LMFDB - Embedded newform 363.2.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,2,Mod(1,363)] 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("363.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	363.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$2.89856959337$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.73205 q^{6} -3.46410 q^{7} -1.73205 q^{8} +1.00000 q^{9} -5.19615 q^{10} -1.00000 q^{12} -1.73205 q^{13} -6.00000 q^{14} +3.00000 q^{15} -5.00000 q^{16} +1.73205 q^{17} +1.73205 q^{18} +6.92820 q^{19} -3.00000 q^{20} +3.46410 q^{21} -6.00000 q^{23} +1.73205 q^{24} +4.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} -3.46410 q^{28} -1.73205 q^{29} +5.19615 q^{30} +4.00000 q^{31} -5.19615 q^{32} +3.00000 q^{34} +10.3923 q^{35} +1.00000 q^{36} -11.0000 q^{37} +12.0000 q^{38} +1.73205 q^{39} +5.19615 q^{40} -1.73205 q^{41} +6.00000 q^{42} +3.46410 q^{43} -3.00000 q^{45} -10.3923 q^{46} +5.00000 q^{48} +5.00000 q^{49} +6.92820 q^{50} -1.73205 q^{51} -1.73205 q^{52} -9.00000 q^{53} -1.73205 q^{54} +6.00000 q^{56} -6.92820 q^{57} -3.00000 q^{58} -6.00000 q^{59} +3.00000 q^{60} +6.92820 q^{62} -3.46410 q^{63} +1.00000 q^{64} +5.19615 q^{65} -2.00000 q^{67} +1.73205 q^{68} +6.00000 q^{69} +18.0000 q^{70} -6.00000 q^{71} -1.73205 q^{72} +6.92820 q^{73} -19.0526 q^{74} -4.00000 q^{75} +6.92820 q^{76} +3.00000 q^{78} +15.0000 q^{80} +1.00000 q^{81} -3.00000 q^{82} +3.46410 q^{84} -5.19615 q^{85} +6.00000 q^{86} +1.73205 q^{87} +9.00000 q^{89} -5.19615 q^{90} +6.00000 q^{91} -6.00000 q^{92} -4.00000 q^{93} -20.7846 q^{95} +5.19615 q^{96} -7.00000 q^{97} +8.66025 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} - 6 q^{5} + 2 q^{9} - 2 q^{12} - 12 q^{14} + 6 q^{15} - 10 q^{16} - 6 q^{20} - 12 q^{23} + 8 q^{25} - 6 q^{26} - 2 q^{27} + 8 q^{31} + 6 q^{34} + 2 q^{36} - 22 q^{37} + 24 q^{38}+ \cdots - 14 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^5 + 2 * q^9 - 2 * q^12 - 12 * q^14 + 6 * q^15 - 10 * q^16 - 6 * q^20 - 12 * q^23 + 8 * q^25 - 6 * q^26 - 2 * q^27 + 8 * q^31 + 6 * q^34 + 2 * q^36 - 22 * q^37 + 24 * q^38 + 12 * q^42 - 6 * q^45 + 10 * q^48 + 10 * q^49 - 18 * q^53 + 12 * q^56 - 6 * q^58 - 12 * q^59 + 6 * q^60 + 2 * q^64 - 4 * q^67 + 12 * q^69 + 36 * q^70 - 12 * q^71 - 8 * q^75 + 6 * q^78 + 30 * q^80 + 2 * q^81 - 6 * q^82 + 12 * q^86 + 18 * q^89 + 12 * q^91 - 12 * q^92 - 8 * q^93 - 14 * q^97

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−1.73205	−0.707107
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	−1.73205	−0.612372
$9$	1.00000	0.333333
$10$	−5.19615	−1.64317
$11$	0	0
$11$	0	0
$12$	−1.00000	−0.288675
$13$	−1.73205	−0.480384	−0.240192	−	0.970725i	$-0.577210\pi$
$13$	−1.73205	−0.480384	−0.240192	+	0.970725i	$0.577210\pi$
$14$	−6.00000	−1.60357
$15$	3.00000	0.774597
$16$	−5.00000	−1.25000
$17$	1.73205	0.420084	0.210042	−	0.977692i	$-0.432640\pi$
$17$	1.73205	0.420084	0.210042	+	0.977692i	$0.432640\pi$
$18$	1.73205	0.408248
$19$	6.92820	1.58944	0.794719	−	0.606977i	$-0.207618\pi$
$19$	6.92820	1.58944	0.794719	+	0.606977i	$0.207618\pi$
$20$	−3.00000	−0.670820
$21$	3.46410	0.755929
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.73205	0.353553
$25$	4.00000	0.800000
$26$	−3.00000	−0.588348
$27$	−1.00000	−0.192450
$28$	−3.46410	−0.654654
$29$	−1.73205	−0.321634	−0.160817	−	0.986984i	$-0.551413\pi$
$29$	−1.73205	−0.321634	−0.160817	+	0.986984i	$0.551413\pi$
$30$	5.19615	0.948683
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	3.00000	0.514496
$35$	10.3923	1.75662
$36$	1.00000	0.166667
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	12.0000	1.94666
$39$	1.73205	0.277350
$40$	5.19615	0.821584
$41$	−1.73205	−0.270501	−0.135250	−	0.990811i	$-0.543184\pi$
$41$	−1.73205	−0.270501	−0.135250	+	0.990811i	$0.543184\pi$
$42$	6.00000	0.925820
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	−10.3923	−1.53226
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	5.00000	0.721688
$49$	5.00000	0.714286
$50$	6.92820	0.979796
$51$	−1.73205	−0.242536
$52$	−1.73205	−0.240192
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	−1.73205	−0.235702
$55$	0	0
$56$	6.00000	0.801784
$57$	−6.92820	−0.917663
$58$	−3.00000	−0.393919
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	3.00000	0.387298
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	6.92820	0.879883
$63$	−3.46410	−0.436436
$64$	1.00000	0.125000
$65$	5.19615	0.644503
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	1.73205	0.210042
$69$	6.00000	0.722315
$70$	18.0000	2.15141
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−1.73205	−0.204124
$73$	6.92820	0.810885	0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820	0.810885	0.405442	+	0.914121i	$0.367117\pi$
