LMFDB - Embedded newform 384.5.e.c.257.6

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

$q$-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	$-0.866921\pi$
$7$	−89.5593			−1.82774			−0.913870	+	0.406006i	$0.866921\pi$
$8$		0			0
$9$	−15.8529	−	79.4335i	−0.195715	−	0.980661i
$10$		0			0
$11$			131.070i			1.08322i	0.840629	+	0.541611i	$0.182185\pi$
$11$			131.070i			1.08322i	−0.840629	+	0.541611i	$0.817815\pi$
$12$		0			0
$13$	−174.660			−1.03349			−0.516745	−	0.856140i	$-0.672856\pi$
$13$	−174.660			−1.03349			−0.516745	+	0.856140i	$0.672856\pi$
$14$		0			0
$15$	−241.547	−	198.104i	−1.07354	−	0.880462i
$16$		0			0
$17$		−	221.593i		−	0.766757i	−0.923591	−	0.383378i	$-0.874761\pi$
$17$		−	221.593i		−	0.766757i	0.923591	−	0.383378i	$-0.125239\pi$
$18$		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	$-0.700227\pi$
$31$	−1130.83			−1.17672			−0.588361	+	0.808598i	$0.700227\pi$
$32$		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	$-0.239580\pi$
$37$	1998.39			1.45974			0.729870	+	0.683586i	$0.239580\pi$
$38$		0			0
$39$	996.840	−	1215.44i	0.655384	−	0.799107i
$40$		0			0
$41$			1965.77i			1.16941i	0.811247	+	0.584703i	$0.198789\pi$
$41$			1965.77i			1.16941i	−0.811247	+	0.584703i	$0.801211\pi$
$42$		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	$0.00816760\pi$
$59$			178.620i			0.0513129i	−0.999671	+	0.0256565i	$0.991832\pi$
$60$		0			0
$61$	−4290.67			−1.15310			−0.576548	−	0.817063i	$-0.695601\pi$
$61$	−4290.67			−1.15310			−0.576548	+	0.817063i	$0.695601\pi$
$62$		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	$-0.579362\pi$
$79$	−3079.92			−0.493499			−0.246749	+	0.969079i	$0.579362\pi$
$80$		0			0
$81$	−6058.37	+	2518.50i	−0.923392	+	0.383859i
$82$		0			0
$83$		−	4983.98i		−	0.723469i	−0.932281	−	0.361734i	$-0.882185\pi$
$83$		−	4983.98i		−	0.723469i	0.932281	−	0.361734i	$-0.117815\pi$
$84$		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	$-0.700882\pi$
$89$		−	12790.6i		−	1.61477i	0.590024	−	0.807386i	$-0.299118\pi$
$90$		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	$-0.441559\pi$
$103$	3873.76			0.365139			0.182570	+	0.983193i	$0.441559\pi$
$104$		0			0
$105$	21632.8	+	17742.1i	1.96216	+	1.60926i
$106$		0			0
$107$			7921.68i			0.691910i	0.938251	+	0.345955i	$0.112445\pi$
$107$			7921.68i			0.691910i	−0.938251	+	0.345955i	$0.887555\pi$
$108$		0			0
$109$	4194.10			0.353009			0.176504	−	0.984300i	$-0.443521\pi$
$109$	4194.10			0.353009			0.176504	+	0.984300i	$0.443521\pi$
$110$		0			0
$111$	−11405.4	+	13906.6i	−0.925691	+	1.12869i
$112$		0			0
$113$		−	13756.0i		−	1.07729i	−0.842532	−	0.538647i	$-0.818936\pi$
$113$		−	13756.0i		−	1.07729i	0.842532	−	0.538647i	$-0.181064\pi$
$114$		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	$-0.570343\pi$
$127$	−7070.76			−0.438388			−0.219194	+	0.975681i	$0.570343\pi$
$128$		0			0
$129$	8829.24	−	10765.4i	0.530571	−	0.646923i
$130$		0			0
$131$		−	26698.2i		−	1.55575i	−0.628419	−	0.777875i	$-0.716298\pi$
$131$		−	26698.2i		−	1.55575i	0.628419	−	0.777875i	$-0.283702\pi$
$132$		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	$0.0925001\pi$
$137$			10755.6i			0.573050i	−0.958073	+	0.286525i	$0.907500\pi$
$138$		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	$-0.550799\pi$
$151$	−7246.71			−0.317824			−0.158912	+	0.987293i	$0.550799\pi$
$152$		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	$-0.485552\pi$
$157$	2236.91			0.0907507			0.0453754	+	0.998970i	$0.485552\pi$
