LMFDB - Embedded newform 1520.2.q.n.961.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(881,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(6))

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("1520.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.q (of order $3$, degree $2$, 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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.1500534351369.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 13x^{6} - 18x^{5} + 147x^{4} - 156x^{3} + 369x^{2} + 180x + 144 $ x^8 - x^7 + 13x^6 - 18x^5 + 147x^4 - 156x^3 + 369x^2 + 180x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.2
Root			$-0.282064 + 0.488549i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.n.881.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.282064 + 0.488549i) q^{3} +(-0.500000 + 0.866025i) q^{5} +1.56413 q^{7} +(1.34088 + 2.32247i) q^{9} +O(q^{10})$	\(q+(-0.282064 + 0.488549i) q^{3} +(-0.500000 + 0.866025i) q^{5} +1.56413 q^{7} +(1.34088 + 2.32247i) q^{9} +4.21586 q^{11} +(-1.32587 - 2.29647i) q^{13} +(-0.282064 - 0.488549i) q^{15} +(-3.84088 + 6.65260i) q^{17} +(2.78206 + 3.35561i) q^{19} +(-0.441184 + 0.764153i) q^{21} +(-2.34088 - 4.05452i) q^{23} +(-0.500000 - 0.866025i) q^{25} -3.20524 q^{27} +(-1.60793 - 2.78502i) q^{29} +8.81002 q^{31} +(-1.18914 + 2.05966i) q^{33} +(-0.782064 + 1.35457i) q^{35} +3.56413 q^{37} +1.49592 q^{39} +(-1.21794 + 2.10953i) q^{41} +(2.23295 - 3.86758i) q^{43} -2.68176 q^{45} +(-1.17206 - 2.03007i) q^{47} -4.55350 q^{49} +(-2.16675 - 3.75292i) q^{51} +(2.57914 + 4.46720i) q^{53} +(-2.10793 + 3.65105i) q^{55} +(-2.42410 + 0.412678i) q^{57} +(0.607932 - 1.05297i) q^{59} +(6.57176 + 11.3826i) q^{61} +(2.09731 + 3.63264i) q^{63} +2.65174 q^{65} +(4.24589 + 7.35409i) q^{67} +2.64111 q^{69} +(-6.01294 + 10.4147i) q^{71} +(-6.19677 + 10.7331i) q^{73} +0.564128 q^{75} +6.59415 q^{77} +(-3.56413 + 6.17325i) q^{79} +(-3.11856 + 5.40150i) q^{81} -4.37829 q^{83} +(-3.84088 - 6.65260i) q^{85} +1.81416 q^{87} +(-7.63588 - 13.2257i) q^{89} +(-2.07383 - 3.59197i) q^{91} +(-2.48499 + 4.30413i) q^{93} +(-4.29708 + 0.731533i) q^{95} +(-4.71794 + 8.17170i) q^{97} +(5.65297 + 9.79123i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9}+O(q^{10}) $ 8 * q + q^3 - 4 * q^5 + 6 * q^7 - 13 * q^9	$ 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9} + 8 q^{11} - q^{13} + q^{15} - 7 q^{17} + 19 q^{19} - 24 q^{21} + 5 q^{23} - 4 q^{25} + 28 q^{27} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 22 q^{37} - 38 q^{39} - 13 q^{41} + 7 q^{43} + 26 q^{45} + 10 q^{47} - 2 q^{49} + 16 q^{51} - 4 q^{55} - 25 q^{57} - 8 q^{59} - 11 q^{61} - 24 q^{63} + 2 q^{65} - 20 q^{67} - 26 q^{69} - 5 q^{71} + 12 q^{73} - 2 q^{75} + 18 q^{77} - 22 q^{79} - 40 q^{81} - 26 q^{83} - 7 q^{85} + 12 q^{87} + 9 q^{89} + 18 q^{91} - 34 q^{93} - 17 q^{95} - 41 q^{97} - 10 q^{99}+O(q^{100}) $ 8 * q + q^3 - 4 * q^5 + 6 * q^7 - 13 * q^9 + 8 * q^11 - q^13 + q^15 - 7 * q^17 + 19 * q^19 - 24 * q^21 + 5 * q^23 - 4 * q^25 + 28 * q^27 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 22 * q^37 - 38 * q^39 - 13 * q^41 + 7 * q^43 + 26 * q^45 + 10 * q^47 - 2 * q^49 + 16 * q^51 - 4 * q^55 - 25 * q^57 - 8 * q^59 - 11 * q^61 - 24 * q^63 + 2 * q^65 - 20 * q^67 - 26 * q^69 - 5 * q^71 + 12 * q^73 - 2 * q^75 + 18 * q^77 - 22 * q^79 - 40 * q^81 - 26 * q^83 - 7 * q^85 + 12 * q^87 + 9 * q^89 + 18 * q^91 - 34 * q^93 - 17 * q^95 - 41 * q^97 - 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.282064	+	0.488549i	−0.162850	+	0.282064i	−0.935890	−	0.352293i	$-0.885402\pi$
$3$	−0.282064	+	0.488549i	−0.162850	+	0.282064i	0.773040	+	0.634357i	$0.218735\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	1.56413			0.591185			0.295592	−	0.955314i	$-0.404483\pi$
$7$	1.56413			0.591185			0.295592	+	0.955314i	$0.404483\pi$
$8$		0			0
$9$	1.34088	+	2.32247i	0.446960	+	0.774157i
$10$		0			0
$11$	4.21586			1.27113			0.635565	−	0.772047i	$-0.280767\pi$
$11$	4.21586			1.27113			0.635565	+	0.772047i	$0.280767\pi$
$12$		0			0
$13$	−1.32587	−	2.29647i	−0.367730	−	0.636926i	0.621481	−	0.783430i	$-0.286531\pi$
$13$	−1.32587	−	2.29647i	−0.367730	−	0.636926i	−0.989210	+	0.146503i	$0.953198\pi$
$14$		0			0
$15$	−0.282064	−	0.488549i	−0.0728286	−	0.126143i
$16$		0			0
$17$	−3.84088	+	6.65260i	−0.931550	+	1.61349i	−0.150877	+	0.988553i	$0.548210\pi$
$17$	−3.84088	+	6.65260i	−0.931550	+	1.61349i	−0.780673	+	0.624940i	$0.785124\pi$
$18$		0			0
$19$	2.78206	+	3.35561i	0.638249	+	0.769830i
$19$	2.78206	+	3.35561i	0.638249	+	0.769830i
$20$		0			0
$21$	−0.441184	+	0.764153i	−0.0962743	+	0.166752i
$22$		0			0
$23$	−2.34088	−	4.05452i	−0.488107	−	0.845426i	0.511799	−	0.859105i	$-0.328979\pi$
$23$	−2.34088	−	4.05452i	−0.488107	−	0.845426i	−0.999906	+	0.0136786i	$0.995646\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	−3.20524			−0.616849
$28$		0			0
$29$	−1.60793	−	2.78502i	−0.298585	−	0.517165i	0.677227	−	0.735774i	$-0.263181\pi$
$29$	−1.60793	−	2.78502i	−0.298585	−	0.517165i	−0.975813	+	0.218609i	$0.929848\pi$
$30$		0			0
$31$	8.81002			1.58233			0.791163	−	0.611606i	$-0.209476\pi$
$31$	8.81002			1.58233			0.791163	+	0.611606i	$0.209476\pi$
$32$		0			0
$33$	−1.18914	+	2.05966i	−0.207003	+	0.358540i
$34$		0			0
$35$	−0.782064	+	1.35457i	−0.132193	+	0.228965i
$36$		0			0
$37$	3.56413			0.585939			0.292970	−	0.956122i	$-0.405357\pi$
