LMFDB - Embedded newform 384.10.d.f.193.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,Mod(193,384)]

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([0, 1, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	384.d (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:	$197.773761087$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 300700 x^{18} + 6140664 x^{17} + 35387063979 x^{16} - 1130222504088 x^{15} + \cdots + 12\!\cdots\!52 $ x^20 - 8x^19 - 300700x^18 + 6140664x^17 + 35387063979x^16 - 1130222504088x^15 - 2027709461599244x^14 + 84850591004671480x^13 + 57031067836759165823x^12 - 2761881829653049004184x^11 - 715774112122919877685260x^10 + 33804686703728196561023832x^9 + 4440127592352179578492608037x^8 - 186062650255909170075184345736x^7 - 13450754766452318411315895909412x^6 + 457952106473696272472169073091112x^5 + 15298236539262291502286172763910964x^4 - 382483105264444074951018570546548352x^3 + 3006361776222337276289989101322653304x^2 - 10187939951443559461324435287012809264*x + 12842550816412602186535854197192532452
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{175}\cdot 3^{32} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.9
Root			$79.0321 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.10.d.f.193.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-81.0000i q^{3} +1825.80i q^{5} -10705.0 q^{7} -6561.00 q^{9} +O(q^{10})$	$q-81.0000i q^{3} +1825.80i q^{5} -10705.0 q^{7} -6561.00 q^{9} +19027.7i q^{11} -43409.9i q^{13} +147890. q^{15} +247009. q^{17} -210886. i q^{19} +867102. i q^{21} -757980. q^{23} -1.38043e6 q^{25} +531441. i q^{27} -3.97665e6i q^{29} +5.47687e6 q^{31} +1.54124e6 q^{33} -1.95451e7i q^{35} -2.85512e6i q^{37} -3.51620e6 q^{39} +2.61047e7 q^{41} -4.12446e7i q^{43} -1.19791e7i q^{45} -2.52327e7 q^{47} +7.42425e7 q^{49} -2.00077e7i q^{51} +8.17211e7i q^{53} -3.47408e7 q^{55} -1.70818e7 q^{57} +5.26397e7i q^{59} -7.03685e7i q^{61} +7.02352e7 q^{63} +7.92578e7 q^{65} -2.56485e7i q^{67} +6.13964e7i q^{69} -1.94484e8 q^{71} +4.12562e8 q^{73} +1.11814e8i q^{75} -2.03690e8i q^{77} -8.17079e7 q^{79} +4.30467e7 q^{81} +5.77186e8i q^{83} +4.50989e8i q^{85} -3.22109e8 q^{87} -4.48369e8 q^{89} +4.64701e8i q^{91} -4.43626e8i q^{93} +3.85036e8 q^{95} +4.52162e8 q^{97} -1.24841e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 131220 q^{9}+O(q^{10}) $ 20 * q - 131220 * q^9	$ 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) $ 20 * q - 131220 * q^9 - 905768 * q^17 - 9682620 * q^25 + 12054096 * q^33 + 74264008 * q^41 + 252775700 * q^49 - 5335632 * q^57 + 245588672 * q^65 - 895193896 * q^73 + 860934420 * q^81 + 882422136 * q^89 + 433683736 * 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
$3$	− 81.0000i	− 0.577350i
$3$	− 81.0000i	− 0.577350i
$4$	0	0
$5$	1825.80i	1.30644i	0.757169	+	0.653219i	$0.226582\pi$
$5$	1825.80i	1.30644i	−0.757169	+	0.653219i	$0.773418\pi$
$6$	0	0
$7$	−10705.0	−1.68517	−0.842585	−	0.538563i	$-0.818967\pi$
$7$	−10705.0	−1.68517	−0.842585	+	0.538563i	$0.818967\pi$
$8$	0	0
$9$	−6561.00	−0.333333
$10$	0	0
$11$	19027.7i	0.391849i	0.980619	+	0.195924i	$0.0627707\pi$
$11$	19027.7i	0.391849i	−0.980619	+	0.195924i	$0.937229\pi$
$12$	0	0
$13$	− 43409.9i	− 0.421544i	−0.977535	−	0.210772i	$-0.932402\pi$
$13$	− 43409.9i	− 0.421544i	0.977535	−	0.210772i	$-0.0675978\pi$
$14$	0	0
$15$	147890.	0.754272
$16$	0	0
$17$	247009.	0.717285	0.358643	−	0.933475i	$-0.383240\pi$
$17$	247009.	0.717285	0.358643	+	0.933475i	$0.383240\pi$
$18$	0	0
$19$	− 210886.i	− 0.371241i	−0.982621	−	0.185621i	$-0.940570\pi$
$19$	− 210886.i	− 0.371241i	0.982621	−	0.185621i	$-0.0594295\pi$
$20$	0	0
$21$	867102.i	0.972933i
$22$	0	0
$23$	−757980.	−0.564784	−0.282392	−	0.959299i	$-0.591128\pi$
$23$	−757980.	−0.564784	−0.282392	+	0.959299i	$0.591128\pi$
$24$	0	0
$25$	−1.38043e6	−0.706778
$26$	0	0
$27$	531441.i	0.192450i
$28$	0	0
$29$	− 3.97665e6i	− 1.04406i	−0.852926	−	0.522031i	$-0.825174\pi$
$29$	− 3.97665e6i	− 1.04406i	0.852926	−	0.522031i	$-0.174826\pi$
$30$	0	0
$31$	5.47687e6	1.06513	0.532567	−	0.846388i	$-0.321227\pi$
$31$	5.47687e6	1.06513	0.532567	+	0.846388i	$0.321227\pi$
$32$	0	0
$33$	1.54124e6	0.226234
$34$	0	0
$35$	− 1.95451e7i	− 2.20157i
$36$	0	0
$37$	− 2.85512e6i	− 0.250447i	−0.992129	−	0.125224i	$-0.960035\pi$
$37$	− 2.85512e6i	− 0.250447i	0.992129	−	0.125224i	$-0.0399648\pi$
$38$	0	0
$39$	−3.51620e6	−0.243379
$40$	0	0
$41$	2.61047e7	1.44275	0.721376	−	0.692543i	$-0.243510\pi$
$41$	2.61047e7	1.44275	0.721376	+	0.692543i	$0.243510\pi$
$42$	0	0
$43$	− 4.12446e7i	− 1.83975i	−0.392212	−	0.919875i	$-0.628290\pi$
$43$	− 4.12446e7i	− 1.83975i	0.392212	−	0.919875i	$-0.371710\pi$
$44$	0	0
$45$	− 1.19791e7i	− 0.435479i
$46$	0	0
$47$	−2.52327e7	−0.754265	−0.377132	−	0.926159i	$-0.623090\pi$
$47$	−2.52327e7	−0.754265	−0.377132	+	0.926159i	$0.623090\pi$
$48$	0	0
$49$	7.42425e7	1.83980
$50$	0	0
$51$	− 2.00077e7i	− 0.414125i
$52$	0	0
$53$	8.17211e7i	1.42263i	0.702872	+	0.711316i	$0.251901\pi$
$53$	8.17211e7i	1.42263i	−0.702872	+	0.711316i	$0.748099\pi$
$54$	0	0
$55$	−3.47408e7	−0.511926
