LMFDB - Embedded newform 8001.2.a.s.1.2

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			1.2
Root			$2.63870$ 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)

$f(q)$	$=$	$q-2.63870 q^{2} +4.96274 q^{4} -0.422965 q^{5} -1.00000 q^{7} -7.81779 q^{8} +O(q^{10})$	\(q-2.63870 q^{2} +4.96274 q^{4} -0.422965 q^{5} -1.00000 q^{7} -7.81779 q^{8} +1.11608 q^{10} -4.71685 q^{11} +5.79847 q^{13} +2.63870 q^{14} +10.7033 q^{16} +7.52411 q^{17} -3.61435 q^{19} -2.09907 q^{20} +12.4463 q^{22} +9.40514 q^{23} -4.82110 q^{25} -15.3004 q^{26} -4.96274 q^{28} -1.54396 q^{29} +9.34218 q^{31} -12.6073 q^{32} -19.8539 q^{34} +0.422965 q^{35} +3.22647 q^{37} +9.53718 q^{38} +3.30665 q^{40} +9.20474 q^{41} +5.26388 q^{43} -23.4085 q^{44} -24.8174 q^{46} +13.4783 q^{47} +1.00000 q^{49} +12.7214 q^{50} +28.7763 q^{52} -7.03165 q^{53} +1.99506 q^{55} +7.81779 q^{56} +4.07404 q^{58} -3.67029 q^{59} -11.6944 q^{61} -24.6512 q^{62} +11.8602 q^{64} -2.45255 q^{65} +8.43069 q^{67} +37.3402 q^{68} -1.11608 q^{70} +0.349766 q^{71} +9.26310 q^{73} -8.51369 q^{74} -17.9371 q^{76} +4.71685 q^{77} -0.0362893 q^{79} -4.52713 q^{80} -24.2886 q^{82} +1.22938 q^{83} -3.18243 q^{85} -13.8898 q^{86} +36.8753 q^{88} -3.45918 q^{89} -5.79847 q^{91} +46.6753 q^{92} -35.5651 q^{94} +1.52874 q^{95} -11.6255 q^{97} -2.63870 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.63870	−1.86584	−0.932922	−	0.360080i	$-0.882750\pi$
$2$	−2.63870	−1.86584	−0.932922	+	0.360080i	$0.882750\pi$
$3$	0	0
$3$	0	0
$4$	4.96274	2.48137
$5$	−0.422965	−0.189156	−0.0945779	−	0.995517i	$-0.530150\pi$
$5$	−0.422965	−0.189156	−0.0945779	+	0.995517i	$0.530150\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−7.81779	−2.76401
$9$	0	0
$10$	1.11608	0.352935
$11$	−4.71685	−1.42218	−0.711092	−	0.703099i	$-0.751799\pi$
$11$	−4.71685	−1.42218	−0.711092	+	0.703099i	$0.751799\pi$
$12$	0	0
$13$	5.79847	1.60820	0.804102	−	0.594491i	$-0.202646\pi$
$13$	5.79847	1.60820	0.804102	+	0.594491i	$0.202646\pi$
$14$	2.63870	0.705222
$15$	0	0
$16$	10.7033	2.67583
$17$	7.52411	1.82486	0.912432	−	0.409228i	$-0.134202\pi$
$17$	7.52411	1.82486	0.912432	+	0.409228i	$0.134202\pi$
$18$	0	0
$19$	−3.61435	−0.829188	−0.414594	−	0.910007i	$-0.636076\pi$
$19$	−3.61435	−0.829188	−0.414594	+	0.910007i	$0.636076\pi$
$20$	−2.09907	−0.469365
$21$	0	0
$22$	12.4463	2.65357
$23$	9.40514	1.96111	0.980554	−	0.196250i	$-0.0628763\pi$
$23$	9.40514	1.96111	0.980554	+	0.196250i	$0.0628763\pi$
$24$	0	0
$25$	−4.82110	−0.964220
$26$	−15.3004	−3.00066
$27$	0	0
$28$	−4.96274	−0.937870
$29$	−1.54396	−0.286706	−0.143353	−	0.989672i	$-0.545788\pi$
$29$	−1.54396	−0.286706	−0.143353	+	0.989672i	$0.545788\pi$
$30$	0	0
$31$	9.34218	1.67791	0.838953	−	0.544204i	$-0.183168\pi$
$31$	9.34218	1.67791	0.838953	+	0.544204i	$0.183168\pi$
$32$	−12.6073	−2.22867
$33$	0	0
$34$	−19.8539	−3.40491
$35$	0.422965	0.0714941
$36$	0	0
$37$	3.22647	0.530429	0.265214	−	0.964189i	$-0.414557\pi$
$37$	3.22647	0.530429	0.265214	+	0.964189i	$0.414557\pi$
$38$	9.53718	1.54713
$39$	0	0
$40$	3.30665	0.522827
$41$	9.20474	1.43754	0.718769	−	0.695248i	$-0.244706\pi$
$41$	9.20474	1.43754	0.718769	+	0.695248i	$0.244706\pi$
$42$	0	0
$43$	5.26388	0.802734	0.401367	−	0.915917i	$-0.368535\pi$
$43$	5.26388	0.802734	0.401367	+	0.915917i	$0.368535\pi$
$44$	−23.4085	−3.52896
$45$	0	0
$46$	−24.8174	−3.65912
$47$	13.4783	1.96601	0.983004	−	0.183582i	$-0.0587692\pi$
$47$	13.4783	1.96601	0.983004	+	0.183582i	$0.0587692\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	12.7214	1.79908
$51$	0	0
$52$	28.7763	3.99055
$53$	−7.03165	−0.965872	−0.482936	−	0.875656i	$-0.660430\pi$
$53$	−7.03165	−0.965872	−0.482936	+	0.875656i	$0.660430\pi$
$54$	0	0
$55$	1.99506	0.269014
$56$	7.81779	1.04470
$57$	0	0
$58$	4.07404	0.534948
$59$	−3.67029	−0.477831	−0.238915	−	0.971040i	$-0.576792\pi$
$59$	−3.67029	−0.477831	−0.238915	+	0.971040i	$0.576792\pi$
$60$	0	0
$61$	−11.6944	−1.49731	−0.748657	−	0.662957i	$-0.769301\pi$
$61$	−11.6944	−1.49731	−0.748657	+	0.662957i	$0.769301\pi$
$62$	−24.6512	−3.13071
$63$	0	0
$64$	11.8602	1.48252
$65$	−2.45255	−0.304201
$66$	0	0
$67$	8.43069	1.02997	0.514986	−	0.857199i	$-0.327797\pi$
$67$	8.43069	1.02997	0.514986	+	0.857199i	$0.327797\pi$
$68$	37.3402	4.52816
$69$	0	0
$70$	−1.11608	−0.133397
$71$	0.349766	0.0415096	0.0207548	−	0.999785i	$-0.493393\pi$
$71$	0.349766	0.0415096	0.0207548	+	0.999785i	$0.493393\pi$
$72$	0	0
$73$	9.26310	1.08416	0.542082	−	0.840326i	$-0.317636\pi$
$73$	9.26310	1.08416	0.542082	+	0.840326i	$0.317636\pi$
$74$	−8.51369	−0.989697
$75$	0	0
$76$	−17.9371	−2.05752
$77$	4.71685	0.537535
$78$	0	0
$79$	−0.0362893	−0.00408286	−0.00204143	−	0.999998i	$-0.500650\pi$
$79$	−0.0362893	−0.00408286	−0.00204143	+	0.999998i	$0.500650\pi$
$80$	−4.52713	−0.506148
