LMFDB - Embedded newform 384.5.e.b.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			$6.34189 + 3.61720i$ of defining polynomial
Character	$\chi$	$=$	384.257
Dual form			384.5.e.b.257.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.08506 + 8.01949i) q^{3} -47.0883i q^{5} -13.4570 q^{7} +(-47.6246 - 65.5202i) q^{9} +O(q^{10})$	\(q+(-4.08506 + 8.01949i) q^{3} -47.0883i q^{5} -13.4570 q^{7} +(-47.6246 - 65.5202i) q^{9} -135.410i q^{11} -80.8573 q^{13} +(377.624 + 192.358i) q^{15} -430.724i q^{17} +43.5828 q^{19} +(54.9727 - 107.918i) q^{21} +873.291i q^{23} -1592.31 q^{25} +(719.988 - 114.271i) q^{27} +707.044i q^{29} -677.729 q^{31} +(1085.92 + 553.159i) q^{33} +633.668i q^{35} +1630.90 q^{37} +(330.307 - 648.435i) q^{39} +1227.00i q^{41} +1431.45 q^{43} +(-3085.23 + 2242.56i) q^{45} -2066.09i q^{47} -2219.91 q^{49} +(3454.19 + 1759.53i) q^{51} +1465.79i q^{53} -6376.24 q^{55} +(-178.038 + 349.512i) q^{57} -470.434i q^{59} -39.9022 q^{61} +(640.885 + 881.706i) q^{63} +3807.43i q^{65} -4525.04 q^{67} +(-7003.35 - 3567.44i) q^{69} +9851.74i q^{71} -3277.79 q^{73} +(6504.67 - 12769.5i) q^{75} +1822.22i q^{77} -6933.92 q^{79} +(-2024.80 + 6240.75i) q^{81} -12738.8i q^{83} -20282.0 q^{85} +(-5670.14 - 2888.32i) q^{87} -9780.60i q^{89} +1088.10 q^{91} +(2768.56 - 5435.04i) q^{93} -2052.24i q^{95} -9324.85 q^{97} +(-8872.11 + 6448.86i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−4.08506	+	8.01949i	−0.453895	+	0.891055i
$3$	−4.08506	+	8.01949i	−0.453895	+	0.891055i
$4$		0			0
$5$		−	47.0883i		−	1.88353i	−0.336270	−	0.941766i	$-0.609165\pi$
$5$		−	47.0883i		−	1.88353i	0.336270	−	0.941766i	$-0.390835\pi$
$6$		0			0
$7$	−13.4570			−0.274633			−0.137316	−	0.990527i	$-0.543848\pi$
$7$	−13.4570			−0.274633			−0.137316	+	0.990527i	$0.543848\pi$
$8$		0			0
$9$	−47.6246	−	65.5202i	−0.587958	−	0.808892i
$10$		0			0
$11$		−	135.410i		−	1.11909i	−0.828799	−	0.559546i	$-0.810975\pi$
$11$		−	135.410i		−	1.11909i	0.828799	−	0.559546i	$-0.189025\pi$
$12$		0			0
$13$	−80.8573			−0.478445			−0.239223	−	0.970965i	$-0.576893\pi$
$13$	−80.8573			−0.478445			−0.239223	+	0.970965i	$0.576893\pi$
$14$		0			0
$15$	377.624	+	192.358i	1.67833	+	0.854926i
$16$		0			0
$17$		−	430.724i		−	1.49039i	−0.666845	−	0.745197i	$-0.732356\pi$
$17$		−	430.724i		−	1.49039i	0.666845	−	0.745197i	$-0.267644\pi$
$18$		0			0
$19$	43.5828			0.120728			0.0603640	−	0.998176i	$-0.480774\pi$
$19$	43.5828			0.120728			0.0603640	+	0.998176i	$0.480774\pi$
$20$		0			0
$21$	54.9727	−	107.918i	0.124655	−	0.244713i
$22$		0			0
$23$			873.291i			1.65083i	0.564524	+	0.825416i	$0.309060\pi$
$23$			873.291i			1.65083i	−0.564524	+	0.825416i	$0.690940\pi$
$24$		0			0
$25$	−1592.31			−2.54769
$26$		0			0
$27$	719.988	−	114.271i	0.987638	−	0.156751i
$28$		0			0
$29$			707.044i			0.840718i	0.907358	+	0.420359i	$0.138096\pi$
$29$			707.044i			0.840718i	−0.907358	+	0.420359i	$0.861904\pi$
$30$		0			0
$31$	−677.729			−0.705233			−0.352616	−	0.935768i	$-0.614708\pi$
$31$	−677.729			−0.705233			−0.352616	+	0.935768i	$0.614708\pi$
$32$		0			0
$33$	1085.92	+	553.159i	0.997173	+	0.507951i
$34$		0			0
$35$			633.668i			0.517280i
$36$		0			0
$37$	1630.90			1.19130			0.595652	−	0.803242i	$-0.296894\pi$
$37$	1630.90			1.19130			0.595652	+	0.803242i	$0.296894\pi$
$38$		0			0
$39$	330.307	−	648.435i	0.217164	−	0.426321i
$40$		0			0
$41$			1227.00i			0.729922i	0.931023	+	0.364961i	$0.118918\pi$
$41$			1227.00i			0.729922i	−0.931023	+	0.364961i	$0.881082\pi$
$42$		0			0
$43$	1431.45			0.774175			0.387088	−	0.922043i	$-0.373481\pi$
$43$	1431.45			0.774175			0.387088	+	0.922043i	$0.373481\pi$
$44$		0			0
$45$	−3085.23	+	2242.56i	−1.52357	+	1.10744i
$46$		0			0
$47$		−	2066.09i		−	0.935307i	−0.883912	−	0.467653i	$-0.845100\pi$
$47$		−	2066.09i		−	0.935307i	0.883912	−	0.467653i	$-0.154900\pi$
$48$		0			0
$49$	−2219.91			−0.924577
$50$		0			0
$51$	3454.19	+	1759.53i	1.32802	+	0.676483i
$52$		0			0
$53$			1465.79i			0.521818i	0.965363	+	0.260909i	$0.0840222\pi$
$53$			1465.79i			0.521818i	−0.965363	+	0.260909i	$0.915978\pi$
$54$		0			0
$55$	−6376.24			−2.10785
$56$		0			0
$57$	−178.038	+	349.512i	−0.0547979	+	0.107575i
$58$		0			0
$59$		−	470.434i		−	0.135143i	−0.997714	−	0.0675716i	$-0.978475\pi$
$59$		−	470.434i		−	0.135143i	0.997714	−	0.0675716i	$-0.0215251\pi$
$60$		0			0
$61$	−39.9022			−0.0107235			−0.00536176	−	0.999986i	$-0.501707\pi$
$61$	−39.9022			−0.0107235			−0.00536176	+	0.999986i	$0.501707\pi$
$62$		0			0
$63$	640.885	+	881.706i	0.161473	+	0.222148i
$64$		0			0
$65$			3807.43i			0.901167i
$66$		0			0
$67$	−4525.04			−1.00803			−0.504014	−	0.863695i	$-0.668144\pi$
$67$	−4525.04			−1.00803			−0.504014	+	0.863695i	$0.668144\pi$
$68$		0			0
$69$	−7003.35	−	3567.44i	−1.47098	−	0.749306i
$70$		0			0
$71$			9851.74i			1.95432i	0.212498	+	0.977161i	$0.431840\pi$
$71$			9851.74i			1.95432i	−0.212498	+	0.977161i	$0.568160\pi$
$72$		0			0
$73$	−3277.79			−0.615086			−0.307543	−	0.951534i	$-0.599507\pi$
$73$	−3277.79			−0.615086			−0.307543	+	0.951534i	$0.599507\pi$
$74$		0			0
$75$	6504.67	−	12769.5i	1.15639	−	2.27013i
$76$		0			0
$77$			1822.22i			0.307340i
$78$		0			0
$79$	−6933.92			−1.11103			−0.555514	−	0.831507i	$-0.687478\pi$
$79$	−6933.92			−1.11103			−0.555514	+	0.831507i	$0.687478\pi$
$80$		0			0
$81$	−2024.80	+	6240.75i	−0.308611	+	0.951188i
$82$		0			0
$83$		−	12738.8i		−	1.84915i	−0.380998	−	0.924576i	$-0.624420\pi$
$83$		−	12738.8i		−	1.84915i	0.380998	−	0.924576i	$-0.375580\pi$
