LMFDB - Embedded newform 880.2.bo.i.81.3

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			81.3
Root			$1.85498 - 1.34772i$ of defining polynomial
Character	$\chi$	$=$	880.81
Dual form			880.2.bo.i.641.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.01756 - 3.13172i) q^{3} +(0.809017 - 0.587785i) q^{5} +(-1.08622 - 3.34304i) q^{7} +(-6.34518 - 4.61004i) q^{9} +O(q^{10})$	\(q+(1.01756 - 3.13172i) q^{3} +(0.809017 - 0.587785i) q^{5} +(-1.08622 - 3.34304i) q^{7} +(-6.34518 - 4.61004i) q^{9} +(1.91497 - 2.70793i) q^{11} +(3.45546 + 2.51054i) q^{13} +(-1.01756 - 3.13172i) q^{15} +(-1.28632 + 0.934565i) q^{17} +(-1.71533 + 5.27925i) q^{19} -11.5747 q^{21} +6.39042 q^{23} +(0.309017 - 0.951057i) q^{25} +(-12.9019 + 9.37380i) q^{27} +(-0.117804 - 0.362562i) q^{29} +(-0.615229 - 0.446990i) q^{31} +(-6.53187 - 8.75262i) q^{33} +(-2.84376 - 2.06611i) q^{35} +(-0.448664 - 1.38085i) q^{37} +(11.3784 - 8.26690i) q^{39} +(-1.89183 + 5.82245i) q^{41} +7.19067 q^{43} -7.84307 q^{45} +(-1.33546 + 4.11012i) q^{47} +(-4.33291 + 3.14804i) q^{49} +(1.61789 + 4.97936i) q^{51} +(5.62189 + 4.08454i) q^{53} +(-0.0424355 - 3.31635i) q^{55} +(14.7877 + 10.7439i) q^{57} +(-3.92793 - 12.0889i) q^{59} +(7.19700 - 5.22893i) q^{61} +(-8.51929 + 26.2197i) q^{63} +4.27118 q^{65} -12.5135 q^{67} +(6.50261 - 20.0130i) q^{69} +(5.95347 - 4.32545i) q^{71} +(2.17304 + 6.68793i) q^{73} +(-2.66400 - 1.93551i) q^{75} +(-11.1328 - 3.46042i) q^{77} +(-5.65811 - 4.11086i) q^{79} +(8.95672 + 27.5660i) q^{81} +(-8.69768 + 6.31923i) q^{83} +(-0.491330 + 1.51216i) q^{85} -1.25531 q^{87} +0.451594 q^{89} +(4.63944 - 14.2787i) q^{91} +(-2.02588 + 1.47189i) q^{93} +(1.71533 + 5.27925i) q^{95} +(2.24871 + 1.63379i) q^{97} +(-24.6345 + 8.35419i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.01756	−	3.13172i	0.587487	−	1.80810i	−0.00156040	−	0.999999i	$-0.500497\pi$
$3$	1.01756	−	3.13172i	0.587487	−	1.80810i	0.589047	−	0.808099i	$-0.299503\pi$
$4$		0			0
$5$	0.809017	−	0.587785i	0.361803	−	0.262866i
$5$	0.809017	−	0.587785i	0.361803	−	0.262866i
$6$		0			0
$7$	−1.08622	−	3.34304i	−0.410552	−	1.26355i	−0.916169	−	0.400791i	$-0.868735\pi$
$7$	−1.08622	−	3.34304i	−0.410552	−	1.26355i	0.505617	−	0.862758i	$-0.331265\pi$
$8$		0			0
$9$	−6.34518	−	4.61004i	−2.11506	−	1.53668i
$10$		0			0
$11$	1.91497	−	2.70793i	0.577386	−	0.816471i
$11$	1.91497	−	2.70793i	0.577386	−	0.816471i
$12$		0			0
$13$	3.45546	+	2.51054i	0.958372	+	0.696298i	0.952772	−	0.303687i	$-0.0982176\pi$
$13$	3.45546	+	2.51054i	0.958372	+	0.696298i	0.00559963	+	0.999984i	$0.498218\pi$
$14$		0			0
$15$	−1.01756	−	3.13172i	−0.262732	−	0.808606i
$16$		0			0
$17$	−1.28632	+	0.934565i	−0.311978	+	0.226665i	−0.732745	−	0.680504i	$-0.761761\pi$
$17$	−1.28632	+	0.934565i	−0.311978	+	0.226665i	0.420767	+	0.907169i	$0.361761\pi$
$18$		0			0
$19$	−1.71533	+	5.27925i	−0.393525	+	1.21114i	0.536580	+	0.843849i	$0.319716\pi$
$19$	−1.71533	+	5.27925i	−0.393525	+	1.21114i	−0.930105	+	0.367295i	$0.880284\pi$
$20$		0			0
$21$	−11.5747			−2.52581
$22$		0			0
$23$	6.39042			1.33249			0.666247	−	0.745731i	$-0.267900\pi$
$23$	6.39042			1.33249			0.666247	+	0.745731i	$0.267900\pi$
$24$		0			0
$25$	0.309017	−	0.951057i	0.0618034	−	0.190211i
$26$		0			0
$27$	−12.9019	+	9.37380i	−2.48298	+	1.80399i
$28$		0			0
$29$	−0.117804	−	0.362562i	−0.0218756	−	0.0673261i	0.939523	−	0.342486i	$-0.111269\pi$
$29$	−0.117804	−	0.362562i	−0.0218756	−	0.0673261i	−0.961398	+	0.275160i	$0.911269\pi$
$30$		0			0
$31$	−0.615229	−	0.446990i	−0.110498	−	0.0802818i	0.531164	−	0.847269i	$-0.321755\pi$
$31$	−0.615229	−	0.446990i	−0.110498	−	0.0802818i	−0.641662	+	0.766987i	$0.721755\pi$
$32$		0			0
$33$	−6.53187	−	8.75262i	−1.13705	−	1.52364i
$34$		0			0
$35$	−2.84376	−	2.06611i	−0.480683	−	0.349236i
$36$		0			0
$37$	−0.448664	−	1.38085i	−0.0737599	−	0.227010i	0.907379	−	0.420313i	$-0.138080\pi$
$37$	−0.448664	−	1.38085i	−0.0737599	−	0.227010i	−0.981139	+	0.193304i	$0.938080\pi$
$38$		0			0
$39$	11.3784	−	8.26690i	1.82200	−	1.32376i
$40$		0			0
$41$	−1.89183	+	5.82245i	−0.295454	+	0.909313i	0.687615	+	0.726076i	$0.258658\pi$
$41$	−1.89183	+	5.82245i	−0.295454	+	0.909313i	−0.983069	+	0.183238i	$0.941342\pi$
$42$		0			0
$43$	7.19067			1.09657			0.548284	−	0.836293i	$-0.315281\pi$
$43$	7.19067			1.09657			0.548284	+	0.836293i	$0.315281\pi$
$44$		0			0
$45$	−7.84307			−1.16918
$46$		0			0
$47$	−1.33546	+	4.11012i	−0.194797	+	0.599523i	0.805182	+	0.593028i	$0.202068\pi$
$47$	−1.33546	+	4.11012i	−0.194797	+	0.599523i	−0.999979	+	0.00649536i	$0.997932\pi$
$48$		0			0
$49$	−4.33291	+	3.14804i	−0.618987	+	0.449720i
$50$		0			0
$51$	1.61789	+	4.97936i	0.226550	+	0.697250i
$52$		0			0
$53$	5.62189	+	4.08454i	0.772226	+	0.561055i	0.902636	−	0.430405i	$-0.141629\pi$
$53$	5.62189	+	4.08454i	0.772226	+	0.561055i	−0.130410	+	0.991460i	$0.541629\pi$
$54$		0			0
$55$	−0.0424355	−	3.31635i	−0.00572200	−	0.447177i
$56$		0			0
$57$	14.7877	+	10.7439i	1.95868	+	1.42306i
$58$		0			0
$59$	−3.92793	−	12.0889i	−0.511373	−	1.57384i	−0.789786	−	0.613382i	$-0.789808\pi$
$59$	−3.92793	−	12.0889i	−0.511373	−	1.57384i	0.278413	−	0.960461i	$-0.410192\pi$
$60$		0			0
$61$	7.19700	−	5.22893i	0.921482	−	0.669496i	−0.0224105	−	0.999749i	$-0.507134\pi$