$37$	3.56413			0.585939			0.292970	+	0.956122i	$0.405357\pi$
$38$		0			0
$39$	1.49592			0.239539
$40$		0			0
$41$	−1.21794	+	2.10953i	−0.190210	+	0.329453i	−0.945320	−	0.326145i	$-0.894250\pi$
$41$	−1.21794	+	2.10953i	−0.190210	+	0.329453i	0.755110	+	0.655598i	$0.227583\pi$
$42$		0			0
$43$	2.23295	−	3.86758i	0.340521	−	0.589800i	−0.644008	−	0.765019i	$-0.722730\pi$
$43$	2.23295	−	3.86758i	0.340521	−	0.589800i	0.984530	+	0.175218i	$0.0560631\pi$
$44$		0			0
$45$	−2.68176			−0.399773
$46$		0			0
$47$	−1.17206	−	2.03007i	−0.170963	−	0.296116i	0.767794	−	0.640697i	$-0.221354\pi$
$47$	−1.17206	−	2.03007i	−0.170963	−	0.296116i	−0.938757	+	0.344581i	$0.888021\pi$
$48$		0			0
$49$	−4.55350			−0.650501
$50$		0			0
$51$	−2.16675	−	3.75292i	−0.303405	−	0.525514i
$52$		0			0
$53$	2.57914	+	4.46720i	0.354272	+	0.613617i	0.986993	−	0.160763i	$-0.0513953\pi$
$53$	2.57914	+	4.46720i	0.354272	+	0.613617i	−0.632721	+	0.774380i	$0.718062\pi$
$54$		0			0
$55$	−2.10793	+	3.65105i	−0.284234	+	0.492307i
$56$		0			0
$57$	−2.42410	+	0.412678i	−0.321080	+	0.0546606i
$58$		0			0
$59$	0.607932	−	1.05297i	0.0791460	−	0.137085i	−0.823736	−	0.566974i	$-0.808114\pi$
$59$	0.607932	−	1.05297i	0.0791460	−	0.137085i	0.902882	+	0.429889i	$0.141447\pi$
$60$		0			0
$61$	6.57176	+	11.3826i	0.841427	+	1.45739i	0.888688	+	0.458512i	$0.151617\pi$
$61$	6.57176	+	11.3826i	0.841427	+	1.45739i	−0.0472612	+	0.998883i	$0.515049\pi$
$62$		0			0
$63$	2.09731	+	3.63264i	0.264236	+	0.457670i
$64$		0			0
$65$	2.65174			0.328907
$66$		0			0
$67$	4.24589	+	7.35409i	0.518718	+	0.898445i	0.999763	+	0.0217499i	$0.00692377\pi$
$67$	4.24589	+	7.35409i	0.518718	+	0.898445i	−0.481046	+	0.876696i	$0.659743\pi$
$68$		0			0
$69$	2.64111			0.317952
$70$		0			0
$71$	−6.01294	+	10.4147i	−0.713605	+	1.23600i	0.249891	+	0.968274i	$0.419605\pi$
$71$	−6.01294	+	10.4147i	−0.713605	+	1.23600i	−0.963495	+	0.267725i	$0.913728\pi$
$72$		0			0
$73$	−6.19677	+	10.7331i	−0.725277	+	1.25622i	0.233583	+	0.972337i	$0.424955\pi$
$73$	−6.19677	+	10.7331i	−0.725277	+	1.25622i	−0.958860	+	0.283880i	$0.908378\pi$
$74$		0			0
$75$	0.564128			0.0651399
$76$		0			0
$77$	6.59415			0.751473
$78$		0			0
$79$	−3.56413	+	6.17325i	−0.400996	+	0.694545i	−0.993846	−	0.110767i	$-0.964669\pi$
$79$	−3.56413	+	6.17325i	−0.400996	+	0.694545i	0.592851	+	0.805312i	$0.298002\pi$
$80$		0			0
$81$	−3.11856	+	5.40150i	−0.346506	+	0.600166i
$82$		0			0
$83$	−4.37829			−0.480579			−0.240290	−	0.970701i	$-0.577242\pi$
$83$	−4.37829			−0.480579			−0.240290	+	0.970701i	$0.577242\pi$
$84$		0			0
$85$	−3.84088	−	6.65260i	−0.416602	−	0.721576i
$86$		0			0
$87$	1.81416			0.194498
$88$		0			0
$89$	−7.63588	−	13.2257i	−0.809402	−	1.40193i	−0.913279	−	0.407335i	$-0.866458\pi$
$89$	−7.63588	−	13.2257i	−0.809402	−	1.40193i	0.103877	−	0.994590i	$-0.466875\pi$
$90$		0			0
$91$	−2.07383	−	3.59197i	−0.217396	−	0.376541i
$92$		0			0
$93$	−2.48499	+	4.30413i	−0.257681	+	0.446317i
$94$		0			0
$95$	−4.29708	+	0.731533i	−0.440871	+	0.0750537i
$96$		0			0
$97$	−4.71794	+	8.17170i	−0.479034	+	0.829711i	−0.999711	−	0.0240428i	$-0.992346\pi$
$97$	−4.71794	+	8.17170i	−0.479034	+	0.829711i	0.520677	+	0.853754i	$0.325680\pi$
$98$		0			0
$99$	5.65297	+	9.79123i	0.568145	+	0.984055i
$100$		0			0
$101$	7.49262	+	12.9776i	0.745543	+	1.29132i	0.949941	+	0.312431i	$0.101143\pi$
$101$	7.49262	+	12.9776i	0.745543	+	1.29132i	−0.204397	+	0.978888i	$0.565523\pi$
$102$		0			0
$103$	−6.68176			−0.658373			−0.329187	−	0.944265i	$-0.606775\pi$
$103$	−6.68176			−0.658373			−0.329187	+	0.944265i	$0.606775\pi$
$104$		0			0
$105$	−0.441184	−	0.764153i	−0.0430552	−	0.0745737i
$106$		0			0
$107$	14.7652			1.42741			0.713704	−	0.700447i	$-0.247016\pi$
$107$	14.7652			1.42741			0.713704	+	0.700447i	$0.247016\pi$
$108$		0			0
$109$	5.79708	−	10.0408i	0.555259	−	0.961737i	−0.442624	−	0.896707i	$-0.645952\pi$
$109$	5.79708	−	10.0408i	0.555259	−	0.961737i	0.997883	−	0.0650298i	$-0.0207143\pi$
$110$		0			0
$111$	−1.00531	+	1.74125i	−0.0954200	+	0.165272i
$112$		0			0
$113$	11.9276			1.12206			0.561029	−	0.827796i	$-0.310405\pi$
$113$	11.9276			1.12206			0.561029	+	0.827796i	$0.310405\pi$
$114$		0			0
$115$	4.68176			0.436576
$116$		0			0
$117$	3.55566	−	6.15858i	0.328721	−	0.569361i
$118$		0			0
$119$	−6.00763	+	10.4055i	−0.550718	+	0.953872i
$120$		0			0
$121$	6.77351			0.615774
$122$		0			0
$123$	−0.687072	−	1.19004i	−0.0619512	−	0.107303i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−0.825868	−	1.43045i	−0.0732839	−	0.126932i	0.827055	−	0.562121i	$-0.190015\pi$
$127$	−0.825868	−	1.43045i	−0.0732839	−	0.126932i	−0.900339	+	0.435190i	$0.856681\pi$
$128$		0			0
$129$	1.25967	+	2.18181i	0.110908	+	0.192098i
$130$		0			0
$131$	−2.34619	+	4.06372i	−0.204988	+	0.355049i	−0.950129	−	0.311858i	$-0.899049\pi$
$131$	−2.34619	+	4.06372i	−0.204988	+	0.355049i	0.745141	+	0.666907i	$0.232382\pi$
$132$		0			0
$133$	4.35150	+	5.24860i	0.377323	+	0.455112i
$134$		0			0
$135$	1.60262	−	2.77582i	0.137932	−	0.238904i
$136$		0			0
$137$	−3.41879	−	5.92152i	−0.292087	−	0.505909i	0.682216	−	0.731151i	$-0.261016\pi$
$137$	−3.41879	−	5.92152i	−0.292087	−	0.505909i	−0.974303	+	0.225241i	$0.927683\pi$
$138$		0			0
$139$	−3.13588	−	5.43151i	−0.265982	−	0.460695i	0.701838	−	0.712336i	$-0.252363\pi$
$139$	−3.13588	−	5.43151i	−0.265982	−	0.460695i	−0.967820	+	0.251642i	$0.919030\pi$
$140$		0			0
