LMFDB - Embedded newform 384.5.e.b.257.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(257,384)]

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, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("384.257");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.e (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:	$39.6940658242$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 32 x^{14} + 356 x^{13} + 1348 x^{12} - 8992 x^{11} + 22064 x^{10} + \cdots + 21479188203 $ x^16 - 4x^15 - 32x^14 + 356x^13 + 1348x^12 - 8992x^11 + 22064x^10 + 391324x^9 + 724325x^8 - 2262056x^7 + 45109352x^6 + 210288484x^5 + 385003542x^4 + 866573964x^3 + 12462543468x^2 + 25672770504*x + 21479188203
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{54}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.2
Root			$-5.66644 + 2.02063i$ of defining polynomial
Character	$\chi$	$=$	384.257
Dual form			384.5.e.b.257.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.99926 + 0.115244i) q^{3} -25.0292i q^{5} -59.0069 q^{7} +(80.9734 - 2.07423i) q^{9} +O(q^{10})$	\(q+(-8.99926 + 0.115244i) q^{3} -25.0292i q^{5} -59.0069 q^{7} +(80.9734 - 2.07423i) q^{9} +219.418i q^{11} +0.112775 q^{13} +(2.88447 + 225.244i) q^{15} -160.826i q^{17} +592.455 q^{19} +(531.019 - 6.80020i) q^{21} +11.6137i q^{23} -1.46145 q^{25} +(-728.462 + 27.9982i) q^{27} +1261.19i q^{29} -217.007 q^{31} +(-25.2867 - 1974.60i) q^{33} +1476.90i q^{35} -1940.21 q^{37} +(-1.01489 + 0.0129966i) q^{39} -1163.10i q^{41} -1084.69 q^{43} +(-51.9162 - 2026.70i) q^{45} -3114.40i q^{47} +1080.81 q^{49} +(18.5342 + 1447.31i) q^{51} -4799.21i q^{53} +5491.86 q^{55} +(-5331.66 + 68.2770i) q^{57} +4250.36i q^{59} +3336.81 q^{61} +(-4777.99 + 122.394i) q^{63} -2.82266i q^{65} +2642.37 q^{67} +(-1.33841 - 104.514i) q^{69} -6539.46i q^{71} +4384.38 q^{73} +(13.1520 - 0.168424i) q^{75} -12947.2i q^{77} -4151.19 q^{79} +(6552.40 - 335.914i) q^{81} -6495.95i q^{83} -4025.34 q^{85} +(-145.345 - 11349.8i) q^{87} -8685.42i q^{89} -6.65449 q^{91} +(1952.90 - 25.0088i) q^{93} -14828.7i q^{95} +274.827 q^{97} +(455.123 + 17767.0i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{3} + 80 q^{7}+O(q^{10}) $ 16 * q - 8 * q^3 + 80 * q^7	$ 16 q - 8 q^{3} + 80 q^{7} + 416 q^{15} + 816 q^{19} + 608 q^{21} - 2000 q^{25} + 280 q^{27} + 592 q^{31} - 496 q^{33} + 2240 q^{37} - 16 q^{39} + 368 q^{43} + 800 q^{45} + 3984 q^{49} - 352 q^{51} + 1920 q^{55} + 560 q^{57} - 3520 q^{61} - 816 q^{63} - 3536 q^{67} - 10784 q^{69} + 3680 q^{73} + 5112 q^{75} - 14448 q^{79} - 624 q^{81} - 11136 q^{85} - 14944 q^{87} - 22944 q^{91} + 13760 q^{93} + 3264 q^{97} - 26976 q^{99}+O(q^{100}) $ 16 * q - 8 * q^3 + 80 * q^7 + 416 * q^15 + 816 * q^19 + 608 * q^21 - 2000 * q^25 + 280 * q^27 + 592 * q^31 - 496 * q^33 + 2240 * q^37 - 16 * q^39 + 368 * q^43 + 800 * q^45 + 3984 * q^49 - 352 * q^51 + 1920 * q^55 + 560 * q^57 - 3520 * q^61 - 816 * q^63 - 3536 * q^67 - 10784 * q^69 + 3680 * q^73 + 5112 * q^75 - 14448 * q^79 - 624 * q^81 - 11136 * q^85 - 14944 * q^87 - 22944 * q^91 + 13760 * q^93 + 3264 * q^97 - 26976 * q^99

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
$3$	−8.99926	+	0.115244i	−0.999918	+	0.0128049i
$3$	−8.99926	+	0.115244i	−0.999918	+	0.0128049i
$4$		0			0
$5$		−	25.0292i		−	1.00117i	−0.865688	−	0.500584i	$-0.833119\pi$
$5$		−	25.0292i		−	1.00117i	0.865688	−	0.500584i	$-0.166881\pi$
$6$		0			0
$7$	−59.0069			−1.20422			−0.602111	−	0.798412i	$-0.705674\pi$
$7$	−59.0069			−1.20422			−0.602111	+	0.798412i	$0.705674\pi$
$8$		0			0
$9$	80.9734	−	2.07423i	0.999672	−	0.0256077i
$10$		0			0
$11$			219.418i			1.81337i	0.421805	+	0.906687i	$0.361397\pi$
$11$			219.418i			1.81337i	−0.421805	+	0.906687i	$0.638603\pi$
$12$		0			0
$13$	0.112775			0.000667306			0.000333653	−	1.00000i	$-0.499894\pi$
$13$	0.112775			0.000667306			0.000333653		1.00000i	$0.499894\pi$
$14$		0			0
$15$	2.88447	+	225.244i	0.0128199	+	1.00109i
$16$		0			0
$17$		−	160.826i		−	0.556491i	−0.960510	−	0.278245i	$-0.910247\pi$
$17$		−	160.826i		−	0.556491i	0.960510	−	0.278245i	$-0.0897528\pi$
$18$		0			0
$19$	592.455			1.64115			0.820575	−	0.571539i	$-0.193653\pi$
$19$	592.455			1.64115			0.820575	+	0.571539i	$0.193653\pi$
$20$		0			0
$21$	531.019	−	6.80020i	1.20412	−	0.0154200i
$22$		0			0
$23$			11.6137i			0.0219540i	0.999940	+	0.0109770i	$0.00349416\pi$
$23$			11.6137i			0.0219540i	−0.999940	+	0.0109770i	$0.996506\pi$
$24$		0			0
$25$	−1.46145			−0.00233833
$26$		0			0
$27$	−728.462	+	27.9982i	−0.999262	+	0.0384063i
$28$		0			0
$29$			1261.19i			1.49963i	0.661648	+	0.749815i	$0.269857\pi$
$29$			1261.19i			1.49963i	−0.661648	+	0.749815i	$0.730143\pi$
$30$		0			0
$31$	−217.007			−0.225813			−0.112907	−	0.993606i	$-0.536016\pi$
$31$	−217.007			−0.225813			−0.112907	+	0.993606i	$0.536016\pi$
$32$		0			0
$33$	−25.2867	−	1974.60i	−0.0232201	−	1.81322i
$34$		0			0
$35$			1476.90i			1.20563i
$36$		0			0
$37$	−1940.21			−1.41725			−0.708624	−	0.705586i	$-0.750684\pi$
$37$	−1940.21			−1.41725			−0.708624	+	0.705586i	$0.750684\pi$
$38$		0			0
$39$	−1.01489	+	0.0129966i	−0.000667252	+	8.54480e-6i
$40$		0			0
$41$		−	1163.10i		−	0.691912i	−0.938251	−	0.345956i	$-0.887555\pi$
$41$		−	1163.10i		−	0.691912i	0.938251	−	0.345956i	$-0.112445\pi$
$42$		0			0
$43$	−1084.69			−0.586634			−0.293317	−	0.956015i	$-0.594759\pi$
$43$	−1084.69			−0.586634			−0.293317	+	0.956015i	$0.594759\pi$
$44$		0			0
$45$	−51.9162	−	2026.70i	−0.0256376	−	1.00084i
$46$		0			0
$47$		−	3114.40i		−	1.40987i	−0.709272	−	0.704935i	$-0.750976\pi$
$47$		−	3114.40i		−	1.40987i	0.709272	−	0.704935i	$-0.249024\pi$
$48$		0			0
$49$	1080.81			0.450152
$50$		0			0
$51$	18.5342	+	1447.31i	0.00712581	+	0.556445i
$52$		0			0
$53$		−	4799.21i		−	1.70851i	−0.519852	−	0.854256i	$-0.674013\pi$
$53$		−	4799.21i		−	1.70851i	0.519852	−	0.854256i	$-0.325987\pi$
$54$		0			0
$55$	5491.86			1.81549
$56$		0			0
