LMFDB - Embedded newform 3808.2.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3808,2,Mod(1,3808)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3808, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3808.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$30.4070330897$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.93059344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 41x^{2} - 8x - 20 $ x^6 - 13x^4 + 41x^2 - 8*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.644180$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.644180 q^{3} -2.53164 q^{5} +1.00000 q^{7} -2.58503 q^{9} +O(q^{10})$	$q+0.644180 q^{3} -2.53164 q^{5} +1.00000 q^{7} -2.58503 q^{9} -2.23245 q^{11} -5.25278 q^{13} -1.63083 q^{15} -1.00000 q^{17} +2.57420 q^{19} +0.644180 q^{21} +3.02032 q^{23} +1.40921 q^{25} -3.59777 q^{27} -0.232452 q^{29} +6.24195 q^{31} -1.43810 q^{33} -2.53164 q^{35} +3.26418 q^{37} -3.38373 q^{39} -6.47116 q^{41} -4.37614 q^{43} +6.54438 q^{45} +10.1380 q^{47} +1.00000 q^{49} -0.644180 q^{51} +6.27501 q^{53} +5.65177 q^{55} +1.65825 q^{57} +1.13610 q^{59} -1.14215 q^{61} -2.58503 q^{63} +13.2981 q^{65} -9.62799 q^{67} +1.94563 q^{69} +7.88135 q^{71} +1.67627 q^{73} +0.907785 q^{75} -2.23245 q^{77} +3.38373 q^{79} +5.43749 q^{81} +15.6628 q^{83} +2.53164 q^{85} -0.149741 q^{87} +16.8164 q^{89} -5.25278 q^{91} +4.02094 q^{93} -6.51696 q^{95} -0.616763 q^{97} +5.77096 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{5} + 6 q^{7} + 8 q^{9}+O(q^{10}) $ 6 * q + 4 * q^5 + 6 * q^7 + 8 * q^9	$ 6 q + 4 q^{5} + 6 q^{7} + 8 q^{9} - 4 q^{11} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 6 q^{19} + 8 q^{25} + 8 q^{29} + 12 q^{31} + 28 q^{33} + 4 q^{35} + 10 q^{37} + 14 q^{39} + 14 q^{41} - 12 q^{43} + 22 q^{45} + 2 q^{47} + 6 q^{49} + 26 q^{53} + 10 q^{55} + 22 q^{57} + 22 q^{59} + 2 q^{61} + 8 q^{63} + 8 q^{65} - 14 q^{67} + 14 q^{69} - 4 q^{71} + 18 q^{73} - 28 q^{75} - 4 q^{77} - 14 q^{79} - 6 q^{81} + 8 q^{83} - 4 q^{85} + 28 q^{87} + 6 q^{89} - 4 q^{91} + 8 q^{93} - 2 q^{95} + 20 q^{97} - 26 q^{99}+O(q^{100}) $ 6 * q + 4 * q^5 + 6 * q^7 + 8 * q^9 - 4 * q^11 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 6 * q^19 + 8 * q^25 + 8 * q^29 + 12 * q^31 + 28 * q^33 + 4 * q^35 + 10 * q^37 + 14 * q^39 + 14 * q^41 - 12 * q^43 + 22 * q^45 + 2 * q^47 + 6 * q^49 + 26 * q^53 + 10 * q^55 + 22 * q^57 + 22 * q^59 + 2 * q^61 + 8 * q^63 + 8 * q^65 - 14 * q^67 + 14 * q^69 - 4 * q^71 + 18 * q^73 - 28 * q^75 - 4 * q^77 - 14 * q^79 - 6 * q^81 + 8 * q^83 - 4 * q^85 + 28 * q^87 + 6 * q^89 - 4 * q^91 + 8 * q^93 - 2 * q^95 + 20 * q^97 - 26 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.644180	0.371917	0.185959	−	0.982558i	$-0.440461\pi$
$3$	0.644180	0.371917	0.185959	+	0.982558i	$0.440461\pi$
$4$	0	0
$5$	−2.53164	−1.13218	−0.566092	−	0.824342i	$-0.691545\pi$
$5$	−2.53164	−1.13218	−0.566092	+	0.824342i	$0.691545\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.58503	−0.861677
$10$	0	0
$11$	−2.23245	−0.673110	−0.336555	−	0.941664i	$-0.609262\pi$
$11$	−2.23245	−0.673110	−0.336555	+	0.941664i	$0.609262\pi$
$12$	0	0
$13$	−5.25278	−1.45686	−0.728429	−	0.685121i	$-0.759749\pi$
$13$	−5.25278	−1.45686	−0.728429	+	0.685121i	$0.759749\pi$
$14$	0	0
$15$	−1.63083	−0.421079
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.57420	0.590563	0.295281	−	0.955410i	$-0.404587\pi$
$19$	2.57420	0.590563	0.295281	+	0.955410i	$0.404587\pi$
$20$	0	0
$21$	0.644180	0.140572
$22$	0	0
$23$	3.02032	0.629781	0.314890	−	0.949128i	$-0.398032\pi$
$23$	3.02032	0.629781	0.314890	+	0.949128i	$0.398032\pi$
$24$	0	0
$25$	1.40921	0.281842
$26$	0	0
$27$	−3.59777	−0.692390
$28$	0	0
$29$	−0.232452	−0.0431653	−0.0215826	−	0.999767i	$-0.506871\pi$
$29$	−0.232452	−0.0431653	−0.0215826	+	0.999767i	$0.506871\pi$
$30$	0	0
$31$	6.24195	1.12109	0.560543	−	0.828125i	$-0.310592\pi$
$31$	6.24195	1.12109	0.560543	+	0.828125i	$0.310592\pi$
$32$	0	0
$33$	−1.43810	−0.250341
$34$	0	0
$35$	−2.53164	−0.427926
$36$	0	0
$37$	3.26418	0.536628	0.268314	−	0.963331i	$-0.413533\pi$
$37$	3.26418	0.536628	0.268314	+	0.963331i	$0.413533\pi$
$38$	0	0
$39$	−3.38373	−0.541831
$40$	0	0
$41$	−6.47116	−1.01063	−0.505313	−	0.862936i	$-0.668623\pi$
$41$	−6.47116	−1.01063	−0.505313	+	0.862936i	$0.668623\pi$
$42$	0	0
$43$	−4.37614	−0.667356	−0.333678	−	0.942687i	$-0.608290\pi$
$43$	−4.37614	−0.667356	−0.333678	+	0.942687i	$0.608290\pi$
$44$	0	0
$45$	6.54438	0.975578
$46$	0	0
$47$	10.1380	1.47878	0.739388	−	0.673280i	$-0.235115\pi$
$47$	10.1380	1.47878	0.739388	+	0.673280i	$0.235115\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.644180	−0.0902032
$52$	0	0
$53$	6.27501	0.861939	0.430970	−	0.902366i	$-0.358172\pi$
$53$	6.27501	0.861939	0.430970	+	0.902366i	$0.358172\pi$
$54$	0	0
