LMFDB - Embedded newform 384.9.g.a.127.14

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			127.14
Character	$\chi$	$=$	384.127
Dual form			384.9.g.a.127.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} -441.863 q^{5} -408.575i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} -441.863 q^{5} -408.575i q^{7} -2187.00 q^{9} +13982.3i q^{11} +50503.9 q^{13} -20663.9i q^{15} +29394.2 q^{17} -465.954i q^{19} +19107.2 q^{21} -349779. i q^{23} -195382. q^{25} -102276. i q^{27} +413282. q^{29} +291268. i q^{31} -653888. q^{33} +180534. i q^{35} -3.18340e6 q^{37} +2.36183e6i q^{39} -3.90588e6 q^{41} -5.13172e6i q^{43} +966354. q^{45} +3.80555e6i q^{47} +5.59787e6 q^{49} +1.37463e6i q^{51} -1.21515e6 q^{53} -6.17826e6i q^{55} +21790.5 q^{57} +5.53322e6i q^{59} +8.14915e6 q^{61} +893553. i q^{63} -2.23158e7 q^{65} -2.31597e7i q^{67} +1.63575e7 q^{69} -1.37847e7i q^{71} +5.32192e6 q^{73} -9.13712e6i q^{75} +5.71282e6 q^{77} -3.42382e7i q^{79} +4.78297e6 q^{81} -2.71409e7i q^{83} -1.29882e7 q^{85} +1.93273e7i q^{87} -3.90336e7 q^{89} -2.06346e7i q^{91} -1.36212e7 q^{93} +205888. i q^{95} +4.04636e6 q^{97} -3.05793e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−441.863	−0.706981	−0.353490	−	0.935438i	$-0.615005\pi$
$5$	−441.863	−0.706981	−0.353490	+	0.935438i	$0.615005\pi$
$6$	0	0
$7$	− 408.575i	− 0.170169i	−0.996374	−	0.0850843i	$-0.972884\pi$
$7$	− 408.575i	− 0.170169i	0.996374	−	0.0850843i	$-0.0271160\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	13982.3i	0.955010i	0.878629	+	0.477505i	$0.158459\pi$
$11$	13982.3i	0.955010i	−0.878629	+	0.477505i	$0.841541\pi$
$12$	0	0
$13$	50503.9	1.76828	0.884141	−	0.467220i	$-0.154745\pi$
$13$	50503.9	1.76828	0.884141	+	0.467220i	$0.154745\pi$
$14$	0	0
$15$	− 20663.9i	− 0.408175i
$16$	0	0
$17$	29394.2	0.351938	0.175969	−	0.984396i	$-0.443694\pi$
$17$	29394.2	0.351938	0.175969	+	0.984396i	$0.443694\pi$
$18$	0	0
$19$	− 465.954i	− 0.00357543i	−0.999998	−	0.00178772i	$-0.999431\pi$
$19$	− 465.954i	− 0.00357543i	0.999998	−	0.00178772i	$-0.000569048\pi$
$20$	0	0
$21$	19107.2	0.0982469
$22$	0	0
$23$	− 349779.i	− 1.24992i	−0.780657	−	0.624960i	$-0.785115\pi$
$23$	− 349779.i	− 1.24992i	0.780657	−	0.624960i	$-0.214885\pi$
$24$	0	0
$25$	−195382.	−0.500178
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	413282.	0.584325	0.292163	−	0.956369i	$-0.405625\pi$
$29$	413282.	0.584325	0.292163	+	0.956369i	$0.405625\pi$
$30$	0	0
$31$	291268.i	0.315388i	0.987488	+	0.157694i	$0.0504060\pi$
$31$	291268.i	0.315388i	−0.987488	+	0.157694i	$0.949594\pi$
$32$	0	0
$33$	−653888.	−0.551375
$34$	0	0
$35$	180534.i	0.120306i
$36$	0	0
$37$	−3.18340e6	−1.69857	−0.849286	−	0.527933i	$-0.822967\pi$
$37$	−3.18340e6	−1.69857	−0.849286	+	0.527933i	$0.822967\pi$
$38$	0	0
$39$	2.36183e6i	1.02092i
$40$	0	0
$41$	−3.90588e6	−1.38224	−0.691121	−	0.722739i	$-0.742883\pi$
$41$	−3.90588e6	−1.38224	−0.691121	+	0.722739i	$0.742883\pi$
$42$	0	0
$43$	− 5.13172e6i	− 1.50103i	−0.660853	−	0.750515i	$-0.729805\pi$
$43$	− 5.13172e6i	− 1.50103i	0.660853	−	0.750515i	$-0.270195\pi$
$44$	0	0
$45$	966354.	0.235660
$46$	0	0
$47$	3.80555e6i	0.779878i	0.920841	+	0.389939i	$0.127504\pi$
$47$	3.80555e6i	0.779878i	−0.920841	+	0.389939i	$0.872496\pi$
$48$	0	0
$49$	5.59787e6	0.971043
$50$	0	0
$51$	1.37463e6i	0.203191i
$52$	0	0
$53$	−1.21515e6	−0.154002	−0.0770009	−	0.997031i	$-0.524534\pi$
$53$	−1.21515e6	−0.154002	−0.0770009	+	0.997031i	$0.524534\pi$
$54$	0	0
$55$	− 6.17826e6i	− 0.675174i
$56$	0	0
$57$	21790.5	0.00206428
$58$	0	0
$59$	5.53322e6i	0.456636i	0.973587	+	0.228318i	$0.0733226\pi$
$59$	5.53322e6i	0.456636i	−0.973587	+	0.228318i	$0.926677\pi$
$60$	0	0
$61$	8.14915e6	0.588563	0.294282	−	0.955719i	$-0.404920\pi$
$61$	8.14915e6	0.588563	0.294282	+	0.955719i	$0.404920\pi$
$62$	0	0
$63$	893553.i	0.0567229i
$64$	0	0
$65$	−2.23158e7	−1.25014
$66$	0	0
$67$	− 2.31597e7i	− 1.14930i	−0.818399	−	0.574651i	$-0.805138\pi$
$67$	− 2.31597e7i	− 1.14930i	0.818399	−	0.574651i	$-0.194862\pi$
$68$	0	0
$69$	1.63575e7	0.721641
$70$	0	0
$71$	− 1.37847e7i	− 0.542454i	−0.962515	−	0.271227i	$-0.912571\pi$
$71$	− 1.37847e7i	− 0.542454i	0.962515	−	0.271227i	$-0.0874294\pi$
$72$	0	0
$73$	5.32192e6	0.187403	0.0937015	−	0.995600i	$-0.470130\pi$
$73$	5.32192e6	0.187403	0.0937015	+	0.995600i	$0.470130\pi$
$74$	0	0
$75$	− 9.13712e6i	− 0.288778i
$76$	0	0
$77$	5.71282e6	0.162513
$78$	0	0
$79$	− 3.42382e7i	− 0.879026i	−0.898236	−	0.439513i	$-0.855151\pi$
$79$	− 3.42382e7i	− 0.879026i	0.898236	−	0.439513i	$-0.144849\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	− 2.71409e7i	− 0.571889i	−0.958246	−	0.285944i	$-0.907693\pi$
$83$	− 2.71409e7i	− 0.571889i	0.958246	−	0.285944i	$-0.0923072\pi$
$84$	0	0
$85$	−1.29882e7	−0.248813
$86$	0	0
$87$	1.93273e7i	0.337360i
$88$	0	0
