LMFDB - Embedded newform 384.5.e.d.257.5

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			257.5
Root			$4.43019 + 3.93201i$ of defining polynomial
Character	$\chi$	$=$	384.257
Dual form			384.5.e.d.257.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.70733 - 6.95891i) q^{3} +34.7105i q^{5} +89.5593 q^{7} +(-15.8529 + 79.4335i) q^{9} +O(q^{10})$	\(q+(-5.70733 - 6.95891i) q^{3} +34.7105i q^{5} +89.5593 q^{7} +(-15.8529 + 79.4335i) q^{9} -131.070i q^{11} +174.660 q^{13} +(241.547 - 198.104i) q^{15} +221.593i q^{17} -317.020 q^{19} +(-511.144 - 623.235i) q^{21} +322.390i q^{23} -579.818 q^{25} +(643.248 - 343.034i) q^{27} +365.630i q^{29} +1130.83 q^{31} +(-912.103 + 748.058i) q^{33} +3108.65i q^{35} -1998.39 q^{37} +(-996.840 - 1215.44i) q^{39} -1965.77i q^{41} -1547.00 q^{43} +(-2757.18 - 550.261i) q^{45} +3113.20i q^{47} +5619.87 q^{49} +(1542.04 - 1264.70i) q^{51} +1268.86i q^{53} +4549.50 q^{55} +(1809.34 + 2206.12i) q^{57} -178.620i q^{59} +4290.67 q^{61} +(-1419.77 + 7114.01i) q^{63} +6062.52i q^{65} +6539.87 q^{67} +(2243.48 - 1839.98i) q^{69} -69.7236i q^{71} +9215.83 q^{73} +(3309.21 + 4034.90i) q^{75} -11738.5i q^{77} +3079.92 q^{79} +(-6058.37 - 2518.50i) q^{81} +4983.98i q^{83} -7691.59 q^{85} +(2544.39 - 2086.77i) q^{87} +12790.6i q^{89} +15642.4 q^{91} +(-6454.01 - 7869.34i) q^{93} -11003.9i q^{95} -5222.78 q^{97} +(10411.3 + 2077.83i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−5.70733	−	6.95891i	−0.634147	−	0.773212i
$3$	−5.70733	−	6.95891i	−0.634147	−	0.773212i
$4$		0			0
$5$			34.7105i			1.38842i	0.719773	+	0.694210i	$0.244246\pi$
$5$			34.7105i			1.38842i	−0.719773	+	0.694210i	$0.755754\pi$
$6$		0			0
$7$	89.5593			1.82774			0.913870	−	0.406006i	$-0.133079\pi$
$7$	89.5593			1.82774			0.913870	+	0.406006i	$0.133079\pi$
$8$		0			0
$9$	−15.8529	+	79.4335i	−0.195715	+	0.980661i
$10$		0			0
$11$		−	131.070i		−	1.08322i	−0.840629	−	0.541611i	$-0.817815\pi$
$11$		−	131.070i		−	1.08322i	0.840629	−	0.541611i	$-0.182185\pi$
$12$		0			0
$13$	174.660			1.03349			0.516745	−	0.856140i	$-0.327144\pi$
$13$	174.660			1.03349			0.516745	+	0.856140i	$0.327144\pi$
$14$		0			0
$15$	241.547	−	198.104i	1.07354	−	0.880462i
$16$		0			0
$17$			221.593i			0.766757i	0.923591	+	0.383378i	$0.125239\pi$
$17$			221.593i			0.766757i	−0.923591	+	0.383378i	$0.874761\pi$
$18$		0			0
$19$	−317.020			−0.878172			−0.439086	−	0.898445i	$-0.644698\pi$
$19$	−317.020			−0.878172			−0.439086	+	0.898445i	$0.644698\pi$
$20$		0			0
$21$	−511.144	−	623.235i	−1.15906	−	1.41323i
$22$		0			0
$23$			322.390i			0.609433i	0.952443	+	0.304716i	$0.0985617\pi$
$23$			322.390i			0.609433i	−0.952443	+	0.304716i	$0.901438\pi$
$24$		0			0
$25$	−579.818			−0.927708
$26$		0			0
$27$	643.248	−	343.034i	0.882371	−	0.470554i
$28$		0			0
$29$			365.630i			0.434757i	0.976087	+	0.217378i	$0.0697505\pi$
$29$			365.630i			0.434757i	−0.976087	+	0.217378i	$0.930249\pi$
$30$		0			0
$31$	1130.83			1.17672			0.588361	−	0.808598i	$-0.299773\pi$
$31$	1130.83			1.17672			0.588361	+	0.808598i	$0.299773\pi$
$32$		0			0
$33$	−912.103	+	748.058i	−0.837561	+	0.686922i
$34$		0			0
$35$			3108.65i			2.53767i
$36$		0			0
$37$	−1998.39			−1.45974			−0.729870	−	0.683586i	$-0.760420\pi$
$37$	−1998.39			−1.45974			−0.729870	+	0.683586i	$0.760420\pi$
$38$		0			0
$39$	−996.840	−	1215.44i	−0.655384	−	0.799107i
$40$		0			0
$41$		−	1965.77i		−	1.16941i	−0.811247	−	0.584703i	$-0.801211\pi$
$41$		−	1965.77i		−	1.16941i	0.811247	−	0.584703i	$-0.198789\pi$
$42$		0			0
$43$	−1547.00			−0.836669			−0.418334	−	0.908293i	$-0.637386\pi$
$43$	−1547.00			−0.836669			−0.418334	+	0.908293i	$0.637386\pi$
$44$		0			0
$45$	−2757.18	−	550.261i	−1.36157	−	0.271734i
$46$		0			0
$47$			3113.20i			1.40932i	0.709544	+	0.704662i	$0.248901\pi$
$47$			3113.20i			1.40932i	−0.709544	+	0.704662i	$0.751099\pi$
$48$		0			0
$49$	5619.87			2.34064
$50$		0			0
$51$	1542.04	−	1264.70i	0.592866	−	0.486237i
$52$		0			0
$53$			1268.86i			0.451713i	0.974161	+	0.225857i	$0.0725181\pi$
$53$			1268.86i			0.451713i	−0.974161	+	0.225857i	$0.927482\pi$
$54$		0			0
$55$	4549.50			1.50397
$56$		0			0
$57$	1809.34	+	2206.12i	0.556891	+	0.679014i
$58$		0			0
$59$		−	178.620i		−	0.0513129i	−0.999671	−	0.0256565i	$-0.991832\pi$
$59$		−	178.620i		−	0.0513129i	0.999671	−	0.0256565i	$-0.00816760\pi$
$60$		0			0
$61$	4290.67			1.15310			0.576548	−	0.817063i	$-0.304399\pi$
$61$	4290.67			1.15310			0.576548	+	0.817063i	$0.304399\pi$
$62$		0			0
$63$	−1419.77	+	7114.01i	−0.357715	+	1.79239i
$64$		0			0
$65$			6062.52i			1.43492i
$66$		0			0
$67$	6539.87			1.45687			0.728433	−	0.685118i	$-0.240249\pi$
$67$	6539.87			1.45687			0.728433	+	0.685118i	$0.240249\pi$
$68$		0			0
$69$	2243.48	−	1839.98i	0.471221	−	0.386470i
$70$		0			0
$71$		−	69.7236i		−	0.0138313i	−0.999976	−	0.00691565i	$-0.997799\pi$
$71$		−	69.7236i		−	0.0138313i	0.999976	−	0.00691565i	$-0.00220134\pi$
$72$		0			0
$73$	9215.83			1.72937			0.864687	−	0.502311i	$-0.167517\pi$
$73$	9215.83			1.72937			0.864687	+	0.502311i	$0.167517\pi$
$74$		0			0
$75$	3309.21	+	4034.90i	0.588304	+	0.717316i
$76$		0			0
$77$		−	11738.5i		−	1.97985i
$78$		0			0
$79$	3079.92			0.493499			0.246749	−	0.969079i	$-0.420638\pi$
$79$	3079.92			0.493499			0.246749	+	0.969079i	$0.420638\pi$
$80$		0			0
$81$	−6058.37	−	2518.50i	−0.923392	−	0.383859i
$82$		0			0
$83$			4983.98i			0.723469i	0.932281	+	0.361734i	$0.117815\pi$
$83$			4983.98i			0.723469i	−0.932281	+	0.361734i	$0.882185\pi$
$84$		0			0
$85$	−7691.59			−1.06458
$86$		0			0
$87$	2544.39	−	2086.77i	0.336159	−	0.275700i
