LMFDB - Embedded newform 2169.2.a.e.1.1

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			1.1
Root			$2.73684$ 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)

$f(q)$	$=$	$q-1.73684 q^{2} +1.01662 q^{4} +2.63180 q^{5} -2.01025 q^{7} +1.70797 q^{8} +O(q^{10})$	\(q-1.73684 q^{2} +1.01662 q^{4} +2.63180 q^{5} -2.01025 q^{7} +1.70797 q^{8} -4.57103 q^{10} +3.39618 q^{11} +5.63669 q^{13} +3.49150 q^{14} -4.99972 q^{16} -0.866432 q^{17} +2.46437 q^{19} +2.67556 q^{20} -5.89863 q^{22} +6.37847 q^{23} +1.92640 q^{25} -9.79005 q^{26} -2.04367 q^{28} +4.52212 q^{29} -3.51511 q^{31} +5.26780 q^{32} +1.50486 q^{34} -5.29060 q^{35} -5.19315 q^{37} -4.28022 q^{38} +4.49504 q^{40} -1.35422 q^{41} +8.49015 q^{43} +3.45264 q^{44} -11.0784 q^{46} +9.44537 q^{47} -2.95888 q^{49} -3.34585 q^{50} +5.73040 q^{52} -9.71877 q^{53} +8.93808 q^{55} -3.43345 q^{56} -7.85421 q^{58} -6.03110 q^{59} +4.45402 q^{61} +6.10520 q^{62} +0.850111 q^{64} +14.8347 q^{65} -10.9216 q^{67} -0.880836 q^{68} +9.18894 q^{70} +3.01063 q^{71} -0.255916 q^{73} +9.01969 q^{74} +2.50534 q^{76} -6.82718 q^{77} -10.3262 q^{79} -13.1583 q^{80} +2.35207 q^{82} +16.8148 q^{83} -2.28028 q^{85} -14.7461 q^{86} +5.80057 q^{88} +17.4574 q^{89} -11.3312 q^{91} +6.48451 q^{92} -16.4051 q^{94} +6.48574 q^{95} +0.273223 q^{97} +5.13911 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.73684	−1.22813	−0.614067	−	0.789254i	$-0.710467\pi$
$2$	−1.73684	−1.22813	−0.614067	+	0.789254i	$0.710467\pi$
$3$	0	0
$3$	0	0
$4$	1.01662	0.508312
$5$	2.63180	1.17698	0.588489	−	0.808505i	$-0.299723\pi$
$5$	2.63180	1.17698	0.588489	+	0.808505i	$0.299723\pi$
$6$	0	0
$7$	−2.01025	−0.759805	−0.379902	−	0.925027i	$-0.624042\pi$
$7$	−2.01025	−0.759805	−0.379902	+	0.925027i	$0.624042\pi$
$8$	1.70797	0.603858
$9$	0	0
$10$	−4.57103	−1.44549
$11$	3.39618	1.02399	0.511993	−	0.858989i	$-0.328907\pi$
$11$	3.39618	1.02399	0.511993	+	0.858989i	$0.328907\pi$
$12$	0	0
$13$	5.63669	1.56334	0.781669	−	0.623694i	$-0.214369\pi$
$13$	5.63669	1.56334	0.781669	+	0.623694i	$0.214369\pi$
$14$	3.49150	0.933141
$15$	0	0
$16$	−4.99972	−1.24993
$17$	−0.866432	−0.210141	−0.105070	−	0.994465i	$-0.533507\pi$
$17$	−0.866432	−0.210141	−0.105070	+	0.994465i	$0.533507\pi$
$18$	0	0
$19$	2.46437	0.565365	0.282682	−	0.959214i	$-0.408776\pi$
$19$	2.46437	0.565365	0.282682	+	0.959214i	$0.408776\pi$
$20$	2.67556	0.598273
$21$	0	0
$22$	−5.89863	−1.25759
$23$	6.37847	1.33000	0.665001	−	0.746842i	$-0.268431\pi$
$23$	6.37847	1.33000	0.665001	+	0.746842i	$0.268431\pi$
$24$	0	0
$25$	1.92640	0.385279
$26$	−9.79005	−1.91999
$27$	0	0
$28$	−2.04367	−0.386218
$29$	4.52212	0.839737	0.419868	−	0.907585i	$-0.362076\pi$
$29$	4.52212	0.839737	0.419868	+	0.907585i	$0.362076\pi$
$30$	0	0
$31$	−3.51511	−0.631332	−0.315666	−	0.948870i	$-0.602228\pi$
$31$	−3.51511	−0.631332	−0.315666	+	0.948870i	$0.602228\pi$
$32$	5.26780	0.931224
$33$	0	0
$34$	1.50486	0.258081
$35$	−5.29060	−0.894274
$36$	0	0
$37$	−5.19315	−0.853748	−0.426874	−	0.904311i	$-0.640385\pi$
$37$	−5.19315	−0.853748	−0.426874	+	0.904311i	$0.640385\pi$
$38$	−4.28022	−0.694344
$39$	0	0
$40$	4.49504	0.710729
$41$	−1.35422	−0.211494	−0.105747	−	0.994393i	$-0.533723\pi$
$41$	−1.35422	−0.211494	−0.105747	+	0.994393i	$0.533723\pi$
$42$	0	0
$43$	8.49015	1.29474	0.647368	−	0.762178i	$-0.275870\pi$
$43$	8.49015	1.29474	0.647368	+	0.762178i	$0.275870\pi$
$44$	3.45264	0.520505
$45$	0	0
$46$	−11.0784	−1.63342
$47$	9.44537	1.37775	0.688875	−	0.724880i	$-0.258105\pi$
$47$	9.44537	1.37775	0.688875	+	0.724880i	$0.258105\pi$
$48$	0	0
$49$	−2.95888	−0.422697
$50$	−3.34585	−0.473174
$51$	0	0
$52$	5.73040	0.794663
$53$	−9.71877	−1.33498	−0.667488	−	0.744620i	$-0.732631\pi$
$53$	−9.71877	−1.33498	−0.667488	+	0.744620i	$0.732631\pi$
$54$	0	0
$55$	8.93808	1.20521
$56$	−3.43345	−0.458814
$57$	0	0
$58$	−7.85421	−1.03131
$59$	−6.03110	−0.785182	−0.392591	−	0.919713i	$-0.628421\pi$
$59$	−6.03110	−0.785182	−0.392591	+	0.919713i	$0.628421\pi$
$60$	0	0
$61$	4.45402	0.570279	0.285140	−	0.958486i	$-0.407960\pi$
$61$	4.45402	0.570279	0.285140	+	0.958486i	$0.407960\pi$
$62$	6.10520	0.775361
$63$	0	0
$64$	0.850111	0.106264
$65$	14.8347	1.84002
$66$	0	0
$67$	−10.9216	−1.33428	−0.667141	−	0.744932i	$-0.732482\pi$
$67$	−10.9216	−1.33428	−0.667141	+	0.744932i	$0.732482\pi$
$68$	−0.880836	−0.106817
$69$	0	0
$70$	9.18894	1.09829
$71$	3.01063	0.357296	0.178648	−	0.983913i	$-0.442828\pi$
$71$	3.01063	0.357296	0.178648	+	0.983913i	$0.442828\pi$
$72$	0	0
$73$	−0.255916	−0.0299527	−0.0149764	−	0.999888i	$-0.504767\pi$
$73$	−0.255916	−0.0299527	−0.0149764	+	0.999888i	$0.504767\pi$
$74$	9.01969	1.04852
$75$	0	0
$76$	2.50534	0.287382
$77$	−6.82718	−0.778030
$78$	0	0
$79$	−10.3262	−1.16179	−0.580893	−	0.813980i	$-0.697297\pi$
$79$	−10.3262	−1.16179	−0.580893	+	0.813980i	$0.697297\pi$
