LMFDB - Embedded newform 48.6.c.c.47.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,6,Mod(47,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	48.c (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:	$7.69842335102$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 3 $ x^2 - x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.1
Root			$0.500000 - 1.65831i$ of defining polynomial
Character	$\chi$	$=$	48.47
Dual form			48.6.c.c.47.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(12.0000 - 9.94987i) q^{3} +79.5990i q^{5} +179.098i q^{7} +(45.0000 - 238.797i) q^{9} +O(q^{10})$	\(q+(12.0000 - 9.94987i) q^{3} +79.5990i q^{5} +179.098i q^{7} +(45.0000 - 238.797i) q^{9} +648.000 q^{11} -242.000 q^{13} +(792.000 + 955.188i) q^{15} -318.396i q^{17} +1253.68i q^{19} +(1782.00 + 2149.17i) q^{21} +1296.00 q^{23} -3211.00 q^{25} +(-1836.00 - 3313.31i) q^{27} -1989.97i q^{29} -3402.86i q^{31} +(7776.00 - 6447.52i) q^{33} -14256.0 q^{35} -12058.0 q^{37} +(-2904.00 + 2407.87i) q^{39} +14805.4i q^{41} -18088.9i q^{43} +(19008.0 + 3581.95i) q^{45} +12960.0 q^{47} -15269.0 q^{49} +(-3168.00 - 3820.75i) q^{51} -26984.1i q^{53} +51580.1i q^{55} +(12474.0 + 15044.2i) q^{57} +8424.00 q^{59} -25762.0 q^{61} +(42768.0 + 8059.40i) q^{63} -19263.0i q^{65} -10208.6i q^{67} +(15552.0 - 12895.0i) q^{69} +55728.0 q^{71} +26026.0 q^{73} +(-38532.0 + 31949.0i) q^{75} +116055. i q^{77} -19163.5i q^{79} +(-54999.0 - 21491.7i) q^{81} -78408.0 q^{83} +25344.0 q^{85} +(-19800.0 - 23879.7i) q^{87} -84215.7i q^{89} -43341.7i q^{91} +(-33858.0 - 40834.3i) q^{93} -99792.0 q^{95} +103090. q^{97} +(29160.0 - 154740. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 24 q^{3} + 90 q^{9}+O(q^{10}) $ 2 * q + 24 * q^3 + 90 * q^9	$ 2 q + 24 q^{3} + 90 q^{9} + 1296 q^{11} - 484 q^{13} + 1584 q^{15} + 3564 q^{21} + 2592 q^{23} - 6422 q^{25} - 3672 q^{27} + 15552 q^{33} - 28512 q^{35} - 24116 q^{37} - 5808 q^{39} + 38016 q^{45} + 25920 q^{47} - 30538 q^{49} - 6336 q^{51} + 24948 q^{57} + 16848 q^{59} - 51524 q^{61} + 85536 q^{63} + 31104 q^{69} + 111456 q^{71} + 52052 q^{73} - 77064 q^{75} - 109998 q^{81} - 156816 q^{83} + 50688 q^{85} - 39600 q^{87} - 67716 q^{93} - 199584 q^{95} + 206180 q^{97} + 58320 q^{99}+O(q^{100}) $ 2 * q + 24 * q^3 + 90 * q^9 + 1296 * q^11 - 484 * q^13 + 1584 * q^15 + 3564 * q^21 + 2592 * q^23 - 6422 * q^25 - 3672 * q^27 + 15552 * q^33 - 28512 * q^35 - 24116 * q^37 - 5808 * q^39 + 38016 * q^45 + 25920 * q^47 - 30538 * q^49 - 6336 * q^51 + 24948 * q^57 + 16848 * q^59 - 51524 * q^61 + 85536 * q^63 + 31104 * q^69 + 111456 * q^71 + 52052 * q^73 - 77064 * q^75 - 109998 * q^81 - 156816 * q^83 + 50688 * q^85 - 39600 * q^87 - 67716 * q^93 - 199584 * q^95 + 206180 * q^97 + 58320 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	12.0000	−	9.94987i	0.769800	−	0.638285i
$3$	12.0000	−	9.94987i	0.769800	−	0.638285i
$4$		0			0
$5$			79.5990i			1.42391i	0.702225	+	0.711955i	$0.252190\pi$
$5$			79.5990i			1.42391i	−0.702225	+	0.711955i	$0.747810\pi$
$6$		0			0
$7$			179.098i			1.38148i	0.723103	+	0.690741i	$0.242715\pi$
$7$			179.098i			1.38148i	−0.723103	+	0.690741i	$0.757285\pi$
$8$		0			0
$9$	45.0000	−	238.797i	0.185185	−	0.982704i
$10$		0			0
$11$	648.000			1.61471			0.807353	−	0.590069i	$-0.200900\pi$
$11$	648.000			1.61471			0.807353	+	0.590069i	$0.200900\pi$
$12$		0			0
$13$	−242.000			−0.397152			−0.198576	−	0.980085i	$-0.563632\pi$
$13$	−242.000			−0.397152			−0.198576	+	0.980085i	$0.563632\pi$
$14$		0			0
$15$	792.000	+	955.188i	0.908860	+	1.09613i
$16$		0			0
$17$		−	318.396i		−	0.267205i	−0.991035	−	0.133603i	$-0.957345\pi$
$17$		−	318.396i		−	0.267205i	0.991035	−	0.133603i	$-0.0426546\pi$
$18$		0			0
$19$			1253.68i			0.796717i	0.917230	+	0.398359i	$0.130420\pi$
$19$			1253.68i			0.796717i	−0.917230	+	0.398359i	$0.869580\pi$
$20$		0			0
$21$	1782.00	+	2149.17i	0.881778	+	1.06346i
$22$		0			0
$23$	1296.00			0.510841			0.255420	−	0.966830i	$-0.417786\pi$
$23$	1296.00			0.510841			0.255420	+	0.966830i	$0.417786\pi$
$24$		0			0
$25$	−3211.00			−1.02752
$26$		0			0
$27$	−1836.00	−	3313.31i	−0.484689	−	0.874686i
$28$		0			0
$29$		−	1989.97i		−	0.439392i	−0.975568	−	0.219696i	$-0.929493\pi$
$29$		−	1989.97i		−	0.439392i	0.975568	−	0.219696i	$-0.0705066\pi$
$30$		0			0
$31$		−	3402.86i		−	0.635974i	−0.948095	−	0.317987i	$-0.896993\pi$
$31$		−	3402.86i		−	0.635974i	0.948095	−	0.317987i	$-0.103007\pi$
$32$		0			0
$33$	7776.00	−	6447.52i	1.24300	−	1.03064i
$34$		0			0
$35$	−14256.0			−1.96711
$36$		0			0
$37$	−12058.0			−1.44801			−0.724004	−	0.689796i	$-0.757700\pi$
$37$	−12058.0			−1.44801			−0.724004	+	0.689796i	$0.757700\pi$
$38$		0			0
$39$	−2904.00	+	2407.87i	−0.305728	+	0.253496i
$40$		0			0
$41$			14805.4i			1.37550i	0.725947	+	0.687750i	$0.241402\pi$
$41$			14805.4i			1.37550i	−0.725947	+	0.687750i	$0.758598\pi$
$42$		0			0
$43$		−	18088.9i		−	1.49190i	−0.666001	−	0.745951i	$-0.731995\pi$
$43$		−	18088.9i		−	1.49190i	0.666001	−	0.745951i	$-0.268005\pi$
$44$		0			0
$45$	19008.0	+	3581.95i	1.39928	+	0.263687i
$46$		0			0
$47$	12960.0			0.855777			0.427888	−	0.903832i	$-0.359258\pi$
$47$	12960.0			0.855777			0.427888	+	0.903832i	$0.359258\pi$
$48$		0			0
$49$	−15269.0			−0.908491
$50$		0			0
$51$	−3168.00	−	3820.75i	−0.170553	−	0.205695i
$52$		0			0
$53$		−	26984.1i		−	1.31952i	−0.751474	−	0.659762i	$-0.770657\pi$
$53$		−	26984.1i		−	1.31952i	0.751474	−	0.659762i	$-0.229343\pi$
$54$		0			0
$55$			51580.1i			2.29920i
$56$		0			0
$57$	12474.0	+	15044.2i	0.508532	+	0.613313i
$58$		0			0
$59$	8424.00			0.315056			0.157528	−	0.987514i	$-0.449647\pi$
$59$	8424.00			0.315056			0.157528	+	0.987514i	$0.449647\pi$
$60$		0			0
$61$	−25762.0			−0.886452			−0.443226	−	0.896410i	$-0.646166\pi$
$61$	−25762.0			−0.886452			−0.443226	+	0.896410i	$0.646166\pi$
$62$		0			0
$63$	42768.0	+	8059.40i	1.35759	+	0.255830i
$64$		0			0
$65$		−	19263.0i		−	0.565509i
$66$		0			0
