LMFDB - Embedded newform 9024.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9024,2,Mod(1,9024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9024 = 2^{6} \cdot 3 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9024.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:	$72.0570027840$
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	$\chi$	$=$	9024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{15} +2.00000 q^{17} -6.00000 q^{19} +3.00000 q^{21} +3.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} -3.00000 q^{29} +2.00000 q^{31} +1.00000 q^{33} -3.00000 q^{35} +7.00000 q^{37} -2.00000 q^{39} +10.0000 q^{41} +10.0000 q^{43} +1.00000 q^{45} -1.00000 q^{47} +2.00000 q^{49} -2.00000 q^{51} -4.00000 q^{53} -1.00000 q^{55} +6.00000 q^{57} -8.00000 q^{59} +10.0000 q^{61} -3.00000 q^{63} +2.00000 q^{65} -10.0000 q^{67} -3.00000 q^{69} -14.0000 q^{71} -10.0000 q^{73} +4.00000 q^{75} +3.00000 q^{77} +17.0000 q^{79} +1.00000 q^{81} -8.00000 q^{83} +2.00000 q^{85} +3.00000 q^{87} +6.00000 q^{89} -6.00000 q^{91} -2.00000 q^{93} -6.00000 q^{95} +1.00000 q^{97} -1.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	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
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	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
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	−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	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	17.0000	1.91265	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	17.0000	1.91265	0.956325	+	0.292306i	$0.0944227\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−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
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	16.0000	1.59206	0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000	1.59206	0.796030	+	0.605257i	$0.206930\pi$
$102$	0	0
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	−5.00000	−0.483368	−0.241684	−	0.970355i	$-0.577700\pi$
$107$	−5.00000	−0.483368	−0.241684	+	0.970355i	$0.577700\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−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	$0.655876\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−10.0000	−0.901670
$124$	0	0
$125$	−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
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	18.0000	1.56080
$134$	0	0
$135$	−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	$0.888983\pi$
$138$	0	0
$139$	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
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−3.00000	−0.249136
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−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
$156$	0	0
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	−9.00000	−0.709299
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	0	0
$181$	−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
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	26.0000	1.84309	0.921546	−	0.388270i	$-0.126927\pi$
$199$	26.0000	1.84309	0.921546	+	0.388270i	$0.126927\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	0	0
$203$	9.00000	0.631676
$204$	0	0
$205$	10.0000	0.698430
$206$	0	0
$207$	3.00000	0.208514
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	14.0000	0.959264
$214$	0	0
$215$	10.0000	0.681994
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	−7.00000	−0.464606	−0.232303	−	0.972643i	$-0.574626\pi$
$227$	−7.00000	−0.464606	−0.232303	+	0.972643i	$0.574626\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	23.0000	1.50678	0.753390	−	0.657574i	$-0.228417\pi$
$233$	23.0000	1.50678	0.753390	+	0.657574i	$0.228417\pi$
$234$	0	0
$235$	−1.00000	−0.0652328
$236$	0	0
$237$	−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	$0.217137\pi$
$240$	0	0
$241$	11.0000	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$241$	11.0000	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	−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
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	−31.0000	−1.93373	−0.966863	−	0.255294i	$-0.917828\pi$
$257$	−31.0000	−1.93373	−0.966863	+	0.255294i	$0.917828\pi$
$258$	0	0
$259$	−21.0000	−1.30488
$260$	0	0
$261$	−3.00000	−0.185695
$262$	0	0
$263$	−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
$268$	0	0