$80$	−13.1583	−1.47114
$81$	0	0
$82$	2.35207	0.259743
$83$	16.8148	1.84567	0.922833	−	0.385201i	$-0.125868\pi$
$83$	16.8148	1.84567	0.922833	+	0.385201i	$0.125868\pi$
$84$	0	0
$85$	−2.28028	−0.247331
$86$	−14.7461	−1.59011
$87$	0	0
$88$	5.80057	0.618343
$89$	17.4574	1.85048	0.925241	−	0.379380i	$-0.123863\pi$
$89$	17.4574	1.85048	0.925241	+	0.379380i	$0.123863\pi$
$90$	0	0
$91$	−11.3312	−1.18783
$92$	6.48451	0.676056
$93$	0	0
$94$	−16.4051	−1.69206
$95$	6.48574	0.665423
$96$	0	0
$97$	0.273223	0.0277416	0.0138708	−	0.999904i	$-0.495585\pi$
$97$	0.273223	0.0277416	0.0138708	+	0.999904i	$0.495585\pi$
$98$	5.13911	0.519128
$99$	0	0
$100$	1.95842	0.195842
$101$	−7.47149	−0.743441	−0.371721	−	0.928345i	$-0.621232\pi$
$101$	−7.47149	−0.743441	−0.371721	+	0.928345i	$0.621232\pi$
$102$	0	0
$103$	14.5145	1.43015	0.715077	−	0.699046i	$-0.246392\pi$
$103$	14.5145	1.43015	0.715077	+	0.699046i	$0.246392\pi$
$104$	9.62730	0.944035
$105$	0	0
$106$	16.8800	1.63953
$107$	−16.7492	−1.61921	−0.809605	−	0.586976i	$-0.800318\pi$
$107$	−16.7492	−1.61921	−0.809605	+	0.586976i	$0.800318\pi$
$108$	0	0
$109$	−6.01965	−0.576578	−0.288289	−	0.957543i	$-0.593086\pi$
$109$	−6.01965	−0.576578	−0.288289	+	0.957543i	$0.593086\pi$
$110$	−15.5240	−1.48016
$111$	0	0
$112$	10.0507	0.949703
$113$	−18.9246	−1.78028	−0.890139	−	0.455689i	$-0.849393\pi$
$113$	−18.9246	−1.78028	−0.890139	+	0.455689i	$0.849393\pi$
$114$	0	0
$115$	16.7869	1.56539
$116$	4.59730	0.426848
$117$	0	0
$118$	10.4751	0.964308
$119$	1.74175	0.159666
$120$	0	0
$121$	0.534035	0.0485487
$122$	−7.73594	−0.700379
$123$	0	0
$124$	−3.57355	−0.320914
$125$	−8.08912	−0.723513
$126$	0	0
$127$	−13.7300	−1.21834	−0.609172	−	0.793038i	$-0.708498\pi$
$127$	−13.7300	−1.21834	−0.609172	+	0.793038i	$0.708498\pi$
$128$	−12.0121	−1.06173
$129$	0	0
$130$	−25.7655	−2.25978
$131$	14.3193	1.25108	0.625542	−	0.780190i	$-0.284878\pi$
$131$	14.3193	1.25108	0.625542	+	0.780190i	$0.284878\pi$
$132$	0	0
$133$	−4.95401	−0.429567
$134$	18.9690	1.63868
$135$	0	0
$136$	−1.47984	−0.126895
$137$	−10.1743	−0.869252	−0.434626	−	0.900611i	$-0.643119\pi$
$137$	−10.1743	−0.869252	−0.434626	+	0.900611i	$0.643119\pi$
$138$	0	0
$139$	8.00345	0.678844	0.339422	−	0.940634i	$-0.389769\pi$
$139$	8.00345	0.678844	0.339422	+	0.940634i	$0.389769\pi$
$140$	−5.37855	−0.454570
$141$	0	0
$142$	−5.22899	−0.438807
$143$	19.1432	1.60084
$144$	0	0
$145$	11.9013	0.988352
$146$	0.444486	0.0367860
$147$	0	0
$148$	−5.27948	−0.433971
$149$	15.5640	1.27505	0.637527	−	0.770428i	$-0.279958\pi$
$149$	15.5640	1.27505	0.637527	+	0.770428i	$0.279958\pi$
$150$	0	0
$151$	−2.43764	−0.198372	−0.0991862	−	0.995069i	$-0.531624\pi$
$151$	−2.43764	−0.198372	−0.0991862	+	0.995069i	$0.531624\pi$
$152$	4.20907	0.341400
$153$	0	0
$154$	11.8577	0.955524
$155$	−9.25108	−0.743065
$156$	0	0
$157$	12.1633	0.970735	0.485367	−	0.874310i	$-0.338686\pi$
$157$	12.1633	0.970735	0.485367	+	0.874310i	$0.338686\pi$
$158$	17.9350	1.42683
$159$	0	0
$160$	13.8638	1.09603
$161$	−12.8223	−1.01054
$162$	0	0
$163$	4.23026	0.331340	0.165670	−	0.986181i	$-0.447021\pi$
$163$	4.23026	0.331340	0.165670	+	0.986181i	$0.447021\pi$
$164$	−1.37674	−0.107505
$165$	0	0
$166$	−29.2047	−2.26672
$167$	−14.3677	−1.11181	−0.555903	−	0.831247i	$-0.687628\pi$
$167$	−14.3677	−1.11181	−0.555903	+	0.831247i	$0.687628\pi$
$168$	0	0
$169$	18.7723	1.44402
$170$	3.96049	0.303756
$171$	0	0
$172$	8.63129	0.658130
$173$	−1.08253	−0.0823030	−0.0411515	−	0.999153i	$-0.513103\pi$
$173$	−1.08253	−0.0823030	−0.0411515	+	0.999153i	$0.513103\pi$
$174$	0	0
$175$	−3.87255	−0.292737
$176$	−16.9800	−1.27991
$177$	0	0
$178$	−30.3208	−2.27264
$179$	24.0238	1.79562	0.897810	−	0.440382i	$-0.145157\pi$
$179$	24.0238	1.79562	0.897810	+	0.440382i	$0.145157\pi$
$180$	0	0
$181$	−4.72624	−0.351298	−0.175649	−	0.984453i	$-0.556202\pi$
$181$	−4.72624	−0.351298	−0.175649	+	0.984453i	$0.556202\pi$
$182$	19.6805	1.45882
$183$	0	0
$184$	10.8942	0.803133
$185$	−13.6674	−1.00484
$186$	0	0
$187$	−2.94256	−0.215181
$188$	9.60239	0.700327
$189$	0	0
$190$	−11.2647	−0.817228
$191$	24.3499	1.76190	0.880948	−	0.473213i	$-0.156906\pi$
$191$	24.3499	1.76190	0.880948	+	0.473213i	$0.156906\pi$
$192$	0	0
$193$	3.22480	0.232126	0.116063	−	0.993242i	$-0.462973\pi$
$193$	3.22480	0.232126	0.116063	+	0.993242i	$0.462973\pi$
$194$	−0.474545	−0.0340704
$195$	0	0
$196$	−3.00807	−0.214862
$197$	3.60952	0.257167	0.128584	−	0.991699i	$-0.458957\pi$
$197$	3.60952	0.257167	0.128584	+	0.991699i	$0.458957\pi$
$198$	0	0
$199$	11.6028	0.822499	0.411249	−	0.911523i	$-0.365093\pi$
$199$	11.6028	0.822499	0.411249	+	0.911523i	$0.365093\pi$
$200$	3.29023	0.232654
$201$	0	0
$202$	12.9768	0.913045
$203$	−9.09061	−0.638036
$204$	0	0
$205$	−3.56405	−0.248924
$206$	−25.2094	−1.75642
$207$	0	0
