LMFDB - Embedded newform 880.2.bo.i.81.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,2,Mod(81,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("880.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.bo (of order $5$, degree $4$, 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:	$7.02683537787$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$3$ over $\Q(\zeta_{5})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 5 x^{10} + 4 x^{9} + 28 x^{8} - 81 x^{7} + 335 x^{6} - 235 x^{5} + 782 x^{4} + \cdots + 1 $ x^12 - x^11 + 5x^10 + 4x^9 + 28x^8 - 81x^7 + 335x^6 - 235x^5 + 782x^4 - 302x^3 + 711x^2 - 43x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			81.2
Root			$0.0307040 - 0.0223078i$ of defining polynomial
Character	$\chi$	$=$	880.81
Dual form			880.2.bo.i.641.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.320745 - 0.987151i) q^{3} +(0.809017 - 0.587785i) q^{5} +(0.804092 + 2.47474i) q^{7} +(1.55546 + 1.13011i) q^{9} +O(q^{10})$	\(q+(0.320745 - 0.987151i) q^{3} +(0.809017 - 0.587785i) q^{5} +(0.804092 + 2.47474i) q^{7} +(1.55546 + 1.13011i) q^{9} +(-1.34374 + 3.03222i) q^{11} +(2.32799 + 1.69139i) q^{13} +(-0.320745 - 0.987151i) q^{15} +(-0.655541 + 0.476279i) q^{17} +(-1.59165 + 4.89860i) q^{19} +2.70085 q^{21} -4.98262 q^{23} +(0.309017 - 0.951057i) q^{25} +(4.13366 - 3.00328i) q^{27} +(2.07948 + 6.39997i) q^{29} +(-3.35001 - 2.43392i) q^{31} +(2.56226 + 2.29904i) q^{33} +(2.10514 + 1.52947i) q^{35} +(1.25193 + 3.85304i) q^{37} +(2.41635 - 1.75558i) q^{39} +(3.01626 - 9.28309i) q^{41} +5.92912 q^{43} +1.92266 q^{45} +(0.299099 - 0.920531i) q^{47} +(0.185338 - 0.134656i) q^{49} +(0.259897 + 0.799882i) q^{51} +(-3.61991 - 2.63002i) q^{53} +(0.695188 + 3.24295i) q^{55} +(4.32514 + 3.14240i) q^{57} +(-2.10365 - 6.47438i) q^{59} +(2.11402 - 1.53592i) q^{61} +(-1.54599 + 4.75807i) q^{63} +2.87756 q^{65} +15.8408 q^{67} +(-1.59815 + 4.91860i) q^{69} +(-1.14351 + 0.830809i) q^{71} +(-4.06139 - 12.4997i) q^{73} +(-0.839721 - 0.610093i) q^{75} +(-8.58445 - 0.887220i) q^{77} +(4.60431 + 3.34522i) q^{79} +(0.143559 + 0.441831i) q^{81} +(6.44760 - 4.68445i) q^{83} +(-0.250394 + 0.770635i) q^{85} +6.98472 q^{87} -14.5196 q^{89} +(-2.31382 + 7.12121i) q^{91} +(-3.47715 + 2.52630i) q^{93} +(1.59165 + 4.89860i) q^{95} +(2.42159 + 1.75939i) q^{97} +(-5.51687 + 3.19793i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{3} + 3 q^{5} + q^{7} - 10 q^{9}+O(q^{10}) $ 12 * q + q^3 + 3 * q^5 + q^7 - 10 * q^9	$ 12 q + q^{3} + 3 q^{5} + q^{7} - 10 q^{9} - 4 q^{11} + 18 q^{13} - q^{15} + 3 q^{17} - 4 q^{19} - 28 q^{21} + 18 q^{23} - 3 q^{25} - 23 q^{27} + 15 q^{29} + 8 q^{31} + 4 q^{33} - 6 q^{35} + 6 q^{37} + 33 q^{39} + 2 q^{41} + 36 q^{43} - 10 q^{45} + 16 q^{47} - 16 q^{49} + 10 q^{51} + 19 q^{53} - 6 q^{55} + 62 q^{57} - 46 q^{59} + 18 q^{61} + 7 q^{63} + 2 q^{65} + 44 q^{67} - q^{69} + 6 q^{71} + 25 q^{73} - 4 q^{75} + 10 q^{77} - 19 q^{79} + 30 q^{81} - 3 q^{85} - 6 q^{87} - 50 q^{89} + 46 q^{91} - 37 q^{93} + 4 q^{95} - 31 q^{97} - 79 q^{99}+O(q^{100}) $ 12 * q + q^3 + 3 * q^5 + q^7 - 10 * q^9 - 4 * q^11 + 18 * q^13 - q^15 + 3 * q^17 - 4 * q^19 - 28 * q^21 + 18 * q^23 - 3 * q^25 - 23 * q^27 + 15 * q^29 + 8 * q^31 + 4 * q^33 - 6 * q^35 + 6 * q^37 + 33 * q^39 + 2 * q^41 + 36 * q^43 - 10 * q^45 + 16 * q^47 - 16 * q^49 + 10 * q^51 + 19 * q^53 - 6 * q^55 + 62 * q^57 - 46 * q^59 + 18 * q^61 + 7 * q^63 + 2 * q^65 + 44 * q^67 - q^69 + 6 * q^71 + 25 * q^73 - 4 * q^75 + 10 * q^77 - 19 * q^79 + 30 * q^81 - 3 * q^85 - 6 * q^87 - 50 * q^89 + 46 * q^91 - 37 * q^93 + 4 * q^95 - 31 * q^97 - 79 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{5}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	0.320745	−	0.987151i	0.185182	−	0.569932i	−0.814769	−	0.579785i	$-0.803136\pi$
$3$	0.320745	−	0.987151i	0.185182	−	0.569932i	0.999951	+	0.00985324i	$0.00313643\pi$
$4$		0			0
$5$	0.809017	−	0.587785i	0.361803	−	0.262866i
$5$	0.809017	−	0.587785i	0.361803	−	0.262866i
$6$		0			0
$7$	0.804092	+	2.47474i	0.303918	+	0.935364i	0.980079	+	0.198610i	$0.0636427\pi$
$7$	0.804092	+	2.47474i	0.303918	+	0.935364i	−0.676160	+	0.736755i	$0.736357\pi$
$8$		0			0
$9$	1.55546	+	1.13011i	0.518487	+	0.376703i
$10$		0			0
$11$	−1.34374	+	3.03222i	−0.405152	+	0.914249i
$11$	−1.34374	+	3.03222i	−0.405152	+	0.914249i
$12$		0			0
$13$	2.32799	+	1.69139i	0.645669	+	0.469106i	0.861793	−	0.507260i	$-0.169342\pi$
$13$	2.32799	+	1.69139i	0.645669	+	0.469106i	−0.216124	+	0.976366i	$0.569342\pi$
$14$		0			0
$15$	−0.320745	−	0.987151i	−0.0828160	−	0.254881i
$16$		0			0
$17$	−0.655541	+	0.476279i	−0.158992	+	0.115515i	−0.664437	−	0.747345i	$-0.731328\pi$
$17$	−0.655541	+	0.476279i	−0.158992	+	0.115515i	0.505445	+	0.862859i	$0.331328\pi$
$18$		0			0
$19$	−1.59165	+	4.89860i	−0.365150	+	1.12381i	0.584738	+	0.811222i	$0.301197\pi$
$19$	−1.59165	+	4.89860i	−0.365150	+	1.12381i	−0.949887	+	0.312592i	$0.898803\pi$
$20$		0			0
$21$	2.70085			0.589374
$22$		0			0
$23$	−4.98262			−1.03895			−0.519474	−	0.854486i	$-0.673872\pi$
$23$	−4.98262			−1.03895			−0.519474	+	0.854486i	$0.673872\pi$
$24$		0			0
$25$	0.309017	−	0.951057i	0.0618034	−	0.190211i
$26$		0			0
$27$	4.13366	−	3.00328i	0.795523	−	0.577981i
$28$		0			0
$29$	2.07948	+	6.39997i	0.386149	+	1.18845i	0.935643	+	0.352947i	$0.114821\pi$
$29$	2.07948	+	6.39997i	0.386149	+	1.18845i	−0.549494	+	0.835498i	$0.685179\pi$
$30$		0			0
$31$	−3.35001	−	2.43392i	−0.601679	−	0.437146i	0.244795	−	0.969575i	$-0.421279\pi$
$31$	−3.35001	−	2.43392i	−0.601679	−	0.437146i	−0.846475	+	0.532429i	$0.821279\pi$
$32$		0			0
$33$	2.56226	+	2.29904i	0.446033	+	0.400212i
