LMFDB - Embedded newform 8001.2.a.n.1.1

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} + \cdots - 1 $ x^12 - 5x^11 - 3x^10 + 41x^9 - 11x^8 - 123x^7 + 44x^6 + 159x^5 - 39x^4 - 71x^3 + 16x^2 + 7*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.98033$ 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)

$f(q)$	$=$	$q-1.98033 q^{2} +1.92172 q^{4} +1.61478 q^{5} +1.00000 q^{7} +0.155028 q^{8} +O(q^{10})$	\(q-1.98033 q^{2} +1.92172 q^{4} +1.61478 q^{5} +1.00000 q^{7} +0.155028 q^{8} -3.19781 q^{10} +3.80491 q^{11} +3.75846 q^{13} -1.98033 q^{14} -4.15044 q^{16} +1.43488 q^{17} -3.83393 q^{19} +3.10315 q^{20} -7.53499 q^{22} +7.74028 q^{23} -2.39247 q^{25} -7.44300 q^{26} +1.92172 q^{28} +1.46417 q^{29} +7.16135 q^{31} +7.90919 q^{32} -2.84155 q^{34} +1.61478 q^{35} +1.16907 q^{37} +7.59245 q^{38} +0.250337 q^{40} +2.32603 q^{41} -8.53051 q^{43} +7.31196 q^{44} -15.3283 q^{46} +5.68514 q^{47} +1.00000 q^{49} +4.73789 q^{50} +7.22270 q^{52} -7.44663 q^{53} +6.14411 q^{55} +0.155028 q^{56} -2.89955 q^{58} -0.528711 q^{59} +3.01696 q^{61} -14.1819 q^{62} -7.36195 q^{64} +6.06910 q^{65} +4.85329 q^{67} +2.75744 q^{68} -3.19781 q^{70} -12.6882 q^{71} +1.64984 q^{73} -2.31515 q^{74} -7.36772 q^{76} +3.80491 q^{77} +1.31624 q^{79} -6.70206 q^{80} -4.60631 q^{82} +6.14878 q^{83} +2.31703 q^{85} +16.8932 q^{86} +0.589870 q^{88} +17.9538 q^{89} +3.75846 q^{91} +14.8746 q^{92} -11.2585 q^{94} -6.19096 q^{95} +12.1396 q^{97} -1.98033 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * 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 $

$2$	−1.98033	−1.40031	−0.700153	−	0.713993i	$-0.746885\pi$
$2$	−1.98033	−1.40031	−0.700153	+	0.713993i	$0.746885\pi$
$3$	0	0
$3$	0	0
$4$	1.92172	0.960858
$5$	1.61478	0.722153	0.361077	−	0.932536i	$-0.382409\pi$
$5$	1.61478	0.722153	0.361077	+	0.932536i	$0.382409\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.155028	0.0548108
$9$	0	0
$10$	−3.19781	−1.01124
$11$	3.80491	1.14722	0.573612	−	0.819127i	$-0.305542\pi$
$11$	3.80491	1.14722	0.573612	+	0.819127i	$0.305542\pi$
$12$	0	0
$13$	3.75846	1.04241	0.521205	−	0.853432i	$-0.325483\pi$
$13$	3.75846	1.04241	0.521205	+	0.853432i	$0.325483\pi$
$14$	−1.98033	−0.529266
$15$	0	0
$16$	−4.15044	−1.03761
$17$	1.43488	0.348011	0.174005	−	0.984745i	$-0.444329\pi$
$17$	1.43488	0.348011	0.174005	+	0.984745i	$0.444329\pi$
$18$	0	0
$19$	−3.83393	−0.879563	−0.439782	−	0.898105i	$-0.644944\pi$
$19$	−3.83393	−0.879563	−0.439782	+	0.898105i	$0.644944\pi$
$20$	3.10315	0.693887
$21$	0	0
$22$	−7.53499	−1.60647
$23$	7.74028	1.61396	0.806980	−	0.590578i	$-0.201100\pi$
$23$	7.74028	1.61396	0.806980	+	0.590578i	$0.201100\pi$
$24$	0	0
$25$	−2.39247	−0.478495
$26$	−7.44300	−1.45969
$27$	0	0
$28$	1.92172	0.363170
$29$	1.46417	0.271890	0.135945	−	0.990716i	$-0.456593\pi$
$29$	1.46417	0.271890	0.135945	+	0.990716i	$0.456593\pi$
$30$	0	0
$31$	7.16135	1.28622	0.643108	−	0.765775i	$-0.277645\pi$
$31$	7.16135	1.28622	0.643108	+	0.765775i	$0.277645\pi$
$32$	7.90919	1.39816
$33$	0	0
$34$	−2.84155	−0.487321
$35$	1.61478	0.272948
$36$	0	0
$37$	1.16907	0.192194	0.0960972	−	0.995372i	$-0.469364\pi$
$37$	1.16907	0.192194	0.0960972	+	0.995372i	$0.469364\pi$
$38$	7.59245	1.23166
$39$	0	0
$40$	0.250337	0.0395818
$41$	2.32603	0.363265	0.181632	−	0.983367i	$-0.441862\pi$
$41$	2.32603	0.363265	0.181632	+	0.983367i	$0.441862\pi$
$42$	0	0
$43$	−8.53051	−1.30089	−0.650445	−	0.759553i	$-0.725418\pi$
$43$	−8.53051	−1.30089	−0.650445	+	0.759553i	$0.725418\pi$
$44$	7.31196	1.10232
$45$	0	0
$46$	−15.3283	−2.26004
$47$	5.68514	0.829263	0.414632	−	0.909989i	$-0.363910\pi$
$47$	5.68514	0.829263	0.414632	+	0.909989i	$0.363910\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.73789	0.670039
$51$	0	0
$52$	7.22270	1.00161
$53$	−7.44663	−1.02287	−0.511437	−	0.859321i	$-0.670887\pi$
$53$	−7.44663	−1.02287	−0.511437	+	0.859321i	$0.670887\pi$
$54$	0	0
$55$	6.14411	0.828472
$56$	0.155028	0.0207165
$57$	0	0
$58$	−2.89955	−0.380729
$59$	−0.528711	−0.0688323	−0.0344161	−	0.999408i	$-0.510957\pi$
$59$	−0.528711	−0.0688323	−0.0344161	+	0.999408i	$0.510957\pi$
$60$	0	0
$61$	3.01696	0.386282	0.193141	−	0.981171i	$-0.438133\pi$
$61$	3.01696	0.386282	0.193141	+	0.981171i	$0.438133\pi$
$62$	−14.1819	−1.80110
$63$	0	0
$64$	−7.36195	−0.920244
$65$	6.06910	0.752780
$66$	0	0
$67$	4.85329	0.592923	0.296462	−	0.955045i	$-0.404193\pi$
$67$	4.85329	0.592923	0.296462	+	0.955045i	$0.404193\pi$
$68$	2.75744	0.334389
$69$	0	0
$70$	−3.19781	−0.382211
$71$	−12.6882	−1.50581	−0.752903	−	0.658131i	$-0.771347\pi$
$71$	−12.6882	−1.50581	−0.752903	+	0.658131i	$0.771347\pi$
$72$	0	0
$73$	1.64984	0.193099	0.0965497	−	0.995328i	$-0.469219\pi$
$73$	1.64984	0.193099	0.0965497	+	0.995328i	$0.469219\pi$
$74$	−2.31515	−0.269131
$75$	0	0
$76$	−7.36772	−0.845135
