LMFDB - Embedded newform 8036.2.a.s.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Root			$-2.45978$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.45978 q^{3} +2.10149 q^{5} +3.05053 q^{9} +O(q^{10})$	$q+2.45978 q^{3} +2.10149 q^{5} +3.05053 q^{9} -2.99553 q^{11} -0.922682 q^{13} +5.16921 q^{15} -6.22399 q^{17} -6.64918 q^{19} -6.34110 q^{23} -0.583733 q^{25} +0.124300 q^{27} -1.64093 q^{29} -3.00257 q^{31} -7.36835 q^{33} +9.04669 q^{37} -2.26960 q^{39} +1.00000 q^{41} +7.65476 q^{43} +6.41067 q^{45} +6.66681 q^{47} -15.3097 q^{51} -7.74187 q^{53} -6.29508 q^{55} -16.3555 q^{57} +13.3223 q^{59} -9.83873 q^{61} -1.93901 q^{65} -8.26139 q^{67} -15.5977 q^{69} -3.01924 q^{71} +16.5270 q^{73} -1.43586 q^{75} +0.193471 q^{79} -8.84585 q^{81} -1.85504 q^{83} -13.0797 q^{85} -4.03632 q^{87} -1.02262 q^{89} -7.38568 q^{93} -13.9732 q^{95} -17.5072 q^{97} -9.13796 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9	$ 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9} - 8 q^{11} - 12 q^{13} + 8 q^{15} - 8 q^{17} - 24 q^{19} + 8 q^{23} + 20 q^{25} - 16 q^{27} - 12 q^{29} - 44 q^{33} + 12 q^{37} + 12 q^{39} + 20 q^{41} + 4 q^{43} - 40 q^{45} - 4 q^{47} + 4 q^{51} - 12 q^{53} + 16 q^{55} + 28 q^{57} - 16 q^{59} - 68 q^{61} - 8 q^{65} + 4 q^{67} - 32 q^{69} + 8 q^{71} - 48 q^{73} - 60 q^{75} - 20 q^{79} + 32 q^{81} + 8 q^{83} - 28 q^{85} - 60 q^{89} - 16 q^{93} + 20 q^{95} - 40 q^{97}+O(q^{100}) $ 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9 - 8 * q^11 - 12 * q^13 + 8 * q^15 - 8 * q^17 - 24 * q^19 + 8 * q^23 + 20 * q^25 - 16 * q^27 - 12 * q^29 - 44 * q^33 + 12 * q^37 + 12 * q^39 + 20 * q^41 + 4 * q^43 - 40 * q^45 - 4 * q^47 + 4 * q^51 - 12 * q^53 + 16 * q^55 + 28 * q^57 - 16 * q^59 - 68 * q^61 - 8 * q^65 + 4 * q^67 - 32 * q^69 + 8 * q^71 - 48 * q^73 - 60 * q^75 - 20 * q^79 + 32 * q^81 + 8 * q^83 - 28 * q^85 - 60 * q^89 - 16 * q^93 + 20 * q^95 - 40 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.45978	1.42016	0.710078	−	0.704123i	$-0.248660\pi$
$3$	2.45978	1.42016	0.710078	+	0.704123i	$0.248660\pi$
$4$	0	0
$5$	2.10149	0.939816	0.469908	−	0.882716i	$-0.344287\pi$
$5$	2.10149	0.939816	0.469908	+	0.882716i	$0.344287\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.05053	1.01684
$10$	0	0
$11$	−2.99553	−0.903186	−0.451593	−	0.892224i	$-0.649144\pi$
$11$	−2.99553	−0.903186	−0.451593	+	0.892224i	$0.649144\pi$
$12$	0	0
$13$	−0.922682	−0.255906	−0.127953	−	0.991780i	$-0.540841\pi$
$13$	−0.922682	−0.255906	−0.127953	+	0.991780i	$0.540841\pi$
$14$	0	0
$15$	5.16921	1.33469
$16$	0	0
$17$	−6.22399	−1.50954	−0.754770	−	0.655990i	$-0.772251\pi$
$17$	−6.22399	−1.50954	−0.754770	+	0.655990i	$0.772251\pi$
$18$	0	0
$19$	−6.64918	−1.52543	−0.762713	−	0.646737i	$-0.776133\pi$
$19$	−6.64918	−1.52543	−0.762713	+	0.646737i	$0.776133\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.34110	−1.32221	−0.661106	−	0.750293i	$-0.729912\pi$
$23$	−6.34110	−1.32221	−0.661106	+	0.750293i	$0.729912\pi$
$24$	0	0
$25$	−0.583733	−0.116747
$26$	0	0
$27$	0.124300	0.0239215
$28$	0	0
$29$	−1.64093	−0.304712	−0.152356	−	0.988326i	$-0.548686\pi$
$29$	−1.64093	−0.304712	−0.152356	+	0.988326i	$0.548686\pi$
$30$	0	0
$31$	−3.00257	−0.539278	−0.269639	−	0.962961i	$-0.586904\pi$
$31$	−3.00257	−0.539278	−0.269639	+	0.962961i	$0.586904\pi$
$32$	0	0
$33$	−7.36835	−1.28267
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.04669	1.48727	0.743634	−	0.668587i	$-0.233101\pi$
$37$	9.04669	1.48727	0.743634	+	0.668587i	$0.233101\pi$
$38$	0	0
$39$	−2.26960	−0.363426
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	7.65476	1.16734	0.583670	−	0.811991i	$-0.301616\pi$
$43$	7.65476	1.16734	0.583670	+	0.811991i	$0.301616\pi$
$44$	0	0
$45$	6.41067	0.955646
$46$	0	0
$47$	6.66681	0.972454	0.486227	−	0.873833i	$-0.338373\pi$
$47$	6.66681	0.972454	0.486227	+	0.873833i	$0.338373\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−15.3097	−2.14378
$52$	0	0
$53$	−7.74187	−1.06343	−0.531714	−	0.846924i	$-0.678452\pi$
$53$	−7.74187	−1.06343	−0.531714	+	0.846924i	$0.678452\pi$
$54$	0	0
$55$	−6.29508	−0.848829
$56$	0	0
$57$	−16.3555	−2.16634
$58$	0	0
$59$	13.3223	1.73442	0.867209	−	0.497944i	$-0.165911\pi$
$59$	13.3223	1.73442	0.867209	+	0.497944i	$0.165911\pi$
$60$	0	0
$61$	−9.83873	−1.25972	−0.629860	−	0.776708i	$-0.716888\pi$
$61$	−9.83873	−1.25972	−0.629860	+	0.776708i	$0.716888\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.93901	−0.240504
$66$	0	0
$67$	−8.26139	−1.00929	−0.504644	−	0.863327i	$-0.668376\pi$
$67$	−8.26139	−1.00929	−0.504644	+	0.863327i	$0.668376\pi$
$68$	0	0
$69$	−15.5977	−1.87775