$67$		−	10208.6i		−	0.277829i	−0.990304	−	0.138915i	$-0.955639\pi$
$67$		−	10208.6i		−	0.277829i	0.990304	−	0.138915i	$-0.0443614\pi$
$68$		0			0
$69$	15552.0	−	12895.0i	0.393245	−	0.326062i
$70$		0			0
$71$	55728.0			1.31198			0.655991	−	0.754769i	$-0.272251\pi$
$71$	55728.0			1.31198			0.655991	+	0.754769i	$0.272251\pi$
$72$		0			0
$73$	26026.0			0.571611			0.285805	−	0.958288i	$-0.407739\pi$
$73$	26026.0			0.571611			0.285805	+	0.958288i	$0.407739\pi$
$74$		0			0
$75$	−38532.0	+	31949.0i	−0.790985	+	0.655850i
$76$		0			0
$77$			116055.i			2.23069i
$78$		0			0
$79$		−	19163.5i		−	0.345467i	−0.984969	−	0.172733i	$-0.944740\pi$
$79$		−	19163.5i		−	0.345467i	0.984969	−	0.172733i	$-0.0552599\pi$
$80$		0			0
$81$	−54999.0	−	21491.7i	−0.931413	−	0.363964i
$82$		0			0
$83$	−78408.0			−1.24930			−0.624648	−	0.780907i	$-0.714757\pi$
$83$	−78408.0			−1.24930			−0.624648	+	0.780907i	$0.714757\pi$
$84$		0			0
$85$	25344.0			0.380477
$86$		0			0
$87$	−19800.0	−	23879.7i	−0.280458	−	0.338244i
$88$		0			0
$89$		−	84215.7i		−	1.12699i	−0.826121	−	0.563493i	$-0.809457\pi$
$89$		−	84215.7i		−	1.12699i	0.826121	−	0.563493i	$-0.190543\pi$
$90$		0			0
$91$		−	43341.7i		−	0.548658i
$92$		0			0
$93$	−33858.0	−	40834.3i	−0.405933	−	0.489573i
$94$		0			0
$95$	−99792.0			−1.13445
$96$		0			0
$97$	103090.			1.11247			0.556234	−	0.831026i	$-0.312246\pi$
$97$	103090.			1.11247			0.556234	+	0.831026i	$0.312246\pi$
$98$		0			0
$99$	29160.0	−	154740.i	0.299020	−	1.58678i
$100$		0			0
$101$		−	83340.1i		−	0.812926i	−0.913667	−	0.406463i	$-0.866762\pi$
$101$		−	83340.1i		−	0.812926i	0.913667	−	0.406463i	$-0.133238\pi$
$102$		0			0
$103$		−	45669.9i		−	0.424167i	−0.977252	−	0.212084i	$-0.931975\pi$
$103$		−	45669.9i		−	0.424167i	0.977252	−	0.212084i	$-0.0680249\pi$
$104$		0			0
$105$	−171072.	+	141845.i	−1.51428	+	1.25557i
$106$		0			0
$107$	21384.0			0.180563			0.0902817	−	0.995916i	$-0.471223\pi$
$107$	21384.0			0.180563			0.0902817	+	0.995916i	$0.471223\pi$
$108$		0			0
$109$	58894.0			0.474794			0.237397	−	0.971413i	$-0.423706\pi$
$109$	58894.0			0.474794			0.237397	+	0.971413i	$0.423706\pi$
$110$		0			0
$111$	−144696.	+	119976.i	−1.11468	+	0.924241i
$112$		0			0
$113$			79917.4i			0.588769i	0.955687	+	0.294385i	$0.0951147\pi$
$113$			79917.4i			0.588769i	−0.955687	+	0.294385i	$0.904885\pi$
$114$		0			0
$115$			103160.i			0.727391i
$116$		0			0
$117$	−10890.0	+	57788.9i	−0.0735467	+	0.390283i
$118$		0			0
$119$	57024.0			0.369139
$120$		0			0
$121$	258853.			1.60727
$122$		0			0
$123$	147312.	+	177665.i	0.877961	+	1.05886i
$124$		0			0
$125$		−	6845.51i		−	0.0391860i
$126$		0			0
$127$		−	140950.i		−	0.775453i	−0.921774	−	0.387727i	$-0.873260\pi$
$127$		−	140950.i		−	0.775453i	0.921774	−	0.387727i	$-0.126740\pi$
$128$		0			0
$129$	−179982.	−	217066.i	−0.952258	−	1.14847i
$130$		0			0
$131$	292248.			1.48790			0.743949	−	0.668236i	$-0.232950\pi$
$131$	292248.			1.48790			0.743949	+	0.668236i	$0.232950\pi$
$132$		0			0
$133$	−224532.			−1.10065
$134$		0			0
$135$	263736.	−	146144.i	1.24547	−	0.690154i
$136$		0			0
$137$			155536.i			0.707996i	0.935246	+	0.353998i	$0.115178\pi$
$137$			155536.i			0.707996i	−0.935246	+	0.353998i	$0.884822\pi$
$138$		0			0
$139$			398851.i			1.75095i	0.483265	+	0.875474i	$0.339451\pi$
$139$			398851.i			1.75095i	−0.483265	+	0.875474i	$0.660549\pi$
$140$		0			0
$141$	155520.	−	128950.i	0.658777	−	0.546229i
$142$		0			0
$143$	−156816.			−0.641284
$144$		0			0
$145$	158400.			0.625655
$146$		0			0
$147$	−183228.	+	151925.i	−0.699356	+	0.579876i
$148$		0			0
$149$		−	213564.i		−	0.788066i	−0.919096	−	0.394033i	$-0.871080\pi$
$149$		−	213564.i		−	0.788066i	0.919096	−	0.394033i	$-0.128920\pi$
$150$		0			0
$151$			199336.i			0.711448i	0.934591	+	0.355724i	$0.115766\pi$
$151$			199336.i			0.711448i	−0.934591	+	0.355724i	$0.884234\pi$
$152$		0			0
$153$	−76032.0	−	14327.8i	−0.262584	−	0.0494825i
$154$		0			0
$155$	270864.			0.905570
$156$		0			0
$157$	−195490.			−0.632959			−0.316479	−	0.948599i	$-0.602501\pi$
$157$	−195490.			−0.632959			−0.316479	+	0.948599i	$0.602501\pi$
$158$		0			0
$159$	−268488.	−	323809.i	−0.842233	−	1.01577i
$160$		0			0
$161$			232111.i			0.705717i
$162$		0			0
$163$		−	450073.i		−	1.32682i	−0.748254	−	0.663412i	$-0.769108\pi$
$163$		−	450073.i		−	1.32682i	0.748254	−	0.663412i	$-0.230892\pi$
$164$		0			0
$165$	513216.	+	618962.i	1.46754	+	1.76992i
$166$		0			0
$167$	−384912.			−1.06800			−0.533999	−	0.845485i	$-0.679311\pi$
$167$	−384912.			−1.06800			−0.533999	+	0.845485i	$0.679311\pi$
$168$		0			0
$169$	−312729.			−0.842270
$170$		0			0
$171$	299376.	+	56415.8i	0.782937	+	0.147540i
$172$		0			0
$173$		−	48794.2i		−	0.123952i	−0.998078	−	0.0619759i	$-0.980260\pi$
$173$		−	48794.2i		−	0.123952i	0.998078	−	0.0619759i	$-0.0197402\pi$
$174$		0			0
$175$		−	575083.i		−	1.41950i
$176$		0			0
$177$	101088.	−	83817.7i	0.242531	−	0.201096i
$178$		0			0
$179$	−607176.			−1.41639			−0.708194	−	0.706018i	$-0.750490\pi$
$179$	−607176.			−1.41639			−0.708194	+	0.706018i	$0.750490\pi$
$180$		0			0
$181$	−511850.			−1.16130			−0.580652	−	0.814152i	$-0.697202\pi$
$181$	−511850.			−1.16130			−0.580652	+	0.814152i	$0.697202\pi$
$182$		0			0
$183$	−309144.	+	256329.i	−0.682391	+	0.565808i
$184$		0			0
$185$		−	959805.i		−	2.06183i
$186$		0			0
$187$		−	206321.i		−	0.431458i
$188$		0			0
$189$	593406.	−	328823.i	1.20836	−	0.669589i
$190$		0			0
$191$	−186624.			−0.370155			−0.185078	−	0.982724i	$-0.559254\pi$
$191$	−186624.			−0.370155			−0.185078	+	0.982724i	$0.559254\pi$
$192$		0			0
$193$	169730.			0.327994			0.163997	−	0.986461i	$-0.447561\pi$
$193$	169730.			0.327994			0.163997	+	0.986461i	$0.447561\pi$
$194$		0			0
$195$	−191664.	−	231155.i	−0.360956	−	0.435329i
$196$		0			0
$197$			715834.i			1.31416i	0.753823	+	0.657078i	$0.228208\pi$
$197$			715834.i			1.31416i	−0.753823	+	0.657078i	$0.771792\pi$
$198$		0			0
$199$			596216.i			1.06726i	0.845717	+	0.533631i	$0.179173\pi$
$199$			596216.i			1.06726i	−0.845717	+	0.533631i	$0.820827\pi$
