LMFDB - Embedded newform 4140.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.73549$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -3.73549 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -3.73549 q^{7} -4.84469 q^{11} -2.84469 q^{13} -0.890804 q^{17} +6.84469 q^{19} -1.00000 q^{23} +1.00000 q^{25} +0.890804 q^{29} +7.73549 q^{31} -3.73549 q^{35} -1.95388 q^{37} -12.3618 q^{41} -3.47098 q^{43} +6.62629 q^{47} +6.95388 q^{49} -12.3618 q^{53} -4.84469 q^{55} -0.890804 q^{59} +8.62629 q^{61} -2.84469 q^{65} +7.73549 q^{67} +12.3618 q^{71} +16.5341 q^{73} +18.0973 q^{77} +13.4710 q^{79} +4.58018 q^{83} -0.890804 q^{85} +15.1604 q^{89} +10.6263 q^{91} +6.84469 q^{95} -17.2526 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 2 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 + 2 * q^7	$ 3 q + 3 q^{5} + 2 q^{7} - 4 q^{11} + 2 q^{13} + 10 q^{19} - 3 q^{23} + 3 q^{25} + 10 q^{31} + 2 q^{35} + 2 q^{37} - 8 q^{41} + 16 q^{43} + 4 q^{47} + 13 q^{49} - 8 q^{53} - 4 q^{55} + 10 q^{61} + 2 q^{65} + 10 q^{67} + 8 q^{71} + 18 q^{73} + 12 q^{77} + 14 q^{79} - 10 q^{83} - 2 q^{89} + 16 q^{91} + 10 q^{95} - 20 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 + 2 * q^7 - 4 * q^11 + 2 * q^13 + 10 * q^19 - 3 * q^23 + 3 * q^25 + 10 * q^31 + 2 * q^35 + 2 * q^37 - 8 * q^41 + 16 * q^43 + 4 * q^47 + 13 * q^49 - 8 * q^53 - 4 * q^55 + 10 * q^61 + 2 * q^65 + 10 * q^67 + 8 * q^71 + 18 * q^73 + 12 * q^77 + 14 * q^79 - 10 * q^83 - 2 * q^89 + 16 * q^91 + 10 * q^95 - 20 * 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	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.73549	−1.41188	−0.705941	−	0.708270i	$-0.749476\pi$
$7$	−3.73549	−1.41188	−0.705941	+	0.708270i	$0.749476\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.84469	−1.46073	−0.730364	−	0.683058i	$-0.760649\pi$
$11$	−4.84469	−1.46073	−0.730364	+	0.683058i	$0.760649\pi$
$12$	0	0
$13$	−2.84469	−0.788974	−0.394487	−	0.918902i	$-0.629078\pi$
$13$	−2.84469	−0.788974	−0.394487	+	0.918902i	$0.629078\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.890804	−0.216052	−0.108026	−	0.994148i	$-0.534453\pi$
$17$	−0.890804	−0.216052	−0.108026	+	0.994148i	$0.534453\pi$
$18$	0	0
$19$	6.84469	1.57028	0.785139	−	0.619319i	$-0.212591\pi$
$19$	6.84469	1.57028	0.785139	+	0.619319i	$0.212591\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.890804	0.165418	0.0827091	−	0.996574i	$-0.473643\pi$
$29$	0.890804	0.165418	0.0827091	+	0.996574i	$0.473643\pi$
$30$	0	0
$31$	7.73549	1.38933	0.694667	−	0.719331i	$-0.255552\pi$
$31$	7.73549	1.38933	0.694667	+	0.719331i	$0.255552\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.73549	−0.631413
$36$	0	0
$37$	−1.95388	−0.321216	−0.160608	−	0.987018i	$-0.551346\pi$
$37$	−1.95388	−0.321216	−0.160608	+	0.987018i	$0.551346\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.3618	−1.93059	−0.965293	−	0.261169i	$-0.915892\pi$
$41$	−12.3618	−1.93059	−0.965293	+	0.261169i	$0.915892\pi$
$42$	0	0
$43$	−3.47098	−0.529319	−0.264660	−	0.964342i	$-0.585260\pi$
$43$	−3.47098	−0.529319	−0.264660	+	0.964342i	$0.585260\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.62629	0.966544	0.483272	−	0.875470i	$-0.339448\pi$
$47$	6.62629	0.966544	0.483272	+	0.875470i	$0.339448\pi$
$48$	0	0
$49$	6.95388	0.993412
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.3618	−1.69802	−0.849011	−	0.528376i	$-0.822801\pi$
$53$	−12.3618	−1.69802	−0.849011	+	0.528376i	$0.822801\pi$
$54$	0	0
$55$	−4.84469	−0.653257
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.890804	−0.115973	−0.0579864	−	0.998317i	$-0.518468\pi$
$59$	−0.890804	−0.115973	−0.0579864	+	0.998317i	$0.518468\pi$
$60$	0	0
$61$	8.62629	1.10448	0.552242	−	0.833684i	$-0.313773\pi$
$61$	8.62629	1.10448	0.552242	+	0.833684i	$0.313773\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.84469	−0.352840
$66$	0	0
$67$	7.73549	0.945040	0.472520	−	0.881320i	$-0.343344\pi$
$67$	7.73549	0.945040	0.472520	+	0.881320i	$0.343344\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.3618	1.46707	0.733537	−	0.679650i	$-0.237868\pi$
$71$	12.3618	1.46707	0.733537	+	0.679650i	$0.237868\pi$
$72$	0	0
$73$	16.5341	1.93516	0.967582	−	0.252555i	$-0.0812709\pi$
$73$	16.5341	1.93516	0.967582	+	0.252555i	$0.0812709\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.0973	2.06238
$78$	0	0
$79$	13.4710	1.51560	0.757802	−	0.652485i	$-0.226273\pi$
