LMFDB - Embedded newform 8016.2.a.z.1.4

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 8x^{6} + 15x^{5} + 19x^{4} - 31x^{3} - 13x^{2} + 14x + 1 $ x^8 - 2x^7 - 8x^6 + 15x^5 + 19x^4 - 31x^3 - 13x^2 + 14*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.688556$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.0425261 q^{5} -1.43576 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.0425261 q^{5} -1.43576 q^{7} +1.00000 q^{9} +1.32740 q^{11} +6.34177 q^{13} +0.0425261 q^{15} -3.59163 q^{17} -2.26764 q^{19} -1.43576 q^{21} -3.90209 q^{23} -4.99819 q^{25} +1.00000 q^{27} -8.64523 q^{29} +1.32647 q^{31} +1.32740 q^{33} -0.0610571 q^{35} +6.05491 q^{37} +6.34177 q^{39} -8.13514 q^{41} +5.53864 q^{43} +0.0425261 q^{45} -10.3504 q^{47} -4.93860 q^{49} -3.59163 q^{51} -7.21232 q^{53} +0.0564492 q^{55} -2.26764 q^{57} +2.51373 q^{59} +1.35856 q^{61} -1.43576 q^{63} +0.269690 q^{65} +1.90823 q^{67} -3.90209 q^{69} -8.26149 q^{71} -1.99542 q^{73} -4.99819 q^{75} -1.90583 q^{77} +5.18426 q^{79} +1.00000 q^{81} +2.42832 q^{83} -0.152738 q^{85} -8.64523 q^{87} +12.0115 q^{89} -9.10523 q^{91} +1.32647 q^{93} -0.0964338 q^{95} -4.77552 q^{97} +1.32740 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + q^5 + 8 * q^9	$ 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + q^5 + 8 * q^9 - 5 * q^11 + q^15 - 7 * q^17 - 24 * q^19 - q^23 + 3 * q^25 + 8 * q^27 - 11 * q^29 - 30 * q^31 - 5 * q^33 - 26 * q^35 + 11 * q^37 + 10 * q^41 - 24 * q^43 + q^45 + 3 * q^47 + 6 * q^49 - 7 * q^51 - 25 * q^53 - 25 * q^55 - 24 * q^57 - 45 * q^59 + 16 * q^61 - 10 * q^65 - 18 * q^67 - q^69 - 21 * q^71 - 8 * q^73 + 3 * q^75 - 18 * q^77 - 10 * q^79 + 8 * q^81 - 7 * q^83 - 11 * q^85 - 11 * q^87 + 26 * q^89 - 15 * q^91 - 30 * q^93 - q^95 - 3 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.0425261	0.0190182	0.00950911	−	0.999955i	$-0.496973\pi$
$5$	0.0425261	0.0190182	0.00950911	+	0.999955i	$0.496973\pi$
$6$	0	0
$7$	−1.43576	−0.542665	−0.271332	−	0.962486i	$-0.587464\pi$
$7$	−1.43576	−0.542665	−0.271332	+	0.962486i	$0.587464\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.32740	0.400227	0.200114	−	0.979773i	$-0.435869\pi$
$11$	1.32740	0.400227	0.200114	+	0.979773i	$0.435869\pi$
$12$	0	0
$13$	6.34177	1.75889	0.879445	−	0.476001i	$-0.157914\pi$
$13$	6.34177	1.75889	0.879445	+	0.476001i	$0.157914\pi$
$14$	0	0
$15$	0.0425261	0.0109802
$16$	0	0
$17$	−3.59163	−0.871099	−0.435549	−	0.900165i	$-0.643446\pi$
$17$	−3.59163	−0.871099	−0.435549	+	0.900165i	$0.643446\pi$
$18$	0	0
$19$	−2.26764	−0.520233	−0.260116	−	0.965577i	$-0.583761\pi$
$19$	−2.26764	−0.520233	−0.260116	+	0.965577i	$0.583761\pi$
$20$	0	0
$21$	−1.43576	−0.313308
$22$	0	0
$23$	−3.90209	−0.813643	−0.406821	−	0.913508i	$-0.633363\pi$
$23$	−3.90209	−0.813643	−0.406821	+	0.913508i	$0.633363\pi$
$24$	0	0
$25$	−4.99819	−0.999638
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.64523	−1.60538	−0.802689	−	0.596397i	$-0.796598\pi$
$29$	−8.64523	−1.60538	−0.802689	+	0.596397i	$0.796598\pi$
$30$	0	0
$31$	1.32647	0.238241	0.119120	−	0.992880i	$-0.461993\pi$
$31$	1.32647	0.238241	0.119120	+	0.992880i	$0.461993\pi$
$32$	0	0
$33$	1.32740	0.231071
$34$	0	0
$35$	−0.0610571	−0.0103205
$36$	0	0
$37$	6.05491	0.995421	0.497710	−	0.867343i	$-0.334174\pi$
$37$	6.05491	0.995421	0.497710	+	0.867343i	$0.334174\pi$
$38$	0	0
$39$	6.34177	1.01550
$40$	0	0
$41$	−8.13514	−1.27050	−0.635248	−	0.772308i	$-0.719102\pi$
$41$	−8.13514	−1.27050	−0.635248	+	0.772308i	$0.719102\pi$
$42$	0	0
$43$	5.53864	0.844634	0.422317	−	0.906448i	$-0.361217\pi$
$43$	5.53864	0.844634	0.422317	+	0.906448i	$0.361217\pi$
$44$	0	0
$45$	0.0425261	0.00633941
$46$	0	0
$47$	−10.3504	−1.50977	−0.754884	−	0.655859i	$-0.772307\pi$
$47$	−10.3504	−1.50977	−0.754884	+	0.655859i	$0.772307\pi$
$48$	0	0
$49$	−4.93860	−0.705515
$50$	0	0
$51$	−3.59163	−0.502929
$52$	0	0
$53$	−7.21232	−0.990688	−0.495344	−	0.868697i	$-0.664958\pi$
$53$	−7.21232	−0.990688	−0.495344	+	0.868697i	$0.664958\pi$
$54$	0	0
$55$	0.0564492	0.00761161
$56$	0	0
$57$	−2.26764	−0.300356
$58$	0	0
$59$	2.51373	0.327260	0.163630	−	0.986522i	$-0.447680\pi$
$59$	2.51373	0.327260	0.163630	+	0.986522i	$0.447680\pi$
$60$	0	0
$61$	1.35856	0.173946	0.0869728	−	0.996211i	$-0.472281\pi$
$61$	1.35856	0.173946	0.0869728	+	0.996211i	$0.472281\pi$
$62$	0	0
$63$	−1.43576	−0.180888
$64$	0	0
$65$	0.269690	0.0334510
$66$	0	0
$67$	1.90823	0.233128	0.116564	−	0.993183i	$-0.462812\pi$
$67$	1.90823	0.233128	0.116564	+	0.993183i	$0.462812\pi$
$68$	0	0
$69$	−3.90209	−0.469757
$70$	0	0
$71$	−8.26149	−0.980458	−0.490229	−	0.871594i	$-0.663087\pi$