$158$		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	$0.186304\pi$
$179$			35401.6i			1.10489i	−0.833551	+	0.552443i	$0.813696\pi$
$180$		0			0
$181$	−9816.91			−0.299652			−0.149826	−	0.988712i	$-0.547871\pi$
$181$	−9816.91			−0.299652			−0.149826	+	0.988712i	$0.547871\pi$
$182$		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	$-0.608430\pi$
$199$	−26460.8			−0.668185			−0.334093	+	0.942540i	$0.608430\pi$
$200$		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	$-0.411809\pi$
$223$	27204.6			0.547057			0.273529	+	0.961864i	$0.411809\pi$
$224$		0			0
$225$	9191.78	+	46057.0i	0.181566	+	0.909767i
$226$		0			0
$227$		−	4140.63i		−	0.0803552i	−0.999193	−	0.0401776i	$-0.987208\pi$
$227$		−	4140.63i		−	0.0803552i	0.999193	−	0.0401776i	$-0.0127924\pi$
$228$		0			0
$229$	−72247.0			−1.37768			−0.688841	−	0.724913i	$-0.741880\pi$
$229$	−72247.0			−1.37768			−0.688841	+	0.724913i	$0.741880\pi$
$230$		0			0
$231$	81687.3	+	66995.6i	1.53084	+	1.25552i
$232$		0			0
$233$		−	89817.7i		−	1.65444i	−0.561881	−	0.827218i	$-0.689922\pi$
$233$		−	89817.7i		−	1.65444i	0.561881	−	0.827218i	$-0.310078\pi$
$234$		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	$0.185686\pi$
$251$			69404.7i			1.10164i	−0.834623	+	0.550822i	$0.814314\pi$
$252$		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	$0.0836676\pi$
$257$			34323.4i			0.519666i	−0.965654	+	0.259833i	$0.916332\pi$
$258$		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	$-0.497960\pi$
$271$	941.207			0.0128158			0.00640791	+	0.999979i	$0.497960\pi$
$272$		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	$-0.593610\pi$
$277$	−44481.9			−0.579727			−0.289863	+	0.957068i	$0.593610\pi$
$278$		0			0
$279$	17926.9	+	89825.8i	0.230302	+	1.15396i
$280$		0			0
$281$			102386.i			1.29667i	0.761356	+	0.648334i	$0.224534\pi$
$281$			102386.i			1.29667i	−0.761356	+	0.648334i	$0.775466\pi$
$282$		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	$-0.836520\pi$
$347$		−	118315.i		−	0.982608i	0.870988	−	0.491304i	$-0.163480\pi$
$348$		0			0
$349$	−135343.			−1.11119			−0.555593	−	0.831455i	$-0.687509\pi$
$349$	−135343.			−1.11119			−0.555593	+	0.831455i	$0.687509\pi$
$350$		0			0
$351$	−112350.	−	59914.2i	−0.911921	−	0.486313i
$352$		0			0
$353$			63358.6i			0.508459i	0.967144	+	0.254229i	$0.0818218\pi$
$353$			63358.6i			0.508459i	−0.967144	+	0.254229i	$0.918178\pi$
$354$		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	$-0.344583\pi$
$367$	126362.			0.938173			0.469087	+	0.883152i	$0.344583\pi$
$368$		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	$-0.154264\pi$
$373$	246215.			1.76969			0.884845	+	0.465886i	$0.154264\pi$
$374$		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	$-0.508799\pi$
$397$	−8712.29			−0.0552779			−0.0276389	+	0.999618i	$0.508799\pi$
$398$		0			0
$399$	−162043.	+	197578.i	−1.01785	+	1.24106i
$400$		0			0
$401$			147800.i			0.919146i	0.888140	+	0.459573i	$0.151998\pi$
$401$			147800.i			0.919146i	−0.888140	+	0.459573i	$0.848002\pi$
$402$		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	$-0.972872\pi$
$419$		−	29888.0i		−	0.170243i	0.996371	−	0.0851215i	$-0.0271278\pi$
$420$		0			0
$421$	107412.			0.606019			0.303010	−	0.952987i	$-0.402009\pi$
$421$	107412.			0.606019			0.303010	+	0.952987i	$0.402009\pi$
$422$		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	$-0.581229\pi$
$439$	−97295.9			−0.504853			−0.252427	+	0.967616i	$0.581229\pi$
$440$		0			0
$441$	−89091.0	−	446406.i	−0.458096	−	2.29537i
$442$		0			0
