LMFDB - Embedded newform 8001.2.a.s.1.6

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 $ x^16 - 4x^15 - 18x^14 + 83x^13 + 112x^12 - 668x^11 - 235x^10 + 2648x^9 - 298x^8 - 5422x^7 + 2075x^6 + 5385x^5 - 3163x^4 - 1882x^3 + 1614x^2 - 200*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.17466$ 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)

$f(q)$	$=$	$q-1.17466 q^{2} -0.620175 q^{4} -3.75278 q^{5} -1.00000 q^{7} +3.07781 q^{8} +O(q^{10})$	\(q-1.17466 q^{2} -0.620175 q^{4} -3.75278 q^{5} -1.00000 q^{7} +3.07781 q^{8} +4.40823 q^{10} +0.236817 q^{11} +2.34303 q^{13} +1.17466 q^{14} -2.37503 q^{16} +5.49213 q^{17} +1.29877 q^{19} +2.32738 q^{20} -0.278179 q^{22} -4.33999 q^{23} +9.08332 q^{25} -2.75226 q^{26} +0.620175 q^{28} -3.35099 q^{29} +10.2207 q^{31} -3.36577 q^{32} -6.45138 q^{34} +3.75278 q^{35} +5.67027 q^{37} -1.52562 q^{38} -11.5503 q^{40} -5.91718 q^{41} +7.58953 q^{43} -0.146868 q^{44} +5.09801 q^{46} -6.69813 q^{47} +1.00000 q^{49} -10.6698 q^{50} -1.45309 q^{52} +11.3704 q^{53} -0.888720 q^{55} -3.07781 q^{56} +3.93627 q^{58} +1.67458 q^{59} +6.14626 q^{61} -12.0058 q^{62} +8.70370 q^{64} -8.79286 q^{65} -12.3793 q^{67} -3.40608 q^{68} -4.40823 q^{70} +16.3585 q^{71} -2.42897 q^{73} -6.66064 q^{74} -0.805466 q^{76} -0.236817 q^{77} -12.9909 q^{79} +8.91297 q^{80} +6.95068 q^{82} +1.57938 q^{83} -20.6107 q^{85} -8.91512 q^{86} +0.728878 q^{88} -3.02589 q^{89} -2.34303 q^{91} +2.69155 q^{92} +7.86802 q^{94} -4.87400 q^{95} -1.63077 q^{97} -1.17466 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * 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 $

$2$	−1.17466	−0.830610	−0.415305	−	0.909682i	$-0.636325\pi$
$2$	−1.17466	−0.830610	−0.415305	+	0.909682i	$0.636325\pi$
$3$	0	0
$3$	0	0
$4$	−0.620175	−0.310087
$5$	−3.75278	−1.67829	−0.839146	−	0.543906i	$-0.816945\pi$
$5$	−3.75278	−1.67829	−0.839146	+	0.543906i	$0.816945\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.07781	1.08817
$9$	0	0
$10$	4.40823	1.39401
$11$	0.236817	0.0714029	0.0357015	−	0.999362i	$-0.488633\pi$
$11$	0.236817	0.0714029	0.0357015	+	0.999362i	$0.488633\pi$
$12$	0	0
$13$	2.34303	0.649839	0.324920	−	0.945742i	$-0.394663\pi$
$13$	2.34303	0.649839	0.324920	+	0.945742i	$0.394663\pi$
$14$	1.17466	0.313941
$15$	0	0
$16$	−2.37503	−0.593759
$17$	5.49213	1.33204	0.666018	−	0.745935i	$-0.267997\pi$
$17$	5.49213	1.33204	0.666018	+	0.745935i	$0.267997\pi$
$18$	0	0
$19$	1.29877	0.297959	0.148979	−	0.988840i	$-0.452401\pi$
$19$	1.29877	0.297959	0.148979	+	0.988840i	$0.452401\pi$
$20$	2.32738	0.520417
$21$	0	0
$22$	−0.278179	−0.0593080
$23$	−4.33999	−0.904950	−0.452475	−	0.891777i	$-0.649459\pi$
$23$	−4.33999	−0.904950	−0.452475	+	0.891777i	$0.649459\pi$
$24$	0	0
$25$	9.08332	1.81666
$26$	−2.75226	−0.539763
$27$	0	0
$28$	0.620175	0.117202
$29$	−3.35099	−0.622263	−0.311131	−	0.950367i	$-0.600708\pi$
$29$	−3.35099	−0.622263	−0.311131	+	0.950367i	$0.600708\pi$
$30$	0	0
$31$	10.2207	1.83569	0.917846	−	0.396937i	$-0.129927\pi$
$31$	10.2207	1.83569	0.917846	+	0.396937i	$0.129927\pi$
$32$	−3.36577	−0.594990
$33$	0	0
$34$	−6.45138	−1.10640
$35$	3.75278	0.634335
$36$	0	0
$37$	5.67027	0.932186	0.466093	−	0.884736i	$-0.345661\pi$
$37$	5.67027	0.932186	0.466093	+	0.884736i	$0.345661\pi$
$38$	−1.52562	−0.247488
$39$	0	0
$40$	−11.5503	−1.82627
$41$	−5.91718	−0.924109	−0.462054	−	0.886852i	$-0.652888\pi$
$41$	−5.91718	−0.924109	−0.462054	+	0.886852i	$0.652888\pi$
$42$	0	0
$43$	7.58953	1.15739	0.578697	−	0.815543i	$-0.303562\pi$
$43$	7.58953	1.15739	0.578697	+	0.815543i	$0.303562\pi$
$44$	−0.146868	−0.0221411
$45$	0	0
$46$	5.09801	0.751660
$47$	−6.69813	−0.977022	−0.488511	−	0.872558i	$-0.662460\pi$
$47$	−6.69813	−0.977022	−0.488511	+	0.872558i	$0.662460\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−10.6698	−1.50894
$51$	0	0
$52$	−1.45309	−0.201507
$53$	11.3704	1.56185	0.780926	−	0.624624i	$-0.214748\pi$
$53$	11.3704	1.56185	0.780926	+	0.624624i	$0.214748\pi$
$54$	0	0
$55$	−0.888720	−0.119835
$56$	−3.07781	−0.411290
$57$	0	0
$58$	3.93627	0.516858
$59$	1.67458	0.218011	0.109006	−	0.994041i	$-0.465233\pi$
$59$	1.67458	0.218011	0.109006	+	0.994041i	$0.465233\pi$
$60$	0	0
$61$	6.14626	0.786948	0.393474	−	0.919336i	$-0.371273\pi$
$61$	6.14626	0.786948	0.393474	+	0.919336i	$0.371273\pi$
$62$	−12.0058	−1.52474
$63$	0	0
$64$	8.70370	1.08796
$65$	−8.79286	−1.09062
$66$	0	0
$67$	−12.3793	−1.51237	−0.756186	−	0.654357i	$-0.772940\pi$
$67$	−12.3793	−1.51237	−0.756186	+	0.654357i	$0.772940\pi$
$68$	−3.40608	−0.413048
$69$	0	0
$70$	−4.40823	−0.526885
$71$	16.3585	1.94140	0.970701	−	0.240291i	$-0.0772428\pi$
$71$	16.3585	1.94140	0.970701	+	0.240291i	$0.0772428\pi$
$72$	0	0
$73$	−2.42897	−0.284289	−0.142144	−	0.989846i	$-0.545400\pi$
$73$	−2.42897	−0.284289	−0.142144	+	0.989846i	$0.545400\pi$