$84$		0			0
$85$	−20282.0			−2.80720
$86$		0			0
$87$	−5670.14	−	2888.32i	−0.749126	−	0.381598i
$88$		0			0
$89$		−	9780.60i		−	1.23477i	−0.786662	−	0.617384i	$-0.788192\pi$
$89$		−	9780.60i		−	1.23477i	0.786662	−	0.617384i	$-0.211808\pi$
$90$		0			0
$91$	1088.10			0.131397
$92$		0			0
$93$	2768.56	−	5435.04i	0.320102	−	0.628401i
$94$		0			0
$95$		−	2052.24i		−	0.227395i
$96$		0			0
$97$	−9324.85			−0.991056			−0.495528	−	0.868592i	$-0.665025\pi$
$97$	−9324.85			−0.991056			−0.495528	+	0.868592i	$0.665025\pi$
$98$		0			0
$99$	−8872.11	+	6448.86i	−0.905225	+	0.657979i
$100$		0			0
$101$		−	4534.01i		−	0.444467i	−0.974993	−	0.222233i	$-0.928665\pi$
$101$		−	4534.01i		−	0.444467i	0.974993	−	0.222233i	$-0.0713347\pi$
$102$		0			0
$103$	−256.703			−0.0241967			−0.0120984	−	0.999927i	$-0.503851\pi$
$103$	−256.703			−0.0241967			−0.0120984	+	0.999927i	$0.503851\pi$
$104$		0			0
$105$	−5081.69	−	2588.57i	−0.460925	−	0.234791i
$106$		0			0
$107$			14130.8i			1.23424i	0.786871	+	0.617118i	$0.211700\pi$
$107$			14130.8i			1.23424i	−0.786871	+	0.617118i	$0.788300\pi$
$108$		0			0
$109$	3260.70			0.274447			0.137223	−	0.990540i	$-0.456182\pi$
$109$	3260.70			0.274447			0.137223	+	0.990540i	$0.456182\pi$
$110$		0			0
$111$	−6662.31	+	13079.0i	−0.540728	+	1.06152i
$112$		0			0
$113$			17389.3i			1.36184i	0.732358	+	0.680920i	$0.238420\pi$
$113$			17389.3i			1.36184i	−0.732358	+	0.680920i	$0.761580\pi$
$114$		0			0
$115$	41121.8			3.10940
$116$		0			0
$117$	3850.79	+	5297.79i	0.281306	+	0.387010i
$118$		0			0
$119$			5796.25i			0.409311i
$120$		0			0
$121$	−3694.93			−0.252369
$122$		0			0
$123$	−9839.91	−	5012.36i	−0.650401	−	0.331308i
$124$		0			0
$125$			45548.8i			2.91512i
$126$		0			0
$127$	−240.146			−0.0148891			−0.00744454	−	0.999972i	$-0.502370\pi$
$127$	−240.146			−0.0148891			−0.00744454	+	0.999972i	$0.502370\pi$
$128$		0			0
$129$	−5847.56	+	11479.5i	−0.351395	+	0.689833i
$130$		0			0
$131$			10373.7i			0.604493i	0.953230	+	0.302247i	$0.0977366\pi$
$131$			10373.7i			0.604493i	−0.953230	+	0.302247i	$0.902263\pi$
$132$		0			0
$133$	−586.494			−0.0331559
$134$		0			0
$135$	−5380.83	−	33903.0i	−0.295245	−	1.86025i
$136$		0			0
$137$		−	17099.0i		−	0.911025i	−0.890230	−	0.455512i	$-0.849456\pi$
$137$		−	17099.0i		−	0.911025i	0.890230	−	0.455512i	$-0.150544\pi$
$138$		0			0
$139$	20249.2			1.04804			0.524021	−	0.851705i	$-0.324431\pi$
$139$	20249.2			1.04804			0.524021	+	0.851705i	$0.324431\pi$
$140$		0			0
$141$	16569.0	+	8440.11i	0.833410	+	0.424531i
$142$		0			0
$143$			10948.9i			0.535425i
$144$		0			0
$145$	33293.5			1.58352
$146$		0			0
$147$	9068.46	−	17802.5i	0.419661	−	0.823849i
$148$		0			0
$149$		−	5185.03i		−	0.233550i	−0.993158	−	0.116775i	$-0.962744\pi$
$149$		−	5185.03i		−	0.233550i	0.993158	−	0.116775i	$-0.0372556\pi$
$150$		0			0
$151$	30009.7			1.31616			0.658078	−	0.752950i	$-0.271370\pi$
$151$	30009.7			1.31616			0.658078	+	0.752950i	$0.271370\pi$
$152$		0			0
$153$	−28221.1	+	20513.0i	−1.20557	+	0.876288i
$154$		0			0
$155$			31913.1i			1.32833i
$156$		0			0
$157$	−21627.7			−0.877429			−0.438714	−	0.898627i	$-0.644566\pi$
$157$	−21627.7			−0.877429			−0.438714	+	0.898627i	$0.644566\pi$
$158$		0			0
$159$	−11754.9	−	5987.82i	−0.464968	−	0.236851i
$160$		0			0
$161$		−	11751.9i		−	0.453373i
$162$		0			0
$163$	−33476.0			−1.25996			−0.629982	−	0.776610i	$-0.716938\pi$
$163$	−33476.0			−1.25996			−0.629982	+	0.776610i	$0.716938\pi$
$164$		0			0
$165$	26047.3	−	51134.2i	0.956742	−	1.87821i
$166$		0			0
$167$			32548.8i			1.16708i	0.812083	+	0.583542i	$0.198334\pi$
$167$			32548.8i			1.16708i	−0.812083	+	0.583542i	$0.801666\pi$
$168$		0			0
$169$	−22023.1			−0.771090
$170$		0			0
$171$	−2075.61	−	2855.55i	−0.0709829	−	0.0976558i
$172$		0			0
$173$			4318.22i			0.144282i	0.997394	+	0.0721410i	$0.0229832\pi$
$173$			4318.22i			0.144282i	−0.997394	+	0.0721410i	$0.977017\pi$
$174$		0			0
$175$	21427.7			0.699680
$176$		0			0
$177$	3772.64	+	1921.75i	0.120420	+	0.0613409i
$178$		0			0
$179$		−	10769.0i		−	0.336101i	−0.985778	−	0.168051i	$-0.946253\pi$
$179$		−	10769.0i		−	0.336101i	0.985778	−	0.168051i	$-0.0537472\pi$
$180$		0			0
$181$	17404.6			0.531260			0.265630	−	0.964075i	$-0.414420\pi$
$181$	17404.6			0.531260			0.265630	+	0.964075i	$0.414420\pi$
$182$		0			0
$183$	163.003	−	319.996i	0.00486736	−	0.00955525i
$184$		0			0
$185$		−	76796.1i		−	2.24386i
$186$		0			0
$187$	−58324.4			−1.66789
$188$		0			0
$189$	−9688.89	+	1537.75i	−0.271238	+	0.0430489i
$190$		0			0
$191$			21212.1i			0.581457i	0.956806	+	0.290729i	$0.0938977\pi$
$191$			21212.1i			0.581457i	−0.956806	+	0.290729i	$0.906102\pi$
$192$		0			0
$193$	62278.9			1.67196			0.835980	−	0.548759i	$-0.184900\pi$
$193$	62278.9			1.67196			0.835980	+	0.548759i	$0.184900\pi$
$194$		0			0
$195$	−30533.7	−	15553.6i	−0.802989	−	0.409036i
$196$		0			0
$197$		−	62929.2i		−	1.62151i	−0.585384	−	0.810756i	$-0.699056\pi$
$197$		−	62929.2i		−	1.62151i	0.585384	−	0.810756i	$-0.300944\pi$
$198$		0			0
$199$	28784.6			0.726867			0.363433	−	0.931620i	$-0.381604\pi$
$199$	28784.6			0.726867			0.363433	+	0.931620i	$0.381604\pi$
$200$		0			0
$201$	18485.1	−	36288.5i	0.457540	−	0.898209i
$202$		0			0
$203$		−	9514.70i		−	0.230889i
$204$		0			0
$205$	57777.3			1.37483
$206$		0			0
$207$	57218.2	−	41590.1i	1.33534	−	0.970620i
$208$		0			0
$209$		−	5901.55i		−	0.135106i
$210$		0			0
$211$	1138.03			0.0255616			0.0127808	−	0.999918i	$-0.495932\pi$
