LMFDB - Embedded newform 3549.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3549,2,Mod(1,3549)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3549, 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("3549.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3549 = 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3549.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:	$28.3389076774$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +6.00000 q^{10} +2.00000 q^{12} +2.00000 q^{14} -3.00000 q^{15} -4.00000 q^{16} +2.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} -6.00000 q^{20} -1.00000 q^{21} +1.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} -2.00000 q^{28} +5.00000 q^{29} +6.00000 q^{30} -5.00000 q^{31} +8.00000 q^{32} -4.00000 q^{34} +3.00000 q^{35} +2.00000 q^{36} -8.00000 q^{37} +2.00000 q^{38} +10.0000 q^{41} +2.00000 q^{42} -9.00000 q^{43} -3.00000 q^{45} -2.00000 q^{46} +7.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} -8.00000 q^{50} +2.00000 q^{51} +9.00000 q^{53} -2.00000 q^{54} -1.00000 q^{57} -10.0000 q^{58} -4.00000 q^{59} -6.00000 q^{60} -8.00000 q^{61} +10.0000 q^{62} -1.00000 q^{63} -8.00000 q^{64} -2.00000 q^{67} +4.00000 q^{68} +1.00000 q^{69} -6.00000 q^{70} +9.00000 q^{73} +16.0000 q^{74} +4.00000 q^{75} -2.00000 q^{76} +15.0000 q^{79} +12.0000 q^{80} +1.00000 q^{81} -20.0000 q^{82} -9.00000 q^{83} -2.00000 q^{84} -6.00000 q^{85} +18.0000 q^{86} +5.00000 q^{87} -9.00000 q^{89} +6.00000 q^{90} +2.00000 q^{92} -5.00000 q^{93} -14.0000 q^{94} +3.00000 q^{95} +8.00000 q^{96} +13.0000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.00000	1.00000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−2.00000	−0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	6.00000	1.89737
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	2.00000	0.577350
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	−3.00000	−0.774597
$16$	−4.00000	−1.00000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−2.00000	−0.471405
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−6.00000	−1.34164
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	6.00000	1.09545
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	−4.00000	−0.685994
$35$	3.00000	0.507093
$36$	2.00000	0.333333
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	2.00000	0.308607
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	−2.00000	−0.294884
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	−4.00000	−0.577350
$49$	1.00000	0.142857
$50$	−8.00000	−1.13137
$51$	2.00000	0.280056
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−2.00000	−0.272166
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−10.0000	−1.31306
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	−6.00000	−0.774597
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	10.0000	1.27000
$63$	−1.00000	−0.125988
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	4.00000	0.485071
$69$	1.00000	0.120386
$70$	−6.00000	−0.717137
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	16.0000	1.85996
$75$	4.00000	0.461880
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	12.0000	1.34164
$81$	1.00000	0.111111
$82$	−20.0000	−2.20863
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	−2.00000	−0.218218
$85$	−6.00000	−0.650791
$86$	18.0000	1.94099
$87$	5.00000	0.536056
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	6.00000	0.632456
$91$	0	0
$92$	2.00000	0.208514
$93$	−5.00000	−0.518476
$94$	−14.0000	−1.44399
$95$	3.00000	0.307794
$96$	8.00000	0.816497
