LMFDB - Embedded newform 6039.2.a.l.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.40042 q^{2} +3.76203 q^{4} +3.90458 q^{5} -3.05635 q^{7} +4.22962 q^{8} +O(q^{10})$	\(q+2.40042 q^{2} +3.76203 q^{4} +3.90458 q^{5} -3.05635 q^{7} +4.22962 q^{8} +9.37264 q^{10} +1.00000 q^{11} -5.54437 q^{13} -7.33653 q^{14} +2.62881 q^{16} +5.45989 q^{17} +5.83487 q^{19} +14.6891 q^{20} +2.40042 q^{22} +7.83643 q^{23} +10.2458 q^{25} -13.3088 q^{26} -11.4981 q^{28} +3.45552 q^{29} -0.504688 q^{31} -2.14898 q^{32} +13.1060 q^{34} -11.9338 q^{35} -5.34533 q^{37} +14.0062 q^{38} +16.5149 q^{40} +12.1681 q^{41} -4.71869 q^{43} +3.76203 q^{44} +18.8107 q^{46} -3.26502 q^{47} +2.34126 q^{49} +24.5941 q^{50} -20.8581 q^{52} -7.66741 q^{53} +3.90458 q^{55} -12.9272 q^{56} +8.29470 q^{58} -2.01048 q^{59} +1.00000 q^{61} -1.21146 q^{62} -10.4161 q^{64} -21.6484 q^{65} +5.54815 q^{67} +20.5403 q^{68} -28.6461 q^{70} -5.61256 q^{71} +4.97921 q^{73} -12.8311 q^{74} +21.9510 q^{76} -3.05635 q^{77} +2.08461 q^{79} +10.2644 q^{80} +29.2086 q^{82} +4.20087 q^{83} +21.3186 q^{85} -11.3269 q^{86} +4.22962 q^{88} +4.09744 q^{89} +16.9455 q^{91} +29.4809 q^{92} -7.83742 q^{94} +22.7827 q^{95} +7.15422 q^{97} +5.62002 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8} + q^{10} + 21 q^{11} + 20 q^{13} - 17 q^{14} + 50 q^{16} - q^{17} + 15 q^{19} + 2 q^{20} - 11 q^{23} + 48 q^{25} + 5 q^{26} - 16 q^{28} + 9 q^{29} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 21 q^{37} - 11 q^{38} - 16 q^{40} - 7 q^{41} + 16 q^{43} + 32 q^{44} - 3 q^{46} - 5 q^{47} + 80 q^{49} + 33 q^{50} + 60 q^{52} - 9 q^{53} - 7 q^{55} - 44 q^{56} - 27 q^{58} - 13 q^{59} + 21 q^{61} + 23 q^{62} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 33 q^{70} - 12 q^{71} + 20 q^{73} + 12 q^{74} + 59 q^{76} + 5 q^{77} + q^{79} + 38 q^{80} + 7 q^{82} + 19 q^{83} + 38 q^{85} + 3 q^{86} + 6 q^{88} - 37 q^{89} + 24 q^{91} - 31 q^{92} - 64 q^{94} + 43 q^{95} + 68 q^{97} + 2 q^{98}+O(q^{100}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8 + q^10 + 21 * q^11 + 20 * q^13 - 17 * q^14 + 50 * q^16 - q^17 + 15 * q^19 + 2 * q^20 - 11 * q^23 + 48 * q^25 + 5 * q^26 - 16 * q^28 + 9 * q^29 + 22 * q^31 - 3 * q^32 + 33 * q^34 + 39 * q^35 + 21 * q^37 - 11 * q^38 - 16 * q^40 - 7 * q^41 + 16 * q^43 + 32 * q^44 - 3 * q^46 - 5 * q^47 + 80 * q^49 + 33 * q^50 + 60 * q^52 - 9 * q^53 - 7 * q^55 - 44 * q^56 - 27 * q^58 - 13 * q^59 + 21 * q^61 + 23 * q^62 + 66 * q^64 - 25 * q^65 + 38 * q^67 + 74 * q^68 - 33 * q^70 - 12 * q^71 + 20 * q^73 + 12 * q^74 + 59 * q^76 + 5 * q^77 + q^79 + 38 * q^80 + 7 * q^82 + 19 * q^83 + 38 * q^85 + 3 * q^86 + 6 * q^88 - 37 * q^89 + 24 * q^91 - 31 * q^92 - 64 * q^94 + 43 * q^95 + 68 * q^97 + 2 * q^98

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 $

$2$	2.40042	1.69736	0.848678	−	0.528910i	$-0.177399\pi$
$2$	2.40042	1.69736	0.848678	+	0.528910i	$0.177399\pi$
$3$	0	0
$3$	0	0
$4$	3.76203	1.88101
$5$	3.90458	1.74618	0.873091	−	0.487558i	$-0.162112\pi$
$5$	3.90458	1.74618	0.873091	+	0.487558i	$0.162112\pi$
$6$	0	0
$7$	−3.05635	−1.15519	−0.577595	−	0.816323i	$-0.696009\pi$
$7$	−3.05635	−1.15519	−0.577595	+	0.816323i	$0.696009\pi$
$8$	4.22962	1.49539
$9$	0	0
$10$	9.37264	2.96389
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.54437	−1.53773	−0.768865	−	0.639411i	$-0.779178\pi$
$13$	−5.54437	−1.53773	−0.768865	+	0.639411i	$0.779178\pi$
$14$	−7.33653	−1.96077
$15$	0	0
$16$	2.62881	0.657202
$17$	5.45989	1.32422	0.662109	−	0.749408i	$-0.269662\pi$
$17$	5.45989	1.32422	0.662109	+	0.749408i	$0.269662\pi$
$18$	0	0
$19$	5.83487	1.33861	0.669306	−	0.742987i	$-0.266592\pi$
$19$	5.83487	1.33861	0.669306	+	0.742987i	$0.266592\pi$
$20$	14.6891	3.28459
$21$	0	0
$22$	2.40042	0.511772
$23$	7.83643	1.63401	0.817004	−	0.576632i	$-0.195633\pi$
$23$	7.83643	1.63401	0.817004	+	0.576632i	$0.195633\pi$
$24$	0	0
$25$	10.2458	2.04915
$26$	−13.3088	−2.61008
$27$	0	0
$28$	−11.4981	−2.17293
$29$	3.45552	0.641673	0.320837	−	0.947135i	$-0.396036\pi$
$29$	3.45552	0.641673	0.320837	+	0.947135i	$0.396036\pi$
$30$	0	0
$31$	−0.504688	−0.0906446	−0.0453223	−	0.998972i	$-0.514431\pi$
$31$	−0.504688	−0.0906446	−0.0453223	+	0.998972i	$0.514431\pi$
$32$	−2.14898	−0.379890
$33$	0	0
$34$	13.1060	2.24767
$35$	−11.9338	−2.01717
$36$	0	0
$37$	−5.34533	−0.878767	−0.439384	−	0.898299i	$-0.644803\pi$
$37$	−5.34533	−0.878767	−0.439384	+	0.898299i	$0.644803\pi$
$38$	14.0062	2.27210
$39$	0	0
$40$	16.5149	2.61123
$41$	12.1681	1.90034	0.950171	−	0.311731i	$-0.100909\pi$
$41$	12.1681	1.90034	0.950171	+	0.311731i	$0.100909\pi$
$42$	0	0
$43$	−4.71869	−0.719594	−0.359797	−	0.933031i	$-0.617154\pi$
$43$	−4.71869	−0.719594	−0.359797	+	0.933031i	$0.617154\pi$
$44$	3.76203	0.567147
$45$	0	0
$46$	18.8107	2.77349
$47$	−3.26502	−0.476252	−0.238126	−	0.971234i	$-0.576533\pi$
$47$	−3.26502	−0.476252	−0.238126	+	0.971234i	$0.576533\pi$
$48$	0	0
$49$	2.34126	0.334466
$50$	24.5941	3.47814
$51$	0	0
$52$	−20.8581	−2.89249
$53$	−7.66741	−1.05320	−0.526600	−	0.850113i	$-0.676534\pi$
$53$	−7.66741	−1.05320	−0.526600	+	0.850113i	$0.676534\pi$
$54$	0	0
$55$	3.90458	0.526494
