LMFDB - Embedded newform 1629.2.a.b.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1629,2,Mod(1,1629)] 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("1629.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1629, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1629 = 3^{2} \cdot 181 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1629.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,1,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.0076304893$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.170701.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 4x + 1 $ x^5 - x^4 - 6x^3 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 543)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.77019$ of defining polynomial
Character	$\chi$	$=$	1629.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.435090 q^{2} -1.81070 q^{4} -1.04050 q^{5} -4.97235 q^{7} -1.65800 q^{8} -0.452712 q^{10} -1.73346 q^{11} +3.86328 q^{13} -2.16342 q^{14} +2.90001 q^{16} -5.71071 q^{17} +5.93875 q^{19} +1.88403 q^{20} -0.754213 q^{22} -5.61586 q^{23} -3.91736 q^{25} +1.68088 q^{26} +9.00342 q^{28} +1.79471 q^{29} +4.50679 q^{31} +4.57776 q^{32} -2.48468 q^{34} +5.17374 q^{35} -0.447175 q^{37} +2.58389 q^{38} +1.72515 q^{40} +4.37007 q^{41} -5.04718 q^{43} +3.13878 q^{44} -2.44340 q^{46} +10.0870 q^{47} +17.7243 q^{49} -1.70440 q^{50} -6.99523 q^{52} -12.6653 q^{53} +1.80367 q^{55} +8.24415 q^{56} +0.780863 q^{58} +10.7052 q^{59} +8.10907 q^{61} +1.96086 q^{62} -3.80829 q^{64} -4.01975 q^{65} +11.9461 q^{67} +10.3404 q^{68} +2.25104 q^{70} +11.7832 q^{71} +1.50353 q^{73} -0.194562 q^{74} -10.7533 q^{76} +8.61939 q^{77} +1.79243 q^{79} -3.01747 q^{80} +1.90137 q^{82} -8.36317 q^{83} +5.94200 q^{85} -2.19598 q^{86} +2.87408 q^{88} +4.53143 q^{89} -19.2096 q^{91} +10.1686 q^{92} +4.38874 q^{94} -6.17928 q^{95} +7.90078 q^{97} +7.71167 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 3 q^{4} - 3 q^{5} + 9 q^{7} - 9 q^{8} - 14 q^{10} + 4 q^{11} + 9 q^{13} - 3 q^{14} + 7 q^{16} - 9 q^{17} + 6 q^{19} + 25 q^{20} - 17 q^{22} + 5 q^{23} + 12 q^{25} - 6 q^{26} + 27 q^{28} + 20 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) $ 5 * q + q^2 + 3 * q^4 - 3 * q^5 + 9 * q^7 - 9 * q^8 - 14 * q^10 + 4 * q^11 + 9 * q^13 - 3 * q^14 + 7 * q^16 - 9 * q^17 + 6 * q^19 + 25 * q^20 - 17 * q^22 + 5 * q^23 + 12 * q^25 - 6 * q^26 + 27 * q^28 + 20 * q^29 + 13 * q^31 - 12 * q^32 + 22 * q^34 + 8 * q^35 - 5 * q^37 + 26 * q^38 - 13 * q^40 + 2 * q^41 + 4 * q^43 + 9 * q^44 - 20 * q^46 + 2 * q^47 + 50 * q^49 - 20 * q^50 + 14 * q^52 - 19 * q^53 + 11 * q^55 - 8 * q^56 - 3 * q^58 + 25 * q^59 + 17 * q^61 + 11 * q^62 - 11 * q^64 + 8 * q^65 - 9 * q^67 + 15 * q^68 - 30 * q^70 - 9 * q^71 + 7 * q^73 + 15 * q^74 - 15 * q^76 + 38 * q^77 + 2 * q^79 + 31 * q^80 + q^82 - 14 * q^83 - 3 * q^85 + 4 * q^86 - 15 * q^88 + 12 * q^89 - 5 * q^91 + 39 * q^92 + 12 * q^94 - 22 * q^95 - 8 * q^97 + 3 * 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 $

$2$	0.435090	0.307655	0.153828	−	0.988098i	$-0.450840\pi$
$2$	0.435090	0.307655	0.153828	+	0.988098i	$0.450840\pi$
$3$	0	0
$3$	0	0
$4$	−1.81070	−0.905348
$5$	−1.04050	−0.465326	−0.232663	−	0.972557i	$-0.574744\pi$
$5$	−1.04050	−0.465326	−0.232663	+	0.972557i	$0.574744\pi$
$6$	0	0
$7$	−4.97235	−1.87937	−0.939686	−	0.342038i	$-0.888883\pi$
$7$	−4.97235	−1.87937	−0.939686	+	0.342038i	$0.888883\pi$
$8$	−1.65800	−0.586191
$9$	0	0
$10$	−0.452712	−0.143160
$11$	−1.73346	−0.522659	−0.261329	−	0.965250i	$-0.584161\pi$
$11$	−1.73346	−0.522659	−0.261329	+	0.965250i	$0.584161\pi$
$12$	0	0
$13$	3.86328	1.07148	0.535741	−	0.844382i	$-0.320032\pi$
$13$	3.86328	1.07148	0.535741	+	0.844382i	$0.320032\pi$
$14$	−2.16342	−0.578199
$15$	0	0
$16$	2.90001	0.725003
$17$	−5.71071	−1.38505	−0.692525	−	0.721394i	$-0.743502\pi$
$17$	−5.71071	−1.38505	−0.692525	+	0.721394i	$0.743502\pi$
$18$	0	0
$19$	5.93875	1.36244	0.681221	−	0.732078i	$-0.261449\pi$
$19$	5.93875	1.36244	0.681221	+	0.732078i	$0.261449\pi$
$20$	1.88403	0.421282
$21$	0	0
$22$	−0.754213	−0.160799
$23$	−5.61586	−1.17099	−0.585493	−	0.810677i	$-0.699099\pi$
$23$	−5.61586	−1.17099	−0.585493	+	0.810677i	$0.699099\pi$
$24$	0	0
$25$	−3.91736	−0.783471
$26$	1.68088	0.329647
$27$	0	0
$28$	9.00342	1.70149
$29$	1.79471	0.333270	0.166635	−	0.986019i	$-0.446710\pi$
$29$	1.79471	0.333270	0.166635	+	0.986019i	$0.446710\pi$
$30$	0	0
$31$	4.50679	0.809443	0.404721	−	0.914440i	$-0.367369\pi$
$31$	4.50679	0.809443	0.404721	+	0.914440i	$0.367369\pi$
$32$	4.57776	0.809242
$33$	0	0
$34$	−2.48468	−0.426118
$35$	5.17374	0.874522
$36$	0	0
$37$	−0.447175	−0.0735151	−0.0367576	−	0.999324i	$-0.511703\pi$
$37$	−0.447175	−0.0735151	−0.0367576	+	0.999324i	$0.511703\pi$
$38$	2.58389	0.419163
$39$	0	0
$40$	1.72515	0.272770
$41$	4.37007	0.682490	0.341245	−	0.939974i	$-0.389151\pi$
$41$	4.37007	0.682490	0.341245	+	0.939974i	$0.389151\pi$
$42$	0	0
$43$	−5.04718	−0.769687	−0.384844	−	0.922982i	$-0.625745\pi$
$43$	−5.04718	−0.769687	−0.384844	+	0.922982i	$0.625745\pi$
$44$	3.13878	0.473188
$45$	0	0
$46$	−2.44340	−0.360260
$47$	10.0870	1.47133	0.735667	−	0.677343i	$-0.236869\pi$