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	8.00000	0.800000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	−4.00000	−0.396059
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	−18.0000	−1.74831
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	2.00000	0.192450
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	4.00000	0.377964
$113$	−21.0000	−1.97551	−0.987757	−	0.156001i	$-0.950140\pi$
$113$	−21.0000	−1.97551	−0.987757	+	0.156001i	$0.950140\pi$
$114$	2.00000	0.187317
$115$	−3.00000	−0.279751
$116$	10.0000	0.928477
$117$	0	0
$118$	8.00000	0.736460
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	16.0000	1.44857
$123$	10.0000	0.901670
$124$	−10.0000	−0.898027
$125$	3.00000	0.268328
$126$	2.00000	0.178174
$127$	−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
$129$	−9.00000	−0.792406
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	4.00000	0.345547
$135$	−3.00000	−0.258199
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	−2.00000	−0.170251
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	6.00000	0.507093
$141$	7.00000	0.589506
$142$	0	0
$143$	0	0
$144$	−4.00000	−0.333333
$145$	−15.0000	−1.24568
$146$	−18.0000	−1.48969
$147$	1.00000	0.0824786
$148$	−16.0000	−1.31519
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	−8.00000	−0.653197
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	15.0000	1.20483
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−30.0000	−2.38667
$159$	9.00000	0.713746
$160$	−24.0000	−1.89737
$161$	−1.00000	−0.0788110
$162$	−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
$165$	0	0
$166$	18.0000	1.39707
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	0	0
$170$	12.0000	0.920358
$171$	−1.00000	−0.0764719
$172$	−18.0000	−1.37249
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	−10.0000	−0.758098
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−4.00000	−0.300658
$178$	18.0000	1.34916
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	−6.00000	−0.447214
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	24.0000	1.76452
$186$	10.0000	0.733236
$187$	0	0
$188$	14.0000	1.02105
$189$	−1.00000	−0.0727393
$190$	−6.00000	−0.435286
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−8.00000	−0.577350
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−26.0000	−1.86669
$195$	0	0
$196$	2.00000	0.142857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	−16.0000	−1.12576
$203$	−5.00000	−0.350931
$204$	4.00000	0.280056
$205$	−30.0000	−2.09529
$206$	−32.0000	−2.22955
$207$	1.00000	0.0695048
$208$	0	0
$209$	0	0
$210$	−6.00000	−0.414039
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	18.0000	1.23625
$213$	0	0
$214$	24.0000	1.64061
$215$	27.0000	1.84138
$216$	0	0
$217$	5.00000	0.339422
$218$	−28.0000	−1.89640
$219$	9.00000	0.608164
$220$	0	0
$221$	0	0
$222$	16.0000	1.07385
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	−8.00000	−0.534522
$225$	4.00000	0.266667
$226$	42.0000	2.79380
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−2.00000	−0.132453
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	0	0
$233$	1.00000	0.0655122	0.0327561	−	0.999463i	$-0.489572\pi$
$233$	1.00000	0.0655122	0.0327561	+	0.999463i	$0.489572\pi$
$234$	0	0
$235$	−21.0000	−1.36989
$236$	−8.00000	−0.520756
$237$	15.0000	0.974355
$238$	4.00000	0.259281
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	12.0000	0.774597
$241$	−15.0000	−0.966235	−0.483117	−	0.875556i	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	22.0000	1.41421
$243$	1.00000	0.0641500
$244$	−16.0000	−1.02430
$245$	−3.00000	−0.191663
$246$	−20.0000	−1.27515
$247$	0	0
$248$	0	0
