LMFDB - Embedded newform 8016.2.a.bd.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 $ x^10 - 26x^8 - 3x^7 + 220x^6 + 42x^5 - 675x^4 - 67x^3 + 628x^2 - 48x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.0299246$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.970075 q^{5} -4.67245 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.970075 q^{5} -4.67245 q^{7} +1.00000 q^{9} +1.74348 q^{11} +4.10684 q^{13} -0.970075 q^{15} -4.80793 q^{17} +1.30923 q^{19} -4.67245 q^{21} +5.16482 q^{23} -4.05895 q^{25} +1.00000 q^{27} -4.44464 q^{29} +1.87153 q^{31} +1.74348 q^{33} +4.53263 q^{35} -3.00478 q^{37} +4.10684 q^{39} +5.81330 q^{41} +4.95725 q^{43} -0.970075 q^{45} +2.05636 q^{47} +14.8318 q^{49} -4.80793 q^{51} -8.71088 q^{53} -1.69131 q^{55} +1.30923 q^{57} +11.1215 q^{59} -2.05217 q^{61} -4.67245 q^{63} -3.98395 q^{65} -2.26244 q^{67} +5.16482 q^{69} -7.89327 q^{71} +3.30245 q^{73} -4.05895 q^{75} -8.14633 q^{77} -6.51432 q^{79} +1.00000 q^{81} +0.0124320 q^{83} +4.66405 q^{85} -4.44464 q^{87} -6.84247 q^{89} -19.1890 q^{91} +1.87153 q^{93} -1.27005 q^{95} -4.95038 q^{97} +1.74348 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9	$ 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9} + q^{11} - 6 q^{13} - 10 q^{15} - 9 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 12 q^{25} + 10 q^{27} - 13 q^{29} - 23 q^{31} + q^{33} - q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 10 q^{45} - 10 q^{47} + 7 q^{49} - 9 q^{51} - 26 q^{53} - 11 q^{55} - 2 q^{57} + 10 q^{59} - 10 q^{61} - q^{63} - 22 q^{65} + 5 q^{67} - 7 q^{69} - 25 q^{71} - 8 q^{73} + 12 q^{75} - 46 q^{77} - 26 q^{79} + 10 q^{81} + 14 q^{83} + 9 q^{85} - 13 q^{87} - 31 q^{89} + 3 q^{91} - 23 q^{93} + 5 q^{95} - 32 q^{97} + q^{99}+O(q^{100}) $ 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9 + q^11 - 6 * q^13 - 10 * q^15 - 9 * q^17 - 2 * q^19 - q^21 - 7 * q^23 + 12 * q^25 + 10 * q^27 - 13 * q^29 - 23 * q^31 + q^33 - q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 10 * q^45 - 10 * q^47 + 7 * q^49 - 9 * q^51 - 26 * q^53 - 11 * q^55 - 2 * q^57 + 10 * q^59 - 10 * q^61 - q^63 - 22 * q^65 + 5 * q^67 - 7 * q^69 - 25 * q^71 - 8 * q^73 + 12 * q^75 - 46 * q^77 - 26 * q^79 + 10 * q^81 + 14 * q^83 + 9 * q^85 - 13 * q^87 - 31 * q^89 + 3 * q^91 - 23 * q^93 + 5 * q^95 - 32 * q^97 + 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.970075	−0.433831	−0.216915	−	0.976190i	$-0.569600\pi$
$5$	−0.970075	−0.433831	−0.216915	+	0.976190i	$0.569600\pi$
$6$	0	0
$7$	−4.67245	−1.76602	−0.883011	−	0.469353i	$-0.844487\pi$
$7$	−4.67245	−1.76602	−0.883011	+	0.469353i	$0.844487\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.74348	0.525679	0.262840	−	0.964840i	$-0.415341\pi$
$11$	1.74348	0.525679	0.262840	+	0.964840i	$0.415341\pi$
$12$	0	0
$13$	4.10684	1.13903	0.569516	−	0.821980i	$-0.307130\pi$
$13$	4.10684	1.13903	0.569516	+	0.821980i	$0.307130\pi$
$14$	0	0
$15$	−0.970075	−0.250472
$16$	0	0
$17$	−4.80793	−1.16609	−0.583047	−	0.812438i	$-0.698140\pi$
$17$	−4.80793	−1.16609	−0.583047	+	0.812438i	$0.698140\pi$
$18$	0	0
$19$	1.30923	0.300357	0.150178	−	0.988659i	$-0.452015\pi$
$19$	1.30923	0.300357	0.150178	+	0.988659i	$0.452015\pi$
$20$	0	0
$21$	−4.67245	−1.01961
$22$	0	0
$23$	5.16482	1.07694	0.538469	−	0.842645i	$-0.319003\pi$
$23$	5.16482	1.07694	0.538469	+	0.842645i	$0.319003\pi$
$24$	0	0
$25$	−4.05895	−0.811791
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.44464	−0.825348	−0.412674	−	0.910879i	$-0.635405\pi$
$29$	−4.44464	−0.825348	−0.412674	+	0.910879i	$0.635405\pi$
$30$	0	0
$31$	1.87153	0.336136	0.168068	−	0.985775i	$-0.446247\pi$
$31$	1.87153	0.336136	0.168068	+	0.985775i	$0.446247\pi$
$32$	0	0
$33$	1.74348	0.303501
$34$	0	0
$35$	4.53263	0.766155
$36$	0	0
$37$	−3.00478	−0.493983	−0.246992	−	0.969018i	$-0.579442\pi$
$37$	−3.00478	−0.493983	−0.246992	+	0.969018i	$0.579442\pi$
$38$	0	0
$39$	4.10684	0.657621
$40$	0	0
$41$	5.81330	0.907886	0.453943	−	0.891031i	$-0.350017\pi$
$41$	5.81330	0.907886	0.453943	+	0.891031i	$0.350017\pi$
$42$	0	0
$43$	4.95725	0.755974	0.377987	−	0.925811i	$-0.376616\pi$
$43$	4.95725	0.755974	0.377987	+	0.925811i	$0.376616\pi$
$44$	0	0
$45$	−0.970075	−0.144610
$46$	0	0
$47$	2.05636	0.299950	0.149975	−	0.988690i	$-0.452081\pi$
$47$	2.05636	0.299950	0.149975	+	0.988690i	$0.452081\pi$
$48$	0	0
$49$	14.8318	2.11883
$50$	0	0
$51$	−4.80793	−0.673245
$52$	0	0
$53$	−8.71088	−1.19653	−0.598266	−	0.801298i	$-0.704143\pi$
$53$	−8.71088	−1.19653	−0.598266	+	0.801298i	$0.704143\pi$
$54$	0	0
$55$	−1.69131	−0.228056
$56$	0	0
$57$	1.30923	0.173411
$58$	0	0
$59$	11.1215	1.44790	0.723950	−	0.689852i	$-0.242325\pi$
$59$	11.1215	1.44790	0.723950	+	0.689852i	$0.242325\pi$
$60$	0	0
