LMFDB - Embedded newform 1680.2.f.e.881.1

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:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	1680.881
Dual form			1680.2.f.e.881.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.61803 - 0.618034i) q^{3} -1.00000 q^{5} +(-2.61803 + 0.381966i) q^{7} +(2.23607 + 2.00000i) q^{9} +O(q^{10})$	\(q+(-1.61803 - 0.618034i) q^{3} -1.00000 q^{5} +(-2.61803 + 0.381966i) q^{7} +(2.23607 + 2.00000i) q^{9} -4.47214i q^{11} +1.23607i q^{13} +(1.61803 + 0.618034i) q^{15} -5.23607 q^{17} +8.47214i q^{19} +(4.47214 + 1.00000i) q^{21} +4.00000i q^{23} +1.00000 q^{25} +(-2.38197 - 4.61803i) q^{27} -7.70820i q^{29} -2.76393i q^{31} +(-2.76393 + 7.23607i) q^{33} +(2.61803 - 0.381966i) q^{35} +0.763932 q^{37} +(0.763932 - 2.00000i) q^{39} +2.47214 q^{41} +4.94427 q^{43} +(-2.23607 - 2.00000i) q^{45} -6.47214 q^{47} +(6.70820 - 2.00000i) q^{49} +(8.47214 + 3.23607i) q^{51} +0.472136i q^{53} +4.47214i q^{55} +(5.23607 - 13.7082i) q^{57} +4.47214 q^{59} -7.23607i q^{61} +(-6.61803 - 4.38197i) q^{63} -1.23607i q^{65} +12.0000 q^{67} +(2.47214 - 6.47214i) q^{69} +7.23607i q^{71} +11.2361i q^{73} +(-1.61803 - 0.618034i) q^{75} +(1.70820 + 11.7082i) q^{77} +8.94427 q^{79} +(1.00000 + 8.94427i) q^{81} +14.6525 q^{83} +5.23607 q^{85} +(-4.76393 + 12.4721i) q^{87} +5.52786 q^{89} +(-0.472136 - 3.23607i) q^{91} +(-1.70820 + 4.47214i) q^{93} -8.47214i q^{95} -0.763932i q^{97} +(8.94427 - 10.0000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 4 q^{5} - 6 q^{7}+O(q^{10}) $ 4 * q - 2 * q^3 - 4 * q^5 - 6 * q^7	$ 4 q - 2 q^{3} - 4 q^{5} - 6 q^{7} + 2 q^{15} - 12 q^{17} + 4 q^{25} - 14 q^{27} - 20 q^{33} + 6 q^{35} + 12 q^{37} + 12 q^{39} - 8 q^{41} - 16 q^{43} - 8 q^{47} + 16 q^{51} + 12 q^{57} - 22 q^{63} + 48 q^{67} - 8 q^{69} - 2 q^{75} - 20 q^{77} + 4 q^{81} - 4 q^{83} + 12 q^{85} - 28 q^{87} + 40 q^{89} + 16 q^{91} + 20 q^{93}+O(q^{100}) $ 4 * q - 2 * q^3 - 4 * q^5 - 6 * q^7 + 2 * q^15 - 12 * q^17 + 4 * q^25 - 14 * q^27 - 20 * q^33 + 6 * q^35 + 12 * q^37 + 12 * q^39 - 8 * q^41 - 16 * q^43 - 8 * q^47 + 16 * q^51 + 12 * q^57 - 22 * q^63 + 48 * q^67 - 8 * q^69 - 2 * q^75 - 20 * q^77 + 4 * q^81 - 4 * q^83 + 12 * q^85 - 28 * q^87 + 40 * q^89 + 16 * q^91 + 20 * q^93

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
$3$	−1.61803	−	0.618034i	−0.934172	−	0.356822i
$3$	−1.61803	−	0.618034i	−0.934172	−	0.356822i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−2.61803	+	0.381966i	−0.989524	+	0.144370i
$7$	−2.61803	+	0.381966i	−0.989524	+	0.144370i
$8$		0			0
$9$	2.23607	+	2.00000i	0.745356	+	0.666667i
$10$		0			0
$11$		−	4.47214i		−	1.34840i	−0.738549	−	0.674200i	$-0.764489\pi$
$11$		−	4.47214i		−	1.34840i	0.738549	−	0.674200i	$-0.235511\pi$
$12$		0			0
$13$			1.23607i			0.342824i	0.985199	+	0.171412i	$0.0548329\pi$
$13$			1.23607i			0.342824i	−0.985199	+	0.171412i	$0.945167\pi$
$14$		0			0
$15$	1.61803	+	0.618034i	0.417775	+	0.159576i
$16$		0			0
$17$	−5.23607			−1.26993			−0.634967	−	0.772540i	$-0.718986\pi$
$17$	−5.23607			−1.26993			−0.634967	+	0.772540i	$0.718986\pi$
$18$		0			0
$19$			8.47214i			1.94364i	0.235722	+	0.971821i	$0.424255\pi$
$19$			8.47214i			1.94364i	−0.235722	+	0.971821i	$0.575745\pi$
$20$		0			0
$21$	4.47214	+	1.00000i	0.975900	+	0.218218i
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−2.38197	−	4.61803i	−0.458410	−	0.888741i
$28$		0			0
$29$		−	7.70820i		−	1.43138i	−0.698419	−	0.715689i	$-0.746113\pi$
$29$		−	7.70820i		−	1.43138i	0.698419	−	0.715689i	$-0.253887\pi$
$30$		0			0
$31$		−	2.76393i		−	0.496417i	−0.968707	−	0.248208i	$-0.920158\pi$
$31$		−	2.76393i		−	0.496417i	0.968707	−	0.248208i	$-0.0798418\pi$
$32$		0			0
$33$	−2.76393	+	7.23607i	−0.481139	+	1.25964i
$34$		0			0
$35$	2.61803	−	0.381966i	0.442529	−	0.0645640i
$36$		0			0
$37$	0.763932			0.125590			0.0627948	−	0.998026i	$-0.479999\pi$
$37$	0.763932			0.125590			0.0627948	+	0.998026i	$0.479999\pi$
$38$		0			0
$39$	0.763932	−	2.00000i	0.122327	−	0.320256i
$40$		0			0
$41$	2.47214			0.386083			0.193041	−	0.981191i	$-0.438165\pi$
$41$	2.47214			0.386083			0.193041	+	0.981191i	$0.438165\pi$
$42$		0			0
$43$	4.94427			0.753994			0.376997	−	0.926214i	$-0.376957\pi$
$43$	4.94427			0.753994			0.376997	+	0.926214i	$0.376957\pi$
$44$		0			0
$45$	−2.23607	−	2.00000i	−0.333333	−	0.298142i
$46$		0			0
$47$	−6.47214			−0.944058			−0.472029	−	0.881583i	$-0.656478\pi$
$47$	−6.47214			−0.944058			−0.472029	+	0.881583i	$0.656478\pi$
$48$		0			0
$49$	6.70820	−	2.00000i	0.958315	−	0.285714i
$50$		0			0
$51$	8.47214	+	3.23607i	1.18634	+	0.453140i
$52$		0			0
$53$			0.472136i			0.0648529i	0.999474	+	0.0324264i	$0.0103235\pi$
$53$			0.472136i			0.0648529i	−0.999474	+	0.0324264i	$0.989677\pi$
$54$		0			0
$55$			4.47214i			0.603023i
$56$		0			0
$57$	5.23607	−	13.7082i	0.693534	−	1.81570i
$58$		0			0
$59$	4.47214			0.582223			0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214			0.582223			0.291111	+	0.956689i	$0.405975\pi$
$60$		0			0
$61$		−	7.23607i		−	0.926484i	−0.886232	−	0.463242i	$-0.846686\pi$
$61$		−	7.23607i		−	0.926484i	0.886232	−	0.463242i	$-0.153314\pi$
$62$		0			0
$63$	−6.61803	−	4.38197i	−0.833794	−	0.552076i
$64$		0			0
$65$		−	1.23607i		−	0.153315i
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		0			0
$69$	2.47214	−	6.47214i	0.297610	−	0.779154i
$70$		0			0
$71$			7.23607i			0.858763i	0.903123	+	0.429382i	$0.141268\pi$
$71$			7.23607i			0.858763i	−0.903123	+	0.429382i	$0.858732\pi$
$72$		0			0
$73$			11.2361i			1.31508i	0.753419	+	0.657541i	$0.228403\pi$
$73$			11.2361i			1.31508i	−0.753419	+	0.657541i	$0.771597\pi$
$74$		0			0
$75$	−1.61803	−	0.618034i	−0.186834	−	0.0713644i
$76$		0			0
$77$	1.70820	+	11.7082i	0.194668	+	1.33427i
$78$		0			0
$79$	8.94427			1.00631			0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427			1.00631			0.503155	+	0.864196i	$0.332173\pi$
$80$		0			0