$200$		0			0
$201$	−101574.	−	122503.i	−0.177334	−	0.213873i
$202$		0			0
$203$	356400.			0.607012
$204$		0			0
$205$	−1.17850e6			−1.95859
$206$		0			0
$207$	58320.0	−	309481.i	0.0946001	−	0.502005i
$208$		0			0
$209$			812387.i			1.28646i
$210$		0			0
$211$			317898.i			0.491567i	0.969325	+	0.245783i	$0.0790452\pi$
$211$			317898.i			0.491567i	−0.969325	+	0.245783i	$0.920955\pi$
$212$		0			0
$213$	668736.	−	554487.i	1.00996	−	0.837418i
$214$		0			0
$215$	1.43986e6			2.12433
$216$		0			0
$217$	609444.			0.878586
$218$		0			0
$219$	312312.	−	258955.i	0.440026	−	0.364850i
$220$		0			0
$221$			77051.8i			0.106121i
$222$		0			0
$223$			134144.i			0.180638i	0.995913	+	0.0903191i	$0.0287887\pi$
$223$			134144.i			0.180638i	−0.995913	+	0.0903191i	$0.971211\pi$
$224$		0			0
$225$	−144495.	+	766777.i	−0.190281	+	1.00975i
$226$		0			0
$227$	−734184.			−0.945671			−0.472836	−	0.881151i	$-0.656770\pi$
$227$	−734184.			−0.945671			−0.472836	+	0.881151i	$0.656770\pi$
$228$		0			0
$229$	317062.			0.399536			0.199768	−	0.979843i	$-0.435981\pi$
$229$	317062.			0.399536			0.199768	+	0.979843i	$0.435981\pi$
$230$		0			0
$231$	1.15474e6	+	1.39266e6i	1.42381	+	1.71718i
$232$		0			0
$233$		−	234180.i		−	0.282592i	−0.989967	−	0.141296i	$-0.954873\pi$
$233$		−	234180.i		−	0.282592i	0.989967	−	0.141296i	$-0.0451270\pi$
$234$		0			0
$235$			1.03160e6i			1.21855i
$236$		0			0
$237$	−190674.	−	229961.i	−0.220506	−	0.265940i
$238$		0			0
$239$	85536.0			0.0968622			0.0484311	−	0.998827i	$-0.484578\pi$
$239$	85536.0			0.0968622			0.0484311	+	0.998827i	$0.484578\pi$
$240$		0			0
$241$	−666446.			−0.739133			−0.369566	−	0.929204i	$-0.620494\pi$
$241$	−666446.			−0.739133			−0.369566	+	0.929204i	$0.620494\pi$
$242$		0			0
$243$	−873828.	+	289332.i	−0.949315	+	0.314327i
$244$		0			0
$245$		−	1.21540e6i		−	1.29361i
$246$		0			0
$247$		−	303392.i		−	0.316418i
$248$		0			0
$249$	−940896.	+	780150.i	−0.961708	+	0.797406i
$250$		0			0
$251$	832680.			0.834245			0.417123	−	0.908850i	$-0.363039\pi$
$251$	832680.			0.834245			0.417123	+	0.908850i	$0.363039\pi$
$252$		0			0
$253$	839808.			0.824857
$254$		0			0
$255$	304128.	−	252170.i	0.292891	−	0.242852i
$256$		0			0
$257$		−	953914.i		−	0.900900i	−0.892802	−	0.450450i	$-0.851264\pi$
$257$		−	953914.i		−	0.900900i	0.892802	−	0.450450i	$-0.148736\pi$
$258$		0			0
$259$		−	2.15956e6i		−	2.00040i
$260$		0			0
$261$	−475200.	−	89548.9i	−0.431793	−	0.0813690i
$262$		0			0
$263$	584496.			0.521065			0.260533	−	0.965465i	$-0.416102\pi$
$263$	584496.			0.521065			0.260533	+	0.965465i	$0.416102\pi$
$264$		0			0
$265$	2.14790e6			1.87888
$266$		0			0
$267$	−837936.	−	1.01059e6i	−0.719337	−	0.867554i
$268$		0			0
$269$			452361.i			0.381158i	0.981672	+	0.190579i	$0.0610365\pi$
$269$			452361.i			0.381158i	−0.981672	+	0.190579i	$0.938964\pi$
$270$		0			0
$271$			1.17291e6i			0.970157i	0.874471	+	0.485078i	$0.161209\pi$
$271$			1.17291e6i			0.970157i	−0.874471	+	0.485078i	$0.838791\pi$
$272$		0			0
$273$	−431244.	−	520100.i	−0.350200	−	0.422357i
$274$		0			0
$275$	−2.08073e6			−1.65914
$276$		0			0
$277$	−1.10244e6			−0.863289			−0.431645	−	0.902044i	$-0.642067\pi$
$277$	−1.10244e6			−0.863289			−0.431645	+	0.902044i	$0.642067\pi$
$278$		0			0
$279$	−812592.	−	153129.i	−0.624974	−	0.117773i
$280$		0			0
$281$			1.77776e6i			1.34310i	0.740959	+	0.671550i	$0.234371\pi$
$281$			1.77776e6i			1.34310i	−0.740959	+	0.671550i	$0.765629\pi$
$282$		0			0
$283$		−	190023.i		−	0.141039i	−0.997510	−	0.0705195i	$-0.977534\pi$
$283$		−	190023.i		−	0.141039i	0.997510	−	0.0705195i	$-0.0224657\pi$
$284$		0			0
$285$	−1.19750e6	+	992918.i	−0.873303	+	0.724104i
$286$		0			0
$287$	−2.65162e6			−1.90023
$288$		0			0
$289$	1.31848e6			0.928601
$290$		0			0
$291$	1.23708e6	−	1.02573e6i	0.856378	−	0.710071i
$292$		0			0
$293$			2.55425e6i			1.73818i	0.494654	+	0.869090i	$0.335295\pi$
$293$			2.55425e6i			1.73818i	−0.494654	+	0.869090i	$0.664705\pi$
$294$		0			0
$295$			670542.i			0.448612i
$296$		0			0
$297$	−1.18973e6	−	2.14702e6i	−0.782630	−	1.41236i
$298$		0			0
$299$	−313632.			−0.202881
$300$		0			0
$301$	3.23968e6			2.06103
$302$		0			0
$303$	−829224.	−	1.00008e6i	−0.518878	−	0.625790i
$304$		0			0
$305$		−	2.05063e6i		−	1.26223i
$306$		0			0
$307$		−	2.86001e6i		−	1.73190i	−0.500134	−	0.865948i	$-0.666716\pi$
$307$		−	2.86001e6i		−	1.73190i	0.500134	−	0.865948i	$-0.333284\pi$
$308$		0			0
$309$	−454410.	−	548039.i	−0.270740	−	0.326524i
$310$		0			0
$311$	2.90952e6			1.70577			0.852885	−	0.522099i	$-0.174851\pi$
$311$	2.90952e6			1.70577			0.852885	+	0.522099i	$0.174851\pi$
$312$		0			0
$313$	−2.56639e6			−1.48068			−0.740341	−	0.672231i	$-0.765336\pi$
$313$	−2.56639e6			−1.48068			−0.740341	+	0.672231i	$0.765336\pi$
$314$		0			0
$315$	−641520.	+	3.40429e6i	−0.364279	+	1.93308i
$316$		0			0
$317$		−	2.89366e6i		−	1.61733i	−0.588267	−	0.808667i	$-0.700189\pi$
$317$		−	2.89366e6i		−	1.61733i	0.588267	−	0.808667i	$-0.299811\pi$
$318$		0			0
$319$		−	1.28950e6i		−	0.709489i
$320$		0			0
$321$	256608.	−	212768.i	0.138998	−	0.115251i
$322$		0			0
$323$	399168.			0.212887
$324$		0			0
$325$	777062.			0.408082
$326$		0			0
$327$	706728.	−	585988.i	0.365496	−	0.303053i
$328$		0			0
$329$			2.32111e6i			1.18224i
$330$		0			0
$331$			3.88338e6i			1.94823i	0.226060	+	0.974113i	$0.427415\pi$
$331$			3.88338e6i			1.94823i	−0.226060	+	0.974113i	$0.572585\pi$
$332$		0			0
$333$	−542610.	+	2.87941e6i	−0.268150	+	1.42296i
$334$		0			0
$335$	812592.			0.395604
$336$		0			0
$337$	515746.			0.247378			0.123689	−	0.992321i	$-0.460527\pi$
$337$	515746.			0.247378			0.123689	+	0.992321i	$0.460527\pi$
$338$		0			0
$339$	795168.	+	959009.i	0.375802	+	0.453235i
$340$		0			0
$341$		−	2.20505e6i		−	1.02691i
$342$		0			0
$343$			275452.i			0.126419i
$344$		0			0
$345$	1.02643e6	+	1.23792e6i	0.464283	+	0.559946i
$346$		0			0
$347$	1.41847e6			0.632408			0.316204	−	0.948691i	$-0.397592\pi$