$61$	−2.05217	−0.262754	−0.131377	−	0.991332i	$-0.541940\pi$
$61$	−2.05217	−0.262754	−0.131377	+	0.991332i	$0.541940\pi$
$62$	0	0
$63$	−4.67245	−0.588674
$64$	0	0
$65$	−3.98395	−0.494148
$66$	0	0
$67$	−2.26244	−0.276401	−0.138201	−	0.990404i	$-0.544132\pi$
$67$	−2.26244	−0.276401	−0.138201	+	0.990404i	$0.544132\pi$
$68$	0	0
$69$	5.16482	0.621771
$70$	0	0
$71$	−7.89327	−0.936759	−0.468380	−	0.883527i	$-0.655162\pi$
$71$	−7.89327	−0.936759	−0.468380	+	0.883527i	$0.655162\pi$
$72$	0	0
$73$	3.30245	0.386523	0.193261	−	0.981147i	$-0.438093\pi$
$73$	3.30245	0.386523	0.193261	+	0.981147i	$0.438093\pi$
$74$	0	0
$75$	−4.05895	−0.468688
$76$	0	0
$77$	−8.14633	−0.928361
$78$	0	0
$79$	−6.51432	−0.732919	−0.366459	−	0.930434i	$-0.619430\pi$
$79$	−6.51432	−0.732919	−0.366459	+	0.930434i	$0.619430\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.0124320	0.00136459	0.000682297	−	1.00000i	$-0.499783\pi$
$83$	0.0124320	0.00136459	0.000682297		1.00000i	$0.499783\pi$
$84$	0	0
$85$	4.66405	0.505888
$86$	0	0
$87$	−4.44464	−0.476515
$88$	0	0
$89$	−6.84247	−0.725301	−0.362650	−	0.931925i	$-0.618128\pi$
$89$	−6.84247	−0.725301	−0.362650	+	0.931925i	$0.618128\pi$
$90$	0	0
$91$	−19.1890	−2.01156
$92$	0	0
$93$	1.87153	0.194068
$94$	0	0
$95$	−1.27005	−0.130304
$96$	0	0
$97$	−4.95038	−0.502635	−0.251317	−	0.967905i	$-0.580864\pi$
$97$	−4.95038	−0.502635	−0.251317	+	0.967905i	$0.580864\pi$
$98$	0	0
$99$	1.74348	0.175226
$100$	0	0
$101$	−2.76512	−0.275140	−0.137570	−	0.990492i	$-0.543929\pi$
$101$	−2.76512	−0.275140	−0.137570	+	0.990492i	$0.543929\pi$
$102$	0	0
$103$	−11.3601	−1.11934	−0.559670	−	0.828716i	$-0.689072\pi$
$103$	−11.3601	−1.11934	−0.559670	+	0.828716i	$0.689072\pi$
$104$	0	0
$105$	4.53263	0.442340
$106$	0	0
$107$	−4.86456	−0.470275	−0.235138	−	0.971962i	$-0.575554\pi$
$107$	−4.86456	−0.470275	−0.235138	+	0.971962i	$0.575554\pi$
$108$	0	0
$109$	−10.0051	−0.958318	−0.479159	−	0.877728i	$-0.659058\pi$
$109$	−10.0051	−0.958318	−0.479159	+	0.877728i	$0.659058\pi$
$110$	0	0
$111$	−3.00478	−0.285201
$112$	0	0
$113$	−3.01490	−0.283618	−0.141809	−	0.989894i	$-0.545292\pi$
$113$	−3.01490	−0.283618	−0.141809	+	0.989894i	$0.545292\pi$
$114$	0	0
$115$	−5.01026	−0.467209
$116$	0	0
$117$	4.10684	0.379678
$118$	0	0
$119$	22.4648	2.05935
$120$	0	0
$121$	−7.96027	−0.723661
$122$	0	0
$123$	5.81330	0.524168
$124$	0	0
$125$	8.78787	0.786011
$126$	0	0
$127$	15.4944	1.37490	0.687451	−	0.726231i	$-0.258730\pi$
$127$	15.4944	1.37490	0.687451	+	0.726231i	$0.258730\pi$
$128$	0	0
$129$	4.95725	0.436462
$130$	0	0
$131$	−7.99090	−0.698168	−0.349084	−	0.937091i	$-0.613507\pi$
$131$	−7.99090	−0.698168	−0.349084	+	0.937091i	$0.613507\pi$
$132$	0	0
$133$	−6.11730	−0.530437
$134$	0	0
$135$	−0.970075	−0.0834908
$136$	0	0
$137$	2.74950	0.234905	0.117453	−	0.993078i	$-0.462527\pi$
$137$	2.74950	0.234905	0.117453	+	0.993078i	$0.462527\pi$
$138$	0	0
$139$	−9.79336	−0.830662	−0.415331	−	0.909670i	$-0.636334\pi$
$139$	−9.79336	−0.830662	−0.415331	+	0.909670i	$0.636334\pi$
$140$	0	0
$141$	2.05636	0.173176
$142$	0	0
$143$	7.16020	0.598766
$144$	0	0
$145$	4.31163	0.358062
$146$	0	0
$147$	14.8318	1.22331
$148$	0	0
$149$	−5.88636	−0.482229	−0.241115	−	0.970497i	$-0.577513\pi$
$149$	−5.88636	−0.482229	−0.241115	+	0.970497i	$0.577513\pi$
$150$	0	0
$151$	21.0628	1.71406	0.857032	−	0.515263i	$-0.172306\pi$
$151$	21.0628	1.71406	0.857032	+	0.515263i	$0.172306\pi$
$152$	0	0
$153$	−4.80793	−0.388698
$154$	0	0
$155$	−1.81552	−0.145826
$156$	0	0
$157$	−10.5419	−0.841332	−0.420666	−	0.907216i	$-0.638204\pi$
$157$	−10.5419	−0.841332	−0.420666	+	0.907216i	$0.638204\pi$
$158$	0	0
$159$	−8.71088	−0.690818
$160$	0	0
$161$	−24.1324	−1.90190
$162$	0	0
$163$	−2.80561	−0.219753	−0.109876	−	0.993945i	$-0.535045\pi$
$163$	−2.80561	−0.219753	−0.109876	+	0.993945i	$0.535045\pi$
$164$	0	0
$165$	−1.69131	−0.131668
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	3.86615	0.297396
$170$	0	0
$171$	1.30923	0.100119
$172$	0	0
$173$	−1.62114	−0.123253	−0.0616265	−	0.998099i	$-0.519629\pi$
$173$	−1.62114	−0.123253	−0.0616265	+	0.998099i	$0.519629\pi$
$174$	0	0
$175$	18.9653	1.43364
$176$	0	0
$177$	11.1215	0.835945
$178$	0	0
$179$	−12.8493	−0.960405	−0.480202	−	0.877158i	$-0.659437\pi$
$179$	−12.8493	−0.960405	−0.480202	+	0.877158i	$0.659437\pi$
$180$	0	0
$181$	−21.9432	−1.63103	−0.815513	−	0.578738i	$-0.803545\pi$
$181$	−21.9432	−1.63103	−0.815513	+	0.578738i	$0.803545\pi$
$182$	0	0
$183$	−2.05217	−0.151701
$184$	0	0
$185$	2.91487	0.214305
$186$	0	0
$187$	−8.38253	−0.612991
$188$	0	0
$189$	−4.67245	−0.339871
$190$	0	0
$191$	−13.1067	−0.948365	−0.474183	−	0.880427i	$-0.657256\pi$
$191$	−13.1067	−0.948365	−0.474183	+	0.880427i	$0.657256\pi$
$192$	0	0
$193$	24.8976	1.79217	0.896085	−	0.443882i	$-0.146399\pi$