$81$	0	0
$82$	−24.2886	−2.68222
$83$	1.22938	0.134942	0.0674710	−	0.997721i	$-0.478507\pi$
$83$	1.22938	0.134942	0.0674710	+	0.997721i	$0.478507\pi$
$84$	0	0
$85$	−3.18243	−0.345184
$86$	−13.8898	−1.49778
$87$	0	0
$88$	36.8753	3.93092
$89$	−3.45918	−0.366673	−0.183336	−	0.983050i	$-0.558690\pi$
$89$	−3.45918	−0.366673	−0.183336	+	0.983050i	$0.558690\pi$
$90$	0	0
$91$	−5.79847	−0.607844
$92$	46.6753	4.86624
$93$	0	0
$94$	−35.5651	−3.66826
$95$	1.52874	0.156846
$96$	0	0
$97$	−11.6255	−1.18039	−0.590195	−	0.807260i	$-0.700949\pi$
$97$	−11.6255	−1.18039	−0.590195	+	0.807260i	$0.700949\pi$
$98$	−2.63870	−0.266549
$99$	0	0
$100$	−23.9259	−2.39259
$101$	−10.5118	−1.04596	−0.522982	−	0.852343i	$-0.675181\pi$
$101$	−10.5118	−1.04596	−0.522982	+	0.852343i	$0.675181\pi$
$102$	0	0
$103$	−10.2251	−1.00751	−0.503757	−	0.863846i	$-0.668049\pi$
$103$	−10.2251	−1.00751	−0.503757	+	0.863846i	$0.668049\pi$
$104$	−45.3312	−4.44509
$105$	0	0
$106$	18.5544	1.80217
$107$	−7.07863	−0.684317	−0.342158	−	0.939642i	$-0.611158\pi$
$107$	−7.07863	−0.684317	−0.342158	+	0.939642i	$0.611158\pi$
$108$	0	0
$109$	−4.43367	−0.424669	−0.212335	−	0.977197i	$-0.568107\pi$
$109$	−4.43367	−0.424669	−0.212335	+	0.977197i	$0.568107\pi$
$110$	−5.26437	−0.501938
$111$	0	0
$112$	−10.7033	−1.01137
$113$	−5.39766	−0.507769	−0.253885	−	0.967234i	$-0.581708\pi$
$113$	−5.39766	−0.507769	−0.253885	+	0.967234i	$0.581708\pi$
$114$	0	0
$115$	−3.97805	−0.370955
$116$	−7.66226	−0.711423
$117$	0	0
$118$	9.68479	0.891557
$119$	−7.52411	−0.689734
$120$	0	0
$121$	11.2487	1.02260
$122$	30.8580	2.79375
$123$	0	0
$124$	46.3628	4.16351
$125$	4.15398	0.371544
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−6.08095	−0.537485
$129$	0	0
$130$	6.47154	0.567592
$131$	−6.92628	−0.605152	−0.302576	−	0.953125i	$-0.597847\pi$
$131$	−6.92628	−0.605152	−0.302576	+	0.953125i	$0.597847\pi$
$132$	0	0
$133$	3.61435	0.313404
$134$	−22.2461	−1.92177
$135$	0	0
$136$	−58.8219	−5.04393
$137$	−5.28779	−0.451766	−0.225883	−	0.974154i	$-0.572527\pi$
$137$	−5.28779	−0.451766	−0.225883	+	0.974154i	$0.572527\pi$
$138$	0	0
$139$	−15.5325	−1.31745	−0.658723	−	0.752386i	$-0.728903\pi$
$139$	−15.5325	−1.31745	−0.658723	+	0.752386i	$0.728903\pi$
$140$	2.09907	0.177403
$141$	0	0
$142$	−0.922928	−0.0774504
$143$	−27.3505	−2.28716
$144$	0	0
$145$	0.653040	0.0542320
$146$	−24.4426	−2.02288
$147$	0	0
$148$	16.0121	1.31619
$149$	16.4858	1.35057	0.675286	−	0.737556i	$-0.264020\pi$
$149$	16.4858	1.35057	0.675286	+	0.737556i	$0.264020\pi$
$150$	0	0
$151$	18.3881	1.49640	0.748200	−	0.663474i	$-0.230919\pi$
$151$	18.3881	1.49640	0.748200	+	0.663474i	$0.230919\pi$
$152$	28.2562	2.29188
$153$	0	0
$154$	−12.4463	−1.00296
$155$	−3.95142	−0.317386
$156$	0	0
$157$	−6.73860	−0.537799	−0.268899	−	0.963168i	$-0.586660\pi$
$157$	−6.73860	−0.537799	−0.268899	+	0.963168i	$0.586660\pi$
$158$	0.0957566	0.00761798
$159$	0	0
$160$	5.33244	0.421566
$161$	−9.40514	−0.741229
$162$	0	0
$163$	−2.90085	−0.227212	−0.113606	−	0.993526i	$-0.536240\pi$
$163$	−2.90085	−0.227212	−0.113606	+	0.993526i	$0.536240\pi$
$164$	45.6807	3.56707
$165$	0	0
$166$	−3.24397	−0.251781
$167$	1.96764	0.152261	0.0761303	−	0.997098i	$-0.475743\pi$
$167$	1.96764	0.152261	0.0761303	+	0.997098i	$0.475743\pi$
$168$	0	0
$169$	20.6222	1.58632
$170$	8.39749	0.644058
$171$	0	0
$172$	26.1233	1.99188
$173$	−9.45948	−0.719191	−0.359596	−	0.933108i	$-0.617085\pi$
$173$	−9.45948	−0.719191	−0.359596	+	0.933108i	$0.617085\pi$
$174$	0	0
$175$	4.82110	0.364441
$176$	−50.4859	−3.80552
$177$	0	0
$178$	9.12775	0.684154
$179$	11.6094	0.867728	0.433864	−	0.900978i	$-0.357150\pi$
$179$	11.6094	0.867728	0.433864	+	0.900978i	$0.357150\pi$
$180$	0	0
$181$	14.5801	1.08373	0.541865	−	0.840466i	$-0.317718\pi$
$181$	14.5801	1.08373	0.541865	+	0.840466i	$0.317718\pi$
$182$	15.3004	1.13414
$183$	0	0
$184$	−73.5274	−5.42051
$185$	−1.36468	−0.100334
$186$	0	0
$187$	−35.4901	−2.59529
$188$	66.8892	4.87840
$189$	0	0
$190$	−4.03389	−0.292649
$191$	−3.84995	−0.278572	−0.139286	−	0.990252i	$-0.544481\pi$
$191$	−3.84995	−0.278572	−0.139286	+	0.990252i	$0.544481\pi$
$192$	0	0
$193$	16.7391	1.20490	0.602452	−	0.798155i	$-0.294190\pi$
$193$	16.7391	1.20490	0.602452	+	0.798155i	$0.294190\pi$
$194$	30.6762	2.20242
$195$	0	0
$196$	4.96274	0.354482
$197$	13.6700	0.973946	0.486973	−	0.873417i	$-0.338101\pi$
$197$	13.6700	0.973946	0.486973	+	0.873417i	$0.338101\pi$
$198$	0	0
$199$	−21.5578	−1.52819	−0.764097	−	0.645101i	$-0.776815\pi$
$199$	−21.5578	−1.52819	−0.764097	+	0.645101i	$0.776815\pi$
$200$	37.6903	2.66511
$201$	0	0
$202$	27.7375	1.95161
$203$	1.54396	0.108365
$204$	0	0
$205$	−3.89328	−0.271919
$206$	26.9811	1.87986
$207$	0	0
$208$	62.0628	4.30328
$209$	17.0483	1.17926
$210$	0	0
$211$	4.72439	0.325240	0.162620	−	0.986689i	$-0.448006\pi$
