LMFDB - Embedded newform 1143.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, 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("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.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:	$9.12690095103$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.456850$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} +0.456850 q^{5} -3.79129 q^{7} -1.73205 q^{8} +O(q^{10})$	$q+1.73205 q^{2} +1.00000 q^{4} +0.456850 q^{5} -3.79129 q^{7} -1.73205 q^{8} +0.791288 q^{10} -2.18890 q^{11} -1.20871 q^{13} -6.56670 q^{14} -5.00000 q^{16} -4.37780 q^{17} +3.58258 q^{19} +0.456850 q^{20} -3.79129 q^{22} -4.79129 q^{25} -2.09355 q^{26} -3.79129 q^{28} -3.10260 q^{29} +3.58258 q^{31} -5.19615 q^{32} -7.58258 q^{34} -1.73205 q^{35} -2.79129 q^{37} +6.20520 q^{38} -0.791288 q^{40} +9.66930 q^{41} -8.37386 q^{43} -2.18890 q^{44} -0.456850 q^{47} +7.37386 q^{49} -8.29875 q^{50} -1.20871 q^{52} -3.10260 q^{53} -1.00000 q^{55} +6.56670 q^{56} -5.37386 q^{58} +8.75560 q^{59} +1.79129 q^{61} +6.20520 q^{62} +1.00000 q^{64} -0.552200 q^{65} -1.20871 q^{67} -4.37780 q^{68} -3.00000 q^{70} -0.361500 q^{71} -13.3739 q^{73} -4.83465 q^{74} +3.58258 q^{76} +8.29875 q^{77} +5.58258 q^{79} -2.28425 q^{80} +16.7477 q^{82} +0.723000 q^{83} -2.00000 q^{85} -14.5040 q^{86} +3.79129 q^{88} -8.29875 q^{89} +4.58258 q^{91} -0.791288 q^{94} +1.63670 q^{95} -8.00000 q^{97} +12.7719 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 6 q^{7}+O(q^{10}) $ 4 * q + 4 * q^4 - 6 * q^7	$ 4 q + 4 q^{4} - 6 q^{7} - 6 q^{10} - 14 q^{13} - 20 q^{16} - 4 q^{19} - 6 q^{22} - 10 q^{25} - 6 q^{28} - 4 q^{31} - 12 q^{34} - 2 q^{37} + 6 q^{40} - 6 q^{43} + 2 q^{49} - 14 q^{52} - 4 q^{55} + 6 q^{58} - 2 q^{61} + 4 q^{64} - 14 q^{67} - 12 q^{70} - 26 q^{73} - 4 q^{76} + 4 q^{79} + 12 q^{82} - 8 q^{85} + 6 q^{88} + 6 q^{94} - 32 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 6 * q^7 - 6 * q^10 - 14 * q^13 - 20 * q^16 - 4 * q^19 - 6 * q^22 - 10 * q^25 - 6 * q^28 - 4 * q^31 - 12 * q^34 - 2 * q^37 + 6 * q^40 - 6 * q^43 + 2 * q^49 - 14 * q^52 - 4 * q^55 + 6 * q^58 - 2 * q^61 + 4 * q^64 - 14 * q^67 - 12 * q^70 - 26 * q^73 - 4 * q^76 + 4 * q^79 + 12 * q^82 - 8 * q^85 + 6 * q^88 + 6 * q^94 - 32 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.456850	0.204310	0.102155	−	0.994769i	$-0.467426\pi$
$5$	0.456850	0.204310	0.102155	+	0.994769i	$0.467426\pi$
$6$	0	0
$7$	−3.79129	−1.43297	−0.716486	−	0.697601i	$-0.754251\pi$
$7$	−3.79129	−1.43297	−0.716486	+	0.697601i	$0.754251\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	0.791288	0.250227
$11$	−2.18890	−0.659979	−0.329989	−	0.943985i	$-0.607045\pi$
$11$	−2.18890	−0.659979	−0.329989	+	0.943985i	$0.607045\pi$
$12$	0	0
$13$	−1.20871	−0.335236	−0.167618	−	0.985852i	$-0.553608\pi$
$13$	−1.20871	−0.335236	−0.167618	+	0.985852i	$0.553608\pi$
$14$	−6.56670	−1.75503
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−4.37780	−1.06177	−0.530886	−	0.847443i	$-0.678141\pi$
$17$	−4.37780	−1.06177	−0.530886	+	0.847443i	$0.678141\pi$
$18$	0	0
$19$	3.58258	0.821899	0.410950	−	0.911658i	$-0.365197\pi$
$19$	3.58258	0.821899	0.410950	+	0.911658i	$0.365197\pi$
$20$	0.456850	0.102155
$21$	0	0
$22$	−3.79129	−0.808305
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−4.79129	−0.958258
$26$	−2.09355	−0.410579
$27$	0	0
$28$	−3.79129	−0.716486
$29$	−3.10260	−0.576139	−0.288069	−	0.957610i	$-0.593013\pi$
$29$	−3.10260	−0.576139	−0.288069	+	0.957610i	$0.593013\pi$
$30$	0	0
$31$	3.58258	0.643450	0.321725	−	0.946833i	$-0.395737\pi$
$31$	3.58258	0.643450	0.321725	+	0.946833i	$0.395737\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	−7.58258	−1.30040
$35$	−1.73205	−0.292770
$36$	0	0
$37$	−2.79129	−0.458885	−0.229442	−	0.973322i	$-0.573690\pi$
$37$	−2.79129	−0.458885	−0.229442	+	0.973322i	$0.573690\pi$
$38$	6.20520	1.00662
$39$	0	0
$40$	−0.791288	−0.125114
$41$	9.66930	1.51009	0.755046	−	0.655672i	$-0.227615\pi$
$41$	9.66930	1.51009	0.755046	+	0.655672i	$0.227615\pi$
$42$	0	0
$43$	−8.37386	−1.27700	−0.638501	−	0.769621i	$-0.720445\pi$
$43$	−8.37386	−1.27700	−0.638501	+	0.769621i	$0.720445\pi$
$44$	−2.18890	−0.329989
$45$	0	0
$46$	0	0
$47$	−0.456850	−0.0666385	−0.0333192	−	0.999445i	$-0.510608\pi$
$47$	−0.456850	−0.0666385	−0.0333192	+	0.999445i	$0.510608\pi$
$48$	0	0
$49$	7.37386	1.05341
$50$	−8.29875	−1.17362
$51$	0	0
$52$	−1.20871	−0.167618
$53$	−3.10260	−0.426175	−0.213088	−	0.977033i	$-0.568352\pi$
$53$	−3.10260	−0.426175	−0.213088	+	0.977033i	$0.568352\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	6.56670	0.877513
$57$	0	0
$58$	−5.37386	−0.705623
$59$	8.75560	1.13988	0.569941	−	0.821685i	$-0.306966\pi$
$59$	8.75560	1.13988	0.569941	+	0.821685i	$0.306966\pi$
$60$	0	0
$61$	1.79129	0.229351	0.114675	−	0.993403i	$-0.463417\pi$
$61$	1.79129	0.229351	0.114675	+	0.993403i	$0.463417\pi$
