LMFDB - Embedded newform 1881.2.a.g.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1881,2,Mod(1,1881)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1881.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1881, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1881.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,2,0,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$15.0198606202$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 627)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	1881.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.24698 q^{2} +3.04892 q^{4} +2.44504 q^{5} -0.911854 q^{7} +2.35690 q^{8} +5.49396 q^{10} +1.00000 q^{11} +2.85086 q^{13} -2.04892 q^{14} -0.801938 q^{16} -1.85086 q^{17} +1.00000 q^{19} +7.45473 q^{20} +2.24698 q^{22} +7.49396 q^{23} +0.978230 q^{25} +6.40581 q^{26} -2.78017 q^{28} +9.78986 q^{29} +3.78986 q^{31} -6.51573 q^{32} -4.15883 q^{34} -2.22952 q^{35} -8.38404 q^{37} +2.24698 q^{38} +5.76271 q^{40} -5.96077 q^{41} +3.48858 q^{43} +3.04892 q^{44} +16.8388 q^{46} +3.57673 q^{47} -6.16852 q^{49} +2.19806 q^{50} +8.69202 q^{52} +9.26875 q^{53} +2.44504 q^{55} -2.14914 q^{56} +21.9976 q^{58} -2.02177 q^{59} -7.21313 q^{61} +8.51573 q^{62} -13.0368 q^{64} +6.97046 q^{65} -14.3327 q^{67} -5.64310 q^{68} -5.00969 q^{70} -9.05861 q^{71} +0.917231 q^{73} -18.8388 q^{74} +3.04892 q^{76} -0.911854 q^{77} -13.9487 q^{79} -1.96077 q^{80} -13.3937 q^{82} +10.5429 q^{83} -4.52542 q^{85} +7.83877 q^{86} +2.35690 q^{88} +4.00969 q^{89} -2.59956 q^{91} +22.8485 q^{92} +8.03684 q^{94} +2.44504 q^{95} +4.40581 q^{97} -13.8605 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 7 q^{5} + q^{7} + 3 q^{8} + 7 q^{10} + 3 q^{11} - 5 q^{13} + 3 q^{14} + 2 q^{16} + 8 q^{17} + 3 q^{19} + 2 q^{22} + 13 q^{23} + 6 q^{25} + 6 q^{26} - 7 q^{28} + 6 q^{29} - 12 q^{31} - 7 q^{32}+ \cdots - 6 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 7 * q^5 + q^7 + 3 * q^8 + 7 * q^10 + 3 * q^11 - 5 * q^13 + 3 * q^14 + 2 * q^16 + 8 * q^17 + 3 * q^19 + 2 * q^22 + 13 * q^23 + 6 * q^25 + 6 * q^26 - 7 * q^28 + 6 * q^29 - 12 * q^31 - 7 * q^32 - 4 * q^34 + 14 * q^35 - 15 * q^37 + 2 * q^38 - 5 * q^41 + 4 * q^43 + 18 * q^46 + 8 * q^47 + 12 * q^49 + 11 * q^50 + 21 * q^52 + 20 * q^53 + 7 * q^55 - 20 * q^56 + 25 * q^58 - 3 * q^59 - q^61 + 13 * q^62 - 11 * q^64 - 14 * q^65 - q^67 - 21 * q^68 + 7 * q^70 + 4 * q^71 - 4 * q^73 - 24 * q^74 + q^77 - 10 * q^79 + 7 * q^80 - 8 * q^82 + 13 * q^83 + 21 * q^85 - 9 * q^86 + 3 * q^88 - 10 * q^89 - 18 * q^91 + 14 * q^92 - 4 * q^94 + 7 * q^95 - 6 * q^98

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 $

$2$	2.24698	1.58885	0.794427	−	0.607359i	$-0.207771\pi$
$2$	2.24698	1.58885	0.794427	+	0.607359i	$0.207771\pi$
$3$	0	0
$3$	0	0
$4$	3.04892	1.52446
$5$	2.44504	1.09346	0.546728	−	0.837310i	$-0.315873\pi$
$5$	2.44504	1.09346	0.546728	+	0.837310i	$0.315873\pi$
$6$	0	0
$7$	−0.911854	−0.344648	−0.172324	−	0.985040i	$-0.555128\pi$
$7$	−0.911854	−0.344648	−0.172324	+	0.985040i	$0.555128\pi$
$8$	2.35690	0.833289
$9$	0	0
$10$	5.49396	1.73734
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.85086	0.790685	0.395342	−	0.918534i	$-0.370626\pi$
$13$	2.85086	0.790685	0.395342	+	0.918534i	$0.370626\pi$
$14$	−2.04892	−0.547596
$15$	0	0
$16$	−0.801938	−0.200484
$17$	−1.85086	−0.448898	−0.224449	−	0.974486i	$-0.572058\pi$
$17$	−1.85086	−0.448898	−0.224449	+	0.974486i	$0.572058\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	7.45473	1.66693
$21$	0	0
$22$	2.24698	0.479058
$23$	7.49396	1.56260	0.781299	−	0.624157i	$-0.214557\pi$
$23$	7.49396	1.56260	0.781299	+	0.624157i	$0.214557\pi$
$24$	0	0
$25$	0.978230	0.195646
$26$	6.40581	1.25628
$27$	0	0
$28$	−2.78017	−0.525402
$29$	9.78986	1.81793	0.908965	−	0.416872i	$-0.136874\pi$
$29$	9.78986	1.81793	0.908965	+	0.416872i	$0.136874\pi$
$30$	0	0
$31$	3.78986	0.680678	0.340339	−	0.940303i	$-0.389458\pi$
$31$	3.78986	0.680678	0.340339	+	0.940303i	$0.389458\pi$
$32$	−6.51573	−1.15183
$33$	0	0
$34$	−4.15883	−0.713234
$35$	−2.22952	−0.376858
$36$	0	0
$37$	−8.38404	−1.37833	−0.689164	−	0.724605i	$-0.742022\pi$
$37$	−8.38404	−1.37833	−0.689164	+	0.724605i	$0.742022\pi$
$38$	2.24698	0.364508
$39$	0	0
$40$	5.76271	0.911164
$41$	−5.96077	−0.930916	−0.465458	−	0.885070i	$-0.654110\pi$
$41$	−5.96077	−0.930916	−0.465458	+	0.885070i	$0.654110\pi$
$42$	0	0
$43$	3.48858	0.532004	0.266002	−	0.963973i	$-0.414297\pi$
$43$	3.48858	0.532004	0.266002	+	0.963973i	$0.414297\pi$
$44$	3.04892	0.459642
$45$	0	0
$46$	16.8388	2.48274
$47$	3.57673	0.521719	0.260860	−	0.965377i	$-0.415994\pi$
$47$	3.57673	0.521719	0.260860	+	0.965377i	$0.415994\pi$
$48$	0	0
$49$	−6.16852	−0.881217
$50$	2.19806	0.310853
$51$	0	0
$52$	8.69202	1.20537
$53$	9.26875	1.27316	0.636580	−	0.771210i	$-0.280348\pi$
$53$	9.26875	1.27316	0.636580	+	0.771210i	$0.280348\pi$
$54$	0	0
$55$	2.44504	0.329689
$56$	−2.14914	−0.287192
$57$	0	0
$58$	21.9976	2.88843
$59$	−2.02177	−0.263212	−0.131606	−	0.991302i	$-0.542013\pi$
