LMFDB - Embedded newform 990.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [990,2,Mod(1,990)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(990, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("990.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	990.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:	$7.90518980011$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	990.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.37228 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +3.37228 q^{14} +1.00000 q^{16} -1.37228 q^{17} +0.627719 q^{19} -1.00000 q^{20} +1.00000 q^{22} -2.74456 q^{23} +1.00000 q^{25} +2.00000 q^{26} +3.37228 q^{28} -1.37228 q^{29} +3.37228 q^{31} +1.00000 q^{32} -1.37228 q^{34} -3.37228 q^{35} +9.37228 q^{37} +0.627719 q^{38} -1.00000 q^{40} +11.4891 q^{41} -4.00000 q^{43} +1.00000 q^{44} -2.74456 q^{46} -2.74456 q^{47} +4.37228 q^{49} +1.00000 q^{50} +2.00000 q^{52} +4.11684 q^{53} -1.00000 q^{55} +3.37228 q^{56} -1.37228 q^{58} +2.74456 q^{59} -5.37228 q^{61} +3.37228 q^{62} +1.00000 q^{64} -2.00000 q^{65} +8.00000 q^{67} -1.37228 q^{68} -3.37228 q^{70} -10.1168 q^{71} -15.4891 q^{73} +9.37228 q^{74} +0.627719 q^{76} +3.37228 q^{77} -1.25544 q^{79} -1.00000 q^{80} +11.4891 q^{82} +2.74456 q^{83} +1.37228 q^{85} -4.00000 q^{86} +1.00000 q^{88} +1.37228 q^{89} +6.74456 q^{91} -2.74456 q^{92} -2.74456 q^{94} -0.627719 q^{95} -12.7446 q^{97} +4.37228 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{11} + 4 q^{13} + q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} - 2 q^{20} + 2 q^{22} + 6 q^{23} + 2 q^{25} + 4 q^{26} + q^{28} + 3 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 + 2 * q^8 - 2 * q^10 + 2 * q^11 + 4 * q^13 + q^14 + 2 * q^16 + 3 * q^17 + 7 * q^19 - 2 * q^20 + 2 * q^22 + 6 * q^23 + 2 * q^25 + 4 * q^26 + q^28 + 3 * q^29 + q^31 + 2 * q^32 + 3 * q^34 - q^35 + 13 * q^37 + 7 * q^38 - 2 * q^40 - 8 * q^43 + 2 * q^44 + 6 * q^46 + 6 * q^47 + 3 * q^49 + 2 * q^50 + 4 * q^52 - 9 * q^53 - 2 * q^55 + q^56 + 3 * q^58 - 6 * q^59 - 5 * q^61 + q^62 + 2 * q^64 - 4 * q^65 + 16 * q^67 + 3 * q^68 - q^70 - 3 * q^71 - 8 * q^73 + 13 * q^74 + 7 * q^76 + q^77 - 14 * q^79 - 2 * q^80 - 6 * q^83 - 3 * q^85 - 8 * q^86 + 2 * q^88 - 3 * q^89 + 2 * q^91 + 6 * q^92 + 6 * q^94 - 7 * q^95 - 14 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.37228	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$7$	3.37228	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	3.37228	0.901280
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.37228	−0.332827	−0.166414	−	0.986056i	$-0.553219\pi$
$17$	−1.37228	−0.332827	−0.166414	+	0.986056i	$0.553219\pi$
$18$	0	0
$19$	0.627719	0.144009	0.0720043	−	0.997404i	$-0.477060\pi$
$19$	0.627719	0.144009	0.0720043	+	0.997404i	$0.477060\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	−2.74456	−0.572281	−0.286140	−	0.958188i	$-0.592372\pi$
$23$	−2.74456	−0.572281	−0.286140	+	0.958188i	$0.592372\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	3.37228	0.637301
$29$	−1.37228	−0.254826	−0.127413	−	0.991850i	$-0.540667\pi$
$29$	−1.37228	−0.254826	−0.127413	+	0.991850i	$0.540667\pi$
$30$	0	0
$31$	3.37228	0.605680	0.302840	−	0.953041i	$-0.402065\pi$
$31$	3.37228	0.605680	0.302840	+	0.953041i	$0.402065\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.37228	−0.235344
$35$	−3.37228	−0.570020
$36$	0	0
$37$	9.37228	1.54079	0.770397	−	0.637565i	$-0.220058\pi$
$37$	9.37228	1.54079	0.770397	+	0.637565i	$0.220058\pi$
$38$	0.627719	0.101829
$39$	0	0
$40$	−1.00000	−0.158114
$41$	11.4891	1.79430	0.897150	−	0.441726i	$-0.145634\pi$
$41$	11.4891	1.79430	0.897150	+	0.441726i	$0.145634\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−2.74456	−0.404664
$47$	−2.74456	−0.400336	−0.200168	−	0.979762i	$-0.564149\pi$
$47$	−2.74456	−0.400336	−0.200168	+	0.979762i	$0.564149\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	4.11684	0.565492	0.282746	−	0.959195i	$-0.408755\pi$
$53$	4.11684	0.565492	0.282746	+	0.959195i	$0.408755\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	3.37228	0.450640
$57$	0	0
$58$	−1.37228	−0.180189
$59$	2.74456	0.357312	0.178656	−	0.983912i	$-0.442825\pi$
$59$	2.74456	0.357312	0.178656	+	0.983912i	$0.442825\pi$
$60$	0	0
$61$	−5.37228	−0.687850	−0.343925	−	0.938997i	$-0.611757\pi$
$61$	−5.37228	−0.687850	−0.343925	+	0.938997i	$0.611757\pi$
$62$	3.37228	0.428280
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−1.37228	−0.166414
$69$	0	0
$70$	−3.37228	−0.403065
$71$	−10.1168	−1.20065	−0.600324	−	0.799757i	$-0.704962\pi$
$71$	−10.1168	−1.20065	−0.600324	+	0.799757i	$0.704962\pi$
$72$	0	0
$73$	−15.4891	−1.81286	−0.906432	−	0.422351i	$-0.861205\pi$
$73$	−15.4891	−1.81286	−0.906432	+	0.422351i	$0.861205\pi$
$74$	9.37228	1.08951
$75$	0	0
$76$	0.627719	0.0720043