$61$	7.19700	−	5.22893i	0.921482	−	0.669496i	0.943892	+	0.330253i	$0.107134\pi$
$62$		0			0
$63$	−8.51929	+	26.2197i	−1.07333	+	3.30337i
$64$		0			0
$65$	4.27118			0.529775
$66$		0			0
$67$	−12.5135			−1.52877			−0.764383	−	0.644763i	$-0.776956\pi$
$67$	−12.5135			−1.52877			−0.764383	+	0.644763i	$0.776956\pi$
$68$		0			0
$69$	6.50261	−	20.0130i	0.782823	−	2.40928i
$70$		0			0
$71$	5.95347	−	4.32545i	0.706547	−	0.513336i	−0.175511	−	0.984477i	$-0.556158\pi$
$71$	5.95347	−	4.32545i	0.706547	−	0.513336i	0.882058	+	0.471141i	$0.156158\pi$
$72$		0			0
$73$	2.17304	+	6.68793i	0.254335	+	0.782763i	0.993960	+	0.109743i	$0.0350029\pi$
$73$	2.17304	+	6.68793i	0.254335	+	0.782763i	−0.739625	+	0.673019i	$0.764997\pi$
$74$		0			0
$75$	−2.66400	−	1.93551i	−0.307612	−	0.223493i
$76$		0			0
$77$	−11.1328	−	3.46042i	−1.26870	−	0.394352i
$78$		0			0
$79$	−5.65811	−	4.11086i	−0.636587	−	0.462508i	0.222089	−	0.975026i	$-0.428712\pi$
$79$	−5.65811	−	4.11086i	−0.636587	−	0.462508i	−0.858676	+	0.512519i	$0.828712\pi$
$80$		0			0
$81$	8.95672	+	27.5660i	0.995191	+	3.06288i
$82$		0			0
$83$	−8.69768	+	6.31923i	−0.954694	+	0.693626i	−0.951912	−	0.306370i	$-0.900886\pi$
$83$	−8.69768	+	6.31923i	−0.954694	+	0.693626i	−0.00278190	+	0.999996i	$0.500886\pi$
$84$		0			0
$85$	−0.491330	+	1.51216i	−0.0532922	+	0.164017i
$86$		0			0
$87$	−1.25531			−0.134584
$88$		0			0
$89$	0.451594			0.0478689			0.0239344	−	0.999714i	$-0.492381\pi$
$89$	0.451594			0.0478689			0.0239344	+	0.999714i	$0.492381\pi$
$90$		0			0
$91$	4.63944	−	14.2787i	0.486345	−	1.49682i
$92$		0			0
$93$	−2.02588	+	1.47189i	−0.210074	+	0.152627i
$94$		0			0
$95$	1.71533	+	5.27925i	0.175990	+	0.541640i
$96$		0			0
$97$	2.24871	+	1.63379i	0.228322	+	0.165886i	0.696065	−	0.717979i	$-0.254933\pi$
$97$	2.24871	+	1.63379i	0.228322	+	0.165886i	−0.467743	+	0.883865i	$0.654933\pi$
$98$		0			0
$99$	−24.6345	+	8.35419i	−2.47586	+	0.839628i
$100$		0			0
$101$	−4.31798	−	3.13719i	−0.429655	−	0.312162i	0.351856	−	0.936054i	$-0.385551\pi$
$101$	−4.31798	−	3.13719i	−0.429655	−	0.312162i	−0.781511	+	0.623892i	$0.785551\pi$
$102$		0			0
$103$	−5.37124	−	16.5310i	−0.529244	−	1.62884i	−0.755769	−	0.654839i	$-0.772737\pi$
$103$	−5.37124	−	16.5310i	−0.529244	−	1.62884i	0.226525	−	0.974005i	$-0.427263\pi$
$104$		0			0
$105$	−9.36416	+	6.80346i	−0.913848	+	0.663950i
$106$		0			0
$107$	−1.08089	+	3.32663i	−0.104493	+	0.321598i	−0.989611	−	0.143769i	$-0.954078\pi$
$107$	−1.08089	+	3.32663i	−0.104493	+	0.321598i	0.885118	+	0.465367i	$0.154078\pi$
$108$		0			0
$109$	4.91209			0.470493			0.235247	−	0.971936i	$-0.424410\pi$
$109$	4.91209			0.470493			0.235247	+	0.971936i	$0.424410\pi$
$110$		0			0
$111$	−4.78096			−0.453789
$112$		0			0
$113$	−5.30638	+	16.3314i	−0.499182	+	1.53633i	0.311153	+	0.950360i	$0.399285\pi$
$113$	−5.30638	+	16.3314i	−0.499182	+	1.53633i	−0.810336	+	0.585966i	$0.800715\pi$
$114$		0			0
$115$	5.16996	−	3.75619i	0.482101	−	0.350267i
$116$		0			0
$117$	−10.3518	−	31.8596i	−0.957026	−	2.94542i
$118$		0			0
$119$	4.52151	+	3.28507i	0.414486	+	0.301142i
$120$		0			0
$121$	−3.66576	−	10.3712i	−0.333251	−	0.942838i
$122$		0			0
$123$	16.3092	+	11.8493i	1.47055	+	1.06842i
$124$		0			0
$125$	−0.309017	−	0.951057i	−0.0276393	−	0.0850651i
$126$		0			0
$127$	12.5303	−	9.10381i	1.11189	−	0.807833i	0.128927	−	0.991654i	$-0.458847\pi$
$127$	12.5303	−	9.10381i	1.11189	−	0.807833i	0.982960	+	0.183822i	$0.0588469\pi$
$128$		0			0
$129$	7.31691	−	22.5191i	0.644218	−	1.98270i
$130$		0			0
$131$	3.64734			0.318670			0.159335	−	0.987225i	$-0.449065\pi$
$131$	3.64734			0.318670			0.159335	+	0.987225i	$0.449065\pi$
$132$		0			0
$133$	19.5120			1.69190
$134$		0			0
$135$	−4.92810	+	15.1671i	−0.424143	+	1.30538i
$136$		0			0
$137$	13.1529	−	9.55615i	1.12373	−	0.816437i	0.138959	−	0.990298i	$-0.455624\pi$
$137$	13.1529	−	9.55615i	1.12373	−	0.816437i	0.984770	+	0.173861i	$0.0556244\pi$
$138$		0			0
$139$	2.31525	+	7.12562i	0.196377	+	0.604387i	0.999958	+	0.00919101i	$0.00292563\pi$
$139$	2.31525	+	7.12562i	0.196377	+	0.604387i	−0.803580	+	0.595196i	$0.797074\pi$
$140$		0			0
$141$	11.5128	+	8.36457i	0.969556	+	0.704423i
$142$		0			0
$143$	13.4155	−	4.54953i	1.12186	−	0.380450i
$144$		0			0
$145$	−0.308414	−	0.224076i	−0.0256124	−	0.0186085i
$146$		0			0
$147$	5.44980	+	16.7727i	0.449492	+	1.38339i
$148$		0			0
$149$	9.46754	−	6.87857i	0.775611	−	0.563514i	−0.128048	−	0.991768i	$-0.540871\pi$
$149$	9.46754	−	6.87857i	0.775611	−	0.563514i	0.903659	+	0.428254i	$0.140871\pi$
$150$		0			0
$151$	−3.89383	+	11.9840i	−0.316875	+	0.975242i	0.658100	+	0.752930i	$0.271360\pi$
$151$	−3.89383	+	11.9840i	−0.316875	+	0.975242i	−0.974976	+	0.222312i	$0.928640\pi$
$152$		0			0
$153$	12.4703			1.00816
$154$		0			0
$155$	−0.760465			−0.0610820
$156$		0			0
$157$	−2.00136	+	6.15954i	−0.159726	+	0.491585i	−0.998609	−	0.0527263i	$-0.983209\pi$
$157$	−2.00136	+	6.15954i	−0.159726	+	0.491585i	0.838883	+	0.544311i	$0.183209\pi$
$158$		0			0
$159$	18.5122	−	13.4499i	1.46811	−	1.06665i
$160$		0			0
$161$	−6.94139	−	21.3634i	−0.547058	−	1.68367i
$162$		0			0
$163$	−17.6153	−	12.7983i	−1.37974	−	1.00244i	−0.996905	−	0.0786197i	$-0.974949\pi$
