LMFDB - Embedded newform 4056.2.a.be.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4056,2,Mod(1,4056)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4056.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4056, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,4,0,2,0,4,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.25488.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 6x + 6 $ x^4 - 9x^2 - 6x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.47535$ of defining polynomial
Character	$\chi$	$=$	4056.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.20740 q^{5} -3.55539 q^{7} +1.00000 q^{9} +5.08003 q^{11} +4.20740 q^{15} +3.25670 q^{17} -3.33477 q^{19} -3.55539 q^{21} +0.384069 q^{23} +12.7022 q^{25} +1.00000 q^{27} -7.28744 q^{29} +6.47535 q^{31} +5.08003 q^{33} -14.9589 q^{35} +3.12737 q^{37} +6.77404 q^{41} +4.15811 q^{43} +4.20740 q^{45} -5.49484 q^{47} +5.64077 q^{49} +3.25670 q^{51} +0.613974 q^{53} +21.3737 q^{55} -3.33477 q^{57} -7.87891 q^{59} -8.31227 q^{61} -3.55539 q^{63} +2.06878 q^{67} +0.384069 q^{69} -2.82529 q^{71} +3.21865 q^{73} +12.7022 q^{75} -18.0615 q^{77} -2.64077 q^{79} +1.00000 q^{81} +1.96926 q^{83} +13.7022 q^{85} -7.28744 q^{87} +14.1601 q^{89} +6.47535 q^{93} -14.0307 q^{95} +8.86766 q^{97} +5.08003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} + 2 q^{7} + 4 q^{9} + 10 q^{11} + 4 q^{15} + 12 q^{17} + 2 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 4 q^{27} - 6 q^{29} + 20 q^{31} + 10 q^{33} - 10 q^{35} + 10 q^{37} + 6 q^{41}+ \cdots + 10 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 + 2 * q^7 + 4 * q^9 + 10 * q^11 + 4 * q^15 + 12 * q^17 + 2 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 4 * q^27 - 6 * q^29 + 20 * q^31 + 10 * q^33 - 10 * q^35 + 10 * q^37 + 6 * q^41 - 8 * q^43 + 4 * q^45 + 14 * q^47 + 18 * q^49 + 12 * q^51 + 2 * q^53 + 10 * q^55 + 2 * q^57 + 8 * q^59 - 2 * q^61 + 2 * q^63 - 10 * q^67 - 2 * q^69 - 6 * q^71 + 8 * q^73 + 2 * q^75 - 28 * q^77 - 6 * q^79 + 4 * q^81 + 30 * q^83 + 6 * q^85 - 6 * q^87 + 36 * q^89 + 20 * q^93 - 34 * q^95 - 12 * q^97 + 10 * q^99

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	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	4.20740	1.88161	0.940804	−	0.338951i	$-0.110072\pi$
$5$	4.20740	1.88161	0.940804	+	0.338951i	$0.110072\pi$
$6$	0	0
$7$	−3.55539	−1.34381	−0.671905	−	0.740638i	$-0.734524\pi$
$7$	−3.55539	−1.34381	−0.671905	+	0.740638i	$0.734524\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.08003	1.53169	0.765844	−	0.643027i	$-0.222322\pi$
$11$	5.08003	1.53169	0.765844	+	0.643027i	$0.222322\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	4.20740	1.08635
$16$	0	0
$17$	3.25670	0.789865	0.394933	−	0.918710i	$-0.370768\pi$
$17$	3.25670	0.789865	0.394933	+	0.918710i	$0.370768\pi$
$18$	0	0
$19$	−3.33477	−0.765050	−0.382525	−	0.923945i	$-0.624945\pi$
$19$	−3.33477	−0.765050	−0.382525	+	0.923945i	$0.624945\pi$
$20$	0	0
$21$	−3.55539	−0.775849
$22$	0	0
$23$	0.384069	0.0800838	0.0400419	−	0.999198i	$-0.487251\pi$
$23$	0.384069	0.0800838	0.0400419	+	0.999198i	$0.487251\pi$
$24$	0	0
$25$	12.7022	2.54045
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.28744	−1.35324	−0.676621	−	0.736331i	$-0.736557\pi$
$29$	−7.28744	−1.35324	−0.676621	+	0.736331i	$0.736557\pi$
$30$	0	0
$31$	6.47535	1.16301	0.581504	−	0.813544i	$-0.302465\pi$
$31$	6.47535	1.16301	0.581504	+	0.813544i	$0.302465\pi$
$32$	0	0
$33$	5.08003	0.884320
$34$	0	0
$35$	−14.9589	−2.52852
$36$	0	0
$37$	3.12737	0.514137	0.257068	−	0.966393i	$-0.417243\pi$
$37$	3.12737	0.514137	0.257068	+	0.966393i	$0.417243\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.77404	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$41$	6.77404	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$42$	0	0
$43$	4.15811	0.634106	0.317053	−	0.948408i	$-0.397307\pi$
$43$	4.15811	0.634106	0.317053	+	0.948408i	$0.397307\pi$
$44$	0	0
$45$	4.20740	0.627203
$46$	0	0
$47$	−5.49484	−0.801505	−0.400752	−	0.916186i	$-0.631251\pi$
$47$	−5.49484	−0.801505	−0.400752	+	0.916186i	$0.631251\pi$
$48$	0	0
$49$	5.64077	0.805824
$50$	0	0
$51$	3.25670	0.456029
$52$	0	0
$53$	0.613974	0.0843358	0.0421679	−	0.999111i	$-0.486574\pi$
$53$	0.613974	0.0843358	0.0421679	+	0.999111i	$0.486574\pi$
$54$	0	0
$55$	21.3737	2.88204
$56$	0	0
$57$	−3.33477	−0.441702
$58$	0	0
$59$	−7.87891	−1.02575	−0.512873	−	0.858464i	$-0.671419\pi$
$59$	−7.87891	−1.02575	−0.512873	+	0.858464i	$0.671419\pi$
$60$	0	0
$61$	−8.31227	−1.06428	−0.532139	−	0.846657i	$-0.678612\pi$
$61$	−8.31227	−1.06428	−0.532139	+	0.846657i	$0.678612\pi$
$62$	0	0
$63$	−3.55539	−0.447936
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.06878	0.252742	0.126371	−	0.991983i	$-0.459667\pi$
