LMFDB - Embedded newform 1472.4.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,4,Mod(1,1472)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1472.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$86.8508115285$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.334189.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 16x^{2} - 5x + 4 $ x^4 - 2x^3 - 16x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.362907$ 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)

$f(q)$	$=$	$q-4.41777 q^{3} -7.80430 q^{5} +27.0572 q^{7} -7.48328 q^{9} +35.2507 q^{11} -58.1812 q^{13} +34.4776 q^{15} +98.3726 q^{17} -35.3289 q^{19} -119.533 q^{21} +23.0000 q^{23} -64.0928 q^{25} +152.339 q^{27} +235.531 q^{29} +55.0241 q^{31} -155.730 q^{33} -211.163 q^{35} -401.458 q^{37} +257.031 q^{39} -59.2600 q^{41} +11.2341 q^{43} +58.4018 q^{45} +103.224 q^{47} +389.093 q^{49} -434.588 q^{51} +351.594 q^{53} -275.107 q^{55} +156.075 q^{57} -547.016 q^{59} -478.070 q^{61} -202.477 q^{63} +454.063 q^{65} +14.3681 q^{67} -101.609 q^{69} -843.177 q^{71} +118.935 q^{73} +283.148 q^{75} +953.786 q^{77} +388.400 q^{79} -470.952 q^{81} -62.9800 q^{83} -767.730 q^{85} -1040.52 q^{87} +678.372 q^{89} -1574.22 q^{91} -243.084 q^{93} +275.717 q^{95} +421.192 q^{97} -263.791 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 7 q^{3} - 14 q^{5} - 16 q^{7} - 33 q^{9} + 8 q^{11} - 111 q^{13} - 10 q^{15} + 98 q^{17} + 96 q^{19} - 180 q^{21} + 92 q^{23} + 184 q^{25} - 155 q^{27} - 21 q^{29} + 193 q^{31} - 418 q^{33} - 752 q^{35}+ \cdots - 1498 q^{99}+O(q^{100}) $ 4 * q + 7 * q^3 - 14 * q^5 - 16 * q^7 - 33 * q^9 + 8 * q^11 - 111 * q^13 - 10 * q^15 + 98 * q^17 + 96 * q^19 - 180 * q^21 + 92 * q^23 + 184 * q^25 - 155 * q^27 - 21 * q^29 + 193 * q^31 - 418 * q^33 - 752 * q^35 - 170 * q^37 + 291 * q^39 - 125 * q^41 + 2 * q^43 - 168 * q^45 + 677 * q^47 + 1220 * q^49 - 340 * q^51 + 230 * q^53 + 972 * q^55 + 1322 * q^57 - 1140 * q^59 - 754 * q^61 + 1092 * q^63 + 1318 * q^65 + 488 * q^67 + 161 * q^69 + 401 * q^71 + 1509 * q^73 + 1401 * q^75 - 736 * q^77 + 838 * q^79 - 932 * q^81 + 142 * q^83 - 112 * q^85 - 2223 * q^87 + 2342 * q^89 + 292 * q^91 + 509 * q^93 + 956 * q^95 + 1062 * q^97 - 1498 * 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
$3$	−4.41777	−0.850201	−0.425100	−	0.905146i	$-0.639761\pi$
$3$	−4.41777	−0.850201	−0.425100	+	0.905146i	$0.639761\pi$
$4$	0	0
$5$	−7.80430	−0.698038	−0.349019	−	0.937116i	$-0.613485\pi$
$5$	−7.80430	−0.698038	−0.349019	+	0.937116i	$0.613485\pi$
$6$	0	0
$7$	27.0572	1.46095	0.730476	−	0.682938i	$-0.239298\pi$
$7$	27.0572	1.46095	0.730476	+	0.682938i	$0.239298\pi$
$8$	0	0
$9$	−7.48328	−0.277158
$10$	0	0
$11$	35.2507	0.966226	0.483113	−	0.875558i	$-0.339506\pi$
$11$	35.2507	0.966226	0.483113	+	0.875558i	$0.339506\pi$
$12$	0	0
$13$	−58.1812	−1.24127	−0.620637	−	0.784098i	$-0.713126\pi$
$13$	−58.1812	−1.24127	−0.620637	+	0.784098i	$0.713126\pi$
$14$	0	0
$15$	34.4776	0.593473
$16$	0	0
$17$	98.3726	1.40346	0.701731	−	0.712442i	$-0.252411\pi$
$17$	98.3726	1.40346	0.701731	+	0.712442i	$0.252411\pi$
$18$	0	0
$19$	−35.3289	−0.426579	−0.213289	−	0.976989i	$-0.568418\pi$
$19$	−35.3289	−0.426579	−0.213289	+	0.976989i	$0.568418\pi$
$20$	0	0
$21$	−119.533	−1.24210
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−64.0928	−0.512743
$26$	0	0
$27$	152.339	1.08584
$28$	0	0
$29$	235.531	1.50817	0.754087	−	0.656774i	$-0.228080\pi$
$29$	235.531	1.50817	0.754087	+	0.656774i	$0.228080\pi$
$30$	0	0
$31$	55.0241	0.318794	0.159397	−	0.987215i	$-0.449045\pi$
$31$	55.0241	0.318794	0.159397	+	0.987215i	$0.449045\pi$
$32$	0	0
$33$	−155.730	−0.821486
$34$	0	0
$35$	−211.163	−1.01980
$36$	0	0
$37$	−401.458	−1.78377	−0.891883	−	0.452266i	$-0.850616\pi$
$37$	−401.458	−1.78377	−0.891883	+	0.452266i	$0.850616\pi$
$38$	0	0
$39$	257.031	1.05533
$40$	0	0
$41$	−59.2600	−0.225728	−0.112864	−	0.993610i	$-0.536002\pi$
$41$	−59.2600	−0.225728	−0.112864	+	0.993610i	$0.536002\pi$
$42$	0	0
$43$	11.2341	0.0398416	0.0199208	−	0.999802i	$-0.493659\pi$
$43$	11.2341	0.0398416	0.0199208	+	0.999802i	$0.493659\pi$
$44$	0	0
$45$	58.4018	0.193467
$46$	0	0
$47$	103.224	0.320355	0.160178	−	0.987088i	$-0.448793\pi$
$47$	103.224	0.320355	0.160178	+	0.987088i	$0.448793\pi$
$48$	0	0
$49$	389.093	1.13438
$50$	0	0
$51$	−434.588	−1.19323
$52$	0	0
$53$	351.594	0.911229	0.455614	−	0.890177i	$-0.349420\pi$
$53$	351.594	0.911229	0.455614	+	0.890177i	$0.349420\pi$
$54$	0	0
$55$	−275.107	−0.674463
$56$	0	0
$57$	156.075	0.362678
$58$	0	0
$59$	−547.016	−1.20704	−0.603520	−	0.797347i	$-0.706236\pi$
$59$	−547.016	−1.20704	−0.603520	+	0.797347i	$0.706236\pi$
$60$	0	0
$61$	−478.070	−1.00345	−0.501726	−	0.865026i	$-0.667302\pi$
$61$	−478.070	−1.00345	−0.501726	+	0.865026i	$0.667302\pi$
