LMFDB - Embedded newform 2169.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,2,Mod(1,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, 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("2169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2169.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:	$17.3195521984$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	7.7.31056073.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 3x^{5} + 11x^{4} + x^{3} - 9x^{2} + 1 $ x^7 - 3x^6 - 3x^5 + 11x^4 + x^3 - 9x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.48734$ of defining polynomial
Character	$\chi$	$=$	2169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.487343 q^{2} -1.76250 q^{4} -0.961999 q^{5} -4.61392 q^{7} +1.83363 q^{8} +O(q^{10})$	\(q-0.487343 q^{2} -1.76250 q^{4} -0.961999 q^{5} -4.61392 q^{7} +1.83363 q^{8} +0.468824 q^{10} -1.93974 q^{11} -3.85571 q^{13} +2.24856 q^{14} +2.63139 q^{16} -5.40289 q^{17} -4.17145 q^{19} +1.69552 q^{20} +0.945321 q^{22} +1.42545 q^{23} -4.07456 q^{25} +1.87905 q^{26} +8.13202 q^{28} +4.85744 q^{29} -7.24699 q^{31} -4.94964 q^{32} +2.63306 q^{34} +4.43859 q^{35} +7.12597 q^{37} +2.03293 q^{38} -1.76395 q^{40} -9.18955 q^{41} +2.93624 q^{43} +3.41879 q^{44} -0.694685 q^{46} -2.48170 q^{47} +14.2883 q^{49} +1.98571 q^{50} +6.79568 q^{52} -5.64997 q^{53} +1.86603 q^{55} -8.46022 q^{56} -2.36724 q^{58} +11.9783 q^{59} -13.9214 q^{61} +3.53177 q^{62} -2.85060 q^{64} +3.70919 q^{65} -7.30682 q^{67} +9.52258 q^{68} -2.16312 q^{70} +14.8844 q^{71} +0.240264 q^{73} -3.47279 q^{74} +7.35217 q^{76} +8.94983 q^{77} -3.15128 q^{79} -2.53139 q^{80} +4.47847 q^{82} +2.46821 q^{83} +5.19758 q^{85} -1.43096 q^{86} -3.55677 q^{88} -5.55181 q^{89} +17.7900 q^{91} -2.51235 q^{92} +1.20944 q^{94} +4.01293 q^{95} +5.27964 q^{97} -6.96330 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8}+O(q^{10}) $ 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8	$ 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8} + 3 q^{10} + 18 q^{11} - q^{13} + 6 q^{14} + 4 q^{16} + 2 q^{17} - 6 q^{19} + 8 q^{20} + 10 q^{22} + 22 q^{23} + 5 q^{25} - 8 q^{26} + 9 q^{28} + 16 q^{29} - 18 q^{31} + 6 q^{32} + 11 q^{34} - 7 q^{35} + 8 q^{37} - 16 q^{38} + 14 q^{40} + 15 q^{41} + 14 q^{43} + 4 q^{44} + 11 q^{46} + 10 q^{47} + 6 q^{49} + 4 q^{50} + 27 q^{52} - 15 q^{53} + 29 q^{55} - 13 q^{56} + 17 q^{58} + 18 q^{59} + 4 q^{61} - 13 q^{62} + 2 q^{64} + 7 q^{65} + 18 q^{67} + 15 q^{68} + 8 q^{70} + 50 q^{71} - 10 q^{74} - 20 q^{76} - 17 q^{77} - 15 q^{79} + 11 q^{80} + 45 q^{82} + 24 q^{83} - 2 q^{85} + 23 q^{86} + 8 q^{88} + 13 q^{89} - 12 q^{91} + 10 q^{92} - 32 q^{94} + 41 q^{95} + q^{97} - 9 q^{98}+O(q^{100}) $ 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8 + 3 * q^10 + 18 * q^11 - q^13 + 6 * q^14 + 4 * q^16 + 2 * q^17 - 6 * q^19 + 8 * q^20 + 10 * q^22 + 22 * q^23 + 5 * q^25 - 8 * q^26 + 9 * q^28 + 16 * q^29 - 18 * q^31 + 6 * q^32 + 11 * q^34 - 7 * q^35 + 8 * q^37 - 16 * q^38 + 14 * q^40 + 15 * q^41 + 14 * q^43 + 4 * q^44 + 11 * q^46 + 10 * q^47 + 6 * q^49 + 4 * q^50 + 27 * q^52 - 15 * q^53 + 29 * q^55 - 13 * q^56 + 17 * q^58 + 18 * q^59 + 4 * q^61 - 13 * q^62 + 2 * q^64 + 7 * q^65 + 18 * q^67 + 15 * q^68 + 8 * q^70 + 50 * q^71 - 10 * q^74 - 20 * q^76 - 17 * q^77 - 15 * q^79 + 11 * q^80 + 45 * q^82 + 24 * q^83 - 2 * q^85 + 23 * q^86 + 8 * q^88 + 13 * q^89 - 12 * q^91 + 10 * q^92 - 32 * q^94 + 41 * q^95 + q^97 - 9 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.487343	−0.344604	−0.172302	−	0.985044i	$-0.555120\pi$
$2$	−0.487343	−0.344604	−0.172302	+	0.985044i	$0.555120\pi$
$3$	0	0
$3$	0	0
$4$	−1.76250	−0.881248
$5$	−0.961999	−0.430219	−0.215110	−	0.976590i	$-0.569011\pi$
$5$	−0.961999	−0.430219	−0.215110	+	0.976590i	$0.569011\pi$
$6$	0	0
$7$	−4.61392	−1.74390	−0.871950	−	0.489596i	$-0.837144\pi$
$7$	−4.61392	−1.74390	−0.871950	+	0.489596i	$0.837144\pi$
$8$	1.83363	0.648285
$9$	0	0
$10$	0.468824	0.148255
$11$	−1.93974	−0.584855	−0.292427	−	0.956288i	$-0.594463\pi$
$11$	−1.93974	−0.584855	−0.292427	+	0.956288i	$0.594463\pi$
$12$	0	0
$13$	−3.85571	−1.06938	−0.534691	−	0.845048i	$-0.679572\pi$
$13$	−3.85571	−1.06938	−0.534691	+	0.845048i	$0.679572\pi$
$14$	2.24856	0.600954
$15$	0	0
$16$	2.63139	0.657847
$17$	−5.40289	−1.31039	−0.655197	−	0.755458i	$-0.727414\pi$
$17$	−5.40289	−1.31039	−0.655197	+	0.755458i	$0.727414\pi$
$18$	0	0
$19$	−4.17145	−0.956997	−0.478498	−	0.878088i	$-0.658819\pi$
$19$	−4.17145	−0.956997	−0.478498	+	0.878088i	$0.658819\pi$
$20$	1.69552	0.379130
$21$	0	0
$22$	0.945321	0.201543
$23$	1.42545	0.297227	0.148614	−	0.988895i	$-0.452519\pi$
$23$	1.42545	0.297227	0.148614	+	0.988895i	$0.452519\pi$
$24$	0	0
$25$	−4.07456	−0.814911
$26$	1.87905	0.368513
$27$	0	0
$28$	8.13202	1.53681
$29$	4.85744	0.902003	0.451002	−	0.892523i	$-0.351067\pi$
$29$	4.85744	0.902003	0.451002	+	0.892523i	$0.351067\pi$
$30$	0	0
$31$	−7.24699	−1.30160	−0.650799	−	0.759250i	$-0.725566\pi$
$31$	−7.24699	−1.30160	−0.650799	+	0.759250i	$0.725566\pi$
$32$	−4.94964	−0.874982
$33$	0	0
$34$	2.63306	0.451567
$35$	4.43859	0.750259
$36$	0	0
$37$	7.12597	1.17150	0.585751	−	0.810491i	$-0.300800\pi$
$37$	7.12597	1.17150	0.585751	+	0.810491i	$0.300800\pi$
$38$	2.03293	0.329785
$39$	0	0
$40$	−1.76395	−0.278905