$163$	−17.6153	−	12.7983i	−1.37974	−	1.00244i	−0.382832	−	0.923818i	$-0.625051\pi$
$164$		0			0
$165$	−10.4291	−	3.24168i	−0.811901	−	0.252365i
$166$		0			0
$167$	15.4426	+	11.2197i	1.19499	+	0.868208i	0.993782	−	0.111342i	$-0.0355149\pi$
$167$	15.4426	+	11.2197i	1.19499	+	0.868208i	0.201203	+	0.979550i	$0.435515\pi$
$168$		0			0
$169$	1.62017	+	4.98638i	0.124629	+	0.383567i
$170$		0			0
$171$	35.2217	−	25.5900i	2.69347	−	1.95692i
$172$		0			0
$173$	−0.446930	+	1.37551i	−0.0339795	+	0.104578i	−0.966608	−	0.256261i	$-0.917509\pi$
$173$	−0.446930	+	1.37551i	−0.0339795	+	0.104578i	0.932628	+	0.360839i	$0.117509\pi$
$174$		0			0
$175$	−3.51508			−0.265715
$176$		0			0
$177$	−41.8560			−3.14609
$178$		0			0
$179$	2.79036	−	8.58784i	0.208561	−	0.641885i	−0.790987	−	0.611833i	$-0.790433\pi$
$179$	2.79036	−	8.58784i	0.208561	−	0.641885i	0.999548	−	0.0300525i	$-0.00956744\pi$
$180$		0			0
$181$	0.928144	−	0.674336i	0.0689884	−	0.0501230i	−0.552757	−	0.833343i	$-0.686424\pi$
$181$	0.928144	−	0.674336i	0.0689884	−	0.0501230i	0.621745	+	0.783220i	$0.286424\pi$
$182$		0			0
$183$	−9.05217	−	27.8597i	−0.669156	−	2.05945i
$184$		0			0
$185$	−1.17462	−	0.853410i	−0.0863596	−	0.0627439i
$186$		0			0
$187$	0.0674714	+	5.27293i	0.00493400	+	0.385595i
$188$		0			0
$189$	45.3513	+	32.9496i	3.29882	+	2.39673i
$190$		0			0
$191$	4.63116	+	14.2532i	0.335099	+	1.03133i	0.966673	+	0.256013i	$0.0824091\pi$
$191$	4.63116	+	14.2532i	0.335099	+	1.03133i	−0.631574	+	0.775316i	$0.717591\pi$
$192$		0			0
$193$	12.2825	−	8.92379i	0.884116	−	0.642348i	−0.0502209	−	0.998738i	$-0.515993\pi$
$193$	12.2825	−	8.92379i	0.884116	−	0.642348i	0.934337	+	0.356390i	$0.115993\pi$
$194$		0			0
$195$	4.34617	−	13.3761i	0.311236	−	0.957885i
$196$		0			0
$197$	−2.95776			−0.210731			−0.105366	−	0.994434i	$-0.533601\pi$
$197$	−2.95776			−0.210731			−0.105366	+	0.994434i	$0.533601\pi$
$198$		0			0
$199$	5.07974			0.360093			0.180046	−	0.983658i	$-0.442375\pi$
$199$	5.07974			0.360093			0.180046	+	0.983658i	$0.442375\pi$
$200$		0			0
$201$	−12.7332	+	39.1887i	−0.898129	+	2.76416i
$202$		0			0
$203$	−1.08410	+	0.787643i	−0.0760887	+	0.0552817i
$204$		0			0
$205$	1.89183	+	5.82245i	0.132131	+	0.406657i
$206$		0			0
$207$	−40.5484	−	29.4601i	−2.81831	−	2.04762i
$208$		0			0
$209$	11.0110	+	14.7546i	0.761649	+	1.02060i
$210$		0			0
$211$	−6.88323	−	5.00096i	−0.473861	−	0.344280i	0.325083	−	0.945685i	$-0.394608\pi$
$211$	−6.88323	−	5.00096i	−0.473861	−	0.344280i	−0.798944	+	0.601405i	$0.794608\pi$
$212$		0			0
$213$	−7.48809	−	23.0460i	−0.513076	−	1.57908i
$214$		0			0
$215$	5.81738	−	4.22657i	0.396742	−	0.288250i
$216$		0			0
$217$	−0.826031	+	2.54226i	−0.0560746	+	0.172580i
$218$		0			0
$219$	23.1559			1.56473
$220$		0			0
$221$	−6.79108			−0.456817
$222$		0			0
$223$	−4.64893	+	14.3079i	−0.311316	+	0.958131i	0.665929	+	0.746015i	$0.268035\pi$
$223$	−4.64893	+	14.3079i	−0.311316	+	0.958131i	−0.977245	+	0.212116i	$0.931965\pi$
$224$		0			0
$225$	−6.34518	+	4.61004i	−0.423012	+	0.307336i
$226$		0			0
$227$	4.85263	+	14.9349i	0.322081	+	0.991262i	0.972741	+	0.231894i	$0.0744921\pi$
$227$	4.85263	+	14.9349i	0.322081	+	0.991262i	−0.650660	+	0.759369i	$0.725508\pi$
$228$		0			0
$229$	−17.3745	−	12.6233i	−1.14814	−	0.834170i	−0.159905	−	0.987132i	$-0.551119\pi$
$229$	−17.3745	−	12.6233i	−1.14814	−	0.834170i	−0.988232	+	0.152962i	$0.951119\pi$
$230$		0			0
$231$	−22.1653	+	31.3436i	−1.45837	+	2.06225i
$232$		0			0
$233$	5.72472	+	4.15926i	0.375039	+	0.272482i	0.759297	−	0.650744i	$-0.225543\pi$
$233$	5.72472	+	4.15926i	0.375039	+	0.272482i	−0.384258	+	0.923226i	$0.625543\pi$
$234$		0			0
$235$	1.33546	+	4.11012i	0.0871158	+	0.268115i
$236$		0			0
$237$	−18.6315	+	13.5366i	−1.21025	+	0.879295i
$238$		0			0
$239$	−4.01246	+	12.3491i	−0.259545	+	0.798797i	0.733355	+	0.679846i	$0.237953\pi$
$239$	−4.01246	+	12.3491i	−0.259545	+	0.798797i	−0.992900	+	0.118951i	$0.962047\pi$
$240$		0			0
$241$	4.91442			0.316565			0.158283	−	0.987394i	$-0.449404\pi$
$241$	4.91442			0.316565			0.158283	+	0.987394i	$0.449404\pi$
$242$		0			0
$243$	47.5998			3.05353
$244$		0			0
$245$	−1.65502	+	5.09364i	−0.105736	+	0.325421i
$246$		0			0
$247$	−19.1810	+	13.9358i	−1.22046	+	0.886716i
$248$		0			0
$249$	10.9397	+	33.6688i	0.693274	+	2.13368i
$250$		0			0
$251$	3.79074	+	2.75413i	0.239269	+	0.173839i	0.700958	−	0.713203i	$-0.252756\pi$
$251$	3.79074	+	2.75413i	0.239269	+	0.173839i	−0.461688	+	0.887042i	$0.652756\pi$
$252$		0			0
$253$	12.2375	−	17.3048i	0.769364	−	1.08794i
$254$		0			0
$255$	4.23570	+	3.07741i	0.265250	+	0.192715i
$256$		0			0
$257$	0.594723	+	1.83037i	0.0370978	+	0.114175i	0.967891	−	0.251372i	$-0.0808818\pi$
$257$	0.594723	+	1.83037i	0.0370978	+	0.114175i	−0.930793	+	0.365547i	$0.880882\pi$
$258$		0			0
$259$	−4.12887	+	2.99980i	−0.256556	+	0.186399i
$260$		0			0
$261$	−0.923942	+	2.84360i	−0.0571905	+	0.176014i
$262$		0			0
$263$	16.6865			1.02894			0.514468	−	0.857510i	$-0.327990\pi$
$263$	16.6865			1.02894			0.514468	+	0.857510i	$0.327990\pi$
$264$		0			0
$265$	6.94904			0.426876
$266$		0			0
$267$	0.459523	−	1.41427i	0.0281223	−	0.0865516i
$268$		0			0
$269$	3.71156	−	2.69661i	0.226298	−	0.164415i	−0.468859	−	0.883273i	$-0.655335\pi$