$47$	10.0870	1.47133	0.735667	+	0.677343i	$0.236869\pi$
$48$	0	0
$49$	17.7243	2.53204
$50$	−1.70440	−0.241039
$51$	0	0
$52$	−6.99523	−0.970064
$53$	−12.6653	−1.73971	−0.869857	−	0.493303i	$-0.835789\pi$
$53$	−12.6653	−1.73971	−0.869857	+	0.493303i	$0.835789\pi$
$54$	0	0
$55$	1.80367	0.243207
$56$	8.24415	1.10167
$57$	0	0
$58$	0.780863	0.102532
$59$	10.7052	1.39369	0.696847	−	0.717219i	$-0.254585\pi$
$59$	10.7052	1.39369	0.696847	+	0.717219i	$0.254585\pi$
$60$	0	0
$61$	8.10907	1.03826	0.519130	−	0.854695i	$-0.326256\pi$
$61$	8.10907	1.03826	0.519130	+	0.854695i	$0.326256\pi$
$62$	1.96086	0.249029
$63$	0	0
$64$	−3.80829	−0.476036
$65$	−4.01975	−0.498589
$66$	0	0
$67$	11.9461	1.45944	0.729722	−	0.683744i	$-0.239649\pi$
$67$	11.9461	1.45944	0.729722	+	0.683744i	$0.239649\pi$
$68$	10.3404	1.25395
$69$	0	0
$70$	2.25104	0.269051
$71$	11.7832	1.39841	0.699203	−	0.714923i	$-0.253538\pi$
$71$	11.7832	1.39841	0.699203	+	0.714923i	$0.253538\pi$
$72$	0	0
$73$	1.50353	0.175975	0.0879875	−	0.996122i	$-0.471956\pi$
$73$	1.50353	0.175975	0.0879875	+	0.996122i	$0.471956\pi$
$74$	−0.194562	−0.0226173
$75$	0	0
$76$	−10.7533	−1.23348
$77$	8.61939	0.982271
$78$	0	0
$79$	1.79243	0.201664	0.100832	−	0.994903i	$-0.467849\pi$
$79$	1.79243	0.201664	0.100832	+	0.994903i	$0.467849\pi$
$80$	−3.01747	−0.337363
$81$	0	0
$82$	1.90137	0.209972
$83$	−8.36317	−0.917977	−0.458989	−	0.888442i	$-0.651788\pi$
$83$	−8.36317	−0.917977	−0.458989	+	0.888442i	$0.651788\pi$
$84$	0	0
$85$	5.94200	0.644501
$86$	−2.19598	−0.236798
$87$	0	0
$88$	2.87408	0.306378
$89$	4.53143	0.480331	0.240165	−	0.970732i	$-0.422798\pi$
$89$	4.53143	0.480331	0.240165	+	0.970732i	$0.422798\pi$
$90$	0	0
$91$	−19.2096	−2.01371
$92$	10.1686	1.06015
$93$	0	0
$94$	4.38874	0.452664
$95$	−6.17928	−0.633980
$96$	0	0
$97$	7.90078	0.802203	0.401101	−	0.916034i	$-0.368627\pi$
$97$	7.90078	0.802203	0.401101	+	0.916034i	$0.368627\pi$
$98$	7.71167	0.778996
$99$	0	0
$100$	7.09314	0.709314
$101$	−9.44623	−0.939935	−0.469968	−	0.882684i	$-0.655734\pi$
$101$	−9.44623	−0.939935	−0.469968	+	0.882684i	$0.655734\pi$
$102$	0	0
$103$	−11.8927	−1.17182	−0.585912	−	0.810375i	$-0.699263\pi$
$103$	−11.8927	−1.17182	−0.585912	+	0.810375i	$0.699263\pi$
$104$	−6.40531	−0.628093
$105$	0	0
$106$	−5.51056	−0.535233
$107$	−0.134599	−0.0130122	−0.00650611	−	0.999979i	$-0.502071\pi$
$107$	−0.134599	−0.0130122	−0.00650611	+	0.999979i	$0.502071\pi$
$108$	0	0
$109$	−5.43310	−0.520397	−0.260198	−	0.965555i	$-0.583788\pi$
$109$	−5.43310	−0.520397	−0.260198	+	0.965555i	$0.583788\pi$
$110$	0.784760	0.0748239
$111$	0	0
$112$	−14.4199	−1.36255
$113$	−18.4453	−1.73519	−0.867594	−	0.497272i	$-0.834335\pi$
$113$	−18.4453	−1.73519	−0.867594	+	0.497272i	$0.834335\pi$
$114$	0	0
$115$	5.84331	0.544891
$116$	−3.24968	−0.301726
$117$	0	0
$118$	4.65772	0.428778
$119$	28.3957	2.60303
$120$	0	0
$121$	−7.99510	−0.726828
$122$	3.52818	0.319426
$123$	0	0
$124$	−8.16042	−0.732827
$125$	9.27852	0.829896
$126$	0	0
$127$	−0.938331	−0.0832634	−0.0416317	−	0.999133i	$-0.513256\pi$
$127$	−0.938331	−0.0832634	−0.0416317	+	0.999133i	$0.513256\pi$
$128$	−10.8125	−0.955697
$129$	0	0
$130$	−1.74896	−0.153394
$131$	14.9166	1.30327	0.651636	−	0.758532i	$-0.274083\pi$
$131$	14.9166	1.30327	0.651636	+	0.758532i	$0.274083\pi$
$132$	0	0
$133$	−29.5295	−2.56054
$134$	5.19762	0.449006
$135$	0	0
$136$	9.46834	0.811904
$137$	−0.172379	−0.0147273	−0.00736367	−	0.999973i	$-0.502344\pi$
$137$	−0.172379	−0.0147273	−0.00736367	+	0.999973i	$0.502344\pi$
$138$	0	0
$139$	22.2427	1.88660	0.943302	−	0.331937i	$-0.107702\pi$
$139$	22.2427	1.88660	0.943302	+	0.331937i	$0.107702\pi$
$140$	−9.36807	−0.791747
$141$	0	0
$142$	5.12675	0.430227
$143$	−6.69686	−0.560019
$144$	0	0
$145$	−1.86740	−0.155079
$146$	0.654172	0.0541397
$147$	0	0
$148$	0.809698	0.0665568
$149$	−7.38710	−0.605175	−0.302587	−	0.953122i	$-0.597850\pi$
$149$	−7.38710	−0.605175	−0.302587	+	0.953122i	$0.597850\pi$
$150$	0	0
$151$	−11.2063	−0.911959	−0.455980	−	0.889990i	$-0.650711\pi$
$151$	−11.2063	−0.911959	−0.455980	+	0.889990i	$0.650711\pi$
$152$	−9.84643	−0.798651
$153$	0	0
$154$	3.75021	0.302201
$155$	−4.68932	−0.376655
$156$	0	0
$157$	−16.0526	−1.28114	−0.640568	−	0.767901i	$-0.721301\pi$
$157$	−16.0526	−1.28114	−0.640568	+	0.767901i	$0.721301\pi$
$158$	0.779870	0.0620431
$159$	0	0
$160$	−4.76317	−0.376562
$161$	27.9240	2.20072
$162$	0	0
$163$	−0.612491	−0.0479740	−0.0239870	−	0.999712i	$-0.507636\pi$
$163$	−0.612491	−0.0479740	−0.0239870	+	0.999712i	$0.507636\pi$
$164$	−7.91287	−0.617891
$165$	0	0
$166$	−3.63874	−0.282421
$167$	16.8554	1.30431	0.652157	−	0.758084i	$-0.273864\pi$
$167$	16.8554	1.30431	0.652157	+	0.758084i	$0.273864\pi$
$168$	0	0
$169$	1.92495	0.148073
$170$	2.58531	0.198284
$171$	0	0
$172$	9.13890	0.696835
$173$	11.2340	0.854105	0.427052	−	0.904227i	$-0.359552\pi$
$173$	11.2340	0.854105	0.427052	+	0.904227i	$0.359552\pi$
$174$	0	0
$175$	19.4785	1.47243
$176$	−5.02707	−0.378929
$177$	0	0
$178$	1.97158	0.147776
$179$	−17.4405	−1.30357	−0.651783	−	0.758405i	$-0.725979\pi$
$179$	−17.4405	−1.30357	−0.651783	+	0.758405i	$0.725979\pi$
$180$	0	0
$181$	−1.00000	−0.0743294
