LMFDB - Embedded newform 8044.2.a.a.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8044,2,Mod(1,8044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8044 = 2^{2} \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8044.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:	$64.2316633859$
Analytic rank:	$1$
Dimension:	$80$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78192 q^{3} +1.93177 q^{5} -4.05350 q^{7} +4.73908 q^{9} +O(q^{10})$	$q-2.78192 q^{3} +1.93177 q^{5} -4.05350 q^{7} +4.73908 q^{9} -3.78673 q^{11} +5.78719 q^{13} -5.37402 q^{15} -0.804834 q^{17} +4.51032 q^{19} +11.2765 q^{21} +1.66156 q^{23} -1.26828 q^{25} -4.83798 q^{27} -4.07357 q^{29} -8.61475 q^{31} +10.5344 q^{33} -7.83041 q^{35} +0.0364641 q^{37} -16.0995 q^{39} +6.36094 q^{41} -3.82755 q^{43} +9.15479 q^{45} -2.64886 q^{47} +9.43088 q^{49} +2.23898 q^{51} -6.91043 q^{53} -7.31508 q^{55} -12.5474 q^{57} +6.28574 q^{59} -5.61794 q^{61} -19.2099 q^{63} +11.1795 q^{65} +1.38984 q^{67} -4.62234 q^{69} +5.32285 q^{71} +16.8804 q^{73} +3.52826 q^{75} +15.3495 q^{77} +14.9221 q^{79} -0.758356 q^{81} +11.4742 q^{83} -1.55475 q^{85} +11.3324 q^{87} -11.8679 q^{89} -23.4584 q^{91} +23.9656 q^{93} +8.71288 q^{95} +13.4866 q^{97} -17.9456 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.78192	−1.60614	−0.803071	−	0.595883i	$-0.796802\pi$
$3$	−2.78192	−1.60614	−0.803071	+	0.595883i	$0.796802\pi$
$4$	0	0
$5$	1.93177	0.863912	0.431956	−	0.901895i	$-0.357824\pi$
$5$	1.93177	0.863912	0.431956	+	0.901895i	$0.357824\pi$
$6$	0	0
$7$	−4.05350	−1.53208	−0.766040	−	0.642793i	$-0.777776\pi$
$7$	−4.05350	−1.53208	−0.766040	+	0.642793i	$0.777776\pi$
$8$	0	0
$9$	4.73908	1.57969
$10$	0	0
$11$	−3.78673	−1.14174	−0.570872	−	0.821039i	$-0.693395\pi$
$11$	−3.78673	−1.14174	−0.570872	+	0.821039i	$0.693395\pi$
$12$	0	0
$13$	5.78719	1.60508	0.802539	−	0.596600i	$-0.203482\pi$
$13$	5.78719	1.60508	0.802539	+	0.596600i	$0.203482\pi$
$14$	0	0
$15$	−5.37402	−1.38757
$16$	0	0
$17$	−0.804834	−0.195201	−0.0976005	−	0.995226i	$-0.531117\pi$
$17$	−0.804834	−0.195201	−0.0976005	+	0.995226i	$0.531117\pi$
$18$	0	0
$19$	4.51032	1.03474	0.517369	−	0.855762i	$-0.326911\pi$
$19$	4.51032	1.03474	0.517369	+	0.855762i	$0.326911\pi$
$20$	0	0
$21$	11.2765	2.46074
$22$	0	0
$23$	1.66156	0.346460	0.173230	−	0.984881i	$-0.444580\pi$
$23$	1.66156	0.346460	0.173230	+	0.984881i	$0.444580\pi$
$24$	0	0
$25$	−1.26828	−0.253657
$26$	0	0
$27$	−4.83798	−0.931070
$28$	0	0
$29$	−4.07357	−0.756444	−0.378222	−	0.925715i	$-0.623464\pi$
$29$	−4.07357	−0.756444	−0.378222	+	0.925715i	$0.623464\pi$
$30$	0	0
$31$	−8.61475	−1.54726	−0.773628	−	0.633640i	$-0.781560\pi$
$31$	−8.61475	−1.54726	−0.773628	+	0.633640i	$0.781560\pi$
$32$	0	0
$33$	10.5344	1.83380
$34$	0	0
$35$	−7.83041	−1.32358
$36$	0	0
$37$	0.0364641	0.00599467	0.00299733	−	0.999996i	$-0.499046\pi$
$37$	0.0364641	0.00599467	0.00299733	+	0.999996i	$0.499046\pi$
$38$	0	0
$39$	−16.0995	−2.57798
$40$	0	0
$41$	6.36094	0.993411	0.496706	−	0.867919i	$-0.334543\pi$
$41$	6.36094	0.993411	0.496706	+	0.867919i	$0.334543\pi$
$42$	0	0
$43$	−3.82755	−0.583696	−0.291848	−	0.956465i	$-0.594270\pi$
$43$	−3.82755	−0.583696	−0.291848	+	0.956465i	$0.594270\pi$
$44$	0	0
$45$	9.15479	1.36472
$46$	0	0
$47$	−2.64886	−0.386376	−0.193188	−	0.981162i	$-0.561883\pi$
$47$	−2.64886	−0.386376	−0.193188	+	0.981162i	$0.561883\pi$
$48$	0	0
$49$	9.43088	1.34727
$50$	0	0
$51$	2.23898	0.313521
$52$	0	0
$53$	−6.91043	−0.949220	−0.474610	−	0.880196i	$-0.657411\pi$
$53$	−6.91043	−0.949220	−0.474610	+	0.880196i	$0.657411\pi$
$54$	0	0
$55$	−7.31508	−0.986365
$56$	0	0
$57$	−12.5474	−1.66194
$58$	0	0
$59$	6.28574	0.818334	0.409167	−	0.912460i	$-0.365819\pi$
$59$	6.28574	0.818334	0.409167	+	0.912460i	$0.365819\pi$
$60$	0	0
$61$	−5.61794	−0.719304	−0.359652	−	0.933087i	$-0.617105\pi$
$61$	−5.61794	−0.719304	−0.359652	+	0.933087i	$0.617105\pi$
$62$	0	0
$63$	−19.2099	−2.42022
$64$	0	0
$65$	11.1795	1.38664
$66$	0	0
$67$	1.38984	0.169795	0.0848977	−	0.996390i	$-0.472944\pi$
$67$	1.38984	0.169795	0.0848977	+	0.996390i	$0.472944\pi$
$68$	0	0
$69$	−4.62234	−0.556464
$70$	0	0
$71$	5.32285	0.631706	0.315853	−	0.948808i	$-0.397709\pi$
$71$	5.32285	0.631706	0.315853	+	0.948808i	$0.397709\pi$
$72$	0	0
$73$	16.8804	1.97570	0.987849	−	0.155420i	$-0.0496730\pi$
$73$	16.8804	1.97570	0.987849	+	0.155420i	$0.0496730\pi$
$74$	0	0
$75$	3.52826	0.407409
$76$	0	0
$77$	15.3495	1.74924
$78$	0	0