$89$	−3.90336e7	−0.622126	−0.311063	−	0.950389i	$-0.600685\pi$
$89$	−3.90336e7	−0.622126	−0.311063	+	0.950389i	$0.600685\pi$
$90$	0	0
$91$	− 2.06346e7i	− 0.300906i
$92$	0	0
$93$	−1.36212e7	−0.182089
$94$	0	0
$95$	205888.i	0.00252776i
$96$	0	0
$97$	4.04636e6	0.0457065	0.0228533	−	0.999739i	$-0.492725\pi$
$97$	4.04636e6	0.0457065	0.0228533	+	0.999739i	$0.492725\pi$
$98$	0	0
$99$	− 3.05793e7i	− 0.318337i
$100$	0	0
$101$	6.48664e7	0.623353	0.311677	−	0.950188i	$-0.399109\pi$
$101$	6.48664e7	0.623353	0.311677	+	0.950188i	$0.399109\pi$
$102$	0	0
$103$	6.83327e7i	0.607127i	0.952811	+	0.303564i	$0.0981765\pi$
$103$	6.83327e7i	0.607127i	−0.952811	+	0.303564i	$0.901823\pi$
$104$	0	0
$105$	−8.44274e6	−0.0694587
$106$	0	0
$107$	1.79487e8i	1.36930i	0.728873	+	0.684649i	$0.240044\pi$
$107$	1.79487e8i	1.36930i	−0.728873	+	0.684649i	$0.759956\pi$
$108$	0	0
$109$	2.24660e8	1.59155	0.795773	−	0.605595i	$-0.207065\pi$
$109$	2.24660e8	1.59155	0.795773	+	0.605595i	$0.207065\pi$
$110$	0	0
$111$	− 1.48873e8i	− 0.980671i
$112$	0	0
$113$	2.25134e8	1.38079	0.690394	−	0.723433i	$-0.257437\pi$
$113$	2.25134e8	1.38079	0.690394	+	0.723433i	$0.257437\pi$
$114$	0	0
$115$	1.54554e8i	0.883669i
$116$	0	0
$117$	−1.10452e8	−0.589427
$118$	0	0
$119$	− 1.20097e7i	− 0.0598888i
$120$	0	0
$121$	1.88541e7	0.0879558
$122$	0	0
$123$	− 1.82660e8i	− 0.798038i
$124$	0	0
$125$	2.58935e8	1.06060
$126$	0	0
$127$	− 3.32041e7i	− 0.127637i	−0.997962	−	0.0638186i	$-0.979672\pi$
$127$	− 3.32041e7i	− 0.127637i	0.997962	−	0.0638186i	$-0.0203279\pi$
$128$	0	0
$129$	2.39987e8	0.866620
$130$	0	0
$131$	− 5.31550e8i	− 1.80492i	−0.430770	−	0.902462i	$-0.641758\pi$
$131$	− 5.31550e8i	− 1.80492i	0.430770	−	0.902462i	$-0.358242\pi$
$132$	0	0
$133$	−190377.	−0.000608426 0
$134$	0	0
$135$	4.51919e7i	0.136058i
$136$	0	0
$137$	−1.91221e8	−0.542816	−0.271408	−	0.962464i	$-0.587489\pi$
$137$	−1.91221e8	−0.542816	−0.271408	+	0.962464i	$0.587489\pi$
$138$	0	0
$139$	− 1.01862e8i	− 0.272868i	−0.990649	−	0.136434i	$-0.956436\pi$
$139$	− 1.01862e8i	− 0.272868i	0.990649	−	0.136434i	$-0.0435642\pi$
$140$	0	0
$141$	−1.77968e8	−0.450263
$142$	0	0
$143$	7.06161e8i	1.68873i
$144$	0	0
$145$	−1.82614e8	−0.413106
$146$	0	0
$147$	2.61786e8i	0.560632i
$148$	0	0
$149$	4.93975e7	0.100221	0.0501107	−	0.998744i	$-0.484043\pi$
$149$	4.93975e7	0.100221	0.0501107	+	0.998744i	$0.484043\pi$
$150$	0	0
$151$	2.93879e8i	0.565275i	0.959227	+	0.282638i	$0.0912094\pi$
$151$	2.93879e8i	0.565275i	−0.959227	+	0.282638i	$0.908791\pi$
$152$	0	0
$153$	−6.42851e7	−0.117313
$154$	0	0
$155$	− 1.28700e8i	− 0.222973i
$156$	0	0
$157$	−4.27912e8	−0.704297	−0.352149	−	0.935944i	$-0.614549\pi$
$157$	−4.27912e8	−0.704297	−0.352149	+	0.935944i	$0.614549\pi$
$158$	0	0
$159$	− 5.68268e7i	− 0.0889129i
$160$	0	0
$161$	−1.42911e8	−0.212697
$162$	0	0
$163$	− 1.09427e9i	− 1.55015i	−0.631868	−	0.775076i	$-0.717712\pi$
$163$	− 1.09427e9i	− 1.55015i	0.631868	−	0.775076i	$-0.282288\pi$
$164$	0	0
$165$	2.88929e8	0.389812
$166$	0	0
$167$	1.34025e9i	1.72313i	0.507645	+	0.861566i	$0.330516\pi$
$167$	1.34025e9i	1.72313i	−0.507645	+	0.861566i	$0.669484\pi$
$168$	0	0
$169$	1.73491e9	2.12682
$170$	0	0
$171$	1.01904e6i	0.00119181i
$172$	0	0
$173$	1.73349e8	0.193524	0.0967622	−	0.995308i	$-0.469151\pi$
$173$	1.73349e8	0.193524	0.0967622	+	0.995308i	$0.469151\pi$
$174$	0	0
$175$	7.98283e7i	0.0851147i
$176$	0	0
$177$	−2.58763e8	−0.263639
$178$	0	0
$179$	1.20920e9i	1.17784i	0.808190	+	0.588922i	$0.200447\pi$
$179$	1.20920e9i	1.17784i	−0.808190	+	0.588922i	$0.799553\pi$
$180$	0	0
$181$	−5.73459e8	−0.534304	−0.267152	−	0.963654i	$-0.586083\pi$
$181$	−5.73459e8	−0.534304	−0.267152	+	0.963654i	$0.586083\pi$
$182$	0	0
$183$	3.81098e8i	0.339807i
$184$	0	0
$185$	1.40662e9	1.20086
$186$	0	0
$187$	4.10999e8i	0.336104i
$188$	0	0
$189$	−4.17874e7	−0.0327490
$190$	0	0
$191$	− 4.78981e8i	− 0.359903i	−0.983676	−	0.179951i	$-0.942406\pi$
$191$	− 4.78981e8i	− 0.359903i	0.983676	−	0.179951i	$-0.0575940\pi$
$192$	0	0
$193$	1.06454e9	0.767240	0.383620	−	0.923491i	$-0.374677\pi$
$193$	1.06454e9	0.767240	0.383620	+	0.923491i	$0.374677\pi$
$194$	0	0
$195$	− 1.04361e9i	− 0.721769i
$196$	0	0
$197$	2.13527e9	1.41771	0.708857	−	0.705352i	$-0.249211\pi$
$197$	2.13527e9	1.41771	0.708857	+	0.705352i	$0.249211\pi$
$198$	0	0
$199$	4.15711e8i	0.265081i	0.991178	+	0.132541i	$0.0423135\pi$
$199$	4.15711e8i	0.265081i	−0.991178	+	0.132541i	$0.957686\pi$
$200$	0	0
$201$	1.08307e9	0.663550
$202$	0	0
$203$	− 1.68857e8i	− 0.0994338i
$204$	0	0
$205$	1.72587e9	0.977218
$206$	0	0
$207$	7.64966e8i	0.416640i
$208$	0	0
$209$	6.51511e6	0.00341457
$210$	0	0
$211$	− 3.72862e9i	− 1.88113i	−0.339616	−	0.940564i	$-0.610297\pi$
$211$	− 3.72862e9i	− 1.88113i	0.339616	−	0.940564i	$-0.389703\pi$
$212$	0	0
$213$	6.44645e8	0.313186
$214$	0	0
$215$	2.26752e9i	1.06120i
$216$	0	0
$217$	1.19005e8	0.0536692
