LMFDB - Embedded newform 552.4.a.g.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [552,4,Mod(1,552)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("552.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	552.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-12,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.5690543232$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1623576.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 22x^{2} + 17x + 91 $ x^4 - x^3 - 22x^2 + 17x + 91
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.01990$ of defining polynomial
Character	$\chi$	$=$	552.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +6.31926 q^{5} -2.85219 q^{7} +9.00000 q^{9} +22.6922 q^{11} -88.2664 q^{13} -18.9578 q^{15} +61.5538 q^{17} -37.0209 q^{19} +8.55656 q^{21} +23.0000 q^{23} -85.0669 q^{25} -27.0000 q^{27} +37.1407 q^{29} +165.392 q^{31} -68.0765 q^{33} -18.0237 q^{35} -85.9061 q^{37} +264.799 q^{39} -285.578 q^{41} +120.281 q^{43} +56.8734 q^{45} +142.899 q^{47} -334.865 q^{49} -184.661 q^{51} -489.906 q^{53} +143.398 q^{55} +111.063 q^{57} -558.479 q^{59} -44.9003 q^{61} -25.6697 q^{63} -557.779 q^{65} -436.743 q^{67} -69.0000 q^{69} +80.7506 q^{71} -473.265 q^{73} +255.201 q^{75} -64.7223 q^{77} -468.810 q^{79} +81.0000 q^{81} -562.534 q^{83} +388.975 q^{85} -111.422 q^{87} +508.096 q^{89} +251.752 q^{91} -496.176 q^{93} -233.945 q^{95} -273.880 q^{97} +204.229 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 14 q^{5} + 14 q^{7} + 36 q^{9} - 16 q^{11} + 4 q^{13} + 42 q^{15} - 22 q^{17} + 134 q^{19} - 42 q^{21} + 92 q^{23} + 24 q^{25} - 108 q^{27} - 180 q^{29} + 276 q^{31} + 48 q^{33} - 96 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 - 14 * q^5 + 14 * q^7 + 36 * q^9 - 16 * q^11 + 4 * q^13 + 42 * q^15 - 22 * q^17 + 134 * q^19 - 42 * q^21 + 92 * q^23 + 24 * q^25 - 108 * q^27 - 180 * q^29 + 276 * q^31 + 48 * q^33 - 96 * q^35 - 244 * q^37 - 12 * q^39 - 416 * q^41 - 242 * q^43 - 126 * q^45 - 384 * q^47 - 296 * q^49 + 66 * q^51 - 1426 * q^53 - 296 * q^55 - 402 * q^57 - 1000 * q^59 - 1140 * q^61 + 126 * q^63 - 1916 * q^65 - 646 * q^67 - 276 * q^69 - 432 * q^71 - 1568 * q^73 - 72 * q^75 - 2472 * q^77 + 778 * q^79 + 324 * q^81 + 112 * q^83 - 1772 * q^85 + 540 * q^87 - 2458 * q^89 - 316 * q^91 - 828 * q^93 + 952 * q^95 - 980 * q^97 - 144 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	6.31926	0.565212	0.282606	−	0.959236i	$-0.408801\pi$
$5$	6.31926	0.565212	0.282606	+	0.959236i	$0.408801\pi$
$6$	0	0
$7$	−2.85219	−0.154004	−0.0770018	−	0.997031i	$-0.524535\pi$
$7$	−2.85219	−0.154004	−0.0770018	+	0.997031i	$0.524535\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	22.6922	0.621995	0.310997	−	0.950411i	$-0.399337\pi$
$11$	22.6922	0.621995	0.310997	+	0.950411i	$0.399337\pi$
$12$	0	0
$13$	−88.2664	−1.88313	−0.941565	−	0.336831i	$-0.890645\pi$
$13$	−88.2664	−1.88313	−0.941565	+	0.336831i	$0.890645\pi$
$14$	0	0
$15$	−18.9578	−0.326325
$16$	0	0
$17$	61.5538	0.878176	0.439088	−	0.898444i	$-0.355302\pi$
$17$	61.5538	0.878176	0.439088	+	0.898444i	$0.355302\pi$
$18$	0	0
$19$	−37.0209	−0.447009	−0.223505	−	0.974703i	$-0.571750\pi$
$19$	−37.0209	−0.447009	−0.223505	+	0.974703i	$0.571750\pi$
$20$	0	0
$21$	8.55656	0.0889141
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−85.0669	−0.680535
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	37.1407	0.237822	0.118911	−	0.992905i	$-0.462060\pi$
$29$	37.1407	0.237822	0.118911	+	0.992905i	$0.462060\pi$
$30$	0	0
$31$	165.392	0.958236	0.479118	−	0.877751i	$-0.340957\pi$
$31$	165.392	0.958236	0.479118	+	0.877751i	$0.340957\pi$
$32$	0	0
$33$	−68.0765	−0.359109
$34$	0	0
$35$	−18.0237	−0.0870447
$36$	0	0
$37$	−85.9061	−0.381699	−0.190850	−	0.981619i	$-0.561124\pi$
$37$	−85.9061	−0.381699	−0.190850	+	0.981619i	$0.561124\pi$
$38$	0	0
$39$	264.799	1.08723
$40$	0	0
$41$	−285.578	−1.08780	−0.543899	−	0.839151i	$-0.683053\pi$
$41$	−285.578	−1.08780	−0.543899	+	0.839151i	$0.683053\pi$
$42$	0	0
$43$	120.281	0.426575	0.213288	−	0.976989i	$-0.431583\pi$
$43$	120.281	0.426575	0.213288	+	0.976989i	$0.431583\pi$
$44$	0	0
$45$	56.8734	0.188404
$46$	0	0
$47$	142.899	0.443487	0.221744	−	0.975105i	$-0.428825\pi$
$47$	142.899	0.443487	0.221744	+	0.975105i	$0.428825\pi$
$48$	0	0
$49$	−334.865	−0.976283
$50$	0	0
$51$	−184.661	−0.507015
$52$	0	0
$53$	−489.906	−1.26969	−0.634847	−	0.772638i	$-0.718937\pi$
$53$	−489.906	−1.26969	−0.634847	+	0.772638i	$0.718937\pi$
$54$	0	0
$55$	143.398	0.351559
$56$	0	0
$57$	111.063	0.258081
$58$	0	0
$59$	−558.479	−1.23234	−0.616168	−	0.787615i	$-0.711316\pi$
$59$	−558.479	−1.23234	−0.616168	+	0.787615i	$0.711316\pi$
$60$	0	0
$61$	−44.9003	−0.0942441	−0.0471221	−	0.998889i	$-0.515005\pi$