$141$	1.32238			0.111365
$142$		0			0
$143$	−5.58968	−	9.68161i	−0.467433	−	0.809617i
$144$		0			0
$145$	3.21586			0.267063
$146$		0			0
$147$	1.28438	−	2.22461i	0.105934	−	0.183483i
$148$		0			0
$149$	0.299147	−	0.518139i	0.0245071	−	0.0424476i	−0.853512	−	0.521074i	$-0.825532\pi$
$149$	0.299147	−	0.518139i	0.0245071	−	0.0424476i	0.878019	+	0.478626i	$0.158865\pi$
$150$		0			0
$151$	21.7800			1.77243			0.886215	−	0.463274i	$-0.153325\pi$
$151$	21.7800			1.77243			0.886215	+	0.463274i	$0.153325\pi$
$152$		0			0
$153$	−20.6006			−1.66546
$154$		0			0
$155$	−4.40501	+	7.62970i	−0.353819	+	0.612832i
$156$		0			0
$157$	6.82056	−	11.8135i	0.544340	−	0.942824i	−0.454309	−	0.890844i	$-0.650114\pi$
$157$	6.82056	−	11.8135i	0.544340	−	0.942824i	0.998648	−	0.0519795i	$-0.0165530\pi$
$158$		0			0
$159$	−2.90993			−0.230772
$160$		0			0
$161$	−3.66144	−	6.34179i	−0.288562	−	0.499803i
$162$		0			0
$163$	−12.8701			−1.00806			−0.504031	−	0.863686i	$-0.668150\pi$
$163$	−12.8701			−1.00806			−0.504031	+	0.863686i	$0.668150\pi$
$164$		0			0
$165$	−1.18914	−	2.05966i	−0.0925747	−	0.160344i
$166$		0			0
$167$	−8.12618	−	14.0750i	−0.628823	−	1.08915i	−0.987788	−	0.155802i	$-0.950204\pi$
$167$	−8.12618	−	14.0750i	−0.628823	−	1.08915i	0.358965	−	0.933351i	$-0.383130\pi$
$168$		0			0
$169$	2.98415	−	5.16870i	0.229550	−	0.397592i
$170$		0			0
$171$	−4.06290	+	10.9607i	−0.310698	+	0.838188i
$172$		0			0
$173$	1.02795	−	1.78046i	0.0781537	−	0.135366i	−0.824300	−	0.566154i	$-0.808431\pi$
$173$	1.02795	−	1.78046i	0.0781537	−	0.135366i	0.902453	+	0.430788i	$0.141764\pi$
$174$		0			0
$175$	−0.782064	−	1.35457i	−0.0591185	−	0.102396i
$176$		0			0
$177$	0.342952	+	0.594009i	0.0257778	+	0.0446485i
$178$		0			0
$179$	−0.250031			−0.0186882			−0.00934410	−	0.999956i	$-0.502974\pi$
$179$	−0.250031			−0.0186882			−0.00934410	+	0.999956i	$0.502974\pi$
$180$		0			0
$181$	−9.57176	−	16.5788i	−0.711463	−	1.23229i	−0.964308	−	0.264783i	$-0.914700\pi$
$181$	−9.57176	−	16.5788i	−0.711463	−	1.23229i	0.252845	−	0.967507i	$-0.418634\pi$
$182$		0			0
$183$	−7.41462			−0.548105
$184$		0			0
$185$	−1.78206	+	3.08663i	−0.131020	+	0.226933i
$186$		0			0
$187$	−16.1926	+	28.0465i	−1.18412	+	2.05096i
$188$		0			0
$189$	−5.01341			−0.364672
$190$		0			0
$191$	−18.3166			−1.32534			−0.662670	−	0.748912i	$-0.730577\pi$
$191$	−18.3166			−1.32534			−0.662670	+	0.748912i	$0.730577\pi$
$192$		0			0
$193$	9.62087	−	16.6638i	0.692526	−	1.19949i	−0.278482	−	0.960441i	$-0.589831\pi$
$193$	9.62087	−	16.6638i	0.692526	−	1.19949i	0.971008	−	0.239048i	$-0.0768353\pi$
$194$		0			0
$195$	−0.747959	+	1.29550i	−0.0535625	+	0.0927729i
$196$		0			0
$197$	8.61109			0.613515			0.306757	−	0.951788i	$-0.400756\pi$
$197$	8.61109			0.613515			0.306757	+	0.951788i	$0.400756\pi$
$198$		0			0
$199$	12.4488	+	21.5620i	0.882473	+	1.52849i	0.848583	+	0.529062i	$0.177456\pi$
$199$	12.4488	+	21.5620i	0.882473	+	1.52849i	0.0338899	+	0.999426i	$0.489210\pi$
$200$		0			0
$201$	−4.79045			−0.337892
$202$		0			0
$203$	−2.51501	−	4.35613i	−0.176519	−	0.305740i
$204$		0			0
$205$	−1.21794	−	2.10953i	−0.0850643	−	0.147336i
$206$		0			0
$207$	6.27768	−	10.8733i	0.436329	−	0.755743i
$208$		0			0
$209$	11.7288	+	14.1468i	0.811298	+	0.978554i
$210$		0			0
$211$	11.8559	−	20.5350i	0.816193	−	1.41369i	−0.0922747	−	0.995734i	$-0.529414\pi$
$211$	11.8559	−	20.5350i	0.816193	−	1.41369i	0.908468	−	0.417955i	$-0.137253\pi$
$212$		0			0
$213$	−3.39207	−	5.87523i	−0.232421	−	0.402564i
$214$		0			0
$215$	2.23295	+	3.86758i	0.152286	+	0.263767i
$216$		0			0
$217$	13.7800			0.935447
$218$		0			0
$219$	−3.49577	−	6.05486i	−0.236222	−	0.409149i
$220$		0			0
$221$	20.3700			1.37023
$222$		0			0
$223$	6.75967	−	11.7081i	0.452661	−	0.784032i	−0.545889	−	0.837857i	$-0.683808\pi$
$223$	6.75967	−	11.7081i	0.452661	−	0.784032i	0.998550	+	0.0538256i	$0.0171415\pi$
$224$		0			0
$225$	1.34088	−	2.32247i	0.0893920	−	0.154831i
$226$		0			0
$227$	−15.3035			−1.01573			−0.507864	−	0.861438i	$-0.669565\pi$
$227$	−15.3035			−1.01573			−0.507864	+	0.861438i	$0.669565\pi$
$228$		0			0
$229$	6.73934			0.445348			0.222674	−	0.974893i	$-0.428521\pi$
$229$	6.73934			0.445348			0.222674	+	0.974893i	$0.428521\pi$
$230$		0			0
$231$	−1.85997	+	3.22157i	−0.122377	+	0.211964i
$232$		0			0
$233$	−14.2665	+	24.7102i	−0.934627	+	1.61882i	−0.159328	+	0.987226i	$0.550933\pi$
$233$	−14.2665	+	24.7102i	−0.934627	+	1.61882i	−0.775299	+	0.631595i	$0.782401\pi$
$234$		0			0
$235$	2.34412			0.152914
$236$		0			0
$237$	−2.01062	−	3.48250i	−0.130604	−	0.226213i
$238$		0			0
$239$	−0.810016			−0.0523956			−0.0261978	−	0.999657i	$-0.508340\pi$
$239$	−0.810016			−0.0523956			−0.0261978	+	0.999657i	$0.508340\pi$
$240$		0			0
$241$	9.89762	+	17.1432i	0.637562	+	1.10429i	0.985966	+	0.166945i	$0.0533904\pi$
$241$	9.89762	+	17.1432i	0.637562	+	1.10429i	−0.348404	+	0.937344i	$0.613276\pi$
$242$		0			0
$243$	−6.56712	−	11.3746i	−0.421281	−	0.729681i
$244$		0			0
$245$	2.27675	−	3.94345i	0.145456	−	0.251938i
$246$		0			0
$247$	4.01741	−	10.8380i	0.255622	−	0.689607i
$248$		0			0
$249$	1.23496	−	2.13901i	0.0782622	−	0.135554i
$250$		0			0
$251$	4.61556	+	7.99438i	0.291332	+	0.504601i	0.974125	−	0.226011i	$-0.0725684\pi$
$251$	4.61556	+	7.99438i	0.291332	+	0.504601i	−0.682793	+	0.730612i	$0.739235\pi$
$252$		0			0
$253$	−9.86883	−	17.0933i	−0.620448	−	1.07465i
$254$		0			0