$59$	−2.02177	−0.263212	−0.131606	+	0.991302i	$0.542013\pi$
$60$	0	0
$61$	−7.21313	−0.923546	−0.461773	−	0.886998i	$-0.652787\pi$
$61$	−7.21313	−0.923546	−0.461773	+	0.886998i	$0.652787\pi$
$62$	8.51573	1.08150
$63$	0	0
$64$	−13.0368	−1.62960
$65$	6.97046	0.864579
$66$	0	0
$67$	−14.3327	−1.75102	−0.875511	−	0.483198i	$-0.839475\pi$
$67$	−14.3327	−1.75102	−0.875511	+	0.483198i	$0.839475\pi$
$68$	−5.64310	−0.684327
$69$	0	0
$70$	−5.00969	−0.598772
$71$	−9.05861	−1.07506	−0.537529	−	0.843245i	$-0.680642\pi$
$71$	−9.05861	−1.07506	−0.537529	+	0.843245i	$0.680642\pi$
$72$	0	0
$73$	0.917231	0.107354	0.0536769	−	0.998558i	$-0.482906\pi$
$73$	0.917231	0.107354	0.0536769	+	0.998558i	$0.482906\pi$
$74$	−18.8388	−2.18996
$75$	0	0
$76$	3.04892	0.349735
$77$	−0.911854	−0.103915
$78$	0	0
$79$	−13.9487	−1.56935	−0.784675	−	0.619907i	$-0.787170\pi$
$79$	−13.9487	−1.56935	−0.784675	+	0.619907i	$0.787170\pi$
$80$	−1.96077	−0.219221
$81$	0	0
$82$	−13.3937	−1.47909
$83$	10.5429	1.15723	0.578616	−	0.815600i	$-0.303593\pi$
$83$	10.5429	1.15723	0.578616	+	0.815600i	$0.303593\pi$
$84$	0	0
$85$	−4.52542	−0.490851
$86$	7.83877	0.845277
$87$	0	0
$88$	2.35690	0.251246
$89$	4.00969	0.425026	0.212513	−	0.977158i	$-0.431835\pi$
$89$	4.00969	0.425026	0.212513	+	0.977158i	$0.431835\pi$
$90$	0	0
$91$	−2.59956	−0.272508
$92$	22.8485	2.38212
$93$	0	0
$94$	8.03684	0.828936
$95$	2.44504	0.250856
$96$	0	0
$97$	4.40581	0.447343	0.223671	−	0.974665i	$-0.428196\pi$
$97$	4.40581	0.447343	0.223671	+	0.974665i	$0.428196\pi$
$98$	−13.8605	−1.40013
$99$	0	0
$100$	2.98254	0.298254
$101$	−7.70709	−0.766884	−0.383442	−	0.923565i	$-0.625261\pi$
$101$	−7.70709	−0.766884	−0.383442	+	0.923565i	$0.625261\pi$
$102$	0	0
$103$	−12.6136	−1.24285	−0.621426	−	0.783473i	$-0.713446\pi$
$103$	−12.6136	−1.24285	−0.621426	+	0.783473i	$0.713446\pi$
$104$	6.71917	0.658869
$105$	0	0
$106$	20.8267	2.02287
$107$	14.6136	1.41275	0.706373	−	0.707840i	$-0.250330\pi$
$107$	14.6136	1.41275	0.706373	+	0.707840i	$0.250330\pi$
$108$	0	0
$109$	3.99031	0.382203	0.191101	−	0.981570i	$-0.438794\pi$
$109$	3.99031	0.382203	0.191101	+	0.981570i	$0.438794\pi$
$110$	5.49396	0.523828
$111$	0	0
$112$	0.731250	0.0690966
$113$	−8.10321	−0.762286	−0.381143	−	0.924516i	$-0.624469\pi$
$113$	−8.10321	−0.762286	−0.381143	+	0.924516i	$0.624469\pi$
$114$	0	0
$115$	18.3230	1.70863
$116$	29.8485	2.77136
$117$	0	0
$118$	−4.54288	−0.418206
$119$	1.68771	0.154712
$120$	0	0
$121$	1.00000	0.0909091
$122$	−16.2078	−1.46738
$123$	0	0
$124$	11.5550	1.03767
$125$	−9.83340	−0.879526
$126$	0	0
$127$	−4.29590	−0.381199	−0.190600	−	0.981668i	$-0.561043\pi$
$127$	−4.29590	−0.381199	−0.190600	+	0.981668i	$0.561043\pi$
$128$	−16.2620	−1.43738
$129$	0	0
$130$	15.6625	1.37369
$131$	−13.7560	−1.20187	−0.600934	−	0.799299i	$-0.705204\pi$
$131$	−13.7560	−1.20187	−0.600934	+	0.799299i	$0.705204\pi$
$132$	0	0
$133$	−0.911854	−0.0790678
$134$	−32.2054	−2.78212
$135$	0	0
$136$	−4.36227	−0.374062
$137$	2.11529	0.180722	0.0903608	−	0.995909i	$-0.471198\pi$
$137$	2.11529	0.180722	0.0903608	+	0.995909i	$0.471198\pi$
$138$	0	0
$139$	18.3884	1.55968	0.779840	−	0.625979i	$-0.215300\pi$
$139$	18.3884	1.55968	0.779840	+	0.625979i	$0.215300\pi$
$140$	−6.79763	−0.574504
$141$	0	0
$142$	−20.3545	−1.70811
$143$	2.85086	0.238400
$144$	0	0
$145$	23.9366	1.98783
$146$	2.06100	0.170569
$147$	0	0
$148$	−25.5623	−2.10120
$149$	−4.45473	−0.364946	−0.182473	−	0.983211i	$-0.558410\pi$
$149$	−4.45473	−0.364946	−0.182473	+	0.983211i	$0.558410\pi$
$150$	0	0
$151$	12.3787	1.00736	0.503681	−	0.863890i	$-0.331979\pi$
$151$	12.3787	1.00736	0.503681	+	0.863890i	$0.331979\pi$
$152$	2.35690	0.191169
$153$	0	0
$154$	−2.04892	−0.165106
$155$	9.26636	0.744292
$156$	0	0
$157$	−11.3690	−0.907343	−0.453672	−	0.891169i	$-0.649886\pi$
$157$	−11.3690	−0.907343	−0.453672	+	0.891169i	$0.649886\pi$
$158$	−31.3424	−2.49347
$159$	0	0
$160$	−15.9312	−1.25947
$161$	−6.83340	−0.538547
$162$	0	0
$163$	13.0640	1.02325	0.511625	−	0.859209i	$-0.329044\pi$
$163$	13.0640	1.02325	0.511625	+	0.859209i	$0.329044\pi$
$164$	−18.1739	−1.41914
$165$	0	0
$166$	23.6896	1.83867
$167$	−14.0683	−1.08864	−0.544319	−	0.838879i	$-0.683212\pi$
$167$	−14.0683	−1.08864	−0.544319	+	0.838879i	$0.683212\pi$
$168$	0	0
$169$	−4.87263	−0.374817
$170$	−10.1685	−0.779890
$171$	0	0
$172$	10.6364	0.811018
$173$	−4.15883	−0.316190	−0.158095	−	0.987424i	$-0.550535\pi$
$173$	−4.15883	−0.316190	−0.158095	+	0.987424i	$0.550535\pi$
$174$	0	0
$175$	−0.892003	−0.0674291
$176$	−0.801938	−0.0604483
$177$	0	0
$178$	9.00969	0.675305
$179$	7.41789	0.554439	0.277220	−	0.960807i	$-0.410587\pi$
$179$	7.41789	0.554439	0.277220	+	0.960807i	$0.410587\pi$
$180$	0	0
$181$	−7.91723	−0.588483	−0.294242	−	0.955731i	$-0.595067\pi$
$181$	−7.91723	−0.588483	−0.294242	+	0.955731i	$0.595067\pi$
$182$	−5.84117	−0.432976
$183$	0	0
$184$	17.6625	1.30210
$185$	−20.4993	−1.50714
$186$	0	0
$187$	−1.85086	−0.135348
$188$	10.9051	0.795340
$189$	0	0
$190$	5.49396	0.398574
$191$	−1.17390	−0.0849404	−0.0424702	−	0.999098i	$-0.513523\pi$
