LMFDB - Embedded newform 5733.2.a.x.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5733,2,Mod(1,5733)] 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("5733.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5733, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,0,3,2,0,0,-3,0,8,-2,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$45.7782354788$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
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			1.2
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.470683 q^{2} -1.77846 q^{4} +0.529317 q^{5} +1.77846 q^{8} -0.249141 q^{10} +2.24914 q^{11} -1.00000 q^{13} +2.71982 q^{16} -1.30777 q^{17} +1.47068 q^{19} -0.941367 q^{20} -1.05863 q^{22} -5.83709 q^{23} -4.71982 q^{25} +0.470683 q^{26} -5.22154 q^{29} +7.02760 q^{31} -4.83709 q^{32} +0.615547 q^{34} -2.36641 q^{37} -0.692226 q^{38} +0.941367 q^{40} +6.49828 q^{41} +11.3940 q^{43} -4.00000 q^{44} +2.74742 q^{46} +8.58451 q^{47} +2.22154 q^{50} +1.77846 q^{52} -11.2767 q^{53} +1.19051 q^{55} +2.45769 q^{58} -12.1725 q^{59} +2.00000 q^{61} -3.30777 q^{62} -3.16291 q^{64} -0.529317 q^{65} -15.9379 q^{67} +2.32582 q^{68} -1.19051 q^{71} -7.64315 q^{73} +1.11383 q^{74} -2.61555 q^{76} -1.33881 q^{79} +1.43965 q^{80} -3.05863 q^{82} -16.3500 q^{83} -0.692226 q^{85} -5.36297 q^{86} +4.00000 q^{88} +6.91033 q^{89} +10.3810 q^{92} -4.04059 q^{94} +0.778457 q^{95} +3.47068 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{8} + 8 q^{10} - 2 q^{11} - 3 q^{13} - q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 4 q^{22} - 10 q^{23} - 5 q^{25} + q^{26} - 24 q^{29} + 4 q^{31} - 7 q^{32} - 14 q^{34}+ \cdots + 10 q^{97}+O(q^{100}) $ 3 * q - q^2 + 3 * q^4 + 2 * q^5 - 3 * q^8 + 8 * q^10 - 2 * q^11 - 3 * q^13 - q^16 + 4 * q^17 + 4 * q^19 - 2 * q^20 - 4 * q^22 - 10 * q^23 - 5 * q^25 + q^26 - 24 * q^29 + 4 * q^31 - 7 * q^32 - 14 * q^34 - 10 * q^38 + 2 * q^40 + 2 * q^41 + 10 * q^43 - 12 * q^44 - 18 * q^46 - 8 * q^47 + 15 * q^50 - 3 * q^52 - 8 * q^53 - 6 * q^55 + 12 * q^58 - 4 * q^59 + 6 * q^61 - 2 * q^62 - 17 * q^64 - 2 * q^65 - 12 * q^67 + 22 * q^68 + 6 * q^71 + 10 * q^73 - 30 * q^74 + 8 * q^76 - 14 * q^79 - 14 * q^80 - 10 * q^82 - 12 * q^83 - 10 * q^85 + 26 * q^86 + 12 * q^88 + 2 * q^89 + 12 * q^92 + 10 * q^94 - 6 * q^95 + 10 * q^97

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 $

$2$	−0.470683	−0.332823	−0.166412	−	0.986056i	$-0.553218\pi$
$2$	−0.470683	−0.332823	−0.166412	+	0.986056i	$0.553218\pi$
$3$	0	0
$3$	0	0
$4$	−1.77846	−0.889229
$5$	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
$7$	0	0
$7$	0	0
$8$	1.77846	0.628780
$9$	0	0
$10$	−0.249141	−0.0787852
$11$	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
$15$	0	0
$16$	2.71982	0.679956
$17$	−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	$0.446043\pi$
$20$	−0.941367	−0.210496
$21$	0	0
$22$	−1.05863	−0.225701
$23$	−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
$26$	0.470683	0.0923086
$27$	0	0
$28$	0	0
$29$	−5.22154	−0.969616	−0.484808	−	0.874621i	$-0.661111\pi$
$29$	−5.22154	−0.969616	−0.484808	+	0.874621i	$0.661111\pi$
$30$	0	0
$31$	7.02760	1.26219	0.631097	−	0.775704i	$-0.282605\pi$
$31$	7.02760	1.26219	0.631097	+	0.775704i	$0.282605\pi$
$32$	−4.83709	−0.855085
$33$	0	0
$34$	0.615547	0.105566
$35$	0	0
$36$	0	0
$37$	−2.36641	−0.389035	−0.194517	−	0.980899i	$-0.562314\pi$
$37$	−2.36641	−0.389035	−0.194517	+	0.980899i	$0.562314\pi$
$38$	−0.692226	−0.112294
$39$	0	0
$40$	0.941367	0.148843
$41$	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	$-0.164902\pi$
$43$	11.3940	1.73757	0.868785	+	0.495190i	$0.164902\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	2.74742	0.405085
$47$	8.58451	1.25218	0.626090	−	0.779751i	$-0.284654\pi$
$47$	8.58451	1.25218	0.626090	+	0.779751i	$0.284654\pi$
$48$	0	0
$49$	0	0
$50$	2.22154	0.314174
$51$	0	0
$52$	1.77846	0.246628
$53$	−11.2767	−1.54898	−0.774490	−	0.632587i	$-0.781993\pi$
$53$	−11.2767	−1.54898	−0.774490	+	0.632587i	$0.781993\pi$
$54$	0	0
$55$	1.19051	0.160528
$56$	0	0
$57$	0	0
$58$	2.45769	0.322711
$59$	−12.1725	−1.58472	−0.792360	−	0.610054i	$-0.791148\pi$
