LMFDB - Embedded newform 5488.2.a.p.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5488,2,Mod(1,5488)] 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("5488.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5488, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5488 = 2^{4} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5488.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,5,0,-11,0,0,0,9,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$43.8219006293$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1279733.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 10x^{2} - 11x - 1 $ x^6 - 2x^5 - 6x^4 + 10x^3 + 10x^2 - 11*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 343)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.54570$ of defining polynomial
Character	$\chi$	$=$	5488.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37250 q^{3} -0.315405 q^{5} +2.62874 q^{9} +1.80936 q^{11} -5.89929 q^{13} -0.748296 q^{15} -4.31210 q^{17} -6.30043 q^{19} +6.44087 q^{23} -4.90052 q^{25} -0.880823 q^{27} +7.97508 q^{29} -5.19058 q^{31} +4.29271 q^{33} -1.41486 q^{37} -13.9960 q^{39} +2.17315 q^{41} -1.04843 q^{43} -0.829115 q^{45} +1.17861 q^{47} -10.2304 q^{51} -4.10342 q^{53} -0.570682 q^{55} -14.9477 q^{57} +8.37022 q^{59} -5.54922 q^{61} +1.86066 q^{65} -9.80360 q^{67} +15.2809 q^{69} -1.21499 q^{71} -8.12513 q^{73} -11.6265 q^{75} +1.13986 q^{79} -9.97596 q^{81} -3.71459 q^{83} +1.36006 q^{85} +18.9209 q^{87} -16.7464 q^{89} -12.3146 q^{93} +1.98719 q^{95} -9.25759 q^{97} +4.75634 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{3} - 11 q^{5} + 9 q^{9} + q^{11} - 7 q^{13} + 5 q^{15} - 26 q^{17} - 3 q^{19} + 9 q^{23} + 15 q^{25} + 8 q^{27} - 2 q^{29} - 2 q^{31} - 18 q^{33} - 2 q^{37} - 7 q^{39} - 28 q^{41} - 5 q^{43}+ \cdots - 12 q^{99}+O(q^{100}) $ 6 * q + 5 * q^3 - 11 * q^5 + 9 * q^9 + q^11 - 7 * q^13 + 5 * q^15 - 26 * q^17 - 3 * q^19 + 9 * q^23 + 15 * q^25 + 8 * q^27 - 2 * q^29 - 2 * q^31 - 18 * q^33 - 2 * q^37 - 7 * q^39 - 28 * q^41 - 5 * q^43 - 3 * q^45 + 18 * q^47 - 13 * q^51 - q^53 - 22 * q^55 - 26 * q^57 + 5 * q^59 + 17 * q^61 - 14 * q^65 + 22 * q^67 + 20 * q^69 + 2 * q^71 - 12 * q^73 - 27 * q^75 + 2 * q^79 - 30 * q^81 + 7 * q^83 + 37 * q^85 - 25 * q^87 - 39 * q^89 - 22 * q^93 + 11 * q^95 - 35 * q^97 - 12 * 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
$3$	2.37250	1.36976	0.684880	−	0.728655i	$-0.259854\pi$
$3$	2.37250	1.36976	0.684880	+	0.728655i	$0.259854\pi$
$4$	0	0
$5$	−0.315405	−0.141053	−0.0705266	−	0.997510i	$-0.522468\pi$
$5$	−0.315405	−0.141053	−0.0705266	+	0.997510i	$0.522468\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.62874	0.876245
$10$	0	0
$11$	1.80936	0.545544	0.272772	−	0.962079i	$-0.412060\pi$
$11$	1.80936	0.545544	0.272772	+	0.962079i	$0.412060\pi$
$12$	0	0
$13$	−5.89929	−1.63617	−0.818084	−	0.575099i	$-0.804964\pi$
$13$	−5.89929	−1.63617	−0.818084	+	0.575099i	$0.804964\pi$
$14$	0	0
$15$	−0.748296	−0.193209
$16$	0	0
$17$	−4.31210	−1.04584	−0.522919	−	0.852382i	$-0.675157\pi$
$17$	−4.31210	−1.04584	−0.522919	+	0.852382i	$0.675157\pi$
$18$	0	0
$19$	−6.30043	−1.44542	−0.722709	−	0.691152i	$-0.757103\pi$
$19$	−6.30043	−1.44542	−0.722709	+	0.691152i	$0.757103\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.44087	1.34301	0.671507	−	0.740999i	$-0.265647\pi$
$23$	6.44087	1.34301	0.671507	+	0.740999i	$0.265647\pi$
$24$	0	0
$25$	−4.90052	−0.980104
$26$	0	0
$27$	−0.880823	−0.169515
$28$	0	0
$29$	7.97508	1.48094	0.740468	−	0.672092i	$-0.234604\pi$
$29$	7.97508	1.48094	0.740468	+	0.672092i	$0.234604\pi$
$30$	0	0
$31$	−5.19058	−0.932256	−0.466128	−	0.884717i	$-0.654351\pi$
$31$	−5.19058	−0.932256	−0.466128	+	0.884717i	$0.654351\pi$
$32$	0	0
$33$	4.29271	0.747265
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.41486	−0.232601	−0.116301	−	0.993214i	$-0.537104\pi$
$37$	−1.41486	−0.232601	−0.116301	+	0.993214i	$0.537104\pi$
$38$	0	0
$39$	−13.9960	−2.24116
$40$	0	0
$41$	2.17315	0.339389	0.169694	−	0.985497i	$-0.445722\pi$
$41$	2.17315	0.339389	0.169694	+	0.985497i	$0.445722\pi$
$42$	0	0
$43$	−1.04843	−0.159885	−0.0799423	−	0.996799i	$-0.525474\pi$
$43$	−1.04843	−0.159885	−0.0799423	+	0.996799i	$0.525474\pi$
$44$	0	0
$45$	−0.829115	−0.123597
$46$	0	0
$47$	1.17861	0.171918	0.0859589	−	0.996299i	$-0.472605\pi$
$47$	1.17861	0.171918	0.0859589	+	0.996299i	$0.472605\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−10.2304	−1.43255
$52$	0	0
$53$	−4.10342	−0.563648	−0.281824	−	0.959466i	$-0.590939\pi$
$53$	−4.10342	−0.563648	−0.281824	+	0.959466i	$0.590939\pi$
$54$	0	0
$55$	−0.570682	−0.0769507
$56$	0	0
$57$	−14.9477	−1.97988
$58$	0	0
$59$	8.37022	1.08971	0.544855	−	0.838530i	$-0.316585\pi$
$59$	8.37022	1.08971	0.544855	+	0.838530i	$0.316585\pi$
