LMFDB - Embedded newform 8016.2.a.bc.1.3

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 31x^{7} + 24x^{6} + 293x^{5} - 101x^{4} - 864x^{3} - 278x^{2} + 24x + 2 $ x^9 - x^8 - 31x^7 + 24x^6 + 293x^5 - 101x^4 - 864x^3 - 278x^2 + 24*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.74256$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.74256 q^{5} -2.35372 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.74256 q^{5} -2.35372 q^{7} +1.00000 q^{9} -4.17855 q^{11} -1.58172 q^{13} -1.74256 q^{15} -2.99517 q^{17} -6.10269 q^{19} -2.35372 q^{21} +2.00484 q^{23} -1.96348 q^{25} +1.00000 q^{27} -3.60231 q^{29} -3.07769 q^{31} -4.17855 q^{33} +4.10151 q^{35} +3.24009 q^{37} -1.58172 q^{39} +6.34036 q^{41} +1.84400 q^{43} -1.74256 q^{45} -2.38296 q^{47} -1.46000 q^{49} -2.99517 q^{51} -0.819425 q^{53} +7.28138 q^{55} -6.10269 q^{57} +6.19250 q^{59} +8.06136 q^{61} -2.35372 q^{63} +2.75625 q^{65} +1.44065 q^{67} +2.00484 q^{69} -10.8793 q^{71} +14.0131 q^{73} -1.96348 q^{75} +9.83513 q^{77} -13.8424 q^{79} +1.00000 q^{81} -0.554472 q^{83} +5.21927 q^{85} -3.60231 q^{87} -7.50050 q^{89} +3.72293 q^{91} -3.07769 q^{93} +10.6343 q^{95} +13.7359 q^{97} -4.17855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} + q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 + q^5 - 2 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} + q^{5} - 2 q^{7} + 9 q^{9} + 9 q^{11} + 10 q^{13} + q^{15} + 7 q^{17} + 2 q^{19} - 2 q^{21} + 3 q^{23} + 18 q^{25} + 9 q^{27} + 5 q^{29} - 12 q^{31} + 9 q^{33} + 6 q^{35} + 15 q^{37} + 10 q^{39} + 14 q^{41} - 6 q^{43} + q^{45} + 3 q^{47} + 27 q^{49} + 7 q^{51} + 9 q^{53} - 19 q^{55} + 2 q^{57} + 9 q^{59} + 30 q^{61} - 2 q^{63} + 28 q^{65} - 16 q^{67} + 3 q^{69} + 3 q^{71} + 32 q^{73} + 18 q^{75} + 18 q^{77} - 24 q^{79} + 9 q^{81} + 3 q^{83} + 37 q^{85} + 5 q^{87} + 46 q^{89} - 33 q^{91} - 12 q^{93} - 11 q^{95} + 43 q^{97} + 9 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 + q^5 - 2 * q^7 + 9 * q^9 + 9 * q^11 + 10 * q^13 + q^15 + 7 * q^17 + 2 * q^19 - 2 * q^21 + 3 * q^23 + 18 * q^25 + 9 * q^27 + 5 * q^29 - 12 * q^31 + 9 * q^33 + 6 * q^35 + 15 * q^37 + 10 * q^39 + 14 * q^41 - 6 * q^43 + q^45 + 3 * q^47 + 27 * q^49 + 7 * q^51 + 9 * q^53 - 19 * q^55 + 2 * q^57 + 9 * q^59 + 30 * q^61 - 2 * q^63 + 28 * q^65 - 16 * q^67 + 3 * q^69 + 3 * q^71 + 32 * q^73 + 18 * q^75 + 18 * q^77 - 24 * q^79 + 9 * q^81 + 3 * q^83 + 37 * q^85 + 5 * q^87 + 46 * q^89 - 33 * q^91 - 12 * q^93 - 11 * q^95 + 43 * q^97 + 9 * 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
$5$	−1.74256	−0.779298	−0.389649	−	0.920964i	$-0.627404\pi$
$5$	−1.74256	−0.779298	−0.389649	+	0.920964i	$0.627404\pi$
$6$	0	0
$7$	−2.35372	−0.889623	−0.444811	−	0.895624i	$-0.646729\pi$
$7$	−2.35372	−0.889623	−0.444811	+	0.895624i	$0.646729\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.17855	−1.25988	−0.629940	−	0.776644i	$-0.716920\pi$
$11$	−4.17855	−1.25988	−0.629940	+	0.776644i	$0.716920\pi$
$12$	0	0
$13$	−1.58172	−0.438691	−0.219345	−	0.975647i	$-0.570392\pi$
$13$	−1.58172	−0.438691	−0.219345	+	0.975647i	$0.570392\pi$
$14$	0	0
$15$	−1.74256	−0.449928
$16$	0	0
$17$	−2.99517	−0.726435	−0.363218	−	0.931704i	$-0.618322\pi$
$17$	−2.99517	−0.726435	−0.363218	+	0.931704i	$0.618322\pi$
$18$	0	0
$19$	−6.10269	−1.40005	−0.700026	−	0.714117i	$-0.746828\pi$
$19$	−6.10269	−1.40005	−0.700026	+	0.714117i	$0.746828\pi$
$20$	0	0
$21$	−2.35372	−0.513624
$22$	0	0
$23$	2.00484	0.418038	0.209019	−	0.977912i	$-0.432973\pi$
$23$	2.00484	0.418038	0.209019	+	0.977912i	$0.432973\pi$
$24$	0	0
$25$	−1.96348	−0.392695
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.60231	−0.668932	−0.334466	−	0.942408i	$-0.608556\pi$
$29$	−3.60231	−0.668932	−0.334466	+	0.942408i	$0.608556\pi$
$30$	0	0
$31$	−3.07769	−0.552769	−0.276384	−	0.961047i	$-0.589136\pi$
$31$	−3.07769	−0.552769	−0.276384	+	0.961047i	$0.589136\pi$
$32$	0	0
$33$	−4.17855	−0.727392
$34$	0	0
$35$	4.10151	0.693281
$36$	0	0
$37$	3.24009	0.532667	0.266334	−	0.963881i	$-0.414188\pi$
$37$	3.24009	0.532667	0.266334	+	0.963881i	$0.414188\pi$
$38$	0	0
$39$	−1.58172	−0.253278
$40$	0	0
$41$	6.34036	0.990198	0.495099	−	0.868837i	$-0.335132\pi$
$41$	6.34036	0.990198	0.495099	+	0.868837i	$0.335132\pi$
$42$	0	0
$43$	1.84400	0.281208	0.140604	−	0.990066i	$-0.455096\pi$
$43$	1.84400	0.281208	0.140604	+	0.990066i	$0.455096\pi$
$44$	0	0
$45$	−1.74256	−0.259766
$46$	0	0
$47$	−2.38296	−0.347590	−0.173795	−	0.984782i	$-0.555603\pi$
$47$	−2.38296	−0.347590	−0.173795	+	0.984782i	$0.555603\pi$
$48$	0	0
$49$	−1.46000	−0.208572
$50$	0	0
$51$	−2.99517	−0.419408
$52$	0	0
$53$	−0.819425	−0.112557	−0.0562783	−	0.998415i	$-0.517923\pi$
$53$	−0.819425	−0.112557	−0.0562783	+	0.998415i	$0.517923\pi$
$54$	0	0
