LMFDB - Embedded newform 5445.2.a.bq.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5445,2,Mod(1,5445)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5445.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,1,0,1,-4,0,0,-3,0,-1,0,0,-7,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.33826$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82709 q^{2} +1.33826 q^{4} -1.00000 q^{5} -1.74724 q^{7} -1.20906 q^{8} -1.82709 q^{10} +5.35772 q^{13} -3.19236 q^{14} -4.88558 q^{16} +7.33070 q^{17} -6.01478 q^{19} -1.33826 q^{20} -8.35772 q^{23} +1.00000 q^{25} +9.78903 q^{26} -2.33826 q^{28} +5.53818 q^{29} -0.0213642 q^{31} -6.50828 q^{32} +13.3939 q^{34} +1.74724 q^{35} -1.12165 q^{37} -10.9896 q^{38} +1.20906 q^{40} -3.65983 q^{41} +1.96950 q^{43} -15.2703 q^{46} -3.81953 q^{47} -3.94716 q^{49} +1.82709 q^{50} +7.17002 q^{52} -1.67088 q^{53} +2.11251 q^{56} +10.1188 q^{58} -5.66332 q^{59} -9.27977 q^{61} -0.0390343 q^{62} -2.12007 q^{64} -5.35772 q^{65} +0.350489 q^{67} +9.81040 q^{68} +3.19236 q^{70} -4.53818 q^{71} -10.1395 q^{73} -2.04935 q^{74} -8.04935 q^{76} -2.56401 q^{79} +4.88558 q^{80} -6.68684 q^{82} -16.0805 q^{83} -7.33070 q^{85} +3.59845 q^{86} +14.5062 q^{89} -9.36121 q^{91} -11.1848 q^{92} -6.97864 q^{94} +6.01478 q^{95} -1.89972 q^{97} -7.21182 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{4} - 4 q^{5} - 3 q^{8} - q^{10} - 7 q^{13} + 5 q^{14} - 9 q^{16} + 8 q^{17} - 11 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 12 q^{26} - 5 q^{28} + 17 q^{29} - 5 q^{31} + 17 q^{34} + 15 q^{37}+ \cdots + 13 q^{98}+O(q^{100}) $ 4 * q + q^2 + q^4 - 4 * q^5 - 3 * q^8 - q^10 - 7 * q^13 + 5 * q^14 - 9 * q^16 + 8 * q^17 - 11 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 12 * q^26 - 5 * q^28 + 17 * q^29 - 5 * q^31 + 17 * q^34 + 15 * q^37 + q^38 + 3 * q^40 + 10 * q^41 + 4 * q^43 - 15 * q^46 + 8 * q^47 - 8 * q^49 + q^50 + 7 * q^52 - 10 * q^53 - 10 * q^56 - 7 * q^58 - 9 * q^59 - 37 * q^61 - 20 * q^62 - 7 * q^64 + 7 * q^65 + 3 * q^67 + 17 * q^68 - 5 * q^70 - 13 * q^71 - 15 * q^73 - 5 * q^74 - 29 * q^76 - 20 * q^79 + 9 * q^80 + 5 * q^82 - 17 * q^83 - 8 * q^85 - 34 * q^86 + 24 * q^89 - 20 * q^91 - 10 * q^92 - 23 * q^94 + 11 * q^95 - 32 * q^97 + 13 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.82709	1.29195	0.645974	−	0.763359i	$-0.276451\pi$
$2$	1.82709	1.29195	0.645974	+	0.763359i	$0.276451\pi$
$3$	0	0
$3$	0	0
$4$	1.33826	0.669131
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.74724	−0.660394	−0.330197	−	0.943912i	$-0.607115\pi$
$7$	−1.74724	−0.660394	−0.330197	+	0.943912i	$0.607115\pi$
$8$	−1.20906	−0.427466
$9$	0	0
$10$	−1.82709	−0.577777
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.35772	1.48596	0.742981	−	0.669312i	$-0.233411\pi$
$13$	5.35772	1.48596	0.742981	+	0.669312i	$0.233411\pi$
$14$	−3.19236	−0.853195
$15$	0	0
$16$	−4.88558	−1.22139
$17$	7.33070	1.77796	0.888978	−	0.457949i	$-0.151416\pi$
$17$	7.33070	1.77796	0.888978	+	0.457949i	$0.151416\pi$
$18$	0	0
$19$	−6.01478	−1.37989	−0.689943	−	0.723864i	$-0.742364\pi$
$19$	−6.01478	−1.37989	−0.689943	+	0.723864i	$0.742364\pi$
$20$	−1.33826	−0.299244
$21$	0	0
$22$	0	0
$23$	−8.35772	−1.74270	−0.871352	−	0.490658i	$-0.836756\pi$
$23$	−8.35772	−1.74270	−0.871352	+	0.490658i	$0.836756\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	9.78903	1.91979
$27$	0	0
$28$	−2.33826	−0.441890
$29$	5.53818	1.02841	0.514207	−	0.857666i	$-0.328086\pi$
$29$	5.53818	1.02841	0.514207	+	0.857666i	$0.328086\pi$
$30$	0	0
$31$	−0.0213642	−0.00383712	−0.00191856	−	0.999998i	$-0.500611\pi$
$31$	−0.0213642	−0.00383712	−0.00191856	+	0.999998i	$0.500611\pi$
$32$	−6.50828	−1.15051
$33$	0	0
$34$	13.3939	2.29703
$35$	1.74724	0.295337
$36$	0	0
$37$	−1.12165	−0.184398	−0.0921988	−	0.995741i	$-0.529390\pi$
$37$	−1.12165	−0.184398	−0.0921988	+	0.995741i	$0.529390\pi$
$38$	−10.9896	−1.78274
$39$	0	0
$40$	1.20906	0.191169
$41$	−3.65983	−0.571569	−0.285785	−	0.958294i	$-0.592254\pi$
$41$	−3.65983	−0.571569	−0.285785	+	0.958294i	$0.592254\pi$
$42$	0	0
$43$	1.96950	0.300346	0.150173	−	0.988660i	$-0.452017\pi$
$43$	1.96950	0.300346	0.150173	+	0.988660i	$0.452017\pi$
$44$	0	0
$45$	0	0
$46$	−15.2703	−2.25148
$47$	−3.81953	−0.557136	−0.278568	−	0.960416i	$-0.589860\pi$
$47$	−3.81953	−0.557136	−0.278568	+	0.960416i	$0.589860\pi$
$48$	0	0
$49$	−3.94716	−0.563880
$50$	1.82709	0.258390
$51$	0	0
$52$	7.17002	0.994303
$53$	−1.67088	−0.229512	−0.114756	−	0.993394i	$-0.536609\pi$
$53$	−1.67088	−0.229512	−0.114756	+	0.993394i	$0.536609\pi$
$54$	0	0
$55$	0	0
$56$	2.11251	0.282296
$57$	0	0
$58$	10.1188	1.32866
$59$	−5.66332	−0.737301	−0.368651	−	0.929568i	$-0.620180\pi$
$59$	−5.66332	−0.737301	−0.368651	+	0.929568i	$0.620180\pi$
$60$	0	0
$61$	−9.27977	−1.18815	−0.594077	−	0.804408i	$-0.702482\pi$
$61$	−9.27977	−1.18815	−0.594077	+	0.804408i	$0.702482\pi$
$62$	−0.0390343	−0.00495737
$63$	0	0
$64$	−2.12007	−0.265008
$65$	−5.35772	−0.664543
$66$	0	0
$67$	0.350489	0.0428190	0.0214095	−	0.999771i	$-0.493185\pi$