$211$	1138.03			0.0255616			0.0127808	+	0.999918i	$0.495932\pi$
$212$		0			0
$213$	−79006.0	−	40244.9i	−1.74141	−	0.887058i
$214$		0			0
$215$		−	67404.5i		−	1.45818i
$216$		0			0
$217$	9120.20			0.193680
$218$		0			0
$219$	13390.0	−	26286.2i	0.279185	−	0.548075i
$220$		0			0
$221$			34827.1i			0.713072i
$222$		0			0
$223$	−67526.2			−1.35788			−0.678942	−	0.734192i	$-0.737561\pi$
$223$	−67526.2			−1.35788			−0.678942	+	0.734192i	$0.737561\pi$
$224$		0			0
$225$	75832.9	+	104328.i	1.49793	+	2.06081i
$226$		0			0
$227$			69701.4i			1.35266i	0.736597	+	0.676332i	$0.236431\pi$
$227$			69701.4i			1.35266i	−0.736597	+	0.676332i	$0.763569\pi$
$228$		0			0
$229$	4886.85			0.0931877			0.0465938	−	0.998914i	$-0.485163\pi$
$229$	4886.85			0.0931877			0.0465938	+	0.998914i	$0.485163\pi$
$230$		0			0
$231$	−14613.3	−	7443.87i	−0.273857	−	0.139500i
$232$		0			0
$233$			18388.2i			0.338710i	0.985555	+	0.169355i	$0.0541684\pi$
$233$			18388.2i			0.338710i	−0.985555	+	0.169355i	$0.945832\pi$
$234$		0			0
$235$	−97288.7			−1.76168
$236$		0			0
$237$	28325.5	−	55606.5i	0.504290	−	0.989986i
$238$		0			0
$239$			44332.4i			0.776114i	0.921635	+	0.388057i	$0.126854\pi$
$239$			44332.4i			0.776114i	−0.921635	+	0.388057i	$0.873146\pi$
$240$		0			0
$241$	−37613.5			−0.647604			−0.323802	−	0.946125i	$-0.604961\pi$
$241$	−37613.5			−0.647604			−0.323802	+	0.946125i	$0.604961\pi$
$242$		0			0
$243$	−41776.2	−	41731.7i	−0.707484	−	0.706730i
$244$		0			0
$245$			104532.i			1.74147i
$246$		0			0
$247$	−3523.99			−0.0577617
$248$		0			0
$249$	102159.	+	52038.8i	1.64770	+	0.839321i
$250$		0			0
$251$			35403.3i			0.561948i	0.959715	+	0.280974i	$0.0906575\pi$
$251$			35403.3i			0.561948i	−0.959715	+	0.280974i	$0.909343\pi$
$252$		0			0
$253$	118252.			1.84744
$254$		0			0
$255$	82853.3	−	162652.i	1.27418	−	2.50137i
$256$		0			0
$257$		−	26838.2i		−	0.406338i	−0.979144	−	0.203169i	$-0.934876\pi$
$257$		−	26838.2i		−	0.406338i	0.979144	−	0.203169i	$-0.0651241\pi$
$258$		0			0
$259$	−21947.0			−0.327171
$260$		0			0
$261$	46325.7	−	33672.7i	0.680050	−	0.494307i
$262$		0			0
$263$			6793.65i			0.0982182i	0.998793	+	0.0491091i	$0.0156382\pi$
$263$			6793.65i			0.0982182i	−0.998793	+	0.0491091i	$0.984362\pi$
$264$		0			0
$265$	69021.3			0.982860
$266$		0			0
$267$	78435.5	+	39954.3i	1.10025	+	0.560456i
$268$		0			0
$269$			39971.6i			0.552392i	0.961101	+	0.276196i	$0.0890738\pi$
$269$			39971.6i			0.552392i	−0.961101	+	0.276196i	$0.910926\pi$
$270$		0			0
$271$	10277.6			0.139944			0.0699720	−	0.997549i	$-0.477709\pi$
$271$	10277.6			0.139944			0.0699720	+	0.997549i	$0.477709\pi$
$272$		0			0
$273$	−4444.94	+	8725.99i	−0.0596404	+	0.117082i
$274$		0			0
$275$			215615.i			2.85110i
$276$		0			0
$277$	−20101.1			−0.261975			−0.130988	−	0.991384i	$-0.541815\pi$
$277$	−20101.1			−0.261975			−0.130988	+	0.991384i	$0.541815\pi$
$278$		0			0
$279$	32276.5	+	44404.9i	0.414647	+	0.570457i
$280$		0			0
$281$		−	131139.i		−	1.66081i	−0.557159	−	0.830406i	$-0.688109\pi$
$281$		−	131139.i		−	1.66081i	0.557159	−	0.830406i	$-0.311891\pi$
$282$		0			0
$283$	71448.1			0.892109			0.446054	−	0.895006i	$-0.352829\pi$
$283$	71448.1			0.892109			0.446054	+	0.895006i	$0.352829\pi$
$284$		0			0
$285$	16457.9	+	8383.51i	0.202621	+	0.103213i
$286$		0			0
$287$		−	16511.7i		−	0.200461i
$288$		0			0
$289$	−102002.			−1.22127
$290$		0			0
$291$	38092.6	−	74780.6i	0.449836	−	0.883085i
$292$		0			0
$293$		−	21604.4i		−	0.251656i	−0.992052	−	0.125828i	$-0.959841\pi$
$293$		−	21604.4i		−	0.251656i	0.992052	−	0.125828i	$-0.0401587\pi$
$294$		0			0
$295$	−22151.9			−0.254547
$296$		0			0
$297$	−15473.5	−	97493.8i	−0.175418	−	1.10526i
$298$		0			0
$299$		−	70611.9i		−	0.789834i
$300$		0			0
$301$	−19263.0			−0.212614
$302$		0			0
$303$	36360.4	+	18521.7i	0.396044	+	0.201742i
$304$		0			0
$305$			1878.93i			0.0201981i
$306$		0			0
$307$	−109205.			−1.15869			−0.579343	−	0.815084i	$-0.696691\pi$
$307$	−109205.			−1.15869			−0.579343	+	0.815084i	$0.696691\pi$
$308$		0			0
$309$	1048.65	−	2058.63i	0.0109828	−	0.0215606i
$310$		0			0
$311$			87204.3i			0.901606i	0.892623	+	0.450803i	$0.148862\pi$
$311$			87204.3i			0.901606i	−0.892623	+	0.450803i	$0.851138\pi$
$312$		0			0
$313$	−45320.5			−0.462601			−0.231300	−	0.972882i	$-0.574298\pi$
$313$	−45320.5			−0.462601			−0.231300	+	0.972882i	$0.574298\pi$
$314$		0			0
$315$	41518.0	−	30178.2i	0.418423	−	0.304139i
$316$		0			0
$317$		−	56611.4i		−	0.563360i	−0.959508	−	0.281680i	$-0.909108\pi$
$317$		−	56611.4i		−	0.563360i	0.959508	−	0.281680i	$-0.0908916\pi$
$318$		0			0
$319$	95741.0			0.940842
$320$		0			0
$321$	−113322.	−	57725.0i	−1.09977	−	0.560214i
$322$		0			0
$323$		−	18772.1i		−	0.179932i
$324$		0			0
$325$	128750.			1.21893
$326$		0			0
$327$	−13320.1	+	26149.2i	−0.124570	+	0.244547i
$328$		0			0
$329$			27803.4i			0.256866i
$330$		0			0
$331$	−80681.1			−0.736404			−0.368202	−	0.929746i	$-0.620026\pi$
$331$	−80681.1			−0.736404			−0.368202	+	0.929746i	$0.620026\pi$
$332$		0			0
$333$	−77670.7	−	106857.i	−0.700437	−	0.963636i
$334$		0			0
$335$			213076.i			1.89865i
$336$		0			0
$337$	96509.7			0.849790			0.424895	−	0.905243i	$-0.360311\pi$
$337$	96509.7			0.849790			0.424895	+	0.905243i	$0.360311\pi$
$338$		0			0
$339$	−139454.	−	71036.5i	−1.21347	−	0.618133i
$340$		0			0
$341$			91771.4i			0.789221i
$342$		0			0
$343$	62183.6			0.528552
$344$		0			0
$345$	−167985.	+	329776.i	−1.41134	+	2.77064i
$346$		0			0
