LMFDB - Embedded newform 8008.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.244558277.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 13x^{4} + 11x^{3} + 29x^{2} + 5x - 4 $ x^6 - x^5 - 13x^4 + 11x^3 + 29x^2 + 5x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.287253$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.287253 q^{3} -0.115270 q^{5} +1.00000 q^{7} -2.91749 q^{9} +O(q^{10})$	$q-0.287253 q^{3} -0.115270 q^{5} +1.00000 q^{7} -2.91749 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.0331118 q^{15} -5.96127 q^{17} +2.56157 q^{19} -0.287253 q^{21} +1.63059 q^{23} -4.98671 q^{25} +1.69982 q^{27} -4.27397 q^{29} +9.90455 q^{31} +0.287253 q^{33} -0.115270 q^{35} +3.82766 q^{37} +0.287253 q^{39} -6.84847 q^{41} +2.08216 q^{43} +0.336300 q^{45} -10.6507 q^{47} +1.00000 q^{49} +1.71239 q^{51} +13.3842 q^{53} +0.115270 q^{55} -0.735820 q^{57} +12.8664 q^{59} -14.3889 q^{61} -2.91749 q^{63} +0.115270 q^{65} -6.75900 q^{67} -0.468391 q^{69} -6.49199 q^{71} -12.6248 q^{73} +1.43245 q^{75} -1.00000 q^{77} -3.16884 q^{79} +8.26418 q^{81} +2.81798 q^{83} +0.687158 q^{85} +1.22771 q^{87} +13.7455 q^{89} -1.00000 q^{91} -2.84511 q^{93} -0.295274 q^{95} +16.6252 q^{97} +2.91749 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9}+O(q^{10}) $ 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9	$ 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9} - 6 q^{11} - 6 q^{13} - q^{15} + 2 q^{19} - q^{21} + 11 q^{23} + 9 q^{25} - q^{27} + 14 q^{29} + 21 q^{31} + q^{33} + q^{35} - 5 q^{37} + q^{39} + 12 q^{43} + 2 q^{45} + 2 q^{47} + 6 q^{49} - 16 q^{51} + 2 q^{53} - q^{55} - 12 q^{57} + 3 q^{59} - 18 q^{61} + 9 q^{63} - q^{65} - 25 q^{67} + 29 q^{69} - 11 q^{71} + 14 q^{73} - 42 q^{75} - 6 q^{77} + 24 q^{79} + 42 q^{81} + 24 q^{83} - 46 q^{85} - 16 q^{87} + 55 q^{89} - 6 q^{91} + 41 q^{93} - 30 q^{95} + 37 q^{97} - 9 q^{99}+O(q^{100}) $ 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9 - 6 * q^11 - 6 * q^13 - q^15 + 2 * q^19 - q^21 + 11 * q^23 + 9 * q^25 - q^27 + 14 * q^29 + 21 * q^31 + q^33 + q^35 - 5 * q^37 + q^39 + 12 * q^43 + 2 * q^45 + 2 * q^47 + 6 * q^49 - 16 * q^51 + 2 * q^53 - q^55 - 12 * q^57 + 3 * q^59 - 18 * q^61 + 9 * q^63 - q^65 - 25 * q^67 + 29 * q^69 - 11 * q^71 + 14 * q^73 - 42 * q^75 - 6 * q^77 + 24 * q^79 + 42 * q^81 + 24 * q^83 - 46 * q^85 - 16 * q^87 + 55 * q^89 - 6 * q^91 + 41 * q^93 - 30 * q^95 + 37 * 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
$3$	−0.287253	−0.165846	−0.0829228	−	0.996556i	$-0.526425\pi$
$3$	−0.287253	−0.165846	−0.0829228	+	0.996556i	$0.526425\pi$
$4$	0	0
$5$	−0.115270	−0.0515505	−0.0257753	−	0.999668i	$-0.508205\pi$
$5$	−0.115270	−0.0515505	−0.0257753	+	0.999668i	$0.508205\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.91749	−0.972495
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.0331118	0.00854942
$16$	0	0
$17$	−5.96127	−1.44582	−0.722910	−	0.690943i	$-0.757196\pi$
$17$	−5.96127	−1.44582	−0.722910	+	0.690943i	$0.757196\pi$
$18$	0	0
$19$	2.56157	0.587665	0.293833	−	0.955857i	$-0.405069\pi$
$19$	2.56157	0.587665	0.293833	+	0.955857i	$0.405069\pi$
$20$	0	0
$21$	−0.287253	−0.0626837
$22$	0	0
$23$	1.63059	0.340001	0.170001	−	0.985444i	$-0.445623\pi$
$23$	1.63059	0.340001	0.170001	+	0.985444i	$0.445623\pi$
$24$	0	0
$25$	−4.98671	−0.997343
$26$	0	0
$27$	1.69982	0.327130
$28$	0	0
$29$	−4.27397	−0.793656	−0.396828	−	0.917893i	$-0.629889\pi$
$29$	−4.27397	−0.793656	−0.396828	+	0.917893i	$0.629889\pi$
$30$	0	0
$31$	9.90455	1.77891	0.889455	−	0.457022i	$-0.151084\pi$
$31$	9.90455	1.77891	0.889455	+	0.457022i	$0.151084\pi$
$32$	0	0
$33$	0.287253	0.0500043
$34$	0	0
$35$	−0.115270	−0.0194843
$36$	0	0
$37$	3.82766	0.629264	0.314632	−	0.949214i	$-0.398119\pi$
$37$	3.82766	0.629264	0.314632	+	0.949214i	$0.398119\pi$
$38$	0	0
$39$	0.287253	0.0459973
$40$	0	0
$41$	−6.84847	−1.06955	−0.534776	−	0.844994i	$-0.679604\pi$
$41$	−6.84847	−1.06955	−0.534776	+	0.844994i	$0.679604\pi$
$42$	0	0
$43$	2.08216	0.317526	0.158763	−	0.987317i	$-0.449249\pi$
$43$	2.08216	0.317526	0.158763	+	0.987317i	$0.449249\pi$
$44$	0	0
$45$	0.336300	0.0501326
$46$	0	0
$47$	−10.6507	−1.55356	−0.776781	−	0.629771i	$-0.783149\pi$
$47$	−10.6507	−1.55356	−0.776781	+	0.629771i	$0.783149\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.71239	0.239783
$52$	0	0
$53$	13.3842	1.83847	0.919234	−	0.393712i	$-0.128809\pi$
$53$	13.3842	1.83847	0.919234	+	0.393712i	$0.128809\pi$
$54$	0	0
$55$	0.115270	0.0155431
$56$	0	0
$57$	−0.735820	−0.0974617
$58$	0	0
$59$	12.8664	1.67507	0.837535	−	0.546384i	$-0.183996\pi$
$59$	12.8664	1.67507	0.837535	+	0.546384i	$0.183996\pi$
$60$	0	0
$61$	−14.3889	−1.84231	−0.921154	−	0.389199i	$-0.872752\pi$