$62$	0	0
$63$	−202.477	−0.404915
$64$	0	0
$65$	454.063	0.866456
$66$	0	0
$67$	14.3681	0.0261992	0.0130996	−	0.999914i	$-0.495830\pi$
$67$	14.3681	0.0261992	0.0130996	+	0.999914i	$0.495830\pi$
$68$	0	0
$69$	−101.609	−0.177279
$70$	0	0
$71$	−843.177	−1.40939	−0.704695	−	0.709510i	$-0.748916\pi$
$71$	−843.177	−1.40939	−0.704695	+	0.709510i	$0.748916\pi$
$72$	0	0
$73$	118.935	0.190689	0.0953445	−	0.995444i	$-0.469605\pi$
$73$	118.935	0.190689	0.0953445	+	0.995444i	$0.469605\pi$
$74$	0	0
$75$	283.148	0.435934
$76$	0	0
$77$	953.786	1.41161
$78$	0	0
$79$	388.400	0.553145	0.276572	−	0.960993i	$-0.410801\pi$
$79$	388.400	0.553145	0.276572	+	0.960993i	$0.410801\pi$
$80$	0	0
$81$	−470.952	−0.646025
$82$	0	0
$83$	−62.9800	−0.0832886	−0.0416443	−	0.999132i	$-0.513260\pi$
$83$	−62.9800	−0.0832886	−0.0416443	+	0.999132i	$0.513260\pi$
$84$	0	0
$85$	−767.730	−0.979671
$86$	0	0
$87$	−1040.52	−1.28225
$88$	0	0
$89$	678.372	0.807947	0.403974	−	0.914771i	$-0.367629\pi$
$89$	678.372	0.807947	0.403974	+	0.914771i	$0.367629\pi$
$90$	0	0
$91$	−1574.22	−1.81344
$92$	0	0
$93$	−243.084	−0.271039
$94$	0	0
$95$	275.717	0.297768
$96$	0	0
$97$	421.192	0.440882	0.220441	−	0.975400i	$-0.429250\pi$
$97$	421.192	0.440882	0.220441	+	0.975400i	$0.429250\pi$
$98$	0	0
$99$	−263.791	−0.267798
$100$	0	0
$101$	1.65795	0.00163339	0.000816695	−	1.00000i	$-0.499740\pi$
$101$	1.65795	0.00163339	0.000816695		1.00000i	$0.499740\pi$
$102$	0	0
$103$	−1057.63	−1.01176	−0.505879	−	0.862604i	$-0.668832\pi$
$103$	−1057.63	−1.01176	−0.505879	+	0.862604i	$0.668832\pi$
$104$	0	0
$105$	932.869	0.867035
$106$	0	0
$107$	−155.626	−0.140607	−0.0703036	−	0.997526i	$-0.522397\pi$
$107$	−155.626	−0.140607	−0.0703036	+	0.997526i	$0.522397\pi$
$108$	0	0
$109$	1815.84	1.59565	0.797826	−	0.602888i	$-0.205983\pi$
$109$	1815.84	1.59565	0.797826	+	0.602888i	$0.205983\pi$
$110$	0	0
$111$	1773.55	1.51656
$112$	0	0
$113$	485.235	0.403956	0.201978	−	0.979390i	$-0.435263\pi$
$113$	485.235	0.403956	0.201978	+	0.979390i	$0.435263\pi$
$114$	0	0
$115$	−179.499	−0.145551
$116$	0	0
$117$	435.386	0.344029
$118$	0	0
$119$	2661.69	2.05039
$120$	0	0
$121$	−88.3874	−0.0664068
$122$	0	0
$123$	261.797	0.191914
$124$	0	0
$125$	1475.74	1.05595
$126$	0	0
$127$	−2468.34	−1.72464	−0.862322	−	0.506360i	$-0.830991\pi$
$127$	−2468.34	−1.72464	−0.862322	+	0.506360i	$0.830991\pi$
$128$	0	0
$129$	−49.6298	−0.0338734
$130$	0	0
$131$	1071.51	0.714647	0.357323	−	0.933981i	$-0.383689\pi$
$131$	1071.51	0.714647	0.357323	+	0.933981i	$0.383689\pi$
$132$	0	0
$133$	−955.901	−0.623212
$134$	0	0
$135$	−1188.90	−0.757959
$136$	0	0
$137$	241.497	0.150602	0.0753009	−	0.997161i	$-0.476008\pi$
$137$	241.497	0.150602	0.0753009	+	0.997161i	$0.476008\pi$
$138$	0	0
$139$	2659.19	1.62266	0.811329	−	0.584590i	$-0.198745\pi$
$139$	2659.19	1.62266	0.811329	+	0.584590i	$0.198745\pi$
$140$	0	0
$141$	−456.018	−0.272366
$142$	0	0
$143$	−2050.93	−1.19935
$144$	0	0
$145$	−1838.16	−1.05276
$146$	0	0
$147$	−1718.93	−0.964453
$148$	0	0
$149$	182.051	0.100095	0.0500477	−	0.998747i	$-0.484063\pi$
$149$	182.051	0.100095	0.0500477	+	0.998747i	$0.484063\pi$
$150$	0	0
$151$	3377.13	1.82005	0.910023	−	0.414557i	$-0.136063\pi$
$151$	3377.13	1.82005	0.910023	+	0.414557i	$0.136063\pi$
$152$	0	0
$153$	−736.150	−0.388981
$154$	0	0
$155$	−429.425	−0.222531
$156$	0	0
$157$	2843.19	1.44530	0.722648	−	0.691216i	$-0.242925\pi$
$157$	2843.19	1.44530	0.722648	+	0.691216i	$0.242925\pi$
$158$	0	0
$159$	−1553.26	−0.774728
$160$	0	0
$161$	622.316	0.304630
$162$	0	0
$163$	2135.85	1.02633	0.513167	−	0.858289i	$-0.328472\pi$
$163$	2135.85	1.02633	0.513167	+	0.858289i	$0.328472\pi$
$164$	0	0
$165$	1215.36	0.573429
$166$	0	0
$167$	−796.566	−0.369103	−0.184551	−	0.982823i	$-0.559083\pi$
$167$	−796.566	−0.369103	−0.184551	+	0.982823i	$0.559083\pi$
$168$	0	0
$169$	1188.05	0.540759
$170$	0	0
$171$	264.376	0.118230
$172$	0	0
$173$	4456.05	1.95831	0.979154	−	0.203121i	$-0.0651085\pi$
$173$	4456.05	1.95831	0.979154	+	0.203121i	$0.0651085\pi$
$174$	0	0
$175$	−1734.17	−0.749093
$176$	0	0
$177$	2416.59	1.02623
$178$	0	0
$179$	−693.459	−0.289562	−0.144781	−	0.989464i	$-0.546248\pi$
$179$	−693.459	−0.289562	−0.144781	+	0.989464i	$0.546248\pi$
$180$	0	0
$181$	974.743	0.400288	0.200144	−	0.979767i	$-0.435859\pi$
$181$	974.743	0.400288	0.200144	+	0.979767i	$0.435859\pi$
$182$	0	0
$183$	2112.01	0.853136
$184$	0	0
$185$	3133.10	1.24514
$186$	0	0
$187$	3467.70	1.35606
$188$	0	0
$189$	4121.88	1.58636
$190$	0	0
$191$	775.067	0.293623	0.146811	−	0.989165i	$-0.453099\pi$
$191$	775.067	0.293623	0.146811	+	0.989165i	$0.453099\pi$
$192$	0	0
$193$	2021.09	0.753789	0.376895	−	0.926256i	$-0.376992\pi$
$193$	2021.09	0.753789	0.376895	+	0.926256i	$0.376992\pi$
$194$	0	0
