LMFDB - Embedded newform 8004.2.a.j.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + \cdots + 3888 $ x^16 - 3x^15 - 49x^14 + 130x^13 + 932x^12 - 2028x^11 - 8965x^10 + 14400x^9 + 46229x^8 - 47547x^7 - 122604x^6 + 65278x^5 + 151028x^4 - 17988x^3 - 67608x^2 - 8424*x + 3888
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$4.22634$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.22634 q^{5} +0.695618 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.22634 q^{5} +0.695618 q^{7} +1.00000 q^{9} +0.705979 q^{11} +2.93006 q^{13} +4.22634 q^{15} -6.80181 q^{17} +1.49499 q^{19} +0.695618 q^{21} +1.00000 q^{23} +12.8620 q^{25} +1.00000 q^{27} -1.00000 q^{29} -9.11330 q^{31} +0.705979 q^{33} +2.93992 q^{35} +10.9094 q^{37} +2.93006 q^{39} -2.16716 q^{41} +10.9369 q^{43} +4.22634 q^{45} -2.16821 q^{47} -6.51612 q^{49} -6.80181 q^{51} -1.12487 q^{53} +2.98371 q^{55} +1.49499 q^{57} -5.06726 q^{59} +13.4066 q^{61} +0.695618 q^{63} +12.3834 q^{65} +2.02926 q^{67} +1.00000 q^{69} -11.3044 q^{71} +9.21704 q^{73} +12.8620 q^{75} +0.491092 q^{77} +2.34097 q^{79} +1.00000 q^{81} +12.9043 q^{83} -28.7468 q^{85} -1.00000 q^{87} +6.64596 q^{89} +2.03820 q^{91} -9.11330 q^{93} +6.31836 q^{95} +9.49623 q^{97} +0.705979 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * 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
$5$	4.22634	1.89008	0.945039	−	0.326958i	$-0.106023\pi$
$5$	4.22634	1.89008	0.945039	+	0.326958i	$0.106023\pi$
$6$	0	0
$7$	0.695618	0.262919	0.131459	−	0.991322i	$-0.458034\pi$
$7$	0.695618	0.262919	0.131459	+	0.991322i	$0.458034\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.705979	0.212861	0.106430	−	0.994320i	$-0.466058\pi$
$11$	0.705979	0.212861	0.106430	+	0.994320i	$0.466058\pi$
$12$	0	0
$13$	2.93006	0.812651	0.406326	−	0.913728i	$-0.366810\pi$
$13$	2.93006	0.812651	0.406326	+	0.913728i	$0.366810\pi$
$14$	0	0
$15$	4.22634	1.09124
$16$	0	0
$17$	−6.80181	−1.64968	−0.824840	−	0.565366i	$-0.808735\pi$
$17$	−6.80181	−1.64968	−0.824840	+	0.565366i	$0.808735\pi$
$18$	0	0
$19$	1.49499	0.342975	0.171488	−	0.985186i	$-0.445143\pi$
$19$	1.49499	0.342975	0.171488	+	0.985186i	$0.445143\pi$
$20$	0	0
$21$	0.695618	0.151796
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	12.8620	2.57240
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−9.11330	−1.63680	−0.818399	−	0.574651i	$-0.805138\pi$
$31$	−9.11330	−1.63680	−0.818399	+	0.574651i	$0.805138\pi$
$32$	0	0
$33$	0.705979	0.122895
$34$	0	0
$35$	2.93992	0.496937
$36$	0	0
$37$	10.9094	1.79349	0.896744	−	0.442550i	$-0.145926\pi$
$37$	10.9094	1.79349	0.896744	+	0.442550i	$0.145926\pi$
$38$	0	0
$39$	2.93006	0.469184
$40$	0	0
$41$	−2.16716	−0.338453	−0.169226	−	0.985577i	$-0.554127\pi$
$41$	−2.16716	−0.338453	−0.169226	+	0.985577i	$0.554127\pi$
$42$	0	0
$43$	10.9369	1.66785	0.833927	−	0.551875i	$-0.186087\pi$
$43$	10.9369	1.66785	0.833927	+	0.551875i	$0.186087\pi$
$44$	0	0
$45$	4.22634	0.630026
$46$	0	0
$47$	−2.16821	−0.316267	−0.158133	−	0.987418i	$-0.550548\pi$
$47$	−2.16821	−0.316267	−0.158133	+	0.987418i	$0.550548\pi$
$48$	0	0
$49$	−6.51612	−0.930874
$50$	0	0
$51$	−6.80181	−0.952444
$52$	0	0
$53$	−1.12487	−0.154513	−0.0772564	−	0.997011i	$-0.524616\pi$
$53$	−1.12487	−0.154513	−0.0772564	+	0.997011i	$0.524616\pi$
$54$	0	0
$55$	2.98371	0.402323
$56$	0	0
$57$	1.49499	0.198017
$58$	0	0
$59$	−5.06726	−0.659702	−0.329851	−	0.944033i	$-0.606998\pi$
$59$	−5.06726	−0.659702	−0.329851	+	0.944033i	$0.606998\pi$
$60$	0	0
$61$	13.4066	1.71655	0.858273	−	0.513194i	$-0.171538\pi$
$61$	13.4066	1.71655	0.858273	+	0.513194i	$0.171538\pi$
$62$	0	0
$63$	0.695618	0.0876396
$64$	0	0
$65$	12.3834	1.53597
$66$	0	0
$67$	2.02926	0.247914	0.123957	−	0.992288i	$-0.460442\pi$
$67$	2.02926	0.247914	0.123957	+	0.992288i	$0.460442\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−11.3044	−1.34159	−0.670794	−	0.741644i	$-0.734046\pi$
$71$	−11.3044	−1.34159	−0.670794	+	0.741644i	$0.734046\pi$
$72$	0	0
$73$	9.21704	1.07877	0.539387	−	0.842058i	$-0.318656\pi$
$73$	9.21704	1.07877	0.539387	+	0.842058i	$0.318656\pi$
$74$	0	0
$75$	12.8620	1.48517
$76$	0	0
$77$	0.491092	0.0559651
$78$	0	0
$79$	2.34097	0.263380	0.131690	−	0.991291i	$-0.457960\pi$
$79$	2.34097	0.263380	0.131690	+	0.991291i	$0.457960\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.9043	1.41643	0.708215	−	0.705996i	$-0.249501\pi$
$83$	12.9043	1.41643	0.708215	+	0.705996i	$0.249501\pi$