$67$	2.06878	0.252742	0.126371	+	0.991983i	$0.459667\pi$
$68$	0	0
$69$	0.384069	0.0462364
$70$	0	0
$71$	−2.82529	−0.335301	−0.167650	−	0.985847i	$-0.553618\pi$
$71$	−2.82529	−0.335301	−0.167650	+	0.985847i	$0.553618\pi$
$72$	0	0
$73$	3.21865	0.376715	0.188357	−	0.982101i	$-0.439684\pi$
$73$	3.21865	0.376715	0.188357	+	0.982101i	$0.439684\pi$
$74$	0	0
$75$	12.7022	1.46673
$76$	0	0
$77$	−18.0615	−2.05830
$78$	0	0
$79$	−2.64077	−0.297109	−0.148555	−	0.988904i	$-0.547462\pi$
$79$	−2.64077	−0.297109	−0.148555	+	0.988904i	$0.547462\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.96926	0.216155	0.108077	−	0.994142i	$-0.465531\pi$
$83$	1.96926	0.216155	0.108077	+	0.994142i	$0.465531\pi$
$84$	0	0
$85$	13.7022	1.48622
$86$	0	0
$87$	−7.28744	−0.781295
$88$	0	0
$89$	14.1601	1.50096	0.750482	−	0.660891i	$-0.229821\pi$
$89$	14.1601	1.50096	0.750482	+	0.660891i	$0.229821\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.47535	0.671463
$94$	0	0
$95$	−14.0307	−1.43952
$96$	0	0
$97$	8.86766	0.900374	0.450187	−	0.892934i	$-0.351357\pi$
$97$	8.86766	0.900374	0.450187	+	0.892934i	$0.351357\pi$
$98$	0	0
$99$	5.08003	0.510563
$100$	0	0
$101$	9.44750	0.940062	0.470031	−	0.882650i	$-0.344243\pi$
$101$	9.44750	0.940062	0.470031	+	0.882650i	$0.344243\pi$
$102$	0	0
$103$	−2.15811	−0.212645	−0.106322	−	0.994332i	$-0.533908\pi$
$103$	−2.15811	−0.212645	−0.106322	+	0.994332i	$0.533908\pi$
$104$	0	0
$105$	−14.9589	−1.45984
$106$	0	0
$107$	−18.7003	−1.80782	−0.903912	−	0.427718i	$-0.859318\pi$
$107$	−18.7003	−1.80782	−0.903912	+	0.427718i	$0.859318\pi$
$108$	0	0
$109$	14.4754	1.38649	0.693244	−	0.720703i	$-0.256181\pi$
$109$	14.4754	1.38649	0.693244	+	0.720703i	$0.256181\pi$
$110$	0	0
$111$	3.12737	0.296837
$112$	0	0
$113$	2.74330	0.258068	0.129034	−	0.991640i	$-0.458812\pi$
$113$	2.74330	0.258068	0.129034	+	0.991640i	$0.458812\pi$
$114$	0	0
$115$	1.61593	0.150686
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.5788	−1.06143
$120$	0	0
$121$	14.8067	1.34607
$122$	0	0
$123$	6.77404	0.610795
$124$	0	0
$125$	32.4064	2.89852
$126$	0	0
$127$	−10.1274	−0.898659	−0.449329	−	0.893366i	$-0.648337\pi$
$127$	−10.1274	−0.898659	−0.449329	+	0.893366i	$0.648337\pi$
$128$	0	0
$129$	4.15811	0.366101
$130$	0	0
$131$	−13.3430	−1.16578	−0.582892	−	0.812550i	$-0.698079\pi$
$131$	−13.3430	−1.16578	−0.582892	+	0.812550i	$0.698079\pi$
$132$	0	0
$133$	11.8564	1.02808
$134$	0	0
$135$	4.20740	0.362116
$136$	0	0
$137$	11.8955	1.01630	0.508151	−	0.861268i	$-0.330329\pi$
$137$	11.8955	1.01630	0.508151	+	0.861268i	$0.330329\pi$
$138$	0	0
$139$	13.4207	1.13833	0.569165	−	0.822223i	$-0.307267\pi$
$139$	13.4207	1.13833	0.569165	+	0.822223i	$0.307267\pi$
$140$	0	0
$141$	−5.49484	−0.462749
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−30.6612	−2.54627
$146$	0	0
$147$	5.64077	0.465243
$148$	0	0
$149$	−5.51144	−0.451515	−0.225757	−	0.974184i	$-0.572486\pi$
$149$	−5.51144	−0.451515	−0.225757	+	0.974184i	$0.572486\pi$
$150$	0	0
$151$	−5.08003	−0.413407	−0.206704	−	0.978404i	$-0.566274\pi$
$151$	−5.08003	−0.413407	−0.206704	+	0.978404i	$0.566274\pi$
$152$	0	0
$153$	3.25670	0.263288
$154$	0	0
$155$	27.2444	2.18832
$156$	0	0
$157$	−12.5670	−1.00296	−0.501478	−	0.865170i	$-0.667210\pi$
$157$	−12.5670	−1.00296	−0.501478	+	0.865170i	$0.667210\pi$
$158$	0	0
$159$	0.613974	0.0486913
$160$	0	0
$161$	−1.36551	−0.107617
$162$	0	0
$163$	8.81799	0.690678	0.345339	−	0.938478i	$-0.387764\pi$
$163$	8.81799	0.690678	0.345339	+	0.938478i	$0.387764\pi$
$164$	0	0
$165$	21.3737	1.66394
$166$	0	0
$167$	14.6695	1.13516	0.567582	−	0.823317i	$-0.307879\pi$
$167$	14.6695	1.13516	0.567582	+	0.823317i	$0.307879\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3.33477	−0.255017
$172$	0	0
$173$	−2.41481	−0.183594	−0.0917972	−	0.995778i	$-0.529261\pi$
$173$	−2.41481	−0.183594	−0.0917972	+	0.995778i	$0.529261\pi$
$174$	0	0
$175$	−45.1614	−3.41388
$176$	0	0
$177$	−7.87891	−0.592215
$178$	0	0
$179$	−5.35728	−0.400422	−0.200211	−	0.979753i	$-0.564163\pi$
$179$	−5.35728	−0.400422	−0.200211	+	0.979753i	$0.564163\pi$
$180$	0	0
$181$	−17.6715	−1.31351	−0.656756	−	0.754103i	$-0.728072\pi$
$181$	−17.6715	−1.31351	−0.656756	+	0.754103i	$0.728072\pi$
$182$	0	0
$183$	−8.31227	−0.614461
$184$	0	0
$185$	13.1581	0.967403
$186$	0	0
$187$	16.5441	1.20983
$188$	0	0
$189$	−3.55539	−0.258616
$190$	0	0
$191$	4.57487	0.331026	0.165513	−	0.986208i	$-0.447072\pi$
$191$	4.57487	0.331026	0.165513	+	0.986208i	$0.447072\pi$
$192$	0	0
$193$	−16.8428	−1.21237	−0.606186	−	0.795323i	$-0.707301\pi$
$193$	−16.8428	−1.21237	−0.606186	+	0.795323i	$0.707301\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.85641	−0.132263	−0.0661317	−	0.997811i	$-0.521066\pi$
