LMFDB - Embedded newform 384.9.g.b.127.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,9,Mod(127,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("384.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	384.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$156.433386263$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.24
Character	$\chi$	$=$	384.127
Dual form			384.9.g.b.127.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} +966.582 q^{5} -1235.54i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} +966.582 q^{5} -1235.54i q^{7} -2187.00 q^{9} -6450.78i q^{11} -25316.0 q^{13} +45202.6i q^{15} -43990.0 q^{17} +79270.3i q^{19} +57780.4 q^{21} +170286. i q^{23} +543656. q^{25} -102276. i q^{27} +347166. q^{29} -1.63275e6i q^{31} +301673. q^{33} -1.19425e6i q^{35} -2.64294e6 q^{37} -1.18391e6i q^{39} +2.82589e6 q^{41} +1.10024e6i q^{43} -2.11392e6 q^{45} -6.17143e6i q^{47} +4.23824e6 q^{49} -2.05721e6i q^{51} -1.03479e7 q^{53} -6.23520e6i q^{55} -3.70711e6 q^{57} -1.46491e7i q^{59} +5.57255e6 q^{61} +2.70212e6i q^{63} -2.44700e7 q^{65} -3.50959e6i q^{67} -7.96349e6 q^{69} +1.43161e6i q^{71} +2.72673e7 q^{73} +2.54243e7i q^{75} -7.97019e6 q^{77} -4.66244e7i q^{79} +4.78297e6 q^{81} +2.20176e7i q^{83} -4.25199e7 q^{85} +1.62354e7i q^{87} -5.90052e7 q^{89} +3.12789e7i q^{91} +7.63562e7 q^{93} +7.66213e7i q^{95} -1.52888e8 q^{97} +1.41078e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 1344 q^{5} - 69984 q^{9}+O(q^{10}) $ 32 * q + 1344 * q^5 - 69984 * q^9	$ 32 q + 1344 q^{5} - 69984 q^{9} + 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} + 275520 q^{29} - 2421440 q^{37} - 4374720 q^{41} - 2939328 q^{45} - 14219104 q^{49} + 6224448 q^{53} + 3100032 q^{57} + 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} - 154517760 q^{77} + 153055008 q^{81} + 384830848 q^{85} - 182669760 q^{89} - 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) $ 32 * q + 1344 * q^5 - 69984 * q^9 + 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 + 275520 * q^29 - 2421440 * q^37 - 4374720 * q^41 - 2939328 * q^45 - 14219104 * q^49 + 6224448 * q^53 + 3100032 * q^57 + 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 - 154517760 * q^77 + 153055008 * q^81 + 384830848 * q^85 - 182669760 * q^89 - 149817600 * q^93 - 149408192 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	46.7654i	0.577350i
$3$	46.7654i	0.577350i
$4$	0	0
$5$	966.582	1.54653	0.773266	−	0.634082i	$-0.218622\pi$
$5$	966.582	1.54653	0.773266	+	0.634082i	$0.218622\pi$
$6$	0	0
$7$	− 1235.54i	− 0.514594i	−0.966332	−	0.257297i	$-0.917168\pi$
$7$	− 1235.54i	− 0.514594i	0.966332	−	0.257297i	$-0.0828318\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	− 6450.78i	− 0.440597i	−0.975433	−	0.220298i	$-0.929297\pi$
$11$	− 6450.78i	− 0.440597i	0.975433	−	0.220298i	$-0.0707031\pi$
$12$	0	0
$13$	−25316.0	−0.886383	−0.443191	−	0.896427i	$-0.646154\pi$
$13$	−25316.0	−0.886383	−0.443191	+	0.896427i	$0.646154\pi$
$14$	0	0
$15$	45202.6i	0.892890i
$16$	0	0
$17$	−43990.0	−0.526694	−0.263347	−	0.964701i	$-0.584826\pi$
$17$	−43990.0	−0.526694	−0.263347	+	0.964701i	$0.584826\pi$
$18$	0	0
$19$	79270.3i	0.608270i	0.952629	+	0.304135i	$0.0983674\pi$
$19$	79270.3i	0.608270i	−0.952629	+	0.304135i	$0.901633\pi$
$20$	0	0
$21$	57780.4	0.297101
$22$	0	0
$23$	170286.i	0.608510i	0.952591	+	0.304255i	$0.0984075\pi$
$23$	170286.i	0.608510i	−0.952591	+	0.304255i	$0.901593\pi$
$24$	0	0
$25$	543656.	1.39176
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	347166.	0.490846	0.245423	−	0.969416i	$-0.421073\pi$
$29$	347166.	0.490846	0.245423	+	0.969416i	$0.421073\pi$
$30$	0	0
$31$	− 1.63275e6i	− 1.76796i	−0.467522	−	0.883982i	$-0.654853\pi$
$31$	− 1.63275e6i	− 1.76796i	0.467522	−	0.883982i	$-0.345147\pi$
$32$	0	0
$33$	301673.	0.254379
$34$	0	0
$35$	− 1.19425e6i	− 0.795835i
$36$	0	0
$37$	−2.64294e6	−1.41020	−0.705100	−	0.709108i	$-0.749098\pi$
$37$	−2.64294e6	−1.41020	−0.705100	+	0.709108i	$0.749098\pi$
$38$	0	0
$39$	− 1.18391e6i	− 0.511753i
$40$	0	0
$41$	2.82589e6	1.00004	0.500022	−	0.866013i	$-0.333325\pi$
$41$	2.82589e6	1.00004	0.500022	+	0.866013i	$0.333325\pi$
$42$	0	0
$43$	1.10024e6i	0.321819i	0.986969	+	0.160910i	$0.0514428\pi$
$43$	1.10024e6i	0.321819i	−0.986969	+	0.160910i	$0.948557\pi$
$44$	0	0
$45$	−2.11392e6	−0.515510
$46$	0	0
$47$	− 6.17143e6i	− 1.26472i	−0.774675	−	0.632360i	$-0.782086\pi$
$47$	− 6.17143e6i	− 1.26472i	0.774675	−	0.632360i	$-0.217914\pi$
$48$	0	0
$49$	4.23824e6	0.735193
$50$	0	0
$51$	− 2.05721e6i	− 0.304087i
$52$	0	0
$53$	−1.03479e7	−1.31144	−0.655722	−	0.755002i	$-0.727636\pi$
$53$	−1.03479e7	−1.31144	−0.655722	+	0.755002i	$0.727636\pi$
$54$	0	0
$55$	− 6.23520e6i	− 0.681397i
$56$	0	0
$57$	−3.70711e6	−0.351185
$58$	0	0
$59$	− 1.46491e7i	− 1.20894i	−0.796629	−	0.604469i	$-0.793385\pi$
$59$	− 1.46491e7i	− 1.20894i	0.796629	−	0.604469i	$-0.206615\pi$
$60$	0	0
$61$	5.57255e6	0.402471	0.201236	−	0.979543i	$-0.435504\pi$
$61$	5.57255e6	0.402471	0.201236	+	0.979543i	$0.435504\pi$
$62$	0	0
$63$	2.70212e6i	0.171531i
$64$	0	0
$65$	−2.44700e7	−1.37082