$84$	0	0
$85$	−28.7468	−3.11803
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	6.64596	0.704471	0.352235	−	0.935911i	$-0.385422\pi$
$89$	6.64596	0.704471	0.352235	+	0.935911i	$0.385422\pi$
$90$	0	0
$91$	2.03820	0.213661
$92$	0	0
$93$	−9.11330	−0.945006
$94$	0	0
$95$	6.31836	0.648250
$96$	0	0
$97$	9.49623	0.964196	0.482098	−	0.876117i	$-0.339875\pi$
$97$	9.49623	0.964196	0.482098	+	0.876117i	$0.339875\pi$
$98$	0	0
$99$	0.705979	0.0709535
$100$	0	0
$101$	−3.11003	−0.309460	−0.154730	−	0.987957i	$-0.549451\pi$
$101$	−3.11003	−0.309460	−0.154730	+	0.987957i	$0.549451\pi$
$102$	0	0
$103$	−2.89095	−0.284854	−0.142427	−	0.989805i	$-0.545491\pi$
$103$	−2.89095	−0.284854	−0.142427	+	0.989805i	$0.545491\pi$
$104$	0	0
$105$	2.93992	0.286907
$106$	0	0
$107$	9.92832	0.959807	0.479904	−	0.877321i	$-0.340672\pi$
$107$	9.92832	0.959807	0.479904	+	0.877321i	$0.340672\pi$
$108$	0	0
$109$	−7.37977	−0.706854	−0.353427	−	0.935462i	$-0.614984\pi$
$109$	−7.37977	−0.706854	−0.353427	+	0.935462i	$0.614984\pi$
$110$	0	0
$111$	10.9094	1.03547
$112$	0	0
$113$	11.1473	1.04865	0.524325	−	0.851518i	$-0.324318\pi$
$113$	11.1473	1.04865	0.524325	+	0.851518i	$0.324318\pi$
$114$	0	0
$115$	4.22634	0.394109
$116$	0	0
$117$	2.93006	0.270884
$118$	0	0
$119$	−4.73146	−0.433732
$120$	0	0
$121$	−10.5016	−0.954690
$122$	0	0
$123$	−2.16716	−0.195406
$124$	0	0
$125$	33.2274	2.97195
$126$	0	0
$127$	15.1490	1.34426	0.672130	−	0.740433i	$-0.265380\pi$
$127$	15.1490	1.34426	0.672130	+	0.740433i	$0.265380\pi$
$128$	0	0
$129$	10.9369	0.962936
$130$	0	0
$131$	−20.2752	−1.77145	−0.885725	−	0.464210i	$-0.846338\pi$
$131$	−20.2752	−1.77145	−0.885725	+	0.464210i	$0.846338\pi$
$132$	0	0
$133$	1.03995	0.0901747
$134$	0	0
$135$	4.22634	0.363746
$136$	0	0
$137$	15.6266	1.33507	0.667536	−	0.744577i	$-0.267349\pi$
$137$	15.6266	1.33507	0.667536	+	0.744577i	$0.267349\pi$
$138$	0	0
$139$	−18.6584	−1.58258	−0.791290	−	0.611440i	$-0.790590\pi$
$139$	−18.6584	−1.58258	−0.791290	+	0.611440i	$0.790590\pi$
$140$	0	0
$141$	−2.16821	−0.182597
$142$	0	0
$143$	2.06856	0.172981
$144$	0	0
$145$	−4.22634	−0.350979
$146$	0	0
$147$	−6.51612	−0.537440
$148$	0	0
$149$	10.1880	0.834634	0.417317	−	0.908761i	$-0.362970\pi$
$149$	10.1880	0.834634	0.417317	+	0.908761i	$0.362970\pi$
$150$	0	0
$151$	6.23078	0.507054	0.253527	−	0.967328i	$-0.418409\pi$
$151$	6.23078	0.507054	0.253527	+	0.967328i	$0.418409\pi$
$152$	0	0
$153$	−6.80181	−0.549894
$154$	0	0
$155$	−38.5159	−3.09368
$156$	0	0
$157$	−7.78900	−0.621630	−0.310815	−	0.950470i	$-0.600602\pi$
$157$	−7.78900	−0.621630	−0.310815	+	0.950470i	$0.600602\pi$
$158$	0	0
$159$	−1.12487	−0.0892080
$160$	0	0
$161$	0.695618	0.0548224
$162$	0	0
$163$	22.8277	1.78800	0.894000	−	0.448066i	$-0.147887\pi$
$163$	22.8277	1.78800	0.894000	+	0.448066i	$0.147887\pi$
$164$	0	0
$165$	2.98371	0.232281
$166$	0	0
$167$	8.77045	0.678678	0.339339	−	0.940664i	$-0.389797\pi$
$167$	8.77045	0.678678	0.339339	+	0.940664i	$0.389797\pi$
$168$	0	0
$169$	−4.41477	−0.339598
$170$	0	0
$171$	1.49499	0.114325
$172$	0	0
$173$	−7.72555	−0.587363	−0.293681	−	0.955903i	$-0.594880\pi$
$173$	−7.72555	−0.587363	−0.293681	+	0.955903i	$0.594880\pi$
$174$	0	0
$175$	8.94702	0.676331
$176$	0	0
$177$	−5.06726	−0.380879
$178$	0	0
$179$	−14.1011	−1.05397	−0.526983	−	0.849876i	$-0.676677\pi$
$179$	−14.1011	−1.05397	−0.526983	+	0.849876i	$0.676677\pi$
$180$	0	0
$181$	−6.96073	−0.517387	−0.258694	−	0.965959i	$-0.583292\pi$
$181$	−6.96073	−0.517387	−0.258694	+	0.965959i	$0.583292\pi$
$182$	0	0
$183$	13.4066	0.991048
$184$	0	0
$185$	46.1067	3.38983
$186$	0	0
$187$	−4.80193	−0.351152
$188$	0	0
$189$	0.695618	0.0505988
$190$	0	0
$191$	6.81656	0.493229	0.246615	−	0.969114i	$-0.420682\pi$
$191$	6.81656	0.493229	0.246615	+	0.969114i	$0.420682\pi$
$192$	0	0
$193$	−14.8527	−1.06912	−0.534559	−	0.845131i	$-0.679522\pi$
$193$	−14.8527	−1.06912	−0.534559	+	0.845131i	$0.679522\pi$
$194$	0	0
$195$	12.3834	0.886795
$196$	0	0
$197$	3.40123	0.242328	0.121164	−	0.992633i	$-0.461337\pi$
$197$	3.40123	0.242328	0.121164	+	0.992633i	$0.461337\pi$
$198$	0	0
$199$	12.0542	0.854498	0.427249	−	0.904134i	$-0.359483\pi$
$199$	12.0542	0.854498	0.427249	+	0.904134i	$0.359483\pi$
$200$	0	0
$201$	2.02926	0.143133
$202$	0	0
$203$	−0.695618	−0.0488228
$204$	0	0
$205$	−9.15914	−0.639702
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	1.05543	0.0730059
$210$	0	0
$211$	2.91873	0.200934	0.100467	−	0.994940i	$-0.467966\pi$
