LMFDB - Embedded newform 1472.2.h.c.735.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,2,Mod(735,1472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1472.735"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1472.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$11.7539791775$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			735.14
Character	$\chi$	$=$	1472.735
Dual form			1472.2.h.c.735.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.678936 q^{3} -2.56187 q^{5} -4.63615 q^{7} -2.53905 q^{9} -0.0824776i q^{11} -5.74743i q^{13} +1.73935 q^{15} +2.55509i q^{17} +4.26667i q^{19} +3.14765 q^{21} +(-2.75394 - 3.92629i) q^{23} +1.56320 q^{25} +3.76066 q^{27} -4.77193i q^{29} +6.46534i q^{31} +0.0559970i q^{33} +11.8772 q^{35} +7.41432 q^{37} +3.90214i q^{39} -4.66545 q^{41} +0.420821i q^{43} +6.50472 q^{45} +9.59174i q^{47} +14.4939 q^{49} -1.73474i q^{51} +9.22547 q^{53} +0.211297i q^{55} -2.89680i q^{57} +2.10623 q^{59} -6.94097 q^{61} +11.7714 q^{63} +14.7242i q^{65} -0.188823i q^{67} +(1.86975 + 2.66570i) q^{69} +3.17595i q^{71} +4.82843 q^{73} -1.06131 q^{75} +0.382378i q^{77} -11.4857 q^{79} +5.06389 q^{81} +7.33792i q^{83} -6.54582i q^{85} +3.23984i q^{87} -11.4857i q^{89} +26.6459i q^{91} -4.38956i q^{93} -10.9307i q^{95} -3.42270i q^{97} +0.209414i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 40 q^{9} + 32 q^{25} + 8 q^{41} + 48 q^{49} - 104 q^{73} + 112 q^{81}+O(q^{100}) $ 32 * q + 40 * q^9 + 32 * q^25 + 8 * q^41 + 48 * q^49 - 104 * q^73 + 112 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times$.

$n$	$645$	$833$	$1151$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	−0.678936			−0.391984			−0.195992	−	0.980605i	$-0.562793\pi$
$3$	−0.678936			−0.391984			−0.195992	+	0.980605i	$0.562793\pi$
$4$		0			0
$5$	−2.56187			−1.14571			−0.572853	−	0.819658i	$-0.694163\pi$
$5$	−2.56187			−1.14571			−0.572853	+	0.819658i	$0.694163\pi$
$6$		0			0
$7$	−4.63615			−1.75230			−0.876150	−	0.482039i	$-0.839896\pi$
$7$	−4.63615			−1.75230			−0.876150	+	0.482039i	$0.839896\pi$
$8$		0			0
$9$	−2.53905			−0.846348
$10$		0			0
$11$		−	0.0824776i		−	0.0248679i	−0.999923	−	0.0124340i	$-0.996042\pi$
$11$		−	0.0824776i		−	0.0248679i	0.999923	−	0.0124340i	$-0.00395796\pi$
$12$		0			0
$13$		−	5.74743i		−	1.59405i	−0.603947	−	0.797025i	$-0.706406\pi$
$13$		−	5.74743i		−	1.59405i	0.603947	−	0.797025i	$-0.293594\pi$
$14$		0			0
$15$	1.73935			0.449098
$16$		0			0
$17$			2.55509i			0.619701i	0.950785	+	0.309850i	$0.100279\pi$
$17$			2.55509i			0.619701i	−0.950785	+	0.309850i	$0.899721\pi$
$18$		0			0
$19$			4.26667i			0.978842i	0.872048	+	0.489421i	$0.162792\pi$
$19$			4.26667i			0.978842i	−0.872048	+	0.489421i	$0.837208\pi$
$20$		0			0
$21$	3.14765			0.686874
$22$		0			0
$23$	−2.75394	−	3.92629i	−0.574237	−	0.818689i
$23$	−2.75394	−	3.92629i	−0.574237	−	0.818689i
$24$		0			0
$25$	1.56320			0.312640
$26$		0			0
$27$	3.76066			0.723739
$28$		0			0
$29$		−	4.77193i		−	0.886126i	−0.896491	−	0.443063i	$-0.853892\pi$
$29$		−	4.77193i		−	0.886126i	0.896491	−	0.443063i	$-0.146108\pi$
$30$		0			0
$31$			6.46534i			1.16121i	0.814186	+	0.580605i	$0.197184\pi$
$31$			6.46534i			1.16121i	−0.814186	+	0.580605i	$0.802816\pi$
$32$		0			0
$33$			0.0559970i			0.00974783i
$34$		0			0
$35$	11.8772			2.00762
$36$		0			0
$37$	7.41432			1.21891			0.609454	−	0.792822i	$-0.291389\pi$
$37$	7.41432			1.21891			0.609454	+	0.792822i	$0.291389\pi$
$38$		0			0
$39$			3.90214i			0.624842i
$40$		0			0
$41$	−4.66545			−0.728621			−0.364310	−	0.931278i	$-0.618695\pi$
$41$	−4.66545			−0.728621			−0.364310	+	0.931278i	$0.618695\pi$
$42$		0			0
$43$			0.420821i			0.0641746i	0.999485	+	0.0320873i	$0.0102155\pi$
$43$			0.420821i			0.0641746i	−0.999485	+	0.0320873i	$0.989785\pi$
$44$		0			0
$45$	6.50472			0.969666
$46$		0			0
$47$			9.59174i			1.39910i	0.714584	+	0.699550i	$0.246616\pi$
$47$			9.59174i			1.39910i	−0.714584	+	0.699550i	$0.753384\pi$
$48$		0			0
$49$	14.4939			2.07055
$50$		0			0
$51$		−	1.73474i		−	0.242913i
$52$		0			0
$53$	9.22547			1.26722			0.633608	−	0.773655i	$-0.281573\pi$
$53$	9.22547			1.26722			0.633608	+	0.773655i	$0.281573\pi$
$54$		0			0
$55$			0.211297i			0.0284913i
$56$		0			0
$57$		−	2.89680i		−	0.383690i
$58$		0			0
$59$	2.10623			0.274208			0.137104	−	0.990557i	$-0.456221\pi$
$59$	2.10623			0.274208			0.137104	+	0.990557i	$0.456221\pi$
$60$		0			0
$61$	−6.94097			−0.888700			−0.444350	−	0.895853i	$-0.646565\pi$
$61$	−6.94097			−0.888700			−0.444350	+	0.895853i	$0.646565\pi$
$62$		0			0
$63$	11.7714			1.48306
$64$		0			0
$65$			14.7242i			1.82631i
$66$		0			0
$67$		−	0.188823i		−	0.0230684i	−0.999933	−	0.0115342i	$-0.996328\pi$
$67$		−	0.188823i		−	0.0230684i	0.999933	−	0.0115342i	$-0.00367153\pi$
$68$		0			0
$69$	1.86975	+	2.66570i	0.225092	+	0.320913i
$70$		0			0
$71$			3.17595i			0.376916i	0.982081	+	0.188458i	$0.0603490\pi$
$71$			3.17595i			0.376916i	−0.982081	+	0.188458i	$0.939651\pi$
$72$		0			0
$73$	4.82843			0.565125			0.282563	−	0.959249i	$-0.408815\pi$
$73$	4.82843			0.565125			0.282563	+	0.959249i	$0.408815\pi$
$74$		0			0
$75$	−1.06131			−0.122550
$76$		0			0
$77$			0.382378i			0.0435761i
$78$		0			0
$79$	−11.4857			−1.29224			−0.646120	−	0.763236i	$-0.723610\pi$
