LMFDB - Embedded newform 1680.2.f.k.881.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(881,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1680.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.f (of order $2$, degree $1$, not 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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + x^{14} - 4 x^{13} + 10 x^{12} - 32 x^{11} + 71 x^{10} - 70 x^{9} + 74 x^{8} + \cdots + 6561 $ x^16 - 2x^15 + x^14 - 4x^13 + 10x^12 - 32x^11 + 71x^10 - 70x^9 + 74x^8 - 210x^7 + 639x^6 - 864x^5 + 810x^4 - 972x^3 + 729x^2 - 4374x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.7
Root			$1.66912 + 0.462633i$ of defining polynomial
Character	$\chi$	$=$	1680.881
Dual form			1680.2.f.k.881.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.462633 - 1.66912i) q^{3} -1.00000 q^{5} +(-1.62879 - 2.08496i) q^{7} +(-2.57194 + 1.54438i) q^{9} +O(q^{10})$	\(q+(-0.462633 - 1.66912i) q^{3} -1.00000 q^{5} +(-1.62879 - 2.08496i) q^{7} +(-2.57194 + 1.54438i) q^{9} -0.196087i q^{11} +2.37720i q^{13} +(0.462633 + 1.66912i) q^{15} -3.36987 q^{17} -2.89268i q^{19} +(-2.72652 + 3.68322i) q^{21} +5.80220i q^{23} +1.00000 q^{25} +(3.76763 + 3.57841i) q^{27} +5.73415i q^{29} -4.66669i q^{31} +(-0.327293 + 0.0907162i) q^{33} +(1.62879 + 2.08496i) q^{35} -7.34774 q^{37} +(3.96785 - 1.09977i) q^{39} +8.59692 q^{41} +0.444609 q^{43} +(2.57194 - 1.54438i) q^{45} +5.43557 q^{47} +(-1.69409 + 6.79191i) q^{49} +(1.55901 + 5.62473i) q^{51} +2.24643i q^{53} +0.196087i q^{55} +(-4.82823 + 1.33825i) q^{57} +4.10762 q^{59} +1.24011i q^{61} +(7.40912 + 2.84692i) q^{63} -2.37720i q^{65} -7.26282 q^{67} +(9.68459 - 2.68429i) q^{69} +8.21455i q^{71} +11.5044i q^{73} +(-0.462633 - 1.66912i) q^{75} +(-0.408833 + 0.319384i) q^{77} +9.71013 q^{79} +(4.22977 - 7.94412i) q^{81} -5.51467 q^{83} +3.36987 q^{85} +(9.57100 - 2.65280i) q^{87} +2.30915 q^{89} +(4.95637 - 3.87196i) q^{91} +(-7.78927 + 2.15896i) q^{93} +2.89268i q^{95} -6.59507i q^{97} +(0.302833 + 0.504324i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) $ 16 * q - 16 * q^5 - 2 * q^7 - 2 * q^9	$ 16 q - 16 q^{5} - 2 q^{7} - 2 q^{9} + 2 q^{21} + 16 q^{25} + 6 q^{27} - 6 q^{33} + 2 q^{35} + 12 q^{37} - 6 q^{39} - 32 q^{41} - 32 q^{43} + 2 q^{45} + 4 q^{47} - 4 q^{49} - 6 q^{51} - 24 q^{59} + 4 q^{63} - 8 q^{69} + 32 q^{77} + 4 q^{79} - 6 q^{81} + 20 q^{83} + 6 q^{87} + 24 q^{89} - 20 q^{91} - 32 q^{93} + 58 q^{99}+O(q^{100}) $ 16 * q - 16 * q^5 - 2 * q^7 - 2 * q^9 + 2 * q^21 + 16 * q^25 + 6 * q^27 - 6 * q^33 + 2 * q^35 + 12 * q^37 - 6 * q^39 - 32 * q^41 - 32 * q^43 + 2 * q^45 + 4 * q^47 - 4 * q^49 - 6 * q^51 - 24 * q^59 + 4 * q^63 - 8 * q^69 + 32 * q^77 + 4 * q^79 - 6 * q^81 + 20 * q^83 + 6 * q^87 + 24 * q^89 - 20 * q^91 - 32 * q^93 + 58 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$1$	$1$	$-1$	$1$

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
$3$	−0.462633	−	1.66912i	−0.267101	−	0.963669i
$3$	−0.462633	−	1.66912i	−0.267101	−	0.963669i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−1.62879	−	2.08496i	−0.615624	−	0.788040i
$7$	−1.62879	−	2.08496i	−0.615624	−	0.788040i
$8$		0			0
$9$	−2.57194	+	1.54438i	−0.857314	+	0.514794i
$10$		0			0
$11$		−	0.196087i		−	0.0591224i	−0.999563	−	0.0295612i	$-0.990589\pi$
$11$		−	0.196087i		−	0.0591224i	0.999563	−	0.0295612i	$-0.00941100\pi$
$12$		0			0
$13$			2.37720i			0.659318i	0.944100	+	0.329659i	$0.106934\pi$
$13$			2.37720i			0.659318i	−0.944100	+	0.329659i	$0.893066\pi$
$14$		0			0
$15$	0.462633	+	1.66912i	0.119451	+	0.430966i
$16$		0			0
$17$	−3.36987			−0.817315			−0.408657	−	0.912688i	$-0.634003\pi$
$17$	−3.36987			−0.817315			−0.408657	+	0.912688i	$0.634003\pi$
$18$		0			0
$19$		−	2.89268i		−	0.663625i	−0.943345	−	0.331813i	$-0.892340\pi$
$19$		−	2.89268i		−	0.663625i	0.943345	−	0.331813i	$-0.107660\pi$
$20$		0			0
$21$	−2.72652	+	3.68322i	−0.594975	+	0.803744i
$22$		0			0
$23$			5.80220i			1.20984i	0.796285	+	0.604921i	$0.206795\pi$
$23$			5.80220i			1.20984i	−0.796285	+	0.604921i	$0.793205\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	3.76763	+	3.57841i	0.725080	+	0.688665i
$28$		0			0
$29$			5.73415i			1.06480i	0.846491	+	0.532402i	$0.178711\pi$
$29$			5.73415i			1.06480i	−0.846491	+	0.532402i	$0.821289\pi$
$30$		0			0
$31$		−	4.66669i		−	0.838162i	−0.907949	−	0.419081i	$-0.862352\pi$
$31$		−	4.66669i		−	0.838162i	0.907949	−	0.419081i	$-0.137648\pi$
$32$		0			0
$33$	−0.327293	+	0.0907162i	−0.0569744	+	0.0157917i
$34$		0			0
$35$	1.62879	+	2.08496i	0.275316	+	0.352422i
$36$		0			0
$37$	−7.34774			−1.20796			−0.603980	−	0.796999i	$-0.706419\pi$
$37$	−7.34774			−1.20796			−0.603980	+	0.796999i	$0.706419\pi$
$38$		0			0
$39$	3.96785	−	1.09977i	0.635364	−	0.176104i
$40$		0			0
$41$	8.59692			1.34261			0.671307	−	0.741180i	$-0.265733\pi$
$41$	8.59692			1.34261			0.671307	+	0.741180i	$0.265733\pi$
$42$		0			0
$43$	0.444609			0.0678022			0.0339011	−	0.999425i	$-0.489207\pi$
$43$	0.444609			0.0678022			0.0339011	+	0.999425i	$0.489207\pi$
$44$		0			0
$45$	2.57194	−	1.54438i	0.383403	−	0.230223i
$46$		0			0
$47$	5.43557			0.792860			0.396430	−	0.918065i	$-0.370249\pi$
$47$	5.43557			0.792860			0.396430	+	0.918065i	$0.370249\pi$
$48$		0			0
$49$	−1.69409	+	6.79191i	−0.242013	+	0.970273i
$50$		0			0
$51$	1.55901	+	5.62473i	0.218306	+	0.787620i
$52$		0			0
$53$			2.24643i			0.308571i	0.988026	+	0.154286i	$0.0493076\pi$
$53$			2.24643i			0.308571i	−0.988026	+	0.154286i	$0.950692\pi$
$54$		0			0
$55$			0.196087i			0.0264403i
$56$		0			0
$57$	−4.82823	+	1.33825i	−0.639515	+	0.177255i
$58$		0			0
$59$	4.10762			0.534767			0.267384	−	0.963590i	$-0.413841\pi$
$59$	4.10762			0.534767			0.267384	+	0.963590i	$0.413841\pi$
$60$		0			0
$61$			1.24011i			0.158780i	0.996844	+	0.0793901i	$0.0252973\pi$
$61$			1.24011i			0.158780i	−0.996844	+	0.0793901i	$0.974703\pi$
$62$		0			0
$63$	7.40912	+	2.84692i	0.933461	+	0.358678i
$64$		0			0
