LMFDB - Embedded newform 3808.2.a.i.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:	$1$
Dimension:	$6$
Coefficient field:	6.6.80686992.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 15x^{3} + 8x^{2} - 15x + 3 $ x^6 - x^5 - 10x^4 + 15x^3 + 8x^2 - 15x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.856749$ 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)

$f(q)$	$=$	$q+0.601565 q^{3} -4.19936 q^{5} +1.00000 q^{7} -2.63812 q^{9} +2.71475 q^{11} +5.83748 q^{13} -2.52619 q^{15} -1.00000 q^{17} -6.47912 q^{19} +0.601565 q^{21} -3.86701 q^{23} +12.6346 q^{25} -3.39169 q^{27} +0.714752 q^{29} +9.96821 q^{31} +1.63310 q^{33} -4.19936 q^{35} +7.80796 q^{37} +3.51162 q^{39} -3.82668 q^{41} -0.681963 q^{43} +11.0784 q^{45} -7.53322 q^{47} +1.00000 q^{49} -0.601565 q^{51} -7.25346 q^{53} -11.4002 q^{55} -3.89761 q^{57} +1.29375 q^{59} -11.1656 q^{61} -2.63812 q^{63} -24.5137 q^{65} -12.4769 q^{67} -2.32625 q^{69} +6.76563 q^{71} +1.60282 q^{73} +7.60055 q^{75} +2.71475 q^{77} +16.3681 q^{79} +5.87404 q^{81} -8.58284 q^{83} +4.19936 q^{85} +0.429969 q^{87} -9.67049 q^{89} +5.83748 q^{91} +5.99652 q^{93} +27.2082 q^{95} +9.18910 q^{97} -7.16184 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} + 2 q^{15} - 6 q^{17} - 10 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} - 8 q^{27} - 14 q^{29} + 8 q^{31} - 8 q^{33} - 6 q^{35} - 4 q^{37} + 14 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 + 2 * q^15 - 6 * q^17 - 10 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 - 8 * q^27 - 14 * q^29 + 8 * q^31 - 8 * q^33 - 6 * q^35 - 4 * q^37 + 14 * q^39 - 2 * q^41 - 8 * q^43 - 8 * q^45 - 2 * q^47 + 6 * q^49 + 2 * q^51 - 10 * q^53 + 2 * q^55 - 18 * q^57 - 14 * q^59 - 20 * q^61 + 4 * q^63 - 20 * q^65 - 6 * q^67 - 14 * q^69 + 20 * q^71 - 6 * q^73 - 10 * q^75 - 2 * q^77 + 38 * q^79 - 18 * q^81 - 16 * q^83 + 6 * q^85 - 4 * q^87 - 2 * q^89 - 4 * q^91 - 16 * q^93 + 42 * q^95 - 12 * q^97 - 4 * q^99

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
$3$	0.601565	0.347313	0.173657	−	0.984806i	$-0.444442\pi$
$3$	0.601565	0.347313	0.173657	+	0.984806i	$0.444442\pi$
$4$	0	0
$5$	−4.19936	−1.87801	−0.939006	−	0.343901i	$-0.888252\pi$
$5$	−4.19936	−1.87801	−0.939006	+	0.343901i	$0.888252\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.63812	−0.879373
$10$	0	0
$11$	2.71475	0.818528	0.409264	−	0.912416i	$-0.365785\pi$
$11$	2.71475	0.818528	0.409264	+	0.912416i	$0.365785\pi$
$12$	0	0
$13$	5.83748	1.61903	0.809513	−	0.587102i	$-0.199731\pi$
$13$	5.83748	1.61903	0.809513	+	0.587102i	$0.199731\pi$
$14$	0	0
$15$	−2.52619	−0.652259
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.47912	−1.48641	−0.743206	−	0.669062i	$-0.766696\pi$
$19$	−6.47912	−1.48641	−0.743206	+	0.669062i	$0.766696\pi$
$20$	0	0
$21$	0.601565	0.131272
$22$	0	0
$23$	−3.86701	−0.806327	−0.403163	−	0.915128i	$-0.632089\pi$
$23$	−3.86701	−0.806327	−0.403163	+	0.915128i	$0.632089\pi$
$24$	0	0
$25$	12.6346	2.52693
$26$	0	0
$27$	−3.39169	−0.652732
$28$	0	0
$29$	0.714752	0.132726	0.0663630	−	0.997796i	$-0.478860\pi$
$29$	0.714752	0.132726	0.0663630	+	0.997796i	$0.478860\pi$
$30$	0	0
$31$	9.96821	1.79034	0.895172	−	0.445722i	$-0.147053\pi$
$31$	9.96821	1.79034	0.895172	+	0.445722i	$0.147053\pi$
$32$	0	0
$33$	1.63310	0.284286
$34$	0	0
$35$	−4.19936	−0.709822
$36$	0	0
$37$	7.80796	1.28362	0.641810	−	0.766864i	$-0.278184\pi$
$37$	7.80796	1.28362	0.641810	+	0.766864i	$0.278184\pi$
$38$	0	0
$39$	3.51162	0.562310
$40$	0	0
$41$	−3.82668	−0.597628	−0.298814	−	0.954311i	$-0.596591\pi$
$41$	−3.82668	−0.597628	−0.298814	+	0.954311i	$0.596591\pi$
$42$	0	0
$43$	−0.681963	−0.103998	−0.0519992	−	0.998647i	$-0.516559\pi$
$43$	−0.681963	−0.103998	−0.0519992	+	0.998647i	$0.516559\pi$
$44$	0	0
$45$	11.0784	1.65147
$46$	0	0
$47$	−7.53322	−1.09883	−0.549416	−	0.835549i	$-0.685150\pi$
$47$	−7.53322	−1.09883	−0.549416	+	0.835549i	$0.685150\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.601565	−0.0842359
$52$	0	0
$53$	−7.25346	−0.996339	−0.498170	−	0.867080i	$-0.665994\pi$
$53$	−7.25346	−0.996339	−0.498170	+	0.867080i	$0.665994\pi$
$54$	0	0
$55$	−11.4002	−1.53721
$56$	0	0
$57$	−3.89761	−0.516251
$58$	0	0
$59$	1.29375	0.168433	0.0842163	−	0.996448i	$-0.473161\pi$
$59$	1.29375	0.168433	0.0842163	+	0.996448i	$0.473161\pi$
$60$	0	0
$61$	−11.1656	−1.42960	−0.714802	−	0.699327i	$-0.753483\pi$
$61$	−11.1656	−1.42960	−0.714802	+	0.699327i	$0.753483\pi$
$62$	0	0
$63$	−2.63812	−0.332372
$64$	0	0
$65$	−24.5137	−3.04055
$66$	0	0
$67$	−12.4769	−1.52429	−0.762145	−	0.647406i	$-0.775854\pi$
$67$	−12.4769	−1.52429	−0.762145	+	0.647406i	$0.775854\pi$
$68$	0	0
$69$	−2.32625	−0.280048