$77$	3.37228	0.384307
$78$	0	0
$79$	−1.25544	−0.141248	−0.0706239	−	0.997503i	$-0.522499\pi$
$79$	−1.25544	−0.141248	−0.0706239	+	0.997503i	$0.522499\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	11.4891	1.26876
$83$	2.74456	0.301255	0.150627	−	0.988591i	$-0.451871\pi$
$83$	2.74456	0.301255	0.150627	+	0.988591i	$0.451871\pi$
$84$	0	0
$85$	1.37228	0.148845
$86$	−4.00000	−0.431331
$87$	0	0
$88$	1.00000	0.106600
$89$	1.37228	0.145462	0.0727308	−	0.997352i	$-0.476829\pi$
$89$	1.37228	0.145462	0.0727308	+	0.997352i	$0.476829\pi$
$90$	0	0
$91$	6.74456	0.707022
$92$	−2.74456	−0.286140
$93$	0	0
$94$	−2.74456	−0.283080
$95$	−0.627719	−0.0644026
$96$	0	0
$97$	−12.7446	−1.29401	−0.647007	−	0.762484i	$-0.723980\pi$
$97$	−12.7446	−1.29401	−0.647007	+	0.762484i	$0.723980\pi$
$98$	4.37228	0.441667
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−9.48913	−0.934991	−0.467496	−	0.883995i	$-0.654844\pi$
$103$	−9.48913	−0.934991	−0.467496	+	0.883995i	$0.654844\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	4.11684	0.399863
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−15.4891	−1.48359	−0.741795	−	0.670627i	$-0.766025\pi$
$109$	−15.4891	−1.48359	−0.741795	+	0.670627i	$0.766025\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	3.37228	0.318651
$113$	−3.25544	−0.306246	−0.153123	−	0.988207i	$-0.548933\pi$
$113$	−3.25544	−0.306246	−0.153123	+	0.988207i	$0.548933\pi$
$114$	0	0
$115$	2.74456	0.255932
$116$	−1.37228	−0.127413
$117$	0	0
$118$	2.74456	0.252657
$119$	−4.62772	−0.424222
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.37228	−0.486383
$123$	0	0
$124$	3.37228	0.302840
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	−22.1168	−1.93236	−0.966179	−	0.257873i	$-0.916978\pi$
$131$	−22.1168	−1.93236	−0.966179	+	0.257873i	$0.916978\pi$
$132$	0	0
$133$	2.11684	0.183554
$134$	8.00000	0.691095
$135$	0	0
$136$	−1.37228	−0.117672
$137$	−8.74456	−0.747098	−0.373549	−	0.927610i	$-0.621859\pi$
$137$	−8.74456	−0.747098	−0.373549	+	0.927610i	$0.621859\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−3.37228	−0.285010
$141$	0	0
$142$	−10.1168	−0.848987
$143$	2.00000	0.167248
$144$	0	0
$145$	1.37228	0.113962
$146$	−15.4891	−1.28189
$147$	0	0
$148$	9.37228	0.770397
$149$	−21.6060	−1.77003	−0.885015	−	0.465563i	$-0.845852\pi$
$149$	−21.6060	−1.77003	−0.885015	+	0.465563i	$0.845852\pi$
$150$	0	0
$151$	−12.2337	−0.995563	−0.497782	−	0.867302i	$-0.665852\pi$
$151$	−12.2337	−0.995563	−0.497782	+	0.867302i	$0.665852\pi$
$152$	0.627719	0.0509147
$153$	0	0
$154$	3.37228	0.271746
$155$	−3.37228	−0.270868
$156$	0	0
$157$	9.37228	0.747989	0.373995	−	0.927431i	$-0.377988\pi$
$157$	9.37228	0.747989	0.373995	+	0.927431i	$0.377988\pi$
$158$	−1.25544	−0.0998772
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−9.25544	−0.729431
$162$	0	0
$163$	−5.88316	−0.460804	−0.230402	−	0.973095i	$-0.574004\pi$
$163$	−5.88316	−0.460804	−0.230402	+	0.973095i	$0.574004\pi$
$164$	11.4891	0.897150
$165$	0	0
$166$	2.74456	0.213019
$167$	4.62772	0.358104	0.179052	−	0.983840i	$-0.442697\pi$
$167$	4.62772	0.358104	0.179052	+	0.983840i	$0.442697\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	1.37228	0.105249
$171$	0	0
$172$	−4.00000	−0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	3.37228	0.254921
$176$	1.00000	0.0753778
$177$	0	0
$178$	1.37228	0.102857
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	6.74456	0.499940
$183$	0	0
$184$	−2.74456	−0.202332
$185$	−9.37228	−0.689064
$186$	0	0
$187$	−1.37228	−0.100351
$188$	−2.74456	−0.200168
$189$	0	0
$190$	−0.627719	−0.0455395
$191$	5.48913	0.397179	0.198590	−	0.980083i	$-0.436364\pi$
$191$	5.48913	0.397179	0.198590	+	0.980083i	$0.436364\pi$
$192$	0	0
$193$	14.8614	1.06975	0.534874	−	0.844932i	$-0.320359\pi$
$193$	14.8614	1.06975	0.534874	+	0.844932i	$0.320359\pi$
$194$	−12.7446	−0.915006
$195$	0	0
$196$	4.37228	0.312306
$197$	20.7446	1.47799	0.738994	−	0.673712i	$-0.235301\pi$
$197$	20.7446	1.47799	0.738994	+	0.673712i	$0.235301\pi$
$198$	0	0
$199$	18.1168	1.28427	0.642135	−	0.766592i	$-0.278049\pi$
$199$	18.1168	1.28427	0.642135	+	0.766592i	$0.278049\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−6.00000	−0.422159
$203$	−4.62772	−0.324802
$204$	0	0
$205$	−11.4891	−0.802435
$206$	−9.48913	−0.661139
$207$	0	0
$208$	2.00000	0.138675
$209$	0.627719	0.0434202
$210$	0	0
$211$	6.11684	0.421101	0.210550	−	0.977583i	$-0.432474\pi$
$211$	6.11684	0.421101	0.210550	+	0.977583i	$0.432474\pi$
$212$	4.11684	0.282746
$213$	0	0
$214$	12.0000	0.820303
$215$	4.00000	0.272798
$216$	0	0
$217$	11.3723	0.772001
$218$	−15.4891	−1.04906
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	−2.74456	−0.184619
$222$	0	0
$223$	−18.7446	−1.25523	−0.627614	−	0.778524i	$-0.715969\pi$
$223$	−18.7446	−1.25523	−0.627614	+	0.778524i	$0.715969\pi$
