LMFDB - Embedded newform 8649.2.a.bm.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 22x^{10} + 168x^{8} - 554x^{6} + 792x^{4} - 388x^{2} + 1 $ x^12 - 22x^10 + 168x^8 - 554x^6 + 792x^4 - 388*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.46512$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.273088 q^{2} -1.92542 q^{4} -3.38873 q^{5} +1.63268 q^{7} -1.07199 q^{8} +O(q^{10})$	$q+0.273088 q^{2} -1.92542 q^{4} -3.38873 q^{5} +1.63268 q^{7} -1.07199 q^{8} -0.925423 q^{10} +3.66258 q^{11} +2.72296 q^{13} +0.445864 q^{14} +3.55810 q^{16} +7.08082 q^{17} +7.48352 q^{19} +6.52475 q^{20} +1.00021 q^{22} +1.90222 q^{23} +6.48352 q^{25} +0.743608 q^{26} -3.14359 q^{28} +6.93896 q^{29} +3.11565 q^{32} +1.93369 q^{34} -5.53271 q^{35} +2.20349 q^{37} +2.04366 q^{38} +3.63268 q^{40} -9.62003 q^{41} +1.30875 q^{43} -7.05202 q^{44} +0.519474 q^{46} -11.4110 q^{47} -4.33437 q^{49} +1.77057 q^{50} -5.24285 q^{52} +6.30841 q^{53} -12.4115 q^{55} -1.75021 q^{56} +1.89495 q^{58} +4.48109 q^{59} +7.37961 q^{61} -6.26535 q^{64} -9.22739 q^{65} -7.70169 q^{67} -13.6336 q^{68} -1.51092 q^{70} +4.48109 q^{71} +10.6888 q^{73} +0.601745 q^{74} -14.4089 q^{76} +5.97981 q^{77} +4.92644 q^{79} -12.0575 q^{80} -2.62711 q^{82} +6.33721 q^{83} -23.9950 q^{85} +0.357403 q^{86} -3.92624 q^{88} -15.6777 q^{89} +4.44571 q^{91} -3.66258 q^{92} -3.11620 q^{94} -25.3597 q^{95} +9.33437 q^{97} -1.18366 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{4} + 8 q^{7}+O(q^{10}) $ 12 * q + 8 * q^4 + 8 * q^7	$ 12 q + 8 q^{4} + 8 q^{7} + 20 q^{10} + 16 q^{19} + 4 q^{25} + 44 q^{28} + 32 q^{40} + 84 q^{49} - 52 q^{64} + 32 q^{67} + 52 q^{70} - 68 q^{76} + 124 q^{82} + 48 q^{94} - 24 q^{97}+O(q^{100}) $ 12 * q + 8 * q^4 + 8 * q^7 + 20 * q^10 + 16 * q^19 + 4 * q^25 + 44 * q^28 + 32 * q^40 + 84 * q^49 - 52 * q^64 + 32 * q^67 + 52 * q^70 - 68 * q^76 + 124 * q^82 + 48 * q^94 - 24 * q^97

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 $

$2$	0.273088	0.193102	0.0965512	−	0.995328i	$-0.469219\pi$
$2$	0.273088	0.193102	0.0965512	+	0.995328i	$0.469219\pi$
$3$	0	0
$3$	0	0
$4$	−1.92542	−0.962711
$5$	−3.38873	−1.51549	−0.757744	−	0.652552i	$-0.773698\pi$
$5$	−3.38873	−1.51549	−0.757744	+	0.652552i	$0.773698\pi$
$6$	0	0
$7$	1.63268	0.617094	0.308547	−	0.951209i	$-0.400157\pi$
$7$	1.63268	0.617094	0.308547	+	0.951209i	$0.400157\pi$
$8$	−1.07199	−0.379004
$9$	0	0
$10$	−0.925423	−0.292644
$11$	3.66258	1.10431	0.552155	−	0.833741i	$-0.313806\pi$
$11$	3.66258	1.10431	0.552155	+	0.833741i	$0.313806\pi$
$12$	0	0
$13$	2.72296	0.755213	0.377607	−	0.925966i	$-0.376747\pi$
$13$	2.72296	0.755213	0.377607	+	0.925966i	$0.376747\pi$
$14$	0.445864	0.119162
$15$	0	0
$16$	3.55810	0.889525
$17$	7.08082	1.71735	0.858676	−	0.512519i	$-0.171288\pi$
$17$	7.08082	1.71735	0.858676	+	0.512519i	$0.171288\pi$
$18$	0	0
$19$	7.48352	1.71684	0.858419	−	0.512949i	$-0.171447\pi$
$19$	7.48352	1.71684	0.858419	+	0.512949i	$0.171447\pi$
$20$	6.52475	1.45898
$21$	0	0
$22$	1.00021	0.213245
$23$	1.90222	0.396641	0.198320	−	0.980137i	$-0.436451\pi$
$23$	1.90222	0.396641	0.198320	+	0.980137i	$0.436451\pi$
$24$	0	0
$25$	6.48352	1.29670
$26$	0.743608	0.145833
$27$	0	0
$28$	−3.14359	−0.594083
$29$	6.93896	1.28853	0.644266	−	0.764801i	$-0.277163\pi$
$29$	6.93896	1.28853	0.644266	+	0.764801i	$0.277163\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	3.11565	0.550774
$33$	0	0
$34$	1.93369	0.331625
$35$	−5.53271	−0.935198
$36$	0	0
$37$	2.20349	0.362251	0.181125	−	0.983460i	$-0.442026\pi$
$37$	2.20349	0.362251	0.181125	+	0.983460i	$0.442026\pi$
$38$	2.04366	0.331526
$39$	0	0
$40$	3.63268	0.574377
$41$	−9.62003	−1.50240	−0.751198	−	0.660077i	$-0.770524\pi$
$41$	−9.62003	−1.50240	−0.751198	+	0.660077i	$0.770524\pi$
$42$	0	0
$43$	1.30875	0.199582	0.0997909	−	0.995008i	$-0.468183\pi$
$43$	1.30875	0.199582	0.0997909	+	0.995008i	$0.468183\pi$
$44$	−7.05202	−1.06313
$45$	0	0
$46$	0.519474	0.0765923
$47$	−11.4110	−1.66446	−0.832230	−	0.554430i	$-0.812936\pi$
$47$	−11.4110	−1.66446	−0.832230	+	0.554430i	$0.812936\pi$
$48$	0	0
$49$	−4.33437	−0.619195
$50$	1.77057	0.250397
$51$	0	0
$52$	−5.24285	−0.727052
$53$	6.30841	0.866527	0.433263	−	0.901267i	$-0.357362\pi$
$53$	6.30841	0.866527	0.433263	+	0.901267i	$0.357362\pi$
$54$	0	0
$55$	−12.4115	−1.67357
$56$	−1.75021	−0.233881
$57$	0	0
$58$	1.89495	0.248819
$59$	4.48109	0.583388	0.291694	−	0.956512i	$-0.405781\pi$
$59$	4.48109	0.583388	0.291694	+	0.956512i	$0.405781\pi$
$60$	0	0
$61$	7.37961	0.944862	0.472431	−	0.881368i	$-0.343377\pi$
$61$	7.37961	0.944862	0.472431	+	0.881368i	$0.343377\pi$
$62$	0	0
$63$	0	0
$64$	−6.26535	−0.783169
$65$	−9.22739	−1.14452
$66$	0	0
$67$	−7.70169	−0.940911	−0.470456	−	0.882424i	$-0.655910\pi$
$67$	−7.70169	−0.940911	−0.470456	+	0.882424i	$0.655910\pi$
$68$	−13.6336	−1.65331
$69$	0	0
$70$	−1.51092	−0.180589
$71$	4.48109	0.531807	0.265904	−	0.964000i	$-0.414330\pi$
