LMFDB - Embedded newform 8001.2.a.w.1.14

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-0.723268$ of defining polynomial
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)

$f(q)$	$=$	$q+0.723268 q^{2} -1.47688 q^{4} +1.71672 q^{5} +1.00000 q^{7} -2.51472 q^{8} +O(q^{10})$	\(q+0.723268 q^{2} -1.47688 q^{4} +1.71672 q^{5} +1.00000 q^{7} -2.51472 q^{8} +1.24165 q^{10} +2.34692 q^{11} +1.51901 q^{13} +0.723268 q^{14} +1.13495 q^{16} -5.56671 q^{17} +2.03784 q^{19} -2.53539 q^{20} +1.69745 q^{22} -2.09892 q^{23} -2.05288 q^{25} +1.09866 q^{26} -1.47688 q^{28} -8.65813 q^{29} +7.63712 q^{31} +5.85031 q^{32} -4.02623 q^{34} +1.71672 q^{35} -7.01965 q^{37} +1.47391 q^{38} -4.31706 q^{40} -5.73325 q^{41} +3.33661 q^{43} -3.46613 q^{44} -1.51808 q^{46} -11.2014 q^{47} +1.00000 q^{49} -1.48478 q^{50} -2.24341 q^{52} -7.41884 q^{53} +4.02900 q^{55} -2.51472 q^{56} -6.26215 q^{58} -2.73842 q^{59} -12.4790 q^{61} +5.52369 q^{62} +1.96145 q^{64} +2.60772 q^{65} +11.2654 q^{67} +8.22138 q^{68} +1.24165 q^{70} +10.4252 q^{71} -2.46825 q^{73} -5.07709 q^{74} -3.00965 q^{76} +2.34692 q^{77} -2.24902 q^{79} +1.94839 q^{80} -4.14668 q^{82} +3.96749 q^{83} -9.55648 q^{85} +2.41327 q^{86} -5.90185 q^{88} +1.28249 q^{89} +1.51901 q^{91} +3.09985 q^{92} -8.10159 q^{94} +3.49840 q^{95} +14.1304 q^{97} +0.723268 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

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 $

$2$	0.723268	0.511428	0.255714	−	0.966752i	$-0.417689\pi$
$2$	0.723268	0.511428	0.255714	+	0.966752i	$0.417689\pi$
$3$	0	0
$3$	0	0
$4$	−1.47688	−0.738441
$5$	1.71672	0.767740	0.383870	−	0.923387i	$-0.374591\pi$
$5$	1.71672	0.767740	0.383870	+	0.923387i	$0.374591\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.51472	−0.889088
$9$	0	0
$10$	1.24165	0.392644
$11$	2.34692	0.707624	0.353812	−	0.935317i	$-0.384885\pi$
$11$	2.34692	0.707624	0.353812	+	0.935317i	$0.384885\pi$
$12$	0	0
$13$	1.51901	0.421299	0.210649	−	0.977562i	$-0.432442\pi$
$13$	1.51901	0.421299	0.210649	+	0.977562i	$0.432442\pi$
$14$	0.723268	0.193302
$15$	0	0
$16$	1.13495	0.283737
$17$	−5.56671	−1.35013	−0.675063	−	0.737760i	$-0.735884\pi$
$17$	−5.56671	−1.35013	−0.675063	+	0.737760i	$0.735884\pi$
$18$	0	0
$19$	2.03784	0.467512	0.233756	−	0.972295i	$-0.424898\pi$
$19$	2.03784	0.467512	0.233756	+	0.972295i	$0.424898\pi$
$20$	−2.53539	−0.566931
$21$	0	0
$22$	1.69745	0.361899
$23$	−2.09892	−0.437654	−0.218827	−	0.975764i	$-0.570223\pi$
$23$	−2.09892	−0.437654	−0.218827	+	0.975764i	$0.570223\pi$
$24$	0	0
$25$	−2.05288	−0.410576
$26$	1.09866	0.215464
$27$	0	0
$28$	−1.47688	−0.279105
$29$	−8.65813	−1.60777	−0.803887	−	0.594782i	$-0.797238\pi$
$29$	−8.65813	−1.60777	−0.803887	+	0.594782i	$0.797238\pi$
$30$	0	0
$31$	7.63712	1.37167	0.685834	−	0.727758i	$-0.259438\pi$
$31$	7.63712	1.37167	0.685834	+	0.727758i	$0.259438\pi$
$32$	5.85031	1.03420
$33$	0	0
$34$	−4.02623	−0.690492
$35$	1.71672	0.290178
$36$	0	0
$37$	−7.01965	−1.15402	−0.577011	−	0.816736i	$-0.695781\pi$
$37$	−7.01965	−1.15402	−0.577011	+	0.816736i	$0.695781\pi$
$38$	1.47391	0.239099
$39$	0	0
$40$	−4.31706	−0.682588
$41$	−5.73325	−0.895384	−0.447692	−	0.894188i	$-0.647754\pi$
$41$	−5.73325	−0.895384	−0.447692	+	0.894188i	$0.647754\pi$
$42$	0	0
$43$	3.33661	0.508828	0.254414	−	0.967095i	$-0.418117\pi$
$43$	3.33661	0.508828	0.254414	+	0.967095i	$0.418117\pi$
$44$	−3.46613	−0.522539
$45$	0	0
$46$	−1.51808	−0.223829
$47$	−11.2014	−1.63389	−0.816943	−	0.576718i	$-0.804333\pi$
$47$	−11.2014	−1.63389	−0.816943	+	0.576718i	$0.804333\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.48478	−0.209980
$51$	0	0
$52$	−2.24341	−0.311104
$53$	−7.41884	−1.01906	−0.509528	−	0.860454i	$-0.670180\pi$
$53$	−7.41884	−1.01906	−0.509528	+	0.860454i	$0.670180\pi$
$54$	0	0
$55$	4.02900	0.543271
$56$	−2.51472	−0.336044
$57$	0	0
$58$	−6.26215	−0.822260
$59$	−2.73842	−0.356512	−0.178256	−	0.983984i	$-0.557046\pi$
$59$	−2.73842	−0.356512	−0.178256	+	0.983984i	$0.557046\pi$
$60$	0	0
$61$	−12.4790	−1.59778	−0.798888	−	0.601480i	$-0.794578\pi$
$61$	−12.4790	−1.59778	−0.798888	+	0.601480i	$0.794578\pi$
$62$	5.52369	0.701509
$63$	0	0
$64$	1.96145	0.245181
$65$	2.60772	0.323448
$66$	0	0
$67$	11.2654	1.37629	0.688144	−	0.725574i	$-0.258426\pi$
$67$	11.2654	1.37629	0.688144	+	0.725574i	$0.258426\pi$
$68$	8.22138	0.996989
$69$	0	0
$70$	1.24165	0.148405
$71$	10.4252	1.23725	0.618624	−	0.785687i	$-0.287690\pi$
$71$	10.4252	1.23725	0.618624	+	0.785687i	$0.287690\pi$
$72$	0	0
$73$	−2.46825	−0.288887	−0.144443	−	0.989513i	$-0.546139\pi$
$73$	−2.46825	−0.288887	−0.144443	+	0.989513i	$0.546139\pi$
$74$	−5.07709	−0.590199
$75$	0	0
$76$	−3.00965	−0.345231
$77$	2.34692	0.267457
$78$	0	0
$79$	−2.24902	−0.253034	−0.126517	−	0.991964i	$-0.540380\pi$
$79$	−2.24902	−0.253034	−0.126517	+	0.991964i	$0.540380\pi$