$218$	0	0
$219$	2.48881e8i	0.108197i
$220$	0	0
$221$	1.48452e9	0.622325
$222$	0	0
$223$	2.27642e9i	0.920518i	0.887785	+	0.460259i	$0.152244\pi$
$223$	2.27642e9i	0.920518i	−0.887785	+	0.460259i	$0.847756\pi$
$224$	0	0
$225$	4.27301e8	0.166726
$226$	0	0
$227$	− 1.52411e9i	− 0.574003i	−0.957930	−	0.287002i	$-0.907342\pi$
$227$	− 1.52411e9i	− 0.574003i	0.957930	−	0.287002i	$-0.0926585\pi$
$228$	0	0
$229$	3.01820e9	1.09750	0.548752	−	0.835985i	$-0.315103\pi$
$229$	3.01820e9	1.09750	0.548752	+	0.835985i	$0.315103\pi$
$230$	0	0
$231$	2.67162e8i	0.0938268i
$232$	0	0
$233$	4.97441e8	0.168779	0.0843895	−	0.996433i	$-0.473106\pi$
$233$	4.97441e8	0.168779	0.0843895	+	0.996433i	$0.473106\pi$
$234$	0	0
$235$	− 1.68153e9i	− 0.551358i
$236$	0	0
$237$	1.60116e9	0.507506
$238$	0	0
$239$	− 1.57621e9i	− 0.483083i	−0.970390	−	0.241542i	$-0.922347\pi$
$239$	− 1.57621e9i	− 0.483083i	0.970390	−	0.241542i	$-0.0776531\pi$
$240$	0	0
$241$	1.54224e9	0.457176	0.228588	−	0.973523i	$-0.426589\pi$
$241$	1.54224e9	0.457176	0.228588	+	0.973523i	$0.426589\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	−2.47349e9	−0.686508
$246$	0	0
$247$	− 2.35325e7i	− 0.00632237i
$248$	0	0
$249$	1.26925e9	0.330180
$250$	0	0
$251$	− 4.80796e9i	− 1.21134i	−0.795716	−	0.605670i	$-0.792905\pi$
$251$	− 4.80796e9i	− 1.21134i	0.795716	−	0.605670i	$-0.207095\pi$
$252$	0	0
$253$	4.89071e9	1.19369
$254$	0	0
$255$	− 6.07398e8i	− 0.143652i
$256$	0	0
$257$	3.68574e9	0.844874	0.422437	−	0.906392i	$-0.361175\pi$
$257$	3.68574e9	0.844874	0.422437	+	0.906392i	$0.361175\pi$
$258$	0	0
$259$	1.30066e9i	0.289044i
$260$	0	0
$261$	−9.03848e8	−0.194775
$262$	0	0
$263$	6.14029e8i	0.128341i	0.997939	+	0.0641706i	$0.0204402\pi$
$263$	6.14029e8i	0.128341i	−0.997939	+	0.0641706i	$0.979560\pi$
$264$	0	0
$265$	5.36929e8	0.108876
$266$	0	0
$267$	− 1.82542e9i	− 0.359185i
$268$	0	0
$269$	4.98817e9	0.952647	0.476324	−	0.879270i	$-0.341969\pi$
$269$	4.98817e9	0.952647	0.476324	+	0.879270i	$0.341969\pi$
$270$	0	0
$271$	8.07780e9i	1.49767i	0.662757	+	0.748834i	$0.269386\pi$
$271$	8.07780e9i	1.49767i	−0.662757	+	0.748834i	$0.730614\pi$
$272$	0	0
$273$	9.64986e8	0.173728
$274$	0	0
$275$	− 2.73189e9i	− 0.477675i
$276$	0	0
$277$	8.79514e9	1.49391	0.746954	−	0.664876i	$-0.231516\pi$
$277$	8.79514e9	1.49391	0.746954	+	0.664876i	$0.231516\pi$
$278$	0	0
$279$	− 6.37002e8i	− 0.105129i
$280$	0	0
$281$	1.07423e10	1.72294	0.861470	−	0.507809i	$-0.169544\pi$
$281$	1.07423e10	1.72294	0.861470	+	0.507809i	$0.169544\pi$
$282$	0	0
$283$	3.36710e9i	0.524941i	0.964940	+	0.262470i	$0.0845372\pi$
$283$	3.36710e9i	0.524941i	−0.964940	+	0.262470i	$0.915463\pi$
$284$	0	0
$285$	−9.62841e6	−0.00145940
$286$	0	0
$287$	1.59585e9i	0.235214i
$288$	0	0
$289$	−6.11174e9	−0.876140
$290$	0	0
$291$	1.89230e8i	0.0263887i
$292$	0	0
$293$	−1.84802e9	−0.250747	−0.125373	−	0.992110i	$-0.540013\pi$
$293$	−1.84802e9	−0.250747	−0.125373	+	0.992110i	$0.540013\pi$
$294$	0	0
$295$	− 2.44493e9i	− 0.322833i
$296$	0	0
$297$	1.43005e9	0.183792
$298$	0	0
$299$	− 1.76652e10i	− 2.21021i
$300$	0	0
$301$	−2.09669e9	−0.255428
$302$	0	0
$303$	3.03350e9i	0.359893i
$304$	0	0
$305$	−3.60081e9	−0.416103
$306$	0	0
$307$	− 1.29789e10i	− 1.46112i	−0.682850	−	0.730559i	$-0.739260\pi$
$307$	− 1.29789e10i	− 1.46112i	0.682850	−	0.730559i	$-0.260740\pi$
$308$	0	0
$309$	−3.19560e9	−0.350525
$310$	0	0
$311$	5.34433e9i	0.571283i	0.958336	+	0.285642i	$0.0922067\pi$
$311$	5.34433e9i	0.571283i	−0.958336	+	0.285642i	$0.907793\pi$
$312$	0	0
$313$	1.08796e9	0.113354	0.0566770	−	0.998393i	$-0.481949\pi$
$313$	1.08796e9	0.113354	0.0566770	+	0.998393i	$0.481949\pi$
$314$	0	0
$315$	− 3.94828e8i	− 0.0401020i
$316$	0	0
$317$	1.02228e10	1.01236	0.506179	−	0.862429i	$-0.331058\pi$
$317$	1.02228e10	1.01236	0.506179	+	0.862429i	$0.331058\pi$
$318$	0	0
$319$	5.77863e9i	0.558036i
$320$	0	0
$321$	−8.39378e9	−0.790565
$322$	0	0
$323$	− 1.36963e7i	− 0.00125833i
$324$	0	0
$325$	−9.86756e9	−0.884456
$326$	0	0
$327$	1.05063e10i	0.918880i
$328$	0	0
$329$	1.55485e9	0.132711
$330$	0	0
$331$	− 1.62844e10i	− 1.35662i	−0.734775	−	0.678311i	$-0.762712\pi$
$331$	− 1.62844e10i	− 1.35662i	0.734775	−	0.678311i	$-0.237288\pi$
$332$	0	0
$333$	6.96209e9	0.566191
$334$	0	0
$335$	1.02334e10i	0.812534i
$336$	0	0
$337$	3.25820e9	0.252614	0.126307	−	0.991991i	$-0.459688\pi$
$337$	3.25820e9	0.252614	0.126307	+	0.991991i	$0.459688\pi$
$338$	0	0
$339$	1.05285e10i	0.797199i
$340$	0	0
$341$	−4.07259e9	−0.301199
$342$	0	0
$343$	− 4.64250e9i	− 0.335410i
$344$	0	0
$345$	−7.22779e9	−0.510186
$346$	0	0
$347$	1.87548e10i	1.29358i	0.762666	+	0.646792i	$0.223890\pi$
$347$	1.87548e10i	1.29358i	−0.762666	+	0.646792i	$0.776110\pi$
$348$	0	0
$349$	2.34804e10	1.58272	0.791361	−	0.611349i	$-0.209373\pi$
$349$	2.34804e10	1.58272	0.791361	+	0.611349i	$0.209373\pi$
