LMFDB - Embedded newform 693.2.i.f.298.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(100,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("693.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.i (of order $3$, degree $2$, 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:	$5.53363286007$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			298.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	693.298
Dual form			693.2.i.f.100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.20711 - 2.09077i) q^{2} +(-1.91421 + 3.31552i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(1.00000 - 2.44949i) q^{7} +4.41421 q^{8} +O(q^{10})$	\(q+(-1.20711 - 2.09077i) q^{2} +(-1.91421 + 3.31552i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(1.00000 - 2.44949i) q^{7} +4.41421 q^{8} +(-2.41421 + 4.18154i) q^{10} +(0.500000 - 0.866025i) q^{11} -0.828427 q^{13} +(-6.32843 + 0.866025i) q^{14} +(-1.50000 - 2.59808i) q^{16} +(-2.20711 + 3.82282i) q^{17} +(-3.62132 - 6.27231i) q^{19} +7.65685 q^{20} -2.41421 q^{22} +(-3.50000 - 6.06218i) q^{23} +(0.500000 - 0.866025i) q^{25} +(1.00000 + 1.73205i) q^{26} +(6.20711 + 8.00436i) q^{28} -3.24264 q^{29} +(-2.82843 + 4.89898i) q^{31} +(0.792893 - 1.37333i) q^{32} +10.6569 q^{34} +(-5.24264 + 0.717439i) q^{35} +(4.74264 + 8.21449i) q^{37} +(-8.74264 + 15.1427i) q^{38} +(-4.41421 - 7.64564i) q^{40} +1.17157 q^{41} -2.75736 q^{43} +(1.91421 + 3.31552i) q^{44} +(-8.44975 + 14.6354i) q^{46} +(4.91421 + 8.51167i) q^{47} +(-5.00000 - 4.89898i) q^{49} -2.41421 q^{50} +(1.58579 - 2.74666i) q^{52} +(-3.58579 + 6.21076i) q^{53} -2.00000 q^{55} +(4.41421 - 10.8126i) q^{56} +(3.91421 + 6.77962i) q^{58} +(-4.32843 + 7.49706i) q^{59} +(2.00000 + 3.46410i) q^{61} +13.6569 q^{62} -9.82843 q^{64} +(0.828427 + 1.43488i) q^{65} +(1.58579 - 2.74666i) q^{67} +(-8.44975 - 14.6354i) q^{68} +(7.82843 + 10.0951i) q^{70} +4.17157 q^{71} +(-0.171573 + 0.297173i) q^{73} +(11.4497 - 19.8315i) q^{74} +27.7279 q^{76} +(-1.62132 - 2.09077i) q^{77} +(-6.65685 - 11.5300i) q^{79} +(-3.00000 + 5.19615i) q^{80} +(-1.41421 - 2.44949i) q^{82} +2.82843 q^{83} +8.82843 q^{85} +(3.32843 + 5.76500i) q^{86} +(2.20711 - 3.82282i) q^{88} +(-7.07107 - 12.2474i) q^{89} +(-0.828427 + 2.02922i) q^{91} +26.7990 q^{92} +(11.8640 - 20.5490i) q^{94} +(-7.24264 + 12.5446i) q^{95} -11.4853 q^{97} +(-4.20711 + 16.3674i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{4} - 4 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^4 - 4 * q^5 + 4 * q^7 + 12 * q^8	$ 4 q - 2 q^{2} - 2 q^{4} - 4 q^{5} + 4 q^{7} + 12 q^{8} - 4 q^{10} + 2 q^{11} + 8 q^{13} - 14 q^{14} - 6 q^{16} - 6 q^{17} - 6 q^{19} + 8 q^{20} - 4 q^{22} - 14 q^{23} + 2 q^{25} + 4 q^{26} + 22 q^{28} + 4 q^{29} + 6 q^{32} + 20 q^{34} - 4 q^{35} + 2 q^{37} - 18 q^{38} - 12 q^{40} + 16 q^{41} - 28 q^{43} + 2 q^{44} - 14 q^{46} + 14 q^{47} - 20 q^{49} - 4 q^{50} + 12 q^{52} - 20 q^{53} - 8 q^{55} + 12 q^{56} + 10 q^{58} - 6 q^{59} + 8 q^{61} + 32 q^{62} - 28 q^{64} - 8 q^{65} + 12 q^{67} - 14 q^{68} + 20 q^{70} + 28 q^{71} - 12 q^{73} + 26 q^{74} + 60 q^{76} + 2 q^{77} - 4 q^{79} - 12 q^{80} + 24 q^{85} + 2 q^{86} + 6 q^{88} + 8 q^{91} + 28 q^{92} + 22 q^{94} - 12 q^{95} - 12 q^{97} - 14 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^4 - 4 * q^5 + 4 * q^7 + 12 * q^8 - 4 * q^10 + 2 * q^11 + 8 * q^13 - 14 * q^14 - 6 * q^16 - 6 * q^17 - 6 * q^19 + 8 * q^20 - 4 * q^22 - 14 * q^23 + 2 * q^25 + 4 * q^26 + 22 * q^28 + 4 * q^29 + 6 * q^32 + 20 * q^34 - 4 * q^35 + 2 * q^37 - 18 * q^38 - 12 * q^40 + 16 * q^41 - 28 * q^43 + 2 * q^44 - 14 * q^46 + 14 * q^47 - 20 * q^49 - 4 * q^50 + 12 * q^52 - 20 * q^53 - 8 * q^55 + 12 * q^56 + 10 * q^58 - 6 * q^59 + 8 * q^61 + 32 * q^62 - 28 * q^64 - 8 * q^65 + 12 * q^67 - 14 * q^68 + 20 * q^70 + 28 * q^71 - 12 * q^73 + 26 * q^74 + 60 * q^76 + 2 * q^77 - 4 * q^79 - 12 * q^80 + 24 * q^85 + 2 * q^86 + 6 * q^88 + 8 * q^91 + 28 * q^92 + 22 * q^94 - 12 * q^95 - 12 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$442$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	−1.20711	−	2.09077i	−0.853553	−	1.47840i	−0.877981	−	0.478696i	$-0.841110\pi$
$2$	−1.20711	−	2.09077i	−0.853553	−	1.47840i	0.0244272	−	0.999702i	$-0.492224\pi$
$3$		0			0
$3$		0			0
$4$	−1.91421	+	3.31552i	−0.957107	+	1.65776i
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$	1.00000	−	2.44949i	0.377964	−	0.925820i
$7$	1.00000	−	2.44949i	0.377964	−	0.925820i
$8$	4.41421			1.56066
$9$		0			0
$10$	−2.41421	+	4.18154i	−0.763441	+	1.32232i
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i
$12$		0			0
$13$	−0.828427			−0.229764			−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427			−0.229764			−0.114882	+	0.993379i	$0.536649\pi$
$14$	−6.32843	+	0.866025i	−1.69134	+	0.231455i
$15$		0			0
$16$	−1.50000	−	2.59808i	−0.375000	−	0.649519i
$17$	−2.20711	+	3.82282i	−0.535302	+	0.927170i	0.463847	+	0.885916i	$0.346469\pi$
$17$	−2.20711	+	3.82282i	−0.535302	+	0.927170i	−0.999149	+	0.0412548i	$0.986864\pi$
$18$		0			0
$19$	−3.62132	−	6.27231i	−0.830788	−	1.43897i	−0.897414	−	0.441189i	$-0.854557\pi$
$19$	−3.62132	−	6.27231i	−0.830788	−	1.43897i	0.0666264	−	0.997778i	$-0.478776\pi$
$20$	7.65685			1.71212
$21$		0			0
$22$	−2.41421			−0.514712
$23$	−3.50000	−	6.06218i	−0.729800	−	1.26405i	−0.956967	−	0.290196i	$-0.906280\pi$
$23$	−3.50000	−	6.06218i	−0.729800	−	1.26405i	0.227167	−	0.973856i	$-0.427054\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$	1.00000	+	1.73205i	0.196116	+	0.339683i
$27$		0			0
$28$	6.20711	+	8.00436i	1.17303	+	1.51268i
$29$	−3.24264			−0.602143			−0.301072	−	0.953602i	$-0.597344\pi$
$29$	−3.24264			−0.602143			−0.301072	+	0.953602i	$0.597344\pi$
$30$		0			0
$31$	−2.82843	+	4.89898i	−0.508001	+	0.879883i	0.491957	+	0.870620i	$0.336282\pi$
$31$	−2.82843	+	4.89898i	−0.508001	+	0.879883i	−0.999957	+	0.00926296i	$0.997051\pi$
$32$	0.792893	−	1.37333i	0.140165	−	0.242773i
$33$		0			0
$34$	10.6569			1.82764