$79$	14.9221	1.67887	0.839435	−	0.543459i	$-0.182886\pi$
$79$	14.9221	1.67887	0.839435	+	0.543459i	$0.182886\pi$
$80$	0	0
$81$	−0.758356	−0.0842618
$82$	0	0
$83$	11.4742	1.25946	0.629731	−	0.776813i	$-0.283165\pi$
$83$	11.4742	1.25946	0.629731	+	0.776813i	$0.283165\pi$
$84$	0	0
$85$	−1.55475	−0.168636
$86$	0	0
$87$	11.3324	1.21496
$88$	0	0
$89$	−11.8679	−1.25800	−0.628999	−	0.777406i	$-0.716535\pi$
$89$	−11.8679	−1.25800	−0.628999	+	0.777406i	$0.716535\pi$
$90$	0	0
$91$	−23.4584	−2.45911
$92$	0	0
$93$	23.9656	2.48511
$94$	0	0
$95$	8.71288	0.893923
$96$	0	0
$97$	13.4866	1.36936	0.684678	−	0.728845i	$-0.259943\pi$
$97$	13.4866	1.36936	0.684678	+	0.728845i	$0.259943\pi$
$98$	0	0
$99$	−17.9456	−1.80360
$100$	0	0
$101$	17.0512	1.69666	0.848329	−	0.529470i	$-0.177609\pi$
$101$	17.0512	1.69666	0.848329	+	0.529470i	$0.177609\pi$
$102$	0	0
$103$	−0.616459	−0.0607415	−0.0303708	−	0.999539i	$-0.509669\pi$
$103$	−0.616459	−0.0607415	−0.0303708	+	0.999539i	$0.509669\pi$
$104$	0	0
$105$	21.7836	2.12586
$106$	0	0
$107$	−7.55056	−0.729940	−0.364970	−	0.931019i	$-0.618921\pi$
$107$	−7.55056	−0.729940	−0.364970	+	0.931019i	$0.618921\pi$
$108$	0	0
$109$	−18.3153	−1.75428	−0.877142	−	0.480232i	$-0.840553\pi$
$109$	−18.3153	−1.75428	−0.877142	+	0.480232i	$0.840553\pi$
$110$	0	0
$111$	−0.101440	−0.00962829
$112$	0	0
$113$	4.91817	0.462662	0.231331	−	0.972875i	$-0.425692\pi$
$113$	4.91817	0.462662	0.231331	+	0.972875i	$0.425692\pi$
$114$	0	0
$115$	3.20975	0.299311
$116$	0	0
$117$	27.4259	2.53553
$118$	0	0
$119$	3.26240	0.299063
$120$	0	0
$121$	3.33935	0.303577
$122$	0	0
$123$	−17.6956	−1.59556
$124$	0	0
$125$	−12.1089	−1.08305
$126$	0	0
$127$	4.85059	0.430420	0.215210	−	0.976568i	$-0.430956\pi$
$127$	4.85059	0.430420	0.215210	+	0.976568i	$0.430956\pi$
$128$	0	0
$129$	10.6479	0.937499
$130$	0	0
$131$	−12.2509	−1.07037	−0.535185	−	0.844735i	$-0.679758\pi$
$131$	−12.2509	−1.07037	−0.535185	+	0.844735i	$0.679758\pi$
$132$	0	0
$133$	−18.2826	−1.58530
$134$	0	0
$135$	−9.34585	−0.804363
$136$	0	0
$137$	−22.4004	−1.91379	−0.956896	−	0.290430i	$-0.906202\pi$
$137$	−22.4004	−1.91379	−0.956896	+	0.290430i	$0.906202\pi$
$138$	0	0
$139$	6.94226	0.588835	0.294417	−	0.955677i	$-0.404874\pi$
$139$	6.94226	0.588835	0.294417	+	0.955677i	$0.404874\pi$
$140$	0	0
$141$	7.36891	0.620574
$142$	0	0
$143$	−21.9145	−1.83259
$144$	0	0
$145$	−7.86919	−0.653500
$146$	0	0
$147$	−26.2360	−2.16391
$148$	0	0
$149$	12.2483	1.00342	0.501712	−	0.865035i	$-0.332704\pi$
$149$	12.2483	1.00342	0.501712	+	0.865035i	$0.332704\pi$
$150$	0	0
$151$	6.21704	0.505936	0.252968	−	0.967475i	$-0.418593\pi$
$151$	6.21704	0.505936	0.252968	+	0.967475i	$0.418593\pi$
$152$	0	0
$153$	−3.81417	−0.308358
$154$	0	0
$155$	−16.6417	−1.33669
$156$	0	0
$157$	1.20485	0.0961571	0.0480786	−	0.998844i	$-0.484690\pi$
$157$	1.20485	0.0961571	0.0480786	+	0.998844i	$0.484690\pi$
$158$	0	0
$159$	19.2243	1.52458
$160$	0	0
$161$	−6.73516	−0.530805
$162$	0	0
$163$	−8.27967	−0.648514	−0.324257	−	0.945969i	$-0.605114\pi$
$163$	−8.27967	−0.648514	−0.324257	+	0.945969i	$0.605114\pi$
$164$	0	0
$165$	20.3500	1.58424
$166$	0	0
$167$	−1.28353	−0.0993225	−0.0496612	−	0.998766i	$-0.515814\pi$
$167$	−1.28353	−0.0993225	−0.0496612	+	0.998766i	$0.515814\pi$
$168$	0	0
$169$	20.4915	1.57627
$170$	0	0
$171$	21.3748	1.63457
$172$	0	0
$173$	20.2033	1.53603	0.768016	−	0.640431i	$-0.221244\pi$
$173$	20.2033	1.53603	0.768016	+	0.640431i	$0.221244\pi$
$174$	0	0
$175$	5.14099	0.388622
$176$	0	0
$177$	−17.4864	−1.31436
$178$	0	0
$179$	−25.0780	−1.87442	−0.937211	−	0.348764i	$-0.886601\pi$
$179$	−25.0780	−1.87442	−0.937211	+	0.348764i	$0.886601\pi$
$180$	0	0
$181$	20.1245	1.49585	0.747923	−	0.663786i	$-0.231051\pi$
$181$	20.1245	1.49585	0.747923	+	0.663786i	$0.231051\pi$
$182$	0	0
$183$	15.6287	1.15530
$184$	0	0
$185$	0.0704401	0.00517886
$186$	0	0
$187$	3.04769	0.222869
$188$	0	0
$189$	19.6108	1.42647
$190$	0	0
$191$	5.34685	0.386884	0.193442	−	0.981112i	$-0.438035\pi$
$191$	5.34685	0.386884	0.193442	+	0.981112i	$0.438035\pi$
$192$	0	0
$193$	−23.4311	−1.68661	−0.843305	−	0.537435i	$-0.819393\pi$
$193$	−23.4311	−1.68661	−0.843305	+	0.537435i	$0.819393\pi$
$194$	0	0
$195$	−31.1004	−2.22715
$196$	0	0
$197$	−25.3515	−1.80622	−0.903110	−	0.429410i	$-0.858722\pi$
$197$	−25.3515	−1.80622	−0.903110	+	0.429410i	$0.858722\pi$
$198$	0	0
$199$	−0.670466	−0.0475280	−0.0237640	−	0.999718i	$-0.507565\pi$
$199$	−0.670466	−0.0475280	−0.0237640	+	0.999718i	$0.507565\pi$
$200$	0	0