$80$	1.94839	0.217836
$81$	0	0
$82$	−4.14668	−0.457924
$83$	3.96749	0.435489	0.217744	−	0.976006i	$-0.430130\pi$
$83$	3.96749	0.435489	0.217744	+	0.976006i	$0.430130\pi$
$84$	0	0
$85$	−9.55648	−1.03655
$86$	2.41327	0.260229
$87$	0	0
$88$	−5.90185	−0.629139
$89$	1.28249	0.135943	0.0679717	−	0.997687i	$-0.478347\pi$
$89$	1.28249	0.135943	0.0679717	+	0.997687i	$0.478347\pi$
$90$	0	0
$91$	1.51901	0.159236
$92$	3.09985	0.323182
$93$	0	0
$94$	−8.10159	−0.835615
$95$	3.49840	0.358928
$96$	0	0
$97$	14.1304	1.43473	0.717363	−	0.696700i	$-0.245349\pi$
$97$	14.1304	1.43473	0.717363	+	0.696700i	$0.245349\pi$
$98$	0.723268	0.0730611
$99$	0	0
$100$	3.03186	0.303186
$101$	−1.97849	−0.196868	−0.0984338	−	0.995144i	$-0.531383\pi$
$101$	−1.97849	−0.196868	−0.0984338	+	0.995144i	$0.531383\pi$
$102$	0	0
$103$	−0.972130	−0.0957868	−0.0478934	−	0.998852i	$-0.515251\pi$
$103$	−0.972130	−0.0957868	−0.0478934	+	0.998852i	$0.515251\pi$
$104$	−3.81989	−0.374572
$105$	0	0
$106$	−5.36581	−0.521173
$107$	19.4565	1.88094	0.940468	−	0.339883i	$-0.110387\pi$
$107$	19.4565	1.88094	0.940468	+	0.339883i	$0.110387\pi$
$108$	0	0
$109$	−4.29292	−0.411187	−0.205594	−	0.978637i	$-0.565913\pi$
$109$	−4.29292	−0.411187	−0.205594	+	0.978637i	$0.565913\pi$
$110$	2.91405	0.277844
$111$	0	0
$112$	1.13495	0.107243
$113$	−13.1791	−1.23978	−0.619892	−	0.784687i	$-0.712824\pi$
$113$	−13.1791	−1.23978	−0.619892	+	0.784687i	$0.712824\pi$
$114$	0	0
$115$	−3.60325	−0.336005
$116$	12.7870	1.18725
$117$	0	0
$118$	−1.98061	−0.182330
$119$	−5.56671	−0.510300
$120$	0	0
$121$	−5.49196	−0.499269
$122$	−9.02569	−0.817147
$123$	0	0
$124$	−11.2791	−1.01290
$125$	−12.1078	−1.08296
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−10.2820	−0.908806
$129$	0	0
$130$	1.88608	0.165420
$131$	14.1117	1.23295	0.616474	−	0.787376i	$-0.288561\pi$
$131$	14.1117	1.23295	0.616474	+	0.787376i	$0.288561\pi$
$132$	0	0
$133$	2.03784	0.176703
$134$	8.14791	0.703872
$135$	0	0
$136$	13.9987	1.20038
$137$	−11.3531	−0.969957	−0.484979	−	0.874526i	$-0.661173\pi$
$137$	−11.3531	−0.969957	−0.484979	+	0.874526i	$0.661173\pi$
$138$	0	0
$139$	6.58453	0.558493	0.279246	−	0.960220i	$-0.409915\pi$
$139$	6.58453	0.558493	0.279246	+	0.960220i	$0.409915\pi$
$140$	−2.53539	−0.214280
$141$	0	0
$142$	7.54025	0.632764
$143$	3.56501	0.298121
$144$	0	0
$145$	−14.8636	−1.23435
$146$	−1.78521	−0.147745
$147$	0	0
$148$	10.3672	0.852178
$149$	3.02130	0.247515	0.123757	−	0.992312i	$-0.460506\pi$
$149$	3.02130	0.247515	0.123757	+	0.992312i	$0.460506\pi$
$150$	0	0
$151$	−11.3128	−0.920625	−0.460312	−	0.887757i	$-0.652263\pi$
$151$	−11.3128	−0.920625	−0.460312	+	0.887757i	$0.652263\pi$
$152$	−5.12460	−0.415660
$153$	0	0
$154$	1.69745	0.136785
$155$	13.1108	1.05308
$156$	0	0
$157$	−14.9291	−1.19148	−0.595738	−	0.803179i	$-0.703140\pi$
$157$	−14.9291	−1.19148	−0.595738	+	0.803179i	$0.703140\pi$
$158$	−1.62664	−0.129409
$159$	0	0
$160$	10.0433	0.793995
$161$	−2.09892	−0.165418
$162$	0	0
$163$	−16.0952	−1.26067	−0.630335	−	0.776323i	$-0.717082\pi$
$163$	−16.0952	−1.26067	−0.630335	+	0.776323i	$0.717082\pi$
$164$	8.46735	0.661189
$165$	0	0
$166$	2.86956	0.222721
$167$	−20.9991	−1.62496	−0.812478	−	0.582991i	$-0.801882\pi$
$167$	−20.9991	−1.62496	−0.812478	+	0.582991i	$0.801882\pi$
$168$	0	0
$169$	−10.6926	−0.822507
$170$	−6.91190	−0.530118
$171$	0	0
$172$	−4.92778	−0.375740
$173$	−15.9978	−1.21629	−0.608145	−	0.793826i	$-0.708086\pi$
$173$	−15.9978	−1.21629	−0.608145	+	0.793826i	$0.708086\pi$
$174$	0	0
$175$	−2.05288	−0.155183
$176$	2.66364	0.200779
$177$	0	0
$178$	0.927583	0.0695253
$179$	−12.3965	−0.926556	−0.463278	−	0.886213i	$-0.653327\pi$
$179$	−12.3965	−0.926556	−0.463278	+	0.886213i	$0.653327\pi$
$180$	0	0
$181$	−5.91821	−0.439897	−0.219948	−	0.975511i	$-0.570589\pi$
$181$	−5.91821	−0.439897	−0.219948	+	0.975511i	$0.570589\pi$
$182$	1.09866	0.0814377
$183$	0	0
$184$	5.27819	0.389113
$185$	−12.0508	−0.885989
$186$	0	0
$187$	−13.0646	−0.955381
$188$	16.5431	1.20653
$189$	0	0
$190$	2.53028	0.183566
$191$	13.0788	0.946349	0.473174	−	0.880969i	$-0.343108\pi$
$191$	13.0788	0.946349	0.473174	+	0.880969i	$0.343108\pi$
$192$	0	0
$193$	−2.17434	−0.156512	−0.0782560	−	0.996933i	$-0.524935\pi$
$193$	−2.17434	−0.156512	−0.0782560	+	0.996933i	$0.524935\pi$
$194$	10.2201	0.733759
$195$	0	0
$196$	−1.47688	−0.105492
$197$	9.83957	0.701040	0.350520	−	0.936555i	$-0.386005\pi$
$197$	9.83957	0.701040	0.350520	+	0.936555i	$0.386005\pi$
$198$	0	0
$199$	0.804086	0.0570001	0.0285001	−	0.999594i	$-0.490927\pi$
$199$	0.804086	0.0570001	0.0285001	+	0.999594i	$0.490927\pi$
$200$	5.16241	0.365038
$201$	0	0
$202$	−1.43098	−0.100684
$203$	−8.65813	−0.607681
$204$	0	0
$205$	−9.84238	−0.687422
$206$	−0.703111	−0.0489881
$207$	0	0
$208$	1.72400	0.119538
$209$	4.78265	0.330823