$70$	0	0
$71$	6.76563	0.802932	0.401466	−	0.915874i	$-0.368501\pi$
$71$	6.76563	0.802932	0.401466	+	0.915874i	$0.368501\pi$
$72$	0	0
$73$	1.60282	0.187596	0.0937979	−	0.995591i	$-0.470099\pi$
$73$	1.60282	0.187596	0.0937979	+	0.995591i	$0.470099\pi$
$74$	0	0
$75$	7.60055	0.877636
$76$	0	0
$77$	2.71475	0.309375
$78$	0	0
$79$	16.3681	1.84156	0.920779	−	0.390085i	$-0.127555\pi$
$79$	16.3681	1.84156	0.920779	+	0.390085i	$0.127555\pi$
$80$	0	0
$81$	5.87404	0.652671
$82$	0	0
$83$	−8.58284	−0.942089	−0.471044	−	0.882109i	$-0.656123\pi$
$83$	−8.58284	−0.942089	−0.471044	+	0.882109i	$0.656123\pi$
$84$	0	0
$85$	4.19936	0.455485
$86$	0	0
$87$	0.429969	0.0460975
$88$	0	0
$89$	−9.67049	−1.02507	−0.512535	−	0.858666i	$-0.671293\pi$
$89$	−9.67049	−1.02507	−0.512535	+	0.858666i	$0.671293\pi$
$90$	0	0
$91$	5.83748	0.611934
$92$	0	0
$93$	5.99652	0.621810
$94$	0	0
$95$	27.2082	2.79150
$96$	0	0
$97$	9.18910	0.933012	0.466506	−	0.884518i	$-0.345513\pi$
$97$	9.18910	0.933012	0.466506	+	0.884518i	$0.345513\pi$
$98$	0	0
$99$	−7.16184	−0.719792
$100$	0	0
$101$	2.32586	0.231432	0.115716	−	0.993282i	$-0.463084\pi$
$101$	2.32586	0.231432	0.115716	+	0.993282i	$0.463084\pi$
$102$	0	0
$103$	−0.0433398	−0.00427039	−0.00213520	−	0.999998i	$-0.500680\pi$
$103$	−0.0433398	−0.00427039	−0.00213520	+	0.999998i	$0.500680\pi$
$104$	0	0
$105$	−2.52619	−0.246531
$106$	0	0
$107$	1.17336	0.113433	0.0567164	−	0.998390i	$-0.481937\pi$
$107$	1.17336	0.113433	0.0567164	+	0.998390i	$0.481937\pi$
$108$	0	0
$109$	−17.7481	−1.69996	−0.849979	−	0.526817i	$-0.823385\pi$
$109$	−17.7481	−1.69996	−0.849979	+	0.526817i	$0.823385\pi$
$110$	0	0
$111$	4.69699	0.445819
$112$	0	0
$113$	−15.3449	−1.44352	−0.721762	−	0.692141i	$-0.756668\pi$
$113$	−15.3449	−1.44352	−0.721762	+	0.692141i	$0.756668\pi$
$114$	0	0
$115$	16.2390	1.51429
$116$	0	0
$117$	−15.4000	−1.42373
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−3.63012	−0.330011
$122$	0	0
$123$	−2.30200	−0.207564
$124$	0	0
$125$	−32.0606	−2.86759
$126$	0	0
$127$	2.17737	0.193210	0.0966052	−	0.995323i	$-0.469202\pi$
$127$	2.17737	0.193210	0.0966052	+	0.995323i	$0.469202\pi$
$128$	0	0
$129$	−0.410245	−0.0361201
$130$	0	0
$131$	−12.3361	−1.07781	−0.538904	−	0.842367i	$-0.681162\pi$
$131$	−12.3361	−1.07781	−0.538904	+	0.842367i	$0.681162\pi$
$132$	0	0
$133$	−6.47912	−0.561811
$134$	0	0
$135$	14.2429	1.22584
$136$	0	0
$137$	−6.22867	−0.532151	−0.266075	−	0.963952i	$-0.585727\pi$
$137$	−6.22867	−0.532151	−0.266075	+	0.963952i	$0.585727\pi$
$138$	0	0
$139$	−21.1112	−1.79063	−0.895315	−	0.445434i	$-0.853049\pi$
$139$	−21.1112	−1.79063	−0.895315	+	0.445434i	$0.853049\pi$
$140$	0	0
$141$	−4.53172	−0.381639
$142$	0	0
$143$	15.8473	1.32522
$144$	0	0
$145$	−3.00150	−0.249261
$146$	0	0
$147$	0.601565	0.0496162
$148$	0	0
$149$	−11.8305	−0.969194	−0.484597	−	0.874738i	$-0.661034\pi$
$149$	−11.8305	−0.969194	−0.484597	+	0.874738i	$0.661034\pi$
$150$	0	0
$151$	0.273196	0.0222324	0.0111162	−	0.999938i	$-0.496462\pi$
$151$	0.273196	0.0222324	0.0111162	+	0.999938i	$0.496462\pi$
$152$	0	0
$153$	2.63812	0.213279
$154$	0	0
$155$	−41.8601	−3.36229
$156$	0	0
$157$	−11.8928	−0.949151	−0.474576	−	0.880215i	$-0.657398\pi$
$157$	−11.8928	−0.949151	−0.474576	+	0.880215i	$0.657398\pi$
$158$	0	0
$159$	−4.36342	−0.346042
$160$	0	0
$161$	−3.86701	−0.304763
$162$	0	0
$163$	−20.8222	−1.63092	−0.815460	−	0.578813i	$-0.803516\pi$
$163$	−20.8222	−1.63092	−0.815460	+	0.578813i	$0.803516\pi$
$164$	0	0
$165$	−6.85797	−0.533892
$166$	0	0
$167$	−8.96118	−0.693437	−0.346718	−	0.937969i	$-0.612704\pi$
$167$	−8.96118	−0.693437	−0.346718	+	0.937969i	$0.612704\pi$
$168$	0	0
$169$	21.0762	1.62125
$170$	0	0
$171$	17.0927	1.30711
$172$	0	0
$173$	20.2835	1.54213	0.771063	−	0.636758i	$-0.219725\pi$
$173$	20.2835	1.54213	0.771063	+	0.636758i	$0.219725\pi$
$174$	0	0
$175$	12.6346	0.955089
$176$	0	0
$177$	0.778277	0.0584989
$178$	0	0
$179$	9.56350	0.714809	0.357405	−	0.933950i	$-0.383662\pi$
$179$	9.56350	0.714809	0.357405	+	0.933950i	$0.383662\pi$
$180$	0	0
$181$	−10.2199	−0.759641	−0.379820	−	0.925060i	$-0.624014\pi$
$181$	−10.2199	−0.759641	−0.379820	+	0.925060i	$0.624014\pi$
$182$	0	0
$183$	−6.71681	−0.496521
$184$	0	0
$185$	−32.7884	−2.41065
$186$	0	0
$187$	−2.71475	−0.198522
$188$	0	0
$189$	−3.39169	−0.246709
$190$	0	0
$191$	24.6797	1.78576	0.892880	−	0.450294i	$-0.148681\pi$
$191$	24.6797	1.78576	0.892880	+	0.450294i	$0.148681\pi$
$192$	0	0
$193$	−18.7981	−1.35312	−0.676558	−	0.736389i	$-0.736529\pi$
$193$	−18.7981	−1.35312	−0.676558	+	0.736389i	$0.736529\pi$
$194$	0	0
$195$	−14.7466	−1.05602
$196$	0	0
$197$	7.43530	0.529744	0.264872	−	0.964284i	$-0.414670\pi$
