LMFDB - Embedded newform 8016.2.a.m.1.1

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1300.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 10x - 10 $ x^3 - 10*x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.57709$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.57709 q^{5} -1.35861 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.57709 q^{5} -1.35861 q^{7} +1.00000 q^{9} +4.35861 q^{11} +4.79557 q^{13} -2.57709 q^{15} +6.79557 q^{17} +1.64139 q^{19} -1.35861 q^{21} +3.64139 q^{23} +1.64139 q^{25} +1.00000 q^{27} -4.35861 q^{29} +1.00000 q^{31} +4.35861 q^{33} +3.50126 q^{35} +1.42291 q^{37} +4.79557 q^{39} +2.71722 q^{41} +5.07583 q^{43} -2.57709 q^{45} -1.35861 q^{47} -5.15418 q^{49} +6.79557 q^{51} -8.21848 q^{53} -11.2325 q^{55} +1.64139 q^{57} -7.01405 q^{59} -12.8739 q^{61} -1.35861 q^{63} -12.3586 q^{65} +11.0140 q^{67} +3.64139 q^{69} -2.07835 q^{71} +1.51279 q^{73} +1.64139 q^{75} -5.92165 q^{77} +0.717220 q^{79} +1.00000 q^{81} +16.9357 q^{83} -17.5128 q^{85} -4.35861 q^{87} -10.9497 q^{89} -6.51531 q^{91} +1.00000 q^{93} -4.23001 q^{95} -0.361129 q^{97} +4.35861 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 10 q^{11} - 4 q^{13} + 3 q^{15} + 2 q^{17} + 8 q^{19} - q^{21} + 14 q^{23} + 8 q^{25} + 3 q^{27} - 10 q^{29} + 3 q^{31} + 10 q^{33} + 9 q^{35} + 15 q^{37} - 4 q^{39} + 2 q^{41} + 6 q^{43} + 3 q^{45} - q^{47} + 6 q^{49} + 2 q^{51} - 17 q^{53} + 8 q^{57} + 5 q^{59} - 8 q^{61} - q^{63} - 34 q^{65} + 7 q^{67} + 14 q^{69} + 6 q^{71} - 20 q^{73} + 8 q^{75} - 30 q^{77} - 4 q^{79} + 3 q^{81} + 37 q^{83} - 28 q^{85} - 10 q^{87} + 7 q^{89} + 8 q^{91} + 3 q^{93} + 18 q^{95} + 5 q^{97} + 10 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 + 10 * q^11 - 4 * q^13 + 3 * q^15 + 2 * q^17 + 8 * q^19 - q^21 + 14 * q^23 + 8 * q^25 + 3 * q^27 - 10 * q^29 + 3 * q^31 + 10 * q^33 + 9 * q^35 + 15 * q^37 - 4 * q^39 + 2 * q^41 + 6 * q^43 + 3 * q^45 - q^47 + 6 * q^49 + 2 * q^51 - 17 * q^53 + 8 * q^57 + 5 * q^59 - 8 * q^61 - q^63 - 34 * q^65 + 7 * q^67 + 14 * q^69 + 6 * q^71 - 20 * q^73 + 8 * q^75 - 30 * q^77 - 4 * q^79 + 3 * q^81 + 37 * q^83 - 28 * q^85 - 10 * q^87 + 7 * q^89 + 8 * q^91 + 3 * q^93 + 18 * q^95 + 5 * q^97 + 10 * 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
$5$	−2.57709	−1.15251	−0.576255	−	0.817270i	$-0.695486\pi$
$5$	−2.57709	−1.15251	−0.576255	+	0.817270i	$0.695486\pi$
$6$	0	0
$7$	−1.35861	−0.513506	−0.256753	−	0.966477i	$-0.582653\pi$
$7$	−1.35861	−0.513506	−0.256753	+	0.966477i	$0.582653\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.35861	1.31417	0.657085	−	0.753816i	$-0.271789\pi$
$11$	4.35861	1.31417	0.657085	+	0.753816i	$0.271789\pi$
$12$	0	0
$13$	4.79557	1.33005	0.665026	−	0.746820i	$-0.268421\pi$
$13$	4.79557	1.33005	0.665026	+	0.746820i	$0.268421\pi$
$14$	0	0
$15$	−2.57709	−0.665402
$16$	0	0
$17$	6.79557	1.64817	0.824084	−	0.566468i	$-0.191691\pi$
$17$	6.79557	1.64817	0.824084	+	0.566468i	$0.191691\pi$
$18$	0	0
$19$	1.64139	0.376561	0.188280	−	0.982115i	$-0.439709\pi$
$19$	1.64139	0.376561	0.188280	+	0.982115i	$0.439709\pi$
$20$	0	0
$21$	−1.35861	−0.296473
$22$	0	0
$23$	3.64139	0.759282	0.379641	−	0.925134i	$-0.376047\pi$
$23$	3.64139	0.759282	0.379641	+	0.925134i	$0.376047\pi$
$24$	0	0
$25$	1.64139	0.328278
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.35861	−0.809374	−0.404687	−	0.914455i	$-0.632619\pi$
$29$	−4.35861	−0.809374	−0.404687	+	0.914455i	$0.632619\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	4.35861	0.758737
$34$	0	0
$35$	3.50126	0.591821
$36$	0	0
$37$	1.42291	0.233925	0.116963	−	0.993136i	$-0.462684\pi$
$37$	1.42291	0.233925	0.116963	+	0.993136i	$0.462684\pi$
$38$	0	0
$39$	4.79557	0.767906
$40$	0	0
$41$	2.71722	0.424358	0.212179	−	0.977231i	$-0.431944\pi$
$41$	2.71722	0.424358	0.212179	+	0.977231i	$0.431944\pi$
$42$	0	0
$43$	5.07583	0.774057	0.387028	−	0.922068i	$-0.373502\pi$
$43$	5.07583	0.774057	0.387028	+	0.922068i	$0.373502\pi$
$44$	0	0
$45$	−2.57709	−0.384170
$46$	0	0
$47$	−1.35861	−0.198174	−0.0990868	−	0.995079i	$-0.531592\pi$
$47$	−1.35861	−0.198174	−0.0990868	+	0.995079i	$0.531592\pi$
$48$	0	0
$49$	−5.15418	−0.736311
$50$	0	0
$51$	6.79557	0.951570
$52$	0	0
$53$	−8.21848	−1.12889	−0.564447	−	0.825469i	$-0.690911\pi$
$53$	−8.21848	−1.12889	−0.564447	+	0.825469i	$0.690911\pi$
$54$	0	0
$55$	−11.2325	−1.51459
$56$	0	0
$57$	1.64139	0.217407
$58$	0	0
$59$	−7.01405	−0.913151	−0.456576	−	0.889685i	$-0.650924\pi$
$59$	−7.01405	−0.913151	−0.456576	+	0.889685i	$0.650924\pi$
$60$	0	0
$61$	−12.8739	−1.64834	−0.824168	−	0.566345i	$-0.808357\pi$
$61$	−12.8739	−1.64834	−0.824168	+	0.566345i	$0.808357\pi$
$62$	0	0
$63$	−1.35861	−0.171169
$64$	0	0
