LMFDB - Embedded newform 16.38.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,38,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 38, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 38);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 38 $
Character orbit:	$[\chi]$	$=$	16.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:	$138.742460999$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 756643680 $ x^2 - x - 756643680
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$27507.7$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.16218e9 q^{3} -1.00278e13 q^{5} +3.32061e15 q^{7} +9.00385e17 q^{9} +O(q^{10})$	$q-1.16218e9 q^{3} -1.00278e13 q^{5} +3.32061e15 q^{7} +9.00385e17 q^{9} +2.20179e19 q^{11} -2.47054e20 q^{13} +1.16541e22 q^{15} +6.99239e22 q^{17} +8.30820e23 q^{19} -3.85916e24 q^{21} +1.05566e25 q^{23} +2.77971e25 q^{25} -5.23100e26 q^{27} -2.42517e26 q^{29} +1.93516e27 q^{31} -2.55889e28 q^{33} -3.32984e28 q^{35} -8.55331e28 q^{37} +2.87122e29 q^{39} +5.59412e29 q^{41} +3.15675e29 q^{43} -9.02888e30 q^{45} -1.56788e30 q^{47} -7.53564e30 q^{49} -8.12643e31 q^{51} +3.66286e31 q^{53} -2.20791e32 q^{55} -9.65565e32 q^{57} -8.04503e32 q^{59} +1.21434e33 q^{61} +2.98983e33 q^{63} +2.47741e33 q^{65} -1.45851e33 q^{67} -1.22687e34 q^{69} +2.91230e34 q^{71} +3.11895e34 q^{73} -3.23053e34 q^{75} +7.31130e34 q^{77} +8.04771e34 q^{79} +2.02509e35 q^{81} -2.48909e35 q^{83} -7.01182e35 q^{85} +2.81849e35 q^{87} -1.00339e36 q^{89} -8.20372e35 q^{91} -2.24901e36 q^{93} -8.33129e36 q^{95} +2.19237e36 q^{97} +1.98246e37 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 423071208 q^{3} - 13507530555540 q^{5} - 31\!\cdots\!56 q^{7}+ \cdots + 99\!\cdots\!06 q^{9}+O(q^{10}) $ 2 * q - 423071208 * q^3 - 13507530555540 * q^5 - 3106174162962256 * q^7 + 996387550079387706 * q^9	$ 2 q - 423071208 q^{3} - 13507530555540 q^{5} - 31\!\cdots\!56 q^{7}+ \cdots + 17\!\cdots\!88 q^{99}+O(q^{100}) $ 2 * q - 423071208 * q^3 - 13507530555540 * q^5 - 3106174162962256 * q^7 + 996387550079387706 * q^9 - 4002878661541032504 * q^11 - 417717305951174459972 * q^13 + 9082218417464777049360 * q^15 + 924888511508274921636 * q^17 + 1502800666863954423438200 * q^19 - 8609275073860282063267776 * q^21 - 4977294427767526272845808 * q^23 - 32853923797109170328639650 * q^25 - 784953884198822073793762320 * q^27 + 1174284943679786618741874780 * q^29 + 5529622970549622212915224256 * q^31 - 44821137705190940628425921184 * q^33 - 10934908922823267414476448480 * q^35 - 115976327866820065024925346644 * q^37 + 160983234676250224910988001488 * q^39 + 479903809827585250071775594644 * q^41 + 3191808379897050809392290082952 * q^43 - 9362941394381803786652901761220 * q^45 + 11214072446606967601132239964704 * q^47 + 15205846893444101799554822389554 * q^49 - 132262288085411921255362031678544 * q^51 + 53695242990369729463598618231628 * q^53 - 130245707009259764446614200122320 * q^55 - 468896146209528218251899751202400 * q^57 - 58132414930279452353014089908760 * q^59 + 949379410209793628730917745257884 * q^61 + 2372845660625082937916074533710832 * q^63 + 3071271884553430462753913899952040 * q^65 - 3280488273924335151706569871726696 * q^67 - 23749969120311478380705506108559168 * q^69 + 36632636242259358919270327956284976 * q^71 - 10722559667485328549057125275606572 * q^73 - 77133181870178710270426539420629400 * q^75 + 240343208006950469127380344317189312 * q^77 - 41775981111170500771329248250783520 * q^79 - 34258358055565679771691854070856398 * q^81 - 173069025965519133350295483333594888 * q^83 - 461083992329885864995977782015209320 * q^85 + 1329023474711207186565010011774976080 * q^87 - 770863557920750582452392321302299020 * q^89 + 276443184133973152010668401471277216 * q^91 + 407692825919874357767105668290130176 * q^93 - 10669607994591022203853874284760756400 * q^95 - 6502221928957848822710145562502442044 * q^97 + 17326559760258527834659445830047870888 * 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
$3$	−1.16218e9	−1.73193	−0.865967	−	0.500101i	$-0.833296\pi$
$3$	−1.16218e9	−1.73193	−0.865967	+	0.500101i	$0.833296\pi$
$4$	0	0
$5$	−1.00278e13	−1.17560	−0.587801	−	0.809006i	$-0.700006\pi$
$5$	−1.00278e13	−1.17560	−0.587801	+	0.809006i	$0.700006\pi$
$6$	0	0
$7$	3.32061e15	0.770734	0.385367	−	0.922763i	$-0.374075\pi$
$7$	3.32061e15	0.770734	0.385367	+	0.922763i	$0.374075\pi$
$8$	0	0
$9$	9.00385e17	1.99959
$10$	0	0
$11$	2.20179e19	1.19402	0.597010	−	0.802234i	$-0.296355\pi$
$11$	2.20179e19	1.19402	0.597010	+	0.802234i	$0.296355\pi$
$12$	0	0
$13$	−2.47054e20	−0.609313	−0.304656	−	0.952462i	$-0.598542\pi$
$13$	−2.47054e20	−0.609313	−0.304656	+	0.952462i	$0.598542\pi$
$14$	0	0
$15$	1.16541e22	2.03607
$16$	0	0
$17$	6.99239e22	1.20593	0.602963	−	0.797769i	$-0.293987\pi$
$17$	6.99239e22	1.20593	0.602963	+	0.797769i	$0.293987\pi$
$18$	0	0
$19$	8.30820e23	1.83048	0.915241	−	0.402907i	$-0.132000\pi$
$19$	8.30820e23	1.83048	0.915241	+	0.402907i	$0.132000\pi$
$20$	0	0
$21$	−3.85916e24	−1.33486
$22$	0	0
$23$	1.05566e25	0.678514	0.339257	−	0.940694i	$-0.389824\pi$
$23$	1.05566e25	0.678514	0.339257	+	0.940694i	$0.389824\pi$
$24$	0	0
$25$	2.77971e25	0.382040
$26$	0	0
$27$	−5.23100e26	−1.73123
$28$	0	0
$29$	−2.42517e26	−0.213983	−0.106991	−	0.994260i	$-0.534122\pi$
$29$	−2.42517e26	−0.213983	−0.106991	+	0.994260i	$0.534122\pi$
