LMFDB - Embedded newform 45.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,4,Mod(1,45)] 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("45.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	45.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$2.65508595026$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	45.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -7.00000 q^{4} -5.00000 q^{5} -24.0000 q^{7} +15.0000 q^{8} +5.00000 q^{10} -52.0000 q^{11} +22.0000 q^{13} +24.0000 q^{14} +41.0000 q^{16} +14.0000 q^{17} -20.0000 q^{19} +35.0000 q^{20} +52.0000 q^{22} +168.000 q^{23} +25.0000 q^{25} -22.0000 q^{26} +168.000 q^{28} -230.000 q^{29} -288.000 q^{31} -161.000 q^{32} -14.0000 q^{34} +120.000 q^{35} -34.0000 q^{37} +20.0000 q^{38} -75.0000 q^{40} -122.000 q^{41} -188.000 q^{43} +364.000 q^{44} -168.000 q^{46} -256.000 q^{47} +233.000 q^{49} -25.0000 q^{50} -154.000 q^{52} +338.000 q^{53} +260.000 q^{55} -360.000 q^{56} +230.000 q^{58} -100.000 q^{59} +742.000 q^{61} +288.000 q^{62} -167.000 q^{64} -110.000 q^{65} -84.0000 q^{67} -98.0000 q^{68} -120.000 q^{70} +328.000 q^{71} -38.0000 q^{73} +34.0000 q^{74} +140.000 q^{76} +1248.00 q^{77} -240.000 q^{79} -205.000 q^{80} +122.000 q^{82} -1212.00 q^{83} -70.0000 q^{85} +188.000 q^{86} -780.000 q^{88} -330.000 q^{89} -528.000 q^{91} -1176.00 q^{92} +256.000 q^{94} +100.000 q^{95} +866.000 q^{97} -233.000 q^{98} +O(q^{100})\)

Coefficient data

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

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

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

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.875000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−24.0000	−1.29588	−0.647939	−	0.761692i	$-0.724369\pi$
$7$	−24.0000	−1.29588	−0.647939	+	0.761692i	$0.724369\pi$
$8$	15.0000	0.662913
$9$	0	0
$10$	5.00000	0.158114
$11$	−52.0000	−1.42533	−0.712663	−	0.701506i	$-0.752511\pi$
$11$	−52.0000	−1.42533	−0.712663	+	0.701506i	$0.752511\pi$
$12$	0	0
$13$	22.0000	0.469362	0.234681	−	0.972072i	$-0.424595\pi$
$13$	22.0000	0.469362	0.234681	+	0.972072i	$0.424595\pi$
$14$	24.0000	0.458162
$15$	0	0
$16$	41.0000	0.640625
$17$	14.0000	0.199735	0.0998676	−	0.995001i	$-0.468158\pi$
$17$	14.0000	0.199735	0.0998676	+	0.995001i	$0.468158\pi$
$18$	0	0
$19$	−20.0000	−0.241490	−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	35.0000	0.391312
$21$	0	0
$22$	52.0000	0.503929
$23$	168.000	1.52306	0.761531	−	0.648129i	$-0.224448\pi$
$23$	168.000	1.52306	0.761531	+	0.648129i	$0.224448\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−22.0000	−0.165944
$27$	0	0
$28$	168.000	1.13389
$29$	−230.000	−1.47276	−0.736378	−	0.676570i	$-0.763465\pi$
$29$	−230.000	−1.47276	−0.736378	+	0.676570i	$0.763465\pi$
$30$	0	0
$31$	−288.000	−1.66859	−0.834296	−	0.551317i	$-0.814125\pi$
$31$	−288.000	−1.66859	−0.834296	+	0.551317i	$0.814125\pi$
$32$	−161.000	−0.889408
$33$	0	0
$34$	−14.0000	−0.0706171
$35$	120.000	0.579534
$36$	0	0
$37$	−34.0000	−0.151069	−0.0755347	−	0.997143i	$-0.524066\pi$
$37$	−34.0000	−0.151069	−0.0755347	+	0.997143i	$0.524066\pi$
$38$	20.0000	0.0853797
$39$	0	0
$40$	−75.0000	−0.296464
$41$	−122.000	−0.464712	−0.232356	−	0.972631i	$-0.574643\pi$
$41$	−122.000	−0.464712	−0.232356	+	0.972631i	$0.574643\pi$
$42$	0	0
$43$	−188.000	−0.666738	−0.333369	−	0.942796i	$-0.608185\pi$
$43$	−188.000	−0.666738	−0.333369	+	0.942796i	$0.608185\pi$
$44$	364.000	1.24716
$45$	0	0
$46$	−168.000	−0.538484
$47$	−256.000	−0.794499	−0.397249	−	0.917711i	$-0.630035\pi$
$47$	−256.000	−0.794499	−0.397249	+	0.917711i	$0.630035\pi$
$48$	0	0
$49$	233.000	0.679300
$50$	−25.0000	−0.0707107
$51$	0	0
$52$	−154.000	−0.410691
$53$	338.000	0.875998	0.437999	−	0.898976i	$-0.355687\pi$
$53$	338.000	0.875998	0.437999	+	0.898976i	$0.355687\pi$
$54$	0	0
$55$	260.000	0.637425
$56$	−360.000	−0.859054
$57$	0	0
$58$	230.000	0.520698
$59$	−100.000	−0.220659	−0.110330	−	0.993895i	$-0.535191\pi$
$59$	−100.000	−0.220659	−0.110330	+	0.993895i	$0.535191\pi$
$60$	0	0
$61$	742.000	1.55743	0.778716	−	0.627376i	$-0.215871\pi$
$61$	742.000	1.55743	0.778716	+	0.627376i	$0.215871\pi$
$62$	288.000	0.589936
$63$	0	0
$64$	−167.000	−0.326172
$65$	−110.000	−0.209905
$66$	0	0
$67$	−84.0000	−0.153168	−0.0765838	−	0.997063i	$-0.524401\pi$
$67$	−84.0000	−0.153168	−0.0765838	+	0.997063i	$0.524401\pi$
$68$	−98.0000	−0.174768
$69$	0	0
$70$	−120.000	−0.204896
$71$	328.000	0.548260	0.274130	−	0.961693i	$-0.411610\pi$
$71$	328.000	0.548260	0.274130	+	0.961693i	$0.411610\pi$
$72$	0	0
$73$	−38.0000	−0.0609255	−0.0304628	−	0.999536i	$-0.509698\pi$
$73$	−38.0000	−0.0609255	−0.0304628	+	0.999536i	$0.509698\pi$
$74$	34.0000	0.0534111
$75$	0	0
$76$	140.000	0.211304
$77$	1248.00	1.84705
$78$	0	0
$79$	−240.000	−0.341799	−0.170899	−	0.985288i	$-0.554667\pi$
