LMFDB - Embedded newform 384.10.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,Mod(1,384)]

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, 0, 0]))

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

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	384.a (trivial)

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:	yes
Analytic conductor:	$197.773761087$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 14124x^{2} - 170336x + 18391464 $ x^4 - 14124x^2 - 170336x + 18391464
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$31.4956$ of defining polynomial
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-81.0000 q^{3} -1657.87 q^{5} +11369.1 q^{7} +6561.00 q^{9} +O(q^{10})$	$q-81.0000 q^{3} -1657.87 q^{5} +11369.1 q^{7} +6561.00 q^{9} +43824.4 q^{11} +20185.9 q^{13} +134287. q^{15} +107896. q^{17} -870857. q^{19} -920899. q^{21} -1.61330e6 q^{23} +795403. q^{25} -531441. q^{27} +711411. q^{29} -4.84203e6 q^{31} -3.54978e6 q^{33} -1.88485e7 q^{35} -354332. q^{37} -1.63505e6 q^{39} +2.36447e7 q^{41} -5.28433e6 q^{43} -1.08773e7 q^{45} +5.51050e7 q^{47} +8.89033e7 q^{49} -8.73958e6 q^{51} -5.31439e7 q^{53} -7.26552e7 q^{55} +7.05394e7 q^{57} -1.35998e8 q^{59} +2.20573e7 q^{61} +7.45928e7 q^{63} -3.34655e7 q^{65} +1.20139e8 q^{67} +1.30677e8 q^{69} +1.30319e8 q^{71} +2.81926e7 q^{73} -6.44276e7 q^{75} +4.98245e8 q^{77} +5.16349e8 q^{79} +4.30467e7 q^{81} +5.45476e8 q^{83} -1.78878e8 q^{85} -5.76243e7 q^{87} -1.34730e8 q^{89} +2.29495e8 q^{91} +3.92204e8 q^{93} +1.44377e9 q^{95} -1.29643e9 q^{97} +2.87532e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 324 q^{3} - 1728 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) $ 4 * q - 324 * q^3 - 1728 * q^5 + 4840 * q^7 + 26244 * q^9	$ 4 q - 324 q^{3} - 1728 q^{5} + 4840 q^{7} + 26244 q^{9} - 15824 q^{11} - 82440 q^{13} + 139968 q^{15} + 165912 q^{17} - 539904 q^{19} - 392040 q^{21} + 729680 q^{23} + 224812 q^{25} - 2125764 q^{27} - 1850864 q^{29} + 3197960 q^{31} + 1281744 q^{33} - 8574912 q^{35} - 4187992 q^{37} + 6677640 q^{39} + 227704 q^{41} - 1600352 q^{43} - 11337408 q^{45} + 18053904 q^{47} + 57728820 q^{49} - 13438872 q^{51} - 29418288 q^{53} + 45906816 q^{55} + 43732224 q^{57} - 38300048 q^{59} + 99764648 q^{61} + 31755240 q^{63} + 314120832 q^{65} + 183717008 q^{67} - 59104080 q^{69} + 181868080 q^{71} + 254539160 q^{73} - 18209772 q^{75} + 230564704 q^{77} + 831578184 q^{79} + 172186884 q^{81} + 687923952 q^{83} - 1041391104 q^{85} + 149919984 q^{87} + 627627272 q^{89} + 1018147632 q^{91} - 259034760 q^{93} + 2167118208 q^{95} - 889385880 q^{97} - 103821264 q^{99}+O(q^{100}) $ 4 * q - 324 * q^3 - 1728 * q^5 + 4840 * q^7 + 26244 * q^9 - 15824 * q^11 - 82440 * q^13 + 139968 * q^15 + 165912 * q^17 - 539904 * q^19 - 392040 * q^21 + 729680 * q^23 + 224812 * q^25 - 2125764 * q^27 - 1850864 * q^29 + 3197960 * q^31 + 1281744 * q^33 - 8574912 * q^35 - 4187992 * q^37 + 6677640 * q^39 + 227704 * q^41 - 1600352 * q^43 - 11337408 * q^45 + 18053904 * q^47 + 57728820 * q^49 - 13438872 * q^51 - 29418288 * q^53 + 45906816 * q^55 + 43732224 * q^57 - 38300048 * q^59 + 99764648 * q^61 + 31755240 * q^63 + 314120832 * q^65 + 183717008 * q^67 - 59104080 * q^69 + 181868080 * q^71 + 254539160 * q^73 - 18209772 * q^75 + 230564704 * q^77 + 831578184 * q^79 + 172186884 * q^81 + 687923952 * q^83 - 1041391104 * q^85 + 149919984 * q^87 + 627627272 * q^89 + 1018147632 * q^91 - 259034760 * q^93 + 2167118208 * q^95 - 889385880 * q^97 - 103821264 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−81.0000	−0.577350
$3$	−81.0000	−0.577350
$4$	0	0
$5$	−1657.87	−1.18627	−0.593137	−	0.805102i	$-0.702111\pi$
$5$	−1657.87	−1.18627	−0.593137	+	0.805102i	$0.702111\pi$
$6$	0	0
$7$	11369.1	1.78972	0.894861	−	0.446345i	$-0.147274\pi$
$7$	11369.1	1.78972	0.894861	+	0.446345i	$0.147274\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	43824.4	0.902504	0.451252	−	0.892396i	$-0.350977\pi$
$11$	43824.4	0.902504	0.451252	+	0.892396i	$0.350977\pi$
$12$	0	0
$13$	20185.9	0.196021	0.0980104	−	0.995185i	$-0.468752\pi$
$13$	20185.9	0.196021	0.0980104	+	0.995185i	$0.468752\pi$
$14$	0	0
$15$	134287.	0.684896
$16$	0	0
$17$	107896.	0.313318	0.156659	−	0.987653i	$-0.449928\pi$
$17$	107896.	0.313318	0.156659	+	0.987653i	$0.449928\pi$
$18$	0	0
$19$	−870857.	−1.53305	−0.766523	−	0.642216i	$-0.778015\pi$
$19$	−870857.	−1.53305	−0.766523	+	0.642216i	$0.778015\pi$
$20$	0	0
$21$	−920899.	−1.03330
$22$	0	0
$23$	−1.61330e6	−1.20210	−0.601049	−	0.799212i	$-0.705250\pi$
$23$	−1.61330e6	−1.20210	−0.601049	+	0.799212i	$0.705250\pi$
$24$	0	0
$25$	795403.	0.407246
$26$	0	0
$27$	−531441.	−0.192450
$28$	0	0
$29$	711411.	0.186780	0.0933899	−	0.995630i	$-0.470230\pi$
$29$	711411.	0.186780	0.0933899	+	0.995630i	$0.470230\pi$
$30$	0	0
$31$	−4.84203e6	−0.941672	−0.470836	−	0.882221i	$-0.656048\pi$
$31$	−4.84203e6	−0.941672	−0.470836	+	0.882221i	$0.656048\pi$
$32$	0	0
$33$	−3.54978e6	−0.521061
$34$	0	0
$35$	−1.88485e7	−2.12310
$36$	0	0
$37$	−354332.	−0.0310815	−0.0155408	−	0.999879i	$-0.504947\pi$
$37$	−354332.	−0.0310815	−0.0155408	+	0.999879i	$0.504947\pi$
$38$	0	0
$39$	−1.63505e6	−0.113173
$40$	0	0
$41$	2.36447e7	1.30679	0.653395	−	0.757017i	$-0.273344\pi$