$193$	24.8976	1.79217	0.896085	+	0.443882i	$0.146399\pi$
$194$	0	0
$195$	−3.98395	−0.285296
$196$	0	0
$197$	−21.3268	−1.51947	−0.759737	−	0.650230i	$-0.774672\pi$
$197$	−21.3268	−1.51947	−0.759737	+	0.650230i	$0.774672\pi$
$198$	0	0
$199$	4.83237	0.342557	0.171279	−	0.985223i	$-0.445210\pi$
$199$	4.83237	0.342557	0.171279	+	0.985223i	$0.445210\pi$
$200$	0	0
$201$	−2.26244	−0.159580
$202$	0	0
$203$	20.7674	1.45758
$204$	0	0
$205$	−5.63934	−0.393869
$206$	0	0
$207$	5.16482	0.358980
$208$	0	0
$209$	2.28261	0.157891
$210$	0	0
$211$	17.8427	1.22834	0.614170	−	0.789173i	$-0.289491\pi$
$211$	17.8427	1.22834	0.614170	+	0.789173i	$0.289491\pi$
$212$	0	0
$213$	−7.89327	−0.540838
$214$	0	0
$215$	−4.80891	−0.327965
$216$	0	0
$217$	−8.74461	−0.593623
$218$	0	0
$219$	3.30245	0.223159
$220$	0	0
$221$	−19.7454	−1.32822
$222$	0	0
$223$	−10.4058	−0.696825	−0.348413	−	0.937341i	$-0.613279\pi$
$223$	−10.4058	−0.696825	−0.348413	+	0.937341i	$0.613279\pi$
$224$	0	0
$225$	−4.05895	−0.270597
$226$	0	0
$227$	0.436006	0.0289387	0.0144694	−	0.999895i	$-0.495394\pi$
$227$	0.436006	0.0289387	0.0144694	+	0.999895i	$0.495394\pi$
$228$	0	0
$229$	−16.5566	−1.09409	−0.547046	−	0.837102i	$-0.684248\pi$
$229$	−16.5566	−1.09409	−0.547046	+	0.837102i	$0.684248\pi$
$230$	0	0
$231$	−8.14633	−0.535989
$232$	0	0
$233$	5.83930	0.382545	0.191273	−	0.981537i	$-0.438739\pi$
$233$	5.83930	0.382545	0.191273	+	0.981537i	$0.438739\pi$
$234$	0	0
$235$	−1.99482	−0.130128
$236$	0	0
$237$	−6.51432	−0.423151
$238$	0	0
$239$	−15.1389	−0.979251	−0.489626	−	0.871933i	$-0.662867\pi$
$239$	−15.1389	−0.979251	−0.489626	+	0.871933i	$0.662867\pi$
$240$	0	0
$241$	−16.6235	−1.07081	−0.535406	−	0.844595i	$-0.679841\pi$
$241$	−16.6235	−1.07081	−0.535406	+	0.844595i	$0.679841\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−14.3880	−0.919215
$246$	0	0
$247$	5.37678	0.342116
$248$	0	0
$249$	0.0124320	0.000787848 0
$250$	0	0
$251$	11.7737	0.743151	0.371575	−	0.928403i	$-0.378818\pi$
$251$	11.7737	0.743151	0.371575	+	0.928403i	$0.378818\pi$
$252$	0	0
$253$	9.00476	0.566124
$254$	0	0
$255$	4.66405	0.292074
$256$	0	0
$257$	26.3979	1.64666	0.823328	−	0.567566i	$-0.192115\pi$
$257$	26.3979	1.64666	0.823328	+	0.567566i	$0.192115\pi$
$258$	0	0
$259$	14.0397	0.872385
$260$	0	0
$261$	−4.44464	−0.275116
$262$	0	0
$263$	6.84159	0.421871	0.210935	−	0.977500i	$-0.432349\pi$
$263$	6.84159	0.421871	0.210935	+	0.977500i	$0.432349\pi$
$264$	0	0
$265$	8.45021	0.519092
$266$	0	0
$267$	−6.84247	−0.418752
$268$	0	0
$269$	−2.95466	−0.180149	−0.0900743	−	0.995935i	$-0.528710\pi$
$269$	−2.95466	−0.180149	−0.0900743	+	0.995935i	$0.528710\pi$
$270$	0	0
$271$	−22.7010	−1.37898	−0.689492	−	0.724293i	$-0.742166\pi$
$271$	−22.7010	−1.37898	−0.689492	+	0.724293i	$0.742166\pi$
$272$	0	0
$273$	−19.1890	−1.16137
$274$	0	0
$275$	−7.07671	−0.426742
$276$	0	0
$277$	−11.4625	−0.688714	−0.344357	−	0.938839i	$-0.611903\pi$
$277$	−11.4625	−0.688714	−0.344357	+	0.938839i	$0.611903\pi$
$278$	0	0
$279$	1.87153	0.112045
$280$	0	0
$281$	−28.7522	−1.71521	−0.857606	−	0.514307i	$-0.828049\pi$
$281$	−28.7522	−1.71521	−0.857606	+	0.514307i	$0.828049\pi$
$282$	0	0
$283$	12.4547	0.740355	0.370178	−	0.928961i	$-0.379297\pi$
$283$	12.4547	0.740355	0.370178	+	0.928961i	$0.379297\pi$
$284$	0	0
$285$	−1.27005	−0.0752311
$286$	0	0
$287$	−27.1624	−1.60335
$288$	0	0
$289$	6.11619	0.359776
$290$	0	0
$291$	−4.95038	−0.290196
$292$	0	0
$293$	−19.1764	−1.12030	−0.560149	−	0.828392i	$-0.689256\pi$
$293$	−19.1764	−1.12030	−0.560149	+	0.828392i	$0.689256\pi$
$294$	0	0
$295$	−10.7887	−0.628144
$296$	0	0
$297$	1.74348	0.101167
$298$	0	0
$299$	21.2111	1.22667
$300$	0	0
$301$	−23.1625	−1.33507
$302$	0	0
$303$	−2.76512	−0.158852
$304$	0	0
$305$	1.99076	0.113991
$306$	0	0
$307$	−23.3528	−1.33281	−0.666406	−	0.745589i	$-0.732168\pi$
$307$	−23.3528	−1.33281	−0.666406	+	0.745589i	$0.732168\pi$
$308$	0	0
$309$	−11.3601	−0.646251
$310$	0	0
$311$	−0.276596	−0.0156843	−0.00784215	−	0.999969i	$-0.502496\pi$
$311$	−0.276596	−0.0156843	−0.00784215	+	0.999969i	$0.502496\pi$
$312$	0	0
$313$	−21.7496	−1.22936	−0.614681	−	0.788776i	$-0.710715\pi$
$313$	−21.7496	−1.22936	−0.614681	+	0.788776i	$0.710715\pi$
$314$	0	0
$315$	4.53263	0.255385
$316$	0	0
$317$	2.06638	0.116059	0.0580297	−	0.998315i	$-0.481518\pi$
$317$	2.06638	0.116059	0.0580297	+	0.998315i	$0.481518\pi$
$318$	0	0
$319$	−7.74914	−0.433868
$320$	0	0
$321$	−4.86456	−0.271513
$322$	0	0
$323$	−6.29466	−0.350245
$324$	0	0
$325$	−16.6695	−0.924656
$326$	0	0
$327$	−10.0051	−0.553285
$328$	0	0
$329$	−9.60823	−0.529719
$330$	0	0
$331$	−18.9131	−1.03956	−0.519779	−	0.854301i	$-0.673986\pi$
$331$	−18.9131	−1.03956	−0.519779	+	0.854301i	$0.673986\pi$
$332$	0	0
$333$	−3.00478	−0.164661