$56$	0	0
$57$	−1.70818e7	−0.214336
$58$	0	0
$59$	5.26397e7i	0.565560i	0.959185	+	0.282780i	$0.0912567\pi$
$59$	5.26397e7i	0.565560i	−0.959185	+	0.282780i	$0.908743\pi$
$60$	0	0
$61$	− 7.03685e7i	− 0.650720i	−0.945590	−	0.325360i	$-0.894515\pi$
$61$	− 7.03685e7i	− 0.650720i	0.945590	−	0.325360i	$-0.105485\pi$
$62$	0	0
$63$	7.02352e7	0.561723
$64$	0	0
$65$	7.92578e7	0.550721
$66$	0	0
$67$	− 2.56485e7i	− 0.155498i	−0.996973	−	0.0777492i	$-0.975227\pi$
$67$	− 2.56485e7i	− 0.155498i	0.996973	−	0.0777492i	$-0.0247733\pi$
$68$	0	0
$69$	6.13964e7i	0.326078i
$70$	0	0
$71$	−1.94484e8	−0.908283	−0.454142	−	0.890930i	$-0.650054\pi$
$71$	−1.94484e8	−0.908283	−0.454142	+	0.890930i	$0.650054\pi$
$72$	0	0
$73$	4.12562e8	1.70034	0.850170	−	0.526508i	$-0.176499\pi$
$73$	4.12562e8	1.70034	0.850170	+	0.526508i	$0.176499\pi$
$74$	0	0
$75$	1.11814e8i	0.408058i
$76$	0	0
$77$	− 2.03690e8i	− 0.660332i
$78$	0	0
$79$	−8.17079e7	−0.236016	−0.118008	−	0.993013i	$-0.537651\pi$
$79$	−8.17079e7	−0.236016	−0.118008	+	0.993013i	$0.537651\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	5.77186e8i	1.33495i	0.744633	+	0.667474i	$0.232625\pi$
$83$	5.77186e8i	1.33495i	−0.744633	+	0.667474i	$0.767375\pi$
$84$	0	0
$85$	4.50989e8i	0.937088i
$86$	0	0
$87$	−3.22109e8	−0.602790
$88$	0	0
$89$	−4.48369e8	−0.757496	−0.378748	−	0.925500i	$-0.623645\pi$
$89$	−4.48369e8	−0.757496	−0.378748	+	0.925500i	$0.623645\pi$
$90$	0	0
$91$	4.64701e8i	0.710374i
$92$	0	0
$93$	− 4.43626e8i	− 0.614956i
$94$	0	0
$95$	3.85036e8	0.485003
$96$	0	0
$97$	4.52162e8	0.518587	0.259293	−	0.965799i	$-0.416510\pi$
$97$	4.52162e8	0.518587	0.259293	+	0.965799i	$0.416510\pi$
$98$	0	0
$99$	− 1.24841e8i	− 0.130616i
$100$	0	0
$101$	9.23117e8i	0.882695i	0.897336	+	0.441347i	$0.145499\pi$
$101$	9.23117e8i	0.882695i	−0.897336	+	0.441347i	$0.854501\pi$
$102$	0	0
$103$	−1.56936e9	−1.37390	−0.686949	−	0.726705i	$-0.741051\pi$
$103$	−1.56936e9	−1.37390	−0.686949	+	0.726705i	$0.741051\pi$
$104$	0	0
$105$	−1.58316e9	−1.27108
$106$	0	0
$107$	4.21445e8i	0.310824i	0.987850	+	0.155412i	$0.0496704\pi$
$107$	4.21445e8i	0.310824i	−0.987850	+	0.155412i	$0.950330\pi$
$108$	0	0
$109$	− 1.32695e9i	− 0.900398i	−0.892928	−	0.450199i	$-0.851353\pi$
$109$	− 1.32695e9i	− 0.900398i	0.892928	−	0.450199i	$-0.148647\pi$
$110$	0	0
$111$	−2.31264e8	−0.144596
$112$	0	0
$113$	−2.20516e9	−1.27229	−0.636146	−	0.771568i	$-0.719473\pi$
$113$	−2.20516e9	−1.27229	−0.636146	+	0.771568i	$0.719473\pi$
$114$	0	0
$115$	− 1.38392e9i	− 0.737855i
$116$	0	0
$117$	2.84812e8i	0.140515i
$118$	0	0
$119$	−2.64422e9	−1.20875
$120$	0	0
$121$	1.99590e9	0.846454
$122$	0	0
$123$	− 2.11448e9i	− 0.832974i
$124$	0	0
$125$	1.04564e9i	0.383076i
$126$	0	0
$127$	2.38346e9	0.813000	0.406500	−	0.913651i	$-0.366749\pi$
$127$	2.38346e9	0.813000	0.406500	+	0.913651i	$0.366749\pi$
$128$	0	0
$129$	−3.34081e9	−1.06218
$130$	0	0
$131$	4.81369e8i	0.142810i	0.997447	+	0.0714048i	$0.0227482\pi$
$131$	4.81369e8i	0.142810i	−0.997447	+	0.0714048i	$0.977252\pi$
$132$	0	0
$133$	2.25752e9i	0.625605i
$134$	0	0
$135$	−9.70306e8	−0.251424
$136$	0	0
$137$	−7.44085e9	−1.80460	−0.902298	−	0.431112i	$-0.858121\pi$
$137$	−7.44085e9	−1.80460	−0.902298	+	0.431112i	$0.858121\pi$
$138$	0	0
$139$	− 7.71197e9i	− 1.75226i	−0.482074	−	0.876131i	$-0.660116\pi$
$139$	− 7.71197e9i	− 1.75226i	0.482074	−	0.876131i	$-0.339884\pi$
$140$	0	0
$141$	2.04385e9i	0.435475i
$142$	0	0
$143$	8.25989e8	0.165182
$144$	0	0
$145$	7.26058e9	1.36400
$146$	0	0
$147$	− 6.01364e9i	− 1.06221i
$148$	0	0
$149$	− 4.89514e9i	− 0.813629i	−0.913511	−	0.406815i	$-0.866639\pi$
$149$	− 4.89514e9i	− 0.813629i	0.913511	−	0.406815i	$-0.133361\pi$
$150$	0	0
$151$	−7.18370e9	−1.12448	−0.562240	−	0.826974i	$-0.690060\pi$
$151$	−7.18370e9	−1.12448	−0.562240	+	0.826974i	$0.690060\pi$
$152$	0	0
$153$	−1.62062e9	−0.239095
$154$	0	0
$155$	9.99967e9i	1.39153i
$156$	0	0
$157$	1.16362e10i	1.52849i	0.644924	+	0.764246i	$0.276889\pi$
$157$	1.16362e10i	1.52849i	−0.644924	+	0.764246i	$0.723111\pi$
$158$	0	0
$159$	6.61941e9	0.821358
$160$	0	0
$161$	8.11414e9	0.951757
$162$	0	0
$163$	7.16605e9i	0.795126i	0.917575	+	0.397563i	$0.130144\pi$
$163$	7.16605e9i	0.795126i	−0.917575	+	0.397563i	$0.869856\pi$
$164$	0	0
$165$	2.81400e9i	0.295561i
$166$	0	0
$167$	9.91993e9	0.986926	0.493463	−	0.869767i	$-0.335731\pi$
$167$	9.91993e9	0.986926	0.493463	+	0.869767i	$0.335731\pi$
$168$	0	0
$169$	8.72008e9	0.822300
$170$	0	0
$171$	1.38362e9i	0.123747i
$172$	0	0
$173$	1.66331e10i	1.41178i	0.708322	+	0.705889i	$0.249452\pi$
$173$	1.66331e10i	1.41178i	−0.708322	+	0.705889i	$0.750548\pi$
$174$	0	0
$175$	1.47774e10	1.19104
$176$	0	0
$177$	4.26381e9	0.326526
$178$	0	0
$179$	2.19114e10i	1.59526i	0.603146	+	0.797631i	$0.293914\pi$
$179$	2.19114e10i	1.59526i	−0.603146	+	0.797631i	$0.706086\pi$
$180$	0	0
$181$	− 1.75135e10i	− 1.21289i	−0.795127	−	0.606443i	$-0.792596\pi$
$181$	− 1.75135e10i	− 1.21289i	0.795127	−	0.606443i	$-0.207404\pi$
$182$	0	0
$183$	−5.69985e9	−0.375693
$184$	0	0
$185$	5.21288e9	0.327193
$186$	0	0
$187$	4.70000e9i	0.281067i
$188$	0	0
$189$	− 5.68905e9i	− 0.324311i