$79$	13.4710	1.51560	0.757802	+	0.652485i	$0.226273\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.58018	0.502740	0.251370	−	0.967891i	$-0.419119\pi$
$83$	4.58018	0.502740	0.251370	+	0.967891i	$0.419119\pi$
$84$	0	0
$85$	−0.890804	−0.0966212
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.1604	1.60699	0.803497	−	0.595309i	$-0.202970\pi$
$89$	15.1604	1.60699	0.803497	+	0.595309i	$0.202970\pi$
$90$	0	0
$91$	10.6263	1.11394
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.84469	0.702250
$96$	0	0
$97$	−17.2526	−1.75173	−0.875867	−	0.482552i	$-0.839710\pi$
$97$	−17.2526	−1.75173	−0.875867	+	0.482552i	$0.839710\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.67241	0.265915	0.132957	−	0.991122i	$-0.457553\pi$
$101$	2.67241	0.265915	0.132957	+	0.991122i	$0.457553\pi$
$102$	0	0
$103$	6.21839	0.612716	0.306358	−	0.951916i	$-0.400889\pi$
$103$	6.21839	0.612716	0.306358	+	0.951916i	$0.400889\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.7986	1.43063	0.715316	−	0.698801i	$-0.246283\pi$
$107$	14.7986	1.43063	0.715316	+	0.698801i	$0.246283\pi$
$108$	0	0
$109$	−12.5341	−1.20054	−0.600272	−	0.799796i	$-0.704941\pi$
$109$	−12.5341	−1.20054	−0.600272	+	0.799796i	$0.704941\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.3618	−1.16290	−0.581449	−	0.813583i	$-0.697514\pi$
$113$	−12.3618	−1.16290	−0.581449	+	0.813583i	$0.697514\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.32759	0.305040
$120$	0	0
$121$	12.4710	1.13373
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.7866	1.04590	0.522948	−	0.852365i	$-0.324832\pi$
$127$	11.7866	1.04590	0.522948	+	0.852365i	$0.324832\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.4710	1.52645	0.763223	−	0.646135i	$-0.223616\pi$
$131$	17.4710	1.52645	0.763223	+	0.646135i	$0.223616\pi$
$132$	0	0
$133$	−25.5683	−2.21705
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.1604	−1.29524	−0.647618	−	0.761965i	$-0.724235\pi$
$137$	−15.1604	−1.29524	−0.647618	+	0.761965i	$0.724235\pi$
$138$	0	0
$139$	−5.51710	−0.467954	−0.233977	−	0.972242i	$-0.575174\pi$
$139$	−5.51710	−0.467954	−0.233977	+	0.972242i	$0.575174\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.7816	1.15248
$144$	0	0
$145$	0.890804	0.0739772
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.15531	−0.0946470	−0.0473235	−	0.998880i	$-0.515069\pi$
$149$	−1.15531	−0.0946470	−0.0473235	+	0.998880i	$0.515069\pi$
$150$	0	0
$151$	−15.4710	−1.25901	−0.629505	−	0.776996i	$-0.716742\pi$
$151$	−15.4710	−1.25901	−0.629505	+	0.776996i	$0.716742\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.73549	0.621329
$156$	0	0
$157$	18.6774	1.49062	0.745311	−	0.666717i	$-0.232301\pi$
$157$	18.6774	1.49062	0.745311	+	0.666717i	$0.232301\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.73549	0.294398
$162$	0	0
$163$	0.0922364	0.00722451	0.00361226	−	0.999993i	$-0.498850\pi$
$163$	0.0922364	0.00722451	0.00361226	+	0.999993i	$0.498850\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.37371	0.415830	0.207915	−	0.978147i	$-0.433332\pi$
$167$	5.37371	0.415830	0.207915	+	0.978147i	$0.433332\pi$
$168$	0	0
$169$	−4.90776	−0.377520
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−3.73549	−0.282376
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.56322	0.714788	0.357394	−	0.933954i	$-0.383665\pi$
$179$	9.56322	0.714788	0.357394	+	0.933954i	$0.383665\pi$
$180$	0	0
$181$	−7.16035	−0.532225	−0.266112	−	0.963942i	$-0.585739\pi$
$181$	−7.16035	−0.532225	−0.266112	+	0.963942i	$0.585739\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.95388	−0.143652
$186$	0	0
$187$	4.31566	0.315593
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.0050	−1.01337	−0.506684	−	0.862132i	$-0.669129\pi$
$191$	−14.0050	−1.01337	−0.506684	+	0.862132i	$0.669129\pi$
$192$	0	0
$193$	26.7236	1.92360	0.961802	−	0.273745i	$-0.0882626\pi$
$193$	26.7236	1.92360	0.961802	+	0.273745i	$0.0882626\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.16035	−0.652648	−0.326324	−	0.945258i	$-0.605810\pi$
$197$	−9.16035	−0.652648	−0.326324	+	0.945258i	$0.605810\pi$
$198$	0	0
$199$	−0.310629	−0.0220199	−0.0110099	−	0.999939i	$-0.503505\pi$
$199$	−0.310629	−0.0220199	−0.0110099	+	0.999939i	$0.503505\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.32759	−0.233551
$204$	0	0
$205$	−12.3618	−0.863384
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−33.1604	−2.29375
$210$	0	0
$211$	5.95388	0.409882	0.204941	−	0.978774i	$-0.434300\pi$
$211$	5.95388	0.409882	0.204941	+	0.978774i	$0.434300\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.47098	−0.236719
