LMFDB - Embedded newform 1359.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1359,2,Mod(1,1359)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1359, 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("1359.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1359 = 3^{2} \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1359.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:	$10.8516696347$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4838537.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 7x^{4} + 3x^{3} + 13x^{2} + 3x - 1 $ x^6 - x^5 - 7x^4 + 3x^3 + 13x^2 + 3x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 151)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.183668$ of defining polynomial
Character	$\chi$	$=$	1359.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.183668 q^{2} -1.96627 q^{4} -3.71611 q^{5} +3.65107 q^{7} +0.728478 q^{8} +O(q^{10})$	\(q-0.183668 q^{2} -1.96627 q^{4} -3.71611 q^{5} +3.65107 q^{7} +0.728478 q^{8} +0.682533 q^{10} +0.912147 q^{11} -1.27865 q^{13} -0.670586 q^{14} +3.79873 q^{16} -1.65822 q^{17} +3.74742 q^{19} +7.30687 q^{20} -0.167533 q^{22} +0.248713 q^{23} +8.80951 q^{25} +0.234848 q^{26} -7.17897 q^{28} -1.93860 q^{29} -5.44267 q^{31} -2.15466 q^{32} +0.304563 q^{34} -13.5678 q^{35} -4.41526 q^{37} -0.688284 q^{38} -2.70711 q^{40} -8.64880 q^{41} +1.41210 q^{43} -1.79352 q^{44} -0.0456807 q^{46} -2.19095 q^{47} +6.33031 q^{49} -1.61803 q^{50} +2.51417 q^{52} +10.4493 q^{53} -3.38964 q^{55} +2.65972 q^{56} +0.356059 q^{58} +0.862276 q^{59} -2.48235 q^{61} +0.999648 q^{62} -7.20172 q^{64} +4.75161 q^{65} -13.4435 q^{67} +3.26051 q^{68} +2.49198 q^{70} -8.78436 q^{71} -14.3562 q^{73} +0.810944 q^{74} -7.36843 q^{76} +3.33031 q^{77} -10.1620 q^{79} -14.1165 q^{80} +1.58851 q^{82} -7.06074 q^{83} +6.16215 q^{85} -0.259358 q^{86} +0.664479 q^{88} -11.7770 q^{89} -4.66844 q^{91} -0.489036 q^{92} +0.402409 q^{94} -13.9259 q^{95} -0.860838 q^{97} -1.16268 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) $ 6 * q - q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	$ 6 q - q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} + 8 q^{10} - 8 q^{11} - q^{13} - 6 q^{14} - 3 q^{16} - 9 q^{17} - 6 q^{19} - 3 q^{20} - 12 q^{22} + 4 q^{23} - 4 q^{25} + 7 q^{26} - 24 q^{28} + 2 q^{29} - 8 q^{31} + 11 q^{32} - 9 q^{34} - 5 q^{35} - 12 q^{37} + 3 q^{38} + 9 q^{40} - 41 q^{41} + q^{43} - 17 q^{46} - 28 q^{47} + 33 q^{49} - 15 q^{50} + 15 q^{52} - 14 q^{53} + q^{55} + 9 q^{56} + q^{58} - 12 q^{59} + 5 q^{61} + 9 q^{62} - 27 q^{64} - 13 q^{65} - 15 q^{67} - 11 q^{68} - 33 q^{70} + 2 q^{71} - 7 q^{73} - 53 q^{74} - 3 q^{76} + 15 q^{77} - 9 q^{79} - 16 q^{80} + 10 q^{82} + 11 q^{83} - 10 q^{85} + 26 q^{86} + 24 q^{88} - 36 q^{89} - 35 q^{91} + 38 q^{92} + 42 q^{94} - 32 q^{95} + 11 q^{97} - 15 q^{98}+O(q^{100}) $ 6 * q - q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 + 8 * q^10 - 8 * q^11 - q^13 - 6 * q^14 - 3 * q^16 - 9 * q^17 - 6 * q^19 - 3 * q^20 - 12 * q^22 + 4 * q^23 - 4 * q^25 + 7 * q^26 - 24 * q^28 + 2 * q^29 - 8 * q^31 + 11 * q^32 - 9 * q^34 - 5 * q^35 - 12 * q^37 + 3 * q^38 + 9 * q^40 - 41 * q^41 + q^43 - 17 * q^46 - 28 * q^47 + 33 * q^49 - 15 * q^50 + 15 * q^52 - 14 * q^53 + q^55 + 9 * q^56 + q^58 - 12 * q^59 + 5 * q^61 + 9 * q^62 - 27 * q^64 - 13 * q^65 - 15 * q^67 - 11 * q^68 - 33 * q^70 + 2 * q^71 - 7 * q^73 - 53 * q^74 - 3 * q^76 + 15 * q^77 - 9 * q^79 - 16 * q^80 + 10 * q^82 + 11 * q^83 - 10 * q^85 + 26 * q^86 + 24 * q^88 - 36 * q^89 - 35 * q^91 + 38 * q^92 + 42 * q^94 - 32 * q^95 + 11 * q^97 - 15 * 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 $

$2$	−0.183668	−0.129873	−0.0649366	−	0.997889i	$-0.520685\pi$
$2$	−0.183668	−0.129873	−0.0649366	+	0.997889i	$0.520685\pi$
$3$	0	0
$3$	0	0
$4$	−1.96627	−0.983133
$5$	−3.71611	−1.66190	−0.830948	−	0.556349i	$-0.812202\pi$
$5$	−3.71611	−1.66190	−0.830948	+	0.556349i	$0.812202\pi$
$6$	0	0
$7$	3.65107	1.37997	0.689987	−	0.723821i	$-0.257616\pi$
$7$	3.65107	1.37997	0.689987	+	0.723821i	$0.257616\pi$
$8$	0.728478	0.257556
$9$	0	0
$10$	0.682533	0.215836
$11$	0.912147	0.275023	0.137511	−	0.990500i	$-0.456090\pi$
$11$	0.912147	0.275023	0.137511	+	0.990500i	$0.456090\pi$
$12$	0	0
$13$	−1.27865	−0.354634	−0.177317	−	0.984154i	$-0.556742\pi$
$13$	−1.27865	−0.354634	−0.177317	+	0.984154i	$0.556742\pi$
$14$	−0.670586	−0.179222
$15$	0	0
$16$	3.79873	0.949683
$17$	−1.65822	−0.402178	−0.201089	−	0.979573i	$-0.564448\pi$
$17$	−1.65822	−0.402178	−0.201089	+	0.979573i	$0.564448\pi$
$18$	0	0
$19$	3.74742	0.859718	0.429859	−	0.902896i	$-0.358563\pi$
$19$	3.74742	0.859718	0.429859	+	0.902896i	$0.358563\pi$
$20$	7.30687	1.63387
$21$	0	0
$22$	−0.167533	−0.0357181
$23$	0.248713	0.0518602	0.0259301	−	0.999664i	$-0.491745\pi$
$23$	0.248713	0.0518602	0.0259301	+	0.999664i	$0.491745\pi$
$24$	0	0
$25$	8.80951	1.76190
$26$	0.234848	0.0460575
$27$	0	0
$28$	−7.17897	−1.35670
$29$	−1.93860	−0.359989	−0.179994	−	0.983668i	$-0.557608\pi$
$29$	−1.93860	−0.359989	−0.179994	+	0.983668i	$0.557608\pi$
$30$	0	0
$31$	−5.44267	−0.977533	−0.488767	−	0.872415i	$-0.662553\pi$
$31$	−5.44267	−0.977533	−0.488767	+	0.872415i	$0.662553\pi$
$32$	−2.15466	−0.380894
$33$	0	0
$34$	0.304563	0.0522322
$35$	−13.5678	−2.29338
$36$	0	0
$37$	−4.41526	−0.725864	−0.362932	−	0.931816i	$-0.618224\pi$