$62$	6.20520	0.788062
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.552200	−0.0684920
$66$	0	0
$67$	−1.20871	−0.147668	−0.0738338	−	0.997271i	$-0.523523\pi$
$67$	−1.20871	−0.147668	−0.0738338	+	0.997271i	$0.523523\pi$
$68$	−4.37780	−0.530886
$69$	0	0
$70$	−3.00000	−0.358569
$71$	−0.361500	−0.0429022	−0.0214511	−	0.999770i	$-0.506829\pi$
$71$	−0.361500	−0.0429022	−0.0214511	+	0.999770i	$0.506829\pi$
$72$	0	0
$73$	−13.3739	−1.56529	−0.782646	−	0.622467i	$-0.786131\pi$
$73$	−13.3739	−1.56529	−0.782646	+	0.622467i	$0.786131\pi$
$74$	−4.83465	−0.562017
$75$	0	0
$76$	3.58258	0.410950
$77$	8.29875	0.945731
$78$	0	0
$79$	5.58258	0.628089	0.314044	−	0.949408i	$-0.398316\pi$
$79$	5.58258	0.628089	0.314044	+	0.949408i	$0.398316\pi$
$80$	−2.28425	−0.255387
$81$	0	0
$82$	16.7477	1.84948
$83$	0.723000	0.0793596	0.0396798	−	0.999212i	$-0.487366\pi$
$83$	0.723000	0.0793596	0.0396798	+	0.999212i	$0.487366\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	−14.5040	−1.56400
$87$	0	0
$88$	3.79129	0.404153
$89$	−8.29875	−0.879666	−0.439833	−	0.898080i	$-0.644962\pi$
$89$	−8.29875	−0.879666	−0.439833	+	0.898080i	$0.644962\pi$
$90$	0	0
$91$	4.58258	0.480384
$92$	0	0
$93$	0	0
$94$	−0.791288	−0.0816151
$95$	1.63670	0.167922
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	12.7719	1.29016
$99$	0	0
$100$	−4.79129	−0.479129
$101$	17.5112	1.74243	0.871215	−	0.490901i	$-0.163332\pi$
$101$	17.5112	1.74243	0.871215	+	0.490901i	$0.163332\pi$
$102$	0	0
$103$	−7.58258	−0.747133	−0.373567	−	0.927603i	$-0.621865\pi$
$103$	−7.58258	−0.747133	−0.373567	+	0.927603i	$0.621865\pi$
$104$	2.09355	0.205290
$105$	0	0
$106$	−5.37386	−0.521956
$107$	11.0399	1.06726	0.533632	−	0.845717i	$-0.320827\pi$
$107$	11.0399	1.06726	0.533632	+	0.845717i	$0.320827\pi$
$108$	0	0
$109$	−6.74773	−0.646315	−0.323158	−	0.946345i	$-0.604744\pi$
$109$	−6.74773	−0.646315	−0.323158	+	0.946345i	$0.604744\pi$
$110$	−1.73205	−0.165145
$111$	0	0
$112$	18.9564	1.79122
$113$	9.47860	0.891672	0.445836	−	0.895115i	$-0.352906\pi$
$113$	9.47860	0.891672	0.445836	+	0.895115i	$0.352906\pi$
$114$	0	0
$115$	0	0
$116$	−3.10260	−0.288069
$117$	0	0
$118$	15.1652	1.39607
$119$	16.5975	1.52149
$120$	0	0
$121$	−6.20871	−0.564428
$122$	3.10260	0.280896
$123$	0	0
$124$	3.58258	0.321725
$125$	−4.47315	−0.400091
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	12.1244	1.07165
$129$	0	0
$130$	−0.956439	−0.0838853
$131$	−7.38505	−0.645235	−0.322618	−	0.946529i	$-0.604563\pi$
$131$	−7.38505	−0.645235	−0.322618	+	0.946529i	$0.604563\pi$
$132$	0	0
$133$	−13.5826	−1.17776
$134$	−2.09355	−0.180855
$135$	0	0
$136$	7.58258	0.650201
$137$	5.10080	0.435791	0.217895	−	0.975972i	$-0.430081\pi$
$137$	5.10080	0.435791	0.217895	+	0.975972i	$0.430081\pi$
$138$	0	0
$139$	−11.1652	−0.947016	−0.473508	−	0.880790i	$-0.657012\pi$
$139$	−11.1652	−0.947016	−0.473508	+	0.880790i	$0.657012\pi$
$140$	−1.73205	−0.146385
$141$	0	0
$142$	−0.626136	−0.0525442
$143$	2.64575	0.221249
$144$	0	0
$145$	−1.41742	−0.117711
$146$	−23.1642	−1.91708
$147$	0	0
$148$	−2.79129	−0.229442
$149$	−11.4967	−0.941847	−0.470923	−	0.882174i	$-0.656079\pi$
$149$	−11.4967	−0.941847	−0.470923	+	0.882174i	$0.656079\pi$
$150$	0	0
$151$	7.37386	0.600077	0.300038	−	0.953927i	$-0.403001\pi$
$151$	7.37386	0.600077	0.300038	+	0.953927i	$0.403001\pi$
$152$	−6.20520	−0.503308
$153$	0	0
$154$	14.3739	1.15828
$155$	1.63670	0.131463
$156$	0	0
$157$	−13.3739	−1.06735	−0.533675	−	0.845689i	$-0.679190\pi$
$157$	−13.3739	−1.06735	−0.533675	+	0.845689i	$0.679190\pi$
$158$	9.66930	0.769249
$159$	0	0
$160$	−2.37386	−0.187670
$161$	0	0
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	9.66930	0.755046
$165$	0	0
$166$	1.25227	0.0971952
$167$	−7.84190	−0.606825	−0.303412	−	0.952859i	$-0.598126\pi$
$167$	−7.84190	−0.606825	−0.303412	+	0.952859i	$0.598126\pi$
$168$	0	0
$169$	−11.5390	−0.887617
$170$	−3.46410	−0.265684
$171$	0	0
$172$	−8.37386	−0.638501
$173$	17.9681	1.36609	0.683043	−	0.730378i	$-0.260656\pi$
$173$	17.9681	1.36609	0.683043	+	0.730378i	$0.260656\pi$
$174$	0	0
$175$	18.1652	1.37316
$176$	10.9445	0.824973
$177$	0	0
$178$	−14.3739	−1.07737
$179$	18.8818	1.41129	0.705644	−	0.708566i	$-0.250658\pi$
$179$	18.8818	1.41129	0.705644	+	0.708566i	$0.250658\pi$
$180$	0	0
$181$	−19.1652	−1.42453	−0.712267	−	0.701908i	$-0.752332\pi$
$181$	−19.1652	−1.42453	−0.712267	+	0.701908i	$0.752332\pi$
$182$	7.93725	0.588348
$183$	0	0
$184$	0	0
$185$	−1.27520	−0.0937546
$186$	0	0
$187$	9.58258	0.700747
$188$	−0.456850	−0.0333192
$189$	0	0
$190$	2.83485	0.205662
$191$	−8.94630	−0.647332	−0.323666	−	0.946171i	$-0.604915\pi$
$191$	−8.94630	−0.647332	−0.323666	+	0.946171i	$0.604915\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−13.8564	−0.994832
$195$	0	0
$196$	7.37386	0.526705
$197$	−25.1624	−1.79275	−0.896374	−	0.443299i	$-0.853808\pi$