$79$	−240.000	−0.341799	−0.170899	+	0.985288i	$0.554667\pi$
$80$	−205.000	−0.286496
$81$	0	0
$82$	122.000	0.164301
$83$	−1212.00	−1.60282	−0.801411	−	0.598114i	$-0.795917\pi$
$83$	−1212.00	−1.60282	−0.801411	+	0.598114i	$0.795917\pi$
$84$	0	0
$85$	−70.0000	−0.0893243
$86$	188.000	0.235727
$87$	0	0
$88$	−780.000	−0.944867
$89$	−330.000	−0.393033	−0.196516	−	0.980501i	$-0.562963\pi$
$89$	−330.000	−0.393033	−0.196516	+	0.980501i	$0.562963\pi$
$90$	0	0
$91$	−528.000	−0.608236
$92$	−1176.00	−1.33268
$93$	0	0
$94$	256.000	0.280898
$95$	100.000	0.107998
$96$	0	0
$97$	866.000	0.906484	0.453242	−	0.891387i	$-0.350267\pi$
$97$	866.000	0.906484	0.453242	+	0.891387i	$0.350267\pi$
$98$	−233.000	−0.240169
$99$	0	0
$100$	−175.000	−0.175000
$101$	1218.00	1.19996	0.599978	−	0.800017i	$-0.295176\pi$
$101$	1218.00	1.19996	0.599978	+	0.800017i	$0.295176\pi$
$102$	0	0
$103$	−88.0000	−0.0841835	−0.0420917	−	0.999114i	$-0.513402\pi$
$103$	−88.0000	−0.0841835	−0.0420917	+	0.999114i	$0.513402\pi$
$104$	330.000	0.311146
$105$	0	0
$106$	−338.000	−0.309712
$107$	−36.0000	−0.0325257	−0.0162629	−	0.999868i	$-0.505177\pi$
$107$	−36.0000	−0.0325257	−0.0162629	+	0.999868i	$0.505177\pi$
$108$	0	0
$109$	−970.000	−0.852378	−0.426189	−	0.904634i	$-0.640144\pi$
$109$	−970.000	−0.852378	−0.426189	+	0.904634i	$0.640144\pi$
$110$	−260.000	−0.225364
$111$	0	0
$112$	−984.000	−0.830172
$113$	−1042.00	−0.867461	−0.433731	−	0.901043i	$-0.642803\pi$
$113$	−1042.00	−0.867461	−0.433731	+	0.901043i	$0.642803\pi$
$114$	0	0
$115$	−840.000	−0.681134
$116$	1610.00	1.28866
$117$	0	0
$118$	100.000	0.0780148
$119$	−336.000	−0.258833
$120$	0	0
$121$	1373.00	1.03156
$122$	−742.000	−0.550635
$123$	0	0
$124$	2016.00	1.46002
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1936.00	1.35269	0.676347	−	0.736583i	$-0.263562\pi$
$127$	1936.00	1.35269	0.676347	+	0.736583i	$0.263562\pi$
$128$	1455.00	1.00473
$129$	0	0
$130$	110.000	0.0742126
$131$	−732.000	−0.488207	−0.244104	−	0.969749i	$-0.578494\pi$
$131$	−732.000	−0.488207	−0.244104	+	0.969749i	$0.578494\pi$
$132$	0	0
$133$	480.000	0.312942
$134$	84.0000	0.0541529
$135$	0	0
$136$	210.000	0.132407
$137$	2214.00	1.38069	0.690346	−	0.723479i	$-0.257458\pi$
$137$	2214.00	1.38069	0.690346	+	0.723479i	$0.257458\pi$
$138$	0	0
$139$	20.0000	0.0122042	0.00610208	−	0.999981i	$-0.498058\pi$
$139$	20.0000	0.0122042	0.00610208	+	0.999981i	$0.498058\pi$
$140$	−840.000	−0.507093
$141$	0	0
$142$	−328.000	−0.193839
$143$	−1144.00	−0.668994
$144$	0	0
$145$	1150.00	0.658637
$146$	38.0000	0.0215404
$147$	0	0
$148$	238.000	0.132186
$149$	1330.00	0.731261	0.365630	−	0.930760i	$-0.380853\pi$
$149$	1330.00	0.731261	0.365630	+	0.930760i	$0.380853\pi$
$150$	0	0
$151$	−1208.00	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−1208.00	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−300.000	−0.160087
$153$	0	0
$154$	−1248.00	−0.653031
$155$	1440.00	0.746217
$156$	0	0
$157$	−3514.00	−1.78629	−0.893146	−	0.449768i	$-0.851507\pi$
$157$	−3514.00	−1.78629	−0.893146	+	0.449768i	$0.851507\pi$
$158$	240.000	0.120844
$159$	0	0
$160$	805.000	0.397755
$161$	−4032.00	−1.97370
$162$	0	0
$163$	−2068.00	−0.993732	−0.496866	−	0.867827i	$-0.665516\pi$
$163$	−2068.00	−0.993732	−0.496866	+	0.867827i	$0.665516\pi$
$164$	854.000	0.406623
$165$	0	0
$166$	1212.00	0.566683
$167$	24.0000	0.0111208	0.00556041	−	0.999985i	$-0.498230\pi$
$167$	24.0000	0.0111208	0.00556041	+	0.999985i	$0.498230\pi$
$168$	0	0
$169$	−1713.00	−0.779700
$170$	70.0000	0.0315809
$171$	0	0
$172$	1316.00	0.583396
$173$	618.000	0.271593	0.135797	−	0.990737i	$-0.456641\pi$
$173$	618.000	0.271593	0.135797	+	0.990737i	$0.456641\pi$
$174$	0	0
$175$	−600.000	−0.259176
$176$	−2132.00	−0.913100
$177$	0	0
$178$	330.000	0.138958
$179$	−3340.00	−1.39466	−0.697328	−	0.716752i	$-0.745628\pi$
$179$	−3340.00	−1.39466	−0.697328	+	0.716752i	$0.745628\pi$
$180$	0	0
$181$	−178.000	−0.0730974	−0.0365487	−	0.999332i	$-0.511636\pi$
$181$	−178.000	−0.0730974	−0.0365487	+	0.999332i	$0.511636\pi$
$182$	528.000	0.215044
$183$	0	0
$184$	2520.00	1.00966
$185$	170.000	0.0675603
$186$	0	0
$187$	−728.000	−0.284688
$188$	1792.00	0.695186
$189$	0	0
$190$	−100.000	−0.0381830
$191$	1888.00	0.715240	0.357620	−	0.933867i	$-0.383588\pi$
$191$	1888.00	0.715240	0.357620	+	0.933867i	$0.383588\pi$
$192$	0	0
$193$	1922.00	0.716832	0.358416	−	0.933562i	$-0.383317\pi$
$193$	1922.00	0.716832	0.358416	+	0.933562i	$0.383317\pi$
$194$	−866.000	−0.320491
$195$	0	0
$196$	−1631.00	−0.594388
$197$	−2526.00	−0.913554	−0.456777	−	0.889581i	$-0.650996\pi$
$197$	−2526.00	−0.913554	−0.456777	+	0.889581i	$0.650996\pi$
$198$	0	0
$199$	−1160.00	−0.413217	−0.206609	−	0.978424i	$-0.566243\pi$
$199$	−1160.00	−0.413217	−0.206609	+	0.978424i	$0.566243\pi$
$200$	375.000	0.132583
$201$	0	0
$202$	−1218.00	−0.424248