$249$	−9.00000	−0.570352
$250$	−6.00000	−0.379473
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	16.0000	1.00393
$255$	−6.00000	−0.375735
$256$	16.0000	1.00000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	18.0000	1.12063
$259$	8.00000	0.497096
$260$	0	0
$261$	5.00000	0.309492
$262$	36.0000	2.22409
$263$	19.0000	1.17159	0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000	1.17159	0.585795	+	0.810459i	$0.300782\pi$
$264$	0	0
$265$	−27.0000	−1.65860
$266$	−2.00000	−0.122628
$267$	−9.00000	−0.550791
$268$	−4.00000	−0.244339
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	6.00000	0.365148
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−8.00000	−0.485071
$273$	0	0
$274$	−24.0000	−1.44989
$275$	0	0
$276$	2.00000	0.120386
$277$	−3.00000	−0.180253	−0.0901263	−	0.995930i	$-0.528727\pi$
$277$	−3.00000	−0.180253	−0.0901263	+	0.995930i	$0.528727\pi$
$278$	40.0000	2.39904
$279$	−5.00000	−0.299342
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	−14.0000	−0.833688
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	0	0
$285$	3.00000	0.177705
$286$	0	0
$287$	−10.0000	−0.590281
$288$	8.00000	0.471405
$289$	−13.0000	−0.764706
$290$	30.0000	1.76166
$291$	13.0000	0.762073
$292$	18.0000	1.05337
$293$	−31.0000	−1.81104	−0.905520	−	0.424304i	$-0.860519\pi$
$293$	−31.0000	−1.81104	−0.905520	+	0.424304i	$0.860519\pi$
$294$	−2.00000	−0.116642
$295$	12.0000	0.698667
$296$	0	0
$297$	0	0
$298$	−8.00000	−0.463428
$299$	0	0
$300$	8.00000	0.461880
$301$	9.00000	0.518751
$302$	40.0000	2.30174
$303$	8.00000	0.459588
$304$	4.00000	0.229416
$305$	24.0000	1.37424
$306$	−4.00000	−0.228665
$307$	17.0000	0.970241	0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000	0.970241	0.485121	+	0.874447i	$0.338776\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	−30.0000	−1.70389
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	0	0
$313$	4.00000	0.226093	0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000	0.226093	0.113047	+	0.993590i	$0.463939\pi$
$314$	4.00000	0.225733
$315$	3.00000	0.169031
$316$	30.0000	1.68763
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−18.0000	−1.00939
$319$	0	0
$320$	24.0000	1.34164
$321$	−12.0000	−0.669775
$322$	2.00000	0.111456
$323$	−2.00000	−0.111283
$324$	2.00000	0.111111
$325$	0	0
$326$	12.0000	0.664619
$327$	14.0000	0.774202
$328$	0	0
$329$	−7.00000	−0.385922
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−18.0000	−0.987878
$333$	−8.00000	−0.438397
$334$	6.00000	0.328305
$335$	6.00000	0.327815
$336$	4.00000	0.218218
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	0	0
$339$	−21.0000	−1.14056
$340$	−12.0000	−0.650791
$341$	0	0
$342$	2.00000	0.108148
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.00000	−0.161515
$346$	48.0000	2.58050
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	10.0000	0.536056
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	8.00000	0.427618
$351$	0	0
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	−18.0000	−0.953998
$357$	−2.00000	−0.105851
$358$	−10.0000	−0.528516
$359$	−14.0000	−0.738892	−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000	−0.738892	−0.369446	+	0.929252i	$0.620452\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−16.0000	−0.840941
$363$	−11.0000	−0.577350
$364$	0	0
$365$	−27.0000	−1.41324
$366$	16.0000	0.836333
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	−4.00000	−0.208514
$369$	10.0000	0.520579
$370$	−48.0000	−2.49540
$371$	−9.00000	−0.467257
$372$	−10.0000	−0.518476
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	0	0
$378$	2.00000	0.102869
