LMFDB - Embedded newform 363.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,2,Mod(1,363)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$2.89856959337$
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			1.1
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)

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

Display 100 coefficients 10 coefficients

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	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	−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	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	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	$-0.422790\pi$
$13$	1.73205	0.480384	0.240192	+	0.970725i	$0.422790\pi$
$14$	−6.00000	−1.60357
$15$	3.00000	0.774597
$16$	−5.00000	−1.25000
$17$	−1.73205	−0.420084	−0.210042	−	0.977692i	$-0.567360\pi$
$17$	−1.73205	−0.420084	−0.210042	+	0.977692i	$0.567360\pi$
$18$	−1.73205	−0.408248
$19$	−6.92820	−1.58944	−0.794719	−	0.606977i	$-0.792382\pi$
$19$	−6.92820	−1.58944	−0.794719	+	0.606977i	$0.792382\pi$
$20$	−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	$-0.448587\pi$
$29$	1.73205	0.321634	0.160817	+	0.986984i	$0.448587\pi$
$30$	−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	$-0.456816\pi$
$41$	1.73205	0.270501	0.135250	+	0.990811i	$0.456816\pi$
$42$	6.00000	0.925820
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	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	$-0.632883\pi$
$73$	−6.92820	−0.810885	−0.405442	+	0.914121i	$0.632883\pi$
$74$	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	$-0.257882\pi$
$101$	13.8564	1.37876	0.689382	+	0.724398i	$0.257882\pi$
$102$	−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	$-0.553551\pi$
$107$	−3.46410	−0.334887	−0.167444	+	0.985882i	$0.553551\pi$
$108$	−1.00000	−0.0962250
$109$	−15.5885	−1.49310	−0.746552	−	0.665327i	$-0.768292\pi$
$109$	−15.5885	−1.49310	−0.746552	+	0.665327i	$0.768292\pi$
$110$	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	$-0.226835\pi$
$131$	17.3205	1.51330	0.756650	+	0.653820i	$0.226835\pi$
$132$	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	$-0.645281\pi$
$139$	−10.3923	−0.881464	−0.440732	+	0.897639i	$0.645281\pi$
$140$	−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	$-0.334570\pi$
$149$	12.1244	0.993266	0.496633	+	0.867961i	$0.334570\pi$
$150$	6.92820	0.565685
$151$	−13.8564	−1.12762	−0.563809	−	0.825905i	$-0.690665\pi$
$151$	−13.8564	−1.12762	−0.563809	+	0.825905i	$0.690665\pi$
$152$	−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	$-0.457208\pi$
$167$	3.46410	0.268060	0.134030	+	0.990977i	$0.457208\pi$
$168$	−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	$-0.789978\pi$
$173$	−20.7846	−1.58022	−0.790112	+	0.612962i	$0.789978\pi$
$174$	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	$-0.440119\pi$
$193$	5.19615	0.374027	0.187014	+	0.982357i	$0.440119\pi$
$194$	12.1244	0.870478
$195$	5.19615	0.372104
$196$	5.00000	0.357143
$197$	19.0526	1.35744	0.678719	−	0.734398i	$-0.262535\pi$
$197$	19.0526	1.35744	0.678719	+	0.734398i	$0.262535\pi$
$198$	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	$-0.296678\pi$
$211$	17.3205	1.19239	0.596196	+	0.802839i	$0.296678\pi$
$212$	−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	$-0.202314\pi$
$227$	24.2487	1.60944	0.804722	+	0.593652i	$0.202314\pi$
$228$	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	$-0.914929\pi$
$233$	−29.4449	−1.92900	−0.964499	+	0.264088i	$0.914929\pi$
$234$	−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	$-0.428064\pi$
$239$	6.92820	0.448148	0.224074	+	0.974572i	$0.428064\pi$
$240$	−15.0000	−0.968246
$241$	20.7846	1.33885	0.669427	−	0.742878i	$-0.266540\pi$
$241$	20.7846	1.33885	0.669427	+	0.742878i	$0.266540\pi$
$242$	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	$-0.640504\pi$
$263$	−13.8564	−0.854423	−0.427211	+	0.904152i	$0.640504\pi$
$264$	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	$-0.466447\pi$
$271$	3.46410	0.210429	0.105215	+	0.994450i	$0.466447\pi$
