LMFDB - Embedded newform 8016.2.a.bg.1.8

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + \cdots + 12144 $ x^13 - 2x^12 - 49x^11 + 99x^10 + 901x^9 - 1879x^8 - 7582x^7 + 16968x^6 + 26911x^5 - 72240x^4 - 14532x^3 + 112850x^2 - 72184x + 12144
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			1.8
Root			$1.25667$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.25667 q^{5} +4.55521 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.25667 q^{5} +4.55521 q^{7} +1.00000 q^{9} +5.23461 q^{11} -1.93555 q^{13} -1.25667 q^{15} +7.59670 q^{17} +4.42731 q^{19} -4.55521 q^{21} -6.61071 q^{23} -3.42078 q^{25} -1.00000 q^{27} +3.73263 q^{29} +1.27190 q^{31} -5.23461 q^{33} +5.72440 q^{35} +6.87153 q^{37} +1.93555 q^{39} +8.71632 q^{41} -5.25563 q^{43} +1.25667 q^{45} +6.55882 q^{47} +13.7500 q^{49} -7.59670 q^{51} +3.93979 q^{53} +6.57819 q^{55} -4.42731 q^{57} -12.3546 q^{59} -0.282357 q^{61} +4.55521 q^{63} -2.43235 q^{65} +8.25155 q^{67} +6.61071 q^{69} +1.66362 q^{71} -3.89727 q^{73} +3.42078 q^{75} +23.8448 q^{77} -6.21505 q^{79} +1.00000 q^{81} -14.1118 q^{83} +9.54655 q^{85} -3.73263 q^{87} -6.93566 q^{89} -8.81684 q^{91} -1.27190 q^{93} +5.56367 q^{95} -3.11931 q^{97} +5.23461 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * 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
$5$	1.25667	0.562000	0.281000	−	0.959708i	$-0.409334\pi$
$5$	1.25667	0.562000	0.281000	+	0.959708i	$0.409334\pi$
$6$	0	0
$7$	4.55521	1.72171	0.860854	−	0.508852i	$-0.169930\pi$
$7$	4.55521	1.72171	0.860854	+	0.508852i	$0.169930\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.23461	1.57829	0.789147	−	0.614204i	$-0.210523\pi$
$11$	5.23461	1.57829	0.789147	+	0.614204i	$0.210523\pi$
$12$	0	0
$13$	−1.93555	−0.536825	−0.268412	−	0.963304i	$-0.586499\pi$
$13$	−1.93555	−0.536825	−0.268412	+	0.963304i	$0.586499\pi$
$14$	0	0
$15$	−1.25667	−0.324471
$16$	0	0
$17$	7.59670	1.84247	0.921235	−	0.389007i	$-0.127182\pi$
$17$	7.59670	1.84247	0.921235	+	0.389007i	$0.127182\pi$
$18$	0	0
$19$	4.42731	1.01569	0.507847	−	0.861447i	$-0.330442\pi$
$19$	4.42731	1.01569	0.507847	+	0.861447i	$0.330442\pi$
$20$	0	0
$21$	−4.55521	−0.994029
$22$	0	0
$23$	−6.61071	−1.37843	−0.689214	−	0.724558i	$-0.742044\pi$
$23$	−6.61071	−1.37843	−0.689214	+	0.724558i	$0.742044\pi$
$24$	0	0
$25$	−3.42078	−0.684156
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.73263	0.693132	0.346566	−	0.938026i	$-0.387348\pi$
$29$	3.73263	0.693132	0.346566	+	0.938026i	$0.387348\pi$
$30$	0	0
$31$	1.27190	0.228440	0.114220	−	0.993455i	$-0.463563\pi$
$31$	1.27190	0.228440	0.114220	+	0.993455i	$0.463563\pi$
$32$	0	0
$33$	−5.23461	−0.911229
$34$	0	0
$35$	5.72440	0.967601
$36$	0	0
$37$	6.87153	1.12967	0.564836	−	0.825203i	$-0.308939\pi$
$37$	6.87153	1.12967	0.564836	+	0.825203i	$0.308939\pi$
$38$	0	0
$39$	1.93555	0.309936
$40$	0	0
$41$	8.71632	1.36126	0.680631	−	0.732627i	$-0.261706\pi$
$41$	8.71632	1.36126	0.680631	+	0.732627i	$0.261706\pi$
$42$	0	0
$43$	−5.25563	−0.801476	−0.400738	−	0.916193i	$-0.631246\pi$
$43$	−5.25563	−0.801476	−0.400738	+	0.916193i	$0.631246\pi$
$44$	0	0
$45$	1.25667	0.187333
$46$	0	0
$47$	6.55882	0.956702	0.478351	−	0.878169i	$-0.341235\pi$
$47$	6.55882	0.956702	0.478351	+	0.878169i	$0.341235\pi$
$48$	0	0
$49$	13.7500	1.96428
$50$	0	0
$51$	−7.59670	−1.06375
$52$	0	0
$53$	3.93979	0.541172	0.270586	−	0.962696i	$-0.412783\pi$
$53$	3.93979	0.541172	0.270586	+	0.962696i	$0.412783\pi$
$54$	0	0
$55$	6.57819	0.887002
$56$	0	0
$57$	−4.42731	−0.586411
$58$	0	0
$59$	−12.3546	−1.60844	−0.804218	−	0.594335i	$-0.797416\pi$
$59$	−12.3546	−1.60844	−0.804218	+	0.594335i	$0.797416\pi$
$60$	0	0
$61$	−0.282357	−0.0361522	−0.0180761	−	0.999837i	$-0.505754\pi$
$61$	−0.282357	−0.0361522	−0.0180761	+	0.999837i	$0.505754\pi$
$62$	0	0
$63$	4.55521	0.573903
$64$	0	0
$65$	−2.43235	−0.301696
$66$	0	0
$67$	8.25155	1.00809	0.504043	−	0.863678i	$-0.331845\pi$
$67$	8.25155	1.00809	0.504043	+	0.863678i	$0.331845\pi$
$68$	0	0
$69$	6.61071	0.795836
$70$	0	0
$71$	1.66362	0.197435	0.0987175	−	0.995115i	$-0.468526\pi$
$71$	1.66362	0.197435	0.0987175	+	0.995115i	$0.468526\pi$
$72$	0	0
$73$	−3.89727	−0.456141	−0.228071	−	0.973645i	$-0.573242\pi$
$73$	−3.89727	−0.456141	−0.228071	+	0.973645i	$0.573242\pi$
$74$	0	0
$75$	3.42078	0.394997
$76$	0	0
$77$	23.8448	2.71736
$78$	0	0
$79$	−6.21505	−0.699248	−0.349624	−	0.936890i	$-0.613691\pi$