$74$	−6.66064	−0.774283
$75$	0	0
$76$	−0.805466	−0.0923933
$77$	−0.236817	−0.0269878
$78$	0	0
$79$	−12.9909	−1.46159	−0.730796	−	0.682596i	$-0.760851\pi$
$79$	−12.9909	−1.46159	−0.730796	+	0.682596i	$0.760851\pi$
$80$	8.91297	0.996500
$81$	0	0
$82$	6.95068	0.767574
$83$	1.57938	0.173359	0.0866795	−	0.996236i	$-0.472374\pi$
$83$	1.57938	0.173359	0.0866795	+	0.996236i	$0.472374\pi$
$84$	0	0
$85$	−20.6107	−2.23555
$86$	−8.91512	−0.961342
$87$	0	0
$88$	0.728878	0.0776986
$89$	−3.02589	−0.320743	−0.160372	−	0.987057i	$-0.551269\pi$
$89$	−3.02589	−0.320743	−0.160372	+	0.987057i	$0.551269\pi$
$90$	0	0
$91$	−2.34303	−0.245616
$92$	2.69155	0.280614
$93$	0	0
$94$	7.86802	0.811524
$95$	−4.87400	−0.500062
$96$	0	0
$97$	−1.63077	−0.165580	−0.0827899	−	0.996567i	$-0.526383\pi$
$97$	−1.63077	−0.165580	−0.0827899	+	0.996567i	$0.526383\pi$
$98$	−1.17466	−0.118659
$99$	0	0
$100$	−5.63325	−0.563325
$101$	−1.75784	−0.174911	−0.0874557	−	0.996168i	$-0.527874\pi$
$101$	−1.75784	−0.174911	−0.0874557	+	0.996168i	$0.527874\pi$
$102$	0	0
$103$	2.67285	0.263364	0.131682	−	0.991292i	$-0.457962\pi$
$103$	2.67285	0.263364	0.131682	+	0.991292i	$0.457962\pi$
$104$	7.21141	0.707136
$105$	0	0
$106$	−13.3564	−1.29729
$107$	−4.65963	−0.450464	−0.225232	−	0.974305i	$-0.572314\pi$
$107$	−4.65963	−0.450464	−0.225232	+	0.974305i	$0.572314\pi$
$108$	0	0
$109$	−7.23814	−0.693288	−0.346644	−	0.937997i	$-0.612679\pi$
$109$	−7.23814	−0.693288	−0.346644	+	0.937997i	$0.612679\pi$
$110$	1.04394	0.0995361
$111$	0	0
$112$	2.37503	0.224420
$113$	10.6875	1.00540	0.502700	−	0.864461i	$-0.332340\pi$
$113$	10.6875	1.00540	0.502700	+	0.864461i	$0.332340\pi$
$114$	0	0
$115$	16.2870	1.51877
$116$	2.07820	0.192956
$117$	0	0
$118$	−1.96706	−0.181082
$119$	−5.49213	−0.503462
$120$	0	0
$121$	−10.9439	−0.994902
$122$	−7.21977	−0.653647
$123$	0	0
$124$	−6.33862	−0.569225
$125$	−15.3238	−1.37060
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−3.49235	−0.308683
$129$	0	0
$130$	10.3286	0.905880
$131$	−11.3710	−0.993492	−0.496746	−	0.867896i	$-0.665472\pi$
$131$	−11.3710	−0.993492	−0.496746	+	0.867896i	$0.665472\pi$
$132$	0	0
$133$	−1.29877	−0.112618
$134$	14.5415	1.25619
$135$	0	0
$136$	16.9037	1.44948
$137$	−12.7425	−1.08867	−0.544334	−	0.838869i	$-0.683217\pi$
$137$	−12.7425	−1.08867	−0.544334	+	0.838869i	$0.683217\pi$
$138$	0	0
$139$	−13.5449	−1.14886	−0.574432	−	0.818552i	$-0.694777\pi$
$139$	−13.5449	−1.14886	−0.574432	+	0.818552i	$0.694777\pi$
$140$	−2.32738	−0.196699
$141$	0	0
$142$	−19.2157	−1.61255
$143$	0.554868	0.0464004
$144$	0	0
$145$	12.5755	1.04434
$146$	2.85321	0.236133
$147$	0	0
$148$	−3.51656	−0.289059
$149$	−6.88699	−0.564204	−0.282102	−	0.959384i	$-0.591032\pi$
$149$	−6.88699	−0.564204	−0.282102	+	0.959384i	$0.591032\pi$
$150$	0	0
$151$	11.9748	0.974498	0.487249	−	0.873263i	$-0.338000\pi$
$151$	11.9748	0.974498	0.487249	+	0.873263i	$0.338000\pi$
$152$	3.99738	0.324230
$153$	0	0
$154$	0.278179	0.0224163
$155$	−38.3560	−3.08083
$156$	0	0
$157$	4.95178	0.395195	0.197598	−	0.980283i	$-0.436686\pi$
$157$	4.95178	0.395195	0.197598	+	0.980283i	$0.436686\pi$
$158$	15.2599	1.21401
$159$	0	0
$160$	12.6310	0.998567
$161$	4.33999	0.342039
$162$	0	0
$163$	−6.88957	−0.539633	−0.269817	−	0.962912i	$-0.586963\pi$
$163$	−6.88957	−0.539633	−0.269817	+	0.962912i	$0.586963\pi$
$164$	3.66969	0.286554
$165$	0	0
$166$	−1.85523	−0.143994
$167$	13.3312	1.03160	0.515799	−	0.856710i	$-0.327495\pi$
$167$	13.3312	1.03160	0.515799	+	0.856710i	$0.327495\pi$
$168$	0	0
$169$	−7.51022	−0.577709
$170$	24.2106	1.85687
$171$	0	0
$172$	−4.70684	−0.358893
$173$	−14.0887	−1.07114	−0.535572	−	0.844490i	$-0.679904\pi$
$173$	−14.0887	−1.07114	−0.535572	+	0.844490i	$0.679904\pi$
$174$	0	0
$175$	−9.08332	−0.686635
$176$	−0.562448	−0.0423961
$177$	0	0
$178$	3.55439	0.266412
$179$	5.24047	0.391691	0.195845	−	0.980635i	$-0.437255\pi$
$179$	5.24047	0.391691	0.195845	+	0.980635i	$0.437255\pi$
$180$	0	0
$181$	−15.7669	−1.17195	−0.585974	−	0.810330i	$-0.699288\pi$
$181$	−15.7669	−1.17195	−0.585974	+	0.810330i	$0.699288\pi$
$182$	2.75226	0.204011
$183$	0	0
$184$	−13.3577	−0.984741
$185$	−21.2792	−1.56448
$186$	0	0
$187$	1.30063	0.0951113
$188$	4.15401	0.302962
$189$	0	0
$190$	5.72529	0.415357
$191$	22.1699	1.60416	0.802078	−	0.597220i	$-0.203728\pi$
$191$	22.1699	1.60416	0.802078	+	0.597220i	$0.203728\pi$
$192$	0	0
$193$	8.60901	0.619690	0.309845	−	0.950787i	$-0.399723\pi$
$193$	8.60901	0.619690	0.309845	+	0.950787i	$0.399723\pi$
$194$	1.91560	0.137532
$195$	0	0
$196$	−0.620175	−0.0442982
$197$	7.69666	0.548364	0.274182	−	0.961678i	$-0.411593\pi$
$197$	7.69666	0.548364	0.274182	+	0.961678i	$0.411593\pi$
$198$	0	0
$199$	−26.9244	−1.90862	−0.954311	−	0.298814i	$-0.903409\pi$
$199$	−26.9244	−1.90862	−0.954311	+	0.298814i	$0.903409\pi$
$200$	27.9568	1.97684
$201$	0	0
