LMFDB - Embedded newform 6909.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6909,2,Mod(1,6909)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6909, 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("6909.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6909 = 3 \cdot 7^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6909.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:	$55.1686427565$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 141)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6909.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} -4.00000 q^{16} -2.00000 q^{17} +2.00000 q^{18} -6.00000 q^{19} +2.00000 q^{20} +2.00000 q^{22} +3.00000 q^{23} -4.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} -2.00000 q^{30} -2.00000 q^{31} -8.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +2.00000 q^{36} -7.00000 q^{37} -12.0000 q^{38} -2.00000 q^{39} -10.0000 q^{41} -10.0000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} +1.00000 q^{47} +4.00000 q^{48} -8.00000 q^{50} +2.00000 q^{51} +4.00000 q^{52} +4.00000 q^{53} -2.00000 q^{54} +1.00000 q^{55} +6.00000 q^{57} +6.00000 q^{58} -8.00000 q^{59} -2.00000 q^{60} +10.0000 q^{61} -4.00000 q^{62} -8.00000 q^{64} +2.00000 q^{65} -2.00000 q^{66} +10.0000 q^{67} -4.00000 q^{68} -3.00000 q^{69} -14.0000 q^{71} +10.0000 q^{73} -14.0000 q^{74} +4.00000 q^{75} -12.0000 q^{76} -4.00000 q^{78} +17.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -20.0000 q^{82} -8.00000 q^{83} -2.00000 q^{85} -20.0000 q^{86} -3.00000 q^{87} -6.00000 q^{89} +2.00000 q^{90} +6.00000 q^{92} +2.00000 q^{93} +2.00000 q^{94} -6.00000 q^{95} +8.00000 q^{96} -1.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−2.00000	−0.816497
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−2.00000	−0.577350
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	−4.00000	−1.00000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	2.00000	0.471405
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	2.00000	0.426401
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−2.00000	−0.365148
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−8.00000	−1.41421
$33$	−1.00000	−0.174078
$34$	−4.00000	−0.685994
$35$	0	0
$36$	2.00000	0.333333
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−12.0000	−1.94666
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	2.00000	0.301511
$45$	1.00000	0.149071
$46$	6.00000	0.884652
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	4.00000	0.577350
$49$	0	0
$50$	−8.00000	−1.13137
$51$	2.00000	0.280056
$52$	4.00000	0.554700
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	−2.00000	−0.272166
$55$	1.00000	0.134840
$56$	0	0
$57$	6.00000	0.794719
$58$	6.00000	0.787839
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	−2.00000	−0.258199
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	−8.00000	−1.00000
$65$	2.00000	0.248069
$66$	−2.00000	−0.246183
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	−4.00000	−0.485071
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−14.0000	−1.62747
$75$	4.00000	0.461880
$76$	−12.0000	−1.37649
$77$	0	0
$78$	−4.00000	−0.452911
$79$	17.0000	1.91265	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	17.0000	1.91265	0.956325	+	0.292306i	$0.0944227\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−20.0000	−2.20863
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	−20.0000	−2.15666
$87$	−3.00000	−0.321634
$88$	0	0
$89$	−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
$91$	0	0
$92$	6.00000	0.625543
$93$	2.00000	0.207390
$94$	2.00000	0.206284
$95$	−6.00000	−0.615587
$96$	8.00000	0.816497