$347$	1.41847e6			0.632408			0.316204	+	0.948691i	$0.397592\pi$
$348$		0			0
$349$	−2.28138e6			−1.00261			−0.501307	−	0.865270i	$-0.667147\pi$
$349$	−2.28138e6			−1.00261			−0.501307	+	0.865270i	$0.667147\pi$
$350$		0			0
$351$	444312.	+	801821.i	0.192495	+	0.347384i
$352$		0			0
$353$		−	21014.1i		−	0.00897583i	−0.999990	−	0.00448792i	$-0.998571\pi$
$353$		−	21014.1i		−	0.00897583i	0.999990	−	0.00448792i	$-0.00142855\pi$
$354$		0			0
$355$			4.43589e6i			1.86814i
$356$		0			0
$357$	684288.	−	567382.i	0.284164	−	0.235616i
$358$		0			0
$359$	−1.98158e6			−0.811477			−0.405739	−	0.913989i	$-0.632986\pi$
$359$	−1.98158e6			−0.811477			−0.405739	+	0.913989i	$0.632986\pi$
$360$		0			0
$361$	904375.			0.365242
$362$		0			0
$363$	3.10624e6	−	2.57555e6i	1.23728	−	1.02590i
$364$		0			0
$365$			2.07164e6i			0.813922i
$366$		0			0
$367$		−	1.22557e6i		−	0.474976i	−0.971390	−	0.237488i	$-0.923676\pi$
$367$		−	1.22557e6i		−	0.474976i	0.971390	−	0.237488i	$-0.0763240\pi$
$368$		0			0
$369$	3.53549e6	+	666244.i	1.35171	+	0.254722i
$370$		0			0
$371$	4.83278e6			1.82290
$372$		0			0
$373$	324566.			0.120790			0.0603950	−	0.998175i	$-0.480764\pi$
$373$	324566.			0.120790			0.0603950	+	0.998175i	$0.480764\pi$
$374$		0			0
$375$	−68112.0	−	82146.2i	−0.0250118	−	0.0301654i
$376$		0			0
$377$			481574.i			0.174506i
$378$		0			0
$379$		−	2.21168e6i		−	0.790904i	−0.918487	−	0.395452i	$-0.870588\pi$
$379$		−	2.21168e6i		−	0.790904i	0.918487	−	0.395452i	$-0.129412\pi$
$380$		0			0
$381$	−1.40243e6	−	1.69140e6i	−0.494960	−	0.596944i
$382$		0			0
$383$	−2.66458e6			−0.928178			−0.464089	−	0.885789i	$-0.653618\pi$
$383$	−2.66458e6			−0.928178			−0.464089	+	0.885789i	$0.653618\pi$
$384$		0			0
$385$	−9.23789e6			−3.17630
$386$		0			0
$387$	−4.31957e6	−	813999.i	−1.46610	−	0.276278i
$388$		0			0
$389$		−	417656.i		−	0.139941i	−0.997549	−	0.0699704i	$-0.977710\pi$
$389$		−	417656.i		−	0.139941i	0.997549	−	0.0699704i	$-0.0222905\pi$
$390$		0			0
$391$		−	412641.i		−	0.136499i
$392$		0			0
$393$	3.50698e6	−	2.90783e6i	1.14538	−	0.949703i
$394$		0			0
$395$	1.52539e6			0.491913
$396$		0			0
$397$	2.01150e6			0.640537			0.320268	−	0.947327i	$-0.396227\pi$
$397$	2.01150e6			0.640537			0.320268	+	0.947327i	$0.396227\pi$
$398$		0			0
$399$	−2.69438e6	+	2.23407e6i	−0.847281	+	0.702528i
$400$		0			0
$401$		−	872723.i		−	0.271029i	−0.990775	−	0.135514i	$-0.956731\pi$
$401$		−	872723.i		−	0.271029i	0.990775	−	0.135514i	$-0.0432687\pi$
$402$		0			0
$403$			823491.i			0.252579i
$404$		0			0
$405$	1.71072e6	−	4.37787e6i	0.518252	−	1.32625i
$406$		0			0
$407$	−7.81358e6			−2.33811
$408$		0			0
$409$	1.07411e6			0.317496			0.158748	−	0.987319i	$-0.449254\pi$
$409$	1.07411e6			0.317496			0.158748	+	0.987319i	$0.449254\pi$
$410$		0			0
$411$	1.54757e6	+	1.86644e6i	0.451903	+	0.545015i
$412$		0			0
$413$			1.50872e6i			0.435245i
$414$		0			0
$415$		−	6.24120e6i		−	1.77888i
$416$		0			0
$417$	3.96851e6	+	4.78621e6i	1.11760	+	1.34788i
$418$		0			0
$419$	−3.65796e6			−1.01790			−0.508949	−	0.860797i	$-0.669966\pi$
$419$	−3.65796e6			−1.01790			−0.508949	+	0.860797i	$0.669966\pi$
$420$		0			0
$421$	−4.65190e6			−1.27916			−0.639580	−	0.768724i	$-0.720892\pi$
$421$	−4.65190e6			−1.27916			−0.639580	+	0.768724i	$0.720892\pi$
$422$		0			0
$423$	583200.	−	3.09481e6i	0.158477	−	0.840975i
$424$		0			0
$425$			1.02237e6i			0.274559i
$426$		0			0
$427$		−	4.61392e6i		−	1.22462i
$428$		0			0
$429$	−1.88179e6	+	1.56030e6i	−0.493661	+	0.409322i
$430$		0			0
$431$	4.98960e6			1.29382			0.646908	−	0.762568i	$-0.276062\pi$
$431$	4.98960e6			1.29382			0.646908	+	0.762568i	$0.276062\pi$
$432$		0			0
$433$	−3.09888e6			−0.794300			−0.397150	−	0.917754i	$-0.630001\pi$
$433$	−3.09888e6			−0.794300			−0.397150	+	0.917754i	$0.630001\pi$
$434$		0			0
$435$	1.90080e6	−	1.57606e6i	0.481630	−	0.399346i
$436$		0			0
$437$			1.62477e6i			0.406995i
$438$		0			0
$439$		−	840864.i		−	0.208240i	−0.994565	−	0.104120i	$-0.966797\pi$
$439$		−	840864.i		−	0.208240i	0.994565	−	0.104120i	$-0.0332026\pi$
$440$		0			0
$441$	−687105.	+	3.64619e6i	−0.168239	+	0.892777i
$442$		0			0
$443$	−4.72457e6			−1.14381			−0.571904	−	0.820321i	$-0.693795\pi$
$443$	−4.72457e6			−1.14381			−0.571904	+	0.820321i	$0.693795\pi$
$444$		0			0
$445$	6.70349e6			1.60473
$446$		0			0
$447$	−2.12494e6	−	2.56277e6i	−0.503010	−	0.606653i
$448$		0			0
$449$		−	3.68512e6i		−	0.862651i	−0.902196	−	0.431326i	$-0.858046\pi$
$449$		−	3.68512e6i		−	0.862651i	0.902196	−	0.431326i	$-0.141954\pi$
$450$		0			0
$451$			9.59391e6i			2.22103i
$452$		0			0
$453$	1.98337e6	+	2.39203e6i	0.454106	+	0.547673i
$454$		0			0
$455$	3.44995e6			0.781240
$456$		0			0
$457$	−4.98364e6			−1.11624			−0.558118	−	0.829762i	$-0.688476\pi$
$457$	−4.98364e6			−1.11624			−0.558118	+	0.829762i	$0.688476\pi$
$458$		0			0
$459$	−1.05494e6	+	584575.i	−0.233721	+	0.129512i
$460$		0			0
$461$		−	1.24915e6i		−	0.273754i	−0.990588	−	0.136877i	$-0.956293\pi$
$461$		−	1.24915e6i		−	0.273754i	0.990588	−	0.136877i	$-0.0437066\pi$
$462$		0			0
$463$			6.43122e6i			1.39425i	0.716949	+	0.697125i	$0.245538\pi$
$463$			6.43122e6i			1.39425i	−0.716949	+	0.697125i	$0.754462\pi$
$464$		0			0
$465$	3.25037e6	−	2.69506e6i	0.697108	−	0.578012i
$466$		0			0
$467$	2.04703e6			0.434343			0.217171	−	0.976134i	$-0.430317\pi$
$467$	2.04703e6			0.434343			0.217171	+	0.976134i	$0.430317\pi$
$468$		0			0
$469$	1.82833e6			0.383816
$470$		0			0
$471$	−2.34588e6	+	1.94510e6i	−0.487252	+	0.404008i
$472$		0			0
$473$		−	1.17216e7i		−	2.40898i
$474$		0			0
$475$		−	4.02558e6i		−	0.818643i
$476$		0			0
$477$	−6.44371e6	−	1.21428e6i	−1.29670	−	0.244356i
$478$		0			0
$479$	7.01395e6			1.39677			0.698384	−	0.715724i	$-0.253903\pi$
$479$	7.01395e6			1.39677			0.698384	+	0.715724i	$0.253903\pi$
$480$		0			0
$481$	2.91804e6			0.575080
$482$		0			0
$483$	2.30947e6	+	2.78533e6i	0.450448	+	0.543261i
$484$		0			0
$485$			8.20586e6i			1.58405i
$486$		0			0