$55$	7.28138	0.981821
$56$	0	0
$57$	−6.10269	−0.808320
$58$	0	0
$59$	6.19250	0.806195	0.403097	−	0.915157i	$-0.367934\pi$
$59$	6.19250	0.806195	0.403097	+	0.915157i	$0.367934\pi$
$60$	0	0
$61$	8.06136	1.03215	0.516076	−	0.856543i	$-0.327392\pi$
$61$	8.06136	1.03215	0.516076	+	0.856543i	$0.327392\pi$
$62$	0	0
$63$	−2.35372	−0.296541
$64$	0	0
$65$	2.75625	0.341871
$66$	0	0
$67$	1.44065	0.176003	0.0880015	−	0.996120i	$-0.471952\pi$
$67$	1.44065	0.176003	0.0880015	+	0.996120i	$0.471952\pi$
$68$	0	0
$69$	2.00484	0.241355
$70$	0	0
$71$	−10.8793	−1.29113	−0.645567	−	0.763704i	$-0.723379\pi$
$71$	−10.8793	−1.29113	−0.645567	+	0.763704i	$0.723379\pi$
$72$	0	0
$73$	14.0131	1.64011	0.820054	−	0.572286i	$-0.193943\pi$
$73$	14.0131	1.64011	0.820054	+	0.572286i	$0.193943\pi$
$74$	0	0
$75$	−1.96348	−0.226723
$76$	0	0
$77$	9.83513	1.12082
$78$	0	0
$79$	−13.8424	−1.55739	−0.778696	−	0.627401i	$-0.784119\pi$
$79$	−13.8424	−1.55739	−0.778696	+	0.627401i	$0.784119\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.554472	−0.0608612	−0.0304306	−	0.999537i	$-0.509688\pi$
$83$	−0.554472	−0.0608612	−0.0304306	+	0.999537i	$0.509688\pi$
$84$	0	0
$85$	5.21927	0.566109
$86$	0	0
$87$	−3.60231	−0.386208
$88$	0	0
$89$	−7.50050	−0.795052	−0.397526	−	0.917591i	$-0.630131\pi$
$89$	−7.50050	−0.795052	−0.397526	+	0.917591i	$0.630131\pi$
$90$	0	0
$91$	3.72293	0.390269
$92$	0	0
$93$	−3.07769	−0.319141
$94$	0	0
$95$	10.6343	1.09106
$96$	0	0
$97$	13.7359	1.39467	0.697333	−	0.716747i	$-0.254370\pi$
$97$	13.7359	1.39467	0.697333	+	0.716747i	$0.254370\pi$
$98$	0	0
$99$	−4.17855	−0.419960
$100$	0	0
$101$	7.80330	0.776458	0.388229	−	0.921563i	$-0.373087\pi$
$101$	7.80330	0.776458	0.388229	+	0.921563i	$0.373087\pi$
$102$	0	0
$103$	0.856480	0.0843915	0.0421957	−	0.999109i	$-0.486565\pi$
$103$	0.856480	0.0843915	0.0421957	+	0.999109i	$0.486565\pi$
$104$	0	0
$105$	4.10151	0.400266
$106$	0	0
$107$	19.8524	1.91921	0.959604	−	0.281354i	$-0.0907837\pi$
$107$	19.8524	1.91921	0.959604	+	0.281354i	$0.0907837\pi$
$108$	0	0
$109$	9.46807	0.906877	0.453438	−	0.891288i	$-0.350197\pi$
$109$	9.46807	0.906877	0.453438	+	0.891288i	$0.350197\pi$
$110$	0	0
$111$	3.24009	0.307536
$112$	0	0
$113$	11.0805	1.04237	0.521183	−	0.853445i	$-0.325491\pi$
$113$	11.0805	1.04237	0.521183	+	0.853445i	$0.325491\pi$
$114$	0	0
$115$	−3.49356	−0.325776
$116$	0	0
$117$	−1.58172	−0.146230
$118$	0	0
$119$	7.04979	0.646253
$120$	0	0
$121$	6.46025	0.587295
$122$	0	0
$123$	6.34036	0.571691
$124$	0	0
$125$	12.1343	1.08532
$126$	0	0
$127$	−7.40732	−0.657293	−0.328647	−	0.944453i	$-0.606592\pi$
$127$	−7.40732	−0.657293	−0.328647	+	0.944453i	$0.606592\pi$
$128$	0	0
$129$	1.84400	0.162355
$130$	0	0
$131$	−15.4253	−1.34771	−0.673856	−	0.738863i	$-0.735363\pi$
$131$	−15.4253	−1.34771	−0.673856	+	0.738863i	$0.735363\pi$
$132$	0	0
$133$	14.3640	1.24552
$134$	0	0
$135$	−1.74256	−0.149976
$136$	0	0
$137$	−6.82783	−0.583341	−0.291671	−	0.956519i	$-0.594211\pi$
$137$	−6.82783	−0.583341	−0.291671	+	0.956519i	$0.594211\pi$
$138$	0	0
$139$	−14.9107	−1.26471	−0.632355	−	0.774678i	$-0.717912\pi$
$139$	−14.9107	−1.26471	−0.632355	+	0.774678i	$0.717912\pi$
$140$	0	0
$141$	−2.38296	−0.200681
$142$	0	0
$143$	6.60930	0.552697
$144$	0	0
$145$	6.27725	0.521298
$146$	0	0
$147$	−1.46000	−0.120419
$148$	0	0
$149$	12.4310	1.01838	0.509192	−	0.860653i	$-0.329944\pi$
$149$	12.4310	1.01838	0.509192	+	0.860653i	$0.329944\pi$
$150$	0	0
$151$	−17.6408	−1.43559	−0.717793	−	0.696257i	$-0.754847\pi$
$151$	−17.6408	−1.43559	−0.717793	+	0.696257i	$0.754847\pi$
$152$	0	0
$153$	−2.99517	−0.242145
$154$	0	0
$155$	5.36306	0.430772
$156$	0	0
$157$	−14.3930	−1.14868	−0.574342	−	0.818616i	$-0.694742\pi$
$157$	−14.3930	−1.14868	−0.574342	+	0.818616i	$0.694742\pi$
$158$	0	0
$159$	−0.819425	−0.0649846
$160$	0	0
$161$	−4.71884	−0.371896
$162$	0	0
$163$	14.9421	1.17035	0.585176	−	0.810906i	$-0.301025\pi$
$163$	14.9421	1.17035	0.585176	+	0.810906i	$0.301025\pi$
$164$	0	0
$165$	7.28138	0.566855
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−10.4982	−0.807550
$170$	0	0
$171$	−6.10269	−0.466684
$172$	0	0
$173$	5.18285	0.394045	0.197023	−	0.980399i	$-0.436873\pi$
$173$	5.18285	0.394045	0.197023	+	0.980399i	$0.436873\pi$
$174$	0	0
$175$	4.62147	0.349350
$176$	0	0
$177$	6.19250	0.465457
$178$	0	0
$179$	18.1668	1.35785	0.678927	−	0.734206i	$-0.262445\pi$
$179$	18.1668	1.35785	0.678927	+	0.734206i	$0.262445\pi$
$180$	0	0
$181$	9.15342	0.680368	0.340184	−	0.940359i	$-0.389511\pi$
$181$	9.15342	0.680368	0.340184	+	0.940359i	$0.389511\pi$
$182$	0	0
$183$	8.06136	0.595913
$184$	0	0
$185$	−5.64606	−0.415106
$186$	0	0
$187$	12.5155	0.915221
$188$	0	0
$189$	−2.35372	−0.171208
$190$	0	0
$191$	−7.83458	−0.566890	−0.283445	−	0.958988i	$-0.591477\pi$
