LMFDB - Embedded newform 243.2.a.d.1.1

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(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
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			1.1
Root			$-2.44949$ 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)

$f(q)$	$=$	$q-2.44949 q^{2} +4.00000 q^{4} +2.44949 q^{5} +2.00000 q^{7} -4.89898 q^{8} +O(q^{10})$	$q-2.44949 q^{2} +4.00000 q^{4} +2.44949 q^{5} +2.00000 q^{7} -4.89898 q^{8} -6.00000 q^{10} -2.44949 q^{11} -1.00000 q^{13} -4.89898 q^{14} +4.00000 q^{16} +7.34847 q^{17} -1.00000 q^{19} +9.79796 q^{20} +6.00000 q^{22} +2.44949 q^{23} +1.00000 q^{25} +2.44949 q^{26} +8.00000 q^{28} +4.89898 q^{29} -1.00000 q^{31} -18.0000 q^{34} +4.89898 q^{35} +8.00000 q^{37} +2.44949 q^{38} -12.0000 q^{40} -4.89898 q^{41} +11.0000 q^{43} -9.79796 q^{44} -6.00000 q^{46} -9.79796 q^{47} -3.00000 q^{49} -2.44949 q^{50} -4.00000 q^{52} -7.34847 q^{53} -6.00000 q^{55} -9.79796 q^{56} -12.0000 q^{58} +2.44949 q^{59} +5.00000 q^{61} +2.44949 q^{62} -8.00000 q^{64} -2.44949 q^{65} -7.00000 q^{67} +29.3939 q^{68} -12.0000 q^{70} -7.34847 q^{71} +11.0000 q^{73} -19.5959 q^{74} -4.00000 q^{76} -4.89898 q^{77} -7.00000 q^{79} +9.79796 q^{80} +12.0000 q^{82} +12.2474 q^{83} +18.0000 q^{85} -26.9444 q^{86} +12.0000 q^{88} -2.00000 q^{91} +9.79796 q^{92} +24.0000 q^{94} -2.44949 q^{95} -7.00000 q^{97} +7.34847 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{4} + 4 q^{7}+O(q^{10}) $ 2 * q + 8 * q^4 + 4 * q^7	$ 2 q + 8 q^{4} + 4 q^{7} - 12 q^{10} - 2 q^{13} + 8 q^{16} - 2 q^{19} + 12 q^{22} + 2 q^{25} + 16 q^{28} - 2 q^{31} - 36 q^{34} + 16 q^{37} - 24 q^{40} + 22 q^{43} - 12 q^{46} - 6 q^{49} - 8 q^{52} - 12 q^{55} - 24 q^{58} + 10 q^{61} - 16 q^{64} - 14 q^{67} - 24 q^{70} + 22 q^{73} - 8 q^{76} - 14 q^{79} + 24 q^{82} + 36 q^{85} + 24 q^{88} - 4 q^{91} + 48 q^{94} - 14 q^{97}+O(q^{100}) $ 2 * q + 8 * q^4 + 4 * q^7 - 12 * q^10 - 2 * q^13 + 8 * q^16 - 2 * q^19 + 12 * q^22 + 2 * q^25 + 16 * q^28 - 2 * q^31 - 36 * q^34 + 16 * q^37 - 24 * q^40 + 22 * q^43 - 12 * q^46 - 6 * q^49 - 8 * q^52 - 12 * q^55 - 24 * q^58 + 10 * q^61 - 16 * q^64 - 14 * q^67 - 24 * q^70 + 22 * q^73 - 8 * q^76 - 14 * q^79 + 24 * q^82 + 36 * q^85 + 24 * q^88 - 4 * q^91 + 48 * q^94 - 14 * 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 $

$2$	−2.44949	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	−2.44949	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$3$	0	0
$3$	0	0
$4$	4.00000	2.00000
$5$	2.44949	1.09545	0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949	1.09545	0.547723	+	0.836660i	$0.315495\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−4.89898	−1.73205
$9$	0	0
$10$	−6.00000	−1.89737
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−4.89898	−1.30931
$15$	0	0
$16$	4.00000	1.00000
$17$	7.34847	1.78227	0.891133	−	0.453743i	$-0.149911\pi$
$17$	7.34847	1.78227	0.891133	+	0.453743i	$0.149911\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	9.79796	2.19089
$21$	0	0
$22$	6.00000	1.27920
$23$	2.44949	0.510754	0.255377	−	0.966842i	$-0.417800\pi$
$23$	2.44949	0.510754	0.255377	+	0.966842i	$0.417800\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.44949	0.480384
$27$	0	0
$28$	8.00000	1.51186
$29$	4.89898	0.909718	0.454859	−	0.890564i	$-0.349690\pi$
$29$	4.89898	0.909718	0.454859	+	0.890564i	$0.349690\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	0	0
$34$	−18.0000	−3.08697
$35$	4.89898	0.828079
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	2.44949	0.397360
$39$	0	0
$40$	−12.0000	−1.89737
$41$	−4.89898	−0.765092	−0.382546	−	0.923936i	$-0.624953\pi$
$41$	−4.89898	−0.765092	−0.382546	+	0.923936i	$0.624953\pi$
$42$	0	0
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	−9.79796	−1.47710
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−9.79796	−1.42918	−0.714590	−	0.699544i	$-0.753387\pi$
$47$	−9.79796	−1.42918	−0.714590	+	0.699544i	$0.753387\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−2.44949	−0.346410
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−7.34847	−1.00939	−0.504695	−	0.863298i	$-0.668395\pi$
$53$	−7.34847	−1.00939	−0.504695	+	0.863298i	$0.668395\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	−9.79796	−1.30931
$57$	0	0
$58$	−12.0000	−1.57568
$59$	2.44949	0.318896	0.159448	−	0.987206i	$-0.449029\pi$
$59$	2.44949	0.318896	0.159448	+	0.987206i	$0.449029\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	2.44949	0.311086
