LMFDB - Embedded newform 8016.2.a.bf.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 $ x^12 - 4x^11 - 31x^10 + 131x^9 + 309x^8 - 1453x^7 - 1072x^6 + 6350x^5 + 1411x^4 - 11022x^3 - 2450x^2 + 6960*x + 3008
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.53322$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.53322 q^{5} +0.915565 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.53322 q^{5} +0.915565 q^{7} +1.00000 q^{9} -3.09767 q^{11} +1.39795 q^{13} -1.53322 q^{15} -1.50046 q^{17} +4.21086 q^{19} -0.915565 q^{21} -4.95660 q^{23} -2.64923 q^{25} -1.00000 q^{27} +1.76793 q^{29} -3.79993 q^{31} +3.09767 q^{33} +1.40377 q^{35} +4.15442 q^{37} -1.39795 q^{39} +8.32859 q^{41} -8.45697 q^{43} +1.53322 q^{45} +6.81577 q^{47} -6.16174 q^{49} +1.50046 q^{51} +1.43276 q^{53} -4.74943 q^{55} -4.21086 q^{57} -12.9348 q^{59} -3.76679 q^{61} +0.915565 q^{63} +2.14337 q^{65} -7.30009 q^{67} +4.95660 q^{69} -0.883952 q^{71} -0.891880 q^{73} +2.64923 q^{75} -2.83612 q^{77} +1.12995 q^{79} +1.00000 q^{81} -4.26402 q^{83} -2.30053 q^{85} -1.76793 q^{87} +17.6794 q^{89} +1.27991 q^{91} +3.79993 q^{93} +6.45620 q^{95} -4.28751 q^{97} -3.09767 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9} - q^{11} + 8 q^{13} - 4 q^{15} + 3 q^{17} - 12 q^{19} + 11 q^{21} - 7 q^{23} + 18 q^{25} - 12 q^{27} + 5 q^{29} - 33 q^{31} + q^{33} - 15 q^{35} + 8 q^{37} - 8 q^{39} - 6 q^{41} - 16 q^{43} + 4 q^{45} - 18 q^{47} + 25 q^{49} - 3 q^{51} + 20 q^{53} - 39 q^{55} + 12 q^{57} - 4 q^{59} + 10 q^{61} - 11 q^{63} - 9 q^{67} + 7 q^{69} - 11 q^{71} + 22 q^{73} - 18 q^{75} + 24 q^{77} - 56 q^{79} + 12 q^{81} - 26 q^{83} + 15 q^{85} - 5 q^{87} - 15 q^{89} - 11 q^{91} + 33 q^{93} - 3 q^{95} + 8 q^{97} - q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9 - q^11 + 8 * q^13 - 4 * q^15 + 3 * q^17 - 12 * q^19 + 11 * q^21 - 7 * q^23 + 18 * q^25 - 12 * q^27 + 5 * q^29 - 33 * q^31 + q^33 - 15 * q^35 + 8 * q^37 - 8 * q^39 - 6 * q^41 - 16 * q^43 + 4 * q^45 - 18 * q^47 + 25 * q^49 - 3 * q^51 + 20 * q^53 - 39 * q^55 + 12 * q^57 - 4 * q^59 + 10 * q^61 - 11 * q^63 - 9 * q^67 + 7 * q^69 - 11 * q^71 + 22 * q^73 - 18 * q^75 + 24 * q^77 - 56 * q^79 + 12 * q^81 - 26 * q^83 + 15 * q^85 - 5 * q^87 - 15 * q^89 - 11 * q^91 + 33 * q^93 - 3 * q^95 + 8 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.53322	0.685678	0.342839	−	0.939394i	$-0.388611\pi$
$5$	1.53322	0.685678	0.342839	+	0.939394i	$0.388611\pi$
$6$	0	0
$7$	0.915565	0.346051	0.173025	−	0.984917i	$-0.444646\pi$
$7$	0.915565	0.346051	0.173025	+	0.984917i	$0.444646\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.09767	−0.933984	−0.466992	−	0.884262i	$-0.654662\pi$
$11$	−3.09767	−0.933984	−0.466992	+	0.884262i	$0.654662\pi$
$12$	0	0
$13$	1.39795	0.387721	0.193861	−	0.981029i	$-0.437899\pi$
$13$	1.39795	0.387721	0.193861	+	0.981029i	$0.437899\pi$
$14$	0	0
$15$	−1.53322	−0.395877
$16$	0	0
$17$	−1.50046	−0.363914	−0.181957	−	0.983306i	$-0.558243\pi$
$17$	−1.50046	−0.363914	−0.181957	+	0.983306i	$0.558243\pi$
$18$	0	0
$19$	4.21086	0.966038	0.483019	−	0.875610i	$-0.339540\pi$
$19$	4.21086	0.966038	0.483019	+	0.875610i	$0.339540\pi$
$20$	0	0
$21$	−0.915565	−0.199793
$22$	0	0
$23$	−4.95660	−1.03352	−0.516761	−	0.856129i	$-0.672863\pi$
$23$	−4.95660	−1.03352	−0.516761	+	0.856129i	$0.672863\pi$
$24$	0	0
$25$	−2.64923	−0.529845
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.76793	0.328297	0.164148	−	0.986436i	$-0.447512\pi$
$29$	1.76793	0.328297	0.164148	+	0.986436i	$0.447512\pi$
$30$	0	0
$31$	−3.79993	−0.682487	−0.341244	−	0.939975i	$-0.610848\pi$
$31$	−3.79993	−0.682487	−0.341244	+	0.939975i	$0.610848\pi$
$32$	0	0
$33$	3.09767	0.539236
$34$	0	0
$35$	1.40377	0.237280
$36$	0	0
$37$	4.15442	0.682982	0.341491	−	0.939885i	$-0.389068\pi$
$37$	4.15442	0.682982	0.341491	+	0.939885i	$0.389068\pi$
$38$	0	0
$39$	−1.39795	−0.223851
$40$	0	0
$41$	8.32859	1.30071	0.650354	−	0.759631i	$-0.274621\pi$
$41$	8.32859	1.30071	0.650354	+	0.759631i	$0.274621\pi$
$42$	0	0
$43$	−8.45697	−1.28968	−0.644838	−	0.764319i	$-0.723075\pi$
$43$	−8.45697	−1.28968	−0.644838	+	0.764319i	$0.723075\pi$
$44$	0	0
$45$	1.53322	0.228559
$46$	0	0
$47$	6.81577	0.994182	0.497091	−	0.867699i	$-0.334402\pi$
$47$	6.81577	0.994182	0.497091	+	0.867699i	$0.334402\pi$
$48$	0	0
$49$	−6.16174	−0.880249
$50$	0	0
$51$	1.50046	0.210106
$52$	0	0
$53$	1.43276	0.196804	0.0984021	−	0.995147i	$-0.468627\pi$
$53$	1.43276	0.196804	0.0984021	+	0.995147i	$0.468627\pi$
$54$	0	0
$55$	−4.74943	−0.640413
$56$	0	0
$57$	−4.21086	−0.557742
$58$	0	0
$59$	−12.9348	−1.68397	−0.841987	−	0.539499i	$-0.818614\pi$
$59$	−12.9348	−1.68397	−0.841987	+	0.539499i	$0.818614\pi$
$60$	0	0
$61$	−3.76679	−0.482288	−0.241144	−	0.970489i	$-0.577523\pi$