$191$	−7.83458	−0.566890	−0.283445	+	0.958988i	$0.591477\pi$
$192$	0	0
$193$	0.321669	0.0231542	0.0115771	−	0.999933i	$-0.496315\pi$
$193$	0.321669	0.0231542	0.0115771	+	0.999933i	$0.496315\pi$
$194$	0	0
$195$	2.75625	0.197379
$196$	0	0
$197$	−2.63393	−0.187660	−0.0938300	−	0.995588i	$-0.529911\pi$
$197$	−2.63393	−0.187660	−0.0938300	+	0.995588i	$0.529911\pi$
$198$	0	0
$199$	−18.7548	−1.32949	−0.664746	−	0.747070i	$-0.731460\pi$
$199$	−18.7548	−1.32949	−0.664746	+	0.747070i	$0.731460\pi$
$200$	0	0
$201$	1.44065	0.101615
$202$	0	0
$203$	8.47883	0.595098
$204$	0	0
$205$	−11.0485	−0.771659
$206$	0	0
$207$	2.00484	0.139346
$208$	0	0
$209$	25.5004	1.76390
$210$	0	0
$211$	7.93311	0.546138	0.273069	−	0.961995i	$-0.411961\pi$
$211$	7.93311	0.546138	0.273069	+	0.961995i	$0.411961\pi$
$212$	0	0
$213$	−10.8793	−0.745436
$214$	0	0
$215$	−3.21329	−0.219144
$216$	0	0
$217$	7.24402	0.491756
$218$	0	0
$219$	14.0131	0.946917
$220$	0	0
$221$	4.73753	0.318681
$222$	0	0
$223$	−6.02360	−0.403370	−0.201685	−	0.979450i	$-0.564642\pi$
$223$	−6.02360	−0.403370	−0.201685	+	0.979450i	$0.564642\pi$
$224$	0	0
$225$	−1.96348	−0.130898
$226$	0	0
$227$	21.5259	1.42873	0.714363	−	0.699775i	$-0.246716\pi$
$227$	21.5259	1.42873	0.714363	+	0.699775i	$0.246716\pi$
$228$	0	0
$229$	−15.9540	−1.05427	−0.527135	−	0.849781i	$-0.676734\pi$
$229$	−15.9540	−1.05427	−0.527135	+	0.849781i	$0.676734\pi$
$230$	0	0
$231$	9.83513	0.647104
$232$	0	0
$233$	15.6310	1.02402	0.512012	−	0.858979i	$-0.328900\pi$
$233$	15.6310	1.02402	0.512012	+	0.858979i	$0.328900\pi$
$234$	0	0
$235$	4.15246	0.270876
$236$	0	0
$237$	−13.8424	−0.899161
$238$	0	0
$239$	15.9972	1.03477	0.517386	−	0.855752i	$-0.326905\pi$
$239$	15.9972	1.03477	0.517386	+	0.855752i	$0.326905\pi$
$240$	0	0
$241$	5.44015	0.350431	0.175216	−	0.984530i	$-0.443938\pi$
$241$	5.44015	0.350431	0.175216	+	0.984530i	$0.443938\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.54414	0.162539
$246$	0	0
$247$	9.65275	0.614190
$248$	0	0
$249$	−0.554472	−0.0351382
$250$	0	0
$251$	2.47689	0.156340	0.0781698	−	0.996940i	$-0.475092\pi$
$251$	2.47689	0.156340	0.0781698	+	0.996940i	$0.475092\pi$
$252$	0	0
$253$	−8.37732	−0.526678
$254$	0	0
$255$	5.21927	0.326843
$256$	0	0
$257$	8.60694	0.536886	0.268443	−	0.963296i	$-0.413491\pi$
$257$	8.60694	0.536886	0.268443	+	0.963296i	$0.413491\pi$
$258$	0	0
$259$	−7.62626	−0.473873
$260$	0	0
$261$	−3.60231	−0.222977
$262$	0	0
$263$	−22.6496	−1.39664	−0.698318	−	0.715788i	$-0.746068\pi$
$263$	−22.6496	−1.39664	−0.698318	+	0.715788i	$0.746068\pi$
$264$	0	0
$265$	1.42790	0.0877151
$266$	0	0
$267$	−7.50050	−0.459023
$268$	0	0
$269$	−10.0487	−0.612681	−0.306340	−	0.951922i	$-0.599105\pi$
$269$	−10.0487	−0.612681	−0.306340	+	0.951922i	$0.599105\pi$
$270$	0	0
$271$	29.6799	1.80292	0.901462	−	0.432859i	$-0.142495\pi$
$271$	29.6799	1.80292	0.901462	+	0.432859i	$0.142495\pi$
$272$	0	0
$273$	3.72293	0.225322
$274$	0	0
$275$	8.20447	0.494748
$276$	0	0
$277$	−0.543641	−0.0326642	−0.0163321	−	0.999867i	$-0.505199\pi$
$277$	−0.543641	−0.0326642	−0.0163321	+	0.999867i	$0.505199\pi$
$278$	0	0
$279$	−3.07769	−0.184256
$280$	0	0
$281$	−5.87817	−0.350662	−0.175331	−	0.984510i	$-0.556100\pi$
$281$	−5.87817	−0.350662	−0.175331	+	0.984510i	$0.556100\pi$
$282$	0	0
$283$	−21.7477	−1.29277	−0.646384	−	0.763012i	$-0.723719\pi$
$283$	−21.7477	−1.29277	−0.646384	+	0.763012i	$0.723719\pi$
$284$	0	0
$285$	10.6343	0.629922
$286$	0	0
$287$	−14.9234	−0.880902
$288$	0	0
$289$	−8.02896	−0.472292
$290$	0	0
$291$	13.7359	0.805211
$292$	0	0
$293$	−21.8062	−1.27393	−0.636966	−	0.770892i	$-0.719811\pi$
$293$	−21.8062	−1.27393	−0.636966	+	0.770892i	$0.719811\pi$
$294$	0	0
$295$	−10.7908	−0.628266
$296$	0	0
$297$	−4.17855	−0.242464
$298$	0	0
$299$	−3.17110	−0.183390
$300$	0	0
$301$	−4.34026	−0.250169
$302$	0	0
$303$	7.80330	0.448288
$304$	0	0
$305$	−14.0474	−0.804353
$306$	0	0
$307$	−4.76165	−0.271761	−0.135881	−	0.990725i	$-0.543386\pi$
$307$	−4.76165	−0.271761	−0.135881	+	0.990725i	$0.543386\pi$
$308$	0	0
$309$	0.856480	0.0487234
$310$	0	0
$311$	6.10533	0.346202	0.173101	−	0.984904i	$-0.444621\pi$
$311$	6.10533	0.346202	0.173101	+	0.984904i	$0.444621\pi$
$312$	0	0
$313$	8.59750	0.485960	0.242980	−	0.970031i	$-0.421875\pi$
$313$	8.59750	0.485960	0.242980	+	0.970031i	$0.421875\pi$
$314$	0	0
$315$	4.10151	0.231094
$316$	0	0
$317$	11.2906	0.634144	0.317072	−	0.948401i	$-0.397300\pi$
$317$	11.2906	0.634144	0.317072	+	0.948401i	$0.397300\pi$
$318$	0	0
$319$	15.0524	0.842774
$320$	0	0
$321$	19.8524	1.10806
$322$	0	0
$323$	18.2786	1.01705
$324$	0	0
$325$	3.10567	0.172272
$326$	0	0
$327$	9.46807	0.523585
$328$	0	0
$329$	5.60882	0.309224
$330$	0	0
$331$	−4.27945	−0.235220	−0.117610	−	0.993060i	$-0.537523\pi$