$77$	3.80491	0.433610
$78$	0	0
$79$	1.31624	0.148088	0.0740442	−	0.997255i	$-0.476409\pi$
$79$	1.31624	0.148088	0.0740442	+	0.997255i	$0.476409\pi$
$80$	−6.70206	−0.749313
$81$	0	0
$82$	−4.60631	−0.508682
$83$	6.14878	0.674917	0.337458	−	0.941340i	$-0.390433\pi$
$83$	6.14878	0.674917	0.337458	+	0.941340i	$0.390433\pi$
$84$	0	0
$85$	2.31703	0.251317
$86$	16.8932	1.82164
$87$	0	0
$88$	0.589870	0.0628803
$89$	17.9538	1.90310	0.951550	−	0.307495i	$-0.0994909\pi$
$89$	17.9538	1.90310	0.951550	+	0.307495i	$0.0994909\pi$
$90$	0	0
$91$	3.75846	0.393994
$92$	14.8746	1.55079
$93$	0	0
$94$	−11.2585	−1.16122
$95$	−6.19096	−0.635179
$96$	0	0
$97$	12.1396	1.23259	0.616294	−	0.787516i	$-0.288633\pi$
$97$	12.1396	1.23259	0.616294	+	0.787516i	$0.288633\pi$
$98$	−1.98033	−0.200044
$99$	0	0
$100$	−4.59766	−0.459766
$101$	−5.90296	−0.587366	−0.293683	−	0.955903i	$-0.594881\pi$
$101$	−5.90296	−0.587366	−0.293683	+	0.955903i	$0.594881\pi$
$102$	0	0
$103$	3.47396	0.342299	0.171150	−	0.985245i	$-0.445252\pi$
$103$	3.47396	0.342299	0.171150	+	0.985245i	$0.445252\pi$
$104$	0.582669	0.0571354
$105$	0	0
$106$	14.7468	1.43234
$107$	6.01089	0.581095	0.290547	−	0.956861i	$-0.406163\pi$
$107$	6.01089	0.581095	0.290547	+	0.956861i	$0.406163\pi$
$108$	0	0
$109$	3.29100	0.315221	0.157610	−	0.987501i	$-0.449621\pi$
$109$	3.29100	0.315221	0.157610	+	0.987501i	$0.449621\pi$
$110$	−12.1674	−1.16011
$111$	0	0
$112$	−4.15044	−0.392180
$113$	15.1604	1.42617	0.713084	−	0.701079i	$-0.247298\pi$
$113$	15.1604	1.42617	0.713084	+	0.701079i	$0.247298\pi$
$114$	0	0
$115$	12.4989	1.16553
$116$	2.81372	0.261248
$117$	0	0
$118$	1.04702	0.0963863
$119$	1.43488	0.131536
$120$	0	0
$121$	3.47737	0.316125
$122$	−5.97458	−0.540913
$123$	0	0
$124$	13.7621	1.23587
$125$	−11.9372	−1.06770
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−1.23928	−0.109538
$129$	0	0
$130$	−12.0188	−1.05412
$131$	10.9863	0.959881	0.479940	−	0.877301i	$-0.340658\pi$
$131$	10.9863	0.959881	0.479940	+	0.877301i	$0.340658\pi$
$132$	0	0
$133$	−3.83393	−0.332444
$134$	−9.61112	−0.830274
$135$	0	0
$136$	0.222448	0.0190748
$137$	−17.1061	−1.46148	−0.730738	−	0.682658i	$-0.760824\pi$
$137$	−17.1061	−1.46148	−0.730738	+	0.682658i	$0.760824\pi$
$138$	0	0
$139$	−10.9303	−0.927093	−0.463546	−	0.886073i	$-0.653423\pi$
$139$	−10.9303	−0.927093	−0.463546	+	0.886073i	$0.653423\pi$
$140$	3.10315	0.262264
$141$	0	0
$142$	25.1268	2.10859
$143$	14.3006	1.19588
$144$	0	0
$145$	2.36432	0.196346
$146$	−3.26723	−0.270398
$147$	0	0
$148$	2.24663	0.184672
$149$	−4.13285	−0.338576	−0.169288	−	0.985567i	$-0.554147\pi$
$149$	−4.13285	−0.338576	−0.169288	+	0.985567i	$0.554147\pi$
$150$	0	0
$151$	−2.11410	−0.172043	−0.0860215	−	0.996293i	$-0.527415\pi$
$151$	−2.11410	−0.172043	−0.0860215	+	0.996293i	$0.527415\pi$
$152$	−0.594368	−0.0482096
$153$	0	0
$154$	−7.53499	−0.607187
$155$	11.5640	0.928845
$156$	0	0
$157$	4.93068	0.393511	0.196755	−	0.980453i	$-0.436960\pi$
$157$	4.93068	0.393511	0.196755	+	0.980453i	$0.436960\pi$
$158$	−2.60659	−0.207369
$159$	0	0
$160$	12.7716	1.00969
$161$	7.74028	0.610020
$162$	0	0
$163$	−14.3689	−1.12546	−0.562729	−	0.826641i	$-0.690249\pi$
$163$	−14.3689	−1.12546	−0.562729	+	0.826641i	$0.690249\pi$
$164$	4.46997	0.349046
$165$	0	0
$166$	−12.1766	−0.945090
$167$	11.8994	0.920805	0.460403	−	0.887710i	$-0.347705\pi$
$167$	11.8994	0.920805	0.460403	+	0.887710i	$0.347705\pi$
$168$	0	0
$169$	1.12604	0.0866183
$170$	−4.58848	−0.351921
$171$	0	0
$172$	−16.3932	−1.24997
$173$	−1.42902	−0.108647	−0.0543234	−	0.998523i	$-0.517300\pi$
$173$	−1.42902	−0.108647	−0.0543234	+	0.998523i	$0.517300\pi$
$174$	0	0
$175$	−2.39247	−0.180854
$176$	−15.7921	−1.19037
$177$	0	0
$178$	−35.5545	−2.66492
$179$	−15.3547	−1.14766	−0.573831	−	0.818974i	$-0.694543\pi$
$179$	−15.3547	−1.14766	−0.573831	+	0.818974i	$0.694543\pi$
$180$	0	0
$181$	5.46045	0.405872	0.202936	−	0.979192i	$-0.434952\pi$
$181$	5.46045	0.405872	0.202936	+	0.979192i	$0.434952\pi$
$182$	−7.44300	−0.551712
$183$	0	0
$184$	1.19996	0.0884625
$185$	1.88780	0.138794
$186$	0	0
$187$	5.45961	0.399246
$188$	10.9252	0.796804
$189$	0	0
$190$	12.2602	0.889446
$191$	−0.675037	−0.0488439	−0.0244220	−	0.999702i	$-0.507775\pi$
$191$	−0.675037	−0.0488439	−0.0244220	+	0.999702i	$0.507775\pi$
$192$	0	0
$193$	−21.6614	−1.55922	−0.779610	−	0.626265i	$-0.784583\pi$
$193$	−21.6614	−1.55922	−0.779610	+	0.626265i	$0.784583\pi$
$194$	−24.0404	−1.72600
$195$	0	0
$196$	1.92172	0.137265
$197$	15.6669	1.11622	0.558109	−	0.829768i	$-0.311527\pi$
$197$	15.6669	1.11622	0.558109	+	0.829768i	$0.311527\pi$
$198$	0	0
$199$	−3.04235	−0.215667	−0.107833	−	0.994169i	$-0.534391\pi$
$199$	−3.04235	−0.215667	−0.107833	+	0.994169i	$0.534391\pi$
$200$	−0.370902	−0.0262267
$201$	0	0
$202$	11.6898	0.822493
$203$	1.46417	0.102765
$204$	0	0