$30$	0	0
$31$	1.93516e27	0.497195	0.248597	−	0.968607i	$-0.420030\pi$
$31$	1.93516e27	0.497195	0.248597	+	0.968607i	$0.420030\pi$
$32$	0	0
$33$	−2.55889e28	−2.06796
$34$	0	0
$35$	−3.32984e28	−0.906077
$36$	0	0
$37$	−8.55331e28	−0.832535	−0.416268	−	0.909242i	$-0.636662\pi$
$37$	−8.55331e28	−0.832535	−0.416268	+	0.909242i	$0.636662\pi$
$38$	0	0
$39$	2.87122e29	1.05529
$40$	0	0
$41$	5.59412e29	0.815137	0.407569	−	0.913175i	$-0.366377\pi$
$41$	5.59412e29	0.815137	0.407569	+	0.913175i	$0.366377\pi$
$42$	0	0
$43$	3.15675e29	0.190579	0.0952893	−	0.995450i	$-0.469622\pi$
$43$	3.15675e29	0.190579	0.0952893	+	0.995450i	$0.469622\pi$
$44$	0	0
$45$	−9.02888e30	−2.35073
$46$	0	0
$47$	−1.56788e30	−0.182601	−0.0913003	−	0.995823i	$-0.529102\pi$
$47$	−1.56788e30	−0.182601	−0.0913003	+	0.995823i	$0.529102\pi$
$48$	0	0
$49$	−7.53564e30	−0.405969
$50$	0	0
$51$	−8.12643e31	−2.08858
$52$	0	0
$53$	3.66286e31	0.462080	0.231040	−	0.972944i	$-0.425787\pi$
$53$	3.66286e31	0.462080	0.231040	+	0.972944i	$0.425787\pi$
$54$	0	0
$55$	−2.20791e32	−1.40369
$56$	0	0
$57$	−9.65565e32	−3.17027
$58$	0	0
$59$	−8.04503e32	−1.39561	−0.697807	−	0.716286i	$-0.745841\pi$
$59$	−8.04503e32	−1.39561	−0.697807	+	0.716286i	$0.745841\pi$
$60$	0	0
$61$	1.21434e33	1.13694	0.568470	−	0.822704i	$-0.307536\pi$
$61$	1.21434e33	1.13694	0.568470	+	0.822704i	$0.307536\pi$
$62$	0	0
$63$	2.98983e33	1.54116
$64$	0	0
$65$	2.47741e33	0.716310
$66$	0	0
$67$	−1.45851e33	−0.240727	−0.120364	−	0.992730i	$-0.538406\pi$
$67$	−1.45851e33	−0.240727	−0.120364	+	0.992730i	$0.538406\pi$
$68$	0	0
$69$	−1.22687e34	−1.17514
$70$	0	0
$71$	2.91230e34	1.64421	0.822105	−	0.569336i	$-0.192800\pi$
$71$	2.91230e34	1.64421	0.822105	+	0.569336i	$0.192800\pi$
$72$	0	0
$73$	3.11895e34	1.05326	0.526629	−	0.850095i	$-0.323456\pi$
$73$	3.11895e34	1.05326	0.526629	+	0.850095i	$0.323456\pi$
$74$	0	0
$75$	−3.23053e34	−0.661669
$76$	0	0
$77$	7.31130e34	0.920272
$78$	0	0
$79$	8.04771e34	0.630334	0.315167	−	0.949036i	$-0.397939\pi$
$79$	8.04771e34	0.630334	0.315167	+	0.949036i	$0.397939\pi$
$80$	0	0
$81$	2.02509e35	0.998784
$82$	0	0
$83$	−2.48909e35	−0.781800	−0.390900	−	0.920433i	$-0.627836\pi$
$83$	−2.48909e35	−0.781800	−0.390900	+	0.920433i	$0.627836\pi$
$84$	0	0
$85$	−7.01182e35	−1.41769
$86$	0	0
$87$	2.81849e35	0.370604
$88$	0	0
$89$	−1.00339e36	−0.866470	−0.433235	−	0.901281i	$-0.642628\pi$
$89$	−1.00339e36	−0.866470	−0.433235	+	0.901281i	$0.642628\pi$
$90$	0	0
$91$	−8.20372e35	−0.469618
$92$	0	0
$93$	−2.24901e36	−0.861108
$94$	0	0
$95$	−8.33129e36	−2.15192
$96$	0	0
$97$	2.19237e36	0.385156	0.192578	−	0.981282i	$-0.438315\pi$
$97$	2.19237e36	0.385156	0.192578	+	0.981282i	$0.438315\pi$
$98$	0	0
$99$	1.98246e37	2.38756
$100$	0	0
$101$	7.59676e36	0.631950	0.315975	−	0.948767i	$-0.397668\pi$
$101$	7.59676e36	0.631950	0.315975	+	0.948767i	$0.397668\pi$
$102$	0	0
$103$	1.64045e37	0.949458	0.474729	−	0.880132i	$-0.342546\pi$
$103$	1.64045e37	0.949458	0.474729	+	0.880132i	$0.342546\pi$
$104$	0	0
$105$	3.86989e37	1.56926
$106$	0	0
$107$	−4.91223e37	−1.40501	−0.702504	−	0.711680i	$-0.747935\pi$
$107$	−4.91223e37	−1.40501	−0.702504	+	0.711680i	$0.747935\pi$
$108$	0	0
$109$	3.28303e37	0.666630	0.333315	−	0.942816i	$-0.391833\pi$
$109$	3.28303e37	0.666630	0.333315	+	0.942816i	$0.391833\pi$
$110$	0	0
$111$	9.94052e37	1.44190
$112$	0	0
$113$	−1.12641e38	−1.17421	−0.587104	−	0.809512i	$-0.699732\pi$
$113$	−1.12641e38	−1.17421	−0.587104	+	0.809512i	$0.699732\pi$
$114$	0	0
$115$	−1.05859e38	−0.797662
$116$	0	0
$117$	−2.22444e38	−1.21838
$118$	0	0
$119$	2.32190e38	0.929448
$120$	0	0
$121$	1.44750e38	0.425684
$122$	0	0
$123$	−6.50139e38	−1.41176
$124$	0	0
$125$	4.50875e38	0.726475
$126$	0	0
$127$	2.82173e38	0.338959	0.169479	−	0.985534i	$-0.445791\pi$
$127$	2.82173e38	0.338959	0.169479	+	0.985534i	$0.445791\pi$
$128$	0	0
$129$	−3.66872e38	−0.330069
$130$	0	0
$131$	−3.02759e38	−0.204917	−0.102459	−	0.994737i	$-0.532671\pi$
$131$	−3.02759e38	−0.204917	−0.102459	+	0.994737i	$0.532671\pi$
$132$	0	0
$133$	2.75883e39	1.41081
$134$	0	0
$135$	5.24554e39	2.03524
$136$	0	0
$137$	2.26804e39	0.670377	0.335189	−	0.942151i	$-0.391200\pi$
$137$	2.26804e39	0.670377	0.335189	+	0.942151i	$0.391200\pi$
$138$	0	0
$139$	4.94024e39	1.11680	0.558398	−	0.829573i	$-0.311416\pi$
$139$	4.94024e39	1.11680	0.558398	+	0.829573i	$0.311416\pi$
$140$	0	0
$141$	1.82216e39	0.316252
$142$	0	0
$143$	−5.43962e39	−0.727532
$144$	0	0
$145$	2.43191e39	0.251559
$146$	0	0
$147$	8.75779e39	0.703111
$148$	0	0
$149$	−5.44741e38	−0.0340600	−0.0170300	−	0.999855i	$-0.505421\pi$
$149$	−5.44741e38	−0.0340600	−0.0170300	+	0.999855i	$0.505421\pi$
$150$	0	0
$151$	3.86430e40	1.88798	0.943989	−	0.329976i	$-0.107041\pi$
$151$	3.86430e40	1.88798	0.943989	+	0.329976i	$0.107041\pi$
$152$	0	0
$153$	6.29584e40	2.41136
$154$	0	0
$155$	−1.94054e40	−0.584503
$156$	0	0
$157$	−8.04588e40	−1.91175	−0.955873	−	0.293781i	$-0.905086\pi$
$157$	−8.04588e40	−1.91175	−0.955873	+	0.293781i	$0.905086\pi$
$158$	0	0
$159$	−4.25691e40	−0.800291
$160$	0	0
$161$	3.50544e40	0.522954
$162$	0	0
$163$	−3.58379e40	−0.425472	−0.212736	−	0.977110i	$-0.568237\pi$
$163$	−3.58379e40	−0.425472	−0.212736	+	0.977110i	$0.568237\pi$
$164$	0	0
$165$	2.56600e41	2.43110
$166$	0	0
$167$	2.31369e41	1.75409	0.877043	−	0.480411i	$-0.159513\pi$
$167$	2.31369e41	1.75409	0.877043	+	0.480411i	$0.159513\pi$