$79$	−6.21505	−0.699248	−0.349624	+	0.936890i	$0.613691\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.1118	−1.54898	−0.774488	−	0.632588i	$-0.781993\pi$
$83$	−14.1118	−1.54898	−0.774488	+	0.632588i	$0.781993\pi$
$84$	0	0
$85$	9.54655	1.03547
$86$	0	0
$87$	−3.73263	−0.400180
$88$	0	0
$89$	−6.93566	−0.735178	−0.367589	−	0.929988i	$-0.619817\pi$
$89$	−6.93566	−0.735178	−0.367589	+	0.929988i	$0.619817\pi$
$90$	0	0
$91$	−8.81684	−0.924256
$92$	0	0
$93$	−1.27190	−0.131890
$94$	0	0
$95$	5.56367	0.570820
$96$	0	0
$97$	−3.11931	−0.316718	−0.158359	−	0.987382i	$-0.550620\pi$
$97$	−3.11931	−0.316718	−0.158359	+	0.987382i	$0.550620\pi$
$98$	0	0
$99$	5.23461	0.526098
$100$	0	0
$101$	11.8680	1.18091	0.590457	−	0.807069i	$-0.298948\pi$
$101$	11.8680	1.18091	0.590457	+	0.807069i	$0.298948\pi$
$102$	0	0
$103$	2.31747	0.228347	0.114174	−	0.993461i	$-0.463578\pi$
$103$	2.31747	0.228347	0.114174	+	0.993461i	$0.463578\pi$
$104$	0	0
$105$	−5.72440	−0.558645
$106$	0	0
$107$	1.55158	0.149997	0.0749985	−	0.997184i	$-0.476105\pi$
$107$	1.55158	0.149997	0.0749985	+	0.997184i	$0.476105\pi$
$108$	0	0
$109$	5.37197	0.514541	0.257271	−	0.966339i	$-0.417177\pi$
$109$	5.37197	0.514541	0.257271	+	0.966339i	$0.417177\pi$
$110$	0	0
$111$	−6.87153	−0.652217
$112$	0	0
$113$	17.3036	1.62779	0.813893	−	0.581014i	$-0.197344\pi$
$113$	17.3036	1.62779	0.813893	+	0.581014i	$0.197344\pi$
$114$	0	0
$115$	−8.30748	−0.774677
$116$	0	0
$117$	−1.93555	−0.178942
$118$	0	0
$119$	34.6046	3.17220
$120$	0	0
$121$	16.4012	1.49101
$122$	0	0
$123$	−8.71632	−0.785925
$124$	0	0
$125$	−10.5821	−0.946496
$126$	0	0
$127$	−7.88097	−0.699323	−0.349662	−	0.936876i	$-0.613703\pi$
$127$	−7.88097	−0.699323	−0.349662	+	0.936876i	$0.613703\pi$
$128$	0	0
$129$	5.25563	0.462732
$130$	0	0
$131$	−20.2297	−1.76748	−0.883739	−	0.467979i	$-0.844982\pi$
$131$	−20.2297	−1.76748	−0.883739	+	0.467979i	$0.844982\pi$
$132$	0	0
$133$	20.1673	1.74873
$134$	0	0
$135$	−1.25667	−0.108157
$136$	0	0
$137$	−3.73853	−0.319404	−0.159702	−	0.987165i	$-0.551053\pi$
$137$	−3.73853	−0.319404	−0.159702	+	0.987165i	$0.551053\pi$
$138$	0	0
$139$	−13.1822	−1.11810	−0.559049	−	0.829135i	$-0.688833\pi$
$139$	−13.1822	−1.11810	−0.559049	+	0.829135i	$0.688833\pi$
$140$	0	0
$141$	−6.55882	−0.552352
$142$	0	0
$143$	−10.1318	−0.847268
$144$	0	0
$145$	4.69069	0.389540
$146$	0	0
$147$	−13.7500	−1.13408
$148$	0	0
$149$	−16.0892	−1.31808	−0.659040	−	0.752108i	$-0.729037\pi$
$149$	−16.0892	−1.31808	−0.659040	+	0.752108i	$0.729037\pi$
$150$	0	0
$151$	−19.2676	−1.56797	−0.783987	−	0.620778i	$-0.786817\pi$
$151$	−19.2676	−1.56797	−0.783987	+	0.620778i	$0.786817\pi$
$152$	0	0
$153$	7.59670	0.614156
$154$	0	0
$155$	1.59836	0.128383
$156$	0	0
$157$	−8.92368	−0.712187	−0.356094	−	0.934450i	$-0.615892\pi$
$157$	−8.92368	−0.712187	−0.356094	+	0.934450i	$0.615892\pi$
$158$	0	0
$159$	−3.93979	−0.312446
$160$	0	0
$161$	−30.1132	−2.37325
$162$	0	0
$163$	2.94760	0.230874	0.115437	−	0.993315i	$-0.463173\pi$
$163$	2.94760	0.230874	0.115437	+	0.993315i	$0.463173\pi$
$164$	0	0
$165$	−6.57819	−0.512111
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−9.25365	−0.711819
$170$	0	0
$171$	4.42731	0.338564
$172$	0	0
$173$	−12.5486	−0.954055	−0.477028	−	0.878888i	$-0.658286\pi$
$173$	−12.5486	−0.954055	−0.477028	+	0.878888i	$0.658286\pi$
$174$	0	0
$175$	−15.5824	−1.17792
$176$	0	0
$177$	12.3546	0.928631
$178$	0	0
$179$	−3.07676	−0.229968	−0.114984	−	0.993367i	$-0.536682\pi$
$179$	−3.07676	−0.229968	−0.114984	+	0.993367i	$0.536682\pi$
$180$	0	0
$181$	2.50830	0.186440	0.0932201	−	0.995646i	$-0.470284\pi$
$181$	2.50830	0.186440	0.0932201	+	0.995646i	$0.470284\pi$
$182$	0	0
$183$	0.282357	0.0208725
$184$	0	0
$185$	8.63525	0.634877
$186$	0	0
$187$	39.7658	2.90796
$188$	0	0
$189$	−4.55521	−0.331343
$190$	0	0
$191$	3.30849	0.239394	0.119697	−	0.992810i	$-0.461808\pi$
$191$	3.30849	0.239394	0.119697	+	0.992810i	$0.461808\pi$
$192$	0	0
$193$	−18.4096	−1.32515	−0.662577	−	0.748994i	$-0.730537\pi$
$193$	−18.4096	−1.32515	−0.662577	+	0.748994i	$0.730537\pi$
$194$	0	0
$195$	2.43235	0.174184
$196$	0	0
$197$	−13.0745	−0.931520	−0.465760	−	0.884911i	$-0.654219\pi$
$197$	−13.0745	−0.931520	−0.465760	+	0.884911i	$0.654219\pi$
$198$	0	0
$199$	18.8563	1.33669	0.668343	−	0.743853i	$-0.267004\pi$
$199$	18.8563	1.33669	0.668343	+	0.743853i	$0.267004\pi$
$200$	0	0
$201$	−8.25155	−0.582019
$202$	0	0
$203$	17.0029	1.19337
$204$	0	0
$205$	10.9536	0.765029
$206$	0	0
$207$	−6.61071	−0.459476
$208$	0	0
