LMFDB - Embedded newform 8001.2.a.x.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56609 q^{2} +0.452631 q^{4} +2.71234 q^{5} +1.00000 q^{7} -2.42332 q^{8} +O(q^{10})$	\(q+1.56609 q^{2} +0.452631 q^{4} +2.71234 q^{5} +1.00000 q^{7} -2.42332 q^{8} +4.24776 q^{10} -3.86806 q^{11} +6.27068 q^{13} +1.56609 q^{14} -4.70039 q^{16} -6.52039 q^{17} -5.34954 q^{19} +1.22769 q^{20} -6.05773 q^{22} -1.10758 q^{23} +2.35677 q^{25} +9.82044 q^{26} +0.452631 q^{28} -1.30703 q^{29} -5.39238 q^{31} -2.51459 q^{32} -10.2115 q^{34} +2.71234 q^{35} -4.76274 q^{37} -8.37784 q^{38} -6.57285 q^{40} +9.81863 q^{41} -1.05281 q^{43} -1.75081 q^{44} -1.73458 q^{46} +1.90199 q^{47} +1.00000 q^{49} +3.69090 q^{50} +2.83831 q^{52} +3.89518 q^{53} -10.4915 q^{55} -2.42332 q^{56} -2.04693 q^{58} -8.23334 q^{59} -10.2372 q^{61} -8.44494 q^{62} +5.46271 q^{64} +17.0082 q^{65} -9.39671 q^{67} -2.95133 q^{68} +4.24776 q^{70} -9.50632 q^{71} -4.83462 q^{73} -7.45887 q^{74} -2.42137 q^{76} -3.86806 q^{77} -10.7547 q^{79} -12.7490 q^{80} +15.3768 q^{82} +6.54722 q^{83} -17.6855 q^{85} -1.64879 q^{86} +9.37354 q^{88} -9.08436 q^{89} +6.27068 q^{91} -0.501327 q^{92} +2.97869 q^{94} -14.5097 q^{95} +2.12125 q^{97} +1.56609 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) $ 22 * q + 10 * q^4 + 22 * q^7	$ 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) $ 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

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 $

$2$	1.56609	1.10739	0.553696	−	0.832719i	$-0.313217\pi$
$2$	1.56609	1.10739	0.553696	+	0.832719i	$0.313217\pi$
$3$	0	0
$3$	0	0
$4$	0.452631	0.226316
$5$	2.71234	1.21299	0.606497	−	0.795086i	$-0.292574\pi$
$5$	2.71234	1.21299	0.606497	+	0.795086i	$0.292574\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.42332	−0.856771
$9$	0	0
$10$	4.24776	1.34326
$11$	−3.86806	−1.16627	−0.583133	−	0.812377i	$-0.698173\pi$
$11$	−3.86806	−1.16627	−0.583133	+	0.812377i	$0.698173\pi$
$12$	0	0
$13$	6.27068	1.73917	0.869587	−	0.493780i	$-0.164385\pi$
$13$	6.27068	1.73917	0.869587	+	0.493780i	$0.164385\pi$
$14$	1.56609	0.418555
$15$	0	0
$16$	−4.70039	−1.17510
$17$	−6.52039	−1.58143	−0.790714	−	0.612186i	$-0.790290\pi$
$17$	−6.52039	−1.58143	−0.790714	+	0.612186i	$0.790290\pi$
$18$	0	0
$19$	−5.34954	−1.22727	−0.613634	−	0.789591i	$-0.710293\pi$
$19$	−5.34954	−1.22727	−0.613634	+	0.789591i	$0.710293\pi$
$20$	1.22769	0.274519
$21$	0	0
$22$	−6.05773	−1.29151
$23$	−1.10758	−0.230947	−0.115474	−	0.993311i	$-0.536839\pi$
$23$	−1.10758	−0.230947	−0.115474	+	0.993311i	$0.536839\pi$
$24$	0	0
$25$	2.35677	0.471353
$26$	9.82044	1.92595
$27$	0	0
$28$	0.452631	0.0855392
$29$	−1.30703	−0.242710	−0.121355	−	0.992609i	$-0.538724\pi$
$29$	−1.30703	−0.242710	−0.121355	+	0.992609i	$0.538724\pi$
$30$	0	0
$31$	−5.39238	−0.968500	−0.484250	−	0.874930i	$-0.660907\pi$
$31$	−5.39238	−0.968500	−0.484250	+	0.874930i	$0.660907\pi$
$32$	−2.51459	−0.444521
$33$	0	0
$34$	−10.2115	−1.75126
$35$	2.71234	0.458468
$36$	0	0
$37$	−4.76274	−0.782989	−0.391495	−	0.920180i	$-0.628042\pi$
$37$	−4.76274	−0.782989	−0.391495	+	0.920180i	$0.628042\pi$
$38$	−8.37784	−1.35907
$39$	0	0
$40$	−6.57285	−1.03926
$41$	9.81863	1.53341	0.766707	−	0.641998i	$-0.221894\pi$
$41$	9.81863	1.53341	0.766707	+	0.641998i	$0.221894\pi$
$42$	0	0
$43$	−1.05281	−0.160551	−0.0802757	−	0.996773i	$-0.525580\pi$
$43$	−1.05281	−0.160551	−0.0802757	+	0.996773i	$0.525580\pi$
$44$	−1.75081	−0.263944
$45$	0	0
$46$	−1.73458	−0.255749
$47$	1.90199	0.277434	0.138717	−	0.990332i	$-0.455702\pi$
$47$	1.90199	0.277434	0.138717	+	0.990332i	$0.455702\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.69090	0.521972
$51$	0	0
$52$	2.83831	0.393602
$53$	3.89518	0.535044	0.267522	−	0.963552i	$-0.413795\pi$
$53$	3.89518	0.535044	0.267522	+	0.963552i	$0.413795\pi$
$54$	0	0
$55$	−10.4915	−1.41467
$56$	−2.42332	−0.323829
$57$	0	0
$58$	−2.04693	−0.268775
$59$	−8.23334	−1.07189	−0.535945	−	0.844253i	$-0.680045\pi$
$59$	−8.23334	−1.07189	−0.535945	+	0.844253i	$0.680045\pi$
$60$	0	0
$61$	−10.2372	−1.31074	−0.655369	−	0.755309i	$-0.727487\pi$
$61$	−10.2372	−1.31074	−0.655369	+	0.755309i	$0.727487\pi$
$62$	−8.44494	−1.07251
$63$	0	0
$64$	5.46271	0.682839
$65$	17.0082	2.10961
$66$	0	0
$67$	−9.39671	−1.14799	−0.573996	−	0.818858i	$-0.694607\pi$
$67$	−9.39671	−1.14799	−0.573996	+	0.818858i	$0.694607\pi$
$68$	−2.95133	−0.357902
$69$	0	0
$70$	4.24776	0.507704
$71$	−9.50632	−1.12819	−0.564097	−	0.825709i	$-0.690775\pi$
$71$	−9.50632	−1.12819	−0.564097	+	0.825709i	$0.690775\pi$
$72$	0	0
$73$	−4.83462	−0.565850	−0.282925	−	0.959142i	$-0.591305\pi$
$73$	−4.83462	−0.565850	−0.282925	+	0.959142i	$0.591305\pi$
$74$	−7.45887	−0.867076
$75$	0	0
$76$	−2.42137	−0.277750
$77$	−3.86806	−0.440807
$78$	0	0
$79$	−10.7547	−1.21000	−0.605000	−	0.796226i	$-0.706827\pi$
