LMFDB - Embedded newform 171.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,2,Mod(1,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("171.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	171.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:	$1.36544187456$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13068.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 1 $ x^4 - x^3 - 6*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.328543$ of defining polynomial
Character	$\chi$	$=$	171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.71519 q^{2} +5.37228 q^{4} -3.22060 q^{5} -2.37228 q^{7} +9.15640 q^{8} +O(q^{10})$	\(q+2.71519 q^{2} +5.37228 q^{4} -3.22060 q^{5} -2.37228 q^{7} +9.15640 q^{8} -8.74456 q^{10} -2.20979 q^{11} +2.00000 q^{13} -6.44121 q^{14} +14.1168 q^{16} -3.22060 q^{17} +1.00000 q^{19} -17.3020 q^{20} -6.00000 q^{22} +1.01082 q^{23} +5.37228 q^{25} +5.43039 q^{26} -12.7446 q^{28} +1.01082 q^{29} +4.74456 q^{31} +20.0172 q^{32} -8.74456 q^{34} +7.64018 q^{35} +10.7446 q^{37} +2.71519 q^{38} -29.4891 q^{40} -5.43039 q^{41} -11.1168 q^{43} -11.8716 q^{44} +2.74456 q^{46} -4.23142 q^{47} -1.37228 q^{49} +14.5868 q^{50} +10.7446 q^{52} +9.84996 q^{53} +7.11684 q^{55} -21.7216 q^{56} +2.74456 q^{58} +10.8608 q^{59} -5.11684 q^{61} +12.8824 q^{62} +26.1168 q^{64} -6.44121 q^{65} -4.00000 q^{67} -17.3020 q^{68} +20.7446 q^{70} +2.02163 q^{71} -5.11684 q^{73} +29.1736 q^{74} +5.37228 q^{76} +5.24224 q^{77} -4.00000 q^{79} -45.4647 q^{80} -14.7446 q^{82} -11.8716 q^{83} +10.3723 q^{85} -30.1844 q^{86} -20.2337 q^{88} +9.84996 q^{89} -4.74456 q^{91} +5.43039 q^{92} -11.4891 q^{94} -3.22060 q^{95} +7.48913 q^{97} -3.72601 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{4} + 2 q^{7}+O(q^{10}) $ 4 * q + 10 * q^4 + 2 * q^7	$ 4 q + 10 q^{4} + 2 q^{7} - 12 q^{10} + 8 q^{13} + 22 q^{16} + 4 q^{19} - 24 q^{22} + 10 q^{25} - 28 q^{28} - 4 q^{31} - 12 q^{34} + 20 q^{37} - 72 q^{40} - 10 q^{43} - 12 q^{46} + 6 q^{49} + 20 q^{52} - 6 q^{55} - 12 q^{58} + 14 q^{61} + 70 q^{64} - 16 q^{67} + 60 q^{70} + 14 q^{73} + 10 q^{76} - 16 q^{79} - 36 q^{82} + 30 q^{85} - 12 q^{88} + 4 q^{91} - 16 q^{97}+O(q^{100}) $ 4 * q + 10 * q^4 + 2 * q^7 - 12 * q^10 + 8 * q^13 + 22 * q^16 + 4 * q^19 - 24 * q^22 + 10 * q^25 - 28 * q^28 - 4 * q^31 - 12 * q^34 + 20 * q^37 - 72 * q^40 - 10 * q^43 - 12 * q^46 + 6 * q^49 + 20 * q^52 - 6 * q^55 - 12 * q^58 + 14 * q^61 + 70 * q^64 - 16 * q^67 + 60 * q^70 + 14 * q^73 + 10 * q^76 - 16 * q^79 - 36 * q^82 + 30 * q^85 - 12 * q^88 + 4 * q^91 - 16 * 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 $

$2$	2.71519	1.91993	0.959966	−	0.280116i	$-0.0903729\pi$
$2$	2.71519	1.91993	0.959966	+	0.280116i	$0.0903729\pi$
$3$	0	0
$3$	0	0
$4$	5.37228	2.68614
$5$	−3.22060	−1.44030	−0.720149	−	0.693820i	$-0.755926\pi$
$5$	−3.22060	−1.44030	−0.720149	+	0.693820i	$0.755926\pi$
$6$	0	0
$7$	−2.37228	−0.896638	−0.448319	−	0.893874i	$-0.647977\pi$
$7$	−2.37228	−0.896638	−0.448319	+	0.893874i	$0.647977\pi$
$8$	9.15640	3.23728
$9$	0	0
$10$	−8.74456	−2.76527
$11$	−2.20979	−0.666276	−0.333138	−	0.942878i	$-0.608107\pi$
$11$	−2.20979	−0.666276	−0.333138	+	0.942878i	$0.608107\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−6.44121	−1.72148
$15$	0	0
$16$	14.1168	3.52921
$17$	−3.22060	−0.781111	−0.390555	−	0.920579i	$-0.627717\pi$
$17$	−3.22060	−0.781111	−0.390555	+	0.920579i	$0.627717\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−17.3020	−3.86884
$21$	0	0
$22$	−6.00000	−1.27920
$23$	1.01082	0.210770	0.105385	−	0.994432i	$-0.466393\pi$
$23$	1.01082	0.210770	0.105385	+	0.994432i	$0.466393\pi$
$24$	0	0
$25$	5.37228	1.07446
$26$	5.43039	1.06499
$27$	0	0
$28$	−12.7446	−2.40850
$29$	1.01082	0.187704	0.0938519	−	0.995586i	$-0.470082\pi$
$29$	1.01082	0.187704	0.0938519	+	0.995586i	$0.470082\pi$
$30$	0	0
$31$	4.74456	0.852149	0.426074	−	0.904688i	$-0.359896\pi$
$31$	4.74456	0.852149	0.426074	+	0.904688i	$0.359896\pi$
$32$	20.0172	3.53857
$33$	0	0
$34$	−8.74456	−1.49968
$35$	7.64018	1.29143
$36$	0	0
$37$	10.7446	1.76640	0.883198	−	0.469001i	$-0.155386\pi$
$37$	10.7446	1.76640	0.883198	+	0.469001i	$0.155386\pi$
$38$	2.71519	0.440463
$39$	0	0
$40$	−29.4891	−4.66264
$41$	−5.43039	−0.848084	−0.424042	−	0.905642i	$-0.639389\pi$
$41$	−5.43039	−0.848084	−0.424042	+	0.905642i	$0.639389\pi$
$42$	0	0
$43$	−11.1168	−1.69530	−0.847651	−	0.530554i	$-0.821984\pi$
$43$	−11.1168	−1.69530	−0.847651	+	0.530554i	$0.821984\pi$
$44$	−11.8716	−1.78971
$45$	0	0
$46$	2.74456	0.404664
$47$	−4.23142	−0.617216	−0.308608	−	0.951189i	$-0.599863\pi$
$47$	−4.23142	−0.617216	−0.308608	+	0.951189i	$0.599863\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	14.5868	2.06288
$51$	0	0
$52$	10.7446	1.49000
$53$	9.84996	1.35300	0.676498	−	0.736444i	$-0.263497\pi$
$53$	9.84996	1.35300	0.676498	+	0.736444i	$0.263497\pi$
$54$	0	0
$55$	7.11684	0.959635
$56$	−21.7216	−2.90267
$57$	0	0
$58$	2.74456	0.360379