$197$	−25.1624	−1.79275	−0.896374	+	0.443299i	$0.853808\pi$
$198$	0	0
$199$	14.7477	1.04544	0.522719	−	0.852505i	$-0.324918\pi$
$199$	14.7477	1.04544	0.522719	+	0.852505i	$0.324918\pi$
$200$	8.29875	0.586811
$201$	0	0
$202$	30.3303	2.13403
$203$	11.7629	0.825591
$204$	0	0
$205$	4.41742	0.308526
$206$	−13.1334	−0.915048
$207$	0	0
$208$	6.04356	0.419046
$209$	−7.84190	−0.542436
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	−3.10260	−0.213088
$213$	0	0
$214$	19.1216	1.30713
$215$	−3.82560	−0.260904
$216$	0	0
$217$	−13.5826	−0.922045
$218$	−11.6874	−0.791571
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	5.29150	0.355945
$222$	0	0
$223$	19.5390	1.30843	0.654215	−	0.756309i	$-0.272999\pi$
$223$	19.5390	1.30843	0.654215	+	0.756309i	$0.272999\pi$
$224$	19.7001	1.31627
$225$	0	0
$226$	16.4174	1.09207
$227$	−2.18890	−0.145282	−0.0726412	−	0.997358i	$-0.523143\pi$
$227$	−2.18890	−0.145282	−0.0726412	+	0.997358i	$0.523143\pi$
$228$	0	0
$229$	17.5826	1.16189	0.580945	−	0.813943i	$-0.302683\pi$
$229$	17.5826	1.16189	0.580945	+	0.813943i	$0.302683\pi$
$230$	0	0
$231$	0	0
$232$	5.37386	0.352811
$233$	19.7955	1.29684	0.648422	−	0.761281i	$-0.275429\pi$
$233$	19.7955	1.29684	0.648422	+	0.761281i	$0.275429\pi$
$234$	0	0
$235$	−0.208712	−0.0136149
$236$	8.75560	0.569941
$237$	0	0
$238$	28.7477	1.86344
$239$	14.2378	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$239$	14.2378	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$240$	0	0
$241$	1.16515	0.0750540	0.0375270	−	0.999296i	$-0.488052\pi$
$241$	1.16515	0.0750540	0.0375270	+	0.999296i	$0.488052\pi$
$242$	−10.7538	−0.691281
$243$	0	0
$244$	1.79129	0.114675
$245$	3.36875	0.215222
$246$	0	0
$247$	−4.33030	−0.275531
$248$	−6.20520	−0.394031
$249$	0	0
$250$	−7.74773	−0.490009
$251$	−17.3205	−1.09326	−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205	−1.09326	−0.546630	+	0.837374i	$0.684090\pi$
$252$	0	0
$253$	0	0
$254$	−1.73205	−0.108679
$255$	0	0
$256$	19.0000	1.18750
$257$	−28.2650	−1.76312	−0.881562	−	0.472069i	$-0.843507\pi$
$257$	−28.2650	−1.76312	−0.881562	+	0.472069i	$0.843507\pi$
$258$	0	0
$259$	10.5826	0.657569
$260$	−0.552200	−0.0342460
$261$	0	0
$262$	−12.7913	−0.790248
$263$	12.0489	0.742967	0.371484	−	0.928439i	$-0.378849\pi$
$263$	12.0489	0.742967	0.371484	+	0.928439i	$0.378849\pi$
$264$	0	0
$265$	−1.41742	−0.0870717
$266$	−23.5257	−1.44245
$267$	0	0
$268$	−1.20871	−0.0738338
$269$	−12.4104	−0.756676	−0.378338	−	0.925668i	$-0.623504\pi$
$269$	−12.4104	−0.756676	−0.378338	+	0.925668i	$0.623504\pi$
$270$	0	0
$271$	21.5826	1.31105	0.655524	−	0.755174i	$-0.272448\pi$
$271$	21.5826	1.31105	0.655524	+	0.755174i	$0.272448\pi$
$272$	21.8890	1.32722
$273$	0	0
$274$	8.83485	0.533733
$275$	10.4877	0.632429
$276$	0	0
$277$	5.16515	0.310344	0.155172	−	0.987887i	$-0.450407\pi$
$277$	5.16515	0.310344	0.155172	+	0.987887i	$0.450407\pi$
$278$	−19.3386	−1.15985
$279$	0	0
$280$	3.00000	0.179284
$281$	−13.3996	−0.799351	−0.399675	−	0.916657i	$-0.630877\pi$
$281$	−13.3996	−0.799351	−0.399675	+	0.916657i	$0.630877\pi$
$282$	0	0
$283$	−8.20871	−0.487957	−0.243979	−	0.969781i	$-0.578453\pi$
$283$	−8.20871	−0.487957	−0.243979	+	0.969781i	$0.578453\pi$
$284$	−0.361500	−0.0214511
$285$	0	0
$286$	4.58258	0.270973
$287$	−36.6591	−2.16392
$288$	0	0
$289$	2.16515	0.127362
$290$	−2.45505	−0.144166
$291$	0	0
$292$	−13.3739	−0.782646
$293$	1.99820	0.116736	0.0583681	−	0.998295i	$-0.481410\pi$
$293$	1.99820	0.116736	0.0583681	+	0.998295i	$0.481410\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	4.83465	0.281008
$297$	0	0
$298$	−19.9129	−1.15352
$299$	0	0
$300$	0	0
$301$	31.7477	1.82991
$302$	12.7719	0.734941
$303$	0	0
$304$	−17.9129	−1.02737
$305$	0.818350	0.0468586
$306$	0	0
$307$	−18.1216	−1.03425	−0.517127	−	0.855909i	$-0.672998\pi$
$307$	−18.1216	−1.03425	−0.517127	+	0.855909i	$0.672998\pi$
$308$	8.29875	0.472865
$309$	0	0
$310$	2.83485	0.161009
$311$	−6.01450	−0.341051	−0.170526	−	0.985353i	$-0.554547\pi$
$311$	−6.01450	−0.341051	−0.170526	+	0.985353i	$0.554547\pi$
$312$	0	0
$313$	−14.4174	−0.814921	−0.407461	−	0.913223i	$-0.633586\pi$
$313$	−14.4174	−0.814921	−0.407461	+	0.913223i	$0.633586\pi$
$314$	−23.1642	−1.30723
$315$	0	0
$316$	5.58258	0.314044
$317$	−24.4394	−1.37265	−0.686327	−	0.727293i	$-0.740778\pi$
$317$	−24.4394	−1.37265	−0.686327	+	0.727293i	$0.740778\pi$
$318$	0	0
$319$	6.79129	0.380239
$320$	0.456850	0.0255387
$321$	0	0
$322$	0	0
$323$	−15.6838	−0.872670
$324$	0	0
$325$	5.79129	0.321243
$326$	−6.92820	−0.383718
$327$	0	0
$328$	−16.7477	−0.924739
$329$	1.73205	0.0954911
$330$	0	0
$331$	−25.7042	−1.41283	−0.706414	−	0.707799i	$-0.749688\pi$
$331$	−25.7042	−1.41283	−0.706414	+	0.707799i	$0.749688\pi$
$332$	0.723000	0.0396798
$333$	0	0
$334$	−13.5826	−0.743205
$335$	−0.552200	−0.0301699
$336$	0	0
$337$	25.9129	1.41156	0.705782	−	0.708429i	$-0.250596\pi$