$71$	4.48109	0.531807	0.265904	+	0.964000i	$0.414330\pi$
$72$	0	0
$73$	10.6888	1.25103	0.625513	−	0.780214i	$-0.284890\pi$
$73$	10.6888	1.25103	0.625513	+	0.780214i	$0.284890\pi$
$74$	0.601745	0.0699515
$75$	0	0
$76$	−14.4089	−1.65282
$77$	5.97981	0.681463
$78$	0	0
$79$	4.92644	0.554268	0.277134	−	0.960831i	$-0.410615\pi$
$79$	4.92644	0.554268	0.277134	+	0.960831i	$0.410615\pi$
$80$	−12.0575	−1.34806
$81$	0	0
$82$	−2.62711	−0.290116
$83$	6.33721	0.695600	0.347800	−	0.937569i	$-0.386929\pi$
$83$	6.33721	0.695600	0.347800	+	0.937569i	$0.386929\pi$
$84$	0	0
$85$	−23.9950	−2.60263
$86$	0.357403	0.0385397
$87$	0	0
$88$	−3.92624	−0.418538
$89$	−15.6777	−1.66183	−0.830914	−	0.556400i	$-0.812182\pi$
$89$	−15.6777	−1.66183	−0.830914	+	0.556400i	$0.812182\pi$
$90$	0	0
$91$	4.44571	0.466037
$92$	−3.66258	−0.381851
$93$	0	0
$94$	−3.11620	−0.321411
$95$	−25.3597	−2.60185
$96$	0	0
$97$	9.33437	0.947761	0.473881	−	0.880589i	$-0.342853\pi$
$97$	9.33437	0.947761	0.473881	+	0.880589i	$0.342853\pi$
$98$	−1.18366	−0.119568
$99$	0	0
$100$	−12.4835	−1.24835
$101$	−5.87826	−0.584909	−0.292454	−	0.956279i	$-0.594472\pi$
$101$	−5.87826	−0.584909	−0.292454	+	0.956279i	$0.594472\pi$
$102$	0	0
$103$	10.3673	1.02152	0.510761	−	0.859723i	$-0.329364\pi$
$103$	10.3673	1.02152	0.510761	+	0.859723i	$0.329364\pi$
$104$	−2.91897	−0.286229
$105$	0	0
$106$	1.72275	0.167328
$107$	3.58936	0.346996	0.173498	−	0.984834i	$-0.444493\pi$
$107$	3.58936	0.346996	0.173498	+	0.984834i	$0.444493\pi$
$108$	0	0
$109$	1.48352	0.142096	0.0710478	−	0.997473i	$-0.477366\pi$
$109$	1.48352	0.142096	0.0710478	+	0.997473i	$0.477366\pi$
$110$	−3.38944	−0.323170
$111$	0	0
$112$	5.80922	0.548920
$113$	−8.56841	−0.806048	−0.403024	−	0.915189i	$-0.632041\pi$
$113$	−8.56841	−0.806048	−0.403024	+	0.915189i	$0.632041\pi$
$114$	0	0
$115$	−6.44613	−0.601104
$116$	−13.3604	−1.24048
$117$	0	0
$118$	1.22373	0.112654
$119$	11.5607	1.05977
$120$	0	0
$121$	2.41451	0.219501
$122$	2.01528	0.182455
$123$	0	0
$124$	0	0
$125$	−5.02726	−0.449652
$126$	0	0
$127$	20.8968	1.85429	0.927146	−	0.374701i	$-0.122255\pi$
$127$	20.8968	1.85429	0.927146	+	0.374701i	$0.122255\pi$
$128$	−7.94229	−0.702006
$129$	0	0
$130$	−2.51989	−0.221009
$131$	−9.50835	−0.830748	−0.415374	−	0.909651i	$-0.636349\pi$
$131$	−9.50835	−0.830748	−0.415374	+	0.909651i	$0.636349\pi$
$132$	0	0
$133$	12.2182	1.05945
$134$	−2.10324	−0.181692
$135$	0	0
$136$	−7.59054	−0.650884
$137$	−7.92691	−0.677242	−0.338621	−	0.940923i	$-0.609960\pi$
$137$	−7.92691	−0.677242	−0.338621	+	0.940923i	$0.609960\pi$
$138$	0	0
$139$	19.8579	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$139$	19.8579	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$140$	10.6528	0.900326
$141$	0	0
$142$	1.22373	0.102693
$143$	9.97306	0.833989
$144$	0	0
$145$	−23.5143	−1.95276
$146$	2.91897	0.241576
$147$	0	0
$148$	−4.24264	−0.348743
$149$	−0.850997	−0.0697164	−0.0348582	−	0.999392i	$-0.511098\pi$
$149$	−0.850997	−0.0697164	−0.0348582	+	0.999392i	$0.511098\pi$
$150$	0	0
$151$	−8.54414	−0.695312	−0.347656	−	0.937622i	$-0.613022\pi$
$151$	−8.54414	−0.695312	−0.347656	+	0.937622i	$0.613022\pi$
$152$	−8.02223	−0.650689
$153$	0	0
$154$	1.63301	0.131592
$155$	0	0
$156$	0	0
$157$	−5.10197	−0.407182	−0.203591	−	0.979056i	$-0.565261\pi$
$157$	−5.10197	−0.407182	−0.203591	+	0.979056i	$0.565261\pi$
$158$	1.34535	0.107031
$159$	0	0
$160$	−10.5581	−0.834691
$161$	3.10571	0.244764
$162$	0	0
$163$	−12.8980	−1.01025	−0.505126	−	0.863046i	$-0.668554\pi$
$163$	−12.8980	−1.01025	−0.505126	+	0.863046i	$0.668554\pi$
$164$	18.5226	1.44637
$165$	0	0
$166$	1.73062	0.134322
$167$	−6.19535	−0.479411	−0.239705	−	0.970846i	$-0.577051\pi$
$167$	−6.19535	−0.479411	−0.239705	+	0.970846i	$0.577051\pi$
$168$	0	0
$169$	−5.58549	−0.429653
$170$	−6.55275	−0.502573
$171$	0	0
$172$	−2.51989	−0.192140
$173$	20.3324	1.54584	0.772922	−	0.634501i	$-0.218794\pi$
$173$	20.3324	1.54584	0.772922	+	0.634501i	$0.218794\pi$
$174$	0	0
$175$	10.5855	0.800188
$176$	13.0318	0.982311
$177$	0	0
$178$	−4.28138	−0.320903
$179$	−9.58479	−0.716401	−0.358200	−	0.933645i	$-0.616610\pi$
$179$	−9.58479	−0.716401	−0.358200	+	0.933645i	$0.616610\pi$
$180$	0	0
$181$	9.38002	0.697211	0.348606	−	0.937269i	$-0.386655\pi$
$181$	9.38002	0.697211	0.348606	+	0.937269i	$0.386655\pi$
$182$	1.21407	0.0899929
$183$	0	0
$184$	−2.03916	−0.150329
$185$	−7.46703	−0.548987
$186$	0	0
$187$	25.9341	1.89649
$188$	21.9709	1.60240
$189$	0	0
$190$	−6.92542	−0.502423
$191$	13.9487	1.00929	0.504646	−	0.863326i	$-0.331623\pi$
$191$	13.9487	1.00929	0.504646	+	0.863326i	$0.331623\pi$
$192$	0	0
$193$	−0.218170	−0.0157042	−0.00785209	−	0.999969i	$-0.502499\pi$
$193$	−0.218170	−0.0157042	−0.00785209	+	0.999969i	$0.502499\pi$
$194$	2.54910	0.183015
$195$	0	0
$196$	8.34549	0.596107
$197$	9.75545	0.695047	0.347524	−	0.937671i	$-0.387023\pi$
$197$	9.75545	0.695047	0.347524	+	0.937671i	$0.387023\pi$
$198$	0	0