$41$	−9.18955	−1.43517	−0.717583	−	0.696473i	$-0.754752\pi$
$41$	−9.18955	−1.43517	−0.717583	+	0.696473i	$0.754752\pi$
$42$	0	0
$43$	2.93624	0.447772	0.223886	−	0.974615i	$-0.428126\pi$
$43$	2.93624	0.447772	0.223886	+	0.974615i	$0.428126\pi$
$44$	3.41879	0.515402
$45$	0	0
$46$	−0.694685	−0.102426
$47$	−2.48170	−0.361994	−0.180997	−	0.983484i	$-0.557932\pi$
$47$	−2.48170	−0.361994	−0.180997	+	0.983484i	$0.557932\pi$
$48$	0	0
$49$	14.2883	2.04118
$50$	1.98571	0.280822
$51$	0	0
$52$	6.79568	0.942391
$53$	−5.64997	−0.776083	−0.388041	−	0.921642i	$-0.626848\pi$
$53$	−5.64997	−0.776083	−0.388041	+	0.921642i	$0.626848\pi$
$54$	0	0
$55$	1.86603	0.251616
$56$	−8.46022	−1.13054
$57$	0	0
$58$	−2.36724	−0.310834
$59$	11.9783	1.55944	0.779718	−	0.626131i	$-0.215362\pi$
$59$	11.9783	1.55944	0.779718	+	0.626131i	$0.215362\pi$
$60$	0	0
$61$	−13.9214	−1.78245	−0.891223	−	0.453565i	$-0.850152\pi$
$61$	−13.9214	−1.78245	−0.891223	+	0.453565i	$0.850152\pi$
$62$	3.53177	0.448535
$63$	0	0
$64$	−2.85060	−0.356325
$65$	3.70919	0.460069
$66$	0	0
$67$	−7.30682	−0.892670	−0.446335	−	0.894866i	$-0.647271\pi$
$67$	−7.30682	−0.892670	−0.446335	+	0.894866i	$0.647271\pi$
$68$	9.52258	1.15478
$69$	0	0
$70$	−2.16312	−0.258542
$71$	14.8844	1.76645	0.883226	−	0.468948i	$-0.155367\pi$
$71$	14.8844	1.76645	0.883226	+	0.468948i	$0.155367\pi$
$72$	0	0
$73$	0.240264	0.0281208	0.0140604	−	0.999901i	$-0.495524\pi$
$73$	0.240264	0.0281208	0.0140604	+	0.999901i	$0.495524\pi$
$74$	−3.47279	−0.403704
$75$	0	0
$76$	7.35217	0.843352
$77$	8.94983	1.01993
$78$	0	0
$79$	−3.15128	−0.354546	−0.177273	−	0.984162i	$-0.556728\pi$
$79$	−3.15128	−0.354546	−0.177273	+	0.984162i	$0.556728\pi$
$80$	−2.53139	−0.283018
$81$	0	0
$82$	4.47847	0.494564
$83$	2.46821	0.270922	0.135461	−	0.990783i	$-0.456749\pi$
$83$	2.46821	0.270922	0.135461	+	0.990783i	$0.456749\pi$
$84$	0	0
$85$	5.19758	0.563757
$86$	−1.43096	−0.154304
$87$	0	0
$88$	−3.55677	−0.379153
$89$	−5.55181	−0.588491	−0.294246	−	0.955730i	$-0.595068\pi$
$89$	−5.55181	−0.588491	−0.294246	+	0.955730i	$0.595068\pi$
$90$	0	0
$91$	17.7900	1.86489
$92$	−2.51235	−0.261931
$93$	0	0
$94$	1.20944	0.124744
$95$	4.01293	0.411718
$96$	0	0
$97$	5.27964	0.536067	0.268033	−	0.963410i	$-0.413626\pi$
$97$	5.27964	0.536067	0.268033	+	0.963410i	$0.413626\pi$
$98$	−6.96330	−0.703400
$99$	0	0
$100$	7.18139	0.718139
$101$	12.4239	1.23623	0.618113	−	0.786089i	$-0.287897\pi$
$101$	12.4239	1.23623	0.618113	+	0.786089i	$0.287897\pi$
$102$	0	0
$103$	−9.80533	−0.966148	−0.483074	−	0.875580i	$-0.660480\pi$
$103$	−9.80533	−0.966148	−0.483074	+	0.875580i	$0.660480\pi$
$104$	−7.06994	−0.693264
$105$	0	0
$106$	2.75347	0.267441
$107$	9.15031	0.884593	0.442297	−	0.896869i	$-0.354164\pi$
$107$	9.15031	0.884593	0.442297	+	0.896869i	$0.354164\pi$
$108$	0	0
$109$	3.24534	0.310847	0.155424	−	0.987848i	$-0.450326\pi$
$109$	3.24534	0.310847	0.155424	+	0.987848i	$0.450326\pi$
$110$	−0.909398	−0.0867077
$111$	0	0
$112$	−12.1410	−1.14722
$113$	0.842874	0.0792909	0.0396455	−	0.999214i	$-0.487377\pi$
$113$	0.842874	0.0792909	0.0396455	+	0.999214i	$0.487377\pi$
$114$	0	0
$115$	−1.37128	−0.127873
$116$	−8.56121	−0.794889
$117$	0	0
$118$	−5.83752	−0.537387
$119$	24.9285	2.28520
$120$	0	0
$121$	−7.23740	−0.657945
$122$	6.78448	0.614238
$123$	0	0
$124$	12.7728	1.14703
$125$	8.72972	0.780810
$126$	0	0
$127$	11.3416	1.00641	0.503203	−	0.864168i	$-0.332155\pi$
$127$	11.3416	1.00641	0.503203	+	0.864168i	$0.332155\pi$
$128$	11.2885	0.997773
$129$	0	0
$130$	−1.80765	−0.158541
$131$	−14.1261	−1.23421	−0.617103	−	0.786882i	$-0.711694\pi$
$131$	−14.1261	−1.23421	−0.617103	+	0.786882i	$0.711694\pi$
$132$	0	0
$133$	19.2468	1.66891
$134$	3.56093	0.307617
$135$	0	0
$136$	−9.90689	−0.849509
$137$	12.9555	1.10686	0.553430	−	0.832896i	$-0.313319\pi$
$137$	12.9555	1.10686	0.553430	+	0.832896i	$0.313319\pi$
$138$	0	0
$139$	8.23606	0.698574	0.349287	−	0.937016i	$-0.386424\pi$
$139$	8.23606	0.698574	0.349287	+	0.937016i	$0.386424\pi$
$140$	−7.82300	−0.661164
$141$	0	0
$142$	−7.25380	−0.608726
$143$	7.47909	0.625433
$144$	0	0
$145$	−4.67285	−0.388059
$146$	−0.117091	−0.00969054
$147$	0	0
$148$	−12.5595	−1.03238
$149$	−4.48606	−0.367513	−0.183756	−	0.982972i	$-0.558826\pi$
$149$	−4.48606	−0.367513	−0.183756	+	0.982972i	$0.558826\pi$
$150$	0	0
$151$	0.0933624	0.00759773	0.00379886	−	0.999993i	$-0.498791\pi$
$151$	0.0933624	0.00759773	0.00379886	+	0.999993i	$0.498791\pi$
$152$	−7.64889	−0.620407
$153$	0	0
$154$	−4.36164	−0.351471
$155$	6.97160	0.559972
$156$	0	0
$157$	−10.1337	−0.808760	−0.404380	−	0.914591i	$-0.632513\pi$
$157$	−10.1337	−0.808760	−0.404380	+	0.914591i	$0.632513\pi$
$158$	1.53575	0.122178
$159$	0	0
$160$	4.76155	0.376434
$161$	−6.57693	−0.518334
$162$	0	0
$163$	7.49804	0.587292	0.293646	−	0.955914i	$-0.405131\pi$
$163$	7.49804	0.587292	0.293646	+	0.955914i	$0.405131\pi$
$164$	16.1965	1.26474
$165$	0	0
$166$	−1.20287	−0.0933606
$167$	−17.7922	−1.37680	−0.688399	−	0.725332i	$-0.741686\pi$
$167$	−17.7922	−1.37680	−0.688399	+	0.725332i	$0.741686\pi$
$168$	0	0
$169$	1.86650	0.143577
$170$	−2.53301	−0.194273
$171$	0	0
$172$	−5.17511	−0.394599
$173$	−18.4927	−1.40597	−0.702987	−	0.711203i	$-0.748151\pi$
$173$	−18.4927	−1.40597	−0.702987	+	0.711203i	$0.748151\pi$
$174$	0	0
$175$	18.7997	1.42112
$176$	−5.10421	−0.384745