$443$		−	206055.i		−	1.04997i	−0.851112	−	0.524984i	$-0.824071\pi$
$443$		−	206055.i		−	1.04997i	0.851112	−	0.524984i	$-0.175929\pi$
$444$		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	$0.0129200\pi$
$449$			16361.2i			0.0811563i	−0.999176	+	0.0405781i	$0.987080\pi$
$450$		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	$-0.0761335\pi$
$463$	416533.			1.94306			0.971532	+	0.236906i	$0.0761335\pi$
$464$		0			0
$465$	273149.	+	224022.i	1.26326	+	1.03606i
$466$		0			0
$467$		−	199710.i		−	0.915726i	−0.889023	−	0.457863i	$-0.848615\pi$
$467$		−	199710.i		−	0.915726i	0.889023	−	0.457863i	$-0.151385\pi$
$468$		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	$-0.661841\pi$
$487$	−230915.			−0.973630			−0.486815	+	0.873505i	$0.661841\pi$
$488$		0			0
$489$	−65544.9	+	79918.6i	−0.274108	+	0.334218i
$490$		0			0
$491$			227334.i			0.942978i	0.881872	+	0.471489i	$0.156283\pi$
$491$			227334.i			0.942978i	−0.881872	+	0.471489i	$0.843717\pi$
$492$		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	$-0.689956\pi$
$521$		−	449049.i		−	1.65432i	0.561970	−	0.827158i	$-0.310044\pi$
$522$		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	$-0.437483\pi$
$541$	114230.			0.390288			0.195144	+	0.980775i	$0.437483\pi$
$542$		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	$0.233445\pi$
$563$			424353.i			1.33878i	−0.742910	+	0.669391i	$0.766555\pi$
$564$		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	$0.118924\pi$
$569$			236332.i			0.729960i	−0.931015	+	0.364980i	$0.881076\pi$
$570$		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	$0.0517635\pi$
$587$			111574.i			0.323808i	−0.986807	+	0.161904i	$0.948237\pi$
$588$		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	$0.144990\pi$
$593$			309389.i			0.879823i	−0.898041	+	0.439912i	$0.855010\pi$
$594$		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	$-0.233666\pi$
$607$	547106.			1.48489			0.742445	+	0.669907i	$0.233666\pi$
$608$		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	$-0.756666\pi$
$613$	−542429.			−1.44352			−0.721759	+	0.692145i	$0.756666\pi$
$614$		0			0
$615$	389427.	−	474827.i	1.02962	−	1.25541i
$616$		0			0
$617$			561014.i			1.47368i	0.676066	+	0.736841i	$0.263683\pi$
$617$			561014.i			1.47368i	−0.676066	+	0.736841i	$0.736317\pi$
$618$		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	$-0.446109\pi$
$631$	134177.			0.336991			0.168496	+	0.985702i	$0.446109\pi$
$632$		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	$0.0859154\pi$
$641$			219119.i			0.533291i	−0.963795	+	0.266646i	$0.914085\pi$
$642$		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	$0.293243\pi$
$659$			691687.i			1.59272i	−0.604825	+	0.796359i	$0.706757\pi$
$660$		0			0
$661$	−26775.9			−0.0612831			−0.0306416	−	0.999530i	$-0.509755\pi$
$661$	−26775.9			−0.0612831			−0.0306416	+	0.999530i	$0.509755\pi$
$662$		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	$0.225800\pi$
$683$			607703.i			1.30272i	−0.758770	+	0.651358i	$0.774200\pi$
$684$		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	$-0.448018\pi$
$709$	163454.			0.325165			0.162582	+	0.986695i	$0.448018\pi$
$710$		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	$-0.526680\pi$
$727$	−88494.9			−0.167436			−0.0837182	+	0.996489i	$0.526680\pi$
$728$		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	$-0.616490\pi$
$733$	−384536.			−0.715698			−0.357849	+	0.933780i	$0.616490\pi$
$734$		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	$-0.457105\pi$