$74$	−19.0526	−2.21481
$75$	−4.00000	−0.461880
$76$	6.92820	0.794719
$77$	0	0
$78$	3.00000	0.339683
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	15.0000	1.67705
$81$	1.00000	0.111111
$82$	−3.00000	−0.331295
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	3.46410	0.377964
$85$	−5.19615	−0.563602
$86$	6.00000	0.646997
$87$	1.73205	0.185695
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	−5.19615	−0.547723
$91$	6.00000	0.628971
$92$	−6.00000	−0.625543
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−20.7846	−2.13246
$96$	5.19615	0.530330
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	8.66025	0.874818
$99$	0	0
$100$	4.00000	0.400000
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	−3.00000	−0.297044
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	3.00000	0.294174
$105$	−10.3923	−1.01419
$106$	−15.5885	−1.51408
$107$	3.46410	0.334887	0.167444	−	0.985882i	$-0.446449\pi$
$107$	3.46410	0.334887	0.167444	+	0.985882i	$0.446449\pi$
$108$	−1.00000	−0.0962250
$109$	15.5885	1.49310	0.746552	−	0.665327i	$-0.231708\pi$
$109$	15.5885	1.49310	0.746552	+	0.665327i	$0.231708\pi$
$110$	0	0
$111$	11.0000	1.04407
$112$	17.3205	1.63663
$113$	−21.0000	−1.97551	−0.987757	−	0.156001i	$-0.950140\pi$
$113$	−21.0000	−1.97551	−0.987757	+	0.156001i	$0.950140\pi$
$114$	−12.0000	−1.12390
$115$	18.0000	1.67851
$116$	−1.73205	−0.160817
$117$	−1.73205	−0.160128
$118$	−10.3923	−0.956689
$119$	−6.00000	−0.550019
$120$	−5.19615	−0.474342
$121$	0	0
$122$	0	0
$123$	1.73205	0.156174
$124$	4.00000	0.359211
$125$	3.00000	0.268328
$126$	−6.00000	−0.534522
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	12.1244	1.07165
$129$	−3.46410	−0.304997
$130$	9.00000	0.789352
$131$	−17.3205	−1.51330	−0.756650	−	0.653820i	$-0.773165\pi$
$131$	−17.3205	−1.51330	−0.756650	+	0.653820i	$0.773165\pi$
$132$	0	0
$133$	−24.0000	−2.08106
$134$	−3.46410	−0.299253
$135$	3.00000	0.258199
$136$	−3.00000	−0.257248
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	10.3923	0.884652
$139$	10.3923	0.881464	0.440732	−	0.897639i	$-0.354719\pi$
$139$	10.3923	0.881464	0.440732	+	0.897639i	$0.354719\pi$
$140$	10.3923	0.878310
$141$	0	0
$142$	−10.3923	−0.872103
$143$	0	0
$144$	−5.00000	−0.416667
$145$	5.19615	0.431517
$146$	12.0000	0.993127
$147$	−5.00000	−0.412393
$148$	−11.0000	−0.904194
$149$	−12.1244	−0.993266	−0.496633	−	0.867961i	$-0.665430\pi$
$149$	−12.1244	−0.993266	−0.496633	+	0.867961i	$0.665430\pi$
$150$	−6.92820	−0.565685
$151$	13.8564	1.12762	0.563809	−	0.825905i	$-0.309335\pi$
$151$	13.8564	1.12762	0.563809	+	0.825905i	$0.309335\pi$
$152$	−12.0000	−0.973329
$153$	1.73205	0.140028
$154$	0	0
$155$	−12.0000	−0.963863
$156$	1.73205	0.138675
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	9.00000	0.713746
$160$	15.5885	1.23238
$161$	20.7846	1.63806
$162$	1.73205	0.136083
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	−1.73205	−0.135250
$165$	0	0
$166$	0	0
$167$	−3.46410	−0.268060	−0.134030	−	0.990977i	$-0.542792\pi$
$167$	−3.46410	−0.268060	−0.134030	+	0.990977i	$0.542792\pi$
$168$	−6.00000	−0.462910
$169$	−10.0000	−0.769231
$170$	−9.00000	−0.690268
$171$	6.92820	0.529813
$172$	3.46410	0.264135
$173$	20.7846	1.58022	0.790112	−	0.612962i	$-0.210022\pi$
$173$	20.7846	1.58022	0.790112	+	0.612962i	$0.210022\pi$
$174$	3.00000	0.227429
$175$	−13.8564	−1.04745
$176$	0	0
$177$	6.00000	0.450988
$178$	15.5885	1.16840
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−3.00000	−0.223607
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	10.3923	0.770329
$183$	0	0
$184$	10.3923	0.766131
$185$	33.0000	2.42621
$186$	−6.92820	−0.508001
$187$	0	0
$188$	0	0
$189$	3.46410	0.251976
$190$	−36.0000	−2.61171
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	−5.19615	−0.374027	−0.187014	−	0.982357i	$-0.559881\pi$
$193$	−5.19615	−0.374027	−0.187014	+	0.982357i	$0.559881\pi$
$194$	−12.1244	−0.870478
$195$	−5.19615	−0.372104
$196$	5.00000	0.357143
$197$	−19.0526	−1.35744	−0.678719	−	0.734398i	$-0.737465\pi$
$197$	−19.0526	−1.35744	−0.678719	+	0.734398i	$0.737465\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	−6.92820	−0.489898
$201$	2.00000	0.141069
$202$	−24.0000	−1.68863
$203$	6.00000	0.421117
$204$	−1.73205	−0.121268
$205$	5.19615	0.362915
$206$	24.2487	1.68949
$207$	−6.00000	−0.417029
$208$	8.66025	0.600481
$209$	0	0
$210$	−18.0000	−1.24212
$211$	−17.3205	−1.19239	−0.596196	−	0.802839i	$-0.703322\pi$
$211$	−17.3205	−1.19239	−0.596196	+	0.802839i	$0.703322\pi$
$212$	−9.00000	−0.618123
$213$	6.00000	0.411113
$214$	6.00000	0.410152
$215$	−10.3923	−0.708749
$216$	1.73205	0.117851
$217$	−13.8564	−0.940634
$218$	27.0000	1.82867
$219$	−6.92820	−0.468165
$220$	0	0
$221$	−3.00000	−0.201802