$211$	4.72439	0.325240	0.162620	+	0.986689i	$0.448006\pi$
$212$	−34.8963	−2.39669
$213$	0	0
$214$	18.6784	1.27683
$215$	−2.22644	−0.151842
$216$	0	0
$217$	−9.34218	−0.634189
$218$	11.6991	0.792366
$219$	0	0
$220$	9.90098	0.667524
$221$	43.6283	2.93476
$222$	0	0
$223$	11.8053	0.790541	0.395271	−	0.918565i	$-0.370651\pi$
$223$	11.8053	0.790541	0.395271	+	0.918565i	$0.370651\pi$
$224$	12.6073	0.842359
$225$	0	0
$226$	14.2428	0.947418
$227$	−0.673970	−0.0447330	−0.0223665	−	0.999750i	$-0.507120\pi$
$227$	−0.673970	−0.0447330	−0.0223665	+	0.999750i	$0.507120\pi$
$228$	0	0
$229$	−10.0442	−0.663739	−0.331869	−	0.943325i	$-0.607679\pi$
$229$	−10.0442	−0.663739	−0.331869	+	0.943325i	$0.607679\pi$
$230$	10.4969	0.692144
$231$	0	0
$232$	12.0703	0.792456
$233$	5.37724	0.352275	0.176137	−	0.984366i	$-0.443640\pi$
$233$	5.37724	0.352275	0.176137	+	0.984366i	$0.443640\pi$
$234$	0	0
$235$	−5.70084	−0.371882
$236$	−18.2147	−1.18568
$237$	0	0
$238$	19.8539	1.28694
$239$	23.6194	1.52781	0.763905	−	0.645329i	$-0.223280\pi$
$239$	23.6194	1.52781	0.763905	+	0.645329i	$0.223280\pi$
$240$	0	0
$241$	1.08698	0.0700183	0.0350091	−	0.999387i	$-0.488854\pi$
$241$	1.08698	0.0700183	0.0350091	+	0.999387i	$0.488854\pi$
$242$	−29.6818	−1.90802
$243$	0	0
$244$	−58.0363	−3.71539
$245$	−0.422965	−0.0270222
$246$	0	0
$247$	−20.9577	−1.33350
$248$	−73.0352	−4.63774
$249$	0	0
$250$	−10.9611	−0.693242
$251$	28.9806	1.82924	0.914619	−	0.404317i	$-0.132491\pi$
$251$	28.9806	1.82924	0.914619	+	0.404317i	$0.132491\pi$
$252$	0	0
$253$	−44.3626	−2.78905
$254$	2.63870	0.165567
$255$	0	0
$256$	−7.67459	−0.479662
$257$	−18.0836	−1.12802	−0.564012	−	0.825767i	$-0.690743\pi$
$257$	−18.0836	−1.12802	−0.564012	+	0.825767i	$0.690743\pi$
$258$	0	0
$259$	−3.22647	−0.200483
$260$	−12.1714	−0.754836
$261$	0	0
$262$	18.2764	1.12912
$263$	−7.84072	−0.483480	−0.241740	−	0.970341i	$-0.577718\pi$
$263$	−7.84072	−0.483480	−0.241740	+	0.970341i	$0.577718\pi$
$264$	0	0
$265$	2.97414	0.182700
$266$	−9.53718	−0.584762
$267$	0	0
$268$	41.8393	2.55574
$269$	14.9176	0.909543	0.454772	−	0.890608i	$-0.349721\pi$
$269$	14.9176	0.909543	0.454772	+	0.890608i	$0.349721\pi$
$270$	0	0
$271$	13.5998	0.826129	0.413065	−	0.910702i	$-0.364458\pi$
$271$	13.5998	0.826129	0.413065	+	0.910702i	$0.364458\pi$
$272$	80.5329	4.88302
$273$	0	0
$274$	13.9529	0.842924
$275$	22.7404	1.37130
$276$	0	0
$277$	−17.8726	−1.07386	−0.536929	−	0.843627i	$-0.680416\pi$
$277$	−17.8726	−1.07386	−0.536929	+	0.843627i	$0.680416\pi$
$278$	40.9855	2.45815
$279$	0	0
$280$	−3.30665	−0.197610
$281$	1.65026	0.0984464	0.0492232	−	0.998788i	$-0.484325\pi$
$281$	1.65026	0.0984464	0.0492232	+	0.998788i	$0.484325\pi$
$282$	0	0
$283$	19.6297	1.16686	0.583431	−	0.812163i	$-0.301710\pi$
$283$	19.6297	1.16686	0.583431	+	0.812163i	$0.301710\pi$
$284$	1.73580	0.103001
$285$	0	0
$286$	72.1697	4.26749
$287$	−9.20474	−0.543339
$288$	0	0
$289$	39.6122	2.33013
$290$	−1.72318	−0.101188
$291$	0	0
$292$	45.9704	2.69021
$293$	−9.48088	−0.553879	−0.276940	−	0.960887i	$-0.589320\pi$
$293$	−9.48088	−0.553879	−0.276940	+	0.960887i	$0.589320\pi$
$294$	0	0
$295$	1.55240	0.0903844
$296$	−25.2239	−1.46611
$297$	0	0
$298$	−43.5012	−2.51996
$299$	54.5354	3.15386
$300$	0	0
$301$	−5.26388	−0.303405
$302$	−48.5206	−2.79205
$303$	0	0
$304$	−38.6855	−2.21876
$305$	4.94632	0.283226
$306$	0	0
$307$	3.89606	0.222360	0.111180	−	0.993800i	$-0.464537\pi$
$307$	3.89606	0.222360	0.111180	+	0.993800i	$0.464537\pi$
$308$	23.4085	1.33382
$309$	0	0
$310$	10.4266	0.592192
$311$	15.9109	0.902225	0.451112	−	0.892467i	$-0.351027\pi$
$311$	15.9109	0.902225	0.451112	+	0.892467i	$0.351027\pi$
$312$	0	0
$313$	−17.7480	−1.00317	−0.501587	−	0.865107i	$-0.667250\pi$
$313$	−17.7480	−1.00317	−0.501587	+	0.865107i	$0.667250\pi$
$314$	17.7811	1.00345
$315$	0	0
$316$	−0.180094	−0.0101311
$317$	−34.9760	−1.96445	−0.982225	−	0.187710i	$-0.939894\pi$
$317$	−34.9760	−1.96445	−0.982225	+	0.187710i	$0.939894\pi$
$318$	0	0
$319$	7.28261	0.407748
$320$	−5.01645	−0.280428
$321$	0	0
$322$	24.8174	1.38302
$323$	−27.1947	−1.51315
$324$	0	0
$325$	−27.9550	−1.55066
$326$	7.65447	0.423942
$327$	0	0
$328$	−71.9607	−3.97336
$329$	−13.4783	−0.743082
$330$	0	0
$331$	−14.0877	−0.774330	−0.387165	−	0.922010i	$-0.626546\pi$
$331$	−14.0877	−0.774330	−0.387165	+	0.922010i	$0.626546\pi$
$332$	6.10109	0.334841
$333$	0	0
$334$	−5.19202	−0.284094
$335$	−3.56589	−0.194825
$336$	0	0
$337$	15.3172	0.834378	0.417189	−	0.908820i	$-0.363015\pi$
$337$	15.3172	0.834378	0.417189	+	0.908820i	$0.363015\pi$
$338$	−54.4158	−2.95983
$339$	0	0
$340$	−15.7936	−0.856528
$341$	−44.0657	−2.38629
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−41.1519	−2.21876
$345$	0	0
$346$	24.9607	1.34190
$347$	10.3137	0.553666	0.276833	−	0.960918i	$-0.410715\pi$