$61$	−3.76679	−0.482288	−0.241144	+	0.970489i	$0.577523\pi$
$62$	0	0
$63$	0.915565	0.115350
$64$	0	0
$65$	2.14337	0.265852
$66$	0	0
$67$	−7.30009	−0.891848	−0.445924	−	0.895071i	$-0.647125\pi$
$67$	−7.30009	−0.891848	−0.445924	+	0.895071i	$0.647125\pi$
$68$	0	0
$69$	4.95660	0.596705
$70$	0	0
$71$	−0.883952	−0.104906	−0.0524529	−	0.998623i	$-0.516704\pi$
$71$	−0.883952	−0.104906	−0.0524529	+	0.998623i	$0.516704\pi$
$72$	0	0
$73$	−0.891880	−0.104387	−0.0521933	−	0.998637i	$-0.516621\pi$
$73$	−0.891880	−0.104387	−0.0521933	+	0.998637i	$0.516621\pi$
$74$	0	0
$75$	2.64923	0.305906
$76$	0	0
$77$	−2.83612	−0.323206
$78$	0	0
$79$	1.12995	0.127129	0.0635645	−	0.997978i	$-0.479753\pi$
$79$	1.12995	0.127129	0.0635645	+	0.997978i	$0.479753\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.26402	−0.468037	−0.234018	−	0.972232i	$-0.575188\pi$
$83$	−4.26402	−0.468037	−0.234018	+	0.972232i	$0.575188\pi$
$84$	0	0
$85$	−2.30053	−0.249528
$86$	0	0
$87$	−1.76793	−0.189542
$88$	0	0
$89$	17.6794	1.87401	0.937007	−	0.349311i	$-0.113584\pi$
$89$	17.6794	1.87401	0.937007	+	0.349311i	$0.113584\pi$
$90$	0	0
$91$	1.27991	0.134171
$92$	0	0
$93$	3.79993	0.394034
$94$	0	0
$95$	6.45620	0.662392
$96$	0	0
$97$	−4.28751	−0.435331	−0.217666	−	0.976023i	$-0.569844\pi$
$97$	−4.28751	−0.435331	−0.217666	+	0.976023i	$0.569844\pi$
$98$	0	0
$99$	−3.09767	−0.311328
$100$	0	0
$101$	−15.5844	−1.55071	−0.775355	−	0.631526i	$-0.782429\pi$
$101$	−15.5844	−1.55071	−0.775355	+	0.631526i	$0.782429\pi$
$102$	0	0
$103$	5.71458	0.563075	0.281537	−	0.959550i	$-0.409156\pi$
$103$	5.71458	0.563075	0.281537	+	0.959550i	$0.409156\pi$
$104$	0	0
$105$	−1.40377	−0.136993
$106$	0	0
$107$	3.17924	0.307349	0.153675	−	0.988122i	$-0.450889\pi$
$107$	3.17924	0.307349	0.153675	+	0.988122i	$0.450889\pi$
$108$	0	0
$109$	−12.5475	−1.20184	−0.600919	−	0.799310i	$-0.705198\pi$
$109$	−12.5475	−1.20184	−0.600919	+	0.799310i	$0.705198\pi$
$110$	0	0
$111$	−4.15442	−0.394320
$112$	0	0
$113$	2.50783	0.235916	0.117958	−	0.993019i	$-0.462365\pi$
$113$	2.50783	0.235916	0.117958	+	0.993019i	$0.462365\pi$
$114$	0	0
$115$	−7.59958	−0.708664
$116$	0	0
$117$	1.39795	0.129240
$118$	0	0
$119$	−1.37376	−0.125933
$120$	0	0
$121$	−1.40441	−0.127674
$122$	0	0
$123$	−8.32859	−0.750964
$124$	0	0
$125$	−11.7280	−1.04898
$126$	0	0
$127$	−17.3437	−1.53900	−0.769501	−	0.638646i	$-0.779495\pi$
$127$	−17.3437	−1.53900	−0.769501	+	0.638646i	$0.779495\pi$
$128$	0	0
$129$	8.45697	0.744595
$130$	0	0
$131$	16.3850	1.43156	0.715782	−	0.698324i	$-0.246071\pi$
$131$	16.3850	1.43156	0.715782	+	0.698324i	$0.246071\pi$
$132$	0	0
$133$	3.85532	0.334298
$134$	0	0
$135$	−1.53322	−0.131959
$136$	0	0
$137$	−18.5673	−1.58631	−0.793154	−	0.609021i	$-0.791563\pi$
$137$	−18.5673	−1.58631	−0.793154	+	0.609021i	$0.791563\pi$
$138$	0	0
$139$	−15.0137	−1.27345	−0.636724	−	0.771092i	$-0.719711\pi$
$139$	−15.0137	−1.27345	−0.636724	+	0.771092i	$0.719711\pi$
$140$	0	0
$141$	−6.81577	−0.573991
$142$	0	0
$143$	−4.33039	−0.362125
$144$	0	0
$145$	2.71064	0.225106
$146$	0	0
$147$	6.16174	0.508212
$148$	0	0
$149$	22.6712	1.85729	0.928647	−	0.370965i	$-0.120973\pi$
$149$	22.6712	1.85729	0.928647	+	0.370965i	$0.120973\pi$
$150$	0	0
$151$	−19.2737	−1.56847	−0.784237	−	0.620462i	$-0.786945\pi$
$151$	−19.2737	−1.56847	−0.784237	+	0.620462i	$0.786945\pi$
$152$	0	0
$153$	−1.50046	−0.121305
$154$	0	0
$155$	−5.82614	−0.467967
$156$	0	0
$157$	15.3895	1.22821	0.614107	−	0.789222i	$-0.289516\pi$
$157$	15.3895	1.22821	0.614107	+	0.789222i	$0.289516\pi$
$158$	0	0
$159$	−1.43276	−0.113625
$160$	0	0
$161$	−4.53809	−0.357651
$162$	0	0
$163$	2.65971	0.208325	0.104162	−	0.994560i	$-0.466784\pi$
$163$	2.65971	0.208325	0.104162	+	0.994560i	$0.466784\pi$
$164$	0	0
$165$	4.74943	0.369742
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−11.0457	−0.849672
$170$	0	0
$171$	4.21086	0.322013
$172$	0	0
$173$	16.5038	1.25476	0.627382	−	0.778712i	$-0.284126\pi$
$173$	16.5038	1.25476	0.627382	+	0.778712i	$0.284126\pi$
$174$	0	0
$175$	−2.42554	−0.183353
$176$	0	0
$177$	12.9348	0.972242
$178$	0	0
$179$	−0.202161	−0.0151102	−0.00755512	−	0.999971i	$-0.502405\pi$
$179$	−0.202161	−0.0151102	−0.00755512	+	0.999971i	$0.502405\pi$
$180$	0	0
$181$	−2.91030	−0.216321	−0.108160	−	0.994133i	$-0.534496\pi$
$181$	−2.91030	−0.216321	−0.108160	+	0.994133i	$0.534496\pi$
$182$	0	0
$183$	3.76679	0.278449
$184$	0	0
$185$	6.36965	0.468306
$186$	0	0
$187$	4.64792	0.339890
$188$	0	0
$189$	−0.915565	−0.0665975
$190$	0	0
$191$	15.2680	1.10475	0.552377	−	0.833594i	$-0.313721\pi$
$191$	15.2680	1.10475	0.552377	+	0.833594i	$0.313721\pi$
$192$	0	0
$193$	8.99644	0.647578	0.323789	−	0.946129i	$-0.395043\pi$