$66$	0	0
$67$	− 3.50959e6i	− 0.174163i	−0.996201	−	0.0870817i	$-0.972246\pi$
$67$	− 3.50959e6i	− 0.174163i	0.996201	−	0.0870817i	$-0.0277541\pi$
$68$	0	0
$69$	−7.96349e6	−0.351323
$70$	0	0
$71$	1.43161e6i	0.0563368i	0.999603	+	0.0281684i	$0.00896746\pi$
$71$	1.43161e6i	0.0563368i	−0.999603	+	0.0281684i	$0.991033\pi$
$72$	0	0
$73$	2.72673e7	0.960177	0.480088	−	0.877220i	$-0.340605\pi$
$73$	2.72673e7	0.960177	0.480088	+	0.877220i	$0.340605\pi$
$74$	0	0
$75$	2.54243e7i	0.803533i
$76$	0	0
$77$	−7.97019e6	−0.226728
$78$	0	0
$79$	− 4.66244e7i	− 1.19703i	−0.801112	−	0.598515i	$-0.795758\pi$
$79$	− 4.66244e7i	− 1.19703i	0.801112	−	0.598515i	$-0.204242\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	2.20176e7i	0.463935i	0.972724	+	0.231967i	$0.0745162\pi$
$83$	2.20176e7i	0.463935i	−0.972724	+	0.231967i	$0.925484\pi$
$84$	0	0
$85$	−4.25199e7	−0.814549
$86$	0	0
$87$	1.62354e7i	0.283390i
$88$	0	0
$89$	−5.90052e7	−0.940439	−0.470219	−	0.882550i	$-0.655825\pi$
$89$	−5.90052e7	−0.940439	−0.470219	+	0.882550i	$0.655825\pi$
$90$	0	0
$91$	3.12789e7i	0.456127i
$92$	0	0
$93$	7.63562e7	1.02073
$94$	0	0
$95$	7.66213e7i	0.940708i
$96$	0	0
$97$	−1.52888e8	−1.72698	−0.863490	−	0.504366i	$-0.831726\pi$
$97$	−1.52888e8	−1.72698	−0.863490	+	0.504366i	$0.831726\pi$
$98$	0	0
$99$	1.41078e7i	0.146866i
$100$	0	0
$101$	−2.48845e7	−0.239135	−0.119568	−	0.992826i	$-0.538151\pi$
$101$	−2.48845e7	−0.239135	−0.119568	+	0.992826i	$0.538151\pi$
$102$	0	0
$103$	2.90364e7i	0.257985i	0.991646	+	0.128992i	$0.0411743\pi$
$103$	2.90364e7i	0.257985i	−0.991646	+	0.128992i	$0.958826\pi$
$104$	0	0
$105$	5.58495e7	0.459476
$106$	0	0
$107$	2.61862e7i	0.199773i	0.994999	+	0.0998865i	$0.0318480\pi$
$107$	2.61862e7i	0.199773i	−0.994999	+	0.0998865i	$0.968152\pi$
$108$	0	0
$109$	1.47946e8	1.04809	0.524043	−	0.851692i	$-0.324423\pi$
$109$	1.47946e8	1.04809	0.524043	+	0.851692i	$0.324423\pi$
$110$	0	0
$111$	− 1.23598e8i	− 0.814179i
$112$	0	0
$113$	−2.76008e8	−1.69281	−0.846403	−	0.532542i	$-0.821237\pi$
$113$	−2.76008e8	−1.69281	−0.846403	+	0.532542i	$0.821237\pi$
$114$	0	0
$115$	1.64595e8i	0.941080i
$116$	0	0
$117$	5.53660e7	0.295461
$118$	0	0
$119$	5.43514e7i	0.271033i
$120$	0	0
$121$	1.72746e8	0.805875
$122$	0	0
$123$	1.32154e8i	0.577376i
$124$	0	0
$125$	1.47917e8	0.605868
$126$	0	0
$127$	− 1.81815e8i	− 0.698899i	−0.936955	−	0.349449i	$-0.886369\pi$
$127$	− 1.81815e8i	− 0.698899i	0.936955	−	0.349449i	$-0.113631\pi$
$128$	0	0
$129$	−5.14530e7	−0.185802
$130$	0	0
$131$	− 2.24273e8i	− 0.761539i	−0.924670	−	0.380769i	$-0.875659\pi$
$131$	− 2.24273e8i	− 0.761539i	0.924670	−	0.380769i	$-0.124341\pi$
$132$	0	0
$133$	9.79416e7	0.313012
$134$	0	0
$135$	− 9.88580e7i	− 0.297630i
$136$	0	0
$137$	−1.01408e8	−0.287867	−0.143933	−	0.989587i	$-0.545975\pi$
$137$	−1.01408e8	−0.287867	−0.143933	+	0.989587i	$0.545975\pi$
$138$	0	0
$139$	− 6.26557e8i	− 1.67842i	−0.543805	−	0.839212i	$-0.683017\pi$
$139$	− 6.26557e8i	− 1.67842i	0.543805	−	0.839212i	$-0.316983\pi$
$140$	0	0
$141$	2.88609e8	0.730186
$142$	0	0
$143$	1.63308e8i	0.390537i
$144$	0	0
$145$	3.35565e8	0.759109
$146$	0	0
$147$	1.98203e8i	0.424464i
$148$	0	0
$149$	−6.46859e8	−1.31239	−0.656197	−	0.754589i	$-0.727836\pi$
$149$	−6.46859e8	−1.31239	−0.656197	+	0.754589i	$0.727836\pi$
$150$	0	0
$151$	− 6.17190e8i	− 1.18716i	−0.804773	−	0.593582i	$-0.797713\pi$
$151$	− 6.17190e8i	− 1.18716i	0.804773	−	0.593582i	$-0.202287\pi$
$152$	0	0
$153$	9.62061e7	0.175565
$154$	0	0
$155$	− 1.57819e9i	− 2.73421i
$156$	0	0
$157$	−8.13699e7	−0.133926	−0.0669630	−	0.997755i	$-0.521331\pi$
$157$	−8.13699e7	−0.133926	−0.0669630	+	0.997755i	$0.521331\pi$
$158$	0	0
$159$	− 4.83925e8i	− 0.757163i
$160$	0	0
$161$	2.10395e8	0.313135
$162$	0	0
$163$	− 2.36835e8i	− 0.335502i	−0.985829	−	0.167751i	$-0.946350\pi$
$163$	− 2.36835e8i	− 0.335502i	0.985829	−	0.167751i	$-0.0536505\pi$
$164$	0	0
$165$	2.91592e8	0.393404
$166$	0	0
$167$	6.75435e8i	0.868396i	0.900817	+	0.434198i	$0.142968\pi$
$167$	6.75435e8i	0.868396i	−0.900817	+	0.434198i	$0.857032\pi$
$168$	0	0
$169$	−1.74832e8	−0.214326
$170$	0	0
$171$	− 1.73364e8i	− 0.202757i
$172$	0	0
$173$	−8.88747e8	−0.992187	−0.496094	−	0.868269i	$-0.665233\pi$
$173$	−8.88747e8	−0.992187	−0.496094	+	0.868269i	$0.665233\pi$
$174$	0	0
$175$	− 6.71708e8i	− 0.716190i
$176$	0	0
$177$	6.85072e8	0.697980
$178$	0	0
$179$	− 8.22985e8i	− 0.801641i	−0.916157	−	0.400820i	$-0.868725\pi$
$179$	− 8.22985e8i	− 0.801641i	0.916157	−	0.400820i	$-0.131275\pi$
$180$	0	0
$181$	1.77648e9	1.65518	0.827591	−	0.561332i	$-0.189711\pi$
$181$	1.77648e9	1.65518	0.827591	+	0.561332i	$0.189711\pi$
$182$	0	0
$183$	2.60603e8i	0.232367i
$184$	0	0
$185$	−2.55462e9	−2.18092
$186$	0	0
$187$	2.83770e8i	0.232060i
$188$	0	0
$189$	−1.26366e8	−0.0990336
$190$	0	0
$191$	− 1.22847e9i	− 0.923060i	−0.887125	−	0.461530i	$-0.847301\pi$
$191$	− 1.22847e9i	− 0.923060i	0.887125	−	0.461530i	$-0.152699\pi$
$192$	0	0
$193$	−3.36810e8	−0.242748	−0.121374	−	0.992607i	$-0.538730\pi$
$193$	−3.36810e8	−0.242748	−0.121374	+	0.992607i	$0.538730\pi$
$194$	0	0
$195$	− 1.14435e9i	− 0.791443i
$196$	0	0
