LMFDB - Embedded newform 5445.2.a.bx.1.2

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			1.2
Root			$-1.37268$ 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)

$f(q)$	$=$	$q-1.37268 q^{2} -0.115749 q^{4} -1.00000 q^{5} -1.37268 q^{7} +2.90425 q^{8} +O(q^{10})$	$q-1.37268 q^{2} -0.115749 q^{4} -1.00000 q^{5} -1.37268 q^{7} +2.90425 q^{8} +1.37268 q^{10} -1.93253 q^{13} +1.88425 q^{14} -3.75510 q^{16} -6.20946 q^{17} +0.812826 q^{19} +0.115749 q^{20} +3.63935 q^{23} +1.00000 q^{25} +2.65275 q^{26} +0.158887 q^{28} +7.83511 q^{29} -3.40786 q^{31} -0.653939 q^{32} +8.52360 q^{34} +1.37268 q^{35} -4.52360 q^{37} -1.11575 q^{38} -2.90425 q^{40} -1.82613 q^{41} -6.46243 q^{43} -4.99567 q^{46} -4.73820 q^{47} -5.11575 q^{49} -1.37268 q^{50} +0.223690 q^{52} +8.39446 q^{53} -3.98660 q^{56} -10.7551 q^{58} -1.11575 q^{59} -2.54488 q^{61} +4.67789 q^{62} +8.40786 q^{64} +1.93253 q^{65} +2.73820 q^{67} +0.718741 q^{68} -1.88425 q^{70} +1.11575 q^{71} -12.4189 q^{73} +6.20946 q^{74} -0.0940841 q^{76} +5.80849 q^{79} +3.75510 q^{80} +2.50670 q^{82} -15.9771 q^{83} +6.20946 q^{85} +8.87085 q^{86} +2.70789 q^{89} +2.65275 q^{91} -0.421253 q^{92} +6.50403 q^{94} -0.812826 q^{95} +14.1630 q^{97} +7.02229 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.37268	−0.970631	−0.485316	−	0.874339i	$-0.661295\pi$
$2$	−1.37268	−0.970631	−0.485316	+	0.874339i	$0.661295\pi$
$3$	0	0
$3$	0	0
$4$	−0.115749	−0.0578747
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.37268	−0.518824	−0.259412	−	0.965767i	$-0.583529\pi$
$7$	−1.37268	−0.518824	−0.259412	+	0.965767i	$0.583529\pi$
$8$	2.90425	1.02681
$9$	0	0
$10$	1.37268	0.434080
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.93253	−0.535989	−0.267994	−	0.963420i	$-0.586361\pi$
$13$	−1.93253	−0.535989	−0.267994	+	0.963420i	$0.586361\pi$
$14$	1.88425	0.503587
$15$	0	0
$16$	−3.75510	−0.938776
$17$	−6.20946	−1.50602	−0.753008	−	0.658012i	$-0.771398\pi$
$17$	−6.20946	−1.50602	−0.753008	+	0.658012i	$0.771398\pi$
$18$	0	0
$19$	0.812826	0.186475	0.0932375	−	0.995644i	$-0.470278\pi$
$19$	0.812826	0.186475	0.0932375	+	0.995644i	$0.470278\pi$
$20$	0.115749	0.0258824
$21$	0	0
$22$	0	0
$23$	3.63935	0.758858	0.379429	−	0.925221i	$-0.376120\pi$
$23$	3.63935	0.758858	0.379429	+	0.925221i	$0.376120\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.65275	0.520247
$27$	0	0
$28$	0.158887	0.0300268
$29$	7.83511	1.45494	0.727472	−	0.686137i	$-0.240695\pi$
$29$	7.83511	1.45494	0.727472	+	0.686137i	$0.240695\pi$
$30$	0	0
$31$	−3.40786	−0.612069	−0.306034	−	0.952020i	$-0.599002\pi$
$31$	−3.40786	−0.612069	−0.306034	+	0.952020i	$0.599002\pi$
$32$	−0.653939	−0.115601
$33$	0	0
$34$	8.52360	1.46179
$35$	1.37268	0.232025
$36$	0	0
$37$	−4.52360	−0.743676	−0.371838	−	0.928298i	$-0.621272\pi$
$37$	−4.52360	−0.743676	−0.371838	+	0.928298i	$0.621272\pi$
$38$	−1.11575	−0.180998
$39$	0	0
$40$	−2.90425	−0.459202
$41$	−1.82613	−0.285194	−0.142597	−	0.989781i	$-0.545545\pi$
$41$	−1.82613	−0.285194	−0.142597	+	0.989781i	$0.545545\pi$
$42$	0	0
$43$	−6.46243	−0.985512	−0.492756	−	0.870168i	$-0.664010\pi$
$43$	−6.46243	−0.985512	−0.492756	+	0.870168i	$0.664010\pi$
$44$	0	0
$45$	0	0
$46$	−4.99567	−0.736571
$47$	−4.73820	−0.691137	−0.345569	−	0.938393i	$-0.612314\pi$
$47$	−4.73820	−0.691137	−0.345569	+	0.938393i	$0.612314\pi$
$48$	0	0
$49$	−5.11575	−0.730821
$50$	−1.37268	−0.194126
$51$	0	0
$52$	0.223690	0.0310202
$53$	8.39446	1.15307	0.576534	−	0.817073i	$-0.304405\pi$
$53$	8.39446	1.15307	0.576534	+	0.817073i	$0.304405\pi$
$54$	0	0
$55$	0	0
$56$	−3.98660	−0.532732
$57$	0	0
$58$	−10.7551	−1.41221
$59$	−1.11575	−0.145258	−0.0726291	−	0.997359i	$-0.523139\pi$
$59$	−1.11575	−0.145258	−0.0726291	+	0.997359i	$0.523139\pi$
$60$	0	0
$61$	−2.54488	−0.325838	−0.162919	−	0.986639i	$-0.552091\pi$
$61$	−2.54488	−0.325838	−0.162919	+	0.986639i	$0.552091\pi$
$62$	4.67789	0.594093
$63$	0	0
$64$	8.40786	1.05098
$65$	1.93253	0.239701
$66$	0	0
$67$	2.73820	0.334524	0.167262	−	0.985912i	$-0.446507\pi$
$67$	2.73820	0.334524	0.167262	+	0.985912i	$0.446507\pi$
$68$	0.718741	0.0871602
$69$	0	0
$70$	−1.88425	−0.225211
$71$	1.11575	0.132415	0.0662075	−	0.997806i	$-0.478910\pi$
$71$	1.11575	0.132415	0.0662075	+	0.997806i	$0.478910\pi$
$72$	0	0
$73$	−12.4189	−1.45353	−0.726763	−	0.686889i	$-0.758976\pi$
$73$	−12.4189	−1.45353	−0.726763	+	0.686889i	$0.758976\pi$
$74$	6.20946	0.721835
$75$	0	0
$76$	−0.0940841	−0.0107922
$77$	0	0
$78$	0	0
$79$	5.80849	0.653507	0.326753	−	0.945110i	$-0.394045\pi$
$79$	5.80849	0.653507	0.326753	+	0.945110i	$0.394045\pi$
$80$	3.75510	0.419833
$81$	0	0
$82$	2.50670	0.276819
$83$	−15.9771	−1.75372	−0.876858	−	0.480750i	$-0.840365\pi$
$83$	−15.9771	−1.75372	−0.876858	+	0.480750i	$0.840365\pi$
$84$	0	0
$85$	6.20946	0.673511
$86$	8.87085	0.956569
$87$	0	0