$56$	−12.9272	−1.72747
$57$	0	0
$58$	8.29470	1.08915
$59$	−2.01048	−0.261742	−0.130871	−	0.991399i	$-0.541777\pi$
$59$	−2.01048	−0.261742	−0.130871	+	0.991399i	$0.541777\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−1.21146	−0.153856
$63$	0	0
$64$	−10.4161	−1.30201
$65$	−21.6484	−2.68516
$66$	0	0
$67$	5.54815	0.677814	0.338907	−	0.940820i	$-0.389943\pi$
$67$	5.54815	0.677814	0.338907	+	0.940820i	$0.389943\pi$
$68$	20.5403	2.49087
$69$	0	0
$70$	−28.6461	−3.42386
$71$	−5.61256	−0.666089	−0.333044	−	0.942911i	$-0.608076\pi$
$71$	−5.61256	−0.666089	−0.333044	+	0.942911i	$0.608076\pi$
$72$	0	0
$73$	4.97921	0.582773	0.291386	−	0.956605i	$-0.405883\pi$
$73$	4.97921	0.582773	0.291386	+	0.956605i	$0.405883\pi$
$74$	−12.8311	−1.49158
$75$	0	0
$76$	21.9510	2.51795
$77$	−3.05635	−0.348303
$78$	0	0
$79$	2.08461	0.234537	0.117269	−	0.993100i	$-0.462586\pi$
$79$	2.08461	0.234537	0.117269	+	0.993100i	$0.462586\pi$
$80$	10.2644	1.14759
$81$	0	0
$82$	29.2086	3.22555
$83$	4.20087	0.461105	0.230553	−	0.973060i	$-0.425947\pi$
$83$	4.20087	0.461105	0.230553	+	0.973060i	$0.425947\pi$
$84$	0	0
$85$	21.3186	2.31232
$86$	−11.3269	−1.22141
$87$	0	0
$88$	4.22962	0.450879
$89$	4.09744	0.434328	0.217164	−	0.976135i	$-0.430319\pi$
$89$	4.09744	0.434328	0.217164	+	0.976135i	$0.430319\pi$
$90$	0	0
$91$	16.9455	1.77637
$92$	29.4809	3.07359
$93$	0	0
$94$	−7.83742	−0.808368
$95$	22.7827	2.33746
$96$	0	0
$97$	7.15422	0.726401	0.363201	−	0.931711i	$-0.381684\pi$
$97$	7.15422	0.726401	0.363201	+	0.931711i	$0.381684\pi$
$98$	5.62002	0.567708
$99$	0	0
$100$	38.5448	3.85448
$101$	6.28383	0.625264	0.312632	−	0.949874i	$-0.398789\pi$
$101$	6.28383	0.625264	0.312632	+	0.949874i	$0.398789\pi$
$102$	0	0
$103$	−5.04101	−0.496706	−0.248353	−	0.968670i	$-0.579889\pi$
$103$	−5.04101	−0.496706	−0.248353	+	0.968670i	$0.579889\pi$
$104$	−23.4505	−2.29951
$105$	0	0
$106$	−18.4050	−1.78766
$107$	15.0276	1.45278	0.726389	−	0.687284i	$-0.241197\pi$
$107$	15.0276	1.45278	0.726389	+	0.687284i	$0.241197\pi$
$108$	0	0
$109$	−10.9249	−1.04642	−0.523210	−	0.852204i	$-0.675266\pi$
$109$	−10.9249	−1.04642	−0.523210	+	0.852204i	$0.675266\pi$
$110$	9.37264	0.893647
$111$	0	0
$112$	−8.03455	−0.759194
$113$	−6.36396	−0.598671	−0.299335	−	0.954148i	$-0.596765\pi$
$113$	−6.36396	−0.598671	−0.299335	+	0.954148i	$0.596765\pi$
$114$	0	0
$115$	30.5980	2.85328
$116$	12.9998	1.20700
$117$	0	0
$118$	−4.82600	−0.444269
$119$	−16.6873	−1.52972
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.40042	0.217324
$123$	0	0
$124$	−1.89865	−0.170504
$125$	20.4825	1.83201
$126$	0	0
$127$	−8.82634	−0.783211	−0.391606	−	0.920133i	$-0.628080\pi$
$127$	−8.82634	−0.783211	−0.391606	+	0.920133i	$0.628080\pi$
$128$	−20.7050	−1.83008
$129$	0	0
$130$	−51.9654	−4.55767
$131$	−5.63711	−0.492517	−0.246258	−	0.969204i	$-0.579201\pi$
$131$	−5.63711	−0.492517	−0.246258	+	0.969204i	$0.579201\pi$
$132$	0	0
$133$	−17.8334	−1.54635
$134$	13.3179	1.15049
$135$	0	0
$136$	23.0932	1.98023
$137$	6.63735	0.567067	0.283533	−	0.958962i	$-0.408493\pi$
$137$	6.63735	0.567067	0.283533	+	0.958962i	$0.408493\pi$
$138$	0	0
$139$	−16.6192	−1.40962	−0.704812	−	0.709395i	$-0.748968\pi$
$139$	−16.6192	−1.40962	−0.704812	+	0.709395i	$0.748968\pi$
$140$	−44.8951	−3.79433
$141$	0	0
$142$	−13.4725	−1.13059
$143$	−5.54437	−0.463643
$144$	0	0
$145$	13.4923	1.12048
$146$	11.9522	0.989172
$147$	0	0
$148$	−20.1093	−1.65297
$149$	9.73673	0.797664	0.398832	−	0.917024i	$-0.369416\pi$
$149$	9.73673	0.797664	0.398832	+	0.917024i	$0.369416\pi$
$150$	0	0
$151$	−4.86318	−0.395760	−0.197880	−	0.980226i	$-0.563406\pi$
$151$	−4.86318	−0.395760	−0.197880	+	0.980226i	$0.563406\pi$
$152$	24.6793	2.00175
$153$	0	0
$154$	−7.33653	−0.591194
$155$	−1.97059	−0.158282
$156$	0	0
$157$	−20.7193	−1.65358	−0.826790	−	0.562510i	$-0.809836\pi$
$157$	−20.7193	−1.65358	−0.826790	+	0.562510i	$0.809836\pi$
$158$	5.00395	0.398093
$159$	0	0
$160$	−8.39088	−0.663357
$161$	−23.9509	−1.88759
$162$	0	0
$163$	7.33960	0.574882	0.287441	−	0.957798i	$-0.407195\pi$
$163$	7.33960	0.574882	0.287441	+	0.957798i	$0.407195\pi$
$164$	45.7768	3.57457
$165$	0	0
$166$	10.0839	0.782660
$167$	16.3048	1.26171	0.630853	−	0.775902i	$-0.282705\pi$
$167$	16.3048	1.26171	0.630853	+	0.775902i	$0.282705\pi$
$168$	0	0
$169$	17.7400	1.36462
$170$	51.1736	3.92484
$171$	0	0
$172$	−17.7519	−1.35357
$173$	25.3219	1.92519	0.962593	−	0.270950i	$-0.0873379\pi$
$173$	25.3219	1.92519	0.962593	+	0.270950i	$0.0873379\pi$
$174$	0	0
$175$	−31.3146	−2.36716
$176$	2.62881	0.198154
$177$	0	0
$178$	9.83559	0.737208
$179$	3.11194	0.232597	0.116299	−	0.993214i	$-0.462897\pi$
$179$	3.11194	0.232597	0.116299	+	0.993214i	$0.462897\pi$
$180$	0	0
$181$	−6.22703	−0.462851	−0.231426	−	0.972853i	$-0.574339\pi$
$181$	−6.22703	−0.462851	−0.231426	+	0.972853i	$0.574339\pi$
$182$	40.6764	3.01514
$183$	0	0
$184$	33.1451	2.44349
$185$	−20.8713	−1.53449
$186$	0	0
$187$	5.45989	0.399267
$188$	−12.2831	−0.895836
$189$	0	0
$190$	54.6882	3.96750
$191$	−23.1436	−1.67461	−0.837305	−	0.546736i	$-0.815870\pi$
$191$	−23.1436	−1.67461	−0.837305	+	0.546736i	$0.815870\pi$
$192$	0	0