$222$	19.0526	1.27872
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	18.0000	1.20268
$225$	4.00000	0.266667
$226$	−36.3731	−2.41950
$227$	−24.2487	−1.60944	−0.804722	−	0.593652i	$-0.797686\pi$
$227$	−24.2487	−1.60944	−0.804722	+	0.593652i	$0.797686\pi$
$228$	−6.92820	−0.458831
$229$	−23.0000	−1.51988	−0.759941	−	0.649992i	$-0.774772\pi$
$229$	−23.0000	−1.51988	−0.759941	+	0.649992i	$0.774772\pi$
$230$	31.1769	2.05574
$231$	0	0
$232$	3.00000	0.196960
$233$	29.4449	1.92900	0.964499	−	0.264088i	$-0.0850709\pi$
$233$	29.4449	1.92900	0.964499	+	0.264088i	$0.0850709\pi$
$234$	−3.00000	−0.196116
$235$	0	0
$236$	−6.00000	−0.390567
$237$	0	0
$238$	−10.3923	−0.673633
$239$	−6.92820	−0.448148	−0.224074	−	0.974572i	$-0.571936\pi$
$239$	−6.92820	−0.448148	−0.224074	+	0.974572i	$0.571936\pi$
$240$	−15.0000	−0.968246
$241$	−20.7846	−1.33885	−0.669427	−	0.742878i	$-0.733460\pi$
$241$	−20.7846	−1.33885	−0.669427	+	0.742878i	$0.733460\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−15.0000	−0.958315
$246$	3.00000	0.191273
$247$	−12.0000	−0.763542
$248$	−6.92820	−0.439941
$249$	0	0
$250$	5.19615	0.328634
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	−3.46410	−0.218218
$253$	0	0
$254$	0	0
$255$	5.19615	0.325396
$256$	19.0000	1.18750
$257$	9.00000	0.561405	0.280702	−	0.959795i	$-0.409433\pi$
$257$	9.00000	0.561405	0.280702	+	0.959795i	$0.409433\pi$
$258$	−6.00000	−0.373544
$259$	38.1051	2.36774
$260$	5.19615	0.322252
$261$	−1.73205	−0.107211
$262$	−30.0000	−1.85341
$263$	13.8564	0.854423	0.427211	−	0.904152i	$-0.359496\pi$
$263$	13.8564	0.854423	0.427211	+	0.904152i	$0.359496\pi$
$264$	0	0
$265$	27.0000	1.65860
$266$	−41.5692	−2.54877
$267$	−9.00000	−0.550791
$268$	−2.00000	−0.122169
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	5.19615	0.316228
$271$	−3.46410	−0.210429	−0.105215	−	0.994450i	$-0.533553\pi$
$271$	−3.46410	−0.210429	−0.105215	+	0.994450i	$0.533553\pi$
$272$	−8.66025	−0.525105
$273$	−6.00000	−0.363137
$274$	−10.3923	−0.627822
$275$	0	0
$276$	6.00000	0.361158
$277$	−5.19615	−0.312207	−0.156103	−	0.987741i	$-0.549893\pi$
$277$	−5.19615	−0.312207	−0.156103	+	0.987741i	$0.549893\pi$
$278$	18.0000	1.07957
$279$	4.00000	0.239474
$280$	−18.0000	−1.07571
$281$	−6.92820	−0.413302	−0.206651	−	0.978415i	$-0.566256\pi$
$281$	−6.92820	−0.413302	−0.206651	+	0.978415i	$0.566256\pi$
$282$	0	0
$283$	−31.1769	−1.85328	−0.926638	−	0.375956i	$-0.877314\pi$
$283$	−31.1769	−1.85328	−0.926638	+	0.375956i	$0.877314\pi$
$284$	−6.00000	−0.356034
$285$	20.7846	1.23117
$286$	0	0
$287$	6.00000	0.354169
$288$	−5.19615	−0.306186
$289$	−14.0000	−0.823529
$290$	9.00000	0.528498
$291$	7.00000	0.410347
$292$	6.92820	0.405442
$293$	19.0526	1.11306	0.556531	−	0.830827i	$-0.312132\pi$
$293$	19.0526	1.11306	0.556531	+	0.830827i	$0.312132\pi$
$294$	−8.66025	−0.505076
$295$	18.0000	1.04800
$296$	19.0526	1.10741
$297$	0	0
$298$	−21.0000	−1.21650
$299$	10.3923	0.601003
$300$	−4.00000	−0.230940
$301$	−12.0000	−0.691669
$302$	24.0000	1.38104
$303$	13.8564	0.796030
$304$	−34.6410	−1.98680
$305$	0	0
$306$	3.00000	0.171499
$307$	3.46410	0.197707	0.0988534	−	0.995102i	$-0.468483\pi$
$307$	3.46410	0.197707	0.0988534	+	0.995102i	$0.468483\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	−20.7846	−1.18049
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−3.00000	−0.169842
$313$	7.00000	0.395663	0.197832	−	0.980236i	$-0.436610\pi$
$313$	7.00000	0.395663	0.197832	+	0.980236i	$0.436610\pi$
$314$	−24.2487	−1.36843
$315$	10.3923	0.585540
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	15.5885	0.874157
$319$	0	0
$320$	−3.00000	−0.167705
$321$	−3.46410	−0.193347
$322$	36.0000	2.00620
$323$	12.0000	0.667698
$324$	1.00000	0.0555556
$325$	−6.92820	−0.384308
$326$	3.46410	0.191859
$327$	−15.5885	−0.862044
$328$	3.00000	0.165647
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	−11.0000	−0.602796
$334$	−6.00000	−0.328305
$335$	6.00000	0.327815
$336$	−17.3205	−0.944911
$337$	−12.1244	−0.660456	−0.330228	−	0.943901i	$-0.607126\pi$
$337$	−12.1244	−0.660456	−0.330228	+	0.943901i	$0.607126\pi$
$338$	−17.3205	−0.942111
$339$	21.0000	1.14056
$340$	−5.19615	−0.281801
$341$	0	0
$342$	12.0000	0.648886
$343$	6.92820	0.374088
$344$	−6.00000	−0.323498
$345$	−18.0000	−0.969087
$346$	36.0000	1.93537
$347$	27.7128	1.48770	0.743851	−	0.668346i	$-0.232997\pi$
$347$	27.7128	1.48770	0.743851	+	0.668346i	$0.232997\pi$
$348$	1.73205	0.0928477
$349$	−1.73205	−0.0927146	−0.0463573	−	0.998925i	$-0.514761\pi$
$349$	−1.73205	−0.0927146	−0.0463573	+	0.998925i	$0.514761\pi$
$350$	−24.0000	−1.28285
$351$	1.73205	0.0924500
$352$	0	0
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	10.3923	0.552345
$355$	18.0000	0.955341
$356$	9.00000	0.476999