$190$	0	0
$191$	−2.97400e10	−1.61693	−0.808465	−	0.588545i	$-0.799701\pi$
$191$	−2.97400e10	−1.61693	−0.808465	+	0.588545i	$0.799701\pi$
$192$	0	0
$193$	2.47186e10	1.28238	0.641189	−	0.767383i	$-0.278442\pi$
$193$	2.47186e10	1.28238	0.641189	+	0.767383i	$0.278442\pi$
$194$	0	0
$195$	− 6.41988e9i	− 0.317959i
$196$	0	0
$197$	1.53688e10i	0.727012i	0.931592	+	0.363506i	$0.118420\pi$
$197$	1.53688e10i	0.727012i	−0.931592	+	0.363506i	$0.881580\pi$
$198$	0	0
$199$	1.51167e10	0.683312	0.341656	−	0.939825i	$-0.389012\pi$
$199$	1.51167e10	0.683312	0.341656	+	0.939825i	$0.389012\pi$
$200$	0	0
$201$	−2.07753e9	−0.0897771
$202$	0	0
$203$	4.25699e10i	1.75942i
$204$	0	0
$205$	4.76621e10i	1.88487i
$206$	0	0
$207$	4.97311e9	0.188261
$208$	0	0
$209$	4.01267e9	0.145470
$210$	0	0
$211$	− 5.30440e10i	− 1.84232i	−0.389181	−	0.921161i	$-0.627242\pi$
$211$	− 5.30440e10i	− 1.84232i	0.389181	−	0.921161i	$-0.372758\pi$
$212$	0	0
$213$	1.57532e10i	0.524398i
$214$	0	0
$215$	7.53044e10	2.40352
$216$	0	0
$217$	−5.86296e10	−1.79493
$218$	0	0
$219$	− 3.34175e10i	− 0.981692i
$220$	0	0
$221$	− 1.07226e10i	− 0.302368i
$222$	0	0
$223$	−2.06381e10	−0.558852	−0.279426	−	0.960167i	$-0.590144\pi$
$223$	−2.06381e10	−0.558852	−0.279426	+	0.960167i	$0.590144\pi$
$224$	0	0
$225$	9.05697e9	0.235593
$226$	0	0
$227$	− 2.82713e9i	− 0.0706690i	−0.999376	−	0.0353345i	$-0.988750\pi$
$227$	− 2.82713e9i	− 0.0706690i	0.999376	−	0.0353345i	$-0.0112497\pi$
$228$	0	0
$229$	7.54651e10i	1.81337i	0.421809	+	0.906685i	$0.361395\pi$
$229$	7.54651e10i	1.81337i	−0.421809	+	0.906685i	$0.638605\pi$
$230$	0	0
$231$	−1.64989e10	−0.381243
$232$	0	0
$233$	2.32352e10	0.516469	0.258234	−	0.966082i	$-0.416859\pi$
$233$	2.32352e10	0.516469	0.258234	+	0.966082i	$0.416859\pi$
$234$	0	0
$235$	− 4.60699e10i	− 0.985399i
$236$	0	0
$237$	6.61834e9i	0.136264i
$238$	0	0
$239$	−1.76079e10	−0.349073	−0.174536	−	0.984651i	$-0.555843\pi$
$239$	−1.76079e10	−0.349073	−0.174536	+	0.984651i	$0.555843\pi$
$240$	0	0
$241$	−8.15600e10	−1.55740	−0.778701	−	0.627396i	$-0.784121\pi$
$241$	−8.15600e10	−1.55740	−0.778701	+	0.627396i	$0.784121\pi$
$242$	0	0
$243$	− 3.48678e9i	− 0.0641500i
$244$	0	0
$245$	1.35552e11i	2.40358i
$246$	0	0
$247$	−9.15452e9	−0.156495
$248$	0	0
$249$	4.67521e10	0.770733
$250$	0	0
$251$	− 3.80656e10i	− 0.605341i	−0.953095	−	0.302671i	$-0.902122\pi$
$251$	− 3.80656e10i	− 0.605341i	0.953095	−	0.302671i	$-0.0978782\pi$
$252$	0	0
$253$	− 1.44226e10i	− 0.221310i
$254$	0	0
$255$	3.65301e10	0.541028
$256$	0	0
$257$	1.29860e10	0.185685	0.0928427	−	0.995681i	$-0.470405\pi$
$257$	1.29860e10	0.185685	0.0928427	+	0.995681i	$0.470405\pi$
$258$	0	0
$259$	3.05639e10i	0.422046i
$260$	0	0
$261$	2.60908e10i	0.348021i
$262$	0	0
$263$	3.14965e10	0.405939	0.202970	−	0.979185i	$-0.434941\pi$
$263$	3.14965e10	0.405939	0.202970	+	0.979185i	$0.434941\pi$
$264$	0	0
$265$	−1.49207e11	−1.85858
$266$	0	0
$267$	3.63179e10i	0.437341i
$268$	0	0
$269$	9.41097e10i	1.09585i	0.836529	+	0.547923i	$0.184581\pi$
$269$	9.41097e10i	1.09585i	−0.836529	+	0.547923i	$0.815419\pi$
$270$	0	0
$271$	1.26849e9	0.0142864	0.00714322	−	0.999974i	$-0.497726\pi$
$271$	1.26849e9	0.0142864	0.00714322	+	0.999974i	$0.497726\pi$
$272$	0	0
$273$	3.76408e10	0.410135
$274$	0	0
$275$	− 2.62663e10i	− 0.276950i
$276$	0	0
$277$	− 1.39472e10i	− 0.142340i	−0.997464	−	0.0711702i	$-0.977327\pi$
$277$	− 1.39472e10i	− 0.142340i	0.997464	−	0.0711702i	$-0.0226734\pi$
$278$	0	0
$279$	−3.59337e10	−0.355045
$280$	0	0
$281$	−1.12754e11	−1.07883	−0.539416	−	0.842040i	$-0.681355\pi$
$281$	−1.12754e11	−1.07883	−0.539416	+	0.842040i	$0.681355\pi$
$282$	0	0
$283$	1.00750e11i	0.933699i	0.884337	+	0.466850i	$0.154611\pi$
$283$	1.00750e11i	0.933699i	−0.884337	+	0.466850i	$0.845389\pi$
$284$	0	0
$285$	− 3.11879e10i	− 0.280017i
$286$	0	0
$287$	−2.79450e11	−2.43128
$288$	0	0
$289$	−5.75746e10	−0.485502
$290$	0	0
$291$	− 3.66251e10i	− 0.299406i
$292$	0	0
$293$	2.17043e11i	1.72044i	0.509919	+	0.860222i	$0.329675\pi$
$293$	2.17043e11i	1.72044i	−0.509919	+	0.860222i	$0.670325\pi$
$294$	0	0
$295$	−9.61096e10	−0.738869
$296$	0	0
$297$	−1.01121e10	−0.0754114
$298$	0	0
$299$	3.29038e10i	0.238081i
$300$	0	0
$301$	4.41521e11i	3.10029i
$302$	0	0
$303$	7.47725e10	0.509624
$304$	0	0
$305$	1.28479e11	0.850124
$306$	0	0
$307$	7.61715e10i	0.489407i	0.969598	+	0.244703i	$0.0786906\pi$
$307$	7.61715e10i	0.489407i	−0.969598	+	0.244703i	$0.921309\pi$
$308$	0	0
$309$	1.27118e11i	0.793221i
$310$	0	0
$311$	1.54019e11	0.933580	0.466790	−	0.884368i	$-0.345410\pi$
$311$	1.54019e11	0.933580	0.466790	+	0.884368i	$0.345410\pi$
$312$	0	0
$313$	2.99387e11	1.76313	0.881563	−	0.472066i	$-0.156492\pi$
$313$	2.99387e11	1.76313	0.881563	+	0.472066i	$0.156492\pi$
$314$	0	0
$315$	1.28236e11i	0.733856i
$316$	0	0
$317$	− 1.33299e11i	− 0.741411i	−0.928750	−	0.370706i	$-0.879116\pi$
$317$	− 1.33299e11i	− 0.741411i	0.928750	−	0.370706i	$-0.120884\pi$
$318$	0	0
$319$	7.56664e10	0.409115
$320$	0	0
$321$	3.41370e10	0.179454
$322$	0	0
$323$	− 5.20906e10i	− 0.266286i
$324$	0	0
$325$	5.99241e10i	0.297938i
$326$	0	0