$195$	−2005.95	−0.736662
$196$	0	0
$197$	−566.907	−0.205028	−0.102514	−	0.994732i	$-0.532689\pi$
$197$	−566.907	−0.205028	−0.102514	+	0.994732i	$0.532689\pi$
$198$	0	0
$199$	1724.07	0.614150	0.307075	−	0.951685i	$-0.400650\pi$
$199$	1724.07	0.614150	0.307075	+	0.951685i	$0.400650\pi$
$200$	0	0
$201$	−63.4751	−0.0222746
$202$	0	0
$203$	6372.82	2.20337
$204$	0	0
$205$	462.483	0.157567
$206$	0	0
$207$	−172.115	−0.0577915
$208$	0	0
$209$	−1245.37	−0.412172
$210$	0	0
$211$	2757.94	0.899832	0.449916	−	0.893071i	$-0.351454\pi$
$211$	2757.94	0.899832	0.449916	+	0.893071i	$0.351454\pi$
$212$	0	0
$213$	3724.97	1.19827
$214$	0	0
$215$	−87.6745	−0.0278109
$216$	0	0
$217$	1488.80	0.465743
$218$	0	0
$219$	−525.428	−0.162124
$220$	0	0
$221$	−5723.43	−1.74208
$222$	0	0
$223$	4257.81	1.27858	0.639292	−	0.768964i	$-0.279227\pi$
$223$	4257.81	1.27858	0.639292	+	0.768964i	$0.279227\pi$
$224$	0	0
$225$	479.624	0.142111
$226$	0	0
$227$	4320.03	1.26313	0.631564	−	0.775324i	$-0.282413\pi$
$227$	4320.03	1.26313	0.631564	+	0.775324i	$0.282413\pi$
$228$	0	0
$229$	−1057.27	−0.305094	−0.152547	−	0.988296i	$-0.548747\pi$
$229$	−1057.27	−0.305094	−0.152547	+	0.988296i	$0.548747\pi$
$230$	0	0
$231$	−4213.61	−1.20015
$232$	0	0
$233$	−3194.98	−0.898327	−0.449163	−	0.893450i	$-0.648278\pi$
$233$	−3194.98	−0.898327	−0.449163	+	0.893450i	$0.648278\pi$
$234$	0	0
$235$	−805.588	−0.223620
$236$	0	0
$237$	−1715.86	−0.470284
$238$	0	0
$239$	−131.327	−0.0355433	−0.0177717	−	0.999842i	$-0.505657\pi$
$239$	−131.327	−0.0355433	−0.0177717	+	0.999842i	$0.505657\pi$
$240$	0	0
$241$	2429.45	0.649356	0.324678	−	0.945825i	$-0.394744\pi$
$241$	2429.45	0.649356	0.324678	+	0.945825i	$0.394744\pi$
$242$	0	0
$243$	−2032.60	−0.536590
$244$	0	0
$245$	−3036.60	−0.791842
$246$	0	0
$247$	2055.47	0.529501
$248$	0	0
$249$	278.232	0.0708121
$250$	0	0
$251$	689.940	0.173501	0.0867503	−	0.996230i	$-0.472352\pi$
$251$	689.940	0.173501	0.0867503	+	0.996230i	$0.472352\pi$
$252$	0	0
$253$	810.766	0.201472
$254$	0	0
$255$	3391.66	0.832917
$256$	0	0
$257$	−7466.72	−1.81230	−0.906151	−	0.422955i	$-0.860993\pi$
$257$	−7466.72	−1.81230	−0.906151	+	0.422955i	$0.860993\pi$
$258$	0	0
$259$	−10862.3	−2.60600
$260$	0	0
$261$	−1762.55	−0.418003
$262$	0	0
$263$	1688.09	0.395786	0.197893	−	0.980224i	$-0.436590\pi$
$263$	1688.09	0.395786	0.197893	+	0.980224i	$0.436590\pi$
$264$	0	0
$265$	−2743.95	−0.636072
$266$	0	0
$267$	−2996.89	−0.686917
$268$	0	0
$269$	3665.13	0.830731	0.415365	−	0.909655i	$-0.363654\pi$
$269$	3665.13	0.830731	0.415365	+	0.909655i	$0.363654\pi$
$270$	0	0
$271$	−289.699	−0.0649371	−0.0324686	−	0.999473i	$-0.510337\pi$
$271$	−289.699	−0.0649371	−0.0324686	+	0.999473i	$0.510337\pi$
$272$	0	0
$273$	6954.55	1.54179
$274$	0	0
$275$	−2259.32	−0.495425
$276$	0	0
$277$	−0.383938	−8.32802e−5 0	−4.16401e−5	−	1.00000i	$-0.500013\pi$
$277$	−0.383938	−8.32802e−5 0	−4.16401e−5		1.00000i	$0.500013\pi$
$278$	0	0
$279$	−411.761	−0.0883565
$280$	0	0
$281$	−3507.39	−0.744602	−0.372301	−	0.928112i	$-0.621431\pi$
$281$	−3507.39	−0.744602	−0.372301	+	0.928112i	$0.621431\pi$
$282$	0	0
$283$	4231.65	0.888854	0.444427	−	0.895815i	$-0.353407\pi$
$283$	4231.65	0.888854	0.444427	+	0.895815i	$0.353407\pi$
$284$	0	0
$285$	−1218.06	−0.253163
$286$	0	0
$287$	−1603.41	−0.329778
$288$	0	0
$289$	4764.17	0.969707
$290$	0	0
$291$	−1860.73	−0.374839
$292$	0	0
$293$	1066.58	0.212663	0.106331	−	0.994331i	$-0.466090\pi$
$293$	1066.58	0.212663	0.106331	+	0.994331i	$0.466090\pi$
$294$	0	0
$295$	4269.08	0.842561
$296$	0	0
$297$	5370.07	1.04917
$298$	0	0
$299$	−1338.17	−0.258823
$300$	0	0
$301$	303.964	0.0582067
$302$	0	0
$303$	−7.32446	−0.00138871
$304$	0	0
$305$	3731.00	0.700448
$306$	0	0
$307$	−2744.52	−0.510222	−0.255111	−	0.966912i	$-0.582112\pi$
$307$	−2744.52	−0.510222	−0.255111	+	0.966912i	$0.582112\pi$
$308$	0	0
$309$	4672.36	0.860198
$310$	0	0
$311$	−2719.23	−0.495798	−0.247899	−	0.968786i	$-0.579740\pi$
$311$	−2719.23	−0.495798	−0.247899	+	0.968786i	$0.579740\pi$
$312$	0	0
$313$	5919.04	1.06890	0.534448	−	0.845202i	$-0.320520\pi$
$313$	5919.04	1.06890	0.534448	+	0.845202i	$0.320520\pi$
$314$	0	0
$315$	1580.19	0.282646
$316$	0	0
$317$	−1327.55	−0.235213	−0.117606	−	0.993060i	$-0.537522\pi$
$317$	−1327.55	−0.235213	−0.117606	+	0.993060i	$0.537522\pi$
$318$	0	0
$319$	8302.64	1.45724
$320$	0	0
$321$	687.522	0.119544
$322$	0	0
$323$	−3475.39	−0.598688
$324$	0	0
$325$	3729.00	0.636454
$326$	0	0
$327$	−8021.98	−1.35663
$328$	0	0
$329$	2792.94	0.468024
$330$	0	0
$331$	3236.92	0.537514	0.268757	−	0.963208i	$-0.413387\pi$
$331$	3236.92	0.537514	0.268757	+	0.963208i	$0.413387\pi$
$332$	0	0
$333$	3004.22	0.494386
$334$	0	0
$335$	−112.133	−0.0182880
$336$	0	0
$337$	12263.3	1.98226	0.991132	−	0.132878i	$-0.0424220\pi$
