LMFDB - Embedded newform 1456.2.a.t.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1456,2,Mod(1,1456)] 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("1456.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1456, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,2,0,2,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$11.6262185343$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	1456.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24914 q^{3} +0.529317 q^{5} +1.00000 q^{7} +2.05863 q^{9} +2.24914 q^{11} +1.00000 q^{13} -1.19051 q^{15} -1.30777 q^{17} +1.47068 q^{19} -2.24914 q^{21} -5.83709 q^{23} -4.71982 q^{25} +2.11727 q^{27} +5.22154 q^{29} +7.02760 q^{31} -5.05863 q^{33} +0.529317 q^{35} -2.36641 q^{37} -2.24914 q^{39} +6.49828 q^{41} -11.3940 q^{43} +1.08967 q^{45} -8.58451 q^{47} +1.00000 q^{49} +2.94137 q^{51} +11.2767 q^{53} +1.19051 q^{55} -3.30777 q^{57} +12.1725 q^{59} -2.00000 q^{61} +2.05863 q^{63} +0.529317 q^{65} +15.9379 q^{67} +13.1284 q^{69} -1.19051 q^{71} +7.64315 q^{73} +10.6155 q^{75} +2.24914 q^{77} +1.33881 q^{79} -10.9379 q^{81} +16.3500 q^{83} -0.692226 q^{85} -11.7440 q^{87} +6.91033 q^{89} +1.00000 q^{91} -15.8061 q^{93} +0.778457 q^{95} -3.47068 q^{97} +4.63016 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{5} + 3 q^{7} + 7 q^{9} - 2 q^{11} + 3 q^{13} + 6 q^{15} + 4 q^{17} + 4 q^{19} + 2 q^{21} - 10 q^{23} - 5 q^{25} + 8 q^{27} + 24 q^{29} + 4 q^{31} - 16 q^{33} + 2 q^{35} + 2 q^{39} + 2 q^{41}+ \cdots - 14 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^5 + 3 * q^7 + 7 * q^9 - 2 * q^11 + 3 * q^13 + 6 * q^15 + 4 * q^17 + 4 * q^19 + 2 * q^21 - 10 * q^23 - 5 * q^25 + 8 * q^27 + 24 * q^29 + 4 * q^31 - 16 * q^33 + 2 * q^35 + 2 * q^39 + 2 * q^41 - 10 * q^43 + 22 * q^45 + 8 * q^47 + 3 * q^49 + 8 * q^51 + 8 * q^53 - 6 * q^55 - 2 * q^57 + 4 * q^59 - 6 * q^61 + 7 * q^63 + 2 * q^65 + 12 * q^67 - 6 * q^69 + 6 * q^71 - 10 * q^73 + 16 * q^75 - 2 * q^77 + 14 * q^79 + 3 * q^81 + 12 * q^83 - 10 * q^85 + 26 * q^87 + 2 * q^89 + 3 * q^91 - 22 * q^93 - 6 * q^95 - 10 * q^97 - 14 * 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
$3$	−2.24914	−1.29854	−0.649271	−	0.760557i	$-0.724926\pi$
$3$	−2.24914	−1.29854	−0.649271	+	0.760557i	$0.724926\pi$
$4$	0	0
$5$	0.529317	0.236718	0.118359	−	0.992971i	$-0.462237\pi$
$5$	0.529317	0.236718	0.118359	+	0.992971i	$0.462237\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.05863	0.686211
$10$	0	0
$11$	2.24914	0.678141	0.339071	−	0.940761i	$-0.389887\pi$
$11$	2.24914	0.678141	0.339071	+	0.940761i	$0.389887\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.19051	−0.307388
$16$	0	0
$17$	−1.30777	−0.317182	−0.158591	−	0.987344i	$-0.550695\pi$
$17$	−1.30777	−0.317182	−0.158591	+	0.987344i	$0.550695\pi$
$18$	0	0
$19$	1.47068	0.337398	0.168699	−	0.985668i	$-0.446043\pi$
$19$	1.47068	0.337398	0.168699	+	0.985668i	$0.446043\pi$
$20$	0	0
$21$	−2.24914	−0.490803
$22$	0	0
$23$	−5.83709	−1.21712	−0.608559	−	0.793509i	$-0.708252\pi$
$23$	−5.83709	−1.21712	−0.608559	+	0.793509i	$0.708252\pi$
$24$	0	0
$25$	−4.71982	−0.943965
$26$	0	0
$27$	2.11727	0.407468
$28$	0	0
$29$	5.22154	0.969616	0.484808	−	0.874621i	$-0.338889\pi$
$29$	5.22154	0.969616	0.484808	+	0.874621i	$0.338889\pi$
$30$	0	0
$31$	7.02760	1.26219	0.631097	−	0.775704i	$-0.282605\pi$
$31$	7.02760	1.26219	0.631097	+	0.775704i	$0.282605\pi$
$32$	0	0
$33$	−5.05863	−0.880595
$34$	0	0
$35$	0.529317	0.0894708
$36$	0	0
$37$	−2.36641	−0.389035	−0.194517	−	0.980899i	$-0.562314\pi$
$37$	−2.36641	−0.389035	−0.194517	+	0.980899i	$0.562314\pi$
$38$	0	0
$39$	−2.24914	−0.360151
$40$	0	0
$41$	6.49828	1.01486	0.507431	−	0.861693i	$-0.330595\pi$
$41$	6.49828	1.01486	0.507431	+	0.861693i	$0.330595\pi$
$42$	0	0
$43$	−11.3940	−1.73757	−0.868785	−	0.495190i	$-0.835098\pi$
$43$	−11.3940	−1.73757	−0.868785	+	0.495190i	$0.835098\pi$
$44$	0	0
$45$	1.08967	0.162438
$46$	0	0
$47$	−8.58451	−1.25218	−0.626090	−	0.779751i	$-0.715346\pi$
$47$	−8.58451	−1.25218	−0.626090	+	0.779751i	$0.715346\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.94137	0.411874
$52$	0	0
$53$	11.2767	1.54898	0.774490	−	0.632587i	$-0.218007\pi$
$53$	11.2767	1.54898	0.774490	+	0.632587i	$0.218007\pi$
$54$	0	0
$55$	1.19051	0.160528
$56$	0	0
$57$	−3.30777	−0.438125
$58$	0	0
$59$	12.1725	1.58472	0.792360	−	0.610054i	$-0.208852\pi$
$59$	12.1725	1.58472	0.792360	+	0.610054i	$0.208852\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	2.05863	0.259363
$64$	0	0
$65$	0.529317	0.0656536
$66$	0	0
$67$	15.9379	1.94713	0.973564	−	0.228415i	$-0.0733542\pi$
$67$	15.9379	1.94713	0.973564	+	0.228415i	$0.0733542\pi$
$68$	0	0
$69$	13.1284	1.58048
$70$	0	0
$71$	−1.19051	−0.141287	−0.0706436	−	0.997502i	$-0.522505\pi$