$191$	−1.17390	−0.0849404	−0.0424702	+	0.999098i	$0.513523\pi$
$192$	0	0
$193$	−18.2664	−1.31484	−0.657421	−	0.753524i	$-0.728352\pi$
$193$	−18.2664	−1.31484	−0.657421	+	0.753524i	$0.728352\pi$
$194$	9.89977	0.710762
$195$	0	0
$196$	−18.8073	−1.34338
$197$	−15.4819	−1.10304	−0.551519	−	0.834162i	$-0.685952\pi$
$197$	−15.4819	−1.10304	−0.551519	+	0.834162i	$0.685952\pi$
$198$	0	0
$199$	11.5429	0.818253	0.409126	−	0.912478i	$-0.365834\pi$
$199$	11.5429	0.818253	0.409126	+	0.912478i	$0.365834\pi$
$200$	2.30559	0.163030
$201$	0	0
$202$	−17.3177	−1.21847
$203$	−8.92692	−0.626547
$204$	0	0
$205$	−14.5743	−1.01792
$206$	−28.3424	−1.97471
$207$	0	0
$208$	−2.28621	−0.158520
$209$	1.00000	0.0691714
$210$	0	0
$211$	4.44504	0.306009	0.153005	−	0.988225i	$-0.451105\pi$
$211$	4.44504	0.306009	0.153005	+	0.988225i	$0.451105\pi$
$212$	28.2597	1.94088
$213$	0	0
$214$	32.8364	2.24465
$215$	8.52973	0.581723
$216$	0	0
$217$	−3.45580	−0.234595
$218$	8.96615	0.607264
$219$	0	0
$220$	7.45473	0.502598
$221$	−5.27652	−0.354937
$222$	0	0
$223$	−7.72455	−0.517274	−0.258637	−	0.965975i	$-0.583273\pi$
$223$	−7.72455	−0.517274	−0.258637	+	0.965975i	$0.583273\pi$
$224$	5.94139	0.396976
$225$	0	0
$226$	−18.2078	−1.21116
$227$	11.7657	0.780916	0.390458	−	0.920621i	$-0.372317\pi$
$227$	11.7657	0.780916	0.390458	+	0.920621i	$0.372317\pi$
$228$	0	0
$229$	15.1414	1.00057	0.500285	−	0.865861i	$-0.333229\pi$
$229$	15.1414	1.00057	0.500285	+	0.865861i	$0.333229\pi$
$230$	41.1715	2.71477
$231$	0	0
$232$	23.0737	1.51486
$233$	7.70841	0.504995	0.252498	−	0.967598i	$-0.418748\pi$
$233$	7.70841	0.504995	0.252498	+	0.967598i	$0.418748\pi$
$234$	0	0
$235$	8.74525	0.570477
$236$	−6.16421	−0.401256
$237$	0	0
$238$	3.79225	0.245815
$239$	6.57002	0.424979	0.212490	−	0.977163i	$-0.431843\pi$
$239$	6.57002	0.424979	0.212490	+	0.977163i	$0.431843\pi$
$240$	0	0
$241$	−5.45712	−0.351524	−0.175762	−	0.984433i	$-0.556239\pi$
$241$	−5.45712	−0.351524	−0.175762	+	0.984433i	$0.556239\pi$
$242$	2.24698	0.144441
$243$	0	0
$244$	−21.9922	−1.40791
$245$	−15.0823	−0.963572
$246$	0	0
$247$	2.85086	0.181396
$248$	8.93230	0.567201
$249$	0	0
$250$	−22.0954	−1.39744
$251$	−11.3894	−0.718894	−0.359447	−	0.933166i	$-0.617035\pi$
$251$	−11.3894	−0.718894	−0.359447	+	0.933166i	$0.617035\pi$
$252$	0	0
$253$	7.49396	0.471141
$254$	−9.65279	−0.605670
$255$	0	0
$256$	−10.4668	−0.654176
$257$	9.08144	0.566485	0.283242	−	0.959048i	$-0.408590\pi$
$257$	9.08144	0.566485	0.283242	+	0.959048i	$0.408590\pi$
$258$	0	0
$259$	7.64502	0.475039
$260$	21.2524	1.31802
$261$	0	0
$262$	−30.9095	−1.90959
$263$	−17.0761	−1.05296	−0.526478	−	0.850189i	$-0.676488\pi$
$263$	−17.0761	−1.05296	−0.526478	+	0.850189i	$0.676488\pi$
$264$	0	0
$265$	22.6625	1.39215
$266$	−2.04892	−0.125627
$267$	0	0
$268$	−43.6993	−2.66936
$269$	−14.6582	−0.893724	−0.446862	−	0.894603i	$-0.647459\pi$
$269$	−14.6582	−0.893724	−0.446862	+	0.894603i	$0.647459\pi$
$270$	0	0
$271$	15.6189	0.948783	0.474392	−	0.880314i	$-0.342668\pi$
$271$	15.6189	0.948783	0.474392	+	0.880314i	$0.342668\pi$
$272$	1.48427	0.0899971
$273$	0	0
$274$	4.75302	0.287140
$275$	0.978230	0.0589895
$276$	0	0
$277$	−24.7657	−1.48803	−0.744013	−	0.668165i	$-0.767080\pi$
$277$	−24.7657	−1.48803	−0.744013	+	0.668165i	$0.767080\pi$
$278$	41.3183	2.47810
$279$	0	0
$280$	−5.25475	−0.314031
$281$	23.9463	1.42852	0.714258	−	0.699882i	$-0.246764\pi$
$281$	23.9463	1.42852	0.714258	+	0.699882i	$0.246764\pi$
$282$	0	0
$283$	5.75063	0.341839	0.170920	−	0.985285i	$-0.445326\pi$
$283$	5.75063	0.341839	0.170920	+	0.985285i	$0.445326\pi$
$284$	−27.6189	−1.63888
$285$	0	0
$286$	6.40581	0.378784
$287$	5.43535	0.320839
$288$	0	0
$289$	−13.5743	−0.798490
$290$	53.7851	3.15837
$291$	0	0
$292$	2.79656	0.163656
$293$	8.27114	0.483205	0.241603	−	0.970375i	$-0.422327\pi$
$293$	8.27114	0.483205	0.241603	+	0.970375i	$0.422327\pi$
$294$	0	0
$295$	−4.94331	−0.287811
$296$	−19.7603	−1.14855
$297$	0	0
$298$	−10.0097	−0.579846
$299$	21.3642	1.23552
$300$	0	0
$301$	−3.18108	−0.183354
$302$	27.8146	1.60055
$303$	0	0
$304$	−0.801938	−0.0459943
$305$	−17.6364	−1.00986
$306$	0	0
$307$	17.1226	0.977238	0.488619	−	0.872497i	$-0.337501\pi$
$307$	17.1226	0.977238	0.488619	+	0.872497i	$0.337501\pi$
$308$	−2.78017	−0.158415
$309$	0	0
$310$	20.8213	1.18257
$311$	−22.8025	−1.29301	−0.646506	−	0.762909i	$-0.723771\pi$
$311$	−22.8025	−1.29301	−0.646506	+	0.762909i	$0.723771\pi$
$312$	0	0
$313$	−8.07069	−0.456182	−0.228091	−	0.973640i	$-0.573248\pi$
$313$	−8.07069	−0.456182	−0.228091	+	0.973640i	$0.573248\pi$
$314$	−25.5459	−1.44164
$315$	0	0
$316$	−42.5284	−2.39241
$317$	6.76271	0.379832	0.189916	−	0.981800i	$-0.439179\pi$
$317$	6.76271	0.379832	0.189916	+	0.981800i	$0.439179\pi$
$318$	0	0
$319$	9.78986	0.548127
$320$	−31.8756	−1.78190
$321$	0	0
$322$	−15.3545	−0.855673
$323$	−1.85086	−0.102984
$324$	0	0
$325$	2.78879	0.154694
$326$	29.3545	1.62580
$327$	0	0
$328$	−14.0489	−0.775722
$329$	−3.26145	−0.179810
$330$	0	0
$331$	22.3739	1.22978	0.614890	−	0.788613i	$-0.289200\pi$
$331$	22.3739	1.22978	0.614890	+	0.788613i	$0.289200\pi$