$59$	−12.1725	−1.58472	−0.792360	+	0.610054i	$0.791148\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−3.30777	−0.420088
$63$	0	0
$64$	−3.16291	−0.395364
$65$	−0.529317	−0.0656536
$66$	0	0
$67$	−15.9379	−1.94713	−0.973564	−	0.228415i	$-0.926646\pi$
$67$	−15.9379	−1.94713	−0.973564	+	0.228415i	$0.926646\pi$
$68$	2.32582	0.282047
$69$	0	0
$70$	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	$-0.647608\pi$
$73$	−7.64315	−0.894562	−0.447281	+	0.894393i	$0.647608\pi$
$74$	1.11383	0.129480
$75$	0	0
$76$	−2.61555	−0.300024
$77$	0	0
$78$	0	0
$79$	−1.33881	−0.150628	−0.0753139	−	0.997160i	$-0.523996\pi$
$79$	−1.33881	−0.150628	−0.0753139	+	0.997160i	$0.523996\pi$
$80$	1.43965	0.160958
$81$	0	0
$82$	−3.05863	−0.337770
$83$	−16.3500	−1.79464	−0.897322	−	0.441377i	$-0.854490\pi$
$83$	−16.3500	−1.79464	−0.897322	+	0.441377i	$0.854490\pi$
$84$	0	0
$85$	−0.692226	−0.0750825
$86$	−5.36297	−0.578304
$87$	0	0
$88$	4.00000	0.426401
$89$	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
$91$	0	0
$92$	10.3810	1.08230
$93$	0	0
$94$	−4.04059	−0.416755
$95$	0.778457	0.0798680
$96$	0	0
$97$	3.47068	0.352395	0.176197	−	0.984355i	$-0.443620\pi$
$97$	3.47068	0.352395	0.176197	+	0.984355i	$0.443620\pi$
$98$	0	0
$99$	0	0
$100$	8.39400	0.839400
$101$	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	$0.184098\pi$
$104$	−1.77846	−0.174392
$105$	0	0
$106$	5.30777	0.515537
$107$	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	$0.376110\pi$
$110$	−0.560352	−0.0534275
$111$	0	0
$112$	0	0
$113$	9.89229	0.930588	0.465294	−	0.885156i	$-0.345949\pi$
$113$	9.89229	0.930588	0.465294	+	0.885156i	$0.345949\pi$
$114$	0	0
$115$	−3.08967	−0.288113
$116$	9.28629	0.862210
$117$	0	0
$118$	5.72938	0.527432
$119$	0	0
$120$	0	0
$121$	−5.94137	−0.540124
$122$	−0.941367	−0.0852273
$123$	0	0
$124$	−12.4983	−1.12238
$125$	−5.14486	−0.460171
$126$	0	0
$127$	0.824101	0.0731271	0.0365635	−	0.999331i	$-0.488359\pi$
$127$	0.824101	0.0731271	0.0365635	+	0.999331i	$0.488359\pi$
$128$	11.1629	0.986671
$129$	0	0
$130$	0.249141	0.0218511
$131$	10.6155	0.927485	0.463742	−	0.885970i	$-0.346506\pi$
$131$	10.6155	0.927485	0.463742	+	0.885970i	$0.346506\pi$
$132$	0	0
$133$	0	0
$134$	7.50172	0.648050
$135$	0	0
$136$	−2.32582	−0.199437
$137$	−11.3630	−0.970804	−0.485402	−	0.874291i	$-0.661327\pi$
$137$	−11.3630	−0.970804	−0.485402	+	0.874291i	$0.661327\pi$
$138$	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
$141$	0	0
$142$	0.560352	0.0470237
$143$	−2.24914	−0.188083
$144$	0	0
$145$	−2.76385	−0.229525
$146$	3.59750	0.297731
$147$	0	0
$148$	4.20855	0.345941
$149$	−9.30777	−0.762523	−0.381261	−	0.924467i	$-0.624510\pi$
$149$	−9.30777	−0.762523	−0.381261	+	0.924467i	$0.624510\pi$
$150$	0	0
$151$	7.07324	0.575612	0.287806	−	0.957689i	$-0.407074\pi$
$151$	7.07324	0.575612	0.287806	+	0.957689i	$0.407074\pi$
$152$	2.61555	0.212149
$153$	0	0
$154$	0	0
$155$	3.71982	0.298783
$156$	0	0
$157$	6.04059	0.482091	0.241046	−	0.970514i	$-0.422510\pi$
$157$	6.04059	0.482091	0.241046	+	0.970514i	$0.422510\pi$
$158$	0.630155	0.0501325
$159$	0	0
$160$	−2.56035	−0.202414
$161$	0	0
$162$	0	0
$163$	6.38101	0.499800	0.249900	−	0.968272i	$-0.419602\pi$
$163$	6.38101	0.499800	0.249900	+	0.968272i	$0.419602\pi$
$164$	−11.5569	−0.902443
$165$	0	0
$166$	7.69566	0.597299
$167$	−16.5845	−1.28335	−0.641674	−	0.766977i	$-0.721760\pi$
$167$	−16.5845	−1.28335	−0.641674	+	0.766977i	$0.721760\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0.325819	0.0249892
$171$	0	0
$172$	−20.2637	−1.54510
$173$	−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
$175$	0	0
$176$	6.11727	0.461106
$177$	0	0
$178$	−3.25258	−0.243791
$179$	−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	$-0.713848\pi$
$181$	−16.7474	−1.24483	−0.622413	+	0.782689i	$0.713848\pi$
$182$	0	0
$183$	0	0
$184$	−10.3810	−0.765299
$185$	−1.25258	−0.0920914
$186$	0	0
$187$	−2.94137	−0.215094
$188$	−15.2672	−1.11347
$189$	0	0
$190$	−0.366407	−0.0265819
$191$	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	$0.517212\pi$
$194$	−1.63359	−0.117285
$195$	0	0
$196$	0	0
$197$	−23.9931	−1.70944	−0.854720	−	0.519090i	$-0.826271\pi$
$197$	−23.9931	−1.70944	−0.854720	+	0.519090i	$0.826271\pi$
$198$	0	0