$55$	5.65177	0.762084
$56$	0	0
$57$	1.65825	0.219641
$58$	0	0
$59$	1.13610	0.147908	0.0739539	−	0.997262i	$-0.476438\pi$
$59$	1.13610	0.147908	0.0739539	+	0.997262i	$0.476438\pi$
$60$	0	0
$61$	−1.14215	−0.146238	−0.0731188	−	0.997323i	$-0.523295\pi$
$61$	−1.14215	−0.146238	−0.0731188	+	0.997323i	$0.523295\pi$
$62$	0	0
$63$	−2.58503	−0.325683
$64$	0	0
$65$	13.2981	1.64943
$66$	0	0
$67$	−9.62799	−1.17625	−0.588123	−	0.808771i	$-0.700133\pi$
$67$	−9.62799	−1.17625	−0.588123	+	0.808771i	$0.700133\pi$
$68$	0	0
$69$	1.94563	0.234227
$70$	0	0
$71$	7.88135	0.935344	0.467672	−	0.883902i	$-0.345093\pi$
$71$	7.88135	0.935344	0.467672	+	0.883902i	$0.345093\pi$
$72$	0	0
$73$	1.67627	0.196192	0.0980962	−	0.995177i	$-0.468725\pi$
$73$	1.67627	0.196192	0.0980962	+	0.995177i	$0.468725\pi$
$74$	0	0
$75$	0.907785	0.104822
$76$	0	0
$77$	−2.23245	−0.254412
$78$	0	0
$79$	3.38373	0.380700	0.190350	−	0.981716i	$-0.439038\pi$
$79$	3.38373	0.380700	0.190350	+	0.981716i	$0.439038\pi$
$80$	0	0
$81$	5.43749	0.604165
$82$	0	0
$83$	15.6628	1.71922	0.859610	−	0.510950i	$-0.170706\pi$
$83$	15.6628	1.71922	0.859610	+	0.510950i	$0.170706\pi$
$84$	0	0
$85$	2.53164	0.274595
$86$	0	0
$87$	−0.149741	−0.0160539
$88$	0	0
$89$	16.8164	1.78254	0.891270	−	0.453474i	$-0.149815\pi$
$89$	16.8164	1.78254	0.891270	+	0.453474i	$0.149815\pi$
$90$	0	0
$91$	−5.25278	−0.550640
$92$	0	0
$93$	4.02094	0.416952
$94$	0	0
$95$	−6.51696	−0.668626
$96$	0	0
$97$	−0.616763	−0.0626228	−0.0313114	−	0.999510i	$-0.509968\pi$
$97$	−0.616763	−0.0626228	−0.0313114	+	0.999510i	$0.509968\pi$
$98$	0	0
$99$	5.77096	0.580003
$100$	0	0
$101$	13.6375	1.35698	0.678490	−	0.734609i	$-0.262635\pi$
$101$	13.6375	1.35698	0.678490	+	0.734609i	$0.262635\pi$
$102$	0	0
$103$	−8.02119	−0.790352	−0.395176	−	0.918605i	$-0.629316\pi$
$103$	−8.02119	−0.790352	−0.395176	+	0.918605i	$0.629316\pi$
$104$	0	0
$105$	−1.63083	−0.159153
$106$	0	0
$107$	8.22058	0.794713	0.397357	−	0.917664i	$-0.369928\pi$
$107$	8.22058	0.794713	0.397357	+	0.917664i	$0.369928\pi$
$108$	0	0
$109$	−5.59453	−0.535858	−0.267929	−	0.963439i	$-0.586339\pi$
$109$	−5.59453	−0.535858	−0.267929	+	0.963439i	$0.586339\pi$
$110$	0	0
$111$	2.10272	0.199581
$112$	0	0
$113$	5.17017	0.486369	0.243184	−	0.969980i	$-0.421808\pi$
$113$	5.17017	0.486369	0.243184	+	0.969980i	$0.421808\pi$
$114$	0	0
$115$	−7.64638	−0.713028
$116$	0	0
$117$	13.5786	1.25534
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−6.01616	−0.546923
$122$	0	0
$123$	−4.16859	−0.375869
$124$	0	0
$125$	9.09059	0.813087
$126$	0	0
$127$	7.13639	0.633253	0.316626	−	0.948550i	$-0.397450\pi$
$127$	7.13639	0.633253	0.316626	+	0.948550i	$0.397450\pi$
$128$	0	0
$129$	−2.81902	−0.248201
$130$	0	0
$131$	−10.5435	−0.921194	−0.460597	−	0.887609i	$-0.652365\pi$
$131$	−10.5435	−0.921194	−0.460597	+	0.887609i	$0.652365\pi$
$132$	0	0
$133$	2.57420	0.223212
$134$	0	0
$135$	9.10825	0.783914
$136$	0	0
$137$	3.58575	0.306352	0.153176	−	0.988199i	$-0.451050\pi$
$137$	3.58575	0.306352	0.153176	+	0.988199i	$0.451050\pi$
$138$	0	0
$139$	−8.90646	−0.755436	−0.377718	−	0.925921i	$-0.623291\pi$
$139$	−8.90646	−0.755436	−0.377718	+	0.925921i	$0.623291\pi$
$140$	0	0
$141$	6.53068	0.549983
$142$	0	0
$143$	11.7266	0.980625
$144$	0	0
$145$	0.588486	0.0488711
$146$	0	0
$147$	0.644180	0.0531311
$148$	0	0
$149$	−15.3248	−1.25546	−0.627730	−	0.778431i	$-0.716016\pi$
$149$	−15.3248	−1.25546	−0.627730	+	0.778431i	$0.716016\pi$
$150$	0	0
$151$	−6.17776	−0.502739	−0.251370	−	0.967891i	$-0.580881\pi$
$151$	−6.17776	−0.502739	−0.251370	+	0.967891i	$0.580881\pi$
$152$	0	0
$153$	2.58503	0.208987
$154$	0	0
$155$	−15.8024	−1.26928
$156$	0	0
$157$	3.03309	0.242067	0.121034	−	0.992648i	$-0.461379\pi$
$157$	3.03309	0.242067	0.121034	+	0.992648i	$0.461379\pi$
$158$	0	0
$159$	4.04224	0.320570
$160$	0	0
$161$	3.02032	0.238035
$162$	0	0
$163$	−18.3161	−1.43462	−0.717312	−	0.696752i	$-0.754628\pi$
$163$	−18.3161	−1.43462	−0.717312	+	0.696752i	$0.754628\pi$
$164$	0	0
$165$	3.64076	0.283433
$166$	0	0
$167$	−5.61198	−0.434268	−0.217134	−	0.976142i	$-0.569671\pi$
$167$	−5.61198	−0.434268	−0.217134	+	0.976142i	$0.569671\pi$
$168$	0	0
$169$	14.5917	1.12243
$170$	0	0
$171$	−6.65440	−0.508874
$172$	0	0
$173$	−23.3613	−1.77613	−0.888064	−	0.459720i	$-0.847950\pi$
$173$	−23.3613	−1.77613	−0.888064	+	0.459720i	$0.847950\pi$
$174$	0	0
$175$	1.40921	0.106526
$176$	0	0
$177$	0.731854	0.0550095
$178$	0	0
$179$	−5.84203	−0.436654	−0.218327	−	0.975876i	$-0.570060\pi$
$179$	−5.84203	−0.436654	−0.218327	+	0.975876i	$0.570060\pi$
$180$	0	0
$181$	24.1189	1.79274	0.896370	−	0.443306i	$-0.146194\pi$
$181$	24.1189	1.79274	0.896370	+	0.443306i	$0.146194\pi$
$182$	0	0
$183$	−0.735751	−0.0543883
$184$	0	0
$185$	−8.26374	−0.607562
$186$	0	0
$187$	2.23245	0.163253
$188$	0	0
$189$	−3.59777	−0.261699