$35$	−5.24264	+	0.717439i	−0.886168	+	0.121269i
$36$		0			0
$37$	4.74264	+	8.21449i	0.779685	+	1.35045i	0.932123	+	0.362142i	$0.117954\pi$
$37$	4.74264	+	8.21449i	0.779685	+	1.35045i	−0.152438	+	0.988313i	$0.548712\pi$
$38$	−8.74264	+	15.1427i	−1.41824	+	2.45647i
$39$		0			0
$40$	−4.41421	−	7.64564i	−0.697948	−	1.20888i
$41$	1.17157			0.182969			0.0914845	−	0.995807i	$-0.470839\pi$
$41$	1.17157			0.182969			0.0914845	+	0.995807i	$0.470839\pi$
$42$		0			0
$43$	−2.75736			−0.420493			−0.210247	−	0.977648i	$-0.567427\pi$
$43$	−2.75736			−0.420493			−0.210247	+	0.977648i	$0.567427\pi$
$44$	1.91421	+	3.31552i	0.288579	+	0.499833i
$45$		0			0
$46$	−8.44975	+	14.6354i	−1.24585	+	2.15787i
$47$	4.91421	+	8.51167i	0.716812	+	1.24155i	0.962257	+	0.272144i	$0.0877326\pi$
$47$	4.91421	+	8.51167i	0.716812	+	1.24155i	−0.245445	+	0.969411i	$0.578934\pi$
$48$		0			0
$49$	−5.00000	−	4.89898i	−0.714286	−	0.699854i
$50$	−2.41421			−0.341421
$51$		0			0
$52$	1.58579	−	2.74666i	0.219909	−	0.380894i
$53$	−3.58579	+	6.21076i	−0.492546	+	0.853114i	−0.999963	−	0.00858626i	$-0.997267\pi$
$53$	−3.58579	+	6.21076i	−0.492546	+	0.853114i	0.507417	+	0.861700i	$0.330600\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$	4.41421	−	10.8126i	0.589874	−	1.44489i
$57$		0			0
$58$	3.91421	+	6.77962i	0.513961	+	0.890207i
$59$	−4.32843	+	7.49706i	−0.563513	+	0.976034i	0.433673	+	0.901070i	$0.357217\pi$
$59$	−4.32843	+	7.49706i	−0.563513	+	0.976034i	−0.997186	+	0.0749632i	$0.976116\pi$
$60$		0			0
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	0.965187	−	0.261562i	$-0.0842377\pi$
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	−0.709113	+	0.705095i	$0.750904\pi$
$62$	13.6569			1.73442
$63$		0			0
$64$	−9.82843			−1.22855
$65$	0.828427	+	1.43488i	0.102754	+	0.177975i
$66$		0			0
$67$	1.58579	−	2.74666i	0.193735	−	0.335558i	−0.752750	−	0.658306i	$-0.771273\pi$
$67$	1.58579	−	2.74666i	0.193735	−	0.335558i	0.946485	+	0.322748i	$0.104607\pi$
$68$	−8.44975	−	14.6354i	−1.02468	−	1.77480i
$69$		0			0
$70$	7.82843	+	10.0951i	0.935676	+	1.20660i
$71$	4.17157			0.495075			0.247537	−	0.968878i	$-0.420379\pi$
$71$	4.17157			0.495075			0.247537	+	0.968878i	$0.420379\pi$
$72$		0			0
$73$	−0.171573	+	0.297173i	−0.0200811	+	0.0347815i	−0.875891	−	0.482508i	$-0.839726\pi$
$73$	−0.171573	+	0.297173i	−0.0200811	+	0.0347815i	0.855810	+	0.517290i	$0.173059\pi$
$74$	11.4497	−	19.8315i	1.33101	−	2.30537i
$75$		0			0
$76$	27.7279			3.18061
$77$	−1.62132	−	2.09077i	−0.184767	−	0.238265i
$78$		0			0
$79$	−6.65685	−	11.5300i	−0.748955	−	1.29723i	−0.948324	−	0.317303i	$-0.897223\pi$
$79$	−6.65685	−	11.5300i	−0.748955	−	1.29723i	0.199370	−	0.979924i	$-0.436111\pi$
$80$	−3.00000	+	5.19615i	−0.335410	+	0.580948i
$81$		0			0
$82$	−1.41421	−	2.44949i	−0.156174	−	0.270501i
$83$	2.82843			0.310460			0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843			0.310460			0.155230	+	0.987878i	$0.450388\pi$
$84$		0			0
$85$	8.82843			0.957577
$86$	3.32843	+	5.76500i	0.358914	+	0.621656i
$87$		0			0
$88$	2.20711	−	3.82282i	0.235278	−	0.407514i
$89$	−7.07107	−	12.2474i	−0.749532	−	1.29823i	−0.948047	−	0.318129i	$-0.896945\pi$
$89$	−7.07107	−	12.2474i	−0.749532	−	1.29823i	0.198516	−	0.980098i	$-0.436388\pi$
$90$		0			0
$91$	−0.828427	+	2.02922i	−0.0868428	+	0.212720i
$92$	26.7990			2.79399
$93$		0			0
$94$	11.8640	−	20.5490i	1.22367	−	2.11947i
$95$	−7.24264	+	12.5446i	−0.743079	+	1.28705i
$96$		0			0
$97$	−11.4853			−1.16615			−0.583077	−	0.812417i	$-0.698151\pi$
$97$	−11.4853			−1.16615			−0.583077	+	0.812417i	$0.698151\pi$
$98$	−4.20711	+	16.3674i	−0.424982	+	1.65336i
$99$		0			0
$100$	1.91421	+	3.31552i	0.191421	+	0.331552i
$101$	2.44975	−	4.24309i	0.243759	−	0.422203i	−0.718023	−	0.696019i	$-0.754953\pi$
$101$	2.44975	−	4.24309i	0.243759	−	0.422203i	0.961782	+	0.273816i	$0.0882861\pi$
$102$		0			0
$103$	−6.24264	−	10.8126i	−0.615106	−	1.06539i	−0.990366	−	0.138475i	$-0.955780\pi$
$103$	−6.24264	−	10.8126i	−0.615106	−	1.06539i	0.375260	−	0.926919i	$-0.377553\pi$
$104$	−3.65685			−0.358584
$105$		0			0
$106$	17.3137			1.68166
$107$	4.82843	+	8.36308i	0.466782	+	0.808490i	0.999280	−	0.0379415i	$-0.0120801\pi$
$107$	4.82843	+	8.36308i	0.466782	+	0.808490i	−0.532498	+	0.846431i	$0.678747\pi$
$108$		0			0
$109$	3.41421	−	5.91359i	0.327022	−	0.566419i	−0.654897	−	0.755718i	$-0.727288\pi$
$109$	3.41421	−	5.91359i	0.327022	−	0.566419i	0.981920	+	0.189299i	$0.0606214\pi$
$110$	2.41421	+	4.18154i	0.230186	+	0.398694i
$111$		0			0
$112$	−7.86396	+	1.07616i	−0.743074	+	0.101687i
$113$	7.65685			0.720296			0.360148	−	0.932895i	$-0.382726\pi$
$113$	7.65685			0.720296			0.360148	+	0.932895i	$0.382726\pi$
$114$		0			0
$115$	−7.00000	+	12.1244i	−0.652753	+	1.13060i
$116$	6.20711	−	10.7510i	0.576315	−	0.998208i
$117$		0			0
$118$	20.8995			1.92395
$119$	7.15685	+	9.22911i	0.656068	+	0.846031i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$	4.82843	−	8.36308i	0.437145	−	0.757158i
$123$		0			0
$124$	−10.8284	−	18.7554i	−0.972421	−	1.68428i
$125$	−12.0000			−1.07331
$126$		0			0
$127$	−16.0711			−1.42608			−0.713038	−	0.701125i	$-0.752681\pi$
$127$	−16.0711			−1.42608			−0.713038	+	0.701125i	$0.752681\pi$
$128$	10.2782	+	17.8023i	0.908471	+	1.57352i
$129$		0			0
$130$	2.00000	−	3.46410i	0.175412	−	0.303822i
$131$	−2.58579	−	4.47871i	−0.225921	−	0.391307i	0.730674	−	0.682726i	$-0.239206\pi$
$131$	−2.58579	−	4.47871i	−0.225921	−	0.391307i	−0.956595	+	0.291419i	$0.905873\pi$
$132$		0			0
$133$	−18.9853	+	2.59808i	−1.64623	+	0.225282i
$134$	−7.65685			−0.661451
$135$		0			0
$136$	−9.74264	+	16.8747i	−0.835425	+	1.44700i
$137$	0.242641	−	0.420266i	0.0207302	−	0.0359057i	−0.855474	−	0.517845i	$-0.826734\pi$
$137$	0.242641	−	0.420266i	0.0207302	−	0.0359057i	0.876204	+	0.481940i	$0.160068\pi$
$138$		0			0
$139$	8.41421			0.713684			0.356842	−	0.934165i	$-0.383853\pi$
$139$	8.41421			0.713684			0.356842	+	0.934165i	$0.383853\pi$
$140$	7.65685	−	18.7554i	0.647122	−	1.58512i
$141$		0			0