$210$	0	0
$211$	−7.35853	−0.506582	−0.253291	−	0.967390i	$-0.581513\pi$
$211$	−7.35853	−0.506582	−0.253291	+	0.967390i	$0.581513\pi$
$212$	10.9568	0.752513
$213$	0	0
$214$	14.0723	0.961963
$215$	5.72802	0.390648
$216$	0	0
$217$	7.63712	0.518441
$218$	−3.10494	−0.210293
$219$	0	0
$220$	−5.95037	−0.401174
$221$	−8.45592	−0.568806
$222$	0	0
$223$	1.21172	0.0811431	0.0405715	−	0.999177i	$-0.487082\pi$
$223$	1.21172	0.0811431	0.0405715	+	0.999177i	$0.487082\pi$
$224$	5.85031	0.390890
$225$	0	0
$226$	−9.53202	−0.634060
$227$	−5.96872	−0.396158	−0.198079	−	0.980186i	$-0.563470\pi$
$227$	−5.96872	−0.396158	−0.198079	+	0.980186i	$0.563470\pi$
$228$	0	0
$229$	9.03764	0.597224	0.298612	−	0.954375i	$-0.403476\pi$
$229$	9.03764	0.597224	0.298612	+	0.954375i	$0.403476\pi$
$230$	−2.60612	−0.171842
$231$	0	0
$232$	21.7728	1.42945
$233$	19.8103	1.29781	0.648907	−	0.760868i	$-0.275226\pi$
$233$	19.8103	1.29781	0.648907	+	0.760868i	$0.275226\pi$
$234$	0	0
$235$	−19.2296	−1.25440
$236$	4.04433	0.263263
$237$	0	0
$238$	−4.02623	−0.260982
$239$	−11.8646	−0.767457	−0.383728	−	0.923446i	$-0.625360\pi$
$239$	−11.8646	−0.767457	−0.383728	+	0.923446i	$0.625360\pi$
$240$	0	0
$241$	−15.5177	−0.999585	−0.499793	−	0.866145i	$-0.666590\pi$
$241$	−15.5177	−0.999585	−0.499793	+	0.866145i	$0.666590\pi$
$242$	−3.97216	−0.255340
$243$	0	0
$244$	18.4301	1.17986
$245$	1.71672	0.109677
$246$	0	0
$247$	3.09551	0.196962
$248$	−19.2052	−1.21953
$249$	0	0
$250$	−8.75719	−0.553854
$251$	27.1523	1.71384	0.856921	−	0.515449i	$-0.172375\pi$
$251$	27.1523	1.71384	0.856921	+	0.515449i	$0.172375\pi$
$252$	0	0
$253$	−4.92599	−0.309695
$254$	−0.723268	−0.0453819
$255$	0	0
$256$	−11.3595	−0.709970
$257$	24.1385	1.50572	0.752858	−	0.658182i	$-0.228674\pi$
$257$	24.1385	1.50572	0.752858	+	0.658182i	$0.228674\pi$
$258$	0	0
$259$	−7.01965	−0.436180
$260$	−3.85130	−0.238847
$261$	0	0
$262$	10.2066	0.630564
$263$	5.82032	0.358896	0.179448	−	0.983767i	$-0.442569\pi$
$263$	5.82032	0.358896	0.179448	+	0.983767i	$0.442569\pi$
$264$	0	0
$265$	−12.7360	−0.782369
$266$	1.47391	0.0903709
$267$	0	0
$268$	−16.6377	−1.01631
$269$	−2.77706	−0.169321	−0.0846603	−	0.996410i	$-0.526981\pi$
$269$	−2.77706	−0.169321	−0.0846603	+	0.996410i	$0.526981\pi$
$270$	0	0
$271$	−6.35708	−0.386165	−0.193083	−	0.981183i	$-0.561849\pi$
$271$	−6.35708	−0.386165	−0.193083	+	0.981183i	$0.561849\pi$
$272$	−6.31793	−0.383081
$273$	0	0
$274$	−8.21131	−0.496063
$275$	−4.81795	−0.290533
$276$	0	0
$277$	31.5334	1.89466	0.947330	−	0.320258i	$-0.103770\pi$
$277$	31.5334	1.89466	0.947330	+	0.320258i	$0.103770\pi$
$278$	4.76238	0.285629
$279$	0	0
$280$	−4.31706	−0.257994
$281$	9.53727	0.568946	0.284473	−	0.958684i	$-0.408181\pi$
$281$	9.53727	0.568946	0.284473	+	0.958684i	$0.408181\pi$
$282$	0	0
$283$	−19.1335	−1.13737	−0.568684	−	0.822556i	$-0.692547\pi$
$283$	−19.1335	−1.13737	−0.568684	+	0.822556i	$0.692547\pi$
$284$	−15.3969	−0.913636
$285$	0	0
$286$	2.57846	0.152467
$287$	−5.73325	−0.338423
$288$	0	0
$289$	13.9883	0.822841
$290$	−10.7503	−0.631282
$291$	0	0
$292$	3.64532	0.213326
$293$	−1.07399	−0.0627433	−0.0313717	−	0.999508i	$-0.509988\pi$
$293$	−1.07399	−0.0627433	−0.0313717	+	0.999508i	$0.509988\pi$
$294$	0	0
$295$	−4.70110	−0.273709
$296$	17.6524	1.02603
$297$	0	0
$298$	2.18521	0.126586
$299$	−3.18828	−0.184383
$300$	0	0
$301$	3.33661	0.192319
$302$	−8.18221	−0.470833
$303$	0	0
$304$	2.31284	0.132651
$305$	−21.4230	−1.22668
$306$	0	0
$307$	−18.7085	−1.06775	−0.533876	−	0.845563i	$-0.679265\pi$
$307$	−18.7085	−1.06775	−0.533876	+	0.845563i	$0.679265\pi$
$308$	−3.46613	−0.197501
$309$	0	0
$310$	9.48261	0.538576
$311$	−25.4377	−1.44244	−0.721219	−	0.692707i	$-0.756418\pi$
$311$	−25.4377	−1.44244	−0.721219	+	0.692707i	$0.756418\pi$
$312$	0	0
$313$	−2.93785	−0.166057	−0.0830287	−	0.996547i	$-0.526459\pi$
$313$	−2.93785	−0.166057	−0.0830287	+	0.996547i	$0.526459\pi$
$314$	−10.7978	−0.609354
$315$	0	0
$316$	3.32153	0.186851
$317$	−24.4090	−1.37095	−0.685473	−	0.728098i	$-0.740405\pi$
$317$	−24.4090	−1.37095	−0.685473	+	0.728098i	$0.740405\pi$
$318$	0	0
$319$	−20.3199	−1.13770
$320$	3.36726	0.188235
$321$	0	0
$322$	−1.51808	−0.0845993
$323$	−11.3441	−0.631201
$324$	0	0
$325$	−3.11835	−0.172975
$326$	−11.6411	−0.644742
$327$	0	0
$328$	14.4175	0.796075
$329$	−11.2014	−0.617551
$330$	0	0
$331$	20.9164	1.14967	0.574835	−	0.818269i	$-0.305066\pi$
$331$	20.9164	1.14967	0.574835	+	0.818269i	$0.305066\pi$
$332$	−5.85952	−0.321583
$333$	0	0
$334$	−15.1880	−0.831049
$335$	19.3395	1.05663
$336$	0	0
$337$	28.8655	1.57240	0.786201	−	0.617971i	$-0.212045\pi$
$337$	28.8655	1.57240	0.786201	+	0.617971i	$0.212045\pi$
$338$	−7.73362	−0.420653
$339$	0	0
$340$	14.1138	0.765428
$341$	17.9237	0.970624
$342$	0	0
$343$	1.00000	0.0539949
$344$	−8.39064	−0.452393
$345$	0	0