$61$	−44.9003	−0.0942441	−0.0471221	+	0.998889i	$0.515005\pi$
$62$	0	0
$63$	−25.6697	−0.0513346
$64$	0	0
$65$	−557.779	−1.06437
$66$	0	0
$67$	−436.743	−0.796368	−0.398184	−	0.917306i	$-0.630359\pi$
$67$	−436.743	−0.796368	−0.398184	+	0.917306i	$0.630359\pi$
$68$	0	0
$69$	−69.0000	−0.120386
$70$	0	0
$71$	80.7506	0.134977	0.0674883	−	0.997720i	$-0.478501\pi$
$71$	80.7506	0.134977	0.0674883	+	0.997720i	$0.478501\pi$
$72$	0	0
$73$	−473.265	−0.758788	−0.379394	−	0.925235i	$-0.623867\pi$
$73$	−473.265	−0.758788	−0.379394	+	0.925235i	$0.623867\pi$
$74$	0	0
$75$	255.201	0.392907
$76$	0	0
$77$	−64.7223	−0.0957895
$78$	0	0
$79$	−468.810	−0.667662	−0.333831	−	0.942633i	$-0.608341\pi$
$79$	−468.810	−0.667662	−0.333831	+	0.942633i	$0.608341\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−562.534	−0.743929	−0.371965	−	0.928247i	$-0.621316\pi$
$83$	−562.534	−0.743929	−0.371965	+	0.928247i	$0.621316\pi$
$84$	0	0
$85$	388.975	0.496356
$86$	0	0
$87$	−111.422	−0.137307
$88$	0	0
$89$	508.096	0.605147	0.302574	−	0.953126i	$-0.402154\pi$
$89$	508.096	0.605147	0.302574	+	0.953126i	$0.402154\pi$
$90$	0	0
$91$	251.752	0.290009
$92$	0	0
$93$	−496.176	−0.553238
$94$	0	0
$95$	−233.945	−0.252655
$96$	0	0
$97$	−273.880	−0.286684	−0.143342	−	0.989673i	$-0.545785\pi$
$97$	−273.880	−0.286684	−0.143342	+	0.989673i	$0.545785\pi$
$98$	0	0
$99$	204.229	0.207332
$100$	0	0
$101$	−850.889	−0.838283	−0.419142	−	0.907921i	$-0.637669\pi$
$101$	−850.889	−0.838283	−0.419142	+	0.907921i	$0.637669\pi$
$102$	0	0
$103$	−953.440	−0.912090	−0.456045	−	0.889957i	$-0.650734\pi$
$103$	−953.440	−0.912090	−0.456045	+	0.889957i	$0.650734\pi$
$104$	0	0
$105$	54.0712	0.0502553
$106$	0	0
$107$	−635.312	−0.573999	−0.286999	−	0.957931i	$-0.592658\pi$
$107$	−635.312	−0.573999	−0.286999	+	0.957931i	$0.592658\pi$
$108$	0	0
$109$	−492.269	−0.432577	−0.216288	−	0.976330i	$-0.569395\pi$
$109$	−492.269	−0.432577	−0.216288	+	0.976330i	$0.569395\pi$
$110$	0	0
$111$	257.718	0.220374
$112$	0	0
$113$	−2263.93	−1.88472	−0.942358	−	0.334607i	$-0.891397\pi$
$113$	−2263.93	−1.88472	−0.942358	+	0.334607i	$0.891397\pi$
$114$	0	0
$115$	145.343	0.117855
$116$	0	0
$117$	−794.398	−0.627710
$118$	0	0
$119$	−175.563	−0.135242
$120$	0	0
$121$	−816.066	−0.613122
$122$	0	0
$123$	856.733	0.628040
$124$	0	0
$125$	−1327.47	−0.949859
$126$	0	0
$127$	−329.042	−0.229904	−0.114952	−	0.993371i	$-0.536671\pi$
$127$	−329.042	−0.229904	−0.114952	+	0.993371i	$0.536671\pi$
$128$	0	0
$129$	−360.844	−0.246283
$130$	0	0
$131$	2156.47	1.43826	0.719128	−	0.694878i	$-0.244542\pi$
$131$	2156.47	1.43826	0.719128	+	0.694878i	$0.244542\pi$
$132$	0	0
$133$	105.591	0.0688410
$134$	0	0
$135$	−170.620	−0.108775
$136$	0	0
$137$	1714.23	1.06903	0.534514	−	0.845160i	$-0.320495\pi$
$137$	1714.23	1.06903	0.534514	+	0.845160i	$0.320495\pi$
$138$	0	0
$139$	1999.27	1.21997	0.609984	−	0.792413i	$-0.291176\pi$
$139$	1999.27	1.21997	0.609984	+	0.792413i	$0.291176\pi$
$140$	0	0
$141$	−428.696	−0.256048
$142$	0	0
$143$	−2002.96	−1.17130
$144$	0	0
$145$	234.702	0.134420
$146$	0	0
$147$	1004.60	0.563657
$148$	0	0
$149$	−1080.84	−0.594269	−0.297135	−	0.954836i	$-0.596031\pi$
$149$	−1080.84	−0.594269	−0.297135	+	0.954836i	$0.596031\pi$
$150$	0	0
$151$	1525.31	0.822038	0.411019	−	0.911627i	$-0.365173\pi$
$151$	1525.31	0.822038	0.411019	+	0.911627i	$0.365173\pi$
$152$	0	0
$153$	553.984	0.292725
$154$	0	0
$155$	1045.16	0.541606
$156$	0	0
$157$	313.469	0.159347	0.0796737	−	0.996821i	$-0.474612\pi$
$157$	313.469	0.159347	0.0796737	+	0.996821i	$0.474612\pi$
$158$	0	0
$159$	1469.72	0.733058
$160$	0	0
$161$	−65.6003	−0.0321120
$162$	0	0
$163$	859.158	0.412850	0.206425	−	0.978462i	$-0.433817\pi$
$163$	859.158	0.412850	0.206425	+	0.978462i	$0.433817\pi$
$164$	0	0
$165$	−430.193	−0.202973
$166$	0	0
$167$	1947.50	0.902406	0.451203	−	0.892421i	$-0.350995\pi$
$167$	1947.50	0.902406	0.451203	+	0.892421i	$0.350995\pi$
$168$	0	0
$169$	5593.96	2.54618
$170$	0	0
$171$	−333.188	−0.149003
$172$	0	0
$173$	−2007.07	−0.882051	−0.441025	−	0.897495i	$-0.645385\pi$
$173$	−2007.07	−0.882051	−0.441025	+	0.897495i	$0.645385\pi$
$174$	0	0
$175$	242.627	0.104805
$176$	0	0
$177$	1675.44	0.711489
$178$	0	0
$179$	−3266.37	−1.36391	−0.681955	−	0.731394i	$-0.738870\pi$
$179$	−3266.37	−1.36391	−0.681955	+	0.731394i	$0.738870\pi$
$180$	0	0
$181$	4305.00	1.76789	0.883946	−	0.467589i	$-0.154877\pi$
$181$	4305.00	1.76789	0.883946	+	0.467589i	$0.154877\pi$
$182$	0	0
$183$	134.701	0.0544119
$184$	0	0
$185$	−542.863	−0.215741
$186$	0	0
$187$	1396.79	0.546221
$188$	0	0
$189$	77.0091	0.0296380
$190$	0	0
$191$	−2323.50	−0.880221	−0.440111	−	0.897944i	$-0.645061\pi$
$191$	−2323.50	−0.880221	−0.440111	+	0.897944i	$0.645061\pi$
