LMFDB - Embedded newform 8024.2.a.bc.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.270435 q^{3} -0.155002 q^{5} -4.46551 q^{7} -2.92686 q^{9} +O(q^{10})$	$q+0.270435 q^{3} -0.155002 q^{5} -4.46551 q^{7} -2.92686 q^{9} -4.65618 q^{11} +3.82056 q^{13} -0.0419180 q^{15} -1.00000 q^{17} -7.06910 q^{19} -1.20763 q^{21} +4.51801 q^{23} -4.97597 q^{25} -1.60283 q^{27} +7.41624 q^{29} +1.74215 q^{31} -1.25919 q^{33} +0.692163 q^{35} -10.3393 q^{37} +1.03321 q^{39} -8.28516 q^{41} -12.4900 q^{43} +0.453670 q^{45} +3.61317 q^{47} +12.9408 q^{49} -0.270435 q^{51} -1.45405 q^{53} +0.721717 q^{55} -1.91173 q^{57} -1.00000 q^{59} +8.19393 q^{61} +13.0699 q^{63} -0.592194 q^{65} -11.3162 q^{67} +1.22183 q^{69} -5.87317 q^{71} -2.61360 q^{73} -1.34568 q^{75} +20.7922 q^{77} -3.41396 q^{79} +8.34713 q^{81} -4.12605 q^{83} +0.155002 q^{85} +2.00561 q^{87} +11.1480 q^{89} -17.0607 q^{91} +0.471139 q^{93} +1.09572 q^{95} -15.5037 q^{97} +13.6280 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * 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
$3$	0.270435	0.156136	0.0780679	−	0.996948i	$-0.475125\pi$
$3$	0.270435	0.156136	0.0780679	+	0.996948i	$0.475125\pi$
$4$	0	0
$5$	−0.155002	−0.0693190	−0.0346595	−	0.999399i	$-0.511035\pi$
$5$	−0.155002	−0.0693190	−0.0346595	+	0.999399i	$0.511035\pi$
$6$	0	0
$7$	−4.46551	−1.68780	−0.843902	−	0.536497i	$-0.819747\pi$
$7$	−4.46551	−1.68780	−0.843902	+	0.536497i	$0.819747\pi$
$8$	0	0
$9$	−2.92686	−0.975622
$10$	0	0
$11$	−4.65618	−1.40389	−0.701945	−	0.712231i	$-0.747685\pi$
$11$	−4.65618	−1.40389	−0.701945	+	0.712231i	$0.747685\pi$
$12$	0	0
$13$	3.82056	1.05963	0.529816	−	0.848113i	$-0.322261\pi$
$13$	3.82056	1.05963	0.529816	+	0.848113i	$0.322261\pi$
$14$	0	0
$15$	−0.0419180	−0.0108232
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−7.06910	−1.62176	−0.810881	−	0.585211i	$-0.801012\pi$
$19$	−7.06910	−1.62176	−0.810881	+	0.585211i	$0.801012\pi$
$20$	0	0
$21$	−1.20763	−0.263527
$22$	0	0
$23$	4.51801	0.942070	0.471035	−	0.882115i	$-0.343881\pi$
$23$	4.51801	0.942070	0.471035	+	0.882115i	$0.343881\pi$
$24$	0	0
$25$	−4.97597	−0.995195
$26$	0	0
$27$	−1.60283	−0.308465
$28$	0	0
$29$	7.41624	1.37716	0.688580	−	0.725160i	$-0.258234\pi$
$29$	7.41624	1.37716	0.688580	+	0.725160i	$0.258234\pi$
$30$	0	0
$31$	1.74215	0.312900	0.156450	−	0.987686i	$-0.449995\pi$
$31$	1.74215	0.312900	0.156450	+	0.987686i	$0.449995\pi$
$32$	0	0
$33$	−1.25919	−0.219198
$34$	0	0
$35$	0.692163	0.116997
$36$	0	0
$37$	−10.3393	−1.69978	−0.849889	−	0.526962i	$-0.823331\pi$
$37$	−10.3393	−1.69978	−0.849889	+	0.526962i	$0.823331\pi$
$38$	0	0
$39$	1.03321	0.165447
$40$	0	0
$41$	−8.28516	−1.29392	−0.646962	−	0.762522i	$-0.723961\pi$
$41$	−8.28516	−1.29392	−0.646962	+	0.762522i	$0.723961\pi$
$42$	0	0
$43$	−12.4900	−1.90470	−0.952351	−	0.305005i	$-0.901342\pi$
$43$	−12.4900	−1.90470	−0.952351	+	0.305005i	$0.901342\pi$
$44$	0	0
$45$	0.453670	0.0676291
$46$	0	0
$47$	3.61317	0.527035	0.263518	−	0.964655i	$-0.415117\pi$
$47$	3.61317	0.527035	0.263518	+	0.964655i	$0.415117\pi$
$48$	0	0
$49$	12.9408	1.84868
$50$	0	0
$51$	−0.270435	−0.0378685
$52$	0	0
$53$	−1.45405	−0.199729	−0.0998643	−	0.995001i	$-0.531841\pi$
$53$	−1.45405	−0.199729	−0.0998643	+	0.995001i	$0.531841\pi$
$54$	0	0
$55$	0.721717	0.0973163
$56$	0	0
$57$	−1.91173	−0.253215
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	8.19393	1.04913	0.524563	−	0.851372i	$-0.324229\pi$
$61$	8.19393	1.04913	0.524563	+	0.851372i	$0.324229\pi$
$62$	0	0
$63$	13.0699	1.64666
$64$	0	0
$65$	−0.592194	−0.0734527
$66$	0	0
$67$	−11.3162	−1.38249	−0.691245	−	0.722621i	$-0.742937\pi$
$67$	−11.3162	−1.38249	−0.691245	+	0.722621i	$0.742937\pi$
$68$	0	0
$69$	1.22183	0.147091
$70$	0	0
$71$	−5.87317	−0.697017	−0.348508	−	0.937306i	$-0.613312\pi$
$71$	−5.87317	−0.697017	−0.348508	+	0.937306i	$0.613312\pi$
$72$	0	0
$73$	−2.61360	−0.305899	−0.152949	−	0.988234i	$-0.548877\pi$
$73$	−2.61360	−0.305899	−0.152949	+	0.988234i	$0.548877\pi$
$74$	0	0
$75$	−1.34568	−0.155386
$76$	0	0
$77$	20.7922	2.36949
$78$	0	0
$79$	−3.41396	−0.384101	−0.192050	−	0.981385i	$-0.561514\pi$
$79$	−3.41396	−0.384101	−0.192050	+	0.981385i	$0.561514\pi$
$80$	0	0
$81$	8.34713	0.927459
$82$	0	0
$83$	−4.12605	−0.452893	−0.226446	−	0.974024i	$-0.572711\pi$
$83$	−4.12605	−0.452893	−0.226446	+	0.974024i	$0.572711\pi$
$84$	0	0
$85$	0.155002	0.0168123
$86$	0	0
$87$	2.00561	0.215024
$88$	0	0
