LMFDB - Embedded newform 1472.2.a.x.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,2,Mod(1,1472)] 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("1472.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1472.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,4,0,2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$11.7539791775$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 736)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.34292 q^{3} +1.14637 q^{5} +1.14637 q^{7} +8.17513 q^{9} +3.14637 q^{11} -2.48929 q^{13} +3.83221 q^{15} +0.853635 q^{17} -5.66442 q^{19} +3.83221 q^{21} -1.00000 q^{23} -3.68585 q^{25} +17.3001 q^{27} +6.88240 q^{29} -8.32150 q^{31} +10.5181 q^{33} +1.31415 q^{35} -8.81079 q^{37} -8.32150 q^{39} -6.48929 q^{41} -2.97858 q^{43} +9.37169 q^{45} -2.94981 q^{47} -5.68585 q^{49} +2.85363 q^{51} -0.393115 q^{53} +3.60688 q^{55} -18.9357 q^{57} +5.70727 q^{59} +14.3503 q^{61} +9.37169 q^{63} -2.85363 q^{65} +7.93260 q^{67} -3.34292 q^{69} +0.657077 q^{71} -1.90383 q^{73} -12.3215 q^{75} +3.60688 q^{77} +16.0575 q^{79} +33.3074 q^{81} +2.75325 q^{83} +0.978577 q^{85} +23.0073 q^{87} -15.7648 q^{89} -2.85363 q^{91} -27.8181 q^{93} -6.49350 q^{95} -14.8108 q^{97} +25.7220 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{3} + 2 q^{5} + 2 q^{7} + 5 q^{9} + 8 q^{11} - 2 q^{15} + 4 q^{17} + 10 q^{19} - 2 q^{21} - 3 q^{23} + q^{25} + 16 q^{27} + 4 q^{29} - 4 q^{31} + 6 q^{33} + 16 q^{35} + 2 q^{37} - 4 q^{39} - 12 q^{41}+ \cdots + 14 q^{99}+O(q^{100}) $ 3 * q + 4 * q^3 + 2 * q^5 + 2 * q^7 + 5 * q^9 + 8 * q^11 - 2 * q^15 + 4 * q^17 + 10 * q^19 - 2 * q^21 - 3 * q^23 + q^25 + 16 * q^27 + 4 * q^29 - 4 * q^31 + 6 * q^33 + 16 * q^35 + 2 * q^37 - 4 * q^39 - 12 * q^41 + 6 * q^43 + 4 * q^45 - 12 * q^47 - 5 * q^49 + 10 * q^51 + 8 * q^53 + 20 * q^55 - 12 * q^57 + 20 * q^59 + 4 * q^61 + 4 * q^63 - 10 * q^65 + 4 * q^67 - 4 * q^69 + 8 * q^71 - 4 * q^73 - 16 * q^75 + 20 * q^77 + 12 * q^79 + 31 * q^81 + 16 * q^83 - 12 * q^85 + 36 * q^87 - 14 * q^89 - 10 * q^91 - 22 * q^93 - 4 * q^95 - 16 * q^97 + 14 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.34292	1.93004	0.965019	−	0.262181i	$-0.0844417\pi$
$3$	3.34292	1.93004	0.965019	+	0.262181i	$0.0844417\pi$
$4$	0	0
$5$	1.14637	0.512670	0.256335	−	0.966588i	$-0.417485\pi$
$5$	1.14637	0.512670	0.256335	+	0.966588i	$0.417485\pi$
$6$	0	0
$7$	1.14637	0.433285	0.216643	−	0.976251i	$-0.430489\pi$
$7$	1.14637	0.433285	0.216643	+	0.976251i	$0.430489\pi$
$8$	0	0
$9$	8.17513	2.72504
$10$	0	0
$11$	3.14637	0.948665	0.474332	−	0.880346i	$-0.342689\pi$
$11$	3.14637	0.948665	0.474332	+	0.880346i	$0.342689\pi$
$12$	0	0
$13$	−2.48929	−0.690404	−0.345202	−	0.938528i	$-0.612190\pi$
$13$	−2.48929	−0.690404	−0.345202	+	0.938528i	$0.612190\pi$
$14$	0	0
$15$	3.83221	0.989473
$16$	0	0
$17$	0.853635	0.207037	0.103518	−	0.994628i	$-0.466990\pi$
$17$	0.853635	0.207037	0.103518	+	0.994628i	$0.466990\pi$
$18$	0	0
$19$	−5.66442	−1.29951	−0.649754	−	0.760145i	$-0.725128\pi$
$19$	−5.66442	−1.29951	−0.649754	+	0.760145i	$0.725128\pi$
$20$	0	0
$21$	3.83221	0.836257
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.68585	−0.737169
$26$	0	0
$27$	17.3001	3.32940
$28$	0	0
$29$	6.88240	1.27803	0.639015	−	0.769194i	$-0.279342\pi$
$29$	6.88240	1.27803	0.639015	+	0.769194i	$0.279342\pi$
$30$	0	0
$31$	−8.32150	−1.49459	−0.747293	−	0.664495i	$-0.768647\pi$
$31$	−8.32150	−1.49459	−0.747293	+	0.664495i	$0.768647\pi$
$32$	0	0
$33$	10.5181	1.83096
$34$	0	0
$35$	1.31415	0.222133
$36$	0	0
$37$	−8.81079	−1.44848	−0.724242	−	0.689545i	$-0.757810\pi$
$37$	−8.81079	−1.44848	−0.724242	+	0.689545i	$0.757810\pi$
$38$	0	0
$39$	−8.32150	−1.33251
$40$	0	0
$41$	−6.48929	−1.01346	−0.506728	−	0.862106i	$-0.669145\pi$
$41$	−6.48929	−1.01346	−0.506728	+	0.862106i	$0.669145\pi$
$42$	0	0
$43$	−2.97858	−0.454229	−0.227114	−	0.973868i	$-0.572929\pi$
$43$	−2.97858	−0.454229	−0.227114	+	0.973868i	$0.572929\pi$
$44$	0	0
$45$	9.37169	1.39705
$46$	0	0
$47$	−2.94981	−0.430274	−0.215137	−	0.976584i	$-0.569020\pi$
$47$	−2.94981	−0.430274	−0.215137	+	0.976584i	$0.569020\pi$
$48$	0	0
$49$	−5.68585	−0.812264
$50$	0	0
$51$	2.85363	0.399589
$52$	0	0
$53$	−0.393115	−0.0539985	−0.0269993	−	0.999635i	$-0.508595\pi$
$53$	−0.393115	−0.0539985	−0.0269993	+	0.999635i	$0.508595\pi$
$54$	0	0
$55$	3.60688	0.486352
$56$	0	0
$57$	−18.9357	−2.50810
$58$	0	0
$59$	5.70727	0.743023	0.371512	−	0.928428i	$-0.378840\pi$
$59$	5.70727	0.743023	0.371512	+	0.928428i	$0.378840\pi$
$60$	0	0
$61$	14.3503	1.83736	0.918682	−	0.394998i	$-0.129255\pi$
$61$	14.3503	1.83736	0.918682	+	0.394998i	$0.129255\pi$
$62$	0	0
$63$	9.37169	1.18072
$64$	0	0
$65$	−2.85363	−0.353950