$197$	2.49946e9	1.65952	0.829759	−	0.558121i	$-0.188478\pi$
$197$	2.49946e9	1.65952	0.829759	+	0.558121i	$0.188478\pi$
$198$	0	0
$199$	− 1.67146e9i	− 1.06582i	−0.846172	−	0.532910i	$-0.821098\pi$
$199$	− 1.67146e9i	− 1.06582i	0.846172	−	0.532910i	$-0.178902\pi$
$200$	0	0
$201$	1.64127e8	0.100553
$202$	0	0
$203$	− 4.28938e8i	− 0.252586i
$204$	0	0
$205$	2.73145e9	1.54660
$206$	0	0
$207$	− 3.72416e8i	− 0.202837i
$208$	0	0
$209$	5.11355e8	0.268002
$210$	0	0
$211$	1.81257e9i	0.914461i	0.889348	+	0.457230i	$0.151159\pi$
$211$	1.81257e9i	0.914461i	−0.889348	+	0.457230i	$0.848841\pi$
$212$	0	0
$213$	−6.69499e7	−0.0325260
$214$	0	0
$215$	1.06347e9i	0.497704i
$216$	0	0
$217$	−2.01733e9	−0.909783
$218$	0	0
$219$	1.27517e9i	0.554358i
$220$	0	0
$221$	1.11365e9	0.466852
$222$	0	0
$223$	2.32221e9i	0.939036i	0.882923	+	0.469518i	$0.155572\pi$
$223$	2.32221e9i	0.939036i	−0.882923	+	0.469518i	$0.844428\pi$
$224$	0	0
$225$	−1.18898e9	−0.463920
$226$	0	0
$227$	− 3.41714e9i	− 1.28694i	−0.765469	−	0.643472i	$-0.777493\pi$
$227$	− 3.41714e9i	− 1.28694i	0.765469	−	0.643472i	$-0.222507\pi$
$228$	0	0
$229$	2.33777e8	0.0850079	0.0425040	−	0.999096i	$-0.486466\pi$
$229$	2.33777e8	0.0850079	0.0425040	+	0.999096i	$0.486466\pi$
$230$	0	0
$231$	− 3.72729e8i	− 0.130902i
$232$	0	0
$233$	−4.76430e9	−1.61650	−0.808249	−	0.588840i	$-0.799585\pi$
$233$	−4.76430e9	−1.61650	−0.808249	+	0.588840i	$0.799585\pi$
$234$	0	0
$235$	− 5.96519e9i	− 1.95593i
$236$	0	0
$237$	2.18041e9	0.691105
$238$	0	0
$239$	− 6.32691e9i	− 1.93910i	−0.244894	−	0.969550i	$-0.578753\pi$
$239$	− 6.32691e9i	− 1.93910i	0.244894	−	0.969550i	$-0.421247\pi$
$240$	0	0
$241$	−2.40868e9	−0.714021	−0.357011	−	0.934100i	$-0.616204\pi$
$241$	−2.40868e9	−0.714021	−0.357011	+	0.934100i	$0.616204\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	4.09661e9	1.13700
$246$	0	0
$247$	− 2.00681e9i	− 0.539160i
$248$	0	0
$249$	−1.02966e9	−0.267853
$250$	0	0
$251$	5.70375e9i	1.43703i	0.695512	+	0.718514i	$0.255177\pi$
$251$	5.70375e9i	1.43703i	−0.695512	+	0.718514i	$0.744823\pi$
$252$	0	0
$253$	1.09848e9	0.268107
$254$	0	0
$255$	− 1.98846e9i	− 0.470280i
$256$	0	0
$257$	−2.37264e9	−0.543875	−0.271937	−	0.962315i	$-0.587664\pi$
$257$	−2.37264e9	−0.543875	−0.271937	+	0.962315i	$0.587664\pi$
$258$	0	0
$259$	3.26546e9i	0.725679i
$260$	0	0
$261$	−7.59253e8	−0.163615
$262$	0	0
$263$	− 3.30702e9i	− 0.691217i	−0.938379	−	0.345608i	$-0.887673\pi$
$263$	− 3.30702e9i	− 0.691217i	0.938379	−	0.345608i	$-0.112327\pi$
$264$	0	0
$265$	−1.00021e10	−2.02819
$266$	0	0
$267$	− 2.75940e9i	− 0.542963i
$268$	0	0
$269$	3.45153e9	0.659178	0.329589	−	0.944125i	$-0.393090\pi$
$269$	3.45153e9	0.659178	0.329589	+	0.944125i	$0.393090\pi$
$270$	0	0
$271$	2.04626e8i	0.0379388i	0.999820	+	0.0189694i	$0.00603852\pi$
$271$	2.04626e8i	0.0379388i	−0.999820	+	0.0189694i	$0.993961\pi$
$272$	0	0
$273$	−1.46277e9	−0.263345
$274$	0	0
$275$	− 3.50700e9i	− 0.613204i
$276$	0	0
$277$	−5.37375e9	−0.912764	−0.456382	−	0.889784i	$-0.650855\pi$
$277$	−5.37375e9	−0.912764	−0.456382	+	0.889784i	$0.650855\pi$
$278$	0	0
$279$	3.57083e9i	0.589321i
$280$	0	0
$281$	1.01055e10	1.62081	0.810406	−	0.585868i	$-0.199246\pi$
$281$	1.01055e10	1.62081	0.810406	+	0.585868i	$0.199246\pi$
$282$	0	0
$283$	− 5.38932e9i	− 0.840210i	−0.907475	−	0.420105i	$-0.861993\pi$
$283$	− 5.38932e9i	− 0.840210i	0.907475	−	0.420105i	$-0.138007\pi$
$284$	0	0
$285$	−3.58322e9	−0.543118
$286$	0	0
$287$	− 3.49149e9i	− 0.514616i
$288$	0	0
$289$	−5.04064e9	−0.722594
$290$	0	0
$291$	− 7.14988e9i	− 0.997072i
$292$	0	0
$293$	−1.56279e9	−0.212046	−0.106023	−	0.994364i	$-0.533812\pi$
$293$	−1.56279e9	−0.212046	−0.106023	+	0.994364i	$0.533812\pi$
$294$	0	0
$295$	− 1.41596e10i	− 1.86966i
$296$	0	0
$297$	−6.59759e8	−0.0847929
$298$	0	0
$299$	− 4.31096e9i	− 0.539373i
$300$	0	0
$301$	1.35939e9	0.165606
$302$	0	0
$303$	− 1.16373e9i	− 0.138065i
$304$	0	0
$305$	5.38633e9	0.622435
$306$	0	0
$307$	− 6.42722e9i	− 0.723551i	−0.932265	−	0.361776i	$-0.882171\pi$
$307$	− 6.42722e9i	− 0.723551i	0.932265	−	0.361776i	$-0.117829\pi$
$308$	0	0
$309$	−1.35790e9	−0.148948
$310$	0	0
$311$	− 1.55207e10i	− 1.65909i	−0.558438	−	0.829546i	$-0.688599\pi$
$311$	− 1.55207e10i	− 1.65909i	0.558438	−	0.829546i	$-0.311401\pi$
$312$	0	0
$313$	−2.72875e9	−0.284306	−0.142153	−	0.989845i	$-0.545402\pi$
$313$	−2.72875e9	−0.284306	−0.142153	+	0.989845i	$0.545402\pi$
$314$	0	0
$315$	2.61182e9i	0.265278i
$316$	0	0
$317$	1.97792e10	1.95872	0.979358	−	0.202135i	$-0.0647878\pi$
$317$	1.97792e10	1.95872	0.979358	+	0.202135i	$0.0647878\pi$
$318$	0	0
$319$	− 2.23949e9i	− 0.216265i
$320$	0	0
$321$	−1.22461e9	−0.115339
$322$	0	0
$323$	− 3.48710e9i	− 0.320372i
$324$	0	0
$325$	−1.37632e10	−1.23363
$326$	0	0
$327$	6.91875e9i	0.605113i
$328$	0	0
$329$	−7.62504e9	−0.650817
$330$	0	0
$331$	− 2.22402e10i	− 1.85279i	−0.376553	−	0.926395i	$-0.622891\pi$
$331$	− 2.22402e10i	− 1.85279i	0.376553	−	0.926395i	$-0.377109\pi$
$332$	0	0
$333$	5.78011e9	0.470066
$334$	0	0
$335$	− 3.39230e9i	− 0.269349i
$336$	0	0
$337$	6.67658e9	0.517648	0.258824	−	0.965925i	$-0.416665\pi$
