LMFDB - Embedded newform 384.9.g.b.127.6

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			127.6
Character	$\chi$	$=$	384.127
Dual form			384.9.g.b.127.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} -407.399 q^{5} -3273.60i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} -407.399 q^{5} -3273.60i q^{7} -2187.00 q^{9} +10496.0i q^{11} -13445.5 q^{13} -19052.2i q^{15} +10188.3 q^{17} +46661.8i q^{19} +153091. q^{21} -347752. i q^{23} -224651. q^{25} -102276. i q^{27} -567090. q^{29} -870569. i q^{31} -490848. q^{33} +1.33366e6i q^{35} -1.06479e6 q^{37} -628782. i q^{39} -795034. q^{41} +326583. i q^{43} +890981. q^{45} -2.23717e6i q^{47} -4.95162e6 q^{49} +476460. i q^{51} +5.05380e6 q^{53} -4.27605e6i q^{55} -2.18216e6 q^{57} -1.57715e7i q^{59} +9.22670e6 q^{61} +7.15935e6i q^{63} +5.47767e6 q^{65} +2.63315e7i q^{67} +1.62628e7 q^{69} -3.08716e7i q^{71} +4.97145e7 q^{73} -1.05059e7i q^{75} +3.43596e7 q^{77} +6.23184e7i q^{79} +4.78297e6 q^{81} -1.35256e6i q^{83} -4.15070e6 q^{85} -2.65202e7i q^{87} +2.12314e7 q^{89} +4.40150e7i q^{91} +4.07125e7 q^{93} -1.90100e7i q^{95} +3.07649e7 q^{97} -2.29547e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−407.399	−0.651838	−0.325919	−	0.945398i	$-0.605674\pi$
$5$	−407.399	−0.651838	−0.325919	+	0.945398i	$0.605674\pi$
$6$	0	0
$7$	− 3273.60i	− 1.36343i	−0.731618	−	0.681715i	$-0.761234\pi$
$7$	− 3273.60i	− 1.36343i	0.731618	−	0.681715i	$-0.238766\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	10496.0i	0.716889i	0.933551	+	0.358444i	$0.116693\pi$
$11$	10496.0i	0.716889i	−0.933551	+	0.358444i	$0.883307\pi$
$12$	0	0
$13$	−13445.5	−0.470763	−0.235382	−	0.971903i	$-0.575634\pi$
$13$	−13445.5	−0.470763	−0.235382	+	0.971903i	$0.575634\pi$
$14$	0	0
$15$	− 19052.2i	− 0.376339i
$16$	0	0
$17$	10188.3	0.121985	0.0609925	−	0.998138i	$-0.480573\pi$
$17$	10188.3	0.121985	0.0609925	+	0.998138i	$0.480573\pi$
$18$	0	0
$19$	46661.8i	0.358053i	0.983844	+	0.179027i	$0.0572948\pi$
$19$	46661.8i	0.358053i	−0.983844	+	0.179027i	$0.942705\pi$
$20$	0	0
$21$	153091.	0.787177
$22$	0	0
$23$	− 347752.i	− 1.24268i	−0.783542	−	0.621339i	$-0.786589\pi$
$23$	− 347752.i	− 1.24268i	0.783542	−	0.621339i	$-0.213411\pi$
$24$	0	0
$25$	−224651.	−0.575107
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	−567090.	−0.801789	−0.400894	−	0.916124i	$-0.631301\pi$
$29$	−567090.	−0.801789	−0.400894	+	0.916124i	$0.631301\pi$
$30$	0	0
$31$	− 870569.i	− 0.942663i	−0.881956	−	0.471331i	$-0.843774\pi$
$31$	− 870569.i	− 0.942663i	0.881956	−	0.471331i	$-0.156226\pi$
$32$	0	0
$33$	−490848.	−0.413896
$34$	0	0
$35$	1.33366e6i	0.888736i
$36$	0	0
$37$	−1.06479e6	−0.568141	−0.284070	−	0.958803i	$-0.591685\pi$
$37$	−1.06479e6	−0.568141	−0.284070	+	0.958803i	$0.591685\pi$
$38$	0	0
$39$	− 628782.i	− 0.271795i
$40$	0	0
$41$	−795034.	−0.281352	−0.140676	−	0.990056i	$-0.544928\pi$
$41$	−795034.	−0.281352	−0.140676	+	0.990056i	$0.544928\pi$
$42$	0	0
$43$	326583.i	0.0955257i	0.998859	+	0.0477629i	$0.0152092\pi$
$43$	326583.i	0.0955257i	−0.998859	+	0.0477629i	$0.984791\pi$
$44$	0	0
$45$	890981.	0.217279
$46$	0	0
$47$	− 2.23717e6i	− 0.458467i	−0.973371	−	0.229233i	$-0.926378\pi$
$47$	− 2.23717e6i	− 0.458467i	0.973371	−	0.229233i	$-0.0736219\pi$
$48$	0	0
$49$	−4.95162e6	−0.858941
$50$	0	0
$51$	476460.i	0.0704280i
$52$	0	0
$53$	5.05380e6	0.640494	0.320247	−	0.947334i	$-0.396234\pi$
$53$	5.05380e6	0.640494	0.320247	+	0.947334i	$0.396234\pi$
$54$	0	0
$55$	− 4.27605e6i	− 0.467296i
$56$	0	0
$57$	−2.18216e6	−0.206722
$58$	0	0
$59$	− 1.57715e7i	− 1.30156i	−0.759266	−	0.650780i	$-0.774442\pi$
$59$	− 1.57715e7i	− 1.30156i	0.759266	−	0.650780i	$-0.225558\pi$
$60$	0	0
$61$	9.22670e6	0.666388	0.333194	−	0.942858i	$-0.391874\pi$
$61$	9.22670e6	0.666388	0.333194	+	0.942858i	$0.391874\pi$
$62$	0	0
$63$	7.15935e6i	0.454477i
$64$	0	0
$65$	5.47767e6	0.306862
$66$	0	0
$67$	2.63315e7i	1.30670i	0.757055	+	0.653352i	$0.226638\pi$
$67$	2.63315e7i	1.30670i	−0.757055	+	0.653352i	$0.773362\pi$
$68$	0	0
$69$	1.62628e7	0.717461
$70$	0	0
$71$	− 3.08716e7i	− 1.21486i	−0.794374	−	0.607429i	$-0.792201\pi$
$71$	− 3.08716e7i	− 1.21486i	0.794374	−	0.607429i	$-0.207799\pi$
$72$	0	0
$73$	4.97145e7	1.75062	0.875310	−	0.483563i	$-0.160658\pi$
$73$	4.97145e7	1.75062	0.875310	+	0.483563i	$0.160658\pi$
$74$	0	0
$75$	− 1.05059e7i	− 0.332038i
$76$	0	0
$77$	3.43596e7	0.977428
$78$	0	0
$79$	6.23184e7i	1.59996i	0.600030	+	0.799978i	$0.295155\pi$
$79$	6.23184e7i	1.59996i	−0.600030	+	0.799978i	$0.704845\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	− 1.35256e6i	− 0.0284999i	−0.999898	−	0.0142499i	$-0.995464\pi$
$83$	− 1.35256e6i	− 0.0284999i	0.999898	−	0.0142499i	$-0.00453605\pi$
$84$	0	0
$85$	−4.15070e6	−0.0795144
$86$	0	0
$87$	− 2.65202e7i	− 0.462913i
$88$	0	0
$89$	2.12314e7	0.338390	0.169195	−	0.985583i	$-0.445883\pi$