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−2.44949	−0.303822
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	29.3939	3.56453
$69$	0	0
$70$	−12.0000	−1.43427
$71$	−7.34847	−0.872103	−0.436051	−	0.899922i	$-0.643623\pi$
$71$	−7.34847	−0.872103	−0.436051	+	0.899922i	$0.643623\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	−19.5959	−2.27798
$75$	0	0
$76$	−4.00000	−0.458831
$77$	−4.89898	−0.558291
$78$	0	0
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	9.79796	1.09545
$81$	0	0
$82$	12.0000	1.32518
$83$	12.2474	1.34433	0.672166	−	0.740400i	$-0.265364\pi$
$83$	12.2474	1.34433	0.672166	+	0.740400i	$0.265364\pi$
$84$	0	0
$85$	18.0000	1.95237
$86$	−26.9444	−2.90549
$87$	0	0
$88$	12.0000	1.27920
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	9.79796	1.02151
$93$	0	0
$94$	24.0000	2.47541
$95$	−2.44949	−0.251312
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	7.34847	0.742307
$99$	0	0
$100$	4.00000	0.400000
$101$	4.89898	0.487467	0.243733	−	0.969842i	$-0.421628\pi$
$101$	4.89898	0.487467	0.243733	+	0.969842i	$0.421628\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	4.89898	0.480384
$105$	0	0
$106$	18.0000	1.74831
$107$	−14.6969	−1.42081	−0.710403	−	0.703795i	$-0.751487\pi$
$107$	−14.6969	−1.42081	−0.710403	+	0.703795i	$0.751487\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	14.6969	1.40130
$111$	0	0
$112$	8.00000	0.755929
$113$	9.79796	0.921714	0.460857	−	0.887474i	$-0.347542\pi$
$113$	9.79796	0.921714	0.460857	+	0.887474i	$0.347542\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	19.5959	1.81944
$117$	0	0
$118$	−6.00000	−0.552345
$119$	14.6969	1.34727
$120$	0	0
$121$	−5.00000	−0.454545
$122$	−12.2474	−1.10883
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−9.79796	−0.876356
$126$	0	0
$127$	−19.0000	−1.68598	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000	−1.68598	−0.842989	+	0.537931i	$0.819206\pi$
$128$	19.5959	1.73205
$129$	0	0
$130$	6.00000	0.526235
$131$	−12.2474	−1.07006	−0.535032	−	0.844832i	$-0.679701\pi$
$131$	−12.2474	−1.07006	−0.535032	+	0.844832i	$0.679701\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	17.1464	1.48123
$135$	0	0
$136$	−36.0000	−3.08697
$137$	−9.79796	−0.837096	−0.418548	−	0.908195i	$-0.637461\pi$
$137$	−9.79796	−0.837096	−0.418548	+	0.908195i	$0.637461\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	19.5959	1.65616
$141$	0	0
$142$	18.0000	1.51053
$143$	2.44949	0.204837
$144$	0	0
$145$	12.0000	0.996546
$146$	−26.9444	−2.22993
$147$	0	0
$148$	32.0000	2.63038
$149$	−12.2474	−1.00335	−0.501675	−	0.865056i	$-0.667283\pi$
$149$	−12.2474	−1.00335	−0.501675	+	0.865056i	$0.667283\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	4.89898	0.397360
$153$	0	0
$154$	12.0000	0.966988
$155$	−2.44949	−0.196748
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	17.1464	1.36410
$159$	0	0
$160$	0	0
$161$	4.89898	0.386094
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	−19.5959	−1.53018
$165$	0	0
$166$	−30.0000	−2.32845
$167$	−4.89898	−0.379094	−0.189547	−	0.981872i	$-0.560702\pi$
$167$	−4.89898	−0.379094	−0.189547	+	0.981872i	$0.560702\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−44.0908	−3.38161
$171$	0	0
$172$	44.0000	3.35497
$173$	−9.79796	−0.744925	−0.372463	−	0.928047i	$-0.621486\pi$
$173$	−9.79796	−0.744925	−0.372463	+	0.928047i	$0.621486\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	−9.79796	−0.738549
$177$	0	0
$178$	0	0
$179$	14.6969	1.09850	0.549250	−	0.835658i	$-0.314913\pi$
$179$	14.6969	1.09850	0.549250	+	0.835658i	$0.314913\pi$
$180$	0	0
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	4.89898	0.363137
$183$	0	0
$184$	−12.0000	−0.884652
$185$	19.5959	1.44072
$186$	0	0
$187$	−18.0000	−1.31629
$188$	−39.1918	−2.85836
$189$	0	0
$190$	6.00000	0.435286
$191$	−9.79796	−0.708955	−0.354478	−	0.935064i	$-0.615341\pi$
$191$	−9.79796	−0.708955	−0.354478	+	0.935064i	$0.615341\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	17.1464	1.23104
$195$	0	0
$196$	−12.0000	−0.857143
$197$	−14.6969	−1.04711	−0.523557	−	0.851991i	$-0.675395\pi$
$197$	−14.6969	−1.04711	−0.523557	+	0.851991i	$0.675395\pi$
$198$	0	0
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	−4.89898	−0.346410
$201$	0	0
$202$	−12.0000	−0.844317
$203$	9.79796	0.687682
$204$	0	0
$205$	−12.0000	−0.838116
$206$	17.1464	1.19465
$207$	0	0
$208$	−4.00000	−0.277350
$209$	2.44949	0.169435
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	−29.3939	−2.01878
$213$	0	0
$214$	36.0000	2.46091