$59$	10.8608	1.41395	0.706976	−	0.707237i	$-0.250059\pi$
$59$	10.8608	1.41395	0.706976	+	0.707237i	$0.250059\pi$
$60$	0	0
$61$	−5.11684	−0.655145	−0.327572	−	0.944826i	$-0.606231\pi$
$61$	−5.11684	−0.655145	−0.327572	+	0.944826i	$0.606231\pi$
$62$	12.8824	1.63607
$63$	0	0
$64$	26.1168	3.26461
$65$	−6.44121	−0.798933
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−17.3020	−2.09817
$69$	0	0
$70$	20.7446	2.47945
$71$	2.02163	0.239924	0.119962	−	0.992779i	$-0.461723\pi$
$71$	2.02163	0.239924	0.119962	+	0.992779i	$0.461723\pi$
$72$	0	0
$73$	−5.11684	−0.598881	−0.299441	−	0.954115i	$-0.596800\pi$
$73$	−5.11684	−0.598881	−0.299441	+	0.954115i	$0.596800\pi$
$74$	29.1736	3.39136
$75$	0	0
$76$	5.37228	0.616243
$77$	5.24224	0.597408
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−45.4647	−5.08311
$81$	0	0
$82$	−14.7446	−1.62826
$83$	−11.8716	−1.30308	−0.651538	−	0.758616i	$-0.725876\pi$
$83$	−11.8716	−1.30308	−0.651538	+	0.758616i	$0.725876\pi$
$84$	0	0
$85$	10.3723	1.12503
$86$	−30.1844	−3.25487
$87$	0	0
$88$	−20.2337	−2.15692
$89$	9.84996	1.04409	0.522047	−	0.852917i	$-0.325169\pi$
$89$	9.84996	1.04409	0.522047	+	0.852917i	$0.325169\pi$
$90$	0	0
$91$	−4.74456	−0.497365
$92$	5.43039	0.566157
$93$	0	0
$94$	−11.4891	−1.18501
$95$	−3.22060	−0.330427
$96$	0	0
$97$	7.48913	0.760405	0.380203	−	0.924903i	$-0.375854\pi$
$97$	7.48913	0.760405	0.380203	+	0.924903i	$0.375854\pi$
$98$	−3.72601	−0.376384
$99$	0	0
$100$	28.8614	2.88614
$101$	12.8824	1.28185	0.640924	−	0.767604i	$-0.278551\pi$
$101$	12.8824	1.28185	0.640924	+	0.767604i	$0.278551\pi$
$102$	0	0
$103$	−7.25544	−0.714899	−0.357450	−	0.933932i	$-0.616354\pi$
$103$	−7.25544	−0.714899	−0.357450	+	0.933932i	$0.616354\pi$
$104$	18.3128	1.79572
$105$	0	0
$106$	26.7446	2.59766
$107$	4.41957	0.427256	0.213628	−	0.976915i	$-0.431472\pi$
$107$	4.41957	0.427256	0.213628	+	0.976915i	$0.431472\pi$
$108$	0	0
$109$	5.25544	0.503380	0.251690	−	0.967808i	$-0.419014\pi$
$109$	5.25544	0.503380	0.251690	+	0.967808i	$0.419014\pi$
$110$	19.3236	1.84243
$111$	0	0
$112$	−33.4891	−3.16442
$113$	−11.8716	−1.11679	−0.558393	−	0.829577i	$-0.688582\pi$
$113$	−11.8716	−1.11679	−0.558393	+	0.829577i	$0.688582\pi$
$114$	0	0
$115$	−3.25544	−0.303571
$116$	5.43039	0.504199
$117$	0	0
$118$	29.4891	2.71469
$119$	7.64018	0.700374
$120$	0	0
$121$	−6.11684	−0.556077
$122$	−13.8932	−1.25783
$123$	0	0
$124$	25.4891	2.28899
$125$	−1.19897	−0.107239
$126$	0	0
$127$	−12.7446	−1.13090	−0.565449	−	0.824784i	$-0.691297\pi$
$127$	−12.7446	−1.13090	−0.565449	+	0.824784i	$0.691297\pi$
$128$	30.8780	2.72925
$129$	0	0
$130$	−17.4891	−1.53390
$131$	−15.0922	−1.31861	−0.659306	−	0.751875i	$-0.729150\pi$
$131$	−15.0922	−1.31861	−0.659306	+	0.751875i	$0.729150\pi$
$132$	0	0
$133$	−2.37228	−0.205703
$134$	−10.8608	−0.938228
$135$	0	0
$136$	−29.4891	−2.52867
$137$	12.0597	1.03033	0.515167	−	0.857090i	$-0.327730\pi$
$137$	12.0597	1.03033	0.515167	+	0.857090i	$0.327730\pi$
$138$	0	0
$139$	3.11684	0.264367	0.132184	−	0.991225i	$-0.457801\pi$
$139$	3.11684	0.264367	0.132184	+	0.991225i	$0.457801\pi$
$140$	41.0452	3.46895
$141$	0	0
$142$	5.48913	0.460637
$143$	−4.41957	−0.369583
$144$	0	0
$145$	−3.25544	−0.270349
$146$	−13.8932	−1.14981
$147$	0	0
$148$	57.7228	4.74479
$149$	20.5226	1.68128	0.840638	−	0.541598i	$-0.182180\pi$
$149$	20.5226	1.68128	0.840638	+	0.541598i	$0.182180\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	9.15640	0.742682
$153$	0	0
$154$	14.2337	1.14698
$155$	−15.2804	−1.22735
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−10.8608	−0.864037
$159$	0	0
$160$	−64.4674	−5.09659
$161$	−2.39794	−0.188984
$162$	0	0
$163$	−21.4891	−1.68316	−0.841579	−	0.540134i	$-0.818374\pi$
$163$	−21.4891	−1.68316	−0.841579	+	0.540134i	$0.818374\pi$
$164$	−29.1736	−2.27807
$165$	0	0
$166$	−32.2337	−2.50182
$167$	−6.44121	−0.498435	−0.249218	−	0.968447i	$-0.580173\pi$
$167$	−6.44121	−0.498435	−0.249218	+	0.968447i	$0.580173\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	28.1628	2.15999
$171$	0	0
$172$	−59.7228	−4.55382
$173$	−1.01082	−0.0768509	−0.0384255	−	0.999261i	$-0.512234\pi$
$173$	−1.01082	−0.0768509	−0.0384255	+	0.999261i	$0.512234\pi$
$174$	0	0
$175$	−12.7446	−0.963398
$176$	−31.1952	−2.35143
$177$	0	0
$178$	26.7446	2.00459
$179$	−4.41957	−0.330334	−0.165167	−	0.986266i	$-0.552816\pi$
$179$	−4.41957	−0.330334	−0.165167	+	0.986266i	$0.552816\pi$
$180$	0	0
$181$	19.4891	1.44862	0.724308	−	0.689477i	$-0.242160\pi$
$181$	19.4891	1.44862	0.724308	+	0.689477i	$0.242160\pi$
$182$	−12.8824	−0.954908
$183$	0	0
$184$	9.25544	0.682320
$185$	−34.6040	−2.54413
$186$	0	0
$187$	7.11684	0.520435
$188$	−22.7324	−1.65793
$189$	0	0
$190$	−8.74456	−0.634397
$191$	−21.5334	−1.55810	−0.779051	−	0.626960i	$-0.784299\pi$
$191$	−21.5334	−1.55810	−0.779051	+	0.626960i	$0.784299\pi$
$192$	0	0
$193$	16.2337	1.16853	0.584263	−	0.811564i	$-0.301384\pi$
$193$	16.2337	1.16853	0.584263	+	0.811564i	$0.301384\pi$
$194$	20.3344	1.45993