$88$	0	0
$89$	2.70789	0.287036	0.143518	−	0.989648i	$-0.454158\pi$
$89$	2.70789	0.287036	0.143518	+	0.989648i	$0.454158\pi$
$90$	0	0
$91$	2.65275	0.278084
$92$	−0.421253	−0.0439187
$93$	0	0
$94$	6.50403	0.670839
$95$	−0.812826	−0.0833941
$96$	0	0
$97$	14.1630	1.43803	0.719015	−	0.694994i	$-0.244593\pi$
$97$	14.1630	1.43803	0.719015	+	0.694994i	$0.244593\pi$
$98$	7.02229	0.709358
$99$	0	0
$100$	−0.115749	−0.0115749
$101$	−11.4056	−1.13490	−0.567451	−	0.823408i	$-0.692070\pi$
$101$	−11.4056	−1.13490	−0.567451	+	0.823408i	$0.692070\pi$
$102$	0	0
$103$	13.8709	1.36674	0.683368	−	0.730074i	$-0.260515\pi$
$103$	13.8709	1.36674	0.683368	+	0.730074i	$0.260515\pi$
$104$	−5.61256	−0.550357
$105$	0	0
$106$	−11.5229	−1.11920
$107$	1.06580	0.103034	0.0515172	−	0.998672i	$-0.483594\pi$
$107$	1.06580	0.103034	0.0515172	+	0.998672i	$0.483594\pi$
$108$	0	0
$109$	11.3115	1.08345	0.541724	−	0.840556i	$-0.317772\pi$
$109$	11.3115	1.08345	0.541724	+	0.840556i	$0.317772\pi$
$110$	0	0
$111$	0	0
$112$	5.15456	0.487060
$113$	3.76850	0.354511	0.177255	−	0.984165i	$-0.443278\pi$
$113$	3.76850	0.354511	0.177255	+	0.984165i	$0.443278\pi$
$114$	0	0
$115$	−3.63935	−0.339371
$116$	−0.906910	−0.0842044
$117$	0	0
$118$	1.53157	0.140992
$119$	8.52360	0.781358
$120$	0	0
$121$	0	0
$122$	3.49330	0.316269
$123$	0	0
$124$	0.394457	0.0354233
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.67956	−0.149037	−0.0745186	−	0.997220i	$-0.523742\pi$
$127$	−1.67956	−0.149037	−0.0745186	+	0.997220i	$0.523742\pi$
$128$	−10.2334	−0.904515
$129$	0	0
$130$	−2.65275	−0.232662
$131$	−11.7943	−1.03047	−0.515235	−	0.857049i	$-0.672295\pi$
$131$	−11.7943	−1.03047	−0.515235	+	0.857049i	$0.672295\pi$
$132$	0	0
$133$	−1.11575	−0.0967477
$134$	−3.75867	−0.324700
$135$	0	0
$136$	−18.0338	−1.54639
$137$	7.40786	0.632896	0.316448	−	0.948610i	$-0.397510\pi$
$137$	7.40786	0.632896	0.316448	+	0.948610i	$0.397510\pi$
$138$	0	0
$139$	−2.65128	−0.224878	−0.112439	−	0.993659i	$-0.535866\pi$
$139$	−2.65128	−0.224878	−0.112439	+	0.993659i	$0.535866\pi$
$140$	−0.158887	−0.0134284
$141$	0	0
$142$	−1.53157	−0.128526
$143$	0	0
$144$	0	0
$145$	−7.83511	−0.650671
$146$	17.0472	1.41084
$147$	0	0
$148$	0.523604	0.0430400
$149$	−8.35337	−0.684335	−0.342167	−	0.939639i	$-0.611161\pi$
$149$	−8.35337	−0.684335	−0.342167	+	0.939639i	$0.611161\pi$
$150$	0	0
$151$	7.42325	0.604096	0.302048	−	0.953293i	$-0.402330\pi$
$151$	7.42325	0.604096	0.302048	+	0.953293i	$0.402330\pi$
$152$	2.36065	0.191474
$153$	0	0
$154$	0	0
$155$	3.40786	0.273726
$156$	0	0
$157$	−8.58421	−0.685095	−0.342547	−	0.939501i	$-0.611290\pi$
$157$	−8.58421	−0.685095	−0.342547	+	0.939501i	$0.611290\pi$
$158$	−7.97320	−0.634314
$159$	0	0
$160$	0.653939	0.0516984
$161$	−4.99567	−0.393714
$162$	0	0
$163$	16.3776	1.28279	0.641394	−	0.767211i	$-0.278356\pi$
$163$	16.3776	1.28279	0.641394	+	0.767211i	$0.278356\pi$
$164$	0.211374	0.0165055
$165$	0	0
$166$	21.9315	1.70221
$167$	−21.4385	−1.65896	−0.829482	−	0.558533i	$-0.811364\pi$
$167$	−21.4385	−1.65896	−0.829482	+	0.558533i	$0.811364\pi$
$168$	0	0
$169$	−9.26531	−0.712716
$170$	−8.52360	−0.653731
$171$	0	0
$172$	0.748023	0.0570362
$173$	−13.7377	−1.04446	−0.522229	−	0.852806i	$-0.674899\pi$
$173$	−13.7377	−1.04446	−0.522229	+	0.852806i	$0.674899\pi$
$174$	0	0
$175$	−1.37268	−0.103765
$176$	0	0
$177$	0	0
$178$	−3.71707	−0.278606
$179$	3.76850	0.281671	0.140836	−	0.990033i	$-0.455021\pi$
$179$	3.76850	0.281671	0.140836	+	0.990033i	$0.455021\pi$
$180$	0	0
$181$	−0.0133979	−0.000995860 0	−0.000497930	−	1.00000i	$-0.500158\pi$
$181$	−0.0133979	−0.000995860 0	−0.000497930		1.00000i	$0.500158\pi$
$182$	−3.64138	−0.269917
$183$	0	0
$184$	10.5696	0.779200
$185$	4.52360	0.332582
$186$	0	0
$187$	0	0
$188$	0.548444	0.0399994
$189$	0	0
$190$	1.11575	0.0809450
$191$	16.6260	1.20301	0.601506	−	0.798868i	$-0.294568\pi$
$191$	16.6260	1.20301	0.601506	+	0.798868i	$0.294568\pi$
$192$	0	0
$193$	−26.7813	−1.92776	−0.963879	−	0.266340i	$-0.914186\pi$
$193$	−26.7813	−1.92776	−0.963879	+	0.266340i	$0.914186\pi$
$194$	−19.4412	−1.39580
$195$	0	0
$196$	0.592145	0.0422961
$197$	2.55719	0.182192	0.0910962	−	0.995842i	$-0.470963\pi$
$197$	2.55719	0.182192	0.0910962	+	0.995842i	$0.470963\pi$
$198$	0	0
$199$	−21.8629	−1.54982	−0.774911	−	0.632071i	$-0.782205\pi$
$199$	−21.8629	−1.54982	−0.774911	+	0.632071i	$0.782205\pi$
$200$	2.90425	0.205361
$201$	0	0
$202$	15.6563	1.10157
$203$	−10.7551	−0.754860
$204$	0	0
$205$	1.82613	0.127543
$206$	−19.0402	−1.32660
$207$	0	0
$208$	7.25687	0.503173
$209$	0	0
$210$	0	0
$211$	−12.8308	−0.883307	−0.441654	−	0.897186i	$-0.645608\pi$
$211$	−12.8308	−0.883307	−0.441654	+	0.897186i	$0.645608\pi$
$212$	−0.971653	−0.0667334
$213$	0	0
$214$	−1.46300	−0.100008
$215$	6.46243	0.440734
$216$	0	0
$217$	4.67789	0.317556
$218$	−15.5271	−1.05163