$66$	0	0
$67$	7.93260	0.969121	0.484560	−	0.874758i	$-0.338980\pi$
$67$	7.93260	0.969121	0.484560	+	0.874758i	$0.338980\pi$
$68$	0	0
$69$	−3.34292	−0.402441
$70$	0	0
$71$	0.657077	0.0779807	0.0389903	−	0.999240i	$-0.487586\pi$
$71$	0.657077	0.0779807	0.0389903	+	0.999240i	$0.487586\pi$
$72$	0	0
$73$	−1.90383	−0.222826	−0.111413	−	0.993774i	$-0.535538\pi$
$73$	−1.90383	−0.222826	−0.111413	+	0.993774i	$0.535538\pi$
$74$	0	0
$75$	−12.3215	−1.42276
$76$	0	0
$77$	3.60688	0.411043
$78$	0	0
$79$	16.0575	1.80661	0.903307	−	0.428994i	$-0.141132\pi$
$79$	16.0575	1.80661	0.903307	+	0.428994i	$0.141132\pi$
$80$	0	0
$81$	33.3074	3.70082
$82$	0	0
$83$	2.75325	0.302208	0.151104	−	0.988518i	$-0.451717\pi$
$83$	2.75325	0.302208	0.151104	+	0.988518i	$0.451717\pi$
$84$	0	0
$85$	0.978577	0.106142
$86$	0	0
$87$	23.0073	2.46665
$88$	0	0
$89$	−15.7648	−1.67107	−0.835533	−	0.549440i	$-0.814841\pi$
$89$	−15.7648	−1.67107	−0.835533	+	0.549440i	$0.814841\pi$
$90$	0	0
$91$	−2.85363	−0.299142
$92$	0	0
$93$	−27.8181	−2.88461
$94$	0	0
$95$	−6.49350	−0.666219
$96$	0	0
$97$	−14.8108	−1.50381	−0.751904	−	0.659273i	$-0.770864\pi$
$97$	−14.8108	−1.50381	−0.751904	+	0.659273i	$0.770864\pi$
$98$	0	0
$99$	25.7220	2.58515
$100$	0	0
$101$	−17.2713	−1.71856	−0.859280	−	0.511506i	$-0.829088\pi$
$101$	−17.2713	−1.71856	−0.859280	+	0.511506i	$0.829088\pi$
$102$	0	0
$103$	16.2253	1.59873	0.799364	−	0.600846i	$-0.205170\pi$
$103$	16.2253	1.59873	0.799364	+	0.600846i	$0.205170\pi$
$104$	0	0
$105$	4.39312	0.428724
$106$	0	0
$107$	−3.70727	−0.358395	−0.179198	−	0.983813i	$-0.557350\pi$
$107$	−3.70727	−0.358395	−0.179198	+	0.983813i	$0.557350\pi$
$108$	0	0
$109$	4.58546	0.439208	0.219604	−	0.975589i	$-0.429524\pi$
$109$	4.58546	0.439208	0.219604	+	0.975589i	$0.429524\pi$
$110$	0	0
$111$	−29.4538	−2.79563
$112$	0	0
$113$	−0.518058	−0.0487348	−0.0243674	−	0.999703i	$-0.507757\pi$
$113$	−0.518058	−0.0487348	−0.0243674	+	0.999703i	$0.507757\pi$
$114$	0	0
$115$	−1.14637	−0.106899
$116$	0	0
$117$	−20.3503	−1.88138
$118$	0	0
$119$	0.978577	0.0897060
$120$	0	0
$121$	−1.10038	−0.100035
$122$	0	0
$123$	−21.6932	−1.95601
$124$	0	0
$125$	−9.95715	−0.890595
$126$	0	0
$127$	8.65708	0.768191	0.384096	−	0.923293i	$-0.374513\pi$
$127$	8.65708	0.768191	0.384096	+	0.923293i	$0.374513\pi$
$128$	0	0
$129$	−9.95715	−0.876679
$130$	0	0
$131$	18.6142	1.62633	0.813166	−	0.582031i	$-0.197742\pi$
$131$	18.6142	1.62633	0.813166	+	0.582031i	$0.197742\pi$
$132$	0	0
$133$	−6.49350	−0.563058
$134$	0	0
$135$	19.8322	1.70689
$136$	0	0
$137$	7.76481	0.663392	0.331696	−	0.943386i	$-0.392379\pi$
$137$	7.76481	0.663392	0.331696	+	0.943386i	$0.392379\pi$
$138$	0	0
$139$	9.30008	0.788822	0.394411	−	0.918934i	$-0.370949\pi$
$139$	9.30008	0.788822	0.394411	+	0.918934i	$0.370949\pi$
$140$	0	0
$141$	−9.86098	−0.830444
$142$	0	0
$143$	−7.83221	−0.654962
$144$	0	0
$145$	7.88975	0.655208
$146$	0	0
$147$	−19.0073	−1.56770
$148$	0	0
$149$	10.3503	0.847927	0.423964	−	0.905679i	$-0.360638\pi$
$149$	10.3503	0.847927	0.423964	+	0.905679i	$0.360638\pi$
$150$	0	0
$151$	−14.6718	−1.19397	−0.596986	−	0.802252i	$-0.703635\pi$
$151$	−14.6718	−1.19397	−0.596986	+	0.802252i	$0.703635\pi$
$152$	0	0
$153$	6.97858	0.564185
$154$	0	0
$155$	−9.53948	−0.766230
$156$	0	0
$157$	19.3288	1.54261	0.771305	−	0.636466i	$-0.219604\pi$
$157$	19.3288	1.54261	0.771305	+	0.636466i	$0.219604\pi$
$158$	0	0
$159$	−1.31415	−0.104219
$160$	0	0
$161$	−1.14637	−0.0903463
$162$	0	0
$163$	1.24254	0.0973232	0.0486616	−	0.998815i	$-0.484504\pi$
$163$	1.24254	0.0973232	0.0486616	+	0.998815i	$0.484504\pi$
$164$	0	0
$165$	12.0575	0.938678
$166$	0	0
$167$	−2.29273	−0.177417	−0.0887084	−	0.996058i	$-0.528274\pi$
$167$	−2.29273	−0.177417	−0.0887084	+	0.996058i	$0.528274\pi$
$168$	0	0
$169$	−6.80344	−0.523342
$170$	0	0
$171$	−46.3074	−3.54122
$172$	0	0
$173$	8.93573	0.679371	0.339686	−	0.940539i	$-0.389679\pi$
$173$	8.93573	0.679371	0.339686	+	0.940539i	$0.389679\pi$
$174$	0	0
$175$	−4.22533	−0.319405
$176$	0	0
$177$	19.0790	1.43406
$178$	0	0
$179$	−17.6932	−1.32245	−0.661226	−	0.750187i	$-0.729963\pi$
$179$	−17.6932	−1.32245	−0.661226	+	0.750187i	$0.729963\pi$
$180$	0	0
$181$	−12.4752	−0.927275	−0.463638	−	0.886025i	$-0.653456\pi$
$181$	−12.4752	−0.927275	−0.463638	+	0.886025i	$0.653456\pi$
$182$	0	0
$183$	47.9718	3.54618
$184$	0	0
$185$	−10.1004	−0.742595
$186$	0	0
$187$	2.68585	0.196409
$188$	0	0
$189$	19.8322	1.44258
$190$	0	0
$191$	−11.2467	−0.813786	−0.406893	−	0.913476i	$-0.633388\pi$
$191$	−11.2467	−0.813786	−0.406893	+	0.913476i	$0.633388\pi$
$192$	0	0
$193$	−1.17513	−0.0845880	−0.0422940	−	0.999105i	$-0.513467\pi$
$193$	−1.17513	−0.0845880	−0.0422940	+	0.999105i	$0.513467\pi$