$142$	−5.03553	−	8.72180i	−0.422573	−	0.731917i
$143$	−0.414214	+	0.717439i	−0.0346383	+	0.0599953i
$144$		0			0
$145$	3.24264	+	5.61642i	0.269287	+	0.466418i
$146$	0.828427			0.0685611
$147$		0			0
$148$	−36.3137			−2.98497
$149$	−11.1066	−	19.2372i	−0.909888	−	1.57597i	−0.814218	−	0.580560i	$-0.802834\pi$
$149$	−11.1066	−	19.2372i	−0.909888	−	1.57597i	−0.0956705	−	0.995413i	$-0.530500\pi$
$150$		0			0
$151$	9.44975	−	16.3674i	0.769010	−	1.33196i	−0.169091	−	0.985600i	$-0.554083\pi$
$151$	9.44975	−	16.3674i	0.769010	−	1.33196i	0.938101	−	0.346363i	$-0.112583\pi$
$152$	−15.9853	−	27.6873i	−1.29658	−	2.24574i
$153$		0			0
$154$	−2.41421	+	5.91359i	−0.194543	+	0.476531i
$155$	11.3137			0.908739
$156$		0			0
$157$	−3.50000	+	6.06218i	−0.279330	+	0.483814i	−0.971219	−	0.238190i	$-0.923446\pi$
$157$	−3.50000	+	6.06218i	−0.279330	+	0.483814i	0.691888	+	0.722005i	$0.256779\pi$
$158$	−16.0711	+	27.8359i	−1.27855	+	2.21451i
$159$		0			0
$160$	−3.17157			−0.250735
$161$	−18.3492	+	2.51104i	−1.44612	+	0.197897i
$162$		0			0
$163$	10.4853	+	18.1610i	0.821271	+	1.42248i	0.904736	+	0.425972i	$0.140068\pi$
$163$	10.4853	+	18.1610i	0.821271	+	1.42248i	−0.0834658	+	0.996511i	$0.526599\pi$
$164$	−2.24264	+	3.88437i	−0.175121	+	0.303318i
$165$		0			0
$166$	−3.41421	−	5.91359i	−0.264994	−	0.458984i
$167$	−5.17157			−0.400188			−0.200094	−	0.979777i	$-0.564125\pi$
$167$	−5.17157			−0.400188			−0.200094	+	0.979777i	$0.564125\pi$
$168$		0			0
$169$	−12.3137			−0.947208
$170$	−10.6569	−	18.4582i	−0.817343	−	1.41568i
$171$		0			0
$172$	5.27817	−	9.14207i	0.402457	−	0.697076i
$173$	−7.41421	−	12.8418i	−0.563692	−	0.976344i	−0.997170	−	0.0751793i	$-0.976047\pi$
$173$	−7.41421	−	12.8418i	−0.563692	−	0.976344i	0.433478	−	0.901164i	$-0.357286\pi$
$174$		0			0
$175$	−1.62132	−	2.09077i	−0.122560	−	0.158047i
$176$	−3.00000			−0.226134
$177$		0			0
$178$	−17.0711	+	29.5680i	−1.27953	+	2.21621i
$179$	8.91421	−	15.4399i	0.666280	−	1.15403i	−0.312657	−	0.949866i	$-0.601219\pi$
$179$	8.91421	−	15.4399i	0.666280	−	1.15403i	0.978937	−	0.204164i	$-0.0654476\pi$
$180$		0			0
$181$	−11.6569			−0.866447			−0.433224	−	0.901286i	$-0.642624\pi$
$181$	−11.6569			−0.866447			−0.433224	+	0.901286i	$0.642624\pi$
$182$	5.24264	−	0.717439i	0.388610	−	0.0531801i
$183$		0			0
$184$	−15.4497	−	26.7597i	−1.13897	−	1.97275i
$185$	9.48528	−	16.4290i	0.697372	−	1.20788i
$186$		0			0
$187$	2.20711	+	3.82282i	0.161400	+	0.279552i
$188$	−37.6274			−2.74426
$189$		0			0
$190$	34.9706			2.53703
$191$	−10.8284	−	18.7554i	−0.783517	−	1.35709i	−0.929881	−	0.367861i	$-0.880090\pi$
$191$	−10.8284	−	18.7554i	−0.783517	−	1.35709i	0.146363	−	0.989231i	$-0.453243\pi$
$192$		0			0
$193$	6.07107	−	10.5154i	0.437005	−	0.756915i	−0.560452	−	0.828187i	$-0.689373\pi$
$193$	6.07107	−	10.5154i	0.437005	−	0.756915i	0.997457	+	0.0712721i	$0.0227059\pi$
$194$	13.8640	+	24.0131i	0.995374	+	1.72404i
$195$		0			0
$196$	25.8137	−	7.19988i	1.84384	−	0.514277i
$197$	16.4142			1.16946			0.584732	−	0.811226i	$-0.301200\pi$
$197$	16.4142			1.16946			0.584732	+	0.811226i	$0.301200\pi$
$198$		0			0
$199$	9.89949	−	17.1464i	0.701757	−	1.21548i	−0.266093	−	0.963947i	$-0.585733\pi$
$199$	9.89949	−	17.1464i	0.701757	−	1.21548i	0.967849	−	0.251531i	$-0.0809339\pi$
$200$	2.20711	−	3.82282i	0.156066	−	0.270314i
$201$		0			0
$202$	−11.8284			−0.832245
$203$	−3.24264	+	7.94282i	−0.227589	+	0.557476i
$204$		0			0
$205$	−1.17157	−	2.02922i	−0.0818262	−	0.141727i
$206$	−15.0711	+	26.1039i	−1.05005	+	1.81874i
$207$		0			0
$208$	1.24264	+	2.15232i	0.0861616	+	0.149236i
$209$	−7.24264			−0.500984
$210$		0			0
$211$	14.9706			1.03062			0.515308	−	0.857005i	$-0.327678\pi$
$211$	14.9706			1.03062			0.515308	+	0.857005i	$0.327678\pi$
$212$	−13.7279	−	23.7775i	−0.942838	−	1.63304i
$213$		0			0
$214$	11.6569	−	20.1903i	0.796846	−	1.38018i
$215$	2.75736	+	4.77589i	0.188050	+	0.325713i
$216$		0			0
$217$	9.17157	+	11.8272i	0.622607	+	0.802881i
$218$	−16.4853			−1.11652
$219$		0			0
$220$	3.82843	−	6.63103i	0.258113	−	0.447064i
$221$	1.82843	−	3.16693i	0.122993	−	0.213031i
$222$		0			0
$223$	−22.9706			−1.53822			−0.769111	−	0.639115i	$-0.779301\pi$
$223$	−22.9706			−1.53822			−0.769111	+	0.639115i	$0.779301\pi$
$224$	−2.57107	−	3.31552i	−0.171787	−	0.221527i
$225$		0			0
$226$	−9.24264	−	16.0087i	−0.614811	−	1.06488i
$227$	7.48528	−	12.9649i	0.496816	−	0.860510i	−0.503178	−	0.864183i	$-0.667836\pi$
$227$	7.48528	−	12.9649i	0.496816	−	0.860510i	0.999993	+	0.00367316i	$0.00116921\pi$
$228$		0			0
$229$	7.82843	+	13.5592i	0.517317	+	0.896019i	0.999798	+	0.0201128i	$0.00640252\pi$
$229$	7.82843	+	13.5592i	0.517317	+	0.896019i	−0.482481	+	0.875907i	$0.660264\pi$
$230$	33.7990			2.22864
$231$		0			0
$232$	−14.3137			−0.939741
$233$	5.20711	+	9.01897i	0.341129	+	0.590853i	0.984643	−	0.174582i	$-0.0558574\pi$
$233$	5.20711	+	9.01897i	0.341129	+	0.590853i	−0.643514	+	0.765435i	$0.722524\pi$
$234$		0			0
$235$	9.82843	−	17.0233i	0.641136	−	1.11048i
$236$	−16.5711	−	28.7019i	−1.07868	−	1.86834i
$237$		0			0
$238$	10.6569	−	26.1039i	0.690781	−	1.69206i
$239$	6.48528			0.419498			0.209749	−	0.977755i	$-0.432735\pi$
$239$	6.48528			0.419498			0.209749	+	0.977755i	$0.432735\pi$
$240$		0			0
$241$	−3.65685	+	6.33386i	−0.235559	+	0.408000i	−0.959435	−	0.281930i	$-0.909025\pi$
$241$	−3.65685	+	6.33386i	−0.235559	+	0.408000i	0.723876	+	0.689930i	$0.242359\pi$
$242$	−1.20711	+	2.09077i	−0.0775958	+	0.134400i
$243$		0			0
$244$	−15.3137			−0.980360
$245$	−3.48528	+	13.5592i	−0.222666	+	0.866268i
$246$		0			0
$247$	3.00000	+	5.19615i	0.190885	+	0.330623i
$248$	−12.4853	+	21.6251i	−0.792816	+	1.37320i
$249$		0			0
$250$	14.4853	+	25.0892i	0.916130	+	1.58678i
$251$	2.51472			0.158728			0.0793638	−	0.996846i	$-0.474711\pi$
$251$	2.51472			0.158728			0.0793638	+	0.996846i	$0.474711\pi$
$252$		0			0
$253$	−7.00000			−0.440086
$254$	19.3995	+	33.6009i	1.21723	+	2.10831i
$255$		0			0
$256$	14.9853	−	25.9553i	0.936580	−	1.62220i