$346$	−11.5707	−0.622045
$347$	10.6913	0.573937	0.286969	−	0.957940i	$-0.407352\pi$
$347$	10.6913	0.573937	0.286969	+	0.957940i	$0.407352\pi$
$348$	0	0
$349$	−28.8884	−1.54636	−0.773180	−	0.634187i	$-0.781335\pi$
$349$	−28.8884	−1.54636	−0.773180	+	0.634187i	$0.781335\pi$
$350$	−1.48478	−0.0793650
$351$	0	0
$352$	13.7302	0.731823
$353$	7.08939	0.377330	0.188665	−	0.982041i	$-0.439584\pi$
$353$	7.08939	0.377330	0.188665	+	0.982041i	$0.439584\pi$
$354$	0	0
$355$	17.8972	0.949885
$356$	−1.89408	−0.100386
$357$	0	0
$358$	−8.96598	−0.473867
$359$	−16.8745	−0.890604	−0.445302	−	0.895380i	$-0.646904\pi$
$359$	−16.8745	−0.890604	−0.445302	+	0.895380i	$0.646904\pi$
$360$	0	0
$361$	−14.8472	−0.781432
$362$	−4.28045	−0.224976
$363$	0	0
$364$	−2.24341	−0.117586
$365$	−4.23729	−0.221790
$366$	0	0
$367$	−5.70904	−0.298010	−0.149005	−	0.988836i	$-0.547607\pi$
$367$	−5.70904	−0.298010	−0.149005	+	0.988836i	$0.547607\pi$
$368$	−2.38216	−0.124179
$369$	0	0
$370$	−8.71593	−0.453120
$371$	−7.41884	−0.385167
$372$	0	0
$373$	−5.60619	−0.290277	−0.145139	−	0.989411i	$-0.546363\pi$
$373$	−5.60619	−0.290277	−0.145139	+	0.989411i	$0.546363\pi$
$374$	−9.44924	−0.488609
$375$	0	0
$376$	28.1683	1.45267
$377$	−13.1518	−0.677353
$378$	0	0
$379$	−24.7151	−1.26953	−0.634764	−	0.772706i	$-0.718903\pi$
$379$	−24.7151	−1.26953	−0.634764	+	0.772706i	$0.718903\pi$
$380$	−5.16672	−0.265047
$381$	0	0
$382$	9.45948	0.483989
$383$	−24.8544	−1.27000	−0.635000	−	0.772512i	$-0.719000\pi$
$383$	−24.8544	−1.27000	−0.635000	+	0.772512i	$0.719000\pi$
$384$	0	0
$385$	4.02900	0.205337
$386$	−1.57263	−0.0800447
$387$	0	0
$388$	−20.8689	−1.05946
$389$	−31.4488	−1.59452	−0.797258	−	0.603638i	$-0.793717\pi$
$389$	−31.4488	−1.59452	−0.797258	+	0.603638i	$0.793717\pi$
$390$	0	0
$391$	11.6841	0.590889
$392$	−2.51472	−0.127013
$393$	0	0
$394$	7.11665	0.358532
$395$	−3.86093	−0.194264
$396$	0	0
$397$	−24.3152	−1.22034	−0.610172	−	0.792269i	$-0.708900\pi$
$397$	−24.3152	−1.22034	−0.610172	+	0.792269i	$0.708900\pi$
$398$	0.581570	0.0291515
$399$	0	0
$400$	−2.32991	−0.116496
$401$	−14.4478	−0.721489	−0.360745	−	0.932665i	$-0.617477\pi$
$401$	−14.4478	−0.721489	−0.360745	+	0.932665i	$0.617477\pi$
$402$	0	0
$403$	11.6009	0.577882
$404$	2.92200	0.145375
$405$	0	0
$406$	−6.26215	−0.310785
$407$	−16.4746	−0.816614
$408$	0	0
$409$	−14.7521	−0.729444	−0.364722	−	0.931116i	$-0.618836\pi$
$409$	−14.7521	−0.729444	−0.364722	+	0.931116i	$0.618836\pi$
$410$	−7.11868	−0.351567
$411$	0	0
$412$	1.43572	0.0707330
$413$	−2.73842	−0.134749
$414$	0	0
$415$	6.81106	0.334342
$416$	8.88671	0.435707
$417$	0	0
$418$	3.45914	0.169192
$419$	31.6597	1.54668	0.773339	−	0.633993i	$-0.218585\pi$
$419$	31.6597	1.54668	0.773339	+	0.633993i	$0.218585\pi$
$420$	0	0
$421$	−4.66224	−0.227223	−0.113612	−	0.993525i	$-0.536242\pi$
$421$	−4.66224	−0.227223	−0.113612	+	0.993525i	$0.536242\pi$
$422$	−5.32219	−0.259080
$423$	0	0
$424$	18.6563	0.906030
$425$	11.4278	0.554329
$426$	0	0
$427$	−12.4790	−0.603902
$428$	−28.7350	−1.38896
$429$	0	0
$430$	4.14290	0.199788
$431$	−37.1901	−1.79138	−0.895692	−	0.444676i	$-0.853319\pi$
$431$	−37.1901	−1.79138	−0.895692	+	0.444676i	$0.853319\pi$
$432$	0	0
$433$	−35.5641	−1.70910	−0.854551	−	0.519367i	$-0.826168\pi$
$433$	−35.5641	−1.70910	−0.854551	+	0.519367i	$0.826168\pi$
$434$	5.52369	0.265145
$435$	0	0
$436$	6.34014	0.303638
$437$	−4.27726	−0.204609
$438$	0	0
$439$	31.5694	1.50673	0.753363	−	0.657605i	$-0.228431\pi$
$439$	31.5694	1.50673	0.753363	+	0.657605i	$0.228431\pi$
$440$	−10.1318	−0.483015
$441$	0	0
$442$	−6.11590	−0.290904
$443$	8.60137	0.408664	0.204332	−	0.978902i	$-0.434498\pi$
$443$	8.60137	0.408664	0.204332	+	0.978902i	$0.434498\pi$
$444$	0	0
$445$	2.20167	0.104369
$446$	0.876402	0.0414989
$447$	0	0
$448$	1.96145	0.0926698
$449$	10.8646	0.512734	0.256367	−	0.966580i	$-0.417475\pi$
$449$	10.8646	0.512734	0.256367	+	0.966580i	$0.417475\pi$
$450$	0	0
$451$	−13.4555	−0.633595
$452$	19.4640	0.915508
$453$	0	0
$454$	−4.31698	−0.202606
$455$	2.60772	0.122252
$456$	0	0
$457$	34.9934	1.63692	0.818461	−	0.574562i	$-0.194827\pi$
$457$	34.9934	1.63692	0.818461	+	0.574562i	$0.194827\pi$
$458$	6.53664	0.305437
$459$	0	0
$460$	5.32158	0.248120
$461$	−35.1287	−1.63611	−0.818053	−	0.575143i	$-0.804946\pi$
$461$	−35.1287	−1.63611	−0.818053	+	0.575143i	$0.804946\pi$
$462$	0	0
$463$	35.6458	1.65660	0.828300	−	0.560286i	$-0.189309\pi$
$463$	35.6458	1.65660	0.828300	+	0.560286i	$0.189309\pi$
$464$	−9.82652	−0.456185
$465$	0	0
$466$	14.3281	0.663739
$467$	5.05279	0.233815	0.116908	−	0.993143i	$-0.462702\pi$
$467$	5.05279	0.233815	0.116908	+	0.993143i	$0.462702\pi$
$468$	0	0
$469$	11.2654	0.520188
$470$	−13.9081	−0.641535
$471$	0	0
$472$	6.88636	0.316971
$473$	7.83076	0.360059
$474$	0	0
$475$	−4.18344	−0.191949
$476$	8.22138	0.376826
$477$	0	0