$350$	0	0
$351$	− 5.16533e9i	− 0.340306i
$352$	0	0
$353$	1.78231e10	1.14785	0.573924	−	0.818909i	$-0.305420\pi$
$353$	1.78231e10	1.14785	0.573924	+	0.818909i	$0.305420\pi$
$354$	0	0
$355$	6.09093e9i	0.383504i
$356$	0	0
$357$	5.61640e8	0.0345768
$358$	0	0
$359$	1.24594e9i	0.0750099i	0.999296	+	0.0375049i	$0.0119410\pi$
$359$	1.24594e9i	0.0750099i	−0.999296	+	0.0375049i	$0.988059\pi$
$360$	0	0
$361$	1.69833e10	0.999987
$362$	0	0
$363$	8.81719e8i	0.0507813i
$364$	0	0
$365$	−2.35156e9	−0.132490
$366$	0	0
$367$	− 1.96071e10i	− 1.08081i	−0.841406	−	0.540404i	$-0.818271\pi$
$367$	− 1.96071e10i	− 1.08081i	0.841406	−	0.540404i	$-0.181729\pi$
$368$	0	0
$369$	8.54217e9	0.460747
$370$	0	0
$371$	4.96479e8i	0.0262063i
$372$	0	0
$373$	−1.68736e10	−0.871711	−0.435856	−	0.900017i	$-0.643554\pi$
$373$	−1.68736e10	−0.871711	−0.435856	+	0.900017i	$0.643554\pi$
$374$	0	0
$375$	1.21092e10i	0.612336i
$376$	0	0
$377$	2.08723e10	1.03325
$378$	0	0
$379$	1.18982e10i	0.576664i	0.957530	+	0.288332i	$0.0931007\pi$
$379$	1.18982e10i	0.576664i	−0.957530	+	0.288332i	$0.906899\pi$
$380$	0	0
$381$	1.55280e9	0.0736913
$382$	0	0
$383$	1.40923e10i	0.654918i	0.944865	+	0.327459i	$0.106192\pi$
$383$	1.40923e10i	0.654918i	−0.944865	+	0.327459i	$0.893808\pi$
$384$	0	0
$385$	−2.52428e9	−0.114893
$386$	0	0
$387$	1.12231e10i	0.500344i
$388$	0	0
$389$	−3.26952e10	−1.42786	−0.713931	−	0.700216i	$-0.753087\pi$
$389$	−3.26952e10	−1.42786	−0.713931	+	0.700216i	$0.753087\pi$
$390$	0	0
$391$	− 1.02815e10i	− 0.439894i
$392$	0	0
$393$	2.48581e10	1.04207
$394$	0	0
$395$	1.51286e10i	0.621455i
$396$	0	0
$397$	2.03713e10	0.820080	0.410040	−	0.912068i	$-0.365515\pi$
$397$	2.03713e10	0.820080	0.410040	+	0.912068i	$0.365515\pi$
$398$	0	0
$399$	− 8.90305e6i	0 0.000351275i
$400$	0	0
$401$	−4.79872e10	−1.85587	−0.927937	−	0.372738i	$-0.878419\pi$
$401$	−4.79872e10	−1.85587	−0.927937	+	0.372738i	$0.878419\pi$
$402$	0	0
$403$	1.47101e10i	0.557695i
$404$	0	0
$405$	−2.11342e9	−0.0785534
$406$	0	0
$407$	− 4.45112e10i	− 1.62215i
$408$	0	0
$409$	−6.04253e9	−0.215936	−0.107968	−	0.994154i	$-0.534434\pi$
$409$	−6.04253e9	−0.215936	−0.107968	+	0.994154i	$0.534434\pi$
$410$	0	0
$411$	− 8.94252e9i	− 0.313395i
$412$	0	0
$413$	2.26074e9	0.0777051
$414$	0	0
$415$	1.19925e10i	0.404314i
$416$	0	0
$417$	4.76361e9	0.157540
$418$	0	0
$419$	1.14993e10i	0.373092i	0.982446	+	0.186546i	$0.0597294\pi$
$419$	1.14993e10i	0.373092i	−0.982446	+	0.186546i	$0.940271\pi$
$420$	0	0
$421$	−1.21975e10	−0.388278	−0.194139	−	0.980974i	$-0.562191\pi$
$421$	−1.21975e10	−0.388278	−0.194139	+	0.980974i	$0.562191\pi$
$422$	0	0
$423$	− 8.32275e9i	− 0.259959i
$424$	0	0
$425$	−5.74310e9	−0.176032
$426$	0	0
$427$	− 3.32954e9i	− 0.100155i
$428$	0	0
$429$	−3.30239e10	−0.974987
$430$	0	0
$431$	1.23665e10i	0.358376i	0.983815	+	0.179188i	$0.0573470\pi$
$431$	1.23665e10i	0.358376i	−0.983815	+	0.179188i	$0.942653\pi$
$432$	0	0
$433$	−5.82670e10	−1.65757	−0.828784	−	0.559569i	$-0.810967\pi$
$433$	−5.82670e10	−1.65757	−0.828784	+	0.559569i	$0.810967\pi$
$434$	0	0
$435$	− 8.54001e9i	− 0.238507i
$436$	0	0
$437$	−1.62981e8	−0.00446900
$438$	0	0
$439$	1.31028e10i	0.352781i	0.984320	+	0.176390i	$0.0564421\pi$
$439$	1.31028e10i	0.352781i	−0.984320	+	0.176390i	$0.943558\pi$
$440$	0	0
$441$	−1.22425e10	−0.323681
$442$	0	0
$443$	8.18524e9i	0.212528i	0.994338	+	0.106264i	$0.0338889\pi$
$443$	8.18524e9i	0.212528i	−0.994338	+	0.106264i	$0.966111\pi$
$444$	0	0
$445$	1.72475e10	0.439831
$446$	0	0
$447$	2.31009e9i	0.0578628i
$448$	0	0
$449$	−4.39174e10	−1.08057	−0.540283	−	0.841483i	$-0.681683\pi$
$449$	−4.39174e10	−1.08057	−0.540283	+	0.841483i	$0.681683\pi$
$450$	0	0
$451$	− 5.46133e10i	− 1.32005i
$452$	0	0
$453$	−1.37433e10	−0.326362
$454$	0	0
$455$	9.11767e9i	0.212735i
$456$	0	0
$457$	2.38106e10	0.545890	0.272945	−	0.962030i	$-0.412002\pi$
$457$	2.38106e10	0.545890	0.272945	+	0.962030i	$0.412002\pi$
$458$	0	0
$459$	− 3.00632e9i	− 0.0677305i
$460$	0	0
$461$	−3.88807e10	−0.860855	−0.430427	−	0.902625i	$-0.641637\pi$
$461$	−3.88807e10	−0.860855	−0.430427	+	0.902625i	$0.641637\pi$
$462$	0	0
$463$	− 5.69790e10i	− 1.23991i	−0.784636	−	0.619957i	$-0.787150\pi$
$463$	− 5.69790e10i	− 1.23991i	0.784636	−	0.619957i	$-0.212850\pi$
$464$	0	0
$465$	6.01872e9	0.128734
$466$	0	0
$467$	8.00144e10i	1.68229i	0.540810	+	0.841145i	$0.318118\pi$
$467$	8.00144e10i	1.68229i	−0.540810	+	0.841145i	$0.681882\pi$
$468$	0	0
$469$	−9.46248e9	−0.195575
$470$	0	0
$471$	− 2.00115e10i	− 0.406626i
$472$	0	0
$473$	7.17533e10	1.43350
$474$	0	0
$475$	9.10391e7i	0.00178835i
$476$	0	0
$477$	2.65753e9	0.0513339
$478$	0	0
$479$	− 9.14708e10i	− 1.73756i	−0.495196	−	0.868781i	$-0.664904\pi$
$479$	− 9.14708e10i	− 1.73756i	0.495196	−	0.868781i	$-0.335096\pi$
$480$	0	0
$481$	−1.60774e11	−3.00355
$482$	0	0
$483$	− 6.68328e9i	− 0.122801i