$70$	0	0
$71$	−3.01924	−0.358317	−0.179159	−	0.983820i	$-0.557338\pi$
$71$	−3.01924	−0.358317	−0.179159	+	0.983820i	$0.557338\pi$
$72$	0	0
$73$	16.5270	1.93434	0.967170	−	0.254130i	$-0.0817890\pi$
$73$	16.5270	1.93434	0.967170	+	0.254130i	$0.0817890\pi$
$74$	0	0
$75$	−1.43586	−0.165799
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.193471	0.0217672	0.0108836	−	0.999941i	$-0.496536\pi$
$79$	0.193471	0.0217672	0.0108836	+	0.999941i	$0.496536\pi$
$80$	0	0
$81$	−8.84585	−0.982872
$82$	0	0
$83$	−1.85504	−0.203618	−0.101809	−	0.994804i	$-0.532463\pi$
$83$	−1.85504	−0.203618	−0.101809	+	0.994804i	$0.532463\pi$
$84$	0	0
$85$	−13.0797	−1.41869
$86$	0	0
$87$	−4.03632	−0.432739
$88$	0	0
$89$	−1.02262	−0.108397	−0.0541986	−	0.998530i	$-0.517260\pi$
$89$	−1.02262	−0.108397	−0.0541986	+	0.998530i	$0.517260\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.38568	−0.765859
$94$	0	0
$95$	−13.9732	−1.43362
$96$	0	0
$97$	−17.5072	−1.77759	−0.888794	−	0.458306i	$-0.848456\pi$
$97$	−17.5072	−1.77759	−0.888794	+	0.458306i	$0.848456\pi$
$98$	0	0
$99$	−9.13796	−0.918400
$100$	0	0
$101$	−12.3992	−1.23376	−0.616882	−	0.787055i	$-0.711605\pi$
$101$	−12.3992	−1.23376	−0.616882	+	0.787055i	$0.711605\pi$
$102$	0	0
$103$	−12.7268	−1.25401	−0.627003	−	0.779017i	$-0.715719\pi$
$103$	−12.7268	−1.25401	−0.627003	+	0.779017i	$0.715719\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.91995	−0.475630	−0.237815	−	0.971311i	$-0.576431\pi$
$107$	−4.91995	−0.475630	−0.237815	+	0.971311i	$0.576431\pi$
$108$	0	0
$109$	−9.01906	−0.863870	−0.431935	−	0.901905i	$-0.642169\pi$
$109$	−9.01906	−0.863870	−0.431935	+	0.901905i	$0.642169\pi$
$110$	0	0
$111$	22.2529	2.11215
$112$	0	0
$113$	0.0393302	0.00369988	0.00184994	−	0.999998i	$-0.499411\pi$
$113$	0.0393302	0.00369988	0.00184994	+	0.999998i	$0.499411\pi$
$114$	0	0
$115$	−13.3258	−1.24263
$116$	0	0
$117$	−2.81467	−0.260216
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.02680	−0.184255
$122$	0	0
$123$	2.45978	0.221791
$124$	0	0
$125$	−11.7342	−1.04954
$126$	0	0
$127$	−4.19318	−0.372085	−0.186042	−	0.982542i	$-0.559566\pi$
$127$	−4.19318	−0.372085	−0.186042	+	0.982542i	$0.559566\pi$
$128$	0	0
$129$	18.8291	1.65781
$130$	0	0
$131$	21.2344	1.85525	0.927627	−	0.373507i	$-0.121845\pi$
$131$	21.2344	1.85525	0.927627	+	0.373507i	$0.121845\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.261215	0.0224818
$136$	0	0
$137$	0.642660	0.0549061	0.0274531	−	0.999623i	$-0.491260\pi$
$137$	0.642660	0.0549061	0.0274531	+	0.999623i	$0.491260\pi$
$138$	0	0
$139$	−1.63836	−0.138964	−0.0694818	−	0.997583i	$-0.522135\pi$
$139$	−1.63836	−0.138964	−0.0694818	+	0.997583i	$0.522135\pi$
$140$	0	0
$141$	16.3989	1.38104
$142$	0	0
$143$	2.76392	0.231131
$144$	0	0
$145$	−3.44839	−0.286373
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.58039	0.702933	0.351467	−	0.936200i	$-0.385683\pi$
$149$	8.58039	0.702933	0.351467	+	0.936200i	$0.385683\pi$
$150$	0	0
$151$	6.84431	0.556982	0.278491	−	0.960439i	$-0.410166\pi$
$151$	6.84431	0.556982	0.278491	+	0.960439i	$0.410166\pi$
$152$	0	0
$153$	−18.9865	−1.53497
$154$	0	0
$155$	−6.30988	−0.506822
$156$	0	0
$157$	−9.23915	−0.737364	−0.368682	−	0.929556i	$-0.620191\pi$
$157$	−9.23915	−0.737364	−0.368682	+	0.929556i	$0.620191\pi$
$158$	0	0
$159$	−19.0433	−1.51023
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.7674	0.921697	0.460848	−	0.887479i	$-0.347545\pi$
$163$	11.7674	0.921697	0.460848	+	0.887479i	$0.347545\pi$
$164$	0	0
$165$	−15.4845	−1.20547
$166$	0	0
$167$	4.24447	0.328447	0.164224	−	0.986423i	$-0.447488\pi$
$167$	4.24447	0.328447	0.164224	+	0.986423i	$0.447488\pi$
$168$	0	0
$169$	−12.1487	−0.934512
$170$	0	0
$171$	−20.2835	−1.55112
$172$	0	0
$173$	23.5768	1.79251	0.896256	−	0.443537i	$-0.146276\pi$
$173$	23.5768	1.79251	0.896256	+	0.443537i	$0.146276\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	32.7700	2.46315
$178$	0	0
$179$	−3.94971	−0.295215	−0.147608	−	0.989046i	$-0.547157\pi$
$179$	−3.94971	−0.295215	−0.147608	+	0.989046i	$0.547157\pi$
$180$	0	0
$181$	−12.1826	−0.905523	−0.452762	−	0.891632i	$-0.649561\pi$
$181$	−12.1826	−0.905523	−0.452762	+	0.891632i	$0.649561\pi$
$182$	0	0
$183$	−24.2012	−1.78900
$184$	0	0
$185$	19.0115	1.39776
$186$	0	0
$187$	18.6442	1.36340
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.7774	−0.779823	−0.389911	−	0.920852i	$-0.627494\pi$
$191$	−10.7774	−0.779823	−0.389911	+	0.920852i	$0.627494\pi$
$192$	0	0
$193$	−23.7731	−1.71122	−0.855612	−	0.517618i	$-0.826819\pi$
$193$	−23.7731	−1.71122	−0.855612	+	0.517618i	$0.826819\pi$
$194$	0	0
$195$	−4.76954	−0.341554
$196$	0	0