$357$	6.00000	0.317554
$358$	20.7846	1.09850
$359$	31.1769	1.64545	0.822727	−	0.568436i	$-0.192451\pi$
$359$	31.1769	1.64545	0.822727	+	0.568436i	$0.192451\pi$
$360$	5.19615	0.273861
$361$	29.0000	1.52632
$362$	−12.1244	−0.637242
$363$	0	0
$364$	6.00000	0.314485
$365$	−20.7846	−1.08792
$366$	0	0
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	30.0000	1.56386
$369$	−1.73205	−0.0901670
$370$	57.1577	2.97149
$371$	31.1769	1.61862
$372$	−4.00000	−0.207390
$373$	20.7846	1.07619	0.538093	−	0.842885i	$-0.319145\pi$
$373$	20.7846	1.07619	0.538093	+	0.842885i	$0.319145\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	3.00000	0.154508
$378$	6.00000	0.308607
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	−20.7846	−1.06623
$381$	0	0
$382$	20.7846	1.06343
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	−12.1244	−0.618718
$385$	0	0
$386$	−9.00000	−0.458088
$387$	3.46410	0.176090
$388$	−7.00000	−0.355371
$389$	−27.0000	−1.36895	−0.684477	−	0.729034i	$-0.739969\pi$
$389$	−27.0000	−1.36895	−0.684477	+	0.729034i	$0.739969\pi$
$390$	−9.00000	−0.455733
$391$	−10.3923	−0.525561
$392$	−8.66025	−0.437409
$393$	17.3205	0.873704
$394$	−33.0000	−1.66252
$395$	0	0
$396$	0	0
$397$	11.0000	0.552074	0.276037	−	0.961147i	$-0.410979\pi$
$397$	11.0000	0.552074	0.276037	+	0.961147i	$0.410979\pi$
$398$	17.3205	0.868199
$399$	24.0000	1.20150
$400$	−20.0000	−1.00000
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	3.46410	0.172774
$403$	−6.92820	−0.345118
$404$	−13.8564	−0.689382
$405$	−3.00000	−0.149071
$406$	10.3923	0.515761
$407$	0	0
$408$	3.00000	0.148522
$409$	1.73205	0.0856444	0.0428222	−	0.999083i	$-0.486365\pi$
$409$	1.73205	0.0856444	0.0428222	+	0.999083i	$0.486365\pi$
$410$	9.00000	0.444478
$411$	6.00000	0.295958
$412$	14.0000	0.689730
$413$	20.7846	1.02274
$414$	−10.3923	−0.510754
$415$	0	0
$416$	9.00000	0.441261
$417$	−10.3923	−0.508913
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	−10.3923	−0.507093
$421$	7.00000	0.341159	0.170580	−	0.985344i	$-0.445436\pi$
$421$	7.00000	0.341159	0.170580	+	0.985344i	$0.445436\pi$
$422$	−30.0000	−1.46038
$423$	0	0
$424$	15.5885	0.757042
$425$	6.92820	0.336067
$426$	10.3923	0.503509
$427$	0	0
$428$	3.46410	0.167444
$429$	0	0
$430$	−18.0000	−0.868037
$431$	−17.3205	−0.834300	−0.417150	−	0.908838i	$-0.636971\pi$
$431$	−17.3205	−0.834300	−0.417150	+	0.908838i	$0.636971\pi$
$432$	5.00000	0.240563
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	−24.0000	−1.15204
$435$	−5.19615	−0.249136
$436$	15.5885	0.746552
$437$	−41.5692	−1.98853
$438$	−12.0000	−0.573382
$439$	−6.92820	−0.330665	−0.165333	−	0.986238i	$-0.552870\pi$
$439$	−6.92820	−0.330665	−0.165333	+	0.986238i	$0.552870\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	−5.19615	−0.247156
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	11.0000	0.522037
$445$	−27.0000	−1.27992
$446$	34.6410	1.64030
$447$	12.1244	0.573462
$448$	−3.46410	−0.163663
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	6.92820	0.326599
$451$	0	0
$452$	−21.0000	−0.987757
$453$	−13.8564	−0.651031
$454$	−42.0000	−1.97116
$455$	−18.0000	−0.843853
$456$	12.0000	0.561951
$457$	−29.4449	−1.37737	−0.688686	−	0.725059i	$-0.741812\pi$
$457$	−29.4449	−1.37737	−0.688686	+	0.725059i	$0.741812\pi$
$458$	−39.8372	−1.86147
$459$	−1.73205	−0.0808452
$460$	18.0000	0.839254
$461$	−15.5885	−0.726027	−0.363013	−	0.931784i	$-0.618252\pi$
$461$	−15.5885	−0.726027	−0.363013	+	0.931784i	$0.618252\pi$
$462$	0	0
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	8.66025	0.402042
$465$	12.0000	0.556487
$466$	51.0000	2.36253
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	−1.73205	−0.0800641
$469$	6.92820	0.319915
$470$	0	0
$471$	14.0000	0.645086
$472$	10.3923	0.478345
$473$	0	0
$474$	0	0
$475$	27.7128	1.27155
$476$	−6.00000	−0.275010
$477$	−9.00000	−0.412082
$478$	−12.0000	−0.548867
$479$	24.2487	1.10795	0.553976	−	0.832533i	$-0.313110\pi$
$479$	24.2487	1.10795	0.553976	+	0.832533i	$0.313110\pi$
$480$	−15.5885	−0.711512
$481$	19.0526	0.868722
$482$	−36.0000	−1.63976
$483$	−20.7846	−0.945732
$484$	0	0
$485$	21.0000	0.953561
$486$	−1.73205	−0.0785674
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	−25.9808	−1.17369
$491$	−20.7846	−0.937996	−0.468998	−	0.883199i	$-0.655385\pi$
$491$	−20.7846	−0.937996	−0.468998	+	0.883199i	$0.655385\pi$
$492$	1.73205	0.0780869
$493$	−3.00000	−0.135113
$494$	−20.7846	−0.935144
$495$	0	0
$496$	−20.0000	−0.898027
$497$	20.7846	0.932317
$498$	0	0
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	3.00000	0.134164
$501$	3.46410	0.154765
$502$	−10.3923	−0.463831
$503$	31.1769	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$504$	6.00000	0.267261
$505$	41.5692	1.84981
