LMFDB - Embedded newform 12.16.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [12,16,Mod(1,12)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 12 = 2^{2} \cdot 3 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	12.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$17.1232206120$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	12.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-2187.00 q^{3} +45702.0 q^{5} +1.21789e6 q^{7} +4.78297e6 q^{9} -2.68959e7 q^{11} -1.62582e8 q^{13} -9.99503e7 q^{15} -7.43273e8 q^{17} -4.00301e9 q^{19} -2.66352e9 q^{21} -3.00975e10 q^{23} -2.84289e10 q^{25} -1.04604e10 q^{27} +1.90219e10 q^{29} -4.62155e9 q^{31} +5.88214e10 q^{33} +5.56599e10 q^{35} +6.49298e11 q^{37} +3.55566e11 q^{39} +7.90231e11 q^{41} +1.38873e12 q^{43} +2.18591e11 q^{45} -3.93384e12 q^{47} -3.26431e12 q^{49} +1.62554e12 q^{51} -1.34722e13 q^{53} -1.22920e12 q^{55} +8.75459e12 q^{57} -2.46726e13 q^{59} +2.36307e13 q^{61} +5.82512e12 q^{63} -7.43031e12 q^{65} +3.23851e13 q^{67} +6.58233e13 q^{69} -7.44512e13 q^{71} +1.76524e14 q^{73} +6.21740e13 q^{75} -3.27562e13 q^{77} -1.37959e14 q^{79} +2.28768e13 q^{81} -4.58795e14 q^{83} -3.39690e13 q^{85} -4.16009e13 q^{87} -3.22394e13 q^{89} -1.98006e14 q^{91} +1.01073e13 q^{93} -1.82946e14 q^{95} +4.78308e14 q^{97} -1.28642e14 q^{99} +O(q^{100})$

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
$3$	−2187.00	−0.577350
$3$	−2187.00	−0.577350
$4$	0	0
$5$	45702.0	0.261614	0.130807	−	0.991408i	$-0.458243\pi$
$5$	45702.0	0.261614	0.130807	+	0.991408i	$0.458243\pi$
$6$	0	0
$7$	1.21789e6	0.558949	0.279474	−	0.960153i	$-0.409840\pi$
$7$	1.21789e6	0.558949	0.279474	+	0.960153i	$0.409840\pi$
$8$	0	0
$9$	4.78297e6	0.333333
$10$	0	0
$11$	−2.68959e7	−0.416142	−0.208071	−	0.978114i	$-0.566718\pi$
$11$	−2.68959e7	−0.416142	−0.208071	+	0.978114i	$0.566718\pi$
$12$	0	0
$13$	−1.62582e8	−0.718616	−0.359308	−	0.933219i	$-0.616987\pi$
$13$	−1.62582e8	−0.718616	−0.359308	+	0.933219i	$0.616987\pi$
$14$	0	0
$15$	−9.99503e7	−0.151043
$16$	0	0
$17$	−7.43273e8	−0.439320	−0.219660	−	0.975576i	$-0.570495\pi$
$17$	−7.43273e8	−0.439320	−0.219660	+	0.975576i	$0.570495\pi$
$18$	0	0
$19$	−4.00301e9	−1.02739	−0.513695	−	0.857973i	$-0.671724\pi$
$19$	−4.00301e9	−1.02739	−0.513695	+	0.857973i	$0.671724\pi$
$20$	0	0
$21$	−2.66352e9	−0.322709
$22$	0	0
$23$	−3.00975e10	−1.84320	−0.921600	−	0.388142i	$-0.873117\pi$
$23$	−3.00975e10	−1.84320	−0.921600	+	0.388142i	$0.873117\pi$
$24$	0	0
$25$	−2.84289e10	−0.931558
$26$	0	0
$27$	−1.04604e10	−0.192450
$28$	0	0
$29$	1.90219e10	0.204771	0.102386	−	0.994745i	$-0.467352\pi$
$29$	1.90219e10	0.204771	0.102386	+	0.994745i	$0.467352\pi$
$30$	0	0
$31$	−4.62155e9	−0.0301700	−0.0150850	−	0.999886i	$-0.504802\pi$
$31$	−4.62155e9	−0.0301700	−0.0150850	+	0.999886i	$0.504802\pi$
$32$	0	0
$33$	5.88214e10	0.240260
$34$	0	0
$35$	5.56599e10	0.146229
$36$	0	0
$37$	6.49298e11	1.12443	0.562213	−	0.826992i	$-0.309950\pi$
$37$	6.49298e11	1.12443	0.562213	+	0.826992i	$0.309950\pi$
$38$	0	0
$39$	3.55566e11	0.414893
$40$	0	0
$41$	7.90231e11	0.633687	0.316844	−	0.948478i	$-0.397377\pi$
$41$	7.90231e11	0.633687	0.316844	+	0.948478i	$0.397377\pi$
$42$	0	0
$43$	1.38873e12	0.779119	0.389560	−	0.921001i	$-0.372627\pi$
$43$	1.38873e12	0.779119	0.389560	+	0.921001i	$0.372627\pi$
$44$	0	0
$45$	2.18591e11	0.0872045
$46$	0	0
$47$	−3.93384e12	−1.13262	−0.566308	−	0.824194i	$-0.691629\pi$
$47$	−3.93384e12	−1.13262	−0.566308	+	0.824194i	$0.691629\pi$
$48$	0	0
$49$	−3.26431e12	−0.687576
$50$	0	0
$51$	1.62554e12	0.253642
$52$	0	0
$53$	−1.34722e13	−1.57532	−0.787662	−	0.616108i	$-0.788709\pi$
$53$	−1.34722e13	−1.57532	−0.787662	+	0.616108i	$0.788709\pi$
$54$	0	0
$55$	−1.22920e12	−0.108868
$56$	0	0
$57$	8.75459e12	0.593164
$58$	0	0
$59$	−2.46726e13	−1.29070	−0.645349	−	0.763888i	$-0.723288\pi$
$59$	−2.46726e13	−1.29070	−0.645349	+	0.763888i	$0.723288\pi$
$60$	0	0
$61$	2.36307e13	0.962726	0.481363	−	0.876521i	$-0.340142\pi$
$61$	2.36307e13	0.962726	0.481363	+	0.876521i	$0.340142\pi$
$62$	0	0
$63$	5.82512e12	0.186316
$64$	0	0
$65$	−7.43031e12	−0.188000
$66$	0	0
$67$	3.23851e13	0.652806	0.326403	−	0.945231i	$-0.394163\pi$
$67$	3.23851e13	0.652806	0.326403	+	0.945231i	$0.394163\pi$
$68$	0	0
$69$	6.58233e13	1.06417
$70$	0	0
$71$	−7.44512e13	−0.971480	−0.485740	−	0.874103i	$-0.661450\pi$
$71$	−7.44512e13	−0.971480	−0.485740	+	0.874103i	$0.661450\pi$
$72$	0	0
$73$	1.76524e14	1.87018	0.935089	−	0.354413i	$-0.115319\pi$
$73$	1.76524e14	1.87018	0.935089	+	0.354413i	$0.115319\pi$
$74$	0	0
$75$	6.21740e13	0.537835
$76$	0	0
$77$	−3.27562e13	−0.232602
$78$	0	0
