LMFDB - Embedded newform 8016.2.a.y.1.5

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 23x^{6} - 3x^{5} + 163x^{4} + 13x^{3} - 418x^{2} + 4x + 269 $ x^8 - 23x^6 - 3x^5 + 163x^4 + 13x^3 - 418x^2 + 4x + 269
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.02139$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.02139 q^{5} +1.14465 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.02139 q^{5} +1.14465 q^{7} +1.00000 q^{9} +1.84687 q^{11} +3.42999 q^{13} +1.02139 q^{15} +4.07277 q^{17} +7.18037 q^{19} +1.14465 q^{21} -2.14465 q^{23} -3.95676 q^{25} +1.00000 q^{27} -2.76331 q^{29} +8.05818 q^{31} +1.84687 q^{33} +1.16913 q^{35} +3.19913 q^{37} +3.42999 q^{39} +7.47616 q^{41} -9.54123 q^{43} +1.02139 q^{45} +3.93722 q^{47} -5.68978 q^{49} +4.07277 q^{51} -4.47200 q^{53} +1.88637 q^{55} +7.18037 q^{57} +11.0694 q^{59} -0.884407 q^{61} +1.14465 q^{63} +3.50335 q^{65} -12.4180 q^{67} -2.14465 q^{69} +0.259610 q^{71} -5.24359 q^{73} -3.95676 q^{75} +2.11402 q^{77} +8.10045 q^{79} +1.00000 q^{81} -3.31054 q^{83} +4.15989 q^{85} -2.76331 q^{87} -1.15863 q^{89} +3.92613 q^{91} +8.05818 q^{93} +7.33396 q^{95} -11.1724 q^{97} +1.84687 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + q^7 + 8 * q^9	$ 8 q + 8 q^{3} + q^{7} + 8 q^{9} + 3 q^{11} - 8 q^{13} - 7 q^{17} + q^{21} - 9 q^{23} + 6 q^{25} + 8 q^{27} + 17 q^{29} + 23 q^{31} + 3 q^{33} + 15 q^{35} + 8 q^{37} - 8 q^{39} - 8 q^{41} + 2 q^{43} + 34 q^{47} + 5 q^{49} - 7 q^{51} + 12 q^{53} + 7 q^{55} + 16 q^{59} - 2 q^{61} + q^{63} - 14 q^{65} - 21 q^{67} - 9 q^{69} + 29 q^{71} - 38 q^{73} + 6 q^{75} + 20 q^{77} + 12 q^{79} + 8 q^{81} + 32 q^{83} + 23 q^{85} + 17 q^{87} + 11 q^{89} + 5 q^{91} + 23 q^{93} + 67 q^{95} + 8 q^{97} + 3 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + q^7 + 8 * q^9 + 3 * q^11 - 8 * q^13 - 7 * q^17 + q^21 - 9 * q^23 + 6 * q^25 + 8 * q^27 + 17 * q^29 + 23 * q^31 + 3 * q^33 + 15 * q^35 + 8 * q^37 - 8 * q^39 - 8 * q^41 + 2 * q^43 + 34 * q^47 + 5 * q^49 - 7 * q^51 + 12 * q^53 + 7 * q^55 + 16 * q^59 - 2 * q^61 + q^63 - 14 * q^65 - 21 * q^67 - 9 * q^69 + 29 * q^71 - 38 * q^73 + 6 * q^75 + 20 * q^77 + 12 * q^79 + 8 * q^81 + 32 * q^83 + 23 * q^85 + 17 * q^87 + 11 * q^89 + 5 * q^91 + 23 * q^93 + 67 * q^95 + 8 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.02139	0.456779	0.228390	−	0.973570i	$-0.426654\pi$
$5$	1.02139	0.456779	0.228390	+	0.973570i	$0.426654\pi$
$6$	0	0
$7$	1.14465	0.432637	0.216318	−	0.976323i	$-0.430595\pi$
$7$	1.14465	0.432637	0.216318	+	0.976323i	$0.430595\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.84687	0.556852	0.278426	−	0.960458i	$-0.410187\pi$
$11$	1.84687	0.556852	0.278426	+	0.960458i	$0.410187\pi$
$12$	0	0
$13$	3.42999	0.951307	0.475653	−	0.879633i	$-0.342212\pi$
$13$	3.42999	0.951307	0.475653	+	0.879633i	$0.342212\pi$
$14$	0	0
$15$	1.02139	0.263722
$16$	0	0
$17$	4.07277	0.987792	0.493896	−	0.869521i	$-0.335572\pi$
$17$	4.07277	0.987792	0.493896	+	0.869521i	$0.335572\pi$
$18$	0	0
$19$	7.18037	1.64729	0.823645	−	0.567105i	$-0.191937\pi$
$19$	7.18037	1.64729	0.823645	+	0.567105i	$0.191937\pi$
$20$	0	0
$21$	1.14465	0.249783
$22$	0	0
$23$	−2.14465	−0.447190	−0.223595	−	0.974682i	$-0.571779\pi$
$23$	−2.14465	−0.447190	−0.223595	+	0.974682i	$0.571779\pi$
$24$	0	0
$25$	−3.95676	−0.791353
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.76331	−0.513133	−0.256567	−	0.966527i	$-0.582591\pi$
$29$	−2.76331	−0.513133	−0.256567	+	0.966527i	$0.582591\pi$
$30$	0	0
$31$	8.05818	1.44729	0.723646	−	0.690171i	$-0.242465\pi$
$31$	8.05818	1.44729	0.723646	+	0.690171i	$0.242465\pi$
$32$	0	0
$33$	1.84687	0.321499
$34$	0	0
$35$	1.16913	0.197620
$36$	0	0
$37$	3.19913	0.525933	0.262967	−	0.964805i	$-0.415299\pi$
$37$	3.19913	0.525933	0.262967	+	0.964805i	$0.415299\pi$
$38$	0	0
$39$	3.42999	0.549237
$40$	0	0
$41$	7.47616	1.16758	0.583790	−	0.811905i	$-0.301569\pi$
$41$	7.47616	1.16758	0.583790	+	0.811905i	$0.301569\pi$
$42$	0	0
$43$	−9.54123	−1.45502	−0.727512	−	0.686095i	$-0.759323\pi$
$43$	−9.54123	−1.45502	−0.727512	+	0.686095i	$0.759323\pi$
$44$	0	0
$45$	1.02139	0.152260
$46$	0	0
$47$	3.93722	0.574302	0.287151	−	0.957885i	$-0.407292\pi$
$47$	3.93722	0.574302	0.287151	+	0.957885i	$0.407292\pi$
$48$	0	0
$49$	−5.68978	−0.812825
$50$	0	0
$51$	4.07277	0.570302
$52$	0	0
$53$	−4.47200	−0.614277	−0.307138	−	0.951665i	$-0.599371\pi$
$53$	−4.47200	−0.614277	−0.307138	+	0.951665i	$0.599371\pi$
$54$	0	0
$55$	1.88637	0.254359
$56$	0	0
$57$	7.18037	0.951064
$58$	0	0
$59$	11.0694	1.44111	0.720557	−	0.693395i	$-0.243886\pi$
$59$	11.0694	1.44111	0.720557	+	0.693395i	$0.243886\pi$
$60$	0	0
$61$	−0.884407	−0.113237	−0.0566183	−	0.998396i	$-0.518032\pi$