$193$	8.99644	0.647578	0.323789	+	0.946129i	$0.395043\pi$
$194$	0	0
$195$	−2.14337	−0.153490
$196$	0	0
$197$	11.6292	0.828547	0.414273	−	0.910152i	$-0.364036\pi$
$197$	11.6292	0.828547	0.414273	+	0.910152i	$0.364036\pi$
$198$	0	0
$199$	−12.6305	−0.895350	−0.447675	−	0.894196i	$-0.647748\pi$
$199$	−12.6305	−0.895350	−0.447675	+	0.894196i	$0.647748\pi$
$200$	0	0
$201$	7.30009	0.514909
$202$	0	0
$203$	1.61866	0.113607
$204$	0	0
$205$	12.7696	0.891867
$206$	0	0
$207$	−4.95660	−0.344508
$208$	0	0
$209$	−13.0439	−0.902264
$210$	0	0
$211$	11.2769	0.776331	0.388166	−	0.921590i	$-0.373109\pi$
$211$	11.2769	0.776331	0.388166	+	0.921590i	$0.373109\pi$
$212$	0	0
$213$	0.883952	0.0605674
$214$	0	0
$215$	−12.9664	−0.884303
$216$	0	0
$217$	−3.47908	−0.236175
$218$	0	0
$219$	0.891880	0.0602677
$220$	0	0
$221$	−2.09756	−0.141097
$222$	0	0
$223$	11.4567	0.767196	0.383598	−	0.923500i	$-0.374685\pi$
$223$	11.4567	0.767196	0.383598	+	0.923500i	$0.374685\pi$
$224$	0	0
$225$	−2.64923	−0.176615
$226$	0	0
$227$	−6.99214	−0.464085	−0.232042	−	0.972706i	$-0.574541\pi$
$227$	−6.99214	−0.464085	−0.232042	+	0.972706i	$0.574541\pi$
$228$	0	0
$229$	4.84467	0.320145	0.160073	−	0.987105i	$-0.448827\pi$
$229$	4.84467	0.320145	0.160073	+	0.987105i	$0.448827\pi$
$230$	0	0
$231$	2.83612	0.186603
$232$	0	0
$233$	−26.4019	−1.72965	−0.864824	−	0.502075i	$-0.832570\pi$
$233$	−26.4019	−1.72965	−0.864824	+	0.502075i	$0.832570\pi$
$234$	0	0
$235$	10.4501	0.681689
$236$	0	0
$237$	−1.12995	−0.0733979
$238$	0	0
$239$	−23.9728	−1.55067	−0.775335	−	0.631550i	$-0.782419\pi$
$239$	−23.9728	−1.55067	−0.775335	+	0.631550i	$0.782419\pi$
$240$	0	0
$241$	6.33066	0.407794	0.203897	−	0.978992i	$-0.434639\pi$
$241$	6.33066	0.407794	0.203897	+	0.978992i	$0.434639\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−9.44733	−0.603568
$246$	0	0
$247$	5.88657	0.374553
$248$	0	0
$249$	4.26402	0.270221
$250$	0	0
$251$	−10.5048	−0.663059	−0.331529	−	0.943445i	$-0.607565\pi$
$251$	−10.5048	−0.663059	−0.331529	+	0.943445i	$0.607565\pi$
$252$	0	0
$253$	15.3539	0.965294
$254$	0	0
$255$	2.30053	0.144065
$256$	0	0
$257$	−25.5749	−1.59532	−0.797661	−	0.603107i	$-0.793929\pi$
$257$	−25.5749	−1.59532	−0.797661	+	0.603107i	$0.793929\pi$
$258$	0	0
$259$	3.80364	0.236346
$260$	0	0
$261$	1.76793	0.109432
$262$	0	0
$263$	20.5255	1.26566	0.632828	−	0.774292i	$-0.281894\pi$
$263$	20.5255	1.26566	0.632828	+	0.774292i	$0.281894\pi$
$264$	0	0
$265$	2.19674	0.134944
$266$	0	0
$267$	−17.6794	−1.08196
$268$	0	0
$269$	11.2640	0.686781	0.343390	−	0.939193i	$-0.388425\pi$
$269$	11.2640	0.686781	0.343390	+	0.939193i	$0.388425\pi$
$270$	0	0
$271$	5.25659	0.319315	0.159658	−	0.987172i	$-0.448961\pi$
$271$	5.25659	0.319315	0.159658	+	0.987172i	$0.448961\pi$
$272$	0	0
$273$	−1.27991	−0.0774638
$274$	0	0
$275$	8.20644	0.494867
$276$	0	0
$277$	−10.4026	−0.625034	−0.312517	−	0.949912i	$-0.601172\pi$
$277$	−10.4026	−0.625034	−0.312517	+	0.949912i	$0.601172\pi$
$278$	0	0
$279$	−3.79993	−0.227496
$280$	0	0
$281$	−21.6130	−1.28932	−0.644661	−	0.764469i	$-0.723001\pi$
$281$	−21.6130	−1.28932	−0.644661	+	0.764469i	$0.723001\pi$
$282$	0	0
$283$	−24.2913	−1.44397	−0.721983	−	0.691911i	$-0.756769\pi$
$283$	−24.2913	−1.44397	−0.721983	+	0.691911i	$0.756769\pi$
$284$	0	0
$285$	−6.45620	−0.382432
$286$	0	0
$287$	7.62537	0.450111
$288$	0	0
$289$	−14.7486	−0.867567
$290$	0	0
$291$	4.28751	0.251339
$292$	0	0
$293$	15.1490	0.885014	0.442507	−	0.896765i	$-0.354089\pi$
$293$	15.1490	0.885014	0.442507	+	0.896765i	$0.354089\pi$
$294$	0	0
$295$	−19.8320	−1.15466
$296$	0	0
$297$	3.09767	0.179745
$298$	0	0
$299$	−6.92907	−0.400718
$300$	0	0
$301$	−7.74290	−0.446293
$302$	0	0
$303$	15.5844	0.895303
$304$	0	0
$305$	−5.77533	−0.330694
$306$	0	0
$307$	−8.37234	−0.477835	−0.238917	−	0.971040i	$-0.576792\pi$
$307$	−8.37234	−0.477835	−0.238917	+	0.971040i	$0.576792\pi$
$308$	0	0
$309$	−5.71458	−0.325091
$310$	0	0
$311$	−4.65962	−0.264223	−0.132111	−	0.991235i	$-0.542176\pi$
$311$	−4.65962	−0.264223	−0.132111	+	0.991235i	$0.542176\pi$
$312$	0	0
$313$	−8.10404	−0.458067	−0.229034	−	0.973419i	$-0.573557\pi$
$313$	−8.10404	−0.458067	−0.229034	+	0.973419i	$0.573557\pi$
$314$	0	0
$315$	1.40377	0.0790932
$316$	0	0
$317$	−12.1018	−0.679706	−0.339853	−	0.940479i	$-0.610377\pi$
$317$	−12.1018	−0.679706	−0.339853	+	0.940479i	$0.610377\pi$
$318$	0	0
$319$	−5.47648	−0.306624
$320$	0	0
$321$	−3.17924	−0.177448
$322$	0	0
$323$	−6.31822	−0.351555
$324$	0	0
$325$	−3.70348	−0.205432
$326$	0	0
$327$	12.5475	0.693881
$328$	0	0
$329$	6.24027	0.344037
$330$	0	0
$331$	11.5342	0.633979	0.316989	−	0.948429i	$-0.397328\pi$
$331$	11.5342	0.633979	0.316989	+	0.948429i	$0.397328\pi$
$332$	0	0
$333$	4.15442	0.227661
$334$	0	0
$335$	−11.1927	−0.611521