$60$	0	0
$61$	−5.54922	−0.710504	−0.355252	−	0.934771i	$-0.615605\pi$
$61$	−5.54922	−0.710504	−0.355252	+	0.934771i	$0.615605\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.86066	0.230787
$66$	0	0
$67$	−9.80360	−1.19770	−0.598850	−	0.800861i	$-0.704375\pi$
$67$	−9.80360	−1.19770	−0.598850	+	0.800861i	$0.704375\pi$
$68$	0	0
$69$	15.2809	1.83961
$70$	0	0
$71$	−1.21499	−0.144193	−0.0720965	−	0.997398i	$-0.522969\pi$
$71$	−1.21499	−0.144193	−0.0720965	+	0.997398i	$0.522969\pi$
$72$	0	0
$73$	−8.12513	−0.950974	−0.475487	−	0.879723i	$-0.657728\pi$
$73$	−8.12513	−0.950974	−0.475487	+	0.879723i	$0.657728\pi$
$74$	0	0
$75$	−11.6265	−1.34251
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.13986	0.128244	0.0641221	−	0.997942i	$-0.479575\pi$
$79$	1.13986	0.128244	0.0641221	+	0.997942i	$0.479575\pi$
$80$	0	0
$81$	−9.97596	−1.10844
$82$	0	0
$83$	−3.71459	−0.407729	−0.203865	−	0.978999i	$-0.565350\pi$
$83$	−3.71459	−0.407729	−0.203865	+	0.978999i	$0.565350\pi$
$84$	0	0
$85$	1.36006	0.147519
$86$	0	0
$87$	18.9209	2.02853
$88$	0	0
$89$	−16.7464	−1.77512	−0.887558	−	0.460696i	$-0.847600\pi$
$89$	−16.7464	−1.77512	−0.887558	+	0.460696i	$0.847600\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−12.3146	−1.27697
$94$	0	0
$95$	1.98719	0.203881
$96$	0	0
$97$	−9.25759	−0.939966	−0.469983	−	0.882676i	$-0.655740\pi$
$97$	−9.25759	−0.939966	−0.469983	+	0.882676i	$0.655740\pi$
$98$	0	0
$99$	4.75634	0.478030
$100$	0	0
$101$	0.909356	0.0904843	0.0452422	−	0.998976i	$-0.485594\pi$
$101$	0.909356	0.0904843	0.0452422	+	0.998976i	$0.485594\pi$
$102$	0	0
$103$	−12.1316	−1.19536	−0.597681	−	0.801734i	$-0.703911\pi$
$103$	−12.1316	−1.19536	−0.597681	+	0.801734i	$0.703911\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.78691	−0.269421	−0.134710	−	0.990885i	$-0.543010\pi$
$107$	−2.78691	−0.269421	−0.134710	+	0.990885i	$0.543010\pi$
$108$	0	0
$109$	−12.8621	−1.23196	−0.615982	−	0.787760i	$-0.711241\pi$
$109$	−12.8621	−1.23196	−0.615982	+	0.787760i	$0.711241\pi$
$110$	0	0
$111$	−3.35674	−0.318608
$112$	0	0
$113$	−4.57651	−0.430522	−0.215261	−	0.976557i	$-0.569060\pi$
$113$	−4.57651	−0.430522	−0.215261	+	0.976557i	$0.569060\pi$
$114$	0	0
$115$	−2.03148	−0.189436
$116$	0	0
$117$	−15.5077	−1.43368
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.72620	−0.702382
$122$	0	0
$123$	5.15578	0.464881
$124$	0	0
$125$	3.12267	0.279300
$126$	0	0
$127$	13.1388	1.16588	0.582941	−	0.812515i	$-0.301902\pi$
$127$	13.1388	1.16588	0.582941	+	0.812515i	$0.301902\pi$
$128$	0	0
$129$	−2.48740	−0.219004
$130$	0	0
$131$	14.1942	1.24015	0.620076	−	0.784542i	$-0.287102\pi$
$131$	14.1942	1.24015	0.620076	+	0.784542i	$0.287102\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.277816	0.0239106
$136$	0	0
$137$	4.66108	0.398223	0.199112	−	0.979977i	$-0.436194\pi$
$137$	4.66108	0.398223	0.199112	+	0.979977i	$0.436194\pi$
$138$	0	0
$139$	5.69800	0.483298	0.241649	−	0.970364i	$-0.422312\pi$
$139$	5.69800	0.483298	0.241649	+	0.970364i	$0.422312\pi$
$140$	0	0
$141$	2.79625	0.235486
$142$	0	0
$143$	−10.6740	−0.892601
$144$	0	0
$145$	−2.51538	−0.208891
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.6438	1.03582	0.517910	−	0.855435i	$-0.326710\pi$
$149$	12.6438	1.03582	0.517910	+	0.855435i	$0.326710\pi$
$150$	0	0
$151$	−9.97216	−0.811523	−0.405762	−	0.913979i	$-0.632994\pi$
$151$	−9.97216	−0.811523	−0.405762	+	0.913979i	$0.632994\pi$
$152$	0	0
$153$	−11.3354	−0.916410
$154$	0	0
$155$	1.63713	0.131498
$156$	0	0
$157$	18.2931	1.45995	0.729974	−	0.683475i	$-0.239532\pi$
$157$	18.2931	1.45995	0.729974	+	0.683475i	$0.239532\pi$
$158$	0	0
$159$	−9.73535	−0.772064
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−7.58009	−0.593719	−0.296859	−	0.954921i	$-0.595939\pi$
$163$	−7.58009	−0.593719	−0.296859	+	0.954921i	$0.595939\pi$
$164$	0	0
$165$	−1.35394	−0.105404
$166$	0	0
$167$	23.0772	1.78576	0.892882	−	0.450291i	$-0.148680\pi$
$167$	23.0772	1.78576	0.892882	+	0.450291i	$0.148680\pi$
$168$	0	0
$169$	21.8016	1.67705
$170$	0	0
$171$	−16.5622	−1.26654
$172$	0	0
$173$	2.25467	0.171419	0.0857095	−	0.996320i	$-0.472684\pi$
$173$	2.25467	0.171419	0.0857095	+	0.996320i	$0.472684\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	19.8583	1.49264
$178$	0	0
$179$	−8.50300	−0.635544	−0.317772	−	0.948167i	$-0.602935\pi$
$179$	−8.50300	−0.635544	−0.317772	+	0.948167i	$0.602935\pi$
$180$	0	0
$181$	12.9734	0.964306	0.482153	−	0.876087i	$-0.339855\pi$
$181$	12.9734	0.964306	0.482153	+	0.876087i	$0.339855\pi$
$182$	0	0
$183$	−13.1655	−0.973221
$184$	0	0
$185$	0.446253	0.0328092
$186$	0	0
$187$	−7.80216	−0.570550
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.0771	1.74216	0.871080	−	0.491141i	$-0.163420\pi$