$79$	−10.7547	−1.21000	−0.605000	+	0.796226i	$0.706827\pi$
$80$	−12.7490	−1.42538
$81$	0	0
$82$	15.3768	1.69809
$83$	6.54722	0.718651	0.359325	−	0.933212i	$-0.383007\pi$
$83$	6.54722	0.718651	0.359325	+	0.933212i	$0.383007\pi$
$84$	0	0
$85$	−17.6855	−1.91826
$86$	−1.64879	−0.177793
$87$	0	0
$88$	9.37354	0.999223
$89$	−9.08436	−0.962940	−0.481470	−	0.876463i	$-0.659897\pi$
$89$	−9.08436	−0.962940	−0.481470	+	0.876463i	$0.659897\pi$
$90$	0	0
$91$	6.27068	0.657346
$92$	−0.501327	−0.0522670
$93$	0	0
$94$	2.97869	0.307228
$95$	−14.5097	−1.48867
$96$	0	0
$97$	2.12125	0.215380	0.107690	−	0.994185i	$-0.465655\pi$
$97$	2.12125	0.215380	0.107690	+	0.994185i	$0.465655\pi$
$98$	1.56609	0.158199
$99$	0	0
$100$	1.06675	0.106675
$101$	3.64043	0.362236	0.181118	−	0.983461i	$-0.442028\pi$
$101$	3.64043	0.362236	0.181118	+	0.983461i	$0.442028\pi$
$102$	0	0
$103$	−11.8268	−1.16533	−0.582664	−	0.812713i	$-0.697990\pi$
$103$	−11.8268	−1.16533	−0.582664	+	0.812713i	$0.697990\pi$
$104$	−15.1958	−1.49007
$105$	0	0
$106$	6.10019	0.592503
$107$	16.8427	1.62825	0.814123	−	0.580693i	$-0.197218\pi$
$107$	16.8427	1.62825	0.814123	+	0.580693i	$0.197218\pi$
$108$	0	0
$109$	4.05037	0.387956	0.193978	−	0.981006i	$-0.437861\pi$
$109$	4.05037	0.387956	0.193978	+	0.981006i	$0.437861\pi$
$110$	−16.4306	−1.56660
$111$	0	0
$112$	−4.70039	−0.444145
$113$	−3.17196	−0.298392	−0.149196	−	0.988808i	$-0.547669\pi$
$113$	−3.17196	−0.298392	−0.149196	+	0.988808i	$0.547669\pi$
$114$	0	0
$115$	−3.00414	−0.280138
$116$	−0.591603	−0.0549290
$117$	0	0
$118$	−12.8941	−1.18700
$119$	−6.52039	−0.597723
$120$	0	0
$121$	3.96192	0.360175
$122$	−16.0323	−1.45150
$123$	0	0
$124$	−2.44076	−0.219187
$125$	−7.16934	−0.641245
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	13.5843	1.20069
$129$	0	0
$130$	26.6363	2.33616
$131$	−5.03257	−0.439698	−0.219849	−	0.975534i	$-0.570556\pi$
$131$	−5.03257	−0.439698	−0.219849	+	0.975534i	$0.570556\pi$
$132$	0	0
$133$	−5.34954	−0.463864
$134$	−14.7161	−1.27128
$135$	0	0
$136$	15.8010	1.35492
$137$	17.7553	1.51694	0.758470	−	0.651708i	$-0.225947\pi$
$137$	17.7553	1.51694	0.758470	+	0.651708i	$0.225947\pi$
$138$	0	0
$139$	5.65169	0.479370	0.239685	−	0.970851i	$-0.422956\pi$
$139$	5.65169	0.479370	0.239685	+	0.970851i	$0.422956\pi$
$140$	1.22769	0.103759
$141$	0	0
$142$	−14.8877	−1.24935
$143$	−24.2554	−2.02834
$144$	0	0
$145$	−3.54511	−0.294405
$146$	−7.57145	−0.626617
$147$	0	0
$148$	−2.15576	−0.177203
$149$	17.3085	1.41797	0.708985	−	0.705223i	$-0.249153\pi$
$149$	17.3085	1.41797	0.708985	+	0.705223i	$0.249153\pi$
$150$	0	0
$151$	2.44239	0.198759	0.0993793	−	0.995050i	$-0.468314\pi$
$151$	2.44239	0.198759	0.0993793	+	0.995050i	$0.468314\pi$
$152$	12.9636	1.05149
$153$	0	0
$154$	−6.05773	−0.488146
$155$	−14.6259	−1.17478
$156$	0	0
$157$	−0.150575	−0.0120172	−0.00600860	−	0.999982i	$-0.501913\pi$
$157$	−0.150575	−0.0120172	−0.00600860	+	0.999982i	$0.501913\pi$
$158$	−16.8428	−1.33994
$159$	0	0
$160$	−6.82041	−0.539201
$161$	−1.10758	−0.0872899
$162$	0	0
$163$	3.50507	0.274538	0.137269	−	0.990534i	$-0.456167\pi$
$163$	3.50507	0.274538	0.137269	+	0.990534i	$0.456167\pi$
$164$	4.44422	0.347035
$165$	0	0
$166$	10.2535	0.795828
$167$	3.63868	0.281569	0.140785	−	0.990040i	$-0.455037\pi$
$167$	3.63868	0.281569	0.140785	+	0.990040i	$0.455037\pi$
$168$	0	0
$169$	26.3214	2.02473
$170$	−27.6970	−2.12427
$171$	0	0
$172$	−0.476532	−0.0363353
$173$	−22.8409	−1.73656	−0.868280	−	0.496075i	$-0.834774\pi$
$173$	−22.8409	−1.73656	−0.868280	+	0.496075i	$0.834774\pi$
$174$	0	0
$175$	2.35677	0.178155
$176$	18.1814	1.37047
$177$	0	0
$178$	−14.2269	−1.06635
$179$	9.22044	0.689168	0.344584	−	0.938755i	$-0.388020\pi$
$179$	9.22044	0.689168	0.344584	+	0.938755i	$0.388020\pi$
$180$	0	0
$181$	−1.02708	−0.0763419	−0.0381710	−	0.999271i	$-0.512153\pi$
$181$	−1.02708	−0.0763419	−0.0381710	+	0.999271i	$0.512153\pi$
$182$	9.82044	0.727939
$183$	0	0
$184$	2.68403	0.197869
$185$	−12.9181	−0.949761
$186$	0	0
$187$	25.2213	1.84436
$188$	0.860901	0.0627877
$189$	0	0
$190$	−22.7235	−1.64854
$191$	1.13699	0.0822693	0.0411347	−	0.999154i	$-0.486903\pi$
$191$	1.13699	0.0822693	0.0411347	+	0.999154i	$0.486903\pi$
$192$	0	0
$193$	1.32098	0.0950862	0.0475431	−	0.998869i	$-0.484861\pi$
$193$	1.32098	0.0950862	0.0475431	+	0.998869i	$0.484861\pi$
$194$	3.32206	0.238510
$195$	0	0
$196$	0.452631	0.0323308
$197$	5.09750	0.363182	0.181591	−	0.983374i	$-0.441875\pi$
$197$	5.09750	0.363182	0.181591	+	0.983374i	$0.441875\pi$
$198$	0	0
$199$	−4.17243	−0.295776	−0.147888	−	0.989004i	$-0.547248\pi$
$199$	−4.17243	−0.295776	−0.147888	+	0.989004i	$0.547248\pi$
$200$	−5.71119	−0.403842
$201$	0	0
$202$	5.70123	0.401137
$203$	−1.30703	−0.0917356
$204$	0	0
$205$	26.6314	1.86002
$206$	−18.5218	−1.29047
$207$	0	0
$208$	−29.4746	−2.04370