$89$	2.12314e7	0.338390	0.169195	+	0.985583i	$0.445883\pi$
$90$	0	0
$91$	4.40150e7i	0.641853i
$92$	0	0
$93$	4.07125e7	0.544247
$94$	0	0
$95$	− 1.90100e7i	− 0.233393i
$96$	0	0
$97$	3.07649e7	0.347511	0.173756	−	0.984789i	$-0.444410\pi$
$97$	3.07649e7	0.347511	0.173756	+	0.984789i	$0.444410\pi$
$98$	0	0
$99$	− 2.29547e7i	− 0.238963i
$100$	0	0
$101$	−1.20174e8	−1.15484	−0.577422	−	0.816446i	$-0.695941\pi$
$101$	−1.20174e8	−1.15484	−0.577422	+	0.816446i	$0.695941\pi$
$102$	0	0
$103$	2.10009e7i	0.186590i	0.995638	+	0.0932952i	$0.0297400\pi$
$103$	2.10009e7i	0.186590i	−0.995638	+	0.0932952i	$0.970260\pi$
$104$	0	0
$105$	−6.23691e7	−0.513112
$106$	0	0
$107$	2.16063e8i	1.64833i	0.566349	+	0.824165i	$0.308355\pi$
$107$	2.16063e8i	1.64833i	−0.566349	+	0.824165i	$0.691645\pi$
$108$	0	0
$109$	−1.19672e8	−0.847785	−0.423892	−	0.905713i	$-0.639337\pi$
$109$	−1.19672e8	−0.847785	−0.423892	+	0.905713i	$0.639337\pi$
$110$	0	0
$111$	− 4.97952e7i	− 0.328016i
$112$	0	0
$113$	−2.17470e8	−1.33379	−0.666893	−	0.745153i	$-0.732376\pi$
$113$	−2.17470e8	−1.33379	−0.666893	+	0.745153i	$0.732376\pi$
$114$	0	0
$115$	1.41674e8i	0.810025i
$116$	0	0
$117$	2.94052e7	0.156921
$118$	0	0
$119$	− 3.33524e7i	− 0.166318i
$120$	0	0
$121$	1.04193e8	0.486070
$122$	0	0
$123$	− 3.71801e7i	− 0.162439i
$124$	0	0
$125$	2.50663e8	1.02671
$126$	0	0
$127$	1.85119e8i	0.711599i	0.934562	+	0.355800i	$0.115791\pi$
$127$	1.85119e8i	0.711599i	−0.934562	+	0.355800i	$0.884209\pi$
$128$	0	0
$129$	−1.52728e7	−0.0551518
$130$	0	0
$131$	− 6.77743e7i	− 0.230133i	−0.993358	−	0.115067i	$-0.963292\pi$
$131$	− 6.77743e7i	− 0.230133i	0.993358	−	0.115067i	$-0.0367082\pi$
$132$	0	0
$133$	1.52752e8	0.488180
$134$	0	0
$135$	4.16671e7i	0.125446i
$136$	0	0
$137$	−3.06009e8	−0.868665	−0.434332	−	0.900753i	$-0.643016\pi$
$137$	−3.06009e8	−0.868665	−0.434332	+	0.900753i	$0.643016\pi$
$138$	0	0
$139$	2.43908e8i	0.653381i	0.945131	+	0.326691i	$0.105934\pi$
$139$	2.43908e8i	0.653381i	−0.945131	+	0.326691i	$0.894066\pi$
$140$	0	0
$141$	1.04622e8	0.264696
$142$	0	0
$143$	− 1.41123e8i	− 0.337485i
$144$	0	0
$145$	2.31032e8	0.522636
$146$	0	0
$147$	− 2.31565e8i	− 0.495910i
$148$	0	0
$149$	1.37070e8	0.278098	0.139049	−	0.990285i	$-0.455595\pi$
$149$	1.37070e8	0.278098	0.139049	+	0.990285i	$0.455595\pi$
$150$	0	0
$151$	5.29907e8i	1.01928i	0.860389	+	0.509638i	$0.170221\pi$
$151$	5.29907e8i	1.01928i	−0.860389	+	0.509638i	$0.829779\pi$
$152$	0	0
$153$	−2.22818e7	−0.0406616
$154$	0	0
$155$	3.54669e8i	0.614464i
$156$	0	0
$157$	−3.78509e7	−0.0622985	−0.0311492	−	0.999515i	$-0.509917\pi$
$157$	−3.78509e7	−0.0622985	−0.0311492	+	0.999515i	$0.509917\pi$
$158$	0	0
$159$	2.36343e8i	0.369789i
$160$	0	0
$161$	−1.13840e9	−1.69431
$162$	0	0
$163$	− 8.77831e8i	− 1.24354i	−0.783199	−	0.621771i	$-0.786413\pi$
$163$	− 8.77831e8i	− 1.24354i	0.783199	−	0.621771i	$-0.213587\pi$
$164$	0	0
$165$	1.99971e8	0.269793
$166$	0	0
$167$	5.82173e8i	0.748491i	0.927330	+	0.374245i	$0.122098\pi$
$167$	5.82173e8i	0.748491i	−0.927330	+	0.374245i	$0.877902\pi$
$168$	0	0
$169$	−6.34950e8	−0.778382
$170$	0	0
$171$	− 1.02049e8i	− 0.119351i
$172$	0	0
$173$	3.58140e8	0.399824	0.199912	−	0.979814i	$-0.435934\pi$
$173$	3.58140e8	0.399824	0.199912	+	0.979814i	$0.435934\pi$
$174$	0	0
$175$	7.35417e8i	0.784118i
$176$	0	0
$177$	7.37559e8	0.751456
$178$	0	0
$179$	− 1.75311e8i	− 0.170764i	−0.996348	−	0.0853821i	$-0.972789\pi$
$179$	− 1.75311e8i	− 0.170764i	0.996348	−	0.0853821i	$-0.0272111\pi$
$180$	0	0
$181$	1.75626e9	1.63635	0.818174	−	0.574971i	$-0.194987\pi$
$181$	1.75626e9	1.63635	0.818174	+	0.574971i	$0.194987\pi$
$182$	0	0
$183$	4.31490e8i	0.384739i
$184$	0	0
$185$	4.33793e8	0.370336
$186$	0	0
$187$	1.06936e8i	0.0874496i
$188$	0	0
$189$	−3.34810e8	−0.262392
$190$	0	0
$191$	1.59158e9i	1.19590i	0.801534	+	0.597949i	$0.204017\pi$
$191$	1.59158e9i	1.19590i	−0.801534	+	0.597949i	$0.795983\pi$
$192$	0	0
$193$	−2.65268e9	−1.91186	−0.955928	−	0.293601i	$-0.905146\pi$
$193$	−2.65268e9	−1.91186	−0.955928	+	0.293601i	$0.905146\pi$
$194$	0	0
$195$	2.56165e8i	0.177167i
$196$	0	0
$197$	−1.20288e9	−0.798650	−0.399325	−	0.916809i	$-0.630756\pi$
$197$	−1.20288e9	−0.798650	−0.399325	+	0.916809i	$0.630756\pi$
$198$	0	0
$199$	− 1.61284e9i	− 1.02844i	−0.857659	−	0.514220i	$-0.828082\pi$
$199$	− 1.61284e9i	− 1.02844i	0.857659	−	0.514220i	$-0.171918\pi$
$200$	0	0
$201$	−1.23140e9	−0.754425
$202$	0	0
$203$	1.85642e9i	1.09318i
$204$	0	0
$205$	3.23896e8	0.183396
$206$	0	0
$207$	7.60535e8i	0.414226i
$208$	0	0
$209$	−4.89761e8	−0.256684
$210$	0	0
$211$	2.74529e9i	1.38503i	0.721404	+	0.692514i	$0.243497\pi$
$211$	2.74529e9i	1.38503i	−0.721404	+	0.692514i	$0.756503\pi$
$212$	0	0
$213$	1.44372e9	0.701399
$214$	0	0
$215$	− 1.33050e8i	− 0.0622673i
$216$	0	0
$217$	−2.84989e9	−1.28525
$218$	0	0
$219$	2.32492e9i	1.01072i