$192$	0	0
$193$	−4809.99	−1.79394	−0.896972	−	0.442088i	$-0.854238\pi$
$193$	−4809.99	−1.79394	−0.896972	+	0.442088i	$0.854238\pi$
$194$	0	0
$195$	1673.34	0.614513
$196$	0	0
$197$	1611.12	0.582679	0.291340	−	0.956620i	$-0.405899\pi$
$197$	1611.12	0.582679	0.291340	+	0.956620i	$0.405899\pi$
$198$	0	0
$199$	−701.349	−0.249836	−0.124918	−	0.992167i	$-0.539867\pi$
$199$	−701.349	−0.249836	−0.124918	+	0.992167i	$0.539867\pi$
$200$	0	0
$201$	1310.23	0.459783
$202$	0	0
$203$	−105.932	−0.0366255
$204$	0	0
$205$	−1804.64	−0.614837
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	−840.084	−0.278037
$210$	0	0
$211$	270.581	0.0882822	0.0441411	−	0.999025i	$-0.485945\pi$
$211$	270.581	0.0882822	0.0441411	+	0.999025i	$0.485945\pi$
$212$	0	0
$213$	−242.252	−0.0779288
$214$	0	0
$215$	760.089	0.241105
$216$	0	0
$217$	−471.729	−0.147572
$218$	0	0
$219$	1419.80	0.438086
$220$	0	0
$221$	−5433.13	−1.65372
$222$	0	0
$223$	4279.20	1.28501	0.642504	−	0.766283i	$-0.277896\pi$
$223$	4279.20	1.28501	0.642504	+	0.766283i	$0.277896\pi$
$224$	0	0
$225$	−765.602	−0.226845
$226$	0	0
$227$	−2864.39	−0.837517	−0.418759	−	0.908098i	$-0.637535\pi$
$227$	−2864.39	−0.837517	−0.418759	+	0.908098i	$0.637535\pi$
$228$	0	0
$229$	−405.112	−0.116902	−0.0584510	−	0.998290i	$-0.518616\pi$
$229$	−405.112	−0.116902	−0.0584510	+	0.998290i	$0.518616\pi$
$230$	0	0
$231$	194.167	0.0553041
$232$	0	0
$233$	1060.59	0.298205	0.149102	−	0.988822i	$-0.452362\pi$
$233$	1060.59	0.298205	0.149102	+	0.988822i	$0.452362\pi$
$234$	0	0
$235$	903.014	0.250664
$236$	0	0
$237$	1406.43	0.385475
$238$	0	0
$239$	2328.30	0.630148	0.315074	−	0.949067i	$-0.397971\pi$
$239$	2328.30	0.630148	0.315074	+	0.949067i	$0.397971\pi$
$240$	0	0
$241$	2780.67	0.743231	0.371616	−	0.928387i	$-0.378804\pi$
$241$	2780.67	0.743231	0.371616	+	0.928387i	$0.378804\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−2116.10	−0.551807
$246$	0	0
$247$	3267.70	0.841776
$248$	0	0
$249$	1687.60	0.429508
$250$	0	0
$251$	6383.54	1.60528	0.802641	−	0.596463i	$-0.203428\pi$
$251$	6383.54	1.60528	0.802641	+	0.596463i	$0.203428\pi$
$252$	0	0
$253$	521.920	0.129695
$254$	0	0
$255$	−1166.92	−0.286571
$256$	0	0
$257$	1499.30	0.363905	0.181953	−	0.983307i	$-0.441758\pi$
$257$	1499.30	0.363905	0.181953	+	0.983307i	$0.441758\pi$
$258$	0	0
$259$	245.020	0.0587831
$260$	0	0
$261$	334.266	0.0792741
$262$	0	0
$263$	6448.80	1.51198	0.755988	−	0.654585i	$-0.227157\pi$
$263$	6448.80	1.51198	0.755988	+	0.654585i	$0.227157\pi$
$264$	0	0
$265$	−3095.85	−0.717646
$266$	0	0
$267$	−1524.29	−0.349382
$268$	0	0
$269$	−1437.57	−0.325836	−0.162918	−	0.986640i	$-0.552091\pi$
$269$	−1437.57	−0.325836	−0.162918	+	0.986640i	$0.552091\pi$
$270$	0	0
$271$	6642.90	1.48903	0.744516	−	0.667605i	$-0.232680\pi$
$271$	6642.90	1.48903	0.744516	+	0.667605i	$0.232680\pi$
$272$	0	0
$273$	−755.257	−0.167437
$274$	0	0
$275$	−1930.35	−0.423289
$276$	0	0
$277$	4316.43	0.936279	0.468139	−	0.883655i	$-0.344925\pi$
$277$	4316.43	0.936279	0.468139	+	0.883655i	$0.344925\pi$
$278$	0	0
$279$	1488.53	0.319412
$280$	0	0
$281$	−727.607	−0.154468	−0.0772338	−	0.997013i	$-0.524609\pi$
$281$	−727.607	−0.154468	−0.0772338	+	0.997013i	$0.524609\pi$
$282$	0	0
$283$	−8460.61	−1.77714	−0.888571	−	0.458740i	$-0.848301\pi$
$283$	−8460.61	−1.77714	−0.888571	+	0.458740i	$0.848301\pi$
$284$	0	0
$285$	701.834	0.145870
$286$	0	0
$287$	814.521	0.167525
$288$	0	0
$289$	−1124.13	−0.228807
$290$	0	0
$291$	821.640	0.165517
$292$	0	0
$293$	−5914.22	−1.17922	−0.589611	−	0.807687i	$-0.700719\pi$
$293$	−5914.22	−1.17922	−0.589611	+	0.807687i	$0.700719\pi$
$294$	0	0
$295$	−3529.18	−0.696531
$296$	0	0
$297$	−612.688	−0.119703
$298$	0	0
$299$	−2030.13	−0.392660
$300$	0	0
$301$	−343.065	−0.0656941
$302$	0	0
$303$	2552.67	0.483983
$304$	0	0
$305$	−283.737	−0.0532679
$306$	0	0
$307$	−9350.48	−1.73831	−0.869154	−	0.494542i	$-0.835336\pi$
$307$	−9350.48	−1.73831	−0.869154	+	0.494542i	$0.835336\pi$
$308$	0	0
$309$	2860.32	0.526595
$310$	0	0
$311$	7867.96	1.43457	0.717284	−	0.696781i	$-0.245385\pi$
$311$	7867.96	1.43457	0.717284	+	0.696781i	$0.245385\pi$
$312$	0	0
$313$	8753.13	1.58069	0.790345	−	0.612662i	$-0.209901\pi$
$313$	8753.13	1.58069	0.790345	+	0.612662i	$0.209901\pi$
$314$	0	0
$315$	−162.214	−0.0290149
$316$	0	0
$317$	−4125.98	−0.731035	−0.365518	−	0.930804i	$-0.619108\pi$
$317$	−4125.98	−0.731035	−0.365518	+	0.930804i	$0.619108\pi$
$318$	0	0
$319$	842.802	0.147924
$320$	0	0
$321$	1905.93	0.331398
$322$	0	0
$323$	−2278.78	−0.392553
$324$	0	0
$325$	7508.55	1.28154
$326$	0	0
$327$	1476.81	0.249748
$328$	0	0
$329$	−407.574	−0.0682987
$330$	0	0
$331$	−1419.98	−0.235798	−0.117899	−	0.993026i	$-0.537616\pi$
$331$	−1419.98	−0.235798	−0.117899	+	0.993026i	$0.537616\pi$