$65$	−12.3586	−1.53290
$66$	0	0
$67$	11.0140	1.34558	0.672790	−	0.739833i	$-0.265096\pi$
$67$	11.0140	1.34558	0.672790	+	0.739833i	$0.265096\pi$
$68$	0	0
$69$	3.64139	0.438372
$70$	0	0
$71$	−2.07835	−0.246655	−0.123327	−	0.992366i	$-0.539357\pi$
$71$	−2.07835	−0.246655	−0.123327	+	0.992366i	$0.539357\pi$
$72$	0	0
$73$	1.51279	0.177059	0.0885293	−	0.996074i	$-0.471783\pi$
$73$	1.51279	0.177059	0.0885293	+	0.996074i	$0.471783\pi$
$74$	0	0
$75$	1.64139	0.189531
$76$	0	0
$77$	−5.92165	−0.674835
$78$	0	0
$79$	0.717220	0.0806936	0.0403468	−	0.999186i	$-0.487154\pi$
$79$	0.717220	0.0806936	0.0403468	+	0.999186i	$0.487154\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.9357	1.85893	0.929467	−	0.368905i	$-0.120267\pi$
$83$	16.9357	1.85893	0.929467	+	0.368905i	$0.120267\pi$
$84$	0	0
$85$	−17.5128	−1.89953
$86$	0	0
$87$	−4.35861	−0.467292
$88$	0	0
$89$	−10.9497	−1.16067	−0.580335	−	0.814378i	$-0.697079\pi$
$89$	−10.9497	−1.16067	−0.580335	+	0.814378i	$0.697079\pi$
$90$	0	0
$91$	−6.51531	−0.682990
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−4.23001	−0.433990
$96$	0	0
$97$	−0.361129	−0.0366671	−0.0183335	−	0.999832i	$-0.505836\pi$
$97$	−0.361129	−0.0366671	−0.0183335	+	0.999832i	$0.505836\pi$
$98$	0	0
$99$	4.35861	0.438057
$100$	0	0
$101$	11.6529	1.15951	0.579754	−	0.814791i	$-0.303148\pi$
$101$	11.6529	1.15951	0.579754	+	0.814791i	$0.303148\pi$
$102$	0	0
$103$	17.1542	1.69025	0.845126	−	0.534568i	$-0.179526\pi$
$103$	17.1542	1.69025	0.845126	+	0.534568i	$0.179526\pi$
$104$	0	0
$105$	3.50126	0.341688
$106$	0	0
$107$	7.71974	0.746295	0.373148	−	0.927772i	$-0.378278\pi$
$107$	7.71974	0.746295	0.373148	+	0.927772i	$0.378278\pi$
$108$	0	0
$109$	−1.51279	−0.144899	−0.0724494	−	0.997372i	$-0.523082\pi$
$109$	−1.51279	−0.144899	−0.0724494	+	0.997372i	$0.523082\pi$
$110$	0	0
$111$	1.42291	0.135057
$112$	0	0
$113$	0.717220	0.0674704	0.0337352	−	0.999431i	$-0.489260\pi$
$113$	0.717220	0.0674704	0.0337352	+	0.999431i	$0.489260\pi$
$114$	0	0
$115$	−9.38419	−0.875080
$116$	0	0
$117$	4.79557	0.443351
$118$	0	0
$119$	−9.23253	−0.846344
$120$	0	0
$121$	7.99748	0.727044
$122$	0	0
$123$	2.71722	0.245003
$124$	0	0
$125$	8.65544	0.774166
$126$	0	0
$127$	10.7197	0.951223	0.475612	−	0.879655i	$-0.342227\pi$
$127$	10.7197	0.951223	0.475612	+	0.879655i	$0.342227\pi$
$128$	0	0
$129$	5.07583	0.446902
$130$	0	0
$131$	−3.64139	−0.318150	−0.159075	−	0.987267i	$-0.550851\pi$
$131$	−3.64139	−0.318150	−0.159075	+	0.987267i	$0.550851\pi$
$132$	0	0
$133$	−2.23001	−0.193366
$134$	0	0
$135$	−2.57709	−0.221801
$136$	0	0
$137$	−16.5128	−1.41078	−0.705391	−	0.708818i	$-0.749229\pi$
$137$	−16.5128	−1.41078	−0.705391	+	0.708818i	$0.749229\pi$
$138$	0	0
$139$	−4.29683	−0.364452	−0.182226	−	0.983257i	$-0.558330\pi$
$139$	−4.29683	−0.364452	−0.182226	+	0.983257i	$0.558330\pi$
$140$	0	0
$141$	−1.35861	−0.114416
$142$	0	0
$143$	20.9020	1.74791
$144$	0	0
$145$	11.2325	0.932811
$146$	0	0
$147$	−5.15418	−0.425110
$148$	0	0
$149$	7.80962	0.639789	0.319894	−	0.947453i	$-0.396353\pi$
$149$	7.80962	0.639789	0.319894	+	0.947453i	$0.396353\pi$
$150$	0	0
$151$	−18.4370	−1.50038	−0.750189	−	0.661223i	$-0.770038\pi$
$151$	−18.4370	−1.50038	−0.750189	+	0.661223i	$0.770038\pi$
$152$	0	0
$153$	6.79557	0.549389
$154$	0	0
$155$	−2.57709	−0.206997
$156$	0	0
$157$	3.71974	0.296867	0.148434	−	0.988922i	$-0.452577\pi$
$157$	3.71974	0.296867	0.148434	+	0.988922i	$0.452577\pi$
$158$	0	0
$159$	−8.21848	−0.651768
$160$	0	0
$161$	−4.94723	−0.389896
$162$	0	0
$163$	−7.93822	−0.621769	−0.310885	−	0.950448i	$-0.600625\pi$
$163$	−7.93822	−0.621769	−0.310885	+	0.950448i	$0.600625\pi$
$164$	0	0
$165$	−11.2325	−0.874451
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	9.99748	0.769037
$170$	0	0
$171$	1.64139	0.125520
$172$	0	0
$173$	−9.56304	−0.727065	−0.363532	−	0.931582i	$-0.618429\pi$
$173$	−9.56304	−0.727065	−0.363532	+	0.931582i	$0.618429\pi$
$174$	0	0
$175$	−2.23001	−0.168573
$176$	0	0
$177$	−7.01405	−0.527208
$178$	0	0
$179$	20.2581	1.51416	0.757081	−	0.653321i	$-0.226625\pi$
$179$	20.2581	1.51416	0.757081	+	0.653321i	$0.226625\pi$
$180$	0	0
$181$	−15.8211	−1.17598	−0.587988	−	0.808869i	$-0.700080\pi$
$181$	−15.8211	−1.17598	−0.587988	+	0.808869i	$0.700080\pi$
$182$	0	0
$183$	−12.8739	−0.951667
$184$	0	0
$185$	−3.66697	−0.269601
$186$	0	0
$187$	29.6192	2.16597
$188$	0	0
$189$	−1.35861	−0.0988243
$190$	0	0
$191$	22.5911	1.63464	0.817319	−	0.576186i	$-0.195460\pi$
$191$	22.5911	1.63464	0.817319	+	0.576186i	$0.195460\pi$
$192$	0	0
$193$	23.6695	1.70377	0.851883	−	0.523731i	$-0.175460\pi$
$193$	23.6695	1.70377	0.851883	+	0.523731i	$0.175460\pi$
$194$	0	0
$195$	−12.3586	−0.885018
$196$	0	0
