LMFDB - Embedded newform 8016.2.a.w.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 7x^{5} + 11x^{4} + 12x^{3} - 14x^{2} - 6x + 3 $ x^7 - 2x^6 - 7x^5 + 11x^4 + 12x^3 - 14x^2 - 6x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.332704$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.61705 q^{5} -2.72774 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.61705 q^{5} -2.72774 q^{7} +1.00000 q^{9} -4.28246 q^{11} -0.0474194 q^{13} +3.61705 q^{15} +3.70725 q^{17} -0.502366 q^{19} -2.72774 q^{21} -2.97951 q^{23} +8.08304 q^{25} +1.00000 q^{27} -6.25576 q^{29} -5.93115 q^{31} -4.28246 q^{33} -9.86637 q^{35} +0.158358 q^{37} -0.0474194 q^{39} -3.89424 q^{41} -4.94980 q^{43} +3.61705 q^{45} -8.39628 q^{47} +0.440569 q^{49} +3.70725 q^{51} -10.7084 q^{53} -15.4899 q^{55} -0.502366 q^{57} +3.54507 q^{59} +3.62786 q^{61} -2.72774 q^{63} -0.171518 q^{65} -0.477808 q^{67} -2.97951 q^{69} -0.963732 q^{71} -12.5153 q^{73} +8.08304 q^{75} +11.6814 q^{77} +1.53153 q^{79} +1.00000 q^{81} +13.4089 q^{83} +13.4093 q^{85} -6.25576 q^{87} +1.01339 q^{89} +0.129348 q^{91} -5.93115 q^{93} -1.81708 q^{95} -1.19560 q^{97} -4.28246 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} - 3 q^{5} - 8 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 - 3 * q^5 - 8 * q^7 + 7 * q^9	$ 7 q + 7 q^{3} - 3 q^{5} - 8 q^{7} + 7 q^{9} - q^{11} - 2 q^{13} - 3 q^{15} + 11 q^{17} - 2 q^{19} - 8 q^{21} - 17 q^{23} + 4 q^{25} + 7 q^{27} - 7 q^{29} - 10 q^{31} - q^{33} - 10 q^{35} - 21 q^{37} - 2 q^{39} + 8 q^{41} + 12 q^{43} - 3 q^{45} - 25 q^{47} - 7 q^{49} + 11 q^{51} - 7 q^{53} - 15 q^{55} - 2 q^{57} - 3 q^{59} - 14 q^{61} - 8 q^{63} + 4 q^{65} - 4 q^{67} - 17 q^{69} - 27 q^{71} - 12 q^{73} + 4 q^{75} + 16 q^{77} - 8 q^{79} + 7 q^{81} - 15 q^{83} - 3 q^{85} - 7 q^{87} + 14 q^{89} + 3 q^{91} - 10 q^{93} - 37 q^{95} + 3 q^{97} - q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 - 3 * q^5 - 8 * q^7 + 7 * q^9 - q^11 - 2 * q^13 - 3 * q^15 + 11 * q^17 - 2 * q^19 - 8 * q^21 - 17 * q^23 + 4 * q^25 + 7 * q^27 - 7 * q^29 - 10 * q^31 - q^33 - 10 * q^35 - 21 * q^37 - 2 * q^39 + 8 * q^41 + 12 * q^43 - 3 * q^45 - 25 * q^47 - 7 * q^49 + 11 * q^51 - 7 * q^53 - 15 * q^55 - 2 * q^57 - 3 * q^59 - 14 * q^61 - 8 * q^63 + 4 * q^65 - 4 * q^67 - 17 * q^69 - 27 * q^71 - 12 * q^73 + 4 * q^75 + 16 * q^77 - 8 * q^79 + 7 * q^81 - 15 * q^83 - 3 * q^85 - 7 * q^87 + 14 * q^89 + 3 * q^91 - 10 * q^93 - 37 * q^95 + 3 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.61705	1.61759	0.808797	−	0.588088i	$-0.200119\pi$
$5$	3.61705	1.61759	0.808797	+	0.588088i	$0.200119\pi$
$6$	0	0
$7$	−2.72774	−1.03099	−0.515495	−	0.856893i	$-0.672392\pi$
$7$	−2.72774	−1.03099	−0.515495	+	0.856893i	$0.672392\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.28246	−1.29121	−0.645605	−	0.763672i	$-0.723395\pi$
$11$	−4.28246	−1.29121	−0.645605	+	0.763672i	$0.723395\pi$
$12$	0	0
$13$	−0.0474194	−0.0131518	−0.00657589	−	0.999978i	$-0.502093\pi$
$13$	−0.0474194	−0.0131518	−0.00657589	+	0.999978i	$0.502093\pi$
$14$	0	0
$15$	3.61705	0.933918
$16$	0	0
$17$	3.70725	0.899140	0.449570	−	0.893245i	$-0.351577\pi$
$17$	3.70725	0.899140	0.449570	+	0.893245i	$0.351577\pi$
$18$	0	0
$19$	−0.502366	−0.115251	−0.0576254	−	0.998338i	$-0.518353\pi$
$19$	−0.502366	−0.115251	−0.0576254	+	0.998338i	$0.518353\pi$
$20$	0	0
$21$	−2.72774	−0.595242
$22$	0	0
$23$	−2.97951	−0.621270	−0.310635	−	0.950529i	$-0.600542\pi$
$23$	−2.97951	−0.621270	−0.310635	+	0.950529i	$0.600542\pi$
$24$	0	0
$25$	8.08304	1.61661
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.25576	−1.16167	−0.580833	−	0.814023i	$-0.697273\pi$
$29$	−6.25576	−1.16167	−0.580833	+	0.814023i	$0.697273\pi$
$30$	0	0
$31$	−5.93115	−1.06527	−0.532633	−	0.846346i	$-0.678797\pi$
$31$	−5.93115	−1.06527	−0.532633	+	0.846346i	$0.678797\pi$
$32$	0	0
$33$	−4.28246	−0.745480
$34$	0	0
$35$	−9.86637	−1.66772
$36$	0	0
$37$	0.158358	0.0260339	0.0130170	−	0.999915i	$-0.495856\pi$
$37$	0.158358	0.0260339	0.0130170	+	0.999915i	$0.495856\pi$
$38$	0	0
$39$	−0.0474194	−0.00759318
$40$	0	0
$41$	−3.89424	−0.608178	−0.304089	−	0.952644i	$-0.598352\pi$
$41$	−3.89424	−0.608178	−0.304089	+	0.952644i	$0.598352\pi$
$42$	0	0
$43$	−4.94980	−0.754838	−0.377419	−	0.926043i	$-0.623188\pi$
$43$	−4.94980	−0.754838	−0.377419	+	0.926043i	$0.623188\pi$
$44$	0	0
$45$	3.61705	0.539198
$46$	0	0
$47$	−8.39628	−1.22472	−0.612362	−	0.790578i	$-0.709780\pi$
$47$	−8.39628	−1.22472	−0.612362	+	0.790578i	$0.709780\pi$
$48$	0	0
$49$	0.440569	0.0629384
$50$	0	0
$51$	3.70725	0.519119
$52$	0	0
$53$	−10.7084	−1.47091	−0.735453	−	0.677576i	$-0.763031\pi$
$53$	−10.7084	−1.47091	−0.735453	+	0.677576i	$0.763031\pi$
$54$	0	0
$55$	−15.4899	−2.08865
$56$	0	0
$57$	−0.502366	−0.0665401
$58$	0	0