$79$	−1.37959e14	−0.808255	−0.404127	−	0.914703i	$-0.632425\pi$
$79$	−1.37959e14	−0.808255	−0.404127	+	0.914703i	$0.632425\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	0	0
$83$	−4.58795e14	−1.85581	−0.927904	−	0.372820i	$-0.878391\pi$
$83$	−4.58795e14	−1.85581	−0.927904	+	0.372820i	$0.878391\pi$
$84$	0	0
$85$	−3.39690e13	−0.114932
$86$	0	0
$87$	−4.16009e13	−0.118225
$88$	0	0
$89$	−3.22394e13	−0.0772613	−0.0386306	−	0.999254i	$-0.512300\pi$
$89$	−3.22394e13	−0.0772613	−0.0386306	+	0.999254i	$0.512300\pi$
$90$	0	0
$91$	−1.98006e14	−0.401669
$92$	0	0
$93$	1.01073e13	0.0174187
$94$	0	0
$95$	−1.82946e14	−0.268779
$96$	0	0
$97$	4.78308e14	0.601063	0.300531	−	0.953772i	$-0.402836\pi$
$97$	4.78308e14	0.601063	0.300531	+	0.953772i	$0.402836\pi$
$98$	0	0
$99$	−1.28642e14	−0.138714
$100$	0	0
$101$	1.63764e15	1.51988	0.759940	−	0.649994i	$-0.225228\pi$
$101$	1.63764e15	1.51988	0.759940	+	0.649994i	$0.225228\pi$
$102$	0	0
$103$	−9.22136e14	−0.738781	−0.369391	−	0.929274i	$-0.620434\pi$
$103$	−9.22136e14	−0.738781	−0.369391	+	0.929274i	$0.620434\pi$
$104$	0	0
$105$	−1.21728e14	−0.0844251
$106$	0	0
$107$	1.94611e15	1.17162	0.585812	−	0.810447i	$-0.300776\pi$
$107$	1.94611e15	1.17162	0.585812	+	0.810447i	$0.300776\pi$
$108$	0	0
$109$	1.30199e15	0.682198	0.341099	−	0.940027i	$-0.389201\pi$
$109$	1.30199e15	0.682198	0.341099	+	0.940027i	$0.389201\pi$
$110$	0	0
$111$	−1.42001e15	−0.649188
$112$	0	0
$113$	3.40347e15	1.36092	0.680462	−	0.732783i	$-0.261779\pi$
$113$	3.40347e15	1.36092	0.680462	+	0.732783i	$0.261779\pi$
$114$	0	0
$115$	−1.37552e15	−0.482206
$116$	0	0
$117$	−7.77624e14	−0.239539
$118$	0	0
$119$	−9.05223e14	−0.245558
$120$	0	0
$121$	−3.45386e15	−0.826826
$122$	0	0
$123$	−1.72823e15	−0.365860
$124$	0	0
$125$	−2.69397e15	−0.505322
$126$	0	0
$127$	−5.69370e15	−0.948127	−0.474063	−	0.880491i	$-0.657213\pi$
$127$	−5.69370e15	−0.948127	−0.474063	+	0.880491i	$0.657213\pi$
$128$	0	0
$129$	−3.03715e15	−0.449825
$130$	0	0
$131$	5.98970e15	0.790442	0.395221	−	0.918586i	$-0.370668\pi$
$131$	5.98970e15	0.790442	0.395221	+	0.918586i	$0.370668\pi$
$132$	0	0
$133$	−4.87522e15	−0.574258
$134$	0	0
$135$	−4.78059e14	−0.0503475
$136$	0	0
$137$	−2.91811e15	−0.275231	−0.137616	−	0.990486i	$-0.543944\pi$
$137$	−2.91811e15	−0.275231	−0.137616	+	0.990486i	$0.543944\pi$
$138$	0	0
$139$	1.63261e16	1.38125	0.690624	−	0.723214i	$-0.257336\pi$
$139$	1.63261e16	1.38125	0.690624	+	0.723214i	$0.257336\pi$
$140$	0	0
$141$	8.60331e15	0.653916
$142$	0	0
$143$	4.37279e15	0.299046
$144$	0	0
$145$	8.69338e14	0.0535709
$146$	0	0
$147$	7.13905e15	0.396972
$148$	0	0
$149$	−1.72951e16	−0.869010	−0.434505	−	0.900669i	$-0.643077\pi$
$149$	−1.72951e16	−0.869010	−0.434505	+	0.900669i	$0.643077\pi$
$150$	0	0
$151$	2.65936e16	1.20906	0.604532	−	0.796581i	$-0.293360\pi$
$151$	2.65936e16	1.20906	0.604532	+	0.796581i	$0.293360\pi$
$152$	0	0
$153$	−3.55505e15	−0.146440
$154$	0	0
$155$	−2.11214e14	−0.00789288
$156$	0	0
$157$	1.10227e16	0.374146	0.187073	−	0.982346i	$-0.440100\pi$
$157$	1.10227e16	0.374146	0.187073	+	0.982346i	$0.440100\pi$
$158$	0	0
$159$	2.94637e16	0.909513
$160$	0	0
$161$	−3.66554e16	−1.03025
$162$	0	0
$163$	−8.74156e15	−0.223966	−0.111983	−	0.993710i	$-0.535720\pi$
$163$	−8.74156e15	−0.223966	−0.111983	+	0.993710i	$0.535720\pi$
$164$	0	0
$165$	2.68825e15	0.0628552
$166$	0	0
$167$	−2.78118e16	−0.594095	−0.297047	−	0.954863i	$-0.596002\pi$
$167$	−2.78118e16	−0.594095	−0.297047	+	0.954863i	$0.596002\pi$
$168$	0	0
$169$	−2.47531e16	−0.483591
$170$	0	0
$171$	−1.91463e16	−0.342463
$172$	0	0
$173$	−9.60288e16	−1.57419	−0.787093	−	0.616835i	$-0.788415\pi$
$173$	−9.60288e16	−1.57419	−0.787093	+	0.616835i	$0.788415\pi$
$174$	0	0
$175$	−3.46232e16	−0.520693
$176$	0	0
$177$	5.39590e16	0.745185
$178$	0	0
$179$	−1.52758e15	−0.0193912	−0.00969561	−	0.999953i	$-0.503086\pi$
$179$	−1.52758e15	−0.0193912	−0.00969561	+	0.999953i	$0.503086\pi$
$180$	0	0
$181$	5.62266e16	0.656678	0.328339	−	0.944560i	$-0.393511\pi$
$181$	5.62266e16	0.656678	0.328339	+	0.944560i	$0.393511\pi$
$182$	0	0
$183$	−5.16803e16	−0.555830
$184$	0	0
$185$	2.96742e16	0.294165
$186$	0	0
$187$	1.99910e16	0.182819
$188$	0	0
$189$	−1.27395e16	−0.107570
$190$	0	0
$191$	2.23414e17	1.74325	0.871625	−	0.490173i	$-0.163066\pi$
$191$	2.23414e17	1.74325	0.871625	+	0.490173i	$0.163066\pi$
$192$	0	0
$193$	2.40199e16	0.173337	0.0866683	−	0.996237i	$-0.472378\pi$
$193$	2.40199e16	0.173337	0.0866683	+	0.996237i	$0.472378\pi$
$194$	0	0
$195$	1.62501e16	0.108542
$196$	0	0
$197$	1.74868e17	1.08196	0.540982	−	0.841034i	$-0.318053\pi$
$197$	1.74868e17	1.08196	0.540982	+	0.841034i	$0.318053\pi$
$198$	0	0