$195$	0	0
$196$	−7.37228	−0.526592
$197$	−23.7432	−1.69163	−0.845816	−	0.533475i	$-0.820886\pi$
$197$	−23.7432	−1.69163	−0.845816	+	0.533475i	$0.820886\pi$
$198$	0	0
$199$	0.883156	0.0626053	0.0313026	−	0.999510i	$-0.490034\pi$
$199$	0.883156	0.0626053	0.0313026	+	0.999510i	$0.490034\pi$
$200$	49.1908	3.47831
$201$	0	0
$202$	34.9783	2.46106
$203$	−2.39794	−0.168302
$204$	0	0
$205$	17.4891	1.22149
$206$	−19.6999	−1.37256
$207$	0	0
$208$	28.2337	1.95765
$209$	−2.20979	−0.152854
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	52.9168	3.63434
$213$	0	0
$214$	12.0000	0.820303
$215$	35.8029	2.44174
$216$	0	0
$217$	−11.2554	−0.764069
$218$	14.2695	0.966455
$219$	0	0
$220$	38.2337	2.57771
$221$	−6.44121	−0.433282
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−47.4864	−3.17282
$225$	0	0
$226$	−32.2337	−2.14415
$227$	19.3236	1.28255	0.641277	−	0.767310i	$-0.278405\pi$
$227$	19.3236	1.28255	0.641277	+	0.767310i	$0.278405\pi$
$228$	0	0
$229$	15.6277	1.03271	0.516354	−	0.856375i	$-0.327289\pi$
$229$	15.6277	1.03271	0.516354	+	0.856375i	$0.327289\pi$
$230$	−8.83915	−0.582836
$231$	0	0
$232$	9.25544	0.607649
$233$	−7.64018	−0.500525	−0.250262	−	0.968178i	$-0.580517\pi$
$233$	−7.64018	−0.500525	−0.250262	+	0.968178i	$0.580517\pi$
$234$	0	0
$235$	13.6277	0.888974
$236$	58.3472	3.79808
$237$	0	0
$238$	20.7446	1.34467
$239$	−12.6943	−0.821123	−0.410562	−	0.911833i	$-0.634667\pi$
$239$	−12.6943	−0.821123	−0.410562	+	0.911833i	$0.634667\pi$
$240$	0	0
$241$	−24.2337	−1.56103	−0.780515	−	0.625138i	$-0.785043\pi$
$241$	−24.2337	−1.56103	−0.780515	+	0.625138i	$0.785043\pi$
$242$	−16.6084	−1.06763
$243$	0	0
$244$	−27.4891	−1.75981
$245$	4.41957	0.282356
$246$	0	0
$247$	2.00000	0.127257
$248$	43.4431	2.75864
$249$	0	0
$250$	−3.25544	−0.205892
$251$	23.9313	1.51053	0.755266	−	0.655418i	$-0.227507\pi$
$251$	23.9313	1.51053	0.755266	+	0.655418i	$0.227507\pi$
$252$	0	0
$253$	−2.23369	−0.140431
$254$	−34.6040	−2.17125
$255$	0	0
$256$	31.6060	1.97537
$257$	5.43039	0.338738	0.169369	−	0.985553i	$-0.445827\pi$
$257$	5.43039	0.338738	0.169369	+	0.985553i	$0.445827\pi$
$258$	0	0
$259$	−25.4891	−1.58382
$260$	−34.6040	−2.14605
$261$	0	0
$262$	−40.9783	−2.53164
$263$	13.0706	0.805966	0.402983	−	0.915208i	$-0.367973\pi$
$263$	13.0706	0.805966	0.402983	+	0.915208i	$0.367973\pi$
$264$	0	0
$265$	−31.7228	−1.94872
$266$	−6.44121	−0.394936
$267$	0	0
$268$	−21.4891	−1.31266
$269$	7.82833	0.477302	0.238651	−	0.971105i	$-0.423295\pi$
$269$	7.82833	0.477302	0.238651	+	0.971105i	$0.423295\pi$
$270$	0	0
$271$	25.4891	1.54835	0.774177	−	0.632969i	$-0.218164\pi$
$271$	25.4891	1.54835	0.774177	+	0.632969i	$0.218164\pi$
$272$	−45.4647	−2.75671
$273$	0	0
$274$	32.7446	1.97817
$275$	−11.8716	−0.715884
$276$	0	0
$277$	9.11684	0.547778	0.273889	−	0.961761i	$-0.411690\pi$
$277$	9.11684	0.547778	0.273889	+	0.961761i	$0.411690\pi$
$278$	8.46284	0.507567
$279$	0	0
$280$	69.9565	4.18070
$281$	24.7540	1.47670	0.738350	−	0.674418i	$-0.235605\pi$
$281$	24.7540	1.47670	0.738350	+	0.674418i	$0.235605\pi$
$282$	0	0
$283$	6.37228	0.378793	0.189396	−	0.981901i	$-0.439347\pi$
$283$	6.37228	0.378793	0.189396	+	0.981901i	$0.439347\pi$
$284$	10.8608	0.644469
$285$	0	0
$286$	−12.0000	−0.709575
$287$	12.8824	0.760425
$288$	0	0
$289$	−6.62772	−0.389866
$290$	−8.83915	−0.519053
$291$	0	0
$292$	−27.4891	−1.60868
$293$	−25.1303	−1.46813	−0.734064	−	0.679080i	$-0.762379\pi$
$293$	−25.1303	−1.46813	−0.734064	+	0.679080i	$0.762379\pi$
$294$	0	0
$295$	−34.9783	−2.03651
$296$	98.3815	5.71831
$297$	0	0
$298$	55.7228	3.22794
$299$	2.02163	0.116914
$300$	0	0
$301$	26.3723	1.52007
$302$	−10.8608	−0.624968
$303$	0	0
$304$	14.1168	0.809657
$305$	16.4793	0.943603
$306$	0	0
$307$	25.4891	1.45474	0.727371	−	0.686245i	$-0.240742\pi$
$307$	25.4891	1.45474	0.727371	+	0.686245i	$0.240742\pi$
$308$	28.1628	1.60472
$309$	0	0
$310$	−41.4891	−2.35642
$311$	12.6943	0.719825	0.359913	−	0.932986i	$-0.382806\pi$
$311$	12.6943	0.719825	0.359913	+	0.932986i	$0.382806\pi$
$312$	0	0
$313$	7.48913	0.423310	0.211655	−	0.977344i	$-0.432115\pi$
$313$	7.48913	0.423310	0.211655	+	0.977344i	$0.432115\pi$
$314$	5.43039	0.306455
$315$	0	0
$316$	−21.4891	−1.20886
$317$	−12.2479	−0.687911	−0.343955	−	0.938986i	$-0.611767\pi$
$317$	−12.2479	−0.687911	−0.343955	+	0.938986i	$0.611767\pi$
$318$	0	0
$319$	−2.23369	−0.125063
$320$	−84.1120	−4.70200
$321$	0	0
$322$	−6.51087	−0.362837
$323$	−3.22060	−0.179199
$324$	0	0
$325$	10.7446	0.596001
$326$	−58.3472	−3.23155
$327$	0	0
$328$	−49.7228	−2.74548
$329$	10.0381	0.553419
$330$	0	0
$331$	18.9783	1.04314	0.521569	−	0.853209i	$-0.325347\pi$
$331$	18.9783	1.04314	0.521569	+	0.853209i	$0.325347\pi$
$332$	−63.7775	−3.50025
$333$	0	0
$334$	−17.4891	−0.956962
$335$	12.8824	0.703841
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−24.4368	−1.32918
$339$	0	0
$340$	55.7228	3.02199
$341$	−10.4845	−0.567766
$342$	0	0
$343$	19.8614	1.07242
$344$	−101.790	−5.48816
$345$	0	0