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	6.00000	0.307794
$381$	−8.00000	−0.409852
$382$	16.0000	0.818631
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	−28.0000	−1.42516
$387$	−9.00000	−0.457496
$388$	26.0000	1.31995
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	−18.0000	−0.907980
$394$	24.0000	1.20910
$395$	−45.0000	−2.26420
$396$	0	0
$397$	17.0000	0.853206	0.426603	−	0.904439i	$-0.359710\pi$
$397$	17.0000	0.853206	0.426603	+	0.904439i	$0.359710\pi$
$398$	20.0000	1.00251
$399$	1.00000	0.0500626
$400$	−16.0000	−0.800000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	16.0000	0.796030
$405$	−3.00000	−0.149071
$406$	10.0000	0.496292
$407$	0	0
$408$	0	0
$409$	−31.0000	−1.53285	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	−31.0000	−1.53285	−0.766426	+	0.642333i	$0.777967\pi$
$410$	60.0000	2.96319
$411$	12.0000	0.591916
$412$	32.0000	1.57653
$413$	4.00000	0.196827
$414$	−2.00000	−0.0982946
$415$	27.0000	1.32538
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	6.00000	0.292770
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	26.0000	1.26566
$423$	7.00000	0.340352
$424$	0	0
$425$	8.00000	0.388057
$426$	0	0
$427$	8.00000	0.387147
$428$	−24.0000	−1.16008
$429$	0	0
$430$	−54.0000	−2.60411
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	−4.00000	−0.192450
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−10.0000	−0.480015
$435$	−15.0000	−0.719195
$436$	28.0000	1.34096
$437$	−1.00000	−0.0478365
$438$	−18.0000	−0.860073
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−31.0000	−1.47285	−0.736427	−	0.676517i	$-0.763489\pi$
$443$	−31.0000	−1.47285	−0.736427	+	0.676517i	$0.763489\pi$
$444$	−16.0000	−0.759326
$445$	27.0000	1.27992
$446$	38.0000	1.79935
$447$	4.00000	0.189194
$448$	8.00000	0.377964
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	−8.00000	−0.377124
$451$	0	0
$452$	−42.0000	−1.97551
$453$	−20.0000	−0.939682
$454$	24.0000	1.12638
$455$	0	0
$456$	0	0
$457$	−42.0000	−1.96468	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−42.0000	−1.96468	−0.982339	+	0.187112i	$0.940087\pi$
$458$	28.0000	1.30835
$459$	2.00000	0.0933520
$460$	−6.00000	−0.279751
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	−20.0000	−0.928477
$465$	15.0000	0.695608
$466$	−2.00000	−0.0926482
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	42.0000	1.93732
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	0	0
$474$	−30.0000	−1.37795
$475$	−4.00000	−0.183533
$476$	−4.00000	−0.183340
$477$	9.00000	0.412082
$478$	12.0000	0.548867
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	−24.0000	−1.09545
$481$	0	0
$482$	30.0000	1.36646
$483$	−1.00000	−0.0455016
$484$	−22.0000	−1.00000
$485$	−39.0000	−1.77090
$486$	−2.00000	−0.0907218
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	6.00000	0.271052
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	20.0000	0.901670
$493$	10.0000	0.450377
$494$	0	0
$495$	0	0
$496$	20.0000	0.898027
$497$	0	0
$498$	18.0000	0.806599
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	6.00000	0.268328
$501$	−3.00000	−0.134030
$502$	−56.0000	−2.49940
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−24.0000	−1.06799
$506$	0	0
$507$	0	0
$508$	−16.0000	−0.709885
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	12.0000	0.531369
$511$	−9.00000	−0.398137
$512$	−32.0000	−1.41421
$513$	−1.00000	−0.0441511
$514$	−24.0000	−1.05859
$515$	−48.0000	−2.11513
$516$	−18.0000	−0.792406
$517$	0	0
$518$	−16.0000	−0.703000
$519$	−24.0000	−1.05348
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	−10.0000	−0.437688
