LMFDB - Embedded newform 384.5.e.b.257.8

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			257.8
Root			$2.27178 + 3.46189i$ of defining polynomial
Character	$\chi$	$=$	384.257
Dual form			384.5.e.b.257.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.32132 + 8.36474i) q^{3} +25.4639i q^{5} +36.1251 q^{7} +(-58.9376 - 55.5640i) q^{9} +O(q^{10})$	\(q+(-3.32132 + 8.36474i) q^{3} +25.4639i q^{5} +36.1251 q^{7} +(-58.9376 - 55.5640i) q^{9} +188.568i q^{11} +223.334 q^{13} +(-212.999 - 84.5740i) q^{15} +271.168i q^{17} +255.641 q^{19} +(-119.983 + 302.177i) q^{21} +308.112i q^{23} -23.4125 q^{25} +(660.529 - 308.452i) q^{27} +1044.02i q^{29} +1130.03 q^{31} +(-1577.32 - 626.294i) q^{33} +919.887i q^{35} -1429.54 q^{37} +(-741.766 + 1868.13i) q^{39} -1585.74i q^{41} +2585.12 q^{43} +(1414.88 - 1500.78i) q^{45} +430.138i q^{47} -1095.98 q^{49} +(-2268.25 - 900.637i) q^{51} -316.477i q^{53} -4801.68 q^{55} +(-849.067 + 2138.37i) q^{57} -5792.65i q^{59} -7292.05 q^{61} +(-2129.13 - 2007.25i) q^{63} +5686.97i q^{65} +4835.75 q^{67} +(-2577.28 - 1023.34i) q^{69} +733.491i q^{71} -7142.35 q^{73} +(77.7603 - 195.839i) q^{75} +6812.02i q^{77} -7045.23 q^{79} +(386.289 + 6549.62i) q^{81} +4733.27i q^{83} -6905.01 q^{85} +(-8732.94 - 3467.52i) q^{87} +581.091i q^{89} +8067.97 q^{91} +(-3753.21 + 9452.43i) q^{93} +6509.64i q^{95} +7163.06 q^{97} +(10477.6 - 11113.7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.32132	+	8.36474i	−0.369036	+	0.929415i
$3$	−3.32132	+	8.36474i	−0.369036	+	0.929415i
$4$		0			0
$5$			25.4639i			1.01856i	0.860602	+	0.509279i	$0.170088\pi$
$5$			25.4639i			1.01856i	−0.860602	+	0.509279i	$0.829912\pi$
$6$		0			0
$7$	36.1251			0.737246			0.368623	−	0.929579i	$-0.379829\pi$
$7$	36.1251			0.737246			0.368623	+	0.929579i	$0.379829\pi$
$8$		0			0
$9$	−58.9376	−	55.5640i	−0.727625	−	0.685975i
$10$		0			0
$11$			188.568i			1.55841i	0.626769	+	0.779205i	$0.284377\pi$
$11$			188.568i			1.55841i	−0.626769	+	0.779205i	$0.715623\pi$
$12$		0			0
$13$	223.334			1.32151			0.660753	−	0.750604i	$-0.270237\pi$
$13$	223.334			1.32151			0.660753	+	0.750604i	$0.270237\pi$
$14$		0			0
$15$	−212.999	−	84.5740i	−0.946663	−	0.375884i
$16$		0			0
$17$			271.168i			0.938298i	0.883119	+	0.469149i	$0.155439\pi$
$17$			271.168i			0.938298i	−0.883119	+	0.469149i	$0.844561\pi$
$18$		0			0
$19$	255.641			0.708148			0.354074	−	0.935217i	$-0.384796\pi$
$19$	255.641			0.708148			0.354074	+	0.935217i	$0.384796\pi$
$20$		0			0
$21$	−119.983	+	302.177i	−0.272070	+	0.685208i
$22$		0			0
$23$			308.112i			0.582442i	0.956656	+	0.291221i	$0.0940616\pi$
$23$			308.112i			0.582442i	−0.956656	+	0.291221i	$0.905938\pi$
$24$		0			0
$25$	−23.4125			−0.0374599
$26$		0			0
$27$	660.529	−	308.452i	0.906075	−	0.423116i
$28$		0			0
$29$			1044.02i			1.24140i	0.784047	+	0.620701i	$0.213152\pi$
$29$			1044.02i			1.24140i	−0.784047	+	0.620701i	$0.786848\pi$
$30$		0			0
$31$	1130.03			1.17589			0.587947	−	0.808900i	$-0.299937\pi$
$31$	1130.03			1.17589			0.587947	+	0.808900i	$0.299937\pi$
$32$		0			0
$33$	−1577.32	−	626.294i	−1.44841	−	0.575109i
$34$		0			0
$35$			919.887i			0.750928i
$36$		0			0
$37$	−1429.54			−1.04422			−0.522110	−	0.852878i	$-0.674855\pi$
$37$	−1429.54			−1.04422			−0.522110	+	0.852878i	$0.674855\pi$
$38$		0			0
$39$	−741.766	+	1868.13i	−0.487683	+	1.22823i
$40$		0			0
$41$		−	1585.74i		−	0.943332i	−0.881777	−	0.471666i	$-0.843653\pi$
$41$		−	1585.74i		−	0.943332i	0.881777	−	0.471666i	$-0.156347\pi$
$42$		0			0
$43$	2585.12			1.39812			0.699060	−	0.715063i	$-0.253602\pi$
$43$	2585.12			1.39812			0.699060	+	0.715063i	$0.253602\pi$
$44$		0			0
$45$	1414.88	−	1500.78i	0.698705	−	0.741128i
$46$		0			0
$47$			430.138i			0.194721i	0.995249	+	0.0973603i	$0.0310399\pi$
$47$			430.138i			0.194721i	−0.995249	+	0.0973603i	$0.968960\pi$
$48$		0			0
$49$	−1095.98			−0.456468
$50$		0			0
$51$	−2268.25	−	900.637i	−0.872068	−	0.346266i
$52$		0			0
$53$		−	316.477i		−	0.112665i	−0.998412	−	0.0563327i	$-0.982059\pi$
$53$		−	316.477i		−	0.112665i	0.998412	−	0.0563327i	$-0.0179407\pi$
$54$		0			0
$55$	−4801.68			−1.58733
$56$		0			0
$57$	−849.067	+	2138.37i	−0.261332	+	0.658163i
$58$		0			0
$59$		−	5792.65i		−	1.66408i	−0.554719	−	0.832038i	$-0.687174\pi$
$59$		−	5792.65i		−	1.66408i	0.554719	−	0.832038i	$-0.312826\pi$
$60$		0			0
$61$	−7292.05			−1.95970			−0.979851	−	0.199732i	$-0.935993\pi$
$61$	−7292.05			−1.95970			−0.979851	+	0.199732i	$0.935993\pi$
$62$		0			0
$63$	−2129.13	−	2007.25i	−0.536439	−	0.505733i
$64$		0			0
$65$			5686.97i			1.34603i
$66$		0			0
$67$	4835.75			1.07724			0.538622	−	0.842547i	$-0.318945\pi$
$67$	4835.75			1.07724			0.538622	+	0.842547i	$0.318945\pi$
$68$		0			0
$69$	−2577.28	−	1023.34i	−0.541331	−	0.214942i
$70$		0			0
$71$			733.491i			0.145505i	0.997350	+	0.0727525i	$0.0231783\pi$
$71$			733.491i			0.145505i	−0.997350	+	0.0727525i	$0.976822\pi$
$72$		0			0
$73$	−7142.35			−1.34028			−0.670140	−	0.742235i	$-0.733766\pi$
$73$	−7142.35			−1.34028			−0.670140	+	0.742235i	$0.733766\pi$
$74$		0			0
$75$	77.7603	−	195.839i	0.0138241	−	0.0348158i
$76$		0			0
$77$			6812.02i			1.14893i
$78$		0			0
$79$	−7045.23			−1.12886			−0.564432	−	0.825480i	$-0.690905\pi$
$79$	−7045.23			−1.12886			−0.564432	+	0.825480i	$0.690905\pi$
$80$		0			0
$81$	386.289	+	6549.62i	0.0588766	+	0.998265i
$82$		0			0
$83$			4733.27i			0.687076i	0.939139	+	0.343538i	$0.111625\pi$
$83$			4733.27i			0.687076i	−0.939139	+	0.343538i	$0.888375\pi$
$84$		0			0
$85$	−6905.01			−0.955711
$86$		0			0
$87$	−8732.94	−	3467.52i	−1.15378	−	0.458122i