$211$	2.91873	0.200934	0.100467	+	0.994940i	$0.467966\pi$
$212$	0	0
$213$	−11.3044	−0.774566
$214$	0	0
$215$	46.2229	3.15237
$216$	0	0
$217$	−6.33938	−0.430345
$218$	0	0
$219$	9.21704	0.622830
$220$	0	0
$221$	−19.9297	−1.34061
$222$	0	0
$223$	0.890415	0.0596266	0.0298133	−	0.999555i	$-0.490509\pi$
$223$	0.890415	0.0596266	0.0298133	+	0.999555i	$0.490509\pi$
$224$	0	0
$225$	12.8620	0.857465
$226$	0	0
$227$	−2.10706	−0.139851	−0.0699254	−	0.997552i	$-0.522276\pi$
$227$	−2.10706	−0.139851	−0.0699254	+	0.997552i	$0.522276\pi$
$228$	0	0
$229$	−18.5927	−1.22864	−0.614319	−	0.789058i	$-0.710569\pi$
$229$	−18.5927	−1.22864	−0.614319	+	0.789058i	$0.710569\pi$
$230$	0	0
$231$	0.491092	0.0323115
$232$	0	0
$233$	−25.9081	−1.69730	−0.848649	−	0.528956i	$-0.822584\pi$
$233$	−25.9081	−1.69730	−0.848649	+	0.528956i	$0.822584\pi$
$234$	0	0
$235$	−9.16362	−0.597768
$236$	0	0
$237$	2.34097	0.152063
$238$	0	0
$239$	−21.7029	−1.40384	−0.701921	−	0.712255i	$-0.747674\pi$
$239$	−21.7029	−1.40384	−0.701921	+	0.712255i	$0.747674\pi$
$240$	0	0
$241$	9.64217	0.621107	0.310553	−	0.950556i	$-0.399486\pi$
$241$	9.64217	0.621107	0.310553	+	0.950556i	$0.399486\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−27.5393	−1.75942
$246$	0	0
$247$	4.38042	0.278719
$248$	0	0
$249$	12.9043	0.817777
$250$	0	0
$251$	−23.9646	−1.51263	−0.756315	−	0.654208i	$-0.773002\pi$
$251$	−23.9646	−1.51263	−0.756315	+	0.654208i	$0.773002\pi$
$252$	0	0
$253$	0.705979	0.0443845
$254$	0	0
$255$	−28.7468	−1.80019
$256$	0	0
$257$	−19.5614	−1.22021	−0.610104	−	0.792321i	$-0.708873\pi$
$257$	−19.5614	−1.22021	−0.610104	+	0.792321i	$0.708873\pi$
$258$	0	0
$259$	7.58875	0.471542
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	2.55052	0.157272	0.0786361	−	0.996903i	$-0.474943\pi$
$263$	2.55052	0.157272	0.0786361	+	0.996903i	$0.474943\pi$
$264$	0	0
$265$	−4.75409	−0.292041
$266$	0	0
$267$	6.64596	0.406726
$268$	0	0
$269$	29.4270	1.79420	0.897099	−	0.441829i	$-0.145670\pi$
$269$	29.4270	1.79420	0.897099	+	0.441829i	$0.145670\pi$
$270$	0	0
$271$	−12.7498	−0.774498	−0.387249	−	0.921975i	$-0.626575\pi$
$271$	−12.7498	−0.774498	−0.387249	+	0.921975i	$0.626575\pi$
$272$	0	0
$273$	2.03820	0.123357
$274$	0	0
$275$	9.08028	0.547562
$276$	0	0
$277$	−0.598050	−0.0359333	−0.0179667	−	0.999839i	$-0.505719\pi$
$277$	−0.598050	−0.0359333	−0.0179667	+	0.999839i	$0.505719\pi$
$278$	0	0
$279$	−9.11330	−0.545599
$280$	0	0
$281$	−16.9309	−1.01001	−0.505005	−	0.863116i	$-0.668509\pi$
$281$	−16.9309	−1.01001	−0.505005	+	0.863116i	$0.668509\pi$
$282$	0	0
$283$	−4.54766	−0.270330	−0.135165	−	0.990823i	$-0.543157\pi$
$283$	−4.54766	−0.270330	−0.135165	+	0.990823i	$0.543157\pi$
$284$	0	0
$285$	6.31836	0.374267
$286$	0	0
$287$	−1.50751	−0.0889857
$288$	0	0
$289$	29.2646	1.72145
$290$	0	0
$291$	9.49623	0.556679
$292$	0	0
$293$	1.42438	0.0832132	0.0416066	−	0.999134i	$-0.486752\pi$
$293$	1.42438	0.0832132	0.0416066	+	0.999134i	$0.486752\pi$
$294$	0	0
$295$	−21.4160	−1.24689
$296$	0	0
$297$	0.705979	0.0409650
$298$	0	0
$299$	2.93006	0.169449
$300$	0	0
$301$	7.60787	0.438510
$302$	0	0
$303$	−3.11003	−0.178667
$304$	0	0
$305$	56.6611	3.24440
$306$	0	0
$307$	−24.9384	−1.42331	−0.711655	−	0.702529i	$-0.752054\pi$
$307$	−24.9384	−1.42331	−0.711655	+	0.702529i	$0.752054\pi$
$308$	0	0
$309$	−2.89095	−0.164460
$310$	0	0
$311$	−17.1542	−0.972727	−0.486364	−	0.873757i	$-0.661677\pi$
$311$	−17.1542	−0.972727	−0.486364	+	0.873757i	$0.661677\pi$
$312$	0	0
$313$	19.1772	1.08396	0.541979	−	0.840392i	$-0.317675\pi$
$313$	19.1772	1.08396	0.541979	+	0.840392i	$0.317675\pi$
$314$	0	0
$315$	2.93992	0.165646
$316$	0	0
$317$	26.0404	1.46258	0.731288	−	0.682069i	$-0.238920\pi$
$317$	26.0404	1.46258	0.731288	+	0.682069i	$0.238920\pi$
$318$	0	0
$319$	−0.705979	−0.0395272
$320$	0	0
$321$	9.92832	0.554145
$322$	0	0
$323$	−10.1687	−0.565800
$324$	0	0
$325$	37.6863	2.09046
$326$	0	0
$327$	−7.37977	−0.408102
$328$	0	0
$329$	−1.50825	−0.0831525
$330$	0	0
$331$	−31.0116	−1.70455	−0.852276	−	0.523092i	$-0.824779\pi$
$331$	−31.0116	−1.70455	−0.852276	+	0.523092i	$0.824779\pi$
$332$	0	0
$333$	10.9094	0.597829
$334$	0	0
$335$	8.57635	0.468576
$336$	0	0
$337$	−24.2760	−1.32240	−0.661200	−	0.750209i	$-0.729953\pi$
$337$	−24.2760	−1.32240	−0.661200	+	0.750209i	$0.729953\pi$
$338$	0	0
$339$	11.1473	0.605439
$340$	0	0
$341$	−6.43380	−0.348410
$342$	0	0
$343$	−9.40205	−0.507663
$344$	0	0