$209$	20.6924	1.43132
$210$	0	0
$211$	−19.4356	−1.33800	−0.669001	−	0.743261i	$-0.733278\pi$
$211$	−19.4356	−1.33800	−0.669001	+	0.743261i	$0.733278\pi$
$212$	1.76308	0.121089
$213$	0	0
$214$	26.3772	1.80311
$215$	−2.85556	−0.194748
$216$	0	0
$217$	−5.39238	−0.366059
$218$	6.34324	0.429619
$219$	0	0
$220$	−4.74877	−0.320162
$221$	−40.8873	−2.75038
$222$	0	0
$223$	−9.68555	−0.648592	−0.324296	−	0.945956i	$-0.605127\pi$
$223$	−9.68555	−0.648592	−0.324296	+	0.945956i	$0.605127\pi$
$224$	−2.51459	−0.168013
$225$	0	0
$226$	−4.96756	−0.330437
$227$	7.36420	0.488779	0.244389	−	0.969677i	$-0.421412\pi$
$227$	7.36420	0.488779	0.244389	+	0.969677i	$0.421412\pi$
$228$	0	0
$229$	8.45662	0.558829	0.279414	−	0.960171i	$-0.409860\pi$
$229$	8.45662	0.558829	0.279414	+	0.960171i	$0.409860\pi$
$230$	−4.70475	−0.310222
$231$	0	0
$232$	3.16735	0.207947
$233$	−25.0699	−1.64238	−0.821191	−	0.570653i	$-0.806690\pi$
$233$	−25.0699	−1.64238	−0.821191	+	0.570653i	$0.806690\pi$
$234$	0	0
$235$	5.15885	0.336526
$236$	−3.72667	−0.242585
$237$	0	0
$238$	−10.2115	−0.661914
$239$	6.88014	0.445039	0.222520	−	0.974928i	$-0.428572\pi$
$239$	6.88014	0.445039	0.222520	+	0.974928i	$0.428572\pi$
$240$	0	0
$241$	30.5872	1.97029	0.985147	−	0.171715i	$-0.0549309\pi$
$241$	30.5872	1.97029	0.985147	+	0.171715i	$0.0549309\pi$
$242$	6.20472	0.398855
$243$	0	0
$244$	−4.63367	−0.296641
$245$	2.71234	0.173285
$246$	0	0
$247$	−33.5452	−2.13443
$248$	13.0674	0.829783
$249$	0	0
$250$	−11.2278	−0.710109
$251$	−2.69562	−0.170146	−0.0850731	−	0.996375i	$-0.527112\pi$
$251$	−2.69562	−0.170146	−0.0850731	+	0.996375i	$0.527112\pi$
$252$	0	0
$253$	4.28421	0.269346
$254$	1.56609	0.0982651
$255$	0	0
$256$	10.3487	0.646795
$257$	26.3516	1.64377	0.821884	−	0.569655i	$-0.192923\pi$
$257$	26.3516	1.64377	0.821884	+	0.569655i	$0.192923\pi$
$258$	0	0
$259$	−4.76274	−0.295942
$260$	7.69844	0.477437
$261$	0	0
$262$	−7.88145	−0.486918
$263$	2.44314	0.150650	0.0753252	−	0.997159i	$-0.476001\pi$
$263$	2.44314	0.150650	0.0753252	+	0.997159i	$0.476001\pi$
$264$	0	0
$265$	10.5650	0.649005
$266$	−8.37784	−0.513679
$267$	0	0
$268$	−4.25324	−0.259808
$269$	−18.8254	−1.14780	−0.573901	−	0.818925i	$-0.694571\pi$
$269$	−18.8254	−1.14780	−0.573901	+	0.818925i	$0.694571\pi$
$270$	0	0
$271$	−26.7289	−1.62366	−0.811832	−	0.583891i	$-0.801529\pi$
$271$	−26.7289	−1.62366	−0.811832	+	0.583891i	$0.801529\pi$
$272$	30.6484	1.85833
$273$	0	0
$274$	27.8064	1.67985
$275$	−9.11612	−0.549723
$276$	0	0
$277$	−22.8748	−1.37442	−0.687208	−	0.726461i	$-0.741164\pi$
$277$	−22.8748	−1.37442	−0.687208	+	0.726461i	$0.741164\pi$
$278$	8.85104	0.530850
$279$	0	0
$280$	−6.57285	−0.392803
$281$	30.4780	1.81817	0.909084	−	0.416613i	$-0.136783\pi$
$281$	30.4780	1.81817	0.909084	+	0.416613i	$0.136783\pi$
$282$	0	0
$283$	−28.6136	−1.70090	−0.850450	−	0.526056i	$-0.823670\pi$
$283$	−28.6136	−1.70090	−0.850450	+	0.526056i	$0.823670\pi$
$284$	−4.30286	−0.255328
$285$	0	0
$286$	−37.9861	−2.24616
$287$	9.81863	0.579576
$288$	0	0
$289$	25.5155	1.50091
$290$	−5.55195	−0.326022
$291$	0	0
$292$	−2.18830	−0.128061
$293$	−7.02634	−0.410483	−0.205242	−	0.978711i	$-0.565798\pi$
$293$	−7.02634	−0.410483	−0.205242	+	0.978711i	$0.565798\pi$
$294$	0	0
$295$	−22.3316	−1.30020
$296$	11.5416	0.670843
$297$	0	0
$298$	27.1067	1.57025
$299$	−6.94531	−0.401658
$300$	0	0
$301$	−1.05281	−0.0606827
$302$	3.82499	0.220104
$303$	0	0
$304$	25.1449	1.44216
$305$	−27.7667	−1.58992
$306$	0	0
$307$	12.0273	0.686435	0.343217	−	0.939256i	$-0.388483\pi$
$307$	12.0273	0.686435	0.343217	+	0.939256i	$0.388483\pi$
$308$	−1.75081	−0.0997614
$309$	0	0
$310$	−22.9055	−1.30095
$311$	−14.6633	−0.831480	−0.415740	−	0.909483i	$-0.636477\pi$
$311$	−14.6633	−0.831480	−0.415740	+	0.909483i	$0.636477\pi$
$312$	0	0
$313$	10.2544	0.579613	0.289807	−	0.957085i	$-0.406409\pi$
$313$	10.2544	0.579613	0.289807	+	0.957085i	$0.406409\pi$
$314$	−0.235814	−0.0133077
$315$	0	0
$316$	−4.86792	−0.273842
$317$	−13.7709	−0.773452	−0.386726	−	0.922195i	$-0.626394\pi$
$317$	−13.7709	−0.773452	−0.386726	+	0.922195i	$0.626394\pi$
$318$	0	0
$319$	5.05568	0.283064
$320$	14.8167	0.828279
$321$	0	0
$322$	−1.73458	−0.0966641
$323$	34.8811	1.94083
$324$	0	0
$325$	14.7785	0.819765
$326$	5.48925	0.304022
$327$	0	0
$328$	−23.7937	−1.31378
$329$	1.90199	0.104860
$330$	0	0
$331$	8.63320	0.474523	0.237262	−	0.971446i	$-0.423750\pi$
$331$	8.63320	0.474523	0.237262	+	0.971446i	$0.423750\pi$
$332$	2.96348	0.162642
$333$	0	0
$334$	5.69849	0.311807
$335$	−25.4870	−1.39251
$336$	0	0
$337$	−27.4787	−1.49686	−0.748431	−	0.663213i	$-0.769192\pi$
$337$	−27.4787	−1.49686	−0.748431	+	0.663213i	$0.769192\pi$
$338$	41.2217	2.24216
$339$	0	0
$340$	−8.00500	−0.434132
$341$	20.8581	1.12953
$342$	0	0
$343$	1.00000	0.0539949
$344$	2.55128	0.137556
$345$	0	0
