LMFDB - Embedded newform 6048.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(1,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, 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("6048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.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:	$48.2935231425$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.22896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 4x + 3 $ x^4 - 9x^2 - 4x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.15690$ of defining polynomial
Character	$\chi$	$=$	6048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.766021 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-0.766021 q^{5} +1.00000 q^{7} +5.54779 q^{11} +6.31381 q^{13} -7.15032 q^{17} -6.15032 q^{19} -7.17923 q^{23} -4.41321 q^{25} -3.16349 q^{29} -0.163489 q^{31} -0.766021 q^{35} +5.31381 q^{37} -0.865425 q^{41} -8.31381 q^{43} -2.36855 q^{47} +1.00000 q^{49} -4.00000 q^{53} -4.24972 q^{55} -2.36855 q^{59} -13.5635 q^{61} -4.83651 q^{65} +7.05092 q^{67} -7.86159 q^{71} -1.98683 q^{73} +5.54779 q^{77} +5.24972 q^{79} +12.5282 q^{83} +5.47730 q^{85} -4.86542 q^{89} +6.31381 q^{91} +4.71128 q^{95} +3.26289 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q - 2 * q^5 + 4 * q^7	$ 4 q - 2 q^{5} + 4 q^{7} - 2 q^{11} - 8 q^{17} - 4 q^{19} - 2 q^{23} + 8 q^{25} - 8 q^{29} + 4 q^{31} - 2 q^{35} - 4 q^{37} - 2 q^{41} - 8 q^{43} - 12 q^{47} + 4 q^{49} - 16 q^{53} + 4 q^{55} - 12 q^{59} - 8 q^{61} - 24 q^{65} + 8 q^{67} + 18 q^{71} + 8 q^{73} - 2 q^{77} - 8 q^{85} - 18 q^{89} - 10 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 4 * q^7 - 2 * q^11 - 8 * q^17 - 4 * q^19 - 2 * q^23 + 8 * q^25 - 8 * q^29 + 4 * q^31 - 2 * q^35 - 4 * q^37 - 2 * q^41 - 8 * q^43 - 12 * q^47 + 4 * q^49 - 16 * q^53 + 4 * q^55 - 12 * q^59 - 8 * q^61 - 24 * q^65 + 8 * q^67 + 18 * q^71 + 8 * q^73 - 2 * q^77 - 8 * q^85 - 18 * q^89 - 10 * q^95 + 8 * 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
$5$	−0.766021	−0.342575	−0.171288	−	0.985221i	$-0.554793\pi$
$5$	−0.766021	−0.342575	−0.171288	+	0.985221i	$0.554793\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.54779	1.67272	0.836360	−	0.548180i	$-0.184679\pi$
$11$	5.54779	1.67272	0.836360	+	0.548180i	$0.184679\pi$
$12$	0	0
$13$	6.31381	1.75114	0.875568	−	0.483096i	$-0.160488\pi$
$13$	6.31381	1.75114	0.875568	+	0.483096i	$0.160488\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.15032	−1.73421	−0.867104	−	0.498128i	$-0.834021\pi$
$17$	−7.15032	−1.73421	−0.867104	+	0.498128i	$0.834021\pi$
$18$	0	0
$19$	−6.15032	−1.41098	−0.705490	−	0.708720i	$-0.749273\pi$
$19$	−6.15032	−1.41098	−0.705490	+	0.708720i	$0.749273\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.17923	−1.49697	−0.748487	−	0.663150i	$-0.769219\pi$
$23$	−7.17923	−1.49697	−0.748487	+	0.663150i	$0.769219\pi$
$24$	0	0
$25$	−4.41321	−0.882642
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.16349	−0.587445	−0.293723	−	0.955891i	$-0.594894\pi$
$29$	−3.16349	−0.587445	−0.293723	+	0.955891i	$0.594894\pi$
$30$	0	0
$31$	−0.163489	−0.0293635	−0.0146817	−	0.999892i	$-0.504674\pi$
$31$	−0.163489	−0.0293635	−0.0146817	+	0.999892i	$0.504674\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.766021	−0.129481
$36$	0	0
$37$	5.31381	0.873585	0.436792	−	0.899562i	$-0.356114\pi$
$37$	5.31381	0.873585	0.436792	+	0.899562i	$0.356114\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.865425	−0.135157	−0.0675783	−	0.997714i	$-0.521527\pi$
$41$	−0.865425	−0.135157	−0.0675783	+	0.997714i	$0.521527\pi$
$42$	0	0
$43$	−8.31381	−1.26784	−0.633922	−	0.773397i	$-0.718556\pi$
$43$	−8.31381	−1.26784	−0.633922	+	0.773397i	$0.718556\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.36855	−0.345489	−0.172745	−	0.984967i	$-0.555264\pi$
$47$	−2.36855	−0.345489	−0.172745	+	0.984967i	$0.555264\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−4.24972	−0.573032
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.36855	−0.308359	−0.154180	−	0.988043i	$-0.549273\pi$
$59$	−2.36855	−0.308359	−0.154180	+	0.988043i	$0.549273\pi$
$60$	0	0
$61$	−13.5635	−1.73663	−0.868316	−	0.496011i	$-0.834797\pi$
$61$	−13.5635	−1.73663	−0.868316	+	0.496011i	$0.834797\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.83651	−0.599895
$66$	0	0
$67$	7.05092	0.861406	0.430703	−	0.902494i	$-0.358266\pi$
$67$	7.05092	0.861406	0.430703	+	0.902494i	$0.358266\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.86159	−0.933000	−0.466500	−	0.884521i	$-0.654485\pi$
$71$	−7.86159	−0.933000	−0.466500	+	0.884521i	$0.654485\pi$