$81$	1.00000	+	8.94427i	0.111111	+	0.993808i
$82$		0			0
$83$	14.6525			1.60832			0.804159	−	0.594414i	$-0.202616\pi$
$83$	14.6525			1.60832			0.804159	+	0.594414i	$0.202616\pi$
$84$		0			0
$85$	5.23607			0.567931
$86$		0			0
$87$	−4.76393	+	12.4721i	−0.510747	+	1.33715i
$88$		0			0
$89$	5.52786			0.585952			0.292976	−	0.956120i	$-0.405354\pi$
$89$	5.52786			0.585952			0.292976	+	0.956120i	$0.405354\pi$
$90$		0			0
$91$	−0.472136	−	3.23607i	−0.0494933	−	0.339232i
$92$		0			0
$93$	−1.70820	+	4.47214i	−0.177132	+	0.463739i
$94$		0			0
$95$		−	8.47214i		−	0.869223i
$96$		0			0
$97$		−	0.763932i		−	0.0775655i	−0.999248	−	0.0387828i	$-0.987652\pi$
$97$		−	0.763932i		−	0.0775655i	0.999248	−	0.0387828i	$-0.0123480\pi$
$98$		0			0
$99$	8.94427	−	10.0000i	0.898933	−	1.00504i
$100$		0			0
$101$	12.4721			1.24102			0.620512	−	0.784197i	$-0.286925\pi$
$101$	12.4721			1.24102			0.620512	+	0.784197i	$0.286925\pi$
$102$		0			0
$103$		−	14.6525i		−	1.44375i	−0.692023	−	0.721876i	$-0.743280\pi$
$103$		−	14.6525i		−	1.44375i	0.692023	−	0.721876i	$-0.256720\pi$
$104$		0			0
$105$	−4.47214	−	1.00000i	−0.436436	−	0.0975900i
$106$		0			0
$107$		−	11.4164i		−	1.10367i	−0.833955	−	0.551833i	$-0.813929\pi$
$107$		−	11.4164i		−	1.10367i	0.833955	−	0.551833i	$-0.186071\pi$
$108$		0			0
$109$	4.47214			0.428353			0.214176	−	0.976795i	$-0.431293\pi$
$109$	4.47214			0.428353			0.214176	+	0.976795i	$0.431293\pi$
$110$		0			0
$111$	−1.23607	−	0.472136i	−0.117322	−	0.0448132i
$112$		0			0
$113$		−	2.94427i		−	0.276974i	−0.990364	−	0.138487i	$-0.955776\pi$
$113$		−	2.94427i		−	0.276974i	0.990364	−	0.138487i	$-0.0442239\pi$
$114$		0			0
$115$		−	4.00000i		−	0.373002i
$116$		0			0
$117$	−2.47214	+	2.76393i	−0.228549	+	0.255526i
$118$		0			0
$119$	13.7082	−	2.00000i	1.25663	−	0.183340i
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$	−4.00000	−	1.52786i	−0.360668	−	0.137763i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	0.291796			0.0258927			0.0129464	−	0.999916i	$-0.495879\pi$
$127$	0.291796			0.0258927			0.0129464	+	0.999916i	$0.495879\pi$
$128$		0			0
$129$	−8.00000	−	3.05573i	−0.704361	−	0.269042i
$130$		0			0
$131$	5.41641			0.473234			0.236617	−	0.971603i	$-0.423961\pi$
$131$	5.41641			0.473234			0.236617	+	0.971603i	$0.423961\pi$
$132$		0			0
$133$	−3.23607	−	22.1803i	−0.280603	−	1.92328i
$134$		0			0
$135$	2.38197	+	4.61803i	0.205007	+	0.397457i
$136$		0			0
$137$			3.52786i			0.301406i	0.988579	+	0.150703i	$0.0481537\pi$
$137$			3.52786i			0.301406i	−0.988579	+	0.150703i	$0.951846\pi$
$138$		0			0
$139$		−	11.5279i		−	0.977781i	−0.872345	−	0.488890i	$-0.837402\pi$
$139$		−	11.5279i		−	0.977781i	0.872345	−	0.488890i	$-0.162598\pi$
$140$		0			0
$141$	10.4721	+	4.00000i	0.881913	+	0.336861i
$142$		0			0
$143$	5.52786			0.462263
$144$		0			0
$145$			7.70820i			0.640131i
$146$		0			0
$147$	−12.0902	−	0.909830i	−0.997180	−	0.0750415i
$148$		0			0
$149$		−	4.29180i		−	0.351598i	−0.984426	−	0.175799i	$-0.943749\pi$
$149$		−	4.29180i		−	0.351598i	0.984426	−	0.175799i	$-0.0562508\pi$
$150$		0			0
$151$	−20.9443			−1.70442			−0.852210	−	0.523199i	$-0.824738\pi$
$151$	−20.9443			−1.70442			−0.852210	+	0.523199i	$0.824738\pi$
$152$		0			0
$153$	−11.7082	−	10.4721i	−0.946552	−	0.846622i
$154$		0			0
$155$			2.76393i			0.222004i
$156$		0			0
$157$			9.23607i			0.737118i	0.929604	+	0.368559i	$0.120149\pi$
$157$			9.23607i			0.737118i	−0.929604	+	0.368559i	$0.879851\pi$
$158$		0			0
$159$	0.291796	−	0.763932i	0.0231409	−	0.0605838i
$160$		0			0
$161$	−1.52786	−	10.4721i	−0.120413	−	0.825320i
$162$		0			0
$163$	10.4721			0.820241			0.410120	−	0.912031i	$-0.365487\pi$
$163$	10.4721			0.820241			0.410120	+	0.912031i	$0.365487\pi$
$164$		0			0
$165$	2.76393	−	7.23607i	0.215172	−	0.563327i
$166$		0			0
$167$	−0.944272			−0.0730700			−0.0365350	−	0.999332i	$-0.511632\pi$
$167$	−0.944272			−0.0730700			−0.0365350	+	0.999332i	$0.511632\pi$
$168$		0			0
$169$	11.4721			0.882472
$170$		0			0
$171$	−16.9443	+	18.9443i	−1.29576	+	1.44870i
$172$		0			0
$173$	9.41641			0.715916			0.357958	−	0.933738i	$-0.383473\pi$
$173$	9.41641			0.715916			0.357958	+	0.933738i	$0.383473\pi$
$174$		0			0
$175$	−2.61803	+	0.381966i	−0.197905	+	0.0288739i
$176$		0			0
$177$	−7.23607	−	2.76393i	−0.543896	−	0.207750i
$178$		0			0
$179$			14.9443i			1.11699i	0.829509	+	0.558494i	$0.188620\pi$
$179$			14.9443i			1.11699i	−0.829509	+	0.558494i	$0.811380\pi$
$180$		0			0
$181$			16.1803i			1.20268i	0.798995	+	0.601338i	$0.205365\pi$
$181$			16.1803i			1.20268i	−0.798995	+	0.601338i	$0.794635\pi$
$182$		0			0
$183$	−4.47214	+	11.7082i	−0.330590	+	0.865495i
$184$		0			0
$185$	−0.763932			−0.0561654
$186$		0			0
$187$			23.4164i			1.71238i
$188$		0			0
$189$	8.00000	+	11.1803i	0.581914	+	0.813250i
$190$		0			0
$191$		−	7.23607i		−	0.523584i	−0.965124	−	0.261792i	$-0.915687\pi$
$191$		−	7.23607i		−	0.523584i	0.965124	−	0.261792i	$-0.0843134\pi$
$192$		0			0
$193$	−6.00000			−0.431889			−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000			−0.431889			−0.215945	+	0.976406i	$0.569283\pi$
$194$		0			0
$195$	−0.763932	+	2.00000i	−0.0547063	+	0.143223i
$196$		0			0
$197$			3.52786i			0.251350i	0.992071	+	0.125675i	$0.0401096\pi$
$197$			3.52786i			0.251350i	−0.992071	+	0.125675i	$0.959890\pi$
$198$		0			0
$199$		−	22.1803i		−	1.57232i	−0.618021	−	0.786161i	$-0.712065\pi$
$199$		−	22.1803i		−	1.57232i	0.618021	−	0.786161i	$-0.287935\pi$
$200$		0			0
$201$	−19.4164	−	7.41641i	−1.36953	−	0.523113i
$202$		0			0
$203$	2.94427	+	20.1803i	0.206647	+	1.41638i
$204$		0			0
$205$	−2.47214			−0.172661
$206$		0			0
$207$	−8.00000	+	8.94427i	−0.556038	+	0.621670i
$208$		0			0
$209$	37.8885			2.62081
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$		0			0
$213$	4.47214	−	11.7082i	0.306426	−	0.802233i
$214$		0			0
$215$	−4.94427			−0.337197
$216$		0			0
$217$	1.05573	+	7.23607i	0.0716675	+	0.491216i
$218$		0			0