$332$	0	0
$333$	−773.155	−0.127233
$334$	0	0
$335$	−2759.90	−0.450117
$336$	0	0
$337$	7170.82	1.15911	0.579554	−	0.814934i	$-0.303227\pi$
$337$	7170.82	1.15911	0.579554	+	0.814934i	$0.303227\pi$
$338$	0	0
$339$	6791.80	1.08814
$340$	0	0
$341$	3753.10	0.596018
$342$	0	0
$343$	1933.40	0.304355
$344$	0	0
$345$	−436.029	−0.0680435
$346$	0	0
$347$	−6441.72	−0.996569	−0.498285	−	0.867014i	$-0.666037\pi$
$347$	−6441.72	−0.996569	−0.498285	+	0.867014i	$0.666037\pi$
$348$	0	0
$349$	5614.80	0.861185	0.430592	−	0.902547i	$-0.358305\pi$
$349$	5614.80	0.861185	0.430592	+	0.902547i	$0.358305\pi$
$350$	0	0
$351$	2383.19	0.362409
$352$	0	0
$353$	−1019.36	−0.153697	−0.0768487	−	0.997043i	$-0.524486\pi$
$353$	−1019.36	−0.153697	−0.0768487	+	0.997043i	$0.524486\pi$
$354$	0	0
$355$	510.285	0.0762904
$356$	0	0
$357$	526.689	0.0780822
$358$	0	0
$359$	−3893.79	−0.572442	−0.286221	−	0.958164i	$-0.592399\pi$
$359$	−3893.79	−0.572442	−0.286221	+	0.958164i	$0.592399\pi$
$360$	0	0
$361$	−5488.45	−0.800183
$362$	0	0
$363$	2448.20	0.353986
$364$	0	0
$365$	−2990.69	−0.428876
$366$	0	0
$367$	−10570.6	−1.50350	−0.751748	−	0.659450i	$-0.770789\pi$
$367$	−10570.6	−1.50350	−0.751748	+	0.659450i	$0.770789\pi$
$368$	0	0
$369$	−2570.20	−0.362599
$370$	0	0
$371$	1397.30	0.195538
$372$	0	0
$373$	9571.31	1.32864	0.664321	−	0.747447i	$-0.268721\pi$
$373$	9571.31	1.32864	0.664321	+	0.747447i	$0.268721\pi$
$374$	0	0
$375$	3982.40	0.548401
$376$	0	0
$377$	−3278.27	−0.447850
$378$	0	0
$379$	5635.85	0.763836	0.381918	−	0.924196i	$-0.375264\pi$
$379$	5635.85	0.763836	0.381918	+	0.924196i	$0.375264\pi$
$380$	0	0
$381$	987.126	0.132735
$382$	0	0
$383$	9400.32	1.25414	0.627068	−	0.778965i	$-0.284255\pi$
$383$	9400.32	1.25414	0.627068	+	0.778965i	$0.284255\pi$
$384$	0	0
$385$	−408.997	−0.0541414
$386$	0	0
$387$	1082.53	0.142192
$388$	0	0
$389$	−6676.76	−0.870245	−0.435122	−	0.900371i	$-0.643295\pi$
$389$	−6676.76	−0.870245	−0.435122	+	0.900371i	$0.643295\pi$
$390$	0	0
$391$	1415.74	0.183112
$392$	0	0
$393$	−6469.40	−0.830377
$394$	0	0
$395$	−2962.54	−0.377371
$396$	0	0
$397$	−4238.68	−0.535853	−0.267926	−	0.963439i	$-0.586338\pi$
$397$	−4238.68	−0.535853	−0.267926	+	0.963439i	$0.586338\pi$
$398$	0	0
$399$	−316.772	−0.0397454
$400$	0	0
$401$	3565.52	0.444023	0.222012	−	0.975044i	$-0.428738\pi$
$401$	3565.52	0.444023	0.222012	+	0.975044i	$0.428738\pi$
$402$	0	0
$403$	−14598.6	−1.80448
$404$	0	0
$405$	511.860	0.0628013
$406$	0	0
$407$	−1949.40	−0.237415
$408$	0	0
$409$	15676.8	1.89527	0.947635	−	0.319355i	$-0.103466\pi$
$409$	15676.8	1.89527	0.947635	+	0.319355i	$0.103466\pi$
$410$	0	0
$411$	−5142.70	−0.617203
$412$	0	0
$413$	1592.89	0.189784
$414$	0	0
$415$	−3554.80	−0.420478
$416$	0	0
$417$	−5997.80	−0.704349
$418$	0	0
$419$	−4706.77	−0.548785	−0.274392	−	0.961618i	$-0.588477\pi$
$419$	−4706.77	−0.548785	−0.274392	+	0.961618i	$0.588477\pi$
$420$	0	0
$421$	−9095.16	−1.05290	−0.526450	−	0.850206i	$-0.676477\pi$
$421$	−9095.16	−1.05290	−0.526450	+	0.850206i	$0.676477\pi$
$422$	0	0
$423$	1286.09	0.147829
$424$	0	0
$425$	−5236.19	−0.597630
$426$	0	0
$427$	128.064	0.0145139
$428$	0	0
$429$	6008.87	0.676249
$430$	0	0
$431$	−11851.8	−1.32455	−0.662274	−	0.749262i	$-0.730409\pi$
$431$	−11851.8	−1.32455	−0.662274	+	0.749262i	$0.730409\pi$
$432$	0	0
$433$	−8987.98	−0.997540	−0.498770	−	0.866734i	$-0.666215\pi$
$433$	−8987.98	−0.997540	−0.498770	+	0.866734i	$0.666215\pi$
$434$	0	0
$435$	−704.105	−0.0776075
$436$	0	0
$437$	−851.480	−0.0932078
$438$	0	0
$439$	−803.064	−0.0873079	−0.0436540	−	0.999047i	$-0.513900\pi$
$439$	−803.064	−0.0873079	−0.0436540	+	0.999047i	$0.513900\pi$
$440$	0	0
$441$	−3013.79	−0.325428
$442$	0	0
$443$	−4011.52	−0.430233	−0.215117	−	0.976588i	$-0.569013\pi$
$443$	−4011.52	−0.430233	−0.215117	+	0.976588i	$0.569013\pi$
$444$	0	0
$445$	3210.79	0.342036
$446$	0	0
$447$	3242.53	0.343101
$448$	0	0
$449$	5140.49	0.540301	0.270150	−	0.962818i	$-0.412927\pi$
$449$	5140.49	0.540301	0.270150	+	0.962818i	$0.412927\pi$
$450$	0	0
$451$	−6480.37	−0.676605
$452$	0	0
$453$	−4575.92	−0.474604
$454$	0	0
$455$	1590.89	0.163917
$456$	0	0
$457$	5322.84	0.544840	0.272420	−	0.962178i	$-0.412176\pi$
$457$	5322.84	0.544840	0.272420	+	0.962178i	$0.412176\pi$
$458$	0	0
$459$	−1661.95	−0.169005
$460$	0	0
$461$	−7664.96	−0.774387	−0.387194	−	0.921998i	$-0.626556\pi$
$461$	−7664.96	−0.774387	−0.387194	+	0.921998i	$0.626556\pi$
$462$	0	0
$463$	2636.18	0.264608	0.132304	−	0.991209i	$-0.457762\pi$
$463$	2636.18	0.264608	0.132304	+	0.991209i	$0.457762\pi$
$464$	0	0
$465$	−3135.47	−0.312697
$466$	0	0
$467$	−3029.09	−0.300149	−0.150074	−	0.988675i	$-0.547951\pi$
$467$	−3029.09	−0.300149	−0.150074	+	0.988675i	$0.547951\pi$