$194$	0	0
$195$	−9.53948	−0.683136
$196$	0	0
$197$	−5.36748	−0.382417	−0.191209	−	0.981549i	$-0.561241\pi$
$197$	−5.36748	−0.382417	−0.191209	+	0.981549i	$0.561241\pi$
$198$	0	0
$199$	−9.78937	−0.693950	−0.346975	−	0.937874i	$-0.612791\pi$
$199$	−9.78937	−0.693950	−0.346975	+	0.937874i	$0.612791\pi$
$200$	0	0
$201$	26.5181	1.87044
$202$	0	0
$203$	7.88975	0.553752
$204$	0	0
$205$	−7.43910	−0.519569
$206$	0	0
$207$	−8.17513	−0.568211
$208$	0	0
$209$	−17.8223	−1.23280
$210$	0	0
$211$	−4.24989	−0.292574	−0.146287	−	0.989242i	$-0.546732\pi$
$211$	−4.24989	−0.292574	−0.146287	+	0.989242i	$0.546732\pi$
$212$	0	0
$213$	2.19656	0.150506
$214$	0	0
$215$	−3.41454	−0.232870
$216$	0	0
$217$	−9.53948	−0.647582
$218$	0	0
$219$	−6.36435	−0.430063
$220$	0	0
$221$	−2.12494	−0.142939
$222$	0	0
$223$	13.2285	0.885843	0.442922	−	0.896560i	$-0.353942\pi$
$223$	13.2285	0.885843	0.442922	+	0.896560i	$0.353942\pi$
$224$	0	0
$225$	−30.1323	−2.00882
$226$	0	0
$227$	22.8683	1.51782	0.758912	−	0.651193i	$-0.225731\pi$
$227$	22.8683	1.51782	0.758912	+	0.651193i	$0.225731\pi$
$228$	0	0
$229$	−1.17092	−0.0773768	−0.0386884	−	0.999251i	$-0.512318\pi$
$229$	−1.17092	−0.0773768	−0.0386884	+	0.999251i	$0.512318\pi$
$230$	0	0
$231$	12.0575	0.793328
$232$	0	0
$233$	−13.4250	−0.879502	−0.439751	−	0.898120i	$-0.644933\pi$
$233$	−13.4250	−0.879502	−0.439751	+	0.898120i	$0.644933\pi$
$234$	0	0
$235$	−3.38156	−0.220589
$236$	0	0
$237$	53.6791	3.48683
$238$	0	0
$239$	−2.30681	−0.149215	−0.0746075	−	0.997213i	$-0.523770\pi$
$239$	−2.30681	−0.149215	−0.0746075	+	0.997213i	$0.523770\pi$
$240$	0	0
$241$	18.2829	1.17770	0.588851	−	0.808241i	$-0.299580\pi$
$241$	18.2829	1.17770	0.588851	+	0.808241i	$0.299580\pi$
$242$	0	0
$243$	59.4439	3.81333
$244$	0	0
$245$	−6.51806	−0.416423
$246$	0	0
$247$	14.1004	0.897186
$248$	0	0
$249$	9.20390	0.583274
$250$	0	0
$251$	25.6974	1.62201	0.811003	−	0.585042i	$-0.198922\pi$
$251$	25.6974	1.62201	0.811003	+	0.585042i	$0.198922\pi$
$252$	0	0
$253$	−3.14637	−0.197810
$254$	0	0
$255$	3.27131	0.204857
$256$	0	0
$257$	17.8181	1.11146	0.555732	−	0.831361i	$-0.312438\pi$
$257$	17.8181	1.11146	0.555732	+	0.831361i	$0.312438\pi$
$258$	0	0
$259$	−10.1004	−0.627607
$260$	0	0
$261$	56.2646	3.48269
$262$	0	0
$263$	−25.8139	−1.59175	−0.795877	−	0.605458i	$-0.792990\pi$
$263$	−25.8139	−1.59175	−0.795877	+	0.605458i	$0.792990\pi$
$264$	0	0
$265$	−0.450654	−0.0276834
$266$	0	0
$267$	−52.7005	−3.22522
$268$	0	0
$269$	−2.09617	−0.127806	−0.0639030	−	0.997956i	$-0.520355\pi$
$269$	−2.09617	−0.127806	−0.0639030	+	0.997956i	$0.520355\pi$
$270$	0	0
$271$	14.1579	0.860033	0.430016	−	0.902821i	$-0.358508\pi$
$271$	14.1579	0.860033	0.430016	+	0.902821i	$0.358508\pi$
$272$	0	0
$273$	−9.53948	−0.577356
$274$	0	0
$275$	−11.5970	−0.699327
$276$	0	0
$277$	1.17513	0.0706070	0.0353035	−	0.999377i	$-0.488760\pi$
$277$	1.17513	0.0706070	0.0353035	+	0.999377i	$0.488760\pi$
$278$	0	0
$279$	−68.0294	−4.07281
$280$	0	0
$281$	12.1825	0.726746	0.363373	−	0.931644i	$-0.381625\pi$
$281$	12.1825	0.726746	0.363373	+	0.931644i	$0.381625\pi$
$282$	0	0
$283$	11.4819	0.682531	0.341265	−	0.939967i	$-0.389145\pi$
$283$	11.4819	0.682531	0.341265	+	0.939967i	$0.389145\pi$
$284$	0	0
$285$	−21.7073	−1.28583
$286$	0	0
$287$	−7.43910	−0.439116
$288$	0	0
$289$	−16.2713	−0.957136
$290$	0	0
$291$	−49.5113	−2.90241
$292$	0	0
$293$	5.17092	0.302089	0.151044	−	0.988527i	$-0.451736\pi$
$293$	5.17092	0.302089	0.151044	+	0.988527i	$0.451736\pi$
$294$	0	0
$295$	6.54262	0.380926
$296$	0	0
$297$	54.4324	3.15849
$298$	0	0
$299$	2.48929	0.143959
$300$	0	0
$301$	−3.41454	−0.196811
$302$	0	0
$303$	−57.7367	−3.31688
$304$	0	0
$305$	16.4507	0.941962
$306$	0	0
$307$	12.8353	0.732552	0.366276	−	0.930506i	$-0.380633\pi$
$307$	12.8353	0.732552	0.366276	+	0.930506i	$0.380633\pi$
$308$	0	0
$309$	54.2400	3.08561
$310$	0	0
$311$	9.30008	0.527359	0.263680	−	0.964610i	$-0.415064\pi$
$311$	9.30008	0.527359	0.263680	+	0.964610i	$0.415064\pi$
$312$	0	0
$313$	−7.56404	−0.427545	−0.213772	−	0.976883i	$-0.568575\pi$
$313$	−7.56404	−0.427545	−0.213772	+	0.976883i	$0.568575\pi$
$314$	0	0
$315$	10.7434	0.605321
$316$	0	0
$317$	−13.2713	−0.745391	−0.372695	−	0.927954i	$-0.621566\pi$
$317$	−13.2713	−0.745391	−0.372695	+	0.927954i	$0.621566\pi$
$318$	0	0
$319$	21.6546	1.21242
$320$	0	0
$321$	−12.3931	−0.691716
$322$	0	0
$323$	−4.83535	−0.269046
$324$	0	0
$325$	9.17513	0.508945
$326$	0	0
$327$	15.3288	0.847687
$328$	0	0
$329$	−3.38156	−0.186431
$330$	0	0
$331$	−10.2295	−0.562266	−0.281133	−	0.959669i	$-0.590710\pi$
$331$	−10.2295	−0.562266	−0.281133	+	0.959669i	$0.590710\pi$
$332$	0	0
$333$	−72.0294	−3.94719
$334$	0	0
$335$	9.09365	0.496839