$168$	0	0
$169$	−1.03365e41	−0.628738
$170$	0	0
$171$	7.48058e41	3.66022
$172$	0	0
$173$	2.42959e41	0.958693	0.479346	−	0.877626i	$-0.340874\pi$
$173$	2.42959e41	0.958693	0.479346	+	0.877626i	$0.340874\pi$
$174$	0	0
$175$	9.23034e40	0.294452
$176$	0	0
$177$	9.34980e41	2.41711
$178$	0	0
$179$	−1.97163e41	−0.414042	−0.207021	−	0.978336i	$-0.566377\pi$
$179$	−1.97163e41	−0.414042	−0.207021	+	0.978336i	$0.566377\pi$
$180$	0	0
$181$	−8.74626e41	−1.49544	−0.747719	−	0.664015i	$-0.768851\pi$
$181$	−8.74626e41	−1.49544	−0.747719	+	0.664015i	$0.768851\pi$
$182$	0	0
$183$	−1.41128e42	−1.96910
$184$	0	0
$185$	8.57709e41	0.978730
$186$	0	0
$187$	1.53958e42	1.43990
$188$	0	0
$189$	−1.73701e42	−1.33432
$190$	0	0
$191$	−6.05957e41	−0.383111	−0.191555	−	0.981482i	$-0.561353\pi$
$191$	−6.05957e41	−0.383111	−0.191555	+	0.981482i	$0.561353\pi$
$192$	0	0
$193$	−1.72991e41	−0.0902016	−0.0451008	−	0.998982i	$-0.514361\pi$
$193$	−1.72991e41	−0.0902016	−0.0451008	+	0.998982i	$0.514361\pi$
$194$	0	0
$195$	−2.87920e42	−1.24060
$196$	0	0
$197$	−7.91100e41	−0.282232	−0.141116	−	0.989993i	$-0.545069\pi$
$197$	−7.91100e41	−0.282232	−0.141116	+	0.989993i	$0.545069\pi$
$198$	0	0
$199$	2.54348e42	0.752744	0.376372	−	0.926469i	$-0.377172\pi$
$199$	2.54348e42	0.752744	0.376372	+	0.926469i	$0.377172\pi$
$200$	0	0
$201$	1.69505e42	0.416923
$202$	0	0
$203$	−8.05304e41	−0.164924
$204$	0	0
$205$	−5.60967e42	−0.958277
$206$	0	0
$207$	9.50500e42	1.35675
$208$	0	0
$209$	1.82929e43	2.18563
$210$	0	0
$211$	−1.20546e43	−1.20761	−0.603807	−	0.797131i	$-0.706350\pi$
$211$	−1.20546e43	−1.20761	−0.603807	+	0.797131i	$0.706350\pi$
$212$	0	0
$213$	−3.38463e43	−2.84766
$214$	0	0
$215$	−3.16553e42	−0.224045
$216$	0	0
$217$	6.42593e42	0.383205
$218$	0	0
$219$	−3.62479e43	−1.82417
$220$	0	0
$221$	−1.72750e43	−0.734786
$222$	0	0
$223$	−2.30674e43	−0.830536	−0.415268	−	0.909699i	$-0.636312\pi$
$223$	−2.30674e43	−0.830536	−0.415268	+	0.909699i	$0.636312\pi$
$224$	0	0
$225$	2.50281e43	0.763926
$226$	0	0
$227$	3.57525e43	0.926463	0.463231	−	0.886237i	$-0.346690\pi$
$227$	3.57525e43	0.926463	0.463231	+	0.886237i	$0.346690\pi$
$228$	0	0
$229$	−6.86230e43	−1.51187	−0.755933	−	0.654649i	$-0.772817\pi$
$229$	−6.86230e43	−1.51187	−0.755933	+	0.654649i	$0.772817\pi$
$230$	0	0
$231$	−8.49707e43	−1.59385
$232$	0	0
$233$	−7.73057e43	−1.23631	−0.618154	−	0.786057i	$-0.712119\pi$
$233$	−7.73057e43	−1.23631	−0.618154	+	0.786057i	$0.712119\pi$
$234$	0	0
$235$	1.57224e43	0.214666
$236$	0	0
$237$	−9.35291e43	−1.09170
$238$	0	0
$239$	8.60047e43	0.859332	0.429666	−	0.902988i	$-0.358631\pi$
$239$	8.60047e43	0.859332	0.429666	+	0.902988i	$0.358631\pi$
$240$	0	0
$241$	1.68453e44	1.44266	0.721328	−	0.692593i	$-0.243532\pi$
$241$	1.68453e44	1.44266	0.721328	+	0.692593i	$0.243532\pi$
$242$	0	0
$243$	1.90988e41	0.00140375
$244$	0	0
$245$	7.55659e43	0.477258
$246$	0	0
$247$	−2.05258e44	−1.11534
$248$	0	0
$249$	2.89278e44	1.35403
$250$	0	0
$251$	1.93198e44	0.779899	0.389949	−	0.920836i	$-0.372492\pi$
$251$	1.93198e44	0.779899	0.389949	+	0.920836i	$0.372492\pi$
$252$	0	0
$253$	2.32434e44	0.810159
$254$	0	0
$255$	8.14902e44	2.45534
$256$	0	0
$257$	−6.84920e43	−0.178598	−0.0892991	−	0.996005i	$-0.528463\pi$
$257$	−6.84920e43	−0.178598	−0.0892991	+	0.996005i	$0.528463\pi$
$258$	0	0
$259$	−2.84023e44	−0.641663
$260$	0	0
$261$	−2.18358e44	−0.427879
$262$	0	0
$263$	−3.42473e44	−0.582700	−0.291350	−	0.956617i	$-0.594104\pi$
$263$	−3.42473e44	−0.582700	−0.291350	+	0.956617i	$0.594104\pi$
$264$	0	0
$265$	−3.67304e44	−0.543222
$266$	0	0
$267$	1.16612e45	1.50067
$268$	0	0
$269$	1.17310e45	1.31497	0.657486	−	0.753467i	$-0.271620\pi$
$269$	1.17310e45	1.31497	0.657486	+	0.753467i	$0.271620\pi$
$270$	0	0
$271$	−1.48009e45	−1.44662	−0.723311	−	0.690522i	$-0.757381\pi$
$271$	−1.48009e45	−1.44662	−0.723311	+	0.690522i	$0.757381\pi$
$272$	0	0
$273$	9.53422e44	0.813348
$274$	0	0
$275$	6.12034e44	0.456164
$276$	0	0
$277$	−2.54630e45	−1.65972	−0.829858	−	0.557975i	$-0.811578\pi$
$277$	−2.54630e45	−1.65972	−0.829858	+	0.557975i	$0.811578\pi$
$278$	0	0
$279$	1.74239e45	0.994188
$280$	0	0
$281$	−2.93334e45	−1.46655	−0.733273	−	0.679934i	$-0.762008\pi$
$281$	−2.93334e45	−1.46655	−0.733273	+	0.679934i	$0.762008\pi$
$282$	0	0
$283$	−4.02101e44	−0.176314	−0.0881568	−	0.996107i	$-0.528098\pi$
$283$	−4.02101e44	−0.176314	−0.0881568	+	0.996107i	$0.528098\pi$
$284$	0	0
$285$	9.68249e45	3.72698
$286$	0	0
$287$	1.85759e45	0.628254
$288$	0	0
$289$	1.52725e45	0.454256
$290$	0	0
$291$	−2.54793e45	−0.667064
$292$	0	0
$293$	5.50714e45	1.27021	0.635105	−	0.772426i	$-0.280957\pi$
$293$	5.50714e45	1.27021	0.635105	+	0.772426i	$0.280957\pi$
$294$	0	0
$295$	8.06739e45	1.64069
$296$	0	0
$297$	−1.15176e46	−2.06713
$298$	0	0
$299$	−2.60805e45	−0.413427
$300$	0	0
$301$	1.04824e45	0.146885
$302$	0	0
$303$	−8.82882e45	−1.09450
$304$	0	0
$305$	−1.21771e46	−1.33659
$306$	0	0
$307$	−6.23539e45	−0.606462	−0.303231	−	0.952917i	$-0.598065\pi$
$307$	−6.23539e45	−0.606462	−0.303231	+	0.952917i	$0.598065\pi$
$308$	0	0
$309$	−1.90651e46	−1.64440
$310$	0	0
$311$	−1.00705e46	−0.770872	−0.385436	−	0.922735i	$-0.625949\pi$
$311$	−1.00705e46	−0.770872	−0.385436	+	0.922735i	$0.625949\pi$
$312$	0	0
$313$	−2.62521e46	−1.78482	−0.892408	−	0.451230i	$-0.850985\pi$
$313$	−2.62521e46	−1.78482	−0.892408	+	0.451230i	$0.850985\pi$
$314$	0	0