$57$	−5331.66	+	68.2770i	−1.64102	+	0.0210148i
$58$		0			0
$59$			4250.36i			1.22102i	0.792009	+	0.610509i	$0.209035\pi$
$59$			4250.36i			1.22102i	−0.792009	+	0.610509i	$0.790965\pi$
$60$		0			0
$61$	3336.81			0.896752			0.448376	−	0.893845i	$-0.352003\pi$
$61$	3336.81			0.896752			0.448376	+	0.893845i	$0.352003\pi$
$62$		0			0
$63$	−4777.99	+	122.394i	−1.20383	+	0.0308374i
$64$		0			0
$65$		−	2.82266i		−	0.000668086i
$66$		0			0
$67$	2642.37			0.588632			0.294316	−	0.955708i	$-0.404908\pi$
$67$	2642.37			0.588632			0.294316	+	0.955708i	$0.404908\pi$
$68$		0			0
$69$	−1.33841	−	104.514i	−0.000281119	−	0.0219522i
$70$		0			0
$71$		−	6539.46i		−	1.29725i	−0.761106	−	0.648627i	$-0.775343\pi$
$71$		−	6539.46i		−	1.29725i	0.761106	−	0.648627i	$-0.224657\pi$
$72$		0			0
$73$	4384.38			0.822740			0.411370	−	0.911468i	$-0.365050\pi$
$73$	4384.38			0.822740			0.411370	+	0.911468i	$0.365050\pi$
$74$		0			0
$75$	13.1520	−	0.168424i	0.00233813	−	2.99421e-5i
$76$		0			0
$77$		−	12947.2i		−	2.18370i
$78$		0			0
$79$	−4151.19			−0.665148			−0.332574	−	0.943077i	$-0.607917\pi$
$79$	−4151.19			−0.665148			−0.332574	+	0.943077i	$0.607917\pi$
$80$		0			0
$81$	6552.40	−	335.914i	0.998688	−	0.0511986i
$82$		0			0
$83$		−	6495.95i		−	0.942945i	−0.881881	−	0.471473i	$-0.843723\pi$
$83$		−	6495.95i		−	0.942945i	0.881881	−	0.471473i	$-0.156277\pi$
$84$		0			0
$85$	−4025.34			−0.557141
$86$		0			0
$87$	−145.345	−	11349.8i	−0.0192026	−	1.49951i
$88$		0			0
$89$		−	8685.42i		−	1.09651i	−0.836313	−	0.548253i	$-0.815293\pi$
$89$		−	8685.42i		−	1.09651i	0.836313	−	0.548253i	$-0.184707\pi$
$90$		0			0
$91$	−6.65449			−0.000803585
$92$		0			0
$93$	1952.90	−	25.0088i	0.225795	−	0.00289152i
$94$		0			0
$95$		−	14828.7i		−	1.64307i
$96$		0			0
$97$	274.827			0.0292090			0.0146045	−	0.999893i	$-0.495351\pi$
$97$	274.827			0.0292090			0.0146045	+	0.999893i	$0.495351\pi$
$98$		0			0
$99$	455.123	+	17767.0i	0.0464363	+	1.81278i
$100$		0			0
$101$		−	5692.09i		−	0.557993i	−0.960292	−	0.278997i	$-0.909998\pi$
$101$		−	5692.09i		−	0.557993i	0.960292	−	0.278997i	$-0.0900018\pi$
$102$		0			0
$103$	−8524.52			−0.803518			−0.401759	−	0.915746i	$-0.631601\pi$
$103$	−8524.52			−0.803518			−0.401759	+	0.915746i	$0.631601\pi$
$104$		0			0
$105$	−170.204	−	13291.0i	−0.0154380	−	1.20553i
$106$		0			0
$107$			3118.98i			0.272423i	0.990680	+	0.136212i	$0.0434927\pi$
$107$			3118.98i			0.272423i	−0.990680	+	0.136212i	$0.956507\pi$
$108$		0			0
$109$	11890.3			1.00078			0.500390	−	0.865800i	$-0.333190\pi$
$109$	11890.3			1.00078			0.500390	+	0.865800i	$0.333190\pi$
$110$		0			0
$111$	17460.5	−	223.598i	1.41713	−	0.0181477i
$112$		0			0
$113$		−	10893.3i		−	0.853105i	−0.904463	−	0.426553i	$-0.859728\pi$
$113$		−	10893.3i		−	0.853105i	0.904463	−	0.426553i	$-0.140272\pi$
$114$		0			0
$115$	290.681			0.0219796
$116$		0			0
$117$	9.13176	−	0.233920i	0.000667087	−	1.70882e-5i
$118$		0			0
$119$			9489.83i			0.670139i
$120$		0			0
$121$	−33503.3			−2.28832
$122$		0			0
$123$	134.041	+	10467.1i	0.00885988	+	0.691856i
$124$		0			0
$125$		−	15606.7i		−	0.998827i
$126$		0			0
$127$	7804.26			0.483865			0.241932	−	0.970293i	$-0.422219\pi$
$127$	7804.26			0.483865			0.241932	+	0.970293i	$0.422219\pi$
$128$		0			0
$129$	9761.37	−	125.004i	0.586586	−	0.00751179i
$130$		0			0
$131$			12622.7i			0.735544i	0.929916	+	0.367772i	$0.119879\pi$
$131$			12622.7i			0.735544i	−0.929916	+	0.367772i	$0.880121\pi$
$132$		0			0
$133$	−34958.9			−1.97631
$134$		0			0
$135$	700.773	+	18232.8i	0.0384512	+	1.00043i
$136$		0			0
$137$			16861.5i			0.898367i	0.893439	+	0.449184i	$0.148285\pi$
$137$			16861.5i			0.898367i	−0.893439	+	0.449184i	$0.851715\pi$
$138$		0			0
$139$	−9216.37			−0.477013			−0.238507	−	0.971141i	$-0.576658\pi$
$139$	−9216.37			−0.477013			−0.238507	+	0.971141i	$0.576658\pi$
$140$		0			0
$141$	358.917	+	28027.3i	0.0180532	+	1.40975i
$142$		0			0
$143$			24.7448i			0.00121008i
$144$		0			0
$145$	31566.6			1.50138
$146$		0			0
$147$	−9726.53	+	124.558i	−0.450115	+	0.00576415i
$148$		0			0
$149$			3916.55i			0.176413i	0.996102	+	0.0882066i	$0.0281136\pi$
$149$			3916.55i			0.176413i	−0.996102	+	0.0882066i	$0.971886\pi$
$150$		0			0
$151$	−15731.8			−0.689960			−0.344980	−	0.938610i	$-0.612114\pi$
$151$	−15731.8			−0.689960			−0.344980	+	0.938610i	$0.612114\pi$
$152$		0			0
$153$	−333.589	−	13022.6i	−0.0142505	−	0.556308i
$154$		0			0
$155$			5431.51i			0.226077i
$156$		0			0
$157$	41175.4			1.67047			0.835234	−	0.549895i	$-0.185332\pi$
$157$	41175.4			1.67047			0.835234	+	0.549895i	$0.185332\pi$
$158$		0			0
$159$	553.081	+	43189.4i	0.0218774	+	1.70837i
$160$		0			0
$161$		−	685.286i		−	0.0264375i
$162$		0			0
$163$	−30022.1			−1.12997			−0.564984	−	0.825102i	$-0.691118\pi$
$163$	−30022.1			−1.12997			−0.564984	+	0.825102i	$0.691118\pi$
$164$		0			0
$165$	−49422.7	+	632.905i	−1.81534	+	0.0232472i
$166$		0			0
$167$		−	32800.4i		−	1.17610i	−0.808823	−	0.588052i	$-0.799895\pi$
$167$		−	32800.4i		−	1.17610i	0.808823	−	0.588052i	$-0.200105\pi$
$168$		0			0
$169$	−28561.0			−1.00000
$170$		0			0
$171$	47973.1	−	1228.89i	1.64061	−	0.0420261i
$172$		0			0
$173$			33725.5i			1.12685i	0.826168	+	0.563424i	$0.190516\pi$
$173$			33725.5i			1.12685i	−0.826168	+	0.563424i	$0.809484\pi$
$174$		0			0
$175$	86.2359			0.00281587
$176$		0			0
$177$	−489.830	−	38250.1i	−0.0156350	−	1.22092i
$178$		0			0
$179$		−	54831.5i		−	1.71129i	−0.517563	−	0.855645i	$-0.673161\pi$
$179$		−	54831.5i		−	1.71129i	0.517563	−	0.855645i	$-0.326839\pi$
$180$		0			0
$181$	45688.3			1.39459			0.697297	−	0.716782i	$-0.254386\pi$
$181$	45688.3			1.39459			0.697297	+	0.716782i	$0.254386\pi$
$182$		0			0
$183$	−30028.9	+	384.548i	−0.896678	+	0.0114828i
$184$		0			0
$185$			48562.0i			1.41890i
$186$		0			0
$187$	35288.1			1.00913
$188$		0			0
$189$	42984.3	−	1652.09i	1.20333	−	0.0462498i
$190$		0			0
$191$		−	23834.3i		−	0.653336i	−0.945139	−	0.326668i	$-0.894074\pi$