$220$	0	0
$221$	−1.36987e8	−0.0574260
$222$	0	0
$223$	2.49333e9i	1.00823i	0.863636	+	0.504116i	$0.168182\pi$
$223$	2.49333e9i	1.00823i	−0.863636	+	0.504116i	$0.831818\pi$
$224$	0	0
$225$	4.91312e8	0.191702
$226$	0	0
$227$	4.50386e9i	1.69622i	0.529823	+	0.848108i	$0.322258\pi$
$227$	4.50386e9i	1.69622i	−0.529823	+	0.848108i	$0.677742\pi$
$228$	0	0
$229$	3.17602e9	1.15489	0.577446	−	0.816429i	$-0.304049\pi$
$229$	3.17602e9	1.15489	0.577446	+	0.816429i	$0.304049\pi$
$230$	0	0
$231$	1.60684e9i	0.564318i
$232$	0	0
$233$	3.87947e7	0.0131628	0.00658141	−	0.999978i	$-0.497905\pi$
$233$	3.87947e7	0.0131628	0.00658141	+	0.999978i	$0.497905\pi$
$234$	0	0
$235$	9.11422e8i	0.298846i
$236$	0	0
$237$	−2.91434e9	−0.923735
$238$	0	0
$239$	2.33943e9i	0.717000i	0.933530	+	0.358500i	$0.116712\pi$
$239$	2.33943e9i	0.717000i	−0.933530	+	0.358500i	$0.883288\pi$
$240$	0	0
$241$	3.11513e9	0.923440	0.461720	−	0.887026i	$-0.347233\pi$
$241$	3.11513e9	0.923440	0.461720	+	0.887026i	$0.347233\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	2.01729e9	0.559891
$246$	0	0
$247$	− 6.27390e8i	− 0.168558i
$248$	0	0
$249$	6.32528e7	0.0164544
$250$	0	0
$251$	− 3.85352e7i	− 0.00970874i	−0.999988	−	0.00485437i	$-0.998455\pi$
$251$	− 3.85352e7i	− 0.00970874i	0.999988	−	0.00485437i	$-0.00154520\pi$
$252$	0	0
$253$	3.65000e9	0.890863
$254$	0	0
$255$	− 1.94109e8i	− 0.0459077i
$256$	0	0
$257$	−6.70732e9	−1.53750	−0.768752	−	0.639547i	$-0.779122\pi$
$257$	−6.70732e9	−1.53750	−0.768752	+	0.639547i	$0.779122\pi$
$258$	0	0
$259$	3.48568e9i	0.774620i
$260$	0	0
$261$	1.24023e9	0.267263
$262$	0	0
$263$	3.10992e9i	0.650019i	0.945711	+	0.325009i	$0.105367\pi$
$263$	3.10992e9i	0.650019i	−0.945711	+	0.325009i	$0.894633\pi$
$264$	0	0
$265$	−2.05891e9	−0.417498
$266$	0	0
$267$	9.92892e8i	0.195370i
$268$	0	0
$269$	8.41767e9	1.60762	0.803808	−	0.594888i	$-0.202804\pi$
$269$	8.41767e9	1.60762	0.803808	+	0.594888i	$0.202804\pi$
$270$	0	0
$271$	1.56535e9i	0.290225i	0.989415	+	0.145113i	$0.0463545\pi$
$271$	1.56535e9i	0.290225i	−0.989415	+	0.145113i	$0.953646\pi$
$272$	0	0
$273$	−2.05838e9	−0.370574
$274$	0	0
$275$	− 2.35793e9i	− 0.412288i
$276$	0	0
$277$	5.42631e9	0.921691	0.460846	−	0.887480i	$-0.347546\pi$
$277$	5.42631e9	0.921691	0.460846	+	0.887480i	$0.347546\pi$
$278$	0	0
$279$	1.90393e9i	0.314221i
$280$	0	0
$281$	7.39636e9	1.18630	0.593148	−	0.805094i	$-0.297885\pi$
$281$	7.39636e9	1.18630	0.593148	+	0.805094i	$0.297885\pi$
$282$	0	0
$283$	− 3.37541e9i	− 0.526236i	−0.964764	−	0.263118i	$-0.915249\pi$
$283$	− 3.37541e9i	− 0.526236i	0.964764	−	0.263118i	$-0.0847508\pi$
$284$	0	0
$285$	8.89009e8	0.134749
$286$	0	0
$287$	2.60262e9i	0.383604i
$288$	0	0
$289$	−6.87196e9	−0.985120
$290$	0	0
$291$	1.43873e9i	0.200636i
$292$	0	0
$293$	8.29070e9	1.12492	0.562459	−	0.826825i	$-0.309856\pi$
$293$	8.29070e9	1.12492	0.562459	+	0.826825i	$0.309856\pi$
$294$	0	0
$295$	6.42528e9i	0.848407i
$296$	0	0
$297$	1.07348e9	0.137965
$298$	0	0
$299$	4.67570e9i	0.585008i
$300$	0	0
$301$	1.06910e9	0.130243
$302$	0	0
$303$	− 5.61996e9i	− 0.666750i
$304$	0	0
$305$	−3.75895e9	−0.434377
$306$	0	0
$307$	9.50863e9i	1.07044i	0.844711	+	0.535222i	$0.179772\pi$
$307$	9.50863e9i	1.07044i	−0.844711	+	0.535222i	$0.820228\pi$
$308$	0	0
$309$	−9.82116e8	−0.107728
$310$	0	0
$311$	1.34808e10i	1.44104i	0.693436	+	0.720519i	$0.256096\pi$
$311$	1.34808e10i	1.44104i	−0.693436	+	0.720519i	$0.743904\pi$
$312$	0	0
$313$	4.39796e9	0.458220	0.229110	−	0.973401i	$-0.426418\pi$
$313$	4.39796e9	0.458220	0.229110	+	0.973401i	$0.426418\pi$
$314$	0	0
$315$	− 2.91671e9i	− 0.296245i
$316$	0	0
$317$	1.56770e9	0.155248	0.0776239	−	0.996983i	$-0.475267\pi$
$317$	1.56770e9	0.155248	0.0776239	+	0.996983i	$0.475267\pi$
$318$	0	0
$319$	− 5.95216e9i	− 0.574793i
$320$	0	0
$321$	−1.01042e10	−0.951664
$322$	0	0
$323$	4.75405e8i	0.0436771i
$324$	0	0
$325$	3.02054e9	0.270739
$326$	0	0
$327$	− 5.59649e9i	− 0.489469i
$328$	0	0
$329$	−7.32360e9	−0.625088
$330$	0	0
$331$	2.12629e10i	1.77137i	0.464284	+	0.885686i	$0.346312\pi$
$331$	2.12629e10i	1.77137i	−0.464284	+	0.885686i	$0.653688\pi$
$332$	0	0
$333$	2.32869e9	0.189380
$334$	0	0
$335$	− 1.07274e10i	− 0.851759i
$336$	0	0
$337$	−2.59216e9	−0.200975	−0.100487	−	0.994938i	$-0.532040\pi$
$337$	−2.59216e9	−0.200975	−0.100487	+	0.994938i	$0.532040\pi$
$338$	0	0
$339$	− 1.01701e10i	− 0.770062i
$340$	0	0
$341$	9.13747e9	0.675785
$342$	0	0
$343$	− 2.66201e9i	− 0.192324i
$344$	0	0
$345$	−6.62543e9	−0.467668
$346$	0	0
$347$	− 2.19276e8i	− 0.0151242i	−0.999971	−	0.00756212i	$-0.997593\pi$
$347$	− 2.19276e8i	− 0.0151242i	0.999971	−	0.00756212i	$-0.00240712\pi$
$348$	0	0
$349$	−1.94017e10	−1.30779	−0.653896	−	0.756585i	$-0.726866\pi$
$349$	−1.94017e10	−1.30779	−0.653896	+	0.756585i	$0.726866\pi$
$350$	0	0
$351$	1.37515e9i	0.0905984i
$352$	0	0