$177$	0	0
$178$	2.70564	0.202796
$179$	17.3387	1.29595	0.647977	−	0.761660i	$-0.275615\pi$
$179$	17.3387	1.29595	0.647977	+	0.761660i	$0.275615\pi$
$180$	0	0
$181$	17.7124	1.31655	0.658275	−	0.752778i	$-0.271287\pi$
$181$	17.7124	1.31655	0.658275	+	0.752778i	$0.271287\pi$
$182$	−8.66981	−0.642649
$183$	0	0
$184$	2.61375	0.192688
$185$	−6.85518	−0.504003
$186$	0	0
$187$	10.4802	0.766390
$188$	4.37400	0.319006
$189$	0	0
$190$	−1.95568	−0.141880
$191$	−13.2440	−0.958303	−0.479151	−	0.877732i	$-0.659056\pi$
$191$	−13.2440	−0.958303	−0.479151	+	0.877732i	$0.659056\pi$
$192$	0	0
$193$	−25.5030	−1.83574	−0.917872	−	0.396877i	$-0.870094\pi$
$193$	−25.5030	−1.83574	−0.917872	+	0.396877i	$0.870094\pi$
$194$	−2.57300	−0.184731
$195$	0	0
$196$	−25.1831	−1.79879
$197$	15.0303	1.07087	0.535433	−	0.844578i	$-0.320149\pi$
$197$	15.0303	1.07087	0.535433	+	0.844578i	$0.320149\pi$
$198$	0	0
$199$	−19.3801	−1.37382	−0.686911	−	0.726742i	$-0.741034\pi$
$199$	−19.3801	−1.37382	−0.686911	+	0.726742i	$0.741034\pi$
$200$	−7.47122	−0.528295
$201$	0	0
$202$	−6.05472	−0.426008
$203$	−22.4118	−1.57300
$204$	0	0
$205$	8.84034	0.617436
$206$	4.77856	0.332938
$207$	0	0
$208$	−10.1459	−0.703489
$209$	8.09154	0.559704
$210$	0	0
$211$	27.9383	1.92335	0.961675	−	0.274192i	$-0.0884105\pi$
$211$	27.9383	1.92335	0.961675	+	0.274192i	$0.0884105\pi$
$212$	9.95805	0.683922
$213$	0	0
$214$	−4.45934	−0.304834
$215$	−2.82466	−0.192640
$216$	0	0
$217$	33.4370	2.26985
$218$	−1.58160	−0.107119
$219$	0	0
$220$	−3.28887	−0.221736
$221$	20.8320	1.40131
$222$	0	0
$223$	−0.387500	−0.0259489	−0.0129745	−	0.999916i	$-0.504130\pi$
$223$	−0.387500	−0.0259489	−0.0129745	+	0.999916i	$0.504130\pi$
$224$	22.8373	1.52588
$225$	0	0
$226$	−0.410769	−0.0273240
$227$	−10.2778	−0.682161	−0.341081	−	0.940034i	$-0.610793\pi$
$227$	−10.2778	−0.682161	−0.341081	+	0.940034i	$0.610793\pi$
$228$	0	0
$229$	−10.5346	−0.696147	−0.348074	−	0.937467i	$-0.613164\pi$
$229$	−10.5346	−0.696147	−0.348074	+	0.937467i	$0.613164\pi$
$230$	0.668286	0.0440655
$231$	0	0
$232$	8.90673	0.584755
$233$	−25.6725	−1.68186	−0.840932	−	0.541141i	$-0.817993\pi$
$233$	−25.6725	−1.68186	−0.840932	+	0.541141i	$0.817993\pi$
$234$	0	0
$235$	2.38740	0.155737
$236$	−21.1116	−1.37425
$237$	0	0
$238$	−12.1488	−0.787487
$239$	25.1312	1.62560	0.812801	−	0.582542i	$-0.197942\pi$
$239$	25.1312	1.62560	0.812801	+	0.582542i	$0.197942\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	3.52710	0.226730
$243$	0	0
$244$	24.5363	1.57078
$245$	−13.7453	−0.878157
$246$	0	0
$247$	16.0839	1.02339
$248$	−13.2883	−0.843806
$249$	0	0
$250$	−4.25437	−0.269070
$251$	−2.52238	−0.159211	−0.0796055	−	0.996826i	$-0.525366\pi$
$251$	−2.52238	−0.159211	−0.0796055	+	0.996826i	$0.525366\pi$
$252$	0	0
$253$	−2.76501	−0.173835
$254$	−5.52726	−0.346811
$255$	0	0
$256$	0.199817	0.0124886
$257$	−3.92348	−0.244740	−0.122370	−	0.992485i	$-0.539049\pi$
$257$	−3.92348	−0.244740	−0.122370	+	0.992485i	$0.539049\pi$
$258$	0	0
$259$	−32.8787	−2.04298
$260$	−6.53744	−0.405435
$261$	0	0
$262$	6.88428	0.425312
$263$	−6.24984	−0.385382	−0.192691	−	0.981260i	$-0.561721\pi$
$263$	−6.24984	−0.385382	−0.192691	+	0.981260i	$0.561721\pi$
$264$	0	0
$265$	5.43527	0.333886
$266$	−9.37978	−0.575111
$267$	0	0
$268$	12.8782	0.786664
$269$	−20.4943	−1.24956	−0.624780	−	0.780801i	$-0.714811\pi$
$269$	−20.4943	−1.24956	−0.624780	+	0.780801i	$0.714811\pi$
$270$	0	0
$271$	15.3179	0.930499	0.465249	−	0.885180i	$-0.345965\pi$
$271$	15.3179	0.930499	0.465249	+	0.885180i	$0.345965\pi$
$272$	−14.2171	−0.862039
$273$	0	0
$274$	−6.31376	−0.381428
$275$	7.90359	0.476605
$276$	0	0
$277$	3.26388	0.196108	0.0980539	−	0.995181i	$-0.468738\pi$
$277$	3.26388	0.196108	0.0980539	+	0.995181i	$0.468738\pi$
$278$	−4.01379	−0.240731
$279$	0	0
$280$	8.13872	0.486382
$281$	12.4791	0.744442	0.372221	−	0.928144i	$-0.378596\pi$
$281$	12.4791	0.744442	0.372221	+	0.928144i	$0.378596\pi$
$282$	0	0
$283$	−15.8115	−0.939899	−0.469949	−	0.882693i	$-0.655728\pi$
$283$	−15.8115	−0.939899	−0.469949	+	0.882693i	$0.655728\pi$
$284$	−26.2337	−1.55668
$285$	0	0
$286$	−3.64488	−0.215526
$287$	42.3999	2.50279
$288$	0	0
$289$	12.1913	0.717133
$290$	2.27728	0.133727
$291$	0	0
$292$	−0.423465	−0.0247814
$293$	−18.7147	−1.09333	−0.546663	−	0.837353i	$-0.684102\pi$
$293$	−18.7147	−1.09333	−0.546663	+	0.837353i	$0.684102\pi$
$294$	0	0
$295$	−11.5231	−0.670899
$296$	13.0664	0.759467
$297$	0	0
$298$	2.18625	0.126646
$299$	−5.49613	−0.317849
$300$	0	0
$301$	−13.5476	−0.780870
$302$	−0.0454995	−0.00261820
$303$	0	0
$304$	−10.9767	−0.629557
$305$	13.3923	0.766843
$306$	0	0
$307$	5.23477	0.298764	0.149382	−	0.988780i	$-0.452272\pi$
$307$	5.23477	0.298764	0.149382	+	0.988780i	$0.452272\pi$
$308$	−15.7740	−0.898809
$309$	0	0
$310$	−3.39756	−0.192969
$311$	26.6878	1.51332	0.756662	−	0.653806i	$-0.226829\pi$
$311$	26.6878	1.51332	0.756662	+	0.653806i	$0.226829\pi$
$312$	0	0
$313$	6.24141	0.352786	0.176393	−	0.984320i	$-0.443557\pi$
$313$	6.24141	0.352786	0.176393	+	0.984320i	$0.443557\pi$
$314$	4.93861	0.278702
$315$	0	0
$316$	5.55412	0.312443
$317$	−6.89526	−0.387276	−0.193638	−	0.981073i	$-0.562029\pi$
$317$	−6.89526	−0.387276	−0.193638	+	0.981073i	$0.562029\pi$
$318$	0	0
$319$	−9.42218	−0.527541
$320$	2.74227	0.153298
$321$	0	0