$337$	25.9129	1.41156	0.705782	+	0.708429i	$0.250596\pi$
$338$	−19.9862	−1.08710
$339$	0	0
$340$	−2.00000	−0.108465
$341$	−7.84190	−0.424663
$342$	0	0
$343$	−1.41742	−0.0765337
$344$	14.5040	0.782001
$345$	0	0
$346$	31.1216	1.67311
$347$	15.8745	0.852188	0.426094	−	0.904679i	$-0.359889\pi$
$347$	15.8745	0.852188	0.426094	+	0.904679i	$0.359889\pi$
$348$	0	0
$349$	−10.4174	−0.557632	−0.278816	−	0.960345i	$-0.589942\pi$
$349$	−10.4174	−0.557632	−0.278816	+	0.960345i	$0.589942\pi$
$350$	31.4630	1.68177
$351$	0	0
$352$	11.3739	0.606229
$353$	27.1805	1.44667	0.723336	−	0.690496i	$-0.242608\pi$
$353$	27.1805	1.44667	0.723336	+	0.690496i	$0.242608\pi$
$354$	0	0
$355$	−0.165151	−0.00876533
$356$	−8.29875	−0.439833
$357$	0	0
$358$	32.7042	1.72847
$359$	−12.4104	−0.654996	−0.327498	−	0.944852i	$-0.606206\pi$
$359$	−12.4104	−0.654996	−0.327498	+	0.944852i	$0.606206\pi$
$360$	0	0
$361$	−6.16515	−0.324482
$362$	−33.1950	−1.74469
$363$	0	0
$364$	4.58258	0.240192
$365$	−6.10985	−0.319804
$366$	0	0
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	0	0
$369$	0	0
$370$	−2.20871	−0.114825
$371$	11.7629	0.610697
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	16.5975	0.858237
$375$	0	0
$376$	0.791288	0.0408076
$377$	3.75015	0.193143
$378$	0	0
$379$	−29.1216	−1.49588	−0.747938	−	0.663769i	$-0.768956\pi$
$379$	−29.1216	−1.49588	−0.747938	+	0.663769i	$0.768956\pi$
$380$	1.63670	0.0839610
$381$	0	0
$382$	−15.4955	−0.792816
$383$	−17.1497	−0.876309	−0.438155	−	0.898900i	$-0.644368\pi$
$383$	−17.1497	−0.876309	−0.438155	+	0.898900i	$0.644368\pi$
$384$	0	0
$385$	3.79129	0.193222
$386$	−27.7128	−1.41055
$387$	0	0
$388$	−8.00000	−0.406138
$389$	−0.723000	−0.0366576	−0.0183288	−	0.999832i	$-0.505835\pi$
$389$	−0.723000	−0.0366576	−0.0183288	+	0.999832i	$0.505835\pi$
$390$	0	0
$391$	0	0
$392$	−12.7719	−0.645079
$393$	0	0
$394$	−43.5826	−2.19566
$395$	2.55040	0.128325
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	25.5438	1.28040
$399$	0	0
$400$	23.9564	1.19782
$401$	35.0224	1.74894	0.874468	−	0.485083i	$-0.161211\pi$
$401$	35.0224	1.74894	0.874468	+	0.485083i	$0.161211\pi$
$402$	0	0
$403$	−4.33030	−0.215708
$404$	17.5112	0.871215
$405$	0	0
$406$	20.3739	1.01114
$407$	6.10985	0.302854
$408$	0	0
$409$	−11.1652	−0.552081	−0.276041	−	0.961146i	$-0.589022\pi$
$409$	−11.1652	−0.552081	−0.276041	+	0.961146i	$0.589022\pi$
$410$	7.65120	0.377866
$411$	0	0
$412$	−7.58258	−0.373567
$413$	−33.1950	−1.63342
$414$	0	0
$415$	0.330303	0.0162139
$416$	6.28065	0.307934
$417$	0	0
$418$	−13.5826	−0.664345
$419$	−29.4648	−1.43945	−0.719724	−	0.694260i	$-0.755732\pi$
$419$	−29.4648	−1.43945	−0.719724	+	0.694260i	$0.755732\pi$
$420$	0	0
$421$	−15.1652	−0.739104	−0.369552	−	0.929210i	$-0.620489\pi$
$421$	−15.1652	−0.739104	−0.369552	+	0.929210i	$0.620489\pi$
$422$	27.7128	1.34904
$423$	0	0
$424$	5.37386	0.260978
$425$	20.9753	1.01745
$426$	0	0
$427$	−6.79129	−0.328653
$428$	11.0399	0.533632
$429$	0	0
$430$	−6.62614	−0.319541
$431$	−27.6374	−1.33124	−0.665622	−	0.746289i	$-0.731834\pi$
$431$	−27.6374	−1.33124	−0.665622	+	0.746289i	$0.731834\pi$
$432$	0	0
$433$	23.7042	1.13915	0.569575	−	0.821940i	$-0.307108\pi$
$433$	23.7042	1.13915	0.569575	+	0.821940i	$0.307108\pi$
$434$	−23.5257	−1.12927
$435$	0	0
$436$	−6.74773	−0.323158
$437$	0	0
$438$	0	0
$439$	22.7913	1.08777	0.543884	−	0.839160i	$-0.316953\pi$
$439$	22.7913	1.08777	0.543884	+	0.839160i	$0.316953\pi$
$440$	1.73205	0.0825723
$441$	0	0
$442$	9.16515	0.435942
$443$	−5.84370	−0.277643	−0.138821	−	0.990317i	$-0.544331\pi$
$443$	−5.84370	−0.277643	−0.138821	+	0.990317i	$0.544331\pi$
$444$	0	0
$445$	−3.79129	−0.179724
$446$	33.8426	1.60249
$447$	0	0
$448$	−3.79129	−0.179122
$449$	6.20520	0.292842	0.146421	−	0.989222i	$-0.453225\pi$
$449$	6.20520	0.292842	0.146421	+	0.989222i	$0.453225\pi$
$450$	0	0
$451$	−21.1652	−0.996628
$452$	9.47860	0.445836
$453$	0	0
$454$	−3.79129	−0.177934
$455$	2.09355	0.0981472
$456$	0	0
$457$	−10.6261	−0.497070	−0.248535	−	0.968623i	$-0.579949\pi$
$457$	−10.6261	−0.497070	−0.248535	+	0.968623i	$0.579949\pi$
$458$	30.4539	1.42302
$459$	0	0
$460$	0	0
$461$	−7.48040	−0.348397	−0.174199	−	0.984711i	$-0.555733\pi$
$461$	−7.48040	−0.348397	−0.174199	+	0.984711i	$0.555733\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	15.5130	0.720173
$465$	0	0
$466$	34.2867	1.58830
$467$	0.190700	0.00882456	0.00441228	−	0.999990i	$-0.498596\pi$
$467$	0.190700	0.00882456	0.00441228	+	0.999990i	$0.498596\pi$
$468$	0	0
$469$	4.58258	0.211604
$470$	−0.361500	−0.0166748
$471$	0	0
$472$	−15.1652	−0.698033
$473$	18.3296	0.842794
$474$	0	0
$475$	−17.1652	−0.787591
$476$	16.5975	0.760746
$477$	0	0
$478$	24.6606	1.12795
$479$	−16.5221	−0.754912	−0.377456	−	0.926027i	$-0.623201\pi$
$479$	−16.5221	−0.754912	−0.377456	+	0.926027i	$0.623201\pi$
$480$	0	0
$481$	3.37386	0.153835