$336$	0	0
$337$	4.87706	0.265670	0.132835	−	0.991138i	$-0.457592\pi$
$337$	4.87706	0.265670	0.132835	+	0.991138i	$0.457592\pi$
$338$	0	0
$339$	−2.50783	−0.136206
$340$	0	0
$341$	11.7709	0.637432
$342$	0	0
$343$	−12.0504	−0.650662
$344$	0	0
$345$	7.59958	0.409148
$346$	0	0
$347$	5.41332	0.290602	0.145301	−	0.989387i	$-0.453585\pi$
$347$	5.41332	0.290602	0.145301	+	0.989387i	$0.453585\pi$
$348$	0	0
$349$	16.3896	0.877313	0.438656	−	0.898655i	$-0.355455\pi$
$349$	16.3896	0.877313	0.438656	+	0.898655i	$0.355455\pi$
$350$	0	0
$351$	−1.39795	−0.0746169
$352$	0	0
$353$	−33.9228	−1.80553	−0.902765	−	0.430133i	$-0.858467\pi$
$353$	−33.9228	−1.80553	−0.902765	+	0.430133i	$0.858467\pi$
$354$	0	0
$355$	−1.35530	−0.0719316
$356$	0	0
$357$	1.37376	0.0727073
$358$	0	0
$359$	−30.9733	−1.63471	−0.817354	−	0.576135i	$-0.804560\pi$
$359$	−30.9733	−1.63471	−0.817354	+	0.576135i	$0.804560\pi$
$360$	0	0
$361$	−1.26863	−0.0667699
$362$	0	0
$363$	1.40441	0.0737126
$364$	0	0
$365$	−1.36745	−0.0715757
$366$	0	0
$367$	19.7551	1.03121	0.515603	−	0.856827i	$-0.327568\pi$
$367$	19.7551	1.03121	0.515603	+	0.856827i	$0.327568\pi$
$368$	0	0
$369$	8.32859	0.433569
$370$	0	0
$371$	1.31178	0.0681043
$372$	0	0
$373$	9.12807	0.472634	0.236317	−	0.971676i	$-0.424060\pi$
$373$	9.12807	0.472634	0.236317	+	0.971676i	$0.424060\pi$
$374$	0	0
$375$	11.7280	0.605630
$376$	0	0
$377$	2.47148	0.127288
$378$	0	0
$379$	13.9800	0.718103	0.359052	−	0.933318i	$-0.383100\pi$
$379$	13.9800	0.718103	0.359052	+	0.933318i	$0.383100\pi$
$380$	0	0
$381$	17.3437	0.888543
$382$	0	0
$383$	34.7750	1.77692	0.888459	−	0.458956i	$-0.151776\pi$
$383$	34.7750	1.77692	0.888459	+	0.458956i	$0.151776\pi$
$384$	0	0
$385$	−4.34841	−0.221615
$386$	0	0
$387$	−8.45697	−0.429892
$388$	0	0
$389$	−16.7127	−0.847369	−0.423684	−	0.905810i	$-0.639263\pi$
$389$	−16.7127	−0.847369	−0.423684	+	0.905810i	$0.639263\pi$
$390$	0	0
$391$	7.43716	0.376113
$392$	0	0
$393$	−16.3850	−0.826513
$394$	0	0
$395$	1.73246	0.0871696
$396$	0	0
$397$	−3.95221	−0.198356	−0.0991780	−	0.995070i	$-0.531621\pi$
$397$	−3.95221	−0.198356	−0.0991780	+	0.995070i	$0.531621\pi$
$398$	0	0
$399$	−3.85532	−0.193007
$400$	0	0
$401$	−19.9460	−0.996055	−0.498028	−	0.867161i	$-0.665942\pi$
$401$	−19.9460	−0.996055	−0.498028	+	0.867161i	$0.665942\pi$
$402$	0	0
$403$	−5.31210	−0.264615
$404$	0	0
$405$	1.53322	0.0761865
$406$	0	0
$407$	−12.8690	−0.637894
$408$	0	0
$409$	−4.18390	−0.206881	−0.103440	−	0.994636i	$-0.532985\pi$
$409$	−4.18390	−0.206881	−0.103440	+	0.994636i	$0.532985\pi$
$410$	0	0
$411$	18.5673	0.915856
$412$	0	0
$413$	−11.8427	−0.582740
$414$	0	0
$415$	−6.53769	−0.320923
$416$	0	0
$417$	15.0137	0.735225
$418$	0	0
$419$	39.5510	1.93219	0.966097	−	0.258179i	$-0.0831225\pi$
$419$	39.5510	1.93219	0.966097	+	0.258179i	$0.0831225\pi$
$420$	0	0
$421$	−15.4917	−0.755019	−0.377510	−	0.926006i	$-0.623220\pi$
$421$	−15.4917	−0.755019	−0.377510	+	0.926006i	$0.623220\pi$
$422$	0	0
$423$	6.81577	0.331394
$424$	0	0
$425$	3.97505	0.192818
$426$	0	0
$427$	−3.44874	−0.166896
$428$	0	0
$429$	4.33039	0.209073
$430$	0	0
$431$	−16.0476	−0.772984	−0.386492	−	0.922293i	$-0.626313\pi$
$431$	−16.0476	−0.772984	−0.386492	+	0.922293i	$0.626313\pi$
$432$	0	0
$433$	21.5403	1.03516	0.517580	−	0.855635i	$-0.326833\pi$
$433$	21.5403	1.03516	0.517580	+	0.855635i	$0.326833\pi$
$434$	0	0
$435$	−2.71064	−0.129965
$436$	0	0
$437$	−20.8716	−0.998423
$438$	0	0
$439$	−20.1392	−0.961191	−0.480595	−	0.876942i	$-0.659579\pi$
$439$	−20.1392	−0.961191	−0.480595	+	0.876942i	$0.659579\pi$
$440$	0	0
$441$	−6.16174	−0.293416
$442$	0	0
$443$	−17.2013	−0.817257	−0.408629	−	0.912701i	$-0.633993\pi$
$443$	−17.2013	−0.817257	−0.408629	+	0.912701i	$0.633993\pi$
$444$	0	0
$445$	27.1065	1.28497
$446$	0	0
$447$	−22.6712	−1.07231
$448$	0	0
$449$	−28.7950	−1.35892	−0.679461	−	0.733712i	$-0.737786\pi$
$449$	−28.7950	−1.35892	−0.679461	+	0.733712i	$0.737786\pi$
$450$	0	0
$451$	−25.7993	−1.21484
$452$	0	0
$453$	19.2737	0.905559
$454$	0	0
$455$	1.96239	0.0919983
$456$	0	0
$457$	31.8010	1.48759	0.743794	−	0.668409i	$-0.233024\pi$
$457$	31.8010	1.48759	0.743794	+	0.668409i	$0.233024\pi$
$458$	0	0
$459$	1.50046	0.0700353
$460$	0	0
$461$	23.3828	1.08905	0.544524	−	0.838745i	$-0.316710\pi$
$461$	23.3828	1.08905	0.544524	+	0.838745i	$0.316710\pi$
$462$	0	0
$463$	−39.4475	−1.83328	−0.916640	−	0.399714i	$-0.869109\pi$
$463$	−39.4475	−1.83328	−0.916640	+	0.399714i	$0.869109\pi$
$464$	0	0
$465$	5.82614	0.270181
$466$	0	0
$467$	31.6227	1.46332	0.731661	−	0.681668i	$-0.238745\pi$
$467$	31.6227	1.46332	0.731661	+	0.681668i	$0.238745\pi$
$468$	0	0
$469$	−6.68370	−0.308625
$470$	0	0
$471$	−15.3895	−0.709110
$472$	0	0
$473$	26.1969	1.20454
$474$	0	0