$202$	2.06486	0.145283
$203$	3.35099	0.235193
$204$	0	0
$205$	22.2059	1.55092
$206$	−3.13969	−0.218753
$207$	0	0
$208$	−5.56477	−0.385848
$209$	0.307571	0.0212751
$210$	0	0
$211$	20.2896	1.39680	0.698398	−	0.715710i	$-0.253897\pi$
$211$	20.2896	1.39680	0.698398	+	0.715710i	$0.253897\pi$
$212$	−7.05166	−0.484310
$213$	0	0
$214$	5.47348	0.374160
$215$	−28.4818	−1.94244
$216$	0	0
$217$	−10.2207	−0.693826
$218$	8.50235	0.575852
$219$	0	0
$220$	0.551162	0.0371593
$221$	12.8682	0.865610
$222$	0	0
$223$	−27.3229	−1.82968	−0.914839	−	0.403818i	$-0.867683\pi$
$223$	−27.3229	−1.82968	−0.914839	+	0.403818i	$0.867683\pi$
$224$	3.36577	0.224885
$225$	0	0
$226$	−12.5542	−0.835095
$227$	20.0510	1.33083	0.665416	−	0.746473i	$-0.268254\pi$
$227$	20.0510	1.33083	0.665416	+	0.746473i	$0.268254\pi$
$228$	0	0
$229$	1.22569	0.0809961	0.0404980	−	0.999180i	$-0.487106\pi$
$229$	1.22569	0.0809961	0.0404980	+	0.999180i	$0.487106\pi$
$230$	−19.1317	−1.26151
$231$	0	0
$232$	−10.3137	−0.677129
$233$	1.48770	0.0974623	0.0487311	−	0.998812i	$-0.484482\pi$
$233$	1.48770	0.0974623	0.0487311	+	0.998812i	$0.484482\pi$
$234$	0	0
$235$	25.1366	1.63973
$236$	−1.03853	−0.0676025
$237$	0	0
$238$	6.45138	0.418181
$239$	19.2034	1.24217	0.621084	−	0.783744i	$-0.286693\pi$
$239$	19.2034	1.24217	0.621084	+	0.783744i	$0.286693\pi$
$240$	0	0
$241$	19.9637	1.28598	0.642989	−	0.765875i	$-0.277694\pi$
$241$	19.9637	1.28598	0.642989	+	0.765875i	$0.277694\pi$
$242$	12.8554	0.826375
$243$	0	0
$244$	−3.81176	−0.244023
$245$	−3.75278	−0.239756
$246$	0	0
$247$	3.04306	0.193625
$248$	31.4574	1.99755
$249$	0	0
$250$	18.0002	1.13844
$251$	−3.91123	−0.246875	−0.123437	−	0.992352i	$-0.539392\pi$
$251$	−3.91123	−0.246875	−0.123437	+	0.992352i	$0.539392\pi$
$252$	0	0
$253$	−1.02778	−0.0646161
$254$	1.17466	0.0737047
$255$	0	0
$256$	−13.3051	−0.831568
$257$	1.56737	0.0977700	0.0488850	−	0.998804i	$-0.484433\pi$
$257$	1.56737	0.0977700	0.0488850	+	0.998804i	$0.484433\pi$
$258$	0	0
$259$	−5.67027	−0.352333
$260$	5.45311	0.338187
$261$	0	0
$262$	13.3571	0.825204
$263$	−12.5052	−0.771105	−0.385553	−	0.922686i	$-0.625989\pi$
$263$	−12.5052	−0.771105	−0.385553	+	0.922686i	$0.625989\pi$
$264$	0	0
$265$	−42.6707	−2.62124
$266$	1.52562	0.0935415
$267$	0	0
$268$	7.67733	0.468967
$269$	8.28407	0.505089	0.252544	−	0.967585i	$-0.418733\pi$
$269$	8.28407	0.505089	0.252544	+	0.967585i	$0.418733\pi$
$270$	0	0
$271$	18.6975	1.13579	0.567895	−	0.823101i	$-0.307758\pi$
$271$	18.6975	1.13579	0.567895	+	0.823101i	$0.307758\pi$
$272$	−13.0440	−0.790908
$273$	0	0
$274$	14.9681	0.904258
$275$	2.15108	0.129715
$276$	0	0
$277$	1.86253	0.111909	0.0559544	−	0.998433i	$-0.482180\pi$
$277$	1.86253	0.111909	0.0559544	+	0.998433i	$0.482180\pi$
$278$	15.9107	0.954258
$279$	0	0
$280$	11.5503	0.690265
$281$	−21.2145	−1.26555	−0.632775	−	0.774336i	$-0.718084\pi$
$281$	−21.2145	−1.26555	−0.632775	+	0.774336i	$0.718084\pi$
$282$	0	0
$283$	−7.20597	−0.428351	−0.214175	−	0.976795i	$-0.568706\pi$
$283$	−7.20597	−0.428351	−0.214175	+	0.976795i	$0.568706\pi$
$284$	−10.1452	−0.602004
$285$	0	0
$286$	−0.651781	−0.0385406
$287$	5.91718	0.349280
$288$	0	0
$289$	13.1635	0.774321
$290$	−14.7719	−0.867438
$291$	0	0
$292$	1.50638	0.0881544
$293$	−15.7001	−0.917209	−0.458604	−	0.888641i	$-0.651651\pi$
$293$	−15.7001	−0.917209	−0.458604	+	0.888641i	$0.651651\pi$
$294$	0	0
$295$	−6.28431	−0.365887
$296$	17.4520	1.01438
$297$	0	0
$298$	8.08987	0.468633
$299$	−10.1687	−0.588072
$300$	0	0
$301$	−7.58953	−0.437453
$302$	−14.0664	−0.809428
$303$	0	0
$304$	−3.08463	−0.176916
$305$	−23.0655	−1.32073
$306$	0	0
$307$	5.96213	0.340277	0.170138	−	0.985420i	$-0.445579\pi$
$307$	5.96213	0.340277	0.170138	+	0.985420i	$0.445579\pi$
$308$	0.146868	0.00836857
$309$	0	0
$310$	45.0552	2.55897
$311$	10.8576	0.615679	0.307840	−	0.951438i	$-0.400394\pi$
$311$	10.8576	0.615679	0.307840	+	0.951438i	$0.400394\pi$
$312$	0	0
$313$	28.1954	1.59370	0.796850	−	0.604177i	$-0.206498\pi$
$313$	28.1954	1.59370	0.796850	+	0.604177i	$0.206498\pi$
$314$	−5.81666	−0.328253
$315$	0	0
$316$	8.05664	0.453221
$317$	−6.35059	−0.356685	−0.178342	−	0.983968i	$-0.557073\pi$
$317$	−6.35059	−0.356685	−0.178342	+	0.983968i	$0.557073\pi$
$318$	0	0
$319$	−0.793570	−0.0444314
$320$	−32.6630	−1.82592
$321$	0	0
$322$	−5.09801	−0.284101
$323$	7.13303	0.396892
$324$	0	0
$325$	21.2825	1.18054
$326$	8.09290	0.448225
$327$	0	0
$328$	−18.2120	−1.00559
$329$	6.69813	0.369280
$330$	0	0
$331$	15.7882	0.867798	0.433899	−	0.900961i	$-0.357137\pi$
$331$	15.7882	0.867798	0.433899	+	0.900961i	$0.357137\pi$
$332$	−0.979489	−0.0537564
$333$	0	0
$334$	−15.6596	−0.856855
$335$	46.4567	2.53820
$336$	0	0
$337$	18.5731	1.01174	0.505871	−	0.862609i	$-0.331171\pi$
$337$	18.5731	1.01174	0.505871	+	0.862609i	$0.331171\pi$
$338$	8.82195	0.479851
$339$	0	0
$340$	12.7822	0.693215
