LMFDB - Embedded newform 84.6.f.b.41.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,6,Mod(41,84)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("84.41");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	84.f (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:	$13.4722408643$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-6}, \sqrt{-161})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 334x^{2} + 24025 $ x^4 + 334*x^2 + 24025
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.4
Root			$10.2391i$ of defining polynomial
Character	$\chi$	$=$	84.41
Dual form			84.6.f.b.41.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(15.5403 + 1.22474i) q^{3} -31.0805 q^{5} +(-49.0000 - 120.025i) q^{7} +(240.000 + 38.0657i) q^{9} +O(q^{10})$	\(q+(15.5403 + 1.22474i) q^{3} -31.0805 q^{5} +(-49.0000 - 120.025i) q^{7} +(240.000 + 38.0657i) q^{9} -609.052i q^{11} -585.428i q^{13} +(-483.000 - 38.0657i) q^{15} +1554.03 q^{17} +2584.21i q^{19} +(-614.473 - 1925.23i) q^{21} -3730.44i q^{23} -2159.00 q^{25} +(3683.04 + 885.491i) q^{27} -5405.33i q^{29} -352.727i q^{31} +(745.933 - 9464.83i) q^{33} +(1522.95 + 3730.44i) q^{35} +12430.0 q^{37} +(717.000 - 9097.71i) q^{39} -3045.89 q^{41} -17900.0 q^{43} +(-7459.33 - 1183.10i) q^{45} +6029.62 q^{47} +(-12005.0 + 11762.4i) q^{49} +(24150.0 + 1903.29i) q^{51} +24895.0i q^{53} +18929.7i q^{55} +(-3165.00 + 40159.3i) q^{57} -22533.4 q^{59} +23174.6i q^{61} +(-7191.16 - 30671.2i) q^{63} +18195.4i q^{65} +6968.00 q^{67} +(4568.84 - 57972.1i) q^{69} -1218.10i q^{71} +43610.7i q^{73} +(-33551.4 - 2644.22i) q^{75} +(-73101.4 + 29843.5i) q^{77} +29846.0 q^{79} +(56151.0 + 18271.6i) q^{81} +86559.3 q^{83} -48300.0 q^{85} +(6620.16 - 84000.4i) q^{87} +50723.4 q^{89} +(-70266.0 + 28686.0i) q^{91} +(432.000 - 5481.47i) q^{93} -80318.7i q^{95} +75243.4i q^{97} +(23184.0 - 146172. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 196 q^{7} + 960 q^{9}+O(q^{10}) $ 4 * q - 196 * q^7 + 960 * q^9	$ 4 q - 196 q^{7} + 960 q^{9} - 1932 q^{15} + 588 q^{21} - 8636 q^{25} + 49720 q^{37} + 2868 q^{39} - 71600 q^{43} - 48020 q^{49} + 96600 q^{51} - 12660 q^{57} - 47040 q^{63} + 27872 q^{67} + 119384 q^{79} + 224604 q^{81} - 193200 q^{85} - 281064 q^{91} + 1728 q^{93} + 92736 q^{99}+O(q^{100}) $ 4 * q - 196 * q^7 + 960 * q^9 - 1932 * q^15 + 588 * q^21 - 8636 * q^25 + 49720 * q^37 + 2868 * q^39 - 71600 * q^43 - 48020 * q^49 + 96600 * q^51 - 12660 * q^57 - 47040 * q^63 + 27872 * q^67 + 119384 * q^79 + 224604 * q^81 - 193200 * q^85 - 281064 * q^91 + 1728 * q^93 + 92736 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	15.5403	+	1.22474i	0.996909	+	0.0785674i
$3$	15.5403	+	1.22474i	0.996909	+	0.0785674i
$4$		0			0
$5$	−31.0805			−0.555986			−0.277993	−	0.960583i	$-0.589669\pi$
$5$	−31.0805			−0.555986			−0.277993	+	0.960583i	$0.589669\pi$
$6$		0			0
$7$	−49.0000	−	120.025i	−0.377964	−	0.925820i
$7$	−49.0000	−	120.025i	−0.377964	−	0.925820i
$8$		0			0
$9$	240.000	+	38.0657i	0.987654	+	0.156649i
$10$		0			0
$11$		−	609.052i		−	1.51765i	−0.651293	−	0.758826i	$-0.725773\pi$
$11$		−	609.052i		−	1.51765i	0.651293	−	0.758826i	$-0.274227\pi$
$12$		0			0
$13$		−	585.428i		−	0.960761i	−0.877060	−	0.480380i	$-0.840499\pi$
$13$		−	585.428i		−	0.960761i	0.877060	−	0.480380i	$-0.159501\pi$
$14$		0			0
$15$	−483.000	−	38.0657i	−0.554267	−	0.0436824i
$16$		0			0
$17$	1554.03			1.30418			0.652088	−	0.758143i	$-0.273893\pi$
$17$	1554.03			1.30418			0.652088	+	0.758143i	$0.273893\pi$
$18$		0			0
$19$			2584.21i			1.64227i	0.570735	+	0.821134i	$0.306658\pi$
$19$			2584.21i			1.64227i	−0.570735	+	0.821134i	$0.693342\pi$
$20$		0			0
$21$	−614.473	−	1925.23i	−0.304057	−	0.952654i
$22$		0			0
$23$		−	3730.44i		−	1.47042i	−0.677841	−	0.735209i	$-0.737084\pi$
$23$		−	3730.44i		−	1.47042i	0.677841	−	0.735209i	$-0.262916\pi$
$24$		0			0
$25$	−2159.00			−0.690880
$26$		0			0
$27$	3683.04	+	885.491i	0.972294	+	0.233762i
$28$		0			0
$29$		−	5405.33i		−	1.19351i	−0.802422	−	0.596757i	$-0.796456\pi$
$29$		−	5405.33i		−	1.19351i	0.802422	−	0.596757i	$-0.203544\pi$
$30$		0			0
$31$		−	352.727i		−	0.0659225i	−0.999457	−	0.0329613i	$-0.989506\pi$
$31$		−	352.727i		−	0.0659225i	0.999457	−	0.0329613i	$-0.0104938\pi$
$32$		0			0
$33$	745.933	−	9464.83i	0.119238	−	1.51296i
$34$		0			0
$35$	1522.95	+	3730.44i	0.210143	+	0.514743i
$36$		0			0
$37$	12430.0			1.49268			0.746340	−	0.665565i	$-0.231809\pi$
$37$	12430.0			1.49268			0.746340	+	0.665565i	$0.231809\pi$
$38$		0			0
$39$	717.000	−	9097.71i	0.0754845	−	0.957791i
$40$		0			0
$41$	−3045.89			−0.282980			−0.141490	−	0.989940i	$-0.545189\pi$
$41$	−3045.89			−0.282980			−0.141490	+	0.989940i	$0.545189\pi$
$42$		0			0
$43$	−17900.0			−1.47632			−0.738162	−	0.674623i	$-0.764306\pi$
$43$	−17900.0			−1.47632			−0.738162	+	0.674623i	$0.764306\pi$
$44$		0			0
$45$	−7459.33	−	1183.10i	−0.549122	−	0.0870946i
$46$		0			0
$47$	6029.62			0.398149			0.199075	−	0.979984i	$-0.436206\pi$
$47$	6029.62			0.398149			0.199075	+	0.979984i	$0.436206\pi$
$48$		0			0
$49$	−12005.0	+	11762.4i	−0.714286	+	0.699854i
$50$		0			0
$51$	24150.0	+	1903.29i	1.30014	+	0.102466i
$52$		0			0
$53$			24895.0i			1.21737i	0.793412	+	0.608685i	$0.208302\pi$
$53$			24895.0i			1.21737i	−0.793412	+	0.608685i	$0.791698\pi$
$54$		0			0
$55$			18929.7i			0.843793i
$56$		0			0
$57$	−3165.00	+	40159.3i	−0.129029	+	1.63719i
$58$		0			0
$59$	−22533.4			−0.842746			−0.421373	−	0.906887i	$-0.638452\pi$
$59$	−22533.4			−0.842746			−0.421373	+	0.906887i	$0.638452\pi$
$60$		0			0
$61$			23174.6i			0.797422i	0.917077	+	0.398711i	$0.130542\pi$
$61$			23174.6i			0.797422i	−0.917077	+	0.398711i	$0.869458\pi$
$62$		0			0
$63$	−7191.16	−	30671.2i	−0.228269	−	0.973598i
$64$		0			0
$65$			18195.4i			0.534169i
$66$		0			0
$67$	6968.00			0.189636			0.0948181	−	0.995495i	$-0.469773\pi$
$67$	6968.00			0.189636			0.0948181	+	0.995495i	$0.469773\pi$
$68$		0			0
$69$	4568.84	−	57972.1i	0.115527	−	1.46587i
$70$		0			0
$71$		−	1218.10i		−	0.0286773i	−0.999897	−	0.0143387i	$-0.995436\pi$