$88$		0			0
$89$			12790.6i			1.61477i	0.590024	+	0.807386i	$0.299118\pi$
$89$			12790.6i			1.61477i	−0.590024	+	0.807386i	$0.700882\pi$
$90$		0			0
$91$	15642.4			1.88895
$92$		0			0
$93$	−6454.01	−	7869.34i	−0.746215	−	0.909856i
$94$		0			0
$95$		−	11003.9i		−	1.21927i
$96$		0			0
$97$	−5222.78			−0.555084			−0.277542	−	0.960714i	$-0.589520\pi$
$97$	−5222.78			−0.555084			−0.277542	+	0.960714i	$0.589520\pi$
$98$		0			0
$99$	10411.3	+	2077.83i	1.06227	+	0.212002i
$100$		0			0
$101$		−	2131.15i		−	0.208916i	−0.994529	−	0.104458i	$-0.966689\pi$
$101$		−	2131.15i		−	0.208916i	0.994529	−	0.104458i	$-0.0333107\pi$
$102$		0			0
$103$	−3873.76			−0.365139			−0.182570	−	0.983193i	$-0.558441\pi$
$103$	−3873.76			−0.365139			−0.182570	+	0.983193i	$0.558441\pi$
$104$		0			0
$105$	21632.8	−	17742.1i	1.96216	−	1.60926i
$106$		0			0
$107$		−	7921.68i		−	0.691910i	−0.938251	−	0.345955i	$-0.887555\pi$
$107$		−	7921.68i		−	0.691910i	0.938251	−	0.345955i	$-0.112445\pi$
$108$		0			0
$109$	−4194.10			−0.353009			−0.176504	−	0.984300i	$-0.556479\pi$
$109$	−4194.10			−0.353009			−0.176504	+	0.984300i	$0.556479\pi$
$110$		0			0
$111$	11405.4	+	13906.6i	0.925691	+	1.12869i
$112$		0			0
$113$			13756.0i			1.07729i	0.842532	+	0.538647i	$0.181064\pi$
$113$			13756.0i			1.07729i	−0.842532	+	0.538647i	$0.818936\pi$
$114$		0			0
$115$	−11190.3			−0.846148
$116$		0			0
$117$	−2768.86	+	13873.8i	−0.202269	+	1.01350i
$118$		0			0
$119$			19845.7i			1.40143i
$120$		0			0
$121$	−2538.31			−0.173370
$122$		0			0
$123$	−13679.6	+	11219.3i	−0.904200	+	0.741576i
$124$		0			0
$125$			1568.30i			0.100371i
$126$		0			0
$127$	7070.76			0.438388			0.219194	−	0.975681i	$-0.429657\pi$
$127$	7070.76			0.438388			0.219194	+	0.975681i	$0.429657\pi$
$128$		0			0
$129$	8829.24	+	10765.4i	0.530571	+	0.646923i
$130$		0			0
$131$			26698.2i			1.55575i	0.628419	+	0.777875i	$0.283702\pi$
$131$			26698.2i			1.55575i	−0.628419	+	0.777875i	$0.716298\pi$
$132$		0			0
$133$	−28392.1			−1.60507
$134$		0			0
$135$	11906.9	+	22327.5i	0.653327	+	1.22510i
$136$		0			0
$137$		−	10755.6i		−	0.573050i	−0.958073	−	0.286525i	$-0.907500\pi$
$137$		−	10755.6i		−	0.573050i	0.958073	−	0.286525i	$-0.0925001\pi$
$138$		0			0
$139$	−7700.04			−0.398532			−0.199266	−	0.979945i	$-0.563856\pi$
$139$	−7700.04			−0.398532			−0.199266	+	0.979945i	$0.563856\pi$
$140$		0			0
$141$	21664.4	−	17768.0i	1.08971	−	0.893718i
$142$		0			0
$143$		−	22892.6i		−	1.11950i
$144$		0			0
$145$	−12691.2			−0.603625
$146$		0			0
$147$	−32074.4	−	39108.1i	−1.48431	−	1.80981i
$148$		0			0
$149$			43698.5i			1.96831i	0.177302	+	0.984156i	$0.443263\pi$
$149$			43698.5i			1.96831i	−0.177302	+	0.984156i	$0.556737\pi$
$150$		0			0
$151$	7246.71			0.317824			0.158912	−	0.987293i	$-0.449201\pi$
$151$	7246.71			0.317824			0.158912	+	0.987293i	$0.449201\pi$
$152$		0			0
$153$	−17601.9	−	3512.88i	−0.751928	−	0.150065i
$154$		0			0
$155$			39251.6i			1.63378i
$156$		0			0
$157$	−2236.91			−0.0907507			−0.0453754	−	0.998970i	$-0.514448\pi$
$157$	−2236.91			−0.0907507			−0.0453754	+	0.998970i	$0.514448\pi$
$158$		0			0
$159$	8829.90	−	7241.81i	0.349270	−	0.286453i
$160$		0			0
$161$			28873.0i			1.11389i
$162$		0			0
$163$	11484.4			0.432246			0.216123	−	0.976366i	$-0.430659\pi$
$163$	11484.4			0.432246			0.216123	+	0.976366i	$0.430659\pi$
$164$		0			0
$165$	−25965.5	−	31659.6i	−0.953736	−	1.16289i
$166$		0			0
$167$		−	19549.6i		−	0.700978i	−0.936567	−	0.350489i	$-0.886015\pi$
$167$		−	19549.6i		−	0.700978i	0.936567	−	0.350489i	$-0.113985\pi$
$168$		0			0
$169$	1945.00			0.0680999
$170$		0			0
$171$	5025.68	−	25182.0i	0.171871	−	0.861189i
$172$		0			0
$173$			8765.86i			0.292889i	0.989219	+	0.146444i	$0.0467829\pi$
$173$			8765.86i			0.292889i	−0.989219	+	0.146444i	$0.953217\pi$
$174$		0			0
$175$	−51928.1			−1.69561
$176$		0			0
$177$	−1243.00	+	1019.44i	−0.0396758	+	0.0325400i
$178$		0			0
$179$		−	35401.6i		−	1.10489i	−0.833551	−	0.552443i	$-0.813696\pi$
$179$		−	35401.6i		−	1.10489i	0.833551	−	0.552443i	$-0.186304\pi$
$180$		0			0
$181$	9816.91			0.299652			0.149826	−	0.988712i	$-0.452129\pi$
$181$	9816.91			0.299652			0.149826	+	0.988712i	$0.452129\pi$
$182$		0			0
$183$	−24488.3	−	29858.4i	−0.731233	−	0.891589i
$184$		0			0
$185$		−	69364.9i		−	2.02673i
$186$		0			0
$187$	29044.1			0.830568
$188$		0			0
$189$	57608.9	−	30721.9i	1.61275	−	0.860051i
$190$		0			0
$191$		−	16215.8i		−	0.444499i	−0.974990	−	0.222249i	$-0.928660\pi$
$191$		−	16215.8i		−	0.444499i	0.974990	−	0.222249i	$-0.0713399\pi$
$192$		0			0
$193$	−71118.6			−1.90928			−0.954638	−	0.297770i	$-0.903757\pi$
$193$	−71118.6			−1.90928			−0.954638	+	0.297770i	$0.903757\pi$
$194$		0			0
$195$	42188.5	−	34600.8i	1.10950	−	0.909948i
$196$		0			0
$197$			17080.3i			0.440113i	0.975487	+	0.220056i	$0.0706242\pi$
$197$			17080.3i			0.440113i	−0.975487	+	0.220056i	$0.929376\pi$
$198$		0			0
$199$	26460.8			0.668185			0.334093	−	0.942540i	$-0.391570\pi$
$199$	26460.8			0.668185			0.334093	+	0.942540i	$0.391570\pi$
$200$		0			0
$201$	−37325.1	−	45510.4i	−0.923867	−	1.12647i
$202$		0			0
$203$			32745.6i			0.794622i
$204$		0			0
$205$	68232.9			1.62363
$206$		0			0
$207$	−25608.6	−	5110.81i	−0.597647	−	0.119275i
$208$		0			0
$209$			41551.8i			0.951256i
$210$		0			0
$211$	−10911.9			−0.245096			−0.122548	−	0.992463i	$-0.539106\pi$
$211$	−10911.9			−0.245096			−0.122548	+	0.992463i	$0.539106\pi$
$212$		0			0
$213$	−485.200	+	397.935i	−0.0106945	+	0.00877108i
$214$		0			0
$215$		−	53697.1i		−	1.16165i
$216$		0			0