$61$	−0.884407	−0.113237	−0.0566183	+	0.998396i	$0.518032\pi$
$62$	0	0
$63$	1.14465	0.144212
$64$	0	0
$65$	3.50335	0.434537
$66$	0	0
$67$	−12.4180	−1.51710	−0.758548	−	0.651617i	$-0.774091\pi$
$67$	−12.4180	−1.51710	−0.758548	+	0.651617i	$0.774091\pi$
$68$	0	0
$69$	−2.14465	−0.258185
$70$	0	0
$71$	0.259610	0.0308101	0.0154050	−	0.999881i	$-0.495096\pi$
$71$	0.259610	0.0308101	0.0154050	+	0.999881i	$0.495096\pi$
$72$	0	0
$73$	−5.24359	−0.613716	−0.306858	−	0.951755i	$-0.599278\pi$
$73$	−5.24359	−0.613716	−0.306858	+	0.951755i	$0.599278\pi$
$74$	0	0
$75$	−3.95676	−0.456888
$76$	0	0
$77$	2.11402	0.240915
$78$	0	0
$79$	8.10045	0.911372	0.455686	−	0.890141i	$-0.349394\pi$
$79$	8.10045	0.911372	0.455686	+	0.890141i	$0.349394\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.31054	−0.363378	−0.181689	−	0.983356i	$-0.558156\pi$
$83$	−3.31054	−0.363378	−0.181689	+	0.983356i	$0.558156\pi$
$84$	0	0
$85$	4.15989	0.451203
$86$	0	0
$87$	−2.76331	−0.296258
$88$	0	0
$89$	−1.15863	−0.122814	−0.0614070	−	0.998113i	$-0.519559\pi$
$89$	−1.15863	−0.122814	−0.0614070	+	0.998113i	$0.519559\pi$
$90$	0	0
$91$	3.92613	0.411570
$92$	0	0
$93$	8.05818	0.835594
$94$	0	0
$95$	7.33396	0.752449
$96$	0	0
$97$	−11.1724	−1.13438	−0.567190	−	0.823587i	$-0.691970\pi$
$97$	−11.1724	−1.13438	−0.567190	+	0.823587i	$0.691970\pi$
$98$	0	0
$99$	1.84687	0.185617
$100$	0	0
$101$	2.80841	0.279447	0.139724	−	0.990191i	$-0.455379\pi$
$101$	2.80841	0.279447	0.139724	+	0.990191i	$0.455379\pi$
$102$	0	0
$103$	−13.7122	−1.35111	−0.675553	−	0.737311i	$-0.736095\pi$
$103$	−13.7122	−1.35111	−0.675553	+	0.737311i	$0.736095\pi$
$104$	0	0
$105$	1.16913	0.114096
$106$	0	0
$107$	12.5037	1.20878	0.604389	−	0.796689i	$-0.293417\pi$
$107$	12.5037	1.20878	0.604389	+	0.796689i	$0.293417\pi$
$108$	0	0
$109$	14.4292	1.38207	0.691033	−	0.722823i	$-0.257156\pi$
$109$	14.4292	1.38207	0.691033	+	0.722823i	$0.257156\pi$
$110$	0	0
$111$	3.19913	0.303648
$112$	0	0
$113$	−7.76624	−0.730586	−0.365293	−	0.930893i	$-0.619031\pi$
$113$	−7.76624	−0.730586	−0.365293	+	0.930893i	$0.619031\pi$
$114$	0	0
$115$	−2.19052	−0.204267
$116$	0	0
$117$	3.42999	0.317102
$118$	0	0
$119$	4.66190	0.427355
$120$	0	0
$121$	−7.58907	−0.689916
$122$	0	0
$123$	7.47616	0.674102
$124$	0	0
$125$	−9.14835	−0.818253
$126$	0	0
$127$	−9.97429	−0.885075	−0.442537	−	0.896750i	$-0.645922\pi$
$127$	−9.97429	−0.885075	−0.442537	+	0.896750i	$0.645922\pi$
$128$	0	0
$129$	−9.54123	−0.840059
$130$	0	0
$131$	0.243431	0.0212687	0.0106343	−	0.999943i	$-0.496615\pi$
$131$	0.243431	0.0212687	0.0106343	+	0.999943i	$0.496615\pi$
$132$	0	0
$133$	8.21901	0.712679
$134$	0	0
$135$	1.02139	0.0879073
$136$	0	0
$137$	−18.4284	−1.57445	−0.787223	−	0.616669i	$-0.788482\pi$
$137$	−18.4284	−1.57445	−0.787223	+	0.616669i	$0.788482\pi$
$138$	0	0
$139$	19.8687	1.68525	0.842623	−	0.538505i	$-0.181011\pi$
$139$	19.8687	1.68525	0.842623	+	0.538505i	$0.181011\pi$
$140$	0	0
$141$	3.93722	0.331574
$142$	0	0
$143$	6.33474	0.529737
$144$	0	0
$145$	−2.82242	−0.234389
$146$	0	0
$147$	−5.68978	−0.469285
$148$	0	0
$149$	3.51817	0.288220	0.144110	−	0.989562i	$-0.453968\pi$
$149$	3.51817	0.288220	0.144110	+	0.989562i	$0.453968\pi$
$150$	0	0
$151$	11.1920	0.910789	0.455395	−	0.890290i	$-0.349498\pi$
$151$	11.1920	0.910789	0.455395	+	0.890290i	$0.349498\pi$
$152$	0	0
$153$	4.07277	0.329264
$154$	0	0
$155$	8.23055	0.661093
$156$	0	0
$157$	−13.1470	−1.04924	−0.524621	−	0.851336i	$-0.675793\pi$
$157$	−13.1470	−1.04924	−0.524621	+	0.851336i	$0.675793\pi$
$158$	0	0
$159$	−4.47200	−0.354653
$160$	0	0
$161$	−2.45487	−0.193471
$162$	0	0
$163$	−11.7308	−0.918830	−0.459415	−	0.888222i	$-0.651941\pi$
$163$	−11.7308	−0.918830	−0.459415	+	0.888222i	$0.651941\pi$
$164$	0	0
$165$	1.88637	0.146854
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−1.23520	−0.0950151
$170$	0	0
$171$	7.18037	0.549097
$172$	0	0
$173$	−14.1694	−1.07728	−0.538639	−	0.842536i	$-0.681061\pi$
$173$	−14.1694	−1.07728	−0.538639	+	0.842536i	$0.681061\pi$
$174$	0	0
$175$	−4.52911	−0.342368
$176$	0	0
$177$	11.0694	0.832028
$178$	0	0
$179$	26.3136	1.96677	0.983386	−	0.181528i	$-0.0581044\pi$
$179$	26.3136	1.96677	0.983386	+	0.181528i	$0.0581044\pi$
$180$	0	0
$181$	−9.03443	−0.671524	−0.335762	−	0.941947i	$-0.608994\pi$
$181$	−9.03443	−0.671524	−0.335762	+	0.941947i	$0.608994\pi$
$182$	0	0
$183$	−0.884407	−0.0653772
$184$	0	0
$185$	3.26755	0.240235
$186$	0	0
$187$	7.52188	0.550054
$188$	0	0
$189$	1.14465	0.0832610
$190$	0	0
$191$	−21.2440	−1.53716	−0.768582	−	0.639751i	$-0.779037\pi$
$191$	−21.2440	−1.53716	−0.768582	+	0.639751i	$0.779037\pi$
$192$	0	0
$193$	6.40529	0.461063	0.230532	−	0.973065i	$-0.425954\pi$
$193$	6.40529	0.461063	0.230532	+	0.973065i	$0.425954\pi$