$482$	2.01810	0.0919220
$483$	0	0
$484$	−6.20871	−0.282214
$485$	−3.65480	−0.165956
$486$	0	0
$487$	−13.9564	−0.632427	−0.316213	−	0.948688i	$-0.602412\pi$
$487$	−13.9564	−0.632427	−0.316213	+	0.948688i	$0.602412\pi$
$488$	−3.10260	−0.140448
$489$	0	0
$490$	5.83485	0.263592
$491$	−19.8709	−0.896762	−0.448381	−	0.893843i	$-0.647999\pi$
$491$	−19.8709	−0.896762	−0.448381	+	0.893843i	$0.647999\pi$
$492$	0	0
$493$	13.5826	0.611728
$494$	−7.50030	−0.337455
$495$	0	0
$496$	−17.9129	−0.804312
$497$	1.37055	0.0614776
$498$	0	0
$499$	24.6261	1.10242	0.551209	−	0.834367i	$-0.314167\pi$
$499$	24.6261	1.10242	0.551209	+	0.834367i	$0.314167\pi$
$500$	−4.47315	−0.200045
$501$	0	0
$502$	−30.0000	−1.33897
$503$	38.4865	1.71603	0.858015	−	0.513625i	$-0.171698\pi$
$503$	38.4865	1.71603	0.858015	+	0.513625i	$0.171698\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	0	0
$508$	−1.00000	−0.0443678
$509$	42.8643	1.89993	0.949964	−	0.312360i	$-0.101120\pi$
$509$	42.8643	1.89993	0.949964	+	0.312360i	$0.101120\pi$
$510$	0	0
$511$	50.7042	2.24302
$512$	8.66025	0.382733
$513$	0	0
$514$	−48.9564	−2.15938
$515$	−3.46410	−0.152647
$516$	0	0
$517$	1.00000	0.0439799
$518$	18.3296	0.805355
$519$	0	0
$520$	0.956439	0.0419426
$521$	−18.9572	−0.830530	−0.415265	−	0.909700i	$-0.636311\pi$
$521$	−18.9572	−0.830530	−0.415265	+	0.909700i	$0.636311\pi$
$522$	0	0
$523$	−24.8348	−1.08595	−0.542976	−	0.839748i	$-0.682703\pi$
$523$	−24.8348	−1.08595	−0.542976	+	0.839748i	$0.682703\pi$
$524$	−7.38505	−0.322618
$525$	0	0
$526$	20.8693	0.909945
$527$	−15.6838	−0.683197
$528$	0	0
$529$	−23.0000	−1.00000
$530$	−2.45505	−0.106641
$531$	0	0
$532$	−13.5826	−0.588879
$533$	−11.6874	−0.506238
$534$	0	0
$535$	5.04356	0.218052
$536$	2.09355	0.0904276
$537$	0	0
$538$	−21.4955	−0.926735
$539$	−16.1407	−0.695227
$540$	0	0
$541$	3.66970	0.157773	0.0788863	−	0.996884i	$-0.474864\pi$
$541$	3.66970	0.157773	0.0788863	+	0.996884i	$0.474864\pi$
$542$	37.3821	1.60570
$543$	0	0
$544$	22.7477	0.975301
$545$	−3.08270	−0.132048
$546$	0	0
$547$	9.04356	0.386675	0.193337	−	0.981132i	$-0.438069\pi$
$547$	9.04356	0.386675	0.193337	+	0.981132i	$0.438069\pi$
$548$	5.10080	0.217895
$549$	0	0
$550$	18.1652	0.774565
$551$	−11.1153	−0.473528
$552$	0	0
$553$	−21.1652	−0.900034
$554$	8.94630	0.380092
$555$	0	0
$556$	−11.1652	−0.473508
$557$	−7.11890	−0.301638	−0.150819	−	0.988561i	$-0.548191\pi$
$557$	−7.11890	−0.301638	−0.150819	+	0.988561i	$0.548191\pi$
$558$	0	0
$559$	10.1216	0.428098
$560$	8.66025	0.365963
$561$	0	0
$562$	−23.2087	−0.979000
$563$	9.47860	0.399476	0.199738	−	0.979849i	$-0.435991\pi$
$563$	9.47860	0.399476	0.199738	+	0.979849i	$0.435991\pi$
$564$	0	0
$565$	4.33030	0.182177
$566$	−14.2179	−0.597623
$567$	0	0
$568$	0.626136	0.0262721
$569$	13.8564	0.580891	0.290445	−	0.956892i	$-0.406197\pi$
$569$	13.8564	0.580891	0.290445	+	0.956892i	$0.406197\pi$
$570$	0	0
$571$	33.8693	1.41739	0.708693	−	0.705517i	$-0.249285\pi$
$571$	33.8693	1.41739	0.708693	+	0.705517i	$0.249285\pi$
$572$	2.64575	0.110624
$573$	0	0
$574$	−63.4955	−2.65025
$575$	0	0
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	3.75015	0.155986
$579$	0	0
$580$	−1.41742	−0.0588553
$581$	−2.74110	−0.113720
$582$	0	0
$583$	6.79129	0.281266
$584$	23.1642	0.958542
$585$	0	0
$586$	3.46099	0.142972
$587$	7.28970	0.300878	0.150439	−	0.988619i	$-0.451931\pi$
$587$	7.28970	0.300878	0.150439	+	0.988619i	$0.451931\pi$
$588$	0	0
$589$	12.8348	0.528851
$590$	6.92820	0.285230
$591$	0	0
$592$	13.9564	0.573606
$593$	−16.2360	−0.666733	−0.333366	−	0.942797i	$-0.608185\pi$
$593$	−16.2360	−0.666733	−0.333366	+	0.942797i	$0.608185\pi$
$594$	0	0
$595$	7.58258	0.310855
$596$	−11.4967	−0.470923
$597$	0	0
$598$	0	0
$599$	38.4865	1.57252	0.786258	−	0.617898i	$-0.212016\pi$
$599$	38.4865	1.57252	0.786258	+	0.617898i	$0.212016\pi$
$600$	0	0
$601$	−20.7477	−0.846317	−0.423159	−	0.906056i	$-0.639079\pi$
$601$	−20.7477	−0.846317	−0.423159	+	0.906056i	$0.639079\pi$
$602$	54.9887	2.24117
$603$	0	0
$604$	7.37386	0.300038
$605$	−2.83645	−0.115318
$606$	0	0
$607$	−21.4955	−0.872474	−0.436237	−	0.899832i	$-0.643689\pi$
$607$	−21.4955	−0.872474	−0.436237	+	0.899832i	$0.643689\pi$
$608$	−18.6156	−0.754963
$609$	0	0
$610$	1.41742	0.0573898
$611$	0.552200	0.0223396
$612$	0	0
$613$	−24.7477	−0.999551	−0.499776	−	0.866155i	$-0.666584\pi$
$613$	−24.7477	−0.999551	−0.499776	+	0.866155i	$0.666584\pi$
$614$	−31.3875	−1.26670
$615$	0	0
$616$	−14.3739	−0.579139
$617$	−42.7889	−1.72262	−0.861308	−	0.508084i	$-0.830354\pi$
$617$	−42.7889	−1.72262	−0.861308	+	0.508084i	$0.830354\pi$
$618$	0	0
$619$	−3.79129	−0.152385	−0.0761924	−	0.997093i	$-0.524276\pi$
$619$	−3.79129	−0.152385	−0.0761924	+	0.997093i	$0.524276\pi$
$620$	1.63670	0.0657315
$621$	0	0
$622$	−10.4174	−0.417701
$623$	31.4630	1.26054
$624$	0	0
$625$	21.9129	0.876515
$626$	−24.9717	−0.998070
$627$	0	0