$197$	−1.85641	−0.132263	−0.0661317	+	0.997811i	$0.521066\pi$
$198$	0	0
$199$	−18.9877	−1.34600	−0.673002	−	0.739641i	$-0.734995\pi$
$199$	−18.9877	−1.34600	−0.673002	+	0.739641i	$0.734995\pi$
$200$	0	0
$201$	2.06878	0.145921
$202$	0	0
$203$	25.9096	1.81850
$204$	0	0
$205$	28.5011	1.99060
$206$	0	0
$207$	0.384069	0.0266946
$208$	0	0
$209$	−16.9408	−1.17182
$210$	0	0
$211$	12.2424	0.842804	0.421402	−	0.906874i	$-0.361538\pi$
$211$	12.2424	0.842804	0.421402	+	0.906874i	$0.361538\pi$
$212$	0	0
$213$	−2.82529	−0.193586
$214$	0	0
$215$	17.4948	1.19314
$216$	0	0
$217$	−23.0224	−1.56286
$218$	0	0
$219$	3.21865	0.217497
$220$	0	0
$221$	0	0
$222$	0	0
$223$	20.7085	1.38675	0.693373	−	0.720579i	$-0.256124\pi$
$223$	20.7085	1.38675	0.693373	+	0.720579i	$0.256124\pi$
$224$	0	0
$225$	12.7022	0.846816
$226$	0	0
$227$	−8.63881	−0.573378	−0.286689	−	0.958024i	$-0.592555\pi$
$227$	−8.63881	−0.573378	−0.286689	+	0.958024i	$0.592555\pi$
$228$	0	0
$229$	20.1111	1.32898	0.664491	−	0.747296i	$-0.268648\pi$
$229$	20.1111	1.32898	0.664491	+	0.747296i	$0.268648\pi$
$230$	0	0
$231$	−18.0615	−1.18836
$232$	0	0
$233$	−4.41089	−0.288967	−0.144484	−	0.989507i	$-0.546152\pi$
$233$	−4.41089	−0.288967	−0.144484	+	0.989507i	$0.546152\pi$
$234$	0	0
$235$	−23.1190	−1.50812
$236$	0	0
$237$	−2.64077	−0.171536
$238$	0	0
$239$	3.51734	0.227518	0.113759	−	0.993508i	$-0.463711\pi$
$239$	3.51734	0.227518	0.113759	+	0.993508i	$0.463711\pi$
$240$	0	0
$241$	−4.74330	−0.305543	−0.152771	−	0.988262i	$-0.548820\pi$
$241$	−4.74330	−0.305543	−0.152771	+	0.988262i	$0.548820\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	23.7330	1.51624
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.96926	0.124797
$250$	0	0
$251$	6.98968	0.441185	0.220592	−	0.975366i	$-0.429201\pi$
$251$	6.98968	0.441185	0.220592	+	0.975366i	$0.429201\pi$
$252$	0	0
$253$	1.95108	0.122663
$254$	0	0
$255$	13.7022	0.858068
$256$	0	0
$257$	−4.71010	−0.293808	−0.146904	−	0.989151i	$-0.546931\pi$
$257$	−4.71010	−0.293808	−0.146904	+	0.989151i	$0.546931\pi$
$258$	0	0
$259$	−11.1190	−0.690902
$260$	0	0
$261$	−7.28744	−0.451081
$262$	0	0
$263$	−4.12933	−0.254625	−0.127313	−	0.991863i	$-0.540635\pi$
$263$	−4.12933	−0.254625	−0.127313	+	0.991863i	$0.540635\pi$
$264$	0	0
$265$	2.58324	0.158687
$266$	0	0
$267$	14.1601	0.866582
$268$	0	0
$269$	16.2587	0.991308	0.495654	−	0.868520i	$-0.334928\pi$
$269$	16.2587	0.991308	0.495654	+	0.868520i	$0.334928\pi$
$270$	0	0
$271$	−32.3171	−1.96313	−0.981563	−	0.191137i	$-0.938783\pi$
$271$	−32.3171	−1.96313	−0.981563	+	0.191137i	$0.938783\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	64.5278	3.89117
$276$	0	0
$277$	−23.7369	−1.42621	−0.713106	−	0.701056i	$-0.752712\pi$
$277$	−23.7369	−1.42621	−0.713106	+	0.701056i	$0.752712\pi$
$278$	0	0
$279$	6.47535	0.387669
$280$	0	0
$281$	−25.5812	−1.52604	−0.763022	−	0.646373i	$-0.776285\pi$
$281$	−25.5812	−1.52604	−0.763022	+	0.646373i	$0.776285\pi$
$282$	0	0
$283$	−11.6486	−0.692439	−0.346220	−	0.938154i	$-0.612535\pi$
$283$	−11.6486	−0.692439	−0.346220	+	0.938154i	$0.612535\pi$
$284$	0	0
$285$	−14.0307	−0.831109
$286$	0	0
$287$	−24.0843	−1.42165
$288$	0	0
$289$	−6.39392	−0.376113
$290$	0	0
$291$	8.86766	0.519831
$292$	0	0
$293$	−9.85016	−0.575452	−0.287726	−	0.957713i	$-0.592899\pi$
$293$	−9.85016	−0.575452	−0.287726	+	0.957713i	$0.592899\pi$
$294$	0	0
$295$	−33.1497	−1.93005
$296$	0	0
$297$	5.08003	0.294773
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−14.7837	−0.852117
$302$	0	0
$303$	9.44750	0.542745
$304$	0	0
$305$	−34.9731	−2.00255
$306$	0	0
$307$	−12.4182	−0.708744	−0.354372	−	0.935105i	$-0.615305\pi$
$307$	−12.4182	−0.708744	−0.354372	+	0.935105i	$0.615305\pi$
$308$	0	0
$309$	−2.15811	−0.122771
$310$	0	0
$311$	7.42267	0.420901	0.210450	−	0.977605i	$-0.432507\pi$
$311$	7.42267	0.420901	0.210450	+	0.977605i	$0.432507\pi$
$312$	0	0
$313$	27.9799	1.58152	0.790758	−	0.612129i	$-0.209687\pi$
$313$	27.9799	1.58152	0.790758	+	0.612129i	$0.209687\pi$
$314$	0	0
$315$	−14.9589	−0.842841
$316$	0	0
$317$	−8.02484	−0.450720	−0.225360	−	0.974276i	$-0.572356\pi$
$317$	−8.02484	−0.450720	−0.225360	+	0.974276i	$0.572356\pi$
$318$	0	0
$319$	−37.0204	−2.07275
$320$	0	0
$321$	−18.7003	−1.04375
$322$	0	0
$323$	−10.8604	−0.604286
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.4754	0.800489
$328$	0	0
$329$	19.5363	1.07707
$330$	0	0
$331$	18.9252	1.04022	0.520112	−	0.854098i	$-0.325890\pi$
$331$	18.9252	1.04022	0.520112	+	0.854098i	$0.325890\pi$
$332$	0	0
$333$	3.12737	0.171379
$334$	0	0
$335$	8.70420	0.475561
$336$	0	0
$337$	13.8296	0.753347	0.376674	−	0.926346i	$-0.377068\pi$
$337$	13.8296	0.753347	0.376674	+	0.926346i	$0.377068\pi$
$338$	0	0
$339$	2.74330	0.148996
$340$	0	0