$191$	24.0771	1.74216	0.871080	+	0.491141i	$0.163420\pi$
$192$	0	0
$193$	5.04598	0.363217	0.181609	−	0.983371i	$-0.441870\pi$
$193$	5.04598	0.363217	0.181609	+	0.983371i	$0.441870\pi$
$194$	0	0
$195$	4.41442	0.316123
$196$	0	0
$197$	−8.75849	−0.624017	−0.312008	−	0.950079i	$-0.601002\pi$
$197$	−8.75849	−0.624017	−0.312008	+	0.950079i	$0.601002\pi$
$198$	0	0
$199$	0.839045	0.0594784	0.0297392	−	0.999558i	$-0.490532\pi$
$199$	0.839045	0.0594784	0.0297392	+	0.999558i	$0.490532\pi$
$200$	0	0
$201$	−23.2590	−1.64056
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.685421	−0.0478719
$206$	0	0
$207$	16.9313	1.17681
$208$	0	0
$209$	−11.3998	−0.788539
$210$	0	0
$211$	0.0296358	0.00204021	0.00102011	−	0.999999i	$-0.499675\pi$
$211$	0.0296358	0.00204021	0.00102011	+	0.999999i	$0.499675\pi$
$212$	0	0
$213$	−2.88256	−0.197510
$214$	0	0
$215$	0.330681	0.0225522
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−19.2768	−1.30261
$220$	0	0
$221$	25.4383	1.71117
$222$	0	0
$223$	−16.0030	−1.07164	−0.535821	−	0.844332i	$-0.679998\pi$
$223$	−16.0030	−1.07164	−0.535821	+	0.844332i	$0.679998\pi$
$224$	0	0
$225$	−12.8822	−0.858811
$226$	0	0
$227$	3.69275	0.245097	0.122548	−	0.992463i	$-0.460893\pi$
$227$	3.69275	0.245097	0.122548	+	0.992463i	$0.460893\pi$
$228$	0	0
$229$	−8.96573	−0.592472	−0.296236	−	0.955115i	$-0.595731\pi$
$229$	−8.96573	−0.592472	−0.296236	+	0.955115i	$0.595731\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.6148	−1.67808	−0.839041	−	0.544068i	$-0.816883\pi$
$233$	−25.6148	−1.67808	−0.839041	+	0.544068i	$0.816883\pi$
$234$	0	0
$235$	−0.371739	−0.0242496
$236$	0	0
$237$	2.70431	0.175664
$238$	0	0
$239$	−4.53752	−0.293508	−0.146754	−	0.989173i	$-0.546883\pi$
$239$	−4.53752	−0.293508	−0.146754	+	0.989173i	$0.546883\pi$
$240$	0	0
$241$	8.99811	0.579619	0.289810	−	0.957084i	$-0.406408\pi$
$241$	8.99811	0.579619	0.289810	+	0.957084i	$0.406408\pi$
$242$	0	0
$243$	−21.0254	−1.34878
$244$	0	0
$245$	0	0
$246$	0	0
$247$	37.1681	2.36495
$248$	0	0
$249$	−8.81285	−0.558492
$250$	0	0
$251$	11.8326	0.746867	0.373434	−	0.927657i	$-0.378180\pi$
$251$	11.8326	0.746867	0.373434	+	0.927657i	$0.378180\pi$
$252$	0	0
$253$	11.6539	0.732672
$254$	0	0
$255$	3.22673	0.202066
$256$	0	0
$257$	−18.9916	−1.18467	−0.592333	−	0.805693i	$-0.701793\pi$
$257$	−18.9916	−1.18467	−0.592333	+	0.805693i	$0.701793\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	20.9644	1.29766
$262$	0	0
$263$	19.0783	1.17642	0.588211	−	0.808708i	$-0.299833\pi$
$263$	19.0783	1.17642	0.588211	+	0.808708i	$0.299833\pi$
$264$	0	0
$265$	1.29424	0.0795044
$266$	0	0
$267$	−39.7308	−2.43148
$268$	0	0
$269$	−23.5539	−1.43610	−0.718052	−	0.695989i	$-0.754966\pi$
$269$	−23.5539	−1.43610	−0.718052	+	0.695989i	$0.754966\pi$
$270$	0	0
$271$	7.17878	0.436080	0.218040	−	0.975940i	$-0.430034\pi$
$271$	7.17878	0.436080	0.218040	+	0.975940i	$0.430034\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.86682	−0.534690
$276$	0	0
$277$	5.28743	0.317691	0.158845	−	0.987303i	$-0.449223\pi$
$277$	5.28743	0.317691	0.158845	+	0.987303i	$0.449223\pi$
$278$	0	0
$279$	−13.6447	−0.816885
$280$	0	0
$281$	15.0127	0.895581	0.447791	−	0.894138i	$-0.352211\pi$
$281$	15.0127	0.895581	0.447791	+	0.894138i	$0.352211\pi$
$282$	0	0
$283$	−11.9400	−0.709759	−0.354879	−	0.934912i	$-0.615478\pi$
$283$	−11.9400	−0.709759	−0.354879	+	0.934912i	$0.615478\pi$
$284$	0	0
$285$	4.71459	0.279268
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.59420	0.0937767
$290$	0	0
$291$	−21.9636	−1.28753
$292$	0	0
$293$	1.37897	0.0805606	0.0402803	−	0.999188i	$-0.487175\pi$
$293$	1.37897	0.0805606	0.0402803	+	0.999188i	$0.487175\pi$
$294$	0	0
$295$	−2.64001	−0.153707
$296$	0	0
$297$	−1.59373	−0.0924776
$298$	0	0
$299$	−37.9965	−2.19740
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.15744	0.123942
$304$	0	0
$305$	1.75025	0.100219
$306$	0	0
$307$	−25.5584	−1.45870	−0.729348	−	0.684143i	$-0.760177\pi$
$307$	−25.5584	−1.45870	−0.729348	+	0.684143i	$0.760177\pi$
$308$	0	0
$309$	−28.7822	−1.63736
$310$	0	0
$311$	−19.8362	−1.12481	−0.562403	−	0.826863i	$-0.690123\pi$
$311$	−19.8362	−1.12481	−0.562403	+	0.826863i	$0.690123\pi$
$312$	0	0
$313$	11.5809	0.654590	0.327295	−	0.944922i	$-0.393863\pi$
$313$	11.5809	0.654590	0.327295	+	0.944922i	$0.393863\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.25523	0.463660	0.231830	−	0.972756i	$-0.425529\pi$
$317$	8.25523	0.463660	0.231830	+	0.972756i	$0.425529\pi$
$318$	0	0
$319$	14.4298	0.807915
$320$	0	0
$321$	−6.61193	−0.369042
$322$	0	0
$323$	27.1681	1.51167
$324$	0	0
$325$	28.9096	1.60362
$326$	0	0
$327$	−30.5153	−1.68750
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.5510	−1.23951	−0.619757	−	0.784794i	$-0.712769\pi$
$331$	−22.5510	−1.23951	−0.619757	+	0.784794i	$0.712769\pi$