$205$	3.75603	0.262333
$206$	−6.87960	−0.479324
$207$	0	0
$208$	−15.5993	−1.08161
$209$	−14.5878	−1.00906
$210$	0	0
$211$	0.545207	0.0375336	0.0187668	−	0.999824i	$-0.494026\pi$
$211$	0.545207	0.0375336	0.0187668	+	0.999824i	$0.494026\pi$
$212$	−14.3103	−0.982836
$213$	0	0
$214$	−11.9036	−0.813711
$215$	−13.7749	−0.939442
$216$	0	0
$217$	7.16135	0.486144
$218$	−6.51727	−0.441405
$219$	0	0
$220$	11.8072	0.796044
$221$	5.39296	0.362770
$222$	0	0
$223$	5.65560	0.378727	0.189363	−	0.981907i	$-0.439358\pi$
$223$	5.65560	0.378727	0.189363	+	0.981907i	$0.439358\pi$
$224$	7.90919	0.528455
$225$	0	0
$226$	−30.0226	−1.99707
$227$	9.83847	0.653002	0.326501	−	0.945197i	$-0.394130\pi$
$227$	9.83847	0.653002	0.326501	+	0.945197i	$0.394130\pi$
$228$	0	0
$229$	10.7073	0.707557	0.353779	−	0.935329i	$-0.384897\pi$
$229$	10.7073	0.707557	0.353779	+	0.935329i	$0.384897\pi$
$230$	−24.7519	−1.63209
$231$	0	0
$232$	0.226988	0.0149025
$233$	26.8215	1.75714	0.878568	−	0.477618i	$-0.158500\pi$
$233$	26.8215	1.75714	0.878568	+	0.477618i	$0.158500\pi$
$234$	0	0
$235$	9.18027	0.598855
$236$	−1.01603	−0.0661380
$237$	0	0
$238$	−2.84155	−0.184190
$239$	12.3563	0.799263	0.399632	−	0.916676i	$-0.369138\pi$
$239$	12.3563	0.799263	0.399632	+	0.916676i	$0.369138\pi$
$240$	0	0
$241$	−18.3187	−1.18001	−0.590005	−	0.807400i	$-0.700874\pi$
$241$	−18.3187	−1.18001	−0.590005	+	0.807400i	$0.700874\pi$
$242$	−6.88635	−0.442671
$243$	0	0
$244$	5.79773	0.371162
$245$	1.61478	0.103165
$246$	0	0
$247$	−14.4097	−0.916865
$248$	1.11021	0.0704986
$249$	0	0
$250$	23.6397	1.49511
$251$	−26.9465	−1.70085	−0.850423	−	0.526100i	$-0.823654\pi$
$251$	−26.9465	−1.70085	−0.850423	+	0.526100i	$0.823654\pi$
$252$	0	0
$253$	29.4511	1.85158
$254$	−1.98033	−0.124257
$255$	0	0
$256$	17.1781	1.07363
$257$	−8.00885	−0.499578	−0.249789	−	0.968300i	$-0.580361\pi$
$257$	−8.00885	−0.499578	−0.249789	+	0.968300i	$0.580361\pi$
$258$	0	0
$259$	1.16907	0.0726427
$260$	11.6631	0.723314
$261$	0	0
$262$	−21.7566	−1.34413
$263$	20.6316	1.27220	0.636099	−	0.771607i	$-0.280547\pi$
$263$	20.6316	1.27220	0.636099	+	0.771607i	$0.280547\pi$
$264$	0	0
$265$	−12.0247	−0.738671
$266$	7.59245	0.465523
$267$	0	0
$268$	9.32664	0.569715
$269$	−9.09438	−0.554494	−0.277247	−	0.960799i	$-0.589422\pi$
$269$	−9.09438	−0.554494	−0.277247	+	0.960799i	$0.589422\pi$
$270$	0	0
$271$	14.3572	0.872140	0.436070	−	0.899913i	$-0.356370\pi$
$271$	14.3572	0.872140	0.436070	+	0.899913i	$0.356370\pi$
$272$	−5.95540	−0.361099
$273$	0	0
$274$	33.8758	2.04651
$275$	−9.10316	−0.548941
$276$	0	0
$277$	−10.8369	−0.651127	−0.325563	−	0.945520i	$-0.605554\pi$
$277$	−10.8369	−0.651127	−0.325563	+	0.945520i	$0.605554\pi$
$278$	21.6456	1.29821
$279$	0	0
$280$	0.250337	0.0149605
$281$	18.6276	1.11123	0.555615	−	0.831440i	$-0.312483\pi$
$281$	18.6276	1.11123	0.555615	+	0.831440i	$0.312483\pi$
$282$	0	0
$283$	−21.9754	−1.30630	−0.653151	−	0.757228i	$-0.726553\pi$
$283$	−21.9754	−1.30630	−0.653151	+	0.757228i	$0.726553\pi$
$284$	−24.3830	−1.44687
$285$	0	0
$286$	−28.3200	−1.67460
$287$	2.32603	0.137301
$288$	0	0
$289$	−14.9411	−0.878889
$290$	−4.68214	−0.274945
$291$	0	0
$292$	3.17053	0.185541
$293$	−25.2100	−1.47279	−0.736393	−	0.676554i	$-0.763473\pi$
$293$	−25.2100	−1.47279	−0.736393	+	0.676554i	$0.763473\pi$
$294$	0	0
$295$	−0.853753	−0.0497074
$296$	0.181240	0.0105343
$297$	0	0
$298$	8.18441	0.474110
$299$	29.0916	1.68241
$300$	0	0
$301$	−8.53051	−0.491690
$302$	4.18662	0.240913
$303$	0	0
$304$	15.9125	0.912644
$305$	4.87173	0.278955
$306$	0	0
$307$	−23.1283	−1.32000	−0.660001	−	0.751264i	$-0.729444\pi$
$307$	−23.1283	−1.32000	−0.660001	+	0.751264i	$0.729444\pi$
$308$	7.31196	0.416638
$309$	0	0
$310$	−22.9006	−1.30067
$311$	21.6858	1.22969	0.614844	−	0.788649i	$-0.289219\pi$
$311$	21.6858	1.22969	0.614844	+	0.788649i	$0.289219\pi$
$312$	0	0
$313$	15.4049	0.870737	0.435369	−	0.900252i	$-0.356618\pi$
$313$	15.4049	0.870737	0.435369	+	0.900252i	$0.356618\pi$
$314$	−9.76438	−0.551036
$315$	0	0
$316$	2.52944	0.142292
$317$	−16.7582	−0.941236	−0.470618	−	0.882337i	$-0.655969\pi$
$317$	−16.7582	−0.941236	−0.470618	+	0.882337i	$0.655969\pi$
$318$	0	0
$319$	5.57105	0.311919
$320$	−11.8880	−0.664557
$321$	0	0
$322$	−15.3283	−0.854215
$323$	−5.50124	−0.306097
$324$	0	0
$325$	−8.99203	−0.498788
$326$	28.4552	1.57599
$327$	0	0
$328$	0.360601	0.0199108
$329$	5.68514	0.313432
$330$	0	0
$331$	−28.3786	−1.55983	−0.779915	−	0.625886i	$-0.784738\pi$
$331$	−28.3786	−1.55983	−0.779915	+	0.625886i	$0.784738\pi$
$332$	11.8162	0.648499
$333$	0	0
$334$	−23.5648	−1.28941
$335$	7.83701	0.428181
$336$	0	0
$337$	23.7608	1.29434	0.647168	−	0.762347i	$-0.275953\pi$
$337$	23.7608	1.29434	0.647168	+	0.762347i	$0.275953\pi$
$338$	−2.22993	−0.121292
$339$	0	0
$340$	4.45267	0.241480
$341$	27.2483	1.47558