$71$	−8.26149	−0.980458	−0.490229	+	0.871594i	$0.663087\pi$
$72$	0	0
$73$	−1.99542	−0.233546	−0.116773	−	0.993159i	$-0.537255\pi$
$73$	−1.99542	−0.233546	−0.116773	+	0.993159i	$0.537255\pi$
$74$	0	0
$75$	−4.99819	−0.577141
$76$	0	0
$77$	−1.90583	−0.217189
$78$	0	0
$79$	5.18426	0.583275	0.291638	−	0.956529i	$-0.405800\pi$
$79$	5.18426	0.583275	0.291638	+	0.956529i	$0.405800\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.42832	0.266543	0.133271	−	0.991080i	$-0.457452\pi$
$83$	2.42832	0.266543	0.133271	+	0.991080i	$0.457452\pi$
$84$	0	0
$85$	−0.152738	−0.0165668
$86$	0	0
$87$	−8.64523	−0.926866
$88$	0	0
$89$	12.0115	1.27321	0.636606	−	0.771189i	$-0.280338\pi$
$89$	12.0115	1.27321	0.636606	+	0.771189i	$0.280338\pi$
$90$	0	0
$91$	−9.10523	−0.954488
$92$	0	0
$93$	1.32647	0.137548
$94$	0	0
$95$	−0.0964338	−0.00989390
$96$	0	0
$97$	−4.77552	−0.484880	−0.242440	−	0.970166i	$-0.577948\pi$
$97$	−4.77552	−0.484880	−0.242440	+	0.970166i	$0.577948\pi$
$98$	0	0
$99$	1.32740	0.133409
$100$	0	0
$101$	3.36079	0.334411	0.167206	−	0.985922i	$-0.446526\pi$
$101$	3.36079	0.334411	0.167206	+	0.985922i	$0.446526\pi$
$102$	0	0
$103$	−14.5605	−1.43469	−0.717347	−	0.696716i	$-0.754644\pi$
$103$	−14.5605	−1.43469	−0.717347	+	0.696716i	$0.754644\pi$
$104$	0	0
$105$	−0.0610571	−0.00595856
$106$	0	0
$107$	8.99330	0.869415	0.434707	−	0.900572i	$-0.356852\pi$
$107$	8.99330	0.869415	0.434707	+	0.900572i	$0.356852\pi$
$108$	0	0
$109$	−5.13840	−0.492170	−0.246085	−	0.969248i	$-0.579144\pi$
$109$	−5.13840	−0.492170	−0.246085	+	0.969248i	$0.579144\pi$
$110$	0	0
$111$	6.05491	0.574706
$112$	0	0
$113$	−2.69638	−0.253654	−0.126827	−	0.991925i	$-0.540479\pi$
$113$	−2.69638	−0.253654	−0.126827	+	0.991925i	$0.540479\pi$
$114$	0	0
$115$	−0.165941	−0.0154740
$116$	0	0
$117$	6.34177	0.586297
$118$	0	0
$119$	5.15671	0.472715
$120$	0	0
$121$	−9.23800	−0.839818
$122$	0	0
$123$	−8.13514	−0.733521
$124$	0	0
$125$	−0.425184	−0.0380296
$126$	0	0
$127$	−15.2503	−1.35325	−0.676623	−	0.736329i	$-0.736557\pi$
$127$	−15.2503	−1.35325	−0.676623	+	0.736329i	$0.736557\pi$
$128$	0	0
$129$	5.53864	0.487650
$130$	0	0
$131$	12.0086	1.04919	0.524597	−	0.851351i	$-0.324216\pi$
$131$	12.0086	1.04919	0.524597	+	0.851351i	$0.324216\pi$
$132$	0	0
$133$	3.25578	0.282312
$134$	0	0
$135$	0.0425261	0.00366006
$136$	0	0
$137$	−8.32804	−0.711513	−0.355756	−	0.934579i	$-0.615777\pi$
$137$	−8.32804	−0.711513	−0.355756	+	0.934579i	$0.615777\pi$
$138$	0	0
$139$	−17.3872	−1.47477	−0.737384	−	0.675474i	$-0.763939\pi$
$139$	−17.3872	−1.47477	−0.737384	+	0.675474i	$0.763939\pi$
$140$	0	0
$141$	−10.3504	−0.871665
$142$	0	0
$143$	8.41808	0.703956
$144$	0	0
$145$	−0.367647	−0.0305315
$146$	0	0
$147$	−4.93860	−0.407329
$148$	0	0
$149$	4.43668	0.363467	0.181733	−	0.983348i	$-0.441829\pi$
$149$	4.43668	0.363467	0.181733	+	0.983348i	$0.441829\pi$
$150$	0	0
$151$	−6.54814	−0.532880	−0.266440	−	0.963852i	$-0.585847\pi$
$151$	−6.54814	−0.532880	−0.266440	+	0.963852i	$0.585847\pi$
$152$	0	0
$153$	−3.59163	−0.290366
$154$	0	0
$155$	0.0564094	0.00453091
$156$	0	0
$157$	24.3666	1.94467	0.972335	−	0.233590i	$-0.0750475\pi$
$157$	24.3666	1.94467	0.972335	+	0.233590i	$0.0750475\pi$
$158$	0	0
$159$	−7.21232	−0.571974
$160$	0	0
$161$	5.60245	0.441535
$162$	0	0
$163$	−2.22713	−0.174442	−0.0872210	−	0.996189i	$-0.527799\pi$
$163$	−2.22713	−0.174442	−0.0872210	+	0.996189i	$0.527799\pi$
$164$	0	0
$165$	0.0564492	0.00439457
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	27.2180	2.09369
$170$	0	0
$171$	−2.26764	−0.173411
$172$	0	0
$173$	8.61490	0.654979	0.327489	−	0.944855i	$-0.393797\pi$
$173$	8.61490	0.654979	0.327489	+	0.944855i	$0.393797\pi$
$174$	0	0
$175$	7.17619	0.542469
$176$	0	0
$177$	2.51373	0.188944
$178$	0	0
$179$	−23.3888	−1.74816	−0.874082	−	0.485778i	$-0.838536\pi$
$179$	−23.3888	−1.74816	−0.874082	+	0.485778i	$0.838536\pi$
$180$	0	0
$181$	20.5359	1.52642	0.763212	−	0.646148i	$-0.223621\pi$
$181$	20.5359	1.52642	0.763212	+	0.646148i	$0.223621\pi$
$182$	0	0
$183$	1.35856	0.100428
$184$	0	0
$185$	0.257491	0.0189311
$186$	0	0
$187$	−4.76755	−0.348638
$188$	0	0
$189$	−1.43576	−0.104436
$190$	0	0
$191$	−5.49934	−0.397919	−0.198959	−	0.980008i	$-0.563756\pi$
$191$	−5.49934	−0.397919	−0.198959	+	0.980008i	$0.563756\pi$
$192$	0	0
$193$	−2.98199	−0.214649	−0.107324	−	0.994224i	$-0.534228\pi$
$193$	−2.98199	−0.214649	−0.107324	+	0.994224i	$0.534228\pi$
$194$	0	0
$195$	0.269690	0.0193129
$196$	0	0
$197$	−1.28940	−0.0918659	−0.0459329	−	0.998945i	$-0.514626\pi$
$197$	−1.28940	−0.0918659	−0.0459329	+	0.998945i	$0.514626\pi$
$198$	0	0
$199$	14.9459	1.05948	0.529742	−	0.848159i	$-0.322289\pi$
