LMFDB - Embedded newform 8016.2.a.bc.1.2

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			1.2
Root			$-2.94765$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.94765 q^{5} -4.65088 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.94765 q^{5} -4.65088 q^{7} +1.00000 q^{9} +5.56158 q^{11} -0.393514 q^{13} -2.94765 q^{15} -0.257869 q^{17} -2.11137 q^{19} -4.65088 q^{21} -5.38988 q^{23} +3.68866 q^{25} +1.00000 q^{27} -0.101053 q^{29} +3.82473 q^{31} +5.56158 q^{33} +13.7092 q^{35} -1.36787 q^{37} -0.393514 q^{39} -5.79375 q^{41} -7.94402 q^{43} -2.94765 q^{45} -5.64008 q^{47} +14.6307 q^{49} -0.257869 q^{51} -4.37285 q^{53} -16.3936 q^{55} -2.11137 q^{57} -7.40379 q^{59} +4.23267 q^{61} -4.65088 q^{63} +1.15994 q^{65} +4.79684 q^{67} -5.38988 q^{69} +7.11410 q^{71} +9.79959 q^{73} +3.68866 q^{75} -25.8663 q^{77} +4.39883 q^{79} +1.00000 q^{81} -9.64462 q^{83} +0.760108 q^{85} -0.101053 q^{87} +10.6901 q^{89} +1.83019 q^{91} +3.82473 q^{93} +6.22358 q^{95} -5.54716 q^{97} +5.56158 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.94765	−1.31823	−0.659115	−	0.752042i	$-0.729069\pi$
$5$	−2.94765	−1.31823	−0.659115	+	0.752042i	$0.729069\pi$
$6$	0	0
$7$	−4.65088	−1.75787	−0.878934	−	0.476944i	$-0.841744\pi$
$7$	−4.65088	−1.75787	−0.878934	+	0.476944i	$0.841744\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.56158	1.67688	0.838440	−	0.544993i	$-0.183468\pi$
$11$	5.56158	1.67688	0.838440	+	0.544993i	$0.183468\pi$
$12$	0	0
$13$	−0.393514	−0.109141	−0.0545706	−	0.998510i	$-0.517379\pi$
$13$	−0.393514	−0.109141	−0.0545706	+	0.998510i	$0.517379\pi$
$14$	0	0
$15$	−2.94765	−0.761081
$16$	0	0
$17$	−0.257869	−0.0625424	−0.0312712	−	0.999511i	$-0.509956\pi$
$17$	−0.257869	−0.0625424	−0.0312712	+	0.999511i	$0.509956\pi$
$18$	0	0
$19$	−2.11137	−0.484381	−0.242191	−	0.970229i	$-0.577866\pi$
$19$	−2.11137	−0.484381	−0.242191	+	0.970229i	$0.577866\pi$
$20$	0	0
$21$	−4.65088	−1.01491
$22$	0	0
$23$	−5.38988	−1.12387	−0.561934	−	0.827182i	$-0.689943\pi$
$23$	−5.38988	−1.12387	−0.561934	+	0.827182i	$0.689943\pi$
$24$	0	0
$25$	3.68866	0.737731
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.101053	−0.0187651	−0.00938257	−	0.999956i	$-0.502987\pi$
$29$	−0.101053	−0.0187651	−0.00938257	+	0.999956i	$0.502987\pi$
$30$	0	0
$31$	3.82473	0.686941	0.343471	−	0.939163i	$-0.388397\pi$
$31$	3.82473	0.686941	0.343471	+	0.939163i	$0.388397\pi$
$32$	0	0
$33$	5.56158	0.968148
$34$	0	0
$35$	13.7092	2.31727
$36$	0	0
$37$	−1.36787	−0.224876	−0.112438	−	0.993659i	$-0.535866\pi$
$37$	−1.36787	−0.224876	−0.112438	+	0.993659i	$0.535866\pi$
$38$	0	0
$39$	−0.393514	−0.0630127
$40$	0	0
$41$	−5.79375	−0.904832	−0.452416	−	0.891807i	$-0.649438\pi$
$41$	−5.79375	−0.904832	−0.452416	+	0.891807i	$0.649438\pi$
$42$	0	0
$43$	−7.94402	−1.21145	−0.605726	−	0.795673i	$-0.707117\pi$
$43$	−7.94402	−1.21145	−0.605726	+	0.795673i	$0.707117\pi$
$44$	0	0
$45$	−2.94765	−0.439410
$46$	0	0
$47$	−5.64008	−0.822691	−0.411345	−	0.911480i	$-0.634941\pi$
$47$	−5.64008	−0.822691	−0.411345	+	0.911480i	$0.634941\pi$
$48$	0	0
$49$	14.6307	2.09010
$50$	0	0
$51$	−0.257869	−0.0361089
$52$	0	0
$53$	−4.37285	−0.600657	−0.300329	−	0.953836i	$-0.597096\pi$
$53$	−4.37285	−0.600657	−0.300329	+	0.953836i	$0.597096\pi$
$54$	0	0
$55$	−16.3936	−2.21052
$56$	0	0
$57$	−2.11137	−0.279658
$58$	0	0
$59$	−7.40379	−0.963891	−0.481946	−	0.876201i	$-0.660070\pi$
$59$	−7.40379	−0.963891	−0.481946	+	0.876201i	$0.660070\pi$
$60$	0	0
$61$	4.23267	0.541938	0.270969	−	0.962588i	$-0.412656\pi$
$61$	4.23267	0.541938	0.270969	+	0.962588i	$0.412656\pi$
$62$	0	0
$63$	−4.65088	−0.585956
$64$	0	0
$65$	1.15994	0.143873
$66$	0	0
$67$	4.79684	0.586027	0.293014	−	0.956108i	$-0.405342\pi$
$67$	4.79684	0.586027	0.293014	+	0.956108i	$0.405342\pi$
$68$	0	0
$69$	−5.38988	−0.648866
$70$	0	0
$71$	7.11410	0.844288	0.422144	−	0.906529i	$-0.361278\pi$
$71$	7.11410	0.844288	0.422144	+	0.906529i	$0.361278\pi$
$72$	0	0
$73$	9.79959	1.14696	0.573478	−	0.819221i	$-0.305594\pi$
$73$	9.79959	1.14696	0.573478	+	0.819221i	$0.305594\pi$
$74$	0	0
$75$	3.68866	0.425929
$76$	0	0
$77$	−25.8663	−2.94773
$78$	0	0
$79$	4.39883	0.494907	0.247453	−	0.968900i	$-0.420406\pi$
$79$	4.39883	0.494907	0.247453	+	0.968900i	$0.420406\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.64462	−1.05863	−0.529317	−	0.848424i	$-0.677552\pi$
$83$	−9.64462	−1.05863	−0.529317	+	0.848424i	$0.677552\pi$
$84$	0	0