$346$	−2.74456	−0.147549
$347$	32.3942	1.73901	0.869505	−	0.493924i	$-0.164438\pi$
$347$	32.3942	1.73901	0.869505	+	0.493924i	$0.164438\pi$
$348$	0	0
$349$	17.8614	0.956099	0.478050	−	0.878333i	$-0.341344\pi$
$349$	17.8614	0.956099	0.478050	+	0.878333i	$0.341344\pi$
$350$	−34.6040	−1.84966
$351$	0	0
$352$	−44.2337	−2.35766
$353$	14.9040	0.793262	0.396631	−	0.917978i	$-0.370179\pi$
$353$	14.9040	0.793262	0.396631	+	0.917978i	$0.370179\pi$
$354$	0	0
$355$	−6.51087	−0.345561
$356$	52.9168	2.80458
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−4.60773	−0.243187	−0.121593	−	0.992580i	$-0.538800\pi$
$359$	−4.60773	−0.243187	−0.121593	+	0.992580i	$0.538800\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	52.9168	2.78124
$363$	0	0
$364$	−25.4891	−1.33599
$365$	16.4793	0.862567
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	14.2695	0.743851
$369$	0	0
$370$	−93.9565	−4.88457
$371$	−23.3669	−1.21315
$372$	0	0
$373$	−24.2337	−1.25477	−0.627386	−	0.778708i	$-0.715875\pi$
$373$	−24.2337	−1.25477	−0.627386	+	0.778708i	$0.715875\pi$
$374$	19.3236	0.999200
$375$	0	0
$376$	−38.7446	−1.99810
$377$	2.02163	0.104119
$378$	0	0
$379$	28.7446	1.47651	0.738255	−	0.674522i	$-0.235650\pi$
$379$	28.7446	1.47651	0.738255	+	0.674522i	$0.235650\pi$
$380$	−17.3020	−0.887573
$381$	0	0
$382$	−58.4674	−2.99145
$383$	−17.3020	−0.884090	−0.442045	−	0.896993i	$-0.645747\pi$
$383$	−17.3020	−0.884090	−0.442045	+	0.896993i	$0.645747\pi$
$384$	0	0
$385$	−16.8832	−0.860445
$386$	44.0776	2.24349
$387$	0	0
$388$	40.2337	2.04256
$389$	12.0597	0.611454	0.305727	−	0.952119i	$-0.401101\pi$
$389$	12.0597	0.611454	0.305727	+	0.952119i	$0.401101\pi$
$390$	0	0
$391$	−3.25544	−0.164635
$392$	−12.5652	−0.634636
$393$	0	0
$394$	−64.4674	−3.24782
$395$	12.8824	0.648184
$396$	0	0
$397$	−17.1168	−0.859070	−0.429535	−	0.903050i	$-0.641322\pi$
$397$	−17.1168	−0.859070	−0.429535	+	0.903050i	$0.641322\pi$
$398$	2.39794	0.120198
$399$	0	0
$400$	75.8397	3.79198
$401$	3.40876	0.170225	0.0851126	−	0.996371i	$-0.472875\pi$
$401$	3.40876	0.170225	0.0851126	+	0.996371i	$0.472875\pi$
$402$	0	0
$403$	9.48913	0.472687
$404$	69.2079	3.44322
$405$	0	0
$406$	−6.51087	−0.323129
$407$	−23.7432	−1.17691
$408$	0	0
$409$	7.48913	0.370313	0.185157	−	0.982709i	$-0.440721\pi$
$409$	7.48913	0.370313	0.185157	+	0.982709i	$0.440721\pi$
$410$	47.4864	2.34519
$411$	0	0
$412$	−38.9783	−1.92032
$413$	−25.7648	−1.26780
$414$	0	0
$415$	38.2337	1.87682
$416$	40.0344	1.96285
$417$	0	0
$418$	−6.00000	−0.293470
$419$	11.8716	0.579965	0.289983	−	0.957032i	$-0.406350\pi$
$419$	11.8716	0.579965	0.289983	+	0.957032i	$0.406350\pi$
$420$	0	0
$421$	−8.97825	−0.437573	−0.218787	−	0.975773i	$-0.570210\pi$
$421$	−8.97825	−0.437573	−0.218787	+	0.975773i	$0.570210\pi$
$422$	−10.8608	−0.528694
$423$	0	0
$424$	90.1902	4.38002
$425$	−17.3020	−0.839269
$426$	0	0
$427$	12.1386	0.587428
$428$	23.7432	1.14767
$429$	0	0
$430$	97.2119	4.68798
$431$	−30.1844	−1.45393	−0.726966	−	0.686674i	$-0.759070\pi$
$431$	−30.1844	−1.45393	−0.726966	+	0.686674i	$0.759070\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	−30.5607	−1.46696
$435$	0	0
$436$	28.2337	1.35215
$437$	1.01082	0.0483539
$438$	0	0
$439$	−18.2337	−0.870246	−0.435123	−	0.900371i	$-0.643295\pi$
$439$	−18.2337	−0.870246	−0.435123	+	0.900371i	$0.643295\pi$
$440$	65.1647	3.10660
$441$	0	0
$442$	−17.4891	−0.831873
$443$	−27.9746	−1.32911	−0.664557	−	0.747238i	$-0.731380\pi$
$443$	−27.9746	−1.32911	−0.664557	+	0.747238i	$0.731380\pi$
$444$	0	0
$445$	−31.7228	−1.50381
$446$	−10.8608	−0.514273
$447$	0	0
$448$	−61.9565	−2.92717
$449$	22.3561	1.05505	0.527524	−	0.849540i	$-0.323120\pi$
$449$	22.3561	1.05505	0.527524	+	0.849540i	$0.323120\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	−63.7775	−2.99984
$453$	0	0
$454$	52.4674	2.46242
$455$	15.2804	0.716354
$456$	0	0
$457$	−8.37228	−0.391639	−0.195819	−	0.980640i	$-0.562737\pi$
$457$	−8.37228	−0.391639	−0.195819	+	0.980640i	$0.562737\pi$
$458$	42.4323	1.98273
$459$	0	0
$460$	−17.4891	−0.815435
$461$	−26.9638	−1.25583	−0.627914	−	0.778282i	$-0.716091\pi$
$461$	−26.9638	−1.25583	−0.627914	+	0.778282i	$0.716091\pi$
$462$	0	0
$463$	−2.37228	−0.110249	−0.0551246	−	0.998479i	$-0.517556\pi$
$463$	−2.37228	−0.110249	−0.0551246	+	0.998479i	$0.517556\pi$
$464$	14.2695	0.662447
$465$	0	0
$466$	−20.7446	−0.960973
$467$	−36.4374	−1.68612	−0.843062	−	0.537816i	$-0.819249\pi$
$467$	−36.4374	−1.68612	−0.843062	+	0.537816i	$0.819249\pi$
$468$	0	0
$469$	9.48913	0.438167
$470$	37.0019	1.70677
$471$	0	0
$472$	99.4456	4.57736
$473$	24.5659	1.12954
$474$	0	0
$475$	5.37228	0.246497
$476$	41.0452	1.88130
$477$	0	0
$478$	−34.4674	−1.57650
$479$	33.5932	1.53491	0.767455	−	0.641103i	$-0.221523\pi$
$479$	33.5932	1.53491	0.767455	+	0.641103i	$0.221523\pi$
$480$	0	0
$481$	21.4891	0.979820
$482$	−65.7992	−2.99707
$483$	0	0
$484$	−32.8614	−1.49370
$485$	−24.1195	−1.09521
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−46.8519	−2.12088