$61$	−14.3889	−1.84231	−0.921154	+	0.389199i	$0.872752\pi$
$62$	0	0
$63$	−2.91749	−0.367569
$64$	0	0
$65$	0.115270	0.0142975
$66$	0	0
$67$	−6.75900	−0.825744	−0.412872	−	0.910789i	$-0.635474\pi$
$67$	−6.75900	−0.825744	−0.412872	+	0.910789i	$0.635474\pi$
$68$	0	0
$69$	−0.468391	−0.0563877
$70$	0	0
$71$	−6.49199	−0.770458	−0.385229	−	0.922821i	$-0.625877\pi$
$71$	−6.49199	−0.770458	−0.385229	+	0.922821i	$0.625877\pi$
$72$	0	0
$73$	−12.6248	−1.47762	−0.738812	−	0.673912i	$-0.764613\pi$
$73$	−12.6248	−1.47762	−0.738812	+	0.673912i	$0.764613\pi$
$74$	0	0
$75$	1.43245	0.165405
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−3.16884	−0.356522	−0.178261	−	0.983983i	$-0.557047\pi$
$79$	−3.16884	−0.356522	−0.178261	+	0.983983i	$0.557047\pi$
$80$	0	0
$81$	8.26418	0.918242
$82$	0	0
$83$	2.81798	0.309313	0.154657	−	0.987968i	$-0.450573\pi$
$83$	2.81798	0.309313	0.154657	+	0.987968i	$0.450573\pi$
$84$	0	0
$85$	0.687158	0.0745327
$86$	0	0
$87$	1.22771	0.131624
$88$	0	0
$89$	13.7455	1.45702	0.728510	−	0.685035i	$-0.240213\pi$
$89$	13.7455	1.45702	0.728510	+	0.685035i	$0.240213\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−2.84511	−0.295024
$94$	0	0
$95$	−0.295274	−0.0302945
$96$	0	0
$97$	16.6252	1.68804	0.844019	−	0.536314i	$-0.180184\pi$
$97$	16.6252	1.68804	0.844019	+	0.536314i	$0.180184\pi$
$98$	0	0
$99$	2.91749	0.293218
$100$	0	0
$101$	13.3659	1.32996	0.664979	−	0.746862i	$-0.268441\pi$
$101$	13.3659	1.32996	0.664979	+	0.746862i	$0.268441\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0.0331118	0.00323138
$106$	0	0
$107$	13.8913	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$107$	13.8913	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$108$	0	0
$109$	−7.71239	−0.738713	−0.369357	−	0.929288i	$-0.620422\pi$
$109$	−7.71239	−0.738713	−0.369357	+	0.929288i	$0.620422\pi$
$110$	0	0
$111$	−1.09951	−0.104361
$112$	0	0
$113$	10.5796	0.995241	0.497620	−	0.867395i	$-0.334207\pi$
$113$	10.5796	0.995241	0.497620	+	0.867395i	$0.334207\pi$
$114$	0	0
$115$	−0.187959	−0.0175272
$116$	0	0
$117$	2.91749	0.269722
$118$	0	0
$119$	−5.96127	−0.546468
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.96724	0.177380
$124$	0	0
$125$	1.15117	0.102964
$126$	0	0
$127$	−6.25037	−0.554630	−0.277315	−	0.960779i	$-0.589445\pi$
$127$	−6.25037	−0.554630	−0.277315	+	0.960779i	$0.589445\pi$
$128$	0	0
$129$	−0.598106	−0.0526603
$130$	0	0
$131$	17.8784	1.56204	0.781021	−	0.624504i	$-0.214699\pi$
$131$	17.8784	1.56204	0.781021	+	0.624504i	$0.214699\pi$
$132$	0	0
$133$	2.56157	0.222117
$134$	0	0
$135$	−0.195938	−0.0168637
$136$	0	0
$137$	15.9046	1.35882	0.679409	−	0.733760i	$-0.262236\pi$
$137$	15.9046	1.35882	0.679409	+	0.733760i	$0.262236\pi$
$138$	0	0
$139$	1.65603	0.140463	0.0702315	−	0.997531i	$-0.477626\pi$
$139$	1.65603	0.140463	0.0702315	+	0.997531i	$0.477626\pi$
$140$	0	0
$141$	3.05944	0.257651
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0.492662	0.0409133
$146$	0	0
$147$	−0.287253	−0.0236922
$148$	0	0
$149$	10.5634	0.865386	0.432693	−	0.901541i	$-0.357563\pi$
$149$	10.5634	0.865386	0.432693	+	0.901541i	$0.357563\pi$
$150$	0	0
$151$	−21.9294	−1.78459	−0.892296	−	0.451452i	$-0.850906\pi$
$151$	−21.9294	−1.78459	−0.892296	+	0.451452i	$0.850906\pi$
$152$	0	0
$153$	17.3919	1.40605
$154$	0	0
$155$	−1.14170	−0.0917037
$156$	0	0
$157$	−18.0117	−1.43749	−0.718745	−	0.695274i	$-0.755283\pi$
$157$	−18.0117	−1.43749	−0.718745	+	0.695274i	$0.755283\pi$
$158$	0	0
$159$	−3.84466	−0.304902
$160$	0	0
$161$	1.63059	0.128508
$162$	0	0
$163$	−15.5890	−1.22102	−0.610512	−	0.792007i	$-0.709036\pi$
$163$	−15.5890	−1.22102	−0.610512	+	0.792007i	$0.709036\pi$
$164$	0	0
$165$	−0.0331118	−0.00257775
$166$	0	0
$167$	−6.76560	−0.523538	−0.261769	−	0.965131i	$-0.584306\pi$
$167$	−6.76560	−0.523538	−0.261769	+	0.965131i	$0.584306\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.47336	−0.571502
$172$	0	0
$173$	1.71059	0.130053	0.0650267	−	0.997884i	$-0.479287\pi$
$173$	1.71059	0.130053	0.0650267	+	0.997884i	$0.479287\pi$
$174$	0	0
$175$	−4.98671	−0.376960
$176$	0	0
$177$	−3.69593	−0.277803
$178$	0	0
$179$	8.82483	0.659599	0.329799	−	0.944051i	$-0.393019\pi$
$179$	8.82483	0.659599	0.329799	+	0.944051i	$0.393019\pi$
$180$	0	0
$181$	5.88642	0.437534	0.218767	−	0.975777i	$-0.429797\pi$
$181$	5.88642	0.437534	0.218767	+	0.975777i	$0.429797\pi$
$182$	0	0
$183$	4.13325	0.305539
$184$	0	0
$185$	−0.441216	−0.0324389
$186$	0	0
$187$	5.96127	0.435931
$188$	0	0
$189$	1.69982	0.123643
$190$	0	0
$191$	15.6022	1.12894	0.564468	−	0.825455i	$-0.309081\pi$
$191$	15.6022	1.12894	0.564468	+	0.825455i	$0.309081\pi$
$192$	0	0
$193$	13.9499	1.00413	0.502067	−	0.864828i	$-0.332573\pi$
$193$	13.9499	1.00413	0.502067	+	0.864828i	$0.332573\pi$
$194$	0	0