$209$	23.1752	1.60306
$210$	0	0
$211$	−7.39616	−0.509173	−0.254586	−	0.967050i	$-0.581939\pi$
$211$	−7.39616	−0.509173	−0.254586	+	0.967050i	$0.581939\pi$
$212$	0	0
$213$	−1.66362	−0.113989
$214$	0	0
$215$	−6.60460	−0.450430
$216$	0	0
$217$	5.79377	0.393307
$218$	0	0
$219$	3.89727	0.263353
$220$	0	0
$221$	−14.7038	−0.989083
$222$	0	0
$223$	1.81997	0.121874	0.0609371	−	0.998142i	$-0.480591\pi$
$223$	1.81997	0.121874	0.0609371	+	0.998142i	$0.480591\pi$
$224$	0	0
$225$	−3.42078	−0.228052
$226$	0	0
$227$	−21.8780	−1.45209	−0.726045	−	0.687647i	$-0.758644\pi$
$227$	−21.8780	−1.45209	−0.726045	+	0.687647i	$0.758644\pi$
$228$	0	0
$229$	1.04962	0.0693609	0.0346804	−	0.999398i	$-0.488959\pi$
$229$	1.04962	0.0693609	0.0346804	+	0.999398i	$0.488959\pi$
$230$	0	0
$231$	−23.8448	−1.56887
$232$	0	0
$233$	−17.9898	−1.17855	−0.589274	−	0.807933i	$-0.700586\pi$
$233$	−17.9898	−1.17855	−0.589274	+	0.807933i	$0.700586\pi$
$234$	0	0
$235$	8.24227	0.537667
$236$	0	0
$237$	6.21505	0.403711
$238$	0	0
$239$	27.1655	1.75719	0.878596	−	0.477565i	$-0.158481\pi$
$239$	27.1655	1.75719	0.878596	+	0.477565i	$0.158481\pi$
$240$	0	0
$241$	21.7168	1.39890	0.699451	−	0.714680i	$-0.253428\pi$
$241$	21.7168	1.39890	0.699451	+	0.714680i	$0.253428\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	17.2792	1.10393
$246$	0	0
$247$	−8.56927	−0.545249
$248$	0	0
$249$	14.1118	0.894302
$250$	0	0
$251$	−30.4390	−1.92130	−0.960648	−	0.277770i	$-0.910405\pi$
$251$	−30.4390	−1.92130	−0.960648	+	0.277770i	$0.910405\pi$
$252$	0	0
$253$	−34.6045	−2.17557
$254$	0	0
$255$	−9.54655	−0.597828
$256$	0	0
$257$	−23.0748	−1.43937	−0.719684	−	0.694301i	$-0.755714\pi$
$257$	−23.0748	−1.43937	−0.719684	+	0.694301i	$0.755714\pi$
$258$	0	0
$259$	31.3013	1.94497
$260$	0	0
$261$	3.73263	0.231044
$262$	0	0
$263$	−21.5164	−1.32676	−0.663379	−	0.748284i	$-0.730878\pi$
$263$	−21.5164	−1.32676	−0.663379	+	0.748284i	$0.730878\pi$
$264$	0	0
$265$	4.95102	0.304139
$266$	0	0
$267$	6.93566	0.424455
$268$	0	0
$269$	16.2955	0.993554	0.496777	−	0.867878i	$-0.334517\pi$
$269$	16.2955	0.993554	0.496777	+	0.867878i	$0.334517\pi$
$270$	0	0
$271$	25.9830	1.57835	0.789177	−	0.614166i	$-0.210507\pi$
$271$	25.9830	1.57835	0.789177	+	0.614166i	$0.210507\pi$
$272$	0	0
$273$	8.81684	0.533619
$274$	0	0
$275$	−17.9064	−1.07980
$276$	0	0
$277$	−8.26377	−0.496522	−0.248261	−	0.968693i	$-0.579859\pi$
$277$	−8.26377	−0.496522	−0.248261	+	0.968693i	$0.579859\pi$
$278$	0	0
$279$	1.27190	0.0761466
$280$	0	0
$281$	−0.956330	−0.0570499	−0.0285249	−	0.999593i	$-0.509081\pi$
$281$	−0.956330	−0.0570499	−0.0285249	+	0.999593i	$0.509081\pi$
$282$	0	0
$283$	−7.81438	−0.464517	−0.232258	−	0.972654i	$-0.574611\pi$
$283$	−7.81438	−0.464517	−0.232258	+	0.972654i	$0.574611\pi$
$284$	0	0
$285$	−5.56367	−0.329563
$286$	0	0
$287$	39.7047	2.34369
$288$	0	0
$289$	40.7098	2.39469
$290$	0	0
$291$	3.11931	0.182857
$292$	0	0
$293$	−18.0569	−1.05490	−0.527448	−	0.849587i	$-0.676851\pi$
$293$	−18.0569	−1.05490	−0.527448	+	0.849587i	$0.676851\pi$
$294$	0	0
$295$	−15.5257	−0.903942
$296$	0	0
$297$	−5.23461	−0.303743
$298$	0	0
$299$	12.7954	0.739974
$300$	0	0
$301$	−23.9405	−1.37991
$302$	0	0
$303$	−11.8680	−0.681801
$304$	0	0
$305$	−0.354830	−0.0203175
$306$	0	0
$307$	26.9495	1.53809	0.769046	−	0.639193i	$-0.220732\pi$
$307$	26.9495	1.53809	0.769046	+	0.639193i	$0.220732\pi$
$308$	0	0
$309$	−2.31747	−0.131836
$310$	0	0
$311$	7.51738	0.426272	0.213136	−	0.977023i	$-0.431632\pi$
$311$	7.51738	0.426272	0.213136	+	0.977023i	$0.431632\pi$
$312$	0	0
$313$	24.9480	1.41015	0.705073	−	0.709135i	$-0.250914\pi$
$313$	24.9480	1.41015	0.705073	+	0.709135i	$0.250914\pi$
$314$	0	0
$315$	5.72440	0.322534
$316$	0	0
$317$	28.0071	1.57303	0.786517	−	0.617569i	$-0.211882\pi$
$317$	28.0071	1.57303	0.786517	+	0.617569i	$0.211882\pi$
$318$	0	0
$319$	19.5389	1.09397
$320$	0	0
$321$	−1.55158	−0.0866008
$322$	0	0
$323$	33.6329	1.87138
$324$	0	0
$325$	6.62108	0.367272
$326$	0	0
$327$	−5.37197	−0.297071
$328$	0	0
$329$	29.8768	1.64716
$330$	0	0
$331$	−7.36380	−0.404751	−0.202375	−	0.979308i	$-0.564866\pi$
$331$	−7.36380	−0.404751	−0.202375	+	0.979308i	$0.564866\pi$
$332$	0	0
$333$	6.87153	0.376558
$334$	0	0
$335$	10.3695	0.566545
$336$	0	0
$337$	12.2355	0.666512	0.333256	−	0.942836i	$-0.391853\pi$
$337$	12.2355	0.666512	0.333256	+	0.942836i	$0.391853\pi$
$338$	0	0
$339$	−17.3036	−0.939803
$340$	0	0
$341$	6.65789	0.360545
$342$	0	0
$343$	30.7475	1.66021