$341$	2.42043	0.131074
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	23.3592	1.25944
$345$	0	0
$346$	16.5494	0.889702
$347$	18.2234	0.978284	0.489142	−	0.872204i	$-0.337310\pi$
$347$	18.2234	0.978284	0.489142	+	0.872204i	$0.337310\pi$
$348$	0	0
$349$	−7.77165	−0.416007	−0.208003	−	0.978128i	$-0.566697\pi$
$349$	−7.77165	−0.416007	−0.208003	+	0.978128i	$0.566697\pi$
$350$	10.6698	0.570326
$351$	0	0
$352$	−0.797071	−0.0424840
$353$	−7.11012	−0.378434	−0.189217	−	0.981935i	$-0.560595\pi$
$353$	−7.11012	−0.378434	−0.189217	+	0.981935i	$0.560595\pi$
$354$	0	0
$355$	−61.3899	−3.25824
$356$	1.87658	0.0994584
$357$	0	0
$358$	−6.15576	−0.325342
$359$	22.1932	1.17131	0.585655	−	0.810561i	$-0.300837\pi$
$359$	22.1932	1.17131	0.585655	+	0.810561i	$0.300837\pi$
$360$	0	0
$361$	−17.3132	−0.911220
$362$	18.5208	0.973431
$363$	0	0
$364$	1.45309	0.0761625
$365$	9.11536	0.477120
$366$	0	0
$367$	18.9581	0.989607	0.494804	−	0.869005i	$-0.335240\pi$
$367$	18.9581	0.989607	0.494804	+	0.869005i	$0.335240\pi$
$368$	10.3076	0.537322
$369$	0	0
$370$	24.9959	1.29947
$371$	−11.3704	−0.590324
$372$	0	0
$373$	28.3350	1.46713	0.733566	−	0.679618i	$-0.237855\pi$
$373$	28.3350	1.46713	0.733566	+	0.679618i	$0.237855\pi$
$374$	−1.52779	−0.0790004
$375$	0	0
$376$	−20.6156	−1.06317
$377$	−7.85146	−0.404371
$378$	0	0
$379$	7.31036	0.375508	0.187754	−	0.982216i	$-0.439879\pi$
$379$	7.31036	0.375508	0.187754	+	0.982216i	$0.439879\pi$
$380$	3.02273	0.155063
$381$	0	0
$382$	−26.0420	−1.33243
$383$	18.0303	0.921306	0.460653	−	0.887580i	$-0.347615\pi$
$383$	18.0303	0.921306	0.460653	+	0.887580i	$0.347615\pi$
$384$	0	0
$385$	0.888720	0.0452934
$386$	−10.1127	−0.514721
$387$	0	0
$388$	1.01136	0.0513442
$389$	−14.9682	−0.758919	−0.379459	−	0.925208i	$-0.623890\pi$
$389$	−14.9682	−0.758919	−0.379459	+	0.925208i	$0.623890\pi$
$390$	0	0
$391$	−23.8358	−1.20543
$392$	3.07781	0.155453
$393$	0	0
$394$	−9.04095	−0.455477
$395$	48.7520	2.45298
$396$	0	0
$397$	37.6493	1.88956	0.944782	−	0.327700i	$-0.106274\pi$
$397$	37.6493	1.88956	0.944782	+	0.327700i	$0.106274\pi$
$398$	31.6270	1.58532
$399$	0	0
$400$	−21.5732	−1.07866
$401$	26.1742	1.30708	0.653538	−	0.756893i	$-0.273284\pi$
$401$	26.1742	1.30708	0.653538	+	0.756893i	$0.273284\pi$
$402$	0	0
$403$	23.9474	1.19290
$404$	1.09017	0.0542378
$405$	0	0
$406$	−3.93627	−0.195354
$407$	1.34281	0.0665608
$408$	0	0
$409$	20.5595	1.01660	0.508301	−	0.861180i	$-0.330274\pi$
$409$	20.5595	1.01660	0.508301	+	0.861180i	$0.330274\pi$
$410$	−26.0843	−1.28821
$411$	0	0
$412$	−1.65763	−0.0816658
$413$	−1.67458	−0.0824005
$414$	0	0
$415$	−5.92704	−0.290947
$416$	−7.88610	−0.386648
$417$	0	0
$418$	−0.361291	−0.0176713
$419$	2.82635	0.138076	0.0690380	−	0.997614i	$-0.478007\pi$
$419$	2.82635	0.138076	0.0690380	+	0.997614i	$0.478007\pi$
$420$	0	0
$421$	−28.5279	−1.39036	−0.695181	−	0.718834i	$-0.744676\pi$
$421$	−28.5279	−1.39036	−0.695181	+	0.718834i	$0.744676\pi$
$422$	−23.8334	−1.16019
$423$	0	0
$424$	34.9961	1.69956
$425$	49.8868	2.41986
$426$	0	0
$427$	−6.14626	−0.297438
$428$	2.88979	0.139683
$429$	0	0
$430$	33.4564	1.61341
$431$	28.7560	1.38513	0.692565	−	0.721356i	$-0.256481\pi$
$431$	28.7560	1.38513	0.692565	+	0.721356i	$0.256481\pi$
$432$	0	0
$433$	−6.63153	−0.318691	−0.159345	−	0.987223i	$-0.550938\pi$
$433$	−6.63153	−0.318691	−0.159345	+	0.987223i	$0.550938\pi$
$434$	12.0058	0.576299
$435$	0	0
$436$	4.48891	0.214980
$437$	−5.63666	−0.269638
$438$	0	0
$439$	26.2024	1.25057	0.625286	−	0.780395i	$-0.284982\pi$
$439$	26.2024	1.25057	0.625286	+	0.780395i	$0.284982\pi$
$440$	−2.73531	−0.130401
$441$	0	0
$442$	−15.1158	−0.718984
$443$	−1.78227	−0.0846780	−0.0423390	−	0.999103i	$-0.513481\pi$
$443$	−1.78227	−0.0846780	−0.0423390	+	0.999103i	$0.513481\pi$
$444$	0	0
$445$	11.3555	0.538301
$446$	32.0951	1.51975
$447$	0	0
$448$	−8.70370	−0.411211
$449$	−11.4181	−0.538853	−0.269426	−	0.963021i	$-0.586834\pi$
$449$	−11.4181	−0.538853	−0.269426	+	0.963021i	$0.586834\pi$
$450$	0	0
$451$	−1.40129	−0.0659841
$452$	−6.62815	−0.311762
$453$	0	0
$454$	−23.5531	−1.10540
$455$	8.79286	0.412216
$456$	0	0
$457$	−10.3267	−0.483065	−0.241532	−	0.970393i	$-0.577650\pi$
$457$	−10.3267	−0.483065	−0.241532	+	0.970393i	$0.577650\pi$
$458$	−1.43977	−0.0672761
$459$	0	0
$460$	−10.1008	−0.470951
$461$	14.8522	0.691736	0.345868	−	0.938283i	$-0.387584\pi$
$461$	14.8522	0.691736	0.345868	+	0.938283i	$0.387584\pi$
$462$	0	0
$463$	−16.0617	−0.746452	−0.373226	−	0.927740i	$-0.621748\pi$
$463$	−16.0617	−0.746452	−0.373226	+	0.927740i	$0.621748\pi$
$464$	7.95871	0.369474
$465$	0	0
$466$	−1.74754	−0.0809531
$467$	34.7302	1.60712	0.803561	−	0.595223i	$-0.202936\pi$
$467$	34.7302	1.60712	0.803561	+	0.595223i	$0.202936\pi$
$468$	0	0
$469$	12.3793	0.571623
$470$	−29.5269	−1.36198
$471$	0	0
$472$	5.15403	0.237234