$199$	−4.00215e16	−0.229559	−0.114780	−	0.993391i	$-0.536616\pi$
$199$	−4.00215e16	−0.229559	−0.114780	+	0.993391i	$0.536616\pi$
$200$	0	0
$201$	−7.08262e16	−0.376898
$202$	0	0
$203$	2.31665e16	0.114457
$204$	0	0
$205$	3.61151e16	0.165781
$206$	0	0
$207$	−1.43956e17	−0.614400
$208$	0	0
$209$	1.07665e17	0.427540
$210$	0	0
$211$	−3.17744e17	−1.17479	−0.587393	−	0.809302i	$-0.699846\pi$
$211$	−3.17744e17	−1.17479	−0.587393	+	0.809302i	$0.699846\pi$
$212$	0	0
$213$	1.62825e17	0.560884
$214$	0	0
$215$	6.34677e16	0.203828
$216$	0	0
$217$	−5.62853e15	−0.0168635
$218$	0	0
$219$	−3.86059e17	−1.07975
$220$	0	0
$221$	1.20843e17	0.315702
$222$	0	0
$223$	−5.18381e17	−1.26579	−0.632896	−	0.774237i	$-0.718134\pi$
$223$	−5.18381e17	−1.26579	−0.632896	+	0.774237i	$0.718134\pi$
$224$	0	0
$225$	−1.35975e17	−0.310519
$226$	0	0
$227$	−1.03081e17	−0.220285	−0.110142	−	0.993916i	$-0.535131\pi$
$227$	−1.03081e17	−0.220285	−0.110142	+	0.993916i	$0.535131\pi$
$228$	0	0
$229$	2.37741e17	0.475705	0.237852	−	0.971301i	$-0.423557\pi$
$229$	2.37741e17	0.475705	0.237852	+	0.971301i	$0.423557\pi$
$230$	0	0
$231$	7.16379e16	0.134293
$232$	0	0
$233$	−5.95188e17	−1.04589	−0.522944	−	0.852367i	$-0.675166\pi$
$233$	−5.95188e17	−1.04589	−0.522944	+	0.852367i	$0.675166\pi$
$234$	0	0
$235$	−1.79784e17	−0.296308
$236$	0	0
$237$	3.01717e17	0.466646
$238$	0	0
$239$	3.52689e17	0.512163	0.256081	−	0.966655i	$-0.417568\pi$
$239$	3.52689e17	0.512163	0.256081	+	0.966655i	$0.417568\pi$
$240$	0	0
$241$	5.75222e17	0.784707	0.392353	−	0.919815i	$-0.371661\pi$
$241$	5.75222e17	0.784707	0.392353	+	0.919815i	$0.371661\pi$
$242$	0	0
$243$	−5.00315e16	−0.0641500
$244$	0	0
$245$	−1.49186e17	−0.179879
$246$	0	0
$247$	6.50817e17	0.738298
$248$	0	0
$249$	1.00338e18	1.07145
$250$	0	0
$251$	1.08097e18	1.08707	0.543537	−	0.839385i	$-0.317085\pi$
$251$	1.08097e18	1.08707	0.543537	+	0.839385i	$0.317085\pi$
$252$	0	0
$253$	8.09501e17	0.767032
$254$	0	0
$255$	7.42903e16	0.0663561
$256$	0	0
$257$	−2.34782e18	−1.97772	−0.988862	−	0.148835i	$-0.952448\pi$
$257$	−2.34782e18	−1.97772	−0.988862	+	0.148835i	$0.952448\pi$
$258$	0	0
$259$	7.90772e17	0.628497
$260$	0	0
$261$	9.09811e16	0.0682570
$262$	0	0
$263$	1.36050e17	0.0963898	0.0481949	−	0.998838i	$-0.484653\pi$
$263$	1.36050e17	0.0963898	0.0481949	+	0.998838i	$0.484653\pi$
$264$	0	0
$265$	−6.15707e17	−0.412126
$266$	0	0
$267$	7.05076e16	0.0446068
$268$	0	0
$269$	−2.30256e18	−1.37743	−0.688713	−	0.725034i	$-0.741824\pi$
$269$	−2.30256e18	−1.37743	−0.688713	+	0.725034i	$0.741824\pi$
$270$	0	0
$271$	−1.06047e18	−0.600107	−0.300053	−	0.953922i	$-0.597004\pi$
$271$	−1.06047e18	−0.600107	−0.300053	+	0.953922i	$0.597004\pi$
$272$	0	0
$273$	4.33040e17	0.231904
$274$	0	0
$275$	7.64622e17	0.387660
$276$	0	0
$277$	6.70093e17	0.321764	0.160882	−	0.986974i	$-0.448566\pi$
$277$	6.70093e17	0.321764	0.160882	+	0.986974i	$0.448566\pi$
$278$	0	0
$279$	−2.21047e16	−0.0100567
$280$	0	0
$281$	−4.40274e18	−1.89857	−0.949283	−	0.314421i	$-0.898189\pi$
$281$	−4.40274e18	−1.89857	−0.949283	+	0.314421i	$0.898189\pi$
$282$	0	0
$283$	2.68593e17	0.109824	0.0549120	−	0.998491i	$-0.482512\pi$
$283$	2.68593e17	0.109824	0.0549120	+	0.998491i	$0.482512\pi$
$284$	0	0
$285$	4.00102e17	0.155180
$286$	0	0
$287$	9.62413e17	0.354199
$288$	0	0
$289$	−2.30997e18	−0.806998
$290$	0	0
$291$	−1.04606e18	−0.347024
$292$	0	0
$293$	2.33452e18	0.735683	0.367842	−	0.929888i	$-0.380097\pi$
$293$	2.33452e18	0.735683	0.367842	+	0.929888i	$0.380097\pi$
$294$	0	0
$295$	−1.12759e18	−0.337664
$296$	0	0
$297$	2.81341e17	0.0800865
$298$	0	0
$299$	4.89331e18	1.32455
$300$	0	0
$301$	1.69132e18	0.435488
$302$	0	0
$303$	−3.58153e18	−0.877503
$304$	0	0
$305$	1.07997e18	0.251862
$306$	0	0
$307$	8.02475e17	0.178194	0.0890971	−	0.996023i	$-0.471602\pi$
$307$	8.02475e17	0.178194	0.0890971	+	0.996023i	$0.471602\pi$
$308$	0	0
$309$	2.01671e18	0.426536
$310$	0	0
$311$	6.07279e18	1.22373	0.611865	−	0.790962i	$-0.290420\pi$
$311$	6.07279e18	1.22373	0.611865	+	0.790962i	$0.290420\pi$
$312$	0	0
$313$	9.67199e17	0.185752	0.0928760	−	0.995678i	$-0.470394\pi$
$313$	9.67199e17	0.185752	0.0928760	+	0.995678i	$0.470394\pi$
$314$	0	0
$315$	2.66220e17	0.0487429
$316$	0	0
$317$	4.52000e18	0.789212	0.394606	−	0.918850i	$-0.370881\pi$
$317$	4.52000e18	0.789212	0.394606	+	0.918850i	$0.370881\pi$
$318$	0	0
$319$	−5.11611e17	−0.0852138
$320$	0	0
$321$	−4.25614e18	−0.676438
$322$	0	0
$323$	2.97533e18	0.451353
$324$	0	0
$325$	4.62202e18	0.669432
$326$	0	0
$327$	−2.84746e18	−0.393867
$328$	0	0
$329$	−4.79098e18	−0.633075
$330$	0	0
$331$	−1.11447e19	−1.40721	−0.703603	−	0.710593i	$-0.748427\pi$