$475$	−11.1555	−0.511851
$476$	0	0
$477$	1.43276	0.0656014
$478$	0	0
$479$	13.4628	0.615129	0.307565	−	0.951527i	$-0.400486\pi$
$479$	13.4628	0.615129	0.307565	+	0.951527i	$0.400486\pi$
$480$	0	0
$481$	5.80766	0.264806
$482$	0	0
$483$	4.53809	0.206490
$484$	0	0
$485$	−6.57372	−0.298497
$486$	0	0
$487$	−34.2506	−1.55204	−0.776022	−	0.630706i	$-0.782765\pi$
$487$	−34.2506	−1.55204	−0.776022	+	0.630706i	$0.782765\pi$
$488$	0	0
$489$	−2.65971	−0.120276
$490$	0	0
$491$	16.2041	0.731279	0.365639	−	0.930757i	$-0.380850\pi$
$491$	16.2041	0.731279	0.365639	+	0.930757i	$0.380850\pi$
$492$	0	0
$493$	−2.65271	−0.119472
$494$	0	0
$495$	−4.74943	−0.213471
$496$	0	0
$497$	−0.809315	−0.0363027
$498$	0	0
$499$	−19.1460	−0.857091	−0.428545	−	0.903520i	$-0.640974\pi$
$499$	−19.1460	−0.857091	−0.428545	+	0.903520i	$0.640974\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−8.19733	−0.365501	−0.182750	−	0.983159i	$-0.558500\pi$
$503$	−8.19733	−0.365501	−0.182750	+	0.983159i	$0.558500\pi$
$504$	0	0
$505$	−23.8944	−1.06329
$506$	0	0
$507$	11.0457	0.490559
$508$	0	0
$509$	15.4558	0.685065	0.342533	−	0.939506i	$-0.388715\pi$
$509$	15.4558	0.685065	0.342533	+	0.939506i	$0.388715\pi$
$510$	0	0
$511$	−0.816574	−0.0361231
$512$	0	0
$513$	−4.21086	−0.185914
$514$	0	0
$515$	8.76173	0.386088
$516$	0	0
$517$	−21.1130	−0.928550
$518$	0	0
$519$	−16.5038	−0.724438
$520$	0	0
$521$	22.9773	1.00666	0.503328	−	0.864096i	$-0.332109\pi$
$521$	22.9773	1.00666	0.503328	+	0.864096i	$0.332109\pi$
$522$	0	0
$523$	1.86529	0.0815636	0.0407818	−	0.999168i	$-0.487015\pi$
$523$	1.86529	0.0815636	0.0407818	+	0.999168i	$0.487015\pi$
$524$	0	0
$525$	2.42554	0.105859
$526$	0	0
$527$	5.70163	0.248367
$528$	0	0
$529$	1.56789	0.0681692
$530$	0	0
$531$	−12.9348	−0.561324
$532$	0	0
$533$	11.6429	0.504312
$534$	0	0
$535$	4.87449	0.210743
$536$	0	0
$537$	0.202161	0.00872390
$538$	0	0
$539$	19.0871	0.822138
$540$	0	0
$541$	−35.4871	−1.52571	−0.762854	−	0.646570i	$-0.776203\pi$
$541$	−35.4871	−1.52571	−0.762854	+	0.646570i	$0.776203\pi$
$542$	0	0
$543$	2.91030	0.124893
$544$	0	0
$545$	−19.2382	−0.824074
$546$	0	0
$547$	−1.16371	−0.0497568	−0.0248784	−	0.999690i	$-0.507920\pi$
$547$	−1.16371	−0.0497568	−0.0248784	+	0.999690i	$0.507920\pi$
$548$	0	0
$549$	−3.76679	−0.160763
$550$	0	0
$551$	7.44452	0.317147
$552$	0	0
$553$	1.03454	0.0439931
$554$	0	0
$555$	−6.36965	−0.270377
$556$	0	0
$557$	−27.6032	−1.16958	−0.584792	−	0.811183i	$-0.698824\pi$
$557$	−27.6032	−1.16958	−0.584792	+	0.811183i	$0.698824\pi$
$558$	0	0
$559$	−11.8224	−0.500034
$560$	0	0
$561$	−4.64792	−0.196235
$562$	0	0
$563$	−26.7132	−1.12583	−0.562913	−	0.826516i	$-0.690319\pi$
$563$	−26.7132	−1.12583	−0.562913	+	0.826516i	$0.690319\pi$
$564$	0	0
$565$	3.84506	0.161763
$566$	0	0
$567$	0.915565	0.0384501
$568$	0	0
$569$	−24.6239	−1.03229	−0.516143	−	0.856502i	$-0.672633\pi$
$569$	−24.6239	−1.03229	−0.516143	+	0.856502i	$0.672633\pi$
$570$	0	0
$571$	16.6623	0.697294	0.348647	−	0.937254i	$-0.386641\pi$
$571$	16.6623	0.697294	0.348647	+	0.937254i	$0.386641\pi$
$572$	0	0
$573$	−15.2680	−0.637830
$574$	0	0
$575$	13.1312	0.547607
$576$	0	0
$577$	−10.9442	−0.455612	−0.227806	−	0.973707i	$-0.573155\pi$
$577$	−10.9442	−0.455612	−0.227806	+	0.973707i	$0.573155\pi$
$578$	0	0
$579$	−8.99644	−0.373879
$580$	0	0
$581$	−3.90398	−0.161965
$582$	0	0
$583$	−4.43821	−0.183812
$584$	0	0
$585$	2.14337	0.0886173
$586$	0	0
$587$	−1.12881	−0.0465908	−0.0232954	−	0.999729i	$-0.507416\pi$
$587$	−1.12881	−0.0465908	−0.0232954	+	0.999729i	$0.507416\pi$
$588$	0	0
$589$	−16.0010	−0.659309
$590$	0	0
$591$	−11.6292	−0.478362
$592$	0	0
$593$	21.3857	0.878205	0.439102	−	0.898437i	$-0.355297\pi$
$593$	21.3857	0.878205	0.439102	+	0.898437i	$0.355297\pi$
$594$	0	0
$595$	−2.10629	−0.0863494
$596$	0	0
$597$	12.6305	0.516930
$598$	0	0
$599$	13.0654	0.533837	0.266919	−	0.963719i	$-0.413994\pi$
$599$	13.0654	0.533837	0.266919	+	0.963719i	$0.413994\pi$
$600$	0	0
$601$	−7.93394	−0.323632	−0.161816	−	0.986821i	$-0.551735\pi$
$601$	−7.93394	−0.323632	−0.161816	+	0.986821i	$0.551735\pi$
$602$	0	0
$603$	−7.30009	−0.297283
$604$	0	0
$605$	−2.15328	−0.0875433
$606$	0	0
$607$	−35.6021	−1.44504	−0.722522	−	0.691348i	$-0.757017\pi$
$607$	−35.6021	−1.44504	−0.722522	+	0.691348i	$0.757017\pi$
$608$	0	0
$609$	−1.61866	−0.0655913
$610$	0	0
$611$	9.52809	0.385465
$612$	0	0
$613$	9.26461	0.374194	0.187097	−	0.982341i	$-0.440092\pi$
$613$	9.26461	0.374194	0.187097	+	0.982341i	$0.440092\pi$
$614$	0	0
$615$	−12.7696	−0.514920
$616$	0	0
$617$	−18.9004	−0.760902	−0.380451	−	0.924801i	$-0.624231\pi$
$617$	−18.9004	−0.760902	−0.380451	+	0.924801i	$0.624231\pi$
$618$	0	0
$619$	1.98374	0.0797333	0.0398666	−	0.999205i	$-0.487307\pi$