$34$		0			0
$35$	2.10514	+	1.52947i	0.355834	+	0.258528i
$36$		0			0
$37$	1.25193	+	3.85304i	0.205816	+	0.633436i	0.999679	+	0.0253395i	$0.00806668\pi$
$37$	1.25193	+	3.85304i	0.205816	+	0.633436i	−0.793863	+	0.608097i	$0.791933\pi$
$38$		0			0
$39$	2.41635	−	1.75558i	0.386925	−	0.281117i
$40$		0			0
$41$	3.01626	−	9.28309i	0.471061	−	1.44978i	−0.380137	−	0.924930i	$-0.624123\pi$
$41$	3.01626	−	9.28309i	0.471061	−	1.44978i	0.851198	−	0.524845i	$-0.175877\pi$
$42$		0			0
$43$	5.92912			0.904182			0.452091	−	0.891972i	$-0.350678\pi$
$43$	5.92912			0.904182			0.452091	+	0.891972i	$0.350678\pi$
$44$		0			0
$45$	1.92266			0.286613
$46$		0			0
$47$	0.299099	−	0.920531i	0.0436280	−	0.134273i	−0.926870	−	0.375383i	$-0.877511\pi$
$47$	0.299099	−	0.920531i	0.0436280	−	0.134273i	0.970498	+	0.241109i	$0.0775113\pi$
$48$		0			0
$49$	0.185338	−	0.134656i	0.0264769	−	0.0192366i
$50$		0			0
$51$	0.259897	+	0.799882i	0.0363929	+	0.112006i
$52$		0			0
$53$	−3.61991	−	2.63002i	−0.497233	−	0.361261i	0.310726	−	0.950499i	$-0.399428\pi$
$53$	−3.61991	−	2.63002i	−0.497233	−	0.361261i	−0.807959	+	0.589239i	$0.799428\pi$
$54$		0			0
$55$	0.695188	+	3.24295i	0.0937391	+	0.437279i
$56$		0			0
$57$	4.32514	+	3.14240i	0.572879	+	0.416221i
$58$		0			0
$59$	−2.10365	−	6.47438i	−0.273872	−	0.842892i	−0.989516	−	0.144426i	$-0.953866\pi$
$59$	−2.10365	−	6.47438i	−0.273872	−	0.842892i	0.715643	−	0.698466i	$-0.246134\pi$
$60$		0			0
$61$	2.11402	−	1.53592i	0.270672	−	0.196655i	−0.444167	−	0.895944i	$-0.646500\pi$
$61$	2.11402	−	1.53592i	0.270672	−	0.196655i	0.714839	+	0.699289i	$0.246500\pi$
$62$		0			0
$63$	−1.54599	+	4.75807i	−0.194777	+	0.599461i
$64$		0			0
$65$	2.87756			0.356917
$66$		0			0
$67$	15.8408			1.93526			0.967629	−	0.252378i	$-0.0812125\pi$
$67$	15.8408			1.93526			0.967629	+	0.252378i	$0.0812125\pi$
$68$		0			0
$69$	−1.59815	+	4.91860i	−0.192395	+	0.592130i
$70$		0			0
$71$	−1.14351	+	0.830809i	−0.135710	+	0.0985989i	−0.653569	−	0.756867i	$-0.726729\pi$
$71$	−1.14351	+	0.830809i	−0.135710	+	0.0985989i	0.517859	+	0.855466i	$0.326729\pi$
$72$		0			0
$73$	−4.06139	−	12.4997i	−0.475349	−	1.46297i	−0.845486	−	0.533998i	$-0.820689\pi$
$73$	−4.06139	−	12.4997i	−0.475349	−	1.46297i	0.370137	−	0.928977i	$-0.379311\pi$
$74$		0			0
$75$	−0.839721	−	0.610093i	−0.0969626	−	0.0704475i
$76$		0			0
$77$	−8.58445	−	0.887220i	−0.978289	−	0.101108i
$78$		0			0
$79$	4.60431	+	3.34522i	0.518025	+	0.376367i	0.815859	−	0.578250i	$-0.196264\pi$
$79$	4.60431	+	3.34522i	0.518025	+	0.376367i	−0.297834	+	0.954618i	$0.596264\pi$
$80$		0			0
$81$	0.143559	+	0.441831i	0.0159510	+	0.0490923i
$82$		0			0
$83$	6.44760	−	4.68445i	0.707716	−	0.514186i	−0.174720	−	0.984618i	$-0.555902\pi$
$83$	6.44760	−	4.68445i	0.707716	−	0.514186i	0.882436	+	0.470432i	$0.155902\pi$
$84$		0			0
$85$	−0.250394	+	0.770635i	−0.0271591	+	0.0835871i
$86$		0			0
$87$	6.98472			0.748841
$88$		0			0
$89$	−14.5196			−1.53907			−0.769535	−	0.638605i	$-0.779512\pi$
$89$	−14.5196			−1.53907			−0.769535	+	0.638605i	$0.779512\pi$
$90$		0			0
$91$	−2.31382	+	7.12121i	−0.242554	+	0.746506i
$92$		0			0
$93$	−3.47715	+	2.52630i	−0.360564	+	0.261965i
$94$		0			0
$95$	1.59165	+	4.89860i	0.163300	+	0.502585i
$96$		0			0
$97$	2.42159	+	1.75939i	0.245876	+	0.178639i	0.703897	−	0.710302i	$-0.251442\pi$
$97$	2.42159	+	1.75939i	0.245876	+	0.178639i	−0.458021	+	0.888941i	$0.651442\pi$
$98$		0			0
$99$	−5.51687	+	3.19793i	−0.554466	+	0.321404i
$100$		0			0
$101$	12.4346	+	9.03428i	1.23729	+	0.898944i	0.997415	−	0.0718593i	$-0.0228932\pi$
$101$	12.4346	+	9.03428i	1.23729	+	0.898944i	0.239876	+	0.970804i	$0.422893\pi$
$102$		0			0
$103$	−1.69746	−	5.22424i	−0.167256	−	0.514760i	0.831940	−	0.554866i	$-0.187230\pi$
$103$	−1.69746	−	5.22424i	−0.167256	−	0.514760i	−0.999195	+	0.0401060i	$0.987230\pi$
$104$		0			0
$105$	2.18504	−	1.58752i	0.213238	−	0.154926i
$106$		0			0
$107$	1.92725	−	5.93147i	0.186315	−	0.573417i	−0.813654	−	0.581349i	$-0.802525\pi$
$107$	1.92725	−	5.93147i	0.186315	−	0.573417i	0.999969	+	0.00793218i	$0.00252492\pi$
$108$		0			0
$109$	−19.1460			−1.83385			−0.916927	−	0.399056i	$-0.869338\pi$
$109$	−19.1460			−1.83385			−0.916927	+	0.399056i	$0.869338\pi$
$110$		0			0
$111$	4.20508			0.399129
$112$		0			0
$113$	−3.35842	+	10.3362i	−0.315934	+	0.972344i	0.659435	+	0.751762i	$0.270796\pi$
$113$	−3.35842	+	10.3362i	−0.315934	+	0.972344i	−0.975368	+	0.220582i	$0.929204\pi$
$114$		0			0
$115$	−4.03103	+	2.92871i	−0.375895	+	0.273104i
$116$		0			0
$117$	1.70965	+	5.26177i	0.158057	+	0.486451i
$118$		0			0
$119$	−1.70578	−	1.23932i	−0.156369	−	0.113609i
$120$		0			0
$121$	−7.38873	−	8.14903i	−0.671703	−	0.740821i
$122$		0			0
$123$	−8.19637	−	5.95501i	−0.739041	−	0.536945i
$124$		0			0
$125$	−0.309017	−	0.951057i	−0.0276393	−	0.0850651i
$126$		0			0
$127$	15.3227	−	11.1326i	1.35967	−	0.987861i	0.361209	−	0.932485i	$-0.382364\pi$
$127$	15.3227	−	11.1326i	1.35967	−	0.987861i	0.998466	−	0.0553760i	$-0.0176358\pi$
$128$		0			0
$129$	1.90173	−	5.85293i	0.167438	−	0.515322i
$130$		0			0
$131$	−14.5409			−1.27045			−0.635224	−	0.772328i	$-0.719092\pi$
$131$	−14.5409			−1.27045			−0.635224	+	0.772328i	$0.719092\pi$
$132$		0			0
$133$	−13.4026			−1.16215
$134$		0			0
$135$	1.57892	−	4.85941i	0.135891	−	0.418231i
$136$		0			0
$137$	16.1674	−	11.7463i	1.38128	−	1.00356i	0.384516	−	0.923118i	$-0.374368\pi$
$137$	16.1674	−	11.7463i	1.38128	−	1.00356i	0.996760	−	0.0804370i	$-0.0256316\pi$
$138$		0			0
$139$	0.639884	+	1.96936i	0.0542742	+	0.167039i	0.974519	−	0.224304i	$-0.0720108\pi$
$139$	0.639884	+	1.96936i	0.0542742	+	0.167039i	−0.920245	+	0.391343i	$0.872011\pi$