$327$	−1.07483e11	−0.519845
$328$	0	0
$329$	2.70115e11	1.27106
$330$	0	0
$331$	3.24454e11i	1.48568i	0.669466	+	0.742842i	$0.266523\pi$
$331$	3.24454e11i	1.48568i	−0.669466	+	0.742842i	$0.733477\pi$
$332$	0	0
$333$	1.87324e10i	0.0834824i
$334$	0	0
$335$	4.68291e10	0.203149
$336$	0	0
$337$	9.02818e10	0.381299	0.190650	−	0.981658i	$-0.438941\pi$
$337$	9.02818e10	0.381299	0.190650	+	0.981658i	$0.438941\pi$
$338$	0	0
$339$	1.78618e11i	0.734559i
$340$	0	0
$341$	1.04212e11i	0.417372i
$342$	0	0
$343$	−3.62779e11	−1.41520
$344$	0	0
$345$	−1.12098e11	−0.426001
$346$	0	0
$347$	− 2.21904e11i	− 0.821643i	−0.911716	−	0.410822i	$-0.865242\pi$
$347$	− 2.21904e11i	− 0.821643i	0.911716	−	0.410822i	$-0.134758\pi$
$348$	0	0
$349$	4.45184e11i	1.60629i	0.595781	+	0.803147i	$0.296843\pi$
$349$	4.45184e11i	1.60629i	−0.595781	+	0.803147i	$0.703157\pi$
$350$	0	0
$351$	2.30698e10	0.0811262
$352$	0	0
$353$	−9.63896e10	−0.330403	−0.165202	−	0.986260i	$-0.552827\pi$
$353$	−9.63896e10	−0.330403	−0.165202	+	0.986260i	$0.552827\pi$
$354$	0	0
$355$	− 3.55089e11i	− 1.18661i
$356$	0	0
$357$	2.14182e11i	0.697871i
$358$	0	0
$359$	−1.73189e11	−0.550295	−0.275148	−	0.961402i	$-0.588727\pi$
$359$	−1.73189e11	−0.550295	−0.275148	+	0.961402i	$0.588727\pi$
$360$	0	0
$361$	2.78215e11	0.862180
$362$	0	0
$363$	− 1.61668e11i	− 0.488701i
$364$	0	0
$365$	7.53256e11i	2.22139i
$366$	0	0
$367$	4.07440e11	1.17237	0.586187	−	0.810175i	$-0.300628\pi$
$367$	4.07440e11	1.17237	0.586187	+	0.810175i	$0.300628\pi$
$368$	0	0
$369$	−1.71273e11	−0.480918
$370$	0	0
$371$	− 8.74821e11i	− 2.39738i
$372$	0	0
$373$	− 3.98143e11i	− 1.06500i	−0.846430	−	0.532500i	$-0.821253\pi$
$373$	− 3.98143e11i	− 1.06500i	0.846430	−	0.532500i	$-0.178747\pi$
$374$	0	0
$375$	8.46964e10	0.221169
$376$	0	0
$377$	−1.72626e11	−0.440119
$378$	0	0
$379$	− 2.64108e10i	− 0.0657514i	−0.999459	−	0.0328757i	$-0.989533\pi$
$379$	− 2.64108e10i	− 0.0657514i	0.999459	−	0.0328757i	$-0.0104665\pi$
$380$	0	0
$381$	− 1.93060e11i	− 0.469386i
$382$	0	0
$383$	5.04020e11	1.19689	0.598444	−	0.801165i	$-0.295786\pi$
$383$	5.04020e11	1.19689	0.598444	+	0.801165i	$0.295786\pi$
$384$	0	0
$385$	3.71898e11	0.862682
$386$	0	0
$387$	2.70606e11i	0.613250i
$388$	0	0
$389$	2.56939e11i	0.568926i	0.958687	+	0.284463i	$0.0918154\pi$
$389$	2.56939e11i	0.568926i	−0.958687	+	0.284463i	$0.908185\pi$
$390$	0	0
$391$	−1.87228e11	−0.405111
$392$	0	0
$393$	3.89909e10	0.0824512
$394$	0	0
$395$	− 1.49182e11i	− 0.308340i
$396$	0	0
$397$	9.51018e11i	1.92146i	0.277484	+	0.960730i	$0.410500\pi$
$397$	9.51018e11i	1.92146i	−0.277484	+	0.960730i	$0.589500\pi$
$398$	0	0
$399$	1.82859e11	0.361193
$400$	0	0
$401$	−8.44303e11	−1.63060	−0.815302	−	0.579036i	$-0.803429\pi$
$401$	−8.44303e11	−1.63060	−0.815302	+	0.579036i	$0.803429\pi$
$402$	0	0
$403$	− 2.37750e11i	− 0.449002i
$404$	0	0
$405$	7.85948e10i	0.145160i
$406$	0	0
$407$	5.43262e10	0.0981375
$408$	0	0
$409$	1.77238e11	0.313186	0.156593	−	0.987663i	$-0.449949\pi$
$409$	1.77238e11	0.313186	0.156593	+	0.987663i	$0.449949\pi$
$410$	0	0
$411$	6.02709e11i	1.04188i
$412$	0	0
$413$	− 5.63505e11i	− 0.953066i
$414$	0	0
$415$	−1.05383e12	−1.74403
$416$	0	0
$417$	−6.24670e11	−1.01167
$418$	0	0
$419$	− 3.88413e11i	− 0.615645i	−0.951444	−	0.307823i	$-0.900400\pi$
$419$	− 3.88413e11i	− 0.615645i	0.951444	−	0.307823i	$-0.0996004\pi$
$420$	0	0
$421$	− 1.98229e11i	− 0.307537i	−0.988107	−	0.153768i	$-0.950859\pi$
$421$	− 1.98229e11i	− 0.307537i	0.988107	−	0.153768i	$-0.0491410\pi$
$422$	0	0
$423$	1.65552e11	0.251422
$424$	0	0
$425$	−3.40977e11	−0.506961
$426$	0	0
$427$	7.53292e11i	1.09657i
$428$	0	0
$429$	− 6.69051e10i	− 0.0953677i
$430$	0	0
$431$	1.02358e12	1.42880	0.714402	−	0.699735i	$-0.246699\pi$
$431$	1.02358e12	1.42880	0.714402	+	0.699735i	$0.246699\pi$
$432$	0	0
$433$	−1.90125e11	−0.259923	−0.129961	−	0.991519i	$-0.541485\pi$
$433$	−1.90125e11	−0.259923	−0.129961	+	0.991519i	$0.541485\pi$
$434$	0	0
$435$	− 5.88107e11i	− 0.787507i
$436$	0	0
$437$	1.59847e11i	0.209671i
$438$	0	0
$439$	1.16798e12	1.50087	0.750436	−	0.660943i	$-0.229843\pi$
$439$	1.16798e12	1.50087	0.750436	+	0.660943i	$0.229843\pi$
$440$	0	0
$441$	−4.87105e11	−0.613266
$442$	0	0
$443$	1.48143e12i	1.82753i	0.406242	+	0.913766i	$0.366839\pi$
$443$	1.48143e12i	1.82753i	−0.406242	+	0.913766i	$0.633161\pi$
$444$	0	0
$445$	− 8.18632e11i	− 0.989621i
$446$	0	0
$447$	−3.96506e11	−0.469749
$448$	0	0
$449$	−7.14080e11	−0.829160	−0.414580	−	0.910013i	$-0.636072\pi$
$449$	−7.14080e11	−0.829160	−0.414580	+	0.910013i	$0.636072\pi$
$450$	0	0
$451$	4.96712e11i	0.565341i
$452$	0	0
$453$	5.81880e11i	0.649219i
$454$	0	0
$455$	−8.48451e11	−0.928059
$456$	0	0
$457$	3.68588e10	0.0395292	0.0197646	−	0.999805i	$-0.493708\pi$
$457$	3.68588e10	0.0395292	0.0197646	+	0.999805i	$0.493708\pi$
$458$	0	0
$459$	1.31270e11i	0.138042i
$460$	0	0
$461$	− 2.85253e11i	− 0.294155i	−0.989125	−	0.147078i	$-0.953013\pi$
$461$	− 2.85253e11i	− 0.294155i	0.989125	−	0.147078i	$-0.0469867\pi$
$462$	0	0
$463$	7.74164e11	0.782922	0.391461	−	0.920195i	$-0.371970\pi$
$463$	7.74164e11	0.782922	0.391461	+	0.920195i	$0.371970\pi$