$346$	−35.7708	−1.92305
$347$	6.58761	0.353641	0.176821	−	0.984243i	$-0.443419\pi$
$347$	6.58761	0.353641	0.176821	+	0.984243i	$0.443419\pi$
$348$	0	0
$349$	−15.1869	−0.812938	−0.406469	−	0.913665i	$-0.633240\pi$
$349$	−15.1869	−0.812938	−0.406469	+	0.913665i	$0.633240\pi$
$350$	3.69090	0.197287
$351$	0	0
$352$	9.72659	0.518429
$353$	15.4860	0.824235	0.412117	−	0.911131i	$-0.364789\pi$
$353$	15.4860	0.824235	0.412117	+	0.911131i	$0.364789\pi$
$354$	0	0
$355$	−25.7843	−1.36849
$356$	−4.11186	−0.217928
$357$	0	0
$358$	14.4400	0.763179
$359$	4.81357	0.254050	0.127025	−	0.991899i	$-0.459457\pi$
$359$	4.81357	0.254050	0.127025	+	0.991899i	$0.459457\pi$
$360$	0	0
$361$	9.61754	0.506186
$362$	−1.60849	−0.0845404
$363$	0	0
$364$	2.83831	0.148768
$365$	−13.1131	−0.686372
$366$	0	0
$367$	5.97569	0.311928	0.155964	−	0.987763i	$-0.450152\pi$
$367$	5.97569	0.311928	0.155964	+	0.987763i	$0.450152\pi$
$368$	5.20608	0.271386
$369$	0	0
$370$	−20.2309	−1.05176
$371$	3.89518	0.202228
$372$	0	0
$373$	−23.0326	−1.19258	−0.596291	−	0.802768i	$-0.703360\pi$
$373$	−23.0326	−1.19258	−0.596291	+	0.802768i	$0.703360\pi$
$374$	39.4988	2.04243
$375$	0	0
$376$	−4.60913	−0.237698
$377$	−8.19598	−0.422114
$378$	0	0
$379$	15.3971	0.790895	0.395447	−	0.918489i	$-0.370590\pi$
$379$	15.3971	0.790895	0.395447	+	0.918489i	$0.370590\pi$
$380$	−6.56756	−0.336909
$381$	0	0
$382$	1.78062	0.0911044
$383$	24.6387	1.25898	0.629488	−	0.777010i	$-0.283264\pi$
$383$	24.6387	1.25898	0.629488	+	0.777010i	$0.283264\pi$
$384$	0	0
$385$	−10.4915	−0.534696
$386$	2.06877	0.105298
$387$	0	0
$388$	0.960143	0.0487439
$389$	−10.7688	−0.545998	−0.272999	−	0.962014i	$-0.588016\pi$
$389$	−10.7688	−0.545998	−0.272999	+	0.962014i	$0.588016\pi$
$390$	0	0
$391$	7.22189	0.365227
$392$	−2.42332	−0.122396
$393$	0	0
$394$	7.98313	0.402184
$395$	−29.1704	−1.46772
$396$	0	0
$397$	12.0438	0.604459	0.302229	−	0.953235i	$-0.402269\pi$
$397$	12.0438	0.604459	0.302229	+	0.953235i	$0.402269\pi$
$398$	−6.53440	−0.327540
$399$	0	0
$400$	−11.0777	−0.553886
$401$	1.75561	0.0876710	0.0438355	−	0.999039i	$-0.486042\pi$
$401$	1.75561	0.0876710	0.0438355	+	0.999039i	$0.486042\pi$
$402$	0	0
$403$	−33.8139	−1.68439
$404$	1.64777	0.0819797
$405$	0	0
$406$	−2.04693	−0.101587
$407$	18.4226	0.913173
$408$	0	0
$409$	−34.3763	−1.69980	−0.849899	−	0.526945i	$-0.823337\pi$
$409$	−34.3763	−1.69980	−0.849899	+	0.526945i	$0.823337\pi$
$410$	41.7072	2.05977
$411$	0	0
$412$	−5.35317	−0.263732
$413$	−8.23334	−0.405136
$414$	0	0
$415$	17.7583	0.871719
$416$	−15.7682	−0.773099
$417$	0	0
$418$	32.4060	1.58503
$419$	−27.1854	−1.32809	−0.664046	−	0.747691i	$-0.731162\pi$
$419$	−27.1854	−1.32809	−0.664046	+	0.747691i	$0.731162\pi$
$420$	0	0
$421$	36.2512	1.76678	0.883388	−	0.468643i	$-0.155257\pi$
$421$	36.2512	1.76678	0.883388	+	0.468643i	$0.155257\pi$
$422$	−30.4379	−1.48169
$423$	0	0
$424$	−9.43925	−0.458410
$425$	−15.3670	−0.745411
$426$	0	0
$427$	−10.2372	−0.495413
$428$	7.62353	0.368497
$429$	0	0
$430$	−4.47206	−0.215662
$431$	−5.80516	−0.279625	−0.139812	−	0.990178i	$-0.544650\pi$
$431$	−5.80516	−0.279625	−0.139812	+	0.990178i	$0.544650\pi$
$432$	0	0
$433$	−38.2348	−1.83745	−0.918724	−	0.394901i	$-0.870779\pi$
$433$	−38.2348	−1.83745	−0.918724	+	0.394901i	$0.870779\pi$
$434$	−8.44494	−0.405370
$435$	0	0
$436$	1.83333	0.0878004
$437$	5.92507	0.283434
$438$	0	0
$439$	−32.8171	−1.56628	−0.783138	−	0.621848i	$-0.786382\pi$
$439$	−32.8171	−1.56628	−0.783138	+	0.621848i	$0.786382\pi$
$440$	25.4242	1.21205
$441$	0	0
$442$	−64.0331	−3.04574
$443$	−32.6179	−1.54972	−0.774861	−	0.632132i	$-0.782180\pi$
$443$	−32.6179	−1.54972	−0.774861	+	0.632132i	$0.782180\pi$
$444$	0	0
$445$	−24.6398	−1.16804
$446$	−15.1684	−0.718245
$447$	0	0
$448$	5.46271	0.258089
$449$	2.06218	0.0973204	0.0486602	−	0.998815i	$-0.484505\pi$
$449$	2.06218	0.0973204	0.0486602	+	0.998815i	$0.484505\pi$
$450$	0	0
$451$	−37.9791	−1.78837
$452$	−1.43573	−0.0675308
$453$	0	0
$454$	11.5330	0.541269
$455$	17.0082	0.797356
$456$	0	0
$457$	4.96188	0.232107	0.116053	−	0.993243i	$-0.462976\pi$
$457$	4.96188	0.232107	0.116053	+	0.993243i	$0.462976\pi$
$458$	13.2438	0.618842
$459$	0	0
$460$	−1.35977	−0.0633995
$461$	29.8590	1.39067	0.695336	−	0.718685i	$-0.255256\pi$
$461$	29.8590	1.39067	0.695336	+	0.718685i	$0.255256\pi$
$462$	0	0
$463$	−10.7077	−0.497628	−0.248814	−	0.968551i	$-0.580041\pi$
$463$	−10.7077	−0.497628	−0.248814	+	0.968551i	$0.580041\pi$
$464$	6.14355	0.285207
$465$	0	0
$466$	−39.2616	−1.81876
$467$	−25.0882	−1.16094	−0.580471	−	0.814281i	$-0.697132\pi$
$467$	−25.0882	−1.16094	−0.580471	+	0.814281i	$0.697132\pi$
$468$	0	0
$469$	−9.39671	−0.433900
$470$	8.07921	0.372666
$471$	0	0
$472$	19.9520	0.918365
$473$	4.07232	0.187245
$474$	0	0
$475$	−12.6076	−0.578477
$476$	−2.95133	−0.135274
$477$	0	0