$331$	−4.27945	−0.235220	−0.117610	+	0.993060i	$0.537523\pi$
$332$	0	0
$333$	3.24009	0.177556
$334$	0	0
$335$	−2.51042	−0.137159
$336$	0	0
$337$	−13.7776	−0.750514	−0.375257	−	0.926921i	$-0.622446\pi$
$337$	−13.7776	−0.750514	−0.375257	+	0.926921i	$0.622446\pi$
$338$	0	0
$339$	11.0805	0.601810
$340$	0	0
$341$	12.8603	0.696422
$342$	0	0
$343$	19.9125	1.07517
$344$	0	0
$345$	−3.49356	−0.188087
$346$	0	0
$347$	−0.751533	−0.0403444	−0.0201722	−	0.999797i	$-0.506421\pi$
$347$	−0.751533	−0.0403444	−0.0201722	+	0.999797i	$0.506421\pi$
$348$	0	0
$349$	26.0989	1.39704	0.698522	−	0.715589i	$-0.253841\pi$
$349$	26.0989	1.39704	0.698522	+	0.715589i	$0.253841\pi$
$350$	0	0
$351$	−1.58172	−0.0844261
$352$	0	0
$353$	6.68881	0.356010	0.178005	−	0.984030i	$-0.443036\pi$
$353$	6.68881	0.356010	0.178005	+	0.984030i	$0.443036\pi$
$354$	0	0
$355$	18.9578	1.00618
$356$	0	0
$357$	7.04979	0.373115
$358$	0	0
$359$	7.23167	0.381673	0.190837	−	0.981622i	$-0.438880\pi$
$359$	7.23167	0.381673	0.190837	+	0.981622i	$0.438880\pi$
$360$	0	0
$361$	18.2428	0.960146
$362$	0	0
$363$	6.46025	0.339075
$364$	0	0
$365$	−24.4187	−1.27813
$366$	0	0
$367$	−25.6815	−1.34057	−0.670283	−	0.742106i	$-0.733827\pi$
$367$	−25.6815	−1.34057	−0.670283	+	0.742106i	$0.733827\pi$
$368$	0	0
$369$	6.34036	0.330066
$370$	0	0
$371$	1.92870	0.100133
$372$	0	0
$373$	3.43407	0.177809	0.0889047	−	0.996040i	$-0.471663\pi$
$373$	3.43407	0.177809	0.0889047	+	0.996040i	$0.471663\pi$
$374$	0	0
$375$	12.1343	0.626612
$376$	0	0
$377$	5.69786	0.293455
$378$	0	0
$379$	19.7864	1.01636	0.508179	−	0.861252i	$-0.330319\pi$
$379$	19.7864	1.01636	0.508179	+	0.861252i	$0.330319\pi$
$380$	0	0
$381$	−7.40732	−0.379488
$382$	0	0
$383$	−9.59335	−0.490197	−0.245099	−	0.969498i	$-0.578820\pi$
$383$	−9.59335	−0.490197	−0.245099	+	0.969498i	$0.578820\pi$
$384$	0	0
$385$	−17.1383	−0.873450
$386$	0	0
$387$	1.84400	0.0937359
$388$	0	0
$389$	5.08056	0.257595	0.128797	−	0.991671i	$-0.458888\pi$
$389$	5.08056	0.257595	0.128797	+	0.991671i	$0.458888\pi$
$390$	0	0
$391$	−6.00484	−0.303678
$392$	0	0
$393$	−15.4253	−0.778102
$394$	0	0
$395$	24.1213	1.21367
$396$	0	0
$397$	−25.8925	−1.29951	−0.649754	−	0.760144i	$-0.725128\pi$
$397$	−25.8925	−1.29951	−0.649754	+	0.760144i	$0.725128\pi$
$398$	0	0
$399$	14.3640	0.719100
$400$	0	0
$401$	22.8118	1.13916	0.569582	−	0.821934i	$-0.307105\pi$
$401$	22.8118	1.13916	0.569582	+	0.821934i	$0.307105\pi$
$402$	0	0
$403$	4.86805	0.242495
$404$	0	0
$405$	−1.74256	−0.0865886
$406$	0	0
$407$	−13.5389	−0.671096
$408$	0	0
$409$	−5.07007	−0.250699	−0.125349	−	0.992113i	$-0.540005\pi$
$409$	−5.07007	−0.250699	−0.125349	+	0.992113i	$0.540005\pi$
$410$	0	0
$411$	−6.82783	−0.336792
$412$	0	0
$413$	−14.5754	−0.717209
$414$	0	0
$415$	0.966202	0.0474290
$416$	0	0
$417$	−14.9107	−0.730181
$418$	0	0
$419$	37.2875	1.82161	0.910807	−	0.412832i	$-0.135460\pi$
$419$	37.2875	1.82161	0.910807	+	0.412832i	$0.135460\pi$
$420$	0	0
$421$	2.19031	0.106749	0.0533745	−	0.998575i	$-0.483002\pi$
$421$	2.19031	0.106749	0.0533745	+	0.998575i	$0.483002\pi$
$422$	0	0
$423$	−2.38296	−0.115863
$424$	0	0
$425$	5.88094	0.285268
$426$	0	0
$427$	−18.9742	−0.918225
$428$	0	0
$429$	6.60930	0.319100
$430$	0	0
$431$	−5.67523	−0.273366	−0.136683	−	0.990615i	$-0.543644\pi$
$431$	−5.67523	−0.273366	−0.136683	+	0.990615i	$0.543644\pi$
$432$	0	0
$433$	7.30770	0.351186	0.175593	−	0.984463i	$-0.443816\pi$
$433$	7.30770	0.351186	0.175593	+	0.984463i	$0.443816\pi$
$434$	0	0
$435$	6.27725	0.300971
$436$	0	0
$437$	−12.2349	−0.585275
$438$	0	0
$439$	−24.6805	−1.17794	−0.588969	−	0.808156i	$-0.700466\pi$
$439$	−24.6805	−1.17794	−0.588969	+	0.808156i	$0.700466\pi$
$440$	0	0
$441$	−1.46000	−0.0695238
$442$	0	0
$443$	38.0162	1.80621	0.903103	−	0.429425i	$-0.141284\pi$
$443$	38.0162	1.80621	0.903103	+	0.429425i	$0.141284\pi$
$444$	0	0
$445$	13.0701	0.619582
$446$	0	0
$447$	12.4310	0.587964
$448$	0	0
$449$	−19.7515	−0.932130	−0.466065	−	0.884751i	$-0.654329\pi$
$449$	−19.7515	−0.932130	−0.466065	+	0.884751i	$0.654329\pi$
$450$	0	0
$451$	−26.4935	−1.24753
$452$	0	0
$453$	−17.6408	−0.828835
$454$	0	0
$455$	−6.48744	−0.304136
$456$	0	0
$457$	37.9031	1.77303	0.886515	−	0.462699i	$-0.153119\pi$
$457$	37.9031	1.77303	0.886515	+	0.462699i	$0.153119\pi$
$458$	0	0
$459$	−2.99517	−0.139803
$460$	0	0
$461$	−28.1505	−1.31110	−0.655550	−	0.755152i	$-0.727563\pi$
$461$	−28.1505	−1.31110	−0.655550	+	0.755152i	$0.727563\pi$
$462$	0	0
$463$	5.23657	0.243364	0.121682	−	0.992569i	$-0.461171\pi$
$463$	5.23657	0.243364	0.121682	+	0.992569i	$0.461171\pi$
$464$	0	0
$465$	5.36306	0.248706
$466$	0	0
$467$	5.44349	0.251895	0.125947	−	0.992037i	$-0.459803\pi$
$467$	5.44349	0.251895	0.125947	+	0.992037i	$0.459803\pi$
$468$	0	0
$469$	−3.39088	−0.156576
$470$	0	0
$471$	−14.3930	−0.663193