$315$	−2.99814e46	−1.81179
$316$	0	0
$317$	1.16596e46	0.626741	0.313371	−	0.949631i	$-0.398542\pi$
$317$	1.16596e46	0.626741	0.313371	+	0.949631i	$0.398542\pi$
$318$	0	0
$319$	−5.33971e45	−0.255500
$320$	0	0
$321$	5.70891e46	2.43338
$322$	0	0
$323$	5.80942e46	2.20742
$324$	0	0
$325$	−6.86739e45	−0.232782
$326$	0	0
$327$	−3.81548e46	−1.15456
$328$	0	0
$329$	−5.20632e45	−0.140736
$330$	0	0
$331$	−2.95352e46	−0.713712	−0.356856	−	0.934159i	$-0.616151\pi$
$331$	−2.95352e46	−0.713712	−0.356856	+	0.934159i	$0.616151\pi$
$332$	0	0
$333$	−7.70128e46	−1.66473
$334$	0	0
$335$	1.46256e46	0.282999
$336$	0	0
$337$	−1.12287e46	−0.194615	−0.0973074	−	0.995254i	$-0.531023\pi$
$337$	−1.12287e46	−0.194615	−0.0973074	+	0.995254i	$0.531023\pi$
$338$	0	0
$339$	1.30909e47	2.03365
$340$	0	0
$341$	4.26083e46	0.593660
$342$	0	0
$343$	−8.66606e46	−1.08363
$344$	0	0
$345$	1.23028e47	1.38150
$346$	0	0
$347$	3.38484e46	0.341541	0.170770	−	0.985311i	$-0.445374\pi$
$347$	3.38484e46	0.341541	0.170770	+	0.985311i	$0.445374\pi$
$348$	0	0
$349$	8.62093e46	0.782138	0.391069	−	0.920361i	$-0.372105\pi$
$349$	8.62093e46	0.782138	0.391069	+	0.920361i	$0.372105\pi$
$350$	0	0
$351$	1.29234e47	1.05486
$352$	0	0
$353$	−1.39430e47	−1.02453	−0.512263	−	0.858828i	$-0.671193\pi$
$353$	−1.39430e47	−1.02453	−0.512263	+	0.858828i	$0.671193\pi$
$354$	0	0
$355$	−2.92040e47	−1.93294
$356$	0	0
$357$	−2.69847e47	−1.60974
$358$	0	0
$359$	2.55899e47	1.37664	0.688321	−	0.725407i	$-0.258348\pi$
$359$	2.55899e47	1.37664	0.688321	+	0.725407i	$0.258348\pi$
$360$	0	0
$361$	4.84254e47	2.35066
$362$	0	0
$363$	−1.68225e47	−0.737257
$364$	0	0
$365$	−3.12762e47	−1.23821
$366$	0	0
$367$	3.41199e47	1.22091	0.610457	−	0.792049i	$-0.290986\pi$
$367$	3.41199e47	1.22091	0.610457	+	0.792049i	$0.290986\pi$
$368$	0	0
$369$	5.03687e47	1.62994
$370$	0	0
$371$	1.21629e47	0.356141
$372$	0	0
$373$	4.46224e47	1.18288	0.591438	−	0.806351i	$-0.298560\pi$
$373$	4.46224e47	1.18288	0.591438	+	0.806351i	$0.298560\pi$
$374$	0	0
$375$	−5.23999e47	−1.25821
$376$	0	0
$377$	5.99148e46	0.130382
$378$	0	0
$379$	7.46655e47	1.47331	0.736655	−	0.676268i	$-0.236404\pi$
$379$	7.46655e47	1.47331	0.736655	+	0.676268i	$0.236404\pi$
$380$	0	0
$381$	−3.27937e47	−0.587054
$382$	0	0
$383$	6.01186e47	0.976858	0.488429	−	0.872604i	$-0.337570\pi$
$383$	6.01186e47	0.976858	0.488429	+	0.872604i	$0.337570\pi$
$384$	0	0
$385$	−7.33162e47	−1.08187
$386$	0	0
$387$	2.84229e47	0.381080
$388$	0	0
$389$	5.11575e47	0.623508	0.311754	−	0.950163i	$-0.399084\pi$
$389$	5.11575e47	0.623508	0.311754	+	0.950163i	$0.399084\pi$
$390$	0	0
$391$	7.38158e47	0.818237
$392$	0	0
$393$	3.51861e47	0.354903
$394$	0	0
$395$	−8.07007e47	−0.741022
$396$	0	0
$397$	1.07709e48	0.900795	0.450398	−	0.892828i	$-0.351282\pi$
$397$	1.07709e48	0.900795	0.450398	+	0.892828i	$0.351282\pi$
$398$	0	0
$399$	−3.20627e48	−2.44344
$400$	0	0
$401$	−1.61629e48	−1.12292	−0.561458	−	0.827505i	$-0.689759\pi$
$401$	−1.61629e48	−1.12292	−0.561458	+	0.827505i	$0.689759\pi$
$402$	0	0
$403$	−4.78090e47	−0.302947
$404$	0	0
$405$	−2.03072e48	−1.17417
$406$	0	0
$407$	−1.88326e48	−0.994064
$408$	0	0
$409$	−7.70085e47	−0.371242	−0.185621	−	0.982621i	$-0.559430\pi$
$409$	−7.70085e47	−0.371242	−0.185621	+	0.982621i	$0.559430\pi$
$410$	0	0
$411$	−2.63587e48	−1.16105
$412$	0	0
$413$	−2.67144e48	−1.07565
$414$	0	0
$415$	2.49601e48	0.919086
$416$	0	0
$417$	−5.74146e48	−1.93422
$418$	0	0
$419$	2.95178e47	0.0910175	0.0455087	−	0.998964i	$-0.485509\pi$
$419$	2.95178e47	0.0910175	0.0455087	+	0.998964i	$0.485509\pi$
$420$	0	0
$421$	6.03774e48	1.70473	0.852365	−	0.522947i	$-0.175167\pi$
$421$	6.03774e48	1.70473	0.852365	+	0.522947i	$0.175167\pi$
$422$	0	0
$423$	−1.41169e48	−0.365127
$424$	0	0
$425$	1.94368e48	0.460712
$426$	0	0
$427$	4.03234e48	0.876278
$428$	0	0
$429$	6.32184e48	1.26004
$430$	0	0
$431$	7.52239e48	1.37571	0.687854	−	0.725849i	$-0.258553\pi$
$431$	7.52239e48	1.37571	0.687854	+	0.725849i	$0.258553\pi$
$432$	0	0
$433$	−9.12185e48	−1.53129	−0.765644	−	0.643265i	$-0.777579\pi$
$433$	−9.12185e48	−1.53129	−0.765644	+	0.643265i	$0.777579\pi$
$434$	0	0
$435$	−2.82632e48	−0.435683
$436$	0	0
$437$	8.77063e48	1.24201
$438$	0	0
$439$	−9.21085e48	−1.19869	−0.599344	−	0.800492i	$-0.704572\pi$
$439$	−9.21085e48	−1.19869	−0.599344	+	0.800492i	$0.704572\pi$
$440$	0	0
$441$	−6.78498e48	−0.811773
$442$	0	0
$443$	1.02129e48	0.112377	0.0561887	−	0.998420i	$-0.482105\pi$
$443$	1.02129e48	0.112377	0.0561887	+	0.998420i	$0.482105\pi$
$444$	0	0
$445$	1.00618e49	1.01862
$446$	0	0
$447$	6.33089e47	0.0589896
$448$	0	0
$449$	1.62214e49	1.39165	0.695825	−	0.718211i	$-0.255039\pi$
$449$	1.62214e49	1.39165	0.695825	+	0.718211i	$0.255039\pi$
$450$	0	0
$451$	1.23171e49	0.973290
$452$	0	0
$453$	−4.49102e49	−3.26985
$454$	0	0
$455$	8.22652e48	0.552084
$456$	0	0
$457$	1.55439e49	0.961858	0.480929	−	0.876759i	$-0.340299\pi$
$457$	1.55439e49	0.961858	0.480929	+	0.876759i	$0.340299\pi$
$458$	0	0
$459$	−3.65772e49	−2.08774
$460$	0	0
$461$	5.47057e48	0.288115	0.144057	−	0.989569i	$-0.453985\pi$
$461$	5.47057e48	0.288115	0.144057	+	0.989569i	$0.453985\pi$
$462$	0	0
$463$	−9.95251e48	−0.483821	−0.241910	−	0.970299i	$-0.577774\pi$
$463$	−9.95251e48	−0.483821	−0.241910	+	0.970299i	$0.577774\pi$
$464$	0	0
$465$	2.25527e49	1.01232
$466$	0	0
$467$	8.97489e48	0.372106	0.186053	−	0.982540i	$-0.440430\pi$
