LMFDB - Embedded newform 8016.2.a.u.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:	$0$
Dimension:	$5$
Coefficient field:	5.5.38569.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 5x^{3} + 4x - 1 $ x^5 - 5x^3 + 4x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			$0.275834$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.81885 q^{5} +0.619101 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.81885 q^{5} +0.619101 q^{7} +1.00000 q^{9} +4.67008 q^{11} +2.78748 q^{13} +1.81885 q^{15} -0.683131 q^{17} +5.92392 q^{19} +0.619101 q^{21} +7.31476 q^{23} -1.69178 q^{25} +1.00000 q^{27} +7.09331 q^{29} +7.05098 q^{31} +4.67008 q^{33} +1.12605 q^{35} -7.99322 q^{37} +2.78748 q^{39} +0.0577645 q^{41} +8.15534 q^{43} +1.81885 q^{45} -1.68571 q^{47} -6.61671 q^{49} -0.683131 q^{51} -10.0094 q^{53} +8.49418 q^{55} +5.92392 q^{57} -11.9921 q^{59} +4.90480 q^{61} +0.619101 q^{63} +5.07002 q^{65} -11.0368 q^{67} +7.31476 q^{69} -15.2933 q^{71} -5.47053 q^{73} -1.69178 q^{75} +2.89125 q^{77} +8.58093 q^{79} +1.00000 q^{81} -1.86623 q^{83} -1.24251 q^{85} +7.09331 q^{87} -5.14998 q^{89} +1.72573 q^{91} +7.05098 q^{93} +10.7747 q^{95} +10.4675 q^{97} +4.67008 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9} + 7 q^{11} - 8 q^{13} - q^{15} + 5 q^{17} + 20 q^{19} + 4 q^{21} - q^{23} - 6 q^{25} + 5 q^{27} - 5 q^{29} + 18 q^{31} + 7 q^{33} + 6 q^{35} - 17 q^{37} - 8 q^{39} + 6 q^{41} + 10 q^{43} - q^{45} + 3 q^{47} - 13 q^{49} + 5 q^{51} + 15 q^{53} + 9 q^{55} + 20 q^{57} + 17 q^{59} - 2 q^{61} + 4 q^{63} + 26 q^{65} + 24 q^{67} - q^{69} - q^{71} + 2 q^{73} - 6 q^{75} + 26 q^{77} + 6 q^{79} + 5 q^{81} + 7 q^{83} - 11 q^{85} - 5 q^{87} + 15 q^{91} + 18 q^{93} - q^{95} - 13 q^{97} + 7 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9 + 7 * q^11 - 8 * q^13 - q^15 + 5 * q^17 + 20 * q^19 + 4 * q^21 - q^23 - 6 * q^25 + 5 * q^27 - 5 * q^29 + 18 * q^31 + 7 * q^33 + 6 * q^35 - 17 * q^37 - 8 * q^39 + 6 * q^41 + 10 * q^43 - q^45 + 3 * q^47 - 13 * q^49 + 5 * q^51 + 15 * q^53 + 9 * q^55 + 20 * q^57 + 17 * q^59 - 2 * q^61 + 4 * q^63 + 26 * q^65 + 24 * q^67 - q^69 - q^71 + 2 * q^73 - 6 * q^75 + 26 * q^77 + 6 * q^79 + 5 * q^81 + 7 * q^83 - 11 * q^85 - 5 * q^87 + 15 * q^91 + 18 * q^93 - q^95 - 13 * q^97 + 7 * 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
$5$	1.81885	0.813415	0.406707	−	0.913559i	$-0.366677\pi$
$5$	1.81885	0.813415	0.406707	+	0.913559i	$0.366677\pi$
$6$	0	0
$7$	0.619101	0.233998	0.116999	−	0.993132i	$-0.462673\pi$
$7$	0.619101	0.233998	0.116999	+	0.993132i	$0.462673\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.67008	1.40808	0.704042	−	0.710159i	$-0.251377\pi$
$11$	4.67008	1.40808	0.704042	+	0.710159i	$0.251377\pi$
$12$	0	0
$13$	2.78748	0.773109	0.386554	−	0.922267i	$-0.373665\pi$
$13$	2.78748	0.773109	0.386554	+	0.922267i	$0.373665\pi$
$14$	0	0
$15$	1.81885	0.469625
$16$	0	0
$17$	−0.683131	−0.165684	−0.0828418	−	0.996563i	$-0.526400\pi$
$17$	−0.683131	−0.165684	−0.0828418	+	0.996563i	$0.526400\pi$
$18$	0	0
$19$	5.92392	1.35904	0.679520	−	0.733657i	$-0.262188\pi$
$19$	5.92392	1.35904	0.679520	+	0.733657i	$0.262188\pi$
$20$	0	0
$21$	0.619101	0.135099
$22$	0	0
$23$	7.31476	1.52523	0.762617	−	0.646851i	$-0.223914\pi$
$23$	7.31476	1.52523	0.762617	+	0.646851i	$0.223914\pi$
$24$	0	0
$25$	−1.69178	−0.338357
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.09331	1.31720	0.658598	−	0.752495i	$-0.271150\pi$
$29$	7.09331	1.31720	0.658598	+	0.752495i	$0.271150\pi$
$30$	0	0
$31$	7.05098	1.26639	0.633197	−	0.773991i	$-0.281742\pi$
$31$	7.05098	1.26639	0.633197	+	0.773991i	$0.281742\pi$
$32$	0	0
$33$	4.67008	0.812957
$34$	0	0
$35$	1.12605	0.190337
$36$	0	0
$37$	−7.99322	−1.31408	−0.657039	−	0.753857i	$-0.728191\pi$
$37$	−7.99322	−1.31408	−0.657039	+	0.753857i	$0.728191\pi$
$38$	0	0
$39$	2.78748	0.446355
$40$	0	0
$41$	0.0577645	0.00902130	0.00451065	−	0.999990i	$-0.498564\pi$
$41$	0.0577645	0.00902130	0.00451065	+	0.999990i	$0.498564\pi$
$42$	0	0
$43$	8.15534	1.24368	0.621839	−	0.783146i	$-0.286386\pi$
$43$	8.15534	1.24368	0.621839	+	0.783146i	$0.286386\pi$
$44$	0	0
$45$	1.81885	0.271138
$46$	0	0
$47$	−1.68571	−0.245887	−0.122943	−	0.992414i	$-0.539233\pi$
$47$	−1.68571	−0.245887	−0.122943	+	0.992414i	$0.539233\pi$
$48$	0	0
$49$	−6.61671	−0.945245
$50$	0	0
$51$	−0.683131	−0.0956575
$52$	0	0
$53$	−10.0094	−1.37490	−0.687451	−	0.726231i	$-0.741270\pi$
$53$	−10.0094	−1.37490	−0.687451	+	0.726231i	$0.741270\pi$
$54$	0	0
$55$	8.49418	1.14536
$56$	0	0
$57$	5.92392	0.784642
$58$	0	0
$59$	−11.9921	−1.56124	−0.780621	−	0.625005i	$-0.785097\pi$
$59$	−11.9921	−1.56124	−0.780621	+	0.625005i	$0.785097\pi$
$60$	0	0
$61$	4.90480	0.627995	0.313998	−	0.949424i	$-0.398332\pi$