$71$	−1.19051	−0.141287	−0.0706436	+	0.997502i	$0.522505\pi$
$72$	0	0
$73$	7.64315	0.894562	0.447281	−	0.894393i	$-0.352392\pi$
$73$	7.64315	0.894562	0.447281	+	0.894393i	$0.352392\pi$
$74$	0	0
$75$	10.6155	1.22578
$76$	0	0
$77$	2.24914	0.256313
$78$	0	0
$79$	1.33881	0.150628	0.0753139	−	0.997160i	$-0.476004\pi$
$79$	1.33881	0.150628	0.0753139	+	0.997160i	$0.476004\pi$
$80$	0	0
$81$	−10.9379	−1.21533
$82$	0	0
$83$	16.3500	1.79464	0.897322	−	0.441377i	$-0.145510\pi$
$83$	16.3500	1.79464	0.897322	+	0.441377i	$0.145510\pi$
$84$	0	0
$85$	−0.692226	−0.0750825
$86$	0	0
$87$	−11.7440	−1.25909
$88$	0	0
$89$	6.91033	0.732494	0.366247	−	0.930518i	$-0.380643\pi$
$89$	6.91033	0.732494	0.366247	+	0.930518i	$0.380643\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−15.8061	−1.63901
$94$	0	0
$95$	0.778457	0.0798680
$96$	0	0
$97$	−3.47068	−0.352395	−0.176197	−	0.984355i	$-0.556380\pi$
$97$	−3.47068	−0.352395	−0.176197	+	0.984355i	$0.556380\pi$
$98$	0	0
$99$	4.63016	0.465348
$100$	0	0
$101$	7.75086	0.771239	0.385620	−	0.922658i	$-0.373988\pi$
$101$	7.75086	0.771239	0.385620	+	0.922658i	$0.373988\pi$
$102$	0	0
$103$	16.9966	1.67472	0.837361	−	0.546651i	$-0.184098\pi$
$103$	16.9966	1.67472	0.837361	+	0.546651i	$0.184098\pi$
$104$	0	0
$105$	−1.19051	−0.116182
$106$	0	0
$107$	5.55691	0.537207	0.268604	−	0.963251i	$-0.413438\pi$
$107$	5.55691	0.537207	0.268604	+	0.963251i	$0.413438\pi$
$108$	0	0
$109$	7.92332	0.758917	0.379458	−	0.925209i	$-0.376110\pi$
$109$	7.92332	0.758917	0.379458	+	0.925209i	$0.376110\pi$
$110$	0	0
$111$	5.32238	0.505178
$112$	0	0
$113$	−9.89229	−0.930588	−0.465294	−	0.885156i	$-0.654051\pi$
$113$	−9.89229	−0.930588	−0.465294	+	0.885156i	$0.654051\pi$
$114$	0	0
$115$	−3.08967	−0.288113
$116$	0	0
$117$	2.05863	0.190321
$118$	0	0
$119$	−1.30777	−0.119883
$120$	0	0
$121$	−5.94137	−0.540124
$122$	0	0
$123$	−14.6155	−1.31784
$124$	0	0
$125$	−5.14486	−0.460171
$126$	0	0
$127$	−0.824101	−0.0731271	−0.0365635	−	0.999331i	$-0.511641\pi$
$127$	−0.824101	−0.0731271	−0.0365635	+	0.999331i	$0.511641\pi$
$128$	0	0
$129$	25.6267	2.25631
$130$	0	0
$131$	−10.6155	−0.927485	−0.463742	−	0.885970i	$-0.653494\pi$
$131$	−10.6155	−0.927485	−0.463742	+	0.885970i	$0.653494\pi$
$132$	0	0
$133$	1.47068	0.127524
$134$	0	0
$135$	1.12070	0.0964549
$136$	0	0
$137$	11.3630	0.970804	0.485402	−	0.874291i	$-0.338673\pi$
$137$	11.3630	0.970804	0.485402	+	0.874291i	$0.338673\pi$
$138$	0	0
$139$	13.9233	1.18096	0.590480	−	0.807052i	$-0.298938\pi$
$139$	13.9233	1.18096	0.590480	+	0.807052i	$0.298938\pi$
$140$	0	0
$141$	19.3078	1.62601
$142$	0	0
$143$	2.24914	0.188083
$144$	0	0
$145$	2.76385	0.229525
$146$	0	0
$147$	−2.24914	−0.185506
$148$	0	0
$149$	9.30777	0.762523	0.381261	−	0.924467i	$-0.375490\pi$
$149$	9.30777	0.762523	0.381261	+	0.924467i	$0.375490\pi$
$150$	0	0
$151$	−7.07324	−0.575612	−0.287806	−	0.957689i	$-0.592926\pi$
$151$	−7.07324	−0.575612	−0.287806	+	0.957689i	$0.592926\pi$
$152$	0	0
$153$	−2.69223	−0.217654
$154$	0	0
$155$	3.71982	0.298783
$156$	0	0
$157$	−6.04059	−0.482091	−0.241046	−	0.970514i	$-0.577490\pi$
$157$	−6.04059	−0.482091	−0.241046	+	0.970514i	$0.577490\pi$
$158$	0	0
$159$	−25.3630	−2.01141
$160$	0	0
$161$	−5.83709	−0.460027
$162$	0	0
$163$	−6.38101	−0.499800	−0.249900	−	0.968272i	$-0.580398\pi$
$163$	−6.38101	−0.499800	−0.249900	+	0.968272i	$0.580398\pi$
$164$	0	0
$165$	−2.67762	−0.208452
$166$	0	0
$167$	16.5845	1.28335	0.641674	−	0.766977i	$-0.278240\pi$
$167$	16.5845	1.28335	0.641674	+	0.766977i	$0.278240\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.02760	0.231526
$172$	0	0
$173$	−23.3009	−1.77153	−0.885767	−	0.464130i	$-0.846367\pi$
$173$	−23.3009	−1.77153	−0.885767	+	0.464130i	$0.846367\pi$
$174$	0	0
$175$	−4.71982	−0.356785
$176$	0	0
$177$	−27.3776	−2.05782
$178$	0	0
$179$	−21.0422	−1.57277	−0.786384	−	0.617738i	$-0.788049\pi$
$179$	−21.0422	−1.57277	−0.786384	+	0.617738i	$0.788049\pi$
$180$	0	0
$181$	16.7474	1.24483	0.622413	−	0.782689i	$-0.286152\pi$
$181$	16.7474	1.24483	0.622413	+	0.782689i	$0.286152\pi$
$182$	0	0
$183$	4.49828	0.332523
$184$	0	0
$185$	−1.25258	−0.0920914
$186$	0	0
$187$	−2.94137	−0.215094
$188$	0	0
$189$	2.11727	0.154008
$190$	0	0
$191$	7.43965	0.538314	0.269157	−	0.963096i	$-0.413255\pi$
$191$	7.43965	0.538314	0.269157	+	0.963096i	$0.413255\pi$
$192$	0	0
$193$	−1.50172	−0.108096	−0.0540480	−	0.998538i	$-0.517212\pi$
$193$	−1.50172	−0.108096	−0.0540480	+	0.998538i	$0.517212\pi$
$194$	0	0
$195$	−1.19051	−0.0852540
$196$	0	0
$197$	23.9931	1.70944	0.854720	−	0.519090i	$-0.173729\pi$
$197$	23.9931	1.70944	0.854720	+	0.519090i	$0.173729\pi$
$198$	0	0
$199$	2.01461	0.142812	0.0714059	−	0.997447i	$-0.477251\pi$
$199$	2.01461	0.142812	0.0714059	+	0.997447i	$0.477251\pi$
$200$	0	0