$197$	7.43530	0.529744	0.264872	+	0.964284i	$0.414670\pi$
$198$	0	0
$199$	−23.3389	−1.65445	−0.827225	−	0.561871i	$-0.810082\pi$
$199$	−23.3389	−1.65445	−0.827225	+	0.561871i	$0.810082\pi$
$200$	0	0
$201$	−7.50564	−0.529407
$202$	0	0
$203$	0.714752	0.0501657
$204$	0	0
$205$	16.0696	1.12235
$206$	0	0
$207$	10.2016	0.709062
$208$	0	0
$209$	−17.5892	−1.21667
$210$	0	0
$211$	−26.5094	−1.82498	−0.912492	−	0.409095i	$-0.865845\pi$
$211$	−26.5094	−1.82498	−0.912492	+	0.409095i	$0.865845\pi$
$212$	0	0
$213$	4.06996	0.278869
$214$	0	0
$215$	2.86381	0.195310
$216$	0	0
$217$	9.96821	0.676686
$218$	0	0
$219$	0.964199	0.0651545
$220$	0	0
$221$	−5.83748	−0.392672
$222$	0	0
$223$	5.63310	0.377220	0.188610	−	0.982052i	$-0.439602\pi$
$223$	5.63310	0.377220	0.188610	+	0.982052i	$0.439602\pi$
$224$	0	0
$225$	−33.3317	−2.22211
$226$	0	0
$227$	−2.60774	−0.173081	−0.0865407	−	0.996248i	$-0.527581\pi$
$227$	−2.60774	−0.173081	−0.0865407	+	0.996248i	$0.527581\pi$
$228$	0	0
$229$	−8.18328	−0.540766	−0.270383	−	0.962753i	$-0.587150\pi$
$229$	−8.18328	−0.540766	−0.270383	+	0.962753i	$0.587150\pi$
$230$	0	0
$231$	1.63310	0.107450
$232$	0	0
$233$	−8.27277	−0.541967	−0.270984	−	0.962584i	$-0.587349\pi$
$233$	−8.27277	−0.541967	−0.270984	+	0.962584i	$0.587349\pi$
$234$	0	0
$235$	31.6347	2.06362
$236$	0	0
$237$	9.84648	0.639598
$238$	0	0
$239$	−9.06562	−0.586406	−0.293203	−	0.956050i	$-0.594721\pi$
$239$	−9.06562	−0.586406	−0.293203	+	0.956050i	$0.594721\pi$
$240$	0	0
$241$	−2.34859	−0.151286	−0.0756431	−	0.997135i	$-0.524101\pi$
$241$	−2.34859	−0.151286	−0.0756431	+	0.997135i	$0.524101\pi$
$242$	0	0
$243$	13.7087	0.879413
$244$	0	0
$245$	−4.19936	−0.268287
$246$	0	0
$247$	−37.8218	−2.40654
$248$	0	0
$249$	−5.16313	−0.327200
$250$	0	0
$251$	15.4228	0.973478	0.486739	−	0.873547i	$-0.338186\pi$
$251$	15.4228	0.973478	0.486739	+	0.873547i	$0.338186\pi$
$252$	0	0
$253$	−10.4980	−0.660001
$254$	0	0
$255$	2.52619	0.158196
$256$	0	0
$257$	12.4305	0.775396	0.387698	−	0.921786i	$-0.373270\pi$
$257$	12.4305	0.775396	0.387698	+	0.921786i	$0.373270\pi$
$258$	0	0
$259$	7.80796	0.485163
$260$	0	0
$261$	−1.88560	−0.116716
$262$	0	0
$263$	22.7654	1.40377	0.701886	−	0.712289i	$-0.252341\pi$
$263$	22.7654	1.40377	0.701886	+	0.712289i	$0.252341\pi$
$264$	0	0
$265$	30.4599	1.87114
$266$	0	0
$267$	−5.81742	−0.356021
$268$	0	0
$269$	−21.4176	−1.30585	−0.652927	−	0.757421i	$-0.726459\pi$
$269$	−21.4176	−1.30585	−0.652927	+	0.757421i	$0.726459\pi$
$270$	0	0
$271$	4.38599	0.266430	0.133215	−	0.991087i	$-0.457470\pi$
$271$	4.38599	0.266430	0.133215	+	0.991087i	$0.457470\pi$
$272$	0	0
$273$	3.51162	0.212533
$274$	0	0
$275$	34.2999	2.06836
$276$	0	0
$277$	12.2418	0.735536	0.367768	−	0.929918i	$-0.380122\pi$
$277$	12.2418	0.735536	0.367768	+	0.929918i	$0.380122\pi$
$278$	0	0
$279$	−26.2973	−1.57438
$280$	0	0
$281$	10.6538	0.635554	0.317777	−	0.948165i	$-0.397064\pi$
$281$	10.6538	0.635554	0.317777	+	0.948165i	$0.397064\pi$
$282$	0	0
$283$	7.29060	0.433381	0.216691	−	0.976240i	$-0.430474\pi$
$283$	7.29060	0.433381	0.216691	+	0.976240i	$0.430474\pi$
$284$	0	0
$285$	16.3675	0.969526
$286$	0	0
$287$	−3.82668	−0.225882
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	5.52784	0.324048
$292$	0	0
$293$	15.9027	0.929044	0.464522	−	0.885562i	$-0.346226\pi$
$293$	15.9027	0.929044	0.464522	+	0.885562i	$0.346226\pi$
$294$	0	0
$295$	−5.43294	−0.316318
$296$	0	0
$297$	−9.20761	−0.534279
$298$	0	0
$299$	−22.5736	−1.30546
$300$	0	0
$301$	−0.681963	−0.0393077
$302$	0	0
$303$	1.39915	0.0803793
$304$	0	0
$305$	46.8882	2.68481
$306$	0	0
$307$	33.5581	1.91526	0.957632	−	0.287994i	$-0.0929882\pi$
$307$	33.5581	1.91526	0.957632	+	0.287994i	$0.0929882\pi$
$308$	0	0
$309$	−0.0260717	−0.00148317
$310$	0	0
$311$	−26.5609	−1.50613	−0.753065	−	0.657947i	$-0.771425\pi$
$311$	−26.5609	−1.50613	−0.753065	+	0.657947i	$0.771425\pi$
$312$	0	0
$313$	−24.3573	−1.37676	−0.688378	−	0.725353i	$-0.741677\pi$
$313$	−24.3573	−1.37676	−0.688378	+	0.725353i	$0.741677\pi$
$314$	0	0
$315$	11.0784	0.624198
$316$	0	0
$317$	−13.6081	−0.764305	−0.382152	−	0.924099i	$-0.624817\pi$
$317$	−13.6081	−0.764305	−0.382152	+	0.924099i	$0.624817\pi$
$318$	0	0
$319$	1.94037	0.108640
$320$	0	0
$321$	0.705851	0.0393967
$322$	0	0
$323$	6.47912	0.360508
$324$	0	0
$325$	73.7545	4.09116
$326$	0	0
$327$	−10.6766	−0.590418
$328$	0	0
$329$	−7.53322	−0.415320
$330$	0	0
$331$	17.1597	0.943184	0.471592	−	0.881817i	$-0.343680\pi$
$331$	17.1597	0.943184	0.471592	+	0.881817i	$0.343680\pi$
$332$	0	0
$333$	−20.5983	−1.12878
$334$	0	0
$335$	52.3948	2.86264
$336$	0	0
$337$	17.1331	0.933300	0.466650	−	0.884442i	$-0.345461\pi$