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−8.00000	−0.800000
$101$	16.0000	1.59206	0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000	1.59206	0.796030	+	0.605257i	$0.206930\pi$
$102$	4.00000	0.396059
$103$	−11.0000	−1.08386	−0.541931	−	0.840423i	$-0.682307\pi$
$103$	−11.0000	−1.08386	−0.541931	+	0.840423i	$0.682307\pi$
$104$	0	0
$105$	0	0
$106$	8.00000	0.777029
$107$	5.00000	0.483368	0.241684	−	0.970355i	$-0.422300\pi$
$107$	5.00000	0.483368	0.241684	+	0.970355i	$0.422300\pi$
$108$	−2.00000	−0.192450
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	2.00000	0.190693
$111$	7.00000	0.664411
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	12.0000	1.12390
$115$	3.00000	0.279751
$116$	6.00000	0.557086
$117$	2.00000	0.184900
$118$	−16.0000	−1.47292
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	20.0000	1.81071
$123$	10.0000	0.901670
$124$	−4.00000	−0.359211
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	4.00000	0.350823
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	20.0000	1.72774
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	−6.00000	−0.510754
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	−1.00000	−0.0842152
$142$	−28.0000	−2.34971
$143$	2.00000	0.167248
$144$	−4.00000	−0.333333
$145$	3.00000	0.249136
$146$	20.0000	1.65521
$147$	0	0
$148$	−14.0000	−1.15079
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	8.00000	0.653197
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−2.00000	−0.160644
$156$	−4.00000	−0.320256
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	34.0000	2.70489
$159$	−4.00000	−0.317221
$160$	−8.00000	−0.632456
$161$	0	0
$162$	2.00000	0.157135
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−20.0000	−1.56174
$165$	−1.00000	−0.0778499
$166$	−16.0000	−1.24184
$167$	−21.0000	−1.62503	−0.812514	−	0.582941i	$-0.801902\pi$
$167$	−21.0000	−1.62503	−0.812514	+	0.582941i	$0.801902\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−4.00000	−0.306786
$171$	−6.00000	−0.458831
$172$	−20.0000	−1.52499
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−4.00000	−0.301511
$177$	8.00000	0.601317
$178$	−12.0000	−0.899438
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	2.00000	0.149071
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−7.00000	−0.514650
$186$	4.00000	0.293294
$187$	−2.00000	−0.146254
$188$	2.00000	0.145865
$189$	0	0
$190$	−12.0000	−0.870572
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	8.00000	0.577350
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−2.00000	−0.143592
$195$	−2.00000	−0.143223
$196$	0	0
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	2.00000	0.142134
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	32.0000	2.25151
$203$	0	0
$204$	4.00000	0.280056
$205$	−10.0000	−0.698430
$206$	−22.0000	−1.53281
$207$	3.00000	0.208514
$208$	−8.00000	−0.554700
$209$	−6.00000	−0.415029
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	8.00000	0.549442
$213$	14.0000	0.959264
$214$	10.0000	0.683586
$215$	−10.0000	−0.681994
$216$	0	0
$217$	0	0
$218$	12.0000	0.812743
$219$	−10.0000	−0.675737
$220$	2.00000	0.134840
$221$	−4.00000	−0.269069
$222$	14.0000	0.939618
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	−20.0000	−1.33038
$227$	−7.00000	−0.464606	−0.232303	−	0.972643i	$-0.574626\pi$
$227$	−7.00000	−0.464606	−0.232303	+	0.972643i	$0.574626\pi$
$228$	12.0000	0.794719
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	0	0
$233$	23.0000	1.50678	0.753390	−	0.657574i	$-0.228417\pi$
$233$	23.0000	1.50678	0.753390	+	0.657574i	$0.228417\pi$
$234$	4.00000	0.261488
$235$	1.00000	0.0652328
$236$	−16.0000	−1.04151
$237$	−17.0000	−1.10427
