LMFDB - Embedded newform 384.5.e.b.257.14

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			257.14
Root			$2.18273 - 4.51404i$ of defining polynomial
Character	$\chi$	$=$	384.257
Dual form			384.5.e.b.257.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.57180 + 6.14911i) q^{3} +9.58700i q^{5} -22.6799 q^{7} +(5.37699 + 80.8213i) q^{9} +O(q^{10})$	\(q+(6.57180 + 6.14911i) q^{3} +9.58700i q^{5} -22.6799 q^{7} +(5.37699 + 80.8213i) q^{9} -2.72041i q^{11} -220.010 q^{13} +(-58.9515 + 63.0038i) q^{15} -172.904i q^{17} -275.669 q^{19} +(-149.048 - 139.461i) q^{21} -420.082i q^{23} +533.089 q^{25} +(-461.642 + 564.205i) q^{27} -207.156i q^{29} -1088.41 q^{31} +(16.7281 - 17.8780i) q^{33} -217.432i q^{35} -1264.09 q^{37} +(-1445.86 - 1352.87i) q^{39} -1233.43i q^{41} +1388.36 q^{43} +(-774.834 + 51.5492i) q^{45} +1436.76i q^{47} -1886.62 q^{49} +(1063.21 - 1136.29i) q^{51} -3630.53i q^{53} +26.0806 q^{55} +(-1811.64 - 1695.12i) q^{57} +5985.78i q^{59} -2174.91 q^{61} +(-121.950 - 1833.02i) q^{63} -2109.24i q^{65} +5983.22 q^{67} +(2583.13 - 2760.69i) q^{69} -7557.44i q^{71} -3322.96 q^{73} +(3503.35 + 3278.02i) q^{75} +61.6987i q^{77} -279.697 q^{79} +(-6503.18 + 869.151i) q^{81} +4339.74i q^{83} +1657.63 q^{85} +(1273.82 - 1361.39i) q^{87} -12561.3i q^{89} +4989.81 q^{91} +(-7152.83 - 6692.77i) q^{93} -2642.84i q^{95} -16573.3 q^{97} +(219.868 - 14.6276i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	6.57180	+	6.14911i	0.730199	+	0.683234i
$3$	6.57180	+	6.14911i	0.730199	+	0.683234i
$4$		0			0
$5$			9.58700i			0.383480i	0.981446	+	0.191740i	$0.0614130\pi$
$5$			9.58700i			0.383480i	−0.981446	+	0.191740i	$0.938587\pi$
$6$		0			0
$7$	−22.6799			−0.462855			−0.231428	−	0.972852i	$-0.574340\pi$
$7$	−22.6799			−0.462855			−0.231428	+	0.972852i	$0.574340\pi$
$8$		0			0
$9$	5.37699	+	80.8213i	0.0663826	+	0.997794i
$10$		0			0
$11$		−	2.72041i		−	0.0224828i	−0.999937	−	0.0112414i	$-0.996422\pi$
$11$		−	2.72041i		−	0.0224828i	0.999937	−	0.0112414i	$-0.00357832\pi$
$12$		0			0
$13$	−220.010			−1.30184			−0.650918	−	0.759148i	$-0.725616\pi$
$13$	−220.010			−1.30184			−0.650918	+	0.759148i	$0.725616\pi$
$14$		0			0
$15$	−58.9515	+	63.0038i	−0.262007	+	0.280017i
$16$		0			0
$17$		−	172.904i		−	0.598285i	−0.954208	−	0.299143i	$-0.903299\pi$
$17$		−	172.904i		−	0.598285i	0.954208	−	0.299143i	$-0.0967006\pi$
$18$		0			0
$19$	−275.669			−0.763626			−0.381813	−	0.924240i	$-0.624700\pi$
$19$	−275.669			−0.763626			−0.381813	+	0.924240i	$0.624700\pi$
$20$		0			0
$21$	−149.048	−	139.461i	−0.337977	−	0.316238i
$22$		0			0
$23$		−	420.082i		−	0.794106i	−0.917796	−	0.397053i	$-0.870033\pi$
$23$		−	420.082i		−	0.794106i	0.917796	−	0.397053i	$-0.129967\pi$
$24$		0			0
$25$	533.089			0.852943
$26$		0			0
$27$	−461.642	+	564.205i	−0.633254	+	0.773944i
$28$		0			0
$29$		−	207.156i		−	0.246321i	−0.992387	−	0.123160i	$-0.960697\pi$
$29$		−	207.156i		−	0.246321i	0.992387	−	0.123160i	$-0.0393030\pi$
$30$		0			0
$31$	−1088.41			−1.13258			−0.566292	−	0.824205i	$-0.691622\pi$
$31$	−1088.41			−1.13258			−0.566292	+	0.824205i	$0.691622\pi$
$32$		0			0
$33$	16.7281	−	17.8780i	0.0153610	−	0.0164169i
$34$		0			0
$35$		−	217.432i		−	0.177496i
$36$		0			0
$37$	−1264.09			−0.923370			−0.461685	−	0.887044i	$-0.652755\pi$
$37$	−1264.09			−0.923370			−0.461685	+	0.887044i	$0.652755\pi$
$38$		0			0
$39$	−1445.86	−	1352.87i	−0.950600	−	0.889458i
$40$		0			0
$41$		−	1233.43i		−	0.733745i	−0.930271	−	0.366873i	$-0.880428\pi$
$41$		−	1233.43i		−	0.733745i	0.930271	−	0.366873i	$-0.119572\pi$
$42$		0			0
$43$	1388.36			0.750873			0.375436	−	0.926848i	$-0.377493\pi$
$43$	1388.36			0.750873			0.375436	+	0.926848i	$0.377493\pi$
$44$		0			0
$45$	−774.834	+	51.5492i	−0.382634	+	0.0254564i
$46$		0			0
$47$			1436.76i			0.650412i	0.945643	+	0.325206i	$0.105434\pi$
$47$			1436.76i			0.650412i	−0.945643	+	0.325206i	$0.894566\pi$
$48$		0			0
$49$	−1886.62			−0.785765
$50$		0			0
$51$	1063.21	−	1136.29i	0.408769	−	0.436868i
$52$		0			0
$53$		−	3630.53i		−	1.29246i	−0.763141	−	0.646231i	$-0.776344\pi$
$53$		−	3630.53i		−	1.29246i	0.763141	−	0.646231i	$-0.223656\pi$
$54$		0			0
$55$	26.0806			0.00862169
$56$		0			0
$57$	−1811.64	−	1695.12i	−0.557599	−	0.521735i
$58$		0			0
$59$			5985.78i			1.71956i	0.510667	+	0.859779i	$0.329399\pi$
$59$			5985.78i			1.71956i	−0.510667	+	0.859779i	$0.670601\pi$
$60$		0			0
$61$	−2174.91			−0.584496			−0.292248	−	0.956343i	$-0.594403\pi$
$61$	−2174.91			−0.584496			−0.292248	+	0.956343i	$0.594403\pi$
$62$		0			0
$63$	−121.950	−	1833.02i	−0.0307255	−	0.461834i
$64$		0			0
$65$		−	2109.24i		−	0.499228i
$66$		0			0
$67$	5983.22			1.33286			0.666431	−	0.745567i	$-0.267821\pi$
$67$	5983.22			1.33286			0.666431	+	0.745567i	$0.267821\pi$
$68$		0			0
$69$	2583.13	−	2760.69i	0.542560	−	0.579856i
$70$		0			0
$71$		−	7557.44i		−	1.49919i	−0.661894	−	0.749597i	$-0.730247\pi$
$71$		−	7557.44i		−	1.49919i	0.661894	−	0.749597i	$-0.269753\pi$
$72$		0			0
$73$	−3322.96			−0.623561			−0.311781	−	0.950154i	$-0.600925\pi$
$73$	−3322.96			−0.623561			−0.311781	+	0.950154i	$0.600925\pi$
$74$		0			0
$75$	3503.35	+	3278.02i	0.622819	+	0.582760i
$76$		0			0
$77$			61.6987i			0.0104063i
$78$		0			0
$79$	−279.697			−0.0448161			−0.0224081	−	0.999749i	$-0.507133\pi$
$79$	−279.697			−0.0448161			−0.0224081	+	0.999749i	$0.507133\pi$
$80$		0			0
$81$	−6503.18	+	869.151i	−0.991187	+	0.132472i
$82$		0			0
$83$			4339.74i			0.629953i	0.949100	+	0.314976i	$0.101997\pi$
$83$			4339.74i			0.629953i	−0.949100	+	0.314976i	$0.898003\pi$
$84$		0			0
$85$	1657.63			0.229430