$353$	−2.82874e10	−1.82177	−0.910886	−	0.412659i	$-0.864600\pi$
$353$	−2.82874e10	−1.82177	−0.910886	+	0.412659i	$0.864600\pi$
$354$	0	0
$355$	1.25771e10i	0.791891i
$356$	0	0
$357$	1.55974e9	0.0960237
$358$	0	0
$359$	1.18790e10i	0.715157i	0.933883	+	0.357578i	$0.116398\pi$
$359$	1.18790e10i	0.715157i	−0.933883	+	0.357578i	$0.883602\pi$
$360$	0	0
$361$	1.48062e10	0.871798
$362$	0	0
$363$	4.87265e9i	0.280633i
$364$	0	0
$365$	−2.02536e10	−1.14112
$366$	0	0
$367$	1.72004e10i	0.948143i	0.880486	+	0.474071i	$0.157216\pi$
$367$	1.72004e10i	0.948143i	−0.880486	+	0.474071i	$0.842784\pi$
$368$	0	0
$369$	1.73874e9	0.0937841
$370$	0	0
$371$	− 1.65441e10i	− 0.873268i
$372$	0	0
$373$	−1.31466e10	−0.679169	−0.339585	−	0.940575i	$-0.610287\pi$
$373$	−1.31466e10	−0.679169	−0.339585	+	0.940575i	$0.610287\pi$
$374$	0	0
$375$	1.17223e10i	0.592774i
$376$	0	0
$377$	7.62479e9	0.377453
$378$	0	0
$379$	− 1.39565e10i	− 0.676425i	−0.941070	−	0.338212i	$-0.890178\pi$
$379$	− 1.39565e10i	− 0.676425i	0.941070	−	0.338212i	$-0.109822\pi$
$380$	0	0
$381$	−8.65715e9	−0.410842
$382$	0	0
$383$	8.85512e8i	0.0411528i	0.999788	+	0.0205764i	$0.00655013\pi$
$383$	8.85512e8i	0.0411528i	−0.999788	+	0.0205764i	$0.993450\pi$
$384$	0	0
$385$	−1.39980e10	−0.637125
$386$	0	0
$387$	− 7.14238e8i	− 0.0318419i
$388$	0	0
$389$	1.15118e10	0.502741	0.251370	−	0.967891i	$-0.419119\pi$
$389$	1.15118e10	0.502741	0.251370	+	0.967891i	$0.419119\pi$
$390$	0	0
$391$	− 3.54301e9i	− 0.151588i
$392$	0	0
$393$	3.16949e9	0.132868
$394$	0	0
$395$	− 2.53884e10i	− 1.04291i
$396$	0	0
$397$	5.50949e9	0.221794	0.110897	−	0.993832i	$-0.464628\pi$
$397$	5.50949e9	0.221794	0.110897	+	0.993832i	$0.464628\pi$
$398$	0	0
$399$	7.14350e9i	0.281851i
$400$	0	0
$401$	2.62380e10	1.01473	0.507367	−	0.861730i	$-0.330619\pi$
$401$	2.62380e10	1.01473	0.507367	+	0.861730i	$0.330619\pi$
$402$	0	0
$403$	1.17052e10i	0.443771i
$404$	0	0
$405$	−1.94858e9	−0.0724265
$406$	0	0
$407$	− 1.11760e10i	− 0.407294i
$408$	0	0
$409$	5.24246e10	1.87345	0.936724	−	0.350068i	$-0.113841\pi$
$409$	5.24246e10	1.87345	0.936724	+	0.350068i	$0.113841\pi$
$410$	0	0
$411$	− 1.43106e10i	− 0.501524i
$412$	0	0
$413$	−5.16294e10	−1.77459
$414$	0	0
$415$	5.51030e8i	0.0185773i
$416$	0	0
$417$	−1.14064e10	−0.377230
$418$	0	0
$419$	1.02975e10i	0.334099i	0.985949	+	0.167049i	$0.0534239\pi$
$419$	1.02975e10i	0.334099i	−0.985949	+	0.167049i	$0.946576\pi$
$420$	0	0
$421$	3.56112e10	1.13359	0.566797	−	0.823857i	$-0.308182\pi$
$421$	3.56112e10	1.13359	0.566797	+	0.823857i	$0.308182\pi$
$422$	0	0
$423$	4.89270e9i	0.152822i
$424$	0	0
$425$	−2.28881e9	−0.0701544
$426$	0	0
$427$	− 3.02045e10i	− 0.908574i
$428$	0	0
$429$	6.59968e9	0.194847
$430$	0	0
$431$	3.77460e9i	0.109386i	0.998503	+	0.0546931i	$0.0174180\pi$
$431$	3.77460e9i	0.109386i	−0.998503	+	0.0546931i	$0.982582\pi$
$432$	0	0
$433$	−1.26932e10	−0.361094	−0.180547	−	0.983566i	$-0.557787\pi$
$433$	−1.26932e10	−0.361094	−0.180547	+	0.983566i	$0.557787\pi$
$434$	0	0
$435$	1.08043e10i	0.301744i
$436$	0	0
$437$	1.62268e10	0.444945
$438$	0	0
$439$	− 3.54955e10i	− 0.955687i	−0.878445	−	0.477843i	$-0.841419\pi$
$439$	− 3.54955e10i	− 0.955687i	0.878445	−	0.477843i	$-0.158581\pi$
$440$	0	0
$441$	1.08292e10	0.286314
$442$	0	0
$443$	− 2.11741e10i	− 0.549782i	−0.961475	−	0.274891i	$-0.911358\pi$
$443$	− 2.11741e10i	− 0.549782i	0.961475	−	0.274891i	$-0.0886418\pi$
$444$	0	0
$445$	−8.64963e9	−0.220576
$446$	0	0
$447$	6.41014e9i	0.160560i
$448$	0	0
$449$	4.46138e10	1.09770	0.548850	−	0.835921i	$-0.315066\pi$
$449$	4.46138e10	1.09770	0.548850	+	0.835921i	$0.315066\pi$
$450$	0	0
$451$	− 8.34466e9i	− 0.201698i
$452$	0	0
$453$	−2.47813e10	−0.588480
$454$	0	0
$455$	− 1.79317e10i	− 0.418384i
$456$	0	0
$457$	3.97116e10	0.910442	0.455221	−	0.890379i	$-0.349560\pi$
$457$	3.97116e10	0.910442	0.455221	+	0.890379i	$0.349560\pi$
$458$	0	0
$459$	− 1.04202e9i	− 0.0234760i
$460$	0	0
$461$	−6.69457e10	−1.48224	−0.741121	−	0.671371i	$-0.765706\pi$
$461$	−6.69457e10	−1.48224	−0.741121	+	0.671371i	$0.765706\pi$
$462$	0	0
$463$	− 3.68160e10i	− 0.801148i	−0.916264	−	0.400574i	$-0.868811\pi$
$463$	− 3.68160e10i	− 0.801148i	0.916264	−	0.400574i	$-0.131189\pi$
$464$	0	0
$465$	−1.65862e10	−0.354761
$466$	0	0
$467$	− 6.52960e9i	− 0.137284i	−0.997641	−	0.0686419i	$-0.978133\pi$
$467$	− 6.52960e9i	− 0.137284i	0.997641	−	0.0686419i	$-0.0218666\pi$
$468$	0	0
$469$	8.61988e10	1.78160
$470$	0	0
$471$	− 1.77011e9i	− 0.0359680i
$472$	0	0
$473$	−3.42781e9	−0.0684813
$474$	0	0
$475$	− 1.04826e10i	− 0.205919i
$476$	0	0
$477$	−1.10527e10	−0.213498
$478$	0	0
$479$	− 2.10502e10i	− 0.399866i	−0.979810	−	0.199933i	$-0.935928\pi$
$479$	− 2.10502e10i	− 0.399866i	0.979810	−	0.199933i	$-0.0640725\pi$
$480$	0	0
$481$	1.43166e10	0.267460
$482$	0	0
$483$	− 5.32377e10i	− 0.978208i
$484$	0	0
$485$	−1.25336e10	−0.226521