$181$	−1.00000	−0.0743294
$182$	−8.35791	−0.619530
$183$	0	0
$184$	9.31107	0.686421
$185$	0.465286	0.0342085
$186$	0	0
$187$	9.89931	0.723909
$188$	−18.2644	−1.33207
$189$	0	0
$190$	−2.68854	−0.195047
$191$	21.6779	1.56856	0.784279	−	0.620408i	$-0.213033\pi$
$191$	21.6779	1.56856	0.784279	+	0.620408i	$0.213033\pi$
$192$	0	0
$193$	0.0598354	0.00430705	0.00215352	−	0.999998i	$-0.499315\pi$
$193$	0.0598354	0.00430705	0.00215352	+	0.999998i	$0.499315\pi$
$194$	3.43755	0.246802
$195$	0	0
$196$	−32.0933	−2.29238
$197$	−16.7826	−1.19571	−0.597857	−	0.801603i	$-0.703981\pi$
$197$	−16.7826	−1.19571	−0.597857	+	0.801603i	$0.703981\pi$
$198$	0	0
$199$	−8.93685	−0.633517	−0.316758	−	0.948506i	$-0.602594\pi$
$199$	−8.93685	−0.633517	−0.316758	+	0.948506i	$0.602594\pi$
$200$	6.49497	0.459264
$201$	0	0
$202$	−4.10996	−0.289176
$203$	−8.92395	−0.626339
$204$	0	0
$205$	−4.54706	−0.317581
$206$	−5.17440	−0.360518
$207$	0	0
$208$	11.2036	0.776828
$209$	−10.2946	−0.712093
$210$	0	0
$211$	1.80003	0.123919	0.0619595	−	0.998079i	$-0.480265\pi$
$211$	1.80003	0.123919	0.0619595	+	0.998079i	$0.480265\pi$
$212$	22.9330	1.57505
$213$	0	0
$214$	−0.0585629	−0.00400328
$215$	5.25159	0.358156
$216$	0	0
$217$	−22.4093	−1.52124
$218$	−2.36389	−0.160103
$219$	0	0
$220$	−3.26590	−0.220187
$221$	−22.0621	−1.48406
$222$	0	0
$223$	0.738244	0.0494364	0.0247182	−	0.999694i	$-0.492131\pi$
$223$	0.738244	0.0494364	0.0247182	+	0.999694i	$0.492131\pi$
$224$	−22.7622	−1.52087
$225$	0	0
$226$	−8.02538	−0.533840
$227$	8.49001	0.563502	0.281751	−	0.959488i	$-0.409085\pi$
$227$	8.49001	0.563502	0.281751	+	0.959488i	$0.409085\pi$
$228$	0	0
$229$	14.6709	0.969477	0.484739	−	0.874659i	$-0.338915\pi$
$229$	14.6709	0.969477	0.484739	+	0.874659i	$0.338915\pi$
$230$	2.54237	0.167639
$231$	0	0
$232$	−2.97563	−0.195360
$233$	19.5033	1.27770	0.638851	−	0.769331i	$-0.279410\pi$
$233$	19.5033	1.27770	0.638851	+	0.769331i	$0.279410\pi$
$234$	0	0
$235$	−10.4955	−0.684651
$236$	−19.3838	−1.26178
$237$	0	0
$238$	12.3547	0.800835
$239$	25.4664	1.64728	0.823642	−	0.567110i	$-0.191938\pi$
$239$	25.4664	1.64728	0.823642	+	0.567110i	$0.191938\pi$
$240$	0	0
$241$	19.3367	1.24559	0.622793	−	0.782387i	$-0.285998\pi$
$241$	19.3367	1.24559	0.622793	+	0.782387i	$0.285998\pi$
$242$	−3.47859	−0.223612
$243$	0	0
$244$	−14.6831	−0.939987
$245$	−18.4421	−1.17823
$246$	0	0
$247$	22.9431	1.45983
$248$	−7.47224	−0.474488
$249$	0	0
$250$	4.03700	0.255322
$251$	−10.1497	−0.640646	−0.320323	−	0.947308i	$-0.603791\pi$
$251$	−10.1497	−0.640646	−0.320323	+	0.947308i	$0.603791\pi$
$252$	0	0
$253$	9.73488	0.612027
$254$	−0.408259	−0.0256164
$255$	0	0
$256$	2.91217	0.182011
$257$	16.6988	1.04165	0.520823	−	0.853665i	$-0.325625\pi$
$257$	16.6988	1.04165	0.520823	+	0.853665i	$0.325625\pi$
$258$	0	0
$259$	2.22351	0.138162
$260$	7.27855	0.451396
$261$	0	0
$262$	6.49009	0.400959
$263$	10.9961	0.678051	0.339025	−	0.940777i	$-0.389903\pi$
$263$	10.9961	0.678051	0.339025	+	0.940777i	$0.389903\pi$
$264$	0	0
$265$	13.1783	0.809535
$266$	−12.8480	−0.787763
$267$	0	0
$268$	−21.6307	−1.32131
$269$	5.15463	0.314283	0.157142	−	0.987576i	$-0.449772\pi$
$269$	5.15463	0.314283	0.157142	+	0.987576i	$0.449772\pi$
$270$	0	0
$271$	−17.3231	−1.05231	−0.526153	−	0.850390i	$-0.676366\pi$
$271$	−17.3231	−1.05231	−0.526153	+	0.850390i	$0.676366\pi$
$272$	−16.5611	−1.00417
$273$	0	0
$274$	−0.0750005	−0.00453094
$275$	6.79059	0.409488
$276$	0	0
$277$	11.3082	0.679445	0.339722	−	0.940526i	$-0.389667\pi$
$277$	11.3082	0.679445	0.339722	+	0.940526i	$0.389667\pi$
$278$	9.67760	0.580424
$279$	0	0
$280$	−8.57805	−0.512636
$281$	−27.4835	−1.63953	−0.819764	−	0.572701i	$-0.805896\pi$
$281$	−27.4835	−1.63953	−0.819764	+	0.572701i	$0.805896\pi$
$282$	0	0
$283$	26.5476	1.57809	0.789046	−	0.614334i	$-0.210575\pi$
$283$	26.5476	1.57809	0.789046	+	0.614334i	$0.210575\pi$
$284$	−21.3358	−1.26604
$285$	0	0
$286$	−2.91374	−0.172293
$287$	−21.7295	−1.28265
$288$	0	0
$289$	15.6122	0.918365
$290$	−0.812489	−0.0477110
$291$	0	0
$292$	−2.72244	−0.159319
$293$	−7.32979	−0.428211	−0.214105	−	0.976811i	$-0.568684\pi$
$293$	−7.32979	−0.428211	−0.214105	+	0.976811i	$0.568684\pi$
$294$	0	0
$295$	−11.1387	−0.648523
$296$	0.741415	0.0430939
$297$	0	0
$298$	−3.21406	−0.186185
$299$	−21.6956	−1.25469
$300$	0	0
$301$	25.0963	1.44653
$302$	−4.87577	−0.280569
$303$	0	0
$304$	17.2225	0.987775
$305$	−8.43750	−0.483130
$306$	0	0
$307$	15.5854	0.889507	0.444754	−	0.895653i	$-0.353291\pi$
$307$	15.5854	0.889507	0.444754	+	0.895653i	$0.353291\pi$
$308$	−15.6071	−0.889297
$309$	0	0
$310$	−2.04028	−0.115880
$311$	8.97394	0.508865	0.254433	−	0.967090i	$-0.418111\pi$
$311$	8.97394	0.508865	0.254433	+	0.967090i	$0.418111\pi$
$312$	0	0
$313$	−13.1963	−0.745899	−0.372950	−	0.927852i	$-0.621654\pi$
$313$	−13.1963	−0.745899	−0.372950	+	0.927852i	$0.621654\pi$
$314$	−6.98433	−0.394149
$315$	0	0
$316$	−3.24555	−0.182576
$317$	−5.98526	−0.336166	−0.168083	−	0.985773i	$-0.553758\pi$
$317$	−5.98526	−0.336166	−0.168083	+	0.985773i	$0.553758\pi$
$318$	0	0
$319$	−3.11107	−0.174187
$320$	3.96253	0.221512
$321$	0	0
$322$	12.1495	0.677063
$323$	−33.9145	−1.88705