$342$	0	0
$343$	1.00000	0.0539949
$344$	−1.32247	−0.0713029
$345$	0	0
$346$	2.82994	0.152139
$347$	19.1426	1.02763	0.513815	−	0.857901i	$-0.328232\pi$
$347$	19.1426	1.02763	0.513815	+	0.857901i	$0.328232\pi$
$348$	0	0
$349$	−26.4945	−1.41822	−0.709110	−	0.705098i	$-0.750903\pi$
$349$	−26.4945	−1.41822	−0.709110	+	0.705098i	$0.750903\pi$
$350$	4.73789	0.253251
$351$	0	0
$352$	30.0938	1.60400
$353$	8.51096	0.452993	0.226496	−	0.974012i	$-0.427273\pi$
$353$	8.51096	0.452993	0.226496	+	0.974012i	$0.427273\pi$
$354$	0	0
$355$	−20.4886	−1.08742
$356$	34.5021	1.82861
$357$	0	0
$358$	30.4074	1.60708
$359$	−8.65537	−0.456813	−0.228406	−	0.973566i	$-0.573351\pi$
$359$	−8.65537	−0.456813	−0.228406	+	0.973566i	$0.573351\pi$
$360$	0	0
$361$	−4.30100	−0.226368
$362$	−10.8135	−0.568345
$363$	0	0
$364$	7.22270	0.378572
$365$	2.66414	0.139447
$366$	0	0
$367$	−35.4941	−1.85278	−0.926390	−	0.376566i	$-0.877105\pi$
$367$	−35.4941	−1.85278	−0.926390	+	0.376566i	$0.877105\pi$
$368$	−32.1256	−1.67466
$369$	0	0
$370$	−3.73847	−0.194354
$371$	−7.44663	−0.386610
$372$	0	0
$373$	3.10559	0.160801	0.0804007	−	0.996763i	$-0.474380\pi$
$373$	3.10559	0.160801	0.0804007	+	0.996763i	$0.474380\pi$
$374$	−10.8118	−0.559067
$375$	0	0
$376$	0.881359	0.0454526
$377$	5.50304	0.283421
$378$	0	0
$379$	−11.8843	−0.610457	−0.305228	−	0.952279i	$-0.598733\pi$
$379$	−11.8843	−0.610457	−0.305228	+	0.952279i	$0.598733\pi$
$380$	−11.8973	−0.610317
$381$	0	0
$382$	1.33680	0.0683965
$383$	−18.8556	−0.963474	−0.481737	−	0.876316i	$-0.659994\pi$
$383$	−18.8556	−0.963474	−0.481737	+	0.876316i	$0.659994\pi$
$384$	0	0
$385$	6.14411	0.313133
$386$	42.8967	2.18339
$387$	0	0
$388$	23.3288	1.18434
$389$	−37.9395	−1.92361	−0.961804	−	0.273738i	$-0.911740\pi$
$389$	−37.9395	−1.92361	−0.961804	+	0.273738i	$0.911740\pi$
$390$	0	0
$391$	11.1064	0.561675
$392$	0.155028	0.00783012
$393$	0	0
$394$	−31.0256	−1.56305
$395$	2.12544	0.106943
$396$	0	0
$397$	37.5899	1.88658	0.943292	−	0.331964i	$-0.107711\pi$
$397$	37.5899	1.88658	0.943292	+	0.331964i	$0.107711\pi$
$398$	6.02487	0.301999
$399$	0	0
$400$	9.92982	0.496491
$401$	12.2179	0.610131	0.305066	−	0.952331i	$-0.401322\pi$
$401$	12.2179	0.610131	0.305066	+	0.952331i	$0.401322\pi$
$402$	0	0
$403$	26.9157	1.34076
$404$	−11.3438	−0.564375
$405$	0	0
$406$	−2.89955	−0.143902
$407$	4.44822	0.220490
$408$	0	0
$409$	−4.62035	−0.228462	−0.114231	−	0.993454i	$-0.536440\pi$
$409$	−4.62035	−0.228462	−0.114231	+	0.993454i	$0.536440\pi$
$410$	−7.43819	−0.367346
$411$	0	0
$412$	6.67597	0.328901
$413$	−0.528711	−0.0260162
$414$	0	0
$415$	9.92895	0.487393
$416$	29.7264	1.45746
$417$	0	0
$418$	28.8886	1.41299
$419$	−14.3399	−0.700549	−0.350275	−	0.936647i	$-0.613912\pi$
$419$	−14.3399	−0.700549	−0.350275	+	0.936647i	$0.613912\pi$
$420$	0	0
$421$	29.3586	1.43085	0.715424	−	0.698690i	$-0.246233\pi$
$421$	29.3586	1.43085	0.715424	+	0.698690i	$0.246233\pi$
$422$	−1.07969	−0.0525586
$423$	0	0
$424$	−1.15444	−0.0560646
$425$	−3.43292	−0.166521
$426$	0	0
$427$	3.01696	0.146001
$428$	11.5512	0.558350
$429$	0	0
$430$	27.2789	1.31551
$431$	4.28577	0.206438	0.103219	−	0.994659i	$-0.467086\pi$
$431$	4.28577	0.206438	0.103219	+	0.994659i	$0.467086\pi$
$432$	0	0
$433$	−8.28129	−0.397973	−0.198987	−	0.980002i	$-0.563765\pi$
$433$	−8.28129	−0.397973	−0.198987	+	0.980002i	$0.563765\pi$
$434$	−14.1819	−0.680751
$435$	0	0
$436$	6.32437	0.302882
$437$	−29.6757	−1.41958
$438$	0	0
$439$	2.97523	0.142000	0.0710000	−	0.997476i	$-0.477381\pi$
$439$	2.97523	0.142000	0.0710000	+	0.997476i	$0.477381\pi$
$440$	0.952512	0.0454092
$441$	0	0
$442$	−10.6799	−0.507989
$443$	10.2682	0.487857	0.243928	−	0.969793i	$-0.421564\pi$
$443$	10.2682	0.487857	0.243928	+	0.969793i	$0.421564\pi$
$444$	0	0
$445$	28.9915	1.37433
$446$	−11.2000	−0.530334
$447$	0	0
$448$	−7.36195	−0.347819
$449$	−21.7543	−1.02665	−0.513325	−	0.858194i	$-0.671587\pi$
$449$	−21.7543	−1.02665	−0.513325	+	0.858194i	$0.671587\pi$
$450$	0	0
$451$	8.85034	0.416746
$452$	29.1339	1.37034
$453$	0	0
$454$	−19.4834	−0.914403
$455$	6.06910	0.284524
$456$	0	0
$457$	−5.68542	−0.265953	−0.132976	−	0.991119i	$-0.542453\pi$
$457$	−5.68542	−0.265953	−0.132976	+	0.991119i	$0.542453\pi$
$458$	−21.2040	−0.990797
$459$	0	0
$460$	24.0193	1.11991
$461$	−5.41069	−0.252001	−0.126001	−	0.992030i	$-0.540214\pi$
$461$	−5.41069	−0.252001	−0.126001	+	0.992030i	$0.540214\pi$
$462$	0	0
$463$	−12.1700	−0.565590	−0.282795	−	0.959180i	$-0.591262\pi$
$463$	−12.1700	−0.565590	−0.282795	+	0.959180i	$0.591262\pi$
$464$	−6.07696	−0.282116
$465$	0	0
$466$	−53.1155	−2.46053
$467$	−4.36138	−0.201821	−0.100910	−	0.994896i	$-0.532176\pi$
$467$	−4.36138	−0.201821	−0.100910	+	0.994896i	$0.532176\pi$
$468$	0	0
$469$	4.85329	0.224104
$470$	−18.1800	−0.838580
$471$	0	0
$472$	−0.0819652	−0.00377275