$65$		−	2.37720i		−	0.294856i
$66$		0			0
$67$	−7.26282			−0.887295			−0.443648	−	0.896201i	$-0.646316\pi$
$67$	−7.26282			−0.887295			−0.443648	+	0.896201i	$0.646316\pi$
$68$		0			0
$69$	9.68459	−	2.68429i	1.16589	−	0.323150i
$70$		0			0
$71$			8.21455i			0.974888i	0.873154	+	0.487444i	$0.162071\pi$
$71$			8.21455i			0.974888i	−0.873154	+	0.487444i	$0.837929\pi$
$72$		0			0
$73$			11.5044i			1.34649i	0.739421	+	0.673243i	$0.235099\pi$
$73$			11.5044i			1.34649i	−0.739421	+	0.673243i	$0.764901\pi$
$74$		0			0
$75$	−0.462633	−	1.66912i	−0.0534202	−	0.192734i
$76$		0			0
$77$	−0.408833	+	0.319384i	−0.0465908	+	0.0363972i
$78$		0			0
$79$	9.71013			1.09248			0.546238	−	0.837630i	$-0.316060\pi$
$79$	9.71013			1.09248			0.546238	+	0.837630i	$0.316060\pi$
$80$		0			0
$81$	4.22977	−	7.94412i	0.469975	−	0.882680i
$82$		0			0
$83$	−5.51467			−0.605314			−0.302657	−	0.953100i	$-0.597874\pi$
$83$	−5.51467			−0.605314			−0.302657	+	0.953100i	$0.597874\pi$
$84$		0			0
$85$	3.36987			0.365514
$86$		0			0
$87$	9.57100	−	2.65280i	1.02612	−	0.284410i
$88$		0			0
$89$	2.30915			0.244770			0.122385	−	0.992483i	$-0.460946\pi$
$89$	2.30915			0.244770			0.122385	+	0.992483i	$0.460946\pi$
$90$		0			0
$91$	4.95637	−	3.87196i	0.519569	−	0.405892i
$92$		0			0
$93$	−7.78927	+	2.15896i	−0.807710	+	0.223874i
$94$		0			0
$95$			2.89268i			0.296782i
$96$		0			0
$97$		−	6.59507i		−	0.669628i	−0.942284	−	0.334814i	$-0.891326\pi$
$97$		−	6.59507i		−	0.669628i	0.942284	−	0.334814i	$-0.108674\pi$
$98$		0			0
$99$	0.302833	+	0.504324i	0.0304358	+	0.0506865i
$100$		0			0
$101$	−0.709225			−0.0705705			−0.0352853	−	0.999377i	$-0.511234\pi$
$101$	−0.709225			−0.0705705			−0.0352853	+	0.999377i	$0.511234\pi$
$102$		0			0
$103$			11.6159i			1.14455i	0.820062	+	0.572275i	$0.193939\pi$
$103$			11.6159i			1.14455i	−0.820062	+	0.572275i	$0.806061\pi$
$104$		0			0
$105$	2.72652	−	3.68322i	0.266081	−	0.359445i
$106$		0			0
$107$		−	6.85555i		−	0.662751i	−0.943499	−	0.331375i	$-0.892487\pi$
$107$		−	6.85555i		−	0.662751i	0.943499	−	0.331375i	$-0.107513\pi$
$108$		0			0
$109$	18.3995			1.76235			0.881174	−	0.472792i	$-0.156754\pi$
$109$	18.3995			1.76235			0.881174	+	0.472792i	$0.156754\pi$
$110$		0			0
$111$	3.39930	+	12.2643i	0.322647	+	1.16407i
$112$		0			0
$113$			8.13112i			0.764911i	0.923974	+	0.382456i	$0.124922\pi$
$113$			8.13112i			0.764911i	−0.923974	+	0.382456i	$0.875078\pi$
$114$		0			0
$115$		−	5.80220i		−	0.541058i
$116$		0			0
$117$	−3.67131	−	6.11403i	−0.339413	−	0.565242i
$118$		0			0
$119$	5.48881	+	7.02604i	0.503159	+	0.644076i
$120$		0			0
$121$	10.9615			0.996505
$122$		0			0
$123$	−3.97722	−	14.3493i	−0.358613	−	1.29383i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−15.7323			−1.39601			−0.698006	−	0.716092i	$-0.745929\pi$
$127$	−15.7323			−1.39601			−0.698006	+	0.716092i	$0.745929\pi$
$128$		0			0
$129$	−0.205691	−	0.742107i	−0.0181100	−	0.0653389i
$130$		0			0
$131$	−22.2013			−1.93974			−0.969869	−	0.243626i	$-0.921663\pi$
$131$	−22.2013			−1.93974			−0.969869	+	0.243626i	$0.921663\pi$
$132$		0			0
$133$	−6.03110	+	4.71156i	−0.522963	+	0.408544i
$134$		0			0
$135$	−3.76763	−	3.57841i	−0.324266	−	0.307980i
$136$		0			0
$137$			6.54886i			0.559507i	0.960072	+	0.279754i	$0.0902528\pi$
$137$			6.54886i			0.559507i	−0.960072	+	0.279754i	$0.909747\pi$
$138$		0			0
$139$			0.422584i			0.0358431i	0.999839	+	0.0179216i	$0.00570492\pi$
$139$			0.422584i			0.0358431i	−0.999839	+	0.0179216i	$0.994295\pi$
$140$		0			0
$141$	−2.51467	−	9.07264i	−0.211774	−	0.764054i
$142$		0			0
$143$	0.466139			0.0389805
$144$		0			0
$145$		−	5.73415i		−	0.476195i
$146$		0			0
$147$	12.1203	−	0.314510i	0.999663	−	0.0259404i
$148$		0			0
$149$			24.2743i			1.98863i	0.106497	+	0.994313i	$0.466036\pi$
$149$			24.2743i			1.98863i	−0.106497	+	0.994313i	$0.533964\pi$
$150$		0			0
$151$	−19.9630			−1.62456			−0.812282	−	0.583265i	$-0.801775\pi$
$151$	−19.9630			−1.62456			−0.812282	+	0.583265i	$0.801775\pi$
$152$		0			0
$153$	8.66712	−	5.20437i	0.700695	−	0.420748i
$154$		0			0
$155$			4.66669i			0.374837i
$156$		0			0
$157$		−	6.46198i		−	0.515722i	−0.966182	−	0.257861i	$-0.916982\pi$
$157$		−	6.46198i		−	0.515722i	0.966182	−	0.257861i	$-0.0830177\pi$
$158$		0			0
$159$	3.74957	−	1.03927i	0.297361	−	0.0824197i
$160$		0			0
$161$	12.0973	−	9.45056i	0.953404	−	0.744809i
$162$		0			0
$163$	−10.1360			−0.793913			−0.396956	−	0.917837i	$-0.629934\pi$
$163$	−10.1360			−0.793913			−0.396956	+	0.917837i	$0.629934\pi$
$164$		0			0
$165$	0.327293	−	0.0907162i	0.0254797	−	0.00706224i
$166$		0			0
$167$	21.8061			1.68741			0.843704	−	0.536808i	$-0.180370\pi$
$167$	21.8061			1.68741			0.843704	+	0.536808i	$0.180370\pi$
$168$		0			0
$169$	7.34890			0.565300
$170$		0			0
$171$	4.46739	+	7.43979i	0.341630	+	0.568935i
$172$		0			0
$173$	−15.9989			−1.21637			−0.608186	−	0.793795i	$-0.708102\pi$
$173$	−15.9989			−1.21637			−0.608186	+	0.793795i	$0.708102\pi$
$174$		0			0
$175$	−1.62879	−	2.08496i	−0.123125	−	0.157608i
$176$		0			0
$177$	−1.90032	−	6.85613i	−0.142837	−	0.515338i
$178$		0			0
$179$		−	12.6765i		−	0.947488i	−0.880663	−	0.473744i	$-0.842902\pi$
$179$		−	12.6765i		−	0.947488i	0.880663	−	0.473744i	$-0.157098\pi$
$180$		0			0
$181$			12.1474i			0.902912i	0.892293	+	0.451456i	$0.149095\pi$
$181$			12.1474i			0.902912i	−0.892293	+	0.451456i	$0.850905\pi$
$182$		0			0
$183$	2.06990	−	0.573717i	0.153011	−	0.0424103i
$184$		0			0
$185$	7.34774			0.540216
$186$		0			0
$187$			0.660788i			0.0483216i
$188$		0			0
$189$	1.32416	−	13.6838i	0.0963182	−	0.995351i
$190$		0			0
$191$			19.1461i			1.38536i	0.721244	+	0.692682i	$0.243571\pi$
$191$			19.1461i			1.38536i	−0.721244	+	0.692682i	$0.756429\pi$
$192$		0			0
$193$	21.6408			1.55774			0.778868	−	0.627188i	$-0.215794\pi$
$193$	21.6408			1.55774			0.778868	+	0.627188i	$0.215794\pi$
$194$		0			0
$195$	−3.96785	+	1.09977i	−0.284143	+	0.0787563i
$196$		0			0