$619$	1.98374	0.0797333	0.0398666	+	0.999205i	$0.487307\pi$
$620$	0	0
$621$	4.95660	0.198902
$622$	0	0
$623$	16.1866	0.648504
$624$	0	0
$625$	−4.73548	−0.189419
$626$	0	0
$627$	13.0439	0.520923
$628$	0	0
$629$	−6.23352	−0.248547
$630$	0	0
$631$	28.1079	1.11896	0.559480	−	0.828844i	$-0.311001\pi$
$631$	28.1079	1.11896	0.559480	+	0.828844i	$0.311001\pi$
$632$	0	0
$633$	−11.2769	−0.448215
$634$	0	0
$635$	−26.5917	−1.05526
$636$	0	0
$637$	−8.61379	−0.341291
$638$	0	0
$639$	−0.883952	−0.0349686
$640$	0	0
$641$	−13.6872	−0.540611	−0.270306	−	0.962775i	$-0.587125\pi$
$641$	−13.6872	−0.540611	−0.270306	+	0.962775i	$0.587125\pi$
$642$	0	0
$643$	−11.8120	−0.465820	−0.232910	−	0.972498i	$-0.574825\pi$
$643$	−11.8120	−0.465820	−0.232910	+	0.972498i	$0.574825\pi$
$644$	0	0
$645$	12.9664	0.510553
$646$	0	0
$647$	−12.9647	−0.509696	−0.254848	−	0.966981i	$-0.582025\pi$
$647$	−12.9647	−0.509696	−0.254848	+	0.966981i	$0.582025\pi$
$648$	0	0
$649$	40.0679	1.57280
$650$	0	0
$651$	3.47908	0.136356
$652$	0	0
$653$	−47.8071	−1.87084	−0.935419	−	0.353541i	$-0.884978\pi$
$653$	−47.8071	−1.87084	−0.935419	+	0.353541i	$0.884978\pi$
$654$	0	0
$655$	25.1219	0.981592
$656$	0	0
$657$	−0.891880	−0.0347956
$658$	0	0
$659$	−16.2125	−0.631550	−0.315775	−	0.948834i	$-0.602265\pi$
$659$	−16.2125	−0.631550	−0.315775	+	0.948834i	$0.602265\pi$
$660$	0	0
$661$	−41.9378	−1.63119	−0.815595	−	0.578623i	$-0.803590\pi$
$661$	−41.9378	−1.63119	−0.815595	+	0.578623i	$0.803590\pi$
$662$	0	0
$663$	2.09756	0.0814625
$664$	0	0
$665$	5.91106	0.229221
$666$	0	0
$667$	−8.76294	−0.339302
$668$	0	0
$669$	−11.4567	−0.442941
$670$	0	0
$671$	11.6683	0.450449
$672$	0	0
$673$	−17.9537	−0.692064	−0.346032	−	0.938223i	$-0.612471\pi$
$673$	−17.9537	−0.692064	−0.346032	+	0.938223i	$0.612471\pi$
$674$	0	0
$675$	2.64923	0.101969
$676$	0	0
$677$	−9.80016	−0.376651	−0.188325	−	0.982107i	$-0.560306\pi$
$677$	−9.80016	−0.376651	−0.188325	+	0.982107i	$0.560306\pi$
$678$	0	0
$679$	−3.92550	−0.150647
$680$	0	0
$681$	6.99214	0.267939
$682$	0	0
$683$	42.7950	1.63751	0.818753	−	0.574145i	$-0.194666\pi$
$683$	42.7950	1.63751	0.818753	+	0.574145i	$0.194666\pi$
$684$	0	0
$685$	−28.4678	−1.08770
$686$	0	0
$687$	−4.84467	−0.184836
$688$	0	0
$689$	2.00292	0.0763051
$690$	0	0
$691$	−47.2115	−1.79601	−0.898005	−	0.439985i	$-0.854984\pi$
$691$	−47.2115	−1.79601	−0.898005	+	0.439985i	$0.854984\pi$
$692$	0	0
$693$	−2.83612	−0.107735
$694$	0	0
$695$	−23.0194	−0.873176
$696$	0	0
$697$	−12.4967	−0.473346
$698$	0	0
$699$	26.4019	0.998613
$700$	0	0
$701$	−0.0922506	−0.00348426	−0.00174213	−	0.999998i	$-0.500555\pi$
$701$	−0.0922506	−0.00348426	−0.00174213	+	0.999998i	$0.500555\pi$
$702$	0	0
$703$	17.4937	0.659787
$704$	0	0
$705$	−10.4501	−0.393573
$706$	0	0
$707$	−14.2686	−0.536624
$708$	0	0
$709$	−33.0225	−1.24019	−0.620093	−	0.784528i	$-0.712905\pi$
$709$	−33.0225	−1.24019	−0.620093	+	0.784528i	$0.712905\pi$
$710$	0	0
$711$	1.12995	0.0423763
$712$	0	0
$713$	18.8347	0.705366
$714$	0	0
$715$	−6.63945	−0.248301
$716$	0	0
$717$	23.9728	0.895279
$718$	0	0
$719$	2.96188	0.110459	0.0552297	−	0.998474i	$-0.482411\pi$
$719$	2.96188	0.110459	0.0552297	+	0.998474i	$0.482411\pi$
$720$	0	0
$721$	5.23207	0.194852
$722$	0	0
$723$	−6.33066	−0.235440
$724$	0	0
$725$	−4.68365	−0.173946
$726$	0	0
$727$	−14.9351	−0.553914	−0.276957	−	0.960882i	$-0.589326\pi$
$727$	−14.9351	−0.553914	−0.276957	+	0.960882i	$0.589326\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.6893	0.469331
$732$	0	0
$733$	−28.0289	−1.03527	−0.517635	−	0.855602i	$-0.673187\pi$
$733$	−28.0289	−1.03527	−0.517635	+	0.855602i	$0.673187\pi$
$734$	0	0
$735$	9.44733	0.348470
$736$	0	0
$737$	22.6133	0.832971
$738$	0	0
$739$	34.2689	1.26060	0.630300	−	0.776351i	$-0.282932\pi$
$739$	34.2689	1.26060	0.630300	+	0.776351i	$0.282932\pi$
$740$	0	0
$741$	−5.88657	−0.216248
$742$	0	0
$743$	40.7965	1.49668	0.748340	−	0.663316i	$-0.230851\pi$
$743$	40.7965	1.49668	0.748340	+	0.663316i	$0.230851\pi$
$744$	0	0
$745$	34.7600	1.27351
$746$	0	0
$747$	−4.26402	−0.156012
$748$	0	0
$749$	2.91080	0.106358
$750$	0	0
$751$	−41.0708	−1.49869	−0.749347	−	0.662177i	$-0.769633\pi$
$751$	−41.0708	−1.49869	−0.749347	+	0.662177i	$0.769633\pi$
$752$	0	0
$753$	10.5048	0.382817
$754$	0	0
$755$	−29.5509	−1.07547
$756$	0	0
$757$	−33.5015	−1.21763	−0.608817	−	0.793311i	$-0.708356\pi$
$757$	−33.5015	−1.21763	−0.608817	+	0.793311i	$0.708356\pi$
$758$	0	0
$759$	−15.3539	−0.557313
$760$	0	0
$761$	−15.9152	−0.576926	−0.288463	−	0.957491i	$-0.593144\pi$
$761$	−15.9152	−0.576926	−0.288463	+	0.957491i	$0.593144\pi$
$762$	0	0
$763$	−11.4881	−0.415897
$764$	0	0
$765$	−2.30053	−0.0831760
$766$	0	0
$767$	−18.0822	−0.652912
$768$	0	0
