LMFDB - Embedded newform 2169.2.a.f.1.7

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 8x^{7} + 27x^{6} + 15x^{5} - 71x^{4} + 7x^{3} + 46x^{2} - 12x - 4 $ x^9 - 3x^8 - 8x^7 + 27x^6 + 15x^5 - 71x^4 + 7x^3 + 46x^2 - 12x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 723)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.900190$ 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)

$f(q)$	$=$	$q+0.900190 q^{2} -1.18966 q^{4} -2.66512 q^{5} +3.59926 q^{7} -2.87130 q^{8} +O(q^{10})$	\(q+0.900190 q^{2} -1.18966 q^{4} -2.66512 q^{5} +3.59926 q^{7} -2.87130 q^{8} -2.39912 q^{10} -0.709205 q^{11} -0.335398 q^{13} +3.24002 q^{14} -0.205399 q^{16} +5.42712 q^{17} +1.03077 q^{19} +3.17059 q^{20} -0.638419 q^{22} -5.35800 q^{23} +2.10289 q^{25} -0.301922 q^{26} -4.28189 q^{28} -3.29353 q^{29} -6.05692 q^{31} +5.55770 q^{32} +4.88544 q^{34} -9.59248 q^{35} +8.93642 q^{37} +0.927886 q^{38} +7.65237 q^{40} -8.14498 q^{41} -5.21956 q^{43} +0.843711 q^{44} -4.82322 q^{46} -6.97401 q^{47} +5.95468 q^{49} +1.89300 q^{50} +0.399009 q^{52} -3.96679 q^{53} +1.89012 q^{55} -10.3346 q^{56} -2.96480 q^{58} -9.11677 q^{59} -0.537001 q^{61} -5.45238 q^{62} +5.41378 q^{64} +0.893878 q^{65} -12.6703 q^{67} -6.45642 q^{68} -8.63505 q^{70} +1.77843 q^{71} -5.47439 q^{73} +8.04448 q^{74} -1.22626 q^{76} -2.55261 q^{77} +4.97542 q^{79} +0.547415 q^{80} -7.33203 q^{82} -5.60617 q^{83} -14.4640 q^{85} -4.69860 q^{86} +2.03634 q^{88} -16.0610 q^{89} -1.20719 q^{91} +6.37419 q^{92} -6.27794 q^{94} -2.74712 q^{95} +17.1785 q^{97} +5.36034 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{2} + 7 q^{4} - 12 q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10}) $ 9 * q - 3 * q^2 + 7 * q^4 - 12 * q^5 - 5 * q^7 - 6 * q^8	$ 9 q - 3 q^{2} + 7 q^{4} - 12 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{10} - 3 q^{11} + 18 q^{13} - q^{16} - 16 q^{17} + 3 q^{19} - 17 q^{20} + 8 q^{22} - 9 q^{23} + 15 q^{25} - 14 q^{26} - 3 q^{28} - 15 q^{29} - 13 q^{31} - 10 q^{32} + q^{34} - q^{35} + 19 q^{37} - 12 q^{38} + 24 q^{40} - 18 q^{41} - 11 q^{43} - 14 q^{44} + q^{46} - 19 q^{47} + 14 q^{49} - 21 q^{50} + 10 q^{52} - 31 q^{53} - q^{55} - q^{56} + 12 q^{58} - 11 q^{59} + 2 q^{61} - 11 q^{62} - 34 q^{64} - 33 q^{65} - 12 q^{67} - 24 q^{68} - 42 q^{70} - 6 q^{71} + 18 q^{73} - 8 q^{74} + q^{76} - 20 q^{77} - 32 q^{79} - 7 q^{80} - 27 q^{82} - 13 q^{83} + 12 q^{85} + 2 q^{86} - 25 q^{88} - 24 q^{89} - 34 q^{91} + 7 q^{92} - 21 q^{94} - 14 q^{95} + 23 q^{97} - 8 q^{98}+O(q^{100}) $ 9 * q - 3 * q^2 + 7 * q^4 - 12 * q^5 - 5 * q^7 - 6 * q^8 + 5 * q^10 - 3 * q^11 + 18 * q^13 - q^16 - 16 * q^17 + 3 * q^19 - 17 * q^20 + 8 * q^22 - 9 * q^23 + 15 * q^25 - 14 * q^26 - 3 * q^28 - 15 * q^29 - 13 * q^31 - 10 * q^32 + q^34 - q^35 + 19 * q^37 - 12 * q^38 + 24 * q^40 - 18 * q^41 - 11 * q^43 - 14 * q^44 + q^46 - 19 * q^47 + 14 * q^49 - 21 * q^50 + 10 * q^52 - 31 * q^53 - q^55 - q^56 + 12 * q^58 - 11 * q^59 + 2 * q^61 - 11 * q^62 - 34 * q^64 - 33 * q^65 - 12 * q^67 - 24 * q^68 - 42 * q^70 - 6 * q^71 + 18 * q^73 - 8 * q^74 + q^76 - 20 * q^77 - 32 * q^79 - 7 * q^80 - 27 * q^82 - 13 * q^83 + 12 * q^85 + 2 * q^86 - 25 * q^88 - 24 * q^89 - 34 * q^91 + 7 * q^92 - 21 * q^94 - 14 * q^95 + 23 * q^97 - 8 * 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 $

$2$	0.900190	0.636531	0.318265	−	0.948002i	$-0.396900\pi$
$2$	0.900190	0.636531	0.318265	+	0.948002i	$0.396900\pi$
$3$	0	0
$3$	0	0
$4$	−1.18966	−0.594829
$5$	−2.66512	−1.19188	−0.595940	−	0.803029i	$-0.703220\pi$
$5$	−2.66512	−1.19188	−0.595940	+	0.803029i	$0.703220\pi$
$6$	0	0
$7$	3.59926	1.36039	0.680196	−	0.733030i	$-0.261894\pi$
$7$	3.59926	1.36039	0.680196	+	0.733030i	$0.261894\pi$
$8$	−2.87130	−1.01516
$9$	0	0
$10$	−2.39912	−0.758668
$11$	−0.709205	−0.213833	−0.106917	−	0.994268i	$-0.534098\pi$
$11$	−0.709205	−0.213833	−0.106917	+	0.994268i	$0.534098\pi$
$12$	0	0
$13$	−0.335398	−0.0930227	−0.0465114	−	0.998918i	$-0.514810\pi$
$13$	−0.335398	−0.0930227	−0.0465114	+	0.998918i	$0.514810\pi$
$14$	3.24002	0.865932
$15$	0	0
$16$	−0.205399	−0.0513498
$17$	5.42712	1.31627	0.658135	−	0.752900i	$-0.271346\pi$
$17$	5.42712	1.31627	0.658135	+	0.752900i	$0.271346\pi$
$18$	0	0
$19$	1.03077	0.236474	0.118237	−	0.992985i	$-0.462276\pi$
$19$	1.03077	0.236474	0.118237	+	0.992985i	$0.462276\pi$
$20$	3.17059	0.708964
$21$	0	0
$22$	−0.638419	−0.136111
$23$	−5.35800	−1.11722	−0.558610	−	0.829430i	$-0.688665\pi$
$23$	−5.35800	−1.11722	−0.558610	+	0.829430i	$0.688665\pi$
$24$	0	0
$25$	2.10289	0.420578
$26$	−0.301922	−0.0592118
$27$	0	0
$28$	−4.28189	−0.809201
$29$	−3.29353	−0.611593	−0.305797	−	0.952097i	$-0.598923\pi$
$29$	−3.29353	−0.611593	−0.305797	+	0.952097i	$0.598923\pi$
$30$	0	0
$31$	−6.05692	−1.08785	−0.543927	−	0.839132i	$-0.683063\pi$
$31$	−6.05692	−1.08785	−0.543927	+	0.839132i	$0.683063\pi$
$32$	5.55770	0.982472
$33$	0	0
$34$	4.88544	0.837846
$35$	−9.59248	−1.62142
$36$	0	0
$37$	8.93642	1.46914	0.734569	−	0.678534i	$-0.237384\pi$
$37$	8.93642	1.46914	0.734569	+	0.678534i	$0.237384\pi$
$38$	0.927886	0.150523
$39$	0	0
$40$	7.65237	1.20995
$41$	−8.14498	−1.27203	−0.636016	−	0.771676i	$-0.719419\pi$
$41$	−8.14498	−1.27203	−0.636016	+	0.771676i	$0.719419\pi$
$42$	0	0