$506$	0	0
$507$	10.0000	0.444116
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	9.00000	0.398527
$511$	−24.0000	−1.06170
$512$	8.66025	0.382733
$513$	−6.92820	−0.305888
$514$	15.5885	0.687577
$515$	−42.0000	−1.85074
$516$	−3.46410	−0.152499
$517$	0	0
$518$	66.0000	2.89987
$519$	−20.7846	−0.912343
$520$	−9.00000	−0.394676
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−3.00000	−0.131306
$523$	−41.5692	−1.81770	−0.908848	−	0.417128i	$-0.863037\pi$
$523$	−41.5692	−1.81770	−0.908848	+	0.417128i	$0.863037\pi$
$524$	−17.3205	−0.756650
$525$	13.8564	0.604743
$526$	24.0000	1.04645
$527$	6.92820	0.301797
$528$	0	0
$529$	13.0000	0.565217
$530$	46.7654	2.03136
$531$	−6.00000	−0.260378
$532$	−24.0000	−1.04053
$533$	3.00000	0.129944
$534$	−15.5885	−0.674579
$535$	−10.3923	−0.449299
$536$	3.46410	0.149626
$537$	−12.0000	−0.517838
$538$	−36.3731	−1.56815
$539$	0	0
$540$	3.00000	0.129099
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	−6.00000	−0.257722
$543$	7.00000	0.300399
$544$	−9.00000	−0.385872
$545$	−46.7654	−2.00321
$546$	−10.3923	−0.444750
$547$	−34.6410	−1.48114	−0.740571	−	0.671978i	$-0.765445\pi$
$547$	−34.6410	−1.48114	−0.740571	+	0.671978i	$0.765445\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	−10.3923	−0.442326
$553$	0	0
$554$	−9.00000	−0.382373
$555$	−33.0000	−1.40077
$556$	10.3923	0.440732
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	6.92820	0.293294
$559$	−6.00000	−0.253773
$560$	−51.9615	−2.19578
$561$	0	0
$562$	−12.0000	−0.506189
$563$	17.3205	0.729972	0.364986	−	0.931013i	$-0.381074\pi$
$563$	17.3205	0.729972	0.364986	+	0.931013i	$0.381074\pi$
$564$	0	0
$565$	63.0000	2.65043
$566$	−54.0000	−2.26979
$567$	−3.46410	−0.145479
$568$	10.3923	0.436051
$569$	−27.7128	−1.16178	−0.580891	−	0.813982i	$-0.697296\pi$
$569$	−27.7128	−1.16178	−0.580891	+	0.813982i	$0.697296\pi$
$570$	36.0000	1.50787
$571$	3.46410	0.144968	0.0724841	−	0.997370i	$-0.476907\pi$
$571$	3.46410	0.144968	0.0724841	+	0.997370i	$0.476907\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	10.3923	0.433766
$575$	−24.0000	−1.00087
$576$	1.00000	0.0416667
$577$	13.0000	0.541197	0.270599	−	0.962692i	$-0.412778\pi$
$577$	13.0000	0.541197	0.270599	+	0.962692i	$0.412778\pi$
$578$	−24.2487	−1.00861
$579$	5.19615	0.215945
$580$	5.19615	0.215758
$581$	0	0
$582$	12.1244	0.502571
$583$	0	0
$584$	−12.0000	−0.496564
$585$	5.19615	0.214834
$586$	33.0000	1.36322
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−5.00000	−0.206197
$589$	27.7128	1.14189
$590$	31.1769	1.28353
$591$	19.0526	0.783718
$592$	55.0000	2.26049
$593$	−22.5167	−0.924648	−0.462324	−	0.886711i	$-0.652984\pi$
$593$	−22.5167	−0.924648	−0.462324	+	0.886711i	$0.652984\pi$
$594$	0	0
$595$	18.0000	0.737928
$596$	−12.1244	−0.496633
$597$	−10.0000	−0.409273
$598$	18.0000	0.736075
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	6.92820	0.282843
$601$	−22.5167	−0.918474	−0.459237	−	0.888314i	$-0.651877\pi$
$601$	−22.5167	−0.918474	−0.459237	+	0.888314i	$0.651877\pi$
$602$	−20.7846	−0.847117
$603$	−2.00000	−0.0814463
$604$	13.8564	0.563809
$605$	0	0
$606$	24.0000	0.974933
$607$	20.7846	0.843621	0.421811	−	0.906684i	$-0.361395\pi$
$607$	20.7846	0.843621	0.421811	+	0.906684i	$0.361395\pi$
$608$	−36.0000	−1.45999
$609$	−6.00000	−0.243132
$610$	0	0
$611$	0	0
$612$	1.73205	0.0700140
$613$	19.0526	0.769526	0.384763	−	0.923015i	$-0.374283\pi$
$613$	19.0526	0.769526	0.384763	+	0.923015i	$0.374283\pi$
$614$	6.00000	0.242140
$615$	−5.19615	−0.209529
$616$	0	0
$617$	−9.00000	−0.362326	−0.181163	−	0.983453i	$-0.557986\pi$
$617$	−9.00000	−0.362326	−0.181163	+	0.983453i	$0.557986\pi$
$618$	−24.2487	−0.975426
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	−12.0000	−0.481932
$621$	6.00000	0.240772
$622$	20.7846	0.833387
$623$	−31.1769	−1.24908
$624$	−8.66025	−0.346688
$625$	−29.0000	−1.16000
$626$	12.1244	0.484587
$627$	0	0
$628$	−14.0000	−0.558661
$629$	−19.0526	−0.759675
$630$	18.0000	0.717137
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	0	0
$633$	17.3205	0.688428
$634$	10.3923	0.412731
$635$	0	0
$636$	9.00000	0.356873
$637$	−8.66025	−0.343132
$638$	0	0
$639$	−6.00000	−0.237356
$640$	−36.3731	−1.43777
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	−6.00000	−0.236801
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	20.7846	0.819028
$645$	10.3923	0.409197
$646$	20.7846	0.817760
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	−1.73205	−0.0680414
$649$	0	0
$650$	−12.0000	−0.470679
$651$	13.8564	0.543075
$652$	2.00000	0.0783260
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	−27.0000	−1.05578
$655$	51.9615	2.03030
$656$	8.66025	0.338126
$657$	6.92820	0.270295
$658$	0	0
$659$	13.8564	0.539769	0.269884	−	0.962893i	$-0.413014\pi$