$71$		−	1218.10i		−	0.0286773i	0.999897	−	0.0143387i	$-0.00456429\pi$
$72$		0			0
$73$			43610.7i			0.957825i	0.877863	+	0.478912i	$0.158969\pi$
$73$			43610.7i			0.957825i	−0.877863	+	0.478912i	$0.841031\pi$
$74$		0			0
$75$	−33551.4	−	2644.22i	−0.688744	−	0.0542807i
$76$		0			0
$77$	−73101.4	+	29843.5i	−1.40507	+	0.573619i
$78$		0			0
$79$	29846.0			0.538045			0.269022	−	0.963134i	$-0.413299\pi$
$79$	29846.0			0.538045			0.269022	+	0.963134i	$0.413299\pi$
$80$		0			0
$81$	56151.0	+	18271.6i	0.950922	+	0.309430i
$82$		0			0
$83$	86559.3			1.37917			0.689586	−	0.724204i	$-0.257792\pi$
$83$	86559.3			1.37917			0.689586	+	0.724204i	$0.257792\pi$
$84$		0			0
$85$	−48300.0			−0.725103
$86$		0			0
$87$	6620.16	−	84000.4i	0.0937713	−	1.18982i
$88$		0			0
$89$	50723.4			0.678787			0.339394	−	0.940644i	$-0.389778\pi$
$89$	50723.4			0.678787			0.339394	+	0.940644i	$0.389778\pi$
$90$		0			0
$91$	−70266.0	+	28686.0i	−0.889491	+	0.363133i
$92$		0			0
$93$	432.000	−	5481.47i	0.00517936	−	0.0657188i
$94$		0			0
$95$		−	80318.7i		−	0.913077i
$96$		0			0
$97$			75243.4i			0.811969i	0.913880	+	0.405984i	$0.133071\pi$
$97$			75243.4i			0.811969i	−0.913880	+	0.405984i	$0.866929\pi$
$98$		0			0
$99$	23184.0	−	146172.i	0.237739	−	1.49892i
$100$		0			0
$101$	−47646.5			−0.464758			−0.232379	−	0.972625i	$-0.574651\pi$
$101$	−47646.5			−0.464758			−0.232379	+	0.972625i	$0.574651\pi$
$102$		0			0
$103$		−	10121.3i		−	0.0940033i	−0.998895	−	0.0470016i	$-0.985033\pi$
$103$		−	10121.3i		−	0.0940033i	0.998895	−	0.0470016i	$-0.0149666\pi$
$104$		0			0
$105$	19098.2	+	59837.3i	0.169051	+	0.529662i
$106$		0			0
$107$		−	123485.i		−	1.04269i	−0.853346	−	0.521345i	$-0.825430\pi$
$107$		−	123485.i		−	1.04269i	0.853346	−	0.521345i	$-0.174570\pi$
$108$		0			0
$109$	−19814.0			−0.159737			−0.0798686	−	0.996805i	$-0.525450\pi$
$109$	−19814.0			−0.159737			−0.0798686	+	0.996805i	$0.525450\pi$
$110$		0			0
$111$	193166.	+	15223.6i	1.48807	+	0.117276i
$112$		0			0
$113$			74228.2i			0.546856i	0.961893	+	0.273428i	$0.0881575\pi$
$113$			74228.2i			0.546856i	−0.961893	+	0.273428i	$0.911843\pi$
$114$		0			0
$115$			115944.i			0.817531i
$116$		0			0
$117$	22284.7	−	140503.i	0.150502	−	0.948899i
$118$		0			0
$119$	−76147.3	−	186522.i	−0.492932	−	1.20743i
$120$		0			0
$121$	−209893.			−1.30327
$122$		0			0
$123$	−47334.0	−	3730.44i	−0.282105	−	0.0222330i
$124$		0			0
$125$	164230.			0.940105
$126$		0			0
$127$	171464.			0.943330			0.471665	−	0.881778i	$-0.343653\pi$
$127$	171464.			0.943330			0.471665	+	0.881778i	$0.343653\pi$
$128$		0			0
$129$	−278171.	−	21922.9i	−1.47176	−	0.115991i
$130$		0			0
$131$	250727.			1.27650			0.638252	−	0.769827i	$-0.279658\pi$
$131$	250727.			1.27650			0.638252	+	0.769827i	$0.279658\pi$
$132$		0			0
$133$	310170.	−	126626.i	1.52044	−	0.620719i
$134$		0			0
$135$	−114471.	−	27521.5i	−0.540581	−	0.129968i
$136$		0			0
$137$		−	92423.6i		−	0.420709i	−0.977625	−	0.210354i	$-0.932538\pi$
$137$		−	92423.6i		−	0.420709i	0.977625	−	0.210354i	$-0.0674618\pi$
$138$		0			0
$139$		−	269133.i		−	1.18149i	−0.806859	−	0.590744i	$-0.798834\pi$
$139$		−	269133.i		−	1.18149i	0.806859	−	0.590744i	$-0.201166\pi$
$140$		0			0
$141$	93702.0	+	7384.75i	0.396918	+	0.0312815i
$142$		0			0
$143$	−356556.			−1.45810
$144$		0			0
$145$			168001.i			0.663577i
$146$		0			0
$147$	−200967.	+	168089.i	−0.767063	+	0.641571i
$148$		0			0
$149$			424585.i			1.56675i	0.621551	+	0.783374i	$0.286503\pi$
$149$			424585.i			1.56675i	−0.621551	+	0.783374i	$0.713497\pi$
$150$		0			0
$151$	−281000.			−1.00291			−0.501457	−	0.865182i	$-0.667203\pi$
$151$	−281000.			−1.00291			−0.501457	+	0.865182i	$0.667203\pi$
$152$		0			0
$153$	372966.	+	59155.2i	1.28808	+	0.204298i
$154$		0			0
$155$			10962.9i			0.0366520i
$156$		0			0
$157$		−	244197.i		−	0.790663i	−0.918539	−	0.395331i	$-0.870630\pi$
$157$		−	244197.i		−	0.790663i	0.918539	−	0.395331i	$-0.129370\pi$
$158$		0			0
$159$	−30490.0	+	386875.i	−0.0956455	+	1.21361i
$160$		0			0
$161$	−447746.	+	182792.i	−1.36134	+	0.555766i
$162$		0			0
$163$	156608.			0.461684			0.230842	−	0.972991i	$-0.425852\pi$
$163$	156608.			0.461684			0.230842	+	0.972991i	$0.425852\pi$
$164$		0			0
$165$	−23184.0	+	294172.i	−0.0662947	+	0.841185i
$166$		0			0
$167$	550561.			1.52762			0.763808	−	0.645444i	$-0.223328\pi$
$167$	550561.			1.52762			0.763808	+	0.645444i	$0.223328\pi$
$168$		0			0
$169$	28567.0			0.0769392
$170$		0			0
$171$	−98369.9	+	620211.i	−0.257260	+	1.62199i
$172$		0			0
$173$	−198698.			−0.504752			−0.252376	−	0.967629i	$-0.581212\pi$
$173$	−198698.			−0.504752			−0.252376	+	0.967629i	$0.581212\pi$
$174$		0			0
$175$	105791.	+	259134.i	0.261128	+	0.639631i
$176$		0			0
$177$	−350175.	−	27597.7i	−0.840141	−	0.0662124i
$178$		0			0
$179$			659298.i			1.53798i	0.639263	+	0.768988i	$0.279239\pi$
$179$			659298.i			1.53798i	−0.639263	+	0.768988i	$0.720761\pi$
$180$		0			0
$181$		−	8712.84i		−	0.0197680i	−0.999951	−	0.00988401i	$-0.996854\pi$
$181$		−	8712.84i		−	0.0197680i	0.999951	−	0.00988401i	$-0.00314623\pi$
$182$		0			0
$183$	−28383.0	+	360140.i	−0.0626514	+	0.794957i
$184$		0			0
$185$	−386331.			−0.829909
$186$		0			0
$187$		−	946483.i		−	1.97929i
$188$		0			0
$189$	−74188.2	−	485446.i	−0.151071	−	0.988523i
$190$		0			0
$191$		−	194897.i		−	0.386563i	−0.981143	−	0.193282i	$-0.938087\pi$
$191$		−	194897.i		−	0.386563i	0.981143	−	0.193282i	$-0.0619131\pi$
$192$		0			0
$193$	177124.			0.342282			0.171141	−	0.985247i	$-0.445255\pi$
$193$	177124.			0.342282			0.171141	+	0.985247i	$0.445255\pi$
$194$		0			0
$195$	−22284.7	+	282762.i	−0.0419683	+	0.532518i
$196$		0			0
$197$		−	198170.i		−	0.363808i	−0.983316	−	0.181904i	$-0.941774\pi$
$197$		−	198170.i		−	0.363808i	0.983316	−	0.181904i	$-0.0582261\pi$
$198$		0			0
$199$		−	780692.i		−	1.39748i	−0.715374	−	0.698742i	$-0.753744\pi$
$199$		−	780692.i		−	1.39748i	0.715374	−	0.698742i	$-0.246256\pi$
$200$		0			0
$201$	108285.	+	8534.02i	0.189050	+	0.0148992i
$202$		0			0
$203$	−648775.	+	264861.i	−1.10498	+	0.451106i