$195$	−0.0331118	−0.00237118
$196$	0	0
$197$	4.63080	0.329931	0.164965	−	0.986299i	$-0.447249\pi$
$197$	4.63080	0.329931	0.164965	+	0.986299i	$0.447249\pi$
$198$	0	0
$199$	−12.5869	−0.892259	−0.446129	−	0.894968i	$-0.647198\pi$
$199$	−12.5869	−0.892259	−0.446129	+	0.894968i	$0.647198\pi$
$200$	0	0
$201$	1.94154	0.136946
$202$	0	0
$203$	−4.27397	−0.299974
$204$	0	0
$205$	0.789426	0.0551359
$206$	0	0
$207$	−4.75722	−0.330650
$208$	0	0
$209$	−2.56157	−0.177188
$210$	0	0
$211$	11.7016	0.805574	0.402787	−	0.915294i	$-0.368042\pi$
$211$	11.7016	0.805574	0.402787	+	0.915294i	$0.368042\pi$
$212$	0	0
$213$	1.86484	0.127777
$214$	0	0
$215$	−0.240011	−0.0163686
$216$	0	0
$217$	9.90455	0.672365
$218$	0	0
$219$	3.62652	0.245057
$220$	0	0
$221$	5.96127	0.400998
$222$	0	0
$223$	17.6129	1.17945	0.589723	−	0.807606i	$-0.299237\pi$
$223$	17.6129	1.17945	0.589723	+	0.807606i	$0.299237\pi$
$224$	0	0
$225$	14.5487	0.969911
$226$	0	0
$227$	−20.4255	−1.35569	−0.677846	−	0.735204i	$-0.737086\pi$
$227$	−20.4255	−1.35569	−0.677846	+	0.735204i	$0.737086\pi$
$228$	0	0
$229$	−3.90607	−0.258120	−0.129060	−	0.991637i	$-0.541196\pi$
$229$	−3.90607	−0.258120	−0.129060	+	0.991637i	$0.541196\pi$
$230$	0	0
$231$	0.287253	0.0188999
$232$	0	0
$233$	0.102756	0.00673174	0.00336587	−	0.999994i	$-0.498929\pi$
$233$	0.102756	0.00673174	0.00336587	+	0.999994i	$0.498929\pi$
$234$	0	0
$235$	1.22771	0.0800869
$236$	0	0
$237$	0.910258	0.0591276
$238$	0	0
$239$	10.3614	0.670223	0.335111	−	0.942179i	$-0.391226\pi$
$239$	10.3614	0.670223	0.335111	+	0.942179i	$0.391226\pi$
$240$	0	0
$241$	−28.1851	−1.81556	−0.907782	−	0.419443i	$-0.862225\pi$
$241$	−28.1851	−1.81556	−0.907782	+	0.419443i	$0.862225\pi$
$242$	0	0
$243$	−7.47336	−0.479416
$244$	0	0
$245$	−0.115270	−0.00736436
$246$	0	0
$247$	−2.56157	−0.162989
$248$	0	0
$249$	−0.809473	−0.0512982
$250$	0	0
$251$	−5.37645	−0.339358	−0.169679	−	0.985499i	$-0.554273\pi$
$251$	−5.37645	−0.339358	−0.169679	+	0.985499i	$0.554273\pi$
$252$	0	0
$253$	−1.63059	−0.102514
$254$	0	0
$255$	−0.197388	−0.0123609
$256$	0	0
$257$	15.1917	0.947634	0.473817	−	0.880623i	$-0.342876\pi$
$257$	15.1917	0.947634	0.473817	+	0.880623i	$0.342876\pi$
$258$	0	0
$259$	3.82766	0.237839
$260$	0	0
$261$	12.4692	0.771826
$262$	0	0
$263$	17.8548	1.10097	0.550487	−	0.834844i	$-0.314442\pi$
$263$	17.8548	1.10097	0.550487	+	0.834844i	$0.314442\pi$
$264$	0	0
$265$	−1.54281	−0.0947739
$266$	0	0
$267$	−3.94844	−0.241640
$268$	0	0
$269$	10.6834	0.651381	0.325691	−	0.945476i	$-0.394403\pi$
$269$	10.6834	0.651381	0.325691	+	0.945476i	$0.394403\pi$
$270$	0	0
$271$	26.6735	1.62030	0.810150	−	0.586222i	$-0.199385\pi$
$271$	26.6735	1.62030	0.810150	+	0.586222i	$0.199385\pi$
$272$	0	0
$273$	0.287253	0.0173853
$274$	0	0
$275$	4.98671	0.300710
$276$	0	0
$277$	−5.25866	−0.315962	−0.157981	−	0.987442i	$-0.550499\pi$
$277$	−5.25866	−0.315962	−0.157981	+	0.987442i	$0.550499\pi$
$278$	0	0
$279$	−28.8964	−1.72998
$280$	0	0
$281$	−5.06808	−0.302336	−0.151168	−	0.988508i	$-0.548304\pi$
$281$	−5.06808	−0.302336	−0.151168	+	0.988508i	$0.548304\pi$
$282$	0	0
$283$	−25.6354	−1.52387	−0.761933	−	0.647656i	$-0.775749\pi$
$283$	−25.6354	−1.52387	−0.761933	+	0.647656i	$0.775749\pi$
$284$	0	0
$285$	0.0848183	0.00502420
$286$	0	0
$287$	−6.84847	−0.404252
$288$	0	0
$289$	18.5367	1.09039
$290$	0	0
$291$	−4.77565	−0.279954
$292$	0	0
$293$	29.4704	1.72168	0.860838	−	0.508879i	$-0.169940\pi$
$293$	29.4704	1.72168	0.860838	+	0.508879i	$0.169940\pi$
$294$	0	0
$295$	−1.48312	−0.0863507
$296$	0	0
$297$	−1.69982	−0.0986333
$298$	0	0
$299$	−1.63059	−0.0942994
$300$	0	0
$301$	2.08216	0.120014
$302$	0	0
$303$	−3.83940	−0.220568
$304$	0	0
$305$	1.65861	0.0949719
$306$	0	0
$307$	23.1581	1.32170	0.660851	−	0.750517i	$-0.270196\pi$
$307$	23.1581	1.32170	0.660851	+	0.750517i	$0.270196\pi$
$308$	0	0
$309$	−2.29802	−0.130730
$310$	0	0
$311$	3.49999	0.198466	0.0992330	−	0.995064i	$-0.468361\pi$
$311$	3.49999	0.198466	0.0992330	+	0.995064i	$0.468361\pi$
$312$	0	0
$313$	12.1931	0.689197	0.344598	−	0.938750i	$-0.388015\pi$
$313$	12.1931	0.689197	0.344598	+	0.938750i	$0.388015\pi$
$314$	0	0
$315$	0.336300	0.0189484
$316$	0	0
$317$	−30.6592	−1.72199	−0.860997	−	0.508610i	$-0.830159\pi$
$317$	−30.6592	−1.72199	−0.860997	+	0.508610i	$0.830159\pi$
$318$	0	0
$319$	4.27397	0.239296
$320$	0	0
$321$	−3.99031	−0.222717
$322$	0	0
$323$	−15.2702	−0.849658
$324$	0	0
$325$	4.98671	0.276613
$326$	0	0
$327$	2.21541	0.122512
$328$	0	0
$329$	−10.6507	−0.587191
$330$	0	0
$331$	−13.4671	−0.740217	−0.370109	−	0.928988i	$-0.620680\pi$
$331$	−13.4671	−0.740217	−0.370109	+	0.928988i	$0.620680\pi$
$332$	0	0
$333$	−11.1671	−0.611956
$334$	0	0
$335$	0.779113	0.0425675
$336$	0	0
$337$	14.7100	0.801306	0.400653	−	0.916230i	$-0.368783\pi$