$86$		0			0
$87$	1273.82	−	1361.39i	0.168295	−	0.179863i
$88$		0			0
$89$		−	12561.3i		−	1.58582i	−0.609336	−	0.792912i	$-0.708564\pi$
$89$		−	12561.3i		−	1.58582i	0.609336	−	0.792912i	$-0.291436\pi$
$90$		0			0
$91$	4989.81			0.602561
$92$		0			0
$93$	−7152.83	−	6692.77i	−0.827012	−	0.773820i
$94$		0			0
$95$		−	2642.84i		−	0.292835i
$96$		0			0
$97$	−16573.3			−1.76143			−0.880714	−	0.473648i	$-0.842937\pi$
$97$	−16573.3			−1.76143			−0.880714	+	0.473648i	$0.842937\pi$
$98$		0			0
$99$	219.868	−	14.6276i	0.0224332	−	0.00149246i
$100$		0			0
$101$			1477.67i			0.144855i	0.997374	+	0.0724277i	$0.0230747\pi$
$101$			1477.67i			0.144855i	−0.997374	+	0.0724277i	$0.976925\pi$
$102$		0			0
$103$	−8937.51			−0.842446			−0.421223	−	0.906957i	$-0.638399\pi$
$103$	−8937.51			−0.842446			−0.421223	+	0.906957i	$0.638399\pi$
$104$		0			0
$105$	1337.01	−	1428.92i	0.121271	−	0.129607i
$106$		0			0
$107$			2595.38i			0.226690i	0.993556	+	0.113345i	$0.0361566\pi$
$107$			2595.38i			0.226690i	−0.993556	+	0.113345i	$0.963843\pi$
$108$		0			0
$109$	−14318.9			−1.20519			−0.602597	−	0.798045i	$-0.705868\pi$
$109$	−14318.9			−1.20519			−0.602597	+	0.798045i	$0.705868\pi$
$110$		0			0
$111$	−8307.36	−	7773.05i	−0.674244	−	0.630878i
$112$		0			0
$113$			1553.40i			0.121654i	0.998148	+	0.0608270i	$0.0193738\pi$
$113$			1553.40i			0.121654i	−0.998148	+	0.0608270i	$0.980626\pi$
$114$		0			0
$115$	4027.32			0.304524
$116$		0			0
$117$	−1182.99	−	17781.5i	−0.0864192	−	1.29896i
$118$		0			0
$119$			3921.46i			0.276919i
$120$		0			0
$121$	14633.6			0.999495
$122$		0			0
$123$	7584.47	−	8105.82i	0.501320	−	0.535780i
$124$		0			0
$125$			11102.6i			0.710566i
$126$		0			0
$127$	−11644.5			−0.721959			−0.360979	−	0.932574i	$-0.617558\pi$
$127$	−11644.5			−0.721959			−0.360979	+	0.932574i	$0.617558\pi$
$128$		0			0
$129$	9124.04	+	8537.20i	0.548287	+	0.513022i
$130$		0			0
$131$			28808.7i			1.67873i	0.543567	+	0.839366i	$0.317074\pi$
$131$			28808.7i			1.67873i	−0.543567	+	0.839366i	$0.682926\pi$
$132$		0			0
$133$	6252.14			0.353448
$134$		0			0
$135$	−5409.03	−	4425.77i	−0.296792	−	0.242840i
$136$		0			0
$137$		−	21776.3i		−	1.16023i	−0.814536	−	0.580113i	$-0.803008\pi$
$137$		−	21776.3i		−	1.16023i	0.814536	−	0.580113i	$-0.196992\pi$
$138$		0			0
$139$	−18061.1			−0.934789			−0.467395	−	0.884049i	$-0.654807\pi$
$139$	−18061.1			−0.934789			−0.467395	+	0.884049i	$0.654807\pi$
$140$		0			0
$141$	−8834.79	+	9442.09i	−0.444384	+	0.474931i
$142$		0			0
$143$			598.519i			0.0292689i
$144$		0			0
$145$	1986.00			0.0944591
$146$		0			0
$147$	−12398.5	−	11601.0i	−0.573765	−	0.536861i
$148$		0			0
$149$		−	36383.8i		−	1.63884i	−0.573195	−	0.819419i	$-0.694296\pi$
$149$		−	36383.8i		−	1.63884i	0.573195	−	0.819419i	$-0.305704\pi$
$150$		0			0
$151$	−620.295			−0.0272047			−0.0136024	−	0.999907i	$-0.504330\pi$
$151$	−620.295			−0.0272047			−0.0136024	+	0.999907i	$0.504330\pi$
$152$		0			0
$153$	13974.4	−	929.706i	0.596966	−	0.0397157i
$154$		0			0
$155$		−	10434.6i		−	0.434323i
$156$		0			0
$157$	−18826.6			−0.763789			−0.381894	−	0.924206i	$-0.624728\pi$
$157$	−18826.6			−0.763789			−0.381894	+	0.924206i	$0.624728\pi$
$158$		0			0
$159$	22324.5	−	23859.1i	0.883055	−	0.943756i
$160$		0			0
$161$			9527.42i			0.367556i
$162$		0			0
$163$	−12306.6			−0.463192			−0.231596	−	0.972812i	$-0.574395\pi$
$163$	−12306.6			−0.463192			−0.231596	+	0.972812i	$0.574395\pi$
$164$		0			0
$165$	171.396	+	160.372i	0.00629555	+	0.00589063i
$166$		0			0
$167$			53134.8i			1.90523i	0.304185	+	0.952613i	$0.401616\pi$
$167$			53134.8i			1.90523i	−0.304185	+	0.952613i	$0.598384\pi$
$168$		0			0
$169$	19843.5			0.694776
$170$		0			0
$171$	−1482.27	−	22279.9i	−0.0506915	−	0.761941i
$172$		0			0
$173$			56705.1i			1.89465i	0.320267	+	0.947327i	$0.396227\pi$
$173$			56705.1i			1.89465i	−0.320267	+	0.947327i	$0.603773\pi$
$174$		0			0
$175$	−12090.4			−0.394789
$176$		0			0
$177$	−36807.2	+	39337.3i	−1.17486	+	1.25562i
$178$		0			0
$179$			3555.51i			0.110967i	0.998460	+	0.0554837i	$0.0176701\pi$
$179$			3555.51i			0.110967i	−0.998460	+	0.0554837i	$0.982330\pi$
$180$		0			0
$181$	38795.9			1.18421			0.592104	−	0.805861i	$-0.298297\pi$
$181$	38795.9			1.18421			0.592104	+	0.805861i	$0.298297\pi$
$182$		0			0
$183$	−14293.1	−	13373.8i	−0.426799	−	0.399348i
$184$		0			0
$185$		−	12118.9i		−	0.354094i
$186$		0			0
$187$	−470.372			−0.0134511
$188$		0			0
$189$	10470.0	−	12796.1i	0.293105	−	0.358224i
$190$		0			0
$191$			18658.8i			0.511466i	0.966747	+	0.255733i	$0.0823168\pi$
$191$			18658.8i			0.511466i	−0.966747	+	0.255733i	$0.917683\pi$
$192$		0			0
$193$	−30562.4			−0.820488			−0.410244	−	0.911976i	$-0.634556\pi$
$193$	−30562.4			−0.820488			−0.410244	+	0.911976i	$0.634556\pi$
$194$		0			0
$195$	12969.9	−	13861.5i	0.341089	−	0.364536i
$196$		0			0
$197$			74919.3i			1.93046i	0.261400	+	0.965230i	$0.415816\pi$
$197$			74919.3i			1.93046i	−0.261400	+	0.965230i	$0.584184\pi$
$198$		0			0
$199$	62035.4			1.56651			0.783256	−	0.621699i	$-0.213557\pi$
$199$	62035.4			1.56651			0.783256	+	0.621699i	$0.213557\pi$
$200$		0			0
$201$	39320.5	+	36791.4i	0.973255	+	0.910657i
$202$		0			0
$203$			4698.27i			0.114011i
$204$		0			0
$205$	11824.9			0.281377
$206$		0			0
$207$	33951.6	−	2258.78i	0.792354	−	0.0527148i
$208$		0			0
$209$			749.934i			0.0171684i
$210$		0			0
$211$	5427.63			0.121912			0.0609559	−	0.998140i	$-0.480585\pi$
$211$	5427.63			0.121912			0.0609559	+	0.998140i	$0.480585\pi$
$212$		0			0
$213$	46471.5	−	49666.0i	1.02430	−	1.09471i
$214$		0			0
$215$			13310.2i			0.287945i