$472$	0	0
$473$	−7.70524	−0.354288
$474$	0	0
$475$	11.9825	0.549794
$476$	0	0
$477$	−0.819425	−0.0375189
$478$	0	0
$479$	−15.5356	−0.709839	−0.354920	−	0.934897i	$-0.615492\pi$
$479$	−15.5356	−0.709839	−0.354920	+	0.934897i	$0.615492\pi$
$480$	0	0
$481$	−5.12492	−0.233676
$482$	0	0
$483$	−4.71884	−0.214714
$484$	0	0
$485$	−23.9356	−1.08686
$486$	0	0
$487$	20.1891	0.914856	0.457428	−	0.889247i	$-0.348771\pi$
$487$	20.1891	0.914856	0.457428	+	0.889247i	$0.348771\pi$
$488$	0	0
$489$	14.9421	0.675703
$490$	0	0
$491$	14.1222	0.637325	0.318663	−	0.947868i	$-0.396766\pi$
$491$	14.1222	0.637325	0.318663	+	0.947868i	$0.396766\pi$
$492$	0	0
$493$	10.7895	0.485936
$494$	0	0
$495$	7.28138	0.327274
$496$	0	0
$497$	25.6068	1.14862
$498$	0	0
$499$	8.72698	0.390673	0.195337	−	0.980736i	$-0.437420\pi$
$499$	8.72698	0.390673	0.195337	+	0.980736i	$0.437420\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	25.5925	1.14111	0.570557	−	0.821258i	$-0.306727\pi$
$503$	25.5925	1.14111	0.570557	+	0.821258i	$0.306727\pi$
$504$	0	0
$505$	−13.5977	−0.605092
$506$	0	0
$507$	−10.4982	−0.466239
$508$	0	0
$509$	−27.4336	−1.21597	−0.607987	−	0.793947i	$-0.708023\pi$
$509$	−27.4336	−1.21597	−0.607987	+	0.793947i	$0.708023\pi$
$510$	0	0
$511$	−32.9829	−1.45908
$512$	0	0
$513$	−6.10269	−0.269440
$514$	0	0
$515$	−1.49247	−0.0657661
$516$	0	0
$517$	9.95731	0.437922
$518$	0	0
$519$	5.18285	0.227502
$520$	0	0
$521$	32.4202	1.42035	0.710177	−	0.704023i	$-0.248615\pi$
$521$	32.4202	1.42035	0.710177	+	0.704023i	$0.248615\pi$
$522$	0	0
$523$	−30.9080	−1.35151	−0.675755	−	0.737126i	$-0.736182\pi$
$523$	−30.9080	−1.35151	−0.675755	+	0.737126i	$0.736182\pi$
$524$	0	0
$525$	4.62147	0.201698
$526$	0	0
$527$	9.21820	0.401551
$528$	0	0
$529$	−18.9806	−0.825244
$530$	0	0
$531$	6.19250	0.268732
$532$	0	0
$533$	−10.0287	−0.434391
$534$	0	0
$535$	−34.5941	−1.49563
$536$	0	0
$537$	18.1668	0.783957
$538$	0	0
$539$	6.10068	0.262775
$540$	0	0
$541$	15.2116	0.653997	0.326999	−	0.945025i	$-0.393963\pi$
$541$	15.2116	0.653997	0.326999	+	0.945025i	$0.393963\pi$
$542$	0	0
$543$	9.15342	0.392811
$544$	0	0
$545$	−16.4987	−0.706727
$546$	0	0
$547$	18.1213	0.774812	0.387406	−	0.921909i	$-0.373371\pi$
$547$	18.1213	0.774812	0.387406	+	0.921909i	$0.373371\pi$
$548$	0	0
$549$	8.06136	0.344050
$550$	0	0
$551$	21.9838	0.936540
$552$	0	0
$553$	32.5812	1.38549
$554$	0	0
$555$	−5.64606	−0.239662
$556$	0	0
$557$	11.6524	0.493730	0.246865	−	0.969050i	$-0.420600\pi$
$557$	11.6524	0.493730	0.246865	+	0.969050i	$0.420600\pi$
$558$	0	0
$559$	−2.91670	−0.123363
$560$	0	0
$561$	12.5155	0.528403
$562$	0	0
$563$	12.5841	0.530355	0.265178	−	0.964200i	$-0.414569\pi$
$563$	12.5841	0.530355	0.265178	+	0.964200i	$0.414569\pi$
$564$	0	0
$565$	−19.3085	−0.812313
$566$	0	0
$567$	−2.35372	−0.0988470
$568$	0	0
$569$	−39.4763	−1.65493	−0.827465	−	0.561517i	$-0.810218\pi$
$569$	−39.4763	−1.65493	−0.827465	+	0.561517i	$0.810218\pi$
$570$	0	0
$571$	12.7627	0.534102	0.267051	−	0.963682i	$-0.413951\pi$
$571$	12.7627	0.534102	0.267051	+	0.963682i	$0.413951\pi$
$572$	0	0
$573$	−7.83458	−0.327294
$574$	0	0
$575$	−3.93646	−0.164162
$576$	0	0
$577$	8.21156	0.341852	0.170926	−	0.985284i	$-0.445324\pi$
$577$	8.21156	0.341852	0.170926	+	0.985284i	$0.445324\pi$
$578$	0	0
$579$	0.321669	0.0133681
$580$	0	0
$581$	1.30507	0.0541435
$582$	0	0
$583$	3.42400	0.141808
$584$	0	0
$585$	2.75625	0.113957
$586$	0	0
$587$	−8.50643	−0.351098	−0.175549	−	0.984471i	$-0.556170\pi$
$587$	−8.50643	−0.351098	−0.175549	+	0.984471i	$0.556170\pi$
$588$	0	0
$589$	18.7822	0.773905
$590$	0	0
$591$	−2.63393	−0.108346
$592$	0	0
$593$	22.5548	0.926216	0.463108	−	0.886302i	$-0.346734\pi$
$593$	22.5548	0.926216	0.463108	+	0.886302i	$0.346734\pi$
$594$	0	0
$595$	−12.2847	−0.503624
$596$	0	0
$597$	−18.7548	−0.767582
$598$	0	0
$599$	4.87353	0.199127	0.0995635	−	0.995031i	$-0.468255\pi$
$599$	4.87353	0.199127	0.0995635	+	0.995031i	$0.468255\pi$
$600$	0	0
$601$	−14.0874	−0.574637	−0.287319	−	0.957835i	$-0.592764\pi$
$601$	−14.0874	−0.574637	−0.287319	+	0.957835i	$0.592764\pi$
$602$	0	0
$603$	1.44065	0.0586676
$604$	0	0
$605$	−11.2574	−0.457678
$606$	0	0
$607$	12.8622	0.522061	0.261030	−	0.965331i	$-0.415938\pi$
$607$	12.8622	0.522061	0.261030	+	0.965331i	$0.415938\pi$
$608$	0	0
$609$	8.47883	0.343580
$610$	0	0
$611$	3.76918	0.152485
$612$	0	0
$613$	−33.2461	−1.34280	−0.671398	−	0.741097i	$-0.734306\pi$
$613$	−33.2461	−1.34280	−0.671398	+	0.741097i	$0.734306\pi$
$614$	0	0
$615$	−11.0485	−0.445517
$616$	0	0
$617$	−5.59295	−0.225164	−0.112582	−	0.993642i	$-0.535912\pi$
$617$	−5.59295	−0.225164	−0.112582	+	0.993642i	$0.535912\pi$
$618$	0	0
$619$	−9.16730	−0.368465	−0.184232	−	0.982883i	$-0.558980\pi$
$619$	−9.16730	−0.368465	−0.184232	+	0.982883i	$0.558980\pi$