$489$	0	0
$490$	12.0000	0.542105
$491$	−1.01082	−0.0456175	−0.0228087	−	0.999740i	$-0.507261\pi$
$491$	−1.01082	−0.0456175	−0.0228087	+	0.999740i	$0.507261\pi$
$492$	0	0
$493$	−3.25544	−0.146618
$494$	5.43039	0.244325
$495$	0	0
$496$	66.9783	3.00741
$497$	−4.79588	−0.215125
$498$	0	0
$499$	−11.1168	−0.497658	−0.248829	−	0.968547i	$-0.580046\pi$
$499$	−11.1168	−0.497658	−0.248829	+	0.968547i	$0.580046\pi$
$500$	−6.44121	−0.288059
$501$	0	0
$502$	64.9783	2.90012
$503$	−31.5715	−1.40770	−0.703852	−	0.710346i	$-0.748538\pi$
$503$	−31.5715	−1.40770	−0.703852	+	0.710346i	$0.748538\pi$
$504$	0	0
$505$	−41.4891	−1.84624
$506$	−6.06490	−0.269618
$507$	0	0
$508$	−68.4674	−3.03775
$509$	24.7540	1.09720	0.548601	−	0.836084i	$-0.315161\pi$
$509$	24.7540	1.09720	0.548601	+	0.836084i	$0.315161\pi$
$510$	0	0
$511$	12.1386	0.536980
$512$	24.0604	1.06333
$513$	0	0
$514$	14.7446	0.650355
$515$	23.3669	1.02967
$516$	0	0
$517$	9.35053	0.411236
$518$	−69.2079	−3.04082
$519$	0	0
$520$	−58.9783	−2.58637
$521$	16.2912	0.713729	0.356864	−	0.934156i	$-0.383846\pi$
$521$	16.2912	0.713729	0.356864	+	0.934156i	$0.383846\pi$
$522$	0	0
$523$	−18.2337	−0.797304	−0.398652	−	0.917102i	$-0.630522\pi$
$523$	−18.2337	−0.797304	−0.398652	+	0.917102i	$0.630522\pi$
$524$	−81.0795	−3.54198
$525$	0	0
$526$	35.4891	1.54740
$527$	−15.2804	−0.665623
$528$	0	0
$529$	−21.9783	−0.955576
$530$	−86.1336	−3.74140
$531$	0	0
$532$	−12.7446	−0.552547
$533$	−10.8608	−0.470433
$534$	0	0
$535$	−14.2337	−0.615376
$536$	−36.6256	−1.58198
$537$	0	0
$538$	21.2554	0.916387
$539$	3.03245	0.130617
$540$	0	0
$541$	21.1168	0.907884	0.453942	−	0.891031i	$-0.350017\pi$
$541$	21.1168	0.907884	0.453942	+	0.891031i	$0.350017\pi$
$542$	69.2079	2.97274
$543$	0	0
$544$	−64.4674	−2.76402
$545$	−16.9257	−0.725016
$546$	0	0
$547$	22.2337	0.950644	0.475322	−	0.879812i	$-0.342332\pi$
$547$	22.2337	0.950644	0.475322	+	0.879812i	$0.342332\pi$
$548$	64.7884	2.76762
$549$	0	0
$550$	−32.2337	−1.37445
$551$	1.01082	0.0430622
$552$	0	0
$553$	9.48913	0.403519
$554$	24.7540	1.05170
$555$	0	0
$556$	16.7446	0.710128
$557$	−20.5226	−0.869570	−0.434785	−	0.900534i	$-0.643176\pi$
$557$	−20.5226	−0.869570	−0.434785	+	0.900534i	$0.643176\pi$
$558$	0	0
$559$	−22.2337	−0.940385
$560$	107.855	4.55771
$561$	0	0
$562$	67.2119	2.83516
$563$	−38.6472	−1.62879	−0.814393	−	0.580313i	$-0.802930\pi$
$563$	−38.6472	−1.62879	−0.814393	+	0.580313i	$0.802930\pi$
$564$	0	0
$565$	38.2337	1.60850
$566$	17.3020	0.727257
$567$	0	0
$568$	18.5109	0.776699
$569$	3.03245	0.127127	0.0635634	−	0.997978i	$-0.479753\pi$
$569$	3.03245	0.127127	0.0635634	+	0.997978i	$0.479753\pi$
$570$	0	0
$571$	2.51087	0.105077	0.0525384	−	0.998619i	$-0.483269\pi$
$571$	2.51087	0.105077	0.0525384	+	0.998619i	$0.483269\pi$
$572$	−23.7432	−0.992753
$573$	0	0
$574$	34.9783	1.45996
$575$	5.43039	0.226463
$576$	0	0
$577$	−41.1168	−1.71172	−0.855858	−	0.517210i	$-0.826970\pi$
$577$	−41.1168	−1.71172	−0.855858	+	0.517210i	$0.826970\pi$
$578$	−17.9955	−0.748516
$579$	0	0
$580$	−17.4891	−0.726196
$581$	28.1628	1.16839
$582$	0	0
$583$	−21.7663	−0.901469
$584$	−46.8519	−1.93874
$585$	0	0
$586$	−68.2337	−2.81871
$587$	−1.83348	−0.0756758	−0.0378379	−	0.999284i	$-0.512047\pi$
$587$	−1.83348	−0.0756758	−0.0378379	+	0.999284i	$0.512047\pi$
$588$	0	0
$589$	4.74456	0.195496
$590$	−94.9728	−3.90997
$591$	0	0
$592$	151.679	6.23398
$593$	4.04326	0.166037	0.0830185	−	0.996548i	$-0.473544\pi$
$593$	4.04326	0.166037	0.0830185	+	0.996548i	$0.473544\pi$
$594$	0	0
$595$	−24.6060	−1.00875
$596$	110.253	4.51614
$597$	0	0
$598$	5.48913	0.224467
$599$	12.8824	0.526361	0.263181	−	0.964747i	$-0.415229\pi$
$599$	12.8824	0.526361	0.263181	+	0.964747i	$0.415229\pi$
$600$	0	0
$601$	−25.2554	−1.03019	−0.515095	−	0.857133i	$-0.672244\pi$
$601$	−25.2554	−1.03019	−0.515095	+	0.857133i	$0.672244\pi$
$602$	71.6059	2.91844
$603$	0	0
$604$	−21.4891	−0.874380
$605$	19.6999	0.800916
$606$	0	0
$607$	28.7446	1.16671	0.583353	−	0.812219i	$-0.301740\pi$
$607$	28.7446	1.16671	0.583353	+	0.812219i	$0.301740\pi$
$608$	20.0172	0.811804
$609$	0	0
$610$	44.7446	1.81165
$611$	−8.46284	−0.342370
$612$	0	0
$613$	2.60597	0.105254	0.0526271	−	0.998614i	$-0.483241\pi$
$613$	2.60597	0.105254	0.0526271	+	0.998614i	$0.483241\pi$
$614$	69.2079	2.79300
$615$	0	0
$616$	48.0000	1.93398
$617$	3.22060	0.129657	0.0648283	−	0.997896i	$-0.479350\pi$
$617$	3.22060	0.129657	0.0648283	+	0.997896i	$0.479350\pi$
$618$	0	0
$619$	6.97825	0.280480	0.140240	−	0.990118i	$-0.455213\pi$
$619$	6.97825	0.280480	0.140240	+	0.990118i	$0.455213\pi$
$620$	−82.0903	−3.29683
$621$	0	0
$622$	34.4674	1.38202
$623$	−23.3669	−0.936174
$624$	0	0
$625$	−23.0000	−0.920000
$626$	20.3344	0.812727
$627$	0	0
$628$	10.7446	0.428755
$629$	−34.6040	−1.37975
$630$	0	0
$631$	15.1168	0.601792	0.300896	−	0.953657i	$-0.402714\pi$
$631$	15.1168	0.601792	0.300896	+	0.953657i	$0.402714\pi$
$632$	−36.6256	−1.45689
$633$	0	0
$634$	−33.2554	−1.32074
