LMFDB - Embedded newform 8016.2.a.bf.1.12

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 $ x^12 - 4x^11 - 31x^10 + 131x^9 + 309x^8 - 1453x^7 - 1072x^6 + 6350x^5 + 1411x^4 - 11022x^3 - 2450x^2 + 6960*x + 3008
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$3.77676$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.77676 q^{5} +1.75359 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.77676 q^{5} +1.75359 q^{7} +1.00000 q^{9} -2.41460 q^{11} -1.58092 q^{13} -3.77676 q^{15} +0.811189 q^{17} -3.41955 q^{19} -1.75359 q^{21} -6.93574 q^{23} +9.26390 q^{25} -1.00000 q^{27} -0.520367 q^{29} +3.53773 q^{31} +2.41460 q^{33} +6.62290 q^{35} -9.20908 q^{37} +1.58092 q^{39} -3.31600 q^{41} -6.00002 q^{43} +3.77676 q^{45} -12.4567 q^{47} -3.92491 q^{49} -0.811189 q^{51} +12.1117 q^{53} -9.11937 q^{55} +3.41955 q^{57} +10.0728 q^{59} -13.5319 q^{61} +1.75359 q^{63} -5.97076 q^{65} +14.1817 q^{67} +6.93574 q^{69} -11.5259 q^{71} +3.71440 q^{73} -9.26390 q^{75} -4.23423 q^{77} -12.9098 q^{79} +1.00000 q^{81} -13.0558 q^{83} +3.06367 q^{85} +0.520367 q^{87} +0.129594 q^{89} -2.77230 q^{91} -3.53773 q^{93} -12.9148 q^{95} -11.6438 q^{97} -2.41460 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9} - q^{11} + 8 q^{13} - 4 q^{15} + 3 q^{17} - 12 q^{19} + 11 q^{21} - 7 q^{23} + 18 q^{25} - 12 q^{27} + 5 q^{29} - 33 q^{31} + q^{33} - 15 q^{35} + 8 q^{37} - 8 q^{39} - 6 q^{41} - 16 q^{43} + 4 q^{45} - 18 q^{47} + 25 q^{49} - 3 q^{51} + 20 q^{53} - 39 q^{55} + 12 q^{57} - 4 q^{59} + 10 q^{61} - 11 q^{63} - 9 q^{67} + 7 q^{69} - 11 q^{71} + 22 q^{73} - 18 q^{75} + 24 q^{77} - 56 q^{79} + 12 q^{81} - 26 q^{83} + 15 q^{85} - 5 q^{87} - 15 q^{89} - 11 q^{91} + 33 q^{93} - 3 q^{95} + 8 q^{97} - q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9 - q^11 + 8 * q^13 - 4 * q^15 + 3 * q^17 - 12 * q^19 + 11 * q^21 - 7 * q^23 + 18 * q^25 - 12 * q^27 + 5 * q^29 - 33 * q^31 + q^33 - 15 * q^35 + 8 * q^37 - 8 * q^39 - 6 * q^41 - 16 * q^43 + 4 * q^45 - 18 * q^47 + 25 * q^49 - 3 * q^51 + 20 * q^53 - 39 * q^55 + 12 * q^57 - 4 * q^59 + 10 * q^61 - 11 * q^63 - 9 * q^67 + 7 * q^69 - 11 * q^71 + 22 * q^73 - 18 * q^75 + 24 * q^77 - 56 * q^79 + 12 * q^81 - 26 * q^83 + 15 * q^85 - 5 * q^87 - 15 * q^89 - 11 * q^91 + 33 * q^93 - 3 * q^95 + 8 * 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
$5$	3.77676	1.68902	0.844509	−	0.535542i	$-0.179892\pi$
$5$	3.77676	1.68902	0.844509	+	0.535542i	$0.179892\pi$
$6$	0	0
$7$	1.75359	0.662796	0.331398	−	0.943491i	$-0.392480\pi$
$7$	1.75359	0.662796	0.331398	+	0.943491i	$0.392480\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.41460	−0.728030	−0.364015	−	0.931393i	$-0.618594\pi$
$11$	−2.41460	−0.728030	−0.364015	+	0.931393i	$0.618594\pi$
$12$	0	0
$13$	−1.58092	−0.438469	−0.219235	−	0.975672i	$-0.570356\pi$
$13$	−1.58092	−0.438469	−0.219235	+	0.975672i	$0.570356\pi$
$14$	0	0
$15$	−3.77676	−0.975155
$16$	0	0
$17$	0.811189	0.196742	0.0983712	−	0.995150i	$-0.468637\pi$
$17$	0.811189	0.196742	0.0983712	+	0.995150i	$0.468637\pi$
$18$	0	0
$19$	−3.41955	−0.784498	−0.392249	−	0.919859i	$-0.628303\pi$
$19$	−3.41955	−0.784498	−0.392249	+	0.919859i	$0.628303\pi$
$20$	0	0
$21$	−1.75359	−0.382665
$22$	0	0
$23$	−6.93574	−1.44620	−0.723101	−	0.690742i	$-0.757284\pi$
$23$	−6.93574	−1.44620	−0.723101	+	0.690742i	$0.757284\pi$
$24$	0	0
$25$	9.26390	1.85278
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.520367	−0.0966297	−0.0483149	−	0.998832i	$-0.515385\pi$
$29$	−0.520367	−0.0966297	−0.0483149	+	0.998832i	$0.515385\pi$
$30$	0	0
$31$	3.53773	0.635394	0.317697	−	0.948192i	$-0.397090\pi$
$31$	3.53773	0.635394	0.317697	+	0.948192i	$0.397090\pi$
$32$	0	0
$33$	2.41460	0.420328
$34$	0	0
$35$	6.62290	1.11947
$36$	0	0
$37$	−9.20908	−1.51396	−0.756982	−	0.653436i	$-0.773327\pi$
$37$	−9.20908	−1.51396	−0.756982	+	0.653436i	$0.773327\pi$
$38$	0	0
$39$	1.58092	0.253150
$40$	0	0
$41$	−3.31600	−0.517873	−0.258936	−	0.965894i	$-0.583372\pi$
$41$	−3.31600	−0.517873	−0.258936	+	0.965894i	$0.583372\pi$
$42$	0	0
$43$	−6.00002	−0.914995	−0.457498	−	0.889211i	$-0.651254\pi$
$43$	−6.00002	−0.914995	−0.457498	+	0.889211i	$0.651254\pi$
$44$	0	0
$45$	3.77676	0.563006
$46$	0	0
$47$	−12.4567	−1.81699	−0.908496	−	0.417893i	$-0.862769\pi$
$47$	−12.4567	−1.81699	−0.908496	+	0.417893i	$0.862769\pi$
$48$	0	0
$49$	−3.92491	−0.560701
$50$	0	0
$51$	−0.811189	−0.113589
$52$	0	0
$53$	12.1117	1.66368	0.831838	−	0.555019i	$-0.187289\pi$
$53$	12.1117	1.66368	0.831838	+	0.555019i	$0.187289\pi$
$54$	0	0
$55$	−9.11937	−1.22966
$56$	0	0
$57$	3.41955	0.452930
$58$	0	0
$59$	10.0728	1.31137	0.655683	−	0.755036i	$-0.272381\pi$
$59$	10.0728	1.31137	0.655683	+	0.755036i	$0.272381\pi$
$60$	0	0