$345$	4.22634	0.227539
$346$	0	0
$347$	12.5571	0.674098	0.337049	−	0.941487i	$-0.390571\pi$
$347$	12.5571	0.674098	0.337049	+	0.941487i	$0.390571\pi$
$348$	0	0
$349$	26.5819	1.42289	0.711447	−	0.702740i	$-0.248040\pi$
$349$	26.5819	1.42289	0.711447	+	0.702740i	$0.248040\pi$
$350$	0	0
$351$	2.93006	0.156395
$352$	0	0
$353$	2.94759	0.156884	0.0784422	−	0.996919i	$-0.475005\pi$
$353$	2.94759	0.156884	0.0784422	+	0.996919i	$0.475005\pi$
$354$	0	0
$355$	−47.7763	−2.53570
$356$	0	0
$357$	−4.73146	−0.250415
$358$	0	0
$359$	−10.4319	−0.550573	−0.275286	−	0.961362i	$-0.588773\pi$
$359$	−10.4319	−0.550573	−0.275286	+	0.961362i	$0.588773\pi$
$360$	0	0
$361$	−16.7650	−0.882368
$362$	0	0
$363$	−10.5016	−0.551191
$364$	0	0
$365$	38.9544	2.03897
$366$	0	0
$367$	−5.00509	−0.261263	−0.130632	−	0.991431i	$-0.541701\pi$
$367$	−5.00509	−0.261263	−0.130632	+	0.991431i	$0.541701\pi$
$368$	0	0
$369$	−2.16716	−0.112818
$370$	0	0
$371$	−0.782480	−0.0406243
$372$	0	0
$373$	8.76104	0.453630	0.226815	−	0.973938i	$-0.427169\pi$
$373$	8.76104	0.453630	0.226815	+	0.973938i	$0.427169\pi$
$374$	0	0
$375$	33.2274	1.71586
$376$	0	0
$377$	−2.93006	−0.150906
$378$	0	0
$379$	−5.18877	−0.266529	−0.133265	−	0.991081i	$-0.542546\pi$
$379$	−5.18877	−0.266529	−0.133265	+	0.991081i	$0.542546\pi$
$380$	0	0
$381$	15.1490	0.776109
$382$	0	0
$383$	15.1249	0.772847	0.386424	−	0.922321i	$-0.373710\pi$
$383$	15.1249	0.772847	0.386424	+	0.922321i	$0.373710\pi$
$384$	0	0
$385$	2.07552	0.105778
$386$	0	0
$387$	10.9369	0.555951
$388$	0	0
$389$	20.1526	1.02178	0.510888	−	0.859648i	$-0.329317\pi$
$389$	20.1526	1.02178	0.510888	+	0.859648i	$0.329317\pi$
$390$	0	0
$391$	−6.80181	−0.343982
$392$	0	0
$393$	−20.2752	−1.02275
$394$	0	0
$395$	9.89376	0.497809
$396$	0	0
$397$	−32.7190	−1.64212	−0.821060	−	0.570842i	$-0.806617\pi$
$397$	−32.7190	−1.64212	−0.821060	+	0.570842i	$0.806617\pi$
$398$	0	0
$399$	1.03995	0.0520624
$400$	0	0
$401$	−29.6852	−1.48241	−0.741203	−	0.671280i	$-0.765745\pi$
$401$	−29.6852	−1.48241	−0.741203	+	0.671280i	$0.765745\pi$
$402$	0	0
$403$	−26.7025	−1.33015
$404$	0	0
$405$	4.22634	0.210009
$406$	0	0
$407$	7.70178	0.381763
$408$	0	0
$409$	0.692258	0.0342299	0.0171150	−	0.999854i	$-0.494552\pi$
$409$	0.692258	0.0342299	0.0171150	+	0.999854i	$0.494552\pi$
$410$	0	0
$411$	15.6266	0.770804
$412$	0	0
$413$	−3.52488	−0.173448
$414$	0	0
$415$	54.5380	2.67717
$416$	0	0
$417$	−18.6584	−0.913704
$418$	0	0
$419$	−37.2573	−1.82014	−0.910070	−	0.414454i	$-0.863972\pi$
$419$	−37.2573	−1.82014	−0.910070	+	0.414454i	$0.863972\pi$
$420$	0	0
$421$	−19.2567	−0.938514	−0.469257	−	0.883062i	$-0.655478\pi$
$421$	−19.2567	−0.938514	−0.469257	+	0.883062i	$0.655478\pi$
$422$	0	0
$423$	−2.16821	−0.105422
$424$	0	0
$425$	−87.4847	−4.24363
$426$	0	0
$427$	9.32591	0.451312
$428$	0	0
$429$	2.06856	0.0998709
$430$	0	0
$431$	4.74403	0.228512	0.114256	−	0.993451i	$-0.463552\pi$
$431$	4.74403	0.228512	0.114256	+	0.993451i	$0.463552\pi$
$432$	0	0
$433$	18.8800	0.907313	0.453657	−	0.891177i	$-0.350119\pi$
$433$	18.8800	0.907313	0.453657	+	0.891177i	$0.350119\pi$
$434$	0	0
$435$	−4.22634	−0.202638
$436$	0	0
$437$	1.49499	0.0715153
$438$	0	0
$439$	−10.2819	−0.490730	−0.245365	−	0.969431i	$-0.578908\pi$
$439$	−10.2819	−0.490730	−0.245365	+	0.969431i	$0.578908\pi$
$440$	0	0
$441$	−6.51612	−0.310291
$442$	0	0
$443$	0.989031	0.0469903	0.0234951	−	0.999724i	$-0.492521\pi$
$443$	0.989031	0.0469903	0.0234951	+	0.999724i	$0.492521\pi$
$444$	0	0
$445$	28.0881	1.33150
$446$	0	0
$447$	10.1880	0.481876
$448$	0	0
$449$	28.5709	1.34834	0.674171	−	0.738575i	$-0.264501\pi$
$449$	28.5709	1.34834	0.674171	+	0.738575i	$0.264501\pi$
$450$	0	0
$451$	−1.52997	−0.0720433
$452$	0	0
$453$	6.23078	0.292748
$454$	0	0
$455$	8.61413	0.403837
$456$	0	0
$457$	8.16371	0.381882	0.190941	−	0.981601i	$-0.438846\pi$
$457$	8.16371	0.381882	0.190941	+	0.981601i	$0.438846\pi$
$458$	0	0
$459$	−6.80181	−0.317481
$460$	0	0
$461$	−2.15436	−0.100338	−0.0501692	−	0.998741i	$-0.515976\pi$
$461$	−2.15436	−0.100338	−0.0501692	+	0.998741i	$0.515976\pi$
$462$	0	0
$463$	−4.28392	−0.199091	−0.0995453	−	0.995033i	$-0.531739\pi$
$463$	−4.28392	−0.199091	−0.0995453	+	0.995033i	$0.531739\pi$
$464$	0	0
$465$	−38.5159	−1.78613
$466$	0	0
$467$	−34.8913	−1.61458	−0.807289	−	0.590156i	$-0.799066\pi$
$467$	−34.8913	−1.61458	−0.807289	+	0.590156i	$0.799066\pi$
$468$	0	0
$469$	1.41159	0.0651812
$470$	0	0
$471$	−7.78900	−0.358898
$472$	0	0