$337$	14.7100	0.801306	0.400653	+	0.916230i	$0.368783\pi$
$338$	0	0
$339$	−3.03901	−0.165056
$340$	0	0
$341$	−9.90455	−0.536362
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.0539917	0.00290681
$346$	0	0
$347$	16.0242	0.860224	0.430112	−	0.902775i	$-0.358474\pi$
$347$	16.0242	0.860224	0.430112	+	0.902775i	$0.358474\pi$
$348$	0	0
$349$	−32.2852	−1.72819	−0.864095	−	0.503330i	$-0.832108\pi$
$349$	−32.2852	−1.72819	−0.864095	+	0.503330i	$0.832108\pi$
$350$	0	0
$351$	−1.69982	−0.0907294
$352$	0	0
$353$	−33.2173	−1.76798	−0.883990	−	0.467505i	$-0.845153\pi$
$353$	−33.2173	−1.76798	−0.883990	+	0.467505i	$0.845153\pi$
$354$	0	0
$355$	0.748335	0.0397175
$356$	0	0
$357$	1.71239	0.0906294
$358$	0	0
$359$	5.32488	0.281037	0.140518	−	0.990078i	$-0.455123\pi$
$359$	5.32488	0.281037	0.140518	+	0.990078i	$0.455123\pi$
$360$	0	0
$361$	−12.4383	−0.654649
$362$	0	0
$363$	−0.287253	−0.0150769
$364$	0	0
$365$	1.45527	0.0761723
$366$	0	0
$367$	10.4334	0.544621	0.272310	−	0.962209i	$-0.412212\pi$
$367$	10.4334	0.544621	0.272310	+	0.962209i	$0.412212\pi$
$368$	0	0
$369$	19.9803	1.04013
$370$	0	0
$371$	13.3842	0.694875
$372$	0	0
$373$	11.5332	0.597167	0.298583	−	0.954384i	$-0.403486\pi$
$373$	11.5332	0.597167	0.298583	+	0.954384i	$0.403486\pi$
$374$	0	0
$375$	−0.330678	−0.0170761
$376$	0	0
$377$	4.27397	0.220120
$378$	0	0
$379$	14.4503	0.742261	0.371131	−	0.928581i	$-0.378970\pi$
$379$	14.4503	0.742261	0.371131	+	0.928581i	$0.378970\pi$
$380$	0	0
$381$	1.79544	0.0919830
$382$	0	0
$383$	8.20043	0.419023	0.209511	−	0.977806i	$-0.432813\pi$
$383$	8.20043	0.419023	0.209511	+	0.977806i	$0.432813\pi$
$384$	0	0
$385$	0.115270	0.00587473
$386$	0	0
$387$	−6.07467	−0.308793
$388$	0	0
$389$	24.8649	1.26070	0.630351	−	0.776310i	$-0.282911\pi$
$389$	24.8649	1.26070	0.630351	+	0.776310i	$0.282911\pi$
$390$	0	0
$391$	−9.72037	−0.491580
$392$	0	0
$393$	−5.13562	−0.259058
$394$	0	0
$395$	0.365273	0.0183789
$396$	0	0
$397$	24.0136	1.20521	0.602604	−	0.798040i	$-0.294130\pi$
$397$	24.0136	1.20521	0.602604	+	0.798040i	$0.294130\pi$
$398$	0	0
$399$	−0.735820	−0.0368371
$400$	0	0
$401$	1.76568	0.0881740	0.0440870	−	0.999028i	$-0.485962\pi$
$401$	1.76568	0.0881740	0.0440870	+	0.999028i	$0.485962\pi$
$402$	0	0
$403$	−9.90455	−0.493381
$404$	0	0
$405$	−0.952616	−0.0473359
$406$	0	0
$407$	−3.82766	−0.189730
$408$	0	0
$409$	27.9268	1.38089	0.690445	−	0.723385i	$-0.257415\pi$
$409$	27.9268	1.38089	0.690445	+	0.723385i	$0.257415\pi$
$410$	0	0
$411$	−4.56863	−0.225354
$412$	0	0
$413$	12.8664	0.633117
$414$	0	0
$415$	−0.324830	−0.0159453
$416$	0	0
$417$	−0.475701	−0.0232952
$418$	0	0
$419$	11.8510	0.578961	0.289481	−	0.957184i	$-0.406517\pi$
$419$	11.8510	0.578961	0.289481	+	0.957184i	$0.406517\pi$
$420$	0	0
$421$	−11.9756	−0.583654	−0.291827	−	0.956471i	$-0.594263\pi$
$421$	−11.9756	−0.583654	−0.291827	+	0.956471i	$0.594263\pi$
$422$	0	0
$423$	31.0732	1.51083
$424$	0	0
$425$	29.7271	1.44198
$426$	0	0
$427$	−14.3889	−0.696327
$428$	0	0
$429$	−0.287253	−0.0138687
$430$	0	0
$431$	1.65137	0.0795438	0.0397719	−	0.999209i	$-0.487337\pi$
$431$	1.65137	0.0795438	0.0397719	+	0.999209i	$0.487337\pi$
$432$	0	0
$433$	29.2014	1.40333	0.701665	−	0.712507i	$-0.252440\pi$
$433$	29.2014	1.40333	0.701665	+	0.712507i	$0.252440\pi$
$434$	0	0
$435$	−0.141519	−0.00678530
$436$	0	0
$437$	4.17687	0.199807
$438$	0	0
$439$	24.1437	1.15231	0.576157	−	0.817339i	$-0.304552\pi$
$439$	24.1437	1.15231	0.576157	+	0.817339i	$0.304552\pi$
$440$	0	0
$441$	−2.91749	−0.138928
$442$	0	0
$443$	21.6224	1.02731	0.513656	−	0.857996i	$-0.328291\pi$
$443$	21.6224	1.02731	0.513656	+	0.857996i	$0.328291\pi$
$444$	0	0
$445$	−1.58445	−0.0751101
$446$	0	0
$447$	−3.03436	−0.143520
$448$	0	0
$449$	−8.90848	−0.420417	−0.210209	−	0.977657i	$-0.567414\pi$
$449$	−8.90848	−0.420417	−0.210209	+	0.977657i	$0.567414\pi$
$450$	0	0
$451$	6.84847	0.322482
$452$	0	0
$453$	6.29929	0.295967
$454$	0	0
$455$	0.115270	0.00540396
$456$	0	0
$457$	40.0334	1.87269	0.936343	−	0.351087i	$-0.114188\pi$
$457$	40.0334	1.87269	0.936343	+	0.351087i	$0.114188\pi$
$458$	0	0
$459$	−10.1331	−0.472970
$460$	0	0
$461$	1.99114	0.0927365	0.0463682	−	0.998924i	$-0.485235\pi$
$461$	1.99114	0.0927365	0.0463682	+	0.998924i	$0.485235\pi$
$462$	0	0
$463$	−26.1779	−1.21659	−0.608296	−	0.793710i	$-0.708147\pi$
$463$	−26.1779	−1.21659	−0.608296	+	0.793710i	$0.708147\pi$
$464$	0	0
$465$	0.327957	0.0152087
$466$	0	0
$467$	3.13824	0.145220	0.0726102	−	0.997360i	$-0.476867\pi$
$467$	3.13824	0.145220	0.0726102	+	0.997360i	$0.476867\pi$
$468$	0	0
$469$	−6.75900	−0.312102
$470$	0	0
$471$	5.17391	0.238401
$472$	0	0
$473$	−2.08216	−0.0957378
$474$	0	0
$475$	−12.7738	−0.586104
$476$	0	0
$477$	−39.0483	−1.78790