$347$		−	111073.i		−	0.922466i	−0.887279	−	0.461233i	$-0.847407\pi$
$347$		−	111073.i		−	0.922466i	0.887279	−	0.461233i	$-0.152593\pi$
$348$		0			0
$349$	−162219.			−1.33184			−0.665919	−	0.746024i	$-0.731960\pi$
$349$	−162219.			−1.33184			−0.665919	+	0.746024i	$0.731960\pi$
$350$		0			0
$351$	−58216.3	+	9239.66i	−0.472531	+	0.0749966i
$352$		0			0
$353$		−	181489.i		−	1.45647i	−0.685327	−	0.728236i	$-0.740341\pi$
$353$		−	181489.i		−	1.45647i	0.685327	−	0.728236i	$-0.259659\pi$
$354$		0			0
$355$	463902.			3.68103
$356$		0			0
$357$	−46483.0	−	23678.0i	−0.364719	−	0.185784i
$358$		0			0
$359$			143379.i			1.11249i	0.831017	+	0.556247i	$0.187759\pi$
$359$			143379.i			1.11249i	−0.831017	+	0.556247i	$0.812241\pi$
$360$		0			0
$361$	−128422.			−0.985425
$362$		0			0
$363$	15094.0	−	29631.5i	0.114549	−	0.224874i
$364$		0			0
$365$			154346.i			1.15853i
$366$		0			0
$367$	−106490.			−0.790635			−0.395318	−	0.918544i	$-0.629366\pi$
$367$	−106490.			−0.790635			−0.395318	+	0.918544i	$0.629366\pi$
$368$		0			0
$369$	80393.2	−	58435.3i	0.590428	−	0.429163i
$370$		0			0
$371$		−	19725.1i		−	0.143308i
$372$		0			0
$373$	−173884.			−1.24981			−0.624903	−	0.780702i	$-0.714862\pi$
$373$	−173884.			−1.24981			−0.624903	+	0.780702i	$0.714862\pi$
$374$		0			0
$375$	−365278.	−	186070.i	−2.59754	−	1.32316i
$376$		0			0
$377$		−	57169.7i		−	0.402238i
$378$		0			0
$379$	16828.6			0.117158			0.0585788	−	0.998283i	$-0.481343\pi$
$379$	16828.6			0.117158			0.0585788	+	0.998283i	$0.481343\pi$
$380$		0			0
$381$	981.011	−	1925.85i	0.00675809	−	0.0132670i
$382$		0			0
$383$		−	149500.i		−	1.01916i	−0.860422	−	0.509582i	$-0.829800\pi$
$383$		−	149500.i		−	1.01916i	0.860422	−	0.509582i	$-0.170200\pi$
$384$		0			0
$385$	85805.1			0.578884
$386$		0			0
$387$	−68172.2	−	93788.9i	−0.455183	−	0.626224i
$388$		0			0
$389$			70496.4i			0.465873i	0.972492	+	0.232937i	$0.0748334\pi$
$389$			70496.4i			0.465873i	−0.972492	+	0.232937i	$0.925167\pi$
$390$		0			0
$391$	376147.			2.46039
$392$		0			0
$393$	−83191.9	−	42377.2i	−0.538637	−	0.274377i
$394$		0			0
$395$			326506.i			2.09265i
$396$		0			0
$397$	−155197.			−0.984698			−0.492349	−	0.870398i	$-0.663862\pi$
$397$	−155197.			−0.984698			−0.492349	+	0.870398i	$0.663862\pi$
$398$		0			0
$399$	2395.86	−	4703.39i	0.0150493	−	0.0295437i
$400$		0			0
$401$			59448.0i			0.369699i	0.982767	+	0.184850i	$0.0591798\pi$
$401$			59448.0i			0.369699i	−0.982767	+	0.184850i	$0.940820\pi$
$402$		0			0
$403$	54799.3			0.337415
$404$		0			0
$405$	293866.	+	95344.2i	1.79159	+	0.581279i
$406$		0			0
$407$		−	220840.i		−	1.33318i
$408$		0			0
$409$	166697.			0.996507			0.498254	−	0.867031i	$-0.333975\pi$
$409$	166697.			0.996507			0.498254	+	0.867031i	$0.333975\pi$
$410$		0			0
$411$	137125.	+	69850.5i	0.811773	+	0.413510i
$412$		0			0
$413$			6330.63i			0.0371148i
$414$		0			0
$415$	−599848.			−3.48293
$416$		0			0
$417$	−82719.3	+	162389.i	−0.475701	+	0.933863i
$418$		0			0
$419$		−	229117.i		−	1.30506i	−0.757765	−	0.652528i	$-0.773709\pi$
$419$		−	229117.i		−	1.30506i	0.757765	−	0.652528i	$-0.226291\pi$
$420$		0			0
$421$	−56063.1			−0.316310			−0.158155	−	0.987414i	$-0.550555\pi$
$421$	−56063.1			−0.316310			−0.158155	+	0.987414i	$0.550555\pi$
$422$		0			0
$423$	−135371.	+	98396.8i	−0.756562	+	0.549921i
$424$		0			0
$425$			685844.i			3.79706i
$426$		0			0
$427$	536.965			0.00294503
$428$		0			0
$429$	−87804.7	−	44726.9i	−0.477093	−	0.243027i
$430$		0			0
$431$			161763.i			0.870813i	0.900234	+	0.435406i	$0.143395\pi$
$431$			161763.i			0.870813i	−0.900234	+	0.435406i	$0.856605\pi$
$432$		0			0
$433$	−248951.			−1.32782			−0.663909	−	0.747814i	$-0.731104\pi$
$433$	−248951.			−1.32782			−0.663909	+	0.747814i	$0.731104\pi$
$434$		0			0
$435$	−136006.	+	266997.i	−0.718752	+	1.41100i
$436$		0			0
$437$			38060.4i			0.199302i
$438$		0			0
$439$	−267787.			−1.38950			−0.694752	−	0.719249i	$-0.744486\pi$
$439$	−267787.			−1.38950			−0.694752	+	0.719249i	$0.744486\pi$
$440$		0			0
$441$	105722.	+	145449.i	0.543612	+	0.747882i
$442$		0			0
$443$		−	73935.0i		−	0.376741i	−0.982098	−	0.188370i	$-0.939679\pi$
$443$		−	73935.0i		−	0.376741i	0.982098	−	0.188370i	$-0.0603205\pi$
$444$		0			0
$445$	−460552.			−2.32573
$446$		0			0
$447$	41581.4	+	21181.2i	0.208106	+	0.106007i
$448$		0			0
$449$		−	81901.6i		−	0.406256i	−0.979152	−	0.203128i	$-0.934889\pi$
$449$		−	81901.6i		−	0.406256i	0.979152	−	0.203128i	$-0.0651108\pi$
$450$		0			0
$451$	166148.			0.816850
$452$		0			0
$453$	−122591.	+	240662.i	−0.597397	+	1.17277i
$454$		0			0
$455$		−	51236.6i		−	0.247490i
$456$		0			0
$457$	174674.			0.836365			0.418183	−	0.908363i	$-0.362667\pi$
$457$	174674.			0.836365			0.418183	+	0.908363i	$0.362667\pi$
$458$		0			0
$459$	−49219.3	−	310116.i	−0.233620	−	1.47197i
$460$		0			0
$461$		−	365747.i		−	1.72099i	−0.509458	−	0.860495i	$-0.670154\pi$
$461$		−	365747.i		−	1.72099i	0.509458	−	0.860495i	$-0.329846\pi$
$462$		0			0
$463$	99999.2			0.466482			0.233241	−	0.972419i	$-0.425067\pi$
$463$	99999.2			0.466482			0.233241	+	0.972419i	$0.425067\pi$
$464$		0			0
$465$	−255927.	−	130367.i	−1.18361	−	0.602922i
$466$		0			0
$467$		−	341556.i		−	1.56613i	−0.621938	−	0.783067i	$-0.713654\pi$
$467$		−	341556.i		−	1.56613i	0.621938	−	0.783067i	$-0.286346\pi$
$468$		0			0
$469$	60893.5			0.276838
$470$		0			0
$471$	88350.6	−	173444.i	0.398261	−	0.781837i
$472$		0			0
$473$		−	193833.i		−	0.866374i
$474$		0			0
$475$	−69397.1			−0.307577
$476$		0			0
$477$	96038.6	−	69807.4i	0.422094	−	0.306807i