$41$	2.36447e7	1.30679	0.653395	+	0.757017i	$0.273344\pi$
$42$	0	0
$43$	−5.28433e6	−0.235712	−0.117856	−	0.993031i	$-0.537602\pi$
$43$	−5.28433e6	−0.235712	−0.117856	+	0.993031i	$0.537602\pi$
$44$	0	0
$45$	−1.08773e7	−0.395425
$46$	0	0
$47$	5.51050e7	1.64722	0.823608	−	0.567160i	$-0.191958\pi$
$47$	5.51050e7	1.64722	0.823608	+	0.567160i	$0.191958\pi$
$48$	0	0
$49$	8.89033e7	2.20311
$50$	0	0
$51$	−8.73958e6	−0.180894
$52$	0	0
$53$	−5.31439e7	−0.925150	−0.462575	−	0.886580i	$-0.653074\pi$
$53$	−5.31439e7	−0.925150	−0.462575	+	0.886580i	$0.653074\pi$
$54$	0	0
$55$	−7.26552e7	−1.07062
$56$	0	0
$57$	7.05394e7	0.885105
$58$	0	0
$59$	−1.35998e8	−1.46116	−0.730579	−	0.682829i	$-0.760750\pi$
$59$	−1.35998e8	−1.46116	−0.730579	+	0.682829i	$0.760750\pi$
$60$	0	0
$61$	2.20573e7	0.203971	0.101986	−	0.994786i	$-0.467480\pi$
$61$	2.20573e7	0.203971	0.101986	+	0.994786i	$0.467480\pi$
$62$	0	0
$63$	7.45928e7	0.596574
$64$	0	0
$65$	−3.34655e7	−0.232534
$66$	0	0
$67$	1.20139e8	0.728361	0.364181	−	0.931328i	$-0.381349\pi$
$67$	1.20139e8	0.728361	0.364181	+	0.931328i	$0.381349\pi$
$68$	0	0
$69$	1.30677e8	0.694032
$70$	0	0
$71$	1.30319e8	0.608620	0.304310	−	0.952573i	$-0.401574\pi$
$71$	1.30319e8	0.608620	0.304310	+	0.952573i	$0.401574\pi$
$72$	0	0
$73$	2.81926e7	0.116193	0.0580967	−	0.998311i	$-0.481497\pi$
$73$	2.81926e7	0.116193	0.0580967	+	0.998311i	$0.481497\pi$
$74$	0	0
$75$	−6.44276e7	−0.235124
$76$	0	0
$77$	4.98245e8	1.61523
$78$	0	0
$79$	5.16349e8	1.49149	0.745747	−	0.666230i	$-0.232093\pi$
$79$	5.16349e8	1.49149	0.745747	+	0.666230i	$0.232093\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	5.45476e8	1.26161	0.630804	−	0.775943i	$-0.282725\pi$
$83$	5.45476e8	1.26161	0.630804	+	0.775943i	$0.282725\pi$
$84$	0	0
$85$	−1.78878e8	−0.371681
$86$	0	0
$87$	−5.76243e7	−0.107837
$88$	0	0
$89$	−1.34730e8	−0.227620	−0.113810	−	0.993503i	$-0.536306\pi$
$89$	−1.34730e8	−0.227620	−0.113810	+	0.993503i	$0.536306\pi$
$90$	0	0
$91$	2.29495e8	0.350823
$92$	0	0
$93$	3.92204e8	0.543675
$94$	0	0
$95$	1.44377e9	1.81861
$96$	0	0
$97$	−1.29643e9	−1.48688	−0.743440	−	0.668802i	$-0.766807\pi$
$97$	−1.29643e9	−1.48688	−0.743440	+	0.668802i	$0.766807\pi$
$98$	0	0
$99$	2.87532e8	0.300835
$100$	0	0
$101$	−1.45555e9	−1.39182	−0.695908	−	0.718131i	$-0.744998\pi$
$101$	−1.45555e9	−1.39182	−0.695908	+	0.718131i	$0.744998\pi$
$102$	0	0
$103$	−1.57323e9	−1.37729	−0.688643	−	0.725100i	$-0.741793\pi$
$103$	−1.57323e9	−1.37729	−0.688643	+	0.725100i	$0.741793\pi$
$104$	0	0
$105$	1.52673e9	1.22577
$106$	0	0
$107$	−2.65618e8	−0.195898	−0.0979490	−	0.995191i	$-0.531228\pi$
$107$	−2.65618e8	−0.195898	−0.0979490	+	0.995191i	$0.531228\pi$
$108$	0	0
$109$	−2.60050e9	−1.76456	−0.882282	−	0.470721i	$-0.843994\pi$
$109$	−2.60050e9	−1.76456	−0.882282	+	0.470721i	$0.843994\pi$
$110$	0	0
$111$	2.87009e7	0.0179449
$112$	0	0
$113$	−2.25685e8	−0.130212	−0.0651058	−	0.997878i	$-0.520738\pi$
$113$	−2.25685e8	−0.130212	−0.0651058	+	0.997878i	$0.520738\pi$
$114$	0	0
$115$	2.67464e9	1.42602
$116$	0	0
$117$	1.32439e8	0.0653403
$118$	0	0
$119$	1.22668e9	0.560753
$120$	0	0
$121$	−4.37365e8	−0.185486
$122$	0	0
$123$	−1.91522e9	−0.754476
$124$	0	0
$125$	1.91935e9	0.703168
$126$	0	0
$127$	−5.02556e9	−1.71422	−0.857112	−	0.515130i	$-0.827744\pi$
$127$	−5.02556e9	−1.71422	−0.857112	+	0.515130i	$0.827744\pi$
$128$	0	0
$129$	4.28031e8	0.136088
$130$	0	0
$131$	−1.83634e9	−0.544794	−0.272397	−	0.962185i	$-0.587816\pi$
$131$	−1.83634e9	−0.544794	−0.272397	+	0.962185i	$0.587816\pi$
$132$	0	0
$133$	−9.90087e9	−2.74373
$134$	0	0
$135$	8.81059e8	0.228299
$136$	0	0
$137$	5.09529e9	1.23574	0.617869	−	0.786281i	$-0.287996\pi$
$137$	5.09529e9	1.23574	0.617869	+	0.786281i	$0.287996\pi$
$138$	0	0
$139$	4.22920e9	0.960929	0.480464	−	0.877014i	$-0.340468\pi$
$139$	4.22920e9	0.960929	0.480464	+	0.877014i	$0.340468\pi$
$140$	0	0
$141$	−4.46350e9	−0.951020
$142$	0	0
$143$	8.84634e8	0.176910
$144$	0	0
$145$	−1.17943e9	−0.221572
$146$	0	0
$147$	−7.20117e9	−1.27196
$148$	0	0
$149$	4.14480e9	0.688914	0.344457	−	0.938802i	$-0.388063\pi$
$149$	4.14480e9	0.688914	0.344457	+	0.938802i	$0.388063\pi$
$150$	0	0
$151$	−3.72372e9	−0.582882	−0.291441	−	0.956589i	$-0.594135\pi$
$151$	−3.72372e9	−0.582882	−0.291441	+	0.956589i	$0.594135\pi$
$152$	0	0
$153$	7.07906e8	0.104439
$154$	0	0
$155$	8.02745e9	1.11708
$156$	0	0
$157$	7.47987e9	0.982529	0.491265	−	0.871010i	$-0.336535\pi$
$157$	7.47987e9	0.982529	0.491265	+	0.871010i	$0.336535\pi$
$158$	0	0
$159$	4.30466e9	0.534136
$160$	0	0
$161$	−1.83418e10	−2.15142
$162$	0	0
$163$	−1.58179e10	−1.75511	−0.877557	−	0.479473i	$-0.840828\pi$
$163$	−1.58179e10	−1.75511	−0.877557	+	0.479473i	$0.840828\pi$
$164$	0	0
$165$	5.88507e9	0.618121
$166$	0	0
$167$	9.10811e9	0.906159	0.453079	−	0.891470i	$-0.350325\pi$
$167$	9.10811e9	0.906159	0.453079	+	0.891470i	$0.350325\pi$
$168$	0	0
$169$	−1.01970e10	−0.961576
$170$	0	0
$171$	−5.71369e9	−0.511016
$172$	0	0
$173$	1.33337e10	1.13173	0.565865	−	0.824498i	$-0.308542\pi$
$173$	1.33337e10	1.13173	0.565865	+	0.824498i	$0.308542\pi$
$174$	0	0
$175$	9.04303e9	0.728858
$176$	0	0
$177$	1.10158e10	0.843600
$178$	0	0
$179$	−9.36604e9	−0.681895	−0.340947	−	0.940082i	$-0.610748\pi$
$179$	−9.36604e9	−0.681895	−0.340947	+	0.940082i	$0.610748\pi$