$331$	−1.11447e19	−1.40721	−0.703603	+	0.710593i	$0.748427\pi$
$332$	0	0
$333$	3.10557e18	0.374809
$334$	0	0
$335$	1.48006e18	0.170783
$336$	0	0
$337$	−1.24812e19	−1.37731	−0.688654	−	0.725090i	$-0.741798\pi$
$337$	−1.24812e19	−1.37731	−0.688654	+	0.725090i	$0.741798\pi$
$338$	0	0
$339$	−7.44340e18	−0.785730
$340$	0	0
$341$	1.24301e17	0.0125550
$342$	0	0
$343$	−9.75756e18	−0.943269
$344$	0	0
$345$	3.00826e18	0.278402
$346$	0	0
$347$	−6.60645e18	−0.585459	−0.292730	−	0.956195i	$-0.594564\pi$
$347$	−6.60645e18	−0.585459	−0.292730	+	0.956195i	$0.594564\pi$
$348$	0	0
$349$	2.27753e19	1.93319	0.966594	−	0.256314i	$-0.0825081\pi$
$349$	2.27753e19	1.93319	0.966594	+	0.256314i	$0.0825081\pi$
$350$	0	0
$351$	1.70066e18	0.138298
$352$	0	0
$353$	−9.54105e18	−0.743509	−0.371754	−	0.928331i	$-0.621244\pi$
$353$	−9.54105e18	−0.743509	−0.371754	+	0.928331i	$0.621244\pi$
$354$	0	0
$355$	−3.40257e18	−0.254152
$356$	0	0
$357$	1.97972e18	0.141773
$358$	0	0
$359$	−3.43625e18	−0.235980	−0.117990	−	0.993015i	$-0.537645\pi$
$359$	−3.43625e18	−0.235980	−0.117990	+	0.993015i	$0.537645\pi$
$360$	0	0
$361$	8.43000e17	0.0555295
$362$	0	0
$363$	7.55359e18	0.477368
$364$	0	0
$365$	8.06751e18	0.489264
$366$	0	0
$367$	1.47345e19	0.857710	0.428855	−	0.903373i	$-0.358917\pi$
$367$	1.47345e19	0.857710	0.428855	+	0.903373i	$0.358917\pi$
$368$	0	0
$369$	3.77965e18	0.211229
$370$	0	0
$371$	−1.64076e19	−0.880525
$372$	0	0
$373$	1.80124e19	0.928445	0.464223	−	0.885719i	$-0.346334\pi$
$373$	1.80124e19	0.928445	0.464223	+	0.885719i	$0.346334\pi$
$374$	0	0
$375$	5.89172e18	0.291748
$376$	0	0
$377$	−3.09261e18	−0.147152
$378$	0	0
$379$	7.34337e18	0.335816	0.167908	−	0.985803i	$-0.446299\pi$
$379$	7.34337e18	0.335816	0.167908	+	0.985803i	$0.446299\pi$
$380$	0	0
$381$	1.24521e19	0.547401
$382$	0	0
$383$	−1.60188e19	−0.677077	−0.338538	−	0.940953i	$-0.609933\pi$
$383$	−1.60188e19	−0.677077	−0.338538	+	0.940953i	$0.609933\pi$
$384$	0	0
$385$	−1.49702e18	−0.0608518
$386$	0	0
$387$	6.64224e18	0.259706
$388$	0	0
$389$	3.03795e19	1.14277	0.571386	−	0.820682i	$-0.306406\pi$
$389$	3.03795e19	1.14277	0.571386	+	0.820682i	$0.306406\pi$
$390$	0	0
$391$	2.23707e19	0.809755
$392$	0	0
$393$	−1.30995e19	−0.456362
$394$	0	0
$395$	−6.30502e18	−0.211450
$396$	0	0
$397$	−2.38726e19	−0.770852	−0.385426	−	0.922739i	$-0.625946\pi$
$397$	−2.38726e19	−0.770852	−0.385426	+	0.922739i	$0.625946\pi$
$398$	0	0
$399$	1.06621e19	0.331548
$400$	0	0
$401$	−5.45495e19	−1.63384	−0.816918	−	0.576754i	$-0.804319\pi$
$401$	−5.45495e19	−1.63384	−0.816918	+	0.576754i	$0.804319\pi$
$402$	0	0
$403$	7.51380e17	0.0216806
$404$	0	0
$405$	1.04552e18	0.0290682
$406$	0	0
$407$	−1.74635e19	−0.467921
$408$	0	0
$409$	−4.07053e19	−1.05130	−0.525649	−	0.850701i	$-0.676178\pi$
$409$	−4.07053e19	−1.05130	−0.525649	+	0.850701i	$0.676178\pi$
$410$	0	0
$411$	6.38191e18	0.158905
$412$	0	0
$413$	−3.00485e19	−0.721434
$414$	0	0
$415$	−2.09678e19	−0.485504
$416$	0	0
$417$	−3.57052e19	−0.797463
$418$	0	0
$419$	5.06879e19	1.09219	0.546096	−	0.837723i	$-0.316113\pi$
$419$	5.06879e19	1.09219	0.546096	+	0.837723i	$0.316113\pi$
$420$	0	0
$421$	9.70673e18	0.201817	0.100908	−	0.994896i	$-0.467825\pi$
$421$	9.70673e18	0.201817	0.100908	+	0.994896i	$0.467825\pi$
$422$	0	0
$423$	−1.88154e19	−0.377539
$424$	0	0
$425$	2.11304e19	0.409252
$426$	0	0
$427$	2.87795e19	0.538114
$428$	0	0
$429$	−9.56329e18	−0.172654
$430$	0	0
$431$	4.07116e19	0.709805	0.354902	−	0.934903i	$-0.384514\pi$
$431$	4.07116e19	0.709805	0.354902	+	0.934903i	$0.384514\pi$
$432$	0	0
$433$	−1.01839e20	−1.71496	−0.857482	−	0.514514i	$-0.827972\pi$
$433$	−1.01839e20	−1.71496	−0.857482	+	0.514514i	$0.827972\pi$
$434$	0	0
$435$	−1.90124e18	−0.0309292
$436$	0	0
$437$	1.20481e20	1.89368
$438$	0	0
$439$	9.94692e19	1.51079	0.755396	−	0.655268i	$-0.227444\pi$
$439$	9.94692e19	1.51079	0.755396	+	0.655268i	$0.227444\pi$
$440$	0	0
$441$	−1.56131e19	−0.229192
$442$	0	0
$443$	−1.67145e19	−0.237173	−0.118586	−	0.992944i	$-0.537836\pi$
$443$	−1.67145e19	−0.237173	−0.118586	+	0.992944i	$0.537836\pi$
$444$	0	0
$445$	−1.47341e18	−0.0202126
$446$	0	0
$447$	3.78243e19	0.501723
$448$	0	0
$449$	9.78231e19	1.25486	0.627428	−	0.778674i	$-0.284108\pi$
$449$	9.78231e19	1.25486	0.627428	+	0.778674i	$0.284108\pi$
$450$	0	0
$451$	−2.12540e19	−0.263704
$452$	0	0
$453$	−5.81601e19	−0.698054
$454$	0	0
$455$	−9.04929e18	−0.105082
$456$	0	0
$457$	3.13259e19	0.351991	0.175996	−	0.984391i	$-0.443686\pi$
$457$	3.13259e19	0.351991	0.175996	+	0.984391i	$0.443686\pi$
$458$	0	0
$459$	7.77489e18	0.0845472
$460$	0	0
$461$	7.42122e19	0.781121	0.390560	−	0.920577i	$-0.372281\pi$