$197$	8.07835	0.575559	0.287779	−	0.957697i	$-0.407083\pi$
$197$	8.07835	0.575559	0.287779	+	0.957697i	$0.407083\pi$
$198$	0	0
$199$	−11.1542	−0.790699	−0.395349	−	0.918531i	$-0.629376\pi$
$199$	−11.1542	−0.790699	−0.395349	+	0.918531i	$0.629376\pi$
$200$	0	0
$201$	11.0140	0.776871
$202$	0	0
$203$	5.92165	0.415618
$204$	0	0
$205$	−7.00252	−0.489077
$206$	0	0
$207$	3.64139	0.253094
$208$	0	0
$209$	7.15418	0.494865
$210$	0	0
$211$	−20.3084	−1.39809	−0.699043	−	0.715080i	$-0.746390\pi$
$211$	−20.3084	−1.39809	−0.699043	+	0.715080i	$0.746390\pi$
$212$	0	0
$213$	−2.07835	−0.142406
$214$	0	0
$215$	−13.0809	−0.892108
$216$	0	0
$217$	−1.35861	−0.0922285
$218$	0	0
$219$	1.51279	0.102225
$220$	0	0
$221$	32.5886	2.19215
$222$	0	0
$223$	17.2300	1.15381	0.576903	−	0.816812i	$-0.304261\pi$
$223$	17.2300	1.15381	0.576903	+	0.816812i	$0.304261\pi$
$224$	0	0
$225$	1.64139	0.109426
$226$	0	0
$227$	−10.4485	−0.693491	−0.346745	−	0.937959i	$-0.612713\pi$
$227$	−10.4485	−0.693491	−0.346745	+	0.937959i	$0.612713\pi$
$228$	0	0
$229$	−20.6167	−1.36239	−0.681195	−	0.732102i	$-0.738540\pi$
$229$	−20.6167	−1.36239	−0.681195	+	0.732102i	$0.738540\pi$
$230$	0	0
$231$	−5.92165	−0.389616
$232$	0	0
$233$	16.5630	1.08508	0.542540	−	0.840030i	$-0.317463\pi$
$233$	16.5630	1.08508	0.542540	+	0.840030i	$0.317463\pi$
$234$	0	0
$235$	3.50126	0.228397
$236$	0	0
$237$	0.717220	0.0465884
$238$	0	0
$239$	−17.4625	−1.12956	−0.564779	−	0.825242i	$-0.691039\pi$
$239$	−17.4625	−1.12956	−0.564779	+	0.825242i	$0.691039\pi$
$240$	0	0
$241$	6.66697	0.429457	0.214729	−	0.976674i	$-0.431113\pi$
$241$	6.66697	0.429457	0.214729	+	0.976674i	$0.431113\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	13.2828	0.848606
$246$	0	0
$247$	7.87140	0.500845
$248$	0	0
$249$	16.9357	1.07326
$250$	0	0
$251$	9.66949	0.610333	0.305166	−	0.952299i	$-0.401288\pi$
$251$	9.66949	0.610333	0.305166	+	0.952299i	$0.401288\pi$
$252$	0	0
$253$	15.8714	0.997826
$254$	0	0
$255$	−17.5128	−1.09669
$256$	0	0
$257$	15.0758	0.940404	0.470202	−	0.882559i	$-0.344181\pi$
$257$	15.0758	0.940404	0.470202	+	0.882559i	$0.344181\pi$
$258$	0	0
$259$	−1.93318	−0.120122
$260$	0	0
$261$	−4.35861	−0.269791
$262$	0	0
$263$	0.793050	0.0489016	0.0244508	−	0.999701i	$-0.492216\pi$
$263$	0.793050	0.0489016	0.0244508	+	0.999701i	$0.492216\pi$
$264$	0	0
$265$	21.1798	1.30106
$266$	0	0
$267$	−10.9497	−0.670114
$268$	0	0
$269$	18.1682	1.10774	0.553868	−	0.832604i	$-0.313151\pi$
$269$	18.1682	1.10774	0.553868	+	0.832604i	$0.313151\pi$
$270$	0	0
$271$	−12.7956	−0.777275	−0.388638	−	0.921391i	$-0.627054\pi$
$271$	−12.7956	−0.777275	−0.388638	+	0.921391i	$0.627054\pi$
$272$	0	0
$273$	−6.51531	−0.394324
$274$	0	0
$275$	7.15418	0.431413
$276$	0	0
$277$	−9.34456	−0.561460	−0.280730	−	0.959787i	$-0.590577\pi$
$277$	−9.34456	−0.561460	−0.280730	+	0.959787i	$0.590577\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	28.4625	1.69793	0.848966	−	0.528447i	$-0.177225\pi$
$281$	28.4625	1.69793	0.848966	+	0.528447i	$0.177225\pi$
$282$	0	0
$283$	0.588619	0.0349898	0.0174949	−	0.999847i	$-0.494431\pi$
$283$	0.588619	0.0349898	0.0174949	+	0.999847i	$0.494431\pi$
$284$	0	0
$285$	−4.23001	−0.250564
$286$	0	0
$287$	−3.69164	−0.217911
$288$	0	0
$289$	29.1798	1.71646
$290$	0	0
$291$	−0.361129	−0.0211698
$292$	0	0
$293$	−7.28278	−0.425465	−0.212732	−	0.977111i	$-0.568236\pi$
$293$	−7.28278	−0.425465	−0.212732	+	0.977111i	$0.568236\pi$
$294$	0	0
$295$	18.0758	1.05242
$296$	0	0
$297$	4.35861	0.252912
$298$	0	0
$299$	17.4625	1.00988
$300$	0	0
$301$	−6.89607	−0.397483
$302$	0	0
$303$	11.6529	0.669443
$304$	0	0
$305$	33.1772	1.89972
$306$	0	0
$307$	18.5268	1.05738	0.528691	−	0.848814i	$-0.322683\pi$
$307$	18.5268	1.05738	0.528691	+	0.848814i	$0.322683\pi$
$308$	0	0
$309$	17.1542	0.975867
$310$	0	0
$311$	25.4370	1.44240	0.721199	−	0.692728i	$-0.243591\pi$
$311$	25.4370	1.44240	0.721199	+	0.692728i	$0.243591\pi$
$312$	0	0
$313$	−22.2300	−1.25651	−0.628257	−	0.778006i	$-0.716231\pi$
$313$	−22.2300	−1.25651	−0.628257	+	0.778006i	$0.716231\pi$
$314$	0	0
$315$	3.50126	0.197274
$316$	0	0
$317$	24.5153	1.37692	0.688458	−	0.725276i	$-0.258288\pi$
$317$	24.5153	1.37692	0.688458	+	0.725276i	$0.258288\pi$
$318$	0	0
$319$	−18.9975	−1.06365
$320$	0	0
$321$	7.71974	0.430874
$322$	0	0
$323$	11.1542	0.620635
$324$	0	0
$325$	7.87140	0.436627
$326$	0	0
$327$	−1.51279	−0.0836574
$328$	0	0
$329$	1.84582	0.101763
$330$	0	0
$331$	20.5153	1.12762	0.563812	−	0.825903i	$-0.309334\pi$
$331$	20.5153	1.12762	0.563812	+	0.825903i	$0.309334\pi$
$332$	0	0
$333$	1.42291	0.0779750
$334$	0	0
$335$	−28.3842	−1.55079
$336$	0	0
$337$	3.15166	0.171682	0.0858409	−	0.996309i	$-0.472642\pi$
$337$	3.15166	0.171682	0.0858409	+	0.996309i	$0.472642\pi$