$208$	−28.1819	−1.95406
$209$	8.36944	0.578926
$210$	0	0
$211$	25.9439	1.78605	0.893025	−	0.450007i	$-0.148579\pi$
$211$	25.9439	1.78605	0.893025	+	0.450007i	$0.148579\pi$
$212$	−9.88034	−0.678585
$213$	0	0
$214$	29.0908	1.98861
$215$	22.3444	1.52388
$216$	0	0
$217$	7.06626	0.479689
$218$	10.4552	0.708115
$219$	0	0
$220$	9.08667	0.612623
$221$	−4.88381	−0.328521
$222$	0	0
$223$	−4.40090	−0.294706	−0.147353	−	0.989084i	$-0.547075\pi$
$223$	−4.40090	−0.294706	−0.147353	+	0.989084i	$0.547075\pi$
$224$	−10.5896	−0.707548
$225$	0	0
$226$	32.8691	2.18642
$227$	23.0781	1.53175	0.765873	−	0.642991i	$-0.222307\pi$
$227$	23.0781	1.53175	0.765873	+	0.642991i	$0.222307\pi$
$228$	0	0
$229$	3.21172	0.212236	0.106118	−	0.994354i	$-0.466158\pi$
$229$	3.21172	0.212236	0.106118	+	0.994354i	$0.466158\pi$
$230$	−29.1562	−1.92250
$231$	0	0
$232$	7.72364	0.507082
$233$	−7.13210	−0.467239	−0.233620	−	0.972328i	$-0.575057\pi$
$233$	−7.13210	−0.467239	−0.233620	+	0.972328i	$0.575057\pi$
$234$	0	0
$235$	24.8584	1.62158
$236$	−6.13136	−0.399117
$237$	0	0
$238$	−3.02514	−0.196091
$239$	−3.66241	−0.236902	−0.118451	−	0.992960i	$-0.537793\pi$
$239$	−3.66241	−0.236902	−0.118451	+	0.992960i	$0.537793\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−0.927536	−0.0596243
$243$	0	0
$244$	4.52807	0.289880
$245$	−7.78719	−0.497506
$246$	0	0
$247$	13.8909	0.883856
$248$	−6.00370	−0.381235
$249$	0	0
$250$	14.0495	0.888571
$251$	−1.41613	−0.0893854	−0.0446927	−	0.999001i	$-0.514231\pi$
$251$	−1.41613	−0.0893854	−0.0446927	+	0.999001i	$0.514231\pi$
$252$	0	0
$253$	21.6624	1.36190
$254$	23.8469	1.49629
$255$	0	0
$256$	19.1629	1.19768
$257$	11.0690	0.690466	0.345233	−	0.938517i	$-0.387800\pi$
$257$	11.0690	0.690466	0.345233	+	0.938517i	$0.387800\pi$
$258$	0	0
$259$	10.4395	0.648682
$260$	15.0813	0.935302
$261$	0	0
$262$	−24.8704	−1.53650
$263$	3.31569	0.204454	0.102227	−	0.994761i	$-0.467403\pi$
$263$	3.31569	0.204454	0.102227	+	0.994761i	$0.467403\pi$
$264$	0	0
$265$	−25.5779	−1.57124
$266$	8.60433	0.527565
$267$	0	0
$268$	−11.1031	−0.678231
$269$	14.3455	0.874663	0.437332	−	0.899300i	$-0.355924\pi$
$269$	14.3455	0.874663	0.437332	+	0.899300i	$0.355924\pi$
$270$	0	0
$271$	−1.55240	−0.0943014	−0.0471507	−	0.998888i	$-0.515014\pi$
$271$	−1.55240	−0.0943014	−0.0471507	+	0.998888i	$0.515014\pi$
$272$	4.33192	0.262661
$273$	0	0
$274$	17.6712	1.06756
$275$	6.54239	0.394521
$276$	0	0
$277$	18.0089	1.08205	0.541024	−	0.841007i	$-0.318037\pi$
$277$	18.0089	1.08205	0.541024	+	0.841007i	$0.318037\pi$
$278$	−13.9007	−0.833711
$279$	0	0
$280$	−9.03618	−0.540015
$281$	11.5476	0.688873	0.344436	−	0.938810i	$-0.388070\pi$
$281$	11.5476	0.688873	0.344436	+	0.938810i	$0.388070\pi$
$282$	0	0
$283$	−1.84755	−0.109825	−0.0549126	−	0.998491i	$-0.517488\pi$
$283$	−1.84755	−0.109825	−0.0549126	+	0.998491i	$0.517488\pi$
$284$	3.06068	0.181618
$285$	0	0
$286$	−33.2488	−1.96604
$287$	2.72233	0.160694
$288$	0	0
$289$	−16.2493	−0.955841
$290$	−20.6708	−1.21383
$291$	0	0
$292$	−0.260171	−0.0152253
$293$	3.03213	0.177139	0.0885694	−	0.996070i	$-0.471771\pi$
$293$	3.03213	0.177139	0.0885694	+	0.996070i	$0.471771\pi$
$294$	0	0
$295$	−15.8727	−0.924143
$296$	−8.86974	−0.515543
$297$	0	0
$298$	−27.0322	−1.56594
$299$	35.9535	2.07924
$300$	0	0
$301$	−17.0674	−0.983746
$302$	4.23380	0.243628
$303$	0	0
$304$	−12.3212	−0.706667
$305$	11.7221	0.671207
$306$	0	0
$307$	8.40693	0.479809	0.239905	−	0.970796i	$-0.422884\pi$
$307$	8.40693	0.479809	0.239905	+	0.970796i	$0.422884\pi$
$308$	−6.94068	−0.395482
$309$	0	0
$310$	16.0677	0.912583
$311$	−2.94070	−0.166752	−0.0833759	−	0.996518i	$-0.526570\pi$
$311$	−2.94070	−0.166752	−0.0833759	+	0.996518i	$0.526570\pi$
$312$	0	0
$313$	16.5110	0.933258	0.466629	−	0.884453i	$-0.345468\pi$
$313$	16.5110	0.933258	0.466629	+	0.884453i	$0.345468\pi$
$314$	−21.1257	−1.19219
$315$	0	0
$316$	−10.4978	−0.590550
$317$	20.3799	1.14465	0.572325	−	0.820027i	$-0.306041\pi$
$317$	20.3799	1.14465	0.572325	+	0.820027i	$0.306041\pi$
$318$	0	0
$319$	15.3579	0.859879
$320$	2.23733	0.125070
$321$	0	0
$322$	22.2704	1.24108
$323$	−2.13521	−0.118806
$324$	0	0
$325$	10.8585	0.602322
$326$	−7.34731	−0.406930
$327$	0	0
$328$	−2.31297	−0.127713
$329$	−18.9876	−1.04682
$330$	0	0
$331$	9.08522	0.499369	0.249684	−	0.968327i	$-0.419673\pi$
$331$	9.08522	0.499369	0.249684	+	0.968327i	$0.419673\pi$
$332$	17.0943	0.938174
$333$	0	0
$334$	24.9545	1.36545
$335$	−28.7434	−1.57042
$336$	0	0
$337$	−34.1864	−1.86225	−0.931125	−	0.364700i	$-0.881171\pi$
$337$	−34.1864	−1.86225	−0.931125	+	0.364700i	$0.881171\pi$
$338$	−32.6046	−1.77345
$339$	0	0
$340$	−2.31819	−0.125721
$341$	−11.9379	−0.646476
$342$	0	0
$343$	20.0199	1.08097
$344$	14.5009	0.781837
$345$	0	0
$346$	1.88018	0.101079
$347$	9.13414	0.490346	0.245173	−	0.969479i	$-0.421155\pi$