$67$	0.350489	0.0428190	0.0214095	+	0.999771i	$0.493185\pi$
$68$	9.81040	1.18969
$69$	0	0
$70$	3.19236	0.381560
$71$	−4.53818	−0.538583	−0.269292	−	0.963059i	$-0.586790\pi$
$71$	−4.53818	−0.538583	−0.269292	+	0.963059i	$0.586790\pi$
$72$	0	0
$73$	−10.1395	−1.18674	−0.593371	−	0.804929i	$-0.702203\pi$
$73$	−10.1395	−1.18674	−0.593371	+	0.804929i	$0.702203\pi$
$74$	−2.04935	−0.238232
$75$	0	0
$76$	−8.04935	−0.923324
$77$	0	0
$78$	0	0
$79$	−2.56401	−0.288474	−0.144237	−	0.989543i	$-0.546073\pi$
$79$	−2.56401	−0.288474	−0.144237	+	0.989543i	$0.546073\pi$
$80$	4.88558	0.546224
$81$	0	0
$82$	−6.68684	−0.738438
$83$	−16.0805	−1.76506	−0.882532	−	0.470252i	$-0.844163\pi$
$83$	−16.0805	−1.76506	−0.882532	+	0.470252i	$0.844163\pi$
$84$	0	0
$85$	−7.33070	−0.795127
$86$	3.59845	0.388031
$87$	0	0
$88$	0	0
$89$	14.5062	1.53765	0.768825	−	0.639459i	$-0.220841\pi$
$89$	14.5062	1.53765	0.768825	+	0.639459i	$0.220841\pi$
$90$	0	0
$91$	−9.36121	−0.981321
$92$	−11.1848	−1.16610
$93$	0	0
$94$	−6.97864	−0.719791
$95$	6.01478	0.617104
$96$	0	0
$97$	−1.89972	−0.192887	−0.0964435	−	0.995338i	$-0.530747\pi$
$97$	−1.89972	−0.192887	−0.0964435	+	0.995338i	$0.530747\pi$
$98$	−7.21182	−0.728504
$99$	0	0
$100$	1.33826	0.133826
$101$	6.71011	0.667681	0.333841	−	0.942630i	$-0.391655\pi$
$101$	6.71011	0.667681	0.333841	+	0.942630i	$0.391655\pi$
$102$	0	0
$103$	−4.24678	−0.418448	−0.209224	−	0.977868i	$-0.567094\pi$
$103$	−4.24678	−0.418448	−0.209224	+	0.977868i	$0.567094\pi$
$104$	−6.47778	−0.635199
$105$	0	0
$106$	−3.05284	−0.296518
$107$	−15.7230	−1.52000	−0.759999	−	0.649924i	$-0.774801\pi$
$107$	−15.7230	−1.52000	−0.759999	+	0.649924i	$0.774801\pi$
$108$	0	0
$109$	−4.23895	−0.406018	−0.203009	−	0.979177i	$-0.565072\pi$
$109$	−4.23895	−0.406018	−0.203009	+	0.979177i	$0.565072\pi$
$110$	0	0
$111$	0	0
$112$	8.53627	0.806602
$113$	7.19643	0.676983	0.338492	−	0.940969i	$-0.390083\pi$
$113$	7.19643	0.676983	0.338492	+	0.940969i	$0.390083\pi$
$114$	0	0
$115$	8.35772	0.779361
$116$	7.41153	0.688144
$117$	0	0
$118$	−10.3474	−0.952555
$119$	−12.8085	−1.17415
$120$	0	0
$121$	0	0
$122$	−16.9550	−1.53503
$123$	0	0
$124$	−0.0285909	−0.00256754
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.6870	−1.12579	−0.562897	−	0.826527i	$-0.690313\pi$
$127$	−12.6870	−1.12579	−0.562897	+	0.826527i	$0.690313\pi$
$128$	9.14301	0.808136
$129$	0	0
$130$	−9.78903	−0.858555
$131$	−0.970102	−0.0847582	−0.0423791	−	0.999102i	$-0.513494\pi$
$131$	−0.970102	−0.0847582	−0.0423791	+	0.999102i	$0.513494\pi$
$132$	0	0
$133$	10.5093	0.911268
$134$	0.640375	0.0553199
$135$	0	0
$136$	−8.86324	−0.760016
$137$	−4.22035	−0.360569	−0.180284	−	0.983615i	$-0.557702\pi$
$137$	−4.22035	−0.360569	−0.180284	+	0.983615i	$0.557702\pi$
$138$	0	0
$139$	−8.84846	−0.750516	−0.375258	−	0.926920i	$-0.622446\pi$
$139$	−8.84846	−0.750516	−0.375258	+	0.926920i	$0.622446\pi$
$140$	2.33826	0.197619
$141$	0	0
$142$	−8.29167	−0.695821
$143$	0	0
$144$	0	0
$145$	−5.53818	−0.459921
$146$	−18.5258	−1.53321
$147$	0	0
$148$	−1.50106	−0.123386
$149$	9.55357	0.782659	0.391329	−	0.920251i	$-0.372015\pi$
$149$	9.55357	0.782659	0.391329	+	0.920251i	$0.372015\pi$
$150$	0	0
$151$	6.75158	0.549436	0.274718	−	0.961525i	$-0.411416\pi$
$151$	6.75158	0.549436	0.274718	+	0.961525i	$0.411416\pi$
$152$	7.27222	0.589855
$153$	0	0
$154$	0	0
$155$	0.0213642	0.00171601
$156$	0	0
$157$	−12.6604	−1.01041	−0.505206	−	0.862999i	$-0.668584\pi$
$157$	−12.6604	−1.01041	−0.505206	+	0.862999i	$0.668584\pi$
$158$	−4.68468	−0.372693
$159$	0	0
$160$	6.50828	0.514525
$161$	14.6029	1.15087
$162$	0	0
$163$	−23.4058	−1.83328	−0.916640	−	0.399713i	$-0.869110\pi$
$163$	−23.4058	−1.83328	−0.916640	+	0.399713i	$0.869110\pi$
$164$	−4.89781	−0.382454
$165$	0	0
$166$	−29.3805	−2.28037
$167$	3.05191	0.236164	0.118082	−	0.993004i	$-0.462325\pi$
$167$	3.05191	0.236164	0.118082	+	0.993004i	$0.462325\pi$
$168$	0	0
$169$	15.7051	1.20809
$170$	−13.3939	−1.02726
$171$	0	0
$172$	2.63570	0.200971
$173$	16.0074	1.21702	0.608511	−	0.793545i	$-0.291767\pi$
$173$	16.0074	1.21702	0.608511	+	0.793545i	$0.291767\pi$
$174$	0	0
$175$	−1.74724	−0.132079
$176$	0	0
$177$	0	0
$178$	26.5041	1.98657
$179$	19.2991	1.44248	0.721241	−	0.692684i	$-0.243572\pi$
$179$	19.2991	1.44248	0.721241	+	0.692684i	$0.243572\pi$
$180$	0	0
$181$	−12.1911	−0.906154	−0.453077	−	0.891471i	$-0.649674\pi$
$181$	−12.1911	−0.906154	−0.453077	+	0.891471i	$0.649674\pi$
$182$	−17.1038	−1.26782
$183$	0	0
$184$	10.1050	0.744947
$185$	1.12165	0.0824652
$186$	0	0
$187$	0	0
$188$	−5.11153	−0.372797
$189$	0	0
$190$	10.9896	0.797266
$191$	13.4244	0.971353	0.485676	−	0.874139i	$-0.338573\pi$
$191$	13.4244	0.971353	0.485676	+	0.874139i	$0.338573\pi$
$192$	0	0
$193$	−7.57782	−0.545463	−0.272732	−	0.962090i	$-0.587927\pi$
$193$	−7.57782	−0.545463	−0.272732	+	0.962090i	$0.587927\pi$
$194$	−3.47096	−0.249200
$195$	0	0
$196$	−5.28233	−0.377309
$197$	−6.89118	−0.490976	−0.245488	−	0.969400i	$-0.578948\pi$
$197$	−6.89118	−0.490976	−0.245488	+	0.969400i	$0.578948\pi$