$338$	0	0
$339$	0.717220	0.0389540
$340$	0	0
$341$	4.35861	0.236032
$342$	0	0
$343$	16.5128	0.891607
$344$	0	0
$345$	−9.38419	−0.505228
$346$	0	0
$347$	5.85987	0.314574	0.157287	−	0.987553i	$-0.449725\pi$
$347$	5.85987	0.314574	0.157287	+	0.987553i	$0.449725\pi$
$348$	0	0
$349$	24.8854	1.33209	0.666044	−	0.745913i	$-0.267986\pi$
$349$	24.8854	1.33209	0.666044	+	0.745913i	$0.267986\pi$
$350$	0	0
$351$	4.79557	0.255969
$352$	0	0
$353$	−1.71722	−0.0913984	−0.0456992	−	0.998955i	$-0.514552\pi$
$353$	−1.71722	−0.0913984	−0.0456992	+	0.998955i	$0.514552\pi$
$354$	0	0
$355$	5.35609	0.284272
$356$	0	0
$357$	−9.23253	−0.488637
$358$	0	0
$359$	21.4625	1.13275	0.566375	−	0.824148i	$-0.308346\pi$
$359$	21.4625	1.13275	0.566375	+	0.824148i	$0.308346\pi$
$360$	0	0
$361$	−16.3058	−0.858202
$362$	0	0
$363$	7.99748	0.419759
$364$	0	0
$365$	−3.89859	−0.204062
$366$	0	0
$367$	−22.3365	−1.16595	−0.582977	−	0.812489i	$-0.698112\pi$
$367$	−22.3365	−1.16595	−0.582977	+	0.812489i	$0.698112\pi$
$368$	0	0
$369$	2.71722	0.141453
$370$	0	0
$371$	11.1657	0.579695
$372$	0	0
$373$	−0.218479	−0.0113124	−0.00565622	−	0.999984i	$-0.501800\pi$
$373$	−0.218479	−0.0113124	−0.00565622	+	0.999984i	$0.501800\pi$
$374$	0	0
$375$	8.65544	0.446965
$376$	0	0
$377$	−20.9020	−1.07651
$378$	0	0
$379$	−2.21848	−0.113956	−0.0569778	−	0.998375i	$-0.518146\pi$
$379$	−2.21848	−0.113956	−0.0569778	+	0.998375i	$0.518146\pi$
$380$	0	0
$381$	10.7197	0.549189
$382$	0	0
$383$	31.0537	1.58677	0.793384	−	0.608721i	$-0.208317\pi$
$383$	31.0537	1.58677	0.793384	+	0.608721i	$0.208317\pi$
$384$	0	0
$385$	15.2606	0.777753
$386$	0	0
$387$	5.07583	0.258019
$388$	0	0
$389$	−2.51531	−0.127531	−0.0637656	−	0.997965i	$-0.520311\pi$
$389$	−2.51531	−0.127531	−0.0637656	+	0.997965i	$0.520311\pi$
$390$	0	0
$391$	24.7453	1.25142
$392$	0	0
$393$	−3.64139	−0.183684
$394$	0	0
$395$	−1.84834	−0.0930001
$396$	0	0
$397$	−15.5128	−0.778565	−0.389282	−	0.921119i	$-0.627277\pi$
$397$	−15.5128	−0.778565	−0.389282	+	0.921119i	$0.627277\pi$
$398$	0	0
$399$	−2.23001	−0.111640
$400$	0	0
$401$	−28.9020	−1.44330	−0.721649	−	0.692259i	$-0.756615\pi$
$401$	−28.9020	−1.44330	−0.721649	+	0.692259i	$0.756615\pi$
$402$	0	0
$403$	4.79557	0.238884
$404$	0	0
$405$	−2.57709	−0.128057
$406$	0	0
$407$	6.20191	0.307417
$408$	0	0
$409$	9.43948	0.466752	0.233376	−	0.972387i	$-0.425023\pi$
$409$	9.43948	0.466752	0.233376	+	0.972387i	$0.425023\pi$
$410$	0	0
$411$	−16.5128	−0.814516
$412$	0	0
$413$	9.52936	0.468909
$414$	0	0
$415$	−43.6448	−2.14244
$416$	0	0
$417$	−4.29683	−0.210417
$418$	0	0
$419$	−23.7428	−1.15991	−0.579956	−	0.814648i	$-0.696930\pi$
$419$	−23.7428	−1.15991	−0.579956	+	0.814648i	$0.696930\pi$
$420$	0	0
$421$	−34.6951	−1.69093	−0.845467	−	0.534028i	$-0.820678\pi$
$421$	−34.6951	−1.69093	−0.845467	+	0.534028i	$0.820678\pi$
$422$	0	0
$423$	−1.35861	−0.0660579
$424$	0	0
$425$	11.1542	0.541057
$426$	0	0
$427$	17.4906	0.846431
$428$	0	0
$429$	20.9020	1.00916
$430$	0	0
$431$	−17.8739	−0.860956	−0.430478	−	0.902601i	$-0.641655\pi$
$431$	−17.8739	−0.860956	−0.430478	+	0.902601i	$0.641655\pi$
$432$	0	0
$433$	−32.4625	−1.56005	−0.780025	−	0.625748i	$-0.784794\pi$
$433$	−32.4625	−1.56005	−0.780025	+	0.625748i	$0.784794\pi$
$434$	0	0
$435$	11.2325	0.538558
$436$	0	0
$437$	5.97694	0.285916
$438$	0	0
$439$	−4.12860	−0.197047	−0.0985237	−	0.995135i	$-0.531412\pi$
$439$	−4.12860	−0.197047	−0.0985237	+	0.995135i	$0.531412\pi$
$440$	0	0
$441$	−5.15418	−0.245437
$442$	0	0
$443$	20.4485	0.971537	0.485768	−	0.874088i	$-0.338540\pi$
$443$	20.4485	0.971537	0.485768	+	0.874088i	$0.338540\pi$
$444$	0	0
$445$	28.2185	1.33768
$446$	0	0
$447$	7.80962	0.369382
$448$	0	0
$449$	−31.4098	−1.48232	−0.741159	−	0.671329i	$-0.765724\pi$
$449$	−31.4098	−1.48232	−0.741159	+	0.671329i	$0.765724\pi$
$450$	0	0
$451$	11.8433	0.557679
$452$	0	0
$453$	−18.4370	−0.866244
$454$	0	0
$455$	16.7905	0.787152
$456$	0	0
$457$	−16.1517	−0.755543	−0.377771	−	0.925899i	$-0.623309\pi$
$457$	−16.1517	−0.755543	−0.377771	+	0.925899i	$0.623309\pi$
$458$	0	0
$459$	6.79557	0.317190
$460$	0	0
$461$	−0.487211	−0.0226917	−0.0113458	−	0.999936i	$-0.503612\pi$
$461$	−0.487211	−0.0226917	−0.0113458	+	0.999936i	$0.503612\pi$
$462$	0	0
$463$	−4.30836	−0.200226	−0.100113	−	0.994976i	$-0.531920\pi$
$463$	−4.30836	−0.200226	−0.100113	+	0.994976i	$0.531920\pi$
$464$	0	0
$465$	−2.57709	−0.119510
$466$	0	0
$467$	−2.87392	−0.132989	−0.0664945	−	0.997787i	$-0.521181\pi$
$467$	−2.87392	−0.132989	−0.0664945	+	0.997787i	$0.521181\pi$
$468$	0	0
$469$	−14.9638	−0.690964
$470$	0	0
$471$	3.71974	0.171397
$472$	0	0
$473$	22.1236	1.01724
$474$	0	0
$475$	2.69416	0.123617
$476$	0	0
$477$	−8.21848	−0.376298