$199$	14.9459	1.05948	0.529742	+	0.848159i	$0.322289\pi$
$200$	0	0
$201$	1.90823	0.134596
$202$	0	0
$203$	12.4124	0.871183
$204$	0	0
$205$	−0.345955	−0.0241626
$206$	0	0
$207$	−3.90209	−0.271214
$208$	0	0
$209$	−3.01008	−0.208211
$210$	0	0
$211$	−0.629617	−0.0433446	−0.0216723	−	0.999765i	$-0.506899\pi$
$211$	−0.629617	−0.0433446	−0.0216723	+	0.999765i	$0.506899\pi$
$212$	0	0
$213$	−8.26149	−0.566068
$214$	0	0
$215$	0.235536	0.0160634
$216$	0	0
$217$	−1.90448	−0.129285
$218$	0	0
$219$	−1.99542	−0.134838
$220$	0	0
$221$	−22.7773	−1.53217
$222$	0	0
$223$	−0.817955	−0.0547743	−0.0273872	−	0.999625i	$-0.508719\pi$
$223$	−0.817955	−0.0547743	−0.0273872	+	0.999625i	$0.508719\pi$
$224$	0	0
$225$	−4.99819	−0.333213
$226$	0	0
$227$	13.3312	0.884824	0.442412	−	0.896812i	$-0.354123\pi$
$227$	13.3312	0.884824	0.442412	+	0.896812i	$0.354123\pi$
$228$	0	0
$229$	20.9616	1.38518	0.692592	−	0.721330i	$-0.256469\pi$
$229$	20.9616	1.38518	0.692592	+	0.721330i	$0.256469\pi$
$230$	0	0
$231$	−1.90583	−0.125394
$232$	0	0
$233$	−28.2648	−1.85169	−0.925843	−	0.377908i	$-0.876643\pi$
$233$	−28.2648	−1.85169	−0.925843	+	0.377908i	$0.876643\pi$
$234$	0	0
$235$	−0.440164	−0.0287131
$236$	0	0
$237$	5.18426	0.336754
$238$	0	0
$239$	14.9546	0.967330	0.483665	−	0.875253i	$-0.339305\pi$
$239$	14.9546	0.967330	0.483665	+	0.875253i	$0.339305\pi$
$240$	0	0
$241$	−19.8485	−1.27855	−0.639277	−	0.768977i	$-0.720766\pi$
$241$	−19.8485	−1.27855	−0.639277	+	0.768977i	$0.720766\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.210019	−0.0134176
$246$	0	0
$247$	−14.3809	−0.915032
$248$	0	0
$249$	2.42832	0.153889
$250$	0	0
$251$	−16.1941	−1.02216	−0.511081	−	0.859532i	$-0.670755\pi$
$251$	−16.1941	−1.02216	−0.511081	+	0.859532i	$0.670755\pi$
$252$	0	0
$253$	−5.17965	−0.325642
$254$	0	0
$255$	−0.152738	−0.00956482
$256$	0	0
$257$	9.68627	0.604213	0.302107	−	0.953274i	$-0.402310\pi$
$257$	9.68627	0.604213	0.302107	+	0.953274i	$0.402310\pi$
$258$	0	0
$259$	−8.69337	−0.540180
$260$	0	0
$261$	−8.64523	−0.535126
$262$	0	0
$263$	0.644582	0.0397466	0.0198733	−	0.999803i	$-0.493674\pi$
$263$	0.644582	0.0397466	0.0198733	+	0.999803i	$0.493674\pi$
$264$	0	0
$265$	−0.306711	−0.0188411
$266$	0	0
$267$	12.0115	0.735089
$268$	0	0
$269$	−4.21396	−0.256930	−0.128465	−	0.991714i	$-0.541005\pi$
$269$	−4.21396	−0.256930	−0.128465	+	0.991714i	$0.541005\pi$
$270$	0	0
$271$	4.68253	0.284443	0.142222	−	0.989835i	$-0.454575\pi$
$271$	4.68253	0.284443	0.142222	+	0.989835i	$0.454575\pi$
$272$	0	0
$273$	−9.10523	−0.551074
$274$	0	0
$275$	−6.63462	−0.400083
$276$	0	0
$277$	18.9057	1.13593	0.567965	−	0.823052i	$-0.307731\pi$
$277$	18.9057	1.13593	0.567965	+	0.823052i	$0.307731\pi$
$278$	0	0
$279$	1.32647	0.0794135
$280$	0	0
$281$	−10.3747	−0.618900	−0.309450	−	0.950916i	$-0.600145\pi$
$281$	−10.3747	−0.618900	−0.309450	+	0.950916i	$0.600145\pi$
$282$	0	0
$283$	−24.1749	−1.43705	−0.718524	−	0.695502i	$-0.755182\pi$
$283$	−24.1749	−1.43705	−0.718524	+	0.695502i	$0.755182\pi$
$284$	0	0
$285$	−0.0964338	−0.00571225
$286$	0	0
$287$	11.6801	0.689453
$288$	0	0
$289$	−4.10017	−0.241187
$290$	0	0
$291$	−4.77552	−0.279946
$292$	0	0
$293$	−21.8758	−1.27800	−0.638998	−	0.769209i	$-0.720651\pi$
$293$	−21.8758	−1.27800	−0.638998	+	0.769209i	$0.720651\pi$
$294$	0	0
$295$	0.106899	0.00622391
$296$	0	0
$297$	1.32740	0.0770238
$298$	0	0
$299$	−24.7462	−1.43111
$300$	0	0
$301$	−7.95213	−0.458353
$302$	0	0
$303$	3.36079	0.193072
$304$	0	0
$305$	0.0577741	0.00330814
$306$	0	0
$307$	−7.43863	−0.424545	−0.212273	−	0.977211i	$-0.568086\pi$
$307$	−7.43863	−0.424545	−0.212273	+	0.977211i	$0.568086\pi$
$308$	0	0
$309$	−14.5605	−0.828321
$310$	0	0
$311$	−19.4234	−1.10140	−0.550699	−	0.834704i	$-0.685639\pi$
$311$	−19.4234	−1.10140	−0.550699	+	0.834704i	$0.685639\pi$
$312$	0	0
$313$	−22.3630	−1.26403	−0.632017	−	0.774955i	$-0.717773\pi$
$313$	−22.3630	−1.26403	−0.632017	+	0.774955i	$0.717773\pi$
$314$	0	0
$315$	−0.0610571	−0.00344018
$316$	0	0
$317$	9.19060	0.516195	0.258098	−	0.966119i	$-0.416904\pi$
$317$	9.19060	0.516195	0.258098	+	0.966119i	$0.416904\pi$
$318$	0	0
$319$	−11.4757	−0.642516
$320$	0	0
$321$	8.99330	0.501957
$322$	0	0
$323$	8.14454	0.453174
$324$	0	0
$325$	−31.6974	−1.75825
$326$	0	0
$327$	−5.13840	−0.284154
$328$	0	0
$329$	14.8607	0.819298
$330$	0	0
$331$	0.474828	0.0260989	0.0130495	−	0.999915i	$-0.495846\pi$
$331$	0.474828	0.0260989	0.0130495	+	0.999915i	$0.495846\pi$
$332$	0	0
$333$	6.05491	0.331807
$334$	0	0
$335$	0.0811496	0.00443368
$336$	0	0
$337$	−3.54822	−0.193284	−0.0966419	−	0.995319i	$-0.530810\pi$
$337$	−3.54822	−0.193284	−0.0966419	+	0.995319i	$0.530810\pi$