$487$			4.58508e6i			0.876041i	0.898965	+	0.438021i	$0.144320\pi$
$487$			4.58508e6i			0.876041i	−0.898965	+	0.438021i	$0.855680\pi$
$488$		0			0
$489$	−4.47817e6	−	5.40087e6i	−0.846892	−	1.02139i
$490$		0			0
$491$	1.73210e6			0.324243			0.162121	−	0.986771i	$-0.448166\pi$
$491$	1.73210e6			0.324243			0.162121	+	0.986771i	$0.448166\pi$
$492$		0			0
$493$	−633600.			−0.117408
$494$		0			0
$495$	1.23172e7	+	2.32111e6i	2.25943	+	0.425777i
$496$		0			0
$497$			9.98076e6i			1.81248i
$498$		0			0
$499$		−	7.81171e6i		−	1.40441i	−0.711974	−	0.702206i	$-0.752199\pi$
$499$		−	7.81171e6i		−	1.40441i	0.711974	−	0.702206i	$-0.247801\pi$
$500$		0			0
$501$	−4.61894e6	+	3.82983e6i	−0.822145	+	0.681687i
$502$		0			0
$503$	−1.49688e6			−0.263795			−0.131898	−	0.991263i	$-0.542107\pi$
$503$	−1.49688e6			−0.263795			−0.131898	+	0.991263i	$0.542107\pi$
$504$		0			0
$505$	6.63379e6			1.15753
$506$		0			0
$507$	−3.75275e6	+	3.11161e6i	−0.648380	+	0.537608i
$508$		0			0
$509$		−	5.41663e6i		−	0.926691i	−0.886178	−	0.463345i	$-0.846649\pi$
$509$		−	5.41663e6i		−	0.926691i	0.886178	−	0.463345i	$-0.153351\pi$
$510$		0			0
$511$			4.66120e6i			0.789669i
$512$		0			0
$513$	4.15384e6	−	2.30176e6i	0.696878	−	0.386160i
$514$		0			0
$515$	3.63528e6			0.603976
$516$		0			0
$517$	8.39808e6			1.38183
$518$		0			0
$519$	−485496.	−	585530.i	−0.0791165	−	0.0954181i
$520$		0			0
$521$			1.00462e7i			1.62146i	0.585418	+	0.810731i	$0.300930\pi$
$521$			1.00462e7i			1.62146i	−0.585418	+	0.810731i	$0.699070\pi$
$522$		0			0
$523$		−	261662.i		−	0.0418298i	−0.999781	−	0.0209149i	$-0.993342\pi$
$523$		−	261662.i		−	0.0418298i	0.999781	−	0.0209149i	$-0.00665791\pi$
$524$		0			0
$525$	−5.72200e6	−	6.90099e6i	−0.906045	−	1.09273i
$526$		0			0
$527$	−1.08346e6			−0.169936
$528$		0			0
$529$	−4.75673e6			−0.739042
$530$		0			0
$531$	379080.	−	2.01163e6i	0.0583438	−	0.309607i
$532$		0			0
$533$		−	3.58291e6i		−	0.546283i
$534$		0			0
$535$			1.70214e6i			0.257106i
$536$		0			0
$537$	−7.28611e6	+	6.04132e6i	−1.09034	+	0.904058i
$538$		0			0
$539$	−9.89431e6			−1.46694
$540$		0			0
$541$	9.21600e6			1.35378			0.676892	−	0.736083i	$-0.263327\pi$
$541$	9.21600e6			1.35378			0.676892	+	0.736083i	$0.263327\pi$
$542$		0			0
$543$	−6.14220e6	+	5.09284e6i	−0.893973	+	0.741243i
$544$		0			0
$545$			4.68790e6i			0.676063i
$546$		0			0
$547$		−	4.34724e6i		−	0.621220i	−0.950537	−	0.310610i	$-0.899467\pi$
$547$		−	4.34724e6i		−	0.621220i	0.950537	−	0.310610i	$-0.100533\pi$
$548$		0			0
$549$	−1.15929e6	+	6.15189e6i	−0.164158	+	0.871119i
$550$		0			0
$551$	2.49480e6			0.350071
$552$		0			0
$553$	3.43213e6			0.477256
$554$		0			0
$555$	−9.54994e6	−	1.15177e7i	−1.31604	−	1.58720i
$556$		0			0
$557$		−	9.11544e6i		−	1.24491i	−0.782654	−	0.622457i	$-0.786134\pi$
$557$		−	9.11544e6i		−	1.24491i	0.782654	−	0.622457i	$-0.213866\pi$
$558$		0			0
$559$			4.37751e6i			0.592512i
$560$		0			0
$561$	−2.05286e6	−	2.47585e6i	−0.275393	−	0.332137i
$562$		0			0
$563$	891000.			0.118470			0.0592348	−	0.998244i	$-0.481134\pi$
$563$	891000.			0.118470			0.0592348	+	0.998244i	$0.481134\pi$
$564$		0			0
$565$	−6.36134e6			−0.838355
$566$		0			0
$567$	3.84912e6	−	9.85020e6i	0.502810	−	1.28673i
$568$		0			0
$569$		−	1.01544e7i		−	1.31485i	−0.753521	−	0.657424i	$-0.771646\pi$
$569$		−	1.01544e7i		−	1.31485i	0.753521	−	0.657424i	$-0.228354\pi$
$570$		0			0
$571$		−	4.46831e6i		−	0.573526i	−0.958002	−	0.286763i	$-0.907421\pi$
$571$		−	4.46831e6i		−	0.573526i	0.958002	−	0.286763i	$-0.0925792\pi$
$572$		0			0
$573$	−2.23949e6	+	1.85689e6i	−0.284946	+	0.236265i
$574$		0			0
$575$	−4.16146e6			−0.524899
$576$		0			0
$577$	4.65538e6			0.582124			0.291062	−	0.956704i	$-0.405991\pi$
$577$	4.65538e6			0.582124			0.291062	+	0.956704i	$0.405991\pi$
$578$		0			0
$579$	2.03676e6	−	1.68879e6i	0.252490	−	0.209353i
$580$		0			0
$581$		−	1.40427e7i		−	1.72588i
$582$		0			0
$583$		−	1.74857e7i		−	2.13064i
$584$		0			0
$585$	−4.59994e6	−	866833.i	−0.555728	−	0.104724i
$586$		0			0
$587$	−1.33067e7			−1.59395			−0.796975	−	0.604013i	$-0.793568\pi$
$587$	−1.33067e7			−1.59395			−0.796975	+	0.604013i	$0.793568\pi$
$588$		0			0
$589$	4.26611e6			0.506691
$590$		0			0
$591$	7.12246e6	+	8.59001e6i	0.838805	+	1.01164i
$592$		0			0
$593$			4.29102e6i			0.501100i	0.968104	+	0.250550i	$0.0806114\pi$
$593$			4.29102e6i			0.501100i	−0.968104	+	0.250550i	$0.919389\pi$
$594$		0			0
$595$			4.53905e6i			0.525621i
$596$		0			0
$597$	5.93228e6	+	7.15460e6i	0.681217	+	0.821579i
$598$		0			0
$599$	−3.07800e6			−0.350511			−0.175255	−	0.984523i	$-0.556075\pi$
$599$	−3.07800e6			−0.350511			−0.175255	+	0.984523i	$0.556075\pi$
$600$		0			0
$601$	7.36716e6			0.831982			0.415991	−	0.909369i	$-0.363435\pi$
$601$	7.36716e6			0.831982			0.415991	+	0.909369i	$0.363435\pi$
$602$		0			0
$603$	−2.43778e6	−	459386.i	−0.273024	−	0.0514499i
$604$		0			0
$605$			2.06044e7i			2.28861i
$606$		0			0
$607$			2.76187e6i			0.304250i	0.988361	+	0.152125i	$0.0486117\pi$
$607$			2.76187e6i			0.304250i	−0.988361	+	0.152125i	$0.951388\pi$
$608$		0			0
$609$	4.27680e6	−	3.54614e6i	0.467278	−	0.387447i
$610$		0			0
$611$	−3.13632e6			−0.339874
$612$		0			0
$613$	−1.65456e7			−1.77840			−0.889202	−	0.457515i	$-0.848740\pi$
$613$	−1.65456e7			−1.77840			−0.889202	+	0.457515i	$0.848740\pi$
$614$		0			0
$615$	−1.41420e7	+	1.17259e7i	−1.50772	+	1.25014i
$616$		0			0
$617$			7.52577e6i			0.795862i	0.917415	+	0.397931i	$0.130272\pi$
$617$			7.52577e6i			0.795862i	−0.917415	+	0.397931i	$0.869728\pi$
$618$		0			0
$619$			1.14169e7i			1.19763i	0.800887	+	0.598816i	$0.204362\pi$
$619$			1.14169e7i			1.19763i	−0.800887	+	0.598816i	$0.795638\pi$
$620$		0			0
$621$	−2.37946e6	−	4.29405e6i	−0.247599	−	0.446825i
$622$		0			0
$623$	1.50828e7			1.55691
$624$		0			0
$625$	−9.48948e6			−0.971723
$626$		0			0
$627$	8.08315e6	+	9.74865e6i	0.821130	+	0.990320i
$628$		0			0
$629$			3.83922e6i			0.386916i
$630$		0			0
$631$		−	1.69023e7i		−	1.68995i	−0.534806	−	0.844975i	$-0.679615\pi$