$61$	−13.5319	−1.73258	−0.866289	−	0.499544i	$-0.833501\pi$
$61$	−13.5319	−1.73258	−0.866289	+	0.499544i	$0.833501\pi$
$62$	0	0
$63$	1.75359	0.220932
$64$	0	0
$65$	−5.97076	−0.740582
$66$	0	0
$67$	14.1817	1.73257	0.866286	−	0.499548i	$-0.166500\pi$
$67$	14.1817	1.73257	0.866286	+	0.499548i	$0.166500\pi$
$68$	0	0
$69$	6.93574	0.834965
$70$	0	0
$71$	−11.5259	−1.36787	−0.683934	−	0.729544i	$-0.739732\pi$
$71$	−11.5259	−1.36787	−0.683934	+	0.729544i	$0.739732\pi$
$72$	0	0
$73$	3.71440	0.434738	0.217369	−	0.976089i	$-0.430252\pi$
$73$	3.71440	0.434738	0.217369	+	0.976089i	$0.430252\pi$
$74$	0	0
$75$	−9.26390	−1.06970
$76$	0	0
$77$	−4.23423	−0.482536
$78$	0	0
$79$	−12.9098	−1.45246	−0.726232	−	0.687450i	$-0.758730\pi$
$79$	−12.9098	−1.45246	−0.726232	+	0.687450i	$0.758730\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.0558	−1.43306	−0.716532	−	0.697554i	$-0.754272\pi$
$83$	−13.0558	−1.43306	−0.716532	+	0.697554i	$0.754272\pi$
$84$	0	0
$85$	3.06367	0.332301
$86$	0	0
$87$	0.520367	0.0557892
$88$	0	0
$89$	0.129594	0.0137369	0.00686844	−	0.999976i	$-0.497814\pi$
$89$	0.129594	0.0137369	0.00686844	+	0.999976i	$0.497814\pi$
$90$	0	0
$91$	−2.77230	−0.290616
$92$	0	0
$93$	−3.53773	−0.366845
$94$	0	0
$95$	−12.9148	−1.32503
$96$	0	0
$97$	−11.6438	−1.18225	−0.591123	−	0.806582i	$-0.701315\pi$
$97$	−11.6438	−1.18225	−0.591123	+	0.806582i	$0.701315\pi$
$98$	0	0
$99$	−2.41460	−0.242677
$100$	0	0
$101$	9.16093	0.911547	0.455773	−	0.890096i	$-0.349363\pi$
$101$	9.16093	0.911547	0.455773	+	0.890096i	$0.349363\pi$
$102$	0	0
$103$	2.06620	0.203589	0.101794	−	0.994805i	$-0.467542\pi$
$103$	2.06620	0.203589	0.101794	+	0.994805i	$0.467542\pi$
$104$	0	0
$105$	−6.62290	−0.646329
$106$	0	0
$107$	−3.49830	−0.338194	−0.169097	−	0.985599i	$-0.554085\pi$
$107$	−3.49830	−0.338194	−0.169097	+	0.985599i	$0.554085\pi$
$108$	0	0
$109$	8.78182	0.841146	0.420573	−	0.907259i	$-0.361829\pi$
$109$	8.78182	0.841146	0.420573	+	0.907259i	$0.361829\pi$
$110$	0	0
$111$	9.20908	0.874087
$112$	0	0
$113$	−0.230019	−0.0216384	−0.0108192	−	0.999941i	$-0.503444\pi$
$113$	−0.230019	−0.0216384	−0.0108192	+	0.999941i	$0.503444\pi$
$114$	0	0
$115$	−26.1946	−2.44266
$116$	0	0
$117$	−1.58092	−0.146156
$118$	0	0
$119$	1.42250	0.130400
$120$	0	0
$121$	−5.16969	−0.469972
$122$	0	0
$123$	3.31600	0.298994
$124$	0	0
$125$	16.1037	1.44036
$126$	0	0
$127$	11.4655	1.01740	0.508698	−	0.860945i	$-0.330127\pi$
$127$	11.4655	1.01740	0.508698	+	0.860945i	$0.330127\pi$
$128$	0	0
$129$	6.00002	0.528273
$130$	0	0
$131$	−18.3513	−1.60336	−0.801680	−	0.597754i	$-0.796060\pi$
$131$	−18.3513	−1.60336	−0.801680	+	0.597754i	$0.796060\pi$
$132$	0	0
$133$	−5.99650	−0.519962
$134$	0	0
$135$	−3.77676	−0.325052
$136$	0	0
$137$	−16.8506	−1.43964	−0.719822	−	0.694159i	$-0.755776\pi$
$137$	−16.8506	−1.43964	−0.719822	+	0.694159i	$0.755776\pi$
$138$	0	0
$139$	−11.9744	−1.01566	−0.507830	−	0.861458i	$-0.669552\pi$
$139$	−11.9744	−1.01566	−0.507830	+	0.861458i	$0.669552\pi$
$140$	0	0
$141$	12.4567	1.04904
$142$	0	0
$143$	3.81730	0.319219
$144$	0	0
$145$	−1.96530	−0.163209
$146$	0	0
$147$	3.92491	0.323721
$148$	0	0
$149$	8.00922	0.656141	0.328071	−	0.944653i	$-0.393602\pi$
$149$	8.00922	0.656141	0.328071	+	0.944653i	$0.393602\pi$
$150$	0	0
$151$	−11.3118	−0.920545	−0.460272	−	0.887778i	$-0.652248\pi$
$151$	−11.3118	−0.920545	−0.460272	+	0.887778i	$0.652248\pi$
$152$	0	0
$153$	0.811189	0.0655808
$154$	0	0
$155$	13.3611	1.07319
$156$	0	0
$157$	−16.7028	−1.33303	−0.666514	−	0.745492i	$-0.732215\pi$
$157$	−16.7028	−1.33303	−0.666514	+	0.745492i	$0.732215\pi$
$158$	0	0
$159$	−12.1117	−0.960523
$160$	0	0
$161$	−12.1625	−0.958537
$162$	0	0
$163$	15.1384	1.18573	0.592864	−	0.805302i	$-0.297997\pi$
$163$	15.1384	1.18573	0.592864	+	0.805302i	$0.297997\pi$
$164$	0	0
$165$	9.11937	0.709942
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−10.5007	−0.807745
$170$	0	0
$171$	−3.41955	−0.261499
$172$	0	0
$173$	−19.6112	−1.49101	−0.745507	−	0.666498i	$-0.767792\pi$
$173$	−19.6112	−1.49101	−0.745507	+	0.666498i	$0.767792\pi$
$174$	0	0
$175$	16.2451	1.22802
$176$	0	0
$177$	−10.0728	−0.757117
$178$	0	0
$179$	9.58562	0.716463	0.358231	−	0.933633i	$-0.383380\pi$
$179$	9.58562	0.716463	0.358231	+	0.933633i	$0.383380\pi$
$180$	0	0
$181$	11.6532	0.866176	0.433088	−	0.901352i	$-0.357424\pi$
$181$	11.6532	0.866176	0.433088	+	0.901352i	$0.357424\pi$
$182$	0	0
$183$	13.5319	1.00030
$184$	0	0
$185$	−34.7805	−2.55711
$186$	0	0
$187$	−1.95870	−0.143234
$188$	0	0
$189$	−1.75359	−0.127555
$190$	0	0
$191$	4.42341	0.320067	0.160034	−	0.987112i	$-0.448840\pi$
$191$	4.42341	0.320067	0.160034	+	0.987112i	$0.448840\pi$
$192$	0	0
$193$	18.6988	1.34597	0.672983	−	0.739658i	$-0.265013\pi$
