LMFDB - Embedded newform 8016.2.a.m.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:	$3$
Coefficient field:	3.3.1300.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 10x - 10 $ x^3 - 10*x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.15347$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.15347 q^{5} -3.36258 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.15347 q^{5} -3.36258 q^{7} +1.00000 q^{9} +6.36258 q^{11} -6.66951 q^{13} +2.15347 q^{15} -4.66951 q^{17} -0.362579 q^{19} -3.36258 q^{21} +1.63742 q^{23} -0.362579 q^{25} +1.00000 q^{27} -6.36258 q^{29} +1.00000 q^{31} +6.36258 q^{33} -7.24120 q^{35} +6.15347 q^{37} -6.66951 q^{39} +6.72516 q^{41} +11.0877 q^{43} +2.15347 q^{45} -3.36258 q^{47} +4.30693 q^{49} -4.66951 q^{51} -1.48395 q^{53} +13.7016 q^{55} -0.362579 q^{57} +11.1856 q^{59} +14.0642 q^{61} -3.36258 q^{63} -14.3626 q^{65} -7.18556 q^{67} +1.63742 q^{69} +13.3947 q^{71} -5.94436 q^{73} -0.362579 q^{75} -21.3947 q^{77} +4.72516 q^{79} +1.00000 q^{81} +14.2091 q^{83} -10.0556 q^{85} -6.36258 q^{87} +9.97645 q^{89} +22.4268 q^{91} +1.00000 q^{93} -0.780801 q^{95} +19.1198 q^{97} +6.36258 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 10 q^{11} - 4 q^{13} + 3 q^{15} + 2 q^{17} + 8 q^{19} - q^{21} + 14 q^{23} + 8 q^{25} + 3 q^{27} - 10 q^{29} + 3 q^{31} + 10 q^{33} + 9 q^{35} + 15 q^{37} - 4 q^{39} + 2 q^{41} + 6 q^{43} + 3 q^{45} - q^{47} + 6 q^{49} + 2 q^{51} - 17 q^{53} + 8 q^{57} + 5 q^{59} - 8 q^{61} - q^{63} - 34 q^{65} + 7 q^{67} + 14 q^{69} + 6 q^{71} - 20 q^{73} + 8 q^{75} - 30 q^{77} - 4 q^{79} + 3 q^{81} + 37 q^{83} - 28 q^{85} - 10 q^{87} + 7 q^{89} + 8 q^{91} + 3 q^{93} + 18 q^{95} + 5 q^{97} + 10 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 + 10 * q^11 - 4 * q^13 + 3 * q^15 + 2 * q^17 + 8 * q^19 - q^21 + 14 * q^23 + 8 * q^25 + 3 * q^27 - 10 * q^29 + 3 * q^31 + 10 * q^33 + 9 * q^35 + 15 * q^37 - 4 * q^39 + 2 * q^41 + 6 * q^43 + 3 * q^45 - q^47 + 6 * q^49 + 2 * q^51 - 17 * q^53 + 8 * q^57 + 5 * q^59 - 8 * q^61 - q^63 - 34 * q^65 + 7 * q^67 + 14 * q^69 + 6 * q^71 - 20 * q^73 + 8 * q^75 - 30 * q^77 - 4 * q^79 + 3 * q^81 + 37 * q^83 - 28 * q^85 - 10 * q^87 + 7 * q^89 + 8 * q^91 + 3 * q^93 + 18 * q^95 + 5 * q^97 + 10 * 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
$5$	2.15347	0.963060	0.481530	−	0.876430i	$-0.340081\pi$
$5$	2.15347	0.963060	0.481530	+	0.876430i	$0.340081\pi$
$6$	0	0
$7$	−3.36258	−1.27094	−0.635468	−	0.772128i	$-0.719193\pi$
$7$	−3.36258	−1.27094	−0.635468	+	0.772128i	$0.719193\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.36258	1.91839	0.959195	−	0.282746i	$-0.0912454\pi$
$11$	6.36258	1.91839	0.959195	+	0.282746i	$0.0912454\pi$
$12$	0	0
$13$	−6.66951	−1.84979	−0.924895	−	0.380222i	$-0.875847\pi$
$13$	−6.66951	−1.84979	−0.924895	+	0.380222i	$0.875847\pi$
$14$	0	0
$15$	2.15347	0.556023
$16$	0	0
$17$	−4.66951	−1.13252	−0.566262	−	0.824226i	$-0.691611\pi$
$17$	−4.66951	−1.13252	−0.566262	+	0.824226i	$0.691611\pi$
$18$	0	0
$19$	−0.362579	−0.0831812	−0.0415906	−	0.999135i	$-0.513243\pi$
$19$	−0.362579	−0.0831812	−0.0415906	+	0.999135i	$0.513243\pi$
$20$	0	0
$21$	−3.36258	−0.733775
$22$	0	0
$23$	1.63742	0.341426	0.170713	−	0.985321i	$-0.445393\pi$
$23$	1.63742	0.341426	0.170713	+	0.985321i	$0.445393\pi$
$24$	0	0
$25$	−0.362579	−0.0725157
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.36258	−1.18150	−0.590751	−	0.806854i	$-0.701168\pi$
$29$	−6.36258	−1.18150	−0.590751	+	0.806854i	$0.701168\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	6.36258	1.10758
$34$	0	0
$35$	−7.24120	−1.22399
$36$	0	0
$37$	6.15347	1.01162	0.505812	−	0.862644i	$-0.331193\pi$
$37$	6.15347	1.01162	0.505812	+	0.862644i	$0.331193\pi$
$38$	0	0
$39$	−6.66951	−1.06798
$40$	0	0
$41$	6.72516	1.05029	0.525147	−	0.851012i	$-0.324011\pi$
$41$	6.72516	1.05029	0.525147	+	0.851012i	$0.324011\pi$
$42$	0	0
$43$	11.0877	1.69086	0.845432	−	0.534083i	$-0.179343\pi$
$43$	11.0877	1.69086	0.845432	+	0.534083i	$0.179343\pi$
$44$	0	0
$45$	2.15347	0.321020
$46$	0	0
$47$	−3.36258	−0.490482	−0.245241	−	0.969462i	$-0.578867\pi$
$47$	−3.36258	−0.490482	−0.245241	+	0.969462i	$0.578867\pi$
$48$	0	0
$49$	4.30693	0.615276
$50$	0	0
$51$	−4.66951	−0.653863
$52$	0	0
$53$	−1.48395	−0.203837	−0.101918	−	0.994793i	$-0.532498\pi$
$53$	−1.48395	−0.203837	−0.101918	+	0.994793i	$0.532498\pi$
$54$	0	0
$55$	13.7016	1.84752
$56$	0	0
$57$	−0.362579	−0.0480247
$58$	0	0
$59$	11.1856	1.45624	0.728118	−	0.685452i	$-0.240395\pi$
$59$	11.1856	1.45624	0.728118	+	0.685452i	$0.240395\pi$
$60$	0	0
$61$	14.0642	1.80073	0.900367	−	0.435131i	$-0.143298\pi$
$61$	14.0642	1.80073	0.900367	+	0.435131i	$0.143298\pi$
$62$	0	0
$63$	−3.36258	−0.423645