$468$	0	0
$469$	1245.67	0.122644
$470$	0	0
$471$	−940.406	−0.0919992
$472$	0	0
$473$	2729.44	0.265328
$474$	0	0
$475$	3149.25	0.304205
$476$	0	0
$477$	−4409.16	−0.423231
$478$	0	0
$479$	−5133.59	−0.489687	−0.244843	−	0.969563i	$-0.578737\pi$
$479$	−5133.59	−0.489687	−0.244843	+	0.969563i	$0.578737\pi$
$480$	0	0
$481$	7582.62	0.718790
$482$	0	0
$483$	196.801	0.0185399
$484$	0	0
$485$	−1730.72	−0.162037
$486$	0	0
$487$	1779.51	0.165580	0.0827899	−	0.996567i	$-0.473617\pi$
$487$	1779.51	0.165580	0.0827899	+	0.996567i	$0.473617\pi$
$488$	0	0
$489$	−2577.48	−0.238359
$490$	0	0
$491$	−1336.02	−0.122798	−0.0613989	−	0.998113i	$-0.519556\pi$
$491$	−1336.02	−0.122798	−0.0613989	+	0.998113i	$0.519556\pi$
$492$	0	0
$493$	2286.15	0.208850
$494$	0	0
$495$	1290.58	0.117186
$496$	0	0
$497$	−230.316	−0.0207869
$498$	0	0
$499$	−14269.2	−1.28012	−0.640058	−	0.768326i	$-0.721090\pi$
$499$	−14269.2	−1.28012	−0.640058	+	0.768326i	$0.721090\pi$
$500$	0	0
$501$	−5842.49	−0.521004
$502$	0	0
$503$	270.476	0.0239760	0.0119880	−	0.999928i	$-0.496184\pi$
$503$	270.476	0.0239760	0.0119880	+	0.999928i	$0.496184\pi$
$504$	0	0
$505$	−5376.99	−0.473808
$506$	0	0
$507$	−16781.9	−1.47004
$508$	0	0
$509$	9398.27	0.818411	0.409205	−	0.912442i	$-0.365806\pi$
$509$	9398.27	0.818411	0.409205	+	0.912442i	$0.365806\pi$
$510$	0	0
$511$	1349.84	0.116856
$512$	0	0
$513$	999.564	0.0860269
$514$	0	0
$515$	−6025.04	−0.515524
$516$	0	0
$517$	3242.68	0.275847
$518$	0	0
$519$	6021.21	0.509252
$520$	0	0
$521$	3173.60	0.266868	0.133434	−	0.991058i	$-0.457400\pi$
$521$	3173.60	0.266868	0.133434	+	0.991058i	$0.457400\pi$
$522$	0	0
$523$	−15479.0	−1.29416	−0.647082	−	0.762420i	$-0.724011\pi$
$523$	−15479.0	−1.29416	−0.647082	+	0.762420i	$0.724011\pi$
$524$	0	0
$525$	−727.880	−0.0605091
$526$	0	0
$527$	10180.5	0.841499
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−5026.31	−0.410778
$532$	0	0
$533$	25206.9	2.04847
$534$	0	0
$535$	−4014.70	−0.324431
$536$	0	0
$537$	9799.11	0.787454
$538$	0	0
$539$	−7598.81	−0.607243
$540$	0	0
$541$	−2803.49	−0.222794	−0.111397	−	0.993776i	$-0.535533\pi$
$541$	−2803.49	−0.222794	−0.111397	+	0.993776i	$0.535533\pi$
$542$	0	0
$543$	−12915.0	−1.02069
$544$	0	0
$545$	−3110.78	−0.244498
$546$	0	0
$547$	−11514.9	−0.900077	−0.450039	−	0.893009i	$-0.648590\pi$
$547$	−11514.9	−0.900077	−0.450039	+	0.893009i	$0.648590\pi$
$548$	0	0
$549$	−404.102	−0.0314147
$550$	0	0
$551$	−1374.98	−0.106309
$552$	0	0
$553$	1337.14	0.102822
$554$	0	0
$555$	1628.59	0.124558
$556$	0	0
$557$	8639.23	0.657192	0.328596	−	0.944471i	$-0.393425\pi$
$557$	8639.23	0.657192	0.328596	+	0.944471i	$0.393425\pi$
$558$	0	0
$559$	−10616.8	−0.803296
$560$	0	0
$561$	−4190.37	−0.315361
$562$	0	0
$563$	12296.8	0.920512	0.460256	−	0.887786i	$-0.347758\pi$
$563$	12296.8	0.920512	0.460256	+	0.887786i	$0.347758\pi$
$564$	0	0
$565$	−14306.4	−1.06526
$566$	0	0
$567$	−231.027	−0.0171115
$568$	0	0
$569$	−11047.3	−0.813930	−0.406965	−	0.913444i	$-0.633413\pi$
$569$	−11047.3	−0.813930	−0.406965	+	0.913444i	$0.633413\pi$
$570$	0	0
$571$	−8028.80	−0.588432	−0.294216	−	0.955739i	$-0.595059\pi$
$571$	−8028.80	−0.588432	−0.294216	+	0.955739i	$0.595059\pi$
$572$	0	0
$573$	6970.49	0.508196
$574$	0	0
$575$	−1956.54	−0.141901
$576$	0	0
$577$	8795.59	0.634602	0.317301	−	0.948325i	$-0.397223\pi$
$577$	8795.59	0.634602	0.317301	+	0.948325i	$0.397223\pi$
$578$	0	0
$579$	14430.0	1.03573
$580$	0	0
$581$	1604.45	0.114568
$582$	0	0
$583$	−11117.0	−0.789743
$584$	0	0
$585$	−5020.01	−0.354789
$586$	0	0
$587$	−3235.31	−0.227488	−0.113744	−	0.993510i	$-0.536284\pi$
$587$	−3235.31	−0.227488	−0.113744	+	0.993510i	$0.536284\pi$
$588$	0	0
$589$	−6122.96	−0.428340
$590$	0	0
$591$	−4833.37	−0.336410
$592$	0	0
$593$	239.789	0.0166053	0.00830265	−	0.999966i	$-0.497357\pi$
$593$	239.789	0.0166053	0.00830265	+	0.999966i	$0.497357\pi$
$594$	0	0
$595$	−1109.43	−0.0764406
$596$	0	0
$597$	2104.05	0.144243
$598$	0	0
$599$	−1913.61	−0.130531	−0.0652653	−	0.997868i	$-0.520789\pi$
$599$	−1913.61	−0.130531	−0.0652653	+	0.997868i	$0.520789\pi$
$600$	0	0
$601$	7618.63	0.517089	0.258544	−	0.965999i	$-0.416757\pi$
$601$	7618.63	0.517089	0.258544	+	0.965999i	$0.416757\pi$
$602$	0	0
$603$	−3930.69	−0.265456
$604$	0	0
$605$	−5156.94	−0.346544
$606$	0	0
$607$	17540.4	1.17289	0.586443	−	0.809990i	$-0.300528\pi$
$607$	17540.4	1.17289	0.586443	+	0.809990i	$0.300528\pi$
$608$	0	0
$609$	317.796	0.0211458
$610$	0	0
$611$	−12613.1	−0.835145
$612$	0	0
$613$	−10741.9	−0.707764	−0.353882	−	0.935290i	$-0.615139\pi$
$613$	−10741.9	−0.707764	−0.353882	+	0.935290i	$0.615139\pi$
$614$	0	0
$615$	5413.92	0.354976
$616$	0	0
$617$	−9650.00	−0.629650	−0.314825	−	0.949150i	$-0.601946\pi$