$478$	0	0
$479$	19.3109	0.882336	0.441168	−	0.897424i	$-0.354564\pi$
$479$	19.3109	0.882336	0.441168	+	0.897424i	$0.354564\pi$
$480$	0	0
$481$	6.82367	0.311132
$482$	0	0
$483$	−4.94723	−0.225107
$484$	0	0
$485$	0.930662	0.0422592
$486$	0	0
$487$	−13.0758	−0.592522	−0.296261	−	0.955107i	$-0.595740\pi$
$487$	−13.0758	−0.592522	−0.296261	+	0.955107i	$0.595740\pi$
$488$	0	0
$489$	−7.93822	−0.358979
$490$	0	0
$491$	11.2828	0.509185	0.254592	−	0.967048i	$-0.418059\pi$
$491$	11.2828	0.509185	0.254592	+	0.967048i	$0.418059\pi$
$492$	0	0
$493$	−29.6192	−1.33398
$494$	0	0
$495$	−11.2325	−0.504865
$496$	0	0
$497$	2.82367	0.126659
$498$	0	0
$499$	−1.33303	−0.0596747	−0.0298374	−	0.999555i	$-0.509499\pi$
$499$	−1.33303	−0.0596747	−0.0298374	+	0.999555i	$0.509499\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−25.3058	−1.12833	−0.564166	−	0.825662i	$-0.690802\pi$
$503$	−25.3058	−1.12833	−0.564166	+	0.825662i	$0.690802\pi$
$504$	0	0
$505$	−30.0306	−1.33634
$506$	0	0
$507$	9.99748	0.444004
$508$	0	0
$509$	38.8970	1.72408	0.862039	−	0.506842i	$-0.169187\pi$
$509$	38.8970	1.72408	0.862039	+	0.506842i	$0.169187\pi$
$510$	0	0
$511$	−2.05529	−0.0909207
$512$	0	0
$513$	1.64139	0.0724691
$514$	0	0
$515$	−44.2079	−1.94803
$516$	0	0
$517$	−5.92165	−0.260434
$518$	0	0
$519$	−9.56304	−0.419771
$520$	0	0
$521$	23.9497	1.04926	0.524629	−	0.851331i	$-0.324204\pi$
$521$	23.9497	1.04926	0.524629	+	0.851331i	$0.324204\pi$
$522$	0	0
$523$	34.9523	1.52836	0.764178	−	0.645005i	$-0.223145\pi$
$523$	34.9523	1.52836	0.764178	+	0.645005i	$0.223145\pi$
$524$	0	0
$525$	−2.23001	−0.0973256
$526$	0	0
$527$	6.79557	0.296020
$528$	0	0
$529$	−9.74028	−0.423490
$530$	0	0
$531$	−7.01405	−0.304384
$532$	0	0
$533$	13.0306	0.564419
$534$	0	0
$535$	−19.8945	−0.860112
$536$	0	0
$537$	20.2581	0.874202
$538$	0	0
$539$	−22.4651	−0.967638
$540$	0	0
$541$	18.8352	0.809788	0.404894	−	0.914364i	$-0.367308\pi$
$541$	18.8352	0.809788	0.404894	+	0.914364i	$0.367308\pi$
$542$	0	0
$543$	−15.8211	−0.678950
$544$	0	0
$545$	3.89859	0.166997
$546$	0	0
$547$	13.4460	0.574908	0.287454	−	0.957794i	$-0.407191\pi$
$547$	13.4460	0.574908	0.287454	+	0.957794i	$0.407191\pi$
$548$	0	0
$549$	−12.8739	−0.549445
$550$	0	0
$551$	−7.15418	−0.304778
$552$	0	0
$553$	−0.974422	−0.0414366
$554$	0	0
$555$	−3.66697	−0.155654
$556$	0	0
$557$	20.3867	0.863812	0.431906	−	0.901919i	$-0.357841\pi$
$557$	20.3867	0.863812	0.431906	+	0.901919i	$0.357841\pi$
$558$	0	0
$559$	24.3415	1.02954
$560$	0	0
$561$	29.6192	1.25053
$562$	0	0
$563$	40.1064	1.69029	0.845143	−	0.534541i	$-0.179515\pi$
$563$	40.1064	1.69029	0.845143	+	0.534541i	$0.179515\pi$
$564$	0	0
$565$	−1.84834	−0.0777603
$566$	0	0
$567$	−1.35861	−0.0570563
$568$	0	0
$569$	4.23001	0.177331	0.0886656	−	0.996061i	$-0.471740\pi$
$569$	4.23001	0.177331	0.0886656	+	0.996061i	$0.471740\pi$
$570$	0	0
$571$	33.8377	1.41606	0.708032	−	0.706180i	$-0.249583\pi$
$571$	33.8377	1.41606	0.708032	+	0.706180i	$0.249583\pi$
$572$	0	0
$573$	22.5911	0.943758
$574$	0	0
$575$	5.97694	0.249256
$576$	0	0
$577$	−1.76747	−0.0735808	−0.0367904	−	0.999323i	$-0.511713\pi$
$577$	−1.76747	−0.0735808	−0.0367904	+	0.999323i	$0.511713\pi$
$578$	0	0
$579$	23.6695	0.983670
$580$	0	0
$581$	−23.0090	−0.954575
$582$	0	0
$583$	−35.8211	−1.48356
$584$	0	0
$585$	−12.3586	−0.510966
$586$	0	0
$587$	23.0924	0.953125	0.476563	−	0.879141i	$-0.341883\pi$
$587$	23.0924	0.953125	0.476563	+	0.879141i	$0.341883\pi$
$588$	0	0
$589$	1.64139	0.0676323
$590$	0	0
$591$	8.07835	0.332299
$592$	0	0
$593$	28.6670	1.17721	0.588606	−	0.808420i	$-0.299677\pi$
$593$	28.6670	1.17721	0.588606	+	0.808420i	$0.299677\pi$
$594$	0	0
$595$	23.7930	0.975420
$596$	0	0
$597$	−11.1542	−0.456510
$598$	0	0
$599$	−22.8211	−0.932447	−0.466223	−	0.884667i	$-0.654386\pi$
$599$	−22.8211	−0.932447	−0.466223	+	0.884667i	$0.654386\pi$
$600$	0	0
$601$	3.20695	0.130814	0.0654071	−	0.997859i	$-0.479165\pi$
$601$	3.20695	0.130814	0.0654071	+	0.997859i	$0.479165\pi$
$602$	0	0
$603$	11.0140	0.448527
$604$	0	0
$605$	−20.6102	−0.837925
$606$	0	0
$607$	−20.5937	−0.835871	−0.417935	−	0.908477i	$-0.637246\pi$
$607$	−20.5937	−0.835871	−0.417935	+	0.908477i	$0.637246\pi$
$608$	0	0
$609$	5.92165	0.239957
$610$	0	0
$611$	−6.51531	−0.263581
$612$	0	0
$613$	23.2044	0.937218	0.468609	−	0.883406i	$-0.344755\pi$
$613$	23.2044	0.937218	0.468609	+	0.883406i	$0.344755\pi$
$614$	0	0
$615$	−7.00252	−0.282369
$616$	0	0
$617$	12.4370	0.500693	0.250347	−	0.968156i	$-0.419455\pi$
$617$	12.4370	0.500693	0.250347	+	0.968156i	$0.419455\pi$
$618$	0	0
$619$	−14.2350	−0.572155	−0.286077	−	0.958207i	$-0.592351\pi$
$619$	−14.2350	−0.572155	−0.286077	+	0.958207i	$0.592351\pi$
$620$	0	0
