LMFDB - Embedded newform 5445.2.a.bx.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$43.4785439006$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.27433728.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 15x^{2} - 3 $ x^6 - 9x^4 + 15x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 605)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.480901$ 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)

$f(q)$	$=$	$q-0.480901 q^{2} -1.76873 q^{4} -1.00000 q^{5} -0.480901 q^{7} +1.81239 q^{8} +O(q^{10})$	$q-0.480901 q^{2} -1.76873 q^{4} -1.00000 q^{5} -0.480901 q^{7} +1.81239 q^{8} +0.480901 q^{10} +4.79559 q^{13} +0.231266 q^{14} +2.66589 q^{16} +2.50230 q^{17} +5.75739 q^{19} +1.76873 q^{20} -4.43462 q^{23} +1.00000 q^{25} -2.30620 q^{26} +0.850586 q^{28} +9.01248 q^{29} +7.97209 q^{31} -4.90680 q^{32} -1.20336 q^{34} +0.480901 q^{35} +5.20336 q^{37} -2.76873 q^{38} -1.81239 q^{40} -8.45124 q^{41} -8.53158 q^{43} +2.13261 q^{46} -9.60168 q^{47} -6.76873 q^{49} -0.480901 q^{50} -8.48212 q^{52} -6.10051 q^{53} -0.871579 q^{56} -4.33411 q^{58} -2.76873 q^{59} -4.02534 q^{61} -3.83379 q^{62} -2.97209 q^{64} -4.79559 q^{65} +7.60168 q^{67} -4.42590 q^{68} -0.231266 q^{70} +2.76873 q^{71} +5.00460 q^{73} -2.50230 q^{74} -10.1833 q^{76} +3.62478 q^{79} -2.66589 q^{80} +4.06421 q^{82} -1.71459 q^{83} -2.50230 q^{85} +4.10284 q^{86} +15.7408 q^{89} -2.30620 q^{91} +7.84367 q^{92} +4.61746 q^{94} -5.75739 q^{95} -3.63798 q^{97} +3.25509 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{4} - 6 q^{5}+O(q^{10}) $ 6 * q + 6 * q^4 - 6 * q^5	$ 6 q + 6 q^{4} - 6 q^{5} + 18 q^{14} + 18 q^{16} - 6 q^{20} - 12 q^{23} + 6 q^{25} + 36 q^{26} + 24 q^{34} - 42 q^{47} - 24 q^{49} - 24 q^{53} + 30 q^{56} - 24 q^{58} + 30 q^{64} + 30 q^{67} - 18 q^{70} - 18 q^{80} + 42 q^{82} + 6 q^{86} + 30 q^{89} + 36 q^{91} - 36 q^{92} + 24 q^{97}+O(q^{100}) $ 6 * q + 6 * q^4 - 6 * q^5 + 18 * q^14 + 18 * q^16 - 6 * q^20 - 12 * q^23 + 6 * q^25 + 36 * q^26 + 24 * q^34 - 42 * q^47 - 24 * q^49 - 24 * q^53 + 30 * q^56 - 24 * q^58 + 30 * q^64 + 30 * q^67 - 18 * q^70 - 18 * q^80 + 42 * q^82 + 6 * q^86 + 30 * q^89 + 36 * q^91 - 36 * q^92 + 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 $

$2$	−0.480901	−0.340048	−0.170024	−	0.985440i	$-0.554385\pi$
$2$	−0.480901	−0.340048	−0.170024	+	0.985440i	$0.554385\pi$
$3$	0	0
$3$	0	0
$4$	−1.76873	−0.884367
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.480901	−0.181763	−0.0908817	−	0.995862i	$-0.528969\pi$
$7$	−0.480901	−0.181763	−0.0908817	+	0.995862i	$0.528969\pi$
$8$	1.81239	0.640776
$9$	0	0
$10$	0.480901	0.152074
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.79559	1.33006	0.665028	−	0.746818i	$-0.268419\pi$
$13$	4.79559	1.33006	0.665028	+	0.746818i	$0.268419\pi$
$14$	0.231266	0.0618084
$15$	0	0
$16$	2.66589	0.666472
$17$	2.50230	0.606897	0.303448	−	0.952848i	$-0.401862\pi$
$17$	2.50230	0.606897	0.303448	+	0.952848i	$0.401862\pi$
$18$	0	0
$19$	5.75739	1.32084	0.660418	−	0.750898i	$-0.270379\pi$
$19$	5.75739	1.32084	0.660418	+	0.750898i	$0.270379\pi$
$20$	1.76873	0.395501
$21$	0	0
$22$	0	0
$23$	−4.43462	−0.924683	−0.462342	−	0.886702i	$-0.652991\pi$
$23$	−4.43462	−0.924683	−0.462342	+	0.886702i	$0.652991\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.30620	−0.452284
$27$	0	0
$28$	0.850586	0.160746
$29$	9.01248	1.67358	0.836788	−	0.547527i	$-0.184431\pi$
$29$	9.01248	1.67358	0.836788	+	0.547527i	$0.184431\pi$
$30$	0	0
$31$	7.97209	1.43183	0.715915	−	0.698187i	$-0.246010\pi$
$31$	7.97209	1.43183	0.715915	+	0.698187i	$0.246010\pi$
$32$	−4.90680	−0.867409
$33$	0	0
$34$	−1.20336	−0.206374
$35$	0.480901	0.0812871
$36$	0	0
$37$	5.20336	0.855427	0.427713	−	0.903914i	$-0.359319\pi$
$37$	5.20336	0.855427	0.427713	+	0.903914i	$0.359319\pi$
$38$	−2.76873	−0.449148
$39$	0	0
$40$	−1.81239	−0.286564
$41$	−8.45124	−1.31986	−0.659931	−	0.751326i	$-0.729415\pi$
$41$	−8.45124	−1.31986	−0.659931	+	0.751326i	$0.729415\pi$
$42$	0	0
$43$	−8.53158	−1.30105	−0.650527	−	0.759483i	$-0.725452\pi$
$43$	−8.53158	−1.30105	−0.650527	+	0.759483i	$0.725452\pi$
$44$	0	0
$45$	0	0
$46$	2.13261	0.314437
$47$	−9.60168	−1.40055	−0.700274	−	0.713874i	$-0.746939\pi$
$47$	−9.60168	−1.40055	−0.700274	+	0.713874i	$0.746939\pi$
$48$	0	0
$49$	−6.76873	−0.966962
$50$	−0.480901	−0.0680097
$51$	0	0
$52$	−8.48212	−1.17626
$53$	−6.10051	−0.837970	−0.418985	−	0.907993i	$-0.637614\pi$
$53$	−6.10051	−0.837970	−0.418985	+	0.907993i	$0.637614\pi$
$54$	0	0
$55$	0	0
$56$	−0.871579	−0.116470
$57$	0	0
$58$	−4.33411	−0.569097
$59$	−2.76873	−0.360459	−0.180229	−	0.983625i	$-0.557684\pi$
$59$	−2.76873	−0.360459	−0.180229	+	0.983625i	$0.557684\pi$
$60$	0	0
$61$	−4.02534	−0.515392	−0.257696	−	0.966226i	$-0.582963\pi$
$61$	−4.02534	−0.515392	−0.257696	+	0.966226i	$0.582963\pi$
$62$	−3.83379	−0.486891
$63$	0	0
$64$	−2.97209	−0.371512
$65$	−4.79559	−0.594820
$66$	0	0
$67$	7.60168	0.928693	0.464346	−	0.885654i	$-0.346289\pi$
$67$	7.60168	0.928693	0.464346	+	0.885654i	$0.346289\pi$
