LMFDB - Embedded newform 8001.2.a.t.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8001,2,Mod(1,8001)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [16,2,0,12,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + 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			1.9
Root			$0.102239$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.102239 q^{2} -1.98955 q^{4} -0.280585 q^{5} -1.00000 q^{7} -0.407889 q^{8} -0.0286868 q^{10} -0.974733 q^{11} -1.41349 q^{13} -0.102239 q^{14} +3.93739 q^{16} -6.28548 q^{17} +1.00698 q^{19} +0.558236 q^{20} -0.0996561 q^{22} -1.39508 q^{23} -4.92127 q^{25} -0.144515 q^{26} +1.98955 q^{28} +2.24230 q^{29} -9.08093 q^{31} +1.21833 q^{32} -0.642624 q^{34} +0.280585 q^{35} +8.73158 q^{37} +0.102954 q^{38} +0.114447 q^{40} +3.37544 q^{41} -4.53376 q^{43} +1.93928 q^{44} -0.142632 q^{46} +3.98353 q^{47} +1.00000 q^{49} -0.503148 q^{50} +2.81221 q^{52} -12.4108 q^{53} +0.273495 q^{55} +0.407889 q^{56} +0.229251 q^{58} +2.70787 q^{59} -6.02128 q^{61} -0.928429 q^{62} -7.75022 q^{64} +0.396604 q^{65} -1.29868 q^{67} +12.5053 q^{68} +0.0286868 q^{70} -1.77583 q^{71} -14.2675 q^{73} +0.892711 q^{74} -2.00344 q^{76} +0.974733 q^{77} -9.13520 q^{79} -1.10477 q^{80} +0.345103 q^{82} -2.94615 q^{83} +1.76361 q^{85} -0.463529 q^{86} +0.397583 q^{88} +8.92572 q^{89} +1.41349 q^{91} +2.77558 q^{92} +0.407274 q^{94} -0.282544 q^{95} -1.25797 q^{97} +0.102239 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.102239	0.0722942	0.0361471	−	0.999346i	$-0.488492\pi$
$2$	0.102239	0.0722942	0.0361471	+	0.999346i	$0.488492\pi$
$3$	0	0
$3$	0	0
$4$	−1.98955	−0.994774
$5$	−0.280585	−0.125481	−0.0627406	−	0.998030i	$-0.519984\pi$
$5$	−0.280585	−0.125481	−0.0627406	+	0.998030i	$0.519984\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.407889	−0.144211
$9$	0	0
$10$	−0.0286868	−0.00907157
$11$	−0.974733	−0.293893	−0.146947	−	0.989144i	$-0.546945\pi$
$11$	−0.974733	−0.293893	−0.146947	+	0.989144i	$0.546945\pi$
$12$	0	0
$13$	−1.41349	−0.392032	−0.196016	−	0.980601i	$-0.562800\pi$
$13$	−1.41349	−0.392032	−0.196016	+	0.980601i	$0.562800\pi$
$14$	−0.102239	−0.0273246
$15$	0	0
$16$	3.93739	0.984348
$17$	−6.28548	−1.52445	−0.762227	−	0.647310i	$-0.775894\pi$
$17$	−6.28548	−1.52445	−0.762227	+	0.647310i	$0.775894\pi$
$18$	0	0
$19$	1.00698	0.231018	0.115509	−	0.993306i	$-0.463150\pi$
$19$	1.00698	0.231018	0.115509	+	0.993306i	$0.463150\pi$
$20$	0.558236	0.124825
$21$	0	0
$22$	−0.0996561	−0.0212468
$23$	−1.39508	−0.290895	−0.145447	−	0.989366i	$-0.546462\pi$
$23$	−1.39508	−0.290895	−0.145447	+	0.989366i	$0.546462\pi$
$24$	0	0
$25$	−4.92127	−0.984254
$26$	−0.144515	−0.0283416
$27$	0	0
$28$	1.98955	0.375989
$29$	2.24230	0.416384	0.208192	−	0.978088i	$-0.433242\pi$
$29$	2.24230	0.416384	0.208192	+	0.978088i	$0.433242\pi$
$30$	0	0
$31$	−9.08093	−1.63098	−0.815492	−	0.578769i	$-0.803533\pi$
$31$	−9.08093	−1.63098	−0.815492	+	0.578769i	$0.803533\pi$
$32$	1.21833	0.215373
$33$	0	0
$34$	−0.642624	−0.110209
$35$	0.280585	0.0474275
$36$	0	0
$37$	8.73158	1.43546	0.717731	−	0.696320i	$-0.245181\pi$
$37$	8.73158	1.43546	0.717731	+	0.696320i	$0.245181\pi$
$38$	0.102954	0.0167013
$39$	0	0
$40$	0.114447	0.0180957
$41$	3.37544	0.527155	0.263577	−	0.964638i	$-0.415098\pi$
$41$	3.37544	0.527155	0.263577	+	0.964638i	$0.415098\pi$
$42$	0	0
$43$	−4.53376	−0.691392	−0.345696	−	0.938346i	$-0.612357\pi$
$43$	−4.53376	−0.691392	−0.345696	+	0.938346i	$0.612357\pi$
$44$	1.93928	0.292357
$45$	0	0
$46$	−0.142632	−0.0210300
$47$	3.98353	0.581057	0.290529	−	0.956866i	$-0.406169\pi$
$47$	3.98353	0.581057	0.290529	+	0.956866i	$0.406169\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.503148	−0.0711559
$51$	0	0
$52$	2.81221	0.389983
$53$	−12.4108	−1.70475	−0.852375	−	0.522930i	$-0.824839\pi$
$53$	−12.4108	−1.70475	−0.852375	+	0.522930i	$0.824839\pi$
$54$	0	0
$55$	0.273495	0.0368781
$56$	0.407889	0.0545065
$57$	0	0
$58$	0.229251	0.0301021
$59$	2.70787	0.352534	0.176267	−	0.984342i	$-0.443598\pi$
$59$	2.70787	0.352534	0.176267	+	0.984342i	$0.443598\pi$
$60$	0	0
$61$	−6.02128	−0.770946	−0.385473	−	0.922719i	$-0.625962\pi$
$61$	−6.02128	−0.770946	−0.385473	+	0.922719i	$0.625962\pi$
$62$	−0.928429	−0.117911
$63$	0	0
$64$	−7.75022	−0.968778
$65$	0.396604	0.0491927
$66$	0	0
$67$	−1.29868	−0.158658	−0.0793292	−	0.996848i	$-0.525278\pi$
$67$	−1.29868	−0.158658	−0.0793292	+	0.996848i	$0.525278\pi$
$68$	12.5053	1.51649
$69$	0	0
$70$	0.0286868	0.00342873
$71$	−1.77583	−0.210752	−0.105376	−	0.994432i	$-0.533605\pi$
$71$	−1.77583	−0.210752	−0.105376	+	0.994432i	$0.533605\pi$
$72$	0	0
$73$	−14.2675	−1.66989	−0.834943	−	0.550336i	$-0.814500\pi$
$73$	−14.2675	−1.66989	−0.834943	+	0.550336i	$0.814500\pi$
$74$	0.892711	0.103776
$75$	0	0
$76$	−2.00344	−0.229811
$77$	0.974733	0.111081
$78$	0	0
$79$	−9.13520	−1.02779	−0.513895	−	0.857853i	$-0.671798\pi$