$344$	0	0
$345$	8.30748	0.447260
$346$	0	0
$347$	−19.3771	−1.04022	−0.520109	−	0.854100i	$-0.674109\pi$
$347$	−19.3771	−1.04022	−0.520109	+	0.854100i	$0.674109\pi$
$348$	0	0
$349$	−7.13099	−0.381713	−0.190857	−	0.981618i	$-0.561127\pi$
$349$	−7.13099	−0.381713	−0.190857	+	0.981618i	$0.561127\pi$
$350$	0	0
$351$	1.93555	0.103312
$352$	0	0
$353$	10.7743	0.573457	0.286729	−	0.958012i	$-0.407432\pi$
$353$	10.7743	0.573457	0.286729	+	0.958012i	$0.407432\pi$
$354$	0	0
$355$	2.09062	0.110959
$356$	0	0
$357$	−34.6046	−1.83147
$358$	0	0
$359$	21.8570	1.15357	0.576784	−	0.816896i	$-0.304307\pi$
$359$	21.8570	1.15357	0.576784	+	0.816896i	$0.304307\pi$
$360$	0	0
$361$	0.601031	0.0316332
$362$	0	0
$363$	−16.4012	−0.860838
$364$	0	0
$365$	−4.89759	−0.256352
$366$	0	0
$367$	0.0341568	0.00178297	0.000891485	−	1.00000i	$-0.499716\pi$
$367$	0.0341568	0.00178297	0.000891485		1.00000i	$0.499716\pi$
$368$	0	0
$369$	8.71632	0.453754
$370$	0	0
$371$	17.9466	0.931740
$372$	0	0
$373$	−4.34690	−0.225074	−0.112537	−	0.993648i	$-0.535898\pi$
$373$	−4.34690	−0.225074	−0.112537	+	0.993648i	$0.535898\pi$
$374$	0	0
$375$	10.5821	0.546460
$376$	0	0
$377$	−7.22469	−0.372090
$378$	0	0
$379$	−32.2351	−1.65581	−0.827904	−	0.560870i	$-0.810467\pi$
$379$	−32.2351	−1.65581	−0.827904	+	0.560870i	$0.810467\pi$
$380$	0	0
$381$	7.88097	0.403754
$382$	0	0
$383$	−23.6117	−1.20650	−0.603250	−	0.797552i	$-0.706128\pi$
$383$	−23.6117	−1.20650	−0.603250	+	0.797552i	$0.706128\pi$
$384$	0	0
$385$	29.9650	1.52716
$386$	0	0
$387$	−5.25563	−0.267159
$388$	0	0
$389$	−31.0376	−1.57367	−0.786834	−	0.617164i	$-0.788281\pi$
$389$	−31.0376	−1.57367	−0.786834	+	0.617164i	$0.788281\pi$
$390$	0	0
$391$	−50.2195	−2.53971
$392$	0	0
$393$	20.2297	1.02045
$394$	0	0
$395$	−7.81028	−0.392978
$396$	0	0
$397$	3.95462	0.198476	0.0992382	−	0.995064i	$-0.468359\pi$
$397$	3.95462	0.198476	0.0992382	+	0.995064i	$0.468359\pi$
$398$	0	0
$399$	−20.1673	−1.00963
$400$	0	0
$401$	19.9560	0.996556	0.498278	−	0.867017i	$-0.333966\pi$
$401$	19.9560	0.996556	0.498278	+	0.867017i	$0.333966\pi$
$402$	0	0
$403$	−2.46182	−0.122632
$404$	0	0
$405$	1.25667	0.0624445
$406$	0	0
$407$	35.9698	1.78296
$408$	0	0
$409$	38.6158	1.90943	0.954714	−	0.297524i	$-0.0961607\pi$
$409$	38.6158	1.90943	0.954714	+	0.297524i	$0.0961607\pi$
$410$	0	0
$411$	3.73853	0.184408
$412$	0	0
$413$	−56.2780	−2.76926
$414$	0	0
$415$	−17.7340	−0.870526
$416$	0	0
$417$	13.1822	0.645534
$418$	0	0
$419$	−4.87110	−0.237969	−0.118984	−	0.992896i	$-0.537964\pi$
$419$	−4.87110	−0.237969	−0.118984	+	0.992896i	$0.537964\pi$
$420$	0	0
$421$	10.1671	0.495512	0.247756	−	0.968822i	$-0.420307\pi$
$421$	10.1671	0.495512	0.247756	+	0.968822i	$0.420307\pi$
$422$	0	0
$423$	6.55882	0.318901
$424$	0	0
$425$	−25.9866	−1.26054
$426$	0	0
$427$	−1.28620	−0.0622435
$428$	0	0
$429$	10.1318	0.489170
$430$	0	0
$431$	34.5825	1.66578	0.832890	−	0.553438i	$-0.186685\pi$
$431$	34.5825	1.66578	0.832890	+	0.553438i	$0.186685\pi$
$432$	0	0
$433$	−30.2208	−1.45232	−0.726160	−	0.687526i	$-0.758697\pi$
$433$	−30.2208	−1.45232	−0.726160	+	0.687526i	$0.758697\pi$
$434$	0	0
$435$	−4.69069	−0.224901
$436$	0	0
$437$	−29.2676	−1.40006
$438$	0	0
$439$	−3.35552	−0.160150	−0.0800751	−	0.996789i	$-0.525516\pi$
$439$	−3.35552	−0.160150	−0.0800751	+	0.996789i	$0.525516\pi$
$440$	0	0
$441$	13.7500	0.654760
$442$	0	0
$443$	−24.9007	−1.18307	−0.591533	−	0.806281i	$-0.701477\pi$
$443$	−24.9007	−1.18307	−0.591533	+	0.806281i	$0.701477\pi$
$444$	0	0
$445$	−8.71584	−0.413170
$446$	0	0
$447$	16.0892	0.760993
$448$	0	0
$449$	23.2157	1.09562	0.547808	−	0.836604i	$-0.315463\pi$
$449$	23.2157	1.09562	0.547808	+	0.836604i	$0.315463\pi$
$450$	0	0
$451$	45.6266	2.14847
$452$	0	0
$453$	19.2676	0.905270
$454$	0	0
$455$	−11.0799	−0.519432
$456$	0	0
$457$	−1.56399	−0.0731602	−0.0365801	−	0.999331i	$-0.511646\pi$
$457$	−1.56399	−0.0731602	−0.0365801	+	0.999331i	$0.511646\pi$
$458$	0	0
$459$	−7.59670	−0.354583
$460$	0	0
$461$	−19.9153	−0.927550	−0.463775	−	0.885953i	$-0.653505\pi$
$461$	−19.9153	−0.927550	−0.463775	+	0.885953i	$0.653505\pi$
$462$	0	0
$463$	−26.9153	−1.25086	−0.625431	−	0.780280i	$-0.715077\pi$
$463$	−26.9153	−1.25086	−0.625431	+	0.780280i	$0.715077\pi$
$464$	0	0
$465$	−1.59836	−0.0741221
$466$	0	0
$467$	39.6163	1.83322	0.916611	−	0.399780i	$-0.130913\pi$
$467$	39.6163	1.83322	0.916611	+	0.399780i	$0.130913\pi$
$468$	0	0
$469$	37.5875	1.73563
$470$	0	0
$471$	8.92368	0.411182