$193$	18.6988	1.34597	0.672983	+	0.739658i	$0.265013\pi$
$194$	0	0
$195$	5.97076	0.427575
$196$	0	0
$197$	13.0189	0.927557	0.463778	−	0.885951i	$-0.346493\pi$
$197$	13.0189	0.927557	0.463778	+	0.885951i	$0.346493\pi$
$198$	0	0
$199$	−1.19991	−0.0850596	−0.0425298	−	0.999095i	$-0.513542\pi$
$199$	−1.19991	−0.0850596	−0.0425298	+	0.999095i	$0.513542\pi$
$200$	0	0
$201$	−14.1817	−1.00030
$202$	0	0
$203$	−0.912512	−0.0640458
$204$	0	0
$205$	−12.5237	−0.874696
$206$	0	0
$207$	−6.93574	−0.482067
$208$	0	0
$209$	8.25685	0.571139
$210$	0	0
$211$	0.264859	0.0182336	0.00911681	−	0.999958i	$-0.497098\pi$
$211$	0.264859	0.0182336	0.00911681	+	0.999958i	$0.497098\pi$
$212$	0	0
$213$	11.5259	0.789739
$214$	0	0
$215$	−22.6606	−1.54544
$216$	0	0
$217$	6.20373	0.421137
$218$	0	0
$219$	−3.71440	−0.250996
$220$	0	0
$221$	−1.28243	−0.0862654
$222$	0	0
$223$	10.3649	0.694086	0.347043	−	0.937849i	$-0.387186\pi$
$223$	10.3649	0.694086	0.347043	+	0.937849i	$0.387186\pi$
$224$	0	0
$225$	9.26390	0.617594
$226$	0	0
$227$	8.46824	0.562057	0.281028	−	0.959699i	$-0.409324\pi$
$227$	8.46824	0.562057	0.281028	+	0.959699i	$0.409324\pi$
$228$	0	0
$229$	0.0378672	0.00250233	0.00125117	−	0.999999i	$-0.499602\pi$
$229$	0.0378672	0.00250233	0.00125117	+	0.999999i	$0.499602\pi$
$230$	0	0
$231$	4.23423	0.278592
$232$	0	0
$233$	1.89120	0.123896	0.0619482	−	0.998079i	$-0.480269\pi$
$233$	1.89120	0.123896	0.0619482	+	0.998079i	$0.480269\pi$
$234$	0	0
$235$	−47.0458	−3.06893
$236$	0	0
$237$	12.9098	0.838580
$238$	0	0
$239$	18.5619	1.20067	0.600334	−	0.799749i	$-0.295034\pi$
$239$	18.5619	1.20067	0.600334	+	0.799749i	$0.295034\pi$
$240$	0	0
$241$	28.3402	1.82555	0.912776	−	0.408462i	$-0.133935\pi$
$241$	28.3402	1.82555	0.912776	+	0.408462i	$0.133935\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−14.8234	−0.947035
$246$	0	0
$247$	5.40604	0.343978
$248$	0	0
$249$	13.0558	0.827380
$250$	0	0
$251$	13.7155	0.865717	0.432858	−	0.901462i	$-0.357505\pi$
$251$	13.7155	0.865717	0.432858	+	0.901462i	$0.357505\pi$
$252$	0	0
$253$	16.7471	1.05288
$254$	0	0
$255$	−3.06367	−0.191854
$256$	0	0
$257$	24.7495	1.54383	0.771917	−	0.635724i	$-0.219298\pi$
$257$	24.7495	1.54383	0.771917	+	0.635724i	$0.219298\pi$
$258$	0	0
$259$	−16.1490	−1.00345
$260$	0	0
$261$	−0.520367	−0.0322099
$262$	0	0
$263$	6.68918	0.412472	0.206236	−	0.978502i	$-0.433879\pi$
$263$	6.68918	0.412472	0.206236	+	0.978502i	$0.433879\pi$
$264$	0	0
$265$	45.7431	2.80998
$266$	0	0
$267$	−0.129594	−0.00793099
$268$	0	0
$269$	−13.4921	−0.822628	−0.411314	−	0.911494i	$-0.634930\pi$
$269$	−13.4921	−0.822628	−0.411314	+	0.911494i	$0.634930\pi$
$270$	0	0
$271$	23.0913	1.40270	0.701349	−	0.712818i	$-0.252582\pi$
$271$	23.0913	1.40270	0.701349	+	0.712818i	$0.252582\pi$
$272$	0	0
$273$	2.77230	0.167787
$274$	0	0
$275$	−22.3687	−1.34888
$276$	0	0
$277$	10.5004	0.630906	0.315453	−	0.948941i	$-0.397843\pi$
$277$	10.5004	0.630906	0.315453	+	0.948941i	$0.397843\pi$
$278$	0	0
$279$	3.53773	0.211798
$280$	0	0
$281$	7.39023	0.440864	0.220432	−	0.975402i	$-0.429253\pi$
$281$	7.39023	0.440864	0.220432	+	0.975402i	$0.429253\pi$
$282$	0	0
$283$	10.2756	0.610819	0.305410	−	0.952221i	$-0.401207\pi$
$283$	10.2756	0.610819	0.305410	+	0.952221i	$0.401207\pi$
$284$	0	0
$285$	12.9148	0.765007
$286$	0	0
$287$	−5.81492	−0.343244
$288$	0	0
$289$	−16.3420	−0.961292
$290$	0	0
$291$	11.6438	0.682570
$292$	0	0
$293$	−11.5362	−0.673950	−0.336975	−	0.941514i	$-0.609404\pi$
$293$	−11.5362	−0.673950	−0.336975	+	0.941514i	$0.609404\pi$
$294$	0	0
$295$	38.0425	2.21492
$296$	0	0
$297$	2.41460	0.140109
$298$	0	0
$299$	10.9649	0.634115
$300$	0	0
$301$	−10.5216	−0.606455
$302$	0	0
$303$	−9.16093	−0.526282
$304$	0	0
$305$	−51.1066	−2.92635
$306$	0	0
$307$	30.0628	1.71577	0.857887	−	0.513839i	$-0.171777\pi$
$307$	30.0628	1.71577	0.857887	+	0.513839i	$0.171777\pi$
$308$	0	0
$309$	−2.06620	−0.117542
$310$	0	0
$311$	−3.22673	−0.182971	−0.0914856	−	0.995806i	$-0.529162\pi$
$311$	−3.22673	−0.182971	−0.0914856	+	0.995806i	$0.529162\pi$
$312$	0	0
$313$	17.2220	0.973448	0.486724	−	0.873556i	$-0.338192\pi$
$313$	17.2220	0.973448	0.486724	+	0.873556i	$0.338192\pi$
$314$	0	0
$315$	6.62290	0.373158
$316$	0	0
$317$	8.51767	0.478400	0.239200	−	0.970970i	$-0.423115\pi$
$317$	8.51767	0.478400	0.239200	+	0.970970i	$0.423115\pi$
$318$	0	0
$319$	1.25648	0.0703494
$320$	0	0
$321$	3.49830	0.195256
$322$	0	0
$323$	−2.77390	−0.154344
$324$	0	0
$325$	−14.6455	−0.812387
$326$	0	0
$327$	−8.78182	−0.485636
$328$	0	0
$329$	−21.8439	−1.20430
$330$	0	0
$331$	−15.5876	−0.856770	−0.428385	−	0.903596i	$-0.640917\pi$
$331$	−15.5876	−0.856770	−0.428385	+	0.903596i	$0.640917\pi$
$332$	0	0
$333$	−9.20908	−0.504655
$334$	0	0
$335$	53.5609	2.92635