$338$	0	0
$339$	−2.69638	−0.146447
$340$	0	0
$341$	1.76076	0.0953503
$342$	0	0
$343$	17.1409	0.925523
$344$	0	0
$345$	−0.165941	−0.00893394
$346$	0	0
$347$	−31.5298	−1.69261	−0.846305	−	0.532699i	$-0.821178\pi$
$347$	−31.5298	−1.69261	−0.846305	+	0.532699i	$0.821178\pi$
$348$	0	0
$349$	−21.7445	−1.16396	−0.581978	−	0.813205i	$-0.697721\pi$
$349$	−21.7445	−1.16396	−0.581978	+	0.813205i	$0.697721\pi$
$350$	0	0
$351$	6.34177	0.338498
$352$	0	0
$353$	−35.3529	−1.88164	−0.940821	−	0.338903i	$-0.889944\pi$
$353$	−35.3529	−1.88164	−0.940821	+	0.338903i	$0.889944\pi$
$354$	0	0
$355$	−0.351328	−0.0186466
$356$	0	0
$357$	5.15671	0.272922
$358$	0	0
$359$	−32.1273	−1.69561	−0.847806	−	0.530306i	$-0.822077\pi$
$359$	−32.1273	−1.69561	−0.847806	+	0.530306i	$0.822077\pi$
$360$	0	0
$361$	−13.8578	−0.729358
$362$	0	0
$363$	−9.23800	−0.484869
$364$	0	0
$365$	−0.0848571	−0.00444162
$366$	0	0
$367$	−22.4070	−1.16964	−0.584818	−	0.811164i	$-0.698834\pi$
$367$	−22.4070	−1.16964	−0.584818	+	0.811164i	$0.698834\pi$
$368$	0	0
$369$	−8.13514	−0.423499
$370$	0	0
$371$	10.3551	0.537612
$372$	0	0
$373$	−25.3807	−1.31416	−0.657082	−	0.753819i	$-0.728209\pi$
$373$	−25.3807	−1.31416	−0.657082	+	0.753819i	$0.728209\pi$
$374$	0	0
$375$	−0.425184	−0.0219564
$376$	0	0
$377$	−54.8260	−2.82368
$378$	0	0
$379$	34.9058	1.79299	0.896496	−	0.443052i	$-0.146104\pi$
$379$	34.9058	1.79299	0.896496	+	0.443052i	$0.146104\pi$
$380$	0	0
$381$	−15.2503	−0.781297
$382$	0	0
$383$	19.8940	1.01654	0.508269	−	0.861198i	$-0.330286\pi$
$383$	19.8940	1.01654	0.508269	+	0.861198i	$0.330286\pi$
$384$	0	0
$385$	−0.0810474	−0.00413056
$386$	0	0
$387$	5.53864	0.281545
$388$	0	0
$389$	35.8056	1.81542	0.907708	−	0.419601i	$-0.137830\pi$
$389$	35.8056	1.81542	0.907708	+	0.419601i	$0.137830\pi$
$390$	0	0
$391$	14.0149	0.708763
$392$	0	0
$393$	12.0086	0.605752
$394$	0	0
$395$	0.220466	0.0110929
$396$	0	0
$397$	−12.4834	−0.626523	−0.313261	−	0.949667i	$-0.601422\pi$
$397$	−12.4834	−0.626523	−0.313261	+	0.949667i	$0.601422\pi$
$398$	0	0
$399$	3.25578	0.162993
$400$	0	0
$401$	5.25866	0.262605	0.131302	−	0.991342i	$-0.458084\pi$
$401$	5.25866	0.262605	0.131302	+	0.991342i	$0.458084\pi$
$402$	0	0
$403$	8.41214	0.419039
$404$	0	0
$405$	0.0425261	0.00211314
$406$	0	0
$407$	8.03731	0.398394
$408$	0	0
$409$	39.9463	1.97522	0.987608	−	0.156942i	$-0.0501635\pi$
$409$	39.9463	1.97522	0.987608	+	0.156942i	$0.0501635\pi$
$410$	0	0
$411$	−8.32804	−0.410792
$412$	0	0
$413$	−3.60911	−0.177593
$414$	0	0
$415$	0.103267	0.00506917
$416$	0	0
$417$	−17.3872	−0.851457
$418$	0	0
$419$	−25.2130	−1.23174	−0.615868	−	0.787850i	$-0.711194\pi$
$419$	−25.2130	−1.23174	−0.615868	+	0.787850i	$0.711194\pi$
$420$	0	0
$421$	36.1158	1.76018	0.880088	−	0.474810i	$-0.157483\pi$
$421$	36.1158	1.76018	0.880088	+	0.474810i	$0.157483\pi$
$422$	0	0
$423$	−10.3504	−0.503256
$424$	0	0
$425$	17.9517	0.870784
$426$	0	0
$427$	−1.95056	−0.0943942
$428$	0	0
$429$	8.41808	0.406429
$430$	0	0
$431$	−12.6943	−0.611462	−0.305731	−	0.952118i	$-0.598901\pi$
$431$	−12.6943	−0.611462	−0.305731	+	0.952118i	$0.598901\pi$
$432$	0	0
$433$	−20.1696	−0.969287	−0.484644	−	0.874712i	$-0.661051\pi$
$433$	−20.1696	−0.969287	−0.484644	+	0.874712i	$0.661051\pi$
$434$	0	0
$435$	−0.367647	−0.0176273
$436$	0	0
$437$	8.84855	0.423283
$438$	0	0
$439$	14.6147	0.697520	0.348760	−	0.937212i	$-0.386603\pi$
$439$	14.6147	0.697520	0.348760	+	0.937212i	$0.386603\pi$
$440$	0	0
$441$	−4.93860	−0.235172
$442$	0	0
$443$	−2.15892	−0.102573	−0.0512867	−	0.998684i	$-0.516332\pi$
$443$	−2.15892	−0.102573	−0.0512867	+	0.998684i	$0.516332\pi$
$444$	0	0
$445$	0.510800	0.0242142
$446$	0	0
$447$	4.43668	0.209848
$448$	0	0
$449$	−12.5476	−0.592157	−0.296078	−	0.955164i	$-0.595679\pi$
$449$	−12.5476	−0.592157	−0.296078	+	0.955164i	$0.595679\pi$
$450$	0	0
$451$	−10.7986	−0.508487
$452$	0	0
$453$	−6.54814	−0.307658
$454$	0	0
$455$	−0.387210	−0.0181527
$456$	0	0
$457$	−22.0493	−1.03142	−0.515710	−	0.856763i	$-0.672472\pi$
$457$	−22.0493	−1.03142	−0.515710	+	0.856763i	$0.672472\pi$
$458$	0	0
$459$	−3.59163	−0.167643
$460$	0	0
$461$	22.0305	1.02606	0.513031	−	0.858370i	$-0.328523\pi$
$461$	22.0305	1.02606	0.513031	+	0.858370i	$0.328523\pi$
$462$	0	0
$463$	−2.57748	−0.119785	−0.0598927	−	0.998205i	$-0.519076\pi$
$463$	−2.57748	−0.119785	−0.0598927	+	0.998205i	$0.519076\pi$
$464$	0	0
$465$	0.0564094	0.00261592
$466$	0	0
$467$	−16.9217	−0.783042	−0.391521	−	0.920169i	$-0.628051\pi$
$467$	−16.9217	−0.783042	−0.391521	+	0.920169i	$0.628051\pi$
$468$	0	0
$469$	−2.73976	−0.126510
$470$	0	0
$471$	24.3666	1.12276
$472$	0	0