$486$	0	0
$487$	− 7.19798e10i	− 1.27966i	−0.768516	−	0.639831i	$-0.779004\pi$
$487$	− 7.19798e10i	− 1.27966i	0.768516	−	0.639831i	$-0.220996\pi$
$488$	0	0
$489$	4.10521e10	0.717959
$490$	0	0
$491$	4.10927e10i	0.707032i	0.935429	+	0.353516i	$0.115014\pi$
$491$	4.10927e10i	0.707032i	−0.935429	+	0.353516i	$0.884986\pi$
$492$	0	0
$493$	−5.77768e9	−0.0978061
$494$	0	0
$495$	9.35171e9i	0.155765i
$496$	0	0
$497$	−1.01061e11	−1.65637
$498$	0	0
$499$	− 7.87766e10i	− 1.27056i	−0.772282	−	0.635280i	$-0.780885\pi$
$499$	− 7.87766e10i	− 1.27056i	0.772282	−	0.635280i	$-0.219115\pi$
$500$	0	0
$501$	−2.72255e10	−0.432141
$502$	0	0
$503$	− 5.83657e10i	− 0.911771i	−0.890038	−	0.455886i	$-0.849323\pi$
$503$	− 5.83657e10i	− 0.911771i	0.890038	−	0.455886i	$-0.150677\pi$
$504$	0	0
$505$	4.89586e10	0.752772
$506$	0	0
$507$	− 2.96937e10i	− 0.449399i
$508$	0	0
$509$	−1.24022e11	−1.84768	−0.923842	−	0.382774i	$-0.874969\pi$
$509$	−1.24022e11	−1.84768	−0.923842	+	0.382774i	$0.874969\pi$
$510$	0	0
$511$	− 1.62745e11i	− 2.38685i
$512$	0	0
$513$	4.77238e9	0.0689074
$514$	0	0
$515$	− 8.55575e9i	− 0.121627i
$516$	0	0
$517$	2.34813e10	0.328670
$518$	0	0
$519$	1.67486e10i	0.230838i
$520$	0	0
$521$	3.90408e10	0.529868	0.264934	−	0.964267i	$-0.414650\pi$
$521$	3.90408e10	0.529868	0.264934	+	0.964267i	$0.414650\pi$
$522$	0	0
$523$	1.01521e11i	1.35691i	0.734644	+	0.678453i	$0.237349\pi$
$523$	1.01521e11i	1.35691i	−0.734644	+	0.678453i	$0.762651\pi$
$524$	0	0
$525$	−3.43920e10	−0.452711
$526$	0	0
$527$	− 8.86962e9i	− 0.114991i
$528$	0	0
$529$	−4.26208e10	−0.544250
$530$	0	0
$531$	3.44922e10i	0.433853i
$532$	0	0
$533$	1.06896e10	0.132450
$534$	0	0
$535$	− 8.80236e10i	− 1.07444i
$536$	0	0
$537$	8.19848e9	0.0985907
$538$	0	0
$539$	− 5.19721e10i	− 0.615765i
$540$	0	0
$541$	−9.37074e10	−1.09392	−0.546959	−	0.837159i	$-0.684215\pi$
$541$	−9.37074e10	−1.09392	−0.546959	+	0.837159i	$0.684215\pi$
$542$	0	0
$543$	8.21324e10i	0.944746i
$544$	0	0
$545$	4.87541e10	0.552619
$546$	0	0
$547$	− 4.19572e10i	− 0.468659i	−0.972157	−	0.234329i	$-0.924711\pi$
$547$	− 4.19572e10i	− 0.468659i	0.972157	−	0.234329i	$-0.0752894\pi$
$548$	0	0
$549$	−2.01788e10	−0.222129
$550$	0	0
$551$	− 2.64615e10i	− 0.287083i
$552$	0	0
$553$	2.04005e11	2.18143
$554$	0	0
$555$	2.02865e10i	0.213813i
$556$	0	0
$557$	1.60858e10	0.167118	0.0835588	−	0.996503i	$-0.473371\pi$
$557$	1.60858e10	0.167118	0.0835588	+	0.996503i	$0.473371\pi$
$558$	0	0
$559$	− 4.39107e9i	− 0.0449700i
$560$	0	0
$561$	−5.00091e9	−0.0504891
$562$	0	0
$563$	− 9.73474e10i	− 0.968926i	−0.874812	−	0.484463i	$-0.839015\pi$
$563$	− 9.73474e10i	− 0.968926i	0.874812	−	0.484463i	$-0.160985\pi$
$564$	0	0
$565$	8.85972e10	0.869413
$566$	0	0
$567$	− 1.56575e10i	− 0.151492i
$568$	0	0
$569$	1.23860e11	1.18163	0.590817	−	0.806805i	$-0.298805\pi$
$569$	1.23860e11	1.18163	0.590817	+	0.806805i	$0.298805\pi$
$570$	0	0
$571$	6.86051e10i	0.645375i	0.946506	+	0.322687i	$0.104586\pi$
$571$	6.86051e10i	0.645375i	−0.946506	+	0.322687i	$0.895414\pi$
$572$	0	0
$573$	−7.44307e10	−0.690452
$574$	0	0
$575$	7.81230e10i	0.714673i
$576$	0	0
$577$	1.90364e11	1.71744	0.858722	−	0.512443i	$-0.171259\pi$
$577$	1.90364e11	1.71744	0.858722	+	0.512443i	$0.171259\pi$
$578$	0	0
$579$	− 1.24053e11i	− 1.10381i
$580$	0	0
$581$	−4.42772e9	−0.0388576
$582$	0	0
$583$	5.30446e10i	0.459163i
$584$	0	0
$585$	−1.19797e10	−0.102287
$586$	0	0
$587$	− 9.70732e10i	− 0.817611i	−0.912622	−	0.408806i	$-0.865945\pi$
$587$	− 9.70732e10i	− 0.817611i	0.912622	−	0.408806i	$-0.134055\pi$
$588$	0	0
$589$	4.06223e10	0.337523
$590$	0	0
$591$	− 5.62530e10i	− 0.461101i
$592$	0	0
$593$	−1.79803e11	−1.45404	−0.727022	−	0.686614i	$-0.759096\pi$
$593$	−1.79803e11	−1.45404	−0.727022	+	0.686614i	$0.759096\pi$
$594$	0	0
$595$	1.35877e10i	0.108412i
$596$	0	0
$597$	7.54250e10	0.593770
$598$	0	0
$599$	1.77390e11i	1.37791i	0.724803	+	0.688956i	$0.241931\pi$
$599$	1.77390e11i	1.37791i	−0.724803	+	0.688956i	$0.758069\pi$
$600$	0	0
$601$	5.99124e9	0.0459218	0.0229609	−	0.999736i	$-0.492691\pi$
$601$	5.99124e9	0.0459218	0.0229609	+	0.999736i	$0.492691\pi$
$602$	0	0
$603$	− 5.75871e10i	− 0.435568i
$604$	0	0
$605$	−4.24483e10	−0.316839
$606$	0	0
$607$	− 4.43542e10i	− 0.326723i	−0.986566	−	0.163361i	$-0.947766\pi$
$607$	− 4.43542e10i	− 0.326723i	0.986566	−	0.163361i	$-0.0522337\pi$
$608$	0	0
$609$	−8.68163e10	−0.631149
$610$	0	0
$611$	3.00798e10i	0.215829i
$612$	0	0
$613$	−1.09894e11	−0.778271	−0.389136	−	0.921180i	$-0.627226\pi$
$613$	−1.09894e11	−0.778271	−0.389136	+	0.921180i	$0.627226\pi$
$614$	0	0
$615$	1.51471e10i	0.105884i
$616$	0	0
$617$	1.71291e11	1.18193	0.590967	−	0.806696i	$-0.298746\pi$
$617$	1.71291e11	1.18193	0.590967	+	0.806696i	$0.298746\pi$
$618$	0	0
$619$	9.88692e10i	0.673440i	0.941605	+	0.336720i	$0.109318\pi$
$619$	9.88692e10i	0.673440i	−0.941605	+	0.336720i	$0.890682\pi$