$337$	12263.3	1.98226	0.991132	+	0.132878i	$0.0424220\pi$
$338$	0	0
$339$	−2143.66	−0.343444
$340$	0	0
$341$	1939.64	0.308028
$342$	0	0
$343$	1247.15	0.196326
$344$	0	0
$345$	792.986	0.123748
$346$	0	0
$347$	−10377.8	−1.60551	−0.802753	−	0.596311i	$-0.796632\pi$
$347$	−10377.8	−1.60551	−0.802753	+	0.596311i	$0.796632\pi$
$348$	0	0
$349$	−4837.57	−0.741975	−0.370987	−	0.928638i	$-0.620981\pi$
$349$	−4837.57	−0.741975	−0.370987	+	0.928638i	$0.620981\pi$
$350$	0	0
$351$	−8863.28	−1.34783
$352$	0	0
$353$	6761.01	1.01941	0.509706	−	0.860349i	$-0.329754\pi$
$353$	6761.01	1.01941	0.509706	+	0.860349i	$0.329754\pi$
$354$	0	0
$355$	6580.41	0.983809
$356$	0	0
$357$	−11758.7	−1.74325
$358$	0	0
$359$	−4539.67	−0.667394	−0.333697	−	0.942680i	$-0.608296\pi$
$359$	−4539.67	−0.667394	−0.333697	+	0.942680i	$0.608296\pi$
$360$	0	0
$361$	−5610.87	−0.818030
$362$	0	0
$363$	390.476	0.0564591
$364$	0	0
$365$	−928.206	−0.133108
$366$	0	0
$367$	9564.79	1.36043	0.680215	−	0.733012i	$-0.261886\pi$
$367$	9564.79	1.36043	0.680215	+	0.733012i	$0.261886\pi$
$368$	0	0
$369$	443.459	0.0625625
$370$	0	0
$371$	9513.15	1.33126
$372$	0	0
$373$	5028.99	0.698100	0.349050	−	0.937104i	$-0.386504\pi$
$373$	5028.99	0.698100	0.349050	+	0.937104i	$0.386504\pi$
$374$	0	0
$375$	−6519.48	−0.897772
$376$	0	0
$377$	−13703.5	−1.87206
$378$	0	0
$379$	560.190	0.0759235	0.0379618	−	0.999279i	$-0.487913\pi$
$379$	560.190	0.0759235	0.0379618	+	0.999279i	$0.487913\pi$
$380$	0	0
$381$	10904.6	1.46629
$382$	0	0
$383$	−5273.52	−0.703563	−0.351781	−	0.936082i	$-0.614424\pi$
$383$	−5273.52	−0.703563	−0.351781	+	0.936082i	$0.614424\pi$
$384$	0	0
$385$	−7443.64	−0.985358
$386$	0	0
$387$	−84.0681	−0.0110424
$388$	0	0
$389$	9371.31	1.22145	0.610725	−	0.791843i	$-0.290878\pi$
$389$	9371.31	1.22145	0.610725	+	0.791843i	$0.290878\pi$
$390$	0	0
$391$	2262.57	0.292642
$392$	0	0
$393$	−4733.71	−0.607593
$394$	0	0
$395$	−3031.19	−0.386116
$396$	0	0
$397$	−6200.86	−0.783909	−0.391955	−	0.919985i	$-0.628201\pi$
$397$	−6200.86	−0.783909	−0.391955	+	0.919985i	$0.628201\pi$
$398$	0	0
$399$	4222.95	0.529855
$400$	0	0
$401$	2600.97	0.323906	0.161953	−	0.986798i	$-0.448221\pi$
$401$	2600.97	0.323906	0.161953	+	0.986798i	$0.448221\pi$
$402$	0	0
$403$	−3201.37	−0.395711
$404$	0	0
$405$	3675.45	0.450950
$406$	0	0
$407$	−14151.7	−1.72352
$408$	0	0
$409$	6260.14	0.756831	0.378416	−	0.925636i	$-0.376469\pi$
$409$	6260.14	0.756831	0.378416	+	0.925636i	$0.376469\pi$
$410$	0	0
$411$	−1066.88	−0.128042
$412$	0	0
$413$	−14800.7	−1.76343
$414$	0	0
$415$	491.515	0.0581387
$416$	0	0
$417$	−11747.7	−1.37959
$418$	0	0
$419$	−6352.30	−0.740645	−0.370322	−	0.928903i	$-0.620753\pi$
$419$	−6352.30	−0.740645	−0.370322	+	0.928903i	$0.620753\pi$
$420$	0	0
$421$	−3405.54	−0.394242	−0.197121	−	0.980379i	$-0.563159\pi$
$421$	−3405.54	−0.394242	−0.197121	+	0.980379i	$0.563159\pi$
$422$	0	0
$423$	−772.450	−0.0887892
$424$	0	0
$425$	−6304.98	−0.719615
$426$	0	0
$427$	−12935.2	−1.46600
$428$	0	0
$429$	9060.53	1.01969
$430$	0	0
$431$	17077.6	1.90858	0.954290	−	0.298881i	$-0.0966135\pi$
$431$	17077.6	1.90858	0.954290	+	0.298881i	$0.0966135\pi$
$432$	0	0
$433$	15884.5	1.76296	0.881480	−	0.472222i	$-0.156548\pi$
$433$	15884.5	1.76296	0.881480	+	0.472222i	$0.156548\pi$
$434$	0	0
$435$	8120.56	0.895060
$436$	0	0
$437$	−812.564	−0.0889479
$438$	0	0
$439$	5977.96	0.649914	0.324957	−	0.945729i	$-0.394650\pi$
$439$	5977.96	0.649914	0.324957	+	0.945729i	$0.394650\pi$
$440$	0	0
$441$	−2911.69	−0.314404
$442$	0	0
$443$	−1747.71	−0.187440	−0.0937202	−	0.995599i	$-0.529876\pi$
$443$	−1747.71	−0.187440	−0.0937202	+	0.995599i	$0.529876\pi$
$444$	0	0
$445$	−5294.22	−0.563978
$446$	0	0
$447$	−804.261	−0.0851012
$448$	0	0
$449$	10320.1	1.08472	0.542358	−	0.840148i	$-0.317532\pi$
$449$	10320.1	1.08472	0.542358	+	0.840148i	$0.317532\pi$
$450$	0	0
$451$	−2088.96	−0.218105
$452$	0	0
$453$	−14919.4	−1.54741
$454$	0	0
$455$	12285.7	1.26585
$456$	0	0
$457$	−11934.9	−1.22164	−0.610821	−	0.791769i	$-0.709161\pi$
$457$	−11934.9	−1.22164	−0.610821	+	0.791769i	$0.709161\pi$
$458$	0	0
$459$	14986.0	1.52394
$460$	0	0
$461$	5673.34	0.573175	0.286588	−	0.958054i	$-0.407479\pi$
$461$	5673.34	0.573175	0.286588	+	0.958054i	$0.407479\pi$
$462$	0	0
$463$	7559.91	0.758831	0.379416	−	0.925226i	$-0.376125\pi$
$463$	7559.91	0.758831	0.379416	+	0.925226i	$0.376125\pi$
$464$	0	0
$465$	1897.10	0.189196
$466$	0	0
$467$	1203.47	0.119250	0.0596252	−	0.998221i	$-0.481009\pi$
$467$	1203.47	0.119250	0.0596252	+	0.998221i	$0.481009\pi$
$468$	0	0
$469$	388.761	0.0382758
$470$	0	0
$471$	−12560.6	−1.22879
$472$	0	0
$473$	396.011	0.0384960
$474$	0	0
$475$	2264.33	0.218725
$476$	0	0
$477$	−2631.07	−0.252555
$478$	0	0
$479$	−7928.98	−0.756335	−0.378167	−	0.925737i	$-0.623446\pi$