$198$	0	0
$199$	0.639398	0.0453257	0.0226629	−	0.999743i	$-0.492786\pi$
$199$	0.639398	0.0453257	0.0226629	+	0.999743i	$0.492786\pi$
$200$	−1.20906	−0.0854932
$201$	0	0
$202$	12.2600	0.862610
$203$	−9.67652	−0.679159
$204$	0	0
$205$	3.65983	0.255614
$206$	−7.75926	−0.540613
$207$	0	0
$208$	−26.1755	−1.81495
$209$	0	0
$210$	0	0
$211$	3.82149	0.263082	0.131541	−	0.991311i	$-0.458008\pi$
$211$	3.82149	0.263082	0.131541	+	0.991311i	$0.458008\pi$
$212$	−2.23607	−0.153574
$213$	0	0
$214$	−28.7273	−1.96376
$215$	−1.96950	−0.134319
$216$	0	0
$217$	0.0373284	0.00253401
$218$	−7.74496	−0.524555
$219$	0	0
$220$	0	0
$221$	39.2758	2.64198
$222$	0	0
$223$	−26.7201	−1.78931	−0.894657	−	0.446755i	$-0.852580\pi$
$223$	−26.7201	−1.78931	−0.894657	+	0.446755i	$0.852580\pi$
$224$	11.3715	0.759792
$225$	0	0
$226$	13.1485	0.874627
$227$	12.2373	0.812219	0.406110	−	0.913824i	$-0.366885\pi$
$227$	12.2373	0.812219	0.406110	+	0.913824i	$0.366885\pi$
$228$	0	0
$229$	−6.68664	−0.441865	−0.220933	−	0.975289i	$-0.570910\pi$
$229$	−6.68664	−0.441865	−0.220933	+	0.975289i	$0.570910\pi$
$230$	15.2703	1.00689
$231$	0	0
$232$	−6.69598	−0.439612
$233$	−6.51742	−0.426970	−0.213485	−	0.976946i	$-0.568482\pi$
$233$	−6.51742	−0.426970	−0.213485	+	0.976946i	$0.568482\pi$
$234$	0	0
$235$	3.81953	0.249159
$236$	−7.57900	−0.493351
$237$	0	0
$238$	−23.4023	−1.51694
$239$	−10.8419	−0.701303	−0.350651	−	0.936506i	$-0.614040\pi$
$239$	−10.8419	−0.701303	−0.350651	+	0.936506i	$0.614040\pi$
$240$	0	0
$241$	−27.1803	−1.75084	−0.875420	−	0.483363i	$-0.839415\pi$
$241$	−27.1803	−1.75084	−0.875420	+	0.483363i	$0.839415\pi$
$242$	0	0
$243$	0	0
$244$	−12.4188	−0.795030
$245$	3.94716	0.252175
$246$	0	0
$247$	−32.2255	−2.05046
$248$	0.0258305	0.00164024
$249$	0	0
$250$	−1.82709	−0.115555
$251$	−5.54443	−0.349961	−0.174981	−	0.984572i	$-0.555986\pi$
$251$	−5.54443	−0.349961	−0.174981	+	0.984572i	$0.555986\pi$
$252$	0	0
$253$	0	0
$254$	−23.1804	−1.45447
$255$	0	0
$256$	20.9452	1.30908
$257$	18.9439	1.18169	0.590845	−	0.806785i	$-0.298794\pi$
$257$	18.9439	1.18169	0.590845	+	0.806785i	$0.298794\pi$
$258$	0	0
$259$	1.95979	0.121775
$260$	−7.17002	−0.444666
$261$	0	0
$262$	−1.77247	−0.109503
$263$	3.20873	0.197859	0.0989293	−	0.995094i	$-0.468458\pi$
$263$	3.20873	0.197859	0.0989293	+	0.995094i	$0.468458\pi$
$264$	0	0
$265$	1.67088	0.102641
$266$	19.2014	1.17731
$267$	0	0
$268$	0.469045	0.0286515
$269$	3.24423	0.197804	0.0989020	−	0.995097i	$-0.468467\pi$
$269$	3.24423	0.197804	0.0989020	+	0.995097i	$0.468467\pi$
$270$	0	0
$271$	2.02977	0.123300	0.0616499	−	0.998098i	$-0.480364\pi$
$271$	2.02977	0.123300	0.0616499	+	0.998098i	$0.480364\pi$
$272$	−35.8147	−2.17159
$273$	0	0
$274$	−7.71096	−0.465836
$275$	0	0
$276$	0	0
$277$	−11.3166	−0.679947	−0.339973	−	0.940435i	$-0.610418\pi$
$277$	−11.3166	−0.679947	−0.339973	+	0.940435i	$0.610418\pi$
$278$	−16.1669	−0.969628
$279$	0	0
$280$	−2.11251	−0.126247
$281$	−10.3177	−0.615503	−0.307751	−	0.951467i	$-0.599576\pi$
$281$	−10.3177	−0.615503	−0.307751	+	0.951467i	$0.599576\pi$
$282$	0	0
$283$	2.10589	0.125182	0.0625910	−	0.998039i	$-0.480064\pi$
$283$	2.10589	0.125182	0.0625910	+	0.998039i	$0.480064\pi$
$284$	−6.07327	−0.360382
$285$	0	0
$286$	0	0
$287$	6.39459	0.377461
$288$	0	0
$289$	36.7392	2.16113
$290$	−10.1188	−0.594194
$291$	0	0
$292$	−13.5693	−0.794085
$293$	−23.3755	−1.36561	−0.682805	−	0.730601i	$-0.739240\pi$
$293$	−23.3755	−1.36561	−0.682805	+	0.730601i	$0.739240\pi$
$294$	0	0
$295$	5.66332	0.329731
$296$	1.35614	0.0788238
$297$	0	0
$298$	17.4552	1.01115
$299$	−44.7783	−2.58959
$300$	0	0
$301$	−3.44118	−0.198347
$302$	12.3357	0.709843
$303$	0	0
$304$	29.3857	1.68539
$305$	9.27977	0.531358
$306$	0	0
$307$	32.0518	1.82929	0.914646	−	0.404256i	$-0.132469\pi$
$307$	32.0518	1.82929	0.914646	+	0.404256i	$0.132469\pi$
$308$	0	0
$309$	0	0
$310$	0.0390343	0.00221700
$311$	−1.63433	−0.0926743	−0.0463371	−	0.998926i	$-0.514755\pi$
$311$	−1.63433	−0.0926743	−0.0463371	+	0.998926i	$0.514755\pi$
$312$	0	0
$313$	30.4130	1.71904	0.859522	−	0.511099i	$-0.170761\pi$
$313$	30.4130	1.71904	0.859522	+	0.511099i	$0.170761\pi$
$314$	−23.1318	−1.30540
$315$	0	0
$316$	−3.43132	−0.193027
$317$	29.4340	1.65318	0.826590	−	0.562805i	$-0.190278\pi$
$317$	29.4340	1.65318	0.826590	+	0.562805i	$0.190278\pi$
$318$	0	0
$319$	0	0
$320$	2.12007	0.118515
$321$	0	0
$322$	26.6809	1.48687
$323$	−44.0926	−2.45338
$324$	0	0
$325$	5.35772	0.297193
$326$	−42.7645	−2.36850
$327$	0	0
$328$	4.42494	0.244327
$329$	6.67364	0.367929
$330$	0	0
$331$	−22.7902	−1.25266	−0.626332	−	0.779557i	$-0.715444\pi$
$331$	−22.7902	−1.25266	−0.626332	+	0.779557i	$0.715444\pi$
$332$	−21.5199	−1.18106
$333$	0	0
$334$	5.57611	0.305111
$335$	−0.350489	−0.0191492
$336$	0	0
$337$	14.4901	0.789325	0.394662	−	0.918826i	$-0.370862\pi$
$337$	14.4901	0.789325	0.394662	+	0.918826i	$0.370862\pi$
$338$	28.6947	1.56078
$339$	0	0
$340$	−9.81040	−0.532043
$341$	0	0
$342$	0	0
$343$	19.1273	1.03278
$344$	−2.38124	−0.128388
