LMFDB - Embedded newform 5586.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5586,2,Mod(1,5586)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5586, 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("5586.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5586.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:	$44.6044345691$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 798)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5586.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} +4.00000 q^{29} -1.00000 q^{30} +1.00000 q^{32} +1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} +1.00000 q^{38} +6.00000 q^{39} +1.00000 q^{40} -12.0000 q^{41} -7.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} -9.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} -2.00000 q^{51} -6.00000 q^{52} -1.00000 q^{54} -1.00000 q^{55} -1.00000 q^{57} +4.00000 q^{58} -2.00000 q^{59} -1.00000 q^{60} -3.00000 q^{61} +1.00000 q^{64} -6.00000 q^{65} +1.00000 q^{66} -2.00000 q^{67} +2.00000 q^{68} +1.00000 q^{69} +2.00000 q^{71} +1.00000 q^{72} +1.00000 q^{73} +8.00000 q^{74} +4.00000 q^{75} +1.00000 q^{76} +6.00000 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} -1.00000 q^{83} +2.00000 q^{85} -7.00000 q^{86} -4.00000 q^{87} -1.00000 q^{88} -18.0000 q^{89} +1.00000 q^{90} -1.00000 q^{92} -9.00000 q^{94} +1.00000 q^{95} -1.00000 q^{96} +4.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	−1.00000	−0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	1.00000	0.162221
$39$	6.00000	0.960769
$40$	1.00000	0.158114
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	−1.00000	−0.150756
$45$	1.00000	0.149071
$46$	−1.00000	−0.147442
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−2.00000	−0.280056
$52$	−6.00000	−0.832050
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−1.00000	−0.136083
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−1.00000	−0.132453
$58$	4.00000	0.525226
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	−1.00000	−0.129099
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.00000	−0.744208
$66$	1.00000	0.123091
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	2.00000	0.242536
$69$	1.00000	0.120386
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	1.00000	0.117851
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	8.00000	0.929981
$75$	4.00000	0.461880
$76$	1.00000	0.114708
$77$	0	0
$78$	6.00000	0.679366
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−12.0000	−1.32518
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	−7.00000	−0.754829
$87$	−4.00000	−0.428845
$88$	−1.00000	−0.106600
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−9.00000	−0.928279
$95$	1.00000	0.102598
$96$	−1.00000	−0.102062
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	−4.00000	−0.400000
$101$	−17.0000	−1.69156	−0.845782	−	0.533529i	$-0.820865\pi$
$101$	−17.0000	−1.69156	−0.845782	+	0.533529i	$0.820865\pi$
$102$	−2.00000	−0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−1.00000	−0.0962250
$109$	−20.0000	−1.91565	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000	−1.91565	−0.957826	+	0.287348i	$0.907226\pi$
$110$	−1.00000	−0.0953463
$111$	−8.00000	−0.759326
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	−1.00000	−0.0936586
$115$	−1.00000	−0.0932505
$116$	4.00000	0.371391
$117$	−6.00000	−0.554700
$118$	−2.00000	−0.184115
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−10.0000	−0.909091
$122$	−3.00000	−0.271607