$219$	6.94427	−	18.1803i	0.469250	−	1.22851i
$220$		0			0
$221$		−	6.47214i		−	0.435363i
$222$		0			0
$223$			17.7082i			1.18583i	0.805265	+	0.592915i	$0.202023\pi$
$223$			17.7082i			1.18583i	−0.805265	+	0.592915i	$0.797977\pi$
$224$		0			0
$225$	2.23607	+	2.00000i	0.149071	+	0.133333i
$226$		0			0
$227$	0.763932			0.0507039			0.0253520	−	0.999679i	$-0.491929\pi$
$227$	0.763932			0.0507039			0.0253520	+	0.999679i	$0.491929\pi$
$228$		0			0
$229$			8.76393i			0.579137i	0.957157	+	0.289568i	$0.0935118\pi$
$229$			8.76393i			0.579137i	−0.957157	+	0.289568i	$0.906488\pi$
$230$		0			0
$231$	4.47214	−	20.0000i	0.294245	−	1.31590i
$232$		0			0
$233$			11.5279i			0.755215i	0.925966	+	0.377608i	$0.123253\pi$
$233$			11.5279i			0.755215i	−0.925966	+	0.377608i	$0.876747\pi$
$234$		0			0
$235$	6.47214			0.422196
$236$		0			0
$237$	−14.4721	−	5.52786i	−0.940066	−	0.359073i
$238$		0			0
$239$		−	0.180340i		−	0.0116652i	−0.999983	−	0.00583261i	$-0.998143\pi$
$239$		−	0.180340i		−	0.0116652i	0.999983	−	0.00583261i	$-0.00185659\pi$
$240$		0			0
$241$		−	17.8885i		−	1.15230i	−0.817343	−	0.576151i	$-0.804554\pi$
$241$		−	17.8885i		−	1.15230i	0.817343	−	0.576151i	$-0.195446\pi$
$242$		0			0
$243$	3.90983	−	15.0902i	0.250816	−	0.968035i
$244$		0			0
$245$	−6.70820	+	2.00000i	−0.428571	+	0.127775i
$246$		0			0
$247$	−10.4721			−0.666326
$248$		0			0
$249$	−23.7082	−	9.05573i	−1.50245	−	0.573883i
$250$		0			0
$251$	−12.4721			−0.787234			−0.393617	−	0.919274i	$-0.628776\pi$
$251$	−12.4721			−0.787234			−0.393617	+	0.919274i	$0.628776\pi$
$252$		0			0
$253$	17.8885			1.12464
$254$		0			0
$255$	−8.47214	−	3.23607i	−0.530546	−	0.202650i
$256$		0			0
$257$	−14.1803			−0.884545			−0.442273	−	0.896881i	$-0.645828\pi$
$257$	−14.1803			−0.884545			−0.442273	+	0.896881i	$0.645828\pi$
$258$		0			0
$259$	−2.00000	+	0.291796i	−0.124274	+	0.0181313i
$260$		0			0
$261$	15.4164	−	17.2361i	0.954252	−	1.06689i
$262$		0			0
$263$			12.9443i			0.798178i	0.916912	+	0.399089i	$0.130674\pi$
$263$			12.9443i			0.798178i	−0.916912	+	0.399089i	$0.869326\pi$
$264$		0			0
$265$		−	0.472136i		−	0.0290031i
$266$		0			0
$267$	−8.94427	−	3.41641i	−0.547381	−	0.209081i
$268$		0			0
$269$	−4.47214			−0.272671			−0.136335	−	0.990663i	$-0.543533\pi$
$269$	−4.47214			−0.272671			−0.136335	+	0.990663i	$0.543533\pi$
$270$		0			0
$271$			31.7082i			1.92614i	0.269258	+	0.963068i	$0.413222\pi$
$271$			31.7082i			1.92614i	−0.269258	+	0.963068i	$0.586778\pi$
$272$		0			0
$273$	−1.23607	+	5.52786i	−0.0748102	+	0.334562i
$274$		0			0
$275$		−	4.47214i		−	0.269680i
$276$		0			0
$277$	−17.1246			−1.02892			−0.514459	−	0.857515i	$-0.672007\pi$
$277$	−17.1246			−1.02892			−0.514459	+	0.857515i	$0.672007\pi$
$278$		0			0
$279$	5.52786	−	6.18034i	0.330945	−	0.370007i
$280$		0			0
$281$			20.0000i			1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$281$			20.0000i			1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$282$		0			0
$283$			13.2361i			0.786803i	0.919367	+	0.393401i	$0.128702\pi$
$283$			13.2361i			0.786803i	−0.919367	+	0.393401i	$0.871298\pi$
$284$		0			0
$285$	−5.23607	+	13.7082i	−0.310158	+	0.812004i
$286$		0			0
$287$	−6.47214	+	0.944272i	−0.382038	+	0.0557386i
$288$		0			0
$289$	10.4164			0.612730
$290$		0			0
$291$	−0.472136	+	1.23607i	−0.0276771	+	0.0724596i
$292$		0			0
$293$	2.58359			0.150935			0.0754675	−	0.997148i	$-0.475955\pi$
$293$	2.58359			0.150935			0.0754675	+	0.997148i	$0.475955\pi$
$294$		0			0
$295$	−4.47214			−0.260378
$296$		0			0
$297$	−20.6525	+	10.6525i	−1.19838	+	0.618119i
$298$		0			0
$299$	−4.94427			−0.285935
$300$		0			0
$301$	−12.9443	+	1.88854i	−0.746095	+	0.108854i
$302$		0			0
$303$	−20.1803	−	7.70820i	−1.15933	−	0.442825i
$304$		0			0
$305$			7.23607i			0.414336i
$306$		0			0
$307$		−	18.1803i		−	1.03761i	−0.854894	−	0.518803i	$-0.826378\pi$
$307$		−	18.1803i		−	1.03761i	0.854894	−	0.518803i	$-0.173622\pi$
$308$		0			0
$309$	−9.05573	+	23.7082i	−0.515162	+	1.34871i
$310$		0			0
$311$	17.5279			0.993914			0.496957	−	0.867775i	$-0.334451\pi$
$311$	17.5279			0.993914			0.496957	+	0.867775i	$0.334451\pi$
$312$		0			0
$313$			5.70820i			0.322647i	0.986902	+	0.161323i	$0.0515762\pi$
$313$			5.70820i			0.322647i	−0.986902	+	0.161323i	$0.948424\pi$
$314$		0			0
$315$	6.61803	+	4.38197i	0.372884	+	0.246896i
$316$		0			0
$317$		−	10.9443i		−	0.614692i	−0.951598	−	0.307346i	$-0.900559\pi$
$317$		−	10.9443i		−	0.614692i	0.951598	−	0.307346i	$-0.0994408\pi$
$318$		0			0
$319$	−34.4721			−1.93007
$320$		0			0
$321$	−7.05573	+	18.4721i	−0.393812	+	1.03101i
$322$		0			0
$323$		−	44.3607i		−	2.46829i
$324$		0			0
$325$			1.23607i			0.0685647i
$326$		0			0
$327$	−7.23607	−	2.76393i	−0.400155	−	0.152846i
$328$		0			0
$329$	16.9443	−	2.47214i	0.934168	−	0.136293i
$330$		0			0
$331$	−3.05573			−0.167958			−0.0839790	−	0.996468i	$-0.526763\pi$
$331$	−3.05573			−0.167958			−0.0839790	+	0.996468i	$0.526763\pi$
$332$		0			0
$333$	1.70820	+	1.52786i	0.0936090	+	0.0837264i
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$	21.4164			1.16663			0.583313	−	0.812247i	$-0.301756\pi$
$337$	21.4164			1.16663			0.583313	+	0.812247i	$0.301756\pi$
$338$		0			0
$339$	−1.81966	+	4.76393i	−0.0988304	+	0.258741i
$340$		0			0
$341$	−12.3607			−0.669368
$342$		0			0
$343$	−16.7984	+	7.79837i	−0.907027	+	0.421073i
$344$		0			0
$345$	−2.47214	+	6.47214i	−0.133095	+	0.348448i
$346$		0			0
$347$		−	2.47214i		−	0.132711i	−0.997796	−	0.0663556i	$-0.978863\pi$
$347$		−	2.47214i		−	0.132711i	0.997796	−	0.0663556i	$-0.0211372\pi$
$348$		0			0
$349$		−	20.1803i		−	1.08023i	−0.841592	−	0.540114i	$-0.818381\pi$
$349$		−	20.1803i		−	1.08023i	0.841592	−	0.540114i	$-0.181619\pi$
$350$		0			0
$351$	5.70820	−	2.94427i	0.304681	−	0.157154i
$352$		0			0
$353$	27.7082			1.47476			0.737379	−	0.675479i	$-0.236063\pi$
$353$	27.7082			1.47476			0.737379	+	0.675479i	$0.236063\pi$
$354$		0			0
$355$		−	7.23607i		−	0.384051i