$620$	0	0
$621$	−3.55667e10	−0.239154
$622$	0	0
$623$	− 6.95029e10i	− 0.461371i
$624$	0	0
$625$	−1.43654e10	−0.0941450
$626$	0	0
$627$	− 2.29039e10i	− 0.148197i
$628$	0	0
$629$	−1.08484e10	−0.0693046
$630$	0	0
$631$	− 8.72235e9i	− 0.0550194i	−0.999622	−	0.0275097i	$-0.991242\pi$
$631$	− 8.72235e9i	− 0.0550194i	0.999622	−	0.0275097i	$-0.00875772\pi$
$632$	0	0
$633$	−1.28385e11	−0.799647
$634$	0	0
$635$	− 7.54172e10i	− 0.463848i
$636$	0	0
$637$	6.65769e10	0.404358
$638$	0	0
$639$	6.75162e10i	0.404953i
$640$	0	0
$641$	−2.21791e11	−1.31375	−0.656873	−	0.754001i	$-0.728121\pi$
$641$	−2.21791e11	−1.31375	−0.656873	+	0.754001i	$0.728121\pi$
$642$	0	0
$643$	1.41105e11i	0.825465i	0.910852	+	0.412732i	$0.135426\pi$
$643$	1.41105e11i	0.825465i	−0.910852	+	0.412732i	$0.864574\pi$
$644$	0	0
$645$	6.22212e9	0.0359501
$646$	0	0
$647$	3.37906e10i	0.192832i	0.995341	+	0.0964159i	$0.0307379\pi$
$647$	3.37906e10i	0.192832i	−0.995341	+	0.0964159i	$0.969262\pi$
$648$	0	0
$649$	1.65537e11	0.933074
$650$	0	0
$651$	− 1.33276e11i	− 0.742042i
$652$	0	0
$653$	−1.53145e11	−0.842270	−0.421135	−	0.906998i	$-0.638368\pi$
$653$	−1.53145e11	−0.842270	−0.421135	+	0.906998i	$0.638368\pi$
$654$	0	0
$655$	2.76112e10i	0.150010i
$656$	0	0
$657$	−1.08726e11	−0.583540
$658$	0	0
$659$	− 2.23374e11i	− 1.18438i	−0.805799	−	0.592189i	$-0.798264\pi$
$659$	− 2.23374e11i	− 1.18438i	0.805799	−	0.592189i	$-0.201736\pi$
$660$	0	0
$661$	4.03808e10	0.211529	0.105764	−	0.994391i	$-0.466271\pi$
$661$	4.03808e10	0.211529	0.105764	+	0.994391i	$0.466271\pi$
$662$	0	0
$663$	− 6.40623e9i	− 0.0331549i
$664$	0	0
$665$	−6.22310e10	−0.318215
$666$	0	0
$667$	1.97207e11i	0.996366i
$668$	0	0
$669$	−1.16602e11	−0.582103
$670$	0	0
$671$	9.68432e10i	0.477726i
$672$	0	0
$673$	−1.83374e10	−0.0893874	−0.0446937	−	0.999001i	$-0.514231\pi$
$673$	−1.83374e10	−0.0893874	−0.0446937	+	0.999001i	$0.514231\pi$
$674$	0	0
$675$	2.29764e10i	0.110679i
$676$	0	0
$677$	−1.67626e11	−0.797971	−0.398986	−	0.916957i	$-0.630638\pi$
$677$	−1.67626e11	−0.797971	−0.398986	+	0.916957i	$0.630638\pi$
$678$	0	0
$679$	− 1.00712e11i	− 0.473808i
$680$	0	0
$681$	−2.10625e11	−0.979311
$682$	0	0
$683$	− 3.66433e11i	− 1.68388i	−0.539570	−	0.841941i	$-0.681413\pi$
$683$	− 3.66433e11i	− 1.68388i	0.539570	−	0.841941i	$-0.318587\pi$
$684$	0	0
$685$	1.24668e11	0.566229
$686$	0	0
$687$	1.48528e11i	0.666778i
$688$	0	0
$689$	−6.79508e10	−0.301521
$690$	0	0
$691$	4.53924e11i	1.99100i	0.0947667	+	0.995500i	$0.469790\pi$
$691$	4.53924e11i	1.99100i	−0.0947667	+	0.995500i	$0.530210\pi$
$692$	0	0
$693$	−7.51444e10	−0.325809
$694$	0	0
$695$	− 9.93678e10i	− 0.425899i
$696$	0	0
$697$	−8.10005e9	−0.0343207
$698$	0	0
$699$	1.81425e9i	0.00759955i
$700$	0	0
$701$	9.27521e10	0.384106	0.192053	−	0.981385i	$-0.438485\pi$
$701$	9.27521e10	0.384106	0.192053	+	0.981385i	$0.438485\pi$
$702$	0	0
$703$	− 4.96849e10i	− 0.203425i
$704$	0	0
$705$	−4.26230e10	−0.172539
$706$	0	0
$707$	3.93400e11i	1.57455i
$708$	0	0
$709$	3.88967e11	1.53931	0.769657	−	0.638457i	$-0.220427\pi$
$709$	3.88967e11	1.53931	0.769657	+	0.638457i	$0.220427\pi$
$710$	0	0
$711$	− 1.36290e11i	− 0.533319i
$712$	0	0
$713$	−3.02742e11	−1.17143
$714$	0	0
$715$	5.74935e10i	0.219986i
$716$	0	0
$717$	−1.09405e11	−0.413960
$718$	0	0
$719$	3.55192e10i	0.132907i	0.997790	+	0.0664534i	$0.0211684\pi$
$719$	3.55192e10i	0.132907i	−0.997790	+	0.0664534i	$0.978832\pi$
$720$	0	0
$721$	6.87485e10	0.254403
$722$	0	0
$723$	1.45680e11i	0.533148i
$724$	0	0
$725$	1.27397e11	0.461114
$726$	0	0
$727$	− 1.88598e11i	− 0.675150i	−0.941299	−	0.337575i	$-0.890393\pi$
$727$	− 1.88598e11i	− 0.675150i	0.941299	−	0.337575i	$-0.109607\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	3.32733e9i	0.0116527i
$732$	0	0
$733$	−2.35526e11	−0.815875	−0.407937	−	0.913010i	$-0.633752\pi$
$733$	−2.35526e11	−0.815875	−0.407937	+	0.913010i	$0.633752\pi$
$734$	0	0
$735$	9.43391e10i	0.323253i
$736$	0	0
$737$	−2.76375e11	−0.936761
$738$	0	0
$739$	− 1.38779e11i	− 0.465313i	−0.972559	−	0.232657i	$-0.925258\pi$
$739$	− 1.38779e11i	− 0.465313i	0.972559	−	0.232657i	$-0.0747419\pi$
$740$	0	0
$741$	2.93401e10	0.0973172
$742$	0	0
$743$	3.55684e11i	1.16710i	0.812076	+	0.583552i	$0.198338\pi$
$743$	3.55684e11i	1.16710i	−0.812076	+	0.583552i	$0.801662\pi$
$744$	0	0
$745$	−5.58422e10	−0.181275
$746$	0	0
$747$	2.95804e9i	0.00949996i
$748$	0	0
$749$	7.07301e11	2.24738
$750$	0	0
$751$	− 2.58057e11i	− 0.811252i	−0.914039	−	0.405626i	$-0.867053\pi$
$751$	− 2.58057e11i	− 0.811252i	0.914039	−	0.405626i	$-0.132947\pi$
$752$	0	0
$753$	1.80211e9	0.00560534
$754$	0	0
$755$	− 2.15884e11i	− 0.664404i
$756$	0	0
$757$	6.18945e11	1.88481	0.942407	−	0.334467i	$-0.108556\pi$
$757$	6.18945e11	1.88481	0.942407	+	0.334467i	$0.108556\pi$
$758$	0	0
$759$	1.70694e11i	0.514340i
$760$	0	0