$61$	4.90480	0.627995	0.313998	+	0.949424i	$0.398332\pi$
$62$	0	0
$63$	0.619101	0.0779994
$64$	0	0
$65$	5.07002	0.628858
$66$	0	0
$67$	−11.0368	−1.34837	−0.674183	−	0.738565i	$-0.735504\pi$
$67$	−11.0368	−1.34837	−0.674183	+	0.738565i	$0.735504\pi$
$68$	0	0
$69$	7.31476	0.880594
$70$	0	0
$71$	−15.2933	−1.81498	−0.907488	−	0.420078i	$-0.862003\pi$
$71$	−15.2933	−1.81498	−0.907488	+	0.420078i	$0.862003\pi$
$72$	0	0
$73$	−5.47053	−0.640277	−0.320139	−	0.947371i	$-0.603729\pi$
$73$	−5.47053	−0.640277	−0.320139	+	0.947371i	$0.603729\pi$
$74$	0	0
$75$	−1.69178	−0.195350
$76$	0	0
$77$	2.89125	0.329489
$78$	0	0
$79$	8.58093	0.965430	0.482715	−	0.875778i	$-0.339651\pi$
$79$	8.58093	0.965430	0.482715	+	0.875778i	$0.339651\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.86623	−0.204846	−0.102423	−	0.994741i	$-0.532660\pi$
$83$	−1.86623	−0.204846	−0.102423	+	0.994741i	$0.532660\pi$
$84$	0	0
$85$	−1.24251	−0.134769
$86$	0	0
$87$	7.09331	0.760483
$88$	0	0
$89$	−5.14998	−0.545897	−0.272948	−	0.962029i	$-0.587999\pi$
$89$	−5.14998	−0.545897	−0.272948	+	0.962029i	$0.587999\pi$
$90$	0	0
$91$	1.72573	0.180906
$92$	0	0
$93$	7.05098	0.731153
$94$	0	0
$95$	10.7747	1.10546
$96$	0	0
$97$	10.4675	1.06282	0.531408	−	0.847116i	$-0.321663\pi$
$97$	10.4675	1.06282	0.531408	+	0.847116i	$0.321663\pi$
$98$	0	0
$99$	4.67008	0.469361
$100$	0	0
$101$	−4.04032	−0.402027	−0.201014	−	0.979588i	$-0.564424\pi$
$101$	−4.04032	−0.402027	−0.201014	+	0.979588i	$0.564424\pi$
$102$	0	0
$103$	−12.5828	−1.23982	−0.619910	−	0.784673i	$-0.712831\pi$
$103$	−12.5828	−1.23982	−0.619910	+	0.784673i	$0.712831\pi$
$104$	0	0
$105$	1.12605	0.109891
$106$	0	0
$107$	−3.37816	−0.326579	−0.163289	−	0.986578i	$-0.552210\pi$
$107$	−3.37816	−0.326579	−0.163289	+	0.986578i	$0.552210\pi$
$108$	0	0
$109$	−16.0236	−1.53478	−0.767390	−	0.641180i	$-0.778445\pi$
$109$	−16.0236	−1.53478	−0.767390	+	0.641180i	$0.778445\pi$
$110$	0	0
$111$	−7.99322	−0.758683
$112$	0	0
$113$	6.17791	0.581169	0.290584	−	0.956849i	$-0.406150\pi$
$113$	6.17791	0.581169	0.290584	+	0.956849i	$0.406150\pi$
$114$	0	0
$115$	13.3045	1.24065
$116$	0	0
$117$	2.78748	0.257703
$118$	0	0
$119$	−0.422927	−0.0387696
$120$	0	0
$121$	10.8097	0.982698
$122$	0	0
$123$	0.0577645	0.00520845
$124$	0	0
$125$	−12.1714	−1.08864
$126$	0	0
$127$	−8.59133	−0.762357	−0.381179	−	0.924501i	$-0.624482\pi$
$127$	−8.59133	−0.762357	−0.381179	+	0.924501i	$0.624482\pi$
$128$	0	0
$129$	8.15534	0.718037
$130$	0	0
$131$	5.93595	0.518626	0.259313	−	0.965793i	$-0.416504\pi$
$131$	5.93595	0.518626	0.259313	+	0.965793i	$0.416504\pi$
$132$	0	0
$133$	3.66750	0.318013
$134$	0	0
$135$	1.81885	0.156542
$136$	0	0
$137$	18.7914	1.60545	0.802727	−	0.596346i	$-0.203382\pi$
$137$	18.7914	1.60545	0.802727	+	0.596346i	$0.203382\pi$
$138$	0	0
$139$	−6.45851	−0.547803	−0.273902	−	0.961758i	$-0.588314\pi$
$139$	−6.45851	−0.547803	−0.273902	+	0.961758i	$0.588314\pi$
$140$	0	0
$141$	−1.68571	−0.141963
$142$	0	0
$143$	13.0178	1.08860
$144$	0	0
$145$	12.9017	1.07143
$146$	0	0
$147$	−6.61671	−0.545737
$148$	0	0
$149$	−16.6797	−1.36645	−0.683226	−	0.730207i	$-0.739424\pi$
$149$	−16.6797	−1.36645	−0.683226	+	0.730207i	$0.739424\pi$
$150$	0	0
$151$	21.4080	1.74216	0.871079	−	0.491143i	$-0.163421\pi$
$151$	21.4080	1.74216	0.871079	+	0.491143i	$0.163421\pi$
$152$	0	0
$153$	−0.683131	−0.0552279
$154$	0	0
$155$	12.8247	1.03010
$156$	0	0
$157$	−7.23856	−0.577700	−0.288850	−	0.957374i	$-0.593273\pi$
$157$	−7.23856	−0.577700	−0.288850	+	0.957374i	$0.593273\pi$
$158$	0	0
$159$	−10.0094	−0.793800
$160$	0	0
$161$	4.52858	0.356902
$162$	0	0
$163$	8.00975	0.627372	0.313686	−	0.949527i	$-0.398436\pi$
$163$	8.00975	0.627372	0.313686	+	0.949527i	$0.398436\pi$
$164$	0	0
$165$	8.49418	0.661271
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−5.22994	−0.402303
$170$	0	0
$171$	5.92392	0.453013
$172$	0	0
$173$	17.3124	1.31624	0.658119	−	0.752914i	$-0.271353\pi$
$173$	17.3124	1.31624	0.658119	+	0.752914i	$0.271353\pi$
$174$	0	0
$175$	−1.04738	−0.0791748
$176$	0	0
$177$	−11.9921	−0.901384
$178$	0	0
$179$	3.93567	0.294166	0.147083	−	0.989124i	$-0.453012\pi$
$179$	3.93567	0.294166	0.147083	+	0.989124i	$0.453012\pi$
$180$	0	0
$181$	4.48440	0.333323	0.166661	−	0.986014i	$-0.446701\pi$
$181$	4.48440	0.333323	0.166661	+	0.986014i	$0.446701\pi$
$182$	0	0
$183$	4.90480	0.362573
$184$	0	0
$185$	−14.5385	−1.06889
$186$	0	0
$187$	−3.19028	−0.233296
$188$	0	0
$189$	0.619101	0.0450330
$190$	0	0
$191$	−15.9079	−1.15105	−0.575526	−	0.817784i	$-0.695203\pi$
$191$	−15.9079	−1.15105	−0.575526	+	0.817784i	$0.695203\pi$
$192$	0	0
$193$	−22.1335	−1.59320	−0.796602	−	0.604505i	$-0.793371\pi$
$193$	−22.1335	−1.59320	−0.796602	+	0.604505i	$0.793371\pi$