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−7.36415	−0.493140	−0.246570	−	0.969125i	$-0.579304\pi$
$223$	−7.36415	−0.493140	−0.246570	+	0.969125i	$0.579304\pi$
$224$	0.897649	0.0599767
$225$	0	0
$226$	−5.17295	−0.344099
$227$	1.37268	0.0911080	0.0455540	−	0.998962i	$-0.485495\pi$
$227$	1.37268	0.0911080	0.0455540	+	0.998962i	$0.485495\pi$
$228$	0	0
$229$	29.4968	1.94920	0.974602	−	0.223944i	$-0.0718933\pi$
$229$	29.4968	1.94920	0.974602	+	0.223944i	$0.0718933\pi$
$230$	4.99567	0.329405
$231$	0	0
$232$	22.7551	1.49395
$233$	21.9984	1.44116	0.720582	−	0.693370i	$-0.243875\pi$
$233$	21.9984	1.44116	0.720582	+	0.693370i	$0.243875\pi$
$234$	0	0
$235$	4.73820	0.309086
$236$	0.129147	0.00840677
$237$	0	0
$238$	−11.7002	−0.758410
$239$	18.7225	1.21106	0.605528	−	0.795824i	$-0.292962\pi$
$239$	18.7225	1.21106	0.605528	+	0.795824i	$0.292962\pi$
$240$	0	0
$241$	26.2630	1.69175	0.845875	−	0.533382i	$-0.179079\pi$
$241$	26.2630	1.69175	0.845875	+	0.533382i	$0.179079\pi$
$242$	0	0
$243$	0	0
$244$	0.294568	0.0188578
$245$	5.11575	0.326833
$246$	0	0
$247$	−1.57081	−0.0999485
$248$	−9.89725	−0.628476
$249$	0	0
$250$	1.37268	0.0868159
$251$	22.6260	1.42814	0.714069	−	0.700075i	$-0.246850\pi$
$251$	22.6260	1.42814	0.714069	+	0.700075i	$0.246850\pi$
$252$	0	0
$253$	0	0
$254$	2.30550	0.144660
$255$	0	0
$256$	−2.76850	−0.173031
$257$	−17.2102	−1.07354	−0.536770	−	0.843728i	$-0.680356\pi$
$257$	−17.2102	−1.07354	−0.536770	+	0.843728i	$0.680356\pi$
$258$	0	0
$259$	6.20946	0.385837
$260$	−0.223690	−0.0138726
$261$	0	0
$262$	16.1898	1.00021
$263$	5.71441	0.352366	0.176183	−	0.984357i	$-0.443625\pi$
$263$	5.71441	0.352366	0.176183	+	0.984357i	$0.443625\pi$
$264$	0	0
$265$	−8.39446	−0.515667
$266$	1.53157	0.0939064
$267$	0	0
$268$	−0.316945	−0.0193605
$269$	2.87085	0.175039	0.0875195	−	0.996163i	$-0.472106\pi$
$269$	2.87085	0.175039	0.0875195	+	0.996163i	$0.472106\pi$
$270$	0	0
$271$	7.64694	0.464519	0.232259	−	0.972654i	$-0.425388\pi$
$271$	7.64694	0.464519	0.232259	+	0.972654i	$0.425388\pi$
$272$	23.3172	1.41381
$273$	0	0
$274$	−10.1686	−0.614309
$275$	0	0
$276$	0	0
$277$	29.3970	1.76630	0.883148	−	0.469094i	$-0.155420\pi$
$277$	29.3970	1.76630	0.883148	+	0.469094i	$0.155420\pi$
$278$	3.63935	0.218274
$279$	0	0
$280$	3.98660	0.238245
$281$	24.2241	1.44509	0.722544	−	0.691325i	$-0.242973\pi$
$281$	24.2241	1.44509	0.722544	+	0.691325i	$0.242973\pi$
$282$	0	0
$283$	−5.11903	−0.304295	−0.152147	−	0.988358i	$-0.548619\pi$
$283$	−5.11903	−0.304295	−0.152147	+	0.988358i	$0.548619\pi$
$284$	−0.129147	−0.00766348
$285$	0	0
$286$	0	0
$287$	2.50670	0.147966
$288$	0	0
$289$	21.5574	1.26808
$290$	10.7551	0.631561
$291$	0	0
$292$	1.43748	0.0841223
$293$	10.9923	0.642179	0.321089	−	0.947049i	$-0.395951\pi$
$293$	10.9923	0.642179	0.321089	+	0.947049i	$0.395951\pi$
$294$	0	0
$295$	1.11575	0.0649614
$296$	−13.1377	−0.763611
$297$	0	0
$298$	11.4665	0.664237
$299$	−7.03318	−0.406739
$300$	0	0
$301$	8.87085	0.511307
$302$	−10.1898	−0.586354
$303$	0	0
$304$	−3.05224	−0.175058
$305$	2.54488	0.145719
$306$	0	0
$307$	14.3515	0.819081	0.409540	−	0.912292i	$-0.365689\pi$
$307$	14.3515	0.819081	0.409540	+	0.912292i	$0.365689\pi$
$308$	0	0
$309$	0	0
$310$	−4.67789	−0.265687
$311$	−27.3811	−1.55264	−0.776319	−	0.630341i	$-0.782915\pi$
$311$	−27.3811	−1.55264	−0.776319	+	0.630341i	$0.782915\pi$
$312$	0	0
$313$	3.87085	0.218794	0.109397	−	0.993998i	$-0.465108\pi$
$313$	3.87085	0.218794	0.109397	+	0.993998i	$0.465108\pi$
$314$	11.7834	0.664974
$315$	0	0
$316$	−0.672330	−0.0378215
$317$	15.5102	0.871140	0.435570	−	0.900155i	$-0.356547\pi$
$317$	15.5102	0.871140	0.435570	+	0.900155i	$0.356547\pi$
$318$	0	0
$319$	0	0
$320$	−8.40786	−0.470013
$321$	0	0
$322$	6.85745	0.382151
$323$	−5.04721	−0.280834
$324$	0	0
$325$	−1.93253	−0.107198
$326$	−22.4811	−1.24512
$327$	0	0
$328$	−5.30355	−0.292839
$329$	6.50403	0.358579
$330$	0	0
$331$	26.8575	1.47622	0.738110	−	0.674681i	$-0.235719\pi$
$331$	26.8575	1.47622	0.738110	+	0.674681i	$0.235719\pi$
$332$	1.84934	0.101496
$333$	0	0
$334$	29.4283	1.61024
$335$	−2.73820	−0.149604
$336$	0	0
$337$	1.93253	0.105272	0.0526359	−	0.998614i	$-0.483238\pi$
$337$	1.93253	0.105272	0.0526359	+	0.998614i	$0.483238\pi$
$338$	12.7183	0.691785
$339$	0	0
$340$	−0.718741	−0.0389792
$341$	0	0
$342$	0	0
$343$	16.6310	0.897992
$344$	−18.7685	−1.01193
$345$	0	0
$346$	18.8575	1.01378
$347$	−3.80027	−0.204009	−0.102004	−	0.994784i	$-0.532526\pi$
$347$	−3.80027	−0.204009	−0.102004	+	0.994784i	$0.532526\pi$
$348$	0	0
$349$	17.1909	0.920208	0.460104	−	0.887865i	$-0.347812\pi$
$349$	17.1909	0.920208	0.460104	+	0.887865i	$0.347812\pi$
$350$	1.88425	0.100717
$351$	0	0
$352$	0	0
$353$	1.99207	0.106027	0.0530135	−	0.998594i	$-0.483117\pi$
$353$	1.99207	0.106027	0.0530135	+	0.998594i	$0.483117\pi$