$85$	0.760108	0.0824453
$86$	0	0
$87$	−0.101053	−0.0108341
$88$	0	0
$89$	10.6901	1.13315	0.566575	−	0.824010i	$-0.308268\pi$
$89$	10.6901	1.13315	0.566575	+	0.824010i	$0.308268\pi$
$90$	0	0
$91$	1.83019	0.191856
$92$	0	0
$93$	3.82473	0.396606
$94$	0	0
$95$	6.22358	0.638526
$96$	0	0
$97$	−5.54716	−0.563229	−0.281614	−	0.959528i	$-0.590870\pi$
$97$	−5.54716	−0.563229	−0.281614	+	0.959528i	$0.590870\pi$
$98$	0	0
$99$	5.56158	0.558960
$100$	0	0
$101$	14.0543	1.39846	0.699230	−	0.714897i	$-0.253526\pi$
$101$	14.0543	1.39846	0.699230	+	0.714897i	$0.253526\pi$
$102$	0	0
$103$	−6.68601	−0.658792	−0.329396	−	0.944192i	$-0.606845\pi$
$103$	−6.68601	−0.658792	−0.329396	+	0.944192i	$0.606845\pi$
$104$	0	0
$105$	13.7092	1.33788
$106$	0	0
$107$	−18.5825	−1.79644	−0.898220	−	0.439546i	$-0.855139\pi$
$107$	−18.5825	−1.79644	−0.898220	+	0.439546i	$0.855139\pi$
$108$	0	0
$109$	4.80580	0.460312	0.230156	−	0.973154i	$-0.426076\pi$
$109$	4.80580	0.460312	0.230156	+	0.973154i	$0.426076\pi$
$110$	0	0
$111$	−1.36787	−0.129832
$112$	0	0
$113$	11.0810	1.04242	0.521208	−	0.853430i	$-0.325482\pi$
$113$	11.0810	1.04242	0.521208	+	0.853430i	$0.325482\pi$
$114$	0	0
$115$	15.8875	1.48152
$116$	0	0
$117$	−0.393514	−0.0363804
$118$	0	0
$119$	1.19932	0.109941
$120$	0	0
$121$	19.9312	1.81193
$122$	0	0
$123$	−5.79375	−0.522405
$124$	0	0
$125$	3.86538	0.345730
$126$	0	0
$127$	−20.5991	−1.82787	−0.913937	−	0.405856i	$-0.866974\pi$
$127$	−20.5991	−1.82787	−0.913937	+	0.405856i	$0.866974\pi$
$128$	0	0
$129$	−7.94402	−0.699432
$130$	0	0
$131$	5.89118	0.514715	0.257358	−	0.966316i	$-0.417148\pi$
$131$	5.89118	0.514715	0.257358	+	0.966316i	$0.417148\pi$
$132$	0	0
$133$	9.81972	0.851478
$134$	0	0
$135$	−2.94765	−0.253694
$136$	0	0
$137$	19.9620	1.70547	0.852736	−	0.522342i	$-0.174941\pi$
$137$	19.9620	1.70547	0.852736	+	0.522342i	$0.174941\pi$
$138$	0	0
$139$	9.24509	0.784158	0.392079	−	0.919931i	$-0.371756\pi$
$139$	9.24509	0.784158	0.392079	+	0.919931i	$0.371756\pi$
$140$	0	0
$141$	−5.64008	−0.474981
$142$	0	0
$143$	−2.18856	−0.183017
$144$	0	0
$145$	0.297870	0.0247368
$146$	0	0
$147$	14.6307	1.20672
$148$	0	0
$149$	−2.57188	−0.210696	−0.105348	−	0.994435i	$-0.533596\pi$
$149$	−2.57188	−0.210696	−0.105348	+	0.994435i	$0.533596\pi$
$150$	0	0
$151$	9.44717	0.768800	0.384400	−	0.923167i	$-0.374408\pi$
$151$	9.44717	0.768800	0.384400	+	0.923167i	$0.374408\pi$
$152$	0	0
$153$	−0.257869	−0.0208475
$154$	0	0
$155$	−11.2740	−0.905547
$156$	0	0
$157$	11.0957	0.885533	0.442766	−	0.896637i	$-0.353997\pi$
$157$	11.0957	0.885533	0.442766	+	0.896637i	$0.353997\pi$
$158$	0	0
$159$	−4.37285	−0.346790
$160$	0	0
$161$	25.0677	1.97561
$162$	0	0
$163$	−11.4423	−0.896233	−0.448116	−	0.893975i	$-0.647905\pi$
$163$	−11.4423	−0.896233	−0.448116	+	0.893975i	$0.647905\pi$
$164$	0	0
$165$	−16.3936	−1.27624
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.8451	−0.988088
$170$	0	0
$171$	−2.11137	−0.161460
$172$	0	0
$173$	14.0924	1.07142	0.535712	−	0.844401i	$-0.320043\pi$
$173$	14.0924	1.07142	0.535712	+	0.844401i	$0.320043\pi$
$174$	0	0
$175$	−17.1555	−1.29683
$176$	0	0
$177$	−7.40379	−0.556503
$178$	0	0
$179$	6.89743	0.515538	0.257769	−	0.966207i	$-0.417013\pi$
$179$	6.89743	0.515538	0.257769	+	0.966207i	$0.417013\pi$
$180$	0	0
$181$	−19.0876	−1.41877	−0.709384	−	0.704822i	$-0.751027\pi$
$181$	−19.0876	−1.41877	−0.709384	+	0.704822i	$0.751027\pi$
$182$	0	0
$183$	4.23267	0.312888
$184$	0	0
$185$	4.03200	0.296439
$186$	0	0
$187$	−1.43416	−0.104876
$188$	0	0
$189$	−4.65088	−0.338302
$190$	0	0
$191$	−1.60862	−0.116396	−0.0581979	−	0.998305i	$-0.518535\pi$
$191$	−1.60862	−0.116396	−0.0581979	+	0.998305i	$0.518535\pi$
$192$	0	0
$193$	18.7119	1.34692	0.673458	−	0.739226i	$-0.264808\pi$
$193$	18.7119	1.34692	0.673458	+	0.739226i	$0.264808\pi$
$194$	0	0
$195$	1.15994	0.0830652
$196$	0	0
$197$	−1.93111	−0.137586	−0.0687928	−	0.997631i	$-0.521915\pi$
$197$	−1.93111	−0.137586	−0.0687928	+	0.997631i	$0.521915\pi$
$198$	0	0
$199$	19.3394	1.37093	0.685465	−	0.728105i	$-0.259599\pi$
$199$	19.3394	1.37093	0.685465	+	0.728105i	$0.259599\pi$
$200$	0	0
$201$	4.79684	0.338343
$202$	0	0
$203$	0.469987	0.0329866
$204$	0	0
$205$	17.0780	1.19278
$206$	0	0
$207$	−5.38988	−0.374623
$208$	0	0
$209$	−11.7426	−0.812250
$210$	0	0
$211$	15.9760	1.09983	0.549916	−	0.835220i	$-0.314660\pi$
$211$	15.9760	1.09983	0.549916	+	0.835220i	$0.314660\pi$
$212$	0	0
$213$	7.11410	0.487450