$473$	7.72119	0.355020
$474$	0	0
$475$	19.2286	0.882268
$476$	0	0
$477$	−1.12487	−0.0515043
$478$	0	0
$479$	−36.2133	−1.65463	−0.827314	−	0.561739i	$-0.810132\pi$
$479$	−36.2133	−1.65463	−0.827314	+	0.561739i	$0.810132\pi$
$480$	0	0
$481$	31.9650	1.45748
$482$	0	0
$483$	0.695618	0.0316517
$484$	0	0
$485$	40.1343	1.82241
$486$	0	0
$487$	−7.38649	−0.334714	−0.167357	−	0.985896i	$-0.553523\pi$
$487$	−7.38649	−0.334714	−0.167357	+	0.985896i	$0.553523\pi$
$488$	0	0
$489$	22.8277	1.03230
$490$	0	0
$491$	29.6416	1.33771	0.668853	−	0.743395i	$-0.266786\pi$
$491$	29.6416	1.33771	0.668853	+	0.743395i	$0.266786\pi$
$492$	0	0
$493$	6.80181	0.306338
$494$	0	0
$495$	2.98371	0.134108
$496$	0	0
$497$	−7.86356	−0.352729
$498$	0	0
$499$	6.54447	0.292971	0.146485	−	0.989213i	$-0.453204\pi$
$499$	6.54447	0.292971	0.146485	+	0.989213i	$0.453204\pi$
$500$	0	0
$501$	8.77045	0.391835
$502$	0	0
$503$	−34.4277	−1.53505	−0.767527	−	0.641017i	$-0.778513\pi$
$503$	−34.4277	−1.53505	−0.767527	+	0.641017i	$0.778513\pi$
$504$	0	0
$505$	−13.1441	−0.584903
$506$	0	0
$507$	−4.41477	−0.196067
$508$	0	0
$509$	10.4694	0.464048	0.232024	−	0.972710i	$-0.425465\pi$
$509$	10.4694	0.464048	0.232024	+	0.972710i	$0.425465\pi$
$510$	0	0
$511$	6.41154	0.283630
$512$	0	0
$513$	1.49499	0.0660056
$514$	0	0
$515$	−12.2181	−0.538396
$516$	0	0
$517$	−1.53071	−0.0673207
$518$	0	0
$519$	−7.72555	−0.339114
$520$	0	0
$521$	−14.6679	−0.642614	−0.321307	−	0.946975i	$-0.604122\pi$
$521$	−14.6679	−0.642614	−0.321307	+	0.946975i	$0.604122\pi$
$522$	0	0
$523$	−33.5365	−1.46645	−0.733224	−	0.679987i	$-0.761985\pi$
$523$	−33.5365	−1.46645	−0.733224	+	0.679987i	$0.761985\pi$
$524$	0	0
$525$	8.94702	0.390480
$526$	0	0
$527$	61.9869	2.70019
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.06726	−0.219901
$532$	0	0
$533$	−6.34989	−0.275044
$534$	0	0
$535$	41.9605	1.81411
$536$	0	0
$537$	−14.1011	−0.608507
$538$	0	0
$539$	−4.60024	−0.198146
$540$	0	0
$541$	−19.8448	−0.853195	−0.426597	−	0.904442i	$-0.640288\pi$
$541$	−19.8448	−0.853195	−0.426597	+	0.904442i	$0.640288\pi$
$542$	0	0
$543$	−6.96073	−0.298714
$544$	0	0
$545$	−31.1894	−1.33601
$546$	0	0
$547$	−0.592763	−0.0253447	−0.0126724	−	0.999920i	$-0.504034\pi$
$547$	−0.592763	−0.0253447	−0.0126724	+	0.999920i	$0.504034\pi$
$548$	0	0
$549$	13.4066	0.572182
$550$	0	0
$551$	−1.49499	−0.0636889
$552$	0	0
$553$	1.62842	0.0692476
$554$	0	0
$555$	46.1067	1.95712
$556$	0	0
$557$	−6.32303	−0.267915	−0.133958	−	0.990987i	$-0.542769\pi$
$557$	−6.32303	−0.267915	−0.133958	+	0.990987i	$0.542769\pi$
$558$	0	0
$559$	32.0456	1.35538
$560$	0	0
$561$	−4.80193	−0.202738
$562$	0	0
$563$	−22.8228	−0.961867	−0.480933	−	0.876757i	$-0.659702\pi$
$563$	−22.8228	−0.961867	−0.480933	+	0.876757i	$0.659702\pi$
$564$	0	0
$565$	47.1123	1.98203
$566$	0	0
$567$	0.695618	0.0292132
$568$	0	0
$569$	35.8006	1.50084	0.750419	−	0.660962i	$-0.229852\pi$
$569$	35.8006	1.50084	0.750419	+	0.660962i	$0.229852\pi$
$570$	0	0
$571$	24.1526	1.01075	0.505377	−	0.862899i	$-0.331354\pi$
$571$	24.1526	1.01075	0.505377	+	0.862899i	$0.331354\pi$
$572$	0	0
$573$	6.81656	0.284766
$574$	0	0
$575$	12.8620	0.536381
$576$	0	0
$577$	34.2863	1.42736	0.713678	−	0.700474i	$-0.247028\pi$
$577$	34.2863	1.42736	0.713678	+	0.700474i	$0.247028\pi$
$578$	0	0
$579$	−14.8527	−0.617255
$580$	0	0
$581$	8.97646	0.372406
$582$	0	0
$583$	−0.794134	−0.0328897
$584$	0	0
$585$	12.3834	0.511991
$586$	0	0
$587$	−23.0081	−0.949648	−0.474824	−	0.880081i	$-0.657488\pi$
$587$	−23.0081	−0.949648	−0.474824	+	0.880081i	$0.657488\pi$
$588$	0	0
$589$	−13.6243	−0.561381
$590$	0	0
$591$	3.40123	0.139908
$592$	0	0
$593$	−14.1127	−0.579539	−0.289770	−	0.957096i	$-0.593579\pi$
$593$	−14.1127	−0.579539	−0.289770	+	0.957096i	$0.593579\pi$
$594$	0	0
$595$	−19.9968	−0.819788
$596$	0	0
$597$	12.0542	0.493344
$598$	0	0
$599$	−2.50251	−0.102250	−0.0511249	−	0.998692i	$-0.516281\pi$
$599$	−2.50251	−0.102250	−0.0511249	+	0.998692i	$0.516281\pi$
$600$	0	0
$601$	−14.0139	−0.571640	−0.285820	−	0.958283i	$-0.592266\pi$
$601$	−14.0139	−0.571640	−0.285820	+	0.958283i	$0.592266\pi$
$602$	0	0
$603$	2.02926	0.0826379
$604$	0	0
$605$	−44.3833	−1.80444
$606$	0	0
$607$	−25.3436	−1.02866	−0.514332	−	0.857591i	$-0.671960\pi$
$607$	−25.3436	−1.02866	−0.514332	+	0.857591i	$0.671960\pi$
$608$	0	0
$609$	−0.695618	−0.0281879
$610$	0	0
$611$	−6.35299	−0.257014
$612$	0	0
$613$	−18.6346	−0.752644	−0.376322	−	0.926489i	$-0.622811\pi$