$332$	0	0
$333$	−3.71929	−0.203816
$334$	0	0
$335$	3.09210	0.168939
$336$	0	0
$337$	−7.84006	−0.427075	−0.213538	−	0.976935i	$-0.568499\pi$
$337$	−7.84006	−0.427075	−0.213538	+	0.976935i	$0.568499\pi$
$338$	0	0
$339$	−10.8578	−0.589712
$340$	0	0
$341$	−9.39165	−0.508586
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4.81968	−0.259483
$346$	0	0
$347$	25.7782	1.38384	0.691922	−	0.721972i	$-0.256764\pi$
$347$	25.7782	1.38384	0.691922	+	0.721972i	$0.256764\pi$
$348$	0	0
$349$	31.4253	1.68216	0.841080	−	0.540910i	$-0.181920\pi$
$349$	31.4253	1.68216	0.841080	+	0.540910i	$0.181920\pi$
$350$	0	0
$351$	5.19623	0.277354
$352$	0	0
$353$	7.76310	0.413188	0.206594	−	0.978427i	$-0.433762\pi$
$353$	7.76310	0.413188	0.206594	+	0.978427i	$0.433762\pi$
$354$	0	0
$355$	0.383214	0.0203389
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.5711	−0.716253	−0.358126	−	0.933673i	$-0.616584\pi$
$359$	−13.5711	−0.716253	−0.358126	+	0.933673i	$0.616584\pi$
$360$	0	0
$361$	20.6955	1.08923
$362$	0	0
$363$	−18.3304	−0.962096
$364$	0	0
$365$	2.56270	0.134138
$366$	0	0
$367$	−0.510059	−0.0266249	−0.0133124	−	0.999911i	$-0.504238\pi$
$367$	−0.510059	−0.0266249	−0.0133124	+	0.999911i	$0.504238\pi$
$368$	0	0
$369$	5.71263	0.297388
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−14.1579	−0.733066	−0.366533	−	0.930405i	$-0.619455\pi$
$373$	−14.1579	−0.733066	−0.366533	+	0.930405i	$0.619455\pi$
$374$	0	0
$375$	7.40852	0.382574
$376$	0	0
$377$	−47.0473	−2.42306
$378$	0	0
$379$	22.3862	1.14990	0.574951	−	0.818188i	$-0.305021\pi$
$379$	22.3862	1.14990	0.574951	+	0.818188i	$0.305021\pi$
$380$	0	0
$381$	31.1718	1.59698
$382$	0	0
$383$	−3.98108	−0.203423	−0.101712	−	0.994814i	$-0.532432\pi$
$383$	−3.98108	−0.203423	−0.101712	+	0.994814i	$0.532432\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.75605	−0.140098
$388$	0	0
$389$	1.28096	0.0649473	0.0324736	−	0.999473i	$-0.489661\pi$
$389$	1.28096	0.0649473	0.0324736	+	0.999473i	$0.489661\pi$
$390$	0	0
$391$	−27.7737	−1.40457
$392$	0	0
$393$	33.6757	1.69871
$394$	0	0
$395$	−0.359517	−0.0180893
$396$	0	0
$397$	−13.0394	−0.654429	−0.327214	−	0.944950i	$-0.606110\pi$
$397$	−13.0394	−0.654429	−0.327214	+	0.944950i	$0.606110\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.5407	−1.17557	−0.587784	−	0.809018i	$-0.699999\pi$
$401$	−23.5407	−1.17557	−0.587784	+	0.809018i	$0.699999\pi$
$402$	0	0
$403$	30.6207	1.52533
$404$	0	0
$405$	3.14646	0.156349
$406$	0	0
$407$	−2.55999	−0.126894
$408$	0	0
$409$	31.1044	1.53802	0.769008	−	0.639239i	$-0.220751\pi$
$409$	31.1044	1.53802	0.769008	+	0.639239i	$0.220751\pi$
$410$	0	0
$411$	11.0584	0.545471
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.17160	0.0575115
$416$	0	0
$417$	13.5185	0.662003
$418$	0	0
$419$	9.06838	0.443020	0.221510	−	0.975158i	$-0.428902\pi$
$419$	9.06838	0.443020	0.221510	+	0.975158i	$0.428902\pi$
$420$	0	0
$421$	−25.6040	−1.24786	−0.623932	−	0.781478i	$-0.714466\pi$
$421$	−25.6040	−1.24786	−0.623932	+	0.781478i	$0.714466\pi$
$422$	0	0
$423$	3.09825	0.150642
$424$	0	0
$425$	21.1315	1.02503
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−25.3239	−1.22265
$430$	0	0
$431$	1.87741	0.0904317	0.0452158	−	0.998977i	$-0.485602\pi$
$431$	1.87741	0.0904317	0.0452158	+	0.998977i	$0.485602\pi$
$432$	0	0
$433$	20.0260	0.962389	0.481195	−	0.876614i	$-0.340203\pi$
$433$	20.0260	0.962389	0.481195	+	0.876614i	$0.340203\pi$
$434$	0	0
$435$	−5.96773	−0.286131
$436$	0	0
$437$	−40.5802	−1.94122
$438$	0	0
$439$	−14.7909	−0.705933	−0.352966	−	0.935636i	$-0.614827\pi$
$439$	−14.7909	−0.705933	−0.352966	+	0.935636i	$0.614827\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−29.9458	−1.42277	−0.711385	−	0.702802i	$-0.751932\pi$
$443$	−29.9458	−1.42277	−0.711385	+	0.702802i	$0.751932\pi$
$444$	0	0
$445$	5.28189	0.250386
$446$	0	0
$447$	29.9974	1.41883
$448$	0	0
$449$	8.34110	0.393641	0.196820	−	0.980440i	$-0.436938\pi$
$449$	8.34110	0.393641	0.196820	+	0.980440i	$0.436938\pi$
$450$	0	0
$451$	3.93201	0.185151
$452$	0	0
$453$	−23.6589	−1.11159
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.5910	1.19710	0.598548	−	0.801087i	$-0.295745\pi$
$457$	25.5910	1.19710	0.598548	+	0.801087i	$0.295745\pi$
$458$	0	0
$459$	3.79820	0.177285
$460$	0	0
$461$	−7.02658	−0.327260	−0.163630	−	0.986522i	$-0.552320\pi$
$461$	−7.02658	−0.327260	−0.163630	+	0.986522i	$0.552320\pi$
$462$	0	0
$463$	26.6142	1.23687	0.618434	−	0.785837i	$-0.287767\pi$
$463$	26.6142	1.23687	0.618434	+	0.785837i	$0.287767\pi$
$464$	0	0
$465$	3.88409	0.180120
$466$	0	0
$467$	19.6657	0.910020	0.455010	−	0.890486i	$-0.349636\pi$
$467$	19.6657	0.910020	0.455010	+	0.890486i	$0.349636\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	43.4003	1.99978
$472$	0	0