$201$	−35.8466	−2.52843
$202$	0	0
$203$	5.22154	0.366480
$204$	0	0
$205$	3.43965	0.240235
$206$	0	0
$207$	−12.0164	−0.835199
$208$	0	0
$209$	3.30777	0.228803
$210$	0	0
$211$	−10.1008	−0.695370	−0.347685	−	0.937611i	$-0.613032\pi$
$211$	−10.1008	−0.695370	−0.347685	+	0.937611i	$0.613032\pi$
$212$	0	0
$213$	2.67762	0.183467
$214$	0	0
$215$	−6.03104	−0.411313
$216$	0	0
$217$	7.02760	0.477064
$218$	0	0
$219$	−17.1905	−1.16163
$220$	0	0
$221$	−1.30777	−0.0879704
$222$	0	0
$223$	−10.1414	−0.679120	−0.339560	−	0.940584i	$-0.610278\pi$
$223$	−10.1414	−0.679120	−0.339560	+	0.940584i	$0.610278\pi$
$224$	0	0
$225$	−9.71639	−0.647759
$226$	0	0
$227$	5.38445	0.357379	0.178689	−	0.983906i	$-0.442814\pi$
$227$	5.38445	0.357379	0.178689	+	0.983906i	$0.442814\pi$
$228$	0	0
$229$	3.32238	0.219549	0.109775	−	0.993957i	$-0.464987\pi$
$229$	3.32238	0.219549	0.109775	+	0.993957i	$0.464987\pi$
$230$	0	0
$231$	−5.05863	−0.332834
$232$	0	0
$233$	13.7198	0.898816	0.449408	−	0.893327i	$-0.351635\pi$
$233$	13.7198	0.898816	0.449408	+	0.893327i	$0.351635\pi$
$234$	0	0
$235$	−4.54392	−0.296413
$236$	0	0
$237$	−3.01117	−0.195597
$238$	0	0
$239$	3.50172	0.226507	0.113254	−	0.993566i	$-0.463873\pi$
$239$	3.50172	0.226507	0.113254	+	0.993566i	$0.463873\pi$
$240$	0	0
$241$	1.58795	0.102289	0.0511444	−	0.998691i	$-0.483713\pi$
$241$	1.58795	0.102289	0.0511444	+	0.998691i	$0.483713\pi$
$242$	0	0
$243$	18.2491	1.17068
$244$	0	0
$245$	0.529317	0.0338168
$246$	0	0
$247$	1.47068	0.0935773
$248$	0	0
$249$	−36.7734	−2.33042
$250$	0	0
$251$	4.92676	0.310974	0.155487	−	0.987838i	$-0.450305\pi$
$251$	4.92676	0.310974	0.155487	+	0.987838i	$0.450305\pi$
$252$	0	0
$253$	−13.1284	−0.825378
$254$	0	0
$255$	1.55691	0.0974978
$256$	0	0
$257$	−8.01461	−0.499938	−0.249969	−	0.968254i	$-0.580420\pi$
$257$	−8.01461	−0.499938	−0.249969	+	0.968254i	$0.580420\pi$
$258$	0	0
$259$	−2.36641	−0.147041
$260$	0	0
$261$	10.7492	0.665361
$262$	0	0
$263$	−1.60256	−0.0988179	−0.0494090	−	0.998779i	$-0.515734\pi$
$263$	−1.60256	−0.0988179	−0.0494090	+	0.998779i	$0.515734\pi$
$264$	0	0
$265$	5.96896	0.366671
$266$	0	0
$267$	−15.5423	−0.951174
$268$	0	0
$269$	−11.8207	−0.720719	−0.360359	−	0.932814i	$-0.617346\pi$
$269$	−11.8207	−0.720719	−0.360359	+	0.932814i	$0.617346\pi$
$270$	0	0
$271$	−21.8827	−1.32928	−0.664641	−	0.747163i	$-0.731415\pi$
$271$	−21.8827	−1.32928	−0.664641	+	0.747163i	$0.731415\pi$
$272$	0	0
$273$	−2.24914	−0.136124
$274$	0	0
$275$	−10.6155	−0.640142
$276$	0	0
$277$	−10.2181	−0.613946	−0.306973	−	0.951718i	$-0.599316\pi$
$277$	−10.2181	−0.613946	−0.306973	+	0.951718i	$0.599316\pi$
$278$	0	0
$279$	14.4672	0.866131
$280$	0	0
$281$	1.54231	0.0920063	0.0460031	−	0.998941i	$-0.485352\pi$
$281$	1.54231	0.0920063	0.0460031	+	0.998941i	$0.485352\pi$
$282$	0	0
$283$	−15.8466	−0.941985	−0.470993	−	0.882137i	$-0.656104\pi$
$283$	−15.8466	−0.941985	−0.470993	+	0.882137i	$0.656104\pi$
$284$	0	0
$285$	−1.75086	−0.103712
$286$	0	0
$287$	6.49828	0.383581
$288$	0	0
$289$	−15.2897	−0.899396
$290$	0	0
$291$	7.80605	0.457599
$292$	0	0
$293$	11.0828	0.647464	0.323732	−	0.946149i	$-0.395062\pi$
$293$	11.0828	0.647464	0.323732	+	0.946149i	$0.395062\pi$
$294$	0	0
$295$	6.44309	0.375131
$296$	0	0
$297$	4.76203	0.276321
$298$	0	0
$299$	−5.83709	−0.337568
$300$	0	0
$301$	−11.3940	−0.656740
$302$	0	0
$303$	−17.4328	−1.00149
$304$	0	0
$305$	−1.05863	−0.0606172
$306$	0	0
$307$	−20.4121	−1.16498	−0.582489	−	0.812839i	$-0.697921\pi$
$307$	−20.4121	−1.16498	−0.582489	+	0.812839i	$0.697921\pi$
$308$	0	0
$309$	−38.2277	−2.17470
$310$	0	0
$311$	1.92332	0.109062	0.0545308	−	0.998512i	$-0.482634\pi$
$311$	1.92332	0.109062	0.0545308	+	0.998512i	$0.482634\pi$
$312$	0	0
$313$	−14.3664	−0.812037	−0.406019	−	0.913865i	$-0.633083\pi$
$313$	−14.3664	−0.812037	−0.406019	+	0.913865i	$0.633083\pi$
$314$	0	0
$315$	1.08967	0.0613959
$316$	0	0
$317$	15.5569	0.873763	0.436882	−	0.899519i	$-0.356083\pi$
$317$	15.5569	0.873763	0.436882	+	0.899519i	$0.356083\pi$
$318$	0	0
$319$	11.7440	0.657537
$320$	0	0
$321$	−12.4983	−0.697586
$322$	0	0
$323$	−1.92332	−0.107016
$324$	0	0
$325$	−4.71982	−0.261809
$326$	0	0
$327$	−17.8207	−0.985485
$328$	0	0
$329$	−8.58451	−0.473279
$330$	0	0
$331$	−31.5500	−1.73415	−0.867073	−	0.498180i	$-0.834002\pi$
$331$	−31.5500	−1.73415	−0.867073	+	0.498180i	$0.834002\pi$
$332$	0	0
$333$	−4.87156	−0.266960
$334$	0	0
$335$	8.43621	0.460919
$336$	0	0
$337$	−8.42666	−0.459029	−0.229515	−	0.973305i	$-0.573714\pi$
$337$	−8.42666	−0.459029	−0.229515	+	0.973305i	$0.573714\pi$
$338$	0	0
$339$	22.2491	1.20841
$340$	0	0
$341$	15.8061	0.855946
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	6.94910	0.374127
$346$	0	0
$347$	−19.3484	−1.03867	−0.519337	−	0.854569i	$-0.673821\pi$