$216$	0	0
$217$	−28.8958	−1.96158
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.53406	0.170459
$222$	0	0
$223$	−11.9078	−0.797403	−0.398701	−	0.917081i	$-0.630539\pi$
$223$	−11.9078	−0.797403	−0.398701	+	0.917081i	$0.630539\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.37874	−0.489744	−0.244872	−	0.969555i	$-0.578746\pi$
$227$	−7.37874	−0.489744	−0.244872	+	0.969555i	$0.578746\pi$
$228$	0	0
$229$	23.1604	1.53048	0.765240	−	0.643746i	$-0.222621\pi$
$229$	23.1604	1.53048	0.765240	+	0.643746i	$0.222621\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.9420	1.50298	0.751489	−	0.659746i	$-0.229336\pi$
$233$	22.9420	1.50298	0.751489	+	0.659746i	$0.229336\pi$
$234$	0	0
$235$	6.62629	0.432252
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−0.361783	−0.0234018	−0.0117009	−	0.999932i	$-0.503725\pi$
$239$	−0.361783	−0.0234018	−0.0117009	+	0.999932i	$0.503725\pi$
$240$	0	0
$241$	28.5341	1.83804	0.919020	−	0.394211i	$-0.128982\pi$
$241$	28.5341	1.83804	0.919020	+	0.394211i	$0.128982\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.95388	0.444267
$246$	0	0
$247$	−19.4710	−1.23891
$248$	0	0
$249$	0	0
$250$	0	0
$251$	30.8497	1.94722	0.973609	−	0.228224i	$-0.0732920\pi$
$251$	30.8497	1.94722	0.973609	+	0.228224i	$0.0732920\pi$
$252$	0	0
$253$	4.84469	0.304583
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.7524	0.795476	0.397738	−	0.917499i	$-0.369795\pi$
$257$	12.7524	0.795476	0.397738	+	0.917499i	$0.369795\pi$
$258$	0	0
$259$	7.29870	0.453519
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.36178	−0.392284	−0.196142	−	0.980575i	$-0.562841\pi$
$263$	−6.36178	−0.392284	−0.196142	+	0.980575i	$0.562841\pi$
$264$	0	0
$265$	−12.3618	−0.759378
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.5802	−0.645085	−0.322542	−	0.946555i	$-0.604537\pi$
$269$	−10.5802	−0.645085	−0.322542	+	0.946555i	$0.604537\pi$
$270$	0	0
$271$	−4.26451	−0.259051	−0.129525	−	0.991576i	$-0.541345\pi$
$271$	−4.26451	−0.259051	−0.129525	+	0.991576i	$0.541345\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.84469	−0.292146
$276$	0	0
$277$	−29.3787	−1.76520	−0.882599	−	0.470127i	$-0.844208\pi$
$277$	−29.3787	−1.76520	−0.882599	+	0.470127i	$0.844208\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.7524	1.11868	0.559339	−	0.828939i	$-0.311055\pi$
$281$	18.7524	1.11868	0.559339	+	0.828939i	$0.311055\pi$
$282$	0	0
$283$	−12.8958	−0.766578	−0.383289	−	0.923628i	$-0.625209\pi$
$283$	−12.8958	−0.766578	−0.383289	+	0.923628i	$0.625209\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	46.1773	2.72576
$288$	0	0
$289$	−16.2065	−0.953322
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.79857	−0.514018	−0.257009	−	0.966409i	$-0.582737\pi$
$293$	−8.79857	−0.514018	−0.257009	+	0.966409i	$0.582737\pi$
$294$	0	0
$295$	−0.890804	−0.0518646
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.84469	0.164512
$300$	0	0
$301$	12.9658	0.747337
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.62629	0.493940
$306$	0	0
$307$	10.0050	0.571018	0.285509	−	0.958376i	$-0.407837\pi$
$307$	10.0050	0.571018	0.285509	+	0.958376i	$0.407837\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.5290	1.05068	0.525342	−	0.850891i	$-0.323937\pi$
$311$	18.5290	1.05068	0.525342	+	0.850891i	$0.323937\pi$
$312$	0	0
$313$	2.39067	0.135128	0.0675642	−	0.997715i	$-0.478477\pi$
$313$	2.39067	0.135128	0.0675642	+	0.997715i	$0.478477\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.3157	0.916379	0.458190	−	0.888855i	$-0.348498\pi$
$317$	16.3157	0.916379	0.458190	+	0.888855i	$0.348498\pi$
$318$	0	0
$319$	−4.31566	−0.241631
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.09727	−0.339261
$324$	0	0
$325$	−2.84469	−0.157795
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−24.7524	−1.36465
$330$	0	0
$331$	−24.8958	−1.36840	−0.684200	−	0.729295i	$-0.739848\pi$
$331$	−24.8958	−1.36840	−0.684200	+	0.729295i	$0.739848\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.73549	0.422635
$336$	0	0
$337$	−27.4710	−1.49644	−0.748220	−	0.663451i	$-0.769091\pi$
$337$	−27.4710	−1.49644	−0.748220	+	0.663451i	$0.769091\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−37.4760	−2.02944
$342$	0	0
$343$	0.172274	0.00930193
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.25259	0.0672424	0.0336212	−	0.999435i	$-0.489296\pi$
$347$	1.25259	0.0672424	0.0336212	+	0.999435i	$0.489296\pi$
$348$	0	0