$322$	3.20522	0.178620
$323$	22.5379	1.25404
$324$	0	0
$325$	15.7103	0.871451
$326$	−3.65412	−0.202383
$327$	0	0
$328$	−16.8502	−0.930397
$329$	11.4504	0.631281
$330$	0	0
$331$	−13.5697	−0.745860	−0.372930	−	0.927859i	$-0.621647\pi$
$331$	−13.5697	−0.745860	−0.372930	+	0.927859i	$0.621647\pi$
$332$	−4.35022	−0.238749
$333$	0	0
$334$	8.67089	0.474450
$335$	7.02916	0.384044
$336$	0	0
$337$	−20.2288	−1.10193	−0.550966	−	0.834528i	$-0.685740\pi$
$337$	−20.2288	−1.10193	−0.550966	+	0.834528i	$0.685740\pi$
$338$	−0.909627	−0.0494772
$339$	0	0
$340$	−9.16072	−0.496810
$341$	14.0573	0.761245
$342$	0	0
$343$	−33.6276	−1.81572
$344$	5.38397	0.290284
$345$	0	0
$346$	9.01229	0.484504
$347$	−1.52458	−0.0818437	−0.0409219	−	0.999162i	$-0.513029\pi$
$347$	−1.52458	−0.0818437	−0.0409219	+	0.999162i	$0.513029\pi$
$348$	0	0
$349$	16.5585	0.886358	0.443179	−	0.896433i	$-0.353851\pi$
$349$	16.5585	0.886358	0.443179	+	0.896433i	$0.353851\pi$
$350$	−9.16191	−0.489724
$351$	0	0
$352$	9.60104	0.511737
$353$	−34.6802	−1.84584	−0.922920	−	0.384992i	$-0.874204\pi$
$353$	−34.6802	−1.84584	−0.922920	+	0.384992i	$0.874204\pi$
$354$	0	0
$355$	−14.3188	−0.759961
$356$	9.78505	0.518607
$357$	0	0
$358$	−8.44989	−0.446590
$359$	−18.1341	−0.957084	−0.478542	−	0.878065i	$-0.658835\pi$
$359$	−18.1341	−0.957084	−0.478542	+	0.878065i	$0.658835\pi$
$360$	0	0
$361$	−1.59899	−0.0841573
$362$	−8.63200	−0.453688
$363$	0	0
$364$	−31.3547	−1.64343
$365$	−0.231134	−0.0120981
$366$	0	0
$367$	−10.9879	−0.573566	−0.286783	−	0.957996i	$-0.592586\pi$
$367$	−10.9879	−0.573566	−0.286783	+	0.957996i	$0.592586\pi$
$368$	3.75092	0.195530
$369$	0	0
$370$	3.34082	0.173681
$371$	26.0685	1.35341
$372$	0	0
$373$	5.23646	0.271134	0.135567	−	0.990768i	$-0.456714\pi$
$373$	5.23646	0.271134	0.135567	+	0.990768i	$0.456714\pi$
$374$	−5.10747	−0.264101
$375$	0	0
$376$	−4.55052	−0.234675
$377$	−18.7289	−0.964586
$378$	0	0
$379$	8.07101	0.414580	0.207290	−	0.978280i	$-0.433536\pi$
$379$	8.07101	0.414580	0.207290	+	0.978280i	$0.433536\pi$
$380$	−7.07278	−0.362826
$381$	0	0
$382$	6.45438	0.330235
$383$	−2.46503	−0.125957	−0.0629785	−	0.998015i	$-0.520060\pi$
$383$	−2.46503	−0.125957	−0.0629785	+	0.998015i	$0.520060\pi$
$384$	0	0
$385$	−8.60973	−0.438792
$386$	12.4287	0.632604
$387$	0	0
$388$	−9.30535	−0.472408
$389$	28.7159	1.45595	0.727977	−	0.685601i	$-0.240461\pi$
$389$	28.7159	1.45595	0.727977	+	0.685601i	$0.240461\pi$
$390$	0	0
$391$	−7.70157	−0.389485
$392$	26.1994	1.32327
$393$	0	0
$394$	−7.32492	−0.369024
$395$	3.03153	0.152533
$396$	0	0
$397$	12.0605	0.605301	0.302651	−	0.953102i	$-0.402129\pi$
$397$	12.0605	0.605301	0.302651	+	0.953102i	$0.402129\pi$
$398$	9.44478	0.473424
$399$	0	0
$400$	−10.7217	−0.536087
$401$	31.7082	1.58343	0.791716	−	0.610890i	$-0.209188\pi$
$401$	31.7082	1.58343	0.791716	+	0.610890i	$0.209188\pi$
$402$	0	0
$403$	27.9423	1.39190
$404$	−21.8971	−1.08942
$405$	0	0
$406$	10.9223	0.542063
$407$	−13.8225	−0.685158
$408$	0	0
$409$	−14.1893	−0.701617	−0.350809	−	0.936447i	$-0.614093\pi$
$409$	−14.1893	−0.701617	−0.350809	+	0.936447i	$0.614093\pi$
$410$	−4.30828	−0.212771
$411$	0	0
$412$	17.2819	0.851416
$413$	−55.2668	−2.71950
$414$	0	0
$415$	−2.37442	−0.116556
$416$	19.0844	0.935689
$417$	0	0
$418$	−3.94336	−0.192876
$419$	−31.1014	−1.51941	−0.759703	−	0.650271i	$-0.774655\pi$
$419$	−31.1014	−1.51941	−0.759703	+	0.650271i	$0.774655\pi$
$420$	0	0
$421$	−27.4415	−1.33742	−0.668709	−	0.743524i	$-0.733153\pi$
$421$	−27.4415	−1.33742	−0.668709	+	0.743524i	$0.733153\pi$
$422$	−13.6155	−0.662794
$423$	0	0
$424$	−10.3599	−0.503123
$425$	22.0144	1.06786
$426$	0	0
$427$	64.2321	3.10841
$428$	−16.1274	−0.779546
$429$	0	0
$430$	1.37658	0.0663846
$431$	14.1005	0.679199	0.339599	−	0.940570i	$-0.389709\pi$
$431$	14.1005	0.679199	0.339599	+	0.940570i	$0.389709\pi$
$432$	0	0
$433$	−17.7403	−0.852546	−0.426273	−	0.904595i	$-0.640174\pi$
$433$	−17.7403	−0.852546	−0.426273	+	0.904595i	$0.640174\pi$
$434$	−16.2953	−0.782200
$435$	0	0
$436$	−5.71991	−0.273934
$437$	−5.94621	−0.284446
$438$	0	0
$439$	−2.52447	−0.120486	−0.0602431	−	0.998184i	$-0.519188\pi$
$439$	−2.52447	−0.120486	−0.0602431	+	0.998184i	$0.519188\pi$
$440$	3.42161	0.163119
$441$	0	0
$442$	−10.1523	−0.482897
$443$	16.8382	0.800008	0.400004	−	0.916513i	$-0.369009\pi$
$443$	16.8382	0.800008	0.400004	+	0.916513i	$0.369009\pi$
$444$	0	0
$445$	5.34084	0.253180
$446$	0.188846	0.00894211
$447$	0	0
$448$	13.1524	0.621394
$449$	30.5688	1.44263	0.721314	−	0.692608i	$-0.243538\pi$
$449$	30.5688	1.44263	0.721314	+	0.692608i	$0.243538\pi$
$450$	0	0
$451$	17.8254	0.839364
$452$	−1.48556	−0.0698750
$453$	0	0
$454$	5.00882	0.235075
$455$	−17.1139	−0.802313
$456$	0	0
$457$	−32.7043	−1.52984	−0.764922	−	0.644123i	$-0.777223\pi$
$457$	−32.7043	−1.52984	−0.764922	+	0.644123i	$0.777223\pi$
$458$	5.13398	0.239895
$459$	0	0
$460$	2.41688	0.112688
$461$	−12.3932	−0.577210	−0.288605	−	0.957448i	$-0.593191\pi$
$461$	−12.3932	−0.577210	−0.288605	+	0.957448i	$0.593191\pi$
$462$	0	0
$463$	12.1379	0.564094	0.282047	−	0.959401i	$-0.408987\pi$
$463$	12.1379	0.564094	0.282047	+	0.959401i	$0.408987\pi$
$464$	12.7818	0.593380
$465$	0	0
$466$	12.5113	0.579577
$467$	−5.38203	−0.249051	−0.124525	−	0.992216i	$-0.539741\pi$
$467$	−5.38203	−0.249051	−0.124525	+	0.992216i	$0.539741\pi$