$190$	0	0
$191$	−10.9353	−0.791253	−0.395627	−	0.918411i	$-0.629473\pi$
$191$	−10.9353	−0.791253	−0.395627	+	0.918411i	$0.629473\pi$
$192$	0	0
$193$	5.38605	0.387696	0.193848	−	0.981032i	$-0.437903\pi$
$193$	5.38605	0.387696	0.193848	+	0.981032i	$0.437903\pi$
$194$	0	0
$195$	8.56640	0.613453
$196$	0	0
$197$	20.2473	1.44256	0.721279	−	0.692644i	$-0.243554\pi$
$197$	20.2473	1.44256	0.721279	+	0.692644i	$0.243554\pi$
$198$	0	0
$199$	4.73197	0.335441	0.167720	−	0.985835i	$-0.446359\pi$
$199$	4.73197	0.335441	0.167720	+	0.985835i	$0.446359\pi$
$200$	0	0
$201$	−6.20216	−0.437467
$202$	0	0
$203$	−0.232452	−0.0163149
$204$	0	0
$205$	16.3827	1.14421
$206$	0	0
$207$	−7.80763	−0.542668
$208$	0	0
$209$	−5.74678	−0.397513
$210$	0	0
$211$	12.0289	0.828106	0.414053	−	0.910253i	$-0.364113\pi$
$211$	12.0289	0.828106	0.414053	+	0.910253i	$0.364113\pi$
$212$	0	0
$213$	5.07701	0.347871
$214$	0	0
$215$	11.0788	0.755570
$216$	0	0
$217$	6.24195	0.423731
$218$	0	0
$219$	1.07982	0.0729674
$220$	0	0
$221$	5.25278	0.353340
$222$	0	0
$223$	26.3066	1.76162	0.880810	−	0.473470i	$-0.156999\pi$
$223$	26.3066	1.76162	0.880810	+	0.473470i	$0.156999\pi$
$224$	0	0
$225$	−3.64285	−0.242857
$226$	0	0
$227$	−12.9452	−0.859205	−0.429603	−	0.903018i	$-0.641346\pi$
$227$	−12.9452	−0.859205	−0.429603	+	0.903018i	$0.641346\pi$
$228$	0	0
$229$	11.9309	0.788415	0.394207	−	0.919021i	$-0.371019\pi$
$229$	11.9309	0.788415	0.394207	+	0.919021i	$0.371019\pi$
$230$	0	0
$231$	−1.43810	−0.0946201
$232$	0	0
$233$	17.9552	1.17628	0.588141	−	0.808758i	$-0.299860\pi$
$233$	17.9552	1.17628	0.588141	+	0.808758i	$0.299860\pi$
$234$	0	0
$235$	−25.6657	−1.67425
$236$	0	0
$237$	2.17973	0.141589
$238$	0	0
$239$	19.0273	1.23077	0.615386	−	0.788226i	$-0.289000\pi$
$239$	19.0273	1.23077	0.615386	+	0.788226i	$0.289000\pi$
$240$	0	0
$241$	−24.8273	−1.59927	−0.799633	−	0.600489i	$-0.794973\pi$
$241$	−24.8273	−1.59927	−0.799633	+	0.600489i	$0.794973\pi$
$242$	0	0
$243$	14.2960	0.917090
$244$	0	0
$245$	−2.53164	−0.161741
$246$	0	0
$247$	−13.5217	−0.860366
$248$	0	0
$249$	10.0897	0.639408
$250$	0	0
$251$	10.6601	0.672861	0.336431	−	0.941708i	$-0.390780\pi$
$251$	10.6601	0.672861	0.336431	+	0.941708i	$0.390780\pi$
$252$	0	0
$253$	−6.74273	−0.423912
$254$	0	0
$255$	1.63083	0.102127
$256$	0	0
$257$	20.0555	1.25103	0.625513	−	0.780214i	$-0.284890\pi$
$257$	20.0555	1.25103	0.625513	+	0.780214i	$0.284890\pi$
$258$	0	0
$259$	3.26418	0.202826
$260$	0	0
$261$	0.600897	0.0371946
$262$	0	0
$263$	−14.8216	−0.913938	−0.456969	−	0.889483i	$-0.651065\pi$
$263$	−14.8216	−0.913938	−0.456969	+	0.889483i	$0.651065\pi$
$264$	0	0
$265$	−15.8861	−0.975874
$266$	0	0
$267$	10.8328	0.662957
$268$	0	0
$269$	27.1783	1.65709	0.828546	−	0.559921i	$-0.189168\pi$
$269$	27.1783	1.65709	0.828546	+	0.559921i	$0.189168\pi$
$270$	0	0
$271$	18.4417	1.12026	0.560128	−	0.828406i	$-0.310752\pi$
$271$	18.4417	1.12026	0.560128	+	0.828406i	$0.310752\pi$
$272$	0	0
$273$	−3.38373	−0.204793
$274$	0	0
$275$	−3.14599	−0.189711
$276$	0	0
$277$	−9.25080	−0.555826	−0.277913	−	0.960606i	$-0.589643\pi$
$277$	−9.25080	−0.555826	−0.277913	+	0.960606i	$0.589643\pi$
$278$	0	0
$279$	−16.1356	−0.966015
$280$	0	0
$281$	2.18341	0.130251	0.0651257	−	0.997877i	$-0.479255\pi$
$281$	2.18341	0.130251	0.0651257	+	0.997877i	$0.479255\pi$
$282$	0	0
$283$	−9.32228	−0.554152	−0.277076	−	0.960848i	$-0.589365\pi$
$283$	−9.32228	−0.554152	−0.277076	+	0.960848i	$0.589365\pi$
$284$	0	0
$285$	−4.19809	−0.248674
$286$	0	0
$287$	−6.47116	−0.381980
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−0.397307	−0.0232905
$292$	0	0
$293$	5.03911	0.294388	0.147194	−	0.989108i	$-0.452976\pi$
$293$	5.03911	0.294388	0.147194	+	0.989108i	$0.452976\pi$
$294$	0	0
$295$	−2.87620	−0.167459
$296$	0	0
$297$	8.03184	0.466055
$298$	0	0
$299$	−15.8651	−0.917501
$300$	0	0
$301$	−4.37614	−0.252237
$302$	0	0
$303$	8.78500	0.504685
$304$	0	0
$305$	2.89152	0.165568
$306$	0	0
$307$	−8.14430	−0.464820	−0.232410	−	0.972618i	$-0.574661\pi$
$307$	−8.14430	−0.464820	−0.232410	+	0.972618i	$0.574661\pi$
$308$	0	0
$309$	−5.16709	−0.293946
$310$	0	0
$311$	2.13704	0.121180	0.0605901	−	0.998163i	$-0.480702\pi$
$311$	2.13704	0.121180	0.0605901	+	0.998163i	$0.480702\pi$
$312$	0	0
$313$	−8.04392	−0.454669	−0.227335	−	0.973817i	$-0.573001\pi$
$313$	−8.04392	−0.454669	−0.227335	+	0.973817i	$0.573001\pi$
$314$	0	0
$315$	6.54438	0.368734
$316$	0	0
$317$	26.3395	1.47938	0.739688	−	0.672950i	$-0.234973\pi$
$317$	26.3395	1.47938	0.739688	+	0.672950i	$0.234973\pi$
$318$	0	0
$319$	0.518939	0.0290550
$320$	0	0
$321$	5.29553	0.295568
$322$	0	0
$323$	−2.57420	−0.143232
$324$	0	0
$325$	−7.40227	−0.410604
$326$	0	0
$327$	−3.60388	−0.199295
$328$	0	0
$329$	10.1380	0.558925
$330$	0	0
$331$	15.9085	0.874408	0.437204	−	0.899362i	$-0.355969\pi$