$255$	4.33350			0.271374
$256$		0			0
$257$	11.2203	+	19.4342i	0.699905	+	1.21227i	0.968499	+	0.249018i	$0.0801078\pi$
$257$	11.2203	+	19.4342i	0.699905	+	1.21227i	−0.268594	+	0.963254i	$0.586559\pi$
$258$		0			0
$259$	5.57475			0.346398
$260$		0			0
$261$	4.31209	−	7.46875i	0.266912	−	0.462304i
$262$		0			0
$263$	6.13997	−	10.6347i	0.378606	−	0.655766i	−0.612253	−	0.790662i	$-0.709737\pi$
$263$	6.13997	−	10.6347i	0.378606	−	0.655766i	0.990860	+	0.134896i	$0.0430701\pi$
$264$		0			0
$265$	−5.15828			−0.316871
$266$		0			0
$267$	8.61523			0.527244
$268$		0			0
$269$	4.89000	−	8.46972i	0.298148	−	0.516408i	−0.677564	−	0.735464i	$-0.736964\pi$
$269$	4.89000	−	8.46972i	0.298148	−	0.516408i	0.975712	+	0.219056i	$0.0702977\pi$
$270$		0			0
$271$	11.9768	−	20.7444i	0.727537	−	1.26013i	−0.230385	−	0.973100i	$-0.573998\pi$
$271$	11.9768	−	20.7444i	0.727537	−	1.26013i	0.957921	−	0.287031i	$-0.0926682\pi$
$272$		0			0
$273$	2.33981			0.141612
$274$		0			0
$275$	−2.10793	−	3.65105i	−0.127113	−	0.220166i
$276$		0			0
$277$	−0.500062			−0.0300458			−0.0150229	−	0.999887i	$-0.504782\pi$
$277$	−0.500062			−0.0300458			−0.0150229	+	0.999887i	$0.504782\pi$
$278$		0			0
$279$	11.8132	+	20.4610i	0.707236	+	1.22497i
$280$		0			0
$281$	−2.16468	−	3.74933i	−0.129134	−	0.223666i	0.794207	−	0.607647i	$-0.207886\pi$
$281$	−2.16468	−	3.74933i	−0.129134	−	0.223666i	−0.923341	+	0.383980i	$0.874553\pi$
$282$		0			0
$283$	13.0506	−	22.6043i	0.775777	−	1.34369i	−0.158580	−	0.987346i	$-0.550691\pi$
$283$	13.0506	−	22.6043i	0.775777	−	1.34369i	0.934357	−	0.356339i	$-0.115975\pi$
$284$		0			0
$285$	0.854661	−	2.30567i	0.0506257	−	0.136576i
$286$		0			0
$287$	−1.90501	+	3.29957i	−0.112449	+	0.194767i
$288$		0			0
$289$	−21.0047	−	36.3812i	−1.23557	−	2.14007i
$290$		0			0
$291$	−2.66152	−	4.60989i	−0.156021	−	0.270236i
$292$		0			0
$293$	−9.11349			−0.532416			−0.266208	−	0.963916i	$-0.585771\pi$
$293$	−9.11349			−0.532416			−0.266208	+	0.963916i	$0.585771\pi$
$294$		0			0
$295$	0.607932	+	1.05297i	0.0353952	+	0.0613062i
$296$		0			0
$297$	−13.5129			−0.784095
$298$		0			0
$299$	−6.20740	+	10.7515i	−0.358983	+	0.621777i
$300$		0			0
$301$	3.49262	−	6.04939i	0.201311	−	0.348681i
$302$		0			0
$303$	−8.45359			−0.485646
$304$		0			0
$305$	−13.1435			−0.752595
$306$		0			0
$307$	10.8656	−	18.8198i	0.620132	−	1.07410i	−0.369329	−	0.929299i	$-0.620412\pi$
$307$	10.8656	−	18.8198i	0.620132	−	1.07410i	0.989461	−	0.144801i	$-0.0462543\pi$
$308$		0			0
$309$	1.88468	−	3.26437i	0.107216	−	0.185703i
$310$		0			0
$311$	−12.0876			−0.685425			−0.342713	−	0.939440i	$-0.611346\pi$
$311$	−12.0876			−0.685425			−0.342713	+	0.939440i	$0.611346\pi$
$312$		0			0
$313$	−6.16675	−	10.6811i	−0.348565	−	0.603733i	0.637430	−	0.770509i	$-0.279998\pi$
$313$	−6.16675	−	10.6811i	−0.348565	−	0.603733i	−0.985995	+	0.166776i	$0.946664\pi$
$314$		0			0
$315$	−4.19462			−0.236340
$316$		0			0
$317$	−5.71055	−	9.89097i	−0.320737	−	0.555532i	0.659904	−	0.751350i	$-0.270597\pi$
$317$	−5.71055	−	9.89097i	−0.320737	−	0.555532i	−0.980640	+	0.195818i	$0.937264\pi$
$318$		0			0
$319$	−6.77882	−	11.7413i	−0.379541	−	0.657385i
$320$		0			0
$321$	−4.16474	+	7.21354i	−0.232453	+	0.402620i
$322$		0			0
$323$	−33.0091	+	5.61946i	−1.83668	+	0.312675i
$324$		0			0
$325$	−1.32587	+	2.29647i	−0.0735459	+	0.127385i
$326$		0			0
$327$	3.27029	+	5.66431i	0.180848	+	0.313237i
$328$		0			0
$329$	−1.83325	−	3.17529i	−0.101070	−	0.175059i
$330$		0			0
$331$	17.4042			0.956620			0.478310	−	0.878191i	$-0.341250\pi$
$331$	17.4042			0.956620			0.478310	+	0.878191i	$0.341250\pi$
$332$		0			0
$333$	4.77907	+	8.27759i	0.261891	+	0.453609i
$334$		0			0
$335$	−8.49178			−0.463955
$336$		0			0
$337$	−4.31410	+	7.47224i	−0.235004	+	0.407039i	−0.959274	−	0.282478i	$-0.908844\pi$
$337$	−4.31410	+	7.47224i	−0.235004	+	0.407039i	0.724270	+	0.689517i	$0.242177\pi$
$338$		0			0
$339$	−3.36436	+	5.82724i	−0.182727	+	0.316492i
$340$		0			0
$341$	37.1418			2.01134
$342$		0			0
$343$	−18.0712			−0.975751
$344$		0			0
$345$	−1.32056	+	2.28727i	−0.0710963	+	0.123142i
$346$		0			0
$347$	11.0430	−	19.1270i	0.592817	−	1.02679i	−0.401034	−	0.916063i	$-0.631349\pi$
$347$	11.0430	−	19.1270i	0.592817	−	1.02679i	0.993851	−	0.110726i	$-0.0353177\pi$
$348$		0			0
$349$	0.489438			0.0261990			0.0130995	−	0.999914i	$-0.495830\pi$
$349$	0.489438			0.0261990			0.0130995	+	0.999914i	$0.495830\pi$
$350$		0			0
$351$	4.24973	+	7.36074i	0.226834	+	0.392887i
$352$		0			0
$353$	−29.7418			−1.58300			−0.791498	−	0.611171i	$-0.790699\pi$
$353$	−29.7418			−1.58300			−0.791498	+	0.611171i	$0.790699\pi$
$354$		0			0
$355$	−6.01294	−	10.4147i	−0.319134	−	0.552756i
$356$		0			0
$357$	−3.38907	−	5.87004i	−0.179369	−	0.310676i
$358$		0			0
$359$	7.14850	−	12.3816i	0.377283	−	0.653474i	−0.613383	−	0.789786i	$-0.710192\pi$
$359$	7.14850	−	12.3816i	0.377283	−	0.653474i	0.990666	+	0.136312i	$0.0435250\pi$
$360$		0			0
$361$	−3.52024	+	18.6710i	−0.185276	+	0.982687i
$362$		0			0
$363$	−1.91056	+	3.30919i	−0.100279	+	0.173688i
$364$		0			0
$365$	−6.19677	−	10.7331i	−0.324354	−	0.561797i
$366$		0			0
$367$	6.10586	+	10.5757i	0.318723	+	0.552045i	0.980222	−	0.197901i	$-0.0634126\pi$
$367$	6.10586	+	10.5757i	0.318723	+	0.552045i	−0.661499	+	0.749946i	$0.730079\pi$
$368$		0			0
$369$	−6.53242			−0.340064
$370$		0			0
$371$	4.03410	+	6.98727i	0.209440	+	0.362761i