$194$	0	0
$195$	3.50335	0.250880
$196$	0	0
$197$	−11.1588	−0.795031	−0.397515	−	0.917595i	$-0.630127\pi$
$197$	−11.1588	−0.795031	−0.397515	+	0.917595i	$0.630127\pi$
$198$	0	0
$199$	16.3274	1.15742	0.578708	−	0.815535i	$-0.303557\pi$
$199$	16.3274	1.15742	0.578708	+	0.815535i	$0.303557\pi$
$200$	0	0
$201$	−12.4180	−0.875896
$202$	0	0
$203$	−3.16302	−0.222000
$204$	0	0
$205$	7.63607	0.533326
$206$	0	0
$207$	−2.14465	−0.149063
$208$	0	0
$209$	13.2612	0.917297
$210$	0	0
$211$	−4.82804	−0.332376	−0.166188	−	0.986094i	$-0.553146\pi$
$211$	−4.82804	−0.332376	−0.166188	+	0.986094i	$0.553146\pi$
$212$	0	0
$213$	0.259610	0.0177882
$214$	0	0
$215$	−9.74532	−0.664625
$216$	0	0
$217$	9.22379	0.626152
$218$	0	0
$219$	−5.24359	−0.354329
$220$	0	0
$221$	13.9696	0.939694
$222$	0	0
$223$	14.7337	0.986645	0.493322	−	0.869847i	$-0.335782\pi$
$223$	14.7337	0.986645	0.493322	+	0.869847i	$0.335782\pi$
$224$	0	0
$225$	−3.95676	−0.263784
$226$	0	0
$227$	−21.8923	−1.45304	−0.726520	−	0.687145i	$-0.758864\pi$
$227$	−21.8923	−1.45304	−0.726520	+	0.687145i	$0.758864\pi$
$228$	0	0
$229$	18.5361	1.22490	0.612451	−	0.790509i	$-0.290184\pi$
$229$	18.5361	1.22490	0.612451	+	0.790509i	$0.290184\pi$
$230$	0	0
$231$	2.11402	0.139092
$232$	0	0
$233$	13.9469	0.913694	0.456847	−	0.889545i	$-0.348979\pi$
$233$	13.9469	0.913694	0.456847	+	0.889545i	$0.348979\pi$
$234$	0	0
$235$	4.02144	0.262330
$236$	0	0
$237$	8.10045	0.526181
$238$	0	0
$239$	11.5048	0.744182	0.372091	−	0.928196i	$-0.378641\pi$
$239$	11.5048	0.744182	0.372091	+	0.928196i	$0.378641\pi$
$240$	0	0
$241$	−12.0664	−0.777264	−0.388632	−	0.921393i	$-0.627052\pi$
$241$	−12.0664	−0.777264	−0.388632	+	0.921393i	$0.627052\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−5.81148	−0.371282
$246$	0	0
$247$	24.6286	1.56708
$248$	0	0
$249$	−3.31054	−0.209797
$250$	0	0
$251$	7.60035	0.479730	0.239865	−	0.970806i	$-0.422897\pi$
$251$	7.60035	0.479730	0.239865	+	0.970806i	$0.422897\pi$
$252$	0	0
$253$	−3.96089	−0.249019
$254$	0	0
$255$	4.15989	0.260502
$256$	0	0
$257$	10.0637	0.627754	0.313877	−	0.949464i	$-0.398372\pi$
$257$	10.0637	0.627754	0.313877	+	0.949464i	$0.398372\pi$
$258$	0	0
$259$	3.66188	0.227538
$260$	0	0
$261$	−2.76331	−0.171044
$262$	0	0
$263$	29.0844	1.79342	0.896712	−	0.442615i	$-0.145949\pi$
$263$	29.0844	1.79342	0.896712	+	0.442615i	$0.145949\pi$
$264$	0	0
$265$	−4.56766	−0.280589
$266$	0	0
$267$	−1.15863	−0.0709068
$268$	0	0
$269$	−11.1474	−0.679669	−0.339835	−	0.940485i	$-0.610371\pi$
$269$	−11.1474	−0.679669	−0.339835	+	0.940485i	$0.610371\pi$
$270$	0	0
$271$	1.24441	0.0755928	0.0377964	−	0.999285i	$-0.487966\pi$
$271$	1.24441	0.0755928	0.0377964	+	0.999285i	$0.487966\pi$
$272$	0	0
$273$	3.92613	0.237620
$274$	0	0
$275$	−7.30762	−0.440666
$276$	0	0
$277$	10.5406	0.633321	0.316660	−	0.948539i	$-0.397438\pi$
$277$	10.5406	0.633321	0.316660	+	0.948539i	$0.397438\pi$
$278$	0	0
$279$	8.05818	0.482431
$280$	0	0
$281$	−21.9069	−1.30686	−0.653428	−	0.756989i	$-0.726670\pi$
$281$	−21.9069	−1.30686	−0.653428	+	0.756989i	$0.726670\pi$
$282$	0	0
$283$	−1.02635	−0.0610100	−0.0305050	−	0.999535i	$-0.509712\pi$
$283$	−1.02635	−0.0610100	−0.0305050	+	0.999535i	$0.509712\pi$
$284$	0	0
$285$	7.33396	0.434426
$286$	0	0
$287$	8.55758	0.505138
$288$	0	0
$289$	−0.412526	−0.0242662
$290$	0	0
$291$	−11.1724	−0.654935
$292$	0	0
$293$	19.5305	1.14098	0.570491	−	0.821304i	$-0.306753\pi$
$293$	19.5305	1.14098	0.570491	+	0.821304i	$0.306753\pi$
$294$	0	0
$295$	11.3062	0.658272
$296$	0	0
$297$	1.84687	0.107166
$298$	0	0
$299$	−7.35612	−0.425415
$300$	0	0
$301$	−10.9214	−0.629497
$302$	0	0
$303$	2.80841	0.161339
$304$	0	0
$305$	−0.903324	−0.0517242
$306$	0	0
$307$	4.61461	0.263370	0.131685	−	0.991292i	$-0.457961\pi$
$307$	4.61461	0.263370	0.131685	+	0.991292i	$0.457961\pi$
$308$	0	0
$309$	−13.7122	−0.780062
$310$	0	0
$311$	22.9291	1.30019	0.650094	−	0.759854i	$-0.274729\pi$
$311$	22.9291	1.30019	0.650094	+	0.759854i	$0.274729\pi$
$312$	0	0
$313$	−3.67954	−0.207980	−0.103990	−	0.994578i	$-0.533161\pi$
$313$	−3.67954	−0.207980	−0.103990	+	0.994578i	$0.533161\pi$
$314$	0	0
$315$	1.16913	0.0658732
$316$	0	0
$317$	−28.2921	−1.58904	−0.794522	−	0.607236i	$-0.792278\pi$
$317$	−28.2921	−1.58904	−0.794522	+	0.607236i	$0.792278\pi$
$318$	0	0
$319$	−5.10347	−0.285739
$320$	0	0
$321$	12.5037	0.697888
$322$	0	0
$323$	29.2440	1.62718
$324$	0	0
$325$	−13.5716	−0.752819
$326$	0	0
$327$	14.4292	0.797936
$328$	0	0
$329$	4.50673	0.248464
$330$	0	0
$331$	−13.2253	−0.726926	−0.363463	−	0.931609i	$-0.618406\pi$
$331$	−13.2253	−0.726926	−0.363463	+	0.931609i	$0.618406\pi$
$332$	0	0
$333$	3.19913	0.175311
$334$	0	0
$335$	−12.6836	−0.692979
$336$	0	0