$224$	3.37228	0.225320
$225$	0	0
$226$	−3.25544	−0.216548
$227$	2.74456	0.182163	0.0910815	−	0.995843i	$-0.470968\pi$
$227$	2.74456	0.182163	0.0910815	+	0.995843i	$0.470968\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	2.74456	0.180971
$231$	0	0
$232$	−1.37228	−0.0900947
$233$	−1.37228	−0.0899011	−0.0449506	−	0.998989i	$-0.514313\pi$
$233$	−1.37228	−0.0899011	−0.0449506	+	0.998989i	$0.514313\pi$
$234$	0	0
$235$	2.74456	0.179036
$236$	2.74456	0.178656
$237$	0	0
$238$	−4.62772	−0.299970
$239$	14.7446	0.953746	0.476873	−	0.878972i	$-0.341770\pi$
$239$	14.7446	0.953746	0.476873	+	0.878972i	$0.341770\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−5.37228	−0.343925
$245$	−4.37228	−0.279335
$246$	0	0
$247$	1.25544	0.0798816
$248$	3.37228	0.214140
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−2.74456	−0.173235	−0.0866176	−	0.996242i	$-0.527606\pi$
$251$	−2.74456	−0.173235	−0.0866176	+	0.996242i	$0.527606\pi$
$252$	0	0
$253$	−2.74456	−0.172549
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	31.6060	1.96390
$260$	−2.00000	−0.124035
$261$	0	0
$262$	−22.1168	−1.36638
$263$	24.8614	1.53302	0.766510	−	0.642232i	$-0.221992\pi$
$263$	24.8614	1.53302	0.766510	+	0.642232i	$0.221992\pi$
$264$	0	0
$265$	−4.11684	−0.252896
$266$	2.11684	0.129792
$267$	0	0
$268$	8.00000	0.488678
$269$	8.74456	0.533165	0.266583	−	0.963812i	$-0.414105\pi$
$269$	8.74456	0.533165	0.266583	+	0.963812i	$0.414105\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−1.37228	−0.0832068
$273$	0	0
$274$	−8.74456	−0.528278
$275$	1.00000	0.0603023
$276$	0	0
$277$	−12.7446	−0.765747	−0.382873	−	0.923801i	$-0.625065\pi$
$277$	−12.7446	−0.765747	−0.382873	+	0.923801i	$0.625065\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	−3.37228	−0.201532
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	5.25544	0.312403	0.156202	−	0.987725i	$-0.450075\pi$
$283$	5.25544	0.312403	0.156202	+	0.987725i	$0.450075\pi$
$284$	−10.1168	−0.600324
$285$	0	0
$286$	2.00000	0.118262
$287$	38.7446	2.28702
$288$	0	0
$289$	−15.1168	−0.889226
$290$	1.37228	0.0805831
$291$	0	0
$292$	−15.4891	−0.906432
$293$	−23.4891	−1.37225	−0.686125	−	0.727484i	$-0.740690\pi$
$293$	−23.4891	−1.37225	−0.686125	+	0.727484i	$0.740690\pi$
$294$	0	0
$295$	−2.74456	−0.159795
$296$	9.37228	0.544753
$297$	0	0
$298$	−21.6060	−1.25160
$299$	−5.48913	−0.317444
$300$	0	0
$301$	−13.4891	−0.777500
$302$	−12.2337	−0.703970
$303$	0	0
$304$	0.627719	0.0360021
$305$	5.37228	0.307616
$306$	0	0
$307$	5.25544	0.299944	0.149972	−	0.988690i	$-0.452082\pi$
$307$	5.25544	0.299944	0.149972	+	0.988690i	$0.452082\pi$
$308$	3.37228	0.192154
$309$	0	0
$310$	−3.37228	−0.191533
$311$	−19.3723	−1.09850	−0.549251	−	0.835658i	$-0.685087\pi$
$311$	−19.3723	−1.09850	−0.549251	+	0.835658i	$0.685087\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	9.37228	0.528908
$315$	0	0
$316$	−1.25544	−0.0706239
$317$	24.3505	1.36766	0.683831	−	0.729640i	$-0.260313\pi$
$317$	24.3505	1.36766	0.683831	+	0.729640i	$0.260313\pi$
$318$	0	0
$319$	−1.37228	−0.0768330
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−9.25544	−0.515785
$323$	−0.861407	−0.0479299
$324$	0	0
$325$	2.00000	0.110940
$326$	−5.88316	−0.325838
$327$	0	0
$328$	11.4891	0.634381
$329$	−9.25544	−0.510269
$330$	0	0
$331$	30.9783	1.70272	0.851359	−	0.524583i	$-0.175779\pi$
$331$	30.9783	1.70272	0.851359	+	0.524583i	$0.175779\pi$
$332$	2.74456	0.150627
$333$	0	0
$334$	4.62772	0.253217
$335$	−8.00000	−0.437087
$336$	0	0
$337$	24.1168	1.31373	0.656864	−	0.754009i	$-0.271883\pi$
$337$	24.1168	1.31373	0.656864	+	0.754009i	$0.271883\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	1.37228	0.0744224
$341$	3.37228	0.182619
$342$	0	0
$343$	−8.86141	−0.478471
$344$	−4.00000	−0.215666
$345$	0	0
$346$	6.00000	0.322562
$347$	32.2337	1.73040	0.865198	−	0.501431i	$-0.167193\pi$
$347$	32.2337	1.73040	0.865198	+	0.501431i	$0.167193\pi$
$348$	0	0
$349$	19.4891	1.04323	0.521614	−	0.853181i	$-0.325330\pi$
$349$	19.4891	1.04323	0.521614	+	0.853181i	$0.325330\pi$
$350$	3.37228	0.180256
$351$	0	0
$352$	1.00000	0.0533002
$353$	0.510875	0.0271911	0.0135956	−	0.999908i	$-0.495672\pi$
$353$	0.510875	0.0271911	0.0135956	+	0.999908i	$0.495672\pi$
$354$	0	0
$355$	10.1168	0.536946
$356$	1.37228	0.0727308
$357$	0	0
$358$	12.0000	0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−18.6060	−0.979262
$362$	−10.0000	−0.525588
$363$	0	0
$364$	6.74456	0.353511
$365$	15.4891	0.810738
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	−2.74456	−0.143070
$369$	0	0
$370$	−9.37228	−0.487242
$371$	13.8832	0.720778
$372$	0	0
$373$	31.4891	1.63045	0.815223	−	0.579148i	$-0.196615\pi$
$373$	31.4891	1.63045	0.815223	+	0.579148i	$0.196615\pi$
$374$	−1.37228	−0.0709590
$375$	0	0
$376$	−2.74456	−0.141540
$377$	−2.74456	−0.141352