$472$	0	0
$473$	−27.5112	−1.26497
$474$	0	0
$475$	−15.1448	−0.694892
$476$	0	0
$477$	3.93979	0.180391
$478$	0	0
$479$	−38.0876	−1.74027	−0.870134	−	0.492815i	$-0.835968\pi$
$479$	−38.0876	−1.74027	−0.870134	+	0.492815i	$0.835968\pi$
$480$	0	0
$481$	−13.3002	−0.606436
$482$	0	0
$483$	30.1132	1.37020
$484$	0	0
$485$	−3.91994	−0.177995
$486$	0	0
$487$	4.70083	0.213015	0.106508	−	0.994312i	$-0.466033\pi$
$487$	4.70083	0.213015	0.106508	+	0.994312i	$0.466033\pi$
$488$	0	0
$489$	−2.94760	−0.133295
$490$	0	0
$491$	−26.9587	−1.21663	−0.608315	−	0.793696i	$-0.708154\pi$
$491$	−26.9587	−1.21663	−0.608315	+	0.793696i	$0.708154\pi$
$492$	0	0
$493$	28.3556	1.27707
$494$	0	0
$495$	6.57819	0.295667
$496$	0	0
$497$	7.57813	0.339926
$498$	0	0
$499$	29.4450	1.31814	0.659069	−	0.752082i	$-0.270950\pi$
$499$	29.4450	1.31814	0.659069	+	0.752082i	$0.270950\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−27.2438	−1.21474	−0.607371	−	0.794418i	$-0.707776\pi$
$503$	−27.2438	−1.21474	−0.607371	+	0.794418i	$0.707776\pi$
$504$	0	0
$505$	14.9142	0.663674
$506$	0	0
$507$	9.25365	0.410969
$508$	0	0
$509$	5.63468	0.249753	0.124876	−	0.992172i	$-0.460147\pi$
$509$	5.63468	0.249753	0.124876	+	0.992172i	$0.460147\pi$
$510$	0	0
$511$	−17.7529	−0.785342
$512$	0	0
$513$	−4.42731	−0.195470
$514$	0	0
$515$	2.91230	0.128331
$516$	0	0
$517$	34.3329	1.50996
$518$	0	0
$519$	12.5486	0.550824
$520$	0	0
$521$	−6.45179	−0.282658	−0.141329	−	0.989963i	$-0.545138\pi$
$521$	−6.45179	−0.282658	−0.141329	+	0.989963i	$0.545138\pi$
$522$	0	0
$523$	−4.82181	−0.210843	−0.105422	−	0.994428i	$-0.533619\pi$
$523$	−4.82181	−0.210843	−0.105422	+	0.994428i	$0.533619\pi$
$524$	0	0
$525$	15.5824	0.680070
$526$	0	0
$527$	9.66222	0.420893
$528$	0	0
$529$	20.7015	0.900063
$530$	0	0
$531$	−12.3546	−0.536145
$532$	0	0
$533$	−16.8709	−0.730759
$534$	0	0
$535$	1.94983	0.0842984
$536$	0	0
$537$	3.07676	0.132772
$538$	0	0
$539$	71.9757	3.10021
$540$	0	0
$541$	35.7479	1.53692	0.768461	−	0.639896i	$-0.221023\pi$
$541$	35.7479	1.53692	0.768461	+	0.639896i	$0.221023\pi$
$542$	0	0
$543$	−2.50830	−0.107641
$544$	0	0
$545$	6.75080	0.289172
$546$	0	0
$547$	−2.22907	−0.0953084	−0.0476542	−	0.998864i	$-0.515175\pi$
$547$	−2.22907	−0.0953084	−0.0476542	+	0.998864i	$0.515175\pi$
$548$	0	0
$549$	−0.282357	−0.0120507
$550$	0	0
$551$	16.5255	0.704009
$552$	0	0
$553$	−28.3109	−1.20390
$554$	0	0
$555$	−8.63525	−0.366546
$556$	0	0
$557$	14.8215	0.628008	0.314004	−	0.949422i	$-0.398330\pi$
$557$	14.8215	0.628008	0.314004	+	0.949422i	$0.398330\pi$
$558$	0	0
$559$	10.1725	0.430252
$560$	0	0
$561$	−39.7658	−1.67891
$562$	0	0
$563$	21.1799	0.892627	0.446314	−	0.894877i	$-0.352737\pi$
$563$	21.1799	0.892627	0.446314	+	0.894877i	$0.352737\pi$
$564$	0	0
$565$	21.7449	0.914817
$566$	0	0
$567$	4.55521	0.191301
$568$	0	0
$569$	−7.59884	−0.318560	−0.159280	−	0.987233i	$-0.550917\pi$
$569$	−7.59884	−0.318560	−0.159280	+	0.987233i	$0.550917\pi$
$570$	0	0
$571$	10.3154	0.431688	0.215844	−	0.976428i	$-0.430750\pi$
$571$	10.3154	0.431688	0.215844	+	0.976428i	$0.430750\pi$
$572$	0	0
$573$	−3.30849	−0.138214
$574$	0	0
$575$	22.6138	0.943059
$576$	0	0
$577$	16.2672	0.677211	0.338605	−	0.940928i	$-0.390045\pi$
$577$	16.2672	0.677211	0.338605	+	0.940928i	$0.390045\pi$
$578$	0	0
$579$	18.4096	0.765078
$580$	0	0
$581$	−64.2825	−2.66689
$582$	0	0
$583$	20.6233	0.854129
$584$	0	0
$585$	−2.43235	−0.100565
$586$	0	0
$587$	26.5861	1.09733	0.548664	−	0.836043i	$-0.315137\pi$
$587$	26.5861	1.09733	0.548664	+	0.836043i	$0.315137\pi$
$588$	0	0
$589$	5.63108	0.232025
$590$	0	0
$591$	13.0745	0.537813
$592$	0	0
$593$	11.4438	0.469939	0.234969	−	0.972003i	$-0.424501\pi$
$593$	11.4438	0.469939	0.234969	+	0.972003i	$0.424501\pi$
$594$	0	0
$595$	43.4866	1.78277
$596$	0	0
$597$	−18.8563	−0.771736
$598$	0	0
$599$	−17.6317	−0.720410	−0.360205	−	0.932873i	$-0.617293\pi$
$599$	−17.6317	−0.720410	−0.360205	+	0.932873i	$0.617293\pi$
$600$	0	0
$601$	−5.54770	−0.226295	−0.113148	−	0.993578i	$-0.536093\pi$
$601$	−5.54770	−0.226295	−0.113148	+	0.993578i	$0.536093\pi$
$602$	0	0
$603$	8.25155	0.336029
$604$	0	0
$605$	20.6109	0.837951
$606$	0	0
$607$	−21.5641	−0.875259	−0.437630	−	0.899155i	$-0.644182\pi$
$607$	−21.5641	−0.875259	−0.437630	+	0.899155i	$0.644182\pi$
$608$	0	0
$609$	−17.0029	−0.688993
$610$	0	0
$611$	−12.6949	−0.513581
$612$	0	0
$613$	47.5125	1.91901	0.959506	−	0.281687i	$-0.0908939\pi$