$257$	10.7279	+	18.5813i	0.669189	+	1.15907i	0.978131	+	0.207989i	$0.0666918\pi$
$257$	10.7279	+	18.5813i	0.669189	+	1.15907i	−0.308942	+	0.951081i	$0.599975\pi$
$258$		0			0
$259$	24.8640	−	3.40256i	1.54497	−	0.211424i
$260$	−6.34315			−0.393385
$261$		0			0
$262$	−6.24264	+	10.8126i	−0.385672	+	0.668003i
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	−0.442943	−	0.896550i	$-0.646065\pi$
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	0.997906	−	0.0646755i	$-0.0206012\pi$
$264$		0			0
$265$	14.3431			0.881092
$266$	28.3492	+	36.5577i	1.73820	+	2.24150i
$267$		0			0
$268$	6.07107	+	10.5154i	0.370849	+	0.642330i
$269$	−13.8995	+	24.0746i	−0.847467	+	1.46786i	0.0359941	+	0.999352i	$0.488540\pi$
$269$	−13.8995	+	24.0746i	−0.847467	+	1.46786i	−0.883461	+	0.468504i	$0.844793\pi$
$270$		0			0
$271$	11.4853	+	19.8931i	0.697681	+	1.20842i	0.969268	+	0.246005i	$0.0791181\pi$
$271$	11.4853	+	19.8931i	0.697681	+	1.20842i	−0.271587	+	0.962414i	$0.587549\pi$
$272$	13.2426			0.802953
$273$		0			0
$274$	−1.17157			−0.0707773
$275$	−0.500000	−	0.866025i	−0.0301511	−	0.0522233i
$276$		0			0
$277$	12.8284	−	22.2195i	0.770785	−	1.33504i	−0.166348	−	0.986067i	$-0.553197\pi$
$277$	12.8284	−	22.2195i	0.770785	−	1.33504i	0.937133	−	0.348972i	$-0.113469\pi$
$278$	−10.1569	−	17.5922i	−0.609168	−	1.05511i
$279$		0			0
$280$	−23.1421	+	3.16693i	−1.38301	+	0.189260i
$281$	−14.4142			−0.859880			−0.429940	−	0.902857i	$-0.641465\pi$
$281$	−14.4142			−0.859880			−0.429940	+	0.902857i	$0.641465\pi$
$282$		0			0
$283$	−10.1716	+	17.6177i	−0.604637	+	1.04726i	0.387471	+	0.921882i	$0.373349\pi$
$283$	−10.1716	+	17.6177i	−0.604637	+	1.04726i	−0.992109	+	0.125381i	$0.959985\pi$
$284$	−7.98528	+	13.8309i	−0.473839	+	0.820714i
$285$		0			0
$286$	2.00000			0.118262
$287$	1.17157	−	2.86976i	0.0691558	−	0.169396i
$288$		0			0
$289$	−1.24264	−	2.15232i	−0.0730965	−	0.126607i
$290$	7.82843	−	13.5592i	0.459701	−	0.796226i
$291$		0			0
$292$	−0.656854	−	1.13770i	−0.0384395	−	0.0665791i
$293$	−5.72792			−0.334629			−0.167314	−	0.985904i	$-0.553509\pi$
$293$	−5.72792			−0.334629			−0.167314	+	0.985904i	$0.553509\pi$
$294$		0			0
$295$	17.3137			1.00804
$296$	20.9350	+	36.2605i	1.21682	+	2.10760i
$297$		0			0
$298$	−26.8137	+	46.4427i	−1.55328	+	2.69035i
$299$	2.89949	+	5.02207i	0.167682	+	0.290434i
$300$		0			0
$301$	−2.75736	+	6.75412i	−0.158932	+	0.389301i
$302$	−45.6274			−2.62556
$303$		0			0
$304$	−10.8640	+	18.8169i	−0.623091	+	1.07923i
$305$	4.00000	−	6.92820i	0.229039	−	0.396708i
$306$		0			0
$307$	−17.3137			−0.988146			−0.494073	−	0.869421i	$-0.664492\pi$
$307$	−17.3137			−0.988146			−0.494073	+	0.869421i	$0.664492\pi$
$308$	10.0355	−	1.37333i	0.571828	−	0.0782528i
$309$		0			0
$310$	−13.6569	−	23.6544i	−0.775657	−	1.34348i
$311$	14.8137	−	25.6581i	0.840008	−	1.45494i	−0.0498782	−	0.998755i	$-0.515883\pi$
$311$	14.8137	−	25.6581i	0.840008	−	1.45494i	0.889887	−	0.456182i	$-0.150783\pi$
$312$		0			0
$313$	−2.39949	−	4.15605i	−0.135627	−	0.234914i	0.790210	−	0.612837i	$-0.209972\pi$
$313$	−2.39949	−	4.15605i	−0.135627	−	0.234914i	−0.925837	+	0.377923i	$0.876638\pi$
$314$	16.8995			0.953694
$315$		0			0
$316$	50.9706			2.86732
$317$	−12.6569	−	21.9223i	−0.710880	−	1.23128i	−0.964527	−	0.263983i	$-0.914964\pi$
$317$	−12.6569	−	21.9223i	−0.710880	−	1.23128i	0.253648	−	0.967297i	$-0.418370\pi$
$318$		0			0
$319$	−1.62132	+	2.80821i	−0.0907765	+	0.157230i
$320$	9.82843	+	17.0233i	0.549426	+	0.951633i
$321$		0			0
$322$	27.3995	+	35.3330i	1.52691	+	1.96903i
$323$	31.9706			1.77889
$324$		0			0
$325$	−0.414214	+	0.717439i	−0.0229764	+	0.0397964i
$326$	25.3137	−	43.8446i	1.40200	−	2.42833i
$327$		0			0
$328$	5.17157			0.285552
$329$	25.7635	−	3.52565i	1.42039	−	0.194375i
$330$		0			0
$331$	−11.2426	−	19.4728i	−0.617951	−	1.07032i	−0.989859	−	0.142053i	$-0.954629\pi$
$331$	−11.2426	−	19.4728i	−0.617951	−	1.07032i	0.371908	−	0.928270i	$-0.378704\pi$
$332$	−5.41421	+	9.37769i	−0.297144	+	0.514668i
$333$		0			0
$334$	6.24264	+	10.8126i	0.341582	+	0.591638i
$335$	−6.34315			−0.346563
$336$		0			0
$337$	3.85786			0.210151			0.105076	−	0.994464i	$-0.466492\pi$
$337$	3.85786			0.210151			0.105076	+	0.994464i	$0.466492\pi$
$338$	14.8640	+	25.7451i	0.808493	+	1.40035i
$339$		0			0
$340$	−16.8995	+	29.2708i	−0.916504	+	1.58743i
$341$	2.82843	+	4.89898i	0.153168	+	0.265295i
$342$		0			0
$343$	−17.0000	+	7.34847i	−0.917914	+	0.396780i
$344$	−12.1716			−0.656247
$345$		0			0
$346$	−17.8995	+	31.0028i	−0.962283	+	1.66672i
$347$	7.48528	−	12.9649i	0.401831	−	0.695992i	−0.592116	−	0.805853i	$-0.701707\pi$
$347$	7.48528	−	12.9649i	0.401831	−	0.695992i	0.993947	+	0.109861i	$0.0350406\pi$
$348$		0			0
$349$	10.9706			0.587241			0.293620	−	0.955922i	$-0.405140\pi$
$349$	10.9706			0.587241			0.293620	+	0.955922i	$0.405140\pi$
$350$	−2.41421	+	5.91359i	−0.129045	+	0.316095i
$351$		0			0
$352$	−0.792893	−	1.37333i	−0.0422614	−	0.0731988i
$353$	6.65685	−	11.5300i	0.354309	−	0.613680i	−0.632691	−	0.774405i	$-0.718050\pi$
$353$	6.65685	−	11.5300i	0.354309	−	0.613680i	0.986999	+	0.160724i	$0.0513829\pi$
$354$		0			0
$355$	−4.17157	−	7.22538i	−0.221404	−	0.383483i
$356$	54.1421			2.86953
$357$		0			0
$358$	−43.0416			−2.27482
$359$	−4.00000	−	6.92820i	−0.211112	−	0.365657i	0.740951	−	0.671559i	$-0.234375\pi$
$359$	−4.00000	−	6.92820i	−0.211112	−	0.365657i	−0.952063	+	0.305903i	$0.901042\pi$
$360$		0			0
$361$	−16.7279	+	28.9736i	−0.880417	+	1.52493i
$362$	14.0711	+	24.3718i	0.739559	+	1.28095i
$363$		0			0
$364$	−5.14214	−	6.63103i	−0.269521	−	0.347560i
$365$	0.686292			0.0359221
$366$		0			0
$367$	2.07107	−	3.58719i	0.108109	−	0.187250i	−0.806895	−	0.590695i	$-0.798854\pi$
$367$	2.07107	−	3.58719i	0.108109	−	0.187250i	0.915004	+	0.403445i	$0.132187\pi$
$368$	−10.5000	+	18.1865i	−0.547350	+	0.948039i
$369$		0			0
$370$	−45.7990			−2.38098
$371$	11.6274	+	14.9941i	0.603665	+	0.778455i
$372$		0			0
$373$	0.828427	+	1.43488i	0.0428943	+	0.0742952i	0.886675	−	0.462392i	$-0.153009\pi$