$336$	0	0
$337$	18.9786	1.03383	0.516914	−	0.856037i	$-0.327081\pi$
$337$	18.9786	1.03383	0.516914	+	0.856037i	$0.327081\pi$
$338$	0	0
$339$	−1.73183	−0.0940600
$340$	0	0
$341$	−26.1825	−1.41786
$342$	0	0
$343$	−14.5426	−0.785227
$344$	0	0
$345$	−3.83221	−0.206319
$346$	0	0
$347$	−20.6430	−1.10817	−0.554087	−	0.832459i	$-0.686933\pi$
$347$	−20.6430	−1.10817	−0.554087	+	0.832459i	$0.686933\pi$
$348$	0	0
$349$	−23.9185	−1.28033	−0.640164	−	0.768238i	$-0.721134\pi$
$349$	−23.9185	−1.28033	−0.640164	+	0.768238i	$0.721134\pi$
$350$	0	0
$351$	−43.0649	−2.29863
$352$	0	0
$353$	14.0105	0.745703	0.372851	−	0.927891i	$-0.378380\pi$
$353$	14.0105	0.745703	0.372851	+	0.927891i	$0.378380\pi$
$354$	0	0
$355$	0.753250	0.0399784
$356$	0	0
$357$	3.27131	0.173136
$358$	0	0
$359$	−22.1004	−1.16641	−0.583207	−	0.812324i	$-0.698202\pi$
$359$	−22.1004	−1.16641	−0.583207	+	0.812324i	$0.698202\pi$
$360$	0	0
$361$	13.0857	0.688721
$362$	0	0
$363$	−3.67850	−0.193071
$364$	0	0
$365$	−2.18248	−0.114236
$366$	0	0
$367$	17.1464	0.895033	0.447516	−	0.894276i	$-0.352309\pi$
$367$	17.1464	0.895033	0.447516	+	0.894276i	$0.352309\pi$
$368$	0	0
$369$	−53.0508	−2.76171
$370$	0	0
$371$	−0.450654	−0.0233968
$372$	0	0
$373$	−25.2614	−1.30799	−0.653994	−	0.756500i	$-0.726908\pi$
$373$	−25.2614	−1.30799	−0.653994	+	0.756500i	$0.726908\pi$
$374$	0	0
$375$	−33.2860	−1.71888
$376$	0	0
$377$	−17.1323	−0.882358
$378$	0	0
$379$	19.7648	1.01525	0.507625	−	0.861578i	$-0.330524\pi$
$379$	19.7648	1.01525	0.507625	+	0.861578i	$0.330524\pi$
$380$	0	0
$381$	28.9399	1.48264
$382$	0	0
$383$	−19.4145	−0.992037	−0.496018	−	0.868312i	$-0.665205\pi$
$383$	−19.4145	−0.992037	−0.496018	+	0.868312i	$0.665205\pi$
$384$	0	0
$385$	4.13481	0.210729
$386$	0	0
$387$	−24.3503	−1.23779
$388$	0	0
$389$	−24.7862	−1.25671	−0.628356	−	0.777926i	$-0.716272\pi$
$389$	−24.7862	−1.25671	−0.628356	+	0.777926i	$0.716272\pi$
$390$	0	0
$391$	−0.853635	−0.0431702
$392$	0	0
$393$	62.2259	3.13888
$394$	0	0
$395$	18.4078	0.926197
$396$	0	0
$397$	−26.1537	−1.31262	−0.656309	−	0.754493i	$-0.727883\pi$
$397$	−26.1537	−1.31262	−0.656309	+	0.754493i	$0.727883\pi$
$398$	0	0
$399$	−21.7073	−1.08672
$400$	0	0
$401$	−14.6676	−0.732463	−0.366231	−	0.930524i	$-0.619352\pi$
$401$	−14.6676	−0.732463	−0.366231	+	0.930524i	$0.619352\pi$
$402$	0	0
$403$	20.7146	1.03187
$404$	0	0
$405$	38.1825	1.89730
$406$	0	0
$407$	−27.7220	−1.37413
$408$	0	0
$409$	−3.94667	−0.195150	−0.0975752	−	0.995228i	$-0.531109\pi$
$409$	−3.94667	−0.195150	−0.0975752	+	0.995228i	$0.531109\pi$
$410$	0	0
$411$	25.9572	1.28037
$412$	0	0
$413$	6.54262	0.321941
$414$	0	0
$415$	3.15623	0.154933
$416$	0	0
$417$	31.0894	1.52246
$418$	0	0
$419$	0.235192	0.0114899	0.00574495	−	0.999983i	$-0.498171\pi$
$419$	0.235192	0.0114899	0.00574495	+	0.999983i	$0.498171\pi$
$420$	0	0
$421$	8.95402	0.436392	0.218196	−	0.975905i	$-0.429983\pi$
$421$	8.95402	0.436392	0.218196	+	0.975905i	$0.429983\pi$
$422$	0	0
$423$	−24.1151	−1.17252
$424$	0	0
$425$	−3.14637	−0.152621
$426$	0	0
$427$	16.4507	0.796103
$428$	0	0
$429$	−26.1825	−1.26410
$430$	0	0
$431$	19.3288	0.931038	0.465519	−	0.885038i	$-0.345868\pi$
$431$	19.3288	0.931038	0.465519	+	0.885038i	$0.345868\pi$
$432$	0	0
$433$	−32.1249	−1.54383	−0.771913	−	0.635728i	$-0.780700\pi$
$433$	−32.1249	−1.54383	−0.771913	+	0.635728i	$0.780700\pi$
$434$	0	0
$435$	26.3748	1.26458
$436$	0	0
$437$	5.66442	0.270966
$438$	0	0
$439$	12.3790	0.590819	0.295410	−	0.955371i	$-0.404544\pi$
$439$	12.3790	0.590819	0.295410	+	0.955371i	$0.404544\pi$
$440$	0	0
$441$	−46.4826	−2.21346
$442$	0	0
$443$	−33.0220	−1.56892	−0.784462	−	0.620177i	$-0.787061\pi$
$443$	−33.0220	−1.56892	−0.784462	+	0.620177i	$0.787061\pi$
$444$	0	0
$445$	−18.0722	−0.856706
$446$	0	0
$447$	34.6002	1.63653
$448$	0	0
$449$	5.21377	0.246053	0.123027	−	0.992403i	$-0.460740\pi$
$449$	5.21377	0.246053	0.123027	+	0.992403i	$0.460740\pi$
$450$	0	0
$451$	−20.4177	−0.961431
$452$	0	0
$453$	−49.0466	−2.30441
$454$	0	0
$455$	−3.27131	−0.153361
$456$	0	0
$457$	5.88975	0.275511	0.137755	−	0.990466i	$-0.456011\pi$
$457$	5.88975	0.275511	0.137755	+	0.990466i	$0.456011\pi$
$458$	0	0
$459$	14.7679	0.689309
$460$	0	0
$461$	30.0680	1.40041	0.700204	−	0.713943i	$-0.253093\pi$
$461$	30.0680	1.40041	0.700204	+	0.713943i	$0.253093\pi$
$462$	0	0
$463$	−22.6858	−1.05430	−0.527150	−	0.849772i	$-0.676740\pi$
$463$	−22.6858	−1.05430	−0.527150	+	0.849772i	$0.676740\pi$
$464$	0	0
$465$	−31.8898	−1.47885
$466$	0	0
$467$	31.8223	1.47256	0.736281	−	0.676676i	$-0.236580\pi$
$467$	31.8223	1.47256	0.736281	+	0.676676i	$0.236580\pi$
$468$	0	0
$469$	9.09365	0.419906
$470$	0	0
$471$	64.6148	2.97730
$472$	0	0
$473$	−9.37169	−0.430911