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−36.0000	−1.57267
$525$	−4.00000	−0.174574
$526$	−38.0000	−1.65688
$527$	−10.0000	−0.435607
$528$	0	0
$529$	−22.0000	−0.956522
$530$	54.0000	2.34561
$531$	−4.00000	−0.173585
$532$	2.00000	0.0867110
$533$	0	0
$534$	18.0000	0.778936
$535$	36.0000	1.55642
$536$	0	0
$537$	5.00000	0.215766
$538$	−20.0000	−0.862261
$539$	0	0
$540$	−6.00000	−0.258199
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−40.0000	−1.71815
$543$	8.00000	0.343313
$544$	16.0000	0.685994
$545$	−42.0000	−1.79908
$546$	0	0
$547$	13.0000	0.555840	0.277920	−	0.960604i	$-0.410355\pi$
$547$	13.0000	0.555840	0.277920	+	0.960604i	$0.410355\pi$
$548$	24.0000	1.02523
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−5.00000	−0.213007
$552$	0	0
$553$	−15.0000	−0.637865
$554$	6.00000	0.254916
$555$	24.0000	1.01874
$556$	−40.0000	−1.69638
$557$	32.0000	1.35588	0.677942	−	0.735116i	$-0.262872\pi$
$557$	32.0000	1.35588	0.677942	+	0.735116i	$0.262872\pi$
$558$	10.0000	0.423334
$559$	0	0
$560$	−12.0000	−0.507093
$561$	0	0
$562$	−20.0000	−0.843649
$563$	26.0000	1.09577	0.547885	−	0.836554i	$-0.315433\pi$
$563$	26.0000	1.09577	0.547885	+	0.836554i	$0.315433\pi$
$564$	14.0000	0.589506
$565$	63.0000	2.65043
$566$	48.0000	2.01759
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	5.00000	0.209611	0.104805	−	0.994493i	$-0.466578\pi$
$569$	5.00000	0.209611	0.104805	+	0.994493i	$0.466578\pi$
$570$	−6.00000	−0.251312
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	20.0000	0.834784
$575$	4.00000	0.166812
$576$	−8.00000	−0.333333
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	26.0000	1.08146
$579$	14.0000	0.581820
$580$	−30.0000	−1.24568
$581$	9.00000	0.373383
$582$	−26.0000	−1.07773
$583$	0	0
$584$	0	0
$585$	0	0
$586$	62.0000	2.56120
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	2.00000	0.0824786
$589$	5.00000	0.206021
$590$	−24.0000	−0.988064
$591$	−12.0000	−0.493614
$592$	32.0000	1.31519
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	8.00000	0.327693
$597$	−10.0000	−0.409273
$598$	0	0
$599$	−35.0000	−1.43006	−0.715031	−	0.699093i	$-0.753587\pi$
$599$	−35.0000	−1.43006	−0.715031	+	0.699093i	$0.753587\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−18.0000	−0.733625
$603$	−2.00000	−0.0814463
$604$	−40.0000	−1.62758
$605$	33.0000	1.34164
$606$	−16.0000	−0.649956
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	−8.00000	−0.324443
$609$	−5.00000	−0.202610
$610$	−48.0000	−1.94346
$611$	0	0
$612$	4.00000	0.161690
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−34.0000	−1.37213
$615$	−30.0000	−1.20972
$616$	0	0
$617$	48.0000	1.93241	0.966204	−	0.257780i	$-0.0829910\pi$
$617$	48.0000	1.93241	0.966204	+	0.257780i	$0.0829910\pi$
$618$	−32.0000	−1.28723
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	30.0000	1.20483
$621$	1.00000	0.0401286
$622$	4.00000	0.160385
$623$	9.00000	0.360577
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−8.00000	−0.319744
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−16.0000	−0.637962
$630$	−6.00000	−0.239046
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	4.00000	0.158860
$635$	24.0000	0.952411
$636$	18.0000	0.713746
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−17.0000	−0.671460	−0.335730	−	0.941958i	$-0.608983\pi$
$641$	−17.0000	−0.671460	−0.335730	+	0.941958i	$0.608983\pi$
$642$	24.0000	0.947204
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	−2.00000	−0.0788110
$645$	27.0000	1.06312
$646$	4.00000	0.157378
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	5.00000	0.195965
$652$	−12.0000	−0.469956
