LMFDB - Embedded newform 384.9.g.a.127.29

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			127.29
Character	$\chi$	$=$	384.127
Dual form			384.9.g.a.127.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654i q^{3} +909.886 q^{5} +3388.15i q^{7} -2187.00 q^{9} +O(q^{10})$	$q-46.7654i q^{3} +909.886 q^{5} +3388.15i q^{7} -2187.00 q^{9} -2606.80i q^{11} -17282.5 q^{13} -42551.2i q^{15} +192.193 q^{17} +43536.8i q^{19} +158448. q^{21} +170200. i q^{23} +437267. q^{25} +102276. i q^{27} -1.07400e6 q^{29} -278276. i q^{31} -121908. q^{33} +3.08283e6i q^{35} -3.51333e6 q^{37} +808223. i q^{39} +20363.4 q^{41} +3.76618e6i q^{43} -1.98992e6 q^{45} -6.48421e6i q^{47} -5.71473e6 q^{49} -8987.96i q^{51} +746510. q^{53} -2.37189e6i q^{55} +2.03602e6 q^{57} -1.79604e7i q^{59} -1.96877e7 q^{61} -7.40988e6i q^{63} -1.57251e7 q^{65} +1.79496e7i q^{67} +7.95946e6 q^{69} -1.59590e7i q^{71} +2.27427e6 q^{73} -2.04490e7i q^{75} +8.83221e6 q^{77} -5.12517e7i q^{79} +4.78297e6 q^{81} -4.37064e7i q^{83} +174873. q^{85} +5.02259e7i q^{87} +7.31669e7 q^{89} -5.85557e7i q^{91} -1.30137e7 q^{93} +3.96135e7i q^{95} -3.05720e7 q^{97} +5.70107e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	909.886	1.45582	0.727909	−	0.685674i	$-0.240492\pi$
$5$	909.886	1.45582	0.727909	+	0.685674i	$0.240492\pi$
$6$	0	0
$7$	3388.15i	1.41114i	0.708640	+	0.705570i	$0.249309\pi$
$7$	3388.15i	1.41114i	−0.708640	+	0.705570i	$0.750691\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	− 2606.80i	− 0.178048i	−0.996029	−	0.0890239i	$-0.971625\pi$
$11$	− 2606.80i	− 0.178048i	0.996029	−	0.0890239i	$-0.0283748\pi$
$12$	0	0
$13$	−17282.5	−0.605109	−0.302554	−	0.953132i	$-0.597839\pi$
$13$	−17282.5	−0.605109	−0.302554	+	0.953132i	$0.597839\pi$
$14$	0	0
$15$	− 42551.2i	− 0.840517i
$16$	0	0
$17$	192.193	0.00230113	0.00115057	−	0.999999i	$-0.499634\pi$
$17$	192.193	0.00230113	0.00115057	+	0.999999i	$0.499634\pi$
$18$	0	0
$19$	43536.8i	0.334074i	0.985951	+	0.167037i	$0.0534199\pi$
$19$	43536.8i	0.334074i	−0.985951	+	0.167037i	$0.946580\pi$
$20$	0	0
$21$	158448.	0.814722
$22$	0	0
$23$	170200.i	0.608202i	0.952640	+	0.304101i	$0.0983560\pi$
$23$	170200.i	0.608202i	−0.952640	+	0.304101i	$0.901644\pi$
$24$	0	0
$25$	437267.	1.11940
$26$	0	0
$27$	102276.i	0.192450i
$28$	0	0
$29$	−1.07400e6	−1.51849	−0.759243	−	0.650807i	$-0.774431\pi$
$29$	−1.07400e6	−1.51849	−0.759243	+	0.650807i	$0.774431\pi$
$30$	0	0
$31$	− 278276.i	− 0.301321i	−0.988586	−	0.150660i	$-0.951860\pi$
$31$	− 278276.i	− 0.301321i	0.988586	−	0.150660i	$-0.0481400\pi$
$32$	0	0
$33$	−121908.	−0.102796
$34$	0	0
$35$	3.08283e6i	2.05436i
$36$	0	0
$37$	−3.51333e6	−1.87461	−0.937307	−	0.348504i	$-0.886690\pi$
$37$	−3.51333e6	−1.87461	−0.937307	+	0.348504i	$0.886690\pi$
$38$	0	0
$39$	808223.i	0.349360i
$40$	0	0
$41$	20363.4	0.00720635	0.00360318	−	0.999994i	$-0.498853\pi$
$41$	20363.4	0.00720635	0.00360318	+	0.999994i	$0.498853\pi$
$42$	0	0
$43$	3.76618e6i	1.10161i	0.834634	+	0.550805i	$0.185679\pi$
$43$	3.76618e6i	1.10161i	−0.834634	+	0.550805i	$0.814321\pi$
$44$	0	0
$45$	−1.98992e6	−0.485272
$46$	0	0
$47$	− 6.48421e6i	− 1.32882i	−0.747369	−	0.664409i	$-0.768683\pi$
$47$	− 6.48421e6i	− 1.32882i	0.747369	−	0.664409i	$-0.231317\pi$
$48$	0	0
$49$	−5.71473e6	−0.991315
$50$	0	0
$51$	− 8987.96i	− 0.00132856i
$52$	0	0
$53$	746510.	0.0946089	0.0473045	−	0.998881i	$-0.484937\pi$
$53$	746510.	0.0946089	0.0473045	+	0.998881i	$0.484937\pi$
$54$	0	0
$55$	− 2.37189e6i	− 0.259205i
$56$	0	0
$57$	2.03602e6	0.192878
$58$	0	0
$59$	− 1.79604e7i	− 1.48221i	−0.671392	−	0.741103i	$-0.734303\pi$
$59$	− 1.79604e7i	− 1.48221i	0.671392	−	0.741103i	$-0.265697\pi$
$60$	0	0
$61$	−1.96877e7	−1.42192	−0.710961	−	0.703232i	$-0.751740\pi$
$61$	−1.96877e7	−1.42192	−0.710961	+	0.703232i	$0.751740\pi$
$62$	0	0
$63$	− 7.40988e6i	− 0.470380i
$64$	0	0
$65$	−1.57251e7	−0.880928
$66$	0	0
$67$	1.79496e7i	0.890749i	0.895344	+	0.445374i	$0.146929\pi$
$67$	1.79496e7i	0.890749i	−0.895344	+	0.445374i	$0.853071\pi$
$68$	0	0
$69$	7.95946e6	0.351146
$70$	0	0
$71$	− 1.59590e7i	− 0.628018i	−0.949420	−	0.314009i	$-0.898328\pi$
$71$	− 1.59590e7i	− 0.628018i	0.949420	−	0.314009i	$-0.101672\pi$
$72$	0	0
$73$	2.27427e6	0.0800847	0.0400424	−	0.999198i	$-0.487251\pi$
$73$	2.27427e6	0.0800847	0.0400424	+	0.999198i	$0.487251\pi$
$74$	0	0
$75$	− 2.04490e7i	− 0.646288i
$76$	0	0
$77$	8.83221e6	0.251250
$78$	0	0
$79$	− 5.12517e7i	− 1.31583i	−0.753092	−	0.657915i	$-0.771438\pi$
$79$	− 5.12517e7i	− 1.31583i	0.753092	−	0.657915i	$-0.228562\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	− 4.37064e7i	− 0.920943i	−0.887675	−	0.460471i	$-0.847680\pi$
$83$	− 4.37064e7i	− 0.920943i	0.887675	−	0.460471i	$-0.152320\pi$
$84$	0	0
$85$	174873.	0.00335003
$86$	0	0
$87$	5.02259e7i	0.876699i
$88$	0	0
$89$	7.31669e7	1.16615	0.583075	−	0.812418i	$-0.301849\pi$