$331$	15.9085	0.874408	0.437204	+	0.899362i	$0.355969\pi$
$332$	0	0
$333$	−8.43802	−0.462401
$334$	0	0
$335$	24.3746	1.33173
$336$	0	0
$337$	12.0768	0.657864	0.328932	−	0.944354i	$-0.393311\pi$
$337$	12.0768	0.657864	0.328932	+	0.944354i	$0.393311\pi$
$338$	0	0
$339$	3.33052	0.180889
$340$	0	0
$341$	−13.9348	−0.754614
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−4.92564	−0.265188
$346$	0	0
$347$	−4.40403	−0.236421	−0.118210	−	0.992989i	$-0.537716\pi$
$347$	−4.40403	−0.236421	−0.118210	+	0.992989i	$0.537716\pi$
$348$	0	0
$349$	−16.9043	−0.904866	−0.452433	−	0.891798i	$-0.649444\pi$
$349$	−16.9043	−0.904866	−0.452433	+	0.891798i	$0.649444\pi$
$350$	0	0
$351$	18.8983	1.00871
$352$	0	0
$353$	−11.8874	−0.632704	−0.316352	−	0.948642i	$-0.602458\pi$
$353$	−11.8874	−0.632704	−0.316352	+	0.948642i	$0.602458\pi$
$354$	0	0
$355$	−19.9527	−1.05898
$356$	0	0
$357$	−0.644180	−0.0340936
$358$	0	0
$359$	−19.0842	−1.00723	−0.503613	−	0.863929i	$-0.667996\pi$
$359$	−19.0842	−1.00723	−0.503613	+	0.863929i	$0.667996\pi$
$360$	0	0
$361$	−12.3735	−0.651236
$362$	0	0
$363$	−3.87549	−0.203410
$364$	0	0
$365$	−4.24371	−0.222126
$366$	0	0
$367$	22.6353	1.18155	0.590777	−	0.806835i	$-0.298821\pi$
$367$	22.6353	1.18155	0.590777	+	0.806835i	$0.298821\pi$
$368$	0	0
$369$	16.7282	0.870833
$370$	0	0
$371$	6.27501	0.325782
$372$	0	0
$373$	−36.7066	−1.90059	−0.950297	−	0.311343i	$-0.899221\pi$
$373$	−36.7066	−1.90059	−0.950297	+	0.311343i	$0.899221\pi$
$374$	0	0
$375$	5.85598	0.302401
$376$	0	0
$377$	1.22102	0.0628857
$378$	0	0
$379$	18.0207	0.925660	0.462830	−	0.886447i	$-0.346834\pi$
$379$	18.0207	0.925660	0.462830	+	0.886447i	$0.346834\pi$
$380$	0	0
$381$	4.59712	0.235518
$382$	0	0
$383$	21.0899	1.07764	0.538821	−	0.842421i	$-0.318870\pi$
$383$	21.0899	1.07764	0.538821	+	0.842421i	$0.318870\pi$
$384$	0	0
$385$	5.65177	0.288041
$386$	0	0
$387$	11.3125	0.575045
$388$	0	0
$389$	1.72499	0.0874603	0.0437302	−	0.999043i	$-0.486076\pi$
$389$	1.72499	0.0874603	0.0437302	+	0.999043i	$0.486076\pi$
$390$	0	0
$391$	−3.02032	−0.152744
$392$	0	0
$393$	−6.79194	−0.342608
$394$	0	0
$395$	−8.56640	−0.431022
$396$	0	0
$397$	25.7157	1.29063	0.645316	−	0.763916i	$-0.276726\pi$
$397$	25.7157	1.29063	0.645316	+	0.763916i	$0.276726\pi$
$398$	0	0
$399$	1.65825	0.0830163
$400$	0	0
$401$	−3.46437	−0.173002	−0.0865011	−	0.996252i	$-0.527569\pi$
$401$	−3.46437	−0.173002	−0.0865011	+	0.996252i	$0.527569\pi$
$402$	0	0
$403$	−32.7875	−1.63326
$404$	0	0
$405$	−13.7658	−0.684027
$406$	0	0
$407$	−7.28713	−0.361210
$408$	0	0
$409$	−24.0357	−1.18849	−0.594244	−	0.804285i	$-0.702549\pi$
$409$	−24.0357	−1.18849	−0.594244	+	0.804285i	$0.702549\pi$
$410$	0	0
$411$	2.30987	0.113938
$412$	0	0
$413$	1.13610	0.0559039
$414$	0	0
$415$	−39.6527	−1.94648
$416$	0	0
$417$	−5.73736	−0.280960
$418$	0	0
$419$	2.87998	0.140696	0.0703481	−	0.997523i	$-0.477589\pi$
$419$	2.87998	0.140696	0.0703481	+	0.997523i	$0.477589\pi$
$420$	0	0
$421$	−17.1075	−0.833767	−0.416883	−	0.908960i	$-0.636878\pi$
$421$	−17.1075	−0.833767	−0.416883	+	0.908960i	$0.636878\pi$
$422$	0	0
$423$	−26.2070	−1.27423
$424$	0	0
$425$	−1.40921	−0.0683567
$426$	0	0
$427$	−1.14215	−0.0552726
$428$	0	0
$429$	7.55402	0.364712
$430$	0	0
$431$	−9.13203	−0.439874	−0.219937	−	0.975514i	$-0.570585\pi$
$431$	−9.13203	−0.439874	−0.219937	+	0.975514i	$0.570585\pi$
$432$	0	0
$433$	23.5045	1.12955	0.564776	−	0.825244i	$-0.308962\pi$
$433$	23.5045	1.12955	0.564776	+	0.825244i	$0.308962\pi$
$434$	0	0
$435$	0.379091	0.0181760
$436$	0	0
$437$	7.77492	0.371925
$438$	0	0
$439$	4.41378	0.210658	0.105329	−	0.994437i	$-0.466410\pi$
$439$	4.41378	0.210658	0.105329	+	0.994437i	$0.466410\pi$
$440$	0	0
$441$	−2.58503	−0.123097
$442$	0	0
$443$	−39.7219	−1.88724	−0.943622	−	0.331026i	$-0.892605\pi$
$443$	−39.7219	−1.88724	−0.943622	+	0.331026i	$0.892605\pi$
$444$	0	0
$445$	−42.5732	−2.01816
$446$	0	0
$447$	−9.87195	−0.466927
$448$	0	0
$449$	18.3900	0.867876	0.433938	−	0.900943i	$-0.357124\pi$
$449$	18.3900	0.867876	0.433938	+	0.900943i	$0.357124\pi$
$450$	0	0
$451$	14.4466	0.680262
$452$	0	0
$453$	−3.97959	−0.186977
$454$	0	0
$455$	13.2981	0.623427
$456$	0	0
$457$	30.5510	1.42911	0.714557	−	0.699577i	$-0.246628\pi$
$457$	30.5510	1.42911	0.714557	+	0.699577i	$0.246628\pi$
$458$	0	0
$459$	3.59777	0.167929
$460$	0	0
$461$	−32.9266	−1.53354	−0.766772	−	0.641920i	$-0.778138\pi$
$461$	−32.9266	−1.53354	−0.766772	+	0.641920i	$0.778138\pi$
$462$	0	0
$463$	27.9500	1.29895	0.649474	−	0.760384i	$-0.274989\pi$
$463$	27.9500	1.29895	0.649474	+	0.760384i	$0.274989\pi$
$464$	0	0
$465$	−10.1796	−0.472066
$466$	0	0
$467$	−23.5747	−1.09091	−0.545453	−	0.838141i	$-0.683643\pi$
$467$	−23.5747	−1.09091	−0.545453	+	0.838141i	$0.683643\pi$
$468$	0	0
$469$	−9.62799	−0.444579