$373$	0.828427	+	1.43488i	0.0428943	+	0.0742952i	−0.843781	+	0.536687i	$0.819675\pi$
$374$	5.32843	−	9.22911i	0.275526	−	0.477226i
$375$		0			0
$376$	21.6924	+	37.5723i	1.11870	+	1.93764i
$377$	2.68629			0.138351
$378$		0			0
$379$	18.8284			0.967151			0.483576	−	0.875303i	$-0.339338\pi$
$379$	18.8284			0.967151			0.483576	+	0.875303i	$0.339338\pi$
$380$	−27.7279	−	48.0262i	−1.42241	−	2.46369i
$381$		0			0
$382$	−26.1421	+	45.2795i	−1.33755	+	2.31670i
$383$	1.15685	+	2.00373i	0.0591125	+	0.102386i	0.894067	−	0.447933i	$-0.147840\pi$
$383$	1.15685	+	2.00373i	0.0591125	+	0.102386i	−0.834955	+	0.550319i	$0.814506\pi$
$384$		0			0
$385$	−2.00000	+	4.89898i	−0.101929	+	0.249675i
$386$	−29.3137			−1.49203
$387$		0			0
$388$	21.9853	−	38.0796i	1.11613	−	1.93320i
$389$	−5.89949	+	10.2182i	−0.299116	+	0.518085i	−0.975934	−	0.218066i	$-0.930025\pi$
$389$	−5.89949	+	10.2182i	−0.299116	+	0.518085i	0.676818	+	0.736151i	$0.263359\pi$
$390$		0			0
$391$	30.8995			1.56265
$392$	−22.0711	−	21.6251i	−1.11476	−	1.09223i
$393$		0			0
$394$	−19.8137	−	34.3183i	−0.998200	−	1.72893i
$395$	−13.3137	+	23.0600i	−0.669885	+	1.16028i
$396$		0			0
$397$	−8.74264	−	15.1427i	−0.438781	−	0.759990i	0.558815	−	0.829292i	$-0.311256\pi$
$397$	−8.74264	−	15.1427i	−0.438781	−	0.759990i	−0.997596	+	0.0693020i	$0.977923\pi$
$398$	−47.7990			−2.39595
$399$		0			0
$400$	−3.00000			−0.150000
$401$	3.75736	+	6.50794i	0.187634	+	0.324991i	0.944461	−	0.328624i	$-0.106585\pi$
$401$	3.75736	+	6.50794i	0.187634	+	0.324991i	−0.756827	+	0.653615i	$0.773252\pi$
$402$		0			0
$403$	2.34315	−	4.05845i	0.116720	−	0.202166i
$404$	9.37868	+	16.2443i	0.466607	+	0.808187i
$405$		0			0
$406$	20.5208	−	2.80821i	1.01843	−	0.139369i
$407$	9.48528			0.470168
$408$		0			0
$409$	3.89949	−	6.75412i	0.192818	−	0.333970i	−0.753365	−	0.657602i	$-0.771571\pi$
$409$	3.89949	−	6.75412i	0.192818	−	0.333970i	0.946183	+	0.323632i	$0.104904\pi$
$410$	−2.82843	+	4.89898i	−0.139686	+	0.241943i
$411$		0			0
$412$	47.7990			2.35489
$413$	14.0355	+	18.0995i	0.690643	+	0.890618i
$414$		0			0
$415$	−2.82843	−	4.89898i	−0.138842	−	0.240481i
$416$	−0.656854	+	1.13770i	−0.0322049	+	0.0557806i
$417$		0			0
$418$	8.74264	+	15.1427i	0.427617	+	0.740654i
$419$	−14.7990			−0.722978			−0.361489	−	0.932376i	$-0.617731\pi$
$419$	−14.7990			−0.722978			−0.361489	+	0.932376i	$0.617731\pi$
$420$		0			0
$421$	−7.00000			−0.341159			−0.170580	−	0.985344i	$-0.554564\pi$
$421$	−7.00000			−0.341159			−0.170580	+	0.985344i	$0.554564\pi$
$422$	−18.0711	−	31.3000i	−0.879686	−	1.52366i
$423$		0			0
$424$	−15.8284	+	27.4156i	−0.768696	+	1.33142i
$425$	2.20711	+	3.82282i	0.107060	+	0.185434i
$426$		0			0
$427$	10.4853	−	1.43488i	0.507418	−	0.0694386i
$428$	−36.9706			−1.78704
$429$		0			0
$430$	6.65685	−	11.5300i	0.321022	−	0.556026i
$431$	3.48528	−	6.03668i	0.167880	−	0.290777i	−0.769794	−	0.638292i	$-0.779641\pi$
$431$	3.48528	−	6.03668i	0.167880	−	0.290777i	0.937674	+	0.347515i	$0.112975\pi$
$432$		0			0
$433$	0.857864			0.0412263			0.0206132	−	0.999788i	$-0.493438\pi$
$433$	0.857864			0.0412263			0.0206132	+	0.999788i	$0.493438\pi$
$434$	13.6569	−	33.4523i	0.655550	−	1.60576i
$435$		0			0
$436$	13.0711	+	22.6398i	0.625991	+	1.08425i
$437$	−25.3492	+	43.9062i	−1.21262	+	2.10032i
$438$		0			0
$439$	−15.4497	−	26.7597i	−0.737376	−	1.27717i	−0.953673	−	0.300845i	$-0.902731\pi$
$439$	−15.4497	−	26.7597i	−0.737376	−	1.27717i	0.216297	−	0.976328i	$-0.430602\pi$
$440$	−8.82843			−0.420879
$441$		0			0
$442$	−8.82843			−0.419925
$443$	16.8137	+	29.1222i	0.798843	+	1.38364i	0.920370	+	0.391049i	$0.127888\pi$
$443$	16.8137	+	29.1222i	0.798843	+	1.38364i	−0.121526	+	0.992588i	$0.538779\pi$
$444$		0			0
$445$	−14.1421	+	24.4949i	−0.670402	+	1.16117i
$446$	27.7279	+	48.0262i	1.31296	+	2.27411i
$447$		0			0
$448$	−9.82843	+	24.0746i	−0.464350	+	1.13742i
$449$	4.00000			0.188772			0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000			0.188772			0.0943858	+	0.995536i	$0.469911\pi$
$450$		0			0
$451$	0.585786	−	1.01461i	0.0275836	−	0.0477762i
$452$	−14.6569	+	25.3864i	−0.689400	+	1.19408i
$453$		0			0
$454$	−36.1421			−1.69623
$455$	4.34315	−	0.594346i	0.203610	−	0.0278634i
$456$		0			0
$457$	−9.41421	−	16.3059i	−0.440378	−	0.762758i	0.557339	−	0.830285i	$-0.311822\pi$
$457$	−9.41421	−	16.3059i	−0.440378	−	0.762758i	−0.997717	+	0.0675273i	$0.978489\pi$
$458$	18.8995	−	32.7349i	0.883115	−	1.52960i
$459$		0			0
$460$	−26.7990	−	46.4172i	−1.24951	−	2.16421i
$461$	−6.75736			−0.314722			−0.157361	−	0.987541i	$-0.550299\pi$
$461$	−6.75736			−0.314722			−0.157361	+	0.987541i	$0.550299\pi$
$462$		0			0
$463$	18.6274			0.865689			0.432845	−	0.901468i	$-0.357510\pi$
$463$	18.6274			0.865689			0.432845	+	0.901468i	$0.357510\pi$
$464$	4.86396	+	8.42463i	0.225804	+	0.391104i
$465$		0			0
$466$	12.5711	−	21.7737i	0.582343	−	1.00865i
$467$	−4.15685	−	7.19988i	−0.192356	−	0.333171i	0.753674	−	0.657248i	$-0.228280\pi$
$467$	−4.15685	−	7.19988i	−0.192356	−	0.333171i	−0.946031	+	0.324077i	$0.894946\pi$
$468$		0			0
$469$	−5.14214	−	6.63103i	−0.237442	−	0.306193i
$470$	−47.4558			−2.18897
$471$		0			0
$472$	−19.1066	+	33.0936i	−0.879453	+	1.52326i
$473$	−1.37868	+	2.38794i	−0.0633918	+	0.109798i
$474$		0			0
$475$	−7.24264			−0.332315
$476$	−44.2990	+	6.06218i	−2.03044	+	0.277859i
$477$		0			0
$478$	−7.82843	−	13.5592i	−0.358064	−	0.620185i
$479$	7.00000	−	12.1244i	0.319838	−	0.553976i	−0.660616	−	0.750724i	$-0.729705\pi$
$479$	7.00000	−	12.1244i	0.319838	−	0.553976i	0.980454	+	0.196748i	$0.0630381\pi$
$480$		0			0
$481$	−3.92893	−	6.80511i	−0.179144	−	0.310286i
$482$	17.6569			0.804248
$483$		0			0
$484$	3.82843			0.174019
$485$	11.4853	+	19.8931i	0.521520	+	0.903299i
$486$		0			0
$487$	16.3137	−	28.2562i	0.739245	−	1.28041i	−0.213591	−	0.976923i	$-0.568516\pi$
$487$	16.3137	−	28.2562i	0.739245	−	1.28041i	0.952836	−	0.303486i	$-0.0981506\pi$
$488$	8.82843	+	15.2913i	0.399644	+	0.692204i
$489$		0			0
$490$	32.5563	−	9.08052i	1.47075	−	0.410216i