$336$	0	0
$337$	−24.7917	−1.35049	−0.675246	−	0.737593i	$-0.735962\pi$
$337$	−24.7917	−1.35049	−0.675246	+	0.737593i	$0.735962\pi$
$338$	0	0
$339$	0.230019	0.0124929
$340$	0	0
$341$	−8.54220	−0.462586
$342$	0	0
$343$	−19.1579	−1.03443
$344$	0	0
$345$	26.1946	1.41027
$346$	0	0
$347$	−20.0028	−1.07380	−0.536902	−	0.843645i	$-0.680406\pi$
$347$	−20.0028	−1.07380	−0.536902	+	0.843645i	$0.680406\pi$
$348$	0	0
$349$	12.8599	0.688374	0.344187	−	0.938901i	$-0.388155\pi$
$349$	12.8599	0.688374	0.344187	+	0.938901i	$0.388155\pi$
$350$	0	0
$351$	1.58092	0.0843834
$352$	0	0
$353$	−17.4529	−0.928922	−0.464461	−	0.885594i	$-0.653752\pi$
$353$	−17.4529	−0.928922	−0.464461	+	0.885594i	$0.653752\pi$
$354$	0	0
$355$	−43.5304	−2.31035
$356$	0	0
$357$	−1.42250	−0.0752865
$358$	0	0
$359$	29.6090	1.56270	0.781352	−	0.624091i	$-0.214531\pi$
$359$	29.6090	1.56270	0.781352	+	0.624091i	$0.214531\pi$
$360$	0	0
$361$	−7.30668	−0.384562
$362$	0	0
$363$	5.16969	0.271338
$364$	0	0
$365$	14.0284	0.734281
$366$	0	0
$367$	−30.1359	−1.57308	−0.786540	−	0.617540i	$-0.788129\pi$
$367$	−30.1359	−1.57308	−0.786540	+	0.617540i	$0.788129\pi$
$368$	0	0
$369$	−3.31600	−0.172624
$370$	0	0
$371$	21.2391	1.10268
$372$	0	0
$373$	10.0710	0.521458	0.260729	−	0.965412i	$-0.416037\pi$
$373$	10.0710	0.521458	0.260729	+	0.965412i	$0.416037\pi$
$374$	0	0
$375$	−16.1037	−0.831593
$376$	0	0
$377$	0.822660	0.0423691
$378$	0	0
$379$	10.7931	0.554407	0.277203	−	0.960811i	$-0.410592\pi$
$379$	10.7931	0.554407	0.277203	+	0.960811i	$0.410592\pi$
$380$	0	0
$381$	−11.4655	−0.587394
$382$	0	0
$383$	−35.0045	−1.78865	−0.894323	−	0.447422i	$-0.852342\pi$
$383$	−35.0045	−1.78865	−0.894323	+	0.447422i	$0.852342\pi$
$384$	0	0
$385$	−15.9917	−0.815011
$386$	0	0
$387$	−6.00002	−0.304998
$388$	0	0
$389$	12.6554	0.641655	0.320828	−	0.947138i	$-0.396039\pi$
$389$	12.6554	0.641655	0.320828	+	0.947138i	$0.396039\pi$
$390$	0	0
$391$	−5.62620	−0.284529
$392$	0	0
$393$	18.3513	0.925700
$394$	0	0
$395$	−48.7571	−2.45324
$396$	0	0
$397$	23.1156	1.16014	0.580069	−	0.814567i	$-0.303025\pi$
$397$	23.1156	1.16014	0.580069	+	0.814567i	$0.303025\pi$
$398$	0	0
$399$	5.99650	0.300200
$400$	0	0
$401$	−11.3533	−0.566956	−0.283478	−	0.958979i	$-0.591488\pi$
$401$	−11.3533	−0.566956	−0.283478	+	0.958979i	$0.591488\pi$
$402$	0	0
$403$	−5.59287	−0.278601
$404$	0	0
$405$	3.77676	0.187669
$406$	0	0
$407$	22.2363	1.10221
$408$	0	0
$409$	−33.5045	−1.65669	−0.828346	−	0.560217i	$-0.810718\pi$
$409$	−33.5045	−1.65669	−0.828346	+	0.560217i	$0.810718\pi$
$410$	0	0
$411$	16.8506	0.831178
$412$	0	0
$413$	17.6636	0.869168
$414$	0	0
$415$	−49.3087	−2.42047
$416$	0	0
$417$	11.9744	0.586391
$418$	0	0
$419$	−26.2541	−1.28260	−0.641299	−	0.767291i	$-0.721604\pi$
$419$	−26.2541	−1.28260	−0.641299	+	0.767291i	$0.721604\pi$
$420$	0	0
$421$	−27.5746	−1.34390	−0.671951	−	0.740596i	$-0.734543\pi$
$421$	−27.5746	−1.34390	−0.671951	+	0.740596i	$0.734543\pi$
$422$	0	0
$423$	−12.4567	−0.605664
$424$	0	0
$425$	7.51478	0.364520
$426$	0	0
$427$	−23.7294	−1.14835
$428$	0	0
$429$	−3.81730	−0.184301
$430$	0	0
$431$	24.4575	1.17808	0.589039	−	0.808105i	$-0.299507\pi$
$431$	24.4575	1.17808	0.589039	+	0.808105i	$0.299507\pi$
$432$	0	0
$433$	−2.62253	−0.126031	−0.0630153	−	0.998013i	$-0.520072\pi$
$433$	−2.62253	−0.126031	−0.0630153	+	0.998013i	$0.520072\pi$
$434$	0	0
$435$	1.96530	0.0942289
$436$	0	0
$437$	23.7171	1.13454
$438$	0	0
$439$	−29.1576	−1.39161	−0.695807	−	0.718228i	$-0.744953\pi$
$439$	−29.1576	−1.39161	−0.695807	+	0.718228i	$0.744953\pi$
$440$	0	0
$441$	−3.92491	−0.186900
$442$	0	0
$443$	40.5738	1.92772	0.963859	−	0.266411i	$-0.0858379\pi$
$443$	40.5738	1.92772	0.963859	+	0.266411i	$0.0858379\pi$
$444$	0	0
$445$	0.489443	0.0232018
$446$	0	0
$447$	−8.00922	−0.378823
$448$	0	0
$449$	−27.1601	−1.28176	−0.640881	−	0.767640i	$-0.721431\pi$
$449$	−27.1601	−1.28176	−0.640881	+	0.767640i	$0.721431\pi$
$450$	0	0
$451$	8.00683	0.377027
$452$	0	0
$453$	11.3118	0.531477
$454$	0	0
$455$	−10.4703	−0.490855
$456$	0	0
$457$	−39.8219	−1.86279	−0.931396	−	0.364007i	$-0.881408\pi$
$457$	−39.8219	−1.86279	−0.931396	+	0.364007i	$0.881408\pi$
$458$	0	0
$459$	−0.811189	−0.0378631
$460$	0	0
$461$	29.6135	1.37924	0.689618	−	0.724173i	$-0.257778\pi$
$461$	29.6135	1.37924	0.689618	+	0.724173i	$0.257778\pi$
$462$	0	0
$463$	−2.07428	−0.0964001	−0.0482000	−	0.998838i	$-0.515348\pi$
$463$	−2.07428	−0.0964001	−0.0482000	+	0.998838i	$0.515348\pi$
$464$	0	0
$465$	−13.3611	−0.619608
$466$	0	0
$467$	−9.02001	−0.417396	−0.208698	−	0.977980i	$-0.566923\pi$
$467$	−9.02001	−0.417396	−0.208698	+	0.977980i	$0.566923\pi$
$468$	0	0
$469$	24.8690	1.14834
$470$	0	0
$471$	16.7028	0.769625
$472$	0	0
$473$	14.4877	0.666144