$37$	−4.41526	−0.725864	−0.362932	+	0.931816i	$0.618224\pi$
$38$	−0.688284	−0.111654
$39$	0	0
$40$	−2.70711	−0.428031
$41$	−8.64880	−1.35072	−0.675358	−	0.737490i	$-0.736011\pi$
$41$	−8.64880	−1.35072	−0.675358	+	0.737490i	$0.736011\pi$
$42$	0	0
$43$	1.41210	0.215343	0.107671	−	0.994187i	$-0.465661\pi$
$43$	1.41210	0.215343	0.107671	+	0.994187i	$0.465661\pi$
$44$	−1.79352	−0.270384
$45$	0	0
$46$	−0.0456807	−0.00673526
$47$	−2.19095	−0.319583	−0.159791	−	0.987151i	$-0.551082\pi$
$47$	−2.19095	−0.319583	−0.159791	+	0.987151i	$0.551082\pi$
$48$	0	0
$49$	6.33031	0.904330
$50$	−1.61803	−0.228824
$51$	0	0
$52$	2.51417	0.348652
$53$	10.4493	1.43532	0.717661	−	0.696393i	$-0.245213\pi$
$53$	10.4493	1.43532	0.717661	+	0.696393i	$0.245213\pi$
$54$	0	0
$55$	−3.38964	−0.457059
$56$	2.65972	0.355421
$57$	0	0
$58$	0.356059	0.0467529
$59$	0.862276	0.112259	0.0561294	−	0.998424i	$-0.482124\pi$
$59$	0.862276	0.112259	0.0561294	+	0.998424i	$0.482124\pi$
$60$	0	0
$61$	−2.48235	−0.317833	−0.158916	−	0.987292i	$-0.550800\pi$
$61$	−2.48235	−0.317833	−0.158916	+	0.987292i	$0.550800\pi$
$62$	0.999648	0.126955
$63$	0	0
$64$	−7.20172	−0.900215
$65$	4.75161	0.589365
$66$	0	0
$67$	−13.4435	−1.64239	−0.821195	−	0.570647i	$-0.806692\pi$
$67$	−13.4435	−1.64239	−0.821195	+	0.570647i	$0.806692\pi$
$68$	3.26051	0.395395
$69$	0	0
$70$	2.49198	0.297848
$71$	−8.78436	−1.04251	−0.521256	−	0.853400i	$-0.674536\pi$
$71$	−8.78436	−1.04251	−0.521256	+	0.853400i	$0.674536\pi$
$72$	0	0
$73$	−14.3562	−1.68026	−0.840131	−	0.542383i	$-0.817522\pi$
$73$	−14.3562	−1.68026	−0.840131	+	0.542383i	$0.817522\pi$
$74$	0.810944	0.0942703
$75$	0	0
$76$	−7.36843	−0.845217
$77$	3.33031	0.379524
$78$	0	0
$79$	−10.1620	−1.14331	−0.571655	−	0.820494i	$-0.693698\pi$
$79$	−10.1620	−1.14331	−0.571655	+	0.820494i	$0.693698\pi$
$80$	−14.1165	−1.57828
$81$	0	0
$82$	1.58851	0.175422
$83$	−7.06074	−0.775017	−0.387509	−	0.921866i	$-0.626664\pi$
$83$	−7.06074	−0.775017	−0.387509	+	0.921866i	$0.626664\pi$
$84$	0	0
$85$	6.16215	0.668378
$86$	−0.259358	−0.0279673
$87$	0	0
$88$	0.664479	0.0708337
$89$	−11.7770	−1.24836	−0.624180	−	0.781280i	$-0.714567\pi$
$89$	−11.7770	−1.24836	−0.624180	+	0.781280i	$0.714567\pi$
$90$	0	0
$91$	−4.66844	−0.489386
$92$	−0.489036	−0.0509855
$93$	0	0
$94$	0.402409	0.0415053
$95$	−13.9259	−1.42876
$96$	0	0
$97$	−0.860838	−0.0874048	−0.0437024	−	0.999045i	$-0.513915\pi$
$97$	−0.860838	−0.0874048	−0.0437024	+	0.999045i	$0.513915\pi$
$98$	−1.16268	−0.117448
$99$	0	0
$100$	−17.3218	−1.73218
$101$	7.72970	0.769134	0.384567	−	0.923097i	$-0.374351\pi$
$101$	7.72970	0.769134	0.384567	+	0.923097i	$0.374351\pi$
$102$	0	0
$103$	1.97534	0.194636	0.0973179	−	0.995253i	$-0.468974\pi$
$103$	1.97534	0.194636	0.0973179	+	0.995253i	$0.468974\pi$
$104$	−0.931469	−0.0913381
$105$	0	0
$106$	−1.91921	−0.186410
$107$	0.319598	0.0308967	0.0154484	−	0.999881i	$-0.495082\pi$
$107$	0.319598	0.0308967	0.0154484	+	0.999881i	$0.495082\pi$
$108$	0	0
$109$	12.6271	1.20945	0.604727	−	0.796433i	$-0.293282\pi$
$109$	12.6271	1.20945	0.604727	+	0.796433i	$0.293282\pi$
$110$	0.622570	0.0593597
$111$	0	0
$112$	13.8694	1.31054
$113$	13.1810	1.23997	0.619984	−	0.784615i	$-0.287139\pi$
$113$	13.1810	1.23997	0.619984	+	0.784615i	$0.287139\pi$
$114$	0	0
$115$	−0.924246	−0.0861864
$116$	3.81180	0.353917
$117$	0	0
$118$	−0.158373	−0.0145794
$119$	−6.05429	−0.554996
$120$	0	0
$121$	−10.1680	−0.924363
$122$	0.455930	0.0412780
$123$	0	0
$124$	10.7017	0.961045
$125$	−14.1566	−1.26620
$126$	0	0
$127$	0.830524	0.0736971	0.0368485	−	0.999321i	$-0.488268\pi$
$127$	0.830524	0.0736971	0.0368485	+	0.999321i	$0.488268\pi$
$128$	5.63206	0.497808
$129$	0	0
$130$	−0.872722	−0.0765428
$131$	−9.31948	−0.814247	−0.407124	−	0.913373i	$-0.633468\pi$
$131$	−9.31948	−0.814247	−0.407124	+	0.913373i	$0.633468\pi$
$132$	0	0
$133$	13.6821	1.18639
$134$	2.46916	0.213303
$135$	0	0
$136$	−1.20798	−0.103583
$137$	−4.27610	−0.365332	−0.182666	−	0.983175i	$-0.558473\pi$
$137$	−4.27610	−0.365332	−0.182666	+	0.983175i	$0.558473\pi$
$138$	0	0
$139$	11.8522	1.00529	0.502645	−	0.864493i	$-0.332360\pi$
$139$	11.8522	1.00529	0.502645	+	0.864493i	$0.332360\pi$
$140$	26.6779	2.25469
$141$	0	0
$142$	1.61341	0.135394
$143$	−1.16632	−0.0975323
$144$	0	0
$145$	7.20405	0.598264
$146$	2.63678	0.218221
$147$	0	0
$148$	8.68158	0.713621
$149$	21.2817	1.74347	0.871734	−	0.489979i	$-0.162996\pi$
$149$	21.2817	1.74347	0.871734	+	0.489979i	$0.162996\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	2.72992	0.221425
$153$	0	0
$154$	−0.611673	−0.0492900
$155$	20.2256	1.62456
$156$	0	0
$157$	−15.8157	−1.26223	−0.631115	−	0.775689i	$-0.717403\pi$
$157$	−15.8157	−1.26223	−0.631115	+	0.775689i	$0.717403\pi$
$158$	1.86643	0.148485
$159$	0	0
$160$	8.00698	0.633007
$161$	0.908069	0.0715658
$162$	0	0
$163$	−13.1212	−1.02773	−0.513866	−	0.857871i	$-0.671787\pi$
$163$	−13.1212	−1.02773	−0.513866	+	0.857871i	$0.671787\pi$
$164$	17.0058	1.32793
$165$	0	0
$166$	1.29684	0.100654
$167$	−20.1890	−1.56227	−0.781134	−	0.624363i	$-0.785359\pi$
$167$	−20.1890	−1.56227	−0.781134	+	0.624363i	$0.785359\pi$
$168$	0	0
$169$	−11.3651	−0.874235
$170$	−1.13179	−0.0868045
$171$	0	0
$172$	−2.77656	−0.211711
$173$	−0.672028	−0.0510933	−0.0255467	−	0.999674i	$-0.508133\pi$
$173$	−0.672028	−0.0510933	−0.0255467	+	0.999674i	$0.508133\pi$
$174$	0	0
$175$	32.1641	2.43138
$176$	3.46500	0.261184
$177$	0	0