$89$	11.1480	1.18168	0.590840	−	0.806789i	$-0.298796\pi$
$89$	11.1480	1.18168	0.590840	+	0.806789i	$0.298796\pi$
$90$	0	0
$91$	−17.0607	−1.78845
$92$	0	0
$93$	0.471139	0.0488548
$94$	0	0
$95$	1.09572	0.112419
$96$	0	0
$97$	−15.5037	−1.57416	−0.787081	−	0.616850i	$-0.788409\pi$
$97$	−15.5037	−1.57416	−0.787081	+	0.616850i	$0.788409\pi$
$98$	0	0
$99$	13.6280	1.36967
$100$	0	0
$101$	12.1722	1.21118	0.605588	−	0.795778i	$-0.292938\pi$
$101$	12.1722	1.21118	0.605588	+	0.795778i	$0.292938\pi$
$102$	0	0
$103$	−9.88400	−0.973900	−0.486950	−	0.873430i	$-0.661891\pi$
$103$	−9.88400	−0.973900	−0.486950	+	0.873430i	$0.661891\pi$
$104$	0	0
$105$	0.187185	0.0182674
$106$	0	0
$107$	1.38640	0.134028	0.0670140	−	0.997752i	$-0.478653\pi$
$107$	1.38640	0.134028	0.0670140	+	0.997752i	$0.478653\pi$
$108$	0	0
$109$	−17.3828	−1.66497	−0.832483	−	0.554050i	$-0.813082\pi$
$109$	−17.3828	−1.66497	−0.832483	+	0.554050i	$0.813082\pi$
$110$	0	0
$111$	−2.79612	−0.265396
$112$	0	0
$113$	6.31341	0.593916	0.296958	−	0.954891i	$-0.404028\pi$
$113$	6.31341	0.593916	0.296958	+	0.954891i	$0.404028\pi$
$114$	0	0
$115$	−0.700300	−0.0653033
$116$	0	0
$117$	−11.1823	−1.03380
$118$	0	0
$119$	4.46551	0.409353
$120$	0	0
$121$	10.6800	0.970909
$122$	0	0
$123$	−2.24060	−0.202028
$124$	0	0
$125$	1.54630	0.138305
$126$	0	0
$127$	8.39536	0.744967	0.372484	−	0.928039i	$-0.378506\pi$
$127$	8.39536	0.744967	0.372484	+	0.928039i	$0.378506\pi$
$128$	0	0
$129$	−3.37772	−0.297392
$130$	0	0
$131$	−12.4736	−1.08983	−0.544914	−	0.838492i	$-0.683438\pi$
$131$	−12.4736	−1.08983	−0.544914	+	0.838492i	$0.683438\pi$
$132$	0	0
$133$	31.5671	2.73722
$134$	0	0
$135$	0.248442	0.0213825
$136$	0	0
$137$	17.9956	1.53747	0.768733	−	0.639570i	$-0.220888\pi$
$137$	17.9956	1.53747	0.768733	+	0.639570i	$0.220888\pi$
$138$	0	0
$139$	0.860412	0.0729792	0.0364896	−	0.999334i	$-0.488382\pi$
$139$	0.860412	0.0729792	0.0364896	+	0.999334i	$0.488382\pi$
$140$	0	0
$141$	0.977129	0.0822891
$142$	0	0
$143$	−17.7892	−1.48761
$144$	0	0
$145$	−1.14953	−0.0954634
$146$	0	0
$147$	3.49965	0.288646
$148$	0	0
$149$	6.92181	0.567057	0.283529	−	0.958964i	$-0.408495\pi$
$149$	6.92181	0.567057	0.283529	+	0.958964i	$0.408495\pi$
$150$	0	0
$151$	15.3290	1.24746	0.623728	−	0.781641i	$-0.285617\pi$
$151$	15.3290	1.24746	0.623728	+	0.781641i	$0.285617\pi$
$152$	0	0
$153$	2.92686	0.236623
$154$	0	0
$155$	−0.270037	−0.0216899
$156$	0	0
$157$	16.2996	1.30085	0.650426	−	0.759569i	$-0.274590\pi$
$157$	16.2996	1.30085	0.650426	+	0.759569i	$0.274590\pi$
$158$	0	0
$159$	−0.393225	−0.0311848
$160$	0	0
$161$	−20.1752	−1.59003
$162$	0	0
$163$	22.7466	1.78165	0.890827	−	0.454342i	$-0.150126\pi$
$163$	22.7466	1.78165	0.890827	+	0.454342i	$0.150126\pi$
$164$	0	0
$165$	0.195178	0.0151946
$166$	0	0
$167$	−5.09921	−0.394588	−0.197294	−	0.980344i	$-0.563215\pi$
$167$	−5.09921	−0.394588	−0.197294	+	0.980344i	$0.563215\pi$
$168$	0	0
$169$	1.59667	0.122821
$170$	0	0
$171$	20.6903	1.58223
$172$	0	0
$173$	10.5671	0.803405	0.401703	−	0.915770i	$-0.368419\pi$
$173$	10.5671	0.803405	0.401703	+	0.915770i	$0.368419\pi$
$174$	0	0
$175$	22.2203	1.67969
$176$	0	0
$177$	−0.270435	−0.0203272
$178$	0	0
$179$	0.317599	0.0237385	0.0118692	−	0.999930i	$-0.496222\pi$
$179$	0.317599	0.0237385	0.0118692	+	0.999930i	$0.496222\pi$
$180$	0	0
$181$	−11.0522	−0.821506	−0.410753	−	0.911747i	$-0.634734\pi$
$181$	−11.0522	−0.821506	−0.410753	+	0.911747i	$0.634734\pi$
$182$	0	0
$183$	2.21593	0.163806
$184$	0	0
$185$	1.60262	0.117827
$186$	0	0
$187$	4.65618	0.340493
$188$	0	0
$189$	7.15747	0.520629
$190$	0	0
$191$	8.28865	0.599746	0.299873	−	0.953979i	$-0.403056\pi$
$191$	8.28865	0.599746	0.299873	+	0.953979i	$0.403056\pi$
$192$	0	0
$193$	−16.8397	−1.21215	−0.606074	−	0.795409i	$-0.707256\pi$
$193$	−16.8397	−1.21215	−0.606074	+	0.795409i	$0.707256\pi$
$194$	0	0
$195$	−0.160150	−0.0114686
$196$	0	0
$197$	22.6197	1.61159	0.805794	−	0.592196i	$-0.201739\pi$
$197$	22.6197	1.61159	0.805794	+	0.592196i	$0.201739\pi$
$198$	0	0
$199$	24.7156	1.75204	0.876020	−	0.482275i	$-0.160189\pi$
$199$	24.7156	1.75204	0.876020	+	0.482275i	$0.160189\pi$
$200$	0	0
$201$	−3.06029	−0.215856
$202$	0	0
$203$	−33.1173	−2.32438
$204$	0	0
$205$	1.28422	0.0896936
$206$	0	0
$207$	−13.2236	−0.919104
$208$	0	0
$209$	32.9150	2.27678
$210$	0	0
$211$	23.9773	1.65066	0.825332	−	0.564648i	$-0.190988\pi$
$211$	23.9773	1.65066	0.825332	+	0.564648i	$0.190988\pi$
$212$	0	0