$613$	−18.6346	−0.752644	−0.376322	+	0.926489i	$0.622811\pi$
$614$	0	0
$615$	−9.15914	−0.369332
$616$	0	0
$617$	24.2469	0.976143	0.488072	−	0.872804i	$-0.337700\pi$
$617$	24.2469	0.976143	0.488072	+	0.872804i	$0.337700\pi$
$618$	0	0
$619$	44.6116	1.79309	0.896546	−	0.442950i	$-0.146068\pi$
$619$	44.6116	1.79309	0.896546	+	0.442950i	$0.146068\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	4.62305	0.185219
$624$	0	0
$625$	76.1205	3.04482
$626$	0	0
$627$	1.05543	0.0421500
$628$	0	0
$629$	−74.2034	−2.95868
$630$	0	0
$631$	31.6825	1.26126	0.630631	−	0.776083i	$-0.282796\pi$
$631$	31.6825	1.26126	0.630631	+	0.776083i	$0.282796\pi$
$632$	0	0
$633$	2.91873	0.116009
$634$	0	0
$635$	64.0250	2.54076
$636$	0	0
$637$	−19.0926	−0.756476
$638$	0	0
$639$	−11.3044	−0.447196
$640$	0	0
$641$	35.9623	1.42043	0.710213	−	0.703987i	$-0.248599\pi$
$641$	35.9623	1.42043	0.710213	+	0.703987i	$0.248599\pi$
$642$	0	0
$643$	−31.4634	−1.24079	−0.620397	−	0.784288i	$-0.713028\pi$
$643$	−31.4634	−1.24079	−0.620397	+	0.784288i	$0.713028\pi$
$644$	0	0
$645$	46.2229	1.82002
$646$	0	0
$647$	−11.8638	−0.466414	−0.233207	−	0.972427i	$-0.574922\pi$
$647$	−11.8638	−0.466414	−0.233207	+	0.972427i	$0.574922\pi$
$648$	0	0
$649$	−3.57738	−0.140424
$650$	0	0
$651$	−6.33938	−0.248460
$652$	0	0
$653$	15.5072	0.606844	0.303422	−	0.952856i	$-0.401871\pi$
$653$	15.5072	0.606844	0.303422	+	0.952856i	$0.401871\pi$
$654$	0	0
$655$	−85.6899	−3.34818
$656$	0	0
$657$	9.21704	0.359591
$658$	0	0
$659$	5.80769	0.226235	0.113118	−	0.993582i	$-0.463916\pi$
$659$	5.80769	0.226235	0.113118	+	0.993582i	$0.463916\pi$
$660$	0	0
$661$	8.82689	0.343326	0.171663	−	0.985156i	$-0.445086\pi$
$661$	8.82689	0.343326	0.171663	+	0.985156i	$0.445086\pi$
$662$	0	0
$663$	−19.9297	−0.774004
$664$	0	0
$665$	4.39517	0.170437
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	0.890415	0.0344254
$670$	0	0
$671$	9.46481	0.365385
$672$	0	0
$673$	21.3368	0.822473	0.411237	−	0.911529i	$-0.365097\pi$
$673$	21.3368	0.822473	0.411237	+	0.911529i	$0.365097\pi$
$674$	0	0
$675$	12.8620	0.495058
$676$	0	0
$677$	0.748433	0.0287646	0.0143823	−	0.999897i	$-0.495422\pi$
$677$	0.748433	0.0287646	0.0143823	+	0.999897i	$0.495422\pi$
$678$	0	0
$679$	6.60575	0.253505
$680$	0	0
$681$	−2.10706	−0.0807429
$682$	0	0
$683$	−13.8840	−0.531258	−0.265629	−	0.964075i	$-0.585580\pi$
$683$	−13.8840	−0.531258	−0.265629	+	0.964075i	$0.585580\pi$
$684$	0	0
$685$	66.0434	2.52339
$686$	0	0
$687$	−18.5927	−0.709355
$688$	0	0
$689$	−3.29593	−0.125565
$690$	0	0
$691$	−0.657470	−0.0250113	−0.0125057	−	0.999922i	$-0.503981\pi$
$691$	−0.657470	−0.0250113	−0.0125057	+	0.999922i	$0.503981\pi$
$692$	0	0
$693$	0.491092	0.0186550
$694$	0	0
$695$	−78.8566	−2.99120
$696$	0	0
$697$	14.7406	0.558339
$698$	0	0
$699$	−25.9081	−0.979936
$700$	0	0
$701$	−17.4436	−0.658836	−0.329418	−	0.944184i	$-0.606852\pi$
$701$	−17.4436	−0.658836	−0.329418	+	0.944184i	$0.606852\pi$
$702$	0	0
$703$	16.3094	0.615122
$704$	0	0
$705$	−9.16362	−0.345122
$706$	0	0
$707$	−2.16340	−0.0813629
$708$	0	0
$709$	−28.3024	−1.06292	−0.531460	−	0.847083i	$-0.678356\pi$
$709$	−28.3024	−1.06292	−0.531460	+	0.847083i	$0.678356\pi$
$710$	0	0
$711$	2.34097	0.0877934
$712$	0	0
$713$	−9.11330	−0.341296
$714$	0	0
$715$	8.74243	0.326948
$716$	0	0
$717$	−21.7029	−0.810508
$718$	0	0
$719$	−31.1323	−1.16104	−0.580520	−	0.814246i	$-0.697151\pi$
$719$	−31.1323	−1.16104	−0.580520	+	0.814246i	$0.697151\pi$
$720$	0	0
$721$	−2.01100	−0.0748934
$722$	0	0
$723$	9.64217	0.358596
$724$	0	0
$725$	−12.8620	−0.477682
$726$	0	0
$727$	11.0340	0.409227	0.204613	−	0.978843i	$-0.434406\pi$
$727$	11.0340	0.409227	0.204613	+	0.978843i	$0.434406\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−74.3904	−2.75143
$732$	0	0
$733$	29.5537	1.09159	0.545796	−	0.837918i	$-0.316228\pi$
$733$	29.5537	1.09159	0.545796	+	0.837918i	$0.316228\pi$
$734$	0	0
$735$	−27.5393	−1.01580
$736$	0	0
$737$	1.43262	0.0527711
$738$	0	0
$739$	48.7783	1.79434	0.897169	−	0.441688i	$-0.145620\pi$
$739$	48.7783	1.79434	0.897169	+	0.441688i	$0.145620\pi$
$740$	0	0
$741$	4.38042	0.160919
$742$	0	0
$743$	−23.7435	−0.871063	−0.435531	−	0.900174i	$-0.643440\pi$
$743$	−23.7435	−0.871063	−0.435531	+	0.900174i	$0.643440\pi$
$744$	0	0
$745$	43.0580	1.57752
$746$	0	0
$747$	12.9043	0.472144
$748$	0	0
$749$	6.90632	0.252352
$750$	0	0
$751$	5.16660	0.188532	0.0942659	−	0.995547i	$-0.469950\pi$