$89$	7.31669e7	1.16615	0.583075	+	0.812418i	$0.301849\pi$
$90$	0	0
$91$	− 5.85557e7i	− 0.853893i
$92$	0	0
$93$	−1.30137e7	−0.173967
$94$	0	0
$95$	3.96135e7i	0.486350i
$96$	0	0
$97$	−3.05720e7	−0.345332	−0.172666	−	0.984980i	$-0.555238\pi$
$97$	−3.05720e7	−0.345332	−0.172666	+	0.984980i	$0.555238\pi$
$98$	0	0
$99$	5.70107e6i	0.0593493i
$100$	0	0
$101$	1.37405e8	1.32043	0.660217	−	0.751075i	$-0.270464\pi$
$101$	1.37405e8	1.32043	0.660217	+	0.751075i	$0.270464\pi$
$102$	0	0
$103$	2.15130e7i	0.191140i	0.995423	+	0.0955699i	$0.0304673\pi$
$103$	2.15130e7i	0.191140i	−0.995423	+	0.0955699i	$0.969533\pi$
$104$	0	0
$105$	1.44170e8	1.18609
$106$	0	0
$107$	− 1.53392e8i	− 1.17022i	−0.810953	−	0.585111i	$-0.801051\pi$
$107$	− 1.53392e8i	− 1.17022i	0.810953	−	0.585111i	$-0.198949\pi$
$108$	0	0
$109$	−2.69536e8	−1.90946	−0.954730	−	0.297473i	$-0.903856\pi$
$109$	−2.69536e8	−1.90946	−0.954730	+	0.297473i	$0.903856\pi$
$110$	0	0
$111$	1.64302e8i	1.08231i
$112$	0	0
$113$	−2.22215e7	−0.136288	−0.0681442	−	0.997675i	$-0.521708\pi$
$113$	−2.22215e7	−0.136288	−0.0681442	+	0.997675i	$0.521708\pi$
$114$	0	0
$115$	1.54862e8i	0.885431i
$116$	0	0
$117$	3.77969e7	0.201703
$118$	0	0
$119$	651177.i	0.00324722i
$120$	0	0
$121$	2.07563e8	0.968299
$122$	0	0
$123$	− 952303.i	− 0.00416059i
$124$	0	0
$125$	4.24392e7	0.173831
$126$	0	0
$127$	3.26078e8i	1.25345i	0.779241	+	0.626724i	$0.215605\pi$
$127$	3.26078e8i	1.25345i	−0.779241	+	0.626724i	$0.784395\pi$
$128$	0	0
$129$	1.76127e8	0.636015
$130$	0	0
$131$	− 4.51070e8i	− 1.53165i	−0.643051	−	0.765823i	$-0.722332\pi$
$131$	− 4.51070e8i	− 1.53165i	0.643051	−	0.765823i	$-0.277668\pi$
$132$	0	0
$133$	−1.47509e8	−0.471425
$134$	0	0
$135$	9.30594e7i	0.280172i
$136$	0	0
$137$	−5.76361e8	−1.63611	−0.818055	−	0.575139i	$-0.804948\pi$
$137$	−5.76361e8	−1.63611	−0.818055	+	0.575139i	$0.804948\pi$
$138$	0	0
$139$	4.27296e8i	1.14464i	0.820029	+	0.572321i	$0.193957\pi$
$139$	4.27296e8i	1.14464i	−0.820029	+	0.572321i	$0.806043\pi$
$140$	0	0
$141$	−3.03237e8	−0.767194
$142$	0	0
$143$	4.50520e7i	0.107738i
$144$	0	0
$145$	−9.77214e8	−2.21064
$146$	0	0
$147$	2.67252e8i	0.572336i
$148$	0	0
$149$	−1.50195e8	−0.304727	−0.152363	−	0.988325i	$-0.548688\pi$
$149$	−1.50195e8	−0.304727	−0.152363	+	0.988325i	$0.548688\pi$
$150$	0	0
$151$	− 7.05079e8i	− 1.35622i	−0.734961	−	0.678110i	$-0.762799\pi$
$151$	− 7.05079e8i	− 1.35622i	0.734961	−	0.678110i	$-0.237201\pi$
$152$	0	0
$153$	−420325.	−0.000767043 0
$154$	0	0
$155$	− 2.53199e8i	− 0.438668i
$156$	0	0
$157$	−2.75539e8	−0.453508	−0.226754	−	0.973952i	$-0.572811\pi$
$157$	−2.75539e8	−0.453508	−0.226754	+	0.973952i	$0.572811\pi$
$158$	0	0
$159$	− 3.49108e7i	− 0.0546225i
$160$	0	0
$161$	−5.76662e8	−0.858258
$162$	0	0
$163$	− 5.17113e8i	− 0.732546i	−0.930507	−	0.366273i	$-0.880634\pi$
$163$	− 5.17113e8i	− 0.732546i	0.930507	−	0.366273i	$-0.119366\pi$
$164$	0	0
$165$	−1.10922e8	−0.149652
$166$	0	0
$167$	− 1.03507e9i	− 1.33078i	−0.746497	−	0.665389i	$-0.768266\pi$
$167$	− 1.03507e9i	− 1.33078i	0.746497	−	0.665389i	$-0.231734\pi$
$168$	0	0
$169$	−5.17046e8	−0.633843
$170$	0	0
$171$	− 9.52150e7i	− 0.111358i
$172$	0	0
$173$	1.56191e9	1.74370	0.871851	−	0.489771i	$-0.162920\pi$
$173$	1.56191e9	1.74370	0.871851	+	0.489771i	$0.162920\pi$
$174$	0	0
$175$	1.48153e9i	1.57964i
$176$	0	0
$177$	−8.39926e8	−0.855752
$178$	0	0
$179$	3.82436e8i	0.372517i	0.982501	+	0.186259i	$0.0596362\pi$
$179$	3.82436e8i	0.372517i	−0.982501	+	0.186259i	$0.940364\pi$
$180$	0	0
$181$	−1.05321e9	−0.981296	−0.490648	−	0.871358i	$-0.663240\pi$
$181$	−1.05321e9	−0.981296	−0.490648	+	0.871358i	$0.663240\pi$
$182$	0	0
$183$	9.20702e8i	0.820947i
$184$	0	0
$185$	−3.19673e9	−2.72910
$186$	0	0
$187$	− 501008.i	0 0.000409711i
$188$	0	0
$189$	−3.46526e8	−0.271574
$190$	0	0
$191$	3.53145e8i	0.265350i	0.991160	+	0.132675i	$0.0423567\pi$
$191$	3.53145e8i	0.265350i	−0.991160	+	0.132675i	$0.957643\pi$
$192$	0	0
$193$	2.41226e9	1.73858	0.869292	−	0.494299i	$-0.164575\pi$
$193$	2.41226e9	1.73858	0.869292	+	0.494299i	$0.164575\pi$
$194$	0	0
$195$	7.35391e8i	0.508604i
$196$	0	0
$197$	−2.67345e9	−1.77504	−0.887518	−	0.460772i	$-0.847573\pi$
$197$	−2.67345e9	−1.77504	−0.887518	+	0.460772i	$0.847573\pi$
$198$	0	0
$199$	2.61653e9i	1.66845i	0.551425	+	0.834224i	$0.314084\pi$
$199$	2.61653e9i	1.66845i	−0.551425	+	0.834224i	$0.685916\pi$
$200$	0	0
$201$	8.39419e8	0.514274
$202$	0	0
$203$	− 3.63886e9i	− 2.14280i
$204$	0	0
$205$	1.85284e7	0.0104911
$206$	0	0
$207$	− 3.72227e8i	− 0.202734i
$208$	0	0
$209$	1.13492e8	0.0594811
$210$	0	0
$211$	− 1.10751e9i	− 0.558749i	−0.960182	−	0.279374i	$-0.909873\pi$
$211$	− 1.10751e9i	− 0.558749i	0.960182	−	0.279374i	$-0.0901270\pi$
$212$	0	0
$213$	−7.46328e8	−0.362586
$214$	0	0
$215$	3.42680e9i	1.60374i
$216$	0	0
$217$	9.42839e8	0.425205
