LMFDB - Embedded newform 45.3.g.a.28.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,3,Mod(28,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("45.28");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	45.g (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.22616118962$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			28.2
Root			$1.58114 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	45.28
Dual form			45.3.g.a.37.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.58114 - 1.58114i) q^{2} -1.00000i q^{4} +(1.58114 - 4.74342i) q^{5} +(-5.00000 + 5.00000i) q^{7} +(4.74342 + 4.74342i) q^{8} +O(q^{10})$	\(q+(1.58114 - 1.58114i) q^{2} -1.00000i q^{4} +(1.58114 - 4.74342i) q^{5} +(-5.00000 + 5.00000i) q^{7} +(4.74342 + 4.74342i) q^{8} +(-5.00000 - 10.0000i) q^{10} -15.8114 q^{11} +(10.0000 + 10.0000i) q^{13} +15.8114i q^{14} +19.0000 q^{16} +(-3.16228 + 3.16228i) q^{17} -18.0000i q^{19} +(-4.74342 - 1.58114i) q^{20} +(-25.0000 + 25.0000i) q^{22} +(-3.16228 - 3.16228i) q^{23} +(-20.0000 - 15.0000i) q^{25} +31.6228 q^{26} +(5.00000 + 5.00000i) q^{28} -47.4342i q^{29} +8.00000 q^{31} +(11.0680 - 11.0680i) q^{32} +10.0000i q^{34} +(15.8114 + 31.6228i) q^{35} +(10.0000 - 10.0000i) q^{37} +(-28.4605 - 28.4605i) q^{38} +(30.0000 - 15.0000i) q^{40} +31.6228 q^{41} +(10.0000 + 10.0000i) q^{43} +15.8114i q^{44} -10.0000 q^{46} +(-41.1096 + 41.1096i) q^{47} -1.00000i q^{49} +(-55.3399 + 7.90569i) q^{50} +(10.0000 - 10.0000i) q^{52} +(25.2982 + 25.2982i) q^{53} +(-25.0000 + 75.0000i) q^{55} -47.4342 q^{56} +(-75.0000 - 75.0000i) q^{58} +47.4342i q^{59} -58.0000 q^{61} +(12.6491 - 12.6491i) q^{62} +41.0000i q^{64} +(63.2456 - 31.6228i) q^{65} +(70.0000 - 70.0000i) q^{67} +(3.16228 + 3.16228i) q^{68} +(75.0000 + 25.0000i) q^{70} -63.2456 q^{71} +(55.0000 + 55.0000i) q^{73} -31.6228i q^{74} -18.0000 q^{76} +(79.0569 - 79.0569i) q^{77} -12.0000i q^{79} +(30.0416 - 90.1249i) q^{80} +(50.0000 - 50.0000i) q^{82} +(53.7587 + 53.7587i) q^{83} +(10.0000 + 20.0000i) q^{85} +31.6228 q^{86} +(-75.0000 - 75.0000i) q^{88} -100.000 q^{91} +(-3.16228 + 3.16228i) q^{92} +130.000i q^{94} +(-85.3815 - 28.4605i) q^{95} +(-5.00000 + 5.00000i) q^{97} +(-1.58114 - 1.58114i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{7}+O(q^{10}) $ 4 * q - 20 * q^7	$ 4 q - 20 q^{7} - 20 q^{10} + 40 q^{13} + 76 q^{16} - 100 q^{22} - 80 q^{25} + 20 q^{28} + 32 q^{31} + 40 q^{37} + 120 q^{40} + 40 q^{43} - 40 q^{46} + 40 q^{52} - 100 q^{55} - 300 q^{58} - 232 q^{61} + 280 q^{67} + 300 q^{70} + 220 q^{73} - 72 q^{76} + 200 q^{82} + 40 q^{85} - 300 q^{88} - 400 q^{91} - 20 q^{97}+O(q^{100}) $ 4 * q - 20 * q^7 - 20 * q^10 + 40 * q^13 + 76 * q^16 - 100 * q^22 - 80 * q^25 + 20 * q^28 + 32 * q^31 + 40 * q^37 + 120 * q^40 + 40 * q^43 - 40 * q^46 + 40 * q^52 - 100 * q^55 - 300 * q^58 - 232 * q^61 + 280 * q^67 + 300 * q^70 + 220 * q^73 - 72 * q^76 + 200 * q^82 + 40 * q^85 - 300 * q^88 - 400 * q^91 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$	1.58114	−	1.58114i	0.790569	−	0.790569i	−0.191017	−	0.981587i	$-0.561179\pi$
$2$	1.58114	−	1.58114i	0.790569	−	0.790569i	0.981587	+	0.191017i	$0.0611786\pi$
$3$		0			0
$3$		0			0
$4$		−	1.00000i		−	0.250000i
$5$	1.58114	−	4.74342i	0.316228	−	0.948683i
$5$	1.58114	−	4.74342i	0.316228	−	0.948683i
$6$		0			0
$7$	−5.00000	+	5.00000i	−0.714286	+	0.714286i	−0.967429	−	0.253143i	$-0.918536\pi$
$7$	−5.00000	+	5.00000i	−0.714286	+	0.714286i	0.253143	+	0.967429i	$0.418536\pi$
$8$	4.74342	+	4.74342i	0.592927	+	0.592927i
$9$		0			0
$10$	−5.00000	−	10.0000i	−0.500000	−	1.00000i
$11$	−15.8114			−1.43740			−0.718699	−	0.695321i	$-0.755262\pi$
$11$	−15.8114			−1.43740			−0.718699	+	0.695321i	$0.755262\pi$
$12$		0			0
$13$	10.0000	+	10.0000i	0.769231	+	0.769231i	0.977971	−	0.208740i	$-0.0669363\pi$
$13$	10.0000	+	10.0000i	0.769231	+	0.769231i	−0.208740	+	0.977971i	$0.566936\pi$
$14$			15.8114i			1.12938i
$15$		0			0
$16$	19.0000			1.18750
$17$	−3.16228	+	3.16228i	−0.186016	+	0.186016i	−0.793971	−	0.607955i	$-0.791990\pi$