$345$	0	0
$346$	29.2470	1.57233
$347$	4.38728	0.235522	0.117761	−	0.993042i	$-0.462428\pi$
$347$	4.38728	0.235522	0.117761	+	0.993042i	$0.462428\pi$
$348$	0	0
$349$	−14.4061	−0.771140	−0.385570	−	0.922679i	$-0.625995\pi$
$349$	−14.4061	−0.771140	−0.385570	+	0.922679i	$0.625995\pi$
$350$	−3.19236	−0.170639
$351$	0	0
$352$	0	0
$353$	6.01065	0.319914	0.159957	−	0.987124i	$-0.448864\pi$
$353$	6.01065	0.319914	0.159957	+	0.987124i	$0.448864\pi$
$354$	0	0
$355$	4.53818	0.240862
$356$	19.4130	1.02889
$357$	0	0
$358$	35.2612	1.86361
$359$	−2.45715	−0.129683	−0.0648417	−	0.997896i	$-0.520654\pi$
$359$	−2.45715	−0.129683	−0.0648417	+	0.997896i	$0.520654\pi$
$360$	0	0
$361$	17.1776	0.904085
$362$	−22.2742	−1.17070
$363$	0	0
$364$	−12.5277	−0.656632
$365$	10.1395	0.530727
$366$	0	0
$367$	1.47990	0.0772500	0.0386250	−	0.999254i	$-0.487702\pi$
$367$	1.47990	0.0772500	0.0386250	+	0.999254i	$0.487702\pi$
$368$	40.8323	2.12853
$369$	0	0
$370$	2.04935	0.106541
$371$	2.91942	0.151569
$372$	0	0
$373$	24.3331	1.25992	0.629961	−	0.776627i	$-0.283071\pi$
$373$	24.3331	1.25992	0.629961	+	0.776627i	$0.283071\pi$
$374$	0	0
$375$	0	0
$376$	4.61803	0.238157
$377$	29.6720	1.52819
$378$	0	0
$379$	15.9311	0.818328	0.409164	−	0.912461i	$-0.365820\pi$
$379$	15.9311	0.818328	0.409164	+	0.912461i	$0.365820\pi$
$380$	8.04935	0.412923
$381$	0	0
$382$	24.5275	1.25494
$383$	3.91994	0.200300	0.100150	−	0.994972i	$-0.468068\pi$
$383$	3.91994	0.200300	0.100150	+	0.994972i	$0.468068\pi$
$384$	0	0
$385$	0	0
$386$	−13.8454	−0.704710
$387$	0	0
$388$	−2.54232	−0.129067
$389$	35.2910	1.78933	0.894663	−	0.446742i	$-0.147416\pi$
$389$	35.2910	1.78933	0.894663	+	0.446742i	$0.147416\pi$
$390$	0	0
$391$	−61.2679	−3.09845
$392$	4.77234	0.241040
$393$	0	0
$394$	−12.5908	−0.634316
$395$	2.56401	0.129009
$396$	0	0
$397$	23.2231	1.16553	0.582767	−	0.812639i	$-0.301970\pi$
$397$	23.2231	1.16553	0.582767	+	0.812639i	$0.301970\pi$
$398$	1.16824	0.0585585
$399$	0	0
$400$	−4.88558	−0.244279
$401$	−5.69460	−0.284375	−0.142187	−	0.989840i	$-0.545414\pi$
$401$	−5.69460	−0.284375	−0.142187	+	0.989840i	$0.545414\pi$
$402$	0	0
$403$	−0.114463	−0.00570182
$404$	8.97989	0.446766
$405$	0	0
$406$	−17.6799	−0.877438
$407$	0	0
$408$	0	0
$409$	−25.4530	−1.25857	−0.629284	−	0.777176i	$-0.716652\pi$
$409$	−25.4530	−1.25857	−0.629284	+	0.777176i	$0.716652\pi$
$410$	6.68684	0.330239
$411$	0	0
$412$	−5.68331	−0.279996
$413$	9.89517	0.486909
$414$	0	0
$415$	16.0805	0.789361
$416$	−34.8695	−1.70962
$417$	0	0
$418$	0	0
$419$	13.4850	0.658786	0.329393	−	0.944193i	$-0.393156\pi$
$419$	13.4850	0.658786	0.329393	+	0.944193i	$0.393156\pi$
$420$	0	0
$421$	2.94967	0.143758	0.0718791	−	0.997413i	$-0.477100\pi$
$421$	2.94967	0.143758	0.0718791	+	0.997413i	$0.477100\pi$
$422$	6.98220	0.339888
$423$	0	0
$424$	2.02018	0.0981088
$425$	7.33070	0.355591
$426$	0	0
$427$	16.2140	0.784649
$428$	−21.0415	−1.01708
$429$	0	0
$430$	−3.59845	−0.173533
$431$	11.5227	0.555031	0.277515	−	0.960721i	$-0.410489\pi$
$431$	11.5227	0.555031	0.277515	+	0.960721i	$0.410489\pi$
$432$	0	0
$433$	11.1638	0.536500	0.268250	−	0.963349i	$-0.413555\pi$
$433$	11.1638	0.536500	0.268250	+	0.963349i	$0.413555\pi$
$434$	0.0682023	0.00327382
$435$	0	0
$436$	−5.67283	−0.271679
$437$	50.2698	2.40473
$438$	0	0
$439$	−0.993624	−0.0474231	−0.0237115	−	0.999719i	$-0.507548\pi$
$439$	−0.993624	−0.0474231	−0.0237115	+	0.999719i	$0.507548\pi$
$440$	0	0
$441$	0	0
$442$	71.7605	3.41330
$443$	10.9825	0.521796	0.260898	−	0.965366i	$-0.415981\pi$
$443$	10.9825	0.521796	0.260898	+	0.965366i	$0.415981\pi$
$444$	0	0
$445$	−14.5062	−0.687658
$446$	−48.8201	−2.31170
$447$	0	0
$448$	3.70426	0.175010
$449$	−29.0203	−1.36955	−0.684776	−	0.728753i	$-0.740100\pi$
$449$	−29.0203	−1.36955	−0.684776	+	0.728753i	$0.740100\pi$
$450$	0	0
$451$	0	0
$452$	9.63070	0.452990
$453$	0	0
$454$	22.3587	1.04935
$455$	9.36121	0.438860
$456$	0	0
$457$	3.55670	0.166375	0.0831877	−	0.996534i	$-0.473490\pi$
$457$	3.55670	0.166375	0.0831877	+	0.996534i	$0.473490\pi$
$458$	−12.2171	−0.570867
$459$	0	0
$460$	11.1848	0.521494
$461$	25.1829	1.17288	0.586441	−	0.809992i	$-0.300528\pi$
$461$	25.1829	1.17288	0.586441	+	0.809992i	$0.300528\pi$
$462$	0	0
$463$	−17.8743	−0.830690	−0.415345	−	0.909664i	$-0.636339\pi$
$463$	−17.8743	−0.830690	−0.415345	+	0.909664i	$0.636339\pi$
$464$	−27.0572	−1.25610
$465$	0	0
$466$	−11.9079	−0.551624
$467$	25.2252	1.16728	0.583642	−	0.812011i	$-0.301627\pi$
$467$	25.2252	1.16728	0.583642	+	0.812011i	$0.301627\pi$
$468$	0	0
$469$	−0.612387	−0.0282774
$470$	6.97864	0.321900
$471$	0	0
$472$	6.84727	0.315171
$473$	0	0
$474$	0	0
$475$	−6.01478	−0.275977
$476$	−17.1411	−0.785661
$477$	0	0
$478$	−19.8091	−0.906047
$479$	−5.23574	−0.239227	−0.119613	−	0.992821i	$-0.538166\pi$
$479$	−5.23574	−0.239227	−0.119613	+	0.992821i	$0.538166\pi$
$480$	0	0
$481$	−6.00947	−0.274008
$482$	−49.6610	−2.26199
$483$	0	0
$484$	0	0
$485$	1.89972	0.0862617
$486$	0	0
$487$	9.04508	0.409872	0.204936	−	0.978775i	$-0.434301\pi$