$197$			21.8641i			1.55775i	0.627177	+	0.778877i	$0.284210\pi$
$197$			21.8641i			1.55775i	−0.627177	+	0.778877i	$0.715790\pi$
$198$		0			0
$199$			5.31869i			0.377032i	0.982070	+	0.188516i	$0.0603678\pi$
$199$			5.31869i			0.377032i	−0.982070	+	0.188516i	$0.939632\pi$
$200$		0			0
$201$	3.36002	+	12.1225i	0.236997	+	0.855058i
$202$		0			0
$203$	11.9555	−	9.33972i	0.839109	−	0.655520i
$204$		0			0
$205$	−8.59692			−0.600435
$206$		0			0
$207$	−8.96081	−	14.9229i	−0.622819	−	1.03722i
$208$		0			0
$209$	−0.567216			−0.0392351
$210$		0			0
$211$	−6.72971			−0.463292			−0.231646	−	0.972800i	$-0.574411\pi$
$211$	−6.72971			−0.463292			−0.231646	+	0.972800i	$0.574411\pi$
$212$		0			0
$213$	13.7111	−	3.80032i	0.939469	−	0.260394i
$214$		0			0
$215$	−0.444609			−0.0303221
$216$		0			0
$217$	−9.72984	+	7.60105i	−0.660505	+	0.515993i
$218$		0			0
$219$	19.2022	−	5.32230i	1.29757	−	0.359648i
$220$		0			0
$221$		−	8.01088i		−	0.538870i
$222$		0			0
$223$			8.90230i			0.596142i	0.954544	+	0.298071i	$0.0963433\pi$
$223$			8.90230i			0.596142i	−0.954544	+	0.298071i	$0.903657\pi$
$224$		0			0
$225$	−2.57194	+	1.54438i	−0.171463	+	0.102959i
$226$		0			0
$227$	−26.1415			−1.73507			−0.867537	−	0.497373i	$-0.834298\pi$
$227$	−26.1415			−1.73507			−0.867537	+	0.497373i	$0.834298\pi$
$228$		0			0
$229$		−	17.8960i		−	1.18260i	−0.806451	−	0.591301i	$-0.798614\pi$
$229$		−	17.8960i		−	1.18260i	0.806451	−	0.591301i	$-0.201386\pi$
$230$		0			0
$231$	0.722231	+	0.534635i	0.0475193	+	0.0351764i
$232$		0			0
$233$		−	26.2856i		−	1.72203i	−0.508582	−	0.861014i	$-0.669830\pi$
$233$		−	26.2856i		−	1.72203i	0.508582	−	0.861014i	$-0.330170\pi$
$234$		0			0
$235$	−5.43557			−0.354578
$236$		0			0
$237$	−4.49222	−	16.2074i	−0.291801	−	1.05278i
$238$		0			0
$239$			18.3474i			1.18679i	0.804910	+	0.593397i	$0.202213\pi$
$239$			18.3474i			1.18679i	−0.804910	+	0.593397i	$0.797787\pi$
$240$		0			0
$241$		−	22.3756i		−	1.44134i	−0.693278	−	0.720670i	$-0.743834\pi$
$241$		−	22.3756i		−	1.44134i	0.693278	−	0.720670i	$-0.256166\pi$
$242$		0			0
$243$	−15.2165	−	3.38481i	−0.976142	−	0.217135i
$244$		0			0
$245$	1.69409	−	6.79191i	0.108232	−	0.433919i
$246$		0			0
$247$	6.87648			0.437540
$248$		0			0
$249$	2.55127	+	9.20466i	0.161680	+	0.583322i
$250$		0			0
$251$	−21.4170			−1.35183			−0.675914	−	0.736981i	$-0.736251\pi$
$251$	−21.4170			−1.35183			−0.675914	+	0.736981i	$0.736251\pi$
$252$		0			0
$253$	1.13774			0.0715288
$254$		0			0
$255$	−1.55901	−	5.62473i	−0.0976292	−	0.352234i
$256$		0			0
$257$	−4.19079			−0.261414			−0.130707	−	0.991421i	$-0.541725\pi$
$257$	−4.19079			−0.261414			−0.130707	+	0.991421i	$0.541725\pi$
$258$		0			0
$259$	11.9679	+	15.3197i	0.743650	+	0.951921i
$260$		0			0
$261$	−8.85571	−	14.7479i	−0.548155	−	0.912872i
$262$		0			0
$263$			20.3154i			1.25270i	0.779541	+	0.626352i	$0.215453\pi$
$263$			20.3154i			1.25270i	−0.779541	+	0.626352i	$0.784547\pi$
$264$		0			0
$265$		−	2.24643i		−	0.137997i
$266$		0			0
$267$	−1.06829	−	3.85426i	−0.0653782	−	0.235877i
$268$		0			0
$269$	−26.4004			−1.60966			−0.804831	−	0.593504i	$-0.797744\pi$
$269$	−26.4004			−1.60966			−0.804831	+	0.593504i	$0.797744\pi$
$270$		0			0
$271$		−	1.59460i		−	0.0968648i	−0.998826	−	0.0484324i	$-0.984577\pi$
$271$		−	1.59460i		−	0.0968648i	0.998826	−	0.0484324i	$-0.0154225\pi$
$272$		0			0
$273$	−8.75576	−	6.48149i	−0.529923	−	0.392278i
$274$		0			0
$275$		−	0.196087i		−	0.0118245i
$276$		0			0
$277$	−3.27229			−0.196613			−0.0983064	−	0.995156i	$-0.531343\pi$
$277$	−3.27229			−0.196613			−0.0983064	+	0.995156i	$0.531343\pi$
$278$		0			0
$279$	7.20714	+	12.0024i	0.431480	+	0.718568i
$280$		0			0
$281$			24.6152i			1.46842i	0.678921	+	0.734211i	$0.262448\pi$
$281$			24.6152i			1.46842i	−0.678921	+	0.734211i	$0.737552\pi$
$282$		0			0
$283$			6.38730i			0.379686i	0.981815	+	0.189843i	$0.0607978\pi$
$283$			6.38730i			0.379686i	−0.981815	+	0.189843i	$0.939202\pi$
$284$		0			0
$285$	4.82823	−	1.33825i	0.286000	−	0.0792708i
$286$		0			0
$287$	−14.0026	−	17.9242i	−0.826546	−	1.05803i
$288$		0			0
$289$	−5.64395			−0.331997
$290$		0			0
$291$	−11.0080	+	3.05110i	−0.645300	+	0.178858i
$292$		0			0
$293$	−9.71110			−0.567329			−0.283664	−	0.958924i	$-0.591550\pi$
$293$	−9.71110			−0.567329			−0.283664	+	0.958924i	$0.591550\pi$
$294$		0			0
$295$	−4.10762			−0.239155
$296$		0			0
$297$	0.701679	−	0.738782i	0.0407155	−	0.0428685i
$298$		0			0
$299$	−13.7930			−0.797671
$300$		0			0
$301$	−0.724174	−	0.926991i	−0.0417407	−	0.0534309i
$302$		0			0
$303$	0.328111	+	1.18378i	0.0188495	+	0.0680066i
$304$		0			0
$305$		−	1.24011i		−	0.0710087i
$306$		0			0
$307$		−	22.1102i		−	1.26190i	−0.775825	−	0.630949i	$-0.782666\pi$
$307$		−	22.1102i		−	1.26190i	0.775825	−	0.630949i	$-0.217334\pi$
$308$		0			0
$309$	19.3884	−	5.37390i	1.10297	−	0.305711i
$310$		0			0
$311$	−15.6135			−0.885359			−0.442679	−	0.896680i	$-0.645972\pi$
$311$	−15.6135			−0.885359			−0.442679	+	0.896680i	$0.645972\pi$
$312$		0			0
$313$		−	1.04206i		−	0.0589009i	−0.999566	−	0.0294504i	$-0.990624\pi$
$313$		−	1.04206i		−	0.0589009i	0.999566	−	0.0294504i	$-0.00937572\pi$
$314$		0			0
$315$	−7.40912	−	2.84692i	−0.417457	−	0.160406i
$316$		0			0
$317$			15.6828i			0.880836i	0.897793	+	0.440418i	$0.145170\pi$
$317$			15.6828i			0.880836i	−0.897793	+	0.440418i	$0.854830\pi$
$318$		0			0
$319$	1.12439			0.0629538
$320$		0			0
$321$	−11.4428	+	3.17160i	−0.638672	+	0.177021i
$322$		0			0
$323$			9.74795i			0.542391i
$324$		0			0
$325$			2.37720i			0.131864i
$326$		0			0
$327$	−8.51219	−	30.7110i	−0.470725	−	1.69832i
$328$		0			0
$329$	−8.85340	−	11.3329i	−0.488104	−	0.624805i
$330$		0			0
$331$	31.2221			1.71612			0.858061	−	0.513548i	$-0.171669\pi$
$331$	31.2221			1.71612			0.858061	+	0.513548i	$0.171669\pi$
$332$		0			0
$333$	18.8980	−	11.3477i	1.03560	−	0.621850i
$334$		0			0
$335$	7.26282			0.396810
$336$		0			0
$337$	−27.7126			−1.50960			−0.754800	−	0.655955i	$-0.772266\pi$
