LMFDB - Embedded newform 8016.2.a.o.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:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.36234$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.87806 q^{5} +3.28324 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.87806 q^{5} +3.28324 q^{7} +1.00000 q^{9} +4.43662 q^{13} -1.87806 q^{15} +6.72468 q^{17} -1.69324 q^{19} +3.28324 q^{21} +1.69324 q^{23} -1.47289 q^{25} +1.00000 q^{27} +9.75612 q^{29} -1.59000 q^{31} -6.16612 q^{35} +4.53985 q^{37} +4.43662 q^{39} +3.33821 q^{41} -7.16130 q^{43} -1.87806 q^{45} -4.47289 q^{47} +3.77965 q^{49} +6.72468 q^{51} -0.184825 q^{53} -1.69324 q^{57} +13.4760 q^{59} -5.75612 q^{61} +3.28324 q^{63} -8.33224 q^{65} +8.45418 q^{67} +1.69324 q^{69} -11.1426 q^{71} -10.6294 q^{73} -1.47289 q^{75} +10.7876 q^{79} +1.00000 q^{81} -14.6656 q^{83} -12.6294 q^{85} +9.75612 q^{87} +8.16612 q^{89} +14.5665 q^{91} -1.59000 q^{93} +3.18000 q^{95} +10.2290 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9} + 8 q^{13} + 5 q^{15} + 10 q^{17} + 2 q^{19} - q^{21} - 2 q^{23} + 5 q^{25} + 4 q^{27} + 14 q^{29} - q^{31} - 5 q^{35} + 5 q^{37} + 8 q^{39} + 14 q^{41} - 2 q^{43} + 5 q^{45} - 7 q^{47} + 13 q^{49} + 10 q^{51} + 3 q^{53} + 2 q^{57} + 5 q^{59} + 2 q^{61} - q^{63} + 6 q^{65} + 7 q^{67} - 2 q^{69} - 2 q^{71} + 2 q^{73} + 5 q^{75} + 10 q^{79} + 4 q^{81} - 13 q^{83} - 6 q^{85} + 14 q^{87} + 13 q^{89} + 30 q^{91} - q^{93} + 2 q^{95} + 5 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9 + 8 * q^13 + 5 * q^15 + 10 * q^17 + 2 * q^19 - q^21 - 2 * q^23 + 5 * q^25 + 4 * q^27 + 14 * q^29 - q^31 - 5 * q^35 + 5 * q^37 + 8 * q^39 + 14 * q^41 - 2 * q^43 + 5 * q^45 - 7 * q^47 + 13 * q^49 + 10 * q^51 + 3 * q^53 + 2 * q^57 + 5 * q^59 + 2 * q^61 - q^63 + 6 * q^65 + 7 * q^67 - 2 * q^69 - 2 * q^71 + 2 * q^73 + 5 * q^75 + 10 * q^79 + 4 * q^81 - 13 * q^83 - 6 * q^85 + 14 * q^87 + 13 * q^89 + 30 * q^91 - q^93 + 2 * q^95 + 5 * q^97

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
$5$	−1.87806	−0.839895	−0.419947	−	0.907548i	$-0.637951\pi$
$5$	−1.87806	−0.839895	−0.419947	+	0.907548i	$0.637951\pi$
$6$	0	0
$7$	3.28324	1.24095	0.620474	−	0.784227i	$-0.286940\pi$
$7$	3.28324	1.24095	0.620474	+	0.784227i	$0.286940\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	4.43662	1.23050	0.615248	−	0.788333i	$-0.289056\pi$
$13$	4.43662	1.23050	0.615248	+	0.788333i	$0.289056\pi$
$14$	0	0
$15$	−1.87806	−0.484913
$16$	0	0
$17$	6.72468	1.63097	0.815487	−	0.578775i	$-0.196469\pi$
$17$	6.72468	1.63097	0.815487	+	0.578775i	$0.196469\pi$
$18$	0	0
$19$	−1.69324	−0.388455	−0.194228	−	0.980957i	$-0.562220\pi$
$19$	−1.69324	−0.388455	−0.194228	+	0.980957i	$0.562220\pi$
$20$	0	0
$21$	3.28324	0.716461
$22$	0	0
$23$	1.69324	0.353064	0.176532	−	0.984295i	$-0.443512\pi$
$23$	1.69324	0.353064	0.176532	+	0.984295i	$0.443512\pi$
$24$	0	0
$25$	−1.47289	−0.294577
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.75612	1.81167	0.905833	−	0.423634i	$-0.139246\pi$
$29$	9.75612	1.81167	0.905833	+	0.423634i	$0.139246\pi$
$30$	0	0
$31$	−1.59000	−0.285573	−0.142786	−	0.989754i	$-0.545606\pi$
$31$	−1.59000	−0.285573	−0.142786	+	0.989754i	$0.545606\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.16612	−1.04226
$36$	0	0
$37$	4.53985	0.746348	0.373174	−	0.927761i	$-0.378270\pi$
$37$	4.53985	0.746348	0.373174	+	0.927761i	$0.378270\pi$
$38$	0	0
$39$	4.43662	0.710428
$40$	0	0
$41$	3.33821	0.521340	0.260670	−	0.965428i	$-0.416056\pi$
$41$	3.33821	0.521340	0.260670	+	0.965428i	$0.416056\pi$
$42$	0	0
$43$	−7.16130	−1.09209	−0.546044	−	0.837757i	$-0.683867\pi$
$43$	−7.16130	−1.09209	−0.546044	+	0.837757i	$0.683867\pi$
$44$	0	0
$45$	−1.87806	−0.279965
$46$	0	0
$47$	−4.47289	−0.652437	−0.326219	−	0.945294i	$-0.605775\pi$
$47$	−4.47289	−0.652437	−0.326219	+	0.945294i	$0.605775\pi$
$48$	0	0
$49$	3.77965	0.539950
$50$	0	0
$51$	6.72468	0.941644
$52$	0	0
$53$	−0.184825	−0.0253877	−0.0126938	−	0.999919i	$-0.504041\pi$
$53$	−0.184825	−0.0253877	−0.0126938	+	0.999919i	$0.504041\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.69324	−0.224275
$58$	0	0
$59$	13.4760	1.75442	0.877212	−	0.480104i	$-0.159401\pi$
$59$	13.4760	1.75442	0.877212	+	0.480104i	$0.159401\pi$
$60$	0	0
$61$	−5.75612	−0.736996	−0.368498	−	0.929629i	$-0.620128\pi$
$61$	−5.75612	−0.736996	−0.368498	+	0.929629i	$0.620128\pi$
$62$	0	0
$63$	3.28324	0.413649
$64$	0	0
$65$	−8.33224	−1.03349
$66$	0	0
$67$	8.45418	1.03284	0.516421	−	0.856335i	$-0.327264\pi$
$67$	8.45418	1.03284	0.516421	+	0.856335i	$0.327264\pi$
$68$	0	0
$69$	1.69324	0.203842
$70$	0	0
$71$	−11.1426	−1.32238	−0.661191	−	0.750217i	$-0.729949\pi$