$217$	101276.			2.15074
$218$		0			0
$219$	−52597.7	−	64132.2i	−1.09668	−	1.33717i
$220$		0			0
$221$			38703.3i			0.792435i
$222$		0			0
$223$	−27204.6			−0.547057			−0.273529	−	0.961864i	$-0.588191\pi$
$223$	−27204.6			−0.547057			−0.273529	+	0.961864i	$0.588191\pi$
$224$		0			0
$225$	9191.78	−	46057.0i	0.181566	−	0.909767i
$226$		0			0
$227$			4140.63i			0.0803552i	0.999193	+	0.0401776i	$0.0127924\pi$
$227$			4140.63i			0.0803552i	−0.999193	+	0.0401776i	$0.987208\pi$
$228$		0			0
$229$	72247.0			1.37768			0.688841	−	0.724913i	$-0.258120\pi$
$229$	72247.0			1.37768			0.688841	+	0.724913i	$0.258120\pi$
$230$		0			0
$231$	−81687.3	+	66995.6i	−1.53084	+	1.25552i
$232$		0			0
$233$			89817.7i			1.65444i	0.561881	+	0.827218i	$0.310078\pi$
$233$			89817.7i			1.65444i	−0.561881	+	0.827218i	$0.689922\pi$
$234$		0			0
$235$	−108061.			−1.95673
$236$		0			0
$237$	−17578.1	−	21432.9i	−0.312951	−	0.381579i
$238$		0			0
$239$			35286.1i			0.617743i	0.951104	+	0.308872i	$0.0999513\pi$
$239$			35286.1i			0.617743i	−0.951104	+	0.308872i	$0.900049\pi$
$240$		0			0
$241$	−18958.7			−0.326418			−0.163209	−	0.986592i	$-0.552185\pi$
$241$	−18958.7			−0.326418			−0.163209	+	0.986592i	$0.552185\pi$
$242$		0			0
$243$	17051.1	+	56533.6i	0.288762	+	0.957401i
$244$		0			0
$245$			195068.i			3.24978i
$246$		0			0
$247$	−55370.7			−0.907582
$248$		0			0
$249$	34683.1	−	28445.2i	0.559395	−	0.458786i
$250$		0			0
$251$		−	69404.7i		−	1.10164i	−0.834623	−	0.550822i	$-0.814314\pi$
$251$		−	69404.7i		−	1.10164i	0.834623	−	0.550822i	$-0.185686\pi$
$252$		0			0
$253$	42255.6			0.660151
$254$		0			0
$255$	43898.4	+	53525.1i	0.675100	+	0.823146i
$256$		0			0
$257$		−	34323.4i		−	0.519666i	−0.965654	−	0.259833i	$-0.916332\pi$
$257$		−	34323.4i		−	0.519666i	0.965654	−	0.259833i	$-0.0836676\pi$
$258$		0			0
$259$	−178974.			−2.66803
$260$		0			0
$261$	−29043.3	−	5796.30i	−0.426349	−	0.0850882i
$262$		0			0
$263$		−	100398.i		−	1.45149i	−0.687966	−	0.725743i	$-0.741496\pi$
$263$		−	100398.i		−	1.45149i	0.687966	−	0.725743i	$-0.258504\pi$
$264$		0			0
$265$	−44042.8			−0.627167
$266$		0			0
$267$	89008.7	−	73000.1i	1.24856	−	1.02400i
$268$		0			0
$269$		−	85438.5i		−	1.18073i	−0.807138	−	0.590363i	$-0.798985\pi$
$269$		−	85438.5i		−	1.18073i	0.807138	−	0.590363i	$-0.201015\pi$
$270$		0			0
$271$	−941.207			−0.0128158			−0.00640791	−	0.999979i	$-0.502040\pi$
$271$	−941.207			−0.0128158			−0.00640791	+	0.999979i	$0.502040\pi$
$272$		0			0
$273$	−89276.2	−	108854.i	−1.19787	−	1.46056i
$274$		0			0
$275$			75996.6i			1.00491i
$276$		0			0
$277$	44481.9			0.579727			0.289863	−	0.957068i	$-0.406390\pi$
$277$	44481.9			0.579727			0.289863	+	0.957068i	$0.406390\pi$
$278$		0			0
$279$	−17926.9	+	89825.8i	−0.230302	+	1.15396i
$280$		0			0
$281$		−	102386.i		−	1.29667i	−0.761356	−	0.648334i	$-0.775466\pi$
$281$		−	102386.i		−	1.29667i	0.761356	−	0.648334i	$-0.224534\pi$
$282$		0			0
$283$	138886.			1.73414			0.867071	−	0.498185i	$-0.166000\pi$
$283$	138886.			1.73414			0.867071	+	0.498185i	$0.166000\pi$
$284$		0			0
$285$	−76575.3	+	62803.0i	−0.942756	+	0.773198i
$286$		0			0
$287$		−	176053.i		−	2.13737i
$288$		0			0
$289$	34417.7			0.412084
$290$		0			0
$291$	29808.1	+	36344.9i	0.352005	+	0.429197i
$292$		0			0
$293$		−	90846.9i		−	1.05822i	−0.848554	−	0.529108i	$-0.822526\pi$
$293$		−	90846.9i		−	1.05822i	0.848554	−	0.529108i	$-0.177474\pi$
$294$		0			0
$295$	6200.00			0.0712439
$296$		0			0
$297$	−44961.4	−	84310.5i	−0.509715	−	0.955804i
$298$		0			0
$299$			56308.5i			0.629842i
$300$		0			0
$301$	−138548.			−1.52921
$302$		0			0
$303$	−14830.5	+	12163.2i	−0.161536	+	0.132483i
$304$		0			0
$305$			148931.i			1.60098i
$306$		0			0
$307$	31650.9			0.335822			0.167911	−	0.985802i	$-0.446298\pi$
$307$	31650.9			0.335822			0.167911	+	0.985802i	$0.446298\pi$
$308$		0			0
$309$	22108.8	+	26957.2i	0.231552	+	0.282330i
$310$		0			0
$311$		−	174051.i		−	1.79952i	−0.436387	−	0.899759i	$-0.643742\pi$
$311$		−	174051.i		−	1.79952i	0.436387	−	0.899759i	$-0.356258\pi$
$312$		0			0
$313$	−120101.			−1.22591			−0.612954	−	0.790119i	$-0.710019\pi$
$313$	−120101.			−1.22591			−0.612954	+	0.790119i	$0.710019\pi$
$314$		0			0
$315$	−246931.	−	49281.0i	−2.48859	−	0.496659i
$316$		0			0
$317$			73382.8i			0.730257i	0.930957	+	0.365129i	$0.118975\pi$
$317$			73382.8i			0.730257i	−0.930957	+	0.365129i	$0.881025\pi$
$318$		0			0
$319$	47923.1			0.470938
$320$		0			0
$321$	−55126.3	+	45211.6i	−0.534994	+	0.438773i
$322$		0			0
$323$		−	70249.4i		−	0.673344i
$324$		0			0
$325$	−101271.			−0.958777
$326$		0			0
$327$	23937.1	+	29186.3i	0.223860	+	0.272951i
$328$		0			0
$329$			278816.i			2.57588i
$330$		0			0
$331$	378.606			0.00345567			0.00172783	−	0.999999i	$-0.499450\pi$
$331$	378.606			0.00345567			0.00172783	+	0.999999i	$0.499450\pi$
$332$		0			0
$333$	31680.2	−	158739.i	0.285693	−	1.43151i
$334$		0			0
$335$			227002.i			2.02274i
$336$		0			0
$337$	−29442.6			−0.259248			−0.129624	−	0.991563i	$-0.541377\pi$
$337$	−29442.6			−0.259248			−0.129624	+	0.991563i	$0.541377\pi$
$338$		0			0
$339$	95726.5	−	78509.8i	0.832977	−	0.683163i
$340$		0			0
$341$		−	148218.i		−	1.27465i
$342$		0			0
$343$	288279.			2.45033
$344$		0			0
$345$	63866.8	+	77872.4i	0.536583	+	0.654252i
$346$		0			0
$347$			118315.i			0.982608i	0.870988	+	0.491304i	$0.163480\pi$
$347$			118315.i			0.982608i	−0.870988	+	0.491304i	$0.836520\pi$
$348$		0			0
$349$	135343.			1.11119			0.555593	−	0.831455i	$-0.312491\pi$
$349$	135343.			1.11119			0.555593	+	0.831455i	$0.312491\pi$
$350$		0			0