$628$	−13.3739	−0.533675
$629$	12.2197	0.487232
$630$	0	0
$631$	−34.7042	−1.38155	−0.690776	−	0.723069i	$-0.742731\pi$
$631$	−34.7042	−1.38155	−0.690776	+	0.723069i	$0.742731\pi$
$632$	−9.66930	−0.384624
$633$	0	0
$634$	−42.3303	−1.68115
$635$	−0.456850	−0.0181295
$636$	0	0
$637$	−8.91288	−0.353141
$638$	11.7629	0.465696
$639$	0	0
$640$	5.53901	0.218949
$641$	26.7237	1.05552	0.527761	−	0.849393i	$-0.323032\pi$
$641$	26.7237	1.05552	0.527761	+	0.849393i	$0.323032\pi$
$642$	0	0
$643$	−6.00000	−0.236617	−0.118308	−	0.992977i	$-0.537747\pi$
$643$	−6.00000	−0.236617	−0.118308	+	0.992977i	$0.537747\pi$
$644$	0	0
$645$	0	0
$646$	−27.1652	−1.06880
$647$	24.2487	0.953315	0.476658	−	0.879089i	$-0.341848\pi$
$647$	24.2487	0.953315	0.476658	+	0.879089i	$0.341848\pi$
$648$	0	0
$649$	−19.1652	−0.752298
$650$	10.0308	0.393441
$651$	0	0
$652$	−4.00000	−0.156652
$653$	28.6265	1.12024	0.560121	−	0.828411i	$-0.310755\pi$
$653$	28.6265	1.12024	0.560121	+	0.828411i	$0.310755\pi$
$654$	0	0
$655$	−3.37386	−0.131828
$656$	−48.3465	−1.88761
$657$	0	0
$658$	3.00000	0.116952
$659$	−45.0333	−1.75425	−0.877125	−	0.480263i	$-0.840541\pi$
$659$	−45.0333	−1.75425	−0.877125	+	0.480263i	$0.840541\pi$
$660$	0	0
$661$	−15.7042	−0.610821	−0.305411	−	0.952221i	$-0.598794\pi$
$661$	−15.7042	−0.610821	−0.305411	+	0.952221i	$0.598794\pi$
$662$	−44.5209	−1.73035
$663$	0	0
$664$	−1.25227	−0.0485976
$665$	−6.20520	−0.240627
$666$	0	0
$667$	0	0
$668$	−7.84190	−0.303412
$669$	0	0
$670$	−0.956439	−0.0369505
$671$	−3.92095	−0.151367
$672$	0	0
$673$	−26.7042	−1.02937	−0.514685	−	0.857379i	$-0.672091\pi$
$673$	−26.7042	−1.02937	−0.514685	+	0.857379i	$0.672091\pi$
$674$	44.8824	1.72881
$675$	0	0
$676$	−11.5390	−0.443808
$677$	−4.91010	−0.188711	−0.0943553	−	0.995539i	$-0.530079\pi$
$677$	−4.91010	−0.188711	−0.0943553	+	0.995539i	$0.530079\pi$
$678$	0	0
$679$	30.3303	1.16397
$680$	3.46410	0.132842
$681$	0	0
$682$	−13.5826	−0.520104
$683$	41.7599	1.59790	0.798949	−	0.601398i	$-0.205389\pi$
$683$	41.7599	1.59790	0.798949	+	0.601398i	$0.205389\pi$
$684$	0	0
$685$	2.33030	0.0890363
$686$	−2.45505	−0.0937343
$687$	0	0
$688$	41.8693	1.59625
$689$	3.75015	0.142869
$690$	0	0
$691$	−22.7913	−0.867021	−0.433511	−	0.901148i	$-0.642725\pi$
$691$	−22.7913	−0.867021	−0.433511	+	0.901148i	$0.642725\pi$
$692$	17.9681	0.683043
$693$	0	0
$694$	27.4955	1.04371
$695$	−5.10080	−0.193484
$696$	0	0
$697$	−42.3303	−1.60337
$698$	−18.0435	−0.682957
$699$	0	0
$700$	18.1652	0.686578
$701$	−16.2360	−0.613226	−0.306613	−	0.951834i	$-0.599196\pi$
$701$	−16.2360	−0.613226	−0.306613	+	0.951834i	$0.599196\pi$
$702$	0	0
$703$	−10.0000	−0.377157
$704$	−2.18890	−0.0824973
$705$	0	0
$706$	47.0780	1.77181
$707$	−66.3900	−2.49685
$708$	0	0
$709$	41.1216	1.54435	0.772177	−	0.635408i	$-0.219168\pi$
$709$	41.1216	1.54435	0.772177	+	0.635408i	$0.219168\pi$
$710$	−0.286051	−0.0107353
$711$	0	0
$712$	14.3739	0.538683
$713$	0	0
$714$	0	0
$715$	1.20871	0.0452033
$716$	18.8818	0.705644
$717$	0	0
$718$	−21.4955	−0.802203
$719$	22.5366	0.840472	0.420236	−	0.907415i	$-0.361947\pi$
$719$	22.5366	0.840472	0.420236	+	0.907415i	$0.361947\pi$
$720$	0	0
$721$	28.7477	1.07062
$722$	−10.6784	−0.397407
$723$	0	0
$724$	−19.1652	−0.712267
$725$	14.8655	0.552089
$726$	0	0
$727$	−13.2087	−0.489884	−0.244942	−	0.969538i	$-0.578769\pi$
$727$	−13.2087	−0.489884	−0.244942	+	0.969538i	$0.578769\pi$
$728$	−7.93725	−0.294174
$729$	0	0
$730$	−10.5826	−0.391679
$731$	36.6591	1.35589
$732$	0	0
$733$	49.0345	1.81113	0.905565	−	0.424208i	$-0.139448\pi$
$733$	49.0345	1.81113	0.905565	+	0.424208i	$0.139448\pi$
$734$	3.46410	0.127862
$735$	0	0
$736$	0	0
$737$	2.64575	0.0974575
$738$	0	0
$739$	31.9129	1.17393	0.586967	−	0.809611i	$-0.300322\pi$
$739$	31.9129	1.17393	0.586967	+	0.809611i	$0.300322\pi$
$740$	−1.27520	−0.0468773
$741$	0	0
$742$	20.3739	0.747948
$743$	−10.5830	−0.388253	−0.194126	−	0.980977i	$-0.562187\pi$
$743$	−10.5830	−0.388253	−0.194126	+	0.980977i	$0.562187\pi$
$744$	0	0
$745$	−5.25227	−0.192428
$746$	6.92820	0.253660
$747$	0	0
$748$	9.58258	0.350374
$749$	−41.8553	−1.52936
$750$	0	0
$751$	−14.3303	−0.522920	−0.261460	−	0.965214i	$-0.584204\pi$
$751$	−14.3303	−0.522920	−0.261460	+	0.965214i	$0.584204\pi$
$752$	2.28425	0.0832981
$753$	0	0
$754$	6.49545	0.236550
$755$	3.36875	0.122601
$756$	0	0
$757$	9.79129	0.355870	0.177935	−	0.984042i	$-0.443058\pi$
$757$	9.79129	0.355870	0.177935	+	0.984042i	$0.443058\pi$
$758$	−50.4401	−1.83207
$759$	0	0
$760$	−2.83485	−0.102831
$761$	−51.6000	−1.87050	−0.935250	−	0.353989i	$-0.884825\pi$
$761$	−51.6000	−1.87050	−0.935250	+	0.353989i	$0.884825\pi$
$762$	0	0
$763$	25.5826	0.926151
$764$	−8.94630	−0.323666
$765$	0	0
$766$	−29.7042	−1.07326
$767$	−10.5830	−0.382130
$768$	0	0
$769$	0.417424	0.0150527	0.00752635	−	0.999972i	$-0.497604\pi$
$769$	0.417424	0.0150527	0.00752635	+	0.999972i	$0.497604\pi$