$468$	0	0
$469$	33.7131	1.55673
$470$	−1.16348	−0.0536675
$471$	0	0
$472$	21.9637	1.01096
$473$	−5.69555	−0.261882
$474$	0	0
$475$	16.9968	0.779868
$476$	−43.9365	−2.01382
$477$	0	0
$478$	−12.2475	−0.560188
$479$	25.3752	1.15942	0.579711	−	0.814822i	$-0.303166\pi$
$479$	25.3752	1.15942	0.579711	+	0.814822i	$0.303166\pi$
$480$	0	0
$481$	−27.4757	−1.25278
$482$	0.487343	0.0221979
$483$	0	0
$484$	12.7559	0.579813
$485$	−5.07901	−0.230626
$486$	0	0
$487$	−10.4243	−0.472369	−0.236185	−	0.971708i	$-0.575897\pi$
$487$	−10.4243	−0.472369	−0.236185	+	0.971708i	$0.575897\pi$
$488$	−25.5266	−1.15553
$489$	0	0
$490$	6.69869	0.302616
$491$	−34.2858	−1.54730	−0.773649	−	0.633614i	$-0.781571\pi$
$491$	−34.2858	−1.54730	−0.773649	+	0.633614i	$0.781571\pi$
$492$	0	0
$493$	−26.2442	−1.18198
$494$	−7.83839	−0.352666
$495$	0	0
$496$	−19.0696	−0.856252
$497$	−68.6754	−3.08051
$498$	0	0
$499$	15.9672	0.714792	0.357396	−	0.933953i	$-0.383665\pi$
$499$	15.9672	0.714792	0.357396	+	0.933953i	$0.383665\pi$
$500$	−15.3861	−0.688087
$501$	0	0
$502$	1.22926	0.0548647
$503$	4.53174	0.202060	0.101030	−	0.994883i	$-0.467786\pi$
$503$	4.53174	0.202060	0.101030	+	0.994883i	$0.467786\pi$
$504$	0	0
$505$	−11.9518	−0.531849
$506$	1.34751	0.0599041
$507$	0	0
$508$	−19.9896	−0.886893
$509$	4.08798	0.181197	0.0905983	−	0.995888i	$-0.471122\pi$
$509$	4.08798	0.181197	0.0905983	+	0.995888i	$0.471122\pi$
$510$	0	0
$511$	−1.10856	−0.0490398
$512$	−22.6744	−1.00208
$513$	0	0
$514$	1.91208	0.0843384
$515$	9.43272	0.415655
$516$	0	0
$517$	4.81387	0.211714
$518$	16.0232	0.704019
$519$	0	0
$520$	6.80127	0.298256
$521$	−34.2884	−1.50220	−0.751102	−	0.660186i	$-0.770477\pi$
$521$	−34.2884	−1.50220	−0.751102	+	0.660186i	$0.770477\pi$
$522$	0	0
$523$	33.7724	1.47676	0.738382	−	0.674382i	$-0.235590\pi$
$523$	33.7724	1.47676	0.738382	+	0.674382i	$0.235590\pi$
$524$	24.8973	1.08764
$525$	0	0
$526$	3.04582	0.132804
$527$	39.1547	1.70561
$528$	0	0
$529$	−20.9681	−0.911656
$530$	−2.64884	−0.115058
$531$	0	0
$532$	−33.9223	−1.47072
$533$	35.4322	1.53474
$534$	0	0
$535$	−8.80259	−0.380569
$536$	−13.3980	−0.578705
$537$	0	0
$538$	9.98777	0.430603
$539$	−27.7156	−1.19380
$540$	0	0
$541$	−31.0330	−1.33421	−0.667106	−	0.744963i	$-0.732467\pi$
$541$	−31.0330	−1.33421	−0.667106	+	0.744963i	$0.732467\pi$
$542$	−7.46510	−0.320653
$543$	0	0
$544$	26.7424	1.14657
$545$	−3.12202	−0.133733
$546$	0	0
$547$	−33.0104	−1.41142	−0.705712	−	0.708499i	$-0.749373\pi$
$547$	−33.0104	−1.41142	−0.705712	+	0.708499i	$0.749373\pi$
$548$	−22.8340	−0.975418
$549$	0	0
$550$	−3.85176	−0.164240
$551$	−20.2626	−0.863214
$552$	0	0
$553$	14.5398	0.618293
$554$	−1.59063	−0.0675795
$555$	0	0
$556$	−14.5160	−0.615617
$557$	−15.4826	−0.656019	−0.328010	−	0.944674i	$-0.606378\pi$
$557$	−15.4826	−0.656019	−0.328010	+	0.944674i	$0.606378\pi$
$558$	0	0
$559$	−11.3213	−0.478840
$560$	11.6797	0.493555
$561$	0	0
$562$	−6.08161	−0.256537
$563$	−6.21072	−0.261751	−0.130875	−	0.991399i	$-0.541779\pi$
$563$	−6.21072	−0.261751	−0.130875	+	0.991399i	$0.541779\pi$
$564$	0	0
$565$	−0.810845	−0.0341125
$566$	7.70565	0.323893
$567$	0	0
$568$	27.2924	1.14516
$569$	20.3014	0.851078	0.425539	−	0.904940i	$-0.360084\pi$
$569$	20.3014	0.851078	0.425539	+	0.904940i	$0.360084\pi$
$570$	0	0
$571$	−38.5209	−1.61205	−0.806024	−	0.591882i	$-0.798385\pi$
$571$	−38.5209	−1.61205	−0.806024	+	0.591882i	$0.798385\pi$
$572$	−13.1819	−0.551161
$573$	0	0
$574$	−20.6633	−0.862469
$575$	−5.80809	−0.242214
$576$	0	0
$577$	19.3481	0.805472	0.402736	−	0.915316i	$-0.368059\pi$
$577$	19.3481	0.805472	0.402736	+	0.915316i	$0.368059\pi$
$578$	−5.94133	−0.247127
$579$	0	0
$580$	8.23588	0.341976
$581$	−11.3881	−0.472460
$582$	0	0
$583$	10.9595	0.453896
$584$	0.440555	0.0182303
$585$	0	0
$586$	9.12050	0.376764
$587$	20.0183	0.826243	0.413122	−	0.910676i	$-0.364438\pi$
$587$	20.0183	0.826243	0.413122	+	0.910676i	$0.364438\pi$
$588$	0	0
$589$	30.2305	1.24562
$590$	5.61569	0.231194
$591$	0	0
$592$	18.7512	0.770668
$593$	−12.5904	−0.517027	−0.258513	−	0.966008i	$-0.583233\pi$
$593$	−12.5904	−0.517027	−0.258513	+	0.966008i	$0.583233\pi$
$594$	0	0
$595$	−23.9812	−0.983135
$596$	7.90667	0.323870
$597$	0	0
$598$	2.67850	0.109532
$599$	−24.4282	−0.998107	−0.499054	−	0.866571i	$-0.666319\pi$
$599$	−24.4282	−0.998107	−0.499054	+	0.866571i	$0.666319\pi$
$600$	0	0
$601$	9.02256	0.368038	0.184019	−	0.982923i	$-0.441089\pi$
$601$	9.02256	0.368038	0.184019	+	0.982923i	$0.441089\pi$
$602$	6.60233	0.269091
$603$	0	0
$604$	−0.164551	−0.00669548
$605$	6.96237	0.283061
$606$	0	0
$607$	−19.0044	−0.771364	−0.385682	−	0.922632i	$-0.626034\pi$
$607$	−19.0044	−0.771364	−0.385682	+	0.922632i	$0.626034\pi$
$608$	20.6472	0.837355
$609$	0	0
$610$	−6.52666	−0.264257
$611$	9.56873	0.387110
$612$	0	0
$613$	3.38617	0.136766	0.0683831	−	0.997659i	$-0.478216\pi$
$613$	3.38617	0.136766	0.0683831	+	0.997659i	$0.478216\pi$
$614$	−2.55113	−0.102955
$615$	0	0
$616$	16.4106	0.661204
$617$	33.9920	1.36847	0.684234	−	0.729263i	$-0.260137\pi$
$617$	33.9920	1.36847	0.684234	+	0.729263i	$0.260137\pi$
$618$	0	0
$619$	29.6477	1.19164	0.595822	−	0.803117i	$-0.296826\pi$
$619$	29.6477	1.19164	0.595822	+	0.803117i	$0.296826\pi$
$620$	−12.2874	−0.493475
$621$	0	0
$622$	−13.0061	−0.521497
$623$	25.6156	1.02627
$624$	0	0