$59$	3.54507	0.461529	0.230764	−	0.973010i	$-0.425877\pi$
$59$	3.54507	0.461529	0.230764	+	0.973010i	$0.425877\pi$
$60$	0	0
$61$	3.62786	0.464500	0.232250	−	0.972656i	$-0.425391\pi$
$61$	3.62786	0.464500	0.232250	+	0.972656i	$0.425391\pi$
$62$	0	0
$63$	−2.72774	−0.343663
$64$	0	0
$65$	−0.171518	−0.0212742
$66$	0	0
$67$	−0.477808	−0.0583736	−0.0291868	−	0.999574i	$-0.509292\pi$
$67$	−0.477808	−0.0583736	−0.0291868	+	0.999574i	$0.509292\pi$
$68$	0	0
$69$	−2.97951	−0.358691
$70$	0	0
$71$	−0.963732	−0.114374	−0.0571870	−	0.998363i	$-0.518213\pi$
$71$	−0.963732	−0.114374	−0.0571870	+	0.998363i	$0.518213\pi$
$72$	0	0
$73$	−12.5153	−1.46481	−0.732405	−	0.680870i	$-0.761602\pi$
$73$	−12.5153	−1.46481	−0.732405	+	0.680870i	$0.761602\pi$
$74$	0	0
$75$	8.08304	0.933349
$76$	0	0
$77$	11.6814	1.33122
$78$	0	0
$79$	1.53153	0.172311	0.0861554	−	0.996282i	$-0.472542\pi$
$79$	1.53153	0.172311	0.0861554	+	0.996282i	$0.472542\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.4089	1.47182	0.735911	−	0.677078i	$-0.236754\pi$
$83$	13.4089	1.47182	0.735911	+	0.677078i	$0.236754\pi$
$84$	0	0
$85$	13.4093	1.45444
$86$	0	0
$87$	−6.25576	−0.670688
$88$	0	0
$89$	1.01339	0.107419	0.0537095	−	0.998557i	$-0.482896\pi$
$89$	1.01339	0.107419	0.0537095	+	0.998557i	$0.482896\pi$
$90$	0	0
$91$	0.129348	0.0135593
$92$	0	0
$93$	−5.93115	−0.615031
$94$	0	0
$95$	−1.81708	−0.186429
$96$	0	0
$97$	−1.19560	−0.121395	−0.0606973	−	0.998156i	$-0.519332\pi$
$97$	−1.19560	−0.121395	−0.0606973	+	0.998156i	$0.519332\pi$
$98$	0	0
$99$	−4.28246	−0.430403
$100$	0	0
$101$	2.15983	0.214911	0.107456	−	0.994210i	$-0.465730\pi$
$101$	2.15983	0.214911	0.107456	+	0.994210i	$0.465730\pi$
$102$	0	0
$103$	13.6082	1.34086	0.670428	−	0.741975i	$-0.266111\pi$
$103$	13.6082	1.34086	0.670428	+	0.741975i	$0.266111\pi$
$104$	0	0
$105$	−9.86637	−0.962859
$106$	0	0
$107$	3.76340	0.363821	0.181911	−	0.983315i	$-0.441772\pi$
$107$	3.76340	0.363821	0.181911	+	0.983315i	$0.441772\pi$
$108$	0	0
$109$	−11.9192	−1.14165	−0.570824	−	0.821072i	$-0.693376\pi$
$109$	−11.9192	−1.14165	−0.570824	+	0.821072i	$0.693376\pi$
$110$	0	0
$111$	0.158358	0.0150307
$112$	0	0
$113$	−5.99961	−0.564396	−0.282198	−	0.959356i	$-0.591063\pi$
$113$	−5.99961	−0.564396	−0.282198	+	0.959356i	$0.591063\pi$
$114$	0	0
$115$	−10.7770	−1.00496
$116$	0	0
$117$	−0.0474194	−0.00438393
$118$	0	0
$119$	−10.1124	−0.927003
$120$	0	0
$121$	7.33944	0.667222
$122$	0	0
$123$	−3.89424	−0.351132
$124$	0	0
$125$	11.1515	0.997421
$126$	0	0
$127$	−3.12683	−0.277461	−0.138731	−	0.990330i	$-0.544302\pi$
$127$	−3.12683	−0.277461	−0.138731	+	0.990330i	$0.544302\pi$
$128$	0	0
$129$	−4.94980	−0.435806
$130$	0	0
$131$	17.0808	1.49236	0.746180	−	0.665745i	$-0.231886\pi$
$131$	17.0808	1.49236	0.746180	+	0.665745i	$0.231886\pi$
$132$	0	0
$133$	1.37033	0.118822
$134$	0	0
$135$	3.61705	0.311306
$136$	0	0
$137$	−5.91181	−0.505080	−0.252540	−	0.967586i	$-0.581266\pi$
$137$	−5.91181	−0.505080	−0.252540	+	0.967586i	$0.581266\pi$
$138$	0	0
$139$	−19.2110	−1.62946	−0.814729	−	0.579842i	$-0.803114\pi$
$139$	−19.2110	−1.62946	−0.814729	+	0.579842i	$0.803114\pi$
$140$	0	0
$141$	−8.39628	−0.707094
$142$	0	0
$143$	0.203072	0.0169817
$144$	0	0
$145$	−22.6274	−1.87910
$146$	0	0
$147$	0.440569	0.0363375
$148$	0	0
$149$	12.4216	1.01762	0.508810	−	0.860879i	$-0.330086\pi$
$149$	12.4216	1.01762	0.508810	+	0.860879i	$0.330086\pi$
$150$	0	0
$151$	19.4910	1.58615	0.793076	−	0.609123i	$-0.208478\pi$
$151$	19.4910	1.58615	0.793076	+	0.609123i	$0.208478\pi$
$152$	0	0
$153$	3.70725	0.299713
$154$	0	0
$155$	−21.4533	−1.72317
$156$	0	0
$157$	−9.17743	−0.732439	−0.366219	−	0.930529i	$-0.619348\pi$
$157$	−9.17743	−0.732439	−0.366219	+	0.930529i	$0.619348\pi$
$158$	0	0
$159$	−10.7084	−0.849228
$160$	0	0
$161$	8.12733	0.640523
$162$	0	0
$163$	−4.11130	−0.322022	−0.161011	−	0.986953i	$-0.551475\pi$
$163$	−4.11130	−0.322022	−0.161011	+	0.986953i	$0.551475\pi$
$164$	0	0
$165$	−15.4899	−1.20588
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.9978	−0.999827
$170$	0	0
$171$	−0.502366	−0.0384169
$172$	0	0
$173$	1.79124	0.136185	0.0680925	−	0.997679i	$-0.478309\pi$
$173$	1.79124	0.136185	0.0680925	+	0.997679i	$0.478309\pi$
$174$	0	0
$175$	−22.0484	−1.66670
$176$	0	0
$177$	3.54507	0.266464
$178$	0	0
$179$	−8.69277	−0.649728	−0.324864	−	0.945761i	$-0.605319\pi$
$179$	−8.69277	−0.649728	−0.324864	+	0.945761i	$0.605319\pi$
$180$	0	0
$181$	−11.5836	−0.861003	−0.430501	−	0.902590i	$-0.641663\pi$
$181$	−11.5836	−0.861003	−0.430501	+	0.902590i	$0.641663\pi$
$182$	0	0
$183$	3.62786	0.268179
$184$	0	0
$185$	0.572789	0.0421123
$186$	0	0
$187$	−15.8761	−1.16098
$188$	0	0
$189$	−2.72774	−0.198414
$190$	0	0
$191$	−3.96251	−0.286717	−0.143359	−	0.989671i	$-0.545790\pi$
$191$	−3.96251	−0.286717	−0.143359	+	0.989671i	$0.545790\pi$