$470$	0	0
$471$	1.95386	0.0900290
$472$	0	0
$473$	9.76953	0.449204
$474$	0	0
$475$	3.62759	0.166445
$476$	0	0
$477$	−16.2211	−0.742714
$478$	0	0
$479$	−16.8208	−0.768563	−0.384281	−	0.923216i	$-0.625551\pi$
$479$	−16.8208	−0.768563	−0.384281	+	0.923216i	$0.625551\pi$
$480$	0	0
$481$	−17.1460	−0.781791
$482$	0	0
$483$	1.94563	0.0885293
$484$	0	0
$485$	1.56142	0.0709006
$486$	0	0
$487$	−26.3615	−1.19456	−0.597278	−	0.802035i	$-0.703751\pi$
$487$	−26.3615	−1.19456	−0.597278	+	0.802035i	$0.703751\pi$
$488$	0	0
$489$	−11.7988	−0.533562
$490$	0	0
$491$	14.1855	0.640182	0.320091	−	0.947387i	$-0.396286\pi$
$491$	14.1855	0.640182	0.320091	+	0.947387i	$0.396286\pi$
$492$	0	0
$493$	0.232452	0.0104691
$494$	0	0
$495$	−14.6100	−0.656671
$496$	0	0
$497$	7.88135	0.353527
$498$	0	0
$499$	17.3945	0.778683	0.389341	−	0.921094i	$-0.372703\pi$
$499$	17.3945	0.778683	0.389341	+	0.921094i	$0.372703\pi$
$500$	0	0
$501$	−3.61513	−0.161512
$502$	0	0
$503$	37.3654	1.66604	0.833021	−	0.553241i	$-0.186609\pi$
$503$	37.3654	1.66604	0.833021	+	0.553241i	$0.186609\pi$
$504$	0	0
$505$	−34.5252	−1.53635
$506$	0	0
$507$	9.39965	0.417453
$508$	0	0
$509$	18.1939	0.806430	0.403215	−	0.915105i	$-0.367893\pi$
$509$	18.1939	0.806430	0.403215	+	0.915105i	$0.367893\pi$
$510$	0	0
$511$	1.67627	0.0741538
$512$	0	0
$513$	−9.26138	−0.408900
$514$	0	0
$515$	20.3068	0.894824
$516$	0	0
$517$	−22.6325	−0.995378
$518$	0	0
$519$	−15.0489	−0.660573
$520$	0	0
$521$	6.70351	0.293686	0.146843	−	0.989160i	$-0.453089\pi$
$521$	6.70351	0.293686	0.146843	+	0.989160i	$0.453089\pi$
$522$	0	0
$523$	4.02474	0.175990	0.0879948	−	0.996121i	$-0.471954\pi$
$523$	4.02474	0.175990	0.0879948	+	0.996121i	$0.471954\pi$
$524$	0	0
$525$	0.907785	0.0396190
$526$	0	0
$527$	−6.24195	−0.271903
$528$	0	0
$529$	−13.8776	−0.603376
$530$	0	0
$531$	−2.93686	−0.127449
$532$	0	0
$533$	33.9915	1.47234
$534$	0	0
$535$	−20.8116	−0.899762
$536$	0	0
$537$	−3.76332	−0.162399
$538$	0	0
$539$	−2.23245	−0.0961585
$540$	0	0
$541$	−13.9857	−0.601291	−0.300646	−	0.953736i	$-0.597202\pi$
$541$	−13.9857	−0.601291	−0.300646	+	0.953736i	$0.597202\pi$
$542$	0	0
$543$	15.5369	0.666752
$544$	0	0
$545$	14.1633	0.606691
$546$	0	0
$547$	−36.1152	−1.54418	−0.772088	−	0.635516i	$-0.780788\pi$
$547$	−36.1152	−1.54418	−0.772088	+	0.635516i	$0.780788\pi$
$548$	0	0
$549$	2.95250	0.126010
$550$	0	0
$551$	−0.598379	−0.0254918
$552$	0	0
$553$	3.38373	0.143891
$554$	0	0
$555$	−5.32334	−0.225963
$556$	0	0
$557$	4.42618	0.187543	0.0937716	−	0.995594i	$-0.470108\pi$
$557$	4.42618	0.187543	0.0937716	+	0.995594i	$0.470108\pi$
$558$	0	0
$559$	22.9869	0.972242
$560$	0	0
$561$	1.43810	0.0607167
$562$	0	0
$563$	29.1911	1.23026	0.615129	−	0.788426i	$-0.289104\pi$
$563$	29.1911	1.23026	0.615129	+	0.788426i	$0.289104\pi$
$564$	0	0
$565$	−13.0890	−0.550659
$566$	0	0
$567$	5.43749	0.228353
$568$	0	0
$569$	−2.84863	−0.119421	−0.0597103	−	0.998216i	$-0.519018\pi$
$569$	−2.84863	−0.119421	−0.0597103	+	0.998216i	$0.519018\pi$
$570$	0	0
$571$	2.68846	0.112508	0.0562542	−	0.998416i	$-0.482084\pi$
$571$	2.68846	0.112508	0.0562542	+	0.998416i	$0.482084\pi$
$572$	0	0
$573$	−7.04433	−0.294281
$574$	0	0
$575$	4.25627	0.177499
$576$	0	0
$577$	−8.47099	−0.352652	−0.176326	−	0.984332i	$-0.556421\pi$
$577$	−8.47099	−0.352652	−0.176326	+	0.984332i	$0.556421\pi$
$578$	0	0
$579$	3.46958	0.144191
$580$	0	0
$581$	15.6628	0.649804
$582$	0	0
$583$	−14.0087	−0.580180
$584$	0	0
$585$	−34.3761	−1.42128
$586$	0	0
$587$	22.1521	0.914316	0.457158	−	0.889385i	$-0.348867\pi$
$587$	22.1521	0.914316	0.457158	+	0.889385i	$0.348867\pi$
$588$	0	0
$589$	16.0680	0.662072
$590$	0	0
$591$	13.0429	0.536513
$592$	0	0
$593$	40.7679	1.67414	0.837068	−	0.547099i	$-0.184268\pi$
$593$	40.7679	1.67414	0.837068	+	0.547099i	$0.184268\pi$
$594$	0	0
$595$	2.53164	0.103787
$596$	0	0
$597$	3.04824	0.124756
$598$	0	0
$599$	−8.20893	−0.335408	−0.167704	−	0.985837i	$-0.553635\pi$
$599$	−8.20893	−0.335408	−0.167704	+	0.985837i	$0.553635\pi$
$600$	0	0
$601$	17.1112	0.697981	0.348990	−	0.937126i	$-0.386525\pi$
$601$	17.1112	0.697981	0.348990	+	0.937126i	$0.386525\pi$
$602$	0	0
$603$	24.8887	1.01355
$604$	0	0
$605$	15.2308	0.619218
$606$	0	0
$607$	7.07815	0.287293	0.143647	−	0.989629i	$-0.454117\pi$
$607$	7.07815	0.287293	0.143647	+	0.989629i	$0.454117\pi$
$608$	0	0
$609$	−0.149741	−0.00606781
$610$	0	0
$611$	−53.2525	−2.15437
$612$	0	0
$613$	−12.7724	−0.515871	−0.257935	−	0.966162i	$-0.583042\pi$
$613$	−12.7724	−0.515871	−0.257935	+	0.966162i	$0.583042\pi$
$614$	0	0
$615$	10.5534	0.425553
$616$	0	0
$617$	−14.2332	−0.573009	−0.286504	−	0.958079i	$-0.592493\pi$
$617$	−14.2332	−0.573009	−0.286504	+	0.958079i	$0.592493\pi$
$618$	0	0
$619$	23.6954	0.952399	0.476199	−	0.879337i	$-0.342014\pi$
$619$	23.6954	0.952399	0.476199	+	0.879337i	$0.342014\pi$