$79$	−9.13520	−1.02779	−0.513895	+	0.857853i	$0.671798\pi$
$80$	−1.10477	−0.123517
$81$	0	0
$82$	0.345103	0.0381102
$83$	−2.94615	−0.323382	−0.161691	−	0.986841i	$-0.551695\pi$
$83$	−2.94615	−0.323382	−0.161691	+	0.986841i	$0.551695\pi$
$84$	0	0
$85$	1.76361	0.191290
$86$	−0.463529	−0.0499837
$87$	0	0
$88$	0.397583	0.0423825
$89$	8.92572	0.946124	0.473062	−	0.881029i	$-0.343149\pi$
$89$	8.92572	0.946124	0.473062	+	0.881029i	$0.343149\pi$
$90$	0	0
$91$	1.41349	0.148174
$92$	2.77558	0.289374
$93$	0	0
$94$	0.407274	0.0420071
$95$	−0.282544	−0.0289884
$96$	0	0
$97$	−1.25797	−0.127727	−0.0638636	−	0.997959i	$-0.520342\pi$
$97$	−1.25797	−0.127727	−0.0638636	+	0.997959i	$0.520342\pi$
$98$	0.102239	0.0103277
$99$	0	0
$100$	9.79110	0.979110
$101$	9.32514	0.927886	0.463943	−	0.885865i	$-0.346434\pi$
$101$	9.32514	0.927886	0.463943	+	0.885865i	$0.346434\pi$
$102$	0	0
$103$	11.5869	1.14169	0.570846	−	0.821057i	$-0.306615\pi$
$103$	11.5869	1.14169	0.570846	+	0.821057i	$0.306615\pi$
$104$	0.576548	0.0565351
$105$	0	0
$106$	−1.26887	−0.123244
$107$	0.203921	0.0197138	0.00985689	−	0.999951i	$-0.496862\pi$
$107$	0.203921	0.0197138	0.00985689	+	0.999951i	$0.496862\pi$
$108$	0	0
$109$	17.5817	1.68402	0.842012	−	0.539459i	$-0.181371\pi$
$109$	17.5817	1.68402	0.842012	+	0.539459i	$0.181371\pi$
$110$	0.0279620	0.00266607
$111$	0	0
$112$	−3.93739	−0.372049
$113$	16.0844	1.51309	0.756546	−	0.653941i	$-0.226886\pi$
$113$	16.0844	1.51309	0.756546	+	0.653941i	$0.226886\pi$
$114$	0	0
$115$	0.391438	0.0365018
$116$	−4.46116	−0.414208
$117$	0	0
$118$	0.276851	0.0254862
$119$	6.28548	0.576189
$120$	0	0
$121$	−10.0499	−0.913627
$122$	−0.615612	−0.0557349
$123$	0	0
$124$	18.0669	1.62246
$125$	2.78376	0.248987
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−3.22905	−0.285410
$129$	0	0
$130$	0.0405486	0.00355634
$131$	16.7647	1.46474	0.732370	−	0.680907i	$-0.238414\pi$
$131$	16.7647	1.46474	0.732370	+	0.680907i	$0.238414\pi$
$132$	0	0
$133$	−1.00698	−0.0873167
$134$	−0.132776	−0.0114701
$135$	0	0
$136$	2.56378	0.219842
$137$	−16.3690	−1.39849	−0.699247	−	0.714880i	$-0.746481\pi$
$137$	−16.3690	−1.39849	−0.699247	+	0.714880i	$0.746481\pi$
$138$	0	0
$139$	1.38049	0.117092	0.0585459	−	0.998285i	$-0.481354\pi$
$139$	1.38049	0.117092	0.0585459	+	0.998285i	$0.481354\pi$
$140$	−0.558236	−0.0471796
$141$	0	0
$142$	−0.181560	−0.0152362
$143$	1.37778	0.115215
$144$	0	0
$145$	−0.629154	−0.0522484
$146$	−1.45870	−0.120723
$147$	0	0
$148$	−17.3719	−1.42796
$149$	−16.3718	−1.34123	−0.670616	−	0.741804i	$-0.733970\pi$
$149$	−16.3718	−1.34123	−0.670616	+	0.741804i	$0.733970\pi$
$150$	0	0
$151$	−22.0724	−1.79623	−0.898113	−	0.439765i	$-0.855062\pi$
$151$	−22.0724	−1.79623	−0.898113	+	0.439765i	$0.855062\pi$
$152$	−0.410738	−0.0333153
$153$	0	0
$154$	0.0996561	0.00803052
$155$	2.54797	0.204658
$156$	0	0
$157$	5.31714	0.424354	0.212177	−	0.977231i	$-0.431945\pi$
$157$	5.31714	0.424354	0.212177	+	0.977231i	$0.431945\pi$
$158$	−0.933978	−0.0743033
$159$	0	0
$160$	−0.341846	−0.0270253
$161$	1.39508	0.109948
$162$	0	0
$163$	14.8072	1.15979	0.579896	−	0.814690i	$-0.303093\pi$
$163$	14.8072	1.15979	0.579896	+	0.814690i	$0.303093\pi$
$164$	−6.71559	−0.524399
$165$	0	0
$166$	−0.301213	−0.0233787
$167$	−6.35017	−0.491391	−0.245696	−	0.969347i	$-0.579016\pi$
$167$	−6.35017	−0.491391	−0.245696	+	0.969347i	$0.579016\pi$
$168$	0	0
$169$	−11.0020	−0.846311
$170$	0.180310	0.0138292
$171$	0	0
$172$	9.02014	0.687779
$173$	−2.22737	−0.169344	−0.0846719	−	0.996409i	$-0.526984\pi$
$173$	−2.22737	−0.169344	−0.0846719	+	0.996409i	$0.526984\pi$
$174$	0	0
$175$	4.92127	0.372013
$176$	−3.83791	−0.289293
$177$	0	0
$178$	0.912560	0.0683993
$179$	−3.65043	−0.272846	−0.136423	−	0.990651i	$-0.543561\pi$
$179$	−3.65043	−0.272846	−0.136423	+	0.990651i	$0.543561\pi$
$180$	0	0
$181$	8.51775	0.633119	0.316560	−	0.948573i	$-0.397472\pi$
$181$	8.51775	0.633119	0.316560	+	0.948573i	$0.397472\pi$
$182$	0.144515	0.0107121
$183$	0	0
$184$	0.569038	0.0419501
$185$	−2.44995	−0.180124
$186$	0	0
$187$	6.12667	0.448026
$188$	−7.92542	−0.578021
$189$	0	0
$190$	−0.0288872	−0.00209570
$191$	17.7340	1.28319	0.641593	−	0.767045i	$-0.278274\pi$
$191$	17.7340	1.28319	0.641593	+	0.767045i	$0.278274\pi$
$192$	0	0
$193$	−2.13995	−0.154037	−0.0770186	−	0.997030i	$-0.524540\pi$
$193$	−2.13995	−0.154037	−0.0770186	+	0.997030i	$0.524540\pi$
$194$	−0.128614	−0.00923394
$195$	0	0
$196$	−1.98955	−0.142111
$197$	17.0094	1.21187	0.605936	−	0.795513i	$-0.292799\pi$
$197$	17.0094	1.21187	0.605936	+	0.795513i	$0.292799\pi$
$198$	0	0
$199$	−7.03395	−0.498624	−0.249312	−	0.968423i	$-0.580204\pi$
$199$	−7.03395	−0.498624	−0.249312	+	0.968423i	$0.580204\pi$
$200$	2.00733	0.141940
$201$	0	0
$202$	0.953397	0.0670808
$203$	−2.24230	−0.157378
$204$	0	0
$205$	−0.947095	−0.0661480
$206$	1.18464	0.0825378