$659$	13.8564	0.539769	0.269884	+	0.962893i	$0.413014\pi$
$660$	0	0
$661$	−41.0000	−1.59472	−0.797358	−	0.603507i	$-0.793769\pi$
$661$	−41.0000	−1.59472	−0.797358	+	0.603507i	$0.793769\pi$
$662$	34.6410	1.34636
$663$	3.00000	0.116510
$664$	0	0
$665$	72.0000	2.79204
$666$	−19.0526	−0.738272
$667$	10.3923	0.402392
$668$	−3.46410	−0.134030
$669$	−20.0000	−0.773245
$670$	10.3923	0.401490
$671$	0	0
$672$	−18.0000	−0.694365
$673$	−20.7846	−0.801188	−0.400594	−	0.916256i	$-0.631196\pi$
$673$	−20.7846	−0.801188	−0.400594	+	0.916256i	$0.631196\pi$
$674$	−21.0000	−0.808890
$675$	−4.00000	−0.153960
$676$	−10.0000	−0.384615
$677$	32.9090	1.26479	0.632397	−	0.774644i	$-0.282071\pi$
$677$	32.9090	1.26479	0.632397	+	0.774644i	$0.282071\pi$
$678$	36.3731	1.39690
$679$	24.2487	0.930580
$680$	9.00000	0.345134
$681$	24.2487	0.929213
$682$	0	0
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	6.92820	0.264906
$685$	18.0000	0.687745
$686$	12.0000	0.458162
$687$	23.0000	0.877505
$688$	−17.3205	−0.660338
$689$	15.5885	0.593873
$690$	−31.1769	−1.18688
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	20.7846	0.790112
$693$	0	0
$694$	48.0000	1.82206
$695$	−31.1769	−1.18261
$696$	−3.00000	−0.113715
$697$	−3.00000	−0.113633
$698$	−3.00000	−0.113552
$699$	−29.4449	−1.11371
$700$	−13.8564	−0.523723
$701$	−39.8372	−1.50463	−0.752315	−	0.658804i	$-0.771063\pi$
$701$	−39.8372	−1.50463	−0.752315	+	0.658804i	$0.771063\pi$
$702$	3.00000	0.113228
$703$	−76.2102	−2.87432
$704$	0	0
$705$	0	0
$706$	36.3731	1.36892
$707$	48.0000	1.80523
$708$	6.00000	0.225494
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	31.1769	1.17005
$711$	0	0
$712$	−15.5885	−0.584202
$713$	−24.0000	−0.898807
$714$	10.3923	0.388922
$715$	0	0
$716$	12.0000	0.448461
$717$	6.92820	0.258738
$718$	54.0000	2.01526
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	15.0000	0.559017
$721$	−48.4974	−1.80614
$722$	50.2295	1.86935
$723$	20.7846	0.772988
$724$	−7.00000	−0.260153
$725$	−6.92820	−0.257307
$726$	0	0
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	−10.3923	−0.385164
$729$	1.00000	0.0370370
$730$	−36.0000	−1.33242
$731$	6.00000	0.221918
$732$	0	0
$733$	8.66025	0.319874	0.159937	−	0.987127i	$-0.448871\pi$
$733$	8.66025	0.319874	0.159937	+	0.987127i	$0.448871\pi$
$734$	45.0333	1.66221
$735$	15.0000	0.553283
$736$	31.1769	1.14920
$737$	0	0
$738$	−3.00000	−0.110432
$739$	−27.7128	−1.01943	−0.509716	−	0.860343i	$-0.670250\pi$
$739$	−27.7128	−1.01943	−0.509716	+	0.860343i	$0.670250\pi$
$740$	33.0000	1.21310
$741$	12.0000	0.440831
$742$	54.0000	1.98240
$743$	−51.9615	−1.90628	−0.953142	−	0.302524i	$-0.902171\pi$
$743$	−51.9615	−1.90628	−0.953142	+	0.302524i	$0.902171\pi$
$744$	6.92820	0.254000
$745$	36.3731	1.33261
$746$	36.0000	1.31805
$747$	0	0
$748$	0	0
$749$	−12.0000	−0.438470
$750$	−5.19615	−0.189737
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	5.19615	0.189233
$755$	−41.5692	−1.51286
$756$	3.46410	0.125988
$757$	−17.0000	−0.617876	−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000	−0.617876	−0.308938	+	0.951082i	$0.599973\pi$
$758$	24.2487	0.880753
$759$	0	0
$760$	36.0000	1.30586
$761$	32.9090	1.19295	0.596475	−	0.802632i	$-0.296568\pi$
$761$	32.9090	1.19295	0.596475	+	0.802632i	$0.296568\pi$
$762$	0	0
$763$	−54.0000	−1.95493
$764$	12.0000	0.434145
$765$	−5.19615	−0.187867
$766$	10.3923	0.375489
$767$	10.3923	0.375244
$768$	−19.0000	−0.685603
$769$	25.9808	0.936890	0.468445	−	0.883493i	$-0.344814\pi$
$769$	25.9808	0.936890	0.468445	+	0.883493i	$0.344814\pi$
$770$	0	0
$771$	−9.00000	−0.324127
$772$	−5.19615	−0.187014
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	6.00000	0.215666
$775$	16.0000	0.574737
$776$	12.1244	0.435239
$777$	−38.1051	−1.36701
$778$	−46.7654	−1.67662
$779$	−12.0000	−0.429945
$780$	−5.19615	−0.186052
$781$	0	0
$782$	−18.0000	−0.643679
$783$	1.73205	0.0618984
$784$	−25.0000	−0.892857
$785$	42.0000	1.49904
$786$	30.0000	1.07006
$787$	−6.92820	−0.246964	−0.123482	−	0.992347i	$-0.539406\pi$
$787$	−6.92820	−0.246964	−0.123482	+	0.992347i	$0.539406\pi$
$788$	−19.0526	−0.678719
$789$	−13.8564	−0.493301
$790$	0	0
$791$	72.7461	2.58655
$792$	0	0
$793$	0	0
$794$	19.0526	0.676150
$795$	−27.0000	−0.957591
$796$	10.0000	0.354441
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	41.5692	1.47153
$799$	0	0
$800$	−20.7846	−0.734847
$801$	9.00000	0.317999
$802$	5.19615	0.183483
$803$	0	0
$804$	2.00000	0.0705346
$805$	−62.3538	−2.19768
$806$	−12.0000	−0.422682
$807$	21.0000	0.739235
$808$	24.0000	0.844317
$809$	20.7846	0.730748	0.365374	−	0.930861i	$-0.380941\pi$
$809$	20.7846	0.730748	0.365374	+	0.930861i	$0.380941\pi$
$810$	−5.19615	−0.182574