$479$	−7928.98	−0.756335	−0.378167	+	0.925737i	$0.623446\pi$
$480$	0	0
$481$	23357.3	2.21414
$482$	0	0
$483$	−2749.25	−0.258996
$484$	0	0
$485$	−3287.11	−0.307753
$486$	0	0
$487$	−748.445	−0.0696412	−0.0348206	−	0.999394i	$-0.511086\pi$
$487$	−748.445	−0.0696412	−0.0348206	+	0.999394i	$0.511086\pi$
$488$	0	0
$489$	−9435.70	−0.872591
$490$	0	0
$491$	−2201.18	−0.202318	−0.101159	−	0.994870i	$-0.532255\pi$
$491$	−2201.18	−0.202318	−0.101159	+	0.994870i	$0.532255\pi$
$492$	0	0
$493$	23169.8	2.11667
$494$	0	0
$495$	2058.70	0.186933
$496$	0	0
$497$	−22814.0	−2.05905
$498$	0	0
$499$	−12268.4	−1.10062	−0.550308	−	0.834961i	$-0.685490\pi$
$499$	−12268.4	−1.10062	−0.550308	+	0.834961i	$0.685490\pi$
$500$	0	0
$501$	3519.05	0.313811
$502$	0	0
$503$	−10161.0	−0.900711	−0.450356	−	0.892849i	$-0.648703\pi$
$503$	−10161.0	−0.900711	−0.450356	+	0.892849i	$0.648703\pi$
$504$	0	0
$505$	−12.9392	−0.00114017
$506$	0	0
$507$	−5248.52	−0.459754
$508$	0	0
$509$	−18108.1	−1.57687	−0.788434	−	0.615119i	$-0.789108\pi$
$509$	−18108.1	−1.57687	−0.788434	+	0.615119i	$0.789108\pi$
$510$	0	0
$511$	3218.05	0.278588
$512$	0	0
$513$	−5381.98	−0.463197
$514$	0	0
$515$	8254.04	0.706246
$516$	0	0
$517$	3638.70	0.309536
$518$	0	0
$519$	−19685.8	−1.66495
$520$	0	0
$521$	−10987.7	−0.923950	−0.461975	−	0.886893i	$-0.652859\pi$
$521$	−10987.7	−0.923950	−0.461975	+	0.886893i	$0.652859\pi$
$522$	0	0
$523$	20489.8	1.71311	0.856553	−	0.516059i	$-0.172601\pi$
$523$	20489.8	1.71311	0.856553	+	0.516059i	$0.172601\pi$
$524$	0	0
$525$	7661.19	0.636879
$526$	0	0
$527$	5412.87	0.447416
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	4093.47	0.334542
$532$	0	0
$533$	3447.82	0.280190
$534$	0	0
$535$	1214.55	0.0981491
$536$	0	0
$537$	3063.54	0.246186
$538$	0	0
$539$	13715.8	1.09607
$540$	0	0
$541$	−1261.91	−0.100284	−0.0501420	−	0.998742i	$-0.515967\pi$
$541$	−1261.91	−0.100284	−0.0501420	+	0.998742i	$0.515967\pi$
$542$	0	0
$543$	−4306.20	−0.340325
$544$	0	0
$545$	−14171.4	−1.11383
$546$	0	0
$547$	−5337.23	−0.417191	−0.208596	−	0.978002i	$-0.566889\pi$
$547$	−5337.23	−0.417191	−0.208596	+	0.978002i	$0.566889\pi$
$548$	0	0
$549$	3577.53	0.278115
$550$	0	0
$551$	−8321.05	−0.643355
$552$	0	0
$553$	10509.0	0.808118
$554$	0	0
$555$	−13841.3	−1.05862
$556$	0	0
$557$	10891.2	0.828503	0.414251	−	0.910163i	$-0.364043\pi$
$557$	10891.2	0.828503	0.414251	+	0.910163i	$0.364043\pi$
$558$	0	0
$559$	−653.615	−0.0494543
$560$	0	0
$561$	−15319.5	−1.15293
$562$	0	0
$563$	19620.8	1.46877	0.734385	−	0.678733i	$-0.237471\pi$
$563$	19620.8	1.46877	0.734385	+	0.678733i	$0.237471\pi$
$564$	0	0
$565$	−3786.92	−0.281977
$566$	0	0
$567$	−12742.7	−0.943812
$568$	0	0
$569$	25207.8	1.85723	0.928617	−	0.371041i	$-0.120999\pi$
$569$	25207.8	1.85723	0.928617	+	0.371041i	$0.120999\pi$
$570$	0	0
$571$	18889.0	1.38438	0.692190	−	0.721715i	$-0.256646\pi$
$571$	18889.0	1.38438	0.692190	+	0.721715i	$0.256646\pi$
$572$	0	0
$573$	−3424.07	−0.249638
$574$	0	0
$575$	−1474.14	−0.106914
$576$	0	0
$577$	−8214.44	−0.592672	−0.296336	−	0.955084i	$-0.595765\pi$
$577$	−8214.44	−0.592672	−0.296336	+	0.955084i	$0.595765\pi$
$578$	0	0
$579$	−8928.72	−0.640872
$580$	0	0
$581$	−1704.06	−0.121681
$582$	0	0
$583$	12393.9	0.880453
$584$	0	0
$585$	−3397.88	−0.240146
$586$	0	0
$587$	3022.88	0.212551	0.106276	−	0.994337i	$-0.466107\pi$
$587$	3022.88	0.212551	0.106276	+	0.994337i	$0.466107\pi$
$588$	0	0
$589$	−1943.94	−0.135991
$590$	0	0
$591$	2504.47	0.174315
$592$	0	0
$593$	−16386.9	−1.13479	−0.567394	−	0.823446i	$-0.692048\pi$
$593$	−16386.9	−1.13479	−0.567394	+	0.823446i	$0.692048\pi$
$594$	0	0
$595$	−20772.6	−1.43125
$596$	0	0
$597$	−7616.53	−0.522151
$598$	0	0
$599$	4768.20	0.325247	0.162624	−	0.986688i	$-0.448004\pi$
$599$	4768.20	0.325247	0.162624	+	0.986688i	$0.448004\pi$
$600$	0	0
$601$	1849.74	0.125545	0.0627725	−	0.998028i	$-0.480006\pi$
$601$	1849.74	0.125545	0.0627725	+	0.998028i	$0.480006\pi$
$602$	0	0
$603$	−107.521	−0.00726132
$604$	0	0
$605$	689.802	0.0463545
$606$	0	0
$607$	−18350.2	−1.22704	−0.613519	−	0.789680i	$-0.710246\pi$
$607$	−18350.2	−1.22704	−0.613519	+	0.789680i	$0.710246\pi$
$608$	0	0
$609$	−28153.7	−1.87331
$610$	0	0
$611$	−6005.67	−0.397649
$612$	0	0
$613$	−13252.1	−0.873160	−0.436580	−	0.899666i	$-0.643810\pi$
$613$	−13252.1	−0.873160	−0.436580	+	0.899666i	$0.643810\pi$
$614$	0	0
$615$	−2043.15	−0.133964
$616$	0	0
$617$	−5203.51	−0.339523	−0.169761	−	0.985485i	$-0.554300\pi$
$617$	−5203.51	−0.339523	−0.169761	+	0.985485i	$0.554300\pi$
$618$	0	0
$619$	−3974.35	−0.258066	−0.129033	−	0.991640i	$-0.541187\pi$
$619$	−3974.35	−0.258066	−0.129033	+	0.991640i	$0.541187\pi$
$620$	0	0
$621$	3503.80	0.226414
$622$	0	0
$623$	18354.9	1.18037
$624$	0	0
$625$	−3505.50	−0.224352
$626$	0	0