$140$		0			0
$141$	−0.812769	−	0.590512i	−0.0684475	−	0.0497300i
$142$		0			0
$143$	−8.25687	+	4.78621i	−0.690474	+	0.400243i
$144$		0			0
$145$	5.44414	+	3.95540i	0.452111	+	0.328478i
$146$		0			0
$147$	−0.0734796	−	0.226147i	−0.00606049	−	0.0186523i
$148$		0			0
$149$	−0.408306	+	0.296651i	−0.0334497	+	0.0243026i	−0.604384	−	0.796693i	$-0.706581\pi$
$149$	−0.408306	+	0.296651i	−0.0334497	+	0.0243026i	0.570935	+	0.820995i	$0.306581\pi$
$150$		0			0
$151$	−3.74492	+	11.5257i	−0.304757	+	0.937947i	0.675010	+	0.737808i	$0.264139\pi$
$151$	−3.74492	+	11.5257i	−0.304757	+	0.937947i	−0.979768	+	0.200139i	$0.935861\pi$
$152$		0			0
$153$	−1.55791			−0.125950
$154$		0			0
$155$	−4.14084			−0.332600
$156$		0			0
$157$	−3.67275	+	11.3036i	−0.293117	+	0.902123i	0.690730	+	0.723113i	$0.257289\pi$
$157$	−3.67275	+	11.3036i	−0.293117	+	0.902123i	−0.983847	+	0.179010i	$0.942711\pi$
$158$		0			0
$159$	−3.75729	+	2.72983i	−0.297973	+	0.216490i
$160$		0			0
$161$	−4.00649	−	12.3307i	−0.315756	−	0.971796i
$162$		0			0
$163$	0.0510648	+	0.0371008i	0.00399970	+	0.00290596i	0.589783	−	0.807562i	$-0.299213\pi$
$163$	0.0510648	+	0.0371008i	0.00399970	+	0.00290596i	−0.585784	+	0.810468i	$0.699213\pi$
$164$		0			0
$165$	3.42426	+	0.353904i	0.266578	+	0.0275514i
$166$		0			0
$167$	−6.10284	−	4.43397i	−0.472252	−	0.343111i	0.326066	−	0.945347i	$-0.394277\pi$
$167$	−6.10284	−	4.43397i	−0.472252	−	0.343111i	−0.798318	+	0.602236i	$0.794277\pi$
$168$		0			0
$169$	−1.45846	−	4.48866i	−0.112189	−	0.345282i
$170$		0			0
$171$	−8.01169	+	5.82084i	−0.612669	+	0.445130i
$172$		0			0
$173$	−0.436524	+	1.34348i	−0.0331883	+	0.102143i	−0.966278	−	0.257500i	$-0.917101\pi$
$173$	−0.436524	+	1.34348i	−0.0331883	+	0.102143i	0.933090	+	0.359643i	$0.117101\pi$
$174$		0			0
$175$	2.60210			0.196700
$176$		0			0
$177$	−7.06593			−0.531108
$178$		0			0
$179$	4.88918	−	15.0473i	0.365434	−	1.12469i	−0.584274	−	0.811556i	$-0.698621\pi$
$179$	4.88918	−	15.0473i	0.365434	−	1.12469i	0.949709	−	0.313135i	$-0.101379\pi$
$180$		0			0
$181$	−15.3530	+	11.1546i	−1.14118	+	0.829115i	−0.987283	−	0.158973i	$-0.949182\pi$
$181$	−15.3530	+	11.1546i	−1.14118	+	0.829115i	−0.153895	+	0.988087i	$0.549182\pi$
$182$		0			0
$183$	−0.838128	−	2.57949i	−0.0619562	−	0.190682i
$184$		0			0
$185$	3.27759	+	2.38131i	0.240973	+	0.175077i
$186$		0			0
$187$	−0.563306	−	2.62774i	−0.0411930	−	0.192159i
$188$		0			0
$189$	10.7562	+	7.81482i	0.782397	+	0.568445i
$190$		0			0
$191$	−3.20344	−	9.85919i	−0.231793	−	0.713386i	−0.997531	−	0.0702325i	$-0.977626\pi$
$191$	−3.20344	−	9.85919i	−0.231793	−	0.713386i	0.765738	−	0.643153i	$-0.222374\pi$
$192$		0			0
$193$	0.492410	−	0.357757i	0.0354444	−	0.0257519i	−0.569922	−	0.821699i	$-0.693027\pi$
$193$	0.492410	−	0.357757i	0.0354444	−	0.0257519i	0.605366	+	0.795947i	$0.293027\pi$
$194$		0			0
$195$	0.922962	−	2.84058i	0.0660947	−	0.203418i
$196$		0			0
$197$	11.1521			0.794553			0.397277	−	0.917699i	$-0.369955\pi$
$197$	11.1521			0.794553			0.397277	+	0.917699i	$0.369955\pi$
$198$		0			0
$199$	−8.91481			−0.631954			−0.315977	−	0.948767i	$-0.602332\pi$
$199$	−8.91481			−0.631954			−0.315977	+	0.948767i	$0.602332\pi$
$200$		0			0
$201$	5.08084	−	15.6372i	0.358375	−	1.10297i
$202$		0			0
$203$	−14.1662	+	10.2923i	−0.994271	+	0.722380i
$204$		0			0
$205$	−3.01626	−	9.28309i	−0.210665	−	0.648359i
$206$		0			0
$207$	−7.75027	−	5.63090i	−0.538681	−	0.391375i
$208$		0			0
$209$	−12.7149	−	11.4087i	−0.879505	−	0.789154i
$210$		0			0
$211$	14.0806	+	10.2302i	0.969351	+	0.704275i	0.955304	−	0.295627i	$-0.0955284\pi$
$211$	14.0806	+	10.2302i	0.969351	+	0.704275i	0.0140475	+	0.999901i	$0.495528\pi$
$212$		0			0
$213$	0.453359	+	1.39530i	0.0310637	+	0.0956041i
$214$		0			0
$215$	4.79676	−	3.48505i	0.327136	−	0.237678i
$216$		0			0
$217$	3.32962	−	10.2475i	0.226029	−	0.695646i
$218$		0			0
$219$	−13.6417			−0.921822
$220$		0			0
$221$	−2.33167			−0.156845
$222$		0			0
$223$	0.474865	−	1.46148i	0.0317993	−	0.0978683i	−0.933897	−	0.357542i	$-0.883615\pi$
$223$	0.474865	−	1.46148i	0.0317993	−	0.0978683i	0.965696	+	0.259674i	$0.0836150\pi$
$224$		0			0
$225$	1.55546	−	1.13011i	0.103697	−	0.0753406i
$226$		0			0
$227$	−3.12047	−	9.60382i	−0.207113	−	0.637428i	−0.999620	−	0.0275653i	$-0.991225\pi$
$227$	−3.12047	−	9.60382i	−0.207113	−	0.637428i	0.792507	−	0.609863i	$-0.208775\pi$
$228$		0			0
$229$	−1.93578	−	1.40643i	−0.127920	−	0.0929395i	0.521985	−	0.852954i	$-0.325192\pi$
$229$	−1.93578	−	1.40643i	−0.127920	−	0.0929395i	−0.649906	+	0.760015i	$0.725192\pi$
$230$		0			0
$231$	−3.62924	+	8.18958i	−0.238786	+	0.538835i
$232$		0			0
$233$	6.70453	+	4.87113i	0.439229	+	0.319118i	0.785328	−	0.619079i	$-0.212494\pi$
$233$	6.70453	+	4.87113i	0.439229	+	0.319118i	−0.346100	+	0.938198i	$0.612494\pi$
$234$		0			0
$235$	−0.299099	−	0.920531i	−0.0195111	−	0.0600489i
$236$		0			0
$237$	4.77905	−	3.47218i	0.310433	−	0.225542i
$238$		0			0
$239$	−1.40975	+	4.33878i	−0.0911894	+	0.280652i	−0.986242	−	0.165309i	$-0.947138\pi$
$239$	−1.40975	+	4.33878i	−0.0911894	+	0.280652i	0.895052	+	0.445961i	$0.147138\pi$
$240$		0			0
$241$	23.4148			1.50828			0.754139	−	0.656715i	$-0.228054\pi$
$241$	23.4148			1.50828			0.754139	+	0.656715i	$0.228054\pi$
$242$		0			0
$243$	15.8106			1.01425
$244$		0			0
$245$	0.0707928	−	0.217878i	0.00452279	−	0.0139197i
$246$		0			0
$247$	−11.9908	+	8.71180i	−0.762954	+	0.554319i
$248$		0			0
$249$	−2.55623	−	7.86727i	−0.161995	−	0.498568i
$250$		0			0
$251$	−22.4458	−	16.3079i	−1.41677	−	1.02934i	−0.992294	−	0.123903i	$-0.960459\pi$
$251$	−22.4458	−	16.3079i	−1.41677	−	1.02934i	−0.424474	−	0.905440i	$-0.639541\pi$