$478$	−8.58128	−0.392499
$479$	23.1220	1.05647	0.528236	−	0.849098i	$-0.322854\pi$
$479$	23.1220	1.05647	0.528236	+	0.849098i	$0.322854\pi$
$480$	0	0
$481$	−10.6629	−0.486188
$482$	−11.2235	−0.511216
$483$	0	0
$484$	8.11098	0.368681
$485$	24.2579	1.10150
$486$	0	0
$487$	3.35799	0.152165	0.0760826	−	0.997102i	$-0.475759\pi$
$487$	3.35799	0.152165	0.0760826	+	0.997102i	$0.475759\pi$
$488$	31.3813	1.42056
$489$	0	0
$490$	1.24165	0.0560919
$491$	−8.66904	−0.391228	−0.195614	−	0.980681i	$-0.562670\pi$
$491$	−8.66904	−0.391228	−0.195614	+	0.980681i	$0.562670\pi$
$492$	0	0
$493$	48.1973	2.17070
$494$	2.23888	0.100732
$495$	0	0
$496$	8.66774	0.389193
$497$	10.4252	0.467636
$498$	0	0
$499$	15.7783	0.706333	0.353167	−	0.935560i	$-0.385105\pi$
$499$	15.7783	0.706333	0.353167	+	0.935560i	$0.385105\pi$
$500$	17.8818	0.799699
$501$	0	0
$502$	19.6384	0.876506
$503$	−30.8916	−1.37739	−0.688694	−	0.725052i	$-0.741816\pi$
$503$	−30.8916	−1.37739	−0.688694	+	0.725052i	$0.741816\pi$
$504$	0	0
$505$	−3.39652	−0.151143
$506$	−3.56282	−0.158387
$507$	0	0
$508$	1.47688	0.0655261
$509$	−24.4520	−1.08381	−0.541907	−	0.840439i	$-0.682297\pi$
$509$	−24.4520	−1.08381	−0.541907	+	0.840439i	$0.682297\pi$
$510$	0	0
$511$	−2.46825	−0.109189
$512$	12.3480	0.545708
$513$	0	0
$514$	17.4586	0.770066
$515$	−1.66887	−0.0735394
$516$	0	0
$517$	−26.2887	−1.15618
$518$	−5.07709	−0.223074
$519$	0	0
$520$	−6.55768	−0.287573
$521$	12.7629	0.559152	0.279576	−	0.960124i	$-0.409806\pi$
$521$	12.7629	0.559152	0.279576	+	0.960124i	$0.409806\pi$
$522$	0	0
$523$	13.9186	0.608618	0.304309	−	0.952573i	$-0.401574\pi$
$523$	13.9186	0.608618	0.304309	+	0.952573i	$0.401574\pi$
$524$	−20.8414	−0.910459
$525$	0	0
$526$	4.20965	0.183550
$527$	−42.5137	−1.85192
$528$	0	0
$529$	−18.5945	−0.808459
$530$	−9.21158	−0.400126
$531$	0	0
$532$	−3.00965	−0.130485
$533$	−8.70890	−0.377224
$534$	0	0
$535$	33.4014	1.44407
$536$	−28.3293	−1.22364
$537$	0	0
$538$	−2.00856	−0.0865953
$539$	2.34692	0.101089
$540$	0	0
$541$	−1.91298	−0.0822454	−0.0411227	−	0.999154i	$-0.513093\pi$
$541$	−1.91298	−0.0822454	−0.0411227	+	0.999154i	$0.513093\pi$
$542$	−4.59788	−0.197496
$543$	0	0
$544$	−32.5670	−1.39630
$545$	−7.36974	−0.315685
$546$	0	0
$547$	16.1959	0.692489	0.346244	−	0.938144i	$-0.387457\pi$
$547$	16.1959	0.692489	0.346244	+	0.938144i	$0.387457\pi$
$548$	16.7671	0.716257
$549$	0	0
$550$	−3.48467	−0.148587
$551$	−17.6439	−0.751654
$552$	0	0
$553$	−2.24902	−0.0956379
$554$	22.8071	0.968982
$555$	0	0
$556$	−9.72458	−0.412414
$557$	0.461853	0.0195693	0.00978467	−	0.999952i	$-0.496885\pi$
$557$	0.461853	0.0195693	0.00978467	+	0.999952i	$0.496885\pi$
$558$	0	0
$559$	5.06836	0.214369
$560$	1.94839	0.0823344
$561$	0	0
$562$	6.89800	0.290975
$563$	7.34035	0.309359	0.154679	−	0.987965i	$-0.450566\pi$
$563$	7.34035	0.309359	0.154679	+	0.987965i	$0.450566\pi$
$564$	0	0
$565$	−22.6248	−0.951831
$566$	−13.8387	−0.581682
$567$	0	0
$568$	−26.2166	−1.10002
$569$	−11.8234	−0.495663	−0.247831	−	0.968803i	$-0.579718\pi$
$569$	−11.8234	−0.495663	−0.247831	+	0.968803i	$0.579718\pi$
$570$	0	0
$571$	−8.98707	−0.376097	−0.188049	−	0.982160i	$-0.560216\pi$
$571$	−8.98707	−0.376097	−0.188049	+	0.982160i	$0.560216\pi$
$572$	−5.26510	−0.220145
$573$	0	0
$574$	−4.14668	−0.173079
$575$	4.30882	0.179690
$576$	0	0
$577$	36.7222	1.52876	0.764382	−	0.644763i	$-0.223044\pi$
$577$	36.7222	1.52876	0.764382	+	0.644763i	$0.223044\pi$
$578$	10.1173	0.420824
$579$	0	0
$580$	21.9517	0.911496
$581$	3.96749	0.164599
$582$	0	0
$583$	−17.4114	−0.721108
$584$	6.20696	0.256846
$585$	0	0
$586$	−0.776785	−0.0320887
$587$	5.00428	0.206549	0.103274	−	0.994653i	$-0.467068\pi$
$587$	5.00428	0.206549	0.103274	+	0.994653i	$0.467068\pi$
$588$	0	0
$589$	15.5632	0.641272
$590$	−3.40016	−0.139982
$591$	0	0
$592$	−7.96693	−0.327439
$593$	43.5640	1.78896	0.894479	−	0.447109i	$-0.147547\pi$
$593$	43.5640	1.78896	0.894479	+	0.447109i	$0.147547\pi$
$594$	0	0
$595$	−9.55648	−0.391777
$596$	−4.46211	−0.182775
$597$	0	0
$598$	−2.30599	−0.0942988
$599$	−28.3930	−1.16011	−0.580053	−	0.814578i	$-0.696968\pi$
$599$	−28.3930	−1.16011	−0.580053	+	0.814578i	$0.696968\pi$
$600$	0	0
$601$	−19.2281	−0.784332	−0.392166	−	0.919895i	$-0.628274\pi$
$601$	−19.2281	−0.784332	−0.392166	+	0.919895i	$0.628274\pi$
$602$	2.41327	0.0983573
$603$	0	0
$604$	16.7077	0.679828
$605$	−9.42814	−0.383308
$606$	0	0
$607$	−6.75558	−0.274200	−0.137100	−	0.990557i	$-0.543778\pi$
$607$	−6.75558	−0.274200	−0.137100	+	0.990557i	$0.543778\pi$
$608$	11.9220	0.483501
$609$	0	0
$610$	−15.4946	−0.627356
$611$	−17.0150	−0.688354
$612$	0	0
$613$	−18.5594	−0.749607	−0.374804	−	0.927104i	$-0.622290\pi$
$613$	−18.5594	−0.749607	−0.374804	+	0.927104i	$0.622290\pi$
$614$	−13.5313	−0.546078
$615$	0	0
$616$	−5.90185	−0.237792