$473$	7.35201	0.338046
$474$	0	0
$475$	11.3341	0.520044
$476$	0	0
$477$	−7.21232	−0.330229
$478$	0	0
$479$	35.2779	1.61189	0.805945	−	0.591991i	$-0.201658\pi$
$479$	35.2779	1.61189	0.805945	+	0.591991i	$0.201658\pi$
$480$	0	0
$481$	38.3988	1.75084
$482$	0	0
$483$	5.60245	0.254920
$484$	0	0
$485$	−0.203084	−0.00922156
$486$	0	0
$487$	33.8704	1.53482	0.767408	−	0.641160i	$-0.221546\pi$
$487$	33.8704	1.53482	0.767408	+	0.641160i	$0.221546\pi$
$488$	0	0
$489$	−2.22713	−0.100714
$490$	0	0
$491$	12.1142	0.546707	0.273354	−	0.961914i	$-0.411867\pi$
$491$	12.1142	0.546707	0.273354	+	0.961914i	$0.411867\pi$
$492$	0	0
$493$	31.0505	1.39844
$494$	0	0
$495$	0.0564492	0.00253720
$496$	0	0
$497$	11.8615	0.532060
$498$	0	0
$499$	−3.65298	−0.163530	−0.0817649	−	0.996652i	$-0.526056\pi$
$499$	−3.65298	−0.163530	−0.0817649	+	0.996652i	$0.526056\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	19.4850	0.868795	0.434397	−	0.900721i	$-0.356961\pi$
$503$	19.4850	0.868795	0.434397	+	0.900721i	$0.356961\pi$
$504$	0	0
$505$	0.142921	0.00635991
$506$	0	0
$507$	27.2180	1.20879
$508$	0	0
$509$	−31.1745	−1.38179	−0.690893	−	0.722957i	$-0.742782\pi$
$509$	−31.1745	−1.38179	−0.690893	+	0.722957i	$0.742782\pi$
$510$	0	0
$511$	2.86493	0.126737
$512$	0	0
$513$	−2.26764	−0.100119
$514$	0	0
$515$	−0.619203	−0.0272853
$516$	0	0
$517$	−13.7392	−0.604250
$518$	0	0
$519$	8.61490	0.378152
$520$	0	0
$521$	8.94186	0.391750	0.195875	−	0.980629i	$-0.437245\pi$
$521$	8.94186	0.391750	0.195875	+	0.980629i	$0.437245\pi$
$522$	0	0
$523$	−35.2059	−1.53945	−0.769724	−	0.638377i	$-0.779606\pi$
$523$	−35.2059	−1.53945	−0.769724	+	0.638377i	$0.779606\pi$
$524$	0	0
$525$	7.17619	0.313194
$526$	0	0
$527$	−4.76418	−0.207531
$528$	0	0
$529$	−7.77368	−0.337986
$530$	0	0
$531$	2.51373	0.109087
$532$	0	0
$533$	−51.5912	−2.23466
$534$	0	0
$535$	0.382449	0.0165347
$536$	0	0
$537$	−23.3888	−1.00930
$538$	0	0
$539$	−6.55552	−0.282366
$540$	0	0
$541$	−23.7159	−1.01963	−0.509813	−	0.860285i	$-0.670286\pi$
$541$	−23.7159	−1.01963	−0.509813	+	0.860285i	$0.670286\pi$
$542$	0	0
$543$	20.5359	0.881281
$544$	0	0
$545$	−0.218516	−0.00936020
$546$	0	0
$547$	−5.69889	−0.243667	−0.121834	−	0.992551i	$-0.538877\pi$
$547$	−5.69889	−0.243667	−0.121834	+	0.992551i	$0.538877\pi$
$548$	0	0
$549$	1.35856	0.0579819
$550$	0	0
$551$	19.6043	0.835170
$552$	0	0
$553$	−7.44334	−0.316523
$554$	0	0
$555$	0.257491	0.0109299
$556$	0	0
$557$	18.7746	0.795505	0.397752	−	0.917493i	$-0.369790\pi$
$557$	18.7746	0.795505	0.397752	+	0.917493i	$0.369790\pi$
$558$	0	0
$559$	35.1247	1.48562
$560$	0	0
$561$	−4.76755	−0.201286
$562$	0	0
$563$	−0.441505	−0.0186072	−0.00930360	−	0.999957i	$-0.502961\pi$
$563$	−0.441505	−0.0186072	−0.00930360	+	0.999957i	$0.502961\pi$
$564$	0	0
$565$	−0.114666	−0.00482405
$566$	0	0
$567$	−1.43576	−0.0602961
$568$	0	0
$569$	42.1645	1.76763	0.883813	−	0.467840i	$-0.154968\pi$
$569$	42.1645	1.76763	0.883813	+	0.467840i	$0.154968\pi$
$570$	0	0
$571$	23.1003	0.966715	0.483358	−	0.875423i	$-0.339417\pi$
$571$	23.1003	0.966715	0.483358	+	0.875423i	$0.339417\pi$
$572$	0	0
$573$	−5.49934	−0.229738
$574$	0	0
$575$	19.5034	0.813348
$576$	0	0
$577$	2.40176	0.0999865	0.0499933	−	0.998750i	$-0.484080\pi$
$577$	2.40176	0.0999865	0.0499933	+	0.998750i	$0.484080\pi$
$578$	0	0
$579$	−2.98199	−0.123927
$580$	0	0
$581$	−3.48648	−0.144644
$582$	0	0
$583$	−9.57366	−0.396500
$584$	0	0
$585$	0.269690	0.0111503
$586$	0	0
$587$	46.5092	1.91964	0.959820	−	0.280618i	$-0.0905393\pi$
$587$	46.5092	1.91964	0.959820	+	0.280618i	$0.0905393\pi$
$588$	0	0
$589$	−3.00795	−0.123940
$590$	0	0
$591$	−1.28940	−0.0530388
$592$	0	0
$593$	−23.7393	−0.974856	−0.487428	−	0.873163i	$-0.662065\pi$
$593$	−23.7393	−0.974856	−0.487428	+	0.873163i	$0.662065\pi$
$594$	0	0
$595$	0.219295	0.00899020
$596$	0	0
$597$	14.9459	0.611694
$598$	0	0
$599$	17.9809	0.734681	0.367341	−	0.930086i	$-0.380268\pi$
$599$	17.9809	0.734681	0.367341	+	0.930086i	$0.380268\pi$
$600$	0	0
$601$	−7.96849	−0.325041	−0.162521	−	0.986705i	$-0.551962\pi$
$601$	−7.96849	−0.325041	−0.162521	+	0.986705i	$0.551962\pi$
$602$	0	0
$603$	1.90823	0.0777093
$604$	0	0
$605$	−0.392856	−0.0159719
$606$	0	0
$607$	44.0914	1.78961	0.894807	−	0.446452i	$-0.147313\pi$
$607$	44.0914	1.78961	0.894807	+	0.446452i	$0.147313\pi$
$608$	0	0
$609$	12.4124	0.502978
$610$	0	0
$611$	−65.6401	−2.65551
$612$	0	0
$613$	45.8059	1.85008	0.925042	−	0.379865i	$-0.124030\pi$
$613$	45.8059	1.85008	0.925042	+	0.379865i	$0.124030\pi$
$614$	0	0
$615$	−0.345955	−0.0139503
$616$	0	0
$617$	28.7726	1.15834	0.579170	−	0.815207i	$-0.303377\pi$
$617$	28.7726	1.15834	0.579170	+	0.815207i	$0.303377\pi$