$349$	9.64325	0.516192	0.258096	−	0.966119i	$-0.416905\pi$
$349$	9.64325	0.516192	0.258096	+	0.966119i	$0.416905\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.7524	0.678744	0.339372	−	0.940652i	$-0.389785\pi$
$353$	12.7524	0.678744	0.339372	+	0.940652i	$0.389785\pi$
$354$	0	0
$355$	12.3618	0.656095
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.62629	0.349722	0.174861	−	0.984593i	$-0.444052\pi$
$359$	6.62629	0.349722	0.174861	+	0.984593i	$0.444052\pi$
$360$	0	0
$361$	27.8497	1.46577
$362$	0	0
$363$	0	0
$364$	0	0
$365$	16.5341	0.865432
$366$	0	0
$367$	−9.60933	−0.501603	−0.250802	−	0.968039i	$-0.580694\pi$
$367$	−9.60933	−0.501603	−0.250802	+	0.968039i	$0.580694\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	46.1773	2.39741
$372$	0	0
$373$	29.8839	1.54733	0.773665	−	0.633595i	$-0.218421\pi$
$373$	29.8839	1.54733	0.773665	+	0.633595i	$0.218421\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.53406	−0.130511
$378$	0	0
$379$	2.52902	0.129907	0.0649535	−	0.997888i	$-0.479310\pi$
$379$	2.52902	0.129907	0.0649535	+	0.997888i	$0.479310\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−25.2115	−1.28825	−0.644124	−	0.764921i	$-0.722778\pi$
$383$	−25.2115	−1.28825	−0.644124	+	0.764921i	$0.722778\pi$
$384$	0	0
$385$	18.0973	0.922322
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−35.0681	−1.77802	−0.889012	−	0.457884i	$-0.848608\pi$
$389$	−35.0681	−1.77802	−0.889012	+	0.457884i	$0.848608\pi$
$390$	0	0
$391$	0.890804	0.0450499
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13.4710	0.677799
$396$	0	0
$397$	8.84972	0.444155	0.222077	−	0.975029i	$-0.428716\pi$
$397$	8.84972	0.444155	0.222077	+	0.975029i	$0.428716\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.0681	1.15197	0.575983	−	0.817461i	$-0.304619\pi$
$401$	23.0681	1.15197	0.575983	+	0.817461i	$0.304619\pi$
$402$	0	0
$403$	−22.0050	−1.09615
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.46594	0.469209
$408$	0	0
$409$	5.29870	0.262004	0.131002	−	0.991382i	$-0.458181\pi$
$409$	5.29870	0.262004	0.131002	+	0.991382i	$0.458181\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.32759	0.163740
$414$	0	0
$415$	4.58018	0.224832
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.53406	−0.123797	−0.0618984	−	0.998082i	$-0.519715\pi$
$419$	−2.53406	−0.123797	−0.0618984	+	0.998082i	$0.519715\pi$
$420$	0	0
$421$	22.4079	1.09209	0.546047	−	0.837754i	$-0.316132\pi$
$421$	22.4079	1.09209	0.546047	+	0.837754i	$0.316132\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.890804	−0.0432103
$426$	0	0
$427$	−32.2234	−1.55940
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.5733	−1.52083	−0.760416	−	0.649436i	$-0.775005\pi$
$431$	−31.5733	−1.52083	−0.760416	+	0.649436i	$0.775005\pi$
$432$	0	0
$433$	7.20647	0.346321	0.173160	−	0.984894i	$-0.444602\pi$
$433$	7.20647	0.346321	0.173160	+	0.984894i	$0.444602\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.84469	−0.327426
$438$	0	0
$439$	−4.72357	−0.225443	−0.112722	−	0.993627i	$-0.535957\pi$
$439$	−4.72357	−0.225443	−0.112722	+	0.993627i	$0.535957\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.87888	0.374337	0.187168	−	0.982328i	$-0.440069\pi$
$443$	7.87888	0.374337	0.187168	+	0.982328i	$0.440069\pi$
$444$	0	0
$445$	15.1604	0.718670
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.70633	0.222105	0.111053	−	0.993815i	$-0.464578\pi$
$449$	4.70633	0.222105	0.111053	+	0.993815i	$0.464578\pi$
$450$	0	0
$451$	59.8890	2.82006
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.6263	0.498168
$456$	0	0
$457$	−6.04612	−0.282825	−0.141413	−	0.989951i	$-0.545164\pi$
$457$	−6.04612	−0.282825	−0.141413	+	0.989951i	$0.545164\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.2526	0.896682	0.448341	−	0.893863i	$-0.352015\pi$
$461$	19.2526	0.896682	0.448341	+	0.893863i	$0.352015\pi$
$462$	0	0
$463$	−26.4418	−1.22886	−0.614428	−	0.788973i	$-0.710613\pi$
$463$	−26.4418	−1.22886	−0.614428	+	0.788973i	$0.710613\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.20143	−0.425792	−0.212896	−	0.977075i	$-0.568289\pi$
$467$	−9.20143	−0.425792	−0.212896	+	0.977075i	$0.568289\pi$
$468$	0	0
$469$	−28.8958	−1.33429
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.8158	0.773191
$474$	0	0
$475$	6.84469	0.314056
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.7524	0.582674	0.291337	−	0.956620i	$-0.405900\pi$
$479$	12.7524	0.582674	0.291337	+	0.956620i	$0.405900\pi$
$480$	0	0
$481$	5.55818	0.253431