$191$		−	23834.3i		−	0.653336i	0.945139	−	0.326668i	$-0.105926\pi$
$192$		0			0
$193$	47703.4			1.28066			0.640331	−	0.768099i	$-0.278797\pi$
$193$	47703.4			1.28066			0.640331	+	0.768099i	$0.278797\pi$
$194$		0			0
$195$	0.325295	+	25.4019i	8.55478e−6	+	0.000668031i
$196$		0			0
$197$		−	23866.2i		−	0.614965i	−0.951554	−	0.307482i	$-0.900513\pi$
$197$		−	23866.2i		−	0.614965i	0.951554	−	0.307482i	$-0.0994865\pi$
$198$		0			0
$199$	52663.7			1.32986			0.664929	−	0.746907i	$-0.268462\pi$
$199$	52663.7			1.32986			0.664929	+	0.746907i	$0.268462\pi$
$200$		0			0
$201$	−23779.4	+	304.517i	−0.588583	+	0.00753737i
$202$		0			0
$203$		−	74418.8i		−	1.80589i
$204$		0			0
$205$	−29111.6			−0.692721
$206$		0			0
$207$	24.0893	+	940.398i	0.000562192	+	0.0219468i
$208$		0			0
$209$			129995.i			2.97602i
$210$		0			0
$211$	34108.9			0.766130			0.383065	−	0.923721i	$-0.374869\pi$
$211$	34108.9			0.766130			0.383065	+	0.923721i	$0.374869\pi$
$212$		0			0
$213$	753.635	+	58850.3i	0.0166112	+	1.29715i
$214$		0			0
$215$			27148.8i			0.587319i
$216$		0			0
$217$	12804.9			0.271930
$218$		0			0
$219$	−39456.2	+	505.275i	−0.822673	+	0.0105351i
$220$		0			0
$221$		−	18.1371i		−	0.000371350i
$222$		0			0
$223$	−58821.7			−1.18284			−0.591422	−	0.806362i	$-0.701433\pi$
$223$	−58821.7			−1.18284			−0.591422	+	0.806362i	$0.701433\pi$
$224$		0			0
$225$	−118.339	+	3.03138i	−0.00233756	+	5.98792e-5i
$226$		0			0
$227$		−	42343.8i		−	0.821748i	−0.911692	−	0.410874i	$-0.865224\pi$
$227$		−	42343.8i		−	0.821748i	0.911692	−	0.410874i	$-0.134776\pi$
$228$		0			0
$229$	−91545.0			−1.74568			−0.872838	−	0.488010i	$-0.837723\pi$
$229$	−91545.0			−1.74568			−0.872838	+	0.488010i	$0.837723\pi$
$230$		0			0
$231$	1492.09	+	116515.i	0.0279621	+	2.18353i
$232$		0			0
$233$			49016.4i			0.902880i	0.892302	+	0.451440i	$0.149089\pi$
$233$			49016.4i			0.902880i	−0.892302	+	0.451440i	$0.850911\pi$
$234$		0			0
$235$	−77951.0			−1.41152
$236$		0			0
$237$	37357.7	−	478.401i	0.665094	−	0.00851716i
$238$		0			0
$239$			35317.4i			0.618291i	0.951015	+	0.309145i	$0.100043\pi$
$239$			35317.4i			0.618291i	−0.951015	+	0.309145i	$0.899957\pi$
$240$		0			0
$241$	−5583.38			−0.0961309			−0.0480655	−	0.998844i	$-0.515306\pi$
$241$	−5583.38			−0.0961309			−0.0480655	+	0.998844i	$0.515306\pi$
$242$		0			0
$243$	−58928.0	+	3778.11i	−0.997951	+	0.0639826i
$244$		0			0
$245$		−	27051.9i		−	0.450678i
$246$		0			0
$247$	66.8140			0.00109515
$248$		0			0
$249$	748.620	+	58458.8i	0.0120743	+	0.942868i
$250$		0			0
$251$		−	116699.i		−	1.85234i	−0.377105	−	0.926171i	$-0.623080\pi$
$251$		−	116699.i		−	1.85234i	0.377105	−	0.926171i	$-0.376920\pi$
$252$		0			0
$253$	−2548.25			−0.0398108
$254$		0			0
$255$	36225.1	−	463.897i	0.557095	−	0.00713414i
$256$		0			0
$257$			61721.4i			0.934479i	0.884131	+	0.467240i	$0.154751\pi$
$257$			61721.4i			0.934479i	−0.884131	+	0.467240i	$0.845249\pi$
$258$		0			0
$259$	114486.			1.70668
$260$		0			0
$261$	2615.99	+	102123.i	0.0384021	+	1.49914i
$262$		0			0
$263$		−	77838.7i		−	1.12534i	−0.826681	−	0.562671i	$-0.809774\pi$
$263$		−	77838.7i		−	1.12534i	0.826681	−	0.562671i	$-0.190226\pi$
$264$		0			0
$265$	−120121.			−1.71051
$266$		0			0
$267$	1000.94	+	78162.4i	0.0140406	+	1.09642i
$268$		0			0
$269$		−	79264.6i		−	1.09541i	−0.836673	−	0.547703i	$-0.815503\pi$
$269$		−	79264.6i		−	1.09541i	0.836673	−	0.547703i	$-0.184497\pi$
$270$		0			0
$271$	69505.5			0.946413			0.473207	−	0.880952i	$-0.343096\pi$
$271$	69505.5			0.946413			0.473207	+	0.880952i	$0.343096\pi$
$272$		0			0
$273$	59.8855	−	0.766891i	0.000803519	−	1.02898e-5i
$274$		0			0
$275$		−	320.670i		−	0.00424026i
$276$		0			0
$277$	−28221.2			−0.367804			−0.183902	−	0.982945i	$-0.558873\pi$
$277$	−28221.2			−0.367804			−0.183902	+	0.982945i	$0.558873\pi$
$278$		0			0
$279$	−17571.8	+	450.121i	−0.225739	+	0.00578257i
$280$		0			0
$281$		−	96283.0i		−	1.21937i	−0.792642	−	0.609687i	$-0.791295\pi$
$281$		−	96283.0i		−	1.21937i	0.792642	−	0.609687i	$-0.208705\pi$
$282$		0			0
$283$	−37534.3			−0.468658			−0.234329	−	0.972157i	$-0.575289\pi$
$283$	−37534.3			−0.468658			−0.234329	+	0.972157i	$0.575289\pi$
$284$		0			0
$285$	1708.92	+	133447.i	0.0210393	+	1.64293i
$286$		0			0
$287$			68631.2i			0.833216i
$288$		0			0
$289$	57656.1			0.690318
$290$		0			0
$291$	−2473.24	+	31.6723i	−0.0292066	+	0.000374018i
$292$		0			0
$293$			55043.1i			0.641161i	0.947221	+	0.320581i	$0.103878\pi$
$293$			55043.1i			0.641161i	−0.947221	+	0.320581i	$0.896122\pi$
$294$		0			0
$295$	106383.			1.22245
$296$		0			0
$297$	−6143.32	−	159838.i	−0.0696450	−	1.81204i
$298$		0			0
$299$			1.30973i			1.46500e-5i
$300$		0			0
$301$	64003.9			0.706437
$302$		0			0
$303$	655.980	+	51224.6i	0.00714505	+	0.557948i
$304$		0			0
$305$		−	83517.8i		−	0.897800i
$306$		0			0
$307$	105346.			1.11774			0.558872	−	0.829254i	$-0.311234\pi$
$307$	105346.			1.11774			0.558872	+	0.829254i	$0.311234\pi$
$308$		0			0
$309$	76714.4	−	982.401i	0.803452	−	0.0102890i
$310$		0			0
$311$			4905.52i			0.0507183i	0.999678	+	0.0253591i	$0.00807293\pi$
$311$			4905.52i			0.0507183i	−0.999678	+	0.0253591i	$0.991927\pi$
$312$		0			0
$313$	132759.			1.35512			0.677558	−	0.735470i	$-0.263038\pi$
$313$	132759.			1.35512			0.677558	+	0.735470i	$0.263038\pi$
$314$		0			0
$315$	3063.42	+	119589.i	0.0308734	+	1.20523i
$316$		0			0
$317$			35271.0i			0.350993i	0.984480	+	0.175497i	$0.0561531\pi$
$317$			35271.0i			0.350993i	−0.984480	+	0.175497i	$0.943847\pi$
$318$		0			0
$319$	−276728.			−2.71939
$320$		0			0
$321$	−359.444	−	28068.5i	−0.00348836	−	0.272401i
$322$		0			0
$323$		−	95282.1i		−	0.913285i
$324$		0			0
$325$	−0.164815			−1.56038e−6
$326$		0			0
$327$	−107004.	+	1370.28i	−1.00070	+	0.0128149i
$328$		0			0
$329$			183771.i			1.69780i
$330$		0			0
$331$	−106590.			−0.972884			−0.486442	−	0.873713i	$-0.661705\pi$
$331$	−106590.			−0.972884			−0.486442	+	0.873713i	$0.661705\pi$