$218$	0	0
$219$	− 1.06357e8i	− 0.0462369i
$220$	0	0
$221$	−3.32157e6	−0.00139243
$222$	0	0
$223$	4.43394e9i	1.79296i	0.443086	+	0.896479i	$0.353884\pi$
$223$	4.43394e9i	1.79296i	−0.443086	+	0.896479i	$0.646116\pi$
$224$	0	0
$225$	−9.56304e8	−0.373135
$226$	0	0
$227$	1.27528e9i	0.480288i	0.970737	+	0.240144i	$0.0771946\pi$
$227$	1.27528e9i	0.480288i	−0.970737	+	0.240144i	$0.922805\pi$
$228$	0	0
$229$	7.18369e7	0.0261219	0.0130610	−	0.999915i	$-0.495842\pi$
$229$	7.18369e7	0.0261219	0.0130610	+	0.999915i	$0.495842\pi$
$230$	0	0
$231$	− 4.13042e8i	− 0.145059i
$232$	0	0
$233$	5.30303e8	0.179929	0.0899643	−	0.995945i	$-0.471325\pi$
$233$	5.30303e8	0.179929	0.0899643	+	0.995945i	$0.471325\pi$
$234$	0	0
$235$	− 5.89989e9i	− 1.93452i
$236$	0	0
$237$	−2.39681e9	−0.759695
$238$	0	0
$239$	− 1.02466e9i	− 0.314043i	−0.987595	−	0.157021i	$-0.949811\pi$
$239$	− 1.02466e9i	− 0.314043i	0.987595	−	0.157021i	$-0.0501891\pi$
$240$	0	0
$241$	−5.39176e9	−1.59832	−0.799158	−	0.601121i	$-0.794721\pi$
$241$	−5.39176e9	−1.59832	−0.799158	+	0.601121i	$0.794721\pi$
$242$	0	0
$243$	− 2.23677e8i	− 0.0641500i
$244$	0	0
$245$	−5.19976e9	−1.44317
$246$	0	0
$247$	− 7.52426e8i	− 0.202151i
$248$	0	0
$249$	−2.04395e9	−0.531707
$250$	0	0
$251$	− 5.76938e8i	− 0.145356i	−0.997355	−	0.0726782i	$-0.976845\pi$
$251$	− 5.76938e8i	− 0.145356i	0.997355	−	0.0726782i	$-0.0231546\pi$
$252$	0	0
$253$	4.43677e8	0.108289
$254$	0	0
$255$	− 8.17802e6i	− 0.00193414i
$256$	0	0
$257$	3.70584e9	0.849482	0.424741	−	0.905315i	$-0.360365\pi$
$257$	3.70584e9	0.849482	0.424741	+	0.905315i	$0.360365\pi$
$258$	0	0
$259$	− 1.19037e10i	− 2.64534i
$260$	0	0
$261$	2.34883e9	0.506162
$262$	0	0
$263$	7.37323e9i	1.54111i	0.637372	+	0.770557i	$0.280022\pi$
$263$	7.37323e9i	1.54111i	−0.637372	+	0.770557i	$0.719978\pi$
$264$	0	0
$265$	6.79239e8	0.137733
$266$	0	0
$267$	− 3.42168e9i	− 0.673277i
$268$	0	0
$269$	−9.98678e8	−0.190729	−0.0953644	−	0.995442i	$-0.530402\pi$
$269$	−9.98678e8	−0.190729	−0.0953644	+	0.995442i	$0.530402\pi$
$270$	0	0
$271$	4.56985e8i	0.0847275i	0.999102	+	0.0423638i	$0.0134888\pi$
$271$	4.56985e8i	0.0847275i	−0.999102	+	0.0423638i	$0.986511\pi$
$272$	0	0
$273$	−2.73838e9	−0.492995
$274$	0	0
$275$	− 1.13987e9i	− 0.199308i
$276$	0	0
$277$	−2.55281e9	−0.433610	−0.216805	−	0.976215i	$-0.569564\pi$
$277$	−2.55281e9	−0.433610	−0.216805	+	0.976215i	$0.569564\pi$
$278$	0	0
$279$	6.08589e8i	0.100440i
$280$	0	0
$281$	−4.76804e9	−0.764742	−0.382371	−	0.924009i	$-0.624892\pi$
$281$	−4.76804e9	−0.764742	−0.382371	+	0.924009i	$0.624892\pi$
$282$	0	0
$283$	7.50117e9i	1.16945i	0.811230	+	0.584727i	$0.198798\pi$
$283$	7.50117e9i	1.16945i	−0.811230	+	0.584727i	$0.801202\pi$
$284$	0	0
$285$	1.85254e9	0.280795
$286$	0	0
$287$	6.89943e7i	0.0101692i
$288$	0	0
$289$	−6.97572e9	−0.999995
$290$	0	0
$291$	1.42971e9i	0.199377i
$292$	0	0
$293$	−1.15364e10	−1.56531	−0.782656	−	0.622455i	$-0.786135\pi$
$293$	−1.15364e10	−1.56531	−0.782656	+	0.622455i	$0.786135\pi$
$294$	0	0
$295$	− 1.63419e10i	− 2.15782i
$296$	0	0
$297$	2.66613e8	0.0342653
$298$	0	0
$299$	− 2.94148e9i	− 0.368028i
$300$	0	0
$301$	−1.27604e10	−1.55453
$302$	0	0
$303$	− 6.42579e9i	− 0.762353i
$304$	0	0
$305$	−1.79136e10	−2.07006
$306$	0	0
$307$	− 6.53393e9i	− 0.735565i	−0.929912	−	0.367783i	$-0.880117\pi$
$307$	− 6.53393e9i	− 0.735565i	0.929912	−	0.367783i	$-0.119883\pi$
$308$	0	0
$309$	1.00606e9	0.110355
$310$	0	0
$311$	5.26651e9i	0.562965i	0.959566	+	0.281482i	$0.0908261\pi$
$311$	5.26651e9i	0.562965i	−0.959566	+	0.281482i	$0.909174\pi$
$312$	0	0
$313$	−4.07975e9	−0.425066	−0.212533	−	0.977154i	$-0.568171\pi$
$313$	−4.07975e9	−0.425066	−0.212533	+	0.977154i	$0.568171\pi$
$314$	0	0
$315$	− 6.74214e9i	− 0.684787i
$316$	0	0
$317$	−8.90224e9	−0.881581	−0.440791	−	0.897610i	$-0.645302\pi$
$317$	−8.90224e9	−0.881581	−0.440791	+	0.897610i	$0.645302\pi$
$318$	0	0
$319$	2.79969e9i	0.270363i
$320$	0	0
$321$	−7.17344e9	−0.675628
$322$	0	0
$323$	8.36746e6i	0 0.000768747i
$324$	0	0
$325$	−7.55708e9	−0.677361
$326$	0	0
$327$	1.26049e10i	1.10243i
$328$	0	0
$329$	2.19695e10	1.87515
$330$	0	0
$331$	8.98771e9i	0.748751i	0.927277	+	0.374375i	$0.122143\pi$
$331$	8.98771e9i	0.748751i	−0.927277	+	0.374375i	$0.877857\pi$
$332$	0	0
$333$	7.68365e9	0.624871
$334$	0	0
$335$	1.63321e10i	1.29677i
$336$	0	0
$337$	−2.13401e10	−1.65454	−0.827269	−	0.561805i	$-0.810107\pi$
$337$	−2.13401e10	−1.65454	−0.827269	+	0.561805i	$0.810107\pi$
$338$	0	0
$339$	1.03919e9i	0.0786861i
$340$	0	0
$341$	−7.25409e8	−0.0536495
$342$	0	0
$343$	1.69631e8i	0.0122554i
$344$	0	0
$345$	7.24220e9	0.511204
$346$	0	0
$347$	2.42381e10i	1.67179i	0.548890	+	0.835895i	$0.315051\pi$
$347$	2.42381e10i	1.67179i	−0.548890	+	0.835895i	$0.684949\pi$
$348$	0	0
$349$	8.96748e8	0.0604461	0.0302231	−	0.999543i	$-0.490378\pi$