$199$	2.01461	0.142812	0.0714059	−	0.997447i	$-0.477251\pi$
$199$	2.01461	0.142812	0.0714059	+	0.997447i	$0.477251\pi$
$200$	−8.39400	−0.593546
$201$	0	0
$202$	−3.64820	−0.256687
$203$	0	0
$204$	0	0
$205$	3.43965	0.240235
$206$	−8.00000	−0.557386
$207$	0	0
$208$	−2.71982	−0.188586
$209$	3.30777	0.228803
$210$	0	0
$211$	10.1008	0.695370	0.347685	−	0.937611i	$-0.386968\pi$
$211$	10.1008	0.695370	0.347685	+	0.937611i	$0.386968\pi$
$212$	20.0552	1.37740
$213$	0	0
$214$	−2.61555	−0.178795
$215$	6.03104	0.411313
$216$	0	0
$217$	0	0
$218$	−3.72938	−0.252585
$219$	0	0
$220$	−2.11727	−0.142746
$221$	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
$225$	0	0
$226$	−4.65613	−0.309721
$227$	−5.38445	−0.357379	−0.178689	−	0.983906i	$-0.557186\pi$
$227$	−5.38445	−0.357379	−0.178689	+	0.983906i	$0.557186\pi$
$228$	0	0
$229$	−3.32238	−0.219549	−0.109775	−	0.993957i	$-0.535013\pi$
$229$	−3.32238	−0.219549	−0.109775	+	0.993957i	$0.535013\pi$
$230$	1.45426	0.0958908
$231$	0	0
$232$	−9.28629	−0.609675
$233$	−13.7198	−0.898816	−0.449408	−	0.893327i	$-0.648365\pi$
$233$	−13.7198	−0.898816	−0.449408	+	0.893327i	$0.648365\pi$
$234$	0	0
$235$	4.54392	0.296413
$236$	21.6482	1.40918
$237$	0	0
$238$	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	$-0.516287\pi$
$241$	−1.58795	−0.102289	−0.0511444	+	0.998691i	$0.516287\pi$
$242$	2.79650	0.179766
$243$	0	0
$244$	−3.55691	−0.227708
$245$	0	0
$246$	0	0
$247$	−1.47068	−0.0935773
$248$	12.4983	0.793642
$249$	0	0
$250$	2.42160	0.153156
$251$	−4.92676	−0.310974	−0.155487	−	0.987838i	$-0.549695\pi$
$251$	−4.92676	−0.310974	−0.155487	+	0.987838i	$0.549695\pi$
$252$	0	0
$253$	−13.1284	−0.825378
$254$	−0.387890	−0.0243384
$255$	0	0
$256$	1.07162	0.0669764
$257$	−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
$259$	0	0
$260$	0.941367	0.0583811
$261$	0	0
$262$	−4.99656	−0.308689
$263$	−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
$267$	0	0
$268$	28.3449	1.73144
$269$	−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	$0.731415\pi$
$272$	−3.55691	−0.215670
$273$	0	0
$274$	5.34836	0.323106
$275$	−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	$0.599316\pi$
$278$	−6.55348	−0.393051
$279$	0	0
$280$	0	0
$281$	−1.54231	−0.0920063	−0.0460031	−	0.998941i	$-0.514648\pi$
$281$	−1.54231	−0.0920063	−0.0460031	+	0.998941i	$0.514648\pi$
$282$	0	0
$283$	−15.8466	−0.941985	−0.470993	−	0.882137i	$-0.656104\pi$
$283$	−15.8466	−0.941985	−0.470993	+	0.882137i	$0.656104\pi$
$284$	2.11727	0.125637
$285$	0	0
$286$	1.05863	0.0625983
$287$	0	0
$288$	0	0
$289$	−15.2897	−0.899396
$290$	1.30090	0.0763914
$291$	0	0
$292$	13.5930	0.795470
$293$	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
$296$	−4.20855	−0.244617
$297$	0	0
$298$	4.38101	0.253785
$299$	5.83709	0.337568
$300$	0	0
$301$	0	0
$302$	−3.32926	−0.191577
$303$	0	0
$304$	4.00000	0.229416
$305$	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
$309$	0	0
$310$	−1.75086	−0.0994421
$311$	−1.92332	−0.109062	−0.0545308	−	0.998512i	$-0.517366\pi$
$311$	−1.92332	−0.109062	−0.0545308	+	0.998512i	$0.517366\pi$
$312$	0	0
$313$	14.3664	0.812037	0.406019	−	0.913865i	$-0.366917\pi$
$313$	14.3664	0.812037	0.406019	+	0.913865i	$0.366917\pi$
$314$	−2.84320	−0.160451
$315$	0	0
$316$	2.38101	0.133943
$317$	−15.5569	−0.873763	−0.436882	−	0.899519i	$-0.643917\pi$
$317$	−15.5569	−0.873763	−0.436882	+	0.899519i	$0.643917\pi$
$318$	0	0
$319$	−11.7440	−0.657537
$320$	−1.67418	−0.0935895
$321$	0	0
$322$	0	0
$323$	−1.92332	−0.107016
$324$	0	0
$325$	4.71982	0.261809
$326$	−3.00344	−0.166345
$327$	0	0
$328$	11.5569	0.638124
$329$	0	0
$330$	0	0
$331$	31.5500	1.73415	0.867073	−	0.498180i	$-0.165998\pi$
$331$	31.5500	1.73415	0.867073	+	0.498180i	$0.165998\pi$
$332$	29.0777	1.59585
$333$	0	0
$334$	7.80605	0.427128
$335$	−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	$0.573714\pi$
$338$	−0.470683	−0.0256018
$339$	0	0
$340$	1.23109	0.0667655
$341$	15.8061	0.855946
$342$	0	0
$343$	0	0
$344$	20.2637	1.09255
$345$	0	0
$346$	10.9673	0.589608
$347$	−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	$-0.760233\pi$
$349$	−27.2553	−1.45894	−0.729470	+	0.684013i	$0.760233\pi$
$350$	0	0