$197$	11.3119	0.805940	0.402970	−	0.915213i	$-0.367978\pi$
$197$	11.3119	0.805940	0.402970	+	0.915213i	$0.367978\pi$
$198$	0	0
$199$	2.50514	0.177585	0.0887924	−	0.996050i	$-0.471699\pi$
$199$	2.50514	0.177585	0.0887924	+	0.996050i	$0.471699\pi$
$200$	0	0
$201$	−20.3212	−1.43335
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.10149	0.146775
$206$	0	0
$207$	−19.3437	−1.34448
$208$	0	0
$209$	19.9178	1.37774
$210$	0	0
$211$	17.0504	1.17380	0.586898	−	0.809661i	$-0.300349\pi$
$211$	17.0504	1.17380	0.586898	+	0.809661i	$0.300349\pi$
$212$	0	0
$213$	−7.42667	−0.508867
$214$	0	0
$215$	16.0864	1.09708
$216$	0	0
$217$	0	0
$218$	0	0
$219$	40.6529	2.74707
$220$	0	0
$221$	5.74276	0.386300
$222$	0	0
$223$	28.1839	1.88734	0.943668	−	0.330894i	$-0.107350\pi$
$223$	28.1839	1.88734	0.943668	+	0.330894i	$0.107350\pi$
$224$	0	0
$225$	−1.78070	−0.118713
$226$	0	0
$227$	−11.8695	−0.787810	−0.393905	−	0.919151i	$-0.628876\pi$
$227$	−11.8695	−0.787810	−0.393905	+	0.919151i	$0.628876\pi$
$228$	0	0
$229$	19.8424	1.31122	0.655611	−	0.755099i	$-0.272411\pi$
$229$	19.8424	1.31122	0.655611	+	0.755099i	$0.272411\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.4298	0.748792	0.374396	−	0.927269i	$-0.377850\pi$
$233$	11.4298	0.748792	0.374396	+	0.927269i	$0.377850\pi$
$234$	0	0
$235$	14.0102	0.913927
$236$	0	0
$237$	0.475897	0.0309128
$238$	0	0
$239$	4.81103	0.311199	0.155600	−	0.987820i	$-0.450269\pi$
$239$	4.81103	0.311199	0.155600	+	0.987820i	$0.450269\pi$
$240$	0	0
$241$	10.1967	0.656825	0.328413	−	0.944534i	$-0.393486\pi$
$241$	10.1967	0.656825	0.328413	+	0.944534i	$0.393486\pi$
$242$	0	0
$243$	−22.1318	−1.41975
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.13507	0.390365
$248$	0	0
$249$	−4.56301	−0.289169
$250$	0	0
$251$	3.18649	0.201129	0.100565	−	0.994931i	$-0.467935\pi$
$251$	3.18649	0.201129	0.100565	+	0.994931i	$0.467935\pi$
$252$	0	0
$253$	18.9950	1.19420
$254$	0	0
$255$	−32.1731	−2.01476
$256$	0	0
$257$	−23.6691	−1.47643	−0.738217	−	0.674563i	$-0.764332\pi$
$257$	−23.6691	−1.47643	−0.738217	+	0.674563i	$0.764332\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−5.00570	−0.309845
$262$	0	0
$263$	−17.5035	−1.07931	−0.539655	−	0.841887i	$-0.681445\pi$
$263$	−17.5035	−1.07931	−0.539655	+	0.841887i	$0.681445\pi$
$264$	0	0
$265$	−16.2695	−0.999426
$266$	0	0
$267$	−2.51541	−0.153941
$268$	0	0
$269$	−24.3533	−1.48485	−0.742425	−	0.669929i	$-0.766324\pi$
$269$	−24.3533	−1.48485	−0.742425	+	0.669929i	$0.766324\pi$
$270$	0	0
$271$	−6.21685	−0.377647	−0.188823	−	0.982011i	$-0.560467\pi$
$271$	−6.21685	−0.377647	−0.188823	+	0.982011i	$0.560467\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.74859	0.105444
$276$	0	0
$277$	−2.77913	−0.166982	−0.0834908	−	0.996509i	$-0.526607\pi$
$277$	−2.77913	−0.166982	−0.0834908	+	0.996509i	$0.526607\pi$
$278$	0	0
$279$	−9.15945	−0.548362
$280$	0	0
$281$	21.2802	1.26947	0.634736	−	0.772729i	$-0.281109\pi$
$281$	21.2802	1.26947	0.634736	+	0.772729i	$0.281109\pi$
$282$	0	0
$283$	16.5832	0.985771	0.492886	−	0.870094i	$-0.335942\pi$
$283$	16.5832	0.985771	0.492886	+	0.870094i	$0.335942\pi$
$284$	0	0
$285$	−34.3710	−2.03596
$286$	0	0
$287$	0	0
$288$	0	0
$289$	21.7381	1.27871
$290$	0	0
$291$	−43.0640	−2.52445
$292$	0	0
$293$	7.44832	0.435135	0.217568	−	0.976045i	$-0.430188\pi$
$293$	7.44832	0.435135	0.217568	+	0.976045i	$0.430188\pi$
$294$	0	0
$295$	27.9968	1.63003
$296$	0	0
$297$	−0.372344	−0.0216056
$298$	0	0
$299$	5.85082	0.338362
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−30.4993	−1.75214
$304$	0	0
$305$	−20.6760	−1.18391
$306$	0	0
$307$	−16.2722	−0.928703	−0.464351	−	0.885651i	$-0.653713\pi$
$307$	−16.2722	−0.928703	−0.464351	+	0.885651i	$0.653713\pi$
$308$	0	0
$309$	−31.3051	−1.78088
$310$	0	0
$311$	−17.1922	−0.974882	−0.487441	−	0.873156i	$-0.662070\pi$
$311$	−17.1922	−0.974882	−0.487441	+	0.873156i	$0.662070\pi$
$312$	0	0
$313$	−25.4997	−1.44133	−0.720665	−	0.693283i	$-0.756163\pi$
$313$	−25.4997	−1.44133	−0.720665	+	0.693283i	$0.756163\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.5574	0.592964	0.296482	−	0.955038i	$-0.404186\pi$
$317$	10.5574	0.592964	0.296482	+	0.955038i	$0.404186\pi$
$318$	0	0
$319$	4.91544	0.275212
$320$	0	0
$321$	−12.1020	−0.675469
$322$	0	0
$323$	41.3844	2.30269
$324$	0	0
$325$	0.538600	0.0298762
$326$	0	0
$327$	−22.1849	−1.22683
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.40506	0.461984	0.230992	−	0.972956i	$-0.425803\pi$
$331$	8.40506	0.461984	0.230992	+	0.972956i	$0.425803\pi$
$332$	0	0
$333$	27.5972	1.51232
$334$	0	0
$335$	−17.3612	−0.948545
$336$	0	0
$337$	12.1490	0.661798	0.330899	−	0.943666i	$-0.392648\pi$