$216$		0			0
$217$	24685.1			0.524222
$218$		0			0
$219$	−21837.8	−	20433.2i	−0.455324	−	0.426038i
$220$		0			0
$221$			38040.7i			0.778869i
$222$		0			0
$223$	63912.0			1.28520			0.642602	−	0.766200i	$-0.277855\pi$
$223$	63912.0			1.28520			0.642602	+	0.766200i	$0.277855\pi$
$224$		0			0
$225$	2866.42	+	43085.0i	0.0566206	+	0.851062i
$226$		0			0
$227$		−	37246.6i		−	0.722828i	−0.932405	−	0.361414i	$-0.882294\pi$
$227$		−	37246.6i		−	0.722828i	0.932405	−	0.361414i	$-0.117706\pi$
$228$		0			0
$229$	71947.8			1.37198			0.685988	−	0.727612i	$-0.259370\pi$
$229$	71947.8			1.37198			0.685988	+	0.727612i	$0.259370\pi$
$230$		0			0
$231$	−379.392	+	405.471i	−0.00710991	+	0.00759865i
$232$		0			0
$233$		−	47812.4i		−	0.880701i	−0.897826	−	0.440350i	$-0.854854\pi$
$233$		−	47812.4i		−	0.880701i	0.897826	−	0.440350i	$-0.145146\pi$
$234$		0			0
$235$	−13774.2			−0.249420
$236$		0			0
$237$	−1838.11	−	1719.89i	−0.0327247	−	0.0306199i
$238$		0			0
$239$			102574.i			1.79574i	0.440265	+	0.897868i	$0.354884\pi$
$239$			102574.i			1.79574i	−0.440265	+	0.897868i	$0.645116\pi$
$240$		0			0
$241$	28901.0			0.497598			0.248799	−	0.968555i	$-0.419964\pi$
$241$	28901.0			0.497598			0.248799	+	0.968555i	$0.419964\pi$
$242$		0			0
$243$	−48082.0	−	34276.8i	−0.814274	−	0.580481i
$244$		0			0
$245$		−	18087.0i		−	0.301325i
$246$		0			0
$247$	60650.0			0.994115
$248$		0			0
$249$	−26685.5	+	28519.9i	−0.430405	+	0.459991i
$250$		0			0
$251$			58191.6i			0.923661i	0.886968	+	0.461831i	$0.152807\pi$
$251$			58191.6i			0.923661i	−0.886968	+	0.461831i	$0.847193\pi$
$252$		0			0
$253$	−1142.80			−0.0178537
$254$		0			0
$255$	10893.6	+	10193.0i	0.167530	+	0.156755i
$256$		0			0
$257$			35458.1i			0.536846i	0.963301	+	0.268423i	$0.0865024\pi$
$257$			35458.1i			0.536846i	−0.963301	+	0.268423i	$0.913498\pi$
$258$		0			0
$259$	28669.5			0.427387
$260$		0			0
$261$	16742.6	−	1113.87i	0.245778	−	0.0163514i
$262$		0			0
$263$			27842.2i			0.402525i	0.979537	+	0.201262i	$0.0645044\pi$
$263$			27842.2i			0.402525i	−0.979537	+	0.201262i	$0.935496\pi$
$264$		0			0
$265$	34805.9			0.495634
$266$		0			0
$267$	77240.8	−	82550.4i	1.08349	−	1.15797i
$268$		0			0
$269$			96883.4i			1.33889i	0.742862	+	0.669445i	$0.233468\pi$
$269$			96883.4i			1.33889i	−0.742862	+	0.669445i	$0.766532\pi$
$270$		0			0
$271$	−90782.8			−1.23613			−0.618066	−	0.786126i	$-0.712084\pi$
$271$	−90782.8			−1.23613			−0.618066	+	0.786126i	$0.712084\pi$
$272$		0			0
$273$	32792.0	+	30682.9i	0.439990	+	0.411690i
$274$		0			0
$275$		−	1450.22i		−	0.0191765i
$276$		0			0
$277$	−18108.2			−0.236002			−0.118001	−	0.993013i	$-0.537649\pi$
$277$	−18108.2			−0.236002			−0.118001	+	0.993013i	$0.537649\pi$
$278$		0			0
$279$	−5852.39	−	87967.0i	−0.0751838	−	1.13009i
$280$		0			0
$281$			3961.67i			0.0501725i	0.999685	+	0.0250863i	$0.00798605\pi$
$281$			3961.67i			0.0501725i	−0.999685	+	0.0250863i	$0.992014\pi$
$282$		0			0
$283$	−93471.4			−1.16709			−0.583547	−	0.812079i	$-0.698336\pi$
$283$	−93471.4			−1.16709			−0.583547	+	0.812079i	$0.698336\pi$
$284$		0			0
$285$	16251.1	−	17368.2i	0.200075	−	0.213828i
$286$		0			0
$287$			27974.0i			0.339618i
$288$		0			0
$289$	53625.0			0.642055
$290$		0			0
$291$	−108916.	−	101911.i	−1.28619	−	1.20347i
$292$		0			0
$293$			95579.8i			1.11335i	0.830731	+	0.556674i	$0.187923\pi$
$293$			95579.8i			1.11335i	−0.830731	+	0.556674i	$0.812077\pi$
$294$		0			0
$295$	−57385.7			−0.659416
$296$		0			0
$297$	1534.87	+	1255.86i	0.0174004	+	0.0142373i
$298$		0			0
$299$			92422.3i			1.03379i
$300$		0			0
$301$	−31488.0			−0.347545
$302$		0			0
$303$	−9086.35	+	9710.95i	−0.0989702	+	0.105773i
$304$		0			0
$305$		−	20850.9i		−	0.224143i
$306$		0			0
$307$	33950.8			0.360225			0.180112	−	0.983646i	$-0.442354\pi$
$307$	33950.8			0.360225			0.180112	+	0.983646i	$0.442354\pi$
$308$		0			0
$309$	−58735.5	−	54957.7i	−0.615153	−	0.575588i
$310$		0			0
$311$			18040.3i			0.186519i	0.995642	+	0.0932597i	$0.0297287\pi$
$311$			18040.3i			0.186519i	−0.995642	+	0.0932597i	$0.970271\pi$
$312$		0			0
$313$	16221.1			0.165574			0.0827869	−	0.996567i	$-0.473618\pi$
$313$	16221.1			0.165574			0.0827869	+	0.996567i	$0.473618\pi$
$314$		0			0
$315$	17573.2	−	1169.13i	0.177104	−	0.0117826i
$316$		0			0
$317$		−	50002.3i		−	0.497590i	−0.968556	−	0.248795i	$-0.919965\pi$
$317$		−	50002.3i		−	0.497590i	0.968556	−	0.248795i	$-0.0800345\pi$
$318$		0			0
$319$	−563.550			−0.00553797
$320$		0			0
$321$	−15959.2	+	17056.3i	−0.154882	+	0.165529i
$322$		0			0
$323$			47664.4i			0.456866i
$324$		0			0
$325$	−117285.			−1.11039
$326$		0			0
$327$	−94101.0	−	88048.6i	−0.880033	−	0.823430i
$328$		0			0
$329$		−	32585.6i		−	0.301047i
$330$		0			0
$331$	−151167.			−1.37976			−0.689878	−	0.723926i	$-0.742336\pi$
$331$	−151167.			−1.37976			−0.689878	+	0.723926i	$0.742336\pi$
$332$		0			0
$333$	−6797.02	−	102166.i	−0.0612957	−	0.921333i
$334$		0			0
$335$			57361.1i			0.511126i
$336$		0			0
$337$	−64914.6			−0.571588			−0.285794	−	0.958291i	$-0.592257\pi$
$337$	−64914.6			−0.571588			−0.285794	+	0.958291i	$0.592257\pi$
$338$		0			0
$339$	−9552.02	+	10208.6i	−0.0831182	+	0.0888317i
$340$		0			0
$341$			2960.93i			0.0254636i
$342$		0			0
$343$	97242.8			0.826551
$344$		0			0
$345$	26466.8	+	24764.4i	0.222363	+	0.208061i
$346$		0			0
$347$			16589.8i			0.137779i	0.997624	+	0.0688894i	$0.0219456\pi$
$347$			16589.8i			0.137779i	−0.997624	+	0.0688894i	$0.978054\pi$
$348$		0			0
$349$	16900.0			0.138751			0.0693756	−	0.997591i	$-0.477899\pi$
$349$	16900.0			0.138751			0.0693756	+	0.997591i	$0.477899\pi$