$487$	9.04508	0.409872	0.204936	+	0.978775i	$0.434301\pi$
$488$	11.2198	0.507895
$489$	0	0
$490$	7.21182	0.325797
$491$	−34.8112	−1.57101	−0.785505	−	0.618856i	$-0.787597\pi$
$491$	−34.8112	−1.57101	−0.785505	+	0.618856i	$0.787597\pi$
$492$	0	0
$493$	40.5988	1.82848
$494$	−58.8789	−2.64909
$495$	0	0
$496$	0.104377	0.00468664
$497$	7.92928	0.355677
$498$	0	0
$499$	19.5992	0.877380	0.438690	−	0.898638i	$-0.355443\pi$
$499$	19.5992	0.877380	0.438690	+	0.898638i	$0.355443\pi$
$500$	−1.33826	−0.0598489
$501$	0	0
$502$	−10.1302	−0.452132
$503$	18.4160	0.821129	0.410564	−	0.911832i	$-0.365332\pi$
$503$	18.4160	0.821129	0.410564	+	0.911832i	$0.365332\pi$
$504$	0	0
$505$	−6.71011	−0.298596
$506$	0	0
$507$	0	0
$508$	−16.9786	−0.753303
$509$	−1.23838	−0.0548901	−0.0274451	−	0.999623i	$-0.508737\pi$
$509$	−1.23838	−0.0548901	−0.0274451	+	0.999623i	$0.508737\pi$
$510$	0	0
$511$	17.7162	0.783717
$512$	19.9828	0.883126
$513$	0	0
$514$	34.6123	1.52668
$515$	4.24678	0.187136
$516$	0	0
$517$	0	0
$518$	3.58071	0.157327
$519$	0	0
$520$	6.47778	0.284070
$521$	−33.1850	−1.45386	−0.726931	−	0.686711i	$-0.759054\pi$
$521$	−33.1850	−1.45386	−0.726931	+	0.686711i	$0.759054\pi$
$522$	0	0
$523$	−13.1508	−0.575046	−0.287523	−	0.957774i	$-0.592832\pi$
$523$	−13.1508	−0.575046	−0.287523	+	0.957774i	$0.592832\pi$
$524$	−1.29825	−0.0567143
$525$	0	0
$526$	5.86264	0.255623
$527$	−0.156615	−0.00682224
$528$	0	0
$529$	46.8514	2.03702
$530$	3.05284	0.132607
$531$	0	0
$532$	14.0641	0.609758
$533$	−19.6083	−0.849331
$534$	0	0
$535$	15.7230	0.679764
$536$	−0.423761	−0.0183037
$537$	0	0
$538$	5.92750	0.255553
$539$	0	0
$540$	0	0
$541$	−32.6519	−1.40381	−0.701907	−	0.712269i	$-0.747668\pi$
$541$	−32.6519	−1.40381	−0.701907	+	0.712269i	$0.747668\pi$
$542$	3.70858	0.159297
$543$	0	0
$544$	−47.7103	−2.04556
$545$	4.23895	0.181577
$546$	0	0
$547$	−27.7048	−1.18457	−0.592285	−	0.805728i	$-0.701774\pi$
$547$	−27.7048	−1.18457	−0.592285	+	0.805728i	$0.701774\pi$
$548$	−5.64793	−0.241268
$549$	0	0
$550$	0	0
$551$	−33.3110	−1.41909
$552$	0	0
$553$	4.47994	0.190506
$554$	−20.6764	−0.878456
$555$	0	0
$556$	−11.8415	−0.502193
$557$	−23.4252	−0.992558	−0.496279	−	0.868163i	$-0.665301\pi$
$557$	−23.4252	−0.992558	−0.496279	+	0.868163i	$0.665301\pi$
$558$	0	0
$559$	10.5520	0.446303
$560$	−8.53627	−0.360723
$561$	0	0
$562$	−18.8514	−0.795198
$563$	22.0856	0.930796	0.465398	−	0.885101i	$-0.345911\pi$
$563$	22.0856	0.930796	0.465398	+	0.885101i	$0.345911\pi$
$564$	0	0
$565$	−7.19643	−0.302756
$566$	3.84765	0.161729
$567$	0	0
$568$	5.48692	0.230226
$569$	−13.7351	−0.575805	−0.287902	−	0.957660i	$-0.592958\pi$
$569$	−13.7351	−0.575805	−0.287902	+	0.957660i	$0.592958\pi$
$570$	0	0
$571$	26.5201	1.10983	0.554915	−	0.831907i	$-0.312751\pi$
$571$	26.5201	1.10983	0.554915	+	0.831907i	$0.312751\pi$
$572$	0	0
$573$	0	0
$574$	11.6835	0.487660
$575$	−8.35772	−0.348541
$576$	0	0
$577$	5.68765	0.236780	0.118390	−	0.992967i	$-0.462227\pi$
$577$	5.68765	0.236780	0.118390	+	0.992967i	$0.462227\pi$
$578$	67.1259	2.79207
$579$	0	0
$580$	−7.41153	−0.307747
$581$	28.0965	1.16564
$582$	0	0
$583$	0	0
$584$	12.2593	0.507292
$585$	0	0
$586$	−42.7091	−1.76430
$587$	−13.5460	−0.559104	−0.279552	−	0.960131i	$-0.590186\pi$
$587$	−13.5460	−0.559104	−0.279552	+	0.960131i	$0.590186\pi$
$588$	0	0
$589$	0.128501	0.00529479
$590$	10.3474	0.425996
$591$	0	0
$592$	5.47990	0.225222
$593$	21.1786	0.869702	0.434851	−	0.900502i	$-0.356801\pi$
$593$	21.1786	0.869702	0.434851	+	0.900502i	$0.356801\pi$
$594$	0	0
$595$	12.8085	0.525097
$596$	12.7852	0.523701
$597$	0	0
$598$	−81.8140	−3.34562
$599$	33.8404	1.38268	0.691341	−	0.722528i	$-0.257020\pi$
$599$	33.8404	1.38268	0.691341	+	0.722528i	$0.257020\pi$
$600$	0	0
$601$	−29.3177	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$601$	−29.3177	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$602$	−6.28736	−0.256254
$603$	0	0
$604$	9.03538	0.367644
$605$	0	0
$606$	0	0
$607$	−7.17486	−0.291219	−0.145609	−	0.989342i	$-0.546514\pi$
$607$	−7.17486	−0.291219	−0.145609	+	0.989342i	$0.546514\pi$
$608$	39.1459	1.58758
$609$	0	0
$610$	16.9550	0.686487
$611$	−20.4640	−0.827884
$612$	0	0
$613$	27.5044	1.11089	0.555447	−	0.831552i	$-0.312547\pi$
$613$	27.5044	1.11089	0.555447	+	0.831552i	$0.312547\pi$
$614$	58.5615	2.36335
$615$	0	0
$616$	0	0
$617$	−24.0712	−0.969068	−0.484534	−	0.874772i	$-0.661011\pi$
$617$	−24.0712	−0.969068	−0.484534	+	0.874772i	$0.661011\pi$
$618$	0	0
$619$	26.9729	1.08413	0.542067	−	0.840335i	$-0.317642\pi$
$619$	26.9729	1.08413	0.542067	+	0.840335i	$0.317642\pi$
$620$	0.0285909	0.00114824
$621$	0	0
$622$	−2.98607	−0.119730
$623$	−25.3457	−1.01546
$624$	0	0
$625$	1.00000	0.0400000
$626$	55.5673	2.22092
$627$	0	0
$628$	−16.9430	−0.676098
$629$	−8.22246	−0.327851
$630$	0	0
$631$	9.73627	0.387595	0.193797	−	0.981042i	$-0.437920\pi$
$631$	9.73627	0.387595	0.193797	+	0.981042i	$0.437920\pi$
$632$	3.10004	0.123313
$633$	0	0
$634$	53.7786	2.13582
$635$	12.6870	0.503470
$636$	0	0
$637$	−21.1478	−0.837904
