LMFDB - Embedded newform 546.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [546,2,Mod(1,546)] 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("546.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	546.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,1,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.35983195036$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	546.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} -1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} +2.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +1.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} -2.00000 q^{60} -10.0000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} -2.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +4.00000 q^{77} -1.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +8.00000 q^{83} -1.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} -1.00000 q^{91} -4.00000 q^{92} +12.0000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.00000	0.267261
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−2.00000	−0.447214
$21$	−1.00000	−0.218218
$22$	4.00000	0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	2.00000	0.365148
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	2.00000	0.342997
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	4.00000	0.648886
$39$	1.00000	0.160128
$40$	2.00000	0.316228
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	1.00000	0.154303
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−4.00000	−0.603023
$45$	−2.00000	−0.298142
$46$	4.00000	0.589768
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	1.00000	0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	8.00000	1.07872
$56$	1.00000	0.133631
$57$	−4.00000	−0.529813
$58$	2.00000	0.262613
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−2.00000	−0.258199
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	4.00000	0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−2.00000	−0.242536
$69$	−4.00000	−0.481543
$70$	−2.00000	−0.239046
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−1.00000	−0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	4.00000	0.455842
$78$	−1.00000	−0.113228
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−1.00000	−0.109109
$85$	4.00000	0.433861
$86$	−4.00000	−0.431331
$87$	−2.00000	−0.214423
$88$	4.00000	0.426401
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	2.00000	0.210819
$91$	−1.00000	−0.104828
$92$	−4.00000	−0.417029
$93$	0	0
$94$	12.0000	1.23771
$95$	8.00000	0.820783
$96$	−1.00000	−0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−1.00000	−0.101015
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	2.00000	0.198030
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−1.00000	−0.0980581
$105$	2.00000	0.195180
$106$	−6.00000	−0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	1.00000	0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−8.00000	−0.762770
$111$	−2.00000	−0.189832
$112$	−1.00000	−0.0944911
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	4.00000	0.374634
$115$	8.00000	0.746004
$116$	−2.00000	−0.185695
$117$	1.00000	0.0924500
$118$	0	0
$119$	2.00000	0.183340
$120$	2.00000	0.182574
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	2.00000	0.180334
$124$	0	0
$125$	12.0000	1.07331
$126$	1.00000	0.0890871
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	4.00000	0.352180
$130$	2.00000	0.175412
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−4.00000	−0.348155
$133$	4.00000	0.346844
$134$	−4.00000	−0.345547
$135$	−2.00000	−0.172133
$136$	2.00000	0.171499
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	4.00000	0.340503
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	2.00000	0.169031
$141$	−12.0000	−1.01058
$142$	8.00000	0.671345
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	6.00000	0.496564
$147$	1.00000	0.0824786