$199$	19.8579	1.40769	0.703843	−	0.710356i	$-0.251466\pi$
$199$	19.8579	1.40769	0.703843	+	0.710356i	$0.251466\pi$
$200$	−6.95025	−0.491457
$201$	0	0
$202$	−1.60528	−0.112947
$203$	11.3291	0.795145
$204$	0	0
$205$	32.5997	2.27686
$206$	2.83119	0.197259
$207$	0	0
$208$	9.68856	0.671781
$209$	27.4090	1.89592
$210$	0	0
$211$	8.95282	0.616337	0.308169	−	0.951332i	$-0.400284\pi$
$211$	8.95282	0.616337	0.308169	+	0.951332i	$0.400284\pi$
$212$	−12.1464	−0.834215
$213$	0	0
$214$	0.980211	0.0670058
$215$	−4.43499	−0.302464
$216$	0	0
$217$	0	0
$218$	0.405132	0.0274390
$219$	0	0
$220$	23.8974	1.61116
$221$	19.2808	1.29697
$222$	0	0
$223$	−26.0342	−1.74338	−0.871689	−	0.490060i	$-0.836975\pi$
$223$	−26.0342	−1.74338	−0.871689	+	0.490060i	$0.836975\pi$
$224$	5.08684	0.339879
$225$	0	0
$226$	−2.33993	−0.155650
$227$	16.9919	1.12779	0.563896	−	0.825846i	$-0.309302\pi$
$227$	16.9919	1.12779	0.563896	+	0.825846i	$0.309302\pi$
$228$	0	0
$229$	−9.64982	−0.637678	−0.318839	−	0.947809i	$-0.603293\pi$
$229$	−9.64982	−0.637678	−0.318839	+	0.947809i	$0.603293\pi$
$230$	−1.76036	−0.116075
$231$	0	0
$232$	−7.43847	−0.488359
$233$	9.66076	0.632897	0.316449	−	0.948610i	$-0.397509\pi$
$233$	9.66076	0.632897	0.316449	+	0.948610i	$0.397509\pi$
$234$	0	0
$235$	38.6687	2.52247
$236$	−8.62799	−0.561634
$237$	0	0
$238$	3.15709	0.204643
$239$	−24.0196	−1.55370	−0.776849	−	0.629687i	$-0.783183\pi$
$239$	−24.0196	−1.55370	−0.776849	+	0.629687i	$0.783183\pi$
$240$	0	0
$241$	−17.1736	−1.10625	−0.553125	−	0.833098i	$-0.686565\pi$
$241$	−17.1736	−1.10625	−0.553125	+	0.833098i	$0.686565\pi$
$242$	0.659373	0.0423861
$243$	0	0
$244$	−14.2089	−0.909629
$245$	14.6880	0.938383
$246$	0	0
$247$	20.3773	1.29658
$248$	0	0
$249$	0	0
$250$	−1.37289	−0.0868289
$251$	10.2153	0.644786	0.322393	−	0.946606i	$-0.395513\pi$
$251$	10.2153	0.644786	0.322393	+	0.946606i	$0.395513\pi$
$252$	0	0
$253$	6.96704	0.438014
$254$	5.70667	0.358068
$255$	0	0
$256$	10.3618	0.647610
$257$	−21.9227	−1.36750	−0.683751	−	0.729715i	$-0.739653\pi$
$257$	−21.9227	−1.36750	−0.683751	+	0.729715i	$0.739653\pi$
$258$	0	0
$259$	3.59758	0.223543
$260$	17.7666	1.10184
$261$	0	0
$262$	−2.59662	−0.160420
$263$	−14.5927	−0.899826	−0.449913	−	0.893072i	$-0.648545\pi$
$263$	−14.5927	−0.899826	−0.449913	+	0.893072i	$0.648545\pi$
$264$	0	0
$265$	−21.3775	−1.31321
$266$	3.33664	0.204582
$267$	0	0
$268$	14.8290	0.905826
$269$	−26.6078	−1.62231	−0.811153	−	0.584834i	$-0.801160\pi$
$269$	−26.6078	−1.62231	−0.811153	+	0.584834i	$0.801160\pi$
$270$	0	0
$271$	9.33341	0.566965	0.283482	−	0.958977i	$-0.408510\pi$
$271$	9.33341	0.566965	0.283482	+	0.958977i	$0.408510\pi$
$272$	25.1943	1.52763
$273$	0	0
$274$	−2.16474	−0.130777
$275$	23.7464	1.43196
$276$	0	0
$277$	−7.68814	−0.461936	−0.230968	−	0.972961i	$-0.574189\pi$
$277$	−7.68814	−0.461936	−0.230968	+	0.972961i	$0.574189\pi$
$278$	5.42294	0.325246
$279$	0	0
$280$	5.93098	0.354444
$281$	19.6338	1.17126	0.585628	−	0.810580i	$-0.300848\pi$
$281$	19.6338	1.17126	0.585628	+	0.810580i	$0.300848\pi$
$282$	0	0
$283$	−17.4977	−1.04013	−0.520067	−	0.854126i	$-0.674093\pi$
$283$	−17.4977	−1.04013	−0.520067	+	0.854126i	$0.674093\pi$
$284$	−8.62799	−0.511977
$285$	0	0
$286$	2.72352	0.161045
$287$	−15.7064	−0.927119
$288$	0	0
$289$	33.1380	1.94930
$290$	−6.42147	−0.377082
$291$	0	0
$292$	−20.5804	−1.20438
$293$	−0.200623	−0.0117205	−0.00586027	−	0.999983i	$-0.501865\pi$
$293$	−0.200623	−0.0117205	−0.00586027	+	0.999983i	$0.501865\pi$
$294$	0	0
$295$	−15.1852	−0.884117
$296$	−2.36211	−0.137295
$297$	0	0
$298$	−0.232397	−0.0134624
$299$	5.17967	0.299548
$300$	0	0
$301$	2.13676	0.123161
$302$	−2.33330	−0.134267
$303$	0	0
$304$	26.6271	1.52717
$305$	−25.0075	−1.43193
$306$	0	0
$307$	−10.7489	−0.613471	−0.306735	−	0.951795i	$-0.599237\pi$
$307$	−10.7489	−0.613471	−0.306735	+	0.951795i	$0.599237\pi$
$308$	−11.5137	−0.656052
$309$	0	0
$310$	0	0
$311$	−14.2535	−0.808243	−0.404122	−	0.914705i	$-0.632423\pi$
$311$	−14.2535	−0.808243	−0.404122	+	0.914705i	$0.632423\pi$
$312$	0	0
$313$	17.8653	1.00981	0.504903	−	0.863176i	$-0.331528\pi$
$313$	17.8653	1.00981	0.504903	+	0.863176i	$0.331528\pi$
$314$	−1.39329	−0.0786278
$315$	0	0
$316$	−9.48549	−0.533600
$317$	1.20403	0.0676251	0.0338125	−	0.999428i	$-0.489235\pi$
$317$	1.20403	0.0676251	0.0338125	+	0.999428i	$0.489235\pi$
$318$	0	0
$319$	25.4145	1.42294
$320$	21.2316	1.18688
$321$	0	0
$322$	0.848133	0.0472646
$323$	52.9895	2.94841
$324$	0	0
$325$	17.6544	0.979288
$326$	−3.52230	−0.195082
$327$	0	0
$328$	10.3125	0.569415
$329$	−18.6304	−1.02713
$330$	0	0
$331$	−13.8336	−0.760363	−0.380182	−	0.924912i	$-0.624139\pi$
$331$	−13.8336	−0.760363	−0.380182	+	0.924912i	$0.624139\pi$
$332$	−12.2018	−0.669662
$333$	0	0
$334$	−1.69188	−0.0925754
$335$	26.0990	1.42594
$336$	0	0
$337$	32.1050	1.74887	0.874436	−	0.485140i	$-0.161231\pi$
$337$	32.1050	1.74887	0.874436	+	0.485140i	$0.161231\pi$