$341$	32.8950	1.78136
$342$	0	0
$343$	4.83260	0.260936
$344$	0	0
$345$	1.61593	0.0869988
$346$	0	0
$347$	−16.9093	−0.907737	−0.453869	−	0.891069i	$-0.649956\pi$
$347$	−16.9093	−0.907737	−0.453869	+	0.891069i	$0.649956\pi$
$348$	0	0
$349$	15.7462	0.842874	0.421437	−	0.906858i	$-0.361526\pi$
$349$	15.7462	0.842874	0.421437	+	0.906858i	$0.361526\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.9960	−1.17073	−0.585363	−	0.810771i	$-0.699048\pi$
$353$	−21.9960	−1.17073	−0.585363	+	0.810771i	$0.699048\pi$
$354$	0	0
$355$	−11.8871	−0.630904
$356$	0	0
$357$	−11.5788	−0.612816
$358$	0	0
$359$	−24.5606	−1.29626	−0.648130	−	0.761530i	$-0.724449\pi$
$359$	−24.5606	−1.29626	−0.648130	+	0.761530i	$0.724449\pi$
$360$	0	0
$361$	−7.87928	−0.414699
$362$	0	0
$363$	14.8067	0.777152
$364$	0	0
$365$	13.5422	0.708830
$366$	0	0
$367$	8.20778	0.428443	0.214221	−	0.976785i	$-0.431279\pi$
$367$	8.20778	0.428443	0.214221	+	0.976785i	$0.431279\pi$
$368$	0	0
$369$	6.77404	0.352642
$370$	0	0
$371$	−2.18291	−0.113331
$372$	0	0
$373$	−10.1190	−0.523942	−0.261971	−	0.965076i	$-0.584373\pi$
$373$	−10.1190	−0.523942	−0.261971	+	0.965076i	$0.584373\pi$
$374$	0	0
$375$	32.4064	1.67346
$376$	0	0
$377$	0	0
$378$	0	0
$379$	21.1614	1.08699	0.543493	−	0.839413i	$-0.317101\pi$
$379$	21.1614	1.08699	0.543493	+	0.839413i	$0.317101\pi$
$380$	0	0
$381$	−10.1274	−0.518841
$382$	0	0
$383$	26.1151	1.33442	0.667209	−	0.744871i	$-0.267489\pi$
$383$	26.1151	1.33442	0.667209	+	0.744871i	$0.267489\pi$
$384$	0	0
$385$	−75.9919	−3.87291
$386$	0	0
$387$	4.15811	0.211369
$388$	0	0
$389$	9.18493	0.465695	0.232847	−	0.972513i	$-0.425196\pi$
$389$	9.18493	0.465695	0.232847	+	0.972513i	$0.425196\pi$
$390$	0	0
$391$	1.25080	0.0632554
$392$	0	0
$393$	−13.3430	−0.673066
$394$	0	0
$395$	−11.1108	−0.559044
$396$	0	0
$397$	−11.7912	−0.591783	−0.295892	−	0.955222i	$-0.595617\pi$
$397$	−11.7912	−0.591783	−0.295892	+	0.955222i	$0.595617\pi$
$398$	0	0
$399$	11.8564	0.593563
$400$	0	0
$401$	−30.4822	−1.52221	−0.761104	−	0.648630i	$-0.775342\pi$
$401$	−30.4822	−1.52221	−0.761104	+	0.648630i	$0.775342\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	4.20740	0.209068
$406$	0	0
$407$	15.8871	0.787497
$408$	0	0
$409$	16.8096	0.831182	0.415591	−	0.909552i	$-0.363575\pi$
$409$	16.8096	0.831182	0.415591	+	0.909552i	$0.363575\pi$
$410$	0	0
$411$	11.8955	0.586762
$412$	0	0
$413$	28.0126	1.37841
$414$	0	0
$415$	8.28548	0.406718
$416$	0	0
$417$	13.4207	0.657215
$418$	0	0
$419$	−24.5788	−1.20075	−0.600376	−	0.799718i	$-0.704982\pi$
$419$	−24.5788	−1.20075	−0.600376	+	0.799718i	$0.704982\pi$
$420$	0	0
$421$	11.1157	0.541748	0.270874	−	0.962615i	$-0.412687\pi$
$421$	11.1157	0.541748	0.270874	+	0.962615i	$0.412687\pi$
$422$	0	0
$423$	−5.49484	−0.267168
$424$	0	0
$425$	41.3674	2.00661
$426$	0	0
$427$	29.5533	1.43019
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.5606	1.76106	0.880531	−	0.473988i	$-0.157186\pi$
$431$	36.5606	1.76106	0.880531	+	0.473988i	$0.157186\pi$
$432$	0	0
$433$	−19.1765	−0.921566	−0.460783	−	0.887513i	$-0.652431\pi$
$433$	−19.1765	−0.921566	−0.460783	+	0.887513i	$0.652431\pi$
$434$	0	0
$435$	−30.6612	−1.47009
$436$	0	0
$437$	−1.28078	−0.0612681
$438$	0	0
$439$	23.8213	1.13693	0.568463	−	0.822709i	$-0.307538\pi$
$439$	23.8213	1.13693	0.568463	+	0.822709i	$0.307538\pi$
$440$	0	0
$441$	5.64077	0.268608
$442$	0	0
$443$	6.48020	0.307884	0.153942	−	0.988080i	$-0.450803\pi$
$443$	6.48020	0.307884	0.153942	+	0.988080i	$0.450803\pi$
$444$	0	0
$445$	59.5771	2.82423
$446$	0	0
$447$	−5.51144	−0.260682
$448$	0	0
$449$	7.11077	0.335578	0.167789	−	0.985823i	$-0.446337\pi$
$449$	7.11077	0.335578	0.167789	+	0.985823i	$0.446337\pi$
$450$	0	0
$451$	34.4123	1.62041
$452$	0	0
$453$	−5.08003	−0.238681
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.71558	−0.0802514	−0.0401257	−	0.999195i	$-0.512776\pi$
$457$	−1.71558	−0.0802514	−0.0401257	+	0.999195i	$0.512776\pi$
$458$	0	0
$459$	3.25670	0.152010
$460$	0	0
$461$	32.5129	1.51428	0.757139	−	0.653254i	$-0.226597\pi$
$461$	32.5129	1.51428	0.757139	+	0.653254i	$0.226597\pi$
$462$	0	0
$463$	3.37282	0.156748	0.0783741	−	0.996924i	$-0.475027\pi$
$463$	3.37282	0.156748	0.0783741	+	0.996924i	$0.475027\pi$
$464$	0	0
$465$	27.2444	1.26343
$466$	0	0
$467$	−19.3241	−0.894212	−0.447106	−	0.894481i	$-0.647545\pi$
$467$	−19.3241	−0.894212	−0.447106	+	0.894481i	$0.647545\pi$
$468$	0	0
$469$	−7.35532	−0.339637
$470$	0	0
$471$	−12.5670	−0.579057
$472$	0	0
$473$	21.1233	0.971252
$474$	0	0
$475$	−42.3591	−1.94357
$476$	0	0
$477$	0.613974	0.0281119
$478$	0	0
$479$	−8.72313	−0.398570	−0.199285	−	0.979942i	$-0.563862\pi$
$479$	−8.72313	−0.398570	−0.199285	+	0.979942i	$0.563862\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−1.36551	−0.0621329