$474$	0	0
$475$	20.8782	0.957957
$476$	0	0
$477$	−3.21377	−0.147148
$478$	0	0
$479$	−24.5609	−1.12222	−0.561108	−	0.827742i	$-0.689625\pi$
$479$	−24.5609	−1.12222	−0.561108	+	0.827742i	$0.689625\pi$
$480$	0	0
$481$	21.9326	1.00004
$482$	0	0
$483$	−3.83221	−0.174372
$484$	0	0
$485$	−16.9786	−0.770957
$486$	0	0
$487$	40.1783	1.82065	0.910326	−	0.413893i	$-0.135831\pi$
$487$	40.1783	1.82065	0.910326	+	0.413893i	$0.135831\pi$
$488$	0	0
$489$	4.15371	0.187837
$490$	0	0
$491$	−17.2425	−0.778145	−0.389072	−	0.921207i	$-0.627204\pi$
$491$	−17.2425	−0.778145	−0.389072	+	0.921207i	$0.627204\pi$
$492$	0	0
$493$	5.87506	0.264599
$494$	0	0
$495$	29.4868	1.32533
$496$	0	0
$497$	0.753250	0.0337879
$498$	0	0
$499$	−7.59281	−0.339901	−0.169950	−	0.985453i	$-0.554361\pi$
$499$	−7.59281	−0.339901	−0.169950	+	0.985453i	$0.554361\pi$
$500$	0	0
$501$	−7.66442	−0.342421
$502$	0	0
$503$	39.3864	1.75615	0.878076	−	0.478521i	$-0.158827\pi$
$503$	39.3864	1.75615	0.878076	+	0.478521i	$0.158827\pi$
$504$	0	0
$505$	−19.7992	−0.881054
$506$	0	0
$507$	−22.7434	−1.01007
$508$	0	0
$509$	−4.98279	−0.220858	−0.110429	−	0.993884i	$-0.535223\pi$
$509$	−4.98279	−0.220858	−0.110429	+	0.993884i	$0.535223\pi$
$510$	0	0
$511$	−2.18248	−0.0965473
$512$	0	0
$513$	−97.9950	−4.32658
$514$	0	0
$515$	18.6002	0.819621
$516$	0	0
$517$	−9.28117	−0.408186
$518$	0	0
$519$	29.8715	1.31121
$520$	0	0
$521$	1.22219	0.0535452	0.0267726	−	0.999642i	$-0.491477\pi$
$521$	1.22219	0.0535452	0.0267726	+	0.999642i	$0.491477\pi$
$522$	0	0
$523$	−10.2829	−0.449638	−0.224819	−	0.974401i	$-0.572179\pi$
$523$	−10.2829	−0.449638	−0.224819	+	0.974401i	$0.572179\pi$
$524$	0	0
$525$	−14.1249	−0.616463
$526$	0	0
$527$	−7.10352	−0.309434
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	46.6577	2.02477
$532$	0	0
$533$	16.1537	0.699695
$534$	0	0
$535$	−4.24989	−0.183739
$536$	0	0
$537$	−59.1470	−2.55238
$538$	0	0
$539$	−17.8898	−0.770566
$540$	0	0
$541$	−11.2180	−0.482299	−0.241149	−	0.970488i	$-0.577524\pi$
$541$	−11.2180	−0.482299	−0.241149	+	0.970488i	$0.577524\pi$
$542$	0	0
$543$	−41.7037	−1.78968
$544$	0	0
$545$	5.25662	0.225169
$546$	0	0
$547$	−0.657077	−0.0280946	−0.0140473	−	0.999901i	$-0.504472\pi$
$547$	−0.657077	−0.0280946	−0.0140473	+	0.999901i	$0.504472\pi$
$548$	0	0
$549$	117.315	5.00690
$550$	0	0
$551$	−38.9848	−1.66081
$552$	0	0
$553$	18.4078	0.782780
$554$	0	0
$555$	−33.7648	−1.43324
$556$	0	0
$557$	24.7862	1.05023	0.525113	−	0.851032i	$-0.324023\pi$
$557$	24.7862	1.05023	0.525113	+	0.851032i	$0.324023\pi$
$558$	0	0
$559$	7.41454	0.313602
$560$	0	0
$561$	8.97858	0.379076
$562$	0	0
$563$	10.3931	0.438018	0.219009	−	0.975723i	$-0.429718\pi$
$563$	10.3931	0.438018	0.219009	+	0.975723i	$0.429718\pi$
$564$	0	0
$565$	−0.593884	−0.0249849
$566$	0	0
$567$	38.1825	1.60351
$568$	0	0
$569$	25.0460	1.04998	0.524991	−	0.851108i	$-0.324069\pi$
$569$	25.0460	1.04998	0.524991	+	0.851108i	$0.324069\pi$
$570$	0	0
$571$	−4.12494	−0.172623	−0.0863117	−	0.996268i	$-0.527508\pi$
$571$	−4.12494	−0.172623	−0.0863117	+	0.996268i	$0.527508\pi$
$572$	0	0
$573$	−37.5970	−1.57064
$574$	0	0
$575$	3.68585	0.153710
$576$	0	0
$577$	−3.71775	−0.154772	−0.0773860	−	0.997001i	$-0.524657\pi$
$577$	−3.71775	−0.154772	−0.0773860	+	0.997001i	$0.524657\pi$
$578$	0	0
$579$	−3.92839	−0.163258
$580$	0	0
$581$	3.15623	0.130943
$582$	0	0
$583$	−1.23688	−0.0512265
$584$	0	0
$585$	−23.3288	−0.964529
$586$	0	0
$587$	−28.9070	−1.19312	−0.596559	−	0.802569i	$-0.703466\pi$
$587$	−28.9070	−1.19312	−0.596559	+	0.802569i	$0.703466\pi$
$588$	0	0
$589$	47.1365	1.94223
$590$	0	0
$591$	−17.9431	−0.738080
$592$	0	0
$593$	14.3074	0.587535	0.293768	−	0.955877i	$-0.405091\pi$
$593$	14.3074	0.587535	0.293768	+	0.955877i	$0.405091\pi$
$594$	0	0
$595$	1.12181	0.0459896
$596$	0	0
$597$	−32.7251	−1.33935
$598$	0	0
$599$	−46.6577	−1.90638	−0.953191	−	0.302369i	$-0.902222\pi$
$599$	−46.6577	−1.90638	−0.953191	+	0.302369i	$0.902222\pi$
$600$	0	0
$601$	14.9315	0.609069	0.304535	−	0.952501i	$-0.401499\pi$
$601$	14.9315	0.609069	0.304535	+	0.952501i	$0.401499\pi$
$602$	0	0
$603$	64.8500	2.64090
$604$	0	0
$605$	−1.26144	−0.0512849
$606$	0	0
$607$	−13.4208	−0.544734	−0.272367	−	0.962193i	$-0.587806\pi$
$607$	−13.4208	−0.544734	−0.272367	+	0.962193i	$0.587806\pi$
$608$	0	0
$609$	26.3748	1.06876
$610$	0	0
$611$	7.34292	0.297063
$612$	0	0
$613$	−21.7894	−0.880064	−0.440032	−	0.897982i	$-0.645033\pi$
$613$	−21.7894	−0.880064	−0.440032	+	0.897982i	$0.645033\pi$
$614$	0	0
$615$	−24.8683	−1.00279
$616$	0	0
$617$	23.0852	0.929376	0.464688	−	0.885474i	$-0.346166\pi$
$617$	23.0852	0.929376	0.464688	+	0.885474i	$0.346166\pi$
$618$	0	0
$619$	22.9786	0.923587	0.461793	−	0.886987i	$-0.347206\pi$