$123$	12.0000	1.08200
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	1.00000	0.0883883
$129$	7.00000	0.616316
$130$	−6.00000	−0.526235
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−2.00000	−0.172774
$135$	−1.00000	−0.0860663
$136$	2.00000	0.171499
$137$	−11.0000	−0.939793	−0.469897	−	0.882721i	$-0.655709\pi$
$137$	−11.0000	−0.939793	−0.469897	+	0.882721i	$0.655709\pi$
$138$	1.00000	0.0851257
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	2.00000	0.167836
$143$	6.00000	0.501745
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	1.00000	0.0827606
$147$	0	0
$148$	8.00000	0.657596
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	4.00000	0.326599
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	1.00000	0.0811107
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	6.00000	0.480384
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	12.0000	0.954669
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	−12.0000	−0.937043
$165$	1.00000	0.0778499
$166$	−1.00000	−0.0776151
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	2.00000	0.153393
$171$	1.00000	0.0764719
$172$	−7.00000	−0.533745
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	2.00000	0.150329
$178$	−18.0000	−1.34916
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	1.00000	0.0745356
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	3.00000	0.221766
$184$	−1.00000	−0.0737210
$185$	8.00000	0.588172
$186$	0	0
$187$	−2.00000	−0.146254
$188$	−9.00000	−0.656392
$189$	0	0
$190$	1.00000	0.0725476
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	−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	$0.477068\pi$
$194$	4.00000	0.287183
$195$	6.00000	0.429669
$196$	0	0
$197$	17.0000	1.21120	0.605600	−	0.795769i	$-0.292933\pi$
$197$	17.0000	1.21120	0.605600	+	0.795769i	$0.292933\pi$
$198$	−1.00000	−0.0710669
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	−4.00000	−0.282843
$201$	2.00000	0.141069
$202$	−17.0000	−1.19612
$203$	0	0
$204$	−2.00000	−0.140028
$205$	−12.0000	−0.838116
$206$	8.00000	0.557386
$207$	−1.00000	−0.0695048
$208$	−6.00000	−0.416025
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	0	0
$213$	−2.00000	−0.137038
$214$	0	0
$215$	−7.00000	−0.477396
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−20.0000	−1.35457
$219$	−1.00000	−0.0675737
$220$	−1.00000	−0.0674200
$221$	−12.0000	−0.807207
$222$	−8.00000	−0.536925
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	4.00000	0.266076
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−1.00000	−0.0662266
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	4.00000	0.262613
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	−6.00000	−0.392232
$235$	−9.00000	−0.587095
$236$	−2.00000	−0.130189
$237$	−12.0000	−0.779484
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	−1.00000	−0.0645497
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	−10.0000	−0.642824
$243$	−1.00000	−0.0641500
$244$	−3.00000	−0.192055
$245$	0	0
$246$	12.0000	0.765092
$247$	−6.00000	−0.381771
$248$	0	0
$249$	1.00000	0.0633724
$250$	−9.00000	−0.569210
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	−18.0000	−1.12942
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	7.00000	0.435801
$259$	0	0
$260$	−6.00000	−0.372104
$261$	4.00000	0.247594
$262$	4.00000	0.247121
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	0	0
$267$	18.0000	1.10158
$268$	−2.00000	−0.122169
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	−1.00000	−0.0608581