$349$	8.96748e8	0.0604461	0.0302231	+	0.999543i	$0.490378\pi$
$350$	0	0
$351$	− 1.76758e9i	− 0.116453i
$352$	0	0
$353$	6.78743e9	0.437126	0.218563	−	0.975823i	$-0.429863\pi$
$353$	6.78743e9	0.437126	0.218563	+	0.975823i	$0.429863\pi$
$354$	0	0
$355$	− 1.45209e10i	− 0.914279i
$356$	0	0
$357$	3.04525e7	0.00187478
$358$	0	0
$359$	1.71336e10i	1.03150i	0.856738	+	0.515751i	$0.172487\pi$
$359$	1.71336e10i	1.03150i	−0.856738	+	0.515751i	$0.827513\pi$
$360$	0	0
$361$	1.50881e10	0.888395
$362$	0	0
$363$	− 9.70678e9i	− 0.559048i
$364$	0	0
$365$	2.06932e9	0.116589
$366$	0	0
$367$	1.15134e10i	0.634656i	0.948316	+	0.317328i	$0.102786\pi$
$367$	1.15134e10i	0.634656i	−0.948316	+	0.317328i	$0.897214\pi$
$368$	0	0
$369$	−4.45348e7	−0.00240212
$370$	0	0
$371$	2.52929e9i	0.133506i
$372$	0	0
$373$	−1.58051e10	−0.816512	−0.408256	−	0.912868i	$-0.633863\pi$
$373$	−1.58051e10	−0.816512	−0.408256	+	0.912868i	$0.633863\pi$
$374$	0	0
$375$	− 1.98469e9i	− 0.100361i
$376$	0	0
$377$	1.85614e10	0.918849
$378$	0	0
$379$	2.45175e10i	1.18828i	0.804360	+	0.594142i	$0.202508\pi$
$379$	2.45175e10i	1.18828i	−0.804360	+	0.594142i	$0.797492\pi$
$380$	0	0
$381$	1.52492e10	0.723679
$382$	0	0
$383$	5.24358e9i	0.243687i	0.992549	+	0.121844i	$0.0388807\pi$
$383$	5.24358e9i	0.243687i	−0.992549	+	0.121844i	$0.961119\pi$
$384$	0	0
$385$	8.03631e9	0.365775
$386$	0	0
$387$	− 8.23665e9i	− 0.367203i
$388$	0	0
$389$	3.26057e10	1.42395	0.711977	−	0.702203i	$-0.247800\pi$
$389$	3.26057e10	1.42395	0.711977	+	0.702203i	$0.247800\pi$
$390$	0	0
$391$	3.27112e7i	0.00139955i
$392$	0	0
$393$	−2.10944e10	−0.884297
$394$	0	0
$395$	− 4.66332e10i	− 1.91561i
$396$	0	0
$397$	3.57643e10	1.43975	0.719876	−	0.694103i	$-0.244199\pi$
$397$	3.57643e10	1.43975	0.719876	+	0.694103i	$0.244199\pi$
$398$	0	0
$399$	6.89832e9i	0.272177i
$400$	0	0
$401$	−3.21622e10	−1.24385	−0.621925	−	0.783077i	$-0.713649\pi$
$401$	−3.21622e10	−1.24385	−0.621925	+	0.783077i	$0.713649\pi$
$402$	0	0
$403$	4.80931e9i	0.182332i
$404$	0	0
$405$	4.35196e9	0.161757
$406$	0	0
$407$	9.15854e9i	0.333771i
$408$	0	0
$409$	1.14513e10	0.409226	0.204613	−	0.978843i	$-0.434406\pi$
$409$	1.14513e10	0.409226	0.204613	+	0.978843i	$0.434406\pi$
$410$	0	0
$411$	2.69538e10i	0.944609i
$412$	0	0
$413$	6.08525e10	2.09160
$414$	0	0
$415$	− 3.97678e10i	− 1.34072i
$416$	0	0
$417$	1.99827e10	0.660860
$418$	0	0
$419$	4.79690e9i	0.155634i	0.996968	+	0.0778170i	$0.0247950\pi$
$419$	4.79690e9i	0.155634i	−0.996968	+	0.0778170i	$0.975205\pi$
$420$	0	0
$421$	−3.69529e10	−1.17630	−0.588152	−	0.808750i	$-0.700144\pi$
$421$	−3.69529e10	−1.17630	−0.588152	+	0.808750i	$0.700144\pi$
$422$	0	0
$423$	1.41810e10i	0.442940i
$424$	0	0
$425$	8.40396e7	0.00257590
$426$	0	0
$427$	− 6.67048e10i	− 2.00653i
$428$	0	0
$429$	2.10687e9	0.0622027
$430$	0	0
$431$	− 5.85350e10i	− 1.69632i	−0.529743	−	0.848158i	$-0.677712\pi$
$431$	− 5.85350e10i	− 1.69632i	0.529743	−	0.848158i	$-0.322288\pi$
$432$	0	0
$433$	3.22405e10	0.917170	0.458585	−	0.888651i	$-0.348356\pi$
$433$	3.22405e10	0.917170	0.458585	+	0.888651i	$0.348356\pi$
$434$	0	0
$435$	4.56998e10i	1.27631i
$436$	0	0
$437$	−7.40996e9	−0.203184
$438$	0	0
$439$	− 2.20279e10i	− 0.593083i	−0.955020	−	0.296541i	$-0.904167\pi$
$439$	− 2.20279e10i	− 0.593083i	0.955020	−	0.296541i	$-0.0958333\pi$
$440$	0	0
$441$	1.24981e10	0.330438
$442$	0	0
$443$	4.97540e10i	1.29185i	0.763399	+	0.645927i	$0.223529\pi$
$443$	4.97540e10i	1.29185i	−0.763399	+	0.645927i	$0.776471\pi$
$444$	0	0
$445$	6.65735e10	1.69770
$446$	0	0
$447$	7.02393e9i	0.175934i
$448$	0	0
$449$	−6.64008e10	−1.63376	−0.816880	−	0.576808i	$-0.804298\pi$
$449$	−6.64008e10	−1.63376	−0.816880	+	0.576808i	$0.804298\pi$
$450$	0	0
$451$	− 5.30833e7i	− 0.00128308i
$452$	0	0
$453$	−3.29733e10	−0.783014
$454$	0	0
$455$	− 5.32790e10i	− 1.24311i
$456$	0	0
$457$	3.58250e10	0.821337	0.410669	−	0.911785i	$-0.365295\pi$
$457$	3.58250e10	0.821337	0.410669	+	0.911785i	$0.365295\pi$
$458$	0	0
$459$	1.96567e7i	0 0.000442853i
$460$	0	0
$461$	1.25630e9	0.0278157	0.0139078	−	0.999903i	$-0.495573\pi$
$461$	1.25630e9	0.0278157	0.0139078	+	0.999903i	$0.495573\pi$
$462$	0	0
$463$	− 1.30243e10i	− 0.283420i	−0.989908	−	0.141710i	$-0.954740\pi$
$463$	− 1.30243e10i	− 0.283420i	0.989908	−	0.141710i	$-0.0452601\pi$
$464$	0	0
$465$	−1.18410e10	−0.253265
$466$	0	0
$467$	3.27697e10i	0.688977i	0.938791	+	0.344489i	$0.111948\pi$
$467$	3.27697e10i	0.688977i	−0.938791	+	0.344489i	$0.888052\pi$
$468$	0	0
$469$	−6.08158e10	−1.25697
$470$	0	0
$471$	1.28857e10i	0.261833i
$472$	0	0
$473$	9.81768e9	0.196139
$474$	0	0
$475$	1.90372e10i	0.373964i
$476$	0	0
$477$	−1.63262e9	−0.0315363
$478$	0	0
$479$	− 4.37719e10i	− 0.831483i	−0.909483	−	0.415741i	$-0.863522\pi$
$479$	− 4.37719e10i	− 0.831483i	0.909483	−	0.415741i	$-0.136478\pi$
$480$	0	0
$481$	6.07192e10	1.13435
$482$	0	0
$483$	2.69678e10i	0.495515i