$337$	17.1331	0.933300	0.466650	+	0.884442i	$0.345461\pi$
$338$	0	0
$339$	−9.23093	−0.501355
$340$	0	0
$341$	27.0612	1.46545
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	9.76878	0.525934
$346$	0	0
$347$	13.2188	0.709621	0.354811	−	0.934938i	$-0.384545\pi$
$347$	13.2188	0.709621	0.354811	+	0.934938i	$0.384545\pi$
$348$	0	0
$349$	−10.9389	−0.585544	−0.292772	−	0.956182i	$-0.594578\pi$
$349$	−10.9389	−0.585544	−0.292772	+	0.956182i	$0.594578\pi$
$350$	0	0
$351$	−19.7989	−1.05679
$352$	0	0
$353$	−15.4555	−0.822615	−0.411307	−	0.911497i	$-0.634928\pi$
$353$	−15.4555	−0.822615	−0.411307	+	0.911497i	$0.634928\pi$
$354$	0	0
$355$	−28.4113	−1.50792
$356$	0	0
$357$	−0.601565	−0.0318382
$358$	0	0
$359$	8.87902	0.468617	0.234308	−	0.972162i	$-0.424717\pi$
$359$	8.87902	0.468617	0.234308	+	0.972162i	$0.424717\pi$
$360$	0	0
$361$	22.9790	1.20942
$362$	0	0
$363$	−2.18375	−0.114617
$364$	0	0
$365$	−6.73082	−0.352307
$366$	0	0
$367$	−13.9183	−0.726531	−0.363266	−	0.931686i	$-0.618338\pi$
$367$	−13.9183	−0.726531	−0.363266	+	0.931686i	$0.618338\pi$
$368$	0	0
$369$	10.0953	0.525538
$370$	0	0
$371$	−7.25346	−0.376581
$372$	0	0
$373$	−7.31656	−0.378837	−0.189419	−	0.981896i	$-0.560660\pi$
$373$	−7.31656	−0.378837	−0.189419	+	0.981896i	$0.560660\pi$
$374$	0	0
$375$	−19.2865	−0.995952
$376$	0	0
$377$	4.17235	0.214887
$378$	0	0
$379$	14.3982	0.739585	0.369792	−	0.929114i	$-0.379429\pi$
$379$	14.3982	0.739585	0.369792	+	0.929114i	$0.379429\pi$
$380$	0	0
$381$	1.30983	0.0671046
$382$	0	0
$383$	−15.2999	−0.781788	−0.390894	−	0.920436i	$-0.627834\pi$
$383$	−15.2999	−0.781788	−0.390894	+	0.920436i	$0.627834\pi$
$384$	0	0
$385$	−11.4002	−0.581009
$386$	0	0
$387$	1.79910	0.0914534
$388$	0	0
$389$	−27.4474	−1.39164	−0.695819	−	0.718217i	$-0.744959\pi$
$389$	−27.4474	−1.39164	−0.695819	+	0.718217i	$0.744959\pi$
$390$	0	0
$391$	3.86701	0.195563
$392$	0	0
$393$	−7.42095	−0.374338
$394$	0	0
$395$	−68.7357	−3.45847
$396$	0	0
$397$	−22.2906	−1.11873	−0.559367	−	0.828920i	$-0.688956\pi$
$397$	−22.2906	−1.11873	−0.559367	+	0.828920i	$0.688956\pi$
$398$	0	0
$399$	−3.89761	−0.195125
$400$	0	0
$401$	4.00679	0.200089	0.100045	−	0.994983i	$-0.468101\pi$
$401$	4.00679	0.200089	0.100045	+	0.994983i	$0.468101\pi$
$402$	0	0
$403$	58.1892	2.89861
$404$	0	0
$405$	−24.6672	−1.22572
$406$	0	0
$407$	21.1967	1.05068
$408$	0	0
$409$	−11.8248	−0.584698	−0.292349	−	0.956312i	$-0.594437\pi$
$409$	−11.8248	−0.584698	−0.292349	+	0.956312i	$0.594437\pi$
$410$	0	0
$411$	−3.74694	−0.184823
$412$	0	0
$413$	1.29375	0.0636615
$414$	0	0
$415$	36.0424	1.76925
$416$	0	0
$417$	−12.6998	−0.621910
$418$	0	0
$419$	−19.3712	−0.946347	−0.473174	−	0.880969i	$-0.656892\pi$
$419$	−19.3712	−0.946347	−0.473174	+	0.880969i	$0.656892\pi$
$420$	0	0
$421$	11.3651	0.553900	0.276950	−	0.960884i	$-0.410676\pi$
$421$	11.3651	0.553900	0.276950	+	0.960884i	$0.410676\pi$
$422$	0	0
$423$	19.8735	0.966284
$424$	0	0
$425$	−12.6346	−0.612870
$426$	0	0
$427$	−11.1656	−0.540339
$428$	0	0
$429$	9.53318	0.460266
$430$	0	0
$431$	2.46761	0.118861	0.0594304	−	0.998232i	$-0.481072\pi$
$431$	2.46761	0.118861	0.0594304	+	0.998232i	$0.481072\pi$
$432$	0	0
$433$	37.5583	1.80494	0.902469	−	0.430754i	$-0.141752\pi$
$433$	37.5583	1.80494	0.902469	+	0.430754i	$0.141752\pi$
$434$	0	0
$435$	−1.80560	−0.0865717
$436$	0	0
$437$	25.0548	1.19853
$438$	0	0
$439$	2.33859	0.111615	0.0558074	−	0.998442i	$-0.482227\pi$
$439$	2.33859	0.111615	0.0558074	+	0.998442i	$0.482227\pi$
$440$	0	0
$441$	−2.63812	−0.125625
$442$	0	0
$443$	11.8358	0.562334	0.281167	−	0.959659i	$-0.409279\pi$
$443$	11.8358	0.562334	0.281167	+	0.959659i	$0.409279\pi$
$444$	0	0
$445$	40.6099	1.92509
$446$	0	0
$447$	−7.11682	−0.336614
$448$	0	0
$449$	−13.5460	−0.639273	−0.319637	−	0.947540i	$-0.603561\pi$
$449$	−13.5460	−0.639273	−0.319637	+	0.947540i	$0.603561\pi$
$450$	0	0
$451$	−10.3885	−0.489175
$452$	0	0
$453$	0.164345	0.00772160
$454$	0	0
$455$	−24.5137	−1.14922
$456$	0	0
$457$	−25.7469	−1.20439	−0.602194	−	0.798350i	$-0.705707\pi$
$457$	−25.7469	−1.20439	−0.602194	+	0.798350i	$0.705707\pi$
$458$	0	0
$459$	3.39169	0.158311
$460$	0	0
$461$	30.3762	1.41476	0.707381	−	0.706833i	$-0.249877\pi$
$461$	30.3762	1.41476	0.707381	+	0.706833i	$0.249877\pi$
$462$	0	0
$463$	35.9718	1.67175	0.835875	−	0.548921i	$-0.184961\pi$
$463$	35.9718	1.67175	0.835875	+	0.548921i	$0.184961\pi$
$464$	0	0
$465$	−25.1816	−1.16777
$466$	0	0
$467$	−2.02508	−0.0937094	−0.0468547	−	0.998902i	$-0.514920\pi$
$467$	−2.02508	−0.0937094	−0.0468547	+	0.998902i	$0.514920\pi$
$468$	0	0
$469$	−12.4769	−0.576128
$470$	0	0
$471$	−7.15431	−0.329653
$472$	0	0
$473$	−1.85136	−0.0851257
$474$	0	0