$193$	12.6861	0.913162	0.456581	−	0.889682i	$-0.349074\pi$
$193$	12.6861	0.913162	0.456581	+	0.889682i	$0.349074\pi$
$194$	17.1732	1.23296
$195$	0	0
$196$	8.80790	0.629136
$197$	−22.0266	−1.56933	−0.784664	−	0.619922i	$-0.787164\pi$
$197$	−22.0266	−1.56933	−0.784664	+	0.619922i	$0.787164\pi$
$198$	0	0
$199$	−15.1778	−1.07592	−0.537962	−	0.842969i	$-0.680806\pi$
$199$	−15.1778	−1.07592	−0.537962	+	0.842969i	$0.680806\pi$
$200$	43.3356	3.06429
$201$	0	0
$202$	15.0838	1.06130
$203$	−10.5613	−0.741255
$204$	0	0
$205$	47.5114	3.31834
$206$	−12.1006	−0.843086
$207$	0	0
$208$	−14.5751	−1.01060
$209$	5.83487	0.403606
$210$	0	0
$211$	−25.7691	−1.77402	−0.887008	−	0.461754i	$-0.847220\pi$
$211$	−25.7691	−1.77402	−0.887008	+	0.461754i	$0.847220\pi$
$212$	−28.8450	−1.98109
$213$	0	0
$214$	36.0727	2.46588
$215$	−18.4245	−1.25654
$216$	0	0
$217$	1.54250	0.104712
$218$	−26.2245	−1.77615
$219$	0	0
$220$	14.6891	0.990342
$221$	−30.2716	−2.03629
$222$	0	0
$223$	−28.1180	−1.88292	−0.941459	−	0.337127i	$-0.890545\pi$
$223$	−28.1180	−1.88292	−0.941459	+	0.337127i	$0.890545\pi$
$224$	6.56804	0.438846
$225$	0	0
$226$	−15.2762	−1.01616
$227$	0.658880	0.0437314	0.0218657	−	0.999761i	$-0.493039\pi$
$227$	0.658880	0.0437314	0.0218657	+	0.999761i	$0.493039\pi$
$228$	0	0
$229$	−8.51326	−0.562572	−0.281286	−	0.959624i	$-0.590761\pi$
$229$	−8.51326	−0.562572	−0.281286	+	0.959624i	$0.590761\pi$
$230$	73.4481	4.84302
$231$	0	0
$232$	14.6155	0.959555
$233$	6.97393	0.456877	0.228439	−	0.973558i	$-0.426638\pi$
$233$	6.97393	0.456877	0.228439	+	0.973558i	$0.426638\pi$
$234$	0	0
$235$	−12.7485	−0.831622
$236$	−7.56348	−0.492340
$237$	0	0
$238$	−40.0566	−2.59649
$239$	1.06405	0.0688276	0.0344138	−	0.999408i	$-0.489044\pi$
$239$	1.06405	0.0688276	0.0344138	+	0.999408i	$0.489044\pi$
$240$	0	0
$241$	22.8037	1.46891	0.734456	−	0.678656i	$-0.237437\pi$
$241$	22.8037	1.46891	0.734456	+	0.678656i	$0.237437\pi$
$242$	2.40042	0.154305
$243$	0	0
$244$	3.76203	0.240839
$245$	9.14165	0.584039
$246$	0	0
$247$	−32.3507	−2.05842
$248$	−2.13464	−0.135549
$249$	0	0
$250$	49.1666	3.10957
$251$	−22.7712	−1.43730	−0.718651	−	0.695370i	$-0.755240\pi$
$251$	−22.7712	−1.43730	−0.718651	+	0.695370i	$0.755240\pi$
$252$	0	0
$253$	7.83643	0.492672
$254$	−21.1870	−1.32939
$255$	0	0
$256$	−28.8687	−1.80429
$257$	−12.0057	−0.748896	−0.374448	−	0.927248i	$-0.622168\pi$
$257$	−12.0057	−0.748896	−0.374448	+	0.927248i	$0.622168\pi$
$258$	0	0
$259$	16.3372	1.01514
$260$	−81.4420	−5.05082
$261$	0	0
$262$	−13.5315	−0.835976
$263$	15.8133	0.975092	0.487546	−	0.873097i	$-0.337892\pi$
$263$	15.8133	0.975092	0.487546	+	0.873097i	$0.337892\pi$
$264$	0	0
$265$	−29.9380	−1.83908
$266$	−42.8077	−2.62471
$267$	0	0
$268$	20.8723	1.27498
$269$	19.0522	1.16164	0.580818	−	0.814033i	$-0.302733\pi$
$269$	19.0522	1.16164	0.580818	+	0.814033i	$0.302733\pi$
$270$	0	0
$271$	20.4576	1.24271	0.621356	−	0.783528i	$-0.286582\pi$
$271$	20.4576	1.24271	0.621356	+	0.783528i	$0.286582\pi$
$272$	14.3530	0.870278
$273$	0	0
$274$	15.9324	0.962514
$275$	10.2458	0.617842
$276$	0	0
$277$	−11.0227	−0.662291	−0.331146	−	0.943580i	$-0.607435\pi$
$277$	−11.0227	−0.662291	−0.331146	+	0.943580i	$0.607435\pi$
$278$	−39.8931	−2.39263
$279$	0	0
$280$	−50.4752	−3.01647
$281$	12.9918	0.775029	0.387514	−	0.921864i	$-0.373334\pi$
$281$	12.9918	0.775029	0.387514	+	0.921864i	$0.373334\pi$
$282$	0	0
$283$	−28.7769	−1.71061	−0.855305	−	0.518125i	$-0.826630\pi$
$283$	−28.7769	−1.71061	−0.855305	+	0.518125i	$0.826630\pi$
$284$	−21.1146	−1.25292
$285$	0	0
$286$	−13.3088	−0.786967
$287$	−37.1900	−2.19526
$288$	0	0
$289$	12.8104	0.753553
$290$	32.3873	1.90185
$291$	0	0
$292$	18.7319	1.09620
$293$	−21.8789	−1.27818	−0.639088	−	0.769134i	$-0.720688\pi$
$293$	−21.8789	−1.27818	−0.639088	+	0.769134i	$0.720688\pi$
$294$	0	0
$295$	−7.85007	−0.457049
$296$	−22.6087	−1.31410
$297$	0	0
$298$	23.3723	1.35392
$299$	−43.4480	−2.51267
$300$	0	0
$301$	14.4220	0.831268
$302$	−11.6737	−0.671745
$303$	0	0
$304$	15.3387	0.879738
$305$	3.90458	0.223576
$306$	0	0
$307$	−21.4140	−1.22216	−0.611080	−	0.791569i	$-0.709265\pi$
$307$	−21.4140	−1.22216	−0.611080	+	0.791569i	$0.709265\pi$
$308$	−11.4981	−0.655163
$309$	0	0
$310$	−4.73026	−0.268661
$311$	−12.1431	−0.688574	−0.344287	−	0.938864i	$-0.611879\pi$
$311$	−12.1431	−0.688574	−0.344287	+	0.938864i	$0.611879\pi$
$312$	0	0
$313$	11.4623	0.647886	0.323943	−	0.946077i	$-0.394991\pi$
$313$	11.4623	0.647886	0.323943	+	0.946077i	$0.394991\pi$
$314$	−49.7351	−2.80671
$315$	0	0
$316$	7.84237	0.441168
$317$	−23.7449	−1.33365	−0.666824	−	0.745215i	$-0.732347\pi$
$317$	−23.7449	−1.33365	−0.666824	+	0.745215i	$0.732347\pi$
$318$	0	0
$319$	3.45552	0.193472
$320$	−40.6704	−2.27355
$321$	0	0
$322$	−57.4922	−3.20391
$323$	31.8578	1.77261
$324$	0	0
$325$	−56.8062	−3.15104
$326$	17.6182	0.975779
$327$	0	0
$328$	51.4665	2.84176
$329$	9.97903	0.550162
$330$	0	0
$331$	−4.65951	−0.256110	−0.128055	−	0.991767i	$-0.540873\pi$
$331$	−4.65951	−0.256110	−0.128055	+	0.991767i	$0.540873\pi$
$332$	15.8038	0.867346
$333$	0	0
$334$	39.1385	2.14156
$335$	21.6632	1.18359
$336$	0	0
$337$	1.69410	0.0922838	0.0461419	−	0.998935i	$-0.485307\pi$
