LMFDB - Embedded newform 243.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [243,2,Mod(1,243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("243.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 243 = 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	243.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:	$1.94036476912$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	243.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} -1.73205 q^{8} +6.00000 q^{10} -3.46410 q^{11} +5.00000 q^{13} -1.73205 q^{14} -5.00000 q^{16} -1.00000 q^{19} +3.46410 q^{20} -6.00000 q^{22} -6.92820 q^{23} +7.00000 q^{25} +8.66025 q^{26} -1.00000 q^{28} -3.46410 q^{29} +5.00000 q^{31} -5.19615 q^{32} -3.46410 q^{35} -1.00000 q^{37} -1.73205 q^{38} -6.00000 q^{40} +3.46410 q^{41} -1.00000 q^{43} -3.46410 q^{44} -12.0000 q^{46} -3.46410 q^{47} -6.00000 q^{49} +12.1244 q^{50} +5.00000 q^{52} -10.3923 q^{53} -12.0000 q^{55} +1.73205 q^{56} -6.00000 q^{58} +3.46410 q^{59} +2.00000 q^{61} +8.66025 q^{62} +1.00000 q^{64} +17.3205 q^{65} +8.00000 q^{67} -6.00000 q^{70} +10.3923 q^{71} +2.00000 q^{73} -1.73205 q^{74} -1.00000 q^{76} +3.46410 q^{77} -1.00000 q^{79} -17.3205 q^{80} +6.00000 q^{82} +6.92820 q^{83} -1.73205 q^{86} +6.00000 q^{88} +10.3923 q^{89} -5.00000 q^{91} -6.92820 q^{92} -6.00000 q^{94} -3.46410 q^{95} +17.0000 q^{97} -10.3923 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{7} + 12 q^{10} + 10 q^{13} - 10 q^{16} - 2 q^{19} - 12 q^{22} + 14 q^{25} - 2 q^{28} + 10 q^{31} - 2 q^{37} - 12 q^{40} - 2 q^{43} - 24 q^{46} - 12 q^{49} + 10 q^{52} - 24 q^{55} - 12 q^{58}+ \cdots + 34 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^7 + 12 * q^10 + 10 * q^13 - 10 * q^16 - 2 * q^19 - 12 * q^22 + 14 * q^25 - 2 * q^28 + 10 * q^31 - 2 * q^37 - 12 * q^40 - 2 * q^43 - 24 * q^46 - 12 * q^49 + 10 * q^52 - 24 * q^55 - 12 * q^58 + 4 * q^61 + 2 * q^64 + 16 * q^67 - 12 * q^70 + 4 * q^73 - 2 * q^76 - 2 * q^79 + 12 * q^82 + 12 * q^88 - 10 * q^91 - 12 * q^94 + 34 * q^97

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	6.00000	1.89737
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	−1.73205	−0.462910
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	3.46410	0.774597
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	8.66025	1.69842
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	−1.73205	−0.280976
$39$	0	0
$40$	−6.00000	−0.948683
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−3.46410	−0.522233
$45$	0	0
$46$	−12.0000	−1.76930
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	12.1244	1.71464
$51$	0	0
$52$	5.00000	0.693375
$53$	−10.3923	−1.42749	−0.713746	−	0.700404i	$-0.753003\pi$
$53$	−10.3923	−1.42749	−0.713746	+	0.700404i	$0.753003\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	1.73205	0.231455
$57$	0	0
$58$	−6.00000	−0.787839
$59$	3.46410	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.66025	1.09985
$63$	0	0
$64$	1.00000	0.125000
$65$	17.3205	2.14834
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	0	0
$70$	−6.00000	−0.717137
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−1.73205	−0.201347
$75$	0	0
$76$	−1.00000	−0.114708
$77$	3.46410	0.394771
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−17.3205	−1.93649
$81$	0	0
$82$	6.00000	0.662589
$83$	6.92820	0.760469	0.380235	−	0.924890i	$-0.375843\pi$
$83$	6.92820	0.760469	0.380235	+	0.924890i	$0.375843\pi$
$84$	0	0
$85$	0	0
$86$	−1.73205	−0.186772
$87$	0	0
$88$	6.00000	0.639602
$89$	10.3923	1.10158	0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923	1.10158	0.550791	+	0.834643i	$0.314326\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	−6.92820	−0.722315
$93$	0	0
$94$	−6.00000	−0.618853
$95$	−3.46410	−0.355409
$96$	0	0
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	−10.3923	−1.04978
$99$	0	0
$100$	7.00000	0.700000
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−8.66025	−0.849208
$105$	0	0
$106$	−18.0000	−1.74831
$107$	10.3923	1.00466	0.502331	−	0.864675i	$-0.332476\pi$
$107$	10.3923	1.00466	0.502331	+	0.864675i	$0.332476\pi$
$108$	0	0