$347$	10.3137	0.553666	0.276833	+	0.960918i	$0.410715\pi$
$348$	0	0
$349$	1.86866	0.100027	0.0500135	−	0.998749i	$-0.484074\pi$
$349$	1.86866	0.100027	0.0500135	+	0.998749i	$0.484074\pi$
$350$	−12.7214	−0.679990
$351$	0	0
$352$	59.4666	3.16958
$353$	3.03828	0.161711	0.0808557	−	0.996726i	$-0.474235\pi$
$353$	3.03828	0.161711	0.0808557	+	0.996726i	$0.474235\pi$
$354$	0	0
$355$	−0.147939	−0.00785178
$356$	−17.1670	−0.909851
$357$	0	0
$358$	−30.6338	−1.61905
$359$	9.82165	0.518367	0.259184	−	0.965828i	$-0.416547\pi$
$359$	9.82165	0.518367	0.259184	+	0.965828i	$0.416547\pi$
$360$	0	0
$361$	−5.93650	−0.312448
$362$	−38.4725	−2.02207
$363$	0	0
$364$	−28.7763	−1.50829
$365$	−3.91797	−0.205076
$366$	0	0
$367$	2.79683	0.145994	0.0729968	−	0.997332i	$-0.476744\pi$
$367$	2.79683	0.145994	0.0729968	+	0.997332i	$0.476744\pi$
$368$	100.666	5.24759
$369$	0	0
$370$	3.60099	0.187207
$371$	7.03165	0.365065
$372$	0	0
$373$	−12.5087	−0.647676	−0.323838	−	0.946113i	$-0.604973\pi$
$373$	−12.5087	−0.647676	−0.323838	+	0.946113i	$0.604973\pi$
$374$	93.6477	4.84241
$375$	0	0
$376$	−105.370	−5.43406
$377$	−8.95259	−0.461082
$378$	0	0
$379$	19.2299	0.987774	0.493887	−	0.869526i	$-0.335576\pi$
$379$	19.2299	0.987774	0.493887	+	0.869526i	$0.335576\pi$
$380$	7.58675	0.389192
$381$	0	0
$382$	10.1589	0.519773
$383$	−7.17718	−0.366737	−0.183368	−	0.983044i	$-0.558700\pi$
$383$	−7.17718	−0.366737	−0.183368	+	0.983044i	$0.558700\pi$
$384$	0	0
$385$	−1.99506	−0.101678
$386$	−44.1694	−2.24816
$387$	0	0
$388$	−57.6943	−2.92899
$389$	−22.0544	−1.11820	−0.559101	−	0.829099i	$-0.688854\pi$
$389$	−22.0544	−1.11820	−0.559101	+	0.829099i	$0.688854\pi$
$390$	0	0
$391$	70.7653	3.57876
$392$	−7.81779	−0.394858
$393$	0	0
$394$	−36.0710	−1.81723
$395$	0.0153491	0.000772297 0
$396$	0	0
$397$	4.32853	0.217243	0.108621	−	0.994083i	$-0.465356\pi$
$397$	4.32853	0.217243	0.108621	+	0.994083i	$0.465356\pi$
$398$	56.8847	2.85137
$399$	0	0
$400$	−51.6018	−2.58009
$401$	1.73045	0.0864147	0.0432073	−	0.999066i	$-0.486242\pi$
$401$	1.73045	0.0864147	0.0432073	+	0.999066i	$0.486242\pi$
$402$	0	0
$403$	54.1703	2.69842
$404$	−52.1674	−2.59543
$405$	0	0
$406$	−4.07404	−0.202191
$407$	−15.2188	−0.754367
$408$	0	0
$409$	−8.81849	−0.436046	−0.218023	−	0.975944i	$-0.569961\pi$
$409$	−8.81849	−0.436046	−0.218023	+	0.975944i	$0.569961\pi$
$410$	10.2732	0.507358
$411$	0	0
$412$	−50.7447	−2.50001
$413$	3.67029	0.180603
$414$	0	0
$415$	−0.519985	−0.0255250
$416$	−73.1029	−3.58416
$417$	0	0
$418$	−44.9854	−2.20031
$419$	24.9330	1.21806	0.609028	−	0.793149i	$-0.291560\pi$
$419$	24.9330	1.21806	0.609028	+	0.793149i	$0.291560\pi$
$420$	0	0
$421$	8.48516	0.413541	0.206771	−	0.978389i	$-0.433705\pi$
$421$	8.48516	0.413541	0.206771	+	0.978389i	$0.433705\pi$
$422$	−12.4662	−0.606847
$423$	0	0
$424$	54.9720	2.66968
$425$	−36.2745	−1.75957
$426$	0	0
$427$	11.6944	0.565932
$428$	−35.1294	−1.69804
$429$	0	0
$430$	5.87490	0.283313
$431$	34.6664	1.66982	0.834910	−	0.550387i	$-0.185520\pi$
$431$	34.6664	1.66982	0.834910	+	0.550387i	$0.185520\pi$
$432$	0	0
$433$	3.00742	0.144528	0.0722638	−	0.997386i	$-0.476978\pi$
$433$	3.00742	0.144528	0.0722638	+	0.997386i	$0.476978\pi$
$434$	24.6512	1.18330
$435$	0	0
$436$	−22.0032	−1.05376
$437$	−33.9934	−1.62613
$438$	0	0
$439$	−2.24827	−0.107304	−0.0536521	−	0.998560i	$-0.517086\pi$
$439$	−2.24827	−0.107304	−0.0536521	+	0.998560i	$0.517086\pi$
$440$	−15.5970	−0.743556
$441$	0	0
$442$	−115.122	−5.47579
$443$	−32.1187	−1.52600	−0.763002	−	0.646396i	$-0.776275\pi$
$443$	−32.1187	−1.52600	−0.763002	+	0.646396i	$0.776275\pi$
$444$	0	0
$445$	1.46311	0.0693583
$446$	−31.1507	−1.47503
$447$	0	0
$448$	−11.8602	−0.560341
$449$	8.22636	0.388226	0.194113	−	0.980979i	$-0.437817\pi$
$449$	8.22636	0.388226	0.194113	+	0.980979i	$0.437817\pi$
$450$	0	0
$451$	−43.4174	−2.04444
$452$	−26.7872	−1.25996
$453$	0	0
$454$	1.77841	0.0834647
$455$	2.45255	0.114977
$456$	0	0
$457$	−1.21215	−0.0567019	−0.0283509	−	0.999598i	$-0.509026\pi$
$457$	−1.21215	−0.0567019	−0.0283509	+	0.999598i	$0.509026\pi$
$458$	26.5036	1.23843
$459$	0	0
$460$	−19.7420	−0.920476
$461$	−15.0219	−0.699638	−0.349819	−	0.936817i	$-0.613757\pi$
$461$	−15.0219	−0.699638	−0.349819	+	0.936817i	$0.613757\pi$
$462$	0	0
$463$	27.3058	1.26901	0.634504	−	0.772919i	$-0.281204\pi$
$463$	27.3058	1.26901	0.634504	+	0.772919i	$0.281204\pi$
$464$	−16.5255	−0.767176
$465$	0	0
$466$	−14.1889	−0.657289
$467$	−37.0583	−1.71485	−0.857426	−	0.514608i	$-0.827938\pi$
$467$	−37.0583	−1.71485	−0.857426	+	0.514608i	$0.827938\pi$
$468$	0	0
$469$	−8.43069	−0.389293
$470$	15.0428	0.693873
$471$	0	0
$472$	28.6935	1.32073
$473$	−24.8289	−1.14163
$474$	0	0
$475$	17.4251	0.799520
$476$	−37.3402	−1.71149
$477$	0	0
$478$	−62.3244	−2.85065