$207$	0	0
$208$	−5.56547	−0.385896
$209$	−0.981541	−0.0678946
$210$	0	0
$211$	2.25672	0.155359	0.0776797	−	0.996978i	$-0.475249\pi$
$211$	2.25672	0.155359	0.0776797	+	0.996978i	$0.475249\pi$
$212$	24.6918	1.69584
$213$	0	0
$214$	0.0208488	0.00142519
$215$	1.27210	0.0867568
$216$	0	0
$217$	9.08093	0.616454
$218$	1.79754	0.121745
$219$	0	0
$220$	−0.544131	−0.0366853
$221$	8.88447	0.597634
$222$	0	0
$223$	−16.8223	−1.12650	−0.563252	−	0.826285i	$-0.690450\pi$
$223$	−16.8223	−1.12650	−0.563252	+	0.826285i	$0.690450\pi$
$224$	−1.21833	−0.0814034
$225$	0	0
$226$	1.64446	0.109388
$227$	26.9785	1.79063	0.895314	−	0.445435i	$-0.146951\pi$
$227$	26.9785	1.79063	0.895314	+	0.445435i	$0.146951\pi$
$228$	0	0
$229$	−11.5853	−0.765576	−0.382788	−	0.923836i	$-0.625036\pi$
$229$	−11.5853	−0.765576	−0.382788	+	0.923836i	$0.625036\pi$
$230$	0.0400204	0.00263887
$231$	0	0
$232$	−0.914608	−0.0600470
$233$	10.3197	0.676064	0.338032	−	0.941135i	$-0.390239\pi$
$233$	10.3197	0.676064	0.338032	+	0.941135i	$0.390239\pi$
$234$	0	0
$235$	−1.11772	−0.0729118
$236$	−5.38743	−0.350692
$237$	0	0
$238$	0.642624	0.0416551
$239$	15.1850	0.982233	0.491116	−	0.871094i	$-0.336589\pi$
$239$	15.1850	0.982233	0.491116	+	0.871094i	$0.336589\pi$
$240$	0	0
$241$	5.19032	0.334338	0.167169	−	0.985928i	$-0.446537\pi$
$241$	5.19032	0.334338	0.167169	+	0.985928i	$0.446537\pi$
$242$	−1.02750	−0.0660499
$243$	0	0
$244$	11.9796	0.766917
$245$	−0.280585	−0.0179259
$246$	0	0
$247$	−1.42336	−0.0905665
$248$	3.70401	0.235205
$249$	0	0
$250$	0.284610	0.0180003
$251$	21.6309	1.36533	0.682664	−	0.730732i	$-0.260821\pi$
$251$	21.6309	1.36533	0.682664	+	0.730732i	$0.260821\pi$
$252$	0	0
$253$	1.35983	0.0854919
$254$	−0.102239	−0.00641507
$255$	0	0
$256$	15.1703	0.948144
$257$	−10.2922	−0.642010	−0.321005	−	0.947077i	$-0.604021\pi$
$257$	−10.2922	−0.642010	−0.321005	+	0.947077i	$0.604021\pi$
$258$	0	0
$259$	−8.73158	−0.542554
$260$	−0.789062	−0.0489356
$261$	0	0
$262$	1.71401	0.105892
$263$	14.2342	0.877719	0.438859	−	0.898556i	$-0.355383\pi$
$263$	14.2342	0.877719	0.438859	+	0.898556i	$0.355383\pi$
$264$	0	0
$265$	3.48227	0.213914
$266$	−0.102954	−0.00631249
$267$	0	0
$268$	2.58378	0.157829
$269$	24.6333	1.50192	0.750958	−	0.660350i	$-0.229592\pi$
$269$	24.6333	1.50192	0.750958	+	0.660350i	$0.229592\pi$
$270$	0	0
$271$	−4.57306	−0.277794	−0.138897	−	0.990307i	$-0.544356\pi$
$271$	−4.57306	−0.277794	−0.138897	+	0.990307i	$0.544356\pi$
$272$	−24.7484	−1.50059
$273$	0	0
$274$	−1.67355	−0.101103
$275$	4.79693	0.289266
$276$	0	0
$277$	−24.5754	−1.47659	−0.738296	−	0.674477i	$-0.764369\pi$
$277$	−24.5754	−1.47659	−0.738296	+	0.674477i	$0.764369\pi$
$278$	0.141141	0.00846505
$279$	0	0
$280$	−0.114447	−0.00683954
$281$	−13.4450	−0.802061	−0.401030	−	0.916065i	$-0.631348\pi$
$281$	−13.4450	−0.802061	−0.401030	+	0.916065i	$0.631348\pi$
$282$	0	0
$283$	−25.9931	−1.54513	−0.772564	−	0.634937i	$-0.781026\pi$
$283$	−25.9931	−1.54513	−0.772564	+	0.634937i	$0.781026\pi$
$284$	3.53310	0.209651
$285$	0	0
$286$	0.140863	0.00832941
$287$	−3.37544	−0.199246
$288$	0	0
$289$	22.5073	1.32396
$290$	−0.0643243	−0.00377726
$291$	0	0
$292$	28.3859	1.66116
$293$	15.0055	0.876628	0.438314	−	0.898822i	$-0.355576\pi$
$293$	15.0055	0.876628	0.438314	+	0.898822i	$0.355576\pi$
$294$	0	0
$295$	−0.759786	−0.0442364
$296$	−3.56151	−0.207009
$297$	0	0
$298$	−1.67385	−0.0969634
$299$	1.97193	0.114040
$300$	0	0
$301$	4.53376	0.261322
$302$	−2.25667	−0.129857
$303$	0	0
$304$	3.96489	0.227402
$305$	1.68948	0.0967393
$306$	0	0
$307$	15.2999	0.873209	0.436605	−	0.899653i	$-0.356181\pi$
$307$	15.2999	0.873209	0.436605	+	0.899653i	$0.356181\pi$
$308$	−1.93928	−0.110501
$309$	0	0
$310$	0.260503	0.0147956
$311$	7.23014	0.409984	0.204992	−	0.978764i	$-0.434283\pi$
$311$	7.23014	0.409984	0.204992	+	0.978764i	$0.434283\pi$
$312$	0	0
$313$	1.04458	0.0590433	0.0295217	−	0.999564i	$-0.490602\pi$
$313$	1.04458	0.0590433	0.0295217	+	0.999564i	$0.490602\pi$
$314$	0.543622	0.0306784
$315$	0	0
$316$	18.1749	1.02242
$317$	−4.47560	−0.251375	−0.125687	−	0.992070i	$-0.540114\pi$
$317$	−4.47560	−0.251375	−0.125687	+	0.992070i	$0.540114\pi$
$318$	0	0
$319$	−2.18564	−0.122372
$320$	2.17459	0.121563
$321$	0	0
$322$	0.142632	0.00794859
$323$	−6.32938	−0.352176
$324$	0	0
$325$	6.95617	0.385859
$326$	1.51388	0.0838463
$327$	0	0
$328$	−1.37680	−0.0760212
$329$	−3.98353	−0.219619
$330$	0	0
$331$	17.2234	0.946683	0.473342	−	0.880879i	$-0.343048\pi$
$331$	17.2234	0.946683	0.473342	+	0.880879i	$0.343048\pi$
$332$	5.86151	0.321692
$333$	0	0
$334$	−0.649238	−0.0355247
$335$	0.364388	0.0199087
$336$	0	0
$337$	6.92934	0.377465	0.188733	−	0.982029i	$-0.439562\pi$
$337$	6.92934	0.377465	0.188733	+	0.982029i	$0.439562\pi$
$338$	−1.12484	−0.0611834
$339$	0	0
$340$	−3.50878	−0.190291
$341$	8.85148	0.479335
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	1.84927	0.0997061