$354$	0	0
$355$	−1.11575	−0.0592178
$356$	−0.313437	−0.0166121
$357$	0	0
$358$	−5.17295	−0.273399
$359$	15.5761	0.822077	0.411039	−	0.911618i	$-0.365166\pi$
$359$	15.5761	0.822077	0.411039	+	0.911618i	$0.365166\pi$
$360$	0	0
$361$	−18.3393	−0.965227
$362$	0.0183911	0.000966613 0
$363$	0	0
$364$	−0.307054	−0.0160940
$365$	12.4189	0.650036
$366$	0	0
$367$	18.7720	0.979891	0.489945	−	0.871753i	$-0.337017\pi$
$367$	18.7720	0.979891	0.489945	+	0.871753i	$0.337017\pi$
$368$	−13.6661	−0.712397
$369$	0	0
$370$	−6.20946	−0.322815
$371$	−11.5229	−0.598239
$372$	0	0
$373$	−2.62664	−0.136003	−0.0680013	−	0.997685i	$-0.521662\pi$
$373$	−2.62664	−0.136003	−0.0680013	+	0.997685i	$0.521662\pi$
$374$	0	0
$375$	0	0
$376$	−13.7609	−0.709664
$377$	−15.1416	−0.779833
$378$	0	0
$379$	2.66069	0.136670	0.0683351	−	0.997662i	$-0.478231\pi$
$379$	2.66069	0.136670	0.0683351	+	0.997662i	$0.478231\pi$
$380$	0.0940841	0.00482641
$381$	0	0
$382$	−22.8221	−1.16768
$383$	−21.9315	−1.12065	−0.560323	−	0.828274i	$-0.689323\pi$
$383$	−21.9315	−1.12065	−0.560323	+	0.828274i	$0.689323\pi$
$384$	0	0
$385$	0	0
$386$	36.7621	1.87114
$387$	0	0
$388$	−1.63935	−0.0832256
$389$	3.42125	0.173464	0.0867322	−	0.996232i	$-0.472358\pi$
$389$	3.42125	0.173464	0.0867322	+	0.996232i	$0.472358\pi$
$390$	0	0
$391$	−22.5984	−1.14285
$392$	−14.8574	−0.750412
$393$	0	0
$394$	−3.51021	−0.176842
$395$	−5.80849	−0.292257
$396$	0	0
$397$	5.51021	0.276549	0.138275	−	0.990394i	$-0.455844\pi$
$397$	5.51021	0.276549	0.138275	+	0.990394i	$0.455844\pi$
$398$	30.0108	1.50431
$399$	0	0
$400$	−3.75510	−0.187755
$401$	17.9260	0.895181	0.447591	−	0.894239i	$-0.352282\pi$
$401$	17.9260	0.895181	0.447591	+	0.894239i	$0.352282\pi$
$402$	0	0
$403$	6.58580	0.328062
$404$	1.32019	0.0656821
$405$	0	0
$406$	14.7633	0.732691
$407$	0	0
$408$	0	0
$409$	−29.3261	−1.45008	−0.725042	−	0.688704i	$-0.758180\pi$
$409$	−29.3261	−1.45008	−0.725042	+	0.688704i	$0.758180\pi$
$410$	−2.50670	−0.123797
$411$	0	0
$412$	−1.60554	−0.0790994
$413$	1.53157	0.0753635
$414$	0	0
$415$	15.9771	0.784285
$416$	1.26376	0.0619609
$417$	0	0
$418$	0	0
$419$	−2.65275	−0.129595	−0.0647977	−	0.997898i	$-0.520640\pi$
$419$	−2.65275	−0.129595	−0.0647977	+	0.997898i	$0.520640\pi$
$420$	0	0
$421$	−11.1630	−0.544049	−0.272025	−	0.962290i	$-0.587693\pi$
$421$	−11.1630	−0.544049	−0.272025	+	0.962290i	$0.587693\pi$
$422$	17.6126	0.857366
$423$	0	0
$424$	24.3796	1.18398
$425$	−6.20946	−0.301203
$426$	0	0
$427$	3.49330	0.169053
$428$	−0.123365	−0.00596309
$429$	0	0
$430$	−8.87085	−0.427791
$431$	4.55918	0.219608	0.109804	−	0.993953i	$-0.464978\pi$
$431$	4.55918	0.219608	0.109804	+	0.993953i	$0.464978\pi$
$432$	0	0
$433$	20.1630	0.968970	0.484485	−	0.874800i	$-0.339007\pi$
$433$	20.1630	0.968970	0.484485	+	0.874800i	$0.339007\pi$
$434$	−6.42125	−0.308230
$435$	0	0
$436$	−1.30930	−0.0627042
$437$	2.95816	0.141508
$438$	0	0
$439$	34.4868	1.64596	0.822982	−	0.568067i	$-0.192309\pi$
$439$	34.4868	1.64596	0.822982	+	0.568067i	$0.192309\pi$
$440$	0	0
$441$	0	0
$442$	−16.4722	−0.783501
$443$	−31.4586	−1.49464	−0.747321	−	0.664463i	$-0.768660\pi$
$443$	−31.4586	−1.49464	−0.747321	+	0.664463i	$0.768660\pi$
$444$	0	0
$445$	−2.70789	−0.128367
$446$	10.1086	0.478657
$447$	0	0
$448$	−11.5413	−0.545275
$449$	4.77643	0.225414	0.112707	−	0.993628i	$-0.464048\pi$
$449$	4.77643	0.225414	0.112707	+	0.993628i	$0.464048\pi$
$450$	0	0
$451$	0	0
$452$	−0.436202	−0.0205172
$453$	0	0
$454$	−1.88425	−0.0884323
$455$	−2.65275	−0.124363
$456$	0	0
$457$	−1.53157	−0.0716437	−0.0358218	−	0.999358i	$-0.511405\pi$
$457$	−1.53157	−0.0716437	−0.0358218	+	0.999358i	$0.511405\pi$
$458$	−40.4897	−1.89196
$459$	0	0
$460$	0.421253	0.0196410
$461$	18.5220	0.862655	0.431327	−	0.902195i	$-0.358045\pi$
$461$	18.5220	0.862655	0.431327	+	0.902195i	$0.358045\pi$
$462$	0	0
$463$	9.68306	0.450010	0.225005	−	0.974358i	$-0.427760\pi$
$463$	9.68306	0.450010	0.225005	+	0.974358i	$0.427760\pi$
$464$	−29.4217	−1.36587
$465$	0	0
$466$	−30.1968	−1.39884
$467$	31.6980	1.46681	0.733404	−	0.679793i	$-0.237930\pi$
$467$	31.6980	1.46681	0.733404	+	0.679793i	$0.237930\pi$
$468$	0	0
$469$	−3.75867	−0.173559
$470$	−6.50403	−0.300009
$471$	0	0
$472$	−3.24041	−0.149152
$473$	0	0
$474$	0	0
$475$	0.812826	0.0372950
$476$	−0.986602	−0.0452208
$477$	0	0
$478$	−25.7000	−1.17549
$479$	30.8345	1.40886	0.704432	−	0.709771i	$-0.251202\pi$
$479$	30.8345	1.40886	0.704432	+	0.709771i	$0.251202\pi$
$480$	0	0
$481$	8.74202	0.398602
$482$	−36.0507	−1.64207
$483$	0	0
$484$	0	0
$485$	−14.1630	−0.643107
$486$	0	0
$487$	29.9921	1.35907	0.679535	−	0.733643i	$-0.262182\pi$
$487$	29.9921	1.35907	0.679535	+	0.733643i	$0.262182\pi$
$488$	−7.39095	−0.334573
$489$	0	0
$490$	−7.02229	−0.317235
$491$	34.5114	1.55748	0.778739	−	0.627348i	$-0.215860\pi$