$467$	8.97489e48	0.372106	0.186053	+	0.982540i	$0.440430\pi$
$468$	0	0
$469$	−4.84314e48	−0.185537
$470$	0	0
$471$	9.35079e49	3.31102
$472$	0	0
$473$	6.95051e48	0.227555
$474$	0	0
$475$	2.30944e49	0.699318
$476$	0	0
$477$	3.29799e49	0.923972
$478$	0	0
$479$	2.92109e49	0.757421	0.378711	−	0.925515i	$-0.376368\pi$
$479$	2.92109e49	0.757421	0.378711	+	0.925515i	$0.376368\pi$
$480$	0	0
$481$	2.11313e49	0.507274
$482$	0	0
$483$	−4.07396e49	−0.905722
$484$	0	0
$485$	−2.19846e49	−0.452790
$486$	0	0
$487$	−7.31257e48	−0.139568	−0.0697838	−	0.997562i	$-0.522231\pi$
$487$	−7.31257e48	−0.139568	−0.0697838	+	0.997562i	$0.522231\pi$
$488$	0	0
$489$	4.16501e49	0.736890
$490$	0	0
$491$	1.60707e49	0.263649	0.131825	−	0.991273i	$-0.457916\pi$
$491$	1.60707e49	0.263649	0.131825	+	0.991273i	$0.457916\pi$
$492$	0	0
$493$	−1.69577e49	−0.258047
$494$	0	0
$495$	−1.98797e50	−2.80682
$496$	0	0
$497$	9.67064e49	1.26725
$498$	0	0
$499$	−2.11587e49	−0.257412	−0.128706	−	0.991683i	$-0.541082\pi$
$499$	−2.11587e49	−0.257412	−0.128706	+	0.991683i	$0.541082\pi$
$500$	0	0
$501$	−2.68894e50	−3.03796
$502$	0	0
$503$	3.29392e48	0.0345705	0.0172853	−	0.999851i	$-0.494498\pi$
$503$	3.29392e48	0.0345705	0.0172853	+	0.999851i	$0.494498\pi$
$504$	0	0
$505$	−7.61787e49	−0.742922
$506$	0	0
$507$	1.20129e50	1.08893
$508$	0	0
$509$	2.11519e50	1.78267	0.891333	−	0.453349i	$-0.149771\pi$
$509$	2.11519e50	1.78267	0.891333	+	0.453349i	$0.149771\pi$
$510$	0	0
$511$	1.03568e50	0.811783
$512$	0	0
$513$	−4.34602e50	−3.16899
$514$	0	0
$515$	−1.64501e50	−1.11618
$516$	0	0
$517$	−3.45214e49	−0.218029
$518$	0	0
$519$	−2.82362e50	−1.66039
$520$	0	0
$521$	2.89219e50	1.58391	0.791954	−	0.610581i	$-0.209064\pi$
$521$	2.89219e50	1.58391	0.791954	+	0.610581i	$0.209064\pi$
$522$	0	0
$523$	4.61612e49	0.235503	0.117751	−	0.993043i	$-0.462431\pi$
$523$	4.61612e49	0.235503	0.117751	+	0.993043i	$0.462431\pi$
$524$	0	0
$525$	−1.07273e50	−0.509971
$526$	0	0
$527$	1.35314e50	0.599580
$528$	0	0
$529$	−1.30622e50	−0.539619
$530$	0	0
$531$	−7.24363e50	−2.79066
$532$	0	0
$533$	−1.38205e50	−0.496674
$534$	0	0
$535$	4.92588e50	1.65173
$536$	0	0
$537$	2.29140e50	0.717094
$538$	0	0
$539$	−1.65919e50	−0.484735
$540$	0	0
$541$	4.12287e50	1.12473	0.562367	−	0.826888i	$-0.309891\pi$
$541$	4.12287e50	1.12473	0.562367	+	0.826888i	$0.309891\pi$
$542$	0	0
$543$	1.01648e51	2.59000
$544$	0	0
$545$	−3.29215e50	−0.783691
$546$	0	0
$547$	3.85047e50	0.856540	0.428270	−	0.903651i	$-0.359123\pi$
$547$	3.85047e50	0.856540	0.428270	+	0.903651i	$0.359123\pi$
$548$	0	0
$549$	1.09337e51	2.27342
$550$	0	0
$551$	−2.01488e50	−0.391691
$552$	0	0
$553$	2.67233e50	0.485820
$554$	0	0
$555$	−9.96815e50	−1.69510
$556$	0	0
$557$	−5.73633e50	−0.912668	−0.456334	−	0.889809i	$-0.650838\pi$
$557$	−5.73633e50	−0.912668	−0.456334	+	0.889809i	$0.650838\pi$
$558$	0	0
$559$	−7.79889e49	−0.116122
$560$	0	0
$561$	−1.78927e51	−2.49381
$562$	0	0
$563$	6.88670e50	0.898682	0.449341	−	0.893360i	$-0.351659\pi$
$563$	6.88670e50	0.898682	0.449341	+	0.893360i	$0.351659\pi$
$564$	0	0
$565$	1.12954e51	1.38040
$566$	0	0
$567$	6.72454e50	0.769797
$568$	0	0
$569$	−4.67962e50	−0.501919	−0.250960	−	0.967998i	$-0.580746\pi$
$569$	−4.67962e50	−0.501919	−0.250960	+	0.967998i	$0.580746\pi$
$570$	0	0
$571$	−8.40413e50	−0.844743	−0.422371	−	0.906423i	$-0.638802\pi$
$571$	−8.40413e50	−0.844743	−0.422371	+	0.906423i	$0.638802\pi$
$572$	0	0
$573$	7.04233e50	0.663523
$574$	0	0
$575$	2.93443e50	0.259220
$576$	0	0
$577$	2.43834e50	0.201996	0.100998	−	0.994887i	$-0.467796\pi$
$577$	2.43834e50	0.201996	0.100998	+	0.994887i	$0.467796\pi$
$578$	0	0
$579$	2.01048e50	0.156223
$580$	0	0
$581$	−8.26531e50	−0.602560
$582$	0	0
$583$	8.06486e50	0.551732
$584$	0	0
$585$	2.23062e51	1.43233
$586$	0	0
$587$	1.26888e51	0.764921	0.382460	−	0.923972i	$-0.375077\pi$
$587$	1.26888e51	0.764921	0.382460	+	0.923972i	$0.375077\pi$
$588$	0	0
$589$	1.60777e51	0.910105
$590$	0	0
$591$	9.19403e50	0.488807
$592$	0	0
$593$	3.44917e51	1.72267	0.861335	−	0.508037i	$-0.169629\pi$
$593$	3.44917e51	1.72267	0.861335	+	0.508037i	$0.169629\pi$
$594$	0	0
$595$	−2.32836e51	−1.09266
$596$	0	0
$597$	−2.95599e51	−1.30370
$598$	0	0
$599$	1.03421e51	0.428759	0.214379	−	0.976750i	$-0.431227\pi$
$599$	1.03421e51	0.428759	0.214379	+	0.976750i	$0.431227\pi$
$600$	0	0
$601$	3.31491e50	0.129209	0.0646046	−	0.997911i	$-0.479421\pi$
$601$	3.31491e50	0.129209	0.0646046	+	0.997911i	$0.479421\pi$
$602$	0	0
$603$	−1.31322e51	−0.481357
$604$	0	0
$605$	−1.45152e51	−0.500436
$606$	0	0
$607$	1.31827e51	0.427577	0.213788	−	0.976880i	$-0.431420\pi$
$607$	1.31827e51	0.427577	0.213788	+	0.976880i	$0.431420\pi$
$608$	0	0
$609$	9.35911e50	0.285637
$610$	0	0
$611$	3.87351e50	0.111261
$612$	0	0
$613$	−4.00784e51	−1.08366	−0.541829	−	0.840489i	$-0.682268\pi$
$613$	−4.00784e51	−1.08366	−0.541829	+	0.840489i	$0.682268\pi$
$614$	0	0
$615$	6.51946e51	1.65967
$616$	0	0
$617$	−3.89908e51	−0.934731	−0.467366	−	0.884064i	$-0.654797\pi$
$617$	−3.89908e51	−0.934731	−0.467366	+	0.884064i	$0.654797\pi$
$618$	0	0
$619$	−6.80067e51	−1.53559	−0.767794	−	0.640697i	$-0.778646\pi$
$619$	−6.80067e51	−1.53559	−0.767794	+	0.640697i	$0.778646\pi$
$620$	0	0
$621$	−5.52215e51	−1.17466
$622$	0	0
$623$	−3.33186e51	−0.667818
$624$	0	0
$625$	−6.54378e51	−1.23609
$626$	0	0
$627$	−2.12597e52	−3.78537
$628$	0	0
$629$	−5.98081e51	−1.00398
$630$	0	0