$347$	−19.3484	−1.03867	−0.519337	+	0.854569i	$0.673821\pi$
$348$	0	0
$349$	27.2553	1.45894	0.729470	−	0.684013i	$-0.239767\pi$
$349$	27.2553	1.45894	0.729470	+	0.684013i	$0.239767\pi$
$350$	0	0
$351$	2.11727	0.113011
$352$	0	0
$353$	−25.6742	−1.36650	−0.683249	−	0.730185i	$-0.739434\pi$
$353$	−25.6742	−1.36650	−0.683249	+	0.730185i	$0.739434\pi$
$354$	0	0
$355$	−0.630155	−0.0334452
$356$	0	0
$357$	2.94137	0.155674
$358$	0	0
$359$	23.4182	1.23596	0.617982	−	0.786192i	$-0.287951\pi$
$359$	23.4182	1.23596	0.617982	+	0.786192i	$0.287951\pi$
$360$	0	0
$361$	−16.8371	−0.886163
$362$	0	0
$363$	13.3630	0.701374
$364$	0	0
$365$	4.04564	0.211759
$366$	0	0
$367$	14.6854	0.766569	0.383285	−	0.923630i	$-0.374793\pi$
$367$	14.6854	0.766569	0.383285	+	0.923630i	$0.374793\pi$
$368$	0	0
$369$	13.3776	0.696409
$370$	0	0
$371$	11.2767	0.585459
$372$	0	0
$373$	−23.6673	−1.22545	−0.612723	−	0.790298i	$-0.709926\pi$
$373$	−23.6673	−1.22545	−0.612723	+	0.790298i	$0.709926\pi$
$374$	0	0
$375$	11.5715	0.597551
$376$	0	0
$377$	5.22154	0.268923
$378$	0	0
$379$	32.7405	1.68177	0.840884	−	0.541215i	$-0.182035\pi$
$379$	32.7405	1.68177	0.840884	+	0.541215i	$0.182035\pi$
$380$	0	0
$381$	1.85352	0.0949586
$382$	0	0
$383$	22.6155	1.15560	0.577800	−	0.816178i	$-0.303911\pi$
$383$	22.6155	1.15560	0.577800	+	0.816178i	$0.303911\pi$
$384$	0	0
$385$	1.19051	0.0606739
$386$	0	0
$387$	−23.4561	−1.19234
$388$	0	0
$389$	38.0483	1.92913	0.964563	−	0.263852i	$-0.0849930\pi$
$389$	38.0483	1.92913	0.964563	+	0.263852i	$0.0849930\pi$
$390$	0	0
$391$	7.63359	0.386047
$392$	0	0
$393$	23.8759	1.20438
$394$	0	0
$395$	0.708654	0.0356562
$396$	0	0
$397$	−11.7052	−0.587468	−0.293734	−	0.955887i	$-0.594898\pi$
$397$	−11.7052	−0.587468	−0.293734	+	0.955887i	$0.594898\pi$
$398$	0	0
$399$	−3.30777	−0.165596
$400$	0	0
$401$	3.55691	0.177624	0.0888119	−	0.996048i	$-0.471693\pi$
$401$	3.55691	0.177624	0.0888119	+	0.996048i	$0.471693\pi$
$402$	0	0
$403$	7.02760	0.350070
$404$	0	0
$405$	−5.78963	−0.287689
$406$	0	0
$407$	−5.32238	−0.263821
$408$	0	0
$409$	5.26213	0.260196	0.130098	−	0.991501i	$-0.458471\pi$
$409$	5.26213	0.260196	0.130098	+	0.991501i	$0.458471\pi$
$410$	0	0
$411$	−25.5569	−1.26063
$412$	0	0
$413$	12.1725	0.598968
$414$	0	0
$415$	8.65432	0.424824
$416$	0	0
$417$	−31.3155	−1.53353
$418$	0	0
$419$	−26.0337	−1.27183	−0.635915	−	0.771759i	$-0.719377\pi$
$419$	−26.0337	−1.27183	−0.635915	+	0.771759i	$0.719377\pi$
$420$	0	0
$421$	22.2423	1.08402	0.542011	−	0.840372i	$-0.317663\pi$
$421$	22.2423	1.08402	0.542011	+	0.840372i	$0.317663\pi$
$422$	0	0
$423$	−17.6724	−0.859260
$424$	0	0
$425$	6.17246	0.299408
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−5.05863	−0.244233
$430$	0	0
$431$	−27.6742	−1.33302	−0.666509	−	0.745497i	$-0.732212\pi$
$431$	−27.6742	−1.33302	−0.666509	+	0.745497i	$0.732212\pi$
$432$	0	0
$433$	−12.7880	−0.614552	−0.307276	−	0.951620i	$-0.599418\pi$
$433$	−12.7880	−0.614552	−0.307276	+	0.951620i	$0.599418\pi$
$434$	0	0
$435$	−6.21629	−0.298048
$436$	0	0
$437$	−8.58451	−0.410653
$438$	0	0
$439$	18.1656	0.866996	0.433498	−	0.901155i	$-0.357279\pi$
$439$	18.1656	0.866996	0.433498	+	0.901155i	$0.357279\pi$
$440$	0	0
$441$	2.05863	0.0980302
$442$	0	0
$443$	−0.107714	−0.00511767	−0.00255883	−	0.999997i	$-0.500815\pi$
$443$	−0.107714	−0.00511767	−0.00255883	+	0.999997i	$0.500815\pi$
$444$	0	0
$445$	3.65775	0.173394
$446$	0	0
$447$	−20.9345	−0.990167
$448$	0	0
$449$	−22.1725	−1.04638	−0.523192	−	0.852215i	$-0.675259\pi$
$449$	−22.1725	−1.04638	−0.523192	+	0.852215i	$0.675259\pi$
$450$	0	0
$451$	14.6155	0.688219
$452$	0	0
$453$	15.9087	0.747457
$454$	0	0
$455$	0.529317	0.0248147
$456$	0	0
$457$	4.35953	0.203930	0.101965	−	0.994788i	$-0.467487\pi$
$457$	4.35953	0.203930	0.101965	+	0.994788i	$0.467487\pi$
$458$	0	0
$459$	−2.76891	−0.129241
$460$	0	0
$461$	32.3810	1.50813	0.754067	−	0.656797i	$-0.228089\pi$
$461$	32.3810	1.50813	0.754067	+	0.656797i	$0.228089\pi$
$462$	0	0
$463$	−8.36641	−0.388820	−0.194410	−	0.980920i	$-0.562279\pi$
$463$	−8.36641	−0.388820	−0.194410	+	0.980920i	$0.562279\pi$
$464$	0	0
$465$	−8.36641	−0.387983
$466$	0	0
$467$	19.5423	0.904310	0.452155	−	0.891939i	$-0.350655\pi$
$467$	19.5423	0.904310	0.452155	+	0.891939i	$0.350655\pi$
$468$	0	0
$469$	15.9379	0.735945
$470$	0	0
$471$	13.5861	0.626016
$472$	0	0
$473$	−25.6267	−1.17832
$474$	0	0
$475$	−6.94137	−0.318492
$476$	0	0
$477$	23.2147	1.06293
$478$	0	0
$479$	28.5224	1.30322	0.651612	−	0.758553i	$-0.274093\pi$
$479$	28.5224	1.30322	0.651612	+	0.758553i	$0.274093\pi$
$480$	0	0
$481$	−2.36641	−0.107899
$482$	0	0
$483$	13.1284	0.597365
$484$	0	0
$485$	−1.83709	−0.0834180
$486$	0	0
$487$	−24.8241	−1.12489	−0.562444	−	0.826836i	$-0.690139\pi$
$487$	−24.8241	−1.12489	−0.562444	+	0.826836i	$0.690139\pi$