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−17.2526	−0.783400
$486$	0	0
$487$	−30.5341	−1.38363	−0.691815	−	0.722075i	$-0.743189\pi$
$487$	−30.5341	−1.38363	−0.691815	+	0.722075i	$0.743189\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.67241	0.120604	0.0603021	−	0.998180i	$-0.480794\pi$
$491$	2.67241	0.120604	0.0603021	+	0.998180i	$0.480794\pi$
$492$	0	0
$493$	−0.793532	−0.0357389
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−46.1773	−2.07134
$498$	0	0
$499$	−20.8036	−0.931297	−0.465649	−	0.884970i	$-0.654179\pi$
$499$	−20.8036	−0.931297	−0.465649	+	0.884970i	$0.654179\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.32759	−0.148370	−0.0741849	−	0.997245i	$-0.523636\pi$
$503$	−3.32759	−0.148370	−0.0741849	+	0.997245i	$0.523636\pi$
$504$	0	0
$505$	2.67241	0.118921
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16.2184	−0.718868	−0.359434	−	0.933171i	$-0.617030\pi$
$509$	−16.2184	−0.718868	−0.359434	+	0.933171i	$0.617030\pi$
$510$	0	0
$511$	−61.7628	−2.73223
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.21839	0.274015
$516$	0	0
$517$	−32.1023	−1.41186
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.18923	−0.139723	−0.0698614	−	0.997557i	$-0.522256\pi$
$521$	−3.18923	−0.139723	−0.0698614	+	0.997557i	$0.522256\pi$
$522$	0	0
$523$	23.8155	1.04138	0.520690	−	0.853746i	$-0.325675\pi$
$523$	23.8155	1.04138	0.520690	+	0.853746i	$0.325675\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.89080	−0.300168
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	35.1654	1.52318
$534$	0	0
$535$	14.7986	0.639798
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−33.6894	−1.45110
$540$	0	0
$541$	15.7816	0.678504	0.339252	−	0.940695i	$-0.389826\pi$
$541$	15.7816	0.678504	0.339252	+	0.940695i	$0.389826\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.5341	−0.536900
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.09727	0.259753
$552$	0	0
$553$	−50.3207	−2.13985
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.7405	−1.34489	−0.672445	−	0.740147i	$-0.734756\pi$
$557$	−31.7405	−1.34489	−0.672445	+	0.740147i	$0.734756\pi$
$558$	0	0
$559$	9.87384	0.417619
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.9250	−0.924029	−0.462014	−	0.886872i	$-0.652873\pi$
$563$	−21.9250	−0.924029	−0.462014	+	0.886872i	$0.652873\pi$
$564$	0	0
$565$	−12.3618	−0.520064
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.6313	0.613377	0.306689	−	0.951810i	$-0.400779\pi$
$569$	14.6313	0.613377	0.306689	+	0.951810i	$0.400779\pi$
$570$	0	0
$571$	1.24755	0.0522084	0.0261042	−	0.999659i	$-0.491690\pi$
$571$	1.24755	0.0522084	0.0261042	+	0.999659i	$0.491690\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	18.0922	0.753190	0.376595	−	0.926378i	$-0.377095\pi$
$577$	18.0922	0.753190	0.376595	+	0.926378i	$0.377095\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17.1092	−0.709809
$582$	0	0
$583$	59.8890	2.48035
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.5290	−0.517128	−0.258564	−	0.965994i	$-0.583249\pi$
$587$	−12.5290	−0.517128	−0.258564	+	0.965994i	$0.583249\pi$
$588$	0	0
$589$	52.9470	2.18164
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.9470	1.02445	0.512225	−	0.858851i	$-0.328821\pi$
$593$	24.9470	1.02445	0.512225	+	0.858851i	$0.328821\pi$
$594$	0	0
$595$	3.32759	0.136418
$596$	0	0
$597$	0	0
$598$	0	0
$599$	34.5391	1.41123	0.705615	−	0.708596i	$-0.250671\pi$
$599$	34.5391	1.41123	0.705615	+	0.708596i	$0.250671\pi$
$600$	0	0
$601$	37.9300	1.54720	0.773599	−	0.633675i	$-0.218454\pi$
$601$	37.9300	1.54720	0.773599	+	0.633675i	$0.218454\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12.4710	0.507017
$606$	0	0
$607$	26.0973	1.05926	0.529628	−	0.848230i	$-0.322332\pi$
$607$	26.0973	1.05926	0.529628	+	0.848230i	$0.322332\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−18.8497	−0.762578
$612$	0	0
$613$	2.84972	0.115099	0.0575496	−	0.998343i	$-0.481671\pi$
$613$	2.84972	0.115099	0.0575496	+	0.998343i	$0.481671\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.52213	0.383347	0.191673	−	0.981459i	$-0.438609\pi$
$617$	9.52213	0.383347	0.191673	+	0.981459i	$0.438609\pi$
$618$	0	0
$619$	−23.9762	−0.963683	−0.481841	−	0.876258i	$-0.660032\pi$
$619$	−23.9762	−0.963683	−0.481841	+	0.876258i	$0.660032\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−56.6313	−2.26889
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.74053	0.0693993
$630$	0	0