$109$	17.0000	1.62830	0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000	1.62830	0.814152	+	0.580651i	$0.197202\pi$
$110$	−20.7846	−1.98173
$111$	0	0
$112$	5.00000	0.472456
$113$	−17.3205	−1.62938	−0.814688	−	0.579899i	$-0.803092\pi$
$113$	−17.3205	−1.62938	−0.814688	+	0.579899i	$0.803092\pi$
$114$	0	0
$115$	−24.0000	−2.23801
$116$	−3.46410	−0.321634
$117$	0	0
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.46410	0.313625
$123$	0	0
$124$	5.00000	0.449013
$125$	6.92820	0.619677
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	30.0000	2.63117
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	13.8564	1.19701
$135$	0	0
$136$	0	0
$137$	6.92820	0.591916	0.295958	−	0.955201i	$-0.404361\pi$
$137$	6.92820	0.591916	0.295958	+	0.955201i	$0.404361\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	−3.46410	−0.292770
$141$	0	0
$142$	18.0000	1.51053
$143$	−17.3205	−1.44841
$144$	0	0
$145$	−12.0000	−0.996546
$146$	3.46410	0.286691
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−6.92820	−0.567581	−0.283790	−	0.958886i	$-0.591592\pi$
$149$	−6.92820	−0.567581	−0.283790	+	0.958886i	$0.591592\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	1.73205	0.140488
$153$	0	0
$154$	6.00000	0.483494
$155$	17.3205	1.39122
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	−1.73205	−0.137795
$159$	0	0
$160$	−18.0000	−1.42302
$161$	6.92820	0.546019
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	12.0000	0.931381
$167$	24.2487	1.87642	0.938211	−	0.346064i	$-0.112482\pi$
$167$	24.2487	1.87642	0.938211	+	0.346064i	$0.112482\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	−13.8564	−1.05348	−0.526742	−	0.850026i	$-0.676586\pi$
$173$	−13.8564	−1.05348	−0.526742	+	0.850026i	$0.676586\pi$
$174$	0	0
$175$	−7.00000	−0.529150
$176$	17.3205	1.30558
$177$	0	0
$178$	18.0000	1.34916
$179$	−20.7846	−1.55351	−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	−8.66025	−0.641941
$183$	0	0
$184$	12.0000	0.884652
$185$	−3.46410	−0.254686
$186$	0	0
$187$	0	0
$188$	−3.46410	−0.252646
$189$	0	0
$190$	−6.00000	−0.435286
$191$	6.92820	0.501307	0.250654	−	0.968077i	$-0.419354\pi$
$191$	6.92820	0.501307	0.250654	+	0.968077i	$0.419354\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	29.4449	2.11402
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−10.3923	−0.740421	−0.370211	−	0.928948i	$-0.620714\pi$
$197$	−10.3923	−0.740421	−0.370211	+	0.928948i	$0.620714\pi$
$198$	0	0
$199$	−19.0000	−1.34687	−0.673437	−	0.739244i	$-0.735183\pi$
$199$	−19.0000	−1.34687	−0.673437	+	0.739244i	$0.735183\pi$
$200$	−12.1244	−0.857321
$201$	0	0
$202$	−24.0000	−1.68863
$203$	3.46410	0.243132
$204$	0	0
$205$	12.0000	0.838116
$206$	13.8564	0.965422
$207$	0	0
$208$	−25.0000	−1.73344
$209$	3.46410	0.239617
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−10.3923	−0.713746
$213$	0	0
$214$	18.0000	1.23045
$215$	−3.46410	−0.236250
$216$	0	0
$217$	−5.00000	−0.339422
$218$	29.4449	1.99426
$219$	0	0
$220$	−12.0000	−0.809040
$221$	0	0
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	5.19615	0.347183
$225$	0	0
$226$	−30.0000	−1.99557
$227$	−13.8564	−0.919682	−0.459841	−	0.888001i	$-0.652094\pi$
$227$	−13.8564	−0.919682	−0.459841	+	0.888001i	$0.652094\pi$
$228$	0	0
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	−41.5692	−2.74099
$231$	0	0
$232$	6.00000	0.393919
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	3.46410	0.225494
$237$	0	0
$238$	0	0
$239$	−6.92820	−0.448148	−0.224074	−	0.974572i	$-0.571936\pi$
$239$	−6.92820	−0.448148	−0.224074	+	0.974572i	$0.571936\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	1.73205	0.111340
$243$	0	0
$244$	2.00000	0.128037
$245$	−20.7846	−1.32788
$246$	0	0
$247$	−5.00000	−0.318142
$248$	−8.66025	−0.549927
$249$	0	0
$250$	12.0000	0.758947
$251$	20.7846	1.31191	0.655956	−	0.754799i	$-0.272265\pi$
$251$	20.7846	1.31191	0.655956	+	0.754799i	$0.272265\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	29.4449	1.84754
$255$	0	0
$256$	19.0000	1.18750
$257$	3.46410	0.216085	0.108042	−	0.994146i	$-0.465542\pi$