$488$	0	0
$489$	14.3518	0.649011
$490$	0	0
$491$	−29.1690	−1.31638	−0.658190	−	0.752852i	$-0.728678\pi$
$491$	−29.1690	−1.31638	−0.658190	+	0.752852i	$0.728678\pi$
$492$	0	0
$493$	−6.82860	−0.307545
$494$	0	0
$495$	2.45082	0.110156
$496$	0	0
$497$	−1.19051	−0.0534016
$498$	0	0
$499$	33.3009	1.49075	0.745376	−	0.666644i	$-0.232270\pi$
$499$	33.3009	1.49075	0.745376	+	0.666644i	$0.232270\pi$
$500$	0	0
$501$	−37.3009	−1.66648
$502$	0	0
$503$	12.3258	0.549581	0.274791	−	0.961504i	$-0.411391\pi$
$503$	12.3258	0.549581	0.274791	+	0.961504i	$0.411391\pi$
$504$	0	0
$505$	4.10266	0.182566
$506$	0	0
$507$	−2.24914	−0.0998878
$508$	0	0
$509$	23.7052	1.05072	0.525358	−	0.850882i	$-0.323932\pi$
$509$	23.7052	1.05072	0.525358	+	0.850882i	$0.323932\pi$
$510$	0	0
$511$	7.64315	0.338113
$512$	0	0
$513$	3.11383	0.137479
$514$	0	0
$515$	8.99656	0.396436
$516$	0	0
$517$	−19.3078	−0.849155
$518$	0	0
$519$	52.4070	2.30041
$520$	0	0
$521$	43.9018	1.92337	0.961687	−	0.274149i	$-0.0883962\pi$
$521$	43.9018	1.92337	0.961687	+	0.274149i	$0.0883962\pi$
$522$	0	0
$523$	−37.4328	−1.63682	−0.818410	−	0.574634i	$-0.805144\pi$
$523$	−37.4328	−1.63682	−0.818410	+	0.574634i	$0.805144\pi$
$524$	0	0
$525$	10.6155	0.463300
$526$	0	0
$527$	−9.19051	−0.400345
$528$	0	0
$529$	11.0716	0.481375
$530$	0	0
$531$	25.0586	1.08745
$532$	0	0
$533$	6.49828	0.281472
$534$	0	0
$535$	2.94137	0.127166
$536$	0	0
$537$	47.3269	2.04231
$538$	0	0
$539$	2.24914	0.0968773
$540$	0	0
$541$	−34.9751	−1.50370	−0.751848	−	0.659336i	$-0.770837\pi$
$541$	−34.9751	−1.50370	−0.751848	+	0.659336i	$0.770837\pi$
$542$	0	0
$543$	−37.6673	−1.61646
$544$	0	0
$545$	4.19395	0.179649
$546$	0	0
$547$	−6.50783	−0.278255	−0.139127	−	0.990274i	$-0.544430\pi$
$547$	−6.50783	−0.278255	−0.139127	+	0.990274i	$0.544430\pi$
$548$	0	0
$549$	−4.11727	−0.175721
$550$	0	0
$551$	7.67924	0.327146
$552$	0	0
$553$	1.33881	0.0569320
$554$	0	0
$555$	2.81722	0.119585
$556$	0	0
$557$	−43.4328	−1.84031	−0.920153	−	0.391559i	$-0.871936\pi$
$557$	−43.4328	−1.84031	−0.920153	+	0.391559i	$0.871936\pi$
$558$	0	0
$559$	−11.3940	−0.481915
$560$	0	0
$561$	6.61555	0.279309
$562$	0	0
$563$	33.8827	1.42799	0.713993	−	0.700152i	$-0.246885\pi$
$563$	33.8827	1.42799	0.713993	+	0.700152i	$0.246885\pi$
$564$	0	0
$565$	−5.23615	−0.220287
$566$	0	0
$567$	−10.9379	−0.459350
$568$	0	0
$569$	19.2147	0.805521	0.402760	−	0.915305i	$-0.368051\pi$
$569$	19.2147	0.805521	0.402760	+	0.915305i	$0.368051\pi$
$570$	0	0
$571$	−20.8268	−0.871573	−0.435787	−	0.900050i	$-0.643530\pi$
$571$	−20.8268	−0.871573	−0.435787	+	0.900050i	$0.643530\pi$
$572$	0	0
$573$	−16.7328	−0.699023
$574$	0	0
$575$	27.5500	1.14892
$576$	0	0
$577$	28.6448	1.19250	0.596249	−	0.802800i	$-0.296657\pi$
$577$	28.6448	1.19250	0.596249	+	0.802800i	$0.296657\pi$
$578$	0	0
$579$	3.37758	0.140367
$580$	0	0
$581$	16.3500	0.678311
$582$	0	0
$583$	25.3630	1.05043
$584$	0	0
$585$	1.08967	0.0450523
$586$	0	0
$587$	4.32076	0.178337	0.0891685	−	0.996017i	$-0.471579\pi$
$587$	4.32076	0.178337	0.0891685	+	0.996017i	$0.471579\pi$
$588$	0	0
$589$	10.3354	0.425862
$590$	0	0
$591$	−53.9639	−2.21978
$592$	0	0
$593$	−15.9690	−0.655767	−0.327883	−	0.944718i	$-0.606335\pi$
$593$	−15.9690	−0.655767	−0.327883	+	0.944718i	$0.606335\pi$
$594$	0	0
$595$	−0.692226	−0.0283785
$596$	0	0
$597$	−4.53114	−0.185447
$598$	0	0
$599$	16.8697	0.689279	0.344640	−	0.938735i	$-0.388001\pi$
$599$	16.8697	0.689279	0.344640	+	0.938735i	$0.388001\pi$
$600$	0	0
$601$	−15.3415	−0.625792	−0.312896	−	0.949787i	$-0.601299\pi$
$601$	−15.3415	−0.625792	−0.312896	+	0.949787i	$0.601299\pi$
$602$	0	0
$603$	32.8103	1.33614
$604$	0	0
$605$	−3.14486	−0.127857
$606$	0	0
$607$	−35.8353	−1.45451	−0.727254	−	0.686368i	$-0.759204\pi$
$607$	−35.8353	−1.45451	−0.727254	+	0.686368i	$0.759204\pi$
$608$	0	0
$609$	−11.7440	−0.475890
$610$	0	0
$611$	−8.58451	−0.347292
$612$	0	0
$613$	19.6673	0.794355	0.397177	−	0.917742i	$-0.369990\pi$
$613$	19.6673	0.794355	0.397177	+	0.917742i	$0.369990\pi$
$614$	0	0
$615$	−7.73625	−0.311956
$616$	0	0
$617$	−41.4588	−1.66907	−0.834533	−	0.550958i	$-0.814263\pi$
$617$	−41.4588	−1.66907	−0.834533	+	0.550958i	$0.814263\pi$
$618$	0	0
$619$	−10.8793	−0.437276	−0.218638	−	0.975806i	$-0.570161\pi$
$619$	−10.8793	−0.437276	−0.218638	+	0.975806i	$0.570161\pi$
$620$	0	0
$621$	−12.3587	−0.495937
$622$	0	0
$623$	6.91033	0.276857
$624$	0	0
$625$	20.8759	0.835034
$626$	0	0
$627$	−7.43965	−0.297111
$628$	0	0
$629$	3.09472	0.123395
$630$	0	0
$631$	31.4396	1.25159	0.625796	−	0.779987i	$-0.284774\pi$
$631$	31.4396	1.25159	0.625796	+	0.779987i	$0.284774\pi$
$632$	0	0
$633$	22.7182	0.902968
$634$	0	0
$635$	−0.436210	−0.0173105
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−2.45082	−0.0969529
$640$	0	0