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	4.00000	0.258199
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	−20.0000	−1.28565
$243$	−1.00000	−0.0641500
$244$	20.0000	1.28037
$245$	0	0
$246$	20.0000	1.27515
$247$	−12.0000	−0.763542
$248$	0	0
$249$	8.00000	0.506979
$250$	−18.0000	−1.13842
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	−16.0000	−1.00393
$255$	2.00000	0.125245
$256$	16.0000	1.00000
$257$	31.0000	1.93373	0.966863	−	0.255294i	$-0.0821723\pi$
$257$	31.0000	1.93373	0.966863	+	0.255294i	$0.0821723\pi$
$258$	20.0000	1.24515
$259$	0	0
$260$	4.00000	0.248069
$261$	3.00000	0.185695
$262$	12.0000	0.741362
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	6.00000	0.367194
$268$	20.0000	1.22169
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	−2.00000	−0.121716
$271$	15.0000	0.911185	0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000	0.911185	0.455593	+	0.890188i	$0.349427\pi$
$272$	8.00000	0.485071
$273$	0	0
$274$	−44.0000	−2.65814
$275$	−4.00000	−0.241209
$276$	−6.00000	−0.361158
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	4.00000	0.239904
$279$	−2.00000	−0.119737
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	−2.00000	−0.119098
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	−28.0000	−1.66149
$285$	6.00000	0.355409
$286$	4.00000	0.236525
$287$	0	0
$288$	−8.00000	−0.471405
$289$	−13.0000	−0.764706
$290$	6.00000	0.352332
$291$	1.00000	0.0586210
$292$	20.0000	1.17041
$293$	−27.0000	−1.57736	−0.788678	−	0.614806i	$-0.789234\pi$
$293$	−27.0000	−1.57736	−0.788678	+	0.614806i	$0.789234\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	−24.0000	−1.39028
$299$	6.00000	0.346989
$300$	8.00000	0.461880
$301$	0	0
$302$	−28.0000	−1.61122
$303$	−16.0000	−0.919176
$304$	24.0000	1.37649
$305$	10.0000	0.572598
$306$	−4.00000	−0.228665
$307$	−23.0000	−1.31268	−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000	−1.31268	−0.656340	+	0.754466i	$0.727896\pi$
$308$	0	0
$309$	11.0000	0.625768
$310$	−4.00000	−0.227185
$311$	23.0000	1.30421	0.652105	−	0.758129i	$-0.273886\pi$
$311$	23.0000	1.30421	0.652105	+	0.758129i	$0.273886\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	34.0000	1.91265
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	−8.00000	−0.448618
$319$	3.00000	0.167968
$320$	−8.00000	−0.447214
$321$	−5.00000	−0.279073
$322$	0	0
$323$	12.0000	0.667698
$324$	2.00000	0.111111
$325$	−8.00000	−0.443760
$326$	−12.0000	−0.664619
$327$	−6.00000	−0.331801
$328$	0	0
$329$	0	0
$330$	−2.00000	−0.110096
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−16.0000	−0.878114
$333$	−7.00000	−0.383598
$334$	−42.0000	−2.29814
$335$	10.0000	0.546358
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−18.0000	−0.979071
$339$	10.0000	0.543125
$340$	−4.00000	−0.216930
$341$	−2.00000	−0.108306
$342$	−12.0000	−0.648886
$343$	0	0
$344$	0	0
$345$	−3.00000	−0.161515
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−6.00000	−0.321634
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−8.00000	−0.426401
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	16.0000	0.850390
$355$	−14.0000	−0.743043
$356$	−12.0000	−0.635999
$357$	0	0
$358$	10.0000	0.528516
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−24.0000	−1.26141
$363$	10.0000	0.524864
$364$	0	0
$365$	10.0000	0.523424
$366$	−20.0000	−1.04542
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−12.0000	−0.625543
$369$	−10.0000	−0.520579
$370$	−14.0000	−0.727825
$371$	0	0
$372$	4.00000	0.207390
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	−4.00000	−0.206835
$375$	9.00000	0.464758
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	35.0000	1.79783	0.898915	−	0.438124i	$-0.144357\pi$