$257$	3.46410	0.216085	0.108042	+	0.994146i	$0.465542\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	17.3205	1.07417
$261$	0	0
$262$	6.00000	0.370681
$263$	−13.8564	−0.854423	−0.427211	−	0.904152i	$-0.640504\pi$
$263$	−13.8564	−0.854423	−0.427211	+	0.904152i	$0.640504\pi$
$264$	0	0
$265$	−36.0000	−2.21146
$266$	1.73205	0.106199
$267$	0	0
$268$	8.00000	0.488678
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	12.0000	0.724947
$275$	−24.2487	−1.46225
$276$	0	0
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	−22.5167	−1.35046
$279$	0	0
$280$	6.00000	0.358569
$281$	−13.8564	−0.826604	−0.413302	−	0.910594i	$-0.635625\pi$
$281$	−13.8564	−0.826604	−0.413302	+	0.910594i	$0.635625\pi$
$282$	0	0
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	10.3923	0.616670
$285$	0	0
$286$	−30.0000	−1.77394
$287$	−3.46410	−0.204479
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−20.7846	−1.22051
$291$	0	0
$292$	2.00000	0.117041
$293$	13.8564	0.809500	0.404750	−	0.914427i	$-0.367359\pi$
$293$	13.8564	0.809500	0.404750	+	0.914427i	$0.367359\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	1.73205	0.100673
$297$	0	0
$298$	−12.0000	−0.695141
$299$	−34.6410	−2.00334
$300$	0	0
$301$	1.00000	0.0576390
$302$	−27.7128	−1.59469
$303$	0	0
$304$	5.00000	0.286770
$305$	6.92820	0.396708
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	3.46410	0.197386
$309$	0	0
$310$	30.0000	1.70389
$311$	13.8564	0.785725	0.392862	−	0.919597i	$-0.371485\pi$
$311$	13.8564	0.785725	0.392862	+	0.919597i	$0.371485\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	−22.5167	−1.27069
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	27.7128	1.55651	0.778253	−	0.627950i	$-0.216106\pi$
$317$	27.7128	1.55651	0.778253	+	0.627950i	$0.216106\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	3.46410	0.193649
$321$	0	0
$322$	12.0000	0.668734
$323$	0	0
$324$	0	0
$325$	35.0000	1.94145
$326$	−1.73205	−0.0959294
$327$	0	0
$328$	−6.00000	−0.331295
$329$	3.46410	0.190982
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	6.92820	0.380235
$333$	0	0
$334$	42.0000	2.29814
$335$	27.7128	1.51411
$336$	0	0
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	20.7846	1.13053
$339$	0	0
$340$	0	0
$341$	−17.3205	−0.937958
$342$	0	0
$343$	13.0000	0.701934
$344$	1.73205	0.0933859
$345$	0	0
$346$	−24.0000	−1.29025
$347$	24.2487	1.30174	0.650870	−	0.759190i	$-0.274404\pi$
$347$	24.2487	1.30174	0.650870	+	0.759190i	$0.274404\pi$
$348$	0	0
$349$	−1.00000	−0.0535288	−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000	−0.0535288	−0.0267644	+	0.999642i	$0.508520\pi$
$350$	−12.1244	−0.648074
$351$	0	0
$352$	18.0000	0.959403
$353$	17.3205	0.921878	0.460939	−	0.887432i	$-0.347513\pi$
$353$	17.3205	0.921878	0.460939	+	0.887432i	$0.347513\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	10.3923	0.550791
$357$	0	0
$358$	−36.0000	−1.90266
$359$	31.1769	1.64545	0.822727	−	0.568436i	$-0.192451\pi$
$359$	31.1769	1.64545	0.822727	+	0.568436i	$0.192451\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	29.4449	1.54759
$363$	0	0
$364$	−5.00000	−0.262071
$365$	6.92820	0.362639
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	34.6410	1.80579
$369$	0	0
$370$	−6.00000	−0.311925
$371$	10.3923	0.539542
$372$	0	0
$373$	23.0000	1.19089	0.595447	−	0.803394i	$-0.296975\pi$
$373$	23.0000	1.19089	0.595447	+	0.803394i	$0.296975\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	−17.3205	−0.892052
$378$	0	0
$379$	−19.0000	−0.975964	−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000	−0.975964	−0.487982	+	0.872854i	$0.662267\pi$
$380$	−3.46410	−0.177705
$381$	0	0
$382$	12.0000	0.613973
$383$	−17.3205	−0.885037	−0.442518	−	0.896759i	$-0.645915\pi$
$383$	−17.3205	−0.885037	−0.442518	+	0.896759i	$0.645915\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	−17.3205	−0.881591
$387$	0	0
$388$	17.0000	0.863044
$389$	6.92820	0.351274	0.175637	−	0.984455i	$-0.443802\pi$
$389$	6.92820	0.351274	0.175637	+	0.984455i	$0.443802\pi$
$390$	0	0
$391$	0	0
$392$	10.3923	0.524891
$393$	0	0
$394$	−18.0000	−0.906827
$395$	−3.46410	−0.174298
$396$	0	0