$372$		0			0
$373$	−33.5329			−1.73627			−0.868134	−	0.496330i	$-0.834681\pi$
$373$	−33.5329			−1.73627			−0.868134	+	0.496330i	$0.834681\pi$
$374$		0			0
$375$	−0.282064	+	0.488549i	−0.0145657	+	0.0252286i
$376$		0			0
$377$	−4.26381	+	7.38514i	−0.219597	+	0.380354i
$378$		0			0
$379$	1.82941			0.0939707			0.0469854	−	0.998896i	$-0.485039\pi$
$379$	1.82941			0.0939707			0.0469854	+	0.998896i	$0.485039\pi$
$380$		0			0
$381$	0.931791			0.0477371
$382$		0			0
$383$	7.76291	−	13.4458i	0.396666	−	0.687046i	−0.596646	−	0.802504i	$-0.703500\pi$
$383$	7.76291	−	13.4458i	0.396666	−	0.687046i	0.993312	+	0.115459i	$0.0368337\pi$
$384$		0			0
$385$	−3.29708	+	5.71070i	−0.168035	+	0.291044i
$386$		0			0
$387$	11.9765			0.608798
$388$		0			0
$389$	2.81617	+	4.87775i	0.142785	+	0.247312i	0.928545	−	0.371221i	$-0.121061\pi$
$389$	2.81617	+	4.87775i	0.142785	+	0.247312i	−0.785759	+	0.618533i	$0.787727\pi$
$390$		0			0
$391$	35.9642			1.81879
$392$		0			0
$393$	−1.32355	−	2.29246i	−0.0667644	−	0.115639i
$394$		0			0
$395$	−3.56413	−	6.17325i	−0.179331	−	0.310610i
$396$		0			0
$397$	−7.33142	+	12.6984i	−0.367954	+	0.637314i	−0.989245	−	0.146265i	$-0.953275\pi$
$397$	−7.33142	+	12.6984i	−0.367954	+	0.637314i	0.621292	+	0.783579i	$0.286608\pi$
$398$		0			0
$399$	−3.79160	+	0.645482i	−0.189818	+	0.0323145i
$400$		0			0
$401$	−5.54704	+	9.60776i	−0.277006	+	0.479789i	−0.970639	−	0.240540i	$-0.922676\pi$
$401$	−5.54704	+	9.60776i	−0.277006	+	0.479789i	0.693633	+	0.720328i	$0.256009\pi$
$402$		0			0
$403$	−11.6809	−	20.2319i	−0.581868	−	1.00782i
$404$		0			0
$405$	−3.11856	−	5.40150i	−0.154962	−	0.268403i
$406$		0			0
$407$	15.0259			0.744805
$408$		0			0
$409$	4.09085	+	7.08556i	0.202279	+	0.350358i	0.949263	−	0.314485i	$-0.101832\pi$
$409$	4.09085	+	7.08556i	0.202279	+	0.350358i	−0.746983	+	0.664843i	$0.768498\pi$
$410$		0			0
$411$	3.85727			0.190265
$412$		0			0
$413$	0.950884	−	1.64698i	0.0467899	−	0.0810425i
$414$		0			0
$415$	2.18914	−	3.79171i	0.107461	−	0.186128i
$416$		0			0
$417$	3.53808			0.173261
$418$		0			0
$419$	15.0687			0.736154			0.368077	−	0.929795i	$-0.380016\pi$
$419$	15.0687			0.736154			0.368077	+	0.929795i	$0.380016\pi$
$420$		0			0
$421$	4.93380	−	8.54559i	0.240459	−	0.416487i	−0.720386	−	0.693573i	$-0.756035\pi$
$421$	4.93380	−	8.54559i	0.240459	−	0.416487i	0.960845	+	0.277086i	$0.0893688\pi$
$422$		0			0
$423$	3.14318	−	5.44415i	0.152827	−	0.264704i
$424$		0			0
$425$	7.68176			0.372620
$426$		0			0
$427$	10.2791	+	17.8039i	0.497439	+	0.861589i
$428$		0			0
$429$	6.30659			0.304485
$430$		0			0
$431$	−9.25967	−	16.0382i	−0.446023	−	0.772534i	0.552100	−	0.833778i	$-0.313827\pi$
$431$	−9.25967	−	16.0382i	−0.446023	−	0.772534i	−0.998123	+	0.0612440i	$0.980493\pi$
$432$		0			0
$433$	1.45728	+	2.52408i	0.0700324	+	0.121300i	0.898915	−	0.438122i	$-0.144356\pi$
$433$	1.45728	+	2.52408i	0.0700324	+	0.121300i	−0.828883	+	0.559422i	$0.811023\pi$
$434$		0			0
$435$	−0.907080	+	1.57111i	−0.0434911	+	0.0753289i
$436$		0			0
$437$	7.09292	−	19.1350i	0.339300	−	0.915352i
$438$		0			0
$439$	15.8039	−	27.3731i	0.754277	−	1.30645i	−0.191456	−	0.981501i	$-0.561321\pi$
$439$	15.8039	−	27.3731i	0.754277	−	1.30645i	0.945733	−	0.324945i	$-0.105346\pi$
$440$		0			0
$441$	−6.10570	−	10.5754i	−0.290748	−	0.503590i
$442$		0			0
$443$	4.05443	+	7.02248i	0.192632	+	0.333648i	0.946122	−	0.323812i	$-0.104964\pi$
$443$	4.05443	+	7.02248i	0.192632	+	0.333648i	−0.753490	+	0.657460i	$0.771631\pi$
$444$		0			0
$445$	15.2718			0.723951
$446$		0			0
$447$	0.168757	+	0.292296i	0.00798195	+	0.0138251i
$448$		0			0
$449$	−26.4382			−1.24770			−0.623848	−	0.781546i	$-0.714432\pi$
$449$	−26.4382			−1.24770			−0.623848	+	0.781546i	$0.714432\pi$
$450$		0			0
$451$	−5.13465	+	8.89348i	−0.241781	+	0.418778i
$452$		0			0
$453$	−6.14335	+	10.6406i	−0.288640	+	0.499939i
$454$		0			0
$455$	4.14765			0.194445
$456$		0			0
$457$	30.4300			1.42346			0.711729	−	0.702454i	$-0.247913\pi$
$457$	30.4300			1.42346			0.711729	+	0.702454i	$0.247913\pi$
$458$		0			0
$459$	12.3109	−	21.3232i	0.574626	−	0.995281i
$460$		0			0
$461$	−11.0697	+	19.1733i	−0.515567	+	0.892988i	0.484270	+	0.874919i	$0.339085\pi$
$461$	−11.0697	+	19.1733i	−0.515567	+	0.892988i	−0.999837	+	0.0180690i	$0.994248\pi$
$462$		0			0
$463$	−13.5535			−0.629885			−0.314942	−	0.949111i	$-0.601985\pi$
$463$	−13.5535			−0.629885			−0.314942	+	0.949111i	$0.601985\pi$
$464$		0			0
$465$	−2.48499	−	4.30413i	−0.115239	−	0.199599i
$466$		0			0
$467$	−24.6222			−1.13938			−0.569690	−	0.821860i	$-0.692937\pi$
$467$	−24.6222			−1.13938			−0.569690	+	0.821860i	$0.692937\pi$
$468$		0			0
$469$	6.64111	+	11.5027i	0.306658	+	0.531147i
$470$		0			0
$471$	3.84767	+	6.66435i	0.177291	+	0.307077i
$472$		0			0
$473$	9.41380	−	16.3052i	0.432847	−	0.749713i
$474$		0			0
$475$	1.51501	−	4.08714i	0.0695135	−	0.187531i
$476$		0			0
$477$	−6.91663	+	11.9800i	−0.316691	+	0.548525i
$478$		0			0
$479$	−19.9330	−	34.5249i	−0.910760	−	1.57748i	−0.812993	−	0.582274i	$-0.802163\pi$
$479$	−19.9330	−	34.5249i	−0.910760	−	1.57748i	−0.0977672	−	0.995209i	$-0.531170\pi$
$480$		0			0
$481$	−4.72556	−	8.18492i	−0.215467	−	0.373200i
$482$		0			0
$483$	4.13104			0.187969
$484$		0			0
$485$	−4.71794	−	8.17170i	−0.214230	−	0.371058i
$486$		0			0
$487$	−1.59001			−0.0720501			−0.0360251	−	0.999351i	$-0.511470\pi$
$487$	−1.59001			−0.0720501			−0.0360251	+	0.999351i	$0.511470\pi$