$337$	1.69410	0.0922838	0.0461419	+	0.998935i	$0.485307\pi$
$338$	42.5835	2.31624
$339$	0	0
$340$	80.2011	4.34952
$341$	−0.504688	−0.0273304
$342$	0	0
$343$	14.2387	0.768819
$344$	−19.9583	−1.07608
$345$	0	0
$346$	60.7832	3.26773
$347$	−6.77085	−0.363478	−0.181739	−	0.983347i	$-0.558173\pi$
$347$	−6.77085	−0.363478	−0.181739	+	0.983347i	$0.558173\pi$
$348$	0	0
$349$	−28.2464	−1.51200	−0.755999	−	0.654573i	$-0.772848\pi$
$349$	−28.2464	−1.51200	−0.755999	+	0.654573i	$0.772848\pi$
$350$	−75.1682	−4.01791
$351$	0	0
$352$	−2.14898	−0.114541
$353$	−22.2759	−1.18563	−0.592814	−	0.805339i	$-0.701983\pi$
$353$	−22.2759	−1.18563	−0.592814	+	0.805339i	$0.701983\pi$
$354$	0	0
$355$	−21.9147	−1.16311
$356$	15.4147	0.816977
$357$	0	0
$358$	7.46998	0.394800
$359$	−17.0196	−0.898261	−0.449130	−	0.893466i	$-0.648266\pi$
$359$	−17.0196	−0.898261	−0.449130	+	0.893466i	$0.648266\pi$
$360$	0	0
$361$	15.0457	0.791880
$362$	−14.9475	−0.785623
$363$	0	0
$364$	63.7495	3.34138
$365$	19.4417	1.01763
$366$	0	0
$367$	21.8660	1.14140	0.570699	−	0.821159i	$-0.306672\pi$
$367$	21.8660	1.14140	0.570699	+	0.821159i	$0.306672\pi$
$368$	20.6005	1.07387
$369$	0	0
$370$	−50.0999	−2.60457
$371$	23.4343	1.21665
$372$	0	0
$373$	−5.98827	−0.310061	−0.155030	−	0.987910i	$-0.549548\pi$
$373$	−5.98827	−0.310061	−0.155030	+	0.987910i	$0.549548\pi$
$374$	13.1060	0.677697
$375$	0	0
$376$	−13.8098	−0.712184
$377$	−19.1587	−0.986721
$378$	0	0
$379$	−8.87032	−0.455638	−0.227819	−	0.973704i	$-0.573159\pi$
$379$	−8.87032	−0.455638	−0.227819	+	0.973704i	$0.573159\pi$
$380$	85.7093	4.39679
$381$	0	0
$382$	−55.5543	−2.84241
$383$	25.0388	1.27942	0.639712	−	0.768614i	$-0.279053\pi$
$383$	25.0388	1.27942	0.639712	+	0.768614i	$0.279053\pi$
$384$	0	0
$385$	−11.9338	−0.608201
$386$	30.4519	1.54996
$387$	0	0
$388$	26.9144	1.36637
$389$	17.1937	0.871756	0.435878	−	0.900006i	$-0.356438\pi$
$389$	17.1937	0.871756	0.435878	+	0.900006i	$0.356438\pi$
$390$	0	0
$391$	42.7860	2.16378
$392$	9.90265	0.500159
$393$	0	0
$394$	−52.8731	−2.66371
$395$	8.13953	0.409544
$396$	0	0
$397$	−24.0795	−1.20851	−0.604257	−	0.796790i	$-0.706530\pi$
$397$	−24.0795	−1.20851	−0.604257	+	0.796790i	$0.706530\pi$
$398$	−36.4331	−1.82623
$399$	0	0
$400$	26.9341	1.34671
$401$	0.0181059	0.000904168 0	0.000452084	−	1.00000i	$-0.499856\pi$
$401$	0.0181059	0.000904168 0	0.000452084		1.00000i	$0.499856\pi$
$402$	0	0
$403$	2.79817	0.139387
$404$	23.6399	1.17613
$405$	0	0
$406$	−25.3515	−1.25817
$407$	−5.34533	−0.264958
$408$	0	0
$409$	22.1873	1.09709	0.548546	−	0.836120i	$-0.315181\pi$
$409$	22.1873	1.09709	0.548546	+	0.836120i	$0.315181\pi$
$410$	114.047	5.63240
$411$	0	0
$412$	−18.9644	−0.934311
$413$	6.14472	0.302362
$414$	0	0
$415$	16.4026	0.805174
$416$	11.9148	0.584169
$417$	0	0
$418$	14.0062	0.685064
$419$	29.2309	1.42802	0.714012	−	0.700133i	$-0.246876\pi$
$419$	29.2309	1.42802	0.714012	+	0.700133i	$0.246876\pi$
$420$	0	0
$421$	13.7237	0.668853	0.334426	−	0.942422i	$-0.391457\pi$
$421$	13.7237	0.668853	0.334426	+	0.942422i	$0.391457\pi$
$422$	−61.8567	−3.01114
$423$	0	0
$424$	−32.4302	−1.57495
$425$	55.9407	2.71352
$426$	0	0
$427$	−3.05635	−0.147907
$428$	56.5344	2.73270
$429$	0	0
$430$	−44.2266	−2.13280
$431$	−5.92420	−0.285359	−0.142679	−	0.989769i	$-0.545572\pi$
$431$	−5.92420	−0.285359	−0.142679	+	0.989769i	$0.545572\pi$
$432$	0	0
$433$	−15.6146	−0.750392	−0.375196	−	0.926946i	$-0.622425\pi$
$433$	−15.6146	−0.750392	−0.375196	+	0.926946i	$0.622425\pi$
$434$	3.70266	0.177733
$435$	0	0
$436$	−41.1000	−1.96833
$437$	45.7246	2.18730
$438$	0	0
$439$	11.4705	0.547457	0.273728	−	0.961807i	$-0.411743\pi$
$439$	11.4705	0.547457	0.273728	+	0.961807i	$0.411743\pi$
$440$	16.5149	0.787316
$441$	0	0
$442$	−72.6647	−3.45631
$443$	−34.4924	−1.63878	−0.819391	−	0.573235i	$-0.805688\pi$
$443$	−34.4924	−1.63878	−0.819391	+	0.573235i	$0.805688\pi$
$444$	0	0
$445$	15.9988	0.758415
$446$	−67.4950	−3.19598
$447$	0	0
$448$	31.8352	1.50407
$449$	9.16375	0.432464	0.216232	−	0.976342i	$-0.430623\pi$
$449$	9.16375	0.432464	0.216232	+	0.976342i	$0.430623\pi$
$450$	0	0
$451$	12.1681	0.572974
$452$	−23.9414	−1.12611
$453$	0	0
$454$	1.58159	0.0742277
$455$	66.1651	3.10187
$456$	0	0
$457$	10.5400	0.493040	0.246520	−	0.969138i	$-0.420713\pi$
$457$	10.5400	0.493040	0.246520	+	0.969138i	$0.420713\pi$
$458$	−20.4354	−0.954884
$459$	0	0
$460$	115.110	5.36705
$461$	−38.3015	−1.78388	−0.891939	−	0.452157i	$-0.850655\pi$
$461$	−38.3015	−1.78388	−0.891939	+	0.452157i	$0.850655\pi$
$462$	0	0
$463$	1.10261	0.0512428	0.0256214	−	0.999672i	$-0.491844\pi$
$463$	1.10261	0.0512428	0.0256214	+	0.999672i	$0.491844\pi$
$464$	9.08389	0.421709
$465$	0	0
$466$	16.7404	0.775483
$467$	−0.840580	−0.0388974	−0.0194487	−	0.999811i	$-0.506191\pi$
$467$	−0.840580	−0.0388974	−0.0194487	+	0.999811i	$0.506191\pi$
$468$	0	0
$469$	−16.9571	−0.783005
$470$	−30.6018	−1.41156
$471$	0	0
$472$	−8.50355	−0.391408
$473$	−4.71869	−0.216966
$474$	0	0
$475$	59.7826	2.74302
$476$	−62.7782	−2.87743
$477$	0	0
$478$	2.55417	0.116825
$479$	35.4273	1.61871	0.809357	−	0.587317i	$-0.199816\pi$