$620$	0	0
$621$	−10.8664	−0.436054
$622$	0	0
$623$	16.8164	0.673736
$624$	0	0
$625$	−30.0602	−1.20241
$626$	0	0
$627$	−3.70196	−0.147842
$628$	0	0
$629$	−3.26418	−0.130152
$630$	0	0
$631$	22.3675	0.890435	0.445217	−	0.895423i	$-0.353126\pi$
$631$	22.3675	0.890435	0.445217	+	0.895423i	$0.353126\pi$
$632$	0	0
$633$	7.74879	0.307987
$634$	0	0
$635$	−18.0668	−0.716959
$636$	0	0
$637$	−5.25278	−0.208123
$638$	0	0
$639$	−20.3735	−0.805965
$640$	0	0
$641$	11.4522	0.452333	0.226167	−	0.974089i	$-0.427381\pi$
$641$	11.4522	0.452333	0.226167	+	0.974089i	$0.427381\pi$
$642$	0	0
$643$	2.27858	0.0898586	0.0449293	−	0.998990i	$-0.485694\pi$
$643$	2.27858	0.0898586	0.0449293	+	0.998990i	$0.485694\pi$
$644$	0	0
$645$	7.13676	0.281010
$646$	0	0
$647$	−5.52484	−0.217204	−0.108602	−	0.994085i	$-0.534637\pi$
$647$	−5.52484	−0.217204	−0.108602	+	0.994085i	$0.534637\pi$
$648$	0	0
$649$	−2.53629	−0.0995582
$650$	0	0
$651$	4.02094	0.157593
$652$	0	0
$653$	9.20106	0.360065	0.180033	−	0.983661i	$-0.442380\pi$
$653$	9.20106	0.360065	0.180033	+	0.983661i	$0.442380\pi$
$654$	0	0
$655$	26.6925	1.04296
$656$	0	0
$657$	−4.33321	−0.169055
$658$	0	0
$659$	2.93917	0.114494	0.0572469	−	0.998360i	$-0.481768\pi$
$659$	2.93917	0.114494	0.0572469	+	0.998360i	$0.481768\pi$
$660$	0	0
$661$	5.41241	0.210518	0.105259	−	0.994445i	$-0.466433\pi$
$661$	5.41241	0.210518	0.105259	+	0.994445i	$0.466433\pi$
$662$	0	0
$663$	3.38373	0.131413
$664$	0	0
$665$	−6.51696	−0.252717
$666$	0	0
$667$	−0.702081	−0.0271847
$668$	0	0
$669$	16.9462	0.655177
$670$	0	0
$671$	2.54980	0.0984339
$672$	0	0
$673$	−44.8633	−1.72935	−0.864677	−	0.502328i	$-0.832477\pi$
$673$	−44.8633	−1.72935	−0.864677	+	0.502328i	$0.832477\pi$
$674$	0	0
$675$	−5.07001	−0.195145
$676$	0	0
$677$	7.92315	0.304511	0.152256	−	0.988341i	$-0.451346\pi$
$677$	7.92315	0.304511	0.152256	+	0.988341i	$0.451346\pi$
$678$	0	0
$679$	−0.616763	−0.0236692
$680$	0	0
$681$	−8.33906	−0.319553
$682$	0	0
$683$	−39.8206	−1.52369	−0.761847	−	0.647757i	$-0.775707\pi$
$683$	−39.8206	−1.52369	−0.761847	+	0.647757i	$0.775707\pi$
$684$	0	0
$685$	−9.07785	−0.346847
$686$	0	0
$687$	7.68564	0.293225
$688$	0	0
$689$	−32.9612	−1.25572
$690$	0	0
$691$	8.92588	0.339557	0.169778	−	0.985482i	$-0.445695\pi$
$691$	8.92588	0.339557	0.169778	+	0.985482i	$0.445695\pi$
$692$	0	0
$693$	5.77096	0.219221
$694$	0	0
$695$	22.5480	0.855293
$696$	0	0
$697$	6.47116	0.245113
$698$	0	0
$699$	11.5664	0.437480
$700$	0	0
$701$	−41.1490	−1.55418	−0.777088	−	0.629392i	$-0.783304\pi$
$701$	−41.1490	−1.55418	−0.777088	+	0.629392i	$0.783304\pi$
$702$	0	0
$703$	8.40267	0.316913
$704$	0	0
$705$	−16.5333	−0.622682
$706$	0	0
$707$	13.6375	0.512890
$708$	0	0
$709$	−0.447742	−0.0168153	−0.00840766	−	0.999965i	$-0.502676\pi$
$709$	−0.447742	−0.0168153	−0.00840766	+	0.999965i	$0.502676\pi$
$710$	0	0
$711$	−8.74706	−0.328040
$712$	0	0
$713$	18.8527	0.706039
$714$	0	0
$715$	−29.6875	−1.11025
$716$	0	0
$717$	12.2570	0.457746
$718$	0	0
$719$	−23.5887	−0.879712	−0.439856	−	0.898068i	$-0.644970\pi$
$719$	−23.5887	−0.879712	−0.439856	+	0.898068i	$0.644970\pi$
$720$	0	0
$721$	−8.02119	−0.298725
$722$	0	0
$723$	−15.9932	−0.594795
$724$	0	0
$725$	−0.327574	−0.0121658
$726$	0	0
$727$	−22.0040	−0.816082	−0.408041	−	0.912964i	$-0.633788\pi$
$727$	−22.0040	−0.816082	−0.408041	+	0.912964i	$0.633788\pi$
$728$	0	0
$729$	−7.10325	−0.263083
$730$	0	0
$731$	4.37614	0.161858
$732$	0	0
$733$	−25.7809	−0.952238	−0.476119	−	0.879381i	$-0.657957\pi$
$733$	−25.7809	−0.952238	−0.476119	+	0.879381i	$0.657957\pi$
$734$	0	0
$735$	−1.63083	−0.0601542
$736$	0	0
$737$	21.4940	0.791743
$738$	0	0
$739$	−25.3808	−0.933647	−0.466823	−	0.884351i	$-0.654602\pi$
$739$	−25.3808	−0.933647	−0.466823	+	0.884351i	$0.654602\pi$
$740$	0	0
$741$	−8.71041	−0.319985
$742$	0	0
$743$	14.4501	0.530122	0.265061	−	0.964232i	$-0.414608\pi$
$743$	14.4501	0.530122	0.265061	+	0.964232i	$0.414608\pi$
$744$	0	0
$745$	38.7970	1.42141
$746$	0	0
$747$	−40.4890	−1.48141
$748$	0	0
$749$	8.22058	0.300373
$750$	0	0
$751$	49.9082	1.82118	0.910588	−	0.413316i	$-0.135630\pi$
$751$	49.9082	1.82118	0.910588	+	0.413316i	$0.135630\pi$
$752$	0	0
$753$	6.86704	0.250249
$754$	0	0
$755$	15.6399	0.569193
$756$	0	0
$757$	29.6587	1.07796	0.538982	−	0.842317i	$-0.318809\pi$
$757$	29.6587	1.07796	0.538982	+	0.842317i	$0.318809\pi$
$758$	0	0
$759$	−4.34353	−0.157660
$760$	0	0
$761$	5.59792	0.202924	0.101462	−	0.994839i	$-0.467648\pi$
$761$	5.59792	0.202924	0.101462	+	0.994839i	$0.467648\pi$
$762$	0	0
$763$	−5.59453	−0.202535
$764$	0	0
$765$	−6.54438	−0.236612
$766$	0	0
$767$	−5.96769	−0.215481
$768$	0	0
$769$	13.5885	0.490014	0.245007	−	0.969521i	$-0.421210\pi$
$769$	13.5885	0.490014	0.245007	+	0.969521i	$0.421210\pi$
$770$	0	0
$771$	12.9193	0.465278