$88$		0			0
$89$			581.091i			0.0733609i	0.999327	+	0.0366804i	$0.0116784\pi$
$89$			581.091i			0.0733609i	−0.999327	+	0.0366804i	$0.988322\pi$
$90$		0			0
$91$	8067.97			0.974275
$92$		0			0
$93$	−3753.21	+	9452.43i	−0.433947	+	1.09289i
$94$		0			0
$95$			6509.64i			0.721289i
$96$		0			0
$97$	7163.06			0.761299			0.380649	−	0.924719i	$-0.375701\pi$
$97$	7163.06			0.761299			0.380649	+	0.924719i	$0.375701\pi$
$98$		0			0
$99$	10477.6	−	11113.7i	1.06903	−	1.13394i
$100$		0			0
$101$		−	9244.51i		−	0.906236i	−0.891451	−	0.453118i	$-0.850312\pi$
$101$		−	9244.51i		−	0.906236i	0.891451	−	0.453118i	$-0.149688\pi$
$102$		0			0
$103$	−819.090			−0.0772071			−0.0386036	−	0.999255i	$-0.512291\pi$
$103$	−819.090			−0.0772071			−0.0386036	+	0.999255i	$0.512291\pi$
$104$		0			0
$105$	−7694.61	−	3055.24i	−0.697924	−	0.277119i
$106$		0			0
$107$		−	1011.64i		−	0.0883604i	−0.999024	−	0.0441802i	$-0.985932\pi$
$107$		−	1011.64i		−	0.0883604i	0.999024	−	0.0441802i	$-0.0140676\pi$
$108$		0			0
$109$	703.421			0.0592055			0.0296028	−	0.999562i	$-0.490576\pi$
$109$	703.421			0.0592055			0.0296028	+	0.999562i	$0.490576\pi$
$110$		0			0
$111$	4747.95	−	11957.7i	0.385354	−	0.970514i
$112$		0			0
$113$		−	3954.29i		−	0.309679i	−0.987940	−	0.154840i	$-0.950514\pi$
$113$		−	3954.29i		−	0.309679i	0.987940	−	0.154840i	$-0.0494861\pi$
$114$		0			0
$115$	−7845.75			−0.593251
$116$		0			0
$117$	−13162.8	−	12409.3i	−0.961560	−	0.906520i
$118$		0			0
$119$			9795.97i			0.691757i
$120$		0			0
$121$	−20916.8			−1.42864
$122$		0			0
$123$	13264.3	+	5266.76i	0.876747	+	0.348123i
$124$		0			0
$125$			15318.8i			0.980403i
$126$		0			0
$127$	−14940.7			−0.926323			−0.463162	−	0.886274i	$-0.653285\pi$
$127$	−14940.7			−0.926323			−0.463162	+	0.886274i	$0.653285\pi$
$128$		0			0
$129$	−8586.03	+	21623.9i	−0.515957	+	1.29943i
$130$		0			0
$131$		−	3302.98i		−	0.192470i	−0.995359	−	0.0962350i	$-0.969320\pi$
$131$		−	3302.98i		−	0.192470i	0.995359	−	0.0962350i	$-0.0306800\pi$
$132$		0			0
$133$	9235.06			0.522079
$134$		0			0
$135$	7854.40	+	16819.7i	0.430969	+	0.922890i
$136$		0			0
$137$		−	34668.3i		−	1.84710i	−0.383473	−	0.923552i	$-0.625272\pi$
$137$		−	34668.3i		−	1.84710i	0.383473	−	0.923552i	$-0.374728\pi$
$138$		0			0
$139$	−14558.7			−0.753519			−0.376760	−	0.926311i	$-0.622962\pi$
$139$	−14558.7			−0.753519			−0.376760	+	0.926311i	$0.622962\pi$
$140$		0			0
$141$	−3597.99	−	1428.63i	−0.180976	−	0.0718589i
$142$		0			0
$143$			42113.6i			2.05945i
$144$		0			0
$145$	−26584.8			−1.26444
$146$		0			0
$147$	3640.10	−	9167.58i	0.168453	−	0.424248i
$148$		0			0
$149$			27893.4i			1.25640i	0.778051	+	0.628201i	$0.216208\pi$
$149$			27893.4i			1.25640i	−0.778051	+	0.628201i	$0.783792\pi$
$150$		0			0
$151$	18459.2			0.809580			0.404790	−	0.914410i	$-0.367345\pi$
$151$	18459.2			0.809580			0.404790	+	0.914410i	$0.367345\pi$
$152$		0			0
$153$	15067.2	−	15982.0i	0.643649	−	0.682729i
$154$		0			0
$155$			28775.1i			1.19772i
$156$		0			0
$157$	35913.9			1.45701			0.728506	−	0.685039i	$-0.240215\pi$
$157$	35913.9			1.45701			0.728506	+	0.685039i	$0.240215\pi$
$158$		0			0
$159$	2647.25	+	1051.12i	0.104713	+	0.0415776i
$160$		0			0
$161$			11130.6i			0.429404i
$162$		0			0
$163$	49026.8			1.84526			0.922631	−	0.385684i	$-0.126034\pi$
$163$	49026.8			1.84526			0.922631	+	0.385684i	$0.126034\pi$
$164$		0			0
$165$	15947.9	−	40164.8i	0.585782	−	1.47529i
$166$		0			0
$167$		−	36862.9i		−	1.32177i	−0.750487	−	0.660885i	$-0.770181\pi$
$167$		−	36862.9i		−	1.32177i	0.750487	−	0.660885i	$-0.229819\pi$
$168$		0			0
$169$	21317.3			0.746376
$170$		0			0
$171$	−15066.9	−	14204.4i	−0.515266	−	0.485772i
$172$		0			0
$173$			42164.3i			1.40881i	0.709799	+	0.704405i	$0.248786\pi$
$173$			42164.3i			1.40881i	−0.709799	+	0.704405i	$0.751214\pi$
$174$		0			0
$175$	−845.777			−0.0276172
$176$		0			0
$177$	48454.0	+	19239.3i	1.54662	+	0.614104i
$178$		0			0
$179$			31282.8i			0.976335i	0.872750	+	0.488168i	$0.162334\pi$
$179$			31282.8i			0.976335i	−0.872750	+	0.488168i	$0.837666\pi$
$180$		0			0
$181$	−53438.8			−1.63117			−0.815586	−	0.578636i	$-0.803585\pi$
$181$	−53438.8			−1.63117			−0.815586	+	0.578636i	$0.803585\pi$
$182$		0			0
$183$	24219.2	−	60996.1i	0.723200	−	1.82138i
$184$		0			0
$185$		−	36401.6i		−	1.06360i
$186$		0			0
$187$	−51133.5			−1.46225
$188$		0			0
$189$	23861.7	−	11142.8i	0.668001	−	0.311941i
$190$		0			0
$191$		−	37089.3i		−	1.01667i	−0.861158	−	0.508337i	$-0.830261\pi$
$191$		−	37089.3i		−	1.01667i	0.861158	−	0.508337i	$-0.169739\pi$
$192$		0			0
$193$	38238.1			1.02655			0.513277	−	0.858223i	$-0.328431\pi$
$193$	38238.1			1.02655			0.513277	+	0.858223i	$0.328431\pi$
$194$		0			0
$195$	−47570.0	−	18888.3i	−1.25102	−	0.496733i
$196$		0			0
$197$		−	72951.0i		−	1.87974i	−0.341527	−	0.939872i	$-0.610944\pi$
$197$		−	72951.0i		−	1.87974i	0.341527	−	0.939872i	$-0.389056\pi$
$198$		0			0
$199$	3822.20			0.0965177			0.0482589	−	0.998835i	$-0.484633\pi$
$199$	3822.20			0.0965177			0.0482589	+	0.998835i	$0.484633\pi$
$200$		0			0
$201$	−16061.1	+	40449.8i	−0.397542	+	1.00121i
$202$		0			0
$203$			37715.3i			0.915219i
$204$		0			0
$205$	40379.2			0.960838
$206$		0			0
$207$	17119.9	−	18159.4i	0.399541	−	0.423800i
$208$		0			0
$209$			48205.7i			1.10358i
$210$		0			0
$211$	−50444.1			−1.13304			−0.566521	−	0.824047i	$-0.691711\pi$
$211$	−50444.1			−1.13304			−0.566521	+	0.824047i	$0.691711\pi$
$212$		0			0
$213$	−6135.45	−	2436.16i	−0.135235	−	0.0536965i
$214$		0			0
$215$			65827.5i			1.42407i
$216$		0			0
$217$	40822.5			0.866923