$334$	0	0
$335$	2.19474	0.119911
$336$	0	0
$337$	11.8528	0.645664	0.322832	−	0.946456i	$-0.395365\pi$
$337$	11.8528	0.645664	0.322832	+	0.946456i	$0.395365\pi$
$338$	0	0
$339$	−3.01490	−0.163747
$340$	0	0
$341$	3.26297	0.176700
$342$	0	0
$343$	−36.5938	−1.97588
$344$	0	0
$345$	−5.01026	−0.269743
$346$	0	0
$347$	17.7139	0.950932	0.475466	−	0.879734i	$-0.342279\pi$
$347$	17.7139	0.950932	0.475466	+	0.879734i	$0.342279\pi$
$348$	0	0
$349$	19.1345	1.02425	0.512123	−	0.858912i	$-0.328859\pi$
$349$	19.1345	1.02425	0.512123	+	0.858912i	$0.328859\pi$
$350$	0	0
$351$	4.10684	0.219207
$352$	0	0
$353$	14.8175	0.788658	0.394329	−	0.918969i	$-0.370977\pi$
$353$	14.8175	0.788658	0.394329	+	0.918969i	$0.370977\pi$
$354$	0	0
$355$	7.65707	0.406395
$356$	0	0
$357$	22.4648	1.18896
$358$	0	0
$359$	−30.1001	−1.58862	−0.794312	−	0.607510i	$-0.792168\pi$
$359$	−30.1001	−1.58862	−0.794312	+	0.607510i	$0.792168\pi$
$360$	0	0
$361$	−17.2859	−0.909786
$362$	0	0
$363$	−7.96027	−0.417806
$364$	0	0
$365$	−3.20363	−0.167686
$366$	0	0
$367$	34.2087	1.78568	0.892841	−	0.450373i	$-0.148709\pi$
$367$	34.2087	1.78568	0.892841	+	0.450373i	$0.148709\pi$
$368$	0	0
$369$	5.81330	0.302629
$370$	0	0
$371$	40.7012	2.11310
$372$	0	0
$373$	−16.6069	−0.859874	−0.429937	−	0.902859i	$-0.641464\pi$
$373$	−16.6069	−0.859874	−0.429937	+	0.902859i	$0.641464\pi$
$374$	0	0
$375$	8.78787	0.453804
$376$	0	0
$377$	−18.2534	−0.940099
$378$	0	0
$379$	8.27067	0.424836	0.212418	−	0.977179i	$-0.431866\pi$
$379$	8.27067	0.424836	0.212418	+	0.977179i	$0.431866\pi$
$380$	0	0
$381$	15.4944	0.793800
$382$	0	0
$383$	−5.06497	−0.258808	−0.129404	−	0.991592i	$-0.541306\pi$
$383$	−5.06497	−0.258808	−0.129404	+	0.991592i	$0.541306\pi$
$384$	0	0
$385$	7.90256	0.402752
$386$	0	0
$387$	4.95725	0.251991
$388$	0	0
$389$	−12.0992	−0.613453	−0.306727	−	0.951798i	$-0.599234\pi$
$389$	−12.0992	−0.613453	−0.306727	+	0.951798i	$0.599234\pi$
$390$	0	0
$391$	−24.8321	−1.25581
$392$	0	0
$393$	−7.99090	−0.403088
$394$	0	0
$395$	6.31939	0.317963
$396$	0	0
$397$	3.01793	0.151466	0.0757328	−	0.997128i	$-0.475870\pi$
$397$	3.01793	0.151466	0.0757328	+	0.997128i	$0.475870\pi$
$398$	0	0
$399$	−6.11730	−0.306248
$400$	0	0
$401$	−17.7424	−0.886012	−0.443006	−	0.896519i	$-0.646088\pi$
$401$	−17.7424	−0.886012	−0.443006	+	0.896519i	$0.646088\pi$
$402$	0	0
$403$	7.68606	0.382870
$404$	0	0
$405$	−0.970075	−0.0482034
$406$	0	0
$407$	−5.23878	−0.259677
$408$	0	0
$409$	−6.79453	−0.335968	−0.167984	−	0.985790i	$-0.553726\pi$
$409$	−6.79453	−0.335968	−0.167984	+	0.985790i	$0.553726\pi$
$410$	0	0
$411$	2.74950	0.135623
$412$	0	0
$413$	−51.9648	−2.55702
$414$	0	0
$415$	−0.0120600	−0.000592003 0
$416$	0	0
$417$	−9.79336	−0.479583
$418$	0	0
$419$	21.8431	1.06710	0.533551	−	0.845768i	$-0.320857\pi$
$419$	21.8431	1.06710	0.533551	+	0.845768i	$0.320857\pi$
$420$	0	0
$421$	−12.3905	−0.603878	−0.301939	−	0.953327i	$-0.597634\pi$
$421$	−12.3905	−0.603878	−0.301939	+	0.953327i	$0.597634\pi$
$422$	0	0
$423$	2.05636	0.0999835
$424$	0	0
$425$	19.5152	0.946624
$426$	0	0
$427$	9.58868	0.464029
$428$	0	0
$429$	7.16020	0.345698
$430$	0	0
$431$	−36.2512	−1.74616	−0.873080	−	0.487576i	$-0.837881\pi$
$431$	−36.2512	−1.74616	−0.873080	+	0.487576i	$0.837881\pi$
$432$	0	0
$433$	17.1303	0.823227	0.411614	−	0.911358i	$-0.364965\pi$
$433$	17.1303	0.823227	0.411614	+	0.911358i	$0.364965\pi$
$434$	0	0
$435$	4.31163	0.206727
$436$	0	0
$437$	6.76191	0.323466
$438$	0	0
$439$	10.2664	0.489987	0.244993	−	0.969525i	$-0.421214\pi$
$439$	10.2664	0.489987	0.244993	+	0.969525i	$0.421214\pi$
$440$	0	0
$441$	14.8318	0.706277
$442$	0	0
$443$	−39.3040	−1.86739	−0.933695	−	0.358070i	$-0.883435\pi$
$443$	−39.3040	−1.86739	−0.933695	+	0.358070i	$0.883435\pi$
$444$	0	0
$445$	6.63771	0.314658
$446$	0	0
$447$	−5.88636	−0.278415
$448$	0	0
$449$	−8.04873	−0.379843	−0.189922	−	0.981799i	$-0.560823\pi$
$449$	−8.04873	−0.379843	−0.189922	+	0.981799i	$0.560823\pi$
$450$	0	0
$451$	10.1354	0.477257
$452$	0	0
$453$	21.0628	0.989615
$454$	0	0
$455$	18.6148	0.872675
$456$	0	0
$457$	8.51527	0.398328	0.199164	−	0.979966i	$-0.436177\pi$
$457$	8.51527	0.398328	0.199164	+	0.979966i	$0.436177\pi$
$458$	0	0
$459$	−4.80793	−0.224415
$460$	0	0
$461$	−8.80450	−0.410066	−0.205033	−	0.978755i	$-0.565730\pi$
$461$	−8.80450	−0.410066	−0.205033	+	0.978755i	$0.565730\pi$
$462$	0	0
$463$	0.373084	0.0173387	0.00866935	−	0.999962i	$-0.497240\pi$
$463$	0.373084	0.0173387	0.00866935	+	0.999962i	$0.497240\pi$
$464$	0	0
$465$	−1.81552	−0.0841928
$466$	0	0
$467$	30.2668	1.40058	0.700290	−	0.713858i	$-0.253054\pi$
$467$	30.2668	1.40058	0.700290	+	0.713858i	$0.253054\pi$
$468$	0	0
$469$	10.5711	0.488130
$470$	0	0
$471$	−10.5419	−0.485743
$472$	0	0
$473$	8.64287	0.397400