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	0	0
$279$	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	$0.201981\pi$
$282$	0	0
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	0	0
$287$	−30.0000	−1.77084
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−1.00000	−0.0586210
$292$	0	0
$293$	−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
$298$	0	0
$299$	6.00000	0.346989
$300$	0	0
$301$	−30.0000	−1.72917
$302$	0	0
$303$	−16.0000	−0.919176
$304$	0	0
$305$	10.0000	0.572598
$306$	0	0
$307$	−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
$310$	0	0
$311$	−23.0000	−1.30421	−0.652105	−	0.758129i	$-0.726114\pi$
$311$	−23.0000	−1.30421	−0.652105	+	0.758129i	$0.726114\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	0	0
$321$	5.00000	0.279073
$322$	0	0
$323$	−12.0000	−0.667698
$324$	0	0
$325$	−8.00000	−0.443760
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	7.00000	0.383598
$334$	0	0
$335$	−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	$0.615205\pi$
$338$	0	0
$339$	10.0000	0.543125
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	−3.00000	−0.161515
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−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
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−14.0000	−0.743043
$356$	0	0
$357$	6.00000	0.317554
$358$	0	0
$359$	−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
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	−35.0000	−1.79783	−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000	−1.79783	−0.898915	+	0.438124i	$0.855643\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	0	0
$387$	10.0000	0.508329
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	17.0000	0.855363
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	−18.0000	−0.901127
$400$	0	0
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−7.00000	−0.346977
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	22.0000	1.08518
$412$	0	0
$413$	24.0000	1.18096
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	−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	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	0	0
$423$	−1.00000	−0.0486217
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	−30.0000	−1.45180
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	0	0
$437$	−18.0000	−0.861057
$438$	0	0
$439$	−15.0000	−0.715911	−0.357955	−	0.933739i	$-0.616526\pi$
$439$	−15.0000	−0.715911	−0.357955	+	0.933739i	$0.616526\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	−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	$0.447181\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	0	0
$453$	14.0000	0.657777
$454$	0	0
$455$	−6.00000	−0.281284
$456$	0	0
$457$	−13.0000	−0.608114	−0.304057	−	0.952654i	$-0.598341\pi$
$457$	−13.0000	−0.608114	−0.304057	+	0.952654i	$0.598341\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−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	$0.725497\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	0	0
$469$	30.0000	1.38527
$470$	0	0
$471$	−3.00000	−0.138233
$472$	0	0
$473$	−10.0000	−0.459800
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	−4.00000	−0.183147
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	14.0000	0.638345
$482$	0	0
$483$	9.00000	0.409514
$484$	0	0
$485$	1.00000	0.0454077
$486$	0	0
$487$	−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	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	42.0000	1.88396
$498$	0	0
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	0	0
$501$	−21.0000	−0.938211
$502$	0	0
$503$	−21.0000	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$503$	−21.0000	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$504$	0	0
$505$	16.0000	0.711991
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	30.0000	1.32712
$512$	0	0
$513$	6.00000	0.264906
$514$	0	0
$515$	11.0000	0.484718
$516$	0	0
$517$	1.00000	0.0439799
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	13.0000	0.568450	0.284225	−	0.958758i	$-0.408264\pi$
$523$	13.0000	0.568450	0.284225	+	0.958758i	$0.408264\pi$
$524$	0	0
$525$	−12.0000	−0.523723
$526$	0	0
$527$	4.00000	0.174243
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	20.0000	0.866296
$534$	0	0
$535$	−5.00000	−0.216169
$536$	0	0
$537$	5.00000	0.215766
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	12.0000	0.514969