$378$	0	0
$379$	−0.233688	−0.0120037	−0.00600187	−	0.999982i	$-0.501910\pi$
$379$	−0.233688	−0.0120037	−0.00600187	+	0.999982i	$0.501910\pi$
$380$	−0.627719	−0.0322013
$381$	0	0
$382$	5.48913	0.280848
$383$	−32.2337	−1.64706	−0.823532	−	0.567269i	$-0.808000\pi$
$383$	−32.2337	−1.64706	−0.823532	+	0.567269i	$0.808000\pi$
$384$	0	0
$385$	−3.37228	−0.171867
$386$	14.8614	0.756426
$387$	0	0
$388$	−12.7446	−0.647007
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	3.76631	0.190471
$392$	4.37228	0.220834
$393$	0	0
$394$	20.7446	1.04510
$395$	1.25544	0.0631679
$396$	0	0
$397$	24.9783	1.25362	0.626811	−	0.779171i	$-0.284360\pi$
$397$	24.9783	1.25362	0.626811	+	0.779171i	$0.284360\pi$
$398$	18.1168	0.908115
$399$	0	0
$400$	1.00000	0.0500000
$401$	−13.3723	−0.667780	−0.333890	−	0.942612i	$-0.608361\pi$
$401$	−13.3723	−0.667780	−0.333890	+	0.942612i	$0.608361\pi$
$402$	0	0
$403$	6.74456	0.335971
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−4.62772	−0.229670
$407$	9.37228	0.464567
$408$	0	0
$409$	−1.76631	−0.0873385	−0.0436693	−	0.999046i	$-0.513905\pi$
$409$	−1.76631	−0.0873385	−0.0436693	+	0.999046i	$0.513905\pi$
$410$	−11.4891	−0.567407
$411$	0	0
$412$	−9.48913	−0.467496
$413$	9.25544	0.455430
$414$	0	0
$415$	−2.74456	−0.134725
$416$	2.00000	0.0980581
$417$	0	0
$418$	0.627719	0.0307027
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	10.2337	0.498759	0.249380	−	0.968406i	$-0.419773\pi$
$421$	10.2337	0.498759	0.249380	+	0.968406i	$0.419773\pi$
$422$	6.11684	0.297763
$423$	0	0
$424$	4.11684	0.199932
$425$	−1.37228	−0.0665654
$426$	0	0
$427$	−18.1168	−0.876736
$428$	12.0000	0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	34.9783	1.68484	0.842422	−	0.538819i	$-0.181129\pi$
$431$	34.9783	1.68484	0.842422	+	0.538819i	$0.181129\pi$
$432$	0	0
$433$	27.7228	1.33227	0.666137	−	0.745830i	$-0.267947\pi$
$433$	27.7228	1.33227	0.666137	+	0.745830i	$0.267947\pi$
$434$	11.3723	0.545887
$435$	0	0
$436$	−15.4891	−0.741795
$437$	−1.72281	−0.0824133
$438$	0	0
$439$	18.9783	0.905782	0.452891	−	0.891566i	$-0.350393\pi$
$439$	18.9783	0.905782	0.452891	+	0.891566i	$0.350393\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	−2.74456	−0.130546
$443$	−29.4891	−1.40107	−0.700535	−	0.713618i	$-0.747055\pi$
$443$	−29.4891	−1.40107	−0.700535	+	0.713618i	$0.747055\pi$
$444$	0	0
$445$	−1.37228	−0.0650524
$446$	−18.7446	−0.887581
$447$	0	0
$448$	3.37228	0.159325
$449$	−28.9783	−1.36757	−0.683784	−	0.729684i	$-0.739667\pi$
$449$	−28.9783	−1.36757	−0.683784	+	0.729684i	$0.739667\pi$
$450$	0	0
$451$	11.4891	0.541002
$452$	−3.25544	−0.153123
$453$	0	0
$454$	2.74456	0.128809
$455$	−6.74456	−0.316190
$456$	0	0
$457$	−16.3505	−0.764846	−0.382423	−	0.923987i	$-0.624910\pi$
$457$	−16.3505	−0.764846	−0.382423	+	0.923987i	$0.624910\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	2.74456	0.127966
$461$	−16.1168	−0.750636	−0.375318	−	0.926896i	$-0.622467\pi$
$461$	−16.1168	−0.750636	−0.375318	+	0.926896i	$0.622467\pi$
$462$	0	0
$463$	−0.233688	−0.0108604	−0.00543020	−	0.999985i	$-0.501728\pi$
$463$	−0.233688	−0.0108604	−0.00543020	+	0.999985i	$0.501728\pi$
$464$	−1.37228	−0.0637066
$465$	0	0
$466$	−1.37228	−0.0635697
$467$	19.3723	0.896442	0.448221	−	0.893923i	$-0.352058\pi$
$467$	19.3723	0.896442	0.448221	+	0.893923i	$0.352058\pi$
$468$	0	0
$469$	26.9783	1.24574
$470$	2.74456	0.126597
$471$	0	0
$472$	2.74456	0.126329
$473$	−4.00000	−0.183920
$474$	0	0
$475$	0.627719	0.0288017
$476$	−4.62772	−0.212111
$477$	0	0
$478$	14.7446	0.674401
$479$	−5.48913	−0.250805	−0.125402	−	0.992106i	$-0.540022\pi$
$479$	−5.48913	−0.250805	−0.125402	+	0.992106i	$0.540022\pi$
$480$	0	0
$481$	18.7446	0.854678
$482$	−22.0000	−1.00207
$483$	0	0
$484$	1.00000	0.0454545
$485$	12.7446	0.578701
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−5.37228	−0.243192
$489$	0	0
$490$	−4.37228	−0.197520
$491$	7.37228	0.332706	0.166353	−	0.986066i	$-0.446801\pi$
$491$	7.37228	0.332706	0.166353	+	0.986066i	$0.446801\pi$
$492$	0	0
$493$	1.88316	0.0848131
$494$	1.25544	0.0564848
$495$	0	0
$496$	3.37228	0.151420
$497$	−34.1168	−1.53035
$498$	0	0
$499$	−33.4891	−1.49918	−0.749590	−	0.661903i	$-0.769749\pi$
$499$	−33.4891	−1.49918	−0.749590	+	0.661903i	$0.769749\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−2.74456	−0.122496
$503$	34.9783	1.55960	0.779802	−	0.626027i	$-0.215320\pi$
$503$	34.9783	1.55960	0.779802	+	0.626027i	$0.215320\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−2.74456	−0.122011
$507$	0	0
$508$	8.00000	0.354943
$509$	−9.76631	−0.432884	−0.216442	−	0.976295i	$-0.569445\pi$
$509$	−9.76631	−0.432884	−0.216442	+	0.976295i	$0.569445\pi$
$510$	0	0
$511$	−52.2337	−2.31068
$512$	1.00000	0.0441942
$513$	0	0
$514$	−18.0000	−0.793946
$515$	9.48913	0.418141
$516$	0	0
$517$	−2.74456	−0.120706