$252$		0			0
$253$	6.69534	−	15.1084i	0.420933	−	0.949858i
$254$		0			0
$255$	0.680420	+	0.494354i	0.0426096	+	0.0309577i
$256$		0			0
$257$	−7.87507	−	24.2370i	−0.491233	−	1.51186i	−0.822746	−	0.568410i	$-0.807559\pi$
$257$	−7.87507	−	24.2370i	−0.491233	−	1.51186i	0.331512	−	0.943451i	$-0.392441\pi$
$258$		0			0
$259$	−8.52861	+	6.19640i	−0.529942	+	0.385026i
$260$		0			0
$261$	−3.99812	+	12.3049i	−0.247477	+	0.761657i
$262$		0			0
$263$	−16.1271			−0.994442			−0.497221	−	0.867624i	$-0.665646\pi$
$263$	−16.1271			−0.994442			−0.497221	+	0.867624i	$0.665646\pi$
$264$		0			0
$265$	−4.47445			−0.274863
$266$		0			0
$267$	−4.65707	+	14.3330i	−0.285008	+	0.877165i
$268$		0			0
$269$	−14.1769	+	10.3001i	−0.864380	+	0.628009i	−0.929073	−	0.369896i	$-0.879393\pi$
$269$	−14.1769	+	10.3001i	−0.864380	+	0.628009i	0.0646931	+	0.997905i	$0.479393\pi$
$270$		0			0
$271$	−2.26740	−	6.97834i	−0.137735	−	0.423904i	0.858271	−	0.513197i	$-0.171539\pi$
$271$	−2.26740	−	6.97834i	−0.137735	−	0.423904i	−0.996005	+	0.0892933i	$0.971539\pi$
$272$		0			0
$273$	6.28757	+	4.56818i	0.380541	+	0.276479i
$274$		0			0
$275$	2.46858	+	2.21498i	0.148861	+	0.133568i
$276$		0			0
$277$	14.0824	+	10.2315i	0.846131	+	0.614750i	0.924077	−	0.382207i	$-0.124836\pi$
$277$	14.0824	+	10.2315i	0.846131	+	0.614750i	−0.0779455	+	0.996958i	$0.524836\pi$
$278$		0			0
$279$	−2.46021	−	7.57175i	−0.147289	−	0.453309i
$280$		0			0
$281$	−11.1504	+	8.10126i	−0.665179	+	0.483281i	−0.868408	−	0.495851i	$-0.834856\pi$
$281$	−11.1504	+	8.10126i	−0.665179	+	0.483281i	0.203229	+	0.979131i	$0.434856\pi$
$282$		0			0
$283$	−3.58164	+	11.0232i	−0.212906	+	0.655259i	0.786389	+	0.617731i	$0.211948\pi$
$283$	−3.58164	+	11.0232i	−0.212906	+	0.655259i	−0.999296	+	0.0375273i	$0.988052\pi$
$284$		0			0
$285$	5.34617			0.316680
$286$		0			0
$287$	25.3986			1.49923
$288$		0			0
$289$	−5.05040	+	15.5435i	−0.297082	+	0.914325i
$290$		0			0
$291$	2.51350	−	1.82616i	0.147344	−	0.107052i
$292$		0			0
$293$	−4.98906	−	15.3547i	−0.291464	−	0.897034i	−0.984386	−	0.176021i	$-0.943677\pi$
$293$	−4.98906	−	15.3547i	−0.291464	−	0.897034i	0.692922	−	0.721012i	$-0.256323\pi$
$294$		0			0
$295$	−5.50744	−	4.00139i	−0.320655	−	0.232970i
$296$		0			0
$297$	3.55205	+	16.5698i	0.206111	+	0.961476i
$298$		0			0
$299$	−11.5995	−	8.42754i	−0.670817	−	0.487377i
$300$		0			0
$301$	4.76756	+	14.6730i	0.274797	+	0.845739i
$302$		0			0
$303$	12.9065	−	9.37715i	0.741461	−	0.538703i
$304$		0			0
$305$	0.807482	−	2.48518i	0.0462363	−	0.142301i
$306$		0			0
$307$	14.3107			0.816753			0.408377	−	0.912814i	$-0.366095\pi$
$307$	14.3107			0.816753			0.408377	+	0.912814i	$0.366095\pi$
$308$		0			0
$309$	−5.70157			−0.324351
$310$		0			0
$311$	6.91405	−	21.2793i	0.392060	−	1.20664i	−0.539167	−	0.842199i	$-0.681261\pi$
$311$	6.91405	−	21.2793i	0.392060	−	1.20664i	0.931228	−	0.364438i	$-0.118739\pi$
$312$		0			0
$313$	−9.33349	+	6.78118i	−0.527560	+	0.383295i	−0.819444	−	0.573159i	$-0.805718\pi$
$313$	−9.33349	+	6.78118i	−0.527560	+	0.383295i	0.291884	+	0.956454i	$0.405718\pi$
$314$		0			0
$315$	1.54599	+	4.75807i	0.0871068	+	0.268087i
$316$		0			0
$317$	20.4935	+	14.8894i	1.15103	+	0.836272i	0.988618	−	0.150451i	$-0.0480724\pi$
$317$	20.4935	+	14.8894i	1.15103	+	0.836272i	0.162413	+	0.986723i	$0.448072\pi$
$318$		0			0
$319$	−22.2004	−	2.29446i	−1.24298	−	0.128465i
$320$		0			0
$321$	−5.23711	−	3.80498i	−0.292307	−	0.212373i
$322$		0			0
$323$	−1.28970	−	3.96930i	−0.0717610	−	0.220858i
$324$		0			0
$325$	2.32799	−	1.69139i	0.129134	−	0.0938212i
$326$		0			0
$327$	−6.14098	+	18.9000i	−0.339597	+	1.04517i
$328$		0			0
$329$	2.51858			0.138854
$330$		0			0
$331$	20.9669			1.15244			0.576221	−	0.817294i	$-0.304527\pi$
$331$	20.9669			1.15244			0.576221	+	0.817294i	$0.304527\pi$
$332$		0			0
$333$	−2.40703	+	7.40807i	−0.131904	+	0.405960i
$334$		0			0
$335$	12.8154	−	9.31097i	0.700183	−	0.508713i
$336$		0			0
$337$	−5.60649	−	17.2550i	−0.305405	−	0.939940i	−0.979526	−	0.201319i	$-0.935477\pi$
$337$	−5.60649	−	17.2550i	−0.305405	−	0.939940i	0.674121	−	0.738621i	$-0.264523\pi$
$338$		0			0
$339$	9.12616	+	6.63054i	0.495665	+	0.360122i
$340$		0			0
$341$	11.8817	−	6.88741i	0.643432	−	0.372974i
$342$		0			0
$343$	15.2183	+	11.0567i	0.821708	+	0.597006i
$344$		0			0
$345$	1.59815	+	4.91860i	0.0860415	+	0.264809i
$346$		0			0
$347$	15.3632	−	11.1620i	0.824740	−	0.599209i	−0.0933260	−	0.995636i	$-0.529750\pi$
$347$	15.3632	−	11.1620i	0.824740	−	0.599209i	0.918066	+	0.396427i	$0.129750\pi$
$348$		0			0
$349$	4.98048	−	15.3283i	0.266599	−	0.820507i	−0.724722	−	0.689041i	$-0.758032\pi$
$349$	4.98048	−	15.3283i	0.266599	−	0.820507i	0.991321	−	0.131465i	$-0.0419682\pi$
$350$		0			0
$351$	14.7028			0.784779
$352$		0			0
$353$	28.0252			1.49163			0.745816	−	0.666152i	$-0.232060\pi$
$353$	28.0252			1.49163			0.745816	+	0.666152i	$0.232060\pi$
$354$		0			0
$355$	−0.436782	+	1.34428i	−0.0231820	+	0.0713468i
$356$		0			0
$357$	−1.77052	+	1.28636i	−0.0937059	+	0.0680813i
$358$		0			0
$359$	−1.61893	−	4.98255i	−0.0854438	−	0.262969i	0.899202	−	0.437534i	$-0.144148\pi$
$359$	−1.61893	−	4.98255i	−0.0854438	−	0.262969i	−0.984646	+	0.174565i	$0.944148\pi$
$360$		0			0
$361$	−6.09156	−	4.42578i	−0.320609	−	0.232936i
$362$		0			0
$363$	−10.4142	+	4.68004i	−0.546605	+	0.245638i
$364$		0			0
$365$	−10.6328	−	7.72522i	−0.556549	−	0.404356i
$366$		0			0
$367$	8.12960	+	25.0203i	0.424362	+	1.30605i	0.903604	+	0.428368i	$0.140911\pi$
$367$	8.12960	+	25.0203i	0.424362	+	1.30605i	−0.479242	+	0.877683i	$0.659089\pi$
$368$		0			0
$369$	15.1826	−	11.0308i	0.790373	−	0.574240i
$370$		0			0