$337$	25.8742	1.40946	0.704729	−	0.709476i	$-0.251068\pi$
$337$	25.8742	1.40946	0.704729	+	0.709476i	$0.251068\pi$
$338$	0	0
$339$	−7.76624	−0.421804
$340$	0	0
$341$	14.8824	0.805928
$342$	0	0
$343$	−14.5253	−0.784295
$344$	0	0
$345$	−2.19052	−0.117934
$346$	0	0
$347$	14.0028	0.751708	0.375854	−	0.926679i	$-0.377349\pi$
$347$	14.0028	0.751708	0.375854	+	0.926679i	$0.377349\pi$
$348$	0	0
$349$	4.56330	0.244268	0.122134	−	0.992514i	$-0.461026\pi$
$349$	4.56330	0.244268	0.122134	+	0.992514i	$0.461026\pi$
$350$	0	0
$351$	3.42999	0.183079
$352$	0	0
$353$	−6.71565	−0.357438	−0.178719	−	0.983900i	$-0.557195\pi$
$353$	−6.71565	−0.357438	−0.178719	+	0.983900i	$0.557195\pi$
$354$	0	0
$355$	0.265164	0.0140734
$356$	0	0
$357$	4.66190	0.246734
$358$	0	0
$359$	18.4224	0.972297	0.486149	−	0.873876i	$-0.338401\pi$
$359$	18.4224	0.972297	0.486149	+	0.873876i	$0.338401\pi$
$360$	0	0
$361$	32.5578	1.71357
$362$	0	0
$363$	−7.58907	−0.398323
$364$	0	0
$365$	−5.35575	−0.280333
$366$	0	0
$367$	−27.3601	−1.42819	−0.714093	−	0.700051i	$-0.753161\pi$
$367$	−27.3601	−1.42819	−0.714093	+	0.700051i	$0.753161\pi$
$368$	0	0
$369$	7.47616	0.389193
$370$	0	0
$371$	−5.11887	−0.265759
$372$	0	0
$373$	−27.7644	−1.43759	−0.718794	−	0.695223i	$-0.755305\pi$
$373$	−27.7644	−1.43759	−0.718794	+	0.695223i	$0.755305\pi$
$374$	0	0
$375$	−9.14835	−0.472419
$376$	0	0
$377$	−9.47811	−0.488147
$378$	0	0
$379$	30.5040	1.56689	0.783443	−	0.621464i	$-0.213462\pi$
$379$	30.5040	1.56689	0.783443	+	0.621464i	$0.213462\pi$
$380$	0	0
$381$	−9.97429	−0.510998
$382$	0	0
$383$	0.396471	0.0202587	0.0101294	−	0.999949i	$-0.496776\pi$
$383$	0.396471	0.0202587	0.0101294	+	0.999949i	$0.496776\pi$
$384$	0	0
$385$	2.15924	0.110045
$386$	0	0
$387$	−9.54123	−0.485008
$388$	0	0
$389$	30.8541	1.56437	0.782183	−	0.623049i	$-0.214106\pi$
$389$	30.8541	1.56437	0.782183	+	0.623049i	$0.214106\pi$
$390$	0	0
$391$	−8.73467	−0.441731
$392$	0	0
$393$	0.243431	0.0122795
$394$	0	0
$395$	8.27372	0.416296
$396$	0	0
$397$	−1.60516	−0.0805609	−0.0402804	−	0.999188i	$-0.512825\pi$
$397$	−1.60516	−0.0805609	−0.0402804	+	0.999188i	$0.512825\pi$
$398$	0	0
$399$	8.21901	0.411465
$400$	0	0
$401$	−1.14027	−0.0569424	−0.0284712	−	0.999595i	$-0.509064\pi$
$401$	−1.14027	−0.0569424	−0.0284712	+	0.999595i	$0.509064\pi$
$402$	0	0
$403$	27.6394	1.37682
$404$	0	0
$405$	1.02139	0.0507533
$406$	0	0
$407$	5.90837	0.292867
$408$	0	0
$409$	1.11165	0.0549675	0.0274837	−	0.999622i	$-0.491251\pi$
$409$	1.11165	0.0549675	0.0274837	+	0.999622i	$0.491251\pi$
$410$	0	0
$411$	−18.4284	−0.909006
$412$	0	0
$413$	12.6706	0.623479
$414$	0	0
$415$	−3.38135	−0.165984
$416$	0	0
$417$	19.8687	0.972977
$418$	0	0
$419$	−30.3831	−1.48431	−0.742155	−	0.670228i	$-0.766196\pi$
$419$	−30.3831	−1.48431	−0.742155	+	0.670228i	$0.766196\pi$
$420$	0	0
$421$	1.60031	0.0779941	0.0389971	−	0.999239i	$-0.487584\pi$
$421$	1.60031	0.0779941	0.0389971	+	0.999239i	$0.487584\pi$
$422$	0	0
$423$	3.93722	0.191434
$424$	0	0
$425$	−16.1150	−0.781692
$426$	0	0
$427$	−1.01234	−0.0489904
$428$	0	0
$429$	6.33474	0.305844
$430$	0	0
$431$	19.3590	0.932490	0.466245	−	0.884656i	$-0.345606\pi$
$431$	19.3590	0.932490	0.466245	+	0.884656i	$0.345606\pi$
$432$	0	0
$433$	−26.1462	−1.25651	−0.628253	−	0.778009i	$-0.716230\pi$
$433$	−26.1462	−1.25651	−0.628253	+	0.778009i	$0.716230\pi$
$434$	0	0
$435$	−2.82242	−0.135324
$436$	0	0
$437$	−15.3994	−0.736652
$438$	0	0
$439$	−27.4889	−1.31197	−0.655986	−	0.754773i	$-0.727747\pi$
$439$	−27.4889	−1.31197	−0.655986	+	0.754773i	$0.727747\pi$
$440$	0	0
$441$	−5.68978	−0.270942
$442$	0	0
$443$	29.3214	1.39310	0.696551	−	0.717508i	$-0.254717\pi$
$443$	29.3214	1.39310	0.696551	+	0.717508i	$0.254717\pi$
$444$	0	0
$445$	−1.18341	−0.0560990
$446$	0	0
$447$	3.51817	0.166404
$448$	0	0
$449$	5.86233	0.276661	0.138330	−	0.990386i	$-0.455826\pi$
$449$	5.86233	0.276661	0.138330	+	0.990386i	$0.455826\pi$
$450$	0	0
$451$	13.8075	0.650169
$452$	0	0
$453$	11.1920	0.525844
$454$	0	0
$455$	4.01011	0.187997
$456$	0	0
$457$	−7.49999	−0.350835	−0.175417	−	0.984494i	$-0.556127\pi$
$457$	−7.49999	−0.350835	−0.175417	+	0.984494i	$0.556127\pi$
$458$	0	0
$459$	4.07277	0.190101
$460$	0	0
$461$	−20.1096	−0.936598	−0.468299	−	0.883570i	$-0.655133\pi$
$461$	−20.1096	−0.936598	−0.468299	+	0.883570i	$0.655133\pi$
$462$	0	0
$463$	−11.2158	−0.521242	−0.260621	−	0.965441i	$-0.583927\pi$
$463$	−11.2158	−0.521242	−0.260621	+	0.965441i	$0.583927\pi$
$464$	0	0
$465$	8.23055	0.381682
$466$	0	0
$467$	22.4001	1.03655	0.518277	−	0.855213i	$-0.326574\pi$
$467$	22.4001	1.03655	0.518277	+	0.855213i	$0.326574\pi$
$468$	0	0
$469$	−14.2142	−0.656352
$470$	0	0
$471$	−13.1470	−0.605780
$472$	0	0
$473$	−17.6214	−0.810233