$473$	1.79733	0.0826413
$474$	0	0
$475$	11.7972	0.541292
$476$	3.40608	0.156117
$477$	0	0
$478$	−22.5575	−1.03176
$479$	1.51745	0.0693338	0.0346669	−	0.999399i	$-0.488963\pi$
$479$	1.51745	0.0693338	0.0346669	+	0.999399i	$0.488963\pi$
$480$	0	0
$481$	13.2856	0.605771
$482$	−23.4506	−1.06815
$483$	0	0
$484$	6.78714	0.308506
$485$	6.11992	0.277891
$486$	0	0
$487$	11.4155	0.517286	0.258643	−	0.965973i	$-0.416725\pi$
$487$	11.4155	0.517286	0.258643	+	0.965973i	$0.416725\pi$
$488$	18.9170	0.856334
$489$	0	0
$490$	4.40823	0.199144
$491$	−23.6645	−1.06797	−0.533983	−	0.845495i	$-0.679305\pi$
$491$	−23.6645	−1.06797	−0.533983	+	0.845495i	$0.679305\pi$
$492$	0	0
$493$	−18.4040	−0.828877
$494$	−3.57456	−0.160827
$495$	0	0
$496$	−24.2745	−1.08996
$497$	−16.3585	−0.733781
$498$	0	0
$499$	−19.0298	−0.851891	−0.425945	−	0.904749i	$-0.640058\pi$
$499$	−19.0298	−0.851891	−0.425945	+	0.904749i	$0.640058\pi$
$500$	9.50343	0.425006
$501$	0	0
$502$	4.59437	0.205057
$503$	1.94722	0.0868224	0.0434112	−	0.999057i	$-0.486177\pi$
$503$	1.94722	0.0868224	0.0434112	+	0.999057i	$0.486177\pi$
$504$	0	0
$505$	6.59677	0.293552
$506$	1.20729	0.0536707
$507$	0	0
$508$	0.620175	0.0275158
$509$	36.4390	1.61513	0.807566	−	0.589778i	$-0.200785\pi$
$509$	36.4390	1.61513	0.807566	+	0.589778i	$0.200785\pi$
$510$	0	0
$511$	2.42897	0.107451
$512$	22.6136	0.999391
$513$	0	0
$514$	−1.84113	−0.0812087
$515$	−10.0306	−0.442002
$516$	0	0
$517$	−1.58623	−0.0697623
$518$	6.66064	0.292652
$519$	0	0
$520$	−27.0628	−1.18678
$521$	21.1250	0.925502	0.462751	−	0.886488i	$-0.346862\pi$
$521$	21.1250	0.925502	0.462751	+	0.886488i	$0.346862\pi$
$522$	0	0
$523$	1.12611	0.0492413	0.0246206	−	0.999697i	$-0.492162\pi$
$523$	1.12611	0.0492413	0.0246206	+	0.999697i	$0.492162\pi$
$524$	7.05203	0.308069
$525$	0	0
$526$	14.6894	0.640487
$527$	56.1334	2.44521
$528$	0	0
$529$	−4.16451	−0.181066
$530$	50.1236	2.17723
$531$	0	0
$532$	0.805466	0.0349214
$533$	−13.8641	−0.600522
$534$	0	0
$535$	17.4866	0.756010
$536$	−38.1012	−1.64572
$537$	0	0
$538$	−9.73096	−0.419531
$539$	0.236817	0.0102004
$540$	0	0
$541$	−38.0060	−1.63400	−0.817002	−	0.576635i	$-0.804366\pi$
$541$	−38.0060	−1.63400	−0.817002	+	0.576635i	$0.804366\pi$
$542$	−21.9631	−0.943398
$543$	0	0
$544$	−18.4852	−0.792548
$545$	27.1631	1.16354
$546$	0	0
$547$	−12.0457	−0.515035	−0.257517	−	0.966274i	$-0.582904\pi$
$547$	−12.0457	−0.515035	−0.257517	+	0.966274i	$0.582904\pi$
$548$	7.90259	0.337582
$549$	0	0
$550$	−2.52679	−0.107743
$551$	−4.35217	−0.185409
$552$	0	0
$553$	12.9909	0.552430
$554$	−2.18784	−0.0929526
$555$	0	0
$556$	8.40021	0.356248
$557$	0.211358	0.00895553	0.00447776	−	0.999990i	$-0.498575\pi$
$557$	0.211358	0.00895553	0.00447776	+	0.999990i	$0.498575\pi$
$558$	0	0
$559$	17.7825	0.752119
$560$	−8.91297	−0.376642
$561$	0	0
$562$	24.9198	1.05118
$563$	19.7090	0.830635	0.415318	−	0.909676i	$-0.363670\pi$
$563$	19.7090	0.830635	0.415318	+	0.909676i	$0.363670\pi$
$564$	0	0
$565$	−40.1080	−1.68736
$566$	8.46456	0.355792
$567$	0	0
$568$	50.3485	2.11258
$569$	12.2524	0.513647	0.256824	−	0.966458i	$-0.417324\pi$
$569$	12.2524	0.513647	0.256824	+	0.966458i	$0.417324\pi$
$570$	0	0
$571$	−30.3969	−1.27207	−0.636036	−	0.771660i	$-0.719427\pi$
$571$	−30.3969	−1.27207	−0.636036	+	0.771660i	$0.719427\pi$
$572$	−0.344115	−0.0143882
$573$	0	0
$574$	−6.95068	−0.290116
$575$	−39.4215	−1.64399
$576$	0	0
$577$	13.0374	0.542754	0.271377	−	0.962473i	$-0.412521\pi$
$577$	13.0374	0.542754	0.271377	+	0.962473i	$0.412521\pi$
$578$	−15.4626	−0.643159
$579$	0	0
$580$	−7.79901	−0.323836
$581$	−1.57938	−0.0655236
$582$	0	0
$583$	2.69271	0.111521
$584$	−7.47590	−0.309355
$585$	0	0
$586$	18.4423	0.761843
$587$	18.3713	0.758263	0.379132	−	0.925343i	$-0.376223\pi$
$587$	18.3713	0.758263	0.379132	+	0.925343i	$0.376223\pi$
$588$	0	0
$589$	13.2744	0.546961
$590$	7.38192	0.303909
$591$	0	0
$592$	−13.4671	−0.553494
$593$	−30.9081	−1.26924	−0.634621	−	0.772823i	$-0.718844\pi$
$593$	−30.9081	−1.26924	−0.634621	+	0.772823i	$0.718844\pi$
$594$	0	0
$595$	20.6107	0.844957
$596$	4.27114	0.174953
$597$	0	0
$598$	11.9448	0.488458
$599$	−41.1317	−1.68059	−0.840297	−	0.542126i	$-0.817619\pi$
$599$	−41.1317	−1.68059	−0.840297	+	0.542126i	$0.817619\pi$
$600$	0	0
$601$	8.15562	0.332675	0.166337	−	0.986069i	$-0.446806\pi$
$601$	8.15562	0.332675	0.166337	+	0.986069i	$0.446806\pi$
$602$	8.91512	0.363353
$603$	0	0
$604$	−7.42649	−0.302180
$605$	41.0701	1.66974
$606$	0	0
$607$	−43.6397	−1.77128	−0.885641	−	0.464371i	$-0.846280\pi$
$607$	−43.6397	−1.77128	−0.885641	+	0.464371i	$0.846280\pi$
$608$	−4.37137	−0.177283
$609$	0	0
$610$	27.0942	1.09701
$611$	−15.6939	−0.634907
$612$	0	0
$613$	16.0347	0.647634	0.323817	−	0.946120i	$-0.395034\pi$
$613$	16.0347	0.647634	0.323817	+	0.946120i	$0.395034\pi$
$614$	−7.00347	−0.282637
$615$	0	0