$478$		0			0
$479$		−	168022.i		−	0.732309i	−0.930554	−	0.366155i	$-0.880674\pi$
$479$		−	168022.i		−	0.732309i	0.930554	−	0.366155i	$-0.119326\pi$
$480$		0			0
$481$	−131870.			−0.569974
$482$		0			0
$483$	94244.2	+	48007.1i	0.403980	+	0.205784i
$484$		0			0
$485$			439091.i			1.86669i
$486$		0			0
$487$	−173451.			−0.731339			−0.365669	−	0.930745i	$-0.619160\pi$
$487$	−173451.			−0.731339			−0.365669	+	0.930745i	$0.619160\pi$
$488$		0			0
$489$	136751.	−	268460.i	0.571892	−	1.12270i
$490$		0			0
$491$			46403.4i			0.192481i	0.995358	+	0.0962403i	$0.0306817\pi$
$491$			46403.4i			0.192481i	−0.995358	+	0.0962403i	$0.969318\pi$
$492$		0			0
$493$	304541.			1.25300
$494$		0			0
$495$	303666.	+	417772.i	1.23932	+	1.70502i
$496$		0			0
$497$		−	132575.i		−	0.536721i
$498$		0			0
$499$	269136.			1.08086			0.540432	−	0.841388i	$-0.318261\pi$
$499$	269136.			1.08086			0.540432	+	0.841388i	$0.318261\pi$
$500$		0			0
$501$	−261025.	−	132964.i	−1.03994	−	0.529734i
$502$		0			0
$503$			494532.i			1.95460i	0.211854	+	0.977301i	$0.432050\pi$
$503$			494532.i			1.95460i	−0.211854	+	0.977301i	$0.567950\pi$
$504$		0			0
$505$	−213499.			−0.837167
$506$		0			0
$507$	89965.7	−	176614.i	0.349994	−	0.687084i
$508$		0			0
$509$		−	291389.i		−	1.12470i	−0.826899	−	0.562350i	$-0.809897\pi$
$509$		−	291389.i		−	1.12470i	0.826899	−	0.562350i	$-0.190103\pi$
$510$		0			0
$511$	44109.3			0.168923
$512$		0			0
$513$	31379.1	−	4980.25i	0.119236	−	0.0189242i
$514$		0			0
$515$			12087.7i			0.0455753i
$516$		0			0
$517$	−279770.			−1.04669
$518$		0			0
$519$	−34629.9	−	17640.2i	−0.128563	−	0.0654890i
$520$		0			0
$521$		−	47916.2i		−	0.176525i	−0.996097	−	0.0882626i	$-0.971869\pi$
$521$		−	47916.2i		−	0.176525i	0.996097	−	0.0882626i	$-0.0281315\pi$
$522$		0			0
$523$	−256372.			−0.937275			−0.468638	−	0.883390i	$-0.655255\pi$
$523$	−256372.			−0.937275			−0.468638	+	0.883390i	$0.655255\pi$
$524$		0			0
$525$	−87533.4	+	171839.i	−0.317581	+	0.623453i
$526$		0			0
$527$			291914.i			1.05107i
$528$		0			0
$529$	−482796.			−1.72525
$530$		0			0
$531$	−30822.9	+	22404.2i	−0.109316	+	0.0794585i
$532$		0			0
$533$		−	99211.8i		−	0.349228i
$534$		0			0
$535$	665393.			2.32472
$536$		0			0
$537$	86362.1	+	43992.1i	0.299485	+	0.152555i
$538$		0			0
$539$			300598.i			1.03469i
$540$		0			0
$541$	555503.			1.89798			0.948991	−	0.315303i	$-0.102106\pi$
$541$	555503.			1.89798			0.948991	+	0.315303i	$0.102106\pi$
$542$		0			0
$543$	−71098.8	+	139576.i	−0.241136	+	0.473382i
$544$		0			0
$545$		−	153541.i		−	0.516929i
$546$		0			0
$547$	−441029.			−1.47398			−0.736992	−	0.675902i	$-0.763754\pi$
$547$	−441029.			−1.47398			−0.736992	+	0.675902i	$0.763754\pi$
$548$		0			0
$549$	1900.33	+	2614.40i	0.00630498	+	0.00867417i
$550$		0			0
$551$			30815.0i			0.101498i
$552$		0			0
$553$	93309.9			0.305125
$554$		0			0
$555$	615866.	+	313717.i	1.99940	+	1.01848i
$556$		0			0
$557$		−	284934.i		−	0.918403i	−0.888332	−	0.459201i	$-0.848136\pi$
$557$		−	284934.i		−	0.918403i	0.888332	−	0.459201i	$-0.151864\pi$
$558$		0			0
$559$	−115743.			−0.370401
$560$		0			0
$561$	238259.	−	467732.i	0.757047	−	1.48618i
$562$		0			0
$563$		−	47991.2i		−	0.151407i	−0.997130	−	0.0757033i	$-0.975880\pi$
$563$		−	47991.2i		−	0.151407i	0.997130	−	0.0757033i	$-0.0241202\pi$
$564$		0			0
$565$	818834.			2.56507
$566$		0			0
$567$	27247.7	−	83981.8i	0.0847548	−	0.261228i
$568$		0			0
$569$			115826.i			0.357752i	0.983872	+	0.178876i	$0.0572461\pi$
$569$			115826.i			0.357752i	−0.983872	+	0.178876i	$0.942754\pi$
$570$		0			0
$571$	−364260.			−1.11722			−0.558611	−	0.829430i	$-0.688666\pi$
$571$	−364260.			−1.11722			−0.558611	+	0.829430i	$0.688666\pi$
$572$		0			0
$573$	−170111.	−	86652.8i	−0.518110	−	0.263921i
$574$		0			0
$575$		−	1.39055e6i		−	4.20581i
$576$		0			0
$577$	−24454.8			−0.0734536			−0.0367268	−	0.999325i	$-0.511693\pi$
$577$	−24454.8			−0.0734536			−0.0367268	+	0.999325i	$0.511693\pi$
$578$		0			0
$579$	−254413.	+	499445.i	−0.758895	+	1.48981i
$580$		0			0
$581$			171426.i			0.507838i
$582$		0			0
$583$	198482.			0.583962
$584$		0			0
$585$	249464.	−	181327.i	0.728946	−	0.529848i
$586$		0			0
$587$		−	469246.i		−	1.36183i	−0.732361	−	0.680917i	$-0.761582\pi$
$587$		−	469246.i		−	1.36183i	0.732361	−	0.680917i	$-0.238418\pi$
$588$		0			0
$589$	−29537.3			−0.0851413
$590$		0			0
$591$	504661.	+	257070.i	1.44486	+	0.735997i
$592$		0			0
$593$			420993.i			1.19720i	0.801049	+	0.598599i	$0.204276\pi$
$593$			420993.i			1.19720i	−0.801049	+	0.598599i	$0.795724\pi$
$594$		0			0
$595$	272936.			0.770950
$596$		0			0
$597$	−117587.	+	230838.i	−0.329921	+	0.647678i
$598$		0			0
$599$		−	451331.i		−	1.25789i	−0.777452	−	0.628943i	$-0.783488\pi$
$599$		−	451331.i		−	1.25789i	0.777452	−	0.628943i	$-0.216512\pi$
$600$		0			0
$601$	−52086.7			−0.144204			−0.0721021	−	0.997397i	$-0.522971\pi$
$601$	−52086.7			−0.144204			−0.0721021	+	0.997397i	$0.522971\pi$
$602$		0			0
$603$	215503.	+	296482.i	0.592678	+	0.815386i
$604$		0			0
$605$			173988.i			0.475344i
$606$		0			0
$607$	−433006.			−1.17521			−0.587606	−	0.809147i	$-0.699929\pi$
$607$	−433006.			−1.17521			−0.587606	+	0.809147i	$0.699929\pi$
$608$		0			0
$609$	76303.1	+	38868.1i	0.205735	+	0.104799i
$610$		0			0
$611$			167059.i			0.447493i
$612$		0			0
$613$	136805.			0.364067			0.182033	−	0.983292i	$-0.441732\pi$
$613$	136805.			0.364067			0.182033	+	0.983292i	$0.441732\pi$
$614$		0			0
$615$	−236024.	+	463344.i	−0.624029	+	1.22505i