$203$	5520.00	1.90851
$204$	0	0
$205$	610.000	0.207826
$206$	88.0000	0.0297634
$207$	0	0
$208$	902.000	0.300685
$209$	1040.00	0.344202
$210$	0	0
$211$	−4468.00	−1.45777	−0.728886	−	0.684635i	$-0.759961\pi$
$211$	−4468.00	−1.45777	−0.728886	+	0.684635i	$0.759961\pi$
$212$	−2366.00	−0.766498
$213$	0	0
$214$	36.0000	0.0114996
$215$	940.000	0.298174
$216$	0	0
$217$	6912.00	2.16229
$218$	970.000	0.301361
$219$	0	0
$220$	−1820.00	−0.557747
$221$	308.000	0.0937481
$222$	0	0
$223$	6032.00	1.81136	0.905678	−	0.423965i	$-0.139362\pi$
$223$	6032.00	1.81136	0.905678	+	0.423965i	$0.139362\pi$
$224$	3864.00	1.15256
$225$	0	0
$226$	1042.00	0.306694
$227$	−2636.00	−0.770738	−0.385369	−	0.922763i	$-0.625926\pi$
$227$	−2636.00	−0.770738	−0.385369	+	0.922763i	$0.625926\pi$
$228$	0	0
$229$	4830.00	1.39378	0.696889	−	0.717179i	$-0.254567\pi$
$229$	4830.00	1.39378	0.696889	+	0.717179i	$0.254567\pi$
$230$	840.000	0.240817
$231$	0	0
$232$	−3450.00	−0.976309
$233$	−2682.00	−0.754093	−0.377046	−	0.926194i	$-0.623060\pi$
$233$	−2682.00	−0.754093	−0.377046	+	0.926194i	$0.623060\pi$
$234$	0	0
$235$	1280.00	0.355311
$236$	700.000	0.193077
$237$	0	0
$238$	336.000	0.0915111
$239$	−2320.00	−0.627901	−0.313950	−	0.949439i	$-0.601653\pi$
$239$	−2320.00	−0.627901	−0.313950	+	0.949439i	$0.601653\pi$
$240$	0	0
$241$	2002.00	0.535104	0.267552	−	0.963543i	$-0.413785\pi$
$241$	2002.00	0.535104	0.267552	+	0.963543i	$0.413785\pi$
$242$	−1373.00	−0.364710
$243$	0	0
$244$	−5194.00	−1.36275
$245$	−1165.00	−0.303792
$246$	0	0
$247$	−440.000	−0.113346
$248$	−4320.00	−1.10613
$249$	0	0
$250$	125.000	0.0316228
$251$	−132.000	−0.0331943	−0.0165971	−	0.999862i	$-0.505283\pi$
$251$	−132.000	−0.0331943	−0.0165971	+	0.999862i	$0.505283\pi$
$252$	0	0
$253$	−8736.00	−2.17086
$254$	−1936.00	−0.478250
$255$	0	0
$256$	−119.000	−0.0290527
$257$	7614.00	1.84805	0.924024	−	0.382335i	$-0.124880\pi$
$257$	7614.00	1.84805	0.924024	+	0.382335i	$0.124880\pi$
$258$	0	0
$259$	816.000	0.195767
$260$	770.000	0.183667
$261$	0	0
$262$	732.000	0.172607
$263$	4888.00	1.14603	0.573017	−	0.819543i	$-0.305773\pi$
$263$	4888.00	1.14603	0.573017	+	0.819543i	$0.305773\pi$
$264$	0	0
$265$	−1690.00	−0.391758
$266$	−480.000	−0.110642
$267$	0	0
$268$	588.000	0.134022
$269$	−1270.00	−0.287856	−0.143928	−	0.989588i	$-0.545973\pi$
$269$	−1270.00	−0.287856	−0.143928	+	0.989588i	$0.545973\pi$
$270$	0	0
$271$	1072.00	0.240293	0.120146	−	0.992756i	$-0.461664\pi$
$271$	1072.00	0.240293	0.120146	+	0.992756i	$0.461664\pi$
$272$	574.000	0.127955
$273$	0	0
$274$	−2214.00	−0.488148
$275$	−1300.00	−0.285065
$276$	0	0
$277$	−5394.00	−1.17001	−0.585007	−	0.811028i	$-0.698908\pi$
$277$	−5394.00	−1.17001	−0.585007	+	0.811028i	$0.698908\pi$
$278$	−20.0000	−0.00431482
$279$	0	0
$280$	1800.00	0.384181
$281$	−2442.00	−0.518425	−0.259213	−	0.965820i	$-0.583463\pi$
$281$	−2442.00	−0.518425	−0.259213	+	0.965820i	$0.583463\pi$
$282$	0	0
$283$	2772.00	0.582255	0.291128	−	0.956684i	$-0.405970\pi$
$283$	2772.00	0.582255	0.291128	+	0.956684i	$0.405970\pi$
$284$	−2296.00	−0.479727
$285$	0	0
$286$	1144.00	0.236525
$287$	2928.00	0.602210
$288$	0	0
$289$	−4717.00	−0.960106
$290$	−1150.00	−0.232863
$291$	0	0
$292$	266.000	0.0533098
$293$	−4542.00	−0.905619	−0.452810	−	0.891607i	$-0.649578\pi$
$293$	−4542.00	−0.905619	−0.452810	+	0.891607i	$0.649578\pi$
$294$	0	0
$295$	500.000	0.0986818
$296$	−510.000	−0.100146
$297$	0	0
$298$	−1330.00	−0.258540
$299$	3696.00	0.714867
$300$	0	0
$301$	4512.00	0.864011
$302$	1208.00	0.230174
$303$	0	0
$304$	−820.000	−0.154705
$305$	−3710.00	−0.696505
$306$	0	0
$307$	5116.00	0.951093	0.475546	−	0.879691i	$-0.342250\pi$
$307$	5116.00	0.951093	0.475546	+	0.879691i	$0.342250\pi$
$308$	−8736.00	−1.61617
$309$	0	0
$310$	−1440.00	−0.263827
$311$	2808.00	0.511984	0.255992	−	0.966679i	$-0.417598\pi$
$311$	2808.00	0.511984	0.255992	+	0.966679i	$0.417598\pi$
$312$	0	0
$313$	−7318.00	−1.32153	−0.660763	−	0.750594i	$-0.729767\pi$
$313$	−7318.00	−1.32153	−0.660763	+	0.750594i	$0.729767\pi$
$314$	3514.00	0.631549
$315$	0	0
$316$	1680.00	0.299074
$317$	−2246.00	−0.397943	−0.198971	−	0.980005i	$-0.563760\pi$
$317$	−2246.00	−0.397943	−0.198971	+	0.980005i	$0.563760\pi$
$318$	0	0
$319$	11960.0	2.09916
$320$	835.000	0.145868
$321$	0	0
$322$	4032.00	0.697809
$323$	−280.000	−0.0482341
$324$	0	0
$325$	550.000	0.0938723
$326$	2068.00	0.351337
$327$	0	0
$328$	−1830.00	−0.308064
$329$	6144.00	1.02957
$330$	0	0
$331$	1332.00	0.221188	0.110594	−	0.993866i	$-0.464725\pi$
$331$	1332.00	0.221188	0.110594	+	0.993866i	$0.464725\pi$
$332$	8484.00	1.40247
$333$	0	0
$334$	−24.0000	−0.00393180
$335$	420.000	0.0684987
$336$	0	0
$337$	−11534.0	−1.86438	−0.932191	−	0.361966i	$-0.882106\pi$
$337$	−11534.0	−1.86438	−0.932191	+	0.361966i	$0.882106\pi$
$338$	1713.00	0.275665
$339$	0	0
$340$	490.000	0.0781588