$484$	0	0
$485$	−2.78170e10	−0.502740
$486$	0	0
$487$	2.58409e9i	0.0459400i	0.999736	+	0.0229700i	$0.00731222\pi$
$487$	2.58409e9i	0.0459400i	−0.999736	+	0.0229700i	$0.992688\pi$
$488$	0	0
$489$	−2.41830e10	−0.422936
$490$	0	0
$491$	3.22146e10i	0.554276i	0.960830	+	0.277138i	$0.0893860\pi$
$491$	3.22146e10i	0.554276i	−0.960830	+	0.277138i	$0.910614\pi$
$492$	0	0
$493$	−2.06414e8	−0.00349424
$494$	0	0
$495$	5.18732e9i	0.0864017i
$496$	0	0
$497$	5.40714e10	0.886221
$498$	0	0
$499$	7.88352e10i	1.27150i	0.771893	+	0.635752i	$0.219310\pi$
$499$	7.88352e10i	1.27150i	−0.771893	+	0.635752i	$0.780690\pi$
$500$	0	0
$501$	−4.84057e10	−0.768325
$502$	0	0
$503$	− 4.42656e10i	− 0.691503i	−0.938326	−	0.345751i	$-0.887624\pi$
$503$	− 4.42656e10i	− 0.691503i	0.938326	−	0.345751i	$-0.112376\pi$
$504$	0	0
$505$	1.25023e11	1.92231
$506$	0	0
$507$	2.41798e10i	0.365950i
$508$	0	0
$509$	9.11890e10	1.35854	0.679268	−	0.733890i	$-0.262297\pi$
$509$	9.11890e10	1.35854	0.679268	+	0.733890i	$0.262297\pi$
$510$	0	0
$511$	7.70555e9i	0.113011i
$512$	0	0
$513$	−4.45277e9	−0.0642925
$514$	0	0
$515$	1.95743e10i	0.278265i
$516$	0	0
$517$	−1.69030e10	−0.236593
$518$	0	0
$519$	− 7.30434e10i	− 1.00673i
$520$	0	0
$521$	4.97962e10	0.675842	0.337921	−	0.941175i	$-0.390276\pi$
$521$	4.97962e10	0.675842	0.337921	+	0.941175i	$0.390276\pi$
$522$	0	0
$523$	6.46212e9i	0.0863711i	0.999067	+	0.0431856i	$0.0137507\pi$
$523$	6.46212e9i	0.0863711i	−0.999067	+	0.0431856i	$0.986249\pi$
$524$	0	0
$525$	6.92841e10	0.912003
$526$	0	0
$527$	− 5.34826e7i	0 0.000693378i
$528$	0	0
$529$	4.93430e10	0.630090
$530$	0	0
$531$	3.92794e10i	0.494068i
$532$	0	0
$533$	−3.51931e8	−0.00436063
$534$	0	0
$535$	− 1.39569e11i	− 1.70363i
$536$	0	0
$537$	1.78847e10	0.215073
$538$	0	0
$539$	1.48972e10i	0.176502i
$540$	0	0
$541$	−2.40351e10	−0.280581	−0.140290	−	0.990110i	$-0.544804\pi$
$541$	−2.40351e10	−0.280581	−0.140290	+	0.990110i	$0.544804\pi$
$542$	0	0
$543$	4.92537e10i	0.566552i
$544$	0	0
$545$	−2.45247e11	−2.77983
$546$	0	0
$547$	− 8.14216e10i	− 0.909474i	−0.890626	−	0.454737i	$-0.849733\pi$
$547$	− 8.14216e10i	− 0.909474i	0.890626	−	0.454737i	$-0.150267\pi$
$548$	0	0
$549$	4.30570e10	0.473974
$550$	0	0
$551$	− 4.67584e10i	− 0.507287i
$552$	0	0
$553$	1.73648e11	1.85682
$554$	0	0
$555$	1.49496e11i	1.57564i
$556$	0	0
$557$	9.09868e10	0.945274	0.472637	−	0.881257i	$-0.343302\pi$
$557$	9.09868e10	0.945274	0.472637	+	0.881257i	$0.343302\pi$
$558$	0	0
$559$	− 6.50891e10i	− 0.666594i
$560$	0	0
$561$	−2.34298e7	−0.000236547 0
$562$	0	0
$563$	− 3.93250e10i	− 0.391413i	−0.980663	−	0.195706i	$-0.937300\pi$
$563$	− 3.93250e10i	− 0.391413i	0.980663	−	0.195706i	$-0.0627000\pi$
$564$	0	0
$565$	−2.02190e10	−0.198411
$566$	0	0
$567$	1.62054e10i	0.156793i
$568$	0	0
$569$	1.33087e11	1.26966	0.634829	−	0.772653i	$-0.281071\pi$
$569$	1.33087e11	1.26966	0.634829	+	0.772653i	$0.281071\pi$
$570$	0	0
$571$	− 1.06912e11i	− 1.00573i	−0.864364	−	0.502867i	$-0.832279\pi$
$571$	− 1.06912e11i	− 1.00573i	0.864364	−	0.502867i	$-0.167721\pi$
$572$	0	0
$573$	1.65149e10	0.153200
$574$	0	0
$575$	7.44228e10i	0.680824i
$576$	0	0
$577$	−5.31659e10	−0.479656	−0.239828	−	0.970815i	$-0.577091\pi$
$577$	−5.31659e10	−0.479656	−0.239828	+	0.970815i	$0.577091\pi$
$578$	0	0
$579$	− 1.12810e11i	− 1.00377i
$580$	0	0
$581$	1.48084e11	1.29958
$582$	0	0
$583$	− 1.94600e9i	− 0.0168449i
$584$	0	0
$585$	3.43908e10	0.293643
$586$	0	0
$587$	− 2.75698e10i	− 0.232210i	−0.993237	−	0.116105i	$-0.962959\pi$
$587$	− 2.75698e10i	− 0.232210i	0.993237	−	0.116105i	$-0.0370410\pi$
$588$	0	0
$589$	1.21152e10	0.100663
$590$	0	0
$591$	1.25025e11i	1.02482i
$592$	0	0
$593$	1.16852e11	0.944971	0.472485	−	0.881338i	$-0.343357\pi$
$593$	1.16852e11	0.944971	0.472485	+	0.881338i	$0.343357\pi$
$594$	0	0
$595$	5.92497e8i	0.00472735i
$596$	0	0
$597$	1.22363e11	0.963279
$598$	0	0
$599$	− 6.40065e10i	− 0.497184i	−0.968608	−	0.248592i	$-0.920032\pi$
$599$	− 6.40065e10i	− 0.497184i	0.968608	−	0.248592i	$-0.0799678\pi$
$600$	0	0
$601$	1.46926e11	1.12616	0.563082	−	0.826401i	$-0.309616\pi$
$601$	1.46926e11	1.12616	0.563082	+	0.826401i	$0.309616\pi$
$602$	0	0
$603$	− 3.92557e10i	− 0.296916i
$604$	0	0
$605$	1.88859e11	1.40967
$606$	0	0
$607$	− 1.73354e11i	− 1.27696i	−0.769637	−	0.638482i	$-0.779563\pi$
$607$	− 1.73354e11i	− 1.27696i	0.769637	−	0.638482i	$-0.220437\pi$
$608$	0	0
$609$	−1.70173e11	−1.23714
$610$	0	0
$611$	1.12063e11i	0.804080i
$612$	0	0
$613$	−5.87360e10	−0.415970	−0.207985	−	0.978132i	$-0.566691\pi$
$613$	−5.87360e10	−0.415970	−0.207985	+	0.978132i	$0.566691\pi$
$614$	0	0
$615$	− 8.66487e8i	− 0.00605706i
$616$	0	0
$617$	7.26853e9	0.0501540	0.0250770	−	0.999686i	$-0.492017\pi$
$617$	7.26853e9	0.0501540	0.0250770	+	0.999686i	$0.492017\pi$
$618$	0	0
$619$	− 2.14695e11i	− 1.46238i	−0.682176	−	0.731188i	$-0.738966\pi$