$616$		0			0
$617$			473739.i			1.24443i	0.782848	+	0.622213i	$0.213766\pi$
$617$			473739.i			1.24443i	−0.782848	+	0.622213i	$0.786234\pi$
$618$		0			0
$619$	−148551.			−0.387700			−0.193850	−	0.981031i	$-0.562097\pi$
$619$	−148551.			−0.387700			−0.193850	+	0.981031i	$0.562097\pi$
$620$		0			0
$621$	99791.9	+	628759.i	0.258769	+	1.63043i
$622$		0			0
$623$			131618.i			0.339108i
$624$		0			0
$625$	1.14962e6			2.94304
$626$		0			0
$627$	47327.5	+	24108.2i	0.120387	+	0.0613239i
$628$		0			0
$629$		−	702465.i		−	1.77551i
$630$		0			0
$631$	−376817.			−0.946393			−0.473197	−	0.880957i	$-0.656900\pi$
$631$	−376817.			−0.946393			−0.473197	+	0.880957i	$0.656900\pi$
$632$		0			0
$633$	−4648.91	+	9126.40i	−0.0116023	+	0.0227768i
$634$		0			0
$635$			11308.1i			0.0280441i
$636$		0			0
$637$	179496.			0.442360
$638$		0			0
$639$	645488.	−	469185.i	1.58084	−	1.14906i
$640$		0			0
$641$		−	53679.4i		−	0.130645i	−0.997864	−	0.0653223i	$-0.979192\pi$
$641$		−	53679.4i		−	0.130645i	0.997864	−	0.0653223i	$-0.0208076\pi$
$642$		0			0
$643$	−298361.			−0.721640			−0.360820	−	0.932636i	$-0.617503\pi$
$643$	−298361.			−0.721640			−0.360820	+	0.932636i	$0.617503\pi$
$644$		0			0
$645$	540550.	+	275352.i	1.29932	+	0.661863i
$646$		0			0
$647$			239784.i			0.572811i	0.958109	+	0.286405i	$0.0924604\pi$
$647$			239784.i			0.572811i	−0.958109	+	0.286405i	$0.907540\pi$
$648$		0			0
$649$	−63701.5			−0.151238
$650$		0			0
$651$	−37256.6	+	73139.4i	−0.0879105	+	0.172580i
$652$		0			0
$653$			334473.i			0.784394i	0.919881	+	0.392197i	$0.128285\pi$
$653$			334473.i			0.784394i	−0.919881	+	0.392197i	$0.871715\pi$
$654$		0			0
$655$	488480.			1.13858
$656$		0			0
$657$	156103.	+	214762.i	0.361644	+	0.497538i
$658$		0			0
$659$			403774.i			0.929753i	0.885375	+	0.464877i	$0.153901\pi$
$659$			403774.i			0.929753i	−0.885375	+	0.464877i	$0.846099\pi$
$660$		0			0
$661$	434314.			0.994033			0.497016	−	0.867741i	$-0.334429\pi$
$661$	434314.			0.994033			0.497016	+	0.867741i	$0.334429\pi$
$662$		0			0
$663$	−279296.	−	142271.i	−0.635386	−	0.323660i
$664$		0			0
$665$			27617.0i			0.0624501i
$666$		0			0
$667$	−617455.			−1.38789
$668$		0			0
$669$	275848.	−	541526.i	0.616337	−	1.20995i
$670$		0			0
$671$			5403.17i			0.0120006i
$672$		0			0
$673$	202502.			0.447095			0.223547	−	0.974693i	$-0.428236\pi$
$673$	202502.			0.447095			0.223547	+	0.974693i	$0.428236\pi$
$674$		0			0
$675$	−1.14644e6	+	181955.i	−2.51620	+	0.399352i
$676$		0			0
$677$		−	572509.i		−	1.24912i	−0.780976	−	0.624561i	$-0.785278\pi$
$677$		−	572509.i		−	1.24912i	0.780976	−	0.624561i	$-0.214722\pi$
$678$		0			0
$679$	125485.			0.272177
$680$		0			0
$681$	−558970.	−	284734.i	−1.20530	−	0.613968i
$682$		0			0
$683$		−	500197.i		−	1.07226i	−0.844136	−	0.536130i	$-0.819886\pi$
$683$		−	500197.i		−	1.07226i	0.844136	−	0.536130i	$-0.180114\pi$
$684$		0			0
$685$	−805163.			−1.71594
$686$		0			0
$687$	−19963.1	+	39190.1i	−0.0422975	+	0.0830353i
$688$		0			0
$689$		−	118519.i		−	0.249661i
$690$		0			0
$691$	199477.			0.417769			0.208885	−	0.977940i	$-0.433017\pi$
$691$	199477.			0.417769			0.208885	+	0.977940i	$0.433017\pi$
$692$		0			0
$693$	119392.	−	86782.3i	0.248605	−	0.180703i
$694$		0			0
$695$		−	953501.i		−	1.97402i
$696$		0			0
$697$	528497.			1.08787
$698$		0			0
$699$	−147464.	−	75116.9i	−0.301809	−	0.153739i
$700$		0			0
$701$		−	465306.i		−	0.946896i	−0.880822	−	0.473448i	$-0.843009\pi$
$701$		−	465306.i		−	0.946896i	0.880822	−	0.473448i	$-0.156991\pi$
$702$		0			0
$703$	71079.0			0.143824
$704$		0			0
$705$	397430.	−	780207.i	0.799618	−	1.56975i
$706$		0			0
$707$			61014.2i			0.122065i
$708$		0			0
$709$	−609830.			−1.21316			−0.606578	−	0.795024i	$-0.707458\pi$
$709$	−609830.			−1.21316			−0.606578	+	0.795024i	$0.707458\pi$
$710$		0			0
$711$	330225.	+	454312.i	0.653237	+	0.898700i
$712$		0			0
$713$		−	591854.i		−	1.16422i
$714$		0			0
$715$	515565.			1.00849
$716$		0			0
$717$	−355524.	−	181101.i	−0.691561	−	0.352275i
$718$		0			0
$719$			149703.i			0.289583i	0.989462	+	0.144792i	$0.0462512\pi$
$719$			149703.i			0.289583i	−0.989462	+	0.144792i	$0.953749\pi$
$720$		0			0
$721$	3454.46			0.00664522
$722$		0			0
$723$	153653.	−	301641.i	0.293945	−	0.577051i
$724$		0			0
$725$		−	1.12583e6i		−	2.14189i
$726$		0			0
$727$	583781.			1.10454			0.552270	−	0.833666i	$-0.313762\pi$
$727$	583781.			1.10454			0.552270	+	0.833666i	$0.313762\pi$
$728$		0			0
$729$	505325.	−	164548.i	0.950859	−	0.309626i
$730$		0			0
$731$		−	616560.i		−	1.15383i
$732$		0			0
$733$	−362613.			−0.674894			−0.337447	−	0.941344i	$-0.609563\pi$
$733$	−362613.			−0.674894			−0.337447	+	0.941344i	$0.609563\pi$
$734$		0			0
$735$	−838291.	−	427018.i	−1.55174	−	0.790445i
$736$		0			0
$737$			612737.i			1.12808i
$738$		0			0
$739$	−507748.			−0.929736			−0.464868	−	0.885380i	$-0.653898\pi$
$739$	−507748.			−0.929736			−0.464868	+	0.885380i	$0.653898\pi$
$740$		0			0
$741$	14395.7	−	28260.6i	0.0262178	−	0.0514689i
$742$		0			0
$743$		−	332553.i		−	0.602398i	−0.953561	−	0.301199i	$-0.902613\pi$
$743$		−	332553.i		−	0.602398i	0.953561	−	0.301199i	$-0.0973868\pi$
$744$		0			0
$745$	−244154.			−0.439898
$746$		0			0
$747$	−834649.	+	606680.i	−1.49576	+	1.08722i
$748$		0			0
$749$		−	190158.i		−	0.338962i
$750$		0			0
$751$	504118.			0.893824			0.446912	−	0.894578i	$-0.352524\pi$
$751$	504118.			0.893824			0.446912	+	0.894578i	$0.352524\pi$
$752$		0			0
$753$	−283916.	−	144624.i	−0.500726	−	0.255066i