$488$		0			0
$489$	3.63018	−	6.28766i	0.164162	−	0.284338i
$490$		0			0
$491$	11.2085	−	19.4137i	0.505832	−	0.876126i	−0.494146	−	0.869379i	$-0.664519\pi$
$491$	11.2085	−	19.4137i	0.505832	−	0.876126i	0.999977	−	0.00674693i	$-0.00214763\pi$
$492$		0			0
$493$	24.7035			1.11259
$494$		0			0
$495$	−11.3059			−0.508164
$496$		0			0
$497$	−9.40501	+	16.2900i	−0.421872	+	0.730704i
$498$		0			0
$499$	3.25942	−	5.64549i	0.145912	−	0.252727i	−0.783801	−	0.621012i	$-0.786722\pi$
$499$	3.25942	−	5.64549i	0.145912	−	0.252727i	0.929713	+	0.368285i	$0.120055\pi$
$500$		0			0
$501$	9.16842			0.409615
$502$		0			0
$503$	9.74904	+	16.8858i	0.434688	+	0.752902i	0.997270	−	0.0738393i	$-0.0235252\pi$
$503$	9.74904	+	16.8858i	0.434688	+	0.752902i	−0.562582	+	0.826742i	$0.690192\pi$
$504$		0			0
$505$	−14.9852			−0.666834
$506$		0			0
$507$	1.68344	+	2.91581i	0.0747642	+	0.129495i
$508$		0			0
$509$	1.06089	+	1.83751i	0.0470230	+	0.0814462i	0.888579	−	0.458724i	$-0.151693\pi$
$509$	1.06089	+	1.83751i	0.0470230	+	0.0814462i	−0.841556	+	0.540170i	$0.818360\pi$
$510$		0			0
$511$	−9.69254	+	16.7880i	−0.428773	+	0.742656i
$512$		0			0
$513$	−8.91718	−	10.7555i	−0.393703	−	0.474868i
$514$		0			0
$515$	3.34088	−	5.78657i	0.147217	−	0.254987i
$516$		0			0
$517$	−4.94125	−	8.55849i	−0.217316	−	0.376402i
$518$		0			0
$519$	0.579896	+	1.00441i	0.0254546	+	0.0440887i
$520$		0			0
$521$	30.9129			1.35432			0.677159	−	0.735837i	$-0.263211\pi$
$521$	30.9129			1.35432			0.677159	+	0.735837i	$0.263211\pi$
$522$		0			0
$523$	16.8915	+	29.2569i	0.738612	+	1.27931i	0.953120	+	0.302592i	$0.0978520\pi$
$523$	16.8915	+	29.2569i	0.738612	+	1.27931i	−0.214508	+	0.976722i	$0.568815\pi$
$524$		0			0
$525$	0.882368			0.0385097
$526$		0			0
$527$	−33.8382	+	58.6095i	−1.47402	+	2.55307i
$528$		0			0
$529$	0.540564	−	0.936284i	0.0235028	−	0.0407080i
$530$		0			0
$531$	3.26066			0.141500
$532$		0			0
$533$	6.45929			0.279783
$534$		0			0
$535$	−7.38261	+	12.7871i	−0.319178	+	0.552833i
$536$		0			0
$537$	0.0705248	−	0.122152i	0.00304337	−	0.00527127i
$538$		0			0
$539$	−19.1970			−0.826871
$540$		0			0
$541$	−18.1412	−	31.4215i	−0.779951	−	1.35091i	−0.931969	−	0.362537i	$-0.881911\pi$
$541$	−18.1412	−	31.4215i	−0.779951	−	1.35091i	0.152018	−	0.988378i	$-0.451423\pi$
$542$		0			0
$543$	10.7994			0.463446
$544$		0			0
$545$	5.79708	+	10.0408i	0.248319	+	0.430102i
$546$		0			0
$547$	−11.4309	−	19.7989i	−0.488749	−	0.846539i	0.511167	−	0.859482i	$-0.329213\pi$
$547$	−11.4309	−	19.7989i	−0.488749	−	0.846539i	−0.999916	+	0.0129426i	$0.995880\pi$
$548$		0			0
$549$	−17.6239	+	30.5254i	−0.752168	+	1.30279i
$550$		0			0
$551$	4.87207	−	13.1437i	0.207557	−	0.559940i
$552$		0			0
$553$	−5.57475	+	9.65575i	−0.237063	+	0.410604i
$554$		0			0
$555$	−1.00531	−	1.74125i	−0.0426731	−	0.0739120i
$556$		0			0
$557$	−17.4695	−	30.2580i	−0.740205	−	1.28207i	−0.952402	−	0.304846i	$-0.901395\pi$
$557$	−17.4695	−	30.2580i	−0.740205	−	1.28207i	0.212197	−	0.977227i	$-0.431938\pi$
$558$		0			0
$559$	−11.8424			−0.500879
$560$		0			0
$561$	−9.13471	−	15.8218i	−0.385668	−	0.667997i
$562$		0			0
$563$	3.34826			0.141112			0.0705562	−	0.997508i	$-0.477523\pi$
$563$	3.34826			0.141112			0.0705562	+	0.997508i	$0.477523\pi$
$564$		0			0
$565$	−5.96382	+	10.3296i	−0.250900	+	0.434571i
$566$		0			0
$567$	−4.87782	+	8.44864i	−0.204849	+	0.354809i
$568$		0			0
$569$	20.3270			0.852153			0.426076	−	0.904687i	$-0.359895\pi$
$569$	20.3270			0.852153			0.426076	+	0.904687i	$0.359895\pi$
$570$		0			0
$571$	−9.53644			−0.399088			−0.199544	−	0.979889i	$-0.563946\pi$
$571$	−9.53644			−0.399088			−0.199544	+	0.979889i	$0.563946\pi$
$572$		0			0
$573$	5.16644	−	8.94854i	0.215831	−	0.373831i
$574$		0			0
$575$	−2.34088	+	4.05452i	−0.0976214	+	0.169085i
$576$		0			0
$577$	8.51471			0.354472			0.177236	−	0.984168i	$-0.443284\pi$
$577$	8.51471			0.354472			0.177236	+	0.984168i	$0.443284\pi$
$578$		0			0
$579$	5.42740	+	9.40054i	0.225555	+	0.390673i
$580$		0			0
$581$	−6.84820			−0.284111
$582$		0			0
$583$	10.8733	+	18.8331i	0.450326	+	0.779988i
$584$		0			0
$585$	3.55566	+	6.15858i	0.147008	+	0.254626i
$586$		0			0
$587$	17.5847	−	30.4576i	0.725798	−	1.25712i	−0.232847	−	0.972513i	$-0.574804\pi$
$587$	17.5847	−	30.4576i	0.725798	−	1.25712i	0.958645	−	0.284606i	$-0.0918627\pi$
$588$		0			0
$589$	24.5100	+	29.5630i	1.00992	+	1.21812i
$590$		0			0
$591$	−2.42888	+	4.20694i	−0.0999107	+	0.173050i
$592$		0			0
$593$	−9.24589	−	16.0143i	−0.379683	−	0.657630i	0.611333	−	0.791374i	$-0.290634\pi$
$593$	−9.24589	−	16.0143i	−0.379683	−	0.657630i	−0.991016	+	0.133743i	$0.957300\pi$
$594$		0			0
$595$	−6.00763	−	10.4055i	−0.246289	−	0.426585i
$596$		0			0
$597$	−14.0454			−0.574842
$598$		0			0
$599$	−1.66351	−	2.88128i	−0.0679691	−	0.117726i	0.830038	−	0.557707i	$-0.188319\pi$
$599$	−1.66351	−	2.88128i	−0.0679691	−	0.117726i	−0.898007	+	0.439981i	$0.854985\pi$
$600$		0			0
$601$	2.03663			0.0830758			0.0415379	−	0.999137i	$-0.486774\pi$
$601$	2.03663			0.0830758			0.0415379	+	0.999137i	$0.486774\pi$
$602$		0			0
$603$	−11.3864	+	19.7219i	−0.463692	+	0.803138i
$604$		0			0
$605$	−3.38676	+	5.86603i	−0.137691	+	0.238488i
$606$		0			0
$607$	13.1948			0.535560			0.267780	−	0.963480i	$-0.413710\pi$
$607$	13.1948			0.535560			0.267780	+	0.963480i	$0.413710\pi$
$608$		0			0
$609$	2.83758			0.114984