$625$	11.9748	0.478992
$626$	−3.04171	−0.121571
$627$	0	0
$628$	17.8607	0.712718
$629$	−38.5008	−1.53513
$630$	0	0
$631$	14.6241	0.582176	0.291088	−	0.956696i	$-0.405983\pi$
$631$	14.6241	0.582176	0.291088	+	0.956696i	$0.405983\pi$
$632$	−5.77827	−0.229847
$633$	0	0
$634$	3.36036	0.133457
$635$	−10.9106	−0.432975
$636$	0	0
$637$	−55.0915	−2.18280
$638$	4.59184	0.181793
$639$	0	0
$640$	−10.8595	−0.429261
$641$	−4.17547	−0.164921	−0.0824606	−	0.996594i	$-0.526278\pi$
$641$	−4.17547	−0.164921	−0.0824606	+	0.996594i	$0.526278\pi$
$642$	0	0
$643$	23.1928	0.914635	0.457317	−	0.889304i	$-0.348810\pi$
$643$	23.1928	0.914635	0.457317	+	0.889304i	$0.348810\pi$
$644$	11.5918	0.456781
$645$	0	0
$646$	−10.9837	−0.432148
$647$	31.0975	1.22257	0.611284	−	0.791411i	$-0.290653\pi$
$647$	31.0975	1.22257	0.611284	+	0.791411i	$0.290653\pi$
$648$	0	0
$649$	−23.2347	−0.912043
$650$	−7.65632	−0.300305
$651$	0	0
$652$	−13.2153	−0.517550
$653$	31.5439	1.23441	0.617204	−	0.786803i	$-0.288265\pi$
$653$	31.5439	1.23441	0.617204	+	0.786803i	$0.288265\pi$
$654$	0	0
$655$	13.5893	0.530979
$656$	−24.1813	−0.944120
$657$	0	0
$658$	−5.58027	−0.217542
$659$	−25.2754	−0.984590	−0.492295	−	0.870428i	$-0.663842\pi$
$659$	−25.2754	−0.984590	−0.492295	+	0.870428i	$0.663842\pi$
$660$	0	0
$661$	−21.7628	−0.846476	−0.423238	−	0.906018i	$-0.639107\pi$
$661$	−21.7628	−0.846476	−0.423238	+	0.906018i	$0.639107\pi$
$662$	6.61312	0.257026
$663$	0	0
$664$	4.52578	0.175634
$665$	−18.5154	−0.717995
$666$	0	0
$667$	6.92404	0.268100
$668$	31.3586	1.21330
$669$	0	0
$670$	−3.42561	−0.132343
$671$	27.0038	1.04247
$672$	0	0
$673$	−37.2710	−1.43669	−0.718346	−	0.695686i	$-0.755101\pi$
$673$	−37.2710	−1.43669	−0.718346	+	0.695686i	$0.755101\pi$
$674$	9.85835	0.379730
$675$	0	0
$676$	−3.28970	−0.126527
$677$	33.5515	1.28949	0.644745	−	0.764398i	$-0.276964\pi$
$677$	33.5515	1.28949	0.644745	+	0.764398i	$0.276964\pi$
$678$	0	0
$679$	−24.3599	−0.934846
$680$	9.53043	0.365475
$681$	0	0
$682$	−6.85073	−0.262328
$683$	−8.94737	−0.342362	−0.171181	−	0.985240i	$-0.554758\pi$
$683$	−8.94737	−0.342362	−0.171181	+	0.985240i	$0.554758\pi$
$684$	0	0
$685$	−12.4631	−0.476192
$686$	16.3882	0.625704
$687$	0	0
$688$	7.72638	0.294566
$689$	21.7846	0.829929
$690$	0	0
$691$	7.04969	0.268183	0.134092	−	0.990969i	$-0.457188\pi$
$691$	7.04969	0.268183	0.134092	+	0.990969i	$0.457188\pi$
$692$	32.5933	1.23901
$693$	0	0
$694$	0.742994	0.0282036
$695$	−7.92309	−0.300540
$696$	0	0
$697$	49.6502	1.88063
$698$	−8.06969	−0.305442
$699$	0	0
$700$	−33.1344	−1.25236
$701$	5.19179	0.196091	0.0980456	−	0.995182i	$-0.468741\pi$
$701$	5.19179	0.196091	0.0980456	+	0.995182i	$0.468741\pi$
$702$	0	0
$703$	−29.7256	−1.12112
$704$	5.52943	0.208398
$705$	0	0
$706$	16.9012	0.636083
$707$	−57.3230	−2.15586
$708$	0	0
$709$	−16.3067	−0.612410	−0.306205	−	0.951966i	$-0.599059\pi$
$709$	−16.3067	−0.612410	−0.306205	+	0.951966i	$0.599059\pi$
$710$	6.97816	0.261886
$711$	0	0
$712$	−10.1800	−0.381510
$713$	−10.3302	−0.386870
$714$	0	0
$715$	−7.19488	−0.269073
$716$	−30.5594	−1.14206
$717$	0	0
$718$	8.83756	0.329815
$719$	27.4968	1.02546	0.512729	−	0.858551i	$-0.328635\pi$
$719$	27.4968	1.02546	0.512729	+	0.858551i	$0.328635\pi$
$720$	0	0
$721$	45.2410	1.68486
$722$	0.779256	0.0290009
$723$	0	0
$724$	−31.2180	−1.16021
$725$	−19.7919	−0.735053
$726$	0	0
$727$	−15.2319	−0.564919	−0.282460	−	0.959279i	$-0.591150\pi$
$727$	−15.2319	−0.564919	−0.282460	+	0.959279i	$0.591150\pi$
$728$	32.6201	1.20898
$729$	0	0
$730$	0.112642	0.00416905
$731$	−15.8642	−0.586758
$732$	0	0
$733$	6.76475	0.249862	0.124931	−	0.992165i	$-0.460129\pi$
$733$	6.76475	0.249862	0.124931	+	0.992165i	$0.460129\pi$
$734$	5.35490	0.197653
$735$	0	0
$736$	−7.05548	−0.260068
$737$	14.1734	0.522082
$738$	0	0
$739$	17.9519	0.660372	0.330186	−	0.943916i	$-0.392889\pi$
$739$	17.9519	0.660372	0.330186	+	0.943916i	$0.392889\pi$
$740$	12.0822	0.444151
$741$	0	0
$742$	−12.7043	−0.466390
$743$	17.2096	0.631357	0.315679	−	0.948866i	$-0.397768\pi$
$743$	17.2096	0.631357	0.315679	+	0.948866i	$0.397768\pi$
$744$	0	0
$745$	4.31559	0.158111
$746$	−2.55195	−0.0934336
$747$	0	0
$748$	−18.4714	−0.675380
$749$	−42.2188	−1.54264
$750$	0	0
$751$	4.98853	0.182034	0.0910170	−	0.995849i	$-0.470988\pi$
$751$	4.98853	0.182034	0.0910170	+	0.995849i	$0.470988\pi$
$752$	−6.53032	−0.238136
$753$	0	0
$754$	9.12739	0.332400
$755$	−0.0898146	−0.00326869
$756$	0	0
$757$	−19.5585	−0.710864	−0.355432	−	0.934702i	$-0.615666\pi$
$757$	−19.5585	−0.710864	−0.355432	+	0.934702i	$0.615666\pi$
$758$	−3.93335	−0.142866
$759$	0	0
$760$	7.35823	0.266911
$761$	41.0235	1.48710	0.743550	−	0.668681i	$-0.233141\pi$
$761$	41.0235	1.48710	0.743550	+	0.668681i	$0.233141\pi$
$762$	0	0
$763$	−14.9738	−0.542087
$764$	23.3425	0.844503
$765$	0	0
$766$	1.20131	0.0434053
$767$	−46.1847	−1.66763
$768$	0	0
$769$	11.7068	0.422159	0.211079	−	0.977469i	$-0.432302\pi$
$769$	11.7068	0.422159	0.211079	+	0.977469i	$0.432302\pi$
$770$	4.19589	0.151209
$771$	0	0
$772$	44.9489	1.61775
$773$	6.66278	0.239643	0.119822	−	0.992795i	$-0.461768\pi$
$773$	6.66278	0.239643	0.119822	+	0.992795i	$0.461768\pi$
$774$	0	0
$775$	29.5283	1.06069
$776$	9.68090	0.347524
$777$	0	0
$778$	−13.9945	−0.501727
$779$	38.3338	1.37345
$780$	0	0
$781$	−28.8719	−1.03312
$782$	3.75331	0.134218