$214$	0	0
$215$	23.4162	1.59697
$216$	0	0
$217$	−17.7883	−1.20755
$218$	0	0
$219$	9.79959	0.662195
$220$	0	0
$221$	0.101475	0.00682595
$222$	0	0
$223$	−25.3254	−1.69592	−0.847958	−	0.530064i	$-0.822168\pi$
$223$	−25.3254	−1.69592	−0.847958	+	0.530064i	$0.822168\pi$
$224$	0	0
$225$	3.68866	0.245910
$226$	0	0
$227$	−6.37245	−0.422954	−0.211477	−	0.977383i	$-0.567827\pi$
$227$	−6.37245	−0.422954	−0.211477	+	0.977383i	$0.567827\pi$
$228$	0	0
$229$	1.56881	0.103670	0.0518351	−	0.998656i	$-0.483493\pi$
$229$	1.56881	0.103670	0.0518351	+	0.998656i	$0.483493\pi$
$230$	0	0
$231$	−25.8663	−1.70187
$232$	0	0
$233$	−25.4918	−1.67002	−0.835012	−	0.550232i	$-0.814539\pi$
$233$	−25.4918	−1.67002	−0.835012	+	0.550232i	$0.814539\pi$
$234$	0	0
$235$	16.6250	1.08450
$236$	0	0
$237$	4.39883	0.285735
$238$	0	0
$239$	2.23855	0.144800	0.0723998	−	0.997376i	$-0.476934\pi$
$239$	2.23855	0.144800	0.0723998	+	0.997376i	$0.476934\pi$
$240$	0	0
$241$	9.73714	0.627224	0.313612	−	0.949551i	$-0.398461\pi$
$241$	9.73714	0.627224	0.313612	+	0.949551i	$0.398461\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−43.1262	−2.75523
$246$	0	0
$247$	0.830853	0.0528659
$248$	0	0
$249$	−9.64462	−0.611203
$250$	0	0
$251$	12.7392	0.804090	0.402045	−	0.915620i	$-0.368300\pi$
$251$	12.7392	0.804090	0.402045	+	0.915620i	$0.368300\pi$
$252$	0	0
$253$	−29.9763	−1.88459
$254$	0	0
$255$	0.760108	0.0475998
$256$	0	0
$257$	11.4381	0.713486	0.356743	−	0.934203i	$-0.383887\pi$
$257$	11.4381	0.713486	0.356743	+	0.934203i	$0.383887\pi$
$258$	0	0
$259$	6.36179	0.395302
$260$	0	0
$261$	−0.101053	−0.00625505
$262$	0	0
$263$	23.7134	1.46223	0.731116	−	0.682253i	$-0.239000\pi$
$263$	23.7134	1.46223	0.731116	+	0.682253i	$0.239000\pi$
$264$	0	0
$265$	12.8896	0.791805
$266$	0	0
$267$	10.6901	0.654224
$268$	0	0
$269$	20.3877	1.24306	0.621529	−	0.783391i	$-0.286512\pi$
$269$	20.3877	1.24306	0.621529	+	0.783391i	$0.286512\pi$
$270$	0	0
$271$	−0.753640	−0.0457804	−0.0228902	−	0.999738i	$-0.507287\pi$
$271$	−0.753640	−0.0457804	−0.0228902	+	0.999738i	$0.507287\pi$
$272$	0	0
$273$	1.83019	0.110768
$274$	0	0
$275$	20.5148	1.23709
$276$	0	0
$277$	−26.2696	−1.57839	−0.789195	−	0.614143i	$-0.789502\pi$
$277$	−26.2696	−1.57839	−0.789195	+	0.614143i	$0.789502\pi$
$278$	0	0
$279$	3.82473	0.228980
$280$	0	0
$281$	18.7488	1.11846	0.559230	−	0.829013i	$-0.311097\pi$
$281$	18.7488	1.11846	0.559230	+	0.829013i	$0.311097\pi$
$282$	0	0
$283$	30.8657	1.83478	0.917389	−	0.397992i	$-0.130293\pi$
$283$	30.8657	1.83478	0.917389	+	0.397992i	$0.130293\pi$
$284$	0	0
$285$	6.22358	0.368653
$286$	0	0
$287$	26.9460	1.59057
$288$	0	0
$289$	−16.9335	−0.996088
$290$	0	0
$291$	−5.54716	−0.325180
$292$	0	0
$293$	−9.54329	−0.557525	−0.278763	−	0.960360i	$-0.589924\pi$
$293$	−9.54329	−0.557525	−0.278763	+	0.960360i	$0.589924\pi$
$294$	0	0
$295$	21.8238	1.27063
$296$	0	0
$297$	5.56158	0.322716
$298$	0	0
$299$	2.12099	0.122660
$300$	0	0
$301$	36.9467	2.12957
$302$	0	0
$303$	14.0543	0.807401
$304$	0	0
$305$	−12.4764	−0.714399
$306$	0	0
$307$	18.4454	1.05273	0.526366	−	0.850258i	$-0.323554\pi$
$307$	18.4454	1.05273	0.526366	+	0.850258i	$0.323554\pi$
$308$	0	0
$309$	−6.68601	−0.380354
$310$	0	0
$311$	2.89773	0.164315	0.0821575	−	0.996619i	$-0.473819\pi$
$311$	2.89773	0.164315	0.0821575	+	0.996619i	$0.473819\pi$
$312$	0	0
$313$	2.39954	0.135630	0.0678149	−	0.997698i	$-0.478397\pi$
$313$	2.39954	0.135630	0.0678149	+	0.997698i	$0.478397\pi$
$314$	0	0
$315$	13.7092	0.772425
$316$	0	0
$317$	10.3953	0.583860	0.291930	−	0.956440i	$-0.405703\pi$
$317$	10.3953	0.583860	0.291930	+	0.956440i	$0.405703\pi$
$318$	0	0
$319$	−0.562017	−0.0314669
$320$	0	0
$321$	−18.5825	−1.03718
$322$	0	0
$323$	0.544456	0.0302944
$324$	0	0
$325$	−1.45154	−0.0805169
$326$	0	0
$327$	4.80580	0.265762
$328$	0	0
$329$	26.2313	1.44618
$330$	0	0
$331$	−13.0622	−0.717965	−0.358983	−	0.933344i	$-0.616876\pi$
$331$	−13.0622	−0.717965	−0.358983	+	0.933344i	$0.616876\pi$
$332$	0	0
$333$	−1.36787	−0.0749587
$334$	0	0
$335$	−14.1394	−0.772519
$336$	0	0
$337$	27.6914	1.50845	0.754224	−	0.656618i	$-0.228013\pi$
$337$	27.6914	1.50845	0.754224	+	0.656618i	$0.228013\pi$
$338$	0	0
$339$	11.0810	0.601839
$340$	0	0
$341$	21.2715	1.15192
$342$	0	0
$343$	−35.4894	−1.91625
$344$	0	0
$345$	15.8875	0.855355
$346$	0	0
$347$	30.6196	1.64374	0.821872	−	0.569672i	$-0.192930\pi$
$347$	30.6196	1.64374	0.821872	+	0.569672i	$0.192930\pi$
$348$	0	0