$638$	0	0
$639$	0	0
$640$	−9.14301	−0.361409
$641$	−25.1165	−0.992040	−0.496020	−	0.868311i	$-0.665206\pi$
$641$	−25.1165	−0.992040	−0.496020	+	0.868311i	$0.665206\pi$
$642$	0	0
$643$	−26.9257	−1.06185	−0.530924	−	0.847420i	$-0.678155\pi$
$643$	−26.9257	−1.06185	−0.530924	+	0.847420i	$0.678155\pi$
$644$	19.5425	0.770083
$645$	0	0
$646$	−80.5612	−3.16964
$647$	26.8011	1.05366	0.526829	−	0.849971i	$-0.323381\pi$
$647$	26.8011	1.05366	0.526829	+	0.849971i	$0.323381\pi$
$648$	0	0
$649$	0	0
$650$	9.78903	0.383957
$651$	0	0
$652$	−31.3230	−1.22670
$653$	−28.7633	−1.12559	−0.562797	−	0.826595i	$-0.690275\pi$
$653$	−28.7633	−1.12559	−0.562797	+	0.826595i	$0.690275\pi$
$654$	0	0
$655$	0.970102	0.0379050
$656$	17.8804	0.698112
$657$	0	0
$658$	12.1933	0.475346
$659$	36.5327	1.42311	0.711556	−	0.702629i	$-0.247991\pi$
$659$	36.5327	1.42311	0.711556	+	0.702629i	$0.247991\pi$
$660$	0	0
$661$	−35.7750	−1.39149	−0.695743	−	0.718291i	$-0.744925\pi$
$661$	−35.7750	−1.39149	−0.695743	+	0.718291i	$0.744925\pi$
$662$	−41.6398	−1.61838
$663$	0	0
$664$	19.4422	0.754505
$665$	−10.5093	−0.407532
$666$	0	0
$667$	−46.2865	−1.79222
$668$	4.08425	0.158024
$669$	0	0
$670$	−0.640375	−0.0247398
$671$	0	0
$672$	0	0
$673$	11.2915	0.435254	0.217627	−	0.976032i	$-0.430168\pi$
$673$	11.2915	0.435254	0.217627	+	0.976032i	$0.430168\pi$
$674$	26.4747	1.01977
$675$	0	0
$676$	21.0175	0.808367
$677$	−43.8574	−1.68558	−0.842788	−	0.538246i	$-0.819087\pi$
$677$	−43.8574	−1.68558	−0.842788	+	0.538246i	$0.819087\pi$
$678$	0	0
$679$	3.31926	0.127381
$680$	8.86324	0.339890
$681$	0	0
$682$	0	0
$683$	18.5355	0.709241	0.354620	−	0.935010i	$-0.384610\pi$
$683$	18.5355	0.709241	0.354620	+	0.935010i	$0.384610\pi$
$684$	0	0
$685$	4.22035	0.161251
$686$	34.9473	1.33429
$687$	0	0
$688$	−9.62214	−0.366841
$689$	−8.95208	−0.341047
$690$	0	0
$691$	−28.2055	−1.07299	−0.536494	−	0.843904i	$-0.680251\pi$
$691$	−28.2055	−1.07299	−0.536494	+	0.843904i	$0.680251\pi$
$692$	21.4221	0.814347
$693$	0	0
$694$	8.01596	0.304282
$695$	8.84846	0.335641
$696$	0	0
$697$	−26.8291	−1.01623
$698$	−26.3212	−0.996274
$699$	0	0
$700$	−2.33826	−0.0883780
$701$	0.315386	0.0119120	0.00595599	−	0.999982i	$-0.498104\pi$
$701$	0.315386	0.0119120	0.00595599	+	0.999982i	$0.498104\pi$
$702$	0	0
$703$	6.74647	0.254448
$704$	0	0
$705$	0	0
$706$	10.9820	0.413313
$707$	−11.7242	−0.440933
$708$	0	0
$709$	39.0246	1.46560	0.732800	−	0.680444i	$-0.238213\pi$
$709$	39.0246	1.46560	0.732800	+	0.680444i	$0.238213\pi$
$710$	8.29167	0.311181
$711$	0	0
$712$	−17.5388	−0.657294
$713$	0.178556	0.00668697
$714$	0	0
$715$	0	0
$716$	25.8272	0.965209
$717$	0	0
$718$	−4.48943	−0.167544
$719$	−28.4691	−1.06172	−0.530859	−	0.847460i	$-0.678131\pi$
$719$	−28.4691	−1.06172	−0.530859	+	0.847460i	$0.678131\pi$
$720$	0	0
$721$	7.42014	0.276341
$722$	31.3851	1.16803
$723$	0	0
$724$	−16.3148	−0.606335
$725$	5.53818	0.205683
$726$	0	0
$727$	45.8400	1.70011	0.850057	−	0.526691i	$-0.176568\pi$
$727$	45.8400	1.70011	0.850057	+	0.526691i	$0.176568\pi$
$728$	11.3182	0.419482
$729$	0	0
$730$	18.5258	0.685672
$731$	14.4378	0.534002
$732$	0	0
$733$	−37.9681	−1.40239	−0.701193	−	0.712972i	$-0.747349\pi$
$733$	−37.9681	−1.40239	−0.701193	+	0.712972i	$0.747349\pi$
$734$	2.70391	0.0998030
$735$	0	0
$736$	54.3944	2.00500
$737$	0	0
$738$	0	0
$739$	−21.1226	−0.777008	−0.388504	−	0.921447i	$-0.627008\pi$
$739$	−21.1226	−0.777008	−0.388504	+	0.921447i	$0.627008\pi$
$740$	1.50106	0.0551800
$741$	0	0
$742$	5.33404	0.195819
$743$	7.98035	0.292771	0.146385	−	0.989228i	$-0.453236\pi$
$743$	7.98035	0.292771	0.146385	+	0.989228i	$0.453236\pi$
$744$	0	0
$745$	−9.55357	−0.350016
$746$	44.4588	1.62775
$747$	0	0
$748$	0	0
$749$	27.4718	1.00380
$750$	0	0
$751$	−9.62482	−0.351215	−0.175607	−	0.984460i	$-0.556189\pi$
$751$	−9.62482	−0.351215	−0.175607	+	0.984460i	$0.556189\pi$
$752$	18.6606	0.680483
$753$	0	0
$754$	54.2134	1.97434
$755$	−6.75158	−0.245715
$756$	0	0
$757$	−46.6153	−1.69426	−0.847130	−	0.531385i	$-0.821672\pi$
$757$	−46.6153	−1.69426	−0.847130	+	0.531385i	$0.821672\pi$
$758$	29.1076	1.05724
$759$	0	0
$760$	−7.27222	−0.263791
$761$	−21.9943	−0.797293	−0.398646	−	0.917105i	$-0.630520\pi$
$761$	−21.9943	−0.797293	−0.398646	+	0.917105i	$0.630520\pi$
$762$	0	0
$763$	7.40646	0.268132
$764$	17.9653	0.649962
$765$	0	0
$766$	7.16209	0.258777
$767$	−30.3424	−1.09560
$768$	0	0
$769$	−48.0330	−1.73211	−0.866057	−	0.499946i	$-0.833353\pi$
$769$	−48.0330	−1.73211	−0.866057	+	0.499946i	$0.833353\pi$
$770$	0	0
$771$	0	0
$772$	−10.1411	−0.364986
$773$	17.0851	0.614509	0.307255	−	0.951627i	$-0.400590\pi$
$773$	17.0851	0.614509	0.307255	+	0.951627i	$0.400590\pi$
$774$	0	0
$775$	−0.0213642	−0.000767425 0
$776$	2.29687	0.0824527
$777$	0	0
$778$	64.4799	2.31172
$779$	22.0131	0.788700
$780$	0	0
$781$	0	0
$782$	−111.942	−4.00304
$783$	0	0
$784$	19.2842	0.688720
$785$	12.6604	0.451870
$786$	0	0
$787$	22.4918	0.801746	0.400873	−	0.916134i	$-0.368707\pi$
$787$	22.4918	0.801746	0.400873	+	0.916134i	$0.368707\pi$