$613$	47.5125	1.91901	0.959506	+	0.281687i	$0.0908939\pi$
$614$	0	0
$615$	−10.9536	−0.441690
$616$	0	0
$617$	−3.79509	−0.152785	−0.0763923	−	0.997078i	$-0.524340\pi$
$617$	−3.79509	−0.152785	−0.0763923	+	0.997078i	$0.524340\pi$
$618$	0	0
$619$	−10.0986	−0.405898	−0.202949	−	0.979189i	$-0.565053\pi$
$619$	−10.0986	−0.405898	−0.202949	+	0.979189i	$0.565053\pi$
$620$	0	0
$621$	6.61071	0.265279
$622$	0	0
$623$	−31.5934	−1.26576
$624$	0	0
$625$	3.80561	0.152224
$626$	0	0
$627$	−23.1752	−0.925529
$628$	0	0
$629$	52.2009	2.08139
$630$	0	0
$631$	30.5310	1.21542	0.607709	−	0.794160i	$-0.292089\pi$
$631$	30.5310	1.21542	0.607709	+	0.794160i	$0.292089\pi$
$632$	0	0
$633$	7.39616	0.293971
$634$	0	0
$635$	−9.90379	−0.393020
$636$	0	0
$637$	−26.6137	−1.05447
$638$	0	0
$639$	1.66362	0.0658117
$640$	0	0
$641$	−3.23785	−0.127887	−0.0639437	−	0.997954i	$-0.520368\pi$
$641$	−3.23785	−0.127887	−0.0639437	+	0.997954i	$0.520368\pi$
$642$	0	0
$643$	30.2108	1.19140	0.595698	−	0.803209i	$-0.296876\pi$
$643$	30.2108	1.19140	0.595698	+	0.803209i	$0.296876\pi$
$644$	0	0
$645$	6.60460	0.260056
$646$	0	0
$647$	31.9382	1.25562	0.627810	−	0.778367i	$-0.283952\pi$
$647$	31.9382	1.25562	0.627810	+	0.778367i	$0.283952\pi$
$648$	0	0
$649$	−64.6717	−2.53859
$650$	0	0
$651$	−5.79377	−0.227076
$652$	0	0
$653$	−21.2765	−0.832613	−0.416307	−	0.909224i	$-0.636676\pi$
$653$	−21.2765	−0.832613	−0.416307	+	0.909224i	$0.636676\pi$
$654$	0	0
$655$	−25.4221	−0.993324
$656$	0	0
$657$	−3.89727	−0.152047
$658$	0	0
$659$	20.4493	0.796593	0.398297	−	0.917257i	$-0.369601\pi$
$659$	20.4493	0.796593	0.398297	+	0.917257i	$0.369601\pi$
$660$	0	0
$661$	22.9118	0.891165	0.445582	−	0.895241i	$-0.352997\pi$
$661$	22.9118	0.891165	0.445582	+	0.895241i	$0.352997\pi$
$662$	0	0
$663$	14.7038	0.571047
$664$	0	0
$665$	25.3437	0.982786
$666$	0	0
$667$	−24.6753	−0.955432
$668$	0	0
$669$	−1.81997	−0.0703641
$670$	0	0
$671$	−1.47803	−0.0570588
$672$	0	0
$673$	29.8721	1.15149	0.575743	−	0.817630i	$-0.304713\pi$
$673$	29.8721	1.15149	0.575743	+	0.817630i	$0.304713\pi$
$674$	0	0
$675$	3.42078	0.131666
$676$	0	0
$677$	−10.0786	−0.387353	−0.193676	−	0.981065i	$-0.562041\pi$
$677$	−10.0786	−0.387353	−0.193676	+	0.981065i	$0.562041\pi$
$678$	0	0
$679$	−14.2091	−0.545295
$680$	0	0
$681$	21.8780	0.838365
$682$	0	0
$683$	−17.1938	−0.657901	−0.328951	−	0.944347i	$-0.606695\pi$
$683$	−17.1938	−0.657901	−0.328951	+	0.944347i	$0.606695\pi$
$684$	0	0
$685$	−4.69810	−0.179505
$686$	0	0
$687$	−1.04962	−0.0400455
$688$	0	0
$689$	−7.62566	−0.290515
$690$	0	0
$691$	−4.30843	−0.163900	−0.0819502	−	0.996636i	$-0.526115\pi$
$691$	−4.30843	−0.163900	−0.0819502	+	0.996636i	$0.526115\pi$
$692$	0	0
$693$	23.8448	0.905788
$694$	0	0
$695$	−16.5657	−0.628371
$696$	0	0
$697$	66.2153	2.50808
$698$	0	0
$699$	17.9898	0.680435
$700$	0	0
$701$	−20.4059	−0.770720	−0.385360	−	0.922766i	$-0.625923\pi$
$701$	−20.4059	−0.770720	−0.385360	+	0.922766i	$0.625923\pi$
$702$	0	0
$703$	30.4224	1.14740
$704$	0	0
$705$	−8.24227	−0.310422
$706$	0	0
$707$	54.0614	2.03319
$708$	0	0
$709$	−8.65515	−0.325051	−0.162525	−	0.986704i	$-0.551964\pi$
$709$	−8.65515	−0.325051	−0.162525	+	0.986704i	$0.551964\pi$
$710$	0	0
$711$	−6.21505	−0.233083
$712$	0	0
$713$	−8.40815	−0.314888
$714$	0	0
$715$	−12.7324	−0.476165
$716$	0	0
$717$	−27.1655	−1.01452
$718$	0	0
$719$	37.7603	1.40822	0.704111	−	0.710090i	$-0.251346\pi$
$719$	37.7603	1.40822	0.704111	+	0.710090i	$0.251346\pi$
$720$	0	0
$721$	10.5566	0.393148
$722$	0	0
$723$	−21.7168	−0.807657
$724$	0	0
$725$	−12.7685	−0.474210
$726$	0	0
$727$	−50.5464	−1.87466	−0.937332	−	0.348438i	$-0.886712\pi$
$727$	−50.5464	−1.87466	−0.937332	+	0.348438i	$0.886712\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−39.9254	−1.47669
$732$	0	0
$733$	45.7422	1.68953	0.844763	−	0.535140i	$-0.179741\pi$
$733$	45.7422	1.68953	0.844763	+	0.535140i	$0.179741\pi$
$734$	0	0
$735$	−17.2792	−0.637352
$736$	0	0
$737$	43.1936	1.59106
$738$	0	0
$739$	−9.01671	−0.331685	−0.165842	−	0.986152i	$-0.553034\pi$
$739$	−9.01671	−0.331685	−0.165842	+	0.986152i	$0.553034\pi$
$740$	0	0
$741$	8.56927	0.314800
$742$	0	0
$743$	−24.8707	−0.912416	−0.456208	−	0.889873i	$-0.650793\pi$
$743$	−24.8707	−0.912416	−0.456208	+	0.889873i	$0.650793\pi$
$744$	0	0
$745$	−20.2188	−0.740761
$746$	0	0
$747$	−14.1118	−0.516326
$748$	0	0
$749$	7.06778	0.258251
$750$	0	0
$751$	50.0545	1.82652	0.913258	−	0.407382i	$-0.133558\pi$
$751$	50.0545	1.82652	0.913258	+	0.407382i	$0.133558\pi$