$478$	10.7749	0.492832
$479$	2.55321	0.116659	0.0583295	−	0.998297i	$-0.481423\pi$
$479$	2.55321	0.116659	0.0583295	+	0.998297i	$0.481423\pi$
$480$	0	0
$481$	−29.8656	−1.36175
$482$	47.9022	2.18189
$483$	0	0
$484$	1.79329	0.0815132
$485$	5.75354	0.261255
$486$	0	0
$487$	−10.2712	−0.465434	−0.232717	−	0.972544i	$-0.574762\pi$
$487$	−10.2712	−0.465434	−0.232717	+	0.972544i	$0.574762\pi$
$488$	24.8080	1.12300
$489$	0	0
$490$	4.24776	0.191894
$491$	5.39330	0.243396	0.121698	−	0.992567i	$-0.461166\pi$
$491$	5.39330	0.243396	0.121698	+	0.992567i	$0.461166\pi$
$492$	0	0
$493$	8.52236	0.383828
$494$	−52.5348	−2.36365
$495$	0	0
$496$	25.3463	1.13808
$497$	−9.50632	−0.426417
$498$	0	0
$499$	19.4530	0.870835	0.435418	−	0.900229i	$-0.356601\pi$
$499$	19.4530	0.870835	0.435418	+	0.900229i	$0.356601\pi$
$500$	−3.24507	−0.145124
$501$	0	0
$502$	−4.22158	−0.188419
$503$	2.74700	0.122483	0.0612414	−	0.998123i	$-0.480494\pi$
$503$	2.74700	0.122483	0.0612414	+	0.998123i	$0.480494\pi$
$504$	0	0
$505$	9.87407	0.439390
$506$	6.70945	0.298271
$507$	0	0
$508$	0.452631	0.0200823
$509$	23.0785	1.02294	0.511469	−	0.859302i	$-0.329102\pi$
$509$	23.0785	1.02294	0.511469	+	0.859302i	$0.329102\pi$
$510$	0	0
$511$	−4.83462	−0.213871
$512$	−10.9615	−0.484435
$513$	0	0
$514$	41.2689	1.82029
$515$	−32.0782	−1.41353
$516$	0	0
$517$	−7.35703	−0.323562
$518$	−7.45887	−0.327724
$519$	0	0
$520$	−41.2162	−1.80745
$521$	−31.2600	−1.36952	−0.684762	−	0.728766i	$-0.740094\pi$
$521$	−31.2600	−1.36952	−0.684762	+	0.728766i	$0.740094\pi$
$522$	0	0
$523$	18.7351	0.819230	0.409615	−	0.912258i	$-0.365663\pi$
$523$	18.7351	0.819230	0.409615	+	0.912258i	$0.365663\pi$
$524$	−2.27790	−0.0995105
$525$	0	0
$526$	3.82617	0.166829
$527$	35.1604	1.53161
$528$	0	0
$529$	−21.7733	−0.946663
$530$	16.5458	0.718702
$531$	0	0
$532$	−2.42137	−0.104980
$533$	61.5695	2.66687
$534$	0	0
$535$	45.6831	1.97505
$536$	22.7712	0.983566
$537$	0	0
$538$	−29.4822	−1.27107
$539$	−3.86806	−0.166609
$540$	0	0
$541$	0.127636	0.00548749	0.00274375	−	0.999996i	$-0.499127\pi$
$541$	0.127636	0.00548749	0.00274375	+	0.999996i	$0.499127\pi$
$542$	−41.8598	−1.79803
$543$	0	0
$544$	16.3961	0.702977
$545$	10.9860	0.470588
$546$	0	0
$547$	4.43794	0.189752	0.0948762	−	0.995489i	$-0.469754\pi$
$547$	4.43794	0.189752	0.0948762	+	0.995489i	$0.469754\pi$
$548$	8.03661	0.343307
$549$	0	0
$550$	−14.2766	−0.608758
$551$	6.99201	0.297870
$552$	0	0
$553$	−10.7547	−0.457337
$554$	−35.8240	−1.52202
$555$	0	0
$556$	2.55813	0.108489
$557$	−26.2148	−1.11076	−0.555379	−	0.831598i	$-0.687427\pi$
$557$	−26.2148	−1.11076	−0.555379	+	0.831598i	$0.687427\pi$
$558$	0	0
$559$	−6.60181	−0.279227
$560$	−12.7490	−0.538745
$561$	0	0
$562$	47.7313	2.01342
$563$	16.8260	0.709130	0.354565	−	0.935031i	$-0.384629\pi$
$563$	16.8260	0.709130	0.354565	+	0.935031i	$0.384629\pi$
$564$	0	0
$565$	−8.60341	−0.361948
$566$	−44.8114	−1.88356
$567$	0	0
$568$	23.0368	0.966604
$569$	−29.1074	−1.22025	−0.610124	−	0.792306i	$-0.708880\pi$
$569$	−29.1074	−1.22025	−0.610124	+	0.792306i	$0.708880\pi$
$570$	0	0
$571$	−11.0301	−0.461595	−0.230797	−	0.973002i	$-0.574133\pi$
$571$	−11.0301	−0.461595	−0.230797	+	0.973002i	$0.574133\pi$
$572$	−10.9787	−0.459045
$573$	0	0
$574$	15.3768	0.641817
$575$	−2.61032	−0.108858
$576$	0	0
$577$	27.8261	1.15842	0.579209	−	0.815179i	$-0.303362\pi$
$577$	27.8261	1.15842	0.579209	+	0.815179i	$0.303362\pi$
$578$	39.9595	1.66210
$579$	0	0
$580$	−1.60463	−0.0666285
$581$	6.54722	0.271625
$582$	0	0
$583$	−15.0668	−0.624003
$584$	11.7158	0.484804
$585$	0	0
$586$	−11.0039	−0.454566
$587$	21.8437	0.901587	0.450794	−	0.892628i	$-0.351141\pi$
$587$	21.8437	0.901587	0.450794	+	0.892628i	$0.351141\pi$
$588$	0	0
$589$	28.8467	1.18861
$590$	−34.9732	−1.43983
$591$	0	0
$592$	22.3867	0.920088
$593$	36.0876	1.48194	0.740970	−	0.671538i	$-0.234366\pi$
$593$	36.0876	1.48194	0.740970	+	0.671538i	$0.234366\pi$
$594$	0	0
$595$	−17.6855	−0.725035
$596$	7.83438	0.320909
$597$	0	0
$598$	−10.8770	−0.444792
$599$	8.54344	0.349076	0.174538	−	0.984650i	$-0.444157\pi$
$599$	8.54344	0.349076	0.174538	+	0.984650i	$0.444157\pi$
$600$	0	0
$601$	33.8015	1.37879	0.689396	−	0.724384i	$-0.257876\pi$
$601$	33.8015	1.37879	0.689396	+	0.724384i	$0.257876\pi$
$602$	−1.64879	−0.0671995
$603$	0	0
$604$	1.10550	0.0449822
$605$	10.7461	0.436890
$606$	0	0
$607$	30.8645	1.25275	0.626375	−	0.779522i	$-0.284538\pi$
$607$	30.8645	1.25275	0.626375	+	0.779522i	$0.284538\pi$
$608$	13.4519	0.545546
$609$	0	0
$610$	−43.4851	−1.76066
$611$	11.9268	0.482507
$612$	0	0
$613$	1.75536	0.0708985	0.0354493	−	0.999371i	$-0.488714\pi$
$613$	1.75536	0.0708985	0.0354493	+	0.999371i	$0.488714\pi$
$614$	18.8358	0.760152
$615$	0	0
$616$	9.37354	0.377671
$617$	−42.8696	−1.72586	−0.862932	−	0.505320i	$-0.831375\pi$