$491$	1.51472			0.0683583			0.0341791	−	0.999416i	$-0.489118\pi$
$491$	1.51472			0.0683583			0.0341791	+	0.999416i	$0.489118\pi$
$492$		0			0
$493$	7.15685	−	12.3960i	0.322329	−	0.558289i
$494$	7.24264	−	12.5446i	0.325862	−	0.564409i
$495$		0			0
$496$	16.9706			0.762001
$497$	4.17157	−	10.2182i	0.187121	−	0.458350i
$498$		0			0
$499$	15.8995	+	27.5387i	0.711759	+	1.23280i	0.964196	+	0.265190i	$0.0854346\pi$
$499$	15.8995	+	27.5387i	0.711759	+	1.23280i	−0.252437	+	0.967613i	$0.581232\pi$
$500$	22.9706	−	39.7862i	1.02727	−	1.77929i
$501$		0			0
$502$	−3.03553	−	5.25770i	−0.135483	−	0.234663i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−9.79899			−0.436049
$506$	8.44975	+	14.6354i	0.375637	+	0.650623i
$507$		0			0
$508$	30.7635	−	53.2839i	1.36491	−	2.36409i
$509$	−6.00000	−	10.3923i	−0.265945	−	0.460631i	0.701866	−	0.712309i	$-0.252351\pi$
$509$	−6.00000	−	10.3923i	−0.265945	−	0.460631i	−0.967811	+	0.251679i	$0.919017\pi$
$510$		0			0
$511$	0.556349	+	0.717439i	0.0246114	+	0.0317376i
$512$	−31.2426			−1.38074
$513$		0			0
$514$	25.8995	−	44.8592i	1.14238	−	1.97866i
$515$	−12.4853	+	21.6251i	−0.550167	+	0.952918i
$516$		0			0
$517$	9.82843			0.432254
$518$	−37.1274	−	47.8776i	−1.63129	−	2.10362i
$519$		0			0
$520$	3.65685	+	6.33386i	0.160364	+	0.277758i
$521$	15.5858	−	26.9954i	0.682826	−	1.18269i	−0.291289	−	0.956635i	$-0.594084\pi$
$521$	15.5858	−	26.9954i	0.682826	−	1.18269i	0.974115	−	0.226054i	$-0.0725825\pi$
$522$		0			0
$523$	11.1421	+	19.2987i	0.487212	+	0.843875i	0.999892	−	0.0147044i	$-0.00468072\pi$
$523$	11.1421	+	19.2987i	0.487212	+	0.843875i	−0.512680	+	0.858580i	$0.671347\pi$
$524$	19.7990			0.864923
$525$		0			0
$526$	−43.4558			−1.89476
$527$	−12.4853	−	21.6251i	−0.543867	−	0.942006i
$528$		0			0
$529$	−13.0000	+	22.5167i	−0.565217	+	0.978985i
$530$	−17.3137	−	29.9882i	−0.752059	−	1.30261i
$531$		0			0
$532$	27.7279	−	67.9193i	1.20216	−	2.94467i
$533$	−0.970563			−0.0420397
$534$		0			0
$535$	9.65685	−	16.7262i	0.417502	−	0.723135i
$536$	7.00000	−	12.1244i	0.302354	−	0.523692i
$537$		0			0
$538$	67.1127			2.89343
$539$	−6.74264	+	1.88064i	−0.290426	+	0.0810048i
$540$		0			0
$541$	−9.82843	−	17.0233i	−0.422557	−	0.731890i	0.573632	−	0.819113i	$-0.305534\pi$
$541$	−9.82843	−	17.0233i	−0.422557	−	0.731890i	−0.996189	+	0.0872230i	$0.972201\pi$
$542$	27.7279	−	48.0262i	1.19102	−	2.06290i
$543$		0			0
$544$	3.50000	+	6.06218i	0.150061	+	0.259914i
$545$	−13.6569			−0.584995
$546$		0			0
$547$	−24.2132			−1.03528			−0.517641	−	0.855598i	$-0.673190\pi$
$547$	−24.2132			−1.03528			−0.517641	+	0.855598i	$0.673190\pi$
$548$	0.928932	+	1.60896i	0.0396820	+	0.0687313i
$549$		0			0
$550$	−1.20711	+	2.09077i	−0.0514712	+	0.0891507i
$551$	11.7426	+	20.3389i	0.500253	+	0.866464i
$552$		0			0
$553$	−34.8995	+	4.77589i	−1.48408	+	0.203091i
$554$	−61.9411			−2.63163
$555$		0			0
$556$	−16.1066	+	27.8975i	−0.683072	+	1.18312i
$557$	−4.27817	+	7.41002i	−0.181272	+	0.313972i	−0.942314	−	0.334730i	$-0.891355\pi$
$557$	−4.27817	+	7.41002i	−0.181272	+	0.313972i	0.761042	+	0.648703i	$0.224688\pi$
$558$		0			0
$559$	2.28427			0.0966144
$560$	9.72792	+	12.5446i	0.411080	+	0.530107i
$561$		0			0
$562$	17.3995	+	30.1368i	0.733953	+	1.27124i
$563$	12.4142	−	21.5020i	0.523197	−	0.906203i	−0.476439	−	0.879208i	$-0.658073\pi$
$563$	12.4142	−	21.5020i	0.523197	−	0.906203i	0.999636	−	0.0269957i	$-0.00859404\pi$
$564$		0			0
$565$	−7.65685	−	13.2621i	−0.322126	−	0.557939i
$566$	49.1127			2.06436
$567$		0			0
$568$	18.4142			0.772643
$569$	0.621320	+	1.07616i	0.0260471	+	0.0451149i	0.878755	−	0.477273i	$-0.158375\pi$
$569$	0.621320	+	1.07616i	0.0260471	+	0.0451149i	−0.852708	+	0.522388i	$0.825041\pi$
$570$		0			0
$571$	1.96447	−	3.40256i	0.0822103	−	0.142392i	−0.821989	−	0.569504i	$-0.807135\pi$
$571$	1.96447	−	3.40256i	0.0822103	−	0.142392i	0.904199	+	0.427111i	$0.140469\pi$
$572$	−1.58579	−	2.74666i	−0.0663051	−	0.114844i
$573$		0			0
$574$	−7.41421	+	1.01461i	−0.309463	+	0.0423491i
$575$	−7.00000			−0.291920
$576$		0			0
$577$	−3.82843	+	6.63103i	−0.159380	+	0.276053i	−0.934645	−	0.355582i	$-0.884283\pi$
$577$	−3.82843	+	6.63103i	−0.159380	+	0.276053i	0.775266	+	0.631635i	$0.217616\pi$
$578$	−3.00000	+	5.19615i	−0.124784	+	0.216131i
$579$		0			0
$580$	−24.8284			−1.03094
$581$	2.82843	−	6.92820i	0.117343	−	0.287430i
$582$		0			0
$583$	3.58579	+	6.21076i	0.148508	+	0.257224i
$584$	−0.757359	+	1.31178i	−0.0313398	+	0.0542820i
$585$		0			0
$586$	6.91421	+	11.9758i	0.285624	+	0.494714i
$587$	−8.68629			−0.358522			−0.179261	−	0.983802i	$-0.557371\pi$
$587$	−8.68629			−0.358522			−0.179261	+	0.983802i	$0.557371\pi$
$588$		0			0
$589$	40.9706			1.68816
$590$	−20.8995	−	36.1990i	−0.860419	−	1.49029i
$591$		0			0
$592$	14.2279	−	24.6435i	0.584764	−	1.01284i
$593$	−11.1360	−	19.2882i	−0.457302	−	0.792071i	0.541515	−	0.840691i	$-0.317851\pi$
$593$	−11.1360	−	19.2882i	−0.457302	−	0.792071i	−0.998817	+	0.0486202i	$0.984518\pi$
$594$		0			0
$595$	8.82843	−	21.6251i	0.361930	−	0.886544i
$596$	85.0416			3.48344
$597$		0			0
$598$	7.00000	−	12.1244i	0.286251	−	0.495802i
$599$	−0.686292	+	1.18869i	−0.0280411	+	0.0485686i	−0.879705	−	0.475519i	$-0.842260\pi$
$599$	−0.686292	+	1.18869i	−0.0280411	+	0.0485686i	0.851664	+	0.524088i	$0.175594\pi$
$600$		0			0
$601$	−14.4853			−0.590867			−0.295433	−	0.955363i	$-0.595464\pi$
$601$	−14.4853			−0.590867			−0.295433	+	0.955363i	$0.595464\pi$
$602$	17.4497	−	2.38794i	0.711199	−	0.0973253i
$603$		0			0
$604$	36.1777	+	62.6616i	1.47205	+	2.54966i
$605$	−1.00000	+	1.73205i	−0.0406558	+	0.0704179i
$606$		0			0
$607$	9.34315	+	16.1828i	0.379227	+	0.656840i	0.990950	−	0.134232i	$-0.0428568\pi$
$607$	9.34315	+	16.1828i	0.379227	+	0.656840i	−0.611723	+	0.791072i	$0.709523\pi$
$608$	−11.4853			−0.465790
$609$		0			0
$610$	−19.3137			−0.781989
$611$	−4.07107	−	7.05130i	−0.164698	−	0.285265i
$612$		0			0
$613$	−12.5858	+	21.7992i	−0.508335	+	0.880462i	0.491618	+	0.870811i	$0.336406\pi$