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	−32.9090	−1.64958
$399$	0	0
$400$	−35.0000	−1.75000
$401$	3.46410	0.172989	0.0864945	−	0.996252i	$-0.472434\pi$
$401$	3.46410	0.172989	0.0864945	+	0.996252i	$0.472434\pi$
$402$	0	0
$403$	25.0000	1.24534
$404$	−13.8564	−0.689382
$405$	0	0
$406$	6.00000	0.297775
$407$	3.46410	0.171709
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	20.7846	1.02648
$411$	0	0
$412$	8.00000	0.394132
$413$	−3.46410	−0.170457
$414$	0	0
$415$	24.0000	1.17811
$416$	−25.9808	−1.27381
$417$	0	0
$418$	6.00000	0.293470
$419$	−38.1051	−1.86156	−0.930778	−	0.365584i	$-0.880869\pi$
$419$	−38.1051	−1.86156	−0.930778	+	0.365584i	$0.880869\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	8.66025	0.421575
$423$	0	0
$424$	18.0000	0.874157
$425$	0	0
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	10.3923	0.502331
$429$	0	0
$430$	−6.00000	−0.289346
$431$	−20.7846	−1.00116	−0.500580	−	0.865690i	$-0.666880\pi$
$431$	−20.7846	−1.00116	−0.500580	+	0.865690i	$0.666880\pi$
$432$	0	0
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	−8.66025	−0.415705
$435$	0	0
$436$	17.0000	0.814152
$437$	6.92820	0.331421
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	20.7846	0.990867
$441$	0	0
$442$	0	0
$443$	−13.8564	−0.658338	−0.329169	−	0.944271i	$-0.606769\pi$
$443$	−13.8564	−0.658338	−0.329169	+	0.944271i	$0.606769\pi$
$444$	0	0
$445$	36.0000	1.70656
$446$	−32.9090	−1.55828
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−10.3923	−0.490443	−0.245222	−	0.969467i	$-0.578861\pi$
$449$	−10.3923	−0.490443	−0.245222	+	0.969467i	$0.578861\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	−17.3205	−0.814688
$453$	0	0
$454$	−24.0000	−1.12638
$455$	−17.3205	−0.811998
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	8.66025	0.404667
$459$	0	0
$460$	−24.0000	−1.11901
$461$	27.7128	1.29071	0.645357	−	0.763881i	$-0.276709\pi$
$461$	27.7128	1.29071	0.645357	+	0.763881i	$0.276709\pi$
$462$	0	0
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	17.3205	0.804084
$465$	0	0
$466$	0	0
$467$	−10.3923	−0.480899	−0.240449	−	0.970662i	$-0.577295\pi$
$467$	−10.3923	−0.480899	−0.240449	+	0.970662i	$0.577295\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	−20.7846	−0.958723
$471$	0	0
$472$	−6.00000	−0.276172
$473$	3.46410	0.159280
$474$	0	0
$475$	−7.00000	−0.321182
$476$	0	0
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−13.8564	−0.633115	−0.316558	−	0.948573i	$-0.602527\pi$
$479$	−13.8564	−0.633115	−0.316558	+	0.948573i	$0.602527\pi$
$480$	0	0
$481$	−5.00000	−0.227980
$482$	−32.9090	−1.49896
$483$	0	0
$484$	1.00000	0.0454545
$485$	58.8897	2.67404
$486$	0	0
$487$	−19.0000	−0.860972	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	−19.0000	−0.860972	−0.430486	+	0.902597i	$0.641658\pi$
$488$	−3.46410	−0.156813
$489$	0	0
$490$	−36.0000	−1.62631
$491$	−38.1051	−1.71966	−0.859830	−	0.510581i	$-0.829431\pi$
$491$	−38.1051	−1.71966	−0.859830	+	0.510581i	$0.829431\pi$
$492$	0	0
$493$	0	0
$494$	−8.66025	−0.389643
$495$	0	0
$496$	−25.0000	−1.12253
$497$	−10.3923	−0.466159
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	6.92820	0.309839
$501$	0	0
$502$	36.0000	1.60676
$503$	41.5692	1.85348	0.926740	−	0.375703i	$-0.122599\pi$
$503$	41.5692	1.85348	0.926740	+	0.375703i	$0.122599\pi$
$504$	0	0
$505$	−48.0000	−2.13597
$506$	41.5692	1.84798
$507$	0	0
$508$	17.0000	0.754253
$509$	−27.7128	−1.22835	−0.614174	−	0.789170i	$-0.710511\pi$
$509$	−27.7128	−1.22835	−0.614174	+	0.789170i	$0.710511\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	8.66025	0.382733
$513$	0	0
$514$	6.00000	0.264649
$515$	27.7128	1.22117
$516$	0	0
$517$	12.0000	0.527759
$518$	1.73205	0.0761019
$519$	0	0
$520$	−30.0000	−1.31559
$521$	−20.7846	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	3.46410	0.151330
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	25.0000	1.08696
$530$	−62.3538	−2.70848
$531$	0	0
$532$	1.00000	0.0433555
$533$	17.3205	0.750234
$534$	0	0
$535$	36.0000	1.55642
$536$	−13.8564	−0.598506
$537$	0	0
$538$	0	0
$539$	20.7846	0.895257
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	−27.7128	−1.19037