$617$	−42.8696	−1.72586	−0.862932	+	0.505320i	$0.831375\pi$
$618$	0	0
$619$	36.8448	1.48092	0.740458	−	0.672103i	$-0.234609\pi$
$619$	36.8448	1.48092	0.740458	+	0.672103i	$0.234609\pi$
$620$	−6.62016	−0.265872
$621$	0	0
$622$	−22.9640	−0.920774
$623$	−9.08436	−0.363957
$624$	0	0
$625$	−31.2295	−1.24918
$626$	16.0593	0.641859
$627$	0	0
$628$	−0.0681549	−0.00271968
$629$	31.0549	1.23824
$630$	0	0
$631$	−29.5692	−1.17713	−0.588565	−	0.808450i	$-0.700307\pi$
$631$	−29.5692	−1.17713	−0.588565	+	0.808450i	$0.700307\pi$
$632$	26.0621	1.03669
$633$	0	0
$634$	−21.5665	−0.856514
$635$	2.71234	0.107636
$636$	0	0
$637$	6.27068	0.248453
$638$	7.91764	0.313462
$639$	0	0
$640$	36.8451	1.45643
$641$	17.5984	0.695096	0.347548	−	0.937662i	$-0.387014\pi$
$641$	17.5984	0.695096	0.347548	+	0.937662i	$0.387014\pi$
$642$	0	0
$643$	3.06784	0.120984	0.0604920	−	0.998169i	$-0.480733\pi$
$643$	3.06784	0.120984	0.0604920	+	0.998169i	$0.480733\pi$
$644$	−0.501327	−0.0197551
$645$	0	0
$646$	54.6268	2.14926
$647$	−14.7451	−0.579689	−0.289845	−	0.957074i	$-0.593604\pi$
$647$	−14.7451	−0.579689	−0.289845	+	0.957074i	$0.593604\pi$
$648$	0	0
$649$	31.8471	1.25011
$650$	23.1445	0.907801
$651$	0	0
$652$	1.58650	0.0621323
$653$	−12.2262	−0.478448	−0.239224	−	0.970964i	$-0.576893\pi$
$653$	−12.2262	−0.478448	−0.239224	+	0.970964i	$0.576893\pi$
$654$	0	0
$655$	−13.6500	−0.533351
$656$	−46.1514	−1.80191
$657$	0	0
$658$	2.97869	0.116121
$659$	−15.3263	−0.597027	−0.298513	−	0.954405i	$-0.596491\pi$
$659$	−15.3263	−0.597027	−0.298513	+	0.954405i	$0.596491\pi$
$660$	0	0
$661$	11.1871	0.435126	0.217563	−	0.976046i	$-0.430189\pi$
$661$	11.1871	0.435126	0.217563	+	0.976046i	$0.430189\pi$
$662$	13.5203	0.525483
$663$	0	0
$664$	−15.8660	−0.615720
$665$	−14.5097	−0.562664
$666$	0	0
$667$	1.44765	0.0560532
$668$	1.64698	0.0637235
$669$	0	0
$670$	−39.9149	−1.54205
$671$	39.5981	1.52867
$672$	0	0
$673$	45.1599	1.74079	0.870394	−	0.492356i	$-0.163864\pi$
$673$	45.1599	1.74079	0.870394	+	0.492356i	$0.163864\pi$
$674$	−43.0341	−1.65761
$675$	0	0
$676$	11.9139	0.458227
$677$	14.4541	0.555516	0.277758	−	0.960651i	$-0.410409\pi$
$677$	14.4541	0.555516	0.277758	+	0.960651i	$0.410409\pi$
$678$	0	0
$679$	2.12125	0.0814061
$680$	42.8575	1.64351
$681$	0	0
$682$	32.6656	1.25083
$683$	49.1376	1.88020	0.940099	−	0.340902i	$-0.110732\pi$
$683$	49.1376	1.88020	0.940099	+	0.340902i	$0.110732\pi$
$684$	0	0
$685$	48.1584	1.84004
$686$	1.56609	0.0597935
$687$	0	0
$688$	4.94859	0.188663
$689$	24.4254	0.930535
$690$	0	0
$691$	42.6691	1.62321	0.811604	−	0.584207i	$-0.198595\pi$
$691$	42.6691	1.62321	0.811604	+	0.584207i	$0.198595\pi$
$692$	−10.3385	−0.393010
$693$	0	0
$694$	10.3168	0.391619
$695$	15.3293	0.581472
$696$	0	0
$697$	−64.0213	−2.42498
$698$	−23.7841	−0.900240
$699$	0	0
$700$	1.06675	0.0403192
$701$	42.6336	1.61025	0.805124	−	0.593107i	$-0.202099\pi$
$701$	42.6336	1.61025	0.805124	+	0.593107i	$0.202099\pi$
$702$	0	0
$703$	25.4784	0.960938
$704$	−21.1301	−0.796371
$705$	0	0
$706$	24.2524	0.912750
$707$	3.64043	0.136912
$708$	0	0
$709$	−25.5370	−0.959062	−0.479531	−	0.877525i	$-0.659193\pi$
$709$	−25.5370	−0.959062	−0.479531	+	0.877525i	$0.659193\pi$
$710$	−40.3805	−1.51545
$711$	0	0
$712$	22.0143	0.825020
$713$	5.97252	0.223673
$714$	0	0
$715$	−65.7888	−2.46036
$716$	4.17346	0.155969
$717$	0	0
$718$	7.53847	0.281333
$719$	−23.5075	−0.876682	−0.438341	−	0.898809i	$-0.644434\pi$
$719$	−23.5075	−0.876682	−0.438341	+	0.898809i	$0.644434\pi$
$720$	0	0
$721$	−11.8268	−0.440452
$722$	15.0619	0.560546
$723$	0	0
$724$	−0.464886	−0.0172774
$725$	−3.08037	−0.114402
$726$	0	0
$727$	47.3060	1.75448	0.877241	−	0.480050i	$-0.159381\pi$
$727$	47.3060	1.75448	0.877241	+	0.480050i	$0.159381\pi$
$728$	−15.1958	−0.563195
$729$	0	0
$730$	−20.5363	−0.760083
$731$	6.86470	0.253900
$732$	0	0
$733$	−2.04894	−0.0756794	−0.0378397	−	0.999284i	$-0.512048\pi$
$733$	−2.04894	−0.0756794	−0.0378397	+	0.999284i	$0.512048\pi$
$734$	9.35845	0.345427
$735$	0	0
$736$	2.78512	0.102661
$737$	36.3471	1.33886
$738$	0	0
$739$	−36.7744	−1.35277	−0.676384	−	0.736549i	$-0.736454\pi$
$739$	−36.7744	−1.35277	−0.676384	+	0.736549i	$0.736454\pi$
$740$	−5.84715	−0.214946
$741$	0	0
$742$	6.10019	0.223945
$743$	40.4334	1.48336	0.741679	−	0.670756i	$-0.234030\pi$
$743$	40.4334	1.48336	0.741679	+	0.670756i	$0.234030\pi$
$744$	0	0
$745$	46.9466	1.71999
$746$	−36.0711	−1.32066
$747$	0	0
$748$	11.4159	0.417408
$749$	16.8427	0.615419
$750$	0	0
$751$	1.94227	0.0708745	0.0354373	−	0.999372i	$-0.488718\pi$
$751$	1.94227	0.0708745	0.0354373	+	0.999372i	$0.488718\pi$
$752$	−8.94011	−0.326012
$753$	0	0
$754$	−12.8356	−0.467446
$755$	6.62457	0.241093
$756$	0	0
$757$	38.0524	1.38304	0.691519	−	0.722358i	$-0.256942\pi$
$757$	38.0524	1.38304	0.691519	+	0.722358i	$0.256942\pi$
$758$	24.1132	0.875830
$759$	0	0