$337$	12.1490	0.661798	0.330899	+	0.943666i	$0.392648\pi$
$338$	0	0
$339$	0.0967438	0.00525440
$340$	0	0
$341$	8.99430	0.487069
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−32.7785	−1.76474
$346$	0	0
$347$	5.65522	0.303588	0.151794	−	0.988412i	$-0.451495\pi$
$347$	5.65522	0.303588	0.151794	+	0.988412i	$0.451495\pi$
$348$	0	0
$349$	−13.5239	−0.723917	−0.361959	−	0.932194i	$-0.617892\pi$
$349$	−13.5239	−0.723917	−0.361959	+	0.932194i	$0.617892\pi$
$350$	0	0
$351$	−0.114689	−0.00612166
$352$	0	0
$353$	−25.9341	−1.38033	−0.690165	−	0.723652i	$-0.742462\pi$
$353$	−25.9341	−1.38033	−0.690165	+	0.723652i	$0.742462\pi$
$354$	0	0
$355$	−6.34490	−0.336752
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.31446	0.333265	0.166632	−	0.986019i	$-0.446711\pi$
$359$	6.31446	0.333265	0.166632	+	0.986019i	$0.446711\pi$
$360$	0	0
$361$	25.2115	1.32692
$362$	0	0
$363$	−4.98549	−0.261670
$364$	0	0
$365$	34.7314	1.81792
$366$	0	0
$367$	−4.45266	−0.232427	−0.116213	−	0.993224i	$-0.537076\pi$
$367$	−4.45266	−0.232427	−0.116213	+	0.993224i	$0.537076\pi$
$368$	0	0
$369$	3.05053	0.158804
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−36.9550	−1.91346	−0.956728	−	0.290983i	$-0.906018\pi$
$373$	−36.9550	−1.91346	−0.956728	+	0.290983i	$0.906018\pi$
$374$	0	0
$375$	−28.8635	−1.49051
$376$	0	0
$377$	1.51405	0.0779777
$378$	0	0
$379$	17.6482	0.906527	0.453263	−	0.891377i	$-0.350260\pi$
$379$	17.6482	0.906527	0.453263	+	0.891377i	$0.350260\pi$
$380$	0	0
$381$	−10.3143	−0.528418
$382$	0	0
$383$	36.4077	1.86035	0.930174	−	0.367119i	$-0.119656\pi$
$383$	36.4077	1.86035	0.930174	+	0.367119i	$0.119656\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	23.3511	1.18700
$388$	0	0
$389$	20.9868	1.06407	0.532035	−	0.846722i	$-0.321427\pi$
$389$	20.9868	1.06407	0.532035	+	0.846722i	$0.321427\pi$
$390$	0	0
$391$	39.4670	1.99593
$392$	0	0
$393$	52.2319	2.63475
$394$	0	0
$395$	0.406578	0.0204572
$396$	0	0
$397$	−25.5486	−1.28225	−0.641123	−	0.767438i	$-0.721531\pi$
$397$	−25.5486	−1.28225	−0.641123	+	0.767438i	$0.721531\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.6613	1.48121	0.740607	−	0.671938i	$-0.234538\pi$
$401$	29.6613	1.48121	0.740607	+	0.671938i	$0.234538\pi$
$402$	0	0
$403$	2.77042	0.138004
$404$	0	0
$405$	−18.5895	−0.923718
$406$	0	0
$407$	−27.0996	−1.34328
$408$	0	0
$409$	4.74246	0.234499	0.117250	−	0.993102i	$-0.462592\pi$
$409$	4.74246	0.234499	0.117250	+	0.993102i	$0.462592\pi$
$410$	0	0
$411$	1.58080	0.0779753
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−3.89836	−0.191363
$416$	0	0
$417$	−4.03000	−0.197350
$418$	0	0
$419$	−14.3836	−0.702685	−0.351342	−	0.936247i	$-0.614275\pi$
$419$	−14.3836	−0.702685	−0.351342	+	0.936247i	$0.614275\pi$
$420$	0	0
$421$	−9.89043	−0.482030	−0.241015	−	0.970521i	$-0.577480\pi$
$421$	−9.89043	−0.482030	−0.241015	+	0.970521i	$0.577480\pi$
$422$	0	0
$423$	20.3373	0.988834
$424$	0	0
$425$	3.63315	0.176234
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.79865	0.328242
$430$	0	0
$431$	13.1907	0.635375	0.317688	−	0.948195i	$-0.397094\pi$
$431$	13.1907	0.635375	0.317688	+	0.948195i	$0.397094\pi$
$432$	0	0
$433$	18.8403	0.905406	0.452703	−	0.891661i	$-0.350460\pi$
$433$	18.8403	0.905406	0.452703	+	0.891661i	$0.350460\pi$
$434$	0	0
$435$	−8.48230	−0.406695
$436$	0	0
$437$	42.1631	2.01693
$438$	0	0
$439$	16.2902	0.777487	0.388743	−	0.921346i	$-0.372909\pi$
$439$	16.2902	0.777487	0.388743	+	0.921346i	$0.372909\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.7111	1.41162	0.705809	−	0.708402i	$-0.250584\pi$
$443$	29.7111	1.41162	0.705809	+	0.708402i	$0.250584\pi$
$444$	0	0
$445$	−2.14902	−0.101873
$446$	0	0
$447$	21.1059	0.998275
$448$	0	0
$449$	2.26145	0.106724	0.0533622	−	0.998575i	$-0.483006\pi$
$449$	2.26145	0.106724	0.0533622	+	0.998575i	$0.483006\pi$
$450$	0	0
$451$	−2.99553	−0.141054
$452$	0	0
$453$	16.8355	0.791002
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.65074	−0.264330	−0.132165	−	0.991228i	$-0.542193\pi$
$457$	−5.65074	−0.264330	−0.132165	+	0.991228i	$0.542193\pi$
$458$	0	0
$459$	−0.773642	−0.0361105
$460$	0	0
$461$	15.1407	0.705171	0.352585	−	0.935780i	$-0.385303\pi$
$461$	15.1407	0.705171	0.352585	+	0.935780i	$0.385303\pi$
$462$	0	0
$463$	−34.3527	−1.59650	−0.798252	−	0.602324i	$-0.794242\pi$
$463$	−34.3527	−1.59650	−0.798252	+	0.602324i	$0.794242\pi$
$464$	0	0
$465$	−15.5209	−0.719766
$466$	0	0
$467$	−30.7199	−1.42155	−0.710773	−	0.703421i	$-0.751655\pi$
$467$	−30.7199	−1.42155	−0.710773	+	0.703421i	$0.751655\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−22.7263	−1.04717
$472$	0	0