$269$	3.71156	−	2.69661i	0.226298	−	0.164415i	0.695157	+	0.718858i	$0.255335\pi$
$270$		0			0
$271$	−6.25535	−	19.2520i	−0.379985	−	1.16948i	−0.940053	−	0.341028i	$-0.889225\pi$
$271$	−6.25535	−	19.2520i	−0.379985	−	1.16948i	0.560068	−	0.828447i	$-0.310775\pi$
$272$		0			0
$273$	−39.9960	−	29.0588i	−2.42067	−	1.75872i
$274$		0			0
$275$	−1.98363	−	2.65804i	−0.119618	−	0.160286i
$276$		0			0
$277$	−3.88252	−	2.82082i	−0.233278	−	0.169487i	0.465005	−	0.885308i	$-0.346052\pi$
$277$	−3.88252	−	2.82082i	−0.233278	−	0.169487i	−0.698283	+	0.715821i	$0.746052\pi$
$278$		0			0
$279$	1.84309	+	5.67246i	0.110343	+	0.339601i
$280$		0			0
$281$	17.4731	−	12.6950i	1.04236	−	0.757319i	0.0716156	−	0.997432i	$-0.477185\pi$
$281$	17.4731	−	12.6950i	1.04236	−	0.757319i	0.970745	+	0.240113i	$0.0771845\pi$
$282$		0			0
$283$	3.78668	−	11.6542i	0.225095	−	0.692771i	−0.773187	−	0.634178i	$-0.781339\pi$
$283$	3.78668	−	11.6542i	0.225095	−	0.692771i	0.998282	−	0.0585928i	$-0.0186614\pi$
$284$		0			0
$285$	18.2786			1.08273
$286$		0			0
$287$	21.5196			1.27026
$288$		0			0
$289$	−4.47209	+	13.7637i	−0.263064	+	0.809627i
$290$		0			0
$291$	7.40475	−	5.37986i	0.434074	−	0.315373i
$292$		0			0
$293$	4.42491	+	13.6185i	0.258506	+	0.795600i	0.993119	+	0.117113i	$0.0373640\pi$
$293$	4.42491	+	13.6185i	0.258506	+	0.795600i	−0.734612	+	0.678487i	$0.762636\pi$
$294$		0			0
$295$	−10.2835	−	7.47137i	−0.598726	−	0.435000i
$296$		0			0
$297$	0.676747	+	52.8881i	0.0392688	+	3.06888i
$298$		0			0
$299$	22.0818	+	16.0434i	1.27702	+	0.927813i
$300$		0			0
$301$	−7.81064	−	24.0387i	−0.450198	−	1.38557i
$302$		0			0
$303$	−14.2186	+	10.3304i	−0.816836	+	0.593466i
$304$		0			0
$305$	2.74901	−	8.46059i	0.157408	−	0.484452i
$306$		0			0
$307$	−0.568487			−0.0324453			−0.0162226	−	0.999868i	$-0.505164\pi$
$307$	−0.568487			−0.0324453			−0.0162226	+	0.999868i	$0.505164\pi$
$308$		0			0
$309$	−57.2358			−3.25603
$310$		0			0
$311$	−3.15342	+	9.70522i	−0.178814	+	0.550333i	−0.999787	−	0.0206341i	$-0.993432\pi$
$311$	−3.15342	+	9.70522i	−0.178814	+	0.550333i	0.820973	+	0.570967i	$0.193432\pi$
$312$		0			0
$313$	7.55319	−	5.48771i	0.426931	−	0.310184i	−0.353489	−	0.935439i	$-0.615005\pi$
$313$	7.55319	−	5.48771i	0.426931	−	0.310184i	0.780421	+	0.625255i	$0.215005\pi$
$314$		0			0
$315$	8.51929	+	26.2197i	0.480008	+	1.47731i
$316$		0			0
$317$	22.8858	+	16.6275i	1.28539	+	0.933894i	0.999702	−	0.0244273i	$-0.00777624\pi$
$317$	22.8858	+	16.6275i	1.28539	+	0.933894i	0.285693	+	0.958321i	$0.407776\pi$
$318$		0			0
$319$	−1.20738	−	0.375293i	−0.0676004	−	0.0210124i
$320$		0			0
$321$	9.31821	+	6.77007i	0.520092	+	0.377869i
$322$		0			0
$323$	−2.72734	−	8.39389i	−0.151753	−	0.467049i
$324$		0			0
$325$	3.45546	−	2.51054i	0.191674	−	0.139260i
$326$		0			0
$327$	4.99833	−	15.3833i	0.276408	−	0.850697i
$328$		0			0
$329$	15.1909			0.837501
$330$		0			0
$331$	5.31846			0.292329			0.146164	−	0.989260i	$-0.453307\pi$
$331$	5.31846			0.292329			0.146164	+	0.989260i	$0.453307\pi$
$332$		0			0
$333$	−3.51890	+	10.8301i	−0.192835	+	0.593484i
$334$		0			0
$335$	−10.1236	+	7.35524i	−0.553113	+	0.401860i
$336$		0			0
$337$	−5.44920	−	16.7709i	−0.296837	−	0.913570i	−0.982598	−	0.185743i	$-0.940531\pi$
$337$	−5.44920	−	16.7709i	−0.296837	−	0.913570i	0.685762	−	0.727826i	$-0.259469\pi$
$338$		0			0
$339$	45.7457	+	33.2362i	2.48456	+	1.80514i
$340$		0			0
$341$	−2.38856	+	0.810023i	−0.129348	+	0.0438652i
$342$		0			0
$343$	−4.67580	−	3.39717i	−0.252469	−	0.183430i
$344$		0			0
$345$	−6.50261	−	20.0130i	−0.350089	−	1.07746i
$346$		0			0
$347$	−9.56975	+	6.95283i	−0.513731	+	0.373247i	−0.814237	−	0.580533i	$-0.802844\pi$
$347$	−9.56975	+	6.95283i	−0.513731	+	0.373247i	0.300506	+	0.953780i	$0.402844\pi$
$348$		0			0
$349$	−2.88972	+	8.89363i	−0.154683	+	0.476065i	−0.998129	−	0.0611492i	$-0.980523\pi$
$349$	−2.88972	+	8.89363i	−0.154683	+	0.476065i	0.843446	+	0.537215i	$0.180523\pi$
$350$		0			0
$351$	−68.1154			−3.63573
$352$		0			0
$353$	−17.7336			−0.943863			−0.471932	−	0.881635i	$-0.656443\pi$
$353$	−17.7336			−0.943863			−0.471932	+	0.881635i	$0.656443\pi$
$354$		0			0
$355$	2.27402	−	6.99873i	0.120693	−	0.371454i
$356$		0			0
$357$	14.8888	−	10.8173i	0.787999	−	0.572514i
$358$		0			0
$359$	5.83547	+	17.9597i	0.307984	+	0.947878i	0.978547	+	0.206025i	$0.0660528\pi$
$359$	5.83547	+	17.9597i	0.307984	+	0.947878i	−0.670562	+	0.741853i	$0.733947\pi$
$360$		0			0
$361$	−9.55683	−	6.94344i	−0.502991	−	0.365444i
$362$		0			0
$363$	−36.2098	+	0.926821i	−1.90052	+	0.0486455i
$364$		0			0
$365$	5.68909	+	4.13337i	0.297781	+	0.216350i
$366$		0			0
$367$	8.15607	+	25.1018i	0.425744	+	1.31030i	0.902281	+	0.431149i	$0.141892\pi$
$367$	8.15607	+	25.1018i	0.425744	+	1.31030i	−0.476537	+	0.879154i	$0.658108\pi$
$368$		0			0
$369$	38.8457	−	28.2231i	2.02223	−	1.46923i
$370$		0			0
$371$	7.54817	−	23.2309i	0.391882	−	1.20609i
$372$		0			0
$373$	−19.2368			−0.996043			−0.498022	−	0.867165i	$-0.665940\pi$
$373$	−19.2368			−0.996043			−0.498022	+	0.867165i	$0.665940\pi$
$374$		0			0
$375$	−3.29288			−0.170044
$376$		0			0
$377$	0.503160	−	1.54857i	0.0259141	−	0.0797553i
$378$		0			0
$379$	−27.8840	+	20.2589i	−1.43230	+	1.04063i	−0.442720	+	0.896660i	$0.645986\pi$