$178$	2.16307	0.162129
$179$	2.07329	0.154965	0.0774823	−	0.996994i	$-0.475312\pi$
$179$	2.07329	0.154965	0.0774823	+	0.996994i	$0.475312\pi$
$180$	0	0
$181$	2.78471	0.206985	0.103493	−	0.994630i	$-0.466998\pi$
$181$	2.78471	0.206985	0.103493	+	0.994630i	$0.466998\pi$
$182$	0.857446	0.0635581
$183$	0	0
$184$	0.181182	0.0133569
$185$	16.4076	1.20631
$186$	0	0
$187$	−1.51254	−0.110608
$188$	4.30799	0.314193
$189$	0	0
$190$	2.55774	0.185558
$191$	−12.6722	−0.916930	−0.458465	−	0.888712i	$-0.651601\pi$
$191$	−12.6722	−0.916930	−0.458465	+	0.888712i	$0.651601\pi$
$192$	0	0
$193$	−6.20943	−0.446964	−0.223482	−	0.974708i	$-0.571742\pi$
$193$	−6.20943	−0.446964	−0.223482	+	0.974708i	$0.571742\pi$
$194$	0.158109	0.0113515
$195$	0	0
$196$	−12.4471	−0.889077
$197$	14.8505	1.05805	0.529027	−	0.848605i	$-0.322557\pi$
$197$	14.8505	1.05805	0.529027	+	0.848605i	$0.322557\pi$
$198$	0	0
$199$	15.0571	1.06737	0.533685	−	0.845683i	$-0.320807\pi$
$199$	15.0571	1.06737	0.533685	+	0.845683i	$0.320807\pi$
$200$	6.41753	0.453788
$201$	0	0
$202$	−1.41970	−0.0998899
$203$	−7.07796	−0.496775
$204$	0	0
$205$	32.1399	2.24475
$206$	−0.362807	−0.0252780
$207$	0	0
$208$	−4.85725	−0.336790
$209$	3.41820	0.236442
$210$	0	0
$211$	−14.1331	−0.972961	−0.486481	−	0.873691i	$-0.661720\pi$
$211$	−14.1331	−0.972961	−0.486481	+	0.873691i	$0.661720\pi$
$212$	−20.5461	−1.41111
$213$	0	0
$214$	−0.0587001	−0.00401266
$215$	−5.24751	−0.357878
$216$	0	0
$217$	−19.8716	−1.34897
$218$	−2.31919	−0.157076
$219$	0	0
$220$	6.66493	0.449350
$221$	2.12029	0.142626
$222$	0	0
$223$	−9.31277	−0.623629	−0.311815	−	0.950143i	$-0.600937\pi$
$223$	−9.31277	−0.623629	−0.311815	+	0.950143i	$0.600937\pi$
$224$	−7.86683	−0.525625
$225$	0	0
$226$	−2.42094	−0.161039
$227$	−0.485548	−0.0322270	−0.0161135	−	0.999870i	$-0.505129\pi$
$227$	−0.485548	−0.0322270	−0.0161135	+	0.999870i	$0.505129\pi$
$228$	0	0
$229$	15.3956	1.01737	0.508685	−	0.860953i	$-0.330132\pi$
$229$	15.3956	1.01737	0.508685	+	0.860953i	$0.330132\pi$
$230$	0.169755	0.0111933
$231$	0	0
$232$	−1.41223	−0.0927172
$233$	20.5505	1.34631	0.673154	−	0.739502i	$-0.264939\pi$
$233$	20.5505	1.34631	0.673154	+	0.739502i	$0.264939\pi$
$234$	0	0
$235$	8.14182	0.531114
$236$	−1.69546	−0.110365
$237$	0	0
$238$	1.11198	0.0720791
$239$	2.85049	0.184383	0.0921915	−	0.995741i	$-0.470613\pi$
$239$	2.85049	0.184383	0.0921915	+	0.995741i	$0.470613\pi$
$240$	0	0
$241$	11.0331	0.710707	0.355354	−	0.934732i	$-0.384360\pi$
$241$	11.0331	0.710707	0.355354	+	0.934732i	$0.384360\pi$
$242$	1.86754	0.120050
$243$	0	0
$244$	4.88097	0.312472
$245$	−23.5242	−1.50290
$246$	0	0
$247$	−4.79165	−0.304885
$248$	−3.96487	−0.251769
$249$	0	0
$250$	2.60011	0.164446
$251$	14.4565	0.912486	0.456243	−	0.889855i	$-0.349195\pi$
$251$	14.4565	0.912486	0.456243	+	0.889855i	$0.349195\pi$
$252$	0	0
$253$	0.226863	0.0142627
$254$	−0.152541	−0.00957127
$255$	0	0
$256$	13.3690	0.835563
$257$	−28.0657	−1.75069	−0.875345	−	0.483500i	$-0.839365\pi$
$257$	−28.0657	−1.75069	−0.875345	+	0.483500i	$0.839365\pi$
$258$	0	0
$259$	−16.1204	−1.00167
$260$	−9.34294	−0.579424
$261$	0	0
$262$	1.71170	0.105749
$263$	10.9878	0.677537	0.338768	−	0.940870i	$-0.389990\pi$
$263$	10.9878	0.677537	0.338768	+	0.940870i	$0.389990\pi$
$264$	0	0
$265$	−38.8308	−2.38536
$266$	−2.51297	−0.154080
$267$	0	0
$268$	26.4336	1.61469
$269$	14.6293	0.891965	0.445983	−	0.895042i	$-0.352854\pi$
$269$	14.6293	0.891965	0.445983	+	0.895042i	$0.352854\pi$
$270$	0	0
$271$	−31.9043	−1.93805	−0.969025	−	0.246964i	$-0.920567\pi$
$271$	−31.9043	−1.93805	−0.969025	+	0.246964i	$0.920567\pi$
$272$	−6.29915	−0.381942
$273$	0	0
$274$	0.785384	0.0474468
$275$	8.03556	0.484562
$276$	0	0
$277$	14.7225	0.884588	0.442294	−	0.896870i	$-0.354165\pi$
$277$	14.7225	0.884588	0.442294	+	0.896870i	$0.354165\pi$
$278$	−2.17687	−0.130560
$279$	0	0
$280$	−9.88384	−0.590672
$281$	−27.3492	−1.63152	−0.815758	−	0.578394i	$-0.803680\pi$
$281$	−27.3492	−1.63152	−0.815758	+	0.578394i	$0.803680\pi$
$282$	0	0
$283$	26.2831	1.56237	0.781183	−	0.624302i	$-0.214616\pi$
$283$	26.2831	1.56237	0.781183	+	0.624302i	$0.214616\pi$
$284$	17.2724	1.02493
$285$	0	0
$286$	0.214216	0.0126668
$287$	−31.5774	−1.86395
$288$	0	0
$289$	−14.2503	−0.838253
$290$	−1.32316	−0.0776985
$291$	0	0
$292$	28.2280	1.65192
$293$	−16.2252	−0.947884	−0.473942	−	0.880556i	$-0.657169\pi$
$293$	−16.2252	−0.947884	−0.473942	+	0.880556i	$0.657169\pi$
$294$	0	0
$295$	−3.20432	−0.186563
$296$	−3.21642	−0.186951
$297$	0	0
$298$	−3.90879	−0.226430
$299$	−0.318017	−0.0183914
$300$	0	0
$301$	5.15567	0.297168
$302$	−0.183668	−0.0105689
$303$	0	0
$304$	14.2355	0.816460
$305$	9.22471	0.528205
$306$	0	0
$307$	0.0943867	0.00538693	0.00269347	−	0.999996i	$-0.499143\pi$
$307$	0.0943867	0.00538693	0.00269347	+	0.999996i	$0.499143\pi$
$308$	−6.54828	−0.373123
$309$	0	0
$310$	−3.71480	−0.210987
$311$	13.8783	0.786968	0.393484	−	0.919332i	$-0.371270\pi$
$311$	13.8783	0.786968	0.393484	+	0.919332i	$0.371270\pi$
$312$	0	0
$313$	−13.4879	−0.762379	−0.381190	−	0.924497i	$-0.624486\pi$
$313$	−13.4879	−0.762379	−0.381190	+	0.924497i	$0.624486\pi$
$314$	2.90485	0.163930
$315$	0	0
$316$	19.9811	1.12403
$317$	2.90664	0.163253	0.0816267	−	0.996663i	$-0.473988\pi$
$317$	2.90664	0.163253	0.0816267	+	0.996663i	$0.473988\pi$
$318$	0	0
$319$	−1.76829	−0.0990050
$320$	26.7624	1.49607
$321$	0	0