$474$	0	0
$475$	−5.31409	−0.243827
$476$	0	0
$477$	−8.71088	−0.398844
$478$	0	0
$479$	−16.0770	−0.734578	−0.367289	−	0.930107i	$-0.619714\pi$
$479$	−16.0770	−0.734578	−0.367289	+	0.930107i	$0.619714\pi$
$480$	0	0
$481$	−12.3402	−0.562663
$482$	0	0
$483$	−24.1324	−1.09806
$484$	0	0
$485$	4.80224	0.218058
$486$	0	0
$487$	−13.1557	−0.596140	−0.298070	−	0.954544i	$-0.596343\pi$
$487$	−13.1557	−0.596140	−0.298070	+	0.954544i	$0.596343\pi$
$488$	0	0
$489$	−2.80561	−0.126874
$490$	0	0
$491$	−34.2502	−1.54569	−0.772844	−	0.634596i	$-0.781166\pi$
$491$	−34.2502	−1.54569	−0.772844	+	0.634596i	$0.781166\pi$
$492$	0	0
$493$	21.3695	0.962434
$494$	0	0
$495$	−1.69131	−0.0760186
$496$	0	0
$497$	36.8810	1.65434
$498$	0	0
$499$	34.0026	1.52217	0.761083	−	0.648655i	$-0.224668\pi$
$499$	34.0026	1.52217	0.761083	+	0.648655i	$0.224668\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−22.7942	−1.01634	−0.508172	−	0.861256i	$-0.669678\pi$
$503$	−22.7942	−1.01634	−0.508172	+	0.861256i	$0.669678\pi$
$504$	0	0
$505$	2.68237	0.119364
$506$	0	0
$507$	3.86615	0.171702
$508$	0	0
$509$	37.9529	1.68223	0.841117	−	0.540853i	$-0.181898\pi$
$509$	37.9529	1.68223	0.841117	+	0.540853i	$0.181898\pi$
$510$	0	0
$511$	−15.4306	−0.682608
$512$	0	0
$513$	1.30923	0.0578037
$514$	0	0
$515$	11.0201	0.485604
$516$	0	0
$517$	3.58522	0.157678
$518$	0	0
$519$	−1.62114	−0.0711601
$520$	0	0
$521$	−38.9359	−1.70581	−0.852907	−	0.522062i	$-0.825163\pi$
$521$	−38.9359	−1.70581	−0.852907	+	0.522062i	$0.825163\pi$
$522$	0	0
$523$	−13.3686	−0.584567	−0.292284	−	0.956332i	$-0.594415\pi$
$523$	−13.3686	−0.584567	−0.292284	+	0.956332i	$0.594415\pi$
$524$	0	0
$525$	18.9653	0.827712
$526$	0	0
$527$	−8.99816	−0.391966
$528$	0	0
$529$	3.67533	0.159797
$530$	0	0
$531$	11.1215	0.482633
$532$	0	0
$533$	23.8743	1.03411
$534$	0	0
$535$	4.71899	0.204020
$536$	0	0
$537$	−12.8493	−0.554490
$538$	0	0
$539$	25.8590	1.11383
$540$	0	0
$541$	15.8797	0.682724	0.341362	−	0.939932i	$-0.389112\pi$
$541$	15.8797	0.682724	0.341362	+	0.939932i	$0.389112\pi$
$542$	0	0
$543$	−21.9432	−0.941674
$544$	0	0
$545$	9.70574	0.415748
$546$	0	0
$547$	−11.6791	−0.499362	−0.249681	−	0.968328i	$-0.580326\pi$
$547$	−11.6791	−0.499362	−0.249681	+	0.968328i	$0.580326\pi$
$548$	0	0
$549$	−2.05217	−0.0875846
$550$	0	0
$551$	−5.81903	−0.247899
$552$	0	0
$553$	30.4379	1.29435
$554$	0	0
$555$	2.91487	0.123729
$556$	0	0
$557$	−1.93554	−0.0820114	−0.0410057	−	0.999159i	$-0.513056\pi$
$557$	−1.93554	−0.0820114	−0.0410057	+	0.999159i	$0.513056\pi$
$558$	0	0
$559$	20.3586	0.861079
$560$	0	0
$561$	−8.38253	−0.353911
$562$	0	0
$563$	−20.4173	−0.860487	−0.430244	−	0.902713i	$-0.641572\pi$
$563$	−20.4173	−0.860487	−0.430244	+	0.902713i	$0.641572\pi$
$564$	0	0
$565$	2.92468	0.123042
$566$	0	0
$567$	−4.67245	−0.196225
$568$	0	0
$569$	12.6345	0.529666	0.264833	−	0.964294i	$-0.414683\pi$
$569$	12.6345	0.529666	0.264833	+	0.964294i	$0.414683\pi$
$570$	0	0
$571$	−15.4590	−0.646941	−0.323470	−	0.946238i	$-0.604850\pi$
$571$	−15.4590	−0.646941	−0.323470	+	0.946238i	$0.604850\pi$
$572$	0	0
$573$	−13.1067	−0.547539
$574$	0	0
$575$	−20.9638	−0.874249
$576$	0	0
$577$	31.1270	1.29583	0.647917	−	0.761711i	$-0.275640\pi$
$577$	31.1270	1.29583	0.647917	+	0.761711i	$0.275640\pi$
$578$	0	0
$579$	24.8976	1.03471
$580$	0	0
$581$	−0.0580881	−0.00240990
$582$	0	0
$583$	−15.1872	−0.628992
$584$	0	0
$585$	−3.98395	−0.164716
$586$	0	0
$587$	18.2275	0.752328	0.376164	−	0.926553i	$-0.377243\pi$
$587$	18.2275	0.752328	0.376164	+	0.926553i	$0.377243\pi$
$588$	0	0
$589$	2.45025	0.100961
$590$	0	0
$591$	−21.3268	−0.877269
$592$	0	0
$593$	−23.1957	−0.952532	−0.476266	−	0.879301i	$-0.658010\pi$
$593$	−23.1957	−0.952532	−0.476266	+	0.879301i	$0.658010\pi$
$594$	0	0
$595$	−21.7926	−0.893408
$596$	0	0
$597$	4.83237	0.197776
$598$	0	0
$599$	11.1000	0.453532	0.226766	−	0.973949i	$-0.427185\pi$
$599$	11.1000	0.453532	0.226766	+	0.973949i	$0.427185\pi$
$600$	0	0
$601$	28.7184	1.17145	0.585725	−	0.810510i	$-0.300810\pi$
$601$	28.7184	1.17145	0.585725	+	0.810510i	$0.300810\pi$
$602$	0	0
$603$	−2.26244	−0.0921337
$604$	0	0
$605$	7.72207	0.313947
$606$	0	0
$607$	31.1027	1.26242	0.631209	−	0.775613i	$-0.282559\pi$
$607$	31.1027	1.26242	0.631209	+	0.775613i	$0.282559\pi$
$608$	0	0
$609$	20.7674	0.841536
$610$	0	0
$611$	8.44513	0.341653
$612$	0	0
$613$	−3.54402	−0.143142	−0.0715709	−	0.997436i	$-0.522801\pi$
$613$	−3.54402	−0.143142	−0.0715709	+	0.997436i	$0.522801\pi$
$614$	0	0
$615$	−5.63934	−0.227400
$616$	0	0
$617$	−29.1227	−1.17243	−0.586217	−	0.810154i	$-0.699383\pi$
$617$	−29.1227	−1.17243	−0.586217	+	0.810154i	$0.699383\pi$
$618$	0	0
$619$	−3.24641	−0.130484	−0.0652421	−	0.997869i	$-0.520782\pi$
$619$	−3.24641	−0.130484	−0.0652421	+	0.997869i	$0.520782\pi$