$17$	−3.16228	+	3.16228i	−0.186016	+	0.186016i	0.607955	+	0.793971i	$0.291990\pi$
$18$		0			0
$19$		−	18.0000i		−	0.947368i	−0.880695	−	0.473684i	$-0.842924\pi$
$19$		−	18.0000i		−	0.947368i	0.880695	−	0.473684i	$-0.157076\pi$
$20$	−4.74342	−	1.58114i	−0.237171	−	0.0790569i
$21$		0			0
$22$	−25.0000	+	25.0000i	−1.13636	+	1.13636i
$23$	−3.16228	−	3.16228i	−0.137490	−	0.137490i	0.635012	−	0.772502i	$-0.280995\pi$
$23$	−3.16228	−	3.16228i	−0.137490	−	0.137490i	−0.772502	+	0.635012i	$0.780995\pi$
$24$		0			0
$25$	−20.0000	−	15.0000i	−0.800000	−	0.600000i
$26$	31.6228			1.21626
$27$		0			0
$28$	5.00000	+	5.00000i	0.178571	+	0.178571i
$29$		−	47.4342i		−	1.63566i	−0.575459	−	0.817830i	$-0.695177\pi$
$29$		−	47.4342i		−	1.63566i	0.575459	−	0.817830i	$-0.304823\pi$
$30$		0			0
$31$	8.00000			0.258065			0.129032	−	0.991640i	$-0.458813\pi$
$31$	8.00000			0.258065			0.129032	+	0.991640i	$0.458813\pi$
$32$	11.0680	−	11.0680i	0.345874	−	0.345874i
$33$		0			0
$34$			10.0000i			0.294118i
$35$	15.8114	+	31.6228i	0.451754	+	0.903508i
$36$		0			0
$37$	10.0000	−	10.0000i	0.270270	−	0.270270i	−0.558939	−	0.829209i	$-0.688791\pi$
$37$	10.0000	−	10.0000i	0.270270	−	0.270270i	0.829209	+	0.558939i	$0.188791\pi$
$38$	−28.4605	−	28.4605i	−0.748960	−	0.748960i
$39$		0			0
$40$	30.0000	−	15.0000i	0.750000	−	0.375000i
$41$	31.6228			0.771287			0.385644	−	0.922648i	$-0.373979\pi$
$41$	31.6228			0.771287			0.385644	+	0.922648i	$0.373979\pi$
$42$		0			0
$43$	10.0000	+	10.0000i	0.232558	+	0.232558i	0.813760	−	0.581202i	$-0.197417\pi$
$43$	10.0000	+	10.0000i	0.232558	+	0.232558i	−0.581202	+	0.813760i	$0.697417\pi$
$44$			15.8114i			0.359350i
$45$		0			0
$46$	−10.0000			−0.217391
$47$	−41.1096	+	41.1096i	−0.874673	+	0.874673i	−0.992977	−	0.118305i	$-0.962254\pi$
$47$	−41.1096	+	41.1096i	−0.874673	+	0.874673i	0.118305	+	0.992977i	$0.462254\pi$
$48$		0			0
$49$		−	1.00000i		−	0.0204082i
$50$	−55.3399	+	7.90569i	−1.10680	+	0.158114i
$51$		0			0
$52$	10.0000	−	10.0000i	0.192308	−	0.192308i
$53$	25.2982	+	25.2982i	0.477325	+	0.477325i	0.904275	−	0.426950i	$-0.140412\pi$
$53$	25.2982	+	25.2982i	0.477325	+	0.477325i	−0.426950	+	0.904275i	$0.640412\pi$
$54$		0			0
$55$	−25.0000	+	75.0000i	−0.454545	+	1.36364i
$56$	−47.4342			−0.847039
$57$		0			0
$58$	−75.0000	−	75.0000i	−1.29310	−	1.29310i
$59$			47.4342i			0.803969i	0.915646	+	0.401984i	$0.131679\pi$
$59$			47.4342i			0.803969i	−0.915646	+	0.401984i	$0.868321\pi$
$60$		0			0
$61$	−58.0000			−0.950820			−0.475410	−	0.879764i	$-0.657700\pi$
$61$	−58.0000			−0.950820			−0.475410	+	0.879764i	$0.657700\pi$
$62$	12.6491	−	12.6491i	0.204018	−	0.204018i
$63$		0			0
$64$			41.0000i			0.640625i
$65$	63.2456	−	31.6228i	0.973009	−	0.486504i
$66$		0			0
$67$	70.0000	−	70.0000i	1.04478	−	1.04478i	0.0458267	−	0.998949i	$-0.485408\pi$
$67$	70.0000	−	70.0000i	1.04478	−	1.04478i	0.998949	−	0.0458267i	$-0.0145922\pi$
$68$	3.16228	+	3.16228i	0.0465041	+	0.0465041i
$69$		0			0
$70$	75.0000	+	25.0000i	1.07143	+	0.357143i
$71$	−63.2456			−0.890782			−0.445391	−	0.895336i	$-0.646935\pi$
$71$	−63.2456			−0.890782			−0.445391	+	0.895336i	$0.646935\pi$
$72$		0			0
$73$	55.0000	+	55.0000i	0.753425	+	0.753425i	0.975117	−	0.221692i	$-0.0711580\pi$
$73$	55.0000	+	55.0000i	0.753425	+	0.753425i	−0.221692	+	0.975117i	$0.571158\pi$
$74$		−	31.6228i		−	0.427335i
$75$		0			0
$76$	−18.0000			−0.236842
$77$	79.0569	−	79.0569i	1.02671	−	1.02671i
$78$		0			0
$79$		−	12.0000i		−	0.151899i	−0.997112	−	0.0759494i	$-0.975801\pi$
$79$		−	12.0000i		−	0.151899i	0.997112	−	0.0759494i	$-0.0241987\pi$
$80$	30.0416	−	90.1249i	0.375520	−	1.12656i
$81$		0			0
$82$	50.0000	−	50.0000i	0.609756	−	0.609756i
$83$	53.7587	+	53.7587i	0.647695	+	0.647695i	0.952436	−	0.304740i	$-0.0985696\pi$
$83$	53.7587	+	53.7587i	0.647695	+	0.647695i	−0.304740	+	0.952436i	$0.598570\pi$
$84$		0			0
$85$	10.0000	+	20.0000i	0.117647	+	0.235294i
$86$	31.6228			0.367707
$87$		0			0