$322$	−0.166784	−0.00929449
$323$	−6.21406	−0.345760
$324$	0	0
$325$	−11.2643	−0.624830
$326$	2.40995	0.133475
$327$	0	0
$328$	−6.30046	−0.347885
$329$	−7.99931	−0.441016
$330$	0	0
$331$	−11.1032	−0.610288	−0.305144	−	0.952306i	$-0.598705\pi$
$331$	−11.1032	−0.610288	−0.305144	+	0.952306i	$0.598705\pi$
$332$	13.8833	0.761945
$333$	0	0
$334$	3.70808	0.202897
$335$	49.9578	2.72948
$336$	0	0
$337$	−0.880266	−0.0479512	−0.0239756	−	0.999713i	$-0.507632\pi$
$337$	−0.880266	−0.0479512	−0.0239756	+	0.999713i	$0.507632\pi$
$338$	2.08740	0.113540
$339$	0	0
$340$	−12.1164	−0.657105
$341$	−4.96452	−0.268844
$342$	0	0
$343$	−2.44508	−0.132022
$344$	1.02868	0.0554628
$345$	0	0
$346$	0.123430	0.00663565
$347$	−34.5128	−1.85275	−0.926373	−	0.376608i	$-0.877090\pi$
$347$	−34.5128	−1.85275	−0.926373	+	0.376608i	$0.877090\pi$
$348$	0	0
$349$	−28.8423	−1.54389	−0.771947	−	0.635687i	$-0.780717\pi$
$349$	−28.8423	−1.54389	−0.771947	+	0.635687i	$0.780717\pi$
$350$	−5.90753	−0.315771
$351$	0	0
$352$	−1.96537	−0.104755
$353$	−23.9585	−1.27518	−0.637591	−	0.770375i	$-0.720069\pi$
$353$	−23.9585	−1.27518	−0.637591	+	0.770375i	$0.720069\pi$
$354$	0	0
$355$	32.6437	1.73255
$356$	23.1567	1.22730
$357$	0	0
$358$	−0.380797	−0.0201258
$359$	1.51542	0.0799807	0.0399904	−	0.999200i	$-0.487267\pi$
$359$	1.51542	0.0799807	0.0399904	+	0.999200i	$0.487267\pi$
$360$	0	0
$361$	−4.95681	−0.260885
$362$	−0.511463	−0.0268819
$363$	0	0
$364$	9.17940	0.481131
$365$	53.3492	2.79242
$366$	0	0
$367$	−7.67846	−0.400812	−0.200406	−	0.979713i	$-0.564226\pi$
$367$	−7.67846	−0.400812	−0.200406	+	0.979713i	$0.564226\pi$
$368$	0.944794	0.0492508
$369$	0	0
$370$	−3.01356	−0.156668
$371$	38.1511	1.98071
$372$	0	0
$373$	23.5390	1.21880	0.609402	−	0.792861i	$-0.291409\pi$
$373$	23.5390	1.21880	0.609402	+	0.792861i	$0.291409\pi$
$374$	0.277806	0.0143650
$375$	0	0
$376$	−1.59606	−0.0823105
$377$	2.47879	0.127664
$378$	0	0
$379$	−23.0030	−1.18159	−0.590793	−	0.806823i	$-0.701185\pi$
$379$	−23.0030	−1.18159	−0.590793	+	0.806823i	$0.701185\pi$
$380$	27.3819	1.40466
$381$	0	0
$382$	2.32749	0.119085
$383$	−20.8953	−1.06770	−0.533851	−	0.845579i	$-0.679256\pi$
$383$	−20.8953	−1.06770	−0.533851	+	0.845579i	$0.679256\pi$
$384$	0	0
$385$	−12.3758	−0.630730
$386$	1.14048	0.0580487
$387$	0	0
$388$	1.69264	0.0859306
$389$	−15.5215	−0.786971	−0.393486	−	0.919331i	$-0.628731\pi$
$389$	−15.5215	−0.786971	−0.393486	+	0.919331i	$0.628731\pi$
$390$	0	0
$391$	−0.412422	−0.0208571
$392$	4.61149	0.232916
$393$	0	0
$394$	−2.72757	−0.137413
$395$	37.7630	1.90006
$396$	0	0
$397$	−32.1277	−1.61244	−0.806222	−	0.591614i	$-0.798491\pi$
$397$	−32.1277	−1.61244	−0.806222	+	0.591614i	$0.798491\pi$
$398$	−2.76552	−0.138623
$399$	0	0
$400$	33.4650	1.67325
$401$	−1.10536	−0.0551989	−0.0275994	−	0.999619i	$-0.508786\pi$
$401$	−1.10536	−0.0551989	−0.0275994	+	0.999619i	$0.508786\pi$
$402$	0	0
$403$	6.95928	0.346667
$404$	−15.1986	−0.756161
$405$	0	0
$406$	1.30000	0.0645178
$407$	−4.02736	−0.199629
$408$	0	0
$409$	30.4605	1.50617	0.753086	−	0.657922i	$-0.228564\pi$
$409$	30.4605	1.50617	0.753086	+	0.657922i	$0.228564\pi$
$410$	−5.90309	−0.291533
$411$	0	0
$412$	−3.88404	−0.191353
$413$	3.14823	0.154914
$414$	0	0
$415$	26.2385	1.28800
$416$	2.75506	0.135078
$417$	0	0
$418$	−0.627816	−0.0307075
$419$	−12.4139	−0.606459	−0.303230	−	0.952917i	$-0.598065\pi$
$419$	−12.4139	−0.606459	−0.303230	+	0.952917i	$0.598065\pi$
$420$	0	0
$421$	14.3449	0.699126	0.349563	−	0.936913i	$-0.386330\pi$
$421$	14.3449	0.699126	0.349563	+	0.936913i	$0.386330\pi$
$422$	2.59580	0.126362
$423$	0	0
$424$	7.61208	0.369675
$425$	−14.6081	−0.708598
$426$	0	0
$427$	−9.06324	−0.438601
$428$	−0.628415	−0.0303756
$429$	0	0
$430$	0.963803	0.0464787
$431$	40.0541	1.92934	0.964668	−	0.263468i	$-0.0848662\pi$
$431$	40.0541	1.92934	0.964668	+	0.263468i	$0.0848662\pi$
$432$	0	0
$433$	30.1463	1.44874	0.724369	−	0.689412i	$-0.242131\pi$
$433$	30.1463	1.44874	0.724369	+	0.689412i	$0.242131\pi$
$434$	3.64978	0.175195
$435$	0	0
$436$	−24.8282	−1.18905
$437$	0.932033	0.0445852
$438$	0	0
$439$	18.4456	0.880359	0.440179	−	0.897910i	$-0.354915\pi$
$439$	18.4456	0.880359	0.440179	+	0.897910i	$0.354915\pi$
$440$	−2.46928	−0.117718
$441$	0	0
$442$	−0.389430	−0.0185233
$443$	−4.84442	−0.230165	−0.115083	−	0.993356i	$-0.536713\pi$
$443$	−4.84442	−0.230165	−0.115083	+	0.993356i	$0.536713\pi$
$444$	0	0
$445$	43.7647	2.07465
$446$	1.71046	0.0809928
$447$	0	0
$448$	−26.2940	−1.24227
$449$	21.8751	1.03235	0.516174	−	0.856484i	$-0.327356\pi$
$449$	21.8751	1.03235	0.516174	+	0.856484i	$0.327356\pi$
$450$	0	0
$451$	−7.88897	−0.371477
$452$	−25.9174	−1.21905
$453$	0	0
$454$	0.0891800	0.00418542
$455$	17.3485	0.813309
$456$	0	0
$457$	0.383056	0.0179186	0.00895930	−	0.999960i	$-0.497148\pi$
$457$	0.383056	0.0179186	0.00895930	+	0.999960i	$0.497148\pi$
$458$	−2.82768	−0.132129
$459$	0	0
$460$	1.81731	0.0847327
$461$	−34.9368	−1.62717	−0.813585	−	0.581447i	$-0.802487\pi$
$461$	−34.9368	−1.62717	−0.813585	+	0.581447i	$0.802487\pi$
$462$	0	0
$463$	13.6404	0.633925	0.316963	−	0.948438i	$-0.397337\pi$
$463$	13.6404	0.633925	0.316963	+	0.948438i	$0.397337\pi$
$464$	−7.36422	−0.341875
$465$	0	0
$466$	−3.77448	−0.174849
$467$	30.3922	1.40638	0.703192	−	0.711000i	$-0.251757\pi$
$467$	30.3922	1.40638	0.703192	+	0.711000i	$0.251757\pi$
$468$	0	0
$469$	−49.0833	−2.26646