$631$		−	1.69023e7i		−	1.68995i	0.534806	−	0.844975i	$-0.320385\pi$
$632$		0			0
$633$	3.16305e6	+	3.81478e6i	0.313759	+	0.378408i
$634$		0			0
$635$	1.12195e7			1.10418
$636$		0			0
$637$	3.69510e6			0.360809
$638$		0			0
$639$	2.50776e6	−	1.33077e7i	0.242959	−	1.28929i
$640$		0			0
$641$		−	1.09089e7i		−	1.04866i	−0.851515	−	0.524331i	$-0.824316\pi$
$641$		−	1.09089e7i		−	1.04866i	0.851515	−	0.524331i	$-0.175684\pi$
$642$		0			0
$643$			4.25805e6i			0.406147i	0.979164	+	0.203073i	$0.0650930\pi$
$643$			4.25805e6i			0.406147i	−0.979164	+	0.203073i	$0.934907\pi$
$644$		0			0
$645$	1.72783e7	−	1.43264e7i	1.63531	−	1.35593i
$646$		0			0
$647$	1.18338e7			1.11138			0.555690	−	0.831390i	$-0.312454\pi$
$647$	1.18338e7			1.11138			0.555690	+	0.831390i	$0.312454\pi$
$648$		0			0
$649$	5.45875e6			0.508723
$650$		0			0
$651$	7.31333e6	−	6.06389e6i	0.676336	−	0.560788i
$652$		0			0
$653$			1.55813e7i			1.42995i	0.699152	+	0.714973i	$0.253561\pi$
$653$			1.55813e7i			1.42995i	−0.699152	+	0.714973i	$0.746439\pi$
$654$		0			0
$655$			2.32626e7i			2.11863i
$656$		0			0
$657$	1.17117e6	−	6.21493e6i	0.105854	−	0.561724i
$658$		0			0
$659$	1.04710e7			0.939239			0.469619	−	0.882869i	$-0.344391\pi$
$659$	1.04710e7			0.939239			0.469619	+	0.882869i	$0.344391\pi$
$660$		0			0
$661$	−8.68177e6			−0.772867			−0.386433	−	0.922317i	$-0.626293\pi$
$661$	−8.68177e6			−0.772867			−0.386433	+	0.922317i	$0.626293\pi$
$662$		0			0
$663$	766656.	+	924622.i	0.0677356	+	0.0816922i
$664$		0			0
$665$		−	1.78725e7i		−	1.56723i
$666$		0			0
$667$		−	2.57901e6i		−	0.224460i
$668$		0			0
$669$	1.33472e6	+	1.60973e6i	0.115299	+	0.139055i
$670$		0			0
$671$	−1.66938e7			−1.43136
$672$		0			0
$673$	−1.94048e7			−1.65147			−0.825737	−	0.564055i	$-0.809241\pi$
$673$	−1.94048e7			−1.65147			−0.825737	+	0.564055i	$0.809241\pi$
$674$		0			0
$675$	5.89540e6	+	1.06390e7i	0.498028	+	0.898758i
$676$		0			0
$677$		−	8.69914e6i		−	0.729465i	−0.931112	−	0.364732i	$-0.881160\pi$
$677$		−	8.69914e6i		−	0.729465i	0.931112	−	0.364732i	$-0.118840\pi$
$678$		0			0
$679$			1.84632e7i			1.53685i
$680$		0			0
$681$	−8.81021e6	+	7.30504e6i	−0.727978	+	0.603608i
$682$		0			0
$683$	1.25511e7			1.02951			0.514755	−	0.857337i	$-0.327883\pi$
$683$	1.25511e7			1.02951			0.514755	+	0.857337i	$0.327883\pi$
$684$		0			0
$685$	−1.23805e7			−1.00812
$686$		0			0
$687$	3.80474e6	−	3.15473e6i	0.307563	−	0.255017i
$688$		0			0
$689$			6.53014e6i			0.524052i
$690$		0			0
$691$			6.02180e6i			0.479768i	0.970802	+	0.239884i	$0.0771094\pi$
$691$			6.02180e6i			0.479768i	−0.970802	+	0.239884i	$0.922891\pi$
$692$		0			0
$693$	2.77137e7	+	5.22249e6i	2.19210	+	0.413090i
$694$		0			0
$695$	−3.17481e7			−2.49319
$696$		0			0
$697$	4.71398e6			0.367541
$698$		0			0
$699$	−2.33006e6	−	2.81016e6i	−0.180374	−	0.217540i
$700$		0			0
$701$		−	4.60918e6i		−	0.354265i	−0.984187	−	0.177133i	$-0.943318\pi$
$701$		−	4.60918e6i		−	0.354265i	0.984187	−	0.177133i	$-0.0566822\pi$
$702$		0			0
$703$		−	1.51169e7i		−	1.15365i
$704$		0			0
$705$	1.02643e7	+	1.23792e7i	0.777781	+	0.938039i
$706$		0			0
$707$	1.49260e7			1.12304
$708$		0			0
$709$	−5.92601e6			−0.442738			−0.221369	−	0.975190i	$-0.571053\pi$
$709$	−5.92601e6			−0.442738			−0.221369	+	0.975190i	$0.571053\pi$
$710$		0			0
$711$	−4.57618e6	−	862356.i	−0.339491	−	0.0639753i
$712$		0			0
$713$		−	4.41010e6i		−	0.324881i
$714$		0			0
$715$		−	1.24824e7i		−	0.913131i
$716$		0			0
$717$	1.02643e6	−	851072.i	0.0745645	−	0.0618256i
$718$		0			0
$719$	1.25012e7			0.901841			0.450921	−	0.892564i	$-0.351096\pi$
$719$	1.25012e7			0.901841			0.450921	+	0.892564i	$0.351096\pi$
$720$		0			0
$721$	8.17938e6			0.585979
$722$		0			0
$723$	−7.99735e6	+	6.63105e6i	−0.568985	+	0.471777i
$724$		0			0
$725$			6.38981e6i			0.451485i
$726$		0			0
$727$		−	1.56959e7i		−	1.10142i	−0.834698	−	0.550708i	$-0.814358\pi$
$727$		−	1.56959e7i		−	1.10142i	0.834698	−	0.550708i	$-0.185642\pi$
$728$		0			0
$729$	−7.60712e6	+	1.21665e7i	−0.530153	+	0.847902i
$730$		0			0
$731$	−5.75942e6			−0.398644
$732$		0			0
$733$	−1.47330e7			−1.01282			−0.506410	−	0.862293i	$-0.669028\pi$
$733$	−1.47330e7			−1.01282			−0.506410	+	0.862293i	$0.669028\pi$
$734$		0			0
$735$	−1.20930e7	−	1.45848e7i	−0.825691	−	0.995821i
$736$		0			0
$737$		−	6.61515e6i		−	0.448612i
$738$		0			0
$739$		−	6.73569e6i		−	0.453702i	−0.973929	−	0.226851i	$-0.927157\pi$
$739$		−	6.73569e6i		−	0.453702i	0.973929	−	0.226851i	$-0.0728431\pi$
$740$		0			0
$741$	−3.01871e6	−	3.64070e6i	−0.201965	−	0.243579i
$742$		0			0
$743$	−1.46409e7			−0.972962			−0.486481	−	0.873691i	$-0.661720\pi$
$743$	−1.46409e7			−0.972962			−0.486481	+	0.873691i	$0.661720\pi$
$744$		0			0
$745$	1.69995e7			1.12214
$746$		0			0
$747$	−3.52836e6	+	1.87236e7i	−0.231351	+	1.22769i
$748$		0			0
$749$			3.82983e6i			0.249445i
$750$		0			0
$751$		−	2.66574e7i		−	1.72472i	−0.506295	−	0.862360i	$-0.668985\pi$
$751$		−	2.66574e7i		−	1.72472i	0.506295	−	0.862360i	$-0.331015\pi$
$752$		0			0
$753$	9.99216e6	−	8.28506e6i	0.642202	−	0.532486i
$754$		0			0
$755$	−1.58669e7			−1.01304
$756$		0			0
$757$	−2.47914e7			−1.57239			−0.786197	−	0.617976i	$-0.787953\pi$
$757$	−2.47914e7			−1.57239			−0.786197	+	0.617976i	$0.787953\pi$
$758$		0			0
$759$	1.00777e7	−	8.35598e6i	0.634975	−	0.526494i
$760$		0			0
$761$			3.05085e7i			1.90968i	0.297126	+	0.954838i	$0.403972\pi$
$761$			3.05085e7i			1.90968i	−0.297126	+	0.954838i	$0.596028\pi$
$762$		0			0
$763$			1.05478e7i			0.655918i
$764$		0			0
$765$	1.14048e6	−	6.05207e6i	0.0704586	−	0.373896i
$766$		0			0
$767$	−2.03861e6			−0.125125
$768$		0			0
$769$	−2.46605e6			−0.150378			−0.0751892	−	0.997169i	$-0.523956\pi$
$769$	−2.46605e6			−0.150378			−0.0751892	+	0.997169i	$0.523956\pi$
$770$		0			0
$771$	−9.49133e6	−	1.14470e7i	−0.575031	−	0.693513i
$772$		0			0
$773$		−	1.72684e7i		−	1.03945i	−0.854333	−	0.519726i	$-0.826034\pi$
$773$		−	1.72684e7i		−	1.03945i	0.854333	−	0.519726i	$-0.173966\pi$