$72$	0	0
$73$	−1.98683	−0.232541	−0.116270	−	0.993218i	$-0.537094\pi$
$73$	−1.98683	−0.232541	−0.116270	+	0.993218i	$0.537094\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.54779	0.632229
$78$	0	0
$79$	5.24972	0.590640	0.295320	−	0.955398i	$-0.404574\pi$
$79$	5.24972	0.590640	0.295320	+	0.955398i	$0.404574\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.5282	1.37515	0.687575	−	0.726113i	$-0.258675\pi$
$83$	12.5282	1.37515	0.687575	+	0.726113i	$0.258675\pi$
$84$	0	0
$85$	5.47730	0.594096
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.86542	−0.515734	−0.257867	−	0.966180i	$-0.583020\pi$
$89$	−4.86542	−0.515734	−0.257867	+	0.966180i	$0.583020\pi$
$90$	0	0
$91$	6.31381	0.661867
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.71128	0.483367
$96$	0	0
$97$	3.26289	0.331297	0.165648	−	0.986185i	$-0.447028\pi$
$97$	3.26289	0.331297	0.165648	+	0.986185i	$0.447028\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.1503	−1.50751	−0.753757	−	0.657154i	$-0.771760\pi$
$101$	−15.1503	−1.50751	−0.753757	+	0.657154i	$0.771760\pi$
$102$	0	0
$103$	1.42638	0.140546	0.0702728	−	0.997528i	$-0.477613\pi$
$103$	1.42638	0.140546	0.0702728	+	0.997528i	$0.477613\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.2459	−1.37720	−0.688601	−	0.725140i	$-0.741775\pi$
$107$	−14.2459	−1.37720	−0.688601	+	0.725140i	$0.741775\pi$
$108$	0	0
$109$	−17.7270	−1.69794	−0.848970	−	0.528441i	$-0.822777\pi$
$109$	−17.7270	−1.69794	−0.848970	+	0.528441i	$0.822777\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.09557	0.667495	0.333748	−	0.942662i	$-0.391687\pi$
$113$	7.09557	0.667495	0.333748	+	0.942662i	$0.391687\pi$
$114$	0	0
$115$	5.49945	0.512826
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.15032	−0.655469
$120$	0	0
$121$	19.7779	1.79799
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.21072	0.644946
$126$	0	0
$127$	5.37789	0.477211	0.238605	−	0.971117i	$-0.423310\pi$
$127$	5.37789	0.477211	0.238605	+	0.971117i	$0.423310\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.69553	−0.759732	−0.379866	−	0.925042i	$-0.624030\pi$
$131$	−8.69553	−0.759732	−0.379866	+	0.925042i	$0.624030\pi$
$132$	0	0
$133$	−6.15032	−0.533300
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0917	1.54568	0.772841	−	0.634600i	$-0.218835\pi$
$137$	18.0917	1.54568	0.772841	+	0.634600i	$0.218835\pi$
$138$	0	0
$139$	11.8905	1.00854	0.504270	−	0.863546i	$-0.331762\pi$
$139$	11.8905	1.00854	0.504270	+	0.863546i	$0.331762\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	35.0277	2.92916
$144$	0	0
$145$	2.42330	0.201244
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.8327	−1.62476	−0.812378	−	0.583131i	$-0.801827\pi$
$149$	−19.8327	−1.62476	−0.812378	+	0.583131i	$0.801827\pi$
$150$	0	0
$151$	15.5635	1.26654	0.633271	−	0.773930i	$-0.281712\pi$
$151$	15.5635	1.26654	0.633271	+	0.773930i	$0.281712\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.125236	0.0100592
$156$	0	0
$157$	−12.6408	−1.00884	−0.504422	−	0.863457i	$-0.668294\pi$
$157$	−12.6408	−1.00884	−0.504422	+	0.863457i	$0.668294\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.17923	−0.565803
$162$	0	0
$163$	24.6144	1.92795	0.963976	−	0.265989i	$-0.0856985\pi$
$163$	24.6144	1.92795	0.963976	+	0.265989i	$0.0856985\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.09557	0.239543	0.119771	−	0.992802i	$-0.461784\pi$
$167$	3.09557	0.239543	0.119771	+	0.992802i	$0.461784\pi$
$168$	0	0
$169$	26.8642	2.06647
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15.9163	−1.21010	−0.605048	−	0.796189i	$-0.706846\pi$
$173$	−15.9163	−1.21010	−0.605048	+	0.796189i	$0.706846\pi$
$174$	0	0
$175$	−4.41321	−0.333607
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.6824	1.24690	0.623449	−	0.781864i	$-0.285731\pi$
$179$	16.6824	1.24690	0.623449	+	0.781864i	$0.285731\pi$
$180$	0	0
$181$	17.8642	1.32783	0.663917	−	0.747807i	$-0.268893\pi$
$181$	17.8642	1.32783	0.663917	+	0.747807i	$0.268893\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.07049	−0.299268
$186$	0	0
$187$	−39.6684	−2.90084
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.58936	0.115002	0.0575011	−	0.998345i	$-0.481687\pi$
$191$	1.58936	0.115002	0.0575011	+	0.998345i	$0.481687\pi$
$192$	0	0
$193$	−7.36472	−0.530124	−0.265062	−	0.964231i	$-0.585392\pi$
$193$	−7.36472	−0.530124	−0.265062	+	0.964231i	$0.585392\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.73085	−0.693294	−0.346647	−	0.937996i	$-0.612680\pi$
$197$	−9.73085	−0.693294	−0.346647	+	0.937996i	$0.612680\pi$