$43$	−5.21956	−0.795976	−0.397988	−	0.917391i	$-0.630291\pi$
$43$	−5.21956	−0.795976	−0.397988	+	0.917391i	$0.630291\pi$
$44$	0.843711	0.127194
$45$	0	0
$46$	−4.82322	−0.711145
$47$	−6.97401	−1.01726	−0.508632	−	0.860984i	$-0.669849\pi$
$47$	−6.97401	−1.01726	−0.508632	+	0.860984i	$0.669849\pi$
$48$	0	0
$49$	5.95468	0.850669
$50$	1.89300	0.267711
$51$	0	0
$52$	0.399009	0.0553326
$53$	−3.96679	−0.544880	−0.272440	−	0.962173i	$-0.587831\pi$
$53$	−3.96679	−0.544880	−0.272440	+	0.962173i	$0.587831\pi$
$54$	0	0
$55$	1.89012	0.254864
$56$	−10.3346	−1.38101
$57$	0	0
$58$	−2.96480	−0.389298
$59$	−9.11677	−1.18690	−0.593451	−	0.804870i	$-0.702235\pi$
$59$	−9.11677	−1.18690	−0.593451	+	0.804870i	$0.702235\pi$
$60$	0	0
$61$	−0.537001	−0.0687559	−0.0343780	−	0.999409i	$-0.510945\pi$
$61$	−0.537001	−0.0687559	−0.0343780	+	0.999409i	$0.510945\pi$
$62$	−5.45238	−0.692453
$63$	0	0
$64$	5.41378	0.676723
$65$	0.893878	0.110872
$66$	0	0
$67$	−12.6703	−1.54793	−0.773963	−	0.633231i	$-0.781728\pi$
$67$	−12.6703	−1.54793	−0.773963	+	0.633231i	$0.781728\pi$
$68$	−6.45642	−0.782956
$69$	0	0
$70$	−8.63505	−1.03209
$71$	1.77843	0.211060	0.105530	−	0.994416i	$-0.466346\pi$
$71$	1.77843	0.211060	0.105530	+	0.994416i	$0.466346\pi$
$72$	0	0
$73$	−5.47439	−0.640729	−0.320364	−	0.947294i	$-0.603805\pi$
$73$	−5.47439	−0.640729	−0.320364	+	0.947294i	$0.603805\pi$
$74$	8.04448	0.935152
$75$	0	0
$76$	−1.22626	−0.140662
$77$	−2.55261	−0.290897
$78$	0	0
$79$	4.97542	0.559778	0.279889	−	0.960032i	$-0.409702\pi$
$79$	4.97542	0.559778	0.279889	+	0.960032i	$0.409702\pi$
$80$	0.547415	0.0612029
$81$	0	0
$82$	−7.33203	−0.809687
$83$	−5.60617	−0.615357	−0.307679	−	0.951490i	$-0.599552\pi$
$83$	−5.60617	−0.615357	−0.307679	+	0.951490i	$0.599552\pi$
$84$	0	0
$85$	−14.4640	−1.56884
$86$	−4.69860	−0.506663
$87$	0	0
$88$	2.03634	0.217074
$89$	−16.0610	−1.70247	−0.851233	−	0.524788i	$-0.824145\pi$
$89$	−16.0610	−1.70247	−0.851233	+	0.524788i	$0.824145\pi$
$90$	0	0
$91$	−1.20719	−0.126547
$92$	6.37419	0.664555
$93$	0	0
$94$	−6.27794	−0.647520
$95$	−2.74712	−0.281849
$96$	0	0
$97$	17.1785	1.74421	0.872104	−	0.489320i	$-0.162755\pi$
$97$	17.1785	1.74421	0.872104	+	0.489320i	$0.162755\pi$
$98$	5.36034	0.541477
$99$	0	0
$100$	−2.50172	−0.250172
$101$	8.92264	0.887836	0.443918	−	0.896067i	$-0.353588\pi$
$101$	8.92264	0.887836	0.443918	+	0.896067i	$0.353588\pi$
$102$	0	0
$103$	−16.0294	−1.57943	−0.789714	−	0.613475i	$-0.789771\pi$
$103$	−16.0294	−1.57943	−0.789714	+	0.613475i	$0.789771\pi$
$104$	0.963028	0.0944327
$105$	0	0
$106$	−3.57086	−0.346833
$107$	8.07473	0.780614	0.390307	−	0.920685i	$-0.372369\pi$
$107$	8.07473	0.780614	0.390307	+	0.920685i	$0.372369\pi$
$108$	0	0
$109$	−3.30664	−0.316718	−0.158359	−	0.987382i	$-0.550620\pi$
$109$	−3.30664	−0.316718	−0.158359	+	0.987382i	$0.550620\pi$
$110$	1.70147	0.162228
$111$	0	0
$112$	−0.739286	−0.0698560
$113$	−7.83740	−0.737281	−0.368641	−	0.929572i	$-0.620177\pi$
$113$	−7.83740	−0.737281	−0.368641	+	0.929572i	$0.620177\pi$
$114$	0	0
$115$	14.2797	1.33159
$116$	3.91817	0.363793
$117$	0	0
$118$	−8.20683	−0.755500
$119$	19.5336	1.79064
$120$	0	0
$121$	−10.4970	−0.954275
$122$	−0.483403	−0.0437653
$123$	0	0
$124$	7.20566	0.647087
$125$	7.72116	0.690602
$126$	0	0
$127$	−5.06217	−0.449195	−0.224598	−	0.974452i	$-0.572107\pi$
$127$	−5.06217	−0.449195	−0.224598	+	0.974452i	$0.572107\pi$
$128$	−6.24196	−0.551717
$129$	0	0
$130$	0.804660	0.0705734
$131$	14.9980	1.31038	0.655190	−	0.755464i	$-0.272589\pi$
$131$	14.9980	1.31038	0.655190	+	0.755464i	$0.272589\pi$
$132$	0	0
$133$	3.71000	0.321698
$134$	−11.4057	−0.985302
$135$	0	0
$136$	−15.5829	−1.33622
$137$	−13.1423	−1.12282	−0.561410	−	0.827538i	$-0.689741\pi$
$137$	−13.1423	−1.12282	−0.561410	+	0.827538i	$0.689741\pi$
$138$	0	0
$139$	17.1362	1.45348	0.726738	−	0.686914i	$-0.241035\pi$
$139$	17.1362	1.45348	0.726738	+	0.686914i	$0.241035\pi$
$140$	11.4118	0.964470
$141$	0	0
$142$	1.60092	0.134346
$143$	0.237866	0.0198913
$144$	0	0
$145$	8.77767	0.728945
$146$	−4.92799	−0.407844
$147$	0	0
$148$	−10.6313	−0.873886
$149$	−17.9326	−1.46909	−0.734547	−	0.678557i	$-0.762606\pi$
$149$	−17.9326	−1.46909	−0.734547	+	0.678557i	$0.762606\pi$
$150$	0	0
$151$	−10.7872	−0.877854	−0.438927	−	0.898523i	$-0.644641\pi$
$151$	−10.7872	−0.877854	−0.438927	+	0.898523i	$0.644641\pi$
$152$	−2.95964	−0.240059
$153$	0	0
$154$	−2.29784	−0.185165
$155$	16.1424	1.29659
$156$	0	0
$157$	22.3697	1.78529	0.892647	−	0.450756i	$-0.148846\pi$
$157$	22.3697	1.78529	0.892647	+	0.450756i	$0.148846\pi$
$158$	4.47882	0.356316
$159$	0	0
$160$	−14.8120	−1.17099
$161$	−19.2848	−1.51986
$162$	0	0
$163$	−0.00741019	−0.000580411 0	−0.000290205	−	1.00000i	$-0.500092\pi$
$163$	−0.00741019	−0.000580411 0	−0.000290205		1.00000i	$0.500092\pi$
$164$	9.68974	0.756641
$165$	0	0
$166$	−5.04662	−0.391694
$167$	−18.6203	−1.44088	−0.720439	−	0.693518i	$-0.756060\pi$
$167$	−18.6203	−1.44088	−0.720439	+	0.693518i	$0.756060\pi$
$168$	0	0
$169$	−12.8875	−0.991347
$170$	−13.0203	−0.998612
$171$	0	0
$172$	6.20949	0.473470
$173$	0.563448	0.0428381	0.0214191	−	0.999771i	$-0.493182\pi$
$173$	0.563448	0.0428381	0.0214191	+	0.999771i	$0.493182\pi$
$174$	0	0
$175$	7.56884	0.572151
$176$	0.145670	0.0109803
$177$	0	0
$178$	−14.4580	−1.08367