$180$	0	0
$181$	−4.21356e9	−0.291807	−0.145904	−	0.989299i	$-0.546609\pi$
$181$	−4.21356e9	−0.291807	−0.145904	+	0.989299i	$0.546609\pi$
$182$	0	0
$183$	−1.78664e9	−0.117763
$184$	0	0
$185$	5.87436e8	0.0368712
$186$	0	0
$187$	4.72849e9	0.282771
$188$	0	0
$189$	−6.04202e9	−0.344432
$190$	0	0
$191$	3.02725e10	1.64588	0.822939	−	0.568130i	$-0.192333\pi$
$191$	3.02725e10	1.64588	0.822939	+	0.568130i	$0.192333\pi$
$192$	0	0
$193$	1.16337e10	0.603545	0.301773	−	0.953380i	$-0.402422\pi$
$193$	1.16337e10	0.603545	0.301773	+	0.953380i	$0.402422\pi$
$194$	0	0
$195$	2.71071e9	0.134254
$196$	0	0
$197$	1.89560e10	0.896705	0.448352	−	0.893857i	$-0.352011\pi$
$197$	1.89560e10	0.896705	0.448352	+	0.893857i	$0.352011\pi$
$198$	0	0
$199$	2.63284e10	1.19010	0.595052	−	0.803687i	$-0.297131\pi$
$199$	2.63284e10	1.19010	0.595052	+	0.803687i	$0.297131\pi$
$200$	0	0
$201$	−9.73124e9	−0.420519
$202$	0	0
$203$	8.08812e9	0.334284
$204$	0	0
$205$	−3.91998e10	−1.55021
$206$	0	0
$207$	−1.05849e10	−0.400699
$208$	0	0
$209$	−3.81648e10	−1.38358
$210$	0	0
$211$	−5.38261e10	−1.86949	−0.934743	−	0.355325i	$-0.884370\pi$
$211$	−5.38261e10	−1.86949	−0.934743	+	0.355325i	$0.884370\pi$
$212$	0	0
$213$	−1.05559e10	−0.351387
$214$	0	0
$215$	8.76072e9	0.279619
$216$	0	0
$217$	−5.50496e10	−1.68533
$218$	0	0
$219$	−2.28360e9	−0.0670843
$220$	0	0
$221$	2.17798e9	0.0614169
$222$	0	0
$223$	−2.63731e10	−0.714150	−0.357075	−	0.934076i	$-0.616226\pi$
$223$	−2.63731e10	−0.714150	−0.357075	+	0.934076i	$0.616226\pi$
$224$	0	0
$225$	5.21864e9	0.135749
$226$	0	0
$227$	9.65262e9	0.241284	0.120642	−	0.992696i	$-0.461505\pi$
$227$	9.65262e9	0.241284	0.120642	+	0.992696i	$0.461505\pi$
$228$	0	0
$229$	4.68891e10	1.12671	0.563355	−	0.826215i	$-0.309510\pi$
$229$	4.68891e10	1.12671	0.563355	+	0.826215i	$0.309510\pi$
$230$	0	0
$231$	−4.03579e10	−0.932555
$232$	0	0
$233$	−8.65849e9	−0.192460	−0.0962300	−	0.995359i	$-0.530678\pi$
$233$	−8.65849e9	−0.192460	−0.0962300	+	0.995359i	$0.530678\pi$
$234$	0	0
$235$	−9.13568e10	−1.95405
$236$	0	0
$237$	−4.18242e10	−0.861114
$238$	0	0
$239$	3.15630e9	0.0625732	0.0312866	−	0.999510i	$-0.490040\pi$
$239$	3.15630e9	0.0625732	0.0312866	+	0.999510i	$0.490040\pi$
$240$	0	0
$241$	6.97082e10	1.33109	0.665545	−	0.746358i	$-0.268199\pi$
$241$	6.97082e10	1.33109	0.665545	+	0.746358i	$0.268199\pi$
$242$	0	0
$243$	−3.48678e9	−0.0641500
$244$	0	0
$245$	−1.47390e11	−2.61349
$246$	0	0
$247$	−1.75790e10	−0.300509
$248$	0	0
$249$	−4.41836e10	−0.728389
$250$	0	0
$251$	−8.46962e10	−1.34689	−0.673445	−	0.739238i	$-0.735186\pi$
$251$	−8.46962e10	−1.34689	−0.673445	+	0.739238i	$0.735186\pi$
$252$	0	0
$253$	−7.07020e10	−1.08490
$254$	0	0
$255$	1.44891e10	0.214590
$256$	0	0
$257$	−1.32043e11	−1.88806	−0.944031	−	0.329856i	$-0.893000\pi$
$257$	−1.32043e11	−1.88806	−0.944031	+	0.329856i	$0.893000\pi$
$258$	0	0
$259$	−4.02844e9	−0.0556273
$260$	0	0
$261$	4.66757e9	0.0622599
$262$	0	0
$263$	−6.87758e10	−0.886411	−0.443205	−	0.896420i	$-0.646159\pi$
$263$	−6.87758e10	−0.886411	−0.443205	+	0.896420i	$0.646159\pi$
$264$	0	0
$265$	8.81057e10	1.09748
$266$	0	0
$267$	1.09132e10	0.131416
$268$	0	0
$269$	−1.19295e11	−1.38912	−0.694558	−	0.719437i	$-0.744400\pi$
$269$	−1.19295e11	−1.38912	−0.694558	+	0.719437i	$0.744400\pi$
$270$	0	0
$271$	1.61696e11	1.82112	0.910558	−	0.413381i	$-0.135652\pi$
$271$	1.61696e11	1.82112	0.910558	+	0.413381i	$0.135652\pi$
$272$	0	0
$273$	−1.85891e10	−0.202548
$274$	0	0
$275$	3.48581e10	0.367542
$276$	0	0
$277$	6.99145e10	0.713523	0.356762	−	0.934195i	$-0.383881\pi$
$277$	6.99145e10	0.713523	0.356762	+	0.934195i	$0.383881\pi$
$278$	0	0
$279$	−3.17686e10	−0.313891
$280$	0	0
$281$	−1.57703e11	−1.50891	−0.754454	−	0.656353i	$-0.772098\pi$
$281$	−1.57703e11	−1.50891	−0.754454	+	0.656353i	$0.772098\pi$
$282$	0	0
$283$	1.21847e11	1.12922	0.564608	−	0.825359i	$-0.309027\pi$
$283$	1.21847e11	1.12922	0.564608	+	0.825359i	$0.309027\pi$
$284$	0	0
$285$	−1.16945e11	−1.04998
$286$	0	0
$287$	2.68819e11	2.33879
$288$	0	0
$289$	−1.06946e11	−0.901832
$290$	0	0
$291$	1.05011e11	0.858451
$292$	0	0
$293$	−1.45429e11	−1.15278	−0.576391	−	0.817174i	$-0.695540\pi$
$293$	−1.45429e11	−1.15278	−0.576391	+	0.817174i	$0.695540\pi$
$294$	0	0
$295$	2.25466e11	1.73333
$296$	0	0
$297$	−2.32901e10	−0.173687
$298$	0	0
$299$	−3.25659e10	−0.235636
$300$	0	0
$301$	−6.00782e10	−0.421859
$302$	0	0
$303$	1.17900e11	0.803565
$304$	0	0
$305$	−3.65681e10	−0.241966
$306$	0	0
$307$	−2.72299e11	−1.74953	−0.874767	−	0.484543i	$-0.838986\pi$
$307$	−2.72299e11	−1.74953	−0.874767	+	0.484543i	$0.838986\pi$
$308$	0	0
$309$	1.27432e11	0.795177
$310$	0	0
$311$	6.04740e10	0.366561	0.183281	−	0.983061i	$-0.441328\pi$
$311$	6.04740e10	0.366561	0.183281	+	0.983061i	$0.441328\pi$
$312$	0	0
$313$	−2.88478e11	−1.69888	−0.849442	−	0.527682i	$-0.823061\pi$
$313$	−2.88478e11	−1.69888	−0.849442	+	0.527682i	$0.823061\pi$
$314$	0	0
$315$	−1.23665e11	−0.707700
$316$	0	0
$317$	1.93523e11	1.07638	0.538191	−	0.842823i	$-0.319108\pi$
$317$	1.93523e11	1.07638	0.538191	+	0.842823i	$0.319108\pi$
$318$	0	0
$319$	3.11772e10	0.168570
$320$	0	0
$321$	2.15150e10	0.113102
$322$	0	0
$323$	−9.39620e10	−0.480331
$324$	0	0
$325$	1.60559e10	0.0798287
$326$	0	0
$327$	2.10640e11	1.01877