$71$	−11.1426	−1.32238	−0.661191	+	0.750217i	$0.729949\pi$
$72$	0	0
$73$	−10.6294	−1.24407	−0.622036	−	0.782988i	$-0.713694\pi$
$73$	−10.6294	−1.24407	−0.622036	+	0.782988i	$0.713694\pi$
$74$	0	0
$75$	−1.47289	−0.170074
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.7876	1.21370	0.606848	−	0.794818i	$-0.292434\pi$
$79$	10.7876	1.21370	0.606848	+	0.794818i	$0.292434\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.6656	−1.60976	−0.804881	−	0.593436i	$-0.797771\pi$
$83$	−14.6656	−1.60976	−0.804881	+	0.593436i	$0.797771\pi$
$84$	0	0
$85$	−12.6294	−1.36985
$86$	0	0
$87$	9.75612	1.04597
$88$	0	0
$89$	8.16612	0.865607	0.432804	−	0.901488i	$-0.357524\pi$
$89$	8.16612	0.865607	0.432804	+	0.901488i	$0.357524\pi$
$90$	0	0
$91$	14.5665	1.52698
$92$	0	0
$93$	−1.59000	−0.164875
$94$	0	0
$95$	3.18000	0.326261
$96$	0	0
$97$	10.2290	1.03860	0.519299	−	0.854592i	$-0.326193\pi$
$97$	10.2290	1.03860	0.519299	+	0.854592i	$0.326193\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.56165	0.951420	0.475710	−	0.879602i	$-0.342191\pi$
$101$	9.56165	0.951420	0.475710	+	0.879602i	$0.342191\pi$
$102$	0	0
$103$	−18.8034	−1.85275	−0.926377	−	0.376597i	$-0.877094\pi$
$103$	−18.8034	−1.85275	−0.926377	+	0.376597i	$0.877094\pi$
$104$	0	0
$105$	−6.16612	−0.601752
$106$	0	0
$107$	4.07253	0.393707	0.196853	−	0.980433i	$-0.436928\pi$
$107$	4.07253	0.393707	0.196853	+	0.980433i	$0.436928\pi$
$108$	0	0
$109$	−1.18409	−0.113415	−0.0567074	−	0.998391i	$-0.518060\pi$
$109$	−1.18409	−0.113415	−0.0567074	+	0.998391i	$0.518060\pi$
$110$	0	0
$111$	4.53985	0.430904
$112$	0	0
$113$	−19.9302	−1.87487	−0.937436	−	0.348158i	$-0.886807\pi$
$113$	−19.9302	−1.87487	−0.937436	+	0.348158i	$0.886807\pi$
$114$	0	0
$115$	−3.18000	−0.296537
$116$	0	0
$117$	4.43662	0.410166
$118$	0	0
$119$	22.0787	2.02395
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	3.33821	0.300996
$124$	0	0
$125$	12.1565	1.08731
$126$	0	0
$127$	−6.42583	−0.570201	−0.285100	−	0.958498i	$-0.592027\pi$
$127$	−6.42583	−0.570201	−0.285100	+	0.958498i	$0.592027\pi$
$128$	0	0
$129$	−7.16130	−0.630517
$130$	0	0
$131$	−1.24697	−0.108948	−0.0544742	−	0.998515i	$-0.517348\pi$
$131$	−1.24697	−0.108948	−0.0544742	+	0.998515i	$0.517348\pi$
$132$	0	0
$133$	−5.55930	−0.482052
$134$	0	0
$135$	−1.87806	−0.161638
$136$	0	0
$137$	21.0020	1.79432	0.897159	−	0.441708i	$-0.145627\pi$
$137$	21.0020	1.79432	0.897159	+	0.441708i	$0.145627\pi$
$138$	0	0
$139$	4.62553	0.392332	0.196166	−	0.980571i	$-0.437151\pi$
$139$	4.62553	0.392332	0.196166	+	0.980571i	$0.437151\pi$
$140$	0	0
$141$	−4.47289	−0.376685
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−18.3226	−1.52161
$146$	0	0
$147$	3.77965	0.311740
$148$	0	0
$149$	5.87806	0.481550	0.240775	−	0.970581i	$-0.422598\pi$
$149$	5.87806	0.481550	0.240775	+	0.970581i	$0.422598\pi$
$150$	0	0
$151$	−7.66080	−0.623427	−0.311714	−	0.950176i	$-0.600903\pi$
$151$	−7.66080	−0.623427	−0.311714	+	0.950176i	$0.600903\pi$
$152$	0	0
$153$	6.72468	0.543658
$154$	0	0
$155$	2.98612	0.239851
$156$	0	0
$157$	1.38647	0.110653	0.0553263	−	0.998468i	$-0.482380\pi$
$157$	1.38647	0.110653	0.0553263	+	0.998468i	$0.482380\pi$
$158$	0	0
$159$	−0.184825	−0.0146576
$160$	0	0
$161$	5.55930	0.438134
$162$	0	0
$163$	9.20165	0.720729	0.360364	−	0.932812i	$-0.382652\pi$
$163$	9.20165	0.720729	0.360364	+	0.932812i	$0.382652\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	6.68359	0.514122
$170$	0	0
$171$	−1.69324	−0.129485
$172$	0	0
$173$	5.75612	0.437630	0.218815	−	0.975766i	$-0.429781\pi$
$173$	5.75612	0.437630	0.218815	+	0.975766i	$0.429781\pi$
$174$	0	0
$175$	−4.83583	−0.365555
$176$	0	0
$177$	13.4760	1.01292
$178$	0	0
$179$	7.38647	0.552091	0.276045	−	0.961145i	$-0.410976\pi$
$179$	7.38647	0.552091	0.276045	+	0.961145i	$0.410976\pi$
$180$	0	0
$181$	−16.1968	−1.20390	−0.601950	−	0.798534i	$-0.705609\pi$
$181$	−16.1968	−1.20390	−0.601950	+	0.798534i	$0.705609\pi$
$182$	0	0
$183$	−5.75612	−0.425505
$184$	0	0
$185$	−8.52613	−0.626853
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.28324	0.238820
$190$	0	0
$191$	7.07677	0.512057	0.256028	−	0.966669i	$-0.417586\pi$
$191$	7.07677	0.512057	0.256028	+	0.966669i	$0.417586\pi$
$192$	0	0
$193$	−10.8732	−0.782673	−0.391336	−	0.920248i	$-0.627987\pi$
$193$	−10.8732	−0.782673	−0.391336	+	0.920248i	$0.627987\pi$
$194$	0	0
$195$	−8.33224	−0.596684
$196$	0	0
$197$	−13.1613	−0.937704	−0.468852	−	0.883277i	$-0.655332\pi$
$197$	−13.1613	−0.937704	−0.468852	+	0.883277i	$0.655332\pi$
$198$	0	0
$199$	−4.87324	−0.345455	−0.172727	−	0.984970i	$-0.555258\pi$