$379$	−27.8840	+	20.2589i	−1.43230	+	1.04063i	−0.989582	+	0.143969i	$0.954014\pi$
$380$		0			0
$381$	−15.7602	−	48.5051i	−0.807422	−	2.48499i
$382$		0			0
$383$	−18.0549	−	13.1177i	−0.922564	−	0.670282i	0.0215967	−	0.999767i	$-0.493125\pi$
$383$	−18.0549	−	13.1177i	−0.922564	−	0.670282i	−0.944161	+	0.329485i	$0.893125\pi$
$384$		0			0
$385$	−11.0406	+	3.74415i	−0.562681	+	0.190819i
$386$		0			0
$387$	−45.6261	−	33.1493i	−2.31930	−	1.68507i
$388$		0			0
$389$	−4.97726	−	15.3184i	−0.252357	−	0.776675i	−0.994339	−	0.106255i	$-0.966114\pi$
$389$	−4.97726	−	15.3184i	−0.252357	−	0.776675i	0.741982	−	0.670420i	$-0.233886\pi$
$390$		0			0
$391$	−8.22011	+	5.97226i	−0.415709	+	0.302030i
$392$		0			0
$393$	3.71138	−	11.4224i	0.187214	−	0.576186i
$394$		0			0
$395$	−6.99381			−0.351897
$396$		0			0
$397$	16.8311			0.844731			0.422365	−	0.906426i	$-0.361200\pi$
$397$	16.8311			0.844731			0.422365	+	0.906426i	$0.361200\pi$
$398$		0			0
$399$	19.8545	−	61.1060i	0.993970	−	3.05912i
$400$		0			0
$401$	−4.26126	+	3.09599i	−0.212797	+	0.154606i	−0.689078	−	0.724687i	$-0.741984\pi$
$401$	−4.26126	+	3.09599i	−0.212797	+	0.154606i	0.476281	+	0.879293i	$0.341984\pi$
$402$		0			0
$403$	−1.00371	−	3.08911i	−0.0499985	−	0.153880i
$404$		0			0
$405$	23.4490	+	17.0367i	1.16519	+	0.846560i
$406$		0			0
$407$	−4.59841	−	1.42933i	−0.227935	−	0.0708493i
$408$		0			0
$409$	−18.8805	−	13.7175i	−0.933581	−	0.678286i	0.0132860	−	0.999912i	$-0.495771\pi$
$409$	−18.8805	−	13.7175i	−0.933581	−	0.678286i	−0.946867	+	0.321625i	$0.895771\pi$
$410$		0			0
$411$	−16.5433	−	50.9151i	−0.816022	−	2.51146i
$412$		0			0
$413$	−36.1471	+	26.2624i	−1.77868	+	1.29229i
$414$		0			0
$415$	−3.32222	+	10.2247i	−0.163081	+	0.501913i
$416$		0			0
$417$	24.6713			1.20816
$418$		0			0
$419$	−9.21755			−0.450307			−0.225153	−	0.974323i	$-0.572288\pi$
$419$	−9.21755			−0.450307			−0.225153	+	0.974323i	$0.572288\pi$
$420$		0			0
$421$	6.48628	−	19.9627i	0.316122	−	0.972924i	−0.659168	−	0.751996i	$-0.729091\pi$
$421$	6.48628	−	19.9627i	0.316122	−	0.972924i	0.975290	−	0.220928i	$-0.0709086\pi$
$422$		0			0
$423$	27.4216	−	19.9229i	1.33328	−	0.968686i
$424$		0			0
$425$	0.491330	+	1.51216i	0.0238330	+	0.0733504i
$426$		0			0
$427$	−25.2980	−	18.3801i	−1.22426	−	0.889475i
$428$		0			0
$429$	−0.596834	−	46.6428i	−0.0288154	−	2.25194i
$430$		0			0
$431$	29.7863	+	21.6410i	1.43476	+	1.04241i	0.989106	+	0.147205i	$0.0470279\pi$
$431$	29.7863	+	21.6410i	1.43476	+	1.04241i	0.445651	+	0.895207i	$0.352972\pi$
$432$		0			0
$433$	3.59506	+	11.0645i	0.172768	+	0.531724i	0.999524	−	0.0308355i	$-0.00981681\pi$
$433$	3.59506	+	11.0645i	0.172768	+	0.531724i	−0.826757	+	0.562559i	$0.809817\pi$
$434$		0			0
$435$	−1.01557	+	0.737855i	−0.0486928	+	0.0353774i
$436$		0			0
$437$	−10.9617	+	33.7366i	−0.524369	+	1.61384i
$438$		0			0
$439$	12.4765			0.595471			0.297735	−	0.954648i	$-0.403769\pi$
$439$	12.4765			0.595471			0.297735	+	0.954648i	$0.403769\pi$
$440$		0			0
$441$	42.0057			2.00027
$442$		0			0
$443$	−2.54233	+	7.82449i	−0.120790	+	0.371753i	−0.993111	−	0.117181i	$-0.962614\pi$
$443$	−2.54233	+	7.82449i	−0.120790	+	0.371753i	0.872321	+	0.488934i	$0.162614\pi$
$444$		0			0
$445$	0.365347	−	0.265440i	0.0173191	−	0.0125831i
$446$		0			0
$447$	−11.9080	−	36.6490i	−0.563228	−	1.73344i
$448$		0			0
$449$	27.7003	+	20.1255i	1.30726	+	0.949779i	0.999998	−	0.00184649i	$-0.000587755\pi$
$449$	27.7003	+	20.1255i	1.30726	+	0.949779i	0.307260	+	0.951625i	$0.400588\pi$
$450$		0			0
$451$	12.1440	+	16.2728i	0.571837	+	0.766254i
$452$		0			0
$453$	33.5682	+	24.3887i	1.57717	+	1.14588i
$454$		0			0
$455$	−4.63944	−	14.2787i	−0.217500	−	0.669397i
$456$		0			0
$457$	−19.7657	+	14.3606i	−0.924600	+	0.671761i	−0.944665	−	0.328038i	$-0.893613\pi$
$457$	−19.7657	+	14.3606i	−0.924600	+	0.671761i	0.0200648	+	0.999799i	$0.493613\pi$
$458$		0			0
$459$	7.83557	−	24.1154i	0.365733	−	1.12561i
$460$		0			0
$461$	−15.4406			−0.719141			−0.359571	−	0.933118i	$-0.617077\pi$
$461$	−15.4406			−0.719141			−0.359571	+	0.933118i	$0.617077\pi$
$462$		0			0
$463$	25.6507			1.19209			0.596046	−	0.802951i	$-0.296738\pi$
$463$	25.6507			1.19209			0.596046	+	0.802951i	$0.296738\pi$
$464$		0			0
$465$	−0.773816	+	2.38156i	−0.0358849	+	0.110442i
$466$		0			0
$467$	−6.31075	+	4.58503i	−0.292027	+	0.212170i	−0.724146	−	0.689647i	$-0.757766\pi$
$467$	−6.31075	+	4.58503i	−0.292027	+	0.212170i	0.432119	+	0.901816i	$0.357766\pi$
$468$		0			0
$469$	13.5924	+	41.8330i	0.627638	+	1.93167i
$470$		0			0
$471$	17.2534	+	12.5354i	0.794997	+	0.577599i
$472$		0			0
$473$	13.7699	−	19.4718i	0.633142	−	0.895316i
$474$		0			0
$475$	4.49080	+	3.26276i	0.206052	+	0.149706i
$476$		0			0
$477$	−16.8420	−	51.8343i	−0.771141	−	2.37333i
$478$		0			0
$479$	−31.4994	+	22.8856i	−1.43924	+	1.04567i	−0.451042	+	0.892503i	$0.648948\pi$
$479$	−31.4994	+	22.8856i	−1.43924	+	1.04567i	−0.988200	+	0.153168i	$0.951052\pi$
$480$		0			0
$481$	1.91633	−	5.89784i	0.0873769	−	0.268918i
$482$		0			0
$483$	−73.9674			−3.36563
$484$		0			0
$485$	2.77956			0.126213
$486$		0			0
$487$	5.50829	−	16.9528i	0.249605	−	0.768204i	−0.745240	−	0.666796i	$-0.767665\pi$
$487$	5.50829	−	16.9528i	0.249605	−	0.768204i	0.994845	−	0.101408i	$-0.0323347\pi$