$751$	5.16660	0.188532	0.0942659	+	0.995547i	$0.469950\pi$
$752$	0	0
$753$	−23.9646	−0.873317
$754$	0	0
$755$	26.3334	0.958372
$756$	0	0
$757$	0.538380	0.0195678	0.00978389	−	0.999952i	$-0.496886\pi$
$757$	0.538380	0.0195678	0.00978389	+	0.999952i	$0.496886\pi$
$758$	0	0
$759$	0.705979	0.0256254
$760$	0	0
$761$	−50.1455	−1.81777	−0.908886	−	0.417044i	$-0.863066\pi$
$761$	−50.1455	−1.81777	−0.908886	+	0.417044i	$0.863066\pi$
$762$	0	0
$763$	−5.13350	−0.185845
$764$	0	0
$765$	−28.7468	−1.03934
$766$	0	0
$767$	−14.8474	−0.536107
$768$	0	0
$769$	36.1197	1.30251	0.651255	−	0.758859i	$-0.274243\pi$
$769$	36.1197	1.30251	0.651255	+	0.758859i	$0.274243\pi$
$770$	0	0
$771$	−19.5614	−0.704488
$772$	0	0
$773$	−6.42381	−0.231048	−0.115524	−	0.993305i	$-0.536855\pi$
$773$	−6.42381	−0.231048	−0.115524	+	0.993305i	$0.536855\pi$
$774$	0	0
$775$	−117.215	−4.21049
$776$	0	0
$777$	7.58875	0.272245
$778$	0	0
$779$	−3.23989	−0.116081
$780$	0	0
$781$	−7.98068	−0.285571
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−32.9190	−1.17493
$786$	0	0
$787$	−12.1635	−0.433582	−0.216791	−	0.976218i	$-0.569559\pi$
$787$	−12.1635	−0.433582	−0.216791	+	0.976218i	$0.569559\pi$
$788$	0	0
$789$	2.55052	0.0908011
$790$	0	0
$791$	7.75427	0.275710
$792$	0	0
$793$	39.2822	1.39495
$794$	0	0
$795$	−4.75409	−0.168610
$796$	0	0
$797$	−0.452690	−0.0160351	−0.00801755	−	0.999968i	$-0.502552\pi$
$797$	−0.452690	−0.0160351	−0.00801755	+	0.999968i	$0.502552\pi$
$798$	0	0
$799$	14.7478	0.521739
$800$	0	0
$801$	6.64596	0.234824
$802$	0	0
$803$	6.50704	0.229628
$804$	0	0
$805$	2.93992	0.103619
$806$	0	0
$807$	29.4270	1.03588
$808$	0	0
$809$	−34.7499	−1.22174	−0.610871	−	0.791730i	$-0.709181\pi$
$809$	−34.7499	−1.22174	−0.610871	+	0.791730i	$0.709181\pi$
$810$	0	0
$811$	−12.8338	−0.450657	−0.225328	−	0.974283i	$-0.572345\pi$
$811$	−12.8338	−0.450657	−0.225328	+	0.974283i	$0.572345\pi$
$812$	0	0
$813$	−12.7498	−0.447157
$814$	0	0
$815$	96.4775	3.37946
$816$	0	0
$817$	16.3505	0.572033
$818$	0	0
$819$	2.03820	0.0712205
$820$	0	0
$821$	−8.09811	−0.282626	−0.141313	−	0.989965i	$-0.545132\pi$
$821$	−8.09811	−0.282626	−0.141313	+	0.989965i	$0.545132\pi$
$822$	0	0
$823$	52.1215	1.81684	0.908421	−	0.418056i	$-0.137288\pi$
$823$	52.1215	1.81684	0.908421	+	0.418056i	$0.137288\pi$
$824$	0	0
$825$	9.08028	0.316135
$826$	0	0
$827$	−10.4426	−0.363125	−0.181562	−	0.983379i	$-0.558115\pi$
$827$	−10.4426	−0.363125	−0.181562	+	0.983379i	$0.558115\pi$
$828$	0	0
$829$	49.4044	1.71588	0.857942	−	0.513746i	$-0.171743\pi$
$829$	49.4044	1.71588	0.857942	+	0.513746i	$0.171743\pi$
$830$	0	0
$831$	−0.598050	−0.0207461
$832$	0	0
$833$	44.3214	1.53564
$834$	0	0
$835$	37.0669	1.28275
$836$	0	0
$837$	−9.11330	−0.315002
$838$	0	0
$839$	14.1529	0.488611	0.244306	−	0.969698i	$-0.421440\pi$
$839$	14.1529	0.488611	0.244306	+	0.969698i	$0.421440\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−16.9309	−0.583130
$844$	0	0
$845$	−18.6584	−0.641867
$846$	0	0
$847$	−7.30510	−0.251006
$848$	0	0
$849$	−4.54766	−0.156075
$850$	0	0
$851$	10.9094	0.373968
$852$	0	0
$853$	−47.2167	−1.61667	−0.808334	−	0.588724i	$-0.799630\pi$
$853$	−47.2167	−1.61667	−0.808334	+	0.588724i	$0.799630\pi$
$854$	0	0
$855$	6.31836	0.216083
$856$	0	0
$857$	26.4901	0.904884	0.452442	−	0.891794i	$-0.350553\pi$
$857$	26.4901	0.904884	0.452442	+	0.891794i	$0.350553\pi$
$858$	0	0
$859$	5.24114	0.178825	0.0894127	−	0.995995i	$-0.471501\pi$
$859$	5.24114	0.178825	0.0894127	+	0.995995i	$0.471501\pi$
$860$	0	0
$861$	−1.50751	−0.0513759
$862$	0	0
$863$	21.8415	0.743492	0.371746	−	0.928335i	$-0.378759\pi$
$863$	21.8415	0.743492	0.371746	+	0.928335i	$0.378759\pi$
$864$	0	0
$865$	−32.6508	−1.11016
$866$	0	0
$867$	29.2646	0.993878
$868$	0	0
$869$	1.65268	0.0560633
$870$	0	0
$871$	5.94585	0.201467
$872$	0	0
$873$	9.49623	0.321399
$874$	0	0
$875$	23.1136	0.781382
$876$	0	0
$877$	−22.8446	−0.771408	−0.385704	−	0.922623i	$-0.626041\pi$
$877$	−22.8446	−0.771408	−0.385704	+	0.922623i	$0.626041\pi$
$878$	0	0
$879$	1.42438	0.0480432
$880$	0	0
$881$	18.3047	0.616703	0.308351	−	0.951273i	$-0.400223\pi$
$881$	18.3047	0.616703	0.308351	+	0.951273i	$0.400223\pi$
$882$	0	0
$883$	−37.2872	−1.25481	−0.627406	−	0.778692i	$-0.715883\pi$
$883$	−37.2872	−1.25481	−0.627406	+	0.778692i	$0.715883\pi$
$884$	0	0
$885$	−21.4160	−0.719891
$886$	0	0
$887$	12.5057	0.419901	0.209950	−	0.977712i	$-0.432670\pi$
$887$	12.5057	0.419901	0.209950	+	0.977712i	$0.432670\pi$