$473$	−32.4578	−1.49241
$474$	0	0
$475$	9.17257	0.420867
$476$	2.75744	0.126387
$477$	0	0
$478$	−24.4696	−1.11921
$479$	20.4512	0.934439	0.467219	−	0.884141i	$-0.345256\pi$
$479$	20.4512	0.934439	0.467219	+	0.884141i	$0.345256\pi$
$480$	0	0
$481$	4.39392	0.200345
$482$	36.2771	1.65237
$483$	0	0
$484$	6.68252	0.303751
$485$	19.6028	0.890118
$486$	0	0
$487$	−37.0815	−1.68032	−0.840161	−	0.542338i	$-0.817539\pi$
$487$	−37.0815	−1.68032	−0.840161	+	0.542338i	$0.817539\pi$
$488$	0.467714	0.0211724
$489$	0	0
$490$	−3.19781	−0.144462
$491$	32.7380	1.47744	0.738722	−	0.674011i	$-0.235430\pi$
$491$	32.7380	1.47744	0.738722	+	0.674011i	$0.235430\pi$
$492$	0	0
$493$	2.10092	0.0946206
$494$	28.5359	1.28389
$495$	0	0
$496$	−29.7228	−1.33459
$497$	−12.6882	−0.569141
$498$	0	0
$499$	1.17215	0.0524727	0.0262364	−	0.999656i	$-0.491648\pi$
$499$	1.17215	0.0524727	0.0262364	+	0.999656i	$0.491648\pi$
$500$	−22.9400	−1.02591
$501$	0	0
$502$	53.3629	2.38170
$503$	2.79242	0.124508	0.0622540	−	0.998060i	$-0.480171\pi$
$503$	2.79242	0.124508	0.0622540	+	0.998060i	$0.480171\pi$
$504$	0	0
$505$	−9.53200	−0.424168
$506$	−58.3230	−2.59277
$507$	0	0
$508$	1.92172	0.0852624
$509$	−31.4714	−1.39495	−0.697473	−	0.716611i	$-0.745692\pi$
$509$	−31.4714	−1.39495	−0.697473	+	0.716611i	$0.745692\pi$
$510$	0	0
$511$	1.64984	0.0729847
$512$	−31.5398	−1.39387
$513$	0	0
$514$	15.8602	0.699562
$515$	5.60969	0.247193
$516$	0	0
$517$	21.6315	0.951351
$518$	−2.31515	−0.101722
$519$	0	0
$520$	0.940884	0.0412605
$521$	29.2099	1.27971	0.639855	−	0.768496i	$-0.278994\pi$
$521$	29.2099	1.27971	0.639855	+	0.768496i	$0.278994\pi$
$522$	0	0
$523$	0.776146	0.0339385	0.0169693	−	0.999856i	$-0.494598\pi$
$523$	0.776146	0.0339385	0.0169693	+	0.999856i	$0.494598\pi$
$524$	21.1126	0.922309
$525$	0	0
$526$	−40.8574	−1.78147
$527$	10.2757	0.447617
$528$	0	0
$529$	36.9120	1.60487
$530$	23.8129	1.03437
$531$	0	0
$532$	−7.36772	−0.319431
$533$	8.74229	0.378671
$534$	0	0
$535$	9.70629	0.419640
$536$	0.752397	0.0324986
$537$	0	0
$538$	18.0099	0.776462
$539$	3.80491	0.163889
$540$	0	0
$541$	−33.9914	−1.46141	−0.730703	−	0.682696i	$-0.760807\pi$
$541$	−33.9914	−1.46141	−0.730703	+	0.682696i	$0.760807\pi$
$542$	−28.4321	−1.22126
$543$	0	0
$544$	11.3488	0.486575
$545$	5.31425	0.227638
$546$	0	0
$547$	21.1826	0.905703	0.452851	−	0.891586i	$-0.350407\pi$
$547$	21.1826	0.905703	0.452851	+	0.891586i	$0.350407\pi$
$548$	−32.8731	−1.40427
$549$	0	0
$550$	18.0273	0.768686
$551$	−5.61353	−0.239144
$552$	0	0
$553$	1.31624	0.0559722
$554$	21.4607	0.911777
$555$	0	0
$556$	−21.0049	−0.890805
$557$	−35.3545	−1.49802	−0.749009	−	0.662559i	$-0.769470\pi$
$557$	−35.3545	−1.49802	−0.749009	+	0.662559i	$0.769470\pi$
$558$	0	0
$559$	−32.0616	−1.35606
$560$	−6.70206	−0.283214
$561$	0	0
$562$	−36.8889	−1.55606
$563$	−5.54742	−0.233796	−0.116898	−	0.993144i	$-0.537295\pi$
$563$	−5.54742	−0.233796	−0.116898	+	0.993144i	$0.537295\pi$
$564$	0	0
$565$	24.4807	1.02991
$566$	43.5186	1.82922
$567$	0	0
$568$	−1.96702	−0.0825345
$569$	34.0908	1.42916	0.714579	−	0.699554i	$-0.246618\pi$
$569$	34.0908	1.42916	0.714579	+	0.699554i	$0.246618\pi$
$570$	0	0
$571$	−19.5696	−0.818964	−0.409482	−	0.912318i	$-0.634290\pi$
$571$	−19.5696	−0.818964	−0.409482	+	0.912318i	$0.634290\pi$
$572$	27.4817	1.14907
$573$	0	0
$574$	−4.60631	−0.192264
$575$	−18.5184	−0.772272
$576$	0	0
$577$	42.0474	1.75046	0.875229	−	0.483709i	$-0.160711\pi$
$577$	42.0474	1.75046	0.875229	+	0.483709i	$0.160711\pi$
$578$	29.5884	1.23071
$579$	0	0
$580$	4.54355	0.188661
$581$	6.14878	0.255095
$582$	0	0
$583$	−28.3338	−1.17347
$584$	0.255772	0.0105839
$585$	0	0
$586$	49.9242	2.06235
$587$	−40.7894	−1.68356	−0.841779	−	0.539823i	$-0.818491\pi$
$587$	−40.7894	−1.68356	−0.841779	+	0.539823i	$0.818491\pi$
$588$	0	0
$589$	−27.4561	−1.13131
$590$	1.69072	0.0696056
$591$	0	0
$592$	−4.85217	−0.199423
$593$	−40.5999	−1.66724	−0.833620	−	0.552339i	$-0.813735\pi$
$593$	−40.5999	−1.66724	−0.833620	+	0.552339i	$0.813735\pi$
$594$	0	0
$595$	2.31703	0.0949889
$596$	−7.94216	−0.325323
$597$	0	0
$598$	−57.6110	−2.35589
$599$	1.48212	0.0605579	0.0302789	−	0.999541i	$-0.490360\pi$
$599$	1.48212	0.0605579	0.0302789	+	0.999541i	$0.490360\pi$
$600$	0	0
$601$	−34.8568	−1.42184	−0.710918	−	0.703275i	$-0.751720\pi$
$601$	−34.8568	−1.42184	−0.710918	+	0.703275i	$0.751720\pi$
$602$	16.8932	0.688517
$603$	0	0
$604$	−4.06270	−0.165309
$605$	5.61520	0.228290
$606$	0	0
$607$	22.3801	0.908381	0.454190	−	0.890905i	$-0.349929\pi$
$607$	22.3801	0.908381	0.454190	+	0.890905i	$0.349929\pi$
$608$	−30.3233	−1.22977
$609$	0	0
$610$	−9.64765	−0.390622
$611$	21.3674	0.864432
$612$	0	0
$613$	22.2355	0.898082	0.449041	−	0.893511i	$-0.351766\pi$
$613$	22.2355	0.898082	0.449041	+	0.893511i	$0.351766\pi$
$614$	45.8017	1.84841
$615$	0	0
$616$	0.589870	0.0237665