$620$	0	0
$621$	2.00484	0.0804515
$622$	0	0
$623$	17.6541	0.707296
$624$	0	0
$625$	−11.3274	−0.453095
$626$	0	0
$627$	25.5004	1.01839
$628$	0	0
$629$	−9.70462	−0.386948
$630$	0	0
$631$	40.0201	1.59318	0.796588	−	0.604523i	$-0.206636\pi$
$631$	40.0201	1.59318	0.796588	+	0.604523i	$0.206636\pi$
$632$	0	0
$633$	7.93311	0.315313
$634$	0	0
$635$	12.9077	0.512227
$636$	0	0
$637$	2.30932	0.0914984
$638$	0	0
$639$	−10.8793	−0.430378
$640$	0	0
$641$	24.5897	0.971233	0.485617	−	0.874172i	$-0.338595\pi$
$641$	24.5897	0.971233	0.485617	+	0.874172i	$0.338595\pi$
$642$	0	0
$643$	17.8388	0.703493	0.351747	−	0.936095i	$-0.385588\pi$
$643$	17.8388	0.703493	0.351747	+	0.936095i	$0.385588\pi$
$644$	0	0
$645$	−3.21329	−0.126523
$646$	0	0
$647$	−10.7564	−0.422877	−0.211438	−	0.977391i	$-0.567815\pi$
$647$	−10.7564	−0.422877	−0.211438	+	0.977391i	$0.567815\pi$
$648$	0	0
$649$	−25.8756	−1.01571
$650$	0	0
$651$	7.24402	0.283915
$652$	0	0
$653$	4.24085	0.165957	0.0829786	−	0.996551i	$-0.473557\pi$
$653$	4.24085	0.165957	0.0829786	+	0.996551i	$0.473557\pi$
$654$	0	0
$655$	26.8795	1.05027
$656$	0	0
$657$	14.0131	0.546703
$658$	0	0
$659$	−8.31678	−0.323976	−0.161988	−	0.986793i	$-0.551791\pi$
$659$	−8.31678	−0.323976	−0.161988	+	0.986793i	$0.551791\pi$
$660$	0	0
$661$	−14.2659	−0.554880	−0.277440	−	0.960743i	$-0.589486\pi$
$661$	−14.2659	−0.554880	−0.277440	+	0.960743i	$0.589486\pi$
$662$	0	0
$663$	4.73753	0.183990
$664$	0	0
$665$	−25.0302	−0.970629
$666$	0	0
$667$	−7.22206	−0.279639
$668$	0	0
$669$	−6.02360	−0.232886
$670$	0	0
$671$	−33.6848	−1.30039
$672$	0	0
$673$	−48.0157	−1.85087	−0.925435	−	0.378905i	$-0.876301\pi$
$673$	−48.0157	−1.85087	−0.925435	+	0.378905i	$0.876301\pi$
$674$	0	0
$675$	−1.96348	−0.0755742
$676$	0	0
$677$	42.2818	1.62502	0.812510	−	0.582947i	$-0.198100\pi$
$677$	42.2818	1.62502	0.812510	+	0.582947i	$0.198100\pi$
$678$	0	0
$679$	−32.3304	−1.24073
$680$	0	0
$681$	21.5259	0.824876
$682$	0	0
$683$	−20.9990	−0.803503	−0.401751	−	0.915749i	$-0.631598\pi$
$683$	−20.9990	−0.803503	−0.401751	+	0.915749i	$0.631598\pi$
$684$	0	0
$685$	11.8979	0.454596
$686$	0	0
$687$	−15.9540	−0.608684
$688$	0	0
$689$	1.29610	0.0493776
$690$	0	0
$691$	−15.3381	−0.583490	−0.291745	−	0.956496i	$-0.594236\pi$
$691$	−15.3381	−0.583490	−0.291745	+	0.956496i	$0.594236\pi$
$692$	0	0
$693$	9.83513	0.373606
$694$	0	0
$695$	25.9829	0.985586
$696$	0	0
$697$	−18.9905	−0.719315
$698$	0	0
$699$	15.6310	0.591220
$700$	0	0
$701$	17.6107	0.665146	0.332573	−	0.943077i	$-0.392083\pi$
$701$	17.6107	0.665146	0.332573	+	0.943077i	$0.392083\pi$
$702$	0	0
$703$	−19.7732	−0.745762
$704$	0	0
$705$	4.15246	0.156391
$706$	0	0
$707$	−18.3668	−0.690754
$708$	0	0
$709$	−45.6193	−1.71327	−0.856635	−	0.515923i	$-0.827449\pi$
$709$	−45.6193	−1.71327	−0.856635	+	0.515923i	$0.827449\pi$
$710$	0	0
$711$	−13.8424	−0.519131
$712$	0	0
$713$	−6.17028	−0.231079
$714$	0	0
$715$	−11.5171	−0.430716
$716$	0	0
$717$	15.9972	0.597426
$718$	0	0
$719$	−25.6154	−0.955292	−0.477646	−	0.878552i	$-0.658510\pi$
$719$	−25.6154	−0.955292	−0.477646	+	0.878552i	$0.658510\pi$
$720$	0	0
$721$	−2.01591	−0.0750766
$722$	0	0
$723$	5.44015	0.202322
$724$	0	0
$725$	7.07305	0.262687
$726$	0	0
$727$	−9.99950	−0.370861	−0.185430	−	0.982657i	$-0.559368\pi$
$727$	−9.99950	−0.370861	−0.185430	+	0.982657i	$0.559368\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.52310	−0.204279
$732$	0	0
$733$	13.0990	0.483822	0.241911	−	0.970298i	$-0.422226\pi$
$733$	13.0990	0.483822	0.241911	+	0.970298i	$0.422226\pi$
$734$	0	0
$735$	2.54414	0.0938421
$736$	0	0
$737$	−6.01981	−0.221742
$738$	0	0
$739$	−2.03676	−0.0749233	−0.0374617	−	0.999298i	$-0.511927\pi$
$739$	−2.03676	−0.0749233	−0.0374617	+	0.999298i	$0.511927\pi$
$740$	0	0
$741$	9.65275	0.354603
$742$	0	0
$743$	−21.6744	−0.795155	−0.397577	−	0.917569i	$-0.630149\pi$
$743$	−21.6744	−0.795155	−0.397577	+	0.917569i	$0.630149\pi$
$744$	0	0
$745$	−21.6617	−0.793624
$746$	0	0
$747$	−0.554472	−0.0202871
$748$	0	0
$749$	−46.7271	−1.70737
$750$	0	0
$751$	13.6865	0.499428	0.249714	−	0.968320i	$-0.419663\pi$
$751$	13.6865	0.499428	0.249714	+	0.968320i	$0.419663\pi$
$752$	0	0
$753$	2.47689	0.0902627
$754$	0	0
$755$	30.7401	1.11875
$756$	0	0
$757$	−31.4662	−1.14366	−0.571830	−	0.820372i	$-0.693766\pi$
$757$	−31.4662	−1.14366	−0.571830	+	0.820372i	$0.693766\pi$
$758$	0	0
$759$	−8.37732	−0.304078
$760$	0	0
$761$	31.0429	1.12530	0.562652	−	0.826694i	$-0.309781\pi$
$761$	31.0429	1.12530	0.562652	+	0.826694i	$0.309781\pi$
$762$	0	0
$763$	−22.2852	−0.806778
$764$	0	0
$765$	5.21927	0.188703
$766$	0	0
$767$	−9.79481	−0.353670
$768$	0	0
$769$	25.5535	0.921482	0.460741	−	0.887535i	$-0.347584\pi$
$769$	25.5535	0.921482	0.460741	+	0.887535i	$0.347584\pi$
$770$	0	0