$64$	0	0
$65$	−14.3626	−1.78146
$66$	0	0
$67$	−7.18556	−0.877856	−0.438928	−	0.898522i	$-0.644642\pi$
$67$	−7.18556	−0.877856	−0.438928	+	0.898522i	$0.644642\pi$
$68$	0	0
$69$	1.63742	0.197122
$70$	0	0
$71$	13.3947	1.58965	0.794827	−	0.606835i	$-0.207561\pi$
$71$	13.3947	1.58965	0.794827	+	0.606835i	$0.207561\pi$
$72$	0	0
$73$	−5.94436	−0.695734	−0.347867	−	0.937544i	$-0.613094\pi$
$73$	−5.94436	−0.695734	−0.347867	+	0.937544i	$0.613094\pi$
$74$	0	0
$75$	−0.362579	−0.0418670
$76$	0	0
$77$	−21.3947	−2.43815
$78$	0	0
$79$	4.72516	0.531622	0.265811	−	0.964025i	$-0.414360\pi$
$79$	4.72516	0.531622	0.265811	+	0.964025i	$0.414360\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.2091	1.55965	0.779826	−	0.625996i	$-0.215307\pi$
$83$	14.2091	1.55965	0.779826	+	0.625996i	$0.215307\pi$
$84$	0	0
$85$	−10.0556	−1.09069
$86$	0	0
$87$	−6.36258	−0.682140
$88$	0	0
$89$	9.97645	1.05750	0.528751	−	0.848777i	$-0.322661\pi$
$89$	9.97645	1.05750	0.528751	+	0.848777i	$0.322661\pi$
$90$	0	0
$91$	22.4268	2.35096
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−0.780801	−0.0801085
$96$	0	0
$97$	19.1198	1.94132	0.970662	−	0.240447i	$-0.0772941\pi$
$97$	19.1198	1.94132	0.970662	+	0.240447i	$0.0772941\pi$
$98$	0	0
$99$	6.36258	0.639463
$100$	0	0
$101$	12.9343	1.28701	0.643504	−	0.765443i	$-0.277480\pi$
$101$	12.9343	1.28701	0.643504	+	0.765443i	$0.277480\pi$
$102$	0	0
$103$	7.69307	0.758020	0.379010	−	0.925393i	$-0.376265\pi$
$103$	7.69307	0.758020	0.379010	+	0.925393i	$0.376265\pi$
$104$	0	0
$105$	−7.24120	−0.706669
$106$	0	0
$107$	−9.75725	−0.943269	−0.471634	−	0.881794i	$-0.656336\pi$
$107$	−9.75725	−0.943269	−0.471634	+	0.881794i	$0.656336\pi$
$108$	0	0
$109$	5.94436	0.569366	0.284683	−	0.958622i	$-0.408112\pi$
$109$	5.94436	0.569366	0.284683	+	0.958622i	$0.408112\pi$
$110$	0	0
$111$	6.15347	0.584061
$112$	0	0
$113$	4.72516	0.444505	0.222253	−	0.974989i	$-0.428659\pi$
$113$	4.72516	0.444505	0.222253	+	0.974989i	$0.428659\pi$
$114$	0	0
$115$	3.52613	0.328814
$116$	0	0
$117$	−6.66951	−0.616597
$118$	0	0
$119$	15.7016	1.43936
$120$	0	0
$121$	29.4824	2.68022
$122$	0	0
$123$	6.72516	0.606387
$124$	0	0
$125$	−11.5481	−1.03290
$126$	0	0
$127$	−6.75725	−0.599609	−0.299804	−	0.954001i	$-0.596921\pi$
$127$	−6.75725	−0.599609	−0.299804	+	0.954001i	$0.596921\pi$
$128$	0	0
$129$	11.0877	0.976221
$130$	0	0
$131$	−1.63742	−0.143062	−0.0715311	−	0.997438i	$-0.522789\pi$
$131$	−1.63742	−0.143062	−0.0715311	+	0.997438i	$0.522789\pi$
$132$	0	0
$133$	1.21920	0.105718
$134$	0	0
$135$	2.15347	0.185341
$136$	0	0
$137$	−9.05564	−0.773676	−0.386838	−	0.922148i	$-0.626433\pi$
$137$	−9.05564	−0.773676	−0.386838	+	0.922148i	$0.626433\pi$
$138$	0	0
$139$	17.9107	1.51917	0.759584	−	0.650410i	$-0.225403\pi$
$139$	17.9107	1.51917	0.759584	+	0.650410i	$0.225403\pi$
$140$	0	0
$141$	−3.36258	−0.283180
$142$	0	0
$143$	−42.4353	−3.54862
$144$	0	0
$145$	−13.7016	−1.13786
$146$	0	0
$147$	4.30693	0.355230
$148$	0	0
$149$	−21.8551	−1.79044	−0.895219	−	0.445627i	$-0.852981\pi$
$149$	−21.8551	−1.79044	−0.895219	+	0.445627i	$0.852981\pi$
$150$	0	0
$151$	−4.96791	−0.404283	−0.202141	−	0.979356i	$-0.564790\pi$
$151$	−4.96791	−0.404283	−0.202141	+	0.979356i	$0.564790\pi$
$152$	0	0
$153$	−4.66951	−0.377508
$154$	0	0
$155$	2.15347	0.172971
$156$	0	0
$157$	−13.7572	−1.09795	−0.548974	−	0.835839i	$-0.684981\pi$
$157$	−13.7572	−1.09795	−0.548974	+	0.835839i	$0.684981\pi$
$158$	0	0
$159$	−1.48395	−0.117685
$160$	0	0
$161$	−5.50596	−0.433930
$162$	0	0
$163$	16.2733	1.27462	0.637311	−	0.770606i	$-0.280047\pi$
$163$	16.2733	1.27462	0.637311	+	0.770606i	$0.280047\pi$
$164$	0	0
$165$	13.7016	1.06667
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	31.4824	2.42172
$170$	0	0
$171$	−0.362579	−0.0277271
$172$	0	0
$173$	−23.0321	−1.75110	−0.875549	−	0.483130i	$-0.839500\pi$
$173$	−23.0321	−1.75110	−0.875549	+	0.483130i	$0.839500\pi$
$174$	0	0
$175$	1.21920	0.0921628
$176$	0	0
$177$	11.1856	0.840758
$178$	0	0
$179$	−19.5903	−1.46425	−0.732125	−	0.681171i	$-0.761471\pi$
$179$	−19.5903	−1.46425	−0.732125	+	0.681171i	$0.761471\pi$
$180$	0	0
$181$	10.5582	0.784787	0.392393	−	0.919798i	$-0.371647\pi$
$181$	10.5582	0.784787	0.392393	+	0.919798i	$0.371647\pi$
$182$	0	0
$183$	14.0642	1.03965
$184$	0	0
$185$	13.2513	0.974254
$186$	0	0
$187$	−29.7101	−2.17262
$188$	0	0
$189$	−3.36258	−0.244592
$190$	0	0
$191$	−0.339026	−0.0245311	−0.0122655	−	0.999925i	$-0.503904\pi$
$191$	−0.339026	−0.0245311	−0.0122655	+	0.999925i	$0.503904\pi$
$192$	0	0
$193$	−14.7337	−1.06055	−0.530277	−	0.847824i	$-0.677912\pi$
$193$	−14.7337	−1.06055	−0.530277	+	0.847824i	$0.677912\pi$
$194$	0	0
$195$	−14.3626	−1.02853
$196$	0	0