$337$	−27.7126			−1.50960			−0.754800	+	0.655955i	$0.772266\pi$
$338$		0			0
$339$	13.5718	−	3.76172i	0.737121	−	0.204309i
$340$		0			0
$341$	−0.915076			−0.0495541
$342$		0			0
$343$	16.9202	−	7.53048i	0.913603	−	0.406608i
$344$		0			0
$345$	−9.68459	+	2.68429i	−0.521401	+	0.144517i
$346$		0			0
$347$		−	21.0936i		−	1.13236i	−0.824281	−	0.566181i	$-0.808420\pi$
$347$		−	21.0936i		−	1.13236i	0.824281	−	0.566181i	$-0.191580\pi$
$348$		0			0
$349$			34.2739i			1.83464i	0.398148	+	0.917321i	$0.369653\pi$
$349$			34.2739i			1.83464i	−0.398148	+	0.917321i	$0.630347\pi$
$350$		0			0
$351$	−8.50660	+	8.95642i	−0.454049	+	0.478058i
$352$		0			0
$353$	−36.0053			−1.91637			−0.958183	−	0.286155i	$-0.907623\pi$
$353$	−36.0053			−1.91637			−0.958183	+	0.286155i	$0.907623\pi$
$354$		0			0
$355$		−	8.21455i		−	0.435983i
$356$		0			0
$357$	9.18803	−	12.4120i	0.486282	−	0.656912i
$358$		0			0
$359$		−	4.46402i		−	0.235602i	−0.993037	−	0.117801i	$-0.962416\pi$
$359$		−	4.46402i		−	0.235602i	0.993037	−	0.117801i	$-0.0375845\pi$
$360$		0			0
$361$	10.6324			0.559602
$362$		0			0
$363$	−5.07117	−	18.2962i	−0.266167	−	0.960300i
$364$		0			0
$365$		−	11.5044i		−	0.602167i
$366$		0			0
$367$		−	0.327460i		−	0.0170933i	−0.999963	−	0.00854663i	$-0.997279\pi$
$367$		−	0.327460i		−	0.0170933i	0.999963	−	0.00854663i	$-0.00272051\pi$
$368$		0			0
$369$	−22.1108	+	13.2769i	−1.15104	+	0.691169i
$370$		0			0
$371$	4.68372	−	3.65897i	0.243167	−	0.189964i
$372$		0			0
$373$	−10.3788			−0.537393			−0.268696	−	0.963225i	$-0.586593\pi$
$373$	−10.3788			−0.537393			−0.268696	+	0.963225i	$0.586593\pi$
$374$		0			0
$375$	0.462633	+	1.66912i	0.0238902	+	0.0861931i
$376$		0			0
$377$	−13.6312			−0.702045
$378$		0			0
$379$	−11.7360			−0.602837			−0.301419	−	0.953492i	$-0.597460\pi$
$379$	−11.7360			−0.602837			−0.301419	+	0.953492i	$0.597460\pi$
$380$		0			0
$381$	7.27825	+	26.2591i	0.372876	+	1.34529i
$382$		0			0
$383$	11.9198			0.609073			0.304536	−	0.952501i	$-0.401499\pi$
$383$	11.9198			0.609073			0.304536	+	0.952501i	$0.401499\pi$
$384$		0			0
$385$	0.408833	−	0.319384i	0.0208360	−	0.0162773i
$386$		0			0
$387$	−1.14351	+	0.686646i	−0.0581278	+	0.0349042i
$388$		0			0
$389$		−	10.4525i		−	0.529965i	−0.964253	−	0.264982i	$-0.914634\pi$
$389$		−	10.4525i		−	0.529965i	0.964253	−	0.264982i	$-0.0853662\pi$
$390$		0			0
$391$		−	19.5527i		−	0.988822i
$392$		0			0
$393$	10.2711	+	37.0567i	0.518106	+	1.86927i
$394$		0			0
$395$	−9.71013			−0.488570
$396$		0			0
$397$		−	19.7169i		−	0.989563i	−0.869017	−	0.494781i	$-0.835248\pi$
$397$		−	19.7169i		−	0.989563i	0.869017	−	0.494781i	$-0.164752\pi$
$398$		0			0
$399$	10.6544	+	7.88693i	0.533385	+	0.394841i
$400$		0			0
$401$		−	11.2315i		−	0.560872i	−0.959873	−	0.280436i	$-0.909521\pi$
$401$		−	11.2315i		−	0.560872i	0.959873	−	0.280436i	$-0.0904790\pi$
$402$		0			0
$403$	11.0937			0.552615
$404$		0			0
$405$	−4.22977	+	7.94412i	−0.210179	+	0.394746i
$406$		0			0
$407$			1.44079i			0.0714175i
$408$		0			0
$409$			1.61432i			0.0798229i	0.999203	+	0.0399114i	$0.0127076\pi$
$409$			1.61432i			0.0798229i	−0.999203	+	0.0399114i	$0.987292\pi$
$410$		0			0
$411$	10.9309	−	3.02972i	0.539179	−	0.149445i
$412$		0			0
$413$	−6.69045	−	8.56422i	−0.329216	−	0.421418i
$414$		0			0
$415$	5.51467			0.270705
$416$		0			0
$417$	0.705345	−	0.195501i	0.0345409	−	0.00957374i
$418$		0			0
$419$	−29.8474			−1.45814			−0.729071	−	0.684438i	$-0.760048\pi$
$419$	−29.8474			−1.45814			−0.729071	+	0.684438i	$0.760048\pi$
$420$		0			0
$421$	−6.87115			−0.334879			−0.167440	−	0.985882i	$-0.553550\pi$
$421$	−6.87115			−0.334879			−0.167440	+	0.985882i	$0.553550\pi$
$422$		0			0
$423$	−13.9800	+	8.39460i	−0.679730	+	0.408159i
$424$		0			0
$425$	−3.36987			−0.163463
$426$		0			0
$427$	2.58558	−	2.01988i	0.125125	−	0.0977489i
$428$		0			0
$429$	−0.215651	−	0.778042i	−0.0104117	−	0.0375642i
$430$		0			0
$431$		−	35.3572i		−	1.70310i	−0.524275	−	0.851549i	$-0.675664\pi$
$431$		−	35.3572i		−	1.70310i	0.524275	−	0.851549i	$-0.324336\pi$
$432$		0			0
$433$			17.5668i			0.844207i	0.906548	+	0.422104i	$0.138708\pi$
$433$			17.5668i			0.844207i	−0.906548	+	0.422104i	$0.861292\pi$
$434$		0			0
$435$	−9.57100	+	2.65280i	−0.458894	+	0.127192i
$436$		0			0
$437$	16.7839			0.802882
$438$		0			0
$439$			5.33236i			0.254500i	0.991871	+	0.127250i	$0.0406150\pi$
$439$			5.33236i			0.254500i	−0.991871	+	0.127250i	$0.959385\pi$
$440$		0			0
$441$	−6.13219	−	20.0847i	−0.292009	−	0.956416i
$442$		0			0
$443$			8.57620i			0.407468i	0.979026	+	0.203734i	$0.0653077\pi$
$443$			8.57620i			0.407468i	−0.979026	+	0.203734i	$0.934692\pi$
$444$		0			0
$445$	−2.30915			−0.109464
$446$		0			0
$447$	40.5167	−	11.2301i	1.91638	−	0.531164i
$448$		0			0
$449$		−	5.56437i		−	0.262599i	−0.991343	−	0.131299i	$-0.958085\pi$
$449$		−	5.56437i		−	0.262599i	0.991343	−	0.131299i	$-0.0419149\pi$
$450$		0			0
$451$		−	1.68574i		−	0.0793786i
$452$		0			0
$453$	9.23552	+	33.3206i	0.433923	+	1.56554i
$454$		0			0
$455$	−4.95637	+	3.87196i	−0.232358	+	0.181520i
$456$		0			0
$457$	−18.4095			−0.861160			−0.430580	−	0.902552i	$-0.641691\pi$
$457$	−18.4095			−0.861160			−0.430580	+	0.902552i	$0.641691\pi$
$458$		0			0
$459$	−12.6964	−	12.0588i	−0.592618	−	0.562856i
$460$		0			0
$461$	−12.2709			−0.571511			−0.285755	−	0.958303i	$-0.592244\pi$
$461$	−12.2709			−0.571511			−0.285755	+	0.958303i	$0.592244\pi$
$462$		0			0
$463$	21.6465			1.00600			0.502999	−	0.864287i	$-0.332230\pi$
$463$	21.6465			1.00600			0.502999	+	0.864287i	$0.332230\pi$
$464$		0			0
$465$	7.78927	−	2.15896i	0.361219	−	0.100119i
$466$		0			0
$467$	13.4338			0.621644			0.310822	−	0.950468i	$-0.399396\pi$
$467$	13.4338			0.621644			0.310822	+	0.950468i	$0.399396\pi$
$468$		0			0
$469$	11.8296	+	15.1427i	0.546241	+	0.699224i
$470$		0			0
$471$	−10.7858	+	2.98952i	−0.496985	+	0.137750i
$472$		0			0
$473$		−	0.0871820i		−	0.00400863i
$474$		0			0
$475$		−	2.89268i		−	0.132725i