$484$	0	0
$485$	37.3098	1.69415
$486$	0	0
$487$	−11.6431	−0.527599	−0.263800	−	0.964578i	$-0.584976\pi$
$487$	−11.6431	−0.527599	−0.263800	+	0.964578i	$0.584976\pi$
$488$	0	0
$489$	8.81799	0.398763
$490$	0	0
$491$	7.77208	0.350749	0.175375	−	0.984502i	$-0.443886\pi$
$491$	7.77208	0.350749	0.175375	+	0.984502i	$0.443886\pi$
$492$	0	0
$493$	−23.7330	−1.06888
$494$	0	0
$495$	21.3737	0.960679
$496$	0	0
$497$	10.0450	0.450580
$498$	0	0
$499$	20.3434	0.910695	0.455348	−	0.890314i	$-0.349515\pi$
$499$	20.3434	0.910695	0.455348	+	0.890314i	$0.349515\pi$
$500$	0	0
$501$	14.6695	0.655387
$502$	0	0
$503$	−21.1522	−0.943130	−0.471565	−	0.881831i	$-0.656311\pi$
$503$	−21.1522	−0.943130	−0.471565	+	0.881831i	$0.656311\pi$
$504$	0	0
$505$	39.7495	1.76883
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.0803	−1.68788	−0.843939	−	0.536438i	$-0.819769\pi$
$509$	−38.0803	−1.68788	−0.843939	+	0.536438i	$0.819769\pi$
$510$	0	0
$511$	−11.4436	−0.506233
$512$	0	0
$513$	−3.33477	−0.147234
$514$	0	0
$515$	−9.08003	−0.400114
$516$	0	0
$517$	−27.9140	−1.22765
$518$	0	0
$519$	−2.41481	−0.105998
$520$	0	0
$521$	20.7433	0.908781	0.454390	−	0.890803i	$-0.349857\pi$
$521$	20.7433	0.908781	0.454390	+	0.890803i	$0.349857\pi$
$522$	0	0
$523$	−4.94247	−0.216119	−0.108060	−	0.994144i	$-0.534464\pi$
$523$	−4.94247	−0.216119	−0.108060	+	0.994144i	$0.534464\pi$
$524$	0	0
$525$	−45.1614	−1.97100
$526$	0	0
$527$	21.0883	0.918619
$528$	0	0
$529$	−22.8525	−0.993587
$530$	0	0
$531$	−7.87891	−0.341915
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−78.6796	−3.40162
$536$	0	0
$537$	−5.35728	−0.231184
$538$	0	0
$539$	28.6553	1.23427
$540$	0	0
$541$	43.7196	1.87965	0.939827	−	0.341650i	$-0.110986\pi$
$541$	43.7196	1.87965	0.939827	+	0.341650i	$0.110986\pi$
$542$	0	0
$543$	−17.6715	−0.758357
$544$	0	0
$545$	60.9036	2.60883
$546$	0	0
$547$	39.1028	1.67191	0.835957	−	0.548795i	$-0.184913\pi$
$547$	39.1028	1.67191	0.835957	+	0.548795i	$0.184913\pi$
$548$	0	0
$549$	−8.31227	−0.354759
$550$	0	0
$551$	24.3020	1.03530
$552$	0	0
$553$	9.38894	0.399259
$554$	0	0
$555$	13.1581	0.558531
$556$	0	0
$557$	−23.9916	−1.01656	−0.508279	−	0.861192i	$-0.669718\pi$
$557$	−23.9916	−1.01656	−0.508279	+	0.861192i	$0.669718\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	16.5441	0.698494
$562$	0	0
$563$	10.4763	0.441523	0.220761	−	0.975328i	$-0.429146\pi$
$563$	10.4763	0.441523	0.220761	+	0.975328i	$0.429146\pi$
$564$	0	0
$565$	11.5422	0.485583
$566$	0	0
$567$	−3.55539	−0.149312
$568$	0	0
$569$	30.7964	1.29105	0.645526	−	0.763738i	$-0.276638\pi$
$569$	30.7964	1.29105	0.645526	+	0.763738i	$0.276638\pi$
$570$	0	0
$571$	9.52126	0.398452	0.199226	−	0.979954i	$-0.436157\pi$
$571$	9.52126	0.398452	0.199226	+	0.979954i	$0.436157\pi$
$572$	0	0
$573$	4.57487	0.191118
$574$	0	0
$575$	4.87853	0.203449
$576$	0	0
$577$	−47.7495	−1.98784	−0.993918	−	0.110124i	$-0.964875\pi$
$577$	−47.7495	−1.98784	−0.993918	+	0.110124i	$0.964875\pi$
$578$	0	0
$579$	−16.8428	−0.699964
$580$	0	0
$581$	−7.00149	−0.290471
$582$	0	0
$583$	3.11901	0.129176
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−39.4845	−1.62970	−0.814851	−	0.579671i	$-0.803181\pi$
$587$	−39.4845	−1.62970	−0.814851	+	0.579671i	$0.803181\pi$
$588$	0	0
$589$	−21.5938	−0.889758
$590$	0	0
$591$	−1.85641	−0.0763624
$592$	0	0
$593$	20.0166	0.821983	0.410992	−	0.911639i	$-0.365183\pi$
$593$	20.0166	0.821983	0.410992	+	0.911639i	$0.365183\pi$
$594$	0	0
$595$	−48.7168	−1.99719
$596$	0	0
$597$	−18.9877	−0.777116
$598$	0	0
$599$	−19.8517	−0.811120	−0.405560	−	0.914068i	$-0.632923\pi$
$599$	−19.8517	−0.811120	−0.405560	+	0.914068i	$0.632923\pi$
$600$	0	0
$601$	11.7022	0.477344	0.238672	−	0.971100i	$-0.423288\pi$
$601$	11.7022	0.477344	0.238672	+	0.971100i	$0.423288\pi$
$602$	0	0
$603$	2.06878	0.0842473
$604$	0	0
$605$	62.2979	2.53277
$606$	0	0
$607$	−43.1955	−1.75325	−0.876626	−	0.481173i	$-0.840211\pi$
$607$	−43.1955	−1.75325	−0.876626	+	0.481173i	$0.840211\pi$
$608$	0	0
$609$	25.9096	1.04991
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−31.6417	−1.27800	−0.638998	−	0.769208i	$-0.720651\pi$
$613$	−31.6417	−1.27800	−0.638998	+	0.769208i	$0.720651\pi$
$614$	0	0
$615$	28.5011	1.14928
$616$	0	0
$617$	−15.8027	−0.636193	−0.318096	−	0.948058i	$-0.603044\pi$
$617$	−15.8027	−0.636193	−0.318096	+	0.948058i	$0.603044\pi$
$618$	0	0
$619$	−36.0663	−1.44963	−0.724814	−	0.688945i	$-0.758074\pi$
$619$	−36.0663	−1.44963	−0.724814	+	0.688945i	$0.758074\pi$
$620$	0	0
$621$	0.384069	0.0154121
$622$	0	0
$623$	−50.3445	−2.01701
$624$	0	0
$625$	72.8358	2.91343
$626$	0	0
$627$	−16.9408	−0.676549
$628$	0	0
$629$	10.1849	0.406099
$630$	0	0
$631$	17.9063	0.712837	0.356418	−	0.934326i	$-0.383998\pi$