$619$	22.9786	0.923587	0.461793	+	0.886987i	$0.347206\pi$
$620$	0	0
$621$	−17.3001	−0.694228
$622$	0	0
$623$	−18.0722	−0.724049
$624$	0	0
$625$	7.01469	0.280588
$626$	0	0
$627$	−59.5787	−2.37935
$628$	0	0
$629$	−7.52119	−0.299890
$630$	0	0
$631$	37.0852	1.47634	0.738170	−	0.674615i	$-0.235690\pi$
$631$	37.0852	1.47634	0.738170	+	0.674615i	$0.235690\pi$
$632$	0	0
$633$	−14.2070	−0.564679
$634$	0	0
$635$	9.92417	0.393829
$636$	0	0
$637$	14.1537	0.560790
$638$	0	0
$639$	5.37169	0.212501
$640$	0	0
$641$	29.9227	1.18188	0.590938	−	0.806717i	$-0.298758\pi$
$641$	29.9227	1.18188	0.590938	+	0.806717i	$0.298758\pi$
$642$	0	0
$643$	−18.2744	−0.720674	−0.360337	−	0.932822i	$-0.617338\pi$
$643$	−18.2744	−0.720674	−0.360337	+	0.932822i	$0.617338\pi$
$644$	0	0
$645$	−11.4145	−0.449447
$646$	0	0
$647$	19.7360	0.775904	0.387952	−	0.921680i	$-0.373183\pi$
$647$	19.7360	0.775904	0.387952	+	0.921680i	$0.373183\pi$
$648$	0	0
$649$	17.9572	0.704880
$650$	0	0
$651$	−31.8898	−1.24986
$652$	0	0
$653$	−25.8463	−1.01144	−0.505722	−	0.862697i	$-0.668774\pi$
$653$	−25.8463	−1.01144	−0.505722	+	0.862697i	$0.668774\pi$
$654$	0	0
$655$	21.3387	0.833772
$656$	0	0
$657$	−15.5640	−0.607211
$658$	0	0
$659$	13.9473	0.543309	0.271655	−	0.962395i	$-0.412429\pi$
$659$	13.9473	0.543309	0.271655	+	0.962395i	$0.412429\pi$
$660$	0	0
$661$	−0.978577	−0.0380622	−0.0190311	−	0.999819i	$-0.506058\pi$
$661$	−0.978577	−0.0380622	−0.0190311	+	0.999819i	$0.506058\pi$
$662$	0	0
$663$	−7.10352	−0.275878
$664$	0	0
$665$	−7.44392	−0.288663
$666$	0	0
$667$	−6.88240	−0.266488
$668$	0	0
$669$	44.2217	1.70971
$670$	0	0
$671$	45.1512	1.74304
$672$	0	0
$673$	−24.3032	−0.936820	−0.468410	−	0.883511i	$-0.655173\pi$
$673$	−24.3032	−0.936820	−0.468410	+	0.883511i	$0.655173\pi$
$674$	0	0
$675$	−63.7654	−2.45433
$676$	0	0
$677$	23.6069	0.907286	0.453643	−	0.891183i	$-0.350124\pi$
$677$	23.6069	0.907286	0.453643	+	0.891183i	$0.350124\pi$
$678$	0	0
$679$	−16.9786	−0.651578
$680$	0	0
$681$	76.4471	2.92946
$682$	0	0
$683$	−20.9070	−0.799983	−0.399991	−	0.916519i	$-0.630987\pi$
$683$	−20.9070	−0.799983	−0.399991	+	0.916519i	$0.630987\pi$
$684$	0	0
$685$	8.90131	0.340101
$686$	0	0
$687$	−3.91431	−0.149340
$688$	0	0
$689$	0.978577	0.0372808
$690$	0	0
$691$	48.1151	1.83038	0.915192	−	0.403018i	$-0.132039\pi$
$691$	48.1151	1.83038	0.915192	+	0.403018i	$0.132039\pi$
$692$	0	0
$693$	29.4868	1.12011
$694$	0	0
$695$	10.6613	0.404406
$696$	0	0
$697$	−5.53948	−0.209823
$698$	0	0
$699$	−44.8788	−1.69747
$700$	0	0
$701$	−3.38156	−0.127720	−0.0638598	−	0.997959i	$-0.520341\pi$
$701$	−3.38156	−0.127720	−0.0638598	+	0.997959i	$0.520341\pi$
$702$	0	0
$703$	49.9080	1.88232
$704$	0	0
$705$	−11.3043	−0.425744
$706$	0	0
$707$	−19.7992	−0.744627
$708$	0	0
$709$	−31.1119	−1.16843	−0.584217	−	0.811598i	$-0.698598\pi$
$709$	−31.1119	−1.16843	−0.584217	+	0.811598i	$0.698598\pi$
$710$	0	0
$711$	131.273	4.92310
$712$	0	0
$713$	8.32150	0.311643
$714$	0	0
$715$	−8.97858	−0.335780
$716$	0	0
$717$	−7.71148	−0.287990
$718$	0	0
$719$	−20.9870	−0.782683	−0.391341	−	0.920246i	$-0.627989\pi$
$719$	−20.9870	−0.782683	−0.391341	+	0.920246i	$0.627989\pi$
$720$	0	0
$721$	18.6002	0.692706
$722$	0	0
$723$	61.1182	2.27301
$724$	0	0
$725$	−25.3675	−0.942125
$726$	0	0
$727$	5.92417	0.219715	0.109858	−	0.993947i	$-0.464960\pi$
$727$	5.92417	0.219715	0.109858	+	0.993947i	$0.464960\pi$
$728$	0	0
$729$	98.7942	3.65904
$730$	0	0
$731$	−2.54262	−0.0940421
$732$	0	0
$733$	2.00987	0.0742361	0.0371180	−	0.999311i	$-0.488182\pi$
$733$	2.00987	0.0742361	0.0371180	+	0.999311i	$0.488182\pi$
$734$	0	0
$735$	−21.7894	−0.803713
$736$	0	0
$737$	24.9588	0.919371
$738$	0	0
$739$	−19.6785	−0.723885	−0.361943	−	0.932200i	$-0.617886\pi$
$739$	−19.6785	−0.723885	−0.361943	+	0.932200i	$0.617886\pi$
$740$	0	0
$741$	47.1365	1.73160
$742$	0	0
$743$	36.1396	1.32583	0.662917	−	0.748693i	$-0.269318\pi$
$743$	36.1396	1.32583	0.662917	+	0.748693i	$0.269318\pi$
$744$	0	0
$745$	11.8652	0.434707
$746$	0	0
$747$	22.5082	0.823532
$748$	0	0
$749$	−4.24989	−0.155287
$750$	0	0
$751$	−1.57246	−0.0573799	−0.0286900	−	0.999588i	$-0.509134\pi$
$751$	−1.57246	−0.0573799	−0.0286900	+	0.999588i	$0.509134\pi$
$752$	0	0
$753$	85.9044	3.13053
$754$	0	0
$755$	−16.8192	−0.612114
$756$	0	0
$757$	35.8041	1.30132	0.650660	−	0.759369i	$-0.274492\pi$
$757$	35.8041	1.30132	0.650660	+	0.759369i	$0.274492\pi$
$758$	0	0
$759$	−10.5181	−0.381781
$760$	0	0
$761$	38.6044	1.39941	0.699704	−	0.714433i	$-0.253315\pi$
$761$	38.6044	1.39941	0.699704	+	0.714433i	$0.253315\pi$
$762$	0	0
$763$	5.25662	0.190302
$764$	0	0
$765$	8.00000	0.289241
$766$	0	0
$767$	−14.2070	−0.512986
$768$	0	0