$148$	−2.00000	−0.164399
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	1.00000	0.0816497
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000	0.324443
$153$	−2.00000	−0.161690
$154$	−4.00000	−0.322329
$155$	0	0
$156$	1.00000	0.0800641
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	−8.00000	−0.636446
$159$	6.00000	0.475831
$160$	2.00000	0.158114
$161$	4.00000	0.315244
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	2.00000	0.156174
$165$	8.00000	0.622799
$166$	−8.00000	−0.620920
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	1.00000	0.0771517
$169$	1.00000	0.0769231
$170$	−4.00000	−0.306786
$171$	−4.00000	−0.305888
$172$	4.00000	0.304997
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	2.00000	0.151620
$175$	1.00000	0.0755929
$176$	−4.00000	−0.301511
$177$	0	0
$178$	6.00000	0.449719
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	−2.00000	−0.149071
$181$	−26.0000	−1.93256	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000	−1.93256	−0.966282	+	0.257485i	$0.917106\pi$
$182$	1.00000	0.0741249
$183$	−10.0000	−0.739221
$184$	4.00000	0.294884
$185$	4.00000	0.294086
$186$	0	0
$187$	8.00000	0.585018
$188$	−12.0000	−0.875190
$189$	−1.00000	−0.0727393
$190$	−8.00000	−0.580381
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.00000	0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−2.00000	−0.143592
$195$	−2.00000	−0.143223
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	4.00000	0.284268
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	6.00000	0.422159
$203$	2.00000	0.140372
$204$	−2.00000	−0.140028
$205$	−4.00000	−0.279372
$206$	8.00000	0.557386
$207$	−4.00000	−0.278019
$208$	1.00000	0.0693375
$209$	16.0000	1.10674
$210$	−2.00000	−0.138013
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000	0.412082
$213$	−8.00000	−0.548151
$214$	0	0
$215$	−8.00000	−0.545595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−14.0000	−0.948200
$219$	−6.00000	−0.405442
$220$	8.00000	0.539360
$221$	−2.00000	−0.134535
$222$	2.00000	0.134231
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	1.00000	0.0668153
$225$	−1.00000	−0.0666667
$226$	−10.0000	−0.665190
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	−4.00000	−0.264906
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	−8.00000	−0.527504
$231$	4.00000	0.263181
$232$	2.00000	0.131306
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	−1.00000	−0.0653720
$235$	24.0000	1.56559
$236$	0	0
$237$	8.00000	0.519656
$238$	−2.00000	−0.129641
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	−2.00000	−0.129099
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	−2.00000	−0.127775
$246$	−2.00000	−0.127515
$247$	−4.00000	−0.254514
$248$	0	0
$249$	8.00000	0.506979
$250$	−12.0000	−0.758947
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	−1.00000	−0.0629941
$253$	16.0000	1.00591
$254$	8.00000	0.501965
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−4.00000	−0.249029
$259$	2.00000	0.124274
$260$	−2.00000	−0.124035
$261$	−2.00000	−0.123797
$262$	−12.0000	−0.741362
$263$	−20.0000	−1.23325	−0.616626	−	0.787256i	$-0.711501\pi$
$263$	−20.0000	−1.23325	−0.616626	+	0.787256i	$0.711501\pi$
$264$	4.00000	0.246183
$265$	−12.0000	−0.737154
$266$	−4.00000	−0.245256
$267$	−6.00000	−0.367194
$268$	4.00000	0.244339
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	2.00000	0.121716
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	−2.00000	−0.121268
$273$	−1.00000	−0.0605228
$274$	−22.0000	−1.32907
$275$	4.00000	0.241209
$276$	−4.00000	−0.240772
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	−2.00000	−0.119523
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	12.0000	0.714590
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	−8.00000	−0.474713
$285$	8.00000	0.473879
$286$	4.00000	0.236525
$287$	−2.00000	−0.118056
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	2.00000	0.117242