$761$	1.28206e11	0.382269	0.191135	−	0.981564i	$-0.438783\pi$
$761$	1.28206e11	0.382269	0.191135	+	0.981564i	$0.438783\pi$
$762$	0	0
$763$	3.91757e11i	1.15590i
$764$	0	0
$765$	9.07759e9	0.0265048
$766$	0	0
$767$	2.12055e11i	0.612727i
$768$	0	0
$769$	4.45860e11	1.27495	0.637476	−	0.770471i	$-0.279979\pi$
$769$	4.45860e11	1.27495	0.637476	+	0.770471i	$0.279979\pi$
$770$	0	0
$771$	− 3.13670e11i	− 0.887679i
$772$	0	0
$773$	1.06021e11	0.296943	0.148471	−	0.988917i	$-0.452565\pi$
$773$	1.06021e11	0.296943	0.148471	+	0.988917i	$0.452565\pi$
$774$	0	0
$775$	1.95574e11i	0.542132i
$776$	0	0
$777$	−1.63009e11	−0.447227
$778$	0	0
$779$	− 3.70978e10i	− 0.100739i
$780$	0	0
$781$	3.24027e11	0.870919
$782$	0	0
$783$	5.79996e10i	0.154304i
$784$	0	0
$785$	1.54204e10	0.0406085
$786$	0	0
$787$	4.81138e11i	1.25421i	0.778934	+	0.627106i	$0.215761\pi$
$787$	4.81138e11i	1.25421i	−0.778934	+	0.627106i	$0.784239\pi$
$788$	0	0
$789$	−1.45436e11	−0.375289
$790$	0	0
$791$	7.11910e11i	1.81852i
$792$	0	0
$793$	−1.24057e11	−0.313711
$794$	0	0
$795$	− 9.62859e10i	− 0.241043i
$796$	0	0
$797$	−5.48975e11	−1.36057	−0.680283	−	0.732950i	$-0.738143\pi$
$797$	−5.48975e11	−1.36057	−0.680283	+	0.732950i	$0.738143\pi$
$798$	0	0
$799$	− 2.27930e10i	− 0.0559261i
$800$	0	0
$801$	−4.64330e10	−0.112797
$802$	0	0
$803$	5.21802e11i	1.25500i
$804$	0	0
$805$	4.63783e11	1.10441
$806$	0	0
$807$	3.93655e11i	0.928158i
$808$	0	0
$809$	−1.33330e11	−0.311268	−0.155634	−	0.987815i	$-0.549742\pi$
$809$	−1.33330e11	−0.311268	−0.155634	+	0.987815i	$0.549742\pi$
$810$	0	0
$811$	2.64471e11i	0.611357i	0.952135	+	0.305679i	$0.0988833\pi$
$811$	2.64471e11i	0.611357i	−0.952135	+	0.305679i	$0.901117\pi$
$812$	0	0
$813$	−7.32044e10	−0.167562
$814$	0	0
$815$	3.57627e11i	0.810588i
$816$	0	0
$817$	−1.52390e10	−0.0342033
$818$	0	0
$819$	− 9.62609e10i	− 0.213951i
$820$	0	0
$821$	−1.60809e11	−0.353947	−0.176974	−	0.984216i	$-0.556631\pi$
$821$	−1.60809e11	−0.353947	−0.176974	+	0.984216i	$0.556631\pi$
$822$	0	0
$823$	− 4.42006e11i	− 0.963448i	−0.876323	−	0.481724i	$-0.840011\pi$
$823$	− 4.42006e11i	− 0.963448i	0.876323	−	0.481724i	$-0.159989\pi$
$824$	0	0
$825$	1.10270e11	0.238035
$826$	0	0
$827$	7.82275e11i	1.67239i	0.548432	+	0.836195i	$0.315225\pi$
$827$	7.82275e11i	1.67239i	−0.548432	+	0.836195i	$0.684775\pi$
$828$	0	0
$829$	8.11654e11	1.71851	0.859256	−	0.511546i	$-0.170927\pi$
$829$	8.11654e11	1.71851	0.859256	+	0.511546i	$0.170927\pi$
$830$	0	0
$831$	2.53763e11i	0.532139i
$832$	0	0
$833$	−5.04487e10	−0.104778
$834$	0	0
$835$	− 2.37177e11i	− 0.487895i
$836$	0	0
$837$	−8.90382e10	−0.181416
$838$	0	0
$839$	5.92405e11i	1.19556i	0.801661	+	0.597779i	$0.203950\pi$
$839$	5.92405e11i	1.19556i	−0.801661	+	0.597779i	$0.796050\pi$
$840$	0	0
$841$	−1.78655e11	−0.357135
$842$	0	0
$843$	3.45894e11i	0.684908i
$844$	0	0
$845$	2.58678e11	0.507379
$846$	0	0
$847$	− 3.41087e11i	− 0.662723i
$848$	0	0
$849$	1.57852e11	0.303822
$850$	0	0
$851$	3.70282e11i	0.706016i
$852$	0	0
$853$	−3.20328e10	−0.0605060	−0.0302530	−	0.999542i	$-0.509631\pi$
$853$	−3.20328e10	−0.0605060	−0.0302530	+	0.999542i	$0.509631\pi$
$854$	0	0
$855$	4.15748e10i	0.0777976i
$856$	0	0
$857$	−6.98922e11	−1.29570	−0.647851	−	0.761767i	$-0.724332\pi$
$857$	−6.98922e11	−1.29570	−0.647851	+	0.761767i	$0.724332\pi$
$858$	0	0
$859$	2.26501e10i	0.0416004i	0.999784	+	0.0208002i	$0.00662139\pi$
$859$	2.26501e10i	0.0416004i	−0.999784	+	0.0208002i	$0.993379\pi$
$860$	0	0
$861$	−1.21712e11	−0.221474
$862$	0	0
$863$	− 7.17224e11i	− 1.29304i	−0.762897	−	0.646520i	$-0.776224\pi$
$863$	− 7.17224e11i	− 1.29304i	0.762897	−	0.646520i	$-0.223776\pi$
$864$	0	0
$865$	−1.45906e11	−0.260620
$866$	0	0
$867$	− 3.21370e11i	− 0.568759i
$868$	0	0
$869$	−6.54092e11	−1.14699
$870$	0	0
$871$	− 3.54040e11i	− 0.615148i
$872$	0	0
$873$	−6.72829e10	−0.115837
$874$	0	0
$875$	− 8.20569e11i	− 1.39985i
$876$	0	0
$877$	9.27817e11	1.56843	0.784213	−	0.620491i	$-0.213067\pi$
$877$	9.27817e11	1.56843	0.784213	+	0.620491i	$0.213067\pi$
$878$	0	0
$879$	3.87718e11i	0.649471i
$880$	0	0
$881$	6.05841e11	1.00567	0.502835	−	0.864383i	$-0.332290\pi$
$881$	6.05841e11	1.00567	0.502835	+	0.864383i	$0.332290\pi$
$882$	0	0
$883$	7.40530e11i	1.21815i	0.793114	+	0.609073i	$0.208459\pi$
$883$	7.40530e11i	1.21815i	−0.793114	+	0.609073i	$0.791541\pi$
$884$	0	0
$885$	−3.00481e11	−0.489828
$886$	0	0
$887$	3.31969e11i	0.536294i	0.963378	+	0.268147i	$0.0864114\pi$
$887$	3.31969e11i	0.536294i	−0.963378	+	0.268147i	$0.913589\pi$
$888$	0	0
$889$	6.06004e11	0.970216
$890$	0	0
$891$	5.02019e10i	0.0796543i
$892$	0	0
$893$	1.04391e11	0.164156
$894$	0	0
$895$	7.14215e10i	0.111311i
$896$	0	0
$897$	−2.18661e11	−0.337754
$898$	0	0
$899$	4.93691e11i	0.755816i
$900$	0	0
$901$	5.14897e10	0.0781306
$902$	0	0
$903$	4.99970e10i	0.0751956i
$904$	0	0
$905$	−7.15500e11	−1.06663