$198$	0	0
$199$	−8.37789	−0.593893	−0.296947	−	0.954894i	$-0.595968\pi$
$199$	−8.37789	−0.593893	−0.296947	+	0.954894i	$0.595968\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.16349	−0.222033
$204$	0	0
$205$	0.662934	0.0463013
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−34.1207	−2.36018
$210$	0	0
$211$	−5.23655	−0.360499	−0.180250	−	0.983621i	$-0.557691\pi$
$211$	−5.23655	−0.360499	−0.180250	+	0.983621i	$0.557691\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.36855	0.434332
$216$	0	0
$217$	−0.163489	−0.0110983
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−45.1457	−3.03683
$222$	0	0
$223$	−7.31381	−0.489769	−0.244884	−	0.969552i	$-0.578750\pi$
$223$	−7.31381	−0.489769	−0.244884	+	0.969552i	$0.578750\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.76860	−0.581992	−0.290996	−	0.956724i	$-0.593987\pi$
$227$	−8.76860	−0.581992	−0.290996	+	0.956724i	$0.593987\pi$
$228$	0	0
$229$	−19.6917	−1.30126	−0.650632	−	0.759393i	$-0.725496\pi$
$229$	−19.6917	−1.30126	−0.650632	+	0.759393i	$0.725496\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.3585	−1.20270	−0.601352	−	0.798985i	$-0.705371\pi$
$233$	−18.3585	−1.20270	−0.601352	+	0.798985i	$0.705371\pi$
$234$	0	0
$235$	1.81436	0.118356
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.708701	0.0458421	0.0229210	−	0.999737i	$-0.492703\pi$
$239$	0.708701	0.0458421	0.0229210	+	0.999737i	$0.492703\pi$
$240$	0	0
$241$	4.42330	0.284930	0.142465	−	0.989800i	$-0.454497\pi$
$241$	4.42330	0.284930	0.142465	+	0.989800i	$0.454497\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.766021	−0.0489393
$246$	0	0
$247$	−38.8319	−2.47082
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.9546	1.19640	0.598202	−	0.801346i	$-0.295882\pi$
$251$	18.9546	1.19640	0.598202	+	0.801346i	$0.295882\pi$
$252$	0	0
$253$	−39.8288	−2.50402
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.2849	−1.14058	−0.570290	−	0.821443i	$-0.693169\pi$
$257$	−18.2849	−1.14058	−0.570290	+	0.821443i	$0.693169\pi$
$258$	0	0
$259$	5.31381	0.330184
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.5440	−1.02014	−0.510072	−	0.860132i	$-0.670381\pi$
$263$	−16.5440	−1.02014	−0.510072	+	0.860132i	$0.670381\pi$
$264$	0	0
$265$	3.06409	0.188225
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0.852255	0.0519629	0.0259815	−	0.999662i	$-0.491729\pi$
$269$	0.852255	0.0519629	0.0259815	+	0.999662i	$0.491729\pi$
$270$	0	0
$271$	18.9283	1.14981	0.574905	−	0.818220i	$-0.305039\pi$
$271$	18.9283	1.14981	0.574905	+	0.818220i	$0.305039\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−24.4836	−1.47641
$276$	0	0
$277$	14.8773	0.893893	0.446946	−	0.894561i	$-0.352511\pi$
$277$	14.8773	0.893893	0.446946	+	0.894561i	$0.352511\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.6692	−0.755781	−0.377890	−	0.925850i	$-0.623350\pi$
$281$	−12.6692	−0.755781	−0.377890	+	0.925850i	$0.623350\pi$
$282$	0	0
$283$	−7.06409	−0.419916	−0.209958	−	0.977710i	$-0.567333\pi$
$283$	−7.06409	−0.419916	−0.209958	+	0.977710i	$0.567333\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.865425	−0.0510844
$288$	0	0
$289$	34.1271	2.00747
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.27849	0.0746901	0.0373451	−	0.999302i	$-0.488110\pi$
$293$	1.27849	0.0746901	0.0373451	+	0.999302i	$0.488110\pi$
$294$	0	0
$295$	1.81436	0.105636
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−45.3283	−2.62140
$300$	0	0
$301$	−8.31381	−0.479200
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.3900	0.594927
$306$	0	0
$307$	−32.2320	−1.83958	−0.919788	−	0.392416i	$-0.871639\pi$
$307$	−32.2320	−1.83958	−0.919788	+	0.392416i	$0.871639\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.36230	−0.190658	−0.0953292	−	0.995446i	$-0.530390\pi$
$311$	−3.36230	−0.190658	−0.0953292	+	0.995446i	$0.530390\pi$
$312$	0	0
$313$	−4.72394	−0.267013	−0.133506	−	0.991048i	$-0.542624\pi$
$313$	−4.72394	−0.267013	−0.133506	+	0.991048i	$0.542624\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.2327	0.574727	0.287363	−	0.957822i	$-0.407221\pi$
$317$	10.2327	0.574727	0.287363	+	0.957822i	$0.407221\pi$
$318$	0	0
$319$	−17.5504	−0.982632
$320$	0	0
$321$	0	0
$322$	0	0
$323$	43.9767	2.44693
$324$	0	0
$325$	−27.8642	−1.54563
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.36855	−0.130583
$330$	0	0
$331$	−3.80119	−0.208932	−0.104466	−	0.994528i	$-0.533313\pi$
$331$	−3.80119	−0.208932	−0.104466	+	0.994528i	$0.533313\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.40115	−0.295096
$336$	0	0