$351$	0	0
$352$	−10.8793	−0.579868
$353$	−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
$356$	−12.2897	−0.651354
$357$	0	0
$358$	9.90422	0.523454
$359$	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
$362$	7.88273	0.414307
$363$	0	0
$364$	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	$0.374793\pi$
$368$	−15.8759	−0.827586
$369$	0	0
$370$	0.589568	0.0306502
$371$	0	0
$372$	0	0
$373$	−23.6673	−1.22545	−0.612723	−	0.790298i	$-0.709926\pi$
$373$	−23.6673	−1.22545	−0.612723	+	0.790298i	$0.709926\pi$
$374$	1.38445	0.0715883
$375$	0	0
$376$	15.2672	0.787345
$377$	5.22154	0.268923
$378$	0	0
$379$	−32.7405	−1.68177	−0.840884	−	0.541215i	$-0.817965\pi$
$379$	−32.7405	−1.68177	−0.840884	+	0.541215i	$0.817965\pi$
$380$	−1.38445	−0.0710209
$381$	0	0
$382$	−3.50172	−0.179164
$383$	−22.6155	−1.15560	−0.577800	−	0.816178i	$-0.696089\pi$
$383$	−22.6155	−1.15560	−0.577800	+	0.816178i	$0.696089\pi$
$384$	0	0
$385$	0	0
$386$	0.706834	0.0359769
$387$	0	0
$388$	−6.17246	−0.313359
$389$	−38.0483	−1.92913	−0.964563	−	0.263852i	$-0.915007\pi$
$389$	−38.0483	−1.92913	−0.964563	+	0.263852i	$0.915007\pi$
$390$	0	0
$391$	7.63359	0.386047
$392$	0	0
$393$	0	0
$394$	11.2932	0.568941
$395$	−0.708654	−0.0356562
$396$	0	0
$397$	11.7052	0.587468	0.293734	−	0.955887i	$-0.405102\pi$
$397$	11.7052	0.587468	0.293734	+	0.955887i	$0.405102\pi$
$398$	−0.948243	−0.0475311
$399$	0	0
$400$	−12.8371	−0.641855
$401$	−3.55691	−0.177624	−0.0888119	−	0.996048i	$-0.528307\pi$
$401$	−3.55691	−0.177624	−0.0888119	+	0.996048i	$0.528307\pi$
$402$	0	0
$403$	−7.02760	−0.350070
$404$	−13.7846	−0.685808
$405$	0	0
$406$	0	0
$407$	−5.32238	−0.263821
$408$	0	0
$409$	−5.26213	−0.260196	−0.130098	−	0.991501i	$-0.541529\pi$
$409$	−5.26213	−0.260196	−0.130098	+	0.991501i	$0.541529\pi$
$410$	−1.61899	−0.0799560
$411$	0	0
$412$	−30.2277	−1.48921
$413$	0	0
$414$	0	0
$415$	−8.65432	−0.424824
$416$	4.83709	0.237158
$417$	0	0
$418$	−1.55691	−0.0761512
$419$	26.0337	1.27183	0.635915	−	0.771759i	$-0.280623\pi$
$419$	26.0337	1.27183	0.635915	+	0.771759i	$0.280623\pi$
$420$	0	0
$421$	22.2423	1.08402	0.542011	−	0.840372i	$-0.317663\pi$
$421$	22.2423	1.08402	0.542011	+	0.840372i	$0.317663\pi$
$422$	−4.75430	−0.231436
$423$	0	0
$424$	−20.0552	−0.973966
$425$	6.17246	0.299408
$426$	0	0
$427$	0	0
$428$	−9.88273	−0.477700
$429$	0	0
$430$	−2.83871	−0.136895
$431$	−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	$-0.400582\pi$
$433$	12.7880	0.614552	0.307276	+	0.951620i	$0.400582\pi$
$434$	0	0
$435$	0	0
$436$	−14.0913	−0.674850
$437$	−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	$0.357279\pi$
$440$	2.11727	0.100937
$441$	0	0
$442$	−0.615547	−0.0292786
$443$	−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
$446$	4.77340	0.226027
$447$	0	0
$448$	0	0
$449$	22.1725	1.04638	0.523192	−	0.852215i	$-0.324741\pi$
$449$	22.1725	1.04638	0.523192	+	0.852215i	$0.324741\pi$
$450$	0	0
$451$	14.6155	0.688219
$452$	−17.5930	−0.827505
$453$	0	0
$454$	2.53437	0.118944
$455$	0	0
$456$	0	0
$457$	4.35953	0.203930	0.101965	−	0.994788i	$-0.467487\pi$
$457$	4.35953	0.203930	0.101965	+	0.994788i	$0.467487\pi$
$458$	1.56379	0.0730711
$459$	0	0
$460$	5.49484	0.256198
$461$	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	$-0.437721\pi$
$463$	8.36641	0.388820	0.194410	+	0.980920i	$0.437721\pi$
$464$	−14.2017	−0.659296
$465$	0	0
$466$	6.45769	0.299147
$467$	−19.5423	−0.904310	−0.452155	−	0.891939i	$-0.649345\pi$
$467$	−19.5423	−0.904310	−0.452155	+	0.891939i	$0.649345\pi$
$468$	0	0
$469$	0	0
$470$	−2.13875	−0.0986532
$471$	0	0
$472$	−21.6482	−0.996439
$473$	25.6267	1.17832
$474$	0	0
$475$	−6.94137	−0.318492
$476$	0	0
$477$	0	0
$478$	−1.64820	−0.0753870
$479$	−28.5224	−1.30322	−0.651612	−	0.758553i	$-0.725907\pi$
$479$	−28.5224	−1.30322	−0.651612	+	0.758553i	$0.725907\pi$
$480$	0	0
$481$	2.36641	0.107899
$482$	0.747422	0.0340441
$483$	0	0
$484$	10.5665	0.480294
$485$	1.83709	0.0834180
$486$	0	0
$487$	24.8241	1.12489	0.562444	−	0.826836i	$-0.309861\pi$
$487$	24.8241	1.12489	0.562444	+	0.826836i	$0.309861\pi$
$488$	3.55691	0.161014
$489$	0	0
$490$	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
$494$	0.692226	0.0311447