$618$	0	0
$619$	−2.02444	−0.0813692	−0.0406846	−	0.999172i	$-0.512954\pi$
$619$	−2.02444	−0.0813692	−0.0406846	+	0.999172i	$0.512954\pi$
$620$	0	0
$621$	−3.90209	−0.156586
$622$	0	0
$623$	−17.2455	−0.690927
$624$	0	0
$625$	24.9729	0.998915
$626$	0	0
$627$	−3.01008	−0.120211
$628$	0	0
$629$	−21.7470	−0.867110
$630$	0	0
$631$	−10.8068	−0.430213	−0.215107	−	0.976591i	$-0.569010\pi$
$631$	−10.8068	−0.430213	−0.215107	+	0.976591i	$0.569010\pi$
$632$	0	0
$633$	−0.629617	−0.0250250
$634$	0	0
$635$	−0.648536	−0.0257364
$636$	0	0
$637$	−31.3195	−1.24092
$638$	0	0
$639$	−8.26149	−0.326819
$640$	0	0
$641$	−38.8620	−1.53496	−0.767479	−	0.641074i	$-0.778489\pi$
$641$	−38.8620	−1.53496	−0.767479	+	0.641074i	$0.778489\pi$
$642$	0	0
$643$	−41.2457	−1.62657	−0.813285	−	0.581865i	$-0.802323\pi$
$643$	−41.2457	−1.62657	−0.813285	+	0.581865i	$0.802323\pi$
$644$	0	0
$645$	0.235536	0.00927424
$646$	0	0
$647$	−25.3249	−0.995625	−0.497813	−	0.867285i	$-0.665863\pi$
$647$	−25.3249	−0.995625	−0.497813	+	0.867285i	$0.665863\pi$
$648$	0	0
$649$	3.33674	0.130978
$650$	0	0
$651$	−1.90448	−0.0746426
$652$	0	0
$653$	24.7451	0.968349	0.484174	−	0.874972i	$-0.339120\pi$
$653$	24.7451	0.968349	0.484174	+	0.874972i	$0.339120\pi$
$654$	0	0
$655$	0.510677	0.0199538
$656$	0	0
$657$	−1.99542	−0.0778486
$658$	0	0
$659$	−40.7120	−1.58591	−0.792956	−	0.609278i	$-0.791459\pi$
$659$	−40.7120	−1.58591	−0.792956	+	0.609278i	$0.791459\pi$
$660$	0	0
$661$	−20.9326	−0.814183	−0.407091	−	0.913387i	$-0.633457\pi$
$661$	−20.9326	−0.814183	−0.407091	+	0.913387i	$0.633457\pi$
$662$	0	0
$663$	−22.7773	−0.884597
$664$	0	0
$665$	0.138456	0.00536907
$666$	0	0
$667$	33.7345	1.30620
$668$	0	0
$669$	−0.817955	−0.0316240
$670$	0	0
$671$	1.80336	0.0696178
$672$	0	0
$673$	−9.57119	−0.368942	−0.184471	−	0.982838i	$-0.559057\pi$
$673$	−9.57119	−0.368942	−0.184471	+	0.982838i	$0.559057\pi$
$674$	0	0
$675$	−4.99819	−0.192380
$676$	0	0
$677$	36.7478	1.41233	0.706166	−	0.708047i	$-0.250423\pi$
$677$	36.7478	1.41233	0.706166	+	0.708047i	$0.250423\pi$
$678$	0	0
$679$	6.85648	0.263127
$680$	0	0
$681$	13.3312	0.510854
$682$	0	0
$683$	23.7496	0.908754	0.454377	−	0.890810i	$-0.349862\pi$
$683$	23.7496	0.908754	0.454377	+	0.890810i	$0.349862\pi$
$684$	0	0
$685$	−0.354159	−0.0135317
$686$	0	0
$687$	20.9616	0.799736
$688$	0	0
$689$	−45.7388	−1.74251
$690$	0	0
$691$	−11.2159	−0.426673	−0.213336	−	0.976979i	$-0.568433\pi$
$691$	−11.2159	−0.426673	−0.213336	+	0.976979i	$0.568433\pi$
$692$	0	0
$693$	−1.90583	−0.0723964
$694$	0	0
$695$	−0.739411	−0.0280475
$696$	0	0
$697$	29.2184	1.10673
$698$	0	0
$699$	−28.2648	−1.06907
$700$	0	0
$701$	−41.4109	−1.56407	−0.782034	−	0.623235i	$-0.785818\pi$
$701$	−41.4109	−1.56407	−0.782034	+	0.623235i	$0.785818\pi$
$702$	0	0
$703$	−13.7304	−0.517850
$704$	0	0
$705$	−0.440164	−0.0165775
$706$	0	0
$707$	−4.82528	−0.181473
$708$	0	0
$709$	28.3485	1.06465	0.532326	−	0.846539i	$-0.321318\pi$
$709$	28.3485	1.06465	0.532326	+	0.846539i	$0.321318\pi$
$710$	0	0
$711$	5.18426	0.194425
$712$	0	0
$713$	−5.17600	−0.193843
$714$	0	0
$715$	0.357988	0.0133880
$716$	0	0
$717$	14.9546	0.558489
$718$	0	0
$719$	22.5573	0.841244	0.420622	−	0.907236i	$-0.361812\pi$
$719$	22.5573	0.841244	0.420622	+	0.907236i	$0.361812\pi$
$720$	0	0
$721$	20.9054	0.778558
$722$	0	0
$723$	−19.8485	−0.738173
$724$	0	0
$725$	43.2105	1.60480
$726$	0	0
$727$	46.6369	1.72967	0.864834	−	0.502058i	$-0.167423\pi$
$727$	46.6369	1.72967	0.864834	+	0.502058i	$0.167423\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−19.8928	−0.735760
$732$	0	0
$733$	−4.84754	−0.179048	−0.0895240	−	0.995985i	$-0.528535\pi$
$733$	−4.84754	−0.179048	−0.0895240	+	0.995985i	$0.528535\pi$
$734$	0	0
$735$	−0.210019	−0.00774668
$736$	0	0
$737$	2.53300	0.0933041
$738$	0	0
$739$	24.1894	0.889822	0.444911	−	0.895575i	$-0.353235\pi$
$739$	24.1894	0.889822	0.444911	+	0.895575i	$0.353235\pi$
$740$	0	0
$741$	−14.3809	−0.528294
$742$	0	0
$743$	−0.571040	−0.0209494	−0.0104747	−	0.999945i	$-0.503334\pi$
$743$	−0.571040	−0.0209494	−0.0104747	+	0.999945i	$0.503334\pi$
$744$	0	0
$745$	0.188674	0.00691250
$746$	0	0
$747$	2.42832	0.0888476
$748$	0	0
$749$	−12.9122	−0.471801
$750$	0	0
$751$	−48.9235	−1.78525	−0.892623	−	0.450804i	$-0.851137\pi$
$751$	−48.9235	−1.78525	−0.892623	+	0.450804i	$0.851137\pi$
$752$	0	0
$753$	−16.1941	−0.590146
$754$	0	0
$755$	−0.278466	−0.0101344
$756$	0	0
$757$	7.51246	0.273045	0.136523	−	0.990637i	$-0.456407\pi$
$757$	7.51246	0.273045	0.136523	+	0.990637i	$0.456407\pi$
$758$	0	0
$759$	−5.17965	−0.188009
$760$	0	0
$761$	3.70719	0.134386	0.0671928	−	0.997740i	$-0.478596\pi$