$332$		0			0
$333$	−157106.	+	4024.44i	−1.41678	+	0.0362925i
$334$		0			0
$335$		−	66136.4i		−	0.589319i
$336$		0			0
$337$	−66337.7			−0.584118			−0.292059	−	0.956400i	$-0.594340\pi$
$337$	−66337.7			−0.584118			−0.292059	+	0.956400i	$0.594340\pi$
$338$		0			0
$339$	1255.39	+	98031.6i	0.0109239	+	0.853035i
$340$		0			0
$341$		−	47615.2i		−	0.409484i
$342$		0			0
$343$	77900.1			0.662140
$344$		0			0
$345$	−2615.91	+	33.4993i	−0.0219778	+	0.000281447i
$346$		0			0
$347$			9211.86i			0.0765047i	0.999268	+	0.0382524i	$0.0121791\pi$
$347$			9211.86i			0.0765047i	−0.999268	+	0.0382524i	$0.987821\pi$
$348$		0			0
$349$	39622.4			0.325304			0.162652	−	0.986683i	$-0.447995\pi$
$349$	39622.4			0.325304			0.162652	+	0.986683i	$0.447995\pi$
$350$		0			0
$351$	−82.1521	+	3.15749i	−0.000666814	+	2.56288e-5i
$352$		0			0
$353$		−	43436.9i		−	0.348586i	−0.984694	−	0.174293i	$-0.944236\pi$
$353$		−	43436.9i		−	0.348586i	0.984694	−	0.174293i	$-0.0557639\pi$
$354$		0			0
$355$	−163678.			−1.29877
$356$		0			0
$357$	−1093.65	−	85401.5i	−0.00858106	−	0.670084i
$358$		0			0
$359$			38920.2i			0.301986i	0.988535	+	0.150993i	$0.0482470\pi$
$359$			38920.2i			0.301986i	−0.988535	+	0.150993i	$0.951753\pi$
$360$		0			0
$361$	220682.			1.69337
$362$		0			0
$363$	301505.	−	3861.06i	2.28813	−	0.0293018i
$364$		0			0
$365$		−	109738.i		−	0.823702i
$366$		0			0
$367$	118239.			0.877866			0.438933	−	0.898520i	$-0.355357\pi$
$367$	118239.			0.877866			0.438933	+	0.898520i	$0.355357\pi$
$368$		0			0
$369$	−2412.54	−	94180.6i	−0.0177183	−	0.691686i
$370$		0			0
$371$			283187.i			2.05743i
$372$		0			0
$373$	263000.			1.89033			0.945167	−	0.326586i	$-0.105898\pi$
$373$	263000.			1.89033			0.945167	+	0.326586i	$0.105898\pi$
$374$		0			0
$375$	1798.58	+	140449.i	0.0127899	+	0.998746i
$376$		0			0
$377$			142.230i			0.00100071i
$378$		0			0
$379$	63440.1			0.441657			0.220829	−	0.975313i	$-0.429124\pi$
$379$	63440.1			0.441657			0.220829	+	0.975313i	$0.429124\pi$
$380$		0			0
$381$	−70232.5	+	899.395i	−0.483825	+	0.00619585i
$382$		0			0
$383$		−	251370.i		−	1.71363i	−0.515627	−	0.856813i	$-0.672441\pi$
$383$		−	251370.i		−	1.71363i	0.515627	−	0.856813i	$-0.327559\pi$
$384$		0			0
$385$	−324058.			−2.18626
$386$		0			0
$387$	−87830.7	+	2249.88i	−0.586441	+	0.0150223i
$388$		0			0
$389$		−	215823.i		−	1.42626i	−0.701032	−	0.713130i	$-0.747277\pi$
$389$		−	215823.i		−	1.42626i	0.701032	−	0.713130i	$-0.252723\pi$
$390$		0			0
$391$	1867.78			0.0122172
$392$		0			0
$393$	−1454.69	−	113595.i	−0.00941857	−	0.735483i
$394$		0			0
$395$			103901.i			0.665926i
$396$		0			0
$397$	−204653.			−1.29848			−0.649242	−	0.760582i	$-0.724914\pi$
$397$	−204653.			−1.29848			−0.649242	+	0.760582i	$0.724914\pi$
$398$		0			0
$399$	314605.	−	4028.82i	1.97615	−	0.0253065i
$400$		0			0
$401$		−	73107.8i		−	0.454648i	−0.973819	−	0.227324i	$-0.927002\pi$
$401$		−	73107.8i		−	0.454648i	0.973819	−	0.227324i	$-0.0729976\pi$
$402$		0			0
$403$	−24.4729			−0.000150687
$404$		0			0
$405$	−8407.67	−	164001.i	−0.0512585	−	0.999855i
$406$		0			0
$407$		−	425718.i		−	2.57000i
$408$		0			0
$409$	−104755.			−0.626221			−0.313110	−	0.949717i	$-0.601371\pi$
$409$	−104755.			−0.626221			−0.313110	+	0.949717i	$0.601371\pi$
$410$		0			0
$411$	−1943.18	−	151741.i	−0.0115035	−	0.898293i
$412$		0			0
$413$		−	250801.i		−	1.47038i
$414$		0			0
$415$	−162589.			−0.944047
$416$		0			0
$417$	82940.5	−	1062.13i	0.476974	−	0.00610811i
$418$		0			0
$419$		−	57233.8i		−	0.326005i	−0.986626	−	0.163003i	$-0.947882\pi$
$419$		−	57233.8i		−	0.326005i	0.986626	−	0.163003i	$-0.0521179\pi$
$420$		0			0
$421$	26782.1			0.151106			0.0755528	−	0.997142i	$-0.475928\pi$
$421$	26782.1			0.151106			0.0755528	+	0.997142i	$0.475928\pi$
$422$		0			0
$423$	−6459.97	−	252184.i	−0.0361035	−	1.40941i
$424$		0			0
$425$			235.040i			0.00130126i
$426$		0			0
$427$	−196895.			−1.07989
$428$		0			0
$429$	−2.85170	−	222.685i	−1.54949e−5	−	0.00120998i
$430$		0			0
$431$		−	202264.i		−	1.08884i	−0.838812	−	0.544421i	$-0.816749\pi$
$431$		−	202264.i		−	1.08884i	0.838812	−	0.544421i	$-0.183251\pi$
$432$		0			0
$433$	−150338.			−0.801850			−0.400925	−	0.916111i	$-0.631311\pi$
$433$	−150338.			−0.801850			−0.400925	+	0.916111i	$0.631311\pi$
$434$		0			0
$435$	−284076.	+	3637.86i	−1.50126	+	0.0192251i
$436$		0			0
$437$			6880.57i			0.0360298i
$438$		0			0
$439$	−115380.			−0.598690			−0.299345	−	0.954145i	$-0.596768\pi$
$439$	−115380.			−0.598690			−0.299345	+	0.954145i	$0.596768\pi$
$440$		0			0
$441$	87517.2	−	2241.85i	0.450004	−	0.0115274i
$442$		0			0
$443$			76763.3i			0.391152i	0.980689	+	0.195576i	$0.0626577\pi$
$443$			76763.3i			0.391152i	−0.980689	+	0.195576i	$0.937342\pi$
$444$		0			0
$445$	−217389.			−1.09779
$446$		0			0
$447$	−451.359	−	35246.0i	−0.00225895	−	0.176399i
$448$		0			0
$449$		−	102877.i		−	0.510302i	−0.966901	−	0.255151i	$-0.917875\pi$
$449$		−	102877.i		−	0.510302i	0.966901	−	0.255151i	$-0.0821252\pi$
$450$		0			0
$451$	255206.			1.25470
$452$		0			0
$453$	141574.	−	1813.00i	0.689903	−	0.00883487i
$454$		0			0
$455$			166.557i			0.000804524i
$456$		0			0
$457$	238938.			1.14407			0.572035	−	0.820229i	$-0.306154\pi$
$457$	238938.			1.14407			0.572035	+	0.820229i	$0.306154\pi$
$458$		0			0
$459$	4502.84	+	117156.i	0.0213728	+	0.556080i
$460$		0			0
$461$			238281.i			1.12121i	0.828082	+	0.560607i	$0.189432\pi$
$461$			238281.i			1.12121i	−0.828082	+	0.560607i	$0.810568\pi$
$462$		0			0
$463$	133548.			0.622980			0.311490	−	0.950249i	$-0.399172\pi$
$463$	133548.			0.622980			0.311490	+	0.950249i	$0.399172\pi$
$464$		0			0
$465$	−625.949	−	48879.5i	−0.00289490	−	0.226059i
$466$		0			0
$467$		−	258049.i		−	1.18323i	−0.806221	−	0.591614i	$-0.798491\pi$
$467$		−	258049.i		−	1.18323i	0.806221	−	0.591614i	$-0.201509\pi$
$468$		0			0
$469$	−155918.			−0.708843
$470$		0			0
$471$	−370548.	+	4745.22i	−1.67033	+	0.0213902i