$351$	112350.	−	59914.2i	0.911921	−	0.486313i
$352$		0			0
$353$		−	63358.6i		−	0.508459i	−0.967144	−	0.254229i	$-0.918178\pi$
$353$		−	63358.6i		−	0.508459i	0.967144	−	0.254229i	$-0.0818218\pi$
$354$		0			0
$355$	2420.14			0.0192036
$356$		0			0
$357$	138104.	−	113266.i	1.08360	−	0.888714i
$358$		0			0
$359$			45672.7i			0.354379i	0.984177	+	0.177189i	$0.0567005\pi$
$359$			45672.7i			0.354379i	−0.984177	+	0.177189i	$0.943300\pi$
$360$		0			0
$361$	−29819.2			−0.228813
$362$		0			0
$363$	14487.0	+	17663.9i	0.109942	+	0.134052i
$364$		0			0
$365$			319886.i			2.40110i
$366$		0			0
$367$	−126362.			−0.938173			−0.469087	−	0.883152i	$-0.655417\pi$
$367$	−126362.			−0.938173			−0.469087	+	0.883152i	$0.655417\pi$
$368$		0			0
$369$	156148.	+	31163.2i	1.14679	+	0.228870i
$370$		0			0
$371$			113638.i			0.825614i
$372$		0			0
$373$	−246215.			−1.76969			−0.884845	−	0.465886i	$-0.845736\pi$
$373$	−246215.			−1.76969			−0.884845	+	0.465886i	$0.845736\pi$
$374$		0			0
$375$	10913.6	−	8950.79i	0.0776082	−	0.0636500i
$376$		0			0
$377$			63860.9i			0.449316i
$378$		0			0
$379$	115326.			0.802875			0.401437	−	0.915887i	$-0.368511\pi$
$379$	115326.			0.802875			0.401437	+	0.915887i	$0.368511\pi$
$380$		0			0
$381$	−40355.1	−	49204.8i	−0.278003	−	0.338967i
$382$		0			0
$383$			80778.7i			0.550680i	0.961347	+	0.275340i	$0.0887904\pi$
$383$			80778.7i			0.550680i	−0.961347	+	0.275340i	$0.911210\pi$
$384$		0			0
$385$	407450.			2.74886
$386$		0			0
$387$	24524.4	−	122884.i	0.163748	−	0.820488i
$388$		0			0
$389$			30491.5i			0.201502i	0.994912	+	0.100751i	$0.0321246\pi$
$389$			30491.5i			0.201502i	−0.994912	+	0.100751i	$0.967875\pi$
$390$		0			0
$391$	−71439.2			−0.467287
$392$		0			0
$393$	185791.	−	152375.i	1.20292	−	0.986574i
$394$		0			0
$395$			106906.i			0.685183i
$396$		0			0
$397$	8712.29			0.0552779			0.0276389	−	0.999618i	$-0.491201\pi$
$397$	8712.29			0.0552779			0.0276389	+	0.999618i	$0.491201\pi$
$398$		0			0
$399$	162043.	+	197578.i	1.01785	+	1.24106i
$400$		0			0
$401$		−	147800.i		−	0.919146i	−0.888140	−	0.459573i	$-0.848002\pi$
$401$		−	147800.i		−	0.919146i	0.888140	−	0.459573i	$-0.151998\pi$
$402$		0			0
$403$	197510.			1.21613
$404$		0			0
$405$	87418.4	−	210289.i	0.532958	−	1.28205i
$406$		0			0
$407$			261928.i			1.58122i
$408$		0			0
$409$	−130253.			−0.778647			−0.389323	−	0.921101i	$-0.627291\pi$
$409$	−130253.			−0.778647			−0.389323	+	0.921101i	$0.627291\pi$
$410$		0			0
$411$	−74847.1	+	61385.6i	−0.443089	+	0.363398i
$412$		0			0
$413$		−	15997.1i		−	0.0937867i
$414$		0			0
$415$	−172996.			−1.00448
$416$		0			0
$417$	43946.7	+	53583.9i	0.252728	+	0.308150i
$418$		0			0
$419$			29888.0i			0.170243i	0.996371	+	0.0851215i	$0.0271278\pi$
$419$			29888.0i			0.170243i	−0.996371	+	0.0851215i	$0.972872\pi$
$420$		0			0
$421$	−107412.			−0.606019			−0.303010	−	0.952987i	$-0.597991\pi$
$421$	−107412.			−0.606019			−0.303010	+	0.952987i	$0.597991\pi$
$422$		0			0
$423$	−247292.	−	49353.1i	−1.38207	−	0.275825i
$424$		0			0
$425$		−	128483.i		−	0.711326i
$426$		0			0
$427$	384270.			2.10756
$428$		0			0
$429$	−159308.	+	130656.i	−0.865610	+	0.709927i
$430$		0			0
$431$		−	228557.i		−	1.23038i	−0.788378	−	0.615192i	$-0.789079\pi$
$431$		−	228557.i		−	1.23038i	0.788378	−	0.615192i	$-0.210921\pi$
$432$		0			0
$433$	152954.			0.815805			0.407902	−	0.913026i	$-0.366260\pi$
$433$	152954.			0.815805			0.407902	+	0.913026i	$0.366260\pi$
$434$		0			0
$435$	72432.9	+	88317.0i	0.382787	+	0.466730i
$436$		0			0
$437$		−	102204.i		−	0.535187i
$438$		0			0
$439$	97295.9			0.504853			0.252427	−	0.967616i	$-0.418771\pi$
$439$	97295.9			0.504853			0.252427	+	0.967616i	$0.418771\pi$
$440$		0			0
$441$	−89091.0	+	446406.i	−0.458096	+	2.29537i
$442$		0			0
$443$			206055.i			1.04997i	0.851112	+	0.524984i	$0.175929\pi$
$443$			206055.i			1.04997i	−0.851112	+	0.524984i	$0.824071\pi$
$444$		0			0
$445$	−443968.			−2.24198
$446$		0			0
$447$	304094.	−	249402.i	1.52192	−	1.24820i
$448$		0			0
$449$		−	16361.2i		−	0.0811563i	−0.999176	−	0.0405781i	$-0.987080\pi$
$449$		−	16361.2i		−	0.0811563i	0.999176	−	0.0405781i	$-0.0129200\pi$
$450$		0			0
$451$	−257654.			−1.26673
$452$		0			0
$453$	−41359.3	−	50429.2i	−0.201547	−	0.245746i
$454$		0			0
$455$			542955.i			2.62265i
$456$		0			0
$457$	85649.4			0.410102			0.205051	−	0.978751i	$-0.434264\pi$
$457$	85649.4			0.410102			0.205051	+	0.978751i	$0.434264\pi$
$458$		0			0
$459$	76013.9	+	142539.i	0.360801	+	0.676564i
$460$		0			0
$461$		−	206782.i		−	0.972994i	−0.873682	−	0.486497i	$-0.838274\pi$
$461$		−	206782.i		−	0.972994i	0.873682	−	0.486497i	$-0.161726\pi$
$462$		0			0
$463$	−416533.			−1.94306			−0.971532	−	0.236906i	$-0.923867\pi$
$463$	−416533.			−1.94306			−0.971532	+	0.236906i	$0.923867\pi$
$464$		0			0
$465$	273149.	−	224022.i	1.26326	−	1.03606i
$466$		0			0
$467$			199710.i			0.915726i	0.889023	+	0.457863i	$0.151385\pi$
$467$			199710.i			0.915726i	−0.889023	+	0.457863i	$0.848615\pi$
$468$		0			0
$469$	585706.			2.66277
$470$		0			0
$471$	12766.8	+	15566.5i	0.0575493	+	0.0701696i
$472$		0			0
$473$			202765.i			0.906298i
$474$		0			0
$475$	183814.			0.814688
$476$		0			0
$477$	−100790.	−	20115.1i	−0.442977	−	0.0884068i
$478$		0			0
$479$		−	352232.i		−	1.53517i	−0.640945	−	0.767587i	$-0.721457\pi$
$479$		−	352232.i		−	1.53517i	0.640945	−	0.767587i	$-0.278543\pi$
$480$		0			0
$481$	−349037.			−1.50863
$482$		0			0
$483$	200925.	−	164788.i	0.861270	−	0.706367i
$484$		0			0
$485$		−	181285.i		−	0.770689i
$486$		0			0