$324$	0	0
$325$	−15.1339	−0.839475
$326$	−0.266489	−0.0147595
$327$	0	0
$328$	−7.24556	−0.400069
$329$	−50.1559	−2.76518
$330$	0	0
$331$	17.4456	0.958898	0.479449	−	0.877570i	$-0.340836\pi$
$331$	17.4456	0.958898	0.479449	+	0.877570i	$0.340836\pi$
$332$	15.1432	0.831089
$333$	0	0
$334$	7.33364	0.401279
$335$	−12.4299	−0.679118
$336$	0	0
$337$	17.5337	0.955122	0.477561	−	0.878599i	$-0.341521\pi$
$337$	17.5337	0.955122	0.477561	+	0.878599i	$0.341521\pi$
$338$	0.837528	0.0455555
$339$	0	0
$340$	−10.7592	−0.583498
$341$	−7.81235	−0.423062
$342$	0	0
$343$	−53.3249	−2.87927
$344$	8.36820	0.451183
$345$	0	0
$346$	4.88780	0.262770
$347$	−6.66721	−0.357914	−0.178957	−	0.983857i	$-0.557272\pi$
$347$	−6.66721	−0.357914	−0.178957	+	0.983857i	$0.557272\pi$
$348$	0	0
$349$	−26.7680	−1.43286	−0.716430	−	0.697659i	$-0.754225\pi$
$349$	−26.7680	−1.43286	−0.716430	+	0.697659i	$0.754225\pi$
$350$	8.47490	0.453002
$351$	0	0
$352$	−7.93538	−0.422957
$353$	21.2070	1.12873	0.564367	−	0.825524i	$-0.309120\pi$
$353$	21.2070	1.12873	0.564367	+	0.825524i	$0.309120\pi$
$354$	0	0
$355$	−12.2604	−0.650715
$356$	−8.20505	−0.434867
$357$	0	0
$358$	−7.58821	−0.401049
$359$	3.01897	0.159335	0.0796676	−	0.996821i	$-0.474614\pi$
$359$	3.01897	0.159335	0.0796676	+	0.996821i	$0.474614\pi$
$360$	0	0
$361$	16.2687	0.856249
$362$	−0.435090	−0.0228678
$363$	0	0
$364$	34.7828	1.82311
$365$	−1.56443	−0.0818858
$366$	0	0
$367$	4.71001	0.245860	0.122930	−	0.992415i	$-0.460771\pi$
$367$	4.71001	0.245860	0.122930	+	0.992415i	$0.460771\pi$
$368$	−16.2861	−0.848970
$369$	0	0
$370$	0.202442	0.0105244
$371$	62.9764	3.26957
$372$	0	0
$373$	−0.00531301	−0.000275097 0	−0.000137549	−	1.00000i	$-0.500044\pi$
$373$	−0.00531301	−0.000275097 0	−0.000137549		1.00000i	$0.500044\pi$
$374$	4.30709	0.222714
$375$	0	0
$376$	−16.7242	−0.862482
$377$	6.93349	0.357093
$378$	0	0
$379$	−31.6246	−1.62445	−0.812225	−	0.583345i	$-0.801744\pi$
$379$	−31.6246	−1.62445	−0.812225	+	0.583345i	$0.801744\pi$
$380$	11.1888	0.573973
$381$	0	0
$382$	9.43185	0.482576
$383$	30.1387	1.54002	0.770008	−	0.638035i	$-0.220252\pi$
$383$	30.1387	1.54002	0.770008	+	0.638035i	$0.220252\pi$
$384$	0	0
$385$	−8.96849	−0.457077
$386$	0.0260338	0.00132509
$387$	0	0
$388$	−14.3059	−0.726273
$389$	21.2233	1.07607	0.538033	−	0.842924i	$-0.319168\pi$
$389$	21.2233	1.07607	0.538033	+	0.842924i	$0.319168\pi$
$390$	0	0
$391$	32.0705	1.62188
$392$	−29.3868	−1.48426
$393$	0	0
$394$	−7.30197	−0.367868
$395$	−1.86503	−0.0938398
$396$	0	0
$397$	−2.32490	−0.116683	−0.0583416	−	0.998297i	$-0.518581\pi$
$397$	−2.32490	−0.116683	−0.0583416	+	0.998297i	$0.518581\pi$
$398$	−3.88834	−0.194905
$399$	0	0
$400$	−11.3604	−0.568019
$401$	18.6616	0.931918	0.465959	−	0.884806i	$-0.345709\pi$
$401$	18.6616	0.931918	0.465959	+	0.884806i	$0.345709\pi$
$402$	0	0
$403$	17.4110	0.867303
$404$	17.1043	0.850969
$405$	0	0
$406$	−3.88273	−0.192696
$407$	0.775161	0.0384233
$408$	0	0
$409$	−36.3105	−1.79544	−0.897720	−	0.440567i	$-0.854777\pi$
$409$	−36.3105	−1.79544	−0.897720	+	0.440567i	$0.854777\pi$
$410$	−1.97838	−0.0977054
$411$	0	0
$412$	21.5341	1.06091
$413$	−53.2299	−2.61927
$414$	0	0
$415$	8.70189	0.427159
$416$	17.6852	0.867088
$417$	0	0
$418$	−4.47908	−0.219079
$419$	34.3587	1.67853	0.839266	−	0.543721i	$-0.182985\pi$
$419$	34.3587	1.67853	0.839266	+	0.543721i	$0.182985\pi$
$420$	0	0
$421$	−11.2616	−0.548855	−0.274428	−	0.961608i	$-0.588488\pi$
$421$	−11.2616	−0.548855	−0.274428	+	0.961608i	$0.588488\pi$
$422$	0.783175	0.0381244
$423$	0	0
$424$	20.9991	1.01980
$425$	22.3709	1.08515
$426$	0	0
$427$	−40.3211	−1.95128
$428$	0.243719	0.0117806
$429$	0	0
$430$	2.28492	0.110189
$431$	20.3604	0.980728	0.490364	−	0.871518i	$-0.336864\pi$
$431$	20.3604	0.980728	0.490364	+	0.871518i	$0.336864\pi$
$432$	0	0
$433$	8.55947	0.411342	0.205671	−	0.978621i	$-0.434062\pi$
$433$	8.55947	0.411342	0.205671	+	0.978621i	$0.434062\pi$
$434$	−9.75008	−0.468019
$435$	0	0
$436$	9.83769	0.471140
$437$	−33.3512	−1.59540
$438$	0	0
$439$	2.08992	0.0997465	0.0498733	−	0.998756i	$-0.484118\pi$
$439$	2.08992	0.0997465	0.0498733	+	0.998756i	$0.484118\pi$
$440$	−2.99048	−0.142566
$441$	0	0
$442$	−9.59900	−0.456578
$443$	−12.6084	−0.599042	−0.299521	−	0.954090i	$-0.596827\pi$
$443$	−12.6084	−0.599042	−0.299521	+	0.954090i	$0.596827\pi$
$444$	0	0
$445$	−4.71496	−0.223511
$446$	0.321203	0.0152094
$447$	0	0
$448$	18.9361	0.894649
$449$	26.9674	1.27267	0.636336	−	0.771412i	$-0.280449\pi$
$449$	26.9674	1.27267	0.636336	+	0.771412i	$0.280449\pi$
$450$	0	0
$451$	−7.57535	−0.356709
$452$	33.3989	1.57095
$453$	0	0
$454$	3.69392	0.173364
$455$	19.9876	0.937034
$456$	0	0
$457$	23.1095	1.08102	0.540508	−	0.841339i	$-0.318232\pi$
$457$	23.1095	1.08102	0.540508	+	0.841339i	$0.318232\pi$
$458$	6.38315	0.298265
$459$	0	0
$460$	−10.5805	−0.493316
$461$	10.7806	0.502103	0.251052	−	0.967974i	$-0.419224\pi$
$461$	10.7806	0.502103	0.251052	+	0.967974i	$0.419224\pi$
$462$	0	0
$463$	14.1337	0.656849	0.328425	−	0.944530i	$-0.393482\pi$
$463$	14.1337	0.656849	0.328425	+	0.944530i	$0.393482\pi$
$464$	5.20470	0.241622
$465$	0	0
$466$	8.48568	0.393092