$518$	31.6060	1.38869
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	−12.5109	−0.548111	−0.274056	−	0.961714i	$-0.588365\pi$
$521$	−12.5109	−0.548111	−0.274056	+	0.961714i	$0.588365\pi$
$522$	0	0
$523$	30.9783	1.35458	0.677292	−	0.735714i	$-0.263153\pi$
$523$	30.9783	1.35458	0.677292	+	0.735714i	$0.263153\pi$
$524$	−22.1168	−0.966179
$525$	0	0
$526$	24.8614	1.08401
$527$	−4.62772	−0.201587
$528$	0	0
$529$	−15.4674	−0.672495
$530$	−4.11684	−0.178824
$531$	0	0
$532$	2.11684	0.0917768
$533$	22.9783	0.995299
$534$	0	0
$535$	−12.0000	−0.518805
$536$	8.00000	0.345547
$537$	0	0
$538$	8.74456	0.377005
$539$	4.37228	0.188327
$540$	0	0
$541$	−20.1168	−0.864891	−0.432445	−	0.901660i	$-0.642349\pi$
$541$	−20.1168	−0.864891	−0.432445	+	0.901660i	$0.642349\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	−1.37228	−0.0588361
$545$	15.4891	0.663481
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−8.74456	−0.373549
$549$	0	0
$550$	1.00000	0.0426401
$551$	−0.861407	−0.0366972
$552$	0	0
$553$	−4.23369	−0.180035
$554$	−12.7446	−0.541465
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−4.97825	−0.210935	−0.105468	−	0.994423i	$-0.533634\pi$
$557$	−4.97825	−0.210935	−0.105468	+	0.994423i	$0.533634\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	−3.37228	−0.142505
$561$	0	0
$562$	−18.0000	−0.759284
$563$	8.23369	0.347009	0.173504	−	0.984833i	$-0.444491\pi$
$563$	8.23369	0.347009	0.173504	+	0.984833i	$0.444491\pi$
$564$	0	0
$565$	3.25544	0.136957
$566$	5.25544	0.220903
$567$	0	0
$568$	−10.1168	−0.424493
$569$	15.2554	0.639541	0.319771	−	0.947495i	$-0.396394\pi$
$569$	15.2554	0.639541	0.319771	+	0.947495i	$0.396394\pi$
$570$	0	0
$571$	15.3723	0.643310	0.321655	−	0.946857i	$-0.395761\pi$
$571$	15.3723	0.643310	0.321655	+	0.946857i	$0.395761\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	38.7446	1.61717
$575$	−2.74456	−0.114456
$576$	0	0
$577$	36.9783	1.53942	0.769712	−	0.638391i	$-0.220400\pi$
$577$	36.9783	1.53942	0.769712	+	0.638391i	$0.220400\pi$
$578$	−15.1168	−0.628778
$579$	0	0
$580$	1.37228	0.0569809
$581$	9.25544	0.383980
$582$	0	0
$583$	4.11684	0.170502
$584$	−15.4891	−0.640945
$585$	0	0
$586$	−23.4891	−0.970327
$587$	−24.8614	−1.02614	−0.513070	−	0.858347i	$-0.671492\pi$
$587$	−24.8614	−1.02614	−0.513070	+	0.858347i	$0.671492\pi$
$588$	0	0
$589$	2.11684	0.0872230
$590$	−2.74456	−0.112992
$591$	0	0
$592$	9.37228	0.385198
$593$	12.5109	0.513760	0.256880	−	0.966443i	$-0.417305\pi$
$593$	12.5109	0.513760	0.256880	+	0.966443i	$0.417305\pi$
$594$	0	0
$595$	4.62772	0.189718
$596$	−21.6060	−0.885015
$597$	0	0
$598$	−5.48913	−0.224467
$599$	39.6060	1.61826	0.809128	−	0.587632i	$-0.199940\pi$
$599$	39.6060	1.61826	0.809128	+	0.587632i	$0.199940\pi$
$600$	0	0
$601$	−16.5109	−0.673493	−0.336746	−	0.941595i	$-0.609326\pi$
$601$	−16.5109	−0.673493	−0.336746	+	0.941595i	$0.609326\pi$
$602$	−13.4891	−0.549776
$603$	0	0
$604$	−12.2337	−0.497782
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−5.88316	−0.238790	−0.119395	−	0.992847i	$-0.538095\pi$
$607$	−5.88316	−0.238790	−0.119395	+	0.992847i	$0.538095\pi$
$608$	0.627719	0.0254574
$609$	0	0
$610$	5.37228	0.217517
$611$	−5.48913	−0.222066
$612$	0	0
$613$	20.5109	0.828426	0.414213	−	0.910180i	$-0.364057\pi$
$613$	20.5109	0.828426	0.414213	+	0.910180i	$0.364057\pi$
$614$	5.25544	0.212092
$615$	0	0
$616$	3.37228	0.135873
$617$	2.23369	0.0899249	0.0449624	−	0.998989i	$-0.485683\pi$
$617$	2.23369	0.0899249	0.0449624	+	0.998989i	$0.485683\pi$
$618$	0	0
$619$	−44.4674	−1.78729	−0.893647	−	0.448770i	$-0.851862\pi$
$619$	−44.4674	−1.78729	−0.893647	+	0.448770i	$0.851862\pi$
$620$	−3.37228	−0.135434
$621$	0	0
$622$	−19.3723	−0.776758
$623$	4.62772	0.185406
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	9.37228	0.373995
$629$	−12.8614	−0.512818
$630$	0	0
$631$	42.1168	1.67665	0.838323	−	0.545175i	$-0.183537\pi$
$631$	42.1168	1.67665	0.838323	+	0.545175i	$0.183537\pi$
$632$	−1.25544	−0.0499386
$633$	0	0
$634$	24.3505	0.967083
$635$	−8.00000	−0.317470
$636$	0	0
$637$	8.74456	0.346472
$638$	−1.37228	−0.0543291
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	27.0951	1.07019	0.535096	−	0.844791i	$-0.320275\pi$
$641$	27.0951	1.07019	0.535096	+	0.844791i	$0.320275\pi$
$642$	0	0
$643$	−5.88316	−0.232009	−0.116005	−	0.993249i	$-0.537009\pi$
$643$	−5.88316	−0.232009	−0.116005	+	0.993249i	$0.537009\pi$
$644$	−9.25544	−0.364715
$645$	0	0
$646$	−0.861407	−0.0338916
$647$	37.7228	1.48304	0.741518	−	0.670933i	$-0.234106\pi$
$647$	37.7228	1.48304	0.741518	+	0.670933i	$0.234106\pi$
$648$	0	0
$649$	2.74456	0.107734
$650$	2.00000	0.0784465
$651$	0	0
$652$	−5.88316	−0.230402
$653$	−10.6277	−0.415895	−0.207947	−	0.978140i	$-0.566678\pi$
$653$	−10.6277	−0.415895	−0.207947	+	0.978140i	$0.566678\pi$
$654$	0	0
$655$	22.1168	0.864177