$79$	−11.4857			−1.29224			−0.646120	+	0.763236i	$0.723610\pi$
$80$		0			0
$81$	5.06389			0.562654
$82$		0			0
$83$			7.33792i			0.805442i	0.915323	+	0.402721i	$0.131935\pi$
$83$			7.33792i			0.805442i	−0.915323	+	0.402721i	$0.868065\pi$
$84$		0			0
$85$		−	6.54582i		−	0.709994i
$86$		0			0
$87$			3.23984i			0.347347i
$88$		0			0
$89$		−	11.4857i		−	1.21748i	−0.793370	−	0.608740i	$-0.791675\pi$
$89$		−	11.4857i		−	1.21748i	0.793370	−	0.608740i	$-0.208325\pi$
$90$		0			0
$91$			26.6459i			2.79325i
$92$		0			0
$93$		−	4.38956i		−	0.455176i
$94$		0			0
$95$		−	10.9307i		−	1.12146i
$96$		0			0
$97$		−	3.42270i		−	0.347523i	−0.984788	−	0.173761i	$-0.944408\pi$
$97$		−	3.42270i		−	0.347523i	0.984788	−	0.173761i	$-0.0555922\pi$
$98$		0			0
$99$			0.209414i			0.0210469i
$100$		0			0
$101$		−	13.2351i		−	1.31694i	−0.752606	−	0.658471i	$-0.771203\pi$
$101$		−	13.2351i		−	1.31694i	0.752606	−	0.658471i	$-0.228797\pi$
$102$		0			0
$103$	4.92186			0.484965			0.242483	−	0.970156i	$-0.422038\pi$
$103$	4.92186			0.484965			0.242483	+	0.970156i	$0.422038\pi$
$104$		0			0
$105$	−8.06389			−0.786955
$106$		0			0
$107$		−	15.4743i		−	1.49596i	−0.663722	−	0.747979i	$-0.731024\pi$
$107$		−	15.4743i		−	1.49596i	0.663722	−	0.747979i	$-0.268976\pi$
$108$		0			0
$109$	−15.6393			−1.49797			−0.748985	−	0.662587i	$-0.769459\pi$
$109$	−15.6393			−1.49797			−0.748985	+	0.662587i	$0.769459\pi$
$110$		0			0
$111$	−5.03385			−0.477792
$112$		0			0
$113$		−	9.79820i		−	0.921737i	−0.887468	−	0.460869i	$-0.847538\pi$
$113$		−	9.79820i		−	0.921737i	0.887468	−	0.460869i	$-0.152462\pi$
$114$		0			0
$115$	7.05526	+	10.0587i	0.657906	+	0.937976i
$116$		0			0
$117$			14.5930i			1.34912i
$118$		0			0
$119$		−	11.8458i		−	1.08590i
$120$		0			0
$121$	10.9932			0.999382
$122$		0			0
$123$	3.16754			0.285608
$124$		0			0
$125$	8.80465			0.787511
$126$		0			0
$127$		−	7.94365i		−	0.704885i	−0.935833	−	0.352442i	$-0.885351\pi$
$127$		−	7.94365i		−	0.704885i	0.935833	−	0.352442i	$-0.114649\pi$
$128$		0			0
$129$		−	0.285711i		−	0.0251554i
$130$		0			0
$131$	5.62797			0.491718			0.245859	−	0.969306i	$-0.420930\pi$
$131$	5.62797			0.491718			0.245859	+	0.969306i	$0.420930\pi$
$132$		0			0
$133$		−	19.7809i		−	1.71522i
$134$		0			0
$135$	−9.63434			−0.829192
$136$		0			0
$137$			21.9113i			1.87201i	0.351988	+	0.936005i	$0.385506\pi$
$137$			21.9113i			1.87201i	−0.351988	+	0.936005i	$0.614494\pi$
$138$		0			0
$139$	−1.65443			−0.140327			−0.0701635	−	0.997536i	$-0.522352\pi$
$139$	−1.65443			−0.140327			−0.0701635	+	0.997536i	$0.522352\pi$
$140$		0			0
$141$		−	6.51218i		−	0.548425i
$142$		0			0
$143$	−0.474034			−0.0396407
$144$		0			0
$145$			12.2251i			1.01524i
$146$		0			0
$147$	−9.84042			−0.811625
$148$		0			0
$149$	−14.4078			−1.18034			−0.590168	−	0.807281i	$-0.700938\pi$
$149$	−14.4078			−1.18034			−0.590168	+	0.807281i	$0.700938\pi$
$150$		0			0
$151$			21.5849i			1.75656i	0.478149	+	0.878279i	$0.341308\pi$
$151$			21.5849i			1.75656i	−0.478149	+	0.878279i	$0.658692\pi$
$152$		0			0
$153$		−	6.48749i		−	0.524483i
$154$		0			0
$155$		−	16.5634i		−	1.33040i
$156$		0			0
$157$	15.2030			1.21333			0.606666	−	0.794957i	$-0.292506\pi$
$157$	15.2030			1.21333			0.606666	+	0.794957i	$0.292506\pi$
$158$		0			0
$159$	−6.26351			−0.496728
$160$		0			0
$161$	12.7677	+	18.2029i	1.00624	+	1.43459i
$162$		0			0
$163$	21.7177			1.70106			0.850529	−	0.525928i	$-0.176282\pi$
$163$	21.7177			1.70106			0.850529	+	0.525928i	$0.176282\pi$
$164$		0			0
$165$		−	0.143457i		−	0.0111681i
$166$		0			0
$167$		−	2.37428i		−	0.183727i	−0.995772	−	0.0918637i	$-0.970718\pi$
$167$		−	2.37428i		−	0.183727i	0.995772	−	0.0918637i	$-0.0292824\pi$
$168$		0			0
$169$	−20.0329			−1.54099
$170$		0			0
$171$		−	10.8333i		−	0.828441i
$172$		0			0
$173$			21.4211i			1.62862i	0.580433	+	0.814308i	$0.302883\pi$
$173$			21.4211i			1.62862i	−0.580433	+	0.814308i	$0.697117\pi$
$174$		0			0
$175$	−7.24724			−0.547840
$176$		0			0
$177$	−1.43000			−0.107485
$178$		0			0
$179$	−15.4108			−1.15185			−0.575927	−	0.817501i	$-0.695359\pi$
$179$	−15.4108			−1.15185			−0.575927	+	0.817501i	$0.695359\pi$
$180$		0			0
$181$	2.39692			0.178162			0.0890808	−	0.996024i	$-0.471607\pi$
$181$	2.39692			0.178162			0.0890808	+	0.996024i	$0.471607\pi$
$182$		0			0
$183$	4.71248			0.348356
$184$		0			0
$185$	−18.9946			−1.39651
$186$		0			0
$187$	0.210738			0.0154107
$188$		0			0
$189$	−17.4350			−1.26821
$190$		0			0
$191$	−10.8997			−0.788671			−0.394336	−	0.918966i	$-0.629025\pi$
$191$	−10.8997			−0.788671			−0.394336	+	0.918966i	$0.629025\pi$
$192$		0			0
$193$	8.63951			0.621886			0.310943	−	0.950429i	$-0.399355\pi$
$193$	8.63951			0.621886			0.310943	+	0.950429i	$0.399355\pi$
$194$		0			0
$195$		−	9.99679i		−	0.715885i
$196$		0			0
$197$			24.6529i			1.75645i	0.478248	+	0.878225i	$0.341272\pi$
$197$			24.6529i			1.75645i	−0.478248	+	0.878225i	$0.658728\pi$
$198$		0			0
$199$	24.9964			1.77195			0.885975	−	0.463733i	$-0.153490\pi$
$199$	24.9964			1.77195			0.885975	+	0.463733i	$0.153490\pi$
$200$		0			0
$201$			0.128199i			0.00904245i
$202$		0			0
$203$			22.1234i			1.55276i
$204$		0			0
$205$	11.9523			0.834785
$206$		0			0
$207$	6.99239	+	9.96904i	0.486005	+	0.692896i