$473$	−1.89700	−0.0872241
$474$	0	0
$475$	30.8754	1.41666
$476$	0	0
$477$	−10.7868	−0.493894
$478$	0	0
$479$	2.33577	0.106724	0.0533620	−	0.998575i	$-0.483006\pi$
$479$	2.33577	0.106724	0.0533620	+	0.998575i	$0.483006\pi$
$480$	0	0
$481$	8.34666	0.380575
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.91989	0.132585
$486$	0	0
$487$	−29.0051	−1.31435	−0.657173	−	0.753739i	$-0.728248\pi$
$487$	−29.0051	−1.31435	−0.657173	+	0.753739i	$0.728248\pi$
$488$	0	0
$489$	−17.9837	−0.813253
$490$	0	0
$491$	24.3816	1.10032	0.550162	−	0.835058i	$-0.314566\pi$
$491$	24.3816	1.10032	0.550162	+	0.835058i	$0.314566\pi$
$492$	0	0
$493$	−34.3894	−1.54882
$494$	0	0
$495$	−1.50017	−0.0674277
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−18.0540	−0.808209	−0.404105	−	0.914713i	$-0.632417\pi$
$499$	−18.0540	−0.808209	−0.404105	+	0.914713i	$0.632417\pi$
$500$	0	0
$501$	54.7505	2.44607
$502$	0	0
$503$	20.8224	0.928426	0.464213	−	0.885724i	$-0.346337\pi$
$503$	20.8224	0.928426	0.464213	+	0.885724i	$0.346337\pi$
$504$	0	0
$505$	−0.286815	−0.0127631
$506$	0	0
$507$	51.7242	2.29715
$508$	0	0
$509$	10.3112	0.457038	0.228519	−	0.973539i	$-0.426612\pi$
$509$	10.3112	0.457038	0.228519	+	0.973539i	$0.426612\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.54957	0.245019
$514$	0	0
$515$	3.82636	0.168610
$516$	0	0
$517$	2.13253	0.0937887
$518$	0	0
$519$	5.34919	0.234803
$520$	0	0
$521$	−14.6474	−0.641716	−0.320858	−	0.947127i	$-0.603971\pi$
$521$	−14.6474	−0.641716	−0.320858	+	0.947127i	$0.603971\pi$
$522$	0	0
$523$	23.4188	1.02403	0.512016	−	0.858976i	$-0.328899\pi$
$523$	23.4188	1.02403	0.512016	+	0.858976i	$0.328899\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.3823	0.974988
$528$	0	0
$529$	18.4847	0.803685
$530$	0	0
$531$	22.0031	0.954853
$532$	0	0
$533$	−12.8200	−0.555297
$534$	0	0
$535$	0.879005	0.0380027
$536$	0	0
$537$	−20.1733	−0.870544
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.92261	0.383613	0.191807	−	0.981433i	$-0.438565\pi$
$541$	8.92261	0.383613	0.191807	+	0.981433i	$0.438565\pi$
$542$	0	0
$543$	30.7794	1.32087
$544$	0	0
$545$	4.05676	0.173773
$546$	0	0
$547$	−31.6625	−1.35379	−0.676896	−	0.736079i	$-0.736675\pi$
$547$	−31.6625	−1.35379	−0.676896	+	0.736079i	$0.736675\pi$
$548$	0	0
$549$	−14.5874	−0.622576
$550$	0	0
$551$	−50.2465	−2.14057
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1.05873	0.0449407
$556$	0	0
$557$	14.9280	0.632520	0.316260	−	0.948673i	$-0.397573\pi$
$557$	14.9280	0.632520	0.316260	+	0.948673i	$0.397573\pi$
$558$	0	0
$559$	6.18501	0.261598
$560$	0	0
$561$	−18.5106	−0.781517
$562$	0	0
$563$	−29.8296	−1.25717	−0.628584	−	0.777742i	$-0.716365\pi$
$563$	−29.8296	−1.25717	−0.628584	+	0.777742i	$0.716365\pi$
$564$	0	0
$565$	1.44345	0.0607265
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.43144	0.185776	0.0928878	−	0.995677i	$-0.470390\pi$
$569$	4.43144	0.185776	0.0928878	+	0.995677i	$0.470390\pi$
$570$	0	0
$571$	42.5487	1.78061	0.890305	−	0.455366i	$-0.150491\pi$
$571$	42.5487	1.78061	0.890305	+	0.455366i	$0.150491\pi$
$572$	0	0
$573$	57.1229	2.38634
$574$	0	0
$575$	−31.5636	−1.31629
$576$	0	0
$577$	−1.07428	−0.0447231	−0.0223615	−	0.999750i	$-0.507118\pi$
$577$	−1.07428	−0.0447231	−0.0223615	+	0.999750i	$0.507118\pi$
$578$	0	0
$579$	11.9716	0.497521
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7.42458	−0.307495
$584$	0	0
$585$	4.89119	0.202226
$586$	0	0
$587$	−18.3363	−0.756821	−0.378411	−	0.925638i	$-0.623529\pi$
$587$	−18.3363	−0.756821	−0.378411	+	0.925638i	$0.623529\pi$
$588$	0	0
$589$	32.7029	1.34750
$590$	0	0
$591$	−20.7795	−0.854754
$592$	0	0
$593$	−41.6503	−1.71037	−0.855187	−	0.518320i	$-0.826558\pi$
$593$	−41.6503	−1.71037	−0.855187	+	0.518320i	$0.826558\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.99063	0.0814711
$598$	0	0
$599$	−27.6544	−1.12993	−0.564964	−	0.825116i	$-0.691110\pi$
$599$	−27.6544	−1.12993	−0.564964	+	0.825116i	$0.691110\pi$
$600$	0	0
$601$	−25.5296	−1.04137	−0.520687	−	0.853747i	$-0.674324\pi$
$601$	−25.5296	−1.04137	−0.520687	+	0.853747i	$0.674324\pi$
$602$	0	0
$603$	−25.7711	−1.04948
$604$	0	0
$605$	2.43688	0.0990733
$606$	0	0
$607$	6.41487	0.260371	0.130186	−	0.991490i	$-0.458443\pi$
$607$	6.41487	0.260371	0.130186	+	0.991490i	$0.458443\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.95296	−0.281287
$612$	0	0
$613$	−45.2341	−1.82699	−0.913495	−	0.406851i	$-0.866627\pi$
$613$	−45.2341	−1.82699	−0.913495	+	0.406851i	$0.866627\pi$
$614$	0	0
$615$	−1.62616	−0.0655730
$616$	0	0
$617$	31.7057	1.27642	0.638211	−	0.769862i	$-0.279675\pi$
$617$	31.7057	1.27642	0.638211	+	0.769862i	$0.279675\pi$
$618$	0	0
$619$	43.6533	1.75458	0.877288	−	0.479965i	$-0.159351\pi$
$619$	43.6533	1.75458	0.877288	+	0.479965i	$0.159351\pi$
$620$	0	0