$656$	11.4891	0.448575
$657$	0	0
$658$	−9.25544	−0.360815
$659$	−12.8614	−0.501009	−0.250505	−	0.968115i	$-0.580597\pi$
$659$	−12.8614	−0.501009	−0.250505	+	0.968115i	$0.580597\pi$
$660$	0	0
$661$	8.51087	0.331035	0.165517	−	0.986207i	$-0.447071\pi$
$661$	8.51087	0.331035	0.165517	+	0.986207i	$0.447071\pi$
$662$	30.9783	1.20400
$663$	0	0
$664$	2.74456	0.106510
$665$	−2.11684	−0.0820877
$666$	0	0
$667$	3.76631	0.145832
$668$	4.62772	0.179052
$669$	0	0
$670$	−8.00000	−0.309067
$671$	−5.37228	−0.207395
$672$	0	0
$673$	14.8614	0.572865	0.286433	−	0.958100i	$-0.407531\pi$
$673$	14.8614	0.572865	0.286433	+	0.958100i	$0.407531\pi$
$674$	24.1168	0.928946
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−3.25544	−0.125117	−0.0625583	−	0.998041i	$-0.519926\pi$
$677$	−3.25544	−0.125117	−0.0625583	+	0.998041i	$0.519926\pi$
$678$	0	0
$679$	−42.9783	−1.64935
$680$	1.37228	0.0526246
$681$	0	0
$682$	3.37228	0.129131
$683$	28.6277	1.09541	0.547705	−	0.836672i	$-0.315502\pi$
$683$	28.6277	1.09541	0.547705	+	0.836672i	$0.315502\pi$
$684$	0	0
$685$	8.74456	0.334113
$686$	−8.86141	−0.338330
$687$	0	0
$688$	−4.00000	−0.152499
$689$	8.23369	0.313679
$690$	0	0
$691$	40.2337	1.53056	0.765281	−	0.643697i	$-0.222600\pi$
$691$	40.2337	1.53056	0.765281	+	0.643697i	$0.222600\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	32.2337	1.22357
$695$	4.00000	0.151729
$696$	0	0
$697$	−15.7663	−0.597192
$698$	19.4891	0.737674
$699$	0	0
$700$	3.37228	0.127460
$701$	37.3723	1.41153	0.705766	−	0.708445i	$-0.250603\pi$
$701$	37.3723	1.41153	0.705766	+	0.708445i	$0.250603\pi$
$702$	0	0
$703$	5.88316	0.221887
$704$	1.00000	0.0376889
$705$	0	0
$706$	0.510875	0.0192270
$707$	−20.2337	−0.760966
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	10.1168	0.379678
$711$	0	0
$712$	1.37228	0.0514284
$713$	−9.25544	−0.346619
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	−13.8832	−0.517754	−0.258877	−	0.965910i	$-0.583352\pi$
$719$	−13.8832	−0.517754	−0.258877	+	0.965910i	$0.583352\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	−18.6060	−0.692442
$723$	0	0
$724$	−10.0000	−0.371647
$725$	−1.37228	−0.0509652
$726$	0	0
$727$	−24.2337	−0.898778	−0.449389	−	0.893336i	$-0.648358\pi$
$727$	−24.2337	−0.898778	−0.449389	+	0.893336i	$0.648358\pi$
$728$	6.74456	0.249970
$729$	0	0
$730$	15.4891	0.573278
$731$	5.48913	0.203023
$732$	0	0
$733$	46.2337	1.70768	0.853840	−	0.520535i	$-0.174268\pi$
$733$	46.2337	1.70768	0.853840	+	0.520535i	$0.174268\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	−2.74456	−0.101166
$737$	8.00000	0.294684
$738$	0	0
$739$	−20.4674	−0.752905	−0.376452	−	0.926436i	$-0.622856\pi$
$739$	−20.4674	−0.752905	−0.376452	+	0.926436i	$0.622856\pi$
$740$	−9.37228	−0.344532
$741$	0	0
$742$	13.8832	0.509667
$743$	4.62772	0.169775	0.0848873	−	0.996391i	$-0.472947\pi$
$743$	4.62772	0.169775	0.0848873	+	0.996391i	$0.472947\pi$
$744$	0	0
$745$	21.6060	0.791581
$746$	31.4891	1.15290
$747$	0	0
$748$	−1.37228	−0.0501756
$749$	40.4674	1.47865
$750$	0	0
$751$	8.86141	0.323357	0.161679	−	0.986843i	$-0.448309\pi$
$751$	8.86141	0.323357	0.161679	+	0.986843i	$0.448309\pi$
$752$	−2.74456	−0.100084
$753$	0	0
$754$	−2.74456	−0.0999511
$755$	12.2337	0.445229
$756$	0	0
$757$	−20.9783	−0.762467	−0.381234	−	0.924479i	$-0.624501\pi$
$757$	−20.9783	−0.762467	−0.381234	+	0.924479i	$0.624501\pi$
$758$	−0.233688	−0.00848793
$759$	0	0
$760$	−0.627719	−0.0227697
$761$	−4.97825	−0.180461	−0.0902307	−	0.995921i	$-0.528760\pi$
$761$	−4.97825	−0.180461	−0.0902307	+	0.995921i	$0.528760\pi$
$762$	0	0
$763$	−52.2337	−1.89099
$764$	5.48913	0.198590
$765$	0	0
$766$	−32.2337	−1.16465
$767$	5.48913	0.198201
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	−3.37228	−0.121529
$771$	0	0
$772$	14.8614	0.534874
$773$	33.6060	1.20872	0.604361	−	0.796710i	$-0.293428\pi$
$773$	33.6060	1.20872	0.604361	+	0.796710i	$0.293428\pi$
$774$	0	0
$775$	3.37228	0.121136
$776$	−12.7446	−0.457503
$777$	0	0
$778$	−6.00000	−0.215110
$779$	7.21194	0.258395
$780$	0	0
$781$	−10.1168	−0.362009
$782$	3.76631	0.134683
$783$	0	0
$784$	4.37228	0.156153
$785$	−9.37228	−0.334511
$786$	0	0
$787$	−44.4674	−1.58509	−0.792545	−	0.609813i	$-0.791245\pi$
$787$	−44.4674	−1.58509	−0.792545	+	0.609813i	$0.791245\pi$
$788$	20.7446	0.738994
$789$	0	0
$790$	1.25544	0.0446665
$791$	−10.9783	−0.390342
$792$	0	0
$793$	−10.7446	−0.381551
$794$	24.9783	0.886445
$795$	0	0
$796$	18.1168	0.642135
$797$	−11.4891	−0.406966	−0.203483	−	0.979079i	$-0.565226\pi$
$797$	−11.4891	−0.406966	−0.203483	+	0.979079i	$0.565226\pi$
$798$	0	0
$799$	3.76631	0.133243
$800$	1.00000	0.0353553
$801$	0	0
$802$	−13.3723	−0.472192
$803$	−15.4891	−0.546599
$804$	0	0
$805$	9.25544	0.326211
$806$	6.74456	0.237567
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	44.8614	1.57530	0.787649	−	0.616125i	$-0.211298\pi$