$761$	3.70719	0.134386	0.0671928	+	0.997740i	$0.478596\pi$
$762$	0	0
$763$	7.37750	0.267083
$764$	0	0
$765$	−0.152738	−0.00552225
$766$	0	0
$767$	15.9415	0.575615
$768$	0	0
$769$	39.1055	1.41018	0.705089	−	0.709118i	$-0.250907\pi$
$769$	39.1055	1.41018	0.705089	+	0.709118i	$0.250907\pi$
$770$	0	0
$771$	9.68627	0.348843
$772$	0	0
$773$	−6.53276	−0.234967	−0.117484	−	0.993075i	$-0.537483\pi$
$773$	−6.53276	−0.234967	−0.117484	+	0.993075i	$0.537483\pi$
$774$	0	0
$775$	−6.62994	−0.238154
$776$	0	0
$777$	−8.69337	−0.311873
$778$	0	0
$779$	18.4476	0.660953
$780$	0	0
$781$	−10.9663	−0.392406
$782$	0	0
$783$	−8.64523	−0.308955
$784$	0	0
$785$	1.03622	0.0369842
$786$	0	0
$787$	−3.20358	−0.114195	−0.0570977	−	0.998369i	$-0.518185\pi$
$787$	−3.20358	−0.114195	−0.0570977	+	0.998369i	$0.518185\pi$
$788$	0	0
$789$	0.644582	0.0229477
$790$	0	0
$791$	3.87135	0.137649
$792$	0	0
$793$	8.61566	0.305951
$794$	0	0
$795$	−0.306711	−0.0108779
$796$	0	0
$797$	27.5555	0.976065	0.488032	−	0.872826i	$-0.337715\pi$
$797$	27.5555	0.976065	0.488032	+	0.872826i	$0.337715\pi$
$798$	0	0
$799$	37.1750	1.31516
$800$	0	0
$801$	12.0115	0.424404
$802$	0	0
$803$	−2.64872	−0.0934713
$804$	0	0
$805$	0.238250	0.00839722
$806$	0	0
$807$	−4.21396	−0.148338
$808$	0	0
$809$	−24.2387	−0.852189	−0.426095	−	0.904679i	$-0.640111\pi$
$809$	−24.2387	−0.852189	−0.426095	+	0.904679i	$0.640111\pi$
$810$	0	0
$811$	−24.8023	−0.870927	−0.435463	−	0.900206i	$-0.643416\pi$
$811$	−24.8023	−0.870927	−0.435463	+	0.900206i	$0.643416\pi$
$812$	0	0
$813$	4.68253	0.164223
$814$	0	0
$815$	−0.0947109	−0.00331758
$816$	0	0
$817$	−12.5596	−0.439406
$818$	0	0
$819$	−9.10523	−0.318163
$820$	0	0
$821$	21.0477	0.734569	0.367284	−	0.930109i	$-0.380288\pi$
$821$	21.0477	0.734569	0.367284	+	0.930109i	$0.380288\pi$
$822$	0	0
$823$	50.2834	1.75277	0.876384	−	0.481612i	$-0.159949\pi$
$823$	50.2834	1.75277	0.876384	+	0.481612i	$0.159949\pi$
$824$	0	0
$825$	−6.63462	−0.230988
$826$	0	0
$827$	−15.5012	−0.539029	−0.269514	−	0.962996i	$-0.586863\pi$
$827$	−15.5012	−0.539029	−0.269514	+	0.962996i	$0.586863\pi$
$828$	0	0
$829$	32.2024	1.11843	0.559217	−	0.829021i	$-0.311102\pi$
$829$	32.2024	1.11843	0.559217	+	0.829021i	$0.311102\pi$
$830$	0	0
$831$	18.9057	0.655830
$832$	0	0
$833$	17.7377	0.614573
$834$	0	0
$835$	−0.0425261	−0.00147167
$836$	0	0
$837$	1.32647	0.0458494
$838$	0	0
$839$	−28.7256	−0.991716	−0.495858	−	0.868403i	$-0.665146\pi$
$839$	−28.7256	−0.991716	−0.495858	+	0.868403i	$0.665146\pi$
$840$	0	0
$841$	45.7400	1.57724
$842$	0	0
$843$	−10.3747	−0.357322
$844$	0	0
$845$	1.15747	0.0398183
$846$	0	0
$847$	13.2635	0.455740
$848$	0	0
$849$	−24.1749	−0.829681
$850$	0	0
$851$	−23.6268	−0.809917
$852$	0	0
$853$	−48.4773	−1.65983	−0.829915	−	0.557889i	$-0.811611\pi$
$853$	−48.4773	−1.65983	−0.829915	+	0.557889i	$0.811611\pi$
$854$	0	0
$855$	−0.0964338	−0.00329797
$856$	0	0
$857$	34.1133	1.16529	0.582644	−	0.812728i	$-0.302018\pi$
$857$	34.1133	1.16529	0.582644	+	0.812728i	$0.302018\pi$
$858$	0	0
$859$	34.5614	1.17922	0.589610	−	0.807688i	$-0.299281\pi$
$859$	34.5614	1.17922	0.589610	+	0.807688i	$0.299281\pi$
$860$	0	0
$861$	11.6801	0.398056
$862$	0	0
$863$	−29.0199	−0.987848	−0.493924	−	0.869505i	$-0.664438\pi$
$863$	−29.0199	−0.987848	−0.493924	+	0.869505i	$0.664438\pi$
$864$	0	0
$865$	0.366358	0.0124565
$866$	0	0
$867$	−4.10017	−0.139249
$868$	0	0
$869$	6.88161	0.233443
$870$	0	0
$871$	12.1016	0.410046
$872$	0	0
$873$	−4.77552	−0.161627
$874$	0	0
$875$	0.610460	0.0206373
$876$	0	0
$877$	7.40606	0.250085	0.125042	−	0.992151i	$-0.460093\pi$
$877$	7.40606	0.250085	0.125042	+	0.992151i	$0.460093\pi$
$878$	0	0
$879$	−21.8758	−0.737851
$880$	0	0
$881$	16.5531	0.557689	0.278844	−	0.960336i	$-0.410049\pi$
$881$	16.5531	0.557689	0.278844	+	0.960336i	$0.410049\pi$
$882$	0	0
$883$	6.16318	0.207408	0.103704	−	0.994608i	$-0.466931\pi$
$883$	6.16318	0.207408	0.103704	+	0.994608i	$0.466931\pi$
$884$	0	0
$885$	0.106899	0.00359338
$886$	0	0
$887$	−43.7337	−1.46843	−0.734217	−	0.678914i	$-0.762451\pi$
$887$	−43.7337	−1.46843	−0.734217	+	0.678914i	$0.762451\pi$
$888$	0	0
$889$	21.8957	0.734360
$890$	0	0
$891$	1.32740	0.0444697
$892$	0	0
$893$	23.4711	0.785430
$894$	0	0
$895$	−0.994635	−0.0332470
$896$	0	0
$897$	−24.7462	−0.826250
$898$	0	0
$899$	−11.4676	−0.382466
$900$	0	0
$901$	25.9040	0.862987
$902$	0	0
$903$	−7.95213	−0.264630
$904$	0	0
$905$	0.873312	0.0290299
$906$	0	0
$907$	37.8585	1.25707	0.628535	−	0.777781i	$-0.283655\pi$
$907$	37.8585	1.25707	0.628535	+	0.777781i	$0.283655\pi$
$908$	0	0
$909$	3.36079	0.111470
$910$	0	0
$911$	−38.7345	−1.28333	−0.641666	−	0.766984i	$-0.721757\pi$