$473$	−22.9301	−1.05433
$474$	0	0
$475$	3.88135	0.178088
$476$	0	0
$477$	−23.6168	−1.08134
$478$	0	0
$479$	9.91797	0.453163	0.226582	−	0.973992i	$-0.427245\pi$
$479$	9.91797	0.453163	0.226582	+	0.973992i	$0.427245\pi$
$480$	0	0
$481$	−8.34722	−0.380600
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−36.7913	−1.67061
$486$	0	0
$487$	6.95675	0.315240	0.157620	−	0.987500i	$-0.449618\pi$
$487$	6.95675	0.315240	0.157620	+	0.987500i	$0.449618\pi$
$488$	0	0
$489$	28.9453	1.30895
$490$	0	0
$491$	−6.26318	−0.282653	−0.141327	−	0.989963i	$-0.545137\pi$
$491$	−6.26318	−0.282653	−0.141327	+	0.989963i	$0.545137\pi$
$492$	0	0
$493$	10.2131	0.459975
$494$	0	0
$495$	−19.2034	−0.863126
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−11.9396	−0.534491	−0.267246	−	0.963628i	$-0.586113\pi$
$499$	−11.9396	−0.534491	−0.267246	+	0.963628i	$0.586113\pi$
$500$	0	0
$501$	10.4405	0.466447
$502$	0	0
$503$	12.4862	0.556731	0.278366	−	0.960475i	$-0.410207\pi$
$503$	12.4862	0.556731	0.278366	+	0.960475i	$0.410207\pi$
$504$	0	0
$505$	−26.0568	−1.15951
$506$	0	0
$507$	−29.8831	−1.32715
$508$	0	0
$509$	−35.0082	−1.55171	−0.775856	−	0.630910i	$-0.782682\pi$
$509$	−35.0082	−1.55171	−0.775856	+	0.630910i	$0.782682\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−0.826492	−0.0364905
$514$	0	0
$515$	−26.7452	−1.17853
$516$	0	0
$517$	−19.9706	−0.878307
$518$	0	0
$519$	57.9939	2.54565
$520$	0	0
$521$	−30.3464	−1.32950	−0.664749	−	0.747067i	$-0.731462\pi$
$521$	−30.3464	−1.32950	−0.664749	+	0.747067i	$0.731462\pi$
$522$	0	0
$523$	35.2273	1.54038	0.770191	−	0.637813i	$-0.220161\pi$
$523$	35.2273	1.54038	0.770191	+	0.637813i	$0.220161\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.6880	0.814062
$528$	0	0
$529$	17.2096	0.748243
$530$	0	0
$531$	40.6402	1.76363
$532$	0	0
$533$	−0.922682	−0.0399658
$534$	0	0
$535$	−10.3392	−0.447004
$536$	0	0
$537$	−9.71544	−0.419252
$538$	0	0
$539$	0	0
$540$	0	0
$541$	34.5207	1.48416	0.742079	−	0.670312i	$-0.233840\pi$
$541$	34.5207	1.48416	0.742079	+	0.670312i	$0.233840\pi$
$542$	0	0
$543$	−29.9665	−1.28598
$544$	0	0
$545$	−18.9535	−0.811878
$546$	0	0
$547$	1.86234	0.0796281	0.0398141	−	0.999207i	$-0.487323\pi$
$547$	1.86234	0.0796281	0.0398141	+	0.999207i	$0.487323\pi$
$548$	0	0
$549$	−30.0134	−1.28094
$550$	0	0
$551$	10.9108	0.464816
$552$	0	0
$553$	0	0
$554$	0	0
$555$	46.7643	1.98503
$556$	0	0
$557$	7.47891	0.316892	0.158446	−	0.987368i	$-0.449352\pi$
$557$	7.47891	0.316892	0.158446	+	0.987368i	$0.449352\pi$
$558$	0	0
$559$	−7.06291	−0.298729
$560$	0	0
$561$	45.8606	1.93623
$562$	0	0
$563$	−32.4596	−1.36801	−0.684005	−	0.729477i	$-0.739763\pi$
$563$	−32.4596	−1.36801	−0.684005	+	0.729477i	$0.739763\pi$
$564$	0	0
$565$	0.0826521	0.00347720
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.6801	−1.49579	−0.747894	−	0.663819i	$-0.768935\pi$
$569$	−35.6801	−1.49579	−0.747894	+	0.663819i	$0.768935\pi$
$570$	0	0
$571$	9.06761	0.379468	0.189734	−	0.981836i	$-0.439237\pi$
$571$	9.06761	0.379468	0.189734	+	0.981836i	$0.439237\pi$
$572$	0	0
$573$	−26.5100	−1.10747
$574$	0	0
$575$	3.70151	0.154364
$576$	0	0
$577$	13.1045	0.545548	0.272774	−	0.962078i	$-0.412059\pi$
$577$	13.1045	0.545548	0.272774	+	0.962078i	$0.412059\pi$
$578$	0	0
$579$	−58.4766	−2.43021
$580$	0	0
$581$	0	0
$582$	0	0
$583$	23.1910	0.960474
$584$	0	0
$585$	−5.91501	−0.244555
$586$	0	0
$587$	34.7207	1.43308	0.716538	−	0.697548i	$-0.245726\pi$
$587$	34.7207	1.43308	0.716538	+	0.697548i	$0.245726\pi$
$588$	0	0
$589$	19.9646	0.822629
$590$	0	0
$591$	27.8248	1.14456
$592$	0	0
$593$	−0.944182	−0.0387729	−0.0193864	−	0.999812i	$-0.506171\pi$
$593$	−0.944182	−0.0387729	−0.0193864	+	0.999812i	$0.506171\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.16211	0.252198
$598$	0	0
$599$	−9.72293	−0.397268	−0.198634	−	0.980074i	$-0.563651\pi$
$599$	−9.72293	−0.397268	−0.198634	+	0.980074i	$0.563651\pi$
$600$	0	0
$601$	−2.61321	−0.106595	−0.0532976	−	0.998579i	$-0.516973\pi$
$601$	−2.61321	−0.106595	−0.0532976	+	0.998579i	$0.516973\pi$
$602$	0	0
$603$	−25.2016	−1.02629
$604$	0	0
$605$	−4.25930	−0.173165
$606$	0	0
$607$	6.08841	0.247121	0.123560	−	0.992337i	$-0.460569\pi$
$607$	6.08841	0.247121	0.123560	+	0.992337i	$0.460569\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.15134	−0.248857
$612$	0	0
$613$	−2.42610	−0.0979894	−0.0489947	−	0.998799i	$-0.515602\pi$
$613$	−2.42610	−0.0979894	−0.0489947	+	0.998799i	$0.515602\pi$
$614$	0	0
$615$	5.16921	0.208443
$616$	0	0
$617$	2.75936	0.111087	0.0555437	−	0.998456i	$-0.482311\pi$
$617$	2.75936	0.111087	0.0555437	+	0.998456i	$0.482311\pi$
$618$	0	0