$491$	34.5114	1.55748	0.778739	+	0.627348i	$0.215860\pi$
$492$	0	0
$493$	−48.6518	−2.19117
$494$	2.15622	0.0970131
$495$	0	0
$496$	12.7968	0.574595
$497$	−1.53157	−0.0687002
$498$	0	0
$499$	17.8629	0.799654	0.399827	−	0.916591i	$-0.369070\pi$
$499$	17.8629	0.799654	0.399827	+	0.916591i	$0.369070\pi$
$500$	0.115749	0.00517647
$501$	0	0
$502$	−31.0582	−1.38620
$503$	5.11903	0.228246	0.114123	−	0.993467i	$-0.463594\pi$
$503$	5.11903	0.228246	0.114123	+	0.993467i	$0.463594\pi$
$504$	0	0
$505$	11.4056	0.507543
$506$	0	0
$507$	0	0
$508$	0.194408	0.00862548
$509$	18.1078	0.802615	0.401307	−	0.915943i	$-0.368556\pi$
$509$	18.1078	0.802615	0.401307	+	0.915943i	$0.368556\pi$
$510$	0	0
$511$	17.0472	0.754124
$512$	24.2671	1.07246
$513$	0	0
$514$	23.6241	1.04201
$515$	−13.8709	−0.611223
$516$	0	0
$517$	0	0
$518$	−8.52360	−0.374506
$519$	0	0
$520$	5.61256	0.246127
$521$	34.5708	1.51457	0.757287	−	0.653082i	$-0.226524\pi$
$521$	34.5708	1.51457	0.757287	+	0.653082i	$0.226524\pi$
$522$	0	0
$523$	−32.7917	−1.43388	−0.716940	−	0.697135i	$-0.754458\pi$
$523$	−32.7917	−1.43388	−0.716940	+	0.697135i	$0.754458\pi$
$524$	1.36518	0.0596381
$525$	0	0
$526$	−7.84406	−0.342017
$527$	21.1609	0.921785
$528$	0	0
$529$	−9.75510	−0.424135
$530$	11.5229	0.500523
$531$	0	0
$532$	0.129147	0.00559925
$533$	3.52907	0.152861
$534$	0	0
$535$	−1.06580	−0.0460784
$536$	7.95240	0.343491
$537$	0	0
$538$	−3.94076	−0.169898
$539$	0	0
$540$	0	0
$541$	−4.37244	−0.187986	−0.0939929	−	0.995573i	$-0.529963\pi$
$541$	−4.37244	−0.187986	−0.0939929	+	0.995573i	$0.529963\pi$
$542$	−10.4968	−0.450877
$543$	0	0
$544$	4.06061	0.174097
$545$	−11.3115	−0.484533
$546$	0	0
$547$	−31.6473	−1.35314	−0.676571	−	0.736377i	$-0.736535\pi$
$547$	−31.6473	−1.35314	−0.676571	+	0.736377i	$0.736535\pi$
$548$	−0.857455	−0.0366287
$549$	0	0
$550$	0	0
$551$	6.36858	0.271311
$552$	0	0
$553$	−7.97320	−0.339055
$554$	−40.3527	−1.71442
$555$	0	0
$556$	0.306884	0.0130148
$557$	1.52068	0.0644331	0.0322166	−	0.999481i	$-0.489743\pi$
$557$	1.52068	0.0644331	0.0322166	+	0.999481i	$0.489743\pi$
$558$	0	0
$559$	12.4889	0.528223
$560$	−5.15456	−0.217820
$561$	0	0
$562$	−33.2519	−1.40265
$563$	0.653939	0.0275602	0.0137801	−	0.999905i	$-0.495614\pi$
$563$	0.653939	0.0275602	0.0137801	+	0.999905i	$0.495614\pi$
$564$	0	0
$565$	−3.76850	−0.158542
$566$	7.02680	0.295358
$567$	0	0
$568$	3.24041	0.135965
$569$	32.6606	1.36921	0.684603	−	0.728916i	$-0.259976\pi$
$569$	32.6606	1.36921	0.684603	+	0.728916i	$0.259976\pi$
$570$	0	0
$571$	−26.1457	−1.09416	−0.547082	−	0.837079i	$-0.684262\pi$
$571$	−26.1457	−1.09416	−0.547082	+	0.837079i	$0.684262\pi$
$572$	0	0
$573$	0	0
$574$	−3.44090	−0.143620
$575$	3.63935	0.151772
$576$	0	0
$577$	5.73377	0.238700	0.119350	−	0.992852i	$-0.461919\pi$
$577$	5.73377	0.238700	0.119350	+	0.992852i	$0.461919\pi$
$578$	−29.5914	−1.23084
$579$	0	0
$580$	0.906910	0.0376574
$581$	21.9315	0.909870
$582$	0	0
$583$	0	0
$584$	−36.0676	−1.49249
$585$	0	0
$586$	−15.0890	−0.623319
$587$	−43.4586	−1.79373	−0.896864	−	0.442307i	$-0.854160\pi$
$587$	−43.4586	−1.79373	−0.896864	+	0.442307i	$0.854160\pi$
$588$	0	0
$589$	−2.76999	−0.114136
$590$	−1.53157	−0.0630536
$591$	0	0
$592$	16.9866	0.698145
$593$	9.23707	0.379321	0.189661	−	0.981850i	$-0.439261\pi$
$593$	9.23707	0.379321	0.189661	+	0.981850i	$0.439261\pi$
$594$	0	0
$595$	−8.52360	−0.349434
$596$	0.966898	0.0396057
$597$	0	0
$598$	9.65430	0.394794
$599$	36.0865	1.47445	0.737227	−	0.675645i	$-0.236135\pi$
$599$	36.0865	1.47445	0.737227	+	0.675645i	$0.236135\pi$
$600$	0	0
$601$	23.1290	0.943452	0.471726	−	0.881745i	$-0.343631\pi$
$601$	23.1290	0.943452	0.471726	+	0.881745i	$0.343631\pi$
$602$	−12.1768	−0.496291
$603$	0	0
$604$	−0.859237	−0.0349619
$605$	0	0
$606$	0	0
$607$	−21.9984	−0.892888	−0.446444	−	0.894812i	$-0.647310\pi$
$607$	−21.9984	−0.892888	−0.446444	+	0.894812i	$0.647310\pi$
$608$	−0.531538	−0.0215567
$609$	0	0
$610$	−3.49330	−0.141440
$611$	9.15673	0.370442
$612$	0	0
$613$	−8.42425	−0.340252	−0.170126	−	0.985422i	$-0.554418\pi$
$613$	−8.42425	−0.340252	−0.170126	+	0.985422i	$0.554418\pi$
$614$	−19.7000	−0.795026
$615$	0	0
$616$	0	0
$617$	17.8709	0.719453	0.359727	−	0.933058i	$-0.382870\pi$
$617$	17.8709	0.719453	0.359727	+	0.933058i	$0.382870\pi$
$618$	0	0
$619$	29.5102	1.18612	0.593058	−	0.805160i	$-0.297921\pi$
$619$	29.5102	1.18612	0.593058	+	0.805160i	$0.297921\pi$
$620$	−0.394457	−0.0158418
$621$	0	0
$622$	37.5854	1.50704
$623$	−3.71707	−0.148921
$624$	0	0
$625$	1.00000	0.0400000
$626$	−5.31344	−0.212368
$627$	0	0
$628$	0.993617	0.0396496
$629$	28.0891	1.11999
$630$	0	0
$631$	23.0204	0.916428	0.458214	−	0.888842i	$-0.348489\pi$
$631$	23.0204	0.916428	0.458214	+	0.888842i	$0.348489\pi$
$632$	16.8693	0.671025
$633$	0	0
$634$	−21.2906	−0.845556
$635$	1.67956	0.0666515
$636$	0	0