$356$		0			0
$357$	−23.4164	−	5.23607i	−1.23933	−	0.277122i
$358$		0			0
$359$			12.1803i			0.642854i	0.946934	+	0.321427i	$0.104162\pi$
$359$			12.1803i			0.642854i	−0.946934	+	0.321427i	$0.895838\pi$
$360$		0			0
$361$	−52.7771			−2.77774
$362$		0			0
$363$	14.5623	+	5.56231i	0.764323	+	0.291945i
$364$		0			0
$365$		−	11.2361i		−	0.588123i
$366$		0			0
$367$		−	8.18034i		−	0.427010i	−0.976942	−	0.213505i	$-0.931512\pi$
$367$		−	8.18034i		−	0.427010i	0.976942	−	0.213505i	$-0.0684880\pi$
$368$		0			0
$369$	5.52786	+	4.94427i	0.287769	+	0.257389i
$370$		0			0
$371$	−0.180340	−	1.23607i	−0.00936278	−	0.0641735i
$372$		0			0
$373$	−32.1803			−1.66623			−0.833117	−	0.553096i	$-0.813446\pi$
$373$	−32.1803			−1.66623			−0.833117	+	0.553096i	$0.813446\pi$
$374$		0			0
$375$	1.61803	+	0.618034i	0.0835549	+	0.0319151i
$376$		0			0
$377$	9.52786			0.490710
$378$		0			0
$379$	17.8885			0.918873			0.459436	−	0.888211i	$-0.348051\pi$
$379$	17.8885			0.918873			0.459436	+	0.888211i	$0.348051\pi$
$380$		0			0
$381$	−0.472136	−	0.180340i	−0.0241883	−	0.00923909i
$382$		0			0
$383$	−13.8885			−0.709671			−0.354836	−	0.934929i	$-0.615463\pi$
$383$	−13.8885			−0.709671			−0.354836	+	0.934929i	$0.615463\pi$
$384$		0			0
$385$	−1.70820	−	11.7082i	−0.0870581	−	0.596705i
$386$		0			0
$387$	11.0557	+	9.88854i	0.561994	+	0.502663i
$388$		0			0
$389$			30.1803i			1.53020i	0.643909	+	0.765102i	$0.277311\pi$
$389$			30.1803i			1.53020i	−0.643909	+	0.765102i	$0.722689\pi$
$390$		0			0
$391$		−	20.9443i		−	1.05920i
$392$		0			0
$393$	−8.76393	−	3.34752i	−0.442082	−	0.168860i
$394$		0			0
$395$	−8.94427			−0.450035
$396$		0			0
$397$		−	23.1246i		−	1.16059i	−0.814406	−	0.580295i	$-0.802937\pi$
$397$		−	23.1246i		−	1.16059i	0.814406	−	0.580295i	$-0.197063\pi$
$398$		0			0
$399$	−8.47214	+	37.8885i	−0.424137	+	1.89680i
$400$		0			0
$401$			14.4721i			0.722704i	0.932429	+	0.361352i	$0.117685\pi$
$401$			14.4721i			0.722704i	−0.932429	+	0.361352i	$0.882315\pi$
$402$		0			0
$403$	3.41641			0.170183
$404$		0			0
$405$	−1.00000	−	8.94427i	−0.0496904	−	0.444444i
$406$		0			0
$407$		−	3.41641i		−	0.169345i
$408$		0			0
$409$		−	7.41641i		−	0.366718i	−0.983046	−	0.183359i	$-0.941303\pi$
$409$		−	7.41641i		−	0.366718i	0.983046	−	0.183359i	$-0.0586970\pi$
$410$		0			0
$411$	2.18034	−	5.70820i	0.107548	−	0.281565i
$412$		0			0
$413$	−11.7082	+	1.70820i	−0.576123	+	0.0840552i
$414$		0			0
$415$	−14.6525			−0.719262
$416$		0			0
$417$	−7.12461	+	18.6525i	−0.348894	+	0.913416i
$418$		0			0
$419$	36.8328			1.79940			0.899700	−	0.436508i	$-0.143785\pi$
$419$	36.8328			1.79940			0.899700	+	0.436508i	$0.143785\pi$
$420$		0			0
$421$	−3.52786			−0.171938			−0.0859688	−	0.996298i	$-0.527399\pi$
$421$	−3.52786			−0.171938			−0.0859688	+	0.996298i	$0.527399\pi$
$422$		0			0
$423$	−14.4721	−	12.9443i	−0.703659	−	0.629372i
$424$		0			0
$425$	−5.23607			−0.253987
$426$		0			0
$427$	2.76393	+	18.9443i	0.133756	+	0.916778i
$428$		0			0
$429$	−8.94427	−	3.41641i	−0.431834	−	0.164946i
$430$		0			0
$431$			39.5967i			1.90731i	0.300905	+	0.953654i	$0.402711\pi$
$431$			39.5967i			1.90731i	−0.300905	+	0.953654i	$0.597289\pi$
$432$		0			0
$433$		−	26.6525i		−	1.28084i	−0.768026	−	0.640418i	$-0.778761\pi$
$433$		−	26.6525i		−	1.28084i	0.768026	−	0.640418i	$-0.221239\pi$
$434$		0			0
$435$	4.76393	−	12.4721i	0.228413	−	0.597993i
$436$		0			0
$437$	−33.8885			−1.62111
$438$		0			0
$439$			10.1803i			0.485881i	0.970041	+	0.242941i	$0.0781120\pi$
$439$			10.1803i			0.485881i	−0.970041	+	0.242941i	$0.921888\pi$
$440$		0			0
$441$	19.0000	+	8.94427i	0.904762	+	0.425918i
$442$		0			0
$443$			18.4721i			0.877638i	0.898576	+	0.438819i	$0.144603\pi$
$443$			18.4721i			0.877638i	−0.898576	+	0.438819i	$0.855397\pi$
$444$		0			0
$445$	−5.52786			−0.262046
$446$		0			0
$447$	−2.65248	+	6.94427i	−0.125458	+	0.328453i
$448$		0			0
$449$		−	10.4721i		−	0.494211i	−0.968989	−	0.247105i	$-0.920521\pi$
$449$		−	10.4721i		−	0.494211i	0.968989	−	0.247105i	$-0.0794794\pi$
$450$		0			0
$451$		−	11.0557i		−	0.520594i
$452$		0			0
$453$	33.8885	+	12.9443i	1.59222	+	0.608175i
$454$		0			0
$455$	0.472136	+	3.23607i	0.0221341	+	0.151709i
$456$		0			0
$457$	26.9443			1.26040			0.630200	−	0.776433i	$-0.282973\pi$
$457$	26.9443			1.26040			0.630200	+	0.776433i	$0.282973\pi$
$458$		0			0
$459$	12.4721	+	24.1803i	0.582149	+	1.12864i
$460$		0			0
$461$	6.94427			0.323427			0.161713	−	0.986838i	$-0.448298\pi$
$461$	6.94427			0.323427			0.161713	+	0.986838i	$0.448298\pi$
$462$		0			0
$463$	22.1803			1.03081			0.515404	−	0.856947i	$-0.327642\pi$
$463$	22.1803			1.03081			0.515404	+	0.856947i	$0.327642\pi$
$464$		0			0
$465$	1.70820	−	4.47214i	0.0792161	−	0.207390i
$466$		0			0
$467$	−33.7082			−1.55983			−0.779915	−	0.625886i	$-0.784738\pi$
$467$	−33.7082			−1.55983			−0.779915	+	0.625886i	$0.784738\pi$
$468$		0			0
$469$	−31.4164	+	4.58359i	−1.45067	+	0.211651i
$470$		0			0
$471$	5.70820	−	14.9443i	0.263020	−	0.688596i
$472$		0			0
$473$		−	22.1115i		−	1.01669i
$474$		0			0
$475$			8.47214i			0.388728i
$476$		0			0
$477$	−0.944272	+	1.05573i	−0.0432352	+	0.0483385i
$478$		0			0
$479$	37.8885			1.73117			0.865586	−	0.500761i	$-0.166946\pi$
$479$	37.8885			1.73117			0.865586	+	0.500761i	$0.166946\pi$
$480$		0			0
$481$			0.944272i			0.0430551i
$482$		0			0
$483$	−4.00000	+	17.8885i	−0.182006	+	0.813957i
$484$		0			0
$485$			0.763932i			0.0346884i
$486$		0			0
$487$	−17.5967			−0.797385			−0.398692	−	0.917085i	$-0.630536\pi$
$487$	−17.5967			−0.797385			−0.398692	+	0.917085i	$0.630536\pi$
$488$		0			0
$489$	−16.9443	−	6.47214i	−0.766246	−	0.292680i
$490$		0			0
$491$		−	13.4164i		−	0.605474i	−0.953074	−	0.302737i	$-0.902100\pi$
$491$		−	13.4164i		−	0.605474i	0.953074	−	0.302737i	$-0.0979004\pi$
$492$		0			0
$493$			40.3607i			1.81775i
$494$		0			0
$495$	−8.94427	+	10.0000i	−0.402015	+	0.449467i
$496$		0			0