$619$	−35.0572	−1.40907	−0.704534	−	0.709670i	$-0.748844\pi$
$619$	−35.0572	−1.40907	−0.704534	+	0.709670i	$0.748844\pi$
$620$	0	0
$621$	−0.788199	−0.0316293
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−21.7406	−0.869624
$626$	0	0
$627$	48.9935	1.95661
$628$	0	0
$629$	−56.3065	−2.24509
$630$	0	0
$631$	−25.2717	−1.00605	−0.503026	−	0.864271i	$-0.667780\pi$
$631$	−25.2717	−1.00605	−0.503026	+	0.864271i	$0.667780\pi$
$632$	0	0
$633$	41.9402	1.66697
$634$	0	0
$635$	−8.81193	−0.349691
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9.21028	−0.364353
$640$	0	0
$641$	4.47393	0.176710	0.0883548	−	0.996089i	$-0.471839\pi$
$641$	4.47393	0.176710	0.0883548	+	0.996089i	$0.471839\pi$
$642$	0	0
$643$	−31.3398	−1.23592	−0.617961	−	0.786209i	$-0.712041\pi$
$643$	−31.3398	−1.23592	−0.617961	+	0.786209i	$0.712041\pi$
$644$	0	0
$645$	39.5691	1.55803
$646$	0	0
$647$	−28.1114	−1.10518	−0.552588	−	0.833455i	$-0.686360\pi$
$647$	−28.1114	−1.10518	−0.552588	+	0.833455i	$0.686360\pi$
$648$	0	0
$649$	−39.9074	−1.56650
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.97196	0.194568	0.0972839	−	0.995257i	$-0.468985\pi$
$653$	4.97196	0.194568	0.0972839	+	0.995257i	$0.468985\pi$
$654$	0	0
$655$	44.6238	1.74360
$656$	0	0
$657$	50.4162	1.96692
$658$	0	0
$659$	−22.9246	−0.893014	−0.446507	−	0.894780i	$-0.647332\pi$
$659$	−22.9246	−0.893014	−0.446507	+	0.894780i	$0.647332\pi$
$660$	0	0
$661$	−45.1921	−1.75777	−0.878884	−	0.477036i	$-0.841711\pi$
$661$	−45.1921	−1.75777	−0.878884	+	0.477036i	$0.841711\pi$
$662$	0	0
$663$	14.1260	0.548607
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.4053	0.402894
$668$	0	0
$669$	69.3264	2.68031
$670$	0	0
$671$	29.4722	1.13776
$672$	0	0
$673$	47.6558	1.83700	0.918498	−	0.395426i	$-0.129403\pi$
$673$	47.6558	1.83700	0.918498	+	0.395426i	$0.129403\pi$
$674$	0	0
$675$	−0.0725580	−0.00279276
$676$	0	0
$677$	−10.8310	−0.416270	−0.208135	−	0.978100i	$-0.566739\pi$
$677$	−10.8310	−0.416270	−0.208135	+	0.978100i	$0.566739\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−29.1965	−1.11881
$682$	0	0
$683$	−14.1592	−0.541787	−0.270893	−	0.962609i	$-0.587319\pi$
$683$	−14.1592	−0.541787	−0.270893	+	0.962609i	$0.587319\pi$
$684$	0	0
$685$	1.35054	0.0516016
$686$	0	0
$687$	48.8080	1.86214
$688$	0	0
$689$	7.14329	0.272138
$690$	0	0
$691$	−2.50530	−0.0953061	−0.0476530	−	0.998864i	$-0.515174\pi$
$691$	−2.50530	−0.0953061	−0.0476530	+	0.998864i	$0.515174\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.44299	−0.130600
$696$	0	0
$697$	−6.22399	−0.235750
$698$	0	0
$699$	28.1149	1.06340
$700$	0	0
$701$	−11.8134	−0.446185	−0.223092	−	0.974797i	$-0.571615\pi$
$701$	−11.8134	−0.446185	−0.223092	+	0.974797i	$0.571615\pi$
$702$	0	0
$703$	−60.1531	−2.26872
$704$	0	0
$705$	34.4622	1.29792
$706$	0	0
$707$	0	0
$708$	0	0
$709$	26.5103	0.995614	0.497807	−	0.867288i	$-0.334139\pi$
$709$	26.5103	0.995614	0.497807	+	0.867288i	$0.334139\pi$
$710$	0	0
$711$	0.590190	0.0221339
$712$	0	0
$713$	19.0396	0.713040
$714$	0	0
$715$	5.80836	0.217220
$716$	0	0
$717$	11.8341	0.441952
$718$	0	0
$719$	17.0081	0.634294	0.317147	−	0.948376i	$-0.397275\pi$
$719$	17.0081	0.634294	0.317147	+	0.948376i	$0.397275\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	25.0816	0.932794
$724$	0	0
$725$	0.957864	0.0355742
$726$	0	0
$727$	2.07621	0.0770025	0.0385013	−	0.999259i	$-0.487742\pi$
$727$	2.07621	0.0770025	0.0385013	+	0.999259i	$0.487742\pi$
$728$	0	0
$729$	−27.9018	−1.03340
$730$	0	0
$731$	−47.6432	−1.76215
$732$	0	0
$733$	31.9911	1.18162	0.590808	−	0.806812i	$-0.298809\pi$
$733$	31.9911	1.18162	0.590808	+	0.806812i	$0.298809\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.7472	0.911576
$738$	0	0
$739$	10.3436	0.380494	0.190247	−	0.981736i	$-0.439071\pi$
$739$	10.3436	0.380494	0.190247	+	0.981736i	$0.439071\pi$
$740$	0	0
$741$	15.0910	0.554380
$742$	0	0
$743$	24.2982	0.891415	0.445707	−	0.895179i	$-0.352952\pi$
$743$	24.2982	0.891415	0.445707	+	0.895179i	$0.352952\pi$
$744$	0	0
$745$	18.0316	0.660628
$746$	0	0
$747$	−5.65887	−0.207047
$748$	0	0
$749$	0	0
$750$	0	0
$751$	37.0572	1.35224	0.676118	−	0.736793i	$-0.263661\pi$
$751$	37.0572	1.35224	0.676118	+	0.736793i	$0.263661\pi$
$752$	0	0
$753$	7.83807	0.285635
$754$	0	0
$755$	14.3833	0.523460
$756$	0	0
$757$	18.8750	0.686023	0.343011	−	0.939331i	$-0.388553\pi$
$757$	18.8750	0.686023	0.343011	+	0.939331i	$0.388553\pi$
$758$	0	0
$759$	46.7235	1.69596
$760$	0	0
$761$	−23.3925	−0.847978	−0.423989	−	0.905667i	$-0.639370\pi$
$761$	−23.3925	−0.847978	−0.423989	+	0.905667i	$0.639370\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−39.8999	−1.44259