$337$	−11.1911	−0.609621	−0.304810	−	0.952413i	$-0.598593\pi$
$337$	−11.1911	−0.609621	−0.304810	+	0.952413i	$0.598593\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.907001	−0.0491169
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.66036	−0.303864	−0.151932	−	0.988391i	$-0.548549\pi$
$347$	−5.66036	−0.303864	−0.151932	+	0.988391i	$0.548549\pi$
$348$	0	0
$349$	−30.6276	−1.63946	−0.819729	−	0.572751i	$-0.805876\pi$
$349$	−30.6276	−1.63946	−0.819729	+	0.572751i	$0.805876\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−29.3521	−1.56225	−0.781126	−	0.624373i	$-0.785355\pi$
$353$	−29.3521	−1.56225	−0.781126	+	0.624373i	$0.785355\pi$
$354$	0	0
$355$	6.02215	0.319622
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.5351	0.714357	0.357178	−	0.934036i	$-0.383739\pi$
$359$	13.5351	0.714357	0.357178	+	0.934036i	$0.383739\pi$
$360$	0	0
$361$	18.8264	0.990864
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.52195	0.0796627
$366$	0	0
$367$	−9.74262	−0.508560	−0.254280	−	0.967131i	$-0.581839\pi$
$367$	−9.74262	−0.508560	−0.254280	+	0.967131i	$0.581839\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	−16.3360	−0.845844	−0.422922	−	0.906166i	$-0.638996\pi$
$373$	−16.3360	−0.845844	−0.422922	+	0.906166i	$0.638996\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.9737	−1.02870
$378$	0	0
$379$	32.9414	1.69209	0.846044	−	0.533114i	$-0.178978\pi$
$379$	32.9414	1.69209	0.846044	+	0.533114i	$0.178978\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−31.2944	−1.59907	−0.799534	−	0.600621i	$-0.794920\pi$
$383$	−31.2944	−1.59907	−0.799534	+	0.600621i	$0.794920\pi$
$384$	0	0
$385$	−4.24972	−0.216586
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.39106	0.273338	0.136669	−	0.990617i	$-0.456360\pi$
$389$	5.39106	0.273338	0.136669	+	0.990617i	$0.456360\pi$
$390$	0	0
$391$	51.3338	2.59606
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.02140	−0.202339
$396$	0	0
$397$	24.0817	1.20862	0.604312	−	0.796748i	$-0.293448\pi$
$397$	24.0817	1.20862	0.604312	+	0.796748i	$0.293448\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.2642	1.51132	0.755661	−	0.654963i	$-0.227316\pi$
$401$	30.2642	1.51132	0.755661	+	0.654963i	$0.227316\pi$
$402$	0	0
$403$	−1.03224	−0.0514194
$404$	0	0
$405$	0	0
$406$	0	0
$407$	29.4799	1.46126
$408$	0	0
$409$	−24.0055	−1.18700	−0.593498	−	0.804835i	$-0.702254\pi$
$409$	−24.0055	−1.18700	−0.593498	+	0.804835i	$0.702254\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.36855	−0.116549
$414$	0	0
$415$	−9.59688	−0.471092
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−21.7308	−1.06162	−0.530811	−	0.847490i	$-0.678112\pi$
$419$	−21.7308	−1.06162	−0.530811	+	0.847490i	$0.678112\pi$
$420$	0	0
$421$	18.0246	0.878464	0.439232	−	0.898374i	$-0.355251\pi$
$421$	18.0246	0.878464	0.439232	+	0.898374i	$0.355251\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	31.5559	1.53068
$426$	0	0
$427$	−13.5635	−0.656385
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.8523	1.00442	0.502209	−	0.864746i	$-0.332521\pi$
$431$	20.8523	1.00442	0.502209	+	0.864746i	$0.332521\pi$
$432$	0	0
$433$	−2.20432	−0.105933	−0.0529663	−	0.998596i	$-0.516868\pi$
$433$	−2.20432	−0.105933	−0.0529663	+	0.998596i	$0.516868\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	44.1546	2.11220
$438$	0	0
$439$	36.6013	1.74688	0.873442	−	0.486929i	$-0.161883\pi$
$439$	36.6013	1.74688	0.873442	+	0.486929i	$0.161883\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	28.5075	1.35443	0.677217	−	0.735784i	$-0.263186\pi$
$443$	28.5075	1.35443	0.677217	+	0.735784i	$0.263186\pi$
$444$	0	0
$445$	3.72702	0.176678
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−39.7232	−1.87465	−0.937327	−	0.348452i	$-0.886707\pi$
$449$	−39.7232	−1.87465	−0.937327	+	0.348452i	$0.886707\pi$
$450$	0	0
$451$	−4.80119	−0.226079
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.83651	−0.226739
$456$	0	0
$457$	−10.6917	−0.500137	−0.250068	−	0.968228i	$-0.580453\pi$
$457$	−10.6917	−0.500137	−0.250068	+	0.968228i	$0.580453\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.7113	0.592023	0.296012	−	0.955184i	$-0.404343\pi$
$461$	12.7113	0.592023	0.296012	+	0.955184i	$0.404343\pi$
$462$	0	0
$463$	14.6276	0.679803	0.339901	−	0.940461i	$-0.389606\pi$
$463$	14.6276	0.679803	0.339901	+	0.940461i	$0.389606\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.7963	1.65645	0.828227	−	0.560393i	$-0.189350\pi$
$467$	35.7963	1.65645	0.828227	+	0.560393i	$0.189350\pi$
$468$	0	0
$469$	7.05092	0.325581
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−46.1232	−2.12075