$484$	0	0
$485$	−1.78794e9	−0.0323136
$486$	0	0
$487$	− 8.68484e10i	− 1.54400i	−0.635625	−	0.771998i	$-0.719258\pi$
$487$	− 8.68484e10i	− 1.54400i	0.635625	−	0.771998i	$-0.280742\pi$
$488$	0	0
$489$	5.11740e10	0.894981
$490$	0	0
$491$	3.57345e9i	0.0614840i	0.999527	+	0.0307420i	$0.00978702\pi$
$491$	3.57345e9i	0.0614840i	−0.999527	+	0.0307420i	$0.990213\pi$
$492$	0	0
$493$	1.21481e10	0.205646
$494$	0	0
$495$	1.35119e10i	0.225058i
$496$	0	0
$497$	−5.63207e9	−0.0923086
$498$	0	0
$499$	5.59047e10i	0.901666i	0.892608	+	0.450833i	$0.148873\pi$
$499$	5.59047e10i	0.901666i	−0.892608	+	0.450833i	$0.851127\pi$
$500$	0	0
$501$	−6.26771e10	−0.994851
$502$	0	0
$503$	1.08704e11i	1.69814i	0.528281	+	0.849070i	$0.322837\pi$
$503$	1.08704e11i	1.69814i	−0.528281	+	0.849070i	$0.677163\pi$
$504$	0	0
$505$	−2.86620e10	−0.440699
$506$	0	0
$507$	8.11338e10i	1.22792i
$508$	0	0
$509$	4.50653e10	0.671384	0.335692	−	0.941972i	$-0.391030\pi$
$509$	4.50653e10	0.671384	0.335692	+	0.941972i	$0.391030\pi$
$510$	0	0
$511$	− 2.17440e9i	− 0.0318901i
$512$	0	0
$513$	−4.76558e7	−0.000688092 0
$514$	0	0
$515$	− 3.01937e10i	− 0.429227i
$516$	0	0
$517$	−5.32104e10	−0.744791
$518$	0	0
$519$	8.10671e9i	0.111731i
$520$	0	0
$521$	−1.35453e11	−1.83839	−0.919197	−	0.393798i	$-0.871161\pi$
$521$	−1.35453e11	−1.83839	−0.919197	+	0.393798i	$0.871161\pi$
$522$	0	0
$523$	1.21536e10i	0.162442i	0.996696	+	0.0812210i	$0.0258820\pi$
$523$	1.21536e10i	0.162442i	−0.996696	+	0.0812210i	$0.974118\pi$
$524$	0	0
$525$	−3.73320e9	−0.0491410
$526$	0	0
$527$	8.56158e9i	0.110997i
$528$	0	0
$529$	−4.40342e10	−0.562299
$530$	0	0
$531$	− 1.21012e10i	− 0.152212i
$532$	0	0
$533$	−1.97262e11	−2.44419
$534$	0	0
$535$	− 7.93086e10i	− 0.968067i
$536$	0	0
$537$	−5.65489e10	−0.680028
$538$	0	0
$539$	7.82711e10i	0.927355i
$540$	0	0
$541$	6.14285e10	0.717102	0.358551	−	0.933510i	$-0.383271\pi$
$541$	6.14285e10	0.717102	0.358551	+	0.933510i	$0.383271\pi$
$542$	0	0
$543$	− 2.68180e10i	− 0.308480i
$544$	0	0
$545$	−9.92688e10	−1.12519
$546$	0	0
$547$	− 1.48785e11i	− 1.66192i	−0.556333	−	0.830960i	$-0.687792\pi$
$547$	− 1.48785e11i	− 1.66192i	0.556333	−	0.830960i	$-0.312208\pi$
$548$	0	0
$549$	−1.78222e10	−0.196188
$550$	0	0
$551$	− 1.92570e8i	− 0.00208921i
$552$	0	0
$553$	−1.39888e10	−0.149583
$554$	0	0
$555$	6.57813e10i	0.693315i
$556$	0	0
$557$	1.18006e11	1.22598	0.612990	−	0.790091i	$-0.289967\pi$
$557$	1.18006e11	1.22598	0.612990	+	0.790091i	$0.289967\pi$
$558$	0	0
$559$	− 2.59172e11i	− 2.65424i
$560$	0	0
$561$	−1.92205e10	−0.194050
$562$	0	0
$563$	5.93137e10i	0.590366i	0.955441	+	0.295183i	$0.0953807\pi$
$563$	5.93137e10i	0.590366i	−0.955441	+	0.295183i	$0.904619\pi$
$564$	0	0
$565$	−9.94783e10	−0.976191
$566$	0	0
$567$	− 1.95420e9i	− 0.0189076i
$568$	0	0
$569$	1.68389e11	1.60644	0.803219	−	0.595683i	$-0.203119\pi$
$569$	1.68389e11	1.60644	0.803219	+	0.595683i	$0.203119\pi$
$570$	0	0
$571$	− 6.53680e10i	− 0.614923i	−0.951561	−	0.307461i	$-0.900521\pi$
$571$	− 6.53680e10i	− 0.614923i	0.951561	−	0.307461i	$-0.0994795\pi$
$572$	0	0
$573$	2.23997e10	0.207790
$574$	0	0
$575$	6.83405e10i	0.625183i
$576$	0	0
$577$	2.36475e10	0.213345	0.106672	−	0.994294i	$-0.465980\pi$
$577$	2.36475e10	0.213345	0.106672	+	0.994294i	$0.465980\pi$
$578$	0	0
$579$	4.97834e10i	0.442966i
$580$	0	0
$581$	−1.10891e10	−0.0973175
$582$	0	0
$583$	− 1.69906e10i	− 0.147073i
$584$	0	0
$585$	4.88046e10	0.416714
$586$	0	0
$587$	4.51763e10i	0.380503i	0.981735	+	0.190251i	$0.0609303\pi$
$587$	4.51763e10i	0.380503i	−0.981735	+	0.190251i	$0.939070\pi$
$588$	0	0
$589$	1.35717e8	0.00112765
$590$	0	0
$591$	9.98569e10i	0.818518i
$592$	0	0
$593$	−7.02734e10	−0.568293	−0.284146	−	0.958781i	$-0.591710\pi$
$593$	−7.02734e10	−0.568293	−0.284146	+	0.958781i	$0.591710\pi$
$594$	0	0
$595$	5.30666e9i	0.0423402i
$596$	0	0
$597$	−1.94409e10	−0.153045
$598$	0	0
$599$	− 1.40943e11i	− 1.09480i	−0.836870	−	0.547401i	$-0.815617\pi$
$599$	− 1.40943e11i	− 1.09480i	0.836870	−	0.547401i	$-0.184383\pi$
$600$	0	0
$601$	−8.64830e10	−0.662877	−0.331439	−	0.943477i	$-0.607534\pi$
$601$	−8.64830e10	−0.662877	−0.331439	+	0.943477i	$0.607534\pi$
$602$	0	0
$603$	5.06503e10i	0.383101i
$604$	0	0
$605$	−8.33093e9	−0.0621830
$606$	0	0
$607$	− 1.73522e11i	− 1.27820i	−0.769124	−	0.639100i	$-0.779307\pi$
$607$	− 1.73522e11i	− 1.27820i	0.769124	−	0.639100i	$-0.220693\pi$
$608$	0	0
$609$	7.89664e9	0.0574081
$610$	0	0
$611$	1.92195e11i	1.37904i
$612$	0	0
$613$	1.35596e11	0.960296	0.480148	−	0.877187i	$-0.340583\pi$
$613$	1.35596e11	0.960296	0.480148	+	0.877187i	$0.340583\pi$
$614$	0	0
$615$	8.07107e10i	0.564197i
$616$	0	0
$617$	1.78273e11	1.23011	0.615056	−	0.788484i	$-0.289133\pi$
$617$	1.78273e11	1.23011	0.615056	+	0.788484i	$0.289133\pi$
$618$	0	0
$619$	− 2.47992e10i	− 0.168917i	−0.996427	−	0.0844587i	$-0.973084\pi$