$349$	26.7482	1.43180	0.715898	−	0.698204i	$-0.246017\pi$
$349$	26.7482	1.43180	0.715898	+	0.698204i	$0.246017\pi$
$350$	0	0
$351$	−0.393514	−0.0210042
$352$	0	0
$353$	25.7882	1.37257	0.686283	−	0.727335i	$-0.259241\pi$
$353$	25.7882	1.37257	0.686283	+	0.727335i	$0.259241\pi$
$354$	0	0
$355$	−20.9699	−1.11297
$356$	0	0
$357$	1.19932	0.0634746
$358$	0	0
$359$	26.2661	1.38627	0.693135	−	0.720808i	$-0.256229\pi$
$359$	26.2661	1.38627	0.693135	+	0.720808i	$0.256229\pi$
$360$	0	0
$361$	−14.5421	−0.765375
$362$	0	0
$363$	19.9312	1.04612
$364$	0	0
$365$	−28.8858	−1.51195
$366$	0	0
$367$	10.6384	0.555323	0.277661	−	0.960679i	$-0.410441\pi$
$367$	10.6384	0.555323	0.277661	+	0.960679i	$0.410441\pi$
$368$	0	0
$369$	−5.79375	−0.301611
$370$	0	0
$371$	20.3376	1.05588
$372$	0	0
$373$	−13.9912	−0.724435	−0.362217	−	0.932094i	$-0.617980\pi$
$373$	−13.9912	−0.724435	−0.362217	+	0.932094i	$0.617980\pi$
$374$	0	0
$375$	3.86538	0.199608
$376$	0	0
$377$	0.0397659	0.00204805
$378$	0	0
$379$	12.4396	0.638979	0.319490	−	0.947590i	$-0.396489\pi$
$379$	12.4396	0.638979	0.319490	+	0.947590i	$0.396489\pi$
$380$	0	0
$381$	−20.5991	−1.05532
$382$	0	0
$383$	2.12077	0.108366	0.0541830	−	0.998531i	$-0.482745\pi$
$383$	2.12077	0.108366	0.0541830	+	0.998531i	$0.482745\pi$
$384$	0	0
$385$	76.2448	3.88579
$386$	0	0
$387$	−7.94402	−0.403817
$388$	0	0
$389$	25.6819	1.30212	0.651062	−	0.759025i	$-0.274324\pi$
$389$	25.6819	1.30212	0.651062	+	0.759025i	$0.274324\pi$
$390$	0	0
$391$	1.38988	0.0702894
$392$	0	0
$393$	5.89118	0.297171
$394$	0	0
$395$	−12.9662	−0.652401
$396$	0	0
$397$	24.9235	1.25087	0.625437	−	0.780274i	$-0.284921\pi$
$397$	24.9235	1.25087	0.625437	+	0.780274i	$0.284921\pi$
$398$	0	0
$399$	9.81972	0.491601
$400$	0	0
$401$	20.9408	1.04574	0.522868	−	0.852414i	$-0.324862\pi$
$401$	20.9408	1.04574	0.522868	+	0.852414i	$0.324862\pi$
$402$	0	0
$403$	−1.50508	−0.0749736
$404$	0	0
$405$	−2.94765	−0.146470
$406$	0	0
$407$	−7.60751	−0.377091
$408$	0	0
$409$	14.0301	0.693742	0.346871	−	0.937913i	$-0.387244\pi$
$409$	14.0301	0.693742	0.346871	+	0.937913i	$0.387244\pi$
$410$	0	0
$411$	19.9620	0.984655
$412$	0	0
$413$	34.4341	1.69439
$414$	0	0
$415$	28.4290	1.39552
$416$	0	0
$417$	9.24509	0.452734
$418$	0	0
$419$	17.6278	0.861173	0.430586	−	0.902549i	$-0.358307\pi$
$419$	17.6278	0.861173	0.430586	+	0.902549i	$0.358307\pi$
$420$	0	0
$421$	−3.22755	−0.157301	−0.0786505	−	0.996902i	$-0.525061\pi$
$421$	−3.22755	−0.157301	−0.0786505	+	0.996902i	$0.525061\pi$
$422$	0	0
$423$	−5.64008	−0.274230
$424$	0	0
$425$	−0.951190	−0.0461395
$426$	0	0
$427$	−19.6856	−0.952654
$428$	0	0
$429$	−2.18856	−0.105665
$430$	0	0
$431$	1.25080	0.0602491	0.0301246	−	0.999546i	$-0.490410\pi$
$431$	1.25080	0.0602491	0.0301246	+	0.999546i	$0.490410\pi$
$432$	0	0
$433$	10.8700	0.522381	0.261191	−	0.965287i	$-0.415885\pi$
$433$	10.8700	0.522381	0.261191	+	0.965287i	$0.415885\pi$
$434$	0	0
$435$	0.297870	0.0142818
$436$	0	0
$437$	11.3800	0.544381
$438$	0	0
$439$	−40.2789	−1.92241	−0.961204	−	0.275838i	$-0.911045\pi$
$439$	−40.2789	−1.92241	−0.961204	+	0.275838i	$0.911045\pi$
$440$	0	0
$441$	14.6307	0.696699
$442$	0	0
$443$	21.0259	0.998970	0.499485	−	0.866323i	$-0.333523\pi$
$443$	21.0259	0.998970	0.499485	+	0.866323i	$0.333523\pi$
$444$	0	0
$445$	−31.5107	−1.49375
$446$	0	0
$447$	−2.57188	−0.121646
$448$	0	0
$449$	−29.3177	−1.38359	−0.691794	−	0.722095i	$-0.743179\pi$
$449$	−29.3177	−1.38359	−0.691794	+	0.722095i	$0.743179\pi$
$450$	0	0
$451$	−32.2224	−1.51729
$452$	0	0
$453$	9.44717	0.443867
$454$	0	0
$455$	−5.39475	−0.252910
$456$	0	0
$457$	−35.2568	−1.64924	−0.824622	−	0.565685i	$-0.808612\pi$
$457$	−35.2568	−1.64924	−0.824622	+	0.565685i	$0.808612\pi$
$458$	0	0
$459$	−0.257869	−0.0120363
$460$	0	0
$461$	7.33399	0.341578	0.170789	−	0.985308i	$-0.445368\pi$
$461$	7.33399	0.341578	0.170789	+	0.985308i	$0.445368\pi$
$462$	0	0
$463$	6.77568	0.314892	0.157446	−	0.987528i	$-0.449674\pi$
$463$	6.77568	0.314892	0.157446	+	0.987528i	$0.449674\pi$
$464$	0	0
$465$	−11.2740	−0.522818
$466$	0	0
$467$	21.8308	1.01021	0.505104	−	0.863058i	$-0.331454\pi$
$467$	21.8308	1.01021	0.505104	+	0.863058i	$0.331454\pi$
$468$	0	0
$469$	−22.3095	−1.03016
$470$	0	0
$471$	11.0957	0.511263
$472$	0	0
$473$	−44.1814	−2.03146
$474$	0	0
$475$	−7.78812	−0.357343
$476$	0	0
$477$	−4.37285	−0.200219
$478$	0	0
$479$	−18.8655	−0.861985	−0.430992	−	0.902356i	$-0.641836\pi$