$272$	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	$-0.450107\pi$
$277$	5.19615	0.312207	0.156103	+	0.987741i	$0.450107\pi$
$278$	18.0000	1.07957
$279$	4.00000	0.239474
$280$	−18.0000	−1.07571
$281$	6.92820	0.413302	0.206651	−	0.978415i	$-0.433744\pi$
$281$	6.92820	0.413302	0.206651	+	0.978415i	$0.433744\pi$
$282$	0	0
$283$	31.1769	1.85328	0.926638	−	0.375956i	$-0.122686\pi$
$283$	31.1769	1.85328	0.926638	+	0.375956i	$0.122686\pi$
$284$	−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	$-0.687868\pi$
$293$	−19.0526	−1.11306	−0.556531	+	0.830827i	$0.687868\pi$
$294$	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	$-0.531517\pi$
$307$	−3.46410	−0.197707	−0.0988534	+	0.995102i	$0.531517\pi$
$308$	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	$-0.392874\pi$
$337$	12.1244	0.660456	0.330228	+	0.943901i	$0.392874\pi$
$338$	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	$-0.767003\pi$
$347$	−27.7128	−1.48770	−0.743851	+	0.668346i	$0.767003\pi$
$348$	−1.73205	−0.0928477
$349$	1.73205	0.0927146	0.0463573	−	0.998925i	$-0.485239\pi$
$349$	1.73205	0.0927146	0.0463573	+	0.998925i	$0.485239\pi$
$350$	−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	$-0.807549\pi$
$359$	−31.1769	−1.64545	−0.822727	+	0.568436i	$0.807549\pi$
$360$	−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	$-0.680855\pi$
$373$	−20.7846	−1.07619	−0.538093	+	0.842885i	$0.680855\pi$
$374$	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	$-0.513635\pi$
$409$	−1.73205	−0.0856444	−0.0428222	+	0.999083i	$0.513635\pi$
$410$	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	$-0.363029\pi$
$431$	17.3205	0.834300	0.417150	+	0.908838i	$0.363029\pi$
$432$	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	$-0.447130\pi$
$439$	6.92820	0.330665	0.165333	+	0.986238i	$0.447130\pi$
$440$	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	$-0.258188\pi$
$457$	29.4449	1.37737	0.688686	+	0.725059i	$0.258188\pi$
$458$	39.8372	1.86147
$459$	1.73205	0.0808452
$460$	18.0000	0.839254
$461$	15.5885	0.726027	0.363013	−	0.931784i	$-0.381748\pi$
$461$	15.5885	0.726027	0.363013	+	0.931784i	$0.381748\pi$
$462$	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	$-0.686890\pi$
$479$	−24.2487	−1.10795	−0.553976	+	0.832533i	$0.686890\pi$
$480$	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	$-0.344615\pi$
$491$	20.7846	0.937996	0.468998	+	0.883199i	$0.344615\pi$
$492$	−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	$-0.744620\pi$
$503$	−31.1769	−1.39011	−0.695055	+	0.718957i	$0.744620\pi$
$504$	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	$-0.136963\pi$
$523$	41.5692	1.81770	0.908848	+	0.417128i	$0.136963\pi$
$524$	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	$-0.234555\pi$
$547$	34.6410	1.48114	0.740571	+	0.671978i	$0.234555\pi$
$548$	−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	$-0.618926\pi$
$563$	−17.3205	−0.729972	−0.364986	+	0.931013i	$0.618926\pi$
$564$	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	$-0.302704\pi$
$569$	27.7128	1.16178	0.580891	+	0.813982i	$0.302704\pi$
$570$	36.0000	1.50787
$571$	−3.46410	−0.144968	−0.0724841	−	0.997370i	$-0.523093\pi$
$571$	−3.46410	−0.144968	−0.0724841	+	0.997370i	$0.523093\pi$
$572$	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	$-0.347016\pi$
$593$	22.5167	0.924648	0.462324	+	0.886711i	$0.347016\pi$
$594$	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	$-0.348123\pi$
$601$	22.5167	0.918474	0.459237	+	0.888314i	$0.348123\pi$
$602$	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	$-0.638605\pi$
$607$	−20.7846	−0.843621	−0.421811	+	0.906684i	$0.638605\pi$
$608$	−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	$-0.625717\pi$
$613$	−19.0526	−0.769526	−0.384763	+	0.923015i	$0.625717\pi$
$614$	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	$-0.586986\pi$
$659$	−13.8564	−0.539769	−0.269884	+	0.962893i	$0.586986\pi$
$660$	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	$-0.368804\pi$