$478$	0	0
$479$	4.74038	0.216593	0.108297	−	0.994119i	$-0.465460\pi$
$479$	4.74038	0.216593	0.108297	+	0.994119i	$0.465460\pi$
$480$	0	0
$481$	−3.82766	−0.174526
$482$	0	0
$483$	−0.468391	−0.0213125
$484$	0	0
$485$	−1.91640	−0.0870192
$486$	0	0
$487$	35.1732	1.59385	0.796924	−	0.604079i	$-0.206459\pi$
$487$	35.1732	1.59385	0.796924	+	0.604079i	$0.206459\pi$
$488$	0	0
$489$	4.47798	0.202501
$490$	0	0
$491$	−24.1591	−1.09028	−0.545142	−	0.838344i	$-0.683524\pi$
$491$	−24.1591	−1.09028	−0.545142	+	0.838344i	$0.683524\pi$
$492$	0	0
$493$	25.4782	1.14748
$494$	0	0
$495$	−0.336300	−0.0151156
$496$	0	0
$497$	−6.49199	−0.291206
$498$	0	0
$499$	−27.5404	−1.23288	−0.616438	−	0.787404i	$-0.711425\pi$
$499$	−27.5404	−1.23288	−0.616438	+	0.787404i	$0.711425\pi$
$500$	0	0
$501$	1.94344	0.0868265
$502$	0	0
$503$	−19.5891	−0.873435	−0.436717	−	0.899599i	$-0.643859\pi$
$503$	−19.5891	−0.873435	−0.436717	+	0.899599i	$0.643859\pi$
$504$	0	0
$505$	−1.54069	−0.0685600
$506$	0	0
$507$	−0.287253	−0.0127574
$508$	0	0
$509$	30.2158	1.33929	0.669645	−	0.742681i	$-0.266446\pi$
$509$	30.2158	1.33929	0.669645	+	0.742681i	$0.266446\pi$
$510$	0	0
$511$	−12.6248	−0.558489
$512$	0	0
$513$	4.35420	0.192243
$514$	0	0
$515$	−0.922164	−0.0406354
$516$	0	0
$517$	10.6507	0.468417
$518$	0	0
$519$	−0.491371	−0.0215688
$520$	0	0
$521$	−26.3725	−1.15540	−0.577700	−	0.816249i	$-0.696050\pi$
$521$	−26.3725	−1.15540	−0.577700	+	0.816249i	$0.696050\pi$
$522$	0	0
$523$	1.77017	0.0774041	0.0387021	−	0.999251i	$-0.487678\pi$
$523$	1.77017	0.0774041	0.0387021	+	0.999251i	$0.487678\pi$
$524$	0	0
$525$	1.43245	0.0625172
$526$	0	0
$527$	−59.0437	−2.57198
$528$	0	0
$529$	−20.3412	−0.884399
$530$	0	0
$531$	−37.5377	−1.62900
$532$	0	0
$533$	6.84847	0.296640
$534$	0	0
$535$	−1.60125	−0.0692282
$536$	0	0
$537$	−2.53496	−0.109392
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	27.6239	1.18764	0.593822	−	0.804597i	$-0.297618\pi$
$541$	27.6239	1.18764	0.593822	+	0.804597i	$0.297618\pi$
$542$	0	0
$543$	−1.69089	−0.0725631
$544$	0	0
$545$	0.889011	0.0380810
$546$	0	0
$547$	−22.9165	−0.979839	−0.489919	−	0.871768i	$-0.662974\pi$
$547$	−22.9165	−0.979839	−0.489919	+	0.871768i	$0.662974\pi$
$548$	0	0
$549$	41.9794	1.79164
$550$	0	0
$551$	−10.9481	−0.466404
$552$	0	0
$553$	−3.16884	−0.134753
$554$	0	0
$555$	0.126741	0.00537984
$556$	0	0
$557$	14.8411	0.628838	0.314419	−	0.949284i	$-0.398190\pi$
$557$	14.8411	0.628838	0.314419	+	0.949284i	$0.398190\pi$
$558$	0	0
$559$	−2.08216	−0.0880659
$560$	0	0
$561$	−1.71239	−0.0722972
$562$	0	0
$563$	−10.9507	−0.461516	−0.230758	−	0.973011i	$-0.574121\pi$
$563$	−10.9507	−0.461516	−0.230758	+	0.973011i	$0.574121\pi$
$564$	0	0
$565$	−1.21951	−0.0513052
$566$	0	0
$567$	8.26418	0.347063
$568$	0	0
$569$	−2.51093	−0.105264	−0.0526319	−	0.998614i	$-0.516761\pi$
$569$	−2.51093	−0.105264	−0.0526319	+	0.998614i	$0.516761\pi$
$570$	0	0
$571$	12.8918	0.539504	0.269752	−	0.962930i	$-0.413058\pi$
$571$	12.8918	0.539504	0.269752	+	0.962930i	$0.413058\pi$
$572$	0	0
$573$	−4.48178	−0.187229
$574$	0	0
$575$	−8.13128	−0.339098
$576$	0	0
$577$	23.1617	0.964235	0.482118	−	0.876107i	$-0.339868\pi$
$577$	23.1617	0.964235	0.482118	+	0.876107i	$0.339868\pi$
$578$	0	0
$579$	−4.00715	−0.166531
$580$	0	0
$581$	2.81798	0.116909
$582$	0	0
$583$	−13.3842	−0.554319
$584$	0	0
$585$	−0.336300	−0.0139043
$586$	0	0
$587$	2.21369	0.0913688	0.0456844	−	0.998956i	$-0.485453\pi$
$587$	2.21369	0.0913688	0.0456844	+	0.998956i	$0.485453\pi$
$588$	0	0
$589$	25.3713	1.04540
$590$	0	0
$591$	−1.33021	−0.0547176
$592$	0	0
$593$	4.49384	0.184540	0.0922699	−	0.995734i	$-0.470588\pi$
$593$	4.49384	0.184540	0.0922699	+	0.995734i	$0.470588\pi$
$594$	0	0
$595$	0.687158	0.0281707
$596$	0	0
$597$	3.61561	0.147977
$598$	0	0
$599$	35.3870	1.44587	0.722936	−	0.690915i	$-0.242792\pi$
$599$	35.3870	1.44587	0.722936	+	0.690915i	$0.242792\pi$
$600$	0	0
$601$	16.6321	0.678437	0.339219	−	0.940708i	$-0.389837\pi$
$601$	16.6321	0.678437	0.339219	+	0.940708i	$0.389837\pi$
$602$	0	0
$603$	19.7193	0.803032
$604$	0	0
$605$	−0.115270	−0.00468641
$606$	0	0
$607$	−37.8146	−1.53485	−0.767423	−	0.641141i	$-0.778461\pi$
$607$	−37.8146	−1.53485	−0.767423	+	0.641141i	$0.778461\pi$
$608$	0	0
$609$	1.22771	0.0497493
$610$	0	0
$611$	10.6507	0.430881
$612$	0	0
$613$	−31.1614	−1.25860	−0.629298	−	0.777164i	$-0.716658\pi$
$613$	−31.1614	−1.25860	−0.629298	+	0.777164i	$0.716658\pi$
$614$	0	0
$615$	−0.226765	−0.00914405
$616$	0	0
$617$	−4.88500	−0.196663	−0.0983314	−	0.995154i	$-0.531351\pi$
$617$	−4.88500	−0.196663	−0.0983314	+	0.995154i	$0.531351\pi$
$618$	0	0
$619$	16.2956	0.654976	0.327488	−	0.944855i	$-0.393798\pi$
$619$	16.2956	0.654976	0.327488	+	0.944855i	$0.393798\pi$
$620$	0	0