$371$	3.59787	−	11.0731i	0.186792	−	0.574887i
$372$		0			0
$373$	−36.0081			−1.86443			−0.932215	−	0.361905i	$-0.882127\pi$
$373$	−36.0081			−1.86443			−0.932215	+	0.361905i	$0.882127\pi$
$374$		0			0
$375$	−1.03795			−0.0535996
$376$		0			0
$377$	−5.98382	+	18.4163i	−0.308182	+	0.948487i
$378$		0			0
$379$	−11.4027	+	8.28455i	−0.585717	+	0.425549i	−0.840781	−	0.541376i	$-0.817904\pi$
$379$	−11.4027	+	8.28455i	−0.585717	+	0.425549i	0.255063	+	0.966924i	$0.417904\pi$
$380$		0			0
$381$	−6.07490	−	18.6966i	−0.311226	−	0.957856i
$382$		0			0
$383$	1.73227	+	1.25857i	0.0885148	+	0.0643098i	0.631162	−	0.775651i	$-0.282578\pi$
$383$	1.73227	+	1.25857i	0.0885148	+	0.0643098i	−0.542648	+	0.839960i	$0.682578\pi$
$384$		0			0
$385$	−7.46646	+	4.32804i	−0.380526	+	0.220577i
$386$		0			0
$387$	9.22251	+	6.70054i	0.468806	+	0.340608i
$388$		0			0
$389$	−9.04878	−	27.8493i	−0.458791	−	1.41201i	−0.866626	−	0.498959i	$-0.833716\pi$
$389$	−9.04878	−	27.8493i	−0.458791	−	1.41201i	0.407834	−	0.913056i	$-0.366284\pi$
$390$		0			0
$391$	3.26631	−	2.37312i	0.165185	−	0.120014i
$392$		0			0
$393$	−4.66393	+	14.3541i	−0.235264	+	0.724069i
$394$		0			0
$395$	5.69123			0.286357
$396$		0			0
$397$	9.09620			0.456525			0.228263	−	0.973600i	$-0.426696\pi$
$397$	9.09620			0.456525			0.228263	+	0.973600i	$0.426696\pi$
$398$		0			0
$399$	−4.29881	+	13.2304i	−0.215210	+	0.662348i
$400$		0			0
$401$	−26.1810	+	19.0216i	−1.30741	+	0.949892i	−0.999999	−	0.00168265i	$-0.999464\pi$
$401$	−26.1810	+	19.0216i	−1.30741	+	0.949892i	−0.307416	+	0.951575i	$0.599464\pi$
$402$		0			0
$403$	−3.68209	−	11.3323i	−0.183418	−	0.564503i
$404$		0			0
$405$	0.375843	+	0.273066i	0.0186758	+	0.0135688i
$406$		0			0
$407$	−13.3655	−	1.38135i	−0.662505	−	0.0684712i
$408$		0			0
$409$	16.8777	+	12.2624i	0.834548	+	0.606335i	0.920842	−	0.389935i	$-0.127502\pi$
$409$	16.8777	+	12.2624i	0.834548	+	0.606335i	−0.0862941	+	0.996270i	$0.527502\pi$
$410$		0			0
$411$	−6.40977	−	19.7273i	−0.316171	−	0.973074i
$412$		0			0
$413$	14.3309	−	10.4120i	0.705177	−	0.512341i
$414$		0			0
$415$	2.46276	−	7.57961i	0.120892	−	0.372068i
$416$		0			0
$417$	2.14930			0.105251
$418$		0			0
$419$	36.4444			1.78043			0.890213	−	0.455545i	$-0.150556\pi$
$419$	36.4444			1.78043			0.890213	+	0.455545i	$0.150556\pi$
$420$		0			0
$421$	−10.0210	+	30.8414i	−0.488392	+	1.50312i	0.338616	+	0.940925i	$0.390041\pi$
$421$	−10.0210	+	30.8414i	−0.488392	+	1.50312i	−0.827007	+	0.562191i	$0.809959\pi$
$422$		0			0
$423$	1.50554	−	1.09384i	0.0732017	−	0.0531841i
$424$		0			0
$425$	0.250394	+	0.770635i	0.0121459	+	0.0373813i
$426$		0			0
$427$	5.50088	+	3.99662i	0.266206	+	0.193410i
$428$		0			0
$429$	2.07636	+	9.68593i	0.100248	+	0.467641i
$430$		0			0
$431$	23.3589	+	16.9712i	1.12516	+	0.817476i	0.984983	−	0.172650i	$-0.0552330\pi$
$431$	23.3589	+	16.9712i	1.12516	+	0.817476i	0.140176	+	0.990127i	$0.455233\pi$
$432$		0			0
$433$	2.33475	+	7.18563i	0.112201	+	0.345320i	0.991353	−	0.131222i	$-0.0418901\pi$
$433$	2.33475	+	7.18563i	0.112201	+	0.345320i	−0.879152	+	0.476542i	$0.841890\pi$
$434$		0			0
$435$	5.65076	−	4.10552i	0.270933	−	0.196844i
$436$		0			0
$437$	7.93059	−	24.4079i	0.379372	−	1.16759i
$438$		0			0
$439$	1.96072			0.0935801			0.0467900	−	0.998905i	$-0.485101\pi$
$439$	1.96072			0.0935801			0.0467900	+	0.998905i	$0.485101\pi$
$440$		0			0
$441$	0.440462			0.0209744
$442$		0			0
$443$	−1.86431	+	5.73776i	−0.0885761	+	0.272609i	−0.985526	−	0.169522i	$-0.945778\pi$
$443$	−1.86431	+	5.73776i	−0.0885761	+	0.272609i	0.896950	+	0.442131i	$0.145778\pi$
$444$		0			0
$445$	−11.7466	+	8.53438i	−0.556841	+	0.404568i
$446$		0			0
$447$	0.161878	+	0.498209i	0.00765656	+	0.0235645i
$448$		0			0
$449$	−7.94777	−	5.77440i	−0.375079	−	0.272511i	0.384235	−	0.923235i	$-0.374465\pi$
$449$	−7.94777	−	5.77440i	−0.375079	−	0.272511i	−0.759314	+	0.650725i	$0.774465\pi$
$450$		0			0
$451$	24.0953	+	21.6200i	1.13460	+	1.01805i
$452$		0			0
$453$	10.1764	+	7.39361i	0.478130	+	0.347382i
$454$		0			0
$455$	2.31382	+	7.12121i	0.108474	+	0.333848i
$456$		0			0
$457$	6.33363	−	4.60165i	0.296275	−	0.215256i	−0.429710	−	0.902967i	$-0.641384\pi$
$457$	6.33363	−	4.60165i	0.296275	−	0.215256i	0.725985	+	0.687711i	$0.241384\pi$
$458$		0			0
$459$	−1.27939	+	3.93754i	−0.0597166	+	0.183789i
$460$		0			0
$461$	−3.19972			−0.149026			−0.0745130	−	0.997220i	$-0.523740\pi$
$461$	−3.19972			−0.149026			−0.0745130	+	0.997220i	$0.523740\pi$
$462$		0			0
$463$	21.2096			0.985696			0.492848	−	0.870115i	$-0.335956\pi$
$463$	21.2096			0.985696			0.492848	+	0.870115i	$0.335956\pi$
$464$		0			0
$465$	−1.32815	+	4.08763i	−0.0615916	+	0.189559i
$466$		0			0
$467$	−25.3195	+	18.3957i	−1.17165	+	0.851252i	−0.991205	−	0.132334i	$-0.957753\pi$
$467$	−25.3195	+	18.3957i	−1.17165	+	0.851252i	−0.180443	+	0.983586i	$0.557753\pi$
$468$		0			0
$469$	12.7374	+	39.2018i	0.588160	+	1.81017i
$470$		0			0
$471$	9.98031	+	7.25112i	0.459869	+	0.334114i
$472$		0			0
$473$	−7.96718	+	17.9784i	−0.366331	+	0.826647i
$474$		0			0
$475$	4.16699	+	3.02750i	0.191195	+	0.138911i
$476$		0			0
$477$	−2.65842	−	8.18178i	−0.121721	−	0.374618i
$478$		0			0
$479$	14.5641	−	10.5814i	0.665451	−	0.483478i	−0.203049	−	0.979169i	$-0.565085\pi$
$479$	14.5641	−	10.5814i	0.665451	−	0.483478i	0.868499	+	0.495690i	$0.165085\pi$
$480$		0			0
$481$	−3.60250	+	11.0873i	−0.164260	+	0.505540i
$482$		0			0
$483$	−13.4573			−0.612330
$484$		0			0
$485$	2.99325			0.135917
$486$		0			0
$487$	−6.51634	+	20.0552i	−0.295284	+	0.908789i	0.687843	+	0.725860i	$0.258558\pi$
$487$	−6.51634	+	20.0552i	−0.295284	+	0.908789i	−0.983126	+	0.182929i	$0.941442\pi$
$488$		0			0