$770$	6.56670	0.236648
$771$	0	0
$772$	−16.0000	−0.575853
$773$	37.9542	1.36512	0.682559	−	0.730830i	$-0.260867\pi$
$773$	37.9542	1.36512	0.682559	+	0.730830i	$0.260867\pi$
$774$	0	0
$775$	−17.1652	−0.616590
$776$	13.8564	0.497416
$777$	0	0
$778$	−1.25227	−0.0448962
$779$	34.6410	1.24114
$780$	0	0
$781$	0.791288	0.0283145
$782$	0	0
$783$	0	0
$784$	−36.8693	−1.31676
$785$	−6.10985	−0.218070
$786$	0	0
$787$	3.58258	0.127705	0.0638525	−	0.997959i	$-0.479661\pi$
$787$	3.58258	0.127705	0.0638525	+	0.997959i	$0.479661\pi$
$788$	−25.1624	−0.896374
$789$	0	0
$790$	4.41742	0.157165
$791$	−35.9361	−1.27774
$792$	0	0
$793$	−2.16515	−0.0768868
$794$	−58.8897	−2.08992
$795$	0	0
$796$	14.7477	0.522719
$797$	12.4104	0.439599	0.219800	−	0.975545i	$-0.429460\pi$
$797$	12.4104	0.439599	0.219800	+	0.975545i	$0.429460\pi$
$798$	0	0
$799$	2.00000	0.0707549
$800$	24.8963	0.880216
$801$	0	0
$802$	60.6606	2.14200
$803$	29.2741	1.03306
$804$	0	0
$805$	0	0
$806$	−7.50030	−0.264187
$807$	0	0
$808$	−30.3303	−1.06702
$809$	−28.2849	−0.994445	−0.497222	−	0.867623i	$-0.665647\pi$
$809$	−28.2849	−0.994445	−0.497222	+	0.867623i	$0.665647\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	11.7629	0.412795
$813$	0	0
$814$	10.5826	0.370919
$815$	−1.82740	−0.0640111
$816$	0	0
$817$	−30.0000	−1.04957
$818$	−19.3386	−0.676159
$819$	0	0
$820$	4.41742	0.154263
$821$	34.9470	1.21966	0.609829	−	0.792533i	$-0.291238\pi$
$821$	34.9470	1.21966	0.609829	+	0.792533i	$0.291238\pi$
$822$	0	0
$823$	21.5826	0.752321	0.376161	−	0.926554i	$-0.377244\pi$
$823$	21.5826	0.752321	0.376161	+	0.926554i	$0.377244\pi$
$824$	13.1334	0.457524
$825$	0	0
$826$	−57.4955	−2.00052
$827$	2.93180	0.101949	0.0509744	−	0.998700i	$-0.483767\pi$
$827$	2.93180	0.101949	0.0509744	+	0.998700i	$0.483767\pi$
$828$	0	0
$829$	−19.1652	−0.665633	−0.332817	−	0.942992i	$-0.607999\pi$
$829$	−19.1652	−0.665633	−0.332817	+	0.942992i	$0.607999\pi$
$830$	0.572101	0.0198579
$831$	0	0
$832$	−1.20871	−0.0419046
$833$	−32.2813	−1.11848
$834$	0	0
$835$	−3.58258	−0.123980
$836$	−7.84190	−0.271218
$837$	0	0
$838$	−51.0345	−1.76296
$839$	3.46410	0.119594	0.0597970	−	0.998211i	$-0.480955\pi$
$839$	3.46410	0.119594	0.0597970	+	0.998211i	$0.480955\pi$
$840$	0	0
$841$	−19.3739	−0.668064
$842$	−26.2668	−0.905214
$843$	0	0
$844$	16.0000	0.550743
$845$	−5.27160	−0.181349
$846$	0	0
$847$	23.5390	0.808810
$848$	15.5130	0.532719
$849$	0	0
$850$	36.3303	1.24612
$851$	0	0
$852$	0	0
$853$	18.7477	0.641910	0.320955	−	0.947094i	$-0.395996\pi$
$853$	18.7477	0.641910	0.320955	+	0.947094i	$0.395996\pi$
$854$	−11.7629	−0.402517
$855$	0	0
$856$	−19.1216	−0.653563
$857$	−27.7128	−0.946652	−0.473326	−	0.880887i	$-0.656947\pi$
$857$	−27.7128	−0.946652	−0.473326	+	0.880887i	$0.656947\pi$
$858$	0	0
$859$	4.62614	0.157842	0.0789209	−	0.996881i	$-0.474853\pi$
$859$	4.62614	0.157842	0.0789209	+	0.996881i	$0.474853\pi$
$860$	−3.82560	−0.130452
$861$	0	0
$862$	−47.8693	−1.63044
$863$	−32.4720	−1.10536	−0.552680	−	0.833393i	$-0.686395\pi$
$863$	−32.4720	−1.10536	−0.552680	+	0.833393i	$0.686395\pi$
$864$	0	0
$865$	8.20871	0.279105
$866$	41.0568	1.39517
$867$	0	0
$868$	−13.5826	−0.461023
$869$	−12.2197	−0.414525
$870$	0	0
$871$	1.46099	0.0495036
$872$	11.6874	0.395786
$873$	0	0
$874$	0	0
$875$	16.9590	0.573319
$876$	0	0
$877$	49.6170	1.67545	0.837724	−	0.546093i	$-0.183886\pi$
$877$	49.6170	1.67545	0.837724	+	0.546093i	$0.183886\pi$
$878$	39.4757	1.33224
$879$	0	0
$880$	5.00000	0.168550
$881$	17.3404	0.584213	0.292107	−	0.956386i	$-0.405644\pi$
$881$	17.3404	0.584213	0.292107	+	0.956386i	$0.405644\pi$
$882$	0	0
$883$	−0.417424	−0.0140474	−0.00702372	−	0.999975i	$-0.502236\pi$
$883$	−0.417424	−0.0140474	−0.00702372	+	0.999975i	$0.502236\pi$
$884$	5.29150	0.177972
$885$	0	0
$886$	−10.1216	−0.340041
$887$	−9.28790	−0.311857	−0.155929	−	0.987768i	$-0.549837\pi$
$887$	−9.28790	−0.311857	−0.155929	+	0.987768i	$0.549837\pi$
$888$	0	0
$889$	3.79129	0.127156
$890$	−6.56670	−0.220116
$891$	0	0
$892$	19.5390	0.654215
$893$	−1.63670	−0.0547701
$894$	0	0
$895$	8.62614	0.288340
$896$	−45.9669	−1.53565
$897$	0	0
$898$	10.7477	0.358656
$899$	−11.1153	−0.370716
$900$	0	0
$901$	13.5826	0.452501
$902$	−36.6591	−1.22062
$903$	0	0
$904$	−16.4174	−0.546035
$905$	−8.75560	−0.291046
$906$	0	0
$907$	14.3303	0.475830	0.237915	−	0.971286i	$-0.423536\pi$
$907$	14.3303	0.475830	0.237915	+	0.971286i	$0.423536\pi$
$908$	−2.18890	−0.0726412
$909$	0	0
$910$	3.62614	0.120205
$911$	−7.00365	−0.232041	−0.116021	−	0.993247i	$-0.537014\pi$
$911$	−7.00365	−0.232041	−0.116021	+	0.993247i	$0.537014\pi$
$912$	0	0
$913$	−1.58258	−0.0523756
$914$	−18.4050	−0.608784
$915$	0	0
$916$	17.5826	0.580945
$917$	27.9989	0.924604
$918$	0	0
$919$	36.7477	1.21220	0.606098	−	0.795390i	$-0.292734\pi$
$919$	36.7477	1.21220	0.606098	+	0.795390i	$0.292734\pi$
$920$	0	0
$921$	0	0
