891	0.809454	+	0.587183i	$0.199763\pi$
$68$	0	0
$69$	14.8284	1.78513
$70$	0	0
$71$	11.5349	1.36894	0.684470	−	0.729041i	$-0.260034\pi$
$71$	11.5349	1.36894	0.684470	+	0.729041i	$0.260034\pi$
$72$	0	0
$73$	0.828427	0.0969601	0.0484800	−	0.998824i	$-0.484562\pi$
$73$	0.828427	0.0969601	0.0484800	+	0.998824i	$0.484562\pi$
$74$	0	0
$75$	2.61313	0.301738
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	−8.28772	−0.932441	−0.466221	−	0.884668i	$-0.654385\pi$
$79$	−8.28772	−0.932441	−0.466221	+	0.884668i	$0.654385\pi$
$80$	0	0
$81$	−5.82843	−0.647603
$82$	0	0
$83$	3.69552	0.405636	0.202818	−	0.979216i	$-0.434990\pi$
$83$	3.69552	0.405636	0.202818	+	0.979216i	$0.434990\pi$
$84$	0	0
$85$	−7.65685	−0.830502
$86$	0	0
$87$	23.0698	2.47334
$88$	0	0
$89$	−11.6569	−1.23562	−0.617812	−	0.786326i	$-0.711981\pi$
$89$	−11.6569	−1.23562	−0.617812	+	0.786326i	$0.711981\pi$
$90$	0	0
$91$	3.69552	0.387396
$92$	0	0
$93$	22.1421	2.29603
$94$	0	0
$95$	1.08239	0.111051
$96$	0	0
$97$	−5.31371	−0.539525	−0.269763	−	0.962927i	$-0.586945\pi$
$97$	−5.31371	−0.539525	−0.269763	+	0.962927i	$0.586945\pi$
$98$	0	0
$99$	4.14386	0.416474
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−1.71644	−0.169126	−0.0845631	−	0.996418i	$-0.526949\pi$
$103$	−1.71644	−0.169126	−0.0845631	+	0.996418i	$0.526949\pi$
$104$	0	0
$105$	9.65685	0.942412
$106$	0	0
$107$	−3.88123	−0.375212	−0.187606	−	0.982244i	$-0.560073\pi$
$107$	−3.88123	−0.375212	−0.187606	+	0.982244i	$0.560073\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−27.3994	−2.60063
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	5.67459	0.529159
$116$	0	0
$117$	3.82843	0.353938
$118$	0	0
$119$	−28.2960	−2.59389
$120$	0	0
$121$	−9.82843	−0.893493
$122$	0	0
$123$	−5.22625	−0.471235
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.94269	−0.616065	−0.308032	−	0.951376i	$-0.599670\pi$
$127$	−6.94269	−0.616065	−0.308032	+	0.951376i	$0.599670\pi$
$128$	0	0
$129$	−18.1421	−1.59733
$130$	0	0
$131$	−17.8435	−1.55900	−0.779499	−	0.626404i	$-0.784526\pi$
$131$	−17.8435	−1.55900	−0.779499	+	0.626404i	$0.784526\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	2.16478	0.186315
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	3.95815	0.335726	0.167863	−	0.985810i	$-0.446313\pi$
$139$	3.95815	0.335726	0.167863	+	0.985810i	$0.446313\pi$
$140$	0	0
$141$	−28.9706	−2.43976
$142$	0	0
$143$	1.08239	0.0905142
$144$	0	0
$145$	8.82843	0.733161
$146$	0	0
$147$	17.3952	1.43473
$148$	0	0
$149$	0.343146	0.0281116	0.0140558	−	0.999901i	$-0.495526\pi$
$149$	0.343146	0.0281116	0.0140558	+	0.999901i	$0.495526\pi$
$150$	0	0
$151$	3.24718	0.264251	0.132126	−	0.991233i	$-0.457820\pi$
$151$	3.24718	0.264251	0.132126	+	0.991233i	$0.457820\pi$
$152$	0	0
$153$	−29.3137	−2.36987
$154$	0	0
$155$	8.47343	0.680602
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	−24.3379	−1.93012
$160$	0	0
$161$	20.9706	1.65271
$162$	0	0
$163$	−4.59220	−0.359689	−0.179844	−	0.983695i	$-0.557559\pi$
$163$	−4.59220	−0.359689	−0.179844	+	0.983695i	$0.557559\pi$
$164$	0	0
$165$	2.82843	0.220193
$166$	0	0
$167$	−23.3324	−1.80552	−0.902759	−	0.430148i	$-0.858462\pi$
$167$	−23.3324	−1.80552	−0.902759	+	0.430148i	$0.858462\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.14386	0.316889
$172$	0	0
$173$	−14.9706	−1.13819	−0.569095	−	0.822272i	$-0.692706\pi$
$173$	−14.9706	−1.13819	−0.569095	+	0.822272i	$0.692706\pi$
$174$	0	0
$175$	3.69552	0.279355
$176$	0	0
$177$	24.4853	1.84043
$178$	0	0
$179$	8.65914	0.647214	0.323607	−	0.946192i	$-0.395104\pi$
$179$	8.65914	0.647214	0.323607	+	0.946192i	$0.395104\pi$
$180$	0	0
$181$	20.1421	1.49715	0.748577	−	0.663048i	$-0.230738\pi$
$181$	20.1421	1.49715	0.748577	+	0.663048i	$0.230738\pi$
$182$	0	0
$183$	23.0698	1.70537
$184$	0	0
$185$	−10.4853	−0.770893
$186$	0	0
$187$	−8.28772	−0.606058
$188$	0	0
$189$	8.00000	0.581914
$190$	0	0