$754$		0			0
$755$		−	1.41310e6i		−	2.47902i
$756$		0			0
$757$	−456478.			−0.796578			−0.398289	−	0.917260i	$-0.630396\pi$
$757$	−456478.			−0.796578			−0.398289	+	0.917260i	$0.630396\pi$
$758$		0			0
$759$	−483068.	+	948325.i	−0.838542	+	1.64617i
$760$		0			0
$761$		−	297610.i		−	0.513899i	−0.966425	−	0.256950i	$-0.917283\pi$
$761$		−	297610.i		−	0.513899i	0.966425	−	0.256950i	$-0.0827174\pi$
$762$		0			0
$763$	−43879.3			−0.0753721
$764$		0			0
$765$	965924.	+	1.32888e6i	1.65052	+	2.27072i
$766$		0			0
$767$			38038.0i			0.0646587i
$768$		0			0
$769$	58486.7			0.0989018			0.0494509	−	0.998777i	$-0.484253\pi$
$769$	58486.7			0.0989018			0.0494509	+	0.998777i	$0.484253\pi$
$770$		0			0
$771$	215229.	+	109636.i	0.362070	+	0.184435i
$772$		0			0
$773$		−	328151.i		−	0.549180i	−0.961561	−	0.274590i	$-0.911458\pi$
$773$		−	328151.i		−	0.549180i	0.961561	−	0.274590i	$-0.0885422\pi$
$774$		0			0
$775$	1.07915e6			1.79671
$776$		0			0
$777$	89654.7	−	176004.i	0.148502	−	0.291528i
$778$		0			0
$779$			53476.0i			0.0881220i
$780$		0			0
$781$	1.33403e6			2.18707
$782$		0			0
$783$	80794.8	+	509064.i	0.131783	+	0.830326i
$784$		0			0
$785$			1.01841e6i			1.65266i
$786$		0			0
$787$	668331.			1.07905			0.539526	−	0.841969i	$-0.318603\pi$
$787$	668331.			1.07905			0.539526	+	0.841969i	$0.318603\pi$
$788$		0			0
$789$	−54481.7	−	27752.5i	−0.0875178	−	0.0445808i
$790$		0			0
$791$		−	234009.i		−	0.374006i
$792$		0			0
$793$	3226.39			0.00513062
$794$		0			0
$795$	−281956.	+	553516.i	−0.446116	+	0.875782i
$796$		0			0
$797$			247744.i			0.390020i	0.980801	+	0.195010i	$0.0624739\pi$
$797$			247744.i			0.390020i	−0.980801	+	0.195010i	$0.937526\pi$
$798$		0			0
$799$	−889915.			−1.39397
$800$		0			0
$801$	−640827.	+	465797.i	−0.998794	+	0.725992i
$802$		0			0
$803$			443847.i			0.688338i
$804$		0			0
$805$	−553376.			−0.853942
$806$		0			0
$807$	−320552.	−	163286.i	−0.492211	−	0.250728i
$808$		0			0
$809$		−	402635.i		−	0.615197i	−0.951516	−	0.307599i	$-0.900475\pi$
$809$		−	402635.i		−	0.615197i	0.951516	−	0.307599i	$-0.0995254\pi$
$810$		0			0
$811$	257418.			0.391378			0.195689	−	0.980666i	$-0.437306\pi$
$811$	257418.			0.391378			0.195689	+	0.980666i	$0.437306\pi$
$812$		0			0
$813$	−41984.7	+	82421.3i	−0.0635199	+	0.124698i
$814$		0			0
$815$			1.57633e6i			2.37318i
$816$		0			0
$817$	62386.6			0.0934646
$818$		0			0
$819$	−51820.2	−	71292.4i	−0.0772558	−	0.106286i
$820$		0			0
$821$			924513.i			1.37160i	0.727791	+	0.685799i	$0.240547\pi$
$821$			924513.i			1.37160i	−0.727791	+	0.685799i	$0.759453\pi$
$822$		0			0
$823$	−607274.			−0.896572			−0.448286	−	0.893890i	$-0.647965\pi$
$823$	−607274.			−0.896572			−0.448286	+	0.893890i	$0.647965\pi$
$824$		0			0
$825$	−1.72912e6	−	880798.i	−2.54049	−	1.29410i
$826$		0			0
$827$			588258.i			0.860116i	0.902801	+	0.430058i	$0.141507\pi$
$827$			588258.i			0.860116i	−0.902801	+	0.430058i	$0.858493\pi$
$828$		0			0
$829$	517960.			0.753680			0.376840	−	0.926278i	$-0.377011\pi$
$829$	517960.			0.753680			0.376840	+	0.926278i	$0.377011\pi$
$830$		0			0
$831$	82114.2	−	161201.i	0.118909	−	0.233434i
$832$		0			0
$833$			956167.i			1.37798i
$834$		0			0
$835$	1.53267e6			2.19824
$836$		0			0
$837$	−487957.	+	77444.8i	−0.696515	+	0.110546i
$838$		0			0
$839$		−	610245.i		−	0.866923i	−0.901172	−	0.433462i	$-0.857292\pi$
$839$		−	610245.i		−	0.866923i	0.901172	−	0.433462i	$-0.142708\pi$
$840$		0			0
$841$	207369.			0.293193
$842$		0			0
$843$	1.05167e6	+	535712.i	1.47987	+	0.753835i
$844$		0			0
$845$			1.03703e6i			1.45237i
$846$		0			0
$847$	49722.7			0.0693087
$848$		0			0
$849$	−291870.	+	572978.i	−0.404924	+	0.794918i
$850$		0			0
$851$			1.42425e6i			1.96664i
$852$		0			0
$853$	−1.21432e6			−1.66891			−0.834456	−	0.551074i	$-0.814218\pi$
$853$	−1.21432e6			−1.66891			−0.834456	+	0.551074i	$0.814218\pi$
$854$		0			0
$855$	−134463.	+	97737.0i	−0.183938	+	0.133699i
$856$		0			0
$857$			471294.i			0.641697i	0.947131	+	0.320848i	$0.103968\pi$
$857$			471294.i			0.641697i	−0.947131	+	0.320848i	$0.896032\pi$
$858$		0			0
$859$	83030.5			0.112526			0.0562628	−	0.998416i	$-0.482082\pi$
$859$	83030.5			0.112526			0.0562628	+	0.998416i	$0.482082\pi$
$860$		0			0
$861$	132416.	+	67451.4i	0.178621	+	0.0909882i
$862$		0			0
$863$		−	1.01156e6i		−	1.35822i	−0.734038	−	0.679108i	$-0.762367\pi$
$863$		−	1.01156e6i		−	1.35822i	0.734038	−	0.679108i	$-0.237633\pi$
$864$		0			0
$865$	203337.			0.271760
$866$		0			0
$867$	416684.	−	818003.i	0.554330	−	1.08822i
$868$		0			0
$869$			938924.i			1.24334i
$870$		0			0
$871$	365883.			0.482287
$872$		0			0
$873$	444092.	+	610966.i	0.582699	+	0.801657i
$874$		0			0
$875$		−	612951.i		−	0.800589i
$876$		0			0
$877$	1.31317e6			1.70734			0.853670	−	0.520814i	$-0.174371\pi$
$877$	1.31317e6			1.70734			0.853670	+	0.520814i	$0.174371\pi$
$878$		0			0
$879$	173256.	+	88255.3i	0.224239	+	0.114225i
$880$		0			0
$881$		−	1.01689e6i		−	1.31015i	−0.755563	−	0.655076i	$-0.772637\pi$
$881$		−	1.01689e6i		−	1.31015i	0.755563	−	0.655076i	$-0.227363\pi$
$882$		0			0
$883$	−78120.5			−0.100194			−0.0500972	−	0.998744i	$-0.515953\pi$
$883$	−78120.5			−0.100194			−0.0500972	+	0.998744i	$0.515953\pi$
$884$		0			0
$885$	90491.9	−	177647.i	0.115538	−	0.226815i
$886$		0			0
$887$		−	1.12684e6i		−	1.43224i	−0.697978	−	0.716119i	$-0.745917\pi$
$887$		−	1.12684e6i		−	1.43224i	0.697978	−	0.716119i	$-0.254083\pi$
$888$		0			0
$889$	3231.65			0.00408903
$890$		0			0