$328$	0	0
$329$	6.26495e11	2.94806
$330$	0	0
$331$	−8.96795e10	−0.410645	−0.205323	−	0.978694i	$-0.565824\pi$
$331$	−8.96795e10	−0.410645	−0.205323	+	0.978694i	$0.565824\pi$
$332$	0	0
$333$	−2.32477e9	−0.0103605
$334$	0	0
$335$	−1.99174e11	−0.864036
$336$	0	0
$337$	−3.95162e11	−1.66894	−0.834470	−	0.551053i	$-0.814226\pi$
$337$	−3.95162e11	−1.66894	−0.834470	+	0.551053i	$0.814226\pi$
$338$	0	0
$339$	1.82805e10	0.0751776
$340$	0	0
$341$	−2.12199e11	−0.849864
$342$	0	0
$343$	5.51967e11	2.15323
$344$	0	0
$345$	−2.16646e11	−0.823312
$346$	0	0
$347$	−2.50145e11	−0.926209	−0.463105	−	0.886304i	$-0.653265\pi$
$347$	−2.50145e11	−0.926209	−0.463105	+	0.886304i	$0.653265\pi$
$348$	0	0
$349$	−4.24673e11	−1.53229	−0.766143	−	0.642670i	$-0.777827\pi$
$349$	−4.24673e11	−1.53229	−0.766143	+	0.642670i	$0.777827\pi$
$350$	0	0
$351$	−1.07276e10	−0.0377242
$352$	0	0
$353$	−5.87300e10	−0.201314	−0.100657	−	0.994921i	$-0.532094\pi$
$353$	−5.87300e10	−0.201314	−0.100657	+	0.994921i	$0.532094\pi$
$354$	0	0
$355$	−2.16052e11	−0.721990
$356$	0	0
$357$	−9.93614e10	−0.323751
$358$	0	0
$359$	2.50784e11	0.796847	0.398424	−	0.917202i	$-0.369557\pi$
$359$	2.50784e11	0.796847	0.398424	+	0.917202i	$0.369557\pi$
$360$	0	0
$361$	4.35703e11	1.35023
$362$	0	0
$363$	3.54266e10	0.107090
$364$	0	0
$365$	−4.67396e10	−0.137837
$366$	0	0
$367$	−5.85367e11	−1.68434	−0.842172	−	0.539210i	$-0.818723\pi$
$367$	−5.85367e11	−1.68434	−0.842172	+	0.539210i	$0.818723\pi$
$368$	0	0
$369$	1.55133e11	0.435597
$370$	0	0
$371$	−6.04200e11	−1.65576
$372$	0	0
$373$	−2.67463e11	−0.715440	−0.357720	−	0.933829i	$-0.616446\pi$
$373$	−2.67463e11	−0.715440	−0.357720	+	0.933829i	$0.616446\pi$
$374$	0	0
$375$	−1.55467e11	−0.405974
$376$	0	0
$377$	1.43604e10	0.0366127
$378$	0	0
$379$	3.76417e11	0.937114	0.468557	−	0.883433i	$-0.344774\pi$
$379$	3.76417e11	0.937114	0.468557	+	0.883433i	$0.344774\pi$
$380$	0	0
$381$	4.07070e11	0.989708
$382$	0	0
$383$	−9.26258e10	−0.219957	−0.109978	−	0.993934i	$-0.535078\pi$
$383$	−9.26258e10	−0.219957	−0.109978	+	0.993934i	$0.535078\pi$
$384$	0	0
$385$	−8.26025e11	−1.91611
$386$	0	0
$387$	−3.46705e10	−0.0785707
$388$	0	0
$389$	−1.89039e11	−0.418579	−0.209290	−	0.977854i	$-0.567115\pi$
$389$	−1.89039e11	−0.418579	−0.209290	+	0.977854i	$0.567115\pi$
$390$	0	0
$391$	−1.74069e11	−0.376639
$392$	0	0
$393$	1.48744e11	0.314537
$394$	0	0
$395$	−8.56038e11	−1.76932
$396$	0	0
$397$	8.13053e10	0.164271	0.0821356	−	0.996621i	$-0.473826\pi$
$397$	8.13053e10	0.164271	0.0821356	+	0.996621i	$0.473826\pi$
$398$	0	0
$399$	8.01971e11	1.58409
$400$	0	0
$401$	−3.01070e10	−0.0581457	−0.0290729	−	0.999577i	$-0.509255\pi$
$401$	−3.01070e10	−0.0581457	−0.0290729	+	0.999577i	$0.509255\pi$
$402$	0	0
$403$	−9.77405e10	−0.184587
$404$	0	0
$405$	−7.13658e10	−0.131808
$406$	0	0
$407$	−1.55284e10	−0.0280512
$408$	0	0
$409$	−6.44378e11	−1.13864	−0.569319	−	0.822117i	$-0.692793\pi$
$409$	−6.44378e11	−1.13864	−0.569319	+	0.822117i	$0.692793\pi$
$410$	0	0
$411$	−4.12718e11	−0.713453
$412$	0	0
$413$	−1.54617e12	−2.61507
$414$	0	0
$415$	−9.04327e11	−1.49661
$416$	0	0
$417$	−3.42565e11	−0.554793
$418$	0	0
$419$	1.03965e12	1.64787	0.823934	−	0.566686i	$-0.191775\pi$
$419$	1.03965e12	1.64787	0.823934	+	0.566686i	$0.191775\pi$
$420$	0	0
$421$	−1.53352e11	−0.237913	−0.118957	−	0.992899i	$-0.537955\pi$
$421$	−1.53352e11	−0.237913	−0.118957	+	0.992899i	$0.537955\pi$
$422$	0	0
$423$	3.61544e11	0.549072
$424$	0	0
$425$	8.58209e10	0.127598
$426$	0	0
$427$	2.50772e11	0.365052
$428$	0	0
$429$	−7.16554e10	−0.102139
$430$	0	0
$431$	1.12199e11	0.156618	0.0783088	−	0.996929i	$-0.475048\pi$
$431$	1.12199e11	0.156618	0.0783088	+	0.996929i	$0.475048\pi$
$432$	0	0
$433$	−2.08676e11	−0.285283	−0.142642	−	0.989774i	$-0.545560\pi$
$433$	−2.08676e11	−0.285283	−0.142642	+	0.989774i	$0.545560\pi$
$434$	0	0
$435$	9.55335e10	0.127925
$436$	0	0
$437$	1.40495e12	1.84287
$438$	0	0
$439$	−4.48266e10	−0.0576030	−0.0288015	−	0.999585i	$-0.509169\pi$
$439$	−4.48266e10	−0.0576030	−0.0288015	+	0.999585i	$0.509169\pi$
$440$	0	0
$441$	5.83294e11	0.734369
$442$	0	0
$443$	5.61453e11	0.692622	0.346311	−	0.938120i	$-0.387434\pi$
$443$	5.61453e11	0.692622	0.346311	+	0.938120i	$0.387434\pi$
$444$	0	0
$445$	2.23365e11	0.270020
$446$	0	0
$447$	−3.35729e11	−0.397745
$448$	0	0
$449$	−1.46625e12	−1.70255	−0.851275	−	0.524719i	$-0.824170\pi$
$449$	−1.46625e12	−1.70255	−0.851275	+	0.524719i	$0.824170\pi$
$450$	0	0
$451$	1.03621e12	1.17938
$452$	0	0
$453$	3.01621e11	0.336527
$454$	0	0
$455$	−3.80473e11	−0.416172
$456$	0	0
$457$	−6.66517e11	−0.714806	−0.357403	−	0.933950i	$-0.616338\pi$
$457$	−6.66517e11	−0.714806	−0.357403	+	0.933950i	$0.616338\pi$
$458$	0	0
$459$	−5.73404e10	−0.0602981
$460$	0	0
$461$	−8.73305e10	−0.0900558	−0.0450279	−	0.998986i	$-0.514338\pi$
$461$	−8.73305e10	−0.0900558	−0.0450279	+	0.998986i	$0.514338\pi$
$462$	0	0
$463$	2.21546e11	0.224052	0.112026	−	0.993705i	$-0.464266\pi$
$463$	2.21546e11	0.224052	0.112026	+	0.993705i	$0.464266\pi$
$464$	0	0
$465$	−6.50223e11	−0.644947
$466$	0	0
$467$	−6.40407e11	−0.623060	−0.311530	−	0.950236i	$-0.600842\pi$
$467$	−6.40407e11	−0.623060	−0.311530	+	0.950236i	$0.600842\pi$
$468$	0	0
$469$	1.36587e12	1.30356
$470$	0	0
$471$	−6.05869e11	−0.567263