$350$		0			0
$351$	101566.	−	124131.i	0.824393	−	1.00755i
$352$		0			0
$353$		−	182159.i		−	1.46184i	−0.682462	−	0.730921i	$-0.739091\pi$
$353$		−	182159.i		−	1.46184i	0.682462	−	0.730921i	$-0.260909\pi$
$354$		0			0
$355$	72453.2			0.574911
$356$		0			0
$357$	−24113.4	+	25771.0i	−0.189201	+	0.202206i
$358$		0			0
$359$			4618.58i			0.0358360i	0.999839	+	0.0179180i	$0.00570378\pi$
$359$			4618.58i			0.0358360i	−0.999839	+	0.0179180i	$0.994296\pi$
$360$		0			0
$361$	−54327.7			−0.416876
$362$		0			0
$363$	96169.0	+	89983.6i	0.729830	+	0.682889i
$364$		0			0
$365$		−	31857.2i		−	0.239123i
$366$		0			0
$367$	−175198.			−1.30076			−0.650379	−	0.759610i	$-0.725390\pi$
$367$	−175198.			−1.30076			−0.650379	+	0.759610i	$0.725390\pi$
$368$		0			0
$369$	99687.1	−	6632.12i	0.732127	−	0.0487079i
$370$		0			0
$371$			82340.0i			0.598223i
$372$		0			0
$373$	−22897.4			−0.164577			−0.0822884	−	0.996609i	$-0.526223\pi$
$373$	−22897.4			−0.164577			−0.0822884	+	0.996609i	$0.526223\pi$
$374$		0			0
$375$	−68271.1	+	72964.0i	−0.485483	+	0.518855i
$376$		0			0
$377$			45576.4i			0.320669i
$378$		0			0
$379$	154938.			1.07865			0.539323	−	0.842099i	$-0.318680\pi$
$379$	154938.			1.07865			0.539323	+	0.842099i	$0.318680\pi$
$380$		0			0
$381$	−76525.1	−	71603.1i	−0.527174	−	0.493267i
$382$		0			0
$383$		−	261943.i		−	1.78570i	−0.450351	−	0.892852i	$-0.648701\pi$
$383$		−	261943.i		−	1.78570i	0.450351	−	0.892852i	$-0.351299\pi$
$384$		0			0
$385$	−591.506			−0.00399059
$386$		0			0
$387$	7465.22	+	112209.i	0.0498449	+	0.749217i
$388$		0			0
$389$		−	16337.9i		−	0.107969i	−0.998542	−	0.0539843i	$-0.982808\pi$
$389$		−	16337.9i		−	0.107969i	0.998542	−	0.0539843i	$-0.0171921\pi$
$390$		0			0
$391$	−72634.0			−0.475102
$392$		0			0
$393$	−177148.	+	189325.i	−1.14697	+	1.22581i
$394$		0			0
$395$		−	2681.46i		−	0.0171861i
$396$		0			0
$397$	156295.			0.991661			0.495831	−	0.868419i	$-0.334864\pi$
$397$	156295.			0.991661			0.495831	+	0.868419i	$0.334864\pi$
$398$		0			0
$399$	41087.8	+	38445.1i	0.258088	+	0.241488i
$400$		0			0
$401$		−	271550.i		−	1.68873i	−0.535765	−	0.844367i	$-0.679977\pi$
$401$		−	271550.i		−	1.68873i	0.535765	−	0.844367i	$-0.320023\pi$
$402$		0			0
$403$	239462.			1.47444
$404$		0			0
$405$	−8332.55	−	62345.9i	−0.0508005	−	0.380100i
$406$		0			0
$407$			3438.86i			0.0207599i
$408$		0			0
$409$	−298248.			−1.78292			−0.891458	−	0.453103i	$-0.850317\pi$
$409$	−298248.			−1.78292			−0.891458	+	0.453103i	$0.850317\pi$
$410$		0			0
$411$	133905.	−	143109.i	0.792706	−	0.847197i
$412$		0			0
$413$		−	135757.i		−	0.795906i
$414$		0			0
$415$	−41605.1			−0.241574
$416$		0			0
$417$	−118694.	−	111059.i	−0.682583	−	0.638680i
$418$		0			0
$419$			92035.5i			0.524236i	0.965036	+	0.262118i	$0.0844210\pi$
$419$			92035.5i			0.524236i	−0.965036	+	0.262118i	$0.915579\pi$
$420$		0			0
$421$	−185236.			−1.04511			−0.522554	−	0.852606i	$-0.675021\pi$
$421$	−185236.			−1.04511			−0.522554	+	0.852606i	$0.675021\pi$
$422$		0			0
$423$	−116121.	+	7725.44i	−0.648977	+	0.0431760i
$424$		0			0
$425$		−	92173.5i		−	0.510303i
$426$		0			0
$427$	49326.8			0.270537
$428$		0			0
$429$	−3680.36	+	3933.34i	−0.0199975	+	0.0213721i
$430$		0			0
$431$			111959.i			0.602707i	0.953512	+	0.301354i	$0.0974384\pi$
$431$			111959.i			0.602707i	−0.953512	+	0.301354i	$0.902562\pi$
$432$		0			0
$433$	−82885.2			−0.442080			−0.221040	−	0.975265i	$-0.570945\pi$
$433$	−82885.2			−0.442080			−0.221040	+	0.975265i	$0.570945\pi$
$434$		0			0
$435$	13051.6	+	12212.1i	0.0689740	+	0.0645377i
$436$		0			0
$437$			115804.i			0.606400i
$438$		0			0
$439$	201309.			1.04456			0.522281	−	0.852773i	$-0.325081\pi$
$439$	201309.			1.04456			0.522281	+	0.852773i	$0.325081\pi$
$440$		0			0
$441$	−10144.3	−	152479.i	−0.0521611	−	0.784032i
$442$		0			0
$443$		−	40110.1i		−	0.204384i	−0.994765	−	0.102192i	$-0.967414\pi$
$443$		−	40110.1i		−	0.204384i	0.994765	−	0.102192i	$-0.0325856\pi$
$444$		0			0
$445$	120425.			0.608132
$446$		0			0
$447$	223728.	−	239107.i	1.11971	−	1.19668i
$448$		0			0
$449$			23050.1i			0.114335i	0.998365	+	0.0571677i	$0.0182070\pi$
$449$			23050.1i			0.114335i	−0.998365	+	0.0571677i	$0.981793\pi$
$450$		0			0
$451$	−3355.43			−0.0164966
$452$		0			0
$453$	−4076.45	−	3814.26i	−0.0198649	−	0.0185872i
$454$		0			0
$455$			47837.3i			0.231070i
$456$		0			0
$457$	−158527.			−0.759049			−0.379524	−	0.925182i	$-0.623912\pi$
$457$	−158527.			−0.759049			−0.379524	+	0.925182i	$0.623912\pi$
$458$		0			0
$459$	97553.6	+	79820.0i	0.463039	+	0.378867i
$460$		0			0
$461$		−	320969.i		−	1.51029i	−0.655556	−	0.755147i	$-0.727566\pi$
$461$		−	320969.i		−	1.51029i	0.655556	−	0.755147i	$-0.272434\pi$
$462$		0			0
$463$	−369712.			−1.72465			−0.862326	−	0.506353i	$-0.830993\pi$
$463$	−369712.			−1.72465			−0.862326	+	0.506353i	$0.830993\pi$
$464$		0			0
$465$	64163.5	−	68574.1i	0.296744	−	0.317143i
$466$		0			0
$467$			266718.i			1.22298i	0.791254	+	0.611488i	$0.209429\pi$
$467$			266718.i			1.22298i	−0.791254	+	0.611488i	$0.790571\pi$
$468$		0			0
$469$	−135699.			−0.616922
$470$		0			0
$471$	−123725.	−	115767.i	−0.557718	−	0.521846i
$472$		0			0
$473$		−	3776.92i		−	0.0168817i
$474$		0			0
$475$	−146956.			−0.651329
$476$		0			0
$477$	293424.	−	19521.3i	1.28961	−	0.0857970i
$478$		0			0
$479$		−	233327.i		−	1.01694i	−0.861081	−	0.508468i	$-0.830212\pi$
$479$		−	233327.i		−	1.01694i	0.861081	−	0.508468i	$-0.169788\pi$
$480$		0			0
$481$	278113.			1.20208
$482$		0			0
$483$	−58585.1	+	62612.2i	−0.251127	+	0.268389i
$484$		0			0
$485$		−	158888.i		−	0.675473i
$486$		0			0