$470$	−1.49540	−0.0689775
$471$	0	0
$472$	0.628149	0.0289129
$473$	1.28804	0.0592241
$474$	0	0
$475$	33.0130	1.51474
$476$	11.9043	0.545634
$477$	0	0
$478$	−0.523545	−0.0239464
$479$	23.2203	1.06096	0.530482	−	0.847696i	$-0.322011\pi$
$479$	23.2203	1.06096	0.530482	+	0.847696i	$0.322011\pi$
$480$	0	0
$481$	5.64558	0.257416
$482$	−2.02644	−0.0923019
$483$	0	0
$484$	19.9930	0.908771
$485$	3.19897	0.145258
$486$	0	0
$487$	23.9343	1.08457	0.542284	−	0.840195i	$-0.317560\pi$
$487$	23.9343	1.08457	0.542284	+	0.840195i	$0.317560\pi$
$488$	−1.80834	−0.0818597
$489$	0	0
$490$	4.32065	0.195187
$491$	−14.5262	−0.655557	−0.327778	−	0.944755i	$-0.606300\pi$
$491$	−14.5262	−0.655557	−0.327778	+	0.944755i	$0.606300\pi$
$492$	0	0
$493$	3.21463	0.144780
$494$	0.880075	0.0395964
$495$	0	0
$496$	−20.6753	−0.928347
$497$	−32.0723	−1.43864
$498$	0	0
$499$	19.6649	0.880322	0.440161	−	0.897919i	$-0.354921\pi$
$499$	19.6649	0.880322	0.440161	+	0.897919i	$0.354921\pi$
$500$	27.8356	1.24484
$501$	0	0
$502$	−2.65520	−0.118508
$503$	−7.56135	−0.337144	−0.168572	−	0.985689i	$-0.553916\pi$
$503$	−7.56135	−0.337144	−0.168572	+	0.985689i	$0.553916\pi$
$504$	0	0
$505$	−28.7244	−1.27822
$506$	−0.0416675	−0.00185235
$507$	0	0
$508$	−1.63303	−0.0724540
$509$	4.07748	0.180731	0.0903656	−	0.995909i	$-0.471196\pi$
$509$	4.07748	0.180731	0.0903656	+	0.995909i	$0.471196\pi$
$510$	0	0
$511$	−52.4154	−2.31872
$512$	−13.7196	−0.606326
$513$	0	0
$514$	5.15478	0.227368
$515$	−7.34058	−0.323465
$516$	0	0
$517$	−1.99847	−0.0878925
$518$	2.96081	0.130091
$519$	0	0
$520$	3.46145	0.151794
$521$	33.7144	1.47706	0.738528	−	0.674223i	$-0.235521\pi$
$521$	33.7144	1.47706	0.738528	+	0.674223i	$0.235521\pi$
$522$	0	0
$523$	−36.7350	−1.60631	−0.803154	−	0.595772i	$-0.796846\pi$
$523$	−36.7350	−1.60631	−0.803154	+	0.595772i	$0.796846\pi$
$524$	18.3246	0.800513
$525$	0	0
$526$	−2.01811	−0.0879939
$527$	9.02517	0.393142
$528$	0	0
$529$	−22.9381	−0.997311
$530$	7.13199	0.309794
$531$	0	0
$532$	−26.9027	−1.16638
$533$	11.0588	0.479010
$534$	0	0
$535$	−1.18766	−0.0513472
$536$	−9.79333	−0.423007
$537$	0	0
$538$	−2.68694	−0.115842
$539$	5.77417	0.248711
$540$	0	0
$541$	17.3851	0.747442	0.373721	−	0.927541i	$-0.378082\pi$
$541$	17.3851	0.747442	0.373721	+	0.927541i	$0.378082\pi$
$542$	5.85982	0.251701
$543$	0	0
$544$	3.57291	0.153187
$545$	−46.9236	−2.00999
$546$	0	0
$547$	27.2857	1.16665	0.583326	−	0.812238i	$-0.301751\pi$
$547$	27.2857	1.16665	0.583326	+	0.812238i	$0.301751\pi$
$548$	8.40795	0.359170
$549$	0	0
$550$	−1.47588	−0.0629317
$551$	−7.26475	−0.309489
$552$	0	0
$553$	−37.1020	−1.57774
$554$	−2.70406	−0.114884
$555$	0	0
$556$	−23.3045	−0.988333
$557$	−22.0118	−0.932669	−0.466335	−	0.884608i	$-0.654426\pi$
$557$	−22.0118	−0.932669	−0.466335	+	0.884608i	$0.654426\pi$
$558$	0	0
$559$	−1.80558	−0.0763679
$560$	−51.5404	−2.17798
$561$	0	0
$562$	5.02318	0.211890
$563$	34.7867	1.46609	0.733043	−	0.680183i	$-0.238100\pi$
$563$	34.7867	1.46609	0.733043	+	0.680183i	$0.238100\pi$
$564$	0	0
$565$	−48.9822	−2.06070
$566$	−4.82737	−0.202910
$567$	0	0
$568$	−6.39922	−0.268505
$569$	−4.70112	−0.197081	−0.0985407	−	0.995133i	$-0.531417\pi$
$569$	−4.70112	−0.197081	−0.0985407	+	0.995133i	$0.531417\pi$
$570$	0	0
$571$	−10.1822	−0.426112	−0.213056	−	0.977040i	$-0.568342\pi$
$571$	−10.1822	−0.426112	−0.213056	+	0.977040i	$0.568342\pi$
$572$	2.29329	0.0958873
$573$	0	0
$574$	5.79977	0.242078
$575$	2.19104	0.0913726
$576$	0	0
$577$	20.5479	0.855422	0.427711	−	0.903916i	$-0.359320\pi$
$577$	20.5479	0.855422	0.427711	+	0.903916i	$0.359320\pi$
$578$	2.61733	0.108867
$579$	0	0
$580$	−14.1651	−0.588173
$581$	−25.7793	−1.06950
$582$	0	0
$583$	9.53129	0.394746
$584$	−10.4582	−0.432761
$585$	0	0
$586$	2.98005	0.123105
$587$	−8.40929	−0.347088	−0.173544	−	0.984826i	$-0.555522\pi$
$587$	−8.40929	−0.347088	−0.173544	+	0.984826i	$0.555522\pi$
$588$	0	0
$589$	−20.3960	−0.840403
$590$	0.588532	0.0242295
$591$	0	0
$592$	−16.7724	−0.689341
$593$	7.48556	0.307395	0.153697	−	0.988118i	$-0.450882\pi$
$593$	7.48556	0.307395	0.153697	+	0.988118i	$0.450882\pi$
$594$	0	0
$595$	22.4984	0.922345
$596$	−41.8456	−1.71406
$597$	0	0
$598$	0.0584097	0.00238855
$599$	−4.69324	−0.191760	−0.0958802	−	0.995393i	$-0.530567\pi$
$599$	−4.69324	−0.191760	−0.0958802	+	0.995393i	$0.530567\pi$
$600$	0	0
$601$	−1.89605	−0.0773416	−0.0386708	−	0.999252i	$-0.512312\pi$
$601$	−1.89605	−0.0773416	−0.0386708	+	0.999252i	$0.512312\pi$
$602$	−0.946933	−0.0385941
$603$	0	0
$604$	−1.96627	−0.0800062
$605$	37.7854	1.53620
$606$	0	0
$607$	−19.6483	−0.797501	−0.398751	−	0.917059i	$-0.630556\pi$
$607$	−19.6483	−0.797501	−0.398751	+	0.917059i	$0.630556\pi$
$608$	−8.07444	−0.327462
$609$	0	0
$610$	−1.69429	−0.0685997
$611$	2.80146	0.113335
$612$	0	0
$613$	−17.3399	−0.700352	−0.350176	−	0.936684i	$-0.613878\pi$
$613$	−17.3399	−0.700352	−0.350176	+	0.936684i	$0.613878\pi$
$614$	−0.0173359	−0.000699619 0
$615$	0	0
$616$	2.42606	0.0977487
$617$	−6.60055	−0.265728	−0.132864	−	0.991134i	$-0.542417\pi$
$617$	−6.60055	−0.265728	−0.132864	+	0.991134i	$0.542417\pi$
$618$	0	0
$619$	−6.67297	−0.268209	−0.134105	−	0.990967i	$-0.542816\pi$
$619$	−6.67297	−0.268209	−0.134105	+	0.990967i	$0.542816\pi$
$620$	−39.7689	−1.59716
$621$	0	0
$622$	−2.54901	−0.102206
$623$	−42.9987	−1.72271
$624$	0	0
$625$	8.55986	0.342394
$626$	2.47730	0.0990127
$627$	0	0