$811$	44.8614	1.57530	0.787649	+	0.616125i	$0.211298\pi$
$812$	−4.62772	−0.162401
$813$	0	0
$814$	9.37228	0.328498
$815$	5.88316	0.206078
$816$	0	0
$817$	−2.51087	−0.0878444
$818$	−1.76631	−0.0617577
$819$	0	0
$820$	−11.4891	−0.401218
$821$	−11.4891	−0.400973	−0.200487	−	0.979696i	$-0.564252\pi$
$821$	−11.4891	−0.400973	−0.200487	+	0.979696i	$0.564252\pi$
$822$	0	0
$823$	−28.0000	−0.976019	−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000	−0.976019	−0.488009	+	0.872838i	$0.662277\pi$
$824$	−9.48913	−0.330569
$825$	0	0
$826$	9.25544	0.322038
$827$	−46.9783	−1.63359	−0.816797	−	0.576925i	$-0.804252\pi$
$827$	−46.9783	−1.63359	−0.816797	+	0.576925i	$0.804252\pi$
$828$	0	0
$829$	−24.7446	−0.859414	−0.429707	−	0.902968i	$-0.641383\pi$
$829$	−24.7446	−0.859414	−0.429707	+	0.902968i	$0.641383\pi$
$830$	−2.74456	−0.0952652
$831$	0	0
$832$	2.00000	0.0693375
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−4.62772	−0.160149
$836$	0.627719	0.0217101
$837$	0	0
$838$	12.0000	0.414533
$839$	10.9783	0.379011	0.189506	−	0.981880i	$-0.439311\pi$
$839$	10.9783	0.379011	0.189506	+	0.981880i	$0.439311\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	10.2337	0.352676
$843$	0	0
$844$	6.11684	0.210550
$845$	9.00000	0.309609
$846$	0	0
$847$	3.37228	0.115873
$848$	4.11684	0.141373
$849$	0	0
$850$	−1.37228	−0.0470689
$851$	−25.7228	−0.881767
$852$	0	0
$853$	−38.4674	−1.31710	−0.658549	−	0.752538i	$-0.728829\pi$
$853$	−38.4674	−1.31710	−0.658549	+	0.752538i	$0.728829\pi$
$854$	−18.1168	−0.619946
$855$	0	0
$856$	12.0000	0.410152
$857$	−36.3505	−1.24171	−0.620855	−	0.783925i	$-0.713215\pi$
$857$	−36.3505	−1.24171	−0.620855	+	0.783925i	$0.713215\pi$
$858$	0	0
$859$	−42.7446	−1.45843	−0.729213	−	0.684287i	$-0.760114\pi$
$859$	−42.7446	−1.45843	−0.729213	+	0.684287i	$0.760114\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	34.9783	1.19136
$863$	−21.2554	−0.723544	−0.361772	−	0.932267i	$-0.617828\pi$
$863$	−21.2554	−0.723544	−0.361772	+	0.932267i	$0.617828\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	27.7228	0.942060
$867$	0	0
$868$	11.3723	0.386000
$869$	−1.25544	−0.0425878
$870$	0	0
$871$	16.0000	0.542139
$872$	−15.4891	−0.524528
$873$	0	0
$874$	−1.72281	−0.0582750
$875$	−3.37228	−0.114004
$876$	0	0
$877$	36.9783	1.24867	0.624333	−	0.781158i	$-0.285371\pi$
$877$	36.9783	1.24867	0.624333	+	0.781158i	$0.285371\pi$
$878$	18.9783	0.640485
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	3.37228	0.113486	0.0567432	−	0.998389i	$-0.481928\pi$
$883$	3.37228	0.113486	0.0567432	+	0.998389i	$0.481928\pi$
$884$	−2.74456	−0.0923096
$885$	0	0
$886$	−29.4891	−0.990707
$887$	10.9783	0.368614	0.184307	−	0.982869i	$-0.440996\pi$
$887$	10.9783	0.368614	0.184307	+	0.982869i	$0.440996\pi$
$888$	0	0
$889$	26.9783	0.904821
$890$	−1.37228	−0.0459990
$891$	0	0
$892$	−18.7446	−0.627614
$893$	−1.72281	−0.0576517
$894$	0	0
$895$	−12.0000	−0.401116
$896$	3.37228	0.112660
$897$	0	0
$898$	−28.9783	−0.967017
$899$	−4.62772	−0.154343
$900$	0	0
$901$	−5.64947	−0.188211
$902$	11.4891	0.382546
$903$	0	0
$904$	−3.25544	−0.108274
$905$	10.0000	0.332411
$906$	0	0
$907$	−0.394031	−0.0130836	−0.00654179	−	0.999979i	$-0.502082\pi$
$907$	−0.394031	−0.0130836	−0.00654179	+	0.999979i	$0.502082\pi$
$908$	2.74456	0.0910815
$909$	0	0
$910$	−6.74456	−0.223580
$911$	8.39403	0.278107	0.139053	−	0.990285i	$-0.455594\pi$
$911$	8.39403	0.278107	0.139053	+	0.990285i	$0.455594\pi$
$912$	0	0
$913$	2.74456	0.0908318
$914$	−16.3505	−0.540828
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−74.5842	−2.46299
$918$	0	0
$919$	18.9783	0.626035	0.313017	−	0.949747i	$-0.398660\pi$
$919$	18.9783	0.626035	0.313017	+	0.949747i	$0.398660\pi$
$920$	2.74456	0.0904856
$921$	0	0
$922$	−16.1168	−0.530780
$923$	−20.2337	−0.666000
$924$	0	0
$925$	9.37228	0.308159
$926$	−0.233688	−0.00767946
$927$	0	0
$928$	−1.37228	−0.0450473
$929$	−24.3505	−0.798915	−0.399458	−	0.916752i	$-0.630801\pi$
$929$	−24.3505	−0.798915	−0.399458	+	0.916752i	$0.630801\pi$
$930$	0	0
$931$	2.74456	0.0899494
$932$	−1.37228	−0.0449506
$933$	0	0
$934$	19.3723	0.633880
$935$	1.37228	0.0448784
$936$	0	0
$937$	−28.5109	−0.931410	−0.465705	−	0.884940i	$-0.654199\pi$
$937$	−28.5109	−0.931410	−0.465705	+	0.884940i	$0.654199\pi$
$938$	26.9783	0.880871
$939$	0	0
$940$	2.74456	0.0895178
$941$	15.0951	0.492086	0.246043	−	0.969259i	$-0.420870\pi$
$941$	15.0951	0.492086	0.246043	+	0.969259i	$0.420870\pi$
$942$	0	0
$943$	−31.5326	−1.02684
$944$	2.74456	0.0893279
$945$	0	0
$946$	−4.00000	−0.130051
$947$	−48.8614	−1.58778	−0.793891	−	0.608060i	$-0.791948\pi$
$947$	−48.8614	−1.58778	−0.793891	+	0.608060i	$0.791948\pi$
$948$	0	0
$949$	−30.9783	−1.00560
$950$	0.627719	0.0203659
$951$	0	0
$952$	−4.62772	−0.149985