$616$	−0.728878	−0.0293673
$617$	−24.7835	−0.997747	−0.498873	−	0.866675i	$-0.666253\pi$
$617$	−24.7835	−0.997747	−0.498873	+	0.866675i	$0.666253\pi$
$618$	0	0
$619$	−35.4634	−1.42539	−0.712697	−	0.701472i	$-0.752527\pi$
$619$	−35.4634	−1.42539	−0.712697	+	0.701472i	$0.752527\pi$
$620$	23.7874	0.955325
$621$	0	0
$622$	−12.7540	−0.511389
$623$	3.02589	0.121230
$624$	0	0
$625$	12.0902	0.483606
$626$	−33.1200	−1.32374
$627$	0	0
$628$	−3.07097	−0.122545
$629$	31.1418	1.24171
$630$	0	0
$631$	−11.2435	−0.447595	−0.223798	−	0.974636i	$-0.571845\pi$
$631$	−11.2435	−0.447595	−0.223798	+	0.974636i	$0.571845\pi$
$632$	−39.9836	−1.59046
$633$	0	0
$634$	7.45978	0.296266
$635$	3.75278	0.148924
$636$	0	0
$637$	2.34303	0.0928342
$638$	0.932175	0.0369051
$639$	0	0
$640$	13.1060	0.518060
$641$	−21.0411	−0.831072	−0.415536	−	0.909577i	$-0.636406\pi$
$641$	−21.0411	−0.831072	−0.415536	+	0.909577i	$0.636406\pi$
$642$	0	0
$643$	−15.7595	−0.621495	−0.310748	−	0.950492i	$-0.600579\pi$
$643$	−15.7595	−0.621495	−0.310748	+	0.950492i	$0.600579\pi$
$644$	−2.69155	−0.106062
$645$	0	0
$646$	−8.37888	−0.329663
$647$	9.26870	0.364390	0.182195	−	0.983262i	$-0.441680\pi$
$647$	9.26870	0.364390	0.182195	+	0.983262i	$0.441680\pi$
$648$	0	0
$649$	0.396568	0.0155666
$650$	−24.9997	−0.980568
$651$	0	0
$652$	4.27274	0.167333
$653$	−48.5168	−1.89861	−0.949305	−	0.314357i	$-0.898211\pi$
$653$	−48.5168	−1.89861	−0.949305	+	0.314357i	$0.898211\pi$
$654$	0	0
$655$	42.6730	1.66737
$656$	14.0535	0.548697
$657$	0	0
$658$	−7.86802	−0.306727
$659$	−8.35752	−0.325563	−0.162781	−	0.986662i	$-0.552047\pi$
$659$	−8.35752	−0.325563	−0.162781	+	0.986662i	$0.552047\pi$
$660$	0	0
$661$	19.1646	0.745417	0.372708	−	0.927949i	$-0.378429\pi$
$661$	19.1646	0.745417	0.372708	+	0.927949i	$0.378429\pi$
$662$	−18.5458	−0.720802
$663$	0	0
$664$	4.86102	0.188644
$665$	4.87400	0.189006
$666$	0	0
$667$	14.5432	0.563117
$668$	−8.26766	−0.319885
$669$	0	0
$670$	−54.5708	−2.10826
$671$	1.45554	0.0561904
$672$	0	0
$673$	34.6506	1.33568	0.667841	−	0.744304i	$-0.267219\pi$
$673$	34.6506	1.33568	0.667841	+	0.744304i	$0.267219\pi$
$674$	−21.8171	−0.840364
$675$	0	0
$676$	4.65765	0.179140
$677$	21.4559	0.824617	0.412308	−	0.911044i	$-0.364723\pi$
$677$	21.4559	0.824617	0.412308	+	0.911044i	$0.364723\pi$
$678$	0	0
$679$	1.63077	0.0625833
$680$	−63.4360	−2.43266
$681$	0	0
$682$	−2.84318	−0.108871
$683$	16.1925	0.619587	0.309794	−	0.950804i	$-0.399740\pi$
$683$	16.1925	0.619587	0.309794	+	0.950804i	$0.399740\pi$
$684$	0	0
$685$	47.8198	1.82710
$686$	1.17466	0.0448487
$687$	0	0
$688$	−18.0254	−0.687212
$689$	26.6413	1.01495
$690$	0	0
$691$	6.16757	0.234625	0.117313	−	0.993095i	$-0.462572\pi$
$691$	6.16757	0.234625	0.117313	+	0.993095i	$0.462572\pi$
$692$	8.73745	0.332148
$693$	0	0
$694$	−21.4063	−0.812572
$695$	50.8310	1.92813
$696$	0	0
$697$	−32.4979	−1.23095
$698$	9.12904	0.345539
$699$	0	0
$700$	5.63325	0.212917
$701$	40.4749	1.52871	0.764357	−	0.644793i	$-0.223056\pi$
$701$	40.4749	1.52871	0.764357	+	0.644793i	$0.223056\pi$
$702$	0	0
$703$	7.36439	0.277753
$704$	2.06118	0.0776837
$705$	0	0
$706$	8.35197	0.314331
$707$	1.75784	0.0661103
$708$	0	0
$709$	−36.3485	−1.36510	−0.682548	−	0.730841i	$-0.739128\pi$
$709$	−36.3485	−1.36510	−0.682548	+	0.730841i	$0.739128\pi$
$710$	72.1123	2.70633
$711$	0	0
$712$	−9.31311	−0.349024
$713$	−44.3577	−1.66121
$714$	0	0
$715$	−2.08230	−0.0778735
$716$	−3.25000	−0.121458
$717$	0	0
$718$	−26.0694	−0.972901
$719$	−12.1881	−0.454541	−0.227270	−	0.973832i	$-0.572980\pi$
$719$	−12.1881	−0.454541	−0.227270	+	0.973832i	$0.572980\pi$
$720$	0	0
$721$	−2.67285	−0.0995422
$722$	20.3371	0.756869
$723$	0	0
$724$	9.77826	0.363406
$725$	−30.4381	−1.13044
$726$	0	0
$727$	−5.50233	−0.204070	−0.102035	−	0.994781i	$-0.532535\pi$
$727$	−5.50233	−0.204070	−0.102035	+	0.994781i	$0.532535\pi$
$728$	−7.21141	−0.267272
$729$	0	0
$730$	−10.7074	−0.396300
$731$	41.6827	1.54169
$732$	0	0
$733$	34.9339	1.29031	0.645156	−	0.764051i	$-0.276792\pi$
$733$	34.9339	1.29031	0.645156	+	0.764051i	$0.276792\pi$
$734$	−22.2694	−0.821977
$735$	0	0
$736$	14.6074	0.538436
$737$	−2.93162	−0.107988
$738$	0	0
$739$	48.9297	1.79991	0.899954	−	0.435984i	$-0.143600\pi$
$739$	48.9297	1.79991	0.899954	+	0.435984i	$0.143600\pi$
$740$	13.1968	0.485126
$741$	0	0
$742$	13.3564	0.490329
$743$	−53.0965	−1.94792	−0.973962	−	0.226713i	$-0.927202\pi$
$743$	−53.0965	−1.94792	−0.973962	+	0.226713i	$0.927202\pi$
$744$	0	0
$745$	25.8453	0.946899
$746$	−33.2840	−1.21861
$747$	0	0
$748$	−0.806616	−0.0294928
$749$	4.65963	0.170259
$750$	0	0
$751$	−15.2108	−0.555049	−0.277524	−	0.960719i	$-0.589514\pi$
$751$	−15.2108	−0.555049	−0.277524	+	0.960719i	$0.589514\pi$
$752$	15.9083	0.580115
$753$	0	0
$754$	9.22279	0.335874
$755$	−44.9389	−1.63549
$756$	0	0