$192$	0	0
$193$	6.81160	0.490310	0.245155	−	0.969484i	$-0.421161\pi$
$193$	6.81160	0.490310	0.245155	+	0.969484i	$0.421161\pi$
$194$	0	0
$195$	−0.171518	−0.0122827
$196$	0	0
$197$	−7.39262	−0.526702	−0.263351	−	0.964700i	$-0.584828\pi$
$197$	−7.39262	−0.526702	−0.263351	+	0.964700i	$0.584828\pi$
$198$	0	0
$199$	16.2256	1.15020	0.575100	−	0.818083i	$-0.304963\pi$
$199$	16.2256	1.15020	0.575100	+	0.818083i	$0.304963\pi$
$200$	0	0
$201$	−0.477808	−0.0337020
$202$	0	0
$203$	17.0641	1.19766
$204$	0	0
$205$	−14.0856	−0.983784
$206$	0	0
$207$	−2.97951	−0.207090
$208$	0	0
$209$	2.15136	0.148813
$210$	0	0
$211$	−9.20559	−0.633739	−0.316869	−	0.948469i	$-0.602632\pi$
$211$	−9.20559	−0.633739	−0.316869	+	0.948469i	$0.602632\pi$
$212$	0	0
$213$	−0.963732	−0.0660338
$214$	0	0
$215$	−17.9037	−1.22102
$216$	0	0
$217$	16.1786	1.09828
$218$	0	0
$219$	−12.5153	−0.845708
$220$	0	0
$221$	−0.175796	−0.0118253
$222$	0	0
$223$	1.77142	0.118623	0.0593114	−	0.998240i	$-0.481109\pi$
$223$	1.77142	0.118623	0.0593114	+	0.998240i	$0.481109\pi$
$224$	0	0
$225$	8.08304	0.538869
$226$	0	0
$227$	−3.39753	−0.225502	−0.112751	−	0.993623i	$-0.535966\pi$
$227$	−3.39753	−0.225502	−0.112751	+	0.993623i	$0.535966\pi$
$228$	0	0
$229$	11.6919	0.772622	0.386311	−	0.922369i	$-0.373749\pi$
$229$	11.6919	0.772622	0.386311	+	0.922369i	$0.373749\pi$
$230$	0	0
$231$	11.6814	0.768582
$232$	0	0
$233$	20.2352	1.32565	0.662825	−	0.748775i	$-0.269357\pi$
$233$	20.2352	1.32565	0.662825	+	0.748775i	$0.269357\pi$
$234$	0	0
$235$	−30.3698	−1.98110
$236$	0	0
$237$	1.53153	0.0994837
$238$	0	0
$239$	−8.35872	−0.540680	−0.270340	−	0.962765i	$-0.587136\pi$
$239$	−8.35872	−0.540680	−0.270340	+	0.962765i	$0.587136\pi$
$240$	0	0
$241$	−7.49687	−0.482916	−0.241458	−	0.970411i	$-0.577626\pi$
$241$	−7.49687	−0.482916	−0.241458	+	0.970411i	$0.577626\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.59356	0.101809
$246$	0	0
$247$	0.0238219	0.00151575
$248$	0	0
$249$	13.4089	0.849757
$250$	0	0
$251$	−9.63798	−0.608344	−0.304172	−	0.952617i	$-0.598380\pi$
$251$	−9.63798	−0.608344	−0.304172	+	0.952617i	$0.598380\pi$
$252$	0	0
$253$	12.7596	0.802190
$254$	0	0
$255$	13.4093	0.839723
$256$	0	0
$257$	16.0721	1.00255	0.501275	−	0.865288i	$-0.332865\pi$
$257$	16.0721	1.00255	0.501275	+	0.865288i	$0.332865\pi$
$258$	0	0
$259$	−0.431960	−0.0268407
$260$	0	0
$261$	−6.25576	−0.387222
$262$	0	0
$263$	−13.8779	−0.855745	−0.427872	−	0.903839i	$-0.640737\pi$
$263$	−13.8779	−0.855745	−0.427872	+	0.903839i	$0.640737\pi$
$264$	0	0
$265$	−38.7327	−2.37933
$266$	0	0
$267$	1.01339	0.0620184
$268$	0	0
$269$	−11.7466	−0.716202	−0.358101	−	0.933683i	$-0.616576\pi$
$269$	−11.7466	−0.716202	−0.358101	+	0.933683i	$0.616576\pi$
$270$	0	0
$271$	−27.0126	−1.64090	−0.820450	−	0.571718i	$-0.806277\pi$
$271$	−27.0126	−1.64090	−0.820450	+	0.571718i	$0.806277\pi$
$272$	0	0
$273$	0.129348	0.00782849
$274$	0	0
$275$	−34.6153	−2.08738
$276$	0	0
$277$	−27.7942	−1.66999	−0.834997	−	0.550255i	$-0.814531\pi$
$277$	−27.7942	−1.66999	−0.834997	+	0.550255i	$0.814531\pi$
$278$	0	0
$279$	−5.93115	−0.355089
$280$	0	0
$281$	16.0329	0.956443	0.478221	−	0.878239i	$-0.341282\pi$
$281$	16.0329	0.956443	0.478221	+	0.878239i	$0.341282\pi$
$282$	0	0
$283$	9.97967	0.593230	0.296615	−	0.954997i	$-0.404142\pi$
$283$	9.97967	0.593230	0.296615	+	0.954997i	$0.404142\pi$
$284$	0	0
$285$	−1.81708	−0.107635
$286$	0	0
$287$	10.6225	0.627025
$288$	0	0
$289$	−3.25630	−0.191547
$290$	0	0
$291$	−1.19560	−0.0700872
$292$	0	0
$293$	−6.84400	−0.399831	−0.199915	−	0.979813i	$-0.564067\pi$
$293$	−6.84400	−0.399831	−0.199915	+	0.979813i	$0.564067\pi$
$294$	0	0
$295$	12.8227	0.746566
$296$	0	0
$297$	−4.28246	−0.248493
$298$	0	0
$299$	0.141287	0.00817081
$300$	0	0
$301$	13.5018	0.778230
$302$	0	0
$303$	2.15983	0.124079
$304$	0	0
$305$	13.1222	0.751373
$306$	0	0
$307$	1.88679	0.107685	0.0538424	−	0.998549i	$-0.482853\pi$
$307$	1.88679	0.107685	0.0538424	+	0.998549i	$0.482853\pi$
$308$	0	0
$309$	13.6082	0.774144
$310$	0	0
$311$	−18.8665	−1.06982	−0.534910	−	0.844909i	$-0.679654\pi$
$311$	−18.8665	−1.06982	−0.534910	+	0.844909i	$0.679654\pi$
$312$	0	0
$313$	28.2178	1.59497	0.797483	−	0.603341i	$-0.206164\pi$
$313$	28.2178	1.59497	0.797483	+	0.603341i	$0.206164\pi$
$314$	0	0
$315$	−9.86637	−0.555907
$316$	0	0
$317$	−17.9184	−1.00640	−0.503198	−	0.864171i	$-0.667843\pi$
$317$	−17.9184	−1.00640	−0.503198	+	0.864171i	$0.667843\pi$
$318$	0	0
$319$	26.7900	1.49995
$320$	0	0
$321$	3.76340	0.210052
$322$	0	0
$323$	−1.86240	−0.103627
$324$	0	0
$325$	−0.383293	−0.0212613
$326$	0	0
$327$	−11.9192	−0.659131
$328$	0	0
$329$	22.9029	1.26268
$330$	0	0
$331$	−6.98539	−0.383952	−0.191976	−	0.981400i	$-0.561490\pi$
$331$	−6.98539	−0.383952	−0.191976	+	0.981400i	$0.561490\pi$