$197$	−7.39467	−0.526848	−0.263424	−	0.964680i	$-0.584852\pi$
$197$	−7.39467	−0.526848	−0.263424	+	0.964680i	$0.584852\pi$
$198$	0	0
$199$	−1.69307	−0.120018	−0.0600091	−	0.998198i	$-0.519113\pi$
$199$	−1.69307	−0.120018	−0.0600091	+	0.998198i	$0.519113\pi$
$200$	0	0
$201$	−7.18556	−0.506830
$202$	0	0
$203$	21.3947	1.50161
$204$	0	0
$205$	14.4824	1.01150
$206$	0	0
$207$	1.63742	0.113809
$208$	0	0
$209$	−2.30693	−0.159574
$210$	0	0
$211$	−1.38613	−0.0954252	−0.0477126	−	0.998861i	$-0.515193\pi$
$211$	−1.38613	−0.0954252	−0.0477126	+	0.998861i	$0.515193\pi$
$212$	0	0
$213$	13.3947	0.917788
$214$	0	0
$215$	23.8771	1.62840
$216$	0	0
$217$	−3.36258	−0.228267
$218$	0	0
$219$	−5.94436	−0.401682
$220$	0	0
$221$	31.1434	2.09493
$222$	0	0
$223$	13.7808	0.922831	0.461415	−	0.887184i	$-0.347342\pi$
$223$	13.7808	0.922831	0.461415	+	0.887184i	$0.347342\pi$
$224$	0	0
$225$	−0.362579	−0.0241719
$226$	0	0
$227$	−0.264755	−0.0175724	−0.00878621	−	0.999961i	$-0.502797\pi$
$227$	−0.264755	−0.0175724	−0.00878621	+	0.999961i	$0.502797\pi$
$228$	0	0
$229$	17.2277	1.13844	0.569221	−	0.822185i	$-0.307245\pi$
$229$	17.2277	1.13844	0.569221	+	0.822185i	$0.307245\pi$
$230$	0	0
$231$	−21.3947	−1.40767
$232$	0	0
$233$	30.0321	1.96747	0.983734	−	0.179632i	$-0.0574907\pi$
$233$	30.0321	1.96747	0.983734	+	0.179632i	$0.0574907\pi$
$234$	0	0
$235$	−7.24120	−0.472364
$236$	0	0
$237$	4.72516	0.306932
$238$	0	0
$239$	10.9208	0.706408	0.353204	−	0.935546i	$-0.385092\pi$
$239$	10.9208	0.706408	0.353204	+	0.935546i	$0.385092\pi$
$240$	0	0
$241$	−10.2513	−0.660344	−0.330172	−	0.943921i	$-0.607107\pi$
$241$	−10.2513	−0.660344	−0.330172	+	0.943921i	$0.607107\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	9.27484	0.592548
$246$	0	0
$247$	2.41822	0.153868
$248$	0	0
$249$	14.2091	0.900466
$250$	0	0
$251$	−28.7337	−1.81365	−0.906827	−	0.421502i	$-0.861503\pi$
$251$	−28.7337	−1.81365	−0.906827	+	0.421502i	$0.861503\pi$
$252$	0	0
$253$	10.4182	0.654988
$254$	0	0
$255$	−10.0556	−0.629709
$256$	0	0
$257$	21.0877	1.31542	0.657708	−	0.753273i	$-0.271526\pi$
$257$	21.0877	1.31542	0.657708	+	0.753273i	$0.271526\pi$
$258$	0	0
$259$	−20.6915	−1.28571
$260$	0	0
$261$	−6.36258	−0.393834
$262$	0	0
$263$	10.8129	0.666752	0.333376	−	0.942794i	$-0.391812\pi$
$263$	10.8129	0.666752	0.333376	+	0.942794i	$0.391812\pi$
$264$	0	0
$265$	−3.19565	−0.196307
$266$	0	0
$267$	9.97645	0.610549
$268$	0	0
$269$	−9.49249	−0.578768	−0.289384	−	0.957213i	$-0.593450\pi$
$269$	−9.49249	−0.578768	−0.289384	+	0.957213i	$0.593450\pi$
$270$	0	0
$271$	−1.33049	−0.0808213	−0.0404107	−	0.999183i	$-0.512867\pi$
$271$	−1.33049	−0.0808213	−0.0404107	+	0.999183i	$0.512867\pi$
$272$	0	0
$273$	22.4268	1.35733
$274$	0	0
$275$	−2.30693	−0.139113
$276$	0	0
$277$	−29.5481	−1.77538	−0.887688	−	0.460446i	$-0.847690\pi$
$277$	−29.5481	−1.77538	−0.887688	+	0.460446i	$0.847690\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	0.0791962	0.00472445	0.00236222	−	0.999997i	$-0.499248\pi$
$281$	0.0791962	0.00472445	0.00236222	+	0.999997i	$0.499248\pi$
$282$	0	0
$283$	−0.856620	−0.0509208	−0.0254604	−	0.999676i	$-0.508105\pi$
$283$	−0.856620	−0.0509208	−0.0254604	+	0.999676i	$0.508105\pi$
$284$	0	0
$285$	−0.780801	−0.0462507
$286$	0	0
$287$	−22.6139	−1.33485
$288$	0	0
$289$	4.80435	0.282609
$290$	0	0
$291$	19.1198	1.12082
$292$	0	0
$293$	−3.27484	−0.191318	−0.0956592	−	0.995414i	$-0.530496\pi$
$293$	−3.27484	−0.191318	−0.0956592	+	0.995414i	$0.530496\pi$
$294$	0	0
$295$	24.0877	1.40244
$296$	0	0
$297$	6.36258	0.369194
$298$	0	0
$299$	−10.9208	−0.631566
$300$	0	0
$301$	−37.2834	−2.14898
$302$	0	0
$303$	12.9343	0.743054
$304$	0	0
$305$	30.2868	1.73421
$306$	0	0
$307$	−7.12992	−0.406926	−0.203463	−	0.979083i	$-0.565220\pi$
$307$	−7.12992	−0.406926	−0.203463	+	0.979083i	$0.565220\pi$
$308$	0	0
$309$	7.69307	0.437643
$310$	0	0
$311$	11.9679	0.678638	0.339319	−	0.940671i	$-0.389803\pi$
$311$	11.9679	0.678638	0.339319	+	0.940671i	$0.389803\pi$
$312$	0	0
$313$	−18.7808	−1.06155	−0.530777	−	0.847512i	$-0.678100\pi$
$313$	−18.7808	−1.06155	−0.530777	+	0.847512i	$0.678100\pi$
$314$	0	0
$315$	−7.24120	−0.407996
$316$	0	0
$317$	−4.42676	−0.248632	−0.124316	−	0.992243i	$-0.539674\pi$
$317$	−4.42676	−0.248632	−0.124316	+	0.992243i	$0.539674\pi$
$318$	0	0
$319$	−40.4824	−2.26658
$320$	0	0
$321$	−9.75725	−0.544597
$322$	0	0
$323$	1.69307	0.0942047
$324$	0	0
$325$	2.41822	0.134139
$326$	0	0
$327$	5.94436	0.328724
$328$	0	0
$329$	11.3069	0.623371
$330$	0	0
$331$	−8.42676	−0.463177	−0.231588	−	0.972814i	$-0.574392\pi$
$331$	−8.42676	−0.463177	−0.231588	+	0.972814i	$0.574392\pi$
$332$	0	0
$333$	6.15347	0.337208
$334$	0	0
$335$	−15.4739	−0.845428
$336$	0	0
$337$	15.1755	0.826661	0.413330	−	0.910581i	$-0.364365\pi$