$332$	32.1444	1.76415
$333$	0	0
$334$	−31.6112	−1.72969
$335$	−35.0441	−1.91467
$336$	0	0
$337$	21.1323	1.15115	0.575574	−	0.817750i	$-0.304779\pi$
$337$	21.1323	1.15115	0.575574	+	0.817750i	$0.304779\pi$
$338$	−10.9487	−0.595530
$339$	0	0
$340$	−13.7976	−0.748281
$341$	3.78986	0.205232
$342$	0	0
$343$	12.0078	0.648359
$344$	8.22223	0.443313
$345$	0	0
$346$	−9.34481	−0.502380
$347$	29.9288	1.60666	0.803332	−	0.595531i	$-0.203058\pi$
$347$	29.9288	1.60666	0.803332	+	0.595531i	$0.203058\pi$
$348$	0	0
$349$	−29.7603	−1.59303	−0.796517	−	0.604617i	$-0.793326\pi$
$349$	−29.7603	−1.59303	−0.796517	+	0.604617i	$0.793326\pi$
$350$	−2.00431	−0.107135
$351$	0	0
$352$	−6.51573	−0.347290
$353$	−17.9782	−0.956885	−0.478442	−	0.878119i	$-0.658798\pi$
$353$	−17.9782	−0.956885	−0.478442	+	0.878119i	$0.658798\pi$
$354$	0	0
$355$	−22.1487	−1.17553
$356$	12.2252	0.647935
$357$	0	0
$358$	16.6679	0.880924
$359$	−23.3163	−1.23059	−0.615295	−	0.788297i	$-0.710963\pi$
$359$	−23.3163	−1.23059	−0.615295	+	0.788297i	$0.710963\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−17.7899	−0.935014
$363$	0	0
$364$	−7.92585	−0.415428
$365$	2.24267	0.117387
$366$	0	0
$367$	14.2784	0.745328	0.372664	−	0.927966i	$-0.378444\pi$
$367$	14.2784	0.745328	0.372664	+	0.927966i	$0.378444\pi$
$368$	−6.00969	−0.313277
$369$	0	0
$370$	−46.0616	−2.39463
$371$	−8.45175	−0.438793
$372$	0	0
$373$	15.1540	0.784647	0.392323	−	0.919827i	$-0.371671\pi$
$373$	15.1540	0.784647	0.392323	+	0.919827i	$0.371671\pi$
$374$	−4.15883	−0.215048
$375$	0	0
$376$	8.42998	0.434743
$377$	27.9095	1.43741
$378$	0	0
$379$	−8.05861	−0.413943	−0.206971	−	0.978347i	$-0.566361\pi$
$379$	−8.05861	−0.413943	−0.206971	+	0.978347i	$0.566361\pi$
$380$	7.45473	0.382420
$381$	0	0
$382$	−2.63773	−0.134958
$383$	5.09783	0.260487	0.130244	−	0.991482i	$-0.458424\pi$
$383$	5.09783	0.260487	0.130244	+	0.991482i	$0.458424\pi$
$384$	0	0
$385$	−2.22952	−0.113627
$386$	−41.0441	−2.08909
$387$	0	0
$388$	13.4330	0.681955
$389$	11.7181	0.594131	0.297066	−	0.954857i	$-0.403992\pi$
$389$	11.7181	0.594131	0.297066	+	0.954857i	$0.403992\pi$
$390$	0	0
$391$	−13.8702	−0.701448
$392$	−14.5386	−0.734308
$393$	0	0
$394$	−34.7875	−1.75257
$395$	−34.1051	−1.71602
$396$	0	0
$397$	−6.75196	−0.338871	−0.169435	−	0.985541i	$-0.554194\pi$
$397$	−6.75196	−0.338871	−0.169435	+	0.985541i	$0.554194\pi$
$398$	25.9366	1.30008
$399$	0	0
$400$	−0.784479	−0.0392240
$401$	−26.0761	−1.30218	−0.651088	−	0.759002i	$-0.725687\pi$
$401$	−26.0761	−1.30218	−0.651088	+	0.759002i	$0.725687\pi$
$402$	0	0
$403$	10.8043	0.538202
$404$	−23.4983	−1.16908
$405$	0	0
$406$	−20.0586	−0.995492
$407$	−8.38404	−0.415582
$408$	0	0
$409$	22.4698	1.11106	0.555530	−	0.831497i	$-0.312515\pi$
$409$	22.4698	1.11106	0.555530	+	0.831497i	$0.312515\pi$
$410$	−32.7482	−1.61732
$411$	0	0
$412$	−38.4577	−1.89468
$413$	1.84356	0.0907156
$414$	0	0
$415$	25.7778	1.26538
$416$	−18.5754	−0.910734
$417$	0	0
$418$	2.24698	0.109903
$419$	25.8984	1.26522	0.632611	−	0.774470i	$-0.281983\pi$
$419$	25.8984	1.26522	0.632611	+	0.774470i	$0.281983\pi$
$420$	0	0
$421$	−30.1371	−1.46879	−0.734395	−	0.678722i	$-0.762534\pi$
$421$	−30.1371	−1.46879	−0.734395	+	0.678722i	$0.762534\pi$
$422$	9.98792	0.486204
$423$	0	0
$424$	21.8455	1.06091
$425$	−1.81056	−0.0878251
$426$	0	0
$427$	6.57732	0.318299
$428$	44.5555	2.15367
$429$	0	0
$430$	19.1661	0.924273
$431$	38.5090	1.85491	0.927457	−	0.373929i	$-0.121990\pi$
$431$	38.5090	1.85491	0.927457	+	0.373929i	$0.121990\pi$
$432$	0	0
$433$	−27.4330	−1.31834	−0.659172	−	0.751992i	$-0.729093\pi$
$433$	−27.4330	−1.31834	−0.659172	+	0.751992i	$0.729093\pi$
$434$	−7.76510	−0.372737
$435$	0	0
$436$	12.1661	0.582652
$437$	7.49396	0.358485
$438$	0	0
$439$	−19.7362	−0.941955	−0.470978	−	0.882145i	$-0.656099\pi$
$439$	−19.7362	−0.941955	−0.470978	+	0.882145i	$0.656099\pi$
$440$	5.76271	0.274726
$441$	0	0
$442$	−11.8562	−0.563943
$443$	−41.0237	−1.94909	−0.974547	−	0.224183i	$-0.928029\pi$
$443$	−41.0237	−1.94909	−0.974547	+	0.224183i	$0.928029\pi$
$444$	0	0
$445$	9.80386	0.464747
$446$	−17.3569	−0.821873
$447$	0	0
$448$	11.8877	0.561641
$449$	−0.353912	−0.0167021	−0.00835107	−	0.999965i	$-0.502658\pi$
$449$	−0.353912	−0.0167021	−0.00835107	+	0.999965i	$0.502658\pi$
$450$	0	0
$451$	−5.96077	−0.280682
$452$	−24.7060	−1.16207
$453$	0	0
$454$	26.4373	1.24076
$455$	−6.35604	−0.297976
$456$	0	0
$457$	2.05131	0.0959562	0.0479781	−	0.998848i	$-0.484722\pi$
$457$	2.05131	0.0959562	0.0479781	+	0.998848i	$0.484722\pi$
$458$	34.0224	1.58976
$459$	0	0
$460$	55.8654	2.60474
$461$	33.5187	1.56112	0.780561	−	0.625080i	$-0.214933\pi$
$461$	33.5187	1.56112	0.780561	+	0.625080i	$0.214933\pi$
$462$	0	0
$463$	−13.8834	−0.645215	−0.322608	−	0.946533i	$-0.604559\pi$
$463$	−13.8834	−0.645215	−0.322608	+	0.946533i	$0.604559\pi$
$464$	−7.85086	−0.364467
$465$	0	0
$466$	17.3207	0.802364
$467$	8.08575	0.374164	0.187082	−	0.982344i	$-0.440097\pi$
$467$	8.08575	0.374164	0.187082	+	0.982344i	$0.440097\pi$
$468$	0	0
$469$	13.0694	0.603487