$194$	0	0
$195$	5.07002	0.363071
$196$	0	0
$197$	9.75941	0.695329	0.347665	−	0.937619i	$-0.386975\pi$
$197$	9.75941	0.695329	0.347665	+	0.937619i	$0.386975\pi$
$198$	0	0
$199$	−2.36374	−0.167561	−0.0837804	−	0.996484i	$-0.526699\pi$
$199$	−2.36374	−0.167561	−0.0837804	+	0.996484i	$0.526699\pi$
$200$	0	0
$201$	−11.0368	−0.778479
$202$	0	0
$203$	4.39148	0.308221
$204$	0	0
$205$	0.105065	0.00733806
$206$	0	0
$207$	7.31476	0.508411
$208$	0	0
$209$	27.6652	1.91364
$210$	0	0
$211$	−6.75868	−0.465286	−0.232643	−	0.972562i	$-0.574737\pi$
$211$	−6.75868	−0.465286	−0.232643	+	0.972562i	$0.574737\pi$
$212$	0	0
$213$	−15.2933	−1.04788
$214$	0	0
$215$	14.8333	1.01163
$216$	0	0
$217$	4.36527	0.296334
$218$	0	0
$219$	−5.47053	−0.369664
$220$	0	0
$221$	−1.90422	−0.128091
$222$	0	0
$223$	−8.39260	−0.562010	−0.281005	−	0.959706i	$-0.590668\pi$
$223$	−8.39260	−0.562010	−0.281005	+	0.959706i	$0.590668\pi$
$224$	0	0
$225$	−1.69178	−0.112786
$226$	0	0
$227$	−24.8113	−1.64678	−0.823392	−	0.567474i	$-0.807921\pi$
$227$	−24.8113	−1.64678	−0.823392	+	0.567474i	$0.807921\pi$
$228$	0	0
$229$	−24.3000	−1.60579	−0.802894	−	0.596121i	$-0.796708\pi$
$229$	−24.3000	−1.60579	−0.802894	+	0.596121i	$0.796708\pi$
$230$	0	0
$231$	2.89125	0.190230
$232$	0	0
$233$	−9.67691	−0.633955	−0.316978	−	0.948433i	$-0.602668\pi$
$233$	−9.67691	−0.633955	−0.316978	+	0.948433i	$0.602668\pi$
$234$	0	0
$235$	−3.06606	−0.200008
$236$	0	0
$237$	8.58093	0.557391
$238$	0	0
$239$	26.1035	1.68849	0.844247	−	0.535954i	$-0.180048\pi$
$239$	26.1035	1.68849	0.844247	+	0.535954i	$0.180048\pi$
$240$	0	0
$241$	30.1384	1.94138	0.970692	−	0.240328i	$-0.0772549\pi$
$241$	30.1384	1.94138	0.970692	+	0.240328i	$0.0772549\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−12.0348	−0.768876
$246$	0	0
$247$	16.5128	1.05069
$248$	0	0
$249$	−1.86623	−0.118268
$250$	0	0
$251$	18.4290	1.16323	0.581615	−	0.813464i	$-0.302421\pi$
$251$	18.4290	1.16323	0.581615	+	0.813464i	$0.302421\pi$
$252$	0	0
$253$	34.1606	2.14766
$254$	0	0
$255$	−1.24251	−0.0778092
$256$	0	0
$257$	4.06747	0.253722	0.126861	−	0.991921i	$-0.459510\pi$
$257$	4.06747	0.253722	0.126861	+	0.991921i	$0.459510\pi$
$258$	0	0
$259$	−4.94861	−0.307492
$260$	0	0
$261$	7.09331	0.439065
$262$	0	0
$263$	1.51519	0.0934309	0.0467154	−	0.998908i	$-0.485125\pi$
$263$	1.51519	0.0934309	0.0467154	+	0.998908i	$0.485125\pi$
$264$	0	0
$265$	−18.2057	−1.11836
$266$	0	0
$267$	−5.14998	−0.315174
$268$	0	0
$269$	−9.06255	−0.552554	−0.276277	−	0.961078i	$-0.589101\pi$
$269$	−9.06255	−0.552554	−0.276277	+	0.961078i	$0.589101\pi$
$270$	0	0
$271$	10.1878	0.618867	0.309434	−	0.950921i	$-0.399861\pi$
$271$	10.1878	0.618867	0.309434	+	0.950921i	$0.399861\pi$
$272$	0	0
$273$	1.72573	0.104446
$274$	0	0
$275$	−7.90077	−0.476434
$276$	0	0
$277$	−19.6810	−1.18252	−0.591259	−	0.806481i	$-0.701369\pi$
$277$	−19.6810	−1.18252	−0.591259	+	0.806481i	$0.701369\pi$
$278$	0	0
$279$	7.05098	0.422131
$280$	0	0
$281$	17.3377	1.03428	0.517140	−	0.855901i	$-0.326997\pi$
$281$	17.3377	1.03428	0.517140	+	0.855901i	$0.326997\pi$
$282$	0	0
$283$	1.34287	0.0798255	0.0399128	−	0.999203i	$-0.487292\pi$
$283$	1.34287	0.0798255	0.0399128	+	0.999203i	$0.487292\pi$
$284$	0	0
$285$	10.7747	0.638239
$286$	0	0
$287$	0.0357621	0.00211097
$288$	0	0
$289$	−16.5333	−0.972549
$290$	0	0
$291$	10.4675	0.613617
$292$	0	0
$293$	−8.09037	−0.472645	−0.236322	−	0.971675i	$-0.575942\pi$
$293$	−8.09037	−0.472645	−0.236322	+	0.971675i	$0.575942\pi$
$294$	0	0
$295$	−21.8119	−1.26994
$296$	0	0
$297$	4.67008	0.270986
$298$	0	0
$299$	20.3898	1.17917
$300$	0	0
$301$	5.04897	0.291018
$302$	0	0
$303$	−4.04032	−0.232110
$304$	0	0
$305$	8.92110	0.510821
$306$	0	0
$307$	7.90434	0.451125	0.225562	−	0.974229i	$-0.427578\pi$
$307$	7.90434	0.451125	0.225562	+	0.974229i	$0.427578\pi$
$308$	0	0
$309$	−12.5828	−0.715810
$310$	0	0
$311$	−12.4753	−0.707412	−0.353706	−	0.935357i	$-0.615079\pi$
$311$	−12.4753	−0.707412	−0.353706	+	0.935357i	$0.615079\pi$
$312$	0	0
$313$	−17.1389	−0.968750	−0.484375	−	0.874860i	$-0.660953\pi$
$313$	−17.1389	−0.968750	−0.484375	+	0.874860i	$0.660953\pi$
$314$	0	0
$315$	1.12605	0.0634458
$316$	0	0
$317$	4.78055	0.268502	0.134251	−	0.990947i	$-0.457137\pi$
$317$	4.78055	0.268502	0.134251	+	0.990947i	$0.457137\pi$
$318$	0	0
$319$	33.1264	1.85472
$320$	0	0
$321$	−3.37816	−0.188550
$322$	0	0
$323$	−4.04681	−0.225171
$324$	0	0
$325$	−4.71582	−0.261586
$326$	0	0
$327$	−16.0236	−0.886106
$328$	0	0
$329$	−1.04363	−0.0575370
$330$	0	0
$331$	32.6964	1.79716	0.898578	−	0.438814i	$-0.144601\pi$
$331$	32.6964	1.79716	0.898578	+	0.438814i	$0.144601\pi$
$332$	0	0
$333$	−7.99322	−0.438026
$334$	0	0
$335$	−20.0744	−1.09678
$336$	0	0