$337$	15.1755	0.826661	0.413330	+	0.910581i	$0.364365\pi$
$338$	0	0
$339$	4.72516	0.256635
$340$	0	0
$341$	6.36258	0.344553
$342$	0	0
$343$	9.05564	0.488959
$344$	0	0
$345$	3.52613	0.189841
$346$	0	0
$347$	−2.87862	−0.154533	−0.0772663	−	0.997010i	$-0.524619\pi$
$347$	−2.87862	−0.154533	−0.0772663	+	0.997010i	$0.524619\pi$
$348$	0	0
$349$	1.23266	0.0659830	0.0329915	−	0.999456i	$-0.489497\pi$
$349$	1.23266	0.0659830	0.0329915	+	0.999456i	$0.489497\pi$
$350$	0	0
$351$	−6.66951	−0.355992
$352$	0	0
$353$	−5.72516	−0.304719	−0.152360	−	0.988325i	$-0.548687\pi$
$353$	−5.72516	−0.304719	−0.152360	+	0.988325i	$0.548687\pi$
$354$	0	0
$355$	28.8450	1.53093
$356$	0	0
$357$	15.7016	0.831017
$358$	0	0
$359$	−6.92080	−0.365266	−0.182633	−	0.983181i	$-0.558462\pi$
$359$	−6.92080	−0.365266	−0.182633	+	0.983181i	$0.558462\pi$
$360$	0	0
$361$	−18.8685	−0.993081
$362$	0	0
$363$	29.4824	1.54743
$364$	0	0
$365$	−12.8010	−0.670034
$366$	0	0
$367$	32.9850	1.72180	0.860901	−	0.508772i	$-0.169900\pi$
$367$	32.9850	1.72180	0.860901	+	0.508772i	$0.169900\pi$
$368$	0	0
$369$	6.72516	0.350098
$370$	0	0
$371$	4.98991	0.259063
$372$	0	0
$373$	6.51605	0.337388	0.168694	−	0.985668i	$-0.446045\pi$
$373$	6.51605	0.337388	0.168694	+	0.985668i	$0.446045\pi$
$374$	0	0
$375$	−11.5481	−0.596343
$376$	0	0
$377$	42.4353	2.18553
$378$	0	0
$379$	4.51605	0.231974	0.115987	−	0.993251i	$-0.462997\pi$
$379$	4.51605	0.231974	0.115987	+	0.993251i	$0.462997\pi$
$380$	0	0
$381$	−6.75725	−0.346184
$382$	0	0
$383$	−20.2598	−1.03523	−0.517614	−	0.855614i	$-0.673180\pi$
$383$	−20.2598	−1.03523	−0.517614	+	0.855614i	$0.673180\pi$
$384$	0	0
$385$	−46.0727	−2.34808
$386$	0	0
$387$	11.0877	0.563621
$388$	0	0
$389$	26.4268	1.33989	0.669945	−	0.742411i	$-0.266318\pi$
$389$	26.4268	1.33989	0.669945	+	0.742411i	$0.266318\pi$
$390$	0	0
$391$	−7.64596	−0.386673
$392$	0	0
$393$	−1.63742	−0.0825970
$394$	0	0
$395$	10.1755	0.511984
$396$	0	0
$397$	−8.05564	−0.404301	−0.202151	−	0.979354i	$-0.564793\pi$
$397$	−8.05564	−0.404301	−0.202151	+	0.979354i	$0.564793\pi$
$398$	0	0
$399$	1.21920	0.0610363
$400$	0	0
$401$	34.4353	1.71962	0.859808	−	0.510617i	$-0.170583\pi$
$401$	34.4353	1.71962	0.859808	+	0.510617i	$0.170583\pi$
$402$	0	0
$403$	−6.66951	−0.332232
$404$	0	0
$405$	2.15347	0.107007
$406$	0	0
$407$	39.1519	1.94069
$408$	0	0
$409$	−25.5145	−1.26161	−0.630805	−	0.775941i	$-0.717275\pi$
$409$	−25.5145	−1.26161	−0.630805	+	0.775941i	$0.717275\pi$
$410$	0	0
$411$	−9.05564	−0.446682
$412$	0	0
$413$	−37.6123	−1.85078
$414$	0	0
$415$	30.5989	1.50204
$416$	0	0
$417$	17.9107	0.877092
$418$	0	0
$419$	−12.8364	−0.627101	−0.313551	−	0.949571i	$-0.601519\pi$
$419$	−12.8364	−0.627101	−0.313551	+	0.949571i	$0.601519\pi$
$420$	0	0
$421$	18.6224	0.907601	0.453800	−	0.891103i	$-0.350068\pi$
$421$	18.6224	0.907601	0.453800	+	0.891103i	$0.350068\pi$
$422$	0	0
$423$	−3.36258	−0.163494
$424$	0	0
$425$	1.69307	0.0821257
$426$	0	0
$427$	−47.2919	−2.28862
$428$	0	0
$429$	−42.4353	−2.04880
$430$	0	0
$431$	9.06418	0.436606	0.218303	−	0.975881i	$-0.429948\pi$
$431$	9.06418	0.436606	0.218303	+	0.975881i	$0.429948\pi$
$432$	0	0
$433$	−4.07920	−0.196034	−0.0980168	−	0.995185i	$-0.531250\pi$
$433$	−4.07920	−0.196034	−0.0980168	+	0.995185i	$0.531250\pi$
$434$	0	0
$435$	−13.7016	−0.656942
$436$	0	0
$437$	−0.593694	−0.0284002
$438$	0	0
$439$	−9.58178	−0.457313	−0.228657	−	0.973507i	$-0.573433\pi$
$439$	−9.58178	−0.457313	−0.228657	+	0.973507i	$0.573433\pi$
$440$	0	0
$441$	4.30693	0.205092
$442$	0	0
$443$	10.2648	0.487693	0.243847	−	0.969814i	$-0.421591\pi$
$443$	10.2648	0.487693	0.243847	+	0.969814i	$0.421591\pi$
$444$	0	0
$445$	21.4840	1.01844
$446$	0	0
$447$	−21.8551	−1.03371
$448$	0	0
$449$	−3.58515	−0.169194	−0.0845969	−	0.996415i	$-0.526960\pi$
$449$	−3.58515	−0.169194	−0.0845969	+	0.996415i	$0.526960\pi$
$450$	0	0
$451$	42.7893	2.01487
$452$	0	0
$453$	−4.96791	−0.233413
$454$	0	0
$455$	48.2953	2.26412
$456$	0	0
$457$	−28.1755	−1.31799	−0.658996	−	0.752146i	$-0.729019\pi$
$457$	−28.1755	−1.31799	−0.658996	+	0.752146i	$0.729019\pi$
$458$	0	0
$459$	−4.66951	−0.217954
$460$	0	0
$461$	−7.94436	−0.370006	−0.185003	−	0.982738i	$-0.559229\pi$
$461$	−7.94436	−0.370006	−0.185003	+	0.982738i	$0.559229\pi$
$462$	0	0
$463$	14.6139	0.679164	0.339582	−	0.940576i	$-0.389714\pi$
$463$	14.6139	0.679164	0.339582	+	0.940576i	$0.389714\pi$
$464$	0	0
$465$	2.15347	0.0998647
$466$	0	0
$467$	24.0642	1.11356	0.556779	−	0.830661i	$-0.312037\pi$
$467$	24.0642	1.11356	0.556779	+	0.830661i	$0.312037\pi$
$468$	0	0
$469$	24.1620	1.11570
$470$	0	0
$471$	−13.7572	−0.633901
$472$	0	0
$473$	70.5466	3.24374
$474$	0	0
$475$	0.131463	0.00603195
$476$	0	0