$332$	0	0
$333$	0.158358	0.00867797
$334$	0	0
$335$	−1.72826	−0.0944247
$336$	0	0
$337$	11.8306	0.644455	0.322227	−	0.946662i	$-0.395568\pi$
$337$	11.8306	0.644455	0.322227	+	0.946662i	$0.395568\pi$
$338$	0	0
$339$	−5.99961	−0.325854
$340$	0	0
$341$	25.3999	1.37548
$342$	0	0
$343$	17.8924	0.966100
$344$	0	0
$345$	−10.7770	−0.580216
$346$	0	0
$347$	−34.1259	−1.83197	−0.915987	−	0.401208i	$-0.868591\pi$
$347$	−34.1259	−1.83197	−0.915987	+	0.401208i	$0.868591\pi$
$348$	0	0
$349$	28.2829	1.51395	0.756975	−	0.653444i	$-0.226677\pi$
$349$	28.2829	1.51395	0.756975	+	0.653444i	$0.226677\pi$
$350$	0	0
$351$	−0.0474194	−0.00253106
$352$	0	0
$353$	−0.561669	−0.0298946	−0.0149473	−	0.999888i	$-0.504758\pi$
$353$	−0.561669	−0.0298946	−0.0149473	+	0.999888i	$0.504758\pi$
$354$	0	0
$355$	−3.48586	−0.185010
$356$	0	0
$357$	−10.1124	−0.535206
$358$	0	0
$359$	1.11628	0.0589149	0.0294574	−	0.999566i	$-0.490622\pi$
$359$	1.11628	0.0589149	0.0294574	+	0.999566i	$0.490622\pi$
$360$	0	0
$361$	−18.7476	−0.986717
$362$	0	0
$363$	7.33944	0.385221
$364$	0	0
$365$	−45.2686	−2.36947
$366$	0	0
$367$	6.04257	0.315420	0.157710	−	0.987485i	$-0.449589\pi$
$367$	6.04257	0.315420	0.157710	+	0.987485i	$0.449589\pi$
$368$	0	0
$369$	−3.89424	−0.202726
$370$	0	0
$371$	29.2096	1.51649
$372$	0	0
$373$	29.5233	1.52866	0.764328	−	0.644828i	$-0.223071\pi$
$373$	29.5233	1.52866	0.764328	+	0.644828i	$0.223071\pi$
$374$	0	0
$375$	11.1515	0.575861
$376$	0	0
$377$	0.296644	0.0152780
$378$	0	0
$379$	30.8528	1.58480	0.792401	−	0.610000i	$-0.208831\pi$
$379$	30.8528	1.58480	0.792401	+	0.610000i	$0.208831\pi$
$380$	0	0
$381$	−3.12683	−0.160192
$382$	0	0
$383$	11.1831	0.571427	0.285714	−	0.958315i	$-0.407769\pi$
$383$	11.1831	0.571427	0.285714	+	0.958315i	$0.407769\pi$
$384$	0	0
$385$	42.2523	2.15338
$386$	0	0
$387$	−4.94980	−0.251613
$388$	0	0
$389$	24.2164	1.22782	0.613911	−	0.789375i	$-0.289595\pi$
$389$	24.2164	1.22782	0.613911	+	0.789375i	$0.289595\pi$
$390$	0	0
$391$	−11.0458	−0.558609
$392$	0	0
$393$	17.0808	0.861614
$394$	0	0
$395$	5.53962	0.278729
$396$	0	0
$397$	0.297814	0.0149469	0.00747343	−	0.999972i	$-0.497621\pi$
$397$	0.297814	0.0149469	0.00747343	+	0.999972i	$0.497621\pi$
$398$	0	0
$399$	1.37033	0.0686021
$400$	0	0
$401$	1.84713	0.0922413	0.0461207	−	0.998936i	$-0.485314\pi$
$401$	1.84713	0.0922413	0.0461207	+	0.998936i	$0.485314\pi$
$402$	0	0
$403$	0.281252	0.0140101
$404$	0	0
$405$	3.61705	0.179733
$406$	0	0
$407$	−0.678162	−0.0336152
$408$	0	0
$409$	−28.7688	−1.42252	−0.711262	−	0.702927i	$-0.751876\pi$
$409$	−28.7688	−1.42252	−0.711262	+	0.702927i	$0.751876\pi$
$410$	0	0
$411$	−5.91181	−0.291608
$412$	0	0
$413$	−9.67003	−0.475831
$414$	0	0
$415$	48.5008	2.38081
$416$	0	0
$417$	−19.2110	−0.940768
$418$	0	0
$419$	31.6216	1.54482	0.772408	−	0.635127i	$-0.219052\pi$
$419$	31.6216	1.54482	0.772408	+	0.635127i	$0.219052\pi$
$420$	0	0
$421$	−4.86687	−0.237197	−0.118598	−	0.992942i	$-0.537840\pi$
$421$	−4.86687	−0.237197	−0.118598	+	0.992942i	$0.537840\pi$
$422$	0	0
$423$	−8.39628	−0.408241
$424$	0	0
$425$	29.9658	1.45356
$426$	0	0
$427$	−9.89587	−0.478895
$428$	0	0
$429$	0.203072	0.00980439
$430$	0	0
$431$	19.5696	0.942635	0.471317	−	0.881964i	$-0.343779\pi$
$431$	19.5696	0.942635	0.471317	+	0.881964i	$0.343779\pi$
$432$	0	0
$433$	−20.2099	−0.971224	−0.485612	−	0.874174i	$-0.661403\pi$
$433$	−20.2099	−0.971224	−0.485612	+	0.874174i	$0.661403\pi$
$434$	0	0
$435$	−22.6274	−1.08490
$436$	0	0
$437$	1.49681	0.0716019
$438$	0	0
$439$	−24.6452	−1.17625	−0.588126	−	0.808769i	$-0.700134\pi$
$439$	−24.6452	−1.17625	−0.588126	+	0.808769i	$0.700134\pi$
$440$	0	0
$441$	0.440569	0.0209795
$442$	0	0
$443$	−26.1398	−1.24194	−0.620969	−	0.783835i	$-0.713261\pi$
$443$	−26.1398	−1.24194	−0.620969	+	0.783835i	$0.713261\pi$
$444$	0	0
$445$	3.66548	0.173760
$446$	0	0
$447$	12.4216	0.587523
$448$	0	0
$449$	−0.704072	−0.0332272	−0.0166136	−	0.999862i	$-0.505289\pi$
$449$	−0.704072	−0.0332272	−0.0166136	+	0.999862i	$0.505289\pi$
$450$	0	0
$451$	16.6769	0.785285
$452$	0	0
$453$	19.4910	0.915765
$454$	0	0
$455$	0.467857	0.0219335
$456$	0	0
$457$	40.5289	1.89586	0.947931	−	0.318476i	$-0.103171\pi$
$457$	40.5289	1.89586	0.947931	+	0.318476i	$0.103171\pi$
$458$	0	0
$459$	3.70725	0.173040
$460$	0	0
$461$	−15.2707	−0.711228	−0.355614	−	0.934633i	$-0.615728\pi$
$461$	−15.2707	−0.711228	−0.355614	+	0.934633i	$0.615728\pi$
$462$	0	0
$463$	−22.1528	−1.02953	−0.514763	−	0.857333i	$-0.672120\pi$
$463$	−22.1528	−1.02953	−0.514763	+	0.857333i	$0.672120\pi$
$464$	0	0
$465$	−21.4533	−0.994871
$466$	0	0
$467$	−20.2118	−0.935291	−0.467646	−	0.883916i	$-0.654898\pi$
$467$	−20.2118	−0.935291	−0.467646	+	0.883916i	$0.654898\pi$
$468$	0	0
$469$	1.30334	0.0601825
$470$	0	0
$471$	−9.17743	−0.422874
$472$	0	0