$628$	31.0979	1.24094
$629$	7.32149	0.291927
$630$	0	0
$631$	16.7963	0.668651	0.334325	−	0.942458i	$-0.391492\pi$
$631$	16.7963	0.668651	0.334325	+	0.942458i	$0.391492\pi$
$632$	−7.40276	−0.294466
$633$	0	0
$634$	−0.533859	−0.0212022
$635$	−3.08632	−0.122477
$636$	0	0
$637$	−8.09426	−0.320706
$638$	0.324778	0.0128581
$639$	0	0
$640$	−20.9294	−0.827306
$641$	−20.5409	−0.811316	−0.405658	−	0.914025i	$-0.632958\pi$
$641$	−20.5409	−0.811316	−0.405658	+	0.914025i	$0.632958\pi$
$642$	0	0
$643$	1.06457	0.0419825	0.0209913	−	0.999780i	$-0.493318\pi$
$643$	1.06457	0.0419825	0.0209913	+	0.999780i	$0.493318\pi$
$644$	−1.78550	−0.0703587
$645$	0	0
$646$	1.14133	0.0449049
$647$	−29.6687	−1.16640	−0.583198	−	0.812330i	$-0.698199\pi$
$647$	−29.6687	−1.16640	−0.583198	+	0.812330i	$0.698199\pi$
$648$	0	0
$649$	0.786522	0.0308737
$650$	2.06889	0.0811487
$651$	0	0
$652$	25.7998	1.01040
$653$	20.1399	0.788137	0.394069	−	0.919081i	$-0.371067\pi$
$653$	20.1399	0.788137	0.394069	+	0.919081i	$0.371067\pi$
$654$	0	0
$655$	34.6323	1.35319
$656$	−32.8545	−1.28275
$657$	0	0
$658$	1.46922	0.0572762
$659$	−36.5416	−1.42346	−0.711730	−	0.702453i	$-0.752088\pi$
$659$	−36.5416	−1.42346	−0.711730	+	0.702453i	$0.752088\pi$
$660$	0	0
$661$	22.1043	0.859758	0.429879	−	0.902886i	$-0.358556\pi$
$661$	22.1043	0.859758	0.429879	+	0.902886i	$0.358556\pi$
$662$	2.03931	0.0792600
$663$	0	0
$664$	−5.14360	−0.199610
$665$	−50.8443	−1.97166
$666$	0	0
$667$	−0.482155	−0.0186691
$668$	39.6969	1.53592
$669$	0	0
$670$	−9.17567	−0.354487
$671$	−2.26427	−0.0874111
$672$	0	0
$673$	30.5471	1.17750	0.588751	−	0.808314i	$-0.299620\pi$
$673$	30.5471	1.17750	0.588751	+	0.808314i	$0.299620\pi$
$674$	0.161677	0.00622757
$675$	0	0
$676$	22.3467	0.859489
$677$	−24.5063	−0.941854	−0.470927	−	0.882172i	$-0.656080\pi$
$677$	−24.5063	−0.941854	−0.470927	+	0.882172i	$0.656080\pi$
$678$	0	0
$679$	−3.14298	−0.120616
$680$	4.48899	0.172145
$681$	0	0
$682$	0.911825	0.0349156
$683$	36.8637	1.41055	0.705275	−	0.708933i	$-0.250823\pi$
$683$	36.8637	1.41055	0.705275	+	0.708933i	$0.250823\pi$
$684$	0	0
$685$	15.8905	0.607144
$686$	0.449085	0.0171461
$687$	0	0
$688$	5.36418	0.204507
$689$	−13.3610	−0.509014
$690$	0	0
$691$	−12.4203	−0.472492	−0.236246	−	0.971693i	$-0.575917\pi$
$691$	−12.4203	−0.472492	−0.236246	+	0.971693i	$0.575917\pi$
$692$	1.32139	0.0502315
$693$	0	0
$694$	6.33892	0.240622
$695$	−44.0441	−1.67069
$696$	0	0
$697$	14.3416	0.543228
$698$	5.29743	0.200511
$699$	0	0
$700$	−63.2432	−2.39037
$701$	−28.1266	−1.06233	−0.531164	−	0.847269i	$-0.678245\pi$
$701$	−28.1266	−1.06233	−0.531164	+	0.847269i	$0.678245\pi$
$702$	0	0
$703$	−16.5459	−0.624039
$704$	−6.56903	−0.247579
$705$	0	0
$706$	4.40042	0.165612
$707$	28.2217	1.06138
$708$	0	0
$709$	1.43051	0.0537241	0.0268620	−	0.999639i	$-0.491449\pi$
$709$	1.43051	0.0537241	0.0268620	+	0.999639i	$0.491449\pi$
$710$	−5.99562	−0.225012
$711$	0	0
$712$	−8.57929	−0.321523
$713$	−1.35366	−0.0506951
$714$	0	0
$715$	4.33417	0.162089
$716$	−4.07663	−0.152351
$717$	0	0
$718$	−0.278335	−0.0103874
$719$	−16.8995	−0.630244	−0.315122	−	0.949051i	$-0.602045\pi$
$719$	−16.8995	−0.630244	−0.315122	+	0.949051i	$0.602045\pi$
$720$	0	0
$721$	7.21209	0.268592
$722$	0.910409	0.0338819
$723$	0	0
$724$	−5.47547	−0.203494
$725$	−17.0781	−0.634265
$726$	0	0
$727$	26.9422	0.999233	0.499616	−	0.866247i	$-0.333474\pi$
$727$	26.9422	0.999233	0.499616	+	0.866247i	$0.333474\pi$
$728$	−3.40086	−0.126044
$729$	0	0
$730$	−9.79856	−0.362661
$731$	−2.34157	−0.0866062
$732$	0	0
$733$	−9.89979	−0.365657	−0.182829	−	0.983145i	$-0.558525\pi$
$733$	−9.89979	−0.365657	−0.182829	+	0.983145i	$0.558525\pi$
$734$	1.41029	0.0520548
$735$	0	0
$736$	−0.535893	−0.0197533
$737$	−12.2625	−0.451694
$738$	0	0
$739$	36.7085	1.35034	0.675171	−	0.737661i	$-0.264070\pi$
$739$	36.7085	1.35034	0.675171	+	0.737661i	$0.264070\pi$
$740$	−32.2617	−1.18596
$741$	0	0
$742$	−7.00716	−0.257241
$743$	40.2612	1.47704	0.738521	−	0.674231i	$-0.235525\pi$
$743$	40.2612	1.47704	0.738521	+	0.674231i	$0.235525\pi$
$744$	0	0
$745$	−79.0854	−2.89746
$746$	−4.32338	−0.158290
$747$	0	0
$748$	2.97406	0.108742
$749$	1.16688	0.0426367
$750$	0	0
$751$	20.0555	0.731836	0.365918	−	0.930647i	$-0.380755\pi$
$751$	20.0555	0.731836	0.365918	+	0.930647i	$0.380755\pi$
$752$	−8.32284	−0.303503
$753$	0	0
$754$	−0.455276	−0.0165802
$755$	−3.71611	−0.135243
$756$	0	0
$757$	38.8419	1.41173	0.705867	−	0.708345i	$-0.250558\pi$
$757$	38.8419	1.41173	0.705867	+	0.708345i	$0.250558\pi$
$758$	4.22493	0.153456
$759$	0	0
$760$	−10.1447	−0.367986
$761$	−9.26836	−0.335978	−0.167989	−	0.985789i	$-0.553727\pi$
$761$	−9.26836	−0.335978	−0.167989	+	0.985789i	$0.553727\pi$
$762$	0	0
$763$	46.1023	1.66901
$764$	24.9170	0.901464
$765$	0	0
$766$	3.83781	0.138666
$767$	−1.10255	−0.0398108
$768$	0	0
$769$	−8.24686	−0.297389	−0.148695	−	0.988883i	$-0.547507\pi$
$769$	−8.24686	−0.297389	−0.148695	+	0.988883i	$0.547507\pi$
$770$	2.27305	0.0819149
$771$	0	0
$772$	12.2094	0.439425
$773$	17.9148	0.644352	0.322176	−	0.946680i	$-0.395586\pi$
$773$	17.9148	0.644352	0.322176	+	0.946680i	$0.395586\pi$
$774$	0	0
$775$	−47.9473	−1.72232
$776$	−0.627102	−0.0225116
$777$	0	0
$778$	2.85081	0.102206
$779$	−32.4107	−1.16123
$780$	0	0
$781$	−8.01263	−0.286714
$782$	0.0757488	0.00270877
$783$	0	0
$784$	24.0472	0.858827
$785$	58.7730	2.09770
$786$	0	0