$641$	3.04221	0.120160	0.0600799	−	0.998194i	$-0.480864\pi$
$641$	3.04221	0.120160	0.0600799	+	0.998194i	$0.480864\pi$
$642$	0	0
$643$	−8.02922	−0.316641	−0.158321	−	0.987388i	$-0.550608\pi$
$643$	−8.02922	−0.316641	−0.158321	+	0.987388i	$0.550608\pi$
$644$	0	0
$645$	13.5646	0.534107
$646$	0	0
$647$	−7.07324	−0.278078	−0.139039	−	0.990287i	$-0.544401\pi$
$647$	−7.07324	−0.278078	−0.139039	+	0.990287i	$0.544401\pi$
$648$	0	0
$649$	27.3776	1.07466
$650$	0	0
$651$	−15.8061	−0.619488
$652$	0	0
$653$	−15.7586	−0.616681	−0.308341	−	0.951276i	$-0.599774\pi$
$653$	−15.7586	−0.616681	−0.308341	+	0.951276i	$0.599774\pi$
$654$	0	0
$655$	−5.61899	−0.219552
$656$	0	0
$657$	15.7344	0.613859
$658$	0	0
$659$	12.2181	0.475950	0.237975	−	0.971271i	$-0.423516\pi$
$659$	12.2181	0.475950	0.237975	+	0.971271i	$0.423516\pi$
$660$	0	0
$661$	3.73443	0.145253	0.0726263	−	0.997359i	$-0.476862\pi$
$661$	3.73443	0.145253	0.0726263	+	0.997359i	$0.476862\pi$
$662$	0	0
$663$	2.94137	0.114233
$664$	0	0
$665$	0.778457	0.0301873
$666$	0	0
$667$	−30.4786	−1.18014
$668$	0	0
$669$	22.8095	0.881866
$670$	0	0
$671$	−4.49828	−0.173654
$672$	0	0
$673$	−5.65775	−0.218090	−0.109045	−	0.994037i	$-0.534779\pi$
$673$	−5.65775	−0.218090	−0.109045	+	0.994037i	$0.534779\pi$
$674$	0	0
$675$	−9.99312	−0.384636
$676$	0	0
$677$	9.39906	0.361235	0.180618	−	0.983553i	$-0.442190\pi$
$677$	9.39906	0.361235	0.180618	+	0.983553i	$0.442190\pi$
$678$	0	0
$679$	−3.47068	−0.133193
$680$	0	0
$681$	−12.1104	−0.464071
$682$	0	0
$683$	20.7328	0.793319	0.396660	−	0.917966i	$-0.370169\pi$
$683$	20.7328	0.793319	0.396660	+	0.917966i	$0.370169\pi$
$684$	0	0
$685$	6.01461	0.229806
$686$	0	0
$687$	−7.47250	−0.285094
$688$	0	0
$689$	11.2767	0.429610
$690$	0	0
$691$	−16.0862	−0.611949	−0.305975	−	0.952040i	$-0.598982\pi$
$691$	−16.0862	−0.611949	−0.305975	+	0.952040i	$0.598982\pi$
$692$	0	0
$693$	4.63016	0.175885
$694$	0	0
$695$	7.36984	0.279554
$696$	0	0
$697$	−8.49828	−0.321895
$698$	0	0
$699$	−30.8578	−1.16715
$700$	0	0
$701$	−6.98013	−0.263636	−0.131818	−	0.991274i	$-0.542081\pi$
$701$	−6.98013	−0.263636	−0.131818	+	0.991274i	$0.542081\pi$
$702$	0	0
$703$	−3.48024	−0.131260
$704$	0	0
$705$	10.2199	0.384905
$706$	0	0
$707$	7.75086	0.291501
$708$	0	0
$709$	8.39239	0.315183	0.157591	−	0.987504i	$-0.449627\pi$
$709$	8.39239	0.315183	0.157591	+	0.987504i	$0.449627\pi$
$710$	0	0
$711$	2.75612	0.103362
$712$	0	0
$713$	−41.0207	−1.53624
$714$	0	0
$715$	1.19051	0.0445225
$716$	0	0
$717$	−7.87586	−0.294129
$718$	0	0
$719$	5.16129	0.192484	0.0962418	−	0.995358i	$-0.469318\pi$
$719$	5.16129	0.192484	0.0962418	+	0.995358i	$0.469318\pi$
$720$	0	0
$721$	16.9966	0.632985
$722$	0	0
$723$	−3.57152	−0.132826
$724$	0	0
$725$	−24.6448	−0.915284
$726$	0	0
$727$	40.4362	1.49970	0.749848	−	0.661610i	$-0.230127\pi$
$727$	40.4362	1.49970	0.749848	+	0.661610i	$0.230127\pi$
$728$	0	0
$729$	−8.23109	−0.304855
$730$	0	0
$731$	14.9008	0.551125
$732$	0	0
$733$	−39.1311	−1.44534	−0.722670	−	0.691193i	$-0.757085\pi$
$733$	−39.1311	−1.44534	−0.722670	+	0.691193i	$0.757085\pi$
$734$	0	0
$735$	−1.19051	−0.0439125
$736$	0	0
$737$	35.8466	1.32043
$738$	0	0
$739$	7.13531	0.262477	0.131238	−	0.991351i	$-0.458105\pi$
$739$	7.13531	0.262477	0.131238	+	0.991351i	$0.458105\pi$
$740$	0	0
$741$	−3.30777	−0.121514
$742$	0	0
$743$	−13.8827	−0.509308	−0.254654	−	0.967032i	$-0.581962\pi$
$743$	−13.8827	−0.509308	−0.254654	+	0.967032i	$0.581962\pi$
$744$	0	0
$745$	4.92676	0.180502
$746$	0	0
$747$	33.6586	1.23150
$748$	0	0
$749$	5.55691	0.203045
$750$	0	0
$751$	37.3251	1.36201	0.681005	−	0.732278i	$-0.261543\pi$
$751$	37.3251	1.36201	0.681005	+	0.732278i	$0.261543\pi$
$752$	0	0
$753$	−11.0810	−0.403813
$754$	0	0
$755$	−3.74398	−0.136258
$756$	0	0
$757$	−7.10428	−0.258209	−0.129105	−	0.991631i	$-0.541210\pi$
$757$	−7.10428	−0.258209	−0.129105	+	0.991631i	$0.541210\pi$
$758$	0	0
$759$	29.5277	1.07179
$760$	0	0
$761$	−25.9621	−0.941125	−0.470562	−	0.882367i	$-0.655949\pi$
$761$	−25.9621	−0.941125	−0.470562	+	0.882367i	$0.655949\pi$
$762$	0	0
$763$	7.92332	0.286843
$764$	0	0
$765$	−1.42504	−0.0515224
$766$	0	0
$767$	12.1725	0.439522
$768$	0	0
$769$	−21.4638	−0.774005	−0.387002	−	0.922079i	$-0.626489\pi$
$769$	−21.4638	−0.774005	−0.387002	+	0.922079i	$0.626489\pi$
$770$	0	0
$771$	18.0260	0.649190
$772$	0	0
$773$	−40.4914	−1.45637	−0.728187	−	0.685378i	$-0.759637\pi$
$773$	−40.4914	−1.45637	−0.728187	+	0.685378i	$0.759637\pi$
$774$	0	0
$775$	−33.1690	−1.19147
$776$	0	0
$777$	5.32238	0.190939
$778$	0	0
$779$	9.55691	0.342412
$780$	0	0
$781$	−2.67762	−0.0958127
$782$	0	0
$783$	11.0554	0.395088
$784$	0	0
$785$	−3.19738	−0.114119
$786$	0	0
$787$	5.02072	0.178969	0.0894847	−	0.995988i	$-0.471478\pi$
$787$	5.02072	0.178969	0.0894847	+	0.995988i	$0.471478\pi$