$495$	0	0
$496$	19.1138	0.858236
$497$	0	0
$498$	0	0
$499$	−33.3009	−1.49075	−0.745376	−	0.666644i	$-0.767730\pi$
$499$	−33.3009	−1.49075	−0.745376	+	0.666644i	$0.767730\pi$
$500$	9.14992	0.409197
$501$	0	0
$502$	2.31894	0.103500
$503$	−12.3258	−0.549581	−0.274791	−	0.961504i	$-0.588609\pi$
$503$	−12.3258	−0.549581	−0.274791	+	0.961504i	$0.588609\pi$
$504$	0	0
$505$	4.10266	0.182566
$506$	6.17934	0.274705
$507$	0	0
$508$	−1.46563	−0.0650267
$509$	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
$511$	0	0
$512$	−22.8302	−1.00896
$513$	0	0
$514$	3.77234	0.166391
$515$	8.99656	0.396436
$516$	0	0
$517$	19.3078	0.849155
$518$	0	0
$519$	0	0
$520$	−0.941367	−0.0412817
$521$	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	$0.805144\pi$
$524$	−18.8793	−0.824746
$525$	0	0
$526$	0.754297	0.0328889
$527$	−9.19051	−0.400345
$528$	0	0
$529$	11.0716	0.481375
$530$	2.80949	0.122037
$531$	0	0
$532$	0	0
$533$	−6.49828	−0.281472
$534$	0	0
$535$	2.94137	0.127166
$536$	−28.3449	−1.22431
$537$	0	0
$538$	5.56379	0.239872
$539$	0	0
$540$	0	0
$541$	−34.9751	−1.50370	−0.751848	−	0.659336i	$-0.770837\pi$
$541$	−34.9751	−1.50370	−0.751848	+	0.659336i	$0.770837\pi$
$542$	10.2998	0.442416
$543$	0	0
$544$	6.32582	0.271217
$545$	4.19395	0.179649
$546$	0	0
$547$	6.50783	0.278255	0.139127	−	0.990274i	$-0.455570\pi$
$547$	6.50783	0.278255	0.139127	+	0.990274i	$0.455570\pi$
$548$	20.2086	0.863267
$549$	0	0
$550$	4.99656	0.213054
$551$	−7.67924	−0.327146
$552$	0	0
$553$	0	0
$554$	4.80949	0.204336
$555$	0	0
$556$	−24.7620	−1.05014
$557$	43.4328	1.84031	0.920153	−	0.391559i	$-0.128064\pi$
$557$	43.4328	1.84031	0.920153	+	0.391559i	$0.128064\pi$
$558$	0	0
$559$	−11.3940	−0.481915
$560$	0	0
$561$	0	0
$562$	0.725938	0.0306218
$563$	−33.8827	−1.42799	−0.713993	−	0.700152i	$-0.753115\pi$
$563$	−33.8827	−1.42799	−0.713993	+	0.700152i	$0.753115\pi$
$564$	0	0
$565$	5.23615	0.220287
$566$	7.45875	0.313515
$567$	0	0
$568$	−2.11727	−0.0888385
$569$	−19.2147	−0.805521	−0.402760	−	0.915305i	$-0.631949\pi$
$569$	−19.2147	−0.805521	−0.402760	+	0.915305i	$0.631949\pi$
$570$	0	0
$571$	20.8268	0.871573	0.435787	−	0.900050i	$-0.356470\pi$
$571$	20.8268	0.871573	0.435787	+	0.900050i	$0.356470\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	0	0
$575$	27.5500	1.14892
$576$	0	0
$577$	−28.6448	−1.19250	−0.596249	−	0.802800i	$-0.703343\pi$
$577$	−28.6448	−1.19250	−0.596249	+	0.802800i	$0.703343\pi$
$578$	7.19662	0.299340
$579$	0	0
$580$	4.91539	0.204100
$581$	0	0
$582$	0	0
$583$	−25.3630	−1.05043
$584$	−13.5930	−0.562483
$585$	0	0
$586$	−5.21649	−0.215491
$587$	−4.32076	−0.178337	−0.0891685	−	0.996017i	$-0.528421\pi$
$587$	−4.32076	−0.178337	−0.0891685	+	0.996017i	$0.528421\pi$
$588$	0	0
$589$	10.3354	0.425862
$590$	3.03265	0.124852
$591$	0	0
$592$	−6.43621	−0.264527
$593$	−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	0
$596$	16.5535	0.678057
$597$	0	0
$598$	−2.74742	−0.112350
$599$	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	$-0.398701\pi$
$601$	15.3415	0.625792	0.312896	+	0.949787i	$0.398701\pi$
$602$	0	0
$603$	0	0
$604$	−12.5795	−0.511851
$605$	−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	$0.759204\pi$
$608$	−7.11383	−0.288504
$609$	0	0
$610$	−0.498281	−0.0201748
$611$	−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	$0.369990\pi$
$614$	9.60761	0.387732
$615$	0	0
$616$	0	0
$617$	41.4588	1.66907	0.834533	−	0.550958i	$-0.185737\pi$
$617$	41.4588	1.66907	0.834533	+	0.550958i	$0.185737\pi$
$618$	0	0
$619$	−10.8793	−0.437276	−0.218638	−	0.975806i	$-0.570161\pi$
$619$	−10.8793	−0.437276	−0.218638	+	0.975806i	$0.570161\pi$
$620$	−6.61555	−0.265687
$621$	0	0
$622$	0.905275	0.0362982
$623$	0	0
$624$	0	0
$625$	20.8759	0.835034
$626$	−6.76203	−0.270265
$627$	0	0
$628$	−10.7429	−0.428689
$629$	3.09472	0.123395
$630$	0	0
$631$	−31.4396	−1.25159	−0.625796	−	0.779987i	$-0.715226\pi$
$631$	−31.4396	−1.25159	−0.625796	+	0.779987i	$0.715226\pi$
$632$	−2.38101	−0.0947117
$633$	0	0
$634$	7.32238	0.290809
$635$	0.436210	0.0173105
$636$	0	0
$637$	0	0
$638$	5.52770	0.218844
$639$	0	0
$640$	5.90871	0.233562
$641$	−3.04221	−0.120160	−0.0600799	−	0.998194i	$-0.519136\pi$