$617$	−9650.00	−0.629650	−0.314825	+	0.949150i	$0.601946\pi$
$618$	0	0
$619$	−7375.27	−0.478897	−0.239449	−	0.970909i	$-0.576967\pi$
$619$	−7375.27	−0.478897	−0.239449	+	0.970909i	$0.576967\pi$
$620$	0	0
$621$	−621.000	−0.0401286
$622$	0	0
$623$	−1449.19	−0.0931949
$624$	0	0
$625$	2244.74	0.143664
$626$	0	0
$627$	2520.25	0.160525
$628$	0	0
$629$	−5287.85	−0.335199
$630$	0	0
$631$	−8675.39	−0.547324	−0.273662	−	0.961826i	$-0.588235\pi$
$631$	−8675.39	−0.547324	−0.273662	+	0.961826i	$0.588235\pi$
$632$	0	0
$633$	−811.742	−0.0509698
$634$	0	0
$635$	−2079.30	−0.129944
$636$	0	0
$637$	29557.3	1.83847
$638$	0	0
$639$	726.756	0.0449922
$640$	0	0
$641$	931.473	0.0573962	0.0286981	−	0.999588i	$-0.490864\pi$
$641$	931.473	0.0573962	0.0286981	+	0.999588i	$0.490864\pi$
$642$	0	0
$643$	−13303.0	−0.815896	−0.407948	−	0.913005i	$-0.633755\pi$
$643$	−13303.0	−0.815896	−0.407948	+	0.913005i	$0.633755\pi$
$644$	0	0
$645$	−2280.27	−0.139202
$646$	0	0
$647$	23484.7	1.42701	0.713507	−	0.700648i	$-0.247106\pi$
$647$	23484.7	1.42701	0.713507	+	0.700648i	$0.247106\pi$
$648$	0	0
$649$	−12673.1	−0.766506
$650$	0	0
$651$	1415.19	0.0852006
$652$	0	0
$653$	3951.23	0.236789	0.118395	−	0.992967i	$-0.462225\pi$
$653$	3951.23	0.236789	0.118395	+	0.992967i	$0.462225\pi$
$654$	0	0
$655$	13627.3	0.812919
$656$	0	0
$657$	−4259.39	−0.252929
$658$	0	0
$659$	17903.6	1.05831	0.529153	−	0.848526i	$-0.322510\pi$
$659$	17903.6	1.05831	0.529153	+	0.848526i	$0.322510\pi$
$660$	0	0
$661$	24420.0	1.43695	0.718477	−	0.695551i	$-0.244840\pi$
$661$	24420.0	1.43695	0.718477	+	0.695551i	$0.244840\pi$
$662$	0	0
$663$	16299.4	0.954776
$664$	0	0
$665$	667.254	0.0389098
$666$	0	0
$667$	854.235	0.0495894
$668$	0	0
$669$	−12837.6	−0.741899
$670$	0	0
$671$	−1018.88	−0.0586194
$672$	0	0
$673$	−3564.95	−0.204188	−0.102094	−	0.994775i	$-0.532554\pi$
$673$	−3564.95	−0.204188	−0.102094	+	0.994775i	$0.532554\pi$
$674$	0	0
$675$	2296.81	0.130969
$676$	0	0
$677$	−2573.49	−0.146097	−0.0730483	−	0.997328i	$-0.523273\pi$
$677$	−2573.49	−0.146097	−0.0730483	+	0.997328i	$0.523273\pi$
$678$	0	0
$679$	781.157	0.0441503
$680$	0	0
$681$	8593.18	0.483541
$682$	0	0
$683$	1122.55	0.0628888	0.0314444	−	0.999506i	$-0.489989\pi$
$683$	1122.55	0.0628888	0.0314444	+	0.999506i	$0.489989\pi$
$684$	0	0
$685$	10832.7	0.604227
$686$	0	0
$687$	1215.34	0.0674934
$688$	0	0
$689$	43242.3	2.39100
$690$	0	0
$691$	−11878.2	−0.653935	−0.326968	−	0.945036i	$-0.606027\pi$
$691$	−11878.2	−0.653935	−0.326968	+	0.945036i	$0.606027\pi$
$692$	0	0
$693$	−582.501	−0.0319298
$694$	0	0
$695$	12633.9	0.689541
$696$	0	0
$697$	−17578.4	−0.955278
$698$	0	0
$699$	−3181.78	−0.172169
$700$	0	0
$701$	24060.3	1.29635	0.648177	−	0.761490i	$-0.275532\pi$
$701$	24060.3	1.29635	0.648177	+	0.761490i	$0.275532\pi$
$702$	0	0
$703$	3180.32	0.170623
$704$	0	0
$705$	−2709.04	−0.144721
$706$	0	0
$707$	2426.90	0.129099
$708$	0	0
$709$	8958.27	0.474520	0.237260	−	0.971446i	$-0.423751\pi$
$709$	8958.27	0.474520	0.237260	+	0.971446i	$0.423751\pi$
$710$	0	0
$711$	−4219.29	−0.222554
$712$	0	0
$713$	3804.02	0.199806
$714$	0	0
$715$	−12657.2	−0.662031
$716$	0	0
$717$	−6984.91	−0.363816
$718$	0	0
$719$	1583.22	0.0821201	0.0410600	−	0.999157i	$-0.486927\pi$
$719$	1583.22	0.0821201	0.0410600	+	0.999157i	$0.486927\pi$
$720$	0	0
$721$	2719.39	0.140465
$722$	0	0
$723$	−8342.01	−0.429105
$724$	0	0
$725$	−3159.44	−0.161847
$726$	0	0
$727$	9360.49	0.477526	0.238763	−	0.971078i	$-0.423258\pi$
$727$	9360.49	0.477526	0.238763	+	0.971078i	$0.423258\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	7403.77	0.374608
$732$	0	0
$733$	29110.0	1.46685	0.733425	−	0.679770i	$-0.237920\pi$
$733$	29110.0	1.46685	0.733425	+	0.679770i	$0.237920\pi$
$734$	0	0
$735$	6348.30	0.318586
$736$	0	0
$737$	−9910.65	−0.495337
$738$	0	0
$739$	8095.45	0.402971	0.201486	−	0.979491i	$-0.435423\pi$
$739$	8095.45	0.402971	0.201486	+	0.979491i	$0.435423\pi$
$740$	0	0
$741$	−9803.10	−0.486000
$742$	0	0
$743$	26058.3	1.28666	0.643328	−	0.765590i	$-0.277553\pi$
$743$	26058.3	1.28666	0.643328	+	0.765590i	$0.277553\pi$
$744$	0	0
$745$	−6830.13	−0.335888
$746$	0	0
$747$	−5062.81	−0.247976
$748$	0	0
$749$	1812.03	0.0883979
$750$	0	0
$751$	−5375.84	−0.261208	−0.130604	−	0.991435i	$-0.541692\pi$
$751$	−5375.84	−0.261208	−0.130604	+	0.991435i	$0.541692\pi$
$752$	0	0
$753$	−19150.6	−0.926810
$754$	0	0
$755$	9638.82	0.464626
$756$	0	0
$757$	9027.75	0.433447	0.216723	−	0.976233i	$-0.430463\pi$
$757$	9027.75	0.433447	0.216723	+	0.976233i	$0.430463\pi$
$758$	0	0
$759$	−1565.76	−0.0748794
$760$	0	0
$761$	35567.6	1.69425	0.847126	−	0.531392i	$-0.178331\pi$
$761$	35567.6	1.69425	0.847126	+	0.531392i	$0.178331\pi$
$762$	0	0
$763$	1404.04	0.0666184
$764$	0	0
$765$	3500.77	0.165452
$766$	0	0