$292$	−6.00000	−0.351123
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	2.00000	0.116248
$297$	−4.00000	−0.232104
$298$	−18.0000	−1.04271
$299$	−4.00000	−0.231326
$300$	−1.00000	−0.0577350
$301$	−4.00000	−0.230556
$302$	16.0000	0.920697
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	20.0000	1.14520
$306$	2.00000	0.114332
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	4.00000	0.227921
$309$	−8.00000	−0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	−1.00000	−0.0566139
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−22.0000	−1.24153
$315$	2.00000	0.112687
$316$	8.00000	0.450035
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	−6.00000	−0.336463
$319$	8.00000	0.447914
$320$	−2.00000	−0.111803
$321$	0	0
$322$	−4.00000	−0.222911
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	−1.00000	−0.0554700
$326$	−12.0000	−0.664619
$327$	14.0000	0.774202
$328$	−2.00000	−0.110432
$329$	12.0000	0.661581
$330$	−8.00000	−0.440386
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	8.00000	0.439057
$333$	−2.00000	−0.109599
$334$	−12.0000	−0.656611
$335$	−8.00000	−0.437087
$336$	−1.00000	−0.0545545
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	−1.00000	−0.0543928
$339$	10.0000	0.543125
$340$	4.00000	0.216930
$341$	0	0
$342$	4.00000	0.216295
$343$	−1.00000	−0.0539949
$344$	−4.00000	−0.215666
$345$	8.00000	0.430706
$346$	−2.00000	−0.107521
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	−2.00000	−0.107211
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	−1.00000	−0.0534522
$351$	1.00000	0.0533761
$352$	4.00000	0.213201
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	−6.00000	−0.317999
$357$	2.00000	0.105851
$358$	24.0000	1.26844
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	2.00000	0.105409
$361$	−3.00000	−0.157895
$362$	26.0000	1.36653
$363$	5.00000	0.262432
$364$	−1.00000	−0.0524142
$365$	12.0000	0.628109
$366$	10.0000	0.522708
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	−4.00000	−0.208514
$369$	2.00000	0.104116
$370$	−4.00000	−0.207950
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	−8.00000	−0.413670
$375$	12.0000	0.619677
$376$	12.0000	0.618853
$377$	−2.00000	−0.103005
$378$	1.00000	0.0514344
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	8.00000	0.410391
$381$	−8.00000	−0.409852
$382$	12.0000	0.613973
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	−1.00000	−0.0510310
$385$	−8.00000	−0.407718
$386$	−2.00000	−0.101797
$387$	4.00000	0.203331
$388$	2.00000	0.101535
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	2.00000	0.101274
$391$	8.00000	0.404577
$392$	−1.00000	−0.0505076
$393$	12.0000	0.605320
$394$	6.00000	0.302276
$395$	−16.0000	−0.805047
$396$	−4.00000	−0.201008
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	8.00000	0.401004
$399$	4.00000	0.200250
$400$	−1.00000	−0.0500000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	−4.00000	−0.199502
$403$	0	0
$404$	−6.00000	−0.298511
$405$	−2.00000	−0.0993808
$406$	−2.00000	−0.0992583
$407$	8.00000	0.396545
$408$	2.00000	0.0990148
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	4.00000	0.197546
$411$	22.0000	1.08518
$412$	−8.00000	−0.394132
$413$	0	0
$414$	4.00000	0.196589
$415$	−16.0000	−0.785409
$416$	−1.00000	−0.0490290
$417$	4.00000	0.195881
$418$	−16.0000	−0.782586
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	2.00000	0.0975900
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−4.00000	−0.194717
$423$	−12.0000	−0.583460
$424$	−6.00000	−0.291386
$425$	2.00000	0.0970143
$426$	8.00000	0.387601
$427$	10.0000	0.483934
$428$	0	0
$429$	−4.00000	−0.193122
$430$	8.00000	0.385794
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	1.00000	0.0481125
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	14.0000	0.670478
$437$	16.0000	0.765384
$438$	6.00000	0.286691