$757$	19.5018	0.708804	0.354402	−	0.935093i	$-0.384684\pi$
$757$	19.5018	0.708804	0.354402	+	0.935093i	$0.384684\pi$
$758$	−8.58718	−0.311901
$759$	0	0
$760$	−15.0013	−0.544153
$761$	−48.7164	−1.76597	−0.882984	−	0.469404i	$-0.844469\pi$
$761$	−48.7164	−1.76597	−0.882984	+	0.469404i	$0.844469\pi$
$762$	0	0
$763$	7.23814	0.262038
$764$	−13.7492	−0.497428
$765$	0	0
$766$	−21.1795	−0.765246
$767$	3.92358	0.141672
$768$	0	0
$769$	21.1339	0.762108	0.381054	−	0.924553i	$-0.375561\pi$
$769$	21.1339	0.762108	0.381054	+	0.924553i	$0.375561\pi$
$770$	−1.04394	−0.0376211
$771$	0	0
$772$	−5.33909	−0.192158
$773$	27.7575	0.998367	0.499184	−	0.866496i	$-0.333633\pi$
$773$	27.7575	0.998367	0.499184	+	0.866496i	$0.333633\pi$
$774$	0	0
$775$	92.8379	3.33484
$776$	−5.01921	−0.180179
$777$	0	0
$778$	17.5826	0.630366
$779$	−7.68508	−0.275346
$780$	0	0
$781$	3.87398	0.138622
$782$	27.9989	1.00124
$783$	0	0
$784$	−2.37503	−0.0848226
$785$	−18.5829	−0.663253
$786$	0	0
$787$	24.7363	0.881753	0.440876	−	0.897568i	$-0.354668\pi$
$787$	24.7363	0.881753	0.440876	+	0.897568i	$0.354668\pi$
$788$	−4.77327	−0.170041
$789$	0	0
$790$	−57.2670	−2.03747
$791$	−10.6875	−0.380006
$792$	0	0
$793$	14.4009	0.511390
$794$	−44.2251	−1.56949
$795$	0	0
$796$	16.6979	0.591840
$797$	−17.1054	−0.605905	−0.302952	−	0.953006i	$-0.597972\pi$
$797$	−17.1054	−0.605905	−0.302952	+	0.953006i	$0.597972\pi$
$798$	0	0
$799$	−36.7870	−1.30143
$800$	−30.5724	−1.08090
$801$	0	0
$802$	−30.7458	−1.08567
$803$	−0.575220	−0.0202991
$804$	0	0
$805$	−16.2870	−0.574041
$806$	−28.1300	−0.990838
$807$	0	0
$808$	−5.41030	−0.190334
$809$	7.68969	0.270355	0.135178	−	0.990821i	$-0.456839\pi$
$809$	7.68969	0.270355	0.135178	+	0.990821i	$0.456839\pi$
$810$	0	0
$811$	15.3989	0.540728	0.270364	−	0.962758i	$-0.412856\pi$
$811$	15.3989	0.540728	0.270364	+	0.962758i	$0.412856\pi$
$812$	−2.07820	−0.0729304
$813$	0	0
$814$	−1.57735	−0.0552861
$815$	25.8550	0.905662
$816$	0	0
$817$	9.85708	0.344856
$818$	−24.1504	−0.844399
$819$	0	0
$820$	−13.7715	−0.480922
$821$	−27.5047	−0.959919	−0.479960	−	0.877291i	$-0.659349\pi$
$821$	−27.5047	−0.959919	−0.479960	+	0.877291i	$0.659349\pi$
$822$	0	0
$823$	22.8519	0.796568	0.398284	−	0.917262i	$-0.369606\pi$
$823$	22.8519	0.796568	0.398284	+	0.917262i	$0.369606\pi$
$824$	8.22654	0.286585
$825$	0	0
$826$	1.96706	0.0684427
$827$	8.47637	0.294752	0.147376	−	0.989081i	$-0.452917\pi$
$827$	8.47637	0.294752	0.147376	+	0.989081i	$0.452917\pi$
$828$	0	0
$829$	10.7215	0.372373	0.186186	−	0.982514i	$-0.440387\pi$
$829$	10.7215	0.372373	0.186186	+	0.982514i	$0.440387\pi$
$830$	6.96226	0.241664
$831$	0	0
$832$	20.3930	0.707001
$833$	5.49213	0.190291
$834$	0	0
$835$	−50.0289	−1.73132
$836$	−0.190748	−0.00659715
$837$	0	0
$838$	−3.32000	−0.114687
$839$	42.2170	1.45749	0.728746	−	0.684784i	$-0.240103\pi$
$839$	42.2170	1.45749	0.728746	+	0.684784i	$0.240103\pi$
$840$	0	0
$841$	−17.7709	−0.612789
$842$	33.5105	1.15485
$843$	0	0
$844$	−12.5831	−0.433129
$845$	28.1842	0.969564
$846$	0	0
$847$	10.9439	0.376037
$848$	−27.0052	−0.927362
$849$	0	0
$850$	−58.6000	−2.00996
$851$	−24.6089	−0.843582
$852$	0	0
$853$	9.69852	0.332071	0.166035	−	0.986120i	$-0.446903\pi$
$853$	9.69852	0.332071	0.166035	+	0.986120i	$0.446903\pi$
$854$	7.21977	0.247055
$855$	0	0
$856$	−14.3415	−0.490182
$857$	12.9699	0.443044	0.221522	−	0.975155i	$-0.428898\pi$
$857$	12.9699	0.443044	0.221522	+	0.975155i	$0.428898\pi$
$858$	0	0
$859$	−55.4366	−1.89147	−0.945736	−	0.324935i	$-0.894658\pi$
$859$	−55.4366	−1.89147	−0.945736	+	0.324935i	$0.894658\pi$
$860$	17.6637	0.602327
$861$	0	0
$862$	−33.7786	−1.15050
$863$	−7.24768	−0.246714	−0.123357	−	0.992362i	$-0.539366\pi$
$863$	−7.24768	−0.246714	−0.123357	+	0.992362i	$0.539366\pi$
$864$	0	0
$865$	52.8717	1.79769
$866$	7.78979	0.264708
$867$	0	0
$868$	6.33862	0.215147
$869$	−3.07647	−0.104362
$870$	0	0
$871$	−29.0051	−0.982799
$872$	−22.2776	−0.754416
$873$	0	0
$874$	6.62115	0.223964
$875$	15.3238	0.518039
$876$	0	0
$877$	−36.5313	−1.23357	−0.616787	−	0.787130i	$-0.711566\pi$
$877$	−36.5313	−1.23357	−0.616787	+	0.787130i	$0.711566\pi$
$878$	−30.7789	−1.03874
$879$	0	0
$880$	2.11074	0.0711530
$881$	31.2009	1.05119	0.525593	−	0.850736i	$-0.323843\pi$
$881$	31.2009	1.05119	0.525593	+	0.850736i	$0.323843\pi$
$882$	0	0
$883$	−22.5534	−0.758983	−0.379491	−	0.925195i	$-0.623901\pi$
$883$	−22.5534	−0.758983	−0.379491	+	0.925195i	$0.623901\pi$
$884$	−7.98054	−0.268415
$885$	0	0
$886$	2.09356	0.0703344
$887$	−5.16427	−0.173399	−0.0866996	−	0.996235i	$-0.527632\pi$
$887$	−5.16427	−0.173399	−0.0866996	+	0.996235i	$0.527632\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−13.3388	−0.447118
$891$	0	0
$892$	16.9450	0.567360
$893$	−8.69935	−0.291113
$894$	0	0
$895$	−19.6663	−0.657372
$896$	3.49235	0.116671
$897$	0	0
$898$	13.4124	0.447576
$899$	−34.2494	−1.14228