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	18.0000	0.766826
$552$	0	0
$553$	−51.0000	−2.16874
$554$	0	0
$555$	−7.00000	−0.297133
$556$	0	0
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	4.00000	0.165663
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	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
$590$	0	0
$591$	8.00000	0.329076
$592$	0	0
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	0	0
$597$	−26.0000	−1.06411
$598$	0	0
$599$	−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	$-0.454400\pi$
$601$	7.00000	0.285536	0.142768	+	0.989756i	$0.454400\pi$
$602$	0	0
$603$	−10.0000	−0.407231
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	−9.00000	−0.364698
$610$	0	0
$611$	−2.00000	−0.0809113
$612$	0	0
$613$	−27.0000	−1.09052	−0.545260	−	0.838267i	$-0.683569\pi$
$613$	−27.0000	−1.09052	−0.545260	+	0.838267i	$0.683569\pi$
$614$	0	0
$615$	−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	$0.242000\pi$
$618$	0	0
$619$	19.0000	0.763674	0.381837	−	0.924230i	$-0.375291\pi$
$619$	19.0000	0.763674	0.381837	+	0.924230i	$0.375291\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	0	0
$623$	−18.0000	−0.721155
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−6.00000	−0.239617
$628$	0	0
$629$	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
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	−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	$0.391017\pi$
$642$	0	0
$643$	−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
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	0	0
$651$	6.00000	0.235159
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	18.0000	0.698010
$666$	0	0
$667$	−9.00000	−0.348481
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	−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	$0.332924\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	0	0
$679$	−3.00000	−0.115129
$680$	0	0
$681$	7.00000	0.268241
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−22.0000	−0.840577
$686$	0	0
$687$	4.00000	0.152610
$688$	0	0
$689$	−8.00000	−0.304776
$690$	0	0
$691$	−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
$693$	3.00000	0.113961
$694$	0	0
$695$	2.00000	0.0758643
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	−23.0000	−0.869940
$700$	0	0
$701$	1.00000	0.0377695	0.0188847	−	0.999822i	$-0.493988\pi$
$701$	1.00000	0.0377695	0.0188847	+	0.999822i	$0.493988\pi$
$702$	0	0
$703$	−42.0000	−1.58406
$704$	0	0
$705$	1.00000	0.0376622
$706$	0	0
$707$	−48.0000	−1.80523
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	17.0000	0.637550
$712$	0	0
$713$	6.00000	0.224702
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	0	0
$721$	−33.0000	−1.22898
$722$	0	0
$723$	−11.0000	−0.409094
$724$	0	0
$725$	12.0000	0.445669
$726$	0	0
$727$	10.0000	0.370879	0.185440	−	0.982656i	$-0.440629\pi$
$727$	10.0000	0.370879	0.185440	+	0.982656i	$0.440629\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	20.0000	0.739727
$732$	0	0
$733$	−41.0000	−1.51437	−0.757185	−	0.653201i	$-0.773426\pi$
$733$	−41.0000	−1.51437	−0.757185	+	0.653201i	$0.773426\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	10.0000	0.368355
$738$	0	0
$739$	−17.0000	−0.625355	−0.312678	−	0.949859i	$-0.601226\pi$
$739$	−17.0000	−0.625355	−0.312678	+	0.949859i	$0.601226\pi$
$740$	0	0
$741$	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
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	15.0000	0.548088
$750$	0	0
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	−14.0000	−0.509512
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	18.0000	0.651644
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	−16.0000	−0.577727
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	31.0000	1.11644
$772$	0	0
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	21.0000	0.753371
$778$	0	0
$779$	−60.0000	−2.14972
$780$	0	0
$781$	14.0000	0.500959
$782$	0	0
$783$	3.00000	0.107211
$784$	0	0
$785$	3.00000	0.107075
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	30.0000	1.06668
$792$	0	0
$793$	20.0000	0.710221
$794$	0	0
$795$	4.00000	0.141865
$796$	0	0
$797$	−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
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	10.0000	0.352892
$804$	0	0
$805$	−9.00000	−0.317208
$806$	0	0
$807$	−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	$0.118334\pi$