$464$	0	0
$465$	8.09973e11	0.803401
$466$	0	0
$467$	9.96899e11i	0.969896i	0.874543	+	0.484948i	$0.161161\pi$
$467$	9.96899e11i	0.969896i	−0.874543	+	0.484948i	$0.838839\pi$
$468$	0	0
$469$	2.74567e11i	0.262041i
$470$	0	0
$471$	9.42534e11	0.882476
$472$	0	0
$473$	7.84789e11	0.720904
$474$	0	0
$475$	2.91112e11i	0.262385i
$476$	0	0
$477$	− 5.36172e11i	− 0.474211i
$478$	0	0
$479$	−7.25791e11	−0.629944	−0.314972	−	0.949101i	$-0.601995\pi$
$479$	−7.25791e11	−0.629944	−0.314972	+	0.949101i	$0.601995\pi$
$480$	0	0
$481$	−1.23940e11	−0.105575
$482$	0	0
$483$	− 6.57246e11i	− 0.549497i
$484$	0	0
$485$	8.25558e11i	0.677501i
$486$	0	0
$487$	−2.26236e11	−0.182256	−0.0911278	−	0.995839i	$-0.529047\pi$
$487$	−2.26236e11	−0.182256	−0.0911278	+	0.995839i	$0.529047\pi$
$488$	0	0
$489$	5.80450e11	0.459066
$490$	0	0
$491$	1.14799e12i	0.891397i	0.895183	+	0.445699i	$0.147045\pi$
$491$	1.14799e12i	0.891397i	−0.895183	+	0.445699i	$0.852955\pi$
$492$	0	0
$493$	− 9.82267e11i	− 0.748891i
$494$	0	0
$495$	2.27934e11	0.170642
$496$	0	0
$497$	2.08194e12	1.53061
$498$	0	0
$499$	− 1.53957e12i	− 1.11160i	−0.831316	−	0.555799i	$-0.812412\pi$
$499$	− 1.53957e12i	− 1.11160i	0.831316	−	0.555799i	$-0.187588\pi$
$500$	0	0
$501$	− 8.03515e11i	− 0.569802i
$502$	0	0
$503$	1.09194e12	0.760579	0.380289	−	0.924868i	$-0.375824\pi$
$503$	1.09194e12	0.760579	0.380289	+	0.924868i	$0.375824\pi$
$504$	0	0
$505$	−1.68543e12	−1.15319
$506$	0	0
$507$	− 7.06327e11i	− 0.474755i
$508$	0	0
$509$	− 1.07561e12i	− 0.710271i	−0.934815	−	0.355136i	$-0.884435\pi$
$509$	− 1.07561e12i	− 0.710271i	0.934815	−	0.355136i	$-0.115565\pi$
$510$	0	0
$511$	−4.41645e12	−2.86536
$512$	0	0
$513$	1.12073e11	0.0714454
$514$	0	0
$515$	− 2.86534e12i	− 1.79491i
$516$	0	0
$517$	− 4.80120e11i	− 0.295558i
$518$	0	0
$519$	1.34728e12	0.815091
$520$	0	0
$521$	9.75278e11	0.579907	0.289954	−	0.957041i	$-0.406360\pi$
$521$	9.75278e11	0.579907	0.289954	+	0.957041i	$0.406360\pi$
$522$	0	0
$523$	− 1.44784e11i	− 0.0846179i	−0.999105	−	0.0423090i	$-0.986529\pi$
$523$	− 1.44784e11i	− 0.0846179i	0.999105	−	0.0423090i	$-0.0134714\pi$
$524$	0	0
$525$	− 1.19697e12i	− 0.687648i
$526$	0	0
$527$	1.35283e12	0.764006
$528$	0	0
$529$	−1.22662e12	−0.681019
$530$	0	0
$531$	− 3.45369e11i	− 0.188520i
$532$	0	0
$533$	− 1.13320e12i	− 0.608184i
$534$	0	0
$535$	−7.69475e11	−0.406071
$536$	0	0
$537$	1.77483e12	0.921025
$538$	0	0
$539$	1.41266e12i	0.720923i
$540$	0	0
$541$	− 3.70059e11i	− 0.185730i	−0.995679	−	0.0928651i	$-0.970397\pi$
$541$	− 3.70059e11i	− 0.185730i	0.995679	−	0.0928651i	$-0.0296025\pi$
$542$	0	0
$543$	−1.41860e12	−0.700260
$544$	0	0
$545$	2.42274e12	1.17631
$546$	0	0
$547$	− 3.75690e12i	− 1.79427i	−0.441761	−	0.897133i	$-0.645646\pi$
$547$	− 3.75690e12i	− 1.79427i	0.441761	−	0.897133i	$-0.354354\pi$
$548$	0	0
$549$	4.61688e11i	0.216907i
$550$	0	0
$551$	−8.38619e11	−0.387599
$552$	0	0
$553$	8.74679e11	0.397728
$554$	0	0
$555$	− 4.22243e11i	− 0.188905i
$556$	0	0
$557$	− 4.22052e12i	− 1.85788i	−0.370230	−	0.928940i	$-0.620721\pi$
$557$	− 4.22052e12i	− 1.85788i	0.370230	−	0.928940i	$-0.379279\pi$
$558$	0	0
$559$	−1.79042e12	−0.775536
$560$	0	0
$561$	3.80700e11	0.162274
$562$	0	0
$563$	1.18107e12i	0.495438i	0.968832	+	0.247719i	$0.0796809\pi$
$563$	1.18107e12i	0.495438i	−0.968832	+	0.247719i	$0.920319\pi$
$564$	0	0
$565$	− 4.02618e12i	− 1.66217i
$566$	0	0
$567$	−4.60813e11	−0.187241
$568$	0	0
$569$	−1.25136e12	−0.500469	−0.250235	−	0.968185i	$-0.580508\pi$
$569$	−1.25136e12	−0.500469	−0.250235	+	0.968185i	$0.580508\pi$
$570$	0	0
$571$	2.38907e12i	0.940518i	0.882528	+	0.470259i	$0.155840\pi$
$571$	2.38907e12i	0.940518i	−0.882528	+	0.470259i	$0.844160\pi$
$572$	0	0
$573$	2.40894e12i	0.933535i
$574$	0	0
$575$	1.04633e12	0.399177
$576$	0	0
$577$	1.19996e12	0.450687	0.225344	−	0.974279i	$-0.427650\pi$
$577$	1.19996e12	0.450687	0.225344	+	0.974279i	$0.427650\pi$
$578$	0	0
$579$	− 2.00221e12i	− 0.740381i
$580$	0	0
$581$	− 6.17876e12i	− 2.24962i
$582$	0	0
$583$	−1.55496e12	−0.557457
$584$	0	0
$585$	−5.20010e11	−0.183574
$586$	0	0
$587$	5.19650e12i	1.80650i	0.429110	+	0.903252i	$0.358827\pi$
$587$	5.19650e12i	1.80650i	−0.429110	+	0.903252i	$0.641173\pi$
$588$	0	0
$589$	− 1.15499e12i	− 0.395422i
$590$	0	0
$591$	1.24487e12	0.419740
$592$	0	0
$593$	−1.83959e12	−0.610908	−0.305454	−	0.952207i	$-0.598808\pi$
$593$	−1.83959e12	−0.610908	−0.305454	+	0.952207i	$0.598808\pi$
$594$	0	0
$595$	− 4.82781e12i	− 1.57915i
$596$	0	0
$597$	− 1.22445e12i	− 0.394510i
$598$	0	0
$599$	−8.83924e11	−0.280540	−0.140270	−	0.990113i	$-0.544797\pi$
$599$	−8.83924e11	−0.280540	−0.140270	+	0.990113i	$0.544797\pi$
$600$	0	0
$601$	−4.16783e12	−1.30309	−0.651546	−	0.758609i	$-0.725879\pi$
$601$	−4.16783e12	−1.30309	−0.651546	+	0.758609i	$0.725879\pi$
$602$	0	0
$603$	1.68280e11i	0.0518328i
$604$	0	0
$605$	3.64411e12i	1.10584i
$606$	0	0
$607$	−7.35927e11	−0.220032	−0.110016	−	0.993930i	$-0.535090\pi$
$607$	−7.35927e11	−0.220032	−0.110016	+	0.993930i	$0.535090\pi$
$608$	0	0
$609$	3.44816e12	1.01580
$610$	0	0
$611$	1.09535e12i	0.317956i
$612$	0	0