$472$		0			0
$473$		−	238000.i		−	1.06379i
$474$		0			0
$475$	−865.846			−0.00383755
$476$		0			0
$477$	−9954.65	−	388609.i	−0.0437511	−	1.70795i
$478$		0			0
$479$			230624.i			1.00515i	0.864532	+	0.502577i	$0.167615\pi$
$479$			230624.i			1.00515i	−0.864532	+	0.502577i	$0.832385\pi$
$480$		0			0
$481$	−218.807			−0.000945738
$482$		0			0
$483$	78.9752	+	6167.07i	0.000338530	+	0.0264353i
$484$		0			0
$485$		−	6878.71i		−	0.0292431i
$486$		0			0
$487$	79003.8			0.333112			0.166556	−	0.986032i	$-0.446735\pi$
$487$	79003.8			0.333112			0.166556	+	0.986032i	$0.446735\pi$
$488$		0			0
$489$	270177.	−	3459.87i	1.12988	−	0.0144691i
$490$		0			0
$491$			225320.i			0.934625i	0.884092	+	0.467313i	$0.154778\pi$
$491$			225320.i			0.934625i	−0.884092	+	0.467313i	$0.845222\pi$
$492$		0			0
$493$	202832.			0.834530
$494$		0			0
$495$	444695.	−	11391.4i	1.81490	−	0.0464906i
$496$		0			0
$497$			385873.i			1.56218i
$498$		0			0
$499$	127276.			0.511146			0.255573	−	0.966790i	$-0.417736\pi$
$499$	127276.			0.511146			0.255573	+	0.966790i	$0.417736\pi$
$500$		0			0
$501$	3780.05	+	295179.i	0.0150599	+	1.17601i
$502$		0			0
$503$			308204.i			1.21815i	0.793111	+	0.609077i	$0.208460\pi$
$503$			308204.i			1.21815i	−0.793111	+	0.609077i	$0.791540\pi$
$504$		0			0
$505$	−142469.			−0.558645
$506$		0			0
$507$	257028.	−	3291.49i	0.999918	−	0.0128049i
$508$		0			0
$509$		−	114499.i		−	0.441944i	−0.975280	−	0.220972i	$-0.929077\pi$
$509$		−	114499.i		−	0.441944i	0.975280	−	0.220972i	$-0.0709229\pi$
$510$		0			0
$511$	−258709.			−0.990762
$512$		0			0
$513$	−431581.	+	16587.7i	−1.63994	+	0.0630306i
$514$		0			0
$515$			213362.i			0.804457i
$516$		0			0
$517$	683356.			2.55662
$518$		0			0
$519$	−3886.66	−	303504.i	−0.0144292	−	1.12676i
$520$		0			0
$521$		−	321111.i		−	1.18299i	−0.806310	−	0.591494i	$-0.798538\pi$
$521$		−	321111.i		−	1.18299i	0.806310	−	0.591494i	$-0.201462\pi$
$522$		0			0
$523$	132528.			0.484511			0.242255	−	0.970213i	$-0.422113\pi$
$523$	132528.			0.484511			0.242255	+	0.970213i	$0.422113\pi$
$524$		0			0
$525$	−776.059	+	9.93818i	−0.00281563	+	3.60569e-5i
$526$		0			0
$527$			34900.3i			0.125663i
$528$		0			0
$529$	279706.			0.999518
$530$		0			0
$531$	8816.21	+	344167.i	0.0312675	+	1.22062i
$532$		0			0
$533$		−	131.169i		−	0.000461717i
$534$		0			0
$535$	78065.5			0.272742
$536$		0			0
$537$	6319.01	+	493443.i	0.0219129	+	1.71115i
$538$		0			0
$539$			237150.i			0.816293i
$540$		0			0
$541$	−526786.			−1.79986			−0.899932	−	0.436030i	$-0.856384\pi$
$541$	−526786.			−1.79986			−0.899932	+	0.436030i	$0.856384\pi$
$542$		0			0
$543$	−411161.	+	5265.31i	−1.39448	+	0.0178577i
$544$		0			0
$545$		−	297604.i		−	1.00195i
$546$		0			0
$547$	79324.7			0.265115			0.132557	−	0.991175i	$-0.457681\pi$
$547$	79324.7			0.265115			0.132557	+	0.991175i	$0.457681\pi$
$548$		0			0
$549$	270193.	−	6921.30i	0.896458	−	0.0229638i
$550$		0			0
$551$			747198.i			2.46112i
$552$		0			0
$553$	244949.			0.800987
$554$		0			0
$555$	−5596.49	−	437022.i	−0.0181689	−	1.41879i
$556$		0			0
$557$			104134.i			0.335648i	0.985817	+	0.167824i	$0.0536741\pi$
$557$			104134.i			0.335648i	−0.985817	+	0.167824i	$0.946326\pi$
$558$		0			0
$559$	−122.325			−0.000391464
$560$		0			0
$561$	−317567.	+	4066.75i	−1.00904	+	0.0129218i
$562$		0			0
$563$			112937.i			0.356304i	0.984003	+	0.178152i	$0.0570119\pi$
$563$			112937.i			0.356304i	−0.984003	+	0.178152i	$0.942988\pi$
$564$		0			0
$565$	−272651.			−0.854102
$566$		0			0
$567$	−386637.	+	19821.3i	−1.20264	+	0.0616546i
$568$		0			0
$569$		−	449057.i		−	1.38700i	−0.720455	−	0.693501i	$-0.756067\pi$
$569$		−	449057.i		−	1.38700i	0.720455	−	0.693501i	$-0.243933\pi$
$570$		0			0
$571$	−605841.			−1.85818			−0.929088	−	0.369860i	$-0.879406\pi$
$571$	−605841.			−1.85818			−0.929088	+	0.369860i	$0.879406\pi$
$572$		0			0
$573$	2746.77	+	214491.i	0.00836590	+	0.653282i
$574$		0			0
$575$		−	16.9728i		−	5.13356e-5i
$576$		0			0
$577$	94919.8			0.285105			0.142553	−	0.989787i	$-0.454469\pi$
$577$	94919.8			0.285105			0.142553	+	0.989787i	$0.454469\pi$
$578$		0			0
$579$	−429295.	+	5497.53i	−1.28056	+	0.0163988i
$580$		0			0
$581$			383306.i			1.13552i
$582$		0			0
$583$	1.05303e6			3.09817
$584$		0			0
$585$	−5.85484	−	228.561i	−1.71082e−5	−	0.000667867i
$586$		0			0
$587$		−	416501.i		−	1.20876i	−0.796697	−	0.604379i	$-0.793421\pi$
$587$		−	416501.i		−	1.20876i	0.796697	−	0.604379i	$-0.206579\pi$
$588$		0			0
$589$	−128567.			−0.370594
$590$		0			0
$591$	2750.44	+	214778.i	0.00787457	+	0.614914i
$592$		0			0
$593$		−	212187.i		−	0.603405i	−0.953402	−	0.301702i	$-0.902445\pi$
$593$		−	212187.i		−	0.603405i	0.953402	−	0.301702i	$-0.0975549\pi$
$594$		0			0
$595$	237523.			0.670922
$596$		0			0
$597$	−473934.	+	6069.18i	−1.32975	+	0.0170287i
$598$		0			0
$599$		−	98563.1i		−	0.274701i	−0.990522	−	0.137351i	$-0.956141\pi$
$599$		−	98563.1i		−	0.274701i	0.990522	−	0.137351i	$-0.0438587\pi$
$600$		0			0
$601$	456630.			1.26420			0.632100	−	0.774887i	$-0.282193\pi$
$601$	456630.			1.26420			0.632100	+	0.774887i	$0.282193\pi$
$602$		0			0
$603$	213962.	−	5480.86i	0.588438	−	0.0150735i
$604$		0			0
$605$			838562.i			2.29100i
$606$		0			0
$607$	−588973.			−1.59852			−0.799260	−	0.600986i	$-0.794775\pi$
$607$	−588973.			−1.59852			−0.799260	+	0.600986i	$0.794775\pi$
$608$		0			0
$609$	8576.34	+	669715.i	0.0231242	+	1.80574i
$610$		0			0
$611$		−	351.226i		−	0.000940814i
$612$		0			0
$613$	97484.6			0.259427			0.129713	−	0.991552i	$-0.458594\pi$
$613$	97484.6			0.259427			0.129713	+	0.991552i	$0.458594\pi$
$614$		0			0
$615$	261983.	−	3354.94i	0.692664	−	0.00887023i
$616$		0			0
$617$			664979.i			1.74678i	0.487022	+	0.873389i	$0.338083\pi$
$617$			664979.i			1.74678i	−0.487022	+	0.873389i	$0.661917\pi$
$618$		0			0
$619$	−157113.			−0.410046			−0.205023	−	0.978757i	$-0.565727\pi$
$619$	−157113.			−0.410046			−0.205023	+	0.978757i	$0.565727\pi$