$341$	14976.0	2.37829
$342$	0	0
$343$	2640.00	0.415588
$344$	−2820.00	−0.441989
$345$	0	0
$346$	−618.000	−0.0960228
$347$	−11956.0	−1.84966	−0.924830	−	0.380382i	$-0.875793\pi$
$347$	−11956.0	−1.84966	−0.924830	+	0.380382i	$0.875793\pi$
$348$	0	0
$349$	4870.00	0.746949	0.373474	−	0.927640i	$-0.378166\pi$
$349$	4870.00	0.746949	0.373474	+	0.927640i	$0.378166\pi$
$350$	600.000	0.0916324
$351$	0	0
$352$	8372.00	1.26770
$353$	−10722.0	−1.61664	−0.808321	−	0.588742i	$-0.799623\pi$
$353$	−10722.0	−1.61664	−0.808321	+	0.588742i	$0.799623\pi$
$354$	0	0
$355$	−1640.00	−0.245189
$356$	2310.00	0.343904
$357$	0	0
$358$	3340.00	0.493085
$359$	−120.000	−0.0176417	−0.00882083	−	0.999961i	$-0.502808\pi$
$359$	−120.000	−0.0176417	−0.00882083	+	0.999961i	$0.502808\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	178.000	0.0258438
$363$	0	0
$364$	3696.00	0.532206
$365$	190.000	0.0272467
$366$	0	0
$367$	3936.00	0.559830	0.279915	−	0.960025i	$-0.409694\pi$
$367$	3936.00	0.559830	0.279915	+	0.960025i	$0.409694\pi$
$368$	6888.00	0.975711
$369$	0	0
$370$	−170.000	−0.0238862
$371$	−8112.00	−1.13519
$372$	0	0
$373$	3022.00	0.419499	0.209750	−	0.977755i	$-0.432735\pi$
$373$	3022.00	0.419499	0.209750	+	0.977755i	$0.432735\pi$
$374$	728.000	0.100652
$375$	0	0
$376$	−3840.00	−0.526683
$377$	−5060.00	−0.691255
$378$	0	0
$379$	−13340.0	−1.80799	−0.903997	−	0.427539i	$-0.859381\pi$
$379$	−13340.0	−1.80799	−0.903997	+	0.427539i	$0.859381\pi$
$380$	−700.000	−0.0944980
$381$	0	0
$382$	−1888.00	−0.252876
$383$	1008.00	0.134481	0.0672407	−	0.997737i	$-0.478580\pi$
$383$	1008.00	0.134481	0.0672407	+	0.997737i	$0.478580\pi$
$384$	0	0
$385$	−6240.00	−0.826026
$386$	−1922.00	−0.253438
$387$	0	0
$388$	−6062.00	−0.793174
$389$	−9630.00	−1.25517	−0.627584	−	0.778549i	$-0.715956\pi$
$389$	−9630.00	−1.25517	−0.627584	+	0.778549i	$0.715956\pi$
$390$	0	0
$391$	2352.00	0.304209
$392$	3495.00	0.450317
$393$	0	0
$394$	2526.00	0.322990
$395$	1200.00	0.152857
$396$	0	0
$397$	7126.00	0.900866	0.450433	−	0.892810i	$-0.351270\pi$
$397$	7126.00	0.900866	0.450433	+	0.892810i	$0.351270\pi$
$398$	1160.00	0.146094
$399$	0	0
$400$	1025.00	0.128125
$401$	8718.00	1.08568	0.542838	−	0.839837i	$-0.317350\pi$
$401$	8718.00	1.08568	0.542838	+	0.839837i	$0.317350\pi$
$402$	0	0
$403$	−6336.00	−0.783173
$404$	−8526.00	−1.04996
$405$	0	0
$406$	−5520.00	−0.674761
$407$	1768.00	0.215323
$408$	0	0
$409$	−10870.0	−1.31415	−0.657074	−	0.753826i	$-0.728206\pi$
$409$	−10870.0	−1.31415	−0.657074	+	0.753826i	$0.728206\pi$
$410$	−610.000	−0.0734774
$411$	0	0
$412$	616.000	0.0736605
$413$	2400.00	0.285947
$414$	0	0
$415$	6060.00	0.716804
$416$	−3542.00	−0.417454
$417$	0	0
$418$	−1040.00	−0.121694
$419$	9700.00	1.13097	0.565484	−	0.824759i	$-0.308689\pi$
$419$	9700.00	1.13097	0.565484	+	0.824759i	$0.308689\pi$
$420$	0	0
$421$	862.000	0.0997893	0.0498947	−	0.998754i	$-0.484111\pi$
$421$	862.000	0.0997893	0.0498947	+	0.998754i	$0.484111\pi$
$422$	4468.00	0.515400
$423$	0	0
$424$	5070.00	0.580710
$425$	350.000	0.0399470
$426$	0	0
$427$	−17808.0	−2.01824
$428$	252.000	0.0284600
$429$	0	0
$430$	−940.000	−0.105421
$431$	−15792.0	−1.76490	−0.882452	−	0.470402i	$-0.844109\pi$
$431$	−15792.0	−1.76490	−0.882452	+	0.470402i	$0.844109\pi$
$432$	0	0
$433$	11602.0	1.28766	0.643830	−	0.765169i	$-0.277345\pi$
$433$	11602.0	1.28766	0.643830	+	0.765169i	$0.277345\pi$
$434$	−6912.00	−0.764485
$435$	0	0
$436$	6790.00	0.745830
$437$	−3360.00	−0.367805
$438$	0	0
$439$	−440.000	−0.0478361	−0.0239181	−	0.999714i	$-0.507614\pi$
$439$	−440.000	−0.0478361	−0.0239181	+	0.999714i	$0.507614\pi$
$440$	3900.00	0.422557
$441$	0	0
$442$	−308.000	−0.0331449
$443$	10188.0	1.09266	0.546328	−	0.837571i	$-0.316025\pi$
$443$	10188.0	1.09266	0.546328	+	0.837571i	$0.316025\pi$
$444$	0	0
$445$	1650.00	0.175770
$446$	−6032.00	−0.640411
$447$	0	0
$448$	4008.00	0.422679
$449$	13310.0	1.39897	0.699485	−	0.714647i	$-0.253413\pi$
$449$	13310.0	1.39897	0.699485	+	0.714647i	$0.253413\pi$
$450$	0	0
$451$	6344.00	0.662367
$452$	7294.00	0.759029
$453$	0	0
$454$	2636.00	0.272497
$455$	2640.00	0.272011
$456$	0	0
$457$	3226.00	0.330210	0.165105	−	0.986276i	$-0.447204\pi$
$457$	3226.00	0.330210	0.165105	+	0.986276i	$0.447204\pi$
$458$	−4830.00	−0.492775
$459$	0	0
$460$	5880.00	0.595992
$461$	−6582.00	−0.664977	−0.332488	−	0.943107i	$-0.607888\pi$
$461$	−6582.00	−0.664977	−0.332488	+	0.943107i	$0.607888\pi$
$462$	0	0
$463$	15072.0	1.51286	0.756431	−	0.654073i	$-0.226941\pi$
$463$	15072.0	1.51286	0.756431	+	0.654073i	$0.226941\pi$
$464$	−9430.00	−0.943484
$465$	0	0
$466$	2682.00	0.266612
$467$	−476.000	−0.0471663	−0.0235831	−	0.999722i	$-0.507507\pi$
$467$	−476.000	−0.0471663	−0.0235831	+	0.999722i	$0.507507\pi$
$468$	0	0
$469$	2016.00	0.198487
$470$	−1280.00	−0.125621
$471$	0	0
$472$	−1500.00	−0.146278