$617$	−19.5083	−0.785374	−0.392687	−	0.919672i	$-0.628454\pi$
$617$	−19.5083	−0.785374	−0.392687	+	0.919672i	$0.628454\pi$
$618$	0	0
$619$	−41.2182	−1.65670	−0.828349	−	0.560212i	$-0.810720\pi$
$619$	−41.2182	−1.65670	−0.828349	+	0.560212i	$0.810720\pi$
$620$	22.2228	0.892488
$621$	0	0
$622$	−42.9450	−1.72194
$623$	17.9538	0.719304
$624$	0	0
$625$	−7.31369	−0.292548
$626$	−30.5068	−1.21930
$627$	0	0
$628$	9.47536	0.378108
$629$	1.67749	0.0668857
$630$	0	0
$631$	21.6284	0.861014	0.430507	−	0.902587i	$-0.358335\pi$
$631$	21.6284	0.861014	0.430507	+	0.902587i	$0.358335\pi$
$632$	0.204054	0.00811685
$633$	0	0
$634$	33.1868	1.31802
$635$	1.61478	0.0640807
$636$	0	0
$637$	3.75846	0.148916
$638$	−11.0325	−0.436782
$639$	0	0
$640$	−2.00116	−0.0791030
$641$	29.6118	1.16960	0.584799	−	0.811178i	$-0.301173\pi$
$641$	29.6118	1.16960	0.584799	+	0.811178i	$0.301173\pi$
$642$	0	0
$643$	36.4283	1.43659	0.718297	−	0.695737i	$-0.244922\pi$
$643$	36.4283	1.43659	0.718297	+	0.695737i	$0.244922\pi$
$644$	14.8746	0.586142
$645$	0	0
$646$	10.8943	0.428630
$647$	46.9097	1.84421	0.922105	−	0.386939i	$-0.126468\pi$
$647$	46.9097	1.84421	0.922105	+	0.386939i	$0.126468\pi$
$648$	0	0
$649$	−2.01170	−0.0789661
$650$	17.8072	0.698456
$651$	0	0
$652$	−27.6129	−1.08141
$653$	−19.9489	−0.780661	−0.390330	−	0.920675i	$-0.627639\pi$
$653$	−19.9489	−0.780661	−0.390330	+	0.920675i	$0.627639\pi$
$654$	0	0
$655$	17.7406	0.693181
$656$	−9.65404	−0.376927
$657$	0	0
$658$	−11.2585	−0.438901
$659$	8.63867	0.336515	0.168257	−	0.985743i	$-0.446186\pi$
$659$	8.63867	0.336515	0.168257	+	0.985743i	$0.446186\pi$
$660$	0	0
$661$	5.23921	0.203782	0.101891	−	0.994796i	$-0.467511\pi$
$661$	5.23921	0.203782	0.101891	+	0.994796i	$0.467511\pi$
$662$	56.1991	2.18424
$663$	0	0
$664$	0.953236	0.0369927
$665$	−6.19096	−0.240075
$666$	0	0
$667$	11.3331	0.438820
$668$	22.8673	0.884763
$669$	0	0
$670$	−15.5199	−0.599585
$671$	11.4793	0.443152
$672$	0	0
$673$	31.5916	1.21777	0.608883	−	0.793260i	$-0.291618\pi$
$673$	31.5916	1.21777	0.608883	+	0.793260i	$0.291618\pi$
$674$	−47.0544	−1.81247
$675$	0	0
$676$	2.16393	0.0832279
$677$	6.73050	0.258674	0.129337	−	0.991601i	$-0.458715\pi$
$677$	6.73050	0.258674	0.129337	+	0.991601i	$0.458715\pi$
$678$	0	0
$679$	12.1396	0.465875
$680$	0.359205	0.0137749
$681$	0	0
$682$	−53.9607	−2.06626
$683$	34.7414	1.32934	0.664672	−	0.747136i	$-0.268571\pi$
$683$	34.7414	1.32934	0.664672	+	0.747136i	$0.268571\pi$
$684$	0	0
$685$	−27.6227	−1.05541
$686$	−1.98033	−0.0756094
$687$	0	0
$688$	35.4054	1.34982
$689$	−27.9879	−1.06625
$690$	0	0
$691$	31.1319	1.18431	0.592157	−	0.805823i	$-0.298277\pi$
$691$	31.1319	1.18431	0.592157	+	0.805823i	$0.298277\pi$
$692$	−2.74618	−0.104394
$693$	0	0
$694$	−37.9087	−1.43900
$695$	−17.6500	−0.669503
$696$	0	0
$697$	3.33758	0.126420
$698$	52.4680	1.98594
$699$	0	0
$700$	−4.59766	−0.173775
$701$	12.6137	0.476413	0.238207	−	0.971214i	$-0.423440\pi$
$701$	12.6137	0.476413	0.238207	+	0.971214i	$0.423440\pi$
$702$	0	0
$703$	−4.48214	−0.169047
$704$	−28.0116	−1.05573
$705$	0	0
$706$	−16.8545	−0.634328
$707$	−5.90296	−0.222004
$708$	0	0
$709$	50.7989	1.90779	0.953896	−	0.300138i	$-0.0970328\pi$
$709$	50.7989	1.90779	0.953896	+	0.300138i	$0.0970328\pi$
$710$	40.5743	1.52273
$711$	0	0
$712$	2.78335	0.104310
$713$	55.4309	2.07590
$714$	0	0
$715$	23.0924	0.863607
$716$	−29.5073	−1.10274
$717$	0	0
$718$	17.1405	0.639678
$719$	25.6422	0.956294	0.478147	−	0.878280i	$-0.341309\pi$
$719$	25.6422	0.956294	0.478147	+	0.878280i	$0.341309\pi$
$720$	0	0
$721$	3.47396	0.129377
$722$	8.51741	0.316985
$723$	0	0
$724$	10.4934	0.389985
$725$	−3.50300	−0.130098
$726$	0	0
$727$	24.1187	0.894511	0.447256	−	0.894406i	$-0.352401\pi$
$727$	24.1187	0.894511	0.447256	+	0.894406i	$0.352401\pi$
$728$	0.582669	0.0215951
$729$	0	0
$730$	−5.27588	−0.195269
$731$	−12.2403	−0.452724
$732$	0	0
$733$	−5.89742	−0.217826	−0.108913	−	0.994051i	$-0.534737\pi$
$733$	−5.89742	−0.217826	−0.108913	+	0.994051i	$0.534737\pi$
$734$	70.2902	2.59446
$735$	0	0
$736$	61.2194	2.25658
$737$	18.4663	0.680216
$738$	0	0
$739$	6.85584	0.252196	0.126098	−	0.992018i	$-0.459755\pi$
$739$	6.85584	0.252196	0.126098	+	0.992018i	$0.459755\pi$
$740$	3.62782	0.133361
$741$	0	0
$742$	14.7468	0.541372
$743$	36.9652	1.35612	0.678061	−	0.735006i	$-0.262821\pi$
$743$	36.9652	1.35612	0.678061	+	0.735006i	$0.262821\pi$
$744$	0	0
$745$	−6.67365	−0.244504
$746$	−6.15010	−0.225171
$747$	0	0
$748$	10.4918	0.383619
$749$	6.01089	0.219633
$750$	0	0
$751$	29.9659	1.09347	0.546735	−	0.837306i	$-0.315871\pi$
$751$	29.9659	1.09347	0.546735	+	0.837306i	$0.315871\pi$
$752$	−23.5958	−0.860452
$753$	0	0
$754$	−10.8978	−0.396876
$755$	−3.41381	−0.124241
$756$	0	0
$757$	−21.5607	−0.783636	−0.391818	−	0.920043i	$-0.628154\pi$