$627$	5501.75	0.350429
$628$	0	0
$629$	−39492.5	−2.50345
$630$	0	0
$631$	−14357.8	−0.905827	−0.452913	−	0.891555i	$-0.649615\pi$
$631$	−14357.8	−0.905827	−0.452913	+	0.891555i	$0.649615\pi$
$632$	0	0
$633$	−12184.0	−0.765038
$634$	0	0
$635$	19263.7	1.20387
$636$	0	0
$637$	−22637.9	−1.40808
$638$	0	0
$639$	6309.73	0.390625
$640$	0	0
$641$	2587.81	0.159458	0.0797288	−	0.996817i	$-0.474595\pi$
$641$	2587.81	0.159458	0.0797288	+	0.996817i	$0.474595\pi$
$642$	0	0
$643$	−18495.1	−1.13433	−0.567167	−	0.823603i	$-0.691960\pi$
$643$	−18495.1	−1.13433	−0.567167	+	0.823603i	$0.691960\pi$
$644$	0	0
$645$	387.326	0.0236449
$646$	0	0
$647$	−5705.85	−0.346708	−0.173354	−	0.984860i	$-0.555460\pi$
$647$	−5705.85	−0.346708	−0.173354	+	0.984860i	$0.555460\pi$
$648$	0	0
$649$	−19282.7	−1.16627
$650$	0	0
$651$	−6577.18	−0.395975
$652$	0	0
$653$	−17240.3	−1.03318	−0.516588	−	0.856234i	$-0.672798\pi$
$653$	−17240.3	−1.03318	−0.516588	+	0.856234i	$0.672798\pi$
$654$	0	0
$655$	−8362.43	−0.498851
$656$	0	0
$657$	−890.024	−0.0528511
$658$	0	0
$659$	1656.26	0.0979041	0.0489520	−	0.998801i	$-0.484412\pi$
$659$	1656.26	0.0979041	0.0489520	+	0.998801i	$0.484412\pi$
$660$	0	0
$661$	14138.6	0.831965	0.415983	−	0.909373i	$-0.363438\pi$
$661$	14138.6	0.831965	0.415983	+	0.909373i	$0.363438\pi$
$662$	0	0
$663$	25284.8	1.48112
$664$	0	0
$665$	7460.14	0.435025
$666$	0	0
$667$	5417.22	0.314476
$668$	0	0
$669$	−18810.1	−1.08705
$670$	0	0
$671$	−16852.3	−0.969562
$672$	0	0
$673$	19008.2	1.08873	0.544363	−	0.838850i	$-0.316772\pi$
$673$	19008.2	1.08873	0.544363	+	0.838850i	$0.316772\pi$
$674$	0	0
$675$	−9763.86	−0.556757
$676$	0	0
$677$	2957.92	0.167920	0.0839602	−	0.996469i	$-0.473243\pi$
$677$	2957.92	0.167920	0.0839602	+	0.996469i	$0.473243\pi$
$678$	0	0
$679$	11396.3	0.644108
$680$	0	0
$681$	−19084.9	−1.07391
$682$	0	0
$683$	−23345.5	−1.30789	−0.653947	−	0.756540i	$-0.726888\pi$
$683$	−23345.5	−1.30789	−0.653947	+	0.756540i	$0.726888\pi$
$684$	0	0
$685$	−1884.71	−0.105126
$686$	0	0
$687$	4670.78	0.259391
$688$	0	0
$689$	−20456.1	−1.13108
$690$	0	0
$691$	−11901.8	−0.655234	−0.327617	−	0.944811i	$-0.606246\pi$
$691$	−11901.8	−0.655234	−0.327617	+	0.944811i	$0.606246\pi$
$692$	0	0
$693$	−7137.45	−0.391240
$694$	0	0
$695$	−20753.1	−1.13268
$696$	0	0
$697$	−5829.56	−0.316801
$698$	0	0
$699$	14114.7	0.763758
$700$	0	0
$701$	12803.0	0.689817	0.344909	−	0.938636i	$-0.387910\pi$
$701$	12803.0	0.689817	0.344909	+	0.938636i	$0.387910\pi$
$702$	0	0
$703$	14183.1	0.760917
$704$	0	0
$705$	3558.91	0.190122
$706$	0	0
$707$	44.8596	0.00238631
$708$	0	0
$709$	−8087.63	−0.428403	−0.214201	−	0.976790i	$-0.568715\pi$
$709$	−8087.63	−0.428403	−0.214201	+	0.976790i	$0.568715\pi$
$710$	0	0
$711$	−2906.51	−0.153309
$712$	0	0
$713$	1265.56	0.0664732
$714$	0	0
$715$	16006.1	0.837193
$716$	0	0
$717$	580.174	0.0302190
$718$	0	0
$719$	34338.9	1.78112	0.890560	−	0.454867i	$-0.150313\pi$
$719$	34338.9	1.78112	0.890560	+	0.454867i	$0.150313\pi$
$720$	0	0
$721$	−28616.4	−1.47813
$722$	0	0
$723$	−10732.8	−0.552083
$724$	0	0
$725$	−15095.9	−0.773305
$726$	0	0
$727$	−2109.84	−0.107633	−0.0538167	−	0.998551i	$-0.517139\pi$
$727$	−2109.84	−0.107633	−0.0538167	+	0.998551i	$0.517139\pi$
$728$	0	0
$729$	21695.3	1.10223
$730$	0	0
$731$	1105.13	0.0559162
$732$	0	0
$733$	−15133.9	−0.762599	−0.381299	−	0.924452i	$-0.624523\pi$
$733$	−15133.9	−0.762599	−0.381299	+	0.924452i	$0.624523\pi$
$734$	0	0
$735$	13415.0	0.673225
$736$	0	0
$737$	506.486	0.0253143
$738$	0	0
$739$	−29057.4	−1.44641	−0.723204	−	0.690635i	$-0.757331\pi$
$739$	−29057.4	−1.44641	−0.723204	+	0.690635i	$0.757331\pi$
$740$	0	0
$741$	−9080.62	−0.450182
$742$	0	0
$743$	3088.06	0.152476	0.0762382	−	0.997090i	$-0.475709\pi$
$743$	3088.06	0.152476	0.0762382	+	0.997090i	$0.475709\pi$
$744$	0	0
$745$	−1420.78	−0.0698704
$746$	0	0
$747$	471.297	0.0230841
$748$	0	0
$749$	−4210.81	−0.205420
$750$	0	0
$751$	11655.2	0.566317	0.283159	−	0.959073i	$-0.408618\pi$
$751$	11655.2	0.566317	0.283159	+	0.959073i	$0.408618\pi$
$752$	0	0
$753$	−3048.00	−0.147510
$754$	0	0
$755$	−26356.2	−1.27046
$756$	0	0
$757$	38979.8	1.87152	0.935762	−	0.352633i	$-0.114713\pi$
$757$	38979.8	1.87152	0.935762	+	0.352633i	$0.114713\pi$
$758$	0	0
$759$	−3581.78	−0.171292
$760$	0	0
$761$	8875.64	0.422788	0.211394	−	0.977401i	$-0.432200\pi$
$761$	8875.64	0.422788	0.211394	+	0.977401i	$0.432200\pi$
$762$	0	0
$763$	49131.6	2.33117
$764$	0	0
$765$	5745.13	0.271524
$766$	0	0
$767$	31826.0	1.49827
$768$	0	0
$769$	27567.7	1.29274	0.646370	−	0.763024i	$-0.276286\pi$
$769$	27567.7	1.29274	0.646370	+	0.763024i	$0.276286\pi$
$770$	0	0
$771$	32986.3	1.54082
$772$	0	0
$773$	5450.84	0.253626	0.126813	−	0.991927i	$-0.459525\pi$
$773$	5450.84	0.253626	0.126813	+	0.991927i	$0.459525\pi$