$473$	21.1973	0.974654
$474$	0	0
$475$	−4.06065	−0.186315
$476$	0	0
$477$	−10.7084	−0.490302
$478$	0	0
$479$	−32.2890	−1.47532	−0.737661	−	0.675171i	$-0.764070\pi$
$479$	−32.2890	−1.47532	−0.737661	+	0.675171i	$0.764070\pi$
$480$	0	0
$481$	−0.00750925	−0.000342392 0
$482$	0	0
$483$	8.12733	0.369806
$484$	0	0
$485$	−4.32454	−0.196367
$486$	0	0
$487$	−33.6362	−1.52420	−0.762101	−	0.647458i	$-0.775832\pi$
$487$	−33.6362	−1.52420	−0.762101	+	0.647458i	$0.775832\pi$
$488$	0	0
$489$	−4.11130	−0.185919
$490$	0	0
$491$	−0.820863	−0.0370450	−0.0185225	−	0.999828i	$-0.505896\pi$
$491$	−0.820863	−0.0370450	−0.0185225	+	0.999828i	$0.505896\pi$
$492$	0	0
$493$	−23.1917	−1.04450
$494$	0	0
$495$	−15.4899	−0.696217
$496$	0	0
$497$	2.62881	0.117918
$498$	0	0
$499$	34.5903	1.54847	0.774237	−	0.632896i	$-0.218134\pi$
$499$	34.5903	1.54847	0.774237	+	0.632896i	$0.218134\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	28.2747	1.26071	0.630354	−	0.776308i	$-0.282910\pi$
$503$	28.2747	1.26071	0.630354	+	0.776308i	$0.282910\pi$
$504$	0	0
$505$	7.81222	0.347639
$506$	0	0
$507$	−12.9978	−0.577250
$508$	0	0
$509$	−7.01436	−0.310906	−0.155453	−	0.987843i	$-0.549684\pi$
$509$	−7.01436	−0.310906	−0.155453	+	0.987843i	$0.549684\pi$
$510$	0	0
$511$	34.1386	1.51020
$512$	0	0
$513$	−0.502366	−0.0221800
$514$	0	0
$515$	49.2215	2.16896
$516$	0	0
$517$	35.9567	1.58137
$518$	0	0
$519$	1.79124	0.0786265
$520$	0	0
$521$	23.4917	1.02919	0.514594	−	0.857434i	$-0.327943\pi$
$521$	23.4917	1.02919	0.514594	+	0.857434i	$0.327943\pi$
$522$	0	0
$523$	−13.3081	−0.581924	−0.290962	−	0.956735i	$-0.593975\pi$
$523$	−13.3081	−0.581924	−0.290962	+	0.956735i	$0.593975\pi$
$524$	0	0
$525$	−22.0484	−0.962273
$526$	0	0
$527$	−21.9882	−0.957823
$528$	0	0
$529$	−14.1225	−0.614023
$530$	0	0
$531$	3.54507	0.153843
$532$	0	0
$533$	0.184662	0.00799862
$534$	0	0
$535$	13.6124	0.588515
$536$	0	0
$537$	−8.69277	−0.375121
$538$	0	0
$539$	−1.88672	−0.0812666
$540$	0	0
$541$	−1.93696	−0.0832765	−0.0416383	−	0.999133i	$-0.513258\pi$
$541$	−1.93696	−0.0832765	−0.0416383	+	0.999133i	$0.513258\pi$
$542$	0	0
$543$	−11.5836	−0.497100
$544$	0	0
$545$	−43.1122	−1.84672
$546$	0	0
$547$	18.2239	0.779200	0.389600	−	0.920984i	$-0.372613\pi$
$547$	18.2239	0.779200	0.389600	+	0.920984i	$0.372613\pi$
$548$	0	0
$549$	3.62786	0.154833
$550$	0	0
$551$	3.14268	0.133883
$552$	0	0
$553$	−4.17762	−0.177651
$554$	0	0
$555$	0.572789	0.0243135
$556$	0	0
$557$	4.50775	0.190999	0.0954997	−	0.995429i	$-0.469555\pi$
$557$	4.50775	0.190999	0.0954997	+	0.995429i	$0.469555\pi$
$558$	0	0
$559$	0.234717	0.00992746
$560$	0	0
$561$	−15.8761	−0.670291
$562$	0	0
$563$	3.26172	0.137465	0.0687325	−	0.997635i	$-0.478104\pi$
$563$	3.26172	0.137465	0.0687325	+	0.997635i	$0.478104\pi$
$564$	0	0
$565$	−21.7009	−0.912963
$566$	0	0
$567$	−2.72774	−0.114554
$568$	0	0
$569$	18.6572	0.782151	0.391075	−	0.920359i	$-0.372103\pi$
$569$	18.6572	0.782151	0.391075	+	0.920359i	$0.372103\pi$
$570$	0	0
$571$	4.26021	0.178284	0.0891422	−	0.996019i	$-0.471587\pi$
$571$	4.26021	0.178284	0.0891422	+	0.996019i	$0.471587\pi$
$572$	0	0
$573$	−3.96251	−0.165536
$574$	0	0
$575$	−24.0835	−1.00435
$576$	0	0
$577$	9.98097	0.415513	0.207757	−	0.978181i	$-0.433384\pi$
$577$	9.98097	0.415513	0.207757	+	0.978181i	$0.433384\pi$
$578$	0	0
$579$	6.81160	0.283080
$580$	0	0
$581$	−36.5761	−1.51743
$582$	0	0
$583$	45.8581	1.89925
$584$	0	0
$585$	−0.171518	−0.00709141
$586$	0	0
$587$	−36.1902	−1.49373	−0.746864	−	0.664977i	$-0.768441\pi$
$587$	−36.1902	−1.49373	−0.746864	+	0.664977i	$0.768441\pi$
$588$	0	0
$589$	2.97961	0.122773
$590$	0	0
$591$	−7.39262	−0.304092
$592$	0	0
$593$	−2.81631	−0.115652	−0.0578261	−	0.998327i	$-0.518417\pi$
$593$	−2.81631	−0.115652	−0.0578261	+	0.998327i	$0.518417\pi$
$594$	0	0
$595$	−36.5771	−1.49951
$596$	0	0
$597$	16.2256	0.664069
$598$	0	0
$599$	12.4462	0.508537	0.254269	−	0.967134i	$-0.418165\pi$
$599$	12.4462	0.508537	0.254269	+	0.967134i	$0.418165\pi$
$600$	0	0
$601$	0.462497	0.0188657	0.00943283	−	0.999956i	$-0.496997\pi$
$601$	0.462497	0.0188657	0.00943283	+	0.999956i	$0.496997\pi$
$602$	0	0
$603$	−0.477808	−0.0194579
$604$	0	0
$605$	26.5471	1.07929
$606$	0	0
$607$	−10.5665	−0.428883	−0.214441	−	0.976737i	$-0.568793\pi$
$607$	−10.5665	−0.428883	−0.214441	+	0.976737i	$0.568793\pi$
$608$	0	0
$609$	17.0641	0.691472
$610$	0	0
$611$	0.398147	0.0161073
$612$	0	0
$613$	−48.6267	−1.96401	−0.982007	−	0.188846i	$-0.939525\pi$
$613$	−48.6267	−1.96401	−0.982007	+	0.188846i	$0.939525\pi$
$614$	0	0
$615$	−14.0856	−0.567988
$616$	0	0
$617$	1.72932	0.0696197	0.0348098	−	0.999394i	$-0.488917\pi$
$617$	1.72932	0.0696197	0.0348098	+	0.999394i	$0.488917\pi$
$618$	0	0
$619$	48.2593	1.93971	0.969853	−	0.243692i	$-0.0783587\pi$