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	−28.0000	−1.09489
$655$	54.0000	2.10995
$656$	−40.0000	−1.56174
$657$	9.00000	0.351123
$658$	14.0000	0.545777
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	0	0
$661$	−15.0000	−0.583432	−0.291716	−	0.956505i	$-0.594226\pi$
$661$	−15.0000	−0.583432	−0.291716	+	0.956505i	$0.594226\pi$
$662$	40.0000	1.55464
$663$	0	0
$664$	0	0
$665$	−3.00000	−0.116335
$666$	16.0000	0.619987
$667$	5.00000	0.193601
$668$	−6.00000	−0.232147
$669$	−19.0000	−0.734582
$670$	−12.0000	−0.463600
$671$	0	0
$672$	−8.00000	−0.308607
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	46.0000	1.77185
$675$	4.00000	0.153960
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	42.0000	1.61300
$679$	−13.0000	−0.498894
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−2.00000	−0.0764719
$685$	−36.0000	−1.37549
$686$	2.00000	0.0763604
$687$	−14.0000	−0.534133
$688$	36.0000	1.37249
$689$	0	0
$690$	6.00000	0.228416
$691$	−35.0000	−1.33146	−0.665731	−	0.746191i	$-0.731880\pi$
$691$	−35.0000	−1.33146	−0.665731	+	0.746191i	$0.731880\pi$
$692$	−48.0000	−1.82469
$693$	0	0
$694$	24.0000	0.911028
$695$	60.0000	2.27593
$696$	0	0
$697$	20.0000	0.757554
$698$	−2.00000	−0.0757011
$699$	1.00000	0.0378235
$700$	−8.00000	−0.302372
$701$	23.0000	0.868698	0.434349	−	0.900745i	$-0.356978\pi$
$701$	23.0000	0.868698	0.434349	+	0.900745i	$0.356978\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	−21.0000	−0.790906
$706$	−12.0000	−0.451626
$707$	−8.00000	−0.300871
$708$	−8.00000	−0.300658
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	15.0000	0.562544
$712$	0	0
$713$	−5.00000	−0.187251
$714$	4.00000	0.149696
$715$	0	0
$716$	10.0000	0.373718
$717$	−6.00000	−0.224074
$718$	28.0000	1.04495
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	12.0000	0.447214
$721$	−16.0000	−0.595871
$722$	36.0000	1.33978
$723$	−15.0000	−0.557856
$724$	16.0000	0.594635
$725$	20.0000	0.742781
$726$	22.0000	0.816497
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	54.0000	1.99863
$731$	−18.0000	−0.665754
$732$	−16.0000	−0.591377
$733$	−19.0000	−0.701781	−0.350891	−	0.936416i	$-0.614121\pi$
$733$	−19.0000	−0.701781	−0.350891	+	0.936416i	$0.614121\pi$
$734$	−36.0000	−1.32878
$735$	−3.00000	−0.110657
$736$	8.00000	0.294884
$737$	0	0
$738$	−20.0000	−0.736210
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	48.0000	1.76452
$741$	0	0
$742$	18.0000	0.660801
$743$	26.0000	0.953847	0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000	0.953847	0.476924	+	0.878945i	$0.341752\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	52.0000	1.90386
$747$	−9.00000	−0.329293
$748$	0	0
$749$	12.0000	0.438470
$750$	−6.00000	−0.219089
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	−28.0000	−1.02105
$753$	28.0000	1.02038
$754$	0	0
$755$	60.0000	2.18362
$756$	−2.00000	−0.0727393
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	−8.00000	−0.290573
$759$	0	0
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	16.0000	0.579619
$763$	−14.0000	−0.506834
$764$	−16.0000	−0.578860
$765$	−6.00000	−0.216930
$766$	−32.0000	−1.15621
$767$	0	0
$768$	16.0000	0.577350
$769$	−1.00000	−0.0360609	−0.0180305	−	0.999837i	$-0.505740\pi$
$769$	−1.00000	−0.0360609	−0.0180305	+	0.999837i	$0.505740\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	28.0000	1.00774
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	18.0000	0.646997
$775$	−20.0000	−0.718421
$776$	0	0
$777$	8.00000	0.286998
$778$	60.0000	2.15110
$779$	−10.0000	−0.358287
$780$	0	0
$781$	0	0
$782$	−4.00000	−0.143040
$783$	5.00000	0.178685
$784$	−4.00000	−0.142857