$337$	10.3364	0.563057	0.281528	−	0.959553i	$-0.409159\pi$
$337$	10.3364	0.563057	0.281528	+	0.959553i	$0.409159\pi$
$338$	0	0
$339$	6.17791	0.335538
$340$	0	0
$341$	32.9287	1.78319
$342$	0	0
$343$	−8.43012	−0.455184
$344$	0	0
$345$	13.3045	0.716288
$346$	0	0
$347$	−9.69508	−0.520459	−0.260230	−	0.965547i	$-0.583798\pi$
$347$	−9.69508	−0.520459	−0.260230	+	0.965547i	$0.583798\pi$
$348$	0	0
$349$	25.0235	1.33948	0.669739	−	0.742597i	$-0.266406\pi$
$349$	25.0235	1.33948	0.669739	+	0.742597i	$0.266406\pi$
$350$	0	0
$351$	2.78748	0.148785
$352$	0	0
$353$	−2.64539	−0.140800	−0.0704000	−	0.997519i	$-0.522428\pi$
$353$	−2.64539	−0.140800	−0.0704000	+	0.997519i	$0.522428\pi$
$354$	0	0
$355$	−27.8162	−1.47633
$356$	0	0
$357$	−0.422927	−0.0223837
$358$	0	0
$359$	−19.6453	−1.03684	−0.518420	−	0.855126i	$-0.673479\pi$
$359$	−19.6453	−1.03684	−0.518420	+	0.855126i	$0.673479\pi$
$360$	0	0
$361$	16.0928	0.846988
$362$	0	0
$363$	10.8097	0.567361
$364$	0	0
$365$	−9.95008	−0.520811
$366$	0	0
$367$	−33.7060	−1.75944	−0.879720	−	0.475492i	$-0.842270\pi$
$367$	−33.7060	−1.75944	−0.879720	+	0.475492i	$0.842270\pi$
$368$	0	0
$369$	0.0577645	0.00300710
$370$	0	0
$371$	−6.19685	−0.321724
$372$	0	0
$373$	23.7459	1.22952	0.614758	−	0.788715i	$-0.289254\pi$
$373$	23.7459	1.22952	0.614758	+	0.788715i	$0.289254\pi$
$374$	0	0
$375$	−12.1714	−0.628526
$376$	0	0
$377$	19.7725	1.01834
$378$	0	0
$379$	2.74820	0.141166	0.0705828	−	0.997506i	$-0.477514\pi$
$379$	2.74820	0.141166	0.0705828	+	0.997506i	$0.477514\pi$
$380$	0	0
$381$	−8.59133	−0.440147
$382$	0	0
$383$	−3.50394	−0.179043	−0.0895215	−	0.995985i	$-0.528534\pi$
$383$	−3.50394	−0.179043	−0.0895215	+	0.995985i	$0.528534\pi$
$384$	0	0
$385$	5.25876	0.268011
$386$	0	0
$387$	8.15534	0.414559
$388$	0	0
$389$	−19.6535	−0.996474	−0.498237	−	0.867041i	$-0.666019\pi$
$389$	−19.6535	−0.996474	−0.498237	+	0.867041i	$0.666019\pi$
$390$	0	0
$391$	−4.99694	−0.252706
$392$	0	0
$393$	5.93595	0.299429
$394$	0	0
$395$	15.6074	0.785295
$396$	0	0
$397$	−9.07024	−0.455222	−0.227611	−	0.973752i	$-0.573092\pi$
$397$	−9.07024	−0.455222	−0.227611	+	0.973752i	$0.573092\pi$
$398$	0	0
$399$	3.66750	0.183605
$400$	0	0
$401$	4.06991	0.203242	0.101621	−	0.994823i	$-0.467597\pi$
$401$	4.06991	0.203242	0.101621	+	0.994823i	$0.467597\pi$
$402$	0	0
$403$	19.6545	0.979060
$404$	0	0
$405$	1.81885	0.0903794
$406$	0	0
$407$	−37.3290	−1.85033
$408$	0	0
$409$	19.7756	0.977840	0.488920	−	0.872329i	$-0.337391\pi$
$409$	19.7756	0.977840	0.488920	+	0.872329i	$0.337391\pi$
$410$	0	0
$411$	18.7914	0.926910
$412$	0	0
$413$	−7.42434	−0.365328
$414$	0	0
$415$	−3.39440	−0.166625
$416$	0	0
$417$	−6.45851	−0.316274
$418$	0	0
$419$	27.6280	1.34972	0.674859	−	0.737947i	$-0.264204\pi$
$419$	27.6280	1.34972	0.674859	+	0.737947i	$0.264204\pi$
$420$	0	0
$421$	32.8302	1.60005	0.800023	−	0.599970i	$-0.204821\pi$
$421$	32.8302	1.60005	0.800023	+	0.599970i	$0.204821\pi$
$422$	0	0
$423$	−1.68571	−0.0819622
$424$	0	0
$425$	1.15571	0.0560601
$426$	0	0
$427$	3.03657	0.146950
$428$	0	0
$429$	13.0178	0.628504
$430$	0	0
$431$	27.8163	1.33987	0.669933	−	0.742422i	$-0.266323\pi$
$431$	27.8163	1.33987	0.669933	+	0.742422i	$0.266323\pi$
$432$	0	0
$433$	−11.9632	−0.574914	−0.287457	−	0.957794i	$-0.592810\pi$
$433$	−11.9632	−0.574914	−0.287457	+	0.957794i	$0.592810\pi$
$434$	0	0
$435$	12.9017	0.618588
$436$	0	0
$437$	43.3320	2.07285
$438$	0	0
$439$	−6.14418	−0.293246	−0.146623	−	0.989192i	$-0.546840\pi$
$439$	−6.14418	−0.293246	−0.146623	+	0.989192i	$0.546840\pi$
$440$	0	0
$441$	−6.61671	−0.315082
$442$	0	0
$443$	−29.9791	−1.42435	−0.712174	−	0.702003i	$-0.752289\pi$
$443$	−29.9791	−1.42435	−0.712174	+	0.702003i	$0.752289\pi$
$444$	0	0
$445$	−9.36704	−0.444040
$446$	0	0
$447$	−16.6797	−0.788922
$448$	0	0
$449$	1.51936	0.0717032	0.0358516	−	0.999357i	$-0.488586\pi$
$449$	1.51936	0.0717032	0.0358516	+	0.999357i	$0.488586\pi$
$450$	0	0
$451$	0.269765	0.0127027
$452$	0	0
$453$	21.4080	1.00584
$454$	0	0
$455$	3.13885	0.147152
$456$	0	0
$457$	20.4960	0.958764	0.479382	−	0.877606i	$-0.340861\pi$
$457$	20.4960	0.958764	0.479382	+	0.877606i	$0.340861\pi$
$458$	0	0
$459$	−0.683131	−0.0318858
$460$	0	0
$461$	−30.9837	−1.44306	−0.721528	−	0.692385i	$-0.756560\pi$
$461$	−30.9837	−1.44306	−0.721528	+	0.692385i	$0.756560\pi$
$462$	0	0
$463$	10.1283	0.470700	0.235350	−	0.971911i	$-0.424376\pi$
$463$	10.1283	0.470700	0.235350	+	0.971911i	$0.424376\pi$
$464$	0	0
$465$	12.8247	0.594730
$466$	0	0
$467$	7.95900	0.368298	0.184149	−	0.982898i	$-0.441047\pi$
$467$	7.95900	0.368298	0.184149	+	0.982898i	$0.441047\pi$
$468$	0	0
$469$	−6.83292	−0.315515
$470$	0	0
$471$	−7.23856	−0.333535
$472$	0	0
$473$	38.0861	1.75120
$474$	0	0
$475$	−10.0220	−0.459840