$345$	0	0
$346$	−0.227725	−0.0122426
$347$	10.8206	0.580881	0.290441	−	0.956893i	$-0.406198\pi$
$347$	10.8206	0.580881	0.290441	+	0.956893i	$0.406198\pi$
$348$	0	0
$349$	17.2151	0.921502	0.460751	−	0.887529i	$-0.347580\pi$
$349$	17.2151	0.921502	0.460751	+	0.887529i	$0.347580\pi$
$350$	0.503148	0.0268944
$351$	0	0
$352$	−1.18755	−0.0632967
$353$	35.8111	1.90603	0.953016	−	0.302921i	$-0.0979619\pi$
$353$	35.8111	1.90603	0.953016	+	0.302921i	$0.0979619\pi$
$354$	0	0
$355$	0.498270	0.0264454
$356$	−17.7581	−0.941179
$357$	0	0
$358$	−0.373218	−0.0197252
$359$	−12.5791	−0.663899	−0.331950	−	0.943297i	$-0.607706\pi$
$359$	−12.5791	−0.663899	−0.331950	+	0.943297i	$0.607706\pi$
$360$	0	0
$361$	−17.9860	−0.946631
$362$	0.870850	0.0457709
$363$	0	0
$364$	−2.81221	−0.147400
$365$	4.00325	0.209539
$366$	0	0
$367$	−3.95956	−0.206688	−0.103344	−	0.994646i	$-0.532954\pi$
$367$	−3.95956	−0.206688	−0.103344	+	0.994646i	$0.532954\pi$
$368$	−5.49298	−0.286341
$369$	0	0
$370$	−0.250481	−0.0130219
$371$	12.4108	0.644335
$372$	0	0
$373$	30.1501	1.56112	0.780558	−	0.625084i	$-0.214935\pi$
$373$	30.1501	1.56112	0.780558	+	0.625084i	$0.214935\pi$
$374$	0.626387	0.0323897
$375$	0	0
$376$	−1.62484	−0.0837946
$377$	−3.16947	−0.163236
$378$	0	0
$379$	22.0540	1.13284	0.566419	−	0.824117i	$-0.308328\pi$
$379$	22.0540	1.13284	0.566419	+	0.824117i	$0.308328\pi$
$380$	0.562135	0.0288369
$381$	0	0
$382$	1.81311	0.0927669
$383$	−9.13278	−0.466663	−0.233332	−	0.972397i	$-0.574963\pi$
$383$	−9.13278	−0.466663	−0.233332	+	0.972397i	$0.574963\pi$
$384$	0	0
$385$	−0.273495	−0.0139386
$386$	−0.218788	−0.0111360
$387$	0	0
$388$	2.50279	0.127060
$389$	18.5278	0.939397	0.469699	−	0.882827i	$-0.344363\pi$
$389$	18.5278	0.939397	0.469699	+	0.882827i	$0.344363\pi$
$390$	0	0
$391$	8.76876	0.443455
$392$	−0.407889	−0.0206015
$393$	0	0
$394$	1.73904	0.0876114
$395$	2.56320	0.128968
$396$	0	0
$397$	−17.2658	−0.866545	−0.433272	−	0.901263i	$-0.642641\pi$
$397$	−17.2658	−0.866545	−0.433272	+	0.901263i	$0.642641\pi$
$398$	−0.719147	−0.0360476
$399$	0	0
$400$	−19.3770	−0.968849
$401$	−29.4145	−1.46889	−0.734445	−	0.678668i	$-0.762558\pi$
$401$	−29.4145	−1.46889	−0.734445	+	0.678668i	$0.762558\pi$
$402$	0	0
$403$	12.8358	0.639397
$404$	−18.5528	−0.923037
$405$	0	0
$406$	−0.229251	−0.0113775
$407$	−8.51095	−0.421872
$408$	0	0
$409$	36.8255	1.82090	0.910452	−	0.413615i	$-0.135734\pi$
$409$	36.8255	1.82090	0.910452	+	0.413615i	$0.135734\pi$
$410$	−0.0968305	−0.00478212
$411$	0	0
$412$	−23.0527	−1.13573
$413$	−2.70787	−0.133245
$414$	0	0
$415$	0.826646	0.0405784
$416$	−1.72211	−0.0844332
$417$	0	0
$418$	−0.100352	−0.00490839
$419$	−29.4840	−1.44039	−0.720193	−	0.693774i	$-0.755947\pi$
$419$	−29.4840	−1.44039	−0.720193	+	0.693774i	$0.755947\pi$
$420$	0	0
$421$	−4.30802	−0.209960	−0.104980	−	0.994474i	$-0.533478\pi$
$421$	−4.30802	−0.209960	−0.104980	+	0.994474i	$0.533478\pi$
$422$	0.230726	0.0112316
$423$	0	0
$424$	5.06222	0.245843
$425$	30.9326	1.50045
$426$	0	0
$427$	6.02128	0.291390
$428$	−0.405710	−0.0196108
$429$	0	0
$430$	0.130059	0.00627201
$431$	22.0058	1.05998	0.529992	−	0.848003i	$-0.322195\pi$
$431$	22.0058	1.05998	0.529992	+	0.848003i	$0.322195\pi$
$432$	0	0
$433$	15.7052	0.754744	0.377372	−	0.926062i	$-0.376828\pi$
$433$	15.7052	0.754744	0.377372	+	0.926062i	$0.376828\pi$
$434$	0.928429	0.0445660
$435$	0	0
$436$	−34.9797	−1.67522
$437$	−1.40483	−0.0672019
$438$	0	0
$439$	25.5991	1.22178	0.610889	−	0.791716i	$-0.290812\pi$
$439$	25.5991	1.22178	0.610889	+	0.791716i	$0.290812\pi$
$440$	−0.111556	−0.00531821
$441$	0	0
$442$	0.908344	0.0432055
$443$	−17.3757	−0.825545	−0.412772	−	0.910834i	$-0.635439\pi$
$443$	−17.3757	−0.825545	−0.412772	+	0.910834i	$0.635439\pi$
$444$	0	0
$445$	−2.50442	−0.118721
$446$	−1.71990	−0.0814397
$447$	0	0
$448$	7.75022	0.366164
$449$	26.0563	1.22967	0.614836	−	0.788655i	$-0.289222\pi$
$449$	26.0563	1.22967	0.614836	+	0.788655i	$0.289222\pi$
$450$	0	0
$451$	−3.29015	−0.154927
$452$	−32.0006	−1.50518
$453$	0	0
$454$	2.75827	0.129452
$455$	−0.396604	−0.0185931
$456$	0	0
$457$	4.60151	0.215250	0.107625	−	0.994192i	$-0.465675\pi$
$457$	4.60151	0.215250	0.107625	+	0.994192i	$0.465675\pi$
$458$	−1.18447	−0.0553467
$459$	0	0
$460$	−0.778785	−0.0363110
$461$	24.8341	1.15664	0.578320	−	0.815810i	$-0.303708\pi$
$461$	24.8341	1.15664	0.578320	+	0.815810i	$0.303708\pi$
$462$	0	0
$463$	25.8050	1.19926	0.599630	−	0.800277i	$-0.295314\pi$
$463$	25.8050	1.19926	0.599630	+	0.800277i	$0.295314\pi$
$464$	8.82880	0.409867
$465$	0	0
$466$	1.05508	0.0488755
$467$	−13.0335	−0.603119	−0.301559	−	0.953447i	$-0.597507\pi$
$467$	−13.0335	−0.603119	−0.301559	+	0.953447i	$0.597507\pi$
$468$	0	0
$469$	1.29868	0.0599672
$470$	−0.114275	−0.00527110
$471$	0	0
$472$	−1.10451	−0.0508391
$473$	4.41921	0.203195
$474$	0	0
$475$	−4.95565	−0.227381