$631$	−17.8789	−0.711747	−0.355873	−	0.934534i	$-0.615817\pi$
$631$	−17.8789	−0.711747	−0.355873	+	0.934534i	$0.615817\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.7866	0.467739
$636$	0	0
$637$	−19.7816	−0.783776
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.93692	−0.116001	−0.0580007	−	0.998317i	$-0.518473\pi$
$641$	−2.93692	−0.116001	−0.0580007	+	0.998317i	$0.518473\pi$
$642$	0	0
$643$	−0.895840	−0.0353285	−0.0176642	−	0.999844i	$-0.505623\pi$
$643$	−0.895840	−0.0353285	−0.0176642	+	0.999844i	$0.505623\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.4659	0.843913	0.421957	−	0.906616i	$-0.361343\pi$
$647$	21.4659	0.843913	0.421957	+	0.906616i	$0.361343\pi$
$648$	0	0
$649$	4.31566	0.169405
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.84469	0.189587	0.0947936	−	0.995497i	$-0.469781\pi$
$653$	4.84469	0.189587	0.0947936	+	0.995497i	$0.469781\pi$
$654$	0	0
$655$	17.4710	0.682648
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.0973	0.704970	0.352485	−	0.935818i	$-0.385337\pi$
$659$	18.0973	0.704970	0.352485	+	0.935818i	$0.385337\pi$
$660$	0	0
$661$	0.747413	0.0290710	0.0145355	−	0.999894i	$-0.495373\pi$
$661$	0.747413	0.0290710	0.0145355	+	0.999894i	$0.495373\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−25.5683	−0.991494
$666$	0	0
$667$	−0.890804	−0.0344921
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−41.7917	−1.61335
$672$	0	0
$673$	−19.1892	−0.739691	−0.369845	−	0.929093i	$-0.620589\pi$
$673$	−19.1892	−0.739691	−0.369845	+	0.929093i	$0.620589\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.17731	0.160547	0.0802735	−	0.996773i	$-0.474421\pi$
$677$	4.17731	0.160547	0.0802735	+	0.996773i	$0.474421\pi$
$678$	0	0
$679$	64.4469	2.47324
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.0050	0.995055	0.497528	−	0.867448i	$-0.334241\pi$
$683$	26.0050	0.995055	0.497528	+	0.867448i	$0.334241\pi$
$684$	0	0
$685$	−15.1604	−0.579247
$686$	0	0
$687$	0	0
$688$	0	0
$689$	35.1654	1.33969
$690$	0	0
$691$	26.8497	1.02141	0.510706	−	0.859756i	$-0.329384\pi$
$691$	26.8497	1.02141	0.510706	+	0.859756i	$0.329384\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.51710	−0.209275
$696$	0	0
$697$	11.0119	0.417106
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.9027	−0.449560	−0.224780	−	0.974410i	$-0.572166\pi$
$701$	−11.9027	−0.449560	−0.224780	+	0.974410i	$0.572166\pi$
$702$	0	0
$703$	−13.3737	−0.504399
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.98277	−0.375441
$708$	0	0
$709$	−39.3737	−1.47871	−0.739355	−	0.673315i	$-0.764870\pi$
$709$	−39.3737	−1.47871	−0.739355	+	0.673315i	$0.764870\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.73549	−0.289696
$714$	0	0
$715$	13.7816	0.515403
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23.6382	−0.881557	−0.440778	−	0.897616i	$-0.645298\pi$
$719$	−23.6382	−0.881557	−0.440778	+	0.897616i	$0.645298\pi$
$720$	0	0
$721$	−23.2287	−0.865083
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.890804	0.0330836
$726$	0	0
$727$	12.5513	0.465502	0.232751	−	0.972536i	$-0.425227\pi$
$727$	12.5513	0.465502	0.232751	+	0.972536i	$0.425227\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.09196	0.114360
$732$	0	0
$733$	39.1142	1.44472	0.722359	−	0.691519i	$-0.243058\pi$
$733$	39.1142	1.44472	0.722359	+	0.691519i	$0.243058\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−37.4760	−1.38045
$738$	0	0
$739$	35.7456	1.31492	0.657461	−	0.753489i	$-0.271630\pi$
$739$	35.7456	1.31492	0.657461	+	0.753489i	$0.271630\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20.4368	−0.749753	−0.374876	−	0.927075i	$-0.622315\pi$
$743$	−20.4368	−0.749753	−0.374876	+	0.927075i	$0.622315\pi$
$744$	0	0
$745$	−1.15531	−0.0423274
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−55.2799	−2.01988
$750$	0	0
$751$	−0.534057	−0.0194880	−0.00974401	−	0.999953i	$-0.503102\pi$
$751$	−0.534057	−0.0194880	−0.00974401	+	0.999953i	$0.503102\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.4710	−0.563047
$756$	0	0
$757$	−29.5171	−1.07282	−0.536409	−	0.843958i	$-0.680219\pi$
$757$	−29.5171	−1.07282	−0.536409	+	0.843958i	$0.680219\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−38.1434	−1.38270	−0.691348	−	0.722522i	$-0.742983\pi$
$761$	−38.1434	−1.38270	−0.691348	+	0.722522i	$0.742983\pi$
$762$	0	0
$763$	46.8208	1.69503
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.53406	0.0914995
$768$	0	0
$769$	20.8786	0.752902	0.376451	−	0.926437i	$-0.377144\pi$