$474$	0	0
$475$	27.1427	1.24539
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.3193	−0.928413	−0.464207	−	0.885727i	$-0.653660\pi$
$479$	−20.3193	−0.928413	−0.464207	+	0.885727i	$0.653660\pi$
$480$	0	0
$481$	33.5504	1.52976
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.49945	−0.113494
$486$	0	0
$487$	−27.4409	−1.24346	−0.621732	−	0.783230i	$-0.713571\pi$
$487$	−27.4409	−1.24346	−0.621732	+	0.783230i	$0.713571\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.52564	−0.204239	−0.102120	−	0.994772i	$-0.532562\pi$
$491$	−4.52564	−0.204239	−0.102120	+	0.994772i	$0.532562\pi$
$492$	0	0
$493$	22.6200	1.01875
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.86159	−0.352641
$498$	0	0
$499$	−1.87734	−0.0840412	−0.0420206	−	0.999117i	$-0.513380\pi$
$499$	−1.87734	−0.0840412	−0.0420206	+	0.999117i	$0.513380\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−37.7834	−1.68468	−0.842340	−	0.538946i	$-0.818823\pi$
$503$	−37.7834	−1.68468	−0.842340	+	0.538946i	$0.818823\pi$
$504$	0	0
$505$	11.6055	0.516436
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−21.3678	−0.947111	−0.473556	−	0.880764i	$-0.657030\pi$
$509$	−21.3678	−0.947111	−0.473556	+	0.880764i	$0.657030\pi$
$510$	0	0
$511$	−1.98683	−0.0878922
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.09264	−0.0481474
$516$	0	0
$517$	−13.1402	−0.577907
$518$	0	0
$519$	0	0
$520$	0	0
$521$	31.8484	1.39530	0.697652	−	0.716436i	$-0.254228\pi$
$521$	31.8484	1.39530	0.697652	+	0.716436i	$0.254228\pi$
$522$	0	0
$523$	14.2366	0.622521	0.311260	−	0.950325i	$-0.399249\pi$
$523$	14.2366	0.622521	0.311260	+	0.950325i	$0.399249\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.16900	0.0509223
$528$	0	0
$529$	28.5414	1.24093
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.46413	−0.236678
$534$	0	0
$535$	10.9127	0.471795
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.54779	0.238960
$540$	0	0
$541$	−18.5767	−0.798675	−0.399337	−	0.916804i	$-0.630760\pi$
$541$	−18.5767	−0.798675	−0.399337	+	0.916804i	$0.630760\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.5793	0.581672
$546$	0	0
$547$	6.10183	0.260895	0.130448	−	0.991455i	$-0.458359\pi$
$547$	6.10183	0.260895	0.130448	+	0.991455i	$0.458359\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	19.4565	0.828873
$552$	0	0
$553$	5.24972	0.223241
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.5309	−1.29364	−0.646819	−	0.762644i	$-0.723901\pi$
$557$	−30.5309	−1.29364	−0.646819	+	0.762644i	$0.723901\pi$
$558$	0	0
$559$	−52.4918	−2.22017
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15.9899	0.673894	0.336947	−	0.941524i	$-0.390606\pi$
$563$	15.9899	0.673894	0.336947	+	0.941524i	$0.390606\pi$
$564$	0	0
$565$	−5.43536	−0.228667
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.5057	−0.733877	−0.366939	−	0.930245i	$-0.619594\pi$
$569$	−17.5057	−0.733877	−0.366939	+	0.930245i	$0.619594\pi$
$570$	0	0
$571$	30.4918	1.27604	0.638021	−	0.770019i	$-0.279753\pi$
$571$	30.4918	1.27604	0.638021	+	0.770019i	$0.279753\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	31.6835	1.32129
$576$	0	0
$577$	36.9283	1.53734	0.768672	−	0.639644i	$-0.220918\pi$
$577$	36.9283	1.53734	0.768672	+	0.639644i	$0.220918\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.5282	0.519758
$582$	0	0
$583$	−22.1911	−0.919063
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−45.2867	−1.86918	−0.934591	−	0.355723i	$-0.884235\pi$
$587$	−45.2867	−1.86918	−0.934591	+	0.355723i	$0.884235\pi$
$588$	0	0
$589$	1.00551	0.0414313
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.7203	0.891944	0.445972	−	0.895047i	$-0.352858\pi$
$593$	21.7203	0.891944	0.445972	+	0.895047i	$0.352858\pi$
$594$	0	0
$595$	5.47730	0.224547
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.5125	−0.838117	−0.419058	−	0.907959i	$-0.637640\pi$
$599$	−20.5125	−0.838117	−0.419058	+	0.907959i	$0.637640\pi$
$600$	0	0
$601$	−2.31381	−0.0943822	−0.0471911	−	0.998886i	$-0.515027\pi$
$601$	−2.31381	−0.0943822	−0.0471911	+	0.998886i	$0.515027\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−15.1503	−0.615948
$606$	0	0
$607$	21.3647	0.867167	0.433584	−	0.901113i	$-0.357249\pi$
$607$	21.3647	0.867167	0.433584	+	0.901113i	$0.357249\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−14.9546	−0.604998
$612$	0	0
$613$	0.163489	0.00660325	0.00330163	−	0.999995i	$-0.498949\pi$
$613$	0.163489	0.00660325	0.00330163	+	0.999995i	$0.498949\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17.5220	−0.705407	−0.352704	−	0.935735i	$-0.614738\pi$