$474$	0	0
$475$	−31.6784	−1.45350
$476$	0	0
$477$	12.1117	0.554558
$478$	0	0
$479$	25.6751	1.17312	0.586562	−	0.809904i	$-0.300481\pi$
$479$	25.6751	1.17312	0.586562	+	0.809904i	$0.300481\pi$
$480$	0	0
$481$	14.5588	0.663826
$482$	0	0
$483$	12.1625	0.553412
$484$	0	0
$485$	−43.9757	−1.99683
$486$	0	0
$487$	−1.17918	−0.0534337	−0.0267169	−	0.999643i	$-0.508505\pi$
$487$	−1.17918	−0.0534337	−0.0267169	+	0.999643i	$0.508505\pi$
$488$	0	0
$489$	−15.1384	−0.684581
$490$	0	0
$491$	−3.20030	−0.144428	−0.0722138	−	0.997389i	$-0.523006\pi$
$491$	−3.20030	−0.144428	−0.0722138	+	0.997389i	$0.523006\pi$
$492$	0	0
$493$	−0.422116	−0.0190112
$494$	0	0
$495$	−9.11937	−0.409885
$496$	0	0
$497$	−20.2117	−0.906617
$498$	0	0
$499$	−30.2627	−1.35475	−0.677373	−	0.735640i	$-0.736882\pi$
$499$	−30.2627	−1.35475	−0.677373	+	0.735640i	$0.736882\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−8.41219	−0.375081	−0.187540	−	0.982257i	$-0.560052\pi$
$503$	−8.41219	−0.375081	−0.187540	+	0.982257i	$0.560052\pi$
$504$	0	0
$505$	34.5986	1.53962
$506$	0	0
$507$	10.5007	0.466352
$508$	0	0
$509$	−31.2162	−1.38363	−0.691816	−	0.722074i	$-0.743189\pi$
$509$	−31.2162	−1.38363	−0.691816	+	0.722074i	$0.743189\pi$
$510$	0	0
$511$	6.51356	0.288143
$512$	0	0
$513$	3.41955	0.150977
$514$	0	0
$515$	7.80354	0.343865
$516$	0	0
$517$	30.0779	1.32283
$518$	0	0
$519$	19.6112	0.860837
$520$	0	0
$521$	2.42706	0.106331	0.0531656	−	0.998586i	$-0.483069\pi$
$521$	2.42706	0.106331	0.0531656	+	0.998586i	$0.483069\pi$
$522$	0	0
$523$	−10.6211	−0.464429	−0.232214	−	0.972665i	$-0.574597\pi$
$523$	−10.6211	−0.464429	−0.232214	+	0.972665i	$0.574597\pi$
$524$	0	0
$525$	−16.2451	−0.708995
$526$	0	0
$527$	2.86977	0.125009
$528$	0	0
$529$	25.1045	1.09150
$530$	0	0
$531$	10.0728	0.437122
$532$	0	0
$533$	5.24235	0.227071
$534$	0	0
$535$	−13.2123	−0.571215
$536$	0	0
$537$	−9.58562	−0.413650
$538$	0	0
$539$	9.47710	0.408208
$540$	0	0
$541$	−19.6519	−0.844902	−0.422451	−	0.906386i	$-0.638830\pi$
$541$	−19.6519	−0.844902	−0.422451	+	0.906386i	$0.638830\pi$
$542$	0	0
$543$	−11.6532	−0.500087
$544$	0	0
$545$	33.1668	1.42071
$546$	0	0
$547$	−19.2002	−0.820941	−0.410471	−	0.911874i	$-0.634636\pi$
$547$	−19.2002	−0.820941	−0.410471	+	0.911874i	$0.634636\pi$
$548$	0	0
$549$	−13.5319	−0.577526
$550$	0	0
$551$	1.77942	0.0758059
$552$	0	0
$553$	−22.6385	−0.962687
$554$	0	0
$555$	34.7805	1.47635
$556$	0	0
$557$	16.0785	0.681268	0.340634	−	0.940196i	$-0.389358\pi$
$557$	16.0785	0.681268	0.340634	+	0.940196i	$0.389358\pi$
$558$	0	0
$559$	9.48558	0.401197
$560$	0	0
$561$	1.95870	0.0826964
$562$	0	0
$563$	−32.0926	−1.35254	−0.676270	−	0.736654i	$-0.736405\pi$
$563$	−32.0926	−1.35254	−0.676270	+	0.736654i	$0.736405\pi$
$564$	0	0
$565$	−0.868728	−0.0365476
$566$	0	0
$567$	1.75359	0.0736440
$568$	0	0
$569$	11.5180	0.482861	0.241430	−	0.970418i	$-0.422383\pi$
$569$	11.5180	0.482861	0.241430	+	0.970418i	$0.422383\pi$
$570$	0	0
$571$	12.7614	0.534050	0.267025	−	0.963690i	$-0.413959\pi$
$571$	12.7614	0.534050	0.267025	+	0.963690i	$0.413959\pi$
$572$	0	0
$573$	−4.42341	−0.184791
$574$	0	0
$575$	−64.2521	−2.67950
$576$	0	0
$577$	−20.3828	−0.848546	−0.424273	−	0.905534i	$-0.639470\pi$
$577$	−20.3828	−0.848546	−0.424273	+	0.905534i	$0.639470\pi$
$578$	0	0
$579$	−18.6988	−0.777094
$580$	0	0
$581$	−22.8946	−0.949829
$582$	0	0
$583$	−29.2450	−1.21121
$584$	0	0
$585$	−5.97076	−0.246861
$586$	0	0
$587$	41.5389	1.71450	0.857248	−	0.514904i	$-0.172173\pi$
$587$	41.5389	1.71450	0.857248	+	0.514904i	$0.172173\pi$
$588$	0	0
$589$	−12.0974	−0.498466
$590$	0	0
$591$	−13.0189	−0.535525
$592$	0	0
$593$	7.55642	0.310305	0.155152	−	0.987891i	$-0.450413\pi$
$593$	7.55642	0.310305	0.155152	+	0.987891i	$0.450413\pi$
$594$	0	0
$595$	5.37242	0.220248
$596$	0	0
$597$	1.19991	0.0491092
$598$	0	0
$599$	−5.56179	−0.227249	−0.113624	−	0.993524i	$-0.536246\pi$
$599$	−5.56179	−0.227249	−0.113624	+	0.993524i	$0.536246\pi$
$600$	0	0
$601$	−3.53324	−0.144124	−0.0720619	−	0.997400i	$-0.522958\pi$
$601$	−3.53324	−0.144124	−0.0720619	+	0.997400i	$0.522958\pi$
$602$	0	0
$603$	14.1817	0.577524
$604$	0	0
$605$	−19.5247	−0.793791
$606$	0	0
$607$	−15.0346	−0.610236	−0.305118	−	0.952314i	$-0.598696\pi$
$607$	−15.0346	−0.610236	−0.305118	+	0.952314i	$0.598696\pi$
$608$	0	0
$609$	0.912512	0.0369769
$610$	0	0
$611$	19.6930	0.796695
$612$	0	0
$613$	−38.2450	−1.54470	−0.772350	−	0.635197i	$-0.780919\pi$
$613$	−38.2450	−1.54470	−0.772350	+	0.635197i	$0.780919\pi$
$614$	0	0
$615$	12.5237	0.505006
$616$	0	0
$617$	6.92434	0.278764	0.139382	−	0.990239i	$-0.455488\pi$
$617$	6.92434	0.278764	0.139382	+	0.990239i	$0.455488\pi$
$618$	0	0
$619$	33.8045	1.35872	0.679359	−	0.733806i	$-0.262258\pi$