$641$	−3.04221	−0.120160	−0.0600799	+	0.998194i	$0.519136\pi$
$642$	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
$645$	0	0
$646$	0.905275	0.0356176
$647$	7.07324	0.278078	0.139039	−	0.990287i	$-0.455599\pi$
$647$	7.07324	0.278078	0.139039	+	0.990287i	$0.455599\pi$
$648$	0	0
$649$	−27.3776	−1.07466
$650$	−2.22154	−0.0871361
$651$	0	0
$652$	−11.3484	−0.444436
$653$	15.7586	0.616681	0.308341	−	0.951276i	$-0.400226\pi$
$653$	15.7586	0.616681	0.308341	+	0.951276i	$0.400226\pi$
$654$	0	0
$655$	5.61899	0.219552
$656$	17.6742	0.690061
$657$	0	0
$658$	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	$-0.523138\pi$
$661$	−3.73443	−0.145253	−0.0726263	+	0.997359i	$0.523138\pi$
$662$	−14.8501	−0.577165
$663$	0	0
$664$	−29.0777	−1.12844
$665$	0	0
$666$	0	0
$667$	30.4786	1.18014
$668$	29.4948	1.14119
$669$	0	0
$670$	3.97078	0.153405
$671$	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	$0.534779\pi$
$674$	3.96629	0.152776
$675$	0	0
$676$	−1.77846	−0.0684022
$677$	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
$679$	0	0
$680$	−1.23109	−0.0472103
$681$	0	0
$682$	−7.43965	−0.284879
$683$	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
$687$	0	0
$688$	30.9897	1.18147
$689$	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	$0.598982\pi$
$692$	41.4396	1.57530
$693$	0	0
$694$	9.10695	0.345695
$695$	7.36984	0.279554
$696$	0	0
$697$	−8.49828	−0.321895
$698$	12.8286	0.485570
$699$	0	0
$700$	0	0
$701$	6.98013	0.263636	0.131818	−	0.991274i	$-0.457919\pi$
$701$	6.98013	0.263636	0.131818	+	0.991274i	$0.457919\pi$
$702$	0	0
$703$	−3.48024	−0.131260
$704$	−7.11383	−0.268112
$705$	0	0
$706$	12.0844	0.454803
$707$	0	0
$708$	0	0
$709$	8.39239	0.315183	0.157591	−	0.987504i	$-0.449627\pi$
$709$	8.39239	0.315183	0.157591	+	0.987504i	$0.449627\pi$
$710$	0.296604	0.0111313
$711$	0	0
$712$	12.2897	0.460577
$713$	−41.0207	−1.53624
$714$	0	0
$715$	−1.19051	−0.0445225
$716$	37.4227	1.39855
$717$	0	0
$718$	−11.0225	−0.411358
$719$	−5.16129	−0.192484	−0.0962418	−	0.995358i	$-0.530682\pi$
$719$	−5.16129	−0.192484	−0.0962418	+	0.995358i	$0.530682\pi$
$720$	0	0
$721$	0	0
$722$	7.92494	0.294936
$723$	0	0
$724$	29.7846	1.10693
$725$	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
$729$	0	0
$730$	1.90422	0.0704782
$731$	−14.9008	−0.551125
$732$	0	0
$733$	39.1311	1.44534	0.722670	−	0.691193i	$-0.242915\pi$
$733$	39.1311	1.44534	0.722670	+	0.691193i	$0.242915\pi$
$734$	−6.91215	−0.255132
$735$	0	0
$736$	28.2345	1.04074
$737$	−35.8466	−1.32043
$738$	0	0
$739$	−7.13531	−0.262477	−0.131238	−	0.991351i	$-0.541895\pi$
$739$	−7.13531	−0.262477	−0.131238	+	0.991351i	$0.541895\pi$
$740$	2.22766	0.0818903
$741$	0	0
$742$	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
$746$	11.1398	0.407857
$747$	0	0
$748$	5.23109	0.191268
$749$	0	0
$750$	0	0
$751$	−37.3251	−1.36201	−0.681005	−	0.732278i	$-0.738457\pi$
$751$	−37.3251	−1.36201	−0.681005	+	0.732278i	$0.738457\pi$
$752$	23.3484	0.851427
$753$	0	0
$754$	−2.45769	−0.0895039
$755$	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	$0.541210\pi$
$758$	15.4104	0.559732
$759$	0	0
$760$	1.38445	0.0502194
$761$	−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
$763$	0	0
$764$	−13.2311	−0.478684
$765$	0	0
$766$	10.6448	0.384611
$767$	12.1725	0.439522
$768$	0	0
$769$	21.4638	0.774005	0.387002	−	0.922079i	$-0.373511\pi$
$769$	21.4638	0.774005	0.387002	+	0.922079i	$0.373511\pi$
$770$	0	0
$771$	0	0
$772$	2.67074	0.0961221
$773$	−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
$776$	6.17246	0.221578
$777$	0	0
$778$	17.9087	0.642058
$779$	9.55691	0.342412
$780$	0	0
$781$	−2.67762	−0.0958127
$782$	−3.59301	−0.128486
$783$	0	0
$784$	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	$0.471478\pi$
$788$	42.6707	1.52008
$789$	0	0
$790$	0.333552	0.0118672
$791$	0	0
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	−5.50945	−0.195523
$795$	0	0
$796$	−3.58289	−0.126992
$797$	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
$800$	22.8302	0.807170
$801$	0	0
$802$	1.67418	0.0591174
$803$	−17.1905	−0.606640
$804$	0	0
$805$	0	0
$806$	3.30777	0.116511
$807$	0	0
$808$	13.7846	0.484940
$809$	2.57678	0.0905947	0.0452974	−	0.998974i	$-0.485576\pi$