$769$	20.8786	0.752902	0.376451	+	0.926437i	$0.377144\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.31063	−0.298913	−0.149456	−	0.988768i	$-0.547752\pi$
$773$	−8.31063	−0.298913	−0.149456	+	0.988768i	$0.547752\pi$
$774$	0	0
$775$	7.73549	0.277867
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−84.6125	−3.03156
$780$	0	0
$781$	−59.8890	−2.14300
$782$	0	0
$783$	0	0
$784$	0	0
$785$	18.6774	0.666627
$786$	0	0
$787$	−1.95388	−0.0696484	−0.0348242	−	0.999393i	$-0.511087\pi$
$787$	−1.95388	−0.0696484	−0.0348242	+	0.999393i	$0.511087\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	46.1773	1.64188
$792$	0	0
$793$	−24.5391	−0.871409
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.1773	0.573030	0.286515	−	0.958076i	$-0.407503\pi$
$797$	16.1773	0.573030	0.286515	+	0.958076i	$0.407503\pi$
$798$	0	0
$799$	−5.90273	−0.208823
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−80.1023	−2.82675
$804$	0	0
$805$	3.73549	0.131659
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26.8670	0.944592	0.472296	−	0.881440i	$-0.343425\pi$
$809$	26.8670	0.944592	0.472296	+	0.881440i	$0.343425\pi$
$810$	0	0
$811$	2.13835	0.0750878	0.0375439	−	0.999295i	$-0.488047\pi$
$811$	2.13835	0.0750878	0.0375439	+	0.999295i	$0.488047\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.0922364	0.00323090
$816$	0	0
$817$	−23.7578	−0.831179
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.1604	−0.529100	−0.264550	−	0.964372i	$-0.585223\pi$
$821$	−15.1604	−0.529100	−0.264550	+	0.964372i	$0.585223\pi$
$822$	0	0
$823$	−23.1264	−0.806137	−0.403068	−	0.915170i	$-0.632056\pi$
$823$	−23.1264	−0.806137	−0.403068	+	0.915170i	$0.632056\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−47.4299	−1.64930	−0.824650	−	0.565644i	$-0.808628\pi$
$827$	−47.4299	−1.64930	−0.824650	+	0.565644i	$0.808628\pi$
$828$	0	0
$829$	−30.1723	−1.04793	−0.523963	−	0.851741i	$-0.675547\pi$
$829$	−30.1723	−1.04793	−0.523963	+	0.851741i	$0.675547\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.19454	−0.214628
$834$	0	0
$835$	5.37371	0.185965
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18.8497	−0.650765	−0.325382	−	0.945583i	$-0.605493\pi$
$839$	−18.8497	−0.650765	−0.325382	+	0.945583i	$0.605493\pi$
$840$	0	0
$841$	−28.2065	−0.972637
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.90776	−0.168832
$846$	0	0
$847$	−46.5852	−1.60069
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.95388	0.0669782
$852$	0	0
$853$	29.7578	1.01889	0.509443	−	0.860504i	$-0.329851\pi$
$853$	29.7578	1.01889	0.509443	+	0.860504i	$0.329851\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.2865	0.522177	0.261089	−	0.965315i	$-0.415919\pi$
$857$	15.2865	0.522177	0.261089	+	0.965315i	$0.415919\pi$
$858$	0	0
$859$	33.5171	1.14359	0.571794	−	0.820397i	$-0.306248\pi$
$859$	33.5171	1.14359	0.571794	+	0.820397i	$0.306248\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.0443	1.05676	0.528380	−	0.849008i	$-0.322800\pi$
$863$	31.0443	1.05676	0.528380	+	0.849008i	$0.322800\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−65.2627	−2.21388
$870$	0	0
$871$	−22.0050	−0.745612
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.73549	−0.126283
$876$	0	0
$877$	10.6313	0.358994	0.179497	−	0.983758i	$-0.442553\pi$
$877$	10.6313	0.358994	0.179497	+	0.983758i	$0.442553\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−20.5341	−0.691810	−0.345905	−	0.938270i	$-0.612428\pi$
$881$	−20.5341	−0.691810	−0.345905	+	0.938270i	$0.612428\pi$
$882$	0	0
$883$	36.0390	1.21281	0.606404	−	0.795157i	$-0.292612\pi$
$883$	36.0390	1.21281	0.606404	+	0.795157i	$0.292612\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.9759	0.838608	0.419304	−	0.907846i	$-0.362274\pi$
$887$	24.9759	0.838608	0.419304	+	0.907846i	$0.362274\pi$
$888$	0	0
$889$	−44.0289	−1.47668
$890$	0	0
$891$	0	0
$892$	0	0
$893$	45.3549	1.51774
$894$	0	0
$895$	9.56322	0.319663
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.89080	0.229821
$900$	0	0
$901$	11.0119	0.366860
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.16035	−0.238018
$906$	0	0
$907$	41.6194	1.38195	0.690975	−	0.722879i	$-0.257182\pi$
$907$	41.6194	1.38195	0.690975	+	0.722879i	$0.257182\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.81553	−0.126414	−0.0632070	−	0.998000i	$-0.520133\pi$
$911$	−3.81553	−0.126414	−0.0632070	+	0.998000i	$0.520133\pi$
$912$	0	0
$913$	−22.1895	−0.734366
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−65.2627	−2.15516