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−11.0000	−0.664534
$275$	4.00000	0.241209
$276$	1.00000	0.0601929
$277$	−13.0000	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$278$	19.0000	1.13954
$279$	0	0
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	9.00000	0.535942
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	2.00000	0.118678
$285$	−1.00000	−0.0592349
$286$	6.00000	0.354787
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	4.00000	0.234888
$291$	−4.00000	−0.234484
$292$	1.00000	0.0585206
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	−2.00000	−0.116445
$296$	8.00000	0.464991
$297$	1.00000	0.0580259
$298$	15.0000	0.868927
$299$	6.00000	0.346989
$300$	4.00000	0.230940
$301$	0	0
$302$	−8.00000	−0.460348
$303$	17.0000	0.976624
$304$	1.00000	0.0573539
$305$	−3.00000	−0.171780
$306$	2.00000	0.114332
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	6.00000	0.339683
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	12.0000	0.675053
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	2.00000	0.111283
$324$	1.00000	0.0555556
$325$	24.0000	1.33128
$326$	5.00000	0.276924
$327$	20.0000	1.10600
$328$	−12.0000	−0.662589
$329$	0	0
$330$	1.00000	0.0550482
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	−1.00000	−0.0548821
$333$	8.00000	0.438397
$334$	−2.00000	−0.109435
$335$	−2.00000	−0.109272
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	23.0000	1.25104
$339$	−4.00000	−0.217250
$340$	2.00000	0.108465
$341$	0	0
$342$	1.00000	0.0540738
$343$	0	0
$344$	−7.00000	−0.377415
$345$	1.00000	0.0538382
$346$	4.00000	0.215041
$347$	15.0000	0.805242	0.402621	−	0.915367i	$-0.368099\pi$
$347$	15.0000	0.805242	0.402621	+	0.915367i	$0.368099\pi$
$348$	−4.00000	−0.214423
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	−1.00000	−0.0533002
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	2.00000	0.106299
$355$	2.00000	0.106149
$356$	−18.0000	−0.953998
$357$	0	0
$358$	−6.00000	−0.317110
$359$	−11.0000	−0.580558	−0.290279	−	0.956942i	$-0.593748\pi$
$359$	−11.0000	−0.580558	−0.290279	+	0.956942i	$0.593748\pi$
$360$	1.00000	0.0527046
$361$	1.00000	0.0526316
$362$	6.00000	0.315353
$363$	10.0000	0.524864
$364$	0	0
$365$	1.00000	0.0523424
$366$	3.00000	0.156813
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	−1.00000	−0.0521286
$369$	−12.0000	−0.624695
$370$	8.00000	0.415900
$371$	0	0
$372$	0	0
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	−2.00000	−0.103418
$375$	9.00000	0.464758
$376$	−9.00000	−0.464140
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	1.00000	0.0512989
$381$	18.0000	0.922168
$382$	−9.00000	−0.460480
$383$	26.0000	1.32854	0.664269	−	0.747494i	$-0.268743\pi$
$383$	26.0000	1.32854	0.664269	+	0.747494i	$0.268743\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	−7.00000	−0.355830
$388$	4.00000	0.203069
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	6.00000	0.303822
$391$	−2.00000	−0.101144
$392$	0	0
$393$	−4.00000	−0.201773
$394$	17.0000	0.856448
$395$	12.0000	0.603786
$396$	−1.00000	−0.0502519
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	−25.0000	−1.25314
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	2.00000	0.0997509
$403$	0	0
$404$	−17.0000	−0.845782
$405$	1.00000	0.0496904
$406$	0	0
$407$	−8.00000	−0.396545
$408$	−2.00000	−0.0990148
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	−12.0000	−0.592638
$411$	11.0000	0.542590
$412$	8.00000	0.394132
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	−1.00000	−0.0490881