$476$		0			0
$477$	−3.46935	−	5.77770i	−0.158851	−	0.264543i
$478$		0			0
$479$	−5.59941			−0.255844			−0.127922	−	0.991784i	$-0.540831\pi$
$479$	−5.59941			−0.255844			−0.127922	+	0.991784i	$0.540831\pi$
$480$		0			0
$481$		−	17.4671i		−	0.796430i
$482$		0			0
$483$	−21.3708	−	15.8198i	−0.972404	−	0.719826i
$484$		0			0
$485$			6.59507i			0.299467i
$486$		0			0
$487$	−32.9111			−1.49134			−0.745671	−	0.666314i	$-0.767871\pi$
$487$	−32.9111			−1.49134			−0.745671	+	0.666314i	$0.767871\pi$
$488$		0			0
$489$	4.68924	+	16.9182i	0.212055	+	0.765069i
$490$		0			0
$491$		−	6.88974i		−	0.310929i	−0.987841	−	0.155465i	$-0.950313\pi$
$491$		−	6.88974i		−	0.310929i	0.987841	−	0.155465i	$-0.0496875\pi$
$492$		0			0
$493$		−	19.3234i		−	0.870280i
$494$		0			0
$495$	−0.302833	−	0.504324i	−0.0136113	−	0.0226677i
$496$		0			0
$497$	17.1270	−	13.3798i	0.768251	−	0.600165i
$498$		0			0
$499$	24.9765			1.11810			0.559051	−	0.829133i	$-0.311166\pi$
$499$	24.9765			1.11810			0.559051	+	0.829133i	$0.311166\pi$
$500$		0			0
$501$	−10.0882	−	36.3971i	−0.450709	−	1.62610i
$502$		0			0
$503$	20.4820			0.913246			0.456623	−	0.889660i	$-0.349059\pi$
$503$	20.4820			0.913246			0.456623	+	0.889660i	$0.349059\pi$
$504$		0			0
$505$	0.709225			0.0315601
$506$		0			0
$507$	−3.39984	−	12.2662i	−0.150992	−	0.544762i
$508$		0			0
$509$	20.1419			0.892773			0.446386	−	0.894840i	$-0.352711\pi$
$509$	20.1419			0.892773			0.446386	+	0.894840i	$0.352711\pi$
$510$		0			0
$511$	23.9862	−	18.7382i	1.06109	−	0.828930i
$512$		0			0
$513$	10.3512	−	10.8985i	0.457015	−	0.481181i
$514$		0			0
$515$		−	11.6159i		−	0.511859i
$516$		0			0
$517$		−	1.06584i		−	0.0468758i
$518$		0			0
$519$	7.40160	+	26.7041i	0.324894	+	1.17218i
$520$		0			0
$521$	14.1611			0.620410			0.310205	−	0.950670i	$-0.399602\pi$
$521$	14.1611			0.620410			0.310205	+	0.950670i	$0.399602\pi$
$522$		0			0
$523$			7.85547i			0.343496i	0.985141	+	0.171748i	$0.0549414\pi$
$523$			7.85547i			0.343496i	−0.985141	+	0.171748i	$0.945059\pi$
$524$		0			0
$525$	−2.72652	+	3.68322i	−0.118995	+	0.160749i
$526$		0			0
$527$			15.7261i			0.685042i
$528$		0			0
$529$	−10.6655			−0.463719
$530$		0			0
$531$	−10.5646	+	6.34374i	−0.458463	+	0.275295i
$532$		0			0
$533$			20.4366i			0.885209i
$534$		0			0
$535$			6.85555i			0.296391i
$536$		0			0
$537$	−21.1587	+	5.86457i	−0.913064	+	0.253075i
$538$		0			0
$539$	1.33180	+	0.332189i	0.0573649	+	0.0143084i
$540$		0			0
$541$	12.9291			0.555866			0.277933	−	0.960600i	$-0.410351\pi$
$541$	12.9291			0.555866			0.277933	+	0.960600i	$0.410351\pi$
$542$		0			0
$543$	20.2756	−	5.61980i	0.870108	−	0.241169i
$544$		0			0
$545$	−18.3995			−0.788146
$546$		0			0
$547$	1.68245			0.0719366			0.0359683	−	0.999353i	$-0.488548\pi$
$547$	1.68245			0.0719366			0.0359683	+	0.999353i	$0.488548\pi$
$548$		0			0
$549$	−1.91521	−	3.18950i	−0.0817390	−	0.136124i
$550$		0			0
$551$	16.5870			0.706631
$552$		0			0
$553$	−15.8158	−	20.2452i	−0.672554	−	0.860914i
$554$		0			0
$555$	−3.39930	−	12.2643i	−0.144292	−	0.520589i
$556$		0			0
$557$		−	44.7979i		−	1.89815i	−0.315055	−	0.949074i	$-0.602023\pi$
$557$		−	44.7979i		−	1.89815i	0.315055	−	0.949074i	$-0.397977\pi$
$558$		0			0
$559$			1.05693i			0.0447032i
$560$		0			0
$561$	1.10294	−	0.305702i	0.0465660	−	0.0129068i
$562$		0			0
$563$	15.5205			0.654112			0.327056	−	0.945005i	$-0.393943\pi$
$563$	15.5205			0.654112			0.327056	+	0.945005i	$0.393943\pi$
$564$		0			0
$565$		−	8.13112i		−	0.342079i
$566$		0			0
$567$	−23.4526	+	4.12039i	−0.984915	+	0.173040i
$568$		0			0
$569$			12.3342i			0.517077i	0.966001	+	0.258538i	$0.0832409\pi$
$569$			12.3342i			0.517077i	−0.966001	+	0.258538i	$0.916759\pi$
$570$		0			0
$571$	29.9050			1.25149			0.625743	−	0.780029i	$-0.284796\pi$
$571$	29.9050			1.25149			0.625743	+	0.780029i	$0.284796\pi$
$572$		0			0
$573$	31.9572	−	8.85761i	1.33503	−	0.370032i
$574$		0			0
$575$			5.80220i			0.241968i
$576$		0			0
$577$			21.8516i			0.909695i	0.890569	+	0.454847i	$0.150306\pi$
$577$			21.8516i			0.909695i	−0.890569	+	0.454847i	$0.849694\pi$
$578$		0			0
$579$	−10.0117	−	36.1211i	−0.416073	−	1.50114i
$580$		0			0
$581$	8.98224	+	11.4979i	0.372646	+	0.477011i
$582$		0			0
$583$	0.440496			0.0182435
$584$		0			0
$585$	3.67131	+	6.11403i	0.151790	+	0.252784i
$586$		0			0
$587$	2.56025			0.105673			0.0528363	−	0.998603i	$-0.483174\pi$
$587$	2.56025			0.105673			0.0528363	+	0.998603i	$0.483174\pi$
$588$		0			0
$589$	−13.4992			−0.556225
$590$		0			0
$591$	36.4939	−	10.1151i	1.50116	−	0.416078i
$592$		0			0
$593$	14.3072			0.587528			0.293764	−	0.955878i	$-0.405092\pi$
$593$	14.3072			0.587528			0.293764	+	0.955878i	$0.405092\pi$
$594$		0			0
$595$	−5.48881	−	7.02604i	−0.225019	−	0.288040i
$596$		0			0
$597$	8.87755	−	2.46060i	0.363334	−	0.100706i
$598$		0			0
$599$		−	25.7961i		−	1.05400i	−0.849866	−	0.526999i	$-0.823317\pi$
$599$		−	25.7961i		−	1.05400i	0.849866	−	0.526999i	$-0.176683\pi$
$600$		0			0
$601$		−	7.10158i		−	0.289679i	−0.989455	−	0.144840i	$-0.953733\pi$
$601$		−	7.10158i		−	0.289679i	0.989455	−	0.144840i	$-0.0462667\pi$
$602$		0			0
$603$	18.6796	−	11.2166i	0.760691	−	0.456774i
$604$		0			0
$605$	−10.9615			−0.445650
$606$		0			0
$607$			37.7229i			1.53112i	0.643362	+	0.765562i	$0.277539\pi$
$607$			37.7229i			1.53112i	−0.643362	+	0.765562i	$0.722461\pi$
$608$		0			0
$609$	−21.1201	−	15.6343i	−0.855831	−	0.633532i
$610$		0			0
$611$			12.9215i			0.522747i
$612$		0			0
$613$	40.8785			1.65107			0.825533	−	0.564353i	$-0.190874\pi$
$613$	40.8785			1.65107			0.825533	+	0.564353i	$0.190874\pi$
$614$		0			0
$615$	3.97722	+	14.3493i	0.160377	+	0.578620i
$616$		0			0
$617$		−	12.6969i		−	0.511158i	−0.966788	−	0.255579i	$-0.917734\pi$
$617$		−	12.6969i		−	0.511158i	0.966788	−	0.255579i	$-0.0822662\pi$
$618$		0			0
$619$			48.5788i			1.95255i	0.216545	+	0.976273i	$0.430521\pi$
$619$			48.5788i			1.95255i	−0.216545	+	0.976273i	$0.569479\pi$
$620$		0			0