$218$		0			0
$219$	23722.0	−	59743.9i	0.494611	−	1.24568i
$220$		0			0
$221$			60561.2i			1.23997i
$222$		0			0
$223$	26913.7			0.541208			0.270604	−	0.962691i	$-0.412777\pi$
$223$	26913.7			0.541208			0.270604	+	0.962691i	$0.412777\pi$
$224$		0			0
$225$	1379.87	+	1300.89i	0.0272568	+	0.0256966i
$226$		0			0
$227$			29898.2i			0.580221i	0.956993	+	0.290110i	$0.0936920\pi$
$227$			29898.2i			0.580221i	−0.956993	+	0.290110i	$0.906308\pi$
$228$		0			0
$229$	59933.1			1.14287			0.571433	−	0.820649i	$-0.306388\pi$
$229$	59933.1			1.14287			0.571433	+	0.820649i	$0.306388\pi$
$230$		0			0
$231$	−56980.8	−	22624.9i	−1.06784	−	0.423997i
$232$		0			0
$233$		−	87067.8i		−	1.60378i	−0.597469	−	0.801892i	$-0.703827\pi$
$233$		−	87067.8i		−	1.60378i	0.597469	−	0.801892i	$-0.296173\pi$
$234$		0			0
$235$	−10953.0			−0.198334
$236$		0			0
$237$	23399.5	−	58931.5i	0.416591	−	1.04918i
$238$		0			0
$239$			17394.6i			0.304522i	0.988340	+	0.152261i	$0.0486554\pi$
$239$			17394.6i			0.304522i	−0.988340	+	0.152261i	$0.951345\pi$
$240$		0			0
$241$	98390.8			1.69403			0.847014	−	0.531571i	$-0.178398\pi$
$241$	98390.8			1.69403			0.847014	+	0.531571i	$0.178398\pi$
$242$		0			0
$243$	−56068.8	−	18522.2i	−0.949530	−	0.313675i
$244$		0			0
$245$		−	27907.9i		−	0.464939i
$246$		0			0
$247$	57093.5			0.935821
$248$		0			0
$249$	−39592.5	−	15720.7i	−0.638579	−	0.253556i
$250$		0			0
$251$		−	4150.13i		−	0.0658740i	−0.999457	−	0.0329370i	$-0.989514\pi$
$251$		−	4150.13i		−	0.0658740i	0.999457	−	0.0329370i	$-0.0104861\pi$
$252$		0			0
$253$	−58100.0			−0.907684
$254$		0			0
$255$	22933.8	−	57758.6i	0.352691	−	0.888252i
$256$		0			0
$257$		−	58616.0i		−	0.887463i	−0.896160	−	0.443731i	$-0.853654\pi$
$257$		−	58616.0i		−	0.887463i	0.896160	−	0.443731i	$-0.146346\pi$
$258$		0			0
$259$	−51642.1			−0.769847
$260$		0			0
$261$	58009.8	−	61532.0i	0.851571	−	0.903275i
$262$		0			0
$263$			90822.5i			1.31305i	0.754303	+	0.656526i	$0.227975\pi$
$263$			90822.5i			1.31305i	−0.754303	+	0.656526i	$0.772025\pi$
$264$		0			0
$265$	8058.75			0.114756
$266$		0			0
$267$	−4860.68	−	1929.99i	−0.0681827	−	0.0270728i
$268$		0			0
$269$		−	16509.2i		−	0.228151i	−0.993472	−	0.114076i	$-0.963609\pi$
$269$		−	16509.2i		−	0.228151i	0.993472	−	0.114076i	$-0.0363906\pi$
$270$		0			0
$271$	92348.8			1.25746			0.628728	−	0.777625i	$-0.283576\pi$
$271$	92348.8			1.25746			0.628728	+	0.777625i	$0.283576\pi$
$272$		0			0
$273$	−26796.3	+	67486.5i	−0.359542	+	0.905506i
$274$		0			0
$275$		−	4414.83i		−	0.0583779i
$276$		0			0
$277$	30737.6			0.400600			0.200300	−	0.979735i	$-0.435808\pi$
$277$	30737.6			0.400600			0.200300	+	0.979735i	$0.435808\pi$
$278$		0			0
$279$	−66601.5	−	62789.1i	−0.855609	−	0.806633i
$280$		0			0
$281$		−	47241.7i		−	0.598292i	−0.954207	−	0.299146i	$-0.903298\pi$
$281$		−	47241.7i		−	0.598292i	0.954207	−	0.299146i	$-0.0967017\pi$
$282$		0			0
$283$	−36666.6			−0.457823			−0.228911	−	0.973447i	$-0.573517\pi$
$283$	−36666.6			−0.457823			−0.228911	+	0.973447i	$0.573517\pi$
$284$		0			0
$285$	−54451.4	−	21620.6i	−0.670377	−	0.266182i
$286$		0			0
$287$		−	57285.0i		−	0.695468i
$288$		0			0
$289$	9988.86			0.119597
$290$		0			0
$291$	−23790.8	+	59917.1i	−0.280946	+	0.707562i
$292$		0			0
$293$			77039.2i			0.897381i	0.893687	+	0.448690i	$0.148109\pi$
$293$			77039.2i			0.897381i	−0.893687	+	0.448690i	$0.851891\pi$
$294$		0			0
$295$	147504.			1.69496
$296$		0			0
$297$	58164.0	+	124554.i	0.659389	+	1.41204i
$298$		0			0
$299$			68812.0i			0.769701i
$300$		0			0
$301$	93387.8			1.03076
$302$		0			0
$303$	77327.9	+	30704.0i	0.842269	+	0.334433i
$304$		0			0
$305$		−	185684.i		−	1.99607i
$306$		0			0
$307$	148105.			1.57143			0.785714	−	0.618591i	$-0.212296\pi$
$307$	148105.			1.57143			0.785714	+	0.618591i	$0.212296\pi$
$308$		0			0
$309$	2720.46	−	6851.47i	0.0284922	−	0.0717575i
$310$		0			0
$311$			32417.7i			0.335167i	0.985858	+	0.167584i	$0.0535964\pi$
$311$			32417.7i			0.335167i	−0.985858	+	0.167584i	$0.946404\pi$
$312$		0			0
$313$	−52340.4			−0.534255			−0.267127	−	0.963661i	$-0.586074\pi$
$313$	−52340.4			−0.534255			−0.267127	+	0.963661i	$0.586074\pi$
$314$		0			0
$315$	51112.6	−	54216.0i	0.515118	−	0.546394i
$316$		0			0
$317$			120554.i			1.19968i	0.800121	+	0.599839i	$0.204769\pi$
$317$			120554.i			1.19968i	−0.800121	+	0.599839i	$0.795231\pi$
$318$		0			0
$319$	−196868.			−1.93461
$320$		0			0
$321$	8462.08	+	3359.98i	0.0821235	+	0.0326081i
$322$		0			0
$323$			69321.8i			0.664453i
$324$		0			0
$325$	−5228.81			−0.0495035
$326$		0			0
$327$	−2336.29	+	5883.93i	−0.0218490	+	0.0550265i
$328$		0			0
$329$			15538.8i			0.143557i
$330$		0			0
$331$	151450.			1.38234			0.691168	−	0.722694i	$-0.257096\pi$
$331$	151450.			1.38234			0.691168	+	0.722694i	$0.257096\pi$
$332$		0			0
$333$	84253.5	+	79430.7i	0.759800	+	0.716309i
$334$		0			0
$335$			123137.i			1.09724i
$336$		0			0
$337$	−133981.			−1.17973			−0.589864	−	0.807503i	$-0.700819\pi$
$337$	−133981.			−1.17973			−0.589864	+	0.807503i	$0.700819\pi$
$338$		0			0
$339$	33076.6	+	13133.5i	0.287821	+	0.114283i
$340$		0			0
$341$			213088.i			1.83252i
$342$		0			0
$343$	−126329.			−1.07378
$344$		0			0
$345$	26058.3	−	65627.6i	0.218931	−	0.551377i
$346$		0			0
$347$			146907.i			1.22007i	0.792376	+	0.610033i	$0.208844\pi$
$347$			146907.i			1.22007i	−0.792376	+	0.610033i	$0.791156\pi$
$348$		0			0
$349$	−98559.0			−0.809181			−0.404590	−	0.914498i	$-0.632586\pi$
$349$	−98559.0			−0.809181			−0.404590	+	0.914498i	$0.632586\pi$
$350$		0			0
$351$	147519.	−	68887.9i	1.19738	−	0.559151i