$204$		0			0
$205$	94668.0			0.157333
$206$		0			0
$207$	142002.	−	895306.i	0.230340	−	1.45226i
$208$		0			0
$209$	1.57392e6			2.49239
$210$		0			0
$211$	478312.			0.739614			0.369807	−	0.929109i	$-0.379424\pi$
$211$	478312.			0.739614			0.369807	+	0.929109i	$0.379424\pi$
$212$		0			0
$213$	1491.87	−	18929.7i	0.00225310	−	0.0285887i
$214$		0			0
$215$	556342.			0.820815
$216$		0			0
$217$	−42336.0	+	17283.6i	−0.0610324	+	0.0249164i
$218$		0			0
$219$	−53412.0	+	677722.i	−0.0752538	+	0.954864i
$220$		0			0
$221$		−	909771.i		−	1.25300i
$222$		0			0
$223$		−	364117.i		−	0.490319i	−0.969483	−	0.245159i	$-0.921160\pi$
$223$		−	364117.i		−	0.490319i	0.969483	−	0.245159i	$-0.0788403\pi$
$224$		0			0
$225$	−518160.	−	82183.9i	−0.682351	−	0.108226i
$226$		0			0
$227$	−1.25624e6			−1.61812			−0.809058	−	0.587729i	$-0.800022\pi$
$227$	−1.25624e6			−1.61812			−0.809058	+	0.587729i	$0.800022\pi$
$228$		0			0
$229$			467973.i			0.589701i	0.955543	+	0.294850i	$0.0952698\pi$
$229$			467973.i			0.589701i	−0.955543	+	0.294850i	$0.904730\pi$
$230$		0			0
$231$	−1.17257e6	+	374246.i	−1.44580	+	0.461453i
$232$		0			0
$233$		−	1.32986e6i		−	1.60479i	−0.596795	−	0.802394i	$-0.703559\pi$
$233$		−	1.32986e6i		−	1.60479i	0.596795	−	0.802394i	$-0.296441\pi$
$234$		0			0
$235$	−187404.			−0.221365
$236$		0			0
$237$	463815.	+	36553.7i	0.536382	+	0.0422728i
$238$		0			0
$239$		−	700333.i		−	0.793067i	−0.918020	−	0.396534i	$-0.870213\pi$
$239$		−	700333.i		−	0.793067i	0.918020	−	0.396534i	$-0.129787\pi$
$240$		0			0
$241$			1.30647e6i			1.44897i	0.689292	+	0.724483i	$0.257922\pi$
$241$			1.30647e6i			1.44897i	−0.689292	+	0.724483i	$0.742078\pi$
$242$		0			0
$243$	850224.	+	352716.i	0.923671	+	0.383185i
$244$		0			0
$245$	373122.	−	365583.i	0.397133	−	0.389109i
$246$		0			0
$247$	1.51287e6			1.57783
$248$		0			0
$249$	1.34516e6	+	106013.i	1.37491	+	0.108358i
$250$		0			0
$251$	705435.			0.706761			0.353381	−	0.935480i	$-0.385032\pi$
$251$	705435.			0.706761			0.353381	+	0.935480i	$0.385032\pi$
$252$		0			0
$253$	−2.27203e6			−2.23158
$254$		0			0
$255$	−750595.	−	59155.2i	−0.722862	−	0.0569695i
$256$		0			0
$257$	130165.			0.122931			0.0614656	−	0.998109i	$-0.480423\pi$
$257$	130165.			0.122931			0.0614656	+	0.998109i	$0.480423\pi$
$258$		0			0
$259$	−609070.	−	1.49191e6i	−0.564180	−	1.38195i
$260$		0			0
$261$	205758.	−	1.29728e6i	0.186963	−	1.17878i
$262$		0			0
$263$			798162.i			0.711544i	0.934573	+	0.355772i	$0.115782\pi$
$263$			798162.i			0.711544i	−0.934573	+	0.355772i	$0.884218\pi$
$264$		0			0
$265$		−	773750.i		−	0.676840i
$266$		0			0
$267$	788256.	+	62123.3i	0.676689	+	0.0533306i
$268$		0			0
$269$	451880.			0.380752			0.190376	−	0.981711i	$-0.439029\pi$
$269$	451880.			0.380752			0.190376	+	0.981711i	$0.439029\pi$
$270$		0			0
$271$			793306.i			0.656172i	0.944648	+	0.328086i	$0.106404\pi$
$271$			793306.i			0.656172i	−0.944648	+	0.328086i	$0.893596\pi$
$272$		0			0
$273$	−1.12709e6	+	359730.i	−0.915272	+	0.292126i
$274$		0			0
$275$			1.31494e6i			1.04852i
$276$		0			0
$277$	−175942.			−0.137775			−0.0688874	−	0.997624i	$-0.521945\pi$
$277$	−175942.			−0.137775			−0.0688874	+	0.997624i	$0.521945\pi$
$278$		0			0
$279$	13426.8	−	84654.4i	0.0103267	−	0.0651087i
$280$		0			0
$281$			302090.i			0.228229i	0.993468	+	0.114114i	$0.0364030\pi$
$281$			302090.i			0.228229i	−0.993468	+	0.114114i	$0.963597\pi$
$282$		0			0
$283$			9266.42i			0.00687774i	0.999994	+	0.00343887i	$0.00109463\pi$
$283$			9266.42i			0.00687774i	−0.999994	+	0.00343887i	$0.998905\pi$
$284$		0			0
$285$	98369.9	−	1.24817e6i	0.0717381	−	0.910255i
$286$		0			0
$287$	149249.	+	365583.i	0.106956	+	0.261988i
$288$		0			0
$289$	995143.			0.700876
$290$		0			0
$291$	−92154.0	+	1.16930e6i	−0.0637943	+	0.809459i
$292$		0			0
$293$	−1.37249e6			−0.933983			−0.466991	−	0.884262i	$-0.654662\pi$
$293$	−1.37249e6			−0.933983			−0.466991	+	0.884262i	$0.654662\pi$
$294$		0			0
$295$	700350.			0.468554
$296$		0			0
$297$	539310.	−	2.24316e6i	0.354770	−	1.47560i
$298$		0			0
$299$	−2.18391e6			−1.41272
$300$		0			0
$301$	877100.	+	2.14845e6i	0.557998	+	1.36681i
$302$		0			0
$303$	−740439.	−	58354.8i	−0.463322	−	0.0365149i
$304$		0			0
$305$		−	720280.i		−	0.443355i
$306$		0			0
$307$			2.52553e6i			1.52935i	0.644417	+	0.764674i	$0.277100\pi$
$307$			2.52553e6i			1.52935i	−0.644417	+	0.764674i	$0.722900\pi$
$308$		0			0
$309$	12396.0	−	157288.i	0.00738559	−	0.0937127i
$310$		0			0
$311$	535829.			0.314141			0.157071	−	0.987587i	$-0.449795\pi$
$311$	535829.			0.314141			0.157071	+	0.987587i	$0.449795\pi$
$312$		0			0
$313$		−	586619.i		−	0.338450i	−0.985577	−	0.169225i	$-0.945873\pi$
$313$		−	586619.i		−	0.338450i	0.985577	−	0.169225i	$-0.0541265\pi$
$314$		0			0
$315$	223505.	+	953278.i	0.126914	+	0.541306i
$316$		0			0
$317$			2.58885e6i			1.44697i	0.690341	+	0.723484i	$0.257460\pi$
$317$			2.58885e6i			1.44697i	−0.690341	+	0.723484i	$0.742540\pi$
$318$		0			0
$319$	−3.29213e6			−1.81134
$320$		0			0
$321$	151238.	−	1.91899e6i	0.0819215	−	1.03947i
$322$		0			0
$323$			4.01593e6i			2.14181i
$324$		0			0
$325$			1.26394e6i			0.663770i
$326$		0			0
$327$	−307915.	−	24267.1i	−0.159243	−	0.0125501i
$328$		0			0
$329$	−295452.	−	723706.i	−0.150486	−	0.368614i
$330$		0			0
$331$	190888.			0.0957654			0.0478827	−	0.998853i	$-0.484753\pi$
$331$	190888.			0.0957654			0.0478827	+	0.998853i	$0.484753\pi$
$332$		0			0
$333$	2.98320e6	+	473157.i	1.47425	+	0.233827i
$334$		0			0
$335$	−216569.			−0.105435
$336$		0			0
$337$	−1.04749e6			−0.502429			−0.251214	−	0.967932i	$-0.580830\pi$
$337$	−1.04749e6			−0.502429			−0.251214	+	0.967932i	$0.580830\pi$
$338$		0			0
$339$	−90910.6	+	1.15353e6i	−0.0429650	+	0.545165i
$340$		0			0
$341$	−214829.			−0.100048
$342$		0			0
$343$	2.00003e6	+	864540.i	0.917914	+	0.396780i
$344$		0			0
$345$	−142002.	+	1.80180e6i	−0.0642313	+	0.815004i
$346$		0			0
$347$		−	496225.i		−	0.221236i	−0.993863	−	0.110618i	$-0.964717\pi$
$347$		−	496225.i		−	0.221236i	0.993863	−	0.110618i	$-0.0352829\pi$