$637$	9.88636	0.391712
$638$	0	0
$639$	0	0
$640$	10.2334	0.404511
$641$	3.76850	0.148847	0.0744234	−	0.997227i	$-0.476288\pi$
$641$	3.76850	0.148847	0.0744234	+	0.997227i	$0.476288\pi$
$642$	0	0
$643$	28.3011	1.11609	0.558043	−	0.829812i	$-0.311552\pi$
$643$	28.3011	1.11609	0.558043	+	0.829812i	$0.311552\pi$
$644$	0.578246	0.0227861
$645$	0	0
$646$	6.92820	0.272587
$647$	−11.7775	−0.463020	−0.231510	−	0.972832i	$-0.574367\pi$
$647$	−11.7775	−0.463020	−0.231510	+	0.972832i	$0.574367\pi$
$648$	0	0
$649$	0	0
$650$	2.65275	0.104049
$651$	0	0
$652$	−1.89569	−0.0742410
$653$	46.1282	1.80514	0.902569	−	0.430546i	$-0.141679\pi$
$653$	46.1282	1.80514	0.902569	+	0.430546i	$0.141679\pi$
$654$	0	0
$655$	11.7943	0.460840
$656$	6.85733	0.267734
$657$	0	0
$658$	−8.92795	−0.348048
$659$	15.7781	0.614626	0.307313	−	0.951609i	$-0.400570\pi$
$659$	15.7781	0.614626	0.307313	+	0.951609i	$0.400570\pi$
$660$	0	0
$661$	−38.6732	−1.50421	−0.752106	−	0.659042i	$-0.770962\pi$
$661$	−38.6732	−1.50421	−0.752106	+	0.659042i	$0.770962\pi$
$662$	−36.8667	−1.43286
$663$	0	0
$664$	−46.4015	−1.80073
$665$	1.11575	0.0432669
$666$	0	0
$667$	28.5147	1.10410
$668$	2.48150	0.0960121
$669$	0	0
$670$	3.75867	0.145210
$671$	0	0
$672$	0	0
$673$	44.2791	1.70683	0.853416	−	0.521230i	$-0.174527\pi$
$673$	44.2791	1.70683	0.853416	+	0.521230i	$0.174527\pi$
$674$	−2.65275	−0.102180
$675$	0	0
$676$	1.07245	0.0412482
$677$	−6.74004	−0.259041	−0.129520	−	0.991577i	$-0.541344\pi$
$677$	−6.74004	−0.259041	−0.129520	+	0.991577i	$0.541344\pi$
$678$	0	0
$679$	−19.4412	−0.746085
$680$	18.0338	0.691565
$681$	0	0
$682$	0	0
$683$	−33.3303	−1.27535	−0.637675	−	0.770305i	$-0.720104\pi$
$683$	−33.3303	−1.27535	−0.637675	+	0.770305i	$0.720104\pi$
$684$	0	0
$685$	−7.40786	−0.283040
$686$	−22.8291	−0.871619
$687$	0	0
$688$	24.2671	0.925175
$689$	−16.2226	−0.618031
$690$	0	0
$691$	9.00546	0.342584	0.171292	−	0.985220i	$-0.445206\pi$
$691$	9.00546	0.342584	0.171292	+	0.985220i	$0.445206\pi$
$692$	1.59013	0.0604477
$693$	0	0
$694$	5.21655	0.198018
$695$	2.65128	0.100569
$696$	0	0
$697$	11.3393	0.429507
$698$	−23.5976	−0.893183
$699$	0	0
$700$	0.158887	0.00600536
$701$	−45.8938	−1.73339	−0.866693	−	0.498842i	$-0.833759\pi$
$701$	−45.8938	−1.73339	−0.866693	+	0.498842i	$0.833759\pi$
$702$	0	0
$703$	−3.67690	−0.138677
$704$	0	0
$705$	0	0
$706$	−2.73447	−0.102913
$707$	15.6563	0.588814
$708$	0	0
$709$	−8.51021	−0.319608	−0.159804	−	0.987149i	$-0.551086\pi$
$709$	−8.51021	−0.319608	−0.159804	+	0.987149i	$0.551086\pi$
$710$	1.53157	0.0574787
$711$	0	0
$712$	7.86439	0.294731
$713$	−12.4024	−0.464473
$714$	0	0
$715$	0	0
$716$	−0.436202	−0.0163016
$717$	0	0
$718$	−21.3811	−0.797934
$719$	9.60554	0.358226	0.179113	−	0.983828i	$-0.442677\pi$
$719$	9.60554	0.358226	0.179113	+	0.983828i	$0.442677\pi$
$720$	0	0
$721$	−19.0402	−0.709096
$722$	25.1740	0.936880
$723$	0	0
$724$	0.00155080	5.76351e−5 0
$725$	7.83511	0.290989
$726$	0	0
$727$	32.9688	1.22274	0.611372	−	0.791343i	$-0.290618\pi$
$727$	32.9688	1.22274	0.611372	+	0.791343i	$0.290618\pi$
$728$	7.70425	0.285538
$729$	0	0
$730$	−17.0472	−0.630946
$731$	40.1282	1.48420
$732$	0	0
$733$	−27.3704	−1.01095	−0.505475	−	0.862842i	$-0.668683\pi$
$733$	−27.3704	−1.01095	−0.505475	+	0.862842i	$0.668683\pi$
$734$	−25.7680	−0.951113
$735$	0	0
$736$	−2.37991	−0.0877248
$737$	0	0
$738$	0	0
$739$	38.5646	1.41862	0.709312	−	0.704895i	$-0.249006\pi$
$739$	38.5646	1.41862	0.709312	+	0.704895i	$0.249006\pi$
$740$	−0.523604	−0.0192481
$741$	0	0
$742$	15.8173	0.580670
$743$	−34.1644	−1.25337	−0.626684	−	0.779273i	$-0.715588\pi$
$743$	−34.1644	−1.25337	−0.626684	+	0.779273i	$0.715588\pi$
$744$	0	0
$745$	8.35337	0.306044
$746$	3.60554	0.132008
$747$	0	0
$748$	0	0
$749$	−1.46300	−0.0534568
$750$	0	0
$751$	−45.9305	−1.67603	−0.838015	−	0.545648i	$-0.816284\pi$
$751$	−45.9305	−1.67603	−0.838015	+	0.545648i	$0.816284\pi$
$752$	17.7924	0.648823
$753$	0	0
$754$	20.7846	0.756931
$755$	−7.42325	−0.270160
$756$	0	0
$757$	−30.2236	−1.09849	−0.549247	−	0.835660i	$-0.685085\pi$
$757$	−30.2236	−1.09849	−0.549247	+	0.835660i	$0.685085\pi$
$758$	−3.65227	−0.132656
$759$	0	0
$760$	−2.36065	−0.0856296
$761$	13.4308	0.486866	0.243433	−	0.969918i	$-0.421726\pi$
$761$	13.4308	0.486866	0.243433	+	0.969918i	$0.421726\pi$
$762$	0	0
$763$	−15.5271	−0.562119
$764$	−1.92444	−0.0696240
$765$	0	0
$766$	30.1049	1.08773
$767$	2.15622	0.0778567
$768$	0	0
$769$	10.8982	0.393001	0.196500	−	0.980504i	$-0.437042\pi$
$769$	10.8982	0.393001	0.196500	+	0.980504i	$0.437042\pi$
$770$	0	0
$771$	0	0
$772$	3.09992	0.111568
$773$	−13.1496	−0.472957	−0.236478	−	0.971637i	$-0.575993\pi$
$773$	−13.1496	−0.472957	−0.236478	+	0.971637i	$0.575993\pi$
$774$	0	0
$775$	−3.40786	−0.122414
$776$	41.1327	1.47658
$777$	0	0
$778$	−4.69629	−0.168370
$779$	−1.48433	−0.0531816