$475$	−81.8614	−3.75606
$476$	0	0
$477$	19.1355	0.876154
$478$	0	0
$479$	−1.40547	−0.0642175	−0.0321087	−	0.999484i	$-0.510222\pi$
$479$	−1.40547	−0.0642175	−0.0321087	+	0.999484i	$0.510222\pi$
$480$	0	0
$481$	45.5788	2.07821
$482$	0	0
$483$	−2.32625	−0.105848
$484$	0	0
$485$	−38.5884	−1.75221
$486$	0	0
$487$	20.6168	0.934238	0.467119	−	0.884194i	$-0.345292\pi$
$487$	20.6168	0.934238	0.467119	+	0.884194i	$0.345292\pi$
$488$	0	0
$489$	−12.5259	−0.566441
$490$	0	0
$491$	12.6582	0.571255	0.285627	−	0.958341i	$-0.407798\pi$
$491$	12.6582	0.571255	0.285627	+	0.958341i	$0.407798\pi$
$492$	0	0
$493$	−0.714752	−0.0321908
$494$	0	0
$495$	30.0752	1.35178
$496$	0	0
$497$	6.76563	0.303480
$498$	0	0
$499$	1.21180	0.0542475	0.0271238	−	0.999632i	$-0.491365\pi$
$499$	1.21180	0.0542475	0.0271238	+	0.999632i	$0.491365\pi$
$500$	0	0
$501$	−5.39073	−0.240840
$502$	0	0
$503$	−16.3368	−0.728421	−0.364211	−	0.931317i	$-0.618661\pi$
$503$	−16.3368	−0.728421	−0.364211	+	0.931317i	$0.618661\pi$
$504$	0	0
$505$	−9.76713	−0.434631
$506$	0	0
$507$	12.6787	0.563081
$508$	0	0
$509$	−33.4299	−1.48176	−0.740878	−	0.671639i	$-0.765590\pi$
$509$	−33.4299	−1.48176	−0.740878	+	0.671639i	$0.765590\pi$
$510$	0	0
$511$	1.60282	0.0709045
$512$	0	0
$513$	21.9752	0.970229
$514$	0	0
$515$	0.181999	0.00801985
$516$	0	0
$517$	−20.4508	−0.899426
$518$	0	0
$519$	12.2018	0.535601
$520$	0	0
$521$	−28.4945	−1.24837	−0.624183	−	0.781278i	$-0.714568\pi$
$521$	−28.4945	−1.24837	−0.624183	+	0.781278i	$0.714568\pi$
$522$	0	0
$523$	−11.1243	−0.486432	−0.243216	−	0.969972i	$-0.578202\pi$
$523$	−11.1243	−0.486432	−0.243216	+	0.969972i	$0.578202\pi$
$524$	0	0
$525$	7.60055	0.331715
$526$	0	0
$527$	−9.96821	−0.434222
$528$	0	0
$529$	−8.04626	−0.349837
$530$	0	0
$531$	−3.41308	−0.148115
$532$	0	0
$533$	−22.3382	−0.967575
$534$	0	0
$535$	−4.92736	−0.213028
$536$	0	0
$537$	5.75306	0.248263
$538$	0	0
$539$	2.71475	0.116933
$540$	0	0
$541$	7.25734	0.312017	0.156009	−	0.987756i	$-0.450137\pi$
$541$	7.25734	0.312017	0.156009	+	0.987756i	$0.450137\pi$
$542$	0	0
$543$	−6.14794	−0.263833
$544$	0	0
$545$	74.5306	3.19254
$546$	0	0
$547$	38.8270	1.66012	0.830062	−	0.557671i	$-0.188305\pi$
$547$	38.8270	1.66012	0.830062	+	0.557671i	$0.188305\pi$
$548$	0	0
$549$	29.4561	1.25716
$550$	0	0
$551$	−4.63096	−0.197286
$552$	0	0
$553$	16.3681	0.696043
$554$	0	0
$555$	−19.7244	−0.837253
$556$	0	0
$557$	−2.44271	−0.103501	−0.0517504	−	0.998660i	$-0.516480\pi$
$557$	−2.44271	−0.103501	−0.0517504	+	0.998660i	$0.516480\pi$
$558$	0	0
$559$	−3.98095	−0.168376
$560$	0	0
$561$	−1.63310	−0.0689495
$562$	0	0
$563$	−25.3635	−1.06894	−0.534471	−	0.845187i	$-0.679489\pi$
$563$	−25.3635	−1.06894	−0.534471	+	0.845187i	$0.679489\pi$
$564$	0	0
$565$	64.4387	2.71096
$566$	0	0
$567$	5.87404	0.246686
$568$	0	0
$569$	1.12728	0.0472582	0.0236291	−	0.999721i	$-0.492478\pi$
$569$	1.12728	0.0472582	0.0236291	+	0.999721i	$0.492478\pi$
$570$	0	0
$571$	33.9373	1.42023	0.710115	−	0.704086i	$-0.248643\pi$
$571$	33.9373	1.42023	0.710115	+	0.704086i	$0.248643\pi$
$572$	0	0
$573$	14.8464	0.620219
$574$	0	0
$575$	−48.8582	−2.03753
$576$	0	0
$577$	−9.37704	−0.390371	−0.195186	−	0.980766i	$-0.562531\pi$
$577$	−9.37704	−0.390371	−0.195186	+	0.980766i	$0.562531\pi$
$578$	0	0
$579$	−11.3083	−0.469955
$580$	0	0
$581$	−8.58284	−0.356076
$582$	0	0
$583$	−19.6913	−0.815532
$584$	0	0
$585$	64.6701	2.67378
$586$	0	0
$587$	−19.9820	−0.824747	−0.412373	−	0.911015i	$-0.635300\pi$
$587$	−19.9820	−0.824747	−0.412373	+	0.911015i	$0.635300\pi$
$588$	0	0
$589$	−64.5853	−2.66119
$590$	0	0
$591$	4.47282	0.183987
$592$	0	0
$593$	4.26062	0.174963	0.0874814	−	0.996166i	$-0.472118\pi$
$593$	4.26062	0.174963	0.0874814	+	0.996166i	$0.472118\pi$
$594$	0	0
$595$	4.19936	0.172157
$596$	0	0
$597$	−14.0398	−0.574613
$598$	0	0
$599$	41.8610	1.71039	0.855196	−	0.518304i	$-0.173437\pi$
$599$	41.8610	1.71039	0.855196	+	0.518304i	$0.173437\pi$
$600$	0	0
$601$	0.727778	0.0296867	0.0148433	−	0.999890i	$-0.495275\pi$
$601$	0.727778	0.0296867	0.0148433	+	0.999890i	$0.495275\pi$
$602$	0	0
$603$	32.9154	1.34042
$604$	0	0
$605$	15.2442	0.619765
$606$	0	0
$607$	−7.51445	−0.305002	−0.152501	−	0.988303i	$-0.548733\pi$
$607$	−7.51445	−0.305002	−0.152501	+	0.988303i	$0.548733\pi$
$608$	0	0
$609$	0.429969	0.0174232
$610$	0	0
$611$	−43.9750	−1.77904
$612$	0	0
$613$	2.71707	0.109741	0.0548706	−	0.998493i	$-0.482525\pi$
$613$	2.71707	0.109741	0.0548706	+	0.998493i	$0.482525\pi$
$614$	0	0
$615$	9.66692	0.389808
$616$	0	0
$617$	30.5867	1.23138	0.615688	−	0.787990i	$-0.288878\pi$
$617$	30.5867	1.23138	0.615688	+	0.787990i	$0.288878\pi$
$618$	0	0
$619$	7.73567	0.310923	0.155461	−	0.987842i	$-0.450314\pi$