$543$	0	0
$544$	0	0
$545$	58.8897	2.52256
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	6.92820	0.295958
$549$	0	0
$550$	−42.0000	−1.79089
$551$	3.46410	0.147576
$552$	0	0
$553$	1.00000	0.0425243
$554$	29.4449	1.25099
$555$	0	0
$556$	−13.0000	−0.551323
$557$	10.3923	0.440336	0.220168	−	0.975462i	$-0.429339\pi$
$557$	10.3923	0.440336	0.220168	+	0.975462i	$0.429339\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	17.3205	0.731925
$561$	0	0
$562$	−24.0000	−1.01238
$563$	34.6410	1.45994	0.729972	−	0.683477i	$-0.239533\pi$
$563$	34.6410	1.45994	0.729972	+	0.683477i	$0.239533\pi$
$564$	0	0
$565$	−60.0000	−2.52422
$566$	−22.5167	−0.946446
$567$	0	0
$568$	−18.0000	−0.755263
$569$	−24.2487	−1.01656	−0.508279	−	0.861192i	$-0.669718\pi$
$569$	−24.2487	−1.01656	−0.508279	+	0.861192i	$0.669718\pi$
$570$	0	0
$571$	41.0000	1.71580	0.857898	−	0.513820i	$-0.171770\pi$
$571$	41.0000	1.71580	0.857898	+	0.513820i	$0.171770\pi$
$572$	−17.3205	−0.724207
$573$	0	0
$574$	−6.00000	−0.250435
$575$	−48.4974	−2.02248
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	−29.4449	−1.22474
$579$	0	0
$580$	−12.0000	−0.498273
$581$	−6.92820	−0.287430
$582$	0	0
$583$	36.0000	1.49097
$584$	−3.46410	−0.143346
$585$	0	0
$586$	24.0000	0.991431
$587$	6.92820	0.285958	0.142979	−	0.989726i	$-0.454332\pi$
$587$	6.92820	0.285958	0.142979	+	0.989726i	$0.454332\pi$
$588$	0	0
$589$	−5.00000	−0.206021
$590$	20.7846	0.855689
$591$	0	0
$592$	5.00000	0.205499
$593$	20.7846	0.853522	0.426761	−	0.904365i	$-0.359655\pi$
$593$	20.7846	0.853522	0.426761	+	0.904365i	$0.359655\pi$
$594$	0	0
$595$	0	0
$596$	−6.92820	−0.283790
$597$	0	0
$598$	−60.0000	−2.45358
$599$	24.2487	0.990775	0.495388	−	0.868672i	$-0.335026\pi$
$599$	24.2487	0.990775	0.495388	+	0.868672i	$0.335026\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	1.73205	0.0705931
$603$	0	0
$604$	−16.0000	−0.651031
$605$	3.46410	0.140836
$606$	0	0
$607$	−13.0000	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$608$	5.19615	0.210732
$609$	0	0
$610$	12.0000	0.485866
$611$	−17.3205	−0.700713
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	34.6410	1.39800
$615$	0	0
$616$	−6.00000	−0.241747
$617$	−6.92820	−0.278919	−0.139459	−	0.990228i	$-0.544536\pi$
$617$	−6.92820	−0.278919	−0.139459	+	0.990228i	$0.544536\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	17.3205	0.695608
$621$	0	0
$622$	24.0000	0.962312
$623$	−10.3923	−0.416359
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−1.73205	−0.0692267
$627$	0	0
$628$	−13.0000	−0.518756
$629$	0	0
$630$	0	0
$631$	17.0000	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$632$	1.73205	0.0688973
$633$	0	0
$634$	48.0000	1.90632
$635$	58.8897	2.33697
$636$	0	0
$637$	−30.0000	−1.18864
$638$	20.7846	0.822871
$639$	0	0
$640$	42.0000	1.66020
$641$	−45.0333	−1.77871	−0.889355	−	0.457218i	$-0.848846\pi$
$641$	−45.0333	−1.77871	−0.889355	+	0.457218i	$0.848846\pi$
$642$	0	0
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	6.92820	0.273009
$645$	0	0
$646$	0	0
$647$	−20.7846	−0.817127	−0.408564	−	0.912730i	$-0.633970\pi$
$647$	−20.7846	−0.817127	−0.408564	+	0.912730i	$0.633970\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	60.6218	2.37778
$651$	0	0
$652$	−1.00000	−0.0391630
$653$	−17.3205	−0.677804	−0.338902	−	0.940822i	$-0.610055\pi$
$653$	−17.3205	−0.677804	−0.338902	+	0.940822i	$0.610055\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	−17.3205	−0.676252
$657$	0	0
$658$	6.00000	0.233904
$659$	27.7128	1.07954	0.539769	−	0.841813i	$-0.318512\pi$
$659$	27.7128	1.07954	0.539769	+	0.841813i	$0.318512\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−32.9090	−1.27904
$663$	0	0
$664$	−12.0000	−0.465690
$665$	3.46410	0.134332
$666$	0	0
$667$	24.0000	0.929284
$668$	24.2487	0.938211
$669$	0	0
$670$	48.0000	1.85440
$671$	−6.92820	−0.267460
$672$	0	0
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	8.66025	0.333581
$675$	0	0
$676$	12.0000	0.461538
$677$	17.3205	0.665681	0.332841	−	0.942983i	$-0.391993\pi$
$677$	17.3205	0.665681	0.332841	+	0.942983i	$0.391993\pi$