$191$	−11.7206	−0.848073	−0.424037	−	0.905645i	$-0.639387\pi$
$191$	−11.7206	−0.848073	−0.424037	+	0.905645i	$0.639387\pi$
$192$	0	0
$193$	−5.31371	−0.382489	−0.191245	−	0.981542i	$-0.561252\pi$
$193$	−5.31371	−0.382489	−0.191245	+	0.981542i	$0.561252\pi$
$194$	0	0
$195$	2.61313	0.187130
$196$	0	0
$197$	9.31371	0.663574	0.331787	−	0.943354i	$-0.392348\pi$
$197$	9.31371	0.663574	0.331787	+	0.943354i	$0.392348\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	34.6274	2.44243
$202$	0	0
$203$	32.6256	2.28987
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	21.7248	1.50998
$208$	0	0
$209$	1.17157	0.0810394
$210$	0	0
$211$	11.7206	0.806880	0.403440	−	0.915006i	$-0.367814\pi$
$211$	11.7206	0.806880	0.403440	+	0.915006i	$0.367814\pi$
$212$	0	0
$213$	30.1421	2.06531
$214$	0	0
$215$	−6.94269	−0.473488
$216$	0	0
$217$	31.3137	2.12571
$218$	0	0
$219$	2.16478	0.146283
$220$	0	0
$221$	−7.65685	−0.515056
$222$	0	0
$223$	−5.48888	−0.367563	−0.183781	−	0.982967i	$-0.558834\pi$
$223$	−5.48888	−0.367563	−0.183781	+	0.982967i	$0.558834\pi$
$224$	0	0
$225$	3.82843	0.255228
$226$	0	0
$227$	−20.6424	−1.37008	−0.685041	−	0.728504i	$-0.740216\pi$
$227$	−20.6424	−1.37008	−0.685041	+	0.728504i	$0.740216\pi$
$228$	0	0
$229$	−20.6274	−1.36310	−0.681549	−	0.731772i	$-0.738693\pi$
$229$	−20.6274	−1.36310	−0.681549	+	0.731772i	$0.738693\pi$
$230$	0	0
$231$	10.4525	0.687724
$232$	0	0
$233$	5.31371	0.348113	0.174056	−	0.984736i	$-0.444313\pi$
$233$	5.31371	0.348113	0.174056	+	0.984736i	$0.444313\pi$
$234$	0	0
$235$	−11.0866	−0.723207
$236$	0	0
$237$	−21.6569	−1.40676
$238$	0	0
$239$	−17.6578	−1.14219	−0.571095	−	0.820884i	$-0.693481\pi$
$239$	−17.6578	−1.14219	−0.571095	+	0.820884i	$0.693481\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	−21.7248	−1.39364
$244$	0	0
$245$	6.65685	0.425291
$246$	0	0
$247$	1.08239	0.0688710
$248$	0	0
$249$	9.65685	0.611978
$250$	0	0
$251$	−5.22625	−0.329878	−0.164939	−	0.986304i	$-0.552743\pi$
$251$	−5.22625	−0.329878	−0.164939	+	0.986304i	$0.552743\pi$
$252$	0	0
$253$	6.14214	0.386153
$254$	0	0
$255$	−20.0083	−1.25297
$256$	0	0
$257$	7.65685	0.477621	0.238811	−	0.971066i	$-0.423242\pi$
$257$	7.65685	0.477621	0.238811	+	0.971066i	$0.423242\pi$
$258$	0	0
$259$	−38.7485	−2.40772
$260$	0	0
$261$	33.7990	2.09210
$262$	0	0
$263$	0.448342	0.0276459	0.0138230	−	0.999904i	$-0.495600\pi$
$263$	0.448342	0.0276459	0.0138230	+	0.999904i	$0.495600\pi$
$264$	0	0
$265$	−9.31371	−0.572137
$266$	0	0
$267$	−30.4608	−1.86417
$268$	0	0
$269$	15.6569	0.954615	0.477308	−	0.878736i	$-0.341613\pi$
$269$	15.6569	0.954615	0.477308	+	0.878736i	$0.341613\pi$
$270$	0	0
$271$	−17.6578	−1.07264	−0.536318	−	0.844016i	$-0.680185\pi$
$271$	−17.6578	−1.07264	−0.536318	+	0.844016i	$0.680185\pi$
$272$	0	0
$273$	9.65685	0.584459
$274$	0	0
$275$	1.08239	0.0652707
$276$	0	0
$277$	18.9706	1.13983	0.569915	−	0.821703i	$-0.306976\pi$
$277$	18.9706	1.13983	0.569915	+	0.821703i	$0.306976\pi$
$278$	0	0
$279$	32.4399	1.94213
$280$	0	0
$281$	−32.6274	−1.94639	−0.973194	−	0.229985i	$-0.926132\pi$
$281$	−32.6274	−1.94639	−0.973194	+	0.229985i	$0.926132\pi$
$282$	0	0
$283$	7.83938	0.466003	0.233001	−	0.972476i	$-0.425145\pi$
$283$	7.83938	0.466003	0.233001	+	0.972476i	$0.425145\pi$
$284$	0	0
$285$	2.82843	0.167542
$286$	0	0
$287$	−7.39104	−0.436279
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	−13.8854	−0.813976
$292$	0	0
$293$	14.4853	0.846239	0.423120	−	0.906074i	$-0.360935\pi$
$293$	14.4853	0.846239	0.423120	+	0.906074i	$0.360935\pi$
$294$	0	0
$295$	9.37011	0.545549
$296$	0	0
$297$	2.34315	0.135963
$298$	0	0
$299$	5.67459	0.328170
$300$	0	0
$301$	−25.6569	−1.47884
$302$	0	0
$303$	26.1313	1.50120
$304$	0	0
$305$	8.82843	0.505514
$306$	0	0
$307$	2.42742	0.138540	0.0692700	−	0.997598i	$-0.477933\pi$
$307$	2.42742	0.138540	0.0692700	+	0.997598i	$0.477933\pi$
$308$	0	0
$309$	−4.48528	−0.255159
$310$	0	0