$891$	845061.	+	274178.i	1.06447	+	0.345364i
$892$		0			0
$893$		−	90046.0i		−	0.112918i
$894$		0			0
$895$	−507095.			−0.633058
$896$		0			0
$897$	566272.	+	288454.i	0.703785	+	0.358502i
$898$		0			0
$899$		−	479184.i		−	0.592902i
$900$		0			0
$901$	631348.			0.777713
$902$		0			0
$903$	78690.7	−	154480.i	0.0965046	−	0.189451i
$904$		0			0
$905$		−	819553.i		−	1.00064i
$906$		0			0
$907$	−684108.			−0.831592			−0.415796	−	0.909458i	$-0.636497\pi$
$907$	−684108.			−0.831592			−0.415796	+	0.909458i	$0.636497\pi$
$908$		0			0
$909$	−297069.	+	215930.i	−0.359526	+	0.261328i
$910$		0			0
$911$			217276.i			0.261804i	0.991395	+	0.130902i	$0.0417873\pi$
$911$			217276.i			0.261804i	−0.991395	+	0.130902i	$0.958213\pi$
$912$		0			0
$913$	−1.72496e6			−2.06937
$914$		0			0
$915$	−15068.0	−	7675.53i	−0.0179976	−	0.00916782i
$916$		0			0
$917$		−	139599.i		−	0.166014i
$918$		0			0
$919$	−953956.			−1.12953			−0.564764	−	0.825252i	$-0.691033\pi$
$919$	−953956.			−1.12953			−0.564764	+	0.825252i	$0.691033\pi$
$920$		0			0
$921$	446109.	−	875770.i	0.525923	−	1.03245i
$922$		0			0
$923$		−	796585.i		−	0.935037i
$924$		0			0
$925$	−2.59689e6			−3.03508
$926$		0			0
$927$	12225.4	+	16819.3i	0.0142267	+	0.0195725i
$928$		0			0
$929$			1.35448e6i			1.56942i	0.619860	+	0.784712i	$0.287189\pi$
$929$			1.35448e6i			1.56942i	−0.619860	+	0.784712i	$0.712811\pi$
$930$		0			0
$931$	−96749.8			−0.111622
$932$		0			0
$933$	−699334.	−	356235.i	−0.803381	−	0.409235i
$934$		0			0
$935$			2.74640e6i			3.14152i
$936$		0			0
$937$	−915233.			−1.04244			−0.521222	−	0.853421i	$-0.674524\pi$
$937$	−915233.			−1.04244			−0.521222	+	0.853421i	$0.674524\pi$
$938$		0			0
$939$	185137.	−	363448.i	0.209972	−	0.412203i
$940$		0			0
$941$		−	85021.4i		−	0.0960172i	−0.998847	−	0.0480086i	$-0.984713\pi$
$941$		−	85021.4i		−	0.0960172i	0.998847	−	0.0480086i	$-0.0152875\pi$
$942$		0			0
$943$	−1.07153e6			−1.20498
$944$		0			0
$945$	72409.9	+	456233.i	0.0810839	+	0.510885i
$946$		0			0
$947$		−	647558.i		−	0.722069i	−0.932552	−	0.361034i	$-0.882424\pi$
$947$		−	647558.i		−	0.722069i	0.932552	−	0.361034i	$-0.117576\pi$
$948$		0			0
$949$	265033.			0.294285
$950$		0			0
$951$	453995.	+	231261.i	0.501984	+	0.255706i
$952$		0			0
$953$			1.08402e6i			1.19358i	0.802396	+	0.596792i	$0.203558\pi$
$953$			1.08402e6i			1.19358i	−0.802396	+	0.596792i	$0.796442\pi$
$954$		0			0
$955$	998843.			1.09519
$956$		0			0
$957$	−391108.	+	767795.i	−0.427044	+	0.838342i
$958$		0			0
$959$			230102.i			0.250197i
$960$		0			0
$961$	−464205.			−0.502647
$962$		0			0
$963$	925850.	−	672972.i	0.998362	−	0.725678i
$964$		0			0
$965$		−	2.93261e6i		−	3.14919i
$966$		0			0
$967$	−1.36907e6			−1.46410			−0.732052	−	0.681249i	$-0.761437\pi$
$967$	−1.36907e6			−1.46410			−0.732052	+	0.681249i	$0.761437\pi$
$968$		0			0
$969$	150543.	+	76685.3i	0.160329	+	0.0816703i
$970$		0			0
$971$			1.36937e6i			1.45239i	0.687490	+	0.726194i	$0.258713\pi$
$971$			1.36937e6i			1.45239i	−0.687490	+	0.726194i	$0.741287\pi$
$972$		0			0
$973$	−272494.			−0.287827
$974$		0			0
$975$	−525950.	+	1.03251e6i	−0.553267	+	1.08613i
$976$		0			0
$977$		−	410297.i		−	0.429842i	−0.976631	−	0.214921i	$-0.931051\pi$
$977$		−	410297.i		−	0.429842i	0.976631	−	0.214921i	$-0.0689494\pi$
$978$		0			0
$979$	−1.32439e6			−1.38182
$980$		0			0
$981$	−155289.	−	213642.i	−0.161363	−	0.221997i
$982$		0			0
$983$		−	196933.i		−	0.203804i	−0.994794	−	0.101902i	$-0.967507\pi$
$983$		−	196933.i		−	0.203804i	0.994794	−	0.101902i	$-0.0324928\pi$
$984$		0			0
$985$	−2.96323e6			−3.05417
$986$		0			0
$987$	−222969.	−	113579.i	−0.228882	−	0.116590i
$988$		0			0
$989$			1.25007e6i			1.27803i
$990$		0			0
$991$	948066.			0.965365			0.482682	−	0.875795i	$-0.339663\pi$
$991$	948066.			0.965365			0.482682	+	0.875795i	$0.339663\pi$
$992$		0			0
$993$	329587.	−	647022.i	0.334250	−	0.656176i
$994$		0			0
$995$		−	1.35542e6i		−	1.36908i
$996$		0			0
$997$	300322.			0.302132			0.151066	−	0.988524i	$-0.451729\pi$
$997$	300322.			0.302132			0.151066	+	0.988524i	$0.451729\pi$
$998$		0			0
$999$	1.17423e6	−	186364.i	1.17658	−	0.186738i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(-4.08506 + 8.01949i) q^{3} -47.0883i q^{5} -13.4570 q^{7} +(-47.6246 - 65.5202i) q^{9} +O(q^{10})\)	\(q+(-4.08506 + 8.01949i) q^{3} -47.0883i q^{5} -13.4570 q^{7} +(-47.6246 - 65.5202i) q^{9} -135.410i q^{11} -80.8573 q^{13} +(377.624 + 192.358i) q^{15} -430.724i q^{17} +43.5828 q^{19} +(54.9727 - 107.918i) q^{21} +873.291i q^{23} -1592.31 q^{25} +(719.988 - 114.271i) q^{27} +707.044i q^{29} -677.729 q^{31} +(1085.92 + 553.159i) q^{33} +633.668i q^{35} +1630.90 q^{37} +(330.307 - 648.435i) q^{39} +1227.00i q^{41} +1431.45 q^{43} +(-3085.23 + 2242.56i) q^{45} -2066.09i q^{47} -2219.91 q^{49} +(3454.19 + 1759.53i) q^{51} +1465.79i q^{53} -6376.24 q^{55} +(-178.038 + 349.512i) q^{57} -470.434i q^{59} -39.9022 q^{61} +(640.885 + 881.706i) q^{63} +3807.43i q^{65} -4525.04 q^{67} +(-7003.35 - 3567.44i) q^{69} +9851.74i q^{71} -3277.79 q^{73} +(6504.67 - 12769.5i) q^{75} +1822.22i q^{77} -6933.92 q^{79} +(-2024.80 + 6240.75i) q^{81} -12738.8i q^{83} -20282.0 q^{85} +(-5670.14 - 2888.32i) q^{87} -9780.60i q^{89} +1088.10 q^{91} +(2768.56 - 5435.04i) q^{93} -2052.24i q^{95} -9324.85 q^{97} +(-8872.11 + 6448.86i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{3} + 80 q^{7}+O(q^{10}) \) 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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