$619$	− 2.47992e10i	− 0.168917i	0.996427	−	0.0844587i	$-0.0269161\pi$
$620$	0	0
$621$	−3.57739e10	−0.240547
$622$	0	0
$623$	1.59482e10i	0.105866i
$624$	0	0
$625$	−3.80925e10	−0.249643
$626$	0	0
$627$	3.04681e8i	0.00197140i
$628$	0	0
$629$	−9.35734e10	−0.597792
$630$	0	0
$631$	1.32553e11i	0.836125i	0.908418	+	0.418063i	$0.137291\pi$
$631$	1.32553e11i	0.836125i	−0.908418	+	0.418063i	$0.862709\pi$
$632$	0	0
$633$	1.74370e11	1.08607
$634$	0	0
$635$	1.46717e10i	0.0902370i
$636$	0	0
$637$	2.82714e11	1.71708
$638$	0	0
$639$	3.01471e10i	0.180818i
$640$	0	0
$641$	2.12454e11	1.25844	0.629221	−	0.777226i	$-0.283374\pi$
$641$	2.12454e11	1.25844	0.629221	+	0.777226i	$0.283374\pi$
$642$	0	0
$643$	2.56248e11i	1.49905i	0.661976	+	0.749525i	$0.269718\pi$
$643$	2.56248e11i	1.49905i	−0.661976	+	0.749525i	$0.730282\pi$
$644$	0	0
$645$	−1.06041e11	−0.612684
$646$	0	0
$647$	− 7.86616e10i	− 0.448896i	−0.974486	−	0.224448i	$-0.927942\pi$
$647$	− 7.86616e10i	− 0.448896i	0.974486	−	0.224448i	$-0.0720579\pi$
$648$	0	0
$649$	−7.73672e10	−0.436092
$650$	0	0
$651$	5.56529e9i	0.0309859i
$652$	0	0
$653$	−2.88198e11	−1.58503	−0.792517	−	0.609850i	$-0.791230\pi$
$653$	−2.88198e11	−1.58503	−0.792517	+	0.609850i	$0.791230\pi$
$654$	0	0
$655$	2.34872e11i	1.27605i
$656$	0	0
$657$	−1.16390e10	−0.0624677
$658$	0	0
$659$	− 1.93182e11i	− 1.02430i	−0.858897	−	0.512148i	$-0.828850\pi$
$659$	− 1.93182e11i	− 1.02430i	0.858897	−	0.512148i	$-0.171150\pi$
$660$	0	0
$661$	2.68528e10	0.140664	0.0703322	−	0.997524i	$-0.477594\pi$
$661$	2.68528e10	0.140664	0.0703322	+	0.997524i	$0.477594\pi$
$662$	0	0
$663$	6.94242e10i	0.359300i
$664$	0	0
$665$	8.41205e7	0.000430146 0
$666$	0	0
$667$	− 1.44557e11i	− 0.730359i
$668$	0	0
$669$	−1.06458e11	−0.531462
$670$	0	0
$671$	1.13944e11i	0.562084i
$672$	0	0
$673$	−3.05859e11	−1.49094	−0.745471	−	0.666538i	$-0.767776\pi$
$673$	−3.05859e11	−1.49094	−0.745471	+	0.666538i	$0.767776\pi$
$674$	0	0
$675$	1.99829e10i	0.0962594i
$676$	0	0
$677$	−3.23521e11	−1.54010	−0.770049	−	0.637985i	$-0.779768\pi$
$677$	−3.23521e11	−1.54010	−0.770049	+	0.637985i	$0.779768\pi$
$678$	0	0
$679$	− 1.65324e9i	− 0.00777782i
$680$	0	0
$681$	7.12758e10	0.331401
$682$	0	0
$683$	4.95194e10i	0.227558i	0.993506	+	0.113779i	$0.0362956\pi$
$683$	4.95194e10i	0.227558i	−0.993506	+	0.113779i	$0.963704\pi$
$684$	0	0
$685$	8.44934e10	0.383761
$686$	0	0
$687$	1.41147e11i	0.633645i
$688$	0	0
$689$	−6.13697e10	−0.272318
$690$	0	0
$691$	− 3.65642e11i	− 1.60378i	−0.597473	−	0.801889i	$-0.703828\pi$
$691$	− 3.65642e11i	− 1.60378i	0.597473	−	0.801889i	$-0.296172\pi$
$692$	0	0
$693$	−1.24939e10	−0.0541709
$694$	0	0
$695$	4.50090e10i	0.192912i
$696$	0	0
$697$	−1.14810e11	−0.486463
$698$	0	0
$699$	2.32630e10i	0.0974445i
$700$	0	0
$701$	7.67029e10	0.317643	0.158822	−	0.987307i	$-0.449230\pi$
$701$	7.67029e10	0.317643	0.158822	+	0.987307i	$0.449230\pi$
$702$	0	0
$703$	1.48332e9i	0.00607313i
$704$	0	0
$705$	7.86375e10	0.318327
$706$	0	0
$707$	− 2.65028e10i	− 0.106075i
$708$	0	0
$709$	4.43119e11	1.75362	0.876810	−	0.480838i	$-0.159667\pi$
$709$	4.43119e11	1.75362	0.876810	+	0.480838i	$0.159667\pi$
$710$	0	0
$711$	7.48788e10i	0.293009i
$712$	0	0
$713$	1.01879e11	0.394210
$714$	0	0
$715$	− 3.12026e11i	− 1.19390i
$716$	0	0
$717$	7.37120e10	0.278908
$718$	0	0
$719$	4.36762e11i	1.63429i	0.576432	+	0.817145i	$0.304444\pi$
$719$	4.36762e11i	1.63429i	−0.576432	+	0.817145i	$0.695556\pi$
$720$	0	0
$721$	2.79190e10	0.103314
$722$	0	0
$723$	7.21234e10i	0.263951i
$724$	0	0
$725$	−8.07480e10	−0.292267
$726$	0	0
$727$	− 8.38158e10i	− 0.300046i	−0.988682	−	0.150023i	$-0.952065\pi$
$727$	− 8.38158e10i	− 0.300046i	0.988682	−	0.150023i	$-0.0479348\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	− 1.50843e11i	− 0.528270i
$732$	0	0
$733$	2.84254e11	0.984670	0.492335	−	0.870406i	$-0.336144\pi$
$733$	2.84254e11	0.984670	0.492335	+	0.870406i	$0.336144\pi$
$734$	0	0
$735$	− 1.15674e11i	− 0.396356i
$736$	0	0
$737$	3.23826e11	1.09759
$738$	0	0
$739$	− 6.39738e10i	− 0.214499i	−0.994232	−	0.107249i	$-0.965796\pi$
$739$	− 6.39738e10i	− 0.214499i	0.994232	−	0.107249i	$-0.0342043\pi$
$740$	0	0
$741$	1.10051e9	0.00365022
$742$	0	0
$743$	− 1.03461e10i	− 0.0339485i	−0.999856	−	0.0169742i	$-0.994597\pi$
$743$	− 1.03461e10i	− 0.0339485i	0.999856	−	0.0169742i	$-0.00540333\pi$
$744$	0	0
$745$	−2.18269e10	−0.0708546
$746$	0	0
$747$	5.93571e10i	0.190630i
$748$	0	0
$749$	7.33339e10	0.233012
$750$	0	0
$751$	− 5.76822e11i	− 1.81335i	−0.421829	−	0.906675i	$-0.638612\pi$
$751$	− 5.76822e11i	− 1.81335i	0.421829	−	0.906675i	$-0.361388\pi$
$752$	0	0
$753$	2.24846e11	0.699368
$754$	0	0
$755$	− 1.29854e11i	− 0.399639i
$756$	0	0
$757$	1.41065e11	0.429572	0.214786	−	0.976661i	$-0.431095\pi$
$757$	1.41065e11	0.429572	0.214786	+	0.976661i	$0.431095\pi$
$758$	0	0
$759$	2.28716e11i	0.689175i