$470$	19.6504	0.906405
$471$	0	0
$472$	−4.76510	−0.219332
$473$	3.48858	0.160405
$474$	0	0
$475$	0.978230	0.0448843
$476$	5.14569	0.235852
$477$	0	0
$478$	14.7627	0.675231
$479$	−10.1371	−0.463174	−0.231587	−	0.972814i	$-0.574392\pi$
$479$	−10.1371	−0.463174	−0.231587	+	0.972814i	$0.574392\pi$
$480$	0	0
$481$	−23.9017	−1.08982
$482$	−12.2620	−0.558521
$483$	0	0
$484$	3.04892	0.138587
$485$	10.7724	0.489149
$486$	0	0
$487$	−25.3980	−1.15090	−0.575448	−	0.817839i	$-0.695172\pi$
$487$	−25.3980	−1.15090	−0.575448	+	0.817839i	$0.695172\pi$
$488$	−17.0006	−0.769581
$489$	0	0
$490$	−33.8896	−1.53098
$491$	32.5870	1.47063	0.735316	−	0.677725i	$-0.237034\pi$
$491$	32.5870	1.47063	0.735316	+	0.677725i	$0.237034\pi$
$492$	0	0
$493$	−18.1196	−0.816066
$494$	6.40581	0.288211
$495$	0	0
$496$	−3.03923	−0.136465
$497$	8.26013	0.370517
$498$	0	0
$499$	39.3569	1.76186	0.880928	−	0.473250i	$-0.156919\pi$
$499$	39.3569	1.76186	0.880928	+	0.473250i	$0.156919\pi$
$500$	−29.9812	−1.34080
$501$	0	0
$502$	−25.5918	−1.14222
$503$	24.0325	1.07156	0.535779	−	0.844358i	$-0.320018\pi$
$503$	24.0325	1.07156	0.535779	+	0.844358i	$0.320018\pi$
$504$	0	0
$505$	−18.8442	−0.838554
$506$	16.8388	0.748575
$507$	0	0
$508$	−13.0978	−0.581122
$509$	36.4620	1.61615	0.808075	−	0.589079i	$-0.200509\pi$
$509$	36.4620	1.61615	0.808075	+	0.589079i	$0.200509\pi$
$510$	0	0
$511$	−0.836381	−0.0369993
$512$	9.00538	0.397985
$513$	0	0
$514$	20.4058	0.900062
$515$	−30.8407	−1.35900
$516$	0	0
$517$	3.57673	0.157304
$518$	17.1782	0.754767
$519$	0	0
$520$	16.4286	0.720444
$521$	−34.9101	−1.52944	−0.764719	−	0.644364i	$-0.777122\pi$
$521$	−34.9101	−1.52944	−0.764719	+	0.644364i	$0.777122\pi$
$522$	0	0
$523$	−14.2935	−0.625011	−0.312506	−	0.949916i	$-0.601168\pi$
$523$	−14.2935	−0.625011	−0.312506	+	0.949916i	$0.601168\pi$
$524$	−41.9409	−1.83220
$525$	0	0
$526$	−38.3696	−1.67299
$527$	−7.01447	−0.305555
$528$	0	0
$529$	33.1594	1.44171
$530$	50.9221	2.21192
$531$	0	0
$532$	−2.78017	−0.120536
$533$	−16.9933	−0.736061
$534$	0	0
$535$	35.7308	1.54478
$536$	−33.7808	−1.45911
$537$	0	0
$538$	−32.9366	−1.42000
$539$	−6.16852	−0.265697
$540$	0	0
$541$	18.9162	0.813269	0.406635	−	0.913591i	$-0.366702\pi$
$541$	18.9162	0.813269	0.406635	+	0.913591i	$0.366702\pi$
$542$	35.0954	1.50748
$543$	0	0
$544$	12.0597	0.517054
$545$	9.75648	0.417922
$546$	0	0
$547$	−26.0495	−1.11380	−0.556898	−	0.830581i	$-0.688009\pi$
$547$	−26.0495	−1.11380	−0.556898	+	0.830581i	$0.688009\pi$
$548$	6.44935	0.275503
$549$	0	0
$550$	2.19806	0.0937257
$551$	9.78986	0.417062
$552$	0	0
$553$	12.7192	0.540874
$554$	−55.6480	−2.36426
$555$	0	0
$556$	56.0646	2.37767
$557$	26.6920	1.13098	0.565489	−	0.824756i	$-0.308688\pi$
$557$	26.6920	1.13098	0.565489	+	0.824756i	$0.308688\pi$
$558$	0	0
$559$	9.94544	0.420647
$560$	1.78794	0.0755541
$561$	0	0
$562$	53.8068	2.26970
$563$	−11.9202	−0.502377	−0.251189	−	0.967938i	$-0.580821\pi$
$563$	−11.9202	−0.502377	−0.251189	+	0.967938i	$0.580821\pi$
$564$	0	0
$565$	−19.8127	−0.833526
$566$	12.9215	0.543133
$567$	0	0
$568$	−21.3502	−0.895834
$569$	−30.8237	−1.29220	−0.646099	−	0.763254i	$-0.723600\pi$
$569$	−30.8237	−1.29220	−0.646099	+	0.763254i	$0.723600\pi$
$570$	0	0
$571$	37.2959	1.56078	0.780392	−	0.625290i	$-0.215019\pi$
$571$	37.2959	1.56078	0.780392	+	0.625290i	$0.215019\pi$
$572$	8.69202	0.363432
$573$	0	0
$574$	12.2131	0.509766
$575$	7.33081	0.305716
$576$	0	0
$577$	−40.5405	−1.68772	−0.843861	−	0.536562i	$-0.819723\pi$
$577$	−40.5405	−1.68772	−0.843861	+	0.536562i	$0.819723\pi$
$578$	−30.5013	−1.26868
$579$	0	0
$580$	72.9807	3.03036
$581$	−9.61356	−0.398838
$582$	0	0
$583$	9.26875	0.383872
$584$	2.16182	0.0894566
$585$	0	0
$586$	18.5851	0.767743
$587$	8.97690	0.370516	0.185258	−	0.982690i	$-0.440688\pi$
$587$	8.97690	0.370516	0.185258	+	0.982690i	$0.440688\pi$
$588$	0	0
$589$	3.78986	0.156158
$590$	−11.1075	−0.457289
$591$	0	0
$592$	6.72348	0.276333
$593$	−31.6112	−1.29812	−0.649058	−	0.760739i	$-0.724837\pi$
$593$	−31.6112	−1.29812	−0.649058	+	0.760739i	$0.724837\pi$
$594$	0	0
$595$	4.12652	0.169171
$596$	−13.5821	−0.556345
$597$	0	0
$598$	48.0049	1.96307
$599$	11.6504	0.476022	0.238011	−	0.971262i	$-0.423504\pi$
$599$	11.6504	0.476022	0.238011	+	0.971262i	$0.423504\pi$
$600$	0	0
$601$	−23.7114	−0.967208	−0.483604	−	0.875287i	$-0.660673\pi$
$601$	−23.7114	−0.967208	−0.483604	+	0.875287i	$0.660673\pi$
$602$	−7.14782	−0.291323
$603$	0	0
$604$	37.7415	1.53568
$605$	2.44504	0.0994051
$606$	0	0
$607$	41.8039	1.69677	0.848383	−	0.529383i	$-0.177576\pi$
$607$	41.8039	1.69677	0.848383	+	0.529383i	$0.177576\pi$
$608$	−6.51573	−0.264248
$609$	0	0
$610$	−39.6286	−1.60452
$611$	10.1967	0.412516
$612$	0	0
$613$	−3.00298	−0.121289	−0.0606447	−	0.998159i	$-0.519316\pi$
$613$	−3.00298	−0.121289	−0.0606447	+	0.998159i	$0.519316\pi$
$614$	38.4741	1.55269
$615$	0	0
$616$	−2.14914	−0.0865915
$617$	−27.3254	−1.10008	−0.550040	−	0.835138i	$-0.685388\pi$
$617$	−27.3254	−1.10008	−0.550040	+	0.835138i	$0.685388\pi$
$618$	0	0