$479$	−10.1179	−0.462301	−0.231150	−	0.972918i	$-0.574249\pi$
$479$	−10.1179	−0.462301	−0.231150	+	0.972918i	$0.574249\pi$
$480$	0	0
$481$	18.7086	0.853038
$482$	−2.86820	−0.130643
$483$	0	0
$484$	55.8241	2.53746
$485$	4.91718	0.223278
$486$	0	0
$487$	30.8841	1.39949	0.699746	−	0.714392i	$-0.253297\pi$
$487$	30.8841	1.39949	0.699746	+	0.714392i	$0.253297\pi$
$488$	91.4243	4.13858
$489$	0	0
$490$	1.11608	0.0504193
$491$	−11.8119	−0.533063	−0.266531	−	0.963826i	$-0.585878\pi$
$491$	−11.8119	−0.533063	−0.266531	+	0.963826i	$0.585878\pi$
$492$	0	0
$493$	−11.6169	−0.523199
$494$	55.3010	2.48811
$495$	0	0
$496$	99.9924	4.48979
$497$	−0.349766	−0.0156892
$498$	0	0
$499$	−15.1554	−0.678447	−0.339223	−	0.940706i	$-0.610164\pi$
$499$	−15.1554	−0.678447	−0.339223	+	0.940706i	$0.610164\pi$
$500$	20.6151	0.921937
$501$	0	0
$502$	−76.4711	−3.41307
$503$	−23.6479	−1.05441	−0.527205	−	0.849738i	$-0.676760\pi$
$503$	−23.6479	−1.05441	−0.527205	+	0.849738i	$0.676760\pi$
$504$	0	0
$505$	4.44613	0.197850
$506$	117.060	5.20394
$507$	0	0
$508$	−4.96274	−0.220186
$509$	−43.0864	−1.90977	−0.954886	−	0.296972i	$-0.904023\pi$
$509$	−43.0864	−1.90977	−0.954886	+	0.296972i	$0.904023\pi$
$510$	0	0
$511$	−9.26310	−0.409775
$512$	32.4128	1.43246
$513$	0	0
$514$	47.7172	2.10471
$515$	4.32488	0.190577
$516$	0	0
$517$	−63.5750	−2.79602
$518$	8.51369	0.374070
$519$	0	0
$520$	19.1735	0.840814
$521$	24.9343	1.09239	0.546195	−	0.837658i	$-0.316076\pi$
$521$	24.9343	1.09239	0.546195	+	0.837658i	$0.316076\pi$
$522$	0	0
$523$	19.5892	0.856577	0.428288	−	0.903642i	$-0.359117\pi$
$523$	19.5892	0.856577	0.428288	+	0.903642i	$0.359117\pi$
$524$	−34.3733	−1.50161
$525$	0	0
$526$	20.6893	0.902097
$527$	70.2916	3.06195
$528$	0	0
$529$	65.4567	2.84594
$530$	−7.84788	−0.340890
$531$	0	0
$532$	17.9371	0.777670
$533$	53.3734	2.31186
$534$	0	0
$535$	2.99401	0.129442
$536$	−65.9093	−2.84685
$537$	0	0
$538$	−39.3631	−1.69707
$539$	−4.71685	−0.203169
$540$	0	0
$541$	36.9979	1.59066	0.795332	−	0.606174i	$-0.207297\pi$
$541$	36.9979	1.59066	0.795332	+	0.606174i	$0.207297\pi$
$542$	−35.8858	−1.54143
$543$	0	0
$544$	−94.8585	−4.06702
$545$	1.87529	0.0803286
$546$	0	0
$547$	−12.3314	−0.527251	−0.263625	−	0.964625i	$-0.584918\pi$
$547$	−12.3314	−0.527251	−0.263625	+	0.964625i	$0.584918\pi$
$548$	−26.2419	−1.12100
$549$	0	0
$550$	−60.0051	−2.55863
$551$	5.58040	0.237733
$552$	0	0
$553$	0.0362893	0.00154318
$554$	47.1603	2.00365
$555$	0	0
$556$	−77.0835	−3.26907
$557$	11.5262	0.488380	0.244190	−	0.969727i	$-0.421478\pi$
$557$	11.5262	0.488380	0.244190	+	0.969727i	$0.421478\pi$
$558$	0	0
$559$	30.5224	1.29096
$560$	4.52713	0.191306
$561$	0	0
$562$	−4.35455	−0.183686
$563$	10.0097	0.421859	0.210929	−	0.977501i	$-0.432351\pi$
$563$	10.0097	0.421859	0.210929	+	0.977501i	$0.432351\pi$
$564$	0	0
$565$	2.28302	0.0960475
$566$	−51.7968	−2.17718
$567$	0	0
$568$	−2.73440	−0.114733
$569$	−16.4475	−0.689514	−0.344757	−	0.938692i	$-0.612039\pi$
$569$	−16.4475	−0.689514	−0.344757	+	0.938692i	$0.612039\pi$
$570$	0	0
$571$	10.2765	0.430059	0.215030	−	0.976608i	$-0.431015\pi$
$571$	10.2765	0.430059	0.215030	+	0.976608i	$0.431015\pi$
$572$	−135.733	−5.67530
$573$	0	0
$574$	24.2886	1.01378
$575$	−45.3431	−1.89094
$576$	0	0
$577$	26.8471	1.11766	0.558829	−	0.829283i	$-0.311251\pi$
$577$	26.8471	1.11766	0.558829	+	0.829283i	$0.311251\pi$
$578$	−104.525	−4.34765
$579$	0	0
$580$	3.24087	0.134570
$581$	−1.22938	−0.0510033
$582$	0	0
$583$	33.1672	1.37365
$584$	−72.4170	−2.99663
$585$	0	0
$586$	25.0172	1.03345
$587$	36.5029	1.50663	0.753317	−	0.657657i	$-0.228453\pi$
$587$	36.5029	1.50663	0.753317	+	0.657657i	$0.228453\pi$
$588$	0	0
$589$	−33.7659	−1.39130
$590$	−4.09633	−0.168643
$591$	0	0
$592$	34.5339	1.41934
$593$	−36.3039	−1.49082	−0.745412	−	0.666604i	$-0.767747\pi$
$593$	−36.3039	−1.49082	−0.745412	+	0.666604i	$0.767747\pi$
$594$	0	0
$595$	3.18243	0.130467
$596$	81.8149	3.35127
$597$	0	0
$598$	−143.903	−5.88461
$599$	−9.43126	−0.385351	−0.192675	−	0.981263i	$-0.561716\pi$
$599$	−9.43126	−0.385351	−0.192675	+	0.981263i	$0.561716\pi$
$600$	0	0
$601$	3.22289	0.131465	0.0657323	−	0.997837i	$-0.479062\pi$
$601$	3.22289	0.131465	0.0657323	+	0.997837i	$0.479062\pi$
$602$	13.8898	0.566106
$603$	0	0
$604$	91.2552	3.71312
$605$	−4.75779	−0.193432
$606$	0	0
$607$	30.0793	1.22088	0.610441	−	0.792062i	$-0.290992\pi$
$607$	30.0793	1.22088	0.610441	+	0.792062i	$0.290992\pi$
$608$	45.5671	1.84799
$609$	0	0
$610$	−13.0519	−0.528455
$611$	78.1533	3.16175
$612$	0	0
$613$	5.63813	0.227722	0.113861	−	0.993497i	$-0.463678\pi$
$613$	5.63813	0.227722	0.113861	+	0.993497i	$0.463678\pi$
$614$	−10.2805	−0.414889
$615$	0	0
$616$	−36.8753	−1.48575
$617$	40.3851	1.62584	0.812921	−	0.582374i	$-0.197876\pi$
$617$	40.3851	1.62584	0.812921	+	0.582374i	$0.197876\pi$