$760$	0	0
$761$	1.67327e11	0.498917	0.249458	−	0.968386i	$-0.419747\pi$
$761$	1.67327e11	0.498917	0.249458	+	0.968386i	$0.419747\pi$
$762$	0	0
$763$	− 9.17903e10i	− 0.270831i
$764$	0	0
$765$	2.84052e10	0.0829378
$766$	0	0
$767$	2.79449e11i	0.807461i
$768$	0	0
$769$	−2.61655e11	−0.748211	−0.374105	−	0.927386i	$-0.622050\pi$
$769$	−2.61655e11	−0.748211	−0.374105	+	0.927386i	$0.622050\pi$
$770$	0	0
$771$	1.72365e11i	0.487788i
$772$	0	0
$773$	3.07979e11	0.862587	0.431293	−	0.902212i	$-0.358057\pi$
$773$	3.07979e11	0.862587	0.431293	+	0.902212i	$0.358057\pi$
$774$	0	0
$775$	− 5.69085e10i	− 0.157750i
$776$	0	0
$777$	−6.08257e10	−0.166879
$778$	0	0
$779$	1.81996e9i	0.00494211i
$780$	0	0
$781$	1.92741e11	0.518049
$782$	0	0
$783$	− 4.22688e10i	− 0.112453i
$784$	0	0
$785$	1.89078e11	0.497924
$786$	0	0
$787$	− 4.37562e11i	− 1.14062i	−0.821430	−	0.570309i	$-0.806823\pi$
$787$	− 4.37562e11i	− 1.14062i	0.821430	−	0.570309i	$-0.193177\pi$
$788$	0	0
$789$	−2.87153e10	−0.0740978
$790$	0	0
$791$	− 9.19841e10i	− 0.234967i
$792$	0	0
$793$	4.11564e11	1.04075
$794$	0	0
$795$	2.51097e10i	0.0628597i
$796$	0	0
$797$	4.55263e11	1.12831	0.564156	−	0.825668i	$-0.309202\pi$
$797$	4.55263e11	1.12831	0.564156	+	0.825668i	$0.309202\pi$
$798$	0	0
$799$	1.11861e11i	0.274469i
$800$	0	0
$801$	8.53665e10	0.207375
$802$	0	0
$803$	7.44127e10i	0.178972i
$804$	0	0
$805$	6.31470e10	0.150373
$806$	0	0
$807$	2.33274e11i	0.550011i
$808$	0	0
$809$	6.41357e11	1.49729	0.748645	−	0.662972i	$-0.230705\pi$
$809$	6.41357e11	1.49729	0.748645	+	0.662972i	$0.230705\pi$
$810$	0	0
$811$	− 5.10975e11i	− 1.18118i	−0.806971	−	0.590591i	$-0.798895\pi$
$811$	− 5.10975e11i	− 1.18118i	0.806971	−	0.590591i	$-0.201105\pi$
$812$	0	0
$813$	−3.77761e11	−0.864680
$814$	0	0
$815$	4.83517e11i	1.09593i
$816$	0	0
$817$	−2.39115e9	−0.00536683
$818$	0	0
$819$	4.51279e10i	0.100302i
$820$	0	0
$821$	7.83695e11	1.72494	0.862470	−	0.506107i	$-0.168916\pi$
$821$	7.83695e11	1.72494	0.862470	+	0.506107i	$0.168916\pi$
$822$	0	0
$823$	6.34964e10i	0.138404i	0.997603	+	0.0692022i	$0.0220454\pi$
$823$	6.34964e10i	0.138404i	−0.997603	+	0.0692022i	$0.977955\pi$
$824$	0	0
$825$	1.27758e11	0.275786
$826$	0	0
$827$	− 6.17395e11i	− 1.31990i	−0.751309	−	0.659950i	$-0.770577\pi$
$827$	− 6.17395e11i	− 1.31990i	0.751309	−	0.659950i	$-0.229423\pi$
$828$	0	0
$829$	−3.04138e10	−0.0643951	−0.0321975	−	0.999482i	$-0.510251\pi$
$829$	−3.04138e10	−0.0643951	−0.0321975	+	0.999482i	$0.510251\pi$
$830$	0	0
$831$	4.11308e11i	0.862508i
$832$	0	0
$833$	1.64545e11	0.341747
$834$	0	0
$835$	− 5.92205e11i	− 1.21822i
$836$	0	0
$837$	2.97896e10	0.0606965
$838$	0	0
$839$	2.99185e11i	0.603799i	0.953340	+	0.301900i	$0.0976207\pi$
$839$	2.99185e11i	0.603799i	−0.953340	+	0.301900i	$0.902379\pi$
$840$	0	0
$841$	−3.29444e11	−0.658564
$842$	0	0
$843$	5.02365e11i	0.994740i
$844$	0	0
$845$	−7.66593e11	−1.50362
$846$	0	0
$847$	− 7.70331e9i	− 0.0149673i
$848$	0	0
$849$	−1.57464e11	−0.303075
$850$	0	0
$851$	1.11348e12i	2.12308i
$852$	0	0
$853$	9.03892e11	1.70734	0.853671	−	0.520813i	$-0.174371\pi$
$853$	9.03892e11	1.70734	0.853671	+	0.520813i	$0.174371\pi$
$854$	0	0
$855$	− 4.50276e8i	0 0.000842587i
$856$	0	0
$857$	−7.61813e11	−1.41229	−0.706147	−	0.708066i	$-0.749568\pi$
$857$	−7.61813e11	−1.41229	−0.706147	+	0.708066i	$0.749568\pi$
$858$	0	0
$859$	− 7.60448e11i	− 1.39668i	−0.715766	−	0.698340i	$-0.753922\pi$
$859$	− 7.60448e11i	− 1.39668i	0.715766	−	0.698340i	$-0.246078\pi$
$860$	0	0
$861$	−7.46303e10	−0.135801
$862$	0	0
$863$	− 1.15102e11i	− 0.207510i	−0.994603	−	0.103755i	$-0.966914\pi$
$863$	− 1.15102e11i	− 0.207510i	0.994603	−	0.103755i	$-0.0330858\pi$
$864$	0	0
$865$	−7.65963e10	−0.136818
$866$	0	0
$867$	− 2.85818e11i	− 0.505840i
$868$	0	0
$869$	4.78728e11	0.839479
$870$	0	0
$871$	− 1.16966e12i	− 2.03229i
$872$	0	0
$873$	−8.84940e9	−0.0152355
$874$	0	0
$875$	− 1.05794e11i	− 0.180480i
$876$	0	0
$877$	1.65540e11	0.279837	0.139919	−	0.990163i	$-0.455316\pi$
$877$	1.65540e11	0.279837	0.139919	+	0.990163i	$0.455316\pi$
$878$	0	0
$879$	− 8.64232e10i	− 0.144769i
$880$	0	0
$881$	6.92925e11	1.15022	0.575112	−	0.818075i	$-0.304958\pi$
$881$	6.92925e11	1.15022	0.575112	+	0.818075i	$0.304958\pi$
$882$	0	0
$883$	− 7.72114e11i	− 1.27010i	−0.772470	−	0.635051i	$-0.780979\pi$
$883$	− 7.72114e11i	− 1.27010i	0.772470	−	0.635051i	$-0.219021\pi$
$884$	0	0
$885$	1.14338e11	0.186388
$886$	0	0
$887$	8.83378e11i	1.42709i	0.700608	+	0.713546i	$0.252912\pi$
$887$	8.83378e11i	1.42709i	−0.700608	+	0.713546i	$0.747088\pi$
$888$	0	0
$889$	−1.35664e10	−0.0217198
$890$	0	0
$891$	6.68769e10i	0.106112i
$892$	0	0
$893$	1.77321e9	0.00278840
$894$	0	0
$895$	− 5.34302e11i	− 0.832712i
$896$	0	0
$897$	8.26119e11	1.27607
$898$	0	0
$899$	1.20376e11i	0.184289i
$900$	0	0
$901$	−3.57183e10	−0.0541990
$902$	0	0
$903$	− 9.80527e10i	− 0.147472i