$769$	−24.5047	−0.883661	−0.441830	−	0.897099i	$-0.645671\pi$
$769$	−24.5047	−0.883661	−0.441830	+	0.897099i	$0.645671\pi$
$770$	0	0
$771$	25.5749	0.921059
$772$	0	0
$773$	28.0247	1.00798	0.503990	−	0.863710i	$-0.331865\pi$
$773$	28.0247	1.00798	0.503990	+	0.863710i	$0.331865\pi$
$774$	0	0
$775$	10.0669	0.361612
$776$	0	0
$777$	−3.80364	−0.136455
$778$	0	0
$779$	35.0706	1.25653
$780$	0	0
$781$	2.73819	0.0979803
$782$	0	0
$783$	−1.76793	−0.0631808
$784$	0	0
$785$	23.5955	0.842160
$786$	0	0
$787$	13.5173	0.481840	0.240920	−	0.970545i	$-0.422551\pi$
$787$	13.5173	0.481840	0.240920	+	0.970545i	$0.422551\pi$
$788$	0	0
$789$	−20.5255	−0.730727
$790$	0	0
$791$	2.29608	0.0816391
$792$	0	0
$793$	−5.26577	−0.186993
$794$	0	0
$795$	−2.19674	−0.0779102
$796$	0	0
$797$	12.3513	0.437504	0.218752	−	0.975780i	$-0.429801\pi$
$797$	12.3513	0.437504	0.218752	+	0.975780i	$0.429801\pi$
$798$	0	0
$799$	−10.2268	−0.361797
$800$	0	0
$801$	17.6794	0.624671
$802$	0	0
$803$	2.76275	0.0974955
$804$	0	0
$805$	−6.95790	−0.245234
$806$	0	0
$807$	−11.2640	−0.396513
$808$	0	0
$809$	−49.9778	−1.75713	−0.878563	−	0.477626i	$-0.841497\pi$
$809$	−49.9778	−1.75713	−0.878563	+	0.477626i	$0.841497\pi$
$810$	0	0
$811$	41.8896	1.47094	0.735472	−	0.677556i	$-0.236961\pi$
$811$	41.8896	1.47094	0.735472	+	0.677556i	$0.236961\pi$
$812$	0	0
$813$	−5.25659	−0.184357
$814$	0	0
$815$	4.07793	0.142844
$816$	0	0
$817$	−35.6111	−1.24588
$818$	0	0
$819$	1.27991	0.0447237
$820$	0	0
$821$	−20.6369	−0.720232	−0.360116	−	0.932908i	$-0.617263\pi$
$821$	−20.6369	−0.720232	−0.360116	+	0.932908i	$0.617263\pi$
$822$	0	0
$823$	−33.0136	−1.15078	−0.575390	−	0.817879i	$-0.695150\pi$
$823$	−33.0136	−1.15078	−0.575390	+	0.817879i	$0.695150\pi$
$824$	0	0
$825$	−8.20644	−0.285711
$826$	0	0
$827$	4.76225	0.165600	0.0827998	−	0.996566i	$-0.473614\pi$
$827$	4.76225	0.165600	0.0827998	+	0.996566i	$0.473614\pi$
$828$	0	0
$829$	37.3978	1.29888	0.649439	−	0.760414i	$-0.275004\pi$
$829$	37.3978	1.29888	0.649439	+	0.760414i	$0.275004\pi$
$830$	0	0
$831$	10.4026	0.360863
$832$	0	0
$833$	9.24542	0.320335
$834$	0	0
$835$	−1.53322	−0.0530594
$836$	0	0
$837$	3.79993	0.131345
$838$	0	0
$839$	29.8684	1.03117	0.515585	−	0.856838i	$-0.327575\pi$
$839$	29.8684	1.03117	0.515585	+	0.856838i	$0.327575\pi$
$840$	0	0
$841$	−25.8744	−0.892221
$842$	0	0
$843$	21.6130	0.744390
$844$	0	0
$845$	−16.9356	−0.582602
$846$	0	0
$847$	−1.28583	−0.0441817
$848$	0	0
$849$	24.2913	0.833674
$850$	0	0
$851$	−20.5918	−0.705877
$852$	0	0
$853$	21.5896	0.739214	0.369607	−	0.929188i	$-0.379492\pi$
$853$	21.5896	0.739214	0.369607	+	0.929188i	$0.379492\pi$
$854$	0	0
$855$	6.45620	0.220797
$856$	0	0
$857$	−30.7368	−1.04995	−0.524975	−	0.851118i	$-0.675925\pi$
$857$	−30.7368	−1.04995	−0.524975	+	0.851118i	$0.675925\pi$
$858$	0	0
$859$	0.952468	0.0324978	0.0162489	−	0.999868i	$-0.494828\pi$
$859$	0.952468	0.0324978	0.0162489	+	0.999868i	$0.494828\pi$
$860$	0	0
$861$	−7.62537	−0.259872
$862$	0	0
$863$	14.7115	0.500786	0.250393	−	0.968144i	$-0.419440\pi$
$863$	14.7115	0.500786	0.250393	+	0.968144i	$0.419440\pi$
$864$	0	0
$865$	25.3041	0.860365
$866$	0	0
$867$	14.7486	0.500890
$868$	0	0
$869$	−3.50021	−0.118736
$870$	0	0
$871$	−10.2051	−0.345788
$872$	0	0
$873$	−4.28751	−0.145110
$874$	0	0
$875$	−10.7377	−0.363001
$876$	0	0
$877$	−19.3949	−0.654919	−0.327459	−	0.944865i	$-0.606192\pi$
$877$	−19.3949	−0.654919	−0.327459	+	0.944865i	$0.606192\pi$
$878$	0	0
$879$	−15.1490	−0.510963
$880$	0	0
$881$	−40.8536	−1.37639	−0.688197	−	0.725524i	$-0.741597\pi$
$881$	−40.8536	−1.37639	−0.688197	+	0.725524i	$0.741597\pi$
$882$	0	0
$883$	51.0635	1.71843	0.859213	−	0.511619i	$-0.170954\pi$
$883$	51.0635	1.71843	0.859213	+	0.511619i	$0.170954\pi$
$884$	0	0
$885$	19.8320	0.666646
$886$	0	0
$887$	0.844140	0.0283434	0.0141717	−	0.999900i	$-0.495489\pi$
$887$	0.844140	0.0283434	0.0141717	+	0.999900i	$0.495489\pi$
$888$	0	0
$889$	−15.8792	−0.532573
$890$	0	0
$891$	−3.09767	−0.103776
$892$	0	0
$893$	28.7003	0.960418
$894$	0	0
$895$	−0.309958	−0.0103608
$896$	0	0
$897$	6.92907	0.231355
$898$	0	0
$899$	−6.71802	−0.224058
$900$	0	0
$901$	−2.14979	−0.0716198
$902$	0	0
$903$	7.74290	0.257668
$904$	0	0
$905$	−4.46214	−0.148326
$906$	0	0
$907$	−14.2678	−0.473754	−0.236877	−	0.971540i	$-0.576124\pi$
$907$	−14.2678	−0.473754	−0.236877	+	0.971540i	$0.576124\pi$
$908$	0	0
$909$	−15.5844	−0.516903
$910$	0	0
$911$	−2.15685	−0.0714598	−0.0357299	−	0.999361i	$-0.511376\pi$
$911$	−2.15685	−0.0714598	−0.0357299	+	0.999361i	$0.511376\pi$
$912$	0	0
$913$	13.2085	0.437139
$914$	0	0
$915$	5.77533	0.190926
$916$	0	0
$917$	15.0015	0.495394
$918$	0	0
$919$	50.7401	1.67376	0.836881	−	0.547386i	$-0.184377\pi$
$919$	50.7401	1.67376	0.836881	+	0.547386i	$0.184377\pi$