$337$	6.67658e9	0.517648	0.258824	+	0.965925i	$0.416665\pi$
$338$	0	0
$339$	− 1.29076e10i	− 0.977342i
$340$	0	0
$341$	−1.05325e10	−0.778959
$342$	0	0
$343$	− 1.23592e10i	− 0.892919i
$344$	0	0
$345$	−7.69737e9	−0.543333
$346$	0	0
$347$	2.26077e10i	1.55933i	0.626195	+	0.779666i	$0.284611\pi$
$347$	2.26077e10i	1.55933i	−0.626195	+	0.779666i	$0.715389\pi$
$348$	0	0
$349$	1.74243e10	1.17450	0.587250	−	0.809406i	$-0.300211\pi$
$349$	1.74243e10	1.17450	0.587250	+	0.809406i	$0.300211\pi$
$350$	0	0
$351$	2.58921e9i	0.170584i
$352$	0	0
$353$	6.48043e9	0.417355	0.208677	−	0.977985i	$-0.433084\pi$
$353$	6.48043e9	0.417355	0.208677	+	0.977985i	$0.433084\pi$
$354$	0	0
$355$	1.38377e9i	0.0871266i
$356$	0	0
$357$	−2.54176e9	−0.156481
$358$	0	0
$359$	1.62843e10i	0.980373i	0.871618	+	0.490187i	$0.163071\pi$
$359$	1.62843e10i	0.980373i	−0.871618	+	0.490187i	$0.836929\pi$
$360$	0	0
$361$	1.06998e10	0.630008
$362$	0	0
$363$	8.07855e9i	0.465272i
$364$	0	0
$365$	2.63561e10	1.48494
$366$	0	0
$367$	2.34723e10i	1.29387i	0.762544	+	0.646936i	$0.223950\pi$
$367$	2.34723e10i	1.29387i	−0.762544	+	0.646936i	$0.776050\pi$
$368$	0	0
$369$	−6.18021e9	−0.333348
$370$	0	0
$371$	1.27853e10i	0.674861i
$372$	0	0
$373$	6.32175e9	0.326590	0.163295	−	0.986577i	$-0.447788\pi$
$373$	6.32175e9	0.326590	0.163295	+	0.986577i	$0.447788\pi$
$374$	0	0
$375$	6.91739e9i	0.349798i
$376$	0	0
$377$	−8.78885e9	−0.435078
$378$	0	0
$379$	3.85603e10i	1.86889i	0.356107	+	0.934445i	$0.384104\pi$
$379$	3.85603e10i	1.86889i	−0.356107	+	0.934445i	$0.615896\pi$
$380$	0	0
$381$	8.50263e9	0.403509
$382$	0	0
$383$	6.70414e9i	0.311564i	0.987791	+	0.155782i	$0.0497898\pi$
$383$	6.70414e9i	0.311564i	−0.987791	+	0.155782i	$0.950210\pi$
$384$	0	0
$385$	−7.70384e9	−0.350642
$386$	0	0
$387$	− 2.40622e9i	− 0.107273i
$388$	0	0
$389$	6.51675e9	0.284599	0.142299	−	0.989824i	$-0.454550\pi$
$389$	6.51675e9	0.284599	0.142299	+	0.989824i	$0.454550\pi$
$390$	0	0
$391$	− 7.49088e9i	− 0.320498i
$392$	0	0
$393$	1.04882e10	0.439675
$394$	0	0
$395$	− 4.50663e10i	− 1.85124i
$396$	0	0
$397$	6.19185e9	0.249263	0.124632	−	0.992203i	$-0.460225\pi$
$397$	6.19185e9	0.249263	0.124632	+	0.992203i	$0.460225\pi$
$398$	0	0
$399$	4.58028e9i	0.180717i
$400$	0	0
$401$	1.51804e10	0.587090	0.293545	−	0.955945i	$-0.405165\pi$
$401$	1.51804e10	0.587090	0.293545	+	0.955945i	$0.405165\pi$
$402$	0	0
$403$	4.13347e10i	1.56709i
$404$	0	0
$405$	4.62313e9	0.171837
$406$	0	0
$407$	1.70490e10i	0.621329i
$408$	0	0
$409$	−5.09443e10	−1.82055	−0.910275	−	0.414004i	$-0.864130\pi$
$409$	−5.09443e10	−1.82055	−0.910275	+	0.414004i	$0.864130\pi$
$410$	0	0
$411$	− 4.74240e9i	− 0.166200i
$412$	0	0
$413$	−1.80996e10	−0.622112
$414$	0	0
$415$	2.12818e10i	0.717489i
$416$	0	0
$417$	2.93012e10	0.969038
$418$	0	0
$419$	− 4.04147e10i	− 1.31124i	−0.755089	−	0.655622i	$-0.772406\pi$
$419$	− 4.04147e10i	− 1.31124i	0.755089	−	0.655622i	$-0.227594\pi$
$420$	0	0
$421$	−4.49876e10	−1.43207	−0.716035	−	0.698064i	$-0.754045\pi$
$421$	−4.49876e10	−1.43207	−0.716035	+	0.698064i	$0.754045\pi$
$422$	0	0
$423$	1.34969e10i	0.421573i
$424$	0	0
$425$	−2.39154e10	−0.733031
$426$	0	0
$427$	− 6.88511e9i	− 0.207109i
$428$	0	0
$429$	−7.63714e9	−0.225477
$430$	0	0
$431$	3.90187e10i	1.13074i	0.824836	+	0.565371i	$0.191267\pi$
$431$	3.90187e10i	1.13074i	−0.824836	+	0.565371i	$0.808733\pi$
$432$	0	0
$433$	4.86835e10	1.38494	0.692469	−	0.721448i	$-0.256523\pi$
$433$	4.86835e10	1.38494	0.692469	+	0.721448i	$0.256523\pi$
$434$	0	0
$435$	1.56928e10i	0.438272i
$436$	0	0
$437$	−1.34986e10	−0.370138
$438$	0	0
$439$	1.17926e10i	0.317506i	0.987318	+	0.158753i	$0.0507473\pi$
$439$	1.17926e10i	0.317506i	−0.987318	+	0.158753i	$0.949253\pi$
$440$	0	0
$441$	−9.26904e9	−0.245064
$442$	0	0
$443$	6.02368e10i	1.56404i	0.623255	+	0.782019i	$0.285810\pi$
$443$	6.02368e10i	1.56404i	−0.623255	+	0.782019i	$0.714190\pi$
$444$	0	0
$445$	−5.70334e10	−1.45442
$446$	0	0
$447$	− 3.02506e10i	− 0.757711i
$448$	0	0
$449$	5.47118e10	1.34616	0.673078	−	0.739571i	$-0.264972\pi$
$449$	5.47118e10	1.34616	0.673078	+	0.739571i	$0.264972\pi$
$450$	0	0
$451$	− 1.82292e10i	− 0.440616i
$452$	0	0
$453$	2.88631e10	0.685410
$454$	0	0
$455$	3.02336e10i	0.705414i
$456$	0	0
$457$	−2.85856e10	−0.655363	−0.327682	−	0.944788i	$-0.606267\pi$
$457$	−2.85856e10	−0.655363	−0.327682	+	0.944788i	$0.606267\pi$
$458$	0	0
$459$	4.49912e9i	0.101362i
$460$	0	0
$461$	2.13010e10	0.471625	0.235813	−	0.971799i	$-0.424225\pi$
$461$	2.13010e10	0.471625	0.235813	+	0.971799i	$0.424225\pi$
$462$	0	0
$463$	− 8.89516e10i	− 1.93566i	−0.251597	−	0.967832i	$-0.580956\pi$
$463$	− 8.89516e10i	− 1.93566i	0.251597	−	0.967832i	$-0.419044\pi$
$464$	0	0
$465$	7.38046e10	1.57860
$466$	0	0
$467$	− 4.19722e9i	− 0.0882459i	−0.999026	−	0.0441230i	$-0.985951\pi$
$467$	− 4.19722e9i	− 0.0882459i	0.999026	−	0.0441230i	$-0.0140493\pi$
$468$	0	0
$469$	−4.33623e9	−0.0896233
$470$	0	0
$471$	− 3.80529e9i	− 0.0773223i
$472$	0	0
$473$	7.09738e9	0.141793
$474$	0	0
$475$	4.30958e10i	0.846565i
$476$	0	0
$477$	2.26309e10	0.437148
$478$	0	0
$479$	7.76123e9i	0.147431i	0.997279	+	0.0737155i	$0.0234857\pi$