$88$	−75.0000	−	75.0000i	−0.852273	−	0.852273i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−100.000			−1.09890
$92$	−3.16228	+	3.16228i	−0.0343726	+	0.0343726i
$93$		0			0
$94$			130.000i			1.38298i
$95$	−85.3815	−	28.4605i	−0.898753	−	0.299584i
$96$		0			0
$97$	−5.00000	+	5.00000i	−0.0515464	+	0.0515464i	−0.732410	−	0.680864i	$-0.761605\pi$
$97$	−5.00000	+	5.00000i	−0.0515464	+	0.0515464i	0.680864	+	0.732410i	$0.261605\pi$
$98$	−1.58114	−	1.58114i	−0.0161341	−	0.0161341i
$99$		0			0
$100$	−15.0000	+	20.0000i	−0.150000	+	0.200000i
$101$	−15.8114			−0.156548			−0.0782742	−	0.996932i	$-0.524941\pi$
$101$	−15.8114			−0.156548			−0.0782742	+	0.996932i	$0.524941\pi$
$102$		0			0
$103$	−35.0000	−	35.0000i	−0.339806	−	0.339806i	0.516488	−	0.856294i	$-0.327239\pi$
$103$	−35.0000	−	35.0000i	−0.339806	−	0.339806i	−0.856294	+	0.516488i	$0.827239\pi$
$104$			94.8683i			0.912195i
$105$		0			0
$106$	80.0000			0.754717
$107$	−60.0833	+	60.0833i	−0.561526	+	0.561526i	−0.929741	−	0.368215i	$-0.879969\pi$
$107$	−60.0833	+	60.0833i	−0.561526	+	0.561526i	0.368215	+	0.929741i	$0.379969\pi$
$108$		0			0
$109$		−	162.000i		−	1.48624i	−0.669159	−	0.743119i	$-0.733345\pi$
$109$		−	162.000i		−	1.48624i	0.669159	−	0.743119i	$-0.266655\pi$
$110$	79.0569	+	158.114i	0.718699	+	1.43740i
$111$		0			0
$112$	−95.0000	+	95.0000i	−0.848214	+	0.848214i
$113$	−117.004	−	117.004i	−1.03544	−	1.03544i	−0.999349	−	0.0360874i	$-0.988511\pi$
$113$	−117.004	−	117.004i	−1.03544	−	1.03544i	−0.0360874	−	0.999349i	$-0.511489\pi$
$114$		0			0
$115$	−20.0000	+	10.0000i	−0.173913	+	0.0869565i
$116$	−47.4342			−0.408915
$117$		0			0
$118$	75.0000	+	75.0000i	0.635593	+	0.635593i
$119$		−	31.6228i		−	0.265738i
$120$		0			0
$121$	129.000			1.06612
$122$	−91.7061	+	91.7061i	−0.751689	+	0.751689i
$123$		0			0
$124$		−	8.00000i		−	0.0645161i
$125$	−102.774	+	71.1512i	−0.822192	+	0.569210i
$126$		0			0
$127$	55.0000	−	55.0000i	0.433071	−	0.433071i	−0.456601	−	0.889672i	$-0.650933\pi$
$127$	55.0000	−	55.0000i	0.433071	−	0.433071i	0.889672	+	0.456601i	$0.150933\pi$
$128$	109.099	+	109.099i	0.852333	+	0.852333i
$129$		0			0
$130$	50.0000	−	150.000i	0.384615	−	1.15385i
$131$	173.925			1.32767			0.663837	−	0.747877i	$-0.268927\pi$
$131$	173.925			1.32767			0.663837	+	0.747877i	$0.268927\pi$
$132$		0			0
$133$	90.0000	+	90.0000i	0.676692	+	0.676692i
$134$		−	221.359i		−	1.65194i
$135$		0			0
$136$	−30.0000			−0.220588
$137$	15.8114	−	15.8114i	0.115412	−	0.115412i	−0.647042	−	0.762454i	$-0.723994\pi$
$137$	15.8114	−	15.8114i	0.115412	−	0.115412i	0.762454	+	0.647042i	$0.223994\pi$
$138$		0			0
$139$			102.000i			0.733813i	0.930258	+	0.366906i	$0.119583\pi$
$139$			102.000i			0.733813i	−0.930258	+	0.366906i	$0.880417\pi$
$140$	31.6228	−	15.8114i	0.225877	−	0.112938i
$141$		0			0
$142$	−100.000	+	100.000i	−0.704225	+	0.704225i
$143$	−158.114	−	158.114i	−1.10569	−	1.10569i
$144$		0			0
$145$	−225.000	−	75.0000i	−1.55172	−	0.517241i
$146$	173.925			1.19127
$147$		0			0
$148$	−10.0000	−	10.0000i	−0.0675676	−	0.0675676i
$149$			47.4342i			0.318350i	0.987250	+	0.159175i	$0.0508834\pi$
$149$			47.4342i			0.318350i	−0.987250	+	0.159175i	$0.949117\pi$
$150$		0			0
$151$	−22.0000			−0.145695			−0.0728477	−	0.997343i	$-0.523209\pi$
$151$	−22.0000			−0.145695			−0.0728477	+	0.997343i	$0.523209\pi$
$152$	85.3815	−	85.3815i	0.561720	−	0.561720i
$153$		0			0
$154$		−	250.000i		−	1.62338i
$155$	12.6491	−	37.9473i	0.0816072	−	0.244821i
$156$		0			0
$157$	−200.000	+	200.000i	−1.27389	+	1.27389i	−0.329853	+	0.944032i	$0.606999\pi$
$157$	−200.000	+	200.000i	−1.27389	+	1.27389i	−0.944032	+	0.329853i	$0.893001\pi$
$158$	−18.9737	−	18.9737i	−0.120086	−	0.120086i
$159$		0			0
$160$	−35.0000	−	70.0000i	−0.218750	−	0.437500i
$161$	31.6228			0.196415
$162$		0			0
$163$	100.000	+	100.000i	0.613497	+	0.613497i	0.943856	−	0.330359i	$-0.107170\pi$
$163$	100.000	+	100.000i	0.613497	+	0.613497i	−0.330359	+	0.943856i	$0.607170\pi$
$164$		−	31.6228i		−	0.192822i
$165$		0			0
$166$	170.000			1.02410