$213$	−1.58831	−0.108829
$214$	0	0
$215$	1.93597	0.132032
$216$	0	0
$217$	−7.77960	−0.528114
$218$	0	0
$219$	−0.706809	−0.0477617
$220$	0	0
$221$	−3.82056	−0.256999
$222$	0	0
$223$	20.3068	1.35984	0.679922	−	0.733284i	$-0.262014\pi$
$223$	20.3068	1.35984	0.679922	+	0.733284i	$0.262014\pi$
$224$	0	0
$225$	14.5640	0.970934
$226$	0	0
$227$	19.9570	1.32459	0.662295	−	0.749243i	$-0.269583\pi$
$227$	19.9570	1.32459	0.662295	+	0.749243i	$0.269583\pi$
$228$	0	0
$229$	11.4261	0.755055	0.377527	−	0.925998i	$-0.376774\pi$
$229$	11.4261	0.755055	0.377527	+	0.925998i	$0.376774\pi$
$230$	0	0
$231$	5.62295	0.369963
$232$	0	0
$233$	4.72904	0.309810	0.154905	−	0.987929i	$-0.450493\pi$
$233$	4.72904	0.309810	0.154905	+	0.987929i	$0.450493\pi$
$234$	0	0
$235$	−0.560049	−0.0365336
$236$	0	0
$237$	−0.923256	−0.0599719
$238$	0	0
$239$	−18.2042	−1.17753	−0.588767	−	0.808303i	$-0.700387\pi$
$239$	−18.2042	−1.17753	−0.588767	+	0.808303i	$0.700387\pi$
$240$	0	0
$241$	−18.2792	−1.17746	−0.588732	−	0.808328i	$-0.700373\pi$
$241$	−18.2792	−1.17746	−0.588732	+	0.808328i	$0.700373\pi$
$242$	0	0
$243$	7.06586	0.453275
$244$	0	0
$245$	−2.00585	−0.128149
$246$	0	0
$247$	−27.0079	−1.71847
$248$	0	0
$249$	−1.11583	−0.0707128
$250$	0	0
$251$	−31.6152	−1.99553	−0.997766	−	0.0668017i	$-0.978721\pi$
$251$	−31.6152	−1.99553	−0.997766	+	0.0668017i	$0.978721\pi$
$252$	0	0
$253$	−21.0367	−1.32256
$254$	0	0
$255$	0.0419180	0.00262501
$256$	0	0
$257$	8.38821	0.523242	0.261621	−	0.965171i	$-0.415743\pi$
$257$	8.38821	0.523242	0.261621	+	0.965171i	$0.415743\pi$
$258$	0	0
$259$	46.1704	2.86889
$260$	0	0
$261$	−21.7063	−1.34359
$262$	0	0
$263$	−0.329341	−0.0203080	−0.0101540	−	0.999948i	$-0.503232\pi$
$263$	−0.329341	−0.0203080	−0.0101540	+	0.999948i	$0.503232\pi$
$264$	0	0
$265$	0.225380	0.0138450
$266$	0	0
$267$	3.01480	0.184503
$268$	0	0
$269$	1.81648	0.110753	0.0553764	−	0.998466i	$-0.482364\pi$
$269$	1.81648	0.110753	0.0553764	+	0.998466i	$0.482364\pi$
$270$	0	0
$271$	7.28153	0.442321	0.221161	−	0.975237i	$-0.429016\pi$
$271$	7.28153	0.442321	0.221161	+	0.975237i	$0.429016\pi$
$272$	0	0
$273$	−4.61383	−0.279241
$274$	0	0
$275$	23.1690	1.39714
$276$	0	0
$277$	14.9729	0.899636	0.449818	−	0.893120i	$-0.351489\pi$
$277$	14.9729	0.899636	0.449818	+	0.893120i	$0.351489\pi$
$278$	0	0
$279$	−5.09904	−0.305272
$280$	0	0
$281$	−7.92270	−0.472629	−0.236314	−	0.971677i	$-0.575940\pi$
$281$	−7.92270	−0.472629	−0.236314	+	0.971677i	$0.575940\pi$
$282$	0	0
$283$	18.0674	1.07399	0.536996	−	0.843585i	$-0.319559\pi$
$283$	18.0674	1.07399	0.536996	+	0.843585i	$0.319559\pi$
$284$	0	0
$285$	0.296322	0.0175526
$286$	0	0
$287$	36.9975	2.18389
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−4.19275	−0.245783
$292$	0	0
$293$	−14.7678	−0.862743	−0.431371	−	0.902174i	$-0.641970\pi$
$293$	−14.7678	−0.862743	−0.431371	+	0.902174i	$0.641970\pi$
$294$	0	0
$295$	0.155002	0.00902457
$296$	0	0
$297$	7.46308	0.433052
$298$	0	0
$299$	17.2613	0.998248
$300$	0	0
$301$	55.7741	3.21476
$302$	0	0
$303$	3.29178	0.189108
$304$	0	0
$305$	−1.27008	−0.0727243
$306$	0	0
$307$	−12.4597	−0.711110	−0.355555	−	0.934655i	$-0.615708\pi$
$307$	−12.4597	−0.711110	−0.355555	+	0.934655i	$0.615708\pi$
$308$	0	0
$309$	−2.67298	−0.152061
$310$	0	0
$311$	−13.0549	−0.740277	−0.370138	−	0.928977i	$-0.620690\pi$
$311$	−13.0549	−0.740277	−0.370138	+	0.928977i	$0.620690\pi$
$312$	0	0
$313$	−12.7756	−0.722121	−0.361061	−	0.932542i	$-0.617585\pi$
$313$	−12.7756	−0.722121	−0.361061	+	0.932542i	$0.617585\pi$
$314$	0	0
$315$	−2.02587	−0.114145
$316$	0	0
$317$	−2.32696	−0.130695	−0.0653475	−	0.997863i	$-0.520816\pi$
$317$	−2.32696	−0.130695	−0.0653475	+	0.997863i	$0.520816\pi$
$318$	0	0
$319$	−34.5313	−1.93338
$320$	0	0
$321$	0.374930	0.0209266
$322$	0	0
$323$	7.06910	0.393335
$324$	0	0
$325$	−19.0110	−1.05454
$326$	0	0
$327$	−4.70091	−0.259961
$328$	0	0
$329$	−16.1347	−0.889533
$330$	0	0
$331$	−20.6366	−1.13429	−0.567144	−	0.823618i	$-0.691952\pi$
$331$	−20.6366	−1.13429	−0.567144	+	0.823618i	$0.691952\pi$
$332$	0	0
$333$	30.2619	1.65834
$334$	0	0
$335$	1.75403	0.0958328
$336$	0	0
$337$	−3.33801	−0.181833	−0.0909166	−	0.995859i	$-0.528980\pi$
$337$	−3.33801	−0.181833	−0.0909166	+	0.995859i	$0.528980\pi$
$338$	0	0
$339$	1.70737	0.0927315
$340$	0	0
$341$	−8.11177	−0.439277
$342$	0	0
$343$	−26.5287	−1.43241
$344$	0	0
$345$	−0.189386	−0.0101962
$346$	0	0
$347$	−18.7066	−1.00423	−0.502113	−	0.864802i	$-0.667444\pi$