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	−8.00000	−0.381385
$441$	1.00000	0.0476190
$442$	2.00000	0.0951303
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−2.00000	−0.0949158
$445$	12.0000	0.568855
$446$	24.0000	1.13643
$447$	18.0000	0.851371
$448$	−1.00000	−0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	1.00000	0.0471405
$451$	−8.00000	−0.376705
$452$	10.0000	0.470360
$453$	−16.0000	−0.751746
$454$	8.00000	0.375459
$455$	2.00000	0.0937614
$456$	4.00000	0.187317
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	26.0000	1.21490
$459$	−2.00000	−0.0933520
$460$	8.00000	0.373002
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	−4.00000	−0.186097
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−26.0000	−1.20443
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	1.00000	0.0462250
$469$	−4.00000	−0.184703
$470$	−24.0000	−1.10704
$471$	22.0000	1.01371
$472$	0	0
$473$	−16.0000	−0.735681
$474$	−8.00000	−0.367452
$475$	4.00000	0.183533
$476$	2.00000	0.0916698
$477$	6.00000	0.274721
$478$	24.0000	1.09773
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	2.00000	0.0912871
$481$	−2.00000	−0.0911922
$482$	−2.00000	−0.0910975
$483$	4.00000	0.182006
$484$	5.00000	0.227273
$485$	−4.00000	−0.181631
$486$	−1.00000	−0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	10.0000	0.452679
$489$	12.0000	0.542659
$490$	2.00000	0.0903508
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	2.00000	0.0901670
$493$	4.00000	0.180151
$494$	4.00000	0.179969
$495$	8.00000	0.359573
$496$	0	0
$497$	8.00000	0.358849
$498$	−8.00000	−0.358489
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	12.0000	0.536656
$501$	12.0000	0.536120
$502$	4.00000	0.178529
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	1.00000	0.0445435
$505$	12.0000	0.533993
$506$	−16.0000	−0.711287
$507$	1.00000	0.0444116
$508$	−8.00000	−0.354943
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	−4.00000	−0.177123
$511$	6.00000	0.265424
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	18.0000	0.793946
$515$	16.0000	0.705044
$516$	4.00000	0.176090
$517$	48.0000	2.11104
$518$	−2.00000	−0.0878750
$519$	2.00000	0.0877903
$520$	2.00000	0.0877058
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	2.00000	0.0875376
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	12.0000	0.524222
$525$	1.00000	0.0436436
$526$	20.0000	0.872041
$527$	0	0
$528$	−4.00000	−0.174078
$529$	−7.00000	−0.304348
$530$	12.0000	0.521247
$531$	0	0
$532$	4.00000	0.173422
$533$	2.00000	0.0866296
$534$	6.00000	0.259645
$535$	0	0
$536$	−4.00000	−0.172774
$537$	−24.0000	−1.03568
$538$	6.00000	0.258678
$539$	−4.00000	−0.172292
$540$	−2.00000	−0.0860663
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	24.0000	1.03089
$543$	−26.0000	−1.11577
$544$	2.00000	0.0857493
$545$	−28.0000	−1.19939
$546$	1.00000	0.0427960
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	22.0000	0.939793
$549$	−10.0000	−0.426790
$550$	−4.00000	−0.170561
$551$	8.00000	0.340811
$552$	4.00000	0.170251
$553$	−8.00000	−0.340195
$554$	−22.0000	−0.934690
$555$	4.00000	0.169791
$556$	4.00000	0.169638
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	2.00000	0.0845154
$561$	8.00000	0.337760
$562$	−22.0000	−0.928014
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	−12.0000	−0.505291
$565$	−20.0000	−0.841406
$566$	28.0000	1.17693
$567$	−1.00000	−0.0419961
$568$	8.00000	0.335673
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	−8.00000	−0.335083
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	−4.00000	−0.167248
$573$	−12.0000	−0.501307
$574$	2.00000	0.0834784
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	13.0000	0.540729
$579$	2.00000	0.0831172
$580$	4.00000	0.166091
$581$	−8.00000	−0.331896
$582$	−2.00000	−0.0829027
$583$	−24.0000	−0.993978
$584$	6.00000	0.248282
$585$	−2.00000	−0.0826898
$586$	−14.0000	−0.578335