$179$	24.8545	1.85771	0.928856	−	0.370442i	$-0.120794\pi$
$179$	24.8545	1.85771	0.928856	+	0.370442i	$0.120794\pi$
$180$	0	0
$181$	2.62281	0.194952	0.0974759	−	0.995238i	$-0.468923\pi$
$181$	2.62281	0.194952	0.0974759	+	0.995238i	$0.468923\pi$
$182$	−1.08670	−0.0805513
$183$	0	0
$184$	15.3844	1.13415
$185$	−23.8167	−1.75104
$186$	0	0
$187$	−3.84894	−0.281462
$188$	8.29669	0.605098
$189$	0	0
$190$	−2.47293	−0.179405
$191$	3.13534	0.226866	0.113433	−	0.993546i	$-0.463815\pi$
$191$	3.13534	0.226866	0.113433	+	0.993546i	$0.463815\pi$
$192$	0	0
$193$	−7.37593	−0.530931	−0.265466	−	0.964120i	$-0.585526\pi$
$193$	−7.37593	−0.530931	−0.265466	+	0.964120i	$0.585526\pi$
$194$	15.4639	1.11024
$195$	0	0
$196$	−7.08403	−0.506002
$197$	−1.70100	−0.121191	−0.0605955	−	0.998162i	$-0.519300\pi$
$197$	−1.70100	−0.121191	−0.0605955	+	0.998162i	$0.519300\pi$
$198$	0	0
$199$	−8.36159	−0.592738	−0.296369	−	0.955074i	$-0.595776\pi$
$199$	−8.36159	−0.592738	−0.296369	+	0.955074i	$0.595776\pi$
$200$	−6.03802	−0.426952
$201$	0	0
$202$	8.03207	0.565135
$203$	−11.8543	−0.832007
$204$	0	0
$205$	21.7074	1.51611
$206$	−14.4296	−1.00535
$207$	0	0
$208$	0.0688906	0.00477670
$209$	−0.731025	−0.0505660
$210$	0	0
$211$	−1.58232	−0.108932	−0.0544658	−	0.998516i	$-0.517346\pi$
$211$	−1.58232	−0.108932	−0.0544658	+	0.998516i	$0.517346\pi$
$212$	4.71912	0.324110
$213$	0	0
$214$	7.26879	0.496884
$215$	13.9108	0.948708
$216$	0	0
$217$	−21.8004	−1.47991
$218$	−2.97660	−0.201601
$219$	0	0
$220$	−2.24859	−0.151600
$221$	−1.82025	−0.122443
$222$	0	0
$223$	−16.0890	−1.07740	−0.538699	−	0.842498i	$-0.681084\pi$
$223$	−16.0890	−1.07740	−0.538699	+	0.842498i	$0.681084\pi$
$224$	20.0036	1.33655
$225$	0	0
$226$	−7.05515	−0.469302
$227$	6.34837	0.421356	0.210678	−	0.977555i	$-0.432433\pi$
$227$	6.34837	0.421356	0.210678	+	0.977555i	$0.432433\pi$
$228$	0	0
$229$	−7.67314	−0.507056	−0.253528	−	0.967328i	$-0.581591\pi$
$229$	−7.67314	−0.507056	−0.253528	+	0.967328i	$0.581591\pi$
$230$	12.8545	0.847599
$231$	0	0
$232$	9.45671	0.620863
$233$	18.5641	1.21617	0.608086	−	0.793871i	$-0.291937\pi$
$233$	18.5641	1.21617	0.608086	+	0.793871i	$0.291937\pi$
$234$	0	0
$235$	18.5866	1.21246
$236$	10.8458	0.706004
$237$	0	0
$238$	17.5840	1.13980
$239$	12.1272	0.784441	0.392220	−	0.919871i	$-0.371707\pi$
$239$	12.1272	0.784441	0.392220	+	0.919871i	$0.371707\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−9.44932	−0.607425
$243$	0	0
$244$	0.638847	0.0408980
$245$	−15.8700	−1.01389
$246$	0	0
$247$	−0.345717	−0.0219975
$248$	17.3912	1.10434
$249$	0	0
$250$	6.95052	0.439589
$251$	4.30898	0.271980	0.135990	−	0.990710i	$-0.456578\pi$
$251$	4.30898	0.271980	0.135990	+	0.990710i	$0.456578\pi$
$252$	0	0
$253$	3.79992	0.238899
$254$	−4.55692	−0.285927
$255$	0	0
$256$	−16.4465	−1.02791
$257$	−8.55643	−0.533735	−0.266868	−	0.963733i	$-0.585989\pi$
$257$	−8.55643	−0.533735	−0.266868	+	0.963733i	$0.585989\pi$
$258$	0	0
$259$	32.1645	1.99861
$260$	−1.06341	−0.0659498
$261$	0	0
$262$	13.5010	0.834097
$263$	8.51901	0.525305	0.262652	−	0.964890i	$-0.415403\pi$
$263$	8.51901	0.525305	0.262652	+	0.964890i	$0.415403\pi$
$264$	0	0
$265$	10.5720	0.649431
$266$	3.33971	0.204770
$267$	0	0
$268$	15.0733	0.920751
$269$	5.03246	0.306835	0.153417	−	0.988161i	$-0.450972\pi$
$269$	5.03246	0.306835	0.153417	+	0.988161i	$0.450972\pi$
$270$	0	0
$271$	−13.9529	−0.847580	−0.423790	−	0.905761i	$-0.639300\pi$
$271$	−13.9529	−0.847580	−0.423790	+	0.905761i	$0.639300\pi$
$272$	−1.11473	−0.0675903
$273$	0	0
$274$	−11.8306	−0.714710
$275$	−1.49138	−0.0899335
$276$	0	0
$277$	5.55178	0.333574	0.166787	−	0.985993i	$-0.446661\pi$
$277$	5.55178	0.333574	0.166787	+	0.985993i	$0.446661\pi$
$278$	15.4259	0.925182
$279$	0	0
$280$	27.5429	1.64600
$281$	27.9352	1.66647	0.833236	−	0.552918i	$-0.186486\pi$
$281$	27.9352	1.66647	0.833236	+	0.552918i	$0.186486\pi$
$282$	0	0
$283$	18.8343	1.11958	0.559791	−	0.828634i	$-0.310882\pi$
$283$	18.8343	1.11958	0.559791	+	0.828634i	$0.310882\pi$
$284$	−2.11572	−0.125545
$285$	0	0
$286$	0.214125	0.0126615
$287$	−29.3159	−1.73046
$288$	0	0
$289$	12.4536	0.732567
$290$	7.90157	0.463996
$291$	0	0
$292$	6.51265	0.381124
$293$	−14.7622	−0.862417	−0.431208	−	0.902252i	$-0.641913\pi$
$293$	−14.7622	−0.862417	−0.431208	+	0.902252i	$0.641913\pi$
$294$	0	0
$295$	24.2973	1.41465
$296$	−25.6591	−1.49141
$297$	0	0
$298$	−16.1427	−0.935124
$299$	1.79706	0.103927
$300$	0	0
$301$	−18.7866	−1.08284
$302$	−9.71057	−0.558781
$303$	0	0
$304$	−0.211719	−0.0121429
$305$	1.43117	0.0819488
$306$	0	0
$307$	−13.6360	−0.778249	−0.389125	−	0.921185i	$-0.627222\pi$
$307$	−13.6360	−0.778249	−0.389125	+	0.921185i	$0.627222\pi$
$308$	3.03673	0.173034
$309$	0	0
$310$	14.5313	0.825320
$311$	4.52428	0.256549	0.128274	−	0.991739i	$-0.459056\pi$
$311$	4.52428	0.256549	0.128274	+	0.991739i	$0.459056\pi$
$312$	0	0
$313$	−8.42875	−0.476421	−0.238211	−	0.971214i	$-0.576561\pi$
$313$	−8.42875	−0.476421	−0.238211	+	0.971214i	$0.576561\pi$
$314$	20.1370	1.13639
$315$	0	0
$316$	−5.91905	−0.332972
$317$	20.9841	1.17858	0.589291	−	0.807921i	$-0.299407\pi$
$317$	20.9841	1.17858	0.589291	+	0.807921i	$0.299407\pi$
$318$	0	0
$319$	2.33579	0.130779
$320$	−14.4284	−0.806573
$321$	0	0
$322$	−17.3600	−0.967437
$323$	5.59410	0.311264