$201$	−3.86641	−0.272716
$202$	0	0
$203$	16.5122	1.15893
$204$	0	0
$205$	12.2878	0.858220
$206$	0	0
$207$	7.87429	0.547301
$208$	0	0
$209$	−17.0794	−1.18141
$210$	0	0
$211$	19.0120	1.30884	0.654422	−	0.756130i	$-0.272912\pi$
$211$	19.0120	1.30884	0.654422	+	0.756130i	$0.272912\pi$
$212$	0	0
$213$	−14.8078	−1.01461
$214$	0	0
$215$	−7.39393	−0.504262
$216$	0	0
$217$	34.9199	2.37052
$218$	0	0
$219$	−46.9598	−3.17325
$220$	0	0
$221$	−4.65773	−0.313313
$222$	0	0
$223$	11.8597	0.794181	0.397091	−	0.917779i	$-0.370020\pi$
$223$	11.8597	0.794181	0.397091	+	0.917779i	$0.370020\pi$
$224$	0	0
$225$	−6.01050	−0.400700
$226$	0	0
$227$	−6.60187	−0.438182	−0.219091	−	0.975704i	$-0.570309\pi$
$227$	−6.60187	−0.438182	−0.219091	+	0.975704i	$0.570309\pi$
$228$	0	0
$229$	−18.4263	−1.21764	−0.608821	−	0.793308i	$-0.708357\pi$
$229$	−18.4263	−1.21764	−0.608821	+	0.793308i	$0.708357\pi$
$230$	0	0
$231$	−42.7012	−2.80953
$232$	0	0
$233$	−3.01747	−0.197681	−0.0988404	−	0.995103i	$-0.531513\pi$
$233$	−3.01747	−0.197681	−0.0988404	+	0.995103i	$0.531513\pi$
$234$	0	0
$235$	−5.11697	−0.333794
$236$	0	0
$237$	−41.5122	−2.69651
$238$	0	0
$239$	−3.48359	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$239$	−3.48359	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$240$	0	0
$241$	−8.59134	−0.553417	−0.276709	−	0.960954i	$-0.589244\pi$
$241$	−8.59134	−0.553417	−0.276709	+	0.960954i	$0.589244\pi$
$242$	0	0
$243$	16.6236	1.06641
$244$	0	0
$245$	18.2182	1.16392
$246$	0	0
$247$	26.1021	1.66083
$248$	0	0
$249$	−31.9204	−2.02287
$250$	0	0
$251$	−1.95933	−0.123672	−0.0618359	−	0.998086i	$-0.519696\pi$
$251$	−1.95933	−0.123672	−0.0618359	+	0.998086i	$0.519696\pi$
$252$	0	0
$253$	−6.29190	−0.395568
$254$	0	0
$255$	4.32519	0.270854
$256$	0	0
$257$	5.22668	0.326031	0.163016	−	0.986623i	$-0.447878\pi$
$257$	5.22668	0.326031	0.163016	+	0.986623i	$0.447878\pi$
$258$	0	0
$259$	−0.147807	−0.00918431
$260$	0	0
$261$	−19.3050	−1.19495
$262$	0	0
$263$	5.21163	0.321363	0.160681	−	0.987006i	$-0.448631\pi$
$263$	5.21163	0.321363	0.160681	+	0.987006i	$0.448631\pi$
$264$	0	0
$265$	−13.3493	−0.820042
$266$	0	0
$267$	33.0156	2.02052
$268$	0	0
$269$	9.88894	0.602939	0.301470	−	0.953476i	$-0.402523\pi$
$269$	9.88894	0.602939	0.301470	+	0.953476i	$0.402523\pi$
$270$	0	0
$271$	22.0519	1.33956	0.669778	−	0.742561i	$-0.266389\pi$
$271$	22.0519	1.33956	0.669778	+	0.742561i	$0.266389\pi$
$272$	0	0
$273$	65.2593	3.94968
$274$	0	0
$275$	4.80265	0.289611
$276$	0	0
$277$	0.227899	0.0136931	0.00684657	−	0.999977i	$-0.497821\pi$
$277$	0.227899	0.0136931	0.00684657	+	0.999977i	$0.497821\pi$
$278$	0	0
$279$	−40.8260	−2.44419
$280$	0	0
$281$	25.7179	1.53420	0.767100	−	0.641528i	$-0.221699\pi$
$281$	25.7179	1.53420	0.767100	+	0.641528i	$0.221699\pi$
$282$	0	0
$283$	−3.21218	−0.190944	−0.0954722	−	0.995432i	$-0.530436\pi$
$283$	−3.21218	−0.190944	−0.0954722	+	0.995432i	$0.530436\pi$
$284$	0	0
$285$	−24.2385	−1.43577
$286$	0	0
$287$	−25.7841	−1.52199
$288$	0	0
$289$	−16.3522	−0.961897
$290$	0	0
$291$	−37.5186	−2.19938
$292$	0	0
$293$	4.55689	0.266217	0.133108	−	0.991102i	$-0.457504\pi$
$293$	4.55689	0.266217	0.133108	+	0.991102i	$0.457504\pi$
$294$	0	0
$295$	12.1426	0.706968
$296$	0	0
$297$	18.3202	1.06304
$298$	0	0
$299$	9.61579	0.556095
$300$	0	0
$301$	15.5150	0.894269
$302$	0	0
$303$	−47.4351	−2.72507
$304$	0	0
$305$	−10.8525	−0.621415
$306$	0	0
$307$	−8.76531	−0.500263	−0.250131	−	0.968212i	$-0.580474\pi$
$307$	−8.76531	−0.500263	−0.250131	+	0.968212i	$0.580474\pi$
$308$	0	0
$309$	1.71494	0.0975595
$310$	0	0
$311$	−28.5206	−1.61726	−0.808628	−	0.588320i	$-0.799790\pi$
$311$	−28.5206	−1.61726	−0.808628	+	0.588320i	$0.799790\pi$
$312$	0	0
$313$	−0.786313	−0.0444450	−0.0222225	−	0.999753i	$-0.507074\pi$
$313$	−0.786313	−0.0444450	−0.0222225	+	0.999753i	$0.507074\pi$
$314$	0	0
$315$	−37.1090	−2.09085
$316$	0	0
$317$	2.59919	0.145985	0.0729924	−	0.997332i	$-0.476745\pi$
$317$	2.59919	0.145985	0.0729924	+	0.997332i	$0.476745\pi$
$318$	0	0
$319$	15.4255	0.863664
$320$	0	0
$321$	21.0050	1.17239
$322$	0	0
$323$	−3.63006	−0.201982
$324$	0	0
$325$	−7.33980	−0.407139
$326$	0	0
$327$	50.9516	2.81763
$328$	0	0
$329$	10.7371	0.591958
$330$	0	0
$331$	−12.9974	−0.714401	−0.357200	−	0.934028i	$-0.616269\pi$
$331$	−12.9974	−0.714401	−0.357200	+	0.934028i	$0.616269\pi$
$332$	0	0
$333$	0.172806	0.00946974
$334$	0	0
$335$	2.68484	0.146688
$336$	0	0
$337$	−33.1967	−1.80834	−0.904169	−	0.427176i	$-0.859509\pi$