$619$	7.73567	0.310923	0.155461	+	0.987842i	$0.450314\pi$
$620$	0	0
$621$	13.1157	0.526315
$622$	0	0
$623$	−9.67049	−0.387440
$624$	0	0
$625$	71.4610	2.85844
$626$	0	0
$627$	−10.5810	−0.422566
$628$	0	0
$629$	−7.80796	−0.311324
$630$	0	0
$631$	20.6191	0.820835	0.410418	−	0.911898i	$-0.365383\pi$
$631$	20.6191	0.820835	0.410418	+	0.911898i	$0.365383\pi$
$632$	0	0
$633$	−15.9471	−0.633842
$634$	0	0
$635$	−9.14357	−0.362852
$636$	0	0
$637$	5.83748	0.231289
$638$	0	0
$639$	−17.8485	−0.706077
$640$	0	0
$641$	−10.3399	−0.408400	−0.204200	−	0.978929i	$-0.565459\pi$
$641$	−10.3399	−0.408400	−0.204200	+	0.978929i	$0.565459\pi$
$642$	0	0
$643$	15.3243	0.604332	0.302166	−	0.953255i	$-0.402290\pi$
$643$	15.3243	0.604332	0.302166	+	0.953255i	$0.402290\pi$
$644$	0	0
$645$	1.72277	0.0678339
$646$	0	0
$647$	36.5061	1.43520	0.717601	−	0.696454i	$-0.245240\pi$
$647$	36.5061	1.43520	0.717601	+	0.696454i	$0.245240\pi$
$648$	0	0
$649$	3.51222	0.137867
$650$	0	0
$651$	5.99652	0.235022
$652$	0	0
$653$	43.5312	1.70351	0.851754	−	0.523941i	$-0.175539\pi$
$653$	43.5312	1.70351	0.851754	+	0.523941i	$0.175539\pi$
$654$	0	0
$655$	51.8037	2.02414
$656$	0	0
$657$	−4.22843	−0.164967
$658$	0	0
$659$	−10.1417	−0.395064	−0.197532	−	0.980296i	$-0.563293\pi$
$659$	−10.1417	−0.395064	−0.197532	+	0.980296i	$0.563293\pi$
$660$	0	0
$661$	6.37601	0.247998	0.123999	−	0.992282i	$-0.460428\pi$
$661$	6.37601	0.247998	0.123999	+	0.992282i	$0.460428\pi$
$662$	0	0
$663$	−3.51162	−0.136380
$664$	0	0
$665$	27.2082	1.05509
$666$	0	0
$667$	−2.76395	−0.107021
$668$	0	0
$669$	3.38867	0.131014
$670$	0	0
$671$	−30.3117	−1.17017
$672$	0	0
$673$	−8.14535	−0.313980	−0.156990	−	0.987600i	$-0.550179\pi$
$673$	−8.14535	−0.313980	−0.156990	+	0.987600i	$0.550179\pi$
$674$	0	0
$675$	−42.8528	−1.64941
$676$	0	0
$677$	−36.1064	−1.38768	−0.693840	−	0.720129i	$-0.744083\pi$
$677$	−36.1064	−1.38768	−0.693840	+	0.720129i	$0.744083\pi$
$678$	0	0
$679$	9.18910	0.352645
$680$	0	0
$681$	−1.56872	−0.0601135
$682$	0	0
$683$	−30.9620	−1.18473	−0.592363	−	0.805671i	$-0.701805\pi$
$683$	−30.9620	−1.18473	−0.592363	+	0.805671i	$0.701805\pi$
$684$	0	0
$685$	26.1564	0.999386
$686$	0	0
$687$	−4.92277	−0.187815
$688$	0	0
$689$	−42.3419	−1.61310
$690$	0	0
$691$	23.8493	0.907268	0.453634	−	0.891188i	$-0.350127\pi$
$691$	23.8493	0.907268	0.453634	+	0.891188i	$0.350127\pi$
$692$	0	0
$693$	−7.16184	−0.272056
$694$	0	0
$695$	88.6536	3.36282
$696$	0	0
$697$	3.82668	0.144946
$698$	0	0
$699$	−4.97661	−0.188233
$700$	0	0
$701$	−25.2608	−0.954088	−0.477044	−	0.878879i	$-0.658292\pi$
$701$	−25.2608	−0.954088	−0.477044	+	0.878879i	$0.658292\pi$
$702$	0	0
$703$	−50.5887	−1.90799
$704$	0	0
$705$	19.0303	0.716723
$706$	0	0
$707$	2.32586	0.0874730
$708$	0	0
$709$	−1.18527	−0.0445137	−0.0222568	−	0.999752i	$-0.507085\pi$
$709$	−1.18527	−0.0445137	−0.0222568	+	0.999752i	$0.507085\pi$
$710$	0	0
$711$	−43.1811	−1.61942
$712$	0	0
$713$	−38.5471	−1.44360
$714$	0	0
$715$	−66.5486	−2.48878
$716$	0	0
$717$	−5.45356	−0.203667
$718$	0	0
$719$	16.3988	0.611572	0.305786	−	0.952100i	$-0.401081\pi$
$719$	16.3988	0.611572	0.305786	+	0.952100i	$0.401081\pi$
$720$	0	0
$721$	−0.0433398	−0.00161406
$722$	0	0
$723$	−1.41283	−0.0525438
$724$	0	0
$725$	9.03063	0.335389
$726$	0	0
$727$	−21.2325	−0.787470	−0.393735	−	0.919224i	$-0.628817\pi$
$727$	−21.2325	−0.787470	−0.393735	+	0.919224i	$0.628817\pi$
$728$	0	0
$729$	−9.37545	−0.347239
$730$	0	0
$731$	0.681963	0.0252233
$732$	0	0
$733$	−39.3742	−1.45432	−0.727160	−	0.686468i	$-0.759160\pi$
$733$	−39.3742	−1.45432	−0.727160	+	0.686468i	$0.759160\pi$
$734$	0	0
$735$	−2.52619	−0.0931798
$736$	0	0
$737$	−33.8716	−1.24768
$738$	0	0
$739$	5.65369	0.207974	0.103987	−	0.994579i	$-0.466840\pi$
$739$	5.65369	0.207974	0.103987	+	0.994579i	$0.466840\pi$
$740$	0	0
$741$	−22.7522	−0.835824
$742$	0	0
$743$	16.2700	0.596889	0.298444	−	0.954427i	$-0.403532\pi$
$743$	16.2700	0.596889	0.298444	+	0.954427i	$0.403532\pi$
$744$	0	0
$745$	49.6806	1.82016
$746$	0	0
$747$	22.6426	0.828448
$748$	0	0
$749$	1.17336	0.0428736
$750$	0	0
$751$	−10.0946	−0.368357	−0.184179	−	0.982893i	$-0.558963\pi$
$751$	−10.0946	−0.368357	−0.184179	+	0.982893i	$0.558963\pi$
$752$	0	0
$753$	9.27781	0.338102
$754$	0	0
$755$	−1.14725	−0.0417527
$756$	0	0
$757$	4.45496	0.161918	0.0809591	−	0.996717i	$-0.474202\pi$
$757$	4.45496	0.161918	0.0809591	+	0.996717i	$0.474202\pi$
$758$	0	0
$759$	−6.31520	−0.229227
$760$	0	0
$761$	−19.1923	−0.695720	−0.347860	−	0.937547i	$-0.613092\pi$
$761$	−19.1923	−0.695720	−0.347860	+	0.937547i	$0.613092\pi$
$762$	0	0
$763$	−17.7481	−0.642523
$764$	0	0
$765$	−11.0784	−0.400541