$635$	41.0452	1.62883
$636$	0	0
$637$	−2.74456	−0.108744
$638$	−6.06490	−0.240112
$639$	0	0
$640$	−99.4456	−3.93093
$641$	−24.7540	−0.977724	−0.488862	−	0.872361i	$-0.662588\pi$
$641$	−24.7540	−0.977724	−0.488862	+	0.872361i	$0.662588\pi$
$642$	0	0
$643$	−35.1168	−1.38487	−0.692437	−	0.721479i	$-0.743463\pi$
$643$	−35.1168	−1.38487	−0.692437	+	0.721479i	$0.743463\pi$
$644$	−12.8824	−0.507638
$645$	0	0
$646$	−8.74456	−0.344050
$647$	−17.1138	−0.672814	−0.336407	−	0.941717i	$-0.609212\pi$
$647$	−17.1138	−0.672814	−0.336407	+	0.941717i	$0.609212\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	29.1736	1.14428
$651$	0	0
$652$	−115.446	−4.52120
$653$	14.0814	0.551047	0.275524	−	0.961294i	$-0.411149\pi$
$653$	14.0814	0.551047	0.275524	+	0.961294i	$0.411149\pi$
$654$	0	0
$655$	48.6060	1.89919
$656$	−76.6600	−2.99307
$657$	0	0
$658$	27.2554	1.06253
$659$	11.2371	0.437735	0.218867	−	0.975755i	$-0.429764\pi$
$659$	11.2371	0.437735	0.218867	+	0.975755i	$0.429764\pi$
$660$	0	0
$661$	−6.74456	−0.262333	−0.131167	−	0.991360i	$-0.541872\pi$
$661$	−6.74456	−0.262333	−0.131167	+	0.991360i	$0.541872\pi$
$662$	51.5296	2.00276
$663$	0	0
$664$	−108.701	−4.21842
$665$	7.64018	0.296273
$666$	0	0
$667$	1.02175	0.0395623
$668$	−34.6040	−1.33887
$669$	0	0
$670$	34.9783	1.35133
$671$	11.3071	0.436507
$672$	0	0
$673$	−25.2554	−0.973526	−0.486763	−	0.873534i	$-0.661822\pi$
$673$	−25.2554	−0.973526	−0.486763	+	0.873534i	$0.661822\pi$
$674$	38.0127	1.46420
$675$	0	0
$676$	−48.3505	−1.85964
$677$	44.4539	1.70850	0.854252	−	0.519860i	$-0.174016\pi$
$677$	44.4539	1.70850	0.854252	+	0.519860i	$0.174016\pi$
$678$	0	0
$679$	−17.7663	−0.681808
$680$	94.9728	3.64204
$681$	0	0
$682$	−28.4674	−1.09007
$683$	34.2277	1.30968	0.654842	−	0.755765i	$-0.272735\pi$
$683$	34.2277	1.30968	0.654842	+	0.755765i	$0.272735\pi$
$684$	0	0
$685$	−38.8397	−1.48399
$686$	53.9276	2.05896
$687$	0	0
$688$	−156.935	−5.98308
$689$	19.6999	0.750507
$690$	0	0
$691$	27.1168	1.03157	0.515787	−	0.856717i	$-0.327500\pi$
$691$	27.1168	1.03157	0.515787	+	0.856717i	$0.327500\pi$
$692$	−5.43039	−0.206432
$693$	0	0
$694$	87.9565	3.33878
$695$	−10.0381	−0.380767
$696$	0	0
$697$	17.4891	0.662448
$698$	48.4972	1.83565
$699$	0	0
$700$	−68.4674	−2.58782
$701$	−30.5607	−1.15426	−0.577131	−	0.816652i	$-0.695828\pi$
$701$	−30.5607	−1.15426	−0.577131	+	0.816652i	$0.695828\pi$
$702$	0	0
$703$	10.7446	0.405239
$704$	−57.7126	−2.17513
$705$	0	0
$706$	40.4674	1.52301
$707$	−30.5607	−1.14935
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	−17.6783	−0.663454
$711$	0	0
$712$	90.1902	3.38002
$713$	4.79588	0.179607
$714$	0	0
$715$	14.2337	0.532310
$716$	−23.7432	−0.887325
$717$	0	0
$718$	−12.5109	−0.466902
$719$	−38.8354	−1.44832	−0.724158	−	0.689634i	$-0.757771\pi$
$719$	−38.8354	−1.44832	−0.724158	+	0.689634i	$0.757771\pi$
$720$	0	0
$721$	17.2119	0.641006
$722$	2.71519	0.101049
$723$	0	0
$724$	104.701	3.89118
$725$	5.43039	0.201680
$726$	0	0
$727$	−37.3505	−1.38525	−0.692627	−	0.721296i	$-0.743547\pi$
$727$	−37.3505	−1.38525	−0.692627	+	0.721296i	$0.743547\pi$
$728$	−43.4431	−1.61011
$729$	0	0
$730$	44.7446	1.65607
$731$	35.8029	1.32422
$732$	0	0
$733$	18.4674	0.682108	0.341054	−	0.940044i	$-0.389216\pi$
$733$	18.4674	0.682108	0.341054	+	0.940044i	$0.389216\pi$
$734$	21.7216	0.801757
$735$	0	0
$736$	20.2337	0.745824
$737$	8.83915	0.325594
$738$	0	0
$739$	27.1168	0.997509	0.498755	−	0.866743i	$-0.333791\pi$
$739$	27.1168	0.997509	0.498755	+	0.866743i	$0.333791\pi$
$740$	−185.902	−6.83390
$741$	0	0
$742$	−63.4456	−2.32916
$743$	28.5391	1.04700	0.523498	−	0.852027i	$-0.324627\pi$
$743$	28.5391	1.04700	0.523498	+	0.852027i	$0.324627\pi$
$744$	0	0
$745$	−66.0951	−2.42154
$746$	−65.7992	−2.40908
$747$	0	0
$748$	38.2337	1.39796
$749$	−10.4845	−0.383094
$750$	0	0
$751$	24.4674	0.892827	0.446414	−	0.894827i	$-0.352701\pi$
$751$	24.4674	0.892827	0.446414	+	0.894827i	$0.352701\pi$
$752$	−59.7343	−2.17829
$753$	0	0
$754$	5.48913	0.199902
$755$	12.8824	0.468839
$756$	0	0
$757$	−5.11684	−0.185975	−0.0929874	−	0.995667i	$-0.529642\pi$
$757$	−5.11684	−0.185975	−0.0929874	+	0.995667i	$0.529642\pi$
$758$	78.0471	2.83480
$759$	0	0
$760$	−29.4891	−1.06968
$761$	0.822662	0.0298215	0.0149107	−	0.999889i	$-0.495254\pi$
$761$	0.822662	0.0298215	0.0149107	+	0.999889i	$0.495254\pi$
$762$	0	0
$763$	−12.4674	−0.451349
$764$	−115.683	−4.18528
$765$	0	0
$766$	−46.9783	−1.69739
$767$	21.7216	0.784320
$768$	0	0
$769$	6.88316	0.248213	0.124106	−	0.992269i	$-0.460394\pi$
$769$	6.88316	0.248213	0.124106	+	0.992269i	$0.460394\pi$
$770$	−45.8411	−1.65200
$771$	0	0
$772$	87.2119	3.13883
$773$	5.43039	0.195318	0.0976588	−	0.995220i	$-0.468865\pi$
$773$	5.43039	0.195318	0.0976588	+	0.995220i	$0.468865\pi$
$774$	0	0
$775$	25.4891	0.915596
$776$	68.5734	2.46164
$777$	0	0
$778$	32.7446	1.17395
$779$	−5.43039	−0.194564
$780$	0	0
$781$	−4.46738	−0.159855
$782$	−8.83915	−0.316087
$783$	0	0
$784$	−19.3723	−0.691867