$476$	−12.5053	−0.573178
$477$	0	0
$478$	1.55250	0.0710097
$479$	2.00396	0.0915634	0.0457817	−	0.998951i	$-0.485422\pi$
$479$	2.00396	0.0915634	0.0457817	+	0.998951i	$0.485422\pi$
$480$	0	0
$481$	−12.3420	−0.562747
$482$	0.530656	0.0241707
$483$	0	0
$484$	19.9947	0.908852
$485$	0.352966	0.0160274
$486$	0	0
$487$	−16.6173	−0.753003	−0.376502	−	0.926416i	$-0.622873\pi$
$487$	−16.6173	−0.753003	−0.376502	+	0.926416i	$0.622873\pi$
$488$	2.45601	0.111179
$489$	0	0
$490$	−0.0286868	−0.00129594
$491$	20.9773	0.946692	0.473346	−	0.880876i	$-0.343046\pi$
$491$	20.9773	0.946692	0.473346	+	0.880876i	$0.343046\pi$
$492$	0	0
$493$	−14.0939	−0.634758
$494$	−0.145524	−0.00654743
$495$	0	0
$496$	−35.7552	−1.60545
$497$	1.77583	0.0796568
$498$	0	0
$499$	18.9857	0.849916	0.424958	−	0.905213i	$-0.360289\pi$
$499$	18.9857	0.849916	0.424958	+	0.905213i	$0.360289\pi$
$500$	−5.53841	−0.247685
$501$	0	0
$502$	2.21153	0.0987053
$503$	3.24409	0.144647	0.0723234	−	0.997381i	$-0.476959\pi$
$503$	3.24409	0.144647	0.0723234	+	0.997381i	$0.476959\pi$
$504$	0	0
$505$	−2.61649	−0.116432
$506$	0.139028	0.00618057
$507$	0	0
$508$	1.98955	0.0882719
$509$	31.1652	1.38137	0.690686	−	0.723155i	$-0.257309\pi$
$509$	31.1652	1.38137	0.690686	+	0.723155i	$0.257309\pi$
$510$	0	0
$511$	14.2675	0.631158
$512$	8.00910	0.353956
$513$	0	0
$514$	−1.05227	−0.0464136
$515$	−3.25111	−0.143261
$516$	0	0
$517$	−3.88288	−0.170769
$518$	−0.892711	−0.0392235
$519$	0	0
$520$	−0.161770	−0.00709410
$521$	−37.3656	−1.63701	−0.818507	−	0.574496i	$-0.805198\pi$
$521$	−37.3656	−1.63701	−0.818507	+	0.574496i	$0.805198\pi$
$522$	0	0
$523$	8.19492	0.358339	0.179170	−	0.983818i	$-0.442659\pi$
$523$	8.19492	0.358339	0.179170	+	0.983818i	$0.442659\pi$
$524$	−33.3542	−1.45708
$525$	0	0
$526$	1.45530	0.0634540
$527$	57.0780	2.48636
$528$	0	0
$529$	−21.0537	−0.915380
$530$	0.356025	0.0154648
$531$	0	0
$532$	2.00344	0.0868603
$533$	−4.77115	−0.206661
$534$	0	0
$535$	−0.0572171	−0.00247371
$536$	0.529715	0.0228802
$537$	0	0
$538$	2.51849	0.108580
$539$	−0.974733	−0.0419847
$540$	0	0
$541$	−39.2494	−1.68747	−0.843733	−	0.536764i	$-0.819647\pi$
$541$	−39.2494	−1.68747	−0.843733	+	0.536764i	$0.819647\pi$
$542$	−0.467547	−0.0200829
$543$	0	0
$544$	−7.65782	−0.328326
$545$	−4.93316	−0.211313
$546$	0	0
$547$	4.58146	0.195889	0.0979446	−	0.995192i	$-0.468773\pi$
$547$	4.58146	0.195889	0.0979446	+	0.995192i	$0.468773\pi$
$548$	32.5668	1.39119
$549$	0	0
$550$	0.490435	0.0209122
$551$	2.25796	0.0961923
$552$	0	0
$553$	9.13520	0.388468
$554$	−2.51257	−0.106749
$555$	0	0
$556$	−2.74655	−0.116480
$557$	15.9166	0.674410	0.337205	−	0.941431i	$-0.390518\pi$
$557$	15.9166	0.674410	0.337205	+	0.941431i	$0.390518\pi$
$558$	0	0
$559$	6.40843	0.271048
$560$	1.10477	0.0466851
$561$	0	0
$562$	−1.37461	−0.0579843
$563$	−7.61666	−0.321004	−0.160502	−	0.987036i	$-0.551311\pi$
$563$	−7.61666	−0.321004	−0.160502	+	0.987036i	$0.551311\pi$
$564$	0	0
$565$	−4.51303	−0.189865
$566$	−2.65752	−0.111704
$567$	0	0
$568$	0.724341	0.0303927
$569$	−3.85842	−0.161754	−0.0808768	−	0.996724i	$-0.525772\pi$
$569$	−3.85842	−0.161754	−0.0808768	+	0.996724i	$0.525772\pi$
$570$	0	0
$571$	32.1197	1.34417	0.672085	−	0.740474i	$-0.265399\pi$
$571$	32.1197	1.34417	0.672085	+	0.740474i	$0.265399\pi$
$572$	−2.74115	−0.114613
$573$	0	0
$574$	−0.345103	−0.0144043
$575$	6.86557	0.286314
$576$	0	0
$577$	17.3278	0.721364	0.360682	−	0.932689i	$-0.382544\pi$
$577$	17.3278	0.721364	0.360682	+	0.932689i	$0.382544\pi$
$578$	2.30113	0.0957145
$579$	0	0
$580$	1.25173	0.0519753
$581$	2.94615	0.122227
$582$	0	0
$583$	12.0972	0.501014
$584$	5.81956	0.240815
$585$	0	0
$586$	1.53415	0.0633751
$587$	−43.2794	−1.78633	−0.893166	−	0.449728i	$-0.851521\pi$
$587$	−43.2794	−1.78633	−0.893166	+	0.449728i	$0.851521\pi$
$588$	0	0
$589$	−9.14436	−0.376787
$590$	−0.0776800	−0.00319804
$591$	0	0
$592$	34.3796	1.41299
$593$	17.9037	0.735216	0.367608	−	0.929981i	$-0.380177\pi$
$593$	17.9037	0.735216	0.367608	+	0.929981i	$0.380177\pi$
$594$	0	0
$595$	−1.76361	−0.0723009
$596$	32.5725	1.33422
$597$	0	0
$598$	0.201609	0.00824442
$599$	−28.8831	−1.18013	−0.590066	−	0.807355i	$-0.700898\pi$
$599$	−28.8831	−1.18013	−0.590066	+	0.807355i	$0.700898\pi$
$600$	0	0
$601$	12.3997	0.505795	0.252898	−	0.967493i	$-0.418616\pi$
$601$	12.3997	0.505795	0.252898	+	0.967493i	$0.418616\pi$
$602$	0.463529	0.0188920
$603$	0	0
$604$	43.9141	1.78684
$605$	2.81985	0.114643
$606$	0	0
$607$	−12.6815	−0.514726	−0.257363	−	0.966315i	$-0.582854\pi$
$607$	−12.6815	−0.514726	−0.257363	+	0.966315i	$0.582854\pi$
$608$	1.22684	0.0497551
$609$	0	0
$610$	0.172731	0.00699369
$611$	−5.63068	−0.227793
$612$	0	0
$613$	4.45146	0.179793	0.0898965	−	0.995951i	$-0.471346\pi$
$613$	4.45146	0.179793	0.0898965	+	0.995951i	$0.471346\pi$
$614$	1.56425	0.0631280
$615$	0	0
$616$	−0.397583	−0.0160191