$479$	7.76123e9i	0.147431i	−0.997279	+	0.0737155i	$0.976514\pi$
$480$	0	0
$481$	6.69086e10	1.24998
$482$	0	0
$483$	9.83920e9i	0.180789i
$484$	0	0
$485$	−1.47779e11	−2.67083
$486$	0	0
$487$	6.14920e9i	0.109321i	0.998505	+	0.0546604i	$0.0174076\pi$
$487$	6.14920e9i	0.109321i	−0.998505	+	0.0546604i	$0.982592\pi$
$488$	0	0
$489$	1.10757e10	0.193702
$490$	0	0
$491$	− 6.07584e10i	− 1.04540i	−0.852518	−	0.522698i	$-0.824926\pi$
$491$	− 6.07584e10i	− 1.04540i	0.852518	−	0.522698i	$-0.175074\pi$
$492$	0	0
$493$	−1.52718e10	−0.258526
$494$	0	0
$495$	1.36364e10i	0.227132i
$496$	0	0
$497$	1.76881e9	0.0289905
$498$	0	0
$499$	1.16684e11i	1.88195i	0.338474	+	0.940976i	$0.390089\pi$
$499$	1.16684e11i	1.88195i	−0.338474	+	0.940976i	$0.609911\pi$
$500$	0	0
$501$	−3.15870e10	−0.501369
$502$	0	0
$503$	2.16393e10i	0.338042i	0.985612	+	0.169021i	$0.0540606\pi$
$503$	2.16393e10i	0.338042i	−0.985612	+	0.169021i	$0.945939\pi$
$504$	0	0
$505$	−2.40529e10	−0.369830
$506$	0	0
$507$	− 8.17609e9i	− 0.123741i
$508$	0	0
$509$	−9.89628e10	−1.47435	−0.737176	−	0.675701i	$-0.763841\pi$
$509$	−9.89628e10	−1.47435	−0.737176	+	0.675701i	$0.763841\pi$
$510$	0	0
$511$	− 3.36899e10i	− 0.494101i
$512$	0	0
$513$	8.10744e9	0.117062
$514$	0	0
$515$	2.80661e10i	0.398982i
$516$	0	0
$517$	−3.98105e10	−0.557231
$518$	0	0
$519$	− 4.15626e10i	− 0.572840i
$520$	0	0
$521$	−3.25718e10	−0.442070	−0.221035	−	0.975266i	$-0.570944\pi$
$521$	−3.25718e10	−0.442070	−0.221035	+	0.975266i	$0.570944\pi$
$522$	0	0
$523$	9.80064e10i	1.30993i	0.755660	+	0.654964i	$0.227316\pi$
$523$	9.80064e10i	1.30993i	−0.755660	+	0.654964i	$0.772684\pi$
$524$	0	0
$525$	3.14127e10	0.413493
$526$	0	0
$527$	7.18247e10i	0.931176i
$528$	0	0
$529$	4.93137e10	0.629716
$530$	0	0
$531$	3.20377e10i	0.402979i
$532$	0	0
$533$	−7.15401e10	−0.886422
$534$	0	0
$535$	2.53111e10i	0.308955i
$536$	0	0
$537$	3.84872e10	0.462828
$538$	0	0
$539$	− 2.73400e10i	− 0.323924i
$540$	0	0
$541$	2.78807e10	0.325472	0.162736	−	0.986670i	$-0.447968\pi$
$541$	2.78807e10	0.325472	0.162736	+	0.986670i	$0.447968\pi$
$542$	0	0
$543$	8.30777e10i	0.955620i
$544$	0	0
$545$	1.43002e11	1.62090
$546$	0	0
$547$	7.90764e10i	0.883278i	0.897193	+	0.441639i	$0.145603\pi$
$547$	7.90764e10i	0.883278i	−0.897193	+	0.441639i	$0.854397\pi$
$548$	0	0
$549$	−1.21872e10	−0.134157
$550$	0	0
$551$	2.75200e10i	0.298567i
$552$	0	0
$553$	−5.76063e10	−0.615984
$554$	0	0
$555$	− 1.19468e11i	− 1.25915i
$556$	0	0
$557$	1.32879e11	1.38049	0.690246	−	0.723575i	$-0.257502\pi$
$557$	1.32879e11	1.38049	0.690246	+	0.723575i	$0.257502\pi$
$558$	0	0
$559$	− 2.78536e10i	− 0.285255i
$560$	0	0
$561$	−1.32706e10	−0.133980
$562$	0	0
$563$	− 5.55469e9i	− 0.0552874i	−0.999618	−	0.0276437i	$-0.991200\pi$
$563$	− 5.55469e9i	− 0.0552874i	0.999618	−	0.0276437i	$-0.00880038\pi$
$564$	0	0
$565$	−2.66784e11	−2.61798
$566$	0	0
$567$	− 5.90955e9i	− 0.0571771i
$568$	0	0
$569$	2.48359e10	0.236936	0.118468	−	0.992958i	$-0.462202\pi$
$569$	2.48359e10	0.236936	0.118468	+	0.992958i	$0.462202\pi$
$570$	0	0
$571$	1.17066e11i	1.10126i	0.834751	+	0.550628i	$0.185612\pi$
$571$	1.17066e11i	1.10126i	−0.834751	+	0.550628i	$0.814388\pi$
$572$	0	0
$573$	5.74497e10	0.532929
$574$	0	0
$575$	9.25770e10i	0.846899i
$576$	0	0
$577$	−1.05519e11	−0.951976	−0.475988	−	0.879452i	$-0.657909\pi$
$577$	−1.05519e11	−0.951976	−0.475988	+	0.879452i	$0.657909\pi$
$578$	0	0
$579$	− 1.57510e10i	− 0.140150i
$580$	0	0
$581$	2.72036e10	0.238738
$582$	0	0
$583$	6.67521e10i	0.577818i
$584$	0	0
$585$	5.35158e10	0.456940
$586$	0	0
$587$	8.72439e10i	0.734823i	0.930058	+	0.367412i	$0.119756\pi$
$587$	8.72439e10i	0.734823i	−0.930058	+	0.367412i	$0.880244\pi$
$588$	0	0
$589$	1.29429e11	1.07540
$590$	0	0
$591$	1.16888e11i	0.958123i
$592$	0	0
$593$	−5.49633e10	−0.444482	−0.222241	−	0.974992i	$-0.571337\pi$
$593$	−5.49633e10	−0.444482	−0.222241	+	0.974992i	$0.571337\pi$
$594$	0	0
$595$	5.25351e10i	0.419161i
$596$	0	0
$597$	7.81665e10	0.615351
$598$	0	0
$599$	1.77833e11i	1.38135i	0.723165	+	0.690675i	$0.242687\pi$
$599$	1.77833e11i	1.38135i	−0.723165	+	0.690675i	$0.757313\pi$
$600$	0	0
$601$	−3.73412e10	−0.286214	−0.143107	−	0.989707i	$-0.545709\pi$
$601$	−3.73412e10	−0.286214	−0.143107	+	0.989707i	$0.545709\pi$
$602$	0	0
$603$	7.67547e9i	0.0580544i
$604$	0	0
$605$	1.66974e11	1.24631
$606$	0	0
$607$	− 9.46132e10i	− 0.696942i	−0.937320	−	0.348471i	$-0.886701\pi$
$607$	− 9.46132e10i	− 0.696942i	0.937320	−	0.348471i	$-0.113299\pi$
$608$	0	0
$609$	2.00594e10	0.145831
$610$	0	0
$611$	1.56236e11i	1.12103i
$612$	0	0
$613$	−1.57482e11	−1.11530	−0.557648	−	0.830078i	$-0.688296\pi$
$613$	−1.57482e11	−1.11530	−0.557648	+	0.830078i	$0.688296\pi$
$614$	0	0
$615$	1.27737e11i	0.892930i
$616$	0	0
$617$	−1.91709e8	−0.00132282	−0.000661411	−	1.00000i	$-0.500211\pi$
$617$	−1.91709e8	−0.00132282	−0.000661411		1.00000i	$0.500211\pi$
$618$	0	0
$619$	7.29663e10i	0.497004i	0.968631	+	0.248502i	$0.0799382\pi$
$619$	7.29663e10i	0.497004i	−0.968631	+	0.248502i	$0.920062\pi$
$620$	0	0
$621$	1.74162e10	0.117108
$622$	0	0
$623$	7.29033e10i	0.483944i