$474$	0	0
$475$	−28.4110	−1.30359
$476$	0	0
$477$	−4.47200	−0.204759
$478$	0	0
$479$	34.4175	1.57258	0.786288	−	0.617860i	$-0.212000\pi$
$479$	34.4175	1.57258	0.786288	+	0.617860i	$0.212000\pi$
$480$	0	0
$481$	10.9730	0.500324
$482$	0	0
$483$	−2.45487	−0.111701
$484$	0	0
$485$	−11.4113	−0.518162
$486$	0	0
$487$	−2.86151	−0.129667	−0.0648336	−	0.997896i	$-0.520652\pi$
$487$	−2.86151	−0.129667	−0.0648336	+	0.997896i	$0.520652\pi$
$488$	0	0
$489$	−11.7308	−0.530487
$490$	0	0
$491$	−9.65978	−0.435940	−0.217970	−	0.975956i	$-0.569943\pi$
$491$	−9.65978	−0.435940	−0.217970	+	0.975956i	$0.569943\pi$
$492$	0	0
$493$	−11.2543	−0.506869
$494$	0	0
$495$	1.88637	0.0847862
$496$	0	0
$497$	0.297163	0.0133296
$498$	0	0
$499$	−37.5211	−1.67968	−0.839838	−	0.542838i	$-0.817350\pi$
$499$	−37.5211	−1.67968	−0.839838	+	0.542838i	$0.817350\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	18.7105	0.834261	0.417131	−	0.908847i	$-0.363036\pi$
$503$	18.7105	0.834261	0.417131	+	0.908847i	$0.363036\pi$
$504$	0	0
$505$	2.86848	0.127646
$506$	0	0
$507$	−1.23520	−0.0548570
$508$	0	0
$509$	−18.4465	−0.817629	−0.408814	−	0.912618i	$-0.634058\pi$
$509$	−18.4465	−0.817629	−0.408814	+	0.912618i	$0.634058\pi$
$510$	0	0
$511$	−6.00207	−0.265516
$512$	0	0
$513$	7.18037	0.317021
$514$	0	0
$515$	−14.0055	−0.617158
$516$	0	0
$517$	7.27153	0.319801
$518$	0	0
$519$	−14.1694	−0.621967
$520$	0	0
$521$	−41.2554	−1.80743	−0.903717	−	0.428131i	$-0.859172\pi$
$521$	−41.2554	−1.80743	−0.903717	+	0.428131i	$0.859172\pi$
$522$	0	0
$523$	−23.7674	−1.03928	−0.519639	−	0.854386i	$-0.673933\pi$
$523$	−23.7674	−1.03928	−0.519639	+	0.854386i	$0.673933\pi$
$524$	0	0
$525$	−4.52911	−0.197666
$526$	0	0
$527$	32.8191	1.42962
$528$	0	0
$529$	−18.4005	−0.800021
$530$	0	0
$531$	11.0694	0.480372
$532$	0	0
$533$	25.6431	1.11073
$534$	0	0
$535$	12.7712	0.552145
$536$	0	0
$537$	26.3136	1.13552
$538$	0	0
$539$	−10.5083	−0.452624
$540$	0	0
$541$	17.2077	0.739816	0.369908	−	0.929068i	$-0.379389\pi$
$541$	17.2077	0.739816	0.369908	+	0.929068i	$0.379389\pi$
$542$	0	0
$543$	−9.03443	−0.387705
$544$	0	0
$545$	14.7378	0.631299
$546$	0	0
$547$	−15.0682	−0.644271	−0.322136	−	0.946694i	$-0.604401\pi$
$547$	−15.0682	−0.644271	−0.322136	+	0.946694i	$0.604401\pi$
$548$	0	0
$549$	−0.884407	−0.0377456
$550$	0	0
$551$	−19.8416	−0.845280
$552$	0	0
$553$	9.27217	0.394293
$554$	0	0
$555$	3.26755	0.138700
$556$	0	0
$557$	15.0016	0.635637	0.317819	−	0.948151i	$-0.397050\pi$
$557$	15.0016	0.635637	0.317819	+	0.948151i	$0.397050\pi$
$558$	0	0
$559$	−32.7263	−1.38417
$560$	0	0
$561$	7.52188	0.317574
$562$	0	0
$563$	27.9861	1.17947	0.589737	−	0.807595i	$-0.299231\pi$
$563$	27.9861	1.17947	0.589737	+	0.807595i	$0.299231\pi$
$564$	0	0
$565$	−7.93236	−0.333717
$566$	0	0
$567$	1.14465	0.0480708
$568$	0	0
$569$	−5.78081	−0.242344	−0.121172	−	0.992632i	$-0.538665\pi$
$569$	−5.78081	−0.242344	−0.121172	+	0.992632i	$0.538665\pi$
$570$	0	0
$571$	−26.4720	−1.10782	−0.553909	−	0.832577i	$-0.686864\pi$
$571$	−26.4720	−1.10782	−0.553909	+	0.832577i	$0.686864\pi$
$572$	0	0
$573$	−21.2440	−0.887482
$574$	0	0
$575$	8.48587	0.353885
$576$	0	0
$577$	−16.8345	−0.700831	−0.350416	−	0.936594i	$-0.613960\pi$
$577$	−16.8345	−0.700831	−0.350416	+	0.936594i	$0.613960\pi$
$578$	0	0
$579$	6.40529	0.266195
$580$	0	0
$581$	−3.78940	−0.157211
$582$	0	0
$583$	−8.25921	−0.342061
$584$	0	0
$585$	3.50335	0.144846
$586$	0	0
$587$	−9.97410	−0.411675	−0.205838	−	0.978586i	$-0.565992\pi$
$587$	−9.97410	−0.411675	−0.205838	+	0.978586i	$0.565992\pi$
$588$	0	0
$589$	57.8607	2.38411
$590$	0	0
$591$	−11.1588	−0.459011
$592$	0	0
$593$	26.9211	1.10552	0.552758	−	0.833342i	$-0.313575\pi$
$593$	26.9211	1.10552	0.552758	+	0.833342i	$0.313575\pi$
$594$	0	0
$595$	4.76161	0.195207
$596$	0	0
$597$	16.3274	0.668234
$598$	0	0
$599$	4.60938	0.188334	0.0941670	−	0.995556i	$-0.469981\pi$
$599$	4.60938	0.188334	0.0941670	+	0.995556i	$0.469981\pi$
$600$	0	0
$601$	42.4236	1.73050	0.865248	−	0.501344i	$-0.167161\pi$
$601$	42.4236	1.73050	0.865248	+	0.501344i	$0.167161\pi$
$602$	0	0
$603$	−12.4180	−0.505699
$604$	0	0
$605$	−7.75140	−0.315139
$606$	0	0
$607$	−25.3492	−1.02889	−0.514445	−	0.857523i	$-0.672002\pi$
$607$	−25.3492	−1.02889	−0.514445	+	0.857523i	$0.672002\pi$
$608$	0	0
$609$	−3.16302	−0.128172
$610$	0	0
$611$	13.5046	0.546338
$612$	0	0
$613$	−3.55644	−0.143643	−0.0718215	−	0.997417i	$-0.522881\pi$
$613$	−3.55644	−0.143643	−0.0718215	+	0.997417i	$0.522881\pi$
$614$	0	0
$615$	7.63607	0.307916
$616$	0	0
$617$	9.54315	0.384193	0.192096	−	0.981376i	$-0.438471\pi$
$617$	9.54315	0.384193	0.192096	+	0.981376i	$0.438471\pi$
$618$	0	0
$619$	−7.05947	−0.283744	−0.141872	−	0.989885i	$-0.545312\pi$