$473$	9776.00	0.950319
$474$	0	0
$475$	−500.000	−0.0482980
$476$	2352.00	0.226478
$477$	0	0
$478$	2320.00	0.221997
$479$	19680.0	1.87725	0.938624	−	0.344941i	$-0.112101\pi$
$479$	19680.0	1.87725	0.938624	+	0.344941i	$0.112101\pi$
$480$	0	0
$481$	−748.000	−0.0709062
$482$	−2002.00	−0.189188
$483$	0	0
$484$	−9611.00	−0.902611
$485$	−4330.00	−0.405392
$486$	0	0
$487$	−5944.00	−0.553077	−0.276538	−	0.961003i	$-0.589187\pi$
$487$	−5944.00	−0.553077	−0.276538	+	0.961003i	$0.589187\pi$
$488$	11130.0	1.03244
$489$	0	0
$490$	1165.00	0.107407
$491$	−10772.0	−0.990089	−0.495044	−	0.868868i	$-0.664848\pi$
$491$	−10772.0	−0.990089	−0.495044	+	0.868868i	$0.664848\pi$
$492$	0	0
$493$	−3220.00	−0.294161
$494$	440.000	0.0400740
$495$	0	0
$496$	−11808.0	−1.06894
$497$	−7872.00	−0.710478
$498$	0	0
$499$	8140.00	0.730253	0.365127	−	0.930958i	$-0.381026\pi$
$499$	8140.00	0.730253	0.365127	+	0.930958i	$0.381026\pi$
$500$	875.000	0.0782624
$501$	0	0
$502$	132.000	0.0117360
$503$	13768.0	1.22045	0.610223	−	0.792229i	$-0.291080\pi$
$503$	13768.0	1.22045	0.610223	+	0.792229i	$0.291080\pi$
$504$	0	0
$505$	−6090.00	−0.536637
$506$	8736.00	0.767515
$507$	0	0
$508$	−13552.0	−1.18361
$509$	−22150.0	−1.92884	−0.964422	−	0.264368i	$-0.914837\pi$
$509$	−22150.0	−1.92884	−0.964422	+	0.264368i	$0.914837\pi$
$510$	0	0
$511$	912.000	0.0789521
$512$	−11521.0	−0.994455
$513$	0	0
$514$	−7614.00	−0.653384
$515$	440.000	0.0376480
$516$	0	0
$517$	13312.0	1.13242
$518$	−816.000	−0.0692143
$519$	0	0
$520$	−1650.00	−0.139149
$521$	−1562.00	−0.131348	−0.0656741	−	0.997841i	$-0.520920\pi$
$521$	−1562.00	−0.131348	−0.0656741	+	0.997841i	$0.520920\pi$
$522$	0	0
$523$	−668.000	−0.0558501	−0.0279250	−	0.999610i	$-0.508890\pi$
$523$	−668.000	−0.0558501	−0.0279250	+	0.999610i	$0.508890\pi$
$524$	5124.00	0.427181
$525$	0	0
$526$	−4888.00	−0.405184
$527$	−4032.00	−0.333276
$528$	0	0
$529$	16057.0	1.31972
$530$	1690.00	0.138507
$531$	0	0
$532$	−3360.00	−0.273824
$533$	−2684.00	−0.218118
$534$	0	0
$535$	180.000	0.0145459
$536$	−1260.00	−0.101537
$537$	0	0
$538$	1270.00	0.101772
$539$	−12116.0	−0.968225
$540$	0	0
$541$	−6138.00	−0.487788	−0.243894	−	0.969802i	$-0.578425\pi$
$541$	−6138.00	−0.487788	−0.243894	+	0.969802i	$0.578425\pi$
$542$	−1072.00	−0.0849564
$543$	0	0
$544$	−2254.00	−0.177646
$545$	4850.00	0.381195
$546$	0	0
$547$	−10484.0	−0.819494	−0.409747	−	0.912199i	$-0.634383\pi$
$547$	−10484.0	−0.819494	−0.409747	+	0.912199i	$0.634383\pi$
$548$	−15498.0	−1.20811
$549$	0	0
$550$	1300.00	0.100786
$551$	4600.00	0.355656
$552$	0	0
$553$	5760.00	0.442930
$554$	5394.00	0.413663
$555$	0	0
$556$	−140.000	−0.0106786
$557$	−3606.00	−0.274311	−0.137155	−	0.990550i	$-0.543796\pi$
$557$	−3606.00	−0.274311	−0.137155	+	0.990550i	$0.543796\pi$
$558$	0	0
$559$	−4136.00	−0.312941
$560$	4920.00	0.371264
$561$	0	0
$562$	2442.00	0.183291
$563$	−12252.0	−0.917159	−0.458579	−	0.888654i	$-0.651641\pi$
$563$	−12252.0	−0.917159	−0.458579	+	0.888654i	$0.651641\pi$
$564$	0	0
$565$	5210.00	0.387940
$566$	−2772.00	−0.205858
$567$	0	0
$568$	4920.00	0.363448
$569$	14550.0	1.07200	0.536000	−	0.844218i	$-0.319935\pi$
$569$	14550.0	1.07200	0.536000	+	0.844218i	$0.319935\pi$
$570$	0	0
$571$	−25468.0	−1.86655	−0.933277	−	0.359157i	$-0.883064\pi$
$571$	−25468.0	−1.86655	−0.933277	+	0.359157i	$0.883064\pi$
$572$	8008.00	0.585369
$573$	0	0
$574$	−2928.00	−0.212914
$575$	4200.00	0.304612
$576$	0	0
$577$	12866.0	0.928282	0.464141	−	0.885761i	$-0.346363\pi$
$577$	12866.0	0.928282	0.464141	+	0.885761i	$0.346363\pi$
$578$	4717.00	0.339449
$579$	0	0
$580$	−8050.00	−0.576307
$581$	29088.0	2.07706
$582$	0	0
$583$	−17576.0	−1.24858
$584$	−570.000	−0.0403883
$585$	0	0
$586$	4542.00	0.320185
$587$	14844.0	1.04374	0.521872	−	0.853024i	$-0.325234\pi$
$587$	14844.0	1.04374	0.521872	+	0.853024i	$0.325234\pi$
$588$	0	0
$589$	5760.00	0.402948
$590$	−500.000	−0.0348893
$591$	0	0
$592$	−1394.00	−0.0967788
$593$	−20402.0	−1.41283	−0.706416	−	0.707797i	$-0.749689\pi$
$593$	−20402.0	−1.41283	−0.706416	+	0.707797i	$0.749689\pi$
$594$	0	0
$595$	1680.00	0.115753
$596$	−9310.00	−0.639853
$597$	0	0
$598$	−3696.00	−0.252744
$599$	−10760.0	−0.733959	−0.366980	−	0.930229i	$-0.619608\pi$
$599$	−10760.0	−0.733959	−0.366980	+	0.930229i	$0.619608\pi$
$600$	0	0
$601$	14282.0	0.969343	0.484671	−	0.874696i	$-0.338939\pi$
$601$	14282.0	0.969343	0.484671	+	0.874696i	$0.338939\pi$
$602$	−4512.00	−0.305474
$603$	0	0
$604$	8456.00	0.569652
$605$	−6865.00	−0.461326
$606$	0	0
$607$	11056.0	0.739290	0.369645	−	0.929173i	$-0.379479\pi$
$607$	11056.0	0.739290	0.369645	+	0.929173i	$0.379479\pi$
$608$	3220.00	0.214783
$609$	0	0
$610$	3710.00	0.246252
$611$	−5632.00	−0.372907
$612$	0	0
$613$	−16418.0	−1.08176	−0.540878	−	0.841101i	$-0.681908\pi$
$613$	−16418.0	−1.08176	−0.540878	+	0.841101i	$0.681908\pi$