$904$	0	0
$905$	2.53390e11	0.377742
$906$	0	0
$907$	1.20277e12i	1.77727i	0.458619	+	0.888633i	$0.348344\pi$
$907$	1.20277e12i	1.77727i	−0.458619	+	0.888633i	$0.651656\pi$
$908$	0	0
$909$	−1.41863e11	−0.207784
$910$	0	0
$911$	8.65949e10i	0.125724i	0.998022	+	0.0628621i	$0.0200228\pi$
$911$	8.65949e10i	0.125724i	−0.998022	+	0.0628621i	$0.979977\pi$
$912$	0	0
$913$	3.79492e11	0.546159
$914$	0	0
$915$	− 1.68393e11i	− 0.240237i
$916$	0	0
$917$	−2.17178e11	−0.307141
$918$	0	0
$919$	− 1.89815e11i	− 0.266114i	−0.991108	−	0.133057i	$-0.957521\pi$
$919$	− 1.89815e11i	− 0.266114i	0.991108	−	0.133057i	$-0.0424793\pi$
$920$	0	0
$921$	6.06964e11	0.843577
$922$	0	0
$923$	− 6.96179e11i	− 0.959211i
$924$	0	0
$925$	6.21979e11	0.849589
$926$	0	0
$927$	− 1.49444e11i	− 0.202376i
$928$	0	0
$929$	−3.56926e11	−0.479198	−0.239599	−	0.970872i	$-0.577016\pi$
$929$	−3.56926e11	−0.479198	−0.239599	+	0.970872i	$0.577016\pi$
$930$	0	0
$931$	− 2.60835e9i	− 0.00347190i
$932$	0	0
$933$	−2.49929e11	−0.329831
$934$	0	0
$935$	− 1.81605e11i	− 0.237619i
$936$	0	0
$937$	−1.40006e12	−1.81631	−0.908154	−	0.418636i	$-0.862508\pi$
$937$	−1.40006e12	−1.81631	−0.908154	+	0.418636i	$0.862508\pi$
$938$	0	0
$939$	5.08790e10i	0.0654450i
$940$	0	0
$941$	−9.26018e10	−0.118103	−0.0590515	−	0.998255i	$-0.518808\pi$
$941$	−9.26018e10	−0.118103	−0.0590515	+	0.998255i	$0.518808\pi$
$942$	0	0
$943$	1.36620e12i	1.72769i
$944$	0	0
$945$	1.84643e10	0.0231529
$946$	0	0
$947$	1.03974e12i	1.29278i	0.763007	+	0.646390i	$0.223722\pi$
$947$	1.03974e12i	1.29278i	−0.763007	+	0.646390i	$0.776278\pi$
$948$	0	0
$949$	2.68778e11	0.331381
$950$	0	0
$951$	4.78074e11i	0.584485i
$952$	0	0
$953$	−1.29369e12	−1.56841	−0.784205	−	0.620501i	$-0.786929\pi$
$953$	−1.29369e12	−1.56841	−0.784205	+	0.620501i	$0.786929\pi$
$954$	0	0
$955$	2.11644e11i	0.254444i
$956$	0	0
$957$	−2.70240e11	−0.322182
$958$	0	0
$959$	7.81280e10i	0.0923703i
$960$	0	0
$961$	7.68054e11	0.900530
$962$	0	0
$963$	− 3.92538e11i	− 0.456433i
$964$	0	0
$965$	−4.70379e11	−0.542424
$966$	0	0
$967$	1.49704e12i	1.71209i	0.516902	+	0.856045i	$0.327085\pi$
$967$	1.49704e12i	1.71209i	−0.516902	+	0.856045i	$0.672915\pi$
$968$	0	0
$969$	6.40514e8	0.000726497 0
$970$	0	0
$971$	2.15769e11i	0.242724i	0.992608	+	0.121362i	$0.0387261\pi$
$971$	2.15769e11i	0.242724i	−0.992608	+	0.121362i	$0.961274\pi$
$972$	0	0
$973$	−4.16182e10	−0.0464336
$974$	0	0
$975$	− 4.61460e11i	− 0.510641i
$976$	0	0
$977$	9.51586e11	1.04441	0.522203	−	0.852821i	$-0.325110\pi$
$977$	9.51586e11	1.04441	0.522203	+	0.852821i	$0.325110\pi$
$978$	0	0
$979$	− 5.45780e11i	− 0.594137i
$980$	0	0
$981$	−4.91331e11	−0.530515
$982$	0	0
$983$	5.22888e11i	0.560009i	0.959999	+	0.280004i	$0.0903359\pi$
$983$	5.22888e11i	0.560009i	−0.959999	+	0.280004i	$0.909664\pi$
$984$	0	0
$985$	−9.43499e11	−1.00230
$986$	0	0
$987$	7.27133e10i	0.0766206i
$988$	0	0
$989$	−1.79497e12	−1.87617
$990$	0	0
$991$	− 4.96171e11i	− 0.514443i	−0.966352	−	0.257221i	$-0.917193\pi$
$991$	− 4.96171e11i	− 0.514443i	0.966352	−	0.257221i	$-0.0828070\pi$
$992$	0	0
$993$	7.61544e11	0.783246
$994$	0	0
$995$	− 1.83687e11i	− 0.187407i
$996$	0	0
$997$	4.07659e11	0.412588	0.206294	−	0.978490i	$-0.433860\pi$
$997$	4.07659e11	0.412588	0.206294	+	0.978490i	$0.433860\pi$
$998$	0	0
$999$	3.25585e11i	0.326890i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.a.127.14	yes	32
4.3	odd	2	inner	384.9.g.a.127.13	✓	32
8.3	odd	2		384.9.g.b.127.20	yes	32
8.5	even	2		384.9.g.b.127.19	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.13	✓	32	4.3	odd	2	inner
384.9.g.a.127.14	yes	32	1.1	even	1	trivial
384.9.g.b.127.19	yes	32	8.5	even	2
384.9.g.b.127.20	yes	32	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} -441.863 q^{5} -408.575i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} -441.863 q^{5} -408.575i q^{7} -2187.00 q^{9} +13982.3i q^{11} +50503.9 q^{13} -20663.9i q^{15} +29394.2 q^{17} -465.954i q^{19} +19107.2 q^{21} -349779. i q^{23} -195382. q^{25} -102276. i q^{27} +413282. q^{29} +291268. i q^{31} -653888. q^{33} +180534. i q^{35} -3.18340e6 q^{37} +2.36183e6i q^{39} -3.90588e6 q^{41} -5.13172e6i q^{43} +966354. q^{45} +3.80555e6i q^{47} +5.59787e6 q^{49} +1.37463e6i q^{51} -1.21515e6 q^{53} -6.17826e6i q^{55} +21790.5 q^{57} +5.53322e6i q^{59} +8.14915e6 q^{61} +893553. i q^{63} -2.23158e7 q^{65} -2.31597e7i q^{67} +1.63575e7 q^{69} -1.37847e7i q^{71} +5.32192e6 q^{73} -9.13712e6i q^{75} +5.71282e6 q^{77} -3.42382e7i q^{79} +4.78297e6 q^{81} -2.71409e7i q^{83} -1.29882e7 q^{85} +1.93273e7i q^{87} -3.90336e7 q^{89} -2.06346e7i q^{91} -1.36212e7 q^{93} +205888. i q^{95} +4.04636e6 q^{97} -3.05793e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 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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