$337$	−33.1967	−1.80834	−0.904169	+	0.427176i	$0.859509\pi$
$338$	0	0
$339$	−13.6819	−0.743102
$340$	0	0
$341$	32.6218	1.76657
$342$	0	0
$343$	−9.85359	−0.532044
$344$	0	0
$345$	−8.92928	−0.480736
$346$	0	0
$347$	−22.2630	−1.19514	−0.597571	−	0.801816i	$-0.703867\pi$
$347$	−22.2630	−1.19514	−0.597571	+	0.801816i	$0.703867\pi$
$348$	0	0
$349$	−24.5824	−1.31587	−0.657934	−	0.753076i	$-0.728569\pi$
$349$	−24.5824	−1.31587	−0.657934	+	0.753076i	$0.728569\pi$
$350$	0	0
$351$	−27.9983	−1.49444
$352$	0	0
$353$	33.7849	1.79819	0.899093	−	0.437757i	$-0.144227\pi$
$353$	33.7849	1.79819	0.899093	+	0.437757i	$0.144227\pi$
$354$	0	0
$355$	10.2825	0.545738
$356$	0	0
$357$	−9.07573	−0.480339
$358$	0	0
$359$	−16.1364	−0.851646	−0.425823	−	0.904806i	$-0.640015\pi$
$359$	−16.1364	−0.851646	−0.425823	+	0.904806i	$0.640015\pi$
$360$	0	0
$361$	1.34299	0.0706837
$362$	0	0
$363$	−9.28980	−0.487588
$364$	0	0
$365$	32.6089	1.70683
$366$	0	0
$367$	−6.12709	−0.319831	−0.159916	−	0.987131i	$-0.551122\pi$
$367$	−6.12709	−0.319831	−0.159916	+	0.987131i	$0.551122\pi$
$368$	0	0
$369$	30.1450	1.56929
$370$	0	0
$371$	28.0114	1.45428
$372$	0	0
$373$	31.6479	1.63867	0.819334	−	0.573317i	$-0.194344\pi$
$373$	31.6479	1.63867	0.819334	+	0.573317i	$0.194344\pi$
$374$	0	0
$375$	33.6859	1.73953
$376$	0	0
$377$	−23.5745	−1.21415
$378$	0	0
$379$	−19.6259	−1.00812	−0.504058	−	0.863670i	$-0.668160\pi$
$379$	−19.6259	−1.00812	−0.504058	+	0.863670i	$0.668160\pi$
$380$	0	0
$381$	−13.4939	−0.691316
$382$	0	0
$383$	3.84129	0.196281	0.0981403	−	0.995173i	$-0.468711\pi$
$383$	3.84129	0.196281	0.0981403	+	0.995173i	$0.468711\pi$
$384$	0	0
$385$	29.6517	1.51119
$386$	0	0
$387$	−18.1391	−0.922060
$388$	0	0
$389$	−37.9839	−1.92586	−0.962931	−	0.269750i	$-0.913059\pi$
$389$	−37.9839	−1.92586	−0.962931	+	0.269750i	$0.913059\pi$
$390$	0	0
$391$	−1.33728	−0.0676294
$392$	0	0
$393$	34.0811	1.71917
$394$	0	0
$395$	28.8261	1.45040
$396$	0	0
$397$	−4.43936	−0.222805	−0.111402	−	0.993775i	$-0.535534\pi$
$397$	−4.43936	−0.222805	−0.111402	+	0.993775i	$0.535534\pi$
$398$	0	0
$399$	50.8607	2.54622
$400$	0	0
$401$	2.45437	0.122566	0.0612828	−	0.998120i	$-0.480481\pi$
$401$	2.45437	0.122566	0.0612828	+	0.998120i	$0.480481\pi$
$402$	0	0
$403$	−49.8552	−2.48346
$404$	0	0
$405$	−1.46497	−0.0727947
$406$	0	0
$407$	−0.138080	−0.00684437
$408$	0	0
$409$	0.939593	0.0464599	0.0232300	−	0.999730i	$-0.492605\pi$
$409$	0.939593	0.0464599	0.0232300	+	0.999730i	$0.492605\pi$
$410$	0	0
$411$	62.3160	3.07382
$412$	0	0
$413$	−25.4793	−1.25375
$414$	0	0
$415$	22.1655	1.08806
$416$	0	0
$417$	−19.3128	−0.945752
$418$	0	0
$419$	−12.7944	−0.625045	−0.312523	−	0.949910i	$-0.601174\pi$
$419$	−12.7944	−0.625045	−0.312523	+	0.949910i	$0.601174\pi$
$420$	0	0
$421$	−36.1793	−1.76327	−0.881635	−	0.471931i	$-0.843557\pi$
$421$	−36.1793	−1.76327	−0.881635	+	0.471931i	$0.843557\pi$
$422$	0	0
$423$	−12.5531	−0.610355
$424$	0	0
$425$	1.02076	0.0495140
$426$	0	0
$427$	22.7723	1.10203
$428$	0	0
$429$	60.9645	2.94339
$430$	0	0
$431$	25.1726	1.21252	0.606262	−	0.795265i	$-0.292668\pi$
$431$	25.1726	1.21252	0.606262	+	0.795265i	$0.292668\pi$
$432$	0	0
$433$	−9.41591	−0.452500	−0.226250	−	0.974069i	$-0.572647\pi$
$433$	−9.41591	−0.452500	−0.226250	+	0.974069i	$0.572647\pi$
$434$	0	0
$435$	21.8915	1.04961
$436$	0	0
$437$	7.49419	0.358496
$438$	0	0
$439$	−27.5169	−1.31331	−0.656656	−	0.754191i	$-0.728030\pi$
$439$	−27.5169	−1.31331	−0.656656	+	0.754191i	$0.728030\pi$
$440$	0	0
$441$	44.6937	2.12827
$442$	0	0
$443$	−5.36462	−0.254881	−0.127440	−	0.991846i	$-0.540676\pi$
$443$	−5.36462	−0.254881	−0.127440	+	0.991846i	$0.540676\pi$
$444$	0	0
$445$	−22.9260	−1.08680
$446$	0	0
$447$	−34.0739	−1.61164
$448$	0	0
$449$	−26.6019	−1.25542	−0.627712	−	0.778446i	$-0.716008\pi$
$449$	−26.6019	−1.25542	−0.627712	+	0.778446i	$0.716008\pi$
$450$	0	0
$451$	−24.0872	−1.13422
$452$	0	0
$453$	−17.2953	−0.812605
$454$	0	0
$455$	−45.3161	−2.12445
$456$	0	0
$457$	18.1806	0.850455	0.425227	−	0.905087i	$-0.360194\pi$
$457$	18.1806	0.850455	0.425227	+	0.905087i	$0.360194\pi$
$458$	0	0
$459$	3.89378	0.181746
$460$	0	0
$461$	20.2210	0.941786	0.470893	−	0.882190i	$-0.343932\pi$
$461$	20.2210	0.941786	0.470893	+	0.882190i	$0.343932\pi$
$462$	0	0
$463$	−19.0700	−0.886260	−0.443130	−	0.896457i	$-0.646132\pi$
$463$	−19.0700	−0.886260	−0.443130	+	0.896457i	$0.646132\pi$
$464$	0	0
$465$	46.2958	2.14692
$466$	0	0
$467$	−16.3906	−0.758467	−0.379233	−	0.925301i	$-0.623812\pi$