$497$	−2.76393	−	18.9443i	−0.123979	−	0.849767i
$498$		0			0
$499$	−17.8885			−0.800801			−0.400401	−	0.916340i	$-0.631129\pi$
$499$	−17.8885			−0.800801			−0.400401	+	0.916340i	$0.631129\pi$
$500$		0			0
$501$	1.52786	+	0.583592i	0.0682599	+	0.0260730i
$502$		0			0
$503$	−28.3607			−1.26454			−0.632270	−	0.774748i	$-0.717877\pi$
$503$	−28.3607			−1.26454			−0.632270	+	0.774748i	$0.717877\pi$
$504$		0			0
$505$	−12.4721			−0.555003
$506$		0			0
$507$	−18.5623	−	7.09017i	−0.824381	−	0.314886i
$508$		0			0
$509$	−15.5279			−0.688260			−0.344130	−	0.938922i	$-0.611826\pi$
$509$	−15.5279			−0.688260			−0.344130	+	0.938922i	$0.611826\pi$
$510$		0			0
$511$	−4.29180	−	29.4164i	−0.189858	−	1.30131i
$512$		0			0
$513$	39.1246	−	20.1803i	1.72739	−	0.890984i
$514$		0			0
$515$			14.6525i			0.645665i
$516$		0			0
$517$			28.9443i			1.27297i
$518$		0			0
$519$	−15.2361	−	5.81966i	−0.668789	−	0.255455i
$520$		0			0
$521$	36.9443			1.61856			0.809279	−	0.587425i	$-0.199858\pi$
$521$	36.9443			1.61856			0.809279	+	0.587425i	$0.199858\pi$
$522$		0			0
$523$		−	42.5410i		−	1.86019i	−0.367320	−	0.930094i	$-0.619725\pi$
$523$		−	42.5410i		−	1.86019i	0.367320	−	0.930094i	$-0.380275\pi$
$524$		0			0
$525$	4.47214	+	1.00000i	0.195180	+	0.0436436i
$526$		0			0
$527$			14.4721i			0.630416i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$	10.0000	+	8.94427i	0.433963	+	0.388148i
$532$		0			0
$533$			3.05573i			0.132358i
$534$		0			0
$535$			11.4164i			0.493574i
$536$		0			0
$537$	9.23607	−	24.1803i	0.398566	−	1.04346i
$538$		0			0
$539$	−8.94427	−	30.0000i	−0.385257	−	1.29219i
$540$		0			0
$541$	30.9443			1.33040			0.665199	−	0.746666i	$-0.268347\pi$
$541$	30.9443			1.33040			0.665199	+	0.746666i	$0.268347\pi$
$542$		0			0
$543$	10.0000	−	26.1803i	0.429141	−	1.12351i
$544$		0			0
$545$	−4.47214			−0.191565
$546$		0			0
$547$	35.4164			1.51430			0.757148	−	0.653243i	$-0.226592\pi$
$547$	35.4164			1.51430			0.757148	+	0.653243i	$0.226592\pi$
$548$		0			0
$549$	14.4721	−	16.1803i	0.617656	−	0.690560i
$550$		0			0
$551$	65.3050			2.78208
$552$		0			0
$553$	−23.4164	+	3.41641i	−0.995767	+	0.145280i
$554$		0			0
$555$	1.23607	+	0.472136i	0.0524682	+	0.0200411i
$556$		0			0
$557$		−	7.52786i		−	0.318966i	−0.987201	−	0.159483i	$-0.949017\pi$
$557$		−	7.52786i		−	0.318966i	0.987201	−	0.159483i	$-0.0509827\pi$
$558$		0			0
$559$			6.11146i			0.258487i
$560$		0			0
$561$	14.4721	−	37.8885i	0.611014	−	1.59966i
$562$		0			0
$563$	40.1803			1.69340			0.846700	−	0.532071i	$-0.178586\pi$
$563$	40.1803			1.69340			0.846700	+	0.532071i	$0.178586\pi$
$564$		0			0
$565$			2.94427i			0.123866i
$566$		0			0
$567$	−6.03444	−	23.0344i	−0.253423	−	0.967356i
$568$		0			0
$569$			9.52786i			0.399429i	0.979854	+	0.199714i	$0.0640014\pi$
$569$			9.52786i			0.399429i	−0.979854	+	0.199714i	$0.935999\pi$
$570$		0			0
$571$	−18.8328			−0.788129			−0.394064	−	0.919083i	$-0.628931\pi$
$571$	−18.8328			−0.788129			−0.394064	+	0.919083i	$0.628931\pi$
$572$		0			0
$573$	−4.47214	+	11.7082i	−0.186826	+	0.489117i
$574$		0			0
$575$			4.00000i			0.166812i
$576$		0			0
$577$			8.18034i			0.340552i	0.985396	+	0.170276i	$0.0544659\pi$
$577$			8.18034i			0.340552i	−0.985396	+	0.170276i	$0.945534\pi$
$578$		0			0
$579$	9.70820	+	3.70820i	0.403459	+	0.154108i
$580$		0			0
$581$	−38.3607	+	5.59675i	−1.59147	+	0.232192i
$582$		0			0
$583$	2.11146			0.0874476
$584$		0			0
$585$	2.47214	−	2.76393i	0.102210	−	0.114275i
$586$		0			0
$587$	−10.2918			−0.424788			−0.212394	−	0.977184i	$-0.568126\pi$
$587$	−10.2918			−0.424788			−0.212394	+	0.977184i	$0.568126\pi$
$588$		0			0
$589$	23.4164			0.964856
$590$		0			0
$591$	2.18034	−	5.70820i	0.0896872	−	0.234804i
$592$		0			0
$593$	29.0132			1.19143			0.595714	−	0.803197i	$-0.296869\pi$
$593$	29.0132			1.19143			0.595714	+	0.803197i	$0.296869\pi$
$594$		0			0
$595$	−13.7082	+	2.00000i	−0.561982	+	0.0819920i
$596$		0			0
$597$	−13.7082	+	35.8885i	−0.561039	+	1.46882i
$598$		0			0
$599$		−	36.7639i		−	1.50213i	−0.660226	−	0.751067i	$-0.729540\pi$
$599$		−	36.7639i		−	1.50213i	0.660226	−	0.751067i	$-0.270460\pi$
$600$		0			0
$601$		−	5.52786i		−	0.225486i	−0.993624	−	0.112743i	$-0.964036\pi$
$601$		−	5.52786i		−	0.225486i	0.993624	−	0.112743i	$-0.0359637\pi$
$602$		0			0
$603$	26.8328	+	24.0000i	1.09272	+	0.977356i
$604$		0			0
$605$	9.00000			0.365902
$606$		0			0
$607$		−	19.2361i		−	0.780768i	−0.920652	−	0.390384i	$-0.872342\pi$
$607$		−	19.2361i		−	0.780768i	0.920652	−	0.390384i	$-0.127658\pi$
$608$		0			0
$609$	7.70820	−	34.4721i	0.312352	−	1.39688i
$610$		0			0
$611$		−	8.00000i		−	0.323645i
$612$		0			0
$613$	38.0689			1.53759			0.768794	−	0.639497i	$-0.220857\pi$
$613$	38.0689			1.53759			0.768794	+	0.639497i	$0.220857\pi$
$614$		0			0
$615$	4.00000	+	1.52786i	0.161296	+	0.0616094i
$616$		0			0
$617$		−	2.00000i		−	0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$		−	2.00000i		−	0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$		0			0
$619$			14.0000i			0.562708i	0.959604	+	0.281354i	$0.0907834\pi$
$619$			14.0000i			0.562708i	−0.959604	+	0.281354i	$0.909217\pi$
$620$		0			0
$621$	18.4721	−	9.52786i	0.741261	−	0.382340i
$622$		0			0
$623$	−14.4721	+	2.11146i	−0.579814	+	0.0845937i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−61.3050	−	23.4164i	−2.44828	−	0.935161i
$628$		0			0
$629$	−4.00000			−0.159490
$630$		0			0
$631$	5.88854			0.234419			0.117210	−	0.993107i	$-0.462605\pi$
$631$	5.88854			0.234419			0.117210	+	0.993107i	$0.462605\pi$
$632$		0			0
$633$	−12.9443	−	4.94427i	−0.514489	−	0.196517i
$634$		0			0
$635$	−0.291796			−0.0115796
$636$		0			0
$637$	2.47214	+	8.29180i	0.0979496	+	0.328533i
$638$		0			0
$639$	−14.4721	+	16.1803i	−0.572509	+	0.640084i
$640$		0			0
$641$			23.4164i			0.924893i	0.886647	+	0.462446i	$0.153028\pi$
$641$			23.4164i			0.924893i	−0.886647	+	0.462446i	$0.846972\pi$
$642$		0			0
$643$		−	12.2918i		−	0.484741i	−0.970184	−	0.242371i	$-0.922075\pi$