$631$	17.9063	0.712837	0.356418	+	0.934326i	$0.383998\pi$
$632$	0	0
$633$	12.2424	0.486593
$634$	0	0
$635$	−42.6099	−1.69092
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.82529	−0.111767
$640$	0	0
$641$	34.7101	1.37097	0.685483	−	0.728088i	$-0.259591\pi$
$641$	34.7101	1.37097	0.685483	+	0.728088i	$0.259591\pi$
$642$	0	0
$643$	−45.5430	−1.79604	−0.898020	−	0.439955i	$-0.854994\pi$
$643$	−45.5430	−1.79604	−0.898020	+	0.439955i	$0.854994\pi$
$644$	0	0
$645$	17.4948	0.688859
$646$	0	0
$647$	−17.8603	−0.702162	−0.351081	−	0.936345i	$-0.614186\pi$
$647$	−17.8603	−0.702162	−0.351081	+	0.936345i	$0.614186\pi$
$648$	0	0
$649$	−40.0251	−1.57112
$650$	0	0
$651$	−23.0224	−0.902318
$652$	0	0
$653$	−37.6757	−1.47436	−0.737182	−	0.675694i	$-0.763844\pi$
$653$	−37.6757	−1.47436	−0.737182	+	0.675694i	$0.763844\pi$
$654$	0	0
$655$	−56.1394	−2.19355
$656$	0	0
$657$	3.21865	0.125572
$658$	0	0
$659$	34.7475	1.35357	0.676785	−	0.736181i	$-0.263373\pi$
$659$	34.7475	1.35357	0.676785	+	0.736181i	$0.263373\pi$
$660$	0	0
$661$	−39.3234	−1.52950	−0.764752	−	0.644325i	$-0.777139\pi$
$661$	−39.3234	−1.52950	−0.764752	+	0.644325i	$0.777139\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	49.8847	1.93445
$666$	0	0
$667$	−2.79888	−0.108373
$668$	0	0
$669$	20.7085	0.800638
$670$	0	0
$671$	−42.2266	−1.63014
$672$	0	0
$673$	11.1436	0.429554	0.214777	−	0.976663i	$-0.431098\pi$
$673$	11.1436	0.429554	0.214777	+	0.976663i	$0.431098\pi$
$674$	0	0
$675$	12.7022	0.488910
$676$	0	0
$677$	−12.3037	−0.472868	−0.236434	−	0.971648i	$-0.575979\pi$
$677$	−12.3037	−0.472868	−0.236434	+	0.971648i	$0.575979\pi$
$678$	0	0
$679$	−31.5279	−1.20993
$680$	0	0
$681$	−8.63881	−0.331040
$682$	0	0
$683$	−17.5381	−0.671078	−0.335539	−	0.942026i	$-0.608918\pi$
$683$	−17.5381	−0.671078	−0.335539	+	0.942026i	$0.608918\pi$
$684$	0	0
$685$	50.0492	1.91228
$686$	0	0
$687$	20.1111	0.767288
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−13.1181	−0.499035	−0.249518	−	0.968370i	$-0.580272\pi$
$691$	−13.1181	−0.499035	−0.249518	+	0.968370i	$0.580272\pi$
$692$	0	0
$693$	−18.0615	−0.686099
$694$	0	0
$695$	56.4663	2.14189
$696$	0	0
$697$	22.0610	0.835620
$698$	0	0
$699$	−4.41089	−0.166835
$700$	0	0
$701$	−0.768137	−0.0290121	−0.0145061	−	0.999895i	$-0.504618\pi$
$701$	−0.768137	−0.0290121	−0.0145061	+	0.999895i	$0.504618\pi$
$702$	0	0
$703$	−10.4291	−0.393340
$704$	0	0
$705$	−23.1190	−0.870712
$706$	0	0
$707$	−33.5895	−1.26326
$708$	0	0
$709$	−32.0897	−1.20515	−0.602577	−	0.798061i	$-0.705859\pi$
$709$	−32.0897	−1.20515	−0.602577	+	0.798061i	$0.705859\pi$
$710$	0	0
$711$	−2.64077	−0.0990365
$712$	0	0
$713$	2.48698	0.0931381
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.51734	0.131358
$718$	0	0
$719$	−10.9771	−0.409378	−0.204689	−	0.978827i	$-0.565618\pi$
$719$	−10.9771	−0.409378	−0.204689	+	0.978827i	$0.565618\pi$
$720$	0	0
$721$	7.67291	0.285754
$722$	0	0
$723$	−4.74330	−0.176405
$724$	0	0
$725$	−92.5668	−3.43784
$726$	0	0
$727$	40.9056	1.51710	0.758552	−	0.651612i	$-0.225907\pi$
$727$	40.9056	1.51710	0.758552	+	0.651612i	$0.225907\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.5417	0.500858
$732$	0	0
$733$	−6.18829	−0.228570	−0.114285	−	0.993448i	$-0.536458\pi$
$733$	−6.18829	−0.228570	−0.114285	+	0.993448i	$0.536458\pi$
$734$	0	0
$735$	23.7330	0.875404
$736$	0	0
$737$	10.5095	0.387122
$738$	0	0
$739$	−53.3488	−1.96247	−0.981233	−	0.192824i	$-0.938235\pi$
$739$	−53.3488	−1.96247	−0.981233	+	0.192824i	$0.938235\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.5728	1.63522	0.817609	−	0.575774i	$-0.195299\pi$
$743$	44.5728	1.63522	0.817609	+	0.575774i	$0.195299\pi$
$744$	0	0
$745$	−23.1888	−0.849574
$746$	0	0
$747$	1.96926	0.0720515
$748$	0	0
$749$	66.4867	2.42937
$750$	0	0
$751$	−1.50945	−0.0550807	−0.0275403	−	0.999621i	$-0.508767\pi$
$751$	−1.50945	−0.0550807	−0.0275403	+	0.999621i	$0.508767\pi$
$752$	0	0
$753$	6.98968	0.254718
$754$	0	0
$755$	−21.3737	−0.777870
$756$	0	0
$757$	−51.9794	−1.88922	−0.944611	−	0.328192i	$-0.893561\pi$
$757$	−51.9794	−1.88922	−0.944611	+	0.328192i	$0.893561\pi$
$758$	0	0
$759$	1.95108	0.0708198
$760$	0	0
$761$	−10.6695	−0.386771	−0.193385	−	0.981123i	$-0.561947\pi$
$761$	−10.6695	−0.386771	−0.193385	+	0.981123i	$0.561947\pi$
$762$	0	0
$763$	−51.4655	−1.86317
$764$	0	0
$765$	13.7022	0.495406
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.01256	0.216818	0.108409	−	0.994106i	$-0.465424\pi$
$769$	6.01256	0.216818	0.108409	+	0.994106i	$0.465424\pi$
$770$	0	0
$771$	−4.71010	−0.169630
$772$	0	0
$773$	39.8907	1.43477	0.717385	−	0.696677i	$-0.245339\pi$
$773$	39.8907	1.43477	0.717385	+	0.696677i	$0.245339\pi$
$774$	0	0
$775$	82.2515	2.95456
$776$	0	0
$777$	−11.1190	−0.398892