$906$	0	0
$907$	9.22420e11i	1.36301i	0.731812	+	0.681506i	$0.238675\pi$
$907$	9.22420e11i	1.36301i	−0.731812	+	0.681506i	$0.761325\pi$
$908$	0	0
$909$	2.62820e11	0.384948
$910$	0	0
$911$	− 7.95059e11i	− 1.15432i	−0.816631	−	0.577160i	$-0.804161\pi$
$911$	− 7.95059e11i	− 1.15432i	0.816631	−	0.577160i	$-0.195839\pi$
$912$	0	0
$913$	1.41964e10	0.0204313
$914$	0	0
$915$	− 1.75789e11i	− 0.250788i
$916$	0	0
$917$	−2.21866e11	−0.313771
$918$	0	0
$919$	− 2.66604e11i	− 0.373770i	−0.982382	−	0.186885i	$-0.940161\pi$
$919$	− 2.66604e11i	− 0.373770i	0.982382	−	0.186885i	$-0.0598392\pi$
$920$	0	0
$921$	−4.44674e11	−0.618022
$922$	0	0
$923$	4.15083e11i	0.571911i
$924$	0	0
$925$	2.39206e11	0.326742
$926$	0	0
$927$	− 4.59290e10i	− 0.0621968i
$928$	0	0
$929$	−6.32266e11	−0.848862	−0.424431	−	0.905460i	$-0.639526\pi$
$929$	−6.32266e11	−0.848862	−0.424431	+	0.905460i	$0.639526\pi$
$930$	0	0
$931$	− 2.31052e11i	− 0.307547i
$932$	0	0
$933$	−6.30436e11	−0.831983
$934$	0	0
$935$	− 4.35657e10i	− 0.0570030i
$936$	0	0
$937$	−2.56089e11	−0.332225	−0.166113	−	0.986107i	$-0.553122\pi$
$937$	−2.56089e11	−0.332225	−0.166113	+	0.986107i	$0.553122\pi$
$938$	0	0
$939$	2.05672e11i	0.264554i
$940$	0	0
$941$	−9.30732e11	−1.18704	−0.593521	−	0.804818i	$-0.702263\pi$
$941$	−9.30732e11	−1.18704	−0.593521	+	0.804818i	$0.702263\pi$
$942$	0	0
$943$	2.76475e11i	0.349630i
$944$	0	0
$945$	1.36401e11	0.171037
$946$	0	0
$947$	− 1.28083e12i	− 1.59254i	−0.604939	−	0.796272i	$-0.706803\pi$
$947$	− 1.28083e12i	− 1.59254i	0.604939	−	0.796272i	$-0.293197\pi$
$948$	0	0
$949$	−6.68435e11	−0.824127
$950$	0	0
$951$	7.33140e10i	0.0896324i
$952$	0	0
$953$	−1.54398e12	−1.87184	−0.935921	−	0.352210i	$-0.885430\pi$
$953$	−1.54398e12	−1.87184	−0.935921	+	0.352210i	$0.885430\pi$
$954$	0	0
$955$	− 6.48406e11i	− 0.779532i
$956$	0	0
$957$	2.78355e11	0.331857
$958$	0	0
$959$	1.00175e12i	1.18436i
$960$	0	0
$961$	9.50010e10	0.111387
$962$	0	0
$963$	− 4.72529e11i	− 0.549444i
$964$	0	0
$965$	1.08070e12	1.24622
$966$	0	0
$967$	− 1.92379e11i	− 0.220015i	−0.993931	−	0.110008i	$-0.964913\pi$
$967$	− 1.92379e11i	− 0.220015i	0.993931	−	0.110008i	$-0.0350875\pi$
$968$	0	0
$969$	−2.22325e10	−0.0252170
$970$	0	0
$971$	− 1.11611e12i	− 1.25554i	−0.778398	−	0.627771i	$-0.783967\pi$
$971$	− 1.11611e12i	− 1.25554i	0.778398	−	0.627771i	$-0.216033\pi$
$972$	0	0
$973$	7.98456e11	0.890840
$974$	0	0
$975$	1.41257e11i	0.156311i
$976$	0	0
$977$	1.21121e12	1.32936	0.664680	−	0.747128i	$-0.268568\pi$
$977$	1.21121e12	1.32936	0.664680	+	0.747128i	$0.268568\pi$
$978$	0	0
$979$	2.22844e11i	0.242588i
$980$	0	0
$981$	2.61722e11	0.282595
$982$	0	0
$983$	− 1.68429e12i	− 1.80386i	−0.431880	−	0.901931i	$-0.642150\pi$
$983$	− 1.68429e12i	− 1.80386i	0.431880	−	0.901931i	$-0.357850\pi$
$984$	0	0
$985$	4.90051e11	0.520591
$986$	0	0
$987$	− 3.42491e11i	− 0.360895i
$988$	0	0
$989$	1.13570e11	0.118708
$990$	0	0
$991$	1.35131e12i	1.40108i	0.713615	+	0.700538i	$0.247057\pi$
$991$	1.35131e12i	1.40108i	−0.713615	+	0.700538i	$0.752943\pi$
$992$	0	0
$993$	−9.94366e11	−1.02270
$994$	0	0
$995$	6.57069e11i	0.670376i
$996$	0	0
$997$	−1.66118e12	−1.68127	−0.840633	−	0.541604i	$-0.817817\pi$
$997$	−1.66118e12	−1.68127	−0.840633	+	0.541604i	$0.817817\pi$
$998$	0	0
$999$	1.08902e11i	0.109339i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.b.127.6	yes	32
4.3	odd	2	inner	384.9.g.b.127.5	yes	32
8.3	odd	2		384.9.g.a.127.28	yes	32
8.5	even	2		384.9.g.a.127.27	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.27	✓	32	8.5	even	2
384.9.g.a.127.28	yes	32	8.3	odd	2
384.9.g.b.127.5	yes	32	4.3	odd	2	inner
384.9.g.b.127.6	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)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} -407.399 q^{5} -3273.60i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} -407.399 q^{5} -3273.60i q^{7} -2187.00 q^{9} +10496.0i q^{11} -13445.5 q^{13} -19052.2i q^{15} +10188.3 q^{17} +46661.8i q^{19} +153091. q^{21} -347752. i q^{23} -224651. q^{25} -102276. i q^{27} -567090. q^{29} -870569. i q^{31} -490848. q^{33} +1.33366e6i q^{35} -1.06479e6 q^{37} -628782. i q^{39} -795034. q^{41} +326583. i q^{43} +890981. q^{45} -2.23717e6i q^{47} -4.95162e6 q^{49} +476460. i q^{51} +5.05380e6 q^{53} -4.27605e6i q^{55} -2.18216e6 q^{57} -1.57715e7i q^{59} +9.22670e6 q^{61} +7.15935e6i q^{63} +5.47767e6 q^{65} +2.63315e7i q^{67} +1.62628e7 q^{69} -3.08716e7i q^{71} +4.97145e7 q^{73} -1.05059e7i q^{75} +3.43596e7 q^{77} +6.23184e7i q^{79} +4.78297e6 q^{81} -1.35256e6i q^{83} -4.15070e6 q^{85} -2.65202e7i q^{87} +2.12314e7 q^{89} +4.40150e7i q^{91} +4.07125e7 q^{93} -1.90100e7i q^{95} +3.07649e7 q^{97} -2.29547e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 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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