$619$	33.8045	1.35872	0.679359	+	0.733806i	$0.262258\pi$
$620$	0	0
$621$	6.93574	0.278322
$622$	0	0
$623$	0.227254	0.00910475
$624$	0	0
$625$	14.5004	0.580016
$626$	0	0
$627$	−8.25685	−0.329747
$628$	0	0
$629$	−7.47031	−0.297861
$630$	0	0
$631$	−20.7676	−0.826746	−0.413373	−	0.910562i	$-0.635649\pi$
$631$	−20.7676	−0.826746	−0.413373	+	0.910562i	$0.635649\pi$
$632$	0	0
$633$	−0.264859	−0.0105272
$634$	0	0
$635$	43.3023	1.71840
$636$	0	0
$637$	6.20498	0.245850
$638$	0	0
$639$	−11.5259	−0.455956
$640$	0	0
$641$	−42.4228	−1.67560	−0.837800	−	0.545977i	$-0.816158\pi$
$641$	−42.4228	−1.67560	−0.837800	+	0.545977i	$0.816158\pi$
$642$	0	0
$643$	17.1884	0.677844	0.338922	−	0.940814i	$-0.389938\pi$
$643$	17.1884	0.677844	0.338922	+	0.940814i	$0.389938\pi$
$644$	0	0
$645$	22.6606	0.892262
$646$	0	0
$647$	33.9398	1.33431	0.667155	−	0.744919i	$-0.267512\pi$
$647$	33.9398	1.33431	0.667155	+	0.744919i	$0.267512\pi$
$648$	0	0
$649$	−24.3218	−0.954714
$650$	0	0
$651$	−6.20373	−0.243143
$652$	0	0
$653$	−3.87635	−0.151693	−0.0758466	−	0.997119i	$-0.524166\pi$
$653$	−3.87635	−0.151693	−0.0758466	+	0.997119i	$0.524166\pi$
$654$	0	0
$655$	−69.3084	−2.70810
$656$	0	0
$657$	3.71440	0.144913
$658$	0	0
$659$	−23.8087	−0.927457	−0.463728	−	0.885977i	$-0.653489\pi$
$659$	−23.8087	−0.927457	−0.463728	+	0.885977i	$0.653489\pi$
$660$	0	0
$661$	3.05094	0.118668	0.0593340	−	0.998238i	$-0.481102\pi$
$661$	3.05094	0.118668	0.0593340	+	0.998238i	$0.481102\pi$
$662$	0	0
$663$	1.28243	0.0498054
$664$	0	0
$665$	−22.6473	−0.878226
$666$	0	0
$667$	3.60913	0.139746
$668$	0	0
$669$	−10.3649	−0.400731
$670$	0	0
$671$	32.6741	1.26137
$672$	0	0
$673$	−15.3106	−0.590180	−0.295090	−	0.955469i	$-0.595350\pi$
$673$	−15.3106	−0.590180	−0.295090	+	0.955469i	$0.595350\pi$
$674$	0	0
$675$	−9.26390	−0.356568
$676$	0	0
$677$	16.9262	0.650526	0.325263	−	0.945624i	$-0.394547\pi$
$677$	16.9262	0.650526	0.325263	+	0.945624i	$0.394547\pi$
$678$	0	0
$679$	−20.4184	−0.783588
$680$	0	0
$681$	−8.46824	−0.324504
$682$	0	0
$683$	1.54551	0.0591372	0.0295686	−	0.999563i	$-0.490587\pi$
$683$	1.54551	0.0591372	0.0295686	+	0.999563i	$0.490587\pi$
$684$	0	0
$685$	−63.6406	−2.43158
$686$	0	0
$687$	−0.0378672	−0.00144472
$688$	0	0
$689$	−19.1477	−0.729470
$690$	0	0
$691$	30.5564	1.16242	0.581211	−	0.813753i	$-0.302579\pi$
$691$	30.5564	1.16242	0.581211	+	0.813753i	$0.302579\pi$
$692$	0	0
$693$	−4.23423	−0.160845
$694$	0	0
$695$	−45.2246	−1.71547
$696$	0	0
$697$	−2.68991	−0.101887
$698$	0	0
$699$	−1.89120	−0.0715316
$700$	0	0
$701$	−38.5235	−1.45501	−0.727507	−	0.686100i	$-0.759321\pi$
$701$	−38.5235	−1.45501	−0.727507	+	0.686100i	$0.759321\pi$
$702$	0	0
$703$	31.4909	1.18770
$704$	0	0
$705$	47.0458	1.77185
$706$	0	0
$707$	16.0646	0.604170
$708$	0	0
$709$	19.4274	0.729610	0.364805	−	0.931084i	$-0.381136\pi$
$709$	19.4274	0.729610	0.364805	+	0.931084i	$0.381136\pi$
$710$	0	0
$711$	−12.9098	−0.484155
$712$	0	0
$713$	−24.5368	−0.918909
$714$	0	0
$715$	14.4170	0.539166
$716$	0	0
$717$	−18.5619	−0.693206
$718$	0	0
$719$	−2.55899	−0.0954341	−0.0477170	−	0.998861i	$-0.515195\pi$
$719$	−2.55899	−0.0954341	−0.0477170	+	0.998861i	$0.515195\pi$
$720$	0	0
$721$	3.62328	0.134938
$722$	0	0
$723$	−28.3402	−1.05398
$724$	0	0
$725$	−4.82063	−0.179034
$726$	0	0
$727$	−20.2028	−0.749282	−0.374641	−	0.927170i	$-0.622234\pi$
$727$	−20.2028	−0.749282	−0.374641	+	0.927170i	$0.622234\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.86716	−0.180018
$732$	0	0
$733$	16.2646	0.600745	0.300372	−	0.953822i	$-0.402889\pi$
$733$	16.2646	0.600745	0.300372	+	0.953822i	$0.402889\pi$
$734$	0	0
$735$	14.8234	0.546771
$736$	0	0
$737$	−34.2432	−1.26137
$738$	0	0
$739$	47.8478	1.76011	0.880054	−	0.474873i	$-0.157506\pi$
$739$	47.8478	1.76011	0.880054	+	0.474873i	$0.157506\pi$
$740$	0	0
$741$	−5.40604	−0.198596
$742$	0	0
$743$	33.1597	1.21651	0.608256	−	0.793741i	$-0.291870\pi$
$743$	33.1597	1.21651	0.608256	+	0.793741i	$0.291870\pi$
$744$	0	0
$745$	30.2489	1.10823
$746$	0	0
$747$	−13.0558	−0.477688
$748$	0	0
$749$	−6.13460	−0.224154
$750$	0	0
$751$	−43.8667	−1.60072	−0.800359	−	0.599521i	$-0.795358\pi$
$751$	−43.8667	−1.60072	−0.800359	+	0.599521i	$0.795358\pi$
$752$	0	0
$753$	−13.7155	−0.499822
$754$	0	0
$755$	−42.7221	−1.55482
$756$	0	0
$757$	−47.8859	−1.74044	−0.870221	−	0.492661i	$-0.836024\pi$
$757$	−47.8859	−1.74044	−0.870221	+	0.492661i	$0.836024\pi$
$758$	0	0
$759$	−16.7471	−0.607880
$760$	0	0
$761$	14.3472	0.520084	0.260042	−	0.965597i	$-0.416264\pi$
$761$	14.3472	0.520084	0.260042	+	0.965597i	$0.416264\pi$
$762$	0	0
$763$	15.3997	0.557508
$764$	0	0
$765$	3.06367	0.110767
$766$	0	0
$767$	−15.9243	−0.574993
$768$	0	0
$769$	6.42450	0.231673	0.115837	−	0.993268i	$-0.463045\pi$