$774$	0	0
$775$	−3526.65	−0.163459
$776$	0	0
$777$	47987.4	2.21562
$778$	0	0
$779$	2093.59	0.0962909
$780$	0	0
$781$	−29722.6	−1.36179
$782$	0	0
$783$	35880.7	1.63764
$784$	0	0
$785$	−22189.1	−1.00887
$786$	0	0
$787$	20985.1	0.950494	0.475247	−	0.879852i	$-0.342359\pi$
$787$	20985.1	0.950494	0.475247	+	0.879852i	$0.342359\pi$
$788$	0	0
$789$	−7457.58	−0.336498
$790$	0	0
$791$	13129.1	0.590161
$792$	0	0
$793$	27814.7	1.24556
$794$	0	0
$795$	12122.1	0.540789
$796$	0	0
$797$	30421.8	1.35206	0.676032	−	0.736872i	$-0.263698\pi$
$797$	30421.8	1.35206	0.676032	+	0.736872i	$0.263698\pi$
$798$	0	0
$799$	10154.4	0.449607
$800$	0	0
$801$	−5076.45	−0.223929
$802$	0	0
$803$	4192.55	0.184249
$804$	0	0
$805$	−4856.74	−0.212643
$806$	0	0
$807$	−16191.7	−0.706288
$808$	0	0
$809$	−12231.4	−0.531560	−0.265780	−	0.964034i	$-0.585629\pi$
$809$	−12231.4	−0.531560	−0.265780	+	0.964034i	$0.585629\pi$
$810$	0	0
$811$	6733.52	0.291548	0.145774	−	0.989318i	$-0.453433\pi$
$811$	6733.52	0.291548	0.145774	+	0.989318i	$0.453433\pi$
$812$	0	0
$813$	1279.82	0.0552096
$814$	0	0
$815$	−16668.8	−0.716421
$816$	0	0
$817$	−396.889	−0.0169956
$818$	0	0
$819$	11780.3	0.502610
$820$	0	0
$821$	659.373	0.0280296	0.0140148	−	0.999902i	$-0.495539\pi$
$821$	659.373	0.0280296	0.0140148	+	0.999902i	$0.495539\pi$
$822$	0	0
$823$	−37882.3	−1.60449	−0.802245	−	0.596995i	$-0.796361\pi$
$823$	−37882.3	−1.60449	−0.802245	+	0.596995i	$0.796361\pi$
$824$	0	0
$825$	9981.16	0.421211
$826$	0	0
$827$	−36305.0	−1.52654	−0.763271	−	0.646079i	$-0.776408\pi$
$827$	−36305.0	−1.52654	−0.763271	+	0.646079i	$0.776408\pi$
$828$	0	0
$829$	−35900.3	−1.50406	−0.752032	−	0.659127i	$-0.770926\pi$
$829$	−35900.3	−1.50406	−0.752032	+	0.659127i	$0.770926\pi$
$830$	0	0
$831$	1.69615	7.08049e−5 0
$832$	0	0
$833$	38276.1	1.59206
$834$	0	0
$835$	6216.64	0.257648
$836$	0	0
$837$	8382.34	0.346160
$838$	0	0
$839$	40349.3	1.66033	0.830163	−	0.557521i	$-0.188247\pi$
$839$	40349.3	1.66033	0.830163	+	0.557521i	$0.188247\pi$
$840$	0	0
$841$	31086.0	1.27459
$842$	0	0
$843$	15494.8	0.633062
$844$	0	0
$845$	−9271.88	−0.377470
$846$	0	0
$847$	−2391.52	−0.0970172
$848$	0	0
$849$	−18694.5	−0.755704
$850$	0	0
$851$	−9233.54	−0.371941
$852$	0	0
$853$	−19569.0	−0.785498	−0.392749	−	0.919646i	$-0.628476\pi$
$853$	−19569.0	−0.785498	−0.392749	+	0.919646i	$0.628476\pi$
$854$	0	0
$855$	−2063.27	−0.0825290
$856$	0	0
$857$	24551.4	0.978598	0.489299	−	0.872116i	$-0.337253\pi$
$857$	24551.4	0.978598	0.489299	+	0.872116i	$0.337253\pi$
$858$	0	0
$859$	13459.2	0.534599	0.267300	−	0.963613i	$-0.413869\pi$
$859$	13459.2	0.534599	0.267300	+	0.963613i	$0.413869\pi$
$860$	0	0
$861$	7083.51	0.280378
$862$	0	0
$863$	42260.5	1.66694	0.833468	−	0.552568i	$-0.186352\pi$
$863$	42260.5	1.66694	0.833468	+	0.552568i	$0.186352\pi$
$864$	0	0
$865$	−34776.4	−1.36697
$866$	0	0
$867$	−21047.0	−0.824446
$868$	0	0
$869$	13691.4	0.534463
$870$	0	0
$871$	−835.954	−0.0325203
$872$	0	0
$873$	−3151.90	−0.122194
$874$	0	0
$875$	39929.4	1.54270
$876$	0	0
$877$	−37306.1	−1.43642	−0.718208	−	0.695829i	$-0.755037\pi$
$877$	−37306.1	−1.43642	−0.718208	+	0.695829i	$0.755037\pi$
$878$	0	0
$879$	−4711.90	−0.180806
$880$	0	0
$881$	−6563.30	−0.250991	−0.125496	−	0.992094i	$-0.540052\pi$
$881$	−6563.30	−0.250991	−0.125496	+	0.992094i	$0.540052\pi$
$882$	0	0
$883$	−27904.7	−1.06350	−0.531749	−	0.846902i	$-0.678465\pi$
$883$	−27904.7	−1.06350	−0.531749	+	0.846902i	$0.678465\pi$
$884$	0	0
$885$	−18859.8	−0.716346
$886$	0	0
$887$	−4358.15	−0.164974	−0.0824872	−	0.996592i	$-0.526286\pi$
$887$	−4358.15	−0.164974	−0.0824872	+	0.996592i	$0.526286\pi$
$888$	0	0
$889$	−66786.4	−2.51962
$890$	0	0
$891$	−16601.4	−0.624206
$892$	0	0
$893$	−3646.77	−0.136657
$894$	0	0
$895$	5411.96	0.202125
$896$	0	0
$897$	5911.72	0.220052
$898$	0	0
$899$	12959.9	0.480797
$900$	0	0
$901$	34587.2	1.27888
$902$	0	0
$903$	−1342.85	−0.0494874
$904$	0	0
$905$	−7607.19	−0.279416
$906$	0	0
$907$	−52106.6	−1.90758	−0.953788	−	0.300481i	$-0.902853\pi$
$907$	−52106.6	−1.90758	−0.953788	+	0.300481i	$0.902853\pi$
$908$	0	0
$909$	−12.4069	−0.000452708 0
$910$	0	0
$911$	−34691.3	−1.26166	−0.630832	−	0.775920i	$-0.717286\pi$
$911$	−34691.3	−1.26166	−0.630832	+	0.775920i	$0.717286\pi$
$912$	0	0
$913$	−2220.09	−0.0804757
$914$	0	0
$915$	−16482.7	−0.595522
$916$	0	0
$917$	28992.2	1.04406
$918$	0	0
$919$	−5838.08	−0.209555	−0.104777	−	0.994496i	$-0.533413\pi$
$919$	−5838.08	−0.209555	−0.104777	+	0.994496i	$0.533413\pi$
$920$	0	0
$921$	12124.7	0.433791
$922$	0	0
$923$	49057.0	1.74944
$924$	0	0
$925$	25730.6	0.914613
$926$	0	0
$927$	7914.52	0.280417
$928$	0	0
$929$	−31619.5	−1.11669	−0.558344	−	0.829610i	$-0.688563\pi$
$929$	−31619.5	−1.11669	−0.558344	+	0.829610i	$0.688563\pi$
$930$	0	0