$618$	0	0
$619$	−1.11322	−0.0447441	−0.0223721	−	0.999750i	$-0.507122\pi$
$619$	−1.11322	−0.0447441	−0.0223721	+	0.999750i	$0.507122\pi$
$620$	−19.6099	−0.787551
$621$	0	0
$622$	−41.9841	−1.68341
$623$	3.45918	0.138589
$624$	0	0
$625$	22.3485	0.893941
$626$	46.8316	1.87177
$627$	0	0
$628$	−33.4419	−1.33448
$629$	24.2763	0.967960
$630$	0	0
$631$	−31.0627	−1.23659	−0.618294	−	0.785947i	$-0.712176\pi$
$631$	−31.0627	−1.23659	−0.618294	+	0.785947i	$0.712176\pi$
$632$	0.283702	0.0112851
$633$	0	0
$634$	92.2912	3.66535
$635$	0.422965	0.0167849
$636$	0	0
$637$	5.79847	0.229744
$638$	−19.2166	−0.760794
$639$	0	0
$640$	2.57203	0.101668
$641$	−12.1741	−0.480847	−0.240423	−	0.970668i	$-0.577286\pi$
$641$	−12.1741	−0.480847	−0.240423	+	0.970668i	$0.577286\pi$
$642$	0	0
$643$	−22.0575	−0.869863	−0.434931	−	0.900464i	$-0.643227\pi$
$643$	−22.0575	−0.869863	−0.434931	+	0.900464i	$0.643227\pi$
$644$	−46.6753	−1.83926
$645$	0	0
$646$	71.7587	2.82331
$647$	42.4044	1.66709	0.833544	−	0.552452i	$-0.186308\pi$
$647$	42.4044	1.66709	0.833544	+	0.552452i	$0.186308\pi$
$648$	0	0
$649$	17.3122	0.679563
$650$	73.7648	2.89329
$651$	0	0
$652$	−14.3962	−0.563797
$653$	−12.4280	−0.486345	−0.243172	−	0.969983i	$-0.578188\pi$
$653$	−12.4280	−0.486345	−0.243172	+	0.969983i	$0.578188\pi$
$654$	0	0
$655$	2.92957	0.114468
$656$	98.5213	3.84661
$657$	0	0
$658$	35.5651	1.38647
$659$	25.4052	0.989646	0.494823	−	0.868994i	$-0.335233\pi$
$659$	25.4052	0.989646	0.494823	+	0.868994i	$0.335233\pi$
$660$	0	0
$661$	−27.6805	−1.07665	−0.538323	−	0.842739i	$-0.680942\pi$
$661$	−27.6805	−1.07665	−0.538323	+	0.842739i	$0.680942\pi$
$662$	37.1732	1.44478
$663$	0	0
$664$	−9.61103	−0.372980
$665$	−1.52874	−0.0592821
$666$	0	0
$667$	−14.5211	−0.562261
$668$	9.76489	0.377815
$669$	0	0
$670$	9.40931	0.363513
$671$	55.1607	2.12946
$672$	0	0
$673$	15.6461	0.603114	0.301557	−	0.953448i	$-0.402494\pi$
$673$	15.6461	0.603114	0.301557	+	0.953448i	$0.402494\pi$
$674$	−40.4174	−1.55682
$675$	0	0
$676$	102.343	3.93626
$677$	0.906064	0.0348229	0.0174114	−	0.999848i	$-0.494457\pi$
$677$	0.906064	0.0348229	0.0174114	+	0.999848i	$0.494457\pi$
$678$	0	0
$679$	11.6255	0.446146
$680$	24.8796	0.954089
$681$	0	0
$682$	116.276	4.45244
$683$	−41.3193	−1.58104	−0.790519	−	0.612438i	$-0.790189\pi$
$683$	−41.3193	−1.58104	−0.790519	+	0.612438i	$0.790189\pi$
$684$	0	0
$685$	2.23655	0.0854541
$686$	2.63870	0.100746
$687$	0	0
$688$	56.3410	2.14798
$689$	−40.7728	−1.55332
$690$	0	0
$691$	−10.7037	−0.407188	−0.203594	−	0.979055i	$-0.565262\pi$
$691$	−10.7037	−0.407188	−0.203594	+	0.979055i	$0.565262\pi$
$692$	−46.9450	−1.78458
$693$	0	0
$694$	−27.2147	−1.03305
$695$	6.56968	0.249202
$696$	0	0
$697$	69.2575	2.62331
$698$	−4.93083	−0.186635
$699$	0	0
$700$	23.9259	0.904313
$701$	−43.4653	−1.64166	−0.820831	−	0.571172i	$-0.806489\pi$
$701$	−43.4653	−1.64166	−0.820831	+	0.571172i	$0.806489\pi$
$702$	0	0
$703$	−11.6616	−0.439825
$704$	−55.9427	−2.10842
$705$	0	0
$706$	−8.01712	−0.301728
$707$	10.5118	0.395338
$708$	0	0
$709$	−40.0569	−1.50437	−0.752185	−	0.658952i	$-0.771000\pi$
$709$	−40.0569	−1.50437	−0.752185	+	0.658952i	$0.771000\pi$
$710$	0.390366	0.0146502
$711$	0	0
$712$	27.0432	1.01349
$713$	87.8646	3.29055
$714$	0	0
$715$	11.5683	0.432630
$716$	57.6145	2.15316
$717$	0	0
$718$	−25.9164	−0.967192
$719$	32.4491	1.21015	0.605074	−	0.796169i	$-0.293143\pi$
$719$	32.4491	1.21015	0.605074	+	0.796169i	$0.293143\pi$
$720$	0	0
$721$	10.2251	0.380804
$722$	15.6647	0.582978
$723$	0	0
$724$	72.3572	2.68914
$725$	7.44358	0.276447
$726$	0	0
$727$	36.3263	1.34727	0.673635	−	0.739065i	$-0.264732\pi$
$727$	36.3263	1.34727	0.673635	+	0.739065i	$0.264732\pi$
$728$	45.3312	1.68008
$729$	0	0
$730$	10.3383	0.382639
$731$	39.6060	1.46488
$732$	0	0
$733$	−1.30611	−0.0482424	−0.0241212	−	0.999709i	$-0.507679\pi$
$733$	−1.30611	−0.0482424	−0.0241212	+	0.999709i	$0.507679\pi$
$734$	−7.38001	−0.272401
$735$	0	0
$736$	−118.573	−4.37067
$737$	−39.7663	−1.46481
$738$	0	0
$739$	−28.9135	−1.06360	−0.531801	−	0.846870i	$-0.678484\pi$
$739$	−28.9135	−1.06360	−0.531801	+	0.846870i	$0.678484\pi$
$740$	−6.77258	−0.248965
$741$	0	0
$742$	−18.5544	−0.681155
$743$	−25.4690	−0.934367	−0.467184	−	0.884160i	$-0.654731\pi$
$743$	−25.4690	−0.934367	−0.467184	+	0.884160i	$0.654731\pi$
$744$	0	0
$745$	−6.97293	−0.255469
$746$	33.0067	1.20846
$747$	0	0
$748$	−176.128	−6.43988
$749$	7.07863	0.258647
$750$	0	0
$751$	−49.2836	−1.79838	−0.899191	−	0.437555i	$-0.855844\pi$
$751$	−49.2836	−1.79838	−0.899191	+	0.437555i	$0.855844\pi$
$752$	144.262	5.26070
$753$	0	0
$754$	23.6232	0.860306
$755$	−7.77751	−0.283052
$756$	0	0
$757$	39.9940	1.45361	0.726804	−	0.686845i	$-0.241005\pi$
$757$	39.9940	1.45361	0.726804	+	0.686845i	$0.241005\pi$
$758$	−50.7420	−1.84303