$619$	−7.05947	−0.283744	−0.141872	+	0.989885i	$0.545312\pi$
$620$	0	0
$621$	−2.14465	−0.0860618
$622$	0	0
$623$	−1.32622	−0.0531339
$624$	0	0
$625$	10.4398	0.417591
$626$	0	0
$627$	13.2612	0.529602
$628$	0	0
$629$	13.0293	0.519513
$630$	0	0
$631$	37.6316	1.49809	0.749046	−	0.662518i	$-0.230512\pi$
$631$	37.6316	1.49809	0.749046	+	0.662518i	$0.230512\pi$
$632$	0	0
$633$	−4.82804	−0.191897
$634$	0	0
$635$	−10.1876	−0.404284
$636$	0	0
$637$	−19.5159	−0.773246
$638$	0	0
$639$	0.259610	0.0102700
$640$	0	0
$641$	29.6513	1.17115	0.585577	−	0.810617i	$-0.300868\pi$
$641$	29.6513	1.17115	0.585577	+	0.810617i	$0.300868\pi$
$642$	0	0
$643$	−38.2028	−1.50657	−0.753286	−	0.657693i	$-0.771533\pi$
$643$	−38.2028	−1.50657	−0.753286	+	0.657693i	$0.771533\pi$
$644$	0	0
$645$	−9.74532	−0.383722
$646$	0	0
$647$	−10.2800	−0.404149	−0.202075	−	0.979370i	$-0.564768\pi$
$647$	−10.2800	−0.404149	−0.202075	+	0.979370i	$0.564768\pi$
$648$	0	0
$649$	20.4438	0.802488
$650$	0	0
$651$	9.22379	0.361509
$652$	0	0
$653$	2.64514	0.103512	0.0517562	−	0.998660i	$-0.483518\pi$
$653$	2.64514	0.103512	0.0517562	+	0.998660i	$0.483518\pi$
$654$	0	0
$655$	0.248638	0.00971508
$656$	0	0
$657$	−5.24359	−0.204572
$658$	0	0
$659$	−12.8130	−0.499122	−0.249561	−	0.968359i	$-0.580286\pi$
$659$	−12.8130	−0.499122	−0.249561	+	0.968359i	$0.580286\pi$
$660$	0	0
$661$	36.3400	1.41346	0.706731	−	0.707482i	$-0.250169\pi$
$661$	36.3400	1.41346	0.706731	+	0.707482i	$0.250169\pi$
$662$	0	0
$663$	13.9696	0.542532
$664$	0	0
$665$	8.39481	0.325537
$666$	0	0
$667$	5.92633	0.229468
$668$	0	0
$669$	14.7337	0.569640
$670$	0	0
$671$	−1.63338	−0.0630561
$672$	0	0
$673$	27.9291	1.07659	0.538293	−	0.842758i	$-0.319069\pi$
$673$	27.9291	1.07659	0.538293	+	0.842758i	$0.319069\pi$
$674$	0	0
$675$	−3.95676	−0.152296
$676$	0	0
$677$	−10.7492	−0.413125	−0.206562	−	0.978433i	$-0.566228\pi$
$677$	−10.7492	−0.413125	−0.206562	+	0.978433i	$0.566228\pi$
$678$	0	0
$679$	−12.7884	−0.490775
$680$	0	0
$681$	−21.8923	−0.838913
$682$	0	0
$683$	3.30814	0.126582	0.0632912	−	0.997995i	$-0.479840\pi$
$683$	3.30814	0.126582	0.0632912	+	0.997995i	$0.479840\pi$
$684$	0	0
$685$	−18.8226	−0.719174
$686$	0	0
$687$	18.5361	0.707197
$688$	0	0
$689$	−15.3389	−0.584366
$690$	0	0
$691$	19.6114	0.746053	0.373027	−	0.927821i	$-0.378320\pi$
$691$	19.6114	0.746053	0.373027	+	0.927821i	$0.378320\pi$
$692$	0	0
$693$	2.11402	0.0803049
$694$	0	0
$695$	20.2937	0.769785
$696$	0	0
$697$	30.4487	1.15333
$698$	0	0
$699$	13.9469	0.527521
$700$	0	0
$701$	39.0557	1.47512	0.737558	−	0.675284i	$-0.235979\pi$
$701$	39.0557	1.47512	0.737558	+	0.675284i	$0.235979\pi$
$702$	0	0
$703$	22.9709	0.866365
$704$	0	0
$705$	4.02144	0.151456
$706$	0	0
$707$	3.21464	0.120899
$708$	0	0
$709$	17.4690	0.656063	0.328031	−	0.944667i	$-0.393615\pi$
$709$	17.4690	0.656063	0.328031	+	0.944667i	$0.393615\pi$
$710$	0	0
$711$	8.10045	0.303791
$712$	0	0
$713$	−17.2820	−0.647215
$714$	0	0
$715$	6.47024	0.241973
$716$	0	0
$717$	11.5048	0.429654
$718$	0	0
$719$	−27.7436	−1.03466	−0.517332	−	0.855785i	$-0.673075\pi$
$719$	−27.7436	−1.03466	−0.517332	+	0.855785i	$0.673075\pi$
$720$	0	0
$721$	−15.6957	−0.584538
$722$	0	0
$723$	−12.0664	−0.448754
$724$	0	0
$725$	10.9338	0.406069
$726$	0	0
$727$	21.1032	0.782674	0.391337	−	0.920248i	$-0.372013\pi$
$727$	21.1032	0.782674	0.391337	+	0.920248i	$0.372013\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−38.8593	−1.43726
$732$	0	0
$733$	16.0203	0.591725	0.295862	−	0.955231i	$-0.404393\pi$
$733$	16.0203	0.591725	0.295862	+	0.955231i	$0.404393\pi$
$734$	0	0
$735$	−5.81148	−0.214360
$736$	0	0
$737$	−22.9344	−0.844799
$738$	0	0
$739$	−49.5269	−1.82188	−0.910938	−	0.412543i	$-0.864641\pi$
$739$	−49.5269	−1.82188	−0.910938	+	0.412543i	$0.864641\pi$
$740$	0	0
$741$	24.6286	0.904753
$742$	0	0
$743$	−38.8971	−1.42700	−0.713498	−	0.700657i	$-0.752890\pi$
$743$	−38.8971	−1.42700	−0.713498	+	0.700657i	$0.752890\pi$
$744$	0	0
$745$	3.59342	0.131653
$746$	0	0
$747$	−3.31054	−0.121126
$748$	0	0
$749$	14.3123	0.522962
$750$	0	0
$751$	37.0933	1.35355	0.676777	−	0.736188i	$-0.263376\pi$
$751$	37.0933	1.35355	0.676777	+	0.736188i	$0.263376\pi$
$752$	0	0
$753$	7.60035	0.276972
$754$	0	0
$755$	11.4314	0.416030
$756$	0	0
$757$	−10.9647	−0.398520	−0.199260	−	0.979947i	$-0.563854\pi$
$757$	−10.9647	−0.398520	−0.199260	+	0.979947i	$0.563854\pi$
$758$	0	0
$759$	−3.96089	−0.143771
$760$	0	0
$761$	12.6804	0.459666	0.229833	−	0.973230i	$-0.426182\pi$
$761$	12.6804	0.459666	0.229833	+	0.973230i	$0.426182\pi$
$762$	0	0
$763$	16.5164	0.597933
$764$	0	0
$765$	4.15989	0.150401
$766$	0	0
$767$	37.9679	1.37094
$768$	0	0
$769$	41.2875	1.48887	0.744433	−	0.667697i	$-0.232720\pi$