$324$	0	0
$325$	−0.705305	−0.0391233
$326$	−0.00667058	−0.000369449 0
$327$	0	0
$328$	23.3867	1.29131
$329$	−25.1013	−1.38388
$330$	0	0
$331$	22.4405	1.23344	0.616722	−	0.787181i	$-0.288460\pi$
$331$	22.4405	1.23344	0.616722	+	0.787181i	$0.288460\pi$
$332$	6.66942	0.366032
$333$	0	0
$334$	−16.7618	−0.917163
$335$	33.7680	1.84494
$336$	0	0
$337$	28.1009	1.53075	0.765377	−	0.643582i	$-0.222552\pi$
$337$	28.1009	1.53075	0.765377	+	0.643582i	$0.222552\pi$
$338$	−11.6012	−0.631023
$339$	0	0
$340$	17.2072	0.933189
$341$	4.29559	0.232619
$342$	0	0
$343$	−3.76238	−0.203149
$344$	14.9869	0.808041
$345$	0	0
$346$	0.507210	0.0272678
$347$	−13.3266	−0.715411	−0.357706	−	0.933834i	$-0.616441\pi$
$347$	−13.3266	−0.715411	−0.357706	+	0.933834i	$0.616441\pi$
$348$	0	0
$349$	−19.9160	−1.06608	−0.533039	−	0.846091i	$-0.678950\pi$
$349$	−19.9160	−1.06608	−0.533039	+	0.846091i	$0.678950\pi$
$350$	6.81340	0.364191
$351$	0	0
$352$	−3.94155	−0.210085
$353$	2.75493	0.146630	0.0733151	−	0.997309i	$-0.476642\pi$
$353$	2.75493	0.146630	0.0733151	+	0.997309i	$0.476642\pi$
$354$	0	0
$355$	−4.73973	−0.251558
$356$	19.1071	1.01268
$357$	0	0
$358$	22.3738	1.18249
$359$	21.4575	1.13249	0.566243	−	0.824239i	$-0.308397\pi$
$359$	21.4575	1.13249	0.566243	+	0.824239i	$0.308397\pi$
$360$	0	0
$361$	−17.9375	−0.944080
$362$	2.36103	0.124093
$363$	0	0
$364$	1.43614	0.0752741
$365$	14.5899	0.763672
$366$	0	0
$367$	−15.2185	−0.794402	−0.397201	−	0.917732i	$-0.630018\pi$
$367$	−15.2185	−0.794402	−0.397201	+	0.917732i	$0.630018\pi$
$368$	1.10053	0.0573691
$369$	0	0
$370$	−21.4395	−1.11459
$371$	−14.2775	−0.741251
$372$	0	0
$373$	30.6028	1.58456	0.792278	−	0.610161i	$-0.208895\pi$
$373$	30.6028	1.58456	0.792278	+	0.610161i	$0.208895\pi$
$374$	−3.46478	−0.179159
$375$	0	0
$376$	20.0245	1.03268
$377$	1.10464	0.0568920
$378$	0	0
$379$	21.7172	1.11554	0.557769	−	0.829996i	$-0.311657\pi$
$379$	21.7172	1.11554	0.557769	+	0.829996i	$0.311657\pi$
$380$	3.26814	0.167652
$381$	0	0
$382$	2.82241	0.144407
$383$	16.1770	0.826607	0.413303	−	0.910593i	$-0.364375\pi$
$383$	16.1770	0.826607	0.413303	+	0.910593i	$0.364375\pi$
$384$	0	0
$385$	6.80303	0.346714
$386$	−6.63974	−0.337954
$387$	0	0
$388$	−20.4365	−1.03751
$389$	−16.9649	−0.860153	−0.430077	−	0.902792i	$-0.641513\pi$
$389$	−16.9649	−0.860153	−0.430077	+	0.902792i	$0.641513\pi$
$390$	0	0
$391$	−29.0785	−1.47056
$392$	−17.0977	−0.863562
$393$	0	0
$394$	−1.53122	−0.0771418
$395$	−13.2601	−0.667189
$396$	0	0
$397$	−19.1487	−0.961045	−0.480522	−	0.876982i	$-0.659553\pi$
$397$	−19.1487	−0.961045	−0.480522	+	0.876982i	$0.659553\pi$
$398$	−7.52703	−0.377296
$399$	0	0
$400$	−0.431932	−0.0215966
$401$	−9.40245	−0.469536	−0.234768	−	0.972051i	$-0.575433\pi$
$401$	−9.40245	−0.469536	−0.234768	+	0.972051i	$0.575433\pi$
$402$	0	0
$403$	2.03148	0.101195
$404$	−10.6149	−0.528110
$405$	0	0
$406$	−10.6711	−0.529598
$407$	−6.33775	−0.314151
$408$	0	0
$409$	13.7789	0.681325	0.340663	−	0.940186i	$-0.389349\pi$
$409$	13.7789	0.681325	0.340663	+	0.940186i	$0.389349\pi$
$410$	19.5408	0.965050
$411$	0	0
$412$	19.0696	0.939489
$413$	−32.8136	−1.61465
$414$	0	0
$415$	14.9411	0.733432
$416$	−1.86404	−0.0913922
$417$	0	0
$418$	−0.658061	−0.0321868
$419$	−15.5754	−0.760911	−0.380455	−	0.924799i	$-0.624233\pi$
$419$	−15.5754	−0.760911	−0.380455	+	0.924799i	$0.624233\pi$
$420$	0	0
$421$	−36.0108	−1.75506	−0.877529	−	0.479523i	$-0.840810\pi$
$421$	−36.0108	−1.75506	−0.877529	+	0.479523i	$0.840810\pi$
$422$	−1.42439	−0.0693383
$423$	0	0
$424$	11.3898	0.553139
$425$	11.4126	0.553594
$426$	0	0
$427$	−1.93281	−0.0935351
$428$	−9.60616	−0.464331
$429$	0	0
$430$	12.5224	0.603882
$431$	8.60543	0.414509	0.207255	−	0.978287i	$-0.433547\pi$
$431$	8.60543	0.414509	0.207255	+	0.978287i	$0.433547\pi$
$432$	0	0
$433$	24.2836	1.16700	0.583498	−	0.812114i	$-0.301684\pi$
$433$	24.2836	1.16700	0.583498	+	0.812114i	$0.301684\pi$
$434$	−19.6245	−0.942007
$435$	0	0
$436$	3.93376	0.188393
$437$	−5.52285	−0.264194
$438$	0	0
$439$	−23.0802	−1.10156	−0.550779	−	0.834651i	$-0.685669\pi$
$439$	−23.0802	−1.10156	−0.550779	+	0.834651i	$0.685669\pi$
$440$	−5.42709	−0.258727
$441$	0	0
$442$	−1.63857	−0.0779387
$443$	−33.4910	−1.59121	−0.795604	−	0.605817i	$-0.792846\pi$
$443$	−33.4910	−1.59121	−0.795604	+	0.605817i	$0.792846\pi$
$444$	0	0
$445$	42.8047	2.02914
$446$	−14.4832	−0.685797
$447$	0	0
$448$	19.4856	0.920609
$449$	−6.08169	−0.287013	−0.143506	−	0.989649i	$-0.545838\pi$
$449$	−6.08169	−0.287013	−0.143506	+	0.989649i	$0.545838\pi$
$450$	0	0
$451$	5.77646	0.272003
$452$	9.32383	0.438556
$453$	0	0
$454$	5.71474	0.268206
$455$	3.21730	0.150829
$456$	0	0
$457$	−15.3289	−0.717054	−0.358527	−	0.933519i	$-0.616721\pi$
$457$	−15.3289	−0.717054	−0.358527	+	0.933519i	$0.616721\pi$
$458$	−6.90729	−0.322756
$459$	0	0
$460$	−16.9880	−0.792070
$461$	27.1813	1.26596	0.632981	−	0.774168i	$-0.281831\pi$
$461$	27.1813	1.26596	0.632981	+	0.774168i	$0.281831\pi$
$462$	0	0
$463$	−28.8448	−1.34053	−0.670266	−	0.742121i	$-0.733820\pi$
$463$	−28.8448	−1.34053	−0.670266	+	0.742121i	$0.733820\pi$
$464$	0.676489	0.0314052
$465$	0	0
$466$	16.7112	0.774131
$467$	−16.6883	−0.772242	−0.386121	−	0.922448i	$-0.626185\pi$
$467$	−16.6883	−0.772242	−0.386121	+	0.922448i	$0.626185\pi$