$621$	2.77170	0.111224
$622$	0	0
$623$	13.7455	0.550702
$624$	0	0
$625$	24.8009	0.992035
$626$	0	0
$627$	0.735820	0.0293858
$628$	0	0
$629$	−22.8177	−0.909802
$630$	0	0
$631$	−21.0675	−0.838685	−0.419343	−	0.907828i	$-0.637739\pi$
$631$	−21.0675	−0.838685	−0.419343	+	0.907828i	$0.637739\pi$
$632$	0	0
$633$	−3.36133	−0.133601
$634$	0	0
$635$	0.720482	0.0285915
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	18.9403	0.749266
$640$	0	0
$641$	−40.4302	−1.59690	−0.798448	−	0.602064i	$-0.794345\pi$
$641$	−40.4302	−1.59690	−0.798448	+	0.602064i	$0.794345\pi$
$642$	0	0
$643$	23.7462	0.936460	0.468230	−	0.883607i	$-0.344892\pi$
$643$	23.7462	0.936460	0.468230	+	0.883607i	$0.344892\pi$
$644$	0	0
$645$	0.0689440	0.00271467
$646$	0	0
$647$	39.1410	1.53879	0.769396	−	0.638772i	$-0.220557\pi$
$647$	39.1410	1.53879	0.769396	+	0.638772i	$0.220557\pi$
$648$	0	0
$649$	−12.8664	−0.505052
$650$	0	0
$651$	−2.84511	−0.111509
$652$	0	0
$653$	24.3591	0.953246	0.476623	−	0.879108i	$-0.341861\pi$
$653$	24.3591	0.953246	0.476623	+	0.879108i	$0.341861\pi$
$654$	0	0
$655$	−2.06085	−0.0805241
$656$	0	0
$657$	36.8327	1.43698
$658$	0	0
$659$	6.56981	0.255924	0.127962	−	0.991779i	$-0.459157\pi$
$659$	6.56981	0.255924	0.127962	+	0.991779i	$0.459157\pi$
$660$	0	0
$661$	36.2255	1.40901	0.704504	−	0.709700i	$-0.251169\pi$
$661$	36.2255	1.40901	0.704504	+	0.709700i	$0.251169\pi$
$662$	0	0
$663$	−1.71239	−0.0665038
$664$	0	0
$665$	−0.295274	−0.0114502
$666$	0	0
$667$	−6.96908	−0.269844
$668$	0	0
$669$	−5.05935	−0.195606
$670$	0	0
$671$	14.3889	0.555477
$672$	0	0
$673$	−36.2004	−1.39542	−0.697712	−	0.716378i	$-0.745799\pi$
$673$	−36.2004	−1.39542	−0.697712	+	0.716378i	$0.745799\pi$
$674$	0	0
$675$	−8.47649	−0.326260
$676$	0	0
$677$	−19.8264	−0.761992	−0.380996	−	0.924577i	$-0.624419\pi$
$677$	−19.8264	−0.761992	−0.380996	+	0.924577i	$0.624419\pi$
$678$	0	0
$679$	16.6252	0.638018
$680$	0	0
$681$	5.86730	0.224835
$682$	0	0
$683$	−17.5945	−0.673235	−0.336617	−	0.941641i	$-0.609283\pi$
$683$	−17.5945	−0.673235	−0.336617	+	0.941641i	$0.609283\pi$
$684$	0	0
$685$	−1.83332	−0.0700477
$686$	0	0
$687$	1.12203	0.0428081
$688$	0	0
$689$	−13.3842	−0.509899
$690$	0	0
$691$	−5.38560	−0.204878	−0.102439	−	0.994739i	$-0.532665\pi$
$691$	−5.38560	−0.204878	−0.102439	+	0.994739i	$0.532665\pi$
$692$	0	0
$693$	2.91749	0.110826
$694$	0	0
$695$	−0.190892	−0.00724094
$696$	0	0
$697$	40.8256	1.54638
$698$	0	0
$699$	−0.0295168	−0.00111643
$700$	0	0
$701$	−0.982549	−0.0371104	−0.0185552	−	0.999828i	$-0.505907\pi$
$701$	−0.982549	−0.0371104	−0.0185552	+	0.999828i	$0.505907\pi$
$702$	0	0
$703$	9.80484	0.369797
$704$	0	0
$705$	−0.352663	−0.0132821
$706$	0	0
$707$	13.3659	0.502677
$708$	0	0
$709$	−22.1721	−0.832689	−0.416344	−	0.909207i	$-0.636689\pi$
$709$	−22.1721	−0.832689	−0.416344	+	0.909207i	$0.636689\pi$
$710$	0	0
$711$	9.24503	0.346716
$712$	0	0
$713$	16.1503	0.604832
$714$	0	0
$715$	−0.115270	−0.00431087
$716$	0	0
$717$	−2.97634	−0.111153
$718$	0	0
$719$	28.5900	1.06623	0.533114	−	0.846043i	$-0.321022\pi$
$719$	28.5900	1.06623	0.533114	+	0.846043i	$0.321022\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	8.09626	0.301103
$724$	0	0
$725$	21.3130	0.791546
$726$	0	0
$727$	42.1549	1.56344	0.781719	−	0.623631i	$-0.214343\pi$
$727$	42.1549	1.56344	0.781719	+	0.623631i	$0.214343\pi$
$728$	0	0
$729$	−22.6458	−0.838733
$730$	0	0
$731$	−12.4123	−0.459086
$732$	0	0
$733$	42.0529	1.55326	0.776629	−	0.629958i	$-0.216928\pi$
$733$	42.0529	1.55326	0.776629	+	0.629958i	$0.216928\pi$
$734$	0	0
$735$	0.0331118	0.00122135
$736$	0	0
$737$	6.75900	0.248971
$738$	0	0
$739$	6.65950	0.244974	0.122487	−	0.992470i	$-0.460913\pi$
$739$	6.65950	0.244974	0.122487	+	0.992470i	$0.460913\pi$
$740$	0	0
$741$	0.735820	0.0270310
$742$	0	0
$743$	−49.3683	−1.81115	−0.905573	−	0.424190i	$-0.860559\pi$
$743$	−49.3683	−1.81115	−0.905573	+	0.424190i	$0.860559\pi$
$744$	0	0
$745$	−1.21765	−0.0446111
$746$	0	0
$747$	−8.22141	−0.300806
$748$	0	0
$749$	13.8913	0.507576
$750$	0	0
$751$	31.5732	1.15212	0.576061	−	0.817406i	$-0.304589\pi$
$751$	31.5732	1.15212	0.576061	+	0.817406i	$0.304589\pi$
$752$	0	0
$753$	1.54440	0.0562811
$754$	0	0
$755$	2.52781	0.0919966
$756$	0	0
$757$	−49.8434	−1.81159	−0.905795	−	0.423717i	$-0.860725\pi$
$757$	−49.8434	−1.81159	−0.905795	+	0.423717i	$0.860725\pi$
$758$	0	0
$759$	0.468391	0.0170015
$760$	0	0
$761$	31.7809	1.15205	0.576027	−	0.817430i	$-0.304602\pi$
$761$	31.7809	1.15205	0.576027	+	0.817430i	$0.304602\pi$
$762$	0	0
$763$	−7.71239	−0.279207
$764$	0	0
$765$	−2.00477	−0.0724827
$766$	0	0
$767$	−12.8664	−0.464581
$768$	0	0
$769$	12.0128	0.433193	0.216597	−	0.976261i	$-0.430504\pi$
$769$	12.0128	0.433193	0.216597	+	0.976261i	$0.430504\pi$
$770$	0	0