$752$	0	0
$753$	30.4390	1.10926
$754$	0	0
$755$	−24.2130	−0.881201
$756$	0	0
$757$	6.17487	0.224430	0.112215	−	0.993684i	$-0.464206\pi$
$757$	6.17487	0.224430	0.112215	+	0.993684i	$0.464206\pi$
$758$	0	0
$759$	34.6045	1.25606
$760$	0	0
$761$	−29.9236	−1.08473	−0.542365	−	0.840143i	$-0.682471\pi$
$761$	−29.9236	−1.08473	−0.542365	+	0.840143i	$0.682471\pi$
$762$	0	0
$763$	24.4705	0.885890
$764$	0	0
$765$	9.54655	0.345156
$766$	0	0
$767$	23.9130	0.863448
$768$	0	0
$769$	−18.6073	−0.670995	−0.335497	−	0.942041i	$-0.608904\pi$
$769$	−18.6073	−0.670995	−0.335497	+	0.942041i	$0.608904\pi$
$770$	0	0
$771$	23.0748	0.831020
$772$	0	0
$773$	29.9797	1.07830	0.539148	−	0.842211i	$-0.318746\pi$
$773$	29.9797	1.07830	0.539148	+	0.842211i	$0.318746\pi$
$774$	0	0
$775$	−4.35088	−0.156288
$776$	0	0
$777$	−31.3013	−1.12293
$778$	0	0
$779$	38.5898	1.38262
$780$	0	0
$781$	8.70839	0.311611
$782$	0	0
$783$	−3.73263	−0.133393
$784$	0	0
$785$	−11.2141	−0.400250
$786$	0	0
$787$	−37.9894	−1.35417	−0.677087	−	0.735903i	$-0.736758\pi$
$787$	−37.9894	−1.35417	−0.677087	+	0.735903i	$0.736758\pi$
$788$	0	0
$789$	21.5164	0.766004
$790$	0	0
$791$	78.8216	2.80257
$792$	0	0
$793$	0.546517	0.0194074
$794$	0	0
$795$	−4.95102	−0.175595
$796$	0	0
$797$	22.2033	0.786483	0.393242	−	0.919435i	$-0.371354\pi$
$797$	22.2033	0.786483	0.393242	+	0.919435i	$0.371354\pi$
$798$	0	0
$799$	49.8253	1.76269
$800$	0	0
$801$	−6.93566	−0.245059
$802$	0	0
$803$	−20.4007	−0.719925
$804$	0	0
$805$	−37.8424	−1.33377
$806$	0	0
$807$	−16.2955	−0.573629
$808$	0	0
$809$	−15.6829	−0.551380	−0.275690	−	0.961247i	$-0.588906\pi$
$809$	−15.6829	−0.551380	−0.275690	+	0.961247i	$0.588906\pi$
$810$	0	0
$811$	−14.4336	−0.506834	−0.253417	−	0.967357i	$-0.581554\pi$
$811$	−14.4336	−0.506834	−0.253417	+	0.967357i	$0.581554\pi$
$812$	0	0
$813$	−25.9830	−0.911263
$814$	0	0
$815$	3.70417	0.129751
$816$	0	0
$817$	−23.2683	−0.814054
$818$	0	0
$819$	−8.81684	−0.308085
$820$	0	0
$821$	9.25042	0.322842	0.161421	−	0.986886i	$-0.448392\pi$
$821$	9.25042	0.322842	0.161421	+	0.986886i	$0.448392\pi$
$822$	0	0
$823$	15.7006	0.547290	0.273645	−	0.961831i	$-0.411771\pi$
$823$	15.7006	0.547290	0.273645	+	0.961831i	$0.411771\pi$
$824$	0	0
$825$	17.9064	0.623422
$826$	0	0
$827$	−30.4073	−1.05737	−0.528683	−	0.848819i	$-0.677314\pi$
$827$	−30.4073	−1.05737	−0.528683	+	0.848819i	$0.677314\pi$
$828$	0	0
$829$	−38.7371	−1.34539	−0.672697	−	0.739918i	$-0.734864\pi$
$829$	−38.7371	−1.34539	−0.672697	+	0.739918i	$0.734864\pi$
$830$	0	0
$831$	8.26377	0.286667
$832$	0	0
$833$	104.454	3.61913
$834$	0	0
$835$	1.25667	0.0434889
$836$	0	0
$837$	−1.27190	−0.0439632
$838$	0	0
$839$	32.4743	1.12114	0.560568	−	0.828108i	$-0.310583\pi$
$839$	32.4743	1.12114	0.560568	+	0.828108i	$0.310583\pi$
$840$	0	0
$841$	−15.0675	−0.519568
$842$	0	0
$843$	0.956330	0.0329378
$844$	0	0
$845$	−11.6288	−0.400043
$846$	0	0
$847$	74.7108	2.56709
$848$	0	0
$849$	7.81438	0.268189
$850$	0	0
$851$	−45.4257	−1.55717
$852$	0	0
$853$	−21.3706	−0.731716	−0.365858	−	0.930671i	$-0.619224\pi$
$853$	−21.3706	−0.731716	−0.365858	+	0.930671i	$0.619224\pi$
$854$	0	0
$855$	5.56367	0.190273
$856$	0	0
$857$	−38.7853	−1.32488	−0.662440	−	0.749115i	$-0.730479\pi$
$857$	−38.7853	−1.32488	−0.662440	+	0.749115i	$0.730479\pi$
$858$	0	0
$859$	−41.2770	−1.40835	−0.704176	−	0.710025i	$-0.748683\pi$
$859$	−41.2770	−1.40835	−0.704176	+	0.710025i	$0.748683\pi$
$860$	0	0
$861$	−39.7047	−1.35313
$862$	0	0
$863$	−3.03297	−0.103244	−0.0516218	−	0.998667i	$-0.516439\pi$
$863$	−3.03297	−0.103244	−0.0516218	+	0.998667i	$0.516439\pi$
$864$	0	0
$865$	−15.7695	−0.536179
$866$	0	0
$867$	−40.7098	−1.38258
$868$	0	0
$869$	−32.5334	−1.10362
$870$	0	0
$871$	−15.9713	−0.541166
$872$	0	0
$873$	−3.11931	−0.105573
$874$	0	0
$875$	−48.2039	−1.62959
$876$	0	0
$877$	26.2512	0.886439	0.443220	−	0.896413i	$-0.353836\pi$
$877$	26.2512	0.886439	0.443220	+	0.896413i	$0.353836\pi$
$878$	0	0
$879$	18.0569	0.609045
$880$	0	0
$881$	3.69895	0.124621	0.0623104	−	0.998057i	$-0.480153\pi$
$881$	3.69895	0.124621	0.0623104	+	0.998057i	$0.480153\pi$
$882$	0	0
$883$	47.7365	1.60646	0.803230	−	0.595669i	$-0.203113\pi$
$883$	47.7365	1.60646	0.803230	+	0.595669i	$0.203113\pi$
$884$	0	0
$885$	15.5257	0.521891
$886$	0	0
$887$	−1.04390	−0.0350508	−0.0175254	−	0.999846i	$-0.505579\pi$
$887$	−1.04390	−0.0350508	−0.0175254	+	0.999846i	$0.505579\pi$
$888$	0	0
$889$	−35.8995	−1.20403
$890$	0	0