$488$		0			0
$489$	−58.0051	+	42.1432i	−2.62308	+	1.90578i
$490$		0			0
$491$	−1.35228	−	4.16188i	−0.0610274	−	0.187823i	0.915895	−	0.401418i	$-0.131483\pi$
$491$	−1.35228	−	4.16188i	−0.0610274	−	0.187823i	−0.976922	+	0.213595i	$0.931483\pi$
$492$		0			0
$493$	0.490371	+	0.356275i	0.0220852	+	0.0160458i
$494$		0			0
$495$	−15.0193	+	21.2385i	−0.675066	+	0.954599i
$496$		0			0
$497$	−20.9269	−	15.2043i	−0.938700	−	0.682006i
$498$		0			0
$499$	−12.5515	−	38.6296i	−0.561883	−	1.72930i	−0.677037	−	0.735949i	$-0.736736\pi$
$499$	−12.5515	−	38.6296i	−0.561883	−	1.72930i	0.115154	−	0.993348i	$-0.463264\pi$
$500$		0			0
$501$	50.8507	−	36.9452i	2.27184	−	1.65059i
$502$		0			0
$503$	−1.95322	+	6.01140i	−0.0870899	+	0.268035i	−0.985112	−	0.171916i	$-0.945004\pi$
$503$	−1.95322	+	6.01140i	−0.0870899	+	0.268035i	0.898022	+	0.439951i	$0.145004\pi$
$504$		0			0
$505$	−5.33731			−0.237507
$506$		0			0
$507$	17.2645			0.766745
$508$		0			0
$509$	−3.99655	+	12.3001i	−0.177144	+	0.545193i	−0.999725	−	0.0234549i	$-0.992533\pi$
$509$	−3.99655	+	12.3001i	−0.177144	+	0.545193i	0.822581	+	0.568648i	$0.192533\pi$
$510$		0			0
$511$	19.9976	−	14.5291i	0.884642	−	0.642730i
$512$		0			0
$513$	−27.3556	−	84.1918i	−1.20778	−	3.71716i
$514$		0			0
$515$	−14.0621	−	10.2167i	−0.619649	−	0.450201i
$516$		0			0
$517$	8.57255	+	11.4871i	0.377020	+	0.505202i
$518$		0			0
$519$	3.85293	+	2.79932i	0.169125	+	0.122876i
$520$		0			0
$521$	−5.07670	−	15.6245i	−0.222414	−	0.684521i	−0.998544	−	0.0539474i	$-0.982820\pi$
$521$	−5.07670	−	15.6245i	−0.222414	−	0.684521i	0.776129	−	0.630574i	$-0.217180\pi$
$522$		0			0
$523$	−17.2807	+	12.5552i	−0.755632	+	0.548999i	−0.897567	−	0.440877i	$-0.854667\pi$
$523$	−17.2807	+	12.5552i	−0.755632	+	0.548999i	0.141936	+	0.989876i	$0.454667\pi$
$524$		0			0
$525$	−3.57679	+	11.0082i	−0.156104	+	0.480438i
$526$		0			0
$527$	1.20912			0.0526702
$528$		0			0
$529$	17.8375			0.775542
$530$		0			0
$531$	−30.8070	+	94.8143i	−1.33691	+	4.11459i
$532$		0			0
$533$	−21.1546	+	15.3697i	−0.916307	+	0.665736i
$534$		0			0
$535$	1.08089	+	3.32663i	0.0467309	+	0.143823i
$536$		0			0
$537$	−24.0553	−	17.4772i	−1.03806	−	0.754198i
$538$		0			0
$539$	0.227275	+	17.7616i	0.00978941	+	0.765047i
$540$		0			0
$541$	−19.4975	−	14.1658i	−0.838265	−	0.609035i	0.0836204	−	0.996498i	$-0.473352\pi$
$541$	−19.4975	−	14.1658i	−0.838265	−	0.609035i	−0.921885	+	0.387462i	$0.873352\pi$
$542$		0			0
$543$	−1.16739	−	3.59286i	−0.0500976	−	0.154184i
$544$		0			0
$545$	3.97397	−	2.88725i	0.170226	−	0.123676i
$546$		0			0
$547$	−6.85674	+	21.1029i	−0.293173	+	0.902293i	0.690656	+	0.723183i	$0.257322\pi$
$547$	−6.85674	+	21.1029i	−0.293173	+	0.902293i	−0.983829	+	0.179110i	$0.942678\pi$
$548$		0			0
$549$	−69.7719			−2.97779
$550$		0			0
$551$	2.11613			0.0901501
$552$		0			0
$553$	−7.59680	+	23.3806i	−0.323049	+	0.994243i
$554$		0			0
$555$	−3.86788	+	2.81018i	−0.164182	+	0.119285i
$556$		0			0
$557$	10.9315	+	33.6437i	0.463182	+	1.42553i	0.861254	+	0.508174i	$0.169679\pi$
$557$	10.9315	+	33.6437i	0.463182	+	1.42553i	−0.398072	+	0.917354i	$0.630321\pi$
$558$		0			0
$559$	24.8471	+	18.0524i	1.05092	+	0.763537i
$560$		0			0
$561$	16.5820	+	5.15420i	0.700091	+	0.217610i
$562$		0			0
$563$	−5.49638	−	3.99335i	−0.231645	−	0.168300i	0.465908	−	0.884833i	$-0.345728\pi$
$563$	−5.49638	−	3.99335i	−0.231645	−	0.168300i	−0.697553	+	0.716533i	$0.745728\pi$
$564$		0			0
$565$	5.30638	+	16.3314i	0.223241	+	0.687066i
$566$		0			0
$567$	82.4251	−	59.8853i	3.46153	−	2.51495i
$568$		0			0
$569$	11.2890	−	34.7440i	0.473259	−	1.45654i	−0.375032	−	0.927012i	$-0.622368\pi$
$569$	11.2890	−	34.7440i	0.473259	−	1.45654i	0.848291	−	0.529530i	$-0.177632\pi$
$570$		0			0
$571$	−30.4606			−1.27474			−0.637369	−	0.770559i	$-0.719977\pi$
$571$	−30.4606			−1.27474			−0.637369	+	0.770559i	$0.719977\pi$
$572$		0			0
$573$	49.3496			2.06161
$574$		0			0
$575$	1.97475	−	6.07765i	0.0823527	−	0.253456i
$576$		0			0
$577$	−8.50077	+	6.17617i	−0.353892	+	0.257117i	−0.750500	−	0.660871i	$-0.770187\pi$
$577$	−8.50077	+	6.17617i	−0.353892	+	0.257117i	0.396608	+	0.917988i	$0.370187\pi$
$578$		0			0
$579$	−15.4486	−	47.5459i	−0.642022	−	1.97594i
$580$		0			0
$581$	30.5730	+	22.2126i	1.26838	+	0.921534i
$582$		0			0
$583$	21.8264	−	7.40189i	0.903957	−	0.306555i
$584$		0			0
$585$	−27.1014	−	19.6903i	−1.12051	−	0.814095i
$586$		0			0
$587$	6.77626	+	20.8552i	0.279686	+	0.860785i	0.987941	+	0.154829i	$0.0494826\pi$
$587$	6.77626	+	20.8552i	0.279686	+	0.860785i	−0.708255	+	0.705956i	$0.750517\pi$
$588$		0			0
$589$	3.41510	−	2.48121i	0.140717	−	0.102237i
$590$		0			0
$591$	−3.00969	+	9.26286i	−0.123802	+	0.381023i
$592$		0			0
$593$	−21.6482			−0.888987			−0.444493	−	0.895782i	$-0.646616\pi$
$593$	−21.6482			−0.888987			−0.444493	+	0.895782i	$0.646616\pi$
$594$		0			0
$595$	5.58889			0.229122
$596$		0			0
$597$	5.16892	−	15.9083i	0.211550	−	0.651083i
$598$		0			0
$599$	−35.3626	+	25.6924i	−1.44488	+	1.04976i	−0.457880	+	0.889014i	$0.651391\pi$
$599$	−35.3626	+	25.6924i	−1.44488	+	1.04976i	−0.986995	+	0.160750i	$0.948609\pi$
$600$		0			0
$601$	1.26659	+	3.89816i	0.0516653	+	0.159009i	0.973560	−	0.228431i	$-0.0733594\pi$