$468$	0	0
$469$	−45.6038	−2.10579
$470$	16.7315	0.771766
$471$	0	0
$472$	26.1770	1.20489
$473$	3.70174	0.170206
$474$	0	0
$475$	2.16759	0.0994558
$476$	−23.2383	−1.06513
$477$	0	0
$478$	10.9167	0.499320
$479$	−5.73551	−0.262062	−0.131031	−	0.991378i	$-0.541829\pi$
$479$	−5.73551	−0.262062	−0.131031	+	0.991378i	$0.541829\pi$
$480$	0	0
$481$	−2.99726	−0.136663
$482$	0.900190	0.0410025
$483$	0	0
$484$	12.4879	0.567630
$485$	−45.7827	−2.07889
$486$	0	0
$487$	−1.36877	−0.0620249	−0.0310124	−	0.999519i	$-0.509873\pi$
$487$	−1.36877	−0.0620249	−0.0310124	+	0.999519i	$0.509873\pi$
$488$	1.54189	0.0697981
$489$	0	0
$490$	−14.2860	−0.645375
$491$	21.7155	0.980008	0.490004	−	0.871720i	$-0.336995\pi$
$491$	21.7155	0.980008	0.490004	+	0.871720i	$0.336995\pi$
$492$	0	0
$493$	−17.8744	−0.805022
$494$	−0.311211	−0.0140021
$495$	0	0
$496$	1.24409	0.0558612
$497$	6.40102	0.287125
$498$	0	0
$499$	−3.94934	−0.176797	−0.0883984	−	0.996085i	$-0.528175\pi$
$499$	−3.94934	−0.176797	−0.0883984	+	0.996085i	$0.528175\pi$
$500$	−9.18554	−0.410790
$501$	0	0
$502$	3.87890	0.173124
$503$	5.27080	0.235013	0.117507	−	0.993072i	$-0.462510\pi$
$503$	5.27080	0.235013	0.117507	+	0.993072i	$0.462510\pi$
$504$	0	0
$505$	−23.7799	−1.05819
$506$	3.42065	0.152066
$507$	0	0
$508$	6.02225	0.267194
$509$	25.7394	1.14088	0.570439	−	0.821340i	$-0.306773\pi$
$509$	25.7394	1.14088	0.570439	+	0.821340i	$0.306773\pi$
$510$	0	0
$511$	−19.7038	−0.871643
$512$	−2.32107	−0.102578
$513$	0	0
$514$	−7.70241	−0.339739
$515$	42.7205	1.88249
$516$	0	0
$517$	4.94600	0.217525
$518$	28.9542	1.27217
$519$	0	0
$520$	−2.56659	−0.112552
$521$	23.8477	1.04478	0.522392	−	0.852705i	$-0.325040\pi$
$521$	23.8477	1.04478	0.522392	+	0.852705i	$0.325040\pi$
$522$	0	0
$523$	−10.9753	−0.479915	−0.239958	−	0.970783i	$-0.577134\pi$
$523$	−10.9753	−0.479915	−0.239958	+	0.970783i	$0.577134\pi$
$524$	−17.8425	−0.779452
$525$	0	0
$526$	7.66873	0.334373
$527$	−32.8716	−1.43191
$528$	0	0
$529$	5.70818	0.248182
$530$	9.51679	0.413383
$531$	0	0
$532$	−4.41363	−0.191355
$533$	2.73181	0.118328
$534$	0	0
$535$	−21.5202	−0.930398
$536$	36.3803	1.57139
$537$	0	0
$538$	4.53017	0.195310
$539$	−4.22309	−0.181901
$540$	0	0
$541$	−23.2653	−1.00025	−0.500127	−	0.865952i	$-0.666713\pi$
$541$	−23.2653	−1.00025	−0.500127	+	0.865952i	$0.666713\pi$
$542$	−12.5603	−0.539510
$543$	0	0
$544$	30.1623	1.29320
$545$	8.81260	0.377490
$546$	0	0
$547$	−32.9147	−1.40733	−0.703665	−	0.710532i	$-0.748454\pi$
$547$	−32.9147	−1.40733	−0.703665	+	0.710532i	$0.748454\pi$
$548$	15.6348	0.667886
$549$	0	0
$550$	−1.34252	−0.0572454
$551$	−3.39486	−0.144626
$552$	0	0
$553$	17.9078	0.761518
$554$	4.99766	0.212330
$555$	0	0
$556$	−20.3863	−0.864570
$557$	0.975408	0.0413294	0.0206647	−	0.999786i	$-0.493422\pi$
$557$	0.975408	0.0413294	0.0206647	+	0.999786i	$0.493422\pi$
$558$	0	0
$559$	1.75063	0.0740439
$560$	1.97029	0.0832599
$561$	0	0
$562$	25.1470	1.06076
$563$	9.38368	0.395475	0.197738	−	0.980255i	$-0.436641\pi$
$563$	9.38368	0.395475	0.197738	+	0.980255i	$0.436641\pi$
$564$	0	0
$565$	20.8877	0.878750
$566$	16.9544	0.712648
$567$	0	0
$568$	−5.10639	−0.214259
$569$	25.7226	1.07835	0.539174	−	0.842194i	$-0.318737\pi$
$569$	25.7226	1.07835	0.539174	+	0.842194i	$0.318737\pi$
$570$	0	0
$571$	17.5504	0.734463	0.367232	−	0.930130i	$-0.380306\pi$
$571$	17.5504	0.734463	0.367232	+	0.930130i	$0.380306\pi$
$572$	−0.282979	−0.0118319
$573$	0	0
$574$	−26.3899	−1.10149
$575$	−11.2673	−0.469878
$576$	0	0
$577$	2.46696	0.102701	0.0513504	−	0.998681i	$-0.483647\pi$
$577$	2.46696	0.102701	0.0513504	+	0.998681i	$0.483647\pi$
$578$	11.2107	0.466302
$579$	0	0
$580$	−10.4424	−0.433598
$581$	−20.1781	−0.837128
$582$	0	0
$583$	2.81326	0.116513
$584$	15.7186	0.650441
$585$	0	0
$586$	−13.2888	−0.548955
$587$	27.3011	1.12684	0.563418	−	0.826172i	$-0.309486\pi$
$587$	27.3011	1.12684	0.563418	+	0.826172i	$0.309486\pi$
$588$	0	0
$589$	−6.24327	−0.257249
$590$	21.8722	0.900465
$591$	0	0
$592$	−1.83554	−0.0754400
$593$	−39.5470	−1.62400	−0.812001	−	0.583657i	$-0.801621\pi$
$593$	−39.5470	−1.62400	−0.812001	+	0.583657i	$0.801621\pi$
$594$	0	0
$595$	−52.0595	−2.13423
$596$	21.3336	0.873860
$597$	0	0
$598$	1.61770	0.0661527
$599$	−4.96000	−0.202660	−0.101330	−	0.994853i	$-0.532310\pi$
$599$	−4.96000	−0.202660	−0.101330	+	0.994853i	$0.532310\pi$
$600$	0	0
$601$	−26.6058	−1.08527	−0.542637	−	0.839968i	$-0.682574\pi$
$601$	−26.6058	−1.08527	−0.542637	+	0.839968i	$0.682574\pi$
$602$	−16.9115	−0.689261
$603$	0	0
$604$	12.8331	0.522173
$605$	27.9759	1.13738
$606$	0	0
$607$	10.7757	0.437373	0.218687	−	0.975795i	$-0.429823\pi$
$607$	10.7757	0.437373	0.218687	+	0.975795i	$0.429823\pi$
$608$	5.72869	0.232329
$609$	0	0
$610$	1.28833	0.0521629
$611$	2.33907	0.0946287
$612$	0	0
$613$	6.37603	0.257525	0.128763	−	0.991675i	$-0.458899\pi$
$613$	6.37603	0.257525	0.128763	+	0.991675i	$0.458899\pi$
$614$	−12.2750	−0.495379
$615$	0	0
$616$	7.32931	0.295306
$617$	−8.80482	−0.354469	−0.177234	−	0.984169i	$-0.556715\pi$
$617$	−8.80482	−0.354469	−0.177234	+	0.984169i	$0.556715\pi$
$618$	0	0
$619$	−44.7446	−1.79844	−0.899218	−	0.437500i	$-0.855864\pi$
$619$	−44.7446	−1.79844	−0.899218	+	0.437500i	$0.855864\pi$
$620$	−19.2040	−0.771250
$621$	0	0
$622$	4.07272	0.163301