$621$	−20.7626	+	21.8605i	−0.833176	+	0.877233i
$622$		0			0
$623$	−3.76112	−	4.81449i	−0.150686	−	0.192888i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	0.262412	+	0.946753i	0.0104797	+	0.0378097i
$628$		0			0
$629$	24.7609			0.987284
$630$		0			0
$631$	2.07013			0.0824106			0.0412053	−	0.999151i	$-0.486880\pi$
$631$	2.07013			0.0824106			0.0412053	+	0.999151i	$0.486880\pi$
$632$		0			0
$633$	3.11338	+	11.2327i	0.123746	+	0.446460i
$634$		0			0
$635$	15.7323			0.624315
$636$		0			0
$637$	−16.1458	−	4.02720i	−0.639718	−	0.159564i
$638$		0			0
$639$	−12.6864	−	21.1274i	−0.501866	−	0.835785i
$640$		0			0
$641$			21.5974i			0.853045i	0.904477	+	0.426523i	$0.140262\pi$
$641$			21.5974i			0.853045i	−0.904477	+	0.426523i	$0.859738\pi$
$642$		0			0
$643$		−	37.6759i		−	1.48579i	−0.669407	−	0.742896i	$-0.733452\pi$
$643$		−	37.6759i		−	1.48579i	0.669407	−	0.742896i	$-0.266548\pi$
$644$		0			0
$645$	0.205691	+	0.742107i	0.00809906	+	0.0292204i
$646$		0			0
$647$	−30.7561			−1.20915			−0.604574	−	0.796549i	$-0.706657\pi$
$647$	−30.7561			−1.20915			−0.604574	+	0.796549i	$0.706657\pi$
$648$		0			0
$649$		−	0.805451i		−	0.0316167i
$650$		0			0
$651$	17.1884	+	12.7238i	0.673667	+	0.498685i
$652$		0			0
$653$		−	1.26896i		−	0.0496583i	−0.999692	−	0.0248292i	$-0.992096\pi$
$653$		−	1.26896i		−	0.0496583i	0.999692	−	0.0248292i	$-0.00790418\pi$
$654$		0			0
$655$	22.2013			0.867478
$656$		0			0
$657$	−17.7672	−	29.5886i	−0.693163	−	1.15436i
$658$		0			0
$659$			33.1535i			1.29148i	0.763559	+	0.645738i	$0.223450\pi$
$659$			33.1535i			1.29148i	−0.763559	+	0.645738i	$0.776550\pi$
$660$		0			0
$661$		−	6.99036i		−	0.271894i	−0.990716	−	0.135947i	$-0.956592\pi$
$661$		−	6.99036i		−	0.271894i	0.990716	−	0.135947i	$-0.0434076\pi$
$662$		0			0
$663$	−13.3711	+	3.70609i	−0.519292	+	0.143933i
$664$		0			0
$665$	6.03110	−	4.71156i	0.233876	−	0.182706i
$666$		0			0
$667$	−33.2707			−1.28825
$668$		0			0
$669$	14.8590	−	4.11850i	0.574484	−	0.159230i
$670$		0			0
$671$	0.243170			0.00938747
$672$		0			0
$673$	−12.7243			−0.490484			−0.245242	−	0.969462i	$-0.578867\pi$
$673$	−12.7243			−0.490484			−0.245242	+	0.969462i	$0.578867\pi$
$674$		0			0
$675$	3.76763	+	3.57841i	0.145016	+	0.137733i
$676$		0			0
$677$	17.7083			0.680585			0.340293	−	0.940320i	$-0.389474\pi$
$677$	17.7083			0.680585			0.340293	+	0.940320i	$0.389474\pi$
$678$		0			0
$679$	−13.7504	+	10.7420i	−0.527694	+	0.412239i
$680$		0			0
$681$	12.0939	+	43.6334i	0.463440	+	1.67204i
$682$		0			0
$683$			15.5608i			0.595416i	0.954657	+	0.297708i	$0.0962221\pi$
$683$			15.5608i			0.595416i	−0.954657	+	0.297708i	$0.903778\pi$
$684$		0			0
$685$		−	6.54886i		−	0.250219i
$686$		0			0
$687$	−29.8707	+	8.27928i	−1.13964	+	0.315874i
$688$		0			0
$689$	−5.34023			−0.203447
$690$		0			0
$691$			44.5595i			1.69512i	0.530696	+	0.847562i	$0.321931\pi$
$691$			44.5595i			1.69512i	−0.530696	+	0.847562i	$0.678069\pi$
$692$		0			0
$693$	0.558243	−	1.45283i	0.0212059	−	0.0551885i
$694$		0			0
$695$		−	0.422584i		−	0.0160295i
$696$		0			0
$697$	−28.9705			−1.09734
$698$		0			0
$699$	−43.8739	+	12.1606i	−1.65946	+	0.459955i
$700$		0			0
$701$			31.5905i			1.19316i	0.802555	+	0.596578i	$0.203473\pi$
$701$			31.5905i			1.19316i	−0.802555	+	0.596578i	$0.796527\pi$
$702$		0			0
$703$			21.2546i			0.801633i
$704$		0			0
$705$	2.51467	+	9.07264i	0.0947081	+	0.341695i
$706$		0			0
$707$	1.15518	+	1.47870i	0.0434450	+	0.0556124i
$708$		0			0
$709$	−23.2268			−0.872302			−0.436151	−	0.899874i	$-0.643659\pi$
$709$	−23.2268			−0.872302			−0.436151	+	0.899874i	$0.643659\pi$
$710$		0			0
$711$	−24.9739	+	14.9961i	−0.936594	+	0.562399i
$712$		0			0
$713$	27.0770			1.01404
$714$		0			0
$715$	−0.466139			−0.0174326
$716$		0			0
$717$	30.6240	−	8.48809i	1.14368	−	0.316994i
$718$		0			0
$719$	31.0899			1.15946			0.579730	−	0.814809i	$-0.303158\pi$
$719$	31.0899			1.15946			0.579730	+	0.814809i	$0.303158\pi$
$720$		0			0
$721$	24.2187	−	18.9199i	0.901951	−	0.704613i
$722$		0			0
$723$	−37.3477	+	10.3517i	−1.38897	+	0.384984i
$724$		0			0
$725$			5.73415i			0.212961i
$726$		0			0
$727$			25.9826i			0.963641i	0.876270	+	0.481821i	$0.160024\pi$
$727$			25.9826i			0.963641i	−0.876270	+	0.481821i	$0.839976\pi$
$728$		0			0
$729$	1.39001	+	26.9642i	0.0514819	+	0.998674i
$730$		0			0
$731$	−1.49828			−0.0554158
$732$		0			0
$733$		−	2.39842i		−	0.0885875i	−0.999019	−	0.0442938i	$-0.985896\pi$
$733$		−	2.39842i		−	0.0885875i	0.999019	−	0.0442938i	$-0.0141038\pi$
$734$		0			0
$735$	−12.1203	+	0.314510i	−0.447063	+	0.0116009i
$736$		0			0
$737$			1.42414i			0.0524590i
$738$		0			0
$739$	19.6322			0.722183			0.361091	−	0.932530i	$-0.382404\pi$
$739$	19.6322			0.722183			0.361091	+	0.932530i	$0.382404\pi$
$740$		0			0
$741$	−3.18128	−	11.4777i	−0.116867	−	0.421643i
$742$		0			0
$743$			23.0639i			0.846132i	0.906099	+	0.423066i	$0.139046\pi$
$743$			23.0639i			0.846132i	−0.906099	+	0.423066i	$0.860954\pi$
$744$		0			0
$745$		−	24.2743i		−	0.889341i
$746$		0			0
$747$	14.1834	−	8.51675i	0.518944	−	0.311612i
$748$		0			0
$749$	−14.2935	+	11.1662i	−0.522274	+	0.408006i
$750$		0			0
$751$	−5.54259			−0.202252			−0.101126	−	0.994874i	$-0.532245\pi$
$751$	−5.54259			−0.202252			−0.101126	+	0.994874i	$0.532245\pi$
$752$		0			0
$753$	9.90819	+	35.7476i	0.361075	+	1.30271i
$754$		0			0
$755$	19.9630			0.726527
$756$		0			0
$757$	20.0408			0.728397			0.364198	−	0.931321i	$-0.381343\pi$
$757$	20.0408			0.728397			0.364198	+	0.931321i	$0.381343\pi$
$758$		0			0
$759$	−0.526353	−	1.89902i	−0.0191054	−	0.0689301i
$760$		0			0
$761$	23.1864			0.840507			0.420254	−	0.907407i	$-0.361941\pi$
$761$	23.1864			0.840507			0.420254	+	0.907407i	$0.361941\pi$
$762$		0			0
$763$	−29.9688	−	38.3621i	−1.08494	−	1.38880i
$764$		0			0
$765$	−8.66712	+	5.20437i	−0.313360	+	0.188164i
$766$		0			0
$767$			9.76466i			0.352581i
$768$		0			0
$769$			8.09763i			0.292008i	0.989284	+	0.146004i	$0.0466412\pi$