$757$	−21.5607	−0.783636	−0.391818	+	0.920043i	$0.628154\pi$
$758$	23.5349	0.854826
$759$	0	0
$760$	−0.959775	−0.0348147
$761$	−2.26949	−0.0822691	−0.0411345	−	0.999154i	$-0.513097\pi$
$761$	−2.26949	−0.0822691	−0.0411345	+	0.999154i	$0.513097\pi$
$762$	0	0
$763$	3.29100	0.119142
$764$	−1.29723	−0.0469321
$765$	0	0
$766$	37.3403	1.34916
$767$	−1.98714	−0.0717514
$768$	0	0
$769$	37.3401	1.34652	0.673259	−	0.739407i	$-0.264894\pi$
$769$	37.3401	1.34652	0.673259	+	0.739407i	$0.264894\pi$
$770$	−12.1674	−0.438482
$771$	0	0
$772$	−41.6270	−1.49819
$773$	−28.9284	−1.04048	−0.520241	−	0.854020i	$-0.674158\pi$
$773$	−28.9284	−1.04048	−0.520241	+	0.854020i	$0.674158\pi$
$774$	0	0
$775$	−17.1333	−0.615448
$776$	1.88198	0.0675592
$777$	0	0
$778$	75.1328	2.69364
$779$	−8.91783	−0.319514
$780$	0	0
$781$	−48.2773	−1.72750
$782$	−21.9944	−0.786518
$783$	0	0
$784$	−4.15044	−0.148230
$785$	7.96197	0.284175
$786$	0	0
$787$	0.865653	0.0308572	0.0154286	−	0.999881i	$-0.495089\pi$
$787$	0.865653	0.0308572	0.0154286	+	0.999881i	$0.495089\pi$
$788$	30.1073	1.07253
$789$	0	0
$790$	−4.20908	−0.149752
$791$	15.1604	0.539041
$792$	0	0
$793$	11.3391	0.402664
$794$	−74.4405	−2.64180
$795$	0	0
$796$	−5.84654	−0.207225
$797$	−31.7859	−1.12591	−0.562956	−	0.826487i	$-0.690336\pi$
$797$	−31.7859	−1.12591	−0.562956	+	0.826487i	$0.690336\pi$
$798$	0	0
$799$	8.15752	0.288592
$800$	−18.9225	−0.669013
$801$	0	0
$802$	−24.1954	−0.854371
$803$	6.27751	0.221528
$804$	0	0
$805$	12.4989	0.440528
$806$	−53.3020	−1.87748
$807$	0	0
$808$	−0.915126	−0.0321940
$809$	−18.6534	−0.655817	−0.327908	−	0.944710i	$-0.606344\pi$
$809$	−18.6534	−0.655817	−0.327908	+	0.944710i	$0.606344\pi$
$810$	0	0
$811$	46.9249	1.64775	0.823877	−	0.566768i	$-0.191806\pi$
$811$	46.9249	1.64775	0.823877	+	0.566768i	$0.191806\pi$
$812$	2.81372	0.0987423
$813$	0	0
$814$	−8.80896	−0.308754
$815$	−23.2026	−0.812753
$816$	0	0
$817$	32.7053	1.14422
$818$	9.14983	0.319916
$819$	0	0
$820$	7.21803	0.252064
$821$	27.0269	0.943245	0.471623	−	0.881800i	$-0.343668\pi$
$821$	27.0269	0.943245	0.471623	+	0.881800i	$0.343668\pi$
$822$	0	0
$823$	−13.4866	−0.470112	−0.235056	−	0.971982i	$-0.575527\pi$
$823$	−13.4866	−0.470112	−0.235056	+	0.971982i	$0.575527\pi$
$824$	0.538563	0.0187617
$825$	0	0
$826$	1.04702	0.0364306
$827$	0.764123	0.0265711	0.0132856	−	0.999912i	$-0.495771\pi$
$827$	0.764123	0.0265711	0.0132856	+	0.999912i	$0.495771\pi$
$828$	0	0
$829$	21.8018	0.757208	0.378604	−	0.925559i	$-0.376404\pi$
$829$	21.8018	0.757208	0.378604	+	0.925559i	$0.376404\pi$
$830$	−19.6626	−0.682500
$831$	0	0
$832$	−27.6696	−0.959271
$833$	1.43488	0.0497158
$834$	0	0
$835$	19.2150	0.664962
$836$	−28.0335	−0.969560
$837$	0	0
$838$	28.3977	0.980984
$839$	32.4932	1.12179	0.560895	−	0.827887i	$-0.310457\pi$
$839$	32.4932	1.12179	0.560895	+	0.827887i	$0.310457\pi$
$840$	0	0
$841$	−26.8562	−0.926076
$842$	−58.1397	−2.00363
$843$	0	0
$844$	1.04773	0.0360645
$845$	1.81831	0.0625517
$846$	0	0
$847$	3.47737	0.119484
$848$	30.9068	1.06134
$849$	0	0
$850$	6.79833	0.233181
$851$	9.04896	0.310194
$852$	0	0
$853$	−12.8334	−0.439406	−0.219703	−	0.975567i	$-0.570509\pi$
$853$	−12.8334	−0.439406	−0.219703	+	0.975567i	$0.570509\pi$
$854$	−5.97458	−0.204446
$855$	0	0
$856$	0.931859	0.0318503
$857$	−32.2640	−1.10212	−0.551059	−	0.834466i	$-0.685776\pi$
$857$	−32.2640	−1.10212	−0.551059	+	0.834466i	$0.685776\pi$
$858$	0	0
$859$	−2.16562	−0.0738900	−0.0369450	−	0.999317i	$-0.511763\pi$
$859$	−2.16562	−0.0738900	−0.0369450	+	0.999317i	$0.511763\pi$
$860$	−26.4715	−0.902670
$861$	0	0
$862$	−8.48725	−0.289077
$863$	22.1761	0.754885	0.377442	−	0.926033i	$-0.376804\pi$
$863$	22.1761	0.754885	0.377442	+	0.926033i	$0.376804\pi$
$864$	0	0
$865$	−2.30757	−0.0784596
$866$	16.3997	0.557285
$867$	0	0
$868$	13.7621	0.467115
$869$	5.00818	0.169891
$870$	0	0
$871$	18.2409	0.618069
$872$	0.510199	0.0172775
$873$	0	0
$874$	58.7677	1.98785
$875$	−11.9372	−0.403553
$876$	0	0
$877$	1.44534	0.0488056	0.0244028	−	0.999702i	$-0.492232\pi$
$877$	1.44534	0.0488056	0.0244028	+	0.999702i	$0.492232\pi$
$878$	−5.89195	−0.198844
$879$	0	0
$880$	−25.5008	−0.859631
$881$	39.2974	1.32396	0.661981	−	0.749520i	$-0.269716\pi$
$881$	39.2974	1.32396	0.661981	+	0.749520i	$0.269716\pi$
$882$	0	0
$883$	47.7586	1.60721	0.803603	−	0.595166i	$-0.202914\pi$
$883$	47.7586	1.60721	0.803603	+	0.595166i	$0.202914\pi$
$884$	10.3637	0.348570
$885$	0	0
$886$	−20.3344	−0.683149
$887$	−55.3856	−1.85967	−0.929834	−	0.367980i	$-0.880050\pi$
$887$	−55.3856	−1.85967	−0.929834	+	0.367980i	$0.880050\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−57.4128	−1.92448
$891$	0	0
$892$	10.8685	0.363903
$893$	−21.7964	−0.729389
$894$	0	0
$895$	−24.7945	−0.828788
$896$	−1.23928	−0.0414014
$897$	0	0
$898$	43.0808	1.43763
$899$	10.4855	0.349709
$900$	0	0