$476$	0	0
$477$	−10.0094	−0.458300
$478$	0	0
$479$	−20.1206	−0.919332	−0.459666	−	0.888092i	$-0.652031\pi$
$479$	−20.1206	−0.919332	−0.459666	+	0.888092i	$0.652031\pi$
$480$	0	0
$481$	−22.2810	−1.01592
$482$	0	0
$483$	4.52858	0.206057
$484$	0	0
$485$	19.0388	0.864509
$486$	0	0
$487$	−24.0083	−1.08792	−0.543959	−	0.839112i	$-0.683075\pi$
$487$	−24.0083	−1.08792	−0.543959	+	0.839112i	$0.683075\pi$
$488$	0	0
$489$	8.00975	0.362213
$490$	0	0
$491$	13.8846	0.626605	0.313302	−	0.949653i	$-0.398565\pi$
$491$	13.8846	0.626605	0.313302	+	0.949653i	$0.398565\pi$
$492$	0	0
$493$	−4.84566	−0.218238
$494$	0	0
$495$	8.49418	0.381785
$496$	0	0
$497$	−9.46807	−0.424701
$498$	0	0
$499$	−33.2272	−1.48746	−0.743728	−	0.668483i	$-0.766944\pi$
$499$	−33.2272	−1.48746	−0.743728	+	0.668483i	$0.766944\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	41.8540	1.86618	0.933089	−	0.359645i	$-0.117102\pi$
$503$	41.8540	1.86618	0.933089	+	0.359645i	$0.117102\pi$
$504$	0	0
$505$	−7.34874	−0.327015
$506$	0	0
$507$	−5.22994	−0.232270
$508$	0	0
$509$	−20.7891	−0.921459	−0.460730	−	0.887541i	$-0.652412\pi$
$509$	−20.7891	−0.921459	−0.460730	+	0.887541i	$0.652412\pi$
$510$	0	0
$511$	−3.38681	−0.149824
$512$	0	0
$513$	5.92392	0.261547
$514$	0	0
$515$	−22.8862	−1.00849
$516$	0	0
$517$	−7.87243	−0.346229
$518$	0	0
$519$	17.3124	0.759930
$520$	0	0
$521$	−1.82338	−0.0798837	−0.0399418	−	0.999202i	$-0.512717\pi$
$521$	−1.82338	−0.0798837	−0.0399418	+	0.999202i	$0.512717\pi$
$522$	0	0
$523$	−16.0009	−0.699671	−0.349835	−	0.936811i	$-0.613763\pi$
$523$	−16.0009	−0.699671	−0.349835	+	0.936811i	$0.613763\pi$
$524$	0	0
$525$	−1.04738	−0.0457116
$526$	0	0
$527$	−4.81674	−0.209821
$528$	0	0
$529$	30.5058	1.32634
$530$	0	0
$531$	−11.9921	−0.520414
$532$	0	0
$533$	0.161018	0.00697445
$534$	0	0
$535$	−6.14436	−0.265644
$536$	0	0
$537$	3.93567	0.169837
$538$	0	0
$539$	−30.9006	−1.33098
$540$	0	0
$541$	−9.31528	−0.400495	−0.200248	−	0.979745i	$-0.564175\pi$
$541$	−9.31528	−0.400495	−0.200248	+	0.979745i	$0.564175\pi$
$542$	0	0
$543$	4.48440	0.192444
$544$	0	0
$545$	−29.1445	−1.24841
$546$	0	0
$547$	−1.41218	−0.0603806	−0.0301903	−	0.999544i	$-0.509611\pi$
$547$	−1.41218	−0.0603806	−0.0301903	+	0.999544i	$0.509611\pi$
$548$	0	0
$549$	4.90480	0.209332
$550$	0	0
$551$	42.0202	1.79012
$552$	0	0
$553$	5.31246	0.225909
$554$	0	0
$555$	−14.5385	−0.617124
$556$	0	0
$557$	−11.5817	−0.490733	−0.245366	−	0.969430i	$-0.578908\pi$
$557$	−11.5817	−0.490733	−0.245366	+	0.969430i	$0.578908\pi$
$558$	0	0
$559$	22.7329	0.961498
$560$	0	0
$561$	−3.19028	−0.134694
$562$	0	0
$563$	41.5357	1.75052	0.875261	−	0.483651i	$-0.160690\pi$
$563$	41.5357	1.75052	0.875261	+	0.483651i	$0.160690\pi$
$564$	0	0
$565$	11.2367	0.472731
$566$	0	0
$567$	0.619101	0.0259998
$568$	0	0
$569$	−10.7077	−0.448889	−0.224444	−	0.974487i	$-0.572057\pi$
$569$	−10.7077	−0.448889	−0.224444	+	0.974487i	$0.572057\pi$
$570$	0	0
$571$	−6.47801	−0.271096	−0.135548	−	0.990771i	$-0.543280\pi$
$571$	−6.47801	−0.271096	−0.135548	+	0.990771i	$0.543280\pi$
$572$	0	0
$573$	−15.9079	−0.664560
$574$	0	0
$575$	−12.3750	−0.516073
$576$	0	0
$577$	−32.6512	−1.35929	−0.679643	−	0.733543i	$-0.737865\pi$
$577$	−32.6512	−1.35929	−0.679643	+	0.733543i	$0.737865\pi$
$578$	0	0
$579$	−22.1335	−0.919836
$580$	0	0
$581$	−1.15539	−0.0479335
$582$	0	0
$583$	−46.7449	−1.93598
$584$	0	0
$585$	5.07002	0.209619
$586$	0	0
$587$	25.9818	1.07238	0.536192	−	0.844096i	$-0.319862\pi$
$587$	25.9818	1.07238	0.536192	+	0.844096i	$0.319862\pi$
$588$	0	0
$589$	41.7694	1.72108
$590$	0	0
$591$	9.75941	0.401449
$592$	0	0
$593$	2.56680	0.105406	0.0527028	−	0.998610i	$-0.483216\pi$
$593$	2.56680	0.105406	0.0527028	+	0.998610i	$0.483216\pi$
$594$	0	0
$595$	−0.769241	−0.0315358
$596$	0	0
$597$	−2.36374	−0.0967413
$598$	0	0
$599$	−21.6505	−0.884616	−0.442308	−	0.896863i	$-0.645840\pi$
$599$	−21.6505	−0.884616	−0.442308	+	0.896863i	$0.645840\pi$
$600$	0	0
$601$	−11.8211	−0.482193	−0.241096	−	0.970501i	$-0.577507\pi$
$601$	−11.8211	−0.482193	−0.241096	+	0.970501i	$0.577507\pi$
$602$	0	0
$603$	−11.0368	−0.449455
$604$	0	0
$605$	19.6612	0.799341
$606$	0	0
$607$	−10.8427	−0.440093	−0.220046	−	0.975489i	$-0.570621\pi$
$607$	−10.8427	−0.440093	−0.220046	+	0.975489i	$0.570621\pi$
$608$	0	0
$609$	4.39148	0.177952
$610$	0	0
$611$	−4.69890	−0.190097
$612$	0	0
$613$	2.90481	0.117324	0.0586620	−	0.998278i	$-0.481317\pi$
$613$	2.90481	0.117324	0.0586620	+	0.998278i	$0.481317\pi$
$614$	0	0
$615$	0.105065	0.00423663
$616$	0	0
$617$	9.16820	0.369098	0.184549	−	0.982823i	$-0.440918\pi$
$617$	9.16820	0.369098	0.184549	+	0.982823i	$0.440918\pi$
$618$	0	0
$619$	−14.5746	−0.585801	−0.292901	−	0.956143i	$-0.594621\pi$