$467$	−16.3906	−0.758467	−0.379233	+	0.925301i	$0.623812\pi$
$468$	0	0
$469$	−5.63370	−0.260140
$470$	0	0
$471$	−3.35178	−0.154442
$472$	0	0
$473$	14.4939	0.666431
$474$	0	0
$475$	−5.72037	−0.262468
$476$	0	0
$477$	−32.7491	−1.49948
$478$	0	0
$479$	−24.3503	−1.11259	−0.556297	−	0.830984i	$-0.687778\pi$
$479$	−24.3503	−1.11259	−0.556297	+	0.830984i	$0.687778\pi$
$480$	0	0
$481$	0.211025	0.00962190
$482$	0	0
$483$	18.7367	0.852548
$484$	0	0
$485$	26.0529	1.18300
$486$	0	0
$487$	−43.1170	−1.95382	−0.976909	−	0.213657i	$-0.931462\pi$
$487$	−43.1170	−1.95382	−0.976909	+	0.213657i	$0.931462\pi$
$488$	0	0
$489$	23.0334	1.04161
$490$	0	0
$491$	−11.3612	−0.512723	−0.256361	−	0.966581i	$-0.582524\pi$
$491$	−11.3612	−0.512723	−0.256361	+	0.966581i	$0.582524\pi$
$492$	0	0
$493$	3.27855	0.147659
$494$	0	0
$495$	−34.6667	−1.55815
$496$	0	0
$497$	−21.5762	−0.967825
$498$	0	0
$499$	−19.4749	−0.871818	−0.435909	−	0.899991i	$-0.643573\pi$
$499$	−19.4749	−0.871818	−0.435909	+	0.899991i	$0.643573\pi$
$500$	0	0
$501$	3.57068	0.159526
$502$	0	0
$503$	−20.2392	−0.902422	−0.451211	−	0.892417i	$-0.649008\pi$
$503$	−20.2392	−0.902422	−0.451211	+	0.892417i	$0.649008\pi$
$504$	0	0
$505$	32.9389	1.46576
$506$	0	0
$507$	−57.0058	−2.53172
$508$	0	0
$509$	−33.5670	−1.48783	−0.743916	−	0.668273i	$-0.767034\pi$
$509$	−33.5670	−1.48783	−0.743916	+	0.668273i	$0.767034\pi$
$510$	0	0
$511$	−68.4246	−3.02693
$512$	0	0
$513$	−21.8209	−0.963414
$514$	0	0
$515$	−1.19085	−0.0524753
$516$	0	0
$517$	10.0305	0.441142
$518$	0	0
$519$	−56.2041	−2.46709
$520$	0	0
$521$	−43.2539	−1.89499	−0.947495	−	0.319771i	$-0.896394\pi$
$521$	−43.2539	−1.89499	−0.947495	+	0.319771i	$0.896394\pi$
$522$	0	0
$523$	25.9340	1.13401	0.567007	−	0.823713i	$-0.308101\pi$
$523$	25.9340	1.13401	0.567007	+	0.823713i	$0.308101\pi$
$524$	0	0
$525$	−14.3018	−0.624183
$526$	0	0
$527$	6.93345	0.302026
$528$	0	0
$529$	−20.2392	−0.879965
$530$	0	0
$531$	29.7886	1.29272
$532$	0	0
$533$	36.8119	1.59450
$534$	0	0
$535$	−14.5859	−0.630603
$536$	0	0
$537$	69.7651	3.01059
$538$	0	0
$539$	−35.7122	−1.53823
$540$	0	0
$541$	−40.3186	−1.73343	−0.866716	−	0.498802i	$-0.833773\pi$
$541$	−40.3186	−1.73343	−0.866716	+	0.498802i	$0.833773\pi$
$542$	0	0
$543$	−55.9849	−2.40254
$544$	0	0
$545$	−35.3808	−1.51555
$546$	0	0
$547$	−6.61433	−0.282808	−0.141404	−	0.989952i	$-0.545162\pi$
$547$	−6.61433	−0.282808	−0.141404	+	0.989952i	$0.545162\pi$
$548$	0	0
$549$	−26.6239	−1.13628
$550$	0	0
$551$	−18.3731	−0.782721
$552$	0	0
$553$	−60.4869	−2.57216
$554$	0	0
$555$	−0.195959	−0.00831799
$556$	0	0
$557$	−31.6673	−1.34179	−0.670893	−	0.741554i	$-0.734089\pi$
$557$	−31.6673	−1.34179	−0.670893	+	0.741554i	$0.734089\pi$
$558$	0	0
$559$	−22.1507	−0.936877
$560$	0	0
$561$	−8.47844	−0.357960
$562$	0	0
$563$	15.5953	0.657263	0.328632	−	0.944458i	$-0.393413\pi$
$563$	15.5953	0.657263	0.328632	+	0.944458i	$0.393413\pi$
$564$	0	0
$565$	9.50074	0.399699
$566$	0	0
$567$	3.07400	0.129096
$568$	0	0
$569$	−13.4097	−0.562164	−0.281082	−	0.959684i	$-0.590693\pi$
$569$	−13.4097	−0.562164	−0.281082	+	0.959684i	$0.590693\pi$
$570$	0	0
$571$	45.7083	1.91283	0.956416	−	0.292007i	$-0.0943232\pi$
$571$	45.7083	1.91283	0.956416	+	0.292007i	$0.0943232\pi$
$572$	0	0
$573$	−14.8745	−0.621391
$574$	0	0
$575$	−2.10734	−0.0878820
$576$	0	0
$577$	−12.2921	−0.511725	−0.255862	−	0.966713i	$-0.582359\pi$
$577$	−12.2921	−0.511725	−0.255862	+	0.966713i	$0.582359\pi$
$578$	0	0
$579$	65.1835	2.70894
$580$	0	0
$581$	−46.5109	−1.92960
$582$	0	0
$583$	26.1679	1.08377
$584$	0	0
$585$	52.9805	2.19047
$586$	0	0
$587$	34.7857	1.43576	0.717879	−	0.696168i	$-0.245113\pi$
$587$	34.7857	1.43576	0.717879	+	0.696168i	$0.245113\pi$
$588$	0	0
$589$	−38.8553	−1.60100
$590$	0	0
$591$	70.5259	2.90105
$592$	0	0
$593$	20.7913	0.853797	0.426899	−	0.904300i	$-0.359606\pi$
$593$	20.7913	0.853797	0.426899	+	0.904300i	$0.359606\pi$
$594$	0	0
$595$	6.30219	0.258364
$596$	0	0
$597$	1.86518	0.0763368
$598$	0	0
$599$	−0.723039	−0.0295426	−0.0147713	−	0.999891i	$-0.504702\pi$
$599$	−0.723039	−0.0295426	−0.0147713	+	0.999891i	$0.504702\pi$
$600$	0	0
$601$	−8.37858	−0.341770	−0.170885	−	0.985291i	$-0.554663\pi$
$601$	−8.37858	−0.341770	−0.170885	+	0.985291i	$0.554663\pi$
$602$	0	0
$603$	6.58654	0.268225
$604$	0	0
$605$	6.45084	0.262264
$606$	0	0
$607$	−14.0169	−0.568927	−0.284463	−	0.958687i	$-0.591815\pi$
$607$	−14.0169	−0.568927	−0.284463	+	0.958687i	$0.591815\pi$
$608$	0	0
$609$	−45.9357	−1.86141