$379$	35.0000	1.79783	0.898915	+	0.438124i	$0.144357\pi$
$380$	−12.0000	−0.615587
$381$	8.00000	0.409852
$382$	−36.0000	−1.84192
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	0	0
$386$	8.00000	0.407189
$387$	−10.0000	−0.508329
$388$	−2.00000	−0.101535
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−4.00000	−0.202548
$391$	−6.00000	−0.303433
$392$	0	0
$393$	−6.00000	−0.302660
$394$	16.0000	0.806068
$395$	17.0000	0.855363
$396$	2.00000	0.100504
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−52.0000	−2.60652
$399$	0	0
$400$	16.0000	0.800000
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	−20.0000	−0.997509
$403$	−4.00000	−0.199254
$404$	32.0000	1.59206
$405$	1.00000	0.0496904
$406$	0	0
$407$	−7.00000	−0.346977
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	−20.0000	−0.987730
$411$	22.0000	1.08518
$412$	−22.0000	−1.08386
$413$	0	0
$414$	6.00000	0.294884
$415$	−8.00000	−0.392705
$416$	−16.0000	−0.784465
$417$	−2.00000	−0.0979404
$418$	−12.0000	−0.586939
$419$	−25.0000	−1.22133	−0.610665	−	0.791889i	$-0.709098\pi$
$419$	−25.0000	−1.22133	−0.610665	+	0.791889i	$0.709098\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	32.0000	1.55774
$423$	1.00000	0.0486217
$424$	0	0
$425$	8.00000	0.388057
$426$	28.0000	1.35660
$427$	0	0
$428$	10.0000	0.483368
$429$	−2.00000	−0.0965609
$430$	−20.0000	−0.964486
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	4.00000	0.192450
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	12.0000	0.574696
$437$	−18.0000	−0.861057
$438$	−20.0000	−0.955637
$439$	15.0000	0.715911	0.357955	−	0.933739i	$-0.383474\pi$
$439$	15.0000	0.715911	0.357955	+	0.933739i	$0.383474\pi$
$440$	0	0
$441$	0	0
$442$	−8.00000	−0.380521
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	14.0000	0.664411
$445$	−6.00000	−0.284427
$446$	24.0000	1.13643
$447$	12.0000	0.567581
$448$	0	0
$449$	7.00000	0.330350	0.165175	−	0.986264i	$-0.447181\pi$
$449$	7.00000	0.330350	0.165175	+	0.986264i	$0.447181\pi$
$450$	−8.00000	−0.377124
$451$	−10.0000	−0.470882
$452$	−20.0000	−0.940721
$453$	14.0000	0.657777
$454$	−14.0000	−0.657053
$455$	0	0
$456$	0	0
$457$	−13.0000	−0.608114	−0.304057	−	0.952654i	$-0.598341\pi$
$457$	−13.0000	−0.608114	−0.304057	+	0.952654i	$0.598341\pi$
$458$	−8.00000	−0.373815
$459$	2.00000	0.0933520
$460$	6.00000	0.279751
$461$	−31.0000	−1.44381	−0.721907	−	0.691990i	$-0.756734\pi$
$461$	−31.0000	−1.44381	−0.721907	+	0.691990i	$0.756734\pi$
$462$	0	0
$463$	−28.0000	−1.30127	−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000	−1.30127	−0.650635	+	0.759390i	$0.725497\pi$
$464$	−12.0000	−0.557086
$465$	2.00000	0.0927478
$466$	46.0000	2.13091
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	2.00000	0.0922531
$471$	−3.00000	−0.138233
$472$	0	0
$473$	−10.0000	−0.459800
$474$	−34.0000	−1.56167
$475$	24.0000	1.10120
$476$	0	0
$477$	4.00000	0.183147
$478$	48.0000	2.19547
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	8.00000	0.365148
$481$	−14.0000	−0.638345
$482$	−22.0000	−1.00207
$483$	0	0
$484$	−20.0000	−0.909091
$485$	−1.00000	−0.0454077
$486$	−2.00000	−0.0907218
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	6.00000	0.271329
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	20.0000	0.901670
$493$	−6.00000	−0.270226
$494$	−24.0000	−1.07981
$495$	1.00000	0.0449467
$496$	8.00000	0.359211
$497$	0	0
$498$	16.0000	0.716977
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	−18.0000	−0.804984
$501$	21.0000	0.938211
$502$	−36.0000	−1.60676
$503$	21.0000	0.936344	0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000	0.936344	0.468172	+	0.883637i	$0.344913\pi$