$931$	−13746.2	−0.483903
$932$	0	0
$933$	12012.9	0.421528
$934$	0	0
$935$	−27063.0	−0.946583
$936$	0	0
$937$	30060.8	1.04807	0.524036	−	0.851696i	$-0.324426\pi$
$937$	30060.8	1.04807	0.524036	+	0.851696i	$0.324426\pi$
$938$	0	0
$939$	−26149.0	−0.908776
$940$	0	0
$941$	45847.9	1.58831	0.794155	−	0.607715i	$-0.207914\pi$
$941$	45847.9	1.58831	0.794155	+	0.607715i	$0.207914\pi$
$942$	0	0
$943$	−1362.98	−0.0470676
$944$	0	0
$945$	−32168.4	−1.10734
$946$	0	0
$947$	16769.4	0.575431	0.287716	−	0.957716i	$-0.407104\pi$
$947$	16769.4	0.575431	0.287716	+	0.957716i	$0.407104\pi$
$948$	0	0
$949$	−6919.78	−0.236697
$950$	0	0
$951$	5864.80	0.199978
$952$	0	0
$953$	−10747.1	−0.365303	−0.182652	−	0.983178i	$-0.558468\pi$
$953$	−10747.1	−0.365303	−0.182652	+	0.983178i	$0.558468\pi$
$954$	0	0
$955$	−6048.86	−0.204960
$956$	0	0
$957$	−36679.2	−1.23894
$958$	0	0
$959$	6534.23	0.220022
$960$	0	0
$961$	−26763.3	−0.898370
$962$	0	0
$963$	1164.59	0.0389704
$964$	0	0
$965$	−15773.2	−0.526174
$966$	0	0
$967$	12138.2	0.403659	0.201829	−	0.979421i	$-0.435311\pi$
$967$	12138.2	0.403659	0.201829	+	0.979421i	$0.435311\pi$
$968$	0	0
$969$	15353.5	0.509005
$970$	0	0
$971$	12928.1	0.427273	0.213636	−	0.976913i	$-0.431469\pi$
$971$	12928.1	0.427273	0.213636	+	0.976913i	$0.431469\pi$
$972$	0	0
$973$	71950.3	2.37063
$974$	0	0
$975$	−16473.9	−0.541114
$976$	0	0
$977$	14735.6	0.482532	0.241266	−	0.970459i	$-0.422437\pi$
$977$	14735.6	0.482532	0.241266	+	0.970459i	$0.422437\pi$
$978$	0	0
$979$	23913.1	0.780660
$980$	0	0
$981$	−13588.4	−0.442248
$982$	0	0
$983$	−29251.2	−0.949104	−0.474552	−	0.880228i	$-0.657390\pi$
$983$	−29251.2	−0.949104	−0.474552	+	0.880228i	$0.657390\pi$
$984$	0	0
$985$	4424.32	0.143117
$986$	0	0
$987$	−12338.6	−0.397914
$988$	0	0
$989$	258.385	0.00830755
$990$	0	0
$991$	−47986.6	−1.53819	−0.769094	−	0.639136i	$-0.779292\pi$
$991$	−47986.6	−1.53819	−0.769094	+	0.639136i	$0.779292\pi$
$992$	0	0
$993$	−14300.0	−0.456995
$994$	0	0
$995$	−13455.1	−0.428700
$996$	0	0
$997$	−32463.8	−1.03123	−0.515617	−	0.856819i	$-0.672437\pi$
$997$	−32463.8	−1.03123	−0.515617	+	0.856819i	$0.672437\pi$
$998$	0	0
$999$	−61157.9	−1.93689

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bf.1.1		4
4.3	odd	2		1472.4.a.y.1.4		4
8.3	odd	2		23.4.a.b.1.4	✓	4
8.5	even	2		368.4.a.l.1.4		4
24.11	even	2		207.4.a.e.1.1		4
40.3	even	4		575.4.b.g.24.2		8
40.19	odd	2		575.4.a.i.1.1		4
40.27	even	4		575.4.b.g.24.7		8
56.27	even	2		1127.4.a.c.1.4		4
184.91	even	2		529.4.a.g.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.a.b.1.4	✓	4	8.3	odd	2
207.4.a.e.1.1		4	24.11	even	2
368.4.a.l.1.4		4	8.5	even	2
529.4.a.g.1.4		4	184.91	even	2
575.4.a.i.1.1		4	40.19	odd	2
575.4.b.g.24.2		8	40.3	even	4
575.4.b.g.24.7		8	40.27	even	4
1127.4.a.c.1.4		4	56.27	even	2
1472.4.a.y.1.4		4	4.3	odd	2
1472.4.a.bf.1.1		4	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q-4.41777 q^{3} -7.80430 q^{5} +27.0572 q^{7} -7.48328 q^{9} +35.2507 q^{11} -58.1812 q^{13} +34.4776 q^{15} +98.3726 q^{17} -35.3289 q^{19} -119.533 q^{21} +23.0000 q^{23} -64.0928 q^{25} +152.339 q^{27} +235.531 q^{29} +55.0241 q^{31} -155.730 q^{33} -211.163 q^{35} -401.458 q^{37} +257.031 q^{39} -59.2600 q^{41} +11.2341 q^{43} +58.4018 q^{45} +103.224 q^{47} +389.093 q^{49} -434.588 q^{51} +351.594 q^{53} -275.107 q^{55} +156.075 q^{57} -547.016 q^{59} -478.070 q^{61} -202.477 q^{63} +454.063 q^{65} +14.3681 q^{67} -101.609 q^{69} -843.177 q^{71} +118.935 q^{73} +283.148 q^{75} +953.786 q^{77} +388.400 q^{79} -470.952 q^{81} -62.9800 q^{83} -767.730 q^{85} -1040.52 q^{87} +678.372 q^{89} -1574.22 q^{91} -243.084 q^{93} +275.717 q^{95} +421.192 q^{97} -263.791 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 7 q^{3} - 14 q^{5} - 16 q^{7} - 33 q^{9} + 8 q^{11} - 111 q^{13} - 10 q^{15} + 98 q^{17} + 96 q^{19} - 180 q^{21} + 92 q^{23} + 184 q^{25} - 155 q^{27} - 21 q^{29} + 193 q^{31} - 418 q^{33} - 752 q^{35}+ \cdots - 1498 q^{99}+O(q^{100}) \) 4 * q + 7 * q^3 - 14 * q^5 - 16 * q^7 - 33 * q^9 + 8 * q^11 - 111 * q^13 - 10 * q^15 + 98 * q^17 + 96 * q^19 - 180 * q^21 + 92 * q^23 + 184 * q^25 - 155 * q^27 - 21 * q^29 + 193 * q^31 - 418 * q^33 - 752 * q^35 - 170 * q^37 + 291 * q^39 - 125 * q^41 + 2 * q^43 - 168 * q^45 + 677 * q^47 + 1220 * q^49 - 340 * q^51 + 230 * q^53 + 972 * q^55 + 1322 * q^57 - 1140 * q^59 - 754 * q^61 + 1092 * q^63 + 1318 * q^65 + 488 * q^67 + 161 * q^69 + 401 * q^71 + 1509 * q^73 + 1401 * q^75 - 736 * q^77 + 838 * q^79 - 932 * q^81 + 142 * q^83 - 112 * q^85 - 2223 * q^87 + 2342 * q^89 + 292 * q^91 + 509 * q^93 + 956 * q^95 + 1062 * q^97 - 1498 * q^99

Introduction

L-functions

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Database

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