$467$	4.29978	0.198970	0.0994851	−	0.995039i	$-0.468280\pi$
$467$	4.29978	0.198970	0.0994851	+	0.995039i	$0.468280\pi$
$468$	0	0
$469$	−59.4000	−2.74284
$470$	−4.56649	−0.210636
$471$	0	0
$472$	−17.7491	−0.816971
$473$	8.74909	0.402284
$474$	0	0
$475$	−23.2642	−1.06743
$476$	−51.4159	−2.35664
$477$	0	0
$478$	11.0802	0.506796
$479$	4.62278	0.211220	0.105610	−	0.994408i	$-0.466320\pi$
$479$	4.62278	0.211220	0.105610	+	0.994408i	$0.466320\pi$
$480$	0	0
$481$	−1.72756	−0.0787701
$482$	8.41321	0.383211
$483$	0	0
$484$	14.4767	0.658032
$485$	−8.22078	−0.373286
$486$	0	0
$487$	19.1098	0.865947	0.432974	−	0.901407i	$-0.357464\pi$
$487$	19.1098	0.865947	0.432974	+	0.901407i	$0.357464\pi$
$488$	−13.4448	−0.608618
$489$	0	0
$490$	−8.02400	−0.362487
$491$	−8.58025	−0.387221	−0.193611	−	0.981078i	$-0.562020\pi$
$491$	−8.58025	−0.387221	−0.193611	+	0.981078i	$0.562020\pi$
$492$	0	0
$493$	−10.2491	−0.461596
$494$	9.98231	0.449125
$495$	0	0
$496$	13.0697	0.586849
$497$	−58.5901	−2.62812
$498$	0	0
$499$	18.8535	0.844001	0.422000	−	0.906596i	$-0.361328\pi$
$499$	18.8535	0.844001	0.422000	+	0.906596i	$0.361328\pi$
$500$	−16.8006	−0.751345
$501$	0	0
$502$	−4.41605	−0.197098
$503$	−16.9707	−0.756687	−0.378343	−	0.925665i	$-0.623506\pi$
$503$	−16.9707	−0.756687	−0.378343	+	0.925665i	$0.623506\pi$
$504$	0	0
$505$	9.82882	0.437377
$506$	4.23555	0.188293
$507$	0	0
$508$	1.69903	0.0753823
$509$	−7.94385	−0.352105	−0.176052	−	0.984381i	$-0.556333\pi$
$509$	−7.94385	−0.352105	−0.176052	+	0.984381i	$0.556333\pi$
$510$	0	0
$511$	−7.47609	−0.330723
$512$	22.8920	1.01169
$513$	0	0
$514$	7.26551	0.320468
$515$	12.3744	0.545280
$516$	0	0
$517$	−17.4854	−0.769006
$518$	0.967428	0.0425064
$519$	0	0
$520$	6.66474	0.292268
$521$	42.9018	1.87956	0.939780	−	0.341780i	$-0.111030\pi$
$521$	42.9018	1.87956	0.939780	+	0.341780i	$0.111030\pi$
$522$	0	0
$523$	−18.9001	−0.826445	−0.413222	−	0.910630i	$-0.635597\pi$
$523$	−18.9001	−0.826445	−0.413222	+	0.910630i	$0.635597\pi$
$524$	−27.0095	−1.17992
$525$	0	0
$526$	4.78431	0.208606
$527$	−25.7369	−1.12112
$528$	0	0
$529$	8.53783	0.371210
$530$	5.73374	0.249058
$531$	0	0
$532$	53.4690	2.31818
$533$	16.8828	0.731276
$534$	0	0
$535$	0.140051	0.00605493
$536$	−19.8065	−0.855512
$537$	0	0
$538$	2.24273	0.0966910
$539$	−30.7244	−1.32339
$540$	0	0
$541$	20.1381	0.865806	0.432903	−	0.901440i	$-0.357489\pi$
$541$	20.1381	0.865806	0.432903	+	0.901440i	$0.357489\pi$
$542$	−7.53713	−0.323748
$543$	0	0
$544$	−26.1423	−1.12084
$545$	5.65315	0.242154
$546$	0	0
$547$	−26.4146	−1.12940	−0.564702	−	0.825295i	$-0.691009\pi$
$547$	−26.4146	−1.12940	−0.564702	+	0.825295i	$0.691009\pi$
$548$	0.312126	0.0133334
$549$	0	0
$550$	2.95452	0.125981
$551$	10.6584	0.454061
$552$	0	0
$553$	−8.91260	−0.379002
$554$	4.92010	0.209035
$555$	0	0
$556$	−40.2748	−1.70803
$557$	−18.1850	−0.770523	−0.385261	−	0.922807i	$-0.625889\pi$
$557$	−18.1850	−0.770523	−0.385261	+	0.922807i	$0.625889\pi$
$558$	0	0
$559$	−19.4987	−0.824706
$560$	15.0039	0.634031
$561$	0	0
$562$	−11.9578	−0.504410
$563$	−24.2587	−1.02238	−0.511191	−	0.859467i	$-0.670795\pi$
$563$	−24.2587	−1.02238	−0.511191	+	0.859467i	$0.670795\pi$
$564$	0	0
$565$	19.1924	0.807429
$566$	11.5506	0.485509
$567$	0	0
$568$	−19.5365	−0.819732
$569$	−4.03129	−0.169001	−0.0845003	−	0.996423i	$-0.526929\pi$
$569$	−4.03129	−0.169001	−0.0845003	+	0.996423i	$0.526929\pi$
$570$	0	0
$571$	−14.0544	−0.588157	−0.294079	−	0.955781i	$-0.595013\pi$
$571$	−14.0544	−0.588157	−0.294079	+	0.955781i	$0.595013\pi$
$572$	12.1260	0.507013
$573$	0	0
$574$	−9.45431	−0.394615
$575$	21.9993	0.917435
$576$	0	0
$577$	12.6968	0.528574	0.264287	−	0.964444i	$-0.414863\pi$
$577$	12.6968	0.528574	0.264287	+	0.964444i	$0.414863\pi$
$578$	6.79272	0.282540
$579$	0	0
$580$	3.38130	0.140401
$581$	41.5846	1.72522
$582$	0	0
$583$	21.9549	0.909277
$584$	−2.49285	−0.103155
$585$	0	0
$586$	−3.18912	−0.131741
$587$	−15.5407	−0.641433	−0.320716	−	0.947175i	$-0.603924\pi$
$587$	−15.5407	−0.641433	−0.320716	+	0.947175i	$0.603924\pi$
$588$	0	0
$589$	26.7647	1.10282
$590$	−4.84636	−0.199522
$591$	0	0
$592$	−1.29681	−0.0532987
$593$	31.3473	1.28728	0.643641	−	0.765328i	$-0.277423\pi$
$593$	31.3473	1.28728	0.643641	+	0.765328i	$0.277423\pi$
$594$	0	0
$595$	−29.5457	−1.21126
$596$	13.3758	0.547894
$597$	0	0
$598$	−9.43956	−0.386012
$599$	−15.9768	−0.652793	−0.326397	−	0.945233i	$-0.605835\pi$
$599$	−15.9768	−0.652793	−0.326397	+	0.945233i	$0.605835\pi$
$600$	0	0
$601$	13.3949	0.546389	0.273194	−	0.961959i	$-0.411920\pi$
$601$	13.3949	0.546389	0.273194	+	0.961959i	$0.411920\pi$
$602$	10.9192	0.445032
$603$	0	0
$604$	20.2913	0.825641
$605$	8.31892	0.338212
$606$	0	0
$607$	37.9707	1.54118	0.770591	−	0.637330i	$-0.219961\pi$
$607$	37.9707	1.54118	0.770591	+	0.637330i	$0.219961\pi$
$608$	27.1862	1.10255
$609$	0	0
$610$	−3.67108	−0.148637
$611$	38.9688	1.57651
$612$	0	0
$613$	10.4393	0.421641	0.210820	−	0.977525i	$-0.432386\pi$
$613$	10.4393	0.421641	0.210820	+	0.977525i	$0.432386\pi$
$614$	6.78107	0.273662
$615$	0	0
$616$	−14.2909	−0.575798
$617$	36.6275	1.47457	0.737284	−	0.675583i	$-0.236108\pi$
$617$	36.6275	1.47457	0.737284	+	0.675583i	$0.236108\pi$