$769$			8.09763i			0.292008i	−0.989284	+	0.146004i	$0.953359\pi$
$770$		0			0
$771$	1.93879	+	6.99494i	0.0698240	+	0.251917i
$772$		0			0
$773$	−24.7010			−0.888432			−0.444216	−	0.895920i	$-0.646518\pi$
$773$	−24.7010			−0.888432			−0.444216	+	0.895920i	$0.646518\pi$
$774$		0			0
$775$		−	4.66669i		−	0.167632i
$776$		0			0
$777$	20.0337	−	27.0633i	0.718706	−	0.970891i
$778$		0			0
$779$		−	24.8681i		−	0.890992i
$780$		0			0
$781$	1.61077			0.0576377
$782$		0			0
$783$	−20.5191	+	21.6041i	−0.733293	+	0.772069i
$784$		0			0
$785$			6.46198i			0.230638i
$786$		0			0
$787$			6.04084i			0.215333i	0.994187	+	0.107666i	$0.0343378\pi$
$787$			6.04084i			0.215333i	−0.994187	+	0.107666i	$0.965662\pi$
$788$		0			0
$789$	33.9090	−	9.39858i	1.20719	−	0.334598i
$790$		0			0
$791$	16.9530	−	13.2439i	0.602780	−	0.470898i
$792$		0			0
$793$	−2.94800			−0.104687
$794$		0			0
$795$	−3.74957	+	1.03927i	−0.132984	+	0.0368592i
$796$		0			0
$797$	−45.6745			−1.61787			−0.808937	−	0.587895i	$-0.799957\pi$
$797$	−45.6745			−1.61787			−0.808937	+	0.587895i	$0.799957\pi$
$798$		0			0
$799$	−18.3172			−0.648016
$800$		0			0
$801$	−5.93901	+	3.56621i	−0.209845	+	0.126006i
$802$		0			0
$803$	2.25586			0.0796075
$804$		0			0
$805$	−12.0973	+	9.45056i	−0.426375	+	0.333089i
$806$		0			0
$807$	12.2137	+	44.0656i	0.429942	+	1.55118i
$808$		0			0
$809$		−	6.65382i		−	0.233936i	−0.993136	−	0.116968i	$-0.962683\pi$
$809$		−	6.65382i		−	0.233936i	0.993136	−	0.116968i	$-0.0373175\pi$
$810$		0			0
$811$		−	13.1472i		−	0.461662i	−0.972994	−	0.230831i	$-0.925856\pi$
$811$		−	13.1472i		−	0.461662i	0.972994	−	0.230831i	$-0.0741445\pi$
$812$		0			0
$813$	−2.66158	+	0.737712i	−0.0933456	+	0.0258727i
$814$		0			0
$815$	10.1360			0.355049
$816$		0			0
$817$		−	1.28611i		−	0.0449953i
$818$		0			0
$819$	−6.76771	+	17.6130i	−0.236483	+	0.615448i
$820$		0			0
$821$			39.2991i			1.37155i	0.727814	+	0.685775i	$0.240536\pi$
$821$			39.2991i			1.37155i	−0.727814	+	0.685775i	$0.759464\pi$
$822$		0			0
$823$	40.2932			1.40453			0.702266	−	0.711915i	$-0.252172\pi$
$823$	40.2932			1.40453			0.702266	+	0.711915i	$0.252172\pi$
$824$		0			0
$825$	−0.327293	+	0.0907162i	−0.0113949	+	0.00315833i
$826$		0			0
$827$			31.2044i			1.08508i	0.840029	+	0.542541i	$0.182538\pi$
$827$			31.2044i			1.08508i	−0.840029	+	0.542541i	$0.817462\pi$
$828$		0			0
$829$			23.3307i			0.810308i	0.914248	+	0.405154i	$0.132782\pi$
$829$			23.3307i			0.810308i	−0.914248	+	0.405154i	$0.867218\pi$
$830$		0			0
$831$	1.51387	+	5.46185i	0.0525155	+	0.189470i
$832$		0			0
$833$	5.70888	−	22.8879i	0.197801	−	0.793018i
$834$		0			0
$835$	−21.8061			−0.754632
$836$		0			0
$837$	16.6993	−	17.5823i	0.577212	−	0.607734i
$838$		0			0
$839$	−55.7694			−1.92537			−0.962687	−	0.270618i	$-0.912772\pi$
$839$	−55.7694			−1.92537			−0.962687	+	0.270618i	$0.912772\pi$
$840$		0			0
$841$	−3.88047			−0.133809
$842$		0			0
$843$	41.0859	−	11.3878i	1.41507	−	0.392217i
$844$		0			0
$845$	−7.34890			−0.252810
$846$		0			0
$847$	−17.8541	−	22.8544i	−0.613472	−	0.785285i
$848$		0			0
$849$	10.6612	−	2.95497i	0.365891	−	0.101414i
$850$		0			0
$851$		−	42.6330i		−	1.46144i
$852$		0			0
$853$		−	30.0999i		−	1.03060i	−0.857010	−	0.515300i	$-0.827681\pi$
$853$		−	30.0999i		−	1.03060i	0.857010	−	0.515300i	$-0.172319\pi$
$854$		0			0
$855$	−4.46739	−	7.43979i	−0.152782	−	0.254436i
$856$		0			0
$857$	51.3831			1.75521			0.877607	−	0.479382i	$-0.159139\pi$
$857$	51.3831			1.75521			0.877607	+	0.479382i	$0.159139\pi$
$858$		0			0
$859$			15.3624i			0.524160i	0.965046	+	0.262080i	$0.0844084\pi$
$859$			15.3624i			0.524160i	−0.965046	+	0.262080i	$0.915592\pi$
$860$		0			0
$861$	−23.4397	+	31.6643i	−0.798822	+	1.07912i
$862$		0			0
$863$			21.8347i			0.743263i	0.928380	+	0.371632i	$0.121202\pi$
$863$			21.8347i			0.743263i	−0.928380	+	0.371632i	$0.878798\pi$
$864$		0			0
$865$	15.9989			0.543978
$866$		0			0
$867$	2.61107	+	9.42044i	0.0886767	+	0.319935i
$868$		0			0
$869$		−	1.90403i		−	0.0645898i
$870$		0			0
$871$		−	17.2652i		−	0.585010i
$872$		0			0
$873$	10.1853	+	16.9621i	0.344720	+	0.574082i
$874$		0			0
$875$	1.62879	+	2.08496i	0.0550631	+	0.0704844i
$876$		0			0
$877$	24.8639			0.839593			0.419796	−	0.907618i	$-0.362101\pi$
$877$	24.8639			0.839593			0.419796	+	0.907618i	$0.362101\pi$
$878$		0			0
$879$	4.49267	+	16.2090i	0.151534	+	0.546717i
$880$		0			0
$881$	17.5148			0.590088			0.295044	−	0.955484i	$-0.404666\pi$
$881$	17.5148			0.590088			0.295044	+	0.955484i	$0.404666\pi$
$882$		0			0
$883$	−45.7821			−1.54069			−0.770344	−	0.637628i	$-0.779916\pi$
$883$	−45.7821			−1.54069			−0.770344	+	0.637628i	$0.779916\pi$
$884$		0			0
$885$	1.90032	+	6.85613i	0.0638786	+	0.230466i
$886$		0			0
$887$	31.8505			1.06944			0.534718	−	0.845031i	$-0.320418\pi$
$887$	31.8505			1.06944			0.534718	+	0.845031i	$0.320418\pi$
$888$		0			0
$889$	25.6245	+	32.8011i	0.859419	+	1.10011i
$890$		0			0
$891$	−1.55774	−	0.829403i	−0.0521862	−	0.0277861i
$892$		0			0
$893$		−	15.7233i		−	0.526162i
$894$		0			0
$895$			12.6765i			0.423729i
$896$		0			0
$897$	6.38110	+	23.0222i	0.213059	+	0.768690i
$898$		0			0
$899$	26.7595			0.892478
$900$		0			0
$901$		−	7.57020i		−	0.252200i
$902$		0			0
$903$	−1.21223	+	1.63759i	−0.0403407	+	0.0544956i
$904$		0			0
$905$		−	12.1474i		−	0.403795i
$906$		0			0
$907$	−11.0152			−0.365754			−0.182877	−	0.983136i	$-0.558541\pi$
$907$	−11.0152			−0.365754			−0.182877	+	0.983136i	$0.558541\pi$
$908$		0			0
$909$	1.82409	−	1.09531i	0.0605011	−	0.0363293i
$910$		0			0
$911$		−	23.6413i		−	0.783272i	−0.920120	−	0.391636i	$-0.871909\pi$
$911$		−	23.6413i		−	0.783272i	0.920120	−	0.391636i	$-0.128091\pi$
$912$		0			0
$913$			1.08135i			0.0357876i
$914$		0			0
$915$	−2.06990	+	0.573717i	−0.0684288	+	0.0189665i
$916$		0			0
$917$	36.1613	+	46.2888i	1.19415	+	1.52859i
$918$		0			0
$919$	−24.0966			−0.794874			−0.397437	−	0.917629i	$-0.630100\pi$