$788$	0	0
$789$	3.60438	0.128319
$790$	0	0
$791$	−9.89229	−0.351729
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	−13.4250	−0.476137
$796$	0	0
$797$	19.7002	0.697815	0.348908	−	0.937157i	$-0.386553\pi$
$797$	19.7002	0.697815	0.348908	+	0.937157i	$0.386553\pi$
$798$	0	0
$799$	11.2266	0.397169
$800$	0	0
$801$	14.2258	0.502645
$802$	0	0
$803$	17.1905	0.606640
$804$	0	0
$805$	−3.08967	−0.108897
$806$	0	0
$807$	26.5863	0.935883
$808$	0	0
$809$	−2.57678	−0.0905947	−0.0452974	−	0.998974i	$-0.514424\pi$
$809$	−2.57678	−0.0905947	−0.0452974	+	0.998974i	$0.514424\pi$
$810$	0	0
$811$	−41.2311	−1.44782	−0.723910	−	0.689895i	$-0.757657\pi$
$811$	−41.2311	−1.44782	−0.723910	+	0.689895i	$0.757657\pi$
$812$	0	0
$813$	49.2173	1.72613
$814$	0	0
$815$	−3.37758	−0.118311
$816$	0	0
$817$	−16.7570	−0.586252
$818$	0	0
$819$	2.05863	0.0719345
$820$	0	0
$821$	5.26719	0.183826	0.0919130	−	0.995767i	$-0.470702\pi$
$821$	5.26719	0.183826	0.0919130	+	0.995767i	$0.470702\pi$
$822$	0	0
$823$	−36.0191	−1.25555	−0.627774	−	0.778396i	$-0.716034\pi$
$823$	−36.0191	−1.25555	−0.627774	+	0.778396i	$0.716034\pi$
$824$	0	0
$825$	23.8759	0.831251
$826$	0	0
$827$	16.3157	0.567353	0.283676	−	0.958920i	$-0.408446\pi$
$827$	16.3157	0.567353	0.283676	+	0.958920i	$0.408446\pi$
$828$	0	0
$829$	−30.3956	−1.05568	−0.527842	−	0.849343i	$-0.676999\pi$
$829$	−30.3956	−1.05568	−0.527842	+	0.849343i	$0.676999\pi$
$830$	0	0
$831$	22.9820	0.797235
$832$	0	0
$833$	−1.30777	−0.0453117
$834$	0	0
$835$	8.77846	0.303791
$836$	0	0
$837$	14.8793	0.514304
$838$	0	0
$839$	−29.8398	−1.03018	−0.515092	−	0.857135i	$-0.672242\pi$
$839$	−29.8398	−1.03018	−0.515092	+	0.857135i	$0.672242\pi$
$840$	0	0
$841$	−1.73549	−0.0598445
$842$	0	0
$843$	−3.46886	−0.119474
$844$	0	0
$845$	0.529317	0.0182090
$846$	0	0
$847$	−5.94137	−0.204148
$848$	0	0
$849$	35.6413	1.22321
$850$	0	0
$851$	13.8129	0.473501
$852$	0	0
$853$	−0.203497	−0.00696761	−0.00348380	−	0.999994i	$-0.501109\pi$
$853$	−0.203497	−0.00696761	−0.00348380	+	0.999994i	$0.501109\pi$
$854$	0	0
$855$	1.60256	0.0548063
$856$	0	0
$857$	−12.6155	−0.430939	−0.215469	−	0.976511i	$-0.569128\pi$
$857$	−12.6155	−0.430939	−0.215469	+	0.976511i	$0.569128\pi$
$858$	0	0
$859$	27.9671	0.954227	0.477113	−	0.878842i	$-0.341683\pi$
$859$	27.9671	0.954227	0.477113	+	0.878842i	$0.341683\pi$
$860$	0	0
$861$	−14.6155	−0.498097
$862$	0	0
$863$	2.76891	0.0942546	0.0471273	−	0.998889i	$-0.484993\pi$
$863$	2.76891	0.0942546	0.0471273	+	0.998889i	$0.484993\pi$
$864$	0	0
$865$	−12.3336	−0.419353
$866$	0	0
$867$	34.3887	1.16790
$868$	0	0
$869$	3.01117	0.102147
$870$	0	0
$871$	15.9379	0.540036
$872$	0	0
$873$	−7.14486	−0.241817
$874$	0	0
$875$	−5.14486	−0.173928
$876$	0	0
$877$	6.71133	0.226626	0.113313	−	0.993559i	$-0.463854\pi$
$877$	6.71133	0.226626	0.113313	+	0.993559i	$0.463854\pi$
$878$	0	0
$879$	−24.9268	−0.840759
$880$	0	0
$881$	−18.3741	−0.619040	−0.309520	−	0.950893i	$-0.600168\pi$
$881$	−18.3741	−0.619040	−0.309520	+	0.950893i	$0.600168\pi$
$882$	0	0
$883$	−9.93105	−0.334207	−0.167103	−	0.985939i	$-0.553441\pi$
$883$	−9.93105	−0.334207	−0.167103	+	0.985939i	$0.553441\pi$
$884$	0	0
$885$	−14.4914	−0.487123
$886$	0	0
$887$	51.3776	1.72509	0.862545	−	0.505980i	$-0.168869\pi$
$887$	51.3776	1.72509	0.862545	+	0.505980i	$0.168869\pi$
$888$	0	0
$889$	−0.824101	−0.0276394
$890$	0	0
$891$	−24.6009	−0.824162
$892$	0	0
$893$	−12.6251	−0.422483
$894$	0	0
$895$	−11.1380	−0.372302
$896$	0	0
$897$	13.1284	0.438346
$898$	0	0
$899$	36.6949	1.22384
$900$	0	0
$901$	−14.7474	−0.491308
$902$	0	0
$903$	25.6267	0.852804
$904$	0	0
$905$	8.86469	0.294672
$906$	0	0
$907$	−4.34225	−0.144182	−0.0720910	−	0.997398i	$-0.522967\pi$
$907$	−4.34225	−0.144182	−0.0720910	+	0.997398i	$0.522967\pi$
$908$	0	0
$909$	15.9562	0.529233
$910$	0	0
$911$	−31.4853	−1.04315	−0.521577	−	0.853204i	$-0.674656\pi$
$911$	−31.4853	−1.04315	−0.521577	+	0.853204i	$0.674656\pi$
$912$	0	0
$913$	36.7734	1.21702
$914$	0	0
$915$	2.38101	0.0787139
$916$	0	0
$917$	−10.6155	−0.350556
$918$	0	0
$919$	−43.4588	−1.43357	−0.716786	−	0.697293i	$-0.754388\pi$
$919$	−43.4588	−1.43357	−0.716786	+	0.697293i	$0.754388\pi$
$920$	0	0
$921$	45.9096	1.51277
$922$	0	0
$923$	−1.19051	−0.0391860
$924$	0	0
$925$	11.1690	0.367235
$926$	0	0
$927$	34.9897	1.14921
$928$	0	0
$929$	4.40517	0.144529	0.0722645	−	0.997385i	$-0.476977\pi$
$929$	4.40517	0.144529	0.0722645	+	0.997385i	$0.476977\pi$
$930$	0	0
$931$	1.47068	0.0481997
$932$	0	0
$933$	−4.32582	−0.141621
$934$	0	0
$935$	−1.55691	−0.0509165
$936$	0	0
$937$	−34.0990	−1.11397	−0.556983	−	0.830524i	$-0.688041\pi$
$937$	−34.0990	−1.11397	−0.556983	+	0.830524i	$0.688041\pi$
$938$	0	0
$939$	32.3121	1.05446
$940$	0	0
$941$	44.4672	1.44959	0.724795	−	0.688964i	$-0.241934\pi$