$809$	2.57678	0.0905947	0.0452974	+	0.998974i	$0.485576\pi$
$810$	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
$813$	0	0
$814$	2.50516	0.0878057
$815$	3.37758	0.118311
$816$	0	0
$817$	16.7570	0.586252
$818$	2.47680	0.0865992
$819$	0	0
$820$	−6.11727	−0.213624
$821$	−5.26719	−0.183826	−0.0919130	−	0.995767i	$-0.529298\pi$
$821$	−5.26719	−0.183826	−0.0919130	+	0.995767i	$0.529298\pi$
$822$	0	0
$823$	36.0191	1.25555	0.627774	−	0.778396i	$-0.283966\pi$
$823$	36.0191	1.25555	0.627774	+	0.778396i	$0.283966\pi$
$824$	30.2277	1.05303
$825$	0	0
$826$	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	$-0.323001\pi$
$829$	30.3956	1.05568	0.527842	+	0.849343i	$0.323001\pi$
$830$	4.07344	0.141391
$831$	0	0
$832$	3.16291	0.109654
$833$	0	0
$834$	0	0
$835$	−8.77846	−0.303791
$836$	−5.88273	−0.203459
$837$	0	0
$838$	−12.2536	−0.423295
$839$	29.8398	1.03018	0.515092	−	0.857135i	$-0.327758\pi$
$839$	29.8398	1.03018	0.515092	+	0.857135i	$0.327758\pi$
$840$	0	0
$841$	−1.73549	−0.0598445
$842$	−10.4691	−0.360788
$843$	0	0
$844$	−17.9639	−0.618343
$845$	0.529317	0.0182090
$846$	0	0
$847$	0	0
$848$	−30.6707	−1.05324
$849$	0	0
$850$	−2.90528	−0.0996501
$851$	13.8129	0.473501
$852$	0	0
$853$	0.203497	0.00696761	0.00348380	−	0.999994i	$-0.498891\pi$
$853$	0.203497	0.00696761	0.00348380	+	0.999994i	$0.498891\pi$
$854$	0	0
$855$	0	0
$856$	9.88273	0.337785
$857$	−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	$0.341683\pi$
$860$	−10.7259	−0.365751
$861$	0	0
$862$	13.0258	0.443660
$863$	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
$866$	−6.01910	−0.204537
$867$	0	0
$868$	0	0
$869$	−3.01117	−0.102147
$870$	0	0
$871$	15.9379	0.540036
$872$	14.0913	0.477191
$873$	0	0
$874$	4.04059	0.136675
$875$	0	0
$876$	0	0
$877$	6.71133	0.226626	0.113313	−	0.993559i	$-0.463854\pi$
$877$	6.71133	0.226626	0.113313	+	0.993559i	$0.463854\pi$
$878$	−8.55024	−0.288557
$879$	0	0
$880$	3.23797	0.109152
$881$	−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	$-0.446559\pi$
$883$	9.93105	0.334207	0.167103	+	0.985939i	$0.446559\pi$
$884$	−2.32582	−0.0782258
$885$	0	0
$886$	0.0506994	0.00170328
$887$	−51.3776	−1.72509	−0.862545	−	0.505980i	$-0.831131\pi$
$887$	−51.3776	−1.72509	−0.862545	+	0.505980i	$0.831131\pi$
$888$	0	0
$889$	0	0
$890$	−1.72164	−0.0577096
$891$	0	0
$892$	18.0361	0.603893
$893$	12.6251	0.422483
$894$	0	0
$895$	−11.1380	−0.372302
$896$	0	0
$897$	0	0
$898$	−10.4362	−0.348261
$899$	−36.6949	−1.22384
$900$	0	0
$901$	14.7474	0.491308
$902$	−6.87930	−0.229055
$903$	0	0
$904$	17.5930	0.585135
$905$	−8.86469	−0.294672
$906$	0	0
$907$	4.34225	0.144182	0.0720910	−	0.997398i	$-0.477033\pi$
$907$	4.34225	0.144182	0.0720910	+	0.997398i	$0.477033\pi$
$908$	9.57602	0.317791
$909$	0	0
$910$	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
$914$	−2.05196	−0.0678728
$915$	0	0
$916$	5.90871	0.195229
$917$	0	0
$918$	0	0
$919$	43.4588	1.43357	0.716786	−	0.697293i	$-0.245612\pi$
$919$	43.4588	1.43357	0.716786	+	0.697293i	$0.245612\pi$
$920$	−5.49484	−0.181160
$921$	0	0
$922$	−15.2412	−0.501942
$923$	1.19051	0.0391860
$924$	0	0
$925$	11.1690	0.367235
$926$	−3.93793	−0.129408
$927$	0	0
$928$	25.2571	0.829104
$929$	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
$931$	0	0
$932$	24.4001	0.799252
$933$	0	0
$934$	9.19824	0.300976
$935$	−1.55691	−0.0509165
$936$	0	0
$937$	34.0990	1.11397	0.556983	−	0.830524i	$-0.311959\pi$
$937$	34.0990	1.11397	0.556983	+	0.830524i	$0.311959\pi$
$938$	0	0
$939$	0	0
$940$	−8.08117	−0.263579
$941$	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
$944$	−33.1070	−1.07754
$945$	0	0
$946$	−12.0621	−0.392172
$947$	−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
$950$	3.26719	0.106002
$951$	0	0
$952$	0	0
$953$	−35.3060	−1.14367	−0.571836	−	0.820368i	$-0.693769\pi$
$953$	−35.3060	−1.14367	−0.571836	+	0.820368i	$0.693769\pi$
$954$	0	0
$955$	3.93793	0.127428
$956$	−6.22766	−0.201417
$957$	0	0
$958$	13.4250	0.433743
$959$	0	0
$960$	0	0
$961$	18.3871	0.593133
$962$	−1.11383	−0.0359113
$963$	0	0
$964$	2.82410	0.0909582
$965$	−0.794885	−0.0255882
$966$	0	0
$967$	23.7148	0.762616	0.381308	−	0.924448i	$-0.375474\pi$