$624$	0	0
$625$	−6.93917e10	−0.454766
$626$	0	0
$627$	2.39137e10i	0.154731i
$628$	0	0
$629$	1.16263e11	0.742743
$630$	0	0
$631$	− 6.56868e10i	− 0.414344i	−0.978305	−	0.207172i	$-0.933574\pi$
$631$	− 6.56868e10i	− 0.414344i	0.978305	−	0.207172i	$-0.0664259\pi$
$632$	0	0
$633$	−8.47655e10	−0.527964
$634$	0	0
$635$	− 1.75739e11i	− 1.08087i
$636$	0	0
$637$	−1.07295e11	−0.651663
$638$	0	0
$639$	− 3.13094e9i	− 0.0187789i
$640$	0	0
$641$	2.22670e11	1.31895	0.659476	−	0.751726i	$-0.270778\pi$
$641$	2.22670e11	1.31895	0.659476	+	0.751726i	$0.270778\pi$
$642$	0	0
$643$	− 1.98657e11i	− 1.16215i	−0.813851	−	0.581073i	$-0.802633\pi$
$643$	− 1.98657e11i	− 1.16215i	0.813851	−	0.581073i	$-0.197367\pi$
$644$	0	0
$645$	−4.97335e10	−0.287349
$646$	0	0
$647$	− 1.27043e11i	− 0.724992i	−0.931985	−	0.362496i	$-0.881925\pi$
$647$	− 1.27043e11i	− 0.724992i	0.931985	−	0.362496i	$-0.118075\pi$
$648$	0	0
$649$	−9.44983e10	−0.532654
$650$	0	0
$651$	− 9.43411e10i	− 0.525263i
$652$	0	0
$653$	1.86100e11	1.02351	0.511757	−	0.859130i	$-0.328995\pi$
$653$	1.86100e11	1.02351	0.511757	+	0.859130i	$0.328995\pi$
$654$	0	0
$655$	− 2.16778e11i	− 1.17774i
$656$	0	0
$657$	−5.96337e10	−0.320059
$658$	0	0
$659$	− 1.18209e11i	− 0.626770i	−0.949626	−	0.313385i	$-0.898537\pi$
$659$	− 1.18209e11i	− 0.626770i	0.949626	−	0.313385i	$-0.101463\pi$
$660$	0	0
$661$	−9.75157e10	−0.510821	−0.255410	−	0.966833i	$-0.582211\pi$
$661$	−9.75157e10	−0.510821	−0.255410	+	0.966833i	$0.582211\pi$
$662$	0	0
$663$	5.20802e10i	0.269537i
$664$	0	0
$665$	9.46686e10	0.484082
$666$	0	0
$667$	5.91176e10i	0.298685i
$668$	0	0
$669$	−1.08599e11	−0.542153
$670$	0	0
$671$	− 3.59473e10i	− 0.177328i
$672$	0	0
$673$	−7.29217e10	−0.355465	−0.177732	−	0.984079i	$-0.556876\pi$
$673$	−7.29217e10	−0.355465	−0.177732	+	0.984079i	$0.556876\pi$
$674$	0	0
$675$	− 5.56029e10i	− 0.267844i
$676$	0	0
$677$	6.81048e10	0.324207	0.162104	−	0.986774i	$-0.448172\pi$
$677$	6.81048e10	0.324207	0.162104	+	0.986774i	$0.448172\pi$
$678$	0	0
$679$	1.88899e11i	0.888693i
$680$	0	0
$681$	1.59804e11	0.743018
$682$	0	0
$683$	2.74590e10i	0.126183i	0.998008	+	0.0630917i	$0.0200960\pi$
$683$	2.74590e10i	0.126183i	−0.998008	+	0.0630917i	$0.979904\pi$
$684$	0	0
$685$	−9.80195e10	−0.445195
$686$	0	0
$687$	1.09327e10i	0.0490794i
$688$	0	0
$689$	2.61968e11	1.16244
$690$	0	0
$691$	− 3.29254e11i	− 1.44417i	−0.691804	−	0.722085i	$-0.743184\pi$
$691$	− 3.29254e11i	− 1.44417i	0.691804	−	0.722085i	$-0.256816\pi$
$692$	0	0
$693$	1.74308e10	0.0755761
$694$	0	0
$695$	− 6.05619e11i	− 2.59573i
$696$	0	0
$697$	−1.24311e11	−0.526717
$698$	0	0
$699$	− 2.22804e11i	− 0.933286i
$700$	0	0
$701$	1.77453e11	0.734873	0.367436	−	0.930049i	$-0.380236\pi$
$701$	1.77453e11	0.734873	0.367436	+	0.930049i	$0.380236\pi$
$702$	0	0
$703$	− 2.09507e11i	− 0.857782i
$704$	0	0
$705$	2.78964e11	1.12926
$706$	0	0
$707$	3.07458e10i	0.123057i
$708$	0	0
$709$	−2.42297e11	−0.958876	−0.479438	−	0.877576i	$-0.659160\pi$
$709$	−2.42297e11	−0.958876	−0.479438	+	0.877576i	$0.659160\pi$
$710$	0	0
$711$	1.01968e11i	0.399010i
$712$	0	0
$713$	2.78035e11	1.07582
$714$	0	0
$715$	1.57850e11i	0.603978i
$716$	0	0
$717$	2.95880e11	1.11954
$718$	0	0
$719$	6.44501e10i	0.241161i	0.992704	+	0.120581i	$0.0384757\pi$
$719$	6.44501e10i	0.241161i	−0.992704	+	0.120581i	$0.961524\pi$
$720$	0	0
$721$	3.58756e10	0.132757
$722$	0	0
$723$	− 1.12643e11i	− 0.412240i
$724$	0	0
$725$	1.88739e11	0.683140
$726$	0	0
$727$	− 5.78660e10i	− 0.207150i	−0.994622	−	0.103575i	$-0.966972\pi$
$727$	− 5.78660e10i	− 0.207150i	0.994622	−	0.103575i	$-0.0330283\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	− 4.83994e10i	− 0.169500i
$732$	0	0
$733$	−3.63372e11	−1.25874	−0.629369	−	0.777107i	$-0.716687\pi$
$733$	−3.63372e11	−1.25874	−0.629369	+	0.777107i	$0.716687\pi$
$734$	0	0
$735$	1.91580e11i	0.656447i
$736$	0	0
$737$	−2.26396e10	−0.0767358
$738$	0	0
$739$	2.72410e11i	0.913369i	0.889629	+	0.456684i	$0.150963\pi$
$739$	2.72410e11i	0.913369i	−0.889629	+	0.456684i	$0.849037\pi$
$740$	0	0
$741$	9.38490e10	0.311284
$742$	0	0
$743$	− 3.05450e11i	− 1.00227i	−0.865369	−	0.501135i	$-0.832916\pi$
$743$	− 3.05450e11i	− 1.00227i	0.865369	−	0.501135i	$-0.167084\pi$
$744$	0	0
$745$	−6.25242e11	−2.02966
$746$	0	0
$747$	− 4.81524e10i	− 0.154645i
$748$	0	0
$749$	3.23540e10	0.102802
$750$	0	0
$751$	− 4.91711e11i	− 1.54579i	−0.634534	−	0.772895i	$-0.718808\pi$
$751$	− 4.91711e11i	− 1.54579i	0.634534	−	0.772895i	$-0.281192\pi$
$752$	0	0
$753$	−2.66738e11	−0.829669
$754$	0	0
$755$	− 5.96565e11i	− 1.83599i
$756$	0	0
$757$	1.52094e11	0.463158	0.231579	−	0.972816i	$-0.425611\pi$
$757$	1.52094e11	0.463158	0.231579	+	0.972816i	$0.425611\pi$
$758$	0	0
$759$	5.13707e10i	0.154792i
$760$	0	0
$761$	−3.02170e11	−0.900976	−0.450488	−	0.892783i	$-0.648750\pi$
$761$	−3.02170e11	−0.900976	−0.450488	+	0.892783i	$0.648750\pi$
$762$	0	0
$763$	− 1.82793e11i	− 0.539339i
$764$	0	0
$765$	9.29911e10	0.271516
$766$	0	0
$767$	3.70857e11i	1.07158i
$768$	0	0
$769$	−6.65906e11	−1.90418	−0.952089	−	0.305821i	$-0.901069\pi$
$769$	−6.65906e11	−1.90418	−0.952089	+	0.305821i	$0.901069\pi$