$208$		0			0
$209$	0.351905			0.0243418
$210$		0			0
$211$	−1.10255			−0.0759027			−0.0379514	−	0.999280i	$-0.512083\pi$
$211$	−1.10255			−0.0759027			−0.0379514	+	0.999280i	$0.512083\pi$
$212$		0			0
$213$		−	2.15627i		−	0.147745i
$214$		0			0
$215$		−	1.07809i		−	0.0735252i
$216$		0			0
$217$		−	29.9743i		−	2.03479i
$218$		0			0
$219$	−3.27820			−0.221520
$220$		0			0
$221$	14.6852			0.987834
$222$		0			0
$223$		−	19.5930i		−	1.31204i	−0.754742	−	0.656021i	$-0.772238\pi$
$223$		−	19.5930i		−	1.31204i	0.754742	−	0.656021i	$-0.227762\pi$
$224$		0			0
$225$	−3.96904			−0.264603
$226$		0			0
$227$		−	22.2729i		−	1.47831i	−0.673538	−	0.739153i	$-0.735226\pi$
$227$		−	22.2729i		−	1.47831i	0.673538	−	0.739153i	$-0.264774\pi$
$228$		0			0
$229$	10.5155			0.694885			0.347443	−	0.937701i	$-0.387050\pi$
$229$	10.5155			0.694885			0.347443	+	0.937701i	$0.387050\pi$
$230$		0			0
$231$		−	0.259611i		−	0.0170811i
$232$		0			0
$233$	−9.60293			−0.629109			−0.314555	−	0.949239i	$-0.601855\pi$
$233$	−9.60293			−0.629109			−0.314555	+	0.949239i	$0.601855\pi$
$234$		0			0
$235$		−	24.5728i		−	1.60296i
$236$		0			0
$237$	7.79805			0.506538
$238$		0			0
$239$		−	6.80167i		−	0.439964i	−0.975504	−	0.219982i	$-0.929400\pi$
$239$		−	6.80167i		−	0.439964i	0.975504	−	0.219982i	$-0.0705998\pi$
$240$		0			0
$241$		−	4.10612i		−	0.264498i	−0.991217	−	0.132249i	$-0.957780\pi$
$241$		−	4.10612i		−	0.264498i	0.991217	−	0.132249i	$-0.0422199\pi$
$242$		0			0
$243$	−14.7200			−0.944291
$244$		0			0
$245$	−37.1315			−2.37225
$246$		0			0
$247$	24.5224			1.56032
$248$		0			0
$249$		−	4.98198i		−	0.315720i
$250$		0			0
$251$		−	9.89372i		−	0.624486i	−0.950002	−	0.312243i	$-0.898920\pi$
$251$		−	9.89372i		−	0.624486i	0.950002	−	0.312243i	$-0.101080\pi$
$252$		0			0
$253$	−0.323831	+	0.227139i	−0.0203591	+	0.0142801i
$254$		0			0
$255$			4.44420i			0.278307i
$256$		0			0
$257$	−3.47516			−0.216774			−0.108387	−	0.994109i	$-0.534569\pi$
$257$	−3.47516			−0.216774			−0.108387	+	0.994109i	$0.534569\pi$
$258$		0			0
$259$	−34.3739			−2.13589
$260$		0			0
$261$			12.1162i			0.749971i
$262$		0			0
$263$	−9.74221			−0.600730			−0.300365	−	0.953824i	$-0.597109\pi$
$263$	−9.74221			−0.600730			−0.300365	+	0.953824i	$0.597109\pi$
$264$		0			0
$265$	−23.6345			−1.45186
$266$		0			0
$267$			7.79805i			0.477233i
$268$		0			0
$269$			7.59833i			0.463279i	0.972802	+	0.231639i	$0.0744089\pi$
$269$			7.59833i			0.463279i	−0.972802	+	0.231639i	$0.925591\pi$
$270$		0			0
$271$			3.10047i			0.188340i	0.995556	+	0.0941700i	$0.0300197\pi$
$271$			3.10047i			0.188340i	−0.995556	+	0.0941700i	$0.969980\pi$
$272$		0			0
$273$		−	18.0909i		−	1.09491i
$274$		0			0
$275$		−	0.128929i		−	0.00777472i
$276$		0			0
$277$			4.77193i			0.286718i	0.989671	+	0.143359i	$0.0457903\pi$
$277$			4.77193i			0.286718i	−0.989671	+	0.143359i	$0.954210\pi$
$278$		0			0
$279$		−	16.4158i		−	0.982788i
$280$		0			0
$281$		−	26.0257i		−	1.55256i	−0.630387	−	0.776281i	$-0.717104\pi$
$281$		−	26.0257i		−	1.55256i	0.630387	−	0.776281i	$-0.282896\pi$
$282$		0			0
$283$			16.3540i			0.972143i	0.873919	+	0.486072i	$0.161571\pi$
$283$			16.3540i			0.972143i	−0.873919	+	0.486072i	$0.838429\pi$
$284$		0			0
$285$			7.42124i			0.439596i
$286$		0			0
$287$	21.6297			1.27676
$288$		0			0
$289$	10.4715			0.615971
$290$		0			0
$291$			2.32380i			0.136223i
$292$		0			0
$293$	−27.8414			−1.62651			−0.813255	−	0.581908i	$-0.802306\pi$
$293$	−27.8414			−1.62651			−0.813255	+	0.581908i	$0.802306\pi$
$294$		0			0
$295$	−5.39589			−0.314161
$296$		0			0
$297$		−	0.310170i		−	0.0179979i
$298$		0			0
$299$	−22.5661	+	15.8281i	−1.30503	+	0.915362i
$300$		0			0
$301$		−	1.95099i		−	0.112453i
$302$		0			0
$303$			8.98580i			0.516221i
$304$		0			0
$305$	17.7819			1.01819
$306$		0			0
$307$	24.6029			1.40416			0.702081	−	0.712097i	$-0.252254\pi$
$307$	24.6029			1.40416			0.702081	+	0.712097i	$0.252254\pi$
$308$		0			0
$309$	−3.34163			−0.190099
$310$		0			0
$311$			11.9177i			0.675791i	0.941184	+	0.337896i	$0.109715\pi$
$311$			11.9177i			0.675791i	−0.941184	+	0.337896i	$0.890285\pi$
$312$		0			0
$313$			21.7271i			1.22809i	0.789271	+	0.614044i	$0.210458\pi$
$313$			21.7271i			1.22809i	−0.789271	+	0.614044i	$0.789542\pi$
$314$		0			0
$315$	−30.1568			−1.69915
$316$		0			0
$317$			0.110656i			0.00621504i	0.999995	+	0.00310752i	$0.000989156\pi$
$317$			0.110656i			0.00621504i	−0.999995	+	0.00310752i	$0.999011\pi$
$318$		0			0
$319$	−0.393577			−0.0220361
$320$		0			0
$321$			10.5061i			0.586392i
$322$		0			0
$323$	−10.9017			−0.606589
$324$		0			0
$325$		−	8.98439i		−	0.498364i
$326$		0			0
$327$	10.6181			0.587181
$328$		0			0
$329$		−	44.4688i		−	2.45164i
$330$		0			0
$331$	15.8650			0.872021			0.436011	−	0.899942i	$-0.356391\pi$
$331$	15.8650			0.872021			0.436011	+	0.899942i	$0.356391\pi$
$332$		0			0
$333$	−18.8253			−1.03162
$334$		0			0
$335$			0.483741i			0.0264296i
$336$		0			0
$337$			23.8390i			1.29859i	0.760536	+	0.649296i	$0.224936\pi$
$337$			23.8390i			1.29859i	−0.760536	+	0.649296i	$0.775064\pi$
$338$		0			0
$339$			6.65236i			0.361306i
$340$		0			0
$341$	0.533246			0.0288769
$342$		0			0
$343$	−34.7428			−1.87593
$344$		0			0
$345$	−4.79007	−	6.82920i	−0.257889	−	0.367672i
$346$		0			0