$68$	−4.42590	−0.536720
$69$	0	0
$70$	−0.231266	−0.0276415
$71$	2.76873	0.328588	0.164294	−	0.986411i	$-0.447465\pi$
$71$	2.76873	0.328588	0.164294	+	0.986411i	$0.447465\pi$
$72$	0	0
$73$	5.00460	0.585744	0.292872	−	0.956152i	$-0.405389\pi$
$73$	5.00460	0.585744	0.292872	+	0.956152i	$0.405389\pi$
$74$	−2.50230	−0.290886
$75$	0	0
$76$	−10.1833	−1.16810
$77$	0	0
$78$	0	0
$79$	3.62478	0.407819	0.203910	−	0.978990i	$-0.434635\pi$
$79$	3.62478	0.407819	0.203910	+	0.978990i	$0.434635\pi$
$80$	−2.66589	−0.298056
$81$	0	0
$82$	4.06421	0.448817
$83$	−1.71459	−0.188201	−0.0941005	−	0.995563i	$-0.529998\pi$
$83$	−1.71459	−0.188201	−0.0941005	+	0.995563i	$0.529998\pi$
$84$	0	0
$85$	−2.50230	−0.271413
$86$	4.10284	0.442421
$87$	0	0
$88$	0	0
$89$	15.7408	1.66852	0.834262	−	0.551368i	$-0.185894\pi$
$89$	15.7408	1.66852	0.834262	+	0.551368i	$0.185894\pi$
$90$	0	0
$91$	−2.30620	−0.241756
$92$	7.84367	0.817759
$93$	0	0
$94$	4.61746	0.476254
$95$	−5.75739	−0.590696
$96$	0	0
$97$	−3.63798	−0.369381	−0.184691	−	0.982797i	$-0.559128\pi$
$97$	−3.63798	−0.369381	−0.184691	+	0.982797i	$0.559128\pi$
$98$	3.25509	0.328814
$99$	0	0
$100$	−1.76873	−0.176873
$101$	7.69845	0.766025	0.383012	−	0.923743i	$-0.374887\pi$
$101$	7.69845	0.766025	0.383012	+	0.923743i	$0.374887\pi$
$102$	0	0
$103$	9.10284	0.896930	0.448465	−	0.893800i	$-0.351971\pi$
$103$	9.10284	0.896930	0.448465	+	0.893800i	$0.351971\pi$
$104$	8.69147	0.852268
$105$	0	0
$106$	2.93374	0.284950
$107$	16.7913	1.62327	0.811637	−	0.584163i	$-0.198577\pi$
$107$	16.7913	1.62327	0.811637	+	0.584163i	$0.198577\pi$
$108$	0	0
$109$	−17.8817	−1.71276	−0.856380	−	0.516346i	$-0.827292\pi$
$109$	−17.8817	−1.71276	−0.856380	+	0.516346i	$0.827292\pi$
$110$	0	0
$111$	0	0
$112$	−1.28203	−0.121140
$113$	0.462531	0.0435113	0.0217556	−	0.999763i	$-0.493074\pi$
$113$	0.462531	0.0435113	0.0217556	+	0.999763i	$0.493074\pi$
$114$	0	0
$115$	4.43462	0.413531
$116$	−15.9407	−1.48006
$117$	0	0
$118$	1.33149	0.122573
$119$	−1.20336	−0.110312
$120$	0	0
$121$	0	0
$122$	1.93579	0.175258
$123$	0	0
$124$	−14.1005	−1.26626
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.8295	1.40464	0.702319	−	0.711862i	$-0.252148\pi$
$127$	15.8295	1.40464	0.702319	+	0.711862i	$0.252148\pi$
$128$	11.2429	0.993741
$129$	0	0
$130$	2.30620	0.202267
$131$	−9.60460	−0.839158	−0.419579	−	0.907719i	$-0.637822\pi$
$131$	−9.60460	−0.839158	−0.419579	+	0.907719i	$0.637822\pi$
$132$	0	0
$133$	−2.76873	−0.240080
$134$	−3.65565	−0.315800
$135$	0	0
$136$	4.53514	0.388885
$137$	−3.97209	−0.339359	−0.169679	−	0.985499i	$-0.554273\pi$
$137$	−3.97209	−0.339359	−0.169679	+	0.985499i	$0.554273\pi$
$138$	0	0
$139$	9.22149	0.782157	0.391078	−	0.920357i	$-0.372102\pi$
$139$	9.22149	0.782157	0.391078	+	0.920357i	$0.372102\pi$
$140$	−0.850586	−0.0718876
$141$	0	0
$142$	−1.33149	−0.111736
$143$	0	0
$144$	0	0
$145$	−9.01248	−0.748446
$146$	−2.40672	−0.199181
$147$	0	0
$148$	−9.20336	−0.756511
$149$	−7.65012	−0.626722	−0.313361	−	0.949634i	$-0.601455\pi$
$149$	−7.65012	−0.626722	−0.313361	+	0.949634i	$0.601455\pi$
$150$	0	0
$151$	−2.87198	−0.233719	−0.116859	−	0.993148i	$-0.537283\pi$
$151$	−2.87198	−0.233719	−0.116859	+	0.993148i	$0.537283\pi$
$152$	10.4346	0.846360
$153$	0	0
$154$	0	0
$155$	−7.97209	−0.640334
$156$	0	0
$157$	17.4817	1.39519	0.697594	−	0.716493i	$-0.254254\pi$
$157$	17.4817	1.39519	0.697594	+	0.716493i	$0.254254\pi$
$158$	−1.74316	−0.138678
$159$	0	0
$160$	4.90680	0.387917
$161$	2.13261	0.168074
$162$	0	0
$163$	13.1671	1.03132	0.515662	−	0.856792i	$-0.327546\pi$
$163$	13.1671	1.03132	0.515662	+	0.856792i	$0.327546\pi$
$164$	14.9480	1.16724
$165$	0	0
$166$	0.824549	0.0639974
$167$	15.8778	1.22866	0.614331	−	0.789049i	$-0.289426\pi$
$167$	15.8778	1.22866	0.614331	+	0.789049i	$0.289426\pi$
$168$	0	0
$169$	9.99767	0.769051
$170$	1.20336	0.0922934
$171$	0	0
$172$	15.0901	1.15061
$173$	−22.8206	−1.73501	−0.867507	−	0.497425i	$-0.834279\pi$
$173$	−22.8206	−1.73501	−0.867507	+	0.497425i	$0.834279\pi$
$174$	0	0
$175$	−0.480901	−0.0363527
$176$	0	0
$177$	0	0
$178$	−7.56978	−0.567379
$179$	0.462531	0.0345712	0.0172856	−	0.999851i	$-0.494498\pi$
$179$	0.462531	0.0345712	0.0172856	+	0.999851i	$0.494498\pi$
$180$	0	0
$181$	−3.12842	−0.232534	−0.116267	−	0.993218i	$-0.537093\pi$
$181$	−3.12842	−0.232534	−0.116267	+	0.993218i	$0.537093\pi$
$182$	1.10906	0.0822086
$183$	0	0
$184$	−8.03726	−0.592515
$185$	−5.20336	−0.382559
$186$	0	0
$187$	0	0
$188$	16.9828	1.23860
$189$	0	0
$190$	2.76873	0.200865
$191$	5.43695	0.393404	0.196702	−	0.980463i	$-0.436977\pi$
$191$	5.43695	0.393404	0.196702	+	0.980463i	$0.436977\pi$
$192$	0	0
$193$	−3.20675	−0.230827	−0.115414	−	0.993318i	$-0.536819\pi$
$193$	−3.20675	−0.230827	−0.115414	+	0.993318i	$0.536819\pi$
$194$	1.74951	0.125607
$195$	0	0
$196$	11.9721	0.855149
$197$	−19.4048	−1.38253	−0.691267	−	0.722600i	$-0.742947\pi$
$197$	−19.4048	−1.38253	−0.691267	+	0.722600i	$0.742947\pi$
$198$	0	0
$199$	20.3509	1.44264	0.721319	−	0.692603i	$-0.243536\pi$