$347$	−18.7066	−1.00423	−0.502113	+	0.864802i	$0.667444\pi$
$348$	0	0
$349$	−2.47369	−0.132413	−0.0662067	−	0.997806i	$-0.521090\pi$
$349$	−2.47369	−0.132413	−0.0662067	+	0.997806i	$0.521090\pi$
$350$	0	0
$351$	−6.12372	−0.326860
$352$	0	0
$353$	18.5919	0.989549	0.494774	−	0.869021i	$-0.335251\pi$
$353$	18.5919	0.989549	0.494774	+	0.869021i	$0.335251\pi$
$354$	0	0
$355$	0.910353	0.0483165
$356$	0	0
$357$	1.20763	0.0639146
$358$	0	0
$359$	3.84155	0.202749	0.101375	−	0.994848i	$-0.467676\pi$
$359$	3.84155	0.202749	0.101375	+	0.994848i	$0.467676\pi$
$360$	0	0
$361$	30.9721	1.63011
$362$	0	0
$363$	2.88825	0.151594
$364$	0	0
$365$	0.405113	0.0212046
$366$	0	0
$367$	−31.9230	−1.66637	−0.833183	−	0.552997i	$-0.813484\pi$
$367$	−31.9230	−1.66637	−0.833183	+	0.552997i	$0.813484\pi$
$368$	0	0
$369$	24.2495	1.26238
$370$	0	0
$371$	6.49306	0.337103
$372$	0	0
$373$	28.8553	1.49407	0.747036	−	0.664783i	$-0.231476\pi$
$373$	28.8553	1.49407	0.747036	+	0.664783i	$0.231476\pi$
$374$	0	0
$375$	0.418173	0.0215944
$376$	0	0
$377$	28.3342	1.45928
$378$	0	0
$379$	15.6921	0.806051	0.403025	−	0.915189i	$-0.367959\pi$
$379$	15.6921	0.806051	0.403025	+	0.915189i	$0.367959\pi$
$380$	0	0
$381$	2.27040	0.116316
$382$	0	0
$383$	7.01849	0.358628	0.179314	−	0.983792i	$-0.442612\pi$
$383$	7.01849	0.358628	0.179314	+	0.983792i	$0.442612\pi$
$384$	0	0
$385$	−3.22284	−0.164251
$386$	0	0
$387$	36.5564	1.85827
$388$	0	0
$389$	−1.94835	−0.0987854	−0.0493927	−	0.998779i	$-0.515729\pi$
$389$	−1.94835	−0.0987854	−0.0493927	+	0.998779i	$0.515729\pi$
$390$	0	0
$391$	−4.51801	−0.228485
$392$	0	0
$393$	−3.37331	−0.170161
$394$	0	0
$395$	0.529171	0.0266255
$396$	0	0
$397$	−34.6955	−1.74132	−0.870658	−	0.491890i	$-0.836306\pi$
$397$	−34.6955	−1.74132	−0.870658	+	0.491890i	$0.836306\pi$
$398$	0	0
$399$	8.53686	0.427378
$400$	0	0
$401$	−21.3296	−1.06515	−0.532574	−	0.846383i	$-0.678775\pi$
$401$	−21.3296	−1.06515	−0.532574	+	0.846383i	$0.678775\pi$
$402$	0	0
$403$	6.65599	0.331559
$404$	0	0
$405$	−1.29382	−0.0642905
$406$	0	0
$407$	48.1418	2.38630
$408$	0	0
$409$	4.37036	0.216101	0.108050	−	0.994145i	$-0.465539\pi$
$409$	4.37036	0.216101	0.108050	+	0.994145i	$0.465539\pi$
$410$	0	0
$411$	4.86664	0.240054
$412$	0	0
$413$	4.46551	0.219733
$414$	0	0
$415$	0.639546	0.0313941
$416$	0	0
$417$	0.232686	0.0113947
$418$	0	0
$419$	15.4493	0.754748	0.377374	−	0.926061i	$-0.376827\pi$
$419$	15.4493	0.754748	0.377374	+	0.926061i	$0.376827\pi$
$420$	0	0
$421$	−3.18205	−0.155083	−0.0775417	−	0.996989i	$-0.524707\pi$
$421$	−3.18205	−0.155083	−0.0775417	+	0.996989i	$0.524707\pi$
$422$	0	0
$423$	−10.5753	−0.514187
$424$	0	0
$425$	4.97597	0.241370
$426$	0	0
$427$	−36.5901	−1.77072
$428$	0	0
$429$	−4.81083	−0.232269
$430$	0	0
$431$	8.46075	0.407540	0.203770	−	0.979019i	$-0.434681\pi$
$431$	8.46075	0.407540	0.203770	+	0.979019i	$0.434681\pi$
$432$	0	0
$433$	1.81959	0.0874439	0.0437220	−	0.999044i	$-0.486078\pi$
$433$	1.81959	0.0874439	0.0437220	+	0.999044i	$0.486078\pi$
$434$	0	0
$435$	−0.310874	−0.0149053
$436$	0	0
$437$	−31.9382	−1.52781
$438$	0	0
$439$	22.8124	1.08878	0.544388	−	0.838833i	$-0.316762\pi$
$439$	22.8124	1.08878	0.544388	+	0.838833i	$0.316762\pi$
$440$	0	0
$441$	−37.8759	−1.80362
$442$	0	0
$443$	−37.1564	−1.76535	−0.882677	−	0.469981i	$-0.844261\pi$
$443$	−37.1564	−1.76535	−0.882677	+	0.469981i	$0.844261\pi$
$444$	0	0
$445$	−1.72795	−0.0819129
$446$	0	0
$447$	1.87190	0.0885379
$448$	0	0
$449$	−12.8376	−0.605844	−0.302922	−	0.953015i	$-0.597962\pi$
$449$	−12.8376	−0.605844	−0.302922	+	0.953015i	$0.597962\pi$
$450$	0	0
$451$	38.5772	1.81653
$452$	0	0
$453$	4.14550	0.194773
$454$	0	0
$455$	2.64445	0.123974
$456$	0	0
$457$	12.4559	0.582661	0.291330	−	0.956622i	$-0.405902\pi$
$457$	12.4559	0.582661	0.291330	+	0.956622i	$0.405902\pi$
$458$	0	0
$459$	1.60283	0.0748138
$460$	0	0
$461$	−0.417155	−0.0194288	−0.00971441	−	0.999953i	$-0.503092\pi$
$461$	−0.417155	−0.0194288	−0.00971441	+	0.999953i	$0.503092\pi$
$462$	0	0
$463$	−37.6950	−1.75184	−0.875919	−	0.482459i	$-0.839744\pi$
$463$	−37.6950	−1.75184	−0.875919	+	0.482459i	$0.839744\pi$
$464$	0	0
$465$	−0.0730275	−0.00338657
$466$	0	0
$467$	30.3692	1.40532	0.702658	−	0.711527i	$-0.251996\pi$
$467$	30.3692	1.40532	0.702658	+	0.711527i	$0.251996\pi$
$468$	0	0
$469$	50.5325	2.33337
$470$	0	0
$471$	4.40800	0.203110
$472$	0	0
$473$	58.1555	2.67399
$474$	0	0
$475$	35.1756	1.61397
$476$	0	0