$311$	−0.896683	−0.0508462	−0.0254231	−	0.999677i	$-0.508093\pi$
$311$	−0.896683	−0.0508462	−0.0254231	+	0.999677i	$0.508093\pi$
$312$	0	0
$313$	11.6569	0.658884	0.329442	−	0.944176i	$-0.393139\pi$
$313$	11.6569	0.658884	0.329442	+	0.944176i	$0.393139\pi$
$314$	0	0
$315$	14.1480	0.797151
$316$	0	0
$317$	−8.14214	−0.457308	−0.228654	−	0.973508i	$-0.573432\pi$
$317$	−8.14214	−0.457308	−0.228654	+	0.973508i	$0.573432\pi$
$318$	0	0
$319$	9.55582	0.535023
$320$	0	0
$321$	−10.1421	−0.566079
$322$	0	0
$323$	−8.28772	−0.461141
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	36.5838	2.02309
$328$	0	0
$329$	−40.9706	−2.25878
$330$	0	0
$331$	1.45381	0.0799087	0.0399543	−	0.999202i	$-0.487279\pi$
$331$	1.45381	0.0799087	0.0399543	+	0.999202i	$0.487279\pi$
$332$	0	0
$333$	−40.1421	−2.19978
$334$	0	0
$335$	13.2513	0.723998
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	−5.22625	−0.283851
$340$	0	0
$341$	9.17157	0.496669
$342$	0	0
$343$	−1.26810	−0.0684710
$344$	0	0
$345$	14.8284	0.798336
$346$	0	0
$347$	−0.0769232	−0.00412946	−0.00206473	−	0.999998i	$-0.500657\pi$
$347$	−0.0769232	−0.00412946	−0.00206473	+	0.999998i	$0.500657\pi$
$348$	0	0
$349$	34.9706	1.87193	0.935966	−	0.352091i	$-0.114529\pi$
$349$	34.9706	1.87193	0.935966	+	0.352091i	$0.114529\pi$
$350$	0	0
$351$	2.16478	0.115548
$352$	0	0
$353$	6.48528	0.345177	0.172588	−	0.984994i	$-0.444787\pi$
$353$	6.48528	0.345177	0.172588	+	0.984994i	$0.444787\pi$
$354$	0	0
$355$	11.5349	0.612209
$356$	0	0
$357$	−73.9411	−3.91338
$358$	0	0
$359$	−1.97908	−0.104452	−0.0522258	−	0.998635i	$-0.516632\pi$
$359$	−1.97908	−0.104452	−0.0522258	+	0.998635i	$0.516632\pi$
$360$	0	0
$361$	−17.8284	−0.938338
$362$	0	0
$363$	−25.6829	−1.34800
$364$	0	0
$365$	0.828427	0.0433619
$366$	0	0
$367$	−21.3533	−1.11464	−0.557318	−	0.830299i	$-0.688170\pi$
$367$	−21.3533	−1.11464	−0.557318	+	0.830299i	$0.688170\pi$
$368$	0	0
$369$	−7.65685	−0.398600
$370$	0	0
$371$	−34.4190	−1.78694
$372$	0	0
$373$	−29.3137	−1.51781	−0.758903	−	0.651204i	$-0.774264\pi$
$373$	−29.3137	−1.51781	−0.758903	+	0.651204i	$0.774264\pi$
$374$	0	0
$375$	2.61313	0.134941
$376$	0	0
$377$	8.82843	0.454687
$378$	0	0
$379$	35.5014	1.82358	0.911791	−	0.410654i	$-0.134699\pi$
$379$	35.5014	1.82358	0.911791	+	0.410654i	$0.134699\pi$
$380$	0	0
$381$	−18.1421	−0.929450
$382$	0	0
$383$	27.6620	1.41346	0.706731	−	0.707482i	$-0.250169\pi$
$383$	27.6620	1.41346	0.706731	+	0.707482i	$0.250169\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	−26.5796	−1.35112
$388$	0	0
$389$	12.3431	0.625822	0.312911	−	0.949782i	$-0.398696\pi$
$389$	12.3431	0.625822	0.312911	+	0.949782i	$0.398696\pi$
$390$	0	0
$391$	−43.4495	−2.19734
$392$	0	0
$393$	−46.6274	−2.35204
$394$	0	0
$395$	−8.28772	−0.417000
$396$	0	0
$397$	23.4558	1.17722	0.588608	−	0.808419i	$-0.299676\pi$
$397$	23.4558	1.17722	0.588608	+	0.808419i	$0.299676\pi$
$398$	0	0
$399$	10.4525	0.523280
$400$	0	0
$401$	−6.97056	−0.348093	−0.174047	−	0.984737i	$-0.555684\pi$
$401$	−6.97056	−0.348093	−0.174047	+	0.984737i	$0.555684\pi$
$402$	0	0
$403$	8.47343	0.422092
$404$	0	0
$405$	−5.82843	−0.289617
$406$	0	0
$407$	−11.3492	−0.562558
$408$	0	0
$409$	10.6863	0.528403	0.264202	−	0.964467i	$-0.414892\pi$
$409$	10.6863	0.528403	0.264202	+	0.964467i	$0.414892\pi$
$410$	0	0
$411$	−5.22625	−0.257792
$412$	0	0
$413$	34.6274	1.70390
$414$	0	0
$415$	3.69552	0.181406
$416$	0	0
$417$	10.3431	0.506506
$418$	0	0
$419$	−21.2764	−1.03942	−0.519711	−	0.854342i	$-0.673960\pi$
$419$	−21.2764	−1.03942	−0.519711	+	0.854342i	$0.673960\pi$
$420$	0	0
$421$	−8.62742	−0.420475	−0.210237	−	0.977650i	$-0.567424\pi$
$421$	−8.62742	−0.420475	−0.210237	+	0.977650i	$0.567424\pi$
$422$	0	0
$423$	−42.4441	−2.06370
$424$	0	0
$425$	−7.65685	−0.371412
$426$	0	0
$427$	32.6256	1.57886
$428$	0	0
$429$	2.82843	0.136558
$430$	0	0
$431$	−33.3366	−1.60577	−0.802883	−	0.596136i	$-0.796702\pi$