$771$	−4.36387	−0.157161
$772$	0	0
$773$	−2.32357	−0.0835731	−0.0417865	−	0.999127i	$-0.513305\pi$
$773$	−2.32357	−0.0835731	−0.0417865	+	0.999127i	$0.513305\pi$
$774$	0	0
$775$	−49.3912	−1.77418
$776$	0	0
$777$	−1.09951	−0.0394446
$778$	0	0
$779$	−17.5429	−0.628539
$780$	0	0
$781$	6.49199	0.232302
$782$	0	0
$783$	−7.26495	−0.259628
$784$	0	0
$785$	2.07621	0.0741033
$786$	0	0
$787$	26.6235	0.949024	0.474512	−	0.880249i	$-0.342625\pi$
$787$	26.6235	0.949024	0.474512	+	0.880249i	$0.342625\pi$
$788$	0	0
$789$	−5.12884	−0.182592
$790$	0	0
$791$	10.5796	0.376166
$792$	0	0
$793$	14.3889	0.510964
$794$	0	0
$795$	0.443176	0.0157178
$796$	0	0
$797$	20.3246	0.719933	0.359967	−	0.932965i	$-0.382788\pi$
$797$	20.3246	0.719933	0.359967	+	0.932965i	$0.382788\pi$
$798$	0	0
$799$	63.4916	2.24617
$800$	0	0
$801$	−40.1023	−1.41695
$802$	0	0
$803$	12.6248	0.445520
$804$	0	0
$805$	−0.187959	−0.00662467
$806$	0	0
$807$	−3.06885	−0.108029
$808$	0	0
$809$	17.0947	0.601019	0.300509	−	0.953779i	$-0.402843\pi$
$809$	17.0947	0.601019	0.300509	+	0.953779i	$0.402843\pi$
$810$	0	0
$811$	−17.5207	−0.615237	−0.307618	−	0.951510i	$-0.599532\pi$
$811$	−17.5207	−0.615237	−0.307618	+	0.951510i	$0.599532\pi$
$812$	0	0
$813$	−7.66205	−0.268720
$814$	0	0
$815$	1.79695	0.0629444
$816$	0	0
$817$	5.33360	0.186599
$818$	0	0
$819$	2.91749	0.101945
$820$	0	0
$821$	24.1724	0.843622	0.421811	−	0.906684i	$-0.361395\pi$
$821$	24.1724	0.843622	0.421811	+	0.906684i	$0.361395\pi$
$822$	0	0
$823$	−26.6302	−0.928270	−0.464135	−	0.885764i	$-0.653635\pi$
$823$	−26.6302	−0.928270	−0.464135	+	0.885764i	$0.653635\pi$
$824$	0	0
$825$	−1.43245	−0.0498714
$826$	0	0
$827$	33.1017	1.15106	0.575529	−	0.817782i	$-0.304796\pi$
$827$	33.1017	1.15106	0.575529	+	0.817782i	$0.304796\pi$
$828$	0	0
$829$	10.8905	0.378243	0.189121	−	0.981954i	$-0.439436\pi$
$829$	10.8905	0.378243	0.189121	+	0.981954i	$0.439436\pi$
$830$	0	0
$831$	1.51057	0.0524009
$832$	0	0
$833$	−5.96127	−0.206546
$834$	0	0
$835$	0.779874	0.0269886
$836$	0	0
$837$	16.8359	0.581934
$838$	0	0
$839$	−6.72858	−0.232297	−0.116148	−	0.993232i	$-0.537055\pi$
$839$	−6.72858	−0.232297	−0.116148	+	0.993232i	$0.537055\pi$
$840$	0	0
$841$	−10.7332	−0.370111
$842$	0	0
$843$	1.45582	0.0501412
$844$	0	0
$845$	−0.115270	−0.00396542
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	7.36384	0.252726
$850$	0	0
$851$	6.24134	0.213950
$852$	0	0
$853$	−22.3888	−0.766579	−0.383289	−	0.923628i	$-0.625209\pi$
$853$	−22.3888	−0.766579	−0.383289	+	0.923628i	$0.625209\pi$
$854$	0	0
$855$	0.861457	0.0294612
$856$	0	0
$857$	−20.3039	−0.693567	−0.346784	−	0.937945i	$-0.612726\pi$
$857$	−20.3039	−0.693567	−0.346784	+	0.937945i	$0.612726\pi$
$858$	0	0
$859$	−11.6094	−0.396107	−0.198054	−	0.980191i	$-0.563462\pi$
$859$	−11.6094	−0.396107	−0.198054	+	0.980191i	$0.563462\pi$
$860$	0	0
$861$	1.96724	0.0670435
$862$	0	0
$863$	−50.6846	−1.72533	−0.862663	−	0.505780i	$-0.831205\pi$
$863$	−50.6846	−1.72533	−0.862663	+	0.505780i	$0.831205\pi$
$864$	0	0
$865$	−0.197180	−0.00670432
$866$	0	0
$867$	−5.32472	−0.180837
$868$	0	0
$869$	3.16884	0.107495
$870$	0	0
$871$	6.75900	0.229020
$872$	0	0
$873$	−48.5039	−1.64161
$874$	0	0
$875$	1.15117	0.0389167
$876$	0	0
$877$	19.9973	0.675260	0.337630	−	0.941279i	$-0.390375\pi$
$877$	19.9973	0.675260	0.337630	+	0.941279i	$0.390375\pi$
$878$	0	0
$879$	−8.46545	−0.285532
$880$	0	0
$881$	5.42390	0.182736	0.0913679	−	0.995817i	$-0.470876\pi$
$881$	5.42390	0.182736	0.0913679	+	0.995817i	$0.470876\pi$
$882$	0	0
$883$	−28.0604	−0.944306	−0.472153	−	0.881517i	$-0.656523\pi$
$883$	−28.0604	−0.944306	−0.472153	+	0.881517i	$0.656523\pi$
$884$	0	0
$885$	0.426031	0.0143209
$886$	0	0
$887$	5.44069	0.182680	0.0913402	−	0.995820i	$-0.470885\pi$
$887$	5.44069	0.182680	0.0913402	+	0.995820i	$0.470885\pi$
$888$	0	0
$889$	−6.25037	−0.209631
$890$	0	0
$891$	−8.26418	−0.276860
$892$	0	0
$893$	−27.2825	−0.912975
$894$	0	0
$895$	−1.01724	−0.0340027
$896$	0	0
$897$	0.468391	0.0156391
$898$	0	0
$899$	−42.3317	−1.41184
$900$	0	0
$901$	−79.7870	−2.65809
$902$	0	0
$903$	−0.598106	−0.0199037
$904$	0	0
$905$	−0.678530	−0.0225551
$906$	0	0
$907$	−36.3503	−1.20699	−0.603496	−	0.797366i	$-0.706226\pi$
$907$	−36.3503	−1.20699	−0.603496	+	0.797366i	$0.706226\pi$
$908$	0	0
$909$	−38.9949	−1.29338
$910$	0	0
$911$	−44.9078	−1.48786	−0.743931	−	0.668256i	$-0.767041\pi$
$911$	−44.9078	−1.48786	−0.743931	+	0.668256i	$0.767041\pi$
$912$	0	0
$913$	−2.81798	−0.0932615
$914$	0	0
$915$	−0.476441	−0.0157507
$916$	0	0
$917$	17.8784	0.590397
$918$	0	0
$919$	18.5831	0.613000	0.306500	−	0.951871i	$-0.400842\pi$
$919$	18.5831	0.613000	0.306500	+	0.951871i	$0.400842\pi$
$920$	0	0
$921$	−6.65223	−0.219198
$922$	0	0
$923$	6.49199	0.213687
$924$	0	0