$751$	151550.			0.268704			0.134352	+	0.990934i	$0.457105\pi$
$752$		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	$-0.410256\pi$
$757$	318865.			0.556436			0.278218	+	0.960518i	$0.410256\pi$
$758$		0			0
$759$	241167.	−	294053.i	0.418633	−	0.510437i
$760$		0			0
$761$			294118.i			0.507869i	0.967221	+	0.253935i	$0.0817248\pi$
$761$			294118.i			0.507869i	−0.967221	+	0.253935i	$0.918275\pi$
$762$		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	$0.199704\pi$
$809$			768405.i			1.17407i	−0.809562	+	0.587034i	$0.800296\pi$
$810$		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	$-0.853280\pi$
$823$	−1.21328e6			−1.79127			−0.895637	+	0.444786i	$0.853280\pi$
$824$		0			0
$825$	528854.	+	433737.i	0.777012	+	0.637264i
$826$		0			0
$827$		−	1.09147e6i		−	1.59588i	−0.602739	−	0.797938i	$-0.705924\pi$
$827$		−	1.09147e6i		−	1.59588i	0.602739	−	0.797938i	$-0.294076\pi$
$828$		0			0
$829$	750993.			1.09277			0.546383	−	0.837536i	$-0.316005\pi$
$829$	750993.			1.09277			0.546383	+	0.837536i	$0.316005\pi$
$830$		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	$-0.148107\pi$
$853$	1.30051e6			1.78738			0.893691	+	0.448683i	$0.148107\pi$
$854$		0			0
$855$	−874081.	+	174444.i	−1.19569	+	0.238629i
$856$		0			0
$857$		−	102209.i		−	0.139165i	−0.997576	−	0.0695823i	$-0.977833\pi$
$857$		−	102209.i		−	0.139165i	0.997576	−	0.0695823i	$-0.0221666\pi$
$858$		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	$-0.465885\pi$
$877$	164547.			0.213939			0.106969	+	0.994262i	$0.465885\pi$
$878$		0			0
$879$	632195.	+	518493.i	0.818226	+	0.671065i
$880$		0			0
$881$		−	1.03663e6i		−	1.33559i	−0.744345	−	0.667795i	$-0.767238\pi$
$881$		−	1.03663e6i		−	1.33559i	0.744345	−	0.667795i	$-0.232762\pi$
$882$		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	$-0.534784\pi$
$919$	−184215.			−0.218119			−0.109059	+	0.994035i	$0.534784\pi$
$920$		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	$-0.856277\pi$
$929$		−	753145.i		−	0.872664i	0.899786	−	0.436332i	$-0.143723\pi$
$930$		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	$0.0879623\pi$
$947$			489367.i			0.545676i	−0.962060	+	0.272838i	$0.912038\pi$
$948$		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	$-0.771238\pi$
$953$		−	1.19591e6i		−	1.31678i	0.752678	−	0.658389i	$-0.228762\pi$
$954$		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	$-0.368057\pi$
$967$	753201.			0.805486			0.402743	+	0.915313i	$0.368057\pi$
$968$		0			0
$969$	−488859.	−	400936.i	−0.520638	−	0.427000i
$970$		0			0
$971$			185520.i			0.196767i	0.995149	+	0.0983835i	$0.0313672\pi$
$971$			185520.i			0.196767i	−0.995149	+	0.0983835i	$0.968633\pi$
$972$		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	$0.111801\pi$
$977$			656821.i			0.688110i	−0.938950	+	0.344055i	$0.888199\pi$
$978$		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	$-0.691839\pi$
$991$	−1.11339e6			−1.13370			−0.566851	+	0.823820i	$0.691839\pi$
$992$		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	$-0.522586\pi$
$997$	−140946.			−0.141795			−0.0708977	+	0.997484i	$0.522586\pi$
$998$		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.c.257.6	yes	16
3.2	odd	2	inner	384.5.e.c.257.5	yes	16
4.3	odd	2		384.5.e.b.257.11	yes	16
8.3	odd	2		384.5.e.d.257.6	yes	16
8.5	even	2		384.5.e.a.257.11	✓	16
12.11	even	2		384.5.e.b.257.12	yes	16
24.5	odd	2		384.5.e.a.257.12	yes	16
24.11	even	2		384.5.e.d.257.5	yes	16

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

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