$477$	4.25580	0.194860
$478$	0	0
$479$	22.8478	1.04394	0.521972	−	0.852962i	$-0.325196\pi$
$479$	22.8478	1.04394	0.521972	+	0.852962i	$0.325196\pi$
$480$	0	0
$481$	−39.5021	−1.80114
$482$	0	0
$483$	−5.45609	−0.248261
$484$	0	0
$485$	2.40310	0.109119
$486$	0	0
$487$	−0.684325	−0.0310097	−0.0155049	−	0.999880i	$-0.504936\pi$
$487$	−0.684325	−0.0310097	−0.0155049	+	0.999880i	$0.504936\pi$
$488$	0	0
$489$	6.15149	0.278180
$490$	0	0
$491$	−35.2117	−1.58908	−0.794541	−	0.607211i	$-0.792288\pi$
$491$	−35.2117	−1.58908	−0.794541	+	0.607211i	$0.792288\pi$
$492$	0	0
$493$	−7.41624	−0.334011
$494$	0	0
$495$	−2.11237	−0.0949439
$496$	0	0
$497$	26.2267	1.17643
$498$	0	0
$499$	−16.8691	−0.755166	−0.377583	−	0.925976i	$-0.623245\pi$
$499$	−16.8691	−0.755166	−0.377583	+	0.925976i	$0.623245\pi$
$500$	0	0
$501$	−1.37900	−0.0616094
$502$	0	0
$503$	−21.2212	−0.946207	−0.473103	−	0.881007i	$-0.656866\pi$
$503$	−21.2212	−0.946207	−0.473103	+	0.881007i	$0.656866\pi$
$504$	0	0
$505$	−1.88671	−0.0839575
$506$	0	0
$507$	0.431795	0.0191767
$508$	0	0
$509$	−5.99072	−0.265534	−0.132767	−	0.991147i	$-0.542386\pi$
$509$	−5.99072	−0.265534	−0.132767	+	0.991147i	$0.542386\pi$
$510$	0	0
$511$	11.6711	0.516297
$512$	0	0
$513$	11.3306	0.500257
$514$	0	0
$515$	1.53204	0.0675097
$516$	0	0
$517$	−16.8236	−0.739900
$518$	0	0
$519$	2.85773	0.125440
$520$	0	0
$521$	−19.9338	−0.873314	−0.436657	−	0.899628i	$-0.643838\pi$
$521$	−19.9338	−0.873314	−0.436657	+	0.899628i	$0.643838\pi$
$522$	0	0
$523$	−15.1828	−0.663899	−0.331949	−	0.943297i	$-0.607706\pi$
$523$	−15.1828	−0.663899	−0.331949	+	0.943297i	$0.607706\pi$
$524$	0	0
$525$	6.00914	0.262260
$526$	0	0
$527$	−1.74215	−0.0758893
$528$	0	0
$529$	−2.58761	−0.112505
$530$	0	0
$531$	2.92686	0.127015
$532$	0	0
$533$	−31.6539	−1.37108
$534$	0	0
$535$	−0.214894	−0.00929068
$536$	0	0
$537$	0.0858899	0.00370642
$538$	0	0
$539$	−60.2546	−2.59535
$540$	0	0
$541$	29.7759	1.28017	0.640084	−	0.768305i	$-0.278900\pi$
$541$	29.7759	1.28017	0.640084	+	0.768305i	$0.278900\pi$
$542$	0	0
$543$	−2.98891	−0.128267
$544$	0	0
$545$	2.69436	0.115414
$546$	0	0
$547$	17.0819	0.730370	0.365185	−	0.930935i	$-0.381006\pi$
$547$	17.0819	0.730370	0.365185	+	0.930935i	$0.381006\pi$
$548$	0	0
$549$	−23.9825	−1.02355
$550$	0	0
$551$	−52.4261	−2.23343
$552$	0	0
$553$	15.2451	0.648287
$554$	0	0
$555$	0.433404	0.0183970
$556$	0	0
$557$	19.7237	0.835720	0.417860	−	0.908511i	$-0.362780\pi$
$557$	19.7237	0.835720	0.417860	+	0.908511i	$0.362780\pi$
$558$	0	0
$559$	−47.7186	−2.01828
$560$	0	0
$561$	1.25919	0.0531632
$562$	0	0
$563$	−5.60203	−0.236097	−0.118049	−	0.993008i	$-0.537664\pi$
$563$	−5.60203	−0.236097	−0.118049	+	0.993008i	$0.537664\pi$
$564$	0	0
$565$	−0.978591	−0.0411696
$566$	0	0
$567$	−37.2742	−1.56537
$568$	0	0
$569$	−35.2345	−1.47711	−0.738553	−	0.674195i	$-0.764491\pi$
$569$	−35.2345	−1.47711	−0.738553	+	0.674195i	$0.764491\pi$
$570$	0	0
$571$	−25.2293	−1.05581	−0.527907	−	0.849302i	$-0.677023\pi$
$571$	−25.2293	−1.05581	−0.527907	+	0.849302i	$0.677023\pi$
$572$	0	0
$573$	2.24154	0.0936418
$574$	0	0
$575$	−22.4815	−0.937543
$576$	0	0
$577$	−6.25582	−0.260433	−0.130217	−	0.991486i	$-0.541567\pi$
$577$	−6.25582	−0.260433	−0.130217	+	0.991486i	$0.541567\pi$
$578$	0	0
$579$	−4.55404	−0.189260
$580$	0	0
$581$	18.4249	0.764395
$582$	0	0
$583$	6.77030	0.280397
$584$	0	0
$585$	1.73327	0.0716620
$586$	0	0
$587$	−29.4841	−1.21694	−0.608469	−	0.793578i	$-0.708216\pi$
$587$	−29.4841	−1.21694	−0.608469	+	0.793578i	$0.708216\pi$
$588$	0	0
$589$	−12.3154	−0.507449
$590$	0	0
$591$	6.11717	0.251627
$592$	0	0
$593$	4.32214	0.177489	0.0887445	−	0.996054i	$-0.471715\pi$
$593$	4.32214	0.177489	0.0887445	+	0.996054i	$0.471715\pi$
$594$	0	0
$595$	−0.692163	−0.0283759
$596$	0	0
$597$	6.68396	0.273556
$598$	0	0
$599$	−17.4214	−0.711818	−0.355909	−	0.934521i	$-0.615829\pi$
$599$	−17.4214	−0.711818	−0.355909	+	0.934521i	$0.615829\pi$
$600$	0	0
$601$	−38.9412	−1.58844	−0.794222	−	0.607628i	$-0.792121\pi$
$601$	−38.9412	−1.58844	−0.794222	+	0.607628i	$0.792121\pi$
$602$	0	0
$603$	33.1209	1.34879
$604$	0	0
$605$	−1.65542	−0.0673024
$606$	0	0
$607$	−18.2984	−0.742710	−0.371355	−	0.928491i	$-0.621107\pi$
$607$	−18.2984	−0.742710	−0.371355	+	0.928491i	$0.621107\pi$
$608$	0	0
$609$	−8.95608	−0.362919
$610$	0	0
$611$	13.8043	0.558464
$612$	0	0
$613$	2.17819	0.0879763	0.0439882	−	0.999032i	$-0.485994\pi$
$613$	2.17819	0.0879763	0.0439882	+	0.999032i	$0.485994\pi$