$631$	−8.58407e51	−1.35878	−0.679392	−	0.733776i	$-0.737756\pi$
$631$	−8.58407e51	−1.35878	−0.679392	+	0.733776i	$0.737756\pi$
$632$	0	0
$633$	1.40097e52	2.09151
$634$	0	0
$635$	−2.82957e51	−0.398480
$636$	0	0
$637$	1.86171e51	0.247362
$638$	0	0
$639$	2.62220e52	3.28775
$640$	0	0
$641$	4.76661e51	0.564074	0.282037	−	0.959404i	$-0.408990\pi$
$641$	4.76661e51	0.564074	0.282037	+	0.959404i	$0.408990\pi$
$642$	0	0
$643$	1.36823e52	1.52846	0.764231	−	0.644942i	$-0.223119\pi$
$643$	1.36823e52	1.52846	0.764231	+	0.644942i	$0.223119\pi$
$644$	0	0
$645$	3.67892e51	0.388030
$646$	0	0
$647$	−2.16321e51	−0.215461	−0.107731	−	0.994180i	$-0.534358\pi$
$647$	−2.16321e51	−0.215461	−0.107731	+	0.994180i	$0.534358\pi$
$648$	0	0
$649$	−1.77135e52	−1.66639
$650$	0	0
$651$	−7.46811e51	−0.663685
$652$	0	0
$653$	−8.05478e49	−0.00676332	−0.00338166	−	0.999994i	$-0.501076\pi$
$653$	−8.05478e49	−0.00676332	−0.00338166	+	0.999994i	$0.501076\pi$
$654$	0	0
$655$	3.03601e51	0.240901
$656$	0	0
$657$	2.80825e52	2.10609
$658$	0	0
$659$	−1.39232e52	−0.987098	−0.493549	−	0.869718i	$-0.664301\pi$
$659$	−1.39232e52	−0.987098	−0.493549	+	0.869718i	$0.664301\pi$
$660$	0	0
$661$	−1.05998e51	−0.0710511	−0.0355255	−	0.999369i	$-0.511311\pi$
$661$	−1.05998e51	−0.0710511	−0.0355255	+	0.999369i	$0.511311\pi$
$662$	0	0
$663$	2.00767e52	1.27260
$664$	0	0
$665$	−2.76650e52	−1.65856
$666$	0	0
$667$	−2.56015e51	−0.145190
$668$	0	0
$669$	2.68085e52	1.43843
$670$	0	0
$671$	2.67372e52	1.35753
$672$	0	0
$673$	1.75123e51	0.0841522	0.0420761	−	0.999114i	$-0.486603\pi$
$673$	1.75123e51	0.0841522	0.0420761	+	0.999114i	$0.486603\pi$
$674$	0	0
$675$	−1.45407e52	−0.661400
$676$	0	0
$677$	3.00931e52	1.29592	0.647958	−	0.761676i	$-0.275623\pi$
$677$	3.00931e52	1.29592	0.647958	+	0.761676i	$0.275623\pi$
$678$	0	0
$679$	7.28000e51	0.296853
$680$	0	0
$681$	−4.15509e52	−1.60457
$682$	0	0
$683$	−3.45300e52	−1.26303	−0.631515	−	0.775364i	$-0.717567\pi$
$683$	−3.45300e52	−1.26303	−0.631515	+	0.775364i	$0.717567\pi$
$684$	0	0
$685$	−2.27434e52	−0.788097
$686$	0	0
$687$	7.97525e52	2.61845
$688$	0	0
$689$	−9.04925e51	−0.281551
$690$	0	0
$691$	−5.56197e52	−1.64015	−0.820077	−	0.572253i	$-0.806069\pi$
$691$	−5.56197e52	−1.64015	−0.820077	+	0.572253i	$0.806069\pi$
$692$	0	0
$693$	6.58299e52	1.84017
$694$	0	0
$695$	−4.95397e52	−1.31291
$696$	0	0
$697$	3.91163e52	0.982994
$698$	0	0
$699$	8.98433e52	2.14120
$700$	0	0
$701$	2.61308e52	0.590704	0.295352	−	0.955389i	$-0.404563\pi$
$701$	2.61308e52	0.590704	0.295352	+	0.955389i	$0.404563\pi$
$702$	0	0
$703$	−7.10627e52	−1.52394
$704$	0	0
$705$	−1.82723e52	−0.371787
$706$	0	0
$707$	2.52259e52	0.487066
$708$	0	0
$709$	−1.30761e52	−0.239620	−0.119810	−	0.992797i	$-0.538229\pi$
$709$	−1.30761e52	−0.239620	−0.119810	+	0.992797i	$0.538229\pi$
$710$	0	0
$711$	7.24604e52	1.26041
$712$	0	0
$713$	2.04287e52	0.337353
$714$	0	0
$715$	5.45474e52	0.855288
$716$	0	0
$717$	−9.99532e52	−1.48831
$718$	0	0
$719$	5.28485e52	0.747391	0.373695	−	0.927551i	$-0.378091\pi$
$719$	5.28485e52	0.747391	0.373695	+	0.927551i	$0.378091\pi$
$720$	0	0
$721$	5.44732e52	0.731780
$722$	0	0
$723$	−1.95773e53	−2.49859
$724$	0	0
$725$	−6.74126e51	−0.0817500
$726$	0	0
$727$	8.62636e52	0.994125	0.497062	−	0.867715i	$-0.334412\pi$
$727$	8.62636e52	0.994125	0.497062	+	0.867715i	$0.334412\pi$
$728$	0	0
$729$	−9.14085e52	−1.00122
$730$	0	0
$731$	2.20732e52	0.229823
$732$	0	0
$733$	−4.61809e52	−0.457129	−0.228565	−	0.973529i	$-0.573403\pi$
$733$	−4.61809e52	−0.457129	−0.228565	+	0.973529i	$0.573403\pi$
$734$	0	0
$735$	−8.78213e52	−0.826579
$736$	0	0
$737$	−3.21133e52	−0.287433
$738$	0	0
$739$	−9.79613e52	−0.833935	−0.416968	−	0.908921i	$-0.636907\pi$
$739$	−9.79613e52	−0.833935	−0.416968	+	0.908921i	$0.636907\pi$
$740$	0	0
$741$	2.38547e53	1.93169
$742$	0	0
$743$	−3.68344e52	−0.283766	−0.141883	−	0.989883i	$-0.545316\pi$
$743$	−3.68344e52	−0.283766	−0.141883	+	0.989883i	$0.545316\pi$
$744$	0	0
$745$	5.46255e51	0.0400409
$746$	0	0
$747$	−2.24114e53	−1.56328
$748$	0	0
$749$	−1.63116e53	−1.08289
$750$	0	0
$751$	2.91686e53	1.84322	0.921610	−	0.388116i	$-0.126874\pi$
$751$	2.91686e53	1.84322	0.921610	+	0.388116i	$0.126874\pi$
$752$	0	0
$753$	−2.24531e53	−1.35073
$754$	0	0
$755$	−3.87504e53	−2.21951
$756$	0	0
$757$	−2.14424e53	−1.16950	−0.584749	−	0.811214i	$-0.698807\pi$
$757$	−2.14424e53	−1.16950	−0.584749	+	0.811214i	$0.698807\pi$
$758$	0	0
$759$	−2.70131e53	−1.40314
$760$	0	0
$761$	3.02873e53	1.49846	0.749228	−	0.662312i	$-0.230425\pi$
$761$	3.02873e53	1.49846	0.749228	+	0.662312i	$0.230425\pi$
$762$	0	0
$763$	1.09017e53	0.513794
$764$	0	0
$765$	−6.31334e53	−2.83480
$766$	0	0
$767$	1.98756e53	0.850366
$768$	0	0
$769$	2.09535e53	0.854318	0.427159	−	0.904176i	$-0.359514\pi$
$769$	2.09535e53	0.854318	0.427159	+	0.904176i	$0.359514\pi$
$770$	0	0
$771$	7.96002e52	0.309320
$772$	0	0
$773$	−2.74843e53	−1.01804	−0.509018	−	0.860756i	$-0.669991\pi$
$773$	−2.74843e53	−1.01804	−0.509018	+	0.860756i	$0.669991\pi$
$774$	0	0
$775$	5.37919e52	0.189948
$776$	0	0
$777$	3.30086e53	1.11132
$778$	0	0
$779$	4.64771e53	1.49209
$780$	0	0
$781$	6.41229e53	1.96322
$782$	0	0
$783$	1.26861e53	0.370454
$784$	0	0
$785$	8.06824e53	2.24745
$786$	0	0
$787$	4.09272e53	1.08762	0.543812	−	0.839207i	$-0.316980\pi$
$787$	4.09272e53	1.08762	0.543812	+	0.839207i	$0.316980\pi$
$788$	0	0
$789$	3.98016e53	1.00920