$772$	0	0
$773$	−20.2316	−0.727679	−0.363840	−	0.931462i	$-0.618534\pi$
$773$	−20.2316	−0.727679	−0.363840	+	0.931462i	$0.618534\pi$
$774$	0	0
$775$	8.79621	0.315969
$776$	0	0
$777$	2.10272	0.0754347
$778$	0	0
$779$	−16.6581	−0.596837
$780$	0	0
$781$	−17.5947	−0.629589
$782$	0	0
$783$	0.836309	0.0298872
$784$	0	0
$785$	−7.67870	−0.274065
$786$	0	0
$787$	42.0470	1.49882	0.749408	−	0.662109i	$-0.230338\pi$
$787$	42.0470	1.49882	0.749408	+	0.662109i	$0.230338\pi$
$788$	0	0
$789$	−9.54777	−0.339910
$790$	0	0
$791$	5.17017	0.183830
$792$	0	0
$793$	5.99947	0.213047
$794$	0	0
$795$	−10.2335	−0.362945
$796$	0	0
$797$	7.91996	0.280539	0.140270	−	0.990113i	$-0.455203\pi$
$797$	7.91996	0.280539	0.140270	+	0.990113i	$0.455203\pi$
$798$	0	0
$799$	−10.1380	−0.358656
$800$	0	0
$801$	−43.4710	−1.53597
$802$	0	0
$803$	−3.74219	−0.132059
$804$	0	0
$805$	−7.64638	−0.269499
$806$	0	0
$807$	17.5077	0.616302
$808$	0	0
$809$	54.6264	1.92056	0.960281	−	0.279036i	$-0.0900148\pi$
$809$	54.6264	1.92056	0.960281	+	0.279036i	$0.0900148\pi$
$810$	0	0
$811$	−37.4955	−1.31664	−0.658322	−	0.752736i	$-0.728734\pi$
$811$	−37.4955	−1.31664	−0.658322	+	0.752736i	$0.728734\pi$
$812$	0	0
$813$	11.8798	0.416643
$814$	0	0
$815$	46.3697	1.62426
$816$	0	0
$817$	−11.2651	−0.394115
$818$	0	0
$819$	13.5786	0.474474
$820$	0	0
$821$	12.0468	0.420435	0.210217	−	0.977655i	$-0.432583\pi$
$821$	12.0468	0.420435	0.210217	+	0.977655i	$0.432583\pi$
$822$	0	0
$823$	−19.9529	−0.695513	−0.347756	−	0.937585i	$-0.613056\pi$
$823$	−19.9529	−0.695513	−0.347756	+	0.937585i	$0.613056\pi$
$824$	0	0
$825$	−2.02659	−0.0705567
$826$	0	0
$827$	−12.6415	−0.439589	−0.219794	−	0.975546i	$-0.570539\pi$
$827$	−12.6415	−0.439589	−0.219794	+	0.975546i	$0.570539\pi$
$828$	0	0
$829$	−10.1202	−0.351489	−0.175744	−	0.984436i	$-0.556233\pi$
$829$	−10.1202	−0.351489	−0.175744	+	0.984436i	$0.556233\pi$
$830$	0	0
$831$	−5.95918	−0.206722
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	14.2075	0.491672
$836$	0	0
$837$	−22.4571	−0.776230
$838$	0	0
$839$	44.0050	1.51922	0.759610	−	0.650379i	$-0.225390\pi$
$839$	44.0050	1.51922	0.759610	+	0.650379i	$0.225390\pi$
$840$	0	0
$841$	−28.9460	−0.998137
$842$	0	0
$843$	1.40651	0.0484428
$844$	0	0
$845$	−36.9408	−1.27080
$846$	0	0
$847$	−6.01616	−0.206718
$848$	0	0
$849$	−6.00523	−0.206099
$850$	0	0
$851$	9.85889	0.337958
$852$	0	0
$853$	−25.7767	−0.882576	−0.441288	−	0.897365i	$-0.645478\pi$
$853$	−25.7767	−0.882576	−0.441288	+	0.897365i	$0.645478\pi$
$854$	0	0
$855$	16.8465	0.576140
$856$	0	0
$857$	−19.8662	−0.678616	−0.339308	−	0.940675i	$-0.610193\pi$
$857$	−19.8662	−0.678616	−0.339308	+	0.940675i	$0.610193\pi$
$858$	0	0
$859$	14.2553	0.486385	0.243193	−	0.969978i	$-0.421805\pi$
$859$	14.2553	0.486385	0.243193	+	0.969978i	$0.421805\pi$
$860$	0	0
$861$	−4.16859	−0.142065
$862$	0	0
$863$	3.41168	0.116135	0.0580674	−	0.998313i	$-0.481506\pi$
$863$	3.41168	0.116135	0.0580674	+	0.998313i	$0.481506\pi$
$864$	0	0
$865$	59.1425	2.01090
$866$	0	0
$867$	0.644180	0.0218775
$868$	0	0
$869$	−7.55402	−0.256253
$870$	0	0
$871$	50.5737	1.71362
$872$	0	0
$873$	1.59435	0.0539607
$874$	0	0
$875$	9.09059	0.307318
$876$	0	0
$877$	20.4852	0.691738	0.345869	−	0.938283i	$-0.387584\pi$
$877$	20.4852	0.691738	0.345869	+	0.938283i	$0.387584\pi$
$878$	0	0
$879$	3.24609	0.109488
$880$	0	0
$881$	4.91501	0.165591	0.0827955	−	0.996567i	$-0.473615\pi$
$881$	4.91501	0.165591	0.0827955	+	0.996567i	$0.473615\pi$
$882$	0	0
$883$	−22.5908	−0.760241	−0.380121	−	0.924937i	$-0.624118\pi$
$883$	−22.5908	−0.760241	−0.380121	+	0.924937i	$0.624118\pi$
$884$	0	0
$885$	−1.85279	−0.0622809
$886$	0	0
$887$	−32.9356	−1.10587	−0.552934	−	0.833225i	$-0.686492\pi$
$887$	−32.9356	−1.10587	−0.552934	+	0.833225i	$0.686492\pi$
$888$	0	0
$889$	7.13639	0.239347
$890$	0	0
$891$	−12.1389	−0.406670
$892$	0	0
$893$	26.0972	0.873310
$894$	0	0
$895$	14.7899	0.494372
$896$	0	0
$897$	−10.2200	−0.341235
$898$	0	0
$899$	−1.45095	−0.0483920
$900$	0	0
$901$	−6.27501	−0.209051
$902$	0	0
$903$	−2.81902	−0.0938112
$904$	0	0
$905$	−61.0603	−2.02971
$906$	0	0
$907$	−13.4000	−0.444939	−0.222469	−	0.974940i	$-0.571412\pi$
$907$	−13.4000	−0.444939	−0.222469	+	0.974940i	$0.571412\pi$
$908$	0	0
$909$	−35.2533	−1.16928
$910$	0	0
$911$	−28.1036	−0.931113	−0.465557	−	0.885018i	$-0.654146\pi$
$911$	−28.1036	−0.931113	−0.465557	+	0.885018i	$0.654146\pi$
$912$	0	0
$913$	−34.9666	−1.15722
$914$	0	0
$915$	1.86266	0.0615776
$916$	0	0
$917$	−10.5435	−0.348179
$918$	0	0
$919$	49.2243	1.62376	0.811881	−	0.583823i	$-0.198444\pi$
$919$	49.2243	1.62376	0.811881	+	0.583823i	$0.198444\pi$
$920$	0	0
$921$	−5.24640	−0.172875
$922$	0	0
$923$	−41.3989	−1.36266
$924$	0	0
$925$	4.59992	0.151244
$926$	0	0
$927$	20.7350	0.681028
$928$	0	0
$929$	6.33709	0.207913	0.103957	−	0.994582i	$-0.466850\pi$