$920$	0	0
$921$	8.37234	0.275878
$922$	0	0
$923$	−1.23572	−0.0406742
$924$	0	0
$925$	−11.0060	−0.361874
$926$	0	0
$927$	5.71458	0.187692
$928$	0	0
$929$	−20.0006	−0.656199	−0.328100	−	0.944643i	$-0.606408\pi$
$929$	−20.0006	−0.656199	−0.328100	+	0.944643i	$0.606408\pi$
$930$	0	0
$931$	−25.9463	−0.850354
$932$	0	0
$933$	4.65962	0.152549
$934$	0	0
$935$	7.12631	0.233055
$936$	0	0
$937$	8.79635	0.287364	0.143682	−	0.989624i	$-0.454106\pi$
$937$	8.79635	0.287364	0.143682	+	0.989624i	$0.454106\pi$
$938$	0	0
$939$	8.10404	0.264465
$940$	0	0
$941$	−15.3968	−0.501921	−0.250961	−	0.967997i	$-0.580746\pi$
$941$	−15.3968	−0.501921	−0.250961	+	0.967997i	$0.580746\pi$
$942$	0	0
$943$	−41.2815	−1.34431
$944$	0	0
$945$	−1.40377	−0.0456645
$946$	0	0
$947$	33.6342	1.09297	0.546483	−	0.837470i	$-0.315966\pi$
$947$	33.6342	1.09297	0.546483	+	0.837470i	$0.315966\pi$
$948$	0	0
$949$	−1.24680	−0.0404729
$950$	0	0
$951$	12.1018	0.392429
$952$	0	0
$953$	19.0383	0.616712	0.308356	−	0.951271i	$-0.400221\pi$
$953$	19.0383	0.616712	0.308356	+	0.951271i	$0.400221\pi$
$954$	0	0
$955$	23.4093	0.757507
$956$	0	0
$957$	5.47648	0.177029
$958$	0	0
$959$	−16.9995	−0.548943
$960$	0	0
$961$	−16.5605	−0.534211
$962$	0	0
$963$	3.17924	0.102450
$964$	0	0
$965$	13.7935	0.444030
$966$	0	0
$967$	37.3295	1.20044	0.600218	−	0.799836i	$-0.295080\pi$
$967$	37.3295	1.20044	0.600218	+	0.799836i	$0.295080\pi$
$968$	0	0
$969$	6.31822	0.202970
$970$	0	0
$971$	52.8286	1.69535	0.847675	−	0.530516i	$-0.178002\pi$
$971$	52.8286	1.69535	0.847675	+	0.530516i	$0.178002\pi$
$972$	0	0
$973$	−13.7460	−0.440678
$974$	0	0
$975$	3.70348	0.118606
$976$	0	0
$977$	31.0861	0.994532	0.497266	−	0.867598i	$-0.334337\pi$
$977$	31.0861	0.994532	0.497266	+	0.867598i	$0.334337\pi$
$978$	0	0
$979$	−54.7651	−1.75030
$980$	0	0
$981$	−12.5475	−0.400612
$982$	0	0
$983$	2.74694	0.0876136	0.0438068	−	0.999040i	$-0.486051\pi$
$983$	2.74694	0.0876136	0.0438068	+	0.999040i	$0.486051\pi$
$984$	0	0
$985$	17.8302	0.568117
$986$	0	0
$987$	−6.24027	−0.198630
$988$	0	0
$989$	41.9178	1.33291
$990$	0	0
$991$	−36.1120	−1.14713	−0.573567	−	0.819158i	$-0.694441\pi$
$991$	−36.1120	−1.14713	−0.573567	+	0.819158i	$0.694441\pi$
$992$	0	0
$993$	−11.5342	−0.366028
$994$	0	0
$995$	−19.3653	−0.613922
$996$	0	0
$997$	−52.4769	−1.66196	−0.830980	−	0.556302i	$-0.812220\pi$
$997$	−52.4769	−1.66196	−0.830980	+	0.556302i	$0.812220\pi$
$998$	0	0
$999$	−4.15442	−0.131440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bf.1.7		12
4.3	odd	2		4008.2.a.l.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.l.1.7	✓	12	4.3	odd	2
8016.2.a.bf.1.7		12	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.53322 q^{5} +0.915565 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.53322 q^{5} +0.915565 q^{7} +1.00000 q^{9} -3.09767 q^{11} +1.39795 q^{13} -1.53322 q^{15} -1.50046 q^{17} +4.21086 q^{19} -0.915565 q^{21} -4.95660 q^{23} -2.64923 q^{25} -1.00000 q^{27} +1.76793 q^{29} -3.79993 q^{31} +3.09767 q^{33} +1.40377 q^{35} +4.15442 q^{37} -1.39795 q^{39} +8.32859 q^{41} -8.45697 q^{43} +1.53322 q^{45} +6.81577 q^{47} -6.16174 q^{49} +1.50046 q^{51} +1.43276 q^{53} -4.74943 q^{55} -4.21086 q^{57} -12.9348 q^{59} -3.76679 q^{61} +0.915565 q^{63} +2.14337 q^{65} -7.30009 q^{67} +4.95660 q^{69} -0.883952 q^{71} -0.891880 q^{73} +2.64923 q^{75} -2.83612 q^{77} +1.12995 q^{79} +1.00000 q^{81} -4.26402 q^{83} -2.30053 q^{85} -1.76793 q^{87} +17.6794 q^{89} +1.27991 q^{91} +3.79993 q^{93} +6.45620 q^{95} -4.28751 q^{97} -3.09767 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9} - q^{11} + 8 q^{13} - 4 q^{15} + 3 q^{17} - 12 q^{19} + 11 q^{21} - 7 q^{23} + 18 q^{25} - 12 q^{27} + 5 q^{29} - 33 q^{31} + q^{33} - 15 q^{35} + 8 q^{37} - 8 q^{39} - 6 q^{41} - 16 q^{43} + 4 q^{45} - 18 q^{47} + 25 q^{49} - 3 q^{51} + 20 q^{53} - 39 q^{55} + 12 q^{57} - 4 q^{59} + 10 q^{61} - 11 q^{63} - 9 q^{67} + 7 q^{69} - 11 q^{71} + 22 q^{73} - 18 q^{75} + 24 q^{77} - 56 q^{79} + 12 q^{81} - 26 q^{83} + 15 q^{85} - 5 q^{87} - 15 q^{89} - 11 q^{91} + 33 q^{93} - 3 q^{95} + 8 q^{97} - q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9 - q^11 + 8 * q^13 - 4 * q^15 + 3 * q^17 - 12 * q^19 + 11 * q^21 - 7 * q^23 + 18 * q^25 - 12 * q^27 + 5 * q^29 - 33 * q^31 + q^33 - 15 * q^35 + 8 * q^37 - 8 * q^39 - 6 * q^41 - 16 * q^43 + 4 * q^45 - 18 * q^47 + 25 * q^49 - 3 * q^51 + 20 * q^53 - 39 * q^55 + 12 * q^57 - 4 * q^59 + 10 * q^61 - 11 * q^63 - 9 * q^67 + 7 * q^69 - 11 * q^71 + 22 * q^73 - 18 * q^75 + 24 * q^77 - 56 * q^79 + 12 * q^81 - 26 * q^83 + 15 * q^85 - 5 * q^87 - 15 * q^89 - 11 * q^91 + 33 * q^93 - 3 * q^95 + 8 * q^97 - q^99

Introduction

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