$617$	−17.5220	−0.705407	−0.352704	+	0.935735i	$0.614738\pi$
$618$	0	0
$619$	−19.1988	−0.771665	−0.385833	−	0.922569i	$-0.626086\pi$
$619$	−19.1988	−0.771665	−0.385833	+	0.922569i	$0.626086\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.86542	−0.194929
$624$	0	0
$625$	16.5425	0.661700
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−37.9954	−1.51498
$630$	0	0
$631$	−1.18675	−0.0472437	−0.0236218	−	0.999721i	$-0.507520\pi$
$631$	−1.18675	−0.0472437	−0.0236218	+	0.999721i	$0.507520\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.11958	−0.163481
$636$	0	0
$637$	6.31381	0.250162
$638$	0	0
$639$	0	0
$640$	0	0
$641$	40.6591	1.60594	0.802969	−	0.596021i	$-0.203252\pi$
$641$	40.6591	1.60594	0.802969	+	0.596021i	$0.203252\pi$
$642$	0	0
$643$	10.5868	0.417502	0.208751	−	0.977969i	$-0.433060\pi$
$643$	10.5868	0.417502	0.208751	+	0.977969i	$0.433060\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.8303	1.01549	0.507746	−	0.861507i	$-0.330479\pi$
$647$	25.8303	1.01549	0.507746	+	0.861507i	$0.330479\pi$
$648$	0	0
$649$	−13.1402	−0.515799
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.2327	−0.870034	−0.435017	−	0.900422i	$-0.643258\pi$
$653$	−22.2327	−0.870034	−0.435017	+	0.900422i	$0.643258\pi$
$654$	0	0
$655$	6.66096	0.260265
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.9388	1.04939	0.524694	−	0.851291i	$-0.324180\pi$
$659$	26.9388	1.04939	0.524694	+	0.851291i	$0.324180\pi$
$660$	0	0
$661$	16.7301	0.650725	0.325363	−	0.945589i	$-0.394514\pi$
$661$	16.7301	0.650725	0.325363	+	0.945589i	$0.394514\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.71128	0.182695
$666$	0	0
$667$	22.7114	0.879390
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−75.2476	−2.90490
$672$	0	0
$673$	−3.36472	−0.129701	−0.0648503	−	0.997895i	$-0.520657\pi$
$673$	−3.36472	−0.129701	−0.0648503	+	0.997895i	$0.520657\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.1307	0.389356	0.194678	−	0.980867i	$-0.437634\pi$
$677$	10.1307	0.389356	0.194678	+	0.980867i	$0.437634\pi$
$678$	0	0
$679$	3.26289	0.125218
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25.7151	0.983961	0.491981	−	0.870606i	$-0.336273\pi$
$683$	25.7151	0.983961	0.491981	+	0.870606i	$0.336273\pi$
$684$	0	0
$685$	−13.8587	−0.529512
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−25.2552	−0.962148
$690$	0	0
$691$	8.51812	0.324045	0.162022	−	0.986787i	$-0.448198\pi$
$691$	8.51812	0.324045	0.162022	+	0.986787i	$0.448198\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.10838	−0.345501
$696$	0	0
$697$	6.18806	0.234390
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.62519	0.0991520	0.0495760	−	0.998770i	$-0.484213\pi$
$701$	2.62519	0.0991520	0.0495760	+	0.998770i	$0.484213\pi$
$702$	0	0
$703$	−32.6816	−1.23261
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15.1503	−0.569786
$708$	0	0
$709$	−36.2577	−1.36168	−0.680842	−	0.732430i	$-0.738386\pi$
$709$	−36.2577	−1.36168	−0.680842	+	0.732430i	$0.738386\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.17372	0.0439563
$714$	0	0
$715$	−26.8319	−1.00346
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−38.8603	−1.44925	−0.724623	−	0.689145i	$-0.757986\pi$
$719$	−38.8603	−1.44925	−0.724623	+	0.689145i	$0.757986\pi$
$720$	0	0
$721$	1.42638	0.0531212
$722$	0	0
$723$	0	0
$724$	0	0
$725$	13.9611	0.518504
$726$	0	0
$727$	15.7180	0.582950	0.291475	−	0.956578i	$-0.405854\pi$
$727$	15.7180	0.582950	0.291475	+	0.956578i	$0.405854\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	59.4464	2.19870
$732$	0	0
$733$	−14.7503	−0.544814	−0.272407	−	0.962182i	$-0.587820\pi$
$733$	−14.7503	−0.544814	−0.272407	+	0.962182i	$0.587820\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.1170	1.44089
$738$	0	0
$739$	−31.7935	−1.16954	−0.584772	−	0.811198i	$-0.698816\pi$
$739$	−31.7935	−1.16954	−0.584772	+	0.811198i	$0.698816\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.3673	1.51762	0.758809	−	0.651313i	$-0.225781\pi$
$743$	41.3673	1.51762	0.758809	+	0.651313i	$0.225781\pi$
$744$	0	0
$745$	15.1923	0.556601
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−14.2459	−0.520534
$750$	0	0
$751$	−8.42330	−0.307371	−0.153685	−	0.988120i	$-0.549114\pi$
$751$	−8.42330	−0.307371	−0.153685	+	0.988120i	$0.549114\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.9220	−0.433886
$756$	0	0
$757$	−27.8828	−1.01342	−0.506710	−	0.862117i	$-0.669138\pi$
$757$	−27.8828	−1.01342	−0.506710	+	0.862117i	$0.669138\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−17.3415	−0.628628	−0.314314	−	0.949319i	$-0.601774\pi$