$487$	230915.			0.973630			0.486815	−	0.873505i	$-0.338159\pi$
$487$	230915.			0.973630			0.486815	+	0.873505i	$0.338159\pi$
$488$		0			0
$489$	−65544.9	−	79918.6i	−0.274108	−	0.334218i
$490$		0			0
$491$		−	227334.i		−	0.942978i	−0.881872	−	0.471489i	$-0.843717\pi$
$491$		−	227334.i		−	0.942978i	0.881872	−	0.471489i	$-0.156283\pi$
$492$		0			0
$493$	−81021.0			−0.333353
$494$		0			0
$495$	−72122.7	+	361383.i	−0.294348	+	1.47488i
$496$		0			0
$497$		−	6244.40i		−	0.0252800i
$498$		0			0
$499$	−131775.			−0.529214			−0.264607	−	0.964356i	$-0.585242\pi$
$499$	−131775.			−0.529214			−0.264607	+	0.964356i	$0.585242\pi$
$500$		0			0
$501$	−136044.	+	111576.i	−0.542005	+	0.444523i
$502$		0			0
$503$		−	229987.i		−	0.909008i	−0.890745	−	0.454504i	$-0.849817\pi$
$503$		−	229987.i		−	0.909008i	0.890745	−	0.454504i	$-0.150183\pi$
$504$		0			0
$505$	73973.2			0.290063
$506$		0			0
$507$	−11100.8	−	13535.1i	−0.0431854	−	0.0526557i
$508$		0			0
$509$		−	104523.i		−	0.403437i	−0.979444	−	0.201719i	$-0.935347\pi$
$509$		−	104523.i		−	0.403437i	0.979444	−	0.201719i	$-0.0646526\pi$
$510$		0			0
$511$	825363.			3.16085
$512$		0			0
$513$	−203923.	+	108749.i	−0.774874	+	0.413228i
$514$		0			0
$515$		−	134460.i		−	0.506966i
$516$		0			0
$517$	408046.			1.52661
$518$		0			0
$519$	61000.9	−	50029.6i	0.226465	−	0.185735i
$520$		0			0
$521$			449049.i			1.65432i	0.561970	+	0.827158i	$0.310044\pi$
$521$			449049.i			1.65432i	−0.561970	+	0.827158i	$0.689956\pi$
$522$		0			0
$523$	−249983.			−0.913917			−0.456958	−	0.889488i	$-0.651061\pi$
$523$	−249983.			−0.913917			−0.456958	+	0.889488i	$0.651061\pi$
$524$		0			0
$525$	296370.	+	361363.i	1.07527	+	1.31107i
$526$		0			0
$527$			250583.i			0.902259i
$528$		0			0
$529$	175906.			0.628592
$530$		0			0
$531$	14188.4	+	2831.65i	0.0503206	+	0.0100427i
$532$		0			0
$533$		−	343341.i		−	1.20857i
$534$		0			0
$535$	274965.			0.960662
$536$		0			0
$537$	−246357.	+	202049.i	−0.854311	+	0.700660i
$538$		0			0
$539$		−	736595.i		−	2.53543i
$540$		0			0
$541$	−114230.			−0.390288			−0.195144	−	0.980775i	$-0.562517\pi$
$541$	−114230.			−0.390288			−0.195144	+	0.980775i	$0.562517\pi$
$542$		0			0
$543$	−56028.3	−	68315.0i	−0.190024	−	0.231695i
$544$		0			0
$545$		−	145579.i		−	0.490124i
$546$		0			0
$547$	−222189.			−0.742589			−0.371294	−	0.928515i	$-0.621086\pi$
$547$	−222189.			−0.742589			−0.371294	+	0.928515i	$0.621086\pi$
$548$		0			0
$549$	−68019.5	+	340823.i	−0.225678	+	1.13080i
$550$		0			0
$551$		−	115912.i		−	0.381791i
$552$		0			0
$553$	275836.			0.901987
$554$		0			0
$555$	−482704.	+	395888.i	−1.56709	+	1.28525i
$556$		0			0
$557$			143286.i			0.461843i	0.972972	+	0.230922i	$0.0741741\pi$
$557$			143286.i			0.461843i	−0.972972	+	0.230922i	$0.925826\pi$
$558$		0			0
$559$	−270199.			−0.864688
$560$		0			0
$561$	−165764.	−	202115.i	−0.526702	−	0.642205i
$562$		0			0
$563$		−	424353.i		−	1.33878i	−0.742910	−	0.669391i	$-0.766555\pi$
$563$		−	424353.i		−	1.33878i	0.742910	−	0.669391i	$-0.233445\pi$
$564$		0			0
$565$	−477476.			−1.49574
$566$		0			0
$567$	−542583.	−	225555.i	−1.68772	−	0.701595i
$568$		0			0
$569$		−	236332.i		−	0.729960i	−0.931015	−	0.364980i	$-0.881076\pi$
$569$		−	236332.i		−	0.729960i	0.931015	−	0.364980i	$-0.118924\pi$
$570$		0			0
$571$	−63626.8			−0.195150			−0.0975749	−	0.995228i	$-0.531109\pi$
$571$	−63626.8			−0.195150			−0.0975749	+	0.995228i	$0.531109\pi$
$572$		0			0
$573$	−112844.	+	92548.6i	−0.343692	+	0.281878i
$574$		0			0
$575$		−	186927.i		−	0.565376i
$576$		0			0
$577$	−160315.			−0.481528			−0.240764	−	0.970584i	$-0.577398\pi$
$577$	−160315.			−0.481528			−0.240764	+	0.970584i	$0.577398\pi$
$578$		0			0
$579$	405897.	+	494908.i	1.21076	+	1.47628i
$580$		0			0
$581$			446361.i			1.32231i
$582$		0			0
$583$	166310.			0.489306
$584$		0			0
$585$	−481568.	−	96108.4i	−1.40717	−	0.280834i
$586$		0			0
$587$		−	111574.i		−	0.323808i	−0.986807	−	0.161904i	$-0.948237\pi$
$587$		−	111574.i		−	0.323808i	0.986807	−	0.161904i	$-0.0517635\pi$
$588$		0			0
$589$	−358496.			−1.03336
$590$		0			0
$591$	118861.	−	97483.1i	0.340301	−	0.279096i
$592$		0			0
$593$		−	309389.i		−	0.879823i	−0.898041	−	0.439912i	$-0.855010\pi$
$593$		−	309389.i		−	0.879823i	0.898041	−	0.439912i	$-0.144990\pi$
$594$		0			0
$595$	−688853.			−1.94578
$596$		0			0
$597$	−151020.	−	184138.i	−0.423728	−	0.516649i
$598$		0			0
$599$		−	596354.i		−	1.66208i	−0.556216	−	0.831038i	$-0.687747\pi$
$599$		−	596354.i		−	1.66208i	0.556216	−	0.831038i	$-0.312253\pi$
$600$		0			0
$601$	−387022.			−1.07149			−0.535743	−	0.844381i	$-0.679968\pi$
$601$	−387022.			−1.07149			−0.535743	+	0.844381i	$0.679968\pi$
$602$		0			0
$603$	−103676.	+	519485.i	−0.285130	+	1.42869i
$604$		0			0
$605$		−	88105.9i		−	0.240710i
$606$		0			0
$607$	−547106.			−1.48489			−0.742445	−	0.669907i	$-0.766334\pi$
$607$	−547106.			−1.48489			−0.742445	+	0.669907i	$0.766334\pi$
$608$		0			0
$609$	227874.	−	186890.i	0.614412	−	0.503908i
$610$		0			0
$611$			543750.i			1.45652i
$612$		0			0
$613$	542429.			1.44352			0.721759	−	0.692145i	$-0.243334\pi$
$613$	542429.			1.44352			0.721759	+	0.692145i	$0.243334\pi$
$614$		0			0
$615$	−389427.	−	474827.i	−1.02962	−	1.25541i
$616$		0			0
$617$		−	561014.i		−	1.47368i	−0.676066	−	0.736841i	$-0.736317\pi$
$617$		−	561014.i		−	1.47368i	0.676066	−	0.736841i	$-0.263683\pi$
$618$		0			0
$619$	68209.1			0.178017			0.0890084	−	0.996031i	$-0.471630\pi$
$619$	68209.1			0.178017			0.0890084	+	0.996031i	$0.471630\pi$
$620$		0			0
$621$	110591.	+	207377.i	0.286771	+	0.537746i
$622$		0			0
$623$			1.14552e6i			2.95138i