$929$	6.33709	0.207913	0.103957	+	0.994582i	$0.466850\pi$
$930$	0	0
$931$	2.57420	0.0843661
$932$	0	0
$933$	1.37664	0.0450691
$934$	0	0
$935$	−5.65177	−0.184833
$936$	0	0
$937$	−1.88941	−0.0617244	−0.0308622	−	0.999524i	$-0.509825\pi$
$937$	−1.88941	−0.0617244	−0.0308622	+	0.999524i	$0.509825\pi$
$938$	0	0
$939$	−5.18173	−0.169099
$940$	0	0
$941$	12.7804	0.416630	0.208315	−	0.978062i	$-0.433202\pi$
$941$	12.7804	0.416630	0.208315	+	0.978062i	$0.433202\pi$
$942$	0	0
$943$	−19.5450	−0.636472
$944$	0	0
$945$	9.10825	0.296292
$946$	0	0
$947$	59.9664	1.94865	0.974323	−	0.225153i	$-0.0722884\pi$
$947$	59.9664	1.94865	0.974323	+	0.225153i	$0.0722884\pi$
$948$	0	0
$949$	−8.80507	−0.285825
$950$	0	0
$951$	16.9674	0.550206
$952$	0	0
$953$	−42.4980	−1.37664	−0.688322	−	0.725405i	$-0.741653\pi$
$953$	−42.4980	−1.37664	−0.688322	+	0.725405i	$0.741653\pi$
$954$	0	0
$955$	27.6844	0.895845
$956$	0	0
$957$	0.334290	0.0108061
$958$	0	0
$959$	3.58575	0.115790
$960$	0	0
$961$	7.96189	0.256835
$962$	0	0
$963$	−21.2505	−0.684787
$964$	0	0
$965$	−13.6355	−0.438944
$966$	0	0
$967$	−5.63019	−0.181055	−0.0905273	−	0.995894i	$-0.528855\pi$
$967$	−5.63019	−0.181055	−0.0905273	+	0.995894i	$0.528855\pi$
$968$	0	0
$969$	−1.65825	−0.0532707
$970$	0	0
$971$	−33.7682	−1.08367	−0.541836	−	0.840484i	$-0.682271\pi$
$971$	−33.7682	−1.08367	−0.541836	+	0.840484i	$0.682271\pi$
$972$	0	0
$973$	−8.90646	−0.285528
$974$	0	0
$975$	−4.76839	−0.152711
$976$	0	0
$977$	−41.3432	−1.32269	−0.661344	−	0.750083i	$-0.730013\pi$
$977$	−41.3432	−1.32269	−0.661344	+	0.750083i	$0.730013\pi$
$978$	0	0
$979$	−37.5419	−1.19984
$980$	0	0
$981$	14.4620	0.461737
$982$	0	0
$983$	9.91633	0.316282	0.158141	−	0.987417i	$-0.449450\pi$
$983$	9.91633	0.316282	0.158141	+	0.987417i	$0.449450\pi$
$984$	0	0
$985$	−51.2588	−1.63324
$986$	0	0
$987$	6.53068	0.207874
$988$	0	0
$989$	−13.2174	−0.420288
$990$	0	0
$991$	−59.2450	−1.88198	−0.940990	−	0.338434i	$-0.890103\pi$
$991$	−59.2450	−1.88198	−0.940990	+	0.338434i	$0.890103\pi$
$992$	0	0
$993$	10.2479	0.325208
$994$	0	0
$995$	−11.9797	−0.379781
$996$	0	0
$997$	15.0270	0.475911	0.237956	−	0.971276i	$-0.423523\pi$
$997$	15.0270	0.475911	0.237956	+	0.971276i	$0.423523\pi$
$998$	0	0
$999$	−11.7438	−0.371556

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.l.1.4	yes	6
4.3	odd	2		3808.2.a.k.1.3	✓	6
8.3	odd	2		7616.2.a.by.1.4		6
8.5	even	2		7616.2.a.bz.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.k.1.3	✓	6	4.3	odd	2
3808.2.a.l.1.4	yes	6	1.1	even	1	trivial
7616.2.a.by.1.4		6	8.3	odd	2
7616.2.a.bz.1.3		6	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q+0.644180 q^{3} -2.53164 q^{5} +1.00000 q^{7} -2.58503 q^{9} +O(q^{10})\)	\(q+0.644180 q^{3} -2.53164 q^{5} +1.00000 q^{7} -2.58503 q^{9} -2.23245 q^{11} -5.25278 q^{13} -1.63083 q^{15} -1.00000 q^{17} +2.57420 q^{19} +0.644180 q^{21} +3.02032 q^{23} +1.40921 q^{25} -3.59777 q^{27} -0.232452 q^{29} +6.24195 q^{31} -1.43810 q^{33} -2.53164 q^{35} +3.26418 q^{37} -3.38373 q^{39} -6.47116 q^{41} -4.37614 q^{43} +6.54438 q^{45} +10.1380 q^{47} +1.00000 q^{49} -0.644180 q^{51} +6.27501 q^{53} +5.65177 q^{55} +1.65825 q^{57} +1.13610 q^{59} -1.14215 q^{61} -2.58503 q^{63} +13.2981 q^{65} -9.62799 q^{67} +1.94563 q^{69} +7.88135 q^{71} +1.67627 q^{73} +0.907785 q^{75} -2.23245 q^{77} +3.38373 q^{79} +5.43749 q^{81} +15.6628 q^{83} +2.53164 q^{85} -0.149741 q^{87} +16.8164 q^{89} -5.25278 q^{91} +4.02094 q^{93} -6.51696 q^{95} -0.616763 q^{97} +5.77096 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{5} + 6 q^{7} + 8 q^{9}+O(q^{10}) \) 6 * q + 4 * q^5 + 6 * q^7 + 8 * q^9	\( 6 q + 4 q^{5} + 6 q^{7} + 8 q^{9} - 4 q^{11} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 6 q^{19} + 8 q^{25} + 8 q^{29} + 12 q^{31} + 28 q^{33} + 4 q^{35} + 10 q^{37} + 14 q^{39} + 14 q^{41} - 12 q^{43} + 22 q^{45} + 2 q^{47} + 6 q^{49} + 26 q^{53} + 10 q^{55} + 22 q^{57} + 22 q^{59} + 2 q^{61} + 8 q^{63} + 8 q^{65} - 14 q^{67} + 14 q^{69} - 4 q^{71} + 18 q^{73} - 28 q^{75} - 4 q^{77} - 14 q^{79} - 6 q^{81} + 8 q^{83} - 4 q^{85} + 28 q^{87} + 6 q^{89} - 4 q^{91} + 8 q^{93} - 2 q^{95} + 20 q^{97} - 26 q^{99}+O(q^{100}) \) 6 * q + 4 * q^5 + 6 * q^7 + 8 * q^9 - 4 * q^11 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 6 * q^19 + 8 * q^25 + 8 * q^29 + 12 * q^31 + 28 * q^33 + 4 * q^35 + 10 * q^37 + 14 * q^39 + 14 * q^41 - 12 * q^43 + 22 * q^45 + 2 * q^47 + 6 * q^49 + 26 * q^53 + 10 * q^55 + 22 * q^57 + 22 * q^59 + 2 * q^61 + 8 * q^63 + 8 * q^65 - 14 * q^67 + 14 * q^69 - 4 * q^71 + 18 * q^73 - 28 * q^75 - 4 * q^77 - 14 * q^79 - 6 * q^81 + 8 * q^83 - 4 * q^85 + 28 * q^87 + 6 * q^89 - 4 * q^91 + 8 * q^93 - 2 * q^95 + 20 * q^97 - 26 * q^99

Introduction

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