$619$	−14.5746	−0.585801	−0.292901	+	0.956143i	$0.594621\pi$
$620$	0	0
$621$	7.31476	0.293531
$622$	0	0
$623$	−3.18836	−0.127739
$624$	0	0
$625$	−13.6790	−0.547158
$626$	0	0
$627$	27.6652	1.10484
$628$	0	0
$629$	5.46041	0.217721
$630$	0	0
$631$	26.9869	1.07433	0.537166	−	0.843476i	$-0.319495\pi$
$631$	26.9869	1.07433	0.537166	+	0.843476i	$0.319495\pi$
$632$	0	0
$633$	−6.75868	−0.268633
$634$	0	0
$635$	−15.6263	−0.620113
$636$	0	0
$637$	−18.4440	−0.730777
$638$	0	0
$639$	−15.2933	−0.604992
$640$	0	0
$641$	10.2044	0.403051	0.201526	−	0.979483i	$-0.435410\pi$
$641$	10.2044	0.403051	0.201526	+	0.979483i	$0.435410\pi$
$642$	0	0
$643$	7.56619	0.298382	0.149191	−	0.988808i	$-0.452333\pi$
$643$	7.56619	0.298382	0.149191	+	0.988808i	$0.452333\pi$
$644$	0	0
$645$	14.8333	0.584062
$646$	0	0
$647$	23.5945	0.927594	0.463797	−	0.885942i	$-0.346487\pi$
$647$	23.5945	0.927594	0.463797	+	0.885942i	$0.346487\pi$
$648$	0	0
$649$	−56.0042	−2.19836
$650$	0	0
$651$	4.36527	0.171088
$652$	0	0
$653$	42.8469	1.67673	0.838365	−	0.545110i	$-0.183512\pi$
$653$	42.8469	1.67673	0.838365	+	0.545110i	$0.183512\pi$
$654$	0	0
$655$	10.7966	0.421858
$656$	0	0
$657$	−5.47053	−0.213426
$658$	0	0
$659$	14.8220	0.577383	0.288691	−	0.957422i	$-0.406780\pi$
$659$	14.8220	0.577383	0.288691	+	0.957422i	$0.406780\pi$
$660$	0	0
$661$	−25.7271	−1.00067	−0.500333	−	0.865833i	$-0.666789\pi$
$661$	−25.7271	−1.00067	−0.500333	+	0.865833i	$0.666789\pi$
$662$	0	0
$663$	−1.90422	−0.0739536
$664$	0	0
$665$	6.67064	0.258676
$666$	0	0
$667$	51.8859	2.00903
$668$	0	0
$669$	−8.39260	−0.324477
$670$	0	0
$671$	22.9058	0.884270
$672$	0	0
$673$	−32.2272	−1.24227	−0.621134	−	0.783704i	$-0.713328\pi$
$673$	−32.2272	−1.24227	−0.621134	+	0.783704i	$0.713328\pi$
$674$	0	0
$675$	−1.69178	−0.0651168
$676$	0	0
$677$	−3.72394	−0.143123	−0.0715613	−	0.997436i	$-0.522798\pi$
$677$	−3.72394	−0.143123	−0.0715613	+	0.997436i	$0.522798\pi$
$678$	0	0
$679$	6.48045	0.248697
$680$	0	0
$681$	−24.8113	−0.950771
$682$	0	0
$683$	3.93647	0.150625	0.0753124	−	0.997160i	$-0.476005\pi$
$683$	3.93647	0.150625	0.0753124	+	0.997160i	$0.476005\pi$
$684$	0	0
$685$	34.1787	1.30590
$686$	0	0
$687$	−24.3000	−0.927103
$688$	0	0
$689$	−27.9011	−1.06295
$690$	0	0
$691$	48.7531	1.85465	0.927327	−	0.374252i	$-0.122100\pi$
$691$	48.7531	1.85465	0.927327	+	0.374252i	$0.122100\pi$
$692$	0	0
$693$	2.89125	0.109830
$694$	0	0
$695$	−11.7471	−0.445591
$696$	0	0
$697$	−0.0394607	−0.00149468
$698$	0	0
$699$	−9.67691	−0.366014
$700$	0	0
$701$	30.3833	1.14756	0.573782	−	0.819008i	$-0.305476\pi$
$701$	30.3833	1.14756	0.573782	+	0.819008i	$0.305476\pi$
$702$	0	0
$703$	−47.3512	−1.78588
$704$	0	0
$705$	−3.06606	−0.115475
$706$	0	0
$707$	−2.50137	−0.0940736
$708$	0	0
$709$	−25.0956	−0.942486	−0.471243	−	0.882003i	$-0.656195\pi$
$709$	−25.0956	−0.942486	−0.471243	+	0.882003i	$0.656195\pi$
$710$	0	0
$711$	8.58093	0.321810
$712$	0	0
$713$	51.5763	1.93155
$714$	0	0
$715$	23.6774	0.885484
$716$	0	0
$717$	26.1035	0.974852
$718$	0	0
$719$	8.84414	0.329831	0.164915	−	0.986308i	$-0.447265\pi$
$719$	8.84414	0.329831	0.164915	+	0.986308i	$0.447265\pi$
$720$	0	0
$721$	−7.79002	−0.290116
$722$	0	0
$723$	30.1384	1.12086
$724$	0	0
$725$	−12.0003	−0.445682
$726$	0	0
$727$	−11.5581	−0.428667	−0.214333	−	0.976761i	$-0.568758\pi$
$727$	−11.5581	−0.428667	−0.214333	+	0.976761i	$0.568758\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.57116	−0.206057
$732$	0	0
$733$	−48.2694	−1.78287	−0.891435	−	0.453149i	$-0.850300\pi$
$733$	−48.2694	−1.78287	−0.891435	+	0.453149i	$0.850300\pi$
$734$	0	0
$735$	−12.0348	−0.443911
$736$	0	0
$737$	−51.5430	−1.89861
$738$	0	0
$739$	−6.05064	−0.222576	−0.111288	−	0.993788i	$-0.535498\pi$
$739$	−6.05064	−0.222576	−0.111288	+	0.993788i	$0.535498\pi$
$740$	0	0
$741$	16.5128	0.606613
$742$	0	0
$743$	44.4999	1.63254	0.816271	−	0.577669i	$-0.196037\pi$
$743$	44.4999	1.63254	0.816271	+	0.577669i	$0.196037\pi$
$744$	0	0
$745$	−30.3378	−1.11149
$746$	0	0
$747$	−1.86623	−0.0682820
$748$	0	0
$749$	−2.09142	−0.0764188
$750$	0	0
$751$	8.98102	0.327722	0.163861	−	0.986483i	$-0.447605\pi$
$751$	8.98102	0.327722	0.163861	+	0.986483i	$0.447605\pi$
$752$	0	0
$753$	18.4290	0.671592
$754$	0	0
$755$	38.9379	1.41710
$756$	0	0
$757$	−32.2948	−1.17378	−0.586888	−	0.809668i	$-0.699647\pi$
$757$	−32.2948	−1.17378	−0.586888	+	0.809668i	$0.699647\pi$
$758$	0	0
$759$	34.1606	1.23995
$760$	0	0
$761$	28.0185	1.01567	0.507836	−	0.861454i	$-0.330446\pi$
$761$	28.0185	1.01567	0.507836	+	0.861454i	$0.330446\pi$
$762$	0	0
$763$	−9.92021	−0.359136
$764$	0	0
$765$	−1.24251	−0.0449231
$766$	0	0
$767$	−33.4279	−1.20701
$768$	0	0
$769$	−47.3092	−1.70601	−0.853006	−	0.521901i	$-0.825223\pi$