$643$		−	12.2918i		−	0.484741i	0.970184	−	0.242371i	$-0.0779250\pi$
$644$		0			0
$645$	8.00000	+	3.05573i	0.315000	+	0.120319i
$646$		0			0
$647$	34.8328			1.36942			0.684710	−	0.728816i	$-0.259929\pi$
$647$	34.8328			1.36942			0.684710	+	0.728816i	$0.259929\pi$
$648$		0			0
$649$		−	20.0000i		−	0.785069i
$650$		0			0
$651$	2.76393	−	12.3607i	0.108327	−	0.484453i
$652$		0			0
$653$			23.8885i			0.934831i	0.884038	+	0.467415i	$0.154815\pi$
$653$			23.8885i			0.934831i	−0.884038	+	0.467415i	$0.845185\pi$
$654$		0			0
$655$	−5.41641			−0.211637
$656$		0			0
$657$	−22.4721	+	25.1246i	−0.876722	+	0.980204i
$658$		0			0
$659$			31.5279i			1.22815i	0.789247	+	0.614076i	$0.210471\pi$
$659$			31.5279i			1.22815i	−0.789247	+	0.614076i	$0.789529\pi$
$660$		0			0
$661$		−	18.2918i		−	0.711468i	−0.934587	−	0.355734i	$-0.884231\pi$
$661$		−	18.2918i		−	0.711468i	0.934587	−	0.355734i	$-0.115769\pi$
$662$		0			0
$663$	−4.00000	+	10.4721i	−0.155347	+	0.406704i
$664$		0			0
$665$	3.23607	+	22.1803i	0.125489	+	0.860117i
$666$		0			0
$667$	30.8328			1.19385
$668$		0			0
$669$	10.9443	−	28.6525i	0.423130	−	1.10777i
$670$		0			0
$671$	−32.3607			−1.24927
$672$		0			0
$673$	19.5279			0.752744			0.376372	−	0.926469i	$-0.377171\pi$
$673$	19.5279			0.752744			0.376372	+	0.926469i	$0.377171\pi$
$674$		0			0
$675$	−2.38197	−	4.61803i	−0.0916819	−	0.177748i
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	0.291796	+	2.00000i	0.0111981	+	0.0767530i
$680$		0			0
$681$	−1.23607	−	0.472136i	−0.0473662	−	0.0180923i
$682$		0			0
$683$		−	36.0000i		−	1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$		−	36.0000i		−	1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$		0			0
$685$		−	3.52786i		−	0.134793i
$686$		0			0
$687$	5.41641	−	14.1803i	0.206649	−	0.541014i
$688$		0			0
$689$	−0.583592			−0.0222331
$690$		0			0
$691$		−	21.0557i		−	0.800998i	−0.916297	−	0.400499i	$-0.868837\pi$
$691$		−	21.0557i		−	0.800998i	0.916297	−	0.400499i	$-0.131163\pi$
$692$		0			0
$693$	−19.5967	+	29.5967i	−0.744419	+	1.12429i
$694$		0			0
$695$			11.5279i			0.437277i
$696$		0			0
$697$	−12.9443			−0.490299
$698$		0			0
$699$	7.12461	−	18.6525i	0.269478	−	0.705501i
$700$		0			0
$701$		−	44.0689i		−	1.66446i	−0.554431	−	0.832229i	$-0.687064\pi$
$701$		−	44.0689i		−	1.66446i	0.554431	−	0.832229i	$-0.312936\pi$
$702$		0			0
$703$			6.47214i			0.244101i
$704$		0			0
$705$	−10.4721	−	4.00000i	−0.394403	−	0.150649i
$706$		0			0
$707$	−32.6525	+	4.76393i	−1.22802	+	0.179166i
$708$		0			0
$709$	−15.5279			−0.583161			−0.291581	−	0.956546i	$-0.594181\pi$
$709$	−15.5279			−0.583161			−0.291581	+	0.956546i	$0.594181\pi$
$710$		0			0
$711$	20.0000	+	17.8885i	0.750059	+	0.670873i
$712$		0			0
$713$	11.0557			0.414040
$714$		0			0
$715$	−5.52786			−0.206730
$716$		0			0
$717$	−0.111456	+	0.291796i	−0.00416241	+	0.0108973i
$718$		0			0
$719$	−34.4721			−1.28559			−0.642797	−	0.766037i	$-0.722226\pi$
$719$	−34.4721			−1.28559			−0.642797	+	0.766037i	$0.722226\pi$
$720$		0			0
$721$	5.59675	+	38.3607i	0.208434	+	1.42863i
$722$		0			0
$723$	−11.0557	+	28.9443i	−0.411167	+	1.07645i
$724$		0			0
$725$		−	7.70820i		−	0.286276i
$726$		0			0
$727$			26.2918i			0.975109i	0.873093	+	0.487554i	$0.162111\pi$
$727$			26.2918i			0.975109i	−0.873093	+	0.487554i	$0.837889\pi$
$728$		0			0
$729$	−15.6525	+	22.0000i	−0.579721	+	0.814815i
$730$		0			0
$731$	−25.8885			−0.957522
$732$		0			0
$733$		−	20.0689i		−	0.741261i	−0.928780	−	0.370631i	$-0.879142\pi$
$733$		−	20.0689i		−	0.741261i	0.928780	−	0.370631i	$-0.120858\pi$
$734$		0			0
$735$	12.0902	+	0.909830i	0.445953	+	0.0335596i
$736$		0			0
$737$		−	53.6656i		−	1.97680i
$738$		0			0
$739$	−8.94427			−0.329020			−0.164510	−	0.986375i	$-0.552604\pi$
$739$	−8.94427			−0.329020			−0.164510	+	0.986375i	$0.552604\pi$
$740$		0			0
$741$	16.9443	+	6.47214i	0.622463	+	0.237760i
$742$		0			0
$743$		−	10.4721i		−	0.384185i	−0.981377	−	0.192093i	$-0.938473\pi$
$743$		−	10.4721i		−	0.384185i	0.981377	−	0.192093i	$-0.0615274\pi$
$744$		0			0
$745$			4.29180i			0.157239i
$746$		0			0
$747$	32.7639	+	29.3050i	1.19877	+	1.07221i
$748$		0			0
$749$	4.36068	+	29.8885i	0.159336	+	1.09210i
$750$		0			0
$751$	−8.58359			−0.313220			−0.156610	−	0.987661i	$-0.550057\pi$
$751$	−8.58359			−0.313220			−0.156610	+	0.987661i	$0.550057\pi$
$752$		0			0
$753$	20.1803	+	7.70820i	0.735412	+	0.280903i
$754$		0			0
$755$	20.9443			0.762240
$756$		0			0
$757$	−2.65248			−0.0964059			−0.0482029	−	0.998838i	$-0.515349\pi$
$757$	−2.65248			−0.0964059			−0.0482029	+	0.998838i	$0.515349\pi$
$758$		0			0
$759$	−28.9443	−	11.0557i	−1.05061	−	0.401298i
$760$		0			0
$761$	5.88854			0.213460			0.106730	−	0.994288i	$-0.465962\pi$
$761$	5.88854			0.213460			0.106730	+	0.994288i	$0.465962\pi$
$762$		0			0
$763$	−11.7082	+	1.70820i	−0.423865	+	0.0618411i
$764$		0			0
$765$	11.7082	+	10.4721i	0.423311	+	0.378621i
$766$		0			0
$767$			5.52786i			0.199600i
$768$		0			0
$769$			36.0000i			1.29819i	0.760706	+	0.649097i	$0.224853\pi$
$769$			36.0000i			1.29819i	−0.760706	+	0.649097i	$0.775147\pi$
$770$		0			0
$771$	22.9443	+	8.76393i	0.826318	+	0.315625i
$772$		0			0
$773$	−10.5836			−0.380665			−0.190333	−	0.981720i	$-0.560957\pi$
$773$	−10.5836			−0.380665			−0.190333	+	0.981720i	$0.560957\pi$
$774$		0			0
$775$		−	2.76393i		−	0.0992834i
$776$		0			0
$777$	3.41641	+	0.763932i	0.122563	+	0.0274059i
$778$		0			0
$779$			20.9443i			0.750406i
$780$		0			0
$781$	32.3607			1.15796
$782$		0			0
$783$	−35.5967	+	18.3607i	−1.27212	+	0.656157i
$784$		0			0
$785$		−	9.23607i		−	0.329649i
$786$		0			0
$787$			36.2918i			1.29366i	0.762633	+	0.646831i	$0.223906\pi$
$787$			36.2918i			1.29366i	−0.762633	+	0.646831i	$0.776094\pi$
$788$		0			0
$789$	8.00000	−	20.9443i	0.284808	−	0.745636i
$790$		0			0
$791$	1.12461	+	7.70820i	0.0399866	+	0.274072i
$792$		0			0
$793$	8.94427			0.317620