$352$		0			0
$353$			110450.i			0.886374i	0.896429	+	0.443187i	$0.146152\pi$
$353$			110450.i			0.886374i	−0.896429	+	0.443187i	$0.853848\pi$
$354$		0			0
$355$	−18677.6			−0.148205
$356$		0			0
$357$	−81940.7	−	32535.6i	−0.642929	−	0.255283i
$358$		0			0
$359$			103102.i			0.799982i	0.916519	+	0.399991i	$0.130987\pi$
$359$			103102.i			0.799982i	−0.916519	+	0.399991i	$0.869013\pi$
$360$		0			0
$361$	−64968.5			−0.498527
$362$		0			0
$363$	69471.3	−	174963.i	0.527220	−	1.32780i
$364$		0			0
$365$		−	181872.i		−	1.36515i
$366$		0			0
$367$	−145118.			−1.07743			−0.538717	−	0.842487i	$-0.681091\pi$
$367$	−145118.			−1.07743			−0.538717	+	0.842487i	$0.681091\pi$
$368$		0			0
$369$	−88110.1	+	93459.9i	−0.647102	+	0.686392i
$370$		0			0
$371$		−	11432.8i		−	0.0830621i
$372$		0			0
$373$	−76916.3			−0.552841			−0.276421	−	0.961037i	$-0.589148\pi$
$373$	−76916.3			−0.552841			−0.276421	+	0.961037i	$0.589148\pi$
$374$		0			0
$375$	−128138.	−	50878.6i	−0.911201	−	0.361804i
$376$		0			0
$377$			233165.i			1.64052i
$378$		0			0
$379$	−230763.			−1.60652			−0.803262	−	0.595626i	$-0.796904\pi$
$379$	−230763.			−1.60652			−0.803262	+	0.595626i	$0.796904\pi$
$380$		0			0
$381$	49622.8	−	124975.i	0.341847	−	0.860939i
$382$		0			0
$383$			1948.33i			0.0132820i	0.999978	+	0.00664102i	$0.00211392\pi$
$383$			1948.33i			0.0132820i	−0.999978	+	0.00664102i	$0.997886\pi$
$384$		0			0
$385$	−173461.			−1.17025
$386$		0			0
$387$	−152361.	−	143640.i	−1.01731	−	0.959076i
$388$		0			0
$389$			76869.6i			0.507990i	0.967206	+	0.253995i	$0.0817447\pi$
$389$			76869.6i			0.507990i	−0.967206	+	0.253995i	$0.918255\pi$
$390$		0			0
$391$	−83550.1			−0.546504
$392$		0			0
$393$	27628.5	+	10970.3i	0.178885	+	0.0710283i
$394$		0			0
$395$		−	179399.i		−	1.14981i
$396$		0			0
$397$	−161951.			−1.02755			−0.513775	−	0.857925i	$-0.671754\pi$
$397$	−161951.			−1.02755			−0.513775	+	0.857925i	$0.671754\pi$
$398$		0			0
$399$	−30672.6	+	77248.8i	−0.192666	+	0.485228i
$400$		0			0
$401$		−	114906.i		−	0.714585i	−0.933992	−	0.357293i	$-0.883700\pi$
$401$		−	114906.i		−	0.714585i	0.933992	−	0.357293i	$-0.116300\pi$
$402$		0			0
$403$	252375.			1.55395
$404$		0			0
$405$	−166779.	+	9836.45i	−1.01679	+	0.0599692i
$406$		0			0
$407$		−	269564.i		−	1.62732i
$408$		0			0
$409$	−7993.07			−0.0477823			−0.0238912	−	0.999715i	$-0.507606\pi$
$409$	−7993.07			−0.0477823			−0.0238912	+	0.999715i	$0.507606\pi$
$410$		0			0
$411$	289991.	+	115145.i	1.71673	+	0.681648i
$412$		0			0
$413$		−	209260.i		−	1.22683i
$414$		0			0
$415$	−120528.			−0.699827
$416$		0			0
$417$	48354.3	−	121780.i	0.278076	−	0.700332i
$418$		0			0
$419$		−	49105.1i		−	0.279704i	−0.990172	−	0.139852i	$-0.955337\pi$
$419$		−	49105.1i		−	0.279704i	0.990172	−	0.139852i	$-0.0446627\pi$
$420$		0			0
$421$	−15857.5			−0.0894684			−0.0447342	−	0.998999i	$-0.514244\pi$
$421$	−15857.5			−0.0894684			−0.0447342	+	0.998999i	$0.514244\pi$
$422$		0			0
$423$	23900.2	−	25351.3i	0.133573	−	0.141684i
$424$		0			0
$425$		−	6348.71i		−	0.0351486i
$426$		0			0
$427$	−263426.			−1.44478
$428$		0			0
$429$	−352270.	−	139873.i	−1.91408	−	0.760010i
$430$		0			0
$431$			295316.i			1.58977i	0.606763	+	0.794883i	$0.292468\pi$
$431$			295316.i			1.58977i	−0.606763	+	0.794883i	$0.707532\pi$
$432$		0			0
$433$	85443.2			0.455724			0.227862	−	0.973693i	$-0.426826\pi$
$433$	85443.2			0.455724			0.227862	+	0.973693i	$0.426826\pi$
$434$		0			0
$435$	88296.8	−	222375.i	0.466623	−	1.17519i
$436$		0			0
$437$			78766.2i			0.412455i
$438$		0			0
$439$	133472.			0.692564			0.346282	−	0.938131i	$-0.387444\pi$
$439$	133472.			0.692564			0.346282	+	0.938131i	$0.387444\pi$
$440$		0			0
$441$	64594.4	+	60897.0i	0.332137	+	0.313125i
$442$		0			0
$443$		−	124898.i		−	0.636425i	−0.948019	−	0.318212i	$-0.896918\pi$
$443$		−	124898.i		−	0.636425i	0.948019	−	0.318212i	$-0.103082\pi$
$444$		0			0
$445$	−14796.9			−0.0747223
$446$		0			0
$447$	−233321.	−	92642.9i	−1.16772	−	0.463657i
$448$		0			0
$449$		−	155752.i		−	0.772573i	−0.922379	−	0.386287i	$-0.873758\pi$
$449$		−	155752.i		−	0.772573i	0.922379	−	0.386287i	$-0.126242\pi$
$450$		0			0
$451$	299020.			1.47010
$452$		0			0
$453$	−61309.1	+	154407.i	−0.298764	+	0.752436i
$454$		0			0
$455$			205442.i			0.992355i
$456$		0			0
$457$	−4226.95			−0.0202393			−0.0101196	−	0.999949i	$-0.503221\pi$
$457$	−4226.95			−0.0202393			−0.0101196	+	0.999949i	$0.503221\pi$
$458$		0			0
$459$	83642.3	+	179114.i	0.397009	+	0.850169i
$460$		0			0
$461$			209296.i			0.984827i	0.870361	+	0.492414i	$0.163885\pi$
$461$			209296.i			0.984827i	−0.870361	+	0.492414i	$0.836115\pi$
$462$		0			0
$463$	−306346.			−1.42906			−0.714529	−	0.699605i	$-0.753359\pi$
$463$	−306346.			−1.42906			−0.714529	+	0.699605i	$0.753359\pi$
$464$		0			0
$465$	−240696.	−	95571.4i	−1.11317	−	0.442000i
$466$		0			0
$467$		−	128425.i		−	0.588865i	−0.955672	−	0.294433i	$-0.904869\pi$
$467$		−	128425.i		−	0.588865i	0.955672	−	0.294433i	$-0.0951306\pi$
$468$		0			0
$469$	174692.			0.794195
$470$		0			0
$471$	−119282.	+	300410.i	−0.537690	+	1.35417i
$472$		0			0
$473$			487471.i			2.17885i
$474$		0			0
$475$	−5985.19			−0.0265272
$476$		0			0
$477$	−17584.7	+	18652.4i	−0.0772856	+	0.0819782i
$478$		0			0
$479$		−	401383.i		−	1.74939i	−0.484670	−	0.874697i	$-0.661060\pi$
$479$		−	401383.i		−	1.74939i	0.484670	−	0.874697i	$-0.338940\pi$
$480$		0			0
$481$	−319265.			−1.37994
$482$		0			0
$483$	−93104.3	−	36968.2i	−0.399094	−	0.158465i
$484$		0			0
$485$			182400.i			0.775427i
$486$		0			0
$487$	323109.			1.36236			0.681179	−	0.732117i	$-0.261468\pi$