$888$	0	0
$889$	10.5379	0.353431
$890$	0	0
$891$	0.705979	0.0236512
$892$	0	0
$893$	−3.24147	−0.108472
$894$	0	0
$895$	−59.5961	−1.99208
$896$	0	0
$897$	2.93006	0.0978317
$898$	0	0
$899$	9.11330	0.303946
$900$	0	0
$901$	7.65115	0.254897
$902$	0	0
$903$	7.60787	0.253174
$904$	0	0
$905$	−29.4184	−0.977902
$906$	0	0
$907$	38.9299	1.29265	0.646323	−	0.763064i	$-0.276306\pi$
$907$	38.9299	1.29265	0.646323	+	0.763064i	$0.276306\pi$
$908$	0	0
$909$	−3.11003	−0.103153
$910$	0	0
$911$	45.3851	1.50367	0.751837	−	0.659349i	$-0.229168\pi$
$911$	45.3851	1.50367	0.751837	+	0.659349i	$0.229168\pi$
$912$	0	0
$913$	9.11016	0.301502
$914$	0	0
$915$	56.6611	1.87316
$916$	0	0
$917$	−14.1038	−0.465748
$918$	0	0
$919$	−37.3191	−1.23104	−0.615521	−	0.788120i	$-0.711055\pi$
$919$	−37.3191	−1.23104	−0.615521	+	0.788120i	$0.711055\pi$
$920$	0	0
$921$	−24.9384	−0.821748
$922$	0	0
$923$	−33.1226	−1.09024
$924$	0	0
$925$	140.316	4.61356
$926$	0	0
$927$	−2.89095	−0.0949512
$928$	0	0
$929$	−8.78075	−0.288087	−0.144043	−	0.989571i	$-0.546011\pi$
$929$	−8.78075	−0.288087	−0.144043	+	0.989571i	$0.546011\pi$
$930$	0	0
$931$	−9.74156	−0.319267
$932$	0	0
$933$	−17.1542	−0.561604
$934$	0	0
$935$	−20.2946	−0.663705
$936$	0	0
$937$	21.3335	0.696936	0.348468	−	0.937321i	$-0.386702\pi$
$937$	21.3335	0.696936	0.348468	+	0.937321i	$0.386702\pi$
$938$	0	0
$939$	19.1772	0.625823
$940$	0	0
$941$	52.5356	1.71261	0.856305	−	0.516470i	$-0.172754\pi$
$941$	52.5356	1.71261	0.856305	+	0.516470i	$0.172754\pi$
$942$	0	0
$943$	−2.16716	−0.0705723
$944$	0	0
$945$	2.93992	0.0956356
$946$	0	0
$947$	32.4135	1.05330	0.526648	−	0.850083i	$-0.323448\pi$
$947$	32.4135	1.05330	0.526648	+	0.850083i	$0.323448\pi$
$948$	0	0
$949$	27.0064	0.876666
$950$	0	0
$951$	26.0404	0.844419
$952$	0	0
$953$	0.738966	0.0239375	0.0119687	−	0.999928i	$-0.496190\pi$
$953$	0.738966	0.0239375	0.0119687	+	0.999928i	$0.496190\pi$
$954$	0	0
$955$	28.8091	0.932242
$956$	0	0
$957$	−0.705979	−0.0228211
$958$	0	0
$959$	10.8702	0.351016
$960$	0	0
$961$	52.0523	1.67911
$962$	0	0
$963$	9.92832	0.319936
$964$	0	0
$965$	−62.7724	−2.02072
$966$	0	0
$967$	15.5013	0.498490	0.249245	−	0.968441i	$-0.419818\pi$
$967$	15.5013	0.498490	0.249245	+	0.968441i	$0.419818\pi$
$968$	0	0
$969$	−10.1687	−0.326665
$970$	0	0
$971$	−5.20909	−0.167168	−0.0835838	−	0.996501i	$-0.526637\pi$
$971$	−5.20909	−0.167168	−0.0835838	+	0.996501i	$0.526637\pi$
$972$	0	0
$973$	−12.9791	−0.416090
$974$	0	0
$975$	37.6863	1.20693
$976$	0	0
$977$	−35.1132	−1.12337	−0.561685	−	0.827351i	$-0.689847\pi$
$977$	−35.1132	−1.12337	−0.561685	+	0.827351i	$0.689847\pi$
$978$	0	0
$979$	4.69191	0.149954
$980$	0	0
$981$	−7.37977	−0.235618
$982$	0	0
$983$	45.7961	1.46067	0.730335	−	0.683089i	$-0.239364\pi$
$983$	45.7961	1.46067	0.730335	+	0.683089i	$0.239364\pi$
$984$	0	0
$985$	14.3748	0.458019
$986$	0	0
$987$	−1.50825	−0.0480081
$988$	0	0
$989$	10.9369	0.347772
$990$	0	0
$991$	34.3591	1.09145	0.545726	−	0.837964i	$-0.316254\pi$
$991$	34.3591	1.09145	0.545726	+	0.837964i	$0.316254\pi$
$992$	0	0
$993$	−31.0116	−0.984124
$994$	0	0
$995$	50.9451	1.61507
$996$	0	0
$997$	−59.7134	−1.89114	−0.945571	−	0.325415i	$-0.894496\pi$
$997$	−59.7134	−1.89114	−0.945571	+	0.325415i	$0.894496\pi$
$998$	0	0
$999$	10.9094	0.345157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.16	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.16	✓	16	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.22634 q^{5} +0.695618 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.22634 q^{5} +0.695618 q^{7} +1.00000 q^{9} +0.705979 q^{11} +2.93006 q^{13} +4.22634 q^{15} -6.80181 q^{17} +1.49499 q^{19} +0.695618 q^{21} +1.00000 q^{23} +12.8620 q^{25} +1.00000 q^{27} -1.00000 q^{29} -9.11330 q^{31} +0.705979 q^{33} +2.93992 q^{35} +10.9094 q^{37} +2.93006 q^{39} -2.16716 q^{41} +10.9369 q^{43} +4.22634 q^{45} -2.16821 q^{47} -6.51612 q^{49} -6.80181 q^{51} -1.12487 q^{53} +2.98371 q^{55} +1.49499 q^{57} -5.06726 q^{59} +13.4066 q^{61} +0.695618 q^{63} +12.3834 q^{65} +2.02926 q^{67} +1.00000 q^{69} -11.3044 q^{71} +9.21704 q^{73} +12.8620 q^{75} +0.491092 q^{77} +2.34097 q^{79} +1.00000 q^{81} +12.9043 q^{83} -28.7468 q^{85} -1.00000 q^{87} +6.64596 q^{89} +2.03820 q^{91} -9.11330 q^{93} +6.31836 q^{95} +9.49623 q^{97} +0.705979 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Introduction

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