$624$		0			0
$625$	−416822.			−1.06707
$626$		0			0
$627$	289155.	−	237150.i	0.735523	−	0.603236i
$628$		0			0
$629$		−	442827.i		−	1.11927i
$630$		0			0
$631$	−134177.			−0.336991			−0.168496	−	0.985702i	$-0.553891\pi$
$631$	−134177.			−0.336991			−0.168496	+	0.985702i	$0.553891\pi$
$632$		0			0
$633$	62277.8	+	75935.0i	0.155427	+	0.189511i
$634$		0			0
$635$			245430.i			0.608667i
$636$		0			0
$637$	981564.			2.41902
$638$		0			0
$639$	5538.39	+	1105.32i	0.0135638	+	0.00270699i
$640$		0			0
$641$		−	219119.i		−	0.533291i	−0.963795	−	0.266646i	$-0.914085\pi$
$641$		−	219119.i		−	0.533291i	0.963795	−	0.266646i	$-0.0859154\pi$
$642$		0			0
$643$	509154.			1.23148			0.615740	−	0.787949i	$-0.288857\pi$
$643$	509154.			1.23148			0.615740	+	0.787949i	$0.288857\pi$
$644$		0			0
$645$	−373674.	+	306467.i	−0.898200	+	0.736655i
$646$		0			0
$647$			778114.i			1.85881i	0.369064	+	0.929404i	$0.379678\pi$
$647$			778114.i			1.85881i	−0.369064	+	0.929404i	$0.620322\pi$
$648$		0			0
$649$	−23411.7			−0.0555833
$650$		0			0
$651$	−578017.	−	704773.i	−1.36389	−	1.66298i
$652$		0			0
$653$			130029.i			0.304940i	0.988308	+	0.152470i	$0.0487227\pi$
$653$			130029.i			0.304940i	−0.988308	+	0.152470i	$0.951277\pi$
$654$		0			0
$655$	−926708.			−2.16003
$656$		0			0
$657$	−146097.	+	732046.i	−0.338464	+	1.69593i
$658$		0			0
$659$		−	691687.i		−	1.59272i	−0.604825	−	0.796359i	$-0.706757\pi$
$659$		−	691687.i		−	1.59272i	0.604825	−	0.796359i	$-0.293243\pi$
$660$		0			0
$661$	26775.9			0.0612831			0.0306416	−	0.999530i	$-0.490245\pi$
$661$	26775.9			0.0612831			0.0306416	+	0.999530i	$0.490245\pi$
$662$		0			0
$663$	269333.	−	220892.i	0.612720	−	0.502520i
$664$		0			0
$665$		−	985504.i		−	2.22851i
$666$		0			0
$667$	−117876.			−0.264955
$668$		0			0
$669$	155266.	+	189315.i	0.346915	+	0.422992i
$670$		0			0
$671$		−	562378.i		−	1.24906i
$672$		0			0
$673$	559584.			1.23548			0.617740	−	0.786383i	$-0.288049\pi$
$673$	559584.			1.23548			0.617740	+	0.786383i	$0.288049\pi$
$674$		0			0
$675$	−372967.	+	198897.i	−0.818583	+	0.436537i
$676$		0			0
$677$			173500.i			0.378550i	0.981924	+	0.189275i	$0.0606137\pi$
$677$			173500.i			0.378550i	−0.981924	+	0.189275i	$0.939386\pi$
$678$		0			0
$679$	−467749.			−1.01455
$680$		0			0
$681$	28814.2	−	23631.9i	0.0621317	−	0.0509571i
$682$		0			0
$683$		−	607703.i		−	1.30272i	−0.758770	−	0.651358i	$-0.774200\pi$
$683$		−	607703.i		−	1.30272i	0.758770	−	0.651358i	$-0.225800\pi$
$684$		0			0
$685$	373331.			0.795634
$686$		0			0
$687$	−412337.	−	502761.i	−0.873653	−	1.06524i
$688$		0			0
$689$			221619.i			0.466841i
$690$		0			0
$691$	−883564.			−1.85047			−0.925235	−	0.379396i	$-0.876132\pi$
$691$	−883564.			−1.85047			−0.925235	+	0.379396i	$0.876132\pi$
$692$		0			0
$693$	932432.	+	186089.i	1.94156	+	0.387485i
$694$		0			0
$695$		−	267272.i		−	0.553330i
$696$		0			0
$697$	435601.			0.896650
$698$		0			0
$699$	625033.	−	512619.i	1.27923	−	1.04916i
$700$		0			0
$701$			366318.i			0.745456i	0.927941	+	0.372728i	$0.121577\pi$
$701$			366318.i			0.745456i	−0.927941	+	0.372728i	$0.878423\pi$
$702$		0			0
$703$	633529.			1.28190
$704$		0			0
$705$	616736.	+	751983.i	1.24086	+	1.51297i
$706$		0			0
$707$		−	190864.i		−	0.381844i
$708$		0			0
$709$	−163454.			−0.325165			−0.162582	−	0.986695i	$-0.551982\pi$
$709$	−163454.			−0.325165			−0.162582	+	0.986695i	$0.551982\pi$
$710$		0			0
$711$	−48825.7	+	244649.i	−0.0965849	+	0.483955i
$712$		0			0
$713$			364568.i			0.717133i
$714$		0			0
$715$	794614.			1.55433
$716$		0			0
$717$	245553.	−	201389.i	0.477647	−	0.391740i
$718$		0			0
$719$			358095.i			0.692692i	0.938107	+	0.346346i	$0.112578\pi$
$719$			358095.i			0.692692i	−0.938107	+	0.346346i	$0.887422\pi$
$720$		0			0
$721$	−346931.			−0.667380
$722$		0			0
$723$	108203.	+	131932.i	0.206997	+	0.252391i
$724$		0			0
$725$		−	211999.i		−	0.403327i
$726$		0			0
$727$	88494.9			0.167436			0.0837182	−	0.996489i	$-0.473320\pi$
$727$	88494.9			0.167436			0.0837182	+	0.996489i	$0.473320\pi$
$728$		0			0
$729$	296096.	−	441312.i	0.557157	−	0.830407i
$730$		0			0
$731$		−	342804.i		−	0.641521i
$732$		0			0
$733$	384536.			0.715698			0.357849	−	0.933780i	$-0.383510\pi$
$733$	384536.			0.715698			0.357849	+	0.933780i	$0.383510\pi$
$734$		0			0
$735$	1.35746e6	−	1.11332e6i	2.51277	−	2.06084i
$736$		0			0
$737$		−	857179.i		−	1.57811i
$738$		0			0
$739$	465893.			0.853095			0.426548	−	0.904465i	$-0.359730\pi$
$739$	465893.			0.853095			0.426548	+	0.904465i	$0.359730\pi$
$740$		0			0
$741$	316018.	+	385319.i	0.575540	+	0.701753i
$742$		0			0
$743$		−	201374.i		−	0.364775i	−0.983227	−	0.182387i	$-0.941617\pi$
$743$		−	201374.i		−	0.364775i	0.983227	−	0.182387i	$-0.0583825\pi$
$744$		0			0
$745$	−1.51680e6			−2.73284
$746$		0			0
$747$	−395895.	−	79010.4i	−0.709478	−	0.141593i
$748$		0			0
$749$		−	709460.i		−	1.26463i
$750$		0			0
$751$	−151550.			−0.268704			−0.134352	−	0.990934i	$-0.542895\pi$
$751$	−151550.			−0.268704			−0.134352	+	0.990934i	$0.542895\pi$
$752$		0			0
$753$	−482981.	+	396115.i	−0.851805	+	0.698605i
$754$		0			0
$755$			251537.i			0.441274i
$756$		0			0
$757$	−318865.			−0.556436			−0.278218	−	0.960518i	$-0.589744\pi$
$757$	−318865.			−0.556436			−0.278218	+	0.960518i	$0.589744\pi$
$758$		0			0
$759$	−241167.	−	294053.i	−0.418633	−	0.510437i
$760$		0			0
$761$		−	294118.i		−	0.507869i	−0.967221	−	0.253935i	$-0.918275\pi$
$761$		−	294118.i		−	0.507869i	0.967221	−	0.253935i	$-0.0817248\pi$
$762$		0			0
$763$	−375620.			−0.645208
$764$		0			0