$347$	−21.8590			−1.17345			−0.586726	−	0.809786i	$-0.699583\pi$
$347$	−21.8590			−1.17345			−0.586726	+	0.809786i	$0.699583\pi$
$348$		0			0
$349$			17.6528i			0.944935i	0.881348	+	0.472468i	$0.156637\pi$
$349$			17.6528i			0.944935i	−0.881348	+	0.472468i	$0.843363\pi$
$350$		0			0
$351$		−	21.6141i		−	1.15368i
$352$		0			0
$353$	8.68102			0.462044			0.231022	−	0.972948i	$-0.425793\pi$
$353$	8.68102			0.462044			0.231022	+	0.972948i	$0.425793\pi$
$354$		0			0
$355$		−	8.13639i		−	0.431835i
$356$		0			0
$357$			8.04254i			0.425656i
$358$		0			0
$359$	−10.1399			−0.535164			−0.267582	−	0.963535i	$-0.586225\pi$
$359$	−10.1399			−0.535164			−0.267582	+	0.963535i	$0.586225\pi$
$360$		0			0
$361$	0.795505			0.0418687
$362$		0			0
$363$	−7.46368			−0.391742
$364$		0			0
$365$	−12.3698			−0.647467
$366$		0			0
$367$	−9.27643			−0.484226			−0.242113	−	0.970248i	$-0.577840\pi$
$367$	−9.27643			−0.484226			−0.242113	+	0.970248i	$0.577840\pi$
$368$		0			0
$369$	11.8458			0.616667
$370$		0			0
$371$	−42.7706			−2.22054
$372$		0			0
$373$	18.4449			0.955038			0.477519	−	0.878621i	$-0.341536\pi$
$373$	18.4449			0.955038			0.477519	+	0.878621i	$0.341536\pi$
$374$		0			0
$375$	−5.97779			−0.308692
$376$		0			0
$377$	−27.4263			−1.41253
$378$		0			0
$379$		−	33.1301i		−	1.70178i	−0.525346	−	0.850888i	$-0.676064\pi$
$379$		−	33.1301i		−	1.70178i	0.525346	−	0.850888i	$-0.323936\pi$
$380$		0			0
$381$			5.39323i			0.276304i
$382$		0			0
$383$	24.3382			1.24362			0.621812	−	0.783167i	$-0.286397\pi$
$383$	24.3382			1.24362			0.621812	+	0.783167i	$0.286397\pi$
$384$		0			0
$385$		−	0.979605i		−	0.0499253i
$386$		0			0
$387$		−	1.06848i		−	0.0543141i
$388$		0			0
$389$	−1.15612			−0.0586175			−0.0293087	−	0.999570i	$-0.509331\pi$
$389$	−1.15612			−0.0586175			−0.0293087	+	0.999570i	$0.509331\pi$
$390$		0			0
$391$	10.0320	−	7.03658i	0.507342	−	0.355855i
$392$		0			0
$393$	−3.82103			−0.192746
$394$		0			0
$395$	29.4249			1.48053
$396$		0			0
$397$		−	3.93528i		−	0.197506i	−0.995112	−	0.0987530i	$-0.968515\pi$
$397$		−	3.93528i		−	0.197506i	0.995112	−	0.0987530i	$-0.0314854\pi$
$398$		0			0
$399$			13.4300i			0.672341i
$400$		0			0
$401$			12.6228i			0.630353i	0.949033	+	0.315176i	$0.102064\pi$
$401$			12.6228i			0.630353i	−0.949033	+	0.315176i	$0.897936\pi$
$402$		0			0
$403$	37.1591			1.85103
$404$		0			0
$405$	−12.9730			−0.644636
$406$		0			0
$407$		−	0.611515i		−	0.0303117i
$408$		0			0
$409$	11.2510			0.556328			0.278164	−	0.960534i	$-0.410274\pi$
$409$	11.2510			0.556328			0.278164	+	0.960534i	$0.410274\pi$
$410$		0			0
$411$		−	14.8764i		−	0.733798i
$412$		0			0
$413$	−9.76479			−0.480494
$414$		0			0
$415$		−	18.7988i		−	0.922799i
$416$		0			0
$417$	1.12325			0.0550060
$418$		0			0
$419$			0.916971i			0.0447970i	0.999749	+	0.0223985i	$0.00713025\pi$
$419$			0.916971i			0.0447970i	−0.999749	+	0.0223985i	$0.992870\pi$
$420$		0			0
$421$	8.43308			0.411003			0.205502	−	0.978657i	$-0.434117\pi$
$421$	8.43308			0.411003			0.205502	+	0.978657i	$0.434117\pi$
$422$		0			0
$423$		−	24.3539i		−	1.18413i
$424$		0			0
$425$			3.99412i			0.193743i
$426$		0			0
$427$	32.1794			1.55727
$428$		0			0
$429$	0.321839			0.0155385
$430$		0			0
$431$	9.74221			0.469266			0.234633	−	0.972084i	$-0.424611\pi$
$431$	9.74221			0.469266			0.234633	+	0.972084i	$0.424611\pi$
$432$		0			0
$433$			21.7831i			1.04683i	0.852078	+	0.523415i	$0.175342\pi$
$433$			21.7831i			1.04683i	−0.852078	+	0.523415i	$0.824658\pi$
$434$		0			0
$435$		−	8.30006i		−	0.397958i
$436$		0			0
$437$	16.7522	−	11.7502i	0.801367	−	0.562087i
$438$		0			0
$439$		−	3.70641i		−	0.176898i	−0.996081	−	0.0884488i	$-0.971809\pi$
$439$		−	3.70641i		−	0.176898i	0.996081	−	0.0884488i	$-0.0281910\pi$
$440$		0			0
$441$	−36.8006			−1.75241
$442$		0			0
$443$	−0.424823			−0.0201840			−0.0100920	−	0.999949i	$-0.503212\pi$
$443$	−0.424823			−0.0201840			−0.0100920	+	0.999949i	$0.503212\pi$
$444$		0			0
$445$			29.4249i			1.39487i
$446$		0			0
$447$	9.78200			0.462673
$448$		0			0
$449$	−27.8873			−1.31608			−0.658041	−	0.752982i	$-0.728615\pi$
$449$	−27.8873			−1.31608			−0.658041	+	0.752982i	$0.728615\pi$
$450$		0			0
$451$			0.384795i			0.0181193i
$452$		0			0
$453$		−	14.6548i		−	0.688543i
$454$		0			0
$455$		−	68.2635i		−	3.20024i
$456$		0			0
$457$			25.6280i			1.19883i	0.800440	+	0.599413i	$0.204599\pi$
$457$			25.6280i			1.19883i	−0.800440	+	0.599413i	$0.795401\pi$
$458$		0			0
$459$			9.60883i			0.448502i
$460$		0			0
$461$			24.0316i			1.11927i	0.828741	+	0.559633i	$0.189058\pi$
$461$			24.0316i			1.11927i	−0.828741	+	0.559633i	$0.810942\pi$
$462$		0			0
$463$			39.7555i			1.84759i	0.382884	+	0.923796i	$0.374931\pi$
$463$			39.7555i			1.84759i	−0.382884	+	0.923796i	$0.625069\pi$
$464$		0			0
$465$			11.2455i			0.521497i
$466$		0			0
$467$		−	31.4758i		−	1.45653i	−0.685298	−	0.728263i	$-0.740328\pi$
$467$		−	31.4758i		−	1.45653i	0.685298	−	0.728263i	$-0.259672\pi$
$468$		0			0
$469$			0.875412i			0.0404228i
$470$		0			0
$471$	−10.3219			−0.475607
$472$		0			0
$473$	0.0347083			0.00159589
$474$		0			0
$475$			6.66967i			0.306025i
$476$		0			0
$477$	−23.4239			−1.07251
$478$		0			0
$479$	39.1947			1.79085			0.895426	−	0.445210i	$-0.146871\pi$