$489$	0.0530028	−	0.0385088i	0.00239687	−	0.00174143i
$490$		0			0
$491$	6.80014	+	20.9287i	0.306886	+	0.944498i	0.978967	+	0.204020i	$0.0654008\pi$
$491$	6.80014	+	20.9287i	0.306886	+	0.944498i	−0.672081	+	0.740478i	$0.734599\pi$
$492$		0			0
$493$	−4.41135	−	3.20503i	−0.198677	−	0.144348i
$494$		0			0
$495$	−2.58355	+	5.82992i	−0.116122	+	0.262035i
$496$		0			0
$497$	−2.97553	−	2.16185i	−0.133471	−	0.0969721i
$498$		0			0
$499$	−4.98306	−	15.3363i	−0.223072	−	0.686546i	−0.998482	−	0.0550871i	$-0.982456\pi$
$499$	−4.98306	−	15.3363i	−0.223072	−	0.686546i	0.775409	−	0.631459i	$-0.217544\pi$
$500$		0			0
$501$	−6.33445	+	4.60225i	−0.283003	+	0.205613i
$502$		0			0
$503$	−0.978639	+	3.01194i	−0.0436354	+	0.134296i	−0.970501	−	0.241098i	$-0.922493\pi$
$503$	−0.978639	+	3.01194i	−0.0436354	+	0.134296i	0.926865	+	0.375394i	$0.122493\pi$
$504$		0			0
$505$	15.3700			0.683957
$506$		0			0
$507$	−4.89878			−0.217563
$508$		0			0
$509$	8.95663	−	27.5657i	0.396996	−	1.22183i	−0.530401	−	0.847747i	$-0.677959\pi$
$509$	8.95663	−	27.5657i	0.396996	−	1.22183i	0.927396	−	0.374080i	$-0.122041\pi$
$510$		0			0
$511$	27.6677	−	20.1018i	1.22395	−	0.889250i
$512$		0			0
$513$	8.13251	+	25.0293i	0.359059	+	1.10507i
$514$		0			0
$515$	−4.44401	−	3.22876i	−0.195826	−	0.142276i
$516$		0			0
$517$	2.38934	+	2.14389i	0.105083	+	0.0942881i
$518$		0			0
$519$	1.18621	+	0.861831i	0.0520688	+	0.0378302i
$520$		0			0
$521$	−4.89783	−	15.0740i	−0.214578	−	0.660402i	−0.999183	−	0.0404070i	$-0.987135\pi$
$521$	−4.89783	−	15.0740i	−0.214578	−	0.660402i	0.784606	−	0.619995i	$-0.212865\pi$
$522$		0			0
$523$	−19.4915	+	14.1614i	−0.852305	+	0.619236i	−0.925781	−	0.378061i	$-0.876591\pi$
$523$	−19.4915	+	14.1614i	−0.852305	+	0.619236i	0.0734754	+	0.997297i	$0.476591\pi$
$524$		0			0
$525$	0.834609	−	2.56866i	0.0364253	−	0.112106i
$526$		0			0
$527$	3.35529			0.146159
$528$		0			0
$529$	1.82653			0.0794143
$530$		0			0
$531$	4.04460	−	12.4480i	0.175521	−	0.540197i
$532$		0			0
$533$	22.7231	−	16.5093i	0.984248	−	0.715098i
$534$		0			0
$535$	−1.92725	−	5.93147i	−0.0833224	−	0.256440i
$536$		0			0
$537$	−13.2858	−	9.65272i	−0.573326	−	0.416545i
$538$		0			0
$539$	0.159261	+	0.742928i	0.00685985	+	0.0320002i
$540$		0			0
$541$	17.2488	+	12.5320i	0.741584	+	0.538792i	0.893207	−	0.449646i	$-0.148450\pi$
$541$	17.2488	+	12.5320i	0.741584	+	0.538792i	−0.151623	+	0.988438i	$0.548450\pi$
$542$		0			0
$543$	6.08688	+	18.7335i	0.261213	+	0.803931i
$544$		0			0
$545$	−15.4894	+	11.2537i	−0.663494	+	0.482057i
$546$		0			0
$547$	9.05977	−	27.8831i	0.387368	−	1.19220i	−0.547380	−	0.836884i	$-0.684375\pi$
$547$	9.05977	−	27.8831i	0.387368	−	1.19220i	0.934748	−	0.355312i	$-0.115625\pi$
$548$		0			0
$549$	5.02403			0.214420
$550$		0			0
$551$	−34.6607			−1.47659
$552$		0			0
$553$	−4.57628	+	14.0843i	−0.194603	+	0.598927i
$554$		0			0
$555$	3.40198	−	2.47169i	0.144406	−	0.104917i
$556$		0			0
$557$	−0.520009	−	1.60042i	−0.0220335	−	0.0678122i	0.939435	−	0.342727i	$-0.111350\pi$
$557$	−0.520009	−	1.60042i	−0.0220335	−	0.0678122i	−0.961469	+	0.274914i	$0.911350\pi$
$558$		0			0
$559$	13.8029	+	10.0284i	0.583802	+	0.424157i
$560$		0			0
$561$	−2.77465	−	0.286766i	−0.117146	−	0.0121073i
$562$		0			0
$563$	0.929018	+	0.674971i	0.0391534	+	0.0284466i	0.607190	−	0.794557i	$-0.292297\pi$
$563$	0.929018	+	0.674971i	0.0391534	+	0.0284466i	−0.568036	+	0.823003i	$0.692297\pi$
$564$		0			0
$565$	3.35842	+	10.3362i	0.141290	+	0.434846i
$566$		0			0
$567$	−0.977981	+	0.710545i	−0.0410714	+	0.0298401i
$568$		0			0
$569$	−2.06568	+	6.35751i	−0.0865979	+	0.266521i	−0.984973	−	0.172708i	$-0.944748\pi$
$569$	−2.06568	+	6.35751i	−0.0865979	+	0.266521i	0.898375	+	0.439229i	$0.144748\pi$
$570$		0			0
$571$	−17.7964			−0.744756			−0.372378	−	0.928081i	$-0.621458\pi$
$571$	−17.7964			−0.744756			−0.372378	+	0.928081i	$0.621458\pi$
$572$		0			0
$573$	−10.7600			−0.449505
$574$		0			0
$575$	−1.53972	+	4.73876i	−0.0642106	+	0.197620i
$576$		0			0
$577$	37.1992	−	27.0268i	1.54862	−	1.12514i	0.603999	−	0.796985i	$-0.293573\pi$
$577$	37.1992	−	27.0268i	1.54862	−	1.12514i	0.944624	−	0.328156i	$-0.106427\pi$
$578$		0			0
$579$	−0.195222	−	0.600832i	−0.00811315	−	0.0249697i
$580$		0			0
$581$	16.7773	+	12.1894i	0.696039	+	0.505702i
$582$		0			0
$583$	12.8390	−	7.44231i	0.531737	−	0.308229i
$584$		0			0
$585$	4.47593	+	3.25195i	0.185057	+	0.134452i
$586$		0			0
$587$	−1.45930	−	4.49126i	−0.0602318	−	0.185374i	0.916413	−	0.400233i	$-0.131071\pi$
$587$	−1.45930	−	4.49126i	−0.0602318	−	0.185374i	−0.976645	+	0.214859i	$0.931071\pi$
$588$		0			0
$589$	17.2548	−	12.5364i	0.710974	−	0.516553i
$590$		0			0
$591$	3.57697	−	11.0088i	0.147137	−	0.452841i
$592$		0			0
$593$	3.67545			0.150932			0.0754662	−	0.997148i	$-0.475955\pi$
$593$	3.67545			0.150932			0.0754662	+	0.997148i	$0.475955\pi$
$594$		0			0
$595$	−2.10846			−0.0864385
$596$		0			0
$597$	−2.85938	+	8.80026i	−0.117027	+	0.360171i
$598$		0			0
$599$	−32.2566	+	23.4358i	−1.31797	+	0.957562i	−0.318015	+	0.948086i	$0.603016\pi$
$599$	−32.2566	+	23.4358i	−1.31797	+	0.957562i	−0.999955	+	0.00947590i	$0.996984\pi$
$600$		0			0
$601$	−3.81563	−	11.7433i	−0.155643	−	0.479019i	0.842583	−	0.538567i	$-0.181034\pi$
$601$	−3.81563	−	11.7433i	−0.155643	−	0.479019i	−0.998225	+	0.0595481i	$0.981034\pi$
$602$		0			0
$603$	24.6397	+	17.9018i	1.00341	+	0.729017i
$604$		0			0
$605$	−10.7675	−	2.24971i	−0.437761	−	0.0914638i
$606$		0			0
$607$	−15.9430	−	11.5833i	−0.647108	−	0.470151i	0.215177	−	0.976575i	$-0.430967\pi$
$607$	−15.9430	−	11.5833i	−0.647108	−	0.470151i	−0.862285	+	0.506424i	$0.830967\pi$
$608$		0			0