$167$	148.627	−	148.627i	0.889982	−	0.889982i	−0.104539	−	0.994521i	$-0.533337\pi$
$167$	148.627	−	148.627i	0.889982	−	0.889982i	0.994521	+	0.104539i	$0.0333365\pi$
$168$		0			0
$169$			31.0000i			0.183432i
$170$	47.4342	+	15.8114i	0.279024	+	0.0930082i
$171$		0			0
$172$	10.0000	−	10.0000i	0.0581395	−	0.0581395i
$173$	110.680	+	110.680i	0.639767	+	0.639767i	0.950498	−	0.310731i	$-0.100574\pi$
$173$	110.680	+	110.680i	0.639767	+	0.639767i	−0.310731	+	0.950498i	$0.600574\pi$
$174$		0			0
$175$	175.000	−	25.0000i	1.00000	−	0.142857i
$176$	−300.416			−1.70691
$177$		0			0
$178$		0			0
$179$			142.302i			0.794986i	0.917605	+	0.397493i	$0.130120\pi$
$179$			142.302i			0.794986i	−0.917605	+	0.397493i	$0.869880\pi$
$180$		0			0
$181$	218.000			1.20442			0.602210	−	0.798338i	$-0.294287\pi$
$181$	218.000			1.20442			0.602210	+	0.798338i	$0.294287\pi$
$182$	−158.114	+	158.114i	−0.868758	+	0.868758i
$183$		0			0
$184$		−	30.0000i		−	0.163043i
$185$	−31.6228	−	63.2456i	−0.170934	−	0.341868i
$186$		0			0
$187$	50.0000	−	50.0000i	0.267380	−	0.267380i
$188$	41.1096	+	41.1096i	0.218668	+	0.218668i
$189$		0			0
$190$	−180.000	+	90.0000i	−0.947368	+	0.473684i
$191$	−158.114			−0.827821			−0.413911	−	0.910317i	$-0.635837\pi$
$191$	−158.114			−0.827821			−0.413911	+	0.910317i	$0.635837\pi$
$192$		0			0
$193$	−125.000	−	125.000i	−0.647668	−	0.647668i	0.304761	−	0.952429i	$-0.401424\pi$
$193$	−125.000	−	125.000i	−0.647668	−	0.647668i	−0.952429	+	0.304761i	$0.901424\pi$
$194$			15.8114i			0.0815020i
$195$		0			0
$196$	−1.00000			−0.00510204
$197$	−145.465	+	145.465i	−0.738400	+	0.738400i	−0.972268	−	0.233868i	$-0.924862\pi$
$197$	−145.465	+	145.465i	−0.738400	+	0.738400i	0.233868	+	0.972268i	$0.424862\pi$
$198$		0			0
$199$			18.0000i			0.0904523i	0.998977	+	0.0452261i	$0.0144008\pi$
$199$			18.0000i			0.0904523i	−0.998977	+	0.0452261i	$0.985599\pi$
$200$	−23.7171	−	166.020i	−0.118585	−	0.830098i
$201$		0			0
$202$	−25.0000	+	25.0000i	−0.123762	+	0.123762i
$203$	237.171	+	237.171i	1.16833	+	1.16833i
$204$		0			0
$205$	50.0000	−	150.000i	0.243902	−	0.731707i
$206$	−110.680			−0.537280
$207$		0			0
$208$	190.000	+	190.000i	0.913462	+	0.913462i
$209$			284.605i			1.36175i
$210$		0			0
$211$	−298.000			−1.41232			−0.706161	−	0.708051i	$-0.749575\pi$
$211$	−298.000			−1.41232			−0.706161	+	0.708051i	$0.749575\pi$
$212$	25.2982	−	25.2982i	0.119331	−	0.119331i
$213$		0			0
$214$			190.000i			0.887850i
$215$	63.2456	−	31.6228i	0.294165	−	0.147083i
$216$		0			0
$217$	−40.0000	+	40.0000i	−0.184332	+	0.184332i
$218$	−256.144	−	256.144i	−1.17497	−	1.17497i
$219$		0			0
$220$	75.0000	+	25.0000i	0.340909	+	0.113636i
$221$	−63.2456			−0.286179
$222$		0			0
$223$	−215.000	−	215.000i	−0.964126	−	0.964126i	0.0352529	−	0.999378i	$-0.488776\pi$
$223$	−215.000	−	215.000i	−0.964126	−	0.964126i	−0.999378	+	0.0352529i	$0.988776\pi$
$224$			110.680i			0.494106i
$225$		0			0
$226$	−370.000			−1.63717
$227$	271.956	−	271.956i	1.19804	−	1.19804i	0.223292	−	0.974752i	$-0.428320\pi$
$227$	271.956	−	271.956i	1.19804	−	1.19804i	0.974752	−	0.223292i	$-0.0716803\pi$
$228$		0			0
$229$		−	78.0000i		−	0.340611i	−0.985391	−	0.170306i	$-0.945524\pi$
$229$		−	78.0000i		−	0.340611i	0.985391	−	0.170306i	$-0.0544755\pi$
$230$	−15.8114	+	47.4342i	−0.0687452	+	0.206235i
$231$		0			0
$232$	225.000	−	225.000i	0.969828	−	0.969828i
$233$	110.680	+	110.680i	0.475020	+	0.475020i	0.903535	−	0.428515i	$-0.140963\pi$
$233$	110.680	+	110.680i	0.475020	+	0.475020i	−0.428515	+	0.903535i	$0.640963\pi$
$234$		0			0
$235$	130.000	+	260.000i	0.553191	+	1.10638i
$236$	47.4342			0.200992
$237$		0			0
$238$	−50.0000	−	50.0000i	−0.210084	−	0.210084i
$239$		−	379.473i		−	1.58775i	−0.608078	−	0.793877i	$-0.708059\pi$
$239$		−	379.473i		−	1.58775i	0.608078	−	0.793877i	$-0.291941\pi$
$240$		0			0
$241$	212.000			0.879668			0.439834	−	0.898079i	$-0.355037\pi$
$241$	212.000			0.879668			0.439834	+	0.898079i	$0.355037\pi$
$242$	203.967	−	203.967i	0.842838	−	0.842838i
$243$		0			0
$244$			58.0000i			0.237705i