$348$		0			0
$349$			2.65803e6i			1.16814i	0.811702	+	0.584071i	$0.198541\pi$
$349$			2.65803e6i			1.16814i	−0.811702	+	0.584071i	$0.801459\pi$
$350$		0			0
$351$	518391.	−	2.15616e6i	0.224590	−	0.934141i
$352$		0			0
$353$	1.63645e6			0.698983			0.349492	−	0.936940i	$-0.386354\pi$
$353$	1.63645e6			0.698983			0.349492	+	0.936940i	$0.386354\pi$
$354$		0			0
$355$			37859.3i			0.0159442i
$356$		0			0
$357$	−954908.	−	2.99186e6i	−0.396544	−	1.24243i
$358$		0			0
$359$		−	4.25232e6i		−	1.74137i	−0.491844	−	0.870683i	$-0.663677\pi$
$359$		−	4.25232e6i		−	1.74137i	0.491844	−	0.870683i	$-0.336323\pi$
$360$		0			0
$361$	−4.20205e6			−1.69704
$362$		0			0
$363$	−3.26179e6	−	257065.i	−1.29924	−	0.102395i
$364$		0			0
$365$		−	1.35544e6i		−	0.532537i
$366$		0			0
$367$		−	1.39480e6i		−	0.540565i	−0.962781	−	0.270282i	$-0.912883\pi$
$367$		−	1.39480e6i		−	0.540565i	0.962781	−	0.270282i	$-0.0871171\pi$
$368$		0			0
$369$	−731014.	−	115944.i	−0.279486	−	0.0443285i
$370$		0			0
$371$	2.98802e6	−	1.21985e6i	1.12706	−	0.460122i
$372$		0			0
$373$	−642658.			−0.239171			−0.119585	−	0.992824i	$-0.538157\pi$
$373$	−642658.			−0.239171			−0.119585	+	0.992824i	$0.538157\pi$
$374$		0			0
$375$	2.55217e6	+	201139.i	0.937199	+	0.0738616i
$376$		0			0
$377$	−3.16443e6			−1.14668
$378$		0			0
$379$	1.73552e6			0.620627			0.310313	−	0.950634i	$-0.399566\pi$
$379$	1.73552e6			0.620627			0.310313	+	0.950634i	$0.399566\pi$
$380$		0			0
$381$	2.66460e6	+	210000.i	0.940414	+	0.0741150i
$382$		0			0
$383$	3.43384e6			1.19614			0.598072	−	0.801443i	$-0.295934\pi$
$383$	3.43384e6			1.19614			0.598072	+	0.801443i	$0.295934\pi$
$384$		0			0
$385$	2.27203e6	−	927553.i	0.781201	−	0.318924i
$386$		0			0
$387$	−4.29600e6	−	681377.i	−1.45810	−	0.231265i
$388$		0			0
$389$			1.80713e6i			0.605503i	0.953070	+	0.302751i	$0.0979052\pi$
$389$			1.80713e6i			0.605503i	−0.953070	+	0.302751i	$0.902095\pi$
$390$		0			0
$391$		−	5.79721e6i		−	1.91768i
$392$		0			0
$393$	3.89636e6	+	307076.i	1.27256	+	0.100292i
$394$		0			0
$395$	−927630.			−0.299145
$396$		0			0
$397$			5.05399e6i			1.60938i	0.593696	+	0.804689i	$0.297668\pi$
$397$			5.05399e6i			1.60938i	−0.593696	+	0.804689i	$0.702332\pi$
$398$		0			0
$399$	4.97521e6	−	1.58793e6i	1.56451	−	0.499343i
$400$		0			0
$401$		−	1.46820e6i		−	0.455956i	−0.973666	−	0.227978i	$-0.926789\pi$
$401$		−	1.46820e6i		−	0.455956i	0.973666	−	0.227978i	$-0.0732114\pi$
$402$		0			0
$403$	−206496.			−0.0633358
$404$		0			0
$405$	−1.74520e6	−	567890.i	−0.528699	−	0.172039i
$406$		0			0
$407$		−	7.57051e6i		−	2.26537i
$408$		0			0
$409$		−	652157.i		−	0.192772i	−0.995344	−	0.0963860i	$-0.969272\pi$
$409$		−	652157.i		−	0.192772i	0.995344	−	0.0963860i	$-0.0307283\pi$
$410$		0			0
$411$	113195.	−	1.43629e6i	0.0330540	−	0.419408i
$412$		0			0
$413$	1.10414e6	+	2.70457e6i	0.318528	+	0.780231i
$414$		0			0
$415$	−2.69031e6			−0.766800
$416$		0			0
$417$	329619.	−	4.18240e6i	0.0928265	−	1.17784i
$418$		0			0
$419$	−3.79814e6			−1.05690			−0.528452	−	0.848963i	$-0.677227\pi$
$419$	−3.79814e6			−1.05690			−0.528452	+	0.848963i	$0.677227\pi$
$420$		0			0
$421$	101618.			0.0279425			0.0139713	−	0.999902i	$-0.495553\pi$
$421$	101618.			0.0279425			0.0139713	+	0.999902i	$0.495553\pi$
$422$		0			0
$423$	1.44711e6	+	229522.i	0.393234	+	0.0623697i
$424$		0			0
$425$	−3.35514e6			−0.901029
$426$		0			0
$427$	2.78153e6	−	1.13556e6i	0.738269	−	0.301397i
$428$		0			0
$429$	−5.54098e6	−	436690.i	−1.45359	−	0.114559i
$430$		0			0
$431$		−	5.69745e6i		−	1.47736i	−0.674054	−	0.738682i	$-0.735449\pi$
$431$		−	5.69745e6i		−	1.47736i	0.674054	−	0.738682i	$-0.264551\pi$
$432$		0			0
$433$			5.43694e6i			1.39359i	0.717271	+	0.696795i	$0.245391\pi$
$433$			5.43694e6i			1.39359i	−0.717271	+	0.696795i	$0.754609\pi$
$434$		0			0
$435$	−205758.	+	2.61078e6i	−0.0521355	+	0.661525i
$436$		0			0
$437$	9.64025e6			2.41482
$438$		0			0
$439$		−	2.97726e6i		−	0.737319i	−0.929564	−	0.368660i	$-0.879817\pi$
$439$		−	2.97726e6i		−	0.737319i	0.929564	−	0.368660i	$-0.120183\pi$
$440$		0			0
$441$	−3.32895e6	+	2.36601e6i	−0.815099	+	0.579322i
$442$		0			0
$443$			6.59375e6i			1.59633i	0.602438	+	0.798165i	$0.294196\pi$
$443$			6.59375e6i			1.59633i	−0.602438	+	0.798165i	$0.705804\pi$
$444$		0			0
$445$	−1.57651e6			−0.377396
$446$		0			0
$447$	−520009.	+	6.59817e6i	−0.123095	+	1.56190i
$448$		0			0
$449$			3.24792e6i			0.760308i	0.924923	+	0.380154i	$0.124129\pi$
$449$			3.24792e6i			0.760308i	−0.924923	+	0.380154i	$0.875871\pi$
$450$		0			0
$451$			1.85511e6i			0.429465i
$452$		0			0
$453$	−4.36682e6	−	344153.i	−0.999814	−	0.0787964i
$454$		0			0
$455$	2.18391e6	−	891576.i	0.494544	−	0.201897i
$456$		0			0
$457$	6.31289e6			1.41396			0.706981	−	0.707232i	$-0.250057\pi$
$457$	6.31289e6			1.41396			0.706981	+	0.707232i	$0.250057\pi$
$458$		0			0
$459$	5.72355e6	+	1.37608e6i	1.26804	+	0.304867i
$460$		0			0
$461$	−7.53501e6			−1.65132			−0.825661	−	0.564167i	$-0.809197\pi$
$461$	−7.53501e6			−1.65132			−0.825661	+	0.564167i	$0.809197\pi$
$462$		0			0
$463$	−1.68162e6			−0.364565			−0.182282	−	0.983246i	$-0.558349\pi$
$463$	−1.68162e6			−0.364565			−0.182282	+	0.983246i	$0.558349\pi$
$464$		0			0
$465$	−13426.8	+	170367.i	−0.00287965	+	0.0365387i
$466$		0			0
$467$	2.26754e6			0.481131			0.240565	−	0.970633i	$-0.422667\pi$
$467$	2.26754e6			0.481131			0.240565	+	0.970633i	$0.422667\pi$
$468$		0			0
$469$	−341432.	−	836334.i	−0.0716757	−	0.175569i
$470$		0			0
$471$	299079.	−	3.79489e6i	0.0621203	−	0.788218i
$472$		0			0
$473$			1.09020e7i			2.24055i
$474$		0			0
$475$		−	5.57931e6i		−	1.13461i
$476$		0			0
$477$	−947646.	+	5.97480e6i	−0.190700	+	1.20234i
$478$		0			0
$479$	698504.			0.139101			0.0695505	−	0.997578i	$-0.477844\pi$
$479$	698504.			0.139101			0.0695505	+	0.997578i	$0.477844\pi$
$480$		0			0
$481$		−	7.27687e6i		−	1.43411i
$482$		0			0
$483$	−7.18197e6	+	2.29226e6i	−1.40080	+	0.447090i
$484$		0			0
$485$		−	2.33861e6i		−	0.451443i
$486$		0			0
$487$	−5.44122e6			−1.03962			−0.519809	−	0.854283i	$-0.673997\pi$