$623$	−57.8079	−2.31602
$624$	0	0
$625$	−31.0923	−1.24369
$626$	−7.58748	−0.303257
$627$	0	0
$628$	−26.6123	−1.06194
$629$	48.4990	1.93378
$630$	0	0
$631$	−4.75732	−0.189386	−0.0946929	−	0.995507i	$-0.530187\pi$
$631$	−4.75732	−0.189386	−0.0946929	+	0.995507i	$0.530187\pi$
$632$	−14.2859	−0.568263
$633$	0	0
$634$	18.8897	0.750204
$635$	13.4913	0.535387
$636$	0	0
$637$	−1.99719	−0.0791315
$638$	2.10265	0.0832448
$639$	0	0
$640$	16.6356	0.657580
$641$	34.2211	1.35165	0.675826	−	0.737061i	$-0.263787\pi$
$641$	34.2211	1.35165	0.675826	+	0.737061i	$0.263787\pi$
$642$	0	0
$643$	36.7463	1.44913	0.724566	−	0.689205i	$-0.242040\pi$
$643$	36.7463	1.44913	0.724566	+	0.689205i	$0.242040\pi$
$644$	22.9424	0.904056
$645$	0	0
$646$	5.03575	0.198129
$647$	31.0711	1.22153	0.610765	−	0.791812i	$-0.290862\pi$
$647$	31.0711	1.22153	0.610765	+	0.791812i	$0.290862\pi$
$648$	0	0
$649$	6.46566	0.253799
$650$	−0.634908	−0.0249032
$651$	0	0
$652$	0.00881558	0.000345245 0
$653$	−4.02468	−0.157498	−0.0787490	−	0.996894i	$-0.525093\pi$
$653$	−4.02468	−0.157498	−0.0787490	+	0.996894i	$0.525093\pi$
$654$	0	0
$655$	−39.9715	−1.56182
$656$	1.67297	0.0653187
$657$	0	0
$658$	−22.5959	−0.880881
$659$	−26.7110	−1.04051	−0.520256	−	0.854010i	$-0.674164\pi$
$659$	−26.7110	−1.04051	−0.520256	+	0.854010i	$0.674164\pi$
$660$	0	0
$661$	29.9651	1.16551	0.582753	−	0.812649i	$-0.301975\pi$
$661$	29.9651	1.16551	0.582753	+	0.812649i	$0.301975\pi$
$662$	20.2008	0.785125
$663$	0	0
$664$	16.0970	0.624684
$665$	−9.88761	−0.383425
$666$	0	0
$667$	17.6467	0.683284
$668$	22.1517	0.857076
$669$	0	0
$670$	30.3976	1.17436
$671$	0.380844	0.0147023
$672$	0	0
$673$	27.8252	1.07258	0.536291	−	0.844033i	$-0.319825\pi$
$673$	27.8252	1.07258	0.536291	+	0.844033i	$0.319825\pi$
$674$	25.2962	0.974372
$675$	0	0
$676$	15.3317	0.589682
$677$	−25.1005	−0.964689	−0.482344	−	0.875982i	$-0.660215\pi$
$677$	−25.1005	−0.964689	−0.482344	+	0.875982i	$0.660215\pi$
$678$	0	0
$679$	61.8298	2.37281
$680$	41.5303	1.59262
$681$	0	0
$682$	3.86685	0.148069
$683$	21.2513	0.813158	0.406579	−	0.913616i	$-0.366722\pi$
$683$	21.2513	0.813158	0.406579	+	0.913616i	$0.366722\pi$
$684$	0	0
$685$	35.0258	1.33827
$686$	−3.38686	−0.129311
$687$	0	0
$688$	1.07210	0.0408733
$689$	1.33045	0.0506862
$690$	0	0
$691$	29.8715	1.13637	0.568183	−	0.822902i	$-0.307647\pi$
$691$	29.8715	1.13637	0.568183	+	0.822902i	$0.307647\pi$
$692$	−0.670310	−0.0254814
$693$	0	0
$694$	−11.9965	−0.455381
$695$	−45.6702	−1.73237
$696$	0	0
$697$	−44.2038	−1.67434
$698$	−17.9282	−0.678591
$699$	0	0
$700$	−9.00433	−0.340332
$701$	38.9535	1.47125	0.735626	−	0.677387i	$-0.236888\pi$
$701$	38.9535	1.47125	0.735626	+	0.677387i	$0.236888\pi$
$702$	0	0
$703$	9.21137	0.347413
$704$	−3.83948	−0.144706
$705$	0	0
$706$	2.47996	0.0933346
$707$	32.1149	1.20781
$708$	0	0
$709$	−6.30152	−0.236658	−0.118329	−	0.992974i	$-0.537754\pi$
$709$	−6.30152	−0.236658	−0.118329	+	0.992974i	$0.537754\pi$
$710$	−4.26665	−0.160125
$711$	0	0
$712$	46.1160	1.72827
$713$	32.4530	1.21537
$714$	0	0
$715$	−0.633942	−0.0237081
$716$	−29.5683	−1.10502
$717$	0	0
$718$	19.3159	0.720861
$719$	26.2775	0.979985	0.489992	−	0.871727i	$-0.337000\pi$
$719$	26.2775	0.979985	0.489992	+	0.871727i	$0.337000\pi$
$720$	0	0
$721$	−57.6942	−2.14864
$722$	−16.1472	−0.600936
$723$	0	0
$724$	−3.12024	−0.115963
$725$	−6.92592	−0.257222
$726$	0	0
$727$	−19.8372	−0.735723	−0.367861	−	0.929881i	$-0.619910\pi$
$727$	−19.8372	−0.735723	−0.367861	+	0.929881i	$0.619910\pi$
$728$	3.46619	0.128466
$729$	0	0
$730$	13.1337	0.486101
$731$	−28.3272	−1.04772
$732$	0	0
$733$	−39.8548	−1.47207	−0.736036	−	0.676943i	$-0.763305\pi$
$733$	−39.8548	−1.47207	−0.736036	+	0.676943i	$0.763305\pi$
$734$	−13.6996	−0.505661
$735$	0	0
$736$	−29.7782	−1.09764
$737$	8.98585	0.330998
$738$	0	0
$739$	−5.97454	−0.219777	−0.109889	−	0.993944i	$-0.535049\pi$
$739$	−5.97454	−0.219777	−0.109889	+	0.993944i	$0.535049\pi$
$740$	28.3337	1.04157
$741$	0	0
$742$	−12.8525	−0.471829
$743$	−9.92322	−0.364048	−0.182024	−	0.983294i	$-0.558265\pi$
$743$	−9.92322	−0.364048	−0.182024	+	0.983294i	$0.558265\pi$
$744$	0	0
$745$	47.7926	1.75098
$746$	27.5484	1.00862
$747$	0	0
$748$	4.57892	0.167422
$749$	29.0631	1.06194
$750$	0	0
$751$	−30.8990	−1.12752	−0.563760	−	0.825939i	$-0.690646\pi$
$751$	−30.8990	−1.12752	−0.563760	+	0.825939i	$0.690646\pi$
$752$	1.43246	0.0522364
$753$	0	0
$754$	0.994389	0.0362135
$755$	28.7494	1.04630
$756$	0	0
$757$	−39.2438	−1.42634	−0.713170	−	0.700992i	$-0.752741\pi$
$757$	−39.2438	−1.42634	−0.713170	+	0.700992i	$0.752741\pi$
$758$	19.5496	0.710074
$759$	0	0
$760$	7.88781	0.286121
$761$	46.0663	1.66990	0.834951	−	0.550325i	$-0.185496\pi$
$761$	46.0663	1.66990	0.834951	+	0.550325i	$0.185496\pi$
$762$	0	0
$763$	−11.9014	−0.430861
$764$	−3.72999	−0.134946
$765$	0	0
$766$	14.5624	0.526161
$767$	3.05775	0.110409
$768$	0	0
$769$	49.9815	1.80238	0.901189	−	0.433427i	$-0.142696\pi$
$769$	49.9815	1.80238	0.901189	+	0.433427i	$0.142696\pi$
$770$	6.12402	0.220694
$771$	0	0
$772$	8.77484	0.315813
$773$	−41.3442	−1.48705	−0.743524	−	0.668709i	$-0.766847\pi$
$773$	−41.3442	−1.48705	−0.743524	+	0.668709i	$0.766847\pi$
$774$	0	0
$775$	−12.7370	−0.457527
$776$	−49.3245	−1.77065
$777$	0	0
$778$	−15.2716	−0.547514