$199$	−4.87324	−0.345455	−0.172727	+	0.984970i	$0.555258\pi$
$200$	0	0
$201$	8.45418	0.596312
$202$	0	0
$203$	32.0317	2.24818
$204$	0	0
$205$	−6.26936	−0.437871
$206$	0	0
$207$	1.69324	0.117688
$208$	0	0
$209$	0	0
$210$	0	0
$211$	14.3068	0.984918	0.492459	−	0.870336i	$-0.336098\pi$
$211$	14.3068	0.984918	0.492459	+	0.870336i	$0.336098\pi$
$212$	0	0
$213$	−11.1426	−0.763478
$214$	0	0
$215$	13.4494	0.917239
$216$	0	0
$217$	−5.22035	−0.354381
$218$	0	0
$219$	−10.6294	−0.718266
$220$	0	0
$221$	29.8348	2.00691
$222$	0	0
$223$	−3.28324	−0.219862	−0.109931	−	0.993939i	$-0.535063\pi$
$223$	−3.28324	−0.219862	−0.109931	+	0.993939i	$0.535063\pi$
$224$	0	0
$225$	−1.47289	−0.0981924
$226$	0	0
$227$	16.3963	1.08826	0.544129	−	0.839001i	$-0.316860\pi$
$227$	16.3963	1.08826	0.544129	+	0.839001i	$0.316860\pi$
$228$	0	0
$229$	−6.50359	−0.429769	−0.214885	−	0.976639i	$-0.568938\pi$
$229$	−6.50359	−0.429769	−0.214885	+	0.976639i	$0.568938\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.1119	1.12104	0.560519	−	0.828142i	$-0.310602\pi$
$233$	17.1119	1.12104	0.560519	+	0.828142i	$0.310602\pi$
$234$	0	0
$235$	8.40035	0.547979
$236$	0	0
$237$	10.7876	0.700728
$238$	0	0
$239$	4.87324	0.315224	0.157612	−	0.987501i	$-0.449621\pi$
$239$	4.87324	0.315224	0.157612	+	0.987501i	$0.449621\pi$
$240$	0	0
$241$	3.18965	0.205463	0.102732	−	0.994709i	$-0.467242\pi$
$241$	3.18965	0.205463	0.102732	+	0.994709i	$0.467242\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−7.09841	−0.453501
$246$	0	0
$247$	−7.51225	−0.477993
$248$	0	0
$249$	−14.6656	−0.929396
$250$	0	0
$251$	−3.31394	−0.209174	−0.104587	−	0.994516i	$-0.533352\pi$
$251$	−3.31394	−0.209174	−0.104587	+	0.994516i	$0.533352\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−12.6294	−0.790881
$256$	0	0
$257$	0.905670	0.0564942	0.0282471	−	0.999601i	$-0.491007\pi$
$257$	0.905670	0.0564942	0.0282471	+	0.999601i	$0.491007\pi$
$258$	0	0
$259$	14.9054	0.926178
$260$	0	0
$261$	9.75612	0.603889
$262$	0	0
$263$	−27.0926	−1.67060	−0.835301	−	0.549793i	$-0.814706\pi$
$263$	−27.0926	−1.67060	−0.835301	+	0.549793i	$0.814706\pi$
$264$	0	0
$265$	0.347113	0.0213230
$266$	0	0
$267$	8.16612	0.499759
$268$	0	0
$269$	25.4628	1.55250	0.776248	−	0.630427i	$-0.217120\pi$
$269$	25.4628	1.55250	0.776248	+	0.630427i	$0.217120\pi$
$270$	0	0
$271$	20.2743	1.23158	0.615789	−	0.787911i	$-0.288837\pi$
$271$	20.2743	1.23158	0.615789	+	0.787911i	$0.288837\pi$
$272$	0	0
$273$	14.5665	0.881603
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−31.3205	−1.88186	−0.940932	−	0.338595i	$-0.890048\pi$
$277$	−31.3205	−1.88186	−0.940932	+	0.338595i	$0.890048\pi$
$278$	0	0
$279$	−1.59000	−0.0951909
$280$	0	0
$281$	−11.6784	−0.696673	−0.348336	−	0.937370i	$-0.613253\pi$
$281$	−11.6784	−0.696673	−0.348336	+	0.937370i	$0.613253\pi$
$282$	0	0
$283$	23.2842	1.38410	0.692051	−	0.721849i	$-0.256707\pi$
$283$	23.2842	1.38410	0.692051	+	0.721849i	$0.256707\pi$
$284$	0	0
$285$	3.18000	0.188367
$286$	0	0
$287$	10.9601	0.646956
$288$	0	0
$289$	28.2213	1.66008
$290$	0	0
$291$	10.2290	0.599635
$292$	0	0
$293$	10.2439	0.598454	0.299227	−	0.954182i	$-0.403271\pi$
$293$	10.2439	0.598454	0.299227	+	0.954182i	$0.403271\pi$
$294$	0	0
$295$	−25.3087	−1.47353
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.51225	0.434444
$300$	0	0
$301$	−23.5122	−1.35522
$302$	0	0
$303$	9.56165	0.549303
$304$	0	0
$305$	10.8104	0.618999
$306$	0	0
$307$	−12.6414	−0.721481	−0.360740	−	0.932666i	$-0.617476\pi$
$307$	−12.6414	−0.721481	−0.360740	+	0.932666i	$0.617476\pi$
$308$	0	0
$309$	−18.8034	−1.06969
$310$	0	0
$311$	−34.2352	−1.94130	−0.970650	−	0.240497i	$-0.922690\pi$
$311$	−34.2352	−1.94130	−0.970650	+	0.240497i	$0.922690\pi$
$312$	0	0
$313$	32.9713	1.86365	0.931823	−	0.362914i	$-0.118218\pi$
$313$	32.9713	1.86365	0.931823	+	0.362914i	$0.118218\pi$
$314$	0	0
$315$	−6.16612	−0.347422
$316$	0	0
$317$	7.50260	0.421388	0.210694	−	0.977552i	$-0.432428\pi$
$317$	7.50260	0.421388	0.210694	+	0.977552i	$0.432428\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	4.07253	0.227307
$322$	0	0
$323$	−11.3865	−0.633560
$324$	0	0
$325$	−6.53463	−0.362476
$326$	0	0
$327$	−1.18409	−0.0654801
$328$	0	0
$329$	−14.6855	−0.809640
$330$	0	0
$331$	8.35095	0.459010	0.229505	−	0.973308i	$-0.426289\pi$
$331$	8.35095	0.459010	0.229505	+	0.973308i	$0.426289\pi$
$332$	0	0
$333$	4.53985	0.248783
$334$	0	0
$335$	−15.8775	−0.867479
$336$	0	0
$337$	−3.42682	−0.186671	−0.0933354	−	0.995635i	$-0.529753\pi$
$337$	−3.42682	−0.186671	−0.0933354	+	0.995635i	$0.529753\pi$
$338$	0	0
$339$	−19.9302	−1.08246
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−10.5732	−0.570898