$613$	− 3.66116e12i	− 1.04724i	−0.851951	−	0.523621i	$-0.824581\pi$
$613$	− 3.66116e12i	− 1.04724i	0.851951	−	0.523621i	$-0.175419\pi$
$614$	0	0
$615$	3.86063e12	1.08823
$616$	0	0
$617$	1.07297e12	0.298061	0.149030	−	0.988833i	$-0.452385\pi$
$617$	1.07297e12	0.298061	0.149030	+	0.988833i	$0.452385\pi$
$618$	0	0
$619$	2.98195e12i	0.816380i	0.912897	+	0.408190i	$0.133840\pi$
$619$	2.98195e12i	0.816380i	−0.912897	+	0.408190i	$0.866160\pi$
$620$	0	0
$621$	− 4.02822e11i	− 0.108693i
$622$	0	0
$623$	4.79977e12	1.27651
$624$	0	0
$625$	−4.60527e12	−1.20724
$626$	0	0
$627$	− 3.25026e11i	− 0.0839874i
$628$	0	0
$629$	− 7.05238e11i	− 0.179642i
$630$	0	0
$631$	−4.58560e12	−1.15150	−0.575750	−	0.817626i	$-0.695290\pi$
$631$	−4.58560e12	−1.15150	−0.575750	+	0.817626i	$0.695290\pi$
$632$	0	0
$633$	−4.29657e12	−1.06367
$634$	0	0
$635$	4.35172e12i	1.06213i
$636$	0	0
$637$	− 3.22286e12i	− 0.775557i
$638$	0	0
$639$	1.27601e12	0.302761
$640$	0	0
$641$	−3.82224e12	−0.894245	−0.447123	−	0.894473i	$-0.647551\pi$
$641$	−3.82224e12	−0.894245	−0.447123	+	0.894473i	$0.647551\pi$
$642$	0	0
$643$	2.98381e12i	0.688371i	0.938902	+	0.344185i	$0.111845\pi$
$643$	2.98381e12i	0.688371i	−0.938902	+	0.344185i	$0.888155\pi$
$644$	0	0
$645$	− 6.09966e12i	− 1.38767i
$646$	0	0
$647$	−3.86301e12	−0.866674	−0.433337	−	0.901232i	$-0.642664\pi$
$647$	−3.86301e12	−0.866674	−0.433337	+	0.901232i	$0.642664\pi$
$648$	0	0
$649$	−1.00161e12	−0.221614
$650$	0	0
$651$	4.74900e12i	1.03631i
$652$	0	0
$653$	− 1.43653e12i	− 0.309176i	−0.987979	−	0.154588i	$-0.950595\pi$
$653$	− 1.43653e12i	− 0.309176i	0.987979	−	0.154588i	$-0.0494049\pi$
$654$	0	0
$655$	−8.78885e11	−0.186572
$656$	0	0
$657$	−2.70682e12	−0.566780
$658$	0	0
$659$	− 7.02502e12i	− 1.45099i	−0.688229	−	0.725493i	$-0.741612\pi$
$659$	− 7.02502e12i	− 1.45099i	0.688229	−	0.725493i	$-0.258388\pi$
$660$	0	0
$661$	6.50654e12i	1.32570i	0.748754	+	0.662848i	$0.230652\pi$
$661$	6.50654e12i	1.32570i	−0.748754	+	0.662848i	$0.769348\pi$
$662$	0	0
$663$	−8.68531e11	−0.174572
$664$	0	0
$665$	−4.12179e12	−0.817313
$666$	0	0
$667$	3.01422e12i	0.589670i
$668$	0	0
$669$	1.67168e12i	0.322654i
$670$	0	0
$671$	1.33895e12	0.254984
$672$	0	0
$673$	4.49011e12	0.843703	0.421851	−	0.906665i	$-0.361380\pi$
$673$	4.49011e12	0.843703	0.421851	+	0.906665i	$0.361380\pi$
$674$	0	0
$675$	− 7.33615e11i	− 0.136019i
$676$	0	0
$677$	− 7.84775e12i	− 1.43581i	−0.696143	−	0.717903i	$-0.745102\pi$
$677$	− 7.84775e12i	− 1.43581i	0.696143	−	0.717903i	$-0.254898\pi$
$678$	0	0
$679$	−4.84038e12	−0.873907
$680$	0	0
$681$	−2.28997e11	−0.0408007
$682$	0	0
$683$	1.62526e12i	0.285779i	0.989739	+	0.142890i	$0.0456394\pi$
$683$	1.62526e12i	0.285779i	−0.989739	+	0.142890i	$0.954361\pi$
$684$	0	0
$685$	− 1.35855e13i	− 2.35759i
$686$	0	0
$687$	6.11267e12	1.04695
$688$	0	0
$689$	3.54750e12	0.599703
$690$	0	0
$691$	− 6.49894e11i	− 0.108440i	−0.998529	−	0.0542202i	$-0.982733\pi$
$691$	− 6.49894e11i	− 0.108440i	0.998529	−	0.0542202i	$-0.0172673\pi$
$692$	0	0
$693$	1.33641e12i	0.220111i
$694$	0	0
$695$	1.40805e13	2.28922
$696$	0	0
$697$	6.44809e12	1.03487
$698$	0	0
$699$	− 1.88205e12i	− 0.298183i
$700$	0	0
$701$	7.06748e10i	0.0110544i	0.999985	+	0.00552718i	$0.00175936\pi$
$701$	7.06748e10i	0.0110544i	−0.999985	+	0.00552718i	$0.998241\pi$
$702$	0	0
$703$	−6.02104e11	−0.0929763
$704$	0	0
$705$	−3.73166e12	−0.568920
$706$	0	0
$707$	− 9.88193e12i	− 1.48749i
$708$	0	0
$709$	6.17960e12i	0.918443i	0.888322	+	0.459222i	$0.151872\pi$
$709$	6.17960e12i	0.918443i	−0.888322	+	0.459222i	$0.848128\pi$
$710$	0	0
$711$	5.36085e11	0.0786721
$712$	0	0
$713$	−4.15136e12	−0.601571
$714$	0	0
$715$	1.50809e12i	0.215799i
$716$	0	0
$717$	1.42624e12i	0.201537i
$718$	0	0
$719$	−9.77032e11	−0.136342	−0.0681708	−	0.997674i	$-0.521716\pi$
$719$	−9.77032e11	−0.136342	−0.0681708	+	0.997674i	$0.521716\pi$
$720$	0	0
$721$	1.67999e13	2.31525
$722$	0	0
$723$	6.60636e12i	0.899166i
$724$	0	0
$725$	5.48947e12i	0.737920i
$726$	0	0
$727$	−9.97967e12	−1.32499	−0.662493	−	0.749068i	$-0.730502\pi$
$727$	−9.97967e12	−1.32499	−0.662493	+	0.749068i	$0.730502\pi$
$728$	0	0
$729$	−2.82430e11	−0.0370370
$730$	0	0
$731$	− 1.01878e13i	− 1.31963i
$732$	0	0
$733$	9.22525e10i	0.0118035i	0.999983	+	0.00590174i	$0.00187859\pi$
$733$	9.22525e10i	0.0118035i	−0.999983	+	0.00590174i	$0.998121\pi$
$734$	0	0
$735$	1.09797e13	1.38771
$736$	0	0
$737$	4.88032e11	0.0609319
$738$	0	0
$739$	− 1.12201e13i	− 1.38388i	−0.721956	−	0.691939i	$-0.756757\pi$
$739$	− 1.12201e13i	− 1.38388i	0.721956	−	0.691939i	$-0.243243\pi$
$740$	0	0
$741$	7.41516e11i	0.0903522i
$742$	0	0
$743$	4.78624e12	0.576163	0.288081	−	0.957606i	$-0.406983\pi$
$743$	4.78624e12	0.576163	0.288081	+	0.957606i	$0.406983\pi$
$744$	0	0
$745$	8.93755e12	1.06296
$746$	0	0
$747$	− 3.78692e12i	− 0.444983i
$748$	0	0
$749$	− 4.51155e12i	− 0.523791i
$750$	0	0
$751$	−5.59489e12	−0.641818	−0.320909	−	0.947110i	$-0.603988\pi$
$751$	−5.59489e12	−0.641818	−0.320909	+	0.947110i	$0.603988\pi$
$752$	0	0
$753$	−3.08331e12	−0.349494
$754$	0	0
$755$	− 1.31160e13i	− 1.46906i
$756$	0	0