$245$	−4.74342	−	1.58114i	−0.0193609	−	0.00645363i
$246$		0			0
$247$	180.000	−	180.000i	0.728745	−	0.728745i
$248$	37.9473	+	37.9473i	0.153013	+	0.153013i
$249$		0			0
$250$	−50.0000	+	275.000i	−0.200000	+	1.10000i
$251$	363.662			1.44885			0.724426	−	0.689352i	$-0.242105\pi$
$251$	363.662			1.44885			0.724426	+	0.689352i	$0.242105\pi$
$252$		0			0
$253$	50.0000	+	50.0000i	0.197628	+	0.197628i
$254$		−	173.925i		−	0.684745i
$255$		0			0
$256$	181.000			0.707031
$257$	−306.741	+	306.741i	−1.19354	+	1.19354i	−0.217480	+	0.976065i	$0.569784\pi$
$257$	−306.741	+	306.741i	−1.19354	+	1.19354i	−0.976065	+	0.217480i	$0.930216\pi$
$258$		0			0
$259$			100.000i			0.386100i
$260$	−31.6228	−	63.2456i	−0.121626	−	0.243252i
$261$		0			0
$262$	275.000	−	275.000i	1.04962	−	1.04962i
$263$	−287.767	−	287.767i	−1.09417	−	1.09417i	−0.995078	−	0.0990940i	$-0.968406\pi$
$263$	−287.767	−	287.767i	−1.09417	−	1.09417i	−0.0990940	−	0.995078i	$-0.531594\pi$
$264$		0			0
$265$	160.000	−	80.0000i	0.603774	−	0.301887i
$266$	284.605			1.06994
$267$		0			0
$268$	−70.0000	−	70.0000i	−0.261194	−	0.261194i
$269$		−	142.302i		−	0.529006i	−0.964385	−	0.264503i	$-0.914792\pi$
$269$		−	142.302i		−	0.529006i	0.964385	−	0.264503i	$-0.0852078\pi$
$270$		0			0
$271$	−178.000			−0.656827			−0.328413	−	0.944534i	$-0.606514\pi$
$271$	−178.000			−0.656827			−0.328413	+	0.944534i	$0.606514\pi$
$272$	−60.0833	+	60.0833i	−0.220894	+	0.220894i
$273$		0			0
$274$		−	50.0000i		−	0.182482i
$275$	316.228	+	237.171i	1.14992	+	0.862439i
$276$		0			0
$277$	−230.000	+	230.000i	−0.830325	+	0.830325i	−0.987561	−	0.157236i	$-0.949742\pi$
$277$	−230.000	+	230.000i	−0.830325	+	0.830325i	0.157236	+	0.987561i	$0.449742\pi$
$278$	161.276	+	161.276i	0.580130	+	0.580130i
$279$		0			0
$280$	−75.0000	+	225.000i	−0.267857	+	0.803571i
$281$	−158.114			−0.562683			−0.281341	−	0.959608i	$-0.590779\pi$
$281$	−158.114			−0.562683			−0.281341	+	0.959608i	$0.590779\pi$
$282$		0			0
$283$	−350.000	−	350.000i	−1.23675	−	1.23675i	−0.961322	−	0.275427i	$-0.911181\pi$
$283$	−350.000	−	350.000i	−1.23675	−	1.23675i	−0.275427	−	0.961322i	$-0.588819\pi$
$284$			63.2456i			0.222696i
$285$		0			0
$286$	−500.000			−1.74825
$287$	−158.114	+	158.114i	−0.550919	+	0.550919i
$288$		0			0
$289$			269.000i			0.930796i
$290$	−474.342	+	237.171i	−1.63566	+	0.817830i
$291$		0			0
$292$	55.0000	−	55.0000i	0.188356	−	0.188356i
$293$	−202.386	−	202.386i	−0.690736	−	0.690736i	0.271658	−	0.962394i	$-0.412428\pi$
$293$	−202.386	−	202.386i	−0.690736	−	0.690736i	−0.962394	+	0.271658i	$0.912428\pi$
$294$		0			0
$295$	225.000	+	75.0000i	0.762712	+	0.254237i
$296$	94.8683			0.320501
$297$		0			0
$298$	75.0000	+	75.0000i	0.251678	+	0.251678i
$299$		−	63.2456i		−	0.211524i
$300$		0			0
$301$	−100.000			−0.332226
$302$	−34.7851	+	34.7851i	−0.115182	+	0.115182i
$303$		0			0
$304$		−	342.000i		−	1.12500i
$305$	−91.7061	+	275.118i	−0.300676	+	0.902027i
$306$		0			0
$307$	190.000	−	190.000i	0.618893	−	0.618893i	−0.326355	−	0.945247i	$-0.605820\pi$
$307$	190.000	−	190.000i	0.618893	−	0.618893i	0.945247	+	0.326355i	$0.105820\pi$
$308$	−79.0569	−	79.0569i	−0.256678	−	0.256678i
$309$		0			0
$310$	−40.0000	−	80.0000i	−0.129032	−	0.258065i
$311$	−252.982			−0.813448			−0.406724	−	0.913551i	$-0.633329\pi$
$311$	−252.982			−0.813448			−0.406724	+	0.913551i	$0.633329\pi$
$312$		0			0
$313$	145.000	+	145.000i	0.463259	+	0.463259i	0.899722	−	0.436463i	$-0.143769\pi$
$313$	145.000	+	145.000i	0.463259	+	0.463259i	−0.436463	+	0.899722i	$0.643769\pi$
$314$			632.456i			2.01419i
$315$		0			0
$316$	−12.0000			−0.0379747
$317$	15.8114	−	15.8114i	0.0498782	−	0.0498782i	−0.681728	−	0.731606i	$-0.738771\pi$
$317$	15.8114	−	15.8114i	0.0498782	−	0.0498782i	0.731606	+	0.681728i	$0.238771\pi$
$318$		0			0
$319$			750.000i			2.35110i
$320$	194.480	+	64.8267i	0.607750	+	0.202583i
$321$		0			0
$322$	50.0000	−	50.0000i	0.155280	−	0.155280i
$323$	56.9210	+	56.9210i	0.176226	+	0.176226i
$324$		0			0
$325$	−50.0000	−	350.000i	−0.153846	−	1.07692i