$338$	−1.52533	−0.0829671
$339$	0	0
$340$	46.2006	2.50558
$341$	0	0
$342$	0	0
$343$	−18.5054	−0.999195
$344$	−1.40296	−0.0756424
$345$	0	0
$346$	5.55254	0.298506
$347$	24.5349	1.31710	0.658552	−	0.752536i	$-0.271169\pi$
$347$	24.5349	1.31710	0.658552	+	0.752536i	$0.271169\pi$
$348$	0	0
$349$	−0.734647	−0.0393248	−0.0196624	−	0.999807i	$-0.506259\pi$
$349$	−0.734647	−0.0393248	−0.0196624	+	0.999807i	$0.506259\pi$
$350$	2.89077	0.154518
$351$	0	0
$352$	11.4113	0.608225
$353$	−16.2506	−0.864932	−0.432466	−	0.901650i	$-0.642357\pi$
$353$	−16.2506	−0.864932	−0.432466	+	0.901650i	$0.642357\pi$
$354$	0	0
$355$	−15.1852	−0.805947
$356$	30.1861	1.59986
$357$	0	0
$358$	−2.61749	−0.138339
$359$	−24.5087	−1.29352	−0.646759	−	0.762694i	$-0.723876\pi$
$359$	−24.5087	−1.29352	−0.646759	+	0.762694i	$0.723876\pi$
$360$	0	0
$361$	37.0031	1.94753
$362$	2.56157	0.134633
$363$	0	0
$364$	−8.55987	−0.448659
$365$	−36.2214	−1.89591
$366$	0	0
$367$	33.5394	1.75074	0.875371	−	0.483453i	$-0.160617\pi$
$367$	33.5394	1.75074	0.875371	+	0.483453i	$0.160617\pi$
$368$	6.76829	0.352822
$369$	0	0
$370$	−2.03916	−0.106011
$371$	10.2996	0.534728
$372$	0	0
$373$	−9.86507	−0.510794	−0.255397	−	0.966836i	$-0.582206\pi$
$373$	−9.86507	−0.510794	−0.255397	+	0.966836i	$0.582206\pi$
$374$	7.08229	0.366216
$375$	0	0
$376$	12.2324	0.630838
$377$	18.8945	0.973116
$378$	0	0
$379$	−28.8179	−1.48028	−0.740138	−	0.672455i	$-0.765240\pi$
$379$	−28.8179	−1.48028	−0.740138	+	0.672455i	$0.765240\pi$
$380$	48.8281	2.50483
$381$	0	0
$382$	3.80922	0.194897
$383$	−15.9220	−0.813576	−0.406788	−	0.913523i	$-0.633351\pi$
$383$	−15.9220	−0.813576	−0.406788	+	0.913523i	$0.633351\pi$
$384$	0	0
$385$	−20.2640	−1.03275
$386$	−0.0595795	−0.00303252
$387$	0	0
$388$	−17.9726	−0.912421
$389$	−36.2502	−1.83796	−0.918979	−	0.394307i	$-0.870985\pi$
$389$	−36.2502	−1.83796	−0.918979	+	0.394307i	$0.870985\pi$
$390$	0	0
$391$	13.4693	0.681172
$392$	4.64638	0.234678
$393$	0	0
$394$	2.66410	0.134215
$395$	−16.6944	−0.839987
$396$	0	0
$397$	−25.0361	−1.25652	−0.628262	−	0.778002i	$-0.716233\pi$
$397$	−25.0361	−1.25652	−0.628262	+	0.778002i	$0.716233\pi$
$398$	5.42294	0.271828
$399$	0	0
$400$	23.0690	1.15345
$401$	−13.8330	−0.690789	−0.345395	−	0.938458i	$-0.612255\pi$
$401$	−13.8330	−0.690789	−0.345395	+	0.938458i	$0.612255\pi$
$402$	0	0
$403$	0	0
$404$	11.3181	0.563098
$405$	0	0
$406$	3.09383	0.153544
$407$	8.07044	0.400037
$408$	0	0
$409$	−16.6542	−0.823495	−0.411748	−	0.911298i	$-0.635081\pi$
$409$	−16.6542	−0.823495	−0.411748	+	0.911298i	$0.635081\pi$
$410$	8.90259	0.439668
$411$	0	0
$412$	−19.9615	−0.983432
$413$	7.31616	0.360005
$414$	0	0
$415$	−21.4751	−1.05417
$416$	8.48378	0.415951
$417$	0	0
$418$	7.48507	0.366107
$419$	6.07888	0.296973	0.148486	−	0.988914i	$-0.452560\pi$
$419$	6.07888	0.296973	0.148486	+	0.988914i	$0.452560\pi$
$420$	0	0
$421$	28.6545	1.39654	0.698268	−	0.715837i	$-0.253955\pi$
$421$	28.6545	1.39654	0.698268	+	0.715837i	$0.253955\pi$
$422$	2.44491	0.119016
$423$	0	0
$424$	−6.76253	−0.328417
$425$	45.9087	2.22690
$426$	0	0
$427$	12.0485	0.583068
$428$	−6.91103	−0.334057
$429$	0	0
$430$	−1.21114	−0.0584065
$431$	−27.4554	−1.32248	−0.661241	−	0.750174i	$-0.729970\pi$
$431$	−27.4554	−1.32248	−0.661241	+	0.750174i	$0.729970\pi$
$432$	0	0
$433$	−0.164329	−0.00789717	−0.00394859	−	0.999992i	$-0.501257\pi$
$433$	−0.164329	−0.00789717	−0.00394859	+	0.999992i	$0.501257\pi$
$434$	0	0
$435$	0	0
$436$	−2.85641	−0.136797
$437$	14.2353	0.680968
$438$	0	0
$439$	21.3344	1.01823	0.509117	−	0.860697i	$-0.329972\pi$
$439$	21.3344	1.01823	0.509117	+	0.860697i	$0.329972\pi$
$440$	13.3050	0.634290
$441$	0	0
$442$	5.26535	0.250447
$443$	−40.7538	−1.93627	−0.968135	−	0.250430i	$-0.919428\pi$
$443$	−40.7538	−1.93627	−0.968135	+	0.250430i	$0.919428\pi$
$444$	0	0
$445$	53.1274	2.51848
$446$	−7.10962	−0.336650
$447$	0	0
$448$	−10.2293	−0.483289
$449$	21.1006	0.995799	0.497899	−	0.867235i	$-0.334105\pi$
$449$	21.1006	0.995799	0.497899	+	0.867235i	$0.334105\pi$
$450$	0	0
$451$	−35.2341	−1.65911
$452$	16.4978	0.775992
$453$	0	0
$454$	4.64028	0.217779
$455$	−15.0653	−0.706274
$456$	0	0
$457$	−22.5219	−1.05353	−0.526766	−	0.850010i	$-0.676596\pi$
$457$	−22.5219	−1.05353	−0.526766	+	0.850010i	$0.676596\pi$
$458$	−2.63525	−0.123137
$459$	0	0
$460$	12.4115	0.578690
$461$	7.29636	0.339825	0.169913	−	0.985459i	$-0.445651\pi$
$461$	7.29636	0.339825	0.169913	+	0.985459i	$0.445651\pi$
$462$	0	0
$463$	−19.3972	−0.901466	−0.450733	−	0.892659i	$-0.648837\pi$
$463$	−19.3972	−0.901466	−0.450733	+	0.892659i	$0.648837\pi$
$464$	24.6895	1.14618
$465$	0	0
$466$	2.63824	0.122214
$467$	0.698586	0.0323267	0.0161634	−	0.999869i	$-0.494855\pi$
$467$	0.698586	0.0323267	0.0161634	+	0.999869i	$0.494855\pi$
$468$	0	0
$469$	−12.5744	−0.580630
$470$	10.5600	0.487095
$471$	0	0
$472$	−4.80366	−0.221106
$473$	4.79339	0.220400
$474$	0	0
$475$	48.5196	2.22623
$476$	−22.2592	−1.02025