$911$	−38.7345	−1.28333	−0.641666	+	0.766984i	$0.721757\pi$
$912$	0	0
$913$	3.22336	0.106678
$914$	0	0
$915$	0.0577741	0.00190995
$916$	0	0
$917$	−17.2414	−0.569361
$918$	0	0
$919$	−2.34633	−0.0773983	−0.0386992	−	0.999251i	$-0.512321\pi$
$919$	−2.34633	−0.0773983	−0.0386992	+	0.999251i	$0.512321\pi$
$920$	0	0
$921$	−7.43863	−0.245111
$922$	0	0
$923$	−52.3924	−1.72452
$924$	0	0
$925$	−30.2636	−0.995061
$926$	0	0
$927$	−14.5605	−0.478231
$928$	0	0
$929$	6.20862	0.203698	0.101849	−	0.994800i	$-0.467524\pi$
$929$	6.20862	0.203698	0.101849	+	0.994800i	$0.467524\pi$
$930$	0	0
$931$	11.1990	0.367032
$932$	0	0
$933$	−19.4234	−0.635892
$934$	0	0
$935$	−0.202745	−0.00663047
$936$	0	0
$937$	−5.40137	−0.176455	−0.0882276	−	0.996100i	$-0.528120\pi$
$937$	−5.40137	−0.176455	−0.0882276	+	0.996100i	$0.528120\pi$
$938$	0	0
$939$	−22.3630	−0.729790
$940$	0	0
$941$	14.6637	0.478023	0.239011	−	0.971017i	$-0.423177\pi$
$941$	14.6637	0.478023	0.239011	+	0.971017i	$0.423177\pi$
$942$	0	0
$943$	31.7441	1.03373
$944$	0	0
$945$	−0.0610571	−0.00198619
$946$	0	0
$947$	−21.0860	−0.685204	−0.342602	−	0.939481i	$-0.611308\pi$
$947$	−21.0860	−0.685204	−0.342602	+	0.939481i	$0.611308\pi$
$948$	0	0
$949$	−12.6545	−0.410781
$950$	0	0
$951$	9.19060	0.298026
$952$	0	0
$953$	−29.6438	−0.960256	−0.480128	−	0.877198i	$-0.659410\pi$
$953$	−29.6438	−0.960256	−0.480128	+	0.877198i	$0.659410\pi$
$954$	0	0
$955$	−0.233865	−0.00756771
$956$	0	0
$957$	−11.4757	−0.370957
$958$	0	0
$959$	11.9570	0.386113
$960$	0	0
$961$	−29.2405	−0.943241
$962$	0	0
$963$	8.99330	0.289805
$964$	0	0
$965$	−0.126812	−0.00408223
$966$	0	0
$967$	−59.0265	−1.89816	−0.949082	−	0.315030i	$-0.897985\pi$
$967$	−59.0265	−1.89816	−0.949082	+	0.315030i	$0.897985\pi$
$968$	0	0
$969$	8.14454	0.261640
$970$	0	0
$971$	−3.07234	−0.0985961	−0.0492980	−	0.998784i	$-0.515698\pi$
$971$	−3.07234	−0.0985961	−0.0492980	+	0.998784i	$0.515698\pi$
$972$	0	0
$973$	24.9639	0.800304
$974$	0	0
$975$	−31.6974	−1.01513
$976$	0	0
$977$	−6.03476	−0.193069	−0.0965345	−	0.995330i	$-0.530776\pi$
$977$	−6.03476	−0.193069	−0.0965345	+	0.995330i	$0.530776\pi$
$978$	0	0
$979$	15.9440	0.509574
$980$	0	0
$981$	−5.13840	−0.164057
$982$	0	0
$983$	−9.30881	−0.296905	−0.148453	−	0.988920i	$-0.547429\pi$
$983$	−9.30881	−0.296905	−0.148453	+	0.988920i	$0.547429\pi$
$984$	0	0
$985$	−0.0548331	−0.00174713
$986$	0	0
$987$	14.8607	0.473022
$988$	0	0
$989$	−21.6123	−0.687230
$990$	0	0
$991$	−19.7998	−0.628961	−0.314481	−	0.949264i	$-0.601830\pi$
$991$	−19.7998	−0.628961	−0.314481	+	0.949264i	$0.601830\pi$
$992$	0	0
$993$	0.474828	0.0150682
$994$	0	0
$995$	0.635589	0.0201495
$996$	0	0
$997$	22.0272	0.697608	0.348804	−	0.937196i	$-0.386588\pi$
$997$	22.0272	0.697608	0.348804	+	0.937196i	$0.386588\pi$
$998$	0	0
$999$	6.05491	0.191569

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.z.1.4		8
4.3	odd	2		501.2.a.d.1.8	✓	8
12.11	even	2		1503.2.a.f.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.d.1.8	✓	8	4.3	odd	2
1503.2.a.f.1.1		8	12.11	even	2
8016.2.a.z.1.4		8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.0425261 q^{5} -1.43576 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.0425261 q^{5} -1.43576 q^{7} +1.00000 q^{9} +1.32740 q^{11} +6.34177 q^{13} +0.0425261 q^{15} -3.59163 q^{17} -2.26764 q^{19} -1.43576 q^{21} -3.90209 q^{23} -4.99819 q^{25} +1.00000 q^{27} -8.64523 q^{29} +1.32647 q^{31} +1.32740 q^{33} -0.0610571 q^{35} +6.05491 q^{37} +6.34177 q^{39} -8.13514 q^{41} +5.53864 q^{43} +0.0425261 q^{45} -10.3504 q^{47} -4.93860 q^{49} -3.59163 q^{51} -7.21232 q^{53} +0.0564492 q^{55} -2.26764 q^{57} +2.51373 q^{59} +1.35856 q^{61} -1.43576 q^{63} +0.269690 q^{65} +1.90823 q^{67} -3.90209 q^{69} -8.26149 q^{71} -1.99542 q^{73} -4.99819 q^{75} -1.90583 q^{77} +5.18426 q^{79} +1.00000 q^{81} +2.42832 q^{83} -0.152738 q^{85} -8.64523 q^{87} +12.0115 q^{89} -9.10523 q^{91} +1.32647 q^{93} -0.0964338 q^{95} -4.77552 q^{97} +1.32740 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + q^5 + 8 * q^9	\( 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + q^5 + 8 * q^9 - 5 * q^11 + q^15 - 7 * q^17 - 24 * q^19 - q^23 + 3 * q^25 + 8 * q^27 - 11 * q^29 - 30 * q^31 - 5 * q^33 - 26 * q^35 + 11 * q^37 + 10 * q^41 - 24 * q^43 + q^45 + 3 * q^47 + 6 * q^49 - 7 * q^51 - 25 * q^53 - 25 * q^55 - 24 * q^57 - 45 * q^59 + 16 * q^61 - 10 * q^65 - 18 * q^67 - q^69 - 21 * q^71 - 8 * q^73 + 3 * q^75 - 18 * q^77 - 10 * q^79 + 8 * q^81 - 7 * q^83 - 11 * q^85 - 11 * q^87 + 26 * q^89 - 15 * q^91 - 30 * q^93 - q^95 - 3 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more