$618$	0	0
$619$	−35.6460	−1.43274	−0.716368	−	0.697723i	$-0.754197\pi$
$619$	−35.6460	−1.43274	−0.716368	+	0.697723i	$0.754197\pi$
$620$	8.49093	0.341004
$621$	0	0
$622$	3.90447	0.156555
$623$	−22.5319	−0.902721
$624$	0	0
$625$	9.93246	0.397299
$626$	−5.74159	−0.229480
$627$	0	0
$628$	29.0664	1.15987
$629$	2.55369	0.101822
$630$	0	0
$631$	−37.6033	−1.49696	−0.748481	−	0.663156i	$-0.769217\pi$
$631$	−37.6033	−1.49696	−0.748481	+	0.663156i	$0.769217\pi$
$632$	−2.97185	−0.118214
$633$	0	0
$634$	−2.60413	−0.103423
$635$	0.976334	0.0387447
$636$	0	0
$637$	68.4739	2.71304
$638$	−1.35360	−0.0535894
$639$	0	0
$640$	11.2504	0.444711
$641$	−40.6857	−1.60699	−0.803494	−	0.595313i	$-0.797028\pi$
$641$	−40.6857	−1.60699	−0.803494	+	0.595313i	$0.797028\pi$
$642$	0	0
$643$	40.4424	1.59489	0.797447	−	0.603389i	$-0.206183\pi$
$643$	40.4424	1.59489	0.797447	+	0.603389i	$0.206183\pi$
$644$	−50.5619	−1.99242
$645$	0	0
$646$	−14.7559	−0.580562
$647$	−11.3180	−0.444955	−0.222477	−	0.974938i	$-0.571414\pi$
$647$	−11.3180	−0.444955	−0.222477	+	0.974938i	$0.571414\pi$
$648$	0	0
$649$	−18.5570	−0.728427
$650$	−6.58459	−0.258269
$651$	0	0
$652$	1.10904	0.0434332
$653$	29.5565	1.15663	0.578317	−	0.815812i	$-0.303710\pi$
$653$	29.5565	1.15663	0.578317	+	0.815812i	$0.303710\pi$
$654$	0	0
$655$	−15.5208	−0.606447
$656$	12.6733	0.494808
$657$	0	0
$658$	−21.8224	−0.850724
$659$	−13.6711	−0.532549	−0.266274	−	0.963897i	$-0.585793\pi$
$659$	−13.6711	−0.532549	−0.266274	+	0.963897i	$0.585793\pi$
$660$	0	0
$661$	4.01544	0.156183	0.0780913	−	0.996946i	$-0.475117\pi$
$661$	4.01544	0.156183	0.0780913	+	0.996946i	$0.475117\pi$
$662$	7.59043	0.295010
$663$	0	0
$664$	13.8661	0.538110
$665$	30.7255	1.19149
$666$	0	0
$667$	−10.0789	−0.390255
$668$	−30.5201	−1.18086
$669$	0	0
$670$	−5.40813	−0.208934
$671$	−14.0568	−0.542656
$672$	0	0
$673$	5.90376	0.227573	0.113787	−	0.993505i	$-0.463702\pi$
$673$	5.90376	0.227573	0.113787	+	0.993505i	$0.463702\pi$
$674$	7.62875	0.293849
$675$	0	0
$676$	−3.48550	−0.134058
$677$	7.33737	0.281998	0.140999	−	0.990010i	$-0.454969\pi$
$677$	7.33737	0.281998	0.140999	+	0.990010i	$0.454969\pi$
$678$	0	0
$679$	−39.2855	−1.50764
$680$	−9.85183	−0.377800
$681$	0	0
$682$	−3.39908	−0.130157
$683$	−10.2402	−0.391830	−0.195915	−	0.980621i	$-0.562768\pi$
$683$	−10.2402	−0.391830	−0.195915	+	0.980621i	$0.562768\pi$
$684$	0	0
$685$	0.179361	0.00685302
$686$	−23.2012	−0.885824
$687$	0	0
$688$	−14.6369	−0.558026
$689$	−48.9297	−1.86407
$690$	0	0
$691$	−43.0127	−1.63628	−0.818140	−	0.575019i	$-0.804995\pi$
$691$	−43.0127	−1.63628	−0.818140	+	0.575019i	$0.804995\pi$
$692$	−20.3414	−0.773262
$693$	0	0
$694$	−2.90084	−0.110114
$695$	−23.1436	−0.877886
$696$	0	0
$697$	−24.9562	−0.945283
$698$	−11.6465	−0.440827
$699$	0	0
$700$	−35.2696	−1.33307
$701$	−22.5656	−0.852291	−0.426146	−	0.904655i	$-0.640129\pi$
$701$	−22.5656	−0.852291	−0.426146	+	0.904655i	$0.640129\pi$
$702$	0	0
$703$	−2.65566	−0.100160
$704$	6.60153	0.248804
$705$	0	0
$706$	9.22696	0.347261
$707$	46.9700	1.76649
$708$	0	0
$709$	−20.4652	−0.768586	−0.384293	−	0.923211i	$-0.625555\pi$
$709$	−20.4652	−0.768586	−0.384293	+	0.923211i	$0.625555\pi$
$710$	−5.33439	−0.200196
$711$	0	0
$712$	−7.51310	−0.281565
$713$	−25.3095	−0.947847
$714$	0	0
$715$	6.96809	0.260592
$716$	31.5795	1.18018
$717$	0	0
$718$	1.31353	0.0490203
$719$	−11.8034	−0.440192	−0.220096	−	0.975478i	$-0.570637\pi$
$719$	−11.8034	−0.440192	−0.220096	+	0.975478i	$0.570637\pi$
$720$	0	0
$721$	59.1347	2.20229
$722$	7.07837	0.263430
$723$	0	0
$724$	1.81070	0.0672940
$725$	−7.03054	−0.261108
$726$	0	0
$727$	26.6478	0.988313	0.494156	−	0.869373i	$-0.335477\pi$
$727$	26.6478	0.988313	0.494156	+	0.869373i	$0.335477\pi$
$728$	31.8495	1.18042
$729$	0	0
$730$	−0.680667	−0.0251926
$731$	28.8230	1.06606
$732$	0	0
$733$	5.76250	0.212843	0.106421	−	0.994321i	$-0.466061\pi$
$733$	5.76250	0.212843	0.106421	+	0.994321i	$0.466061\pi$
$734$	2.04928	0.0756403
$735$	0	0
$736$	−25.7081	−0.947611
$737$	−20.7081	−0.762791
$738$	0	0
$739$	54.1619	1.99238	0.996188	−	0.0872308i	$-0.0278018\pi$
$739$	54.1619	1.99238	0.996188	+	0.0872308i	$0.0278018\pi$
$740$	−0.842492	−0.0309706
$741$	0	0
$742$	27.4004	1.00590
$743$	−24.1132	−0.884629	−0.442315	−	0.896860i	$-0.645843\pi$
$743$	−24.1132	−0.884629	−0.442315	+	0.896860i	$0.645843\pi$
$744$	0	0
$745$	7.68629	0.281604
$746$	−0.00231164	−8.46351e−5 0
$747$	0	0
$748$	−17.9246	−0.655390
$749$	0.669276	0.0244548
$750$	0	0
$751$	−41.8480	−1.52705	−0.763527	−	0.645775i	$-0.776534\pi$
$751$	−41.8480	−1.52705	−0.763527	+	0.645775i	$0.776534\pi$
$752$	29.2523	1.06672
$753$	0	0
$754$	3.01670	0.109862
$755$	11.6602	0.424359
$756$	0	0
$757$	−2.96428	−0.107739	−0.0538694	−	0.998548i	$-0.517155\pi$
$757$	−2.96428	−0.107739	−0.0538694	+	0.998548i	$0.517155\pi$
$758$	−13.7596	−0.499771
$759$	0	0
$760$	10.2452	0.371633
$761$	−34.8517	−1.26337	−0.631687	−	0.775223i	$-0.717637\pi$
$761$	−34.8517	−1.26337	−0.631687	+	0.775223i	$0.717637\pi$
$762$	0	0
$763$	27.0153	0.978019
$764$	−39.2521	−1.42009
$765$	0	0
$766$	13.1131	0.473794
$767$	41.3571	1.49332
$768$	0	0
$769$	46.4430	1.67478	0.837389	−	0.546607i	$-0.184081\pi$