$472$	0	0
$473$	−2.31583e11	−0.212731
$474$	0	0
$475$	−6.92682e11	−0.624328
$476$	0	0
$477$	−3.48677e11	−0.308383
$478$	0	0
$479$	1.72490e12	1.49711	0.748554	−	0.663073i	$-0.230748\pi$
$479$	1.72490e12	1.49711	0.748554	+	0.663073i	$0.230748\pi$
$480$	0	0
$481$	−7.15250e9	−0.00609263
$482$	0	0
$483$	1.48569e12	1.24212
$484$	0	0
$485$	2.14931e12	1.76385
$486$	0	0
$487$	−7.70139e11	−0.620425	−0.310212	−	0.950667i	$-0.600400\pi$
$487$	−7.70139e11	−0.620425	−0.310212	+	0.950667i	$0.600400\pi$
$488$	0	0
$489$	1.28125e12	1.01332
$490$	0	0
$491$	−7.56286e11	−0.587245	−0.293623	−	0.955921i	$-0.594861\pi$
$491$	−7.56286e11	−0.587245	−0.293623	+	0.955921i	$0.594861\pi$
$492$	0	0
$493$	7.67585e10	0.0585215
$494$	0	0
$495$	−4.76691e11	−0.356873
$496$	0	0
$497$	1.48162e12	1.08926
$498$	0	0
$499$	−1.93577e12	−1.39766	−0.698831	−	0.715287i	$-0.746296\pi$
$499$	−1.93577e12	−1.39766	−0.698831	+	0.715287i	$0.746296\pi$
$500$	0	0
$501$	−7.37757e11	−0.523171
$502$	0	0
$503$	−3.11273e11	−0.216813	−0.108406	−	0.994107i	$-0.534575\pi$
$503$	−3.11273e11	−0.216813	−0.108406	+	0.994107i	$0.534575\pi$
$504$	0	0
$505$	2.41312e12	1.65108
$506$	0	0
$507$	8.25959e11	0.555166
$508$	0	0
$509$	−1.24806e12	−0.824147	−0.412073	−	0.911151i	$-0.635195\pi$
$509$	−1.24806e12	−0.824147	−0.412073	+	0.911151i	$0.635195\pi$
$510$	0	0
$511$	3.20525e11	0.207954
$512$	0	0
$513$	4.62809e11	0.295035
$514$	0	0
$515$	2.60821e12	1.63384
$516$	0	0
$517$	2.41494e12	1.48662
$518$	0	0
$519$	−1.08003e12	−0.653405
$520$	0	0
$521$	−4.11775e11	−0.244844	−0.122422	−	0.992478i	$-0.539066\pi$
$521$	−4.11775e11	−0.244844	−0.122422	+	0.992478i	$0.539066\pi$
$522$	0	0
$523$	−1.28721e12	−0.752303	−0.376152	−	0.926558i	$-0.622753\pi$
$523$	−1.28721e12	−0.752303	−0.376152	+	0.926558i	$0.622753\pi$
$524$	0	0
$525$	−7.32485e11	−0.420806
$526$	0	0
$527$	−5.22436e11	−0.295043
$528$	0	0
$529$	8.01585e11	0.445040
$530$	0	0
$531$	−8.92280e11	−0.487052
$532$	0	0
$533$	4.77288e11	0.256158
$534$	0	0
$535$	4.40359e11	0.232389
$536$	0	0
$537$	7.58649e11	0.393692
$538$	0	0
$539$	3.89614e12	1.98831
$540$	0	0
$541$	−7.91782e11	−0.397391	−0.198695	−	0.980061i	$-0.563670\pi$
$541$	−7.91782e11	−0.397391	−0.198695	+	0.980061i	$0.563670\pi$
$542$	0	0
$543$	3.41299e11	0.168475
$544$	0	0
$545$	4.31128e12	2.09326
$546$	0	0
$547$	−1.11335e12	−0.531726	−0.265863	−	0.964011i	$-0.585657\pi$
$547$	−1.11335e12	−0.531726	−0.265863	+	0.964011i	$0.585657\pi$
$548$	0	0
$549$	1.44718e11	0.0679904
$550$	0	0
$551$	−6.19537e11	−0.286342
$552$	0	0
$553$	5.87043e12	2.66936
$554$	0	0
$555$	−4.75823e10	−0.0212876
$556$	0	0
$557$	−2.80784e12	−1.23601	−0.618007	−	0.786173i	$-0.712060\pi$
$557$	−2.80784e12	−1.23601	−0.618007	+	0.786173i	$0.712060\pi$
$558$	0	0
$559$	−1.06669e11	−0.0462045
$560$	0	0
$561$	−3.83007e11	−0.163258
$562$	0	0
$563$	1.38686e12	0.581760	0.290880	−	0.956760i	$-0.406052\pi$
$563$	1.38686e12	0.581760	0.290880	+	0.956760i	$0.406052\pi$
$564$	0	0
$565$	3.74156e11	0.154467
$566$	0	0
$567$	4.89403e11	0.198858
$568$	0	0
$569$	1.20199e12	0.480722	0.240361	−	0.970684i	$-0.422734\pi$
$569$	1.20199e12	0.480722	0.240361	+	0.970684i	$0.422734\pi$
$570$	0	0
$571$	−7.37383e11	−0.290289	−0.145145	−	0.989410i	$-0.546365\pi$
$571$	−7.37383e11	−0.290289	−0.145145	+	0.989410i	$0.546365\pi$
$572$	0	0
$573$	−2.45207e12	−0.950248
$574$	0	0
$575$	−1.28322e12	−0.489550
$576$	0	0
$577$	−1.69532e12	−0.636739	−0.318369	−	0.947967i	$-0.603135\pi$
$577$	−1.69532e12	−0.636739	−0.318369	+	0.947967i	$0.603135\pi$
$578$	0	0
$579$	−9.42330e11	−0.348457
$580$	0	0
$581$	6.20158e12	2.25793
$582$	0	0
$583$	−2.32900e12	−0.834952
$584$	0	0
$585$	−2.19567e11	−0.0775115
$586$	0	0
$587$	3.27170e12	1.13737	0.568685	−	0.822555i	$-0.307452\pi$
$587$	3.27170e12	1.13737	0.568685	+	0.822555i	$0.307452\pi$
$588$	0	0
$589$	4.21671e12	1.44363
$590$	0	0
$591$	−1.53544e12	−0.517713
$592$	0	0
$593$	−3.07695e12	−1.02182	−0.510911	−	0.859634i	$-0.670692\pi$
$593$	−3.07695e12	−1.02182	−0.510911	+	0.859634i	$0.670692\pi$
$594$	0	0
$595$	−2.03368e12	−0.665206
$596$	0	0
$597$	−2.13260e12	−0.687107
$598$	0	0
$599$	9.53487e11	0.302617	0.151309	−	0.988487i	$-0.451651\pi$
$599$	9.53487e11	0.302617	0.151309	+	0.988487i	$0.451651\pi$
$600$	0	0
$601$	2.02727e12	0.633835	0.316918	−	0.948453i	$-0.397352\pi$
$601$	2.02727e12	0.633835	0.316918	+	0.948453i	$0.397352\pi$
$602$	0	0
$603$	7.88231e11	0.242787
$604$	0	0
$605$	7.25094e11	0.220037
$606$	0	0
$607$	−6.36168e12	−1.90205	−0.951027	−	0.309108i	$-0.899970\pi$
$607$	−6.36168e12	−1.90205	−0.951027	+	0.309108i	$0.899970\pi$
$608$	0	0
$609$	−6.55138e11	−0.192999
$610$	0	0
$611$	1.11234e12	0.322889
$612$	0	0
$613$	5.17898e12	1.48140	0.740699	−	0.671837i	$-0.234494\pi$
$613$	5.17898e12	1.48140	0.740699	+	0.671837i	$0.234494\pi$
$614$	0	0
$615$	3.17518e12	0.895015
$616$	0	0
$617$	−1.62227e11	−0.0450650	−0.0225325	−	0.999746i	$-0.507173\pi$
$617$	−1.62227e11	−0.0450650	−0.0225325	+	0.999746i	$0.507173\pi$
$618$	0	0
$619$	3.57678e12	0.979228	0.489614	−	0.871939i	$-0.337138\pi$
$619$	3.57678e12	0.979228	0.489614	+	0.871939i	$0.337138\pi$
$620$	0	0
$621$	8.57374e11	0.231344
$622$	0	0
$623$	−1.53177e12	−0.407376
$624$	0	0
$625$	−4.73555e12	−1.24140
$626$	0	0
$627$	3.09135e12	0.798811
$628$	0	0