$601$	1.26659	+	3.89816i	0.0516653	+	0.159009i	−0.921895	+	0.387440i	$0.873359\pi$
$602$		0			0
$603$	79.4003	+	57.6877i	3.23343	+	2.34922i
$604$		0			0
$605$	−9.06171	−	6.23581i	−0.368411	−	0.253522i
$606$		0			0
$607$	34.1107	+	24.7829i	1.38451	+	1.00591i	0.996443	+	0.0842744i	$0.0268572\pi$
$607$	34.1107	+	24.7829i	1.38451	+	1.00591i	0.388067	+	0.921631i	$0.373143\pi$
$608$		0			0
$609$	1.36354	+	4.19656i	0.0552536	+	0.170053i
$610$		0			0
$611$	−14.9332	+	10.8496i	−0.604134	+	0.438929i
$612$		0			0
$613$	−3.29326	+	10.1356i	−0.133014	+	0.409374i	−0.995276	−	0.0970885i	$-0.969047\pi$
$613$	−3.29326	+	10.1356i	−0.133014	+	0.409374i	0.862262	+	0.506462i	$0.169047\pi$
$614$		0			0
$615$	20.1593			0.812901
$616$		0			0
$617$	6.71898			0.270496			0.135248	−	0.990812i	$-0.456817\pi$
$617$	6.71898			0.270496			0.135248	+	0.990812i	$0.456817\pi$
$618$		0			0
$619$	10.6056	−	32.6407i	0.426275	−	1.31194i	−0.475493	−	0.879719i	$-0.657730\pi$
$619$	10.6056	−	32.6407i	0.426275	−	1.31194i	0.901768	−	0.432220i	$-0.142270\pi$
$620$		0			0
$621$	−82.4488	+	59.9025i	−3.30855	+	2.40381i
$622$		0			0
$623$	−0.490530	−	1.50970i	−0.0196527	−	0.0604847i
$624$		0			0
$625$	−0.809017	−	0.587785i	−0.0323607	−	0.0235114i
$626$		0			0
$627$	57.4117	−	19.4698i	2.29280	−	0.777547i
$628$		0			0
$629$	1.86762	+	1.35690i	0.0744667	+	0.0541032i
$630$		0			0
$631$	−3.40926	−	10.4926i	−0.135721	−	0.417705i	0.859981	−	0.510327i	$-0.170475\pi$
$631$	−3.40926	−	10.4926i	−0.135721	−	0.417705i	−0.995701	+	0.0926212i	$0.970475\pi$
$632$		0			0
$633$	−22.6657	+	16.4676i	−0.900879	+	0.654527i
$634$		0			0
$635$	4.78616	−	14.7303i	0.189933	−	0.584553i
$636$		0			0
$637$	−22.8755			−0.906358
$638$		0			0
$639$	−57.7163			−2.28322
$640$		0			0
$641$	−7.81307	+	24.0462i	−0.308598	+	0.949767i	0.669712	+	0.742621i	$0.266417\pi$
$641$	−7.81307	+	24.0462i	−0.308598	+	0.949767i	−0.978310	+	0.207146i	$0.933583\pi$
$642$		0			0
$643$	−4.31960	+	3.13838i	−0.170349	+	0.123765i	−0.669693	−	0.742638i	$-0.733574\pi$
$643$	−4.31960	+	3.13838i	−0.170349	+	0.123765i	0.499344	+	0.866404i	$0.333574\pi$
$644$		0			0
$645$	−7.31691	−	22.5191i	−0.288103	−	0.886691i
$646$		0			0
$647$	2.79236	+	2.02877i	0.109779	+	0.0797591i	0.641320	−	0.767273i	$-0.278387\pi$
$647$	2.79236	+	2.02877i	0.109779	+	0.0797591i	−0.531541	+	0.847032i	$0.678387\pi$
$648$		0			0
$649$	−40.2578	−	12.5134i	−1.58026	−	0.491194i
$650$		0			0
$651$	7.12111	+	5.17379i	0.279098	+	0.202777i
$652$		0			0
$653$	3.49945	+	10.7702i	0.136944	+	0.421471i	0.995887	−	0.0906000i	$-0.0288785\pi$
$653$	3.49945	+	10.7702i	0.136944	+	0.421471i	−0.858943	+	0.512071i	$0.828878\pi$
$654$		0			0
$655$	2.95076	−	2.14385i	0.115296	−	0.0837673i
$656$		0			0
$657$	17.0433	−	52.4539i	0.664923	−	2.04642i
$658$		0			0
$659$	−12.0511			−0.469444			−0.234722	−	0.972063i	$-0.575418\pi$
$659$	−12.0511			−0.469444			−0.234722	+	0.972063i	$0.575418\pi$
$660$		0			0
$661$	−11.9301			−0.464027			−0.232013	−	0.972713i	$-0.574531\pi$
$661$	−11.9301			−0.464027			−0.232013	+	0.972713i	$0.574531\pi$
$662$		0			0
$663$	−6.91031	+	21.2677i	−0.268374	+	0.825971i
$664$		0			0
$665$	15.7855	−	11.4688i	0.612136	−	0.444743i
$666$		0			0
$667$	−0.752814	−	2.31692i	−0.0291491	−	0.0897116i
$668$		0			0
$669$	40.0779	+	29.1183i	1.54950	+	1.12578i
$670$		0			0
$671$	−0.377505	−	29.5022i	−0.0145734	−	1.13892i
$672$		0			0
$673$	26.9841	+	19.6051i	1.04016	+	0.755720i	0.970317	−	0.241838i	$-0.0777502\pi$
$673$	26.9841	+	19.6051i	1.04016	+	0.755720i	0.0698428	+	0.997558i	$0.477750\pi$
$674$		0			0
$675$	4.92810	+	15.1671i	0.189683	+	0.583783i
$676$		0			0
$677$	−24.2644	+	17.6291i	−0.932555	+	0.677541i	−0.946617	−	0.322360i	$-0.895524\pi$
$677$	−24.2644	+	17.6291i	−0.932555	+	0.677541i	0.0140621	+	0.999901i	$0.495524\pi$
$678$		0			0
$679$	3.01921	−	9.29218i	0.115867	−	0.356601i
$680$		0			0
$681$	51.7096			1.98152
$682$		0			0
$683$	−13.1505			−0.503190			−0.251595	−	0.967833i	$-0.580955\pi$
$683$	−13.1505			−0.503190			−0.251595	+	0.967833i	$0.580955\pi$
$684$		0			0
$685$	5.02396	−	15.4622i	0.191956	−	0.590779i
$686$		0			0
$687$	−57.2121	+	41.5670i	−2.18278	+	1.58588i
$688$		0			0
$689$	9.17181	+	28.2279i	0.349418	+	1.07540i
$690$		0			0
$691$	−21.1770	−	15.3860i	−0.805613	−	0.585312i	0.106942	−	0.994265i	$-0.465894\pi$
$691$	−21.1770	−	15.3860i	−0.805613	−	0.585312i	−0.912555	+	0.408953i	$0.865894\pi$
$692$		0			0
$693$	54.6868	+	73.2796i	2.07738	+	2.78366i
$694$		0			0
$695$	6.06141	+	4.40387i	0.229923	+	0.167048i
$696$		0			0
$697$	−3.00796	−	9.25755i	−0.113935	−	0.350655i
$698$		0			0
$699$	18.8508	−	13.6959i	0.713004	−	0.518028i
$700$		0			0
$701$	0.437201	−	1.34557i	0.0165129	−	0.0508214i	−0.942461	−	0.334317i	$-0.891494\pi$
$701$	0.437201	−	1.34557i	0.0165129	−	0.0508214i	0.958973	+	0.283496i	$0.0914943\pi$
$702$		0			0
$703$	8.05944			0.303968
$704$		0			0
$705$	14.2306			0.535957
$706$		0			0
$707$	−5.79749	+	17.8428i	−0.218037	+	0.671049i
$708$		0			0
$709$	−2.97859	+	2.16407i	−0.111863	+	0.0812734i	−0.642310	−	0.766445i	$-0.722024\pi$
$709$	−2.97859	+	2.16407i	−0.111863	+	0.0812734i	0.530447	+	0.847718i	$0.322024\pi$
$710$		0			0
$711$	16.9505	+	52.1682i	0.635693	+	1.95646i
$712$		0			0