$504$	0	0
$505$	16.0000	0.711991
$506$	6.00000	0.266733
$507$	9.00000	0.399704
$508$	−16.0000	−0.709885
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	32.0000	1.41421
$513$	6.00000	0.264906
$514$	62.0000	2.73470
$515$	−11.0000	−0.484718
$516$	20.0000	0.880451
$517$	1.00000	0.0439799
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	6.00000	0.262613
$523$	13.0000	0.568450	0.284225	−	0.958758i	$-0.408264\pi$
$523$	13.0000	0.568450	0.284225	+	0.958758i	$0.408264\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−24.0000	−1.04645
$527$	4.00000	0.174243
$528$	4.00000	0.174078
$529$	−14.0000	−0.608696
$530$	8.00000	0.347498
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−20.0000	−0.866296
$534$	12.0000	0.519291
$535$	5.00000	0.216169
$536$	0	0
$537$	−5.00000	−0.215766
$538$	24.0000	1.03471
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	30.0000	1.28861
$543$	12.0000	0.514969
$544$	16.0000	0.685994
$545$	6.00000	0.257012
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−44.0000	−1.87959
$549$	10.0000	0.426790
$550$	−8.00000	−0.341121
$551$	−18.0000	−0.766826
$552$	0	0
$553$	0	0
$554$	52.0000	2.20927
$555$	7.00000	0.297133
$556$	4.00000	0.169638
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	−4.00000	−0.169334
$559$	−20.0000	−0.845910
$560$	0	0
$561$	2.00000	0.0844401
$562$	54.0000	2.27785
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−2.00000	−0.0842152
$565$	−10.0000	−0.420703
$566$	42.0000	1.76539
$567$	0	0
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	12.0000	0.502625
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	4.00000	0.167248
$573$	18.0000	0.751961
$574$	0	0
$575$	−12.0000	−0.500435
$576$	−8.00000	−0.333333
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	−26.0000	−1.08146
$579$	−4.00000	−0.166234
$580$	6.00000	0.249136
$581$	0	0
$582$	2.00000	0.0829027
$583$	4.00000	0.165663
$584$	0	0
$585$	2.00000	0.0826898
$586$	−54.0000	−2.23072
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	−16.0000	−0.658710
$591$	−8.00000	−0.329076
$592$	28.0000	1.15079
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	−24.0000	−0.983078
$597$	26.0000	1.06411
$598$	12.0000	0.490716
$599$	−39.0000	−1.59350	−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000	−1.59350	−0.796748	+	0.604311i	$0.793448\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	0	0
$603$	10.0000	0.407231
$604$	−28.0000	−1.13930
$605$	−10.0000	−0.406558
$606$	−32.0000	−1.29991
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	48.0000	1.94666
$609$	0	0
$610$	20.0000	0.809776
$611$	2.00000	0.0809113
$612$	−4.00000	−0.161690
$613$	27.0000	1.09052	0.545260	−	0.838267i	$-0.316431\pi$
$613$	27.0000	1.09052	0.545260	+	0.838267i	$0.316431\pi$
$614$	−46.0000	−1.85641
$615$	10.0000	0.403239
$616$	0	0
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	22.0000	0.884970
$619$	19.0000	0.763674	0.381837	−	0.924230i	$-0.375291\pi$
$619$	19.0000	0.763674	0.381837	+	0.924230i	$0.375291\pi$
$620$	−4.00000	−0.160644
$621$	−3.00000	−0.120386
$622$	46.0000	1.84443
$623$	0	0
$624$	8.00000	0.320256
$625$	11.0000	0.440000
$626$	−12.0000	−0.479616
$627$	6.00000	0.239617
$628$	6.00000	0.239426
$629$	14.0000	0.558217
$630$	0	0
$631$	−2.00000	−0.0796187	−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000	−0.0796187	−0.0398094	+	0.999207i	$0.512675\pi$
$632$	0	0
$633$	−16.0000	−0.635943
$634$	−6.00000	−0.238290
$635$	−8.00000	−0.317470
$636$	−8.00000	−0.317221
$637$	0	0
$638$	6.00000	0.237542
$639$	−14.0000	−0.553831
$640$	0	0
$641$	17.0000	0.671460	0.335730	−	0.941958i	$-0.391017\pi$
$641$	17.0000	0.671460	0.335730	+	0.941958i	$0.391017\pi$