$215$	26.9444	1.83759
$216$	0	0
$217$	−2.00000	−0.135769
$218$	2.44949	0.165900
$219$	0	0
$220$	−24.0000	−1.61808
$221$	−7.34847	−0.494312
$222$	0	0
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	0	0
$225$	0	0
$226$	−24.0000	−1.59646
$227$	−9.79796	−0.650313	−0.325157	−	0.945660i	$-0.605417\pi$
$227$	−9.79796	−0.650313	−0.325157	+	0.945660i	$0.605417\pi$
$228$	0	0
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	−14.6969	−0.969087
$231$	0	0
$232$	−24.0000	−1.57568
$233$	7.34847	0.481414	0.240707	−	0.970598i	$-0.422621\pi$
$233$	7.34847	0.481414	0.240707	+	0.970598i	$0.422621\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	9.79796	0.637793
$237$	0	0
$238$	−36.0000	−2.33353
$239$	2.44949	0.158444	0.0792222	−	0.996857i	$-0.474756\pi$
$239$	2.44949	0.158444	0.0792222	+	0.996857i	$0.474756\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	12.2474	0.787296
$243$	0	0
$244$	20.0000	1.28037
$245$	−7.34847	−0.469476
$246$	0	0
$247$	1.00000	0.0636285
$248$	4.89898	0.311086
$249$	0	0
$250$	24.0000	1.51789
$251$	7.34847	0.463831	0.231916	−	0.972736i	$-0.425501\pi$
$251$	7.34847	0.463831	0.231916	+	0.972736i	$0.425501\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	46.5403	2.92020
$255$	0	0
$256$	−32.0000	−2.00000
$257$	17.1464	1.06956	0.534782	−	0.844990i	$-0.320394\pi$
$257$	17.1464	1.06956	0.534782	+	0.844990i	$0.320394\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	−9.79796	−0.607644
$261$	0	0
$262$	30.0000	1.85341
$263$	26.9444	1.66146	0.830731	−	0.556674i	$-0.187923\pi$
$263$	26.9444	1.66146	0.830731	+	0.556674i	$0.187923\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	4.89898	0.300376
$267$	0	0
$268$	−28.0000	−1.71037
$269$	22.0454	1.34413	0.672066	−	0.740491i	$-0.265407\pi$
$269$	22.0454	1.34413	0.672066	+	0.740491i	$0.265407\pi$
$270$	0	0
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	29.3939	1.78227
$273$	0	0
$274$	24.0000	1.44989
$275$	−2.44949	−0.147710
$276$	0	0
$277$	11.0000	0.660926	0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$278$	24.4949	1.46911
$279$	0	0
$280$	−24.0000	−1.43427
$281$	12.2474	0.730622	0.365311	−	0.930886i	$-0.380963\pi$
$281$	12.2474	0.730622	0.365311	+	0.930886i	$0.380963\pi$
$282$	0	0
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	−29.3939	−1.74421
$285$	0	0
$286$	−6.00000	−0.354787
$287$	−9.79796	−0.578355
$288$	0	0
$289$	37.0000	2.17647
$290$	−29.3939	−1.72607
$291$	0	0
$292$	44.0000	2.57491
$293$	−4.89898	−0.286201	−0.143101	−	0.989708i	$-0.545707\pi$
$293$	−4.89898	−0.286201	−0.143101	+	0.989708i	$0.545707\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	−39.1918	−2.27798
$297$	0	0
$298$	30.0000	1.73785
$299$	−2.44949	−0.141658
$300$	0	0
$301$	22.0000	1.26806
$302$	−12.2474	−0.704761
$303$	0	0
$304$	−4.00000	−0.229416
$305$	12.2474	0.701287
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	−19.5959	−1.11658
$309$	0	0
$310$	6.00000	0.340777
$311$	24.4949	1.38898	0.694489	−	0.719503i	$-0.255630\pi$
$311$	24.4949	1.38898	0.694489	+	0.719503i	$0.255630\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	−41.6413	−2.34996
$315$	0	0
$316$	−28.0000	−1.57512
$317$	−9.79796	−0.550308	−0.275154	−	0.961400i	$-0.588729\pi$
$317$	−9.79796	−0.550308	−0.275154	+	0.961400i	$0.588729\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	−19.5959	−1.09545
$321$	0	0
$322$	−12.0000	−0.668734
$323$	−7.34847	−0.408880
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	24.4949	1.35665
$327$	0	0
$328$	24.0000	1.32518
$329$	−19.5959	−1.08036
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	48.9898	2.68866
$333$	0	0
$334$	12.0000	0.656611
$335$	−17.1464	−0.936809
$336$	0	0
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	29.3939	1.59882
$339$	0	0
$340$	72.0000	3.90475
$341$	2.44949	0.132647
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−53.8888	−2.90549
$345$	0	0
$346$	24.0000	1.29025
$347$	24.4949	1.31495	0.657477	−	0.753474i	$-0.271623\pi$
$347$	24.4949	1.31495	0.657477	+	0.753474i	$0.271623\pi$
$348$	0	0
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	−4.89898	−0.261861
$351$	0	0
$352$	0	0
$353$	−2.44949	−0.130373	−0.0651866	−	0.997873i	$-0.520764\pi$
$353$	−2.44949	−0.130373	−0.0651866	+	0.997873i	$0.520764\pi$
$354$	0	0
$355$	−18.0000	−0.955341
$356$	0	0
$357$	0	0
$358$	−36.0000	−1.90266
$359$	−29.3939	−1.55135	−0.775675	−	0.631133i	$-0.782590\pi$
$359$	−29.3939	−1.55135	−0.775675	+	0.631133i	$0.782590\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−19.5959	−1.02994