$614$	−5116.00	−0.336262
$615$	0	0
$616$	18720.0	1.22443
$617$	10374.0	0.676891	0.338445	−	0.940986i	$-0.390099\pi$
$617$	10374.0	0.676891	0.338445	+	0.940986i	$0.390099\pi$
$618$	0	0
$619$	−5260.00	−0.341546	−0.170773	−	0.985310i	$-0.554627\pi$
$619$	−5260.00	−0.341546	−0.170773	+	0.985310i	$0.554627\pi$
$620$	−10080.0	−0.652940
$621$	0	0
$622$	−2808.00	−0.181014
$623$	7920.00	0.509323
$624$	0	0
$625$	625.000	0.0400000
$626$	7318.00	0.467230
$627$	0	0
$628$	24598.0	1.56300
$629$	−476.000	−0.0301739
$630$	0	0
$631$	21352.0	1.34708	0.673542	−	0.739149i	$-0.264772\pi$
$631$	21352.0	1.34708	0.673542	+	0.739149i	$0.264772\pi$
$632$	−3600.00	−0.226583
$633$	0	0
$634$	2246.00	0.140694
$635$	−9680.00	−0.604943
$636$	0	0
$637$	5126.00	0.318838
$638$	−11960.0	−0.742164
$639$	0	0
$640$	−7275.00	−0.449328
$641$	29118.0	1.79422	0.897108	−	0.441812i	$-0.145664\pi$
$641$	29118.0	1.79422	0.897108	+	0.441812i	$0.145664\pi$
$642$	0	0
$643$	5772.00	0.354005	0.177003	−	0.984210i	$-0.443360\pi$
$643$	5772.00	0.354005	0.177003	+	0.984210i	$0.443360\pi$
$644$	28224.0	1.72699
$645$	0	0
$646$	280.000	0.0170533
$647$	14264.0	0.866732	0.433366	−	0.901218i	$-0.357326\pi$
$647$	14264.0	0.866732	0.433366	+	0.901218i	$0.357326\pi$
$648$	0	0
$649$	5200.00	0.314511
$650$	−550.000	−0.0331889
$651$	0	0
$652$	14476.0	0.869515
$653$	−6902.00	−0.413623	−0.206812	−	0.978381i	$-0.566309\pi$
$653$	−6902.00	−0.413623	−0.206812	+	0.978381i	$0.566309\pi$
$654$	0	0
$655$	3660.00	0.218333
$656$	−5002.00	−0.297706
$657$	0	0
$658$	−6144.00	−0.364009
$659$	−20140.0	−1.19051	−0.595253	−	0.803539i	$-0.702948\pi$
$659$	−20140.0	−1.19051	−0.595253	+	0.803539i	$0.702948\pi$
$660$	0	0
$661$	−3218.00	−0.189358	−0.0946790	−	0.995508i	$-0.530182\pi$
$661$	−3218.00	−0.189358	−0.0946790	+	0.995508i	$0.530182\pi$
$662$	−1332.00	−0.0782019
$663$	0	0
$664$	−18180.0	−1.06253
$665$	−2400.00	−0.139952
$666$	0	0
$667$	−38640.0	−2.24310
$668$	−168.000	−0.00973071
$669$	0	0
$670$	−420.000	−0.0242179
$671$	−38584.0	−2.21985
$672$	0	0
$673$	−7518.00	−0.430606	−0.215303	−	0.976547i	$-0.569074\pi$
$673$	−7518.00	−0.430606	−0.215303	+	0.976547i	$0.569074\pi$
$674$	11534.0	0.659159
$675$	0	0
$676$	11991.0	0.682237
$677$	18114.0	1.02833	0.514164	−	0.857692i	$-0.328102\pi$
$677$	18114.0	1.02833	0.514164	+	0.857692i	$0.328102\pi$
$678$	0	0
$679$	−20784.0	−1.17469
$680$	−1050.00	−0.0592142
$681$	0	0
$682$	−14976.0	−0.840851
$683$	23868.0	1.33716	0.668582	−	0.743638i	$-0.266901\pi$
$683$	23868.0	1.33716	0.668582	+	0.743638i	$0.266901\pi$
$684$	0	0
$685$	−11070.0	−0.617464
$686$	−2640.00	−0.146932
$687$	0	0
$688$	−7708.00	−0.427129
$689$	7436.00	0.411160
$690$	0	0
$691$	172.000	0.00946916	0.00473458	−	0.999989i	$-0.498493\pi$
$691$	172.000	0.00946916	0.00473458	+	0.999989i	$0.498493\pi$
$692$	−4326.00	−0.237644
$693$	0	0
$694$	11956.0	0.653953
$695$	−100.000	−0.00545787
$696$	0	0
$697$	−1708.00	−0.0928194
$698$	−4870.00	−0.264086
$699$	0	0
$700$	4200.00	0.226779
$701$	22138.0	1.19278	0.596391	−	0.802694i	$-0.296601\pi$
$701$	22138.0	1.19278	0.596391	+	0.802694i	$0.296601\pi$
$702$	0	0
$703$	680.000	0.0364818
$704$	8684.00	0.464901
$705$	0	0
$706$	10722.0	0.571569
$707$	−29232.0	−1.55500
$708$	0	0
$709$	3070.00	0.162618	0.0813091	−	0.996689i	$-0.474090\pi$
$709$	3070.00	0.162618	0.0813091	+	0.996689i	$0.474090\pi$
$710$	1640.00	0.0866875
$711$	0	0
$712$	−4950.00	−0.260546
$713$	−48384.0	−2.54137
$714$	0	0
$715$	5720.00	0.299183
$716$	23380.0	1.22032
$717$	0	0
$718$	120.000	0.00623727
$719$	−15600.0	−0.809154	−0.404577	−	0.914504i	$-0.632581\pi$
$719$	−15600.0	−0.809154	−0.404577	+	0.914504i	$0.632581\pi$
$720$	0	0
$721$	2112.00	0.109092
$722$	6459.00	0.332935
$723$	0	0
$724$	1246.00	0.0639603
$725$	−5750.00	−0.294551
$726$	0	0
$727$	20696.0	1.05581	0.527904	−	0.849304i	$-0.322978\pi$
$727$	20696.0	1.05581	0.527904	+	0.849304i	$0.322978\pi$
$728$	−7920.00	−0.403207
$729$	0	0
$730$	−190.000	−0.00963317
$731$	−2632.00	−0.133171
$732$	0	0
$733$	−30778.0	−1.55090	−0.775451	−	0.631408i	$-0.782478\pi$
$733$	−30778.0	−1.55090	−0.775451	+	0.631408i	$0.782478\pi$
$734$	−3936.00	−0.197930
$735$	0	0
$736$	−27048.0	−1.35462
$737$	4368.00	0.218314
$738$	0	0
$739$	11740.0	0.584388	0.292194	−	0.956359i	$-0.405615\pi$
$739$	11740.0	0.584388	0.292194	+	0.956359i	$0.405615\pi$
$740$	−1190.00	−0.0591152
$741$	0	0
$742$	8112.00	0.401349
$743$	−2632.00	−0.129958	−0.0649789	−	0.997887i	$-0.520698\pi$
$743$	−2632.00	−0.129958	−0.0649789	+	0.997887i	$0.520698\pi$
$744$	0	0
$745$	−6650.00	−0.327030
$746$	−3022.00	−0.148315
$747$	0	0
$748$	5096.00	0.249102
$749$	864.000	0.0421494
$750$	0	0
$751$	−20528.0	−0.997440	−0.498720	−	0.866763i	$-0.666196\pi$
$751$	−20528.0	−0.997440	−0.498720	+	0.866763i	$0.666196\pi$
$752$	−10496.0	−0.508976
$753$	0	0
$754$	5060.00	0.244396
$755$	6040.00	0.291150