$431$	−33.3366	−1.60577	−0.802883	+	0.596136i	$0.796702\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	23.0698	1.10611
$436$	0	0
$437$	6.14214	0.293818
$438$	0	0
$439$	8.65914	0.413278	0.206639	−	0.978417i	$-0.433747\pi$
$439$	8.65914	0.413278	0.206639	+	0.978417i	$0.433747\pi$
$440$	0	0
$441$	25.4853	1.21358
$442$	0	0
$443$	−6.04601	−0.287255	−0.143627	−	0.989632i	$-0.545877\pi$
$443$	−6.04601	−0.287255	−0.143627	+	0.989632i	$0.545877\pi$
$444$	0	0
$445$	−11.6569	−0.552588
$446$	0	0
$447$	0.896683	0.0424117
$448$	0	0
$449$	22.9706	1.08405	0.542024	−	0.840363i	$-0.317658\pi$
$449$	22.9706	1.08405	0.542024	+	0.840363i	$0.317658\pi$
$450$	0	0
$451$	−2.16478	−0.101936
$452$	0	0
$453$	8.48528	0.398673
$454$	0	0
$455$	3.69552	0.173249
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	0	0
$459$	−16.5754	−0.773675
$460$	0	0
$461$	−8.34315	−0.388579	−0.194290	−	0.980944i	$-0.562240\pi$
$461$	−8.34315	−0.388579	−0.194290	+	0.980944i	$0.562240\pi$
$462$	0	0
$463$	0.262632	0.0122056	0.00610278	−	0.999981i	$-0.498057\pi$
$463$	0.262632	0.0122056	0.00610278	+	0.999981i	$0.498057\pi$
$464$	0	0
$465$	22.1421	1.02682
$466$	0	0
$467$	−15.2304	−0.704780	−0.352390	−	0.935853i	$-0.614631\pi$
$467$	−15.2304	−0.704780	−0.352390	+	0.935853i	$0.614631\pi$
$468$	0	0
$469$	48.9706	2.26125
$470$	0	0
$471$	−5.22625	−0.240813
$472$	0	0
$473$	−7.51472	−0.345527
$474$	0	0
$475$	1.08239	0.0496636
$476$	0	0
$477$	−35.6569	−1.63262
$478$	0	0
$479$	16.3897	0.748866	0.374433	−	0.927254i	$-0.377837\pi$
$479$	16.3897	0.748866	0.374433	+	0.927254i	$0.377837\pi$
$480$	0	0
$481$	−10.4853	−0.478088
$482$	0	0
$483$	54.7987	2.49343
$484$	0	0
$485$	−5.31371	−0.241283
$486$	0	0
$487$	13.7766	0.624277	0.312139	−	0.950037i	$-0.398955\pi$
$487$	13.7766	0.624277	0.312139	+	0.950037i	$0.398955\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	−9.55582	−0.431248	−0.215624	−	0.976476i	$-0.569179\pi$
$491$	−9.55582	−0.431248	−0.215624	+	0.976476i	$0.569179\pi$
$492$	0	0
$493$	−67.5980	−3.04446
$494$	0	0
$495$	4.14386	0.186253
$496$	0	0
$497$	42.6274	1.91210
$498$	0	0
$499$	−9.37011	−0.419464	−0.209732	−	0.977759i	$-0.567259\pi$
$499$	−9.37011	−0.419464	−0.209732	+	0.977759i	$0.567259\pi$
$500$	0	0
$501$	−60.9706	−2.72396
$502$	0	0
$503$	41.7331	1.86079	0.930393	−	0.366563i	$-0.119466\pi$
$503$	41.7331	1.86079	0.930393	+	0.366563i	$0.119466\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	2.61313	0.116053
$508$	0	0
$509$	−3.65685	−0.162087	−0.0810436	−	0.996711i	$-0.525825\pi$
$509$	−3.65685	−0.162087	−0.0810436	+	0.996711i	$0.525825\pi$
$510$	0	0
$511$	3.06147	0.135431
$512$	0	0
$513$	2.34315	0.103452
$514$	0	0
$515$	−1.71644	−0.0756355
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	−39.1200	−1.71718
$520$	0	0
$521$	20.1421	0.882443	0.441221	−	0.897398i	$-0.354545\pi$
$521$	20.1421	0.882443	0.441221	+	0.897398i	$0.354545\pi$
$522$	0	0
$523$	40.0936	1.75317	0.876585	−	0.481248i	$-0.159816\pi$
$523$	40.0936	1.75317	0.876585	+	0.481248i	$0.159816\pi$
$524$	0	0
$525$	9.65685	0.421460
$526$	0	0
$527$	−64.8798	−2.82621
$528$	0	0
$529$	9.20101	0.400044
$530$	0	0
$531$	35.8728	1.55675
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	−3.88123	−0.167800
$536$	0	0
$537$	22.6274	0.976445
$538$	0	0
$539$	7.20533	0.310355
$540$	0	0
$541$	5.31371	0.228454	0.114227	−	0.993455i	$-0.463561\pi$
$541$	5.31371	0.228454	0.114227	+	0.993455i	$0.463561\pi$
$542$	0	0
$543$	52.6339	2.25874
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	−33.4454	−1.43002	−0.715010	−	0.699114i	$-0.753578\pi$
$547$	−33.4454	−1.43002	−0.715010	+	0.699114i	$0.753578\pi$
$548$	0	0
$549$	33.7990	1.44251
$550$	0	0
$551$	9.55582	0.407092
$552$	0	0
$553$	−30.6274	−1.30241
$554$	0	0
$555$	−27.3994	−1.16304
$556$	0	0
$557$	−11.8579	−0.502434	−0.251217	−	0.967931i	$-0.580831\pi$
$557$	−11.8579	−0.502434	−0.251217	+	0.967931i	$0.580831\pi$
$558$	0	0
$559$	−6.94269	−0.293645