$477$	−1.48395	−0.0679456
$478$	0	0
$479$	−21.0963	−0.963913	−0.481957	−	0.876195i	$-0.660074\pi$
$479$	−21.0963	−0.963913	−0.481957	+	0.876195i	$0.660074\pi$
$480$	0	0
$481$	−41.0406	−1.87129
$482$	0	0
$483$	−5.50596	−0.250530
$484$	0	0
$485$	41.1739	1.86961
$486$	0	0
$487$	−19.0877	−0.864948	−0.432474	−	0.901646i	$-0.642359\pi$
$487$	−19.0877	−0.864948	−0.432474	+	0.901646i	$0.642359\pi$
$488$	0	0
$489$	16.2733	0.735904
$490$	0	0
$491$	7.27484	0.328309	0.164155	−	0.986435i	$-0.447510\pi$
$491$	7.27484	0.328309	0.164155	+	0.986435i	$0.447510\pi$
$492$	0	0
$493$	29.7101	1.33808
$494$	0	0
$495$	13.7016	0.615841
$496$	0	0
$497$	−45.0406	−2.02035
$498$	0	0
$499$	−18.2513	−0.817040	−0.408520	−	0.912749i	$-0.633955\pi$
$499$	−18.2513	−0.817040	−0.408520	+	0.912749i	$0.633955\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−27.8685	−1.24260	−0.621298	−	0.783574i	$-0.713394\pi$
$503$	−27.8685	−1.24260	−0.621298	+	0.783574i	$0.713394\pi$
$504$	0	0
$505$	27.8535	1.23947
$506$	0	0
$507$	31.4824	1.39818
$508$	0	0
$509$	18.5295	0.821306	0.410653	−	0.911792i	$-0.365301\pi$
$509$	18.5295	0.821306	0.410653	+	0.911792i	$0.365301\pi$
$510$	0	0
$511$	19.9884	0.884233
$512$	0	0
$513$	−0.362579	−0.0160082
$514$	0	0
$515$	16.5668	0.730019
$516$	0	0
$517$	−21.3947	−0.940937
$518$	0	0
$519$	−23.0321	−1.01100
$520$	0	0
$521$	3.02355	0.132464	0.0662321	−	0.997804i	$-0.478902\pi$
$521$	3.02355	0.132464	0.0662321	+	0.997804i	$0.478902\pi$
$522$	0	0
$523$	−7.45885	−0.326153	−0.163076	−	0.986613i	$-0.552142\pi$
$523$	−7.45885	−0.326153	−0.163076	+	0.986613i	$0.552142\pi$
$524$	0	0
$525$	1.21920	0.0532102
$526$	0	0
$527$	−4.66951	−0.203407
$528$	0	0
$529$	−20.3189	−0.883428
$530$	0	0
$531$	11.1856	0.485412
$532$	0	0
$533$	−44.8535	−1.94282
$534$	0	0
$535$	−21.0119	−0.908424
$536$	0	0
$537$	−19.5903	−0.845385
$538$	0	0
$539$	27.4032	1.18034
$540$	0	0
$541$	−25.7438	−1.10681	−0.553406	−	0.832912i	$-0.686672\pi$
$541$	−25.7438	−1.10681	−0.553406	+	0.832912i	$0.686672\pi$
$542$	0	0
$543$	10.5582	0.453097
$544$	0	0
$545$	12.8010	0.548334
$546$	0	0
$547$	24.7472	1.05811	0.529056	−	0.848587i	$-0.322546\pi$
$547$	24.7472	1.05811	0.529056	+	0.848587i	$0.322546\pi$
$548$	0	0
$549$	14.0642	0.600245
$550$	0	0
$551$	2.30693	0.0982787
$552$	0	0
$553$	−15.8887	−0.675657
$554$	0	0
$555$	13.2513	0.562486
$556$	0	0
$557$	−14.0085	−0.593561	−0.296780	−	0.954946i	$-0.595913\pi$
$557$	−14.0085	−0.593561	−0.296780	+	0.954946i	$0.595913\pi$
$558$	0	0
$559$	−73.9498	−3.12774
$560$	0	0
$561$	−29.7101	−1.25436
$562$	0	0
$563$	−11.7658	−0.495869	−0.247934	−	0.968777i	$-0.579752\pi$
$563$	−11.7658	−0.495869	−0.247934	+	0.968777i	$0.579752\pi$
$564$	0	0
$565$	10.1755	0.428085
$566$	0	0
$567$	−3.36258	−0.141215
$568$	0	0
$569$	0.780801	0.0327329	0.0163664	−	0.999866i	$-0.494790\pi$
$569$	0.780801	0.0327329	0.0163664	+	0.999866i	$0.494790\pi$
$570$	0	0
$571$	−32.2262	−1.34862	−0.674312	−	0.738447i	$-0.735560\pi$
$571$	−32.2262	−1.34862	−0.674312	+	0.738447i	$0.735560\pi$
$572$	0	0
$573$	−0.339026	−0.0141630
$574$	0	0
$575$	−0.593694	−0.0247587
$576$	0	0
$577$	−26.7016	−1.11160	−0.555801	−	0.831315i	$-0.687588\pi$
$577$	−26.7016	−1.11160	−0.555801	+	0.831315i	$0.687588\pi$
$578$	0	0
$579$	−14.7337	−0.612312
$580$	0	0
$581$	−47.7793	−1.98222
$582$	0	0
$583$	−9.44177	−0.391038
$584$	0	0
$585$	−14.3626	−0.593820
$586$	0	0
$587$	−10.5802	−0.436693	−0.218346	−	0.975871i	$-0.570066\pi$
$587$	−10.5802	−0.436693	−0.218346	+	0.975871i	$0.570066\pi$
$588$	0	0
$589$	−0.362579	−0.0149398
$590$	0	0
$591$	−7.39467	−0.304176
$592$	0	0
$593$	11.7487	0.482462	0.241231	−	0.970468i	$-0.422449\pi$
$593$	11.7487	0.482462	0.241231	+	0.970468i	$0.422449\pi$
$594$	0	0
$595$	33.8129	1.38619
$596$	0	0
$597$	−1.69307	−0.0692926
$598$	0	0
$599$	3.55823	0.145385	0.0726926	−	0.997354i	$-0.476841\pi$
$599$	3.55823	0.145385	0.0726926	+	0.997354i	$0.476841\pi$
$600$	0	0
$601$	−6.81289	−0.277904	−0.138952	−	0.990299i	$-0.544373\pi$
$601$	−6.81289	−0.277904	−0.138952	+	0.990299i	$0.544373\pi$
$602$	0	0
$603$	−7.18556	−0.292619
$604$	0	0
$605$	63.4894	2.58121
$606$	0	0
$607$	23.8214	0.966882	0.483441	−	0.875377i	$-0.339387\pi$
$607$	23.8214	0.966882	0.483441	+	0.875377i	$0.339387\pi$
$608$	0	0
$609$	21.3947	0.866956
$610$	0	0
$611$	22.4268	0.907290
$612$	0	0
$613$	34.6695	1.40029	0.700144	−	0.714001i	$-0.253119\pi$
$613$	34.6695	1.40029	0.700144	+	0.714001i	$0.253119\pi$
$614$	0	0
$615$	14.4824	0.583987
$616$	0	0
$617$	−1.03209	−0.0415504	−0.0207752	−	0.999784i	$-0.506613\pi$
$617$	−1.03209	−0.0415504	−0.0207752	+	0.999784i	$0.506613\pi$
$618$	0	0
$619$	32.1840	1.29358	0.646792	−	0.762666i	$-0.276110\pi$
$619$	32.1840	1.29358	0.646792	+	0.762666i	$0.276110\pi$