$787$	37.4489	1.33491	0.667454	−	0.744651i	$-0.267384\pi$
$787$	37.4489	1.33491	0.667454	+	0.744651i	$0.267384\pi$
$788$	−29.2000	−1.04021
$789$	0	0
$790$	−6.93587	−0.246767
$791$	48.1249	1.71112
$792$	0	0
$793$	3.17406	0.112714
$794$	5.90085	0.209413
$795$	0	0
$796$	−29.6063	−1.04937
$797$	6.56429	0.232519	0.116259	−	0.993219i	$-0.462910\pi$
$797$	6.56429	0.232519	0.116259	+	0.993219i	$0.462910\pi$
$798$	0	0
$799$	3.63308	0.128529
$800$	−18.9815	−0.671098
$801$	0	0
$802$	0.203019	0.00716886
$803$	−13.0949	−0.462110
$804$	0	0
$805$	−3.37449	−0.118935
$806$	−1.27820	−0.0450227
$807$	0	0
$808$	5.63091	0.198095
$809$	−12.8394	−0.451409	−0.225704	−	0.974196i	$-0.572468\pi$
$809$	−12.8394	−0.451409	−0.225704	+	0.974196i	$0.572468\pi$
$810$	0	0
$811$	−1.31098	−0.0460347	−0.0230174	−	0.999735i	$-0.507327\pi$
$811$	−1.31098	−0.0460347	−0.0230174	+	0.999735i	$0.507327\pi$
$812$	13.9171	0.488396
$813$	0	0
$814$	0.739700	0.0259265
$815$	48.7598	1.70798
$816$	0	0
$817$	5.29173	0.185134
$818$	−5.59463	−0.195612
$819$	0	0
$820$	−63.1956	−2.20689
$821$	46.1086	1.60920	0.804600	−	0.593817i	$-0.202380\pi$
$821$	46.1086	1.60920	0.804600	+	0.593817i	$0.202380\pi$
$822$	0	0
$823$	6.36527	0.221879	0.110940	−	0.993827i	$-0.464614\pi$
$823$	6.36527	0.221879	0.110940	+	0.993827i	$0.464614\pi$
$824$	1.43899	0.0501296
$825$	0	0
$826$	−0.578231	−0.0201192
$827$	−18.4051	−0.640009	−0.320004	−	0.947416i	$-0.603684\pi$
$827$	−18.4051	−0.640009	−0.320004	+	0.947416i	$0.603684\pi$
$828$	0	0
$829$	0.214845	0.00746186	0.00373093	−	0.999993i	$-0.498812\pi$
$829$	0.214845	0.00746186	0.00373093	+	0.999993i	$0.498812\pi$
$830$	−4.81919	−0.167277
$831$	0	0
$832$	9.20849	0.319247
$833$	−10.4971	−0.363702
$834$	0	0
$835$	75.0245	2.59633
$836$	−6.72109	−0.232454
$837$	0	0
$838$	2.28004	0.0787628
$839$	−8.66817	−0.299259	−0.149629	−	0.988742i	$-0.547808\pi$
$839$	−8.66817	−0.299259	−0.149629	+	0.988742i	$0.547808\pi$
$840$	0	0
$841$	−25.2418	−0.870408
$842$	−2.63470	−0.0907977
$843$	0	0
$844$	27.7894	0.956550
$845$	42.2338	1.45289
$846$	0	0
$847$	−37.1240	−1.27560
$848$	39.6941	1.36310
$849$	0	0
$850$	2.68305	0.0920279
$851$	−1.09813	−0.0376435
$852$	0	0
$853$	−22.5241	−0.771211	−0.385605	−	0.922664i	$-0.626007\pi$
$853$	−22.5241	−0.771211	−0.385605	+	0.922664i	$0.626007\pi$
$854$	1.66463	0.0569625
$855$	0	0
$856$	0.232820	0.00795764
$857$	45.0950	1.54041	0.770207	−	0.637794i	$-0.220153\pi$
$857$	45.0950	1.54041	0.770207	+	0.637794i	$0.220153\pi$
$858$	0	0
$859$	−21.0436	−0.718000	−0.359000	−	0.933338i	$-0.616882\pi$
$859$	−21.0436	−0.718000	−0.359000	+	0.933338i	$0.616882\pi$
$860$	10.3180	0.351841
$861$	0	0
$862$	−7.35667	−0.250569
$863$	2.01110	0.0684587	0.0342293	−	0.999414i	$-0.489102\pi$
$863$	2.01110	0.0684587	0.0342293	+	0.999414i	$0.489102\pi$
$864$	0	0
$865$	2.49733	0.0849118
$866$	−5.53693	−0.188152
$867$	0	0
$868$	39.0728	1.32622
$869$	−9.26919	−0.314436
$870$	0	0
$871$	17.1896	0.582448
$872$	9.19854	0.311502
$873$	0	0
$874$	−0.171185	−0.00579042
$875$	−51.6866	−1.74733
$876$	0	0
$877$	0.542916	0.0183330	0.00916648	−	0.999958i	$-0.497082\pi$
$877$	0.542916	0.0183330	0.00916648	+	0.999958i	$0.497082\pi$
$878$	−3.38787	−0.114335
$879$	0	0
$880$	−12.8763	−0.434061
$881$	8.69333	0.292886	0.146443	−	0.989219i	$-0.453218\pi$
$881$	8.69333	0.292886	0.146443	+	0.989219i	$0.453218\pi$
$882$	0	0
$883$	35.6778	1.20065	0.600327	−	0.799755i	$-0.295037\pi$
$883$	35.6778	1.20065	0.600327	+	0.799755i	$0.295037\pi$
$884$	−4.16905	−0.140220
$885$	0	0
$886$	0.889767	0.0298923
$887$	−13.8237	−0.464154	−0.232077	−	0.972697i	$-0.574552\pi$
$887$	−13.8237	−0.464154	−0.232077	+	0.972697i	$0.574552\pi$
$888$	0	0
$889$	3.03230	0.101700
$890$	−8.03820	−0.269441
$891$	0	0
$892$	18.3114	0.613111
$893$	−8.21042	−0.274751
$894$	0	0
$895$	−7.70457	−0.257535
$896$	20.5630	0.686963
$897$	0	0
$898$	−4.01776	−0.134074
$899$	10.5512	0.351901
$900$	0	0
$901$	−17.3273	−0.577255
$902$	1.44896	0.0482449
$903$	0	0
$904$	9.60209	0.319361
$905$	−10.3483	−0.343989
$906$	0	0
$907$	3.22925	0.107225	0.0536127	−	0.998562i	$-0.482926\pi$
$907$	3.22925	0.107225	0.0536127	+	0.998562i	$0.482926\pi$
$908$	0.954717	0.0316834
$909$	0	0
$910$	−3.18637	−0.105627
$911$	36.7510	1.21761	0.608807	−	0.793318i	$-0.291648\pi$
$911$	36.7510	1.21761	0.608807	+	0.793318i	$0.291648\pi$
$912$	0	0
$913$	−6.44043	−0.213147
$914$	−0.0703553	−0.00232715
$915$	0	0
$916$	−30.2718	−1.00021
$917$	−34.0261	−1.12364
$918$	0	0
$919$	−50.3548	−1.66105	−0.830526	−	0.556980i	$-0.811960\pi$
$919$	−50.3548	−1.66105	−0.830526	+	0.556980i	$0.811960\pi$
$920$	−0.673293	−0.0221978
$921$	0	0
$922$	6.41679	0.211326
$923$	11.2321	0.369710
$924$	0	0
$925$	−38.8963	−1.27890
$926$	−2.50532	−0.0823299
$927$	0	0
$928$	4.17703	0.137118
$929$	−49.9006	−1.63719	−0.818593	−	0.574374i	$-0.805245\pi$
$929$	−49.9006	−1.63719	−0.818593	+	0.574374i	$0.805245\pi$
$930$	0	0
$931$	23.7224	0.777469
$932$	−40.4078	−1.32360
$933$	0	0
$934$	−5.58209	−0.182652
$935$	5.62078	0.183819
$936$	0	0
$937$	−39.6375	−1.29490	−0.647451	−	0.762107i	$-0.724165\pi$
$937$	−39.6375	−1.29490	−0.647451	+	0.762107i	$0.724165\pi$
$938$	9.01506	0.294352
$939$	0	0
$940$	−16.0090	−0.522156
$941$	−25.3577	−0.826637	−0.413318	−	0.910587i	$-0.635630\pi$
$941$	−25.3577	−0.826637	−0.413318	+	0.910587i	$0.635630\pi$
$942$	0	0
$943$	−2.15107	−0.0700484
$944$	3.27556	0.106610