$487$	193053.			0.813989			0.406995	−	0.913431i	$-0.366577\pi$
$487$	193053.			0.813989			0.406995	+	0.913431i	$0.366577\pi$
$488$		0			0
$489$	−80876.2	−	75674.3i	−0.338223	−	0.316469i
$490$		0			0
$491$		−	84945.1i		−	0.352351i	−0.984359	−	0.176175i	$-0.943627\pi$
$491$		−	84945.1i		−	0.352351i	0.984359	−	0.176175i	$-0.0563726\pi$
$492$		0			0
$493$	−35818.2			−0.147370
$494$		0			0
$495$	140.235	+	2107.87i	0.000572330	+	0.00860267i
$496$		0			0
$497$			171402.i			0.693910i
$498$		0			0
$499$	111582.			0.448118			0.224059	−	0.974576i	$-0.428069\pi$
$499$	111582.			0.448118			0.224059	+	0.974576i	$0.428069\pi$
$500$		0			0
$501$	−326732.	+	349191.i	−1.30172	+	1.39119i
$502$		0			0
$503$		−	258638.i		−	1.02225i	−0.859507	−	0.511124i	$-0.829229\pi$
$503$		−	258638.i		−	1.02225i	0.859507	−	0.511124i	$-0.170771\pi$
$504$		0			0
$505$	−14166.4			−0.0555492
$506$		0			0
$507$	130407.	+	122020.i	0.507325	+	0.474694i
$508$		0			0
$509$			41792.0i			0.161309i	0.996742	+	0.0806543i	$0.0257010\pi$
$509$			41792.0i			0.161309i	−0.996742	+	0.0806543i	$0.974299\pi$
$510$		0			0
$511$	75364.4			0.288619
$512$		0			0
$513$	127260.	−	155534.i	0.483569	−	0.591003i
$514$		0			0
$515$		−	85683.9i		−	0.323061i
$516$		0			0
$517$	3908.58			0.0146231
$518$		0			0
$519$	−348686.	+	372654.i	−1.29449	+	1.38348i
$520$		0			0
$521$			413559.i			1.52357i	0.647832	+	0.761784i	$0.275676\pi$
$521$			413559.i			1.52357i	−0.647832	+	0.761784i	$0.724324\pi$
$522$		0			0
$523$	−135948.			−0.497013			−0.248507	−	0.968630i	$-0.579940\pi$
$523$	−135948.			−0.497013			−0.248507	+	0.968630i	$0.579940\pi$
$524$		0			0
$525$	−79455.7	−	74345.3i	−0.288275	−	0.269733i
$526$		0			0
$527$			188191.i			0.677608i
$528$		0			0
$529$	103372.			0.369396
$530$		0			0
$531$	−483779.	+	32185.5i	−1.71576	+	0.114149i
$532$		0			0
$533$			271366.i			0.955216i
$534$		0			0
$535$	−24881.9			−0.0869311
$536$		0			0
$537$	−21863.2	+	23366.1i	−0.0758167	+	0.0810283i
$538$		0			0
$539$			5132.39i			0.0176662i
$540$		0			0
$541$	−370481.			−1.26582			−0.632909	−	0.774226i	$-0.718139\pi$
$541$	−370481.			−1.26582			−0.632909	+	0.774226i	$0.718139\pi$
$542$		0			0
$543$	254958.	+	238560.i	0.864709	+	0.809092i
$544$		0			0
$545$		−	137275.i		−	0.462168i
$546$		0			0
$547$	245298.			0.819822			0.409911	−	0.912125i	$-0.365560\pi$
$547$	245298.			0.819822			0.409911	+	0.912125i	$0.365560\pi$
$548$		0			0
$549$	−11694.5	−	175779.i	−0.0388004	−	0.583207i
$550$		0			0
$551$			57106.4i			0.188097i
$552$		0			0
$553$	6343.51			0.0207434
$554$		0			0
$555$	74520.2	−	79642.7i	0.241929	−	0.258559i
$556$		0			0
$557$		−	145458.i		−	0.468843i	−0.972135	−	0.234421i	$-0.924680\pi$
$557$		−	145458.i		−	0.468843i	0.972135	−	0.234421i	$-0.0753195\pi$
$558$		0			0
$559$	−305454.			−0.977513
$560$		0			0
$561$	−3091.19	−	2892.37i	−0.00982199	−	0.00919025i
$562$		0			0
$563$		−	416831.i		−	1.31505i	−0.753431	−	0.657527i	$-0.771603\pi$
$563$		−	416831.i		−	1.31505i	0.753431	−	0.657527i	$-0.228397\pi$
$564$		0			0
$565$	−14892.4			−0.0466519
$566$		0			0
$567$	147491.	−	19712.3i	0.458776	−	0.0613155i
$568$		0			0
$569$			387646.i			1.19732i	0.801002	+	0.598661i	$0.204300\pi$
$569$			387646.i			1.19732i	−0.801002	+	0.598661i	$0.795700\pi$
$570$		0			0
$571$	−538462.			−1.65152			−0.825759	−	0.564024i	$-0.809253\pi$
$571$	−538462.			−1.65152			−0.825759	+	0.564024i	$0.809253\pi$
$572$		0			0
$573$	−114735.	+	122622.i	−0.349451	+	0.373472i
$574$		0			0
$575$		−	223941.i		−	0.677327i
$576$		0			0
$577$	239240.			0.718593			0.359296	−	0.933223i	$-0.383017\pi$
$577$	239240.			0.718593			0.359296	+	0.933223i	$0.383017\pi$
$578$		0			0
$579$	−200850.	−	187931.i	−0.599120	−	0.560585i
$580$		0			0
$581$		−	98425.0i		−	0.291577i
$582$		0			0
$583$	−9876.54			−0.0290581
$584$		0			0
$585$	170471.	−	11341.3i	0.498127	−	0.0331400i
$586$		0			0
$587$		−	225617.i		−	0.654781i	−0.944889	−	0.327390i	$-0.893831\pi$
$587$		−	225617.i		−	0.654781i	0.944889	−	0.327390i	$-0.106169\pi$
$588$		0			0
$589$	300042.			0.864870
$590$		0			0
$591$	−460686.	+	492354.i	−1.31896	+	1.40962i
$592$		0			0
$593$		−	114970.i		−	0.326946i	−0.986548	−	0.163473i	$-0.947730\pi$
$593$		−	114970.i		−	0.326946i	0.986548	−	0.163473i	$-0.0522696\pi$
$594$		0			0
$595$	−37595.0			−0.106193
$596$		0			0
$597$	407684.	+	381462.i	1.14387	+	1.07029i
$598$		0			0
$599$		−	413169.i		−	1.15153i	−0.817616	−	0.575764i	$-0.804705\pi$
$599$		−	413169.i		−	1.15153i	0.817616	−	0.575764i	$-0.195295\pi$
$600$		0			0
$601$	403522.			1.11717			0.558584	−	0.829448i	$-0.311345\pi$
$601$	403522.			1.11717			0.558584	+	0.829448i	$0.311345\pi$
$602$		0			0
$603$	32171.7	+	483572.i	0.0884788	+	1.32992i
$604$		0			0
$605$			140292.i			0.383286i
$606$		0			0
$607$	517445.			1.40439			0.702194	−	0.711986i	$-0.252204\pi$
$607$	517445.			1.40439			0.702194	+	0.711986i	$0.252204\pi$
$608$		0			0
$609$	−28890.2	+	30876.1i	−0.0778961	+	0.0832507i
$610$		0			0
$611$		−	316102.i		−	0.846729i
$612$		0			0
$613$	193190.			0.514119			0.257059	−	0.966396i	$-0.417246\pi$
$613$	193190.			0.514119			0.257059	+	0.966396i	$0.417246\pi$
$614$		0			0
$615$	77710.5	+	72712.3i	0.205461	+	0.192246i
$616$		0			0
$617$		−	147690.i		−	0.387954i	−0.981006	−	0.193977i	$-0.937861\pi$
$617$		−	147690.i		−	0.387954i	0.981006	−	0.193977i	$-0.0621387\pi$
$618$		0			0
$619$	−198636.			−0.518413			−0.259206	−	0.965822i	$-0.583461\pi$
$619$	−198636.			−0.518413			−0.259206	+	0.965822i	$0.583461\pi$
$620$		0			0
$621$	237012.	+	193928.i	0.614593	+	0.502871i
$622$		0			0
$623$			284889.i			0.734007i