$461$	7.42122e19	0.781121	0.390560	+	0.920577i	$0.372281\pi$
$462$	0	0
$463$	−1.20899e20	−1.23187	−0.615937	−	0.787796i	$-0.711222\pi$
$463$	−1.20899e20	−1.23187	−0.615937	+	0.787796i	$0.711222\pi$
$464$	0	0
$465$	4.61925e17	0.00455696
$466$	0	0
$467$	−5.85500e19	−0.559307	−0.279653	−	0.960101i	$-0.590220\pi$
$467$	−5.85500e19	−0.559307	−0.279653	+	0.960101i	$0.590220\pi$
$468$	0	0
$469$	3.94414e19	0.364885
$470$	0	0
$471$	−2.41067e19	−0.216013
$472$	0	0
$473$	−3.73511e19	−0.324224
$474$	0	0
$475$	1.13801e20	0.957073
$476$	0	0
$477$	−6.44372e19	−0.525108
$478$	0	0
$479$	−5.26428e19	−0.415741	−0.207870	−	0.978156i	$-0.566653\pi$
$479$	−5.26428e19	−0.415741	−0.207870	+	0.978156i	$0.566653\pi$
$480$	0	0
$481$	−1.05564e20	−0.808031
$482$	0	0
$483$	8.01654e19	0.594817
$484$	0	0
$485$	2.18596e19	0.157246
$486$	0	0
$487$	−1.55160e20	−1.08221	−0.541107	−	0.840954i	$-0.681995\pi$
$487$	−1.55160e20	−1.08221	−0.541107	+	0.840954i	$0.681995\pi$
$488$	0	0
$489$	1.91178e19	0.129307
$490$	0	0
$491$	−2.32476e20	−1.52499	−0.762493	−	0.646996i	$-0.776025\pi$
$491$	−2.32476e20	−1.52499	−0.762493	+	0.646996i	$0.776025\pi$
$492$	0	0
$493$	−1.41384e19	−0.0899601
$494$	0	0
$495$	−5.87921e18	−0.0362894
$496$	0	0
$497$	−9.06732e19	−0.543008
$498$	0	0
$499$	1.03667e20	0.602400	0.301200	−	0.953561i	$-0.402613\pi$
$499$	1.03667e20	0.602400	0.301200	+	0.953561i	$0.402613\pi$
$500$	0	0
$501$	6.08244e19	0.343001
$502$	0	0
$503$	−2.60380e20	−1.42511	−0.712555	−	0.701616i	$-0.752462\pi$
$503$	−2.60380e20	−1.42511	−0.712555	+	0.701616i	$0.752462\pi$
$504$	0	0
$505$	7.48436e19	0.397621
$506$	0	0
$507$	5.41349e19	0.279202
$508$	0	0
$509$	1.60377e20	0.803077	0.401539	−	0.915842i	$-0.368476\pi$
$509$	1.60377e20	0.803077	0.401539	+	0.915842i	$0.368476\pi$
$510$	0	0
$511$	2.14987e20	1.04533
$512$	0	0
$513$	4.18729e19	0.197721
$514$	0	0
$515$	−4.21435e19	−0.193275
$516$	0	0
$517$	1.05804e20	0.471329
$518$	0	0
$519$	2.10015e20	0.908856
$520$	0	0
$521$	1.24872e20	0.525028	0.262514	−	0.964928i	$-0.415448\pi$
$521$	1.24872e20	0.525028	0.262514	+	0.964928i	$0.415448\pi$
$522$	0	0
$523$	−2.04744e20	−0.836465	−0.418232	−	0.908340i	$-0.637350\pi$
$523$	−2.04744e20	−0.836465	−0.418232	+	0.908340i	$0.637350\pi$
$524$	0	0
$525$	7.57210e19	0.300623
$526$	0	0
$527$	3.43507e18	0.0132543
$528$	0	0
$529$	6.39227e20	2.39738
$530$	0	0
$531$	−1.18008e20	−0.430233
$532$	0	0
$533$	−1.28477e20	−0.455378
$534$	0	0
$535$	8.89410e19	0.306513
$536$	0	0
$537$	3.34081e18	0.0111955
$538$	0	0
$539$	8.77966e19	0.286129
$540$	0	0
$541$	−5.91289e20	−1.87422	−0.937110	−	0.349035i	$-0.886509\pi$
$541$	−5.91289e20	−1.87422	−0.937110	+	0.349035i	$0.886509\pi$
$542$	0	0
$543$	−1.22968e20	−0.379133
$544$	0	0
$545$	5.95038e19	0.178472
$546$	0	0
$547$	2.23972e20	0.653565	0.326783	−	0.945099i	$-0.394035\pi$
$547$	2.23972e20	0.653565	0.326783	+	0.945099i	$0.394035\pi$
$548$	0	0
$549$	1.13025e20	0.320909
$550$	0	0
$551$	−7.61449e19	−0.210380
$552$	0	0
$553$	−1.68019e20	−0.451773
$554$	0	0
$555$	−6.48975e19	−0.169836
$556$	0	0
$557$	−5.23943e20	−1.33466	−0.667329	−	0.744763i	$-0.732562\pi$
$557$	−5.23943e20	−1.33466	−0.667329	+	0.744763i	$0.732562\pi$
$558$	0	0
$559$	−2.25782e20	−0.559887
$560$	0	0
$561$	−4.37203e19	−0.105551
$562$	0	0
$563$	−1.19269e20	−0.280358	−0.140179	−	0.990126i	$-0.544768\pi$
$563$	−1.19269e20	−0.280358	−0.140179	+	0.990126i	$0.544768\pi$
$564$	0	0
$565$	1.55546e20	0.356036
$566$	0	0
$567$	2.78614e19	0.0621054
$568$	0	0
$569$	6.58606e20	1.42983	0.714914	−	0.699212i	$-0.246466\pi$
$569$	6.58606e20	1.42983	0.714914	+	0.699212i	$0.246466\pi$
$570$	0	0
$571$	−5.96008e20	−1.26032	−0.630161	−	0.776465i	$-0.717011\pi$
$571$	−5.96008e20	−1.26032	−0.630161	+	0.776465i	$0.717011\pi$
$572$	0	0
$573$	−4.88606e20	−1.00647
$574$	0	0
$575$	8.55640e20	1.71705
$576$	0	0
$577$	−7.82830e20	−1.53055	−0.765277	−	0.643701i	$-0.777398\pi$
$577$	−7.82830e20	−1.53055	−0.765277	+	0.643701i	$0.777398\pi$
$578$	0	0
$579$	−5.25314e19	−0.100076
$580$	0	0
$581$	−5.58761e20	−1.03730
$582$	0	0
$583$	3.62347e20	0.655558
$584$	0	0
$585$	−3.55390e19	−0.0626665
$586$	0	0
$587$	4.11166e20	0.706694	0.353347	−	0.935492i	$-0.385044\pi$
$587$	4.11166e20	0.706694	0.353347	+	0.935492i	$0.385044\pi$
$588$	0	0
$589$	1.85001e19	0.0309963
$590$	0	0
$591$	−3.82436e20	−0.624672
$592$	0	0
$593$	−6.22544e19	−0.0991425	−0.0495713	−	0.998771i	$-0.515785\pi$
$593$	−6.22544e19	−0.0991425	−0.0495713	+	0.998771i	$0.515785\pi$
$594$	0	0
$595$	−4.13705e19	−0.0642412
$596$	0	0
$597$	8.75269e19	0.132536
$598$	0	0
$599$	1.18581e21	1.75111	0.875553	−	0.483121i	$-0.160497\pi$
$599$	1.18581e21	1.75111	0.875553	+	0.483121i	$0.160497\pi$