$785$	−6.44121	−0.229896
$786$	0	0
$787$	−49.9565	−1.78076	−0.890378	−	0.455221i	$-0.849560\pi$
$787$	−49.9565	−1.78076	−0.890378	+	0.455221i	$0.849560\pi$
$788$	−127.555	−4.54396
$789$	0	0
$790$	34.9783	1.24447
$791$	28.1628	1.00135
$792$	0	0
$793$	−10.2337	−0.363409
$794$	−46.4756	−1.64936
$795$	0	0
$796$	4.74456	0.168167
$797$	−7.45202	−0.263964	−0.131982	−	0.991252i	$-0.542134\pi$
$797$	−7.45202	−0.263964	−0.131982	+	0.991252i	$0.542134\pi$
$798$	0	0
$799$	13.6277	0.482114
$800$	107.538	3.80204
$801$	0	0
$802$	9.25544	0.326821
$803$	11.3071	0.399020
$804$	0	0
$805$	7.72281	0.272193
$806$	25.7648	0.907527
$807$	0	0
$808$	117.957	4.14970
$809$	−3.59691	−0.126461	−0.0632303	−	0.997999i	$-0.520140\pi$
$809$	−3.59691	−0.126461	−0.0632303	+	0.997999i	$0.520140\pi$
$810$	0	0
$811$	−36.7446	−1.29028	−0.645138	−	0.764066i	$-0.723200\pi$
$811$	−36.7446	−1.29028	−0.645138	+	0.764066i	$0.723200\pi$
$812$	−12.8824	−0.452084
$813$	0	0
$814$	−64.4674	−2.25958
$815$	69.2079	2.42425
$816$	0	0
$817$	−11.1168	−0.388929
$818$	20.3344	0.710977
$819$	0	0
$820$	93.9565	3.28110
$821$	−29.3617	−1.02473	−0.512366	−	0.858767i	$-0.671231\pi$
$821$	−29.3617	−1.02473	−0.512366	+	0.858767i	$0.671231\pi$
$822$	0	0
$823$	8.60597	0.299985	0.149993	−	0.988687i	$-0.452075\pi$
$823$	8.60597	0.299985	0.149993	+	0.988687i	$0.452075\pi$
$824$	−66.4337	−2.31433
$825$	0	0
$826$	−69.9565	−2.43410
$827$	−28.5391	−0.992401	−0.496200	−	0.868208i	$-0.665272\pi$
$827$	−28.5391	−0.992401	−0.496200	+	0.868208i	$0.665272\pi$
$828$	0	0
$829$	−1.25544	−0.0436031	−0.0218016	−	0.999762i	$-0.506940\pi$
$829$	−1.25544	−0.0436031	−0.0218016	+	0.999762i	$0.506940\pi$
$830$	103.812	3.60336
$831$	0	0
$832$	52.2337	1.81088
$833$	4.41957	0.153129
$834$	0	0
$835$	20.7446	0.717895
$836$	−11.8716	−0.410588
$837$	0	0
$838$	32.2337	1.11349
$839$	30.1844	1.04208	0.521041	−	0.853532i	$-0.325544\pi$
$839$	30.1844	1.04208	0.521041	+	0.853532i	$0.325544\pi$
$840$	0	0
$841$	−27.9783	−0.964767
$842$	−24.3777	−0.840111
$843$	0	0
$844$	−21.4891	−0.739686
$845$	28.9854	0.997129
$846$	0	0
$847$	14.5109	0.498600
$848$	139.050	4.77501
$849$	0	0
$850$	−46.9783	−1.61134
$851$	10.8608	0.372303
$852$	0	0
$853$	−38.4674	−1.31710	−0.658549	−	0.752538i	$-0.728829\pi$
$853$	−38.4674	−1.31710	−0.658549	+	0.752538i	$0.728829\pi$
$854$	32.9586	1.12782
$855$	0	0
$856$	40.4674	1.38315
$857$	−35.2385	−1.20372	−0.601862	−	0.798600i	$-0.705574\pi$
$857$	−35.2385	−1.20372	−0.601862	+	0.798600i	$0.705574\pi$
$858$	0	0
$859$	3.11684	0.106345	0.0531727	−	0.998585i	$-0.483067\pi$
$859$	3.11684	0.106345	0.0531727	+	0.998585i	$0.483067\pi$
$860$	192.343	6.55886
$861$	0	0
$862$	−81.9565	−2.79145
$863$	−14.9040	−0.507340	−0.253670	−	0.967291i	$-0.581638\pi$
$863$	−14.9040	−0.507340	−0.253670	+	0.967291i	$0.581638\pi$
$864$	0	0
$865$	3.25544	0.110688
$866$	−59.7343	−2.02985
$867$	0	0
$868$	−60.4674	−2.05240
$869$	8.83915	0.299847
$870$	0	0
$871$	−8.00000	−0.271070
$872$	48.1209	1.62958
$873$	0	0
$874$	2.74456	0.0928362
$875$	2.84429	0.0961547
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−49.5080	−1.67081
$879$	0	0
$880$	100.467	3.38675
$881$	−0.822662	−0.0277162	−0.0138581	−	0.999904i	$-0.504411\pi$
$881$	−0.822662	−0.0277162	−0.0138581	+	0.999904i	$0.504411\pi$
$882$	0	0
$883$	3.11684	0.104890	0.0524451	−	0.998624i	$-0.483299\pi$
$883$	3.11684	0.104890	0.0524451	+	0.998624i	$0.483299\pi$
$884$	−34.6040	−1.16386
$885$	0	0
$886$	−75.9565	−2.55181
$887$	21.7216	0.729338	0.364669	−	0.931137i	$-0.381182\pi$
$887$	21.7216	0.729338	0.364669	+	0.931137i	$0.381182\pi$
$888$	0	0
$889$	30.2337	1.01401
$890$	−86.1336	−2.88721
$891$	0	0
$892$	−21.4891	−0.719509
$893$	−4.23142	−0.141599
$894$	0	0
$895$	14.2337	0.475780
$896$	−73.2512	−2.44715
$897$	0	0
$898$	60.7011	2.02562
$899$	4.79588	0.159952
$900$	0	0
$901$	−31.7228	−1.05684
$902$	32.5823	1.08487
$903$	0	0
$904$	−108.701	−3.61534
$905$	−62.7667	−2.08644
$906$	0	0
$907$	23.2554	0.772184	0.386092	−	0.922460i	$-0.373825\pi$
$907$	23.2554	0.772184	0.386092	+	0.922460i	$0.373825\pi$
$908$	103.812	3.44512
$909$	0	0
$910$	41.4891	1.37535
$911$	−37.0019	−1.22593	−0.612964	−	0.790111i	$-0.710023\pi$
$911$	−37.0019	−1.22593	−0.612964	+	0.790111i	$0.710023\pi$
$912$	0	0
$913$	26.2337	0.868208
$914$	−22.7324	−0.751920
$915$	0	0
$916$	83.9565	2.77400
$917$	35.8029	1.18232
$918$	0	0
$919$	18.9783	0.626035	0.313017	−	0.949747i	$-0.398660\pi$
$919$	18.9783	0.626035	0.313017	+	0.949747i	$0.398660\pi$
$920$	−29.8081	−0.982743
$921$	0	0
$922$	−73.2119	−2.41111
$923$	4.04326	0.133086
$924$	0	0
$925$	57.7228	1.89791
$926$	−6.44121	−0.211671
$927$	0	0
$928$	20.2337	0.664203
$929$	4.79588	0.157348	0.0786739	−	0.996900i	$-0.474931\pi$
$929$	4.79588	0.157348	0.0786739	+	0.996900i	$0.474931\pi$
$930$	0	0
$931$	−1.37228	−0.0449747
$932$	−41.0452	−1.34448
$933$	0	0
$934$	−98.9348	−3.23724
$935$	−22.9205	−0.749581
$936$	0	0
$937$	−5.11684	−0.167160	−0.0835800	−	0.996501i	$-0.526635\pi$