$619$	43.2683	1.73910	0.869549	−	0.493846i	$-0.164409\pi$
$619$	43.2683	1.73910	0.869549	+	0.493846i	$0.164409\pi$
$620$	28.2524	1.13464
$621$	0	0
$622$	−51.2368	−2.05441
$623$	−3.65625	−0.146485
$624$	0	0
$625$	−28.9342	−1.15737
$626$	−18.1347	−0.724807
$627$	0	0
$628$	−34.6631	−1.38321
$629$	15.5176	0.618729
$630$	0	0
$631$	43.3478	1.72565	0.862824	−	0.505504i	$-0.168693\pi$
$631$	43.3478	1.72565	0.862824	+	0.505504i	$0.168693\pi$
$632$	−32.8756	−1.30772
$633$	0	0
$634$	15.1957	0.603497
$635$	−10.5036	−0.416825
$636$	0	0
$637$	−17.5856	−0.696765
$638$	21.9976	0.870894
$639$	0	0
$640$	−39.7614	−1.57171
$641$	30.7342	1.21393	0.606965	−	0.794729i	$-0.292387\pi$
$641$	30.7342	1.21393	0.606965	+	0.794729i	$0.292387\pi$
$642$	0	0
$643$	−24.1661	−0.953019	−0.476510	−	0.879169i	$-0.658098\pi$
$643$	−24.1661	−0.953019	−0.476510	+	0.879169i	$0.658098\pi$
$644$	−20.8345	−0.820993
$645$	0	0
$646$	−4.15883	−0.163627
$647$	41.3072	1.62396	0.811978	−	0.583689i	$-0.198391\pi$
$647$	41.3072	1.62396	0.811978	+	0.583689i	$0.198391\pi$
$648$	0	0
$649$	−2.02177	−0.0793614
$650$	6.26636	0.245787
$651$	0	0
$652$	39.8310	1.55990
$653$	45.4271	1.77770	0.888850	−	0.458198i	$-0.151505\pi$
$653$	45.4271	1.77770	0.888850	+	0.458198i	$0.151505\pi$
$654$	0	0
$655$	−33.6340	−1.31419
$656$	4.78017	0.186634
$657$	0	0
$658$	−7.32842	−0.285692
$659$	41.1618	1.60344	0.801718	−	0.597702i	$-0.203919\pi$
$659$	41.1618	1.60344	0.801718	+	0.597702i	$0.203919\pi$
$660$	0	0
$661$	−20.1575	−0.784036	−0.392018	−	0.919958i	$-0.628223\pi$
$661$	−20.1575	−0.784036	−0.392018	+	0.919958i	$0.628223\pi$
$662$	50.2737	1.95394
$663$	0	0
$664$	24.8485	0.964307
$665$	−2.22952	−0.0864571
$666$	0	0
$667$	73.3648	2.84070
$668$	−42.8931	−1.65958
$669$	0	0
$670$	−78.7434	−3.04212
$671$	−7.21313	−0.278460
$672$	0	0
$673$	34.3347	1.32350	0.661752	−	0.749723i	$-0.269813\pi$
$673$	34.3347	1.32350	0.661752	+	0.749723i	$0.269813\pi$
$674$	47.4838	1.82901
$675$	0	0
$676$	−14.8562	−0.571394
$677$	9.15452	0.351837	0.175918	−	0.984405i	$-0.443711\pi$
$677$	9.15452	0.351837	0.175918	+	0.984405i	$0.443711\pi$
$678$	0	0
$679$	−4.01746	−0.154176
$680$	−10.6659	−0.409020
$681$	0	0
$682$	8.51573	0.326084
$683$	−26.8528	−1.02749	−0.513746	−	0.857942i	$-0.671743\pi$
$683$	−26.8528	−1.02749	−0.513746	+	0.857942i	$0.671743\pi$
$684$	0	0
$685$	5.17198	0.197611
$686$	26.9812	1.03015
$687$	0	0
$688$	−2.79763	−0.106658
$689$	26.4239	1.00667
$690$	0	0
$691$	−40.3803	−1.53614	−0.768070	−	0.640366i	$-0.778783\pi$
$691$	−40.3803	−1.53614	−0.768070	+	0.640366i	$0.778783\pi$
$692$	−12.6799	−0.482019
$693$	0	0
$694$	67.2495	2.55276
$695$	44.9603	1.70544
$696$	0	0
$697$	11.0325	0.417887
$698$	−66.8708	−2.53110
$699$	0	0
$700$	−2.71964	−0.102793
$701$	45.5502	1.72041	0.860203	−	0.509952i	$-0.170337\pi$
$701$	45.5502	1.72041	0.860203	+	0.509952i	$0.170337\pi$
$702$	0	0
$703$	−8.38404	−0.316210
$704$	−13.0368	−0.491344
$705$	0	0
$706$	−40.3967	−1.52035
$707$	7.02774	0.264305
$708$	0	0
$709$	11.5894	0.435249	0.217625	−	0.976033i	$-0.430169\pi$
$709$	11.5894	0.435249	0.217625	+	0.976033i	$0.430169\pi$
$710$	−49.7676	−1.86775
$711$	0	0
$712$	9.45042	0.354169
$713$	28.4010	1.06363
$714$	0	0
$715$	6.97046	0.260680
$716$	22.6165	0.845220
$717$	0	0
$718$	−52.3913	−1.95523
$719$	−19.5579	−0.729388	−0.364694	−	0.931127i	$-0.618826\pi$
$719$	−19.5579	−0.729388	−0.364694	+	0.931127i	$0.618826\pi$
$720$	0	0
$721$	11.5017	0.428347
$722$	2.24698	0.0836239
$723$	0	0
$724$	−24.1390	−0.897118
$725$	9.57673	0.355671
$726$	0	0
$727$	35.5652	1.31904	0.659521	−	0.751686i	$-0.270759\pi$
$727$	35.5652	1.31904	0.659521	+	0.751686i	$0.270759\pi$
$728$	−6.12690	−0.227078
$729$	0	0
$730$	5.03923	0.186510
$731$	−6.45686	−0.238816
$732$	0	0
$733$	−25.4198	−0.938902	−0.469451	−	0.882958i	$-0.655548\pi$
$733$	−25.4198	−0.938902	−0.469451	+	0.882958i	$0.655548\pi$
$734$	32.0834	1.18422
$735$	0	0
$736$	−48.8286	−1.79985
$737$	−14.3327	−0.527953
$738$	0	0
$739$	18.9903	0.698570	0.349285	−	0.937017i	$-0.386425\pi$
$739$	18.9903	0.698570	0.349285	+	0.937017i	$0.386425\pi$
$740$	−62.5008	−2.29757
$741$	0	0
$742$	−18.9909	−0.697178
$743$	−30.6510	−1.12448	−0.562238	−	0.826976i	$-0.690059\pi$
$743$	−30.6510	−1.12448	−0.562238	+	0.826976i	$0.690059\pi$
$744$	0	0
$745$	−10.8920	−0.399052
$746$	34.0508	1.24669
$747$	0	0
$748$	−5.64310	−0.206332
$749$	−13.3254	−0.486901
$750$	0	0
$751$	21.5579	0.786660	0.393330	−	0.919397i	$-0.371323\pi$
$751$	21.5579	0.786660	0.393330	+	0.919397i	$0.371323\pi$
$752$	−2.86831	−0.104597
$753$	0	0
$754$	62.7120	2.28384
$755$	30.2664	1.10151
$756$	0	0
$757$	21.0315	0.764401	0.382201	−	0.924079i	$-0.375166\pi$
$757$	21.0315	0.764401	0.382201	+	0.924079i	$0.375166\pi$
$758$	−18.1075	−0.657695
$759$	0	0
$760$	5.76271	0.209035
$761$	−7.30127	−0.264671	−0.132335	−	0.991205i	$-0.542248\pi$
$761$	−7.30127	−0.264671	−0.132335	+	0.991205i	$0.542248\pi$
$762$	0	0
$763$	−3.63858	−0.131725
$764$	−3.57912	−0.129488
$765$	0	0
$766$	11.4547	0.413876
$767$	−5.76377	−0.208118
$768$	0	0
$769$	29.0756	1.04849	0.524246	−	0.851567i	$-0.324347\pi$