$925$	−19.0875	−0.627591
$926$	0	0
$927$	−23.3399	−0.766582
$928$	0	0
$929$	27.1285	0.890057	0.445029	−	0.895516i	$-0.353193\pi$
$929$	27.1285	0.890057	0.445029	+	0.895516i	$0.353193\pi$
$930$	0	0
$931$	2.56157	0.0839522
$932$	0	0
$933$	−1.00538	−0.0329147
$934$	0	0
$935$	−0.687158	−0.0224725
$936$	0	0
$937$	−5.47991	−0.179021	−0.0895104	−	0.995986i	$-0.528530\pi$
$937$	−5.47991	−0.179021	−0.0895104	+	0.995986i	$0.528530\pi$
$938$	0	0
$939$	−3.50252	−0.114300
$940$	0	0
$941$	−43.5989	−1.42129	−0.710643	−	0.703553i	$-0.751596\pi$
$941$	−43.5989	−1.42129	−0.710643	+	0.703553i	$0.751596\pi$
$942$	0	0
$943$	−11.1670	−0.363649
$944$	0	0
$945$	−0.195938	−0.00637388
$946$	0	0
$947$	18.4906	0.600865	0.300433	−	0.953803i	$-0.402869\pi$
$947$	18.4906	0.600865	0.300433	+	0.953803i	$0.402869\pi$
$948$	0	0
$949$	12.6248	0.409819
$950$	0	0
$951$	8.80695	0.285585
$952$	0	0
$953$	47.6508	1.54356	0.771781	−	0.635889i	$-0.219366\pi$
$953$	47.6508	1.54356	0.771781	+	0.635889i	$0.219366\pi$
$954$	0	0
$955$	−1.79847	−0.0581972
$956$	0	0
$957$	−1.22771	−0.0396862
$958$	0	0
$959$	15.9046	0.513585
$960$	0	0
$961$	67.1002	2.16452
$962$	0	0
$963$	−40.5276	−1.30598
$964$	0	0
$965$	−1.60801	−0.0517637
$966$	0	0
$967$	45.7793	1.47216	0.736081	−	0.676893i	$-0.236674\pi$
$967$	45.7793	1.47216	0.736081	+	0.676893i	$0.236674\pi$
$968$	0	0
$969$	4.38642	0.140912
$970$	0	0
$971$	−19.9612	−0.640585	−0.320293	−	0.947319i	$-0.603781\pi$
$971$	−19.9612	−0.640585	−0.320293	+	0.947319i	$0.603781\pi$
$972$	0	0
$973$	1.65603	0.0530900
$974$	0	0
$975$	−1.43245	−0.0458751
$976$	0	0
$977$	32.3841	1.03606	0.518029	−	0.855363i	$-0.326666\pi$
$977$	32.3841	1.03606	0.518029	+	0.855363i	$0.326666\pi$
$978$	0	0
$979$	−13.7455	−0.439308
$980$	0	0
$981$	22.5008	0.718395
$982$	0	0
$983$	42.4144	1.35281	0.676404	−	0.736531i	$-0.263537\pi$
$983$	42.4144	1.35281	0.676404	+	0.736531i	$0.263537\pi$
$984$	0	0
$985$	−0.533794	−0.0170081
$986$	0	0
$987$	3.05944	0.0973831
$988$	0	0
$989$	3.39514	0.107959
$990$	0	0
$991$	−8.32557	−0.264470	−0.132235	−	0.991218i	$-0.542215\pi$
$991$	−8.32557	−0.264470	−0.132235	+	0.991218i	$0.542215\pi$
$992$	0	0
$993$	3.86846	0.122762
$994$	0	0
$995$	1.45089	0.0459964
$996$	0	0
$997$	−40.2962	−1.27619	−0.638097	−	0.769956i	$-0.720278\pi$
$997$	−40.2962	−1.27619	−0.638097	+	0.769956i	$0.720278\pi$
$998$	0	0
$999$	6.50632	0.205851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.k.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.k.1.3	✓	6	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-0.287253 q^{3} -0.115270 q^{5} +1.00000 q^{7} -2.91749 q^{9} +O(q^{10})\)	\(q-0.287253 q^{3} -0.115270 q^{5} +1.00000 q^{7} -2.91749 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.0331118 q^{15} -5.96127 q^{17} +2.56157 q^{19} -0.287253 q^{21} +1.63059 q^{23} -4.98671 q^{25} +1.69982 q^{27} -4.27397 q^{29} +9.90455 q^{31} +0.287253 q^{33} -0.115270 q^{35} +3.82766 q^{37} +0.287253 q^{39} -6.84847 q^{41} +2.08216 q^{43} +0.336300 q^{45} -10.6507 q^{47} +1.00000 q^{49} +1.71239 q^{51} +13.3842 q^{53} +0.115270 q^{55} -0.735820 q^{57} +12.8664 q^{59} -14.3889 q^{61} -2.91749 q^{63} +0.115270 q^{65} -6.75900 q^{67} -0.468391 q^{69} -6.49199 q^{71} -12.6248 q^{73} +1.43245 q^{75} -1.00000 q^{77} -3.16884 q^{79} +8.26418 q^{81} +2.81798 q^{83} +0.687158 q^{85} +1.22771 q^{87} +13.7455 q^{89} -1.00000 q^{91} -2.84511 q^{93} -0.295274 q^{95} +16.6252 q^{97} +2.91749 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9}+O(q^{10}) \) 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9	\( 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9} - 6 q^{11} - 6 q^{13} - q^{15} + 2 q^{19} - q^{21} + 11 q^{23} + 9 q^{25} - q^{27} + 14 q^{29} + 21 q^{31} + q^{33} + q^{35} - 5 q^{37} + q^{39} + 12 q^{43} + 2 q^{45} + 2 q^{47} + 6 q^{49} - 16 q^{51} + 2 q^{53} - q^{55} - 12 q^{57} + 3 q^{59} - 18 q^{61} + 9 q^{63} - q^{65} - 25 q^{67} + 29 q^{69} - 11 q^{71} + 14 q^{73} - 42 q^{75} - 6 q^{77} + 24 q^{79} + 42 q^{81} + 24 q^{83} - 46 q^{85} - 16 q^{87} + 55 q^{89} - 6 q^{91} + 41 q^{93} - 30 q^{95} + 37 q^{97} - 9 q^{99}+O(q^{100}) \) 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9 - 6 * q^11 - 6 * q^13 - q^15 + 2 * q^19 - q^21 + 11 * q^23 + 9 * q^25 - q^27 + 14 * q^29 + 21 * q^31 + q^33 + q^35 - 5 * q^37 + q^39 + 12 * q^43 + 2 * q^45 + 2 * q^47 + 6 * q^49 - 16 * q^51 + 2 * q^53 - q^55 - 12 * q^57 + 3 * q^59 - 18 * q^61 + 9 * q^63 - q^65 - 25 * q^67 + 29 * q^69 - 11 * q^71 + 14 * q^73 - 42 * q^75 - 6 * q^77 + 24 * q^79 + 42 * q^81 + 24 * q^83 - 46 * q^85 - 16 * q^87 + 55 * q^89 - 6 * q^91 + 41 * q^93 - 30 * q^95 + 37 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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