$479$	35.4273	1.61871	0.809357	+	0.587317i	$0.199816\pi$
$480$	0	0
$481$	29.6365	1.35131
$482$	54.7384	2.49327
$483$	0	0
$484$	3.76203	0.171001
$485$	27.9342	1.26843
$486$	0	0
$487$	−5.22639	−0.236830	−0.118415	−	0.992964i	$-0.537781\pi$
$487$	−5.22639	−0.236830	−0.118415	+	0.992964i	$0.537781\pi$
$488$	4.22962	0.191466
$489$	0	0
$490$	21.9438	0.991321
$491$	−15.4077	−0.695338	−0.347669	−	0.937617i	$-0.613027\pi$
$491$	−15.4077	−0.695338	−0.347669	+	0.937617i	$0.613027\pi$
$492$	0	0
$493$	18.8667	0.849715
$494$	−77.6553	−3.49388
$495$	0	0
$496$	−1.32673	−0.0595718
$497$	17.1540	0.769460
$498$	0	0
$499$	−3.91959	−0.175465	−0.0877326	−	0.996144i	$-0.527962\pi$
$499$	−3.91959	−0.175465	−0.0877326	+	0.996144i	$0.527962\pi$
$500$	77.0556	3.44603
$501$	0	0
$502$	−54.6604	−2.43961
$503$	−17.9435	−0.800061	−0.400030	−	0.916502i	$-0.631000\pi$
$503$	−17.9435	−0.800061	−0.400030	+	0.916502i	$0.631000\pi$
$504$	0	0
$505$	24.5357	1.09182
$506$	18.8107	0.836240
$507$	0	0
$508$	−33.2050	−1.47323
$509$	20.0890	0.890431	0.445216	−	0.895423i	$-0.353127\pi$
$509$	20.0890	0.890431	0.445216	+	0.895423i	$0.353127\pi$
$510$	0	0
$511$	−15.2182	−0.673214
$512$	−27.8869	−1.23244
$513$	0	0
$514$	−28.8188	−1.27114
$515$	−19.6830	−0.867338
$516$	0	0
$517$	−3.26502	−0.143595
$518$	39.2162	1.72306
$519$	0	0
$520$	−91.5645	−4.01537
$521$	18.6827	0.818506	0.409253	−	0.912421i	$-0.365789\pi$
$521$	18.6827	0.818506	0.409253	+	0.912421i	$0.365789\pi$
$522$	0	0
$523$	−25.5801	−1.11854	−0.559270	−	0.828986i	$-0.688918\pi$
$523$	−25.5801	−1.11854	−0.559270	+	0.828986i	$0.688918\pi$
$524$	−21.2070	−0.926431
$525$	0	0
$526$	37.9587	1.65508
$527$	−2.75554	−0.120033
$528$	0	0
$529$	38.4096	1.66998
$530$	−71.8640	−3.12157
$531$	0	0
$532$	−67.0898	−2.90871
$533$	−67.4645	−2.92221
$534$	0	0
$535$	58.6767	2.53681
$536$	23.4665	1.01360
$537$	0	0
$538$	45.7334	1.97171
$539$	2.34126	0.100845
$540$	0	0
$541$	18.0242	0.774919	0.387460	−	0.921887i	$-0.373353\pi$
$541$	18.0242	0.774919	0.387460	+	0.921887i	$0.373353\pi$
$542$	49.1069	2.10932
$543$	0	0
$544$	−11.7332	−0.503057
$545$	−42.6573	−1.82724
$546$	0	0
$547$	6.47617	0.276901	0.138451	−	0.990369i	$-0.455788\pi$
$547$	6.47617	0.276901	0.138451	+	0.990369i	$0.455788\pi$
$548$	24.9699	1.06666
$549$	0	0
$550$	24.5941	1.04870
$551$	20.1625	0.858951
$552$	0	0
$553$	−6.37130	−0.270935
$554$	−26.4592	−1.12414
$555$	0	0
$556$	−62.5220	−2.65152
$557$	−29.5658	−1.25274	−0.626372	−	0.779524i	$-0.715461\pi$
$557$	−29.5658	−1.25274	−0.626372	+	0.779524i	$0.715461\pi$
$558$	0	0
$559$	26.1622	1.10654
$560$	−31.3715	−1.32569
$561$	0	0
$562$	31.1859	1.31550
$563$	−25.7670	−1.08595	−0.542976	−	0.839748i	$-0.682702\pi$
$563$	−25.7670	−1.08595	−0.542976	+	0.839748i	$0.682702\pi$
$564$	0	0
$565$	−24.8486	−1.04539
$566$	−69.0767	−2.90351
$567$	0	0
$568$	−23.7390	−0.996066
$569$	16.5438	0.693554	0.346777	−	0.937948i	$-0.387276\pi$
$569$	16.5438	0.693554	0.346777	+	0.937948i	$0.387276\pi$
$570$	0	0
$571$	0.633532	0.0265125	0.0132562	−	0.999912i	$-0.495780\pi$
$571$	0.633532	0.0265125	0.0132562	+	0.999912i	$0.495780\pi$
$572$	−20.8581	−0.872120
$573$	0	0
$574$	−89.2717	−3.72613
$575$	80.2901	3.34833
$576$	0	0
$577$	−5.75419	−0.239550	−0.119775	−	0.992801i	$-0.538217\pi$
$577$	−5.75419	−0.239550	−0.119775	+	0.992801i	$0.538217\pi$
$578$	30.7504	1.27905
$579$	0	0
$580$	50.7586	2.10764
$581$	−12.8393	−0.532665
$582$	0	0
$583$	−7.66741	−0.317552
$584$	21.0602	0.871475
$585$	0	0
$586$	−52.5185	−2.16952
$587$	16.9735	0.700573	0.350286	−	0.936643i	$-0.386084\pi$
$587$	16.9735	0.700573	0.350286	+	0.936643i	$0.386084\pi$
$588$	0	0
$589$	−2.94479	−0.121338
$590$	−18.8435	−0.775774
$591$	0	0
$592$	−14.0518	−0.577527
$593$	−26.2600	−1.07837	−0.539185	−	0.842187i	$-0.681268\pi$
$593$	−26.2600	−1.07837	−0.539185	+	0.842187i	$0.681268\pi$
$594$	0	0
$595$	−65.1570	−2.67118
$596$	36.6299	1.50042
$597$	0	0
$598$	−104.294	−4.26489
$599$	−6.49896	−0.265540	−0.132770	−	0.991147i	$-0.542387\pi$
$599$	−6.49896	−0.265540	−0.132770	+	0.991147i	$0.542387\pi$
$600$	0	0
$601$	4.36919	0.178223	0.0891115	−	0.996022i	$-0.471597\pi$
$601$	4.36919	0.178223	0.0891115	+	0.996022i	$0.471597\pi$
$602$	34.6188	1.41096
$603$	0	0
$604$	−18.2954	−0.744431
$605$	3.90458	0.158744
$606$	0	0
$607$	5.74264	0.233087	0.116543	−	0.993186i	$-0.462819\pi$
$607$	5.74264	0.233087	0.116543	+	0.993186i	$0.462819\pi$
$608$	−12.5390	−0.508525
$609$	0	0
$610$	9.37264	0.379487
$611$	18.1025	0.732347
$612$	0	0
$613$	48.0137	1.93926	0.969629	−	0.244582i	$-0.0786509\pi$
$613$	48.0137	1.93926	0.969629	+	0.244582i	$0.0786509\pi$
$614$	−51.4026	−2.07444
$615$	0	0
$616$	−12.9272	−0.520851
$617$	−22.3596	−0.900165	−0.450083	−	0.892987i	$-0.648605\pi$
$617$	−22.3596	−0.900165	−0.450083	+	0.892987i	$0.648605\pi$
$618$	0	0
$619$	−29.2776	−1.17677	−0.588384	−	0.808582i	$-0.700236\pi$
$619$	−29.2776	−1.17677	−0.588384	+	0.808582i	$0.700236\pi$
$620$	−7.41343	−0.297731
$621$	0	0
$622$	−29.1487	−1.16875
$623$	−12.5232	−0.501731
$624$	0	0
$625$	28.7467	1.14987
$626$	27.5143	1.09969
$627$	0	0
$628$	−77.9466	−3.11041
$629$	−29.1849	−1.16368
$630$	0	0