$487$	−5.44122e6			−1.03962			−0.519809	+	0.854283i	$0.673997\pi$
$488$		0			0
$489$	2.43373e6	+	191805.i	0.460257	+	0.0362733i
$490$		0			0
$491$			253975.i			0.0475430i	0.999717	+	0.0237715i	$0.00756741\pi$
$491$			253975.i			0.0475430i	−0.999717	+	0.0237715i	$0.992433\pi$
$492$		0			0
$493$		−	8.40004e6i		−	1.55655i
$494$		0			0
$495$	−720571.	+	4.54312e6i	−0.132179	+	0.833376i
$496$		0			0
$497$	−146203.	+	59687.1i	−0.0265500	+	0.0108390i
$498$		0			0
$499$	9.03584e6			1.62449			0.812245	−	0.583317i	$-0.198245\pi$
$499$	9.03584e6			1.62449			0.812245	+	0.583317i	$0.198245\pi$
$500$		0			0
$501$	8.55586e6	+	674296.i	1.52289	+	0.120021i
$502$		0			0
$503$	−9.98158e6			−1.75905			−0.879527	−	0.475849i	$-0.842141\pi$
$503$	−9.98158e6			−1.75905			−0.879527	+	0.475849i	$0.842141\pi$
$504$		0			0
$505$	1.48088e6			0.258399
$506$		0			0
$507$	443939.	+	34987.3i	0.0767014	+	0.00604492i
$508$		0			0
$509$	−1.06416e7			−1.82060			−0.910299	−	0.413952i	$-0.864148\pi$
$509$	−1.06416e7			−1.82060			−0.910299	+	0.413952i	$0.864148\pi$
$510$		0			0
$511$	5.23438e6	−	2.13693e6i	0.886773	−	0.362024i
$512$		0			0
$513$	−2.28830e6	+	9.51777e6i	−0.383900	+	1.59677i
$514$		0			0
$515$			314575.i			0.0522645i
$516$		0			0
$517$		−	3.67235e6i		−	0.604252i
$518$		0			0
$519$	−3.08782e6	−	243354.i	−0.503192	−	0.0396571i
$520$		0			0
$521$	7.80836e6			1.26028			0.630138	−	0.776483i	$-0.282998\pi$
$521$	7.80836e6			1.26028			0.630138	+	0.776483i	$0.282998\pi$
$522$		0			0
$523$			1.67110e6i			0.267146i	0.991039	+	0.133573i	$0.0426451\pi$
$523$			1.67110e6i			0.267146i	−0.991039	+	0.133573i	$0.957355\pi$
$524$		0			0
$525$	1.32665e6	+	4.15658e6i	0.210067	+	0.658170i
$526$		0			0
$527$		−	548147.i		−	0.0859746i
$528$		0			0
$529$	−7.47985e6			−1.16213
$530$		0			0
$531$	−5.40801e6	−	857750.i	−0.832341	−	0.132015i
$532$		0			0
$533$			1.78315e6i			0.271876i
$534$		0			0
$535$			3.83799e6i			0.579721i
$536$		0			0
$537$	−807472.	+	1.02457e7i	−0.120835	+	1.53322i
$538$		0			0
$539$	7.16394e6	+	7.31167e6i	1.06214	+	1.08404i
$540$		0			0
$541$	8.67739e6			1.27467			0.637333	−	0.770589i	$-0.280038\pi$
$541$	8.67739e6			1.27467			0.637333	+	0.770589i	$0.280038\pi$
$542$		0			0
$543$	10671.0	−	135400.i	0.00155312	−	0.0197069i
$544$		0			0
$545$	615830.			0.0888116
$546$		0			0
$547$	−1.09886e7			−1.57027			−0.785133	−	0.619327i	$-0.787406\pi$
$547$	−1.09886e7			−1.57027			−0.785133	+	0.619327i	$0.787406\pi$
$548$		0			0
$549$	−882159.	+	5.56191e6i	−0.124915	+	0.787577i
$550$		0			0
$551$	1.39685e7			1.96007
$552$		0			0
$553$	−1.46245e6	−	3.58227e6i	−0.203362	−	0.498133i
$554$		0			0
$555$	−6.00369e6	−	473157.i	−0.827343	−	0.0652038i
$556$		0			0
$557$		−	7.26926e6i		−	0.992778i	−0.868100	−	0.496389i	$-0.834659\pi$
$557$		−	7.26926e6i		−	0.992778i	0.868100	−	0.496389i	$-0.165341\pi$
$558$		0			0
$559$			1.04792e7i			1.41839i
$560$		0			0
$561$	1.15920e6	−	1.47086e7i	0.155507	−	1.97317i
$562$		0			0
$563$	−1.15682e7			−1.53814			−0.769069	−	0.639166i	$-0.779280\pi$
$563$	−1.15682e7			−1.53814			−0.769069	+	0.639166i	$0.779280\pi$
$564$		0			0
$565$		−	2.30705e6i		−	0.304044i
$566$		0			0
$567$	−558356.	−	7.63483e6i	−0.0729380	−	0.997336i
$568$		0			0
$569$		−	4.57025e6i		−	0.591778i	−0.955222	−	0.295889i	$-0.904384\pi$
$569$		−	4.57025e6i		−	0.591778i	0.955222	−	0.295889i	$-0.0956159\pi$
$570$		0			0
$571$	420080.			0.0539190			0.0269595	−	0.999637i	$-0.491417\pi$
$571$	420080.			0.0539190			0.0269595	+	0.999637i	$0.491417\pi$
$572$		0			0
$573$	238699.	−	3.02875e6i	0.0303713	−	0.385368i
$574$		0			0
$575$			8.05402e6i			1.01588i
$576$		0			0
$577$			6.97577e6i			0.872274i	0.899880	+	0.436137i	$0.143654\pi$
$577$			6.97577e6i			0.872274i	−0.899880	+	0.436137i	$0.856346\pi$
$578$		0			0
$579$	2.75255e6	+	216932.i	0.341224	+	0.0268922i
$580$		0			0
$581$	−4.24141e6	−	1.03893e7i	−0.521278	−	1.27687i
$582$		0			0
$583$	1.51623e7			1.84754
$584$		0			0
$585$	−692622.	+	4.36690e6i	−0.0836771	+	0.527574i
$586$		0			0
$587$	3.11045e6			0.372587			0.186293	−	0.982494i	$-0.440352\pi$
$587$	3.11045e6			0.372587			0.186293	+	0.982494i	$0.440352\pi$
$588$		0			0
$589$	911520.			0.108262
$590$		0			0
$591$	242708.	−	3.07962e6i	0.0285835	−	0.362684i
$592$		0			0
$593$	−2.08066e6			−0.242976			−0.121488	−	0.992593i	$-0.538767\pi$
$593$	−2.08066e6			−0.242976			−0.121488	+	0.992593i	$0.538767\pi$
$594$		0			0
$595$	2.36670e6	+	5.79721e6i	0.274063	+	0.671315i
$596$		0			0
$597$	956148.	−	1.21322e7i	0.109797	−	1.39316i
$598$		0			0
$599$		−	5.19064e6i		−	0.591091i	−0.955329	−	0.295545i	$-0.904499\pi$
$599$		−	5.19064e6i		−	0.591091i	0.955329	−	0.295545i	$-0.0955013\pi$
$600$		0			0
$601$		−	1.21494e7i		−	1.37205i	−0.727578	−	0.686025i	$-0.759354\pi$
$601$		−	1.21494e7i		−	1.37205i	0.727578	−	0.686025i	$-0.240646\pi$
$602$		0			0
$603$	1.67232e6	+	265242.i	0.187295	+	0.0297063i
$604$		0			0
$605$	6.52359e6			0.724600
$606$		0			0
$607$		−	4.97850e6i		−	0.548438i	−0.961667	−	0.274219i	$-0.911581\pi$
$607$		−	4.97850e6i		−	0.548438i	0.961667	−	0.274219i	$-0.0884192\pi$
$608$		0			0
$609$	−1.04065e7	+	3.32143e6i	−1.13701	+	0.362896i
$610$		0			0
$611$		−	3.52991e6i		−	0.382526i
$612$		0			0
$613$	−1.46449e7			−1.57411			−0.787057	−	0.616880i	$-0.788396\pi$
$613$	−1.46449e7			−1.57411			−0.787057	+	0.616880i	$0.788396\pi$
$614$		0			0
$615$	1.47117e6	+	115944.i	0.156846	+	0.0123612i
$616$		0			0
$617$		−	2.29514e6i		−	0.242714i	−0.992609	−	0.121357i	$-0.961275\pi$
$617$		−	2.29514e6i		−	0.242714i	0.992609	−	0.121357i	$-0.0387246\pi$
$618$		0			0
$619$		−	1.66046e7i		−	1.74182i	−0.491445	−	0.870909i	$-0.663531\pi$
$619$		−	1.66046e7i		−	1.74182i	0.491445	−	0.870909i	$-0.336469\pi$
$620$		0			0
$621$	3.30327e6	−	1.37394e7i	0.343728	−	1.42968i
$622$		0			0
$623$	−2.48545e6	−	6.08808e6i	−0.256557	−	0.628435i
$624$		0			0
$625$	1.64253e6			0.168195
$626$		0			0
$627$	2.44591e7	+	1.92765e6i	2.48469	+	0.195821i
$628$		0			0
$629$	1.93166e7			1.94672
$630$		0			0