$769$	46.4430	1.67478	0.837389	+	0.546607i	$0.184081\pi$
$770$	−3.90210	−0.140622
$771$	0	0
$772$	−0.108344	−0.00389938
$773$	−12.9256	−0.464900	−0.232450	−	0.972608i	$-0.574674\pi$
$773$	−12.9256	−0.464900	−0.232450	+	0.972608i	$0.574674\pi$
$774$	0	0
$775$	−17.6547	−0.634175
$776$	−13.0995	−0.470244
$777$	0	0
$778$	9.23407	0.331057
$779$	25.9527	0.929853
$780$	0	0
$781$	−20.4257	−0.730889
$782$	13.9536	0.498979
$783$	0	0
$784$	51.4007	1.83574
$785$	16.7028	0.596147
$786$	0	0
$787$	6.82859	0.243413	0.121707	−	0.992566i	$-0.461163\pi$
$787$	6.82859	0.243413	0.121707	+	0.992566i	$0.461163\pi$
$788$	30.3883	1.08254
$789$	0	0
$790$	−0.811456	−0.0288703
$791$	91.7166	3.26107
$792$	0	0
$793$	31.3276	1.11248
$794$	−1.01154	−0.0358982
$795$	0	0
$796$	16.1819	0.573553
$797$	−13.8644	−0.491102	−0.245551	−	0.969384i	$-0.578969\pi$
$797$	−13.8644	−0.491102	−0.245551	+	0.969384i	$0.578969\pi$
$798$	0	0
$799$	−57.6037	−2.03787
$800$	−17.9327	−0.634018
$801$	0	0
$802$	8.11950	0.286710
$803$	−2.60632	−0.0919749
$804$	0	0
$805$	−29.0550	−1.02405
$806$	7.57535	0.266830
$807$	0	0
$808$	15.6618	0.550981
$809$	−26.4249	−0.929050	−0.464525	−	0.885560i	$-0.653775\pi$
$809$	−26.4249	−0.929050	−0.464525	+	0.885560i	$0.653775\pi$
$810$	0	0
$811$	43.2688	1.51937	0.759686	−	0.650290i	$-0.225353\pi$
$811$	43.2688	1.51937	0.759686	+	0.650290i	$0.225353\pi$
$812$	16.1586	0.567055
$813$	0	0
$814$	0.337265	0.0118211
$815$	0.637298	0.0223236
$816$	0	0
$817$	−29.9739	−1.04865
$818$	−15.7984	−0.552377
$819$	0	0
$820$	8.23335	0.287521
$821$	33.0917	1.15491	0.577454	−	0.816423i	$-0.304046\pi$
$821$	33.0917	1.15491	0.577454	+	0.816423i	$0.304046\pi$
$822$	0	0
$823$	11.4473	0.399028	0.199514	−	0.979895i	$-0.436064\pi$
$823$	11.4473	0.399028	0.199514	+	0.979895i	$0.436064\pi$
$824$	19.7181	0.686912
$825$	0	0
$826$	−23.1598	−0.805833
$827$	48.2730	1.67862	0.839309	−	0.543654i	$-0.182960\pi$
$827$	48.2730	1.67862	0.839309	+	0.543654i	$0.182960\pi$
$828$	0	0
$829$	3.38204	0.117463	0.0587315	−	0.998274i	$-0.481294\pi$
$829$	3.38204	0.117463	0.0587315	+	0.998274i	$0.481294\pi$
$830$	3.78611	0.131418
$831$	0	0
$832$	−14.7125	−0.510064
$833$	−101.218	−3.50700
$834$	0	0
$835$	−17.5381	−0.606931
$836$	18.6404	0.644692
$837$	0	0
$838$	14.9491	0.516409
$839$	33.2689	1.14857	0.574285	−	0.818655i	$-0.305280\pi$
$839$	33.2689	1.14857	0.574285	+	0.818655i	$0.305280\pi$
$840$	0	0
$841$	−25.7790	−0.888931
$842$	−4.89980	−0.168858
$843$	0	0
$844$	−3.25930	−0.112190
$845$	−2.00292	−0.0689024
$846$	0	0
$847$	39.7545	1.36598
$848$	−36.7296	−1.26130
$849$	0	0
$850$	9.73336	0.333851
$851$	2.51127	0.0860852
$852$	0	0
$853$	0.145953	0.00499735	0.00249867	−	0.999997i	$-0.499205\pi$
$853$	0.145953	0.00499735	0.00249867	+	0.999997i	$0.499205\pi$
$854$	−17.5433	−0.600321
$855$	0	0
$856$	0.223166	0.00762764
$857$	44.4655	1.51891	0.759457	−	0.650558i	$-0.225465\pi$
$857$	44.4655	1.51891	0.759457	+	0.650558i	$0.225465\pi$
$858$	0	0
$859$	−26.9463	−0.919395	−0.459697	−	0.888076i	$-0.652042\pi$
$859$	−26.9463	−0.919395	−0.459697	+	0.888076i	$0.652042\pi$
$860$	−9.50904	−0.324256
$861$	0	0
$862$	8.85864	0.301726
$863$	−1.43963	−0.0490057	−0.0245029	−	0.999700i	$-0.507800\pi$
$863$	−1.43963	−0.0490057	−0.0245029	+	0.999700i	$0.507800\pi$
$864$	0	0
$865$	−11.6890	−0.397438
$866$	3.72414	0.126552
$867$	0	0
$868$	40.5765	1.37726
$869$	−3.10712	−0.105402
$870$	0	0
$871$	46.1510	1.56377
$872$	9.00806	0.305052
$873$	0	0
$874$	−14.5108	−0.490834
$875$	−46.1361	−1.55968
$876$	0	0
$877$	32.2800	1.09002	0.545009	−	0.838430i	$-0.316526\pi$
$877$	32.2800	1.09002	0.545009	+	0.838430i	$0.316526\pi$
$878$	0.909305	0.0306876
$879$	0	0
$880$	5.23067	0.176326
$881$	14.6381	0.493171	0.246585	−	0.969121i	$-0.420691\pi$
$881$	14.6381	0.493171	0.246585	+	0.969121i	$0.420691\pi$
$882$	0	0
$883$	52.9687	1.78254	0.891270	−	0.453473i	$-0.149815\pi$
$883$	52.9687	1.78254	0.891270	+	0.453473i	$0.149815\pi$
$884$	39.9477	1.34359
$885$	0	0
$886$	−5.48578	−0.184299
$887$	38.3605	1.28802	0.644010	−	0.765017i	$-0.277270\pi$
$887$	38.3605	1.28802	0.644010	+	0.765017i	$0.277270\pi$
$888$	0	0
$889$	4.66571	0.156483
$890$	−2.05144	−0.0687643
$891$	0	0
$892$	−1.33673	−0.0447572
$893$	59.9039	2.00461
$894$	0	0
$895$	18.1469	0.606584
$896$	53.7634	1.79611
$897$	0	0
$898$	11.7333	0.391544
$899$	8.08840	0.269763
$900$	0	0
$901$	72.3279	2.40959
$902$	−3.29596	−0.109744
$903$	0	0
$904$	30.5823	1.01715
$905$	1.04050	0.0345874
$906$	0	0
$907$	−11.2690	−0.374180	−0.187090	−	0.982343i	$-0.559906\pi$
$907$	−11.2690	−0.374180	−0.187090	+	0.982343i	$0.559906\pi$
$908$	−15.3728	−0.510165
$909$	0	0
$910$	8.69642	0.288284
$911$	−40.5127	−1.34225	−0.671124	−	0.741345i	$-0.734188\pi$
$911$	−40.5127	−1.34225	−0.671124	+	0.741345i	$0.734188\pi$
$912$	0	0
$913$	14.4972	0.479789
$914$	10.0547	0.332580
$915$	0	0
$916$	−26.5645	−0.877714
$917$	−74.1708	−2.44933
$918$	0	0
$919$	−7.46636	−0.246293	−0.123146	−	0.992389i	$-0.539298\pi$
$919$	−7.46636	−0.246293	−0.123146	+	0.992389i	$0.539298\pi$
$920$	−9.68819	−0.319410
$921$	0	0
$922$	4.69054	0.154475
$923$	45.5217	1.49837
$924$	0	0
$925$	1.75174	0.0575970