$631$	−44.1200	−1.75639	−0.878195	−	0.478303i	$-0.841252\pi$
$631$	−44.1200	−1.75639	−0.878195	+	0.478303i	$0.841252\pi$
$632$	8.81710	0.350726
$633$	0	0
$634$	−56.9979	−2.26368
$635$	−34.4632	−1.36763
$636$	0	0
$637$	−12.9808	−0.514319
$638$	8.29470	0.328390
$639$	0	0
$640$	−80.8445	−3.19566
$641$	25.1861	0.994790	0.497395	−	0.867524i	$-0.334290\pi$
$641$	25.1861	0.994790	0.497395	+	0.867524i	$0.334290\pi$
$642$	0	0
$643$	−27.5892	−1.08801	−0.544005	−	0.839082i	$-0.683093\pi$
$643$	−27.5892	−1.08801	−0.544005	+	0.839082i	$0.683093\pi$
$644$	−90.1038	−3.55059
$645$	0	0
$646$	76.4721	3.00875
$647$	−1.45014	−0.0570108	−0.0285054	−	0.999594i	$-0.509075\pi$
$647$	−1.45014	−0.0570108	−0.0285054	+	0.999594i	$0.509075\pi$
$648$	0	0
$649$	−2.01048	−0.0789181
$650$	−136.359	−5.34844
$651$	0	0
$652$	27.6118	1.08136
$653$	−3.86759	−0.151350	−0.0756752	−	0.997133i	$-0.524111\pi$
$653$	−3.86759	−0.151350	−0.0756752	+	0.997133i	$0.524111\pi$
$654$	0	0
$655$	−22.0106	−0.860024
$656$	31.9876	1.24891
$657$	0	0
$658$	23.9539	0.933820
$659$	8.00713	0.311914	0.155957	−	0.987764i	$-0.450154\pi$
$659$	8.00713	0.311914	0.155957	+	0.987764i	$0.450154\pi$
$660$	0	0
$661$	−9.68056	−0.376530	−0.188265	−	0.982118i	$-0.560286\pi$
$661$	−9.68056	−0.376530	−0.188265	+	0.982118i	$0.560286\pi$
$662$	−11.1848	−0.434709
$663$	0	0
$664$	17.7681	0.689535
$665$	−69.6319	−2.70021
$666$	0	0
$667$	27.0789	1.04850
$668$	61.3393	2.37329
$669$	0	0
$670$	52.0008	2.00897
$671$	1.00000	0.0386046
$672$	0	0
$673$	36.2068	1.39567	0.697834	−	0.716259i	$-0.254147\pi$
$673$	36.2068	1.39567	0.697834	+	0.716259i	$0.254147\pi$
$674$	4.06657	0.156638
$675$	0	0
$676$	66.7384	2.56686
$677$	9.72560	0.373785	0.186893	−	0.982380i	$-0.440158\pi$
$677$	9.72560	0.373785	0.186893	+	0.982380i	$0.440158\pi$
$678$	0	0
$679$	−21.8658	−0.839132
$680$	90.1694	3.45784
$681$	0	0
$682$	−1.21146	−0.0463894
$683$	30.8902	1.18198	0.590991	−	0.806678i	$-0.298737\pi$
$683$	30.8902	1.18198	0.590991	+	0.806678i	$0.298737\pi$
$684$	0	0
$685$	25.9161	0.990202
$686$	34.1789	1.30496
$687$	0	0
$688$	−12.4045	−0.472918
$689$	42.5110	1.61954
$690$	0	0
$691$	19.0958	0.726439	0.363219	−	0.931704i	$-0.381678\pi$
$691$	19.0958	0.726439	0.363219	+	0.931704i	$0.381678\pi$
$692$	95.2616	3.62130
$693$	0	0
$694$	−16.2529	−0.616952
$695$	−64.8911	−2.46146
$696$	0	0
$697$	66.4366	2.51647
$698$	−67.8034	−2.56640
$699$	0	0
$700$	−117.806	−4.45266
$701$	−21.4093	−0.808618	−0.404309	−	0.914623i	$-0.632488\pi$
$701$	−21.4093	−0.808618	−0.404309	+	0.914623i	$0.632488\pi$
$702$	0	0
$703$	−31.1893	−1.17633
$704$	−10.4161	−0.392571
$705$	0	0
$706$	−53.4717	−2.01243
$707$	−19.2056	−0.722300
$708$	0	0
$709$	29.4953	1.10772	0.553860	−	0.832610i	$-0.313154\pi$
$709$	29.4953	1.10772	0.553860	+	0.832610i	$0.313154\pi$
$710$	−52.6046	−1.97421
$711$	0	0
$712$	17.3306	0.649492
$713$	−3.95495	−0.148114
$714$	0	0
$715$	−21.6484	−0.809605
$716$	11.7072	0.437519
$717$	0	0
$718$	−40.8543	−1.52467
$719$	25.0362	0.933694	0.466847	−	0.884338i	$-0.345390\pi$
$719$	25.0362	0.933694	0.466847	+	0.884338i	$0.345390\pi$
$720$	0	0
$721$	15.4071	0.573790
$722$	36.1161	1.34410
$723$	0	0
$724$	−23.4263	−0.870630
$725$	35.4044	1.31489
$726$	0	0
$727$	−16.7702	−0.621971	−0.310986	−	0.950415i	$-0.600659\pi$
$727$	−16.7702	−0.621971	−0.310986	+	0.950415i	$0.600659\pi$
$728$	71.6730	2.65638
$729$	0	0
$730$	46.6684	1.72727
$731$	−25.7635	−0.952899
$732$	0	0
$733$	−32.1830	−1.18871	−0.594354	−	0.804204i	$-0.702592\pi$
$733$	−32.1830	−1.18871	−0.594354	+	0.804204i	$0.702592\pi$
$734$	52.4877	1.93736
$735$	0	0
$736$	−16.8404	−0.620744
$737$	5.54815	0.204369
$738$	0	0
$739$	2.77895	0.102225	0.0511126	−	0.998693i	$-0.483723\pi$
$739$	2.77895	0.102225	0.0511126	+	0.998693i	$0.483723\pi$
$740$	−78.5184	−2.88639
$741$	0	0
$742$	56.2522	2.06508
$743$	−12.3289	−0.452302	−0.226151	−	0.974092i	$-0.572614\pi$
$743$	−12.3289	−0.452302	−0.226151	+	0.974092i	$0.572614\pi$
$744$	0	0
$745$	38.0178	1.39287
$746$	−14.3744	−0.526283
$747$	0	0
$748$	20.5403	0.751027
$749$	−45.9297	−1.67824
$750$	0	0
$751$	1.48325	0.0541245	0.0270622	−	0.999634i	$-0.491385\pi$
$751$	1.48325	0.0541245	0.0270622	+	0.999634i	$0.491385\pi$
$752$	−8.58310	−0.312993
$753$	0	0
$754$	−45.9889	−1.67482
$755$	−18.9887	−0.691069
$756$	0	0
$757$	−6.42481	−0.233514	−0.116757	−	0.993161i	$-0.537250\pi$
$757$	−6.42481	−0.233514	−0.116757	+	0.993161i	$0.537250\pi$
$758$	−21.2925	−0.773379
$759$	0	0
$760$	96.3622	3.49542
$761$	4.70734	0.170641	0.0853204	−	0.996354i	$-0.472809\pi$
$761$	4.70734	0.170641	0.0853204	+	0.996354i	$0.472809\pi$
$762$	0	0
$763$	33.3904	1.20882
$764$	−87.0668	−3.14997
$765$	0	0
$766$	60.1038	2.17164
$767$	11.1468	0.402489
$768$	0	0
$769$	31.7419	1.14464	0.572321	−	0.820030i	$-0.306043\pi$
$769$	31.7419	1.14464	0.572321	+	0.820030i	$0.306043\pi$
$770$	−28.6461	−1.03233
$771$	0	0
$772$	47.7253	1.71767
$773$	−33.7048	−1.21228	−0.606139	−	0.795359i	$-0.707282\pi$
$773$	−33.7048	−1.21228	−0.606139	+	0.795359i	$0.707282\pi$
$774$	0	0
$775$	−5.17091	−0.185744
$776$	30.2596	1.08626
$777$	0	0
$778$	41.2722	1.47968
$779$	70.9994	2.54382
$780$	0	0
$781$	−5.61256	−0.200833