$624$		0			0
$625$	226740.			0.580455
$626$		0			0
$627$	−4611.42	+	4928.41i	−0.0117300	+	0.0125364i
$628$		0			0
$629$			218567.i			0.552439i
$630$		0			0
$631$	343175.			0.861900			0.430950	−	0.902376i	$-0.358179\pi$
$631$	343175.			0.861900			0.430950	+	0.902376i	$0.358179\pi$
$632$		0			0
$633$	35669.3	+	33375.1i	0.0890199	+	0.0832942i
$634$		0			0
$635$		−	111636.i		−	0.276857i
$636$		0			0
$637$	415076.			1.02294
$638$		0			0
$639$	610802.	−	40636.3i	1.49589	−	0.0995204i
$640$		0			0
$641$			746228.i			1.81616i	0.418792	+	0.908082i	$0.362454\pi$
$641$			746228.i			1.81616i	−0.418792	+	0.908082i	$0.637546\pi$
$642$		0			0
$643$	161983.			0.391784			0.195892	−	0.980625i	$-0.437240\pi$
$643$	161983.			0.391784			0.195892	+	0.980625i	$0.437240\pi$
$644$		0			0
$645$	−81846.1	+	87472.2i	−0.196734	+	0.210257i
$646$		0			0
$647$			220080.i			0.525742i	0.964831	+	0.262871i	$0.0846693\pi$
$647$			220080.i			0.525742i	−0.964831	+	0.262871i	$0.915331\pi$
$648$		0			0
$649$	16283.8			0.0386604
$650$		0			0
$651$	162225.	+	151791.i	0.382787	+	0.358166i
$652$		0			0
$653$		−	101534.i		−	0.238115i	−0.992887	−	0.119057i	$-0.962013\pi$
$653$		−	101534.i		−	0.238115i	0.992887	−	0.119057i	$-0.0379873\pi$
$654$		0			0
$655$	−276189.			−0.643760
$656$		0			0
$657$	−17867.5	−	268566.i	−0.0413936	−	0.622186i
$658$		0			0
$659$		−	691050.i		−	1.59125i	−0.605789	−	0.795625i	$-0.707142\pi$
$659$		−	691050.i		−	1.59125i	0.605789	−	0.795625i	$-0.292858\pi$
$660$		0			0
$661$	58209.1			0.133226			0.0666129	−	0.997779i	$-0.478781\pi$
$661$	58209.1			0.133226			0.0666129	+	0.997779i	$0.478781\pi$
$662$		0			0
$663$	−233917.	+	249996.i	−0.532150	+	0.568730i
$664$		0			0
$665$			59939.3i			0.135540i
$666$		0			0
$667$	−87022.4			−0.195605
$668$		0			0
$669$	420016.	+	393001.i	0.938456	+	0.878096i
$670$		0			0
$671$			5916.66i			0.0131411i
$672$		0			0
$673$	−540244.			−1.19278			−0.596389	−	0.802695i	$-0.703399\pi$
$673$	−540244.			−1.19278			−0.596389	+	0.802695i	$0.703399\pi$
$674$		0			0
$675$	−246097.	+	300772.i	−0.540130	+	0.660130i
$676$		0			0
$677$		−	673444.i		−	1.46935i	−0.678421	−	0.734674i	$-0.737335\pi$
$677$		−	673444.i		−	1.46935i	0.678421	−	0.734674i	$-0.262665\pi$
$678$		0			0
$679$	375880.			0.815286
$680$		0			0
$681$	229033.	−	244777.i	0.493861	−	0.527809i
$682$		0			0
$683$		−	176046.i		−	0.377385i	−0.982036	−	0.188693i	$-0.939575\pi$
$683$		−	176046.i		−	0.377385i	0.982036	−	0.188693i	$-0.0604249\pi$
$684$		0			0
$685$	208769.			0.444924
$686$		0			0
$687$	472826.	+	442415.i	1.00182	+	0.937381i
$688$		0			0
$689$			798753.i			1.68257i
$690$		0			0
$691$	−146155.			−0.306097			−0.153048	−	0.988219i	$-0.548909\pi$
$691$	−146155.			−0.306097			−0.153048	+	0.988219i	$0.548909\pi$
$692$		0			0
$693$	−4986.57	+	331.753i	−0.0103833	+	0.000690795i
$694$		0			0
$695$		−	173151.i		−	0.358473i
$696$		0			0
$697$	−213265.			−0.438989
$698$		0			0
$699$	294003.	−	314213.i	0.601725	−	0.643087i
$700$		0			0
$701$			320326.i			0.651863i	0.945393	+	0.325931i	$0.105678\pi$
$701$			320326.i			0.651863i	−0.945393	+	0.325931i	$0.894322\pi$
$702$		0			0
$703$	348471.			0.705109
$704$		0			0
$705$	−90521.3	−	84699.1i	−0.182126	−	0.170412i
$706$		0			0
$707$		−	33513.4i		−	0.0670471i
$708$		0			0
$709$	269447.			0.536020			0.268010	−	0.963416i	$-0.413634\pi$
$709$	269447.			0.536020			0.268010	+	0.963416i	$0.413634\pi$
$710$		0			0
$711$	−1503.93	−	22605.5i	−0.00297501	−	0.0447173i
$712$		0			0
$713$			457223.i			0.899391i
$714$		0			0
$715$	−5738.00			−0.0112240
$716$		0			0
$717$	−630740.	+	674097.i	−1.22691	+	1.31125i
$718$		0			0
$719$			688905.i			1.33261i	0.745681	+	0.666303i	$0.232124\pi$
$719$			688905.i			1.33261i	−0.745681	+	0.666303i	$0.767876\pi$
$720$		0			0
$721$	202702.			0.389930
$722$		0			0
$723$	189931.	+	177715.i	0.363346	+	0.339976i
$724$		0			0
$725$		−	110433.i		−	0.210098i
$726$		0			0
$727$	947051.			1.79186			0.895931	−	0.444193i	$-0.146510\pi$
$727$	947051.			1.79186			0.895931	+	0.444193i	$0.146510\pi$
$728$		0			0
$729$	−105213.	−	520922.i	−0.197978	−	0.980207i
$730$		0			0
$731$		−	240054.i		−	0.449236i
$732$		0			0
$733$	−482104.			−0.897290			−0.448645	−	0.893710i	$-0.648093\pi$
$733$	−482104.			−0.897290			−0.448645	+	0.893710i	$0.648093\pi$
$734$		0			0
$735$	111219.	−	118864.i	0.205876	−	0.220027i
$736$		0			0
$737$		−	16276.8i		−	0.0299664i
$738$		0			0
$739$	342450.			0.627058			0.313529	−	0.949579i	$-0.398489\pi$
$739$	342450.			0.627058			0.313529	+	0.949579i	$0.398489\pi$
$740$		0			0
$741$	398579.	+	372943.i	0.725902	+	0.679213i
$742$		0			0
$743$			132676.i			0.240333i	0.992754	+	0.120167i	$0.0383428\pi$
$743$			132676.i			0.240333i	−0.992754	+	0.120167i	$0.961657\pi$
$744$		0			0
$745$	348812.			0.628461
$746$		0			0
$747$	−350744.	+	23334.8i	−0.628563	+	0.0418179i
$748$		0			0
$749$		−	58862.9i		−	0.104925i
$750$		0			0
$751$	−1.00065e6			−1.77421			−0.887104	−	0.461570i	$-0.847286\pi$
$751$	−1.00065e6			−1.77421			−0.887104	+	0.461570i	$0.847286\pi$
$752$		0			0
$753$	−357826.	+	382423.i	−0.631077	+	0.674457i
$754$		0			0
$755$		−	5946.76i		−	0.0104325i
$756$		0			0
$757$	883442.			1.54165			0.770826	−	0.637046i	$-0.219844\pi$
$757$	883442.			1.54165			0.770826	+	0.637046i	$0.219844\pi$
$758$		0			0
$759$	−7510.23	−	7027.18i	−0.0130368	−	0.0121982i
$760$		0			0
$761$			216478.i			0.373804i	0.982379	+	0.186902i	$0.0598446\pi$
$761$			216478.i			0.373804i	−0.982379	+	0.186902i	$0.940155\pi$
$762$		0			0
$763$	324752.			0.557831
$764$		0			0
$765$	8913.08	+	133972.i	0.0152302	+	0.228924i
$766$		0			0