$766$	0	0
$767$	−12.2923	−0.443848
$768$	0	0
$769$	−1.63351	−0.0589060	−0.0294530	−	0.999566i	$-0.509377\pi$
$769$	−1.63351	−0.0589060	−0.0294530	+	0.999566i	$0.509377\pi$
$770$	0	0
$771$	−58.2207	−2.09677
$772$	0	0
$773$	−5.51626	−0.198406	−0.0992031	−	0.995067i	$-0.531629\pi$
$773$	−5.51626	−0.198406	−0.0992031	+	0.995067i	$0.531629\pi$
$774$	0	0
$775$	1.75270	0.0629589
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.64918	−0.238231
$780$	0	0
$781$	9.04421	0.323627
$782$	0	0
$783$	−0.203967	−0.00728919
$784$	0	0
$785$	−19.4160	−0.692986
$786$	0	0
$787$	−52.7979	−1.88204	−0.941022	−	0.338347i	$-0.890132\pi$
$787$	−52.7979	−1.88204	−0.941022	+	0.338347i	$0.890132\pi$
$788$	0	0
$789$	−43.0547	−1.53279
$790$	0	0
$791$	0	0
$792$	0	0
$793$	9.07802	0.322370
$794$	0	0
$795$	−40.0194	−1.41934
$796$	0	0
$797$	38.4437	1.36174	0.680872	−	0.732402i	$-0.261601\pi$
$797$	38.4437	1.36174	0.680872	+	0.732402i	$0.261601\pi$
$798$	0	0
$799$	−41.4942	−1.46796
$800$	0	0
$801$	−3.11953	−0.110223
$802$	0	0
$803$	−49.5072	−1.74707
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−59.9039	−2.10872
$808$	0	0
$809$	8.92193	0.313678	0.156839	−	0.987624i	$-0.449870\pi$
$809$	8.92193	0.313678	0.156839	+	0.987624i	$0.449870\pi$
$810$	0	0
$811$	−15.1790	−0.533006	−0.266503	−	0.963834i	$-0.585868\pi$
$811$	−15.1790	−0.533006	−0.266503	+	0.963834i	$0.585868\pi$
$812$	0	0
$813$	−15.2921	−0.536318
$814$	0	0
$815$	24.7292	0.866225
$816$	0	0
$817$	−50.8979	−1.78069
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32.8558	1.14668	0.573338	−	0.819319i	$-0.305648\pi$
$821$	32.8558	1.14668	0.573338	+	0.819319i	$0.305648\pi$
$822$	0	0
$823$	−24.8271	−0.865418	−0.432709	−	0.901534i	$-0.642442\pi$
$823$	−24.8271	−0.865418	−0.432709	+	0.901534i	$0.642442\pi$
$824$	0	0
$825$	4.30116	0.149747
$826$	0	0
$827$	46.9638	1.63309	0.816546	−	0.577281i	$-0.195886\pi$
$827$	46.9638	1.63309	0.816546	+	0.577281i	$0.195886\pi$
$828$	0	0
$829$	−21.7377	−0.754982	−0.377491	−	0.926013i	$-0.623213\pi$
$829$	−21.7377	−0.754982	−0.377491	+	0.926013i	$0.623213\pi$
$830$	0	0
$831$	−6.83605	−0.237140
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8.91973	0.308680
$836$	0	0
$837$	−0.373220	−0.0129004
$838$	0	0
$839$	49.7400	1.71722	0.858608	−	0.512632i	$-0.171330\pi$
$839$	49.7400	1.71722	0.858608	+	0.512632i	$0.171330\pi$
$840$	0	0
$841$	−26.3074	−0.907150
$842$	0	0
$843$	52.3448	1.80285
$844$	0	0
$845$	−25.5303	−0.878269
$846$	0	0
$847$	0	0
$848$	0	0
$849$	40.7912	1.39995
$850$	0	0
$851$	−57.3660	−1.96648
$852$	0	0
$853$	−17.1076	−0.585753	−0.292876	−	0.956150i	$-0.594612\pi$
$853$	−17.1076	−0.585753	−0.292876	+	0.956150i	$0.594612\pi$
$854$	0	0
$855$	−42.6257	−1.45777
$856$	0	0
$857$	24.6162	0.840873	0.420437	−	0.907322i	$-0.361877\pi$
$857$	24.6162	0.840873	0.420437	+	0.907322i	$0.361877\pi$
$858$	0	0
$859$	32.5001	1.10889	0.554445	−	0.832220i	$-0.312931\pi$
$859$	32.5001	1.10889	0.554445	+	0.832220i	$0.312931\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.6514	−0.839143	−0.419572	−	0.907722i	$-0.637820\pi$
$863$	−24.6514	−0.839143	−0.419572	+	0.907722i	$0.637820\pi$
$864$	0	0
$865$	49.5465	1.68463
$866$	0	0
$867$	53.4709	1.81597
$868$	0	0
$869$	−0.579548	−0.0196598
$870$	0	0
$871$	7.62263	0.258283
$872$	0	0
$873$	−53.4063	−1.80753
$874$	0	0
$875$	0	0
$876$	0	0
$877$	55.9313	1.88867	0.944333	−	0.328991i	$-0.106709\pi$
$877$	55.9313	1.88867	0.944333	+	0.328991i	$0.106709\pi$
$878$	0	0
$879$	18.3212	0.617960
$880$	0	0
$881$	4.22704	0.142413	0.0712064	−	0.997462i	$-0.477315\pi$
$881$	4.22704	0.142413	0.0712064	+	0.997462i	$0.477315\pi$
$882$	0	0
$883$	−9.46057	−0.318374	−0.159187	−	0.987248i	$-0.550887\pi$
$883$	−9.46057	−0.318374	−0.159187	+	0.987248i	$0.550887\pi$
$884$	0	0
$885$	68.8659	2.31490
$886$	0	0
$887$	18.4890	0.620800	0.310400	−	0.950606i	$-0.399537\pi$
$887$	18.4890	0.620800	0.310400	+	0.950606i	$0.399537\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	26.4980	0.887717
$892$	0	0
$893$	−44.3288	−1.48341
$894$	0	0
$895$	−8.30029	−0.277448
$896$	0	0
$897$	14.3917	0.480527
$898$	0	0
$899$	4.92700	0.164325
$900$	0	0
$901$	48.1854	1.60529
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−25.6016	−0.851025
$906$	0	0
$907$	−56.4241	−1.87353	−0.936766	−	0.349955i	$-0.886197\pi$
$907$	−56.4241	−1.87353	−0.936766	+	0.349955i	$0.886197\pi$
$908$	0	0
$909$	−37.8241	−1.25455
$910$	0	0
$911$	11.9011	0.394300	0.197150	−	0.980373i	$-0.436831\pi$
$911$	11.9011	0.394300	0.197150	+	0.980373i	$0.436831\pi$
$912$	0	0
$913$	5.55684	0.183905
$914$	0	0
$915$	−50.8585	−1.68133
$916$	0	0
$917$	0	0
$918$	0	0