$620$	0	0
$621$	5.16482	0.207257
$622$	0	0
$623$	31.9711	1.28090
$624$	0	0
$625$	11.7699	0.470795
$626$	0	0
$627$	2.28261	0.0911587
$628$	0	0
$629$	14.4468	0.576031
$630$	0	0
$631$	37.5128	1.49336	0.746681	−	0.665183i	$-0.231646\pi$
$631$	37.5128	1.49336	0.746681	+	0.665183i	$0.231646\pi$
$632$	0	0
$633$	17.8427	0.709183
$634$	0	0
$635$	−15.0307	−0.596475
$636$	0	0
$637$	60.9119	2.41342
$638$	0	0
$639$	−7.89327	−0.312253
$640$	0	0
$641$	4.60388	0.181842	0.0909211	−	0.995858i	$-0.471019\pi$
$641$	4.60388	0.181842	0.0909211	+	0.995858i	$0.471019\pi$
$642$	0	0
$643$	−7.26078	−0.286337	−0.143169	−	0.989698i	$-0.545729\pi$
$643$	−7.26078	−0.286337	−0.143169	+	0.989698i	$0.545729\pi$
$644$	0	0
$645$	−4.80891	−0.189351
$646$	0	0
$647$	29.0344	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$647$	29.0344	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$648$	0	0
$649$	19.3902	0.761131
$650$	0	0
$651$	−8.74461	−0.342728
$652$	0	0
$653$	−27.7782	−1.08705	−0.543523	−	0.839394i	$-0.682910\pi$
$653$	−27.7782	−1.08705	−0.543523	+	0.839394i	$0.682910\pi$
$654$	0	0
$655$	7.75178	0.302887
$656$	0	0
$657$	3.30245	0.128841
$658$	0	0
$659$	−4.81301	−0.187488	−0.0937442	−	0.995596i	$-0.529884\pi$
$659$	−4.81301	−0.187488	−0.0937442	+	0.995596i	$0.529884\pi$
$660$	0	0
$661$	29.3731	1.14248	0.571240	−	0.820783i	$-0.306463\pi$
$661$	29.3731	1.14248	0.571240	+	0.820783i	$0.306463\pi$
$662$	0	0
$663$	−19.7454	−0.766848
$664$	0	0
$665$	5.93424	0.230120
$666$	0	0
$667$	−22.9557	−0.888849
$668$	0	0
$669$	−10.4058	−0.402312
$670$	0	0
$671$	−3.57792	−0.138124
$672$	0	0
$673$	4.76008	0.183487	0.0917437	−	0.995783i	$-0.470756\pi$
$673$	4.76008	0.183487	0.0917437	+	0.995783i	$0.470756\pi$
$674$	0	0
$675$	−4.05895	−0.156229
$676$	0	0
$677$	−2.82348	−0.108515	−0.0542576	−	0.998527i	$-0.517279\pi$
$677$	−2.82348	−0.108515	−0.0542576	+	0.998527i	$0.517279\pi$
$678$	0	0
$679$	23.1304	0.887664
$680$	0	0
$681$	0.436006	0.0167078
$682$	0	0
$683$	−11.6935	−0.447438	−0.223719	−	0.974654i	$-0.571820\pi$
$683$	−11.6935	−0.447438	−0.223719	+	0.974654i	$0.571820\pi$
$684$	0	0
$685$	−2.66722	−0.101909
$686$	0	0
$687$	−16.5566	−0.631675
$688$	0	0
$689$	−35.7742	−1.36289
$690$	0	0
$691$	38.9192	1.48056	0.740278	−	0.672301i	$-0.234694\pi$
$691$	38.9192	1.48056	0.740278	+	0.672301i	$0.234694\pi$
$692$	0	0
$693$	−8.14633	−0.309454
$694$	0	0
$695$	9.50029	0.360367
$696$	0	0
$697$	−27.9500	−1.05868
$698$	0	0
$699$	5.83930	0.220863
$700$	0	0
$701$	21.7642	0.822023	0.411011	−	0.911630i	$-0.365176\pi$
$701$	21.7642	0.822023	0.411011	+	0.911630i	$0.365176\pi$
$702$	0	0
$703$	−3.93394	−0.148371
$704$	0	0
$705$	−1.99482	−0.0751293
$706$	0	0
$707$	12.9199	0.485903
$708$	0	0
$709$	24.1004	0.905109	0.452554	−	0.891737i	$-0.350513\pi$
$709$	24.1004	0.905109	0.452554	+	0.891737i	$0.350513\pi$
$710$	0	0
$711$	−6.51432	−0.244306
$712$	0	0
$713$	9.66608	0.361998
$714$	0	0
$715$	−6.94593	−0.259763
$716$	0	0
$717$	−15.1389	−0.565371
$718$	0	0
$719$	−36.8276	−1.37344	−0.686719	−	0.726923i	$-0.740950\pi$
$719$	−36.8276	−1.37344	−0.686719	+	0.726923i	$0.740950\pi$
$720$	0	0
$721$	53.0794	1.97678
$722$	0	0
$723$	−16.6235	−0.618234
$724$	0	0
$725$	18.0406	0.670010
$726$	0	0
$727$	−53.1050	−1.96955	−0.984777	−	0.173823i	$-0.944388\pi$
$727$	−53.1050	−1.96955	−0.984777	+	0.173823i	$0.944388\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−23.8341	−0.881537
$732$	0	0
$733$	36.6184	1.35253	0.676266	−	0.736658i	$-0.263597\pi$
$733$	36.6184	1.35253	0.676266	+	0.736658i	$0.263597\pi$
$734$	0	0
$735$	−14.3880	−0.530709
$736$	0	0
$737$	−3.94452	−0.145298
$738$	0	0
$739$	13.6761	0.503083	0.251542	−	0.967846i	$-0.419062\pi$
$739$	13.6761	0.503083	0.251542	+	0.967846i	$0.419062\pi$
$740$	0	0
$741$	5.37678	0.197521
$742$	0	0
$743$	17.0434	0.625262	0.312631	−	0.949875i	$-0.398790\pi$
$743$	17.0434	0.625262	0.312631	+	0.949875i	$0.398790\pi$
$744$	0	0
$745$	5.71021	0.209206
$746$	0	0
$747$	0.0124320	0.000454865 0
$748$	0	0
$749$	22.7294	0.830516
$750$	0	0
$751$	21.1103	0.770325	0.385163	−	0.922849i	$-0.374145\pi$
$751$	21.1103	0.770325	0.385163	+	0.922849i	$0.374145\pi$
$752$	0	0
$753$	11.7737	0.429058
$754$	0	0
$755$	−20.4325	−0.743614
$756$	0	0
$757$	35.9849	1.30789	0.653947	−	0.756541i	$-0.273112\pi$
$757$	35.9849	1.30789	0.653947	+	0.756541i	$0.273112\pi$
$758$	0	0
$759$	9.00476	0.326852
$760$	0	0
$761$	−5.38202	−0.195098	−0.0975490	−	0.995231i	$-0.531100\pi$
$761$	−5.38202	−0.195098	−0.0975490	+	0.995231i	$0.531100\pi$
$762$	0	0
$763$	46.7485	1.69241
$764$	0	0
$765$	4.66405	0.168629
$766$	0	0
$767$	45.6744	1.64921
$768$	0	0
$769$	−28.9975	−1.04568	−0.522839	−	0.852432i	$-0.675127\pi$
$769$	−28.9975	−1.04568	−0.522839	+	0.852432i	$0.675127\pi$
$770$	0	0