$477$	0	0
$478$	−6.55946	−0.300023
$479$	35.7597	1.63390	0.816952	−	0.576705i	$-0.195662\pi$
$479$	35.7597	1.63390	0.816952	+	0.576705i	$0.195662\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	−4.68991	−0.213620
$483$	0	0
$484$	−4.64895	−0.211316
$485$	−31.6317	−1.43632
$486$	0	0
$487$	−26.8622	−1.21724	−0.608621	−	0.793461i	$-0.708277\pi$
$487$	−26.8622	−1.21724	−0.608621	+	0.793461i	$0.708277\pi$
$488$	−7.91084	−0.358107
$489$	0	0
$490$	4.01112	0.181204
$491$	1.18742	0.0535874	0.0267937	−	0.999641i	$-0.491470\pi$
$491$	1.18742	0.0535874	0.0267937	+	0.999641i	$0.491470\pi$
$492$	0	0
$493$	49.1335	2.21286
$494$	5.56480	0.250372
$495$	0	0
$496$	0	0
$497$	7.31616	0.328175
$498$	0	0
$499$	0.308538	0.0138121	0.00690604	−	0.999976i	$-0.497802\pi$
$499$	0.308538	0.0138121	0.00690604	+	0.999976i	$0.497802\pi$
$500$	9.67961	0.432885
$501$	0	0
$502$	2.78969	0.124510
$503$	−24.5087	−1.09279	−0.546394	−	0.837528i	$-0.684000\pi$
$503$	−24.5087	−1.09279	−0.546394	+	0.837528i	$0.684000\pi$
$504$	0	0
$505$	19.9199	0.886422
$506$	1.90262	0.0845816
$507$	0	0
$508$	−40.2352	−1.78515
$509$	0.102480	0.00454237	0.00227118	−	0.999997i	$-0.499277\pi$
$509$	0.102480	0.00454237	0.00227118	+	0.999997i	$0.499277\pi$
$510$	0	0
$511$	17.4513	0.772000
$512$	18.7142	0.827061
$513$	0	0
$514$	−5.98683	−0.264068
$515$	−35.1321	−1.54811
$516$	0	0
$517$	−41.7936	−1.83808
$518$	0.982456	0.0431666
$519$	0	0
$520$	9.89163	0.433777
$521$	−0.586908	−0.0257129	−0.0128565	−	0.999917i	$-0.504092\pi$
$521$	−0.586908	−0.0257129	−0.0128565	+	0.999917i	$0.504092\pi$
$522$	0	0
$523$	15.2601	0.667276	0.333638	−	0.942701i	$-0.391724\pi$
$523$	15.2601	0.667276	0.333638	+	0.942701i	$0.391724\pi$
$524$	18.3076	0.799771
$525$	0	0
$526$	−3.98510	−0.173759
$527$	0	0
$528$	0	0
$529$	−19.3816	−0.842676
$530$	−5.83795	−0.253584
$531$	0	0
$532$	−23.5251	−1.01994
$533$	−26.1949	−1.13463
$534$	0	0
$535$	−12.1634	−0.525869
$536$	8.25611	0.356609
$537$	0	0
$538$	−7.26627	−0.313271
$539$	−15.8750	−0.683784
$540$	0	0
$541$	6.89803	0.296569	0.148285	−	0.988945i	$-0.452625\pi$
$541$	6.89803	0.296569	0.148285	+	0.988945i	$0.452625\pi$
$542$	2.54884	0.109482
$543$	0	0
$544$	22.0613	0.945872
$545$	−5.02726	−0.215344
$546$	0	0
$547$	9.63268	0.411863	0.205932	−	0.978566i	$-0.433978\pi$
$547$	9.63268	0.411863	0.205932	+	0.978566i	$0.433978\pi$
$548$	15.2627	0.651988
$549$	0	0
$550$	6.48487	0.276516
$551$	51.9279	2.21220
$552$	0	0
$553$	8.04329	0.342035
$554$	−2.09954	−0.0892009
$555$	0	0
$556$	−38.2348	−1.62152
$557$	−11.2427	−0.476367	−0.238184	−	0.971220i	$-0.576552\pi$
$557$	−11.2427	−0.476367	−0.238184	+	0.971220i	$0.576552\pi$
$558$	0	0
$559$	3.56366	0.150727
$560$	−19.6859	−0.831882
$561$	0	0
$562$	5.36176	0.226172
$563$	−4.68171	−0.197311	−0.0986553	−	0.995122i	$-0.531454\pi$
$563$	−4.68171	−0.197311	−0.0986553	+	0.995122i	$0.531454\pi$
$564$	0	0
$565$	29.0361	1.22156
$566$	−4.77843	−0.200852
$567$	0	0
$568$	−4.80366	−0.201557
$569$	22.0725	0.925326	0.462663	−	0.886534i	$-0.346894\pi$
$569$	22.0725	0.925326	0.462663	+	0.886534i	$0.346894\pi$
$570$	0	0
$571$	10.5244	0.440434	0.220217	−	0.975451i	$-0.429323\pi$
$571$	10.5244	0.440434	0.220217	+	0.975451i	$0.429323\pi$
$572$	−19.2024	−0.802891
$573$	0	0
$574$	−4.28923	−0.179029
$575$	12.3331	0.514326
$576$	0	0
$577$	25.6469	1.06769	0.533847	−	0.845581i	$-0.320746\pi$
$577$	25.6469	1.06769	0.533847	+	0.845581i	$0.320746\pi$
$578$	9.04960	0.376414
$579$	0	0
$580$	45.2750	1.87994
$581$	10.3466	0.429250
$582$	0	0
$583$	23.1051	0.956914
$584$	−11.4582	−0.474144
$585$	0	0
$586$	−0.0547879	−0.00226327
$587$	−34.3086	−1.41607	−0.708034	−	0.706179i	$-0.750417\pi$
$587$	−34.3086	−1.41607	−0.708034	+	0.706179i	$0.750417\pi$
$588$	0	0
$589$	0	0
$590$	−4.14690	−0.170725
$591$	0	0
$592$	7.84022	0.322231
$593$	13.0977	0.537858	0.268929	−	0.963160i	$-0.413330\pi$
$593$	13.0977	0.537858	0.268929	+	0.963160i	$0.413330\pi$
$594$	0	0
$595$	−39.1761	−1.60606
$596$	1.63853	0.0671167
$597$	0	0
$598$	1.41451	0.0578435
$599$	−19.6338	−0.802216	−0.401108	−	0.916031i	$-0.631375\pi$
$599$	−19.6338	−0.802216	−0.401108	+	0.916031i	$0.631375\pi$
$600$	0	0
$601$	31.2383	1.27424	0.637118	−	0.770766i	$-0.280126\pi$
$601$	31.2383	1.27424	0.637118	+	0.770766i	$0.280126\pi$
$602$	0.583523	0.0237826
$603$	0	0
$604$	16.4511	0.669385
$605$	−8.18212	−0.332651
$606$	0	0
$607$	46.0721	1.87001	0.935005	−	0.354635i	$-0.115395\pi$
$607$	46.0721	1.87001	0.935005	+	0.354635i	$0.115395\pi$
$608$	23.3160	0.945589
$609$	0	0
$610$	−6.82926	−0.276509
$611$	−31.0716	−1.25702
$612$	0	0
$613$	15.6741	0.633070	0.316535	−	0.948581i	$-0.397481\pi$
$613$	15.6741	0.633070	0.316535	+	0.948581i	$0.397481\pi$
$614$	−2.93539	−0.118463
$615$	0	0
$616$	−6.41027	−0.258277
$617$	−8.45673	−0.340455	−0.170228	−	0.985405i	$-0.554450\pi$
$617$	−8.45673	−0.340455	−0.170228	+	0.985405i	$0.554450\pi$
$618$	0	0