$918$	0	0
$919$	−39.7917	−1.31261	−0.656303	−	0.754497i	$-0.727881\pi$
$919$	−39.7917	−1.31261	−0.656303	+	0.754497i	$0.727881\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−35.1654	−1.15748
$924$	0	0
$925$	−1.95388	−0.0642432
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−12.6141	−0.413855	−0.206928	−	0.978356i	$-0.566346\pi$
$929$	−12.6141	−0.413855	−0.206928	+	0.978356i	$0.566346\pi$
$930$	0	0
$931$	47.5971	1.55993
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.31566	0.141137
$936$	0	0
$937$	−22.1262	−0.722830	−0.361415	−	0.932405i	$-0.617706\pi$
$937$	−22.1262	−0.722830	−0.361415	+	0.932405i	$0.617706\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11.0392	0.359869	0.179934	−	0.983679i	$-0.442411\pi$
$941$	11.0392	0.359869	0.179934	+	0.983679i	$0.442411\pi$
$942$	0	0
$943$	12.3618	0.402555
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.68937	−0.314862	−0.157431	−	0.987530i	$-0.550321\pi$
$947$	−9.68937	−0.314862	−0.157431	+	0.987530i	$0.550321\pi$
$948$	0	0
$949$	−47.0342	−1.52679
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14.1845	0.459480	0.229740	−	0.973252i	$-0.426212\pi$
$953$	14.1845	0.459480	0.229740	+	0.973252i	$0.426212\pi$
$954$	0	0
$955$	−14.0050	−0.453192
$956$	0	0
$957$	0	0
$958$	0	0
$959$	56.6313	1.82872
$960$	0	0
$961$	28.8378	0.930252
$962$	0	0
$963$	0	0
$964$	0	0
$965$	26.7236	0.860262
$966$	0	0
$967$	12.8447	0.413057	0.206529	−	0.978441i	$-0.433783\pi$
$967$	12.8447	0.413057	0.206529	+	0.978441i	$0.433783\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.1945	0.776440	0.388220	−	0.921567i	$-0.373090\pi$
$971$	24.1945	0.776440	0.388220	+	0.921567i	$0.373090\pi$
$972$	0	0
$973$	20.6091	0.660696
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.4879	−0.591482	−0.295741	−	0.955268i	$-0.595566\pi$
$977$	−18.4879	−0.591482	−0.295741	+	0.955268i	$0.595566\pi$
$978$	0	0
$979$	−73.4471	−2.34738
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.8325	0.919614	0.459807	−	0.888019i	$-0.347919\pi$
$983$	28.8325	0.919614	0.459807	+	0.888019i	$0.347919\pi$
$984$	0	0
$985$	−9.16035	−0.291873
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.47098	0.110371
$990$	0	0
$991$	40.8958	1.29910	0.649550	−	0.760319i	$-0.274957\pi$
$991$	40.8958	1.29910	0.649550	+	0.760319i	$0.274957\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−0.310629	−0.00984759
$996$	0	0
$997$	−4.65518	−0.147431	−0.0737155	−	0.997279i	$-0.523486\pi$
$997$	−4.65518	−0.147431	−0.0737155	+	0.997279i	$0.523486\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.a.s.1.1		3
3.2	odd	2		1380.2.a.j.1.1	✓	3
12.11	even	2		5520.2.a.bv.1.3		3
15.2	even	4		6900.2.f.r.6349.1		6
15.8	even	4		6900.2.f.r.6349.6		6
15.14	odd	2		6900.2.a.x.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.j.1.1	✓	3	3.2	odd	2
4140.2.a.s.1.1		3	1.1	even	1	trivial
5520.2.a.bv.1.3		3	12.11	even	2
6900.2.a.x.1.3		3	15.14	odd	2
6900.2.f.r.6349.1		6	15.2	even	4
6900.2.f.r.6349.6		6	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.73549 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -3.73549 q^{7} -4.84469 q^{11} -2.84469 q^{13} -0.890804 q^{17} +6.84469 q^{19} -1.00000 q^{23} +1.00000 q^{25} +0.890804 q^{29} +7.73549 q^{31} -3.73549 q^{35} -1.95388 q^{37} -12.3618 q^{41} -3.47098 q^{43} +6.62629 q^{47} +6.95388 q^{49} -12.3618 q^{53} -4.84469 q^{55} -0.890804 q^{59} +8.62629 q^{61} -2.84469 q^{65} +7.73549 q^{67} +12.3618 q^{71} +16.5341 q^{73} +18.0973 q^{77} +13.4710 q^{79} +4.58018 q^{83} -0.890804 q^{85} +15.1604 q^{89} +10.6263 q^{91} +6.84469 q^{95} -17.2526 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 2 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 + 2 * q^7	\( 3 q + 3 q^{5} + 2 q^{7} - 4 q^{11} + 2 q^{13} + 10 q^{19} - 3 q^{23} + 3 q^{25} + 10 q^{31} + 2 q^{35} + 2 q^{37} - 8 q^{41} + 16 q^{43} + 4 q^{47} + 13 q^{49} - 8 q^{53} - 4 q^{55} + 10 q^{61} + 2 q^{65} + 10 q^{67} + 8 q^{71} + 18 q^{73} + 12 q^{77} + 14 q^{79} - 10 q^{83} - 2 q^{89} + 16 q^{91} + 10 q^{95} - 20 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 + 2 * q^7 - 4 * q^11 + 2 * q^13 + 10 * q^19 - 3 * q^23 + 3 * q^25 + 10 * q^31 + 2 * q^35 + 2 * q^37 - 8 * q^41 + 16 * q^43 + 4 * q^47 + 13 * q^49 - 8 * q^53 - 4 * q^55 + 10 * q^61 + 2 * q^65 + 10 * q^67 + 8 * q^71 + 18 * q^73 + 12 * q^77 + 14 * q^79 - 10 * q^83 - 2 * q^89 + 16 * q^91 + 10 * q^95 - 20 * q^97

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