$326$	316.228			0.970024
$327$		0			0
$328$	150.000	+	150.000i	0.457317	+	0.457317i
$329$		−	411.096i		−	1.24953i
$330$		0			0
$331$	482.000			1.45619			0.728097	−	0.685474i	$-0.240405\pi$
$331$	482.000			1.45619			0.728097	+	0.685474i	$0.240405\pi$
$332$	53.7587	−	53.7587i	0.161924	−	0.161924i
$333$		0			0
$334$		−	470.000i		−	1.40719i
$335$	−221.359	−	442.719i	−0.660774	−	1.32155i
$336$		0			0
$337$	−155.000	+	155.000i	−0.459941	+	0.459941i	−0.898636	−	0.438695i	$-0.855441\pi$
$337$	−155.000	+	155.000i	−0.459941	+	0.459941i	0.438695	+	0.898636i	$0.355441\pi$
$338$	49.0153	+	49.0153i	0.145016	+	0.145016i
$339$		0			0
$340$	20.0000	−	10.0000i	0.0588235	−	0.0294118i
$341$	−126.491			−0.370942
$342$		0			0
$343$	−240.000	−	240.000i	−0.699708	−	0.699708i
$344$			94.8683i			0.275780i
$345$		0			0
$346$	350.000			1.01156
$347$	−164.438	+	164.438i	−0.473886	+	0.473886i	−0.903170	−	0.429284i	$-0.858766\pi$
$347$	−164.438	+	164.438i	−0.473886	+	0.473886i	0.429284	+	0.903170i	$0.358766\pi$
$348$		0			0
$349$		−	318.000i		−	0.911175i	−0.890191	−	0.455587i	$-0.849429\pi$
$349$		−	318.000i		−	0.911175i	0.890191	−	0.455587i	$-0.150571\pi$
$350$	237.171	−	316.228i	0.677631	−	0.903508i
$351$		0			0
$352$	−175.000	+	175.000i	−0.497159	+	0.497159i
$353$	224.522	+	224.522i	0.636039	+	0.636039i	0.949576	−	0.313537i	$-0.101514\pi$
$353$	224.522	+	224.522i	0.636039	+	0.636039i	−0.313537	+	0.949576i	$0.601514\pi$
$354$		0			0
$355$	−100.000	+	300.000i	−0.281690	+	0.845070i
$356$		0			0
$357$		0			0
$358$	225.000	+	225.000i	0.628492	+	0.628492i
$359$		−	284.605i		−	0.792772i	−0.918084	−	0.396386i	$-0.870264\pi$
$359$		−	284.605i		−	0.792772i	0.918084	−	0.396386i	$-0.129736\pi$
$360$		0			0
$361$	37.0000			0.102493
$362$	344.688	−	344.688i	0.952178	−	0.952178i
$363$		0			0
$364$			100.000i			0.274725i
$365$	347.851	−	173.925i	0.953015	−	0.476508i
$366$		0			0
$367$	−185.000	+	185.000i	−0.504087	+	0.504087i	−0.912705	−	0.408618i	$-0.866011\pi$
$367$	−185.000	+	185.000i	−0.504087	+	0.504087i	0.408618	+	0.912705i	$0.366011\pi$
$368$	−60.0833	−	60.0833i	−0.163270	−	0.163270i
$369$		0			0
$370$	−150.000	−	50.0000i	−0.405405	−	0.135135i
$371$	−252.982			−0.681893
$372$		0			0
$373$	100.000	+	100.000i	0.268097	+	0.268097i	0.828333	−	0.560236i	$-0.189290\pi$
$373$	100.000	+	100.000i	0.268097	+	0.268097i	−0.560236	+	0.828333i	$0.689290\pi$
$374$		−	158.114i		−	0.422764i
$375$		0			0
$376$	−390.000			−1.03723
$377$	474.342	−	474.342i	1.25820	−	1.25820i
$378$		0			0
$379$		−	558.000i		−	1.47230i	−0.676821	−	0.736148i	$-0.736643\pi$
$379$		−	558.000i		−	1.47230i	0.676821	−	0.736148i	$-0.263357\pi$
$380$	−28.4605	+	85.3815i	−0.0748960	+	0.224688i
$381$		0			0
$382$	−250.000	+	250.000i	−0.654450	+	0.654450i
$383$	281.443	+	281.443i	0.734837	+	0.734837i	0.971574	−	0.236737i	$-0.0760778\pi$
$383$	281.443	+	281.443i	0.734837	+	0.734837i	−0.236737	+	0.971574i	$0.576078\pi$
$384$		0			0
$385$	−250.000	−	500.000i	−0.649351	−	1.29870i
$386$	−395.285			−1.02405
$387$		0			0
$388$	5.00000	+	5.00000i	0.0128866	+	0.0128866i
$389$			521.776i			1.34133i	0.741762	+	0.670663i	$0.233990\pi$
$389$			521.776i			1.34133i	−0.741762	+	0.670663i	$0.766010\pi$
$390$		0			0
$391$	20.0000			0.0511509
$392$	4.74342	−	4.74342i	0.0121006	−	0.0121006i
$393$		0			0
$394$			460.000i			1.16751i
$395$	−56.9210	−	18.9737i	−0.144104	−	0.0480346i
$396$		0			0
$397$	−260.000	+	260.000i	−0.654912	+	0.654912i	−0.954172	−	0.299260i	$-0.903260\pi$
$397$	−260.000	+	260.000i	−0.654912	+	0.654912i	0.299260	+	0.954172i	$0.403260\pi$
$398$	28.4605	+	28.4605i	0.0715088	+	0.0715088i
$399$		0			0
$400$	−380.000	−	285.000i	−0.950000	−	0.712500i
$401$	−252.982			−0.630878			−0.315439	−	0.948946i	$-0.602152\pi$
$401$	−252.982			−0.630878			−0.315439	+	0.948946i	$0.602152\pi$
$402$		0			0
$403$	80.0000	+	80.0000i	0.198511	+	0.198511i
$404$			15.8114i			0.0391371i
$405$		0			0
$406$	750.000			1.84729
$407$	−158.114	+	158.114i	−0.388486	+	0.388486i
$408$		0			0