$767$	49294.9	2.32065
$768$	0	0
$769$	−37434.7	−1.75543	−0.877717	−	0.479179i	$-0.840934\pi$
$769$	−37434.7	−1.75543	−0.877717	+	0.479179i	$0.840934\pi$
$770$	0	0
$771$	−4497.90	−0.210101
$772$	0	0
$773$	−34295.9	−1.59578	−0.797891	−	0.602802i	$-0.794051\pi$
$773$	−34295.9	−1.59578	−0.797891	+	0.602802i	$0.794051\pi$
$774$	0	0
$775$	−14069.4	−0.652113
$776$	0	0
$777$	−735.061	−0.0339384
$778$	0	0
$779$	10572.3	0.486256
$780$	0	0
$781$	1832.41	0.0839547
$782$	0	0
$783$	−1002.80	−0.0457689
$784$	0	0
$785$	1980.89	0.0900651
$786$	0	0
$787$	−14951.9	−0.677227	−0.338613	−	0.940926i	$-0.609958\pi$
$787$	−14951.9	−0.677227	−0.338613	+	0.940926i	$0.609958\pi$
$788$	0	0
$789$	−19346.4	−0.872940
$790$	0	0
$791$	6457.16	0.290253
$792$	0	0
$793$	3963.19	0.177474
$794$	0	0
$795$	9287.54	0.414333
$796$	0	0
$797$	−28803.3	−1.28013	−0.640065	−	0.768321i	$-0.721093\pi$
$797$	−28803.3	−1.28013	−0.640065	+	0.768321i	$0.721093\pi$
$798$	0	0
$799$	8795.96	0.389460
$800$	0	0
$801$	4572.87	0.201716
$802$	0	0
$803$	−10739.4	−0.471962
$804$	0	0
$805$	−414.546	−0.0181501
$806$	0	0
$807$	4312.70	0.188122
$808$	0	0
$809$	7507.43	0.326264	0.163132	−	0.986604i	$-0.447840\pi$
$809$	7507.43	0.326264	0.163132	+	0.986604i	$0.447840\pi$
$810$	0	0
$811$	33767.4	1.46206	0.731031	−	0.682344i	$-0.239039\pi$
$811$	33767.4	1.46206	0.731031	+	0.682344i	$0.239039\pi$
$812$	0	0
$813$	−19928.7	−0.859693
$814$	0	0
$815$	5429.25	0.233348
$816$	0	0
$817$	−4452.92	−0.190683
$818$	0	0
$819$	2265.77	0.0966696
$820$	0	0
$821$	46177.1	1.96296	0.981481	−	0.191559i	$-0.0613545\pi$
$821$	46177.1	1.96296	0.981481	+	0.191559i	$0.0613545\pi$
$822$	0	0
$823$	−9956.57	−0.421706	−0.210853	−	0.977518i	$-0.567624\pi$
$823$	−9956.57	−0.421706	−0.210853	+	0.977518i	$0.567624\pi$
$824$	0	0
$825$	5791.06	0.244386
$826$	0	0
$827$	30710.3	1.29130	0.645648	−	0.763635i	$-0.276587\pi$
$827$	30710.3	1.29130	0.645648	+	0.763635i	$0.276587\pi$
$828$	0	0
$829$	32169.8	1.34777	0.673887	−	0.738835i	$-0.264624\pi$
$829$	32169.8	1.34777	0.673887	+	0.738835i	$0.264624\pi$
$830$	0	0
$831$	−12949.3	−0.540561
$832$	0	0
$833$	−20612.2	−0.857348
$834$	0	0
$835$	12306.7	0.510051
$836$	0	0
$837$	−4465.59	−0.184413
$838$	0	0
$839$	23470.9	0.965800	0.482900	−	0.875676i	$-0.339584\pi$
$839$	23470.9	0.965800	0.482900	+	0.875676i	$0.339584\pi$
$840$	0	0
$841$	−23009.6	−0.943441
$842$	0	0
$843$	2182.82	0.0891819
$844$	0	0
$845$	35349.7	1.43913
$846$	0	0
$847$	2327.57	0.0944231
$848$	0	0
$849$	25381.8	1.02603
$850$	0	0
$851$	−1975.84	−0.0795898
$852$	0	0
$853$	−22049.1	−0.885050	−0.442525	−	0.896756i	$-0.645917\pi$
$853$	−22049.1	−0.885050	−0.442525	+	0.896756i	$0.645917\pi$
$854$	0	0
$855$	−2105.50	−0.0842183
$856$	0	0
$857$	42122.4	1.67896	0.839482	−	0.543387i	$-0.182858\pi$
$857$	42122.4	1.67896	0.839482	+	0.543387i	$0.182858\pi$
$858$	0	0
$859$	25283.3	1.00426	0.502128	−	0.864793i	$-0.332551\pi$
$859$	25283.3	1.00426	0.502128	+	0.864793i	$0.332551\pi$
$860$	0	0
$861$	−2443.56	−0.0967205
$862$	0	0
$863$	1331.63	0.0525251	0.0262626	−	0.999655i	$-0.491639\pi$
$863$	1331.63	0.0525251	0.0262626	+	0.999655i	$0.491639\pi$
$864$	0	0
$865$	−12683.2	−0.498546
$866$	0	0
$867$	3372.39	0.132102
$868$	0	0
$869$	−10638.3	−0.415282
$870$	0	0
$871$	38549.7	1.49967
$872$	0	0
$873$	−2464.92	−0.0955612
$874$	0	0
$875$	3786.19	0.146282
$876$	0	0
$877$	−11303.8	−0.435235	−0.217618	−	0.976034i	$-0.569829\pi$
$877$	−11303.8	−0.435235	−0.217618	+	0.976034i	$0.569829\pi$
$878$	0	0
$879$	17742.6	0.680824
$880$	0	0
$881$	−46376.7	−1.77352	−0.886760	−	0.462230i	$-0.847050\pi$
$881$	−46376.7	−1.77352	−0.886760	+	0.462230i	$0.847050\pi$
$882$	0	0
$883$	−50480.9	−1.92392	−0.961958	−	0.273198i	$-0.911919\pi$
$883$	−50480.9	−1.92392	−0.961958	+	0.273198i	$0.911919\pi$
$884$	0	0
$885$	10587.5	0.402142
$886$	0	0
$887$	−9578.96	−0.362604	−0.181302	−	0.983427i	$-0.558031\pi$
$887$	−9578.96	−0.362604	−0.181302	+	0.983427i	$0.558031\pi$
$888$	0	0
$889$	938.490	0.0354060
$890$	0	0
$891$	1838.06	0.0691105
$892$	0	0
$893$	−5290.23	−0.198243
$894$	0	0
$895$	−20641.1	−0.770899
$896$	0	0
$897$	6090.38	0.226702
$898$	0	0
$899$	6142.77	0.227890
$900$	0	0
$901$	−30155.6	−1.11501
$902$	0	0
$903$	1029.19	0.0379285
$904$	0	0
$905$	27204.5	0.999234
$906$	0	0
$907$	42298.9	1.54852	0.774261	−	0.632866i	$-0.218122\pi$
$907$	42298.9	1.54852	0.774261	+	0.632866i	$0.218122\pi$
$908$	0	0
$909$	−7658.00	−0.279428
$910$	0	0
$911$	9558.76	0.347635	0.173818	−	0.984778i	$-0.444390\pi$
$911$	9558.76	0.347635	0.173818	+	0.984778i	$0.444390\pi$
$912$	0	0
$913$	−12765.1	−0.462720
$914$	0	0
$915$	851.210	0.0307542
$916$	0	0
$917$	−6150.65	−0.221497
$918$	0	0
$919$	44677.2	1.60366	0.801830	−	0.597552i	$-0.203860\pi$