$678$	0	0
$679$	−17.0000	−0.652400
$680$	0	0
$681$	0	0
$682$	−30.0000	−1.14876
$683$	41.5692	1.59060	0.795301	−	0.606215i	$-0.207313\pi$
$683$	41.5692	1.59060	0.795301	+	0.606215i	$0.207313\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	22.5167	0.859690
$687$	0	0
$688$	5.00000	0.190623
$689$	−51.9615	−1.97958
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	−13.8564	−0.526742
$693$	0	0
$694$	42.0000	1.59430
$695$	−45.0333	−1.70821
$696$	0	0
$697$	0	0
$698$	−1.73205	−0.0655591
$699$	0	0
$700$	−7.00000	−0.264575
$701$	−41.5692	−1.57005	−0.785024	−	0.619466i	$-0.787349\pi$
$701$	−41.5692	−1.57005	−0.785024	+	0.619466i	$0.787349\pi$
$702$	0	0
$703$	1.00000	0.0377157
$704$	−3.46410	−0.130558
$705$	0	0
$706$	30.0000	1.12906
$707$	13.8564	0.521124
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	62.3538	2.34010
$711$	0	0
$712$	−18.0000	−0.674579
$713$	−34.6410	−1.29732
$714$	0	0
$715$	−60.0000	−2.24387
$716$	−20.7846	−0.776757
$717$	0	0
$718$	54.0000	2.01526
$719$	10.3923	0.387568	0.193784	−	0.981044i	$-0.437924\pi$
$719$	10.3923	0.387568	0.193784	+	0.981044i	$0.437924\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	−31.1769	−1.16028
$723$	0	0
$724$	17.0000	0.631800
$725$	−24.2487	−0.900575
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	8.66025	0.320970
$729$	0	0
$730$	12.0000	0.444140
$731$	0	0
$732$	0	0
$733$	41.0000	1.51437	0.757185	−	0.653201i	$-0.226574\pi$
$733$	41.0000	1.51437	0.757185	+	0.653201i	$0.226574\pi$
$734$	−27.7128	−1.02290
$735$	0	0
$736$	36.0000	1.32698
$737$	−27.7128	−1.02081
$738$	0	0
$739$	−19.0000	−0.698926	−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000	−0.698926	−0.349463	+	0.936950i	$0.613636\pi$
$740$	−3.46410	−0.127343
$741$	0	0
$742$	18.0000	0.660801
$743$	−6.92820	−0.254171	−0.127086	−	0.991892i	$-0.540562\pi$
$743$	−6.92820	−0.254171	−0.127086	+	0.991892i	$0.540562\pi$
$744$	0	0
$745$	−24.0000	−0.879292
$746$	39.8372	1.45854
$747$	0	0
$748$	0	0
$749$	−10.3923	−0.379727
$750$	0	0
$751$	5.00000	0.182453	0.0912263	−	0.995830i	$-0.470921\pi$
$751$	5.00000	0.182453	0.0912263	+	0.995830i	$0.470921\pi$
$752$	17.3205	0.631614
$753$	0	0
$754$	−30.0000	−1.09254
$755$	−55.4256	−2.01715
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−32.9090	−1.19531
$759$	0	0
$760$	6.00000	0.217643
$761$	−27.7128	−1.00459	−0.502294	−	0.864697i	$-0.667511\pi$
$761$	−27.7128	−1.00459	−0.502294	+	0.864697i	$0.667511\pi$
$762$	0	0
$763$	−17.0000	−0.615441
$764$	6.92820	0.250654
$765$	0	0
$766$	−30.0000	−1.08394
$767$	17.3205	0.625407
$768$	0	0
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	20.7846	0.749025
$771$	0	0
$772$	−10.0000	−0.359908
$773$	20.7846	0.747570	0.373785	−	0.927515i	$-0.378060\pi$
$773$	20.7846	0.747570	0.373785	+	0.927515i	$0.378060\pi$
$774$	0	0
$775$	35.0000	1.25724
$776$	−29.4449	−1.05701
$777$	0	0
$778$	12.0000	0.430221
$779$	−3.46410	−0.124114
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	30.0000	1.07143
$785$	−45.0333	−1.60731
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−10.3923	−0.370211
$789$	0	0
$790$	−6.00000	−0.213470
$791$	17.3205	0.615846
$792$	0	0
$793$	10.0000	0.355110
$794$	−1.73205	−0.0614682
$795$	0	0
$796$	−19.0000	−0.673437
$797$	13.8564	0.490819	0.245410	−	0.969419i	$-0.421078\pi$
$797$	13.8564	0.490819	0.245410	+	0.969419i	$0.421078\pi$
$798$	0	0
$799$	0	0
$800$	−36.3731	−1.28598
$801$	0	0
$802$	6.00000	0.211867
$803$	−6.92820	−0.244491
$804$	0	0
$805$	24.0000	0.845889
$806$	43.3013	1.52522
$807$	0	0
$808$	24.0000	0.844317
$809$	−31.1769	−1.09612	−0.548061	−	0.836438i	$-0.684634\pi$
$809$	−31.1769	−1.09612	−0.548061	+	0.836438i	$0.684634\pi$
$810$	0	0
$811$	35.0000	1.22902	0.614508	−	0.788911i	$-0.289355\pi$
$811$	35.0000	1.22902	0.614508	+	0.788911i	$0.289355\pi$
$812$	3.46410	0.121566
$813$	0	0
$814$	6.00000	0.210300
$815$	−3.46410	−0.121342
$816$	0	0
$817$	1.00000	0.0349856
$818$	8.66025	0.302799
$819$	0	0
$820$	12.0000	0.419058
$821$	38.1051	1.32988	0.664939	−	0.746898i	$-0.268458\pi$
$821$	38.1051	1.32988	0.664939	+	0.746898i	$0.268458\pi$
$822$	0	0