$613$	−12.5858	+	21.7992i	−0.508335	+	0.880462i	−0.999953	+	0.00965147i	$0.996928\pi$
$614$	20.8995	+	36.1990i	0.843435	+	1.46087i
$615$		0			0
$616$	−7.15685	−	9.22911i	−0.288358	−	0.371851i
$617$	−13.1716			−0.530268			−0.265134	−	0.964212i	$-0.585416\pi$
$617$	−13.1716			−0.530268			−0.265134	+	0.964212i	$0.585416\pi$
$618$		0			0
$619$	−17.3137	+	29.9882i	−0.695897	+	1.20533i	0.273981	+	0.961735i	$0.411660\pi$
$619$	−17.3137	+	29.9882i	−0.695897	+	1.20533i	−0.969877	+	0.243593i	$0.921674\pi$
$620$	−21.6569	+	37.5108i	−0.869760	+	1.50647i
$621$		0			0
$622$	−71.5269			−2.86797
$623$	−37.0711	+	5.07306i	−1.48522	+	0.203248i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$	−5.79289	+	10.0336i	−0.231531	+	0.401023i
$627$		0			0
$628$	−13.3995	−	23.2086i	−0.534698	−	0.926124i
$629$	−41.8701			−1.66947
$630$		0			0
$631$	26.2843			1.04636			0.523180	−	0.852222i	$-0.324745\pi$
$631$	26.2843			1.04636			0.523180	+	0.852222i	$0.324745\pi$
$632$	−29.3848	−	50.8959i	−1.16886	−	2.02453i
$633$		0			0
$634$	−30.5563	+	52.9251i	−1.21355	+	2.10193i
$635$	16.0711	+	27.8359i	0.637761	+	1.10463i
$636$		0			0
$637$	4.14214	+	4.05845i	0.164117	+	0.160802i
$638$	7.82843			0.309930
$639$		0			0
$640$	20.5563	−	35.6046i	0.812561	−	1.40740i
$641$	−6.24264	+	10.8126i	−0.246569	+	0.427071i	−0.962572	−	0.271027i	$-0.912637\pi$
$641$	−6.24264	+	10.8126i	−0.246569	+	0.427071i	0.716002	+	0.698098i	$0.245970\pi$
$642$		0			0
$643$	2.00000			0.0788723			0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000			0.0788723			0.0394362	+	0.999222i	$0.487444\pi$
$644$	26.7990	−	65.6439i	1.05603	−	2.58673i
$645$		0			0
$646$	−38.5919	−	66.8431i	−1.51838	−	2.62991i
$647$	−9.65685	+	16.7262i	−0.379650	+	0.657573i	−0.991011	−	0.133778i	$-0.957289\pi$
$647$	−9.65685	+	16.7262i	−0.379650	+	0.657573i	0.611361	+	0.791352i	$0.290622\pi$
$648$		0			0
$649$	4.32843	+	7.49706i	0.169906	+	0.294285i
$650$	2.00000			0.0784465
$651$		0			0
$652$	−80.2843			−3.14417
$653$	15.5563	+	26.9444i	0.608767	+	1.05442i	0.991444	+	0.130533i	$0.0416689\pi$
$653$	15.5563	+	26.9444i	0.608767	+	1.05442i	−0.382677	+	0.923882i	$0.624998\pi$
$654$		0			0
$655$	−5.17157	+	8.95743i	−0.202070	+	0.349996i
$656$	−1.75736	−	3.04384i	−0.0686134	−	0.118842i
$657$		0			0
$658$	−38.4706	−	49.6096i	−1.49974	−	1.93398i
$659$	−34.4853			−1.34336			−0.671678	−	0.740843i	$-0.734426\pi$
$659$	−34.4853			−1.34336			−0.671678	+	0.740843i	$0.734426\pi$
$660$		0			0
$661$	−5.98528	+	10.3668i	−0.232800	+	0.403222i	−0.958631	−	0.284651i	$-0.908122\pi$
$661$	−5.98528	+	10.3668i	−0.232800	+	0.403222i	0.725831	+	0.687873i	$0.241456\pi$
$662$	−27.1421	+	47.0116i	−1.05491	+	1.82716i
$663$		0			0
$664$	12.4853			0.484523
$665$	23.4853	+	30.2854i	0.910720	+	1.17442i
$666$		0			0
$667$	11.3492	+	19.6575i	0.439444	+	0.761140i
$668$	9.89949	−	17.1464i	0.383023	−	0.663415i
$669$		0			0
$670$	7.65685	+	13.2621i	0.295810	+	0.512358i
$671$	4.00000			0.154418
$672$		0			0
$673$	27.1716			1.04739			0.523694	−	0.851907i	$-0.324554\pi$
$673$	27.1716			1.04739			0.523694	+	0.851907i	$0.324554\pi$
$674$	−4.65685	−	8.06591i	−0.179375	−	0.310687i
$675$		0			0
$676$	23.5711	−	40.8263i	0.906580	−	1.57024i
$677$	−16.0061	−	27.7234i	−0.615164	−	1.06550i	−0.990356	−	0.138548i	$-0.955756\pi$
$677$	−16.0061	−	27.7234i	−0.615164	−	1.06550i	0.375191	−	0.926947i	$-0.377577\pi$
$678$		0			0
$679$	−11.4853	+	28.1331i	−0.440765	+	1.07965i
$680$	38.9706			1.49445
$681$		0			0
$682$	6.82843	−	11.8272i	0.261474	−	0.452886i
$683$	1.60051	−	2.77216i	0.0612416	−	0.106074i	−0.833779	−	0.552098i	$-0.813827\pi$
$683$	1.60051	−	2.77216i	0.0612416	−	0.106074i	0.895021	+	0.446025i	$0.147161\pi$
$684$		0			0
$685$	−0.970563			−0.0370833
$686$	35.8848	+	26.6727i	1.37009	+	1.01837i
$687$		0			0
$688$	4.13604	+	7.16383i	0.157685	+	0.273118i
$689$	2.97056	−	5.14517i	0.113169	−	0.196015i
$690$		0			0
$691$	5.72792	+	9.92105i	0.217900	+	0.377415i	0.954166	−	0.299278i	$-0.0967459\pi$
$691$	5.72792	+	9.92105i	0.217900	+	0.377415i	−0.736265	+	0.676693i	$0.763413\pi$
$692$	56.7696			2.15805
$693$		0			0
$694$	−36.1421			−1.37194
$695$	−8.41421	−	14.5738i	−0.319169	−	0.552817i
$696$		0			0
$697$	−2.58579	+	4.47871i	−0.0979436	+	0.169643i
$698$	−13.2426	−	22.9369i	−0.501241	−	0.868175i
$699$		0			0
$700$	10.0355	−	1.37333i	0.379308	−	0.0519070i
$701$	2.89949			0.109512			0.0547562	−	0.998500i	$-0.482562\pi$
$701$	2.89949			0.109512			0.0547562	+	0.998500i	$0.482562\pi$
$702$		0			0
$703$	34.3492	−	59.4946i	1.29551	−	2.24388i
$704$	−4.91421	+	8.51167i	−0.185211	+	0.320796i
$705$		0			0
$706$	−32.1421			−1.20969
$707$	−7.94365	−	10.2437i	−0.298752	−	0.385255i
$708$		0			0
$709$	−12.8848	−	22.3171i	−0.483898	−	0.838136i	0.515931	−	0.856630i	$-0.327446\pi$
$709$	−12.8848	−	22.3171i	−0.483898	−	0.838136i	−0.999829	+	0.0184943i	$0.994113\pi$
$710$	−10.0711	+	17.4436i	−0.377960	+	0.654647i
$711$		0			0
$712$	−31.2132	−	54.0629i	−1.16976	−	2.02609i
$713$	39.5980			1.48296
$714$		0			0
$715$	1.65685			0.0619628
$716$	34.1274	+	59.1104i	1.27540	+	2.20906i
$717$		0			0
$718$	−9.65685	+	16.7262i	−0.360391	+	0.624215i
$719$	5.60051	+	9.70036i	0.208864	+	0.361762i	0.951357	−	0.308091i	$-0.0996902\pi$
$719$	5.60051	+	9.70036i	0.208864	+	0.361762i	−0.742493	+	0.669854i	$0.766357\pi$
$720$		0			0
$721$	−32.7279	+	4.47871i	−1.21885	+	0.166796i
$722$	80.7696			3.00593
$723$		0			0
$724$	22.3137	−	38.6485i	0.829282	−	1.43636i
$725$	−1.62132	+	2.80821i	−0.0602143	+	0.104294i
$726$		0			0
$727$	46.0833			1.70913			0.854567	−	0.519342i	$-0.173823\pi$
$727$	46.0833			1.70913			0.854567	+	0.519342i	$0.173823\pi$
$728$	−3.65685	+	8.95743i	−0.135532	+	0.331984i
$729$		0			0
$730$	−0.828427	−	1.43488i	−0.0306615	−	0.0531072i
$731$	6.08579	−	10.5409i	0.225091	−	0.389869i
$732$		0			0
$733$	−4.82843	−	8.36308i	−0.178342	−	0.308897i	0.762971	−	0.646433i	$-0.223740\pi$
$733$	−4.82843	−	8.36308i	−0.178342	−	0.308897i	−0.941313	+	0.337536i	$0.890407\pi$
$734$	−10.0000			−0.369107
$735$		0			0