$479$	39.1947			1.79085			0.895426	+	0.445210i	$0.146871\pi$
$480$		0			0
$481$		−	42.6133i		−	1.94300i
$482$		0			0
$483$	−8.66845	−	12.3586i	−0.394428	−	0.562336i
$484$		0			0
$485$			8.76854i			0.398159i
$486$		0			0
$487$			33.4599i			1.51621i	0.652131	+	0.758107i	$0.273875\pi$
$487$			33.4599i			1.51621i	−0.652131	+	0.758107i	$0.726125\pi$
$488$		0			0
$489$	−14.7449			−0.666788
$490$		0			0
$491$	2.65811			0.119959			0.0599794	−	0.998200i	$-0.480897\pi$
$491$	2.65811			0.119959			0.0599794	+	0.998200i	$0.480897\pi$
$492$		0			0
$493$	12.1927			0.549133
$494$		0			0
$495$		−	0.536493i		−	0.0241136i
$496$		0			0
$497$		−	14.7242i		−	0.660470i
$498$		0			0
$499$	−14.3246			−0.641257			−0.320629	−	0.947205i	$-0.603894\pi$
$499$	−14.3246			−0.641257			−0.320629	+	0.947205i	$0.603894\pi$
$500$		0			0
$501$			1.61199i			0.0720182i
$502$		0			0
$503$	−5.40637			−0.241058			−0.120529	−	0.992710i	$-0.538459\pi$
$503$	−5.40637			−0.241058			−0.120529	+	0.992710i	$0.538459\pi$
$504$		0			0
$505$			33.9067i			1.50883i
$506$		0			0
$507$	13.6011			0.604045
$508$		0			0
$509$		−	6.82300i		−	0.302424i	−0.988501	−	0.151212i	$-0.951682\pi$
$509$		−	6.82300i		−	0.302424i	0.988501	−	0.151212i	$-0.0483177\pi$
$510$		0			0
$511$	−22.3853			−0.990269
$512$		0			0
$513$			16.0455i			0.708426i
$514$		0			0
$515$	−12.6092			−0.555627
$516$		0			0
$517$	0.791104			0.0347927
$518$		0			0
$519$		−	14.5436i		−	0.638392i
$520$		0			0
$521$		−	21.0437i		−	0.921941i	−0.887415	−	0.460970i	$-0.847501\pi$
$521$		−	21.0437i		−	0.921941i	0.887415	−	0.460970i	$-0.152499\pi$
$522$		0			0
$523$		−	10.8726i		−	0.475425i	−0.971336	−	0.237712i	$-0.923602\pi$
$523$		−	10.8726i		−	0.475425i	0.971336	−	0.237712i	$-0.0763975\pi$
$524$		0			0
$525$	4.92041			0.214744
$526$		0			0
$527$	−16.5195			−0.719602
$528$		0			0
$529$	−7.83158	+	21.6256i	−0.340504	+	0.940243i
$530$		0			0
$531$	−5.34781			−0.232075
$532$		0			0
$533$			26.8143i			1.16146i
$534$		0			0
$535$			39.6433i			1.71393i
$536$		0			0
$537$	10.4629			0.451508
$538$		0			0
$539$		−	1.19542i		−	0.0514904i
$540$		0			0
$541$		−	38.0421i		−	1.63556i	−0.575533	−	0.817779i	$-0.695205\pi$
$541$		−	38.0421i		−	1.63556i	0.575533	−	0.817779i	$-0.304795\pi$
$542$		0			0
$543$	−1.62736			−0.0698365
$544$		0			0
$545$	40.0659			1.71623
$546$		0			0
$547$	−0.707120			−0.0302343			−0.0151171	−	0.999886i	$-0.504812\pi$
$547$	−0.707120			−0.0302343			−0.0151171	+	0.999886i	$0.504812\pi$
$548$		0			0
$549$	17.6234			0.752150
$550$		0			0
$551$	20.3603			0.867377
$552$		0			0
$553$	53.2493			2.26439
$554$		0			0
$555$	12.8961			0.547409
$556$		0			0
$557$	18.9903			0.804643			0.402322	−	0.915498i	$-0.368203\pi$
$557$	18.9903			0.804643			0.402322	+	0.915498i	$0.368203\pi$
$558$		0			0
$559$	2.41864			0.102297
$560$		0			0
$561$	−0.143078			−0.00604074
$562$		0			0
$563$			41.5571i			1.75142i	0.482836	+	0.875711i	$0.339607\pi$
$563$			41.5571i			1.75142i	−0.482836	+	0.875711i	$0.660393\pi$
$564$		0			0
$565$			25.1018i			1.05604i
$566$		0			0
$567$	−23.4769			−0.985939
$568$		0			0
$569$			19.7539i			0.828127i	0.910248	+	0.414064i	$0.135891\pi$
$569$			19.7539i			0.828127i	−0.910248	+	0.414064i	$0.864109\pi$
$570$		0			0
$571$			23.7036i			0.991965i	0.868332	+	0.495983i	$0.165192\pi$
$571$			23.7036i			0.991965i	−0.868332	+	0.495983i	$0.834808\pi$
$572$		0			0
$573$	7.40017			0.309147
$574$		0			0
$575$	−4.30497	−	6.13759i	−0.179530	−	0.255955i
$576$		0			0
$577$	−3.13883			−0.130671			−0.0653355	−	0.997863i	$-0.520812\pi$
$577$	−3.13883			−0.130671			−0.0653355	+	0.997863i	$0.520812\pi$
$578$		0			0
$579$	−5.86568			−0.243769
$580$		0			0
$581$		−	34.0197i		−	1.41138i
$582$		0			0
$583$		−	0.760894i		−	0.0315130i
$584$		0			0
$585$		−	37.3854i		−	1.54570i
$586$		0			0
$587$	−21.3071			−0.879438			−0.439719	−	0.898135i	$-0.644922\pi$
$587$	−21.3071			−0.879438			−0.439719	+	0.898135i	$0.644922\pi$
$588$		0			0
$589$	−27.5855			−1.13664
$590$		0			0
$591$		−	16.7378i		−	0.688500i
$592$		0			0
$593$	35.1731			1.44439			0.722194	−	0.691691i	$-0.243134\pi$
$593$	35.1731			1.44439			0.722194	+	0.691691i	$0.243134\pi$
$594$		0			0
$595$			30.3474i			1.24412i
$596$		0			0
$597$	−16.9710			−0.694576
$598$		0			0
$599$			20.9654i			0.856623i	0.903631	+	0.428311i	$0.140891\pi$
$599$			20.9654i			0.856623i	−0.903631	+	0.428311i	$0.859109\pi$
$600$		0			0
$601$	20.1022			0.819988			0.409994	−	0.912088i	$-0.365531\pi$
$601$	20.1022			0.819988			0.409994	+	0.912088i	$0.365531\pi$
$602$		0			0
$603$			0.479431i			0.0195239i
$604$		0			0
$605$	−28.1632			−1.14500
$606$		0			0
$607$		−	20.9566i		−	0.850603i	−0.905052	−	0.425301i	$-0.860168\pi$
$607$		−	20.9566i		−	0.850603i	0.905052	−	0.425301i	$-0.139832\pi$
$608$		0			0
$609$		−	15.0204i		−	0.608657i
$610$		0			0
$611$	55.1279			2.23023
$612$		0			0
$613$	33.0144			1.33344			0.666719	−	0.745309i	$-0.267698\pi$
$613$	33.0144			1.33344			0.666719	+	0.745309i	$0.267698\pi$
$614$		0			0
$615$	−8.11485			−0.327222
$616$		0			0
$617$		−	26.2869i		−	1.05827i	−0.848537	−	0.529136i	$-0.822516\pi$
$617$		−	26.2869i		−	1.05827i	0.848537	−	0.529136i	$-0.177484\pi$
$618$		0			0
$619$			13.6417i			0.548305i	0.961686	+	0.274152i	$0.0883972\pi$