$479$	−18.8655	−0.861985	−0.430992	+	0.902356i	$0.641836\pi$
$480$	0	0
$481$	0.538275	0.0245432
$482$	0	0
$483$	25.0677	1.14062
$484$	0	0
$485$	16.3511	0.742465
$486$	0	0
$487$	22.2690	1.00911	0.504553	−	0.863381i	$-0.331657\pi$
$487$	22.2690	1.00911	0.504553	+	0.863381i	$0.331657\pi$
$488$	0	0
$489$	−11.4423	−0.517440
$490$	0	0
$491$	−26.3929	−1.19110	−0.595549	−	0.803319i	$-0.703065\pi$
$491$	−26.3929	−1.19110	−0.595549	+	0.803319i	$0.703065\pi$
$492$	0	0
$493$	0.0260585	0.00117362
$494$	0	0
$495$	−16.3936	−0.736838
$496$	0	0
$497$	−33.0868	−1.48415
$498$	0	0
$499$	−19.2798	−0.863085	−0.431542	−	0.902093i	$-0.642030\pi$
$499$	−19.2798	−0.863085	−0.431542	+	0.902093i	$0.642030\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	44.5066	1.98445	0.992225	−	0.124456i	$-0.0397186\pi$
$503$	44.5066	1.98445	0.992225	+	0.124456i	$0.0397186\pi$
$504$	0	0
$505$	−41.4273	−1.84349
$506$	0	0
$507$	−12.8451	−0.570473
$508$	0	0
$509$	11.4138	0.505906	0.252953	−	0.967479i	$-0.418598\pi$
$509$	11.4138	0.505906	0.252953	+	0.967479i	$0.418598\pi$
$510$	0	0
$511$	−45.5767	−2.01620
$512$	0	0
$513$	−2.11137	−0.0932192
$514$	0	0
$515$	19.7080	0.868440
$516$	0	0
$517$	−31.3678	−1.37955
$518$	0	0
$519$	14.0924	0.618587
$520$	0	0
$521$	21.1737	0.927638	0.463819	−	0.885930i	$-0.346479\pi$
$521$	21.1737	0.927638	0.463819	+	0.885930i	$0.346479\pi$
$522$	0	0
$523$	−40.6063	−1.77559	−0.887795	−	0.460239i	$-0.847764\pi$
$523$	−40.6063	−1.77559	−0.887795	+	0.460239i	$0.847764\pi$
$524$	0	0
$525$	−17.1555	−0.748727
$526$	0	0
$527$	−0.986278	−0.0429630
$528$	0	0
$529$	6.05084	0.263080
$530$	0	0
$531$	−7.40379	−0.321297
$532$	0	0
$533$	2.27992	0.0987544
$534$	0	0
$535$	54.7748	2.36812
$536$	0	0
$537$	6.89743	0.297646
$538$	0	0
$539$	81.3698	3.50484
$540$	0	0
$541$	3.70757	0.159401	0.0797004	−	0.996819i	$-0.474604\pi$
$541$	3.70757	0.159401	0.0797004	+	0.996819i	$0.474604\pi$
$542$	0	0
$543$	−19.0876	−0.819126
$544$	0	0
$545$	−14.1658	−0.606798
$546$	0	0
$547$	0.265318	0.0113442	0.00567208	−	0.999984i	$-0.498195\pi$
$547$	0.265318	0.0113442	0.00567208	+	0.999984i	$0.498195\pi$
$548$	0	0
$549$	4.23267	0.180646
$550$	0	0
$551$	0.213361	0.00908948
$552$	0	0
$553$	−20.4584	−0.869981
$554$	0	0
$555$	4.03200	0.171149
$556$	0	0
$557$	−10.5449	−0.446802	−0.223401	−	0.974727i	$-0.571716\pi$
$557$	−10.5449	−0.446802	−0.223401	+	0.974727i	$0.571716\pi$
$558$	0	0
$559$	3.12608	0.132219
$560$	0	0
$561$	−1.43416	−0.0605503
$562$	0	0
$563$	31.6916	1.33564	0.667821	−	0.744322i	$-0.267227\pi$
$563$	31.6916	1.33564	0.667821	+	0.744322i	$0.267227\pi$
$564$	0	0
$565$	−32.6630	−1.37414
$566$	0	0
$567$	−4.65088	−0.195319
$568$	0	0
$569$	−40.6165	−1.70273	−0.851365	−	0.524573i	$-0.824225\pi$
$569$	−40.6165	−1.70273	−0.851365	+	0.524573i	$0.824225\pi$
$570$	0	0
$571$	−25.2878	−1.05826	−0.529131	−	0.848540i	$-0.677482\pi$
$571$	−25.2878	−1.05826	−0.529131	+	0.848540i	$0.677482\pi$
$572$	0	0
$573$	−1.60862	−0.0672012
$574$	0	0
$575$	−19.8814	−0.829113
$576$	0	0
$577$	−44.3694	−1.84712	−0.923561	−	0.383452i	$-0.874735\pi$
$577$	−44.3694	−1.84712	−0.923561	+	0.383452i	$0.874735\pi$
$578$	0	0
$579$	18.7119	0.777642
$580$	0	0
$581$	44.8560	1.86094
$582$	0	0
$583$	−24.3200	−1.00723
$584$	0	0
$585$	1.15994	0.0479577
$586$	0	0
$587$	14.9324	0.616325	0.308162	−	0.951334i	$-0.400286\pi$
$587$	14.9324	0.616325	0.308162	+	0.951334i	$0.400286\pi$
$588$	0	0
$589$	−8.07541	−0.332741
$590$	0	0
$591$	−1.93111	−0.0794351
$592$	0	0
$593$	−1.70449	−0.0699948	−0.0349974	−	0.999387i	$-0.511142\pi$
$593$	−1.70449	−0.0699948	−0.0349974	+	0.999387i	$0.511142\pi$
$594$	0	0
$595$	−3.53517	−0.144928
$596$	0	0
$597$	19.3394	0.791507
$598$	0	0
$599$	−28.0511	−1.14614	−0.573069	−	0.819507i	$-0.694247\pi$
$599$	−28.0511	−1.14614	−0.573069	+	0.819507i	$0.694247\pi$
$600$	0	0
$601$	−39.3274	−1.60420	−0.802099	−	0.597191i	$-0.796283\pi$
$601$	−39.3274	−1.60420	−0.802099	+	0.597191i	$0.796283\pi$
$602$	0	0
$603$	4.79684	0.195342
$604$	0	0
$605$	−58.7503	−2.38854
$606$	0	0
$607$	−33.5448	−1.36154	−0.680771	−	0.732496i	$-0.738355\pi$
$607$	−33.5448	−1.36154	−0.680771	+	0.732496i	$0.738355\pi$
$608$	0	0
$609$	0.469987	0.0190448
$610$	0	0
$611$	2.21945	0.0897894
$612$	0	0
$613$	42.5000	1.71656	0.858280	−	0.513182i	$-0.171533\pi$
$613$	42.5000	1.71656	0.858280	+	0.513182i	$0.171533\pi$
$614$	0	0
$615$	17.0780	0.688650
$616$	0	0
$617$	6.95453	0.279979	0.139989	−	0.990153i	$-0.455293\pi$