$416$	−6.00000	−0.294174
$417$	−19.0000	−0.930434
$418$	−1.00000	−0.0489116
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	14.0000	0.681509
$423$	−9.00000	−0.437595
$424$	0	0
$425$	−8.00000	−0.388057
$426$	−2.00000	−0.0969003
$427$	0	0
$428$	0	0
$429$	−6.00000	−0.289683
$430$	−7.00000	−0.337570
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	−1.00000	−0.0481125
$433$	20.0000	0.961139	0.480569	−	0.876957i	$-0.340430\pi$
$433$	20.0000	0.961139	0.480569	+	0.876957i	$0.340430\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	−20.0000	−0.957826
$437$	−1.00000	−0.0478365
$438$	−1.00000	−0.0477818
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	−12.0000	−0.570782
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−8.00000	−0.379663
$445$	−18.0000	−0.853282
$446$	−10.0000	−0.473514
$447$	−15.0000	−0.709476
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	−4.00000	−0.188562
$451$	12.0000	0.565058
$452$	4.00000	0.188144
$453$	8.00000	0.375873
$454$	18.0000	0.844782
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−5.00000	−0.233890	−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000	−0.233890	−0.116945	+	0.993138i	$0.537310\pi$
$458$	−14.0000	−0.654177
$459$	−2.00000	−0.0933520
$460$	−1.00000	−0.0466252
$461$	−17.0000	−0.791769	−0.395884	−	0.918300i	$-0.629562\pi$
$461$	−17.0000	−0.791769	−0.395884	+	0.918300i	$0.629562\pi$
$462$	0	0
$463$	19.0000	0.883005	0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000	0.883005	0.441502	+	0.897260i	$0.354446\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	14.0000	0.648537
$467$	25.0000	1.15686	0.578431	−	0.815731i	$-0.303665\pi$
$467$	25.0000	1.15686	0.578431	+	0.815731i	$0.303665\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	−9.00000	−0.415139
$471$	13.0000	0.599008
$472$	−2.00000	−0.0920575
$473$	7.00000	0.321860
$474$	−12.0000	−0.551178
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	24.0000	1.09773
$479$	29.0000	1.32504	0.662522	−	0.749043i	$-0.269486\pi$
$479$	29.0000	1.32504	0.662522	+	0.749043i	$0.269486\pi$
$480$	−1.00000	−0.0456435
$481$	−48.0000	−2.18861
$482$	8.00000	0.364390
$483$	0	0
$484$	−10.0000	−0.454545
$485$	4.00000	0.181631
$486$	−1.00000	−0.0453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−3.00000	−0.135804
$489$	−5.00000	−0.226108
$490$	0	0
$491$	−21.0000	−0.947717	−0.473858	−	0.880601i	$-0.657139\pi$
$491$	−21.0000	−0.947717	−0.473858	+	0.880601i	$0.657139\pi$
$492$	12.0000	0.541002
$493$	8.00000	0.360302
$494$	−6.00000	−0.269953
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	0	0
$498$	1.00000	0.0448111
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	−9.00000	−0.402492
$501$	2.00000	0.0893534
$502$	−21.0000	−0.937276
$503$	−33.0000	−1.47140	−0.735699	−	0.677309i	$-0.763146\pi$
$503$	−33.0000	−1.47140	−0.735699	+	0.677309i	$0.763146\pi$
$504$	0	0
$505$	−17.0000	−0.756490
$506$	1.00000	0.0444554
$507$	−23.0000	−1.02147
$508$	−18.0000	−0.798621
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	−2.00000	−0.0885615
$511$	0	0
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	6.00000	0.264649
$515$	8.00000	0.352522
$516$	7.00000	0.308158
$517$	9.00000	0.395820
$518$	0	0
$519$	−4.00000	−0.175581
$520$	−6.00000	−0.263117
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	4.00000	0.175075
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−12.0000	−0.523225
$527$	0	0
$528$	1.00000	0.0435194
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−2.00000	−0.0867926