$619$	− 2.14695e11i	− 1.46238i	0.682176	−	0.731188i	$-0.261034\pi$
$620$	0	0
$621$	−1.74073e10	−0.117049
$622$	0	0
$623$	2.47900e11i	1.64560i
$624$	0	0
$625$	−1.32193e11	−0.866338
$626$	0	0
$627$	− 5.30748e9i	− 0.0343414i
$628$	0	0
$629$	−6.75236e8	−0.00431373
$630$	0	0
$631$	8.99857e10i	0.567618i	0.958881	+	0.283809i	$0.0915981\pi$
$631$	8.99857e10i	0.567618i	−0.958881	+	0.283809i	$0.908402\pi$
$632$	0	0
$633$	−5.17930e10	−0.322594
$634$	0	0
$635$	2.96694e11i	1.82479i
$636$	0	0
$637$	9.87650e10	0.599854
$638$	0	0
$639$	3.49023e10i	0.209339i
$640$	0	0
$641$	4.33981e10	0.257062	0.128531	−	0.991705i	$-0.458974\pi$
$641$	4.33981e10	0.257062	0.128531	+	0.991705i	$0.458974\pi$
$642$	0	0
$643$	2.35977e11i	1.38046i	0.723588	+	0.690232i	$0.242492\pi$
$643$	2.35977e11i	1.38046i	−0.723588	+	0.690232i	$0.757508\pi$
$644$	0	0
$645$	1.60255e11	0.925921
$646$	0	0
$647$	2.83184e11i	1.61604i	0.589154	+	0.808020i	$0.299461\pi$
$647$	2.83184e11i	1.61604i	−0.589154	+	0.808020i	$0.700539\pi$
$648$	0	0
$649$	−4.68192e10	−0.263903
$650$	0	0
$651$	− 4.40922e10i	− 0.245492i
$652$	0	0
$653$	−1.83466e11	−1.00903	−0.504514	−	0.863404i	$-0.668328\pi$
$653$	−1.83466e11	−1.00903	−0.504514	+	0.863404i	$0.668328\pi$
$654$	0	0
$655$	− 4.10422e11i	− 2.22980i
$656$	0	0
$657$	−4.97382e9	−0.0266949
$658$	0	0
$659$	− 2.17514e11i	− 1.15331i	−0.816988	−	0.576654i	$-0.804358\pi$
$659$	− 2.17514e11i	− 1.15331i	0.816988	−	0.576654i	$-0.195642\pi$
$660$	0	0
$661$	−1.82614e11	−0.956595	−0.478297	−	0.878198i	$-0.658746\pi$
$661$	−1.82614e11	−0.956595	−0.478297	+	0.878198i	$0.658746\pi$
$662$	0	0
$663$	1.55335e8i	0 0.000803922i
$664$	0	0
$665$	−1.34216e11	−0.686308
$666$	0	0
$667$	− 1.82794e11i	− 0.923547i
$668$	0	0
$669$	2.07355e11	1.03516
$670$	0	0
$671$	5.13218e10i	0.253170i
$672$	0	0
$673$	−1.16177e11	−0.566319	−0.283159	−	0.959073i	$-0.591383\pi$
$673$	−1.16177e11	−0.566319	−0.283159	+	0.959073i	$0.591383\pi$
$674$	0	0
$675$	4.47219e10i	0.215429i
$676$	0	0
$677$	−3.24681e11	−1.54562	−0.772809	−	0.634638i	$-0.781149\pi$
$677$	−3.24681e11	−1.54562	−0.772809	+	0.634638i	$0.781149\pi$
$678$	0	0
$679$	− 1.03582e11i	− 0.487312i
$680$	0	0
$681$	5.96389e10	0.277294
$682$	0	0
$683$	3.20561e11i	1.47308i	0.676392	+	0.736542i	$0.263543\pi$
$683$	3.20561e11i	1.47308i	−0.676392	+	0.736542i	$0.736457\pi$
$684$	0	0
$685$	−5.24423e11	−2.38188
$686$	0	0
$687$	− 3.35948e9i	− 0.0150815i
$688$	0	0
$689$	−1.29016e10	−0.0572487
$690$	0	0
$691$	6.16306e10i	0.270324i	0.990824	+	0.135162i	$0.0431554\pi$
$691$	6.16306e10i	0.270324i	−0.990824	+	0.135162i	$0.956845\pi$
$692$	0	0
$693$	−1.93161e10	−0.0837501
$694$	0	0
$695$	3.88791e11i	1.66639i
$696$	0	0
$697$	3.91370e6	1.65828e−5 0
$698$	0	0
$699$	− 2.47998e10i	− 0.103882i
$700$	0	0
$701$	1.01257e11	0.419328	0.209664	−	0.977774i	$-0.432763\pi$
$701$	1.01257e11	0.419328	0.209664	+	0.977774i	$0.432763\pi$
$702$	0	0
$703$	− 1.52959e11i	− 0.626259i
$704$	0	0
$705$	−2.75911e11	−1.11689
$706$	0	0
$707$	4.65548e11i	1.86332i
$708$	0	0
$709$	2.54755e10	0.100818	0.0504090	−	0.998729i	$-0.483948\pi$
$709$	2.54755e10	0.100818	0.0504090	+	0.998729i	$0.483948\pi$
$710$	0	0
$711$	1.12088e11i	0.438610i
$712$	0	0
$713$	4.73625e10	0.183264
$714$	0	0
$715$	4.09922e10i	0.156847i
$716$	0	0
$717$	−4.79187e10	−0.181313
$718$	0	0
$719$	− 1.06593e11i	− 0.398852i	−0.979913	−	0.199426i	$-0.936092\pi$
$719$	− 1.06593e11i	− 0.398852i	0.979913	−	0.199426i	$-0.0639078\pi$
$720$	0	0
$721$	−7.28890e10	−0.269725
$722$	0	0
$723$	2.52148e11i	0.922788i
$724$	0	0
$725$	−4.69624e11	−1.69980
$726$	0	0
$727$	− 2.11304e11i	− 0.756433i	−0.925717	−	0.378216i	$-0.876538\pi$
$727$	− 2.11304e11i	− 0.756433i	0.925717	−	0.378216i	$-0.123462\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	7.23833e8i	0.00253495i
$732$	0	0
$733$	−2.58593e11	−0.895779	−0.447890	−	0.894089i	$-0.647824\pi$
$733$	−2.58593e11	−0.895779	−0.447890	+	0.894089i	$0.647824\pi$
$734$	0	0
$735$	2.43169e11i	0.833217i
$736$	0	0
$737$	4.67909e10	0.158596
$738$	0	0
$739$	− 5.06620e11i	− 1.69865i	−0.527869	−	0.849326i	$-0.677009\pi$
$739$	− 5.06620e11i	− 1.69865i	0.527869	−	0.849326i	$-0.322991\pi$
$740$	0	0
$741$	−3.51875e10	−0.116712
$742$	0	0
$743$	1.91050e11i	0.626892i	0.949606	+	0.313446i	$0.101483\pi$
$743$	1.91050e11i	0.626892i	−0.949606	+	0.313446i	$0.898517\pi$
$744$	0	0
$745$	−1.36660e11	−0.443627
$746$	0	0
$747$	9.55859e10i	0.306981i
$748$	0	0
$749$	5.19715e11	1.65135
$750$	0	0
$751$	1.45844e11i	0.458487i	0.973369	+	0.229244i	$0.0736253\pi$
$751$	1.45844e11i	0.458487i	−0.973369	+	0.229244i	$0.926375\pi$
$752$	0	0
$753$	−2.69807e10	−0.0839216
$754$	0	0
$755$	− 6.41541e11i	− 1.97441i
$756$	0	0
$757$	−3.58590e11	−1.09198	−0.545990	−	0.837792i	$-0.683846\pi$
$757$	−3.58590e11	−1.09198	−0.545990	+	0.837792i	$0.683846\pi$
$758$	0	0
$759$	− 2.07487e10i	− 0.0625207i