$344$	0	0
$345$	−3.18000	−0.171206
$346$	0	0
$347$	24.3747	1.30850	0.654251	−	0.756277i	$-0.272984\pi$
$347$	24.3747	1.30850	0.654251	+	0.756277i	$0.272984\pi$
$348$	0	0
$349$	−16.0521	−0.859249	−0.429625	−	0.903008i	$-0.641354\pi$
$349$	−16.0521	−0.859249	−0.429625	+	0.903008i	$0.641354\pi$
$350$	0	0
$351$	4.43662	0.236809
$352$	0	0
$353$	−3.67837	−0.195780	−0.0978899	−	0.995197i	$-0.531209\pi$
$353$	−3.67837	−0.195780	−0.0978899	+	0.995197i	$0.531209\pi$
$354$	0	0
$355$	20.9265	1.11066
$356$	0	0
$357$	22.0787	1.16853
$358$	0	0
$359$	14.9361	0.788299	0.394149	−	0.919046i	$-0.371039\pi$
$359$	14.9361	0.788299	0.394149	+	0.919046i	$0.371039\pi$
$360$	0	0
$361$	−16.1330	−0.849103
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	19.9626	1.04489
$366$	0	0
$367$	6.32260	0.330037	0.165018	−	0.986290i	$-0.447232\pi$
$367$	6.32260	0.330037	0.165018	+	0.986290i	$0.447232\pi$
$368$	0	0
$369$	3.33821	0.173780
$370$	0	0
$371$	−0.606824	−0.0315047
$372$	0	0
$373$	33.2383	1.72101	0.860507	−	0.509439i	$-0.170147\pi$
$373$	33.2383	1.72101	0.860507	+	0.509439i	$0.170147\pi$
$374$	0	0
$375$	12.1565	0.627758
$376$	0	0
$377$	43.2842	2.22925
$378$	0	0
$379$	−8.24771	−0.423656	−0.211828	−	0.977307i	$-0.567942\pi$
$379$	−8.24771	−0.423656	−0.211828	+	0.977307i	$0.567942\pi$
$380$	0	0
$381$	−6.42583	−0.329205
$382$	0	0
$383$	−19.0701	−0.974435	−0.487217	−	0.873281i	$-0.661988\pi$
$383$	−19.0701	−0.974435	−0.487217	+	0.873281i	$0.661988\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.16130	−0.364029
$388$	0	0
$389$	11.0610	0.560815	0.280408	−	0.959881i	$-0.409530\pi$
$389$	11.0610	0.560815	0.280408	+	0.959881i	$0.409530\pi$
$390$	0	0
$391$	11.3865	0.575839
$392$	0	0
$393$	−1.24697	−0.0629014
$394$	0	0
$395$	−20.2597	−1.01938
$396$	0	0
$397$	26.0158	1.30570	0.652849	−	0.757488i	$-0.273574\pi$
$397$	26.0158	1.30570	0.652849	+	0.757488i	$0.273574\pi$
$398$	0	0
$399$	−5.55930	−0.278313
$400$	0	0
$401$	22.7405	1.13561	0.567804	−	0.823164i	$-0.307793\pi$
$401$	22.7405	1.13561	0.567804	+	0.823164i	$0.307793\pi$
$402$	0	0
$403$	−7.05423	−0.351396
$404$	0	0
$405$	−1.87806	−0.0933216
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−28.1513	−1.39199	−0.695995	−	0.718047i	$-0.745036\pi$
$409$	−28.1513	−1.39199	−0.695995	+	0.718047i	$0.745036\pi$
$410$	0	0
$411$	21.0020	1.03595
$412$	0	0
$413$	44.2448	2.17715
$414$	0	0
$415$	27.5429	1.35203
$416$	0	0
$417$	4.62553	0.226513
$418$	0	0
$419$	8.25971	0.403513	0.201757	−	0.979436i	$-0.435335\pi$
$419$	8.25971	0.403513	0.201757	+	0.979436i	$0.435335\pi$
$420$	0	0
$421$	−22.1872	−1.08134	−0.540668	−	0.841236i	$-0.681829\pi$
$421$	−22.1872	−1.08134	−0.540668	+	0.841236i	$0.681829\pi$
$422$	0	0
$423$	−4.47289	−0.217479
$424$	0	0
$425$	−9.90468	−0.480448
$426$	0	0
$427$	−18.8987	−0.914573
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.0552	1.49588	0.747938	−	0.663769i	$-0.231044\pi$
$431$	31.0552	1.49588	0.747938	+	0.663769i	$0.231044\pi$
$432$	0	0
$433$	34.6759	1.66642	0.833209	−	0.552959i	$-0.186501\pi$
$433$	34.6759	1.66642	0.833209	+	0.552959i	$0.186501\pi$
$434$	0	0
$435$	−18.3226	−0.878501
$436$	0	0
$437$	−2.86705	−0.137150
$438$	0	0
$439$	−25.4154	−1.21301	−0.606507	−	0.795078i	$-0.707430\pi$
$439$	−25.4154	−1.21301	−0.606507	+	0.795078i	$0.707430\pi$
$440$	0	0
$441$	3.77965	0.179983
$442$	0	0
$443$	25.4034	1.20695	0.603477	−	0.797380i	$-0.293781\pi$
$443$	25.4034	1.20695	0.603477	+	0.797380i	$0.293781\pi$
$444$	0	0
$445$	−15.3365	−0.727019
$446$	0	0
$447$	5.87806	0.278023
$448$	0	0
$449$	−12.1820	−0.574902	−0.287451	−	0.957795i	$-0.592808\pi$
$449$	−12.1820	−0.574902	−0.287451	+	0.957795i	$0.592808\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−7.66080	−0.359936
$454$	0	0
$455$	−27.3567	−1.28250
$456$	0	0
$457$	−11.5833	−0.541844	−0.270922	−	0.962601i	$-0.587329\pi$
$457$	−11.5833	−0.541844	−0.270922	+	0.962601i	$0.587329\pi$
$458$	0	0
$459$	6.72468	0.313881
$460$	0	0
$461$	8.69225	0.404838	0.202419	−	0.979299i	$-0.435120\pi$
$461$	8.69225	0.404838	0.202419	+	0.979299i	$0.435120\pi$
$462$	0	0
$463$	10.0559	0.467339	0.233669	−	0.972316i	$-0.424927\pi$
$463$	10.0559	0.467339	0.233669	+	0.972316i	$0.424927\pi$
$464$	0	0
$465$	2.98612	0.138478
$466$	0	0
$467$	−11.3058	−0.523169	−0.261584	−	0.965181i	$-0.584245\pi$
$467$	−11.3058	−0.523169	−0.261584	+	0.965181i	$0.584245\pi$
$468$	0	0
$469$	27.7571	1.28170
$470$	0	0
$471$	1.38647	0.0638853
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.49394	0.114430
$476$	0	0
$477$	−0.184825	−0.00846255
$478$	0	0
$479$	20.6908	0.945385	0.472693	−	0.881227i	$-0.343282\pi$