$532$	0	0
$533$	72.0000	3.11867
$534$	18.0000	0.778936
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	6.00000	0.258919
$538$	−20.0000	−0.862261
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	−15.0000	−0.644900	−0.322450	−	0.946586i	$-0.604506\pi$
$541$	−15.0000	−0.644900	−0.322450	+	0.946586i	$0.604506\pi$
$542$	−15.0000	−0.644305
$543$	−6.00000	−0.257485
$544$	2.00000	0.0857493
$545$	−20.0000	−0.856706
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	−11.0000	−0.469897
$549$	−3.00000	−0.128037
$550$	4.00000	0.170561
$551$	4.00000	0.170406
$552$	1.00000	0.0425628
$553$	0	0
$554$	−13.0000	−0.552317
$555$	−8.00000	−0.339581
$556$	19.0000	0.805779
$557$	−29.0000	−1.22877	−0.614385	−	0.789007i	$-0.710596\pi$
$557$	−29.0000	−1.22877	−0.614385	+	0.789007i	$0.710596\pi$
$558$	0	0
$559$	42.0000	1.77641
$560$	0	0
$561$	2.00000	0.0844401
$562$	20.0000	0.843649
$563$	38.0000	1.60151	0.800755	−	0.598993i	$-0.204432\pi$
$563$	38.0000	1.60151	0.800755	+	0.598993i	$0.204432\pi$
$564$	9.00000	0.378968
$565$	4.00000	0.168281
$566$	1.00000	0.0420331
$567$	0	0
$568$	2.00000	0.0839181
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	−1.00000	−0.0418854
$571$	3.00000	0.125546	0.0627730	−	0.998028i	$-0.480006\pi$
$571$	3.00000	0.125546	0.0627730	+	0.998028i	$0.480006\pi$
$572$	6.00000	0.250873
$573$	9.00000	0.375980
$574$	0	0
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	31.0000	1.29055	0.645273	−	0.763952i	$-0.276743\pi$
$577$	31.0000	1.29055	0.645273	+	0.763952i	$0.276743\pi$
$578$	−13.0000	−0.540729
$579$	−2.00000	−0.0831172
$580$	4.00000	0.166091
$581$	0	0
$582$	−4.00000	−0.165805
$583$	0	0
$584$	1.00000	0.0413803
$585$	−6.00000	−0.248069
$586$	12.0000	0.495715
$587$	44.0000	1.81607	0.908037	−	0.418890i	$-0.137581\pi$
$587$	44.0000	1.81607	0.908037	+	0.418890i	$0.137581\pi$
$588$	0	0
$589$	0	0
$590$	−2.00000	−0.0823387
$591$	−17.0000	−0.699287
$592$	8.00000	0.328798
$593$	−9.00000	−0.369586	−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000	−0.369586	−0.184793	+	0.982777i	$0.559161\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	15.0000	0.614424
$597$	25.0000	1.02318
$598$	6.00000	0.245358
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	4.00000	0.163299
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	−8.00000	−0.325515
$605$	−10.0000	−0.406558
$606$	17.0000	0.690578
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−3.00000	−0.121466
$611$	54.0000	2.18461
$612$	2.00000	0.0808452
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−32.0000	−1.29141
$615$	12.0000	0.483887
$616$	0	0
$617$	41.0000	1.65060	0.825299	−	0.564696i	$-0.191007\pi$
$617$	41.0000	1.65060	0.825299	+	0.564696i	$0.191007\pi$
$618$	−8.00000	−0.321807
$619$	7.00000	0.281354	0.140677	−	0.990056i	$-0.455072\pi$
$619$	7.00000	0.281354	0.140677	+	0.990056i	$0.455072\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	16.0000	0.641542
$623$	0	0
$624$	6.00000	0.240192
$625$	11.0000	0.440000
$626$	19.0000	0.759393
$627$	1.00000	0.0399362
$628$	−13.0000	−0.518756
$629$	16.0000	0.637962
$630$	0	0
$631$	35.0000	1.39333	0.696664	−	0.717398i	$-0.254667\pi$
$631$	35.0000	1.39333	0.696664	+	0.717398i	$0.254667\pi$
$632$	12.0000	0.477334
$633$	−14.0000	−0.556450
$634$	−18.0000	−0.714871
$635$	−18.0000	−0.714308
$636$	0	0
$637$	0	0
$638$	−4.00000	−0.158362
$639$	2.00000	0.0791188
$640$	1.00000	0.0395285
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	0	0
$645$	7.00000	0.275625
$646$	2.00000	0.0786889