$769$	29.0756	1.04849	0.524246	+	0.851567i	$0.324347\pi$
$770$	−5.00969	−0.180537
$771$	0	0
$772$	−55.6926	−2.00442
$773$	32.5090	1.16927	0.584634	−	0.811297i	$-0.301238\pi$
$773$	32.5090	1.16927	0.584634	+	0.811297i	$0.301238\pi$
$774$	0	0
$775$	3.70735	0.133172
$776$	10.3840	0.372765
$777$	0	0
$778$	26.3303	0.943988
$779$	−5.96077	−0.213567
$780$	0	0
$781$	−9.05861	−0.324142
$782$	−31.1661	−1.11450
$783$	0	0
$784$	4.94677	0.176670
$785$	−27.7976	−0.992140
$786$	0	0
$787$	0.287273	0.0102402	0.00512009	−	0.999987i	$-0.498370\pi$
$787$	0.287273	0.0102402	0.00512009	+	0.999987i	$0.498370\pi$
$788$	−47.2030	−1.68154
$789$	0	0
$790$	−76.6335	−2.72650
$791$	7.38895	0.262721
$792$	0	0
$793$	−20.5636	−0.730234
$794$	−15.1715	−0.538417
$795$	0	0
$796$	35.1933	1.24739
$797$	40.9197	1.44945	0.724726	−	0.689037i	$-0.241966\pi$
$797$	40.9197	1.44945	0.724726	+	0.689037i	$0.241966\pi$
$798$	0	0
$799$	−6.62001	−0.234199
$800$	−6.37388	−0.225351
$801$	0	0
$802$	−58.5924	−2.06897
$803$	0.917231	0.0323684
$804$	0	0
$805$	−16.7079	−0.588878
$806$	24.2771	0.855125
$807$	0	0
$808$	−18.1648	−0.639035
$809$	−11.2336	−0.394951	−0.197476	−	0.980308i	$-0.563274\pi$
$809$	−11.2336	−0.394951	−0.197476	+	0.980308i	$0.563274\pi$
$810$	0	0
$811$	22.7356	0.798354	0.399177	−	0.916874i	$-0.369296\pi$
$811$	22.7356	0.798354	0.399177	+	0.916874i	$0.369296\pi$
$812$	−27.2174	−0.955145
$813$	0	0
$814$	−18.8388	−0.660299
$815$	31.9420	1.11888
$816$	0	0
$817$	3.48858	0.122050
$818$	50.4892	1.76531
$819$	0	0
$820$	−44.4359	−1.55177
$821$	−23.7912	−0.830318	−0.415159	−	0.909749i	$-0.636274\pi$
$821$	−23.7912	−0.830318	−0.415159	+	0.909749i	$0.636274\pi$
$822$	0	0
$823$	3.27844	0.114279	0.0571396	−	0.998366i	$-0.481802\pi$
$823$	3.27844	0.114279	0.0571396	+	0.998366i	$0.481802\pi$
$824$	−29.7289	−1.03565
$825$	0	0
$826$	4.14244	0.144134
$827$	−18.0224	−0.626699	−0.313349	−	0.949638i	$-0.601451\pi$
$827$	−18.0224	−0.626699	−0.313349	+	0.949638i	$0.601451\pi$
$828$	0	0
$829$	27.3279	0.949139	0.474569	−	0.880218i	$-0.342604\pi$
$829$	27.3279	0.949139	0.474569	+	0.880218i	$0.342604\pi$
$830$	57.9221	2.01051
$831$	0	0
$832$	−37.1661	−1.28850
$833$	11.4170	0.395577
$834$	0	0
$835$	−34.3976	−1.19038
$836$	3.04892	0.105449
$837$	0	0
$838$	58.1933	2.01025
$839$	−15.8213	−0.546212	−0.273106	−	0.961984i	$-0.588051\pi$
$839$	−15.8213	−0.546212	−0.273106	+	0.961984i	$0.588051\pi$
$840$	0	0
$841$	66.8413	2.30487
$842$	−67.7174	−2.33369
$843$	0	0
$844$	13.5526	0.466499
$845$	−11.9138	−0.409846
$846$	0	0
$847$	−0.911854	−0.0313317
$848$	−7.43296	−0.255249
$849$	0	0
$850$	−4.06829	−0.139541
$851$	−62.8297	−2.15377
$852$	0	0
$853$	19.3817	0.663615	0.331808	−	0.943347i	$-0.392342\pi$
$853$	19.3817	0.663615	0.331808	+	0.943347i	$0.392342\pi$
$854$	14.7791	0.505730
$855$	0	0
$856$	34.4426	1.17723
$857$	−47.9963	−1.63952	−0.819761	−	0.572706i	$-0.805894\pi$
$857$	−47.9963	−1.63952	−0.819761	+	0.572706i	$0.805894\pi$
$858$	0	0
$859$	−24.4946	−0.835743	−0.417872	−	0.908506i	$-0.637224\pi$
$859$	−24.4946	−0.835743	−0.417872	+	0.908506i	$0.637224\pi$
$860$	26.0064	0.886812
$861$	0	0
$862$	86.5290	2.94719
$863$	4.69441	0.159800	0.0798999	−	0.996803i	$-0.474540\pi$
$863$	4.69441	0.159800	0.0798999	+	0.996803i	$0.474540\pi$
$864$	0	0
$865$	−10.1685	−0.345740
$866$	−61.6413	−2.09466
$867$	0	0
$868$	−10.5364	−0.357630
$869$	−13.9487	−0.473177
$870$	0	0
$871$	−40.8605	−1.38451
$872$	9.40475	0.318485
$873$	0	0
$874$	16.8388	0.569580
$875$	8.96662	0.303127
$876$	0	0
$877$	−0.429976	−0.0145193	−0.00725964	−	0.999974i	$-0.502311\pi$
$877$	−0.429976	−0.0145193	−0.00725964	+	0.999974i	$0.502311\pi$
$878$	−44.3467	−1.49663
$879$	0	0
$880$	−1.96077	−0.0660976
$881$	4.67589	0.157535	0.0787674	−	0.996893i	$-0.474902\pi$
$881$	4.67589	0.157535	0.0787674	+	0.996893i	$0.474902\pi$
$882$	0	0
$883$	43.8044	1.47414	0.737069	−	0.675818i	$-0.236209\pi$
$883$	43.8044	1.47414	0.737069	+	0.675818i	$0.236209\pi$
$884$	−16.0877	−0.541087
$885$	0	0
$886$	−92.1794	−3.09683
$887$	−30.0455	−1.00883	−0.504414	−	0.863462i	$-0.668291\pi$
$887$	−30.0455	−1.00883	−0.504414	+	0.863462i	$0.668291\pi$
$888$	0	0
$889$	3.91723	0.131380
$890$	22.0291	0.738416
$891$	0	0
$892$	−23.5515	−0.788563
$893$	3.57673	0.119691
$894$	0	0
$895$	18.1371	0.606255
$896$	14.8286	0.495389
$897$	0	0
$898$	−0.795233	−0.0265373
$899$	37.1021	1.23743
$900$	0	0
$901$	−17.1551	−0.571520
$902$	−13.3937	−0.445962
$903$	0	0
$904$	−19.0984	−0.635204
$905$	−19.3580	−0.643480
$906$	0	0
$907$	2.81461	0.0934576	0.0467288	−	0.998908i	$-0.485120\pi$
$907$	2.81461	0.0934576	0.0467288	+	0.998908i	$0.485120\pi$
$908$	35.8726	1.19047
$909$	0	0
$910$	−14.2819	−0.473440
$911$	21.9987	0.728849	0.364424	−	0.931233i	$-0.381266\pi$
$911$	21.9987	0.728849	0.364424	+	0.931233i	$0.381266\pi$
$912$	0	0
$913$	10.5429	0.348918
$914$	4.60925	0.152461
$915$	0	0
$916$	46.1648	1.52533
$917$	12.5435	0.414222
$918$	0	0
$919$	−4.60494	−0.151903	−0.0759515	−	0.997112i	$-0.524199\pi$
$919$	−4.60494	−0.151903	−0.0759515	+	0.997112i	$0.524199\pi$