$769$	6.42450	0.231673	0.115837	+	0.993268i	$0.463045\pi$
$770$	0	0
$771$	−24.7495	−0.891332
$772$	0	0
$773$	4.35272	0.156556	0.0782782	−	0.996932i	$-0.475058\pi$
$773$	4.35272	0.156556	0.0782782	+	0.996932i	$0.475058\pi$
$774$	0	0
$775$	32.7732	1.17725
$776$	0	0
$777$	16.1490	0.579342
$778$	0	0
$779$	11.3392	0.406270
$780$	0	0
$781$	27.8304	0.995849
$782$	0	0
$783$	0.520367	0.0185964
$784$	0	0
$785$	−63.0825	−2.25151
$786$	0	0
$787$	4.33561	0.154548	0.0772739	−	0.997010i	$-0.475378\pi$
$787$	4.33561	0.154548	0.0772739	+	0.997010i	$0.475378\pi$
$788$	0	0
$789$	−6.68918	−0.238141
$790$	0	0
$791$	−0.403360	−0.0143418
$792$	0	0
$793$	21.3928	0.759682
$794$	0	0
$795$	−45.7431	−1.62234
$796$	0	0
$797$	−17.2783	−0.612027	−0.306014	−	0.952027i	$-0.598995\pi$
$797$	−17.2783	−0.612027	−0.306014	+	0.952027i	$0.598995\pi$
$798$	0	0
$799$	−10.1047	−0.357479
$800$	0	0
$801$	0.129594	0.00457896
$802$	0	0
$803$	−8.96881	−0.316503
$804$	0	0
$805$	−45.9347	−1.61899
$806$	0	0
$807$	13.4921	0.474945
$808$	0	0
$809$	15.9662	0.561342	0.280671	−	0.959804i	$-0.409443\pi$
$809$	15.9662	0.561342	0.280671	+	0.959804i	$0.409443\pi$
$810$	0	0
$811$	49.3014	1.73121	0.865603	−	0.500731i	$-0.166935\pi$
$811$	49.3014	1.73121	0.865603	+	0.500731i	$0.166935\pi$
$812$	0	0
$813$	−23.0913	−0.809848
$814$	0	0
$815$	57.1740	2.00272
$816$	0	0
$817$	20.5174	0.717812
$818$	0	0
$819$	−2.77230	−0.0968719
$820$	0	0
$821$	38.2189	1.33385	0.666924	−	0.745126i	$-0.267610\pi$
$821$	38.2189	1.33385	0.666924	+	0.745126i	$0.267610\pi$
$822$	0	0
$823$	14.4225	0.502736	0.251368	−	0.967892i	$-0.419119\pi$
$823$	14.4225	0.502736	0.251368	+	0.967892i	$0.419119\pi$
$824$	0	0
$825$	22.3687	0.778777
$826$	0	0
$827$	34.6869	1.20618	0.603090	−	0.797673i	$-0.293936\pi$
$827$	34.6869	1.20618	0.603090	+	0.797673i	$0.293936\pi$
$828$	0	0
$829$	−43.5439	−1.51234	−0.756171	−	0.654375i	$-0.772932\pi$
$829$	−43.5439	−1.51234	−0.756171	+	0.654375i	$0.772932\pi$
$830$	0	0
$831$	−10.5004	−0.364254
$832$	0	0
$833$	−3.18385	−0.110314
$834$	0	0
$835$	−3.77676	−0.130700
$836$	0	0
$837$	−3.53773	−0.122282
$838$	0	0
$839$	−13.1785	−0.454974	−0.227487	−	0.973781i	$-0.573051\pi$
$839$	−13.1785	−0.454974	−0.227487	+	0.973781i	$0.573051\pi$
$840$	0	0
$841$	−28.7292	−0.990663
$842$	0	0
$843$	−7.39023	−0.254533
$844$	0	0
$845$	−39.6585	−1.36430
$846$	0	0
$847$	−9.06554	−0.311496
$848$	0	0
$849$	−10.2756	−0.352657
$850$	0	0
$851$	63.8718	2.18950
$852$	0	0
$853$	13.4897	0.461878	0.230939	−	0.972968i	$-0.425820\pi$
$853$	13.4897	0.461878	0.230939	+	0.972968i	$0.425820\pi$
$854$	0	0
$855$	−12.9148	−0.441677
$856$	0	0
$857$	26.7955	0.915317	0.457659	−	0.889128i	$-0.348688\pi$
$857$	26.7955	0.915317	0.457659	+	0.889128i	$0.348688\pi$
$858$	0	0
$859$	−46.2866	−1.57928	−0.789640	−	0.613571i	$-0.789732\pi$
$859$	−46.2866	−1.57928	−0.789640	+	0.613571i	$0.789732\pi$
$860$	0	0
$861$	5.81492	0.198172
$862$	0	0
$863$	11.0943	0.377653	0.188827	−	0.982010i	$-0.439532\pi$
$863$	11.0943	0.377653	0.188827	+	0.982010i	$0.439532\pi$
$864$	0	0
$865$	−74.0668	−2.51835
$866$	0	0
$867$	16.3420	0.555002
$868$	0	0
$869$	31.1720	1.05744
$870$	0	0
$871$	−22.4202	−0.759680
$872$	0	0
$873$	−11.6438	−0.394082
$874$	0	0
$875$	28.2394	0.954666
$876$	0	0
$877$	3.85241	0.130087	0.0650433	−	0.997882i	$-0.479281\pi$
$877$	3.85241	0.130087	0.0650433	+	0.997882i	$0.479281\pi$
$878$	0	0
$879$	11.5362	0.389105
$880$	0	0
$881$	−50.4047	−1.69818	−0.849089	−	0.528250i	$-0.822849\pi$
$881$	−50.4047	−1.69818	−0.849089	+	0.528250i	$0.822849\pi$
$882$	0	0
$883$	−0.621522	−0.0209159	−0.0104579	−	0.999945i	$-0.503329\pi$
$883$	−0.621522	−0.0209159	−0.0104579	+	0.999945i	$0.503329\pi$
$884$	0	0
$885$	−38.0425	−1.27878
$886$	0	0
$887$	−1.37732	−0.0462458	−0.0231229	−	0.999733i	$-0.507361\pi$
$887$	−1.37732	−0.0462458	−0.0231229	+	0.999733i	$0.507361\pi$
$888$	0	0
$889$	20.1058	0.674326
$890$	0	0
$891$	−2.41460	−0.0808923
$892$	0	0
$893$	42.5962	1.42543
$894$	0	0
$895$	36.2026	1.21012
$896$	0	0
$897$	−10.9649	−0.366106
$898$	0	0
$899$	−1.84092	−0.0613980
$900$	0	0
$901$	9.82491	0.327315
$902$	0	0
$903$	10.5216	0.350137
$904$	0	0
$905$	44.0113	1.46299
$906$	0	0
$907$	−5.57385	−0.185077	−0.0925383	−	0.995709i	$-0.529498\pi$
$907$	−5.57385	−0.185077	−0.0925383	+	0.995709i	$0.529498\pi$
$908$	0	0
$909$	9.16093	0.303849
$910$	0	0
$911$	53.4020	1.76929	0.884644	−	0.466267i	$-0.154401\pi$
$911$	53.4020	1.76929	0.884644	+	0.466267i	$0.154401\pi$
$912$	0	0
$913$	31.5247	1.04331
$914$	0	0
$915$	51.1066	1.68953
$916$	0	0
$917$	−32.1807	−1.06270
$918$	0	0
$919$	−54.9803	−1.81363	−0.906816	−	0.421527i	$-0.861494\pi$
$919$	−54.9803	−1.81363	−0.906816	+	0.421527i	$0.861494\pi$
$920$	0	0
$921$	−30.0628	−0.990602