$620$	0	0
$621$	1.63742	0.0657075
$622$	0	0
$623$	−33.5466	−1.34402
$624$	0	0
$625$	−23.0556	−0.922226
$626$	0	0
$627$	−2.30693	−0.0921301
$628$	0	0
$629$	−28.7337	−1.14569
$630$	0	0
$631$	−10.8044	−0.430115	−0.215057	−	0.976601i	$-0.568994\pi$
$631$	−10.8044	−0.430115	−0.215057	+	0.976601i	$0.568994\pi$
$632$	0	0
$633$	−1.38613	−0.0550938
$634$	0	0
$635$	−14.5515	−0.577459
$636$	0	0
$637$	−28.7252	−1.13813
$638$	0	0
$639$	13.3947	0.529885
$640$	0	0
$641$	11.2277	0.443469	0.221735	−	0.975107i	$-0.428828\pi$
$641$	11.2277	0.443469	0.221735	+	0.975107i	$0.428828\pi$
$642$	0	0
$643$	−0.409683	−0.0161563	−0.00807816	−	0.999967i	$-0.502571\pi$
$643$	−0.409683	−0.0161563	−0.00807816	+	0.999967i	$0.502571\pi$
$644$	0	0
$645$	23.8771	0.940159
$646$	0	0
$647$	6.30693	0.247951	0.123976	−	0.992285i	$-0.460436\pi$
$647$	6.30693	0.247951	0.123976	+	0.992285i	$0.460436\pi$
$648$	0	0
$649$	71.1690	2.79363
$650$	0	0
$651$	−3.36258	−0.131790
$652$	0	0
$653$	−31.1806	−1.22019	−0.610096	−	0.792327i	$-0.708869\pi$
$653$	−31.1806	−1.22019	−0.610096	+	0.792327i	$0.708869\pi$
$654$	0	0
$655$	−3.52613	−0.137777
$656$	0	0
$657$	−5.94436	−0.231911
$658$	0	0
$659$	28.7759	1.12095	0.560474	−	0.828172i	$-0.310619\pi$
$659$	28.7759	1.12095	0.560474	+	0.828172i	$0.310619\pi$
$660$	0	0
$661$	−23.3947	−0.909947	−0.454974	−	0.890505i	$-0.650351\pi$
$661$	−23.3947	−0.909947	−0.454974	+	0.890505i	$0.650351\pi$
$662$	0	0
$663$	31.1434	1.20951
$664$	0	0
$665$	2.62550	0.101813
$666$	0	0
$667$	−10.4182	−0.403395
$668$	0	0
$669$	13.7808	0.532797
$670$	0	0
$671$	89.4845	3.45451
$672$	0	0
$673$	3.59032	0.138397	0.0691983	−	0.997603i	$-0.477956\pi$
$673$	3.59032	0.138397	0.0691983	+	0.997603i	$0.477956\pi$
$674$	0	0
$675$	−0.362579	−0.0139557
$676$	0	0
$677$	26.5111	1.01891	0.509453	−	0.860499i	$-0.329848\pi$
$677$	26.5111	1.01891	0.509453	+	0.860499i	$0.329848\pi$
$678$	0	0
$679$	−64.2919	−2.46730
$680$	0	0
$681$	−0.264755	−0.0101454
$682$	0	0
$683$	19.4589	0.744572	0.372286	−	0.928118i	$-0.378574\pi$
$683$	19.4589	0.744572	0.372286	+	0.928118i	$0.378574\pi$
$684$	0	0
$685$	−19.5010	−0.745096
$686$	0	0
$687$	17.2277	0.657279
$688$	0	0
$689$	9.89725	0.377055
$690$	0	0
$691$	4.29840	0.163519	0.0817593	−	0.996652i	$-0.473946\pi$
$691$	4.29840	0.163519	0.0817593	+	0.996652i	$0.473946\pi$
$692$	0	0
$693$	−21.3947	−0.812716
$694$	0	0
$695$	38.5701	1.46305
$696$	0	0
$697$	−31.4032	−1.18948
$698$	0	0
$699$	30.0321	1.13592
$700$	0	0
$701$	36.9648	1.39614	0.698071	−	0.716029i	$-0.254042\pi$
$701$	36.9648	1.39614	0.698071	+	0.716029i	$0.254042\pi$
$702$	0	0
$703$	−2.23112	−0.0841481
$704$	0	0
$705$	−7.24120	−0.272719
$706$	0	0
$707$	−43.4925	−1.63570
$708$	0	0
$709$	−9.67960	−0.363525	−0.181763	−	0.983342i	$-0.558180\pi$
$709$	−9.67960	−0.363525	−0.181763	+	0.983342i	$0.558180\pi$
$710$	0	0
$711$	4.72516	0.177207
$712$	0	0
$713$	1.63742	0.0613219
$714$	0	0
$715$	−91.3830	−3.41753
$716$	0	0
$717$	10.9208	0.407845
$718$	0	0
$719$	9.45031	0.352437	0.176219	−	0.984351i	$-0.443613\pi$
$719$	9.45031	0.352437	0.176219	+	0.984351i	$0.443613\pi$
$720$	0	0
$721$	−25.8685	−0.963395
$722$	0	0
$723$	−10.2513	−0.381250
$724$	0	0
$725$	2.30693	0.0856774
$726$	0	0
$727$	4.66951	0.173183	0.0865913	−	0.996244i	$-0.472403\pi$
$727$	4.66951	0.173183	0.0865913	+	0.996244i	$0.472403\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−51.7743	−1.91494
$732$	0	0
$733$	−39.4589	−1.45745	−0.728723	−	0.684809i	$-0.759886\pi$
$733$	−39.4589	−1.45745	−0.728723	+	0.684809i	$0.759886\pi$
$734$	0	0
$735$	9.27484	0.342108
$736$	0	0
$737$	−45.7187	−1.68407
$738$	0	0
$739$	−13.4638	−0.495273	−0.247637	−	0.968853i	$-0.579654\pi$
$739$	−13.4638	−0.495273	−0.247637	+	0.968853i	$0.579654\pi$
$740$	0	0
$741$	2.41822	0.0888356
$742$	0	0
$743$	−7.40968	−0.271835	−0.135917	−	0.990720i	$-0.543398\pi$
$743$	−7.40968	−0.271835	−0.135917	+	0.990720i	$0.543398\pi$
$744$	0	0
$745$	−47.0642	−1.72430
$746$	0	0
$747$	14.2091	0.519884
$748$	0	0
$749$	32.8095	1.19883
$750$	0	0
$751$	−20.9293	−0.763723	−0.381861	−	0.924220i	$-0.624717\pi$
$751$	−20.9293	−0.763723	−0.381861	+	0.924220i	$0.624717\pi$
$752$	0	0
$753$	−28.7337	−1.04711
$754$	0	0
$755$	−10.6982	−0.389348
$756$	0	0
$757$	23.3476	0.848582	0.424291	−	0.905526i	$-0.360523\pi$
$757$	23.3476	0.848582	0.424291	+	0.905526i	$0.360523\pi$
$758$	0	0
$759$	10.4182	0.378158
$760$	0	0
$761$	−28.3861	−1.02900	−0.514498	−	0.857491i	$-0.672022\pi$
$761$	−28.3861	−1.02900	−0.514498	+	0.857491i	$0.672022\pi$
$762$	0	0
$763$	−19.9884	−0.723627
$764$	0	0
$765$	−10.0556	−0.363563
$766$	0	0
$767$	−74.6022	−2.69373
$768$	0	0
$769$	53.0877	1.91439	0.957196	−	0.289439i	$-0.0934689\pi$