$621$	3.64139	0.146124
$622$	0	0
$623$	14.8764	0.596012
$624$	0	0
$625$	−30.5128	−1.22051
$626$	0	0
$627$	7.15418	0.285710
$628$	0	0
$629$	9.66949	0.385548
$630$	0	0
$631$	−35.1798	−1.40048	−0.700242	−	0.713906i	$-0.746925\pi$
$631$	−35.1798	−1.40048	−0.700242	+	0.713906i	$0.746925\pi$
$632$	0	0
$633$	−20.3084	−0.807185
$634$	0	0
$635$	−27.6257	−1.09629
$636$	0	0
$637$	−24.7172	−0.979332
$638$	0	0
$639$	−2.07835	−0.0822182
$640$	0	0
$641$	−26.6167	−1.05130	−0.525649	−	0.850702i	$-0.676177\pi$
$641$	−26.6167	−1.05130	−0.525649	+	0.850702i	$0.676177\pi$
$642$	0	0
$643$	−40.2581	−1.58762	−0.793812	−	0.608163i	$-0.791907\pi$
$643$	−40.2581	−1.58762	−0.793812	+	0.608163i	$0.791907\pi$
$644$	0	0
$645$	−13.0809	−0.515059
$646$	0	0
$647$	−3.15418	−0.124004	−0.0620018	−	0.998076i	$-0.519748\pi$
$647$	−3.15418	−0.124004	−0.0620018	+	0.998076i	$0.519748\pi$
$648$	0	0
$649$	−30.5715	−1.20004
$650$	0	0
$651$	−1.35861	−0.0532481
$652$	0	0
$653$	48.5162	1.89859	0.949293	−	0.314393i	$-0.101801\pi$
$653$	48.5162	1.89859	0.949293	+	0.314393i	$0.101801\pi$
$654$	0	0
$655$	9.38419	0.366671
$656$	0	0
$657$	1.51279	0.0590195
$658$	0	0
$659$	−29.2722	−1.14028	−0.570141	−	0.821547i	$-0.693111\pi$
$659$	−29.2722	−1.14028	−0.570141	+	0.821547i	$0.693111\pi$
$660$	0	0
$661$	−7.92165	−0.308117	−0.154058	−	0.988062i	$-0.549234\pi$
$661$	−7.92165	−0.308117	−0.154058	+	0.988062i	$0.549234\pi$
$662$	0	0
$663$	32.5886	1.26564
$664$	0	0
$665$	5.74693	0.222856
$666$	0	0
$667$	−15.8714	−0.614543
$668$	0	0
$669$	17.2300	0.666151
$670$	0	0
$671$	−56.1124	−2.16619
$672$	0	0
$673$	−36.2581	−1.39765	−0.698824	−	0.715294i	$-0.746293\pi$
$673$	−36.2581	−1.39765	−0.698824	+	0.715294i	$0.746293\pi$
$674$	0	0
$675$	1.64139	0.0631771
$676$	0	0
$677$	−41.7206	−1.60345	−0.801727	−	0.597690i	$-0.796085\pi$
$677$	−41.7206	−1.60345	−0.801727	+	0.597690i	$0.796085\pi$
$678$	0	0
$679$	0.490633	0.0188288
$680$	0	0
$681$	−10.4485	−0.400387
$682$	0	0
$683$	−22.9523	−0.878244	−0.439122	−	0.898427i	$-0.644710\pi$
$683$	−22.9523	−0.878244	−0.439122	+	0.898427i	$0.644710\pi$
$684$	0	0
$685$	42.5549	1.62594
$686$	0	0
$687$	−20.6167	−0.786577
$688$	0	0
$689$	−39.4123	−1.50149
$690$	0	0
$691$	29.2325	1.11206	0.556029	−	0.831163i	$-0.312324\pi$
$691$	29.2325	1.11206	0.556029	+	0.831163i	$0.312324\pi$
$692$	0	0
$693$	−5.92165	−0.224945
$694$	0	0
$695$	11.0733	0.420035
$696$	0	0
$697$	18.4651	0.699414
$698$	0	0
$699$	16.5630	0.626472
$700$	0	0
$701$	−6.00504	−0.226807	−0.113404	−	0.993549i	$-0.536175\pi$
$701$	−6.00504	−0.226807	−0.113404	+	0.993549i	$0.536175\pi$
$702$	0	0
$703$	2.33555	0.0880870
$704$	0	0
$705$	3.50126	0.131865
$706$	0	0
$707$	−15.8318	−0.595415
$708$	0	0
$709$	7.96128	0.298992	0.149496	−	0.988762i	$-0.452235\pi$
$709$	7.96128	0.298992	0.149496	+	0.988762i	$0.452235\pi$
$710$	0	0
$711$	0.717220	0.0268979
$712$	0	0
$713$	3.64139	0.136371
$714$	0	0
$715$	−53.8664	−2.01449
$716$	0	0
$717$	−17.4625	−0.652150
$718$	0	0
$719$	1.43444	0.0534956	0.0267478	−	0.999642i	$-0.491485\pi$
$719$	1.43444	0.0534956	0.0267478	+	0.999642i	$0.491485\pi$
$720$	0	0
$721$	−23.3058	−0.867955
$722$	0	0
$723$	6.66697	0.247947
$724$	0	0
$725$	−7.15418	−0.265700
$726$	0	0
$727$	−6.79557	−0.252034	−0.126017	−	0.992028i	$-0.540219\pi$
$727$	−6.79557	−0.252034	−0.126017	+	0.992028i	$0.540219\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	34.4932	1.27578
$732$	0	0
$733$	2.95227	0.109044	0.0545222	−	0.998513i	$-0.482636\pi$
$733$	2.95227	0.109044	0.0545222	+	0.998513i	$0.482636\pi$
$734$	0	0
$735$	13.2828	0.489943
$736$	0	0
$737$	48.0059	1.76832
$738$	0	0
$739$	−32.5499	−1.19737	−0.598684	−	0.800986i	$-0.704309\pi$
$739$	−32.5499	−1.19737	−0.598684	+	0.800986i	$0.704309\pi$
$740$	0	0
$741$	7.87140	0.289163
$742$	0	0
$743$	−47.2581	−1.73373	−0.866866	−	0.498541i	$-0.833869\pi$
$743$	−47.2581	−1.73373	−0.866866	+	0.498541i	$0.833869\pi$
$744$	0	0
$745$	−20.1261	−0.737363
$746$	0	0
$747$	16.9357	0.619645
$748$	0	0
$749$	−10.4881	−0.383227
$750$	0	0
$751$	41.8492	1.52710	0.763550	−	0.645748i	$-0.223454\pi$
$751$	41.8492	1.52710	0.763550	+	0.645748i	$0.223454\pi$
$752$	0	0
$753$	9.66949	0.352376
$754$	0	0
$755$	47.5137	1.72920
$756$	0	0
$757$	−33.9778	−1.23495	−0.617473	−	0.786592i	$-0.711843\pi$
$757$	−33.9778	−1.23495	−0.617473	+	0.786592i	$0.711843\pi$
$758$	0	0
$759$	15.8714	0.576095
$760$	0	0
$761$	−47.3084	−1.71493	−0.857463	−	0.514545i	$-0.827961\pi$
$761$	−47.3084	−1.71493	−0.857463	+	0.514545i	$0.827961\pi$
$762$	0	0
$763$	2.05529	0.0744065
$764$	0	0
$765$	−17.5128	−0.633176
$766$	0	0
$767$	−33.6364	−1.21454
$768$	0	0
$769$	47.0758	1.69760	0.848799	−	0.528716i	$-0.177326\pi$
$769$	47.0758	1.69760	0.848799	+	0.528716i	$0.177326\pi$