$765$	121934.	−	610970.i	0.208354	−	1.04399i
$766$		0			0
$767$		−	31197.8i		−	0.0530314i
$768$		0			0
$769$	−920651.			−1.55683			−0.778417	−	0.627747i	$-0.783977\pi$
$769$	−920651.			−1.55683			−0.778417	+	0.627747i	$0.783977\pi$
$770$		0			0
$771$	−238854.	+	195895.i	−0.401812	+	0.329545i
$772$		0			0
$773$			783795.i			1.31173i	0.754879	+	0.655864i	$0.227695\pi$
$773$			783795.i			1.31173i	−0.754879	+	0.655864i	$0.772305\pi$
$774$		0			0
$775$	−655675.			−1.09165
$776$		0			0
$777$	1.02146e6	+	1.24546e6i	1.69192	+	2.06295i
$778$		0			0
$779$			623190.i			1.02694i
$780$		0			0
$781$	−9138.66			−0.0149824
$782$		0			0
$783$	125424.	+	235191.i	0.204577	+	0.383617i
$784$		0			0
$785$		−	77644.4i		−	0.126000i
$786$		0			0
$787$	30851.2			0.0498106			0.0249053	−	0.999690i	$-0.492072\pi$
$787$	30851.2			0.0498106			0.0249053	+	0.999690i	$0.492072\pi$
$788$		0			0
$789$	−698660.	+	573003.i	−1.12231	+	0.920456i
$790$		0			0
$791$			1.23197e6i			1.96901i
$792$		0			0
$793$	749407.			1.19171
$794$		0			0
$795$	251367.	+	306490.i	0.397716	+	0.484933i
$796$		0			0
$797$		−	1.16858e6i		−	1.83967i	−0.392302	−	0.919836i	$-0.628321\pi$
$797$		−	1.16858e6i		−	1.83967i	0.392302	−	0.919836i	$-0.371679\pi$
$798$		0			0
$799$	−689861.			−1.08061
$800$		0			0
$801$	−1.01600e6	−	202768.i	−1.58354	−	0.316034i
$802$		0			0
$803$		−	1.20792e6i		−	1.87330i
$804$		0			0
$805$	−1.00220e6			−1.54654
$806$		0			0
$807$	−594559.	+	487625.i	−0.912951	+	0.748754i
$808$		0			0
$809$		−	768405.i		−	1.17407i	−0.809562	−	0.587034i	$-0.800296\pi$
$809$		−	768405.i		−	1.17407i	0.809562	−	0.587034i	$-0.199704\pi$
$810$		0			0
$811$	632184.			0.961174			0.480587	−	0.876947i	$-0.340424\pi$
$811$	632184.			0.961174			0.480587	+	0.876947i	$0.340424\pi$
$812$		0			0
$813$	5371.78	+	6549.78i	0.00812712	+	0.00990936i
$814$		0			0
$815$			398627.i			0.600139i
$816$		0			0
$817$	490431.			0.734740
$818$		0			0
$819$	−247977.	+	1.24253e6i	−0.369695	+	1.85242i
$820$		0			0
$821$		−	161462.i		−	0.239543i	−0.992801	−	0.119772i	$-0.961784\pi$
$821$		−	161462.i		−	0.239543i	0.992801	−	0.119772i	$-0.0382162\pi$
$822$		0			0
$823$	1.21328e6			1.79127			0.895637	−	0.444786i	$-0.146720\pi$
$823$	1.21328e6			1.79127			0.895637	+	0.444786i	$0.146720\pi$
$824$		0			0
$825$	528854.	−	433737.i	0.777012	−	0.637264i
$826$		0			0
$827$			1.09147e6i			1.59588i	0.602739	+	0.797938i	$0.294076\pi$
$827$			1.09147e6i			1.59588i	−0.602739	+	0.797938i	$0.705924\pi$
$828$		0			0
$829$	−750993.			−1.09277			−0.546383	−	0.837536i	$-0.683995\pi$
$829$	−750993.			−1.09277			−0.546383	+	0.837536i	$0.683995\pi$
$830$		0			0
$831$	−253872.	−	309545.i	−0.367632	−	0.448252i
$832$		0			0
$833$			1.24532e6i			1.79470i
$834$		0			0
$835$	678575.			0.973252
$836$		0			0
$837$	727404.	−	387913.i	1.03831	−	0.553712i
$838$		0			0
$839$		−	900761.i		−	1.27963i	−0.768527	−	0.639817i	$-0.779010\pi$
$839$		−	900761.i		−	1.27963i	0.768527	−	0.639817i	$-0.220990\pi$
$840$		0			0
$841$	573595.			0.810987
$842$		0			0
$843$	−712496.	+	584351.i	−1.00260	+	0.822278i
$844$		0			0
$845$			67511.9i			0.0945512i
$846$		0			0
$847$	−227329.			−0.316875
$848$		0			0
$849$	−792665.	−	966493.i	−1.09970	−	1.34086i
$850$		0			0
$851$		−	644259.i		−	0.889614i
$852$		0			0
$853$	−1.30051e6			−1.78738			−0.893691	−	0.448683i	$-0.851893\pi$
$853$	−1.30051e6			−1.78738			−0.893691	+	0.448683i	$0.851893\pi$
$854$		0			0
$855$	874081.	+	174444.i	1.19569	+	0.238629i
$856$		0			0
$857$			102209.i			0.139165i	0.997576	+	0.0695823i	$0.0221666\pi$
$857$			102209.i			0.139165i	−0.997576	+	0.0695823i	$0.977833\pi$
$858$		0			0
$859$	−799181.			−1.08308			−0.541538	−	0.840676i	$-0.682158\pi$
$859$	−799181.			−1.08308			−0.541538	+	0.840676i	$0.682158\pi$
$860$		0			0
$861$	−1.22514e6	+	1.00479e6i	−1.65264	+	1.35541i
$862$		0			0
$863$		−	39343.2i		−	0.0528261i	−0.999651	−	0.0264131i	$-0.991591\pi$
$863$		−	39343.2i		−	0.0528261i	0.999651	−	0.0264131i	$-0.00840851\pi$
$864$		0			0
$865$	−304267.			−0.406652
$866$		0			0
$867$	−196433.	−	239510.i	−0.261322	−	0.318629i
$868$		0			0
$869$		−	403685.i		−	0.534568i
$870$		0			0
$871$	1.14225e6			1.50565
$872$		0			0
$873$	82796.1	−	414864.i	0.108638	−	0.544349i
$874$		0			0
$875$			140456.i			0.183452i
$876$		0			0
$877$	−164547.			−0.213939			−0.106969	−	0.994262i	$-0.534115\pi$
$877$	−164547.			−0.213939			−0.106969	+	0.994262i	$0.534115\pi$
$878$		0			0
$879$	−632195.	+	518493.i	−0.818226	+	0.671065i
$880$		0			0
$881$			1.03663e6i			1.33559i	0.744345	+	0.667795i	$0.232762\pi$
$881$			1.03663e6i			1.33559i	−0.744345	+	0.667795i	$0.767238\pi$
$882$		0			0
$883$	72684.9			0.0932230			0.0466115	−	0.998913i	$-0.485158\pi$
$883$	72684.9			0.0932230			0.0466115	+	0.998913i	$0.485158\pi$
$884$		0			0
$885$	−35385.4	−	43145.2i	−0.0451791	−	0.0550866i
$886$		0			0
$887$		−	126797.i		−	0.161162i	−0.996748	−	0.0805811i	$-0.974322\pi$
$887$		−	126797.i		−	0.161162i	0.996748	−	0.0805811i	$-0.0256776\pi$
$888$		0			0
$889$	633252.			0.801260
$890$		0			0
$891$	−330100.	+	794070.i	−0.415805	+	1.00024i
$892$		0			0
$893$		−	986946.i		−	1.23763i
$894$		0			0
$895$	1.22881e6			1.53404
$896$		0			0
$897$	391846.	−	321371.i	0.487002	−	0.399413i
$898$		0			0
$899$			413466.i			0.511588i
$900$		0			0
$901$	−281170.			−0.346354
$902$		0			0
$903$	790740.	+	964145.i	0.969747	+	1.18241i
$904$		0			0
$905$			340750.i			0.416043i
$906$		0			0
$907$	426524.			0.518477			0.259238	−	0.965813i	$-0.416528\pi$
$907$	426524.			0.518477			0.259238	+	0.965813i	$0.416528\pi$
$908$		0			0