$771$	8.60694	0.309971
$772$	0	0
$773$	−5.48231	−0.197185	−0.0985925	−	0.995128i	$-0.531434\pi$
$773$	−5.48231	−0.197185	−0.0985925	+	0.995128i	$0.531434\pi$
$774$	0	0
$775$	6.04296	0.217070
$776$	0	0
$777$	−7.62626	−0.273591
$778$	0	0
$779$	−38.6932	−1.38633
$780$	0	0
$781$	45.4596	1.62667
$782$	0	0
$783$	−3.60231	−0.128736
$784$	0	0
$785$	25.0806	0.895166
$786$	0	0
$787$	42.4826	1.51434	0.757171	−	0.653216i	$-0.226581\pi$
$787$	42.4826	1.51434	0.757171	+	0.653216i	$0.226581\pi$
$788$	0	0
$789$	−22.6496	−0.806348
$790$	0	0
$791$	−26.0804	−0.927312
$792$	0	0
$793$	−12.7508	−0.452795
$794$	0	0
$795$	1.42790	0.0506423
$796$	0	0
$797$	11.1231	0.394002	0.197001	−	0.980403i	$-0.436880\pi$
$797$	11.1231	0.394002	0.197001	+	0.980403i	$0.436880\pi$
$798$	0	0
$799$	7.13737	0.252502
$800$	0	0
$801$	−7.50050	−0.265017
$802$	0	0
$803$	−58.5543	−2.06634
$804$	0	0
$805$	8.22287	0.289818
$806$	0	0
$807$	−10.0487	−0.353731
$808$	0	0
$809$	−35.4616	−1.24676	−0.623381	−	0.781918i	$-0.714242\pi$
$809$	−35.4616	−1.24676	−0.623381	+	0.781918i	$0.714242\pi$
$810$	0	0
$811$	−33.6760	−1.18252	−0.591262	−	0.806479i	$-0.701370\pi$
$811$	−33.6760	−1.18252	−0.591262	+	0.806479i	$0.701370\pi$
$812$	0	0
$813$	29.6799	1.04092
$814$	0	0
$815$	−26.0375	−0.912053
$816$	0	0
$817$	−11.2534	−0.393705
$818$	0	0
$819$	3.72293	0.130090
$820$	0	0
$821$	6.15905	0.214952	0.107476	−	0.994208i	$-0.465723\pi$
$821$	6.15905	0.214952	0.107476	+	0.994208i	$0.465723\pi$
$822$	0	0
$823$	37.0489	1.29144	0.645722	−	0.763572i	$-0.276556\pi$
$823$	37.0489	1.29144	0.645722	+	0.763572i	$0.276556\pi$
$824$	0	0
$825$	8.20447	0.285643
$826$	0	0
$827$	41.5236	1.44392	0.721959	−	0.691936i	$-0.243242\pi$
$827$	41.5236	1.44392	0.721959	+	0.691936i	$0.243242\pi$
$828$	0	0
$829$	−3.62696	−0.125970	−0.0629848	−	0.998014i	$-0.520062\pi$
$829$	−3.62696	−0.125970	−0.0629848	+	0.998014i	$0.520062\pi$
$830$	0	0
$831$	−0.543641	−0.0188587
$832$	0	0
$833$	4.37295	0.151514
$834$	0	0
$835$	−1.74256	−0.0603039
$836$	0	0
$837$	−3.07769	−0.106380
$838$	0	0
$839$	27.4675	0.948284	0.474142	−	0.880448i	$-0.342758\pi$
$839$	27.4675	0.948284	0.474142	+	0.880448i	$0.342758\pi$
$840$	0	0
$841$	−16.0234	−0.552529
$842$	0	0
$843$	−5.87817	−0.202455
$844$	0	0
$845$	18.2937	0.629322
$846$	0	0
$847$	−15.2056	−0.522471
$848$	0	0
$849$	−21.7477	−0.746380
$850$	0	0
$851$	6.49586	0.222675
$852$	0	0
$853$	−10.0323	−0.343500	−0.171750	−	0.985141i	$-0.554942\pi$
$853$	−10.0323	−0.343500	−0.171750	+	0.985141i	$0.554942\pi$
$854$	0	0
$855$	10.6343	0.363686
$856$	0	0
$857$	2.78932	0.0952812	0.0476406	−	0.998865i	$-0.484830\pi$
$857$	2.78932	0.0952812	0.0476406	+	0.998865i	$0.484830\pi$
$858$	0	0
$859$	−10.8774	−0.371132	−0.185566	−	0.982632i	$-0.559412\pi$
$859$	−10.8774	−0.371132	−0.185566	+	0.982632i	$0.559412\pi$
$860$	0	0
$861$	−14.9234	−0.508589
$862$	0	0
$863$	−9.79737	−0.333506	−0.166753	−	0.985999i	$-0.553328\pi$
$863$	−9.79737	−0.333506	−0.166753	+	0.985999i	$0.553328\pi$
$864$	0	0
$865$	−9.03145	−0.307078
$866$	0	0
$867$	−8.02896	−0.272678
$868$	0	0
$869$	57.8412	1.96213
$870$	0	0
$871$	−2.27870	−0.0772109
$872$	0	0
$873$	13.7359	0.464889
$874$	0	0
$875$	−28.5607	−0.965529
$876$	0	0
$877$	1.25262	0.0422981	0.0211490	−	0.999776i	$-0.493268\pi$
$877$	1.25262	0.0422981	0.0211490	+	0.999776i	$0.493268\pi$
$878$	0	0
$879$	−21.8062	−0.735505
$880$	0	0
$881$	29.6654	0.999454	0.499727	−	0.866183i	$-0.333434\pi$
$881$	29.6654	0.999454	0.499727	+	0.866183i	$0.333434\pi$
$882$	0	0
$883$	−20.9774	−0.705946	−0.352973	−	0.935634i	$-0.614829\pi$
$883$	−20.9774	−0.705946	−0.352973	+	0.935634i	$0.614829\pi$
$884$	0	0
$885$	−10.7908	−0.362729
$886$	0	0
$887$	−35.8289	−1.20302	−0.601509	−	0.798866i	$-0.705433\pi$
$887$	−35.8289	−1.20302	−0.601509	+	0.798866i	$0.705433\pi$
$888$	0	0
$889$	17.4348	0.584743
$890$	0	0
$891$	−4.17855	−0.139987
$892$	0	0
$893$	14.5425	0.486645
$894$	0	0
$895$	−31.6569	−1.05817
$896$	0	0
$897$	−3.17110	−0.105880
$898$	0	0
$899$	11.0868	0.369765
$900$	0	0
$901$	2.45432	0.0817651
$902$	0	0
$903$	−4.34026	−0.144435
$904$	0	0
$905$	−15.9504	−0.530210
$906$	0	0
$907$	−43.6465	−1.44926	−0.724630	−	0.689138i	$-0.757989\pi$
$907$	−43.6465	−1.44926	−0.724630	+	0.689138i	$0.757989\pi$
$908$	0	0
$909$	7.80330	0.258819
$910$	0	0
$911$	14.4705	0.479430	0.239715	−	0.970843i	$-0.422946\pi$
$911$	14.4705	0.479430	0.239715	+	0.970843i	$0.422946\pi$
$912$	0	0
$913$	2.31689	0.0766778
$914$	0	0
$915$	−14.0474	−0.464394
$916$	0	0
$917$	36.3068	1.19895
$918$	0	0
$919$	−28.9070	−0.953555	−0.476778	−	0.879024i	$-0.658195\pi$
$919$	−28.9070	−0.953555	−0.476778	+	0.879024i	$0.658195\pi$
$920$	0	0
$921$	−4.76165	−0.156902
$922$	0	0
$923$	17.2080	0.566408
$924$	0	0
$925$	−6.36183	−0.209176
$926$	0	0
$927$	0.856480	0.0281305