$617$	24.0389	0.967769	0.483885	−	0.875132i	$-0.339225\pi$
$617$	24.0389	0.967769	0.483885	+	0.875132i	$0.339225\pi$
$618$	0	0
$619$	17.6987	0.711371	0.355686	−	0.934606i	$-0.384247\pi$
$619$	17.6987	0.711371	0.355686	+	0.934606i	$0.384247\pi$
$620$	−5.06930	−0.203588
$621$	0	0
$622$	0.739205	0.0296394
$623$	−8.92572	−0.357601
$624$	0	0
$625$	23.8253	0.953011
$626$	0.106798	0.00426849
$627$	0	0
$628$	−10.5787	−0.422136
$629$	−54.8822	−2.18830
$630$	0	0
$631$	4.02635	0.160286	0.0801432	−	0.996783i	$-0.474462\pi$
$631$	4.02635	0.160286	0.0801432	+	0.996783i	$0.474462\pi$
$632$	3.72615	0.148218
$633$	0	0
$634$	−0.457583	−0.0181729
$635$	0.280585	0.0111347
$636$	0	0
$637$	−1.41349	−0.0560046
$638$	−0.223459	−0.00884681
$639$	0	0
$640$	0.906021	0.0358136
$641$	28.9089	1.14183	0.570917	−	0.821008i	$-0.306588\pi$
$641$	28.9089	1.14183	0.570917	+	0.821008i	$0.306588\pi$
$642$	0	0
$643$	−4.68023	−0.184570	−0.0922851	−	0.995733i	$-0.529417\pi$
$643$	−4.68023	−0.184570	−0.0922851	+	0.995733i	$0.529417\pi$
$644$	−2.77558	−0.109373
$645$	0	0
$646$	−0.647113	−0.0254603
$647$	−11.1375	−0.437860	−0.218930	−	0.975741i	$-0.570257\pi$
$647$	−11.1375	−0.437860	−0.218930	+	0.975741i	$0.570257\pi$
$648$	0	0
$649$	−2.63945	−0.103607
$650$	0.711195	0.0278954
$651$	0	0
$652$	−29.4597	−1.15373
$653$	−36.8949	−1.44381	−0.721904	−	0.691993i	$-0.756733\pi$
$653$	−36.8949	−1.44381	−0.721904	+	0.691993i	$0.756733\pi$
$654$	0	0
$655$	−4.70392	−0.183797
$656$	13.2904	0.518903
$657$	0	0
$658$	−0.407274	−0.0158772
$659$	22.7628	0.886713	0.443357	−	0.896345i	$-0.353788\pi$
$659$	22.7628	0.886713	0.443357	+	0.896345i	$0.353788\pi$
$660$	0	0
$661$	33.3559	1.29740	0.648698	−	0.761046i	$-0.275314\pi$
$661$	33.3559	1.29740	0.648698	+	0.761046i	$0.275314\pi$
$662$	1.76091	0.0684397
$663$	0	0
$664$	1.20170	0.0466352
$665$	0.282544	0.0109566
$666$	0	0
$667$	−3.12819	−0.121124
$668$	12.6340	0.488823
$669$	0	0
$670$	0.0372548	0.00143928
$671$	5.86914	0.226576
$672$	0	0
$673$	−5.38199	−0.207461	−0.103730	−	0.994605i	$-0.533078\pi$
$673$	−5.38199	−0.207461	−0.103730	+	0.994605i	$0.533078\pi$
$674$	0.708452	0.0272885
$675$	0	0
$676$	21.8891	0.841888
$677$	−35.4290	−1.36165	−0.680823	−	0.732448i	$-0.738378\pi$
$677$	−35.4290	−1.36165	−0.680823	+	0.732448i	$0.738378\pi$
$678$	0	0
$679$	1.25797	0.0482764
$680$	−0.719357	−0.0275861
$681$	0	0
$682$	0.904970	0.0346531
$683$	8.04397	0.307794	0.153897	−	0.988087i	$-0.450818\pi$
$683$	8.04397	0.307794	0.153897	+	0.988087i	$0.450818\pi$
$684$	0	0
$685$	4.59288	0.175485
$686$	−0.102239	−0.00390352
$687$	0	0
$688$	−17.8512	−0.680571
$689$	17.5425	0.668317
$690$	0	0
$691$	−16.1470	−0.614261	−0.307131	−	0.951667i	$-0.599369\pi$
$691$	−16.1470	−0.614261	−0.307131	+	0.951667i	$0.599369\pi$
$692$	4.43146	0.168459
$693$	0	0
$694$	1.10629	0.0419943
$695$	−0.387345	−0.0146928
$696$	0	0
$697$	−21.2162	−0.803622
$698$	1.76006	0.0666192
$699$	0	0
$700$	−9.79110	−0.370069
$701$	31.0098	1.17122	0.585611	−	0.810592i	$-0.300855\pi$
$701$	31.0098	1.17122	0.585611	+	0.810592i	$0.300855\pi$
$702$	0	0
$703$	8.79256	0.331618
$704$	7.55440	0.284717
$705$	0	0
$706$	3.66130	0.137795
$707$	−9.32514	−0.350708
$708$	0	0
$709$	−13.3540	−0.501520	−0.250760	−	0.968049i	$-0.580681\pi$
$709$	−13.3540	−0.501520	−0.250760	+	0.968049i	$0.580681\pi$
$710$	0.0509429	0.00191185
$711$	0	0
$712$	−3.64070	−0.136441
$713$	12.6686	0.474444
$714$	0	0
$715$	−0.386583	−0.0144574
$716$	7.26270	0.271420
$717$	0	0
$718$	−1.28608	−0.0479961
$719$	−6.07809	−0.226674	−0.113337	−	0.993557i	$-0.536154\pi$
$719$	−6.07809	−0.226674	−0.113337	+	0.993557i	$0.536154\pi$
$720$	0	0
$721$	−11.5869	−0.431519
$722$	−1.83888	−0.0684359
$723$	0	0
$724$	−16.9465	−0.629811
$725$	−11.0350	−0.409828
$726$	0	0
$727$	−18.5668	−0.688605	−0.344302	−	0.938859i	$-0.611885\pi$
$727$	−18.5668	−0.688605	−0.344302	+	0.938859i	$0.611885\pi$
$728$	−0.576548	−0.0213683
$729$	0	0
$730$	0.409289	0.0151485
$731$	28.4969	1.05400
$732$	0	0
$733$	31.9202	1.17900	0.589499	−	0.807769i	$-0.299325\pi$
$733$	31.9202	1.17900	0.589499	+	0.807769i	$0.299325\pi$
$734$	−0.404823	−0.0149423
$735$	0	0
$736$	−1.69968	−0.0626509
$737$	1.26586	0.0466286
$738$	0	0
$739$	−47.3446	−1.74160	−0.870799	−	0.491639i	$-0.836398\pi$
$739$	−47.3446	−1.74160	−0.870799	+	0.491639i	$0.836398\pi$
$740$	4.87428	0.179182
$741$	0	0
$742$	1.26887	0.0465817
$743$	1.56541	0.0574293	0.0287147	−	0.999588i	$-0.490859\pi$
$743$	1.56541	0.0574293	0.0287147	+	0.999588i	$0.490859\pi$
$744$	0	0
$745$	4.59369	0.168300
$746$	3.08253	0.112860
$747$	0	0
$748$	−12.1893	−0.445685
$749$	−0.203921	−0.00745111
$750$	0	0
$751$	−54.4033	−1.98520	−0.992601	−	0.121418i	$-0.961256\pi$
$751$	−54.4033	−1.98520	−0.992601	+	0.121418i	$0.961256\pi$
$752$	15.6847	0.571963
$753$	0	0
$754$	−0.324044	−0.0118010
$755$	6.19317	0.225393
$756$	0	0