$614$	0	0
$615$	0.347297	0.0140044
$616$	0	0
$617$	16.5950	0.668091	0.334045	−	0.942557i	$-0.391586\pi$
$617$	16.5950	0.668091	0.334045	+	0.942557i	$0.391586\pi$
$618$	0	0
$619$	−28.4714	−1.14436	−0.572181	−	0.820128i	$-0.693902\pi$
$619$	−28.4714	−1.14436	−0.572181	+	0.820128i	$0.693902\pi$
$620$	0	0
$621$	−7.24161	−0.290596
$622$	0	0
$623$	−49.7813	−1.99445
$624$	0	0
$625$	24.6402	0.985608
$626$	0	0
$627$	8.90137	0.355486
$628$	0	0
$629$	10.3393	0.412257
$630$	0	0
$631$	−7.37945	−0.293771	−0.146886	−	0.989153i	$-0.546925\pi$
$631$	−7.37945	−0.293771	−0.146886	+	0.989153i	$0.546925\pi$
$632$	0	0
$633$	6.48430	0.257728
$634$	0	0
$635$	−1.30130	−0.0516404
$636$	0	0
$637$	49.4411	1.95893
$638$	0	0
$639$	17.1900	0.680025
$640$	0	0
$641$	9.81958	0.387850	0.193925	−	0.981016i	$-0.437878\pi$
$641$	9.81958	0.387850	0.193925	+	0.981016i	$0.437878\pi$
$642$	0	0
$643$	−15.2065	−0.599685	−0.299843	−	0.953989i	$-0.596934\pi$
$643$	−15.2065	−0.599685	−0.299843	+	0.953989i	$0.596934\pi$
$644$	0	0
$645$	0.523554	0.0206149
$646$	0	0
$647$	−12.3530	−0.485648	−0.242824	−	0.970070i	$-0.578074\pi$
$647$	−12.3530	−0.485648	−0.242824	+	0.970070i	$0.578074\pi$
$648$	0	0
$649$	4.65618	0.182771
$650$	0	0
$651$	−2.10388	−0.0824574
$652$	0	0
$653$	−28.5485	−1.11719	−0.558595	−	0.829440i	$-0.688659\pi$
$653$	−28.5485	−1.11719	−0.558595	+	0.829440i	$0.688659\pi$
$654$	0	0
$655$	1.93344	0.0755457
$656$	0	0
$657$	7.64965	0.298441
$658$	0	0
$659$	45.9069	1.78828	0.894140	−	0.447788i	$-0.147788\pi$
$659$	45.9069	1.78828	0.894140	+	0.447788i	$0.147788\pi$
$660$	0	0
$661$	−18.5149	−0.720146	−0.360073	−	0.932924i	$-0.617248\pi$
$661$	−18.5149	−0.720146	−0.360073	+	0.932924i	$0.617248\pi$
$662$	0	0
$663$	−1.03321	−0.0401267
$664$	0	0
$665$	−4.89297	−0.189741
$666$	0	0
$667$	33.5066	1.29738
$668$	0	0
$669$	5.49168	0.212320
$670$	0	0
$671$	−38.1524	−1.47286
$672$	0	0
$673$	6.12698	0.236178	0.118089	−	0.993003i	$-0.462323\pi$
$673$	6.12698	0.236178	0.118089	+	0.993003i	$0.462323\pi$
$674$	0	0
$675$	7.97565	0.306983
$676$	0	0
$677$	8.13791	0.312765	0.156383	−	0.987697i	$-0.450017\pi$
$677$	8.13791	0.312765	0.156383	+	0.987697i	$0.450017\pi$
$678$	0	0
$679$	69.2319	2.65688
$680$	0	0
$681$	5.39706	0.206816
$682$	0	0
$683$	−23.0749	−0.882938	−0.441469	−	0.897277i	$-0.645542\pi$
$683$	−23.0749	−0.882938	−0.441469	+	0.897277i	$0.645542\pi$
$684$	0	0
$685$	−2.78935	−0.106576
$686$	0	0
$687$	3.09001	0.117891
$688$	0	0
$689$	−5.55527	−0.211639
$690$	0	0
$691$	−13.1767	−0.501263	−0.250632	−	0.968083i	$-0.580638\pi$
$691$	−13.1767	−0.501263	−0.250632	+	0.968083i	$0.580638\pi$
$692$	0	0
$693$	−60.8560	−2.31173
$694$	0	0
$695$	−0.133366	−0.00505884
$696$	0	0
$697$	8.28516	0.313823
$698$	0	0
$699$	1.27890	0.0483724
$700$	0	0
$701$	−27.9004	−1.05378	−0.526891	−	0.849933i	$-0.676643\pi$
$701$	−27.9004	−1.05378	−0.526891	+	0.849933i	$0.676643\pi$
$702$	0	0
$703$	73.0898	2.75663
$704$	0	0
$705$	−0.151457	−0.00570420
$706$	0	0
$707$	−54.3550	−2.04423
$708$	0	0
$709$	−8.31572	−0.312303	−0.156152	−	0.987733i	$-0.549909\pi$
$709$	−8.31572	−0.312303	−0.156152	+	0.987733i	$0.549909\pi$
$710$	0	0
$711$	9.99221	0.374737
$712$	0	0
$713$	7.87105	0.294773
$714$	0	0
$715$	2.75736	0.103119
$716$	0	0
$717$	−4.92307	−0.183855
$718$	0	0
$719$	−28.9096	−1.07815	−0.539073	−	0.842259i	$-0.681225\pi$
$719$	−28.9096	−1.07815	−0.539073	+	0.842259i	$0.681225\pi$
$720$	0	0
$721$	44.1371	1.64375
$722$	0	0
$723$	−4.94333	−0.183844
$724$	0	0
$725$	−36.9030	−1.37054
$726$	0	0
$727$	44.1099	1.63594	0.817972	−	0.575258i	$-0.195098\pi$
$727$	44.1099	1.63594	0.817972	+	0.575258i	$0.195098\pi$
$728$	0	0
$729$	−23.1305	−0.856687
$730$	0	0
$731$	12.4900	0.461958
$732$	0	0
$733$	−11.0447	−0.407947	−0.203973	−	0.978976i	$-0.565386\pi$
$733$	−11.0447	−0.407947	−0.203973	+	0.978976i	$0.565386\pi$
$734$	0	0
$735$	−0.542452	−0.0200086
$736$	0	0
$737$	52.6901	1.94086
$738$	0	0
$739$	23.6615	0.870401	0.435200	−	0.900334i	$-0.356678\pi$
$739$	23.6615	0.870401	0.435200	+	0.900334i	$0.356678\pi$
$740$	0	0
$741$	−7.30389	−0.268315
$742$	0	0
$743$	−49.0726	−1.80030	−0.900149	−	0.435582i	$-0.856543\pi$
$743$	−49.0726	−1.80030	−0.900149	+	0.435582i	$0.856543\pi$
$744$	0	0
$745$	−1.07289	−0.0393078
$746$	0	0
$747$	12.0764	0.441852
$748$	0	0
$749$	−6.19097	−0.226213
$750$	0	0
$751$	27.7325	1.01197	0.505986	−	0.862542i	$-0.331129\pi$
$751$	27.7325	1.01197	0.505986	+	0.862542i	$0.331129\pi$