$920$	43.1855	1.42378
$921$	0	0
$922$	75.3159	2.48040
$923$	−25.8248	−0.850033
$924$	0	0
$925$	−8.20152	−0.269664
$926$	−31.1957	−1.02515
$927$	0	0
$928$	−63.7881	−2.09395
$929$	53.8329	1.76620	0.883100	−	0.469184i	$-0.155452\pi$
$929$	53.8329	1.76620	0.883100	+	0.469184i	$0.155452\pi$
$930$	0	0
$931$	−6.16852	−0.202165
$932$	23.5023	0.769844
$933$	0	0
$934$	18.1685	0.594492
$935$	−4.52542	−0.147997
$936$	0	0
$937$	31.2935	1.02231	0.511157	−	0.859487i	$-0.329217\pi$
$937$	31.2935	1.02231	0.511157	+	0.859487i	$0.329217\pi$
$938$	29.3666	0.958853
$939$	0	0
$940$	26.6635	0.869669
$941$	−25.5958	−0.834401	−0.417200	−	0.908815i	$-0.636989\pi$
$941$	−25.5958	−0.834401	−0.417200	+	0.908815i	$0.636989\pi$
$942$	0	0
$943$	−44.6698	−1.45465
$944$	1.62133	0.0527699
$945$	0	0
$946$	7.83877	0.254861
$947$	1.50843	0.0490175	0.0245088	−	0.999700i	$-0.492198\pi$
$947$	1.50843	0.0490175	0.0245088	+	0.999700i	$0.492198\pi$
$948$	0	0
$949$	2.61489	0.0848830
$950$	2.19806	0.0713146
$951$	0	0
$952$	3.97776	0.128920
$953$	37.5338	1.21584	0.607919	−	0.793999i	$-0.292004\pi$
$953$	37.5338	1.21584	0.607919	+	0.793999i	$0.292004\pi$
$954$	0	0
$955$	−2.87023	−0.0928785
$956$	20.0315	0.647864
$957$	0	0
$958$	−22.7778	−0.735916
$959$	−1.92884	−0.0622854
$960$	0	0
$961$	−16.6370	−0.536677
$962$	−53.7066	−1.73157
$963$	0	0
$964$	−16.6383	−0.535884
$965$	−44.6620	−1.43772
$966$	0	0
$967$	28.5144	0.916961	0.458481	−	0.888704i	$-0.348394\pi$
$967$	28.5144	0.916961	0.458481	+	0.888704i	$0.348394\pi$
$968$	2.35690	0.0757535
$969$	0	0
$970$	24.2054	0.777187
$971$	−42.6292	−1.36804	−0.684018	−	0.729465i	$-0.739769\pi$
$971$	−42.6292	−1.36804	−0.684018	+	0.729465i	$0.739769\pi$
$972$	0	0
$973$	−16.7675	−0.537541
$974$	−57.0689	−1.82861
$975$	0	0
$976$	5.78448	0.185157
$977$	−19.7633	−0.632284	−0.316142	−	0.948712i	$-0.602388\pi$
$977$	−19.7633	−0.632284	−0.316142	+	0.948712i	$0.602388\pi$
$978$	0	0
$979$	4.00969	0.128150
$980$	−45.9847	−1.46893
$981$	0	0
$982$	73.2223	2.33662
$983$	−61.2471	−1.95348	−0.976740	−	0.214429i	$-0.931211\pi$
$983$	−61.2471	−1.95348	−0.976740	+	0.214429i	$0.931211\pi$
$984$	0	0
$985$	−37.8538	−1.20612
$986$	−40.7144	−1.29661
$987$	0	0
$988$	8.69202	0.276530
$989$	26.1433	0.831308
$990$	0	0
$991$	7.33167	0.232898	0.116449	−	0.993197i	$-0.462849\pi$
$991$	7.33167	0.232898	0.116449	+	0.993197i	$0.462849\pi$
$992$	−24.6937	−0.784025
$993$	0	0
$994$	18.5603	0.588698
$995$	28.2228	0.894723
$996$	0	0
$997$	−9.39911	−0.297673	−0.148836	−	0.988862i	$-0.547553\pi$
$997$	−9.39911	−0.297673	−0.148836	+	0.988862i	$0.547553\pi$
$998$	88.4341	2.79933
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.g.1.3		3
3.2	odd	2		627.2.a.d.1.1	✓	3
33.32	even	2		6897.2.a.o.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
627.2.a.d.1.1	✓	3	3.2	odd	2
1881.2.a.g.1.3		3	1.1	even	1	trivial
6897.2.a.o.1.3		3	33.32	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q+2.24698 q^{2} +3.04892 q^{4} +2.44504 q^{5} -0.911854 q^{7} +2.35690 q^{8} +5.49396 q^{10} +1.00000 q^{11} +2.85086 q^{13} -2.04892 q^{14} -0.801938 q^{16} -1.85086 q^{17} +1.00000 q^{19} +7.45473 q^{20} +2.24698 q^{22} +7.49396 q^{23} +0.978230 q^{25} +6.40581 q^{26} -2.78017 q^{28} +9.78986 q^{29} +3.78986 q^{31} -6.51573 q^{32} -4.15883 q^{34} -2.22952 q^{35} -8.38404 q^{37} +2.24698 q^{38} +5.76271 q^{40} -5.96077 q^{41} +3.48858 q^{43} +3.04892 q^{44} +16.8388 q^{46} +3.57673 q^{47} -6.16852 q^{49} +2.19806 q^{50} +8.69202 q^{52} +9.26875 q^{53} +2.44504 q^{55} -2.14914 q^{56} +21.9976 q^{58} -2.02177 q^{59} -7.21313 q^{61} +8.51573 q^{62} -13.0368 q^{64} +6.97046 q^{65} -14.3327 q^{67} -5.64310 q^{68} -5.00969 q^{70} -9.05861 q^{71} +0.917231 q^{73} -18.8388 q^{74} +3.04892 q^{76} -0.911854 q^{77} -13.9487 q^{79} -1.96077 q^{80} -13.3937 q^{82} +10.5429 q^{83} -4.52542 q^{85} +7.83877 q^{86} +2.35690 q^{88} +4.00969 q^{89} -2.59956 q^{91} +22.8485 q^{92} +8.03684 q^{94} +2.44504 q^{95} +4.40581 q^{97} -13.8605 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 7 q^{5} + q^{7} + 3 q^{8} + 7 q^{10} + 3 q^{11} - 5 q^{13} + 3 q^{14} + 2 q^{16} + 8 q^{17} + 3 q^{19} + 2 q^{22} + 13 q^{23} + 6 q^{25} + 6 q^{26} - 7 q^{28} + 6 q^{29} - 12 q^{31} - 7 q^{32}+ \cdots - 6 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 7 * q^5 + q^7 + 3 * q^8 + 7 * q^10 + 3 * q^11 - 5 * q^13 + 3 * q^14 + 2 * q^16 + 8 * q^17 + 3 * q^19 + 2 * q^22 + 13 * q^23 + 6 * q^25 + 6 * q^26 - 7 * q^28 + 6 * q^29 - 12 * q^31 - 7 * q^32 - 4 * q^34 + 14 * q^35 - 15 * q^37 + 2 * q^38 - 5 * q^41 + 4 * q^43 + 18 * q^46 + 8 * q^47 + 12 * q^49 + 11 * q^50 + 21 * q^52 + 20 * q^53 + 7 * q^55 - 20 * q^56 + 25 * q^58 - 3 * q^59 - q^61 + 13 * q^62 - 11 * q^64 - 14 * q^65 - q^67 - 21 * q^68 + 7 * q^70 + 4 * q^71 - 4 * q^73 - 24 * q^74 + q^77 - 10 * q^79 + 7 * q^80 - 8 * q^82 + 13 * q^83 + 21 * q^85 - 9 * q^86 + 3 * q^88 - 10 * q^89 - 18 * q^91 + 14 * q^92 - 4 * q^94 + 7 * q^95 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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