$479$	20.6908	0.945385	0.472693	+	0.881227i	$0.343282\pi$
$480$	0	0
$481$	20.1416	0.918378
$482$	0	0
$483$	5.55930	0.252957
$484$	0	0
$485$	−19.2107	−0.872313
$486$	0	0
$487$	3.77891	0.171239	0.0856194	−	0.996328i	$-0.472713\pi$
$487$	3.77891	0.171239	0.0856194	+	0.996328i	$0.472713\pi$
$488$	0	0
$489$	9.20165	0.416113
$490$	0	0
$491$	31.7638	1.43348	0.716740	−	0.697341i	$-0.245633\pi$
$491$	31.7638	1.43348	0.716740	+	0.697341i	$0.245633\pi$
$492$	0	0
$493$	65.6068	2.95478
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−36.5838	−1.64101
$498$	0	0
$499$	−1.35812	−0.0607980	−0.0303990	−	0.999538i	$-0.509678\pi$
$499$	−1.35812	−0.0607980	−0.0303990	+	0.999538i	$0.509678\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−11.5578	−0.515338	−0.257669	−	0.966233i	$-0.582954\pi$
$503$	−11.5578	−0.515338	−0.257669	+	0.966233i	$0.582954\pi$
$504$	0	0
$505$	−17.9574	−0.799092
$506$	0	0
$507$	6.68359	0.296829
$508$	0	0
$509$	−32.0787	−1.42186	−0.710932	−	0.703261i	$-0.751727\pi$
$509$	−32.0787	−1.42186	−0.710932	+	0.703261i	$0.751727\pi$
$510$	0	0
$511$	−34.8987	−1.54383
$512$	0	0
$513$	−1.69324	−0.0747582
$514$	0	0
$515$	35.3139	1.55612
$516$	0	0
$517$	0	0
$518$	0	0
$519$	5.75612	0.252666
$520$	0	0
$521$	4.32955	0.189681	0.0948405	−	0.995492i	$-0.469766\pi$
$521$	4.32955	0.189681	0.0948405	+	0.995492i	$0.469766\pi$
$522$	0	0
$523$	−21.6125	−0.945050	−0.472525	−	0.881317i	$-0.656657\pi$
$523$	−21.6125	−0.945050	−0.472525	+	0.881317i	$0.656657\pi$
$524$	0	0
$525$	−4.83583	−0.211053
$526$	0	0
$527$	−10.6922	−0.465762
$528$	0	0
$529$	−20.1330	−0.875346
$530$	0	0
$531$	13.4760	0.584808
$532$	0	0
$533$	14.8104	0.641508
$534$	0	0
$535$	−7.64847	−0.330672
$536$	0	0
$537$	7.38647	0.318750
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−15.1970	−0.653369	−0.326685	−	0.945133i	$-0.605931\pi$
$541$	−15.1970	−0.653369	−0.326685	+	0.945133i	$0.605931\pi$
$542$	0	0
$543$	−16.1968	−0.695072
$544$	0	0
$545$	2.22378	0.0952565
$546$	0	0
$547$	25.0268	1.07007	0.535035	−	0.844830i	$-0.320298\pi$
$547$	25.0268	1.07007	0.535035	+	0.844830i	$0.320298\pi$
$548$	0	0
$549$	−5.75612	−0.245665
$550$	0	0
$551$	−16.5194	−0.703751
$552$	0	0
$553$	35.4181	1.50613
$554$	0	0
$555$	−8.52613	−0.361914
$556$	0	0
$557$	−31.4652	−1.33322	−0.666612	−	0.745405i	$-0.732256\pi$
$557$	−31.4652	−1.33322	−0.666612	+	0.745405i	$0.732256\pi$
$558$	0	0
$559$	−31.7720	−1.34381
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−30.9520	−1.30447	−0.652235	−	0.758017i	$-0.726168\pi$
$563$	−30.9520	−1.30447	−0.652235	+	0.758017i	$0.726168\pi$
$564$	0	0
$565$	37.4301	1.57469
$566$	0	0
$567$	3.28324	0.137883
$568$	0	0
$569$	−36.8385	−1.54435	−0.772176	−	0.635409i	$-0.780831\pi$
$569$	−36.8385	−1.54435	−0.772176	+	0.635409i	$0.780831\pi$
$570$	0	0
$571$	23.5617	0.986024	0.493012	−	0.870022i	$-0.335896\pi$
$571$	23.5617	0.986024	0.493012	+	0.870022i	$0.335896\pi$
$572$	0	0
$573$	7.07677	0.295636
$574$	0	0
$575$	−2.49394	−0.104005
$576$	0	0
$577$	−7.21219	−0.300247	−0.150124	−	0.988667i	$-0.547967\pi$
$577$	−7.21219	−0.300247	−0.150124	+	0.988667i	$0.547967\pi$
$578$	0	0
$579$	−10.8732	−0.451876
$580$	0	0
$581$	−48.1507	−1.99763
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−8.33224	−0.344496
$586$	0	0
$587$	−1.28880	−0.0531945	−0.0265972	−	0.999646i	$-0.508467\pi$
$587$	−1.28880	−0.0531945	−0.0265972	+	0.999646i	$0.508467\pi$
$588$	0	0
$589$	2.69225	0.110932
$590$	0	0
$591$	−13.1613	−0.541383
$592$	0	0
$593$	−7.80439	−0.320488	−0.160244	−	0.987077i	$-0.551228\pi$
$593$	−7.80439	−0.320488	−0.160244	+	0.987077i	$0.551228\pi$
$594$	0	0
$595$	−41.4652	−1.69991
$596$	0	0
$597$	−4.87324	−0.199448
$598$	0	0
$599$	13.6784	0.558883	0.279441	−	0.960163i	$-0.409851\pi$
$599$	13.6784	0.558883	0.279441	+	0.960163i	$0.409851\pi$
$600$	0	0
$601$	43.7556	1.78483	0.892414	−	0.451217i	$-0.149010\pi$
$601$	43.7556	1.78483	0.892414	+	0.451217i	$0.149010\pi$
$602$	0	0
$603$	8.45418	0.344281
$604$	0	0
$605$	20.6587	0.839895
$606$	0	0
$607$	4.33574	0.175982	0.0879911	−	0.996121i	$-0.471955\pi$
$607$	4.33574	0.175982	0.0879911	+	0.996121i	$0.471955\pi$
$608$	0	0
$609$	32.0317	1.29799
$610$	0	0
$611$	−19.8445	−0.802822
$612$	0	0
$613$	38.2030	1.54301	0.771503	−	0.636226i	$-0.219505\pi$
$613$	38.2030	1.54301	0.771503	+	0.636226i	$0.219505\pi$
$614$	0	0
$615$	−6.26936	−0.252805
$616$	0	0
$617$	−23.4810	−0.945311	−0.472655	−	0.881247i	$-0.656704\pi$
$617$	−23.4810	−0.945311	−0.472655	+	0.881247i	$0.656704\pi$
$618$	0	0
$619$	8.46559	0.340261	0.170130	−	0.985422i	$-0.445581\pi$
$619$	8.46559	0.340261	0.170130	+	0.985422i	$0.445581\pi$
$620$	0	0
$621$	1.69324	0.0679472