$774$		0			0
$775$			1.09266e7i			0.653476i
$776$		0			0
$777$	−2.14874e7	−	2.59147e7i	−1.27682	−	1.53991i
$778$		0			0
$779$	−1.85613e7			−1.09589
$780$		0			0
$781$	3.61117e7			2.11846
$782$		0			0
$783$	−6.59340e6	+	3.65359e6i	−0.384331	+	0.212969i
$784$		0			0
$785$		−	1.55608e7i		−	0.901276i
$786$		0			0
$787$			2.74648e7i			1.58067i	0.612678	+	0.790333i	$0.290092\pi$
$787$			2.74648e7i			1.58067i	−0.612678	+	0.790333i	$0.709908\pi$
$788$		0			0
$789$	7.01395e6	−	5.81566e6i	0.401116	−	0.332588i
$790$		0			0
$791$	−1.43130e7			−0.813374
$792$		0			0
$793$	6.23440e6			0.352056
$794$		0			0
$795$	2.57748e7	−	2.13714e7i	1.44637	−	1.19926i
$796$		0			0
$797$		−	1.49564e7i		−	0.834030i	−0.908900	−	0.417015i	$-0.863076\pi$
$797$		−	1.49564e7i		−	0.834030i	0.908900	−	0.417015i	$-0.136924\pi$
$798$		0			0
$799$		−	4.12641e6i		−	0.228668i
$800$		0			0
$801$	−2.01105e7	−	3.78971e6i	−1.10749	−	0.208701i
$802$		0			0
$803$	1.68648e7			0.922983
$804$		0			0
$805$	−1.84758e7			−1.00488
$806$		0			0
$807$	4.50094e6	+	5.42833e6i	0.243287	+	0.293415i
$808$		0			0
$809$			3.41239e7i			1.83311i	0.399912	+	0.916553i	$0.369041\pi$
$809$			3.41239e7i			1.83311i	−0.399912	+	0.916553i	$0.630959\pi$
$810$		0			0
$811$		−	4.28205e6i		−	0.228612i	−0.993446	−	0.114306i	$-0.963536\pi$
$811$		−	4.28205e6i		−	0.228612i	0.993446	−	0.114306i	$-0.0364645\pi$
$812$		0			0
$813$	1.16703e7	+	1.40749e7i	0.619236	+	0.746827i
$814$		0			0
$815$	3.58253e7			1.88928
$816$		0			0
$817$	2.26777e7			1.18862
$818$		0			0
$819$	−1.03499e7	−	1.95037e6i	−0.539169	−	0.101603i
$820$		0			0
$821$			2.33344e7i			1.20820i	0.796909	+	0.604099i	$0.206467\pi$
$821$			2.33344e7i			1.20820i	−0.796909	+	0.604099i	$0.793533\pi$
$822$		0			0
$823$			6.02216e6i			0.309922i	0.987921	+	0.154961i	$0.0495252\pi$
$823$			6.02216e6i			0.309922i	−0.987921	+	0.154961i	$0.950475\pi$
$824$		0			0
$825$	−2.49687e7	+	2.07030e7i	−1.27721	+	1.05901i
$826$		0			0
$827$	1.90959e7			0.970905			0.485452	−	0.874263i	$-0.338655\pi$
$827$	1.90959e7			0.970905			0.485452	+	0.874263i	$0.338655\pi$
$828$		0			0
$829$	6.41203e6			0.324048			0.162024	−	0.986787i	$-0.448198\pi$
$829$	6.41203e6			0.324048			0.162024	+	0.986787i	$0.448198\pi$
$830$		0			0
$831$	−1.32293e7	+	1.09692e7i	−0.664560	+	0.551024i
$832$		0			0
$833$			4.86159e6i			0.242754i
$834$		0			0
$835$		−	3.06386e7i		−	1.52073i
$836$		0			0
$837$	−1.12747e7	+	6.24765e6i	−0.556278	+	0.308250i
$838$		0			0
$839$	3.85210e7			1.88926			0.944632	−	0.328131i	$-0.106419\pi$
$839$	3.85210e7			1.88926			0.944632	+	0.328131i	$0.106419\pi$
$840$		0			0
$841$	1.65511e7			0.806934
$842$		0			0
$843$	1.76885e7	+	2.13332e7i	0.857280	+	1.03392i
$844$		0			0
$845$		−	2.48929e7i		−	1.19932i
$846$		0			0
$847$			4.63600e7i			2.22042i
$848$		0			0
$849$	−1.89070e6	−	2.28027e6i	−0.0900231	−	0.108572i
$850$		0			0
$851$	−1.56272e7			−0.739701
$852$		0			0
$853$	2.37604e7			1.11810			0.559050	−	0.829134i	$-0.311166\pi$
$853$	2.37604e7			1.11810			0.559050	+	0.829134i	$0.311166\pi$
$854$		0			0
$855$	−4.49064e6	+	2.38300e7i	−0.210084	+	1.11483i
$856$		0			0
$857$		−	2.60405e7i		−	1.21115i	−0.795789	−	0.605574i	$-0.792944\pi$
$857$		−	2.60405e7i		−	1.21115i	0.795789	−	0.605574i	$-0.207056\pi$
$858$		0			0
$859$			1.25374e7i			0.579727i	0.957068	+	0.289864i	$0.0936100\pi$
$859$			1.25374e7i			0.579727i	−0.957068	+	0.289864i	$0.906390\pi$
$860$		0			0
$861$	−3.18194e7	+	2.63832e7i	−1.46280	+	1.21289i
$862$		0			0
$863$	−4.19541e7			−1.91755			−0.958777	−	0.284160i	$-0.908285\pi$
$863$	−4.19541e7			−1.91755			−0.958777	+	0.284160i	$0.908285\pi$
$864$		0			0
$865$	3.88397e6			0.176496
$866$		0			0
$867$	1.58218e7	−	1.31187e7i	0.714838	−	0.592712i
$868$		0			0
$869$		−	1.24179e7i		−	0.557827i
$870$		0			0
$871$			2.47047e6i			0.110340i
$872$		0			0
$873$	4.63905e6	−	2.46176e7i	0.206012	−	1.09323i
$874$		0			0
$875$	1.22602e6			0.0541347
$876$		0			0
$877$	−3.26082e6			−0.143162			−0.0715810	−	0.997435i	$-0.522804\pi$
$877$	−3.26082e6			−0.143162			−0.0715810	+	0.997435i	$0.522804\pi$
$878$		0			0
$879$	2.54145e7	+	3.06510e7i	1.10945	+	1.33805i
$880$		0			0
$881$		−	8.93960e6i		−	0.388042i	−0.980997	−	0.194021i	$-0.937847\pi$
$881$		−	8.93960e6i		−	0.388042i	0.980997	−	0.194021i	$-0.0621530\pi$
$882$		0			0
$883$			6.86428e6i			0.296274i	0.988967	+	0.148137i	$0.0473276\pi$
$883$			6.86428e6i			0.296274i	−0.988967	+	0.148137i	$0.952672\pi$
$884$		0			0
$885$	6.67181e6	+	8.04650e6i	0.286342	+	0.345342i
$886$		0			0
$887$	4.48066e7			1.91220			0.956099	−	0.293043i	$-0.0946678\pi$
$887$	4.48066e7			1.91220			0.956099	+	0.293043i	$0.0946678\pi$
$888$		0			0
$889$	2.52438e7			1.07127
$890$		0			0
$891$	−3.56394e7	−	1.39266e7i	−1.50396	−	0.587695i
$892$		0			0
$893$			1.62477e7i			0.681812i
$894$		0			0
$895$		−	4.83306e7i		−	2.01681i
$896$		0			0
$897$	−3.76358e6	+	3.12060e6i	−0.156178	+	0.129496i
$898$		0			0
$899$	−6.77160e6			−0.279442
$900$		0			0
$901$	−8.59162e6			−0.352584
$902$		0			0
$903$	3.88761e7	−	3.22344e7i	1.58659	−	1.31553i
$904$		0			0
$905$		−	4.07427e7i		−	1.65359i
$906$		0			0
$907$		−	1.59128e6i		−	0.0642287i	−0.999484	−	0.0321144i	$-0.989776\pi$
$907$		−	1.59128e6i		−	0.0642287i	0.999484	−	0.0321144i	$-0.0102241\pi$
$908$		0			0
$909$	−1.99014e7	−	3.75031e6i	−0.798865	−	0.150542i
$910$		0			0
$911$	−2.30818e7			−0.921453			−0.460726	−	0.887542i	$-0.652411\pi$
$911$	−2.30818e7			−0.921453			−0.460726	+	0.887542i	$0.652411\pi$
$912$		0			0
$913$	−5.08084e7			−2.01724
$914$		0			0
$915$	−2.04035e7	−	2.46076e7i	−0.805660	−	0.971663i
$916$		0			0
$917$			5.23410e7i			2.05550i
$918$		0			0
$919$			2.35050e7i			0.918060i	0.888421	+	0.459030i	$0.151803\pi$
$919$			2.35050e7i			0.918060i	−0.888421	+	0.459030i	$0.848197\pi$
$920$		0			0
$921$	−2.84568e7	−	3.43201e7i	−1.10544	−	1.33321i
$922$		0			0
$923$	−1.34862e7			−0.521056
$924$		0			0
$925$	3.87182e7			1.48786
$926$		0			0
$927$	−1.09058e7	−	2.05515e6i	−0.416831	−	0.0785495i