$199$	20.3509	1.44264	0.721319	+	0.692603i	$0.243536\pi$
$200$	1.81239	0.128155
$201$	0	0
$202$	−3.70219	−0.260485
$203$	−4.33411	−0.304195
$204$	0	0
$205$	8.45124	0.590260
$206$	−4.37757	−0.305000
$207$	0	0
$208$	12.7845	0.886446
$209$	0	0
$210$	0	0
$211$	−6.87987	−0.473630	−0.236815	−	0.971555i	$-0.576103\pi$
$211$	−6.87987	−0.473630	−0.236815	+	0.971555i	$0.576103\pi$
$212$	10.7902	0.741073
$213$	0	0
$214$	−8.07494	−0.551991
$215$	8.53158	0.581849
$216$	0	0
$217$	−3.83379	−0.260254
$218$	8.59935	0.582421
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−1.03863	−0.0695521	−0.0347760	−	0.999395i	$-0.511072\pi$
$223$	−1.03863	−0.0695521	−0.0347760	+	0.999395i	$0.511072\pi$
$224$	2.35969	0.157663
$225$	0	0
$226$	−0.222432	−0.0147959
$227$	0.480901	0.0319185	0.0159593	−	0.999873i	$-0.494920\pi$
$227$	0.480901	0.0319185	0.0159593	+	0.999873i	$0.494920\pi$
$228$	0	0
$229$	13.5398	0.894735	0.447368	−	0.894350i	$-0.352362\pi$
$229$	13.5398	0.894735	0.447368	+	0.894350i	$0.352362\pi$
$230$	−2.13261	−0.140620
$231$	0	0
$232$	16.3341	1.07239
$233$	−21.1543	−1.38586	−0.692932	−	0.721003i	$-0.743681\pi$
$233$	−21.1543	−1.38586	−0.692932	+	0.721003i	$0.743681\pi$
$234$	0	0
$235$	9.60168	0.626344
$236$	4.89716	0.318778
$237$	0	0
$238$	0.578696	0.0375113
$239$	2.67639	0.173122	0.0865608	−	0.996247i	$-0.472412\pi$
$239$	2.67639	0.173122	0.0865608	+	0.996247i	$0.472412\pi$
$240$	0	0
$241$	4.56912	0.294323	0.147161	−	0.989112i	$-0.452986\pi$
$241$	4.56912	0.294323	0.147161	+	0.989112i	$0.452986\pi$
$242$	0	0
$243$	0	0
$244$	7.11976	0.455796
$245$	6.76873	0.432439
$246$	0	0
$247$	27.6101	1.75679
$248$	14.4485	0.917482
$249$	0	0
$250$	0.480901	0.0304148
$251$	11.4370	0.721894	0.360947	−	0.932586i	$-0.382453\pi$
$251$	11.4370	0.721894	0.360947	+	0.932586i	$0.382453\pi$
$252$	0	0
$253$	0	0
$254$	−7.61241	−0.477645
$255$	0	0
$256$	0.537469	0.0335918
$257$	20.0447	1.25035	0.625177	−	0.780483i	$-0.285027\pi$
$257$	20.0447	1.25035	0.625177	+	0.780483i	$0.285027\pi$
$258$	0	0
$259$	−2.50230	−0.155485
$260$	8.48212	0.526039
$261$	0	0
$262$	4.61886	0.285354
$263$	−6.55852	−0.404416	−0.202208	−	0.979343i	$-0.564812\pi$
$263$	−6.55852	−0.404416	−0.202208	+	0.979343i	$0.564812\pi$
$264$	0	0
$265$	6.10051	0.374752
$266$	1.33149	0.0816387
$267$	0	0
$268$	−13.4454	−0.821306
$269$	−1.89716	−0.115672	−0.0578358	−	0.998326i	$-0.518420\pi$
$269$	−1.89716	−0.115672	−0.0578358	+	0.998326i	$0.518420\pi$
$270$	0	0
$271$	−11.3541	−0.689713	−0.344856	−	0.938655i	$-0.612072\pi$
$271$	−11.3541	−0.689713	−0.344856	+	0.938655i	$0.612072\pi$
$272$	6.67086	0.404480
$273$	0	0
$274$	1.91018	0.115398
$275$	0	0
$276$	0	0
$277$	22.8340	1.37196	0.685980	−	0.727620i	$-0.259374\pi$
$277$	22.8340	1.37196	0.685980	+	0.727620i	$0.259374\pi$
$278$	−4.43462	−0.265971
$279$	0	0
$280$	0.871579	0.0520868
$281$	22.6115	1.34889	0.674446	−	0.738324i	$-0.264383\pi$
$281$	22.6115	1.34889	0.674446	+	0.738324i	$0.264383\pi$
$282$	0	0
$283$	−27.5667	−1.63867	−0.819335	−	0.573316i	$-0.805657\pi$
$283$	−27.5667	−1.63867	−0.819335	+	0.573316i	$0.805657\pi$
$284$	−4.89716	−0.290593
$285$	0	0
$286$	0	0
$287$	4.06421	0.239903
$288$	0	0
$289$	−10.7385	−0.631676
$290$	4.33411	0.254508
$291$	0	0
$292$	−8.85181	−0.518013
$293$	21.8587	1.27700	0.638501	−	0.769621i	$-0.279555\pi$
$293$	21.8587	1.27700	0.638501	+	0.769621i	$0.279555\pi$
$294$	0	0
$295$	2.76873	0.161202
$296$	9.43050	0.548137
$297$	0	0
$298$	3.67895	0.213116
$299$	−21.2666	−1.22988
$300$	0	0
$301$	4.10284	0.236484
$302$	1.38114	0.0794757
$303$	0	0
$304$	15.3486	0.880301
$305$	4.02534	0.230490
$306$	0	0
$307$	−9.80019	−0.559326	−0.279663	−	0.960098i	$-0.590223\pi$
$307$	−9.80019	−0.559326	−0.279663	+	0.960098i	$0.590223\pi$
$308$	0	0
$309$	0	0
$310$	3.83379	0.217744
$311$	−9.77107	−0.554066	−0.277033	−	0.960860i	$-0.589351\pi$
$311$	−9.77107	−0.554066	−0.277033	+	0.960860i	$0.589351\pi$
$312$	0	0
$313$	−0.897155	−0.0507102	−0.0253551	−	0.999679i	$-0.508072\pi$
$313$	−0.897155	−0.0507102	−0.0253551	+	0.999679i	$0.508072\pi$
$314$	−8.40694	−0.474431
$315$	0	0
$316$	−6.41126	−0.360662
$317$	2.66822	0.149862	0.0749311	−	0.997189i	$-0.476126\pi$
$317$	2.66822	0.149862	0.0749311	+	0.997189i	$0.476126\pi$
$318$	0	0
$319$	0	0
$320$	2.97209	0.166145
$321$	0	0
$322$	−1.02558	−0.0571531
$323$	14.4067	0.801611
$324$	0	0
$325$	4.79559	0.266011
$326$	−6.33205	−0.350700
$327$	0	0
$328$	−15.3169	−0.845736
$329$	4.61746	0.254569
$330$	0	0
$331$	18.9744	1.04293	0.521464	−	0.853273i	$-0.325386\pi$
$331$	18.9744	1.04293	0.521464	+	0.853273i	$0.325386\pi$
$332$	3.03266	0.166439
$333$	0	0
$334$	−7.63565	−0.417804
$335$	−7.60168	−0.415324
$336$	0	0
$337$	−4.79559	−0.261232	−0.130616	−	0.991433i	$-0.541696\pi$
$337$	−4.79559	−0.261232	−0.130616	+	0.991433i	$0.541696\pi$
$338$	−4.80789	−0.261515
$339$	0	0
$340$	4.42590	0.240028
$341$	0	0
$342$	0	0
$343$	6.62140	0.357522
$344$	−15.4625	−0.833684
$345$	0	0
$346$	10.9744	0.589989
$347$	0.258469	0.0138754	0.00693768	−	0.999976i	$-0.497792\pi$