$610$		0			0
$611$	−3.10799	+	5.38320i	−0.125736	+	0.217781i
$612$		0			0
$613$	20.7726	−	35.9792i	0.838998	−	1.45319i	−0.0517358	−	0.998661i	$-0.516475\pi$
$613$	20.7726	−	35.9792i	0.838998	−	1.45319i	0.890734	−	0.454526i	$-0.150191\pi$
$614$		0			0
$615$	1.37414			0.0554108
$616$		0			0
$617$	20.1926	+	34.9747i	0.812925	+	1.40803i	0.910809	+	0.412829i	$0.135459\pi$
$617$	20.1926	+	34.9747i	0.812925	+	1.40803i	−0.0978840	+	0.995198i	$0.531207\pi$
$618$		0			0
$619$	26.9346			1.08259			0.541297	−	0.840832i	$-0.317934\pi$
$619$	26.9346			1.08259			0.541297	+	0.840832i	$0.317934\pi$
$620$		0			0
$621$	7.50308	+	12.9957i	0.301088	+	0.521500i
$622$		0			0
$623$	−11.9435	−	20.6867i	−0.478506	−	0.828797i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−10.2197	+	1.73980i	−0.408135	+	0.0694807i
$628$		0			0
$629$	−13.6894	+	23.7107i	−0.545832	+	0.945408i
$630$		0			0
$631$	−2.48931	−	4.31162i	−0.0990980	−	0.171643i	0.812214	−	0.583360i	$-0.198262\pi$
$631$	−2.48931	−	4.31162i	−0.0990980	−	0.171643i	−0.911312	+	0.411717i	$0.864929\pi$
$632$		0			0
$633$	6.68824	+	11.5844i	0.265834	+	0.460437i
$634$		0			0
$635$	1.65174			0.0655472
$636$		0			0
$637$	6.03735	+	10.4570i	0.239208	+	0.414321i
$638$		0			0
$639$	−32.2505			−1.27581
$640$		0			0
$641$	−10.6809	+	18.4999i	−0.421871	+	0.730702i	−0.996123	−	0.0879770i	$-0.971960\pi$
$641$	−10.6809	+	18.4999i	−0.421871	+	0.730702i	0.574252	+	0.818679i	$0.305293\pi$
$642$		0			0
$643$	−12.6955	+	21.9893i	−0.500663	+	0.867174i	0.499336	+	0.866408i	$0.333577\pi$
$643$	−12.6955	+	21.9893i	−0.500663	+	0.867174i	−1.00000		0.000766009i	$0.999756\pi$
$644$		0			0
$645$	−2.51934			−0.0991988
$646$		0			0
$647$	23.7459			0.933550			0.466775	−	0.884376i	$-0.345416\pi$
$647$	23.7459			0.933550			0.466775	+	0.884376i	$0.345416\pi$
$648$		0			0
$649$	2.56296	−	4.43918i	0.100605	−	0.174253i
$650$		0			0
$651$	−3.88684	+	6.73220i	−0.152337	+	0.263856i
$652$		0			0
$653$	2.48066			0.0970759			0.0485379	−	0.998821i	$-0.484544\pi$
$653$	2.48066			0.0970759			0.0485379	+	0.998821i	$0.484544\pi$
$654$		0			0
$655$	−2.34619	−	4.06372i	−0.0916733	−	0.158783i
$656$		0			0
$657$	−33.2365			−1.29668
$658$		0			0
$659$	1.96259	+	3.39931i	0.0764518	+	0.132418i	0.901717	−	0.432327i	$-0.142308\pi$
$659$	1.96259	+	3.39931i	0.0764518	+	0.132418i	−0.825265	+	0.564746i	$0.808974\pi$
$660$		0			0
$661$	−0.776836	−	1.34552i	−0.0302154	−	0.0523346i	0.850522	−	0.525939i	$-0.176286\pi$
$661$	−0.776836	−	1.34552i	−0.0302154	−	0.0523346i	−0.880738	+	0.473604i	$0.842953\pi$
$662$		0			0
$663$	−5.74564	+	9.95175i	−0.223142	+	0.386494i
$664$		0			0
$665$	−6.72118	+	1.14421i	−0.260636	+	0.0443706i
$666$		0			0
$667$	−7.52795	+	13.0388i	−0.291483	+	0.504864i
$668$		0			0
$669$	3.81332	+	6.60486i	0.147431	+	0.255359i
$670$		0			0
$671$	27.7056	+	47.9876i	1.06956	+	1.85254i
$672$		0			0
$673$	28.0007			1.07935			0.539673	−	0.841875i	$-0.318548\pi$
$673$	28.0007			1.07935			0.539673	+	0.841875i	$0.318548\pi$
$674$		0			0
$675$	1.60262	+	2.77582i	0.0616849	+	0.106841i
$676$		0			0
$677$	37.9859			1.45992			0.729958	−	0.683492i	$-0.239540\pi$
$677$	37.9859			1.45992			0.729958	+	0.683492i	$0.239540\pi$
$678$		0			0
$679$	−7.37946	+	12.7816i	−0.283198	+	0.490513i
$680$		0			0
$681$	4.31656	−	7.47650i	0.165411	−	0.286500i
$682$		0			0
$683$	−3.45712			−0.132283			−0.0661415	−	0.997810i	$-0.521069\pi$
$683$	−3.45712			−0.132283			−0.0661415	+	0.997810i	$0.521069\pi$
$684$		0			0
$685$	6.83758			0.261250
$686$		0			0
$687$	−1.90093	+	3.29250i	−0.0725249	+	0.125617i
$688$		0			0
$689$	6.83920	−	11.8458i	0.260553	−	0.451291i
$690$		0			0
$691$	−18.9877			−0.722326			−0.361163	−	0.932503i	$-0.617620\pi$
$691$	−18.9877			−0.722326			−0.361163	+	0.932503i	$0.617620\pi$
$692$		0			0
$693$	8.84196	+	15.3147i	0.335878	+	0.581759i
$694$		0			0
$695$	6.27177			0.237902
$696$		0			0
$697$	−9.35589	−	16.2049i	−0.354380	−	0.613804i
$698$		0			0
$699$	−8.04811	−	13.9397i	−0.304407	−	0.527249i
$700$		0			0
$701$	3.00861	−	5.21107i	0.113634	−	0.196820i	−0.803599	−	0.595171i	$-0.797084\pi$
$701$	3.00861	−	5.21107i	0.113634	−	0.196820i	0.917233	+	0.398352i	$0.130418\pi$
$702$		0			0
$703$	9.91563	+	11.9598i	0.373975	+	0.451073i
$704$		0			0
$705$	−0.661192	+	1.14522i	−0.0249019	+	0.0431314i
$706$		0			0
$707$	11.7194	+	20.2986i	0.440754	+	0.763408i
$708$		0			0
$709$	5.85796	+	10.1463i	0.220000	+	0.381052i	0.954808	−	0.297224i	$-0.0960608\pi$
$709$	5.85796	+	10.1463i	0.220000	+	0.381052i	−0.734807	+	0.678276i	$0.762727\pi$
$710$		0			0
$711$	−19.1163			−0.716916
$712$		0			0
$713$	−20.6232	−	35.7204i	−0.772344	−	1.33774i
$714$		0			0
$715$	11.1794			0.418084
$716$		0			0
$717$	0.228476	−	0.395732i	0.00853260	−	0.0147789i
$718$		0			0
$719$	−16.7811	+	29.0658i	−0.625831	+	1.08397i	0.362549	+	0.931965i	$0.381907\pi$
$719$	−16.7811	+	29.0658i	−0.625831	+	1.08397i	−0.988380	+	0.152006i	$0.951427\pi$
$720$		0			0
$721$	−10.4511			−0.389220
$722$		0			0
$723$	−11.1671			−0.415307
$724$		0			0
$725$	−1.60793	+	2.78502i	−0.0597171	+	0.103433i
$726$		0			0
$727$	11.1988	−	19.3969i	0.415340	−	0.719390i	−0.580124	−	0.814528i	$-0.696996\pi$
$727$	11.1988	−	19.3969i	0.415340	−	0.719390i	0.995464	+	0.0951384i	$0.0303294\pi$
$728$		0			0
$729$	−11.3019			−0.418590
$730$		0			0
$731$	17.1530	+	29.7098i	0.634425	+	1.09886i
$732$		0			0
$733$	−24.6476			−0.910380			−0.455190	−	0.890394i	$-0.650429\pi$