$761$	−17.3415	−0.628628	−0.314314	+	0.949319i	$0.601774\pi$
$762$	0	0
$763$	−17.7270	−0.641761
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.9546	−0.539979
$768$	0	0
$769$	41.8960	1.51081	0.755404	−	0.655259i	$-0.227440\pi$
$769$	41.8960	1.51081	0.755404	+	0.655259i	$0.227440\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24.0289	0.864260	0.432130	−	0.901811i	$-0.357762\pi$
$773$	24.0289	0.864260	0.432130	+	0.901811i	$0.357762\pi$
$774$	0	0
$775$	0.721511	0.0259174
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.32264	0.190703
$780$	0	0
$781$	−43.6144	−1.56065
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.68311	0.345605
$786$	0	0
$787$	−37.9923	−1.35428	−0.677140	−	0.735854i	$-0.736781\pi$
$787$	−37.9923	−1.35428	−0.677140	+	0.735854i	$0.736781\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.09557	0.252290
$792$	0	0
$793$	−85.6375	−3.04108
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.5470	0.727813	0.363907	−	0.931435i	$-0.381443\pi$
$797$	20.5470	0.727813	0.363907	+	0.931435i	$0.381443\pi$
$798$	0	0
$799$	16.9359	0.599150
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11.0225	−0.388976
$804$	0	0
$805$	5.49945	0.193830
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.13715	0.250929	0.125464	−	0.992098i	$-0.459958\pi$
$809$	7.13715	0.250929	0.125464	+	0.992098i	$0.459958\pi$
$810$	0	0
$811$	37.6276	1.32128	0.660642	−	0.750701i	$-0.270284\pi$
$811$	37.6276	1.32128	0.660642	+	0.750701i	$0.270284\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18.8552	−0.660468
$816$	0	0
$817$	51.1326	1.78890
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.78485	−0.166992	−0.0834961	−	0.996508i	$-0.526609\pi$
$821$	−4.78485	−0.166992	−0.0834961	+	0.996508i	$0.526609\pi$
$822$	0	0
$823$	2.65396	0.0925111	0.0462555	−	0.998930i	$-0.485271\pi$
$823$	2.65396	0.0925111	0.0462555	+	0.998930i	$0.485271\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.2239	1.64214	0.821068	−	0.570831i	$-0.193379\pi$
$827$	47.2239	1.64214	0.821068	+	0.570831i	$0.193379\pi$
$828$	0	0
$829$	−12.3401	−0.428591	−0.214296	−	0.976769i	$-0.568746\pi$
$829$	−12.3401	−0.428591	−0.214296	+	0.976769i	$0.568746\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.15032	−0.247744
$834$	0	0
$835$	−2.37127	−0.0820613
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.18983	−0.248220	−0.124110	−	0.992268i	$-0.539608\pi$
$839$	−7.18983	−0.248220	−0.124110	+	0.992268i	$0.539608\pi$
$840$	0	0
$841$	−18.9923	−0.654908
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−20.5785	−0.707923
$846$	0	0
$847$	19.7779	0.679578
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−38.1491	−1.30773
$852$	0	0
$853$	−29.9923	−1.02692	−0.513459	−	0.858114i	$-0.671636\pi$
$853$	−29.9923	−1.02692	−0.513459	+	0.858114i	$0.671636\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−8.97492	−0.306577	−0.153289	−	0.988181i	$-0.548986\pi$
$857$	−8.97492	−0.306577	−0.153289	+	0.988181i	$0.548986\pi$
$858$	0	0
$859$	−44.5403	−1.51969	−0.759847	−	0.650102i	$-0.774726\pi$
$859$	−44.5403	−1.51969	−0.759847	+	0.650102i	$0.774726\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.7842	1.32023	0.660115	−	0.751165i	$-0.270508\pi$
$863$	38.7842	1.32023	0.660115	+	0.751165i	$0.270508\pi$
$864$	0	0
$865$	12.1923	0.414549
$866$	0	0
$867$	0	0
$868$	0	0
$869$	29.1243	0.987976
$870$	0	0
$871$	44.5181	1.50844
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7.21072	0.243767
$876$	0	0
$877$	−5.66051	−0.191142	−0.0955709	−	0.995423i	$-0.530468\pi$
$877$	−5.66051	−0.191142	−0.0955709	+	0.995423i	$0.530468\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26.7684	0.901852	0.450926	−	0.892561i	$-0.351094\pi$
$881$	26.7684	0.901852	0.450926	+	0.892561i	$0.351094\pi$
$882$	0	0
$883$	11.0377	0.371450	0.185725	−	0.982602i	$-0.440537\pi$
$883$	11.0377	0.371450	0.185725	+	0.982602i	$0.440537\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.617527	0.0207345	0.0103673	−	0.999946i	$-0.496700\pi$
$887$	0.617527	0.0207345	0.0103673	+	0.999946i	$0.496700\pi$
$888$	0	0
$889$	5.37789	0.180369
$890$	0	0
$891$	0	0
$892$	0	0
$893$	14.5674	0.487478
$894$	0	0
$895$	−12.7790	−0.427156
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.517195	0.0172494
$900$	0	0
$901$	28.6013	0.952847
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.6843	−0.454883
$906$	0	0
$907$	−52.9283	−1.75745	−0.878727	−	0.477325i	$-0.841606\pi$