$622$	0	0
$623$	26.8113	1.07417
$624$	0	0
$625$	−15.4662	−0.618647
$626$	0	0
$627$	0	0
$628$	0	0
$629$	30.5291	1.21727
$630$	0	0
$631$	−12.4729	−0.496538	−0.248269	−	0.968691i	$-0.579862\pi$
$631$	−12.4729	−0.496538	−0.248269	+	0.968691i	$0.579862\pi$
$632$	0	0
$633$	14.3068	0.568643
$634$	0	0
$635$	12.0681	0.478908
$636$	0	0
$637$	16.7689	0.664407
$638$	0	0
$639$	−11.1426	−0.440794
$640$	0	0
$641$	35.1792	1.38950	0.694748	−	0.719253i	$-0.255516\pi$
$641$	35.1792	1.38950	0.694748	+	0.719253i	$0.255516\pi$
$642$	0	0
$643$	18.3571	0.723935	0.361967	−	0.932191i	$-0.382105\pi$
$643$	18.3571	0.723935	0.361967	+	0.932191i	$0.382105\pi$
$644$	0	0
$645$	13.4494	0.529568
$646$	0	0
$647$	20.6826	0.813117	0.406558	−	0.913625i	$-0.366729\pi$
$647$	20.6826	0.813117	0.406558	+	0.913625i	$0.366729\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−5.22035	−0.204602
$652$	0	0
$653$	1.88190	0.0736443	0.0368221	−	0.999322i	$-0.488277\pi$
$653$	1.88190	0.0736443	0.0368221	+	0.999322i	$0.488277\pi$
$654$	0	0
$655$	2.34189	0.0915052
$656$	0	0
$657$	−10.6294	−0.414691
$658$	0	0
$659$	−29.2479	−1.13934	−0.569669	−	0.821874i	$-0.692929\pi$
$659$	−29.2479	−1.13934	−0.569669	+	0.821874i	$0.692929\pi$
$660$	0	0
$661$	−46.5230	−1.80954	−0.904768	−	0.425906i	$-0.859956\pi$
$661$	−46.5230	−1.80954	−0.904768	+	0.425906i	$0.859956\pi$
$662$	0	0
$663$	29.8348	1.15869
$664$	0	0
$665$	10.4407	0.404873
$666$	0	0
$667$	16.5194	0.639635
$668$	0	0
$669$	−3.28324	−0.126937
$670$	0	0
$671$	0	0
$672$	0	0
$673$	34.4048	1.32621	0.663103	−	0.748528i	$-0.269239\pi$
$673$	34.4048	1.32621	0.663103	+	0.748528i	$0.269239\pi$
$674$	0	0
$675$	−1.47289	−0.0566914
$676$	0	0
$677$	−27.3768	−1.05218	−0.526088	−	0.850430i	$-0.676342\pi$
$677$	−27.3768	−1.05218	−0.526088	+	0.850430i	$0.676342\pi$
$678$	0	0
$679$	33.5843	1.28885
$680$	0	0
$681$	16.3963	0.628306
$682$	0	0
$683$	−19.4064	−0.742565	−0.371282	−	0.928520i	$-0.621082\pi$
$683$	−19.4064	−0.742565	−0.371282	+	0.928520i	$0.621082\pi$
$684$	0	0
$685$	−39.4430	−1.50704
$686$	0	0
$687$	−6.50359	−0.248127
$688$	0	0
$689$	−0.819998	−0.0312394
$690$	0	0
$691$	18.8075	0.715470	0.357735	−	0.933823i	$-0.383549\pi$
$691$	18.8075	0.715470	0.357735	+	0.933823i	$0.383549\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.68702	−0.329518
$696$	0	0
$697$	22.4484	0.850293
$698$	0	0
$699$	17.1119	0.647231
$700$	0	0
$701$	18.4071	0.695225	0.347612	−	0.937638i	$-0.386992\pi$
$701$	18.4071	0.695225	0.347612	+	0.937638i	$0.386992\pi$
$702$	0	0
$703$	−7.68705	−0.289922
$704$	0	0
$705$	8.40035	0.316376
$706$	0	0
$707$	31.3932	1.18066
$708$	0	0
$709$	−30.0232	−1.12754	−0.563772	−	0.825931i	$-0.690650\pi$
$709$	−30.0232	−1.12754	−0.563772	+	0.825931i	$0.690650\pi$
$710$	0	0
$711$	10.7876	0.404565
$712$	0	0
$713$	−2.69225	−0.100825
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.87324	0.181994
$718$	0	0
$719$	12.0884	0.450820	0.225410	−	0.974264i	$-0.427628\pi$
$719$	12.0884	0.450820	0.225410	+	0.974264i	$0.427628\pi$
$720$	0	0
$721$	−61.7360	−2.29917
$722$	0	0
$723$	3.18965	0.118624
$724$	0	0
$725$	−14.3696	−0.533675
$726$	0	0
$727$	−5.41890	−0.200976	−0.100488	−	0.994938i	$-0.532040\pi$
$727$	−5.41890	−0.200976	−0.100488	+	0.994938i	$0.532040\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−48.1574	−1.78117
$732$	0	0
$733$	12.1436	0.448534	0.224267	−	0.974528i	$-0.428001\pi$
$733$	12.1436	0.448534	0.224267	+	0.974528i	$0.428001\pi$
$734$	0	0
$735$	−7.09841	−0.261829
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−27.9845	−1.02943	−0.514714	−	0.857362i	$-0.672102\pi$
$739$	−27.9845	−1.02943	−0.514714	+	0.857362i	$0.672102\pi$
$740$	0	0
$741$	−7.51225	−0.275969
$742$	0	0
$743$	8.71331	0.319660	0.159830	−	0.987145i	$-0.448905\pi$
$743$	8.71331	0.319660	0.159830	+	0.987145i	$0.448905\pi$
$744$	0	0
$745$	−11.0394	−0.404451
$746$	0	0
$747$	−14.6656	−0.536587
$748$	0	0
$749$	13.3711	0.488569
$750$	0	0
$751$	−44.5125	−1.62428	−0.812142	−	0.583460i	$-0.801698\pi$
$751$	−44.5125	−1.62428	−0.812142	+	0.583460i	$0.801698\pi$
$752$	0	0
$753$	−3.31394	−0.120767
$754$	0	0
$755$	14.3875	0.523613
$756$	0	0
$757$	12.1535	0.441728	0.220864	−	0.975305i	$-0.429112\pi$
$757$	12.1535	0.441728	0.220864	+	0.975305i	$0.429112\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.20972	−0.225102	−0.112551	−	0.993646i	$-0.535902\pi$
$761$	−6.20972	−0.225102	−0.112551	+	0.993646i	$0.535902\pi$
$762$	0	0
$763$	−3.88763	−0.140742
$764$	0	0
$765$	−12.6294	−0.456616
$766$	0	0
$767$	59.7878	2.15881
$768$	0	0
$769$	42.2300	1.52285	0.761426	−	0.648252i	$-0.224500\pi$
$769$	42.2300	1.52285	0.761426	+	0.648252i	$0.224500\pi$