$757$	1.54007e12i	0.170455i	0.996362	+	0.0852276i	$0.0271617\pi$
$757$	1.54007e12i	0.170455i	−0.996362	+	0.0852276i	$0.972838\pi$
$758$	0	0
$759$	−1.16823e12	−0.127773
$760$	0	0
$761$	8.76521e12	0.947395	0.473698	−	0.880688i	$-0.342919\pi$
$761$	8.76521e12	0.947395	0.473698	+	0.880688i	$0.342919\pi$
$762$	0	0
$763$	1.42049e13i	1.51732i
$764$	0	0
$765$	− 2.95894e12i	− 0.312363i
$766$	0	0
$767$	2.28508e12	0.238409
$768$	0	0
$769$	2.15404e12	0.222119	0.111060	−	0.993814i	$-0.464576\pi$
$769$	2.15404e12	0.222119	0.111060	+	0.993814i	$0.464576\pi$
$770$	0	0
$771$	− 1.05187e12i	− 0.107205i
$772$	0	0
$773$	4.52974e12i	0.456315i	0.973624	+	0.228158i	$0.0732702\pi$
$773$	4.52974e12i	0.456315i	−0.973624	+	0.228158i	$0.926730\pi$
$774$	0	0
$775$	−7.56041e12	−0.752814
$776$	0	0
$777$	2.47568e12	0.243668
$778$	0	0
$779$	− 5.50512e12i	− 0.535609i
$780$	0	0
$781$	− 3.70058e12i	− 0.355910i
$782$	0	0
$783$	2.11336e12	0.200930
$784$	0	0
$785$	−2.12454e13	−1.99688
$786$	0	0
$787$	− 5.90810e12i	− 0.548987i	−0.961589	−	0.274493i	$-0.911490\pi$
$787$	− 5.90810e12i	− 0.548987i	0.961589	−	0.274493i	$-0.0885101\pi$
$788$	0	0
$789$	− 2.55122e12i	− 0.234369i
$790$	0	0
$791$	2.36061e13	2.14403
$792$	0	0
$793$	−3.05469e12	−0.274307
$794$	0	0
$795$	1.20857e13i	1.07305i
$796$	0	0
$797$	1.38975e13i	1.22004i	0.792385	+	0.610021i	$0.208839\pi$
$797$	1.38975e13i	1.22004i	−0.792385	+	0.610021i	$0.791161\pi$
$798$	0	0
$799$	−6.23270e12	−0.541023
$800$	0	0
$801$	2.94175e12	0.252499
$802$	0	0
$803$	7.85009e12i	0.666277i
$804$	0	0
$805$	1.48148e13i	1.24341i
$806$	0	0
$807$	7.62289e12	0.632687
$808$	0	0
$809$	2.03083e13	1.66688	0.833440	−	0.552610i	$-0.186368\pi$
$809$	2.03083e13	1.66688	0.833440	+	0.552610i	$0.186368\pi$
$810$	0	0
$811$	1.88632e13i	1.53116i	0.643339	+	0.765581i	$0.277549\pi$
$811$	1.88632e13i	1.53116i	−0.643339	+	0.765581i	$0.722451\pi$
$812$	0	0
$813$	− 1.02747e11i	− 0.00824829i
$814$	0	0
$815$	−1.30838e13	−1.03878
$816$	0	0
$817$	−8.69790e12	−0.682991
$818$	0	0
$819$	− 3.04890e12i	− 0.236791i
$820$	0	0
$821$	4.26507e12i	0.327629i	0.986491	+	0.163814i	$0.0523798\pi$
$821$	4.26507e12i	0.327629i	−0.986491	+	0.163814i	$0.947620\pi$
$822$	0	0
$823$	−2.32425e13	−1.76597	−0.882985	−	0.469402i	$-0.844470\pi$
$823$	−2.32425e13	−1.76597	−0.882985	+	0.469402i	$0.844470\pi$
$824$	0	0
$825$	−2.12757e12	−0.159897
$826$	0	0
$827$	1.89257e13i	1.40694i	0.710723	+	0.703472i	$0.248368\pi$
$827$	1.89257e13i	1.40694i	−0.710723	+	0.703472i	$0.751632\pi$
$828$	0	0
$829$	3.11147e12i	0.228807i	0.993434	+	0.114404i	$0.0364957\pi$
$829$	3.11147e12i	0.228807i	−0.993434	+	0.114404i	$0.963504\pi$
$830$	0	0
$831$	−1.12972e12	−0.0821803
$832$	0	0
$833$	1.83385e13	1.31966
$834$	0	0
$835$	1.81118e13i	1.28936i
$836$	0	0
$837$	2.91063e12i	0.204985i
$838$	0	0
$839$	1.17785e12	0.0820657	0.0410328	−	0.999158i	$-0.486935\pi$
$839$	1.17785e12	0.0820657	0.0410328	+	0.999158i	$0.486935\pi$
$840$	0	0
$841$	−1.30661e12	−0.0900665
$842$	0	0
$843$	9.13307e12i	0.622863i
$844$	0	0
$845$	1.59211e13i	1.07428i
$846$	0	0
$847$	−2.13660e13	−1.42642
$848$	0	0
$849$	8.16077e12	0.539072
$850$	0	0
$851$	2.16412e12i	0.141449i
$852$	0	0
$853$	5.15924e12i	0.333668i	0.985985	+	0.166834i	$0.0533544\pi$
$853$	5.15924e12i	0.333668i	−0.985985	+	0.166834i	$0.946646\pi$
$854$	0	0
$855$	−2.52622e12	−0.161668
$856$	0	0
$857$	−2.48719e13	−1.57505	−0.787527	−	0.616280i	$-0.788639\pi$
$857$	−2.48719e13	−1.57505	−0.787527	+	0.616280i	$0.788639\pi$
$858$	0	0
$859$	− 2.60823e13i	− 1.63447i	−0.576307	−	0.817233i	$-0.695507\pi$
$859$	− 2.60823e13i	− 1.63447i	0.576307	−	0.817233i	$-0.304493\pi$
$860$	0	0
$861$	2.26355e13i	1.40370i
$862$	0	0
$863$	6.95954e12	0.427102	0.213551	−	0.976932i	$-0.431497\pi$
$863$	6.95954e12	0.427102	0.213551	+	0.976932i	$0.431497\pi$
$864$	0	0
$865$	−3.03688e13	−1.84440
$866$	0	0
$867$	4.66354e12i	0.280305i
$868$	0	0
$869$	− 1.55471e12i	− 0.0924827i
$870$	0	0
$871$	−1.11340e12	−0.0655495
$872$	0	0
$873$	−2.96664e12	−0.172862
$874$	0	0
$875$	− 1.11935e13i	− 0.645549i
$876$	0	0
$877$	2.75416e13i	1.57214i	0.618139	+	0.786069i	$0.287887\pi$
$877$	2.75416e13i	1.57214i	−0.618139	+	0.786069i	$0.712113\pi$
$878$	0	0
$879$	1.75805e13	0.993299
$880$	0	0
$881$	−1.92636e13	−1.07732	−0.538662	−	0.842522i	$-0.681070\pi$
$881$	−1.92636e13	−1.07732	−0.538662	+	0.842522i	$0.681070\pi$
$882$	0	0
$883$	7.04766e12i	0.390141i	0.980789	+	0.195071i	$0.0624936\pi$
$883$	7.04766e12i	0.390141i	−0.980789	+	0.195071i	$0.937506\pi$
$884$	0	0
$885$	7.78487e12i	0.426586i
$886$	0	0
$887$	2.27553e13	1.23432	0.617159	−	0.786838i	$-0.288284\pi$
$887$	2.27553e13	1.23432	0.617159	+	0.786838i	$0.288284\pi$
$888$	0	0
$889$	−2.55148e13	−1.37004
$890$	0	0
$891$	8.19079e11i	0.0435388i
$892$	0	0
$893$	5.32122e12i	0.280014i
$894$	0	0
$895$	−4.00059e13	−2.08411
$896$	0	0
$897$	2.66521e12	0.137456
$898$	0	0
$899$	− 2.17796e13i	− 1.11207i
$900$	0	0
$901$	2.01858e13i	1.02043i
$902$	0	0
$903$	3.57632e13	1.78995
$904$	0	0
$905$	3.19762e13	1.58456
$906$	0	0
$907$	1.04356e12i	0.0512018i	0.999672	+	0.0256009i	$0.00814991\pi$
$907$	1.04356e12i	0.0512018i	−0.999672	+	0.0256009i	$0.991850\pi$