$769$	−47.3092	−1.70601	−0.853006	+	0.521901i	$0.825223\pi$
$770$	0	0
$771$	4.06747	0.146486
$772$	0	0
$773$	14.0787	0.506376	0.253188	−	0.967417i	$-0.418521\pi$
$773$	14.0787	0.506376	0.253188	+	0.967417i	$0.418521\pi$
$774$	0	0
$775$	−11.9287	−0.428493
$776$	0	0
$777$	−4.94861	−0.177530
$778$	0	0
$779$	0.342192	0.0122603
$780$	0	0
$781$	−71.4208	−2.55564
$782$	0	0
$783$	7.09331	0.253494
$784$	0	0
$785$	−13.1659	−0.469910
$786$	0	0
$787$	−9.04701	−0.322491	−0.161246	−	0.986914i	$-0.551551\pi$
$787$	−9.04701	−0.322491	−0.161246	+	0.986914i	$0.551551\pi$
$788$	0	0
$789$	1.51519	0.0539423
$790$	0	0
$791$	3.82475	0.135992
$792$	0	0
$793$	13.6720	0.485509
$794$	0	0
$795$	−18.2057	−0.645688
$796$	0	0
$797$	−24.6386	−0.872745	−0.436373	−	0.899766i	$-0.643737\pi$
$797$	−24.6386	−0.872745	−0.436373	+	0.899766i	$0.643737\pi$
$798$	0	0
$799$	1.15156	0.0407394
$800$	0	0
$801$	−5.14998	−0.181966
$802$	0	0
$803$	−25.5478	−0.901564
$804$	0	0
$805$	8.23680	0.290309
$806$	0	0
$807$	−9.06255	−0.319017
$808$	0	0
$809$	28.9517	1.01789	0.508943	−	0.860800i	$-0.330036\pi$
$809$	28.9517	1.01789	0.508943	+	0.860800i	$0.330036\pi$
$810$	0	0
$811$	3.30951	0.116213	0.0581063	−	0.998310i	$-0.481494\pi$
$811$	3.30951	0.116213	0.0581063	+	0.998310i	$0.481494\pi$
$812$	0	0
$813$	10.1878	0.357303
$814$	0	0
$815$	14.5685	0.510314
$816$	0	0
$817$	48.3115	1.69021
$818$	0	0
$819$	1.72573	0.0603020
$820$	0	0
$821$	−9.08580	−0.317097	−0.158548	−	0.987351i	$-0.550681\pi$
$821$	−9.08580	−0.317097	−0.158548	+	0.987351i	$0.550681\pi$
$822$	0	0
$823$	−51.7110	−1.80253	−0.901265	−	0.433268i	$-0.857360\pi$
$823$	−51.7110	−1.80253	−0.901265	+	0.433268i	$0.857360\pi$
$824$	0	0
$825$	−7.90077	−0.275069
$826$	0	0
$827$	−12.4035	−0.431311	−0.215655	−	0.976470i	$-0.569189\pi$
$827$	−12.4035	−0.431311	−0.215655	+	0.976470i	$0.569189\pi$
$828$	0	0
$829$	−17.9171	−0.622287	−0.311144	−	0.950363i	$-0.600712\pi$
$829$	−17.9171	−0.622287	−0.311144	+	0.950363i	$0.600712\pi$
$830$	0	0
$831$	−19.6810	−0.682728
$832$	0	0
$833$	4.52008	0.156612
$834$	0	0
$835$	1.81885	0.0629439
$836$	0	0
$837$	7.05098	0.243718
$838$	0	0
$839$	34.6292	1.19553	0.597766	−	0.801670i	$-0.296055\pi$
$839$	34.6292	1.19553	0.597766	+	0.801670i	$0.296055\pi$
$840$	0	0
$841$	21.3151	0.735004
$842$	0	0
$843$	17.3377	0.597142
$844$	0	0
$845$	−9.51247	−0.327239
$846$	0	0
$847$	6.69228	0.229950
$848$	0	0
$849$	1.34287	0.0460873
$850$	0	0
$851$	−58.4685	−2.00427
$852$	0	0
$853$	22.8267	0.781571	0.390786	−	0.920482i	$-0.372203\pi$
$853$	22.8267	0.781571	0.390786	+	0.920482i	$0.372203\pi$
$854$	0	0
$855$	10.7747	0.368488
$856$	0	0
$857$	35.9758	1.22891	0.614455	−	0.788952i	$-0.289376\pi$
$857$	35.9758	1.22891	0.614455	+	0.788952i	$0.289376\pi$
$858$	0	0
$859$	51.4850	1.75664	0.878322	−	0.478069i	$-0.158663\pi$
$859$	51.4850	1.75664	0.878322	+	0.478069i	$0.158663\pi$
$860$	0	0
$861$	0.0357621	0.00121877
$862$	0	0
$863$	17.1692	0.584447	0.292224	−	0.956350i	$-0.405605\pi$
$863$	17.1692	0.584447	0.292224	+	0.956350i	$0.405605\pi$
$864$	0	0
$865$	31.4887	1.07065
$866$	0	0
$867$	−16.5333	−0.561501
$868$	0	0
$869$	40.0737	1.35941
$870$	0	0
$871$	−30.7650	−1.04243
$872$	0	0
$873$	10.4675	0.354272
$874$	0	0
$875$	−7.53529	−0.254739
$876$	0	0
$877$	20.8184	0.702986	0.351493	−	0.936190i	$-0.385674\pi$
$877$	20.8184	0.702986	0.351493	+	0.936190i	$0.385674\pi$
$878$	0	0
$879$	−8.09037	−0.272882
$880$	0	0
$881$	−18.4316	−0.620977	−0.310489	−	0.950577i	$-0.600493\pi$
$881$	−18.4316	−0.620977	−0.310489	+	0.950577i	$0.600493\pi$
$882$	0	0
$883$	50.6082	1.70310	0.851550	−	0.524273i	$-0.175663\pi$
$883$	50.6082	1.70310	0.851550	+	0.524273i	$0.175663\pi$
$884$	0	0
$885$	−21.8119	−0.733199
$886$	0	0
$887$	−36.8658	−1.23783	−0.618917	−	0.785457i	$-0.712428\pi$
$887$	−36.8658	−1.23783	−0.618917	+	0.785457i	$0.712428\pi$
$888$	0	0
$889$	−5.31890	−0.178390
$890$	0	0
$891$	4.67008	0.156454
$892$	0	0
$893$	−9.98603	−0.334170
$894$	0	0
$895$	7.15839	0.239279
$896$	0	0
$897$	20.3898	0.680795
$898$	0	0
$899$	50.0148	1.66809
$900$	0	0
$901$	6.83775	0.227799
$902$	0	0
$903$	5.04897	0.168019
$904$	0	0
$905$	8.15645	0.271129
$906$	0	0
$907$	8.73808	0.290143	0.145072	−	0.989421i	$-0.453659\pi$
$907$	8.73808	0.290143	0.145072	+	0.989421i	$0.453659\pi$
$908$	0	0
$909$	−4.04032	−0.134009
$910$	0	0
$911$	32.4824	1.07619	0.538095	−	0.842884i	$-0.319144\pi$
$911$	32.4824	1.07619	0.538095	+	0.842884i	$0.319144\pi$
$912$	0	0
$913$	−8.71547	−0.288440
$914$	0	0
$915$	8.92110	0.294922
$916$	0	0
$917$	3.67495	0.121358
$918$	0	0
$919$	8.42216	0.277821	0.138911	−	0.990305i	$-0.455640\pi$
$919$	8.42216	0.277821	0.138911	+	0.990305i	$0.455640\pi$
$920$	0	0
$921$	7.90434	0.260457
$922$	0	0
$923$	−42.6297	−1.40317
$924$	0	0