$928$	0	0
$929$	−16.9643	−0.556580	−0.278290	−	0.960497i	$-0.589768\pi$
$929$	−16.9643	−0.556580	−0.278290	+	0.960497i	$0.589768\pi$
$930$	0	0
$931$	8.90992	0.292011
$932$	0	0
$933$	6.10533	0.199880
$934$	0	0
$935$	−21.8090	−0.713229
$936$	0	0
$937$	−14.2278	−0.464802	−0.232401	−	0.972620i	$-0.574658\pi$
$937$	−14.2278	−0.464802	−0.232401	+	0.972620i	$0.574658\pi$
$938$	0	0
$939$	8.59750	0.280569
$940$	0	0
$941$	41.7691	1.36163	0.680816	−	0.732454i	$-0.261625\pi$
$941$	41.7691	1.36163	0.680816	+	0.732454i	$0.261625\pi$
$942$	0	0
$943$	12.7114	0.413941
$944$	0	0
$945$	4.10151	0.133422
$946$	0	0
$947$	−4.17349	−0.135620	−0.0678101	−	0.997698i	$-0.521601\pi$
$947$	−4.17349	−0.135620	−0.0678101	+	0.997698i	$0.521601\pi$
$948$	0	0
$949$	−22.1648	−0.719500
$950$	0	0
$951$	11.2906	0.366123
$952$	0	0
$953$	9.83077	0.318450	0.159225	−	0.987242i	$-0.449100\pi$
$953$	9.83077	0.318450	0.159225	+	0.987242i	$0.449100\pi$
$954$	0	0
$955$	13.6522	0.441776
$956$	0	0
$957$	15.0524	0.486576
$958$	0	0
$959$	16.0708	0.518953
$960$	0	0
$961$	−21.5278	−0.694446
$962$	0	0
$963$	19.8524	0.639736
$964$	0	0
$965$	−0.560528	−0.0180440
$966$	0	0
$967$	−27.6159	−0.888067	−0.444034	−	0.896010i	$-0.646453\pi$
$967$	−27.6159	−0.888067	−0.444034	+	0.896010i	$0.646453\pi$
$968$	0	0
$969$	18.2786	0.587193
$970$	0	0
$971$	41.6651	1.33710	0.668548	−	0.743669i	$-0.266916\pi$
$971$	41.6651	1.33710	0.668548	+	0.743669i	$0.266916\pi$
$972$	0	0
$973$	35.0957	1.12512
$974$	0	0
$975$	3.10567	0.0994611
$976$	0	0
$977$	−7.45680	−0.238564	−0.119282	−	0.992860i	$-0.538059\pi$
$977$	−7.45680	−0.238564	−0.119282	+	0.992860i	$0.538059\pi$
$978$	0	0
$979$	31.3412	1.00167
$980$	0	0
$981$	9.46807	0.302292
$982$	0	0
$983$	41.1283	1.31179	0.655895	−	0.754852i	$-0.272291\pi$
$983$	41.1283	1.31179	0.655895	+	0.754852i	$0.272291\pi$
$984$	0	0
$985$	4.58980	0.146243
$986$	0	0
$987$	5.60882	0.178531
$988$	0	0
$989$	3.69693	0.117556
$990$	0	0
$991$	14.3322	0.455276	0.227638	−	0.973746i	$-0.426900\pi$
$991$	14.3322	0.455276	0.227638	+	0.973746i	$0.426900\pi$
$992$	0	0
$993$	−4.27945	−0.135804
$994$	0	0
$995$	32.6814	1.03607
$996$	0	0
$997$	−21.9287	−0.694489	−0.347245	−	0.937775i	$-0.612883\pi$
$997$	−21.9287	−0.694489	−0.347245	+	0.937775i	$0.612883\pi$
$998$	0	0
$999$	3.24009	0.102512

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bc.1.3		9
4.3	odd	2		2004.2.a.c.1.3	✓	9
12.11	even	2		6012.2.a.i.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.c.1.3	✓	9	4.3	odd	2
6012.2.a.i.1.7		9	12.11	even	2
8016.2.a.bc.1.3		9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.74256 q^{5} -2.35372 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.74256 q^{5} -2.35372 q^{7} +1.00000 q^{9} -4.17855 q^{11} -1.58172 q^{13} -1.74256 q^{15} -2.99517 q^{17} -6.10269 q^{19} -2.35372 q^{21} +2.00484 q^{23} -1.96348 q^{25} +1.00000 q^{27} -3.60231 q^{29} -3.07769 q^{31} -4.17855 q^{33} +4.10151 q^{35} +3.24009 q^{37} -1.58172 q^{39} +6.34036 q^{41} +1.84400 q^{43} -1.74256 q^{45} -2.38296 q^{47} -1.46000 q^{49} -2.99517 q^{51} -0.819425 q^{53} +7.28138 q^{55} -6.10269 q^{57} +6.19250 q^{59} +8.06136 q^{61} -2.35372 q^{63} +2.75625 q^{65} +1.44065 q^{67} +2.00484 q^{69} -10.8793 q^{71} +14.0131 q^{73} -1.96348 q^{75} +9.83513 q^{77} -13.8424 q^{79} +1.00000 q^{81} -0.554472 q^{83} +5.21927 q^{85} -3.60231 q^{87} -7.50050 q^{89} +3.72293 q^{91} -3.07769 q^{93} +10.6343 q^{95} +13.7359 q^{97} -4.17855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 + q^5 - 2 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} + q^{5} - 2 q^{7} + 9 q^{9} + 9 q^{11} + 10 q^{13} + q^{15} + 7 q^{17} + 2 q^{19} - 2 q^{21} + 3 q^{23} + 18 q^{25} + 9 q^{27} + 5 q^{29} - 12 q^{31} + 9 q^{33} + 6 q^{35} + 15 q^{37} + 10 q^{39} + 14 q^{41} - 6 q^{43} + q^{45} + 3 q^{47} + 27 q^{49} + 7 q^{51} + 9 q^{53} - 19 q^{55} + 2 q^{57} + 9 q^{59} + 30 q^{61} - 2 q^{63} + 28 q^{65} - 16 q^{67} + 3 q^{69} + 3 q^{71} + 32 q^{73} + 18 q^{75} + 18 q^{77} - 24 q^{79} + 9 q^{81} + 3 q^{83} + 37 q^{85} + 5 q^{87} + 46 q^{89} - 33 q^{91} - 12 q^{93} - 11 q^{95} + 43 q^{97} + 9 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 + q^5 - 2 * q^7 + 9 * q^9 + 9 * q^11 + 10 * q^13 + q^15 + 7 * q^17 + 2 * q^19 - 2 * q^21 + 3 * q^23 + 18 * q^25 + 9 * q^27 + 5 * q^29 - 12 * q^31 + 9 * q^33 + 6 * q^35 + 15 * q^37 + 10 * q^39 + 14 * q^41 - 6 * q^43 + q^45 + 3 * q^47 + 27 * q^49 + 7 * q^51 + 9 * q^53 - 19 * q^55 + 2 * q^57 + 9 * q^59 + 30 * q^61 - 2 * q^63 + 28 * q^65 - 16 * q^67 + 3 * q^69 + 3 * q^71 + 32 * q^73 + 18 * q^75 + 18 * q^77 - 24 * q^79 + 9 * q^81 + 3 * q^83 + 37 * q^85 + 5 * q^87 + 46 * q^89 - 33 * q^91 - 12 * q^93 - 11 * q^95 + 43 * q^97 + 9 * q^99

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