$794$		0			0
$795$	−0.291796	+	0.763932i	−0.0103489	+	0.0270939i
$796$		0			0
$797$	−21.4164			−0.758608			−0.379304	−	0.925272i	$-0.623837\pi$
$797$	−21.4164			−0.758608			−0.379304	+	0.925272i	$0.623837\pi$
$798$		0			0
$799$	33.8885			1.19889
$800$		0			0
$801$	12.3607	+	11.0557i	0.436743	+	0.390635i
$802$		0			0
$803$	50.2492			1.77326
$804$		0			0
$805$	1.52786	+	10.4721i	0.0538501	+	0.369094i
$806$		0			0
$807$	7.23607	+	2.76393i	0.254722	+	0.0972950i
$808$		0			0
$809$			36.3607i			1.27837i	0.769052	+	0.639187i	$0.220729\pi$
$809$			36.3607i			1.27837i	−0.769052	+	0.639187i	$0.779271\pi$
$810$		0			0
$811$			36.8328i			1.29338i	0.762755	+	0.646688i	$0.223846\pi$
$811$			36.8328i			1.29338i	−0.762755	+	0.646688i	$0.776154\pi$
$812$		0			0
$813$	19.5967	−	51.3050i	0.687288	−	1.79934i
$814$		0			0
$815$	−10.4721			−0.366823
$816$		0			0
$817$			41.8885i			1.46549i
$818$		0			0
$819$	5.41641	−	8.18034i	0.189265	−	0.285844i
$820$		0			0
$821$		−	31.7082i		−	1.10662i	−0.832974	−	0.553312i	$-0.813364\pi$
$821$		−	31.7082i		−	1.10662i	0.832974	−	0.553312i	$-0.186636\pi$
$822$		0			0
$823$	22.1803			0.773158			0.386579	−	0.922256i	$-0.373657\pi$
$823$	22.1803			0.773158			0.386579	+	0.922256i	$0.373657\pi$
$824$		0			0
$825$	−2.76393	+	7.23607i	−0.0962278	+	0.251928i
$826$		0			0
$827$			38.8328i			1.35035i	0.737658	+	0.675175i	$0.235932\pi$
$827$			38.8328i			1.35035i	−0.737658	+	0.675175i	$0.764068\pi$
$828$		0			0
$829$		−	7.81966i		−	0.271588i	−0.990737	−	0.135794i	$-0.956641\pi$
$829$		−	7.81966i		−	0.271588i	0.990737	−	0.135794i	$-0.0433585\pi$
$830$		0			0
$831$	27.7082	+	10.5836i	0.961187	+	0.367141i
$832$		0			0
$833$	−35.1246	+	10.4721i	−1.21700	+	0.362838i
$834$		0			0
$835$	0.944272			0.0326779
$836$		0			0
$837$	−12.7639	+	6.58359i	−0.441186	+	0.227562i
$838$		0			0
$839$	5.52786			0.190843			0.0954215	−	0.995437i	$-0.469580\pi$
$839$	5.52786			0.190843			0.0954215	+	0.995437i	$0.469580\pi$
$840$		0			0
$841$	−30.4164			−1.04884
$842$		0			0
$843$	12.3607	−	32.3607i	0.425724	−	1.11456i
$844$		0			0
$845$	−11.4721			−0.394653
$846$		0			0
$847$	23.5623	−	3.43769i	0.809610	−	0.118121i
$848$		0			0
$849$	8.18034	−	21.4164i	0.280749	−	0.735009i
$850$		0			0
$851$			3.05573i			0.104749i
$852$		0			0
$853$		−	7.70820i		−	0.263924i	−0.991255	−	0.131962i	$-0.957872\pi$
$853$		−	7.70820i		−	0.263924i	0.991255	−	0.131962i	$-0.0421277\pi$
$854$		0			0
$855$	16.9443	−	18.9443i	0.579482	−	0.647880i
$856$		0			0
$857$	11.3475			0.387624			0.193812	−	0.981039i	$-0.437915\pi$
$857$	11.3475			0.387624			0.193812	+	0.981039i	$0.437915\pi$
$858$		0			0
$859$		−	17.0557i		−	0.581934i	−0.956733	−	0.290967i	$-0.906023\pi$
$859$		−	17.0557i		−	0.581934i	0.956733	−	0.290967i	$-0.0939770\pi$
$860$		0			0
$861$	11.0557	+	2.47214i	0.376778	+	0.0842502i
$862$		0			0
$863$		−	30.4721i		−	1.03728i	−0.854992	−	0.518642i	$-0.826438\pi$
$863$		−	30.4721i		−	1.03728i	0.854992	−	0.518642i	$-0.173562\pi$
$864$		0			0
$865$	−9.41641			−0.320167
$866$		0			0
$867$	−16.8541	−	6.43769i	−0.572395	−	0.218636i
$868$		0			0
$869$		−	40.0000i		−	1.35691i
$870$		0			0
$871$			14.8328i			0.502591i
$872$		0			0
$873$	1.52786	−	1.70820i	0.0517104	−	0.0578139i
$874$		0			0
$875$	2.61803	−	0.381966i	0.0885057	−	0.0129128i
$876$		0			0
$877$	38.6525			1.30520			0.652601	−	0.757702i	$-0.273678\pi$
$877$	38.6525			1.30520			0.652601	+	0.757702i	$0.273678\pi$
$878$		0			0
$879$	−4.18034	−	1.59675i	−0.140999	−	0.0538570i
$880$		0			0
$881$	−15.4164			−0.519392			−0.259696	−	0.965690i	$-0.583622\pi$
$881$	−15.4164			−0.519392			−0.259696	+	0.965690i	$0.583622\pi$
$882$		0			0
$883$	13.8885			0.467387			0.233693	−	0.972310i	$-0.424919\pi$
$883$	13.8885			0.467387			0.233693	+	0.972310i	$0.424919\pi$
$884$		0			0
$885$	7.23607	+	2.76393i	0.243238	+	0.0929086i
$886$		0			0
$887$	−17.5279			−0.588528			−0.294264	−	0.955724i	$-0.595075\pi$
$887$	−17.5279			−0.588528			−0.294264	+	0.955724i	$0.595075\pi$
$888$		0			0
$889$	−0.763932	+	0.111456i	−0.0256215	+	0.00373812i
$890$		0			0
$891$	40.0000	−	4.47214i	1.34005	−	0.149822i
$892$		0			0
$893$		−	54.8328i		−	1.83491i
$894$		0			0
$895$		−	14.9443i		−	0.499532i
$896$		0			0
$897$	8.00000	+	3.05573i	0.267112	+	0.102028i
$898$		0			0
$899$	−21.3050			−0.710560
$900$		0			0
$901$		−	2.47214i		−	0.0823588i
$902$		0			0
$903$	22.1115	+	4.94427i	0.735823	+	0.164535i
$904$		0			0
$905$		−	16.1803i		−	0.537853i
$906$		0			0
$907$	17.5279			0.582003			0.291002	−	0.956723i	$-0.406012\pi$
$907$	17.5279			0.582003			0.291002	+	0.956723i	$0.406012\pi$
$908$		0			0
$909$	27.8885	+	24.9443i	0.925005	+	0.827349i
$910$		0			0
$911$		−	50.6525i		−	1.67819i	−0.543984	−	0.839096i	$-0.683085\pi$
$911$		−	50.6525i		−	1.67819i	0.543984	−	0.839096i	$-0.316915\pi$
$912$		0			0
$913$		−	65.5279i		−	2.16866i
$914$		0			0
$915$	4.47214	−	11.7082i	0.147844	−	0.387061i
$916$		0			0
$917$	−14.1803	+	2.06888i	−0.468276	+	0.0683206i
$918$		0			0
$919$	44.7214			1.47522			0.737611	−	0.675226i	$-0.235954\pi$
$919$	44.7214			1.47522			0.737611	+	0.675226i	$0.235954\pi$
$920$		0			0
$921$	−11.2361	+	29.4164i	−0.370241	+	0.969304i
$922$		0			0
$923$	−8.94427			−0.294404
$924$		0			0
$925$	0.763932			0.0251179
$926$		0			0
$927$	29.3050	−	32.7639i	0.962501	−	1.07611i
$928$		0			0
$929$	−37.8885			−1.24308			−0.621541	−	0.783381i	$-0.713493\pi$
$929$	−37.8885			−1.24308			−0.621541	+	0.783381i	$0.713493\pi$
$930$		0			0
$931$	16.9443	+	56.8328i	0.555326	+	1.86262i
$932$		0			0
$933$	−28.3607	−	10.8328i	−0.928487	−	0.354650i
$934$		0			0
$935$		−	23.4164i		−	0.765798i
$936$		0			0
$937$		−	42.0689i		−	1.37433i	−0.726501	−	0.687165i	$-0.758855\pi$
$937$		−	42.0689i		−	1.37433i	0.726501	−	0.687165i	$-0.241145\pi$
$938$		0			0
$939$	3.52786	−	9.23607i	0.115127	−	0.301408i
$940$		0			0
$941$	−22.0000			−0.717180			−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000			−0.717180			−0.358590	+	0.933495i	$0.616742\pi$