$619$			13.6417i			0.548305i	−0.961686	+	0.274152i	$0.911603\pi$
$620$		0			0
$621$	−10.3566	−	14.7655i	−0.415598	−	0.592517i
$622$		0			0
$623$			53.2493i			2.13339i
$624$		0			0
$625$	−30.3724			−1.21490
$626$		0			0
$627$	−0.238921			−0.00954159
$628$		0			0
$629$			18.9443i			0.755358i
$630$		0			0
$631$	−0.101515			−0.00404124			−0.00202062	−	0.999998i	$-0.500643\pi$
$631$	−0.101515			−0.00404124			−0.00202062	+	0.999998i	$0.500643\pi$
$632$		0			0
$633$	0.748562			0.0297527
$634$		0			0
$635$			20.3506i			0.807590i
$636$		0			0
$637$		−	83.3025i		−	3.30057i
$638$		0			0
$639$		−	8.06389i		−	0.319002i
$640$		0			0
$641$		−	22.0128i		−	0.869454i	−0.900562	−	0.434727i	$-0.856845\pi$
$641$		−	22.0128i		−	0.869454i	0.900562	−	0.434727i	$-0.143155\pi$
$642$		0			0
$643$		−	20.8023i		−	0.820364i	−0.912004	−	0.410182i	$-0.865465\pi$
$643$		−	20.8023i		−	0.820364i	0.912004	−	0.410182i	$-0.134535\pi$
$644$		0			0
$645$			0.731955i			0.0288207i
$646$		0			0
$647$			13.2901i			0.522489i	0.965273	+	0.261244i	$0.0841328\pi$
$647$			13.2901i			0.522489i	−0.965273	+	0.261244i	$0.915867\pi$
$648$		0			0
$649$		−	0.173717i		−	0.00681897i
$650$		0			0
$651$			20.3506i			0.797604i
$652$		0			0
$653$		−	38.7635i		−	1.51693i	−0.651713	−	0.758466i	$-0.725949\pi$
$653$		−	38.7635i		−	1.51693i	0.651713	−	0.758466i	$-0.274051\pi$
$654$		0			0
$655$	−14.4181			−0.563364
$656$		0			0
$657$	−12.2596			−0.478293
$658$		0			0
$659$			12.0441i			0.469173i	0.972095	+	0.234586i	$0.0753736\pi$
$659$			12.0441i			0.469173i	−0.972095	+	0.234586i	$0.924626\pi$
$660$		0			0
$661$	41.4972			1.61405			0.807026	−	0.590515i	$-0.201076\pi$
$661$	41.4972			1.61405			0.807026	+	0.590515i	$0.201076\pi$
$662$		0			0
$663$	−9.97032			−0.387215
$664$		0			0
$665$			50.6763i			1.96514i
$666$		0			0
$667$	−18.7360	+	13.1416i	−0.725462	+	0.508846i
$668$		0			0
$669$			13.3024i			0.514300i
$670$		0			0
$671$			0.572475i			0.0221001i
$672$		0			0
$673$	12.9027			0.497362			0.248681	−	0.968585i	$-0.420003\pi$
$673$	12.9027			0.497362			0.248681	+	0.968585i	$0.420003\pi$
$674$		0			0
$675$	5.87867			0.226270
$676$		0			0
$677$	5.39177			0.207223			0.103611	−	0.994618i	$-0.466960\pi$
$677$	5.39177			0.207223			0.103611	+	0.994618i	$0.466960\pi$
$678$		0			0
$679$			15.8682i			0.608964i
$680$		0			0
$681$			15.1219i			0.579472i
$682$		0			0
$683$	−18.2981			−0.700159			−0.350079	−	0.936720i	$-0.613845\pi$
$683$	−18.2981			−0.700159			−0.350079	+	0.936720i	$0.613845\pi$
$684$		0			0
$685$		−	56.1340i		−	2.14477i
$686$		0			0
$687$	−7.13937			−0.272384
$688$		0			0
$689$		−	53.0227i		−	2.02000i
$690$		0			0
$691$	41.4679			1.57751			0.788757	−	0.614705i	$-0.210725\pi$
$691$	41.4679			1.57751			0.788757	+	0.614705i	$0.210725\pi$
$692$		0			0
$693$		−	0.970876i		−	0.0368805i
$694$		0			0
$695$	4.23844			0.160773
$696$		0			0
$697$		−	11.9207i		−	0.451527i
$698$		0			0
$699$	6.51978			0.246601
$700$		0			0
$701$	44.3405			1.67472			0.837359	−	0.546654i	$-0.184099\pi$
$701$	44.3405			1.67472			0.837359	+	0.546654i	$0.184099\pi$
$702$		0			0
$703$			31.6345i			1.19312i
$704$		0			0
$705$			16.6834i			0.628333i
$706$		0			0
$707$			61.3599i			2.30768i
$708$		0			0
$709$	−21.8376			−0.820128			−0.410064	−	0.912057i	$-0.634494\pi$
$709$	−21.8376			−0.820128			−0.410064	+	0.912057i	$0.634494\pi$
$710$		0			0
$711$	29.1627			1.09369
$712$		0			0
$713$	25.3848	−	17.8052i	0.950669	−	0.666809i
$714$		0			0
$715$	1.21442			0.0454166
$716$		0			0
$717$			4.61790i			0.172459i
$718$		0			0
$719$		−	4.46461i		−	0.166502i	−0.996529	−	0.0832509i	$-0.973470\pi$
$719$		−	4.46461i		−	0.166502i	0.996529	−	0.0832509i	$-0.0265303\pi$
$720$		0			0
$721$	−22.8185			−0.849805
$722$		0			0
$723$			2.78779i			0.103679i
$724$		0			0
$725$		−	7.45950i		−	0.277039i
$726$		0			0
$727$	18.4390			0.683863			0.341932	−	0.939725i	$-0.388919\pi$
$727$	18.4390			0.683863			0.341932	+	0.939725i	$0.388919\pi$
$728$		0			0
$729$	−5.19769			−0.192507
$730$		0			0
$731$	−1.07524			−0.0397690
$732$		0			0
$733$	29.9637			1.10673			0.553367	−	0.832937i	$-0.313343\pi$
$733$	29.9637			1.10673			0.553367	+	0.832937i	$0.313343\pi$
$734$		0			0
$735$	25.2099			0.929882
$736$		0			0
$737$	−0.0155737			−0.000573664
$738$		0			0
$739$	−2.27573			−0.0837141			−0.0418570	−	0.999124i	$-0.513327\pi$
$739$	−2.27573			−0.0837141			−0.0418570	+	0.999124i	$0.513327\pi$
$740$		0			0
$741$	−16.6491			−0.611622
$742$		0			0
$743$	14.4903			0.531599			0.265800	−	0.964028i	$-0.414364\pi$
$743$	14.4903			0.531599			0.265800	+	0.964028i	$0.414364\pi$
$744$		0			0
$745$	36.9110			1.35232
$746$		0			0
$747$		−	18.6313i		−	0.681684i
$748$		0			0
$749$			71.7412i			2.62137i
$750$		0			0
$751$	−38.0771			−1.38945			−0.694726	−	0.719275i	$-0.744474\pi$
$751$	−38.0771			−1.38945			−0.694726	+	0.719275i	$0.744474\pi$
$752$		0			0
$753$			6.71721i			0.244789i
$754$		0			0
$755$		−	55.2979i		−	2.01250i
$756$		0			0
$757$	12.3254			0.447974			0.223987	−	0.974592i	$-0.428093\pi$
$757$	12.3254			0.447974			0.223987	+	0.974592i	$0.428093\pi$
$758$		0			0
$759$	0.219861	−	0.154213i	0.00798044	−	0.00559757i
$760$		0			0
$761$	−22.3706			−0.810935			−0.405467	−	0.914110i	$-0.632891\pi$
$761$	−22.3706			−0.810935			−0.405467	+	0.914110i	$0.632891\pi$