$785$	6.00000	0.214149
$786$	36.0000	1.28408
$787$	7.00000	0.249523	0.124762	−	0.992187i	$-0.460183\pi$
$787$	7.00000	0.249523	0.124762	+	0.992187i	$0.460183\pi$
$788$	−24.0000	−0.854965
$789$	19.0000	0.676418
$790$	90.0000	3.20206
$791$	21.0000	0.746674
$792$	0	0
$793$	0	0
$794$	−34.0000	−1.20661
$795$	−27.0000	−0.957591
$796$	−20.0000	−0.708881
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	−2.00000	−0.0707992
$799$	14.0000	0.495284
$800$	32.0000	1.13137
$801$	−9.00000	−0.317999
$802$	20.0000	0.706225
$803$	0	0
$804$	−4.00000	−0.141069
$805$	3.00000	0.105736
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	−45.0000	−1.58212	−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000	−1.58212	−0.791058	+	0.611741i	$0.790469\pi$
$810$	6.00000	0.210819
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−10.0000	−0.350931
$813$	20.0000	0.701431
$814$	0	0
$815$	18.0000	0.630512
$816$	−8.00000	−0.280056
$817$	9.00000	0.314870
$818$	62.0000	2.16778
$819$	0	0
$820$	−60.0000	−2.09529
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	−24.0000	−0.837096
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	0	0
$826$	−8.00000	−0.278356
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	2.00000	0.0695048
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	−54.0000	−1.87437
$831$	−3.00000	−0.104069
$832$	0	0
$833$	2.00000	0.0692959
$834$	40.0000	1.38509
$835$	9.00000	0.311458
$836$	0	0
$837$	−5.00000	−0.172825
$838$	40.0000	1.38178
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	20.0000	0.689246
$843$	10.0000	0.344418
$844$	−26.0000	−0.894957
$845$	0	0
$846$	−14.0000	−0.481330
$847$	11.0000	0.377964
$848$	−36.0000	−1.23625
$849$	−24.0000	−0.823678
$850$	−16.0000	−0.548795
$851$	−8.00000	−0.274236
$852$	0	0
$853$	−1.00000	−0.0342393	−0.0171197	−	0.999853i	$-0.505450\pi$
$853$	−1.00000	−0.0342393	−0.0171197	+	0.999853i	$0.505450\pi$
$854$	−16.0000	−0.547509
$855$	3.00000	0.102598
$856$	0	0
$857$	−8.00000	−0.273275	−0.136637	−	0.990621i	$-0.543630\pi$
$857$	−8.00000	−0.273275	−0.136637	+	0.990621i	$0.543630\pi$
$858$	0	0
$859$	30.0000	1.02359	0.511793	−	0.859109i	$-0.328981\pi$
$859$	30.0000	1.02359	0.511793	+	0.859109i	$0.328981\pi$
$860$	54.0000	1.84138
$861$	−10.0000	−0.340799
$862$	40.0000	1.36241
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	8.00000	0.272166
$865$	72.0000	2.44807
$866$	−12.0000	−0.407777
$867$	−13.0000	−0.441503
$868$	10.0000	0.339422
$869$	0	0
$870$	30.0000	1.01710
$871$	0	0
$872$	0	0
$873$	13.0000	0.439983
$874$	2.00000	0.0676510
$875$	−3.00000	−0.101419
$876$	18.0000	0.608164
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	−31.0000	−1.04560
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−2.00000	−0.0673435
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	62.0000	2.08293
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	−54.0000	−1.81008
$891$	0	0
$892$	−38.0000	−1.27233
$893$	−7.00000	−0.234246
$894$	−8.00000	−0.267560
$895$	−15.0000	−0.501395
$896$	0	0
$897$	0	0
$898$	68.0000	2.26919
$899$	−25.0000	−0.833797
$900$	8.00000	0.266667
$901$	18.0000	0.599667
$902$	0	0
$903$	9.00000	0.299501
$904$	0	0
$905$	−24.0000	−0.797787
$906$	40.0000	1.32891
$907$	27.0000	0.896520	0.448260	−	0.893903i	$-0.352044\pi$
$907$	27.0000	0.896520	0.448260	+	0.893903i	$0.352044\pi$
$908$	−24.0000	−0.796468
$909$	8.00000	0.265343
$910$	0	0
$911$	−33.0000	−1.09334	−0.546669	−	0.837349i	$-0.684105\pi$
$911$	−33.0000	−1.09334	−0.546669	+	0.837349i	$0.684105\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	84.0000	2.77847
$915$	24.0000	0.793416