$347$	9.13414	0.490346	0.245173	+	0.969479i	$0.421155\pi$
$348$	0	0
$349$	−20.3980	−1.09188	−0.545940	−	0.837824i	$-0.683827\pi$
$349$	−20.3980	−1.09188	−0.545940	+	0.837824i	$0.683827\pi$
$350$	6.72601	0.359520
$351$	0	0
$352$	17.8904	0.953561
$353$	−13.2353	−0.704444	−0.352222	−	0.935916i	$-0.614574\pi$
$353$	−13.2353	−0.704444	−0.352222	+	0.935916i	$0.614574\pi$
$354$	0	0
$355$	7.92339	0.420530
$356$	17.7476	0.940622
$357$	0	0
$358$	−41.7255	−2.20526
$359$	−35.7474	−1.88668	−0.943339	−	0.331831i	$-0.892334\pi$
$359$	−35.7474	−1.88668	−0.943339	+	0.331831i	$0.892334\pi$
$360$	0	0
$361$	−12.9269	−0.680363
$362$	8.20873	0.431441
$363$	0	0
$364$	−11.5196	−0.603789
$365$	−0.673522	−0.0352537
$366$	0	0
$367$	−34.7219	−1.81247	−0.906234	−	0.422777i	$-0.861055\pi$
$367$	−34.7219	−1.81247	−0.906234	+	0.422777i	$0.861055\pi$
$368$	−31.8906	−1.66241
$369$	0	0
$370$	23.7381	1.23408
$371$	19.5372	1.01432
$372$	0	0
$373$	−34.9987	−1.81217	−0.906083	−	0.423100i	$-0.860942\pi$
$373$	−34.9987	−1.81217	−0.906083	+	0.423100i	$0.860942\pi$
$374$	5.11076	0.264271
$375$	0	0
$376$	16.1324	0.831966
$377$	25.4898	1.31279
$378$	0	0
$379$	17.4032	0.893943	0.446972	−	0.894548i	$-0.352502\pi$
$379$	17.4032	0.893943	0.446972	+	0.894548i	$0.352502\pi$
$380$	6.59356	0.338242
$381$	0	0
$382$	−42.2919	−2.16384
$383$	−7.58514	−0.387582	−0.193791	−	0.981043i	$-0.562078\pi$
$383$	−7.58514	−0.387582	−0.193791	+	0.981043i	$0.562078\pi$
$384$	0	0
$385$	−17.9678	−0.915725
$386$	−5.60097	−0.285082
$387$	0	0
$388$	0.277765	0.0141014
$389$	9.68216	0.490905	0.245452	−	0.969409i	$-0.421063\pi$
$389$	9.68216	0.490905	0.245452	+	0.969409i	$0.421063\pi$
$390$	0	0
$391$	−5.52651	−0.279488
$392$	−5.05368	−0.255249
$393$	0	0
$394$	−6.26917	−0.315836
$395$	−27.1765	−1.36740
$396$	0	0
$397$	−8.54563	−0.428893	−0.214446	−	0.976736i	$-0.568795\pi$
$397$	−8.54563	−0.428893	−0.214446	+	0.976736i	$0.568795\pi$
$398$	−20.1522	−1.01014
$399$	0	0
$400$	−9.63145	−0.481573
$401$	−23.0124	−1.14918	−0.574592	−	0.818440i	$-0.694839\pi$
$401$	−23.0124	−1.14918	−0.574592	+	0.818440i	$0.694839\pi$
$402$	0	0
$403$	−19.8136	−0.986986
$404$	−7.59570	−0.377900
$405$	0	0
$406$	15.7890	0.783593
$407$	−17.6369	−0.874227
$408$	0	0
$409$	−24.2236	−1.19778	−0.598891	−	0.800831i	$-0.704392\pi$
$409$	−24.2236	−1.19778	−0.598891	+	0.800831i	$0.704392\pi$
$410$	6.19020	0.305712
$411$	0	0
$412$	14.7558	0.726965
$413$	12.1240	0.596585
$414$	0	0
$415$	44.2533	2.17231
$416$	29.6930	1.45582
$417$	0	0
$418$	−14.5364	−0.710999
$419$	−27.7565	−1.35600	−0.677998	−	0.735064i	$-0.737152\pi$
$419$	−27.7565	−1.35600	−0.677998	+	0.735064i	$0.737152\pi$
$420$	0	0
$421$	−33.1205	−1.61420	−0.807098	−	0.590418i	$-0.798963\pi$
$421$	−33.1205	−1.61420	−0.807098	+	0.590418i	$0.798963\pi$
$422$	−45.0604	−2.19351
$423$	0	0
$424$	−16.5994	−0.806137
$425$	−1.66909	−0.0809628
$426$	0	0
$427$	−8.95372	−0.433301
$428$	−17.0277	−0.823064
$429$	0	0
$430$	−38.8087	−1.87152
$431$	5.07371	0.244392	0.122196	−	0.992506i	$-0.461006\pi$
$431$	5.07371	0.244392	0.122196	+	0.992506i	$0.461006\pi$
$432$	0	0
$433$	11.6416	0.559457	0.279729	−	0.960079i	$-0.409755\pi$
$433$	11.6416	0.559457	0.279729	+	0.960079i	$0.409755\pi$
$434$	−12.2730	−0.589122
$435$	0	0
$436$	−6.11973	−0.293082
$437$	15.7189	0.751937
$438$	0	0
$439$	−14.7945	−0.706105	−0.353052	−	0.935604i	$-0.614856\pi$
$439$	−14.7945	−0.706105	−0.353052	+	0.935604i	$0.614856\pi$
$440$	15.2660	0.727777
$441$	0	0
$442$	8.48241	0.403467
$443$	2.54544	0.120937	0.0604687	−	0.998170i	$-0.480740\pi$
$443$	2.54544	0.120937	0.0604687	+	0.998170i	$0.480740\pi$
$444$	0	0
$445$	45.9445	2.17798
$446$	7.64367	0.361938
$447$	0	0
$448$	−1.70894	−0.0807398
$449$	13.1558	0.620860	0.310430	−	0.950596i	$-0.399527\pi$
$449$	13.1558	0.620860	0.310430	+	0.950596i	$0.399527\pi$
$450$	0	0
$451$	−4.59919	−0.216567
$452$	−19.2392	−0.904937
$453$	0	0
$454$	−40.0830	−1.88119
$455$	−29.8215	−1.39805
$456$	0	0
$457$	5.54809	0.259528	0.129764	−	0.991545i	$-0.458578\pi$
$457$	5.54809	0.259528	0.129764	+	0.991545i	$0.458578\pi$
$458$	−5.57825	−0.260655
$459$	0	0
$460$	17.0660	0.795704
$461$	32.2282	1.50102	0.750508	−	0.660861i	$-0.229809\pi$
$461$	32.2282	1.50102	0.750508	+	0.660861i	$0.229809\pi$
$462$	0	0
$463$	−18.5042	−0.859965	−0.429983	−	0.902837i	$-0.641480\pi$
$463$	−18.5042	−0.859965	−0.429983	+	0.902837i	$0.641480\pi$
$464$	−22.6094	−1.04961
$465$	0	0
$466$	12.3873	0.573832
$467$	−20.9192	−0.968026	−0.484013	−	0.875061i	$-0.660821\pi$
$467$	−20.9192	−0.968026	−0.484013	+	0.875061i	$0.660821\pi$
$468$	0	0
$469$	21.9551	1.01379
$470$	−43.1751	−1.99152
$471$	0	0
$472$	−10.3009	−0.474139
$473$	28.8341	1.32579
$474$	0	0
$475$	4.74735	0.217823
$476$	1.77070	0.0811600
$477$	0	0
$478$	6.36103	0.290947
$479$	9.86001	0.450515	0.225258	−	0.974299i	$-0.427678\pi$