$788$	−9.22220	−0.328527
$789$	0	0
$790$	4.68468	0.166674
$791$	−12.5739	−0.447076
$792$	0	0
$793$	−49.7184	−1.76555
$794$	42.4307	1.50581
$795$	0	0
$796$	0.855682	0.0303288
$797$	3.41778	0.121064	0.0605321	−	0.998166i	$-0.480720\pi$
$797$	3.41778	0.121064	0.0605321	+	0.998166i	$0.480720\pi$
$798$	0	0
$799$	−27.9999	−0.990564
$800$	−6.50828	−0.230103
$801$	0	0
$802$	−10.4046	−0.367398
$803$	0	0
$804$	0	0
$805$	−14.6029	−0.514685
$806$	−0.209135	−0.00736646
$807$	0	0
$808$	−8.11291	−0.285411
$809$	−10.1185	−0.355748	−0.177874	−	0.984053i	$-0.556922\pi$
$809$	−10.1185	−0.355748	−0.177874	+	0.984053i	$0.556922\pi$
$810$	0	0
$811$	−25.3749	−0.891034	−0.445517	−	0.895273i	$-0.646980\pi$
$811$	−25.3749	−0.891034	−0.445517	+	0.895273i	$0.646980\pi$
$812$	−12.9497	−0.454446
$813$	0	0
$814$	0	0
$815$	23.4058	0.819868
$816$	0	0
$817$	−11.8461	−0.414443
$818$	−46.5049	−1.62600
$819$	0	0
$820$	4.89781	0.171039
$821$	1.49915	0.0523206	0.0261603	−	0.999658i	$-0.491672\pi$
$821$	1.49915	0.0523206	0.0261603	+	0.999658i	$0.491672\pi$
$822$	0	0
$823$	−19.1246	−0.666641	−0.333320	−	0.942814i	$-0.608169\pi$
$823$	−19.1246	−0.666641	−0.333320	+	0.942814i	$0.608169\pi$
$824$	5.13460	0.178872
$825$	0	0
$826$	18.0794	0.629062
$827$	−16.6305	−0.578298	−0.289149	−	0.957284i	$-0.593372\pi$
$827$	−16.6305	−0.578298	−0.289149	+	0.957284i	$0.593372\pi$
$828$	0	0
$829$	25.6955	0.892442	0.446221	−	0.894923i	$-0.352769\pi$
$829$	25.6955	0.892442	0.446221	+	0.894923i	$0.352769\pi$
$830$	29.3805	1.01981
$831$	0	0
$832$	−11.3587	−0.393793
$833$	−28.9355	−1.00255
$834$	0	0
$835$	−3.05191	−0.105616
$836$	0	0
$837$	0	0
$838$	24.6383	0.851117
$839$	28.3971	0.980376	0.490188	−	0.871617i	$-0.336928\pi$
$839$	28.3971	0.980376	0.490188	+	0.871617i	$0.336928\pi$
$840$	0	0
$841$	1.67145	0.0576363
$842$	5.38932	0.185728
$843$	0	0
$844$	5.11415	0.176036
$845$	−15.7051	−0.540272
$846$	0	0
$847$	0	0
$848$	8.16320	0.280325
$849$	0	0
$850$	13.3939	0.459406
$851$	9.37441	0.321351
$852$	0	0
$853$	−15.1072	−0.517260	−0.258630	−	0.965976i	$-0.583271\pi$
$853$	−15.1072	−0.517260	−0.258630	+	0.965976i	$0.583271\pi$
$854$	29.6244	1.01373
$855$	0	0
$856$	19.0100	0.649748
$857$	2.37040	0.0809713	0.0404856	−	0.999180i	$-0.487109\pi$
$857$	2.37040	0.0809713	0.0404856	+	0.999180i	$0.487109\pi$
$858$	0	0
$859$	45.0423	1.53682	0.768411	−	0.639956i	$-0.221048\pi$
$859$	45.0423	1.53682	0.768411	+	0.639956i	$0.221048\pi$
$860$	−2.63570	−0.0898768
$861$	0	0
$862$	21.0531	0.717071
$863$	25.0210	0.851724	0.425862	−	0.904788i	$-0.359971\pi$
$863$	25.0210	0.851724	0.425862	+	0.904788i	$0.359971\pi$
$864$	0	0
$865$	−16.0074	−0.544269
$866$	20.3974	0.693130
$867$	0	0
$868$	0.0499551	0.00169559
$869$	0	0
$870$	0	0
$871$	1.87782	0.0636274
$872$	5.12514	0.173559
$873$	0	0
$874$	91.8476	3.10679
$875$	1.74724	0.0590674
$876$	0	0
$877$	48.8755	1.65041	0.825203	−	0.564836i	$-0.191060\pi$
$877$	48.8755	1.65041	0.825203	+	0.564836i	$0.191060\pi$
$878$	−1.81544	−0.0612682
$879$	0	0
$880$	0	0
$881$	−44.5530	−1.50103	−0.750515	−	0.660853i	$-0.770195\pi$
$881$	−44.5530	−1.50103	−0.750515	+	0.660853i	$0.770195\pi$
$882$	0	0
$883$	−44.0547	−1.48256	−0.741279	−	0.671197i	$-0.765780\pi$
$883$	−44.0547	−1.48256	−0.741279	+	0.671197i	$0.765780\pi$
$884$	52.5613	1.76783
$885$	0	0
$886$	20.0661	0.674133
$887$	−12.1293	−0.407261	−0.203631	−	0.979048i	$-0.565274\pi$
$887$	−12.1293	−0.407261	−0.203631	+	0.979048i	$0.565274\pi$
$888$	0	0
$889$	22.1673	0.743467
$890$	−26.5041	−0.888419
$891$	0	0
$892$	−35.7585	−1.19728
$893$	22.9737	0.768785
$894$	0	0
$895$	−19.2991	−0.645098
$896$	−15.9750	−0.533688
$897$	0	0
$898$	−53.0227	−1.76939
$899$	−0.118319	−0.00394615
$900$	0	0
$901$	−12.2487	−0.408063
$902$	0	0
$903$	0	0
$904$	−8.70089	−0.289387
$905$	12.1911	0.405244
$906$	0	0
$907$	32.8672	1.09134	0.545670	−	0.838000i	$-0.316275\pi$
$907$	32.8672	1.09134	0.545670	+	0.838000i	$0.316275\pi$
$908$	16.3767	0.543481
$909$	0	0
$910$	17.1038	0.566985
$911$	46.7054	1.54742	0.773710	−	0.633540i	$-0.218399\pi$
$911$	46.7054	1.54742	0.773710	+	0.633540i	$0.218399\pi$
$912$	0	0
$913$	0	0
$914$	6.49842	0.214949
$915$	0	0
$916$	−8.94847	−0.295666
$917$	1.69500	0.0559738
$918$	0	0
$919$	−7.33826	−0.242067	−0.121033	−	0.992648i	$-0.538621\pi$
$919$	−7.33826	−0.242067	−0.121033	+	0.992648i	$0.538621\pi$
$920$	−10.1050	−0.333150
$921$	0	0
$922$	46.0114	1.51530
$923$	−24.3143	−0.800314
$924$	0	0
$925$	−1.12165	−0.0368795
$926$	−32.6580	−1.07321
$927$	0	0
$928$	−36.0441	−1.18320
$929$	−12.7285	−0.417609	−0.208804	−	0.977957i	$-0.566957\pi$
$929$	−12.7285	−0.417609	−0.208804	+	0.977957i	$0.566957\pi$
$930$	0	0
$931$	23.7413	0.778090
$932$	−8.72201	−0.285699
$933$	0	0
$934$	46.0888	1.50807
$935$	0	0
$936$	0	0
$937$	−19.2533	−0.628979	−0.314490	−	0.949261i	$-0.601833\pi$
$937$	−19.2533	−0.628979	−0.314490	+	0.949261i	$0.601833\pi$
$938$	−1.11889	−0.0365330
$939$	0	0
$940$	5.11153	0.166720
$941$	−41.2694	−1.34535	−0.672673	−	0.739940i	$-0.734854\pi$