$953$	−40.1168	−1.29951	−0.649756	−	0.760143i	$-0.725129\pi$
$953$	−40.1168	−1.29951	−0.649756	+	0.760143i	$0.725129\pi$
$954$	0	0
$955$	−5.48913	−0.177624
$956$	14.7446	0.476873
$957$	0	0
$958$	−5.48913	−0.177346
$959$	−29.4891	−0.952254
$960$	0	0
$961$	−19.6277	−0.633152
$962$	18.7446	0.604349
$963$	0	0
$964$	−22.0000	−0.708572
$965$	−14.8614	−0.478406
$966$	0	0
$967$	47.6060	1.53090	0.765452	−	0.643493i	$-0.222515\pi$
$967$	47.6060	1.53090	0.765452	+	0.643493i	$0.222515\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	12.7446	0.409203
$971$	1.02175	0.0327895	0.0163947	−	0.999866i	$-0.494781\pi$
$971$	1.02175	0.0327895	0.0163947	+	0.999866i	$0.494781\pi$
$972$	0	0
$973$	−13.4891	−0.432442
$974$	20.0000	0.640841
$975$	0	0
$976$	−5.37228	−0.171963
$977$	−14.2337	−0.455376	−0.227688	−	0.973734i	$-0.573117\pi$
$977$	−14.2337	−0.455376	−0.227688	+	0.973734i	$0.573117\pi$
$978$	0	0
$979$	1.37228	0.0438583
$980$	−4.37228	−0.139667
$981$	0	0
$982$	7.37228	0.235259
$983$	13.7228	0.437690	0.218845	−	0.975760i	$-0.429771\pi$
$983$	13.7228	0.437690	0.218845	+	0.975760i	$0.429771\pi$
$984$	0	0
$985$	−20.7446	−0.660977
$986$	1.88316	0.0599719
$987$	0	0
$988$	1.25544	0.0399408
$989$	10.9783	0.349088
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	3.37228	0.107070
$993$	0	0
$994$	−34.1168	−1.08212
$995$	−18.1168	−0.574343
$996$	0	0
$997$	22.2337	0.704148	0.352074	−	0.935972i	$-0.385477\pi$
$997$	22.2337	0.704148	0.352074	+	0.935972i	$0.385477\pi$
$998$	−33.4891	−1.06008
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	990.2.a.m.1.2		2
3.2	odd	2		110.2.a.d.1.1	✓	2
4.3	odd	2		7920.2.a.bq.1.1		2
5.2	odd	4		4950.2.c.bc.199.4		4
5.3	odd	4		4950.2.c.bc.199.1		4
5.4	even	2		4950.2.a.bw.1.1		2
12.11	even	2		880.2.a.n.1.2		2
15.2	even	4		550.2.b.f.199.2		4
15.8	even	4		550.2.b.f.199.3		4
15.14	odd	2		550.2.a.n.1.2		2
21.20	even	2		5390.2.a.bp.1.2		2
24.5	odd	2		3520.2.a.bq.1.2		2
24.11	even	2		3520.2.a.bj.1.1		2
33.32	even	2		1210.2.a.r.1.1		2
60.23	odd	4		4400.2.b.p.4049.4		4
60.47	odd	4		4400.2.b.p.4049.1		4
60.59	even	2		4400.2.a.bl.1.1		2
132.131	odd	2		9680.2.a.bt.1.2		2
165.164	even	2		6050.2.a.cb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.d.1.1	✓	2	3.2	odd	2
550.2.a.n.1.2		2	15.14	odd	2
550.2.b.f.199.2		4	15.2	even	4
550.2.b.f.199.3		4	15.8	even	4
880.2.a.n.1.2		2	12.11	even	2
990.2.a.m.1.2		2	1.1	even	1	trivial
1210.2.a.r.1.1		2	33.32	even	2
3520.2.a.bj.1.1		2	24.11	even	2
3520.2.a.bq.1.2		2	24.5	odd	2
4400.2.a.bl.1.1		2	60.59	even	2
4400.2.b.p.4049.1		4	60.47	odd	4
4400.2.b.p.4049.4		4	60.23	odd	4
4950.2.a.bw.1.1		2	5.4	even	2
4950.2.c.bc.199.1		4	5.3	odd	4
4950.2.c.bc.199.4		4	5.2	odd	4
5390.2.a.bp.1.2		2	21.20	even	2
6050.2.a.cb.1.2		2	165.164	even	2
7920.2.a.bq.1.1		2	4.3	odd	2
9680.2.a.bt.1.2		2	132.131	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	990.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.37228 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +3.37228 q^{14} +1.00000 q^{16} -1.37228 q^{17} +0.627719 q^{19} -1.00000 q^{20} +1.00000 q^{22} -2.74456 q^{23} +1.00000 q^{25} +2.00000 q^{26} +3.37228 q^{28} -1.37228 q^{29} +3.37228 q^{31} +1.00000 q^{32} -1.37228 q^{34} -3.37228 q^{35} +9.37228 q^{37} +0.627719 q^{38} -1.00000 q^{40} +11.4891 q^{41} -4.00000 q^{43} +1.00000 q^{44} -2.74456 q^{46} -2.74456 q^{47} +4.37228 q^{49} +1.00000 q^{50} +2.00000 q^{52} +4.11684 q^{53} -1.00000 q^{55} +3.37228 q^{56} -1.37228 q^{58} +2.74456 q^{59} -5.37228 q^{61} +3.37228 q^{62} +1.00000 q^{64} -2.00000 q^{65} +8.00000 q^{67} -1.37228 q^{68} -3.37228 q^{70} -10.1168 q^{71} -15.4891 q^{73} +9.37228 q^{74} +0.627719 q^{76} +3.37228 q^{77} -1.25544 q^{79} -1.00000 q^{80} +11.4891 q^{82} +2.74456 q^{83} +1.37228 q^{85} -4.00000 q^{86} +1.00000 q^{88} +1.37228 q^{89} +6.74456 q^{91} -2.74456 q^{92} -2.74456 q^{94} -0.627719 q^{95} -12.7446 q^{97} +4.37228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{11} + 4 q^{13} + q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} - 2 q^{20} + 2 q^{22} + 6 q^{23} + 2 q^{25} + 4 q^{26} + q^{28} + 3 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 + 2 * q^8 - 2 * q^10 + 2 * q^11 + 4 * q^13 + q^14 + 2 * q^16 + 3 * q^17 + 7 * q^19 - 2 * q^20 + 2 * q^22 + 6 * q^23 + 2 * q^25 + 4 * q^26 + q^28 + 3 * q^29 + q^31 + 2 * q^32 + 3 * q^34 - q^35 + 13 * q^37 + 7 * q^38 - 2 * q^40 - 8 * q^43 + 2 * q^44 + 6 * q^46 + 6 * q^47 + 3 * q^49 + 2 * q^50 + 4 * q^52 - 9 * q^53 - 2 * q^55 + q^56 + 3 * q^58 - 6 * q^59 - 5 * q^61 + q^62 + 2 * q^64 - 4 * q^65 + 16 * q^67 + 3 * q^68 - q^70 - 3 * q^71 - 8 * q^73 + 13 * q^74 + 7 * q^76 + q^77 - 14 * q^79 - 2 * q^80 - 6 * q^83 - 3 * q^85 - 8 * q^86 + 2 * q^88 - 3 * q^89 + 2 * q^91 + 6 * q^92 + 6 * q^94 - 7 * q^95 - 14 * q^97 + 3 * q^98

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