$487$	323109.			1.36236			0.681179	+	0.732117i	$0.261468\pi$
$488$		0			0
$489$	−162834.	+	410096.i	−0.680968	+	1.71501i
$490$		0			0
$491$			164041.i			0.680439i	0.940346	+	0.340219i	$0.110501\pi$
$491$			164041.i			0.680439i	−0.940346	+	0.340219i	$0.889499\pi$
$492$		0			0
$493$	−283105.			−1.16480
$494$		0			0
$495$	282999.	+	266800.i	1.15498	+	1.08887i
$496$		0			0
$497$			26497.4i			0.107273i
$498$		0			0
$499$	45149.2			0.181321			0.0906607	−	0.995882i	$-0.471102\pi$
$499$	45149.2			0.181321			0.0906607	+	0.995882i	$0.471102\pi$
$500$		0			0
$501$	308348.	+	122433.i	1.22847	+	0.487781i
$502$		0			0
$503$		−	456800.i		−	1.80547i	−0.430199	−	0.902734i	$-0.641557\pi$
$503$		−	456800.i		−	1.80547i	0.430199	−	0.902734i	$-0.358443\pi$
$504$		0			0
$505$	235402.			0.923053
$506$		0			0
$507$	−70801.5	+	178313.i	−0.275440	+	0.693694i
$508$		0			0
$509$		−	182738.i		−	0.705333i	−0.935749	−	0.352666i	$-0.885275\pi$
$509$		−	182738.i		−	0.705333i	0.935749	−	0.352666i	$-0.114725\pi$
$510$		0			0
$511$	−258018.			−0.988116
$512$		0			0
$513$	168858.	−	78853.0i	0.641635	−	0.299629i
$514$		0			0
$515$		−	20857.3i		−	0.0786399i
$516$		0			0
$517$	−81110.0			−0.303454
$518$		0			0
$519$	−352693.	−	140041.i	−1.30937	−	0.519901i
$520$		0			0
$521$		−	323144.i		−	1.19047i	−0.803550	−	0.595237i	$-0.797058\pi$
$521$		−	323144.i		−	1.19047i	0.803550	−	0.595237i	$-0.202942\pi$
$522$		0			0
$523$	52318.2			0.191271			0.0956355	−	0.995416i	$-0.469512\pi$
$523$	52318.2			0.191271			0.0956355	+	0.995416i	$0.469512\pi$
$524$		0			0
$525$	2809.10	−	7074.70i	0.0101917	−	0.0256678i
$526$		0			0
$527$			306429.i			1.10334i
$528$		0			0
$529$	184908.			0.660761
$530$		0			0
$531$	−321863.	+	341405.i	−1.14151	+	1.21082i
$532$		0			0
$533$		−	354151.i		−	1.24662i
$534$		0			0
$535$	25760.3			0.0900001
$536$		0			0
$537$	−261672.	−	103900.i	−0.907421	−	0.360303i
$538$		0			0
$539$		−	206666.i		−	0.711364i
$540$		0			0
$541$	193461.			0.660995			0.330498	−	0.943807i	$-0.392783\pi$
$541$	193461.			0.660995			0.330498	+	0.943807i	$0.392783\pi$
$542$		0			0
$543$	177488.	−	447002.i	0.601961	−	1.51604i
$544$		0			0
$545$			17911.9i			0.0603042i
$546$		0			0
$547$	193567.			0.646929			0.323464	−	0.946240i	$-0.395152\pi$
$547$	193567.			0.646929			0.323464	+	0.946240i	$0.395152\pi$
$548$		0			0
$549$	429776.	+	405175.i	1.42593	+	1.34431i
$550$		0			0
$551$			266894.i			0.879096i
$552$		0			0
$553$	−254510.			−0.832250
$554$		0			0
$555$	304490.	+	120902.i	0.988524	+	0.392506i
$556$		0			0
$557$		−	48934.6i		−	0.157727i	−0.996885	−	0.0788635i	$-0.974871\pi$
$557$		−	48934.6i		−	0.157727i	0.996885	−	0.0788635i	$-0.0251291\pi$
$558$		0			0
$559$	577347.			1.84762
$560$		0			0
$561$	169831.	−	427718.i	0.539624	−	1.35904i
$562$		0			0
$563$			310424.i			0.979352i	0.871905	+	0.489676i	$0.162885\pi$
$563$			310424.i			0.979352i	−0.871905	+	0.489676i	$0.837115\pi$
$564$		0			0
$565$	100692.			0.315426
$566$		0			0
$567$	13954.7	+	236605.i	0.0434065	+	0.735967i
$568$		0			0
$569$		−	46870.8i		−	0.144770i	−0.997377	−	0.0723848i	$-0.976939\pi$
$569$		−	46870.8i		−	0.144770i	0.997377	−	0.0723848i	$-0.0230610\pi$
$570$		0			0
$571$	146352.			0.448875			0.224437	−	0.974488i	$-0.427946\pi$
$571$	146352.			0.448875			0.224437	+	0.974488i	$0.427946\pi$
$572$		0			0
$573$	310242.	+	123185.i	0.944912	+	0.375189i
$574$		0			0
$575$		−	7213.66i		−	0.0218182i
$576$		0			0
$577$	354046.			1.06343			0.531714	−	0.846924i	$-0.321548\pi$
$577$	354046.			1.06343			0.531714	+	0.846924i	$0.321548\pi$
$578$		0			0
$579$	−127001.	+	319852.i	−0.378835	+	0.954095i
$580$		0			0
$581$			170990.i			0.506544i
$582$		0			0
$583$	59677.3			0.175579
$584$		0			0
$585$	315991.	−	335177.i	0.923343	−	0.979405i
$586$		0			0
$587$		−	216485.i		−	0.628278i	−0.949377	−	0.314139i	$-0.898284\pi$
$587$		−	216485.i		−	0.628278i	0.949377	−	0.314139i	$-0.101716\pi$
$588$		0			0
$589$	288883.			0.832706
$590$		0			0
$591$	610216.	+	242294.i	1.74706	+	0.693693i
$592$		0			0
$593$			50177.5i			0.142692i	0.997452	+	0.0713461i	$0.0227295\pi$
$593$			50177.5i			0.142692i	−0.997452	+	0.0713461i	$0.977271\pi$
$594$		0			0
$595$	−249444.			−0.704594
$596$		0			0
$597$	−12694.8	+	31971.7i	−0.0356185	+	0.0897050i
$598$		0			0
$599$		−	69939.9i		−	0.194927i	−0.995239	−	0.0974634i	$-0.968927\pi$
$599$		−	69939.9i		−	0.194927i	0.995239	−	0.0974634i	$-0.0310729\pi$
$600$		0			0
$601$	2201.53			0.00609502			0.00304751	−	0.999995i	$-0.499030\pi$
$601$	2201.53			0.00609502			0.00304751	+	0.999995i	$0.499030\pi$
$602$		0			0
$603$	−285008.	−	268694.i	−0.783830	−	0.738963i
$604$		0			0
$605$		−	532623.i		−	1.45516i
$606$		0			0
$607$	−125161.			−0.339698			−0.169849	−	0.985470i	$-0.554328\pi$
$607$	−125161.			−0.339698			−0.169849	+	0.985470i	$0.554328\pi$
$608$		0			0
$609$	−315478.	−	125265.i	−0.850618	−	0.337749i
$610$		0			0
$611$			96064.5i			0.257324i
$612$		0			0
$613$	−473207.			−1.25930			−0.629651	−	0.776878i	$-0.716802\pi$
$613$	−473207.			−1.25930			−0.629651	+	0.776878i	$0.716802\pi$
$614$		0			0
$615$	−134112.	+	337762.i	−0.354584	+	0.893018i
$616$		0			0
$617$			287389.i			0.754918i	0.926026	+	0.377459i	$0.123202\pi$
$617$			287389.i			0.754918i	−0.926026	+	0.377459i	$0.876798\pi$
$618$		0			0
$619$	−281515.			−0.734718			−0.367359	−	0.930079i	$-0.619738\pi$
$619$	−281515.			−0.734718			−0.367359	+	0.930079i	$0.619738\pi$
$620$		0			0
$621$	95037.7	+	203517.i	0.246441	+	0.527737i
$622$		0			0
$623$			20992.0i			0.0540850i
$624$		0			0
$625$	−404710.			−1.03606