$642$	−10.0000	−0.394669
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	0	0
$645$	10.0000	0.393750
$646$	24.0000	0.944267
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	−16.0000	−0.627572
$651$	0	0
$652$	−12.0000	−0.469956
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−12.0000	−0.469237
$655$	6.00000	0.234439
$656$	40.0000	1.56174
$657$	10.0000	0.390137
$658$	0	0
$659$	−10.0000	−0.389545	−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000	−0.389545	−0.194772	+	0.980848i	$0.562397\pi$
$660$	−2.00000	−0.0778499
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	8.00000	0.310929
$663$	4.00000	0.155347
$664$	0	0
$665$	0	0
$666$	−14.0000	−0.542489
$667$	9.00000	0.348481
$668$	−42.0000	−1.62503
$669$	−12.0000	−0.463947
$670$	20.0000	0.772667
$671$	10.0000	0.386046
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−26.0000	−1.00148
$675$	4.00000	0.153960
$676$	−18.0000	−0.692308
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	20.0000	0.768095
$679$	0	0
$680$	0	0
$681$	7.00000	0.268241
$682$	−4.00000	−0.153168
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	−12.0000	−0.458831
$685$	−22.0000	−0.840577
$686$	0	0
$687$	4.00000	0.152610
$688$	40.0000	1.52499
$689$	8.00000	0.304776
$690$	−6.00000	−0.228416
$691$	−34.0000	−1.29342	−0.646710	−	0.762736i	$-0.723856\pi$
$691$	−34.0000	−1.29342	−0.646710	+	0.762736i	$0.723856\pi$
$692$	0	0
$693$	0	0
$694$	−24.0000	−0.911028
$695$	2.00000	0.0758643
$696$	0	0
$697$	20.0000	0.757554
$698$	−32.0000	−1.21122
$699$	−23.0000	−0.869940
$700$	0	0
$701$	−1.00000	−0.0377695	−0.0188847	−	0.999822i	$-0.506012\pi$
$701$	−1.00000	−0.0377695	−0.0188847	+	0.999822i	$0.506012\pi$
$702$	−4.00000	−0.150970
$703$	42.0000	1.58406
$704$	−8.00000	−0.301511
$705$	−1.00000	−0.0376622
$706$	0	0
$707$	0	0
$708$	16.0000	0.601317
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	−28.0000	−1.05082
$711$	17.0000	0.637550
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	2.00000	0.0747958
$716$	10.0000	0.373718
$717$	−24.0000	−0.896296
$718$	−30.0000	−1.11959
$719$	−2.00000	−0.0745874	−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000	−0.0745874	−0.0372937	+	0.999304i	$0.511874\pi$
$720$	−4.00000	−0.149071
$721$	0	0
$722$	34.0000	1.26535
$723$	11.0000	0.409094
$724$	−24.0000	−0.891953
$725$	−12.0000	−0.445669
$726$	20.0000	0.742270
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	20.0000	0.740233
$731$	20.0000	0.739727
$732$	−20.0000	−0.739221
$733$	−41.0000	−1.51437	−0.757185	−	0.653201i	$-0.773426\pi$
$733$	−41.0000	−1.51437	−0.757185	+	0.653201i	$0.773426\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	−24.0000	−0.884652
$737$	10.0000	0.368355
$738$	−20.0000	−0.736210
$739$	17.0000	0.625355	0.312678	−	0.949859i	$-0.398774\pi$
$739$	17.0000	0.625355	0.312678	+	0.949859i	$0.398774\pi$
$740$	−14.0000	−0.514650
$741$	12.0000	0.440831
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	−32.0000	−1.17160
$747$	−8.00000	−0.292705
$748$	−4.00000	−0.146254
$749$	0	0
$750$	18.0000	0.657267
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	−4.00000	−0.145865
$753$	18.0000	0.655956
$754$	12.0000	0.437014
$755$	−14.0000	−0.509512
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	70.0000	2.54251
$759$	−3.00000	−0.108893
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	16.0000	0.579619
$763$	0	0
$764$	−36.0000	−1.30243
$765$	−2.00000	−0.0723102
$766$	−24.0000	−0.867155
$767$	−16.0000	−0.577727
$768$	−16.0000	−0.577350
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	−31.0000	−1.11644
$772$	8.00000	0.287926
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	−20.0000	−0.718885