$621$	−5.67327	−0.227660
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23.5177	0.940708
$626$	0	0
$627$	−27.0459	−1.08011
$628$	0	0
$629$	6.10101	0.243263
$630$	0	0
$631$	2.88594	0.114887	0.0574437	−	0.998349i	$-0.481705\pi$
$631$	2.88594	0.114887	0.0574437	+	0.998349i	$0.481705\pi$
$632$	0	0
$633$	0.0703107	0.00279460
$634$	0	0
$635$	−4.14404	−0.164451
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.19389	−0.126348
$640$	0	0
$641$	−37.5438	−1.48289	−0.741446	−	0.671013i	$-0.765859\pi$
$641$	−37.5438	−1.48289	−0.741446	+	0.671013i	$0.765859\pi$
$642$	0	0
$643$	13.3115	0.524953	0.262477	−	0.964938i	$-0.415461\pi$
$643$	13.3115	0.524953	0.262477	+	0.964938i	$0.415461\pi$
$644$	0	0
$645$	0.784539	0.0308912
$646$	0	0
$647$	0.323814	0.0127305	0.00636523	−	0.999980i	$-0.497974\pi$
$647$	0.323814	0.0127305	0.00636523	+	0.999980i	$0.497974\pi$
$648$	0	0
$649$	15.1448	0.594485
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20.7854	0.813397	0.406699	−	0.913562i	$-0.366680\pi$
$653$	20.7854	0.813397	0.406699	+	0.913562i	$0.366680\pi$
$654$	0	0
$655$	−4.47691	−0.174927
$656$	0	0
$657$	−21.3588	−0.833287
$658$	0	0
$659$	26.9283	1.04898	0.524488	−	0.851418i	$-0.324257\pi$
$659$	26.9283	1.04898	0.524488	+	0.851418i	$0.324257\pi$
$660$	0	0
$661$	19.3434	0.752370	0.376185	−	0.926545i	$-0.377236\pi$
$661$	19.3434	0.752370	0.376185	+	0.926545i	$0.377236\pi$
$662$	0	0
$663$	60.3523	2.34389
$664$	0	0
$665$	0	0
$666$	0	0
$667$	51.3664	1.98892
$668$	0	0
$669$	−37.9671	−1.46789
$670$	0	0
$671$	−10.0406	−0.387611
$672$	0	0
$673$	−43.9015	−1.69228	−0.846138	−	0.532963i	$-0.821078\pi$
$673$	−43.9015	−1.69228	−0.846138	+	0.532963i	$0.821078\pi$
$674$	0	0
$675$	4.31649	0.166142
$676$	0	0
$677$	47.3308	1.81907	0.909536	−	0.415626i	$-0.136438\pi$
$677$	47.3308	1.81907	0.909536	+	0.415626i	$0.136438\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	8.76104	0.335724
$682$	0	0
$683$	−13.0729	−0.500220	−0.250110	−	0.968217i	$-0.580467\pi$
$683$	−13.0729	−0.500220	−0.250110	+	0.968217i	$0.580467\pi$
$684$	0	0
$685$	−1.47013	−0.0561707
$686$	0	0
$687$	−21.2711	−0.811545
$688$	0	0
$689$	24.2073	0.922224
$690$	0	0
$691$	19.7171	0.750075	0.375038	−	0.927010i	$-0.377630\pi$
$691$	19.7171	0.750075	0.375038	+	0.927010i	$0.377630\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.79718	−0.0681708
$696$	0	0
$697$	−9.37083	−0.354945
$698$	0	0
$699$	−60.7710	−2.29857
$700$	0	0
$701$	32.4637	1.22614	0.613068	−	0.790030i	$-0.289935\pi$
$701$	32.4637	1.22614	0.613068	+	0.790030i	$0.289935\pi$
$702$	0	0
$703$	8.91422	0.336206
$704$	0	0
$705$	−0.881949	−0.0332161
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−45.2338	−1.69879	−0.849396	−	0.527757i	$-0.823033\pi$
$709$	−45.2338	−1.69879	−0.849396	+	0.527757i	$0.823033\pi$
$710$	0	0
$711$	2.99639	0.112373
$712$	0	0
$713$	−33.4318	−1.25203
$714$	0	0
$715$	3.36662	0.125904
$716$	0	0
$717$	−10.7652	−0.402036
$718$	0	0
$719$	32.4165	1.20893	0.604466	−	0.796631i	$-0.293387\pi$
$719$	32.4165	1.20893	0.604466	+	0.796631i	$0.293387\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	21.3480	0.793940
$724$	0	0
$725$	−39.0821	−1.45147
$726$	0	0
$727$	−8.62227	−0.319782	−0.159891	−	0.987135i	$-0.551114\pi$
$727$	−8.62227	−0.319782	−0.159891	+	0.987135i	$0.551114\pi$
$728$	0	0
$729$	−19.9549	−0.739070
$730$	0	0
$731$	4.52095	0.167213
$732$	0	0
$733$	−46.0771	−1.70190	−0.850949	−	0.525249i	$-0.823972\pi$
$733$	−46.0771	−1.70190	−0.850949	+	0.525249i	$0.823972\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.7383	−0.653398
$738$	0	0
$739$	−6.48634	−0.238604	−0.119302	−	0.992858i	$-0.538066\pi$
$739$	−6.48634	−0.238604	−0.119302	+	0.992858i	$0.538066\pi$
$740$	0	0
$741$	88.1811	3.23941
$742$	0	0
$743$	18.3701	0.673935	0.336968	−	0.941516i	$-0.390599\pi$
$743$	18.3701	0.673935	0.336968	+	0.941516i	$0.390599\pi$
$744$	0	0
$745$	−3.98791	−0.146106
$746$	0	0
$747$	−9.76468	−0.357271
$748$	0	0
$749$	0	0
$750$	0	0
$751$	15.7006	0.572922	0.286461	−	0.958092i	$-0.407521\pi$
$751$	15.7006	0.572922	0.286461	+	0.958092i	$0.407521\pi$
$752$	0	0
$753$	28.0728	1.02303
$754$	0	0
$755$	3.14527	0.114468
$756$	0	0
$757$	31.2394	1.13541	0.567707	−	0.823231i	$-0.307831\pi$
$757$	31.2394	1.13541	0.567707	+	0.823231i	$0.307831\pi$
$758$	0	0
$759$	27.6488	1.00359
$760$	0	0
$761$	35.1655	1.27475	0.637374	−	0.770555i	$-0.280021\pi$
$761$	35.1655	1.27475	0.637374	+	0.770555i	$0.280021\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.57523	0.129263
$766$	0	0
$767$	−49.3784	−1.78295
$768$	0	0
$769$	−29.6687	−1.06988	−0.534941	−	0.844889i	$-0.679666\pi$
$769$	−29.6687	−1.06988	−0.534941	+	0.844889i	$0.679666\pi$
$770$	0	0
$771$	−45.0576	−1.62271
$772$	0	0
$773$	−22.2831	−0.801469	−0.400734	−	0.916194i	$-0.631245\pi$