$619$	48.2593	1.93971	0.969853	+	0.243692i	$0.0783587\pi$
$620$	0	0
$621$	−2.97951	−0.119564
$622$	0	0
$623$	−2.76426	−0.110748
$624$	0	0
$625$	−0.0796737	−0.00318695
$626$	0	0
$627$	2.15136	0.0859172
$628$	0	0
$629$	0.587073	0.0234081
$630$	0	0
$631$	−3.15911	−0.125762	−0.0628812	−	0.998021i	$-0.520029\pi$
$631$	−3.15911	−0.125762	−0.0628812	+	0.998021i	$0.520029\pi$
$632$	0	0
$633$	−9.20559	−0.365889
$634$	0	0
$635$	−11.3099	−0.448819
$636$	0	0
$637$	−0.0208915	−0.000827752 0
$638$	0	0
$639$	−0.963732	−0.0381246
$640$	0	0
$641$	30.2179	1.19353	0.596767	−	0.802415i	$-0.296452\pi$
$641$	30.2179	1.19353	0.596767	+	0.802415i	$0.296452\pi$
$642$	0	0
$643$	31.1151	1.22706	0.613531	−	0.789671i	$-0.289749\pi$
$643$	31.1151	1.22706	0.613531	+	0.789671i	$0.289749\pi$
$644$	0	0
$645$	−17.9037	−0.704957
$646$	0	0
$647$	23.7860	0.935125	0.467563	−	0.883960i	$-0.345132\pi$
$647$	23.7860	0.935125	0.467563	+	0.883960i	$0.345132\pi$
$648$	0	0
$649$	−15.1816	−0.595930
$650$	0	0
$651$	16.1786	0.634091
$652$	0	0
$653$	−47.4018	−1.85497	−0.927487	−	0.373855i	$-0.878036\pi$
$653$	−47.4018	−1.85497	−0.927487	+	0.373855i	$0.878036\pi$
$654$	0	0
$655$	61.7822	2.41403
$656$	0	0
$657$	−12.5153	−0.488270
$658$	0	0
$659$	5.52732	0.215314	0.107657	−	0.994188i	$-0.465665\pi$
$659$	5.52732	0.215314	0.107657	+	0.994188i	$0.465665\pi$
$660$	0	0
$661$	−8.39078	−0.326363	−0.163182	−	0.986596i	$-0.552176\pi$
$661$	−8.39078	−0.326363	−0.163182	+	0.986596i	$0.552176\pi$
$662$	0	0
$663$	−0.175796	−0.00682733
$664$	0	0
$665$	4.95653	0.192206
$666$	0	0
$667$	18.6391	0.721708
$668$	0	0
$669$	1.77142	0.0684869
$670$	0	0
$671$	−15.5362	−0.599767
$672$	0	0
$673$	−26.2165	−1.01057	−0.505286	−	0.862952i	$-0.668613\pi$
$673$	−26.2165	−1.01057	−0.505286	+	0.862952i	$0.668613\pi$
$674$	0	0
$675$	8.08304	0.311116
$676$	0	0
$677$	36.4778	1.40196	0.700978	−	0.713183i	$-0.252747\pi$
$677$	36.4778	1.40196	0.700978	+	0.713183i	$0.252747\pi$
$678$	0	0
$679$	3.26128	0.125156
$680$	0	0
$681$	−3.39753	−0.130194
$682$	0	0
$683$	−33.8618	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$683$	−33.8618	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$684$	0	0
$685$	−21.3833	−0.817014
$686$	0	0
$687$	11.6919	0.446074
$688$	0	0
$689$	0.507784	0.0193450
$690$	0	0
$691$	13.0531	0.496562	0.248281	−	0.968688i	$-0.420134\pi$
$691$	13.0531	0.496562	0.248281	+	0.968688i	$0.420134\pi$
$692$	0	0
$693$	11.6814	0.443741
$694$	0	0
$695$	−69.4872	−2.63580
$696$	0	0
$697$	−14.4369	−0.546837
$698$	0	0
$699$	20.2352	0.765364
$700$	0	0
$701$	7.42917	0.280596	0.140298	−	0.990109i	$-0.455194\pi$
$701$	7.42917	0.280596	0.140298	+	0.990109i	$0.455194\pi$
$702$	0	0
$703$	−0.0795538	−0.00300043
$704$	0	0
$705$	−30.3698	−1.14379
$706$	0	0
$707$	−5.89146	−0.221571
$708$	0	0
$709$	−16.6979	−0.627104	−0.313552	−	0.949571i	$-0.601519\pi$
$709$	−16.6979	−0.627104	−0.313552	+	0.949571i	$0.601519\pi$
$710$	0	0
$711$	1.53153	0.0574369
$712$	0	0
$713$	17.6719	0.661818
$714$	0	0
$715$	0.734520	0.0274695
$716$	0	0
$717$	−8.35872	−0.312162
$718$	0	0
$719$	−25.5510	−0.952892	−0.476446	−	0.879204i	$-0.658075\pi$
$719$	−25.5510	−0.952892	−0.476446	+	0.879204i	$0.658075\pi$
$720$	0	0
$721$	−37.1196	−1.38241
$722$	0	0
$723$	−7.49687	−0.278811
$724$	0	0
$725$	−50.5655	−1.87796
$726$	0	0
$727$	24.2347	0.898814	0.449407	−	0.893327i	$-0.351635\pi$
$727$	24.2347	0.898814	0.449407	+	0.893327i	$0.351635\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−18.3502	−0.678705
$732$	0	0
$733$	−41.0374	−1.51575	−0.757875	−	0.652400i	$-0.773762\pi$
$733$	−41.0374	−1.51575	−0.757875	+	0.652400i	$0.773762\pi$
$734$	0	0
$735$	1.59356	0.0587793
$736$	0	0
$737$	2.04619	0.0753725
$738$	0	0
$739$	31.7107	1.16650	0.583248	−	0.812294i	$-0.301782\pi$
$739$	31.7107	1.16650	0.583248	+	0.812294i	$0.301782\pi$
$740$	0	0
$741$	0.0238219	0.000875120 0
$742$	0	0
$743$	−31.7721	−1.16560	−0.582802	−	0.812614i	$-0.698044\pi$
$743$	−31.7721	−1.16560	−0.582802	+	0.812614i	$0.698044\pi$
$744$	0	0
$745$	44.9297	1.64610
$746$	0	0
$747$	13.4089	0.490607
$748$	0	0
$749$	−10.2656	−0.375096
$750$	0	0
$751$	23.5750	0.860264	0.430132	−	0.902766i	$-0.358467\pi$
$751$	23.5750	0.860264	0.430132	+	0.902766i	$0.358467\pi$
$752$	0	0
$753$	−9.63798	−0.351227
$754$	0	0
$755$	70.4997	2.56575
$756$	0	0
$757$	20.9217	0.760411	0.380206	−	0.924902i	$-0.375853\pi$
$757$	20.9217	0.760411	0.380206	+	0.924902i	$0.375853\pi$
$758$	0	0
$759$	12.7596	0.463145
$760$	0	0
$761$	−6.74661	−0.244564	−0.122282	−	0.992495i	$-0.539021\pi$
$761$	−6.74661	−0.244564	−0.122282	+	0.992495i	$0.539021\pi$
$762$	0	0
$763$	32.5124	1.17703
$764$	0	0
$765$	13.4093	0.484814
$766$	0	0
$767$	−0.168105	−0.00606993
$768$	0	0
$769$	11.1193	0.400974	0.200487	−	0.979696i	$-0.435748\pi$
$769$	11.1193	0.400974	0.200487	+	0.979696i	$0.435748\pi$