$770$	0	0
$771$	0.905670	0.0326169
$772$	0	0
$773$	−21.4868	−0.772828	−0.386414	−	0.922325i	$-0.626286\pi$
$773$	−21.4868	−0.772828	−0.386414	+	0.922325i	$0.626286\pi$
$774$	0	0
$775$	2.34189	0.0841231
$776$	0	0
$777$	14.9054	0.534729
$778$	0	0
$779$	−5.65237	−0.202517
$780$	0	0
$781$	0	0
$782$	0	0
$783$	9.75612	0.348655
$784$	0	0
$785$	−2.60388	−0.0929365
$786$	0	0
$787$	34.1291	1.21657	0.608285	−	0.793718i	$-0.291858\pi$
$787$	34.1291	1.21657	0.608285	+	0.793718i	$0.291858\pi$
$788$	0	0
$789$	−27.0926	−0.964522
$790$	0	0
$791$	−65.4355	−2.32662
$792$	0	0
$793$	−25.5377	−0.906871
$794$	0	0
$795$	0.347113	0.0123108
$796$	0	0
$797$	2.57899	0.0913525	0.0456762	−	0.998956i	$-0.485456\pi$
$797$	2.57899	0.0913525	0.0456762	+	0.998956i	$0.485456\pi$
$798$	0	0
$799$	−30.0787	−1.06411
$800$	0	0
$801$	8.16612	0.288536
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−10.4407	−0.367986
$806$	0	0
$807$	25.4628	0.896334
$808$	0	0
$809$	−12.6697	−0.445443	−0.222722	−	0.974882i	$-0.571494\pi$
$809$	−12.6697	−0.445443	−0.222722	+	0.974882i	$0.571494\pi$
$810$	0	0
$811$	−14.0293	−0.492636	−0.246318	−	0.969189i	$-0.579221\pi$
$811$	−14.0293	−0.492636	−0.246318	+	0.969189i	$0.579221\pi$
$812$	0	0
$813$	20.2743	0.711052
$814$	0	0
$815$	−17.2813	−0.605336
$816$	0	0
$817$	12.1258	0.424227
$818$	0	0
$819$	14.5665	0.508994
$820$	0	0
$821$	−49.8065	−1.73826	−0.869129	−	0.494585i	$-0.835320\pi$
$821$	−49.8065	−1.73826	−0.869129	+	0.494585i	$0.835320\pi$
$822$	0	0
$823$	−37.9460	−1.32271	−0.661357	−	0.750071i	$-0.730019\pi$
$823$	−37.9460	−1.32271	−0.661357	+	0.750071i	$0.730019\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.4683	−1.19858	−0.599290	−	0.800532i	$-0.704550\pi$
$827$	−34.4683	−1.19858	−0.599290	+	0.800532i	$0.704550\pi$
$828$	0	0
$829$	−12.6201	−0.438313	−0.219156	−	0.975690i	$-0.570330\pi$
$829$	−12.6201	−0.438313	−0.219156	+	0.975690i	$0.570330\pi$
$830$	0	0
$831$	−31.3205	−1.08650
$832$	0	0
$833$	25.4169	0.880644
$834$	0	0
$835$	−1.87806	−0.0649930
$836$	0	0
$837$	−1.59000	−0.0549585
$838$	0	0
$839$	0.303821	0.0104891	0.00524453	−	0.999986i	$-0.498331\pi$
$839$	0.303821	0.0104891	0.00524453	+	0.999986i	$0.498331\pi$
$840$	0	0
$841$	66.1819	2.28214
$842$	0	0
$843$	−11.6784	−0.402224
$844$	0	0
$845$	−12.5522	−0.431809
$846$	0	0
$847$	−36.1156	−1.24095
$848$	0	0
$849$	23.2842	0.799112
$850$	0	0
$851$	7.68705	0.263509
$852$	0	0
$853$	−18.1416	−0.621157	−0.310578	−	0.950548i	$-0.600523\pi$
$853$	−18.1416	−0.621157	−0.310578	+	0.950548i	$0.600523\pi$
$854$	0	0
$855$	3.18000	0.108754
$856$	0	0
$857$	−5.83536	−0.199332	−0.0996660	−	0.995021i	$-0.531777\pi$
$857$	−5.83536	−0.199332	−0.0996660	+	0.995021i	$0.531777\pi$
$858$	0	0
$859$	31.3216	1.06868	0.534340	−	0.845270i	$-0.320560\pi$
$859$	31.3216	1.06868	0.534340	+	0.845270i	$0.320560\pi$
$860$	0	0
$861$	10.9601	0.373520
$862$	0	0
$863$	52.3206	1.78101	0.890507	−	0.454969i	$-0.150350\pi$
$863$	52.3206	1.78101	0.890507	+	0.454969i	$0.150350\pi$
$864$	0	0
$865$	−10.8104	−0.367563
$866$	0	0
$867$	28.2213	0.958446
$868$	0	0
$869$	0	0
$870$	0	0
$871$	37.5080	1.27091
$872$	0	0
$873$	10.2290	0.346199
$874$	0	0
$875$	39.9126	1.34929
$876$	0	0
$877$	−10.6861	−0.360843	−0.180421	−	0.983589i	$-0.557746\pi$
$877$	−10.6861	−0.360843	−0.180421	+	0.983589i	$0.557746\pi$
$878$	0	0
$879$	10.2439	0.345517
$880$	0	0
$881$	−12.6695	−0.426845	−0.213423	−	0.976960i	$-0.568461\pi$
$881$	−12.6695	−0.426845	−0.213423	+	0.976960i	$0.568461\pi$
$882$	0	0
$883$	−18.6707	−0.628318	−0.314159	−	0.949370i	$-0.601723\pi$
$883$	−18.6707	−0.628318	−0.314159	+	0.949370i	$0.601723\pi$
$884$	0	0
$885$	−25.3087	−0.850743
$886$	0	0
$887$	11.5157	0.386659	0.193330	−	0.981134i	$-0.438071\pi$
$887$	11.5157	0.386659	0.193330	+	0.981134i	$0.438071\pi$
$888$	0	0
$889$	−21.0975	−0.707589
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.57365	0.253443
$894$	0	0
$895$	−13.8722	−0.463698
$896$	0	0
$897$	7.51225	0.250827
$898$	0	0
$899$	−15.5122	−0.517362
$900$	0	0
$901$	−1.24289	−0.0414066
$902$	0	0
$903$	−23.5122	−0.782439
$904$	0	0
$905$	30.4186	1.01115
$906$	0	0
$907$	−1.93711	−0.0643208	−0.0321604	−	0.999483i	$-0.510239\pi$
$907$	−1.93711	−0.0643208	−0.0321604	+	0.999483i	$0.510239\pi$
$908$	0	0
$909$	9.56165	0.317140
$910$	0	0
$911$	10.6481	0.352789	0.176394	−	0.984320i	$-0.443557\pi$
$911$	10.6481	0.352789	0.176394	+	0.984320i	$0.443557\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	10.8104	0.357379
$916$	0	0
$917$	−4.09410	−0.135199
$918$	0	0
$919$	−32.0735	−1.05801	−0.529004	−	0.848620i	$-0.677434\pi$