$629$	−3.82310e10	−0.00973841
$630$	0	0
$631$	−2.62455e12	−0.659056	−0.329528	−	0.944146i	$-0.606890\pi$
$631$	−2.62455e12	−0.659056	−0.329528	+	0.944146i	$0.606890\pi$
$632$	0	0
$633$	4.35992e12	1.07935
$634$	0	0
$635$	8.33172e12	2.03354
$636$	0	0
$637$	1.79459e12	0.431855
$638$	0	0
$639$	8.55025e11	0.202873
$640$	0	0
$641$	−6.39607e12	−1.49642	−0.748208	−	0.663464i	$-0.769085\pi$
$641$	−6.39607e12	−1.49642	−0.748208	+	0.663464i	$0.769085\pi$
$642$	0	0
$643$	7.61888e12	1.75769	0.878844	−	0.477110i	$-0.158316\pi$
$643$	7.61888e12	1.75769	0.878844	+	0.477110i	$0.158316\pi$
$644$	0	0
$645$	−7.09618e11	−0.161438
$646$	0	0
$647$	2.61652e12	0.587022	0.293511	−	0.955956i	$-0.405176\pi$
$647$	2.61652e12	0.587022	0.293511	+	0.955956i	$0.405176\pi$
$648$	0	0
$649$	−5.96002e12	−1.31870
$650$	0	0
$651$	4.45902e12	0.973027
$652$	0	0
$653$	−1.80837e12	−0.389204	−0.194602	−	0.980882i	$-0.562341\pi$
$653$	−1.80837e12	−0.389204	−0.194602	+	0.980882i	$0.562341\pi$
$654$	0	0
$655$	3.04441e12	0.646275
$656$	0	0
$657$	1.84971e11	0.0387312
$658$	0	0
$659$	2.09519e12	0.432752	0.216376	−	0.976310i	$-0.430576\pi$
$659$	2.09519e12	0.432752	0.216376	+	0.976310i	$0.430576\pi$
$660$	0	0
$661$	3.17340e12	0.646573	0.323287	−	0.946301i	$-0.395212\pi$
$661$	3.17340e12	0.646573	0.323287	+	0.946301i	$0.395212\pi$
$662$	0	0
$663$	−1.76416e11	−0.0354591
$664$	0	0
$665$	1.64143e13	3.25481
$666$	0	0
$667$	−1.14772e12	−0.224528
$668$	0	0
$669$	2.13622e12	0.412315
$670$	0	0
$671$	9.66650e11	0.184085
$672$	0	0
$673$	−6.96514e11	−0.130877	−0.0654383	−	0.997857i	$-0.520845\pi$
$673$	−6.96514e11	−0.130877	−0.0654383	+	0.997857i	$0.520845\pi$
$674$	0	0
$675$	−4.22710e11	−0.0783746
$676$	0	0
$677$	−1.28037e12	−0.234254	−0.117127	−	0.993117i	$-0.537369\pi$
$677$	−1.28037e12	−0.234254	−0.117127	+	0.993117i	$0.537369\pi$
$678$	0	0
$679$	−1.47393e13	−2.66110
$680$	0	0
$681$	−7.81862e11	−0.139305
$682$	0	0
$683$	−4.70707e12	−0.827671	−0.413835	−	0.910352i	$-0.635811\pi$
$683$	−4.70707e12	−0.827671	−0.413835	+	0.910352i	$0.635811\pi$
$684$	0	0
$685$	−8.44732e12	−1.46592
$686$	0	0
$687$	−3.79802e12	−0.650507
$688$	0	0
$689$	−1.07276e12	−0.181349
$690$	0	0
$691$	−6.34032e12	−1.05794	−0.528969	−	0.848641i	$-0.677421\pi$
$691$	−6.34032e12	−1.05794	−0.528969	+	0.848641i	$0.677421\pi$
$692$	0	0
$693$	3.26899e12	0.538411
$694$	0	0
$695$	−7.01145e12	−1.13993
$696$	0	0
$697$	2.55117e12	0.409441
$698$	0	0
$699$	7.01338e11	0.111117
$700$	0	0
$701$	3.74948e12	0.586463	0.293231	−	0.956042i	$-0.405269\pi$
$701$	3.74948e12	0.586463	0.293231	+	0.956042i	$0.405269\pi$
$702$	0	0
$703$	3.08572e11	0.0476495
$704$	0	0
$705$	7.39990e12	1.12817
$706$	0	0
$707$	−1.65484e13	−2.49096
$708$	0	0
$709$	5.97312e12	0.887755	0.443878	−	0.896087i	$-0.353602\pi$
$709$	5.97312e12	0.887755	0.443878	+	0.896087i	$0.353602\pi$
$710$	0	0
$711$	3.38776e12	0.497164
$712$	0	0
$713$	7.81165e12	1.13198
$714$	0	0
$715$	−1.46661e12	−0.209863
$716$	0	0
$717$	−2.55661e11	−0.0361266
$718$	0	0
$719$	1.63377e12	0.227988	0.113994	−	0.993481i	$-0.463636\pi$
$719$	1.63377e12	0.227988	0.113994	+	0.993481i	$0.463636\pi$
$720$	0	0
$721$	−1.78862e13	−2.46496
$722$	0	0
$723$	−5.64637e12	−0.768505
$724$	0	0
$725$	5.65858e11	0.0760653
$726$	0	0
$727$	−4.50480e12	−0.598096	−0.299048	−	0.954238i	$-0.596669\pi$
$727$	−4.50480e12	−0.598096	−0.299048	+	0.954238i	$0.596669\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−5.70158e11	−0.0738529
$732$	0	0
$733$	−2.37412e12	−0.303763	−0.151882	−	0.988399i	$-0.548533\pi$
$733$	−2.37412e12	−0.303763	−0.151882	+	0.988399i	$0.548533\pi$
$734$	0	0
$735$	1.19386e13	1.50890
$736$	0	0
$737$	5.26502e12	0.657349
$738$	0	0
$739$	6.70363e12	0.826818	0.413409	−	0.910545i	$-0.364338\pi$
$739$	6.70363e12	0.826818	0.413409	+	0.910545i	$0.364338\pi$
$740$	0	0
$741$	1.42390e12	0.173499
$742$	0	0
$743$	5.57339e12	0.670919	0.335459	−	0.942055i	$-0.391108\pi$
$743$	5.57339e12	0.670919	0.335459	+	0.942055i	$0.391108\pi$
$744$	0	0
$745$	−6.87153e12	−0.817241
$746$	0	0
$747$	3.57887e12	0.420536
$748$	0	0
$749$	−3.01984e12	−0.350603
$750$	0	0
$751$	−1.03177e13	−1.18359	−0.591796	−	0.806088i	$-0.701581\pi$
$751$	−1.03177e13	−1.18359	−0.591796	+	0.806088i	$0.701581\pi$
$752$	0	0
$753$	6.86039e12	0.777627
$754$	0	0
$755$	6.17344e12	0.691458
$756$	0	0
$757$	9.61031e12	1.06367	0.531834	−	0.846849i	$-0.321503\pi$
$757$	9.61031e12	1.06367	0.531834	+	0.846849i	$0.321503\pi$
$758$	0	0
$759$	5.72686e12	0.626367
$760$	0	0
$761$	1.39212e13	1.50468	0.752341	−	0.658774i	$-0.228925\pi$
$761$	1.39212e13	1.50468	0.752341	+	0.658774i	$0.228925\pi$
$762$	0	0
$763$	−2.95654e13	−3.15808
$764$	0	0
$765$	−1.17362e12	−0.123894
$766$	0	0
$767$	−2.74523e12	−0.286417
$768$	0	0
$769$	−6.25086e12	−0.644572	−0.322286	−	0.946642i	$-0.604451\pi$
$769$	−6.25086e12	−0.644572	−0.322286	+	0.946642i	$0.604451\pi$
$770$	0	0
$771$	1.06955e13	1.09007
$772$	0	0
$773$	−1.45050e13	−1.46120	−0.730601	−	0.682805i	$-0.760760\pi$
$773$	−1.45050e13	−1.46120	−0.730601	+	0.682805i	$0.760760\pi$
$774$	0	0
$775$	−3.85136e12	−0.383493
$776$	0	0
$777$	3.26304e11	0.0321165
$778$	0	0
$779$	−2.05911e13	−2.00337
$780$	0	0
$781$	5.71117e12	0.549282
$782$	0	0
$783$	−3.78073e11	−0.0359458
$784$	0	0
$785$	−1.24006e13	−1.16555