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	1.00000	0.0412393
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	−2.00000	−0.0821995
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	4.00000	0.164122
$595$	−4.00000	−0.163984
$596$	18.0000	0.737309
$597$	−8.00000	−0.327418
$598$	4.00000	0.163572
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	1.00000	0.0408248
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	4.00000	0.163028
$603$	4.00000	0.162893
$604$	−16.0000	−0.651031
$605$	−10.0000	−0.406558
$606$	6.00000	0.243733
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	4.00000	0.162221
$609$	2.00000	0.0810441
$610$	−20.0000	−0.809776
$611$	−12.0000	−0.485468
$612$	−2.00000	−0.0808452
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	12.0000	0.484281
$615$	−4.00000	−0.161296
$616$	−4.00000	−0.161165
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	8.00000	0.321807
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	−24.0000	−0.962312
$623$	6.00000	0.240385
$624$	1.00000	0.0400320
$625$	−19.0000	−0.760000
$626$	6.00000	0.239808
$627$	16.0000	0.638978
$628$	22.0000	0.877896
$629$	4.00000	0.159490
$630$	−2.00000	−0.0796819
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	−8.00000	−0.318223
$633$	4.00000	0.158986
$634$	−2.00000	−0.0794301
$635$	16.0000	0.634941
$636$	6.00000	0.237915
$637$	1.00000	0.0396214
$638$	−8.00000	−0.316723
$639$	−8.00000	−0.316475
$640$	2.00000	0.0790569
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	4.00000	0.157622
$645$	−8.00000	−0.315000
$646$	−8.00000	−0.314756
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	1.00000	0.0392232
$651$	0	0
$652$	12.0000	0.469956
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−14.0000	−0.547443
$655$	−24.0000	−0.937758
$656$	2.00000	0.0780869
$657$	−6.00000	−0.234082
$658$	−12.0000	−0.467809
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	8.00000	0.311400
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	12.0000	0.466393
$663$	−2.00000	−0.0776736
$664$	−8.00000	−0.310460
$665$	−8.00000	−0.310227
$666$	2.00000	0.0774984
$667$	8.00000	0.309761
$668$	12.0000	0.464294
$669$	−24.0000	−0.927894
$670$	8.00000	0.309067
$671$	40.0000	1.54418
$672$	1.00000	0.0385758
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	−18.0000	−0.693334
$675$	−1.00000	−0.0384900
$676$	1.00000	0.0384615
$677$	34.0000	1.30673	0.653363	−	0.757045i	$-0.273358\pi$
$677$	34.0000	1.30673	0.653363	+	0.757045i	$0.273358\pi$
$678$	−10.0000	−0.384048
$679$	−2.00000	−0.0767530
$680$	−4.00000	−0.153393
$681$	−8.00000	−0.306561
$682$	0	0
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	−4.00000	−0.152944
$685$	−44.0000	−1.68115
$686$	1.00000	0.0381802
$687$	−26.0000	−0.991962
$688$	4.00000	0.152499
$689$	6.00000	0.228582
$690$	−8.00000	−0.304555
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	2.00000	0.0760286
$693$	4.00000	0.151947
$694$	−16.0000	−0.607352
$695$	−8.00000	−0.303457
$696$	2.00000	0.0758098
$697$	−4.00000	−0.151511
$698$	34.0000	1.28692
$699$	26.0000	0.983410
$700$	1.00000	0.0377964
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	−1.00000	−0.0377426
$703$	8.00000	0.301726
$704$	−4.00000	−0.150756
$705$	24.0000	0.903892
$706$	14.0000	0.526897
$707$	6.00000	0.225653
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	−16.0000	−0.600469
$711$	8.00000	0.300023
$712$	6.00000	0.224860
$713$	0	0
$714$	−2.00000	−0.0748481
$715$	8.00000	0.299183
$716$	−24.0000	−0.896922
$717$	−24.0000	−0.896296
$718$	−24.0000	−0.895672
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	−2.00000	−0.0745356
$721$	8.00000	0.297936
$722$	3.00000	0.111648
$723$	2.00000	0.0743808
$724$	−26.0000	−0.966282
$725$	2.00000	0.0742781
$726$	−5.00000	−0.185567
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	1.00000	0.0370625
$729$	1.00000	0.0370370
$730$	−12.0000	−0.444140
$731$	−8.00000	−0.295891