$770$	0	0
$771$	− 1.10957e11i	− 0.314006i
$772$	0	0
$773$	4.86674e11	1.36308	0.681538	−	0.731782i	$-0.261311\pi$
$773$	4.86674e11	1.36308	0.681538	+	0.731782i	$0.261311\pi$
$774$	0	0
$775$	− 8.87655e11i	− 2.46058i
$776$	0	0
$777$	−1.52710e11	−0.418971
$778$	0	0
$779$	2.24009e11i	0.608297i
$780$	0	0
$781$	9.23501e9	0.0248218
$782$	0	0
$783$	− 3.55067e10i	− 0.0944634i
$784$	0	0
$785$	−7.86507e10	−0.207121
$786$	0	0
$787$	1.20044e11i	0.312927i	0.987684	+	0.156463i	$0.0500093\pi$
$787$	1.20044e11i	0.312927i	−0.987684	+	0.156463i	$0.949991\pi$
$788$	0	0
$789$	1.54654e11	0.399074
$790$	0	0
$791$	3.41018e11i	0.871107i
$792$	0	0
$793$	−1.41075e11	−0.356744
$794$	0	0
$795$	− 4.67753e11i	− 1.17098i
$796$	0	0
$797$	−3.43222e11	−0.850632	−0.425316	−	0.905045i	$-0.639837\pi$
$797$	−3.43222e11	−0.850632	−0.425316	+	0.905045i	$0.639837\pi$
$798$	0	0
$799$	2.71481e11i	0.666120i
$800$	0	0
$801$	1.29044e11	0.313480
$802$	0	0
$803$	− 1.75895e11i	− 0.423051i
$804$	0	0
$805$	2.03364e11	0.484274
$806$	0	0
$807$	1.61412e11i	0.380577i
$808$	0	0
$809$	−5.40630e11	−1.26214	−0.631068	−	0.775727i	$-0.717383\pi$
$809$	−5.40630e11	−1.26214	−0.631068	+	0.775727i	$0.717383\pi$
$810$	0	0
$811$	6.74194e11i	1.55848i	0.626725	+	0.779241i	$0.284395\pi$
$811$	6.74194e11i	1.55848i	−0.626725	+	0.779241i	$0.715605\pi$
$812$	0	0
$813$	−9.56942e9	−0.0219040
$814$	0	0
$815$	− 2.28920e11i	− 0.518865i
$816$	0	0
$817$	−8.72161e10	−0.195753
$818$	0	0
$819$	− 6.84069e10i	− 0.152042i
$820$	0	0
$821$	5.50911e11	1.21258	0.606288	−	0.795245i	$-0.292658\pi$
$821$	5.50911e11	1.21258	0.606288	+	0.795245i	$0.292658\pi$
$822$	0	0
$823$	− 4.30633e11i	− 0.938659i	−0.883023	−	0.469330i	$-0.844496\pi$
$823$	− 4.30633e11i	− 0.938659i	0.883023	−	0.469330i	$-0.155504\pi$
$824$	0	0
$825$	1.64006e11	0.354034
$826$	0	0
$827$	− 2.97622e11i	− 0.636273i	−0.948045	−	0.318136i	$-0.896943\pi$
$827$	− 2.97622e11i	− 0.636273i	0.948045	−	0.318136i	$-0.103057\pi$
$828$	0	0
$829$	−8.30846e10	−0.175915	−0.0879574	−	0.996124i	$-0.528034\pi$
$829$	−8.30846e10	−0.175915	−0.0879574	+	0.996124i	$0.528034\pi$
$830$	0	0
$831$	− 2.51305e11i	− 0.526984i
$832$	0	0
$833$	−1.86440e11	−0.387222
$834$	0	0
$835$	6.52863e11i	1.34300i
$836$	0	0
$837$	−1.66991e11	−0.340245
$838$	0	0
$839$	6.14008e11i	1.23916i	0.784935	+	0.619578i	$0.212696\pi$
$839$	6.14008e11i	1.23916i	−0.784935	+	0.619578i	$0.787304\pi$
$840$	0	0
$841$	−3.79722e11	−0.759070
$842$	0	0
$843$	4.72588e11i	0.935777i
$844$	0	0
$845$	−1.68990e11	−0.331461
$846$	0	0
$847$	− 2.13435e11i	− 0.414698i
$848$	0	0
$849$	2.52033e11	0.485095
$850$	0	0
$851$	− 4.50056e11i	− 0.858120i
$852$	0	0
$853$	5.27721e11	0.996801	0.498400	−	0.866947i	$-0.333921\pi$
$853$	5.27721e11	0.996801	0.498400	+	0.866947i	$0.333921\pi$
$854$	0	0
$855$	− 1.67571e11i	− 0.313569i
$856$	0	0
$857$	−5.34546e11	−0.990973	−0.495487	−	0.868616i	$-0.665010\pi$
$857$	−5.34546e11	−0.990973	−0.495487	+	0.868616i	$0.665010\pi$
$858$	0	0
$859$	− 4.91323e11i	− 0.902390i	−0.892426	−	0.451195i	$-0.850998\pi$
$859$	− 4.91323e11i	− 0.902390i	0.892426	−	0.451195i	$-0.149002\pi$
$860$	0	0
$861$	1.63281e11	0.297114
$862$	0	0
$863$	− 9.07667e10i	− 0.163638i	−0.996647	−	0.0818189i	$-0.973927\pi$
$863$	− 9.07667e10i	− 0.163638i	0.996647	−	0.0818189i	$-0.0260729\pi$
$864$	0	0
$865$	−8.59047e11	−1.53445
$866$	0	0
$867$	− 2.35727e11i	− 0.417190i
$868$	0	0
$869$	−3.00764e11	−0.527407
$870$	0	0
$871$	8.88486e10i	0.154375i
$872$	0	0
$873$	3.34367e11	0.575660
$874$	0	0
$875$	− 1.82757e11i	− 0.311776i
$876$	0	0
$877$	4.51551e11	0.763323	0.381661	−	0.924302i	$-0.375352\pi$
$877$	4.51551e11	0.763323	0.381661	+	0.924302i	$0.375352\pi$
$878$	0	0
$879$	− 7.30844e10i	− 0.122425i
$880$	0	0
$881$	−2.83136e11	−0.469993	−0.234996	−	0.971996i	$-0.575508\pi$
$881$	−2.83136e11	−0.469993	−0.234996	+	0.971996i	$0.575508\pi$
$882$	0	0
$883$	1.00664e12i	1.65589i	0.560808	+	0.827946i	$0.310490\pi$
$883$	1.00664e12i	1.65589i	−0.560808	+	0.827946i	$0.689510\pi$
$884$	0	0
$885$	6.62178e11	1.07945
$886$	0	0
$887$	− 5.46495e11i	− 0.882860i	−0.897296	−	0.441430i	$-0.854471\pi$
$887$	− 5.46495e11i	− 0.882860i	0.897296	−	0.441430i	$-0.145529\pi$
$888$	0	0
$889$	−2.24639e11	−0.359649
$890$	0	0
$891$	− 3.08539e10i	− 0.0489552i
$892$	0	0
$893$	4.89211e11	0.769291
$894$	0	0
$895$	− 7.95483e11i	− 1.23976i
$896$	0	0
$897$	2.01604e11	0.311407
$898$	0	0
$899$	− 5.66836e11i	− 0.867798i
$900$	0	0
$901$	4.55205e11	0.690730
$902$	0	0
$903$	6.35721e10i	0.0956128i
$904$	0	0
$905$	1.71711e12	2.55979
$906$	0	0
$907$	− 3.62390e11i	− 0.535485i	−0.963491	−	0.267742i	$-0.913722\pi$
$907$	− 3.62390e11i	− 0.535485i	0.963491	−	0.267742i	$-0.0862776\pi$
$908$	0	0
$909$	5.44224e10	0.0797118
$910$	0	0
$911$	8.94935e11i	1.29933i	0.760223	+	0.649663i	$0.225090\pi$
$911$	8.94935e11i	1.29933i	−0.760223	+	0.649663i	$0.774910\pi$
$912$	0	0
$913$	1.42030e11	0.204408
$914$	0	0
$915$	2.51894e11i	0.359363i
$916$	0	0
$917$	−2.77098e11	−0.391883
$918$	0	0
$919$	− 8.59958e11i	− 1.20563i	−0.797880	−	0.602817i	$-0.794045\pi$
$919$	− 8.59958e11i	− 1.20563i	0.797880	−	0.602817i	$-0.205955\pi$