$610$	0	0
$611$	−15.3294	−0.620162
$612$	0	0
$613$	27.8874	1.12636	0.563181	−	0.826334i	$-0.309577\pi$
$613$	27.8874	1.12636	0.563181	+	0.826334i	$0.309577\pi$
$614$	0	0
$615$	−34.1838	−1.37842
$616$	0	0
$617$	16.6905	0.671935	0.335968	−	0.941874i	$-0.390937\pi$
$617$	16.6905	0.671935	0.335968	+	0.941874i	$0.390937\pi$
$618$	0	0
$619$	21.4081	0.860465	0.430233	−	0.902718i	$-0.358432\pi$
$619$	21.4081	0.860465	0.430233	+	0.902718i	$0.358432\pi$
$620$	0	0
$621$	−8.03862	−0.322579
$622$	0	0
$623$	48.1067	1.92735
$624$	0	0
$625$	−17.0500	−0.682001
$626$	0	0
$627$	47.5135	1.89751
$628$	0	0
$629$	−0.0293476	−0.00117016
$630$	0	0
$631$	−34.5505	−1.37543	−0.687717	−	0.725979i	$-0.741387\pi$
$631$	−34.5505	−1.37543	−0.687717	+	0.725979i	$0.741387\pi$
$632$	0	0
$633$	−52.8900	−2.10219
$634$	0	0
$635$	9.37019	0.371845
$636$	0	0
$637$	54.5783	2.16247
$638$	0	0
$639$	25.2254	0.997903
$640$	0	0
$641$	−32.3060	−1.27601	−0.638005	−	0.770033i	$-0.720240\pi$
$641$	−32.3060	−1.27601	−0.638005	+	0.770033i	$0.720240\pi$
$642$	0	0
$643$	19.8375	0.782313	0.391157	−	0.920324i	$-0.372075\pi$
$643$	19.8375	0.782313	0.391157	+	0.920324i	$0.372075\pi$
$644$	0	0
$645$	20.5693	0.809916
$646$	0	0
$647$	40.7415	1.60171	0.800856	−	0.598857i	$-0.204378\pi$
$647$	40.7415	1.60171	0.800856	+	0.598857i	$0.204378\pi$
$648$	0	0
$649$	−23.8024	−0.934327
$650$	0	0
$651$	−97.1444	−3.80739
$652$	0	0
$653$	47.8562	1.87276	0.936378	−	0.350993i	$-0.114156\pi$
$653$	47.8562	1.87276	0.936378	+	0.350993i	$0.114156\pi$
$654$	0	0
$655$	−23.6659	−0.924705
$656$	0	0
$657$	79.9974	3.12100
$658$	0	0
$659$	−39.8475	−1.55224	−0.776119	−	0.630586i	$-0.782814\pi$
$659$	−39.8475	−1.55224	−0.776119	+	0.630586i	$0.782814\pi$
$660$	0	0
$661$	−30.9712	−1.20464	−0.602320	−	0.798255i	$-0.705757\pi$
$661$	−30.9712	−1.20464	−0.602320	+	0.798255i	$0.705757\pi$
$662$	0	0
$663$	12.9574	0.503225
$664$	0	0
$665$	−35.3177	−1.36956
$666$	0	0
$667$	−6.76851	−0.262078
$668$	0	0
$669$	−32.9926	−1.27557
$670$	0	0
$671$	21.2737	0.821260
$672$	0	0
$673$	37.9015	1.46100	0.730498	−	0.682914i	$-0.239288\pi$
$673$	37.9015	1.46100	0.730498	+	0.682914i	$0.239288\pi$
$674$	0	0
$675$	6.13594	0.236172
$676$	0	0
$677$	−19.7229	−0.758013	−0.379006	−	0.925394i	$-0.623734\pi$
$677$	−19.7229	−0.758013	−0.379006	+	0.925394i	$0.623734\pi$
$678$	0	0
$679$	−54.6680	−2.09796
$680$	0	0
$681$	18.3659	0.703782
$682$	0	0
$683$	−6.97483	−0.266884	−0.133442	−	0.991057i	$-0.542603\pi$
$683$	−6.97483	−0.266884	−0.133442	+	0.991057i	$0.542603\pi$
$684$	0	0
$685$	−43.2722	−1.65335
$686$	0	0
$687$	51.2604	1.95571
$688$	0	0
$689$	−39.9919	−1.52357
$690$	0	0
$691$	23.3336	0.887652	0.443826	−	0.896113i	$-0.353621\pi$
$691$	23.3336	0.887652	0.443826	+	0.896113i	$0.353621\pi$
$692$	0	0
$693$	72.7427	2.76327
$694$	0	0
$695$	13.4108	0.508701
$696$	0	0
$697$	−5.11950	−0.193915
$698$	0	0
$699$	8.39435	0.317504
$700$	0	0
$701$	−18.9090	−0.714184	−0.357092	−	0.934069i	$-0.616232\pi$
$701$	−18.9090	−0.714184	−0.357092	+	0.934069i	$0.616232\pi$
$702$	0	0
$703$	0.164465	0.00620291
$704$	0	0
$705$	14.2350	0.536121
$706$	0	0
$707$	−69.1171	−2.59941
$708$	0	0
$709$	−19.5503	−0.734228	−0.367114	−	0.930176i	$-0.619654\pi$
$709$	−19.5503	−0.734228	−0.367114	+	0.930176i	$0.619654\pi$
$710$	0	0
$711$	70.7172	2.65210
$712$	0	0
$713$	−14.3140	−0.536062
$714$	0	0
$715$	−42.3337	−1.58319
$716$	0	0
$717$	9.69108	0.361920
$718$	0	0
$719$	−11.6216	−0.433414	−0.216707	−	0.976237i	$-0.569532\pi$
$719$	−11.6216	−0.433414	−0.216707	+	0.976237i	$0.569532\pi$
$720$	0	0
$721$	2.49882	0.0930609
$722$	0	0
$723$	23.9004	0.888867
$724$	0	0
$725$	5.16645	0.191877
$726$	0	0
$727$	36.2788	1.34551	0.672753	−	0.739867i	$-0.265112\pi$
$727$	36.2788	1.34551	0.672753	+	0.739867i	$0.265112\pi$
$728$	0	0
$729$	−43.9706	−1.62854
$730$	0	0
$731$	3.08054	0.113938
$732$	0	0
$733$	27.0377	0.998659	0.499329	−	0.866412i	$-0.333580\pi$
$733$	27.0377	0.998659	0.499329	+	0.866412i	$0.333580\pi$
$734$	0	0
$735$	−50.6817	−1.86942
$736$	0	0
$737$	−5.26294	−0.193863
$738$	0	0
$739$	30.7337	1.13056	0.565279	−	0.824899i	$-0.308768\pi$
$739$	30.7337	1.13056	0.565279	+	0.824899i	$0.308768\pi$
$740$	0	0
$741$	−72.6139	−2.66754
$742$	0	0
$743$	−6.87718	−0.252299	−0.126150	−	0.992011i	$-0.540262\pi$
$743$	−6.87718	−0.252299	−0.126150	+	0.992011i	$0.540262\pi$
$744$	0	0
$745$	23.6609	0.866869
$746$	0	0
$747$	54.3774	1.98956
$748$	0	0
$749$	30.6062	1.11833
$750$	0	0