$941$	−41.2694	−1.34535	−0.672673	+	0.739940i	$0.734854\pi$
$942$	0	0
$943$	30.5878	0.996076
$944$	27.6686	0.900536
$945$	0	0
$946$	0	0
$947$	50.3012	1.63457	0.817285	−	0.576233i	$-0.195478\pi$
$947$	50.3012	1.63457	0.817285	+	0.576233i	$0.195478\pi$
$948$	0	0
$949$	−54.3247	−1.76345
$950$	−10.9896	−0.356548
$951$	0	0
$952$	15.4862	0.501910
$953$	−9.02580	−0.292374	−0.146187	−	0.989257i	$-0.546700\pi$
$953$	−9.02580	−0.292374	−0.146187	+	0.989257i	$0.546700\pi$
$954$	0	0
$955$	−13.4244	−0.434402
$956$	−14.5093	−0.469263
$957$	0	0
$958$	−9.56617	−0.309069
$959$	7.37396	0.238118
$960$	0	0
$961$	−30.9995	−0.999985
$962$	−10.9798	−0.354004
$963$	0	0
$964$	−36.3744	−1.17154
$965$	7.57782	0.243939
$966$	0	0
$967$	30.2503	0.972785	0.486392	−	0.873740i	$-0.338313\pi$
$967$	30.2503	0.972785	0.486392	+	0.873740i	$0.338313\pi$
$968$	0	0
$969$	0	0
$970$	3.47096	0.111446
$971$	−12.8285	−0.411686	−0.205843	−	0.978585i	$-0.565994\pi$
$971$	−12.8285	−0.411686	−0.205843	+	0.978585i	$0.565994\pi$
$972$	0	0
$973$	15.4604	0.495636
$974$	16.5262	0.529533
$975$	0	0
$976$	45.3371	1.45120
$977$	13.1592	0.420999	0.210500	−	0.977594i	$-0.432491\pi$
$977$	13.1592	0.420999	0.210500	+	0.977594i	$0.432491\pi$
$978$	0	0
$979$	0	0
$980$	5.28233	0.168738
$981$	0	0
$982$	−63.6033	−2.02966
$983$	−22.4369	−0.715627	−0.357813	−	0.933793i	$-0.616478\pi$
$983$	−22.4369	−0.715627	−0.357813	+	0.933793i	$0.616478\pi$
$984$	0	0
$985$	6.89118	0.219571
$986$	74.1776	2.36230
$987$	0	0
$988$	−43.1261	−1.37203
$989$	−16.4605	−0.523414
$990$	0	0
$991$	22.9146	0.727907	0.363953	−	0.931417i	$-0.381427\pi$
$991$	22.9146	0.727907	0.363953	+	0.931417i	$0.381427\pi$
$992$	0.139044	0.00441466
$993$	0	0
$994$	14.4875	0.459516
$995$	−0.639398	−0.0202703
$996$	0	0
$997$	16.8394	0.533309	0.266654	−	0.963792i	$-0.414082\pi$
$997$	16.8394	0.533309	0.266654	+	0.963792i	$0.414082\pi$
$998$	35.8095	1.13353
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bq.1.4		4
3.2	odd	2		1815.2.a.q.1.1		4
11.7	odd	10		495.2.n.c.181.1		8
11.8	odd	10		495.2.n.c.361.1		8
11.10	odd	2		5445.2.a.bj.1.1		4
15.14	odd	2		9075.2.a.df.1.4		4
33.8	even	10		165.2.m.c.31.2	yes	8
33.29	even	10		165.2.m.c.16.2	✓	8
33.32	even	2		1815.2.a.u.1.4		4
165.8	odd	20		825.2.bx.e.724.1		16
165.29	even	10		825.2.n.j.676.1		8
165.62	odd	20		825.2.bx.e.49.1		16
165.74	even	10		825.2.n.j.526.1		8
165.107	odd	20		825.2.bx.e.724.4		16
165.128	odd	20		825.2.bx.e.49.4		16
165.164	even	2		9075.2.a.co.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.c.16.2	✓	8	33.29	even	10
165.2.m.c.31.2	yes	8	33.8	even	10
495.2.n.c.181.1		8	11.7	odd	10
495.2.n.c.361.1		8	11.8	odd	10
825.2.n.j.526.1		8	165.74	even	10
825.2.n.j.676.1		8	165.29	even	10
825.2.bx.e.49.1		16	165.62	odd	20
825.2.bx.e.49.4		16	165.128	odd	20
825.2.bx.e.724.1		16	165.8	odd	20
825.2.bx.e.724.4		16	165.107	odd	20
1815.2.a.q.1.1		4	3.2	odd	2
1815.2.a.u.1.4		4	33.32	even	2
5445.2.a.bj.1.1		4	11.10	odd	2
5445.2.a.bq.1.4		4	1.1	even	1	trivial
9075.2.a.co.1.1		4	165.164	even	2
9075.2.a.df.1.4		4	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+1.82709 q^{2} +1.33826 q^{4} -1.00000 q^{5} -1.74724 q^{7} -1.20906 q^{8} -1.82709 q^{10} +5.35772 q^{13} -3.19236 q^{14} -4.88558 q^{16} +7.33070 q^{17} -6.01478 q^{19} -1.33826 q^{20} -8.35772 q^{23} +1.00000 q^{25} +9.78903 q^{26} -2.33826 q^{28} +5.53818 q^{29} -0.0213642 q^{31} -6.50828 q^{32} +13.3939 q^{34} +1.74724 q^{35} -1.12165 q^{37} -10.9896 q^{38} +1.20906 q^{40} -3.65983 q^{41} +1.96950 q^{43} -15.2703 q^{46} -3.81953 q^{47} -3.94716 q^{49} +1.82709 q^{50} +7.17002 q^{52} -1.67088 q^{53} +2.11251 q^{56} +10.1188 q^{58} -5.66332 q^{59} -9.27977 q^{61} -0.0390343 q^{62} -2.12007 q^{64} -5.35772 q^{65} +0.350489 q^{67} +9.81040 q^{68} +3.19236 q^{70} -4.53818 q^{71} -10.1395 q^{73} -2.04935 q^{74} -8.04935 q^{76} -2.56401 q^{79} +4.88558 q^{80} -6.68684 q^{82} -16.0805 q^{83} -7.33070 q^{85} +3.59845 q^{86} +14.5062 q^{89} -9.36121 q^{91} -11.1848 q^{92} -6.97864 q^{94} +6.01478 q^{95} -1.89972 q^{97} -7.21182 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{4} - 4 q^{5} - 3 q^{8} - q^{10} - 7 q^{13} + 5 q^{14} - 9 q^{16} + 8 q^{17} - 11 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 12 q^{26} - 5 q^{28} + 17 q^{29} - 5 q^{31} + 17 q^{34} + 15 q^{37}+ \cdots + 13 q^{98}+O(q^{100}) \) 4 * q + q^2 + q^4 - 4 * q^5 - 3 * q^8 - q^10 - 7 * q^13 + 5 * q^14 - 9 * q^16 + 8 * q^17 - 11 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 12 * q^26 - 5 * q^28 + 17 * q^29 - 5 * q^31 + 17 * q^34 + 15 * q^37 + q^38 + 3 * q^40 + 10 * q^41 + 4 * q^43 - 15 * q^46 + 8 * q^47 - 8 * q^49 + q^50 + 7 * q^52 - 10 * q^53 - 10 * q^56 - 7 * q^58 - 9 * q^59 - 37 * q^61 - 20 * q^62 - 7 * q^64 + 7 * q^65 + 3 * q^67 + 17 * q^68 - 5 * q^70 - 13 * q^71 - 15 * q^73 - 5 * q^74 - 29 * q^76 - 20 * q^79 + 9 * q^80 + 5 * q^82 - 17 * q^83 - 8 * q^85 - 34 * q^86 + 24 * q^89 - 20 * q^91 - 10 * q^92 - 23 * q^94 + 11 * q^95 - 32 * q^97 + 13 * q^98

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