$773$	−22.2831	−0.801469	−0.400734	+	0.916194i	$0.631245\pi$
$774$	0	0
$775$	25.4365	0.913708
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.6918	−0.490558
$780$	0	0
$781$	−2.19836	−0.0786636
$782$	0	0
$783$	−7.02464	−0.251040
$784$	0	0
$785$	−5.76973	−0.205930
$786$	0	0
$787$	−52.4214	−1.86862	−0.934311	−	0.356460i	$-0.883984\pi$
$787$	−52.4214	−1.86862	−0.934311	+	0.356460i	$0.883984\pi$
$788$	0	0
$789$	45.2633	1.61142
$790$	0	0
$791$	0	0
$792$	0	0
$793$	32.7364	1.16250
$794$	0	0
$795$	3.07058	0.108902
$796$	0	0
$797$	21.2475	0.752624	0.376312	−	0.926493i	$-0.377192\pi$
$797$	21.2475	0.752624	0.376312	+	0.926493i	$0.377192\pi$
$798$	0	0
$799$	−5.08228	−0.179798
$800$	0	0
$801$	−44.0219	−1.55544
$802$	0	0
$803$	−14.7013	−0.518798
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−55.8814	−1.96712
$808$	0	0
$809$	10.7340	0.377389	0.188694	−	0.982036i	$-0.439574\pi$
$809$	10.7340	0.377389	0.188694	+	0.982036i	$0.439574\pi$
$810$	0	0
$811$	19.8681	0.697664	0.348832	−	0.937185i	$-0.386578\pi$
$811$	19.8681	0.697664	0.348832	+	0.937185i	$0.386578\pi$
$812$	0	0
$813$	17.0316	0.597325
$814$	0	0
$815$	2.39080	0.0837459
$816$	0	0
$817$	6.60559	0.231100
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.97852	−0.173751	−0.0868757	−	0.996219i	$-0.527688\pi$
$821$	−4.97852	−0.173751	−0.0868757	+	0.996219i	$0.527688\pi$
$822$	0	0
$823$	−12.2946	−0.428563	−0.214282	−	0.976772i	$-0.568741\pi$
$823$	−12.2946	−0.428563	−0.214282	+	0.976772i	$0.568741\pi$
$824$	0	0
$825$	−21.0365	−0.732397
$826$	0	0
$827$	18.4646	0.642076	0.321038	−	0.947066i	$-0.395968\pi$
$827$	18.4646	0.642076	0.321038	+	0.947066i	$0.395968\pi$
$828$	0	0
$829$	−36.3043	−1.26090	−0.630450	−	0.776230i	$-0.717129\pi$
$829$	−36.3043	−1.26090	−0.630450	+	0.776230i	$0.717129\pi$
$830$	0	0
$831$	12.5444	0.435161
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−7.27864	−0.251888
$836$	0	0
$837$	4.57199	0.158031
$838$	0	0
$839$	−23.0411	−0.795469	−0.397734	−	0.917501i	$-0.630203\pi$
$839$	−23.0411	−0.795469	−0.397734	+	0.917501i	$0.630203\pi$
$840$	0	0
$841$	34.6020	1.19317
$842$	0	0
$843$	35.6175	1.22673
$844$	0	0
$845$	−6.87633	−0.236553
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−28.3276	−0.972200
$850$	0	0
$851$	−9.11291	−0.312387
$852$	0	0
$853$	17.4721	0.598235	0.299117	−	0.954216i	$-0.403308\pi$
$853$	17.4721	0.598235	0.299117	+	0.954216i	$0.403308\pi$
$854$	0	0
$855$	5.22379	0.178650
$856$	0	0
$857$	−44.6739	−1.52603	−0.763016	−	0.646380i	$-0.776282\pi$
$857$	−44.6739	−1.52603	−0.763016	+	0.646380i	$0.776282\pi$
$858$	0	0
$859$	−23.0071	−0.784991	−0.392496	−	0.919754i	$-0.628388\pi$
$859$	−23.0071	−0.784991	−0.392496	+	0.919754i	$0.628388\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46.7312	−1.59075	−0.795373	−	0.606120i	$-0.792725\pi$
$863$	−46.7312	−1.59075	−0.795373	+	0.606120i	$0.792725\pi$
$864$	0	0
$865$	−0.711132	−0.0241792
$866$	0	0
$867$	3.78224	0.128452
$868$	0	0
$869$	2.06242	0.0699629
$870$	0	0
$871$	57.8343	1.95964
$872$	0	0
$873$	−24.3357	−0.823640
$874$	0	0
$875$	0	0
$876$	0	0
$877$	26.4926	0.894591	0.447296	−	0.894386i	$-0.352387\pi$
$877$	26.4926	0.894591	0.447296	+	0.894386i	$0.352387\pi$
$878$	0	0
$879$	3.27161	0.110349
$880$	0	0
$881$	−27.8994	−0.939955	−0.469978	−	0.882678i	$-0.655738\pi$
$881$	−27.8994	−0.939955	−0.469978	+	0.882678i	$0.655738\pi$
$882$	0	0
$883$	21.1240	0.710879	0.355439	−	0.934699i	$-0.384331\pi$
$883$	21.1240	0.710879	0.355439	+	0.934699i	$0.384331\pi$
$884$	0	0
$885$	−6.26341	−0.210542
$886$	0	0
$887$	3.45507	0.116010	0.0580049	−	0.998316i	$-0.481526\pi$
$887$	3.45507	0.116010	0.0580049	+	0.998316i	$0.481526\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−18.0501	−0.604702
$892$	0	0
$893$	−7.42575	−0.248493
$894$	0	0
$895$	2.68189	0.0896456
$896$	0	0
$897$	−90.1466	−3.00991
$898$	0	0
$899$	−41.3953	−1.38061
$900$	0	0
$901$	17.6944	0.589485
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.09188	−0.136019
$906$	0	0
$907$	−30.0979	−0.999385	−0.499692	−	0.866203i	$-0.666554\pi$
$907$	−30.0979	−0.999385	−0.499692	+	0.866203i	$0.666554\pi$
$908$	0	0
$909$	2.39046	0.0792864
$910$	0	0
$911$	26.6733	0.883725	0.441863	−	0.897083i	$-0.354318\pi$
$911$	26.6733	0.883725	0.441863	+	0.897083i	$0.354318\pi$
$912$	0	0
$913$	−6.72105	−0.222434
$914$	0	0
$915$	4.15246	0.137276
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8.72809	−0.287913	−0.143957	−	0.989584i	$-0.545983\pi$
$919$	−8.72809	−0.287913	−0.143957	+	0.989584i	$0.545983\pi$
$920$	0	0
$921$	−60.6373	−1.99807
$922$	0	0
$923$	7.16759	0.235924
$924$	0	0
$925$	6.93354	0.227973
$926$	0	0
$927$	−31.8908	−1.04743
$928$	0	0
$929$	34.2412	1.12342	0.561708	−	0.827335i	$-0.310144\pi$