$778$	0	0
$779$	−22.5899	−0.809367
$780$	0	0
$781$	−14.3526	−0.513576
$782$	0	0
$783$	−7.28744	−0.260432
$784$	0	0
$785$	−52.8745	−1.88717
$786$	0	0
$787$	3.50215	0.124838	0.0624190	−	0.998050i	$-0.480118\pi$
$787$	3.50215	0.124838	0.0624190	+	0.998050i	$0.480118\pi$
$788$	0	0
$789$	−4.12933	−0.147008
$790$	0	0
$791$	−9.75350	−0.346794
$792$	0	0
$793$	0	0
$794$	0	0
$795$	2.58324	0.0916179
$796$	0	0
$797$	−24.2869	−0.860287	−0.430144	−	0.902760i	$-0.641537\pi$
$797$	−24.2869	−0.860287	−0.430144	+	0.902760i	$0.641537\pi$
$798$	0	0
$799$	−17.8950	−0.633081
$800$	0	0
$801$	14.1601	0.500321
$802$	0	0
$803$	16.3509	0.577010
$804$	0	0
$805$	−5.74526	−0.202494
$806$	0	0
$807$	16.2587	0.572332
$808$	0	0
$809$	−19.1288	−0.672534	−0.336267	−	0.941767i	$-0.609164\pi$
$809$	−19.1288	−0.672534	−0.336267	+	0.941767i	$0.609164\pi$
$810$	0	0
$811$	55.1857	1.93783	0.968917	−	0.247387i	$-0.0795721\pi$
$811$	55.1857	1.93783	0.968917	+	0.247387i	$0.0795721\pi$
$812$	0	0
$813$	−32.3171	−1.13341
$814$	0	0
$815$	37.1008	1.29959
$816$	0	0
$817$	−13.8664	−0.485122
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.3362	−0.814440	−0.407220	−	0.913330i	$-0.633502\pi$
$821$	−23.3362	−0.814440	−0.407220	+	0.913330i	$0.633502\pi$
$822$	0	0
$823$	−2.05359	−0.0715835	−0.0357918	−	0.999359i	$-0.511395\pi$
$823$	−2.05359	−0.0715835	−0.0357918	+	0.999359i	$0.511395\pi$
$824$	0	0
$825$	64.5278	2.24657
$826$	0	0
$827$	5.96289	0.207350	0.103675	−	0.994611i	$-0.466940\pi$
$827$	5.96289	0.207350	0.103675	+	0.994611i	$0.466940\pi$
$828$	0	0
$829$	14.5631	0.505797	0.252899	−	0.967493i	$-0.418616\pi$
$829$	14.5631	0.505797	0.252899	+	0.967493i	$0.418616\pi$
$830$	0	0
$831$	−23.7369	−0.823424
$832$	0	0
$833$	18.3703	0.636492
$834$	0	0
$835$	61.7207	2.13593
$836$	0	0
$837$	6.47535	0.223821
$838$	0	0
$839$	−49.0737	−1.69421	−0.847105	−	0.531425i	$-0.821657\pi$
$839$	−49.0737	−1.69421	−0.847105	+	0.531425i	$0.821657\pi$
$840$	0	0
$841$	24.1067	0.831267
$842$	0	0
$843$	−25.5812	−0.881062
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−52.6437	−1.80886
$848$	0	0
$849$	−11.6486	−0.399780
$850$	0	0
$851$	1.20112	0.0411740
$852$	0	0
$853$	−54.6929	−1.87265	−0.936324	−	0.351138i	$-0.885795\pi$
$853$	−54.6929	−1.87265	−0.936324	+	0.351138i	$0.885795\pi$
$854$	0	0
$855$	−14.0307	−0.479841
$856$	0	0
$857$	−23.2811	−0.795266	−0.397633	−	0.917545i	$-0.630168\pi$
$857$	−23.2811	−0.795266	−0.397633	+	0.917545i	$0.630168\pi$
$858$	0	0
$859$	43.4229	1.48157	0.740785	−	0.671742i	$-0.234454\pi$
$859$	43.4229	1.48157	0.740785	+	0.671742i	$0.234454\pi$
$860$	0	0
$861$	−24.0843	−0.820792
$862$	0	0
$863$	−7.28977	−0.248147	−0.124073	−	0.992273i	$-0.539596\pi$
$863$	−7.28977	−0.248147	−0.124073	+	0.992273i	$0.539596\pi$
$864$	0	0
$865$	−10.1601	−0.345453
$866$	0	0
$867$	−6.39392	−0.217149
$868$	0	0
$869$	−13.4152	−0.455079
$870$	0	0
$871$	0	0
$872$	0	0
$873$	8.86766	0.300125
$874$	0	0
$875$	−115.217	−3.89506
$876$	0	0
$877$	−33.2464	−1.12265	−0.561326	−	0.827595i	$-0.689708\pi$
$877$	−33.2464	−1.12265	−0.561326	+	0.827595i	$0.689708\pi$
$878$	0	0
$879$	−9.85016	−0.332238
$880$	0	0
$881$	16.1888	0.545415	0.272708	−	0.962097i	$-0.412081\pi$
$881$	16.1888	0.545415	0.272708	+	0.962097i	$0.412081\pi$
$882$	0	0
$883$	−20.0492	−0.674708	−0.337354	−	0.941378i	$-0.609532\pi$
$883$	−20.0492	−0.674708	−0.337354	+	0.941378i	$0.609532\pi$
$884$	0	0
$885$	−33.1497	−1.11432
$886$	0	0
$887$	−54.3120	−1.82362	−0.911810	−	0.410612i	$-0.865315\pi$
$887$	−54.3120	−1.82362	−0.911810	+	0.410612i	$0.865315\pi$
$888$	0	0
$889$	36.0067	1.20763
$890$	0	0
$891$	5.08003	0.170188
$892$	0	0
$893$	18.3240	0.613191
$894$	0	0
$895$	−22.5402	−0.753436
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−47.1887	−1.57383
$900$	0	0
$901$	1.99953	0.0666139
$902$	0	0
$903$	−14.7837	−0.491970
$904$	0	0
$905$	−74.3511	−2.47152
$906$	0	0
$907$	−51.8972	−1.72322	−0.861610	−	0.507571i	$-0.830543\pi$
$907$	−51.8972	−1.72322	−0.861610	+	0.507571i	$0.830543\pi$
$908$	0	0
$909$	9.44750	0.313354
$910$	0	0
$911$	17.0433	0.564669	0.282334	−	0.959316i	$-0.408891\pi$
$911$	17.0433	0.564669	0.282334	+	0.959316i	$0.408891\pi$
$912$	0	0
$913$	10.0039	0.331081
$914$	0	0
$915$	−34.9731	−1.15617
$916$	0	0
$917$	47.4395	1.56659
$918$	0	0
$919$	−16.0693	−0.530078	−0.265039	−	0.964238i	$-0.585385\pi$
$919$	−16.0693	−0.530078	−0.265039	+	0.964238i	$0.585385\pi$
$920$	0	0
$921$	−12.4182	−0.409194
$922$	0	0
$923$	0	0
$924$	0	0
$925$	39.7246	1.30614
$926$	0	0
$927$	−2.15811	−0.0708816
$928$	0	0
$929$	−11.4025	−0.374104	−0.187052	−	0.982350i	$-0.559893\pi$
$929$	−11.4025	−0.374104	−0.187052	+	0.982350i	$0.559893\pi$
$930$	0	0
$931$	−18.8107	−0.616495
$932$	0	0
$933$	7.42267	0.243007
$934$	0	0
$935$	69.6078	2.27642