$769$	41.2875	1.48887	0.744433	+	0.667697i	$0.232720\pi$
$770$	0	0
$771$	10.0637	0.362434
$772$	0	0
$773$	53.6086	1.92817	0.964083	−	0.265600i	$-0.0855700\pi$
$773$	53.6086	1.92817	0.964083	+	0.265600i	$0.0855700\pi$
$774$	0	0
$775$	−31.8843	−1.14532
$776$	0	0
$777$	3.66188	0.131369
$778$	0	0
$779$	53.6816	1.92334
$780$	0	0
$781$	0.479467	0.0171567
$782$	0	0
$783$	−2.76331	−0.0987526
$784$	0	0
$785$	−13.4282	−0.479272
$786$	0	0
$787$	−16.4127	−0.585050	−0.292525	−	0.956258i	$-0.594495\pi$
$787$	−16.4127	−0.585050	−0.292525	+	0.956258i	$0.594495\pi$
$788$	0	0
$789$	29.0844	1.03543
$790$	0	0
$791$	−8.88962	−0.316079
$792$	0	0
$793$	−3.03350	−0.107723
$794$	0	0
$795$	−4.56766	−0.161998
$796$	0	0
$797$	−23.6518	−0.837791	−0.418895	−	0.908034i	$-0.637583\pi$
$797$	−23.6518	−0.837791	−0.418895	+	0.908034i	$0.637583\pi$
$798$	0	0
$799$	16.0354	0.567291
$800$	0	0
$801$	−1.15863	−0.0409380
$802$	0	0
$803$	−9.68422	−0.341749
$804$	0	0
$805$	−2.50738	−0.0883736
$806$	0	0
$807$	−11.1474	−0.392407
$808$	0	0
$809$	−43.4959	−1.52924	−0.764618	−	0.644484i	$-0.777072\pi$
$809$	−43.4959	−1.52924	−0.764618	+	0.644484i	$0.777072\pi$
$810$	0	0
$811$	−44.7765	−1.57232	−0.786158	−	0.618026i	$-0.787933\pi$
$811$	−44.7765	−1.57232	−0.786158	+	0.618026i	$0.787933\pi$
$812$	0	0
$813$	1.24441	0.0436435
$814$	0	0
$815$	−11.9818	−0.419703
$816$	0	0
$817$	−68.5096	−2.39685
$818$	0	0
$819$	3.92613	0.137190
$820$	0	0
$821$	42.4841	1.48271	0.741353	−	0.671115i	$-0.234184\pi$
$821$	42.4841	1.48271	0.741353	+	0.671115i	$0.234184\pi$
$822$	0	0
$823$	16.3816	0.571027	0.285514	−	0.958375i	$-0.407836\pi$
$823$	16.3816	0.571027	0.285514	+	0.958375i	$0.407836\pi$
$824$	0	0
$825$	−7.30762	−0.254419
$826$	0	0
$827$	8.27836	0.287867	0.143933	−	0.989587i	$-0.454025\pi$
$827$	8.27836	0.287867	0.143933	+	0.989587i	$0.454025\pi$
$828$	0	0
$829$	−37.3120	−1.29590	−0.647950	−	0.761683i	$-0.724373\pi$
$829$	−37.3120	−1.29590	−0.647950	+	0.761683i	$0.724373\pi$
$830$	0	0
$831$	10.5406	0.365648
$832$	0	0
$833$	−23.1732	−0.802903
$834$	0	0
$835$	−1.02139	−0.0353467
$836$	0	0
$837$	8.05818	0.278531
$838$	0	0
$839$	−24.5113	−0.846223	−0.423111	−	0.906078i	$-0.639062\pi$
$839$	−24.5113	−0.846223	−0.423111	+	0.906078i	$0.639062\pi$
$840$	0	0
$841$	−21.3641	−0.736694
$842$	0	0
$843$	−21.9069	−0.754513
$844$	0	0
$845$	−1.26162	−0.0434010
$846$	0	0
$847$	−8.68683	−0.298483
$848$	0	0
$849$	−1.02635	−0.0352241
$850$	0	0
$851$	−6.86100	−0.235192
$852$	0	0
$853$	−40.1368	−1.37426	−0.687129	−	0.726535i	$-0.741129\pi$
$853$	−40.1368	−1.37426	−0.687129	+	0.726535i	$0.741129\pi$
$854$	0	0
$855$	7.33396	0.250816
$856$	0	0
$857$	9.36226	0.319809	0.159904	−	0.987133i	$-0.448881\pi$
$857$	9.36226	0.319809	0.159904	+	0.987133i	$0.448881\pi$
$858$	0	0
$859$	−37.0796	−1.26514	−0.632570	−	0.774503i	$-0.718000\pi$
$859$	−37.0796	−1.26514	−0.632570	+	0.774503i	$0.718000\pi$
$860$	0	0
$861$	8.55758	0.291642
$862$	0	0
$863$	−49.4319	−1.68268	−0.841340	−	0.540506i	$-0.818233\pi$
$863$	−49.4319	−1.68268	−0.841340	+	0.540506i	$0.818233\pi$
$864$	0	0
$865$	−14.4725	−0.492079
$866$	0	0
$867$	−0.412526	−0.0140101
$868$	0	0
$869$	14.9605	0.507499
$870$	0	0
$871$	−42.5935	−1.44322
$872$	0	0
$873$	−11.1724	−0.378127
$874$	0	0
$875$	−10.4717	−0.354006
$876$	0	0
$877$	24.8447	0.838946	0.419473	−	0.907768i	$-0.362215\pi$
$877$	24.8447	0.838946	0.419473	+	0.907768i	$0.362215\pi$
$878$	0	0
$879$	19.5305	0.658746
$880$	0	0
$881$	17.0748	0.575266	0.287633	−	0.957741i	$-0.407132\pi$
$881$	17.0748	0.575266	0.287633	+	0.957741i	$0.407132\pi$
$882$	0	0
$883$	−12.2033	−0.410675	−0.205337	−	0.978691i	$-0.565829\pi$
$883$	−12.2033	−0.410675	−0.205337	+	0.978691i	$0.565829\pi$
$884$	0	0
$885$	11.3062	0.380053
$886$	0	0
$887$	5.46779	0.183591	0.0917953	−	0.995778i	$-0.470739\pi$
$887$	5.46779	0.183591	0.0917953	+	0.995778i	$0.470739\pi$
$888$	0	0
$889$	−11.4171	−0.382916
$890$	0	0
$891$	1.84687	0.0618725
$892$	0	0
$893$	28.2707	0.946043
$894$	0	0
$895$	26.8765	0.898381
$896$	0	0
$897$	−7.35612	−0.245614
$898$	0	0
$899$	−22.2672	−0.742654
$900$	0	0
$901$	−18.2134	−0.606778
$902$	0	0
$903$	−10.9214	−0.363440
$904$	0	0
$905$	−9.22768	−0.306739
$906$	0	0
$907$	−53.5640	−1.77856	−0.889282	−	0.457359i	$-0.848796\pi$
$907$	−53.5640	−1.77856	−0.889282	+	0.457359i	$0.848796\pi$
$908$	0	0
$909$	2.80841	0.0931490
$910$	0	0
$911$	49.9092	1.65357	0.826783	−	0.562522i	$-0.190169\pi$
$911$	49.9092	1.65357	0.826783	+	0.562522i	$0.190169\pi$
$912$	0	0
$913$	−6.11413	−0.202348
$914$	0	0
$915$	−0.903324	−0.0298630
$916$	0	0
$917$	0.278643	0.00920160
$918$	0	0
$919$	26.5214	0.874860	0.437430	−	0.899253i	$-0.355889\pi$
$919$	26.5214	0.874860	0.437430	+	0.899253i	$0.355889\pi$