$347$	0.258469	0.0138754	0.00693768	+	0.999976i	$0.497792\pi$
$348$	0	0
$349$	1.34491	0.0719913	0.0359956	−	0.999352i	$-0.488540\pi$
$349$	1.34491	0.0719913	0.0359956	+	0.999352i	$0.488540\pi$
$350$	0.231266	0.0123617
$351$	0	0
$352$	0	0
$353$	−35.4537	−1.88701	−0.943506	−	0.331355i	$-0.892494\pi$
$353$	−35.4537	−1.88701	−0.943506	+	0.331355i	$0.892494\pi$
$354$	0	0
$355$	−2.76873	−0.146949
$356$	−27.8413	−1.47559
$357$	0	0
$358$	−0.222432	−0.0117559
$359$	7.84167	0.413867	0.206934	−	0.978355i	$-0.433652\pi$
$359$	7.84167	0.413867	0.206934	+	0.978355i	$0.433652\pi$
$360$	0	0
$361$	14.1475	0.744608
$362$	1.50446	0.0790727
$363$	0	0
$364$	4.07906	0.213801
$365$	−5.00460	−0.261953
$366$	0	0
$367$	1.06654	0.0556730	0.0278365	−	0.999612i	$-0.491138\pi$
$367$	1.06654	0.0556730	0.0278365	+	0.999612i	$0.491138\pi$
$368$	−11.8222	−0.616276
$369$	0	0
$370$	2.50230	0.130088
$371$	2.93374	0.152312
$372$	0	0
$373$	−37.6388	−1.94886	−0.974431	−	0.224689i	$-0.927863\pi$
$373$	−37.6388	−1.94886	−0.974431	+	0.224689i	$0.927863\pi$
$374$	0	0
$375$	0	0
$376$	−17.4020	−0.897438
$377$	43.2201	2.22595
$378$	0	0
$379$	35.1475	1.80541	0.902704	−	0.430262i	$-0.141579\pi$
$379$	35.1475	1.80541	0.902704	+	0.430262i	$0.141579\pi$
$380$	10.1833	0.522392
$381$	0	0
$382$	−2.61464	−0.133776
$383$	−0.824549	−0.0421325	−0.0210662	−	0.999778i	$-0.506706\pi$
$383$	−0.824549	−0.0421325	−0.0210662	+	0.999778i	$0.506706\pi$
$384$	0	0
$385$	0	0
$386$	1.54213	0.0784924
$387$	0	0
$388$	6.43462	0.326669
$389$	−4.84367	−0.245584	−0.122792	−	0.992432i	$-0.539185\pi$
$389$	−4.84367	−0.245584	−0.122792	+	0.992432i	$0.539185\pi$
$390$	0	0
$391$	−11.0968	−0.561187
$392$	−12.2676	−0.619606
$393$	0	0
$394$	9.33178	0.470128
$395$	−3.62478	−0.182382
$396$	0	0
$397$	−7.33178	−0.367971	−0.183986	−	0.982929i	$-0.558900\pi$
$397$	−7.33178	−0.367971	−0.183986	+	0.982929i	$0.558900\pi$
$398$	−9.78677	−0.490566
$399$	0	0
$400$	2.66589	0.133294
$401$	31.1499	1.55555	0.777775	−	0.628542i	$-0.216348\pi$
$401$	31.1499	1.55555	0.777775	+	0.628542i	$0.216348\pi$
$402$	0	0
$403$	38.2309	1.90442
$404$	−13.6165	−0.677447
$405$	0	0
$406$	2.08428	0.103441
$407$	0	0
$408$	0	0
$409$	−7.23209	−0.357604	−0.178802	−	0.983885i	$-0.557222\pi$
$409$	−7.23209	−0.357604	−0.178802	+	0.983885i	$0.557222\pi$
$410$	−4.06421	−0.200717
$411$	0	0
$412$	−16.1005	−0.793215
$413$	1.33149	0.0655182
$414$	0	0
$415$	1.71459	0.0841660
$416$	−23.5310	−1.15370
$417$	0	0
$418$	0	0
$419$	2.30620	0.112665	0.0563327	−	0.998412i	$-0.482059\pi$
$419$	2.30620	0.112665	0.0563327	+	0.998412i	$0.482059\pi$
$420$	0	0
$421$	6.63798	0.323515	0.161758	−	0.986831i	$-0.448284\pi$
$421$	6.63798	0.323515	0.161758	+	0.986831i	$0.448284\pi$
$422$	3.30853	0.161057
$423$	0	0
$424$	−11.0565	−0.536951
$425$	2.50230	0.121379
$426$	0	0
$427$	1.93579	0.0936794
$428$	−29.6993	−1.43557
$429$	0	0
$430$	−4.10284	−0.197857
$431$	32.8432	1.58200	0.791000	−	0.611816i	$-0.209561\pi$
$431$	32.8432	1.58200	0.791000	+	0.611816i	$0.209561\pi$
$432$	0	0
$433$	2.36202	0.113511	0.0567557	−	0.998388i	$-0.481924\pi$
$433$	2.36202	0.113511	0.0567557	+	0.998388i	$0.481924\pi$
$434$	1.84367	0.0884991
$435$	0	0
$436$	31.6281	1.51471
$437$	−25.5319	−1.22135
$438$	0	0
$439$	30.8847	1.47404	0.737022	−	0.675869i	$-0.236231\pi$
$439$	30.8847	1.47404	0.737022	+	0.675869i	$0.236231\pi$
$440$	0	0
$441$	0	0
$442$	−5.77081	−0.274489
$443$	13.7748	0.654460	0.327230	−	0.944945i	$-0.393885\pi$
$443$	13.7748	0.654460	0.327230	+	0.944945i	$0.393885\pi$
$444$	0	0
$445$	−15.7408	−0.746187
$446$	0.499480	0.0236511
$447$	0	0
$448$	1.42928	0.0675272
$449$	38.9163	1.83657	0.918286	−	0.395917i	$-0.129573\pi$
$449$	38.9163	1.83657	0.918286	+	0.395917i	$0.129573\pi$
$450$	0	0
$451$	0	0
$452$	−0.818095	−0.0384800
$453$	0	0
$454$	−0.231266	−0.0108538
$455$	2.30620	0.108116
$456$	0	0
$457$	−1.33149	−0.0622843	−0.0311422	−	0.999515i	$-0.509914\pi$
$457$	−1.33149	−0.0622843	−0.0311422	+	0.999515i	$0.509914\pi$
$458$	−6.51130	−0.304253
$459$	0	0
$460$	−7.84367	−0.365713
$461$	5.73993	0.267335	0.133668	−	0.991026i	$-0.457325\pi$
$461$	5.73993	0.267335	0.133668	+	0.991026i	$0.457325\pi$
$462$	0	0
$463$	−3.44535	−0.160119	−0.0800595	−	0.996790i	$-0.525511\pi$
$463$	−3.44535	−0.160119	−0.0800595	+	0.996790i	$0.525511\pi$
$464$	24.0263	1.11539
$465$	0	0
$466$	10.1731	0.471261
$467$	27.2164	1.25943	0.629713	−	0.776828i	$-0.283173\pi$
$467$	27.2164	1.25943	0.629713	+	0.776828i	$0.283173\pi$
$468$	0	0
$469$	−3.65565	−0.168802
$470$	−4.61746	−0.212987
$471$	0	0
$472$	−5.01802	−0.230973
$473$	0	0
$474$	0	0
$475$	5.75739	0.264167
$476$	2.12842	0.0975560
$477$	0	0
$478$	−1.28708	−0.0588697
$479$	13.9822	0.638861	0.319431	−	0.947610i	$-0.396508\pi$
$479$	13.9822	0.638861	0.319431	+	0.947610i	$0.396508\pi$
$480$	0	0
$481$	24.9532	1.13777
$482$	−2.19729	−0.100084
$483$	0	0
$484$	0	0
$485$	3.63798	0.165192
$486$	0	0
$487$	−7.45375	−0.337761	−0.168881	−	0.985636i	$-0.554015\pi$