$617$	6.95453	0.279979	0.139989	+	0.990153i	$0.455293\pi$
$618$	0	0
$619$	−8.29546	−0.333423	−0.166711	−	0.986006i	$-0.553315\pi$
$619$	−8.29546	−0.333423	−0.166711	+	0.986006i	$0.553315\pi$
$620$	0	0
$621$	−5.38988	−0.216289
$622$	0	0
$623$	−49.7184	−1.99193
$624$	0	0
$625$	−29.8371	−1.19348
$626$	0	0
$627$	−11.7426	−0.468953
$628$	0	0
$629$	0.352731	0.0140643
$630$	0	0
$631$	−41.7880	−1.66355	−0.831777	−	0.555110i	$-0.812676\pi$
$631$	−41.7880	−1.66355	−0.831777	+	0.555110i	$0.812676\pi$
$632$	0	0
$633$	15.9760	0.634988
$634$	0	0
$635$	60.7190	2.40956
$636$	0	0
$637$	−5.75738	−0.228116
$638$	0	0
$639$	7.11410	0.281429
$640$	0	0
$641$	22.5081	0.889018	0.444509	−	0.895774i	$-0.353378\pi$
$641$	22.5081	0.889018	0.444509	+	0.895774i	$0.353378\pi$
$642$	0	0
$643$	16.1894	0.638446	0.319223	−	0.947680i	$-0.396578\pi$
$643$	16.1894	0.638446	0.319223	+	0.947680i	$0.396578\pi$
$644$	0	0
$645$	23.4162	0.922013
$646$	0	0
$647$	−40.9895	−1.61146	−0.805731	−	0.592282i	$-0.798227\pi$
$647$	−40.9895	−1.61146	−0.805731	+	0.592282i	$0.798227\pi$
$648$	0	0
$649$	−41.1768	−1.61633
$650$	0	0
$651$	−17.7883	−0.697180
$652$	0	0
$653$	−15.8116	−0.618756	−0.309378	−	0.950939i	$-0.600121\pi$
$653$	−15.8116	−0.618756	−0.309378	+	0.950939i	$0.600121\pi$
$654$	0	0
$655$	−17.3652	−0.678513
$656$	0	0
$657$	9.79959	0.382319
$658$	0	0
$659$	−18.9906	−0.739768	−0.369884	−	0.929078i	$-0.620603\pi$
$659$	−18.9906	−0.739768	−0.369884	+	0.929078i	$0.620603\pi$
$660$	0	0
$661$	38.4650	1.49611	0.748057	−	0.663634i	$-0.230987\pi$
$661$	38.4650	1.49611	0.748057	+	0.663634i	$0.230987\pi$
$662$	0	0
$663$	0.101475	0.00394096
$664$	0	0
$665$	−28.9451	−1.12244
$666$	0	0
$667$	0.544666	0.0210896
$668$	0	0
$669$	−25.3254	−0.979137
$670$	0	0
$671$	23.5403	0.908765
$672$	0	0
$673$	39.8458	1.53594	0.767970	−	0.640485i	$-0.221267\pi$
$673$	39.8458	1.53594	0.767970	+	0.640485i	$0.221267\pi$
$674$	0	0
$675$	3.68866	0.141976
$676$	0	0
$677$	23.8601	0.917019	0.458509	−	0.888690i	$-0.348384\pi$
$677$	23.8601	0.917019	0.458509	+	0.888690i	$0.348384\pi$
$678$	0	0
$679$	25.7992	0.990081
$680$	0	0
$681$	−6.37245	−0.244193
$682$	0	0
$683$	16.5117	0.631804	0.315902	−	0.948792i	$-0.397693\pi$
$683$	16.5117	0.631804	0.315902	+	0.948792i	$0.397693\pi$
$684$	0	0
$685$	−58.8412	−2.24821
$686$	0	0
$687$	1.56881	0.0598540
$688$	0	0
$689$	1.72078	0.0655564
$690$	0	0
$691$	18.4923	0.703481	0.351741	−	0.936098i	$-0.385590\pi$
$691$	18.4923	0.703481	0.351741	+	0.936098i	$0.385590\pi$
$692$	0	0
$693$	−25.8663	−0.982578
$694$	0	0
$695$	−27.2513	−1.03370
$696$	0	0
$697$	1.49403	0.0565903
$698$	0	0
$699$	−25.4918	−0.964189
$700$	0	0
$701$	−20.7230	−0.782696	−0.391348	−	0.920243i	$-0.627991\pi$
$701$	−20.7230	−0.782696	−0.391348	+	0.920243i	$0.627991\pi$
$702$	0	0
$703$	2.88807	0.108926
$704$	0	0
$705$	16.6250	0.626134
$706$	0	0
$707$	−65.3651	−2.45831
$708$	0	0
$709$	−16.5876	−0.622961	−0.311480	−	0.950253i	$-0.600825\pi$
$709$	−16.5876	−0.622961	−0.311480	+	0.950253i	$0.600825\pi$
$710$	0	0
$711$	4.39883	0.164969
$712$	0	0
$713$	−20.6148	−0.772032
$714$	0	0
$715$	6.45112	0.241258
$716$	0	0
$717$	2.23855	0.0836001
$718$	0	0
$719$	−28.0418	−1.04578	−0.522890	−	0.852400i	$-0.675146\pi$
$719$	−28.0418	−1.04578	−0.522890	+	0.852400i	$0.675146\pi$
$720$	0	0
$721$	31.0958	1.15807
$722$	0	0
$723$	9.73714	0.362128
$724$	0	0
$725$	−0.372751	−0.0138436
$726$	0	0
$727$	49.0096	1.81766	0.908832	−	0.417162i	$-0.136975\pi$
$727$	49.0096	1.81766	0.908832	+	0.417162i	$0.136975\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.04852	0.0757671
$732$	0	0
$733$	−5.39596	−0.199304	−0.0996522	−	0.995022i	$-0.531773\pi$
$733$	−5.39596	−0.199304	−0.0996522	+	0.995022i	$0.531773\pi$
$734$	0	0
$735$	−43.1262	−1.59073
$736$	0	0
$737$	26.6780	0.982698
$738$	0	0
$739$	−38.9366	−1.43231	−0.716153	−	0.697943i	$-0.754099\pi$
$739$	−38.9366	−1.43231	−0.716153	+	0.697943i	$0.754099\pi$
$740$	0	0
$741$	0.830853	0.0305222
$742$	0	0
$743$	25.0681	0.919659	0.459829	−	0.888007i	$-0.347911\pi$
$743$	25.0681	0.919659	0.459829	+	0.888007i	$0.347911\pi$
$744$	0	0
$745$	7.58100	0.277746
$746$	0	0
$747$	−9.64462	−0.352878
$748$	0	0
$749$	86.4251	3.15790
$750$	0	0
$751$	18.3068	0.668023	0.334012	−	0.942569i	$-0.391597\pi$
$751$	18.3068	0.668023	0.334012	+	0.942569i	$0.391597\pi$
$752$	0	0
$753$	12.7392	0.464241
$754$	0	0
$755$	−27.8470	−1.01346
$756$	0	0
$757$	−5.04280	−0.183284	−0.0916418	−	0.995792i	$-0.529211\pi$