$926$	6.14944	0.202083
$927$	0	0
$928$	8.21578	0.269696
$929$	15.8963	0.521541	0.260770	−	0.965401i	$-0.416023\pi$
$929$	15.8963	0.521541	0.260770	+	0.965401i	$0.416023\pi$
$930$	0	0
$931$	105.260	3.44976
$932$	−35.3145	−1.15676
$933$	0	0
$934$	1.87079	0.0612143
$935$	−10.3002	−0.336854
$936$	0	0
$937$	43.8202	1.43154	0.715771	−	0.698335i	$-0.246075\pi$
$937$	43.8202	1.43154	0.715771	+	0.698335i	$0.246075\pi$
$938$	−25.8444	−0.843849
$939$	0	0
$940$	19.0042	0.619847
$941$	35.6078	1.16078	0.580391	−	0.814338i	$-0.302900\pi$
$941$	35.6078	1.16078	0.580391	+	0.814338i	$0.302900\pi$
$942$	0	0
$943$	−24.5417	−0.799187
$944$	31.0452	1.01043
$945$	0	0
$946$	3.80665	0.123765
$947$	38.6326	1.25539	0.627695	−	0.778459i	$-0.283998\pi$
$947$	38.6326	1.25539	0.627695	+	0.778459i	$0.283998\pi$
$948$	0	0
$949$	5.80857	0.188554
$950$	−10.1220	−0.328402
$951$	0	0
$952$	−47.0799	−1.52587
$953$	−19.8583	−0.643273	−0.321637	−	0.946863i	$-0.604233\pi$
$953$	−19.8583	−0.643273	−0.321637	+	0.946863i	$0.604233\pi$
$954$	0	0
$955$	−22.5559	−0.729892
$956$	−46.1119	−1.49137
$957$	0	0
$958$	2.01133	0.0649830
$959$	0.857129	0.0276782
$960$	0	0
$961$	−10.6889	−0.344803
$962$	−0.751646	−0.0242340
$963$	0	0
$964$	−35.0129	−1.12769
$965$	−0.0622588	−0.00200418
$966$	0	0
$967$	−18.2887	−0.588126	−0.294063	−	0.955786i	$-0.595008\pi$
$967$	−18.2887	−0.588126	−0.294063	+	0.955786i	$0.595008\pi$
$968$	13.2559	0.426060
$969$	0	0
$970$	−3.57678	−0.114844
$971$	29.4102	0.943820	0.471910	−	0.881647i	$-0.343565\pi$
$971$	29.4102	0.943820	0.471910	+	0.881647i	$0.343565\pi$
$972$	0	0
$973$	−110.599	−3.54563
$974$	8.31449	0.266413
$975$	0	0
$976$	23.5164	0.752742
$977$	14.3499	0.459092	0.229546	−	0.973298i	$-0.426276\pi$
$977$	14.3499	0.459092	0.229546	+	0.973298i	$0.426276\pi$
$978$	0	0
$979$	−7.85507	−0.251049
$980$	33.3931	1.06670
$981$	0	0
$982$	−3.73318	−0.119131
$983$	3.02311	0.0964223	0.0482111	−	0.998837i	$-0.484648\pi$
$983$	3.02311	0.0964223	0.0482111	+	0.998837i	$0.484648\pi$
$984$	0	0
$985$	17.4624	0.556397
$986$	−4.45928	−0.142013
$987$	0	0
$988$	−41.5429	−1.32166
$989$	28.3442	0.901293
$990$	0	0
$991$	−16.5867	−0.526894	−0.263447	−	0.964674i	$-0.584859\pi$
$991$	−16.5867	−0.526894	−0.263447	+	0.964674i	$0.584859\pi$
$992$	20.6310	0.655035
$993$	0	0
$994$	−25.4920	−0.808557
$995$	9.29881	0.294792
$996$	0	0
$997$	−25.6850	−0.813453	−0.406727	−	0.913550i	$-0.633330\pi$
$997$	−25.6850	−0.813453	−0.406727	+	0.913550i	$0.633330\pi$
$998$	8.20300	0.259661
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1629.2.a.b.1.3		5
3.2	odd	2		543.2.a.b.1.3	✓	5
12.11	even	2		8688.2.a.z.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
543.2.a.b.1.3	✓	5	3.2	odd	2
1629.2.a.b.1.3		5	1.1	even	1	trivial
8688.2.a.z.1.3		5	12.11	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:	\( N \)	\(=\)	\( 1629 = 3^{2} \cdot 181 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1629.a (trivial)

\(f(q)\)	\(=\)	\(q+0.435090 q^{2} -1.81070 q^{4} -1.04050 q^{5} -4.97235 q^{7} -1.65800 q^{8} -0.452712 q^{10} -1.73346 q^{11} +3.86328 q^{13} -2.16342 q^{14} +2.90001 q^{16} -5.71071 q^{17} +5.93875 q^{19} +1.88403 q^{20} -0.754213 q^{22} -5.61586 q^{23} -3.91736 q^{25} +1.68088 q^{26} +9.00342 q^{28} +1.79471 q^{29} +4.50679 q^{31} +4.57776 q^{32} -2.48468 q^{34} +5.17374 q^{35} -0.447175 q^{37} +2.58389 q^{38} +1.72515 q^{40} +4.37007 q^{41} -5.04718 q^{43} +3.13878 q^{44} -2.44340 q^{46} +10.0870 q^{47} +17.7243 q^{49} -1.70440 q^{50} -6.99523 q^{52} -12.6653 q^{53} +1.80367 q^{55} +8.24415 q^{56} +0.780863 q^{58} +10.7052 q^{59} +8.10907 q^{61} +1.96086 q^{62} -3.80829 q^{64} -4.01975 q^{65} +11.9461 q^{67} +10.3404 q^{68} +2.25104 q^{70} +11.7832 q^{71} +1.50353 q^{73} -0.194562 q^{74} -10.7533 q^{76} +8.61939 q^{77} +1.79243 q^{79} -3.01747 q^{80} +1.90137 q^{82} -8.36317 q^{83} +5.94200 q^{85} -2.19598 q^{86} +2.87408 q^{88} +4.53143 q^{89} -19.2096 q^{91} +10.1686 q^{92} +4.38874 q^{94} -6.17928 q^{95} +7.90078 q^{97} +7.71167 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 3 q^{4} - 3 q^{5} + 9 q^{7} - 9 q^{8} - 14 q^{10} + 4 q^{11} + 9 q^{13} - 3 q^{14} + 7 q^{16} - 9 q^{17} + 6 q^{19} + 25 q^{20} - 17 q^{22} + 5 q^{23} + 12 q^{25} - 6 q^{26} + 27 q^{28} + 20 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) \) 5 * q + q^2 + 3 * q^4 - 3 * q^5 + 9 * q^7 - 9 * q^8 - 14 * q^10 + 4 * q^11 + 9 * q^13 - 3 * q^14 + 7 * q^16 - 9 * q^17 + 6 * q^19 + 25 * q^20 - 17 * q^22 + 5 * q^23 + 12 * q^25 - 6 * q^26 + 27 * q^28 + 20 * q^29 + 13 * q^31 - 12 * q^32 + 22 * q^34 + 8 * q^35 - 5 * q^37 + 26 * q^38 - 13 * q^40 + 2 * q^41 + 4 * q^43 + 9 * q^44 - 20 * q^46 + 2 * q^47 + 50 * q^49 - 20 * q^50 + 14 * q^52 - 19 * q^53 + 11 * q^55 - 8 * q^56 - 3 * q^58 + 25 * q^59 + 17 * q^61 + 11 * q^62 - 11 * q^64 + 8 * q^65 - 9 * q^67 + 15 * q^68 - 30 * q^70 - 9 * q^71 + 7 * q^73 + 15 * q^74 - 15 * q^76 + 38 * q^77 + 2 * q^79 + 31 * q^80 + q^82 - 14 * q^83 - 3 * q^85 + 4 * q^86 - 15 * q^88 + 12 * q^89 - 5 * q^91 + 39 * q^92 + 12 * q^94 - 22 * q^95 - 8 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

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