$560$	0	0
$561$	−21.6569	−0.914353
$562$	0	0
$563$	24.4148	1.02896	0.514481	−	0.857502i	$-0.327985\pi$
$563$	24.4148	1.02896	0.514481	+	0.857502i	$0.327985\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	−21.5391	−0.904555
$568$	0	0
$569$	1.79899	0.0754176	0.0377088	−	0.999289i	$-0.487994\pi$
$569$	1.79899	0.0754176	0.0377088	+	0.999289i	$0.487994\pi$
$570$	0	0
$571$	−21.8017	−0.912372	−0.456186	−	0.889884i	$-0.650785\pi$
$571$	−21.8017	−0.912372	−0.456186	+	0.889884i	$0.650785\pi$
$572$	0	0
$573$	−30.6274	−1.27948
$574$	0	0
$575$	5.67459	0.236647
$576$	0	0
$577$	−13.7990	−0.574459	−0.287230	−	0.957862i	$-0.592734\pi$
$577$	−13.7990	−0.574459	−0.287230	+	0.957862i	$0.592734\pi$
$578$	0	0
$579$	−13.8854	−0.577057
$580$	0	0
$581$	13.6569	0.566582
$582$	0	0
$583$	−10.0811	−0.417516
$584$	0	0
$585$	3.82843	0.158286
$586$	0	0
$587$	−41.5474	−1.71484	−0.857422	−	0.514614i	$-0.827935\pi$
$587$	−41.5474	−1.71484	−0.857422	+	0.514614i	$0.827935\pi$
$588$	0	0
$589$	9.17157	0.377908
$590$	0	0
$591$	24.3379	1.00113
$592$	0	0
$593$	−45.3137	−1.86081	−0.930405	−	0.366532i	$-0.880545\pi$
$593$	−45.3137	−1.86081	−0.930405	+	0.366532i	$0.880545\pi$
$594$	0	0
$595$	−28.2960	−1.16002
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16.9469	0.692430	0.346215	−	0.938155i	$-0.387467\pi$
$599$	16.9469	0.692430	0.346215	+	0.938155i	$0.387467\pi$
$600$	0	0
$601$	10.9706	0.447499	0.223749	−	0.974647i	$-0.428170\pi$
$601$	10.9706	0.447499	0.223749	+	0.974647i	$0.428170\pi$
$602$	0	0
$603$	50.7318	2.06596
$604$	0	0
$605$	−9.82843	−0.399582
$606$	0	0
$607$	−3.50981	−0.142459	−0.0712294	−	0.997460i	$-0.522692\pi$
$607$	−3.50981	−0.142459	−0.0712294	+	0.997460i	$0.522692\pi$
$608$	0	0
$609$	85.2548	3.45470
$610$	0	0
$611$	−11.0866	−0.448514
$612$	0	0
$613$	−37.3137	−1.50709	−0.753543	−	0.657398i	$-0.771657\pi$
$613$	−37.3137	−1.50709	−0.753543	+	0.657398i	$0.771657\pi$
$614$	0	0
$615$	−5.22625	−0.210743
$616$	0	0
$617$	−35.9411	−1.44694	−0.723468	−	0.690358i	$-0.757453\pi$
$617$	−35.9411	−1.44694	−0.723468	+	0.690358i	$0.757453\pi$
$618$	0	0
$619$	−13.3283	−0.535708	−0.267854	−	0.963460i	$-0.586314\pi$
$619$	−13.3283	−0.535708	−0.267854	+	0.963460i	$0.586314\pi$
$620$	0	0
$621$	12.2843	0.492951
$622$	0	0
$623$	−43.0781	−1.72589
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.06147	0.122263
$628$	0	0
$629$	80.2843	3.20114
$630$	0	0
$631$	19.2974	0.768215	0.384108	−	0.923288i	$-0.374509\pi$
$631$	19.2974	0.768215	0.384108	+	0.923288i	$0.374509\pi$
$632$	0	0
$633$	30.6274	1.21733
$634$	0	0
$635$	−6.94269	−0.275512
$636$	0	0
$637$	6.65685	0.263754
$638$	0	0
$639$	44.1605	1.74696
$640$	0	0
$641$	−43.6569	−1.72434	−0.862171	−	0.506617i	$-0.830896\pi$
$641$	−43.6569	−1.72434	−0.862171	+	0.506617i	$0.830896\pi$
$642$	0	0
$643$	8.02509	0.316479	0.158239	−	0.987401i	$-0.449418\pi$
$643$	8.02509	0.316479	0.158239	+	0.987401i	$0.449418\pi$
$644$	0	0
$645$	−18.1421	−0.714346
$646$	0	0
$647$	39.1969	1.54099	0.770494	−	0.637447i	$-0.220009\pi$
$647$	39.1969	1.54099	0.770494	+	0.637447i	$0.220009\pi$
$648$	0	0
$649$	10.1421	0.398114
$650$	0	0
$651$	81.8267	3.20704
$652$	0	0
$653$	18.2843	0.715519	0.357759	−	0.933814i	$-0.383541\pi$
$653$	18.2843	0.715519	0.357759	+	0.933814i	$0.383541\pi$
$654$	0	0
$655$	−17.8435	−0.697205
$656$	0	0
$657$	3.17157	0.123735
$658$	0	0
$659$	21.2764	0.828812	0.414406	−	0.910092i	$-0.363989\pi$
$659$	21.2764	0.828812	0.414406	+	0.910092i	$0.363989\pi$
$660$	0	0
$661$	−29.3137	−1.14017	−0.570086	−	0.821585i	$-0.693090\pi$
$661$	−29.3137	−1.14017	−0.570086	+	0.821585i	$0.693090\pi$
$662$	0	0
$663$	−20.0083	−0.777059
$664$	0	0
$665$	4.00000	0.155113
$666$	0	0
$667$	50.0977	1.93979
$668$	0	0
$669$	−14.3431	−0.554538
$670$	0	0
$671$	9.55582	0.368898
$672$	0	0
$673$	6.97056	0.268695	0.134348	−	0.990934i	$-0.457106\pi$
$673$	6.97056	0.268695	0.134348	+	0.990934i	$0.457106\pi$