$617$	4.21418	0.169656	0.0848282	−	0.996396i	$-0.472966\pi$
$617$	4.21418	0.169656	0.0848282	+	0.996396i	$0.472966\pi$
$618$	0	0
$619$	−9.71760	−0.390583	−0.195292	−	0.980745i	$-0.562565\pi$
$619$	−9.71760	−0.390583	−0.195292	+	0.980745i	$0.562565\pi$
$620$	−19.3631	−0.777640
$621$	0	0
$622$	−18.3983	−0.737703
$623$	1.28249	0.0513818
$624$	0	0
$625$	−10.5213	−0.420852
$626$	−2.12486	−0.0849264
$627$	0	0
$628$	22.0486	0.879835
$629$	39.0763	1.55808
$630$	0	0
$631$	−25.4146	−1.01174	−0.505869	−	0.862611i	$-0.668828\pi$
$631$	−25.4146	−1.01174	−0.505869	+	0.862611i	$0.668828\pi$
$632$	5.65565	0.224970
$633$	0	0
$634$	−17.6543	−0.701141
$635$	−1.71672	−0.0681259
$636$	0	0
$637$	1.51901	0.0601855
$638$	−14.6968	−0.581851
$639$	0	0
$640$	−17.6512	−0.697727
$641$	39.9118	1.57642	0.788212	−	0.615404i	$-0.211007\pi$
$641$	39.9118	1.57642	0.788212	+	0.615404i	$0.211007\pi$
$642$	0	0
$643$	15.7657	0.621739	0.310870	−	0.950453i	$-0.399380\pi$
$643$	15.7657	0.621739	0.310870	+	0.950453i	$0.399380\pi$
$644$	3.09985	0.122151
$645$	0	0
$646$	−8.20481	−0.322814
$647$	1.14451	0.0449952	0.0224976	−	0.999747i	$-0.492838\pi$
$647$	1.14451	0.0449952	0.0224976	+	0.999747i	$0.492838\pi$
$648$	0	0
$649$	−6.42686	−0.252276
$650$	−2.25541	−0.0884643
$651$	0	0
$652$	23.7707	0.930931
$653$	6.64269	0.259948	0.129974	−	0.991517i	$-0.458511\pi$
$653$	6.64269	0.259948	0.129974	+	0.991517i	$0.458511\pi$
$654$	0	0
$655$	24.2259	0.946582
$656$	−6.50695	−0.254054
$657$	0	0
$658$	−8.10159	−0.315833
$659$	−10.2671	−0.399949	−0.199974	−	0.979801i	$-0.564086\pi$
$659$	−10.2671	−0.399949	−0.199974	+	0.979801i	$0.564086\pi$
$660$	0	0
$661$	−17.1707	−0.667861	−0.333930	−	0.942598i	$-0.608375\pi$
$661$	−17.1707	−0.667861	−0.333930	+	0.942598i	$0.608375\pi$
$662$	15.1282	0.587974
$663$	0	0
$664$	−9.97713	−0.387188
$665$	3.49840	0.135662
$666$	0	0
$667$	18.1727	0.703649
$668$	31.0132	1.19994
$669$	0	0
$670$	13.9877	0.540391
$671$	−29.2873	−1.13062
$672$	0	0
$673$	42.7321	1.64720	0.823602	−	0.567169i	$-0.191961\pi$
$673$	42.7321	1.64720	0.823602	+	0.567169i	$0.191961\pi$
$674$	20.8775	0.804170
$675$	0	0
$676$	15.7917	0.607373
$677$	7.68814	0.295479	0.147740	−	0.989026i	$-0.452800\pi$
$677$	7.68814	0.295479	0.147740	+	0.989026i	$0.452800\pi$
$678$	0	0
$679$	14.1304	0.542275
$680$	24.0319	0.921580
$681$	0	0
$682$	12.9637	0.496404
$683$	5.92405	0.226677	0.113339	−	0.993556i	$-0.463845\pi$
$683$	5.92405	0.226677	0.113339	+	0.993556i	$0.463845\pi$
$684$	0	0
$685$	−19.4900	−0.744675
$686$	0.723268	0.0276145
$687$	0	0
$688$	3.78688	0.144373
$689$	−11.2693	−0.429327
$690$	0	0
$691$	22.5056	0.856152	0.428076	−	0.903743i	$-0.359192\pi$
$691$	22.5056	0.856152	0.428076	+	0.903743i	$0.359192\pi$
$692$	23.6269	0.898160
$693$	0	0
$694$	7.73266	0.293528
$695$	11.3038	0.428777
$696$	0	0
$697$	31.9154	1.20888
$698$	−20.8941	−0.790852
$699$	0	0
$700$	3.03186	0.114594
$701$	20.9671	0.791918	0.395959	−	0.918268i	$-0.370412\pi$
$701$	20.9671	0.791918	0.395959	+	0.918268i	$0.370412\pi$
$702$	0	0
$703$	−14.3049	−0.539520
$704$	4.60337	0.173496
$705$	0	0
$706$	5.12753	0.192977
$707$	−1.97849	−0.0744089
$708$	0	0
$709$	−26.1974	−0.983864	−0.491932	−	0.870634i	$-0.663709\pi$
$709$	−26.1974	−0.983864	−0.491932	+	0.870634i	$0.663709\pi$
$710$	12.9445	0.485798
$711$	0	0
$712$	−3.22510	−0.120866
$713$	−16.0297	−0.600316
$714$	0	0
$715$	6.12011	0.228879
$716$	18.3081	0.684207
$717$	0	0
$718$	−12.2048	−0.455480
$719$	−23.0941	−0.861264	−0.430632	−	0.902528i	$-0.641709\pi$
$719$	−23.0941	−0.861264	−0.430632	+	0.902528i	$0.641709\pi$
$720$	0	0
$721$	−0.972130	−0.0362040
$722$	−10.7385	−0.399646
$723$	0	0
$724$	8.74050	0.324838
$725$	17.7741	0.660113
$726$	0	0
$727$	8.37021	0.310434	0.155217	−	0.987880i	$-0.450392\pi$
$727$	8.37021	0.310434	0.155217	+	0.987880i	$0.450392\pi$
$728$	−3.81989	−0.141575
$729$	0	0
$730$	−3.06470	−0.113430
$731$	−18.5740	−0.686982
$732$	0	0
$733$	−15.1625	−0.560038	−0.280019	−	0.959994i	$-0.590341\pi$
$733$	−15.1625	−0.560038	−0.280019	+	0.959994i	$0.590341\pi$
$734$	−4.12917	−0.152410
$735$	0	0
$736$	−12.2793	−0.452622
$737$	26.4390	0.973894
$738$	0	0
$739$	−35.2148	−1.29540	−0.647699	−	0.761897i	$-0.724268\pi$
$739$	−35.2148	−1.29540	−0.647699	+	0.761897i	$0.724268\pi$
$740$	17.7975	0.654251
$741$	0	0
$742$	−5.36581	−0.196985
$743$	−12.3573	−0.453347	−0.226674	−	0.973971i	$-0.572785\pi$
$743$	−12.3573	−0.453347	−0.226674	+	0.973971i	$0.572785\pi$
$744$	0	0
$745$	5.18673	0.190027
$746$	−4.05478	−0.148456
$747$	0	0
$748$	19.2949	0.705493
$749$	19.4565	0.710927
$750$	0	0
$751$	−4.07149	−0.148571	−0.0742853	−	0.997237i	$-0.523668\pi$
$751$	−4.07149	−0.148571	−0.0742853	+	0.997237i	$0.523668\pi$
$752$	−12.7130	−0.463594
$753$	0	0
$754$	−9.51229	−0.346417
$755$	−19.4209	−0.706800
$756$	0	0
$757$	42.2357	1.53508	0.767541	−	0.641000i	$-0.221480\pi$