$941$	44.4672	1.44959	0.724795	+	0.688964i	$0.241934\pi$
$942$	0	0
$943$	−37.9311	−1.23521
$944$	0	0
$945$	1.12070	0.0364565
$946$	0	0
$947$	−57.9311	−1.88251	−0.941253	−	0.337702i	$-0.890350\pi$
$947$	−57.9311	−1.88251	−0.941253	+	0.337702i	$0.890350\pi$
$948$	0	0
$949$	7.64315	0.248107
$950$	0	0
$951$	−34.9897	−1.13462
$952$	0	0
$953$	35.3060	1.14367	0.571836	−	0.820368i	$-0.306231\pi$
$953$	35.3060	1.14367	0.571836	+	0.820368i	$0.306231\pi$
$954$	0	0
$955$	3.93793	0.127428
$956$	0	0
$957$	−26.4139	−0.853839
$958$	0	0
$959$	11.3630	0.366929
$960$	0	0
$961$	18.3871	0.593133
$962$	0	0
$963$	11.4396	0.368638
$964$	0	0
$965$	−0.794885	−0.0255882
$966$	0	0
$967$	−23.7148	−0.762616	−0.381308	−	0.924448i	$-0.624526\pi$
$967$	−23.7148	−0.762616	−0.381308	+	0.924448i	$0.624526\pi$
$968$	0	0
$969$	4.32582	0.138965
$970$	0	0
$971$	23.9379	0.768205	0.384102	−	0.923291i	$-0.374511\pi$
$971$	23.9379	0.768205	0.384102	+	0.923291i	$0.374511\pi$
$972$	0	0
$973$	13.9233	0.446361
$974$	0	0
$975$	10.6155	0.339970
$976$	0	0
$977$	−16.1871	−0.517870	−0.258935	−	0.965895i	$-0.583372\pi$
$977$	−16.1871	−0.517870	−0.258935	+	0.965895i	$0.583372\pi$
$978$	0	0
$979$	15.5423	0.496734
$980$	0	0
$981$	16.3112	0.520777
$982$	0	0
$983$	−45.7243	−1.45838	−0.729190	−	0.684312i	$-0.760103\pi$
$983$	−45.7243	−1.45838	−0.729190	+	0.684312i	$0.760103\pi$
$984$	0	0
$985$	12.7000	0.404654
$986$	0	0
$987$	19.3078	0.614573
$988$	0	0
$989$	66.5078	2.11483
$990$	0	0
$991$	5.40356	0.171650	0.0858248	−	0.996310i	$-0.472647\pi$
$991$	5.40356	0.171650	0.0858248	+	0.996310i	$0.472647\pi$
$992$	0	0
$993$	70.9605	2.25186
$994$	0	0
$995$	1.06637	0.0338061
$996$	0	0
$997$	6.04832	0.191552	0.0957761	−	0.995403i	$-0.469467\pi$
$997$	6.04832	0.191552	0.0957761	+	0.995403i	$0.469467\pi$
$998$	0	0
$999$	−5.01031	−0.158519

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.t.1.1		3
4.3	odd	2		91.2.a.d.1.2	✓	3
8.3	odd	2		5824.2.a.by.1.1		3
8.5	even	2		5824.2.a.bs.1.3		3
12.11	even	2		819.2.a.i.1.2		3
20.19	odd	2		2275.2.a.m.1.2		3
28.3	even	6		637.2.e.i.79.2		6
28.11	odd	6		637.2.e.j.79.2		6
28.19	even	6		637.2.e.i.508.2		6
28.23	odd	6		637.2.e.j.508.2		6
28.27	even	2		637.2.a.j.1.2		3
52.31	even	4		1183.2.c.f.337.3		6
52.47	even	4		1183.2.c.f.337.4		6
52.51	odd	2		1183.2.a.i.1.2		3
84.83	odd	2		5733.2.a.x.1.2		3
364.363	even	2		8281.2.a.bg.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.d.1.2	✓	3	4.3	odd	2
637.2.a.j.1.2		3	28.27	even	2
637.2.e.i.79.2		6	28.3	even	6
637.2.e.i.508.2		6	28.19	even	6
637.2.e.j.79.2		6	28.11	odd	6
637.2.e.j.508.2		6	28.23	odd	6
819.2.a.i.1.2		3	12.11	even	2
1183.2.a.i.1.2		3	52.51	odd	2
1183.2.c.f.337.3		6	52.31	even	4
1183.2.c.f.337.4		6	52.47	even	4
1456.2.a.t.1.1		3	1.1	even	1	trivial
2275.2.a.m.1.2		3	20.19	odd	2
5733.2.a.x.1.2		3	84.83	odd	2
5824.2.a.bs.1.3		3	8.5	even	2
5824.2.a.by.1.1		3	8.3	odd	2
8281.2.a.bg.1.2		3	364.363	even	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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24914 q^{3} +0.529317 q^{5} +1.00000 q^{7} +2.05863 q^{9} +2.24914 q^{11} +1.00000 q^{13} -1.19051 q^{15} -1.30777 q^{17} +1.47068 q^{19} -2.24914 q^{21} -5.83709 q^{23} -4.71982 q^{25} +2.11727 q^{27} +5.22154 q^{29} +7.02760 q^{31} -5.05863 q^{33} +0.529317 q^{35} -2.36641 q^{37} -2.24914 q^{39} +6.49828 q^{41} -11.3940 q^{43} +1.08967 q^{45} -8.58451 q^{47} +1.00000 q^{49} +2.94137 q^{51} +11.2767 q^{53} +1.19051 q^{55} -3.30777 q^{57} +12.1725 q^{59} -2.00000 q^{61} +2.05863 q^{63} +0.529317 q^{65} +15.9379 q^{67} +13.1284 q^{69} -1.19051 q^{71} +7.64315 q^{73} +10.6155 q^{75} +2.24914 q^{77} +1.33881 q^{79} -10.9379 q^{81} +16.3500 q^{83} -0.692226 q^{85} -11.7440 q^{87} +6.91033 q^{89} +1.00000 q^{91} -15.8061 q^{93} +0.778457 q^{95} -3.47068 q^{97} +4.63016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{5} + 3 q^{7} + 7 q^{9} - 2 q^{11} + 3 q^{13} + 6 q^{15} + 4 q^{17} + 4 q^{19} + 2 q^{21} - 10 q^{23} - 5 q^{25} + 8 q^{27} + 24 q^{29} + 4 q^{31} - 16 q^{33} + 2 q^{35} + 2 q^{39} + 2 q^{41}+ \cdots - 14 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^5 + 3 * q^7 + 7 * q^9 - 2 * q^11 + 3 * q^13 + 6 * q^15 + 4 * q^17 + 4 * q^19 + 2 * q^21 - 10 * q^23 - 5 * q^25 + 8 * q^27 + 24 * q^29 + 4 * q^31 - 16 * q^33 + 2 * q^35 + 2 * q^39 + 2 * q^41 - 10 * q^43 + 22 * q^45 + 8 * q^47 + 3 * q^49 + 8 * q^51 + 8 * q^53 - 6 * q^55 - 2 * q^57 + 4 * q^59 - 6 * q^61 + 7 * q^63 + 2 * q^65 + 12 * q^67 - 6 * q^69 + 6 * q^71 - 10 * q^73 + 16 * q^75 - 2 * q^77 + 14 * q^79 + 3 * q^81 + 12 * q^83 - 10 * q^85 + 26 * q^87 + 2 * q^89 + 3 * q^91 - 22 * q^93 - 6 * q^95 - 10 * q^97 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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