$760$	35.1617	1.27545
$761$	24.7232	0.896214	0.448107	−	0.893980i	$-0.352098\pi$
$761$	24.7232	0.896214	0.448107	+	0.893980i	$0.352098\pi$
$762$	0	0
$763$	4.05037	0.146633
$764$	0.514635	0.0186188
$765$	0	0
$766$	38.5863	1.39418
$767$	−51.6287	−1.86420
$768$	0	0
$769$	48.7617	1.75839	0.879196	−	0.476460i	$-0.158080\pi$
$769$	48.7617	1.75839	0.879196	+	0.476460i	$0.158080\pi$
$770$	−16.4306	−0.592118
$771$	0	0
$772$	0.597917	0.0215195
$773$	3.02576	0.108829	0.0544145	−	0.998518i	$-0.482671\pi$
$773$	3.02576	0.108829	0.0544145	+	0.998518i	$0.482671\pi$
$774$	0	0
$775$	−12.7086	−0.456505
$776$	−5.14046	−0.184532
$777$	0	0
$778$	−16.8648	−0.604633
$779$	−52.5251	−1.88191
$780$	0	0
$781$	36.7711	1.31577
$782$	11.3101	0.404449
$783$	0	0
$784$	−4.70039	−0.167871
$785$	−0.408410	−0.0145768
$786$	0	0
$787$	−40.5203	−1.44439	−0.722197	−	0.691688i	$-0.756868\pi$
$787$	−40.5203	−1.44439	−0.722197	+	0.691688i	$0.756868\pi$
$788$	2.30729	0.0821937
$789$	0	0
$790$	−45.6834	−1.62534
$791$	−3.17196	−0.112782
$792$	0	0
$793$	−64.1942	−2.27960
$794$	18.8616	0.669373
$795$	0	0
$796$	−1.88857	−0.0669387
$797$	−54.3528	−1.92528	−0.962638	−	0.270793i	$-0.912714\pi$
$797$	−54.3528	−1.92528	−0.962638	+	0.270793i	$0.912714\pi$
$798$	0	0
$799$	−12.4017	−0.438742
$800$	−5.92630	−0.209526
$801$	0	0
$802$	2.74944	0.0970861
$803$	18.7006	0.659931
$804$	0	0
$805$	−3.00414	−0.105882
$806$	−52.9555	−1.86528
$807$	0	0
$808$	−8.82191	−0.310354
$809$	−28.5678	−1.00439	−0.502195	−	0.864754i	$-0.667474\pi$
$809$	−28.5678	−1.00439	−0.502195	+	0.864754i	$0.667474\pi$
$810$	0	0
$811$	−0.110372	−0.00387567	−0.00193784	−	0.999998i	$-0.500617\pi$
$811$	−0.110372	−0.00387567	−0.00193784	+	0.999998i	$0.500617\pi$
$812$	−0.591603	−0.0207612
$813$	0	0
$814$	28.8514	1.01124
$815$	9.50693	0.333013
$816$	0	0
$817$	5.63202	0.197039
$818$	−53.8363	−1.88234
$819$	0	0
$820$	12.0542	0.420951
$821$	−17.7332	−0.618892	−0.309446	−	0.950917i	$-0.600144\pi$
$821$	−17.7332	−0.618892	−0.309446	+	0.950917i	$0.600144\pi$
$822$	0	0
$823$	−49.0913	−1.71122	−0.855608	−	0.517625i	$-0.826816\pi$
$823$	−49.0913	−1.71122	−0.855608	+	0.517625i	$0.826816\pi$
$824$	28.6600	0.998419
$825$	0	0
$826$	−12.8941	−0.448644
$827$	19.3140	0.671612	0.335806	−	0.941931i	$-0.390991\pi$
$827$	19.3140	0.671612	0.335806	+	0.941931i	$0.390991\pi$
$828$	0	0
$829$	−42.7397	−1.48441	−0.742206	−	0.670172i	$-0.766220\pi$
$829$	−42.7397	−1.48441	−0.742206	+	0.670172i	$0.766220\pi$
$830$	27.8110	0.965334
$831$	0	0
$832$	34.2549	1.18758
$833$	−6.52039	−0.225918
$834$	0	0
$835$	9.86931	0.341542
$836$	9.36600	0.323930
$837$	0	0
$838$	−42.5747	−1.47072
$839$	16.2313	0.560368	0.280184	−	0.959946i	$-0.409604\pi$
$839$	16.2313	0.560368	0.280184	+	0.959946i	$0.409604\pi$
$840$	0	0
$841$	−27.2917	−0.941092
$842$	56.7726	1.95651
$843$	0	0
$844$	−8.79716	−0.302811
$845$	71.3926	2.45598
$846$	0	0
$847$	3.96192	0.136133
$848$	−18.3088	−0.628728
$849$	0	0
$850$	−24.0661	−0.825461
$851$	5.27514	0.180829
$852$	0	0
$853$	−22.4873	−0.769951	−0.384975	−	0.922927i	$-0.625790\pi$
$853$	−22.4873	−0.769951	−0.384975	+	0.922927i	$0.625790\pi$
$854$	−16.0323	−0.548616
$855$	0	0
$856$	−40.8152	−1.39503
$857$	−5.28649	−0.180583	−0.0902916	−	0.995915i	$-0.528780\pi$
$857$	−5.28649	−0.180583	−0.0902916	+	0.995915i	$0.528780\pi$
$858$	0	0
$859$	−1.48519	−0.0506740	−0.0253370	−	0.999679i	$-0.508066\pi$
$859$	−1.48519	−0.0506740	−0.0253370	+	0.999679i	$0.508066\pi$
$860$	−1.29252	−0.0440744
$861$	0	0
$862$	−9.09139	−0.309654
$863$	−7.56318	−0.257454	−0.128727	−	0.991680i	$-0.541089\pi$
$863$	−7.56318	−0.257454	−0.128727	+	0.991680i	$0.541089\pi$
$864$	0	0
$865$	−61.9521	−2.10644
$866$	−59.8791	−2.03477
$867$	0	0
$868$	−2.44076	−0.0828447
$869$	41.5999	1.41118
$870$	0	0
$871$	−58.9238	−1.99656
$872$	−9.81534	−0.332389
$873$	0	0
$874$	9.27917	0.313873
$875$	−7.16934	−0.242368
$876$	0	0
$877$	21.0842	0.711964	0.355982	−	0.934493i	$-0.384146\pi$
$877$	21.0842	0.711964	0.355982	+	0.934493i	$0.384146\pi$
$878$	−51.3945	−1.73448
$879$	0	0
$880$	49.3141	1.66238
$881$	10.0486	0.338544	0.169272	−	0.985569i	$-0.445858\pi$
$881$	10.0486	0.338544	0.169272	+	0.985569i	$0.445858\pi$
$882$	0	0
$883$	−3.38205	−0.113815	−0.0569075	−	0.998379i	$-0.518124\pi$
$883$	−3.38205	−0.113815	−0.0569075	+	0.998379i	$0.518124\pi$
$884$	−18.5069	−0.622453
$885$	0	0
$886$	−51.0825	−1.71615
$887$	−51.7229	−1.73668	−0.868342	−	0.495965i	$-0.834814\pi$
$887$	−51.7229	−1.73668	−0.868342	+	0.495965i	$0.834814\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−38.5881	−1.29348
$891$	0	0
$892$	−4.38398	−0.146787
$893$	−10.1748	−0.340486
$894$	0	0
$895$	25.0089	0.835956
$896$	13.5843	0.453818
$897$	0	0
$898$	3.22956	0.107772
$899$	7.04801	0.235064
$900$	0	0
$901$	−25.3981	−0.846133
$902$	−59.4786	−1.98042
$903$	0	0
$904$	7.68665	0.255654