$673$	20.7846	0.801188	0.400594	+	0.916256i	$0.368804\pi$
$674$	−21.0000	−0.808890
$675$	−4.00000	−0.153960
$676$	−10.0000	−0.384615
$677$	−32.9090	−1.26479	−0.632397	−	0.774644i	$-0.717929\pi$
$677$	−32.9090	−1.26479	−0.632397	+	0.774644i	$0.717929\pi$
$678$	−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	$-0.228937\pi$
$701$	39.8372	1.50463	0.752315	+	0.658804i	$0.228937\pi$
$702$	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	$-0.551129\pi$
$733$	−8.66025	−0.319874	−0.159937	+	0.987127i	$0.551129\pi$
$734$	−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	$-0.329750\pi$
$739$	27.7128	1.01943	0.509716	+	0.860343i	$0.329750\pi$
$740$	33.0000	1.21310
$741$	12.0000	0.440831
$742$	54.0000	1.98240
$743$	51.9615	1.90628	0.953142	−	0.302524i	$-0.0978293\pi$
$743$	51.9615	1.90628	0.953142	+	0.302524i	$0.0978293\pi$
$744$	−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	$-0.703432\pi$
$761$	−32.9090	−1.19295	−0.596475	+	0.802632i	$0.703432\pi$
$762$	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	$-0.655186\pi$
$769$	−25.9808	−0.936890	−0.468445	+	0.883493i	$0.655186\pi$
$770$	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	$-0.460594\pi$
$787$	6.92820	0.246964	0.123482	+	0.992347i	$0.460594\pi$
$788$	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	$-0.619059\pi$
$809$	−20.7846	−0.730748	−0.365374	+	0.930861i	$0.619059\pi$
$810$	5.19615	0.182574
$811$	27.7128	0.973128	0.486564	−	0.873645i	$-0.338250\pi$
$811$	27.7128	0.973128	0.486564	+	0.873645i	$0.338250\pi$
$812$	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	$-0.918228\pi$
$821$	−55.4256	−1.93437	−0.967184	+	0.254078i	$0.918228\pi$
$822$	−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	$-0.269486\pi$
$827$	38.1051	1.32504	0.662522	+	0.749042i	$0.269486\pi$
$828$	−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	$-0.547367\pi$
$853$	−8.66025	−0.296521	−0.148261	+	0.988948i	$0.547367\pi$
$854$	0	0
$855$	20.7846	0.710819
$856$	−6.00000	−0.205076
$857$	−41.5692	−1.41998	−0.709989	−	0.704213i	$-0.751300\pi$
$857$	−41.5692	−1.41998	−0.709989	+	0.704213i	$0.751300\pi$
$858$	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	$-0.395753\pi$
$877$	19.0526	0.643359	0.321680	+	0.946849i	$0.395753\pi$
$878$	−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	$-0.675340\pi$
$887$	−31.1769	−1.04682	−0.523409	+	0.852081i	$0.675340\pi$
$888$	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	$-0.369029\pi$
$919$	24.2487	0.799891	0.399946	+	0.916539i	$0.369029\pi$
$920$	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	$-0.806284\pi$
$937$	−50.2295	−1.64093	−0.820463	+	0.571700i	$0.806284\pi$
$938$	12.0000	0.391814
$939$	−7.00000	−0.228436
$940$	0	0
$941$	−29.4449	−0.959875	−0.479938	−	0.877303i	$-0.659341\pi$
$941$	−29.4449	−0.959875	−0.479938	+	0.877303i	$0.659341\pi$
$942$	−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	$-0.123427\pi$
$953$	57.1577	1.85152	0.925759	+	0.378113i	$0.123427\pi$
$954$	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	$-0.553439\pi$
$967$	−10.3923	−0.334194	−0.167097	+	0.985940i	$0.553439\pi$
$968$	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	$-0.491269\pi$
$997$	1.73205	0.0548546	0.0274273	+	0.999624i	$0.491269\pi$
$998$	−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.1.1	✓	2
3.2	odd	2		1089.2.a.o.1.2		2
4.3	odd	2		5808.2.a.ca.1.1		2
5.4	even	2		9075.2.a.bo.1.2		2
11.2	odd	10		363.2.e.m.202.1		8
11.3	even	5		363.2.e.m.130.1		8
11.4	even	5		363.2.e.m.148.1		8
11.5	even	5		363.2.e.m.124.2		8
11.6	odd	10		363.2.e.m.124.1		8
11.7	odd	10		363.2.e.m.148.2		8
11.8	odd	10		363.2.e.m.130.2		8
11.9	even	5		363.2.e.m.202.2		8
11.10	odd	2	inner	363.2.a.f.1.2	yes	2
33.32	even	2		1089.2.a.o.1.1		2
44.43	even	2		5808.2.a.ca.1.2		2
55.54	odd	2		9075.2.a.bo.1.1		2

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

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

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

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