$732$	−10.0000	−0.369611
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	24.0000	0.885856
$735$	−2.00000	−0.0737711
$736$	4.00000	0.147442
$737$	−16.0000	−0.589368
$738$	−2.00000	−0.0736210
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	4.00000	0.147043
$741$	−4.00000	−0.146944
$742$	6.00000	0.220267
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	26.0000	0.951928
$747$	8.00000	0.292705
$748$	8.00000	0.292509
$749$	0	0
$750$	−12.0000	−0.438178
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	−12.0000	−0.437595
$753$	−4.00000	−0.145768
$754$	2.00000	0.0728357
$755$	32.0000	1.16460
$756$	−1.00000	−0.0363696
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	−28.0000	−1.01701
$759$	16.0000	0.580763
$760$	−8.00000	−0.290191
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	8.00000	0.289809
$763$	−14.0000	−0.506834
$764$	−12.0000	−0.434145
$765$	4.00000	0.144620
$766$	12.0000	0.433578
$767$	0	0
$768$	1.00000	0.0360844
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	8.00000	0.288300
$771$	−18.0000	−0.648254
$772$	2.00000	0.0719816
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	2.00000	0.0717496
$778$	−14.0000	−0.501924
$779$	−8.00000	−0.286630
$780$	−2.00000	−0.0716115
$781$	32.0000	1.14505
$782$	−8.00000	−0.286079
$783$	−2.00000	−0.0714742
$784$	1.00000	0.0357143
$785$	−44.0000	−1.57043
$786$	−12.0000	−0.428026
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	−6.00000	−0.213741
$789$	−20.0000	−0.712019
$790$	16.0000	0.569254
$791$	−10.0000	−0.355559
$792$	4.00000	0.142134
$793$	−10.0000	−0.355110
$794$	−14.0000	−0.496841
$795$	−12.0000	−0.425596
$796$	−8.00000	−0.283552
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	−4.00000	−0.141598
$799$	24.0000	0.849059
$800$	1.00000	0.0353553
$801$	−6.00000	−0.212000
$802$	10.0000	0.353112
$803$	24.0000	0.846942
$804$	4.00000	0.141069
$805$	−8.00000	−0.281963
$806$	0	0
$807$	−6.00000	−0.211210
$808$	6.00000	0.211079
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	2.00000	0.0702728
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	2.00000	0.0701862
$813$	−24.0000	−0.841717
$814$	−8.00000	−0.280400
$815$	−24.0000	−0.840683
$816$	−2.00000	−0.0700140
$817$	−16.0000	−0.559769
$818$	−26.0000	−0.909069
$819$	−1.00000	−0.0349428
$820$	−4.00000	−0.139686
$821$	34.0000	1.18661	0.593304	−	0.804978i	$-0.297823\pi$
$821$	34.0000	1.18661	0.593304	+	0.804978i	$0.297823\pi$
$822$	−22.0000	−0.767338
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	8.00000	0.278693
$825$	4.00000	0.139262
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	−4.00000	−0.139010
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	16.0000	0.555368
$831$	22.0000	0.763172
$832$	1.00000	0.0346688
$833$	−2.00000	−0.0692959
$834$	−4.00000	−0.138509
$835$	−24.0000	−0.830554
$836$	16.0000	0.553372
$837$	0	0
$838$	−20.0000	−0.690889
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	−2.00000	−0.0690066
$841$	−25.0000	−0.862069
$842$	26.0000	0.896019
$843$	22.0000	0.757720
$844$	4.00000	0.137686
$845$	−2.00000	−0.0688021
$846$	12.0000	0.412568
$847$	−5.00000	−0.171802
$848$	6.00000	0.206041
$849$	−28.0000	−0.960958
$850$	−2.00000	−0.0685994
$851$	8.00000	0.274236
$852$	−8.00000	−0.274075
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	−10.0000	−0.342193
$855$	8.00000	0.273594
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	4.00000	0.136558
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	−8.00000	−0.272798
$861$	−2.00000	−0.0681598
$862$	40.0000	1.36241
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−1.00000	−0.0340207
$865$	−4.00000	−0.136004
$866$	30.0000	1.01944
$867$	−13.0000	−0.441503
$868$	0	0
$869$	−32.0000	−1.08553
$870$	−4.00000	−0.135613
$871$	4.00000	0.135535
$872$	−14.0000	−0.474100
$873$	2.00000	0.0676897
$874$	−16.0000	−0.541208
$875$	−12.0000	−0.405674
$876$	−6.00000	−0.202721