$479$	9.86001	0.450515	0.225258	+	0.974299i	$0.427678\pi$
$480$	0	0
$481$	−29.2722	−1.33470
$482$	1.73684	0.0791110
$483$	0	0
$484$	0.542913	0.0246779
$485$	0.719069	0.0326512
$486$	0	0
$487$	−25.9290	−1.17496	−0.587478	−	0.809240i	$-0.699879\pi$
$487$	−25.9290	−1.17496	−0.587478	+	0.809240i	$0.699879\pi$
$488$	7.60734	0.344368
$489$	0	0
$490$	13.5251	0.611003
$491$	22.5594	1.01809	0.509046	−	0.860739i	$-0.329998\pi$
$491$	22.5594	1.01809	0.509046	+	0.860739i	$0.329998\pi$
$492$	0	0
$493$	−3.91811	−0.176463
$494$	−24.1263	−1.08549
$495$	0	0
$496$	17.5746	0.789122
$497$	−6.05213	−0.271475
$498$	0	0
$499$	39.8682	1.78475	0.892373	−	0.451298i	$-0.149039\pi$
$499$	39.8682	1.78475	0.892373	+	0.451298i	$0.149039\pi$
$500$	−8.22360	−0.367771
$501$	0	0
$502$	2.45960	0.109777
$503$	19.7008	0.878416	0.439208	−	0.898386i	$-0.355259\pi$
$503$	19.7008	0.878416	0.439208	+	0.898386i	$0.355259\pi$
$504$	0	0
$505$	−19.6635	−0.875015
$506$	−37.6242	−1.67260
$507$	0	0
$508$	−13.9583	−0.619298
$509$	23.4312	1.03857	0.519284	−	0.854602i	$-0.326199\pi$
$509$	23.4312	1.03857	0.519284	+	0.854602i	$0.326199\pi$
$510$	0	0
$511$	0.514457	0.0227582
$512$	−9.25877	−0.409184
$513$	0	0
$514$	−19.2251	−0.847985
$515$	38.1993	1.68326
$516$	0	0
$517$	32.0782	1.41080
$518$	−18.1319	−0.796668
$519$	0	0
$520$	25.3372	1.11111
$521$	5.76181	0.252430	0.126215	−	0.992003i	$-0.459717\pi$
$521$	5.76181	0.252430	0.126215	+	0.992003i	$0.459717\pi$
$522$	0	0
$523$	42.0766	1.83988	0.919942	−	0.392056i	$-0.128236\pi$
$523$	42.0766	1.83988	0.919942	+	0.392056i	$0.128236\pi$
$524$	14.5574	0.635941
$525$	0	0
$526$	−5.75883	−0.251097
$527$	3.04560	0.132669
$528$	0	0
$529$	17.6849	0.768907
$530$	44.4248	1.92969
$531$	0	0
$532$	−5.03636	−0.218354
$533$	−7.63334	−0.330637
$534$	0	0
$535$	−44.0807	−1.90578
$536$	−18.6537	−0.805717
$537$	0	0
$538$	−24.9160	−1.07420
$539$	−10.0489	−0.432836
$540$	0	0
$541$	7.25557	0.311941	0.155971	−	0.987762i	$-0.450149\pi$
$541$	7.25557	0.311941	0.155971	+	0.987762i	$0.450149\pi$
$542$	2.69627	0.115815
$543$	0	0
$544$	−4.56419	−0.195688
$545$	−15.8426	−0.678620
$546$	0	0
$547$	−27.6574	−1.18254	−0.591272	−	0.806472i	$-0.701374\pi$
$547$	−27.6574	−1.18254	−0.591272	+	0.806472i	$0.701374\pi$
$548$	−10.3435	−0.441851
$549$	0	0
$550$	−11.3631	−0.484524
$551$	11.1442	0.474758
$552$	0	0
$553$	20.7582	0.882730
$554$	−31.2786	−1.32890
$555$	0	0
$556$	8.13650	0.345065
$557$	−26.7523	−1.13353	−0.566766	−	0.823878i	$-0.691806\pi$
$557$	−26.7523	−1.13353	−0.566766	+	0.823878i	$0.691806\pi$
$558$	0	0
$559$	47.8564	2.02411
$560$	26.4515	1.11778
$561$	0	0
$562$	−20.0564	−0.846028
$563$	−20.2482	−0.853362	−0.426681	−	0.904402i	$-0.640317\pi$
$563$	−20.2482	−0.853362	−0.426681	+	0.904402i	$0.640317\pi$
$564$	0	0
$565$	−49.8059	−2.09535
$566$	3.20890	0.134880
$567$	0	0
$568$	5.14206	0.215756
$569$	−41.1274	−1.72415	−0.862076	−	0.506779i	$-0.830836\pi$
$569$	−41.1274	−1.72415	−0.862076	+	0.506779i	$0.830836\pi$
$570$	0	0
$571$	25.2419	1.05634	0.528170	−	0.849139i	$-0.322878\pi$
$571$	25.2419	1.05634	0.528170	+	0.849139i	$0.322878\pi$
$572$	19.4615	0.813725
$573$	0	0
$574$	−4.72827	−0.197354
$575$	12.2875	0.512423
$576$	0	0
$577$	4.90936	0.204379	0.102190	−	0.994765i	$-0.467415\pi$
$577$	4.90936	0.204379	0.102190	+	0.994765i	$0.467415\pi$
$578$	28.2225	1.17390
$579$	0	0
$580$	12.0992	0.502391
$581$	−33.8020	−1.40234
$582$	0	0
$583$	−33.0067	−1.36700
$584$	−0.437097	−0.0180872
$585$	0	0
$586$	−5.26633	−0.217550
$587$	−0.733495	−0.0302746	−0.0151373	−	0.999885i	$-0.504819\pi$
$587$	−0.733495	−0.0302746	−0.0151373	+	0.999885i	$0.504819\pi$
$588$	0	0
$589$	−8.66253	−0.356933
$590$	27.5683	1.13497
$591$	0	0
$592$	25.9643	1.06713
$593$	−43.2540	−1.77623	−0.888115	−	0.459621i	$-0.847985\pi$
$593$	−43.2540	−1.77623	−0.888115	+	0.459621i	$0.847985\pi$
$594$	0	0
$595$	4.58394	0.187923
$596$	15.8227	0.648125
$597$	0	0
$598$	−62.4455	−2.55359
$599$	21.1031	0.862251	0.431126	−	0.902292i	$-0.358117\pi$
$599$	21.1031	0.862251	0.431126	+	0.902292i	$0.358117\pi$
$600$	0	0
$601$	−16.1436	−0.658512	−0.329256	−	0.944241i	$-0.606798\pi$
$601$	−16.1436	−0.658512	−0.329256	+	0.944241i	$0.606798\pi$
$602$	29.6433	1.20817
$603$	0	0
$604$	−2.47816	−0.100835
$605$	1.40548	0.0571408
$606$	0	0
$607$	−32.2628	−1.30951	−0.654753	−	0.755843i	$-0.727227\pi$
$607$	−32.2628	−1.30951	−0.654753	+	0.755843i	$0.727227\pi$
$608$	12.9818	0.526481
$609$	0	0
$610$	−20.3595	−0.824331
$611$	53.2407	2.15389
$612$	0	0
$613$	−44.6321	−1.80267	−0.901337	−	0.433119i	$-0.857413\pi$
$613$	−44.6321	−1.80267	−0.901337	+	0.433119i	$0.857413\pi$
$614$	−14.6015	−0.589270
$615$	0	0
$616$	−11.6606	−0.469820
$617$	−20.2532	−0.815363	−0.407682	−	0.913124i	$-0.633663\pi$
$617$	−20.2532	−0.815363	−0.407682	+	0.913124i	$0.633663\pi$
$618$	0	0