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	1.00000	0.0392837
$649$	2.00000	0.0785069
$650$	24.0000	0.941357
$651$	0	0
$652$	5.00000	0.195815
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	20.0000	0.782062
$655$	4.00000	0.156293
$656$	−12.0000	−0.468521
$657$	1.00000	0.0390137
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	1.00000	0.0389249
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−10.0000	−0.388661
$663$	12.0000	0.466041
$664$	−1.00000	−0.0388075
$665$	0	0
$666$	8.00000	0.309994
$667$	−4.00000	−0.154881
$668$	−2.00000	−0.0773823
$669$	10.0000	0.386622
$670$	−2.00000	−0.0772667
$671$	3.00000	0.115814
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−6.00000	−0.231111
$675$	4.00000	0.153960
$676$	23.0000	0.884615
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	−4.00000	−0.153619
$679$	0	0
$680$	2.00000	0.0766965
$681$	−18.0000	−0.689761
$682$	0	0
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	1.00000	0.0382360
$685$	−11.0000	−0.420288
$686$	0	0
$687$	14.0000	0.534133
$688$	−7.00000	−0.266872
$689$	0	0
$690$	1.00000	0.0380693
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	4.00000	0.152057
$693$	0	0
$694$	15.0000	0.569392
$695$	19.0000	0.720711
$696$	−4.00000	−0.151620
$697$	−24.0000	−0.909065
$698$	−14.0000	−0.529908
$699$	−14.0000	−0.529529
$700$	0	0
$701$	−41.0000	−1.54855	−0.774274	−	0.632850i	$-0.781885\pi$
$701$	−41.0000	−1.54855	−0.774274	+	0.632850i	$0.781885\pi$
$702$	6.00000	0.226455
$703$	8.00000	0.301726
$704$	−1.00000	−0.0376889
$705$	9.00000	0.338960
$706$	−18.0000	−0.677439
$707$	0	0
$708$	2.00000	0.0751646
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	2.00000	0.0750587
$711$	12.0000	0.450035
$712$	−18.0000	−0.674579
$713$	0	0
$714$	0	0
$715$	6.00000	0.224387
$716$	−6.00000	−0.224231
$717$	−24.0000	−0.896296
$718$	−11.0000	−0.410516
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	1.00000	0.0372161
$723$	−8.00000	−0.297523
$724$	6.00000	0.222988
$725$	−16.0000	−0.594225
$726$	10.0000	0.371135
$727$	25.0000	0.927199	0.463599	−	0.886045i	$-0.346558\pi$
$727$	25.0000	0.927199	0.463599	+	0.886045i	$0.346558\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	1.00000	0.0370117
$731$	−14.0000	−0.517809
$732$	3.00000	0.110883
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	20.0000	0.738213
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	2.00000	0.0736709
$738$	−12.0000	−0.441726
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	8.00000	0.294086
$741$	6.00000	0.220416
$742$	0	0
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	16.0000	0.585802
$747$	−1.00000	−0.0365881
$748$	−2.00000	−0.0731272
$749$	0	0
$750$	9.00000	0.328634
$751$	18.0000	0.656829	0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000	0.656829	0.328415	+	0.944534i	$0.393486\pi$
$752$	−9.00000	−0.328196
$753$	21.0000	0.765283
$754$	−24.0000	−0.874028
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−51.0000	−1.85363	−0.926813	−	0.375523i	$-0.877463\pi$
$757$	−51.0000	−1.85363	−0.926813	+	0.375523i	$0.877463\pi$
$758$	−32.0000	−1.16229
$759$	−1.00000	−0.0362977
$760$	1.00000	0.0362738
$761$	−31.0000	−1.12375	−0.561875	−	0.827222i	$-0.689920\pi$
$761$	−31.0000	−1.12375	−0.561875	+	0.827222i	$0.689920\pi$
$762$	18.0000	0.652071
$763$	0	0
$764$	−9.00000	−0.325609
$765$	2.00000	0.0723102
$766$	26.0000	0.939418
$767$	12.0000	0.433295
$768$	−1.00000	−0.0360844
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	2.00000	0.0719816