$823$	23.0000	0.801730	0.400865	−	0.916137i	$-0.368710\pi$
$823$	23.0000	0.801730	0.400865	+	0.916137i	$0.368710\pi$
$824$	−13.8564	−0.482711
$825$	0	0
$826$	−6.00000	−0.208767
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	−19.0000	−0.659897	−0.329949	−	0.943999i	$-0.607031\pi$
$829$	−19.0000	−0.659897	−0.329949	+	0.943999i	$0.607031\pi$
$830$	41.5692	1.44289
$831$	0	0
$832$	5.00000	0.173344
$833$	0	0
$834$	0	0
$835$	84.0000	2.90694
$836$	3.46410	0.119808
$837$	0	0
$838$	−66.0000	−2.27993
$839$	−3.46410	−0.119594	−0.0597970	−	0.998211i	$-0.519045\pi$
$839$	−3.46410	−0.119594	−0.0597970	+	0.998211i	$0.519045\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	−32.9090	−1.13412
$843$	0	0
$844$	5.00000	0.172107
$845$	41.5692	1.43002
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	51.9615	1.78437
$849$	0	0
$850$	0	0
$851$	6.92820	0.237496
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	−3.46410	−0.118539
$855$	0	0
$856$	−18.0000	−0.615227
$857$	38.1051	1.30165	0.650823	−	0.759229i	$-0.274424\pi$
$857$	38.1051	1.30165	0.650823	+	0.759229i	$0.274424\pi$
$858$	0	0
$859$	−49.0000	−1.67186	−0.835929	−	0.548837i	$-0.815071\pi$
$859$	−49.0000	−1.67186	−0.835929	+	0.548837i	$0.815071\pi$
$860$	−3.46410	−0.118125
$861$	0	0
$862$	−36.0000	−1.22616
$863$	10.3923	0.353758	0.176879	−	0.984233i	$-0.443400\pi$
$863$	10.3923	0.353758	0.176879	+	0.984233i	$0.443400\pi$
$864$	0	0
$865$	−48.0000	−1.63205
$866$	−1.73205	−0.0588575
$867$	0	0
$868$	−5.00000	−0.169711
$869$	3.46410	0.117512
$870$	0	0
$871$	40.0000	1.35535
$872$	−29.4449	−0.997129
$873$	0	0
$874$	12.0000	0.405906
$875$	−6.92820	−0.234216
$876$	0	0
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	34.6410	1.16908
$879$	0	0
$880$	60.0000	2.02260
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	0	0
$885$	0	0
$886$	−24.0000	−0.806296
$887$	13.8564	0.465253	0.232626	−	0.972566i	$-0.425268\pi$
$887$	13.8564	0.465253	0.232626	+	0.972566i	$0.425268\pi$
$888$	0	0
$889$	−17.0000	−0.570162
$890$	62.3538	2.09011
$891$	0	0
$892$	−19.0000	−0.636167
$893$	3.46410	0.115922
$894$	0	0
$895$	−72.0000	−2.40669
$896$	−12.1244	−0.405046
$897$	0	0
$898$	−18.0000	−0.600668
$899$	−17.3205	−0.577671
$900$	0	0
$901$	0	0
$902$	−20.7846	−0.692052
$903$	0	0
$904$	30.0000	0.997785
$905$	58.8897	1.95756
$906$	0	0
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	−13.8564	−0.459841
$909$	0	0
$910$	−30.0000	−0.994490
$911$	27.7128	0.918166	0.459083	−	0.888393i	$-0.348178\pi$
$911$	27.7128	0.918166	0.459083	+	0.888393i	$0.348178\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	29.4449	0.973950
$915$	0	0
$916$	5.00000	0.165205
$917$	−3.46410	−0.114395
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	41.5692	1.37050
$921$	0	0
$922$	48.0000	1.58080
$923$	51.9615	1.71033
$924$	0	0
$925$	−7.00000	−0.230159
$926$	−53.6936	−1.76448
$927$	0	0
$928$	18.0000	0.590879
$929$	6.92820	0.227307	0.113653	−	0.993520i	$-0.463745\pi$
$929$	6.92820	0.227307	0.113653	+	0.993520i	$0.463745\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	0	0
$934$	−18.0000	−0.588978
$935$	0	0
$936$	0	0
$937$	−37.0000	−1.20874	−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000	−1.20874	−0.604369	+	0.796705i	$0.706575\pi$
$938$	−13.8564	−0.452428
$939$	0	0
$940$	−12.0000	−0.391397
$941$	3.46410	0.112926	0.0564632	−	0.998405i	$-0.482018\pi$
$941$	3.46410	0.112926	0.0564632	+	0.998405i	$0.482018\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	−17.3205	−0.563735
$945$	0	0
$946$	6.00000	0.195077
$947$	−13.8564	−0.450273	−0.225136	−	0.974327i	$-0.572283\pi$
$947$	−13.8564	−0.450273	−0.225136	+	0.974327i	$0.572283\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	−12.1244	−0.393366
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	−6.92820	−0.224074
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−6.92820	−0.223723
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−8.66025	−0.279218
$963$	0	0
$964$	−19.0000	−0.611949
$965$	−34.6410	−1.11513
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−1.73205	−0.0556702
$969$	0	0
$970$	102.000	3.27502