$757$	42.2357	1.53508	0.767541	+	0.641000i	$0.221480\pi$
$758$	−17.8756	−0.649272
$759$	0	0
$760$	−8.79749	−0.319118
$761$	11.9386	0.432775	0.216387	−	0.976308i	$-0.430573\pi$
$761$	11.9386	0.432775	0.216387	+	0.976308i	$0.430573\pi$
$762$	0	0
$763$	−4.29292	−0.155414
$764$	−19.3159	−0.698823
$765$	0	0
$766$	−17.9764	−0.649514
$767$	−4.15970	−0.150198
$768$	0	0
$769$	−13.7722	−0.496637	−0.248319	−	0.968678i	$-0.579878\pi$
$769$	−13.7722	−0.496637	−0.248319	+	0.968678i	$0.579878\pi$
$770$	2.91405	0.105015
$771$	0	0
$772$	3.21124	0.115575
$773$	23.8983	0.859564	0.429782	−	0.902933i	$-0.358590\pi$
$773$	23.8983	0.859564	0.429782	+	0.902933i	$0.358590\pi$
$774$	0	0
$775$	−15.6781	−0.563173
$776$	−35.5340	−1.27560
$777$	0	0
$778$	−22.7459	−0.815480
$779$	−11.6835	−0.418603
$780$	0	0
$781$	24.4672	0.875506
$782$	8.45072	0.302197
$783$	0	0
$784$	1.13495	0.0405339
$785$	−25.6291	−0.914743
$786$	0	0
$787$	30.1498	1.07472	0.537362	−	0.843351i	$-0.319421\pi$
$787$	30.1498	1.07472	0.537362	+	0.843351i	$0.319421\pi$
$788$	−14.5319	−0.517677
$789$	0	0
$790$	−2.79249	−0.0993522
$791$	−13.1791	−0.468594
$792$	0	0
$793$	−18.9558	−0.673141
$794$	−17.5864	−0.624118
$795$	0	0
$796$	−1.18754	−0.0420913
$797$	21.5779	0.764329	0.382165	−	0.924094i	$-0.375179\pi$
$797$	21.5779	0.764329	0.382165	+	0.924094i	$0.375179\pi$
$798$	0	0
$799$	62.3548	2.20595
$800$	−12.0100	−0.424617
$801$	0	0
$802$	−10.4496	−0.368990
$803$	−5.79279	−0.204423
$804$	0	0
$805$	−3.60325	−0.126998
$806$	8.39056	0.295545
$807$	0	0
$808$	4.97536	0.175032
$809$	−54.5046	−1.91628	−0.958140	−	0.286301i	$-0.907574\pi$
$809$	−54.5046	−1.91628	−0.958140	+	0.286301i	$0.907574\pi$
$810$	0	0
$811$	41.1010	1.44325	0.721626	−	0.692284i	$-0.243395\pi$
$811$	41.1010	1.44325	0.721626	+	0.692284i	$0.243395\pi$
$812$	12.7870	0.448737
$813$	0	0
$814$	−11.9155	−0.417639
$815$	−27.6308	−0.967866
$816$	0	0
$817$	6.79948	0.237884
$818$	−10.6697	−0.373058
$819$	0	0
$820$	14.5360	0.507621
$821$	30.7459	1.07304	0.536519	−	0.843888i	$-0.319739\pi$
$821$	30.7459	1.07304	0.536519	+	0.843888i	$0.319739\pi$
$822$	0	0
$823$	−30.3481	−1.05787	−0.528933	−	0.848663i	$-0.677408\pi$
$823$	−30.3481	−1.05787	−0.528933	+	0.848663i	$0.677408\pi$
$824$	2.44463	0.0851629
$825$	0	0
$826$	−1.98061	−0.0689144
$827$	15.0806	0.524405	0.262202	−	0.965013i	$-0.415551\pi$
$827$	15.0806	0.524405	0.262202	+	0.965013i	$0.415551\pi$
$828$	0	0
$829$	12.4907	0.433820	0.216910	−	0.976192i	$-0.430402\pi$
$829$	12.4907	0.433820	0.216910	+	0.976192i	$0.430402\pi$
$830$	4.92623	0.170992
$831$	0	0
$832$	2.97947	0.103294
$833$	−5.56671	−0.192875
$834$	0	0
$835$	−36.0495	−1.24754
$836$	−7.06341	−0.244293
$837$	0	0
$838$	22.8985	0.791014
$839$	15.6761	0.541198	0.270599	−	0.962692i	$-0.412778\pi$
$839$	15.6761	0.541198	0.270599	+	0.962692i	$0.412778\pi$
$840$	0	0
$841$	45.9631	1.58494
$842$	−3.37205	−0.116208
$843$	0	0
$844$	10.8677	0.374081
$845$	−18.3562	−0.631472
$846$	0	0
$847$	−5.49196	−0.188706
$848$	−8.41999	−0.289144
$849$	0	0
$850$	8.26536	0.283499
$851$	14.7337	0.505063
$852$	0	0
$853$	22.3467	0.765137	0.382569	−	0.923927i	$-0.375040\pi$
$853$	22.3467	0.765137	0.382569	+	0.923927i	$0.375040\pi$
$854$	−9.02569	−0.308853
$855$	0	0
$856$	−48.9278	−1.67232
$857$	13.2804	0.453649	0.226825	−	0.973936i	$-0.427166\pi$
$857$	13.2804	0.453649	0.226825	+	0.973936i	$0.427166\pi$
$858$	0	0
$859$	−52.0838	−1.77707	−0.888537	−	0.458804i	$-0.848278\pi$
$859$	−52.0838	−1.77707	−0.888537	+	0.458804i	$0.848278\pi$
$860$	−8.45961	−0.288470
$861$	0	0
$862$	−26.8984	−0.916163
$863$	15.1414	0.515418	0.257709	−	0.966223i	$-0.417032\pi$
$863$	15.1414	0.515418	0.257709	+	0.966223i	$0.417032\pi$
$864$	0	0
$865$	−27.4637	−0.933795
$866$	−25.7224	−0.874083
$867$	0	0
$868$	−11.2791	−0.382839
$869$	−5.27827	−0.179053
$870$	0	0
$871$	17.1123	0.579828
$872$	10.7955	0.365582
$873$	0	0
$874$	−3.09360	−0.104643
$875$	−12.1078	−0.409319
$876$	0	0
$877$	20.9214	0.706465	0.353233	−	0.935535i	$-0.385082\pi$
$877$	20.9214	0.706465	0.353233	+	0.935535i	$0.385082\pi$
$878$	22.8331	0.770581
$879$	0	0
$880$	4.57271	0.154146
$881$	42.7893	1.44161	0.720804	−	0.693139i	$-0.243773\pi$
$881$	42.7893	1.44161	0.720804	+	0.693139i	$0.243773\pi$
$882$	0	0
$883$	−12.2004	−0.410578	−0.205289	−	0.978701i	$-0.565813\pi$
$883$	−12.2004	−0.410578	−0.205289	+	0.978701i	$0.565813\pi$
$884$	12.4884	0.420030
$885$	0	0
$886$	6.22110	0.209002
$887$	28.3793	0.952885	0.476442	−	0.879206i	$-0.341926\pi$
$887$	28.3793	0.952885	0.476442	+	0.879206i	$0.341926\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	1.59240	0.0533773
$891$	0	0
$892$	−1.78958	−0.0599194
$893$	−22.8266	−0.763862
$894$	0	0
$895$	−21.2813	−0.711354
$896$	−10.2820	−0.343496
$897$	0	0
$898$	7.85804	0.262226
$899$	−66.1231	−2.20533
$900$	0	0
$901$	41.2985	1.37585