$942$		0			0
$943$			9.88854i			0.322015i
$944$		0			0
$945$	−8.00000	−	11.1803i	−0.260240	−	0.363696i
$946$		0			0
$947$		−	60.3607i		−	1.96146i	−0.195372	−	0.980729i	$-0.562591\pi$
$947$		−	60.3607i		−	1.96146i	0.195372	−	0.980729i	$-0.437409\pi$
$948$		0			0
$949$	−13.8885			−0.450841
$950$		0			0
$951$	−6.76393	+	17.7082i	−0.219336	+	0.574228i
$952$		0			0
$953$			22.5836i			0.731554i	0.930702	+	0.365777i	$0.119197\pi$
$953$			22.5836i			0.731554i	−0.930702	+	0.365777i	$0.880803\pi$
$954$		0			0
$955$			7.23607i			0.234154i
$956$		0			0
$957$	55.7771	+	21.3050i	1.80302	+	0.688691i
$958$		0			0
$959$	−1.34752	−	9.23607i	−0.0435138	−	0.298248i
$960$		0			0
$961$	23.3607			0.753570
$962$		0			0
$963$	22.8328	−	25.5279i	0.735777	−	0.822624i
$964$		0			0
$965$	6.00000			0.193147
$966$		0			0
$967$	39.4853			1.26976			0.634881	−	0.772610i	$-0.281049\pi$
$967$	39.4853			1.26976			0.634881	+	0.772610i	$0.281049\pi$
$968$		0			0
$969$	−27.4164	+	71.7771i	−0.880742	+	2.30581i
$970$		0			0
$971$	−58.0000			−1.86131			−0.930654	−	0.365900i	$-0.880761\pi$
$971$	−58.0000			−1.86131			−0.930654	+	0.365900i	$0.880761\pi$
$972$		0			0
$973$	4.40325	+	30.1803i	0.141162	+	0.967537i
$974$		0			0
$975$	0.763932	−	2.00000i	0.0244654	−	0.0640513i
$976$		0			0
$977$			1.41641i			0.0453149i	0.999743	+	0.0226575i	$0.00721271\pi$
$977$			1.41641i			0.0453149i	−0.999743	+	0.0226575i	$0.992787\pi$
$978$		0			0
$979$		−	24.7214i		−	0.790098i
$980$		0			0
$981$	10.0000	+	8.94427i	0.319275	+	0.285569i
$982$		0			0
$983$	−12.5836			−0.401354			−0.200677	−	0.979657i	$-0.564314\pi$
$983$	−12.5836			−0.401354			−0.200677	+	0.979657i	$0.564314\pi$
$984$		0			0
$985$		−	3.52786i		−	0.112407i
$986$		0			0
$987$	−28.9443	−	6.47214i	−0.921306	−	0.206010i
$988$		0			0
$989$			19.7771i			0.628875i
$990$		0			0
$991$	−17.5279			−0.556791			−0.278395	−	0.960467i	$-0.589803\pi$
$991$	−17.5279			−0.556791			−0.278395	+	0.960467i	$0.589803\pi$
$992$		0			0
$993$	4.94427	+	1.88854i	0.156902	+	0.0599311i
$994$		0			0
$995$			22.1803i			0.703164i
$996$		0			0
$997$		−	25.2361i		−	0.799234i	−0.916682	−	0.399617i	$-0.869143\pi$
$997$		−	25.2361i		−	0.799234i	0.916682	−	0.399617i	$-0.130857\pi$
$998$		0			0
$999$	−1.81966	−	3.52786i	−0.0575715	−	0.111617i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.f.e.881.1		4
3.2	odd	2		1680.2.f.i.881.3		4
4.3	odd	2		210.2.b.b.41.4	yes	4
7.6	odd	2		1680.2.f.i.881.4		4
12.11	even	2		210.2.b.a.41.1	✓	4
20.3	even	4		1050.2.d.f.1049.2		4
20.7	even	4		1050.2.d.a.1049.3		4
20.19	odd	2		1050.2.b.a.251.1		4
21.20	even	2	inner	1680.2.f.e.881.2		4
28.27	even	2		210.2.b.a.41.3	yes	4
60.23	odd	4		1050.2.d.c.1049.1		4
60.47	odd	4		1050.2.d.d.1049.4		4
60.59	even	2		1050.2.b.c.251.4		4
84.83	odd	2		210.2.b.b.41.2	yes	4
140.27	odd	4		1050.2.d.c.1049.2		4
140.83	odd	4		1050.2.d.d.1049.3		4
140.139	even	2		1050.2.b.c.251.2		4
420.83	even	4		1050.2.d.a.1049.4		4
420.167	even	4		1050.2.d.f.1049.1		4
420.419	odd	2		1050.2.b.a.251.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.b.a.41.1	✓	4	12.11	even	2
210.2.b.a.41.3	yes	4	28.27	even	2
210.2.b.b.41.2	yes	4	84.83	odd	2
210.2.b.b.41.4	yes	4	4.3	odd	2
1050.2.b.a.251.1		4	20.19	odd	2
1050.2.b.a.251.3		4	420.419	odd	2
1050.2.b.c.251.2		4	140.139	even	2
1050.2.b.c.251.4		4	60.59	even	2
1050.2.d.a.1049.3		4	20.7	even	4
1050.2.d.a.1049.4		4	420.83	even	4
1050.2.d.c.1049.1		4	60.23	odd	4
1050.2.d.c.1049.2		4	140.27	odd	4
1050.2.d.d.1049.3		4	140.83	odd	4
1050.2.d.d.1049.4		4	60.47	odd	4
1050.2.d.f.1049.1		4	420.167	even	4
1050.2.d.f.1049.2		4	20.3	even	4
1680.2.f.e.881.1		4	1.1	even	1	trivial
1680.2.f.e.881.2		4	21.20	even	2	inner
1680.2.f.i.881.3		4	3.2	odd	2
1680.2.f.i.881.4		4	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)

\(f(q)\)	\(=\)	\(q+(-1.61803 - 0.618034i) q^{3} -1.00000 q^{5} +(-2.61803 + 0.381966i) q^{7} +(2.23607 + 2.00000i) q^{9} +O(q^{10})\)	\(q+(-1.61803 - 0.618034i) q^{3} -1.00000 q^{5} +(-2.61803 + 0.381966i) q^{7} +(2.23607 + 2.00000i) q^{9} -4.47214i q^{11} +1.23607i q^{13} +(1.61803 + 0.618034i) q^{15} -5.23607 q^{17} +8.47214i q^{19} +(4.47214 + 1.00000i) q^{21} +4.00000i q^{23} +1.00000 q^{25} +(-2.38197 - 4.61803i) q^{27} -7.70820i q^{29} -2.76393i q^{31} +(-2.76393 + 7.23607i) q^{33} +(2.61803 - 0.381966i) q^{35} +0.763932 q^{37} +(0.763932 - 2.00000i) q^{39} +2.47214 q^{41} +4.94427 q^{43} +(-2.23607 - 2.00000i) q^{45} -6.47214 q^{47} +(6.70820 - 2.00000i) q^{49} +(8.47214 + 3.23607i) q^{51} +0.472136i q^{53} +4.47214i q^{55} +(5.23607 - 13.7082i) q^{57} +4.47214 q^{59} -7.23607i q^{61} +(-6.61803 - 4.38197i) q^{63} -1.23607i q^{65} +12.0000 q^{67} +(2.47214 - 6.47214i) q^{69} +7.23607i q^{71} +11.2361i q^{73} +(-1.61803 - 0.618034i) q^{75} +(1.70820 + 11.7082i) q^{77} +8.94427 q^{79} +(1.00000 + 8.94427i) q^{81} +14.6525 q^{83} +5.23607 q^{85} +(-4.76393 + 12.4721i) q^{87} +5.52786 q^{89} +(-0.472136 - 3.23607i) q^{91} +(-1.70820 + 4.47214i) q^{93} -8.47214i q^{95} -0.763932i q^{97} +(8.94427 - 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 4 q^{5} - 6 q^{7}+O(q^{10}) \) 4 * q - 2 * q^3 - 4 * q^5 - 6 * q^7	\( 4 q - 2 q^{3} - 4 q^{5} - 6 q^{7} + 2 q^{15} - 12 q^{17} + 4 q^{25} - 14 q^{27} - 20 q^{33} + 6 q^{35} + 12 q^{37} + 12 q^{39} - 8 q^{41} - 16 q^{43} - 8 q^{47} + 16 q^{51} + 12 q^{57} - 22 q^{63} + 48 q^{67} - 8 q^{69} - 2 q^{75} - 20 q^{77} + 4 q^{81} - 4 q^{83} + 12 q^{85} - 28 q^{87} + 40 q^{89} + 16 q^{91} + 20 q^{93}+O(q^{100}) \) 4 * q - 2 * q^3 - 4 * q^5 - 6 * q^7 + 2 * q^15 - 12 * q^17 + 4 * q^25 - 14 * q^27 - 20 * q^33 + 6 * q^35 + 12 * q^37 + 12 * q^39 - 8 * q^41 - 16 * q^43 - 8 * q^47 + 16 * q^51 + 12 * q^57 - 22 * q^63 + 48 * q^67 - 8 * q^69 - 2 * q^75 - 20 * q^77 + 4 * q^81 - 4 * q^83 + 12 * q^85 - 28 * q^87 + 40 * q^89 + 16 * q^91 + 20 * q^93

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