$674$	0	0
$675$	2.16478	0.0833226
$676$	0	0
$677$	40.6274	1.56144	0.780719	−	0.624882i	$-0.214853\pi$
$677$	40.6274	1.56144	0.780719	+	0.624882i	$0.214853\pi$
$678$	0	0
$679$	−19.6369	−0.753595
$680$	0	0
$681$	−53.9411	−2.06703
$682$	0	0
$683$	−29.8268	−1.14129	−0.570645	−	0.821197i	$-0.693307\pi$
$683$	−29.8268	−1.14129	−0.570645	+	0.821197i	$0.693307\pi$
$684$	0	0
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	−53.9020	−2.05649
$688$	0	0
$689$	−9.31371	−0.354824
$690$	0	0
$691$	−24.5236	−0.932922	−0.466461	−	0.884542i	$-0.654471\pi$
$691$	−24.5236	−0.932922	−0.466461	+	0.884542i	$0.654471\pi$
$692$	0	0
$693$	15.3137	0.581720
$694$	0	0
$695$	3.95815	0.150141
$696$	0	0
$697$	15.3137	0.580048
$698$	0	0
$699$	13.8854	0.525194
$700$	0	0
$701$	−27.6569	−1.04458	−0.522292	−	0.852766i	$-0.674923\pi$
$701$	−27.6569	−1.04458	−0.522292	+	0.852766i	$0.674923\pi$
$702$	0	0
$703$	−11.3492	−0.428043
$704$	0	0
$705$	−28.9706	−1.09109
$706$	0	0
$707$	36.9552	1.38984
$708$	0	0
$709$	−17.3137	−0.650230	−0.325115	−	0.945674i	$-0.605403\pi$
$709$	−17.3137	−0.650230	−0.325115	+	0.945674i	$0.605403\pi$
$710$	0	0
$711$	−31.7289	−1.18993
$712$	0	0
$713$	48.0833	1.80073
$714$	0	0
$715$	1.08239	0.0404792
$716$	0	0
$717$	−46.1421	−1.72321
$718$	0	0
$719$	47.9329	1.78760	0.893799	−	0.448468i	$-0.148030\pi$
$719$	47.9329	1.78760	0.893799	+	0.448468i	$0.148030\pi$
$720$	0	0
$721$	−6.34315	−0.236231
$722$	0	0
$723$	15.6788	0.583099
$724$	0	0
$725$	8.82843	0.327880
$726$	0	0
$727$	−39.5683	−1.46751	−0.733754	−	0.679416i	$-0.762233\pi$
$727$	−39.5683	−1.46751	−0.733754	+	0.679416i	$0.762233\pi$
$728$	0	0
$729$	−39.2843	−1.45497
$730$	0	0
$731$	53.1592	1.96616
$732$	0	0
$733$	2.68629	0.0992204	0.0496102	−	0.998769i	$-0.484202\pi$
$733$	2.68629	0.0992204	0.0496102	+	0.998769i	$0.484202\pi$
$734$	0	0
$735$	17.3952	0.641632
$736$	0	0
$737$	14.3431	0.528337
$738$	0	0
$739$	−50.2834	−1.84971	−0.924853	−	0.380324i	$-0.875812\pi$
$739$	−50.2834	−1.84971	−0.924853	+	0.380324i	$0.875812\pi$
$740$	0	0
$741$	2.82843	0.103905
$742$	0	0
$743$	−20.2710	−0.743669	−0.371835	−	0.928299i	$-0.621271\pi$
$743$	−20.2710	−0.743669	−0.371835	+	0.928299i	$0.621271\pi$
$744$	0	0
$745$	0.343146	0.0125719
$746$	0	0
$747$	14.1480	0.517649
$748$	0	0
$749$	−14.3431	−0.524087
$750$	0	0
$751$	20.0083	0.730114	0.365057	−	0.930985i	$-0.381049\pi$
$751$	20.0083	0.730114	0.365057	+	0.930985i	$0.381049\pi$
$752$	0	0
$753$	−13.6569	−0.497683
$754$	0	0
$755$	3.24718	0.118177
$756$	0	0
$757$	−11.6569	−0.423676	−0.211838	−	0.977305i	$-0.567945\pi$
$757$	−11.6569	−0.423676	−0.211838	+	0.977305i	$0.567945\pi$
$758$	0	0
$759$	16.0502	0.582584
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	51.7373	1.87301
$764$	0	0
$765$	−29.3137	−1.05984
$766$	0	0
$767$	9.37011	0.338335
$768$	0	0
$769$	−25.3137	−0.912836	−0.456418	−	0.889766i	$-0.650868\pi$
$769$	−25.3137	−0.912836	−0.456418	+	0.889766i	$0.650868\pi$
$770$	0	0
$771$	20.0083	0.720582
$772$	0	0
$773$	8.82843	0.317536	0.158768	−	0.987316i	$-0.449248\pi$
$773$	8.82843	0.317536	0.158768	+	0.987316i	$0.449248\pi$
$774$	0	0
$775$	8.47343	0.304375
$776$	0	0
$777$	−101.255	−3.63250
$778$	0	0
$779$	−2.16478	−0.0775615
$780$	0	0
$781$	12.4853	0.446758
$782$	0	0
$783$	19.1116	0.682994
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−18.4776	−0.658655	−0.329327	−	0.944216i	$-0.606822\pi$
$787$	−18.4776	−0.658655	−0.329327	+	0.944216i	$0.606822\pi$
$788$	0	0
$789$	1.17157	0.0417091
$790$	0	0
$791$	−7.39104	−0.262795
$792$	0	0
$793$	8.82843	0.313507
$794$	0	0
$795$	−24.3379	−0.863176
$796$	0	0
$797$	−13.3137	−0.471596	−0.235798	−	0.971802i	$-0.575770\pi$
$797$	−13.3137	−0.471596	−0.235798	+	0.971802i	$0.575770\pi$
$798$	0	0
$799$	84.8881	3.00313
$800$	0	0
$801$	−44.6274	−1.57683
$802$	0	0
$803$	0.896683	0.0316433
$804$	0	0
$805$	20.9706	0.739115
$806$	0	0
$807$	40.9133	1.44022
$808$	0	0