$752$	0	0
$753$	−8.54986	−0.311574
$754$	0	0
$755$	−2.37603	−0.0864725
$756$	0	0
$757$	25.3425	0.921087	0.460544	−	0.887637i	$-0.347654\pi$
$757$	25.3425	0.921087	0.460544	+	0.887637i	$0.347654\pi$
$758$	0	0
$759$	−5.68905	−0.206499
$760$	0	0
$761$	4.13933	0.150051	0.0750254	−	0.997182i	$-0.476096\pi$
$761$	4.13933	0.150051	0.0750254	+	0.997182i	$0.476096\pi$
$762$	0	0
$763$	77.6229	2.81014
$764$	0	0
$765$	−0.453670	−0.0164025
$766$	0	0
$767$	−3.82056	−0.137952
$768$	0	0
$769$	26.4851	0.955078	0.477539	−	0.878611i	$-0.341529\pi$
$769$	26.4851	0.955078	0.477539	+	0.878611i	$0.341529\pi$
$770$	0	0
$771$	2.26847	0.0816968
$772$	0	0
$773$	28.2647	1.01661	0.508306	−	0.861177i	$-0.330272\pi$
$773$	28.2647	1.01661	0.508306	+	0.861177i	$0.330272\pi$
$774$	0	0
$775$	−8.66890	−0.311396
$776$	0	0
$777$	12.4861	0.447937
$778$	0	0
$779$	58.5686	2.09844
$780$	0	0
$781$	27.3465	0.978536
$782$	0	0
$783$	−11.8870	−0.424806
$784$	0	0
$785$	−2.52648	−0.0901738
$786$	0	0
$787$	−21.5061	−0.766611	−0.383305	−	0.923622i	$-0.625214\pi$
$787$	−21.5061	−0.766611	−0.383305	+	0.923622i	$0.625214\pi$
$788$	0	0
$789$	−0.0890653	−0.00317081
$790$	0	0
$791$	−28.1926	−1.00241
$792$	0	0
$793$	31.3054	1.11169
$794$	0	0
$795$	0.0609507	0.00216170
$796$	0	0
$797$	−24.6824	−0.874295	−0.437147	−	0.899390i	$-0.644011\pi$
$797$	−24.6824	−0.874295	−0.437147	+	0.899390i	$0.644011\pi$
$798$	0	0
$799$	−3.61317	−0.127825
$800$	0	0
$801$	−32.6285	−1.15287
$802$	0	0
$803$	12.1694	0.429448
$804$	0	0
$805$	3.12720	0.110219
$806$	0	0
$807$	0.491240	0.0172925
$808$	0	0
$809$	31.2966	1.10033	0.550165	−	0.835056i	$-0.314565\pi$
$809$	31.2966	1.10033	0.550165	+	0.835056i	$0.314565\pi$
$810$	0	0
$811$	−17.8051	−0.625223	−0.312611	−	0.949881i	$-0.601204\pi$
$811$	−17.8051	−0.625223	−0.312611	+	0.949881i	$0.601204\pi$
$812$	0	0
$813$	1.96918	0.0690622
$814$	0	0
$815$	−3.52577	−0.123503
$816$	0	0
$817$	88.2927	3.08897
$818$	0	0
$819$	49.9345	1.74485
$820$	0	0
$821$	4.02723	0.140551	0.0702756	−	0.997528i	$-0.477612\pi$
$821$	4.02723	0.140551	0.0702756	+	0.997528i	$0.477612\pi$
$822$	0	0
$823$	−3.57535	−0.124629	−0.0623145	−	0.998057i	$-0.519848\pi$
$823$	−3.57535	−0.124629	−0.0623145	+	0.998057i	$0.519848\pi$
$824$	0	0
$825$	6.26572	0.218144
$826$	0	0
$827$	50.3569	1.75108	0.875540	−	0.483145i	$-0.160506\pi$
$827$	50.3569	1.75108	0.875540	+	0.483145i	$0.160506\pi$
$828$	0	0
$829$	2.09043	0.0726038	0.0363019	−	0.999341i	$-0.488442\pi$
$829$	2.09043	0.0726038	0.0363019	+	0.999341i	$0.488442\pi$
$830$	0	0
$831$	4.04921	0.140465
$832$	0	0
$833$	−12.9408	−0.448372
$834$	0	0
$835$	0.790387	0.0273525
$836$	0	0
$837$	−2.79238	−0.0965187
$838$	0	0
$839$	−13.1665	−0.454559	−0.227279	−	0.973830i	$-0.572983\pi$
$839$	−13.1665	−0.454559	−0.227279	+	0.973830i	$0.572983\pi$
$840$	0	0
$841$	26.0006	0.896572
$842$	0	0
$843$	−2.14258	−0.0737943
$844$	0	0
$845$	−0.247487	−0.00851381
$846$	0	0
$847$	−47.6917	−1.63870
$848$	0	0
$849$	4.88605	0.167689
$850$	0	0
$851$	−46.7132	−1.60131
$852$	0	0
$853$	−39.9959	−1.36943	−0.684716	−	0.728810i	$-0.740074\pi$
$853$	−39.9959	−1.36943	−0.684716	+	0.728810i	$0.740074\pi$
$854$	0	0
$855$	−3.20704	−0.109678
$856$	0	0
$857$	43.6931	1.49253	0.746264	−	0.665650i	$-0.231846\pi$
$857$	43.6931	1.49253	0.746264	+	0.665650i	$0.231846\pi$
$858$	0	0
$859$	20.8494	0.711372	0.355686	−	0.934606i	$-0.384247\pi$
$859$	20.8494	0.711372	0.355686	+	0.934606i	$0.384247\pi$
$860$	0	0
$861$	10.0054	0.340984
$862$	0	0
$863$	34.0233	1.15817	0.579084	−	0.815268i	$-0.303410\pi$
$863$	34.0233	1.15817	0.579084	+	0.815268i	$0.303410\pi$
$864$	0	0
$865$	−1.63793	−0.0556912
$866$	0	0
$867$	0.270435	0.00918446
$868$	0	0
$869$	15.8960	0.539236
$870$	0	0
$871$	−43.2341	−1.46493
$872$	0	0
$873$	45.3772	1.53579
$874$	0	0
$875$	−6.90500	−0.233432
$876$	0	0
$877$	−20.8261	−0.703247	−0.351624	−	0.936141i	$-0.614370\pi$
$877$	−20.8261	−0.703247	−0.351624	+	0.936141i	$0.614370\pi$
$878$	0	0
$879$	−3.99373	−0.134705
$880$	0	0
$881$	26.7762	0.902112	0.451056	−	0.892496i	$-0.351047\pi$
$881$	26.7762	0.902112	0.451056	+	0.892496i	$0.351047\pi$
$882$	0	0
$883$	−12.1836	−0.410011	−0.205006	−	0.978761i	$-0.565721\pi$
$883$	−12.1836	−0.410011	−0.205006	+	0.978761i	$0.565721\pi$
$884$	0	0
$885$	0.0419180	0.00140906
$886$	0	0
$887$	43.1881	1.45012	0.725058	−	0.688688i	$-0.241813\pi$
$887$	43.1881	1.45012	0.725058	+	0.688688i	$0.241813\pi$
$888$	0	0
$889$	−37.4896	−1.25736