$771$	26.3979	0.950698
$772$	0	0
$773$	−32.5924	−1.17227	−0.586134	−	0.810214i	$-0.699351\pi$
$773$	−32.5924	−1.17227	−0.586134	+	0.810214i	$0.699351\pi$
$774$	0	0
$775$	−7.59643	−0.272872
$776$	0	0
$777$	14.0397	0.503672
$778$	0	0
$779$	7.61093	0.272690
$780$	0	0
$781$	−13.7618	−0.492435
$782$	0	0
$783$	−4.44464	−0.158838
$784$	0	0
$785$	10.2264	0.364996
$786$	0	0
$787$	48.2759	1.72085	0.860425	−	0.509578i	$-0.170198\pi$
$787$	48.2759	1.72085	0.860425	+	0.509578i	$0.170198\pi$
$788$	0	0
$789$	6.84159	0.243567
$790$	0	0
$791$	14.0870	0.500875
$792$	0	0
$793$	−8.42795	−0.299285
$794$	0	0
$795$	8.45021	0.299698
$796$	0	0
$797$	−29.8547	−1.05751	−0.528753	−	0.848776i	$-0.677340\pi$
$797$	−29.8547	−1.05751	−0.528753	+	0.848776i	$0.677340\pi$
$798$	0	0
$799$	−9.88682	−0.349770
$800$	0	0
$801$	−6.84247	−0.241767
$802$	0	0
$803$	5.75776	0.203187
$804$	0	0
$805$	23.4102	0.825102
$806$	0	0
$807$	−2.95466	−0.104009
$808$	0	0
$809$	23.5350	0.827447	0.413723	−	0.910403i	$-0.364228\pi$
$809$	23.5350	0.827447	0.413723	+	0.910403i	$0.364228\pi$
$810$	0	0
$811$	33.9827	1.19329	0.596646	−	0.802504i	$-0.296499\pi$
$811$	33.9827	1.19329	0.596646	+	0.802504i	$0.296499\pi$
$812$	0	0
$813$	−22.7010	−0.796157
$814$	0	0
$815$	2.72166	0.0953355
$816$	0	0
$817$	6.49016	0.227062
$818$	0	0
$819$	−19.1890	−0.670519
$820$	0	0
$821$	−32.7732	−1.14379	−0.571897	−	0.820326i	$-0.693792\pi$
$821$	−32.7732	−1.14379	−0.571897	+	0.820326i	$0.693792\pi$
$822$	0	0
$823$	5.70260	0.198780	0.0993901	−	0.995049i	$-0.468311\pi$
$823$	5.70260	0.198780	0.0993901	+	0.995049i	$0.468311\pi$
$824$	0	0
$825$	−7.07671	−0.246379
$826$	0	0
$827$	39.8931	1.38722	0.693610	−	0.720351i	$-0.256019\pi$
$827$	39.8931	1.38722	0.693610	+	0.720351i	$0.256019\pi$
$828$	0	0
$829$	28.0450	0.974042	0.487021	−	0.873390i	$-0.338083\pi$
$829$	28.0450	0.974042	0.487021	+	0.873390i	$0.338083\pi$
$830$	0	0
$831$	−11.4625	−0.397629
$832$	0	0
$833$	−71.3103	−2.47076
$834$	0	0
$835$	−0.970075	−0.0335708
$836$	0	0
$837$	1.87153	0.0646894
$838$	0	0
$839$	−37.2047	−1.28445	−0.642224	−	0.766517i	$-0.721988\pi$
$839$	−37.2047	−1.28445	−0.642224	+	0.766517i	$0.721988\pi$
$840$	0	0
$841$	−9.24521	−0.318800
$842$	0	0
$843$	−28.7522	−0.990278
$844$	0	0
$845$	−3.75045	−0.129020
$846$	0	0
$847$	37.1940	1.27800
$848$	0	0
$849$	12.4547	0.427444
$850$	0	0
$851$	−15.5191	−0.531990
$852$	0	0
$853$	−0.701112	−0.0240056	−0.0120028	−	0.999928i	$-0.503821\pi$
$853$	−0.701112	−0.0240056	−0.0120028	+	0.999928i	$0.503821\pi$
$854$	0	0
$855$	−1.27005	−0.0434347
$856$	0	0
$857$	3.52339	0.120357	0.0601784	−	0.998188i	$-0.480833\pi$
$857$	3.52339	0.120357	0.0601784	+	0.998188i	$0.480833\pi$
$858$	0	0
$859$	−13.0212	−0.444278	−0.222139	−	0.975015i	$-0.571304\pi$
$859$	−13.0212	−0.444278	−0.222139	+	0.975015i	$0.571304\pi$
$860$	0	0
$861$	−27.1624	−0.925692
$862$	0	0
$863$	−16.8746	−0.574418	−0.287209	−	0.957868i	$-0.592727\pi$
$863$	−16.8746	−0.574418	−0.287209	+	0.957868i	$0.592727\pi$
$864$	0	0
$865$	1.57263	0.0534710
$866$	0	0
$867$	6.11619	0.207717
$868$	0	0
$869$	−11.3576	−0.385280
$870$	0	0
$871$	−9.29149	−0.314830
$872$	0	0
$873$	−4.95038	−0.167545
$874$	0	0
$875$	−41.0609	−1.38811
$876$	0	0
$877$	12.9850	0.438472	0.219236	−	0.975672i	$-0.429644\pi$
$877$	12.9850	0.438472	0.219236	+	0.975672i	$0.429644\pi$
$878$	0	0
$879$	−19.1764	−0.646805
$880$	0	0
$881$	7.12612	0.240085	0.120042	−	0.992769i	$-0.461697\pi$
$881$	7.12612	0.240085	0.120042	+	0.992769i	$0.461697\pi$
$882$	0	0
$883$	17.5912	0.591992	0.295996	−	0.955189i	$-0.404348\pi$
$883$	17.5912	0.591992	0.295996	+	0.955189i	$0.404348\pi$
$884$	0	0
$885$	−10.7887	−0.362659
$886$	0	0
$887$	29.8787	1.00323	0.501615	−	0.865091i	$-0.332739\pi$
$887$	29.8787	1.00323	0.501615	+	0.865091i	$0.332739\pi$
$888$	0	0
$889$	−72.3967	−2.42811
$890$	0	0
$891$	1.74348	0.0584088
$892$	0	0
$893$	2.69224	0.0900922
$894$	0	0
$895$	12.4648	0.416653
$896$	0	0
$897$	21.2111	0.708217
$898$	0	0
$899$	−8.31825	−0.277429
$900$	0	0
$901$	41.8813	1.39527
$902$	0	0
$903$	−23.1625	−0.770801
$904$	0	0
$905$	21.2866	0.707590
$906$	0	0
$907$	−9.59517	−0.318602	−0.159301	−	0.987230i	$-0.550924\pi$
$907$	−9.59517	−0.318602	−0.159301	+	0.987230i	$0.550924\pi$
$908$	0	0
$909$	−2.76512	−0.0917132
$910$	0	0
$911$	−40.0525	−1.32700	−0.663499	−	0.748177i	$-0.730929\pi$
$911$	−40.0525	−1.32700	−0.663499	+	0.748177i	$0.730929\pi$
$912$	0	0
$913$	0.0216750	0.000717338 0
$914$	0	0
$915$	1.99076	0.0658126
$916$	0	0
$917$	37.3371	1.23298
$918$	0	0
$919$	7.36665	0.243003	0.121502	−	0.992591i	$-0.461229\pi$
$919$	7.36665	0.243003	0.121502	+	0.992591i	$0.461229\pi$
$920$	0	0
$921$	−23.3528	−0.769500
$922$	0	0
$923$	−32.4164	−1.06700
$924$	0	0
$925$	12.1963	0.401011
$926$	0	0