$767$		−	1.31693e6i		−	2.23858i
$768$		0			0
$769$	−637749.			−1.07844			−0.539221	−	0.842164i	$-0.681281\pi$
$769$	−637749.			−1.07844			−0.539221	+	0.842164i	$0.681281\pi$
$770$		0			0
$771$	−218036.	+	233023.i	−0.366791	+	0.392004i
$772$		0			0
$773$		−	432868.i		−	0.724431i	−0.932094	−	0.362215i	$-0.882020\pi$
$773$		−	432868.i		−	0.724431i	0.932094	−	0.362215i	$-0.117980\pi$
$774$		0			0
$775$	−580221.			−0.966030
$776$		0			0
$777$	188410.	+	176292.i	0.312077	+	0.292005i
$778$		0			0
$779$			340017.i			0.560307i
$780$		0			0
$781$	−20559.4			−0.0337060
$782$		0			0
$783$	116878.	+	95631.9i	0.190638	+	0.155984i
$784$		0			0
$785$		−	180491.i		−	0.292898i
$786$		0			0
$787$	−432903.			−0.698942			−0.349471	−	0.936947i	$-0.613639\pi$
$787$	−432903.			−0.698942			−0.349471	+	0.936947i	$0.613639\pi$
$788$		0			0
$789$	−171205.	+	182973.i	−0.275019	+	0.293923i
$790$		0			0
$791$		−	35231.0i		−	0.0563082i
$792$		0			0
$793$	478502.			0.760918
$794$		0			0
$795$	228737.	+	214025.i	0.361911	+	0.338634i
$796$		0			0
$797$		−	69938.3i		−	0.110103i	−0.998484	−	0.0550514i	$-0.982468\pi$
$797$		−	69938.3i		−	0.110103i	0.998484	−	0.0550514i	$-0.0175323\pi$
$798$		0			0
$799$	248422.			0.389132
$800$		0			0
$801$	1.01522e6	−	67542.1i	1.58233	−	0.105271i
$802$		0			0
$803$			9039.83i			0.0140194i
$804$		0			0
$805$	−91339.3			−0.140950
$806$		0			0
$807$	−595746.	+	636698.i	−0.914775	+	0.977657i
$808$		0			0
$809$		−	1.06835e6i		−	1.63236i	−0.577795	−	0.816182i	$-0.696087\pi$
$809$		−	1.06835e6i		−	1.63236i	0.577795	−	0.816182i	$-0.303913\pi$
$810$		0			0
$811$	−571788.			−0.869347			−0.434674	−	0.900588i	$-0.643136\pi$
$811$	−571788.			−0.869347			−0.434674	+	0.900588i	$0.643136\pi$
$812$		0			0
$813$	−596606.	−	558233.i	−0.902623	−	0.844567i
$814$		0			0
$815$		−	117983.i		−	0.177625i
$816$		0			0
$817$	−382729.			−0.573386
$818$		0			0
$819$	26830.2	+	403283.i	0.0399996	+	0.601232i
$820$		0			0
$821$		−	415572.i		−	0.616538i	−0.951299	−	0.308269i	$-0.900250\pi$
$821$		−	415572.i		−	0.616538i	0.951299	−	0.308269i	$-0.0997496\pi$
$822$		0			0
$823$	7708.02			0.0113800			0.00569001	−	0.999984i	$-0.498189\pi$
$823$	7708.02			0.0113800			0.00569001	+	0.999984i	$0.498189\pi$
$824$		0			0
$825$	8917.58	−	9530.58i	0.0131020	−	0.0140027i
$826$		0			0
$827$		−	1.10888e6i		−	1.62134i	−0.585507	−	0.810668i	$-0.699104\pi$
$827$		−	1.10888e6i		−	1.62134i	0.585507	−	0.810668i	$-0.300896\pi$
$828$		0			0
$829$	−943774.			−1.37328			−0.686639	−	0.726998i	$-0.740915\pi$
$829$	−943774.			−1.37328			−0.686639	+	0.726998i	$0.740915\pi$
$830$		0			0
$831$	−119003.	−	111349.i	−0.172328	−	0.161244i
$832$		0			0
$833$			326205.i			0.470112i
$834$		0			0
$835$	−509404.			−0.730616
$836$		0			0
$837$	502458.	−	614088.i	0.717214	−	0.876556i
$838$		0			0
$839$		−	404815.i		−	0.575086i	−0.957768	−	0.287543i	$-0.907162\pi$
$839$		−	404815.i		−	0.575086i	0.957768	−	0.287543i	$-0.0928384\pi$
$840$		0			0
$841$	664367.			0.939326
$842$		0			0
$843$	−24360.8	+	26035.3i	−0.0342796	+	0.0366360i
$844$		0			0
$845$			190239.i			0.266433i
$846$		0			0
$847$	−331889.			−0.462621
$848$		0			0
$849$	−614275.	−	574766.i	−0.852212	−	0.797398i
$850$		0			0
$851$			531023.i			0.733253i
$852$		0			0
$853$	−445369.			−0.612099			−0.306050	−	0.952016i	$-0.599007\pi$
$853$	−445369.			−0.612099			−0.306050	+	0.952016i	$0.599007\pi$
$854$		0			0
$855$	213598.	−	14210.5i	0.292189	−	0.0194392i
$856$		0			0
$857$		−	463544.i		−	0.631145i	−0.948901	−	0.315572i	$-0.897803\pi$
$857$		−	463544.i		−	0.631145i	0.948901	−	0.315572i	$-0.102197\pi$
$858$		0			0
$859$	−209016.			−0.283265			−0.141633	−	0.989919i	$-0.545235\pi$
$859$	−209016.			−0.283265			−0.141633	+	0.989919i	$0.545235\pi$
$860$		0			0
$861$	−172015.	+	183839.i	−0.232038	+	0.247989i
$862$		0			0
$863$		−	1.22573e6i		−	1.64579i	−0.568197	−	0.822893i	$-0.692359\pi$
$863$		−	1.22573e6i		−	1.64579i	0.568197	−	0.822893i	$-0.307641\pi$
$864$		0			0
$865$	−543632.			−0.726562
$866$		0			0
$867$	352413.	+	329746.i	0.468828	+	0.438674i
$868$		0			0
$869$			760.893i			0.00100759i
$870$		0			0
$871$	−1.31637e6			−1.73517
$872$		0			0
$873$	−89114.4	−	1.33947e6i	−0.116928	−	1.75754i
$874$		0			0
$875$		−	251806.i		−	0.328889i
$876$		0			0
$877$	−438960.			−0.570724			−0.285362	−	0.958420i	$-0.592114\pi$
$877$	−438960.			−0.570724			−0.285362	+	0.958420i	$0.592114\pi$
$878$		0			0
$879$	−587730.	+	628131.i	−0.760677	+	0.812966i
$880$		0			0
$881$			630549.i			0.812395i	0.913785	+	0.406198i	$0.133146\pi$
$881$			630549.i			0.812395i	−0.913785	+	0.406198i	$0.866854\pi$
$882$		0			0
$883$	962033.			1.23387			0.616934	−	0.787015i	$-0.288375\pi$
$883$	962033.			1.23387			0.616934	+	0.787015i	$0.288375\pi$
$884$		0			0
$885$	−377127.	−	352870.i	−0.481505	−	0.450535i
$886$		0			0
$887$			750497.i			0.953898i	0.878931	+	0.476949i	$0.158257\pi$
$887$			750497.i			0.953898i	−0.878931	+	0.476949i	$0.841743\pi$
$888$		0			0
$889$	264095.			0.334162
$890$		0			0
$891$	2364.45	+	17691.3i	0.00297834	+	0.0222846i
$892$		0			0
$893$		−	396070.i		−	0.496671i
$894$		0			0
$895$	−34086.6			−0.0425538
$896$		0			0
$897$	−568315.	+	607380.i	−0.706324	+	0.754877i
$898$		0			0
$899$			225471.i			0.278979i
$900$		0			0
$901$	−627735.			−0.773262
$902$		0			0
$903$	−206932.	−	193623.i	−0.253777	−	0.237455i
$904$		0			0
$905$			371936.i			0.454120i
$906$		0			0
$907$	1.09302e6			1.32866			0.664329	−	0.747440i	$-0.268717\pi$
$907$	1.09302e6			1.32866			0.664329	+	0.747440i	$0.268717\pi$
$908$		0			0
$909$	−119427.	+	7945.42i	−0.144536	+	0.00961588i