$736$	−11.1005			−0.409170
$737$	−1.58579	−	2.74666i	−0.0584132	−	0.101175i
$738$		0			0
$739$	19.8284	−	34.3438i	0.729400	−	1.26336i	−0.227736	−	0.973723i	$-0.573132\pi$
$739$	19.8284	−	34.3438i	0.729400	−	1.26336i	0.957137	−	0.289636i	$-0.0935342\pi$
$740$	36.3137	+	62.8972i	1.33492	+	2.31215i
$741$		0			0
$742$	17.3137	−	42.4098i	0.635606	−	1.55691i
$743$	−33.7990			−1.23996			−0.619982	−	0.784616i	$-0.712860\pi$
$743$	−33.7990			−1.23996			−0.619982	+	0.784616i	$0.712860\pi$
$744$		0			0
$745$	−22.2132	+	38.4744i	−0.813829	+	1.40959i
$746$	2.00000	−	3.46410i	0.0732252	−	0.126830i
$747$		0			0
$748$	−16.8995			−0.617907
$749$	25.3137	−	3.46410i	0.924943	−	0.126576i
$750$		0			0
$751$	14.6569	+	25.3864i	0.534836	+	0.926363i	0.999171	+	0.0407039i	$0.0129600\pi$
$751$	14.6569	+	25.3864i	0.534836	+	0.926363i	−0.464335	+	0.885660i	$0.653707\pi$
$752$	14.7426	−	25.5350i	0.537609	−	0.931166i
$753$		0			0
$754$	−3.24264	−	5.61642i	−0.118090	−	0.204538i
$755$	−37.7990			−1.37565
$756$		0			0
$757$	−7.68629			−0.279363			−0.139682	−	0.990196i	$-0.544608\pi$
$757$	−7.68629			−0.279363			−0.139682	+	0.990196i	$0.544608\pi$
$758$	−22.7279	−	39.3659i	−0.825515	−	1.42983i
$759$		0			0
$760$	−31.9706	+	55.3746i	−1.15969	+	2.00865i
$761$	−0.100505	−	0.174080i	−0.00364331	−	0.00631039i	0.864198	−	0.503152i	$-0.167826\pi$
$761$	−0.100505	−	0.174080i	−0.00364331	−	0.00631039i	−0.867841	+	0.496841i	$0.834493\pi$
$762$		0			0
$763$	−11.0711	−	14.2767i	−0.400800	−	0.516850i
$764$	82.9117			2.99964
$765$		0			0
$766$	2.79289	−	4.83743i	0.100911	−	0.174784i
$767$	3.58579	−	6.21076i	0.129475	−	0.224258i
$768$		0			0
$769$	33.7990			1.21882			0.609411	−	0.792854i	$-0.291406\pi$
$769$	33.7990			1.21882			0.609411	+	0.792854i	$0.291406\pi$
$770$	12.6569	−	1.73205i	0.456121	−	0.0624188i
$771$		0			0
$772$	23.2426	+	40.2574i	0.836521	+	1.44890i
$773$	−25.8284	+	44.7361i	−0.928984	+	1.60905i	−0.143959	+	0.989584i	$0.545983\pi$
$773$	−25.8284	+	44.7361i	−0.928984	+	1.60905i	−0.785025	+	0.619464i	$0.787350\pi$
$774$		0			0
$775$	2.82843	+	4.89898i	0.101600	+	0.175977i
$776$	−50.6985			−1.81997
$777$		0			0
$778$	28.4853			1.02125
$779$	−4.24264	−	7.34847i	−0.152008	−	0.263286i
$780$		0			0
$781$	2.08579	−	3.61269i	0.0746353	−	0.129272i
$782$	−37.2990	−	64.6037i	−1.33381	−	2.31023i
$783$		0			0
$784$	−5.22792	+

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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.20711 - 2.09077i) q^{2} +(-1.91421 + 3.31552i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(1.00000 - 2.44949i) q^{7} +4.41421 q^{8} +O(q^{10})\)	\(q+(-1.20711 - 2.09077i) q^{2} +(-1.91421 + 3.31552i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(1.00000 - 2.44949i) q^{7} +4.41421 q^{8} +(-2.41421 + 4.18154i) q^{10} +(0.500000 - 0.866025i) q^{11} -0.828427 q^{13} +(-6.32843 + 0.866025i) q^{14} +(-1.50000 - 2.59808i) q^{16} +(-2.20711 + 3.82282i) q^{17} +(-3.62132 - 6.27231i) q^{19} +7.65685 q^{20} -2.41421 q^{22} +(-3.50000 - 6.06218i) q^{23} +(0.500000 - 0.866025i) q^{25} +(1.00000 + 1.73205i) q^{26} +(6.20711 + 8.00436i) q^{28} -3.24264 q^{29} +(-2.82843 + 4.89898i) q^{31} +(0.792893 - 1.37333i) q^{32} +10.6569 q^{34} +(-5.24264 + 0.717439i) q^{35} +(4.74264 + 8.21449i) q^{37} +(-8.74264 + 15.1427i) q^{38} +(-4.41421 - 7.64564i) q^{40} +1.17157 q^{41} -2.75736 q^{43} +(1.91421 + 3.31552i) q^{44} +(-8.44975 + 14.6354i) q^{46} +(4.91421 + 8.51167i) q^{47} +(-5.00000 - 4.89898i) q^{49} -2.41421 q^{50} +(1.58579 - 2.74666i) q^{52} +(-3.58579 + 6.21076i) q^{53} -2.00000 q^{55} +(4.41421 - 10.8126i) q^{56} +(3.91421 + 6.77962i) q^{58} +(-4.32843 + 7.49706i) q^{59} +(2.00000 + 3.46410i) q^{61} +13.6569 q^{62} -9.82843 q^{64} +(0.828427 + 1.43488i) q^{65} +(1.58579 - 2.74666i) q^{67} +(-8.44975 - 14.6354i) q^{68} +(7.82843 + 10.0951i) q^{70} +4.17157 q^{71} +(-0.171573 + 0.297173i) q^{73} +(11.4497 - 19.8315i) q^{74} +27.7279 q^{76} +(-1.62132 - 2.09077i) q^{77} +(-6.65685 - 11.5300i) q^{79} +(-3.00000 + 5.19615i) q^{80} +(-1.41421 - 2.44949i) q^{82} +2.82843 q^{83} +8.82843 q^{85} +(3.32843 + 5.76500i) q^{86} +(2.20711 - 3.82282i) q^{88} +(-7.07107 - 12.2474i) q^{89} +(-0.828427 + 2.02922i) q^{91} +26.7990 q^{92} +(11.8640 - 20.5490i) q^{94} +(-7.24264 + 12.5446i) q^{95} -11.4853 q^{97} +(-4.20711 + 16.3674i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{4} - 4 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^4 - 4 * q^5 + 4 * q^7 + 12 * q^8	\( 4 q - 2 q^{2} - 2 q^{4} - 4 q^{5} + 4 q^{7} + 12 q^{8} - 4 q^{10} + 2 q^{11} + 8 q^{13} - 14 q^{14} - 6 q^{16} - 6 q^{17} - 6 q^{19} + 8 q^{20} - 4 q^{22} - 14 q^{23} + 2 q^{25} + 4 q^{26} + 22 q^{28} + 4 q^{29} + 6 q^{32} + 20 q^{34} - 4 q^{35} + 2 q^{37} - 18 q^{38} - 12 q^{40} + 16 q^{41} - 28 q^{43} + 2 q^{44} - 14 q^{46} + 14 q^{47} - 20 q^{49} - 4 q^{50} + 12 q^{52} - 20 q^{53} - 8 q^{55} + 12 q^{56} + 10 q^{58} - 6 q^{59} + 8 q^{61} + 32 q^{62} - 28 q^{64} - 8 q^{65} + 12 q^{67} - 14 q^{68} + 20 q^{70} + 28 q^{71} - 12 q^{73} + 26 q^{74} + 60 q^{76} + 2 q^{77} - 4 q^{79} - 12 q^{80} + 24 q^{85} + 2 q^{86} + 6 q^{88} + 8 q^{91} + 28 q^{92} + 22 q^{94} - 12 q^{95} - 12 q^{97} - 14 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^4 - 4 * q^5 + 4 * q^7 + 12 * q^8 - 4 * q^10 + 2 * q^11 + 8 * q^13 - 14 * q^14 - 6 * q^16 - 6 * q^17 - 6 * q^19 + 8 * q^20 - 4 * q^22 - 14 * q^23 + 2 * q^25 + 4 * q^26 + 22 * q^28 + 4 * q^29 + 6 * q^32 + 20 * q^34 - 4 * q^35 + 2 * q^37 - 18 * q^38 - 12 * q^40 + 16 * q^41 - 28 * q^43 + 2 * q^44 - 14 * q^46 + 14 * q^47 - 20 * q^49 - 4 * q^50 + 12 * q^52 - 20 * q^53 - 8 * q^55 + 12 * q^56 + 10 * q^58 - 6 * q^59 + 8 * q^61 + 32 * q^62 - 28 * q^64 - 8 * q^65 + 12 * q^67 - 14 * q^68 + 20 * q^70 + 28 * q^71 - 12 * q^73 + 26 * q^74 + 60 * q^76 + 2 * q^77 - 4 * q^79 - 12 * q^80 + 24 * q^85 + 2 * q^86 + 6 * q^88 + 8 * q^91 + 28 * q^92 + 22 * q^94 - 12 * q^95 - 12 * q^97 - 14 * q^98

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