$811$	−27.7128	−0.973128	−0.486564	−	0.873645i	$-0.661750\pi$
$811$	−27.7128	−0.973128	−0.486564	+	0.873645i	$0.661750\pi$
$812$	6.00000	0.210559
$813$	3.46410	0.121491
$814$	0	0
$815$	−6.00000	−0.210171
$816$	8.66025	0.303170
$817$	24.0000	0.839654
$818$	3.00000	0.104893
$819$	6.00000	0.209657
$820$	5.19615	0.181458
$821$	55.4256	1.93437	0.967184	−	0.254078i	$-0.0817719\pi$
$821$	55.4256	1.93437	0.967184	+	0.254078i	$0.0817719\pi$
$822$	10.3923	0.362473
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	−24.2487	−0.844744
$825$	0	0
$826$	36.0000	1.25260
$827$	−38.1051	−1.32504	−0.662522	−	0.749042i	$-0.730514\pi$
$827$	−38.1051	−1.32504	−0.662522	+	0.749042i	$0.730514\pi$
$828$	−6.00000	−0.208514
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	5.19615	0.180253
$832$	−1.73205	−0.0600481
$833$	8.66025	0.300060
$834$	−18.0000	−0.623289
$835$	10.3923	0.359641
$836$	0	0
$837$	−4.00000	−0.138260
$838$	−51.9615	−1.79498
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	18.0000	0.621059
$841$	−26.0000	−0.896552
$842$	12.1244	0.417833
$843$	6.92820	0.238620
$844$	−17.3205	−0.596196
$845$	30.0000	1.03203
$846$	0	0
$847$	0	0
$848$	45.0000	1.54531
$849$	31.1769	1.06999
$850$	12.0000	0.411597
$851$	66.0000	2.26245
$852$	6.00000	0.205557
$853$	8.66025	0.296521	0.148261	−	0.988948i	$-0.452633\pi$
$853$	8.66025	0.296521	0.148261	+	0.988948i	$0.452633\pi$
$854$	0	0
$855$	−20.7846	−0.710819
$856$	−6.00000	−0.205076
$857$	41.5692	1.41998	0.709989	−	0.704213i	$-0.248700\pi$
$857$	41.5692	1.41998	0.709989	+	0.704213i	$0.248700\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	−10.3923	−0.354375
$861$	−6.00000	−0.204479
$862$	−30.0000	−1.02180
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	5.19615	0.176777
$865$	−62.3538	−2.12009
$866$	−32.9090	−1.11829
$867$	14.0000	0.475465
$868$	−13.8564	−0.470317
$869$	0	0
$870$	−9.00000	−0.305129
$871$	3.46410	0.117377
$872$	−27.0000	−0.914335
$873$	−7.00000	−0.236914
$874$	−72.0000	−2.43544
$875$	−10.3923	−0.351324
$876$	−6.92820	−0.234082
$877$	−19.0526	−0.643359	−0.321680	−	0.946849i	$-0.604247\pi$
$877$	−19.0526	−0.643359	−0.321680	+	0.946849i	$0.604247\pi$
$878$	−12.0000	−0.404980
$879$	−19.0526	−0.642627
$880$	0	0
$881$	−9.00000	−0.303218	−0.151609	−	0.988441i	$-0.548445\pi$
$881$	−9.00000	−0.303218	−0.151609	+	0.988441i	$0.548445\pi$
$882$	8.66025	0.291606
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−3.00000	−0.100901
$885$	−18.0000	−0.605063
$886$	−31.1769	−1.04741
$887$	31.1769	1.04682	0.523409	−	0.852081i	$-0.324660\pi$
$887$	31.1769	1.04682	0.523409	+	0.852081i	$0.324660\pi$
$888$	−19.0526	−0.639362
$889$	0	0
$890$	−46.7654	−1.56758
$891$	0	0
$892$	20.0000	0.669650
$893$	0	0
$894$	21.0000	0.702345
$895$	−36.0000	−1.20335
$896$	−42.0000	−1.40312
$897$	−10.3923	−0.346989
$898$	36.3731	1.21378
$899$	−6.92820	−0.231069
$900$	4.00000	0.133333
$901$	−15.5885	−0.519327
$902$	0	0
$903$	12.0000	0.399335
$904$	36.3731	1.20975
$905$	21.0000	0.698064
$906$	−24.0000	−0.797347
$907$	−22.0000	−0.730498	−0.365249	−	0.930910i	$-0.619016\pi$
$907$	−22.0000	−0.730498	−0.365249	+	0.930910i	$0.619016\pi$
$908$	−24.2487	−0.804722
$909$	−13.8564	−0.459588
$910$	−31.1769	−1.03350
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	34.6410	1.14708
$913$	0	0
$914$	−51.0000	−1.68693
$915$	0	0
$916$	−23.0000	−0.759941
$917$	60.0000	1.98137
$918$	−3.00000	−0.0990148
$919$	−24.2487	−0.799891	−0.399946	−	0.916539i	$-0.630971\pi$
$919$	−24.2487	−0.799891	−0.399946	+	0.916539i	$0.630971\pi$
$920$	−31.1769	−1.02787
$921$	−3.46410	−0.114146
$922$	−27.0000	−0.889198
$923$	10.3923	0.342067
$924$	0	0
$925$	−44.0000	−1.44671
$926$	−58.8897	−1.93524
$927$	14.0000	0.459820
$928$	9.00000	0.295439
$929$	39.0000	1.27955	0.639774	−	0.768563i	$-0.279028\pi$
$929$	39.0000	1.27955	0.639774	+	0.768563i	$0.279028\pi$
$930$	20.7846	0.681554
$931$	34.6410	1.13531
$932$	29.4449	0.964499
$933$	−12.0000	−0.392862
$934$	−31.1769	−1.02014
$935$	0	0
$936$	3.00000	0.0980581
$937$	50.2295	1.64093	0.820463	−	0.571700i	$-0.193716\pi$
$937$	50.2295	1.64093	0.820463	+	0.571700i	$0.193716\pi$
$938$	12.0000	0.391814
$939$	−7.00000	−0.228436
$940$	0	0
$941$	29.4449	0.959875	0.479938	−	0.877303i	$-0.340659\pi$
$941$	29.4449	0.959875	0.479938	+	0.877303i	$0.340659\pi$
$942$	24.2487	0.790066
$943$	10.3923	0.338420
$944$	30.0000	0.976417
$945$	−10.3923	−0.338062
$946$	0	0
$947$	54.0000	1.75476	0.877382	−	0.479792i	$-0.159288\pi$
$947$	54.0000	1.75476	0.877382	+	0.479792i	$0.159288\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	48.0000	1.55733
$951$	−6.00000	−0.194563
$952$	10.3923	0.336817
$953$	−57.1577	−1.85152	−0.925759	−	0.378113i	$-0.876573\pi$
$953$	−57.1577	−1.85152	−0.925759	+	0.378113i	$0.876573\pi$