$909$	169285.	+	33784.9i	0.204876	+	0.0408879i
$910$		0			0
$911$			951534.i			1.14654i	0.819368	+	0.573268i	$0.194324\pi$
$911$			951534.i			1.14654i	−0.819368	+	0.573268i	$0.805676\pi$
$912$		0			0
$913$	653249.			0.783677
$914$		0			0
$915$	1.03640e6	−	849999.i	1.23790	−	1.01526i
$916$		0			0
$917$			2.39107e6i			2.84351i
$918$		0			0
$919$	184215.			0.218119			0.109059	−	0.994035i	$-0.465216\pi$
$919$	184215.			0.218119			0.109059	+	0.994035i	$0.465216\pi$
$920$		0			0
$921$	−180642.	−	220256.i	−0.212961	−	0.259662i
$922$		0			0
$923$		−	12177.9i		−	0.0142945i
$924$		0			0
$925$	1.15870e6			1.35421
$926$		0			0
$927$	61410.3	−	307707.i	0.0714630	−	0.358078i
$928$		0			0
$929$			753145.i			0.872664i	0.899786	+	0.436332i	$0.143723\pi$
$929$			753145.i			0.872664i	−0.899786	+	0.436332i	$0.856277\pi$
$930$		0			0
$931$	−1.78161e6			−2.05548
$932$		0			0
$933$	−1.21121e6	+	993367.i	−1.39141	+	1.14116i
$934$		0			0
$935$			1.00814e6i			1.15318i
$936$		0			0
$937$	904855.			1.03062			0.515312	−	0.857003i	$-0.327676\pi$
$937$	904855.			1.03062			0.515312	+	0.857003i	$0.327676\pi$
$938$		0			0
$939$	685455.	+	835772.i	0.777406	+	0.947887i
$940$		0			0
$941$		−	977193.i		−	1.10357i	−0.833985	−	0.551787i	$-0.813946\pi$
$941$		−	977193.i		−	1.10357i	0.833985	−	0.551787i	$-0.186054\pi$
$942$		0			0
$943$	633745.			0.712675
$944$		0			0
$945$	1.06637e6	+	1.99963e6i	1.19411	+	2.23917i
$946$		0			0
$947$		−	489367.i		−	0.545676i	−0.962060	−	0.272838i	$-0.912038\pi$
$947$		−	489367.i		−	0.545676i	0.962060	−	0.272838i	$-0.0879623\pi$
$948$		0			0
$949$	1.60963e6			1.78729
$950$		0			0
$951$	510664.	−	418820.i	0.564644	−	0.463091i
$952$		0			0
$953$			1.19591e6i			1.31678i	0.752678	+	0.658389i	$0.228762\pi$
$953$			1.19591e6i			1.31678i	−0.752678	+	0.658389i	$0.771238\pi$
$954$		0			0
$955$	562857.			0.617151
$956$		0			0
$957$	−273513.	−	333493.i	−0.298644	−	0.364135i
$958$		0			0
$959$		−	963262.i		−	1.04739i
$960$		0			0
$961$	355254.			0.384674
$962$		0			0
$963$	629247.	+	125581.i	0.678529	+	0.135417i
$964$		0			0
$965$		−	2.46856e6i		−	2.65088i
$966$		0			0
$967$	−753201.			−0.805486			−0.402743	−	0.915313i	$-0.631943\pi$
$967$	−753201.			−0.805486			−0.402743	+	0.915313i	$0.631943\pi$
$968$		0			0
$969$	−488859.	+	400936.i	−0.520638	+	0.427000i
$970$		0			0
$971$		−	185520.i		−	0.196767i	−0.995149	−	0.0983835i	$-0.968633\pi$
$971$		−	185520.i		−	0.196767i	0.995149	−	0.0983835i	$-0.0313672\pi$
$972$		0			0
$973$	−689610.			−0.728414
$974$		0			0
$975$	577985.	+	704734.i	0.608006	+	0.741338i
$976$		0			0
$977$		−	656821.i		−	0.688110i	−0.938950	−	0.344055i	$-0.888199\pi$
$977$		−	656821.i		−	0.688110i	0.938950	−	0.344055i	$-0.111801\pi$
$978$		0			0
$979$	1.67646e6			1.74916
$980$		0			0
$981$	66488.5	−	333152.i	0.0690890	−	0.346182i
$982$		0			0
$983$		−	419323.i		−	0.433952i	−0.976177	−	0.216976i	$-0.930381\pi$
$983$		−	419323.i		−	0.433952i	0.976177	−	0.216976i	$-0.0696194\pi$
$984$		0			0
$985$	−592867.			−0.611061
$986$		0			0
$987$	1.94025e6	−	1.59129e6i	1.99170	−	1.63349i
$988$		0			0
$989$		−	498738.i		−	0.509894i
$990$		0			0
$991$	1.11339e6			1.13370			0.566851	−	0.823820i	$-0.308161\pi$
$991$	1.11339e6			1.13370			0.566851	+	0.823820i	$0.308161\pi$
$992$		0			0
$993$	−2160.83	−	2634.69i	−0.00219140	−	0.00267197i
$994$		0			0
$995$			918467.i			0.927721i
$996$		0			0
$997$	140946.			0.141795			0.0708977	−	0.997484i	$-0.477414\pi$
$997$	140946.			0.141795			0.0708977	+	0.997484i	$0.477414\pi$
$998$		0			0
$999$	−1.28546e6	+	685514.i	−1.28803	+	0.686888i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.e.d.257.5	yes	16
3.2	odd	2	inner	384.5.e.d.257.6	yes	16
4.3	odd	2		384.5.e.a.257.12	yes	16
8.3	odd	2		384.5.e.c.257.5	yes	16
8.5	even	2		384.5.e.b.257.12	yes	16
12.11	even	2		384.5.e.a.257.11	✓	16
24.5	odd	2		384.5.e.b.257.11	yes	16
24.11	even	2		384.5.e.c.257.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.e.a.257.11	✓	16	12.11	even	2
384.5.e.a.257.12	yes	16	4.3	odd	2
384.5.e.b.257.11	yes	16	24.5	odd	2
384.5.e.b.257.12	yes	16	8.5	even	2
384.5.e.c.257.5	yes	16	8.3	odd	2
384.5.e.c.257.6	yes	16	24.11	even	2
384.5.e.d.257.5	yes	16	1.1	even	1	trivial
384.5.e.d.257.6	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-5.70733 - 6.95891i) q^{3} +34.7105i q^{5} +89.5593 q^{7} +(-15.8529 + 79.4335i) q^{9} +O(q^{10})\)	\(q+(-5.70733 - 6.95891i) q^{3} +34.7105i q^{5} +89.5593 q^{7} +(-15.8529 + 79.4335i) q^{9} -131.070i q^{11} +174.660 q^{13} +(241.547 - 198.104i) q^{15} +221.593i q^{17} -317.020 q^{19} +(-511.144 - 623.235i) q^{21} +322.390i q^{23} -579.818 q^{25} +(643.248 - 343.034i) q^{27} +365.630i q^{29} +1130.83 q^{31} +(-912.103 + 748.058i) q^{33} +3108.65i q^{35} -1998.39 q^{37} +(-996.840 - 1215.44i) q^{39} -1965.77i q^{41} -1547.00 q^{43} +(-2757.18 - 550.261i) q^{45} +3113.20i q^{47} +5619.87 q^{49} +(1542.04 - 1264.70i) q^{51} +1268.86i q^{53} +4549.50 q^{55} +(1809.34 + 2206.12i) q^{57} -178.620i q^{59} +4290.67 q^{61} +(-1419.77 + 7114.01i) q^{63} +6062.52i q^{65} +6539.87 q^{67} +(2243.48 - 1839.98i) q^{69} -69.7236i q^{71} +9215.83 q^{73} +(3309.21 + 4034.90i) q^{75} -11738.5i q^{77} +3079.92 q^{79} +(-6058.37 - 2518.50i) q^{81} +4983.98i q^{83} -7691.59 q^{85} +(2544.39 - 2086.77i) q^{87} +12790.6i q^{89} +15642.4 q^{91} +(-6454.01 - 7869.34i) q^{93} -11003.9i q^{95} -5222.78 q^{97} +(10411.3 + 2077.83i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} + 80 q^{7}+O(q^{10}) \) 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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