$619$	−2.51989	−0.101283	−0.0506414	−	0.998717i	$-0.516127\pi$
$619$	−2.51989	−0.101283	−0.0506414	+	0.998717i	$0.516127\pi$
$620$	0	0
$621$	0	0
$622$	−3.89247	−0.156074
$623$	−25.5965	−1.02550
$624$	0	0
$625$	−15.3816	−0.615262
$626$	4.87880	0.194996
$627$	0	0
$628$	9.82345	0.391998
$629$	15.6025	0.622112
$630$	0	0
$631$	−7.75487	−0.308717	−0.154358	−	0.988015i	$-0.549331\pi$
$631$	−7.75487	−0.308717	−0.154358	+	0.988015i	$0.549331\pi$
$632$	−5.28108	−0.210070
$633$	0	0
$634$	0.328806	0.0130586
$635$	−70.8137	−2.81016
$636$	0	0
$637$	−11.8023	−0.467624
$638$	6.94040	0.274773
$639$	0	0
$640$	26.9143	1.06388
$641$	−12.1464	−0.479752	−0.239876	−	0.970803i	$-0.577107\pi$
$641$	−12.1464	−0.479752	−0.239876	+	0.970803i	$0.577107\pi$
$642$	0	0
$643$	−3.34790	−0.132028	−0.0660142	−	0.997819i	$-0.521028\pi$
$643$	−3.34790	−0.132028	−0.0660142	+	0.997819i	$0.521028\pi$
$644$	−5.97981	−0.235638
$645$	0	0
$646$	14.4708	0.569346
$647$	−31.0877	−1.22218	−0.611091	−	0.791560i	$-0.709269\pi$
$647$	−31.0877	−1.22218	−0.611091	+	0.791560i	$0.709269\pi$
$648$	0	0
$649$	16.4123	0.644241
$650$	4.82120	0.189103
$651$	0	0
$652$	24.8342	0.972581
$653$	23.5687	0.922316	0.461158	−	0.887318i	$-0.347434\pi$
$653$	23.5687	0.922316	0.461158	+	0.887318i	$0.347434\pi$
$654$	0	0
$655$	32.2213	1.25899
$656$	−34.2290	−1.33642
$657$	0	0
$658$	−5.08774	−0.198341
$659$	−23.5205	−0.916229	−0.458115	−	0.888893i	$-0.651475\pi$
$659$	−23.5205	−0.916229	−0.458115	+	0.888893i	$0.651475\pi$
$660$	0	0
$661$	20.3673	0.792197	0.396099	−	0.918208i	$-0.370364\pi$
$661$	20.3673	0.792197	0.396099	+	0.918208i	$0.370364\pi$
$662$	−3.77779	−0.146828
$663$	0	0
$664$	−6.79340	−0.263635
$665$	−41.4041	−1.60558
$666$	0	0
$667$	13.1994	0.511084
$668$	11.9287	0.461534
$669$	0	0
$670$	7.12732	0.275352
$671$	27.0284	1.04342
$672$	0	0
$673$	24.7377	0.953568	0.476784	−	0.879021i	$-0.341802\pi$
$673$	24.7377	0.953568	0.476784	+	0.879021i	$0.341802\pi$
$674$	8.76750	0.337712
$675$	0	0
$676$	10.7544	0.413632
$677$	22.5878	0.868120	0.434060	−	0.900884i	$-0.357081\pi$
$677$	22.5878	0.868120	0.434060	+	0.900884i	$0.357081\pi$
$678$	0	0
$679$	15.2400	0.584858
$680$	25.7223	0.986406
$681$	0	0
$682$	0	0
$683$	−0.111678	−0.00427324	−0.00213662	−	0.999998i	$-0.500680\pi$
$683$	−0.111678	−0.00427324	−0.00213662	+	0.999998i	$0.500680\pi$
$684$	0	0
$685$	26.8622	1.02635
$686$	−5.05359	−0.192947
$687$	0	0
$688$	4.65665	0.177533
$689$	17.1775	0.654412
$690$	0	0
$691$	29.3344	1.11593	0.557966	−	0.829864i	$-0.311582\pi$
$691$	29.3344	1.11593	0.557966	+	0.829864i	$0.311582\pi$
$692$	−39.1485	−1.48820
$693$	0	0
$694$	6.70019	0.254336
$695$	−67.2930	−2.55257
$696$	0	0
$697$	−68.1177	−2.58014
$698$	−0.200623	−0.00759371
$699$	0	0
$700$	−20.3816	−0.770350
$701$	48.1816	1.81979	0.909897	−	0.414834i	$-0.136160\pi$
$701$	48.1816	1.81979	0.909897	+	0.414834i	$0.136160\pi$
$702$	0	0
$703$	16.4898	0.621926
$704$	−22.9474	−0.864861
$705$	0	0
$706$	−4.43785	−0.167021
$707$	−9.59730	−0.360943
$708$	0	0
$709$	−2.05141	−0.0770424	−0.0385212	−	0.999258i	$-0.512265\pi$
$709$	−2.05141	−0.0770424	−0.0385212	+	0.999258i	$0.512265\pi$
$710$	−4.14690	−0.155630
$711$	0	0
$712$	16.8062	0.629840
$713$	0	0
$714$	0	0
$715$	−33.7961	−1.26390
$716$	18.4548	0.689687
$717$	0	0
$718$	−6.69303	−0.249782
$719$	11.1402	0.415459	0.207729	−	0.978186i	$-0.433393\pi$
$719$	11.1402	0.415459	0.207729	+	0.978186i	$0.433393\pi$
$720$	0	0
$721$	16.9265	0.630375
$722$	10.1051	0.376073
$723$	0	0
$724$	−18.0605	−0.671213
$725$	44.9889	1.67085
$726$	0	0
$727$	18.1238	0.672175	0.336087	−	0.941831i	$-0.390896\pi$
$727$	18.1238	0.672175	0.336087	+	0.941831i	$0.390896\pi$
$728$	−4.76574	−0.176630
$729$	0	0
$730$	−9.89163	−0.366106
$731$	9.26699	0.342752
$732$	0	0
$733$	−13.7707	−0.508633	−0.254316	−	0.967121i	$-0.581850\pi$
$733$	−13.7707	−0.508633	−0.254316	+	0.967121i	$0.581850\pi$
$734$	9.15920	0.338072
$735$	0	0
$736$	5.92665	0.218459
$737$	−28.2081	−1.03906
$738$	0	0
$739$	−27.9213	−1.02710	−0.513550	−	0.858060i	$-0.671670\pi$
$739$	−27.9213	−1.02710	−0.513550	+	0.858060i	$0.671670\pi$
$740$	14.3772	0.528516
$741$	0	0
$742$	2.81270	0.103257
$743$	−8.72812	−0.320204	−0.160102	−	0.987100i	$-0.551182\pi$
$743$	−8.72812	−0.320204	−0.160102	+	0.987100i	$0.551182\pi$
$744$	0	0
$745$	2.88380	0.105654
$746$	−2.69403	−0.0986356
$747$	0	0
$748$	−49.9341	−1.82577
$749$	5.86026	0.214129
$750$	0	0
$751$	22.0142	0.803311	0.401655	−	0.915791i	$-0.368435\pi$
$751$	22.0142	0.803311	0.401655	+	0.915791i	$0.368435\pi$
$752$	−40.6014	−1.48058
$753$	0	0
$754$	5.15986	0.187911
$755$	28.9538	1.05374
$756$	0	0
$757$	12.5557	0.456346	0.228173	−	0.973621i	$-0.426725\pi$
$757$	12.5557	0.456346	0.228173	+	0.973621i	$0.426725\pi$
$758$	−7.86982	−0.285845
$759$	0	0
$760$	27.1852	0.986111
$761$	33.7174	1.22226	0.611128	−	0.791532i	$-0.290716\pi$
$761$	33.7174	1.22226	0.611128	+	0.791532i	$0.290716\pi$
$762$	0	0
$763$	2.42211	0.0876863