$631$	−1.50064e7			−1.50039			−0.750195	−	0.661217i	$-0.770040\pi$
$631$	−1.50064e7			−1.50039			−0.750195	+	0.661217i	$0.770040\pi$
$632$		0			0
$633$	7.43310e6	+	585810.i	0.737328	+	0.0581096i
$634$		0			0
$635$	−5.32919e6			−0.524478
$636$		0			0
$637$	6.88607e6	+	7.02806e6i	0.672392	+	0.686258i
$638$		0			0
$639$	46368.0	−	292345.i	0.00449227	−	0.0283233i
$640$		0			0
$641$		−	1.46035e7i		−	1.40382i	−0.712267	−	0.701909i	$-0.752331\pi$
$641$		−	1.46035e7i		−	1.40382i	0.712267	−	0.701909i	$-0.247669\pi$
$642$		0			0
$643$			1.04861e7i			1.00020i	0.865969	+	0.500098i	$0.166703\pi$
$643$			1.04861e7i			1.00020i	−0.865969	+	0.500098i	$0.833297\pi$
$644$		0			0
$645$	8.64570e6	+	681377.i	0.818278	+	0.0644893i
$646$		0			0
$647$	−1.82774e7			−1.71654			−0.858270	−	0.513198i	$-0.828461\pi$
$647$	−1.82774e7			−1.71654			−0.858270	+	0.513198i	$0.828461\pi$
$648$		0			0
$649$			1.37240e7i			1.27900i
$650$		0			0
$651$	−679081.	+	216741.i	−0.0628014	+	0.0200442i
$652$		0			0
$653$			9.50905e6i			0.872678i	0.899782	+	0.436339i	$0.143725\pi$
$653$			9.50905e6i			0.872678i	−0.899782	+	0.436339i	$0.856275\pi$
$654$		0			0
$655$	−7.79272e6			−0.709718
$656$		0			0
$657$	−1.66007e6	+	1.04666e7i	−0.150042	+	0.946000i
$658$		0			0
$659$		−	2.57888e6i		−	0.231322i	−0.993289	−	0.115661i	$-0.963101\pi$
$659$		−	2.57888e6i		−	0.231322i	0.993289	−	0.115661i	$-0.0368986\pi$
$660$		0			0
$661$			2.47472e6i			0.220304i	0.993915	+	0.110152i	$0.0351337\pi$
$661$			2.47472e6i			0.220304i	−0.993915	+	0.110152i	$0.964866\pi$
$662$		0			0
$663$	1.11424e6	−	1.41381e7i	0.0984451	−	1.24913i
$664$		0			0
$665$	−9.64025e6	+	3.93562e6i	−0.845345	+	0.345111i
$666$		0			0
$667$	−2.01643e7			−1.75496
$668$		0			0
$669$	445950.	−	5.65847e6i	0.0385231	−	0.488803i
$670$		0			0
$671$	1.41145e7			1.21021
$672$		0			0
$673$	1.81212e7			1.54223			0.771115	−	0.636696i	$-0.219699\pi$
$673$	1.81212e7			1.54223			0.771115	+	0.636696i	$0.219699\pi$
$674$		0			0
$675$	−7.95169e6	−	1.91177e6i	−0.671738	−	0.161502i
$676$		0			0
$677$	229654.			0.0192576			0.00962881	−	0.999954i	$-0.496935\pi$
$677$	229654.			0.0192576			0.00962881	+	0.999954i	$0.496935\pi$
$678$		0			0
$679$	9.03109e6	−	3.68693e6i	0.751737	−	0.306895i
$680$		0			0
$681$	−1.95224e7	−	1.53858e6i	−1.61311	−	0.127131i
$682$		0			0
$683$			2.09109e7i			1.71522i	0.514298	+	0.857611i	$0.328052\pi$
$683$			2.09109e7i			1.71522i	−0.514298	+	0.857611i	$0.671948\pi$
$684$		0			0
$685$			2.87258e6i			0.233908i
$686$		0			0
$687$	−573147.	+	7.27242e6i	−0.0463313	+	0.587878i
$688$		0			0
$689$	1.45742e7			1.16960
$690$		0			0
$691$			1.66001e7i			1.32256i	0.750141	+	0.661278i	$0.229986\pi$
$691$			1.66001e7i			1.32256i	−0.750141	+	0.661278i	$0.770014\pi$
$692$		0			0
$693$	−1.86804e7	+	4.37979e6i	−1.47758	+	0.346434i
$694$		0			0
$695$			8.36479e6i			0.656891i
$696$		0			0
$697$	−4.73340e6			−0.369055
$698$		0			0
$699$	1.62874e6	−	2.06665e7i	0.126084	−	1.59983i
$700$		0			0
$701$		−	1.35573e7i		−	1.04202i	−0.853550	−	0.521011i	$-0.825555\pi$
$701$		−	1.35573e7i		−	1.04202i	0.853550	−	0.521011i	$-0.174445\pi$
$702$		0			0
$703$			3.21218e7i			2.45138i
$704$		0			0
$705$	−2.91231e6	−	229522.i	−0.220681	−	0.0173921i
$706$		0			0
$707$	2.33468e6	+	5.71877e6i	0.175662	+	0.430283i
$708$		0			0
$709$	−4.25581e6			−0.317956			−0.158978	−	0.987282i	$-0.550820\pi$
$709$	−4.25581e6			−0.317956			−0.158978	+	0.987282i	$0.550820\pi$
$710$		0			0
$711$	7.16304e6	+	1.13611e6i	0.531402	+	0.0842842i
$712$		0			0
$713$	−1.31583e6			−0.0969336
$714$		0			0
$715$	1.10820e7			0.810683
$716$		0			0
$717$	857730.	−	1.08834e7i	0.0623093	−	0.790616i
$718$		0			0
$719$	−1.26479e6			−0.0912424			−0.0456212	−	0.998959i	$-0.514527\pi$
$719$	−1.26479e6			−0.0912424			−0.0456212	+	0.998959i	$0.514527\pi$
$720$		0			0
$721$	−1.21481e6	+	495943.i	−0.0870301	+	0.0355299i
$722$		0			0
$723$	−1.60010e6	+	2.03030e7i	−0.113842	+	1.44449i
$724$		0			0
$725$			1.16701e7i			0.824575i
$726$		0			0
$727$		−	6.68678e6i		−	0.469225i	−0.972089	−	0.234613i	$-0.924618\pi$
$727$		−	6.68678e6i		−	0.469225i	0.972089	−	0.234613i	$-0.0753821\pi$
$728$		0			0
$729$	1.27807e7	+	6.52260e6i	0.890710	+	0.454571i
$730$		0			0
$731$	−2.78171e7			−1.92539
$732$		0			0
$733$			2.24438e6i			0.154289i	0.997020	+	0.0771447i	$0.0245803\pi$
$733$			2.24438e6i			0.154289i	−0.997020	+	0.0771447i	$0.975420\pi$
$734$		0			0
$735$	6.24616e6	−	5.22428e6i	0.426476	−	0.356704i
$736$		0			0
$737$		−	4.24387e6i		−	0.287802i
$738$		0			0
$739$	−5.55593e6			−0.374236			−0.187118	−	0.982337i	$-0.559915\pi$
$739$	−5.55593e6			−0.374236			−0.187118	+	0.982337i	$0.559915\pi$
$740$		0			0
$741$	2.35104e7	+	1.85288e6i	1.57295	+	0.123966i
$742$		0			0
$743$			1.23576e7i			0.821224i	0.911810	+	0.410612i	$0.134685\pi$
$743$			1.23576e7i			0.821224i	−0.911810	+	0.410612i	$0.865315\pi$
$744$		0			0
$745$		−	1.31963e7i		−	0.871089i
$746$		0			0
$747$	2.07742e7	+	3.29494e6i	1.36215	+	0.216046i
$748$		0			0
$749$	−1.48213e7	+	6.05078e6i	−0.965344	+	0.394100i
$750$		0			0
$751$	343096.			0.0221981			0.0110991	−	0.999938i	$-0.496467\pi$
$751$	343096.			0.0221981			0.0110991	+	0.999938i	$0.496467\pi$
$752$		0			0
$753$	1.09627e7	+	863978.i	0.704576	+	0.0555284i
$754$		0			0
$755$	8.73363e6			0.557606
$756$		0			0
$757$	−1.75921e7			−1.11578			−0.557891	−	0.829914i	$-0.688389\pi$
$757$	−1.75921e7			−1.11578			−0.557891	+	0.829914i	$0.688389\pi$
$758$		0			0
$759$	−3.53080e7	−	2.78266e6i	−2.22469	−	0.175330i
$760$		0			0
$761$	−7.67720e6			−0.480553			−0.240277	−	0.970704i	$-0.577238\pi$
$761$	−7.67720e6			−0.480553			−0.240277	+	0.970704i	$0.577238\pi$
$762$		0			0
$763$	970886.	+	2.37818e6i	0.0603750	+	0.147888i
$764$		0			0
$765$	−1.15920e7	−	1.83857e6i	−0.716151	−	0.113587i
$766$		0			0
$767$			1.31917e7i			0.809677i
$768$		0			0
$769$		−	2.00721e6i		−	0.122399i	−0.998126	−	0.0611995i	$-0.980507\pi$
$769$		−	2.00721e6i		−	0.122399i	0.998126	−	0.0611995i	$-0.0194926\pi$
$770$		0			0
$771$	2.02280e6	+	159419.i	0.122551	+	0.00965839i
$772$		0			0