$954$	−15.5885	−0.504695
$955$	−36.0000	−1.16493
$956$	−6.92820	−0.224074
$957$	0	0
$958$	42.0000	1.35696
$959$	20.7846	0.671170
$960$	3.00000	0.0968246
$961$	−15.0000	−0.483871
$962$	33.0000	1.06396
$963$	3.46410	0.111629
$964$	−20.7846	−0.669427
$965$	15.5885	0.501810
$966$	−36.0000	−1.15828
$967$	10.3923	0.334194	0.167097	−	0.985940i	$-0.446561\pi$
$967$	10.3923	0.334194	0.167097	+	0.985940i	$0.446561\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	36.3731	1.16787
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	−1.00000	−0.0320750
$973$	−36.0000	−1.15411
$974$	−65.8179	−2.10894
$975$	6.92820	0.221880
$976$	0	0
$977$	−9.00000	−0.287936	−0.143968	−	0.989582i	$-0.545986\pi$
$977$	−9.00000	−0.287936	−0.143968	+	0.989582i	$0.545986\pi$
$978$	−3.46410	−0.110770
$979$	0	0
$980$	−15.0000	−0.479157
$981$	15.5885	0.497701
$982$	−36.0000	−1.14881
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	−3.00000	−0.0956365
$985$	57.1577	1.82120
$986$	−5.19615	−0.165479
$987$	0	0
$988$	−12.0000	−0.381771
$989$	−20.7846	−0.660912
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	−20.7846	−0.659912
$993$	−20.0000	−0.634681
$994$	36.0000	1.14185
$995$	−30.0000	−0.951064
$996$	0	0
$997$	−1.73205	−0.0548546	−0.0274273	−	0.999624i	$-0.508731\pi$
$997$	−1.73205	−0.0548546	−0.0274273	+	0.999624i	$0.508731\pi$
$998$	38.1051	1.20620
$999$	11.0000	0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.2.a.f.1.2	yes	2
3.2	odd	2		1089.2.a.o.1.1		2
4.3	odd	2		5808.2.a.ca.1.2		2
5.4	even	2		9075.2.a.bo.1.1		2
11.2	odd	10		363.2.e.m.202.2		8
11.3	even	5		363.2.e.m.130.2		8
11.4	even	5		363.2.e.m.148.2		8
11.5	even	5		363.2.e.m.124.1		8
11.6	odd	10		363.2.e.m.124.2		8
11.7	odd	10		363.2.e.m.148.1		8
11.8	odd	10		363.2.e.m.130.1		8
11.9	even	5		363.2.e.m.202.1		8
11.10	odd	2	inner	363.2.a.f.1.1	✓	2
33.32	even	2		1089.2.a.o.1.2		2
44.43	even	2		5808.2.a.ca.1.1		2
55.54	odd	2		9075.2.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.f.1.1	✓	2	11.10	odd	2	inner
363.2.a.f.1.2	yes	2	1.1	even	1	trivial
363.2.e.m.124.1		8	11.5	even	5
363.2.e.m.124.2		8	11.6	odd	10
363.2.e.m.130.1		8	11.8	odd	10
363.2.e.m.130.2		8	11.3	even	5
363.2.e.m.148.1		8	11.7	odd	10
363.2.e.m.148.2		8	11.4	even	5
363.2.e.m.202.1		8	11.9	even	5
363.2.e.m.202.2		8	11.2	odd	10
1089.2.a.o.1.1		2	3.2	odd	2
1089.2.a.o.1.2		2	33.32	even	2
5808.2.a.ca.1.1		2	44.43	even	2
5808.2.a.ca.1.2		2	4.3	odd	2
9075.2.a.bo.1.1		2	5.4	even	2
9075.2.a.bo.1.2		2	55.54	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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.73205 q^{6} -3.46410 q^{7} -1.73205 q^{8} +1.00000 q^{9} -5.19615 q^{10} -1.00000 q^{12} -1.73205 q^{13} -6.00000 q^{14} +3.00000 q^{15} -5.00000 q^{16} +1.73205 q^{17} +1.73205 q^{18} +6.92820 q^{19} -3.00000 q^{20} +3.46410 q^{21} -6.00000 q^{23} +1.73205 q^{24} +4.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} -3.46410 q^{28} -1.73205 q^{29} +5.19615 q^{30} +4.00000 q^{31} -5.19615 q^{32} +3.00000 q^{34} +10.3923 q^{35} +1.00000 q^{36} -11.0000 q^{37} +12.0000 q^{38} +1.73205 q^{39} +5.19615 q^{40} -1.73205 q^{41} +6.00000 q^{42} +3.46410 q^{43} -3.00000 q^{45} -10.3923 q^{46} +5.00000 q^{48} +5.00000 q^{49} +6.92820 q^{50} -1.73205 q^{51} -1.73205 q^{52} -9.00000 q^{53} -1.73205 q^{54} +6.00000 q^{56} -6.92820 q^{57} -3.00000 q^{58} -6.00000 q^{59} +3.00000 q^{60} +6.92820 q^{62} -3.46410 q^{63} +1.00000 q^{64} +5.19615 q^{65} -2.00000 q^{67} +1.73205 q^{68} +6.00000 q^{69} +18.0000 q^{70} -6.00000 q^{71} -1.73205 q^{72} +6.92820 q^{73} -19.0526 q^{74} -4.00000 q^{75} +6.92820 q^{76} +3.00000 q^{78} +15.0000 q^{80} +1.00000 q^{81} -3.00000 q^{82} +3.46410 q^{84} -5.19615 q^{85} +6.00000 q^{86} +1.73205 q^{87} +9.00000 q^{89} -5.19615 q^{90} +6.00000 q^{91} -6.00000 q^{92} -4.00000 q^{93} -20.7846 q^{95} +5.19615 q^{96} -7.00000 q^{97} +8.66025 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} - 6 q^{5} + 2 q^{9} - 2 q^{12} - 12 q^{14} + 6 q^{15} - 10 q^{16} - 6 q^{20} - 12 q^{23} + 8 q^{25} - 6 q^{26} - 2 q^{27} + 8 q^{31} + 6 q^{34} + 2 q^{36} - 22 q^{37} + 24 q^{38}+ \cdots - 14 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^5 + 2 * q^9 - 2 * q^12 - 12 * q^14 + 6 * q^15 - 10 * q^16 - 6 * q^20 - 12 * q^23 + 8 * q^25 - 6 * q^26 - 2 * q^27 + 8 * q^31 + 6 * q^34 + 2 * q^36 - 22 * q^37 + 24 * q^38 + 12 * q^42 - 6 * q^45 + 10 * q^48 + 10 * q^49 - 18 * q^53 + 12 * q^56 - 6 * q^58 - 12 * q^59 + 6 * q^60 + 2 * q^64 - 4 * q^67 + 12 * q^69 + 36 * q^70 - 12 * q^71 - 8 * q^75 + 6 * q^78 + 30 * q^80 + 2 * q^81 - 6 * q^82 + 12 * q^86 + 18 * q^89 + 12 * q^91 - 12 * q^92 - 8 * q^93 - 14 * q^97

Introduction

L-functions

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