$786$	0	0
$787$	1.63389e13	1.51823	0.759115	−	0.650957i	$-0.225632\pi$
$787$	1.63389e13	1.51823	0.759115	+	0.650957i	$0.225632\pi$
$788$	0	0
$789$	5.57084e12	0.511770
$790$	0	0
$791$	−2.56584e12	−0.233042
$792$	0	0
$793$	4.45246e11	0.0399826
$794$	0	0
$795$	−7.13656e12	−0.633632
$796$	0	0
$797$	−1.33111e13	−1.16856	−0.584281	−	0.811551i	$-0.698623\pi$
$797$	−1.33111e13	−1.16856	−0.584281	+	0.811551i	$0.698623\pi$
$798$	0	0
$799$	5.94561e12	0.516103
$800$	0	0
$801$	−8.83966e11	−0.0758733
$802$	0	0
$803$	1.23552e12	0.104865
$804$	0	0
$805$	3.04083e13	2.55218
$806$	0	0
$807$	9.66293e12	0.802006
$808$	0	0
$809$	−7.94697e12	−0.652279	−0.326139	−	0.945322i	$-0.605748\pi$
$809$	−7.94697e12	−0.652279	−0.326139	+	0.945322i	$0.605748\pi$
$810$	0	0
$811$	−5.03198e12	−0.408456	−0.204228	−	0.978923i	$-0.565468\pi$
$811$	−5.03198e12	−0.408456	−0.204228	+	0.978923i	$0.565468\pi$
$812$	0	0
$813$	−1.30974e13	−1.05142
$814$	0	0
$815$	2.62240e13	2.08205
$816$	0	0
$817$	4.60189e12	0.361358
$818$	0	0
$819$	1.50572e12	0.116941
$820$	0	0
$821$	−2.35468e13	−1.80879	−0.904394	−	0.426698i	$-0.859677\pi$
$821$	−2.35468e13	−1.80879	−0.904394	+	0.426698i	$0.859677\pi$
$822$	0	0
$823$	7.17638e12	0.545263	0.272632	−	0.962118i	$-0.412106\pi$
$823$	7.17638e12	0.545263	0.272632	+	0.962118i	$0.412106\pi$
$824$	0	0
$825$	−2.82351e12	−0.212200
$826$	0	0
$827$	2.31380e13	1.72009	0.860043	−	0.510221i	$-0.170437\pi$
$827$	2.31380e13	1.72009	0.860043	+	0.510221i	$0.170437\pi$
$828$	0	0
$829$	−5.33375e10	−0.00392227	−0.00196113	−	0.999998i	$-0.500624\pi$
$829$	−5.33375e10	−0.00392227	−0.00196113	+	0.999998i	$0.500624\pi$
$830$	0	0
$831$	−5.66307e12	−0.411953
$832$	0	0
$833$	9.59232e12	0.690273
$834$	0	0
$835$	−1.51001e13	−1.07495
$836$	0	0
$837$	2.57325e12	0.181225
$838$	0	0
$839$	1.67763e13	1.16888	0.584438	−	0.811438i	$-0.301315\pi$
$839$	1.67763e13	1.16888	0.584438	+	0.811438i	$0.301315\pi$
$840$	0	0
$841$	−1.40010e13	−0.965113
$842$	0	0
$843$	1.27740e13	0.871168
$844$	0	0
$845$	1.69053e13	1.14069
$846$	0	0
$847$	−4.97246e12	−0.331968
$848$	0	0
$849$	−9.86963e12	−0.651953
$850$	0	0
$851$	5.71644e11	0.0373631
$852$	0	0
$853$	−1.88628e13	−1.21993	−0.609967	−	0.792427i	$-0.708817\pi$
$853$	−1.88628e13	−1.21993	−0.609967	+	0.792427i	$0.708817\pi$
$854$	0	0
$855$	9.47255e12	0.606205
$856$	0	0
$857$	−5.68742e12	−0.360165	−0.180083	−	0.983651i	$-0.557637\pi$
$857$	−5.68742e12	−0.360165	−0.180083	+	0.983651i	$0.557637\pi$
$858$	0	0
$859$	2.23768e12	0.140226	0.0701131	−	0.997539i	$-0.477664\pi$
$859$	2.23768e12	0.140226	0.0701131	+	0.997539i	$0.477664\pi$
$860$	0	0
$861$	−2.17744e13	−1.35030
$862$	0	0
$863$	−1.97051e13	−1.20929	−0.604645	−	0.796495i	$-0.706685\pi$
$863$	−1.97051e13	−1.20929	−0.604645	+	0.796495i	$0.706685\pi$
$864$	0	0
$865$	−2.21055e13	−1.34254
$866$	0	0
$867$	8.66265e12	0.520673
$868$	0	0
$869$	2.26287e13	1.34608
$870$	0	0
$871$	2.42511e12	0.142774
$872$	0	0
$873$	−8.50587e12	−0.495627
$874$	0	0
$875$	2.18213e13	1.25848
$876$	0	0
$877$	−3.77310e12	−0.215377	−0.107689	−	0.994185i	$-0.534345\pi$
$877$	−3.77310e12	−0.215377	−0.107689	+	0.994185i	$0.534345\pi$
$878$	0	0
$879$	1.17798e13	0.665559
$880$	0	0
$881$	1.71618e13	0.959778	0.479889	−	0.877329i	$-0.340677\pi$
$881$	1.71618e13	0.959778	0.479889	+	0.877329i	$0.340677\pi$
$882$	0	0
$883$	−1.80325e13	−0.998233	−0.499117	−	0.866535i	$-0.666342\pi$
$883$	−1.80325e13	−0.998233	−0.499117	+	0.866535i	$0.666342\pi$
$884$	0	0
$885$	−1.82627e13	−1.00074
$886$	0	0
$887$	2.06193e13	1.11845	0.559226	−	0.829015i	$-0.311098\pi$
$887$	2.06193e13	1.11845	0.559226	+	0.829015i	$0.311098\pi$
$888$	0	0
$889$	−5.71362e13	−3.06799
$890$	0	0
$891$	1.88650e12	0.100278
$892$	0	0
$893$	−4.79885e13	−2.52526
$894$	0	0
$895$	1.55277e13	0.808914
$896$	0	0
$897$	2.63783e12	0.136045
$898$	0	0
$899$	−3.44467e12	−0.175885
$900$	0	0
$901$	−5.73402e12	−0.289866
$902$	0	0
$903$	4.86633e12	0.243560
$904$	0	0
$905$	6.98554e12	0.346163
$906$	0	0
$907$	−1.34788e13	−0.661330	−0.330665	−	0.943748i	$-0.607273\pi$
$907$	−1.34788e13	−0.661330	−0.330665	+	0.943748i	$0.607273\pi$
$908$	0	0
$909$	−9.54988e12	−0.463939
$910$	0	0
$911$	−2.00594e13	−0.964908	−0.482454	−	0.875921i	$-0.660254\pi$
$911$	−2.00594e13	−0.964908	−0.482454	+	0.875921i	$0.660254\pi$
$912$	0	0
$913$	2.39052e13	1.13861
$914$	0	0
$915$	2.96202e12	0.139699
$916$	0	0
$917$	−2.08776e13	−0.975031
$918$	0	0
$919$	2.12498e13	0.982734	0.491367	−	0.870953i	$-0.336497\pi$
$919$	2.12498e13	0.982734	0.491367	+	0.870953i	$0.336497\pi$
$920$	0	0
$921$	2.20562e13	1.01009
$922$	0	0
$923$	2.63061e12	0.119302
$924$	0	0
$925$	−2.81837e11	−0.0126578
$926$	0	0
$927$	−1.03220e13	−0.459095
$928$	0	0
$929$	−4.24671e13	−1.87060	−0.935302	−	0.353850i	$-0.884872\pi$
$929$	−4.24671e13	−1.87060	−0.935302	+	0.353850i	$0.884872\pi$
$930$	0	0
$931$	−7.74220e13	−3.37746
$932$	0	0
$933$	−4.89839e12	−0.211634
$934$	0	0
$935$	−7.83921e12	−0.335444
$936$	0	0
$937$	3.31352e13	1.40430	0.702152	−	0.712028i	$-0.252223\pi$
$937$	3.31352e13	1.40430	0.702152	+	0.712028i	$0.252223\pi$
$938$	0	0
$939$	2.33667e13	0.980851
$940$	0	0
$941$	−1.46081e13	−0.607353	−0.303677	−	0.952775i	$-0.598214\pi$
$941$	−1.46081e13	−0.607353	−0.303677	+	0.952775i	$0.598214\pi$
$942$	0	0
$943$	−3.81460e13	−1.57089