$766$	0	0
$767$	7.55227	0.272697
$768$	0	0
$769$	−5.39903	−0.194694	−0.0973471	−	0.995250i	$-0.531036\pi$
$769$	−5.39903	−0.194694	−0.0973471	+	0.995250i	$0.531036\pi$
$770$	0	0
$771$	7.47778	0.269306
$772$	0	0
$773$	21.8900	0.787327	0.393664	−	0.919255i	$-0.371207\pi$
$773$	21.8900	0.787327	0.393664	+	0.919255i	$0.371207\pi$
$774$	0	0
$775$	125.945	4.52407
$776$	0	0
$777$	4.69699	0.168504
$778$	0	0
$779$	24.7936	0.888321
$780$	0	0
$781$	18.3670	0.657223
$782$	0	0
$783$	−2.42422	−0.0866345
$784$	0	0
$785$	49.9423	1.78252
$786$	0	0
$787$	−24.8816	−0.886934	−0.443467	−	0.896291i	$-0.646252\pi$
$787$	−24.8816	−0.886934	−0.443467	+	0.896291i	$0.646252\pi$
$788$	0	0
$789$	13.6948	0.487549
$790$	0	0
$791$	−15.3449	−0.545601
$792$	0	0
$793$	−65.1788	−2.31457
$794$	0	0
$795$	18.3236	0.649871
$796$	0	0
$797$	−40.8512	−1.44702	−0.723511	−	0.690312i	$-0.757473\pi$
$797$	−40.8512	−1.44702	−0.723511	+	0.690312i	$0.757473\pi$
$798$	0	0
$799$	7.53322	0.266506
$800$	0	0
$801$	25.5119	0.901419
$802$	0	0
$803$	4.35126	0.153552
$804$	0	0
$805$	16.2390	0.572348
$806$	0	0
$807$	−12.8841	−0.453541
$808$	0	0
$809$	−24.4825	−0.860760	−0.430380	−	0.902648i	$-0.641620\pi$
$809$	−24.4825	−0.860760	−0.430380	+	0.902648i	$0.641620\pi$
$810$	0	0
$811$	31.7212	1.11388	0.556942	−	0.830551i	$-0.311975\pi$
$811$	31.7212	1.11388	0.556942	+	0.830551i	$0.311975\pi$
$812$	0	0
$813$	2.63846	0.0925347
$814$	0	0
$815$	87.4399	3.06289
$816$	0	0
$817$	4.41852	0.154585
$818$	0	0
$819$	−15.4000	−0.538119
$820$	0	0
$821$	−15.2705	−0.532943	−0.266472	−	0.963843i	$-0.585858\pi$
$821$	−15.2705	−0.532943	−0.266472	+	0.963843i	$0.585858\pi$
$822$	0	0
$823$	−36.6918	−1.27900	−0.639498	−	0.768792i	$-0.720858\pi$
$823$	−36.6918	−1.27900	−0.639498	+	0.768792i	$0.720858\pi$
$824$	0	0
$825$	20.6336	0.718370
$826$	0	0
$827$	−26.3945	−0.917826	−0.458913	−	0.888481i	$-0.651761\pi$
$827$	−26.3945	−0.917826	−0.458913	+	0.888481i	$0.651761\pi$
$828$	0	0
$829$	42.0709	1.46118	0.730591	−	0.682815i	$-0.239245\pi$
$829$	42.0709	1.46118	0.730591	+	0.682815i	$0.239245\pi$
$830$	0	0
$831$	7.36421	0.255462
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	37.6312	1.30228
$836$	0	0
$837$	−33.8091	−1.16861
$838$	0	0
$839$	50.6353	1.74812	0.874062	−	0.485814i	$-0.161477\pi$
$839$	50.6353	1.74812	0.874062	+	0.485814i	$0.161477\pi$
$840$	0	0
$841$	−28.4891	−0.982384
$842$	0	0
$843$	6.40897	0.220737
$844$	0	0
$845$	−88.5066	−3.04472
$846$	0	0
$847$	−3.63012	−0.124733
$848$	0	0
$849$	4.38577	0.150519
$850$	0	0
$851$	−30.1934	−1.03502
$852$	0	0
$853$	−7.91684	−0.271067	−0.135534	−	0.990773i	$-0.543275\pi$
$853$	−7.91684	−0.271067	−0.135534	+	0.990773i	$0.543275\pi$
$854$	0	0
$855$	−71.7785	−2.45477
$856$	0	0
$857$	38.5967	1.31844	0.659218	−	0.751952i	$-0.270887\pi$
$857$	38.5967	1.31844	0.659218	+	0.751952i	$0.270887\pi$
$858$	0	0
$859$	−40.1499	−1.36990	−0.684949	−	0.728591i	$-0.740176\pi$
$859$	−40.1499	−1.36990	−0.684949	+	0.728591i	$0.740176\pi$
$860$	0	0
$861$	−2.30200	−0.0784519
$862$	0	0
$863$	−17.5729	−0.598187	−0.299094	−	0.954224i	$-0.596684\pi$
$863$	−17.5729	−0.598187	−0.299094	+	0.954224i	$0.596684\pi$
$864$	0	0
$865$	−85.1778	−2.89613
$866$	0	0
$867$	0.601565	0.0204302
$868$	0	0
$869$	44.4354	1.50737
$870$	0	0
$871$	−72.8334	−2.46787
$872$	0	0
$873$	−24.2419	−0.820466
$874$	0	0
$875$	−32.0606	−1.08385
$876$	0	0
$877$	44.0334	1.48690	0.743451	−	0.668791i	$-0.233188\pi$
$877$	44.0334	1.48690	0.743451	+	0.668791i	$0.233188\pi$
$878$	0	0
$879$	9.56649	0.322670
$880$	0	0
$881$	−53.8149	−1.81307	−0.906534	−	0.422132i	$-0.861282\pi$
$881$	−53.8149	−1.81307	−0.906534	+	0.422132i	$0.861282\pi$
$882$	0	0
$883$	−16.4929	−0.555032	−0.277516	−	0.960721i	$-0.589511\pi$
$883$	−16.4929	−0.555032	−0.277516	+	0.960721i	$0.589511\pi$
$884$	0	0
$885$	−3.26827	−0.109862
$886$	0	0
$887$	−20.1825	−0.677663	−0.338831	−	0.940847i	$-0.610032\pi$
$887$	−20.1825	−0.677663	−0.338831	+	0.940847i	$0.610032\pi$
$888$	0	0
$889$	2.17737	0.0730267
$890$	0	0
$891$	15.9466	0.534230
$892$	0	0
$893$	48.8086	1.63332
$894$	0	0
$895$	−40.1606	−1.34242
$896$	0	0
$897$	−13.5795	−0.453405
$898$	0	0
$899$	7.12479	0.237625
$900$	0	0
$901$	7.25346	0.241648
$902$	0	0
$903$	−0.410245	−0.0136521
$904$	0	0
$905$	42.9171	1.42661
$906$	0	0
$907$	4.76682	0.158280	0.0791398	−	0.996864i	$-0.474783\pi$
$907$	4.76682	0.158280	0.0791398	+	0.996864i	$0.474783\pi$
$908$	0	0
$909$	−6.13590	−0.203515
$910$	0	0
$911$	−13.3297	−0.441631	−0.220816	−	0.975316i	$-0.570872\pi$
$911$	−13.3297	−0.441631	−0.220816	+	0.975316i	$0.570872\pi$
$912$	0	0
$913$	−23.3003	−0.771126
$914$	0	0
$915$	28.2063	0.932472
$916$	0	0
$917$	−12.3361	−0.407373