$902$	−9.73194	−0.324038
$903$	0	0
$904$	33.1417	1.10228
$905$	−10.1599	−0.337726
$906$	0	0
$907$	−32.5936	−1.08225	−0.541126	−	0.840942i	$-0.682002\pi$
$907$	−32.5936	−1.08225	−0.541126	+	0.840942i	$0.682002\pi$
$908$	8.81509	0.292539
$909$	0	0
$910$	1.88608	0.0625230
$911$	45.7811	1.51680	0.758398	−	0.651792i	$-0.225983\pi$
$911$	45.7811	1.51680	0.758398	+	0.651792i	$0.225983\pi$
$912$	0	0
$913$	9.31139	0.308162
$914$	25.3096	0.837168
$915$	0	0
$916$	−13.3475	−0.441015
$917$	14.1117	0.466010
$918$	0	0
$919$	−4.25858	−0.140477	−0.0702387	−	0.997530i	$-0.522376\pi$
$919$	−4.25858	−0.140477	−0.0702387	+	0.997530i	$0.522376\pi$
$920$	9.06116	0.298738
$921$	0	0
$922$	−25.4075	−0.836750
$923$	15.8361	0.521251
$924$	0	0
$925$	14.4105	0.473814
$926$	25.7815	0.847231
$927$	0	0
$928$	−50.6527	−1.66276
$929$	−44.4088	−1.45700	−0.728502	−	0.685043i	$-0.759783\pi$
$929$	−44.4088	−1.45700	−0.728502	+	0.685043i	$0.759783\pi$
$930$	0	0
$931$	2.03784	0.0667875
$932$	−29.2575	−0.958360
$933$	0	0
$934$	3.65452	0.119580
$935$	−22.4283	−0.733484
$936$	0	0
$937$	−49.1929	−1.60706	−0.803531	−	0.595263i	$-0.797048\pi$
$937$	−49.1929	−1.60706	−0.803531	+	0.595263i	$0.797048\pi$
$938$	8.14791	0.266039
$939$	0	0
$940$	28.3998	0.926301
$941$	33.3768	1.08805	0.544027	−	0.839068i	$-0.316899\pi$
$941$	33.3768	1.08805	0.544027	+	0.839068i	$0.316899\pi$
$942$	0	0
$943$	12.0336	0.391869
$944$	−3.10797	−0.101156
$945$	0	0
$946$	5.66375	0.184144
$947$	−30.3134	−0.985055	−0.492527	−	0.870297i	$-0.663927\pi$
$947$	−30.3134	−0.985055	−0.492527	+	0.870297i	$0.663927\pi$
$948$	0	0
$949$	−3.74931	−0.121708
$950$	−3.02575	−0.0981682
$951$	0	0
$952$	13.9987	0.453701
$953$	−45.5733	−1.47626	−0.738131	−	0.674657i	$-0.764292\pi$
$953$	−45.5733	−1.47626	−0.738131	+	0.674657i	$0.764292\pi$
$954$	0	0
$955$	22.4526	0.726550
$956$	17.5226	0.566722
$957$	0	0
$958$	16.7234	0.540309
$959$	−11.3531	−0.366609
$960$	0	0
$961$	27.3256	0.881471
$962$	−7.71217	−0.248650
$963$	0	0
$964$	22.9179	0.738135
$965$	−3.73272	−0.120161
$966$	0	0
$967$	16.3461	0.525656	0.262828	−	0.964843i	$-0.415345\pi$
$967$	16.3461	0.525656	0.262828	+	0.964843i	$0.415345\pi$
$968$	13.8107	0.443894
$969$	0	0
$970$	17.5450	0.563336
$971$	27.4250	0.880111	0.440056	−	0.897970i	$-0.354959\pi$
$971$	27.4250	0.880111	0.440056	+	0.897970i	$0.354959\pi$
$972$	0	0
$973$	6.58453	0.211090
$974$	2.42873	0.0778216
$975$	0	0
$976$	−14.1631	−0.453348
$977$	54.9531	1.75811	0.879053	−	0.476725i	$-0.158176\pi$
$977$	54.9531	1.75811	0.879053	+	0.476725i	$0.158176\pi$
$978$	0	0
$979$	3.00990	0.0961968
$980$	−2.53539	−0.0809901
$981$	0	0
$982$	−6.27004	−0.200085
$983$	17.9656	0.573012	0.286506	−	0.958078i	$-0.407506\pi$
$983$	17.9656	0.573012	0.286506	+	0.958078i	$0.407506\pi$
$984$	0	0
$985$	16.8918	0.538217
$986$	34.8596	1.11016
$987$	0	0
$988$	−4.57170	−0.145445
$989$	−7.00327	−0.222691
$990$	0	0
$991$	15.8755	0.504303	0.252152	−	0.967688i	$-0.418862\pi$
$991$	15.8755	0.504303	0.252152	+	0.967688i	$0.418862\pi$
$992$	44.6795	1.41858
$993$	0	0
$994$	7.54025	0.239162
$995$	1.38039	0.0437613
$996$	0	0
$997$	−27.8927	−0.883372	−0.441686	−	0.897170i	$-0.645619\pi$
$997$	−27.8927	−0.883372	−0.441686	+	0.897170i	$0.645619\pi$
$998$	11.4119	0.361239
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.14		20
3.2	odd	2		889.2.a.d.1.7	✓	20
21.20	even	2		6223.2.a.l.1.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.7	✓	20	3.2	odd	2
6223.2.a.l.1.7		20	21.20	even	2
8001.2.a.w.1.14		20	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)

\(f(q)\)	\(=\)	\(q+0.723268 q^{2} -1.47688 q^{4} +1.71672 q^{5} +1.00000 q^{7} -2.51472 q^{8} +O(q^{10})\)	\(q+0.723268 q^{2} -1.47688 q^{4} +1.71672 q^{5} +1.00000 q^{7} -2.51472 q^{8} +1.24165 q^{10} +2.34692 q^{11} +1.51901 q^{13} +0.723268 q^{14} +1.13495 q^{16} -5.56671 q^{17} +2.03784 q^{19} -2.53539 q^{20} +1.69745 q^{22} -2.09892 q^{23} -2.05288 q^{25} +1.09866 q^{26} -1.47688 q^{28} -8.65813 q^{29} +7.63712 q^{31} +5.85031 q^{32} -4.02623 q^{34} +1.71672 q^{35} -7.01965 q^{37} +1.47391 q^{38} -4.31706 q^{40} -5.73325 q^{41} +3.33661 q^{43} -3.46613 q^{44} -1.51808 q^{46} -11.2014 q^{47} +1.00000 q^{49} -1.48478 q^{50} -2.24341 q^{52} -7.41884 q^{53} +4.02900 q^{55} -2.51472 q^{56} -6.26215 q^{58} -2.73842 q^{59} -12.4790 q^{61} +5.52369 q^{62} +1.96145 q^{64} +2.60772 q^{65} +11.2654 q^{67} +8.22138 q^{68} +1.24165 q^{70} +10.4252 q^{71} -2.46825 q^{73} -5.07709 q^{74} -3.00965 q^{76} +2.34692 q^{77} -2.24902 q^{79} +1.94839 q^{80} -4.14668 q^{82} +3.96749 q^{83} -9.55648 q^{85} +2.41327 q^{86} -5.90185 q^{88} +1.28249 q^{89} +1.51901 q^{91} +3.09985 q^{92} -8.10159 q^{94} +3.49840 q^{95} +14.1304 q^{97} +0.723268 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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