$928$		0			0
$929$			3.03737e7i			1.15467i	0.816507	+	0.577336i	$0.195908\pi$
$929$			3.03737e7i			1.15467i	−0.816507	+	0.577336i	$0.804092\pi$
$930$		0			0
$931$		−	1.91425e7i		−	0.723810i
$932$		0			0
$933$	3.49142e7	−	2.89494e7i	1.31310	−	1.08877i
$934$		0			0
$935$	1.64229e7			0.614357
$936$		0			0
$937$	3.17565e7			1.18164			0.590819	−	0.806804i	$-0.298805\pi$
$937$	3.17565e7			1.18164			0.590819	+	0.806804i	$0.298805\pi$
$938$		0			0
$939$	−3.07967e7	+	2.55353e7i	−1.13983	+	0.945097i
$940$		0			0
$941$		−	2.59344e7i		−	0.954777i	−0.878692	−	0.477388i	$-0.841583\pi$
$941$		−	2.59344e7i		−	0.954777i	0.878692	−	0.477388i	$-0.158417\pi$
$942$		0			0
$943$			1.91878e7i			0.702662i
$944$		0			0
$945$	2.61740e7	+	4.72345e7i	0.953434	+	1.72060i
$946$		0			0
$947$	−3.65323e7			−1.32374			−0.661869	−	0.749620i	$-0.730236\pi$
$947$	−3.65323e7			−1.32374			−0.661869	+	0.749620i	$0.730236\pi$
$948$		0			0
$949$	−6.29829e6			−0.227016
$950$		0			0
$951$	−2.87916e7	−	3.47239e7i	−1.03232	−	1.24502i
$952$		0			0
$953$			4.54248e7i			1.62017i	0.586313	+	0.810085i	$0.300579\pi$
$953$			4.54248e7i			1.62017i	−0.586313	+	0.810085i	$0.699421\pi$
$954$		0			0
$955$		−	1.48551e7i		−	0.527068i
$956$		0			0
$957$	−1.28304e7	−	1.54740e7i	−0.452856	−	0.546165i
$958$		0			0
$959$	−2.78562e7			−0.978083
$960$		0			0
$961$	1.70497e7			0.595537
$962$		0			0
$963$	962280.	−	5.10643e6i	0.0334376	−	0.177440i
$964$		0			0
$965$			1.35103e7i			0.467033i
$966$		0			0
$967$			2.02339e7i			0.695847i	0.937523	+	0.347924i	$0.113113\pi$
$967$			2.02339e7i			0.695847i	−0.937523	+	0.347924i	$0.886887\pi$
$968$		0			0
$969$	4.79002e6	−	3.97167e6i	0.163881	−	0.135883i
$970$		0			0
$971$	2.26087e6			0.0769534			0.0384767	−	0.999259i	$-0.487749\pi$
$971$	2.26087e6			0.0769534			0.0384767	+	0.999259i	$0.487749\pi$
$972$		0			0
$973$	−7.14333e7			−2.41890
$974$		0			0
$975$	9.32474e6	−	7.73167e6i	0.314142	−	0.260472i
$976$		0			0
$977$			3.21803e6i			0.107858i	0.998545	+	0.0539291i	$0.0171745\pi$
$977$			3.21803e6i			0.107858i	−0.998545	+	0.0539291i	$0.982825\pi$
$978$		0			0
$979$		−	5.45718e7i		−	1.81975i
$980$		0			0
$981$	2.65023e6	−	1.40637e7i	0.0879247	−	0.466581i
$982$		0			0
$983$	2.51035e6			0.0828611			0.0414306	−	0.999141i	$-0.486808\pi$
$983$	2.51035e6			0.0828611			0.0414306	+	0.999141i	$0.486808\pi$
$984$		0			0
$985$	−5.69796e7			−1.87124
$986$		0			0
$987$	2.30947e7	+	2.78533e7i	0.754605	+	0.910088i
$988$		0			0
$989$		−	2.34432e7i		−	0.762124i
$990$		0			0
$991$		−	4.98270e7i		−	1.61169i	−0.592130	−	0.805843i	$-0.701713\pi$
$991$		−	4.98270e7i		−	1.61169i	0.592130	−	0.805843i	$-0.298287\pi$
$992$		0			0
$993$	3.86391e7	+	4.66005e7i	1.24352	+	1.49975i
$994$		0			0
$995$	−4.74582e7			−1.51969
$996$		0			0
$997$	3.97350e7			1.26601			0.633003	−	0.774150i	$-0.281822\pi$
$997$	3.97350e7			1.26601			0.633003	+	0.774150i	$0.281822\pi$
$998$		0			0
$999$	2.21385e7	+	3.99519e7i	0.701834	+	1.26655i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.6.c.c.47.1	yes	2
3.2	odd	2		48.6.c.a.47.1	✓	2
4.3	odd	2		48.6.c.a.47.2	yes	2
8.3	odd	2		192.6.c.c.191.1		2
8.5	even	2		192.6.c.a.191.2		2
12.11	even	2	inner	48.6.c.c.47.2	yes	2
24.5	odd	2		192.6.c.c.191.2		2
24.11	even	2		192.6.c.a.191.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.6.c.a.47.1	✓	2	3.2	odd	2
48.6.c.a.47.2	yes	2	4.3	odd	2
48.6.c.c.47.1	yes	2	1.1	even	1	trivial
48.6.c.c.47.2	yes	2	12.11	even	2	inner
192.6.c.a.191.1		2	24.11	even	2
192.6.c.a.191.2		2	8.5	even	2
192.6.c.c.191.1		2	8.3	odd	2
192.6.c.c.191.2		2	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	48.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(12.0000 - 9.94987i) q^{3} +79.5990i q^{5} +179.098i q^{7} +(45.0000 - 238.797i) q^{9} +O(q^{10})\)	\(q+(12.0000 - 9.94987i) q^{3} +79.5990i q^{5} +179.098i q^{7} +(45.0000 - 238.797i) q^{9} +648.000 q^{11} -242.000 q^{13} +(792.000 + 955.188i) q^{15} -318.396i q^{17} +1253.68i q^{19} +(1782.00 + 2149.17i) q^{21} +1296.00 q^{23} -3211.00 q^{25} +(-1836.00 - 3313.31i) q^{27} -1989.97i q^{29} -3402.86i q^{31} +(7776.00 - 6447.52i) q^{33} -14256.0 q^{35} -12058.0 q^{37} +(-2904.00 + 2407.87i) q^{39} +14805.4i q^{41} -18088.9i q^{43} +(19008.0 + 3581.95i) q^{45} +12960.0 q^{47} -15269.0 q^{49} +(-3168.00 - 3820.75i) q^{51} -26984.1i q^{53} +51580.1i q^{55} +(12474.0 + 15044.2i) q^{57} +8424.00 q^{59} -25762.0 q^{61} +(42768.0 + 8059.40i) q^{63} -19263.0i q^{65} -10208.6i q^{67} +(15552.0 - 12895.0i) q^{69} +55728.0 q^{71} +26026.0 q^{73} +(-38532.0 + 31949.0i) q^{75} +116055. i q^{77} -19163.5i q^{79} +(-54999.0 - 21491.7i) q^{81} -78408.0 q^{83} +25344.0 q^{85} +(-19800.0 - 23879.7i) q^{87} -84215.7i q^{89} -43341.7i q^{91} +(-33858.0 - 40834.3i) q^{93} -99792.0 q^{95} +103090. q^{97} +(29160.0 - 154740. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 24 q^{3} + 90 q^{9}+O(q^{10}) \) 2 * q + 24 * q^3 + 90 * q^9	\( 2 q + 24 q^{3} + 90 q^{9} + 1296 q^{11} - 484 q^{13} + 1584 q^{15} + 3564 q^{21} + 2592 q^{23} - 6422 q^{25} - 3672 q^{27} + 15552 q^{33} - 28512 q^{35} - 24116 q^{37} - 5808 q^{39} + 38016 q^{45} + 25920 q^{47} - 30538 q^{49} - 6336 q^{51} + 24948 q^{57} + 16848 q^{59} - 51524 q^{61} + 85536 q^{63} + 31104 q^{69} + 111456 q^{71} + 52052 q^{73} - 77064 q^{75} - 109998 q^{81} - 156816 q^{83} + 50688 q^{85} - 39600 q^{87} - 67716 q^{93} - 199584 q^{95} + 206180 q^{97} + 58320 q^{99}+O(q^{100}) \) 2 * q + 24 * q^3 + 90 * q^9 + 1296 * q^11 - 484 * q^13 + 1584 * q^15 + 3564 * q^21 + 2592 * q^23 - 6422 * q^25 - 3672 * q^27 + 15552 * q^33 - 28512 * q^35 - 24116 * q^37 - 5808 * q^39 + 38016 * q^45 + 25920 * q^47 - 30538 * q^49 - 6336 * q^51 + 24948 * q^57 + 16848 * q^59 - 51524 * q^61 + 85536 * q^63 + 31104 * q^69 + 111456 * q^71 + 52052 * q^73 - 77064 * q^75 - 109998 * q^81 - 156816 * q^83 + 50688 * q^85 - 39600 * q^87 - 67716 * q^93 - 199584 * q^95 + 206180 * q^97 + 58320 * q^99

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