$756$	0	0
$757$	21646.0	1.03928	0.519642	−	0.854384i	$-0.326066\pi$
$757$	21646.0	1.03928	0.519642	+	0.854384i	$0.326066\pi$
$758$	13340.0	0.639222
$759$	0	0
$760$	1500.00	0.0715931
$761$	−18282.0	−0.870857	−0.435428	−	0.900223i	$-0.643403\pi$
$761$	−18282.0	−0.870857	−0.435428	+	0.900223i	$0.643403\pi$
$762$	0	0
$763$	23280.0	1.10458
$764$	−13216.0	−0.625835
$765$	0	0
$766$	−1008.00	−0.0475464
$767$	−2200.00	−0.103569
$768$	0	0
$769$	−24190.0	−1.13435	−0.567174	−	0.823598i	$-0.691963\pi$
$769$	−24190.0	−1.13435	−0.567174	+	0.823598i	$0.691963\pi$
$770$	6240.00	0.292044
$771$	0	0
$772$	−13454.0	−0.627228
$773$	25698.0	1.19572	0.597861	−	0.801600i	$-0.296018\pi$
$773$	25698.0	1.19572	0.597861	+	0.801600i	$0.296018\pi$
$774$	0	0
$775$	−7200.00	−0.333718
$776$	12990.0	0.600920
$777$	0	0
$778$	9630.00	0.443769
$779$	2440.00	0.112223
$780$	0	0
$781$	−17056.0	−0.781449
$782$	−2352.00	−0.107554
$783$	0	0
$784$	9553.00	0.435177
$785$	17570.0	0.798854
$786$	0	0
$787$	33436.0	1.51444	0.757220	−	0.653160i	$-0.226557\pi$
$787$	33436.0	1.51444	0.757220	+	0.653160i	$0.226557\pi$
$788$	17682.0	0.799359
$789$	0	0
$790$	−1200.00	−0.0540431
$791$	25008.0	1.12412
$792$	0	0
$793$	16324.0	0.730999
$794$	−7126.00	−0.318504
$795$	0	0
$796$	8120.00	0.361565
$797$	37594.0	1.67083	0.835413	−	0.549623i	$-0.185229\pi$
$797$	37594.0	1.67083	0.835413	+	0.549623i	$0.185229\pi$
$798$	0	0
$799$	−3584.00	−0.158689
$800$	−4025.00	−0.177882
$801$	0	0
$802$	−8718.00	−0.383844
$803$	1976.00	0.0868388
$804$	0	0
$805$	20160.0	0.882667
$806$	6336.00	0.276893
$807$	0	0
$808$	18270.0	0.795466
$809$	−4730.00	−0.205560	−0.102780	−	0.994704i	$-0.532774\pi$
$809$	−4730.00	−0.205560	−0.102780	+	0.994704i	$0.532774\pi$
$810$	0	0
$811$	−8748.00	−0.378772	−0.189386	−	0.981903i	$-0.560650\pi$
$811$	−8748.00	−0.378772	−0.189386	+	0.981903i	$0.560650\pi$
$812$	−38640.0	−1.66995
$813$	0	0
$814$	−1768.00	−0.0761282
$815$	10340.0	0.444410
$816$	0	0
$817$	3760.00	0.161011
$818$	10870.0	0.464622
$819$	0	0
$820$	−4270.00	−0.181847
$821$	−44142.0	−1.87645	−0.938226	−	0.346024i	$-0.887532\pi$
$821$	−44142.0	−1.87645	−0.938226	+	0.346024i	$0.887532\pi$
$822$	0	0
$823$	3992.00	0.169079	0.0845397	−	0.996420i	$-0.473058\pi$
$823$	3992.00	0.169079	0.0845397	+	0.996420i	$0.473058\pi$
$824$	−1320.00	−0.0558063
$825$	0	0
$826$	−2400.00	−0.101098
$827$	14444.0	0.607336	0.303668	−	0.952778i	$-0.401789\pi$
$827$	14444.0	0.607336	0.303668	+	0.952778i	$0.401789\pi$
$828$	0	0
$829$	42150.0	1.76590	0.882949	−	0.469468i	$-0.155554\pi$
$829$	42150.0	1.76590	0.882949	+	0.469468i	$0.155554\pi$
$830$	−6060.00	−0.253429
$831$	0	0
$832$	−3674.00	−0.153093
$833$	3262.00	0.135680
$834$	0	0
$835$	−120.000	−0.00497338
$836$	−7280.00	−0.301177
$837$	0	0
$838$	−9700.00	−0.399858
$839$	−13400.0	−0.551394	−0.275697	−	0.961245i	$-0.588909\pi$
$839$	−13400.0	−0.551394	−0.275697	+	0.961245i	$0.588909\pi$
$840$	0	0
$841$	28511.0	1.16901
$842$	−862.000	−0.0352809
$843$	0	0
$844$	31276.0	1.27555
$845$	8565.00	0.348692
$846$	0	0
$847$	−32952.0	−1.33677
$848$	13858.0	0.561186
$849$	0	0
$850$	−350.000	−0.0141234
$851$	−5712.00	−0.230088
$852$	0	0
$853$	−8658.00	−0.347531	−0.173766	−	0.984787i	$-0.555594\pi$
$853$	−8658.00	−0.347531	−0.173766	+	0.984787i	$0.555594\pi$
$854$	17808.0	0.713556
$855$	0	0
$856$	−540.000	−0.0215617
$857$	−42826.0	−1.70701	−0.853505	−	0.521084i	$-0.825528\pi$
$857$	−42826.0	−1.70701	−0.853505	+	0.521084i	$0.825528\pi$
$858$	0	0
$859$	−35900.0	−1.42595	−0.712976	−	0.701189i	$-0.752653\pi$
$859$	−35900.0	−1.42595	−0.712976	+	0.701189i	$0.752653\pi$
$860$	−6580.00	−0.260902
$861$	0	0
$862$	15792.0	0.623988
$863$	3088.00	0.121804	0.0609019	−	0.998144i	$-0.480602\pi$
$863$	3088.00	0.121804	0.0609019	+	0.998144i	$0.480602\pi$
$864$	0	0
$865$	−3090.00	−0.121460
$866$	−11602.0	−0.455256
$867$	0	0
$868$	−48384.0	−1.89200
$869$	12480.0	0.487175
$870$	0	0
$871$	−1848.00	−0.0718910
$872$	−14550.0	−0.565052
$873$	0	0
$874$	3360.00	0.130039
$875$	3000.00	0.115907
$876$	0	0
$877$	−35274.0	−1.35817	−0.679087	−	0.734058i	$-0.737624\pi$
$877$	−35274.0	−1.35817	−0.679087	+	0.734058i	$0.737624\pi$
$878$	440.000	0.0169126
$879$	0	0
$880$	10660.0	0.408351
$881$	−25042.0	−0.957646	−0.478823	−	0.877911i	$-0.658936\pi$
$881$	−25042.0	−0.957646	−0.478823	+	0.877911i	$0.658936\pi$
$882$	0	0
$883$	12572.0	0.479141	0.239570	−	0.970879i	$-0.422993\pi$
$883$	12572.0	0.479141	0.239570	+	0.970879i	$0.422993\pi$
$884$	−2156.00	−0.0820296
$885$	0	0
$886$	−10188.0	−0.386312
$887$	21864.0	0.827645	0.413823	−	0.910358i	$-0.364193\pi$
$887$	21864.0	0.827645	0.413823	+	0.910358i	$0.364193\pi$
$888$	0	0
$889$	−46464.0	−1.75293
$890$	−1650.00	−0.0621440
$891$	0	0
$892$	−42224.0	−1.58494
$893$	5120.00	0.191864
$894$	0	0
$895$	16700.0	0.623709
$896$	−34920.0	−1.30200
$897$	0	0
$898$	−13310.0	−0.494611