$757$	−18.1815	−0.660817	−0.330409	−	0.943838i	$-0.607187\pi$
$757$	−18.1815	−0.660817	−0.330409	+	0.943838i	$0.607187\pi$
$758$	2.25479	0.0818976
$759$	0	0
$760$	0.115247	0.00418044
$761$	−26.0404	−0.943962	−0.471981	−	0.881609i	$-0.656461\pi$
$761$	−26.0404	−0.943962	−0.471981	+	0.881609i	$0.656461\pi$
$762$	0	0
$763$	−17.5817	−0.636501
$764$	−35.2826	−1.27648
$765$	0	0
$766$	−0.933730	−0.0337370
$767$	−3.82754	−0.138205
$768$	0	0
$769$	−34.5026	−1.24419	−0.622097	−	0.782940i	$-0.713719\pi$
$769$	−34.5026	−1.24419	−0.622097	+	0.782940i	$0.713719\pi$
$770$	−0.0279620	−0.00100768
$771$	0	0
$772$	4.25754	0.153232
$773$	23.2583	0.836542	0.418271	−	0.908322i	$-0.362636\pi$
$773$	23.2583	0.836542	0.418271	+	0.908322i	$0.362636\pi$
$774$	0	0
$775$	44.6897	1.60530
$776$	0.513111	0.0184196
$777$	0	0
$778$	1.89427	0.0679130
$779$	3.39901	0.121782
$780$	0	0
$781$	1.73096	0.0619386
$782$	0.896513	0.0320592
$783$	0	0
$784$	3.93739	0.140621
$785$	−1.49191	−0.0532485
$786$	0	0
$787$	−26.2539	−0.935850	−0.467925	−	0.883768i	$-0.654998\pi$
$787$	−26.2539	−0.935850	−0.467925	+	0.883768i	$0.654998\pi$
$788$	−33.8411	−1.20554
$789$	0	0
$790$	0.262060	0.00932367
$791$	−16.0844	−0.571895
$792$	0	0
$793$	8.51103	0.302235
$794$	−1.76524	−0.0626462
$795$	0	0
$796$	13.9944	0.496018
$797$	−34.9004	−1.23623	−0.618117	−	0.786086i	$-0.712104\pi$
$797$	−34.9004	−1.23623	−0.618117	+	0.786086i	$0.712104\pi$
$798$	0	0
$799$	−25.0384	−0.885795
$800$	−5.99576	−0.211982
$801$	0	0
$802$	−3.00732	−0.106192
$803$	13.9070	0.490768
$804$	0	0
$805$	−0.391438	−0.0137964
$806$	1.31233	0.0462247
$807$	0	0
$808$	−3.80362	−0.133811
$809$	−29.4086	−1.03395	−0.516975	−	0.856001i	$-0.672942\pi$
$809$	−29.4086	−1.03395	−0.516975	+	0.856001i	$0.672942\pi$
$810$	0	0
$811$	41.8547	1.46972	0.734859	−	0.678220i	$-0.237249\pi$
$811$	41.8547	1.46972	0.734859	+	0.678220i	$0.237249\pi$
$812$	4.46116	0.156556
$813$	0	0
$814$	−0.870155	−0.0304989
$815$	−4.15468	−0.145532
$816$	0	0
$817$	−4.56543	−0.159724
$818$	3.76502	0.131641
$819$	0	0
$820$	1.88429	0.0658023
$821$	46.6870	1.62939	0.814694	−	0.579891i	$-0.196905\pi$
$821$	46.6870	1.62939	0.814694	+	0.579891i	$0.196905\pi$
$822$	0	0
$823$	15.4362	0.538071	0.269035	−	0.963130i	$-0.413295\pi$
$823$	15.4362	0.538071	0.269035	+	0.963130i	$0.413295\pi$
$824$	−4.72618	−0.164644
$825$	0	0
$826$	−0.276851	−0.00963287
$827$	22.0580	0.767030	0.383515	−	0.923535i	$-0.374713\pi$
$827$	22.0580	0.767030	0.383515	+	0.923535i	$0.374713\pi$
$828$	0	0
$829$	47.3457	1.64438	0.822191	−	0.569211i	$-0.192751\pi$
$829$	47.3457	1.64438	0.822191	+	0.569211i	$0.192751\pi$
$830$	0.0845158	0.00293359
$831$	0	0
$832$	10.9549	0.379792
$833$	−6.28548	−0.217779
$834$	0	0
$835$	1.78176	0.0616604
$836$	1.95282	0.0675398
$837$	0	0
$838$	−3.01442	−0.104132
$839$	−33.7490	−1.16514	−0.582572	−	0.812779i	$-0.697954\pi$
$839$	−33.7490	−1.16514	−0.582572	+	0.812779i	$0.697954\pi$
$840$	0	0
$841$	−23.9721	−0.826624
$842$	−0.440450	−0.0151789
$843$	0	0
$844$	−4.48986	−0.154547
$845$	3.08700	0.106196
$846$	0	0
$847$	10.0499	0.345319
$848$	−48.8661	−1.67807
$849$	0	0
$850$	3.16253	0.108474
$851$	−12.1813	−0.417568
$852$	0	0
$853$	6.51623	0.223111	0.111556	−	0.993758i	$-0.464417\pi$
$853$	6.51623	0.223111	0.111556	+	0.993758i	$0.464417\pi$
$854$	0.615612	0.0210658
$855$	0	0
$856$	−0.0831771	−0.00284294
$857$	−18.3967	−0.628419	−0.314209	−	0.949354i	$-0.601739\pi$
$857$	−18.3967	−0.628419	−0.314209	+	0.949354i	$0.601739\pi$
$858$	0	0
$859$	43.9167	1.49842	0.749209	−	0.662333i	$-0.230434\pi$
$859$	43.9167	1.49842	0.749209	+	0.662333i	$0.230434\pi$
$860$	−2.53091	−0.0863034
$861$	0	0
$862$	2.24986	0.0766307
$863$	46.1967	1.57255	0.786277	−	0.617875i	$-0.212006\pi$
$863$	46.1967	1.57255	0.786277	+	0.617875i	$0.212006\pi$
$864$	0	0
$865$	0.624966	0.0212495
$866$	1.60569	0.0545636
$867$	0	0
$868$	−18.0669	−0.613232
$869$	8.90438	0.302060
$870$	0	0
$871$	1.83567	0.0621992
$872$	−7.17139	−0.242854
$873$	0	0
$874$	−0.143629	−0.00485831
$875$	−2.78376	−0.0941081
$876$	0	0
$877$	−41.8033	−1.41160	−0.705799	−	0.708412i	$-0.749412\pi$
$877$	−41.8033	−1.41160	−0.705799	+	0.708412i	$0.749412\pi$
$878$	2.61724	0.0883275
$879$	0	0
$880$	1.07686	0.0363009
$881$	17.9451	0.604587	0.302293	−	0.953215i	$-0.402248\pi$
$881$	17.9451	0.604587	0.302293	+	0.953215i	$0.402248\pi$
$882$	0	0
$883$	46.4060	1.56169	0.780843	−	0.624728i	$-0.214790\pi$
$883$	46.4060	1.56169	0.780843	+	0.624728i	$0.214790\pi$
$884$	−17.6761	−0.594511
$885$	0	0
$886$	−1.77648	−0.0596821
$887$	−34.5487	−1.16003	−0.580016	−	0.814605i	$-0.696954\pi$
$887$	−34.5487	−1.16003	−0.580016	+	0.814605i	$0.696954\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−0.256050	−0.00858283
$891$	0	0
$892$	33.4687	1.12062
$893$	4.01135	0.134235
$894$	0	0
$895$	1.02425	0.0342370
$896$	3.22905	0.107875
$897$	0	0
$898$	2.66398	0.0888982