$762$		0			0
$763$	72.5060			2.62489
$764$		0			0
$765$			16.6201i			0.600903i
$766$		0			0
$767$		−	12.1054i		−	0.437101i
$768$		0			0
$769$			31.1276i			1.12249i	0.827650	+	0.561245i	$0.189677\pi$
$769$			31.1276i			1.12249i	−0.827650	+	0.561245i	$0.810323\pi$
$770$		0			0
$771$	2.35941			0.0849721
$772$		0			0
$773$	47.7339			1.71687			0.858434	−	0.512924i	$-0.171438\pi$
$773$	47.7339			1.71687			0.858434	+	0.512924i	$0.171438\pi$
$774$		0			0
$775$			10.1066i			0.363041i
$776$		0			0
$777$	23.3377			0.837235
$778$		0			0
$779$		−	19.9059i		−	0.713204i
$780$		0			0
$781$	0.261945			0.00937312
$782$		0			0
$783$		−	17.9456i		−	0.641324i
$784$		0			0
$785$	−38.9482			−1.39012
$786$		0			0
$787$			42.1873i			1.50381i	0.659269	+	0.751907i	$0.270866\pi$
$787$			42.1873i			1.50381i	−0.659269	+	0.751907i	$0.729134\pi$
$788$		0			0
$789$	6.61434			0.235477
$790$		0			0
$791$			45.4259i			1.61516i
$792$		0			0
$793$			39.8927i			1.41663i
$794$		0			0
$795$	16.0463			0.569104
$796$		0			0
$797$	16.0156			0.567303			0.283651	−	0.958927i	$-0.408454\pi$
$797$	16.0156			0.567303			0.283651	+	0.958927i	$0.408454\pi$
$798$		0			0
$799$	−24.5078			−0.867023
$800$		0			0
$801$			29.1627i			1.03041i
$802$		0			0
$803$		−	0.398237i		−	0.0140535i
$804$		0			0
$805$	−32.7092	−	46.6335i	−1.15285	−	1.64362i
$806$		0			0
$807$		−	5.15879i		−	0.181598i
$808$		0			0
$809$	−24.0923			−0.847040			−0.423520	−	0.905887i	$-0.639206\pi$
$809$	−24.0923			−0.847040			−0.423520	+	0.905887i	$0.639206\pi$
$810$		0			0
$811$	0.407214			0.0142992			0.00714960	−	0.999974i	$-0.497724\pi$
$811$	0.407214			0.0142992			0.00714960	+	0.999974i	$0.497724\pi$
$812$		0			0
$813$		−	2.10502i		−	0.0738263i
$814$		0			0
$815$	−55.6379			−1.94891
$816$		0			0
$817$	−1.79551			−0.0628168
$818$		0			0
$819$		−	67.6552i		−	2.36407i
$820$		0			0
$821$		−	30.6108i		−	1.06832i	−0.845382	−	0.534162i	$-0.820627\pi$
$821$		−	30.6108i		−	1.06832i	0.845382	−	0.534162i	$-0.179373\pi$
$822$		0			0
$823$		−	25.2527i		−	0.880253i	−0.897936	−	0.440126i	$-0.854934\pi$
$823$		−	25.2527i		−	0.880253i	0.897936	−	0.440126i	$-0.145066\pi$
$824$		0			0
$825$			0.0875347i			0.00304757i
$826$		0			0
$827$		−	21.6754i		−	0.753729i	−0.926268	−	0.376864i	$-0.877002\pi$
$827$		−	21.6754i		−	0.753729i	0.926268	−	0.376864i	$-0.122998\pi$
$828$		0			0
$829$		−	5.58090i		−	0.193833i	−0.995293	−	0.0969164i	$-0.969102\pi$
$829$		−	5.58090i		−	0.193833i	0.995293	−	0.0969164i	$-0.0308979\pi$
$830$		0			0
$831$		−	3.23984i		−	0.112389i
$832$		0			0
$833$			37.0332i			1.28312i
$834$		0			0
$835$			6.08261i			0.210497i
$836$		0			0
$837$			24.3139i			0.840413i
$838$		0			0
$839$	−3.36258			−0.116089			−0.0580446	−	0.998314i	$-0.518487\pi$
$839$	−3.36258			−0.116089			−0.0580446	+	0.998314i	$0.518487\pi$
$840$		0			0
$841$	6.22865			0.214781
$842$		0			0
$843$			17.6698i			0.608580i
$844$		0			0
$845$	51.3219			1.76553
$846$		0			0
$847$	−50.9661			−1.75122
$848$		0			0
$849$		−	11.1033i		−	0.381065i
$850$		0			0
$851$	−20.4186	−	29.1108i	−0.699942	−	0.997906i
$852$		0			0
$853$		−	17.2474i		−	0.590540i	−0.955414	−	0.295270i	$-0.904590\pi$
$853$		−	17.2474i		−	0.590540i	0.955414	−	0.295270i	$-0.0954096\pi$
$854$		0			0
$855$			27.7535i			0.949149i
$856$		0			0
$857$	27.2287			0.930113			0.465057	−	0.885281i	$-0.346034\pi$
$857$	27.2287			0.930113			0.465057	+	0.885281i	$0.346034\pi$
$858$		0			0
$859$	29.7496			1.01504			0.507522	−	0.861639i	$-0.330562\pi$
$859$	29.7496			1.01504			0.507522	+	0.861639i	$0.330562\pi$
$860$		0			0
$861$	−14.6852			−0.500470
$862$		0			0
$863$			8.14618i			0.277299i	0.990342	+	0.138650i	$0.0442761\pi$
$863$			8.14618i			0.277299i	−0.990342	+	0.138650i	$0.955724\pi$
$864$		0			0
$865$		−	54.8782i		−	1.86591i
$866$		0			0
$867$	−7.10949			−0.241451
$868$		0			0
$869$			0.947311i			0.0321353i
$870$		0			0
$871$	−1.08525			−0.0367722
$872$		0			0
$873$			8.69040i			0.294125i
$874$		0			0
$875$	−40.8197			−1.37996
$876$		0			0
$877$		−	30.0462i		−	1.01459i	−0.861773	−	0.507294i	$-0.830646\pi$
$877$		−	30.0462i		−	1.01459i	0.861773	−	0.507294i	$-0.169354\pi$
$878$		0			0
$879$	18.9025			0.637566
$880$		0			0
$881$			32.5854i			1.09783i	0.835878	+	0.548915i	$0.184959\pi$
$881$			32.5854i			1.09783i	−0.835878	+	0.548915i	$0.815041\pi$
$882$		0			0
$883$	16.2167			0.545737			0.272869	−	0.962051i	$-0.412028\pi$
$883$	16.2167			0.545737			0.272869	+	0.962051i	$0.412028\pi$
$884$		0			0
$885$	3.66347			0.123146
$886$		0			0
$887$			38.7621i			1.30150i	0.759290	+	0.650752i	$0.225546\pi$
$887$			38.7621i			1.30150i	−0.759290	+	0.650752i	$0.774454\pi$
$888$		0			0
$889$			36.8279i			1.23517i
$890$		0			0
$891$		−	0.417657i		−	0.0139920i
$892$		0			0
$893$	−40.9248			−1.36950
$894$		0			0
$895$	39.4804			1.31969
$896$		0			0
$897$	15.3209	−	10.7463i	0.511551	−	0.358808i
$898$		0			0
$899$	30.8522			1.02898
$900$		0			0
$901$			23.5719i			0.785294i
$902$		0			0
$903$			1.32460i			0.0440798i
$904$		0			0
$905$	−6.14061			−0.204121
$906$		0			0
$907$			1.31020i			0.0435044i	0.999763	+	0.0217522i	$0.00692449\pi$
$907$			1.31020i			0.0435044i	−0.999763	+	0.0217522i	$0.993076\pi$
$908$		0			0
$909$			33.6045i			1.11459i
$910$		0			0