$487$	−7.45375	−0.337761	−0.168881	+	0.985636i	$0.554015\pi$
$488$	−7.29548	−0.330251
$489$	0	0
$490$	−3.25509	−0.147050
$491$	−15.9756	−0.720969	−0.360484	−	0.932765i	$-0.617389\pi$
$491$	−15.9756	−0.720969	−0.360484	+	0.932765i	$0.617389\pi$
$492$	0	0
$493$	22.5519	1.01569
$494$	−13.2777	−0.597392
$495$	0	0
$496$	21.2527	0.954275
$497$	−1.33149	−0.0597253
$498$	0	0
$499$	−24.3509	−1.09010	−0.545048	−	0.838405i	$-0.683489\pi$
$499$	−24.3509	−1.09010	−0.545048	+	0.838405i	$0.683489\pi$
$500$	1.76873	0.0791002
$501$	0	0
$502$	−5.50004	−0.245479
$503$	27.5667	1.22914	0.614569	−	0.788863i	$-0.289330\pi$
$503$	27.5667	1.22914	0.614569	+	0.788863i	$0.289330\pi$
$504$	0	0
$505$	−7.69845	−0.342577
$506$	0	0
$507$	0	0
$508$	−27.9981	−1.24222
$509$	−17.6850	−0.783874	−0.391937	−	0.919992i	$-0.628195\pi$
$509$	−17.6850	−0.783874	−0.391937	+	0.919992i	$0.628195\pi$
$510$	0	0
$511$	−2.40672	−0.106467
$512$	−22.7443	−1.00516
$513$	0	0
$514$	−9.63951	−0.425181
$515$	−9.10284	−0.401119
$516$	0	0
$517$	0	0
$518$	1.20336	0.0528725
$519$	0	0
$520$	−8.69147	−0.381146
$521$	5.38993	0.236137	0.118068	−	0.993005i	$-0.462330\pi$
$521$	5.38993	0.236137	0.118068	+	0.993005i	$0.462330\pi$
$522$	0	0
$523$	37.6737	1.64735	0.823677	−	0.567059i	$-0.191919\pi$
$523$	37.6737	1.64735	0.823677	+	0.567059i	$0.191919\pi$
$524$	16.9880	0.742123
$525$	0	0
$526$	3.15400	0.137521
$527$	19.9486	0.868973
$528$	0	0
$529$	−3.33411	−0.144961
$530$	−2.93374	−0.127434
$531$	0	0
$532$	4.89716	0.212319
$533$	−40.5287	−1.75549
$534$	0	0
$535$	−16.7913	−0.725950
$536$	13.7772	0.595084
$537$	0	0
$538$	0.912344	0.0393339
$539$	0	0
$540$	0	0
$541$	28.9651	1.24531	0.622653	−	0.782498i	$-0.286055\pi$
$541$	28.9651	1.24531	0.622653	+	0.782498i	$0.286055\pi$
$542$	5.46020	0.234536
$543$	0	0
$544$	−12.2783	−0.526428
$545$	17.8817	0.765970
$546$	0	0
$547$	−19.7396	−0.844002	−0.422001	−	0.906595i	$-0.638672\pi$
$547$	−19.7396	−0.844002	−0.422001	+	0.906595i	$0.638672\pi$
$548$	7.02558	0.300118
$549$	0	0
$550$	0	0
$551$	51.8884	2.21052
$552$	0	0
$553$	−1.74316	−0.0741266
$554$	−10.9809	−0.466533
$555$	0	0
$556$	−16.3104	−0.691714
$557$	−16.6801	−0.706757	−0.353378	−	0.935481i	$-0.614967\pi$
$557$	−16.6801	−0.706757	−0.353378	+	0.935481i	$0.614967\pi$
$558$	0	0
$559$	−40.9139	−1.73048
$560$	1.28203	0.0541756
$561$	0	0
$562$	−10.8739	−0.458688
$563$	4.90680	0.206797	0.103399	−	0.994640i	$-0.467028\pi$
$563$	4.90680	0.206797	0.103399	+	0.994640i	$0.467028\pi$
$564$	0	0
$565$	−0.462531	−0.0194588
$566$	13.2568	0.557227
$567$	0	0
$568$	5.01802	0.210551
$569$	22.4334	0.940457	0.470229	−	0.882545i	$-0.344171\pi$
$569$	22.4334	0.940457	0.470229	+	0.882545i	$0.344171\pi$
$570$	0	0
$571$	0.195590	0.00818520	0.00409260	−	0.999992i	$-0.498697\pi$
$571$	0.195590	0.00818520	0.00409260	+	0.999992i	$0.498697\pi$
$572$	0	0
$573$	0	0
$574$	−1.95448	−0.0815785
$575$	−4.43462	−0.184937
$576$	0	0
$577$	−41.2481	−1.71718	−0.858590	−	0.512664i	$-0.828659\pi$
$577$	−41.2481	−1.71718	−0.858590	+	0.512664i	$0.828659\pi$
$578$	5.16415	0.214800
$579$	0	0
$580$	15.9407	0.661901
$581$	0.824549	0.0342081
$582$	0	0
$583$	0	0
$584$	9.07028	0.375331
$585$	0	0
$586$	−10.5119	−0.434242
$587$	1.77480	0.0732538	0.0366269	−	0.999329i	$-0.488339\pi$
$587$	1.77480	0.0732538	0.0366269	+	0.999329i	$0.488339\pi$
$588$	0	0
$589$	45.8984	1.89121
$590$	−1.33149	−0.0548164
$591$	0	0
$592$	13.8716	0.570118
$593$	29.0094	1.19127	0.595636	−	0.803254i	$-0.296900\pi$
$593$	29.0094	1.19127	0.595636	+	0.803254i	$0.296900\pi$
$594$	0	0
$595$	1.20336	0.0493329
$596$	13.5310	0.554252
$597$	0	0
$598$	10.2271	0.418219
$599$	−40.2672	−1.64527	−0.822636	−	0.568568i	$-0.807498\pi$
$599$	−40.2672	−1.64527	−0.822636	+	0.568568i	$0.807498\pi$
$600$	0	0
$601$	−13.6957	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$601$	−13.6957	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$602$	−1.97306	−0.0804160
$603$	0	0
$604$	5.07978	0.206693
$605$	0	0
$606$	0	0
$607$	21.1543	0.858626	0.429313	−	0.903156i	$-0.358756\pi$
$607$	21.1543	0.858626	0.429313	+	0.903156i	$0.358756\pi$
$608$	−28.2504	−1.14570
$609$	0	0
$610$	−1.93579	−0.0783778
$611$	−46.0457	−1.86281
$612$	0	0
$613$	−23.2520	−0.939139	−0.469570	−	0.882895i	$-0.655591\pi$
$613$	−23.2520	−0.939139	−0.469570	+	0.882895i	$0.655591\pi$
$614$	4.71292	0.190198
$615$	0	0
$616$	0	0
$617$	13.1028	0.527501	0.263750	−	0.964591i	$-0.415040\pi$
$617$	13.1028	0.527501	0.263750	+	0.964591i	$0.415040\pi$
$618$	0	0
$619$	16.6682	0.669952	0.334976	−	0.942227i	$-0.391272\pi$
$619$	16.6682	0.669952	0.334976	+	0.942227i	$0.391272\pi$
$620$	14.1005	0.566290
$621$	0	0
$622$	4.69891	0.188409
$623$	−7.56978	−0.303277
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.431443	0.0172439
$627$	0	0
$628$	−30.9204	−1.23386
$629$	13.0204	0.519156
$630$	0	0
$631$	−2.66356	−0.106035	−0.0530173	−	0.998594i	$-0.516884\pi$
$631$	−2.66356	−0.106035	−0.0530173	+	0.998594i	$0.516884\pi$
$632$	6.56950	0.261321
$633$	0	0
$634$	−1.28315	−0.0509604
$635$	−15.8295	−0.628173