$790$	0	0
$791$	−3.74037e53	−0.905002
$792$	0	0
$793$	−3.00007e53	−0.692752
$794$	0	0
$795$	4.26875e53	0.940824
$796$	0	0
$797$	3.09131e51	0.00650374	0.00325187	−	0.999995i	$-0.498965\pi$
$797$	3.09131e51	0.00650374	0.00325187	+	0.999995i	$0.498965\pi$
$798$	0	0
$799$	−1.09632e53	−0.220203
$800$	0	0
$801$	−9.03436e53	−1.73259
$802$	0	0
$803$	6.86728e53	1.25761
$804$	0	0
$805$	−3.51518e53	−0.614786
$806$	0	0
$807$	−1.36335e54	−2.27744
$808$	0	0
$809$	−2.71214e53	−0.432778	−0.216389	−	0.976307i	$-0.569428\pi$
$809$	−2.71214e53	−0.432778	−0.216389	+	0.976307i	$0.569428\pi$
$810$	0	0
$811$	9.27510e52	0.141395	0.0706973	−	0.997498i	$-0.477478\pi$
$811$	9.27510e52	0.141395	0.0706973	+	0.997498i	$0.477478\pi$
$812$	0	0
$813$	1.72013e54	2.50545
$814$	0	0
$815$	3.59375e53	0.500186
$816$	0	0
$817$	2.62269e53	0.348850
$818$	0	0
$819$	−7.38651e53	−0.939046
$820$	0	0
$821$	−9.54561e53	−1.15999	−0.579996	−	0.814619i	$-0.696946\pi$
$821$	−9.54561e53	−1.15999	−0.579996	+	0.814619i	$0.696946\pi$
$822$	0	0
$823$	−1.19456e54	−1.38775	−0.693873	−	0.720097i	$-0.744097\pi$
$823$	−1.19456e54	−1.38775	−0.693873	+	0.720097i	$0.744097\pi$
$824$	0	0
$825$	−7.11296e53	−0.790046
$826$	0	0
$827$	1.05798e54	1.12364	0.561820	−	0.827260i	$-0.310102\pi$
$827$	1.05798e54	1.12364	0.561820	+	0.827260i	$0.310102\pi$
$828$	0	0
$829$	1.68391e54	1.71025	0.855124	−	0.518424i	$-0.173481\pi$
$829$	1.68391e54	1.71025	0.855124	+	0.518424i	$0.173481\pi$
$830$	0	0
$831$	2.95926e54	2.87452
$832$	0	0
$833$	−5.26921e53	−0.489568
$834$	0	0
$835$	−2.32012e54	−2.06211
$836$	0	0
$837$	−1.01228e54	−0.860759
$838$	0	0
$839$	3.78194e53	0.307693	0.153847	−	0.988095i	$-0.450834\pi$
$839$	3.78194e53	0.307693	0.153847	+	0.988095i	$0.450834\pi$
$840$	0	0
$841$	−1.22566e54	−0.954211
$842$	0	0
$843$	3.40908e54	2.53996
$844$	0	0
$845$	1.03652e54	0.739145
$846$	0	0
$847$	4.80657e53	0.328090
$848$	0	0
$849$	4.67315e53	0.305364
$850$	0	0
$851$	−9.02939e53	−0.564887
$852$	0	0
$853$	2.34797e54	1.40649	0.703244	−	0.710948i	$-0.251734\pi$
$853$	2.34797e54	1.40649	0.703244	+	0.710948i	$0.251734\pi$
$854$	0	0
$855$	−7.50137e54	−4.30296
$856$	0	0
$857$	5.17381e53	0.284227	0.142114	−	0.989850i	$-0.454610\pi$
$857$	5.17381e53	0.284227	0.142114	+	0.989850i	$0.454610\pi$
$858$	0	0
$859$	2.26122e54	1.18979	0.594893	−	0.803805i	$-0.297194\pi$
$859$	2.26122e54	1.18979	0.594893	+	0.803805i	$0.297194\pi$
$860$	0	0
$861$	−2.15886e54	−1.08809
$862$	0	0
$863$	1.13614e54	0.548572	0.274286	−	0.961648i	$-0.411558\pi$
$863$	1.13614e54	0.548572	0.274286	+	0.961648i	$0.411558\pi$
$864$	0	0
$865$	−2.43634e54	−1.12704
$866$	0	0
$867$	−1.77495e54	−0.786741
$868$	0	0
$869$	1.77194e54	0.752632
$870$	0	0
$871$	3.60331e53	0.146678
$872$	0	0
$873$	1.97397e54	0.770155
$874$	0	0
$875$	1.49718e54	0.559919
$876$	0	0
$877$	−3.53444e54	−1.26715	−0.633575	−	0.773682i	$-0.718413\pi$
$877$	−3.53444e54	−1.26715	−0.633575	+	0.773682i	$0.718413\pi$
$878$	0	0
$879$	−6.40031e54	−2.19992
$880$	0	0
$881$	−2.74692e54	−0.905300	−0.452650	−	0.891688i	$-0.649521\pi$
$881$	−2.74692e54	−0.905300	−0.452650	+	0.891688i	$0.649521\pi$
$882$	0	0
$883$	−9.56723e53	−0.302352	−0.151176	−	0.988507i	$-0.548306\pi$
$883$	−9.56723e53	−0.302352	−0.151176	+	0.988507i	$0.548306\pi$
$884$	0	0
$885$	−9.37578e54	−2.84156
$886$	0	0
$887$	−2.37072e54	−0.689118	−0.344559	−	0.938765i	$-0.611972\pi$
$887$	−2.37072e54	−0.689118	−0.344559	+	0.938765i	$0.611972\pi$
$888$	0	0
$889$	9.36988e53	0.261247
$890$	0	0
$891$	4.45883e54	1.19257
$892$	0	0
$893$	−1.30263e54	−0.334247
$894$	0	0
$895$	1.97711e54	0.486749
$896$	0	0
$897$	3.03103e54	0.716029
$898$	0	0
$899$	−4.69309e53	−0.106391
$900$	0	0
$901$	2.56121e54	0.557234
$902$	0	0
$903$	−1.21824e54	−0.254396
$904$	0	0
$905$	8.77057e54	1.75804
$906$	0	0
$907$	3.89376e54	0.749264	0.374632	−	0.927174i	$-0.377769\pi$
$907$	3.89376e54	0.749264	0.374632	+	0.927174i	$0.377769\pi$
$908$	0	0
$909$	6.84001e54	1.26364
$910$	0	0
$911$	−4.15122e54	−0.736353	−0.368177	−	0.929756i	$-0.620018\pi$
$911$	−4.15122e54	−0.736353	−0.368177	+	0.929756i	$0.620018\pi$
$912$	0	0
$913$	−5.48046e54	−0.933486
$914$	0	0
$915$	1.41520e55	2.31488
$916$	0	0
$917$	−1.00535e54	−0.157937
$918$	0	0
$919$	−2.84350e54	−0.429058	−0.214529	−	0.976718i	$-0.568822\pi$
$919$	−2.84350e54	−0.429058	−0.214529	+	0.976718i	$0.568822\pi$
$920$	0	0
$921$	7.24666e54	1.05035
$922$	0	0
$923$	−7.19497e54	−1.00184
$924$	0	0
$925$	−2.37757e54	−0.318062
$926$	0	0
$927$	1.47704e55	1.89853
$928$	0	0
$929$	−3.00016e53	−0.0370555	−0.0185278	−	0.999828i	$-0.505898\pi$
$929$	−3.00016e53	−0.0370555	−0.0185278	+	0.999828i	$0.505898\pi$
$930$	0	0
$931$	−6.26076e54	−0.743118
$932$	0	0
$933$	1.17037e55	1.33510
$934$	0	0
$935$	−1.54386e55	−1.69275
$936$	0	0
$937$	5.72228e54	0.603096	0.301548	−	0.953451i	$-0.402497\pi$
$937$	5.72228e54	0.603096	0.301548	+	0.953451i	$0.402497\pi$
$938$	0	0
$939$	3.05097e55	3.09118
$940$	0	0
$941$	−1.79388e55	−1.74737	−0.873684	−	0.486494i	$-0.838275\pi$
$941$	−1.79388e55	−1.74737	−0.873684	+	0.486494i	$0.838275\pi$
$942$	0	0
$943$	5.90549e54	0.553082
$944$	0	0
$945$	1.74184e55	1.56863
$946$	0	0
$947$	1.24204e53	0.0107563	0.00537813	−	0.999986i	$-0.498288\pi$
$947$	1.24204e53	0.0107563	0.00537813	+	0.999986i	$0.498288\pi$
$948$	0	0
$949$	−7.70549e54	−0.641764
$950$	0	0
$951$	−1.35506e55	−1.08547
$952$	0	0