$937$	−5.11684	−0.167160	−0.0835800	+	0.996501i	$0.526635\pi$
$938$	25.7648	0.841251
$939$	0	0
$940$	73.2119	2.38791
$941$	24.7540	0.806957	0.403479	−	0.914989i	$-0.367801\pi$
$941$	24.7540	0.806957	0.403479	+	0.914989i	$0.367801\pi$
$942$	0	0
$943$	−5.48913	−0.178751
$944$	153.320	4.99014
$945$	0	0
$946$	66.7011	2.16864
$947$	−9.84996	−0.320081	−0.160040	−	0.987110i	$-0.551162\pi$
$947$	−9.84996	−0.320081	−0.160040	+	0.987110i	$0.551162\pi$
$948$	0	0
$949$	−10.2337	−0.332200
$950$	14.5868	0.473258
$951$	0	0
$952$	69.9565	2.26730
$953$	9.47365	0.306882	0.153441	−	0.988158i	$-0.450965\pi$
$953$	9.47365	0.306882	0.153441	+	0.988158i	$0.450965\pi$
$954$	0	0
$955$	69.3505	2.24413
$956$	−68.1971	−2.20565
$957$	0	0
$958$	91.2119	2.94692
$959$	−28.6091	−0.923837
$960$	0	0
$961$	−8.48913	−0.273843
$962$	58.3472	1.88119
$963$	0	0
$964$	−130.190	−4.19314
$965$	−52.2823	−1.68303
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−56.0083	−1.80017
$969$	0	0
$970$	−65.4891	−2.10273
$971$	−51.5296	−1.65366	−0.826832	−	0.562448i	$-0.809860\pi$
$971$	−51.5296	−1.65366	−0.826832	+	0.562448i	$0.809860\pi$
$972$	0	0
$973$	−7.39403	−0.237042
$974$	21.7216	0.696004
$975$	0	0
$976$	−72.2337	−2.31214
$977$	−20.7107	−0.662595	−0.331298	−	0.943526i	$-0.607486\pi$
$977$	−20.7107	−0.662595	−0.331298	+	0.943526i	$0.607486\pi$
$978$	0	0
$979$	−21.7663	−0.695654
$980$	23.7432	0.758448
$981$	0	0
$982$	−2.74456	−0.0875825
$983$	52.2823	1.66755	0.833773	−	0.552108i	$-0.186176\pi$
$983$	52.2823	1.66755	0.833773	+	0.552108i	$0.186176\pi$
$984$	0	0
$985$	76.4674	2.43645
$986$	−8.83915	−0.281496
$987$	0	0
$988$	10.7446	0.341830
$989$	−11.2371	−0.357319
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	94.9728	3.01539
$993$	0	0
$994$	−13.0217	−0.413025
$995$	−2.84429	−0.0901702
$996$	0	0
$997$	−19.3505	−0.612837	−0.306419	−	0.951897i	$-0.599131\pi$
$997$	−19.3505	−0.612837	−0.306419	+	0.951897i	$0.599131\pi$
$998$	−30.1844	−0.955470
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.a.e.1.4	yes	4
3.2	odd	2	inner	171.2.a.e.1.1	✓	4
4.3	odd	2		2736.2.a.bf.1.1		4
5.4	even	2		4275.2.a.bp.1.1		4
7.6	odd	2		8379.2.a.bw.1.4		4
12.11	even	2		2736.2.a.bf.1.4		4
15.14	odd	2		4275.2.a.bp.1.4		4
19.18	odd	2		3249.2.a.bf.1.1		4
21.20	even	2		8379.2.a.bw.1.1		4
57.56	even	2		3249.2.a.bf.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.a.e.1.1	✓	4	3.2	odd	2	inner
171.2.a.e.1.4	yes	4	1.1	even	1	trivial
2736.2.a.bf.1.1		4	4.3	odd	2
2736.2.a.bf.1.4		4	12.11	even	2
3249.2.a.bf.1.1		4	19.18	odd	2
3249.2.a.bf.1.4		4	57.56	even	2
4275.2.a.bp.1.1		4	5.4	even	2
4275.2.a.bp.1.4		4	15.14	odd	2
8379.2.a.bw.1.1		4	21.20	even	2
8379.2.a.bw.1.4		4	7.6	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:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

\(f(q)\)	\(=\)	\(q+2.71519 q^{2} +5.37228 q^{4} -3.22060 q^{5} -2.37228 q^{7} +9.15640 q^{8} +O(q^{10})\)	\(q+2.71519 q^{2} +5.37228 q^{4} -3.22060 q^{5} -2.37228 q^{7} +9.15640 q^{8} -8.74456 q^{10} -2.20979 q^{11} +2.00000 q^{13} -6.44121 q^{14} +14.1168 q^{16} -3.22060 q^{17} +1.00000 q^{19} -17.3020 q^{20} -6.00000 q^{22} +1.01082 q^{23} +5.37228 q^{25} +5.43039 q^{26} -12.7446 q^{28} +1.01082 q^{29} +4.74456 q^{31} +20.0172 q^{32} -8.74456 q^{34} +7.64018 q^{35} +10.7446 q^{37} +2.71519 q^{38} -29.4891 q^{40} -5.43039 q^{41} -11.1168 q^{43} -11.8716 q^{44} +2.74456 q^{46} -4.23142 q^{47} -1.37228 q^{49} +14.5868 q^{50} +10.7446 q^{52} +9.84996 q^{53} +7.11684 q^{55} -21.7216 q^{56} +2.74456 q^{58} +10.8608 q^{59} -5.11684 q^{61} +12.8824 q^{62} +26.1168 q^{64} -6.44121 q^{65} -4.00000 q^{67} -17.3020 q^{68} +20.7446 q^{70} +2.02163 q^{71} -5.11684 q^{73} +29.1736 q^{74} +5.37228 q^{76} +5.24224 q^{77} -4.00000 q^{79} -45.4647 q^{80} -14.7446 q^{82} -11.8716 q^{83} +10.3723 q^{85} -30.1844 q^{86} -20.2337 q^{88} +9.84996 q^{89} -4.74456 q^{91} +5.43039 q^{92} -11.4891 q^{94} -3.22060 q^{95} +7.48913 q^{97} -3.72601 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{4} + 2 q^{7}+O(q^{10}) \) 4 * q + 10 * q^4 + 2 * q^7	\( 4 q + 10 q^{4} + 2 q^{7} - 12 q^{10} + 8 q^{13} + 22 q^{16} + 4 q^{19} - 24 q^{22} + 10 q^{25} - 28 q^{28} - 4 q^{31} - 12 q^{34} + 20 q^{37} - 72 q^{40} - 10 q^{43} - 12 q^{46} + 6 q^{49} + 20 q^{52} - 6 q^{55} - 12 q^{58} + 14 q^{61} + 70 q^{64} - 16 q^{67} + 60 q^{70} + 14 q^{73} + 10 q^{76} - 16 q^{79} - 36 q^{82} + 30 q^{85} - 12 q^{88} + 4 q^{91} - 16 q^{97}+O(q^{100}) \) 4 * q + 10 * q^4 + 2 * q^7 - 12 * q^10 + 8 * q^13 + 22 * q^16 + 4 * q^19 - 24 * q^22 + 10 * q^25 - 28 * q^28 - 4 * q^31 - 12 * q^34 + 20 * q^37 - 72 * q^40 - 10 * q^43 - 12 * q^46 + 6 * q^49 + 20 * q^52 - 6 * q^55 - 12 * q^58 + 14 * q^61 + 70 * q^64 - 16 * q^67 + 60 * q^70 + 14 * q^73 + 10 * q^76 - 16 * q^79 - 36 * q^82 + 30 * q^85 - 12 * q^88 + 4 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

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Database

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