$636$	0	0
$637$	−32.4601	−1.28611
$638$	0	0
$639$	0	0
$640$	−11.2429	−0.444414
$641$	0.462531	0.0182689	0.00913445	−	0.999958i	$-0.497092\pi$
$641$	0.462531	0.0182689	0.00913445	+	0.999958i	$0.497092\pi$
$642$	0	0
$643$	−33.4621	−1.31962	−0.659809	−	0.751433i	$-0.729363\pi$
$643$	−33.4621	−1.31962	−0.659809	+	0.751433i	$0.729363\pi$
$644$	−3.77203	−0.148639
$645$	0	0
$646$	−6.92820	−0.272587
$647$	40.2588	1.58274	0.791368	−	0.611340i	$-0.209369\pi$
$647$	40.2588	1.58274	0.791368	+	0.611340i	$0.209369\pi$
$648$	0	0
$649$	0	0
$650$	−2.30620	−0.0904567
$651$	0	0
$652$	−23.2890	−0.912068
$653$	−15.3486	−0.600636	−0.300318	−	0.953839i	$-0.597093\pi$
$653$	−15.3486	−0.600636	−0.300318	+	0.953839i	$0.597093\pi$
$654$	0	0
$655$	9.60460	0.375283
$656$	−22.5301	−0.879652
$657$	0	0
$658$	−2.22054	−0.0865656
$659$	−36.6635	−1.42821	−0.714104	−	0.700039i	$-0.753166\pi$
$659$	−36.6635	−1.42821	−0.714104	+	0.700039i	$0.753166\pi$
$660$	0	0
$661$	−8.03024	−0.312340	−0.156170	−	0.987730i	$-0.549915\pi$
$661$	−8.03024	−0.312340	−0.156170	+	0.987730i	$0.549915\pi$
$662$	−9.12482	−0.354646
$663$	0	0
$664$	−3.10751	−0.120595
$665$	2.76873	0.107367
$666$	0	0
$667$	−39.9670	−1.54753
$668$	−28.0836	−1.08659
$669$	0	0
$670$	3.65565	0.141230
$671$	0	0
$672$	0	0
$673$	−11.7587	−0.453265	−0.226632	−	0.973980i	$-0.572772\pi$
$673$	−11.7587	−0.453265	−0.226632	+	0.973980i	$0.572772\pi$
$674$	2.30620	0.0888316
$675$	0	0
$676$	−17.6832	−0.680124
$677$	27.2948	1.04902	0.524512	−	0.851403i	$-0.324248\pi$
$677$	27.2948	1.04902	0.524512	+	0.851403i	$0.324248\pi$
$678$	0	0
$679$	1.74951	0.0671400
$680$	−4.53514	−0.173915
$681$	0	0
$682$	0	0
$683$	−49.5738	−1.89689	−0.948444	−	0.316945i	$-0.897343\pi$
$683$	−49.5738	−1.89689	−0.948444	+	0.316945i	$0.897343\pi$
$684$	0	0
$685$	3.97209	0.151766
$686$	−3.18424	−0.121575
$687$	0	0
$688$	−22.7443	−0.867116
$689$	−29.2556	−1.11455
$690$	0	0
$691$	−25.3253	−0.963421	−0.481710	−	0.876330i	$-0.659984\pi$
$691$	−25.3253	−0.963421	−0.481710	+	0.876330i	$0.659984\pi$
$692$	40.3635	1.53439
$693$	0	0
$694$	−0.124298	−0.00471829
$695$	−9.22149	−0.349791
$696$	0	0
$697$	−21.1475	−0.801020
$698$	−0.646767	−0.0244805
$699$	0	0
$700$	0.850586	0.0321491
$701$	18.2555	0.689500	0.344750	−	0.938695i	$-0.387964\pi$
$701$	18.2555	0.689500	0.344750	+	0.938695i	$0.387964\pi$
$702$	0	0
$703$	29.9578	1.12988
$704$	0	0
$705$	0	0
$706$	17.0497	0.641675
$707$	−3.70219	−0.139235
$708$	0	0
$709$	4.33178	0.162683	0.0813417	−	0.996686i	$-0.474079\pi$
$709$	4.33178	0.162683	0.0813417	+	0.996686i	$0.474079\pi$
$710$	1.33149	0.0499698
$711$	0	0
$712$	28.5285	1.06915
$713$	−35.3532	−1.32399
$714$	0	0
$715$	0	0
$716$	−0.818095	−0.0305737
$717$	0	0
$718$	−3.77107	−0.140735
$719$	24.1005	0.898797	0.449399	−	0.893331i	$-0.351638\pi$
$719$	24.1005	0.898797	0.449399	+	0.893331i	$0.351638\pi$
$720$	0	0
$721$	−4.37757	−0.163029
$722$	−6.80357	−0.253203
$723$	0	0
$724$	5.53335	0.205645
$725$	9.01248	0.334715
$726$	0	0
$727$	−25.1066	−0.931151	−0.465576	−	0.885008i	$-0.654153\pi$
$727$	−25.1066	−0.931151	−0.465576	+	0.885008i	$0.654153\pi$
$728$	−4.17973	−0.154911
$729$	0	0
$730$	2.40672	0.0890766
$731$	−21.3486	−0.789605
$732$	0	0
$733$	−17.4463	−0.644393	−0.322196	−	0.946673i	$-0.604421\pi$
$733$	−17.4463	−0.644393	−0.322196	+	0.946673i	$0.604421\pi$
$734$	−0.512901	−0.0189315
$735$	0	0
$736$	21.7598	0.802078
$737$	0	0
$738$	0	0
$739$	−5.20019	−0.191292	−0.0956460	−	0.995415i	$-0.530492\pi$
$739$	−5.20019	−0.191292	−0.0956460	+	0.995415i	$0.530492\pi$
$740$	9.20336	0.338322
$741$	0	0
$742$	−1.41084	−0.0517935
$743$	37.1928	1.36447	0.682235	−	0.731133i	$-0.261008\pi$
$743$	37.1928	1.36447	0.682235	+	0.731133i	$0.261008\pi$
$744$	0	0
$745$	7.65012	0.280279
$746$	18.1005	0.662707
$747$	0	0
$748$	0	0
$749$	−8.07494	−0.295052
$750$	0	0
$751$	41.4212	1.51148	0.755740	−	0.654872i	$-0.227277\pi$
$751$	41.4212	1.51148	0.755740	+	0.654872i	$0.227277\pi$
$752$	−25.5970	−0.933427
$753$	0	0
$754$	−20.7846	−0.756931
$755$	2.87198	0.104522
$756$	0	0
$757$	3.91628	0.142340	0.0711698	−	0.997464i	$-0.477327\pi$
$757$	3.91628	0.142340	0.0711698	+	0.997464i	$0.477327\pi$
$758$	−16.9025	−0.613926
$759$	0	0
$760$	−10.4346	−0.378504
$761$	39.1309	1.41849	0.709247	−	0.704960i	$-0.249035\pi$
$761$	39.1309	1.41849	0.709247	+	0.704960i	$0.249035\pi$
$762$	0	0
$763$	8.59935	0.311317
$764$	−9.61653	−0.347914
$765$	0	0
$766$	0.396526	0.0143271
$767$	−13.2777	−0.479430
$768$	0	0
$769$	11.6755	0.421028	0.210514	−	0.977591i	$-0.432486\pi$
$769$	11.6755	0.421028	0.210514	+	0.977591i	$0.432486\pi$
$770$	0	0
$771$	0	0
$772$	5.67189	0.204136
$773$	7.76640	0.279338	0.139669	−	0.990198i	$-0.455396\pi$
$773$	7.76640	0.279338	0.139669	+	0.990198i	$0.455396\pi$
$774$	0	0
$775$	7.97209	0.286366
$776$	−6.59343	−0.236691
$777$	0	0
$778$	2.32933	0.0835104
$779$	−48.6571	−1.74332
$780$	0	0
$781$	0	0
$782$	5.33644	0.190831
$783$	0	0
$784$	−18.0447	−0.644454
$785$	−17.4817	−0.623947
$786$	0	0
$787$	−1.42928	−0.0509484	−0.0254742	−	0.999675i	$-0.508110\pi$