$890$	0	0
$891$	−38.8657	−1.30205
$892$	0	0
$893$	−25.5419	−0.854726
$894$	0	0
$895$	−0.0492285	−0.00164553
$896$	0	0
$897$	4.66807	0.155862
$898$	0	0
$899$	12.9202	0.430913
$900$	0	0
$901$	1.45405	0.0484413
$902$	0	0
$903$	15.0833	0.501940
$904$	0	0
$905$	1.71312	0.0569460
$906$	0	0
$907$	25.3690	0.842365	0.421182	−	0.906976i	$-0.361615\pi$
$907$	25.3690	0.842365	0.421182	+	0.906976i	$0.361615\pi$
$908$	0	0
$909$	−35.6263	−1.18165
$910$	0	0
$911$	12.5592	0.416104	0.208052	−	0.978118i	$-0.433288\pi$
$911$	12.5592	0.416104	0.208052	+	0.978118i	$0.433288\pi$
$912$	0	0
$913$	19.2116	0.635812
$914$	0	0
$915$	−0.343473	−0.0113549
$916$	0	0
$917$	55.7012	1.83942
$918$	0	0
$919$	−10.2721	−0.338845	−0.169423	−	0.985543i	$-0.554190\pi$
$919$	−10.2721	−0.338845	−0.169423	+	0.985543i	$0.554190\pi$
$920$	0	0
$921$	−3.36953	−0.111030
$922$	0	0
$923$	−22.4388	−0.738582
$924$	0	0
$925$	51.4483	1.69161
$926$	0	0
$927$	28.9291	0.950158
$928$	0	0
$929$	20.7364	0.680338	0.340169	−	0.940364i	$-0.389516\pi$
$929$	20.7364	0.680338	0.340169	+	0.940364i	$0.389516\pi$
$930$	0	0
$931$	−91.4797	−2.99813
$932$	0	0
$933$	−3.53051	−0.115584
$934$	0	0
$935$	−0.721717	−0.0236027
$936$	0	0
$937$	36.5409	1.19374	0.596870	−	0.802338i	$-0.296411\pi$
$937$	36.5409	1.19374	0.596870	+	0.802338i	$0.296411\pi$
$938$	0	0
$939$	−3.45498	−0.112749
$940$	0	0
$941$	−49.9086	−1.62697	−0.813487	−	0.581583i	$-0.802434\pi$
$941$	−49.9086	−1.62697	−0.813487	+	0.581583i	$0.802434\pi$
$942$	0	0
$943$	−37.4324	−1.21897
$944$	0	0
$945$	−1.10942	−0.0360895
$946$	0	0
$947$	5.77616	0.187700	0.0938500	−	0.995586i	$-0.470083\pi$
$947$	5.77616	0.187700	0.0938500	+	0.995586i	$0.470083\pi$
$948$	0	0
$949$	−9.98541	−0.324140
$950$	0	0
$951$	−0.629292	−0.0204062
$952$	0	0
$953$	−10.3224	−0.334374	−0.167187	−	0.985925i	$-0.553468\pi$
$953$	−10.3224	−0.334374	−0.167187	+	0.985925i	$0.553468\pi$
$954$	0	0
$955$	−1.28476	−0.0415738
$956$	0	0
$957$	−9.33849	−0.301870
$958$	0	0
$959$	−80.3595	−2.59494
$960$	0	0
$961$	−27.9649	−0.902094
$962$	0	0
$963$	−4.05779	−0.130761
$964$	0	0
$965$	2.61018	0.0840248
$966$	0	0
$967$	17.1270	0.550767	0.275383	−	0.961334i	$-0.411195\pi$
$967$	17.1270	0.550767	0.275383	+	0.961334i	$0.411195\pi$
$968$	0	0
$969$	1.91173	0.0614137
$970$	0	0
$971$	51.2036	1.64320	0.821601	−	0.570063i	$-0.193081\pi$
$971$	51.2036	1.64320	0.821601	+	0.570063i	$0.193081\pi$
$972$	0	0
$973$	−3.84218	−0.123175
$974$	0	0
$975$	−5.14124	−0.164652
$976$	0	0
$977$	34.4477	1.10208	0.551039	−	0.834479i	$-0.314231\pi$
$977$	34.4477	1.10208	0.551039	+	0.834479i	$0.314231\pi$
$978$	0	0
$979$	−51.9068	−1.65895
$980$	0	0
$981$	50.8770	1.62438
$982$	0	0
$983$	−17.6166	−0.561883	−0.280941	−	0.959725i	$-0.590647\pi$
$983$	−17.6166	−0.561883	−0.280941	+	0.959725i	$0.590647\pi$
$984$	0	0
$985$	−3.50610	−0.111714
$986$	0	0
$987$	−4.36338	−0.138888
$988$	0	0
$989$	−56.4297	−1.79436
$990$	0	0
$991$	44.0174	1.39826	0.699129	−	0.714996i	$-0.253571\pi$
$991$	44.0174	1.39826	0.699129	+	0.714996i	$0.253571\pi$
$992$	0	0
$993$	−5.58086	−0.177103
$994$	0	0
$995$	−3.83096	−0.121450
$996$	0	0
$997$	5.22666	0.165530	0.0827650	−	0.996569i	$-0.473625\pi$
$997$	5.22666	0.165530	0.0827650	+	0.996569i	$0.473625\pi$
$998$	0	0
$999$	16.5722	0.524322

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bc.1.18	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bc.1.18	✓	33	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+0.270435 q^{3} -0.155002 q^{5} -4.46551 q^{7} -2.92686 q^{9} +O(q^{10})\)	\(q+0.270435 q^{3} -0.155002 q^{5} -4.46551 q^{7} -2.92686 q^{9} -4.65618 q^{11} +3.82056 q^{13} -0.0419180 q^{15} -1.00000 q^{17} -7.06910 q^{19} -1.20763 q^{21} +4.51801 q^{23} -4.97597 q^{25} -1.60283 q^{27} +7.41624 q^{29} +1.74215 q^{31} -1.25919 q^{33} +0.692163 q^{35} -10.3393 q^{37} +1.03321 q^{39} -8.28516 q^{41} -12.4900 q^{43} +0.453670 q^{45} +3.61317 q^{47} +12.9408 q^{49} -0.270435 q^{51} -1.45405 q^{53} +0.721717 q^{55} -1.91173 q^{57} -1.00000 q^{59} +8.19393 q^{61} +13.0699 q^{63} -0.592194 q^{65} -11.3162 q^{67} +1.22183 q^{69} -5.87317 q^{71} -2.61360 q^{73} -1.34568 q^{75} +20.7922 q^{77} -3.41396 q^{79} +8.34713 q^{81} -4.12605 q^{83} +0.155002 q^{85} +2.00561 q^{87} +11.1480 q^{89} -17.0607 q^{91} +0.471139 q^{93} +1.09572 q^{95} -15.5037 q^{97} +13.6280 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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