$619$	23.9533	0.962766	0.481383	−	0.876510i	$-0.340135\pi$
$619$	23.9533	0.962766	0.481383	+	0.876510i	$0.340135\pi$
$620$	−9.40488	−0.377709
$621$	0	0
$622$	5.10754	0.204794
$623$	−35.0938	−1.40600
$624$	0	0
$625$	−30.9210	−1.23684
$626$	−28.6770	−1.14617
$627$	0	0
$628$	12.3655	0.493436
$629$	4.49951	0.179407
$630$	0	0
$631$	−3.46481	−0.137932	−0.0689659	−	0.997619i	$-0.521970\pi$
$631$	−3.46481	−0.137932	−0.0689659	+	0.997619i	$0.521970\pi$
$632$	−17.6368	−0.701554
$633$	0	0
$634$	−35.3967	−1.40578
$635$	−36.1348	−1.43396
$636$	0	0
$637$	−16.6783	−0.660818
$638$	−26.6743	−1.05605
$639$	0	0
$640$	−31.6135	−1.24963
$641$	28.9401	1.14307	0.571533	−	0.820579i	$-0.306349\pi$
$641$	28.9401	1.14307	0.571533	+	0.820579i	$0.306349\pi$
$642$	0	0
$643$	18.6657	0.736103	0.368052	−	0.929805i	$-0.380025\pi$
$643$	18.6657	0.736103	0.368052	+	0.929805i	$0.380025\pi$
$644$	−13.0355	−0.513671
$645$	0	0
$646$	3.70852	0.145910
$647$	−24.6633	−0.969615	−0.484808	−	0.874621i	$-0.661110\pi$
$647$	−24.6633	−0.969615	−0.484808	+	0.874621i	$0.661110\pi$
$648$	0	0
$649$	−20.4827	−0.804016
$650$	−18.8595	−0.739731
$651$	0	0
$652$	4.30059	0.168424
$653$	−12.4942	−0.488937	−0.244469	−	0.969657i	$-0.578614\pi$
$653$	−12.4942	−0.488937	−0.244469	+	0.969657i	$0.578614\pi$
$654$	0	0
$655$	37.6856	1.47250
$656$	6.77074	0.264353
$657$	0	0
$658$	32.9785	1.28564
$659$	−34.3869	−1.33952	−0.669762	−	0.742576i	$-0.733604\pi$
$659$	−34.3869	−1.33952	−0.669762	+	0.742576i	$0.733604\pi$
$660$	0	0
$661$	19.4957	0.758295	0.379147	−	0.925336i	$-0.376217\pi$
$661$	19.4957	0.758295	0.379147	+	0.925336i	$0.376217\pi$
$662$	−15.7796	−0.613292
$663$	0	0
$664$	28.7192	1.11452
$665$	−13.0380	−0.505591
$666$	0	0
$667$	28.8442	1.11685
$668$	−14.6066	−0.565145
$669$	0	0
$670$	49.9228	1.92869
$671$	15.1267	0.583958
$672$	0	0
$673$	−26.1090	−1.00643	−0.503215	−	0.864161i	$-0.667850\pi$
$673$	−26.1090	−1.00643	−0.503215	+	0.864161i	$0.667850\pi$
$674$	59.3764	2.28709
$675$	0	0
$676$	19.0844	0.734015
$677$	20.5707	0.790597	0.395298	−	0.918553i	$-0.370641\pi$
$677$	20.5707	0.790597	0.395298	+	0.918553i	$0.370641\pi$
$678$	0	0
$679$	−0.549247	−0.0210782
$680$	−3.89465	−0.149353
$681$	0	0
$682$	20.7343	0.793959
$683$	−23.7002	−0.906864	−0.453432	−	0.891291i	$-0.649801\pi$
$683$	−23.7002	−0.906864	−0.453432	+	0.891291i	$0.649801\pi$
$684$	0	0
$685$	−26.7768	−1.02309
$686$	−34.7714	−1.32758
$687$	0	0
$688$	−42.4484	−1.61833
$689$	−54.7818	−2.08702
$690$	0	0
$691$	−7.36209	−0.280067	−0.140034	−	0.990147i	$-0.544721\pi$
$691$	−7.36209	−0.280067	−0.140034	+	0.990147i	$0.544721\pi$
$692$	−1.10052	−0.0418356
$693$	0	0
$694$	−15.8646	−0.602211
$695$	21.0635	0.798985
$696$	0	0
$697$	1.17334	0.0444435
$698$	35.4282	1.34098
$699$	0	0
$700$	−3.93692	−0.148802
$701$	28.4805	1.07569	0.537846	−	0.843043i	$-0.319238\pi$
$701$	28.4805	1.07569	0.537846	+	0.843043i	$0.319238\pi$
$702$	0	0
$703$	−12.7978	−0.482679
$704$	2.88713	0.108813
$705$	0	0
$706$	22.9876	0.865151
$707$	15.0196	0.564870
$708$	0	0
$709$	32.8736	1.23459	0.617297	−	0.786730i	$-0.288228\pi$
$709$	32.8736	1.23459	0.617297	+	0.786730i	$0.288228\pi$
$710$	−13.7617	−0.516467
$711$	0	0
$712$	29.8167	1.11743
$713$	−22.4210	−0.839674
$714$	0	0
$715$	50.3812	1.88415
$716$	24.4231	0.912736
$717$	0	0
$718$	62.0877	2.31709
$719$	13.3988	0.499690	0.249845	−	0.968286i	$-0.419620\pi$
$719$	13.3988	0.499690	0.249845	+	0.968286i	$0.419620\pi$
$720$	0	0
$721$	−29.1778	−1.08664
$722$	22.4520	0.835576
$723$	0	0
$724$	−4.80481	−0.178569
$725$	8.71140	0.323533
$726$	0	0
$727$	24.5791	0.911587	0.455793	−	0.890086i	$-0.349356\pi$
$727$	24.5791	0.911587	0.455793	+	0.890086i	$0.349356\pi$
$728$	−19.3533	−0.717282
$729$	0	0
$730$	1.16980	0.0432963
$731$	−7.35613	−0.272076
$732$	0	0
$733$	2.08207	0.0769029	0.0384515	−	0.999260i	$-0.487758\pi$
$733$	2.08207	0.0769029	0.0384515	+	0.999260i	$0.487758\pi$
$734$	60.3065	2.22595
$735$	0	0
$736$	33.6005	1.23853
$737$	−37.0916	−1.36629
$738$	0	0
$739$	−46.1104	−1.69620	−0.848099	−	0.529838i	$-0.822253\pi$
$739$	−46.1104	−1.69620	−0.848099	+	0.529838i	$0.822253\pi$
$740$	−13.8946	−0.510774
$741$	0	0
$742$	−33.9331	−1.24572
$743$	−16.5456	−0.606998	−0.303499	−	0.952832i	$-0.598155\pi$
$743$	−16.5456	−0.606998	−0.303499	+	0.952832i	$0.598155\pi$
$744$	0	0
$745$	40.9614	1.50071
$746$	60.7873	2.22558
$747$	0	0
$748$	−2.99148	−0.109379
$749$	33.6702	1.23028
$750$	0	0
$751$	36.4121	1.32870	0.664349	−	0.747423i	$-0.268709\pi$
$751$	36.4121	1.32870	0.664349	+	0.747423i	$0.268709\pi$
$752$	−47.2243	−1.72209
$753$	0	0
$754$	−44.2718	−1.61228
$755$	−6.41540	−0.233480
$756$	0	0
$757$	16.3953	0.595898	0.297949	−	0.954582i	$-0.403697\pi$
$757$	16.3953	0.595898	0.297949	+	0.954582i	$0.403697\pi$
$758$	−30.2267	−1.09788
$759$	0	0
$760$	11.0774	0.401821
$761$	−44.2450	−1.60388	−0.801940	−	0.597404i	$-0.796199\pi$