$953$	8.24909e54	0.635605	0.317803	−	0.948157i	$-0.397055\pi$
$953$	8.24909e54	0.635605	0.317803	+	0.948157i	$0.397055\pi$
$954$	0	0
$955$	6.07641e54	0.450386
$956$	0	0
$957$	6.20572e54	0.442509
$958$	0	0
$959$	7.53127e54	0.516683
$960$	0	0
$961$	−1.14041e55	−0.752798
$962$	0	0
$963$	−4.42290e55	−2.80945
$964$	0	0
$965$	1.73472e54	0.106041
$966$	0	0
$967$	5.68031e54	0.334182	0.167091	−	0.985942i	$-0.446563\pi$
$967$	5.68031e54	0.334182	0.167091	+	0.985942i	$0.446563\pi$
$968$	0	0
$969$	−6.75160e55	−3.82311
$970$	0	0
$971$	2.27277e55	1.23880	0.619399	−	0.785077i	$-0.287377\pi$
$971$	2.27277e55	1.23880	0.619399	+	0.785077i	$0.287377\pi$
$972$	0	0
$973$	1.64046e55	0.860753
$974$	0	0
$975$	7.98116e54	0.403163
$976$	0	0
$977$	−7.73374e54	−0.376132	−0.188066	−	0.982156i	$-0.560222\pi$
$977$	−7.73374e54	−0.376132	−0.188066	+	0.982156i	$0.560222\pi$
$978$	0	0
$979$	−2.20925e55	−1.03458
$980$	0	0
$981$	2.95599e55	1.33299
$982$	0	0
$983$	−1.64593e55	−0.714777	−0.357389	−	0.933956i	$-0.616333\pi$
$983$	−1.64593e55	−0.714777	−0.357389	+	0.933956i	$0.616333\pi$
$984$	0	0
$985$	7.93299e54	0.331793
$986$	0	0
$987$	6.05070e54	0.243746
$988$	0	0
$989$	3.33245e54	0.129310
$990$	0	0
$991$	−3.05470e54	−0.114184	−0.0570921	−	0.998369i	$-0.518183\pi$
$991$	−3.05470e54	−0.114184	−0.0570921	+	0.998369i	$0.518183\pi$
$992$	0	0
$993$	3.43254e55	1.23610
$994$	0	0
$995$	−2.55055e55	−0.884927
$996$	0	0
$997$	1.64945e55	0.551416	0.275708	−	0.961241i	$-0.411088\pi$
$997$	1.64945e55	0.551416	0.275708	+	0.961241i	$0.411088\pi$
$998$	0	0
$999$	4.47424e55	1.44131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.38.a.a.1.1		2
4.3	odd	2		2.38.a.a.1.2	✓	2
12.11	even	2		18.38.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.38.a.a.1.2	✓	2	4.3	odd	2
16.38.a.a.1.1		2	1.1	even	1	trivial
18.38.a.f.1.2		2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 38 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-1.16218e9 q^{3} -1.00278e13 q^{5} +3.32061e15 q^{7} +9.00385e17 q^{9} +O(q^{10})\)	\(q-1.16218e9 q^{3} -1.00278e13 q^{5} +3.32061e15 q^{7} +9.00385e17 q^{9} +2.20179e19 q^{11} -2.47054e20 q^{13} +1.16541e22 q^{15} +6.99239e22 q^{17} +8.30820e23 q^{19} -3.85916e24 q^{21} +1.05566e25 q^{23} +2.77971e25 q^{25} -5.23100e26 q^{27} -2.42517e26 q^{29} +1.93516e27 q^{31} -2.55889e28 q^{33} -3.32984e28 q^{35} -8.55331e28 q^{37} +2.87122e29 q^{39} +5.59412e29 q^{41} +3.15675e29 q^{43} -9.02888e30 q^{45} -1.56788e30 q^{47} -7.53564e30 q^{49} -8.12643e31 q^{51} +3.66286e31 q^{53} -2.20791e32 q^{55} -9.65565e32 q^{57} -8.04503e32 q^{59} +1.21434e33 q^{61} +2.98983e33 q^{63} +2.47741e33 q^{65} -1.45851e33 q^{67} -1.22687e34 q^{69} +2.91230e34 q^{71} +3.11895e34 q^{73} -3.23053e34 q^{75} +7.31130e34 q^{77} +8.04771e34 q^{79} +2.02509e35 q^{81} -2.48909e35 q^{83} -7.01182e35 q^{85} +2.81849e35 q^{87} -1.00339e36 q^{89} -8.20372e35 q^{91} -2.24901e36 q^{93} -8.33129e36 q^{95} +2.19237e36 q^{97} +1.98246e37 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 423071208 q^{3} - 13507530555540 q^{5} - 31\!\cdots\!56 q^{7}+ \cdots + 99\!\cdots\!06 q^{9}+O(q^{10}) \) 2 * q - 423071208 * q^3 - 13507530555540 * q^5 - 3106174162962256 * q^7 + 996387550079387706 * q^9	\( 2 q - 423071208 q^{3} - 13507530555540 q^{5} - 31\!\cdots\!56 q^{7}+ \cdots + 17\!\cdots\!88 q^{99}+O(q^{100}) \) 2 * q - 423071208 * q^3 - 13507530555540 * q^5 - 3106174162962256 * q^7 + 996387550079387706 * q^9 - 4002878661541032504 * q^11 - 417717305951174459972 * q^13 + 9082218417464777049360 * q^15 + 924888511508274921636 * q^17 + 1502800666863954423438200 * q^19 - 8609275073860282063267776 * q^21 - 4977294427767526272845808 * q^23 - 32853923797109170328639650 * q^25 - 784953884198822073793762320 * q^27 + 1174284943679786618741874780 * q^29 + 5529622970549622212915224256 * q^31 - 44821137705190940628425921184 * q^33 - 10934908922823267414476448480 * q^35 - 115976327866820065024925346644 * q^37 + 160983234676250224910988001488 * q^39 + 479903809827585250071775594644 * q^41 + 3191808379897050809392290082952 * q^43 - 9362941394381803786652901761220 * q^45 + 11214072446606967601132239964704 * q^47 + 15205846893444101799554822389554 * q^49 - 132262288085411921255362031678544 * q^51 + 53695242990369729463598618231628 * q^53 - 130245707009259764446614200122320 * q^55 - 468896146209528218251899751202400 * q^57 - 58132414930279452353014089908760 * q^59 + 949379410209793628730917745257884 * q^61 + 2372845660625082937916074533710832 * q^63 + 3071271884553430462753913899952040 * q^65 - 3280488273924335151706569871726696 * q^67 - 23749969120311478380705506108559168 * q^69 + 36632636242259358919270327956284976 * q^71 - 10722559667485328549057125275606572 * q^73 - 77133181870178710270426539420629400 * q^75 + 240343208006950469127380344317189312 * q^77 - 41775981111170500771329248250783520 * q^79 - 34258358055565679771691854070856398 * q^81 - 173069025965519133350295483333594888 * q^83 - 461083992329885864995977782015209320 * q^85 + 1329023474711207186565010011774976080 * q^87 - 770863557920750582452392321302299020 * q^89 + 276443184133973152010668401471277216 * q^91 + 407692825919874357767105668290130176 * q^93 - 10669607994591022203853874284760756400 * q^95 - 6502221928957848822710145562502442044 * q^97 + 17326559760258527834659445830047870888 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more