$363$	0	0
$364$	−8.00000	−0.419314
$365$	26.9444	1.41033
$366$	0	0
$367$	5.00000	0.260998	0.130499	−	0.991448i	$-0.458342\pi$
$367$	5.00000	0.260998	0.130499	+	0.991448i	$0.458342\pi$
$368$	9.79796	0.510754
$369$	0	0
$370$	−48.0000	−2.49540
$371$	−14.6969	−0.763027
$372$	0	0
$373$	35.0000	1.81223	0.906116	−	0.423030i	$-0.139034\pi$
$373$	35.0000	1.81223	0.906116	+	0.423030i	$0.139034\pi$
$374$	44.0908	2.27988
$375$	0	0
$376$	48.0000	2.47541
$377$	−4.89898	−0.252310
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	−9.79796	−0.502625
$381$	0	0
$382$	24.0000	1.22795
$383$	−34.2929	−1.75228	−0.876142	−	0.482054i	$-0.839891\pi$
$383$	−34.2929	−1.75228	−0.876142	+	0.482054i	$0.839891\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	−26.9444	−1.37143
$387$	0	0
$388$	−28.0000	−1.42148
$389$	26.9444	1.36613	0.683067	−	0.730355i	$-0.260646\pi$
$389$	26.9444	1.36613	0.683067	+	0.730355i	$0.260646\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	14.6969	0.742307
$393$	0	0
$394$	36.0000	1.81365
$395$	−17.1464	−0.862730
$396$	0	0
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	2.44949	0.122782
$399$	0	0
$400$	4.00000	0.200000
$401$	−34.2929	−1.71250	−0.856252	−	0.516559i	$-0.827213\pi$
$401$	−34.2929	−1.71250	−0.856252	+	0.516559i	$0.827213\pi$
$402$	0	0
$403$	1.00000	0.0498135
$404$	19.5959	0.974933
$405$	0	0
$406$	−24.0000	−1.19110
$407$	−19.5959	−0.971334
$408$	0	0
$409$	−28.0000	−1.38451	−0.692255	−	0.721653i	$-0.743383\pi$
$409$	−28.0000	−1.38451	−0.692255	+	0.721653i	$0.743383\pi$
$410$	29.3939	1.45166
$411$	0	0
$412$	−28.0000	−1.37946
$413$	4.89898	0.241063
$414$	0	0
$415$	30.0000	1.47264
$416$	0	0
$417$	0	0
$418$	−6.00000	−0.293470
$419$	−34.2929	−1.67532	−0.837658	−	0.546195i	$-0.816076\pi$
$419$	−34.2929	−1.67532	−0.837658	+	0.546195i	$0.816076\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	2.44949	0.119239
$423$	0	0
$424$	36.0000	1.74831
$425$	7.34847	0.356453
$426$	0	0
$427$	10.0000	0.483934
$428$	−58.7878	−2.84161
$429$	0	0
$430$	−66.0000	−3.18280
$431$	7.34847	0.353963	0.176982	−	0.984214i	$-0.443367\pi$
$431$	7.34847	0.353963	0.176982	+	0.984214i	$0.443367\pi$
$432$	0	0
$433$	17.0000	0.816968	0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000	0.816968	0.408484	+	0.912766i	$0.366058\pi$
$434$	4.89898	0.235159
$435$	0	0
$436$	−4.00000	−0.191565
$437$	−2.44949	−0.117175
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	29.3939	1.40130
$441$	0	0
$442$	18.0000	0.856173
$443$	12.2474	0.581894	0.290947	−	0.956739i	$-0.406030\pi$
$443$	12.2474	0.581894	0.290947	+	0.956739i	$0.406030\pi$
$444$	0	0
$445$	0	0
$446$	17.1464	0.811907
$447$	0	0
$448$	−16.0000	−0.755929
$449$	−22.0454	−1.04039	−0.520194	−	0.854048i	$-0.674140\pi$
$449$	−22.0454	−1.04039	−0.520194	+	0.854048i	$0.674140\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	39.1918	1.84343
$453$	0	0
$454$	24.0000	1.12638
$455$	−4.89898	−0.229668
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	2.44949	0.114457
$459$	0	0
$460$	24.0000	1.11901
$461$	26.9444	1.25493	0.627463	−	0.778647i	$-0.284094\pi$
$461$	26.9444	1.25493	0.627463	+	0.778647i	$0.284094\pi$
$462$	0	0
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	19.5959	0.909718
$465$	0	0
$466$	−18.0000	−0.833834
$467$	−14.6969	−0.680093	−0.340047	−	0.940409i	$-0.610443\pi$
$467$	−14.6969	−0.680093	−0.340047	+	0.940409i	$0.610443\pi$
$468$	0	0
$469$	−14.0000	−0.646460
$470$	58.7878	2.71168
$471$	0	0
$472$	−12.0000	−0.552345
$473$	−26.9444	−1.23890
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	58.7878	2.69453
$477$	0	0
$478$	−6.00000	−0.274434
$479$	26.9444	1.23112	0.615560	−	0.788090i	$-0.288930\pi$
$479$	26.9444	1.23112	0.615560	+	0.788090i	$0.288930\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	39.1918	1.78514
$483$	0	0
$484$	−20.0000	−0.909091
$485$	−17.1464	−0.778579
$486$	0	0
$487$	35.0000	1.58600	0.793001	−	0.609221i	$-0.208518\pi$
$487$	35.0000	1.58600	0.793001	+	0.609221i	$0.208518\pi$
$488$	−24.4949	−1.10883
$489$	0	0
$490$	18.0000	0.813157
$491$	39.1918	1.76870	0.884351	−	0.466822i	$-0.154601\pi$
$491$	39.1918	1.76870	0.884351	+	0.466822i	$0.154601\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	−2.44949	−0.110208
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−14.6969	−0.659248
$498$	0	0
$499$	2.00000	0.0895323	0.0447661	−	0.998997i	$-0.485746\pi$
$499$	2.00000	0.0895323	0.0447661	+	0.998997i	$0.485746\pi$
$500$	−39.1918	−1.75271