$626$		0			0
$627$	−403228.	−	160107.i	−1.02569	−	0.407262i
$628$		0			0
$629$		−	387645.i		−	0.979789i
$630$		0			0
$631$	730802.			1.83544			0.917722	−	0.397223i	$-0.130026\pi$
$631$	730802.			1.83544			0.917722	+	0.397223i	$0.130026\pi$
$632$		0			0
$633$	167541.	−	421952.i	0.418133	−	1.05307i
$634$		0			0
$635$		−	380448.i		−	0.943514i
$636$		0			0
$637$	−244770.			−0.603225
$638$		0			0
$639$	40755.7	−	43230.2i	0.0998128	−	0.105873i
$640$		0			0
$641$			50278.9i			0.122369i	0.998126	+	0.0611843i	$0.0194877\pi$
$641$			50278.9i			0.122369i	−0.998126	+	0.0611843i	$0.980512\pi$
$642$		0			0
$643$	16539.5			0.0400037			0.0200018	−	0.999800i	$-0.493633\pi$
$643$	16539.5			0.0400037			0.0200018	+	0.999800i	$0.493633\pi$
$644$		0			0
$645$	−550629.	−	218634.i	−1.32355	−	0.525532i
$646$		0			0
$647$		−	405235.i		−	0.968051i	−0.875054	−	0.484026i	$-0.839174\pi$
$647$		−	405235.i		−	0.968051i	0.875054	−	0.484026i	$-0.160826\pi$
$648$		0			0
$649$	1.09231e6			2.59331
$650$		0			0
$651$	−135585.	+	341470.i	−0.319926	+	0.805731i
$652$		0			0
$653$		−	240689.i		−	0.564456i	−0.959347	−	0.282228i	$-0.908926\pi$
$653$		−	240689.i		−	0.564456i	0.959347	−	0.282228i	$-0.0910735\pi$
$654$		0			0
$655$	84106.8			0.196042
$656$		0			0
$657$	420953.	+	396857.i	0.975221	+	0.919398i
$658$		0			0
$659$		−	452953.i		−	1.04300i	−0.853253	−	0.521498i	$-0.825373\pi$
$659$		−	452953.i		−	1.04300i	0.853253	−	0.521498i	$-0.174627\pi$
$660$		0			0
$661$	788032.			1.80360			0.901801	−	0.432151i	$-0.142245\pi$
$661$	788032.			1.80360			0.901801	+	0.432151i	$0.142245\pi$
$662$		0			0
$663$	−506578.	−	201143.i	−1.15244	−	0.457592i
$664$		0			0
$665$			235161.i			0.531768i
$666$		0			0
$667$	−321675.			−0.723045
$668$		0			0
$669$	−89389.2	+	225126.i	−0.199725	+	0.503007i
$670$		0			0
$671$		−	1.37504e6i		−	3.05402i
$672$		0			0
$673$	361685.			0.798547			0.399274	−	0.916832i	$-0.369262\pi$
$673$	361685.			0.798547			0.399274	+	0.916832i	$0.369262\pi$
$674$		0			0
$675$	−15464.6	+	7221.61i	−0.0339415	+	0.0158499i
$676$		0			0
$677$		−	384384.i		−	0.838664i	−0.907833	−	0.419332i	$-0.862264\pi$
$677$		−	384384.i		−	0.838664i	0.907833	−	0.419332i	$-0.137736\pi$
$678$		0			0
$679$	258766.			0.561265
$680$		0			0
$681$	−250090.	−	99301.5i	−0.539266	−	0.214122i
$682$		0			0
$683$		−	4135.23i		−	0.00886459i	−0.999990	−	0.00443230i	$-0.998589\pi$
$683$		−	4135.23i		−	0.00886459i	0.999990	−	0.00443230i	$-0.00141085\pi$
$684$		0			0
$685$	882792.			1.88138
$686$		0			0
$687$	−199057.	+	501324.i	−0.421759	+	1.06220i
$688$		0			0
$689$		−	70680.2i		−	0.148888i
$690$		0			0
$691$	−446884.			−0.935920			−0.467960	−	0.883750i	$-0.655011\pi$
$691$	−446884.			−0.935920			−0.467960	+	0.883750i	$0.655011\pi$
$692$		0			0
$693$	378503.	−	401484.i	0.788139	−	0.835992i
$694$		0			0
$695$		−	370723.i		−	0.767503i
$696$		0			0
$697$	430003.			0.885127
$698$		0			0
$699$	728299.	+	289180.i	1.49058	+	0.591854i
$700$		0			0
$701$			79000.1i			0.160765i	0.996764	+	0.0803825i	$0.0256142\pi$
$701$			79000.1i			0.160765i	−0.996764	+	0.0803825i	$0.974386\pi$
$702$		0			0
$703$	−365449.			−0.739462
$704$		0			0
$705$	36378.5	−	91619.0i	0.0731924	−	0.184335i
$706$		0			0
$707$		−	333959.i		−	0.668119i
$708$		0			0
$709$	2258.67			0.00449325			0.00224662	−	0.999997i	$-0.499285\pi$
$709$	2258.67			0.00449325			0.00224662	+	0.999997i	$0.499285\pi$
$710$		0			0
$711$	415229.	+	391461.i	0.821389	+	0.774372i
$712$		0			0
$713$			348177.i			0.684890i
$714$		0			0
$715$	−1.07238e6			−2.09767
$716$		0			0
$717$	−145501.	−	57773.0i	−0.283027	−	0.112379i
$718$		0			0
$719$			76489.5i			0.147960i	0.997260	+	0.0739800i	$0.0235701\pi$
$719$			76489.5i			0.147960i	−0.997260	+	0.0739800i	$0.976430\pi$
$720$		0			0
$721$	−29589.7			−0.0569207
$722$		0			0
$723$	−326788.	+	823013.i	−0.625157	+	1.57445i
$724$		0			0
$725$		−	24443.0i		−	0.0465028i
$726$		0			0
$727$	−580596.			−1.09851			−0.549256	−	0.835654i	$-0.685089\pi$
$727$	−580596.			−1.09851			−0.549256	+	0.835654i	$0.685089\pi$
$728$		0			0
$729$	341156.	−	407483.i	0.641945	−	0.766751i
$730$		0			0
$731$			701003.i			1.31185i
$732$		0			0
$733$	112182.			0.208793			0.104396	−	0.994536i	$-0.466709\pi$
$733$	112182.			0.208793			0.104396	+	0.994536i	$0.466709\pi$
$734$		0			0
$735$	233443.	+	92691.3i	0.432121	+	0.171579i
$736$		0			0
$737$			911866.i			1.67879i
$738$		0			0
$739$	−289273.			−0.529686			−0.264843	−	0.964291i	$-0.585320\pi$
$739$	−289273.			−0.529686			−0.264843	+	0.964291i	$0.585320\pi$
$740$		0			0
$741$	−189626.	+	477572.i	−0.345351	+	0.869766i
$742$		0			0
$743$			324384.i			0.587600i	0.955867	+	0.293800i	$0.0949199\pi$
$743$			324384.i			0.587600i	−0.955867	+	0.293800i	$0.905080\pi$
$744$		0			0
$745$	−710275.			−1.27972
$746$		0			0
$747$	262999.	−	278968.i	0.471317	−	0.499934i
$748$		0			0
$749$		−	36545.5i		−	0.0651434i
$750$		0			0
$751$	285499.			0.506203			0.253101	−	0.967440i	$-0.418549\pi$
$751$	285499.			0.506203			0.253101	+	0.967440i	$0.418549\pi$
$752$		0			0
$753$	34714.7	+	13783.9i	0.0612243	+	0.0243099i
$754$		0			0
$755$			470045.i			0.824604i
$756$		0			0
$757$	−41740.7			−0.0728396			−0.0364198	−	0.999337i	$-0.511595\pi$
$757$	−41740.7			−0.0728396			−0.0364198	+	0.999337i	$0.511595\pi$
$758$		0			0
$759$	192969.	−	485991.i	0.334968	−	0.843615i
$760$		0			0
$761$		−	984725.i		−	1.70038i	−0.526477	−	0.850189i	$-0.676487\pi$
$761$		−	984725.i		−	1.70038i	0.526477	−	0.850189i	$-0.323513\pi$
$762$		0			0
$763$	25411.1			0.0436490
$764$		0			0
$765$	406965.	+	383670.i	0.695399	+	0.655594i