$919$	9.13243	0.301251	0.150626	−	0.988591i	$-0.451871\pi$
$919$	9.13243	0.301251	0.150626	+	0.988591i	$0.451871\pi$
$920$	0	0
$921$	−40.0261	−1.31890
$922$	0	0
$923$	2.78579	0.0916955
$924$	0	0
$925$	−5.28086	−0.173634
$926$	0	0
$927$	−38.8234	−1.27513
$928$	0	0
$929$	−3.42592	−0.112401	−0.0562003	−	0.998420i	$-0.517899\pi$
$929$	−3.42592	−0.112401	−0.0562003	+	0.998420i	$0.517899\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−42.2892	−1.38449
$934$	0	0
$935$	39.1805	1.28134
$936$	0	0
$937$	−40.9422	−1.33752	−0.668761	−	0.743477i	$-0.733175\pi$
$937$	−40.9422	−1.33752	−0.668761	+	0.743477i	$0.733175\pi$
$938$	0	0
$939$	−62.7238	−2.04691
$940$	0	0
$941$	1.97770	0.0644712	0.0322356	−	0.999480i	$-0.489737\pi$
$941$	1.97770	0.0644712	0.0322356	+	0.999480i	$0.489737\pi$
$942$	0	0
$943$	−6.34110	−0.206495
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.4560	0.827207	0.413604	−	0.910457i	$-0.364270\pi$
$947$	25.4560	0.827207	0.413604	+	0.910457i	$0.364270\pi$
$948$	0	0
$949$	−15.2492	−0.495009
$950$	0	0
$951$	25.9690	0.842102
$952$	0	0
$953$	−40.4963	−1.31180	−0.655902	−	0.754846i	$-0.727711\pi$
$953$	−40.4963	−1.31180	−0.655902	+	0.754846i	$0.727711\pi$
$954$	0	0
$955$	−22.6485	−0.732889
$956$	0	0
$957$	12.0909	0.390844
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−21.9846	−0.709179
$962$	0	0
$963$	−15.0085	−0.483641
$964$	0	0
$965$	−49.9589	−1.60823
$966$	0	0
$967$	23.0567	0.741454	0.370727	−	0.928742i	$-0.379109\pi$
$967$	23.0567	0.741454	0.370727	+	0.928742i	$0.379109\pi$
$968$	0	0
$969$	101.797	3.27018
$970$	0	0
$971$	−3.38223	−0.108541	−0.0542705	−	0.998526i	$-0.517283\pi$
$971$	−3.38223	−0.108541	−0.0542705	+	0.998526i	$0.517283\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.32484	0.0424288
$976$	0	0
$977$	18.6374	0.596263	0.298132	−	0.954525i	$-0.403637\pi$
$977$	18.6374	0.596263	0.298132	+	0.954525i	$0.403637\pi$
$978$	0	0
$979$	3.06328	0.0979028
$980$	0	0
$981$	−27.5129	−0.878421
$982$	0	0
$983$	−53.6211	−1.71025	−0.855123	−	0.518425i	$-0.826519\pi$
$983$	−53.6211	−1.71025	−0.855123	+	0.518425i	$0.826519\pi$
$984$	0	0
$985$	23.7719	0.757435
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−48.5396	−1.54347
$990$	0	0
$991$	46.1420	1.46575	0.732875	−	0.680363i	$-0.238178\pi$
$991$	46.1420	1.46575	0.732875	+	0.680363i	$0.238178\pi$
$992$	0	0
$993$	20.6746	0.656089
$994$	0	0
$995$	5.26454	0.166897
$996$	0	0
$997$	10.7038	0.338993	0.169496	−	0.985531i	$-0.445786\pi$
$997$	10.7038	0.338993	0.169496	+	0.985531i	$0.445786\pi$
$998$	0	0
$999$	1.12450	0.0355777

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.s.1.19	✓	20
7.6	odd	2		8036.2.a.t.1.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.19	✓	20	1.1	even	1	trivial
8036.2.a.t.1.2	yes	20	7.6	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+2.45978 q^{3} +2.10149 q^{5} +3.05053 q^{9} +O(q^{10})\)	\(q+2.45978 q^{3} +2.10149 q^{5} +3.05053 q^{9} -2.99553 q^{11} -0.922682 q^{13} +5.16921 q^{15} -6.22399 q^{17} -6.64918 q^{19} -6.34110 q^{23} -0.583733 q^{25} +0.124300 q^{27} -1.64093 q^{29} -3.00257 q^{31} -7.36835 q^{33} +9.04669 q^{37} -2.26960 q^{39} +1.00000 q^{41} +7.65476 q^{43} +6.41067 q^{45} +6.66681 q^{47} -15.3097 q^{51} -7.74187 q^{53} -6.29508 q^{55} -16.3555 q^{57} +13.3223 q^{59} -9.83873 q^{61} -1.93901 q^{65} -8.26139 q^{67} -15.5977 q^{69} -3.01924 q^{71} +16.5270 q^{73} -1.43586 q^{75} +0.193471 q^{79} -8.84585 q^{81} -1.85504 q^{83} -13.0797 q^{85} -4.03632 q^{87} -1.02262 q^{89} -7.38568 q^{93} -13.9732 q^{95} -17.5072 q^{97} -9.13796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9} - 8 q^{11} - 12 q^{13} + 8 q^{15} - 8 q^{17} - 24 q^{19} + 8 q^{23} + 20 q^{25} - 16 q^{27} - 12 q^{29} - 44 q^{33} + 12 q^{37} + 12 q^{39} + 20 q^{41} + 4 q^{43} - 40 q^{45} - 4 q^{47} + 4 q^{51} - 12 q^{53} + 16 q^{55} + 28 q^{57} - 16 q^{59} - 68 q^{61} - 8 q^{65} + 4 q^{67} - 32 q^{69} + 8 q^{71} - 48 q^{73} - 60 q^{75} - 20 q^{79} + 32 q^{81} + 8 q^{83} - 28 q^{85} - 60 q^{89} - 16 q^{93} + 20 q^{95} - 40 q^{97}+O(q^{100}) \) 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9 - 8 * q^11 - 12 * q^13 + 8 * q^15 - 8 * q^17 - 24 * q^19 + 8 * q^23 + 20 * q^25 - 16 * q^27 - 12 * q^29 - 44 * q^33 + 12 * q^37 + 12 * q^39 + 20 * q^41 + 4 * q^43 - 40 * q^45 - 4 * q^47 + 4 * q^51 - 12 * q^53 + 16 * q^55 + 28 * q^57 - 16 * q^59 - 68 * q^61 - 8 * q^65 + 4 * q^67 - 32 * q^69 + 8 * q^71 - 48 * q^73 - 60 * q^75 - 20 * q^79 + 32 * q^81 + 8 * q^83 - 28 * q^85 - 60 * q^89 - 16 * q^93 + 20 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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