$775$	8.00000	0.287368
$776$	0	0
$777$	0	0
$778$	12.0000	0.430221
$779$	60.0000	2.14972
$780$	−4.00000	−0.143223
$781$	−14.0000	−0.500959
$782$	−12.0000	−0.429119
$783$	−3.00000	−0.107211
$784$	0	0
$785$	3.00000	0.107075
$786$	−12.0000	−0.428026
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	16.0000	0.569976
$789$	12.0000	0.427211
$790$	34.0000	1.20967
$791$	0	0
$792$	0	0
$793$	20.0000	0.710221
$794$	4.00000	0.141955
$795$	−4.00000	−0.141865
$796$	−52.0000	−1.84309
$797$	−29.0000	−1.02723	−0.513616	−	0.858020i	$-0.671695\pi$
$797$	−29.0000	−1.02723	−0.513616	+	0.858020i	$0.671695\pi$
$798$	0	0
$799$	−2.00000	−0.0707549
$800$	32.0000	1.13137
$801$	−6.00000	−0.212000
$802$	8.00000	0.282490
$803$	10.0000	0.352892
$804$	−20.0000	−0.705346
$805$	0	0
$806$	−8.00000	−0.281788
$807$	−12.0000	−0.422420
$808$	0	0
$809$	53.0000	1.86338	0.931690	−	0.363253i	$-0.118334\pi$
$809$	53.0000	1.86338	0.931690	+	0.363253i	$0.118334\pi$
$810$	2.00000	0.0702728
$811$	29.0000	1.01833	0.509164	−	0.860670i	$-0.329955\pi$
$811$	29.0000	1.01833	0.509164	+	0.860670i	$0.329955\pi$
$812$	0	0
$813$	−15.0000	−0.526073
$814$	−14.0000	−0.490700
$815$	−6.00000	−0.210171
$816$	−8.00000	−0.280056
$817$	60.0000	2.09913
$818$	64.0000	2.23771
$819$	0	0
$820$	−20.0000	−0.698430
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	44.0000	1.53468
$823$	−39.0000	−1.35945	−0.679727	−	0.733465i	$-0.737902\pi$
$823$	−39.0000	−1.35945	−0.679727	+	0.733465i	$0.737902\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	−24.0000	−0.834562	−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000	−0.834562	−0.417281	+	0.908778i	$0.637017\pi$
$828$	6.00000	0.208514
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	−16.0000	−0.555368
$831$	−26.0000	−0.901930
$832$	−16.0000	−0.554700
$833$	0	0
$834$	−4.00000	−0.138509
$835$	−21.0000	−0.726735
$836$	−12.0000	−0.415029
$837$	2.00000	0.0691301
$838$	−50.0000	−1.72722
$839$	31.0000	1.07024	0.535119	−	0.844776i	$-0.320267\pi$
$839$	31.0000	1.07024	0.535119	+	0.844776i	$0.320267\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−40.0000	−1.37849
$843$	−27.0000	−0.929929
$844$	32.0000	1.10149
$845$	−9.00000	−0.309609
$846$	2.00000	0.0687614
$847$	0	0
$848$	−16.0000	−0.549442
$849$	−21.0000	−0.720718
$850$	16.0000	0.548795
$851$	−21.0000	−0.719871
$852$	28.0000	0.959264
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	−4.00000	−0.136558
$859$	18.0000	0.614152	0.307076	−	0.951685i	$-0.400649\pi$
$859$	18.0000	0.614152	0.307076	+	0.951685i	$0.400649\pi$
$860$	−20.0000	−0.681994
$861$	0	0
$862$	20.0000	0.681203
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	8.00000	0.272166
$865$	0	0
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	17.0000	0.576686
$870$	−6.00000	−0.203419
$871$	20.0000	0.677674
$872$	0	0
$873$	−1.00000	−0.0338449
$874$	−36.0000	−1.21772
$875$	0	0
$876$	−20.0000	−0.675737
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	30.0000	1.01245
$879$	27.0000	0.910687
$880$	−4.00000	−0.134840
$881$	53.0000	1.78562	0.892808	−	0.450438i	$-0.148732\pi$
$881$	53.0000	1.78562	0.892808	+	0.450438i	$0.148732\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	−8.00000	−0.269069
$885$	8.00000	0.268917
$886$	−24.0000	−0.806296
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	0	0
$890$	−12.0000	−0.402241
$891$	1.00000	0.0335013
$892$	24.0000	0.803579
$893$	−6.00000	−0.200782
$894$	24.0000	0.802680
$895$	5.00000	0.167132
$896$	0	0
$897$	−6.00000	−0.200334
$898$	14.0000	0.467186
$899$	−6.00000	−0.200111
$900$	−8.00000	−0.266667
$901$	−8.00000	−0.266519
$902$	−20.0000	−0.665927
$903$	0	0
$904$	0	0
$905$	−12.0000	−0.398893