$769$	53.0877	1.91439	0.957196	+	0.289439i	$0.0934689\pi$
$770$	0	0
$771$	21.0877	0.759456
$772$	0	0
$773$	14.7117	0.529143	0.264571	−	0.964366i	$-0.414770\pi$
$773$	14.7117	0.529143	0.264571	+	0.964366i	$0.414770\pi$
$774$	0	0
$775$	−0.362579	−0.0130242
$776$	0	0
$777$	−20.6915	−0.742304
$778$	0	0
$779$	−2.43840	−0.0873647
$780$	0	0
$781$	85.2246	3.04958
$782$	0	0
$783$	−6.36258	−0.227380
$784$	0	0
$785$	−29.6258	−1.05739
$786$	0	0
$787$	22.3626	0.797140	0.398570	−	0.917138i	$-0.369507\pi$
$787$	22.3626	0.797140	0.398570	+	0.917138i	$0.369507\pi$
$788$	0	0
$789$	10.8129	0.384949
$790$	0	0
$791$	−15.8887	−0.564938
$792$	0	0
$793$	−93.8013	−3.33098
$794$	0	0
$795$	−3.19565	−0.113338
$796$	0	0
$797$	−3.61725	−0.128129	−0.0640647	−	0.997946i	$-0.520406\pi$
$797$	−3.61725	−0.128129	−0.0640647	+	0.997946i	$0.520406\pi$
$798$	0	0
$799$	15.7016	0.555483
$800$	0	0
$801$	9.97645	0.352500
$802$	0	0
$803$	−37.8214	−1.33469
$804$	0	0
$805$	−11.8569	−0.417901
$806$	0	0
$807$	−9.49249	−0.334152
$808$	0	0
$809$	20.3947	0.717038	0.358519	−	0.933522i	$-0.383282\pi$
$809$	20.3947	0.717038	0.358519	+	0.933522i	$0.383282\pi$
$810$	0	0
$811$	−24.7202	−0.868045	−0.434022	−	0.900902i	$-0.642906\pi$
$811$	−24.7202	−0.868045	−0.434022	+	0.900902i	$0.642906\pi$
$812$	0	0
$813$	−1.33049	−0.0466622
$814$	0	0
$815$	35.0440	1.22754
$816$	0	0
$817$	−4.02018	−0.140648
$818$	0	0
$819$	22.4268	0.783654
$820$	0	0
$821$	−11.0608	−0.386025	−0.193012	−	0.981196i	$-0.561826\pi$
$821$	−11.0608	−0.386025	−0.193012	+	0.981196i	$0.561826\pi$
$822$	0	0
$823$	38.5380	1.34335	0.671676	−	0.740845i	$-0.265575\pi$
$823$	38.5380	1.34335	0.671676	+	0.740845i	$0.265575\pi$
$824$	0	0
$825$	−2.30693	−0.0803172
$826$	0	0
$827$	29.0406	1.00984	0.504921	−	0.863166i	$-0.331522\pi$
$827$	29.0406	1.00984	0.504921	+	0.863166i	$0.331522\pi$
$828$	0	0
$829$	27.9664	0.971312	0.485656	−	0.874150i	$-0.338581\pi$
$829$	27.9664	0.971312	0.485656	+	0.874150i	$0.338581\pi$
$830$	0	0
$831$	−29.5481	−1.02501
$832$	0	0
$833$	−20.1113	−0.696815
$834$	0	0
$835$	2.15347	0.0745238
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−31.1113	−1.07408	−0.537040	−	0.843556i	$-0.680458\pi$
$839$	−31.1113	−1.07408	−0.537040	+	0.843556i	$0.680458\pi$
$840$	0	0
$841$	11.4824	0.395945
$842$	0	0
$843$	0.0791962	0.00272766
$844$	0	0
$845$	67.7963	2.33226
$846$	0	0
$847$	−99.1369	−3.40638
$848$	0	0
$849$	−0.856620	−0.0293991
$850$	0	0
$851$	10.0758	0.345395
$852$	0	0
$853$	31.5145	1.07904	0.539518	−	0.841974i	$-0.318607\pi$
$853$	31.5145	1.07904	0.539518	+	0.841974i	$0.318607\pi$
$854$	0	0
$855$	−0.780801	−0.0267028
$856$	0	0
$857$	−3.04710	−0.104087	−0.0520436	−	0.998645i	$-0.516573\pi$
$857$	−3.04710	−0.104087	−0.0520436	+	0.998645i	$0.516573\pi$
$858$	0	0
$859$	−6.58515	−0.224683	−0.112341	−	0.993670i	$-0.535835\pi$
$859$	−6.58515	−0.224683	−0.112341	+	0.993670i	$0.535835\pi$
$860$	0	0
$861$	−22.6139	−0.770679
$862$	0	0
$863$	27.5145	0.936604	0.468302	−	0.883568i	$-0.344866\pi$
$863$	27.5145	0.936604	0.468302	+	0.883568i	$0.344866\pi$
$864$	0	0
$865$	−49.5989	−1.68641
$866$	0	0
$867$	4.80435	0.163164
$868$	0	0
$869$	30.0642	1.01986
$870$	0	0
$871$	47.9242	1.62385
$872$	0	0
$873$	19.1198	0.647108
$874$	0	0
$875$	38.8315	1.31274
$876$	0	0
$877$	−6.75387	−0.228062	−0.114031	−	0.993477i	$-0.536376\pi$
$877$	−6.75387	−0.228062	−0.114031	+	0.993477i	$0.536376\pi$
$878$	0	0
$879$	−3.27484	−0.110458
$880$	0	0
$881$	23.2547	0.783470	0.391735	−	0.920078i	$-0.371875\pi$
$881$	23.2547	0.783470	0.391735	+	0.920078i	$0.371875\pi$
$882$	0	0
$883$	31.9614	1.07559	0.537794	−	0.843076i	$-0.319258\pi$
$883$	31.9614	1.07559	0.537794	+	0.843076i	$0.319258\pi$
$884$	0	0
$885$	24.0877	0.809700
$886$	0	0
$887$	38.1485	1.28090	0.640451	−	0.767999i	$-0.278747\pi$
$887$	38.1485	1.28090	0.640451	+	0.767999i	$0.278747\pi$
$888$	0	0
$889$	22.7218	0.762064
$890$	0	0
$891$	6.36258	0.213154
$892$	0	0
$893$	1.21920	0.0407989
$894$	0	0
$895$	−42.1871	−1.41016
$896$	0	0
$897$	−10.9208	−0.364635
$898$	0	0
$899$	−6.36258	−0.212204
$900$	0	0
$901$	6.92934	0.230850
$902$	0	0
$903$	−37.2834	−1.24071
$904$	0	0
$905$	22.7368	0.755797
$906$	0	0
$907$	41.2803	1.37069	0.685345	−	0.728219i	$-0.259652\pi$
$907$	41.2803	1.37069	0.685345	+	0.728219i	$0.259652\pi$
$908$	0	0
$909$	12.9343	0.429003
$910$	0	0
$911$	17.5026	0.579886	0.289943	−	0.957044i	$-0.406364\pi$
$911$	17.5026	0.579886	0.289943	+	0.957044i	$0.406364\pi$
$912$	0	0
$913$	90.4066	2.99202
$914$	0	0
$915$	30.2868	1.00125
$916$	0	0
$917$	5.50596	0.181823
$918$	0	0
$919$	27.2482	0.898835	0.449417	−	0.893322i	$-0.351632\pi$
$919$	27.2482	0.898835	0.449417	+	0.893322i	$0.351632\pi$
$920$	0	0
$921$	−7.12992	−0.234939
$922$	0	0
$923$	−89.3359	−2.94053