$809$	−1.51472	−0.0532547	−0.0266273	−	0.999645i	$-0.508477\pi$
$809$	−1.51472	−0.0532547	−0.0266273	+	0.999645i	$0.508477\pi$
$810$	0	0
$811$	−13.3283	−0.468019	−0.234009	−	0.972234i	$-0.575185\pi$
$811$	−13.3283	−0.468019	−0.234009	+	0.972234i	$0.575185\pi$
$812$	0	0
$813$	−46.1421	−1.61828
$814$	0	0
$815$	−4.59220	−0.160858
$816$	0	0
$817$	−7.51472	−0.262907
$818$	0	0
$819$	14.1480	0.494372
$820$	0	0
$821$	−38.2843	−1.33613	−0.668065	−	0.744103i	$-0.732877\pi$
$821$	−38.2843	−1.33613	−0.668065	+	0.744103i	$0.732877\pi$
$822$	0	0
$823$	−29.4872	−1.02786	−0.513930	−	0.857832i	$-0.671811\pi$
$823$	−29.4872	−1.02786	−0.513930	+	0.857832i	$0.671811\pi$
$824$	0	0
$825$	2.82843	0.0984732
$826$	0	0
$827$	8.55035	0.297325	0.148662	−	0.988888i	$-0.452503\pi$
$827$	8.55035	0.297325	0.148662	+	0.988888i	$0.452503\pi$
$828$	0	0
$829$	−17.1127	−0.594349	−0.297174	−	0.954823i	$-0.596044\pi$
$829$	−17.1127	−0.594349	−0.297174	+	0.954823i	$0.596044\pi$
$830$	0	0
$831$	49.5725	1.71965
$832$	0	0
$833$	−50.9706	−1.76603
$834$	0	0
$835$	−23.3324	−0.807452
$836$	0	0
$837$	18.3431	0.634032
$838$	0	0
$839$	−11.9063	−0.411052	−0.205526	−	0.978652i	$-0.565890\pi$
$839$	−11.9063	−0.411052	−0.205526	+	0.978652i	$0.565890\pi$
$840$	0	0
$841$	48.9411	1.68763
$842$	0	0
$843$	−85.2595	−2.93649
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−36.3211	−1.24801
$848$	0	0
$849$	20.4853	0.703053
$850$	0	0
$851$	−59.4997	−2.03962
$852$	0	0
$853$	29.1127	0.996800	0.498400	−	0.866947i	$-0.333921\pi$
$853$	29.1127	0.996800	0.498400	+	0.866947i	$0.333921\pi$
$854$	0	0
$855$	4.14386	0.141717
$856$	0	0
$857$	−35.6569	−1.21801	−0.609007	−	0.793164i	$-0.708432\pi$
$857$	−35.6569	−1.21801	−0.609007	+	0.793164i	$0.708432\pi$
$858$	0	0
$859$	31.7289	1.08258	0.541289	−	0.840837i	$-0.317937\pi$
$859$	31.7289	1.08258	0.541289	+	0.840837i	$0.317937\pi$
$860$	0	0
$861$	−19.3137	−0.658209
$862$	0	0
$863$	−12.3547	−0.420557	−0.210279	−	0.977641i	$-0.567437\pi$
$863$	−12.3547	−0.420557	−0.210279	+	0.977641i	$0.567437\pi$
$864$	0	0
$865$	−14.9706	−0.509014
$866$	0	0
$867$	108.778	3.69428
$868$	0	0
$869$	−8.97056	−0.304305
$870$	0	0
$871$	13.2513	0.449004
$872$	0	0
$873$	−20.3431	−0.688511
$874$	0	0
$875$	3.69552	0.124931
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	0	0
$879$	37.8519	1.27671
$880$	0	0
$881$	−47.1716	−1.58925	−0.794625	−	0.607100i	$-0.792333\pi$
$881$	−47.1716	−1.58925	−0.794625	+	0.607100i	$0.792333\pi$
$882$	0	0
$883$	56.5152	1.90189	0.950943	−	0.309365i	$-0.100117\pi$
$883$	56.5152	1.90189	0.950943	+	0.309365i	$0.100117\pi$
$884$	0	0
$885$	24.4853	0.823064
$886$	0	0
$887$	13.5909	0.456338	0.228169	−	0.973622i	$-0.426726\pi$
$887$	13.5909	0.456338	0.228169	+	0.973622i	$0.426726\pi$
$888$	0	0
$889$	−25.6569	−0.860503
$890$	0	0
$891$	−6.30864	−0.211348
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	8.65914	0.289443
$896$	0	0
$897$	14.8284	0.495107
$898$	0	0
$899$	74.8070	2.49495
$900$	0	0
$901$	71.3137	2.37580
$902$	0	0
$903$	−67.0446	−2.23110
$904$	0	0
$905$	20.1421	0.669547
$906$	0	0
$907$	−30.0125	−0.996548	−0.498274	−	0.867020i	$-0.666033\pi$
$907$	−30.0125	−0.996548	−0.498274	+	0.867020i	$0.666033\pi$
$908$	0	0
$909$	38.2843	1.26981
$910$	0	0
$911$	19.6369	0.650600	0.325300	−	0.945611i	$-0.394535\pi$
$911$	19.6369	0.650600	0.325300	+	0.945611i	$0.394535\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	0	0
$915$	23.0698	0.762664
$916$	0	0
$917$	−65.9411	−2.17757
$918$	0	0
$919$	−15.1535	−0.499868	−0.249934	−	0.968263i	$-0.580409\pi$
$919$	−15.1535	−0.499868	−0.249934	+	0.968263i	$0.580409\pi$
$920$	0	0
$921$	6.34315	0.209014
$922$	0	0
$923$	11.5349	0.379676
$924$	0	0
$925$	−10.4853	−0.344754
$926$	0	0
$927$	−6.57128	−0.215829
$928$	0	0
$929$	6.97056	0.228697	0.114348	−	0.993441i	$-0.463522\pi$
$929$	6.97056	0.228697	0.114348	+	0.993441i	$0.463522\pi$
$930$	0	0
$931$	7.20533	0.236145