$609$	5.61636	+	17.2854i	0.227586	+	0.700439i
$610$		0			0
$611$	2.25327	−	1.63710i	0.0911577	−	0.0662300i
$612$		0			0
$613$	11.6699	−	35.9163i	0.471343	−	1.45065i	−0.379483	−	0.925199i	$-0.623898\pi$
$613$	11.6699	−	35.9163i	0.471343	−	1.45065i	0.850826	−	0.525447i	$-0.176102\pi$
$614$		0			0
$615$	−10.1313			−0.408532
$616$		0			0
$617$	−3.49353			−0.140644			−0.0703222	−	0.997524i	$-0.522403\pi$
$617$	−3.49353			−0.140644			−0.0703222	+	0.997524i	$0.522403\pi$
$618$		0			0
$619$	−2.00503	+	6.17083i	−0.0805888	+	0.248027i	−0.983231	−	0.182365i	$-0.941625\pi$
$619$	−2.00503	+	6.17083i	−0.0805888	+	0.248027i	0.902642	+	0.430392i	$0.141625\pi$
$620$		0			0
$621$	−20.5965	+	14.9642i	−0.826507	+	0.600493i
$622$		0			0
$623$	−11.6751	−	35.9321i	−0.467751	−	1.43959i
$624$		0			0
$625$	−0.809017	−	0.587785i	−0.0323607	−	0.0235114i
$626$		0			0
$627$	−15.3403	+	8.89222i	−0.612633	+	0.355121i
$628$		0			0
$629$	−2.65581	−	1.92956i	−0.105894	−	0.0769366i
$630$		0			0
$631$	−10.1152	−	31.1312i	−0.402678	−	1.23932i	−0.922819	−	0.385235i	$-0.874120\pi$
$631$	−10.1152	−	31.1312i	−0.402678	−	1.23932i	0.520141	−	0.854081i	$-0.325880\pi$
$632$		0			0
$633$	14.6150	−	10.6184i	0.580895	−	0.422045i
$634$		0			0
$635$	5.85277	−	18.0130i	0.232260	−	0.714823i
$636$		0			0
$637$	0.659221			0.0261193
$638$		0			0
$639$	−2.71759			−0.107506
$640$		0			0
$641$	−2.22299	+	6.84167i	−0.0878029	+	0.270230i	−0.985311	−	0.170768i	$-0.945375\pi$
$641$	−2.22299	+	6.84167i	−0.0878029	+	0.270230i	0.897508	+	0.440997i	$0.145375\pi$
$642$		0			0
$643$	6.64278	−	4.82626i	0.261966	−	0.190329i	−0.449047	−	0.893508i	$-0.648237\pi$
$643$	6.64278	−	4.82626i	0.261966	−	0.190329i	0.711013	+	0.703179i	$0.248237\pi$
$644$		0			0
$645$	−1.90173	−	5.85293i	−0.0748807	−	0.230459i
$646$		0			0
$647$	−39.6425	−	28.8019i	−1.55851	−	1.13232i	−0.937221	−	0.348737i	$-0.886611\pi$
$647$	−39.6425	−	28.8019i	−1.55851	−	1.13232i	−0.621286	−	0.783584i	$-0.713389\pi$
$648$		0			0
$649$	22.4585	+	2.32113i	0.881574	+	0.0911124i
$650$		0			0
$651$	−9.04788	−	6.57367i	−0.354614	−	0.257642i
$652$		0			0
$653$	−13.5490	−	41.6995i	−0.530213	−	1.63183i	−0.753770	−	0.657138i	$-0.771767\pi$
$653$	−13.5490	−	41.6995i	−0.530213	−	1.63183i	0.223557	−	0.974691i	$-0.428233\pi$
$654$		0			0
$655$	−11.7639	+	8.54695i	−0.459652	+	0.333957i
$656$		0			0
$657$	7.80865	−	24.0325i	0.304644	−	0.937599i
$658$		0			0
$659$	37.2905			1.45263			0.726315	−	0.687362i	$-0.241231\pi$
$659$	37.2905			1.45263			0.726315	+	0.687362i	$0.241231\pi$
$660$		0			0
$661$	35.5579			1.38304			0.691520	−	0.722357i	$-0.256941\pi$
$661$	35.5579			1.38304			0.691520	+	0.722357i	$0.256941\pi$
$662$		0			0
$663$	−0.747870	+	2.30171i	−0.0290449	+	0.0893909i
$664$		0			0
$665$	−10.8429	+	7.87784i	−0.420471	+	0.305490i
$666$		0			0
$667$	−10.3612	−	31.8886i	−0.401189	−	1.23473i
$668$		0			0
$669$	−1.29040	−	0.937527i	−0.0498896	−	0.0362469i
$670$		0			0
$671$	1.81657	+	8.47404i	0.0701280	+	0.327137i
$672$		0			0
$673$	11.0203	+	8.00673i	0.424802	+	0.308637i	0.779567	−	0.626319i	$-0.215439\pi$
$673$	11.0203	+	8.00673i	0.424802	+	0.308637i	−0.354765	+	0.934955i	$0.615439\pi$
$674$		0			0
$675$	−1.57892	−	4.85941i	−0.0607725	−	0.187039i
$676$		0			0
$677$	−5.57612	+	4.05129i	−0.214308	+	0.155704i	−0.689761	−	0.724038i	$-0.742284\pi$
$677$	−5.57612	+	4.05129i	−0.214308	+	0.155704i	0.475453	+	0.879741i	$0.342284\pi$
$678$		0			0
$679$	−2.40685	+	7.40753i	−0.0923665	+	0.284275i
$680$		0			0
$681$	−10.4813			−0.401644
$682$		0			0
$683$	−8.29642			−0.317454			−0.158727	−	0.987323i	$-0.550739\pi$
$683$	−8.29642			−0.317454			−0.158727	+	0.987323i	$0.550739\pi$
$684$		0			0
$685$	6.17540	−	19.0059i	0.235950	−	0.726180i
$686$		0			0
$687$	−2.00925	+	1.45981i	−0.0766577	+	0.0556951i
$688$		0			0
$689$	−3.97875	−	12.2453i	−0.151578	−	0.466510i
$690$		0			0
$691$	14.8505	+	10.7895i	0.564939	+	0.410452i	0.833263	−	0.552877i	$-0.186470\pi$
$691$	14.8505	+	10.7895i	0.564939	+	0.410452i	−0.268324	+	0.963329i	$0.586470\pi$
$692$		0			0
$693$	−12.3501	−	11.0814i	−0.469142	−	0.420948i
$694$		0			0
$695$	1.67524	+	1.21713i	0.0635454	+	0.0461684i
$696$		0			0
$697$	2.44406	+	7.52203i	0.0925752	+	0.284917i
$698$		0			0
$699$	6.95898	−	5.05600i	0.263213	−	0.191235i
$700$		0			0
$701$	−8.67175	+	26.6889i	−0.327528	+	1.00803i	0.642759	+	0.766068i	$0.277790\pi$
$701$	−8.67175	+	26.6889i	−0.327528	+	1.00803i	−0.970287	+	0.241958i	$0.922210\pi$
$702$		0			0
$703$	−20.8671			−0.787018
$704$		0			0
$705$	−1.00464			−0.0378369
$706$		0			0
$707$	−12.3589	+	38.0369i	−0.464805	+	1.43052i
$708$		0			0
$709$	13.1536	−	9.55667i	0.493995	−	0.358908i	−0.312724	−	0.949844i	$-0.601241\pi$
$709$	13.1536	−	9.55667i	0.493995	−	0.358908i	0.806719	+	0.590936i	$0.201241\pi$
$710$		0			0
$711$	3.38135	+	10.4067i	0.126811	+	0.390283i
$712$		0			0
$713$	16.6918	+	12.1273i	0.625114	+	0.454172i
$714$		0			0
$715$	−3.86669	+	8.72539i	−0.144606	+	0.326311i
$716$		0			0
$717$	3.83086	+	2.78328i	0.143066	+	0.103944i
$718$		0			0
$719$	1.16026	+	3.57092i	0.0432705	+	0.133173i	0.970358	−	0.241672i	$-0.0776959\pi$
$719$	1.16026	+	3.57092i	0.0432705	+	0.133173i	−0.927087	+	0.374845i	$0.877696\pi$
$720$		0			0
$721$	11.5637	−	8.40155i	0.430656	−	0.312890i
$722$		0			0
$723$	7.51017	−	23.1139i	0.279306	−	0.859616i
$724$		0			0
$725$	6.72933			0.249921
$726$		0			0
$727$	−35.1985			−1.30544			−0.652720	−	0.757599i	$-0.726372\pi$
$727$	−35.1985			−1.30544			−0.652720	+	0.757599i	$0.726372\pi$
$728$		0			0
$729$	4.64050	−	14.2820i	0.171871	−	0.528963i
$730$		0			0
$731$	−3.88678	+	2.82391i	−0.143758	+	0.104446i
$732$		0			0