$751$	28.1725	1.02803	0.514014	−	0.857782i	$-0.328158\pi$
$751$	28.1725	1.02803	0.514014	+	0.857782i	$0.328158\pi$
$752$	0	0
$753$	5.45070	0.198634
$754$	0	0
$755$	12.0099	0.437084
$756$	0	0
$757$	−10.1345	−0.368343	−0.184171	−	0.982894i	$-0.558960\pi$
$757$	−10.1345	−0.368343	−0.184171	+	0.982894i	$0.558960\pi$
$758$	0	0
$759$	17.5036	0.635339
$760$	0	0
$761$	25.1013	0.909920	0.454960	−	0.890512i	$-0.349654\pi$
$761$	25.1013	0.909920	0.454960	+	0.890512i	$0.349654\pi$
$762$	0	0
$763$	74.2409	2.68770
$764$	0	0
$765$	−7.36809	−0.266394
$766$	0	0
$767$	36.3768	1.31349
$768$	0	0
$769$	−34.8195	−1.25562	−0.627812	−	0.778365i	$-0.716049\pi$
$769$	−34.8195	−1.25562	−0.627812	+	0.778365i	$0.716049\pi$
$770$	0	0
$771$	−14.5402	−0.523653
$772$	0	0
$773$	27.2164	0.978906	0.489453	−	0.872030i	$-0.337196\pi$
$773$	27.2164	0.978906	0.489453	+	0.872030i	$0.337196\pi$
$774$	0	0
$775$	10.9260	0.392472
$776$	0	0
$777$	0.411189	0.0147513
$778$	0	0
$779$	28.6899	1.02792
$780$	0	0
$781$	−20.1562	−0.721246
$782$	0	0
$783$	19.7079	0.704302
$784$	0	0
$785$	2.32748	0.0830712
$786$	0	0
$787$	−0.631116	−0.0224969	−0.0112484	−	0.999937i	$-0.503581\pi$
$787$	−0.631116	−0.0224969	−0.0112484	+	0.999937i	$0.503581\pi$
$788$	0	0
$789$	−14.4983	−0.516155
$790$	0	0
$791$	−19.9358	−0.708836
$792$	0	0
$793$	−32.5121	−1.15454
$794$	0	0
$795$	37.1368	1.31710
$796$	0	0
$797$	11.0434	0.391177	0.195588	−	0.980686i	$-0.437338\pi$
$797$	11.0434	0.391177	0.195588	+	0.980686i	$0.437338\pi$
$798$	0	0
$799$	2.13189	0.0754209
$800$	0	0
$801$	−56.2431	−1.98725
$802$	0	0
$803$	−63.9214	−2.25574
$804$	0	0
$805$	−13.0107	−0.458568
$806$	0	0
$807$	−27.5103	−0.968407
$808$	0	0
$809$	−1.70202	−0.0598398	−0.0299199	−	0.999552i	$-0.509525\pi$
$809$	−1.70202	−0.0598398	−0.0299199	+	0.999552i	$0.509525\pi$
$810$	0	0
$811$	21.2448	0.746004	0.373002	−	0.927830i	$-0.378328\pi$
$811$	21.2448	0.746004	0.373002	+	0.927830i	$0.378328\pi$
$812$	0	0
$813$	−61.3466	−2.15152
$814$	0	0
$815$	−15.9944	−0.560259
$816$	0	0
$817$	−17.2635	−0.603972
$818$	0	0
$819$	−111.171	−3.88463
$820$	0	0
$821$	1.61621	0.0564063	0.0282031	−	0.999602i	$-0.491021\pi$
$821$	1.61621	0.0564063	0.0282031	+	0.999602i	$0.491021\pi$
$822$	0	0
$823$	−21.6107	−0.753300	−0.376650	−	0.926356i	$-0.622924\pi$
$823$	−21.6107	−0.753300	−0.376650	+	0.926356i	$0.622924\pi$
$824$	0	0
$825$	−13.3606	−0.465156
$826$	0	0
$827$	51.6676	1.79666	0.898329	−	0.439323i	$-0.144782\pi$
$827$	51.6676	1.79666	0.898329	+	0.439323i	$0.144782\pi$
$828$	0	0
$829$	55.8802	1.94080	0.970400	−	0.241502i	$-0.0776401\pi$
$829$	55.8802	1.94080	0.970400	+	0.241502i	$0.0776401\pi$
$830$	0	0
$831$	−0.633998	−0.0219931
$832$	0	0
$833$	−7.59030	−0.262988
$834$	0	0
$835$	−2.47948	−0.0858058
$836$	0	0
$837$	41.6780	1.44060
$838$	0	0
$839$	16.7534	0.578390	0.289195	−	0.957270i	$-0.406612\pi$
$839$	16.7534	0.578390	0.289195	+	0.957270i	$0.406612\pi$
$840$	0	0
$841$	−12.4060	−0.427793
$842$	0	0
$843$	−71.5451	−2.46414
$844$	0	0
$845$	39.5848	1.36176
$846$	0	0
$847$	−13.5361	−0.465104
$848$	0	0
$849$	8.93604	0.306684
$850$	0	0
$851$	0.0605875	0.00207691
$852$	0	0
$853$	−39.4719	−1.35149	−0.675746	−	0.737135i	$-0.736178\pi$
$853$	−39.4719	−1.35149	−0.675746	+	0.737135i	$0.736178\pi$
$854$	0	0
$855$	41.2910	1.41212
$856$	0	0
$857$	−32.2340	−1.10109	−0.550545	−	0.834805i	$-0.685580\pi$
$857$	−32.2340	−1.10109	−0.550545	+	0.834805i	$0.685580\pi$
$858$	0	0
$859$	−30.5192	−1.04130	−0.520650	−	0.853770i	$-0.674310\pi$
$859$	−30.5192	−1.04130	−0.520650	+	0.853770i	$0.674310\pi$
$860$	0	0
$861$	71.7292	2.44453
$862$	0	0
$863$	5.12635	0.174503	0.0872516	−	0.996186i	$-0.472192\pi$
$863$	5.12635	0.174503	0.0872516	+	0.996186i	$0.472192\pi$
$864$	0	0
$865$	39.0281	1.32700
$866$	0	0
$867$	45.4906	1.54494
$868$	0	0
$869$	−56.5061	−1.91684
$870$	0	0
$871$	8.04324	0.272535
$872$	0	0
$873$	63.9141	2.16316
$874$	0	0
$875$	49.0833	1.65932
$876$	0	0
$877$	−20.7845	−0.701843	−0.350921	−	0.936405i	$-0.614132\pi$
$877$	−20.7845	−0.701843	−0.350921	+	0.936405i	$0.614132\pi$
$878$	0	0
$879$	−12.6769	−0.427582
$880$	0	0
$881$	1.16090	0.0391119	0.0195559	−	0.999809i	$-0.493775\pi$
$881$	1.16090	0.0391119	0.0195559	+	0.999809i	$0.493775\pi$
$882$	0	0
$883$	−21.3936	−0.719953	−0.359976	−	0.932961i	$-0.617215\pi$
$883$	−21.3936	−0.719953	−0.359976	+	0.932961i	$0.617215\pi$
$884$	0	0
$885$	−33.7797	−1.13549
$886$	0	0
$887$	−24.5883	−0.825595	−0.412797	−	0.910823i	$-0.635448\pi$
$887$	−24.5883	−0.825595	−0.412797	+	0.910823i	$0.635448\pi$