$764$	−26.8572	−0.971658
$765$	0	0
$766$	−4.34811	−0.157104
$767$	12.2018	0.440582
$768$	0	0
$769$	25.6611	0.925364	0.462682	−	0.886524i	$-0.346887\pi$
$769$	25.6611	0.925364	0.462682	+	0.886524i	$0.346887\pi$
$770$	−5.53385	−0.199426
$771$	0	0
$772$	0.420069	0.0151186
$773$	52.6033	1.89201	0.946004	−	0.324154i	$-0.105080\pi$
$773$	52.6033	1.89201	0.946004	+	0.324154i	$0.105080\pi$
$774$	0	0
$775$	0	0
$776$	−10.0063	−0.359206
$777$	0	0
$778$	−9.89949	−0.354914
$779$	−71.9917	−2.57937
$780$	0	0
$781$	16.4123	0.587280
$782$	3.67830	0.131536
$783$	0	0
$784$	−15.4221	−0.550790
$785$	17.2892	0.617079
$786$	0	0
$787$	6.59820	0.235200	0.117600	−	0.993061i	$-0.462480\pi$
$787$	6.59820	0.235200	0.117600	+	0.993061i	$0.462480\pi$
$788$	−18.7834	−0.669130
$789$	0	0
$790$	−4.55904	−0.162204
$791$	−13.9894	−0.497407
$792$	0	0
$793$	20.0944	0.713572
$794$	−6.83705	−0.242638
$795$	0	0
$796$	−38.2348	−1.35520
$797$	33.2918	1.17926	0.589629	−	0.807674i	$-0.299274\pi$
$797$	33.2918	1.17926	0.589629	+	0.807674i	$0.299274\pi$
$798$	0	0
$799$	−80.7990	−2.85846
$800$	20.2004	0.714191
$801$	0	0
$802$	−3.77764	−0.133393
$803$	39.1485	1.38152
$804$	0	0
$805$	−10.5244	−0.370938
$806$	0	0
$807$	0	0
$808$	6.30141	0.221683
$809$	43.5754	1.53203	0.766014	−	0.642824i	$-0.222237\pi$
$809$	43.5754	1.53203	0.766014	+	0.642824i	$0.222237\pi$
$810$	0	0
$811$	23.2654	0.816957	0.408479	−	0.912768i	$-0.366059\pi$
$811$	23.2654	0.816957	0.408479	+	0.912768i	$0.366059\pi$
$812$	−21.8133	−0.765495
$813$	0	0
$814$	2.20394	0.0772481
$815$	43.7080	1.53102
$816$	0	0
$817$	9.79403	0.342650
$818$	−4.54805	−0.159019
$819$	0	0
$820$	−62.7682	−2.19196
$821$	18.3684	0.641060	0.320530	−	0.947238i	$-0.396139\pi$
$821$	18.3684	0.641060	0.320530	+	0.947238i	$0.396139\pi$
$822$	0	0
$823$	21.5885	0.752527	0.376263	−	0.926513i	$-0.377209\pi$
$823$	21.5885	0.752527	0.376263	+	0.926513i	$0.377209\pi$
$824$	−11.1136	−0.387162
$825$	0	0
$826$	1.99796	0.0695178
$827$	26.5684	0.923874	0.461937	−	0.886913i	$-0.347155\pi$
$827$	26.5684	0.923874	0.461937	+	0.886913i	$0.347155\pi$
$828$	0	0
$829$	−20.7139	−0.719422	−0.359711	−	0.933064i	$-0.617125\pi$
$829$	−20.7139	−0.719422	−0.359711	+	0.933064i	$0.617125\pi$
$830$	−5.86460	−0.203563
$831$	0	0
$832$	−17.0603	−0.591459
$833$	−30.6909	−1.06338
$834$	0	0
$835$	20.9944	0.726541
$836$	−52.7739	−1.82522
$837$	0	0
$838$	1.66007	0.0573462
$839$	−37.3243	−1.28858	−0.644289	−	0.764782i	$-0.722847\pi$
$839$	−37.3243	−1.28858	−0.644289	+	0.764782i	$0.722847\pi$
$840$	0	0
$841$	19.1492	0.660316
$842$	7.82520	0.269674
$843$	0	0
$844$	−17.2380	−0.593355
$845$	18.9278	0.651135
$846$	0	0
$847$	3.94211	0.135452
$848$	22.4460	0.770797
$849$	0	0
$850$	12.5371	0.430019
$851$	4.19152	0.143683
$852$	0	0
$853$	−37.8793	−1.29696	−0.648481	−	0.761231i	$-0.724596\pi$
$853$	−37.8793	−1.29696	−0.648481	+	0.761231i	$0.724596\pi$
$854$	3.29030	0.112592
$855$	0	0
$856$	−3.84774	−0.131513
$857$	5.44376	0.185955	0.0929777	−	0.995668i	$-0.470361\pi$
$857$	5.44376	0.185955	0.0929777	+	0.995668i	$0.470361\pi$
$858$	0	0
$859$	6.02425	0.205545	0.102772	−	0.994705i	$-0.467229\pi$
$859$	6.02425	0.205545	0.102772	+	0.994705i	$0.467229\pi$
$860$	8.53923	0.291185
$861$	0	0
$862$	−7.49775	−0.255374
$863$	45.7325	1.55675	0.778376	−	0.627798i	$-0.216044\pi$
$863$	45.7325	1.55675	0.778376	+	0.627798i	$0.216044\pi$
$864$	0	0
$865$	−68.9011	−2.34271
$866$	−0.0448764	−0.00152496
$867$	0	0
$868$	0	0
$869$	18.0435	0.612084
$870$	0	0
$871$	−20.9714	−0.710589
$872$	−1.59032	−0.0538549
$873$	0	0
$874$	3.88750	0.131497
$875$	−8.20789	−0.277477
$876$	0	0
$877$	10.8037	0.364814	0.182407	−	0.983223i	$-0.441611\pi$
$877$	10.8037	0.364814	0.182407	+	0.983223i	$0.441611\pi$
$878$	5.82616	0.196623
$879$	0	0
$880$	−44.1614	−1.48868
$881$	−26.7967	−0.902803	−0.451402	−	0.892321i	$-0.649076\pi$
$881$	−26.7967	−0.902803	−0.451402	+	0.892321i	$0.649076\pi$
$882$	0	0
$883$	45.2391	1.52242	0.761209	−	0.648507i	$-0.224606\pi$
$883$	45.2391	1.52242	0.761209	+	0.648507i	$0.224606\pi$
$884$	−37.1237	−1.24860
$885$	0	0
$886$	−11.1294	−0.373898
$887$	29.2056	0.980629	0.490315	−	0.871545i	$-0.336882\pi$
$887$	29.2056	0.980629	0.490315	+	0.871545i	$0.336882\pi$
$888$	0	0
$889$	34.1177	1.14427
$890$	14.5085	0.486325
$891$	0	0
$892$	50.1268	1.67837
$893$	−85.3942	−2.85761
$894$	0	0
$895$	32.4803	1.08570
$896$	−12.9672	−0.433203
$897$	0	0
$898$	5.76232	0.192291
$899$	0	0
$900$	0	0
$901$	44.6687	1.48813
$902$	−9.62202	−0.320378
$903$	0	0
$904$	9.18521	0.305496
$905$	−31.7864	−1.05662
$906$	0	0
$907$	−26.4506	−0.878277	−0.439138	−	0.898419i	$-0.644716\pi$
$907$	−26.4506	−0.878277	−0.439138	+	0.898419i	$0.644716\pi$
$908$	−32.7166	−1.08574
$909$	0	0
$910$	−4.11416	−0.136383
$911$	38.8800	1.28815	0.644075	−	0.764962i	$-0.277242\pi$
$911$	38.8800	1.28815	0.644075	+	0.764962i	$0.277242\pi$
$912$	0	0
$913$	23.2106	0.768158
$914$	−6.15048	−0.203440
$915$	0	0
$916$	18.5800	0.613900