$920$	0	0
$921$	4.61461	0.152057
$922$	0	0
$923$	0.890460	0.0293099
$924$	0	0
$925$	−12.6582	−0.416198
$926$	0	0
$927$	−13.7122	−0.450369
$928$	0	0
$929$	−47.5994	−1.56169	−0.780843	−	0.624727i	$-0.785210\pi$
$929$	−47.5994	−1.56169	−0.780843	+	0.624727i	$0.785210\pi$
$930$	0	0
$931$	−40.8547	−1.33896
$932$	0	0
$933$	22.9291	0.750664
$934$	0	0
$935$	7.68277	0.251254
$936$	0	0
$937$	−20.5311	−0.670721	−0.335361	−	0.942090i	$-0.608858\pi$
$937$	−20.5311	−0.670721	−0.335361	+	0.942090i	$0.608858\pi$
$938$	0	0
$939$	−3.67954	−0.120077
$940$	0	0
$941$	31.0644	1.01267	0.506335	−	0.862337i	$-0.331000\pi$
$941$	31.0644	1.01267	0.506335	+	0.862337i	$0.331000\pi$
$942$	0	0
$943$	−16.0337	−0.522130
$944$	0	0
$945$	1.16913	0.0380319
$946$	0	0
$947$	−3.80588	−0.123674	−0.0618372	−	0.998086i	$-0.519696\pi$
$947$	−3.80588	−0.123674	−0.0618372	+	0.998086i	$0.519696\pi$
$948$	0	0
$949$	−17.9854	−0.583832
$950$	0	0
$951$	−28.2921	−0.917435
$952$	0	0
$953$	27.8495	0.902135	0.451068	−	0.892490i	$-0.351043\pi$
$953$	27.8495	0.902135	0.451068	+	0.892490i	$0.351043\pi$
$954$	0	0
$955$	−21.6984	−0.702145
$956$	0	0
$957$	−5.10347	−0.164972
$958$	0	0
$959$	−21.0941	−0.681163
$960$	0	0
$961$	33.9343	1.09465
$962$	0	0
$963$	12.5037	0.402926
$964$	0	0
$965$	6.54230	0.210604
$966$	0	0
$967$	35.9226	1.15519	0.577596	−	0.816323i	$-0.303991\pi$
$967$	35.9226	1.15519	0.577596	+	0.816323i	$0.303991\pi$
$968$	0	0
$969$	29.2440	0.939453
$970$	0	0
$971$	−33.7029	−1.08158	−0.540788	−	0.841159i	$-0.681874\pi$
$971$	−33.7029	−1.08158	−0.540788	+	0.841159i	$0.681874\pi$
$972$	0	0
$973$	22.7428	0.729099
$974$	0	0
$975$	−13.5716	−0.434640
$976$	0	0
$977$	−34.7356	−1.11129	−0.555645	−	0.831420i	$-0.687529\pi$
$977$	−34.7356	−1.11129	−0.555645	+	0.831420i	$0.687529\pi$
$978$	0	0
$979$	−2.13983	−0.0683893
$980$	0	0
$981$	14.4292	0.460689
$982$	0	0
$983$	26.8576	0.856624	0.428312	−	0.903631i	$-0.359108\pi$
$983$	26.8576	0.856624	0.428312	+	0.903631i	$0.359108\pi$
$984$	0	0
$985$	−11.3975	−0.363154
$986$	0	0
$987$	4.50673	0.143451
$988$	0	0
$989$	20.4626	0.650673
$990$	0	0
$991$	32.5441	1.03380	0.516899	−	0.856046i	$-0.327086\pi$
$991$	32.5441	1.03380	0.516899	+	0.856046i	$0.327086\pi$
$992$	0	0
$993$	−13.2253	−0.419691
$994$	0	0
$995$	16.6766	0.528684
$996$	0	0
$997$	24.5380	0.777125	0.388563	−	0.921422i	$-0.372972\pi$
$997$	24.5380	0.777125	0.388563	+	0.921422i	$0.372972\pi$
$998$	0	0
$999$	3.19913	0.101216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.y.1.5		8
4.3	odd	2		4008.2.a.h.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.h.1.5	✓	8	4.3	odd	2
8016.2.a.y.1.5		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.02139 q^{5} +1.14465 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.02139 q^{5} +1.14465 q^{7} +1.00000 q^{9} +1.84687 q^{11} +3.42999 q^{13} +1.02139 q^{15} +4.07277 q^{17} +7.18037 q^{19} +1.14465 q^{21} -2.14465 q^{23} -3.95676 q^{25} +1.00000 q^{27} -2.76331 q^{29} +8.05818 q^{31} +1.84687 q^{33} +1.16913 q^{35} +3.19913 q^{37} +3.42999 q^{39} +7.47616 q^{41} -9.54123 q^{43} +1.02139 q^{45} +3.93722 q^{47} -5.68978 q^{49} +4.07277 q^{51} -4.47200 q^{53} +1.88637 q^{55} +7.18037 q^{57} +11.0694 q^{59} -0.884407 q^{61} +1.14465 q^{63} +3.50335 q^{65} -12.4180 q^{67} -2.14465 q^{69} +0.259610 q^{71} -5.24359 q^{73} -3.95676 q^{75} +2.11402 q^{77} +8.10045 q^{79} +1.00000 q^{81} -3.31054 q^{83} +4.15989 q^{85} -2.76331 q^{87} -1.15863 q^{89} +3.92613 q^{91} +8.05818 q^{93} +7.33396 q^{95} -11.1724 q^{97} +1.84687 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + q^7 + 8 * q^9	\( 8 q + 8 q^{3} + q^{7} + 8 q^{9} + 3 q^{11} - 8 q^{13} - 7 q^{17} + q^{21} - 9 q^{23} + 6 q^{25} + 8 q^{27} + 17 q^{29} + 23 q^{31} + 3 q^{33} + 15 q^{35} + 8 q^{37} - 8 q^{39} - 8 q^{41} + 2 q^{43} + 34 q^{47} + 5 q^{49} - 7 q^{51} + 12 q^{53} + 7 q^{55} + 16 q^{59} - 2 q^{61} + q^{63} - 14 q^{65} - 21 q^{67} - 9 q^{69} + 29 q^{71} - 38 q^{73} + 6 q^{75} + 20 q^{77} + 12 q^{79} + 8 q^{81} + 32 q^{83} + 23 q^{85} + 17 q^{87} + 11 q^{89} + 5 q^{91} + 23 q^{93} + 67 q^{95} + 8 q^{97} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + q^7 + 8 * q^9 + 3 * q^11 - 8 * q^13 - 7 * q^17 + q^21 - 9 * q^23 + 6 * q^25 + 8 * q^27 + 17 * q^29 + 23 * q^31 + 3 * q^33 + 15 * q^35 + 8 * q^37 - 8 * q^39 - 8 * q^41 + 2 * q^43 + 34 * q^47 + 5 * q^49 - 7 * q^51 + 12 * q^53 + 7 * q^55 + 16 * q^59 - 2 * q^61 + q^63 - 14 * q^65 - 21 * q^67 - 9 * q^69 + 29 * q^71 - 38 * q^73 + 6 * q^75 + 20 * q^77 + 12 * q^79 + 8 * q^81 + 32 * q^83 + 23 * q^85 + 17 * q^87 + 11 * q^89 + 5 * q^91 + 23 * q^93 + 67 * q^95 + 8 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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