$773$	−2.04173e7			−1.22900			−0.614498	−	0.788919i	$-0.710641\pi$
$773$	−2.04173e7			−1.22900			−0.614498	+	0.788919i	$0.710641\pi$
$774$		0			0
$775$			761537.i			0.0455446i
$776$		0			0
$777$	−7.63790e6	−	2.39307e7i	−0.453860	−	1.42201i
$778$		0			0
$779$		−	7.87123e6i		−	0.464728i
$780$		0			0
$781$	−741888.			−0.0435222
$782$		0			0
$783$	4.78637e6	−	1.99081e7i	0.278999	−	1.16045i
$784$		0			0
$785$			7.58977e6i			0.439597i
$786$		0			0
$787$			789904.i			0.0454608i	0.999742	+	0.0227304i	$0.00723594\pi$
$787$			789904.i			0.0454608i	−0.999742	+	0.0227304i	$0.992764\pi$
$788$		0			0
$789$	−977545.	+	1.24037e7i	−0.0559042	+	0.709345i
$790$		0			0
$791$	8.90924e6	−	3.63718e6i	0.506290	−	0.206692i
$792$		0			0
$793$	1.35671e7			0.766131
$794$		0			0
$795$	947646.	−	1.20243e7i	0.0531775	−	0.674747i
$796$		0			0
$797$	3.01013e7			1.67857			0.839285	−	0.543692i	$-0.182974\pi$
$797$	3.01013e7			1.67857			0.839285	+	0.543692i	$0.182974\pi$
$798$		0			0
$799$	9.37020e6			0.519256
$800$		0			0
$801$	1.21736e7	+	1.93082e6i	0.670407	+	0.106331i
$802$		0			0
$803$	2.65612e7			1.45365
$804$		0			0
$805$	1.39162e7	−	5.68126e6i	0.756887	−	0.308998i
$806$		0			0
$807$	7.02234e6	+	553438.i	0.379575	+	0.0299147i
$808$		0			0
$809$		−	9.08210e6i		−	0.487882i	−0.969790	−	0.243941i	$-0.921560\pi$
$809$		−	9.08210e6i		−	0.487882i	0.969790	−	0.243941i	$-0.0784404\pi$
$810$		0			0
$811$			2.86726e7i			1.53079i	0.643563	+	0.765393i	$0.277456\pi$
$811$			2.86726e7i			1.53079i	−0.643563	+	0.765393i	$0.722544\pi$
$812$		0			0
$813$	−971598.	+	1.23282e7i	−0.0515537	+	0.654144i
$814$		0			0
$815$	−4.86746e6			−0.256690
$816$		0			0
$817$		−	4.62574e7i		−	2.42452i
$818$		0			0
$819$	−1.79558e7	+	4.20991e6i	−0.935395	+	0.219312i
$820$		0			0
$821$		−	1.73237e6i		−	0.0896981i	−0.998994	−	0.0448490i	$-0.985719\pi$
$821$		−	1.73237e6i		−	0.0896981i	0.998994	−	0.0448490i	$-0.0142807\pi$
$822$		0			0
$823$	−3.10743e7			−1.59920			−0.799599	−	0.600535i	$-0.794954\pi$
$823$	−3.10743e7			−1.59920			−0.799599	+	0.600535i	$0.794954\pi$
$824$		0			0
$825$	−1.61047e6	+	2.04346e7i	−0.0823792	+	1.04527i
$826$		0			0
$827$		−	2.48205e7i		−	1.26197i	−0.775797	−	0.630983i	$-0.782652\pi$
$827$		−	2.48205e7i		−	1.26197i	0.775797	−	0.630983i	$-0.217348\pi$
$828$		0			0
$829$			1.58918e7i			0.803133i	0.915830	+	0.401567i	$0.131534\pi$
$829$			1.58918e7i			0.803133i	−0.915830	+	0.401567i	$0.868466\pi$
$830$		0			0
$831$	−2.73419e6	−	215484.i	−0.137349	−	0.0108246i
$832$		0			0
$833$	−1.86561e7	+	1.82792e7i	−0.931554	+	0.912733i
$834$		0			0
$835$	−1.71117e7			−0.849332
$836$		0			0
$837$	312336.	−	1.29911e6i	0.0154102	−	0.0640961i
$838$		0			0
$839$	1.19152e7			0.584383			0.292191	−	0.956360i	$-0.405616\pi$
$839$	1.19152e7			0.584383			0.292191	+	0.956360i	$0.405616\pi$
$840$		0			0
$841$	−8.70649e6			−0.424476
$842$		0			0
$843$	−369983.	+	4.69455e6i	−0.0179313	+	0.227523i
$844$		0			0
$845$	−887878.			−0.0427771
$846$		0			0
$847$	1.02848e7	+	2.51924e7i	0.492590	+	1.20659i
$848$		0			0
$849$	−11349.0	+	144003.i	−0.000540366	+	0.00685648i
$850$		0			0
$851$		−	4.63694e7i		−	2.19486i
$852$		0			0
$853$		−	8.12799e6i		−	0.382482i	−0.981543	−	0.191241i	$-0.938749\pi$
$853$		−	8.12799e6i		−	0.382482i	0.981543	−	0.191241i	$-0.0612512\pi$
$854$		0			0
$855$	3.05739e6	−	1.92765e7i	0.143033	−	0.901805i
$856$		0			0
$857$	6.27187e6			0.291706			0.145853	−	0.989306i	$-0.453407\pi$
$857$	6.27187e6			0.291706			0.145853	+	0.989306i	$0.453407\pi$
$858$		0			0
$859$		−	1.59011e7i		−	0.735264i	−0.929971	−	0.367632i	$-0.880169\pi$
$859$		−	1.59011e7i		−	0.735264i	0.929971	−	0.367632i	$-0.119831\pi$
$860$		0			0
$861$	1.87162e6	+	5.86405e6i	0.0860418	+	0.269582i
$862$		0			0
$863$		−	4.82582e6i		−	0.220569i	−0.993900	−	0.110284i	$-0.964824\pi$
$863$		−	4.82582e6i		−	0.220569i	0.993900	−	0.110284i	$-0.0351762\pi$
$864$		0			0
$865$	6.17564e6			0.280635
$866$		0			0
$867$	1.54648e7	+	1.21880e6i	0.698709	+	0.0550660i
$868$		0			0
$869$		−	1.81778e7i		−	0.816565i
$870$		0			0
$871$		−	4.07926e6i		−	0.182195i
$872$		0			0
$873$	−2.86420e6	+	1.80584e7i	−0.127194	+	0.801944i
$874$		0			0
$875$	−8.04725e6	−	1.97117e7i	−0.355326	−	0.870368i
$876$		0			0
$877$	2.75228e7			1.20835			0.604177	−	0.796850i	$-0.293502\pi$
$877$	2.75228e7			1.20835			0.604177	+	0.796850i	$0.293502\pi$
$878$		0			0
$879$	−2.13288e7	−	1.68094e6i	−0.931095	−	0.0733806i
$880$		0			0
$881$	1.82884e7			0.793846			0.396923	−	0.917852i	$-0.370078\pi$
$881$	1.82884e7			0.793846			0.396923	+	0.917852i	$0.370078\pi$
$882$		0			0
$883$	3.45138e6			0.148967			0.0744837	−	0.997222i	$-0.476269\pi$
$883$	3.45138e6			0.148967			0.0744837	+	0.997222i	$0.476269\pi$
$884$		0			0
$885$	1.08836e7	+	857750.i	0.467106	+	0.0368131i
$886$		0			0
$887$	−237145.			−0.0101205			−0.00506027	−	0.999987i	$-0.501611\pi$
$887$	−237145.			−0.0101205			−0.00506027	+	0.999987i	$0.501611\pi$
$888$		0			0
$889$	−8.40174e6	−	2.05800e7i	−0.356545	−	0.873354i
$890$		0			0
$891$	1.11283e7	−	3.41989e7i	0.469608	−	1.44317i
$892$		0			0
$893$			1.55818e7i			0.653867i
$894$		0			0
$895$		−	2.04914e7i		−	0.855092i
$896$		0			0
$897$	−3.39385e7	−	2.67473e6i	−1.40835	−	0.110994i
$898$		0			0
$899$	−1.90660e6			−0.0786795
$900$		0			0
$901$			3.86875e7i			1.58766i
$902$		0			0
$903$	1.09991e7	+	3.44617e7i	0.448887	+	1.40643i
$904$		0			0
$905$			270800.i			0.0109907i
$906$		0			0
$907$	3.75768e7			1.51671			0.758353	−	0.651844i	$-0.226004\pi$
$907$	3.75768e7			1.51671			0.758353	+	0.651844i	$0.226004\pi$
$908$		0			0
$909$	−1.14352e7	−	1.81370e6i	−0.459021	−	0.0728040i
$910$		0			0
$911$			1.36704e7i			0.545739i	0.962051	+	0.272870i	$0.0879727\pi$
$911$			1.36704e7i			0.545739i	−0.962051	+	0.272870i	$0.912027\pi$
$912$		0			0
$913$		−	5.27191e7i		−	2.09310i
$914$		0			0
$915$	882159.	−	1.11933e7i	0.0348333	−	0.441985i
$916$		0			0
$917$	−1.22856e7	−	3.00935e7i	−0.482473	−	1.18181i
$918$		0			0
$919$	3.54130e7			1.38316			0.691582	−	0.722298i	$-0.256914\pi$
$919$	3.54130e7			1.38316			0.691582	+	0.722298i	$0.256914\pi$