$917$	−15.5241	−0.512650
$918$	0	0
$919$	−13.9341	−0.459643	−0.229822	−	0.973233i	$-0.573814\pi$
$919$	−13.9341	−0.459643	−0.229822	+	0.973233i	$0.573814\pi$
$920$	6.91016	0.227821
$921$	0	0
$922$	1.99255	0.0656211
$923$	12.2018	0.401628
$924$	0	0
$925$	14.2863	0.469732
$926$	−5.29715	−0.174075
$927$	0	0
$928$	21.6193	0.709690
$929$	−30.1285	−0.988485	−0.494242	−	0.869324i	$-0.664554\pi$
$929$	−30.1285	−0.988485	−0.494242	+	0.869324i	$0.664554\pi$
$930$	0	0
$931$	−32.4363	−1.06306
$932$	−18.6010	−0.609298
$933$	0	0
$934$	0.190776	0.00624237
$935$	−87.8837	−2.87411
$936$	0	0
$937$	−24.9011	−0.813485	−0.406742	−	0.913543i	$-0.633335\pi$
$937$	−24.9011	−0.813485	−0.406742	+	0.913543i	$0.633335\pi$
$938$	−3.43391	−0.112121
$939$	0	0
$940$	−74.4537	−2.42841
$941$	−18.9934	−0.619168	−0.309584	−	0.950872i	$-0.600190\pi$
$941$	−18.9934	−0.619168	−0.309584	+	0.950872i	$0.600190\pi$
$942$	0	0
$943$	−18.2994	−0.595911
$944$	15.9442	0.518938
$945$	0	0
$946$	1.30902	0.0425598
$947$	−28.2390	−0.917645	−0.458823	−	0.888528i	$-0.651729\pi$
$947$	−28.2390	−0.917645	−0.458823	+	0.888528i	$0.651729\pi$
$948$	0	0
$949$	29.1051	0.944791
$950$	13.2501	0.429891
$951$	0	0
$952$	−12.3929	−0.401656
$953$	−49.7707	−1.61223	−0.806116	−	0.591758i	$-0.798434\pi$
$953$	−49.7707	−1.61223	−0.806116	+	0.591758i	$0.798434\pi$
$954$	0	0
$955$	−47.2685	−1.52957
$956$	46.2478	1.49576
$957$	0	0
$958$	9.76556	0.315511
$959$	−12.9421	−0.417921
$960$	0	0
$961$	0	0
$962$	1.63853	0.0528283
$963$	0	0
$964$	33.0665	1.06500
$965$	0.739319	0.0237995
$966$	0	0
$967$	35.9646	1.15654	0.578271	−	0.815845i	$-0.303728\pi$
$967$	35.9646	1.15654	0.578271	+	0.815845i	$0.303728\pi$
$968$	−2.58832	−0.0831917
$969$	0	0
$970$	−8.63824	−0.277357
$971$	−39.4940	−1.26742	−0.633712	−	0.773569i	$-0.718470\pi$
$971$	−39.4940	−1.26742	−0.633712	+	0.773569i	$0.718470\pi$
$972$	0	0
$973$	32.4214	1.03938
$974$	−7.33574	−0.235052
$975$	0	0
$976$	26.2574	0.840478
$977$	−34.1212	−1.09163	−0.545817	−	0.837904i	$-0.683781\pi$
$977$	−34.1212	−1.09163	−0.545817	+	0.837904i	$0.683781\pi$
$978$	0	0
$979$	−57.4207	−1.83517
$980$	−28.2807	−0.903392
$981$	0	0
$982$	0.324269	0.0103478
$983$	−21.0036	−0.669911	−0.334956	−	0.942234i	$-0.608721\pi$
$983$	−21.0036	−0.669911	−0.334956	+	0.942234i	$0.608721\pi$
$984$	0	0
$985$	−33.0586	−1.05334
$986$	13.4178	0.427309
$987$	0	0
$988$	−39.2350	−1.24823
$989$	2.48952	0.0791623
$990$	0	0
$991$	2.75095	0.0873867	0.0436933	−	0.999045i	$-0.486088\pi$
$991$	2.75095	0.0873867	0.0436933	+	0.999045i	$0.486088\pi$
$992$	0	0
$993$	0	0
$994$	1.99796	0.0633713
$995$	−67.2930	−2.13333
$996$	0	0
$997$	−4.06902	−0.128867	−0.0644335	−	0.997922i	$-0.520524\pi$
$997$	−4.06902	−0.128867	−0.0644335	+	0.997922i	$0.520524\pi$
$998$	0.0842581	0.00266714
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bm.1.8	yes	12
3.2	odd	2	inner	8649.2.a.bm.1.5	✓	12
31.30	odd	2	inner	8649.2.a.bm.1.7	yes	12
93.92	even	2	inner	8649.2.a.bm.1.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8649.2.a.bm.1.5	✓	12	3.2	odd	2	inner
8649.2.a.bm.1.6	yes	12	93.92	even	2	inner
8649.2.a.bm.1.7	yes	12	31.30	odd	2	inner
8649.2.a.bm.1.8	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q+0.273088 q^{2} -1.92542 q^{4} -3.38873 q^{5} +1.63268 q^{7} -1.07199 q^{8} +O(q^{10})\)	\(q+0.273088 q^{2} -1.92542 q^{4} -3.38873 q^{5} +1.63268 q^{7} -1.07199 q^{8} -0.925423 q^{10} +3.66258 q^{11} +2.72296 q^{13} +0.445864 q^{14} +3.55810 q^{16} +7.08082 q^{17} +7.48352 q^{19} +6.52475 q^{20} +1.00021 q^{22} +1.90222 q^{23} +6.48352 q^{25} +0.743608 q^{26} -3.14359 q^{28} +6.93896 q^{29} +3.11565 q^{32} +1.93369 q^{34} -5.53271 q^{35} +2.20349 q^{37} +2.04366 q^{38} +3.63268 q^{40} -9.62003 q^{41} +1.30875 q^{43} -7.05202 q^{44} +0.519474 q^{46} -11.4110 q^{47} -4.33437 q^{49} +1.77057 q^{50} -5.24285 q^{52} +6.30841 q^{53} -12.4115 q^{55} -1.75021 q^{56} +1.89495 q^{58} +4.48109 q^{59} +7.37961 q^{61} -6.26535 q^{64} -9.22739 q^{65} -7.70169 q^{67} -13.6336 q^{68} -1.51092 q^{70} +4.48109 q^{71} +10.6888 q^{73} +0.601745 q^{74} -14.4089 q^{76} +5.97981 q^{77} +4.92644 q^{79} -12.0575 q^{80} -2.62711 q^{82} +6.33721 q^{83} -23.9950 q^{85} +0.357403 q^{86} -3.92624 q^{88} -15.6777 q^{89} +4.44571 q^{91} -3.66258 q^{92} -3.11620 q^{94} -25.3597 q^{95} +9.33437 q^{97} -1.18366 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{4} + 8 q^{7}+O(q^{10}) \) 12 * q + 8 * q^4 + 8 * q^7	\( 12 q + 8 q^{4} + 8 q^{7} + 20 q^{10} + 16 q^{19} + 4 q^{25} + 44 q^{28} + 32 q^{40} + 84 q^{49} - 52 q^{64} + 32 q^{67} + 52 q^{70} - 68 q^{76} + 124 q^{82} + 48 q^{94} - 24 q^{97}+O(q^{100}) \) 12 * q + 8 * q^4 + 8 * q^7 + 20 * q^10 + 16 * q^19 + 4 * q^25 + 44 * q^28 + 32 * q^40 + 84 * q^49 - 52 * q^64 + 32 * q^67 + 52 * q^70 - 68 * q^76 + 124 * q^82 + 48 * q^94 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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