$919$	−24.0966			−0.794874			−0.397437	+	0.917629i	$0.630100\pi$
$920$		0			0
$921$	−36.9047	+	10.2289i	−1.21605	+	0.337054i
$922$		0			0
$923$	−19.5277			−0.642761
$924$		0			0
$925$	−7.34774			−0.241592
$926$		0			0
$927$	−17.9394	−	29.8755i	−0.589207	−	0.981239i
$928$		0			0
$929$	37.2676			1.22271			0.611355	−	0.791356i	$-0.290625\pi$
$929$	37.2676			1.22271			0.611355	+	0.791356i	$0.290625\pi$
$930$		0			0
$931$	19.6468	+	4.90046i	0.643898	+	0.160606i
$932$		0			0
$933$	7.22330	+	26.0608i	0.236480	+	0.853193i
$934$		0			0
$935$		−	0.660788i		−	0.0216101i
$936$		0			0
$937$			11.1767i			0.365126i	0.983194	+	0.182563i	$0.0584394\pi$
$937$			11.1767i			0.365126i	−0.983194	+	0.182563i	$0.941561\pi$
$938$		0			0
$939$	−1.73933	+	0.482092i	−0.0567609	+	0.0157325i
$940$		0			0
$941$	−25.8782			−0.843605			−0.421803	−	0.906688i	$-0.638602\pi$
$941$	−25.8782			−0.843605			−0.421803	+	0.906688i	$0.638602\pi$
$942$		0			0
$943$			49.8811i			1.62435i
$944$		0			0
$945$	−1.32416	+	13.6838i	−0.0430748	+	0.445134i
$946$		0			0
$947$			6.08802i			0.197834i	0.995096	+	0.0989170i	$0.0315378\pi$
$947$			6.08802i			0.197834i	−0.995096	+	0.0989170i	$0.968462\pi$
$948$		0			0
$949$	−27.3483			−0.887763
$950$		0			0
$951$	26.1766	−	7.25539i	0.848834	−	0.235272i
$952$		0			0
$953$		−	1.35040i		−	0.0437438i	−0.999761	−	0.0218719i	$-0.993037\pi$
$953$		−	1.35040i		−	0.0437438i	0.999761	−	0.0218719i	$-0.00696259\pi$
$954$		0			0
$955$		−	19.1461i		−	0.619553i
$956$		0			0
$957$	−0.520180	−	1.87675i	−0.0168150	−	0.0606666i
$958$		0			0
$959$	13.6541	−	10.6667i	0.440914	−	0.344446i
$960$		0			0
$961$	9.22204			0.297485
$962$		0			0
$963$	10.5876	+	17.6321i	0.341180	+	0.568186i
$964$		0			0
$965$	−21.6408			−0.696640
$966$		0			0
$967$	3.25722			0.104745			0.0523726	−	0.998628i	$-0.483322\pi$
$967$	3.25722			0.104745			0.0523726	+	0.998628i	$0.483322\pi$
$968$		0			0
$969$	16.2705	−	4.50972i	0.522685	−	0.144873i
$970$		0			0
$971$	13.6431			0.437827			0.218913	−	0.975744i	$-0.429749\pi$
$971$	13.6431			0.437827			0.218913	+	0.975744i	$0.429749\pi$
$972$		0			0
$973$	0.881070	−	0.688301i	0.0282458	−	0.0220659i
$974$		0			0
$975$	3.96785	−	1.09977i	0.127073	−	0.0352209i
$976$		0			0
$977$		−	43.7973i		−	1.40120i	−0.713555	−	0.700599i	$-0.752916\pi$
$977$		−	43.7973i		−	1.40120i	0.713555	−	0.700599i	$-0.247084\pi$
$978$		0			0
$979$		−	0.452795i		−	0.0144714i
$980$		0			0
$981$	−47.3223	+	28.4158i	−1.51089	+	0.907246i
$982$		0			0
$983$	9.68575			0.308927			0.154464	−	0.987998i	$-0.450635\pi$
$983$	9.68575			0.308927			0.154464	+	0.987998i	$0.450635\pi$
$984$		0			0
$985$		−	21.8641i		−	0.696649i
$986$		0			0
$987$	−14.8202	+	20.0204i	−0.471732	+	0.637256i
$988$		0			0
$989$			2.57971i			0.0820300i
$990$		0			0
$991$	−19.1780			−0.609210			−0.304605	−	0.952479i	$-0.598524\pi$
$991$	−19.1780			−0.609210			−0.304605	+	0.952479i	$0.598524\pi$
$992$		0			0
$993$	−14.4444	−	52.1135i	−0.458378	−	1.65377i
$994$		0			0
$995$		−	5.31869i		−	0.168614i
$996$		0			0
$997$		−	0.756399i		−	0.0239554i	−0.999928	−	0.0119777i	$-0.996187\pi$
$997$		−	0.756399i		−	0.0239554i	0.999928	−	0.0119777i	$-0.00381271\pi$
$998$		0			0
$999$	−27.6835	−	26.2932i	−0.875868	−	0.831880i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.f.k.881.7		16
3.2	odd	2		1680.2.f.l.881.9		16
4.3	odd	2		840.2.f.a.41.10	yes	16
7.6	odd	2		1680.2.f.l.881.10		16
12.11	even	2		840.2.f.b.41.8	yes	16
21.20	even	2	inner	1680.2.f.k.881.8		16
28.27	even	2		840.2.f.b.41.7	yes	16
84.83	odd	2		840.2.f.a.41.9	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.f.a.41.9	✓	16	84.83	odd	2
840.2.f.a.41.10	yes	16	4.3	odd	2
840.2.f.b.41.7	yes	16	28.27	even	2
840.2.f.b.41.8	yes	16	12.11	even	2
1680.2.f.k.881.7		16	1.1	even	1	trivial
1680.2.f.k.881.8		16	21.20	even	2	inner
1680.2.f.l.881.9		16	3.2	odd	2
1680.2.f.l.881.10		16	7.6	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.462633 - 1.66912i) q^{3} -1.00000 q^{5} +(-1.62879 - 2.08496i) q^{7} +(-2.57194 + 1.54438i) q^{9} +O(q^{10})\)	\(q+(-0.462633 - 1.66912i) q^{3} -1.00000 q^{5} +(-1.62879 - 2.08496i) q^{7} +(-2.57194 + 1.54438i) q^{9} -0.196087i q^{11} +2.37720i q^{13} +(0.462633 + 1.66912i) q^{15} -3.36987 q^{17} -2.89268i q^{19} +(-2.72652 + 3.68322i) q^{21} +5.80220i q^{23} +1.00000 q^{25} +(3.76763 + 3.57841i) q^{27} +5.73415i q^{29} -4.66669i q^{31} +(-0.327293 + 0.0907162i) q^{33} +(1.62879 + 2.08496i) q^{35} -7.34774 q^{37} +(3.96785 - 1.09977i) q^{39} +8.59692 q^{41} +0.444609 q^{43} +(2.57194 - 1.54438i) q^{45} +5.43557 q^{47} +(-1.69409 + 6.79191i) q^{49} +(1.55901 + 5.62473i) q^{51} +2.24643i q^{53} +0.196087i q^{55} +(-4.82823 + 1.33825i) q^{57} +4.10762 q^{59} +1.24011i q^{61} +(7.40912 + 2.84692i) q^{63} -2.37720i q^{65} -7.26282 q^{67} +(9.68459 - 2.68429i) q^{69} +8.21455i q^{71} +11.5044i q^{73} +(-0.462633 - 1.66912i) q^{75} +(-0.408833 + 0.319384i) q^{77} +9.71013 q^{79} +(4.22977 - 7.94412i) q^{81} -5.51467 q^{83} +3.36987 q^{85} +(9.57100 - 2.65280i) q^{87} +2.30915 q^{89} +(4.95637 - 3.87196i) q^{91} +(-7.78927 + 2.15896i) q^{93} +2.89268i q^{95} -6.59507i q^{97} +(0.302833 + 0.504324i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 16 * q - 16 * q^5 - 2 * q^7 - 2 * q^9	\( 16 q - 16 q^{5} - 2 q^{7} - 2 q^{9} + 2 q^{21} + 16 q^{25} + 6 q^{27} - 6 q^{33} + 2 q^{35} + 12 q^{37} - 6 q^{39} - 32 q^{41} - 32 q^{43} + 2 q^{45} + 4 q^{47} - 4 q^{49} - 6 q^{51} - 24 q^{59} + 4 q^{63} - 8 q^{69} + 32 q^{77} + 4 q^{79} - 6 q^{81} + 20 q^{83} + 6 q^{87} + 24 q^{89} - 20 q^{91} - 32 q^{93} + 58 q^{99}+O(q^{100}) \) 16 * q - 16 * q^5 - 2 * q^7 - 2 * q^9 + 2 * q^21 + 16 * q^25 + 6 * q^27 - 6 * q^33 + 2 * q^35 + 12 * q^37 - 6 * q^39 - 32 * q^41 - 32 * q^43 + 2 * q^45 + 4 * q^47 - 4 * q^49 - 6 * q^51 - 24 * q^59 + 4 * q^63 - 8 * q^69 + 32 * q^77 + 4 * q^79 - 6 * q^81 + 20 * q^83 + 6 * q^87 + 24 * q^89 - 20 * q^91 - 32 * q^93 + 58 * q^99

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