$773$	−20.0000	−0.719350	−0.359675	−	0.933078i	$-0.617112\pi$
$773$	−20.0000	−0.719350	−0.359675	+	0.933078i	$0.617112\pi$
$774$	−7.00000	−0.251610
$775$	0	0
$776$	4.00000	0.143592
$777$	0	0
$778$	−38.0000	−1.36237
$779$	−12.0000	−0.429945
$780$	6.00000	0.214834
$781$	−2.00000	−0.0715656
$782$	−2.00000	−0.0715199
$783$	−4.00000	−0.142948
$784$	0	0
$785$	−13.0000	−0.463990
$786$	−4.00000	−0.142675
$787$	−30.0000	−1.06938	−0.534692	−	0.845047i	$-0.679572\pi$
$787$	−30.0000	−1.06938	−0.534692	+	0.845047i	$0.679572\pi$
$788$	17.0000	0.605600
$789$	12.0000	0.427211
$790$	12.0000	0.426941
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	18.0000	0.639199
$794$	−30.0000	−1.06466
$795$	0	0
$796$	−25.0000	−0.886102
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	−18.0000	−0.636794
$800$	−4.00000	−0.141421
$801$	−18.0000	−0.635999
$802$	−30.0000	−1.05934
$803$	−1.00000	−0.0352892
$804$	2.00000	0.0705346
$805$	0	0
$806$	0	0
$807$	20.0000	0.704033
$808$	−17.0000	−0.598058
$809$	19.0000	0.668004	0.334002	−	0.942572i	$-0.391601\pi$
$809$	19.0000	0.668004	0.334002	+	0.942572i	$0.391601\pi$
$810$	1.00000	0.0351364
$811$	46.0000	1.61528	0.807639	−	0.589677i	$-0.200745\pi$
$811$	46.0000	1.61528	0.807639	+	0.589677i	$0.200745\pi$
$812$	0	0
$813$	15.0000	0.526073
$814$	−8.00000	−0.280400
$815$	5.00000	0.175142
$816$	−2.00000	−0.0700140
$817$	−7.00000	−0.244899
$818$	14.0000	0.489499
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−25.0000	−0.872506	−0.436253	−	0.899824i	$-0.643695\pi$
$821$	−25.0000	−0.872506	−0.436253	+	0.899824i	$0.643695\pi$
$822$	11.0000	0.383669
$823$	13.0000	0.453152	0.226576	−	0.973994i	$-0.427247\pi$
$823$	13.0000	0.453152	0.226576	+	0.973994i	$0.427247\pi$
$824$	8.00000	0.278693
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−1.00000	−0.0347524
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	−1.00000	−0.0347105
$831$	13.0000	0.450965
$832$	−6.00000	−0.208013
$833$	0	0
$834$	−19.0000	−0.657916
$835$	−2.00000	−0.0692129
$836$	−1.00000	−0.0345857
$837$	0	0
$838$	15.0000	0.518166
$839$	50.0000	1.72619	0.863096	−	0.505040i	$-0.168522\pi$
$839$	50.0000	1.72619	0.863096	+	0.505040i	$0.168522\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−18.0000	−0.620321
$843$	−20.0000	−0.688837
$844$	14.0000	0.481900
$845$	23.0000	0.791224
$846$	−9.00000	−0.309426
$847$	0	0
$848$	0	0
$849$	−1.00000	−0.0343199
$850$	−8.00000	−0.274398
$851$	−8.00000	−0.274236
$852$	−2.00000	−0.0685189
$853$	29.0000	0.992941	0.496471	−	0.868054i	$-0.334629\pi$
$853$	29.0000	0.992941	0.496471	+	0.868054i	$0.334629\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	−6.00000	−0.204837
$859$	−7.00000	−0.238837	−0.119418	−	0.992844i	$-0.538103\pi$
$859$	−7.00000	−0.238837	−0.119418	+	0.992844i	$0.538103\pi$
$860$	−7.00000	−0.238698
$861$	0	0
$862$	18.0000	0.613082
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	−1.00000	−0.0340207
$865$	4.00000	0.136004
$866$	20.0000	0.679628
$867$	13.0000	0.441503
$868$	0	0
$869$	−12.0000	−0.407072
$870$	−4.00000	−0.135613
$871$	12.0000	0.406604
$872$	−20.0000	−0.677285
$873$	4.00000	0.135379
$874$	−1.00000	−0.0338255
$875$	0	0
$876$	−1.00000	−0.0337869
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	6.00000	0.202490
$879$	−12.0000	−0.404750
$880$	−1.00000	−0.0337100
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	−12.0000	−0.403604
$885$	2.00000	0.0672293
$886$	−4.00000	−0.134383
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	−8.00000	−0.268462
$889$	0	0