$783$	0	0
$784$	37.5980	1.34279
$785$	9.74865	0.347944
$786$	0	0
$787$	2.88890	0.102978	0.0514891	−	0.998674i	$-0.483603\pi$
$787$	2.88890	0.102978	0.0514891	+	0.998674i	$0.483603\pi$
$788$	−26.4909	−0.943698
$789$	0	0
$790$	−1.47739	−0.0525633
$791$	−3.88896	−0.138275
$792$	0	0
$793$	53.6767	1.90612
$794$	−5.87762	−0.208589
$795$	0	0
$796$	34.1574	1.21068
$797$	−18.0912	−0.640824	−0.320412	−	0.947278i	$-0.603821\pi$
$797$	−18.0912	−0.640824	−0.320412	+	0.947278i	$0.603821\pi$
$798$	0	0
$799$	13.4084	0.474355
$800$	20.1676	0.713033
$801$	0	0
$802$	−15.4528	−0.545656
$803$	−0.466051	−0.0164466
$804$	0	0
$805$	6.32700	0.222997
$806$	−13.6175	−0.479655
$807$	0	0
$808$	22.7809	0.801428
$809$	−8.04121	−0.282714	−0.141357	−	0.989959i	$-0.545147\pi$
$809$	−8.04121	−0.282714	−0.141357	+	0.989959i	$0.545147\pi$
$810$	0	0
$811$	24.5111	0.860701	0.430350	−	0.902662i	$-0.358390\pi$
$811$	24.5111	0.860701	0.430350	+	0.902662i	$0.358390\pi$
$812$	39.5008	1.38621
$813$	0	0
$814$	6.73632	0.236108
$815$	−7.21311	−0.252664
$816$	0	0
$817$	−12.2484	−0.428517
$818$	6.91507	0.241780
$819$	0	0
$820$	−15.5811	−0.544115
$821$	10.6539	0.371823	0.185912	−	0.982566i	$-0.440476\pi$
$821$	10.6539	0.371823	0.185912	+	0.982566i	$0.440476\pi$
$822$	0	0
$823$	25.6473	0.894009	0.447004	−	0.894532i	$-0.352491\pi$
$823$	25.6473	0.894009	0.447004	+	0.894532i	$0.352491\pi$
$824$	−17.9793	−0.626339
$825$	0	0
$826$	26.9339	0.937150
$827$	−18.0150	−0.626442	−0.313221	−	0.949680i	$-0.601408\pi$
$827$	−18.0150	−0.626442	−0.313221	+	0.949680i	$0.601408\pi$
$828$	0	0
$829$	10.1189	0.351444	0.175722	−	0.984440i	$-0.443774\pi$
$829$	10.1189	0.351444	0.175722	+	0.984440i	$0.443774\pi$
$830$	1.15716	0.0401655
$831$	0	0
$832$	10.9911	0.381047
$833$	−77.1981	−2.67476
$834$	0	0
$835$	17.1160	0.592325
$836$	−14.2613	−0.493238
$837$	0	0
$838$	15.1571	0.523593
$839$	46.0365	1.58936	0.794679	−	0.607030i	$-0.207639\pi$
$839$	46.0365	1.58936	0.794679	+	0.607030i	$0.207639\pi$
$840$	0	0
$841$	−5.40531	−0.186390
$842$	13.3734	0.460879
$843$	0	0
$844$	−49.2411	−1.69495
$845$	−1.79557	−0.0617696
$846$	0	0
$847$	33.3928	1.14739
$848$	−14.8673	−0.510544
$849$	0	0
$850$	−10.7286	−0.367987
$851$	10.1577	0.348202
$852$	0	0
$853$	−55.0001	−1.88317	−0.941584	−	0.336780i	$-0.890662\pi$
$853$	−55.0001	−1.88317	−0.941584	+	0.336780i	$0.890662\pi$
$854$	−31.3031	−1.07117
$855$	0	0
$856$	16.7783	0.573469
$857$	31.5202	1.07671	0.538354	−	0.842719i	$-0.319046\pi$
$857$	31.5202	1.07671	0.538354	+	0.842719i	$0.319046\pi$
$858$	0	0
$859$	−37.2208	−1.26996	−0.634979	−	0.772529i	$-0.718991\pi$
$859$	−37.2208	−1.26996	−0.634979	+	0.772529i	$0.718991\pi$
$860$	4.97846	0.169764
$861$	0	0
$862$	−6.87180	−0.234054
$863$	−30.3224	−1.03218	−0.516092	−	0.856533i	$-0.672614\pi$
$863$	−30.3224	−1.03218	−0.516092	+	0.856533i	$0.672614\pi$
$864$	0	0
$865$	17.7900	0.604877
$866$	8.64563	0.293791
$867$	0	0
$868$	−58.9327	−2.00031
$869$	6.11267	0.207358
$870$	0	0
$871$	28.1730	0.954605
$872$	5.95075	0.201518
$873$	0	0
$874$	2.89784	0.0980210
$875$	−40.2783	−1.36165
$876$	0	0
$877$	27.0037	0.911849	0.455925	−	0.890018i	$-0.349309\pi$
$877$	27.0037	0.911849	0.455925	+	0.890018i	$0.349309\pi$
$878$	1.23028	0.0415200
$879$	0	0
$880$	4.91025	0.165525
$881$	0.565506	0.0190524	0.00952619	−	0.999955i	$-0.496968\pi$
$881$	0.565506	0.0190524	0.00952619	+	0.999955i	$0.496968\pi$
$882$	0	0
$883$	23.8838	0.803752	0.401876	−	0.915694i	$-0.368358\pi$
$883$	23.8838	0.803752	0.401876	+	0.915694i	$0.368358\pi$
$884$	−36.7163	−1.23490
$885$	0	0
$886$	−8.20600	−0.275686
$887$	10.1717	0.341533	0.170767	−	0.985311i	$-0.445376\pi$
$887$	10.1717	0.341533	0.170767	+	0.985311i	$0.445376\pi$
$888$	0	0
$889$	−52.3294	−1.75507
$890$	−2.60282	−0.0872468
$891$	0	0
$892$	0.682968	0.0228675
$893$	10.3523	0.346427
$894$	0	0
$895$	−16.6798	−0.557544
$896$	−52.0843	−1.74001
$897$	0	0
$898$	−14.8975	−0.497135
$899$	−35.2018	−1.17405
$900$	0	0
$901$	30.5262	1.01697
$902$	−8.68707	−0.289248
$903$	0	0
$904$	1.54552	0.0514031
$905$	−17.0393	−0.566405
$906$	0	0
$907$	−44.1661	−1.46651	−0.733255	−	0.679954i	$-0.762000\pi$
$907$	−44.1661	−1.46651	−0.733255	+	0.679954i	$0.762000\pi$
$908$	18.1146	0.601154
$909$	0	0
$910$	8.34036	0.276480
$911$	7.07316	0.234344	0.117172	−	0.993112i	$-0.462617\pi$
$911$	7.07316	0.234344	0.117172	+	0.993112i	$0.462617\pi$
$912$	0	0
$913$	−4.78770	−0.158450
$914$	15.9382	0.527190
$915$	0	0
$916$	18.5672	0.613479
$917$	65.1769	2.15233
$918$	0	0
$919$	23.6016	0.778544	0.389272	−	0.921123i	$-0.372727\pi$
$919$	23.6016	0.778544	0.389272	+	0.921123i	$0.372727\pi$
$920$	−2.51442	−0.0828981
$921$	0	0
$922$	6.03976	0.198909
$923$	−57.3899	−1.88901
$924$	0	0
$925$	−29.0352	−0.954670
$926$	−5.91530	−0.194389
$927$	0	0
$928$	−24.0426	−0.789236
$929$	−31.8496	−1.04495	−0.522475	−	0.852654i	$-0.674991\pi$
$929$	−31.8496	−1.04495	−0.522475	+	0.852654i	$0.674991\pi$
$930$	0	0
$931$	−59.6029	−1.95341
$932$	45.2478	1.48214
$933$	0	0
$934$	2.62290	0.0858238
$935$	−10.0820	−0.329716
$936$	0	0
$937$	44.1729	1.44307	0.721533	−	0.692380i	$-0.243438\pi$
$937$	44.1729	1.44307	0.721533	+	0.692380i	$0.243438\pi$
$938$	−16.4299	−0.536454
$939$	0	0
$940$	−4.20778	−0.137243
$941$	34.5303	1.12566	0.562828	−	0.826574i	$-0.309713\pi$
$941$	34.5303	1.12566	0.562828	+	0.826574i	$0.309713\pi$