$770$	0	0
$771$	16.0721	0.578822
$772$	0	0
$773$	−42.9830	−1.54599	−0.772995	−	0.634412i	$-0.781242\pi$
$773$	−42.9830	−1.54599	−0.772995	+	0.634412i	$0.781242\pi$
$774$	0	0
$775$	−47.9417	−1.72212
$776$	0	0
$777$	−0.431960	−0.0154965
$778$	0	0
$779$	1.95633	0.0700930
$780$	0	0
$781$	4.12714	0.147681
$782$	0	0
$783$	−6.25576	−0.223563
$784$	0	0
$785$	−33.1952	−1.18479
$786$	0	0
$787$	3.90798	0.139304	0.0696522	−	0.997571i	$-0.477811\pi$
$787$	3.90798	0.139304	0.0696522	+	0.997571i	$0.477811\pi$
$788$	0	0
$789$	−13.8779	−0.494065
$790$	0	0
$791$	16.3654	0.581886
$792$	0	0
$793$	−0.172031	−0.00610901
$794$	0	0
$795$	−38.7327	−1.37371
$796$	0	0
$797$	20.2466	0.717171	0.358586	−	0.933497i	$-0.383259\pi$
$797$	20.2466	0.717171	0.358586	+	0.933497i	$0.383259\pi$
$798$	0	0
$799$	−31.1271	−1.10120
$800$	0	0
$801$	1.01339	0.0358063
$802$	0	0
$803$	53.5964	1.89138
$804$	0	0
$805$	29.3969	1.03611
$806$	0	0
$807$	−11.7466	−0.413499
$808$	0	0
$809$	−27.8278	−0.978372	−0.489186	−	0.872179i	$-0.662706\pi$
$809$	−27.8278	−0.978372	−0.489186	+	0.872179i	$0.662706\pi$
$810$	0	0
$811$	29.0208	1.01906	0.509529	−	0.860453i	$-0.329820\pi$
$811$	29.0208	1.01906	0.509529	+	0.860453i	$0.329820\pi$
$812$	0	0
$813$	−27.0126	−0.947374
$814$	0	0
$815$	−14.8708	−0.520900
$816$	0	0
$817$	2.48662	0.0869957
$818$	0	0
$819$	0.129348	0.00451978
$820$	0	0
$821$	−44.3461	−1.54769	−0.773845	−	0.633374i	$-0.781669\pi$
$821$	−44.3461	−1.54769	−0.773845	+	0.633374i	$0.781669\pi$
$822$	0	0
$823$	25.6252	0.893239	0.446620	−	0.894724i	$-0.352628\pi$
$823$	25.6252	0.893239	0.446620	+	0.894724i	$0.352628\pi$
$824$	0	0
$825$	−34.6153	−1.20515
$826$	0	0
$827$	14.0006	0.486848	0.243424	−	0.969920i	$-0.421729\pi$
$827$	14.0006	0.486848	0.243424	+	0.969920i	$0.421729\pi$
$828$	0	0
$829$	−12.6296	−0.438643	−0.219321	−	0.975653i	$-0.570384\pi$
$829$	−12.6296	−0.438643	−0.219321	+	0.975653i	$0.570384\pi$
$830$	0	0
$831$	−27.7942	−0.964171
$832$	0	0
$833$	1.63330	0.0565904
$834$	0	0
$835$	3.61705	0.125173
$836$	0	0
$837$	−5.93115	−0.205010
$838$	0	0
$839$	37.8724	1.30750	0.653750	−	0.756710i	$-0.273195\pi$
$839$	37.8724	1.30750	0.653750	+	0.756710i	$0.273195\pi$
$840$	0	0
$841$	10.1345	0.349466
$842$	0	0
$843$	16.0329	0.552202
$844$	0	0
$845$	−47.0135	−1.61731
$846$	0	0
$847$	−20.0201	−0.687898
$848$	0	0
$849$	9.97967	0.342501
$850$	0	0
$851$	−0.471829	−0.0161741
$852$	0	0
$853$	31.1052	1.06502	0.532512	−	0.846423i	$-0.321248\pi$
$853$	31.1052	1.06502	0.532512	+	0.846423i	$0.321248\pi$
$854$	0	0
$855$	−1.81708	−0.0621430
$856$	0	0
$857$	47.6667	1.62826	0.814131	−	0.580681i	$-0.197214\pi$
$857$	47.6667	1.62826	0.814131	+	0.580681i	$0.197214\pi$
$858$	0	0
$859$	−6.93685	−0.236682	−0.118341	−	0.992973i	$-0.537758\pi$
$859$	−6.93685	−0.236682	−0.118341	+	0.992973i	$0.537758\pi$
$860$	0	0
$861$	10.6225	0.362013
$862$	0	0
$863$	−44.3736	−1.51050	−0.755248	−	0.655439i	$-0.772484\pi$
$863$	−44.3736	−1.51050	−0.755248	+	0.655439i	$0.772484\pi$
$864$	0	0
$865$	6.47898	0.220292
$866$	0	0
$867$	−3.25630	−0.110590
$868$	0	0
$869$	−6.55872	−0.222489
$870$	0	0
$871$	0.0226574	0.000767717 0
$872$	0	0
$873$	−1.19560	−0.0404649
$874$	0	0
$875$	−30.4184	−1.02833
$876$	0	0
$877$	−15.2810	−0.516003	−0.258002	−	0.966144i	$-0.583064\pi$
$877$	−15.2810	−0.516003	−0.258002	+	0.966144i	$0.583064\pi$
$878$	0	0
$879$	−6.84400	−0.230842
$880$	0	0
$881$	9.33130	0.314379	0.157190	−	0.987568i	$-0.449757\pi$
$881$	9.33130	0.314379	0.157190	+	0.987568i	$0.449757\pi$
$882$	0	0
$883$	14.6940	0.494492	0.247246	−	0.968953i	$-0.420474\pi$
$883$	14.6940	0.494492	0.247246	+	0.968953i	$0.420474\pi$
$884$	0	0
$885$	12.8227	0.431030
$886$	0	0
$887$	−35.8669	−1.20429	−0.602147	−	0.798385i	$-0.705688\pi$
$887$	−35.8669	−1.20429	−0.602147	+	0.798385i	$0.705688\pi$
$888$	0	0
$889$	8.52918	0.286060
$890$	0	0
$891$	−4.28246	−0.143468
$892$	0	0
$893$	4.21801	0.141150
$894$	0	0
$895$	−31.4422	−1.05100
$896$	0	0
$897$	0.141287	0.00471742
$898$	0	0
$899$	37.1038	1.23748
$900$	0	0
$901$	−39.6986	−1.32255
$902$	0	0
$903$	13.5018	0.449311
$904$	0	0
$905$	−41.8985	−1.39275
$906$	0	0
$907$	−10.0185	−0.332658	−0.166329	−	0.986070i	$-0.553191\pi$
$907$	−10.0185	−0.332658	−0.166329	+	0.986070i	$0.553191\pi$
$908$	0	0
$909$	2.15983	0.0716371
$910$	0	0
$911$	−45.0288	−1.49187	−0.745936	−	0.666018i	$-0.767997\pi$
$911$	−45.0288	−1.49187	−0.745936	+	0.666018i	$0.767997\pi$
$912$	0	0
$913$	−57.4232	−1.90043
$914$	0	0
$915$	13.1222	0.433805
$916$	0	0
$917$	−46.5921	−1.53861
$918$	0	0
$919$	6.14537	0.202717	0.101359	−	0.994850i	$-0.467681\pi$
$919$	6.14537	0.202717	0.101359	+	0.994850i	$0.467681\pi$
$920$	0	0
$921$	1.88679	0.0621718
$922$	0	0
$923$	0.0456996	0.00150422
$924$	0	0
$925$	1.28002	0.0420866
$926$	0	0
$927$	13.6082	0.446952