$620$		0			0
$621$	−325.162	−	8460.11i	−0.000843172	−	0.0219378i
$622$		0			0
$623$			512500.i			1.32044i
$624$		0			0
$625$	−391536.			−1.00233
$626$		0			0
$627$	−14981.2	−	1.16986e6i	−0.0381076	−	2.97577i
$628$		0			0
$629$			312036.i			0.788685i
$630$		0			0
$631$	37893.3			0.0951707			0.0475853	−	0.998867i	$-0.484847\pi$
$631$	37893.3			0.0951707			0.0475853	+	0.998867i	$0.484847\pi$
$632$		0			0
$633$	−306955.	+	3930.85i	−0.766067	+	0.00981022i
$634$		0			0
$635$		−	195334.i		−	0.484430i
$636$		0			0
$637$	121.889			0.000300389
$638$		0			0
$639$	−13564.3	−	529523.i	−0.0332197	−	1.29683i
$640$		0			0
$641$			20542.7i			0.0499967i	0.999687	+	0.0249984i	$0.00795806\pi$
$641$			20542.7i			0.0499967i	−0.999687	+	0.0249984i	$0.992042\pi$
$642$		0			0
$643$	318711.			0.770858			0.385429	−	0.922737i	$-0.374053\pi$
$643$	318711.			0.770858			0.385429	+	0.922737i	$0.374053\pi$
$644$		0			0
$645$	−3128.74	−	244319.i	−0.00752057	−	0.587271i
$646$		0			0
$647$			55597.0i			0.132814i	0.997793	+	0.0664068i	$0.0211535\pi$
$647$			55597.0i			0.132814i	−0.997793	+	0.0664068i	$0.978846\pi$
$648$		0			0
$649$	−932607.			−2.21416
$650$		0			0
$651$	−115235.	+	1475.69i	−0.271907	+	0.00348203i
$652$		0			0
$653$		−	141166.i		−	0.331058i	−0.986205	−	0.165529i	$-0.947067\pi$
$653$		−	141166.i		−	0.331058i	0.986205	−	0.165529i	$-0.0529331\pi$
$654$		0			0
$655$	315935.			0.736403
$656$		0			0
$657$	355019.	−	9094.20i	0.822470	−	0.0210685i
$658$		0			0
$659$		−	536334.i		−	1.23499i	−0.786574	−	0.617496i	$-0.788147\pi$
$659$		−	536334.i		−	1.23499i	0.786574	−	0.617496i	$-0.211853\pi$
$660$		0			0
$661$	36981.2			0.0846404			0.0423202	−	0.999104i	$-0.486525\pi$
$661$	36981.2			0.0846404			0.0423202	+	0.999104i	$0.486525\pi$
$662$		0			0
$663$	2.09019	+	163.220i	4.75510e−6	+	0.000371319i
$664$		0			0
$665$			874995.i			1.97862i
$666$		0			0
$667$	−14647.0			−0.0329229
$668$		0			0
$669$	529352.	−	6778.86i	1.18275	−	0.0151462i
$670$		0			0
$671$			732157.i			1.62615i
$672$		0			0
$673$	114011.			0.251720			0.125860	−	0.992048i	$-0.459831\pi$
$673$	114011.			0.251720			0.125860	+	0.992048i	$0.459831\pi$
$674$		0			0
$675$	1064.61	−	40.9181i	0.00233660	−	8.98065e-5i
$676$		0			0
$677$			425402.i			0.928158i	0.885794	+	0.464079i	$0.153615\pi$
$677$			425402.i			0.928158i	−0.885794	+	0.464079i	$0.846385\pi$
$678$		0			0
$679$	−16216.7			−0.0351741
$680$		0			0
$681$	4879.88	+	381063.i	0.0105224	+	0.821680i
$682$		0			0
$683$			333970.i			0.715923i	0.933736	+	0.357962i	$0.116528\pi$
$683$			333970.i			0.715923i	−0.933736	+	0.357962i	$0.883472\pi$
$684$		0			0
$685$	422029.			0.899417
$686$		0			0
$687$	823837.	−	10550.0i	1.74553	−	0.0223532i
$688$		0			0
$689$		−	541.230i		−	0.00114010i
$690$		0			0
$691$	180196.			0.377388			0.188694	−	0.982036i	$-0.439575\pi$
$691$	180196.			0.377388			0.188694	+	0.982036i	$0.439575\pi$
$692$		0			0
$693$	−26855.4	−	1.04838e6i	−0.0559197	−	2.18299i
$694$		0			0
$695$			230678.i			0.477570i
$696$		0			0
$697$	−187057.			−0.385043
$698$		0			0
$699$	−5648.86	−	441112.i	−0.0115613	−	0.902806i
$700$		0			0
$701$			406441.i			0.827107i	0.910480	+	0.413554i	$0.135713\pi$
$701$			406441.i			0.827107i	−0.910480	+	0.413554i	$0.864287\pi$
$702$		0			0
$703$	−1.14949e6			−2.32592
$704$		0			0
$705$	701502.	−	8983.40i	1.41140	−	0.0180743i
$706$		0			0
$707$			335873.i			0.671948i
$708$		0			0
$709$	−544669.			−1.08353			−0.541764	−	0.840531i	$-0.682243\pi$
$709$	−544669.			−1.08353			−0.541764	+	0.840531i	$0.682243\pi$
$710$		0			0
$711$	−336136.	+	8610.50i	−0.664930	+	0.0170329i
$712$		0			0
$713$		−	2520.24i		−	0.00495751i
$714$		0			0
$715$	619.344			0.00121149
$716$		0			0
$717$	−4070.12	−	317831.i	−0.00791716	−	0.618240i
$718$		0			0
$719$		−	345757.i		−	0.668825i	−0.942427	−	0.334413i	$-0.891462\pi$
$719$		−	345757.i		−	0.668825i	0.942427	−	0.334413i	$-0.108538\pi$
$720$		0			0
$721$	503005.			0.967614
$722$		0			0
$723$	50246.3	−	643.452i	0.0961231	−	0.00123095i
$724$		0			0
$725$		−	1843.17i		−	0.00350662i
$726$		0			0
$727$	−732496.			−1.38591			−0.692957	−	0.720978i	$-0.743693\pi$
$727$	−732496.			−1.38591			−0.692957	+	0.720978i	$0.743693\pi$
$728$		0			0
$729$	529873.	−	40791.3i	0.997050	−	0.0767560i
$730$		0			0
$731$			174445.i			0.326456i
$732$		0			0
$733$	111923.			0.208311			0.104155	−	0.994561i	$-0.466786\pi$
$733$	111923.			0.208311			0.104155	+	0.994561i	$0.466786\pi$
$734$		0			0
$735$	3117.58	+	243447.i	0.00577089	+	0.450641i
$736$		0			0
$737$			579783.i			1.06741i
$738$		0			0
$739$	450315.			0.824570			0.412285	−	0.911055i	$-0.364731\pi$
$739$	450315.			0.824570			0.412285	+	0.911055i	$0.364731\pi$
$740$		0			0
$741$	−601.277	+	7.69992i	−0.00109506	+	1.40233e-5i
$742$		0			0
$743$		−	448643.i		−	0.812687i	−0.913720	−	0.406344i	$-0.866804\pi$
$743$		−	448643.i		−	0.812687i	0.913720	−	0.406344i	$-0.133196\pi$
$744$		0			0
$745$	98028.1			0.176619
$746$		0			0
$747$	−13474.1	−	525999.i	−0.0241467	−	0.942636i
$748$		0			0
$749$		−	184041.i		−	0.328058i
$750$		0			0
$751$	752993.			1.33509			0.667546	−	0.744569i	$-0.267345\pi$
$751$	752993.			1.33509			0.667546	+	0.744569i	$0.267345\pi$
$752$		0			0
$753$	13448.9	+	1.05021e6i	0.0237191	+	1.85219i
$754$		0			0
$755$			393754.i			0.690766i
$756$		0			0
$757$	−524087.			−0.914560			−0.457280	−	0.889323i	$-0.651176\pi$
$757$	−524087.			−0.914560			−0.457280	+	0.889323i	$0.651176\pi$
$758$		0			0
$759$	22932.3	−	293.671i	0.0398075	−	0.000509773i
$760$		0			0
$761$		−	408763.i		−	0.705833i	−0.935655	−	0.352916i	$-0.885190\pi$
$761$		−	408763.i		−	0.705833i	0.935655	−	0.352916i	$-0.114810\pi$
$762$		0			0
$763$	−701608.			−1.20516
$764$		0			0
$765$	−325946.	+	8349.47i	−0.556958	+	0.0142671i
$766$		0			0
$767$			479.334i			0.000814793i
$768$		0			0
$769$	181813.			0.307449			0.153725	−	0.988114i	$-0.450873\pi$
$769$	181813.			0.307449			0.153725	+	0.988114i	$0.450873\pi$