$922$	−12.9564	−0.426698
$923$	0.436950	0.0143824
$924$	0	0
$925$	13.3739	0.439730
$926$	−6.92820	−0.227675
$927$	0	0
$928$	16.1216	0.529217
$929$	−20.9554	−0.687525	−0.343762	−	0.939057i	$-0.611701\pi$
$929$	−20.9554	−0.687525	−0.343762	+	0.939057i	$0.611701\pi$
$930$	0	0
$931$	26.4174	0.865796
$932$	19.7955	0.648422
$933$	0	0
$934$	0.330303	0.0108078
$935$	4.37780	0.143169
$936$	0	0
$937$	−51.4083	−1.67944	−0.839719	−	0.543022i	$-0.817280\pi$
$937$	−51.4083	−1.67944	−0.839719	+	0.543022i	$0.817280\pi$
$938$	7.93725	0.259161
$939$	0	0
$940$	−0.208712	−0.00680744
$941$	24.0580	0.784269	0.392134	−	0.919908i	$-0.371737\pi$
$941$	24.0580	0.784269	0.392134	+	0.919908i	$0.371737\pi$
$942$	0	0
$943$	0	0
$944$	−43.7780	−1.42485
$945$	0	0
$946$	31.7477	1.03221
$947$	−24.2487	−0.787977	−0.393989	−	0.919115i	$-0.628905\pi$
$947$	−24.2487	−0.787977	−0.393989	+	0.919115i	$0.628905\pi$
$948$	0	0
$949$	16.1652	0.524743
$950$	−29.7309	−0.964598
$951$	0	0
$952$	−28.7477	−0.931719
$953$	24.2487	0.785493	0.392746	−	0.919647i	$-0.371525\pi$
$953$	24.2487	0.785493	0.392746	+	0.919647i	$0.371525\pi$
$954$	0	0
$955$	−4.08712	−0.132256
$956$	14.2378	0.460484
$957$	0	0
$958$	−28.6170	−0.924575
$959$	−19.3386	−0.624476
$960$	0	0
$961$	−18.1652	−0.585973
$962$	5.84370	0.188409
$963$	0	0
$964$	1.16515	0.0375270
$965$	−7.30960	−0.235304
$966$	0	0
$967$	50.7913	1.63334	0.816669	−	0.577107i	$-0.195818\pi$
$967$	50.7913	1.63334	0.816669	+	0.577107i	$0.195818\pi$
$968$	10.7538	0.345640
$969$	0	0
$970$	−6.33030	−0.203254
$971$	−39.7617	−1.27601	−0.638007	−	0.770030i	$-0.720241\pi$
$971$	−39.7617	−1.27601	−0.638007	+	0.770030i	$0.720241\pi$
$972$	0	0
$973$	42.3303	1.35705
$974$	−24.1733	−0.774561
$975$	0	0
$976$	−8.95644	−0.286689
$977$	17.1298	0.548031	0.274016	−	0.961725i	$-0.411648\pi$
$977$	17.1298	0.548031	0.274016	+	0.961725i	$0.411648\pi$
$978$	0	0
$979$	18.1652	0.580561
$980$	3.36875	0.107611
$981$	0	0
$982$	−34.4174	−1.09830
$983$	−11.4213	−0.364282	−0.182141	−	0.983272i	$-0.558303\pi$
$983$	−11.4213	−0.364282	−0.182141	+	0.983272i	$0.558303\pi$
$984$	0	0
$985$	−11.4955	−0.366276
$986$	23.5257	0.749211
$987$	0	0
$988$	−4.33030	−0.137765
$989$	0	0
$990$	0	0
$991$	26.9564	0.856300	0.428150	−	0.903708i	$-0.359166\pi$
$991$	26.9564	0.856300	0.428150	+	0.903708i	$0.359166\pi$
$992$	−18.6156	−0.591046
$993$	0	0
$994$	2.37386	0.0752944
$995$	6.73750	0.213593
$996$	0	0
$997$	−11.5826	−0.366824	−0.183412	−	0.983036i	$-0.558714\pi$
$997$	−11.5826	−0.366824	−0.183412	+	0.983036i	$0.558714\pi$
$998$	42.6537	1.35018
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.f.1.4	yes	4
3.2	odd	2	inner	1143.2.a.f.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.2.a.f.1.1	✓	4	3.2	odd	2	inner
1143.2.a.f.1.4	yes	4	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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} +0.456850 q^{5} -3.79129 q^{7} -1.73205 q^{8} +O(q^{10})\)	\(q+1.73205 q^{2} +1.00000 q^{4} +0.456850 q^{5} -3.79129 q^{7} -1.73205 q^{8} +0.791288 q^{10} -2.18890 q^{11} -1.20871 q^{13} -6.56670 q^{14} -5.00000 q^{16} -4.37780 q^{17} +3.58258 q^{19} +0.456850 q^{20} -3.79129 q^{22} -4.79129 q^{25} -2.09355 q^{26} -3.79129 q^{28} -3.10260 q^{29} +3.58258 q^{31} -5.19615 q^{32} -7.58258 q^{34} -1.73205 q^{35} -2.79129 q^{37} +6.20520 q^{38} -0.791288 q^{40} +9.66930 q^{41} -8.37386 q^{43} -2.18890 q^{44} -0.456850 q^{47} +7.37386 q^{49} -8.29875 q^{50} -1.20871 q^{52} -3.10260 q^{53} -1.00000 q^{55} +6.56670 q^{56} -5.37386 q^{58} +8.75560 q^{59} +1.79129 q^{61} +6.20520 q^{62} +1.00000 q^{64} -0.552200 q^{65} -1.20871 q^{67} -4.37780 q^{68} -3.00000 q^{70} -0.361500 q^{71} -13.3739 q^{73} -4.83465 q^{74} +3.58258 q^{76} +8.29875 q^{77} +5.58258 q^{79} -2.28425 q^{80} +16.7477 q^{82} +0.723000 q^{83} -2.00000 q^{85} -14.5040 q^{86} +3.79129 q^{88} -8.29875 q^{89} +4.58258 q^{91} -0.791288 q^{94} +1.63670 q^{95} -8.00000 q^{97} +12.7719 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 6 q^{7}+O(q^{10}) \) 4 * q + 4 * q^4 - 6 * q^7	\( 4 q + 4 q^{4} - 6 q^{7} - 6 q^{10} - 14 q^{13} - 20 q^{16} - 4 q^{19} - 6 q^{22} - 10 q^{25} - 6 q^{28} - 4 q^{31} - 12 q^{34} - 2 q^{37} + 6 q^{40} - 6 q^{43} + 2 q^{49} - 14 q^{52} - 4 q^{55} + 6 q^{58} - 2 q^{61} + 4 q^{64} - 14 q^{67} - 12 q^{70} - 26 q^{73} - 4 q^{76} + 4 q^{79} + 12 q^{82} - 8 q^{85} + 6 q^{88} + 6 q^{94} - 32 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 6 * q^7 - 6 * q^10 - 14 * q^13 - 20 * q^16 - 4 * q^19 - 6 * q^22 - 10 * q^25 - 6 * q^28 - 4 * q^31 - 12 * q^34 - 2 * q^37 + 6 * q^40 - 6 * q^43 + 2 * q^49 - 14 * q^52 - 4 * q^55 + 6 * q^58 - 2 * q^61 + 4 * q^64 - 14 * q^67 - 12 * q^70 - 26 * q^73 - 4 * q^76 + 4 * q^79 + 12 * q^82 - 8 * q^85 + 6 * q^88 + 6 * q^94 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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