$787$	−1.42928	−0.0509484	−0.0254742	+	0.999675i	$0.508110\pi$
$788$	34.3219	1.22267
$789$	0	0
$790$	1.74316	0.0620188
$791$	−0.222432	−0.00790876
$792$	0	0
$793$	−19.3039	−0.685501
$794$	3.52586	0.125128
$795$	0	0
$796$	−35.9953	−1.27582
$797$	36.3788	1.28860	0.644302	−	0.764771i	$-0.277148\pi$
$797$	36.3788	1.28860	0.644302	+	0.764771i	$0.277148\pi$
$798$	0	0
$799$	−24.0263	−0.849989
$800$	−4.90680	−0.173482
$801$	0	0
$802$	−14.9800	−0.528962
$803$	0	0
$804$	0	0
$805$	−2.13261	−0.0751648
$806$	−18.3853	−0.647593
$807$	0	0
$808$	13.9526	0.490850
$809$	−27.8735	−0.979980	−0.489990	−	0.871728i	$-0.663000\pi$
$809$	−27.8735	−0.979980	−0.489990	+	0.871728i	$0.663000\pi$
$810$	0	0
$811$	39.1793	1.37577	0.687885	−	0.725820i	$-0.258539\pi$
$811$	39.1793	1.37577	0.687885	+	0.725820i	$0.258539\pi$
$812$	7.66589	0.269020
$813$	0	0
$814$	0	0
$815$	−13.1671	−0.461222
$816$	0	0
$817$	−49.1196	−1.71848
$818$	3.47792	0.121603
$819$	0	0
$820$	−14.9480	−0.522007
$821$	13.7772	0.480827	0.240414	−	0.970671i	$-0.422717\pi$
$821$	13.7772	0.480827	0.240414	+	0.970671i	$0.422717\pi$
$822$	0	0
$823$	−19.8800	−0.692972	−0.346486	−	0.938055i	$-0.612625\pi$
$823$	−19.8800	−0.692972	−0.346486	+	0.938055i	$0.612625\pi$
$824$	16.4979	0.574731
$825$	0	0
$826$	−0.640313	−0.0222793
$827$	−13.3272	−0.463431	−0.231716	−	0.972784i	$-0.574434\pi$
$827$	−13.3272	−0.463431	−0.231716	+	0.972784i	$0.574434\pi$
$828$	0	0
$829$	51.1899	1.77790	0.888950	−	0.458005i	$-0.151436\pi$
$829$	51.1899	1.77790	0.888950	+	0.458005i	$0.151436\pi$
$830$	−0.824549	−0.0286205
$831$	0	0
$832$	−14.2529	−0.494132
$833$	−16.9374	−0.586846
$834$	0	0
$835$	−15.8778	−0.549474
$836$	0	0
$837$	0	0
$838$	−1.10906	−0.0383117
$839$	16.8739	0.582552	0.291276	−	0.956639i	$-0.405920\pi$
$839$	16.8739	0.582552	0.291276	+	0.956639i	$0.405920\pi$
$840$	0	0
$841$	52.2248	1.80086
$842$	−3.19221	−0.110011
$843$	0	0
$844$	12.1687	0.418862
$845$	−9.99767	−0.343930
$846$	0	0
$847$	0	0
$848$	−16.2633	−0.558484
$849$	0	0
$850$	−1.20336	−0.0412748
$851$	−23.0749	−0.790999
$852$	0	0
$853$	5.42262	0.185667	0.0928335	−	0.995682i	$-0.470408\pi$
$853$	5.42262	0.185667	0.0928335	+	0.995682i	$0.470408\pi$
$854$	−0.930923	−0.0318555
$855$	0	0
$856$	30.4323	1.04015
$857$	−27.8386	−0.950947	−0.475474	−	0.879730i	$-0.657723\pi$
$857$	−27.8386	−0.950947	−0.475474	+	0.879730i	$0.657723\pi$
$858$	0	0
$859$	48.1899	1.64422	0.822109	−	0.569330i	$-0.192797\pi$
$859$	48.1899	1.64422	0.822109	+	0.569330i	$0.192797\pi$
$860$	−15.0901	−0.514568
$861$	0	0
$862$	−15.7943	−0.537956
$863$	−26.8609	−0.914354	−0.457177	−	0.889376i	$-0.651139\pi$
$863$	−26.8609	−0.914354	−0.457177	+	0.889376i	$0.651139\pi$
$864$	0	0
$865$	22.8206	0.775922
$866$	−1.13590	−0.0385993
$867$	0	0
$868$	6.78095	0.230160
$869$	0	0
$870$	0	0
$871$	36.4545	1.23521
$872$	−32.4087	−1.09750
$873$	0	0
$874$	12.2783	0.415320
$875$	0.480901	0.0162574
$876$	0	0
$877$	39.8837	1.34678	0.673389	−	0.739289i	$-0.264838\pi$
$877$	39.8837	1.34678	0.673389	+	0.739289i	$0.264838\pi$
$878$	−14.8525	−0.501246
$879$	0	0
$880$	0	0
$881$	2.82222	0.0950829	0.0475415	−	0.998869i	$-0.484861\pi$
$881$	2.82222	0.0950829	0.0475415	+	0.998869i	$0.484861\pi$
$882$	0	0
$883$	50.5287	1.70043	0.850213	−	0.526439i	$-0.176473\pi$
$883$	50.5287	1.70043	0.850213	+	0.526439i	$0.176473\pi$
$884$	−21.2248	−0.713868
$885$	0	0
$886$	−6.62431	−0.222548
$887$	28.3195	0.950875	0.475437	−	0.879750i	$-0.342290\pi$
$887$	28.3195	0.950875	0.475437	+	0.879750i	$0.342290\pi$
$888$	0	0
$889$	−7.61241	−0.255312
$890$	7.56978	0.253740
$891$	0	0
$892$	1.83707	0.0615096
$893$	−55.2806	−1.84990
$894$	0	0
$895$	−0.462531	−0.0154607
$896$	−5.40672	−0.180626
$897$	0	0
$898$	−18.7149	−0.624523
$899$	71.8483	2.39628
$900$	0	0
$901$	−15.2653	−0.508561
$902$	0	0
$903$	0	0
$904$	0.838286	0.0278810
$905$	3.12842	0.103992
$906$	0	0
$907$	51.0508	1.69511	0.847556	−	0.530705i	$-0.178073\pi$
$907$	51.0508	1.69511	0.847556	+	0.530705i	$0.178073\pi$
$908$	−0.850586	−0.0282277
$909$	0	0
$910$	−1.10906	−0.0367648
$911$	25.1429	0.833021	0.416510	−	0.909131i	$-0.363253\pi$
$911$	25.1429	0.833021	0.416510	+	0.909131i	$0.363253\pi$
$912$	0	0
$913$	0	0
$914$	0.640313	0.0211797
$915$	0	0
$916$	−23.9483	−0.791274
$917$	4.61886	0.152528
$918$	0	0
$919$	−25.6151	−0.844965	−0.422482	−	0.906371i	$-0.638841\pi$
$919$	−25.6151	−0.844965	−0.422482	+	0.906371i	$0.638841\pi$
$920$	8.03726	0.264981
$921$	0	0
$922$	−2.76034	−0.0909069
$923$	13.2777	0.437041
$924$	0	0
$925$	5.20336	0.171085
$926$	1.65687	0.0544482
$927$	0	0
$928$	−44.2225	−1.45167
$929$	8.60774	0.282411	0.141205	−	0.989980i	$-0.454902\pi$
$929$	8.60774	0.282411	0.141205	+	0.989980i	$0.454902\pi$
$930$	0	0
$931$	−38.9702	−1.27720
$932$	37.4163	1.22561
$933$	0	0
$934$	−13.0884	−0.428266
$935$	0	0
$936$	0	0
$937$	5.96640	0.194914	0.0974569	−	0.995240i	$-0.468929\pi$
$937$	5.96640	0.194914	0.0974569	+	0.995240i	$0.468929\pi$
$938$	1.75801	0.0574010
$939$	0	0
$940$	−16.9828	−0.553918