$769$	40.8255	1.47220	0.736102	−	0.676870i	$-0.236664\pi$
$769$	40.8255	1.47220	0.736102	+	0.676870i	$0.236664\pi$
$770$	0	0
$771$	59.5647	2.14517
$772$	0	0
$773$	−18.0722	−0.650013	−0.325006	−	0.945712i	$-0.605366\pi$
$773$	−18.0722	−0.650013	−0.325006	+	0.945712i	$0.605366\pi$
$774$	0	0
$775$	30.6718	1.10176
$776$	0	0
$777$	−33.7648	−1.21131
$778$	0	0
$779$	36.7581	1.31699
$780$	0	0
$781$	2.06740	0.0739775
$782$	0	0
$783$	119.066	4.25508
$784$	0	0
$785$	22.1579	0.790850
$786$	0	0
$787$	−11.7648	−0.419370	−0.209685	−	0.977769i	$-0.567244\pi$
$787$	−11.7648	−0.419370	−0.209685	+	0.977769i	$0.567244\pi$
$788$	0	0
$789$	−86.2940	−3.07215
$790$	0	0
$791$	−0.593884	−0.0211161
$792$	0	0
$793$	−35.7220	−1.26852
$794$	0	0
$795$	−1.50650	−0.0534301
$796$	0	0
$797$	34.9357	1.23749	0.618744	−	0.785593i	$-0.287642\pi$
$797$	34.9357	1.23749	0.618744	+	0.785593i	$0.287642\pi$
$798$	0	0
$799$	−2.51806	−0.0890825
$800$	0	0
$801$	−128.879	−4.55373
$802$	0	0
$803$	−5.99013	−0.211387
$804$	0	0
$805$	−1.31415	−0.0463178
$806$	0	0
$807$	−7.00735	−0.246670
$808$	0	0
$809$	−47.4783	−1.66925	−0.834625	−	0.550819i	$-0.814316\pi$
$809$	−47.4783	−1.66925	−0.834625	+	0.550819i	$0.814316\pi$
$810$	0	0
$811$	21.8855	0.768505	0.384253	−	0.923228i	$-0.374459\pi$
$811$	21.8855	0.768505	0.384253	+	0.923228i	$0.374459\pi$
$812$	0	0
$813$	47.3288	1.65990
$814$	0	0
$815$	1.42440	0.0498947
$816$	0	0
$817$	16.8719	0.590274
$818$	0	0
$819$	−23.3288	−0.815176
$820$	0	0
$821$	24.4653	0.853846	0.426923	−	0.904288i	$-0.359597\pi$
$821$	24.4653	0.853846	0.426923	+	0.904288i	$0.359597\pi$
$822$	0	0
$823$	−15.3429	−0.534821	−0.267410	−	0.963583i	$-0.586168\pi$
$823$	−15.3429	−0.534821	−0.267410	+	0.963583i	$0.586168\pi$
$824$	0	0
$825$	−38.7679	−1.34973
$826$	0	0
$827$	44.9357	1.56257	0.781284	−	0.624175i	$-0.214565\pi$
$827$	44.9357	1.56257	0.781284	+	0.624175i	$0.214565\pi$
$828$	0	0
$829$	28.8009	1.00030	0.500149	−	0.865940i	$-0.333279\pi$
$829$	28.8009	1.00030	0.500149	+	0.865940i	$0.333279\pi$
$830$	0	0
$831$	3.92839	0.136274
$832$	0	0
$833$	−4.85363	−0.168168
$834$	0	0
$835$	−2.62831	−0.0909563
$836$	0	0
$837$	−143.963	−4.97608
$838$	0	0
$839$	−21.1955	−0.731749	−0.365875	−	0.930664i	$-0.619230\pi$
$839$	−21.1955	−0.731749	−0.365875	+	0.930664i	$0.619230\pi$
$840$	0	0
$841$	18.3675	0.633361
$842$	0	0
$843$	40.7251	1.40265
$844$	0	0
$845$	−7.79923	−0.268302
$846$	0	0
$847$	−1.26144	−0.0433437
$848$	0	0
$849$	38.3832	1.31731
$850$	0	0
$851$	8.81079	0.302030
$852$	0	0
$853$	14.8438	0.508241	0.254120	−	0.967173i	$-0.418214\pi$
$853$	14.8438	0.508241	0.254120	+	0.967173i	$0.418214\pi$
$854$	0	0
$855$	−53.0852	−1.81548
$856$	0	0
$857$	−40.1684	−1.37213	−0.686063	−	0.727542i	$-0.740663\pi$
$857$	−40.1684	−1.37213	−0.686063	+	0.727542i	$0.740663\pi$
$858$	0	0
$859$	30.3068	1.03405	0.517027	−	0.855969i	$-0.327039\pi$
$859$	30.3068	1.03405	0.517027	+	0.855969i	$0.327039\pi$
$860$	0	0
$861$	−24.8683	−0.847510
$862$	0	0
$863$	−14.9792	−0.509898	−0.254949	−	0.966955i	$-0.582059\pi$
$863$	−14.9792	−0.509898	−0.254949	+	0.966955i	$0.582059\pi$
$864$	0	0
$865$	10.2436	0.348293
$866$	0	0
$867$	−54.3937	−1.84731
$868$	0	0
$869$	50.5229	1.71387
$870$	0	0
$871$	−19.7465	−0.669085
$872$	0	0
$873$	−121.080	−4.09794
$874$	0	0
$875$	−11.4145	−0.385882
$876$	0	0
$877$	16.7715	0.566335	0.283167	−	0.959071i	$-0.408615\pi$
$877$	16.7715	0.566335	0.283167	+	0.959071i	$0.408615\pi$
$878$	0	0
$879$	17.2860	0.583042
$880$	0	0
$881$	−19.8175	−0.667669	−0.333835	−	0.942632i	$-0.608343\pi$
$881$	−19.8175	−0.667669	−0.333835	+	0.942632i	$0.608343\pi$
$882$	0	0
$883$	12.2008	0.410589	0.205294	−	0.978700i	$-0.434185\pi$
$883$	12.2008	0.410589	0.205294	+	0.978700i	$0.434185\pi$
$884$	0	0
$885$	21.8715	0.735201
$886$	0	0
$887$	−4.04346	−0.135766	−0.0678831	−	0.997693i	$-0.521624\pi$
$887$	−4.04346	−0.135766	−0.0678831	+	0.997693i	$0.521624\pi$
$888$	0	0
$889$	9.92417	0.332846
$890$	0	0
$891$	104.797	3.51084
$892$	0	0
$893$	16.7090	0.559144
$894$	0	0
$895$	−20.2829	−0.677981
$896$	0	0
$897$	8.32150	0.277847
$898$	0	0
$899$	−57.2719	−1.91013
$900$	0	0
$901$	−0.335577	−0.0111797
$902$	0	0
$903$	−11.4145	−0.379852
$904$	0	0
$905$	−14.3012	−0.475386
$906$	0	0
$907$	38.1642	1.26722	0.633611	−	0.773652i	$-0.281572\pi$
$907$	38.1642	1.26722	0.633611	+	0.773652i	$0.281572\pi$
$908$	0	0
$909$	−141.195	−4.68315
$910$	0	0
$911$	−20.5279	−0.680120	−0.340060	−	0.940404i	$-0.610447\pi$
$911$	−20.5279	−0.680120	−0.340060	+	0.940404i	$0.610447\pi$
$912$	0	0
$913$	8.66273	0.286695
$914$	0	0
$915$	54.9933	1.81802
$916$	0	0
$917$	21.3387	0.704666
$918$	0	0
$919$	10.7679	0.355202	0.177601	−	0.984103i	$-0.443166\pi$