$782$	102.705	3.67271
$783$	0	0
$784$	6.15473	0.219812
$785$	−80.9002	−2.88745
$786$	0	0
$787$	10.6021	0.377922	0.188961	−	0.981985i	$-0.439488\pi$
$787$	10.6021	0.377922	0.188961	+	0.981985i	$0.439488\pi$
$788$	−82.8646	−2.95193
$789$	0	0
$790$	19.5383	0.695142
$791$	19.4505	0.691579
$792$	0	0
$793$	−5.54437	−0.196886
$794$	−57.8009	−2.05128
$795$	0	0
$796$	−57.0993	−2.02383
$797$	5.19312	0.183950	0.0919749	−	0.995761i	$-0.470682\pi$
$797$	5.19312	0.183950	0.0919749	+	0.995761i	$0.470682\pi$
$798$	0	0
$799$	−17.8266	−0.630661
$800$	−22.0179	−0.778452
$801$	0	0
$802$	0.0434619	0.00153469
$803$	4.97921	0.175713
$804$	0	0
$805$	−93.5181	−3.29608
$806$	6.71680	0.236589
$807$	0	0
$808$	26.5782	0.935017
$809$	−26.1862	−0.920657	−0.460328	−	0.887749i	$-0.652268\pi$
$809$	−26.1862	−0.920657	−0.460328	+	0.887749i	$0.652268\pi$
$810$	0	0
$811$	−26.7395	−0.938950	−0.469475	−	0.882946i	$-0.655557\pi$
$811$	−26.7395	−0.938950	−0.469475	+	0.882946i	$0.655557\pi$
$812$	−39.7318	−1.39431
$813$	0	0
$814$	−12.8311	−0.449728
$815$	28.6581	1.00385
$816$	0	0
$817$	−27.5330	−0.963256
$818$	53.2589	1.86216
$819$	0	0
$820$	178.739	6.24185
$821$	44.8895	1.56665	0.783327	−	0.621610i	$-0.213521\pi$
$821$	44.8895	1.56665	0.783327	+	0.621610i	$0.213521\pi$
$822$	0	0
$823$	12.5123	0.436152	0.218076	−	0.975932i	$-0.430022\pi$
$823$	12.5123	0.436152	0.218076	+	0.975932i	$0.430022\pi$
$824$	−21.3215	−0.742771
$825$	0	0
$826$	14.7499	0.513216
$827$	12.4709	0.433657	0.216829	−	0.976210i	$-0.430429\pi$
$827$	12.4709	0.433657	0.216829	+	0.976210i	$0.430429\pi$
$828$	0	0
$829$	13.7328	0.476959	0.238480	−	0.971148i	$-0.423351\pi$
$829$	13.7328	0.476959	0.238480	+	0.971148i	$0.423351\pi$
$830$	39.3733	1.36667
$831$	0	0
$832$	57.7506	2.00214
$833$	12.7830	0.442906
$834$	0	0
$835$	63.6636	2.20317
$836$	21.9510	0.759190
$837$	0	0
$838$	70.1666	2.42387
$839$	−25.7292	−0.888270	−0.444135	−	0.895960i	$-0.646489\pi$
$839$	−25.7292	−0.888270	−0.444135	+	0.895960i	$0.646489\pi$
$840$	0	0
$841$	−17.0594	−0.588255
$842$	32.9427	1.13528
$843$	0	0
$844$	−96.9440	−3.33695
$845$	69.2673	2.38287
$846$	0	0
$847$	−3.05635	−0.105017
$848$	−20.1562	−0.692165
$849$	0	0
$850$	134.281	4.60581
$851$	−41.8883	−1.43591
$852$	0	0
$853$	−18.4750	−0.632572	−0.316286	−	0.948664i	$-0.602436\pi$
$853$	−18.4750	−0.632572	−0.316286	+	0.948664i	$0.602436\pi$
$854$	−7.33653	−0.251051
$855$	0	0
$856$	63.5612	2.17248
$857$	29.9179	1.02197	0.510987	−	0.859588i	$-0.329280\pi$
$857$	29.9179	1.02197	0.510987	+	0.859588i	$0.329280\pi$
$858$	0	0
$859$	−6.91499	−0.235936	−0.117968	−	0.993017i	$-0.537638\pi$
$859$	−6.91499	−0.235936	−0.117968	+	0.993017i	$0.537638\pi$
$860$	−69.3136	−2.36357
$861$	0	0
$862$	−14.2206	−0.484355
$863$	−39.9338	−1.35936	−0.679681	−	0.733508i	$-0.737882\pi$
$863$	−39.9338	−1.35936	−0.679681	+	0.733508i	$0.737882\pi$
$864$	0	0
$865$	98.8713	3.36173
$866$	−37.4817	−1.27368
$867$	0	0
$868$	5.80294	0.196965
$869$	2.08461	0.0707156
$870$	0	0
$871$	−30.7610	−1.04230
$872$	−46.2083	−1.56481
$873$	0	0
$874$	109.758	3.71263
$875$	−62.6015	−2.11632
$876$	0	0
$877$	−48.6939	−1.64428	−0.822139	−	0.569287i	$-0.807219\pi$
$877$	−48.6939	−1.64428	−0.822139	+	0.569287i	$0.807219\pi$
$878$	27.5340	0.929229
$879$	0	0
$880$	10.2644	0.346012
$881$	25.0353	0.843460	0.421730	−	0.906721i	$-0.361423\pi$
$881$	25.0353	0.843460	0.421730	+	0.906721i	$0.361423\pi$
$882$	0	0
$883$	41.3439	1.39133	0.695667	−	0.718364i	$-0.255109\pi$
$883$	41.3439	1.39133	0.695667	+	0.718364i	$0.255109\pi$
$884$	−113.883	−3.83029
$885$	0	0
$886$	−82.7963	−2.78160
$887$	22.9598	0.770913	0.385457	−	0.922726i	$-0.374044\pi$
$887$	22.9598	0.770913	0.385457	+	0.922726i	$0.374044\pi$
$888$	0	0
$889$	26.9764	0.904759
$890$	38.4038	1.28730
$891$	0	0
$892$	−105.781	−3.54180
$893$	−19.0510	−0.637516
$894$	0	0
$895$	12.1508	0.406157
$896$	63.2818	2.11410
$897$	0	0
$898$	21.9969	0.734046
$899$	−1.74396	−0.0581642
$900$	0	0
$901$	−41.8632	−1.39467
$902$	29.2086	0.972541
$903$	0	0
$904$	−26.9171	−0.895249
$905$	−24.3139	−0.808222
$906$	0	0
$907$	54.1257	1.79722	0.898608	−	0.438753i	$-0.144580\pi$
$907$	54.1257	1.79722	0.898608	+	0.438753i	$0.144580\pi$
$908$	2.47873	0.0822594
$909$	0	0
$910$	158.824	5.26497
$911$	−13.9020	−0.460594	−0.230297	−	0.973120i	$-0.573970\pi$
$911$	−13.9020	−0.460594	−0.230297	+	0.973120i	$0.573970\pi$
$912$	0	0
$913$	4.20087	0.139029
$914$	25.3004	0.836864
$915$	0	0
$916$	−32.0271	−1.05821
$917$	17.2290	0.568951
$918$	0	0
$919$	−42.6331	−1.40634	−0.703168	−	0.711023i	$-0.748232\pi$
$919$	−42.6331	−1.40634	−0.703168	+	0.711023i	$0.748232\pi$
$920$	129.418	4.26677
$921$	0	0
$922$	−91.9397	−3.02787
$923$	31.1181	1.02427
$924$	0	0
$925$	−54.7670	−1.80073
$926$	2.64674	0.0869772
$927$	0	0
$928$	−7.42585	−0.243765
$929$	45.6604	1.49807	0.749034	−	0.662531i	$-0.230518\pi$
$929$	45.6604	1.49807	0.749034	+	0.662531i	$0.230518\pi$
$930$	0	0
$931$	13.6610	0.447720
$932$	26.2361	0.859393
$933$	0	0
$934$	−2.01775	−0.0660227
$935$	21.3186	0.697192
$936$	0	0
$937$	30.8463	1.00770	0.503852	−	0.863790i	$-0.331916\pi$
$937$	30.8463	1.00770	0.503852	+	0.863790i	$0.331916\pi$