$770$	0	0
$771$	15.0758	0.542943
$772$	0	0
$773$	−16.3982	−0.589804	−0.294902	−	0.955528i	$-0.595287\pi$
$773$	−16.3982	−0.589804	−0.294902	+	0.955528i	$0.595287\pi$
$774$	0	0
$775$	1.64139	0.0589605
$776$	0	0
$777$	−1.93318	−0.0693525
$778$	0	0
$779$	4.46002	0.159797
$780$	0	0
$781$	−9.05871	−0.324146
$782$	0	0
$783$	−4.35861	−0.155764
$784$	0	0
$785$	−9.58610	−0.342143
$786$	0	0
$787$	20.3586	0.725706	0.362853	−	0.931846i	$-0.381803\pi$
$787$	20.3586	0.725706	0.362853	+	0.931846i	$0.381803\pi$
$788$	0	0
$789$	0.793050	0.0282333
$790$	0	0
$791$	−0.974422	−0.0346465
$792$	0	0
$793$	−61.7378	−2.19237
$794$	0	0
$795$	21.1798	0.751169
$796$	0	0
$797$	−17.9728	−0.636629	−0.318315	−	0.947985i	$-0.603117\pi$
$797$	−17.9728	−0.636629	−0.318315	+	0.947985i	$0.603117\pi$
$798$	0	0
$799$	−9.23253	−0.326623
$800$	0	0
$801$	−10.9497	−0.386890
$802$	0	0
$803$	6.59366	0.232685
$804$	0	0
$805$	12.7495	0.449359
$806$	0	0
$807$	18.1682	0.639552
$808$	0	0
$809$	4.92165	0.173036	0.0865180	−	0.996250i	$-0.472426\pi$
$809$	4.92165	0.173036	0.0865180	+	0.996250i	$0.472426\pi$
$810$	0	0
$811$	40.7849	1.43215	0.716077	−	0.698022i	$-0.245936\pi$
$811$	40.7849	1.43215	0.716077	+	0.698022i	$0.245936\pi$
$812$	0	0
$813$	−12.7956	−0.448760
$814$	0	0
$815$	20.4575	0.716595
$816$	0	0
$817$	8.33142	0.291479
$818$	0	0
$819$	−6.51531	−0.227663
$820$	0	0
$821$	49.1551	1.71552	0.857762	−	0.514047i	$-0.171854\pi$
$821$	49.1551	1.71552	0.857762	+	0.514047i	$0.171854\pi$
$822$	0	0
$823$	24.5103	0.854374	0.427187	−	0.904163i	$-0.359505\pi$
$823$	24.5103	0.854374	0.427187	+	0.904163i	$0.359505\pi$
$824$	0	0
$825$	7.15418	0.249077
$826$	0	0
$827$	−18.8237	−0.654563	−0.327281	−	0.944927i	$-0.606133\pi$
$827$	−18.8237	−0.654563	−0.327281	+	0.944927i	$0.606133\pi$
$828$	0	0
$829$	13.2160	0.459009	0.229505	−	0.973308i	$-0.426289\pi$
$829$	13.2160	0.459009	0.229505	+	0.973308i	$0.426289\pi$
$830$	0	0
$831$	−9.34456	−0.324159
$832$	0	0
$833$	−35.0256	−1.21356
$834$	0	0
$835$	−2.57709	−0.0891839
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−46.0256	−1.58898	−0.794490	−	0.607278i	$-0.792262\pi$
$839$	−46.0256	−1.58898	−0.794490	+	0.607278i	$0.792262\pi$
$840$	0	0
$841$	−10.0025	−0.344914
$842$	0	0
$843$	28.4625	0.980302
$844$	0	0
$845$	−25.7644	−0.886322
$846$	0	0
$847$	−10.8655	−0.373342
$848$	0	0
$849$	0.588619	0.0202013
$850$	0	0
$851$	5.18137	0.177615
$852$	0	0
$853$	−3.43948	−0.117765	−0.0588827	−	0.998265i	$-0.518754\pi$
$853$	−3.43948	−0.117765	−0.0588827	+	0.998265i	$0.518754\pi$
$854$	0	0
$855$	−4.23001	−0.144663
$856$	0	0
$857$	−44.8995	−1.53374	−0.766869	−	0.641804i	$-0.778186\pi$
$857$	−44.8995	−1.53374	−0.766869	+	0.641804i	$0.778186\pi$
$858$	0	0
$859$	−34.4098	−1.17405	−0.587023	−	0.809570i	$-0.699700\pi$
$859$	−34.4098	−1.17405	−0.587023	+	0.809570i	$0.699700\pi$
$860$	0	0
$861$	−3.69164	−0.125811
$862$	0	0
$863$	−7.43948	−0.253243	−0.126621	−	0.991951i	$-0.540413\pi$
$863$	−7.43948	−0.253243	−0.126621	+	0.991951i	$0.540413\pi$
$864$	0	0
$865$	24.6448	0.837949
$866$	0	0
$867$	29.1798	0.990996
$868$	0	0
$869$	3.12608	0.106045
$870$	0	0
$871$	52.8186	1.78969
$872$	0	0
$873$	−0.361129	−0.0122224
$874$	0	0
$875$	−11.7594	−0.397539
$876$	0	0
$877$	44.0009	1.48580	0.742902	−	0.669400i	$-0.233449\pi$
$877$	44.0009	1.48580	0.742902	+	0.669400i	$0.233449\pi$
$878$	0	0
$879$	−7.28278	−0.245642
$880$	0	0
$881$	39.6142	1.33464	0.667318	−	0.744773i	$-0.267442\pi$
$881$	39.6142	1.33464	0.667318	+	0.744773i	$0.267442\pi$
$882$	0	0
$883$	−44.2862	−1.49035	−0.745175	−	0.666869i	$-0.767634\pi$
$883$	−44.2862	−1.49035	−0.745175	+	0.666869i	$0.767634\pi$
$884$	0	0
$885$	18.0758	0.607612
$886$	0	0
$887$	−28.0793	−0.942809	−0.471405	−	0.881917i	$-0.656253\pi$
$887$	−28.0793	−0.942809	−0.471405	+	0.881917i	$0.656253\pi$
$888$	0	0
$889$	−14.5639	−0.488459
$890$	0	0
$891$	4.35861	0.146019
$892$	0	0
$893$	−2.23001	−0.0746244
$894$	0	0
$895$	−52.2070	−1.74509
$896$	0	0
$897$	17.4625	0.583057
$898$	0	0
$899$	−4.35861	−0.145368
$900$	0	0
$901$	−55.8492	−1.86061
$902$	0	0
$903$	−6.89607	−0.229487
$904$	0	0
$905$	40.7725	1.35532
$906$	0	0
$907$	−45.5459	−1.51233	−0.756164	−	0.654382i	$-0.772929\pi$
$907$	−45.5459	−1.51233	−0.756164	+	0.654382i	$0.772929\pi$
$908$	0	0
$909$	11.6529	0.386503
$910$	0	0
$911$	−16.3339	−0.541167	−0.270584	−	0.962696i	$-0.587217\pi$
$911$	−16.3339	−0.541167	−0.270584	+	0.962696i	$0.587217\pi$
$912$	0	0
$913$	73.8161	2.44296
$914$	0	0
$915$	33.1772	1.09681
$916$	0	0
$917$	4.94723	0.163372
$918$	0	0
$919$	−46.1090	−1.52099	−0.760497	−	0.649341i	$-0.775045\pi$
$919$	−46.1090	−1.52099	−0.760497	+	0.649341i	$0.775045\pi$
$920$	0	0
$921$	18.5268	0.610480
$922$	0	0