$766$		0			0
$767$		−	1.29370e6i		−	2.19909i
$768$		0			0
$769$	683643.			1.15605			0.578025	−	0.816019i	$-0.303824\pi$
$769$	683643.			1.15605			0.578025	+	0.816019i	$0.303824\pi$
$770$		0			0
$771$	490308.	+	194683.i	0.824821	+	0.327505i
$772$		0			0
$773$			36868.8i			0.0617021i	0.999524	+	0.0308511i	$0.00982175\pi$
$773$			36868.8i			0.0617021i	−0.999524	+	0.0308511i	$0.990178\pi$
$774$		0			0
$775$	−26456.9			−0.0440489
$776$		0			0
$777$	171520.	−	431973.i	0.284101	−	0.715508i
$778$		0			0
$779$		−	405381.i		−	0.668019i
$780$		0			0
$781$	−138313.			−0.226756
$782$		0			0
$783$	322030.	+	689605.i	0.525258	+	1.12480i
$784$		0			0
$785$			914509.i			1.48405i
$786$		0			0
$787$	776279.			1.25334			0.626669	−	0.779285i	$-0.284418\pi$
$787$	776279.			1.25334			0.626669	+	0.779285i	$0.284418\pi$
$788$		0			0
$789$	−759706.	−	301651.i	−1.22037	−	0.484563i
$790$		0			0
$791$		−	142849.i		−	0.228310i
$792$		0			0
$793$	−1.62857e6			−2.58976
$794$		0			0
$795$	−26765.7	+	67409.4i	−0.0423492	+	0.106656i
$796$		0			0
$797$			693607.i			1.09193i	0.837807	+	0.545967i	$0.183838\pi$
$797$			693607.i			1.09193i	−0.837807	+	0.545967i	$0.816162\pi$
$798$		0			0
$799$	−116640.			−0.182706
$800$		0			0
$801$	32287.8	−	34248.2i	0.0503237	−	0.0533792i
$802$		0			0
$803$		−	1.34682e6i		−	2.08870i
$804$		0			0
$805$	−283428.			−0.437372
$806$		0			0
$807$	138096.	+	54832.5i	0.212047	+	0.0841960i
$808$		0			0
$809$			347934.i			0.531618i	0.964026	+	0.265809i	$0.0856390\pi$
$809$			347934.i			0.531618i	−0.964026	+	0.265809i	$0.914361\pi$
$810$		0			0
$811$	394762.			0.600197			0.300098	−	0.953908i	$-0.402980\pi$
$811$	394762.			0.600197			0.300098	+	0.953908i	$0.402980\pi$
$812$		0			0
$813$	−306720.	+	772474.i	−0.464046	+	1.16870i
$814$		0			0
$815$			1.24842e6i			1.87951i
$816$		0			0
$817$	660865.			0.990076
$818$		0			0
$819$	−475507.	−	448289.i	−0.708907	−	0.668328i
$820$		0			0
$821$		−	1.02738e6i		−	1.52421i	−0.647456	−	0.762103i	$-0.724167\pi$
$821$		−	1.02738e6i		−	1.52421i	0.647456	−	0.762103i	$-0.275833\pi$
$822$		0			0
$823$	−295450.			−0.436199			−0.218099	−	0.975927i	$-0.569986\pi$
$823$	−295450.			−0.436199			−0.218099	+	0.975927i	$0.569986\pi$
$824$		0			0
$825$	36928.9	+	14663.1i	0.0542573	+	0.0215435i
$826$		0			0
$827$		−	68670.1i		−	0.100405i	−0.998739	−	0.0502026i	$-0.984013\pi$
$827$		−	68670.1i		−	0.100405i	0.998739	−	0.0502026i	$-0.0159867\pi$
$828$		0			0
$829$	−563087.			−0.819345			−0.409672	−	0.912233i	$-0.634357\pi$
$829$	−563087.			−0.819345			−0.409672	+	0.912233i	$0.634357\pi$
$830$		0			0
$831$	−102090.	+	257112.i	−0.147836	+	0.372324i
$832$		0			0
$833$		−	297195.i		−	0.428303i
$834$		0			0
$835$	938674.			1.34630
$836$		0			0
$837$	746420.	−	348561.i	1.06545	−	0.497540i
$838$		0			0
$839$			591270.i			0.839966i	0.907532	+	0.419983i	$0.137964\pi$
$839$			591270.i			0.839966i	−0.907532	+	0.419983i	$0.862036\pi$
$840$		0			0
$841$	−382695.			−0.541079
$842$		0			0
$843$	395164.	+	156905.i	0.556061	+	0.220791i
$844$		0			0
$845$			542821.i			0.760227i
$846$		0			0
$847$	−755619.			−1.05326
$848$		0			0
$849$	121781.	−	306706.i	0.168953	−	0.425507i
$850$		0			0
$851$		−	440457.i		−	0.608198i
$852$		0			0
$853$	61144.5			0.0840348			0.0420174	−	0.999117i	$-0.486622\pi$
$853$	61144.5			0.0840348			0.0420174	+	0.999117i	$0.486622\pi$
$854$		0			0
$855$	361701.	−	383663.i	0.494786	−	0.524828i
$856$		0			0
$857$		−	439874.i		−	0.598918i	−0.954109	−	0.299459i	$-0.903194\pi$
$857$		−	439874.i		−	0.598918i	0.954109	−	0.299459i	$-0.0968061\pi$
$858$		0			0
$859$	455281.			0.617012			0.308506	−	0.951222i	$-0.400171\pi$
$859$	455281.			0.617012			0.308506	+	0.951222i	$0.400171\pi$
$860$		0			0
$861$	479174.	+	190262.i	0.646379	+	0.256653i
$862$		0			0
$863$			932390.i			1.25192i	0.779856	+	0.625959i	$0.215292\pi$
$863$			932390.i			1.25192i	−0.779856	+	0.625959i	$0.784708\pi$
$864$		0			0
$865$	−1.07367e6			−1.43495
$866$		0			0
$867$	−33176.2	+	83554.2i	−0.0441356	+	0.111155i
$868$		0			0
$869$		−	1.32850e6i		−	1.75923i
$870$		0			0
$871$	1.07999e6			1.42358
$872$		0			0
$873$	−422174.	−	398008.i	−0.553940	−	0.522232i
$874$		0			0
$875$			553392.i			0.722798i
$876$		0			0
$877$	−120801.			−0.157062			−0.0785311	−	0.996912i	$-0.525023\pi$
$877$	−120801.			−0.157062			−0.0785311	+	0.996912i	$0.525023\pi$
$878$		0			0
$879$	−644413.	−	255872.i	−0.834039	−	0.331166i
$880$		0			0
$881$			161464.i			0.208029i	0.994576	+	0.104014i	$0.0331688\pi$
$881$			161464.i			0.208029i	−0.994576	+	0.104014i	$0.966831\pi$
$882$		0			0
$883$	697100.			0.894075			0.447037	−	0.894515i	$-0.352479\pi$
$883$	697100.			0.894075			0.447037	+	0.894515i	$0.352479\pi$
$884$		0			0
$885$	−489907.	+	1.23383e6i	−0.625500	+	1.57532i
$886$		0			0
$887$		−	26817.5i		−	0.0340856i	−0.999855	−	0.0170428i	$-0.994575\pi$
$887$		−	26817.5i		−	0.0340856i	0.999855	−	0.0170428i	$-0.00542515\pi$
$888$		0			0
$889$	−539733.			−0.682929
$890$		0			0
$891$	−1.23505e6	+	72841.6i	−1.55571	+	0.0917538i
$892$		0			0
$893$			109961.i			0.137891i
$894$		0			0
$895$	−796582.			−0.994454
$896$		0			0
$897$	−575594.	−	228547.i	−0.715372	−	0.284047i
$898$		0			0
$899$			1.17978e6i			1.45976i
$900$		0			0
$901$	85818.5			0.105714
$902$		0			0
$903$	−310171.	+	781165.i	−0.380387	+	0.958003i
$904$		0			0
$905$		−	1.36076e6i		−	1.66144i
$906$		0			0
$907$	−419060.			−0.509404			−0.254702	−	0.967020i	$-0.581977\pi$
$907$	−419060.			−0.509404			−0.254702	+	0.967020i	$0.581977\pi$
$908$		0			0
$909$	−513662.	+	544849.i	−0.621655	+	0.659400i
$910$		0			0