$713$	−3.93157	−	2.85645i	−0.147238	−	0.106975i
$714$		0			0
$715$	8.17919	−	11.5661i	0.305885	−	0.432546i
$716$		0			0
$717$	34.5910	+	25.1318i	1.29182	+	0.938565i
$718$		0			0
$719$	2.45788	+	7.56458i	0.0916635	+	0.282111i	0.986370	−	0.164544i	$-0.0526151\pi$
$719$	2.45788	+	7.56458i	0.0916635	+	0.282111i	−0.894706	+	0.446655i	$0.852615\pi$
$720$		0			0
$721$	−49.4293	+	35.9125i	−1.84084	+	1.33745i
$722$		0			0
$723$	5.00070	−	15.3906i	0.185978	−	0.572381i
$724$		0			0
$725$	−0.381220			−0.0141582
$726$		0			0
$727$	35.6495			1.32217			0.661083	−	0.750313i	$-0.270097\pi$
$727$	35.6495			1.32217			0.661083	+	0.750313i	$0.270097\pi$
$728$		0			0
$729$	21.5653	−	66.3711i	0.798714	−	2.45819i
$730$		0			0
$731$	−9.24949	+	6.72015i	−0.342105	+	0.248554i
$732$		0			0
$733$	2.94883	+	9.07558i	0.108918	+	0.335214i	0.990630	−	0.136573i	$-0.0436089\pi$
$733$	2.94883	+	9.07558i	0.108918	+	0.335214i	−0.881712	+	0.471788i	$0.843609\pi$
$734$		0			0
$735$	14.2678	+	10.3661i	0.526274	+	0.382360i
$736$		0			0
$737$	−23.9630	+	33.8856i	−0.882688	+	1.24819i
$738$		0			0
$739$	28.2796	+	20.5463i	1.04028	+	0.755808i	0.970340	−	0.241743i	$-0.0777192\pi$
$739$	28.2796	+	20.5463i	1.04028	+	0.755808i	0.0699400	+	0.997551i	$0.477719\pi$
$740$		0			0
$741$	24.1253	+	74.2500i	0.886265	+	2.72764i
$742$		0			0
$743$	27.3175	−	19.8473i	1.00218	−	0.728128i	0.0396268	−	0.999215i	$-0.487383\pi$
$743$	27.3175	−	19.8473i	1.00218	−	0.728128i	0.962555	+	0.271087i	$0.0873831\pi$
$744$		0			0
$745$	3.61628	−	11.1298i	0.132490	−	0.407763i
$746$		0			0
$747$	84.3203			3.08512
$748$		0			0
$749$	12.2951			0.449255
$750$		0			0
$751$	4.10288	−	12.6274i	0.149716	−	0.460779i	−0.847871	−	0.530202i	$-0.822116\pi$
$751$	4.10288	−	12.6274i	0.149716	−	0.460779i	0.997587	+	0.0694236i	$0.0221160\pi$
$752$		0			0
$753$	12.4825	−	9.06903i	0.454886	−	0.330494i
$754$		0			0
$755$	3.89383	+	11.9840i	0.141711	+	0.436141i
$756$		0			0
$757$	−13.1915	−	9.58418i	−0.479453	−	0.348343i	0.321661	−	0.946855i	$-0.395759\pi$
$757$	−13.1915	−	9.58418i	−0.479453	−	0.348343i	−0.801114	+	0.598512i	$0.795759\pi$
$758$		0			0
$759$	−41.7414	−	55.9329i	−1.51512	−	2.03024i
$760$		0			0
$761$	17.2081	+	12.5025i	0.623795	+	0.453214i	0.854245	−	0.519871i	$-0.174020\pi$
$761$	17.2081	+	12.5025i	0.623795	+	0.453214i	−0.230450	+	0.973084i	$0.574020\pi$
$762$		0			0
$763$	−5.33560	−	16.4213i	−0.193162	−	0.594491i
$764$		0			0
$765$	10.0887	−	7.32986i	0.364757	−	0.265012i
$766$		0			0
$767$	16.7769	−	51.6340i	0.605779	−	1.86440i
$768$		0			0
$769$	−5.81494			−0.209692			−0.104846	−	0.994488i	$-0.533435\pi$
$769$	−5.81494			−0.209692			−0.104846	+	0.994488i	$0.533435\pi$
$770$		0			0
$771$	6.33737			0.228235
$772$		0			0
$773$	5.76962	−	17.7571i	0.207519	−	0.638677i	−0.792082	−	0.610415i	$-0.791003\pi$
$773$	5.76962	−	17.7571i	0.207519	−	0.638677i	0.999601	−	0.0282619i	$-0.00899724\pi$
$774$		0			0
$775$	−0.615229	+	0.446990i	−0.0220997	+	0.0160564i
$776$		0			0
$777$	5.19317	+	15.9829i	0.186304	+	0.573384i
$778$		0			0
$779$	−27.4931	−	19.9749i	−0.985041	−	0.715674i
$780$		0			0
$781$	−0.312278	−	24.4047i	−0.0111742	−	0.873269i
$782$		0			0
$783$	4.91848	+	3.57348i	0.175772	+	0.127706i
$784$		0			0
$785$	2.00136	+	6.15954i	0.0714315	+	0.219843i
$786$		0			0
$787$	−37.4747	+	27.2270i	−1.33583	+	0.970536i	−0.336242	+	0.941775i	$0.609156\pi$
$787$	−37.4747	+	27.2270i	−1.33583	+	0.970536i	−0.999586	+	0.0287609i	$0.990844\pi$
$788$

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)

\(f(q)\)	\(=\)	\(q+(1.01756 - 3.13172i) q^{3} +(0.809017 - 0.587785i) q^{5} +(-1.08622 - 3.34304i) q^{7} +(-6.34518 - 4.61004i) q^{9} +O(q^{10})\)	\(q+(1.01756 - 3.13172i) q^{3} +(0.809017 - 0.587785i) q^{5} +(-1.08622 - 3.34304i) q^{7} +(-6.34518 - 4.61004i) q^{9} +(1.91497 - 2.70793i) q^{11} +(3.45546 + 2.51054i) q^{13} +(-1.01756 - 3.13172i) q^{15} +(-1.28632 + 0.934565i) q^{17} +(-1.71533 + 5.27925i) q^{19} -11.5747 q^{21} +6.39042 q^{23} +(0.309017 - 0.951057i) q^{25} +(-12.9019 + 9.37380i) q^{27} +(-0.117804 - 0.362562i) q^{29} +(-0.615229 - 0.446990i) q^{31} +(-6.53187 - 8.75262i) q^{33} +(-2.84376 - 2.06611i) q^{35} +(-0.448664 - 1.38085i) q^{37} +(11.3784 - 8.26690i) q^{39} +(-1.89183 + 5.82245i) q^{41} +7.19067 q^{43} -7.84307 q^{45} +(-1.33546 + 4.11012i) q^{47} +(-4.33291 + 3.14804i) q^{49} +(1.61789 + 4.97936i) q^{51} +(5.62189 + 4.08454i) q^{53} +(-0.0424355 - 3.31635i) q^{55} +(14.7877 + 10.7439i) q^{57} +(-3.92793 - 12.0889i) q^{59} +(7.19700 - 5.22893i) q^{61} +(-8.51929 + 26.2197i) q^{63} +4.27118 q^{65} -12.5135 q^{67} +(6.50261 - 20.0130i) q^{69} +(5.95347 - 4.32545i) q^{71} +(2.17304 + 6.68793i) q^{73} +(-2.66400 - 1.93551i) q^{75} +(-11.1328 - 3.46042i) q^{77} +(-5.65811 - 4.11086i) q^{79} +(8.95672 + 27.5660i) q^{81} +(-8.69768 + 6.31923i) q^{83} +(-0.491330 + 1.51216i) q^{85} -1.25531 q^{87} +0.451594 q^{89} +(4.63944 - 14.2787i) q^{91} +(-2.02588 + 1.47189i) q^{93} +(1.71533 + 5.27925i) q^{95} +(2.24871 + 1.63379i) q^{97} +(-24.6345 + 8.35419i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{3} + 3 q^{5} + q^{7} - 10 q^{9}+O(q^{10}) \) 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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