$501$	0	0
$502$	−18.0000	−0.803379
$503$	14.6969	0.655304	0.327652	−	0.944798i	$-0.393743\pi$
$503$	14.6969	0.655304	0.327652	+	0.944798i	$0.393743\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	14.6969	0.653359
$507$	0	0
$508$	−76.0000	−3.37195
$509$	9.79796	0.434287	0.217143	−	0.976140i	$-0.430326\pi$
$509$	9.79796	0.434287	0.217143	+	0.976140i	$0.430326\pi$
$510$	0	0
$511$	22.0000	0.973223
$512$	39.1918	1.73205
$513$	0	0
$514$	−42.0000	−1.85254
$515$	−17.1464	−0.755562
$516$	0	0
$517$	24.0000	1.05552
$518$	−39.1918	−1.72199
$519$	0	0
$520$	12.0000	0.526235
$521$	−22.0454	−0.965827	−0.482913	−	0.875668i	$-0.660421\pi$
$521$	−22.0454	−0.965827	−0.482913	+	0.875668i	$0.660421\pi$
$522$	0	0
$523$	−25.0000	−1.09317	−0.546587	−	0.837402i	$-0.684073\pi$
$523$	−25.0000	−1.09317	−0.546587	+	0.837402i	$0.684073\pi$
$524$	−48.9898	−2.14013
$525$	0	0
$526$	−66.0000	−2.87774
$527$	−7.34847	−0.320104
$528$	0	0
$529$	−17.0000	−0.739130
$530$	44.0908	1.91518
$531$	0	0
$532$	−8.00000	−0.346844
$533$	4.89898	0.212198
$534$	0	0
$535$	−36.0000	−1.55642
$536$	34.2929	1.48123
$537$	0	0
$538$	−54.0000	−2.32811
$539$	7.34847	0.316521
$540$	0	0
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	17.1464	0.736502
$543$	0	0
$544$	0	0
$545$	−2.44949	−0.104925
$546$	0	0
$547$	−13.0000	−0.555840	−0.277920	−	0.960604i	$-0.589645\pi$
$547$	−13.0000	−0.555840	−0.277920	+	0.960604i	$0.589645\pi$
$548$	−39.1918	−1.67419
$549$	0	0
$550$	6.00000	0.255841
$551$	−4.89898	−0.208704
$552$	0	0
$553$	−14.0000	−0.595341
$554$	−26.9444	−1.14476
$555$	0	0
$556$	−40.0000	−1.69638
$557$	−7.34847	−0.311365	−0.155682	−	0.987807i	$-0.549758\pi$
$557$	−7.34847	−0.311365	−0.155682	+	0.987807i	$0.549758\pi$
$558$	0	0
$559$	−11.0000	−0.465250
$560$	19.5959	0.828079
$561$	0	0
$562$	−30.0000	−1.26547
$563$	−12.2474	−0.516168	−0.258084	−	0.966122i	$-0.583091\pi$
$563$	−12.2474	−0.516168	−0.258084	+	0.966122i	$0.583091\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	−41.6413	−1.75032
$567$	0	0
$568$	36.0000	1.51053
$569$	12.2474	0.513440	0.256720	−	0.966486i	$-0.417358\pi$
$569$	12.2474	0.513440	0.256720	+	0.966486i	$0.417358\pi$
$570$	0	0
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	9.79796	0.409673
$573$	0	0
$574$	24.0000	1.00174
$575$	2.44949	0.102151
$576$	0	0
$577$	−25.0000	−1.04076	−0.520382	−	0.853934i	$-0.674210\pi$
$577$	−25.0000	−1.04076	−0.520382	+	0.853934i	$0.674210\pi$
$578$	−90.6311	−3.76976
$579$	0	0
$580$	48.0000	1.99309
$581$	24.4949	1.01622
$582$	0	0
$583$	18.0000	0.745484
$584$	−53.8888	−2.22993
$585$	0	0
$586$	12.0000	0.495715
$587$	−2.44949	−0.101101	−0.0505506	−	0.998721i	$-0.516098\pi$
$587$	−2.44949	−0.101101	−0.0505506	+	0.998721i	$0.516098\pi$
$588$	0	0
$589$	1.00000	0.0412043
$590$	−14.6969	−0.605063
$591$	0	0
$592$	32.0000	1.31519
$593$	−7.34847	−0.301765	−0.150883	−	0.988552i	$-0.548212\pi$
$593$	−7.34847	−0.301765	−0.150883	+	0.988552i	$0.548212\pi$
$594$	0	0
$595$	36.0000	1.47586
$596$	−48.9898	−2.00670
$597$	0	0
$598$	6.00000	0.245358
$599$	39.1918	1.60134	0.800668	−	0.599109i	$-0.204478\pi$
$599$	39.1918	1.60134	0.800668	+	0.599109i	$0.204478\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	−53.8888	−2.19634
$603$	0	0
$604$	20.0000	0.813788
$605$	−12.2474	−0.497930
$606$	0	0
$607$	44.0000	1.78590	0.892952	−	0.450151i	$-0.148630\pi$
$607$	44.0000	1.78590	0.892952	+	0.450151i	$0.148630\pi$
$608$	0	0
$609$	0	0
$610$	−30.0000	−1.21466
$611$	9.79796	0.396383
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	−4.89898	−0.197707
$615$	0	0
$616$	24.0000	0.966988
$617$	24.4949	0.986127	0.493064	−	0.869993i	$-0.335877\pi$
$617$	24.4949	0.986127	0.493064	+	0.869993i	$0.335877\pi$
$618$	0	0
$619$	−49.0000	−1.96948	−0.984738	−	0.174042i	$-0.944317\pi$
$619$	−49.0000	−1.96948	−0.984738	+	0.174042i	$0.944317\pi$
$620$	−9.79796	−0.393496
$621$	0	0
$622$	−60.0000	−2.40578
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	39.1918	1.56642
$627$	0	0
$628$	68.0000	2.71350
$629$	58.7878	2.34402
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	34.2929	1.36410
$633$	0	0
$634$	24.0000	0.953162
$635$	−46.5403	−1.84690
$636$	0	0
$637$	3.00000	0.118864
$638$	29.3939	1.16371
$639$	0	0
$640$	48.0000	1.89737
$641$	−17.1464	−0.677243	−0.338622	−	0.940923i	$-0.609961\pi$
$641$	−17.1464	−0.677243	−0.338622	+	0.940923i	$0.609961\pi$
$642$	0	0
$643$	38.0000	1.49857	0.749287	−	0.662246i	$-0.230396\pi$
$643$	38.0000	1.49857	0.749287	+	0.662246i	$0.230396\pi$