$922$	0	0
$923$	18.2215	0.599768
$924$	0	0
$925$	−85.3121	−2.80504
$926$	0	0
$927$	2.06620	0.0678630
$928$	0	0
$929$	−50.7144	−1.66389	−0.831943	−	0.554861i	$-0.812772\pi$
$929$	−50.7144	−1.66389	−0.831943	+	0.554861i	$0.812772\pi$
$930$	0	0
$931$	13.4214	0.439869
$932$	0	0
$933$	3.22673	0.105638
$934$	0	0
$935$	−7.39754	−0.241925
$936$	0	0
$937$	41.1997	1.34594	0.672968	−	0.739672i	$-0.265019\pi$
$937$	41.1997	1.34594	0.672968	+	0.739672i	$0.265019\pi$
$938$	0	0
$939$	−17.2220	−0.562020
$940$	0	0
$941$	11.2604	0.367078	0.183539	−	0.983012i	$-0.441245\pi$
$941$	11.2604	0.367078	0.183539	+	0.983012i	$0.441245\pi$
$942$	0	0
$943$	22.9989	0.748949
$944$	0	0
$945$	−6.62290	−0.215443
$946$	0	0
$947$	−46.2686	−1.50353	−0.751763	−	0.659433i	$-0.770796\pi$
$947$	−46.2686	−1.50353	−0.751763	+	0.659433i	$0.770796\pi$
$948$	0	0
$949$	−5.87219	−0.190619
$950$	0	0
$951$	−8.51767	−0.276204
$952$	0	0
$953$	6.24795	0.202391	0.101196	−	0.994867i	$-0.467733\pi$
$953$	6.24795	0.202391	0.101196	+	0.994867i	$0.467733\pi$
$954$	0	0
$955$	16.7062	0.540599
$956$	0	0
$957$	−1.25648	−0.0406162
$958$	0	0
$959$	−29.5491	−0.954190
$960$	0	0
$961$	−18.4845	−0.596274
$962$	0	0
$963$	−3.49830	−0.112731
$964$	0	0
$965$	70.6207	2.27336
$966$	0	0
$967$	6.48597	0.208575	0.104287	−	0.994547i	$-0.466744\pi$
$967$	6.48597	0.208575	0.104287	+	0.994547i	$0.466744\pi$
$968$	0	0
$969$	2.77390	0.0891106
$970$	0	0
$971$	15.0219	0.482077	0.241038	−	0.970516i	$-0.422512\pi$
$971$	15.0219	0.482077	0.241038	+	0.970516i	$0.422512\pi$
$972$	0	0
$973$	−20.9983	−0.673175
$974$	0	0
$975$	14.6455	0.469032
$976$	0	0
$977$	26.0541	0.833543	0.416772	−	0.909011i	$-0.363161\pi$
$977$	26.0541	0.833543	0.416772	+	0.909011i	$0.363161\pi$
$978$	0	0
$979$	−0.312917	−0.0100009
$980$	0	0
$981$	8.78182	0.280382
$982$	0	0
$983$	−3.46591	−0.110545	−0.0552727	−	0.998471i	$-0.517603\pi$
$983$	−3.46591	−0.110545	−0.0552727	+	0.998471i	$0.517603\pi$
$984$	0	0
$985$	49.1692	1.56666
$986$	0	0
$987$	21.8439	0.695300
$988$	0	0
$989$	41.6146	1.32327
$990$	0	0
$991$	31.3162	0.994791	0.497395	−	0.867524i	$-0.334290\pi$
$991$	31.3162	0.994791	0.497395	+	0.867524i	$0.334290\pi$
$992$	0	0
$993$	15.5876	0.494656
$994$	0	0
$995$	−4.53178	−0.143667
$996$	0	0
$997$	15.0051	0.475215	0.237607	−	0.971361i	$-0.423637\pi$
$997$	15.0051	0.475215	0.237607	+	0.971361i	$0.423637\pi$
$998$	0	0
$999$	9.20908	0.291362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bf.1.12		12
4.3	odd	2		4008.2.a.l.1.12	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.l.1.12	✓	12	4.3	odd	2
8016.2.a.bf.1.12		12	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.77676 q^{5} +1.75359 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.77676 q^{5} +1.75359 q^{7} +1.00000 q^{9} -2.41460 q^{11} -1.58092 q^{13} -3.77676 q^{15} +0.811189 q^{17} -3.41955 q^{19} -1.75359 q^{21} -6.93574 q^{23} +9.26390 q^{25} -1.00000 q^{27} -0.520367 q^{29} +3.53773 q^{31} +2.41460 q^{33} +6.62290 q^{35} -9.20908 q^{37} +1.58092 q^{39} -3.31600 q^{41} -6.00002 q^{43} +3.77676 q^{45} -12.4567 q^{47} -3.92491 q^{49} -0.811189 q^{51} +12.1117 q^{53} -9.11937 q^{55} +3.41955 q^{57} +10.0728 q^{59} -13.5319 q^{61} +1.75359 q^{63} -5.97076 q^{65} +14.1817 q^{67} +6.93574 q^{69} -11.5259 q^{71} +3.71440 q^{73} -9.26390 q^{75} -4.23423 q^{77} -12.9098 q^{79} +1.00000 q^{81} -13.0558 q^{83} +3.06367 q^{85} +0.520367 q^{87} +0.129594 q^{89} -2.77230 q^{91} -3.53773 q^{93} -12.9148 q^{95} -11.6438 q^{97} -2.41460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9} - q^{11} + 8 q^{13} - 4 q^{15} + 3 q^{17} - 12 q^{19} + 11 q^{21} - 7 q^{23} + 18 q^{25} - 12 q^{27} + 5 q^{29} - 33 q^{31} + q^{33} - 15 q^{35} + 8 q^{37} - 8 q^{39} - 6 q^{41} - 16 q^{43} + 4 q^{45} - 18 q^{47} + 25 q^{49} - 3 q^{51} + 20 q^{53} - 39 q^{55} + 12 q^{57} - 4 q^{59} + 10 q^{61} - 11 q^{63} - 9 q^{67} + 7 q^{69} - 11 q^{71} + 22 q^{73} - 18 q^{75} + 24 q^{77} - 56 q^{79} + 12 q^{81} - 26 q^{83} + 15 q^{85} - 5 q^{87} - 15 q^{89} - 11 q^{91} + 33 q^{93} - 3 q^{95} + 8 q^{97} - q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 + 4 * q^5 - 11 * q^7 + 12 * q^9 - q^11 + 8 * q^13 - 4 * q^15 + 3 * q^17 - 12 * q^19 + 11 * q^21 - 7 * q^23 + 18 * q^25 - 12 * q^27 + 5 * q^29 - 33 * q^31 + q^33 - 15 * q^35 + 8 * q^37 - 8 * q^39 - 6 * q^41 - 16 * q^43 + 4 * q^45 - 18 * q^47 + 25 * q^49 - 3 * q^51 + 20 * q^53 - 39 * q^55 + 12 * q^57 - 4 * q^59 + 10 * q^61 - 11 * q^63 - 9 * q^67 + 7 * q^69 - 11 * q^71 + 22 * q^73 - 18 * q^75 + 24 * q^77 - 56 * q^79 + 12 * q^81 - 26 * q^83 + 15 * q^85 - 5 * q^87 - 15 * q^89 - 11 * q^91 + 33 * q^93 - 3 * q^95 + 8 * q^97 - q^99

Introduction

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