$919$	44677.2	1.60366	0.801830	+	0.597552i	$0.203860\pi$
$920$	0	0
$921$	28051.5	1.00361
$922$	0	0
$923$	−7127.57	−0.254178
$924$	0	0
$925$	7307.77	0.259760
$926$	0	0
$927$	−8580.96	−0.304030
$928$	0	0
$929$	−46461.2	−1.64084	−0.820421	−	0.571761i	$-0.806261\pi$
$929$	−46461.2	−1.64084	−0.820421	+	0.571761i	$0.806261\pi$
$930$	0	0
$931$	12397.0	0.436407
$932$	0	0
$933$	−23603.9	−0.828249
$934$	0	0
$935$	8826.68	0.308731
$936$	0	0
$937$	−27305.2	−0.951997	−0.475998	−	0.879446i	$-0.657913\pi$
$937$	−27305.2	−0.951997	−0.475998	+	0.879446i	$0.657913\pi$
$938$	0	0
$939$	−26259.4	−0.912612
$940$	0	0
$941$	−11491.8	−0.398112	−0.199056	−	0.979988i	$-0.563787\pi$
$941$	−11491.8	−0.398112	−0.199056	+	0.979988i	$0.563787\pi$
$942$	0	0
$943$	−6568.28	−0.226822
$944$	0	0
$945$	486.641	0.0167518
$946$	0	0
$947$	−18682.9	−0.641089	−0.320544	−	0.947233i	$-0.603866\pi$
$947$	−18682.9	−0.641089	−0.320544	+	0.947233i	$0.603866\pi$
$948$	0	0
$949$	41773.4	1.42890
$950$	0	0
$951$	12377.9	0.422063
$952$	0	0
$953$	−41342.7	−1.40527	−0.702634	−	0.711551i	$-0.747993\pi$
$953$	−41342.7	−1.40527	−0.702634	+	0.711551i	$0.747993\pi$
$954$	0	0
$955$	−14682.8	−0.497512
$956$	0	0
$957$	−2528.41	−0.0854041
$958$	0	0
$959$	−4889.31	−0.164634
$960$	0	0
$961$	−2436.45	−0.0817847
$962$	0	0
$963$	−5717.80	−0.191333
$964$	0	0
$965$	−30395.6	−1.01396
$966$	0	0
$967$	−44316.0	−1.47374	−0.736870	−	0.676034i	$-0.763697\pi$
$967$	−44316.0	−1.47374	−0.736870	+	0.676034i	$0.763697\pi$
$968$	0	0
$969$	6836.33	0.226640
$970$	0	0
$971$	−17365.8	−0.573940	−0.286970	−	0.957940i	$-0.592648\pi$
$971$	−17365.8	−0.573940	−0.286970	+	0.957940i	$0.592648\pi$
$972$	0	0
$973$	−5702.29	−0.187880
$974$	0	0
$975$	−22525.6	−0.739895
$976$	0	0
$977$	23303.8	0.763105	0.381552	−	0.924347i	$-0.375390\pi$
$977$	23303.8	0.763105	0.381552	+	0.924347i	$0.375390\pi$
$978$	0	0
$979$	11529.8	0.376398
$980$	0	0
$981$	−4430.42	−0.144192
$982$	0	0
$983$	−39472.2	−1.28074	−0.640369	−	0.768067i	$-0.721219\pi$
$983$	−39472.2	−1.28074	−0.640369	+	0.768067i	$0.721219\pi$
$984$	0	0
$985$	10181.1	0.329337
$986$	0	0
$987$	1222.72	0.0394323
$988$	0	0
$989$	2766.47	0.0889470
$990$	0	0
$991$	12338.5	0.395505	0.197752	−	0.980252i	$-0.436636\pi$
$991$	12338.5	0.395505	0.197752	+	0.980252i	$0.436636\pi$
$992$	0	0
$993$	4259.94	0.136138
$994$	0	0
$995$	−4432.01	−0.141210
$996$	0	0
$997$	624.989	0.0198532	0.00992658	−	0.999951i	$-0.496840\pi$
$997$	624.989	0.0198532	0.00992658	+	0.999951i	$0.496840\pi$
$998$	0	0
$999$	2319.47	0.0734581

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.4.a.g.1.3	✓	4
3.2	odd	2		1656.4.a.m.1.2		4
4.3	odd	2		1104.4.a.u.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.4.a.g.1.3	✓	4	1.1	even	1	trivial
1104.4.a.u.1.3		4	4.3	odd	2
1656.4.a.m.1.2		4	3.2	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:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	552.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +6.31926 q^{5} -2.85219 q^{7} +9.00000 q^{9} +22.6922 q^{11} -88.2664 q^{13} -18.9578 q^{15} +61.5538 q^{17} -37.0209 q^{19} +8.55656 q^{21} +23.0000 q^{23} -85.0669 q^{25} -27.0000 q^{27} +37.1407 q^{29} +165.392 q^{31} -68.0765 q^{33} -18.0237 q^{35} -85.9061 q^{37} +264.799 q^{39} -285.578 q^{41} +120.281 q^{43} +56.8734 q^{45} +142.899 q^{47} -334.865 q^{49} -184.661 q^{51} -489.906 q^{53} +143.398 q^{55} +111.063 q^{57} -558.479 q^{59} -44.9003 q^{61} -25.6697 q^{63} -557.779 q^{65} -436.743 q^{67} -69.0000 q^{69} +80.7506 q^{71} -473.265 q^{73} +255.201 q^{75} -64.7223 q^{77} -468.810 q^{79} +81.0000 q^{81} -562.534 q^{83} +388.975 q^{85} -111.422 q^{87} +508.096 q^{89} +251.752 q^{91} -496.176 q^{93} -233.945 q^{95} -273.880 q^{97} +204.229 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 14 q^{5} + 14 q^{7} + 36 q^{9} - 16 q^{11} + 4 q^{13} + 42 q^{15} - 22 q^{17} + 134 q^{19} - 42 q^{21} + 92 q^{23} + 24 q^{25} - 108 q^{27} - 180 q^{29} + 276 q^{31} + 48 q^{33} - 96 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 - 14 * q^5 + 14 * q^7 + 36 * q^9 - 16 * q^11 + 4 * q^13 + 42 * q^15 - 22 * q^17 + 134 * q^19 - 42 * q^21 + 92 * q^23 + 24 * q^25 - 108 * q^27 - 180 * q^29 + 276 * q^31 + 48 * q^33 - 96 * q^35 - 244 * q^37 - 12 * q^39 - 416 * q^41 - 242 * q^43 - 126 * q^45 - 384 * q^47 - 296 * q^49 + 66 * q^51 - 1426 * q^53 - 296 * q^55 - 402 * q^57 - 1000 * q^59 - 1140 * q^61 + 126 * q^63 - 1916 * q^65 - 646 * q^67 - 276 * q^69 - 432 * q^71 - 1568 * q^73 - 72 * q^75 - 2472 * q^77 + 778 * q^79 + 324 * q^81 + 112 * q^83 - 1772 * q^85 + 540 * q^87 - 2458 * q^89 - 316 * q^91 - 828 * q^93 + 952 * q^95 - 980 * q^97 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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