$409$		−	348.000i		−	0.850856i	−0.904992	−	0.425428i	$-0.860124\pi$
$409$		−	348.000i		−	0.850856i	0.904992	−	0.425428i	$-0.139876\pi$
$410$	−158.114	−	316.228i	−0.385644	−	0.771287i
$411$		0			0
$412$	−35.0000	+	35.0000i	−0.0849515	+	0.0849515i
$413$	−237.171	−	237.171i	−0.574263	−	0.574263i
$414$		0			0
$415$	340.000	−	170.000i	0.819277	−	0.409639i
$416$	221.359			0.532114
$417$		0			0
$418$	450.000	+	450.000i	1.07656	+	1.07656i
$419$			616.644i			1.47170i	0.677142	+	0.735852i	$0.263218\pi$
$419$			616.644i			1.47170i	−0.677142	+	0.735852i	$0.736782\pi$
$420$		0			0
$421$	338.000			0.802850			0.401425	−	0.915892i	$-0.368515\pi$
$421$	338.000			0.802850			0.401425	+	0.915892i	$0.368515\pi$
$422$	−471.179	+	471.179i	−1.11654	+	1.11654i
$423$		0			0
$424$			240.000i			0.566038i
$425$	110.680	−	15.8114i	0.260423	−	0.0372033i
$426$		0			0
$427$	290.000	−	290.000i	0.679157	−	0.679157i
$428$	60.0833	+	60.0833i	0.140381	+	0.140381i
$429$		0			0
$430$	50.0000	−	150.000i	0.116279	−	0.348837i
$431$	221.359			0.513595			0.256797	−	0.966465i	$-0.417333\pi$
$431$	221.359			0.513595			0.256797	+	0.966465i	$0.417333\pi$
$432$		0			0
$433$	145.000	+	145.000i	0.334873	+	0.334873i	0.854434	−	0.519561i	$-0.173904\pi$
$433$	145.000	+	145.000i	0.334873	+	0.334873i	−0.519561	+	0.854434i	$0.673904\pi$
$434$			126.491i			0.291454i
$435$		0			0
$436$	−162.000			−0.371560
$437$	−56.9210	+	56.9210i	−0.130254	+	0.130254i
$438$		0			0
$439$			78.0000i			0.177677i	0.996046	+	0.0888383i	$0.0283154\pi$
$439$			78.0000i			0.177677i	−0.996046	+	0.0888383i	$0.971685\pi$
$440$	−474.342	+	237.171i	−1.07805	+	0.539025i
$441$		0			0
$442$	−100.000	+	100.000i	−0.226244	+	0.226244i
$443$	196.061	+	196.061i	0.442576	+	0.442576i	0.892877	−	0.450301i	$-0.148683\pi$
$443$	196.061	+	196.061i	0.442576	+	0.442576i	−0.450301	+	0.892877i	$0.648683\pi$
$444$		0			0
$445$		0			0
$446$	−679.890			−1.52442
$447$		0			0
$448$	−205.000	−	205.000i	−0.457589	−	0.457589i
$449$		−	284.605i		−	0.633864i	−0.948448	−	0.316932i	$-0.897347\pi$
$449$		−	284.605i		−	0.633864i	0.948448	−	0.316932i	$-0.102653\pi$
$450$		0			0
$451$	−500.000			−1.10865
$452$	−117.004	+	117.004i	−0.258859	+	0.258859i
$453$		0			0
$454$		−	860.000i		−	1.89427i
$455$	−158.114	+	474.342i	−0.347503	+	1.04251i
$456$		0			0
$457$	−365.000	+	365.000i	−0.798687	+	0.798687i	−0.982889	−	0.184201i	$-0.941030\pi$
$457$	−365.000	+	365.000i	−0.798687	+	0.798687i	0.184201	+	0.982889i	$0.441030\pi$
$458$	−123.329	−	123.329i	−0.269277	−	0.269277i
$459$		0			0
$460$	10.0000	+	20.0000i	0.0217391	+	0.0434783i
$461$	838.004			1.81780			0.908898	−	0.417019i	$-0.136925\pi$
$461$	838.004			1.81780			0.908898	+	0.417019i	$0.136925\pi$
$462$		0			0
$463$	−35.0000	−	35.0000i	−0.0755940	−	0.0755940i	0.668299	−	0.743893i	$-0.267023\pi$
$463$	−35.0000	−	35.0000i	−0.0755940	−	0.0755940i	−0.743893	+	0.668299i	$0.767023\pi$
$464$		−	901.249i		−	1.94235i
$465$		0			0
$466$	350.000			0.751073
$467$	−117.004	+	117.004i	−0.250544	+	0.250544i	−0.821194	−	0.570649i	$-0.806692\pi$
$467$	−117.004	+	117.004i	−0.250544	+	0.250544i	0.570649	+	0.821194i	$0.306692\pi$
$468$		0			0
$469$			700.000i			1.49254i
$470$	616.644	+	205.548i	1.31201	+	0.437336i
$471$		0			0
$472$	−225.000	+	225.000i	−0.476695	+	0.476695i
$473$	−158.114	−	158.114i	−0.334279	−	0.334279i
$474$		0			0
$475$	−270.000	+	360.000i	−0.568421	+	0.757895i
$476$	−31.6228			−0.0664344
$477$		0			0
$478$	−600.000	−	600.000i	−1.25523	−	1.25523i
$479$			379.473i			0.792220i	0.918203	+	0.396110i	$0.129640\pi$
$479$			379.473i			0.792220i	−0.918203	+	0.396110i	$0.870360\pi$
$480$		0			0
$481$	200.000			0.415800
$482$	335.201	−	335.201i	0.695439	−	0.695439i
$483$		0			0
$484$		−	129.000i		−	0.266529i
$485$	15.8114	+	31.6228i	0.0326008	+	0.0652016i
$486$		0			0
$487$	−125.000	+	125.000i	−0.256674	+	0.256674i	−0.823700	−	0.567026i	$-0.808094\pi$
$487$	−125.000	+	125.000i	−0.256674	+	0.256674i	0.567026	+	0.823700i	$0.308094\pi$
$488$	−275.118	−	275.118i	−0.563767	−	0.563767i