$941$	−22.8380	−0.744498	−0.372249	−	0.928133i	$-0.621413\pi$
$941$	−22.8380	−0.744498	−0.372249	+	0.928133i	$0.621413\pi$
$942$	0	0
$943$	37.4781	1.22045
$944$	−7.38114	−0.240236
$945$	0	0
$946$	0	0
$947$	2.30620	0.0749415	0.0374708	−	0.999298i	$-0.488070\pi$
$947$	2.30620	0.0749415	0.0374708	+	0.999298i	$0.488070\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	−2.76873	−0.0898296
$951$	0	0
$952$	−2.18095	−0.0706851
$953$	28.6129	0.926861	0.463431	−	0.886133i	$-0.346618\pi$
$953$	28.6129	0.926861	0.463431	+	0.886133i	$0.346618\pi$
$954$	0	0
$955$	−5.43695	−0.175936
$956$	−4.73383	−0.153103
$957$	0	0
$958$	−6.72404	−0.217244
$959$	1.91018	0.0616830
$960$	0	0
$961$	32.5543	1.05014
$962$	−12.0000	−0.386896
$963$	0	0
$964$	−8.08156	−0.260289
$965$	3.20675	0.103229
$966$	0	0
$967$	31.7422	1.02076	0.510380	−	0.859949i	$-0.329505\pi$
$967$	31.7422	1.02076	0.510380	+	0.859949i	$0.329505\pi$
$968$	0	0
$969$	0	0
$970$	−1.74951	−0.0561733
$971$	37.0023	1.18746	0.593731	−	0.804664i	$-0.297654\pi$
$971$	37.0023	1.18746	0.593731	+	0.804664i	$0.297654\pi$
$972$	0	0
$973$	−4.43462	−0.142168
$974$	3.58451	0.114855
$975$	0	0
$976$	−10.7311	−0.343494
$977$	−33.8716	−1.08365	−0.541824	−	0.840492i	$-0.682266\pi$
$977$	−33.8716	−1.08365	−0.541824	+	0.840492i	$0.682266\pi$
$978$	0	0
$979$	0	0
$980$	−11.9721	−0.382434
$981$	0	0
$982$	7.68268	0.245164
$983$	37.7516	1.20409	0.602044	−	0.798463i	$-0.294353\pi$
$983$	37.7516	1.20409	0.602044	+	0.798463i	$0.294353\pi$
$984$	0	0
$985$	19.4048	0.618288
$986$	−10.8452	−0.345383
$987$	0	0
$988$	−48.8349	−1.55364
$989$	37.8343	1.20306
$990$	0	0
$991$	−44.9354	−1.42742	−0.713710	−	0.700441i	$-0.752987\pi$
$991$	−44.9354	−1.42742	−0.713710	+	0.700441i	$0.752987\pi$
$992$	−39.1175	−1.24198
$993$	0	0
$994$	0.640313	0.0203095
$995$	−20.3509	−0.645167
$996$	0	0
$997$	0.396526	0.0125581	0.00627906	−	0.999980i	$-0.498001\pi$
$997$	0.396526	0.0125581	0.00627906	+	0.999980i	$0.498001\pi$
$998$	11.7104	0.370685
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bx.1.3		6
3.2	odd	2		605.2.a.m.1.4	yes	6
11.10	odd	2	inner	5445.2.a.bx.1.4		6
12.11	even	2		9680.2.a.cw.1.6		6
15.14	odd	2		3025.2.a.bg.1.3		6
33.2	even	10		605.2.g.q.81.4		24
33.5	odd	10		605.2.g.q.366.3		24
33.8	even	10		605.2.g.q.251.3		24
33.14	odd	10		605.2.g.q.251.4		24
33.17	even	10		605.2.g.q.366.4		24
33.20	odd	10		605.2.g.q.81.3		24
33.26	odd	10		605.2.g.q.511.4		24
33.29	even	10		605.2.g.q.511.3		24
33.32	even	2		605.2.a.m.1.3	✓	6
132.131	odd	2		9680.2.a.cw.1.5		6
165.164	even	2		3025.2.a.bg.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.m.1.3	✓	6	33.32	even	2
605.2.a.m.1.4	yes	6	3.2	odd	2
605.2.g.q.81.3		24	33.20	odd	10
605.2.g.q.81.4		24	33.2	even	10
605.2.g.q.251.3		24	33.8	even	10
605.2.g.q.251.4		24	33.14	odd	10
605.2.g.q.366.3		24	33.5	odd	10
605.2.g.q.366.4		24	33.17	even	10
605.2.g.q.511.3		24	33.29	even	10
605.2.g.q.511.4		24	33.26	odd	10
3025.2.a.bg.1.3		6	15.14	odd	2
3025.2.a.bg.1.4		6	165.164	even	2
5445.2.a.bx.1.3		6	1.1	even	1	trivial
5445.2.a.bx.1.4		6	11.10	odd	2	inner
9680.2.a.cw.1.5		6	132.131	odd	2
9680.2.a.cw.1.6		6	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.480901 q^{2} -1.76873 q^{4} -1.00000 q^{5} -0.480901 q^{7} +1.81239 q^{8} +O(q^{10})\)	\(q-0.480901 q^{2} -1.76873 q^{4} -1.00000 q^{5} -0.480901 q^{7} +1.81239 q^{8} +0.480901 q^{10} +4.79559 q^{13} +0.231266 q^{14} +2.66589 q^{16} +2.50230 q^{17} +5.75739 q^{19} +1.76873 q^{20} -4.43462 q^{23} +1.00000 q^{25} -2.30620 q^{26} +0.850586 q^{28} +9.01248 q^{29} +7.97209 q^{31} -4.90680 q^{32} -1.20336 q^{34} +0.480901 q^{35} +5.20336 q^{37} -2.76873 q^{38} -1.81239 q^{40} -8.45124 q^{41} -8.53158 q^{43} +2.13261 q^{46} -9.60168 q^{47} -6.76873 q^{49} -0.480901 q^{50} -8.48212 q^{52} -6.10051 q^{53} -0.871579 q^{56} -4.33411 q^{58} -2.76873 q^{59} -4.02534 q^{61} -3.83379 q^{62} -2.97209 q^{64} -4.79559 q^{65} +7.60168 q^{67} -4.42590 q^{68} -0.231266 q^{70} +2.76873 q^{71} +5.00460 q^{73} -2.50230 q^{74} -10.1833 q^{76} +3.62478 q^{79} -2.66589 q^{80} +4.06421 q^{82} -1.71459 q^{83} -2.50230 q^{85} +4.10284 q^{86} +15.7408 q^{89} -2.30620 q^{91} +7.84367 q^{92} +4.61746 q^{94} -5.75739 q^{95} -3.63798 q^{97} +3.25509 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{4} - 6 q^{5}+O(q^{10}) \) 6 * q + 6 * q^4 - 6 * q^5	\( 6 q + 6 q^{4} - 6 q^{5} + 18 q^{14} + 18 q^{16} - 6 q^{20} - 12 q^{23} + 6 q^{25} + 36 q^{26} + 24 q^{34} - 42 q^{47} - 24 q^{49} - 24 q^{53} + 30 q^{56} - 24 q^{58} + 30 q^{64} + 30 q^{67} - 18 q^{70} - 18 q^{80} + 42 q^{82} + 6 q^{86} + 30 q^{89} + 36 q^{91} - 36 q^{92} + 24 q^{97}+O(q^{100}) \) 6 * q + 6 * q^4 - 6 * q^5 + 18 * q^14 + 18 * q^16 - 6 * q^20 - 12 * q^23 + 6 * q^25 + 36 * q^26 + 24 * q^34 - 42 * q^47 - 24 * q^49 - 24 * q^53 + 30 * q^56 - 24 * q^58 + 30 * q^64 + 30 * q^67 - 18 * q^70 - 18 * q^80 + 42 * q^82 + 6 * q^86 + 30 * q^89 + 36 * q^91 - 36 * q^92 + 24 * q^97

Introduction

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