$929$	34.2412	1.12342	0.561708	+	0.827335i	$0.310144\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−47.0613	−1.54072
$934$	0	0
$935$	2.46084	0.0804780
$936$	0	0
$937$	−27.2835	−0.891312	−0.445656	−	0.895204i	$-0.647030\pi$
$937$	−27.2835	−0.891312	−0.445656	+	0.895204i	$0.647030\pi$
$938$	0	0
$939$	27.4756	0.896632
$940$	0	0
$941$	21.0498	0.686204	0.343102	−	0.939298i	$-0.388522\pi$
$941$	21.0498	0.686204	0.343102	+	0.939298i	$0.388522\pi$
$942$	0	0
$943$	13.9969	0.455803
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35.9602	1.16855	0.584275	−	0.811556i	$-0.301379\pi$
$947$	35.9602	1.16855	0.584275	+	0.811556i	$0.301379\pi$
$948$	0	0
$949$	47.9325	1.55595
$950$	0	0
$951$	19.5855	0.635104
$952$	0	0
$953$	13.1741	0.426750	0.213375	−	0.976970i	$-0.431554\pi$
$953$	13.1741	0.426750	0.213375	+	0.976970i	$0.431554\pi$
$954$	0	0
$955$	−7.59404	−0.245737
$956$	0	0
$957$	34.2347	1.10665
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−4.05787	−0.130899
$962$	0	0
$963$	−7.32605	−0.236079
$964$	0	0
$965$	−1.59152	−0.0512330
$966$	0	0
$967$	3.60884	0.116052	0.0580262	−	0.998315i	$-0.481519\pi$
$967$	3.60884	0.116052	0.0580262	+	0.998315i	$0.481519\pi$
$968$	0	0
$969$	64.4562	2.07063
$970$	0	0
$971$	0.367539	0.0117949	0.00589744	−	0.999983i	$-0.498123\pi$
$971$	0.367539	0.0117949	0.00589744	+	0.999983i	$0.498123\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	68.5879	2.19657
$976$	0	0
$977$	−45.5163	−1.45619	−0.728097	−	0.685474i	$-0.759595\pi$
$977$	−45.5163	−1.45619	−0.728097	+	0.685474i	$0.759595\pi$
$978$	0	0
$979$	−30.3003	−0.968403
$980$	0	0
$981$	−33.8110	−1.07950
$982$	0	0
$983$	13.4129	0.427806	0.213903	−	0.976855i	$-0.431382\pi$
$983$	13.4129	0.427806	0.213903	+	0.976855i	$0.431382\pi$
$984$	0	0
$985$	2.76247	0.0880196
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.75282	−0.214727
$990$	0	0
$991$	−21.8369	−0.693671	−0.346835	−	0.937926i	$-0.612744\pi$
$991$	−21.8369	−0.693671	−0.346835	+	0.937926i	$0.612744\pi$
$992$	0	0
$993$	−53.5021	−1.69784
$994$	0	0
$995$	−0.264639	−0.00838962
$996$	0	0
$997$	24.0708	0.762329	0.381164	−	0.924507i	$-0.375523\pi$
$997$	24.0708	0.762329	0.381164	+	0.924507i	$0.375523\pi$
$998$	0	0
$999$	1.24624	0.0394293

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5488.2.a.p.1.5		6
4.3	odd	2		343.2.a.c.1.5	✓	6
7.6	odd	2		5488.2.a.h.1.2		6
12.11	even	2		3087.2.a.k.1.2		6
20.19	odd	2		8575.2.a.o.1.2		6
28.3	even	6		343.2.c.d.324.2		12
28.11	odd	6		343.2.c.e.324.2		12
28.19	even	6		343.2.c.d.18.2		12
28.23	odd	6		343.2.c.e.18.2		12
28.27	even	2		343.2.a.d.1.5	yes	6
84.83	odd	2		3087.2.a.j.1.2		6
140.139	even	2		8575.2.a.n.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
343.2.a.c.1.5	✓	6	4.3	odd	2
343.2.a.d.1.5	yes	6	28.27	even	2
343.2.c.d.18.2		12	28.19	even	6
343.2.c.d.324.2		12	28.3	even	6
343.2.c.e.18.2		12	28.23	odd	6
343.2.c.e.324.2		12	28.11	odd	6
3087.2.a.j.1.2		6	84.83	odd	2
3087.2.a.k.1.2		6	12.11	even	2
5488.2.a.h.1.2		6	7.6	odd	2
5488.2.a.p.1.5		6	1.1	even	1	trivial
8575.2.a.n.1.2		6	140.139	even	2
8575.2.a.o.1.2		6	20.19	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:	\( N \)	\(=\)	\( 5488 = 2^{4} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5488.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37250 q^{3} -0.315405 q^{5} +2.62874 q^{9} +1.80936 q^{11} -5.89929 q^{13} -0.748296 q^{15} -4.31210 q^{17} -6.30043 q^{19} +6.44087 q^{23} -4.90052 q^{25} -0.880823 q^{27} +7.97508 q^{29} -5.19058 q^{31} +4.29271 q^{33} -1.41486 q^{37} -13.9960 q^{39} +2.17315 q^{41} -1.04843 q^{43} -0.829115 q^{45} +1.17861 q^{47} -10.2304 q^{51} -4.10342 q^{53} -0.570682 q^{55} -14.9477 q^{57} +8.37022 q^{59} -5.54922 q^{61} +1.86066 q^{65} -9.80360 q^{67} +15.2809 q^{69} -1.21499 q^{71} -8.12513 q^{73} -11.6265 q^{75} +1.13986 q^{79} -9.97596 q^{81} -3.71459 q^{83} +1.36006 q^{85} +18.9209 q^{87} -16.7464 q^{89} -12.3146 q^{93} +1.98719 q^{95} -9.25759 q^{97} +4.75634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} - 11 q^{5} + 9 q^{9} + q^{11} - 7 q^{13} + 5 q^{15} - 26 q^{17} - 3 q^{19} + 9 q^{23} + 15 q^{25} + 8 q^{27} - 2 q^{29} - 2 q^{31} - 18 q^{33} - 2 q^{37} - 7 q^{39} - 28 q^{41} - 5 q^{43}+ \cdots - 12 q^{99}+O(q^{100}) \) 6 * q + 5 * q^3 - 11 * q^5 + 9 * q^9 + q^11 - 7 * q^13 + 5 * q^15 - 26 * q^17 - 3 * q^19 + 9 * q^23 + 15 * q^25 + 8 * q^27 - 2 * q^29 - 2 * q^31 - 18 * q^33 - 2 * q^37 - 7 * q^39 - 28 * q^41 - 5 * q^43 - 3 * q^45 + 18 * q^47 - 13 * q^51 - q^53 - 22 * q^55 - 26 * q^57 + 5 * q^59 + 17 * q^61 - 14 * q^65 + 22 * q^67 + 20 * q^69 + 2 * q^71 - 12 * q^73 - 27 * q^75 + 2 * q^79 - 30 * q^81 + 7 * q^83 + 37 * q^85 - 25 * q^87 - 39 * q^89 - 22 * q^93 + 11 * q^95 - 35 * q^97 - 12 * q^99

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