$971$	51.9615	1.66752	0.833762	−	0.552124i	$-0.186182\pi$
$971$	51.9615	1.66752	0.833762	+	0.552124i	$0.186182\pi$
$972$	0	0
$973$	13.0000	0.416761
$974$	−32.9090	−1.05447
$975$	0	0
$976$	−10.0000	−0.320092
$977$	3.46410	0.110826	0.0554132	−	0.998464i	$-0.482352\pi$
$977$	3.46410	0.110826	0.0554132	+	0.998464i	$0.482352\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	−20.7846	−0.663940
$981$	0	0
$982$	−66.0000	−2.10614
$983$	6.92820	0.220975	0.110488	−	0.993877i	$-0.464759\pi$
$983$	6.92820	0.220975	0.110488	+	0.993877i	$0.464759\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	0	0
$988$	−5.00000	−0.159071
$989$	6.92820	0.220304
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	−25.9808	−0.824890
$993$	0	0
$994$	−18.0000	−0.570925
$995$	−65.8179	−2.08657
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	−48.4974	−1.53516
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.a.c.1.2	yes	2
3.2	odd	2	inner	243.2.a.c.1.1	✓	2
4.3	odd	2		3888.2.a.ba.1.2		2
5.4	even	2		6075.2.a.bm.1.1		2
9.2	odd	6		243.2.c.d.82.2		4
9.4	even	3		243.2.c.d.163.1		4
9.5	odd	6		243.2.c.d.163.2		4
9.7	even	3		243.2.c.d.82.1		4
12.11	even	2		3888.2.a.ba.1.1		2
15.14	odd	2		6075.2.a.bm.1.2		2
27.2	odd	18		729.2.e.n.568.2		12
27.4	even	9		729.2.e.n.406.2		12
27.5	odd	18		729.2.e.n.649.1		12
27.7	even	9		729.2.e.n.325.2		12
27.11	odd	18		729.2.e.n.82.1		12
27.13	even	9		729.2.e.n.163.1		12
27.14	odd	18		729.2.e.n.163.2		12
27.16	even	9		729.2.e.n.82.2		12
27.20	odd	18		729.2.e.n.325.1		12
27.22	even	9		729.2.e.n.649.2		12
27.23	odd	18		729.2.e.n.406.1		12
27.25	even	9		729.2.e.n.568.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.c.1.1	✓	2	3.2	odd	2	inner
243.2.a.c.1.2	yes	2	1.1	even	1	trivial
243.2.c.d.82.1		4	9.7	even	3
243.2.c.d.82.2		4	9.2	odd	6
243.2.c.d.163.1		4	9.4	even	3
243.2.c.d.163.2		4	9.5	odd	6
729.2.e.n.82.1		12	27.11	odd	18
729.2.e.n.82.2		12	27.16	even	9
729.2.e.n.163.1		12	27.13	even	9
729.2.e.n.163.2		12	27.14	odd	18
729.2.e.n.325.1		12	27.20	odd	18
729.2.e.n.325.2		12	27.7	even	9
729.2.e.n.406.1		12	27.23	odd	18
729.2.e.n.406.2		12	27.4	even	9
729.2.e.n.568.1		12	27.25	even	9
729.2.e.n.568.2		12	27.2	odd	18
729.2.e.n.649.1		12	27.5	odd	18
729.2.e.n.649.2		12	27.22	even	9
3888.2.a.ba.1.1		2	12.11	even	2
3888.2.a.ba.1.2		2	4.3	odd	2
6075.2.a.bm.1.1		2	5.4	even	2
6075.2.a.bm.1.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} +3.46410 q^{5} -1.00000 q^{7} -1.73205 q^{8} +6.00000 q^{10} -3.46410 q^{11} +5.00000 q^{13} -1.73205 q^{14} -5.00000 q^{16} -1.00000 q^{19} +3.46410 q^{20} -6.00000 q^{22} -6.92820 q^{23} +7.00000 q^{25} +8.66025 q^{26} -1.00000 q^{28} -3.46410 q^{29} +5.00000 q^{31} -5.19615 q^{32} -3.46410 q^{35} -1.00000 q^{37} -1.73205 q^{38} -6.00000 q^{40} +3.46410 q^{41} -1.00000 q^{43} -3.46410 q^{44} -12.0000 q^{46} -3.46410 q^{47} -6.00000 q^{49} +12.1244 q^{50} +5.00000 q^{52} -10.3923 q^{53} -12.0000 q^{55} +1.73205 q^{56} -6.00000 q^{58} +3.46410 q^{59} +2.00000 q^{61} +8.66025 q^{62} +1.00000 q^{64} +17.3205 q^{65} +8.00000 q^{67} -6.00000 q^{70} +10.3923 q^{71} +2.00000 q^{73} -1.73205 q^{74} -1.00000 q^{76} +3.46410 q^{77} -1.00000 q^{79} -17.3205 q^{80} +6.00000 q^{82} +6.92820 q^{83} -1.73205 q^{86} +6.00000 q^{88} +10.3923 q^{89} -5.00000 q^{91} -6.92820 q^{92} -6.00000 q^{94} -3.46410 q^{95} +17.0000 q^{97} -10.3923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{7} + 12 q^{10} + 10 q^{13} - 10 q^{16} - 2 q^{19} - 12 q^{22} + 14 q^{25} - 2 q^{28} + 10 q^{31} - 2 q^{37} - 12 q^{40} - 2 q^{43} - 24 q^{46} - 12 q^{49} + 10 q^{52} - 24 q^{55} - 12 q^{58}+ \cdots + 34 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^7 + 12 * q^10 + 10 * q^13 - 10 * q^16 - 2 * q^19 - 12 * q^22 + 14 * q^25 - 2 * q^28 + 10 * q^31 - 2 * q^37 - 12 * q^40 - 2 * q^43 - 24 * q^46 - 12 * q^49 + 10 * q^52 - 24 * q^55 - 12 * q^58 + 4 * q^61 + 2 * q^64 + 16 * q^67 - 12 * q^70 + 4 * q^73 - 2 * q^76 - 2 * q^79 + 12 * q^82 + 12 * q^88 - 10 * q^91 - 12 * q^94 + 34 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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