$910$		0			0
$911$		−	480340.i		−	0.578779i	−0.957212	−	0.289389i	$-0.906548\pi$
$911$		−	480340.i		−	0.578779i	0.957212	−	0.289389i	$-0.0934522\pi$
$912$		0			0
$913$	11805.9			0.0141631
$914$		0			0
$915$	128214.	−	137028.i	0.153142	−	0.163669i
$916$		0			0
$917$		−	653379.i		−	0.777010i
$918$		0			0
$919$	−598975.			−0.709214			−0.354607	−	0.935015i	$-0.615385\pi$
$919$	−598975.			−0.709214			−0.354607	+	0.935015i	$0.615385\pi$
$920$		0			0
$921$	223118.	+	208767.i	0.263036	+	0.246118i
$922$		0			0
$923$			1.66271e6i			1.95170i
$924$		0			0
$925$	−673875.			−0.787582
$926$		0			0
$927$	−48056.9	−	722341.i	−0.0559237	−	0.840587i
$928$		0			0
$929$		−	967.870i		−	0.00112146i	−1.00000		0.000560732i	$-0.999822\pi$
$929$		−	967.870i		−	0.00112146i	1.00000		0.000560732i	$-0.000178487\pi$
$930$		0			0
$931$	520083.			0.600030
$932$		0			0
$933$	−110932.	+	118557.i	−0.127436	+	0.136196i
$934$		0			0
$935$		−	4509.45i		−	0.00515823i
$936$		0			0
$937$	−766934.			−0.873533			−0.436766	−	0.899575i	$-0.643876\pi$
$937$	−766934.			−0.873533			−0.436766	+	0.899575i	$0.643876\pi$
$938$		0			0
$939$	106602.	+	99745.2i	0.120902	+	0.113126i
$940$		0			0
$941$			642273.i			0.725338i	0.931918	+	0.362669i	$0.118134\pi$
$941$			642273.i			0.725338i	−0.931918	+	0.362669i	$0.881866\pi$
$942$		0			0
$943$	−518140.			−0.582671
$944$		0			0
$945$	122676.	+	100376.i	0.137372	+	0.112400i
$946$		0			0
$947$			405399.i			0.452046i	0.974122	+	0.226023i	$0.0725724\pi$
$947$			405399.i			0.452046i	−0.974122	+	0.226023i	$0.927428\pi$
$948$		0			0
$949$	731085.			0.811774
$950$		0			0
$951$	307470.	−	328605.i	0.339970	−	0.363340i
$952$		0			0
$953$		−	1.18554e6i		−	1.30536i	−0.757633	−	0.652681i	$-0.773644\pi$
$953$		−	1.18554e6i		−	1.30536i	0.757633	−	0.652681i	$-0.226356\pi$
$954$		0			0
$955$	−178882.			−0.196137
$956$		0			0
$957$	−3703.53	−	3465.33i	−0.00404382	−	0.00378373i
$958$		0			0
$959$			493884.i			0.537017i
$960$		0			0
$961$	261122.			0.282746
$962$		0			0
$963$	−209762.	+	13955.3i	−0.226190	+	0.0150483i
$964$		0			0
$965$		−	293001.i		−	0.314641i
$966$		0			0
$967$	915766.			0.979336			0.489668	−	0.871909i	$-0.337118\pi$
$967$	915766.			0.979336			0.489668	+	0.871909i	$0.337118\pi$
$968$		0			0
$969$	−293093.	+	313241.i	−0.312146	+	0.333603i
$970$		0			0
$971$		−	48579.4i		−	0.0515244i	−0.999668	−	0.0257622i	$-0.991799\pi$
$971$		−	48579.4i		−	0.0515244i	0.999668	−	0.0257622i	$-0.00820128\pi$
$972$		0			0
$973$	409623.			0.432672
$974$		0			0
$975$	−770774.	−	721199.i	−0.810807	−	0.758657i
$976$		0			0
$977$		−	732559.i		−	0.767456i	−0.923446	−	0.383728i	$-0.874640\pi$
$977$		−	732559.i		−	0.767456i	0.923446	−	0.383728i	$-0.125360\pi$
$978$		0			0
$979$	−34172.0			−0.0356537
$980$		0			0
$981$	−76992.7	−	1.15727e6i	−0.0800040	−	1.20254i
$982$		0			0
$983$		−	1.34795e6i		−	1.39497i	−0.716599	−	0.697486i	$-0.754302\pi$
$983$		−	1.34795e6i		−	1.39497i	0.716599	−	0.697486i	$-0.245698\pi$
$984$		0			0
$985$	−718251.			−0.740293
$986$		0			0
$987$	200372.	−	214146.i	0.205685	−	0.219824i
$988$		0			0
$989$		−	583227.i		−	0.596272i
$990$		0			0
$991$	1.72075e6			1.75214			0.876071	−	0.482182i	$-0.160156\pi$
$991$	1.72075e6			1.75214			0.876071	+	0.482182i	$0.160156\pi$
$992$		0			0
$993$	−993441.	−	929545.i	−1.00750	−	0.942696i
$994$		0			0
$995$			594734.i			0.600726i
$996$		0			0
$997$	−97990.5			−0.0985811			−0.0492905	−	0.998784i	$-0.515696\pi$
$997$	−97990.5			−0.0985811			−0.0492905	+	0.998784i	$0.515696\pi$
$998$		0			0
$999$	583559.	−	713208.i	0.584728	−	0.714636i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.e.b.257.14	yes	16
3.2	odd	2	inner	384.5.e.b.257.13	yes	16
4.3	odd	2		384.5.e.c.257.3	yes	16
8.3	odd	2		384.5.e.a.257.14	yes	16
8.5	even	2		384.5.e.d.257.3	yes	16
12.11	even	2		384.5.e.c.257.4	yes	16
24.5	odd	2		384.5.e.d.257.4	yes	16
24.11	even	2		384.5.e.a.257.13	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.e.a.257.13	✓	16	24.11	even	2
384.5.e.a.257.14	yes	16	8.3	odd	2
384.5.e.b.257.13	yes	16	3.2	odd	2	inner
384.5.e.b.257.14	yes	16	1.1	even	1	trivial
384.5.e.c.257.3	yes	16	4.3	odd	2
384.5.e.c.257.4	yes	16	12.11	even	2
384.5.e.d.257.3	yes	16	8.5	even	2
384.5.e.d.257.4	yes	16	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(6.57180 + 6.14911i) q^{3} +9.58700i q^{5} -22.6799 q^{7} +(5.37699 + 80.8213i) q^{9} +O(q^{10})\)	\(q+(6.57180 + 6.14911i) q^{3} +9.58700i q^{5} -22.6799 q^{7} +(5.37699 + 80.8213i) q^{9} -2.72041i q^{11} -220.010 q^{13} +(-58.9515 + 63.0038i) q^{15} -172.904i q^{17} -275.669 q^{19} +(-149.048 - 139.461i) q^{21} -420.082i q^{23} +533.089 q^{25} +(-461.642 + 564.205i) q^{27} -207.156i q^{29} -1088.41 q^{31} +(16.7281 - 17.8780i) q^{33} -217.432i q^{35} -1264.09 q^{37} +(-1445.86 - 1352.87i) q^{39} -1233.43i q^{41} +1388.36 q^{43} +(-774.834 + 51.5492i) q^{45} +1436.76i q^{47} -1886.62 q^{49} +(1063.21 - 1136.29i) q^{51} -3630.53i q^{53} +26.0806 q^{55} +(-1811.64 - 1695.12i) q^{57} +5985.78i q^{59} -2174.91 q^{61} +(-121.950 - 1833.02i) q^{63} -2109.24i q^{65} +5983.22 q^{67} +(2583.13 - 2760.69i) q^{69} -7557.44i q^{71} -3322.96 q^{73} +(3503.35 + 3278.02i) q^{75} +61.6987i q^{77} -279.697 q^{79} +(-6503.18 + 869.151i) q^{81} +4339.74i q^{83} +1657.63 q^{85} +(1273.82 - 1361.39i) q^{87} -12561.3i q^{89} +4989.81 q^{91} +(-7152.83 - 6692.77i) q^{93} -2642.84i q^{95} -16573.3 q^{97} +(219.868 - 14.6276i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{3} + 80 q^{7}+O(q^{10}) \) 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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