$944$	0	0
$945$	1.00169e13	0.408591
$946$	0	0
$947$	2.39736e13	0.968633	0.484316	−	0.874893i	$-0.339068\pi$
$947$	2.39736e13	0.968633	0.484316	+	0.874893i	$0.339068\pi$
$948$	0	0
$949$	5.69091e11	0.0227763
$950$	0	0
$951$	−1.56754e13	−0.621449
$952$	0	0
$953$	−2.97904e13	−1.16993	−0.584963	−	0.811060i	$-0.698891\pi$
$953$	−2.97904e13	−1.16993	−0.584963	+	0.811060i	$0.698891\pi$
$954$	0	0
$955$	−5.01878e13	−1.95246
$956$	0	0
$957$	−2.52535e12	−0.0973237
$958$	0	0
$959$	5.79289e13	2.21163
$960$	0	0
$961$	−2.99437e12	−0.113253
$962$	0	0
$963$	−1.74272e12	−0.0652993
$964$	0	0
$965$	−1.92871e13	−0.715970
$966$	0	0
$967$	5.59844e12	0.205896	0.102948	−	0.994687i	$-0.467172\pi$
$967$	5.59844e12	0.205896	0.102948	+	0.994687i	$0.467172\pi$
$968$	0	0
$969$	7.61092e12	0.277320
$970$	0	0
$971$	−2.45423e13	−0.885989	−0.442995	−	0.896524i	$-0.646084\pi$
$971$	−2.45423e13	−0.885989	−0.442995	+	0.896524i	$0.646084\pi$
$972$	0	0
$973$	4.80822e13	1.71980
$974$	0	0
$975$	−1.30053e12	−0.0460891
$976$	0	0
$977$	−4.05611e12	−0.142424	−0.0712121	−	0.997461i	$-0.522687\pi$
$977$	−4.05611e12	−0.142424	−0.0712121	+	0.997461i	$0.522687\pi$
$978$	0	0
$979$	−5.90448e12	−0.205428
$980$	0	0
$981$	−1.70619e13	−0.588188
$982$	0	0
$983$	−1.13646e13	−0.388206	−0.194103	−	0.980981i	$-0.562180\pi$
$983$	−1.13646e13	−0.388206	−0.194103	+	0.980981i	$0.562180\pi$
$984$	0	0
$985$	−3.14266e13	−1.06374
$986$	0	0
$987$	−5.07461e13	−1.70206
$988$	0	0
$989$	8.52521e12	0.283349
$990$	0	0
$991$	−2.34764e13	−0.773213	−0.386607	−	0.922245i	$-0.626353\pi$
$991$	−2.34764e13	−0.773213	−0.386607	+	0.922245i	$0.626353\pi$
$992$	0	0
$993$	7.26404e12	0.237086
$994$	0	0
$995$	−4.36489e13	−1.41179
$996$	0	0
$997$	−1.70375e13	−0.546108	−0.273054	−	0.961999i	$-0.588034\pi$
$997$	−1.70375e13	−0.546108	−0.273054	+	0.961999i	$0.588034\pi$
$998$	0	0
$999$	1.88307e11	0.00598165

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.a.a.1.2	✓	4
4.3	odd	2		384.10.a.e.1.2	yes	4
8.3	odd	2		384.10.a.d.1.3	yes	4
8.5	even	2		384.10.a.h.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.a.a.1.2	✓	4	1.1	even	1	trivial
384.10.a.d.1.3	yes	4	8.3	odd	2
384.10.a.e.1.2	yes	4	4.3	odd	2
384.10.a.h.1.3	yes	4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q-81.0000 q^{3} -1657.87 q^{5} +11369.1 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q-81.0000 q^{3} -1657.87 q^{5} +11369.1 q^{7} +6561.00 q^{9} +43824.4 q^{11} +20185.9 q^{13} +134287. q^{15} +107896. q^{17} -870857. q^{19} -920899. q^{21} -1.61330e6 q^{23} +795403. q^{25} -531441. q^{27} +711411. q^{29} -4.84203e6 q^{31} -3.54978e6 q^{33} -1.88485e7 q^{35} -354332. q^{37} -1.63505e6 q^{39} +2.36447e7 q^{41} -5.28433e6 q^{43} -1.08773e7 q^{45} +5.51050e7 q^{47} +8.89033e7 q^{49} -8.73958e6 q^{51} -5.31439e7 q^{53} -7.26552e7 q^{55} +7.05394e7 q^{57} -1.35998e8 q^{59} +2.20573e7 q^{61} +7.45928e7 q^{63} -3.34655e7 q^{65} +1.20139e8 q^{67} +1.30677e8 q^{69} +1.30319e8 q^{71} +2.81926e7 q^{73} -6.44276e7 q^{75} +4.98245e8 q^{77} +5.16349e8 q^{79} +4.30467e7 q^{81} +5.45476e8 q^{83} -1.78878e8 q^{85} -5.76243e7 q^{87} -1.34730e8 q^{89} +2.29495e8 q^{91} +3.92204e8 q^{93} +1.44377e9 q^{95} -1.29643e9 q^{97} +2.87532e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 324 q^{3} - 1728 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) 4 * q - 324 * q^3 - 1728 * q^5 + 4840 * q^7 + 26244 * q^9	\( 4 q - 324 q^{3} - 1728 q^{5} + 4840 q^{7} + 26244 q^{9} - 15824 q^{11} - 82440 q^{13} + 139968 q^{15} + 165912 q^{17} - 539904 q^{19} - 392040 q^{21} + 729680 q^{23} + 224812 q^{25} - 2125764 q^{27} - 1850864 q^{29} + 3197960 q^{31} + 1281744 q^{33} - 8574912 q^{35} - 4187992 q^{37} + 6677640 q^{39} + 227704 q^{41} - 1600352 q^{43} - 11337408 q^{45} + 18053904 q^{47} + 57728820 q^{49} - 13438872 q^{51} - 29418288 q^{53} + 45906816 q^{55} + 43732224 q^{57} - 38300048 q^{59} + 99764648 q^{61} + 31755240 q^{63} + 314120832 q^{65} + 183717008 q^{67} - 59104080 q^{69} + 181868080 q^{71} + 254539160 q^{73} - 18209772 q^{75} + 230564704 q^{77} + 831578184 q^{79} + 172186884 q^{81} + 687923952 q^{83} - 1041391104 q^{85} + 149919984 q^{87} + 627627272 q^{89} + 1018147632 q^{91} - 259034760 q^{93} + 2167118208 q^{95} - 889385880 q^{97} - 103821264 q^{99}+O(q^{100}) \) 4 * q - 324 * q^3 - 1728 * q^5 + 4840 * q^7 + 26244 * q^9 - 15824 * q^11 - 82440 * q^13 + 139968 * q^15 + 165912 * q^17 - 539904 * q^19 - 392040 * q^21 + 729680 * q^23 + 224812 * q^25 - 2125764 * q^27 - 1850864 * q^29 + 3197960 * q^31 + 1281744 * q^33 - 8574912 * q^35 - 4187992 * q^37 + 6677640 * q^39 + 227704 * q^41 - 1600352 * q^43 - 11337408 * q^45 + 18053904 * q^47 + 57728820 * q^49 - 13438872 * q^51 - 29418288 * q^53 + 45906816 * q^55 + 43732224 * q^57 - 38300048 * q^59 + 99764648 * q^61 + 31755240 * q^63 + 314120832 * q^65 + 183717008 * q^67 - 59104080 * q^69 + 181868080 * q^71 + 254539160 * q^73 - 18209772 * q^75 + 230564704 * q^77 + 831578184 * q^79 + 172186884 * q^81 + 687923952 * q^83 - 1041391104 * q^85 + 149919984 * q^87 + 627627272 * q^89 + 1018147632 * q^91 - 259034760 * q^93 + 2167118208 * q^95 - 889385880 * q^97 - 103821264 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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