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	32.0000	1.07995
$879$	14.0000	0.472208
$880$	8.00000	0.269680
$881$	22.0000	0.741199	0.370599	−	0.928793i	$-0.379152\pi$
$881$	22.0000	0.741199	0.370599	+	0.928793i	$0.379152\pi$
$882$	−1.00000	−0.0336718
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	24.0000	0.806296
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	2.00000	0.0671156
$889$	8.00000	0.268311
$890$	−12.0000	−0.402241
$891$	−4.00000	−0.134005
$892$	−24.0000	−0.803579
$893$	48.0000	1.60626
$894$	−18.0000	−0.602010
$895$	48.0000	1.60446
$896$	1.00000	0.0334077
$897$	−4.00000	−0.133556
$898$	18.0000	0.600668
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	−12.0000	−0.399778
$902$	8.00000	0.266371
$903$	−4.00000	−0.133112
$904$	−10.0000	−0.332595
$905$	52.0000	1.72854
$906$	16.0000	0.531564
$907$	−36.0000	−1.19536	−0.597680	−	0.801735i	$-0.703911\pi$
$907$	−36.0000	−1.19536	−0.597680	+	0.801735i	$0.703911\pi$
$908$	−8.00000	−0.265489
$909$	−6.00000	−0.199007
$910$	−2.00000	−0.0662994
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	−4.00000	−0.132453
$913$	−32.0000	−1.05905
$914$	−18.0000	−0.595387
$915$	20.0000	0.661180
$916$	−26.0000	−0.859064
$917$	−12.0000	−0.396275
$918$	2.00000	0.0660098
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	−8.00000	−0.263752
$921$	−12.0000	−0.395413
$922$	2.00000	0.0658665
$923$	−8.00000	−0.263323
$924$	4.00000	0.131590
$925$	2.00000	0.0657596
$926$	−24.0000	−0.788689
$927$	−8.00000	−0.262754
$928$	2.00000	0.0656532
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	26.0000	0.851658
$933$	24.0000	0.785725
$934$	−4.00000	−0.130884
$935$	−16.0000	−0.523256
$936$	−1.00000	−0.0326860
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	4.00000	0.130605
$939$	−6.00000	−0.195803
$940$	24.0000	0.782794
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	−22.0000	−0.716799
$943$	−8.00000	−0.260516
$944$	0	0
$945$	2.00000	0.0650600
$946$	16.0000	0.520205
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	8.00000	0.259828
$949$	−6.00000	−0.194768
$950$	−4.00000	−0.129777
$951$	2.00000	0.0648544
$952$	−2.00000	−0.0648204
$953$	−46.0000	−1.49009	−0.745043	−	0.667016i	$-0.767571\pi$
$953$	−46.0000	−1.49009	−0.745043	+	0.667016i	$0.767571\pi$
$954$	−6.00000	−0.194257
$955$	24.0000	0.776622
$956$	−24.0000	−0.776215
$957$	8.00000	0.258603
$958$	−12.0000	−0.387702
$959$	−22.0000	−0.710417
$960$	−2.00000	−0.0645497
$961$	−31.0000	−1.00000
$962$	2.00000	0.0644826
$963$	0	0
$964$	2.00000	0.0644157
$965$	−4.00000	−0.128765
$966$	−4.00000	−0.128698
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	−5.00000	−0.160706
$969$	8.00000	0.256997
$970$	4.00000	0.128432
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	−4.00000	−0.128234
$974$	−32.0000	−1.02535
$975$	−1.00000	−0.0320256
$976$	−10.0000	−0.320092
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−12.0000	−0.383718
$979$	24.0000	0.767043
$980$	−2.00000	−0.0638877
$981$	14.0000	0.446986
$982$	8.00000	0.255290
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	−2.00000	−0.0637577
$985$	12.0000	0.382352
$986$	−4.00000	−0.127386
$987$	12.0000	0.381964
$988$	−4.00000	−0.127257
$989$	−16.0000	−0.508770
$990$	−8.00000	−0.254257
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	−12.0000	−0.380808
$994$	−8.00000	−0.253745
$995$	16.0000	0.507234
$996$	8.00000	0.253490
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	20.0000	0.633089
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.2.a.b.1.1	✓	1
3.2	odd	2		1638.2.a.s.1.1		1
4.3	odd	2		4368.2.a.d.1.1		1
7.6	odd	2		3822.2.a.h.1.1		1
13.12	even	2		7098.2.a.be.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.b.1.1	✓	1	1.1	even	1	trivial
1638.2.a.s.1.1		1	3.2	odd	2
3822.2.a.h.1.1		1	7.6	odd	2
4368.2.a.d.1.1		1	4.3	odd	2
7098.2.a.be.1.1		1	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more