$644$	19.5959	0.772187
$645$	0	0
$646$	18.0000	0.708201
$647$	−36.7423	−1.44449	−0.722245	−	0.691637i	$-0.756890\pi$
$647$	−36.7423	−1.44449	−0.722245	+	0.691637i	$0.756890\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	2.44949	0.0960769
$651$	0	0
$652$	−40.0000	−1.56652
$653$	9.79796	0.383424	0.191712	−	0.981451i	$-0.438596\pi$
$653$	9.79796	0.383424	0.191712	+	0.981451i	$0.438596\pi$
$654$	0	0
$655$	−30.0000	−1.17220
$656$	−19.5959	−0.765092
$657$	0	0
$658$	48.0000	1.87123
$659$	19.5959	0.763349	0.381674	−	0.924297i	$-0.375348\pi$
$659$	19.5959	0.763349	0.381674	+	0.924297i	$0.375348\pi$
$660$	0	0
$661$	11.0000	0.427850	0.213925	−	0.976850i	$-0.431375\pi$
$661$	11.0000	0.427850	0.213925	+	0.976850i	$0.431375\pi$
$662$	17.1464	0.666415
$663$	0	0
$664$	−60.0000	−2.32845
$665$	−4.89898	−0.189974
$666$	0	0
$667$	12.0000	0.464642
$668$	−19.5959	−0.758189
$669$	0	0
$670$	42.0000	1.62260
$671$	−12.2474	−0.472808
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	68.5857	2.64182
$675$	0	0
$676$	−48.0000	−1.84615
$677$	−46.5403	−1.78869	−0.894345	−	0.447379i	$-0.852358\pi$
$677$	−46.5403	−1.78869	−0.894345	+	0.447379i	$0.852358\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	−88.1816	−3.38161
$681$	0	0
$682$	−6.00000	−0.229752
$683$	22.0454	0.843544	0.421772	−	0.906702i	$-0.361408\pi$
$683$	22.0454	0.843544	0.421772	+	0.906702i	$0.361408\pi$
$684$	0	0
$685$	−24.0000	−0.916993
$686$	48.9898	1.87044
$687$	0	0
$688$	44.0000	1.67748
$689$	7.34847	0.279954
$690$	0	0
$691$	47.0000	1.78796	0.893982	−	0.448103i	$-0.147900\pi$
$691$	47.0000	1.78796	0.893982	+	0.448103i	$0.147900\pi$
$692$	−39.1918	−1.48985
$693$	0	0
$694$	−60.0000	−2.27757
$695$	−24.4949	−0.929144
$696$	0	0
$697$	−36.0000	−1.36360
$698$	−48.9898	−1.85429
$699$	0	0
$700$	8.00000	0.302372
$701$	14.6969	0.555096	0.277548	−	0.960712i	$-0.410478\pi$
$701$	14.6969	0.555096	0.277548	+	0.960712i	$0.410478\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	19.5959	0.738549
$705$	0	0
$706$	6.00000	0.225813
$707$	9.79796	0.368490
$708$	0	0
$709$	−7.00000	−0.262891	−0.131445	−	0.991323i	$-0.541962\pi$
$709$	−7.00000	−0.262891	−0.131445	+	0.991323i	$0.541962\pi$
$710$	44.0908	1.65470
$711$	0	0
$712$	0	0
$713$	−2.44949	−0.0917341
$714$	0	0
$715$	6.00000	0.224387
$716$	58.7878	2.19700
$717$	0	0
$718$	72.0000	2.68702
$719$	36.7423	1.37026	0.685129	−	0.728422i	$-0.259746\pi$
$719$	36.7423	1.37026	0.685129	+	0.728422i	$0.259746\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	44.0908	1.64089
$723$	0	0
$724$	32.0000	1.18927
$725$	4.89898	0.181944
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	9.79796	0.363137
$729$	0	0
$730$	−66.0000	−2.44277
$731$	80.8332	2.98972
$732$	0	0
$733$	17.0000	0.627909	0.313955	−	0.949438i	$-0.398346\pi$
$733$	17.0000	0.627909	0.313955	+	0.949438i	$0.398346\pi$
$734$	−12.2474	−0.452062
$735$	0	0
$736$	0	0
$737$	17.1464	0.631597
$738$	0	0
$739$	−1.00000	−0.0367856	−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000	−0.0367856	−0.0183928	+	0.999831i	$0.505855\pi$
$740$	78.3837	2.88144
$741$	0	0
$742$	36.0000	1.32160
$743$	31.8434	1.16822	0.584110	−	0.811675i	$-0.301444\pi$
$743$	31.8434	1.16822	0.584110	+	0.811675i	$0.301444\pi$
$744$	0	0
$745$	−30.0000	−1.09911
$746$	−85.7321	−3.13888
$747$	0	0
$748$	−72.0000	−2.63258
$749$	−29.3939	−1.07403
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	−39.1918	−1.42918
$753$	0	0
$754$	12.0000	0.437014
$755$	12.2474	0.445730
$756$	0	0
$757$	−7.00000	−0.254419	−0.127210	−	0.991876i	$-0.540602\pi$
$757$	−7.00000	−0.254419	−0.127210	+	0.991876i	$0.540602\pi$
$758$	−19.5959	−0.711756
$759$	0	0
$760$	12.0000	0.435286
$761$	2.44949	0.0887939	0.0443970	−	0.999014i	$-0.485863\pi$
$761$	2.44949	0.0887939	0.0443970	+	0.999014i	$0.485863\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	−39.1918	−1.41791
$765$	0	0
$766$	84.0000	3.03504
$767$	−2.44949	−0.0884459
$768$	0	0
$769$	−37.0000	−1.33425	−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000	−1.33425	−0.667127	+	0.744944i	$0.732476\pi$
$770$	29.3939	1.05928
$771$	0	0
$772$	44.0000	1.58359
$773$	44.0908	1.58584	0.792918	−	0.609328i	$-0.208561\pi$
$773$	44.0908	1.58584	0.792918	+	0.609328i	$0.208561\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	34.2929	1.23104
$777$	0	0
$778$	−66.0000	−2.36621
$779$	4.89898	0.175524
$780$	0	0
$781$	18.0000	0.644091
$782$	−44.0908	−1.57668
$783$	0	0
$784$	−12.0000	−0.428571
$785$	41.6413	1.48624
$786$	0	0