$920$		0			0
$921$	−3.09313e6	+	3.92474e7i	−0.120157	+	1.52462i
$922$		0			0
$923$	−713112.			−0.0275520
$924$		0			0
$925$	−2.68364e7			−1.03126
$926$		0			0
$927$	385274.	−	2.42911e6i	0.0147255	−	0.0928427i
$928$		0			0
$929$	−5.51872e6			−0.209797			−0.104899	−	0.994483i	$-0.533452\pi$
$929$	−5.51872e6			−0.209797			−0.104899	+	0.994483i	$0.533452\pi$
$930$		0			0
$931$	−3.03967e7	−	3.10235e7i	−1.14935	−	1.17305i
$932$		0			0
$933$	8.32692e6	+	656253.i	0.313170	+	0.0246813i
$934$		0			0
$935$			2.94172e7i			1.10045i
$936$		0			0
$937$		−	4.00821e7i		−	1.49142i	−0.666269	−	0.745712i	$-0.732110\pi$
$937$		−	4.00821e7i		−	1.49142i	0.666269	−	0.745712i	$-0.267890\pi$
$938$		0			0
$939$	718458.	−	9.11621e6i	0.0265912	−	0.337404i
$940$		0			0
$941$	1.05494e7			0.388377			0.194188	−	0.980964i	$-0.437793\pi$
$941$	1.05494e7			0.388377			0.194188	+	0.980964i	$0.437793\pi$
$942$		0			0
$943$			1.13625e7i			0.416098i
$944$		0			0
$945$	2.30581e6	+	1.50879e7i	0.0839931	+	0.549605i
$946$		0			0
$947$		−	1.94865e7i		−	0.706087i	−0.935607	−	0.353043i	$-0.885147\pi$
$947$		−	1.94865e7i		−	0.706087i	0.935607	−	0.353043i	$-0.114853\pi$
$948$		0			0
$949$	2.55309e7			0.920240
$950$		0			0
$951$	−3.17068e6	+	4.02314e7i	−0.113685	+	1.44249i
$952$		0			0
$953$			4.11398e7i			1.46734i	0.679508	+	0.733668i	$0.262193\pi$
$953$			4.11398e7i			1.46734i	−0.679508	+	0.733668i	$0.737807\pi$
$954$		0			0
$955$			6.05749e6i			0.214924i
$956$		0			0
$957$	−5.11606e7	−	4.03202e6i	−1.80574	−	0.142312i
$958$		0			0
$959$	−1.10931e7	+	4.52876e6i	−0.389501	+	0.159013i
$960$		0			0
$961$	2.85047e7			0.995654
$962$		0			0
$963$	4.70056e6	−	2.96365e7i	0.163337	−	1.02982i
$964$		0			0
$965$	−5.50511e6			−0.190304
$966$		0			0
$967$	4.51057e6			0.155119			0.0775595	−	0.996988i	$-0.475287\pi$
$967$	4.51057e6			0.155119			0.0775595	+	0.996988i	$0.475287\pi$
$968$		0			0
$969$	−4.91850e6	+	6.24087e7i	−0.168276	+	2.13519i
$970$		0			0
$971$	4.24296e7			1.44418			0.722090	−	0.691799i	$-0.243182\pi$
$971$	4.24296e7			1.44418			0.722090	+	0.691799i	$0.243182\pi$
$972$		0			0
$973$	−3.23027e7	+	1.31875e7i	−1.09385	+	0.446561i
$974$		0			0
$975$	−1.54800e6	+	1.96420e7i	−0.0521507	+	0.661718i
$976$		0			0
$977$		−	1.89292e7i		−	0.634447i	−0.948351	−	0.317223i	$-0.897249\pi$
$977$		−	1.89292e7i		−	0.634447i	0.948351	−	0.317223i	$-0.102751\pi$
$978$		0			0
$979$		−	3.08932e7i		−	1.03016i
$980$		0			0
$981$	−4.75536e6	−	754234.i	−0.157765	−	0.0250227i
$982$		0			0
$983$	−2.10074e7			−0.693407			−0.346704	−	0.937975i	$-0.612699\pi$
$983$	−2.10074e7			−0.693407			−0.346704	+	0.937975i	$0.612699\pi$
$984$		0			0
$985$			6.15924e6i			0.202272i
$986$		0			0
$987$	−3.70504e6	−	1.16084e7i	−0.121060	−	0.379298i
$988$		0			0
$989$			6.67749e7i			2.17081i
$990$		0			0
$991$	−3.01462e7			−0.975100			−0.487550	−	0.873095i	$-0.662109\pi$
$991$	−3.01462e7			−0.975100			−0.487550	+	0.873095i	$0.662109\pi$
$992$		0			0
$993$	2.96645e6	+	233789.i	0.0954694	+	0.00752404i
$994$		0			0
$995$			2.42643e7i			0.776981i
$996$		0			0
$997$			3.31302e7i			1.05557i	0.849379	+	0.527784i	$0.176977\pi$
$997$			3.31302e7i			1.05557i	−0.849379	+	0.527784i	$0.823023\pi$
$998$		0			0
$999$	4.57802e7	+	1.10066e7i	1.45132	+	0.348932i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.6.f.b.41.4	yes	4
3.2	odd	2	inner	84.6.f.b.41.2	yes	4
4.3	odd	2		336.6.k.b.209.1		4
7.6	odd	2	inner	84.6.f.b.41.1	✓	4
12.11	even	2		336.6.k.b.209.3		4
21.20	even	2	inner	84.6.f.b.41.3	yes	4
28.27	even	2		336.6.k.b.209.4		4
84.83	odd	2		336.6.k.b.209.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.f.b.41.1	✓	4	7.6	odd	2	inner
84.6.f.b.41.2	yes	4	3.2	odd	2	inner
84.6.f.b.41.3	yes	4	21.20	even	2	inner
84.6.f.b.41.4	yes	4	1.1	even	1	trivial
336.6.k.b.209.1		4	4.3	odd	2
336.6.k.b.209.2		4	84.83	odd	2
336.6.k.b.209.3		4	12.11	even	2
336.6.k.b.209.4		4	28.27	even	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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	84.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(15.5403 + 1.22474i) q^{3} -31.0805 q^{5} +(-49.0000 - 120.025i) q^{7} +(240.000 + 38.0657i) q^{9} +O(q^{10})\)	\(q+(15.5403 + 1.22474i) q^{3} -31.0805 q^{5} +(-49.0000 - 120.025i) q^{7} +(240.000 + 38.0657i) q^{9} -609.052i q^{11} -585.428i q^{13} +(-483.000 - 38.0657i) q^{15} +1554.03 q^{17} +2584.21i q^{19} +(-614.473 - 1925.23i) q^{21} -3730.44i q^{23} -2159.00 q^{25} +(3683.04 + 885.491i) q^{27} -5405.33i q^{29} -352.727i q^{31} +(745.933 - 9464.83i) q^{33} +(1522.95 + 3730.44i) q^{35} +12430.0 q^{37} +(717.000 - 9097.71i) q^{39} -3045.89 q^{41} -17900.0 q^{43} +(-7459.33 - 1183.10i) q^{45} +6029.62 q^{47} +(-12005.0 + 11762.4i) q^{49} +(24150.0 + 1903.29i) q^{51} +24895.0i q^{53} +18929.7i q^{55} +(-3165.00 + 40159.3i) q^{57} -22533.4 q^{59} +23174.6i q^{61} +(-7191.16 - 30671.2i) q^{63} +18195.4i q^{65} +6968.00 q^{67} +(4568.84 - 57972.1i) q^{69} -1218.10i q^{71} +43610.7i q^{73} +(-33551.4 - 2644.22i) q^{75} +(-73101.4 + 29843.5i) q^{77} +29846.0 q^{79} +(56151.0 + 18271.6i) q^{81} +86559.3 q^{83} -48300.0 q^{85} +(6620.16 - 84000.4i) q^{87} +50723.4 q^{89} +(-70266.0 + 28686.0i) q^{91} +(432.000 - 5481.47i) q^{93} -80318.7i q^{95} +75243.4i q^{97} +(23184.0 - 146172. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 196 q^{7} + 960 q^{9}+O(q^{10}) \) 4 * q - 196 * q^7 + 960 * q^9	\( 4 q - 196 q^{7} + 960 q^{9} - 1932 q^{15} + 588 q^{21} - 8636 q^{25} + 49720 q^{37} + 2868 q^{39} - 71600 q^{43} - 48020 q^{49} + 96600 q^{51} - 12660 q^{57} - 47040 q^{63} + 27872 q^{67} + 119384 q^{79} + 224604 q^{81} - 193200 q^{85} - 281064 q^{91} + 1728 q^{93} + 92736 q^{99}+O(q^{100}) \) 4 * q - 196 * q^7 + 960 * q^9 - 1932 * q^15 + 588 * q^21 - 8636 * q^25 + 49720 * q^37 + 2868 * q^39 - 71600 * q^43 - 48020 * q^49 + 96600 * q^51 - 12660 * q^57 - 47040 * q^63 + 27872 * q^67 + 119384 * q^79 + 224604 * q^81 - 193200 * q^85 - 281064 * q^91 + 1728 * q^93 + 92736 * q^99

Introduction

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