$779$	−8.39558	−0.300803
$780$	0	0
$781$	−1.26127	−0.0451317
$782$	−26.1762	−0.936059
$783$	0	0
$784$	−1.22309	−0.0436817
$785$	−59.6180	−2.12786
$786$	0	0
$787$	49.9983	1.78225	0.891124	−	0.453760i	$-0.149918\pi$
$787$	49.9983	1.78225	0.891124	+	0.453760i	$0.149918\pi$
$788$	2.02360	0.0720879
$789$	0	0
$790$	−11.9366	−0.424686
$791$	−28.2089	−1.00299
$792$	0	0
$793$	0.180109	0.00639586
$794$	−17.2375	−0.611734
$795$	0	0
$796$	9.94744	0.352577
$797$	−36.8232	−1.30434	−0.652172	−	0.758071i	$-0.726142\pi$
$797$	−36.8232	−1.30434	−0.652172	+	0.758071i	$0.726142\pi$
$798$	0	0
$799$	−37.8488	−1.33900
$800$	11.6872	0.413206
$801$	0	0
$802$	−8.46399	−0.298874
$803$	3.88246	0.137009
$804$	0	0
$805$	51.3965	1.81149
$806$	1.82872	0.0644138
$807$	0	0
$808$	−25.6196	−0.901293
$809$	9.10579	0.320142	0.160071	−	0.987105i	$-0.448828\pi$
$809$	9.10579	0.320142	0.160071	+	0.987105i	$0.448828\pi$
$810$	0	0
$811$	7.56572	0.265668	0.132834	−	0.991138i	$-0.457592\pi$
$811$	7.56572	0.265668	0.132834	+	0.991138i	$0.457592\pi$
$812$	14.1025	0.494902
$813$	0	0
$814$	−5.70518	−0.199966
$815$	0.0197491	0.000691780 0
$816$	0	0
$817$	−5.38016	−0.188228
$818$	12.4037	0.433684
$819$	0	0
$820$	−25.8244	−0.901826
$821$	−0.958450	−0.0334501	−0.0167251	−	0.999860i	$-0.505324\pi$
$821$	−0.958450	−0.0334501	−0.0167251	+	0.999860i	$0.505324\pi$
$822$	0	0
$823$	−1.06844	−0.0372435	−0.0186218	−	0.999827i	$-0.505928\pi$
$823$	−1.06844	−0.0372435	−0.0186218	+	0.999827i	$0.505928\pi$
$824$	46.0253	1.60337
$825$	0	0
$826$	−29.5385	−1.02778
$827$	−34.1649	−1.18803	−0.594015	−	0.804454i	$-0.702458\pi$
$827$	−34.1649	−1.18803	−0.594015	+	0.804454i	$0.702458\pi$
$828$	0	0
$829$	47.8950	1.66346	0.831730	−	0.555180i	$-0.187351\pi$
$829$	47.8950	1.66346	0.831730	+	0.555180i	$0.187351\pi$
$830$	13.4499	0.466852
$831$	0	0
$832$	−1.81577	−0.0629506
$833$	32.3168	1.11971
$834$	0	0
$835$	49.6253	1.71735
$836$	0.869669	0.0300781
$837$	0	0
$838$	−14.0209	−0.484343
$839$	−3.56411	−0.123047	−0.0615233	−	0.998106i	$-0.519596\pi$
$839$	−3.56411	−0.123047	−0.0615233	+	0.998106i	$0.519596\pi$
$840$	0	0
$841$	−18.1527	−0.625954
$842$	−32.4166	−1.11715
$843$	0	0
$844$	1.88242	0.0647956
$845$	34.3468	1.18157
$846$	0	0
$847$	−37.7815	−1.29819
$848$	0.814775	0.0279795
$849$	0	0
$850$	10.2735	0.352379
$851$	−47.8814	−1.64135
$852$	0	0
$853$	32.3673	1.10824	0.554118	−	0.832438i	$-0.313056\pi$
$853$	32.3673	1.10824	0.554118	+	0.832438i	$0.313056\pi$
$854$	−1.73989	−0.0595379
$855$	0	0
$856$	−23.1850	−0.792446
$857$	−33.4060	−1.14113	−0.570563	−	0.821254i	$-0.693275\pi$
$857$	−33.4060	−1.14113	−0.570563	+	0.821254i	$0.693275\pi$
$858$	0	0
$859$	42.5662	1.45234	0.726171	−	0.687514i	$-0.241298\pi$
$859$	42.5662	1.45234	0.726171	+	0.687514i	$0.241298\pi$
$860$	−16.5491	−0.564319
$861$	0	0
$862$	7.74653	0.263848
$863$	−13.4059	−0.456342	−0.228171	−	0.973621i	$-0.573275\pi$
$863$	−13.4059	−0.456342	−0.228171	+	0.973621i	$0.573275\pi$
$864$	0	0
$865$	−1.50166	−0.0510579
$866$	21.8599	0.742829
$867$	0	0
$868$	25.9350	0.880293
$869$	−3.52859	−0.119699
$870$	0	0
$871$	4.24960	0.143992
$872$	9.49434	0.321519
$873$	0	0
$874$	−4.97162	−0.168167
$875$	27.7905	0.939490
$876$	0	0
$877$	−9.45572	−0.319297	−0.159649	−	0.987174i	$-0.551036\pi$
$877$	−9.45572	−0.319297	−0.159649	+	0.987174i	$0.551036\pi$
$878$	−20.7766	−0.701175
$879$	0	0
$880$	−0.388229	−0.0130872
$881$	−11.7121	−0.394592	−0.197296	−	0.980344i	$-0.563216\pi$
$881$	−11.7121	−0.394592	−0.197296	+	0.980344i	$0.563216\pi$
$882$	0	0
$883$	−42.1079	−1.41704	−0.708522	−	0.705689i	$-0.750638\pi$
$883$	−42.1079	−1.41704	−0.708522	+	0.705689i	$0.750638\pi$
$884$	2.16547	0.0728327
$885$	0	0
$886$	−30.1483	−1.01285
$887$	−27.3443	−0.918133	−0.459067	−	0.888402i	$-0.651816\pi$
$887$	−27.3443	−0.918133	−0.459067	+	0.888402i	$0.651816\pi$
$888$	0	0
$889$	−18.2201	−0.611082
$890$	38.5323	1.29161
$891$	0	0
$892$	19.1404	0.640868
$893$	−7.18858	−0.240557
$894$	0	0
$895$	−66.2403	−2.21417
$896$	−22.4664	−0.750551
$897$	0	0
$898$	−5.47468	−0.182692
$899$	19.9486	0.665324
$900$	0	0
$901$	−21.5282	−0.717209
$902$	5.19991	0.173138
$903$	0	0
$904$	22.5035	0.748456
$905$	−6.99011	−0.232359
$906$	0	0
$907$	15.7787	0.523923	0.261961	−	0.965078i	$-0.415631\pi$
$907$	15.7787	0.523923	0.261961	+	0.965078i	$0.415631\pi$
$908$	−7.55239	−0.250635
$909$	0	0
$910$	2.89618	0.0960075
$911$	−57.7207	−1.91237	−0.956187	−	0.292758i	$-0.905427\pi$
$911$	−57.7207	−1.91237	−0.956187	+	0.292758i	$0.905427\pi$
$912$	0	0
$913$	3.97592	0.131584
$914$	−13.7989	−0.456427
$915$	0	0
$916$	9.12841	0.301611
$917$	53.9817	1.78263
$918$	0	0
$919$	−22.9446	−0.756873	−0.378437	−	0.925627i	$-0.623538\pi$
$919$	−22.9446	−0.756873	−0.378437	+	0.925627i	$0.623538\pi$
$920$	−41.0014	−1.35178
$921$	0	0
$922$	24.4684	0.805823
$923$	−0.596481	−0.0196334
$924$	0	0
$925$	18.7923	0.617887
$926$	−25.9658	−0.853290
$927$	0	0
$928$	−18.3044	−0.600873
$929$	34.5057	1.13210	0.566048	−	0.824372i	$-0.308472\pi$
$929$	34.5057	1.13210	0.566048	+	0.824372i	$0.308472\pi$
$930$	0	0
$931$	6.13789	0.201161
$932$	−22.0849	−0.723415
$933$	0	0
$934$	−15.0226	−0.491556
$935$	10.2579	0.335469
$936$	0	0
$937$	−40.6133	−1.32678	−0.663390	−	0.748274i	$-0.730883\pi$
$937$	−40.6133	−1.32678	−0.663390	+	0.748274i	$0.730883\pi$