$900$	0	0
$901$	62.4480	2.08044
$902$	1.64604	0.0548070
$903$	0	0
$904$	32.8943	1.09405
$905$	59.1698	1.96687
$906$	0	0
$907$	36.9133	1.22569	0.612843	−	0.790205i	$-0.290026\pi$
$907$	36.9133	1.22569	0.612843	+	0.790205i	$0.290026\pi$
$908$	−12.4351	−0.412674
$909$	0	0
$910$	−10.3286	−0.342390
$911$	−27.7392	−0.919042	−0.459521	−	0.888167i	$-0.651979\pi$
$911$	−27.7392	−0.919042	−0.459521	+	0.888167i	$0.651979\pi$
$912$	0	0
$913$	0.374023	0.0123783
$914$	12.1304	0.401238
$915$	0	0
$916$	−0.760144	−0.0251159
$917$	11.3710	0.375505
$918$	0	0
$919$	−50.6943	−1.67225	−0.836125	−	0.548539i	$-0.815184\pi$
$919$	−50.6943	−1.67225	−0.836125	+	0.548539i	$0.815184\pi$
$920$	50.1283	1.65268
$921$	0	0
$922$	−17.4463	−0.574563
$923$	38.3285	1.26160
$924$	0	0
$925$	51.5049	1.69347
$926$	18.8671	0.620011
$927$	0	0
$928$	11.2787	0.370240
$929$	−23.3612	−0.766458	−0.383229	−	0.923653i	$-0.625188\pi$
$929$	−23.3612	−0.766458	−0.383229	+	0.923653i	$0.625188\pi$
$930$	0	0
$931$	1.29877	0.0425656
$932$	−0.922632	−0.0302218
$933$	0	0
$934$	−40.7962	−1.33489
$935$	−4.88096	−0.159625
$936$	0	0
$937$	−57.5544	−1.88022	−0.940111	−	0.340869i	$-0.889279\pi$
$937$	−57.5544	−1.88022	−0.940111	+	0.340869i	$0.889279\pi$
$938$	−14.5415	−0.474796
$939$	0	0
$940$	−15.5891	−0.508459
$941$	−18.2078	−0.593557	−0.296779	−	0.954946i	$-0.595912\pi$
$941$	−18.2078	−0.593557	−0.296779	+	0.954946i	$0.595912\pi$
$942$	0	0
$943$	25.6805	0.836272
$944$	−3.97717	−0.129446
$945$	0	0
$946$	−2.11125	−0.0686426
$947$	54.3384	1.76576	0.882880	−	0.469599i	$-0.155601\pi$
$947$	54.3384	1.76576	0.882880	+	0.469599i	$0.155601\pi$
$948$	0	0
$949$	−5.69114	−0.184742
$950$	−13.8577	−0.449602
$951$	0	0
$952$	−16.9037	−0.547853
$953$	−54.8161	−1.77567	−0.887834	−	0.460164i	$-0.847791\pi$
$953$	−54.8161	−1.77567	−0.887834	+	0.460164i	$0.847791\pi$
$954$	0	0
$955$	−83.1985	−2.69224
$956$	−11.9095	−0.385180
$957$	0	0
$958$	−1.78248	−0.0575894
$959$	12.7425	0.411478
$960$	0	0
$961$	73.4627	2.36976
$962$	−15.6061	−0.503160
$963$	0	0
$964$	−12.3810	−0.398765
$965$	−32.3077	−1.04002
$966$	0	0
$967$	−24.6896	−0.793963	−0.396982	−	0.917827i	$-0.629942\pi$
$967$	−24.6896	−0.793963	−0.396982	+	0.917827i	$0.629942\pi$
$968$	−33.6833	−1.08262
$969$	0	0
$970$	−7.18882	−0.230819
$971$	−24.5256	−0.787065	−0.393532	−	0.919311i	$-0.628747\pi$
$971$	−24.5256	−0.787065	−0.393532	+	0.919311i	$0.628747\pi$
$972$	0	0
$973$	13.5449	0.434230
$974$	−13.4093	−0.429663
$975$	0	0
$976$	−14.5976	−0.467257
$977$	−25.0682	−0.802003	−0.401001	−	0.916077i	$-0.631338\pi$
$977$	−25.0682	−0.802003	−0.401001	+	0.916077i	$0.631338\pi$
$978$	0	0
$979$	−0.716580	−0.0229020
$980$	2.32738	0.0743453
$981$	0	0
$982$	27.7978	0.887063
$983$	19.8975	0.634630	0.317315	−	0.948320i	$-0.397219\pi$
$983$	19.8975	0.634630	0.317315	+	0.948320i	$0.397219\pi$
$984$	0	0
$985$	−28.8838	−0.920315
$986$	21.6185	0.688473
$987$	0	0
$988$	−1.88723	−0.0600408
$989$	−32.9385	−1.04738
$990$	0	0
$991$	10.5795	0.336069	0.168035	−	0.985781i	$-0.446258\pi$
$991$	10.5795	0.336069	0.168035	+	0.985781i	$0.446258\pi$
$992$	−34.4005	−1.09222
$993$	0	0
$994$	19.2157	0.609486
$995$	101.041	3.20323
$996$	0	0
$997$	−37.2369	−1.17931	−0.589653	−	0.807657i	$-0.700735\pi$
$997$	−37.2369	−1.17931	−0.589653	+	0.807657i	$0.700735\pi$
$998$	22.3535	0.707589
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.6		16
3.2	odd	2		2667.2.a.n.1.11	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.11	✓	16	3.2	odd	2
8001.2.a.s.1.6		16	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)

\(f(q)\)	\(=\)	\(q-1.17466 q^{2} -0.620175 q^{4} -3.75278 q^{5} -1.00000 q^{7} +3.07781 q^{8} +O(q^{10})\)	\(q-1.17466 q^{2} -0.620175 q^{4} -3.75278 q^{5} -1.00000 q^{7} +3.07781 q^{8} +4.40823 q^{10} +0.236817 q^{11} +2.34303 q^{13} +1.17466 q^{14} -2.37503 q^{16} +5.49213 q^{17} +1.29877 q^{19} +2.32738 q^{20} -0.278179 q^{22} -4.33999 q^{23} +9.08332 q^{25} -2.75226 q^{26} +0.620175 q^{28} -3.35099 q^{29} +10.2207 q^{31} -3.36577 q^{32} -6.45138 q^{34} +3.75278 q^{35} +5.67027 q^{37} -1.52562 q^{38} -11.5503 q^{40} -5.91718 q^{41} +7.58953 q^{43} -0.146868 q^{44} +5.09801 q^{46} -6.69813 q^{47} +1.00000 q^{49} -10.6698 q^{50} -1.45309 q^{52} +11.3704 q^{53} -0.888720 q^{55} -3.07781 q^{56} +3.93627 q^{58} +1.67458 q^{59} +6.14626 q^{61} -12.0058 q^{62} +8.70370 q^{64} -8.79286 q^{65} -12.3793 q^{67} -3.40608 q^{68} -4.40823 q^{70} +16.3585 q^{71} -2.42897 q^{73} -6.66064 q^{74} -0.805466 q^{76} -0.236817 q^{77} -12.9909 q^{79} +8.91297 q^{80} +6.95068 q^{82} +1.57938 q^{83} -20.6107 q^{85} -8.91512 q^{86} +0.728878 q^{88} -3.02589 q^{89} -2.34303 q^{91} +2.69155 q^{92} +7.86802 q^{94} -4.87400 q^{95} -1.63077 q^{97} -1.17466 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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