$936$	0	0
$937$	−33.8213	−1.10489	−0.552447	−	0.833548i	$-0.686306\pi$
$937$	−33.8213	−1.10489	−0.552447	+	0.833548i	$0.686306\pi$
$938$	0	0
$939$	27.9799	0.913088
$940$	0	0
$941$	−48.0429	−1.56615	−0.783077	−	0.621925i	$-0.786351\pi$
$941$	−48.0429	−1.56615	−0.783077	+	0.621925i	$0.786351\pi$
$942$	0	0
$943$	2.60170	0.0847229
$944$	0	0
$945$	−14.9589	−0.486614
$946$	0	0
$947$	12.0779	0.392481	0.196240	−	0.980556i	$-0.437127\pi$
$947$	12.0779	0.392481	0.196240	+	0.980556i	$0.437127\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−8.02484	−0.260223
$952$	0	0
$953$	15.4251	0.499669	0.249834	−	0.968289i	$-0.419624\pi$
$953$	15.4251	0.499669	0.249834	+	0.968289i	$0.419624\pi$
$954$	0	0
$955$	19.2483	0.622862
$956$	0	0
$957$	−37.0204	−1.19670
$958$	0	0
$959$	−42.2931	−1.36572
$960$	0	0
$961$	10.9302	0.352587
$962$	0	0
$963$	−18.7003	−0.602608
$964$	0	0
$965$	−70.8645	−2.28121
$966$	0	0
$967$	49.7289	1.59917	0.799587	−	0.600550i	$-0.205052\pi$
$967$	49.7289	1.59917	0.799587	+	0.600550i	$0.205052\pi$
$968$	0	0
$969$	−10.8604	−0.348885
$970$	0	0
$971$	−30.9897	−0.994506	−0.497253	−	0.867606i	$-0.665658\pi$
$971$	−30.9897	−0.994506	−0.497253	+	0.867606i	$0.665658\pi$
$972$	0	0
$973$	−47.7158	−1.52970
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−22.8765	−0.731886	−0.365943	−	0.930637i	$-0.619253\pi$
$977$	−22.8765	−0.731886	−0.365943	+	0.930637i	$0.619253\pi$
$978$	0	0
$979$	71.9336	2.29901
$980$	0	0
$981$	14.4754	0.462162
$982$	0	0
$983$	28.0019	0.893121	0.446560	−	0.894754i	$-0.352649\pi$
$983$	28.0019	0.893121	0.446560	+	0.894754i	$0.352649\pi$
$984$	0	0
$985$	−7.81065	−0.248868
$986$	0	0
$987$	19.5363	0.621846
$988$	0	0
$989$	1.59700	0.0507816
$990$	0	0
$991$	−16.2855	−0.517325	−0.258663	−	0.965968i	$-0.583282\pi$
$991$	−16.2855	−0.517325	−0.258663	+	0.965968i	$0.583282\pi$
$992$	0	0
$993$	18.9252	0.600574
$994$	0	0
$995$	−79.8890	−2.53265
$996$	0	0
$997$	46.3034	1.46644	0.733222	−	0.679989i	$-0.238015\pi$
$997$	46.3034	1.46644	0.733222	+	0.679989i	$0.238015\pi$
$998$	0	0
$999$	3.12737	0.0989456

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.be.1.4		4
4.3	odd	2		8112.2.a.cs.1.4		4
13.2	odd	12		312.2.bf.b.121.1	yes	8
13.5	odd	4		4056.2.c.p.337.1		8
13.7	odd	12		312.2.bf.b.49.4	✓	8
13.8	odd	4		4056.2.c.p.337.8		8
13.12	even	2		4056.2.a.bd.1.1		4
39.2	even	12		936.2.bi.c.433.4		8
39.20	even	12		936.2.bi.c.361.1		8
52.7	even	12		624.2.bv.g.49.4		8
52.15	even	12		624.2.bv.g.433.1		8
52.51	odd	2		8112.2.a.cq.1.1		4
156.59	odd	12		1872.2.by.m.1297.1		8
156.119	odd	12		1872.2.by.m.433.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.bf.b.49.4	✓	8	13.7	odd	12
312.2.bf.b.121.1	yes	8	13.2	odd	12
624.2.bv.g.49.4		8	52.7	even	12
624.2.bv.g.433.1		8	52.15	even	12
936.2.bi.c.361.1		8	39.20	even	12
936.2.bi.c.433.4		8	39.2	even	12
1872.2.by.m.433.4		8	156.119	odd	12
1872.2.by.m.1297.1		8	156.59	odd	12
4056.2.a.bd.1.1		4	13.12	even	2
4056.2.a.be.1.4		4	1.1	even	1	trivial
4056.2.c.p.337.1		8	13.5	odd	4
4056.2.c.p.337.8		8	13.8	odd	4
8112.2.a.cq.1.1		4	52.51	odd	2
8112.2.a.cs.1.4		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.20740 q^{5} -3.55539 q^{7} +1.00000 q^{9} +5.08003 q^{11} +4.20740 q^{15} +3.25670 q^{17} -3.33477 q^{19} -3.55539 q^{21} +0.384069 q^{23} +12.7022 q^{25} +1.00000 q^{27} -7.28744 q^{29} +6.47535 q^{31} +5.08003 q^{33} -14.9589 q^{35} +3.12737 q^{37} +6.77404 q^{41} +4.15811 q^{43} +4.20740 q^{45} -5.49484 q^{47} +5.64077 q^{49} +3.25670 q^{51} +0.613974 q^{53} +21.3737 q^{55} -3.33477 q^{57} -7.87891 q^{59} -8.31227 q^{61} -3.55539 q^{63} +2.06878 q^{67} +0.384069 q^{69} -2.82529 q^{71} +3.21865 q^{73} +12.7022 q^{75} -18.0615 q^{77} -2.64077 q^{79} +1.00000 q^{81} +1.96926 q^{83} +13.7022 q^{85} -7.28744 q^{87} +14.1601 q^{89} +6.47535 q^{93} -14.0307 q^{95} +8.86766 q^{97} +5.08003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} + 2 q^{7} + 4 q^{9} + 10 q^{11} + 4 q^{15} + 12 q^{17} + 2 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 4 q^{27} - 6 q^{29} + 20 q^{31} + 10 q^{33} - 10 q^{35} + 10 q^{37} + 6 q^{41}+ \cdots + 10 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 + 2 * q^7 + 4 * q^9 + 10 * q^11 + 4 * q^15 + 12 * q^17 + 2 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 4 * q^27 - 6 * q^29 + 20 * q^31 + 10 * q^33 - 10 * q^35 + 10 * q^37 + 6 * q^41 - 8 * q^43 + 4 * q^45 + 14 * q^47 + 18 * q^49 + 12 * q^51 + 2 * q^53 + 10 * q^55 + 2 * q^57 + 8 * q^59 - 2 * q^61 + 2 * q^63 - 10 * q^67 - 2 * q^69 - 6 * q^71 + 8 * q^73 + 2 * q^75 - 28 * q^77 - 6 * q^79 + 4 * q^81 + 30 * q^83 + 6 * q^85 - 6 * q^87 + 36 * q^89 + 20 * q^93 - 34 * q^95 - 12 * q^97 + 10 * q^99

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