$787$	−25.0000	−0.891154	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	−25.0000	−0.891154	−0.445577	+	0.895244i	$0.647001\pi$
$788$	−58.7878	−2.09423
$789$	0	0
$790$	42.0000	1.49429
$791$	19.5959	0.696751
$792$	0	0
$793$	−5.00000	−0.177555
$794$	2.44949	0.0869291
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−41.6413	−1.47501	−0.737506	−	0.675341i	$-0.763997\pi$
$797$	−41.6413	−1.47501	−0.737506	+	0.675341i	$0.763997\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	0	0
$801$	0	0
$802$	84.0000	2.96614
$803$	−26.9444	−0.950847
$804$	0	0
$805$	12.0000	0.422944
$806$	−2.44949	−0.0862796
$807$	0	0
$808$	−24.0000	−0.844317
$809$	−22.0454	−0.775075	−0.387538	−	0.921854i	$-0.626674\pi$
$809$	−22.0454	−0.775075	−0.387538	+	0.921854i	$0.626674\pi$
$810$	0	0
$811$	35.0000	1.22902	0.614508	−	0.788911i	$-0.289355\pi$
$811$	35.0000	1.22902	0.614508	+	0.788911i	$0.289355\pi$
$812$	39.1918	1.37536
$813$	0	0
$814$	48.0000	1.68240
$815$	−24.4949	−0.858019
$816$	0	0
$817$	−11.0000	−0.384841
$818$	68.5857	2.39804
$819$	0	0
$820$	−48.0000	−1.67623
$821$	−39.1918	−1.36780	−0.683902	−	0.729574i	$-0.739719\pi$
$821$	−39.1918	−1.36780	−0.683902	+	0.729574i	$0.739719\pi$
$822$	0	0
$823$	35.0000	1.22002	0.610012	−	0.792392i	$-0.291165\pi$
$823$	35.0000	1.22002	0.610012	+	0.792392i	$0.291165\pi$
$824$	34.2929	1.19465
$825$	0	0
$826$	−12.0000	−0.417533
$827$	22.0454	0.766594	0.383297	−	0.923625i	$-0.374789\pi$
$827$	22.0454	0.766594	0.383297	+	0.923625i	$0.374789\pi$
$828$	0	0
$829$	−37.0000	−1.28506	−0.642532	−	0.766259i	$-0.722116\pi$
$829$	−37.0000	−1.28506	−0.642532	+	0.766259i	$0.722116\pi$
$830$	−73.4847	−2.55069
$831$	0	0
$832$	8.00000	0.277350
$833$	−22.0454	−0.763828
$834$	0	0
$835$	−12.0000	−0.415277
$836$	9.79796	0.338869
$837$	0	0
$838$	84.0000	2.90173
$839$	4.89898	0.169132	0.0845658	−	0.996418i	$-0.473050\pi$
$839$	4.89898	0.169132	0.0845658	+	0.996418i	$0.473050\pi$
$840$	0	0
$841$	−5.00000	−0.172414
$842$	−4.89898	−0.168830
$843$	0	0
$844$	−4.00000	−0.137686
$845$	−29.3939	−1.01118
$846$	0	0
$847$	−10.0000	−0.343604
$848$	−29.3939	−1.00939
$849$	0	0
$850$	−18.0000	−0.617395
$851$	19.5959	0.671739
$852$	0	0
$853$	−13.0000	−0.445112	−0.222556	−	0.974920i	$-0.571440\pi$
$853$	−13.0000	−0.445112	−0.222556	+	0.974920i	$0.571440\pi$
$854$	−24.4949	−0.838198
$855$	0	0
$856$	72.0000	2.46091
$857$	−24.4949	−0.836730	−0.418365	−	0.908279i	$-0.637397\pi$
$857$	−24.4949	−0.836730	−0.418365	+	0.908279i	$0.637397\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	107.778	3.67518
$861$	0	0
$862$	−18.0000	−0.613082
$863$	7.34847	0.250145	0.125072	−	0.992148i	$-0.460084\pi$
$863$	7.34847	0.250145	0.125072	+	0.992148i	$0.460084\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	−41.6413	−1.41503
$867$	0	0
$868$	−8.00000	−0.271538
$869$	17.1464	0.581653
$870$	0	0
$871$	7.00000	0.237186
$872$	4.89898	0.165900
$873$	0	0
$874$	6.00000	0.202953
$875$	−19.5959	−0.662463
$876$	0	0
$877$	8.00000	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$877$	8.00000	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$878$	−34.2929	−1.15733
$879$	0	0
$880$	−24.0000	−0.809040
$881$	−36.7423	−1.23788	−0.618941	−	0.785438i	$-0.712438\pi$
$881$	−36.7423	−1.23788	−0.618941	+	0.785438i	$0.712438\pi$
$882$	0	0
$883$	17.0000	0.572096	0.286048	−	0.958215i	$-0.407658\pi$
$883$	17.0000	0.572096	0.286048	+	0.958215i	$0.407658\pi$
$884$	−29.3939	−0.988623
$885$	0	0
$886$	−30.0000	−1.00787
$887$	17.1464	0.575721	0.287860	−	0.957672i	$-0.407056\pi$
$887$	17.1464	0.575721	0.287860	+	0.957672i	$0.407056\pi$
$888$	0	0
$889$	−38.0000	−1.27448
$890$	0	0
$891$	0	0
$892$	−28.0000	−0.937509
$893$	9.79796	0.327876
$894$	0	0
$895$	36.0000	1.20335
$896$	39.1918	1.30931
$897$	0	0
$898$	54.0000	1.80200
$899$	−4.89898	−0.163390
$900$	0	0
$901$	−54.0000	−1.79900
$902$	−29.3939	−0.978709
$903$	0	0
$904$	−48.0000	−1.59646
$905$	19.5959	0.651390
$906$	0	0
$907$	−7.00000	−0.232431	−0.116216	−	0.993224i	$-0.537076\pi$
$907$	−7.00000	−0.232431	−0.116216	+	0.993224i	$0.537076\pi$
$908$	−39.1918	−1.30063
$909$	0	0
$910$	12.0000	0.397796
$911$	12.2474	0.405776	0.202888	−	0.979202i	$-0.434967\pi$
$911$	12.2474	0.405776	0.202888	+	0.979202i	$0.434967\pi$
$912$	0	0
$913$	−30.0000	−0.992855
$914$	−71.0352	−2.34964
$915$	0	0
$916$	−4.00000	−0.132164
$917$	−24.4949	−0.808893
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	−29.3939	−0.969087
$921$	0	0
$922$	−66.0000	−2.17359
$923$	7.34847	0.241878
$924$	0	0