$920$	0	0
$921$	3.00571e11	0.417743
$922$	0	0
$923$	− 3.62427e10i	− 0.0499359i
$924$	0	0
$925$	−1.43685e12	−1.96266
$926$	0	0
$927$	− 6.35026e10i	− 0.0859949i
$928$	0	0
$929$	−8.18700e11	−1.09916	−0.549581	−	0.835440i	$-0.685213\pi$
$929$	−8.18700e11	−1.09916	−0.549581	+	0.835440i	$0.685213\pi$
$930$	0	0
$931$	3.35967e11i	0.447196i
$932$	0	0
$933$	7.25833e11	0.957877
$934$	0	0
$935$	2.74287e11i	0.358887i
$936$	0	0
$937$	1.08922e12	1.41305	0.706524	−	0.707690i	$-0.250262\pi$
$937$	1.08922e12	1.41305	0.706524	+	0.707690i	$0.250262\pi$
$938$	0	0
$939$	− 1.27611e11i	− 0.164144i
$940$	0	0
$941$	5.35169e10	0.0682547	0.0341273	−	0.999417i	$-0.489135\pi$
$941$	5.35169e10	0.0682547	0.0341273	+	0.999417i	$0.489135\pi$
$942$	0	0
$943$	4.81209e11i	0.608537i
$944$	0	0
$945$	−1.22143e11	−0.153159
$946$	0	0
$947$	− 1.39610e12i	− 1.73587i	−0.496676	−	0.867936i	$-0.665446\pi$
$947$	− 1.39610e12i	− 1.73587i	0.496676	−	0.867936i	$-0.334554\pi$
$948$	0	0
$949$	−6.90299e11	−0.851084
$950$	0	0
$951$	9.24981e11i	1.13086i
$952$	0	0
$953$	9.02554e11	1.09421	0.547106	−	0.837063i	$-0.315729\pi$
$953$	9.02554e11	1.09421	0.547106	+	0.837063i	$0.315729\pi$
$954$	0	0
$955$	− 1.18741e12i	− 1.42754i
$956$	0	0
$957$	1.04731e11	0.124861
$958$	0	0
$959$	1.25294e11i	0.148134i
$960$	0	0
$961$	−1.81299e12	−2.12569
$962$	0	0
$963$	− 5.72691e10i	− 0.0665910i
$964$	0	0
$965$	−3.25554e11	−0.375417
$966$	0	0
$967$	− 3.01714e11i	− 0.345056i	−0.985005	−	0.172528i	$-0.944806\pi$
$967$	− 3.01714e11i	− 0.345056i	0.985005	−	0.172528i	$-0.0551935\pi$
$968$	0	0
$969$	1.63076e11	0.184967
$970$	0	0
$971$	3.17139e11i	0.356757i	0.983962	+	0.178378i	$0.0570851\pi$
$971$	3.17139e11i	0.356757i	−0.983962	+	0.178378i	$0.942915\pi$
$972$	0	0
$973$	−7.74136e11	−0.863706
$974$	0	0
$975$	− 6.43640e11i	− 0.712237i
$976$	0	0
$977$	−1.60066e12	−1.75679	−0.878396	−	0.477933i	$-0.841386\pi$
$977$	−1.60066e12	−1.75679	−0.878396	+	0.477933i	$0.841386\pi$
$978$	0	0
$979$	3.80630e11i	0.414354i
$980$	0	0
$981$	−3.23558e11	−0.349362
$982$	0	0
$983$	3.47527e11i	0.372198i	0.982531	+	0.186099i	$0.0595845\pi$
$983$	3.47527e11i	0.372198i	−0.982531	+	0.186099i	$0.940415\pi$
$984$	0	0
$985$	2.41594e12	2.56650
$986$	0	0
$987$	− 3.56588e11i	− 0.375749i
$988$	0	0
$989$	−1.87355e11	−0.195830
$990$	0	0
$991$	− 8.85224e11i	− 0.917822i	−0.888482	−	0.458911i	$-0.848240\pi$
$991$	− 8.85224e11i	− 0.917822i	0.888482	−	0.458911i	$-0.151760\pi$
$992$	0	0
$993$	1.04007e12	1.06971
$994$	0	0
$995$	− 1.61560e12i	− 1.64832i
$996$	0	0
$997$	−1.12582e12	−1.13943	−0.569716	−	0.821842i	$-0.692947\pi$
$997$	−1.12582e12	−1.13943	−0.569716	+	0.821842i	$0.692947\pi$
$998$	0	0
$999$	2.70309e11i	0.271393i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.b.127.24	yes	32
4.3	odd	2	inner	384.9.g.b.127.23	yes	32
8.3	odd	2		384.9.g.a.127.10	yes	32
8.5	even	2		384.9.g.a.127.9	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.9	✓	32	8.5	even	2
384.9.g.a.127.10	yes	32	8.3	odd	2
384.9.g.b.127.23	yes	32	4.3	odd	2	inner
384.9.g.b.127.24	yes	32	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	384.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} +966.582 q^{5} -1235.54i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} +966.582 q^{5} -1235.54i q^{7} -2187.00 q^{9} -6450.78i q^{11} -25316.0 q^{13} +45202.6i q^{15} -43990.0 q^{17} +79270.3i q^{19} +57780.4 q^{21} +170286. i q^{23} +543656. q^{25} -102276. i q^{27} +347166. q^{29} -1.63275e6i q^{31} +301673. q^{33} -1.19425e6i q^{35} -2.64294e6 q^{37} -1.18391e6i q^{39} +2.82589e6 q^{41} +1.10024e6i q^{43} -2.11392e6 q^{45} -6.17143e6i q^{47} +4.23824e6 q^{49} -2.05721e6i q^{51} -1.03479e7 q^{53} -6.23520e6i q^{55} -3.70711e6 q^{57} -1.46491e7i q^{59} +5.57255e6 q^{61} +2.70212e6i q^{63} -2.44700e7 q^{65} -3.50959e6i q^{67} -7.96349e6 q^{69} +1.43161e6i q^{71} +2.72673e7 q^{73} +2.54243e7i q^{75} -7.97019e6 q^{77} -4.66244e7i q^{79} +4.78297e6 q^{81} +2.20176e7i q^{83} -4.25199e7 q^{85} +1.62354e7i q^{87} -5.90052e7 q^{89} +3.12789e7i q^{91} +7.63562e7 q^{93} +7.66213e7i q^{95} -1.52888e8 q^{97} +1.41078e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 32 * q + 1344 * q^5 - 69984 * q^9	\( 32 q + 1344 q^{5} - 69984 q^{9} + 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} + 275520 q^{29} - 2421440 q^{37} - 4374720 q^{41} - 2939328 q^{45} - 14219104 q^{49} + 6224448 q^{53} + 3100032 q^{57} + 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} - 154517760 q^{77} + 153055008 q^{81} + 384830848 q^{85} - 182669760 q^{89} - 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) \) 32 * q + 1344 * q^5 - 69984 * q^9 + 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 + 275520 * q^29 - 2421440 * q^37 - 4374720 * q^41 - 2939328 * q^45 - 14219104 * q^49 + 6224448 * q^53 + 3100032 * q^57 + 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 - 154517760 * q^77 + 153055008 * q^81 + 384830848 * q^85 - 182669760 * q^89 - 149817600 * q^93 - 149408192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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