$916$	−28.0000	−0.925146
$917$	18.0000	0.594412
$918$	−4.00000	−0.132020
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	17.0000	0.560169
$922$	−20.0000	−0.658665
$923$	0	0
$924$	0	0
$925$	−32.0000	−1.05215
$926$	52.0000	1.70883
$927$	16.0000	0.525509
$928$	40.0000	1.31306
$929$	29.0000	0.951459	0.475730	−	0.879592i	$-0.342184\pi$
$929$	29.0000	0.951459	0.475730	+	0.879592i	$0.342184\pi$
$930$	−30.0000	−0.983739
$931$	−1.00000	−0.0327737
$932$	2.00000	0.0655122
$933$	−2.00000	−0.0654771
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	−4.00000	−0.130605
$939$	4.00000	0.130535
$940$	−42.0000	−1.36989
$941$	−5.00000	−0.162995	−0.0814977	−	0.996674i	$-0.525970\pi$
$941$	−5.00000	−0.162995	−0.0814977	+	0.996674i	$0.525970\pi$
$942$	4.00000	0.130327
$943$	10.0000	0.325645
$944$	16.0000	0.520756
$945$	3.00000	0.0975900
$946$	0	0
$947$	2.00000	0.0649913	0.0324956	−	0.999472i	$-0.489654\pi$
$947$	2.00000	0.0649913	0.0324956	+	0.999472i	$0.489654\pi$
$948$	30.0000	0.974355
$949$	0	0
$950$	8.00000	0.259554
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	−29.0000	−0.939402	−0.469701	−	0.882826i	$-0.655638\pi$
$953$	−29.0000	−0.939402	−0.469701	+	0.882826i	$0.655638\pi$
$954$	−18.0000	−0.582772
$955$	24.0000	0.776622
$956$	−12.0000	−0.388108
$957$	0	0
$958$	−42.0000	−1.35696
$959$	−12.0000	−0.387500
$960$	24.0000	0.774597
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−12.0000	−0.386695
$964$	−30.0000	−0.966235
$965$	−42.0000	−1.35203
$966$	2.00000	0.0643489
$967$	−52.0000	−1.67221	−0.836104	−	0.548572i	$-0.815172\pi$
$967$	−52.0000	−1.67221	−0.836104	+	0.548572i	$0.815172\pi$
$968$	0	0
$969$	−2.00000	−0.0642493
$970$	78.0000	2.50443
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	2.00000	0.0641500
$973$	20.0000	0.641171
$974$	−16.0000	−0.512673
$975$	0	0
$976$	32.0000	1.02430
$977$	−52.0000	−1.66363	−0.831814	−	0.555055i	$-0.812697\pi$
$977$	−52.0000	−1.66363	−0.831814	+	0.555055i	$0.812697\pi$
$978$	12.0000	0.383718
$979$	0	0
$980$	−6.00000	−0.191663
$981$	14.0000	0.446986
$982$	64.0000	2.04232
$983$	−31.0000	−0.988746	−0.494373	−	0.869250i	$-0.664602\pi$
$983$	−31.0000	−0.988746	−0.494373	+	0.869250i	$0.664602\pi$
$984$	0	0
$985$	36.0000	1.14706
$986$	−20.0000	−0.636930
$987$	−7.00000	−0.222812
$988$	0	0
$989$	−9.00000	−0.286183
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−40.0000	−1.27000
$993$	−20.0000	−0.634681
$994$	0	0
$995$	30.0000	0.951064
$996$	−18.0000	−0.570352
$997$	−32.0000	−1.01345	−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000	−1.01345	−0.506725	+	0.862108i	$0.669144\pi$
$998$	−28.0000	−0.886325
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3549.2.a.a.1.1		1
13.5	odd	4		273.2.c.a.64.2	yes	2
13.8	odd	4		273.2.c.a.64.1	✓	2
13.12	even	2		3549.2.a.e.1.1		1
39.5	even	4		819.2.c.a.64.1		2
39.8	even	4		819.2.c.a.64.2		2
52.31	even	4		4368.2.h.e.337.2		2
52.47	even	4		4368.2.h.e.337.1		2
91.34	even	4		1911.2.c.a.883.1		2
91.83	even	4		1911.2.c.a.883.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.c.a.64.1	✓	2	13.8	odd	4
273.2.c.a.64.2	yes	2	13.5	odd	4
819.2.c.a.64.1		2	39.5	even	4
819.2.c.a.64.2		2	39.8	even	4
1911.2.c.a.883.1		2	91.34	even	4
1911.2.c.a.883.2		2	91.83	even	4
3549.2.a.a.1.1		1	1.1	even	1	trivial
3549.2.a.e.1.1		1	13.12	even	2
4368.2.h.e.337.1		2	52.47	even	4
4368.2.h.e.337.2		2	52.31	even	4

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:	\( N \)	\(=\)	\( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3549.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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