$919$	−32.0735	−1.05801	−0.529004	+	0.848620i	$0.677434\pi$
$920$	0	0
$921$	−12.6414	−0.416547
$922$	0	0
$923$	−49.4355	−1.62719
$924$	0	0
$925$	−6.68669	−0.219857
$926$	0	0
$927$	−18.8034	−0.617585
$928$	0	0
$929$	4.27606	0.140293	0.0701465	−	0.997537i	$-0.477653\pi$
$929$	4.27606	0.140293	0.0701465	+	0.997537i	$0.477653\pi$
$930$	0	0
$931$	−6.39984	−0.209746
$932$	0	0
$933$	−34.2352	−1.12081
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−34.8517	−1.13855	−0.569277	−	0.822146i	$-0.692777\pi$
$937$	−34.8517	−1.13855	−0.569277	+	0.822146i	$0.692777\pi$
$938$	0	0
$939$	32.9713	1.07598
$940$	0	0
$941$	−49.6544	−1.61869	−0.809344	−	0.587335i	$-0.800177\pi$
$941$	−49.6544	−1.61869	−0.809344	+	0.587335i	$0.800177\pi$
$942$	0	0
$943$	5.65237	0.184067
$944$	0	0
$945$	−6.16612	−0.200584
$946$	0	0
$947$	56.4580	1.83464	0.917320	−	0.398152i	$-0.130348\pi$
$947$	56.4580	1.83464	0.917320	+	0.398152i	$0.130348\pi$
$948$	0	0
$949$	−47.1584	−1.53083
$950$	0	0
$951$	7.50260	0.243288
$952$	0	0
$953$	−21.3714	−0.692286	−0.346143	−	0.938182i	$-0.612509\pi$
$953$	−21.3714	−0.692286	−0.346143	+	0.938182i	$0.612509\pi$
$954$	0	0
$955$	−13.2906	−0.430074
$956$	0	0
$957$	0	0
$958$	0	0
$959$	68.9544	2.22665
$960$	0	0
$961$	−28.4719	−0.918448
$962$	0	0
$963$	4.07253	0.131236
$964$	0	0
$965$	20.4206	0.657363
$966$	0	0
$967$	58.2894	1.87446	0.937230	−	0.348711i	$-0.113380\pi$
$967$	58.2894	1.87446	0.937230	+	0.348711i	$0.113380\pi$
$968$	0	0
$969$	−11.3865	−0.365786
$970$	0	0
$971$	29.2403	0.938365	0.469182	−	0.883101i	$-0.344549\pi$
$971$	29.2403	0.938365	0.469182	+	0.883101i	$0.344549\pi$
$972$	0	0
$973$	15.1867	0.486863
$974$	0	0
$975$	−6.53463	−0.209276
$976$	0	0
$977$	−5.96609	−0.190872	−0.0954360	−	0.995436i	$-0.530425\pi$
$977$	−5.96609	−0.190872	−0.0954360	+	0.995436i	$0.530425\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−1.18409	−0.0378049
$982$	0	0
$983$	4.58429	0.146216	0.0731080	−	0.997324i	$-0.476708\pi$
$983$	4.58429	0.146216	0.0731080	+	0.997324i	$0.476708\pi$
$984$	0	0
$985$	24.7177	0.787572
$986$	0	0
$987$	−14.6855	−0.467446
$988$	0	0
$989$	−12.1258	−0.385577
$990$	0	0
$991$	−53.6940	−1.70564	−0.852822	−	0.522201i	$-0.825111\pi$
$991$	−53.6940	−1.70564	−0.852822	+	0.522201i	$0.825111\pi$
$992$	0	0
$993$	8.35095	0.265009
$994$	0	0
$995$	9.15224	0.290146
$996$	0	0
$997$	−21.3298	−0.675521	−0.337760	−	0.941232i	$-0.609669\pi$
$997$	−21.3298	−0.675521	−0.337760	+	0.941232i	$0.609669\pi$
$998$	0	0
$999$	4.53985	0.143635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.o.1.1		4
4.3	odd	2		1002.2.a.i.1.1	✓	4
12.11	even	2		3006.2.a.s.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.i.1.1	✓	4	4.3	odd	2
3006.2.a.s.1.4		4	12.11	even	2
8016.2.a.o.1.1		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.87806 q^{5} +3.28324 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.87806 q^{5} +3.28324 q^{7} +1.00000 q^{9} +4.43662 q^{13} -1.87806 q^{15} +6.72468 q^{17} -1.69324 q^{19} +3.28324 q^{21} +1.69324 q^{23} -1.47289 q^{25} +1.00000 q^{27} +9.75612 q^{29} -1.59000 q^{31} -6.16612 q^{35} +4.53985 q^{37} +4.43662 q^{39} +3.33821 q^{41} -7.16130 q^{43} -1.87806 q^{45} -4.47289 q^{47} +3.77965 q^{49} +6.72468 q^{51} -0.184825 q^{53} -1.69324 q^{57} +13.4760 q^{59} -5.75612 q^{61} +3.28324 q^{63} -8.33224 q^{65} +8.45418 q^{67} +1.69324 q^{69} -11.1426 q^{71} -10.6294 q^{73} -1.47289 q^{75} +10.7876 q^{79} +1.00000 q^{81} -14.6656 q^{83} -12.6294 q^{85} +9.75612 q^{87} +8.16612 q^{89} +14.5665 q^{91} -1.59000 q^{93} +3.18000 q^{95} +10.2290 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9} + 8 q^{13} + 5 q^{15} + 10 q^{17} + 2 q^{19} - q^{21} - 2 q^{23} + 5 q^{25} + 4 q^{27} + 14 q^{29} - q^{31} - 5 q^{35} + 5 q^{37} + 8 q^{39} + 14 q^{41} - 2 q^{43} + 5 q^{45} - 7 q^{47} + 13 q^{49} + 10 q^{51} + 3 q^{53} + 2 q^{57} + 5 q^{59} + 2 q^{61} - q^{63} + 6 q^{65} + 7 q^{67} - 2 q^{69} - 2 q^{71} + 2 q^{73} + 5 q^{75} + 10 q^{79} + 4 q^{81} - 13 q^{83} - 6 q^{85} + 14 q^{87} + 13 q^{89} + 30 q^{91} - q^{93} + 2 q^{95} + 5 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9 + 8 * q^13 + 5 * q^15 + 10 * q^17 + 2 * q^19 - q^21 - 2 * q^23 + 5 * q^25 + 4 * q^27 + 14 * q^29 - q^31 - 5 * q^35 + 5 * q^37 + 8 * q^39 + 14 * q^41 - 2 * q^43 + 5 * q^45 - 7 * q^47 + 13 * q^49 + 10 * q^51 + 3 * q^53 + 2 * q^57 + 5 * q^59 + 2 * q^61 - q^63 + 6 * q^65 + 7 * q^67 - 2 * q^69 - 2 * q^71 + 2 * q^73 + 5 * q^75 + 10 * q^79 + 4 * q^81 - 13 * q^83 - 6 * q^85 + 14 * q^87 + 13 * q^89 + 30 * q^91 - q^93 + 2 * q^95 + 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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