$888$	0	0
$889$	−19.6619	−0.659438
$890$	0	0
$891$	2.87169	0.0962053
$892$	0	0
$893$	−11.9472	−0.399798
$894$	0	0
$895$	−48.4449	−1.61933
$896$	0	0
$897$	−26.7504	−0.893168
$898$	0	0
$899$	35.0928	1.17041
$900$	0	0
$901$	5.56175	0.185289
$902$	0	0
$903$	−43.1614	−1.43632
$904$	0	0
$905$	38.8759	1.29228
$906$	0	0
$907$	−7.11893	−0.236380	−0.118190	−	0.992991i	$-0.537709\pi$
$907$	−7.11893	−0.236380	−0.118190	+	0.992991i	$0.537709\pi$
$908$	0	0
$909$	80.8070	2.68020
$910$	0	0
$911$	7.63476	0.252951	0.126475	−	0.991970i	$-0.459634\pi$
$911$	7.63476	0.252951	0.126475	+	0.991970i	$0.459634\pi$
$912$	0	0
$913$	−43.4499	−1.43798
$914$	0	0
$915$	30.1909	0.998081
$916$	0	0
$917$	49.6592	1.63989
$918$	0	0
$919$	19.6829	0.649278	0.324639	−	0.945838i	$-0.394757\pi$
$919$	19.6829	0.649278	0.324639	+	0.945838i	$0.394757\pi$
$920$	0	0
$921$	24.3844	0.803493
$922$	0	0
$923$	30.8043	1.01394
$924$	0	0
$925$	−0.0462469	−0.00152059
$926$	0	0
$927$	−2.92145	−0.0959530
$928$	0	0
$929$	−44.4798	−1.45933	−0.729667	−	0.683803i	$-0.760325\pi$
$929$	−44.4798	−1.45933	−0.729667	+	0.683803i	$0.760325\pi$
$930$	0	0
$931$	42.5363	1.39407
$932$	0	0
$933$	79.3421	2.59754
$934$	0	0
$935$	5.88743	0.192539
$936$	0	0
$937$	−13.6731	−0.446681	−0.223340	−	0.974741i	$-0.571696\pi$
$937$	−13.6731	−0.446681	−0.223340	+	0.974741i	$0.571696\pi$
$938$	0	0
$939$	2.18746	0.0713850
$940$	0	0
$941$	−6.23872	−0.203377	−0.101688	−	0.994816i	$-0.532424\pi$
$941$	−6.23872	−0.203377	−0.101688	+	0.994816i	$0.532424\pi$
$942$	0	0
$943$	10.5691	0.344177
$944$	0	0
$945$	37.8834	1.23235
$946$	0	0
$947$	−12.5716	−0.408522	−0.204261	−	0.978916i	$-0.565479\pi$
$947$	−12.5716	−0.408522	−0.204261	+	0.978916i	$0.565479\pi$
$948$	0	0
$949$	97.6898	3.17115
$950$	0	0
$951$	−7.23073	−0.234473
$952$	0	0
$953$	7.30024	0.236478	0.118239	−	0.992985i	$-0.462275\pi$
$953$	7.30024	0.236478	0.118239	+	0.992985i	$0.462275\pi$
$954$	0	0
$955$	10.3289	0.334234
$956$	0	0
$957$	−42.9126	−1.38717
$958$	0	0
$959$	90.7999	2.93208
$960$	0	0
$961$	43.2140	1.39400
$962$	0	0
$963$	−35.7827	−1.15308
$964$	0	0
$965$	−45.2634	−1.45708
$966$	0	0
$967$	−12.3112	−0.395901	−0.197950	−	0.980212i	$-0.563429\pi$
$967$	−12.3112	−0.395901	−0.197950	+	0.980212i	$0.563429\pi$
$968$	0	0
$969$	10.0985	0.324412
$970$	0	0
$971$	−42.2623	−1.35626	−0.678131	−	0.734941i	$-0.737210\pi$
$971$	−42.2623	−1.35626	−0.678131	+	0.734941i	$0.737210\pi$
$972$	0	0
$973$	−28.1405	−0.902142
$974$	0	0
$975$	20.4187	0.653923
$976$	0	0
$977$	−31.0500	−0.993376	−0.496688	−	0.867929i	$-0.665451\pi$
$977$	−31.0500	−0.993376	−0.496688	+	0.867929i	$0.665451\pi$
$978$	0	0
$979$	44.9407	1.43631
$980$	0	0
$981$	−86.7975	−2.77123
$982$	0	0
$983$	−32.4626	−1.03540	−0.517698	−	0.855564i	$-0.673211\pi$
$983$	−32.4626	−1.03540	−0.517698	+	0.855564i	$0.673211\pi$
$984$	0	0
$985$	−48.9731	−1.56041
$986$	0	0
$987$	−29.8699	−0.950769
$988$	0	0
$989$	−6.35972	−0.202227
$990$	0	0
$991$	−57.8658	−1.83817	−0.919084	−	0.394063i	$-0.871069\pi$
$991$	−57.8658	−1.83817	−0.919084	+	0.394063i	$0.871069\pi$
$992$	0	0
$993$	36.1577	1.14743
$994$	0	0
$995$	−1.29518	−0.0410600
$996$	0	0
$997$	5.26864	0.166859	0.0834297	−	0.996514i	$-0.473413\pi$
$997$	5.26864	0.166859	0.0834297	+	0.996514i	$0.473413\pi$
$998$	0	0
$999$	−0.176413	−0.00558146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.8	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.8	✓	80	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8044 = 2^{2} \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78192 q^{3} +1.93177 q^{5} -4.05350 q^{7} +4.73908 q^{9} +O(q^{10})\)	\(q-2.78192 q^{3} +1.93177 q^{5} -4.05350 q^{7} +4.73908 q^{9} -3.78673 q^{11} +5.78719 q^{13} -5.37402 q^{15} -0.804834 q^{17} +4.51032 q^{19} +11.2765 q^{21} +1.66156 q^{23} -1.26828 q^{25} -4.83798 q^{27} -4.07357 q^{29} -8.61475 q^{31} +10.5344 q^{33} -7.83041 q^{35} +0.0364641 q^{37} -16.0995 q^{39} +6.36094 q^{41} -3.82755 q^{43} +9.15479 q^{45} -2.64886 q^{47} +9.43088 q^{49} +2.23898 q^{51} -6.91043 q^{53} -7.31508 q^{55} -12.5474 q^{57} +6.28574 q^{59} -5.61794 q^{61} -19.2099 q^{63} +11.1795 q^{65} +1.38984 q^{67} -4.62234 q^{69} +5.32285 q^{71} +16.8804 q^{73} +3.52826 q^{75} +15.3495 q^{77} +14.9221 q^{79} -0.758356 q^{81} +11.4742 q^{83} -1.55475 q^{85} +11.3324 q^{87} -11.8679 q^{89} -23.4584 q^{91} +23.9656 q^{93} +8.71288 q^{95} +13.4866 q^{97} -17.9456 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more