$780$	0	0
$781$	0	0
$782$	31.0204	1.10929
$783$	0	0
$784$	19.2102	0.686077
$785$	8.58421	0.306384
$786$	0	0
$787$	11.5413	0.411403	0.205701	−	0.978615i	$-0.434052\pi$
$787$	11.5413	0.411403	0.205701	+	0.978615i	$0.434052\pi$
$788$	−0.295993	−0.0105443
$789$	0	0
$790$	7.97320	0.283674
$791$	−5.17295	−0.183929
$792$	0	0
$793$	4.91806	0.174645
$794$	−7.56375	−0.268427
$795$	0	0
$796$	2.53062	0.0896954
$797$	5.54494	0.196412	0.0982059	−	0.995166i	$-0.468690\pi$
$797$	5.54494	0.196412	0.0982059	+	0.995166i	$0.468690\pi$
$798$	0	0
$799$	29.4217	1.04086
$800$	−0.653939	−0.0231202
$801$	0	0
$802$	−24.6067	−0.868891
$803$	0	0
$804$	0	0
$805$	4.99567	0.176074
$806$	−9.04019	−0.318427
$807$	0	0
$808$	−33.1247	−1.16532
$809$	18.4402	0.648324	0.324162	−	0.946002i	$-0.394918\pi$
$809$	18.4402	0.648324	0.324162	+	0.946002i	$0.394918\pi$
$810$	0	0
$811$	−6.32818	−0.222212	−0.111106	−	0.993809i	$-0.535439\pi$
$811$	−6.32818	−0.222212	−0.111106	+	0.993809i	$0.535439\pi$
$812$	1.24490	0.0436873
$813$	0	0
$814$	0	0
$815$	−16.3776	−0.573681
$816$	0	0
$817$	−5.25283	−0.183773
$818$	40.2554	1.40750
$819$	0	0
$820$	−0.211374	−0.00738150
$821$	7.95240	0.277541	0.138770	−	0.990325i	$-0.455685\pi$
$821$	7.95240	0.277541	0.138770	+	0.990325i	$0.455685\pi$
$822$	0	0
$823$	1.32241	0.0460963	0.0230481	−	0.999734i	$-0.492663\pi$
$823$	1.32241	0.0460963	0.0230481	+	0.999734i	$0.492663\pi$
$824$	40.2844	1.40337
$825$	0	0
$826$	−2.10235	−0.0731502
$827$	−4.52990	−0.157520	−0.0787600	−	0.996894i	$-0.525096\pi$
$827$	−4.52990	−0.157520	−0.0787600	+	0.996894i	$0.525096\pi$
$828$	0	0
$829$	−37.8148	−1.31336	−0.656681	−	0.754168i	$-0.728040\pi$
$829$	−37.8148	−1.31336	−0.656681	+	0.754168i	$0.728040\pi$
$830$	−21.9315	−0.761252
$831$	0	0
$832$	−16.2485	−0.563314
$833$	31.7661	1.10063
$834$	0	0
$835$	21.4385	0.741912
$836$	0	0
$837$	0	0
$838$	3.64138	0.125789
$839$	39.2519	1.35513	0.677563	−	0.735465i	$-0.263036\pi$
$839$	39.2519	1.35513	0.677563	+	0.735465i	$0.263036\pi$
$840$	0	0
$841$	32.3890	1.11686
$842$	15.3232	0.528071
$843$	0	0
$844$	1.48516	0.0511212
$845$	9.26531	0.318736
$846$	0	0
$847$	0	0
$848$	−31.5221	−1.08247
$849$	0	0
$850$	8.52360	0.292357
$851$	−16.4630	−0.564344
$852$	0	0
$853$	−33.3917	−1.14331	−0.571655	−	0.820494i	$-0.693698\pi$
$853$	−33.3917	−1.14331	−0.571655	+	0.820494i	$0.693698\pi$
$854$	−4.79518	−0.164088
$855$	0	0
$856$	3.09534	0.105796
$857$	−16.9781	−0.579961	−0.289980	−	0.957033i	$-0.593649\pi$
$857$	−16.9781	−0.579961	−0.289980	+	0.957033i	$0.593649\pi$
$858$	0	0
$859$	−40.8148	−1.39258	−0.696291	−	0.717759i	$-0.745168\pi$
$859$	−40.8148	−1.39258	−0.696291	+	0.717759i	$0.745168\pi$
$860$	−0.748023	−0.0255074
$861$	0	0
$862$	−6.25829	−0.213158
$863$	−35.0303	−1.19245	−0.596223	−	0.802819i	$-0.703332\pi$
$863$	−35.0303	−1.19245	−0.596223	+	0.802819i	$0.703332\pi$
$864$	0	0
$865$	13.7377	0.467096
$866$	−27.6773	−0.940513
$867$	0	0
$868$	−0.541464	−0.0183785
$869$	0	0
$870$	0	0
$871$	−5.29166	−0.179301
$872$	32.8515	1.11249
$873$	0	0
$874$	−4.06061	−0.137352
$875$	1.37268	0.0464051
$876$	0	0
$877$	26.6626	0.900331	0.450165	−	0.892945i	$-0.351365\pi$
$877$	26.6626	0.900331	0.450165	+	0.892945i	$0.351365\pi$
$878$	−47.3393	−1.59762
$879$	0	0
$880$	0	0
$881$	4.66615	0.157207	0.0786033	−	0.996906i	$-0.474954\pi$
$881$	4.66615	0.157207	0.0786033	+	0.996906i	$0.474954\pi$
$882$	0	0
$883$	6.47093	0.217764	0.108882	−	0.994055i	$-0.465273\pi$
$883$	6.47093	0.217764	0.108882	+	0.994055i	$0.465273\pi$
$884$	−1.38899	−0.0467169
$885$	0	0
$886$	43.1826	1.45075
$887$	18.3508	0.616159	0.308079	−	0.951361i	$-0.400314\pi$
$887$	18.3508	0.616159	0.308079	+	0.951361i	$0.400314\pi$
$888$	0	0
$889$	2.30550	0.0773241
$890$	3.71707	0.124597
$891$	0	0
$892$	0.852396	0.0285403
$893$	−3.85133	−0.128880
$894$	0	0
$895$	−3.76850	−0.125967
$896$	14.0472	0.469284
$897$	0	0
$898$	−6.55652	−0.218794
$899$	−26.7009	−0.890526
$900$	0	0
$901$	−52.1251	−1.73654
$902$	0	0
$903$	0	0
$904$	10.9447	0.364014
$905$	0.0133979	0.000445362 0
$906$	0	0
$907$	−29.7845	−0.988978	−0.494489	−	0.869184i	$-0.664645\pi$
$907$	−29.7845	−0.988978	−0.494489	+	0.869184i	$0.664645\pi$
$908$	−0.158887	−0.00527285
$909$	0	0
$910$	3.64138	0.120711
$911$	−45.8699	−1.51974	−0.759869	−	0.650076i	$-0.774737\pi$
$911$	−45.8699	−1.51974	−0.759869	+	0.650076i	$0.774737\pi$
$912$	0	0
$913$	0	0
$914$	2.10235	0.0695396
$915$	0	0
$916$	−3.41424	−0.112810
$917$	16.1898	0.534633
$918$	0	0
$919$	58.1355	1.91771	0.958856	−	0.283893i	$-0.0916260\pi$
$919$	58.1355	1.91771	0.958856	+	0.283893i	$0.0916260\pi$
$920$	−10.5696	−0.348469
$921$	0	0
$922$	−25.4248	−0.837320
$923$	−2.15622	−0.0709730
$924$	0	0
$925$	−4.52360	−0.148735
$926$	−13.2917	−0.436794
$927$	0	0
$928$	−5.12368	−0.168193
$929$	−39.8361	−1.30698	−0.653490	−	0.756935i	$-0.726696\pi$