$905$	−2.78577	−0.0926023
$906$	0	0
$907$	−24.2150	−0.804046	−0.402023	−	0.915630i	$-0.631693\pi$
$907$	−24.2150	−0.804046	−0.402023	+	0.915630i	$0.631693\pi$
$908$	3.33326	0.110618
$909$	0	0
$910$	26.6363	0.882986
$911$	−8.68695	−0.287812	−0.143906	−	0.989591i	$-0.545966\pi$
$911$	−8.68695	−0.287812	−0.143906	+	0.989591i	$0.545966\pi$
$912$	0	0
$913$	−25.3251	−0.838138
$914$	7.77073	0.257033
$915$	0	0
$916$	3.82773	0.126472
$917$	−5.03257	−0.166190
$918$	0	0
$919$	17.7099	0.584194	0.292097	−	0.956389i	$-0.405647\pi$
$919$	17.7099	0.584194	0.292097	+	0.956389i	$0.405647\pi$
$920$	7.27998	0.240014
$921$	0	0
$922$	46.7618	1.54002
$923$	−59.6111	−1.96212
$924$	0	0
$925$	−11.2247	−0.369064
$926$	−16.7692	−0.551069
$927$	0	0
$928$	3.28665	0.107889
$929$	−4.75017	−0.155848	−0.0779241	−	0.996959i	$-0.524829\pi$
$929$	−4.75017	−0.155848	−0.0779241	+	0.996959i	$0.524829\pi$
$930$	0	0
$931$	−5.34954	−0.175324
$932$	−11.3474	−0.371697
$933$	0	0
$934$	−39.2903	−1.28562
$935$	68.4086	2.23720
$936$	0	0
$937$	41.0032	1.33951	0.669757	−	0.742580i	$-0.266398\pi$
$937$	41.0032	1.33951	0.669757	+	0.742580i	$0.266398\pi$
$938$	−14.7161	−0.480497
$939$	0	0
$940$	2.33505	0.0761611
$941$	47.3225	1.54267	0.771335	−	0.636429i	$-0.219589\pi$
$941$	47.3225	1.54267	0.771335	+	0.636429i	$0.219589\pi$
$942$	0	0
$943$	−10.8750	−0.354138
$944$	38.6999	1.25957
$945$	0	0
$946$	6.37761	0.207354
$947$	41.7727	1.35743	0.678716	−	0.734401i	$-0.262537\pi$
$947$	41.7727	1.35743	0.678716	+	0.734401i	$0.262537\pi$
$948$	0	0
$949$	−30.3164	−0.984112
$950$	−19.7446	−0.640600
$951$	0	0
$952$	15.8010	0.512112
$953$	−2.94403	−0.0953664	−0.0476832	−	0.998863i	$-0.515184\pi$
$953$	−2.94403	−0.0953664	−0.0476832	+	0.998863i	$0.515184\pi$
$954$	0	0
$955$	3.08389	0.0997922
$956$	3.11416	0.100719
$957$	0	0
$958$	3.99855	0.129187
$959$	17.7553	0.573349
$960$	0	0
$961$	−1.92225	−0.0620081
$962$	−46.7722	−1.50800
$963$	0	0
$964$	13.8447	0.445908
$965$	3.58294	0.115339
$966$	0	0
$967$	−53.4766	−1.71969	−0.859846	−	0.510554i	$-0.829440\pi$
$967$	−53.4766	−1.71969	−0.859846	+	0.510554i	$0.829440\pi$
$968$	−9.60099	−0.308588
$969$	0	0
$970$	9.01055	0.289311
$971$	28.2856	0.907729	0.453864	−	0.891071i	$-0.350045\pi$
$971$	28.2856	0.907729	0.453864	+	0.891071i	$0.350045\pi$
$972$	0	0
$973$	5.65169	0.181185
$974$	−16.0857	−0.515418
$975$	0	0
$976$	48.1188	1.54024
$977$	−21.6721	−0.693353	−0.346677	−	0.937985i	$-0.612690\pi$
$977$	−21.6721	−0.693353	−0.346677	+	0.937985i	$0.612690\pi$
$978$	0	0
$979$	35.1389	1.12304
$980$	1.22769	0.0392170
$981$	0	0
$982$	8.44637	0.269535
$983$	−31.5767	−1.00714	−0.503570	−	0.863955i	$-0.667980\pi$
$983$	−31.5767	−1.00714	−0.503570	+	0.863955i	$0.667980\pi$
$984$	0	0
$985$	13.8261	0.440537
$986$	13.3468	0.425047
$987$	0	0
$988$	−15.1836	−0.483055
$989$	1.16607	0.0370789
$990$	0	0
$991$	−40.7400	−1.29415	−0.647074	−	0.762427i	$-0.724008\pi$
$991$	−40.7400	−1.29415	−0.647074	+	0.762427i	$0.724008\pi$
$992$	13.5596	0.430518
$993$	0	0
$994$	−14.8877	−0.472210
$995$	−11.3170	−0.358774
$996$	0	0
$997$	−44.4314	−1.40716	−0.703578	−	0.710618i	$-0.748416\pi$
$997$	−44.4314	−1.40716	−0.703578	+	0.710618i	$0.748416\pi$
$998$	30.4651	0.964356
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.x.1.17	yes	22
3.2	odd	2	inner	8001.2.a.x.1.6	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.x.1.6	✓	22	3.2	odd	2	inner
8001.2.a.x.1.17	yes	22	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56609 q^{2} +0.452631 q^{4} +2.71234 q^{5} +1.00000 q^{7} -2.42332 q^{8} +O(q^{10})\)	\(q+1.56609 q^{2} +0.452631 q^{4} +2.71234 q^{5} +1.00000 q^{7} -2.42332 q^{8} +4.24776 q^{10} -3.86806 q^{11} +6.27068 q^{13} +1.56609 q^{14} -4.70039 q^{16} -6.52039 q^{17} -5.34954 q^{19} +1.22769 q^{20} -6.05773 q^{22} -1.10758 q^{23} +2.35677 q^{25} +9.82044 q^{26} +0.452631 q^{28} -1.30703 q^{29} -5.39238 q^{31} -2.51459 q^{32} -10.2115 q^{34} +2.71234 q^{35} -4.76274 q^{37} -8.37784 q^{38} -6.57285 q^{40} +9.81863 q^{41} -1.05281 q^{43} -1.75081 q^{44} -1.73458 q^{46} +1.90199 q^{47} +1.00000 q^{49} +3.69090 q^{50} +2.83831 q^{52} +3.89518 q^{53} -10.4915 q^{55} -2.42332 q^{56} -2.04693 q^{58} -8.23334 q^{59} -10.2372 q^{61} -8.44494 q^{62} +5.46271 q^{64} +17.0082 q^{65} -9.39671 q^{67} -2.95133 q^{68} +4.24776 q^{70} -9.50632 q^{71} -4.83462 q^{73} -7.45887 q^{74} -2.42137 q^{76} -3.86806 q^{77} -10.7547 q^{79} -12.7490 q^{80} +15.3768 q^{82} +6.54722 q^{83} -17.6855 q^{85} -1.64879 q^{86} +9.37354 q^{88} -9.08436 q^{89} +6.27068 q^{91} -0.501327 q^{92} +2.97869 q^{94} -14.5097 q^{95} +2.12125 q^{97} +1.56609 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) \) 22 * q + 10 * q^4 + 22 * q^7	\( 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) \) 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more