$927$	−11.3601	−0.373113
$928$	0	0
$929$	17.7871	0.583576	0.291788	−	0.956483i	$-0.405750\pi$
$929$	17.7871	0.583576	0.291788	+	0.956483i	$0.405750\pi$
$930$	0	0
$931$	19.4182	0.636406
$932$	0	0
$933$	−0.276596	−0.00905533
$934$	0	0
$935$	8.13169	0.265935
$936$	0	0
$937$	−16.5721	−0.541388	−0.270694	−	0.962665i	$-0.587253\pi$
$937$	−16.5721	−0.541388	−0.270694	+	0.962665i	$0.587253\pi$
$938$	0	0
$939$	−21.7496	−0.709773
$940$	0	0
$941$	−49.3183	−1.60773	−0.803865	−	0.594811i	$-0.797227\pi$
$941$	−49.3183	−1.60773	−0.803865	+	0.594811i	$0.797227\pi$
$942$	0	0
$943$	30.0246	0.977737
$944$	0	0
$945$	4.53263	0.147447
$946$	0	0
$947$	46.7106	1.51789	0.758945	−	0.651155i	$-0.225715\pi$
$947$	46.7106	1.51789	0.758945	+	0.651155i	$0.225715\pi$
$948$	0	0
$949$	13.5626	0.440262
$950$	0	0
$951$	2.06638	0.0670069
$952$	0	0
$953$	1.06215	0.0344065	0.0172033	−	0.999852i	$-0.494524\pi$
$953$	1.06215	0.0344065	0.0172033	+	0.999852i	$0.494524\pi$
$954$	0	0
$955$	12.7145	0.411430
$956$	0	0
$957$	−7.74914	−0.250494
$958$	0	0
$959$	−12.8469	−0.414848
$960$	0	0
$961$	−27.4974	−0.887013
$962$	0	0
$963$	−4.86456	−0.156758
$964$	0	0
$965$	−24.1526	−0.777499
$966$	0	0
$967$	8.83501	0.284115	0.142057	−	0.989858i	$-0.454628\pi$
$967$	8.83501	0.284115	0.142057	+	0.989858i	$0.454628\pi$
$968$	0	0
$969$	−6.29466	−0.202214
$970$	0	0
$971$	−20.8431	−0.668887	−0.334443	−	0.942416i	$-0.608548\pi$
$971$	−20.8431	−0.668887	−0.334443	+	0.942416i	$0.608548\pi$
$972$	0	0
$973$	45.7590	1.46697
$974$	0	0
$975$	−16.6695	−0.533851
$976$	0	0
$977$	−48.0072	−1.53589	−0.767944	−	0.640517i	$-0.778720\pi$
$977$	−48.0072	−1.53589	−0.767944	+	0.640517i	$0.778720\pi$
$978$	0	0
$979$	−11.9297	−0.381275
$980$	0	0
$981$	−10.0051	−0.319439
$982$	0	0
$983$	−4.11611	−0.131284	−0.0656418	−	0.997843i	$-0.520909\pi$
$983$	−4.11611	−0.131284	−0.0656418	+	0.997843i	$0.520909\pi$
$984$	0	0
$985$	20.6886	0.659195
$986$	0	0
$987$	−9.60823	−0.305833
$988$	0	0
$989$	25.6033	0.814137
$990$	0	0
$991$	−27.3392	−0.868458	−0.434229	−	0.900803i	$-0.642979\pi$
$991$	−27.3392	−0.868458	−0.434229	+	0.900803i	$0.642979\pi$
$992$	0	0
$993$	−18.9131	−0.600189
$994$	0	0
$995$	−4.68776	−0.148612
$996$	0	0
$997$	−7.95177	−0.251835	−0.125918	−	0.992041i	$-0.540187\pi$
$997$	−7.95177	−0.251835	−0.125918	+	0.992041i	$0.540187\pi$
$998$	0	0
$999$	−3.00478	−0.0950671

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bd.1.6		10
4.3	odd	2		4008.2.a.j.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.j.1.6	✓	10	4.3	odd	2
8016.2.a.bd.1.6		10	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.970075 q^{5} -4.67245 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.970075 q^{5} -4.67245 q^{7} +1.00000 q^{9} +1.74348 q^{11} +4.10684 q^{13} -0.970075 q^{15} -4.80793 q^{17} +1.30923 q^{19} -4.67245 q^{21} +5.16482 q^{23} -4.05895 q^{25} +1.00000 q^{27} -4.44464 q^{29} +1.87153 q^{31} +1.74348 q^{33} +4.53263 q^{35} -3.00478 q^{37} +4.10684 q^{39} +5.81330 q^{41} +4.95725 q^{43} -0.970075 q^{45} +2.05636 q^{47} +14.8318 q^{49} -4.80793 q^{51} -8.71088 q^{53} -1.69131 q^{55} +1.30923 q^{57} +11.1215 q^{59} -2.05217 q^{61} -4.67245 q^{63} -3.98395 q^{65} -2.26244 q^{67} +5.16482 q^{69} -7.89327 q^{71} +3.30245 q^{73} -4.05895 q^{75} -8.14633 q^{77} -6.51432 q^{79} +1.00000 q^{81} +0.0124320 q^{83} +4.66405 q^{85} -4.44464 q^{87} -6.84247 q^{89} -19.1890 q^{91} +1.87153 q^{93} -1.27005 q^{95} -4.95038 q^{97} +1.74348 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9	\( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9} + q^{11} - 6 q^{13} - 10 q^{15} - 9 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 12 q^{25} + 10 q^{27} - 13 q^{29} - 23 q^{31} + q^{33} - q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 10 q^{45} - 10 q^{47} + 7 q^{49} - 9 q^{51} - 26 q^{53} - 11 q^{55} - 2 q^{57} + 10 q^{59} - 10 q^{61} - q^{63} - 22 q^{65} + 5 q^{67} - 7 q^{69} - 25 q^{71} - 8 q^{73} + 12 q^{75} - 46 q^{77} - 26 q^{79} + 10 q^{81} + 14 q^{83} + 9 q^{85} - 13 q^{87} - 31 q^{89} + 3 q^{91} - 23 q^{93} + 5 q^{95} - 32 q^{97} + q^{99}+O(q^{100}) \) 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9 + q^11 - 6 * q^13 - 10 * q^15 - 9 * q^17 - 2 * q^19 - q^21 - 7 * q^23 + 12 * q^25 + 10 * q^27 - 13 * q^29 - 23 * q^31 + q^33 - q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 10 * q^45 - 10 * q^47 + 7 * q^49 - 9 * q^51 - 26 * q^53 - 11 * q^55 - 2 * q^57 + 10 * q^59 - 10 * q^61 - q^63 - 22 * q^65 + 5 * q^67 - 7 * q^69 - 25 * q^71 - 8 * q^73 + 12 * q^75 - 46 * q^77 - 26 * q^79 + 10 * q^81 + 14 * q^83 + 9 * q^85 - 13 * q^87 - 31 * q^89 + 3 * q^91 - 23 * q^93 + 5 * q^95 - 32 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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