$759$	0	0
$760$	−11.9514	−0.433522
$761$	10.8991	0.395092	0.197546	−	0.980294i	$-0.436703\pi$
$761$	10.8991	0.395092	0.197546	+	0.980294i	$0.436703\pi$
$762$	0	0
$763$	4.43367	0.160510
$764$	−19.1063	−0.691242
$765$	0	0
$766$	18.9384	0.684273
$767$	−21.2820	−0.768450
$768$	0	0
$769$	−39.3464	−1.41887	−0.709433	−	0.704773i	$-0.751049\pi$
$769$	−39.3464	−1.41887	−0.709433	+	0.704773i	$0.751049\pi$
$770$	5.26437	0.189715
$771$	0	0
$772$	83.0716	2.98981
$773$	8.66945	0.311818	0.155909	−	0.987771i	$-0.450169\pi$
$773$	8.66945	0.311818	0.155909	+	0.987771i	$0.450169\pi$
$774$	0	0
$775$	−45.0396	−1.61787
$776$	90.8857	3.26261
$777$	0	0
$778$	58.1950	2.08639
$779$	−33.2691	−1.19199
$780$	0	0
$781$	−1.64979	−0.0590342
$782$	−186.728	−6.67740
$783$	0	0
$784$	10.7033	0.382261
$785$	2.85019	0.101728
$786$	0	0
$787$	22.2895	0.794533	0.397267	−	0.917703i	$-0.369959\pi$
$787$	22.2895	0.794533	0.397267	+	0.917703i	$0.369959\pi$
$788$	67.8406	2.41672
$789$	0	0
$790$	−0.0405017	−0.00144099
$791$	5.39766	0.191919
$792$	0	0
$793$	−67.8096	−2.40799
$794$	−11.4217	−0.405341
$795$	0	0
$796$	−106.986	−3.79202
$797$	24.7808	0.877780	0.438890	−	0.898541i	$-0.355372\pi$
$797$	24.7808	0.877780	0.438890	+	0.898541i	$0.355372\pi$
$798$	0	0
$799$	101.412	3.58770
$800$	60.7809	2.14893
$801$	0	0
$802$	−4.56615	−0.161236
$803$	−43.6926	−1.54188
$804$	0	0
$805$	3.97805	0.140208
$806$	−142.939	−5.03482
$807$	0	0
$808$	82.1791	2.89105
$809$	−13.0611	−0.459202	−0.229601	−	0.973285i	$-0.573742\pi$
$809$	−13.0611	−0.459202	−0.229601	+	0.973285i	$0.573742\pi$
$810$	0	0
$811$	−30.9524	−1.08689	−0.543443	−	0.839446i	$-0.682880\pi$
$811$	−30.9524	−1.08689	−0.543443	+	0.839446i	$0.682880\pi$
$812$	7.66226	0.268893
$813$	0	0
$814$	40.1578	1.40753
$815$	1.22696	0.0429784
$816$	0	0
$817$	−19.0255	−0.665617
$818$	23.2693	0.813594
$819$	0	0
$820$	−19.3214	−0.674731
$821$	−2.32269	−0.0810624	−0.0405312	−	0.999178i	$-0.512905\pi$
$821$	−2.32269	−0.0810624	−0.0405312	+	0.999178i	$0.512905\pi$
$822$	0	0
$823$	24.6050	0.857677	0.428839	−	0.903381i	$-0.358923\pi$
$823$	24.6050	0.857677	0.428839	+	0.903381i	$0.358923\pi$
$824$	79.9380	2.78477
$825$	0	0
$826$	−9.68479	−0.336977
$827$	19.0726	0.663219	0.331609	−	0.943417i	$-0.392408\pi$
$827$	19.0726	0.663219	0.331609	+	0.943417i	$0.392408\pi$
$828$	0	0
$829$	3.83916	0.133340	0.0666698	−	0.997775i	$-0.478763\pi$
$829$	3.83916	0.133340	0.0666698	+	0.997775i	$0.478763\pi$
$830$	1.37208	0.0476257
$831$	0	0
$832$	68.7709	2.38420
$833$	7.52411	0.260695
$834$	0	0
$835$	−0.832244	−0.0288010
$836$	84.6064	2.92617
$837$	0	0
$838$	−65.7907	−2.27270
$839$	−51.1218	−1.76492	−0.882460	−	0.470388i	$-0.844114\pi$
$839$	−51.1218	−1.76492	−0.882460	+	0.470388i	$0.844114\pi$
$840$	0	0
$841$	−26.6162	−0.917800
$842$	−22.3898	−0.771603
$843$	0	0
$844$	23.4459	0.807041
$845$	−8.72247	−0.300062
$846$	0	0
$847$	−11.2487	−0.386508
$848$	−75.2620	−2.58451
$849$	0	0
$850$	95.7175	3.28308
$851$	30.3454	1.04023
$852$	0	0
$853$	40.8795	1.39969	0.699844	−	0.714295i	$-0.253253\pi$
$853$	40.8795	1.39969	0.699844	+	0.714295i	$0.253253\pi$
$854$	−30.8580	−1.05594
$855$	0	0
$856$	55.3392	1.89145
$857$	−18.9221	−0.646366	−0.323183	−	0.946336i	$-0.604753\pi$
$857$	−18.9221	−0.646366	−0.323183	+	0.946336i	$0.604753\pi$
$858$	0	0
$859$	55.5994	1.89703	0.948513	−	0.316739i	$-0.102588\pi$
$859$	55.5994	1.89703	0.948513	+	0.316739i	$0.102588\pi$
$860$	−11.0492	−0.376776
$861$	0	0
$862$	−91.4741	−3.11562
$863$	21.2207	0.722360	0.361180	−	0.932496i	$-0.382374\pi$
$863$	21.2207	0.722360	0.361180	+	0.932496i	$0.382374\pi$
$864$	0	0
$865$	4.00103	0.136039
$866$	−7.93569	−0.269666
$867$	0	0
$868$	−46.3628	−1.57366
$869$	0.171171	0.00580658
$870$	0	0
$871$	48.8850	1.65641
$872$	34.6615	1.17379
$873$	0	0
$874$	89.6985	3.03410
$875$	−4.15398	−0.140430
$876$	0	0
$877$	−33.8670	−1.14361	−0.571804	−	0.820390i	$-0.693756\pi$
$877$	−33.8670	−1.14361	−0.571804	+	0.820390i	$0.693756\pi$
$878$	5.93252	0.200213
$879$	0	0
$880$	21.3538	0.719836
$881$	−35.0212	−1.17989	−0.589946	−	0.807442i	$-0.700851\pi$
$881$	−35.0212	−1.17989	−0.589946	+	0.807442i	$0.700851\pi$
$882$	0	0
$883$	−9.81337	−0.330246	−0.165123	−	0.986273i	$-0.552802\pi$
$883$	−9.81337	−0.330246	−0.165123	+	0.986273i	$0.552802\pi$
$884$	216.516	7.28222
$885$	0	0
$886$	84.7515	2.84728
$887$	−11.1752	−0.375225	−0.187613	−	0.982243i	$-0.560075\pi$
$887$	−11.1752	−0.375225	−0.187613	+	0.982243i	$0.560075\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−3.86072	−0.129412
$891$	0	0
$892$	58.5866	1.96163
$893$	−48.7152	−1.63019
$894$	0	0
$895$	−4.91038	−0.164136
$896$	6.08095	0.203150
$897$	0	0
$898$	−21.7069	−0.724369
$899$	−14.4239	−0.481065
$900$	0	0
$901$	−52.9069	−1.76259
$902$	114.565	3.81461
$903$	0	0
$904$	42.1978	1.40348