$899$	66240.0	2.45743
$900$	0	0
$901$	4732.00	0.174968
$902$	−6344.00	−0.234182
$903$	0	0
$904$	−15630.0	−0.575051
$905$	890.000	0.0326902
$906$	0	0
$907$	31236.0	1.14352	0.571761	−	0.820420i	$-0.306260\pi$
$907$	31236.0	1.14352	0.571761	+	0.820420i	$0.306260\pi$
$908$	18452.0	0.674396
$909$	0	0
$910$	−2640.00	−0.0961705
$911$	−8272.00	−0.300838	−0.150419	−	0.988622i	$-0.548062\pi$
$911$	−8272.00	−0.300838	−0.150419	+	0.988622i	$0.548062\pi$
$912$	0	0
$913$	63024.0	2.28455
$914$	−3226.00	−0.116747
$915$	0	0
$916$	−33810.0	−1.21956
$917$	17568.0	0.632657
$918$	0	0
$919$	20200.0	0.725067	0.362533	−	0.931971i	$-0.381912\pi$
$919$	20200.0	0.725067	0.362533	+	0.931971i	$0.381912\pi$
$920$	−12600.0	−0.451532
$921$	0	0
$922$	6582.00	0.235105
$923$	7216.00	0.257332
$924$	0	0
$925$	−850.000	−0.0302139
$926$	−15072.0	−0.534878
$927$	0	0
$928$	37030.0	1.30988
$929$	−31010.0	−1.09516	−0.547581	−	0.836753i	$-0.684451\pi$
$929$	−31010.0	−1.09516	−0.547581	+	0.836753i	$0.684451\pi$
$930$	0	0
$931$	−4660.00	−0.164044
$932$	18774.0	0.659831
$933$	0	0
$934$	476.000	0.0166758
$935$	3640.00	0.127316
$936$	0	0
$937$	−39174.0	−1.36580	−0.682902	−	0.730510i	$-0.739283\pi$
$937$	−39174.0	−1.36580	−0.682902	+	0.730510i	$0.739283\pi$
$938$	−2016.00	−0.0701756
$939$	0	0
$940$	−8960.00	−0.310897
$941$	4138.00	0.143353	0.0716764	−	0.997428i	$-0.477165\pi$
$941$	4138.00	0.143353	0.0716764	+	0.997428i	$0.477165\pi$
$942$	0	0
$943$	−20496.0	−0.707785
$944$	−4100.00	−0.141360
$945$	0	0
$946$	−9776.00	−0.335989
$947$	−23676.0	−0.812425	−0.406213	−	0.913779i	$-0.633151\pi$
$947$	−23676.0	−0.812425	−0.406213	+	0.913779i	$0.633151\pi$
$948$	0	0
$949$	−836.000	−0.0285961
$950$	500.000	0.0170759
$951$	0	0
$952$	−5040.00	−0.171583
$953$	−18922.0	−0.643173	−0.321586	−	0.946880i	$-0.604216\pi$
$953$	−18922.0	−0.643173	−0.321586	+	0.946880i	$0.604216\pi$
$954$	0	0
$955$	−9440.00	−0.319865
$956$	16240.0	0.549413
$957$	0	0
$958$	−19680.0	−0.663708
$959$	−53136.0	−1.78921
$960$	0	0
$961$	53153.0	1.78420
$962$	748.000	0.0250691
$963$	0	0
$964$	−14014.0	−0.468216
$965$	−9610.00	−0.320577
$966$	0	0
$967$	39656.0	1.31877	0.659385	−	0.751805i	$-0.270817\pi$
$967$	39656.0	1.31877	0.659385	+	0.751805i	$0.270817\pi$
$968$	20595.0	0.683831
$969$	0	0
$970$	4330.00	0.143328
$971$	33228.0	1.09818	0.549092	−	0.835762i	$-0.314974\pi$
$971$	33228.0	1.09818	0.549092	+	0.835762i	$0.314974\pi$
$972$	0	0
$973$	−480.000	−0.0158151
$974$	5944.00	0.195542
$975$	0	0
$976$	30422.0	0.997730
$977$	974.000	0.0318946	0.0159473	−	0.999873i	$-0.494924\pi$
$977$	974.000	0.0318946	0.0159473	+	0.999873i	$0.494924\pi$
$978$	0	0
$979$	17160.0	0.560200
$980$	8155.00	0.265818
$981$	0	0
$982$	10772.0	0.350049
$983$	13608.0	0.441534	0.220767	−	0.975327i	$-0.429144\pi$
$983$	13608.0	0.441534	0.220767	+	0.975327i	$0.429144\pi$
$984$	0	0
$985$	12630.0	0.408554
$986$	3220.00	0.104002
$987$	0	0
$988$	3080.00	0.0991780
$989$	−31584.0	−1.01548
$990$	0	0
$991$	13472.0	0.431839	0.215919	−	0.976411i	$-0.430725\pi$
$991$	13472.0	0.431839	0.215919	+	0.976411i	$0.430725\pi$
$992$	46368.0	1.48406
$993$	0	0
$994$	7872.00	0.251192
$995$	5800.00	0.184796
$996$	0	0
$997$	−3234.00	−0.102730	−0.0513650	−	0.998680i	$-0.516357\pi$
$997$	−3234.00	−0.102730	−0.0513650	+	0.998680i	$0.516357\pi$
$998$	−8140.00	−0.258184
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.a.c.1.1		1
3.2	odd	2		15.4.a.a.1.1	✓	1
4.3	odd	2		720.4.a.n.1.1		1
5.2	odd	4		225.4.b.e.199.1		2
5.3	odd	4		225.4.b.e.199.2		2
5.4	even	2		225.4.a.f.1.1		1
7.6	odd	2		2205.4.a.l.1.1		1
9.2	odd	6		405.4.e.g.271.1		2
9.4	even	3		405.4.e.i.136.1		2
9.5	odd	6		405.4.e.g.136.1		2
9.7	even	3		405.4.e.i.271.1		2
12.11	even	2		240.4.a.e.1.1		1
15.2	even	4		75.4.b.b.49.2		2
15.8	even	4		75.4.b.b.49.1		2
15.14	odd	2		75.4.a.b.1.1		1
21.20	even	2		735.4.a.e.1.1		1
24.5	odd	2		960.4.a.b.1.1		1
24.11	even	2		960.4.a.ba.1.1		1
33.32	even	2		1815.4.a.e.1.1		1
60.23	odd	4		1200.4.f.b.49.1		2
60.47	odd	4		1200.4.f.b.49.2		2
60.59	even	2		1200.4.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.a.1.1	✓	1	3.2	odd	2
45.4.a.c.1.1		1	1.1	even	1	trivial
75.4.a.b.1.1		1	15.14	odd	2
75.4.b.b.49.1		2	15.8	even	4
75.4.b.b.49.2		2	15.2	even	4
225.4.a.f.1.1		1	5.4	even	2
225.4.b.e.199.1		2	5.2	odd	4
225.4.b.e.199.2		2	5.3	odd	4
240.4.a.e.1.1		1	12.11	even	2
405.4.e.g.136.1		2	9.5	odd	6
405.4.e.g.271.1		2	9.2	odd	6
405.4.e.i.136.1		2	9.4	even	3
405.4.e.i.271.1		2	9.7	even	3
720.4.a.n.1.1		1	4.3	odd	2
735.4.a.e.1.1		1	21.20	even	2
960.4.a.b.1.1		1	24.5	odd	2
960.4.a.ba.1.1		1	24.11	even	2
1200.4.a.t.1.1		1	60.59	even	2
1200.4.f.b.49.1		2	60.23	odd	4
1200.4.f.b.49.2		2	60.47	odd	4
1815.4.a.e.1.1		1	33.32	even	2
2205.4.a.l.1.1		1	7.6	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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