$967$	23.7148	0.762616	0.381308	+	0.924448i	$0.375474\pi$
$968$	−10.5665	−0.339619
$969$	0	0
$970$	−0.864688	−0.0277635
$971$	−23.9379	−0.768205	−0.384102	−	0.923291i	$-0.625489\pi$
$971$	−23.9379	−0.768205	−0.384102	+	0.923291i	$0.625489\pi$
$972$	0	0
$973$	0	0
$974$	−11.6843	−0.374389
$975$	0	0
$976$	5.43965	0.174119
$977$	16.1871	0.517870	0.258935	−	0.965895i	$-0.416628\pi$
$977$	16.1871	0.517870	0.258935	+	0.965895i	$0.416628\pi$
$978$	0	0
$979$	15.5423	0.496734
$980$	0	0
$981$	0	0
$982$	13.7294	0.438122
$983$	45.7243	1.45838	0.729190	−	0.684312i	$-0.239897\pi$
$983$	45.7243	1.45838	0.729190	+	0.684312i	$0.239897\pi$
$984$	0	0
$985$	−12.7000	−0.404654
$986$	−3.21411	−0.102358
$987$	0	0
$988$	2.61555	0.0832116
$989$	−66.5078	−2.11483
$990$	0	0
$991$	−5.40356	−0.171650	−0.0858248	−	0.996310i	$-0.527353\pi$
$991$	−5.40356	−0.171650	−0.0858248	+	0.996310i	$0.527353\pi$
$992$	−33.9931	−1.07928
$993$	0	0
$994$	0	0
$995$	1.06637	0.0338061
$996$	0	0
$997$	−6.04832	−0.191552	−0.0957761	−	0.995403i	$-0.530533\pi$
$997$	−6.04832	−0.191552	−0.0957761	+	0.995403i	$0.530533\pi$
$998$	15.6742	0.496158
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.x.1.2		3
3.2	odd	2		637.2.a.j.1.2		3
7.6	odd	2		819.2.a.i.1.2		3
21.2	odd	6		637.2.e.i.508.2		6
21.5	even	6		637.2.e.j.508.2		6
21.11	odd	6		637.2.e.i.79.2		6
21.17	even	6		637.2.e.j.79.2		6
21.20	even	2		91.2.a.d.1.2	✓	3
39.38	odd	2		8281.2.a.bg.1.2		3
84.83	odd	2		1456.2.a.t.1.1		3
105.104	even	2		2275.2.a.m.1.2		3
168.83	odd	2		5824.2.a.bs.1.3		3
168.125	even	2		5824.2.a.by.1.1		3
273.83	odd	4		1183.2.c.f.337.3		6
273.125	odd	4		1183.2.c.f.337.4		6
273.272	even	2		1183.2.a.i.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.d.1.2	✓	3	21.20	even	2
637.2.a.j.1.2		3	3.2	odd	2
637.2.e.i.79.2		6	21.11	odd	6
637.2.e.i.508.2		6	21.2	odd	6
637.2.e.j.79.2		6	21.17	even	6
637.2.e.j.508.2		6	21.5	even	6
819.2.a.i.1.2		3	7.6	odd	2
1183.2.a.i.1.2		3	273.272	even	2
1183.2.c.f.337.3		6	273.83	odd	4
1183.2.c.f.337.4		6	273.125	odd	4
1456.2.a.t.1.1		3	84.83	odd	2
2275.2.a.m.1.2		3	105.104	even	2
5733.2.a.x.1.2		3	1.1	even	1	trivial
5824.2.a.bs.1.3		3	168.83	odd	2
5824.2.a.by.1.1		3	168.125	even	2
8281.2.a.bg.1.2		3	39.38	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:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q-0.470683 q^{2} -1.77846 q^{4} +0.529317 q^{5} +1.77846 q^{8} -0.249141 q^{10} +2.24914 q^{11} -1.00000 q^{13} +2.71982 q^{16} -1.30777 q^{17} +1.47068 q^{19} -0.941367 q^{20} -1.05863 q^{22} -5.83709 q^{23} -4.71982 q^{25} +0.470683 q^{26} -5.22154 q^{29} +7.02760 q^{31} -4.83709 q^{32} +0.615547 q^{34} -2.36641 q^{37} -0.692226 q^{38} +0.941367 q^{40} +6.49828 q^{41} +11.3940 q^{43} -4.00000 q^{44} +2.74742 q^{46} +8.58451 q^{47} +2.22154 q^{50} +1.77846 q^{52} -11.2767 q^{53} +1.19051 q^{55} +2.45769 q^{58} -12.1725 q^{59} +2.00000 q^{61} -3.30777 q^{62} -3.16291 q^{64} -0.529317 q^{65} -15.9379 q^{67} +2.32582 q^{68} -1.19051 q^{71} -7.64315 q^{73} +1.11383 q^{74} -2.61555 q^{76} -1.33881 q^{79} +1.43965 q^{80} -3.05863 q^{82} -16.3500 q^{83} -0.692226 q^{85} -5.36297 q^{86} +4.00000 q^{88} +6.91033 q^{89} +10.3810 q^{92} -4.04059 q^{94} +0.778457 q^{95} +3.47068 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{8} + 8 q^{10} - 2 q^{11} - 3 q^{13} - q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 4 q^{22} - 10 q^{23} - 5 q^{25} + q^{26} - 24 q^{29} + 4 q^{31} - 7 q^{32} - 14 q^{34}+ \cdots + 10 q^{97}+O(q^{100}) \) 3 * q - q^2 + 3 * q^4 + 2 * q^5 - 3 * q^8 + 8 * q^10 - 2 * q^11 - 3 * q^13 - q^16 + 4 * q^17 + 4 * q^19 - 2 * q^20 - 4 * q^22 - 10 * q^23 - 5 * q^25 + q^26 - 24 * q^29 + 4 * q^31 - 7 * q^32 - 14 * q^34 - 10 * q^38 + 2 * q^40 + 2 * q^41 + 10 * q^43 - 12 * q^44 - 18 * q^46 - 8 * q^47 + 15 * q^50 - 3 * q^52 - 8 * q^53 - 6 * q^55 + 12 * q^58 - 4 * q^59 + 6 * q^61 - 2 * q^62 - 17 * q^64 - 2 * q^65 - 12 * q^67 + 22 * q^68 + 6 * q^71 + 10 * q^73 - 30 * q^74 + 8 * q^76 - 14 * q^79 - 14 * q^80 - 10 * q^82 - 12 * q^83 - 10 * q^85 + 26 * q^86 + 12 * q^88 + 2 * q^89 + 12 * q^92 + 10 * q^94 - 6 * q^95 + 10 * q^97

Introduction

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