$600$	0	0
$601$	8.46670e20	1.21943	0.609713	−	0.792622i	$-0.291285\pi$
$601$	8.46670e20	1.21943	0.609713	+	0.792622i	$0.291285\pi$
$602$	0	0
$603$	1.54897e20	0.217602
$604$	0	0
$605$	−1.57848e20	−0.216309
$606$	0	0
$607$	4.74226e19	0.0633972	0.0316986	−	0.999497i	$-0.489908\pi$
$607$	4.74226e19	0.0633972	0.0316986	+	0.999497i	$0.489908\pi$
$608$	0	0
$609$	−5.06652e19	−0.0660815
$610$	0	0
$611$	6.39571e20	0.813916
$612$	0	0
$613$	−1.17006e21	−1.45297	−0.726484	−	0.687183i	$-0.758847\pi$
$613$	−1.17006e21	−1.45297	−0.726484	+	0.687183i	$0.758847\pi$
$614$	0	0
$615$	−7.89838e19	−0.0957138
$616$	0	0
$617$	2.94504e20	0.348299	0.174149	−	0.984719i	$-0.444282\pi$
$617$	2.94504e20	0.348299	0.174149	+	0.984719i	$0.444282\pi$
$618$	0	0
$619$	1.72157e21	1.98722	0.993610	−	0.112871i	$-0.0360047\pi$
$619$	1.72157e21	1.98722	0.993610	+	0.112871i	$0.0360047\pi$
$620$	0	0
$621$	3.14831e20	0.354724
$622$	0	0
$623$	−3.92640e19	−0.0431851
$624$	0	0
$625$	7.44461e20	0.799359
$626$	0	0
$627$	−2.35463e20	−0.246840
$628$	0	0
$629$	−4.82605e20	−0.493983
$630$	0	0
$631$	−1.58818e21	−1.58738	−0.793690	−	0.608323i	$-0.791843\pi$
$631$	−1.58818e21	−1.58738	−0.793690	+	0.608323i	$0.791843\pi$
$632$	0	0
$633$	6.94906e20	0.678263
$634$	0	0
$635$	−2.60213e20	−0.248043
$636$	0	0
$637$	5.30717e20	0.494103
$638$	0	0
$639$	−3.56098e20	−0.323827
$640$	0	0
$641$	−1.19984e21	−1.06583	−0.532916	−	0.846168i	$-0.678904\pi$
$641$	−1.19984e21	−1.06583	−0.532916	+	0.846168i	$0.678904\pi$
$642$	0	0
$643$	−1.82792e21	−1.58626	−0.793132	−	0.609050i	$-0.791551\pi$
$643$	−1.82792e21	−1.58626	−0.793132	+	0.609050i	$0.791551\pi$
$644$	0	0
$645$	−1.38804e20	−0.117680
$646$	0	0
$647$	−7.13338e20	−0.590899	−0.295450	−	0.955358i	$-0.595469\pi$
$647$	−7.13338e20	−0.590899	−0.295450	+	0.955358i	$0.595469\pi$
$648$	0	0
$649$	6.63592e20	0.537114
$650$	0	0
$651$	1.23096e19	0.00973614
$652$	0	0
$653$	−2.11887e21	−1.63778	−0.818889	−	0.573951i	$-0.805410\pi$
$653$	−2.11887e21	−1.63778	−0.818889	+	0.573951i	$0.805410\pi$
$654$	0	0
$655$	2.73741e20	0.206790
$656$	0	0
$657$	8.44310e20	0.623393
$658$	0	0
$659$	−1.47772e19	−0.0106648	−0.00533238	−	0.999986i	$-0.501697\pi$
$659$	−1.47772e19	−0.0106648	−0.00533238	+	0.999986i	$0.501697\pi$
$660$	0	0
$661$	8.14857e19	0.0574871	0.0287436	−	0.999587i	$-0.490849\pi$
$661$	8.14857e19	0.0574871	0.0287436	+	0.999587i	$0.490849\pi$
$662$	0	0
$663$	−2.64283e20	−0.182271
$664$	0	0
$665$	−2.22807e20	−0.150234
$666$	0	0
$667$	−5.72512e20	−0.377434
$668$	0	0
$669$	1.13370e21	0.730805
$670$	0	0
$671$	−6.35569e20	−0.400630
$672$	0	0
$673$	2.43780e21	1.50275	0.751373	−	0.659878i	$-0.229392\pi$
$673$	2.43780e21	1.50275	0.751373	+	0.659878i	$0.229392\pi$
$674$	0	0
$675$	2.97376e20	0.179278
$676$	0	0
$677$	−7.99743e20	−0.471558	−0.235779	−	0.971807i	$-0.575764\pi$
$677$	−7.99743e20	−0.471558	−0.235779	+	0.971807i	$0.575764\pi$
$678$	0	0
$679$	5.82526e20	0.335963
$680$	0	0
$681$	2.25438e20	0.127181
$682$	0	0
$683$	−1.99607e21	−1.10159	−0.550794	−	0.834641i	$-0.685675\pi$
$683$	−1.99607e21	−1.10159	−0.550794	+	0.834641i	$0.685675\pi$
$684$	0	0
$685$	−1.33364e20	−0.0720042
$686$	0	0
$687$	−5.19939e20	−0.274648
$688$	0	0
$689$	2.19034e21	1.13205
$690$	0	0
$691$	1.99970e21	1.01130	0.505651	−	0.862738i	$-0.331253\pi$
$691$	1.99970e21	1.01130	0.505651	+	0.862738i	$0.331253\pi$
$692$	0	0
$693$	−1.56672e20	−0.0775340
$694$	0	0
$695$	7.46135e20	0.361353
$696$	0	0
$697$	−5.87357e20	−0.278392
$698$	0	0
$699$	1.30168e21	0.603844
$700$	0	0
$701$	1.55556e21	0.706320	0.353160	−	0.935563i	$-0.385107\pi$
$701$	1.55556e21	0.706320	0.353160	+	0.935563i	$0.385107\pi$
$702$	0	0
$703$	−2.59915e21	−1.15522
$704$	0	0
$705$	3.93189e20	0.171073
$706$	0	0
$707$	1.99447e21	0.849535
$708$	0	0
$709$	2.51259e21	1.04779	0.523897	−	0.851782i	$-0.324478\pi$
$709$	2.51259e21	1.04779	0.523897	+	0.851782i	$0.324478\pi$
$710$	0	0
$711$	−6.59856e20	−0.269418
$712$	0	0
$713$	1.39097e20	0.0556093
$714$	0	0
$715$	1.99845e20	0.0782345
$716$	0	0
$717$	−7.71331e20	−0.295697
$718$	0	0
$719$	1.18687e21	0.445592	0.222796	−	0.974865i	$-0.428482\pi$
$719$	1.18687e21	0.445592	0.222796	+	0.974865i	$0.428482\pi$
$720$	0	0
$721$	−1.12306e21	−0.412941
$722$	0	0
$723$	−1.25801e21	−0.453051
$724$	0	0
$725$	−5.40771e20	−0.190756
$726$	0	0
$727$	1.85436e21	0.640747	0.320374	−	0.947291i	$-0.396192\pi$
$727$	1.85436e21	0.640747	0.320374	+	0.947291i	$0.396192\pi$
$728$	0	0
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	−1.03220e21	−0.342283
$732$	0	0
$733$	−2.50403e21	−0.813504	−0.406752	−	0.913539i	$-0.633339\pi$
$733$	−2.50403e21	−0.813504	−0.406752	+	0.913539i	$0.633339\pi$