$925$	13.5228	0.444627
$926$	0	0
$927$	−12.5828	−0.413273
$928$	0	0
$929$	24.7866	0.813221	0.406611	−	0.913602i	$-0.366711\pi$
$929$	24.7866	0.813221	0.406611	+	0.913602i	$0.366711\pi$
$930$	0	0
$931$	−39.1969	−1.28463
$932$	0	0
$933$	−12.4753	−0.408425
$934$	0	0
$935$	−5.80264	−0.189767
$936$	0	0
$937$	−40.7372	−1.33083	−0.665413	−	0.746476i	$-0.731744\pi$
$937$	−40.7372	−1.33083	−0.665413	+	0.746476i	$0.731744\pi$
$938$	0	0
$939$	−17.1389	−0.559308
$940$	0	0
$941$	15.5064	0.505494	0.252747	−	0.967532i	$-0.418666\pi$
$941$	15.5064	0.505494	0.252747	+	0.967532i	$0.418666\pi$
$942$	0	0
$943$	0.422534	0.0137596
$944$	0	0
$945$	1.12605	0.0366305
$946$	0	0
$947$	9.98267	0.324393	0.162197	−	0.986758i	$-0.448142\pi$
$947$	9.98267	0.324393	0.162197	+	0.986758i	$0.448142\pi$
$948$	0	0
$949$	−15.2490	−0.495004
$950$	0	0
$951$	4.78055	0.155020
$952$	0	0
$953$	7.25373	0.234971	0.117486	−	0.993075i	$-0.462517\pi$
$953$	7.25373	0.234971	0.117486	+	0.993075i	$0.462517\pi$
$954$	0	0
$955$	−28.9340	−0.936282
$956$	0	0
$957$	33.1264	1.07082
$958$	0	0
$959$	11.6337	0.375673
$960$	0	0
$961$	18.7164	0.603754
$962$	0	0
$963$	−3.37816	−0.108860
$964$	0	0
$965$	−40.2575	−1.29593
$966$	0	0
$967$	−16.7965	−0.540138	−0.270069	−	0.962841i	$-0.587046\pi$
$967$	−16.7965	−0.540138	−0.270069	+	0.962841i	$0.587046\pi$
$968$	0	0
$969$	−4.04681	−0.130002
$970$	0	0
$971$	34.1516	1.09598	0.547988	−	0.836486i	$-0.315394\pi$
$971$	34.1516	1.09598	0.547988	+	0.836486i	$0.315394\pi$
$972$	0	0
$973$	−3.99847	−0.128185
$974$	0	0
$975$	−4.71582	−0.151027
$976$	0	0
$977$	−31.4062	−1.00477	−0.502387	−	0.864643i	$-0.667545\pi$
$977$	−31.4062	−1.00477	−0.502387	+	0.864643i	$0.667545\pi$
$978$	0	0
$979$	−24.0508	−0.768668
$980$	0	0
$981$	−16.0236	−0.511593
$982$	0	0
$983$	12.4807	0.398073	0.199037	−	0.979992i	$-0.436219\pi$
$983$	12.4807	0.398073	0.199037	+	0.979992i	$0.436219\pi$
$984$	0	0
$985$	17.7509	0.565591
$986$	0	0
$987$	−1.04363	−0.0332190
$988$	0	0
$989$	59.6543	1.89690
$990$	0	0
$991$	−44.0705	−1.39995	−0.699973	−	0.714169i	$-0.746805\pi$
$991$	−44.0705	−1.39995	−0.699973	+	0.714169i	$0.746805\pi$
$992$	0	0
$993$	32.6964	1.03759
$994$	0	0
$995$	−4.29928	−0.136296
$996$	0	0
$997$	10.6923	0.338627	0.169314	−	0.985562i	$-0.445845\pi$
$997$	10.6923	0.338627	0.169314	+	0.985562i	$0.445845\pi$
$998$	0	0
$999$	−7.99322	−0.252894

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.u.1.4		5
4.3	odd	2		501.2.a.c.1.3	✓	5
12.11	even	2		1503.2.a.c.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.c.1.3	✓	5	4.3	odd	2
1503.2.a.c.1.3		5	12.11	even	2
8016.2.a.u.1.4		5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.81885 q^{5} +0.619101 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.81885 q^{5} +0.619101 q^{7} +1.00000 q^{9} +4.67008 q^{11} +2.78748 q^{13} +1.81885 q^{15} -0.683131 q^{17} +5.92392 q^{19} +0.619101 q^{21} +7.31476 q^{23} -1.69178 q^{25} +1.00000 q^{27} +7.09331 q^{29} +7.05098 q^{31} +4.67008 q^{33} +1.12605 q^{35} -7.99322 q^{37} +2.78748 q^{39} +0.0577645 q^{41} +8.15534 q^{43} +1.81885 q^{45} -1.68571 q^{47} -6.61671 q^{49} -0.683131 q^{51} -10.0094 q^{53} +8.49418 q^{55} +5.92392 q^{57} -11.9921 q^{59} +4.90480 q^{61} +0.619101 q^{63} +5.07002 q^{65} -11.0368 q^{67} +7.31476 q^{69} -15.2933 q^{71} -5.47053 q^{73} -1.69178 q^{75} +2.89125 q^{77} +8.58093 q^{79} +1.00000 q^{81} -1.86623 q^{83} -1.24251 q^{85} +7.09331 q^{87} -5.14998 q^{89} +1.72573 q^{91} +7.05098 q^{93} +10.7747 q^{95} +10.4675 q^{97} +4.67008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} - q^{5} + 4 q^{7} + 5 q^{9} + 7 q^{11} - 8 q^{13} - q^{15} + 5 q^{17} + 20 q^{19} + 4 q^{21} - q^{23} - 6 q^{25} + 5 q^{27} - 5 q^{29} + 18 q^{31} + 7 q^{33} + 6 q^{35} - 17 q^{37} - 8 q^{39} + 6 q^{41} + 10 q^{43} - q^{45} + 3 q^{47} - 13 q^{49} + 5 q^{51} + 15 q^{53} + 9 q^{55} + 20 q^{57} + 17 q^{59} - 2 q^{61} + 4 q^{63} + 26 q^{65} + 24 q^{67} - q^{69} - q^{71} + 2 q^{73} - 6 q^{75} + 26 q^{77} + 6 q^{79} + 5 q^{81} + 7 q^{83} - 11 q^{85} - 5 q^{87} + 15 q^{91} + 18 q^{93} - q^{95} - 13 q^{97} + 7 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - q^5 + 4 * q^7 + 5 * q^9 + 7 * q^11 - 8 * q^13 - q^15 + 5 * q^17 + 20 * q^19 + 4 * q^21 - q^23 - 6 * q^25 + 5 * q^27 - 5 * q^29 + 18 * q^31 + 7 * q^33 + 6 * q^35 - 17 * q^37 - 8 * q^39 + 6 * q^41 + 10 * q^43 - q^45 + 3 * q^47 - 13 * q^49 + 5 * q^51 + 15 * q^53 + 9 * q^55 + 20 * q^57 + 17 * q^59 - 2 * q^61 + 4 * q^63 + 26 * q^65 + 24 * q^67 - q^69 - q^71 + 2 * q^73 - 6 * q^75 + 26 * q^77 + 6 * q^79 + 5 * q^81 + 7 * q^83 - 11 * q^85 - 5 * q^87 + 15 * q^91 + 18 * q^93 - q^95 - 13 * q^97 + 7 * q^99

Introduction

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