$924$	0	0
$925$	−2.23112	−0.0733586
$926$	0	0
$927$	7.69307	0.252673
$928$	0	0
$929$	−13.2428	−0.434481	−0.217240	−	0.976118i	$-0.569706\pi$
$929$	−13.2428	−0.434481	−0.217240	+	0.976118i	$0.569706\pi$
$930$	0	0
$931$	−1.56160	−0.0511794
$932$	0	0
$933$	11.9679	0.391812
$934$	0	0
$935$	−63.9798	−2.09236
$936$	0	0
$937$	−9.21920	−0.301178	−0.150589	−	0.988596i	$-0.548117\pi$
$937$	−9.21920	−0.301178	−0.150589	+	0.988596i	$0.548117\pi$
$938$	0	0
$939$	−18.7808	−0.612888
$940$	0	0
$941$	−7.17209	−0.233804	−0.116902	−	0.993143i	$-0.537296\pi$
$941$	−7.17209	−0.233804	−0.116902	+	0.993143i	$0.537296\pi$
$942$	0	0
$943$	11.0119	0.358597
$944$	0	0
$945$	−7.24120	−0.235556
$946$	0	0
$947$	−22.2397	−0.722692	−0.361346	−	0.932432i	$-0.617683\pi$
$947$	−22.2397	−0.722692	−0.361346	+	0.932432i	$0.617683\pi$
$948$	0	0
$949$	39.6460	1.28696
$950$	0	0
$951$	−4.42676	−0.143548
$952$	0	0
$953$	−9.34757	−0.302797	−0.151399	−	0.988473i	$-0.548378\pi$
$953$	−9.34757	−0.302797	−0.151399	+	0.988473i	$0.548378\pi$
$954$	0	0
$955$	−0.730082	−0.0236249
$956$	0	0
$957$	−40.4824	−1.30861
$958$	0	0
$959$	30.4503	0.983292
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−9.75725	−0.314423
$964$	0	0
$965$	−31.7285	−1.02138
$966$	0	0
$967$	−25.2277	−0.811269	−0.405635	−	0.914035i	$-0.632949\pi$
$967$	−25.2277	−0.811269	−0.405635	+	0.914035i	$0.632949\pi$
$968$	0	0
$969$	1.69307	0.0543891
$970$	0	0
$971$	12.7386	0.408802	0.204401	−	0.978887i	$-0.434475\pi$
$971$	12.7386	0.408802	0.204401	+	0.978887i	$0.434475\pi$
$972$	0	0
$973$	−60.2262	−1.93076
$974$	0	0
$975$	2.41822	0.0774451
$976$	0	0
$977$	38.9764	1.24697	0.623484	−	0.781836i	$-0.285717\pi$
$977$	38.9764	1.24697	0.623484	+	0.781836i	$0.285717\pi$
$978$	0	0
$979$	63.4759	2.02870
$980$	0	0
$981$	5.94436	0.189789
$982$	0	0
$983$	29.9327	0.954706	0.477353	−	0.878712i	$-0.341596\pi$
$983$	29.9327	0.954706	0.477353	+	0.878712i	$0.341596\pi$
$984$	0	0
$985$	−15.9242	−0.507387
$986$	0	0
$987$	11.3069	0.359904
$988$	0	0
$989$	18.1553	0.577305
$990$	0	0
$991$	−62.2311	−1.97684	−0.988418	−	0.151754i	$-0.951508\pi$
$991$	−62.2311	−1.97684	−0.988418	+	0.151754i	$0.951508\pi$
$992$	0	0
$993$	−8.42676	−0.267415
$994$	0	0
$995$	−3.64596	−0.115585
$996$	0	0
$997$	35.0406	1.10975	0.554874	−	0.831934i	$-0.312766\pi$
$997$	35.0406	1.10975	0.554874	+	0.831934i	$0.312766\pi$
$998$	0	0
$999$	6.15347	0.194687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.m.1.2		3
4.3	odd	2		1002.2.a.g.1.2	✓	3
12.11	even	2		3006.2.a.q.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.g.1.2	✓	3	4.3	odd	2
3006.2.a.q.1.2		3	12.11	even	2
8016.2.a.m.1.2		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.15347 q^{5} -3.36258 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.15347 q^{5} -3.36258 q^{7} +1.00000 q^{9} +6.36258 q^{11} -6.66951 q^{13} +2.15347 q^{15} -4.66951 q^{17} -0.362579 q^{19} -3.36258 q^{21} +1.63742 q^{23} -0.362579 q^{25} +1.00000 q^{27} -6.36258 q^{29} +1.00000 q^{31} +6.36258 q^{33} -7.24120 q^{35} +6.15347 q^{37} -6.66951 q^{39} +6.72516 q^{41} +11.0877 q^{43} +2.15347 q^{45} -3.36258 q^{47} +4.30693 q^{49} -4.66951 q^{51} -1.48395 q^{53} +13.7016 q^{55} -0.362579 q^{57} +11.1856 q^{59} +14.0642 q^{61} -3.36258 q^{63} -14.3626 q^{65} -7.18556 q^{67} +1.63742 q^{69} +13.3947 q^{71} -5.94436 q^{73} -0.362579 q^{75} -21.3947 q^{77} +4.72516 q^{79} +1.00000 q^{81} +14.2091 q^{83} -10.0556 q^{85} -6.36258 q^{87} +9.97645 q^{89} +22.4268 q^{91} +1.00000 q^{93} -0.780801 q^{95} +19.1198 q^{97} +6.36258 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 10 q^{11} - 4 q^{13} + 3 q^{15} + 2 q^{17} + 8 q^{19} - q^{21} + 14 q^{23} + 8 q^{25} + 3 q^{27} - 10 q^{29} + 3 q^{31} + 10 q^{33} + 9 q^{35} + 15 q^{37} - 4 q^{39} + 2 q^{41} + 6 q^{43} + 3 q^{45} - q^{47} + 6 q^{49} + 2 q^{51} - 17 q^{53} + 8 q^{57} + 5 q^{59} - 8 q^{61} - q^{63} - 34 q^{65} + 7 q^{67} + 14 q^{69} + 6 q^{71} - 20 q^{73} + 8 q^{75} - 30 q^{77} - 4 q^{79} + 3 q^{81} + 37 q^{83} - 28 q^{85} - 10 q^{87} + 7 q^{89} + 8 q^{91} + 3 q^{93} + 18 q^{95} + 5 q^{97} + 10 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 + 10 * q^11 - 4 * q^13 + 3 * q^15 + 2 * q^17 + 8 * q^19 - q^21 + 14 * q^23 + 8 * q^25 + 3 * q^27 - 10 * q^29 + 3 * q^31 + 10 * q^33 + 9 * q^35 + 15 * q^37 - 4 * q^39 + 2 * q^41 + 6 * q^43 + 3 * q^45 - q^47 + 6 * q^49 + 2 * q^51 - 17 * q^53 + 8 * q^57 + 5 * q^59 - 8 * q^61 - q^63 - 34 * q^65 + 7 * q^67 + 14 * q^69 + 6 * q^71 - 20 * q^73 + 8 * q^75 - 30 * q^77 - 4 * q^79 + 3 * q^81 + 37 * q^83 - 28 * q^85 - 10 * q^87 + 7 * q^89 + 8 * q^91 + 3 * q^93 + 18 * q^95 + 5 * q^97 + 10 * q^99

Introduction

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