$899$	−20.3621	−0.679115
$900$	0	0
$901$	78.0077	2.59881
$902$	−0.336383	−0.0112003
$903$	0	0
$904$	−6.56064	−0.218204
$905$	−2.38995	−0.0794446
$906$	0	0
$907$	−15.3750	−0.510518	−0.255259	−	0.966873i	$-0.582161\pi$
$907$	−15.3750	−0.510518	−0.255259	+	0.966873i	$0.582161\pi$
$908$	−53.6751	−1.78127
$909$	0	0
$910$	−0.0405486	−0.00134417
$911$	−25.2602	−0.836908	−0.418454	−	0.908238i	$-0.637428\pi$
$911$	−25.2602	−0.836908	−0.418454	+	0.908238i	$0.637428\pi$
$912$	0	0
$913$	2.87171	0.0950399
$914$	0.470456	0.0155613
$915$	0	0
$916$	23.0494	0.761575
$917$	−16.7647	−0.553620
$918$	0	0
$919$	25.4760	0.840376	0.420188	−	0.907437i	$-0.361964\pi$
$919$	25.4760	0.840376	0.420188	+	0.907437i	$0.361964\pi$
$920$	−0.159663	−0.00526395
$921$	0	0
$922$	2.53903	0.0836184
$923$	2.51012	0.0826216
$924$	0	0
$925$	−42.9705	−1.41286
$926$	2.63829	0.0866996
$927$	0	0
$928$	2.73187	0.0896780
$929$	−23.7913	−0.780566	−0.390283	−	0.920695i	$-0.627623\pi$
$929$	−23.7913	−0.780566	−0.390283	+	0.920695i	$0.627623\pi$
$930$	0	0
$931$	1.00698	0.0330026
$932$	−20.5315	−0.672531
$933$	0	0
$934$	−1.33254	−0.0436020
$935$	−1.71905	−0.0562189
$936$	0	0
$937$	−29.6899	−0.969927	−0.484964	−	0.874534i	$-0.661167\pi$
$937$	−29.6899	−0.969927	−0.484964	+	0.874534i	$0.661167\pi$
$938$	0.132776	0.00433528
$939$	0	0
$940$	2.22375	0.0725308
$941$	40.2348	1.31162	0.655809	−	0.754927i	$-0.272328\pi$
$941$	40.2348	1.31162	0.655809	+	0.754927i	$0.272328\pi$
$942$	0	0
$943$	−4.70901	−0.153346
$944$	10.6619	0.347016
$945$	0	0
$946$	0.451817	0.0146899
$947$	−48.3106	−1.56988	−0.784942	−	0.619569i	$-0.787307\pi$
$947$	−48.3106	−1.56988	−0.784942	+	0.619569i	$0.787307\pi$
$948$	0	0
$949$	20.1670	0.654649
$950$	−0.506662	−0.0164383
$951$	0	0
$952$	−2.56378	−0.0830926
$953$	−46.8476	−1.51754	−0.758772	−	0.651357i	$-0.774200\pi$
$953$	−46.8476	−1.51754	−0.758772	+	0.651357i	$0.774200\pi$
$954$	0	0
$955$	−4.97588	−0.161016
$956$	−30.2112	−0.977099
$957$	0	0
$958$	0.204884	0.00661950
$959$	16.3690	0.528581
$960$	0	0
$961$	51.4633	1.66011
$962$	−1.26184	−0.0406833
$963$	0	0
$964$	−10.3264	−0.332591
$965$	0.600438	0.0193288
$966$	0	0
$967$	−11.3136	−0.363822	−0.181911	−	0.983315i	$-0.558228\pi$
$967$	−11.3136	−0.363822	−0.181911	+	0.983315i	$0.558228\pi$
$968$	4.09924	0.131755
$969$	0	0
$970$	0.0360871	0.00115869
$971$	48.0486	1.54195	0.770976	−	0.636864i	$-0.219769\pi$
$971$	48.0486	1.54195	0.770976	+	0.636864i	$0.219769\pi$
$972$	0	0
$973$	−1.38049	−0.0442565
$974$	−1.69895	−0.0544378
$975$	0	0
$976$	−23.7081	−0.758879
$977$	1.06017	0.0339177	0.0169588	−	0.999856i	$-0.494602\pi$
$977$	1.06017	0.0339177	0.0169588	+	0.999856i	$0.494602\pi$
$978$	0	0
$979$	−8.70019	−0.278059
$980$	0.558236	0.0178322
$981$	0	0
$982$	2.14471	0.0684404
$983$	−2.35200	−0.0750172	−0.0375086	−	0.999296i	$-0.511942\pi$
$983$	−2.35200	−0.0750172	−0.0375086	+	0.999296i	$0.511942\pi$
$984$	0	0
$985$	−4.77259	−0.152067
$986$	−1.44095	−0.0458893
$987$	0	0
$988$	2.83185	0.0900931
$989$	6.32497	0.201122
$990$	0	0
$991$	−26.3037	−0.835564	−0.417782	−	0.908547i	$-0.637193\pi$
$991$	−26.3037	−0.835564	−0.417782	+	0.908547i	$0.637193\pi$
$992$	−11.0636	−0.351270
$993$	0	0
$994$	0.181560	0.00575872
$995$	1.97362	0.0625679
$996$	0	0
$997$	−24.0943	−0.763076	−0.381538	−	0.924353i	$-0.624605\pi$
$997$	−24.0943	−0.763076	−0.381538	+	0.924353i	$0.624605\pi$
$998$	1.94109	0.0614440
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.t.1.9		16
3.2	odd	2		889.2.a.c.1.8	✓	16
21.20	even	2		6223.2.a.k.1.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.8	✓	16	3.2	odd	2
6223.2.a.k.1.8		16	21.20	even	2
8001.2.a.t.1.9		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.102239 q^{2} -1.98955 q^{4} -0.280585 q^{5} -1.00000 q^{7} -0.407889 q^{8} -0.0286868 q^{10} -0.974733 q^{11} -1.41349 q^{13} -0.102239 q^{14} +3.93739 q^{16} -6.28548 q^{17} +1.00698 q^{19} +0.558236 q^{20} -0.0996561 q^{22} -1.39508 q^{23} -4.92127 q^{25} -0.144515 q^{26} +1.98955 q^{28} +2.24230 q^{29} -9.08093 q^{31} +1.21833 q^{32} -0.642624 q^{34} +0.280585 q^{35} +8.73158 q^{37} +0.102954 q^{38} +0.114447 q^{40} +3.37544 q^{41} -4.53376 q^{43} +1.93928 q^{44} -0.142632 q^{46} +3.98353 q^{47} +1.00000 q^{49} -0.503148 q^{50} +2.81221 q^{52} -12.4108 q^{53} +0.273495 q^{55} +0.407889 q^{56} +0.229251 q^{58} +2.70787 q^{59} -6.02128 q^{61} -0.928429 q^{62} -7.75022 q^{64} +0.396604 q^{65} -1.29868 q^{67} +12.5053 q^{68} +0.0286868 q^{70} -1.77583 q^{71} -14.2675 q^{73} +0.892711 q^{74} -2.00344 q^{76} +0.974733 q^{77} -9.13520 q^{79} -1.10477 q^{80} +0.345103 q^{82} -2.94615 q^{83} +1.76361 q^{85} -0.463529 q^{86} +0.397583 q^{88} +8.92572 q^{89} +1.41349 q^{91} +2.77558 q^{92} +0.407274 q^{94} -0.282544 q^{95} -1.25797 q^{97} +0.102239 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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