$901$	−10.6851	−0.355971
$902$	−17.5266	−0.583572
$903$	0	0
$904$	2.35029	0.0781694
$905$	8.81744	0.293102
$906$	0	0
$907$	38.6860	1.28455	0.642274	−	0.766475i	$-0.277991\pi$
$907$	38.6860	1.28455	0.642274	+	0.766475i	$0.277991\pi$
$908$	18.9067	0.627442
$909$	0	0
$910$	−12.0188	−0.398421
$911$	15.6858	0.519693	0.259846	−	0.965650i	$-0.416328\pi$
$911$	15.6858	0.519693	0.259846	+	0.965650i	$0.416328\pi$
$912$	0	0
$913$	23.3956	0.774281
$914$	11.2590	0.372415
$915$	0	0
$916$	20.5764	0.679862
$917$	10.9863	0.362801
$918$	0	0
$919$	14.5441	0.479765	0.239882	−	0.970802i	$-0.422891\pi$
$919$	14.5441	0.479765	0.239882	+	0.970802i	$0.422891\pi$
$920$	1.93768	0.0638835
$921$	0	0
$922$	10.7150	0.352879
$923$	−47.6879	−1.56967
$924$	0	0
$925$	−2.79698	−0.0919641
$926$	24.1007	0.791999
$927$	0	0
$928$	11.5804	0.380146
$929$	47.5080	1.55869	0.779343	−	0.626598i	$-0.215553\pi$
$929$	47.5080	1.55869	0.779343	+	0.626598i	$0.215553\pi$
$930$	0	0
$931$	−3.83393	−0.125652
$932$	51.5433	1.68836
$933$	0	0
$934$	8.63698	0.282611
$935$	8.81609	0.288317
$936$	0	0
$937$	−6.22752	−0.203444	−0.101722	−	0.994813i	$-0.532435\pi$
$937$	−6.22752	−0.203444	−0.101722	+	0.994813i	$0.532435\pi$
$938$	−9.61112	−0.313814
$939$	0	0
$940$	17.6419	0.575415
$941$	5.24236	0.170896	0.0854481	−	0.996343i	$-0.472768\pi$
$941$	5.24236	0.170896	0.0854481	+	0.996343i	$0.472768\pi$
$942$	0	0
$943$	18.0041	0.586295
$944$	2.19438	0.0714211
$945$	0	0
$946$	64.2773	2.08984
$947$	−0.576400	−0.0187305	−0.00936524	−	0.999956i	$-0.502981\pi$
$947$	−0.576400	−0.0187305	−0.00936524	+	0.999956i	$0.502981\pi$
$948$	0	0
$949$	6.20087	0.201289
$950$	−18.1647	−0.589342
$951$	0	0
$952$	0.222448	0.00720958
$953$	5.21218	0.168839	0.0844196	−	0.996430i	$-0.473096\pi$
$953$	5.21218	0.168839	0.0844196	+	0.996430i	$0.473096\pi$
$954$	0	0
$955$	−1.09004	−0.0352728
$956$	23.7453	0.767978
$957$	0	0
$958$	−40.5001	−1.30850
$959$	−17.1061	−0.552386
$960$	0	0
$961$	20.2849	0.654353
$962$	−8.70142	−0.280545
$963$	0	0
$964$	−35.2033	−1.13382
$965$	−34.9784	−1.12600
$966$	0	0
$967$	−36.2548	−1.16588	−0.582938	−	0.812516i	$-0.698097\pi$
$967$	−36.2548	−1.16588	−0.582938	+	0.812516i	$0.698097\pi$
$968$	0.539092	0.0173271
$969$	0	0
$970$	−38.8201	−1.24644
$971$	35.5343	1.14035	0.570176	−	0.821523i	$-0.306875\pi$
$971$	35.5343	1.14035	0.570176	+	0.821523i	$0.306875\pi$
$972$	0	0
$973$	−10.9303	−0.350408
$974$	73.4336	2.35296
$975$	0	0
$976$	−12.5217	−0.400810
$977$	−23.2645	−0.744297	−0.372149	−	0.928173i	$-0.621379\pi$
$977$	−23.2645	−0.744297	−0.372149	+	0.928173i	$0.621379\pi$
$978$	0	0
$979$	68.3127	2.18328
$980$	3.10315	0.0991266
$981$	0	0
$982$	−64.8320	−2.06887
$983$	0.552312	0.0176160	0.00880801	−	0.999961i	$-0.497196\pi$
$983$	0.552312	0.0176160	0.00880801	+	0.999961i	$0.497196\pi$
$984$	0	0
$985$	25.2986	0.806080
$986$	−4.16052	−0.132498
$987$	0	0
$988$	−27.6913	−0.880978
$989$	−66.0285	−2.09959
$990$	0	0
$991$	40.1553	1.27558	0.637788	−	0.770212i	$-0.279850\pi$
$991$	40.1553	1.27558	0.637788	+	0.770212i	$0.279850\pi$
$992$	56.6405	1.79834
$993$	0	0
$994$	25.1268	0.796972
$995$	−4.91274	−0.155744
$996$	0	0
$997$	−33.0083	−1.04538	−0.522692	−	0.852522i	$-0.675072\pi$
$997$	−33.0083	−1.04538	−0.522692	+	0.852522i	$0.675072\pi$
$998$	−2.32125	−0.0734779
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.n.1.1		12
3.2	odd	2		889.2.a.a.1.12	✓	12
21.20	even	2		6223.2.a.i.1.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.a.1.12	✓	12	3.2	odd	2
6223.2.a.i.1.12		12	21.20	even	2
8001.2.a.n.1.1		12	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)

\(f(q)\)	\(=\)	\(q-1.98033 q^{2} +1.92172 q^{4} +1.61478 q^{5} +1.00000 q^{7} +0.155028 q^{8} +O(q^{10})\)	\(q-1.98033 q^{2} +1.92172 q^{4} +1.61478 q^{5} +1.00000 q^{7} +0.155028 q^{8} -3.19781 q^{10} +3.80491 q^{11} +3.75846 q^{13} -1.98033 q^{14} -4.15044 q^{16} +1.43488 q^{17} -3.83393 q^{19} +3.10315 q^{20} -7.53499 q^{22} +7.74028 q^{23} -2.39247 q^{25} -7.44300 q^{26} +1.92172 q^{28} +1.46417 q^{29} +7.16135 q^{31} +7.90919 q^{32} -2.84155 q^{34} +1.61478 q^{35} +1.16907 q^{37} +7.59245 q^{38} +0.250337 q^{40} +2.32603 q^{41} -8.53051 q^{43} +7.31196 q^{44} -15.3283 q^{46} +5.68514 q^{47} +1.00000 q^{49} +4.73789 q^{50} +7.22270 q^{52} -7.44663 q^{53} +6.14411 q^{55} +0.155028 q^{56} -2.89955 q^{58} -0.528711 q^{59} +3.01696 q^{61} -14.1819 q^{62} -7.36195 q^{64} +6.06910 q^{65} +4.85329 q^{67} +2.75744 q^{68} -3.19781 q^{70} -12.6882 q^{71} +1.64984 q^{73} -2.31515 q^{74} -7.36772 q^{76} +3.80491 q^{77} +1.31624 q^{79} -6.70206 q^{80} -4.60631 q^{82} +6.14878 q^{83} +2.31703 q^{85} +16.8932 q^{86} +0.589870 q^{88} +17.9538 q^{89} +3.75846 q^{91} +14.8746 q^{92} -11.2585 q^{94} -6.19096 q^{95} +12.1396 q^{97} -1.98033 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more