$760$	0	0
$761$	−4.99260e11	−1.48863	−0.744317	−	0.667826i	$-0.767225\pi$
$761$	−4.99260e11	−1.48863	−0.744317	+	0.667826i	$0.767225\pi$
$762$	0	0
$763$	− 9.13227e11i	− 2.69452i
$764$	0	0
$765$	−3.82448e8	−0.00111668
$766$	0	0
$767$	3.10401e11i	0.896895i
$768$	0	0
$769$	6.21782e11	1.77800	0.889002	−	0.457903i	$-0.151399\pi$
$769$	6.21782e11	1.77800	0.889002	+	0.457903i	$0.151399\pi$
$770$	0	0
$771$	− 1.73305e11i	− 0.490449i
$772$	0	0
$773$	1.46441e11	0.410153	0.205076	−	0.978746i	$-0.434256\pi$
$773$	1.46441e11	0.410153	0.205076	+	0.978746i	$0.434256\pi$
$774$	0	0
$775$	− 1.21681e11i	− 0.337300i
$776$	0	0
$777$	−5.56680e11	−1.52729
$778$	0	0
$779$	8.86559e8i	0.00240745i
$780$	0	0
$781$	−4.16019e10	−0.111817
$782$	0	0
$783$	− 1.09844e11i	− 0.292233i
$784$	0	0
$785$	−2.50709e11	−0.660224
$786$	0	0
$787$	− 6.52873e11i	− 1.70188i	−0.525260	−	0.850942i	$-0.676032\pi$
$787$	− 6.52873e11i	− 1.70188i	0.525260	−	0.850942i	$-0.323968\pi$
$788$	0	0
$789$	3.44812e11	0.889762
$790$	0	0
$791$	− 7.52896e10i	− 0.192322i
$792$	0	0
$793$	3.40253e11	0.860417
$794$	0	0
$795$	− 3.17649e10i	− 0.0795204i
$796$	0	0
$797$	−3.71487e11	−0.920683	−0.460342	−	0.887742i	$-0.652273\pi$
$797$	−3.71487e11	−0.920683	−0.460342	+	0.887742i	$0.652273\pi$
$798$	0	0
$799$	− 1.24622e9i	− 0.00305778i
$800$	0	0
$801$	−1.60016e11	−0.388717
$802$	0	0
$803$	− 5.92855e9i	− 0.0142589i
$804$	0	0
$805$	−5.24697e11	−1.24947
$806$	0	0
$807$	4.67036e10i	0.110117i
$808$	0	0
$809$	−7.65716e11	−1.78761	−0.893806	−	0.448453i	$-0.851975\pi$
$809$	−7.65716e11	−1.78761	−0.893806	+	0.448453i	$0.851975\pi$
$810$	0	0
$811$	3.07838e11i	0.711604i	0.934561	+	0.355802i	$0.115792\pi$
$811$	3.07838e11i	0.711604i	−0.934561	+	0.355802i	$0.884208\pi$
$812$	0	0
$813$	2.13711e10	0.0489175
$814$	0	0
$815$	− 4.70514e11i	− 1.06645i
$816$	0	0
$817$	−1.63968e11	−0.368019
$818$	0	0
$819$	1.28061e11i	0.284631i
$820$	0	0
$821$	2.17243e11	0.478161	0.239080	−	0.971000i	$-0.423154\pi$
$821$	2.17243e11	0.478161	0.239080	+	0.971000i	$0.423154\pi$
$822$	0	0
$823$	4.60542e11i	1.00385i	0.864911	+	0.501926i	$0.167375\pi$
$823$	4.60542e11i	1.00385i	−0.864911	+	0.501926i	$0.832625\pi$
$824$	0	0
$825$	−5.33063e10	−0.115070
$826$	0	0
$827$	− 3.00000e11i	− 0.641357i	−0.947188	−	0.320678i	$-0.896089\pi$
$827$	− 3.00000e11i	− 0.641357i	0.947188	−	0.320678i	$-0.103911\pi$
$828$	0	0
$829$	3.70548e11	0.784561	0.392281	−	0.919846i	$-0.371686\pi$
$829$	3.70548e11	0.784561	0.392281	+	0.919846i	$0.371686\pi$
$830$	0	0
$831$	1.19383e11i	0.250345i
$832$	0	0
$833$	−1.09833e9	−0.00228115
$834$	0	0
$835$	− 9.41800e11i	− 1.93737i
$836$	0	0
$837$	2.84609e10	0.0579892
$838$	0	0
$839$	6.39346e11i	1.29029i	0.764060	+	0.645146i	$0.223203\pi$
$839$	6.39346e11i	1.29029i	−0.764060	+	0.645146i	$0.776797\pi$
$840$	0	0
$841$	6.53222e11	1.30580
$842$	0	0
$843$	2.22979e11i	0.441524i
$844$	0	0
$845$	−4.70452e11	−0.922760
$846$	0	0
$847$	7.03255e11i	1.36641i
$848$	0	0
$849$	3.50795e11	0.675185
$850$	0	0
$851$	− 5.97968e11i	− 1.14014i
$852$	0	0
$853$	−1.41883e10	−0.0267999	−0.0133999	−	0.999910i	$-0.504265\pi$
$853$	−1.41883e10	−0.0267999	−0.0133999	+	0.999910i	$0.504265\pi$
$854$	0	0
$855$	− 8.66348e10i	− 0.162117i
$856$	0	0
$857$	1.57643e11	0.292248	0.146124	−	0.989266i	$-0.453320\pi$
$857$	1.57643e11	0.292248	0.146124	+	0.989266i	$0.453320\pi$
$858$	0	0
$859$	− 6.91837e11i	− 1.27067i	−0.772239	−	0.635333i	$-0.780863\pi$
$859$	− 6.91837e11i	− 1.27067i	0.772239	−	0.635333i	$-0.219137\pi$
$860$	0	0
$861$	3.22654e9	0.00587117
$862$	0	0
$863$	3.91952e11i	0.706625i	0.935505	+	0.353313i	$0.114945\pi$
$863$	3.91952e11i	0.706625i	−0.935505	+	0.353313i	$0.885055\pi$
$864$	0	0
$865$	1.42116e12	2.53851
$866$	0	0
$867$	3.26222e11i	0.577347i
$868$	0	0
$869$	−1.33603e11	−0.234281
$870$	0	0
$871$	− 3.10214e11i	− 0.539000i
$872$	0	0
$873$	6.68609e10	0.115111
$874$	0	0
$875$	1.43790e11i	0.245300i
$876$	0	0
$877$	−6.35624e11	−1.07449	−0.537244	−	0.843427i	$-0.680535\pi$
$877$	−6.35624e11	−1.07449	−0.537244	+	0.843427i	$0.680535\pi$
$878$	0	0
$879$	5.39505e11i	0.903733i
$880$	0	0
$881$	−1.15725e11	−0.192098	−0.0960489	−	0.995377i	$-0.530621\pi$
$881$	−1.15725e11	−0.192098	−0.0960489	+	0.995377i	$0.530621\pi$
$882$	0	0
$883$	4.57316e11i	0.752270i	0.926565	+	0.376135i	$0.122747\pi$
$883$	4.57316e11i	0.752270i	−0.926565	+	0.376135i	$0.877253\pi$
$884$	0	0
$885$	−7.64236e11	−1.24582
$886$	0	0
$887$	− 6.63016e10i	− 0.107110i	−0.998565	−	0.0535549i	$-0.982945\pi$
$887$	− 6.63016e10i	− 0.107110i	0.998565	−	0.0535549i	$-0.0170552\pi$
$888$	0	0
$889$	−1.10480e12	−1.76879
$890$	0	0
$891$	− 1.24682e10i	− 0.0197831i
$892$	0	0
$893$	2.82302e11	0.443923
$894$	0	0
$895$	3.47973e11i	0.542317i
$896$	0	0
$897$	−1.37559e11	−0.212481
$898$	0	0
$899$	2.98867e11i	0.457551i
$900$	0	0
$901$	1.43474e8	0.000217707 0