$923$	−9.96687	−0.328063
$924$	0	0
$925$	2.33555	0.0767924
$926$	0	0
$927$	17.1542	0.563417
$928$	0	0
$929$	−30.7197	−1.00788	−0.503941	−	0.863738i	$-0.668117\pi$
$929$	−30.7197	−1.00788	−0.503941	+	0.863738i	$0.668117\pi$
$930$	0	0
$931$	−8.46002	−0.277266
$932$	0	0
$933$	25.4370	0.832769
$934$	0	0
$935$	−76.3314	−2.49630
$936$	0	0
$937$	−5.76999	−0.188497	−0.0942487	−	0.995549i	$-0.530045\pi$
$937$	−5.76999	−0.188497	−0.0942487	+	0.995549i	$0.530045\pi$
$938$	0	0
$939$	−22.2300	−0.725449
$940$	0	0
$941$	38.1295	1.24299	0.621493	−	0.783420i	$-0.286526\pi$
$941$	38.1295	1.24299	0.621493	+	0.783420i	$0.286526\pi$
$942$	0	0
$943$	9.89446	0.322208
$944$	0	0
$945$	3.50126	0.113896
$946$	0	0
$947$	16.7223	0.543400	0.271700	−	0.962382i	$-0.412414\pi$
$947$	16.7223	0.543400	0.271700	+	0.962382i	$0.412414\pi$
$948$	0	0
$949$	7.25468	0.235497
$950$	0	0
$951$	24.5153	0.794963
$952$	0	0
$953$	47.9778	1.55415	0.777077	−	0.629405i	$-0.216701\pi$
$953$	47.9778	1.55415	0.777077	+	0.629405i	$0.216701\pi$
$954$	0	0
$955$	−58.2194	−1.88393
$956$	0	0
$957$	−18.9975	−0.614101
$958$	0	0
$959$	22.4344	0.724446
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	7.71974	0.248765
$964$	0	0
$965$	−60.9984	−1.96361
$966$	0	0
$967$	12.6167	0.405726	0.202863	−	0.979207i	$-0.434975\pi$
$967$	12.6167	0.405726	0.202863	+	0.979207i	$0.434975\pi$
$968$	0	0
$969$	11.1542	0.358324
$970$	0	0
$971$	35.8327	1.14993	0.574963	−	0.818180i	$-0.305017\pi$
$971$	35.8327	1.14993	0.574963	+	0.818180i	$0.305017\pi$
$972$	0	0
$973$	5.83771	0.187149
$974$	0	0
$975$	7.87140	0.252087
$976$	0	0
$977$	18.0503	0.577479	0.288739	−	0.957408i	$-0.406764\pi$
$977$	18.0503	0.577479	0.288739	+	0.957408i	$0.406764\pi$
$978$	0	0
$979$	−47.7257	−1.52532
$980$	0	0
$981$	−1.51279	−0.0482996
$982$	0	0
$983$	0.431921	0.0137761	0.00688807	−	0.999976i	$-0.497807\pi$
$983$	0.431921	0.0137761	0.00688807	+	0.999976i	$0.497807\pi$
$984$	0	0
$985$	−20.8186	−0.663337
$986$	0	0
$987$	1.84582	0.0587531
$988$	0	0
$989$	18.4831	0.587728
$990$	0	0
$991$	−57.6644	−1.83177	−0.915886	−	0.401439i	$-0.868510\pi$
$991$	−57.6644	−1.83177	−0.915886	+	0.401439i	$0.868510\pi$
$992$	0	0
$993$	20.5153	0.651034
$994$	0	0
$995$	28.7453	0.911288
$996$	0	0
$997$	−12.8237	−0.406130	−0.203065	−	0.979165i	$-0.565090\pi$
$997$	−12.8237	−0.406130	−0.203065	+	0.979165i	$0.565090\pi$
$998$	0	0
$999$	1.42291	0.0450189

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.m.1.1		3
4.3	odd	2		1002.2.a.g.1.1	✓	3
12.11	even	2		3006.2.a.q.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.g.1.1	✓	3	4.3	odd	2
3006.2.a.q.1.3		3	12.11	even	2
8016.2.a.m.1.1		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.57709 q^{5} -1.35861 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.57709 q^{5} -1.35861 q^{7} +1.00000 q^{9} +4.35861 q^{11} +4.79557 q^{13} -2.57709 q^{15} +6.79557 q^{17} +1.64139 q^{19} -1.35861 q^{21} +3.64139 q^{23} +1.64139 q^{25} +1.00000 q^{27} -4.35861 q^{29} +1.00000 q^{31} +4.35861 q^{33} +3.50126 q^{35} +1.42291 q^{37} +4.79557 q^{39} +2.71722 q^{41} +5.07583 q^{43} -2.57709 q^{45} -1.35861 q^{47} -5.15418 q^{49} +6.79557 q^{51} -8.21848 q^{53} -11.2325 q^{55} +1.64139 q^{57} -7.01405 q^{59} -12.8739 q^{61} -1.35861 q^{63} -12.3586 q^{65} +11.0140 q^{67} +3.64139 q^{69} -2.07835 q^{71} +1.51279 q^{73} +1.64139 q^{75} -5.92165 q^{77} +0.717220 q^{79} +1.00000 q^{81} +16.9357 q^{83} -17.5128 q^{85} -4.35861 q^{87} -10.9497 q^{89} -6.51531 q^{91} +1.00000 q^{93} -4.23001 q^{95} -0.361129 q^{97} +4.35861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 10 q^{11} - 4 q^{13} + 3 q^{15} + 2 q^{17} + 8 q^{19} - q^{21} + 14 q^{23} + 8 q^{25} + 3 q^{27} - 10 q^{29} + 3 q^{31} + 10 q^{33} + 9 q^{35} + 15 q^{37} - 4 q^{39} + 2 q^{41} + 6 q^{43} + 3 q^{45} - q^{47} + 6 q^{49} + 2 q^{51} - 17 q^{53} + 8 q^{57} + 5 q^{59} - 8 q^{61} - q^{63} - 34 q^{65} + 7 q^{67} + 14 q^{69} + 6 q^{71} - 20 q^{73} + 8 q^{75} - 30 q^{77} - 4 q^{79} + 3 q^{81} + 37 q^{83} - 28 q^{85} - 10 q^{87} + 7 q^{89} + 8 q^{91} + 3 q^{93} + 18 q^{95} + 5 q^{97} + 10 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 + 10 * q^11 - 4 * q^13 + 3 * q^15 + 2 * q^17 + 8 * q^19 - q^21 + 14 * q^23 + 8 * q^25 + 3 * q^27 - 10 * q^29 + 3 * q^31 + 10 * q^33 + 9 * q^35 + 15 * q^37 - 4 * q^39 + 2 * q^41 + 6 * q^43 + 3 * q^45 - q^47 + 6 * q^49 + 2 * q^51 - 17 * q^53 + 8 * q^57 + 5 * q^59 - 8 * q^61 - q^63 - 34 * q^65 + 7 * q^67 + 14 * q^69 + 6 * q^71 - 20 * q^73 + 8 * q^75 - 30 * q^77 - 4 * q^79 + 3 * q^81 + 37 * q^83 - 28 * q^85 - 10 * q^87 + 7 * q^89 + 8 * q^91 + 3 * q^93 + 18 * q^95 + 5 * q^97 + 10 * q^99

Introduction

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