$919$	10.7679	0.355202	0.177601	+	0.984103i	$0.443166\pi$
$920$	0	0
$921$	42.9076	1.41385
$922$	0	0
$923$	−1.63565	−0.0538382
$924$	0	0
$925$	32.4752	1.06778
$926$	0	0
$927$	132.644	4.35661
$928$	0	0
$929$	18.3461	0.601915	0.300957	−	0.953638i	$-0.402694\pi$
$929$	18.3461	0.601915	0.300957	+	0.953638i	$0.402694\pi$
$930$	0	0
$931$	32.2070	1.05554
$932$	0	0
$933$	31.0894	1.01782
$934$	0	0
$935$	3.07896	0.100693
$936$	0	0
$937$	−2.95402	−0.0965036	−0.0482518	−	0.998835i	$-0.515365\pi$
$937$	−2.95402	−0.0965036	−0.0482518	+	0.998835i	$0.515365\pi$
$938$	0	0
$939$	−25.2860	−0.825177
$940$	0	0
$941$	15.4475	0.503575	0.251787	−	0.967783i	$-0.418982\pi$
$941$	15.4475	0.503575	0.251787	+	0.967783i	$0.418982\pi$
$942$	0	0
$943$	6.48929	0.211320
$944$	0	0
$945$	22.7350	0.739568
$946$	0	0
$947$	−21.2510	−0.690563	−0.345282	−	0.938499i	$-0.612217\pi$
$947$	−21.2510	−0.690563	−0.345282	+	0.938499i	$0.612217\pi$
$948$	0	0
$949$	4.73917	0.153840
$950$	0	0
$951$	−44.3650	−1.43863
$952$	0	0
$953$	−8.82908	−0.286002	−0.143001	−	0.989723i	$-0.545675\pi$
$953$	−8.82908	−0.286002	−0.143001	+	0.989723i	$0.545675\pi$
$954$	0	0
$955$	−12.8929	−0.417204
$956$	0	0
$957$	72.3895	2.34002
$958$	0	0
$959$	8.90131	0.287438
$960$	0	0
$961$	38.2474	1.23379
$962$	0	0
$963$	−30.3074	−0.976643
$964$	0	0
$965$	−1.34713	−0.0433658
$966$	0	0
$967$	−36.7146	−1.18066	−0.590331	−	0.807161i	$-0.701003\pi$
$967$	−36.7146	−1.18066	−0.590331	+	0.807161i	$0.701003\pi$
$968$	0	0
$969$	−16.1642	−0.519269
$970$	0	0
$971$	−42.1642	−1.35311	−0.676557	−	0.736391i	$-0.736529\pi$
$971$	−42.1642	−1.35311	−0.676557	+	0.736391i	$0.736529\pi$
$972$	0	0
$973$	10.6613	0.341785
$974$	0	0
$975$	30.6718	0.982283
$976$	0	0
$977$	5.96702	0.190902	0.0954509	−	0.995434i	$-0.469571\pi$
$977$	5.96702	0.190902	0.0954509	+	0.995434i	$0.469571\pi$
$978$	0	0
$979$	−49.6018	−1.58528
$980$	0	0
$981$	37.4868	1.19686
$982$	0	0
$983$	43.3007	1.38108	0.690539	−	0.723295i	$-0.257373\pi$
$983$	43.3007	1.38108	0.690539	+	0.723295i	$0.257373\pi$
$984$	0	0
$985$	−6.15310	−0.196054
$986$	0	0
$987$	−11.3043	−0.359819
$988$	0	0
$989$	2.97858	0.0947132
$990$	0	0
$991$	37.4868	1.19081	0.595403	−	0.803427i	$-0.296992\pi$
$991$	37.4868	1.19081	0.595403	+	0.803427i	$0.296992\pi$
$992$	0	0
$993$	−34.1966	−1.08519
$994$	0	0
$995$	−11.2222	−0.355767
$996$	0	0
$997$	38.5939	1.22228	0.611140	−	0.791522i	$-0.290711\pi$
$997$	38.5939	1.22228	0.611140	+	0.791522i	$0.290711\pi$
$998$	0	0
$999$	−152.427	−4.82259

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.2.a.x.1.3		3
4.3	odd	2		1472.2.a.w.1.1		3
8.3	odd	2		736.2.a.f.1.3	yes	3
8.5	even	2		736.2.a.e.1.1	✓	3
24.5	odd	2		6624.2.a.z.1.2		3
24.11	even	2		6624.2.a.y.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.2.a.e.1.1	✓	3	8.5	even	2
736.2.a.f.1.3	yes	3	8.3	odd	2
1472.2.a.w.1.1		3	4.3	odd	2
1472.2.a.x.1.3		3	1.1	even	1	trivial
6624.2.a.y.1.2		3	24.11	even	2
6624.2.a.z.1.2		3	24.5	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q+3.34292 q^{3} +1.14637 q^{5} +1.14637 q^{7} +8.17513 q^{9} +3.14637 q^{11} -2.48929 q^{13} +3.83221 q^{15} +0.853635 q^{17} -5.66442 q^{19} +3.83221 q^{21} -1.00000 q^{23} -3.68585 q^{25} +17.3001 q^{27} +6.88240 q^{29} -8.32150 q^{31} +10.5181 q^{33} +1.31415 q^{35} -8.81079 q^{37} -8.32150 q^{39} -6.48929 q^{41} -2.97858 q^{43} +9.37169 q^{45} -2.94981 q^{47} -5.68585 q^{49} +2.85363 q^{51} -0.393115 q^{53} +3.60688 q^{55} -18.9357 q^{57} +5.70727 q^{59} +14.3503 q^{61} +9.37169 q^{63} -2.85363 q^{65} +7.93260 q^{67} -3.34292 q^{69} +0.657077 q^{71} -1.90383 q^{73} -12.3215 q^{75} +3.60688 q^{77} +16.0575 q^{79} +33.3074 q^{81} +2.75325 q^{83} +0.978577 q^{85} +23.0073 q^{87} -15.7648 q^{89} -2.85363 q^{91} -27.8181 q^{93} -6.49350 q^{95} -14.8108 q^{97} +25.7220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{3} + 2 q^{5} + 2 q^{7} + 5 q^{9} + 8 q^{11} - 2 q^{15} + 4 q^{17} + 10 q^{19} - 2 q^{21} - 3 q^{23} + q^{25} + 16 q^{27} + 4 q^{29} - 4 q^{31} + 6 q^{33} + 16 q^{35} + 2 q^{37} - 4 q^{39} - 12 q^{41}+ \cdots + 14 q^{99}+O(q^{100}) \) 3 * q + 4 * q^3 + 2 * q^5 + 2 * q^7 + 5 * q^9 + 8 * q^11 - 2 * q^15 + 4 * q^17 + 10 * q^19 - 2 * q^21 - 3 * q^23 + q^25 + 16 * q^27 + 4 * q^29 - 4 * q^31 + 6 * q^33 + 16 * q^35 + 2 * q^37 - 4 * q^39 - 12 * q^41 + 6 * q^43 + 4 * q^45 - 12 * q^47 - 5 * q^49 + 10 * q^51 + 8 * q^53 + 20 * q^55 - 12 * q^57 + 20 * q^59 + 4 * q^61 + 4 * q^63 - 10 * q^65 + 4 * q^67 - 4 * q^69 + 8 * q^71 - 4 * q^73 - 16 * q^75 + 20 * q^77 + 12 * q^79 + 31 * q^81 + 16 * q^83 - 12 * q^85 + 36 * q^87 - 14 * q^89 - 10 * q^91 - 22 * q^93 - 4 * q^95 - 16 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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