$770$		0			0
$771$	−7113.03	−	555447.i	−0.0119659	−	0.934403i
$772$		0			0
$773$			332218.i			0.555986i	0.960583	+	0.277993i	$0.0896693\pi$
$773$			332218.i			0.555986i	−0.960583	+	0.277993i	$0.910331\pi$
$774$		0			0
$775$	317.145			0.000528025
$776$		0			0
$777$	−1.03029e6	+	13193.8i	−1.70654	+	0.0218539i
$778$		0			0
$779$		−	689088.i		−	1.13553i
$780$		0			0
$781$	1.43488e6			2.35241
$782$		0			0
$783$	−35311.0	−	918728.i	−0.0575953	−	1.49852i
$784$		0			0
$785$		−	1.03059e6i		−	1.67242i
$786$		0			0
$787$	−421772.			−0.680970			−0.340485	−	0.940250i	$-0.610591\pi$
$787$	−421772.			−0.680970			−0.340485	+	0.940250i	$0.610591\pi$
$788$		0			0
$789$	8970.46	+	700491.i	0.0144099	+	1.12525i
$790$		0			0
$791$			642780.i			1.02733i
$792$		0			0
$793$	376.308			0.000598408
$794$		0			0
$795$	1.08100e6	−	13843.2i	1.71037	−	0.0219029i
$796$		0			0
$797$			49979.5i			0.0786820i	0.999226	+	0.0393410i	$0.0125259\pi$
$797$			49979.5i			0.0786820i	−0.999226	+	0.0393410i	$0.987474\pi$
$798$		0			0
$799$	−500876.			−0.784579
$800$		0			0
$801$	−18015.5	−	703288.i	−0.0280790	−	1.09615i
$802$		0			0
$803$			962013.i			1.49194i
$804$		0			0
$805$	−17152.2			−0.0264684
$806$		0			0
$807$	9134.79	+	713323.i	0.0140266	+	1.09532i
$808$		0			0
$809$		−	891165.i		−	1.36164i	−0.732453	−	0.680818i	$-0.761624\pi$
$809$		−	891165.i		−	1.36164i	0.732453	−	0.680818i	$-0.238376\pi$
$810$		0			0
$811$	267631.			0.406906			0.203453	−	0.979085i	$-0.434784\pi$
$811$	267631.			0.406906			0.203453	+	0.979085i	$0.434784\pi$
$812$		0			0
$813$	−625498.	+	8010.11i	−0.946335	+	0.0121187i
$814$		0			0
$815$			751430.i			1.13129i
$816$		0			0
$817$	−642628.			−0.962754
$818$		0			0
$819$	−538.837	+	13.8029i	−0.000803322	+	2.05780e-5i
$820$		0			0
$821$			657856.i			0.975988i	0.872847	+	0.487994i	$0.162271\pi$
$821$			657856.i			0.975988i	−0.872847	+	0.487994i	$0.837729\pi$
$822$		0			0
$823$	214191.			0.316229			0.158115	−	0.987421i	$-0.449458\pi$
$823$	214191.			0.316229			0.158115	+	0.987421i	$0.449458\pi$
$824$		0			0
$825$	36.9553	+	2885.79i	5.42961e−5	+	0.00423991i
$826$		0			0
$827$			409128.i			0.598202i	0.954221	+	0.299101i	$0.0966867\pi$
$827$			409128.i			0.598202i	−0.954221	+	0.299101i	$0.903313\pi$
$828$		0			0
$829$	435720.			0.634013			0.317007	−	0.948423i	$-0.397322\pi$
$829$	435720.			0.634013			0.317007	+	0.948423i	$0.397322\pi$
$830$		0			0
$831$	253970.	−	3252.33i	0.367774	−	0.00470970i
$832$		0			0
$833$		−	173823.i		−	0.250505i
$834$		0			0
$835$	−820967.			−1.17748
$836$		0			0
$837$	158081.	−	6075.80i	0.225647	−	0.00867266i
$838$		0			0
$839$		−	317366.i		−	0.450855i	−0.974260	−	0.225427i	$-0.927622\pi$
$839$		−	317366.i		−	0.450855i	0.974260	−	0.225427i	$-0.0723779\pi$
$840$		0			0
$841$	−883316.			−1.24889
$842$		0			0
$843$	11096.1	+	866476.i	0.0156140	+	1.21927i
$844$		0			0
$845$			714859.i			1.00117i
$846$		0			0
$847$	1.97693e6			2.75565
$848$		0			0
$849$	337781.	−	4325.61i	0.468619	−	0.00600112i
$850$		0			0
$851$		−	22533.0i		−	0.0311142i
$852$		0			0
$853$	−531190.			−0.730049			−0.365025	−	0.930998i	$-0.618939\pi$
$853$	−531190.			−0.730049			−0.365025	+	0.930998i	$0.618939\pi$
$854$		0			0
$855$	−30758.0	−	1.20073e6i	−0.0420752	−	1.64253i
$856$		0			0
$857$		−	919137.i		−	1.25147i	−0.780038	−	0.625733i	$-0.784800\pi$
$857$		−	919137.i		−	1.25147i	0.780038	−	0.625733i	$-0.215200\pi$
$858$		0			0
$859$	397727.			0.539013			0.269506	−	0.962999i	$-0.413139\pi$
$859$	397727.			0.539013			0.269506	+	0.962999i	$0.413139\pi$
$860$		0			0
$861$	−7909.35	−	617630.i	−0.0106693	−	0.833148i
$862$		0			0
$863$		−	592105.i		−	0.795019i	−0.917598	−	0.397509i	$-0.869875\pi$
$863$		−	592105.i		−	0.795019i	0.917598	−	0.397509i	$-0.130125\pi$
$864$		0			0
$865$	844122.			1.12817
$866$		0			0
$867$	−518862.	+	6644.52i	−0.690261	+	0.00883946i
$868$		0			0
$869$		−	910847.i		−	1.20616i
$870$		0			0
$871$	297.992			0.000392797
$872$		0			0
$873$	22253.7	−	570.054i	0.0291994	−	0.000747976i
$874$		0			0
$875$			920902.i			1.20281i
$876$		0			0
$877$	−47500.1			−0.0617583			−0.0308792	−	0.999523i	$-0.509831\pi$
$877$	−47500.1			−0.0617583			−0.0308792	+	0.999523i	$0.509831\pi$
$878$		0			0
$879$	−6343.39	−	495347.i	−0.00821001	−	0.641109i
$880$		0			0
$881$			404360.i			0.520975i	0.965477	+	0.260488i	$0.0838833\pi$
$881$			404360.i			0.520975i	−0.965477	+	0.260488i	$0.916117\pi$
$882$		0			0
$883$	−1.36248e6			−1.74747			−0.873733	−	0.486406i	$-0.838308\pi$
$883$	−1.36248e6			−1.74747			−0.873733	+	0.486406i	$0.838308\pi$
$884$		0			0
$885$	−957371.	+	12260.1i	−1.22234	+	0.0156533i
$886$		0			0
$887$			112.901i			0.000143500i	1.00000		7.17499e-5i	$2.28387e-5\pi$
$887$			112.901i			0.000143500i	−1.00000		7.17499e-5i	$0.999977\pi$
$888$		0			0
$889$	−460505.			−0.582681
$890$		0			0
$891$	73705.7	+	1.43771e6i	0.0928422	+	1.81099i
$892$		0			0
$893$		−	1.84514e6i		−	2.31381i
$894$		0			0
$895$	−1.37239e6			−1.71329
$896$		0			0
$897$	−0.150938	−	11.7866i	−1.87592e−7	−	1.46488e-5i
$898$		0			0
$899$		−	273686.i		−	0.338636i
$900$		0			0
$901$	−771837.			−0.950772
$902$		0			0
$903$	−575988.	+	7376.08i	−0.706379	+	0.00904587i
$904$		0			0
$905$		−	1.14354e6i		−	1.39622i
$906$		0			0
$907$	1.14759e6			1.39499			0.697496	−	0.716589i	$-0.254298\pi$
$907$	1.14759e6			1.39499			0.697496	+	0.716589i	$0.254298\pi$
$908$		0			0
$909$	−11806.7	−	460908.i	−0.0142889	−	0.557810i
$910$		0			0
$911$			1.43436e6i			1.72831i	0.503223	+	0.864156i	$0.332147\pi$
$911$			1.43436e6i			1.72831i	−0.503223	+	0.864156i	$0.667853\pi$
$912$		0			0
$913$	1.42533e6			1.70991
$914$		0			0
$915$	9624.94	+	751599.i	0.0114962	+	0.897726i
$916$		0			0
$917$		−	744824.i		−	0.885758i
$918$		0			0
$919$	245973.			0.291243			0.145622	−	0.989340i	$-0.453482\pi$
$919$	245973.			0.291243			0.145622	+	0.989340i	$0.453482\pi$
$920$		0			0
$921$	−948039.	+	12140.5i	−1.11765	+	0.0143126i
$922$		0			0