$757$	−5.04280	−0.183284	−0.0916418	+	0.995792i	$0.529211\pi$
$758$	0	0
$759$	−29.9763	−1.08807
$760$	0	0
$761$	1.64107	0.0594889	0.0297444	−	0.999558i	$-0.490531\pi$
$761$	1.64107	0.0594889	0.0297444	+	0.999558i	$0.490531\pi$
$762$	0	0
$763$	−22.3512	−0.809168
$764$	0	0
$765$	0.760108	0.0274818
$766$	0	0
$767$	2.91349	0.105200
$768$	0	0
$769$	−18.6826	−0.673713	−0.336856	−	0.941556i	$-0.609364\pi$
$769$	−18.6826	−0.673713	−0.336856	+	0.941556i	$0.609364\pi$
$770$	0	0
$771$	11.4381	0.411931
$772$	0	0
$773$	−14.3488	−0.516091	−0.258046	−	0.966133i	$-0.583078\pi$
$773$	−14.3488	−0.516091	−0.258046	+	0.966133i	$0.583078\pi$
$774$	0	0
$775$	14.1081	0.506778
$776$	0	0
$777$	6.36179	0.228228
$778$	0	0
$779$	12.2327	0.438283
$780$	0	0
$781$	39.5656	1.41577
$782$	0	0
$783$	−0.101053	−0.00361135
$784$	0	0
$785$	−32.7063	−1.16734
$786$	0	0
$787$	−32.2812	−1.15070	−0.575351	−	0.817907i	$-0.695134\pi$
$787$	−32.2812	−1.15070	−0.575351	+	0.817907i	$0.695134\pi$
$788$	0	0
$789$	23.7134	0.844220
$790$	0	0
$791$	−51.5366	−1.83243
$792$	0	0
$793$	−1.66561	−0.0591477
$794$	0	0
$795$	12.8896	0.457149
$796$	0	0
$797$	6.26106	0.221778	0.110889	−	0.993833i	$-0.464630\pi$
$797$	6.26106	0.221778	0.110889	+	0.993833i	$0.464630\pi$
$798$	0	0
$799$	1.45440	0.0514530
$800$	0	0
$801$	10.6901	0.377716
$802$	0	0
$803$	54.5013	1.92331
$804$	0	0
$805$	−73.8909	−2.60431
$806$	0	0
$807$	20.3877	0.717680
$808$	0	0
$809$	−33.5687	−1.18021	−0.590106	−	0.807326i	$-0.700914\pi$
$809$	−33.5687	−1.18021	−0.590106	+	0.807326i	$0.700914\pi$
$810$	0	0
$811$	−13.9961	−0.491469	−0.245735	−	0.969337i	$-0.579029\pi$
$811$	−13.9961	−0.491469	−0.245735	+	0.969337i	$0.579029\pi$
$812$	0	0
$813$	−0.753640	−0.0264313
$814$	0	0
$815$	33.7280	1.18144
$816$	0	0
$817$	16.7728	0.586805
$818$	0	0
$819$	1.83019	0.0639519
$820$	0	0
$821$	−4.67995	−0.163331	−0.0816656	−	0.996660i	$-0.526024\pi$
$821$	−4.67995	−0.163331	−0.0816656	+	0.996660i	$0.526024\pi$
$822$	0	0
$823$	−24.5726	−0.856548	−0.428274	−	0.903649i	$-0.640878\pi$
$823$	−24.5726	−0.856548	−0.428274	+	0.903649i	$0.640878\pi$
$824$	0	0
$825$	20.5148	0.714233
$826$	0	0
$827$	−31.9433	−1.11078	−0.555388	−	0.831591i	$-0.687430\pi$
$827$	−31.9433	−1.11078	−0.555388	+	0.831591i	$0.687430\pi$
$828$	0	0
$829$	17.4234	0.605140	0.302570	−	0.953127i	$-0.402155\pi$
$829$	17.4234	0.605140	0.302570	+	0.953127i	$0.402155\pi$
$830$	0	0
$831$	−26.2696	−0.911284
$832$	0	0
$833$	−3.77280	−0.130720
$834$	0	0
$835$	−2.94765	−0.102008
$836$	0	0
$837$	3.82473	0.132202
$838$	0	0
$839$	6.29713	0.217401	0.108701	−	0.994075i	$-0.465331\pi$
$839$	6.29713	0.217401	0.108701	+	0.994075i	$0.465331\pi$
$840$	0	0
$841$	−28.9898	−0.999648
$842$	0	0
$843$	18.7488	0.645743
$844$	0	0
$845$	37.8630	1.30253
$846$	0	0
$847$	−92.6977	−3.18513
$848$	0	0
$849$	30.8657	1.05931
$850$	0	0
$851$	7.37265	0.252731
$852$	0	0
$853$	50.7750	1.73850	0.869251	−	0.494371i	$-0.164602\pi$
$853$	50.7750	1.73850	0.869251	+	0.494371i	$0.164602\pi$
$854$	0	0
$855$	6.22358	0.212842
$856$	0	0
$857$	8.83360	0.301750	0.150875	−	0.988553i	$-0.451791\pi$
$857$	8.83360	0.301750	0.150875	+	0.988553i	$0.451791\pi$
$858$	0	0
$859$	−0.0835075	−0.00284924	−0.00142462	−	0.999999i	$-0.500453\pi$
$859$	−0.0835075	−0.00284924	−0.00142462	+	0.999999i	$0.500453\pi$
$860$	0	0
$861$	26.9460	0.918318
$862$	0	0
$863$	7.00419	0.238425	0.119213	−	0.992869i	$-0.461963\pi$
$863$	7.00419	0.238425	0.119213	+	0.992869i	$0.461963\pi$
$864$	0	0
$865$	−41.5394	−1.41238
$866$	0	0
$867$	−16.9335	−0.575092
$868$	0	0
$869$	24.4645	0.829900
$870$	0	0
$871$	−1.88762	−0.0639597
$872$	0	0
$873$	−5.54716	−0.187743
$874$	0	0
$875$	−17.9774	−0.607748
$876$	0	0
$877$	−16.7259	−0.564792	−0.282396	−	0.959298i	$-0.591129\pi$
$877$	−16.7259	−0.564792	−0.282396	+	0.959298i	$0.591129\pi$
$878$	0	0
$879$	−9.54329	−0.321887
$880$	0	0
$881$	−42.6420	−1.43665	−0.718324	−	0.695709i	$-0.755090\pi$
$881$	−42.6420	−1.43665	−0.718324	+	0.695709i	$0.755090\pi$
$882$	0	0
$883$	43.4022	1.46060	0.730300	−	0.683126i	$-0.239380\pi$
$883$	43.4022	1.46060	0.730300	+	0.683126i	$0.239380\pi$
$884$	0	0
$885$	21.8238	0.733599
$886$	0	0
$887$	−28.0250	−0.940986	−0.470493	−	0.882404i	$-0.655924\pi$
$887$	−28.0250	−0.940986	−0.470493	+	0.882404i	$0.655924\pi$
$888$	0	0
$889$	95.8039	3.21316
$890$	0	0
$891$	5.56158	0.186320
$892$	0	0
$893$	11.9083	0.398496
$894$	0	0
$895$	−20.3312	−0.679598
$896$	0	0
$897$	2.12099	0.0708179