$733$	−6.26899	−	19.2940i	−0.231550	−	0.712639i	−0.997560	−	0.0698100i	$-0.977761\pi$
$733$	−6.26899	−	19.2940i	−0.231550	−	0.712639i	0.766010	−	0.642829i	$-0.222239\pi$
$734$		0			0
$735$	−0.192372	−	0.139766i	−0.00709575	−	0.00515536i
$736$		0			0
$737$	−21.2859	+	48.0327i	−0.784074	+	1.76931i
$738$		0			0
$739$	12.6479	+	9.18923i	0.465260	+	0.338031i	0.795591	−	0.605834i	$-0.207160\pi$
$739$	12.6479	+	9.18923i	0.465260	+	0.338031i	−0.330331	+	0.943865i	$0.607160\pi$
$740$		0			0
$741$	4.75389	+	14.6310i	0.174638	+	0.537482i
$742$		0			0
$743$	−16.1671	+	11.7461i	−0.593114	+	0.430922i	−0.843428	−	0.537242i	$-0.819466\pi$
$743$	−16.1671	+	11.7461i	−0.593114	+	0.430922i	0.250314	+	0.968165i	$0.419466\pi$
$744$		0			0
$745$	−0.155959	+	0.479992i	−0.00571389	+	0.0175856i
$746$		0			0
$747$	15.3229			0.560637
$748$		0			0
$749$	16.2286			0.592979
$750$		0			0
$751$	−9.71724	+	29.9066i	−0.354587	+	1.09131i	0.601661	+	0.798751i	$0.294506\pi$
$751$	−9.71724	+	29.9066i	−0.354587	+	1.09131i	−0.956248	+	0.292556i	$0.905494\pi$
$752$		0			0
$753$	−23.2977	+	16.9268i	−0.849016	+	0.616846i
$754$		0			0
$755$	3.74492	+	11.5257i	0.136292	+	0.419463i
$756$		0			0
$757$	−36.7173	−	26.6766i	−1.33451	−	0.969579i	−0.999627	−	0.0273202i	$-0.991303\pi$
$757$	−36.7173	−	26.6766i	−1.33451	−	0.969579i	−0.334885	−	0.942259i	$-0.608697\pi$
$758$		0			0
$759$	−12.7668	−	11.4553i	−0.463405	−	0.415800i
$760$		0			0
$761$	−28.8093	−	20.9312i	−1.04434	−	0.758754i	−0.0732080	−	0.997317i	$-0.523324\pi$
$761$	−28.8093	−	20.9312i	−1.04434	−	0.758754i	−0.971127	+	0.238563i	$0.923324\pi$
$762$		0			0
$763$	−15.3951	−	47.3814i	−0.557342	−	1.71532i
$764$		0			0
$765$	−1.26038	+	0.915719i	−0.0455691	+	0.0331079i
$766$		0			0
$767$	6.05338	−	18.6304i	0.218575	−	0.672705i
$768$		0			0
$769$	−18.9907			−0.684821			−0.342411	−	0.939550i	$-0.611243\pi$
$769$	−18.9907			−0.684821			−0.342411	+	0.939550i	$0.611243\pi$
$770$		0			0
$771$	−26.4514			−0.952625
$772$		0			0
$773$	13.4981	−	41.5429i	0.485493	−	1.49419i	−0.345772	−	0.938318i	$-0.612383\pi$
$773$	13.4981	−	41.5429i	0.485493	−	1.49419i	0.831265	−	0.555876i	$-0.187617\pi$
$774$		0			0
$775$	−3.35001	+	2.43392i	−0.120336	+	0.0874291i
$776$		0			0
$777$	3.38128	+	10.4065i	0.121303	+	0.373331i
$778$		0			0
$779$	40.6733	+	29.5509i	1.45727	+	1.05877i
$780$		0			0
$781$	−0.982618	−	4.58377i	−0.0351608	−	0.164020i
$782$		0			0
$783$	27.8167	+	20.2100i	0.994089	+	0.722248i
$784$		0			0
$785$	3.67275	+	11.3036i	0.131086	+	0.403442i
$786$		0			0
$787$	−41.1672	+	29.9097i	−1.46745	+	1.06617i	−0.486108	+	0.873899i	$0.661584\pi$
$787$	−41.1672	+	29.9097i	−1.46745	+	1.06617i	−0.981343	+	0.192266i	\(

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:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.bo (of order \(5\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.320745 - 0.987151i) q^{3} +(0.809017 - 0.587785i) q^{5} +(0.804092 + 2.47474i) q^{7} +(1.55546 + 1.13011i) q^{9} +O(q^{10})\)	\(q+(0.320745 - 0.987151i) q^{3} +(0.809017 - 0.587785i) q^{5} +(0.804092 + 2.47474i) q^{7} +(1.55546 + 1.13011i) q^{9} +(-1.34374 + 3.03222i) q^{11} +(2.32799 + 1.69139i) q^{13} +(-0.320745 - 0.987151i) q^{15} +(-0.655541 + 0.476279i) q^{17} +(-1.59165 + 4.89860i) q^{19} +2.70085 q^{21} -4.98262 q^{23} +(0.309017 - 0.951057i) q^{25} +(4.13366 - 3.00328i) q^{27} +(2.07948 + 6.39997i) q^{29} +(-3.35001 - 2.43392i) q^{31} +(2.56226 + 2.29904i) q^{33} +(2.10514 + 1.52947i) q^{35} +(1.25193 + 3.85304i) q^{37} +(2.41635 - 1.75558i) q^{39} +(3.01626 - 9.28309i) q^{41} +5.92912 q^{43} +1.92266 q^{45} +(0.299099 - 0.920531i) q^{47} +(0.185338 - 0.134656i) q^{49} +(0.259897 + 0.799882i) q^{51} +(-3.61991 - 2.63002i) q^{53} +(0.695188 + 3.24295i) q^{55} +(4.32514 + 3.14240i) q^{57} +(-2.10365 - 6.47438i) q^{59} +(2.11402 - 1.53592i) q^{61} +(-1.54599 + 4.75807i) q^{63} +2.87756 q^{65} +15.8408 q^{67} +(-1.59815 + 4.91860i) q^{69} +(-1.14351 + 0.830809i) q^{71} +(-4.06139 - 12.4997i) q^{73} +(-0.839721 - 0.610093i) q^{75} +(-8.58445 - 0.887220i) q^{77} +(4.60431 + 3.34522i) q^{79} +(0.143559 + 0.441831i) q^{81} +(6.44760 - 4.68445i) q^{83} +(-0.250394 + 0.770635i) q^{85} +6.98472 q^{87} -14.5196 q^{89} +(-2.31382 + 7.12121i) q^{91} +(-3.47715 + 2.52630i) q^{93} +(1.59165 + 4.89860i) q^{95} +(2.42159 + 1.75939i) q^{97} +(-5.51687 + 3.19793i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{3} + 3 q^{5} + q^{7} - 10 q^{9}+O(q^{10}) \) 12 * q + q^3 + 3 * q^5 + q^7 - 10 * q^9	\( 12 q + q^{3} + 3 q^{5} + q^{7} - 10 q^{9} - 4 q^{11} + 18 q^{13} - q^{15} + 3 q^{17} - 4 q^{19} - 28 q^{21} + 18 q^{23} - 3 q^{25} - 23 q^{27} + 15 q^{29} + 8 q^{31} + 4 q^{33} - 6 q^{35} + 6 q^{37} + 33 q^{39} + 2 q^{41} + 36 q^{43} - 10 q^{45} + 16 q^{47} - 16 q^{49} + 10 q^{51} + 19 q^{53} - 6 q^{55} + 62 q^{57} - 46 q^{59} + 18 q^{61} + 7 q^{63} + 2 q^{65} + 44 q^{67} - q^{69} + 6 q^{71} + 25 q^{73} - 4 q^{75} + 10 q^{77} - 19 q^{79} + 30 q^{81} - 3 q^{85} - 6 q^{87} - 50 q^{89} + 46 q^{91} - 37 q^{93} + 4 q^{95} - 31 q^{97} - 79 q^{99}+O(q^{100}) \) 12 * q + q^3 + 3 * q^5 + q^7 - 10 * q^9 - 4 * q^11 + 18 * q^13 - q^15 + 3 * q^17 - 4 * q^19 - 28 * q^21 + 18 * q^23 - 3 * q^25 - 23 * q^27 + 15 * q^29 + 8 * q^31 + 4 * q^33 - 6 * q^35 + 6 * q^37 + 33 * q^39 + 2 * q^41 + 36 * q^43 - 10 * q^45 + 16 * q^47 - 16 * q^49 + 10 * q^51 + 19 * q^53 - 6 * q^55 + 62 * q^57 - 46 * q^59 + 18 * q^61 + 7 * q^63 + 2 * q^65 + 44 * q^67 - q^69 + 6 * q^71 + 25 * q^73 - 4 * q^75 + 10 * q^77 - 19 * q^79 + 30 * q^81 - 3 * q^85 - 6 * q^87 - 50 * q^89 + 46 * q^91 - 37 * q^93 + 4 * q^95 - 31 * q^97 - 79 * q^99

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