$918$	0	0
$919$	18.9560	0.625300	0.312650	−	0.949868i	$-0.398783\pi$
$919$	18.9560	0.625300	0.312650	+	0.949868i	$0.398783\pi$
$920$	0	0
$921$	20.1874	0.665197
$922$	0	0
$923$	39.4942	1.29997
$924$	0	0
$925$	98.6507	3.24362
$926$	0	0
$927$	0.114336	0.00375527
$928$	0	0
$929$	20.0587	0.658104	0.329052	−	0.944312i	$-0.393271\pi$
$929$	20.0587	0.658104	0.329052	+	0.944312i	$0.393271\pi$
$930$	0	0
$931$	−6.47912	−0.212345
$932$	0	0
$933$	−15.9781	−0.523099
$934$	0	0
$935$	11.4002	0.372827
$936$	0	0
$937$	−7.86978	−0.257095	−0.128547	−	0.991703i	$-0.541031\pi$
$937$	−7.86978	−0.257095	−0.128547	+	0.991703i	$0.541031\pi$
$938$	0	0
$939$	−14.6525	−0.478166
$940$	0	0
$941$	−6.23437	−0.203235	−0.101617	−	0.994824i	$-0.532402\pi$
$941$	−6.23437	−0.203235	−0.101617	+	0.994824i	$0.532402\pi$
$942$	0	0
$943$	14.7978	0.481883
$944$	0	0
$945$	14.2429	0.463323
$946$	0	0
$947$	−22.9566	−0.745989	−0.372995	−	0.927833i	$-0.621669\pi$
$947$	−22.9566	−0.745989	−0.372995	+	0.927833i	$0.621669\pi$
$948$	0	0
$949$	9.35643	0.303722
$950$	0	0
$951$	−8.18613	−0.265453
$952$	0	0
$953$	−17.3898	−0.563311	−0.281655	−	0.959516i	$-0.590884\pi$
$953$	−17.3898	−0.563311	−0.281655	+	0.959516i	$0.590884\pi$
$954$	0	0
$955$	−103.639	−3.35368
$956$	0	0
$957$	1.16726	0.0377322
$958$	0	0
$959$	−6.22867	−0.201134
$960$	0	0
$961$	68.3652	2.20533
$962$	0	0
$963$	−3.09546	−0.0997498
$964$	0	0
$965$	78.9400	2.54117
$966$	0	0
$967$	10.1071	0.325023	0.162511	−	0.986707i	$-0.448041\pi$
$967$	10.1071	0.325023	0.162511	+	0.986707i	$0.448041\pi$
$968$	0	0
$969$	3.89761	0.125209
$970$	0	0
$971$	29.8026	0.956412	0.478206	−	0.878248i	$-0.341287\pi$
$971$	29.8026	0.956412	0.478206	+	0.878248i	$0.341287\pi$
$972$	0	0
$973$	−21.1112	−0.676794
$974$	0	0
$975$	44.3681	1.42092
$976$	0	0
$977$	−21.5430	−0.689221	−0.344610	−	0.938746i	$-0.611989\pi$
$977$	−21.5430	−0.689221	−0.344610	+	0.938746i	$0.611989\pi$
$978$	0	0
$979$	−26.2530	−0.839049
$980$	0	0
$981$	46.8216	1.49490
$982$	0	0
$983$	−24.4596	−0.780142	−0.390071	−	0.920785i	$-0.627549\pi$
$983$	−24.4596	−0.780142	−0.390071	+	0.920785i	$0.627549\pi$
$984$	0	0
$985$	−31.2235	−0.994865
$986$	0	0
$987$	−4.53172	−0.144246
$988$	0	0
$989$	2.63716	0.0838567
$990$	0	0
$991$	12.4456	0.395348	0.197674	−	0.980268i	$-0.436661\pi$
$991$	12.4456	0.395348	0.197674	+	0.980268i	$0.436661\pi$
$992$	0	0
$993$	10.3227	0.327580
$994$	0	0
$995$	98.0084	3.10708
$996$	0	0
$997$	−57.1391	−1.80961	−0.904807	−	0.425821i	$-0.859985\pi$
$997$	−57.1391	−1.80961	−0.904807	+	0.425821i	$0.859985\pi$
$998$	0	0
$999$	−26.4822	−0.837860

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.i.1.4	✓	6
4.3	odd	2		3808.2.a.m.1.3	yes	6
8.3	odd	2		7616.2.a.bx.1.4		6
8.5	even	2		7616.2.a.cb.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.i.1.4	✓	6	1.1	even	1	trivial
3808.2.a.m.1.3	yes	6	4.3	odd	2
7616.2.a.bx.1.4		6	8.3	odd	2
7616.2.a.cb.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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+0.601565 q^{3} -4.19936 q^{5} +1.00000 q^{7} -2.63812 q^{9} +2.71475 q^{11} +5.83748 q^{13} -2.52619 q^{15} -1.00000 q^{17} -6.47912 q^{19} +0.601565 q^{21} -3.86701 q^{23} +12.6346 q^{25} -3.39169 q^{27} +0.714752 q^{29} +9.96821 q^{31} +1.63310 q^{33} -4.19936 q^{35} +7.80796 q^{37} +3.51162 q^{39} -3.82668 q^{41} -0.681963 q^{43} +11.0784 q^{45} -7.53322 q^{47} +1.00000 q^{49} -0.601565 q^{51} -7.25346 q^{53} -11.4002 q^{55} -3.89761 q^{57} +1.29375 q^{59} -11.1656 q^{61} -2.63812 q^{63} -24.5137 q^{65} -12.4769 q^{67} -2.32625 q^{69} +6.76563 q^{71} +1.60282 q^{73} +7.60055 q^{75} +2.71475 q^{77} +16.3681 q^{79} +5.87404 q^{81} -8.58284 q^{83} +4.19936 q^{85} +0.429969 q^{87} -9.67049 q^{89} +5.83748 q^{91} +5.99652 q^{93} +27.2082 q^{95} +9.18910 q^{97} -7.16184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} + 2 q^{15} - 6 q^{17} - 10 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} - 8 q^{27} - 14 q^{29} + 8 q^{31} - 8 q^{33} - 6 q^{35} - 4 q^{37} + 14 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 + 2 * q^15 - 6 * q^17 - 10 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 - 8 * q^27 - 14 * q^29 + 8 * q^31 - 8 * q^33 - 6 * q^35 - 4 * q^37 + 14 * q^39 - 2 * q^41 - 8 * q^43 - 8 * q^45 - 2 * q^47 + 6 * q^49 + 2 * q^51 - 10 * q^53 + 2 * q^55 - 18 * q^57 - 14 * q^59 - 20 * q^61 + 4 * q^63 - 20 * q^65 - 6 * q^67 - 14 * q^69 + 20 * q^71 - 6 * q^73 - 10 * q^75 - 2 * q^77 + 38 * q^79 - 18 * q^81 - 16 * q^83 + 6 * q^85 - 4 * q^87 - 2 * q^89 - 4 * q^91 - 16 * q^93 + 42 * q^95 - 12 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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