$898$	0	0
$899$	−0.386502	−0.0128906
$900$	0	0
$901$	1.12762	0.0375665
$902$	0	0
$903$	36.9467	1.22951
$904$	0	0
$905$	56.2635	1.87026
$906$	0	0
$907$	4.56086	0.151441	0.0757205	−	0.997129i	$-0.475874\pi$
$907$	4.56086	0.151441	0.0757205	+	0.997129i	$0.475874\pi$
$908$	0	0
$909$	14.0543	0.466153
$910$	0	0
$911$	2.27413	0.0753454	0.0376727	−	0.999290i	$-0.488006\pi$
$911$	2.27413	0.0753454	0.0376727	+	0.999290i	$0.488006\pi$
$912$	0	0
$913$	−53.6394	−1.77520
$914$	0	0
$915$	−12.4764	−0.412458
$916$	0	0
$917$	−27.3992	−0.904801
$918$	0	0
$919$	24.0412	0.793046	0.396523	−	0.918025i	$-0.370217\pi$
$919$	24.0412	0.793046	0.396523	+	0.918025i	$0.370217\pi$
$920$	0	0
$921$	18.4454	0.607796
$922$	0	0
$923$	−2.79950	−0.0921466
$924$	0	0
$925$	−5.04560	−0.165898
$926$	0	0
$927$	−6.68601	−0.219597
$928$	0	0
$929$	−20.4175	−0.669875	−0.334938	−	0.942240i	$-0.608715\pi$
$929$	−20.4175	−0.669875	−0.334938	+	0.942240i	$0.608715\pi$
$930$	0	0
$931$	−30.8908	−1.01240
$932$	0	0
$933$	2.89773	0.0948673
$934$	0	0
$935$	4.22740	0.138251
$936$	0	0
$937$	−43.5945	−1.42417	−0.712085	−	0.702094i	$-0.752249\pi$
$937$	−43.5945	−1.42417	−0.712085	+	0.702094i	$0.752249\pi$
$938$	0	0
$939$	2.39954	0.0783059
$940$	0	0
$941$	26.3081	0.857619	0.428810	−	0.903395i	$-0.358933\pi$
$941$	26.3081	0.857619	0.428810	+	0.903395i	$0.358933\pi$
$942$	0	0
$943$	31.2276	1.01691
$944$	0	0
$945$	13.7092	0.445960
$946$	0	0
$947$	−50.0868	−1.62760	−0.813801	−	0.581144i	$-0.802605\pi$
$947$	−50.0868	−1.62760	−0.813801	+	0.581144i	$0.802605\pi$
$948$	0	0
$949$	−3.85628	−0.125180
$950$	0	0
$951$	10.3953	0.337092
$952$	0	0
$953$	19.1095	0.619018	0.309509	−	0.950896i	$-0.399835\pi$
$953$	19.1095	0.619018	0.309509	+	0.950896i	$0.399835\pi$
$954$	0	0
$955$	4.74166	0.153437
$956$	0	0
$957$	−0.562017	−0.0181674
$958$	0	0
$959$	−92.8411	−2.99799
$960$	0	0
$961$	−16.3715	−0.528112
$962$	0	0
$963$	−18.5825	−0.598813
$964$	0	0
$965$	−55.1563	−1.77554
$966$	0	0
$967$	30.4846	0.980320	0.490160	−	0.871633i	$-0.336938\pi$
$967$	30.4846	0.980320	0.490160	+	0.871633i	$0.336938\pi$
$968$	0	0
$969$	0.544456	0.0174905
$970$	0	0
$971$	−0.783562	−0.0251457	−0.0125728	−	0.999921i	$-0.504002\pi$
$971$	−0.783562	−0.0251457	−0.0125728	+	0.999921i	$0.504002\pi$
$972$	0	0
$973$	−42.9978	−1.37845
$974$	0	0
$975$	−1.45154	−0.0464864
$976$	0	0
$977$	1.80461	0.0577347	0.0288673	−	0.999583i	$-0.490810\pi$
$977$	1.80461	0.0577347	0.0288673	+	0.999583i	$0.490810\pi$
$978$	0	0
$979$	59.4539	1.90016
$980$	0	0
$981$	4.80580	0.153437
$982$	0	0
$983$	−39.5566	−1.26166	−0.630830	−	0.775921i	$-0.717286\pi$
$983$	−39.5566	−1.26166	−0.630830	+	0.775921i	$0.717286\pi$
$984$	0	0
$985$	5.69223	0.181370
$986$	0	0
$987$	26.2313	0.834953
$988$	0	0
$989$	42.8174	1.36151
$990$	0	0
$991$	33.8643	1.07574	0.537868	−	0.843029i	$-0.319230\pi$
$991$	33.8643	1.07574	0.537868	+	0.843029i	$0.319230\pi$
$992$	0	0
$993$	−13.0622	−0.414518
$994$	0	0
$995$	−57.0057	−1.80720
$996$	0	0
$997$	25.8336	0.818160	0.409080	−	0.912499i	$-0.365850\pi$
$997$	25.8336	0.818160	0.409080	+	0.912499i	$0.365850\pi$
$998$	0	0
$999$	−1.36787	−0.0432774

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bc.1.2		9
4.3	odd	2		2004.2.a.c.1.2	✓	9
12.11	even	2		6012.2.a.i.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.c.1.2	✓	9	4.3	odd	2
6012.2.a.i.1.8		9	12.11	even	2
8016.2.a.bc.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.94765 q^{5} -4.65088 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.94765 q^{5} -4.65088 q^{7} +1.00000 q^{9} +5.56158 q^{11} -0.393514 q^{13} -2.94765 q^{15} -0.257869 q^{17} -2.11137 q^{19} -4.65088 q^{21} -5.38988 q^{23} +3.68866 q^{25} +1.00000 q^{27} -0.101053 q^{29} +3.82473 q^{31} +5.56158 q^{33} +13.7092 q^{35} -1.36787 q^{37} -0.393514 q^{39} -5.79375 q^{41} -7.94402 q^{43} -2.94765 q^{45} -5.64008 q^{47} +14.6307 q^{49} -0.257869 q^{51} -4.37285 q^{53} -16.3936 q^{55} -2.11137 q^{57} -7.40379 q^{59} +4.23267 q^{61} -4.65088 q^{63} +1.15994 q^{65} +4.79684 q^{67} -5.38988 q^{69} +7.11410 q^{71} +9.79959 q^{73} +3.68866 q^{75} -25.8663 q^{77} +4.39883 q^{79} +1.00000 q^{81} -9.64462 q^{83} +0.760108 q^{85} -0.101053 q^{87} +10.6901 q^{89} +1.83019 q^{91} +3.82473 q^{93} +6.22358 q^{95} -5.54716 q^{97} +5.56158 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) 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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