$945$	0	0
$946$	−0.236572	−0.00769163
$947$	−5.89253	−0.191481	−0.0957407	−	0.995406i	$-0.530522\pi$
$947$	−5.89253	−0.191481	−0.0957407	+	0.995406i	$0.530522\pi$
$948$	0	0
$949$	18.3565	0.595878
$950$	−6.06344	−0.196724
$951$	0	0
$952$	−4.41041	−0.142942
$953$	−10.0351	−0.325070	−0.162535	−	0.986703i	$-0.551967\pi$
$953$	−10.0351	−0.325070	−0.162535	+	0.986703i	$0.551967\pi$
$954$	0	0
$955$	47.0914	1.52384
$956$	−5.60482	−0.181273
$957$	0	0
$958$	−4.26484	−0.137791
$959$	−15.6123	−0.504149
$960$	0	0
$961$	−1.37730	−0.0444291
$962$	−1.03691	−0.0334315
$963$	0	0
$964$	−21.6941	−0.698720
$965$	23.0749	0.742809
$966$	0	0
$967$	−45.6262	−1.46724	−0.733620	−	0.679560i	$-0.762171\pi$
$967$	−45.6262	−1.46724	−0.733620	+	0.679560i	$0.762171\pi$
$968$	−7.40716	−0.238075
$969$	0	0
$970$	−0.587550	−0.0188651
$971$	−50.0563	−1.60638	−0.803191	−	0.595721i	$-0.796866\pi$
$971$	−50.0563	−1.60638	−0.803191	+	0.595721i	$0.796866\pi$
$972$	0	0
$973$	43.2731	1.38727
$974$	−4.39598	−0.140856
$975$	0	0
$976$	−9.42980	−0.301840
$977$	−16.8418	−0.538817	−0.269409	−	0.963026i	$-0.586828\pi$
$977$	−16.8418	−0.538817	−0.269409	+	0.963026i	$0.586828\pi$
$978$	0	0
$979$	−10.7424	−0.343327
$980$	46.2547	1.47755
$981$	0	0
$982$	2.66800	0.0851393
$983$	54.2760	1.73114	0.865568	−	0.500792i	$-0.166958\pi$
$983$	54.2760	1.73114	0.865568	+	0.500792i	$0.166958\pi$
$984$	0	0
$985$	−55.1862	−1.75838
$986$	−0.590426	−0.0188030
$987$	0	0
$988$	9.42166	0.299743
$989$	0.351207	0.0111677
$990$	0	0
$991$	−5.73279	−0.182108	−0.0910540	−	0.995846i	$-0.529024\pi$
$991$	−5.73279	−0.182108	−0.0910540	+	0.995846i	$0.529024\pi$
$992$	11.7271	0.372337
$993$	0	0
$994$	5.89067	0.186841
$995$	−55.9539	−1.77386
$996$	0	0
$997$	19.9095	0.630541	0.315270	−	0.949002i	$-0.397905\pi$
$997$	19.9095	0.630541	0.315270	+	0.949002i	$0.397905\pi$
$998$	−3.61182	−0.114330
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1359.2.a.i.1.3		6
3.2	odd	2		151.2.a.c.1.4	✓	6
12.11	even	2		2416.2.a.o.1.6		6
15.14	odd	2		3775.2.a.p.1.3		6
21.20	even	2		7399.2.a.e.1.4		6
24.5	odd	2		9664.2.a.bh.1.6		6
24.11	even	2		9664.2.a.bc.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.a.c.1.4	✓	6	3.2	odd	2
1359.2.a.i.1.3		6	1.1	even	1	trivial
2416.2.a.o.1.6		6	12.11	even	2
3775.2.a.p.1.3		6	15.14	odd	2
7399.2.a.e.1.4		6	21.20	even	2
9664.2.a.bc.1.1		6	24.11	even	2
9664.2.a.bh.1.6		6	24.5	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:	\( N \)	\(=\)	\( 1359 = 3^{2} \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1359.a (trivial)

\(f(q)\)	\(=\)	\(q-0.183668 q^{2} -1.96627 q^{4} -3.71611 q^{5} +3.65107 q^{7} +0.728478 q^{8} +O(q^{10})\)	\(q-0.183668 q^{2} -1.96627 q^{4} -3.71611 q^{5} +3.65107 q^{7} +0.728478 q^{8} +0.682533 q^{10} +0.912147 q^{11} -1.27865 q^{13} -0.670586 q^{14} +3.79873 q^{16} -1.65822 q^{17} +3.74742 q^{19} +7.30687 q^{20} -0.167533 q^{22} +0.248713 q^{23} +8.80951 q^{25} +0.234848 q^{26} -7.17897 q^{28} -1.93860 q^{29} -5.44267 q^{31} -2.15466 q^{32} +0.304563 q^{34} -13.5678 q^{35} -4.41526 q^{37} -0.688284 q^{38} -2.70711 q^{40} -8.64880 q^{41} +1.41210 q^{43} -1.79352 q^{44} -0.0456807 q^{46} -2.19095 q^{47} +6.33031 q^{49} -1.61803 q^{50} +2.51417 q^{52} +10.4493 q^{53} -3.38964 q^{55} +2.65972 q^{56} +0.356059 q^{58} +0.862276 q^{59} -2.48235 q^{61} +0.999648 q^{62} -7.20172 q^{64} +4.75161 q^{65} -13.4435 q^{67} +3.26051 q^{68} +2.49198 q^{70} -8.78436 q^{71} -14.3562 q^{73} +0.810944 q^{74} -7.36843 q^{76} +3.33031 q^{77} -10.1620 q^{79} -14.1165 q^{80} +1.58851 q^{82} -7.06074 q^{83} +6.16215 q^{85} -0.259358 q^{86} +0.664479 q^{88} -11.7770 q^{89} -4.66844 q^{91} -0.489036 q^{92} +0.402409 q^{94} -13.9259 q^{95} -0.860838 q^{97} -1.16268 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 6 * q - q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	\( 6 q - q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} + 8 q^{10} - 8 q^{11} - q^{13} - 6 q^{14} - 3 q^{16} - 9 q^{17} - 6 q^{19} - 3 q^{20} - 12 q^{22} + 4 q^{23} - 4 q^{25} + 7 q^{26} - 24 q^{28} + 2 q^{29} - 8 q^{31} + 11 q^{32} - 9 q^{34} - 5 q^{35} - 12 q^{37} + 3 q^{38} + 9 q^{40} - 41 q^{41} + q^{43} - 17 q^{46} - 28 q^{47} + 33 q^{49} - 15 q^{50} + 15 q^{52} - 14 q^{53} + q^{55} + 9 q^{56} + q^{58} - 12 q^{59} + 5 q^{61} + 9 q^{62} - 27 q^{64} - 13 q^{65} - 15 q^{67} - 11 q^{68} - 33 q^{70} + 2 q^{71} - 7 q^{73} - 53 q^{74} - 3 q^{76} + 15 q^{77} - 9 q^{79} - 16 q^{80} + 10 q^{82} + 11 q^{83} - 10 q^{85} + 26 q^{86} + 24 q^{88} - 36 q^{89} - 35 q^{91} + 38 q^{92} + 42 q^{94} - 32 q^{95} + 11 q^{97} - 15 q^{98}+O(q^{100}) \) 6 * q - q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 + 8 * q^10 - 8 * q^11 - q^13 - 6 * q^14 - 3 * q^16 - 9 * q^17 - 6 * q^19 - 3 * q^20 - 12 * q^22 + 4 * q^23 - 4 * q^25 + 7 * q^26 - 24 * q^28 + 2 * q^29 - 8 * q^31 + 11 * q^32 - 9 * q^34 - 5 * q^35 - 12 * q^37 + 3 * q^38 + 9 * q^40 - 41 * q^41 + q^43 - 17 * q^46 - 28 * q^47 + 33 * q^49 - 15 * q^50 + 15 * q^52 - 14 * q^53 + q^55 + 9 * q^56 + q^58 - 12 * q^59 + 5 * q^61 + 9 * q^62 - 27 * q^64 - 13 * q^65 - 15 * q^67 - 11 * q^68 - 33 * q^70 + 2 * q^71 - 7 * q^73 - 53 * q^74 - 3 * q^76 + 15 * q^77 - 9 * q^79 - 16 * q^80 + 10 * q^82 + 11 * q^83 - 10 * q^85 + 26 * q^86 + 24 * q^88 - 36 * q^89 - 35 * q^91 + 38 * q^92 + 42 * q^94 - 32 * q^95 + 11 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more