$810$	0	0
$811$	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
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	−60.0000	−2.09913
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	−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	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	21.0000	0.726735
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−31.0000	−1.07024	−0.535119	−	0.844776i	$-0.679733\pi$
$839$	−31.0000	−1.07024	−0.535119	+	0.844776i	$0.679733\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−27.0000	−0.929929
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	30.0000	1.03081
$848$	0	0
$849$	−21.0000	−0.720718
$850$	0	0
$851$	21.0000	0.719871
$852$	0	0
$853$	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	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	18.0000	0.614152	0.307076	−	0.951685i	$-0.400649\pi$
$859$	18.0000	0.614152	0.307076	+	0.951685i	$0.400649\pi$
$860$	0	0
$861$	30.0000	1.02240
$862$	0	0
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	−17.0000	−0.576686
$870$	0	0
$871$	−20.0000	−0.677674
$872$	0	0
$873$	1.00000	0.0338449
$874$	0	0
$875$	27.0000	0.912767
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	0	0
$879$	27.0000	0.910687
$880$	0	0
$881$	−53.0000	−1.78562	−0.892808	−	0.450438i	$-0.851268\pi$
$881$	−53.0000	−1.78562	−0.892808	+	0.450438i	$0.851268\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	6.00000	0.200782
$894$	0	0
$895$	−5.00000	−0.167132
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−8.00000	−0.266519
$902$	0	0
$903$	30.0000	0.998337
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	0	0
$909$	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	$0.189316\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	0	0
$915$	−10.0000	−0.330590
$916$	0	0
$917$	−18.0000	−0.594412
$918$	0	0
$919$	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
$922$	0	0
$923$	−28.0000	−0.921631
$924$	0	0
$925$	−28.0000	−0.920634
$926$	0	0
$927$	11.0000	0.361287
$928$	0	0
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	0	0
$933$	23.0000	0.752986
$934$	0	0
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	0	0
$943$	30.0000	0.976934
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	38.0000	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$947$	38.0000	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	−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	$0.332563\pi$
$954$	0	0
$955$	−18.0000	−0.582466
$956$	0	0
$957$	−3.00000	−0.0969762
$958$	0	0
$959$	66.0000	2.13125
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−5.00000	−0.161123
$964$	0	0
$965$	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
$970$	0	0
$971$	−49.0000	−1.57248	−0.786242	−	0.617918i	$-0.787976\pi$
$971$	−49.0000	−1.57248	−0.786242	+	0.617918i	$0.787976\pi$
$972$	0	0
$973$	−6.00000	−0.192351
$974$	0	0
$975$	8.00000	0.256205
$976$	0	0
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	60.0000	1.91370	0.956851	−	0.290578i	$-0.0938475\pi$
$983$	60.0000	1.91370	0.956851	+	0.290578i	$0.0938475\pi$
$984$	0	0
$985$	−8.00000	−0.254901
$986$	0	0
$987$	−3.00000	−0.0954911
$988$	0	0
$989$	30.0000	0.953945
$990$	0	0
$991$	−57.0000	−1.81066	−0.905332	−	0.424704i	$-0.860378\pi$
$991$	−57.0000	−1.81066	−0.905332	+	0.424704i	$0.860378\pi$
$992$	0	0
$993$	4.00000	0.126936
$994$	0	0
$995$	26.0000	0.824255
$996$	0	0
$997$	−40.0000	−1.26681	−0.633406	−	0.773819i	$-0.718344\pi$
$997$	−40.0000	−1.26681	−0.633406	+	0.773819i	$0.718344\pi$
$998$	0	0
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9024.2.a.n.1.1		1
4.3	odd	2		9024.2.a.bq.1.1		1
8.3	odd	2		2256.2.a.e.1.1		1
8.5	even	2		141.2.a.e.1.1	✓	1
24.5	odd	2		423.2.a.b.1.1		1
24.11	even	2		6768.2.a.n.1.1		1
40.29	even	2		3525.2.a.c.1.1		1
56.13	odd	2		6909.2.a.k.1.1		1
376.93	odd	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	8.5	even	2
423.2.a.b.1.1		1	24.5	odd	2
2256.2.a.e.1.1		1	8.3	odd	2
3525.2.a.c.1.1		1	40.29	even	2
6627.2.a.i.1.1		1	376.93	odd	2
6768.2.a.n.1.1		1	24.11	even	2
6909.2.a.k.1.1		1	56.13	odd	2
9024.2.a.n.1.1		1	1.1	even	1	trivial
9024.2.a.bq.1.1		1	4.3	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:	\( N \)	\(=\)	\( 9024 = 2^{6} \cdot 3 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9024.a (trivial)

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