$929$	−39.8361	−1.30698	−0.653490	+	0.756935i	$0.726696\pi$
$930$	0	0
$931$	−4.15821	−0.136280
$932$	−2.54630	−0.0834069
$933$	0	0
$934$	−43.5112	−1.42373
$935$	0	0
$936$	0	0
$937$	−9.67356	−0.316022	−0.158011	−	0.987437i	$-0.550508\pi$
$937$	−9.67356	−0.316022	−0.158011	+	0.987437i	$0.550508\pi$
$938$	5.15945	0.168462
$939$	0	0
$940$	−0.548444	−0.0178883
$941$	3.97147	0.129466	0.0647331	−	0.997903i	$-0.479380\pi$
$941$	3.97147	0.129466	0.0647331	+	0.997903i	$0.479380\pi$
$942$	0	0
$943$	−6.64595	−0.216422
$944$	4.18975	0.136365
$945$	0	0
$946$	0	0
$947$	−2.65275	−0.0862029	−0.0431014	−	0.999071i	$-0.513724\pi$
$947$	−2.65275	−0.0862029	−0.0431014	+	0.999071i	$0.513724\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	−1.11575	−0.0361997
$951$	0	0
$952$	24.7547	0.802303
$953$	−20.8678	−0.675974	−0.337987	−	0.941151i	$-0.609746\pi$
$953$	−20.8678	−0.675974	−0.337987	+	0.941151i	$0.609746\pi$
$954$	0	0
$955$	−16.6260	−0.538003
$956$	−2.16711	−0.0700895
$957$	0	0
$958$	−42.3259	−1.36749
$959$	−10.1686	−0.328362
$960$	0	0
$961$	−19.3865	−0.625372
$962$	−12.0000	−0.386896
$963$	0	0
$964$	−3.03993	−0.0979095
$965$	26.7813	0.862120
$966$	0	0
$967$	−58.5364	−1.88240	−0.941202	−	0.337843i	$-0.890303\pi$
$967$	−58.5364	−1.88240	−0.941202	+	0.337843i	$0.890303\pi$
$968$	0	0
$969$	0	0
$970$	19.4412	0.624220
$971$	56.2653	1.80564	0.902820	−	0.430019i	$-0.141493\pi$
$971$	56.2653	1.80564	0.902820	+	0.430019i	$0.141493\pi$
$972$	0	0
$973$	3.63935	0.116672
$974$	−41.1695	−1.31916
$975$	0	0
$976$	9.55627	0.305889
$977$	−36.9866	−1.18331	−0.591653	−	0.806193i	$-0.701524\pi$
$977$	−36.9866	−1.18331	−0.591653	+	0.806193i	$0.701524\pi$
$978$	0	0
$979$	0	0
$980$	−0.592145	−0.0189154
$981$	0	0
$982$	−47.3731	−1.51174
$983$	19.6642	0.627190	0.313595	−	0.949557i	$-0.398467\pi$
$983$	19.6642	0.627190	0.313595	+	0.949557i	$0.398467\pi$
$984$	0	0
$985$	−2.55719	−0.0814789
$986$	66.7834	2.12682
$987$	0	0
$988$	0.181821	0.00578449
$989$	−23.5191	−0.747863
$990$	0	0
$991$	18.5763	0.590095	0.295047	−	0.955483i	$-0.404665\pi$
$991$	18.5763	0.590095	0.295047	+	0.955483i	$0.404665\pi$
$992$	2.22853	0.0707558
$993$	0	0
$994$	2.10235	0.0666825
$995$	21.8629	0.693101
$996$	0	0
$997$	30.1049	0.953431	0.476716	−	0.879058i	$-0.341827\pi$
$997$	30.1049	0.953431	0.476716	+	0.879058i	$0.341827\pi$
$998$	−24.5201	−0.776169
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bx.1.2		6
3.2	odd	2		605.2.a.m.1.5	yes	6
11.10	odd	2	inner	5445.2.a.bx.1.5		6
12.11	even	2		9680.2.a.cw.1.2		6
15.14	odd	2		3025.2.a.bg.1.2		6
33.2	even	10		605.2.g.q.81.5		24
33.5	odd	10		605.2.g.q.366.2		24
33.8	even	10		605.2.g.q.251.2		24
33.14	odd	10		605.2.g.q.251.5		24
33.17	even	10		605.2.g.q.366.5		24
33.20	odd	10		605.2.g.q.81.2		24
33.26	odd	10		605.2.g.q.511.5		24
33.29	even	10		605.2.g.q.511.2		24
33.32	even	2		605.2.a.m.1.2	✓	6
132.131	odd	2		9680.2.a.cw.1.1		6
165.164	even	2		3025.2.a.bg.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.m.1.2	✓	6	33.32	even	2
605.2.a.m.1.5	yes	6	3.2	odd	2
605.2.g.q.81.2		24	33.20	odd	10
605.2.g.q.81.5		24	33.2	even	10
605.2.g.q.251.2		24	33.8	even	10
605.2.g.q.251.5		24	33.14	odd	10
605.2.g.q.366.2		24	33.5	odd	10
605.2.g.q.366.5		24	33.17	even	10
605.2.g.q.511.2		24	33.29	even	10
605.2.g.q.511.5		24	33.26	odd	10
3025.2.a.bg.1.2		6	15.14	odd	2
3025.2.a.bg.1.5		6	165.164	even	2
5445.2.a.bx.1.2		6	1.1	even	1	trivial
5445.2.a.bx.1.5		6	11.10	odd	2	inner
9680.2.a.cw.1.1		6	132.131	odd	2
9680.2.a.cw.1.2		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)

\(f(q)\)	\(=\)	\(q-1.37268 q^{2} -0.115749 q^{4} -1.00000 q^{5} -1.37268 q^{7} +2.90425 q^{8} +O(q^{10})\)	\(q-1.37268 q^{2} -0.115749 q^{4} -1.00000 q^{5} -1.37268 q^{7} +2.90425 q^{8} +1.37268 q^{10} -1.93253 q^{13} +1.88425 q^{14} -3.75510 q^{16} -6.20946 q^{17} +0.812826 q^{19} +0.115749 q^{20} +3.63935 q^{23} +1.00000 q^{25} +2.65275 q^{26} +0.158887 q^{28} +7.83511 q^{29} -3.40786 q^{31} -0.653939 q^{32} +8.52360 q^{34} +1.37268 q^{35} -4.52360 q^{37} -1.11575 q^{38} -2.90425 q^{40} -1.82613 q^{41} -6.46243 q^{43} -4.99567 q^{46} -4.73820 q^{47} -5.11575 q^{49} -1.37268 q^{50} +0.223690 q^{52} +8.39446 q^{53} -3.98660 q^{56} -10.7551 q^{58} -1.11575 q^{59} -2.54488 q^{61} +4.67789 q^{62} +8.40786 q^{64} +1.93253 q^{65} +2.73820 q^{67} +0.718741 q^{68} -1.88425 q^{70} +1.11575 q^{71} -12.4189 q^{73} +6.20946 q^{74} -0.0940841 q^{76} +5.80849 q^{79} +3.75510 q^{80} +2.50670 q^{82} -15.9771 q^{83} +6.20946 q^{85} +8.87085 q^{86} +2.70789 q^{89} +2.65275 q^{91} -0.421253 q^{92} +6.50403 q^{94} -0.812826 q^{95} +14.1630 q^{97} +7.02229 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{4} - 6 q^{5}+O(q^{10}) \) 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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