$938$	−40.7041	−1.32904
$939$	0	0
$940$	−47.9603	−1.56429
$941$	4.21276	0.137332	0.0686660	−	0.997640i	$-0.478126\pi$
$941$	4.21276	0.137332	0.0686660	+	0.997640i	$0.478126\pi$
$942$	0	0
$943$	95.3546	3.10517
$944$	−5.28516	−0.172017
$945$	0	0
$946$	−11.3269	−0.368268
$947$	−9.29055	−0.301902	−0.150951	−	0.988541i	$-0.548234\pi$
$947$	−9.29055	−0.301902	−0.150951	+	0.988541i	$0.548234\pi$
$948$	0	0
$949$	−27.6066	−0.896148
$950$	143.504	4.65587
$951$	0	0
$952$	−70.5810	−2.28754
$953$	42.1444	1.36519	0.682595	−	0.730797i	$-0.260851\pi$
$953$	42.1444	1.36519	0.682595	+	0.730797i	$0.260851\pi$
$954$	0	0
$955$	−90.3659	−2.92417
$956$	4.00299	0.129466
$957$	0	0
$958$	85.0405	2.74753
$959$	−20.2860	−0.655070
$960$	0	0
$961$	−30.7453	−0.991784
$962$	71.1401	2.29365
$963$	0	0
$964$	85.7880	2.76305
$965$	49.5337	1.59455
$966$	0	0
$967$	−9.83390	−0.316237	−0.158119	−	0.987420i	$-0.550543\pi$
$967$	−9.83390	−0.316237	−0.158119	+	0.987420i	$0.550543\pi$
$968$	4.22962	0.135945
$969$	0	0
$970$	67.0540	2.15297
$971$	−14.8073	−0.475190	−0.237595	−	0.971364i	$-0.576359\pi$
$971$	−14.8073	−0.475190	−0.237595	+	0.971364i	$0.576359\pi$
$972$	0	0
$973$	50.7941	1.62838
$974$	−12.5456	−0.401985
$975$	0	0
$976$	2.62881	0.0841461
$977$	−46.6744	−1.49325	−0.746623	−	0.665247i	$-0.768326\pi$
$977$	−46.6744	−1.49325	−0.746623	+	0.665247i	$0.768326\pi$
$978$	0	0
$979$	4.09744	0.130955
$980$	34.3912	1.09859
$981$	0	0
$982$	−36.9849	−1.18023
$983$	−43.9079	−1.40044	−0.700222	−	0.713925i	$-0.746916\pi$
$983$	−43.9079	−1.40044	−0.700222	+	0.713925i	$0.746916\pi$
$984$	0	0
$985$	−86.0045	−2.74033
$986$	45.2882	1.44227
$987$	0	0
$988$	−121.704	−3.87193
$989$	−36.9777	−1.17582
$990$	0	0
$991$	51.8812	1.64806	0.824031	−	0.566545i	$-0.191720\pi$
$991$	51.8812	1.64806	0.824031	+	0.566545i	$0.191720\pi$
$992$	1.08457	0.0344350
$993$	0	0
$994$	41.1767	1.30605
$995$	−59.2629	−1.87876
$996$	0	0
$997$	17.3273	0.548760	0.274380	−	0.961621i	$-0.411527\pi$
$997$	17.3273	0.548760	0.274380	+	0.961621i	$0.411527\pi$
$998$	−9.40868	−0.297827
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.l.1.18		21
3.2	odd	2		671.2.a.d.1.4	✓	21
33.32	even	2		7381.2.a.j.1.18		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.d.1.4	✓	21	3.2	odd	2
6039.2.a.l.1.18		21	1.1	even	1	trivial
7381.2.a.j.1.18		21	33.32	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+2.40042 q^{2} +3.76203 q^{4} +3.90458 q^{5} -3.05635 q^{7} +4.22962 q^{8} +O(q^{10})\)	\(q+2.40042 q^{2} +3.76203 q^{4} +3.90458 q^{5} -3.05635 q^{7} +4.22962 q^{8} +9.37264 q^{10} +1.00000 q^{11} -5.54437 q^{13} -7.33653 q^{14} +2.62881 q^{16} +5.45989 q^{17} +5.83487 q^{19} +14.6891 q^{20} +2.40042 q^{22} +7.83643 q^{23} +10.2458 q^{25} -13.3088 q^{26} -11.4981 q^{28} +3.45552 q^{29} -0.504688 q^{31} -2.14898 q^{32} +13.1060 q^{34} -11.9338 q^{35} -5.34533 q^{37} +14.0062 q^{38} +16.5149 q^{40} +12.1681 q^{41} -4.71869 q^{43} +3.76203 q^{44} +18.8107 q^{46} -3.26502 q^{47} +2.34126 q^{49} +24.5941 q^{50} -20.8581 q^{52} -7.66741 q^{53} +3.90458 q^{55} -12.9272 q^{56} +8.29470 q^{58} -2.01048 q^{59} +1.00000 q^{61} -1.21146 q^{62} -10.4161 q^{64} -21.6484 q^{65} +5.54815 q^{67} +20.5403 q^{68} -28.6461 q^{70} -5.61256 q^{71} +4.97921 q^{73} -12.8311 q^{74} +21.9510 q^{76} -3.05635 q^{77} +2.08461 q^{79} +10.2644 q^{80} +29.2086 q^{82} +4.20087 q^{83} +21.3186 q^{85} -11.3269 q^{86} +4.22962 q^{88} +4.09744 q^{89} +16.9455 q^{91} +29.4809 q^{92} -7.83742 q^{94} +22.7827 q^{95} +7.15422 q^{97} +5.62002 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8} + q^{10} + 21 q^{11} + 20 q^{13} - 17 q^{14} + 50 q^{16} - q^{17} + 15 q^{19} + 2 q^{20} - 11 q^{23} + 48 q^{25} + 5 q^{26} - 16 q^{28} + 9 q^{29} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 21 q^{37} - 11 q^{38} - 16 q^{40} - 7 q^{41} + 16 q^{43} + 32 q^{44} - 3 q^{46} - 5 q^{47} + 80 q^{49} + 33 q^{50} + 60 q^{52} - 9 q^{53} - 7 q^{55} - 44 q^{56} - 27 q^{58} - 13 q^{59} + 21 q^{61} + 23 q^{62} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 33 q^{70} - 12 q^{71} + 20 q^{73} + 12 q^{74} + 59 q^{76} + 5 q^{77} + q^{79} + 38 q^{80} + 7 q^{82} + 19 q^{83} + 38 q^{85} + 3 q^{86} + 6 q^{88} - 37 q^{89} + 24 q^{91} - 31 q^{92} - 64 q^{94} + 43 q^{95} + 68 q^{97} + 2 q^{98}+O(q^{100}) \) 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8 + q^10 + 21 * q^11 + 20 * q^13 - 17 * q^14 + 50 * q^16 - q^17 + 15 * q^19 + 2 * q^20 - 11 * q^23 + 48 * q^25 + 5 * q^26 - 16 * q^28 + 9 * q^29 + 22 * q^31 - 3 * q^32 + 33 * q^34 + 39 * q^35 + 21 * q^37 - 11 * q^38 - 16 * q^40 - 7 * q^41 + 16 * q^43 + 32 * q^44 - 3 * q^46 - 5 * q^47 + 80 * q^49 + 33 * q^50 + 60 * q^52 - 9 * q^53 - 7 * q^55 - 44 * q^56 - 27 * q^58 - 13 * q^59 + 21 * q^61 + 23 * q^62 + 66 * q^64 - 25 * q^65 + 38 * q^67 + 74 * q^68 - 33 * q^70 - 12 * q^71 + 20 * q^73 + 12 * q^74 + 59 * q^76 + 5 * q^77 + q^79 + 38 * q^80 + 7 * q^82 + 19 * q^83 + 38 * q^85 + 3 * q^86 + 6 * q^88 - 37 * q^89 + 24 * q^91 - 31 * q^92 - 64 * q^94 + 43 * q^95 + 68 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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