$902$	0	0
$903$	5.96744e11i	0.897506i
$904$	0	0
$905$	−9.58300e11	−1.42859
$906$	0	0
$907$	4.20491e11i	0.621337i	0.950518	+	0.310669i	$0.100553\pi$
$907$	4.20491e11i	0.621337i	−0.950518	+	0.310669i	$0.899447\pi$
$908$	0	0
$909$	−3.00505e11	−0.440145
$910$	0	0
$911$	− 7.72518e10i	− 0.112159i	−0.998426	−	0.0560796i	$-0.982140\pi$
$911$	− 7.72518e10i	− 0.112159i	0.998426	−	0.0560796i	$-0.0178601\pi$
$912$	0	0
$913$	−1.13934e11	−0.163972
$914$	0	0
$915$	8.37734e11i	1.19515i
$916$	0	0
$917$	1.52829e12	2.16137
$918$	0	0
$919$	− 1.11253e12i	− 1.55973i	−0.625949	−	0.779864i	$-0.715288\pi$
$919$	− 1.11253e12i	− 1.55973i	0.625949	−	0.779864i	$-0.284712\pi$
$920$	0	0
$921$	−3.05562e11	−0.424679
$922$	0	0
$923$	2.75811e11i	0.380019i
$924$	0	0
$925$	−1.53626e12	−2.09845
$926$	0	0
$927$	− 4.70488e10i	− 0.0637133i
$928$	0	0
$929$	−7.64282e11	−1.02610	−0.513052	−	0.858358i	$-0.671485\pi$
$929$	−7.64282e11	−1.02610	−0.513052	+	0.858358i	$0.671485\pi$
$930$	0	0
$931$	− 2.48801e11i	− 0.331172i
$932$	0	0
$933$	2.46290e11	0.325028
$934$	0	0
$935$	− 4.55860e8i	0 0.000596465i
$936$	0	0
$937$	1.38152e12	1.79225	0.896125	−	0.443801i	$-0.146370\pi$
$937$	1.38152e12	1.79225	0.896125	+	0.443801i	$0.146370\pi$
$938$	0	0
$939$	1.90791e11i	0.245412i
$940$	0	0
$941$	−1.01538e12	−1.29500	−0.647502	−	0.762064i	$-0.724186\pi$
$941$	−1.01538e12	−1.29500	−0.647502	+	0.762064i	$0.724186\pi$
$942$	0	0
$943$	3.46585e9i	0.00438292i
$944$	0	0
$945$	−3.15299e11	−0.395362
$946$	0	0
$947$	− 6.99146e11i	− 0.869297i	−0.900600	−	0.434649i	$-0.856873\pi$
$947$	− 6.99146e11i	− 0.869297i	0.900600	−	0.434649i	$-0.143127\pi$
$948$	0	0
$949$	−3.93050e10	−0.0484600
$950$	0	0
$951$	4.16317e11i	0.508981i
$952$	0	0
$953$	5.97891e11	0.724854	0.362427	−	0.932012i	$-0.381948\pi$
$953$	5.97891e11	0.724854	0.362427	+	0.932012i	$0.381948\pi$
$954$	0	0
$955$	3.21321e11i	0.386301i
$956$	0	0
$957$	1.30929e11	0.156094
$958$	0	0
$959$	− 1.95280e12i	− 2.30878i
$960$	0	0
$961$	7.75454e11	0.909206
$962$	0	0
$963$	3.35469e11i	0.390074i
$964$	0	0
$965$	2.19489e12	2.53106
$966$	0	0
$967$	− 2.21176e11i	− 0.252949i	−0.991970	−	0.126474i	$-0.959634\pi$
$967$	− 2.21176e11i	− 0.252949i	0.991970	−	0.126474i	$-0.0403661\pi$
$968$	0	0
$969$	3.91307e8	0.000443836 0
$970$	0	0
$971$	2.17180e11i	0.244311i	0.992511	+	0.122155i	$0.0389806\pi$
$971$	2.17180e11i	0.244311i	−0.992511	+	0.122155i	$0.961019\pi$
$972$	0	0
$973$	−1.44774e12	−1.61525
$974$	0	0
$975$	3.53410e11i	0.391075i
$976$	0	0
$977$	−1.27513e12	−1.39951	−0.699753	−	0.714385i	$-0.746706\pi$
$977$	−1.27513e12	−1.39951	−0.699753	+	0.714385i	$0.746706\pi$
$978$	0	0
$979$	− 1.90731e11i	− 0.207630i
$980$	0	0
$981$	5.89475e11	0.636487
$982$	0	0
$983$	9.52579e11i	1.02020i	0.860114	+	0.510102i	$0.170392\pi$
$983$	9.52579e11i	1.02020i	−0.860114	+	0.510102i	$0.829608\pi$
$984$	0	0
$985$	−2.43254e12	−2.58413
$986$	0	0
$987$	− 1.02741e12i	− 1.08262i
$988$	0	0
$989$	−6.41004e11	−0.670001
$990$	0	0
$991$	1.04065e12i	1.07897i	0.841995	+	0.539486i	$0.181381\pi$
$991$	1.04065e12i	1.07897i	−0.841995	+	0.539486i	$0.818619\pi$
$992$	0	0
$993$	4.20314e11	0.432291
$994$	0	0
$995$	2.38074e12i	2.42896i
$996$	0	0
$997$	7.26978e11	0.735767	0.367884	−	0.929872i	$-0.380082\pi$
$997$	7.26978e11	0.735767	0.367884	+	0.929872i	$0.380082\pi$
$998$	0	0
$999$	− 3.59329e11i	− 0.360770i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.a.127.29	✓	32
4.3	odd	2	inner	384.9.g.a.127.30	yes	32
8.3	odd	2		384.9.g.b.127.3	yes	32
8.5	even	2		384.9.g.b.127.4	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.29	✓	32	1.1	even	1	trivial
384.9.g.a.127.30	yes	32	4.3	odd	2	inner
384.9.g.b.127.3	yes	32	8.3	odd	2
384.9.g.b.127.4	yes	32	8.5	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	384.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-46.7654i q^{3} +909.886 q^{5} +3388.15i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q-46.7654i q^{3} +909.886 q^{5} +3388.15i q^{7} -2187.00 q^{9} -2606.80i q^{11} -17282.5 q^{13} -42551.2i q^{15} +192.193 q^{17} +43536.8i q^{19} +158448. q^{21} +170200. i q^{23} +437267. q^{25} +102276. i q^{27} -1.07400e6 q^{29} -278276. i q^{31} -121908. q^{33} +3.08283e6i q^{35} -3.51333e6 q^{37} +808223. i q^{39} +20363.4 q^{41} +3.76618e6i q^{43} -1.98992e6 q^{45} -6.48421e6i q^{47} -5.71473e6 q^{49} -8987.96i q^{51} +746510. q^{53} -2.37189e6i q^{55} +2.03602e6 q^{57} -1.79604e7i q^{59} -1.96877e7 q^{61} -7.40988e6i q^{63} -1.57251e7 q^{65} +1.79496e7i q^{67} +7.95946e6 q^{69} -1.59590e7i q^{71} +2.27427e6 q^{73} -2.04490e7i q^{75} +8.83221e6 q^{77} -5.12517e7i q^{79} +4.78297e6 q^{81} -4.37064e7i q^{83} +174873. q^{85} +5.02259e7i q^{87} +7.31669e7 q^{89} -5.85557e7i q^{91} -1.30137e7 q^{93} +3.96135e7i q^{95} -3.05720e7 q^{97} +5.70107e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more