$923$		−	737.486i		−	0.000865666i
$924$		0			0
$925$	2835.53			0.00331399
$926$		0			0
$927$	−690260.	+	17681.8i	−0.803254	+	0.0205763i
$928$		0			0
$929$		−	1.15123e6i		−	1.33392i	−0.745092	−	0.666962i	$-0.767594\pi$
$929$		−	1.15123e6i		−	1.33392i	0.745092	−	0.666962i	$-0.232406\pi$
$930$		0			0
$931$	640334.			0.738767
$932$		0			0
$933$	−565.333	−	44146.1i	−0.000649443	−	0.0507141i
$934$		0			0
$935$		−	883233.i		−	1.01030i
$936$		0			0
$937$	959065.			1.09237			0.546184	−	0.837665i	$-0.316080\pi$
$937$	959065.			1.09237			0.546184	+	0.837665i	$0.316080\pi$
$938$		0			0
$939$	−1.19474e6	+	15299.7i	−1.35500	+	0.0173521i
$940$		0			0
$941$			706010.i			0.797318i	0.917099	+	0.398659i	$0.130524\pi$
$941$			706010.i			0.797318i	−0.917099	+	0.398659i	$0.869476\pi$
$942$		0			0
$943$	13507.9			0.0151902
$944$		0			0
$945$	−41350.5	−	1.07586e6i	−0.0463038	−	1.20474i
$946$		0			0
$947$			1.51964e6i			1.69450i	0.531195	+	0.847250i	$0.321743\pi$
$947$			1.51964e6i			1.69450i	−0.531195	+	0.847250i	$0.678257\pi$
$948$		0			0
$949$	494.448			0.000549020
$950$		0			0
$951$	−4064.77	−	317413.i	−0.00449444	−	0.350964i
$952$		0			0
$953$			672925.i			0.740937i	0.928845	+	0.370468i	$0.120803\pi$
$953$			672925.i			0.740937i	−0.928845	+	0.370468i	$0.879197\pi$
$954$		0			0
$955$	−596555.			−0.654099
$956$		0			0
$957$	2.49034e6	−	31891.3i	2.71917	−	0.0348215i
$958$		0			0
$959$		−	994942.i		−	1.08183i
$960$		0			0
$961$	−876429.			−0.949008
$962$		0			0
$963$	6469.46	+	252554.i	0.00697614	+	0.272334i
$964$		0			0
$965$		−	1.19398e6i		−	1.28216i
$966$		0			0
$967$	280845.			0.300341			0.150170	−	0.988660i	$-0.452018\pi$
$967$	280845.			0.300341			0.150170	+	0.988660i	$0.452018\pi$
$968$		0			0
$969$	10980.7	+	857469.i	0.0116945	+	0.913210i
$970$		0			0
$971$			430675.i			0.456784i	0.973569	+	0.228392i	$0.0733468\pi$
$971$			430675.i			0.456784i	−0.973569	+	0.228392i	$0.926653\pi$
$972$		0			0
$973$	543829.			0.574430
$974$		0			0
$975$	1.48321	−	0.0189940i	1.56025e−6	−	1.99805e-8i
$976$		0			0
$977$			1.01475e6i			1.06309i	0.847029	+	0.531547i	$0.178389\pi$
$977$			1.01475e6i			1.06309i	−0.847029	+	0.531547i	$0.821611\pi$
$978$		0			0
$979$	1.90574e6			1.98837
$980$		0			0
$981$	962796.	−	24663.1i	1.00045	−	0.0256277i
$982$		0			0
$983$		−	77690.9i		−	0.0804013i	−0.999192	−	0.0402006i	$-0.987200\pi$
$983$		−	77690.9i		−	0.0804013i	0.999192	−	0.0402006i	$-0.0127997\pi$
$984$		0			0
$985$	−597352.			−0.615683
$986$		0			0
$987$	−21178.6	−	1.65380e6i	−0.0217401	−	1.69766i
$988$		0			0
$989$		−	12597.2i		−	0.0128789i
$990$		0			0
$991$	500260.			0.509388			0.254694	−	0.967022i	$-0.418025\pi$
$991$	500260.			0.509388			0.254694	+	0.967022i	$0.418025\pi$
$992$		0			0
$993$	959232.	−	12283.9i	0.972804	−	0.0124577i
$994$		0			0
$995$		−	1.31813e6i		−	1.33141i
$996$		0			0
$997$	−975422.			−0.981301			−0.490651	−	0.871356i	$-0.663241\pi$
$997$	−975422.			−0.981301			−0.490651	+	0.871356i	$0.663241\pi$
$998$		0			0
$999$	1.41337e6	−	54322.5i	1.41620	−	0.0544313i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.e.b.257.2	yes	16
3.2	odd	2	inner	384.5.e.b.257.1	yes	16
4.3	odd	2		384.5.e.c.257.15	yes	16
8.3	odd	2		384.5.e.a.257.2	yes	16
8.5	even	2		384.5.e.d.257.15	yes	16
12.11	even	2		384.5.e.c.257.16	yes	16
24.5	odd	2		384.5.e.d.257.16	yes	16
24.11	even	2		384.5.e.a.257.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.e.a.257.1	✓	16	24.11	even	2
384.5.e.a.257.2	yes	16	8.3	odd	2
384.5.e.b.257.1	yes	16	3.2	odd	2	inner
384.5.e.b.257.2	yes	16	1.1	even	1	trivial
384.5.e.c.257.15	yes	16	4.3	odd	2
384.5.e.c.257.16	yes	16	12.11	even	2
384.5.e.d.257.15	yes	16	8.5	even	2
384.5.e.d.257.16	yes	16	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-8.99926 + 0.115244i) q^{3} -25.0292i q^{5} -59.0069 q^{7} +(80.9734 - 2.07423i) q^{9} +O(q^{10})\)	\(q+(-8.99926 + 0.115244i) q^{3} -25.0292i q^{5} -59.0069 q^{7} +(80.9734 - 2.07423i) q^{9} +219.418i q^{11} +0.112775 q^{13} +(2.88447 + 225.244i) q^{15} -160.826i q^{17} +592.455 q^{19} +(531.019 - 6.80020i) q^{21} +11.6137i q^{23} -1.46145 q^{25} +(-728.462 + 27.9982i) q^{27} +1261.19i q^{29} -217.007 q^{31} +(-25.2867 - 1974.60i) q^{33} +1476.90i q^{35} -1940.21 q^{37} +(-1.01489 + 0.0129966i) q^{39} -1163.10i q^{41} -1084.69 q^{43} +(-51.9162 - 2026.70i) q^{45} -3114.40i q^{47} +1080.81 q^{49} +(18.5342 + 1447.31i) q^{51} -4799.21i q^{53} +5491.86 q^{55} +(-5331.66 + 68.2770i) q^{57} +4250.36i q^{59} +3336.81 q^{61} +(-4777.99 + 122.394i) q^{63} -2.82266i q^{65} +2642.37 q^{67} +(-1.33841 - 104.514i) q^{69} -6539.46i q^{71} +4384.38 q^{73} +(13.1520 - 0.168424i) q^{75} -12947.2i q^{77} -4151.19 q^{79} +(6552.40 - 335.914i) q^{81} -6495.95i q^{83} -4025.34 q^{85} +(-145.345 - 11349.8i) q^{87} -8685.42i q^{89} -6.65449 q^{91} +(1952.90 - 25.0088i) q^{93} -14828.7i q^{95} +274.827 q^{97} +(455.123 + 17767.0i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{3} + 80 q^{7}+O(q^{10}) \) 16 * q - 8 * q^3 + 80 * q^7	\( 16 q - 8 q^{3} + 80 q^{7} + 416 q^{15} + 816 q^{19} + 608 q^{21} - 2000 q^{25} + 280 q^{27} + 592 q^{31} - 496 q^{33} + 2240 q^{37} - 16 q^{39} + 368 q^{43} + 800 q^{45} + 3984 q^{49} - 352 q^{51} + 1920 q^{55} + 560 q^{57} - 3520 q^{61} - 816 q^{63} - 3536 q^{67} - 10784 q^{69} + 3680 q^{73} + 5112 q^{75} - 14448 q^{79} - 624 q^{81} - 11136 q^{85} - 14944 q^{87} - 22944 q^{91} + 13760 q^{93} + 3264 q^{97} - 26976 q^{99}+O(q^{100}) \) 16 * q - 8 * q^3 + 80 * q^7 + 416 * q^15 + 816 * q^19 + 608 * q^21 - 2000 * q^25 + 280 * q^27 + 592 * q^31 - 496 * q^33 + 2240 * q^37 - 16 * q^39 + 368 * q^43 + 800 * q^45 + 3984 * q^49 - 352 * q^51 + 1920 * q^55 + 560 * q^57 - 3520 * q^61 - 816 * q^63 - 3536 * q^67 - 10784 * q^69 + 3680 * q^73 + 5112 * q^75 - 14448 * q^79 - 624 * q^81 - 11136 * q^85 - 14944 * q^87 - 22944 * q^91 + 13760 * q^93 + 3264 * q^97 - 26976 * q^99

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