$906$	28.0000	0.930238
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	−14.0000	−0.464606
$909$	16.0000	0.530687
$910$	0	0
$911$	50.0000	1.65657	0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000	1.65657	0.828287	+	0.560304i	$0.189316\pi$
$912$	−24.0000	−0.794719
$913$	−8.00000	−0.264761
$914$	−26.0000	−0.860004
$915$	−10.0000	−0.330590
$916$	−8.00000	−0.264327
$917$	0	0
$918$	4.00000	0.132020
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	23.0000	0.757876
$922$	−62.0000	−2.04186
$923$	−28.0000	−0.921631
$924$	0	0
$925$	28.0000	0.920634
$926$	−56.0000	−1.84027
$927$	−11.0000	−0.361287
$928$	−24.0000	−0.787839
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	4.00000	0.131165
$931$	0	0
$932$	46.0000	1.50678
$933$	−23.0000	−0.752986
$934$	42.0000	1.37428
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	2.00000	0.0652328
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	−6.00000	−0.195491
$943$	−30.0000	−0.976934
$944$	32.0000	1.04151
$945$	0	0
$946$	−20.0000	−0.650256
$947$	−38.0000	−1.23483	−0.617417	−	0.786636i	$-0.711821\pi$
$947$	−38.0000	−1.23483	−0.617417	+	0.786636i	$0.711821\pi$
$948$	−34.0000	−1.10427
$949$	20.0000	0.649227
$950$	48.0000	1.55733
$951$	3.00000	0.0972817
$952$	0	0
$953$	31.0000	1.00419	0.502094	−	0.864813i	$-0.332563\pi$
$953$	31.0000	1.00419	0.502094	+	0.864813i	$0.332563\pi$
$954$	8.00000	0.259010
$955$	−18.0000	−0.582466
$956$	48.0000	1.55243
$957$	−3.00000	−0.0969762
$958$	−24.0000	−0.775405
$959$	0	0
$960$	8.00000	0.258199
$961$	−27.0000	−0.870968
$962$	−28.0000	−0.902756
$963$	5.00000	0.161123
$964$	−22.0000	−0.708572
$965$	4.00000	0.128765
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	−2.00000	−0.0642161
$971$	−49.0000	−1.57248	−0.786242	−	0.617918i	$-0.787976\pi$
$971$	−49.0000	−1.57248	−0.786242	+	0.617918i	$0.787976\pi$
$972$	−2.00000	−0.0641500
$973$	0	0
$974$	−16.0000	−0.512673
$975$	8.00000	0.256205
$976$	−40.0000	−1.28037
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	12.0000	0.383718
$979$	−6.00000	−0.191761
$980$	0	0
$981$	6.00000	0.191565
$982$	72.0000	2.29761
$983$	−60.0000	−1.91370	−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000	−1.91370	−0.956851	+	0.290578i	$0.906153\pi$
$984$	0	0
$985$	8.00000	0.254901
$986$	−12.0000	−0.382158
$987$	0	0
$988$	−24.0000	−0.763542
$989$	−30.0000	−0.953945
$990$	2.00000	0.0635642
$991$	−57.0000	−1.81066	−0.905332	−	0.424704i	$-0.860378\pi$
$991$	−57.0000	−1.81066	−0.905332	+	0.424704i	$0.860378\pi$
$992$	16.0000	0.508001
$993$	−4.00000	−0.126936
$994$	0	0
$995$	−26.0000	−0.824255
$996$	16.0000	0.506979
$997$	−40.0000	−1.26681	−0.633406	−	0.773819i	$-0.718344\pi$
$997$	−40.0000	−1.26681	−0.633406	+	0.773819i	$0.718344\pi$
$998$	−44.0000	−1.39280
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6909.2.a.k.1.1		1
7.6	odd	2		141.2.a.e.1.1	✓	1
21.20	even	2		423.2.a.b.1.1		1
28.27	even	2		2256.2.a.e.1.1		1
35.34	odd	2		3525.2.a.c.1.1		1
56.13	odd	2		9024.2.a.n.1.1		1
56.27	even	2		9024.2.a.bq.1.1		1
84.83	odd	2		6768.2.a.n.1.1		1
329.328	even	2		6627.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
141.2.a.e.1.1	✓	1	7.6	odd	2
423.2.a.b.1.1		1	21.20	even	2
2256.2.a.e.1.1		1	28.27	even	2
3525.2.a.c.1.1		1	35.34	odd	2
6627.2.a.i.1.1		1	329.328	even	2
6768.2.a.n.1.1		1	84.83	odd	2
6909.2.a.k.1.1		1	1.1	even	1	trivial
9024.2.a.n.1.1		1	56.13	odd	2
9024.2.a.bq.1.1		1	56.27	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:	\( N \)	\(=\)	\( 6909 = 3 \cdot 7^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6909.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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