$733$	−24.6476			−0.910380			−0.455190	+	0.890394i	$0.650429\pi$
$734$		0			0
$735$	1.28438	+	2.22461i	0.0473751	+	0.0820560i
$736$		0			0
$737$	17.9001	+	31.0039i	0.659358	+	1.14204i
$738$		0			0
$739$	−23.2868	+	40.3339i	−0.856618	+	1.48371i	0.0185178	+	0.999829i	$0.494105\pi$
$739$	−23.2868	+	40.3339i	−0.856618	+	1.48371i	−0.875136	+	0.483877i	$0.839228\pi$
$740$		0			0
$741$	4.16174	+	5.01972i	0.152885	+	0.184404i
$742$		0			0
$743$	1.64882	−	2.85585i	0.0604895	−	0.104771i	−0.834195	−	0.551470i	$-0.814067\pi$
$743$	1.64882	−	2.85585i	0.0604895	−	0.104771i	0.894684	+	0.446699i	$0.147401\pi$
$744$		0			0
$745$	0.299147	+	0.518139i	0.0109599	+	0.0189831i
$746$		0			0
$747$	−5.87076	−	10.1684i	−0.214800	−	0.372044i
$748$		0			0
$749$	23.0947			0.843862
$750$		0			0
$751$	−13.1873	−	22.8411i	−0.481212	−	0.833484i	0.518556	−	0.855044i	$-0.326470\pi$
$751$	−13.1873	−	22.8411i	−0.481212	−	0.833484i	−0.999768	+	0.0215603i	$0.993137\pi$
$752$		0			0
$753$	−5.20753			−0.189773
$754$		0			0
$755$	−10.8900	+	18.8620i	−0.396328	+	0.686459i
$756$		0			0
$757$	19.3100	−	33.4459i	0.701834	−	1.21561i	−0.265987	−	0.963977i	$-0.585698\pi$
$757$	19.3100	−	33.4459i	0.701834	−	1.21561i	0.967822	−	0.251636i	$-0.0809687\pi$
$758$		0			0
$759$	11.1346			0.404159
$760$		0			0
$761$	3.60461			0.130667			0.0653335	−	0.997863i	$-0.479189\pi$
$761$	3.60461			0.130667			0.0653335	+	0.997863i	$0.479189\pi$
$762$		0			0
$763$	9.06737	−	15.7051i	0.328261	−	0.568564i
$764$		0			0
$765$	10.3003	−	17.8407i	0.372409	−	0.645031i
$766$		0			0
$767$	−3.22415			−0.116417
$768$		0			0
$769$	5.61439	+	9.72441i	0.202460	+	0.350671i	0.949321	−	0.314310i	$-0.101773\pi$
$769$	5.61439	+	9.72441i	0.202460	+	0.350671i	−0.746860	+	0.664981i	$0.768440\pi$
$770$		0			0
$771$	−12.6594			−0.455918
$772$		0			0
$773$	9.91711	+	17.1769i	0.356694	+	0.617811i	0.987406	−	0.158205i	$-0.0505707\pi$
$773$	9.91711	+	17.1769i	0.356694	+	0.617811i	−0.630713	+	0.776016i	$0.717237\pi$
$774$		0			0
$775$	−4.40501	−	7.62970i	−0.158233	−	0.274067i
$776$		0			0
$777$	−1.57244	+	2.72354i	−0.0564109	+	0.0977065i
$778$		0			0
$779$	−10.4671	+	1.78192i	−0.375024	+	0.0638439i
$780$		0			0
$781$	−25.3497	+	43.9070i	−0.907085	+	1.57112i
$782$		0			0
$783$	5.15381	+	8.92666i	0.184182	+	0.319013i
$784$		0			0
$785$	6.82056	+	11.8135i	0.243436	+	0.421644i
$786$		0			0
$787$	−31.8148			−1.13408			−0.567038	−	0.823692i	$-0.691911\pi$
$787$	−31.8148			−1.13408			−0.567038	+	0.823692i	$0.691911\pi$
$788$		0			0
$789$	3.46373	+	5.99935i	0.123312	+	0.213583i
$790$		0			0
$791$	18.6564			0.663344

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.282064 + 0.488549i) q^{3} +(-0.500000 + 0.866025i) q^{5} +1.56413 q^{7} +(1.34088 + 2.32247i) q^{9} +O(q^{10})\)	\(q+(-0.282064 + 0.488549i) q^{3} +(-0.500000 + 0.866025i) q^{5} +1.56413 q^{7} +(1.34088 + 2.32247i) q^{9} +4.21586 q^{11} +(-1.32587 - 2.29647i) q^{13} +(-0.282064 - 0.488549i) q^{15} +(-3.84088 + 6.65260i) q^{17} +(2.78206 + 3.35561i) q^{19} +(-0.441184 + 0.764153i) q^{21} +(-2.34088 - 4.05452i) q^{23} +(-0.500000 - 0.866025i) q^{25} -3.20524 q^{27} +(-1.60793 - 2.78502i) q^{29} +8.81002 q^{31} +(-1.18914 + 2.05966i) q^{33} +(-0.782064 + 1.35457i) q^{35} +3.56413 q^{37} +1.49592 q^{39} +(-1.21794 + 2.10953i) q^{41} +(2.23295 - 3.86758i) q^{43} -2.68176 q^{45} +(-1.17206 - 2.03007i) q^{47} -4.55350 q^{49} +(-2.16675 - 3.75292i) q^{51} +(2.57914 + 4.46720i) q^{53} +(-2.10793 + 3.65105i) q^{55} +(-2.42410 + 0.412678i) q^{57} +(0.607932 - 1.05297i) q^{59} +(6.57176 + 11.3826i) q^{61} +(2.09731 + 3.63264i) q^{63} +2.65174 q^{65} +(4.24589 + 7.35409i) q^{67} +2.64111 q^{69} +(-6.01294 + 10.4147i) q^{71} +(-6.19677 + 10.7331i) q^{73} +0.564128 q^{75} +6.59415 q^{77} +(-3.56413 + 6.17325i) q^{79} +(-3.11856 + 5.40150i) q^{81} -4.37829 q^{83} +(-3.84088 - 6.65260i) q^{85} +1.81416 q^{87} +(-7.63588 - 13.2257i) q^{89} +(-2.07383 - 3.59197i) q^{91} +(-2.48499 + 4.30413i) q^{93} +(-4.29708 + 0.731533i) q^{95} +(-4.71794 + 8.17170i) q^{97} +(5.65297 + 9.79123i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9}+O(q^{10}) \) 8 * q + q^3 - 4 * q^5 + 6 * q^7 - 13 * q^9	\( 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9} + 8 q^{11} - q^{13} + q^{15} - 7 q^{17} + 19 q^{19} - 24 q^{21} + 5 q^{23} - 4 q^{25} + 28 q^{27} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 22 q^{37} - 38 q^{39} - 13 q^{41} + 7 q^{43} + 26 q^{45} + 10 q^{47} - 2 q^{49} + 16 q^{51} - 4 q^{55} - 25 q^{57} - 8 q^{59} - 11 q^{61} - 24 q^{63} + 2 q^{65} - 20 q^{67} - 26 q^{69} - 5 q^{71} + 12 q^{73} - 2 q^{75} + 18 q^{77} - 22 q^{79} - 40 q^{81} - 26 q^{83} - 7 q^{85} + 12 q^{87} + 9 q^{89} + 18 q^{91} - 34 q^{93} - 17 q^{95} - 41 q^{97} - 10 q^{99}+O(q^{100}) \) 8 * q + q^3 - 4 * q^5 + 6 * q^7 - 13 * q^9 + 8 * q^11 - q^13 + q^15 - 7 * q^17 + 19 * q^19 - 24 * q^21 + 5 * q^23 - 4 * q^25 + 28 * q^27 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 22 * q^37 - 38 * q^39 - 13 * q^41 + 7 * q^43 + 26 * q^45 + 10 * q^47 - 2 * q^49 + 16 * q^51 - 4 * q^55 - 25 * q^57 - 8 * q^59 - 11 * q^61 - 24 * q^63 + 2 * q^65 - 20 * q^67 - 26 * q^69 - 5 * q^71 + 12 * q^73 - 2 * q^75 + 18 * q^77 - 22 * q^79 - 40 * q^81 - 26 * q^83 - 7 * q^85 + 12 * q^87 + 9 * q^89 + 18 * q^91 - 34 * q^93 - 17 * q^95 - 41 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more