$938$	−41.0521	−1.34040
$939$	0	0
$940$	−22.1117	−0.721204
$941$	10.1663	0.331413	0.165706	−	0.986175i	$-0.447010\pi$
$941$	10.1663	0.331413	0.165706	+	0.986175i	$0.447010\pi$
$942$	0	0
$943$	43.6408	1.42114
$944$	1.87258	0.0609473
$945$	0	0
$946$	3.33227	0.108341
$947$	−31.3972	−1.02027	−0.510135	−	0.860094i	$-0.670405\pi$
$947$	−31.3972	−1.02027	−0.510135	+	0.860094i	$0.670405\pi$
$948$	0	0
$949$	1.83610	0.0596024
$950$	1.95124	0.0633066
$951$	0	0
$952$	−56.0869	−1.81779
$953$	−54.5622	−1.76744	−0.883722	−	0.468013i	$-0.844970\pi$
$953$	−54.5622	−1.76744	−0.883722	+	0.468013i	$0.844970\pi$
$954$	0	0
$955$	−8.35608	−0.270396
$956$	−14.4272	−0.466608
$957$	0	0
$958$	−5.16305	−0.166811
$959$	−47.3025	−1.52748
$960$	0	0
$961$	5.68623	0.183427
$962$	−2.69810	−0.0869903
$963$	0	0
$964$	−1.18966	−0.0383163
$965$	19.6578	0.632806
$966$	0	0
$967$	−49.7708	−1.60052	−0.800260	−	0.599653i	$-0.795305\pi$
$967$	−49.7708	−1.60052	−0.800260	+	0.599653i	$0.795305\pi$
$968$	30.1401	0.968740
$969$	0	0
$970$	−41.2132	−1.32328
$971$	50.2692	1.61322	0.806608	−	0.591087i	$-0.201301\pi$
$971$	50.2692	1.61322	0.806608	+	0.591087i	$0.201301\pi$
$972$	0	0
$973$	61.6778	1.97730
$974$	−1.23215	−0.0394807
$975$	0	0
$976$	0.110300	0.00353061
$977$	13.3850	0.428224	0.214112	−	0.976809i	$-0.431314\pi$
$977$	13.3850	0.428224	0.214112	+	0.976809i	$0.431314\pi$
$978$	0	0
$979$	11.3906	0.364044
$980$	18.8798	0.603094
$981$	0	0
$982$	19.5481	0.623805
$983$	−32.0295	−1.02158	−0.510792	−	0.859704i	$-0.670648\pi$
$983$	−32.0295	−1.02158	−0.510792	+	0.859704i	$0.670648\pi$
$984$	0	0
$985$	4.53337	0.144445
$986$	−16.0903	−0.512421
$987$	0	0
$988$	0.411285	0.0130847
$989$	27.9664	0.889281
$990$	0	0
$991$	38.7779	1.23182	0.615911	−	0.787816i	$-0.288788\pi$
$991$	38.7779	1.23182	0.615911	+	0.787816i	$0.288788\pi$
$992$	−33.6625	−1.06879
$993$	0	0
$994$	5.76213	0.182764
$995$	22.2847	0.706472
$996$	0	0
$997$	14.7644	0.467592	0.233796	−	0.972286i	$-0.424885\pi$
$997$	14.7644	0.467592	0.233796	+	0.972286i	$0.424885\pi$
$998$	−3.55516	−0.112537
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.f.1.7		9
3.2	odd	2		723.2.a.e.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
723.2.a.e.1.3	✓	9	3.2	odd	2
2169.2.a.f.1.7		9	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.900190 q^{2} -1.18966 q^{4} -2.66512 q^{5} +3.59926 q^{7} -2.87130 q^{8} +O(q^{10})\)	\(q+0.900190 q^{2} -1.18966 q^{4} -2.66512 q^{5} +3.59926 q^{7} -2.87130 q^{8} -2.39912 q^{10} -0.709205 q^{11} -0.335398 q^{13} +3.24002 q^{14} -0.205399 q^{16} +5.42712 q^{17} +1.03077 q^{19} +3.17059 q^{20} -0.638419 q^{22} -5.35800 q^{23} +2.10289 q^{25} -0.301922 q^{26} -4.28189 q^{28} -3.29353 q^{29} -6.05692 q^{31} +5.55770 q^{32} +4.88544 q^{34} -9.59248 q^{35} +8.93642 q^{37} +0.927886 q^{38} +7.65237 q^{40} -8.14498 q^{41} -5.21956 q^{43} +0.843711 q^{44} -4.82322 q^{46} -6.97401 q^{47} +5.95468 q^{49} +1.89300 q^{50} +0.399009 q^{52} -3.96679 q^{53} +1.89012 q^{55} -10.3346 q^{56} -2.96480 q^{58} -9.11677 q^{59} -0.537001 q^{61} -5.45238 q^{62} +5.41378 q^{64} +0.893878 q^{65} -12.6703 q^{67} -6.45642 q^{68} -8.63505 q^{70} +1.77843 q^{71} -5.47439 q^{73} +8.04448 q^{74} -1.22626 q^{76} -2.55261 q^{77} +4.97542 q^{79} +0.547415 q^{80} -7.33203 q^{82} -5.60617 q^{83} -14.4640 q^{85} -4.69860 q^{86} +2.03634 q^{88} -16.0610 q^{89} -1.20719 q^{91} +6.37419 q^{92} -6.27794 q^{94} -2.74712 q^{95} +17.1785 q^{97} +5.36034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{2} + 7 q^{4} - 12 q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10}) \) 9 * q - 3 * q^2 + 7 * q^4 - 12 * q^5 - 5 * q^7 - 6 * q^8	\( 9 q - 3 q^{2} + 7 q^{4} - 12 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{10} - 3 q^{11} + 18 q^{13} - q^{16} - 16 q^{17} + 3 q^{19} - 17 q^{20} + 8 q^{22} - 9 q^{23} + 15 q^{25} - 14 q^{26} - 3 q^{28} - 15 q^{29} - 13 q^{31} - 10 q^{32} + q^{34} - q^{35} + 19 q^{37} - 12 q^{38} + 24 q^{40} - 18 q^{41} - 11 q^{43} - 14 q^{44} + q^{46} - 19 q^{47} + 14 q^{49} - 21 q^{50} + 10 q^{52} - 31 q^{53} - q^{55} - q^{56} + 12 q^{58} - 11 q^{59} + 2 q^{61} - 11 q^{62} - 34 q^{64} - 33 q^{65} - 12 q^{67} - 24 q^{68} - 42 q^{70} - 6 q^{71} + 18 q^{73} - 8 q^{74} + q^{76} - 20 q^{77} - 32 q^{79} - 7 q^{80} - 27 q^{82} - 13 q^{83} + 12 q^{85} + 2 q^{86} - 25 q^{88} - 24 q^{89} - 34 q^{91} + 7 q^{92} - 21 q^{94} - 14 q^{95} + 23 q^{97} - 8 q^{98}+O(q^{100}) \) 9 * q - 3 * q^2 + 7 * q^4 - 12 * q^5 - 5 * q^7 - 6 * q^8 + 5 * q^10 - 3 * q^11 + 18 * q^13 - q^16 - 16 * q^17 + 3 * q^19 - 17 * q^20 + 8 * q^22 - 9 * q^23 + 15 * q^25 - 14 * q^26 - 3 * q^28 - 15 * q^29 - 13 * q^31 - 10 * q^32 + q^34 - q^35 + 19 * q^37 - 12 * q^38 + 24 * q^40 - 18 * q^41 - 11 * q^43 - 14 * q^44 + q^46 - 19 * q^47 + 14 * q^49 - 21 * q^50 + 10 * q^52 - 31 * q^53 - q^55 - q^56 + 12 * q^58 - 11 * q^59 + 2 * q^61 - 11 * q^62 - 34 * q^64 - 33 * q^65 - 12 * q^67 - 24 * q^68 - 42 * q^70 - 6 * q^71 + 18 * q^73 - 8 * q^74 + q^76 - 20 * q^77 - 32 * q^79 - 7 * q^80 - 27 * q^82 - 13 * q^83 + 12 * q^85 + 2 * q^86 - 25 * q^88 - 24 * q^89 - 34 * q^91 + 7 * q^92 - 21 * q^94 - 14 * q^95 + 23 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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