$925$	8.00000	0.263038
$926$	46.5403	1.52941
$927$	0	0
$928$	0	0
$929$	26.9444	0.884017	0.442008	−	0.897011i	$-0.354266\pi$
$929$	26.9444	0.884017	0.442008	+	0.897011i	$0.354266\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	29.3939	0.962828
$933$	0	0
$934$	36.0000	1.17796
$935$	−44.0908	−1.44192
$936$	0	0
$937$	8.00000	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$938$	34.2929	1.11970
$939$	0	0
$940$	−96.0000	−3.13117
$941$	9.79796	0.319404	0.159702	−	0.987165i	$-0.448947\pi$
$941$	9.79796	0.319404	0.159702	+	0.987165i	$0.448947\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	9.79796	0.318896
$945$	0	0
$946$	66.0000	2.14585
$947$	−24.4949	−0.795977	−0.397989	−	0.917390i	$-0.630292\pi$
$947$	−24.4949	−0.795977	−0.397989	+	0.917390i	$0.630292\pi$
$948$	0	0
$949$	−11.0000	−0.357075
$950$	2.44949	0.0794719
$951$	0	0
$952$	−72.0000	−2.33353
$953$	−29.3939	−0.952161	−0.476081	−	0.879402i	$-0.657943\pi$
$953$	−29.3939	−0.952161	−0.476081	+	0.879402i	$0.657943\pi$
$954$	0	0
$955$	−24.0000	−0.776622
$956$	9.79796	0.316889
$957$	0	0
$958$	−66.0000	−2.13236
$959$	−19.5959	−0.632785
$960$	0	0
$961$	−30.0000	−0.967742
$962$	19.5959	0.631798
$963$	0	0
$964$	−64.0000	−2.06130
$965$	26.9444	0.867371
$966$	0	0
$967$	−7.00000	−0.225105	−0.112552	−	0.993646i	$-0.535903\pi$
$967$	−7.00000	−0.225105	−0.112552	+	0.993646i	$0.535903\pi$
$968$	24.4949	0.787296
$969$	0	0
$970$	42.0000	1.34854
$971$	−29.3939	−0.943294	−0.471647	−	0.881787i	$-0.656340\pi$
$971$	−29.3939	−0.943294	−0.471647	+	0.881787i	$0.656340\pi$
$972$	0	0
$973$	−20.0000	−0.641171
$974$	−85.7321	−2.74703
$975$	0	0
$976$	20.0000	0.640184
$977$	24.4949	0.783661	0.391831	−	0.920037i	$-0.371842\pi$
$977$	24.4949	0.783661	0.391831	+	0.920037i	$0.371842\pi$
$978$	0	0
$979$	0	0
$980$	−29.3939	−0.938953
$981$	0	0
$982$	−96.0000	−3.06348
$983$	4.89898	0.156253	0.0781266	−	0.996943i	$-0.475106\pi$
$983$	4.89898	0.156253	0.0781266	+	0.996943i	$0.475106\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	−88.1816	−2.80828
$987$	0	0
$988$	4.00000	0.127257
$989$	26.9444	0.856782
$990$	0	0
$991$	−7.00000	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$992$	0	0
$993$	0	0
$994$	36.0000	1.14185
$995$	−2.44949	−0.0776540
$996$	0	0
$997$	50.0000	1.58352	0.791758	−	0.610835i	$-0.209166\pi$
$997$	50.0000	1.58352	0.791758	+	0.610835i	$0.209166\pi$
$998$	−4.89898	−0.155074
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.44949 q^{2} +4.00000 q^{4} +2.44949 q^{5} +2.00000 q^{7} -4.89898 q^{8} +O(q^{10})\)	\(q-2.44949 q^{2} +4.00000 q^{4} +2.44949 q^{5} +2.00000 q^{7} -4.89898 q^{8} -6.00000 q^{10} -2.44949 q^{11} -1.00000 q^{13} -4.89898 q^{14} +4.00000 q^{16} +7.34847 q^{17} -1.00000 q^{19} +9.79796 q^{20} +6.00000 q^{22} +2.44949 q^{23} +1.00000 q^{25} +2.44949 q^{26} +8.00000 q^{28} +4.89898 q^{29} -1.00000 q^{31} -18.0000 q^{34} +4.89898 q^{35} +8.00000 q^{37} +2.44949 q^{38} -12.0000 q^{40} -4.89898 q^{41} +11.0000 q^{43} -9.79796 q^{44} -6.00000 q^{46} -9.79796 q^{47} -3.00000 q^{49} -2.44949 q^{50} -4.00000 q^{52} -7.34847 q^{53} -6.00000 q^{55} -9.79796 q^{56} -12.0000 q^{58} +2.44949 q^{59} +5.00000 q^{61} +2.44949 q^{62} -8.00000 q^{64} -2.44949 q^{65} -7.00000 q^{67} +29.3939 q^{68} -12.0000 q^{70} -7.34847 q^{71} +11.0000 q^{73} -19.5959 q^{74} -4.00000 q^{76} -4.89898 q^{77} -7.00000 q^{79} +9.79796 q^{80} +12.0000 q^{82} +12.2474 q^{83} +18.0000 q^{85} -26.9444 q^{86} +12.0000 q^{88} -2.00000 q^{91} +9.79796 q^{92} +24.0000 q^{94} -2.44949 q^{95} -7.00000 q^{97} +7.34847 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{4} + 4 q^{7}+O(q^{10}) \) 2 * q + 8 * q^4 + 4 * q^7	\( 2 q + 8 q^{4} + 4 q^{7} - 12 q^{10} - 2 q^{13} + 8 q^{16} - 2 q^{19} + 12 q^{22} + 2 q^{25} + 16 q^{28} - 2 q^{31} - 36 q^{34} + 16 q^{37} - 24 q^{40} + 22 q^{43} - 12 q^{46} - 6 q^{49} - 8 q^{52} - 12 q^{55} - 24 q^{58} + 10 q^{61} - 16 q^{64} - 14 q^{67} - 24 q^{70} + 22 q^{73} - 8 q^{76} - 14 q^{79} + 24 q^{82} + 36 q^{85} + 24 q^{88} - 4 q^{91} + 48 q^{94} - 14 q^{97}+O(q^{100}) \) 2 * q + 8 * q^4 + 4 * q^7 - 12 * q^10 - 2 * q^13 + 8 * q^16 - 2 * q^19 + 12 * q^22 + 2 * q^25 + 16 * q^28 - 2 * q^31 - 36 * q^34 + 16 * q^37 - 24 * q^40 + 22 * q^43 - 12 * q^46 - 6 * q^49 - 8 * q^52 - 12 * q^55 - 24 * q^58 + 10 * q^61 - 16 * q^64 - 14 * q^67 - 24 * q^70 + 22 * q^73 - 8 * q^76 - 14 * q^79 + 24 * q^82 + 36 * q^85 + 24 * q^88 - 4 * q^91 + 48 * q^94 - 14 * q^97

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