$911$	3.87640			0.128431			0.0642155	−	0.997936i	$-0.479545\pi$
$911$	3.87640			0.128431			0.0642155	+	0.997936i	$0.479545\pi$
$912$		0			0
$913$	0.605214			0.0200297
$914$		0			0
$915$	−12.0728			−0.399114
$916$		0			0
$917$	−26.0921			−0.861637
$918$		0			0
$919$	−14.3573			−0.473604			−0.236802	−	0.971558i	$-0.576099\pi$
$919$	−14.3573			−0.473604			−0.236802	+	0.971558i	$0.576099\pi$
$920$		0			0
$921$	−16.7038			−0.550409
$922$		0			0
$923$	18.2536			0.600823
$924$		0			0
$925$	11.5901			0.381080
$926$		0			0
$927$	−12.4968			−0.410450
$928$		0			0
$929$	−23.7498			−0.779207			−0.389604	−	0.920983i	$-0.627388\pi$
$929$	−23.7498			−0.779207			−0.389604	+	0.920983i	$0.627388\pi$
$930$		0			0
$931$			61.8407i			2.02675i
$932$		0			0
$933$		−	8.09137i		−	0.264900i
$934$		0			0
$935$	−0.539884			−0.0176561
$936$		0			0
$937$			9.22456i			0.301353i	0.988583	+	0.150677i	$0.0481452\pi$
$937$			9.22456i			0.301353i	−0.988583	+	0.150677i	$0.951855\pi$
$938$		0			0
$939$		−	14.7513i		−	0.481391i
$940$		0			0
$941$	−36.6211			−1.19381			−0.596906	−	0.802311i	$-0.703604\pi$
$941$	−36.6211			−1.19381			−0.596906	+	0.802311i	$0.703604\pi$
$942$		0			0
$943$	12.8484	+	18.3179i	0.418401	+	0.596514i
$944$		0			0
$945$	44.6662			1.45299
$946$		0			0
$947$	59.7109			1.94034			0.970172	−	0.242418i	$-0.0779404\pi$
$947$	59.7109			1.94034			0.970172	+	0.242418i	$0.0779404\pi$
$948$		0			0
$949$		−	27.7511i		−	0.900838i
$950$		0			0
$951$		−	0.0751282i		−	0.00243620i
$952$		0			0
$953$			24.6344i			0.797986i	0.916954	+	0.398993i	$0.130640\pi$
$953$			24.6344i			0.797986i	−0.916954	+	0.398993i	$0.869360\pi$
$954$		0			0
$955$	27.9236			0.903585
$956$		0			0
$957$	0.267214			0.00863781
$958$		0			0
$959$		−	101.584i		−	3.28032i
$960$		0			0
$961$	−10.8006			−0.348407
$962$		0			0
$963$			39.2900i			1.26610i
$964$		0			0
$965$	−22.1333			−0.712498
$966$		0			0
$967$		−	5.95725i		−	0.191572i	−0.995402	−	0.0957862i	$-0.969463\pi$
$967$		−	5.95725i		−	0.191572i	0.995402	−	0.0957862i	$-0.0305365\pi$
$968$		0			0
$969$	7.40159			0.237773
$970$		0			0
$971$			26.3386i			0.845246i	0.906306	+	0.422623i	$0.138891\pi$
$971$			26.3386i			0.845246i	−0.906306	+	0.422623i	$0.861109\pi$
$972$		0			0
$973$	7.67019			0.245895
$974$		0			0
$975$			6.09983i			0.195351i
$976$		0			0
$977$			7.58482i			0.242660i	0.992612	+	0.121330i	$0.0387159\pi$
$977$			7.58482i			0.242660i	−0.992612	+	0.121330i	$0.961284\pi$
$978$		0			0
$979$	−0.947311			−0.0302762
$980$		0			0
$981$	39.7088			1.26781
$982$		0			0
$983$	−38.8327			−1.23857			−0.619285	−	0.785166i	$-0.712577\pi$
$983$	−38.8327			−1.23857			−0.619285	+	0.785166i	$0.712577\pi$
$984$		0			0
$985$		−	63.1578i		−	2.01237i
$986$		0			0
$987$			30.1915i			0.961005i
$988$		0			0
$989$	1.65227	−	1.15892i	0.0525390	−	0.0368514i
$990$		0			0
$991$		−	31.9580i		−	1.01518i	−0.861599	−	0.507589i	$-0.830537\pi$
$991$		−	31.9580i		−	1.01518i	0.861599	−	0.507589i	$-0.169463\pi$
$992$		0			0
$993$	−10.7713			−0.341818
$994$		0			0
$995$	−64.0377			−2.03013
$996$		0			0
$997$		−	29.7509i		−	0.942220i	−0.882074	−	0.471110i	$-0.843853\pi$
$997$		−	29.7509i		−	0.942220i	0.882074	−	0.471110i	$-0.156147\pi$
$998$		0			0
$999$	27.8827			0.882171

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.2.h.c.735.14	yes	32
4.3	odd	2	inner	1472.2.h.c.735.19	yes	32
8.3	odd	2	inner	1472.2.h.c.735.13	✓	32
8.5	even	2	inner	1472.2.h.c.735.20	yes	32
23.22	odd	2	inner	1472.2.h.c.735.16	yes	32
92.91	even	2	inner	1472.2.h.c.735.17	yes	32
184.45	odd	2	inner	1472.2.h.c.735.18	yes	32
184.91	even	2	inner	1472.2.h.c.735.15	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1472.2.h.c.735.13	✓	32	8.3	odd	2	inner
1472.2.h.c.735.14	yes	32	1.1	even	1	trivial
1472.2.h.c.735.15	yes	32	184.91	even	2	inner
1472.2.h.c.735.16	yes	32	23.22	odd	2	inner
1472.2.h.c.735.17	yes	32	92.91	even	2	inner
1472.2.h.c.735.18	yes	32	184.45	odd	2	inner
1472.2.h.c.735.19	yes	32	4.3	odd	2	inner
1472.2.h.c.735.20	yes	32	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1472.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.678936 q^{3} -2.56187 q^{5} -4.63615 q^{7} -2.53905 q^{9} -0.0824776i q^{11} -5.74743i q^{13} +1.73935 q^{15} +2.55509i q^{17} +4.26667i q^{19} +3.14765 q^{21} +(-2.75394 - 3.92629i) q^{23} +1.56320 q^{25} +3.76066 q^{27} -4.77193i q^{29} +6.46534i q^{31} +0.0559970i q^{33} +11.8772 q^{35} +7.41432 q^{37} +3.90214i q^{39} -4.66545 q^{41} +0.420821i q^{43} +6.50472 q^{45} +9.59174i q^{47} +14.4939 q^{49} -1.73474i q^{51} +9.22547 q^{53} +0.211297i q^{55} -2.89680i q^{57} +2.10623 q^{59} -6.94097 q^{61} +11.7714 q^{63} +14.7242i q^{65} -0.188823i q^{67} +(1.86975 + 2.66570i) q^{69} +3.17595i q^{71} +4.82843 q^{73} -1.06131 q^{75} +0.382378i q^{77} -11.4857 q^{79} +5.06389 q^{81} +7.33792i q^{83} -6.54582i q^{85} +3.23984i q^{87} -11.4857i q^{89} +26.6459i q^{91} -4.38956i q^{93} -10.9307i q^{95} -3.42270i q^{97} +0.209414i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 40 q^{9} + 32 q^{25} + 8 q^{41} + 48 q^{49} - 104 q^{73} + 112 q^{81}+O(q^{100}) \) 32 * q + 40 * q^9 + 32 * q^25 + 8 * q^41 + 48 * q^49 - 104 * q^73 + 112 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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