$734$	0	0
$735$	3.26269e20	0.103853
$736$	0	0
$737$	−8.71027e20	−0.271660
$738$	0	0
$739$	−4.99089e21	−1.52526	−0.762632	−	0.646833i	$-0.776093\pi$
$739$	−4.99089e21	−1.52526	−0.762632	+	0.646833i	$0.776093\pi$
$740$	0	0
$741$	−1.42334e21	−0.426257
$742$	0	0
$743$	8.88433e20	0.260741	0.130370	−	0.991465i	$-0.458383\pi$
$743$	8.88433e20	0.260741	0.130370	+	0.991465i	$0.458383\pi$
$744$	0	0
$745$	−7.90419e20	−0.227345
$746$	0	0
$747$	−2.19440e21	−0.618603
$748$	0	0
$749$	2.37014e21	0.654878
$750$	0	0
$751$	−1.58372e21	−0.428922	−0.214461	−	0.976733i	$-0.568799\pi$
$751$	−1.58372e21	−0.428922	−0.214461	+	0.976733i	$0.568799\pi$
$752$	0	0
$753$	−2.36407e21	−0.627622
$754$	0	0
$755$	1.21538e21	0.316308
$756$	0	0
$757$	−3.79261e21	−0.967651	−0.483826	−	0.875164i	$-0.660753\pi$
$757$	−3.79261e21	−0.967651	−0.483826	+	0.875164i	$0.660753\pi$
$758$	0	0
$759$	−1.77038e21	−0.442846
$760$	0	0
$761$	−1.82013e21	−0.446393	−0.223196	−	0.974773i	$-0.571649\pi$
$761$	−1.82013e21	−0.446393	−0.223196	+	0.974773i	$0.571649\pi$
$762$	0	0
$763$	1.58568e21	0.381314
$764$	0	0
$765$	−1.62473e20	−0.0383107
$766$	0	0
$767$	4.01131e21	0.927516
$768$	0	0
$769$	3.72176e20	0.0843920	0.0421960	−	0.999109i	$-0.486565\pi$
$769$	3.72176e20	0.0843920	0.0421960	+	0.999109i	$0.486565\pi$
$770$	0	0
$771$	5.13467e21	1.14184
$772$	0	0
$773$	4.88332e21	1.06505	0.532524	−	0.846415i	$-0.321244\pi$
$773$	4.88332e21	1.06505	0.532524	+	0.846415i	$0.321244\pi$
$774$	0	0
$775$	1.31386e20	0.0281051
$776$	0	0
$777$	−1.72942e21	−0.362863
$778$	0	0
$779$	−3.16331e21	−0.651044
$780$	0	0
$781$	2.00243e21	0.404273
$782$	0	0
$783$	−1.98976e20	−0.0394082
$784$	0	0
$785$	5.03760e20	0.0978816
$786$	0	0
$787$	7.74617e21	1.47665	0.738323	−	0.674447i	$-0.235618\pi$
$787$	7.74617e21	1.47665	0.738323	+	0.674447i	$0.235618\pi$
$788$	0	0
$789$	−2.97542e20	−0.0556507
$790$	0	0
$791$	4.14505e21	0.760687
$792$	0	0
$793$	−3.84192e21	−0.691830
$794$	0	0
$795$	1.34655e21	0.237941
$796$	0	0
$797$	3.93494e21	0.682339	0.341170	−	0.940002i	$-0.389177\pi$
$797$	3.93494e21	0.682339	0.341170	+	0.940002i	$0.389177\pi$
$798$	0	0
$799$	2.92392e21	0.497581
$800$	0	0
$801$	−1.54200e20	−0.0257538
$802$	0	0
$803$	−4.74778e21	−0.778259
$804$	0	0
$805$	−1.67523e21	−0.269528
$806$	0	0
$807$	5.03569e21	0.795258
$808$	0	0
$809$	−8.17751e21	−1.26767	−0.633836	−	0.773467i	$-0.718521\pi$
$809$	−8.17751e21	−1.26767	−0.633836	+	0.773467i	$0.718521\pi$
$810$	0	0
$811$	9.02522e21	1.37341	0.686707	−	0.726935i	$-0.259056\pi$
$811$	9.02522e21	1.37341	0.686707	+	0.726935i	$0.259056\pi$
$812$	0	0
$813$	2.31925e21	0.346472
$814$	0	0
$815$	−3.99507e20	−0.0585925
$816$	0	0
$817$	−5.55910e21	−0.800459
$818$	0	0
$819$	−9.47058e20	−0.133890
$820$	0	0
$821$	−7.55316e20	−0.104847	−0.0524234	−	0.998625i	$-0.516695\pi$
$821$	−7.55316e20	−0.104847	−0.0524234	+	0.998625i	$0.516695\pi$
$822$	0	0
$823$	1.96808e21	0.268252	0.134126	−	0.990964i	$-0.457177\pi$
$823$	1.96808e21	0.268252	0.134126	+	0.990964i	$0.457177\pi$
$824$	0	0
$825$	−1.67223e21	−0.223816
$826$	0	0
$827$	−1.06562e22	−1.40059	−0.700293	−	0.713856i	$-0.746947\pi$
$827$	−1.06562e22	−1.40059	−0.700293	+	0.713856i	$0.746947\pi$
$828$	0	0
$829$	4.82792e21	0.623162	0.311581	−	0.950220i	$-0.399141\pi$
$829$	4.82792e21	0.623162	0.311581	+	0.950220i	$0.399141\pi$
$830$	0	0
$831$	−1.46549e21	−0.185770
$832$	0	0
$833$	2.42627e21	0.302066
$834$	0	0
$835$	−1.27105e21	−0.155423
$836$	0	0
$837$	4.83431e19	0.00580622
$838$	0	0
$839$	−5.32810e21	−0.628576	−0.314288	−	0.949328i	$-0.601766\pi$
$839$	−5.32810e21	−0.628576	−0.314288	+	0.949328i	$0.601766\pi$
$840$	0	0
$841$	−8.26736e21	−0.958069
$842$	0	0
$843$	9.62880e21	1.09614
$844$	0	0
$845$	−1.13126e21	−0.126514
$846$	0	0
$847$	−4.20641e21	−0.462153
$848$	0	0
$849$	−5.87414e20	−0.0634069
$850$	0	0
$851$	−1.95423e22	−2.07254
$852$	0	0
$853$	1.36706e22	1.42452	0.712261	−	0.701915i	$-0.247671\pi$
$853$	1.36706e22	1.42452	0.712261	+	0.701915i	$0.247671\pi$
$854$	0	0
$855$	−8.75024e20	−0.0895930
$856$	0	0
$857$	−3.23695e20	−0.0325672	−0.0162836	−	0.999867i	$-0.505183\pi$
$857$	−3.23695e20	−0.0325672	−0.0162836	+	0.999867i	$0.505183\pi$
$858$	0	0
$859$	−1.01873e22	−1.00718	−0.503592	−	0.863942i	$-0.667988\pi$
$859$	−1.01873e22	−1.00718	−0.503592	+	0.863942i	$0.667988\pi$
$860$	0	0
$861$	−2.10480e21	−0.204497
$862$	0	0
$863$	−5.77968e21	−0.551853	−0.275926	−	0.961179i	$-0.588985\pi$
$863$	−5.77968e21	−0.551853	−0.275926	+	0.961179i	$0.588985\pi$
$864$	0	0
$865$	−4.38871e21	−0.411828
$866$	0	0
$867$	5.05190e21	0.465920
$868$	0	0
$869$	3.71055e21	0.336349
$870$	0	0
$871$	−5.26522e21	−0.469116
$872$	0	0
$873$	2.28773e21	0.200354