$761$	−44.2450	−1.60388	−0.801940	+	0.597404i	$0.796199\pi$
$762$	0	0
$763$	12.1010	0.438087
$764$	24.7547	0.895593
$765$	0	0
$766$	13.1742	0.476003
$767$	−33.9954	−1.22750
$768$	0	0
$769$	−21.3439	−0.769680	−0.384840	−	0.922983i	$-0.625743\pi$
$769$	−21.3439	−0.769680	−0.384840	+	0.922983i	$0.625743\pi$
$770$	31.2073	1.12463
$771$	0	0
$772$	3.27841	0.117993
$773$	−20.7861	−0.747623	−0.373812	−	0.927505i	$-0.621949\pi$
$773$	−20.7861	−0.747623	−0.373812	+	0.927505i	$0.621949\pi$
$774$	0	0
$775$	−6.77150	−0.243239
$776$	0.466656	0.0167520
$777$	0	0
$778$	−16.8164	−0.602897
$779$	−3.33731	−0.119571
$780$	0	0
$781$	10.2246	0.365866
$782$	9.59868	0.343248
$783$	0	0
$784$	14.7936	0.528342
$785$	32.0114	1.14253
$786$	0	0
$787$	−28.9072	−1.03043	−0.515215	−	0.857061i	$-0.672288\pi$
$787$	−28.9072	−1.03043	−0.515215	+	0.857061i	$0.672288\pi$
$788$	3.66952	0.130721
$789$	0	0
$790$	47.2013	1.67935
$791$	38.0433	1.35266
$792$	0	0
$793$	25.1060	0.891539
$794$	14.8424	0.526738
$795$	0	0
$796$	11.7957	0.418086
$797$	−21.6563	−0.767106	−0.383553	−	0.923519i	$-0.625300\pi$
$797$	−21.6563	−0.767106	−0.383553	+	0.923519i	$0.625300\pi$
$798$	0	0
$799$	−8.18377	−0.289521
$800$	10.1479	0.358781
$801$	0	0
$802$	39.9689	1.41135
$803$	−0.869138	−0.0306712
$804$	0	0
$805$	−33.7459	−1.18939
$806$	34.4131	1.21215
$807$	0	0
$808$	−12.7611	−0.448933
$809$	−4.12546	−0.145043	−0.0725217	−	0.997367i	$-0.523105\pi$
$809$	−4.12546	−0.145043	−0.0725217	+	0.997367i	$0.523105\pi$
$810$	0	0
$811$	−27.4598	−0.964243	−0.482121	−	0.876104i	$-0.660134\pi$
$811$	−27.4598	−0.964243	−0.482121	+	0.876104i	$0.660134\pi$
$812$	−9.24173	−0.324321
$813$	0	0
$814$	30.6325	1.07367
$815$	11.1332	0.389980
$816$	0	0
$817$	20.9228	0.731998
$818$	42.0726	1.47104
$819$	0	0
$820$	−3.62330	−0.126531
$821$	−26.7943	−0.935127	−0.467564	−	0.883959i	$-0.654868\pi$
$821$	−26.7943	−0.935127	−0.467564	+	0.883959i	$0.654868\pi$
$822$	0	0
$823$	29.5890	1.03141	0.515704	−	0.856767i	$-0.327531\pi$
$823$	29.5890	1.03141	0.515704	+	0.856767i	$0.327531\pi$
$824$	24.7903	0.863611
$825$	0	0
$826$	−21.0575	−0.732686
$827$	41.2969	1.43603	0.718017	−	0.696025i	$-0.245050\pi$
$827$	41.2969	1.43603	0.718017	+	0.696025i	$0.245050\pi$
$828$	0	0
$829$	−3.90138	−0.135501	−0.0677503	−	0.997702i	$-0.521582\pi$
$829$	−3.90138	−0.135501	−0.0677503	+	0.997702i	$0.521582\pi$
$830$	−76.8610	−2.66789
$831$	0	0
$832$	4.79182	0.166126
$833$	2.56367	0.0888258
$834$	0	0
$835$	−37.8130	−1.30857
$836$	8.50857	0.294275
$837$	0	0
$838$	48.2088	1.66534
$839$	−18.4300	−0.636274	−0.318137	−	0.948045i	$-0.603057\pi$
$839$	−18.4300	−0.636274	−0.318137	+	0.948045i	$0.603057\pi$
$840$	0	0
$841$	−8.55043	−0.294842
$842$	57.5252	1.98245
$843$	0	0
$844$	26.3752	0.907871
$845$	49.4051	1.69959
$846$	0	0
$847$	−1.07355	−0.0368875
$848$	48.5912	1.66863
$849$	0	0
$850$	2.89895	0.0994332
$851$	−33.1243	−1.13549
$852$	0	0
$853$	37.7897	1.29389	0.646947	−	0.762535i	$-0.276046\pi$
$853$	37.7897	1.29389	0.646947	+	0.762535i	$0.276046\pi$
$854$	15.5512	0.532151
$855$	0	0
$856$	−28.6072	−0.977773
$857$	58.1926	1.98782	0.993910	−	0.110194i	$-0.0351472\pi$
$857$	58.1926	1.98782	0.993910	+	0.110194i	$0.0351472\pi$
$858$	0	0
$859$	35.3869	1.20738	0.603692	−	0.797218i	$-0.293696\pi$
$859$	35.3869	1.20738	0.603692	+	0.797218i	$0.293696\pi$
$860$	22.7159	0.774605
$861$	0	0
$862$	−8.81224	−0.300146
$863$	−27.4547	−0.934568	−0.467284	−	0.884107i	$-0.654767\pi$
$863$	−27.4547	−0.934568	−0.467284	+	0.884107i	$0.654767\pi$
$864$	0	0
$865$	−2.84900	−0.0968689
$866$	−20.2196	−0.687088
$867$	0	0
$868$	7.18373	0.243832
$869$	−35.0696	−1.18965
$870$	0	0
$871$	−61.5615	−2.08593
$872$	−10.2814	−0.348172
$873$	0	0
$874$	−27.3013	−0.923479
$875$	16.2612	0.549729
$876$	0	0
$877$	−14.8517	−0.501505	−0.250752	−	0.968051i	$-0.580678\pi$
$877$	−14.8517	−0.501505	−0.250752	+	0.968051i	$0.580678\pi$
$878$	25.6958	0.867191
$879$	0	0
$880$	−44.6879	−1.50643
$881$	−37.8714	−1.27592	−0.637960	−	0.770070i	$-0.720221\pi$
$881$	−37.8714	−1.27592	−0.637960	+	0.770070i	$0.720221\pi$
$882$	0	0
$883$	−15.3679	−0.517171	−0.258586	−	0.965988i	$-0.583256\pi$
$883$	−15.3679	−0.517171	−0.258586	+	0.965988i	$0.583256\pi$
$884$	−4.96500	−0.166991
$885$	0	0
$886$	−4.42102	−0.148527
$887$	−2.72841	−0.0916111	−0.0458056	−	0.998950i	$-0.514585\pi$
$887$	−2.72841	−0.0916111	−0.0458056	+	0.998950i	$0.514585\pi$
$888$	0	0
$889$	27.6008	0.925703
$890$	−79.7984	−2.67485
$891$	0	0
$892$	−4.47406	−0.149803
$893$	23.2769	0.778931
$894$	0	0
$895$	63.2259	2.11341
$896$	24.1474	0.806707
$897$	0	0
$898$	−22.8495	−0.762499
$899$	−15.8958	−0.530153
$900$	0	0
$901$	8.42066	0.280533
$902$	7.98807	0.265973
$903$	0	0
$904$	−32.3227	−1.07504
$905$	−12.4385	−0.413471
$906$	0	0
$907$	−6.49334	−0.215608	−0.107804	−	0.994172i	$-0.534382\pi$