$905$	−6.16687	−0.204994
$906$	0	0
$907$	34.3028	1.13901	0.569503	−	0.821989i	$-0.307136\pi$
$907$	34.3028	1.13901	0.569503	+	0.821989i	$0.307136\pi$
$908$	−3.34474	−0.110999
$909$	0	0
$910$	−6.47154	−0.214530
$911$	5.84171	0.193544	0.0967722	−	0.995307i	$-0.469148\pi$
$911$	5.84171	0.193544	0.0967722	+	0.995307i	$0.469148\pi$
$912$	0	0
$913$	−5.79880	−0.191912
$914$	3.19849	0.105797
$915$	0	0
$916$	−49.8467	−1.64698
$917$	6.92628	0.228726
$918$	0	0
$919$	13.1506	0.433799	0.216899	−	0.976194i	$-0.430406\pi$
$919$	13.1506	0.433799	0.216899	+	0.976194i	$0.430406\pi$
$920$	31.0995	1.02532
$921$	0	0
$922$	39.6382	1.30542
$923$	2.02811	0.0667559
$924$	0	0
$925$	−15.5551	−0.511450
$926$	−72.0519	−2.36777
$927$	0	0
$928$	19.4651	0.638973
$929$	7.89138	0.258908	0.129454	−	0.991585i	$-0.458678\pi$
$929$	7.89138	0.258908	0.129454	+	0.991585i	$0.458678\pi$
$930$	0	0
$931$	−3.61435	−0.118455
$932$	26.6858	0.874124
$933$	0	0
$934$	97.7857	3.19964
$935$	15.0111	0.490914
$936$	0	0
$937$	53.5894	1.75069	0.875345	−	0.483498i	$-0.160634\pi$
$937$	53.5894	1.75069	0.875345	+	0.483498i	$0.160634\pi$
$938$	22.2461	0.726360
$939$	0	0
$940$	−28.2918	−0.922777
$941$	−29.7963	−0.971333	−0.485666	−	0.874144i	$-0.661423\pi$
$941$	−29.7963	−0.971333	−0.485666	+	0.874144i	$0.661423\pi$
$942$	0	0
$943$	86.5719	2.81917
$944$	−39.2843	−1.27859
$945$	0	0
$946$	65.5161	2.13011
$947$	12.4975	0.406115	0.203057	−	0.979167i	$-0.434912\pi$
$947$	12.4975	0.406115	0.203057	+	0.979167i	$0.434912\pi$
$948$	0	0
$949$	53.7118	1.74356
$950$	−45.9797	−1.49178
$951$	0	0
$952$	58.8219	1.90643
$953$	40.0886	1.29860	0.649298	−	0.760534i	$-0.275063\pi$
$953$	40.0886	1.29860	0.649298	+	0.760534i	$0.275063\pi$
$954$	0	0
$955$	1.62839	0.0526936
$956$	117.217	3.79106
$957$	0	0
$958$	26.6982	0.862580
$959$	5.28779	0.170751
$960$	0	0
$961$	56.2764	1.81537
$962$	−49.3664	−1.59164
$963$	0	0
$964$	5.39438	0.173741
$965$	−7.08004	−0.227915
$966$	0	0
$967$	−25.5952	−0.823086	−0.411543	−	0.911390i	$-0.635010\pi$
$967$	−25.5952	−0.823086	−0.411543	+	0.911390i	$0.635010\pi$
$968$	−87.9396	−2.82648
$969$	0	0
$970$	−12.9750	−0.416601
$971$	27.5137	0.882956	0.441478	−	0.897272i	$-0.354454\pi$
$971$	27.5137	0.882956	0.441478	+	0.897272i	$0.354454\pi$
$972$	0	0
$973$	15.5325	0.497947
$974$	−81.4939	−2.61123
$975$	0	0
$976$	−125.169	−4.00656
$977$	28.3823	0.908031	0.454015	−	0.890994i	$-0.349991\pi$
$977$	28.3823	0.908031	0.454015	+	0.890994i	$0.349991\pi$
$978$	0	0
$979$	16.3164	0.521476
$980$	−2.09907	−0.0670522
$981$	0	0
$982$	31.1680	0.994611
$983$	−22.2818	−0.710680	−0.355340	−	0.934737i	$-0.615635\pi$
$983$	−22.2818	−0.710680	−0.355340	+	0.934737i	$0.615635\pi$
$984$	0	0
$985$	−5.78193	−0.184227
$986$	30.6535	0.976207
$987$	0	0
$988$	−104.007	−3.30892
$989$	49.5075	1.57425
$990$	0	0
$991$	27.9979	0.889382	0.444691	−	0.895684i	$-0.353313\pi$
$991$	27.9979	0.889382	0.444691	+	0.895684i	$0.353313\pi$
$992$	−117.779	−3.73950
$993$	0	0
$994$	0.922928	0.0292735
$995$	9.11821	0.289067
$996$	0	0
$997$	21.6924	0.687006	0.343503	−	0.939152i	$-0.388386\pi$
$997$	21.6924	0.687006	0.343503	+	0.939152i	$0.388386\pi$
$998$	39.9904	1.26588
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.2		16
3.2	odd	2		2667.2.a.n.1.15	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.15	✓	16	3.2	odd	2
8001.2.a.s.1.2		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)

\(f(q)\)	\(=\)	\(q-2.63870 q^{2} +4.96274 q^{4} -0.422965 q^{5} -1.00000 q^{7} -7.81779 q^{8} +O(q^{10})\)	\(q-2.63870 q^{2} +4.96274 q^{4} -0.422965 q^{5} -1.00000 q^{7} -7.81779 q^{8} +1.11608 q^{10} -4.71685 q^{11} +5.79847 q^{13} +2.63870 q^{14} +10.7033 q^{16} +7.52411 q^{17} -3.61435 q^{19} -2.09907 q^{20} +12.4463 q^{22} +9.40514 q^{23} -4.82110 q^{25} -15.3004 q^{26} -4.96274 q^{28} -1.54396 q^{29} +9.34218 q^{31} -12.6073 q^{32} -19.8539 q^{34} +0.422965 q^{35} +3.22647 q^{37} +9.53718 q^{38} +3.30665 q^{40} +9.20474 q^{41} +5.26388 q^{43} -23.4085 q^{44} -24.8174 q^{46} +13.4783 q^{47} +1.00000 q^{49} +12.7214 q^{50} +28.7763 q^{52} -7.03165 q^{53} +1.99506 q^{55} +7.81779 q^{56} +4.07404 q^{58} -3.67029 q^{59} -11.6944 q^{61} -24.6512 q^{62} +11.8602 q^{64} -2.45255 q^{65} +8.43069 q^{67} +37.3402 q^{68} -1.11608 q^{70} +0.349766 q^{71} +9.26310 q^{73} -8.51369 q^{74} -17.9371 q^{76} +4.71685 q^{77} -0.0362893 q^{79} -4.52713 q^{80} -24.2886 q^{82} +1.22938 q^{83} -3.18243 q^{85} -13.8898 q^{86} +36.8753 q^{88} -3.45918 q^{89} -5.79847 q^{91} +46.6753 q^{92} -35.5651 q^{94} +1.52874 q^{95} -11.6255 q^{97} -2.63870 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) 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

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Groups

Database

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