$891$	5.23461	0.175366
$892$	0	0
$893$	29.0379	0.971716
$894$	0	0
$895$	−3.86647	−0.129242
$896$	0	0
$897$	−12.7954	−0.427224
$898$	0	0
$899$	4.74752	0.158339
$900$	0	0
$901$	29.9294	0.997093
$902$	0	0
$903$	23.9405	0.796690
$904$	0	0
$905$	3.15210	0.104779
$906$	0	0
$907$	15.2974	0.507943	0.253971	−	0.967212i	$-0.418263\pi$
$907$	15.2974	0.507943	0.253971	+	0.967212i	$0.418263\pi$
$908$	0	0
$909$	11.8680	0.393638
$910$	0	0
$911$	33.8233	1.12062	0.560308	−	0.828285i	$-0.310683\pi$
$911$	33.8233	1.12062	0.560308	+	0.828285i	$0.310683\pi$
$912$	0	0
$913$	−73.8701	−2.44474
$914$	0	0
$915$	0.354830	0.0117303
$916$	0	0
$917$	−92.1507	−3.04308
$918$	0	0
$919$	18.1404	0.598396	0.299198	−	0.954191i	$-0.403281\pi$
$919$	18.1404	0.598396	0.299198	+	0.954191i	$0.403281\pi$
$920$	0	0
$921$	−26.9495	−0.888018
$922$	0	0
$923$	−3.22001	−0.105988
$924$	0	0
$925$	−23.5060	−0.772872
$926$	0	0
$927$	2.31747	0.0761158
$928$	0	0
$929$	−49.9152	−1.63767	−0.818833	−	0.574032i	$-0.805378\pi$
$929$	−49.9152	−1.63767	−0.818833	+	0.574032i	$0.805378\pi$
$930$	0	0
$931$	60.8753	1.99511
$932$	0	0
$933$	−7.51738	−0.246108
$934$	0	0
$935$	49.9725	1.63427
$936$	0	0
$937$	−45.2081	−1.47689	−0.738443	−	0.674316i	$-0.764438\pi$
$937$	−45.2081	−1.47689	−0.738443	+	0.674316i	$0.764438\pi$
$938$	0	0
$939$	−24.9480	−0.814148
$940$	0	0
$941$	0.416944	0.0135920	0.00679599	−	0.999977i	$-0.497837\pi$
$941$	0.416944	0.0135920	0.00679599	+	0.999977i	$0.497837\pi$
$942$	0	0
$943$	−57.6211	−1.87640
$944$	0	0
$945$	−5.72440	−0.186215
$946$	0	0
$947$	40.5028	1.31616	0.658082	−	0.752946i	$-0.271368\pi$
$947$	40.5028	1.31616	0.658082	+	0.752946i	$0.271368\pi$
$948$	0	0
$949$	7.54336	0.244868
$950$	0	0
$951$	−28.0071	−0.908192
$952$	0	0
$953$	46.8908	1.51894	0.759470	−	0.650542i	$-0.225458\pi$
$953$	46.8908	1.51894	0.759470	+	0.650542i	$0.225458\pi$
$954$	0	0
$955$	4.15768	0.134539
$956$	0	0
$957$	−19.5389	−0.631602
$958$	0	0
$959$	−17.0298	−0.549920
$960$	0	0
$961$	−29.3823	−0.947815
$962$	0	0
$963$	1.55158	0.0499990
$964$	0	0
$965$	−23.1349	−0.744737
$966$	0	0
$967$	28.5237	0.917262	0.458631	−	0.888627i	$-0.348340\pi$
$967$	28.5237	0.917262	0.458631	+	0.888627i	$0.348340\pi$
$968$	0	0
$969$	−33.6329	−1.08044
$970$	0	0
$971$	−32.4613	−1.04173	−0.520867	−	0.853638i	$-0.674391\pi$
$971$	−32.4613	−1.04173	−0.520867	+	0.853638i	$0.674391\pi$
$972$	0	0
$973$	−60.0476	−1.92504
$974$	0	0
$975$	−6.62108	−0.212044
$976$	0	0
$977$	29.7999	0.953382	0.476691	−	0.879071i	$-0.341836\pi$
$977$	29.7999	0.953382	0.476691	+	0.879071i	$0.341836\pi$
$978$	0	0
$979$	−36.3055	−1.16033
$980$	0	0
$981$	5.37197	0.171514
$982$	0	0
$983$	17.3606	0.553717	0.276859	−	0.960911i	$-0.410707\pi$
$983$	17.3606	0.553717	0.276859	+	0.960911i	$0.410707\pi$
$984$	0	0
$985$	−16.4304	−0.523515
$986$	0	0
$987$	−29.8768	−0.950989
$988$	0	0
$989$	34.7434	1.10478
$990$	0	0
$991$	6.60573	0.209838	0.104919	−	0.994481i	$-0.466542\pi$
$991$	6.60573	0.209838	0.104919	+	0.994481i	$0.466542\pi$
$992$	0	0
$993$	7.36380	0.233683
$994$	0	0
$995$	23.6961	0.751218
$996$	0	0
$997$	0.907767	0.0287493	0.0143746	−	0.999897i	$-0.495424\pi$
$997$	0.907767	0.0287493	0.0143746	+	0.999897i	$0.495424\pi$
$998$	0	0
$999$	−6.87153	−0.217406

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bg.1.8		13
4.3	odd	2		4008.2.a.m.1.8	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.m.1.8	✓	13	4.3	odd	2
8016.2.a.bg.1.8		13	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.25667 q^{5} +4.55521 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.25667 q^{5} +4.55521 q^{7} +1.00000 q^{9} +5.23461 q^{11} -1.93555 q^{13} -1.25667 q^{15} +7.59670 q^{17} +4.42731 q^{19} -4.55521 q^{21} -6.61071 q^{23} -3.42078 q^{25} -1.00000 q^{27} +3.73263 q^{29} +1.27190 q^{31} -5.23461 q^{33} +5.72440 q^{35} +6.87153 q^{37} +1.93555 q^{39} +8.71632 q^{41} -5.25563 q^{43} +1.25667 q^{45} +6.55882 q^{47} +13.7500 q^{49} -7.59670 q^{51} +3.93979 q^{53} +6.57819 q^{55} -4.42731 q^{57} -12.3546 q^{59} -0.282357 q^{61} +4.55521 q^{63} -2.43235 q^{65} +8.25155 q^{67} +6.61071 q^{69} +1.66362 q^{71} -3.89727 q^{73} +3.42078 q^{75} +23.8448 q^{77} -6.21505 q^{79} +1.00000 q^{81} -14.1118 q^{83} +9.54655 q^{85} -3.73263 q^{87} -6.93566 q^{89} -8.81684 q^{91} -1.27190 q^{93} +5.56367 q^{95} -3.11931 q^{97} +5.23461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * q^99

Introduction

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