$928$	0	0
$929$	−24.0472	−0.788964	−0.394482	−	0.918904i	$-0.629076\pi$
$929$	−24.0472	−0.788964	−0.394482	+	0.918904i	$0.629076\pi$
$930$	0	0
$931$	−0.221327	−0.00725370
$932$	0	0
$933$	−18.8665	−0.617661
$934$	0	0
$935$	−57.4247	−1.87799
$936$	0	0
$937$	10.2412	0.334567	0.167283	−	0.985909i	$-0.446501\pi$
$937$	10.2412	0.334567	0.167283	+	0.985909i	$0.446501\pi$
$938$	0	0
$939$	28.2178	0.920854
$940$	0	0
$941$	2.37486	0.0774183	0.0387092	−	0.999251i	$-0.487675\pi$
$941$	2.37486	0.0774183	0.0387092	+	0.999251i	$0.487675\pi$
$942$	0	0
$943$	11.6029	0.377843
$944$	0	0
$945$	−9.86637	−0.320953
$946$	0	0
$947$	10.4904	0.340891	0.170445	−	0.985367i	$-0.445479\pi$
$947$	10.4904	0.340891	0.170445	+	0.985367i	$0.445479\pi$
$948$	0	0
$949$	0.593470	0.0192648
$950$	0	0
$951$	−17.9184	−0.581043
$952$	0	0
$953$	43.4610	1.40784	0.703920	−	0.710279i	$-0.251431\pi$
$953$	43.4610	1.40784	0.703920	+	0.710279i	$0.251431\pi$
$954$	0	0
$955$	−14.3326	−0.463792
$956$	0	0
$957$	26.7900	0.865998
$958$	0	0
$959$	16.1259	0.520732
$960$	0	0
$961$	4.17852	0.134791
$962$	0	0
$963$	3.76340	0.121274
$964$	0	0
$965$	24.6379	0.793122
$966$	0	0
$967$	−14.9457	−0.480623	−0.240311	−	0.970696i	$-0.577250\pi$
$967$	−14.9457	−0.480623	−0.240311	+	0.970696i	$0.577250\pi$
$968$	0	0
$969$	−1.86240	−0.0598288
$970$	0	0
$971$	−7.84860	−0.251873	−0.125937	−	0.992038i	$-0.540194\pi$
$971$	−7.84860	−0.251873	−0.125937	+	0.992038i	$0.540194\pi$
$972$	0	0
$973$	52.4027	1.67995
$974$	0	0
$975$	−0.383293	−0.0122752
$976$	0	0
$977$	−43.6046	−1.39503	−0.697517	−	0.716568i	$-0.745712\pi$
$977$	−43.6046	−1.39503	−0.697517	+	0.716568i	$0.745712\pi$
$978$	0	0
$979$	−4.33979	−0.138700
$980$	0	0
$981$	−11.9192	−0.380549
$982$	0	0
$983$	−49.1287	−1.56696	−0.783481	−	0.621416i	$-0.786558\pi$
$983$	−49.1287	−1.56696	−0.783481	+	0.621416i	$0.786558\pi$
$984$	0	0
$985$	−26.7395	−0.851990
$986$	0	0
$987$	22.9029	0.729007
$988$	0	0
$989$	14.7480	0.468959
$990$	0	0
$991$	4.17577	0.132648	0.0663239	−	0.997798i	$-0.478873\pi$
$991$	4.17577	0.132648	0.0663239	+	0.997798i	$0.478873\pi$
$992$	0	0
$993$	−6.98539	−0.221675
$994$	0	0
$995$	58.6887	1.86056
$996$	0	0
$997$	−4.40338	−0.139456	−0.0697282	−	0.997566i	$-0.522213\pi$
$997$	−4.40338	−0.139456	−0.0697282	+	0.997566i	$0.522213\pi$
$998$	0	0
$999$	0.158358	0.00501023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.w.1.7		7
4.3	odd	2		4008.2.a.g.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.g.1.7	✓	7	4.3	odd	2
8016.2.a.w.1.7		7	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.61705 q^{5} -2.72774 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.61705 q^{5} -2.72774 q^{7} +1.00000 q^{9} -4.28246 q^{11} -0.0474194 q^{13} +3.61705 q^{15} +3.70725 q^{17} -0.502366 q^{19} -2.72774 q^{21} -2.97951 q^{23} +8.08304 q^{25} +1.00000 q^{27} -6.25576 q^{29} -5.93115 q^{31} -4.28246 q^{33} -9.86637 q^{35} +0.158358 q^{37} -0.0474194 q^{39} -3.89424 q^{41} -4.94980 q^{43} +3.61705 q^{45} -8.39628 q^{47} +0.440569 q^{49} +3.70725 q^{51} -10.7084 q^{53} -15.4899 q^{55} -0.502366 q^{57} +3.54507 q^{59} +3.62786 q^{61} -2.72774 q^{63} -0.171518 q^{65} -0.477808 q^{67} -2.97951 q^{69} -0.963732 q^{71} -12.5153 q^{73} +8.08304 q^{75} +11.6814 q^{77} +1.53153 q^{79} +1.00000 q^{81} +13.4089 q^{83} +13.4093 q^{85} -6.25576 q^{87} +1.01339 q^{89} +0.129348 q^{91} -5.93115 q^{93} -1.81708 q^{95} -1.19560 q^{97} -4.28246 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} - 3 q^{5} - 8 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 - 3 * q^5 - 8 * q^7 + 7 * q^9	\( 7 q + 7 q^{3} - 3 q^{5} - 8 q^{7} + 7 q^{9} - q^{11} - 2 q^{13} - 3 q^{15} + 11 q^{17} - 2 q^{19} - 8 q^{21} - 17 q^{23} + 4 q^{25} + 7 q^{27} - 7 q^{29} - 10 q^{31} - q^{33} - 10 q^{35} - 21 q^{37} - 2 q^{39} + 8 q^{41} + 12 q^{43} - 3 q^{45} - 25 q^{47} - 7 q^{49} + 11 q^{51} - 7 q^{53} - 15 q^{55} - 2 q^{57} - 3 q^{59} - 14 q^{61} - 8 q^{63} + 4 q^{65} - 4 q^{67} - 17 q^{69} - 27 q^{71} - 12 q^{73} + 4 q^{75} + 16 q^{77} - 8 q^{79} + 7 q^{81} - 15 q^{83} - 3 q^{85} - 7 q^{87} + 14 q^{89} + 3 q^{91} - 10 q^{93} - 37 q^{95} + 3 q^{97} - q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 - 3 * q^5 - 8 * q^7 + 7 * q^9 - q^11 - 2 * q^13 - 3 * q^15 + 11 * q^17 - 2 * q^19 - 8 * q^21 - 17 * q^23 + 4 * q^25 + 7 * q^27 - 7 * q^29 - 10 * q^31 - q^33 - 10 * q^35 - 21 * q^37 - 2 * q^39 + 8 * q^41 + 12 * q^43 - 3 * q^45 - 25 * q^47 - 7 * q^49 + 11 * q^51 - 7 * q^53 - 15 * q^55 - 2 * q^57 - 3 * q^59 - 14 * q^61 - 8 * q^63 + 4 * q^65 - 4 * q^67 - 17 * q^69 - 27 * q^71 - 12 * q^73 + 4 * q^75 + 16 * q^77 - 8 * q^79 + 7 * q^81 - 15 * q^83 - 3 * q^85 - 7 * q^87 + 14 * q^89 + 3 * q^91 - 10 * q^93 - 37 * q^95 + 3 * q^97 - q^99

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