$907$	−52.9283	−1.75745	−0.878727	+	0.477325i	$0.841606\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	57.2268	1.89601	0.948005	−	0.318256i	$-0.103097\pi$
$911$	57.2268	1.89601	0.948005	+	0.318256i	$0.103097\pi$
$912$	0	0
$913$	69.5038	2.30024
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.69553	−0.287152
$918$	0	0
$919$	−17.7887	−0.586794	−0.293397	−	0.955991i	$-0.594786\pi$
$919$	−17.7887	−0.586794	−0.293397	+	0.955991i	$0.594786\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−49.6366	−1.63381
$924$	0	0
$925$	−23.4510	−0.771063
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.62453	−0.151726	−0.0758630	−	0.997118i	$-0.524171\pi$
$929$	−4.62453	−0.151726	−0.0758630	+	0.997118i	$0.524171\pi$
$930$	0	0
$931$	−6.15032	−0.201569
$932$	0	0
$933$	0	0
$934$	0	0
$935$	30.3869	0.993757
$936$	0	0
$937$	7.43536	0.242903	0.121451	−	0.992597i	$-0.461245\pi$
$937$	7.43536	0.242903	0.121451	+	0.992597i	$0.461245\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	28.8238	0.939631	0.469815	−	0.882765i	$-0.344321\pi$
$941$	28.8238	0.939631	0.469815	+	0.882765i	$0.344321\pi$
$942$	0	0
$943$	6.21309	0.202326
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.5571	−1.05796	−0.528982	−	0.848633i	$-0.677426\pi$
$947$	−32.5571	−1.05796	−0.528982	+	0.848633i	$0.677426\pi$
$948$	0	0
$949$	−12.5445	−0.407211
$950$	0	0
$951$	0	0
$952$	0	0
$953$	37.7834	1.22393	0.611963	−	0.790886i	$-0.290380\pi$
$953$	37.7834	1.22393	0.611963	+	0.790886i	$0.290380\pi$
$954$	0	0
$955$	−1.21749	−0.0393969
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18.0917	0.584213
$960$	0	0
$961$	−30.9733	−0.999138
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.64153	0.181607
$966$	0	0
$967$	−54.6074	−1.75606	−0.878028	−	0.478608i	$-0.841141\pi$
$967$	−54.6074	−1.75606	−0.878028	+	0.478608i	$0.841141\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.554552	0.0177964	0.00889820	−	0.999960i	$-0.497168\pi$
$971$	0.554552	0.0177964	0.00889820	+	0.999960i	$0.497168\pi$
$972$	0	0
$973$	11.8905	0.381192
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.4551	0.398475	0.199238	−	0.979951i	$-0.436153\pi$
$977$	12.4551	0.398475	0.199238	+	0.979951i	$0.436153\pi$
$978$	0	0
$979$	−26.9923	−0.862679
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.47694	−0.110897	−0.0554485	−	0.998462i	$-0.517659\pi$
$983$	−3.47694	−0.110897	−0.0554485	+	0.998462i	$0.517659\pi$
$984$	0	0
$985$	7.45404	0.237505
$986$	0	0
$987$	0	0
$988$	0	0
$989$	59.6868	1.89793
$990$	0	0
$991$	3.96049	0.125809	0.0629046	−	0.998020i	$-0.479964\pi$
$991$	3.96049	0.125809	0.0629046	+	0.998020i	$0.479964\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.41764	0.203453
$996$	0	0
$997$	−10.6013	−0.335746	−0.167873	−	0.985809i	$-0.553690\pi$
$997$	−10.6013	−0.335746	−0.167873	+	0.985809i	$0.553690\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.bo.1.2	yes	4
3.2	odd	2		6048.2.a.bv.1.3	yes	4
4.3	odd	2		6048.2.a.bm.1.2	✓	4
12.11	even	2		6048.2.a.br.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.bm.1.2	✓	4	4.3	odd	2
6048.2.a.bo.1.2	yes	4	1.1	even	1	trivial
6048.2.a.br.1.3	yes	4	12.11	even	2
6048.2.a.bv.1.3	yes	4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.a (trivial)

\(f(q)\)	\(=\)	\(q-0.766021 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-0.766021 q^{5} +1.00000 q^{7} +5.54779 q^{11} +6.31381 q^{13} -7.15032 q^{17} -6.15032 q^{19} -7.17923 q^{23} -4.41321 q^{25} -3.16349 q^{29} -0.163489 q^{31} -0.766021 q^{35} +5.31381 q^{37} -0.865425 q^{41} -8.31381 q^{43} -2.36855 q^{47} +1.00000 q^{49} -4.00000 q^{53} -4.24972 q^{55} -2.36855 q^{59} -13.5635 q^{61} -4.83651 q^{65} +7.05092 q^{67} -7.86159 q^{71} -1.98683 q^{73} +5.54779 q^{77} +5.24972 q^{79} +12.5282 q^{83} +5.47730 q^{85} -4.86542 q^{89} +6.31381 q^{91} +4.71128 q^{95} +3.26289 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q - 2 * q^5 + 4 * q^7	\( 4 q - 2 q^{5} + 4 q^{7} - 2 q^{11} - 8 q^{17} - 4 q^{19} - 2 q^{23} + 8 q^{25} - 8 q^{29} + 4 q^{31} - 2 q^{35} - 4 q^{37} - 2 q^{41} - 8 q^{43} - 12 q^{47} + 4 q^{49} - 16 q^{53} + 4 q^{55} - 12 q^{59} - 8 q^{61} - 24 q^{65} + 8 q^{67} + 18 q^{71} + 8 q^{73} - 2 q^{77} - 8 q^{85} - 18 q^{89} - 10 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 4 * q^7 - 2 * q^11 - 8 * q^17 - 4 * q^19 - 2 * q^23 + 8 * q^25 - 8 * q^29 + 4 * q^31 - 2 * q^35 - 4 * q^37 - 2 * q^41 - 8 * q^43 - 12 * q^47 + 4 * q^49 - 16 * q^53 + 4 * q^55 - 12 * q^59 - 8 * q^61 - 24 * q^65 + 8 * q^67 + 18 * q^71 + 8 * q^73 - 2 * q^77 - 8 * q^85 - 18 * q^89 - 10 * q^95 + 8 * q^97

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