$890$	−18.0000	−0.603361
$891$	−1.00000	−0.0335013
$892$	−10.0000	−0.334825
$893$	−9.00000	−0.301174
$894$	−15.0000	−0.501675
$895$	−6.00000	−0.200558
$896$	0	0
$897$	−6.00000	−0.200334
$898$	−20.0000	−0.667409
$899$	0	0
$900$	−4.00000	−0.133333
$901$	0	0
$902$	12.0000	0.399556
$903$	0	0
$904$	4.00000	0.133038
$905$	6.00000	0.199447
$906$	8.00000	0.265782
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	18.0000	0.597351
$909$	−17.0000	−0.563854
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	−1.00000	−0.0331133
$913$	1.00000	0.0330952
$914$	−5.00000	−0.165385
$915$	3.00000	0.0991769
$916$	−14.0000	−0.462573
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	−23.0000	−0.758700	−0.379350	−	0.925253i	$-0.623852\pi$
$919$	−23.0000	−0.758700	−0.379350	+	0.925253i	$0.623852\pi$
$920$	−1.00000	−0.0329690
$921$	32.0000	1.05444
$922$	−17.0000	−0.559865
$923$	−12.0000	−0.394985
$924$	0	0
$925$	−32.0000	−1.05215
$926$	19.0000	0.624379
$927$	8.00000	0.262754
$928$	4.00000	0.131306
$929$	3.00000	0.0984268	0.0492134	−	0.998788i	$-0.484329\pi$
$929$	3.00000	0.0984268	0.0492134	+	0.998788i	$0.484329\pi$
$930$	0	0
$931$	0	0
$932$	14.0000	0.458585
$933$	−16.0000	−0.523816
$934$	25.0000	0.818025
$935$	−2.00000	−0.0654070
$936$	−6.00000	−0.196116
$937$	17.0000	0.555366	0.277683	−	0.960673i	$-0.410434\pi$
$937$	17.0000	0.555366	0.277683	+	0.960673i	$0.410434\pi$
$938$	0	0
$939$	−19.0000	−0.620042
$940$	−9.00000	−0.293548
$941$	58.0000	1.89075	0.945373	−	0.325991i	$-0.105698\pi$
$941$	58.0000	1.89075	0.945373	+	0.325991i	$0.105698\pi$
$942$	13.0000	0.423563
$943$	12.0000	0.390774
$944$	−2.00000	−0.0650945
$945$	0	0
$946$	7.00000	0.227590
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	−12.0000	−0.389742
$949$	−6.00000	−0.194768
$950$	−4.00000	−0.129777
$951$	18.0000	0.583690
$952$	0	0
$953$	20.0000	0.647864	0.323932	−	0.946080i	$-0.394995\pi$
$953$	20.0000	0.647864	0.323932	+	0.946080i	$0.394995\pi$
$954$	0	0
$955$	−9.00000	−0.291233
$956$	24.0000	0.776215
$957$	4.00000	0.129302
$958$	29.0000	0.936947
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	−31.0000	−1.00000
$962$	−48.0000	−1.54758
$963$	0	0
$964$	8.00000	0.257663
$965$	2.00000	0.0643823
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−10.0000	−0.321412
$969$	−2.00000	−0.0642493
$970$	4.00000	0.128432
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	2.00000	0.0640841
$975$	−24.0000	−0.768615
$976$	−3.00000	−0.0960277
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	−5.00000	−0.159882
$979$	18.0000	0.575282
$980$	0	0
$981$	−20.0000	−0.638551
$982$	−21.0000	−0.670137
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	12.0000	0.382546
$985$	17.0000	0.541665
$986$	8.00000	0.254772
$987$	0	0
$988$	−6.00000	−0.190885
$989$	7.00000	0.222587
$990$	−1.00000	−0.0317821
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	0	0
$993$	10.0000	0.317340
$994$	0	0
$995$	−25.0000	−0.792553
$996$	1.00000	0.0316862
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	−5.00000	−0.158272
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5586.2.a.v.1.1		1
7.2	even	3		798.2.j.a.571.1	yes	2
7.4	even	3		798.2.j.a.457.1	✓	2
7.6	odd	2		5586.2.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.j.a.457.1	✓	2	7.4	even	3
798.2.j.a.571.1	yes	2	7.2	even	3
5586.2.a.v.1.1		1	1.1	even	1	trivial
5586.2.a.ba.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5586.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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