$911$			1.04118e6i			1.25455i	0.778798	+	0.627274i	$0.215830\pi$
$911$			1.04118e6i			1.25455i	−0.778798	+	0.627274i	$0.784170\pi$
$912$		0			0
$913$	−892541.			−1.07075
$914$		0			0
$915$	1.55320e6	+	616718.i	1.85518	+	0.736621i
$916$		0			0
$917$		−	119320.i		−	0.141898i
$918$		0			0
$919$	−526352.			−0.623225			−0.311613	−	0.950209i	$-0.600869\pi$
$919$	−526352.			−0.623225			−0.311613	+	0.950209i	$0.600869\pi$
$920$		0			0
$921$	−491906.	+	1.23886e6i	−0.579913	+	1.46051i
$922$		0			0
$923$			163814.i			0.192286i
$924$		0			0
$925$	33469.0			0.0391164
$926$		0			0
$927$	48275.2	+	45511.9i	0.0561778	+	0.0529621i
$928$		0			0
$929$			356554.i			0.413137i	0.978432	+	0.206568i	$0.0662296\pi$
$929$			356554.i			0.413137i	−0.978432	+	0.206568i	$0.933770\pi$
$930$		0			0
$931$	−280177.			−0.323247
$932$		0			0
$933$	−271166.	−	107670.i	−0.311509	−	0.123689i
$934$		0			0
$935$		−	1.30206e6i		−	1.48939i
$936$		0			0
$937$	−157213.			−0.179064			−0.0895320	−	0.995984i	$-0.528537\pi$
$937$	−157213.			−0.179064			−0.0895320	+	0.995984i	$0.528537\pi$
$938$		0			0
$939$	173839.	−	437814.i	0.197159	−	0.496545i
$940$		0			0
$941$			1.08113e6i			1.22095i	0.792035	+	0.610476i	$0.209022\pi$
$941$			1.08113e6i			1.22095i	−0.792035	+	0.610476i	$0.790978\pi$
$942$		0			0
$943$	488586.			0.549437
$944$		0			0
$945$	283741.	+	607612.i	0.317730	+	0.680397i
$946$		0			0
$947$		−	1.46083e6i		−	1.62892i	−0.580221	−	0.814459i	$-0.697034\pi$
$947$		−	1.46083e6i		−	1.62892i	0.580221	−	0.814459i	$-0.302966\pi$
$948$		0			0
$949$	−1.59513e6			−1.77119
$950$		0			0
$951$	−1.00841e6	−	400400.i	−1.11500	−	0.442724i
$952$		0			0
$953$			37347.7i			0.0411224i	0.999789	+	0.0205612i	$0.00654529\pi$
$953$			37347.7i			0.0411224i	−0.999789	+	0.0205612i	$0.993455\pi$
$954$		0			0
$955$	944439.			1.03554
$956$		0			0
$957$	653863.	−	1.64675e6i	0.713942	−	1.79806i
$958$		0			0
$959$		−	1.25240e6i		−	1.36177i
$960$		0			0
$961$	353454.			0.382725
$962$		0			0
$963$	−56210.6	+	59623.5i	−0.0606130	+	0.0642932i
$964$		0			0
$965$			973693.i			1.04560i
$966$		0			0
$967$	−106656.			−0.114060			−0.0570301	−	0.998372i	$-0.518163\pi$
$967$	−106656.			−0.114060			−0.0570301	+	0.998372i	$0.518163\pi$
$968$		0			0
$969$	−579858.	−	230240.i	−0.617553	−	0.245207i
$970$		0			0
$971$		−	474453.i		−	0.503216i	−0.967829	−	0.251608i	$-0.919041\pi$
$971$		−	474453.i		−	0.503216i	0.967829	−	0.251608i	$-0.0809594\pi$
$972$		0			0
$973$	−525936.			−0.555529
$974$		0			0
$975$	17366.6	−	43737.6i	0.0182686	−	0.0460093i
$976$		0			0
$977$			1.51550e6i			1.58770i	0.608114	+	0.793850i	$0.291926\pi$
$977$			1.51550e6i			1.58770i	−0.608114	+	0.793850i	$0.708074\pi$
$978$		0			0
$979$	−109575.			−0.114326
$980$		0			0
$981$	−41457.9	−	39084.8i	−0.0430794	−	0.0406135i
$982$		0			0
$983$			651454.i			0.674181i	0.941472	+	0.337090i	$0.109443\pi$
$983$			651454.i			0.674181i	−0.941472	+	0.337090i	$0.890557\pi$
$984$		0			0
$985$	1.85762e6			1.91463
$986$		0			0
$987$	−129978.	−	51609.2i	−0.133424	−	0.0529777i
$988$		0			0
$989$			796508.i			0.814325i
$990$		0			0
$991$	−914173.			−0.930853			−0.465426	−	0.885087i	$-0.654099\pi$
$991$	−914173.			−0.930853			−0.465426	+	0.885087i	$0.654099\pi$
$992$		0			0
$993$	−503015.	+	1.26684e6i	−0.510132	+	1.28476i
$994$		0			0
$995$			97328.2i			0.0983089i
$996$		0			0
$997$	82047.6			0.0825421			0.0412711	−	0.999148i	$-0.486859\pi$
$997$	82047.6			0.0825421			0.0412711	+	0.999148i	$0.486859\pi$
$998$		0			0
$999$	−944250.	+	440943.i	−0.946142	+	0.441826i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.e.b.257.8	yes	16
3.2	odd	2	inner	384.5.e.b.257.7	yes	16
4.3	odd	2		384.5.e.c.257.9	yes	16
8.3	odd	2		384.5.e.a.257.8	yes	16
8.5	even	2		384.5.e.d.257.9	yes	16
12.11	even	2		384.5.e.c.257.10	yes	16
24.5	odd	2		384.5.e.d.257.10	yes	16
24.11	even	2		384.5.e.a.257.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.e.a.257.7	✓	16	24.11	even	2
384.5.e.a.257.8	yes	16	8.3	odd	2
384.5.e.b.257.7	yes	16	3.2	odd	2	inner
384.5.e.b.257.8	yes	16	1.1	even	1	trivial
384.5.e.c.257.9	yes	16	4.3	odd	2
384.5.e.c.257.10	yes	16	12.11	even	2
384.5.e.d.257.9	yes	16	8.5	even	2
384.5.e.d.257.10	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)

\(f(q)\)	\(=\)	\(q+(-3.32132 + 8.36474i) q^{3} +25.4639i q^{5} +36.1251 q^{7} +(-58.9376 - 55.5640i) q^{9} +O(q^{10})\)	\(q+(-3.32132 + 8.36474i) q^{3} +25.4639i q^{5} +36.1251 q^{7} +(-58.9376 - 55.5640i) q^{9} +188.568i q^{11} +223.334 q^{13} +(-212.999 - 84.5740i) q^{15} +271.168i q^{17} +255.641 q^{19} +(-119.983 + 302.177i) q^{21} +308.112i q^{23} -23.4125 q^{25} +(660.529 - 308.452i) q^{27} +1044.02i q^{29} +1130.03 q^{31} +(-1577.32 - 626.294i) q^{33} +919.887i q^{35} -1429.54 q^{37} +(-741.766 + 1868.13i) q^{39} -1585.74i q^{41} +2585.12 q^{43} +(1414.88 - 1500.78i) q^{45} +430.138i q^{47} -1095.98 q^{49} +(-2268.25 - 900.637i) q^{51} -316.477i q^{53} -4801.68 q^{55} +(-849.067 + 2138.37i) q^{57} -5792.65i q^{59} -7292.05 q^{61} +(-2129.13 - 2007.25i) q^{63} +5686.97i q^{65} +4835.75 q^{67} +(-2577.28 - 1023.34i) q^{69} +733.491i q^{71} -7142.35 q^{73} +(77.7603 - 195.839i) q^{75} +6812.02i q^{77} -7045.23 q^{79} +(386.289 + 6549.62i) q^{81} +4733.27i q^{83} -6905.01 q^{85} +(-8732.94 - 3467.52i) q^{87} +581.091i q^{89} +8067.97 q^{91} +(-3753.21 + 9452.43i) q^{93} +6509.64i q^{95} +7163.06 q^{97} +(10477.6 - 11113.7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{3} + 80 q^{7}+O(q^{10}) \) 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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