$908$	0	0
$909$	− 6.05657e12i	− 0.294232i
$910$	0	0
$911$	−3.94274e13	−1.89656	−0.948278	−	0.317442i	$-0.897176\pi$
$911$	−3.94274e13	−1.89656	−0.948278	+	0.317442i	$0.897176\pi$
$912$	0	0
$913$	−1.09825e13	−0.523098
$914$	0	0
$915$	− 1.04068e13i	− 0.490819i
$916$	0	0
$917$	− 5.15304e12i	− 0.240659i
$918$	0	0
$919$	−3.11620e13	−1.44114	−0.720569	−	0.693383i	$-0.756119\pi$
$919$	−3.11620e13	−1.44114	−0.720569	+	0.693383i	$0.756119\pi$
$920$	0	0
$921$	6.16989e12	0.282559
$922$	0	0
$923$	8.44252e12i	0.382882i
$924$	0	0
$925$	3.94128e12i	0.177011i
$926$	0	0
$927$	1.02966e13	0.457966
$928$	0	0
$929$	−1.41893e13	−0.625015	−0.312508	−	0.949915i	$-0.601169\pi$
$929$	−1.41893e13	−0.625015	−0.312508	+	0.949915i	$0.601169\pi$
$930$	0	0
$931$	− 1.56567e13i	− 0.683009i
$932$	0	0
$933$	− 1.24755e13i	− 0.539003i
$934$	0	0
$935$	−8.58127e12	−0.367197
$936$	0	0
$937$	−1.55222e12	−0.0657848	−0.0328924	−	0.999459i	$-0.510472\pi$
$937$	−1.55222e12	−0.0657848	−0.0328924	+	0.999459i	$0.510472\pi$
$938$	0	0
$939$	− 2.42504e13i	− 1.01794i
$940$	0	0
$941$	1.00682e13i	0.418598i	0.977852	+	0.209299i	$0.0671182\pi$
$941$	1.00682e13i	0.418598i	−0.977852	+	0.209299i	$0.932882\pi$
$942$	0	0
$943$	−1.97869e13	−0.814844
$944$	0	0
$945$	1.03871e13	0.423692
$946$	0	0
$947$	2.94044e13i	1.18806i	0.804443	+	0.594030i	$0.202464\pi$
$947$	2.94044e13i	1.18806i	−0.804443	+	0.594030i	$0.797536\pi$
$948$	0	0
$949$	− 1.79092e13i	− 0.716769i
$950$	0	0
$951$	−1.07972e13	−0.428054
$952$	0	0
$953$	9.28196e12	0.364520	0.182260	−	0.983250i	$-0.441659\pi$
$953$	9.28196e12	0.364520	0.182260	+	0.983250i	$0.441659\pi$
$954$	0	0
$955$	− 5.42994e13i	− 2.11242i
$956$	0	0
$957$	− 6.12898e12i	− 0.236203i
$958$	0	0
$959$	7.96540e13	3.04105
$960$	0	0
$961$	3.55645e12	0.134512
$962$	0	0
$963$	− 2.76510e12i	− 0.103608i
$964$	0	0
$965$	4.51312e13i	1.67535i
$966$	0	0
$967$	1.09434e13	0.402470	0.201235	−	0.979543i	$-0.435505\pi$
$967$	1.09434e13	0.402470	0.201235	+	0.979543i	$0.435505\pi$
$968$	0	0
$969$	−4.21934e12	−0.153740
$970$	0	0
$971$	2.26615e13i	0.818091i	0.912514	+	0.409046i	$0.134138\pi$
$971$	2.26615e13i	0.818091i	−0.912514	+	0.409046i	$0.865862\pi$
$972$	0	0
$973$	8.25563e13i	2.95286i
$974$	0	0
$975$	4.85385e12	0.172015
$976$	0	0
$977$	4.01329e13	1.40921	0.704603	−	0.709602i	$-0.251125\pi$
$977$	4.01329e13	1.40921	0.704603	+	0.709602i	$0.251125\pi$
$978$	0	0
$979$	− 8.53142e12i	− 0.296824i
$980$	0	0
$981$	8.70610e12i	0.300133i
$982$	0	0
$983$	1.06363e13	0.363330	0.181665	−	0.983360i	$-0.441851\pi$
$983$	1.06363e13	0.363330	0.181665	+	0.983360i	$0.441851\pi$
$984$	0	0
$985$	−2.80603e13	−0.949795
$986$	0	0
$987$	− 2.18793e13i	− 0.733849i
$988$	0	0
$989$	3.12626e13i	1.03906i
$990$	0	0
$991$	1.72671e13	0.568708	0.284354	−	0.958719i	$-0.408221\pi$
$991$	1.72671e13	0.568708	0.284354	+	0.958719i	$0.408221\pi$
$992$	0	0
$993$	2.62807e13	0.857761
$994$	0	0
$995$	2.76001e13i	0.892704i
$996$	0	0
$997$	− 2.13433e13i	− 0.684121i	−0.939678	−	0.342060i	$-0.888875\pi$
$997$	− 2.13433e13i	− 0.684121i	0.939678	−	0.342060i	$-0.111125\pi$
$998$	0	0
$999$	1.51733e12	0.0481986

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.d.f.193.9	yes	20
4.3	odd	2	inner	384.10.d.f.193.19	yes	20
8.3	odd	2	inner	384.10.d.f.193.2	✓	20
8.5	even	2	inner	384.10.d.f.193.12	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.d.f.193.2	✓	20	8.3	odd	2	inner
384.10.d.f.193.9	yes	20	1.1	even	1	trivial
384.10.d.f.193.12	yes	20	8.5	even	2	inner
384.10.d.f.193.19	yes	20	4.3	odd	2	inner

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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-81.0000i q^{3} +1825.80i q^{5} -10705.0 q^{7} -6561.00 q^{9} +O(q^{10})\)	\(q-81.0000i q^{3} +1825.80i q^{5} -10705.0 q^{7} -6561.00 q^{9} +19027.7i q^{11} -43409.9i q^{13} +147890. q^{15} +247009. q^{17} -210886. i q^{19} +867102. i q^{21} -757980. q^{23} -1.38043e6 q^{25} +531441. i q^{27} -3.97665e6i q^{29} +5.47687e6 q^{31} +1.54124e6 q^{33} -1.95451e7i q^{35} -2.85512e6i q^{37} -3.51620e6 q^{39} +2.61047e7 q^{41} -4.12446e7i q^{43} -1.19791e7i q^{45} -2.52327e7 q^{47} +7.42425e7 q^{49} -2.00077e7i q^{51} +8.17211e7i q^{53} -3.47408e7 q^{55} -1.70818e7 q^{57} +5.26397e7i q^{59} -7.03685e7i q^{61} +7.02352e7 q^{63} +7.92578e7 q^{65} -2.56485e7i q^{67} +6.13964e7i q^{69} -1.94484e8 q^{71} +4.12562e8 q^{73} +1.11814e8i q^{75} -2.03690e8i q^{77} -8.17079e7 q^{79} +4.30467e7 q^{81} +5.77186e8i q^{83} +4.50989e8i q^{85} -3.22109e8 q^{87} -4.48369e8 q^{89} +4.64701e8i q^{91} -4.43626e8i q^{93} +3.85036e8 q^{95} +4.52162e8 q^{97} -1.24841e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 131220 q^{9}+O(q^{10}) \) 20 * q - 131220 * q^9	\( 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) \) 20 * q - 131220 * q^9 - 905768 * q^17 - 9682620 * q^25 + 12054096 * q^33 + 74264008 * q^41 + 252775700 * q^49 - 5335632 * q^57 + 245588672 * q^65 - 895193896 * q^73 + 860934420 * q^81 + 882422136 * q^89 + 433683736 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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