$932$	0	0
$933$	−2.34315	−0.0767111
$934$	0	0
$935$	−8.28772	−0.271037
$936$	0	0
$937$	47.2548	1.54375	0.771874	−	0.635775i	$-0.219320\pi$
$937$	47.2548	1.54375	0.771874	+	0.635775i	$0.219320\pi$
$938$	0	0
$939$	30.4608	0.994052
$940$	0	0
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	0	0
$943$	−11.3492	−0.369580
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	25.8686	0.840617	0.420309	−	0.907381i	$-0.361922\pi$
$947$	25.8686	0.840617	0.420309	+	0.907381i	$0.361922\pi$
$948$	0	0
$949$	0.828427	0.0268919
$950$	0	0
$951$	−21.2764	−0.689935
$952$	0	0
$953$	−48.3431	−1.56599	−0.782994	−	0.622029i	$-0.786309\pi$
$953$	−48.3431	−1.56599	−0.782994	+	0.622029i	$0.786309\pi$
$954$	0	0
$955$	−11.7206	−0.379270
$956$	0	0
$957$	24.9706	0.807184
$958$	0	0
$959$	−7.39104	−0.238669
$960$	0	0
$961$	40.7990	1.31610
$962$	0	0
$963$	−14.8590	−0.478824
$964$	0	0
$965$	−5.31371	−0.171054
$966$	0	0
$967$	29.3015	0.942273	0.471137	−	0.882060i	$-0.343844\pi$
$967$	29.3015	0.942273	0.471137	+	0.882060i	$0.343844\pi$
$968$	0	0
$969$	−21.6569	−0.695718
$970$	0	0
$971$	−41.2848	−1.32489	−0.662445	−	0.749110i	$-0.730481\pi$
$971$	−41.2848	−1.32489	−0.662445	+	0.749110i	$0.730481\pi$
$972$	0	0
$973$	14.6274	0.468933
$974$	0	0
$975$	2.61313	0.0836870
$976$	0	0
$977$	4.14214	0.132519	0.0662593	−	0.997802i	$-0.478894\pi$
$977$	4.14214	0.132519	0.0662593	+	0.997802i	$0.478894\pi$
$978$	0	0
$979$	−12.6173	−0.403250
$980$	0	0
$981$	53.5980	1.71125
$982$	0	0
$983$	34.1563	1.08942	0.544709	−	0.838625i	$-0.316640\pi$
$983$	34.1563	1.08942	0.544709	+	0.838625i	$0.316640\pi$
$984$	0	0
$985$	9.31371	0.296759
$986$	0	0
$987$	−107.061	−3.40780
$988$	0	0
$989$	−39.3970	−1.25275
$990$	0	0
$991$	61.4469	1.95193	0.975963	−	0.217937i	$-0.0699327\pi$
$991$	61.4469	1.95193	0.975963	+	0.217937i	$0.0699327\pi$
$992$	0	0
$993$	3.79899	0.120557
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	0	0
$999$	−22.6984	−0.718145

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4160.2.a.bu.1.4		4
4.3	odd	2	inner	4160.2.a.bu.1.1		4
8.3	odd	2		2080.2.a.q.1.4	yes	4
8.5	even	2		2080.2.a.q.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2080.2.a.q.1.1	✓	4	8.5	even	2
2080.2.a.q.1.4	yes	4	8.3	odd	2
4160.2.a.bu.1.1		4	4.3	odd	2	inner
4160.2.a.bu.1.4		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:	\( N \)	\(=\)	\( 4160 = 2^{6} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4160.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61313 q^{3} +1.00000 q^{5} +3.69552 q^{7} +3.82843 q^{9} +1.08239 q^{11} +1.00000 q^{13} +2.61313 q^{15} -7.65685 q^{17} +1.08239 q^{19} +9.65685 q^{21} +5.67459 q^{23} +1.00000 q^{25} +2.16478 q^{27} +8.82843 q^{29} +8.47343 q^{31} +2.82843 q^{33} +3.69552 q^{35} -10.4853 q^{37} +2.61313 q^{39} -2.00000 q^{41} -6.94269 q^{43} +3.82843 q^{45} -11.0866 q^{47} +6.65685 q^{49} -20.0083 q^{51} -9.31371 q^{53} +1.08239 q^{55} +2.82843 q^{57} +9.37011 q^{59} +8.82843 q^{61} +14.1480 q^{63} +1.00000 q^{65} +13.2513 q^{67} +14.8284 q^{69} +11.5349 q^{71} +0.828427 q^{73} +2.61313 q^{75} +4.00000 q^{77} -8.28772 q^{79} -5.82843 q^{81} +3.69552 q^{83} -7.65685 q^{85} +23.0698 q^{87} -11.6569 q^{89} +3.69552 q^{91} +22.1421 q^{93} +1.08239 q^{95} -5.31371 q^{97} +4.14386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{9} + 4 q^{13} - 8 q^{17} + 16 q^{21} + 4 q^{25} + 24 q^{29} - 8 q^{37} - 8 q^{41} + 4 q^{45} + 4 q^{49} + 8 q^{53} + 24 q^{61} + 4 q^{65} + 48 q^{69} - 8 q^{73} + 16 q^{77} - 12 q^{81}+ \cdots + 24 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^9 + 4 * q^13 - 8 * q^17 + 16 * q^21 + 4 * q^25 + 24 * q^29 - 8 * q^37 - 8 * q^41 + 4 * q^45 + 4 * q^49 + 8 * q^53 + 24 * q^61 + 4 * q^65 + 48 * q^69 - 8 * q^73 + 16 * q^77 - 12 * q^81 - 8 * q^85 - 24 * q^89 + 32 * q^93 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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