$942$	0	0
$943$	−13.0993	−0.426571
$944$	31.5194	1.02587
$945$	0	0
$946$	2.77569	0.0902454
$947$	3.46191	0.112497	0.0562485	−	0.998417i	$-0.482086\pi$
$947$	3.46191	0.112497	0.0562485	+	0.998417i	$0.482086\pi$
$948$	0	0
$949$	−0.926389	−0.0300719
$950$	−8.28329	−0.268745
$951$	0	0
$952$	45.7097	1.48146
$953$	−60.2596	−1.95200	−0.976000	−	0.217771i	$-0.930122\pi$
$953$	−60.2596	−1.95200	−0.976000	+	0.217771i	$0.930122\pi$
$954$	0	0
$955$	12.7407	0.412280
$956$	−44.2936	−1.43256
$957$	0	0
$958$	−12.3664	−0.399541
$959$	−59.7755	−1.93025
$960$	0	0
$961$	21.5188	0.694156
$962$	13.3901	0.431713
$963$	0	0
$964$	1.76250	0.0567662
$965$	24.5338	0.789772
$966$	0	0
$967$	−39.9168	−1.28364	−0.641819	−	0.766856i	$-0.721820\pi$
$967$	−39.9168	−1.28364	−0.641819	+	0.766856i	$0.721820\pi$
$968$	−13.2707	−0.426536
$969$	0	0
$970$	2.47522	0.0794746
$971$	−43.7104	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$971$	−43.7104	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$972$	0	0
$973$	−38.0006	−1.21824
$974$	5.08020	0.162780
$975$	0	0
$976$	−36.6325	−1.17258
$977$	0.955388	0.0305656	0.0152828	−	0.999883i	$-0.495135\pi$
$977$	0.955388	0.0305656	0.0152828	+	0.999883i	$0.495135\pi$
$978$	0	0
$979$	10.7691	0.344182
$980$	24.2261	0.773874
$981$	0	0
$982$	16.7090	0.533205
$983$	48.7488	1.55485	0.777423	−	0.628978i	$-0.216526\pi$
$983$	48.7488	1.55485	0.777423	+	0.628978i	$0.216526\pi$
$984$	0	0
$985$	−14.4591	−0.460707
$986$	12.7899	0.407315
$987$	0	0
$988$	−28.3478	−0.901865
$989$	4.18547	0.133090
$990$	0	0
$991$	0.488986	0.0155332	0.00776658	−	0.999970i	$-0.497528\pi$
$991$	0.488986	0.0155332	0.00776658	+	0.999970i	$0.497528\pi$
$992$	35.8700	1.13887
$993$	0	0
$994$	33.4685	1.06156
$995$	18.6437	0.591044
$996$	0	0
$997$	39.5339	1.25205	0.626026	−	0.779802i	$-0.284680\pi$
$997$	39.5339	1.25205	0.626026	+	0.779802i	$0.284680\pi$
$998$	−7.78153	−0.246320
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.e.1.2		7
3.2	odd	2		241.2.a.a.1.6	✓	7
12.11	even	2		3856.2.a.j.1.5		7
15.14	odd	2		6025.2.a.f.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.a.1.6	✓	7	3.2	odd	2
2169.2.a.e.1.2		7	1.1	even	1	trivial
3856.2.a.j.1.5		7	12.11	even	2
6025.2.a.f.1.2		7	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:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2169.a (trivial)

\(f(q)\)	\(=\)	\(q-0.487343 q^{2} -1.76250 q^{4} -0.961999 q^{5} -4.61392 q^{7} +1.83363 q^{8} +O(q^{10})\)	\(q-0.487343 q^{2} -1.76250 q^{4} -0.961999 q^{5} -4.61392 q^{7} +1.83363 q^{8} +0.468824 q^{10} -1.93974 q^{11} -3.85571 q^{13} +2.24856 q^{14} +2.63139 q^{16} -5.40289 q^{17} -4.17145 q^{19} +1.69552 q^{20} +0.945321 q^{22} +1.42545 q^{23} -4.07456 q^{25} +1.87905 q^{26} +8.13202 q^{28} +4.85744 q^{29} -7.24699 q^{31} -4.94964 q^{32} +2.63306 q^{34} +4.43859 q^{35} +7.12597 q^{37} +2.03293 q^{38} -1.76395 q^{40} -9.18955 q^{41} +2.93624 q^{43} +3.41879 q^{44} -0.694685 q^{46} -2.48170 q^{47} +14.2883 q^{49} +1.98571 q^{50} +6.79568 q^{52} -5.64997 q^{53} +1.86603 q^{55} -8.46022 q^{56} -2.36724 q^{58} +11.9783 q^{59} -13.9214 q^{61} +3.53177 q^{62} -2.85060 q^{64} +3.70919 q^{65} -7.30682 q^{67} +9.52258 q^{68} -2.16312 q^{70} +14.8844 q^{71} +0.240264 q^{73} -3.47279 q^{74} +7.35217 q^{76} +8.94983 q^{77} -3.15128 q^{79} -2.53139 q^{80} +4.47847 q^{82} +2.46821 q^{83} +5.19758 q^{85} -1.43096 q^{86} -3.55677 q^{88} -5.55181 q^{89} +17.7900 q^{91} -2.51235 q^{92} +1.20944 q^{94} +4.01293 q^{95} +5.27964 q^{97} -6.96330 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8}+O(q^{10}) \) 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8	\( 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8} + 3 q^{10} + 18 q^{11} - q^{13} + 6 q^{14} + 4 q^{16} + 2 q^{17} - 6 q^{19} + 8 q^{20} + 10 q^{22} + 22 q^{23} + 5 q^{25} - 8 q^{26} + 9 q^{28} + 16 q^{29} - 18 q^{31} + 6 q^{32} + 11 q^{34} - 7 q^{35} + 8 q^{37} - 16 q^{38} + 14 q^{40} + 15 q^{41} + 14 q^{43} + 4 q^{44} + 11 q^{46} + 10 q^{47} + 6 q^{49} + 4 q^{50} + 27 q^{52} - 15 q^{53} + 29 q^{55} - 13 q^{56} + 17 q^{58} + 18 q^{59} + 4 q^{61} - 13 q^{62} + 2 q^{64} + 7 q^{65} + 18 q^{67} + 15 q^{68} + 8 q^{70} + 50 q^{71} - 10 q^{74} - 20 q^{76} - 17 q^{77} - 15 q^{79} + 11 q^{80} + 45 q^{82} + 24 q^{83} - 2 q^{85} + 23 q^{86} + 8 q^{88} + 13 q^{89} - 12 q^{91} + 10 q^{92} - 32 q^{94} + 41 q^{95} + q^{97} - 9 q^{98}+O(q^{100}) \) 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8 + 3 * q^10 + 18 * q^11 - q^13 + 6 * q^14 + 4 * q^16 + 2 * q^17 - 6 * q^19 + 8 * q^20 + 10 * q^22 + 22 * q^23 + 5 * q^25 - 8 * q^26 + 9 * q^28 + 16 * q^29 - 18 * q^31 + 6 * q^32 + 11 * q^34 - 7 * q^35 + 8 * q^37 - 16 * q^38 + 14 * q^40 + 15 * q^41 + 14 * q^43 + 4 * q^44 + 11 * q^46 + 10 * q^47 + 6 * q^49 + 4 * q^50 + 27 * q^52 - 15 * q^53 + 29 * q^55 - 13 * q^56 + 17 * q^58 + 18 * q^59 + 4 * q^61 - 13 * q^62 + 2 * q^64 + 7 * q^65 + 18 * q^67 + 15 * q^68 + 8 * q^70 + 50 * q^71 - 10 * q^74 - 20 * q^76 - 17 * q^77 - 15 * q^79 + 11 * q^80 + 45 * q^82 + 24 * q^83 - 2 * q^85 + 23 * q^86 + 8 * q^88 + 13 * q^89 - 12 * q^91 + 10 * q^92 - 32 * q^94 + 41 * q^95 + q^97 - 9 * q^98

Introduction

L-functions

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Database

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