$874$	0	0
$875$	−3.28096e21	−0.282449
$876$	0	0
$877$	9.30066e21	0.787076	0.393538	−	0.919308i	$-0.371251\pi$
$877$	9.30066e21	0.787076	0.393538	+	0.919308i	$0.371251\pi$
$878$	0	0
$879$	−5.10560e21	−0.424747
$880$	0	0
$881$	1.97068e22	1.61175	0.805875	−	0.592086i	$-0.201695\pi$
$881$	1.97068e22	1.61175	0.805875	+	0.592086i	$0.201695\pi$
$882$	0	0
$883$	2.36048e22	1.89800	0.948998	−	0.315282i	$-0.102099\pi$
$883$	2.36048e22	1.89800	0.948998	+	0.315282i	$0.102099\pi$
$884$	0	0
$885$	2.46603e21	0.194950
$886$	0	0
$887$	1.02201e22	0.794378	0.397189	−	0.917737i	$-0.369986\pi$
$887$	1.02201e22	0.794378	0.397189	+	0.917737i	$0.369986\pi$
$888$	0	0
$889$	−6.93429e21	−0.529954
$890$	0	0
$891$	−6.15292e20	−0.0462380
$892$	0	0
$893$	1.57472e22	1.16364
$894$	0	0
$895$	−6.98133e19	−0.00507301
$896$	0	0
$897$	−1.07017e22	−0.764730
$898$	0	0
$899$	−8.79107e19	−0.00617794
$900$	0	0
$901$	1.00135e22	0.692071
$902$	0	0
$903$	−3.69891e21	−0.251429
$904$	0	0
$905$	2.56967e21	0.171796
$906$	0	0
$907$	−1.87262e22	−1.23139	−0.615694	−	0.787985i	$-0.711124\pi$
$907$	−1.87262e22	−1.23139	−0.615694	+	0.787985i	$0.711124\pi$
$908$	0	0
$909$	7.83280e21	0.506626
$910$	0	0
$911$	1.29391e22	0.823218	0.411609	−	0.911361i	$-0.364967\pi$
$911$	1.29391e22	0.823218	0.411609	+	0.911361i	$0.364967\pi$
$912$	0	0
$913$	1.23397e22	0.772279
$914$	0	0
$915$	−2.36189e21	−0.145413
$916$	0	0
$917$	7.29478e21	0.441817
$918$	0	0
$919$	−1.75633e22	−1.04650	−0.523252	−	0.852178i	$-0.675281\pi$
$919$	−1.75633e22	−1.04650	−0.523252	+	0.852178i	$0.675281\pi$
$920$	0	0
$921$	−1.75501e21	−0.102880
$922$	0	0
$923$	1.21044e22	0.698121
$924$	0	0
$925$	−1.84588e22	−1.04747
$926$	0	0
$927$	−4.41055e21	−0.246260
$928$	0	0
$929$	−3.04375e22	−1.67221	−0.836106	−	0.548568i	$-0.815173\pi$
$929$	−3.04375e22	−1.67221	−0.836106	+	0.548568i	$0.815173\pi$
$930$	0	0
$931$	1.30671e22	0.706409
$932$	0	0
$933$	−1.32812e22	−0.706521
$934$	0	0
$935$	9.13629e20	0.0478281
$936$	0	0
$937$	2.83243e21	0.145919	0.0729594	−	0.997335i	$-0.476756\pi$
$937$	2.83243e21	0.145919	0.0729594	+	0.997335i	$0.476756\pi$
$938$	0	0
$939$	−2.11526e21	−0.107244
$940$	0	0
$941$	1.86046e22	0.928322	0.464161	−	0.885751i	$-0.346356\pi$
$941$	1.86046e22	0.928322	0.464161	+	0.885751i	$0.346356\pi$
$942$	0	0
$943$	−2.37840e22	−1.16801
$944$	0	0
$945$	−5.82222e20	−0.0281417
$946$	0	0
$947$	−4.56986e20	−0.0217409	−0.0108705	−	0.999941i	$-0.503460\pi$
$947$	−4.56986e20	−0.0217409	−0.0108705	+	0.999941i	$0.503460\pi$
$948$	0	0
$949$	−2.86996e22	−1.34394
$950$	0	0
$951$	−9.88523e21	−0.455652
$952$	0	0
$953$	1.14890e22	0.521295	0.260648	−	0.965434i	$-0.416064\pi$
$953$	1.14890e22	0.521295	0.260648	+	0.965434i	$0.416064\pi$
$954$	0	0
$955$	1.02105e22	0.456058
$956$	0	0
$957$	1.11889e21	0.0491982
$958$	0	0
$959$	−3.55393e21	−0.153840
$960$	0	0
$961$	−2.34439e22	−0.999090
$962$	0	0
$963$	9.30817e21	0.390542
$964$	0	0
$965$	1.09776e21	0.0453472
$966$	0	0
$967$	1.55259e22	0.631478	0.315739	−	0.948846i	$-0.397748\pi$
$967$	1.55259e22	0.631478	0.315739	+	0.948846i	$0.397748\pi$
$968$	0	0
$969$	−6.50705e21	−0.260589
$970$	0	0
$971$	−1.48854e22	−0.586972	−0.293486	−	0.955963i	$-0.594815\pi$
$971$	−1.48854e22	−0.586972	−0.293486	+	0.955963i	$0.594815\pi$
$972$	0	0
$973$	1.98834e22	0.772046
$974$	0	0
$975$	−1.01084e22	−0.386497
$976$	0	0
$977$	−2.87451e22	−1.08232	−0.541158	−	0.840921i	$-0.682014\pi$
$977$	−2.87451e22	−1.08232	−0.541158	+	0.840921i	$0.682014\pi$
$978$	0	0
$979$	8.67109e20	0.0321517
$980$	0	0
$981$	6.22740e21	0.227399
$982$	0	0
$983$	4.57350e22	1.64474	0.822371	−	0.568952i	$-0.192651\pi$
$983$	4.57350e22	1.64474	0.822371	+	0.568952i	$0.192651\pi$
$984$	0	0
$985$	7.99180e21	0.283056
$986$	0	0
$987$	1.04779e22	0.365506
$988$	0	0
$989$	−4.17973e22	−1.43607
$990$	0	0
$991$	1.05966e22	0.358603	0.179301	−	0.983794i	$-0.442616\pi$
$991$	1.05966e22	0.358603	0.179301	+	0.983794i	$0.442616\pi$
$992$	0	0
$993$	2.43734e22	0.812451
$994$	0	0
$995$	−1.82906e21	−0.0600558
$996$	0	0
$997$	3.01439e21	0.0974958	0.0487479	−	0.998811i	$-0.484477\pi$
$997$	3.01439e21	0.0974958	0.0487479	+	0.998811i	$0.484477\pi$
$998$	0	0
$999$	−6.79189e21	−0.216396

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.16.a.a.1.1	✓	1
3.2	odd	2		36.16.a.a.1.1		1
4.3	odd	2		48.16.a.f.1.1		1
12.11	even	2		144.16.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.16.a.a.1.1	✓	1	1.1	even	1	trivial
36.16.a.a.1.1		1	3.2	odd	2
48.16.a.f.1.1		1	4.3	odd	2
144.16.a.g.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	12.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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