$907$	−6.49334	−0.215608	−0.107804	+	0.994172i	$0.534382\pi$
$908$	23.4617	0.778605
$909$	0	0
$910$	51.7952	1.71699
$911$	32.2049	1.06699	0.533497	−	0.845802i	$-0.320877\pi$
$911$	32.2049	1.06699	0.533497	+	0.845802i	$0.320877\pi$
$912$	0	0
$913$	57.1061	1.88994
$914$	−9.63615	−0.318736
$915$	0	0
$916$	3.26511	0.107882
$917$	−28.7855	−0.950580
$918$	0	0
$919$	−25.5609	−0.843177	−0.421588	−	0.906787i	$-0.638527\pi$
$919$	−25.5609	−0.843177	−0.421588	+	0.906787i	$0.638527\pi$
$920$	28.6715	0.945271
$921$	0	0
$922$	−55.9753	−1.84345
$923$	16.9700	0.558574
$924$	0	0
$925$	−10.0041	−0.328932
$926$	32.1390	1.05615
$927$	0	0
$928$	23.8216	0.781983
$929$	43.9109	1.44067	0.720336	−	0.693626i	$-0.243988\pi$
$929$	43.9109	1.44067	0.720336	+	0.693626i	$0.243988\pi$
$930$	0	0
$931$	−7.29177	−0.238978
$932$	−7.25067	−0.237503
$933$	0	0
$934$	36.3334	1.18886
$935$	−7.74424	−0.253264
$936$	0	0
$937$	37.5543	1.22684	0.613422	−	0.789755i	$-0.289792\pi$
$937$	37.5543	1.22684	0.613422	+	0.789755i	$0.289792\pi$
$938$	−38.1326	−1.24507
$939$	0	0
$940$	25.2716	0.824270
$941$	20.6958	0.674663	0.337332	−	0.941386i	$-0.390476\pi$
$941$	20.6958	0.674663	0.337332	+	0.941386i	$0.390476\pi$
$942$	0	0
$943$	−8.63787	−0.281288
$944$	30.1538	0.981423
$945$	0	0
$946$	−50.0802	−1.62825
$947$	14.0412	0.456276	0.228138	−	0.973629i	$-0.426736\pi$
$947$	14.0412	0.456276	0.228138	+	0.973629i	$0.426736\pi$
$948$	0	0
$949$	−1.44252	−0.0468262
$950$	−8.24540	−0.267516
$951$	0	0
$952$	2.97485	0.0964155
$953$	32.4866	1.05235	0.526173	−	0.850378i	$-0.323627\pi$
$953$	32.4866	1.05235	0.526173	+	0.850378i	$0.323627\pi$
$954$	0	0
$955$	64.0842	2.07371
$956$	−3.72330	−0.120420
$957$	0	0
$958$	−17.1253	−0.553293
$959$	20.4530	0.660461
$960$	0	0
$961$	−18.6440	−0.601419
$962$	50.8412	1.63919
$963$	0	0
$964$	−1.01662	−0.0327433
$965$	8.48704	0.273208
$966$	0	0
$967$	−53.7501	−1.72849	−0.864244	−	0.503073i	$-0.832203\pi$
$967$	−53.7501	−1.72849	−0.864244	+	0.503073i	$0.832203\pi$
$968$	0.912116	0.0293165
$969$	0	0
$970$	−1.24891	−0.0401001
$971$	−34.1747	−1.09672	−0.548359	−	0.836243i	$-0.684747\pi$
$971$	−34.1747	−1.09672	−0.548359	+	0.836243i	$0.684747\pi$
$972$	0	0
$973$	−16.0890	−0.515789
$974$	45.0346	1.44300
$975$	0	0
$976$	−22.2689	−0.712810
$977$	8.90298	0.284832	0.142416	−	0.989807i	$-0.454513\pi$
$977$	8.90298	0.284832	0.142416	+	0.989807i	$0.454513\pi$
$978$	0	0
$979$	59.2885	1.89487
$980$	−7.91665	−0.252888
$981$	0	0
$982$	−39.1822	−1.25035
$983$	−22.5713	−0.719912	−0.359956	−	0.932969i	$-0.617208\pi$
$983$	−22.5713	−0.719912	−0.359956	+	0.932969i	$0.617208\pi$
$984$	0	0
$985$	9.49955	0.302681
$986$	6.80514	0.216720
$987$	0	0
$988$	14.1218	0.449275
$989$	54.1541	1.72200
$990$	0	0
$991$	46.7740	1.48583	0.742913	−	0.669388i	$-0.233444\pi$
$991$	46.7740	1.48583	0.742913	+	0.669388i	$0.233444\pi$
$992$	−18.5169	−0.587912
$993$	0	0
$994$	10.5116	0.333408
$995$	30.5362	0.968063
$996$	0	0
$997$	−20.7833	−0.658214	−0.329107	−	0.944293i	$-0.606748\pi$
$997$	−20.7833	−0.658214	−0.329107	+	0.944293i	$0.606748\pi$
$998$	−69.2448	−2.19191
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.e.1.1		7
3.2	odd	2		241.2.a.a.1.7	✓	7
12.11	even	2		3856.2.a.j.1.6		7
15.14	odd	2		6025.2.a.f.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.a.1.7	✓	7	3.2	odd	2
2169.2.a.e.1.1		7	1.1	even	1	trivial
3856.2.a.j.1.6		7	12.11	even	2
6025.2.a.f.1.1		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)

\(f(q)\)	\(=\)	\(q-1.73684 q^{2} +1.01662 q^{4} +2.63180 q^{5} -2.01025 q^{7} +1.70797 q^{8} +O(q^{10})\)	\(q-1.73684 q^{2} +1.01662 q^{4} +2.63180 q^{5} -2.01025 q^{7} +1.70797 q^{8} -4.57103 q^{10} +3.39618 q^{11} +5.63669 q^{13} +3.49150 q^{14} -4.99972 q^{16} -0.866432 q^{17} +2.46437 q^{19} +2.67556 q^{20} -5.89863 q^{22} +6.37847 q^{23} +1.92640 q^{25} -9.79005 q^{26} -2.04367 q^{28} +4.52212 q^{29} -3.51511 q^{31} +5.26780 q^{32} +1.50486 q^{34} -5.29060 q^{35} -5.19315 q^{37} -4.28022 q^{38} +4.49504 q^{40} -1.35422 q^{41} +8.49015 q^{43} +3.45264 q^{44} -11.0784 q^{46} +9.44537 q^{47} -2.95888 q^{49} -3.34585 q^{50} +5.73040 q^{52} -9.71877 q^{53} +8.93808 q^{55} -3.43345 q^{56} -7.85421 q^{58} -6.03110 q^{59} +4.45402 q^{61} +6.10520 q^{62} +0.850111 q^{64} +14.8347 q^{65} -10.9216 q^{67} -0.880836 q^{68} +9.18894 q^{70} +3.01063 q^{71} -0.255916 q^{73} +9.01969 q^{74} +2.50534 q^{76} -6.82718 q^{77} -10.3262 q^{79} -13.1583 q^{80} +2.35207 q^{82} +16.8148 q^{83} -2.28028 q^{85} -14.7461 q^{86} +5.80057 q^{88} +17.4574 q^{89} -11.3312 q^{91} +6.48451 q^{92} -16.4051 q^{94} +6.48574 q^{95} +0.273223 q^{97} +5.13911 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8}+O(q^{10}) \) 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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