$489$		0			0
$490$	−10.0000	+	5.00000i	−0.0204082	+	0.0102041i
$491$	458.530			0.933870			0.466935	−	0.884292i	$-0.345358\pi$
$491$	458.530			0.933870			0.466935	+	0.884292i	$0.345358\pi$
$492$		0			0
$493$	150.000	+	150.000i	0.304260	+	0.304260i
$494$		−	569.210i		−	1.15225i
$495$		0			0
$496$	152.000			0.306452
$497$	316.228	−	316.228i	0.636273	−	0.636273i
$498$		0			0
$499$		−	222.000i		−	0.444890i	−0.974945	−	0.222445i	$-0.928596\pi$
$499$		−	222.000i		−	0.444890i	0.974945	−	0.222445i	$-0.0714037\pi$
$500$	71.1512	+	102.774i	0.142302	+	0.205548i
$501$		0			0
$502$	575.000	−	575.000i	1.14542	−	1.14542i
$503$	−458.530	−	458.530i	−0.911591	−	0.911591i	0.0848065	−	0.996397i	$-0.472973\pi$
$503$	−458.530	−	458.530i	−0.911591	−	0.911591i	−0.996397	+	0.0848065i	$0.972973\pi$
$504$		0			0
$505$	−25.0000	+	75.0000i	−0.0495050	+	0.148515i
$506$	158.114			0.312478
$507$		0			0
$508$	−55.0000	−	55.0000i	−0.108268	−	0.108268i
$509$		−	237.171i		−	0.465954i	−0.972482	−	0.232977i	$-0.925153\pi$
$509$		−	237.171i		−	0.465954i	0.972482	−	0.232977i	$-0.0748468\pi$
$510$		0			0
$511$	−550.000			−1.07632
$512$	−150.208	+	150.208i	−0.293375	+	0.293375i
$513$		0			0
$514$			970.000i			1.88716i
$515$	−221.359	+	110.680i	−0.429824	+	0.214912i
$516$		0			0
$517$	650.000	−	650.000i	1.25725	−	1.25725i
$518$	158.114	+	158.114i	0.305239	+	0.305239i
$519$		0			0
$520$	450.000	+	150.000i	0.865385	+	0.288462i
$521$	790.569			1.51741			0.758704	−	0.651436i	$-0.225833\pi$
$521$	790.569			1.51741			0.758704	+	0.651436i	$0.225833\pi$
$522$		0			0
$523$	370.000	+	370.000i	0.707457	+	0.707457i	0.966000	−	0.258543i	$-0.0832423\pi$
$523$	370.000	+	370.000i	0.707457	+	0.707457i	−0.258543	+	0.966000i	$0.583242\pi$
$524$		−	173.925i		−	0.331918i
$525$		0			0
$526$	−910.000			−1.73004
$527$	−25.2982	+	25.2982i	−0.0480042	+	0.0480042i
$528$		0			0
$529$		−	509.000i		−	0.962193i
$530$	126.491	−	379.473i	0.238662	−	0.715987i
$531$		0			0
$532$	90.0000	−	90.0000i	0.169173	−	0.169173i
$533$	316.228	+	316.228i	0.593298	+	0.593298i
$534$		0			0
$535$	190.000	+	380.000i	0.355140	+	0.710280i
$536$	664.078			1.23895
$537$		0			0
$538$	−225.000	−	225.000i	−0.418216	−	0.418216i
$539$			15.8114i			0.0293347i
$540$		0			0
$541$	362.000			0.669131			0.334566	−	0.942372i	$-0.391410\pi$
$541$	362.000			0.669131			0.334566	+	0.942372i	$0.391410\pi$
$542$	−281.443	+	281.443i	−0.519267	+	0.519267i
$543$		0			0
$544$			70.0000i			0.128676i
$545$	−768.433	−	256.144i	−1.40997	−	0.469990i
$546$		0			0
$547$	490.000	−	490.000i	0.895795	−	0.895795i	−0.0992657	−	0.995061i	$-0.531649\pi$
$547$	490.000	−	490.000i	0.895795	−	0.895795i	0.995061	+	0.0992657i	$0.0316494\pi$
$548$	−15.8114	−	15.8114i	−0.0288529	−	0.0288529i
$549$		0			0
$550$	875.000	−	125.000i	1.59091	−	0.227273i
$551$	−853.815			−1.54957
$552$		0			0
$553$	60.0000	+	60.0000i	0.108499	+	0.108499i
$554$			727.324i			1.31286i
$555$		0			0
$556$	102.000			0.183453
$557$	252.982	−	252.982i	0.454187	−	0.454187i	−0.442555	−	0.896742i	$-0.645928\pi$
$557$	252.982	−	252.982i	0.454187	−	0.454187i	0.896742	+	0.442555i	$0.145928\pi$
$558$		0			0
$559$			200.000i			0.357782i
$560$	300.416	+	600.833i	0.536458	+	1.07292i
$561$		0			0
$562$	−250.000	+	250.000i	−0.444840	+	0.444840i
$563$	−202.386	−	202.386i	−0.359477	−	0.359477i	0.504143	−	0.863620i	$-0.331808\pi$
$563$	−202.386	−	202.386i	−0.359477	−	0.359477i	−0.863620	+	0.504143i	$0.831808\pi$
$564$		0			0
$565$	−740.000	+	370.000i	−1.30973	+	0.654867i
$566$	−1106.80			−1.95547
$567$		0			0
$568$	−300.000	−	300.000i	−0.528169	−	0.528169i
$569$		−	189.737i		−	0.333456i	−0.986003	−	0.166728i	$-0.946680\pi$
$569$		−	189.737i		−	0.333456i	0.986003	−	0.166728i	$-0.0533202\pi$
$570$		0			0
$571$	−658.000			−1.15236			−0.576182	−	0.817321i	$-0.695458\pi$
$571$	−658.000			−1.15236			−0.576182	+	0.817321i	$0.695458\pi$
$572$	−158.114	+	158.114i	−0.276423	+	0.276423i
$573$		0			0
$574$			500.000i			0.871080i
$575$	15.8114	+	110.680i	0.0274981	+	0.192486i
$576$		0			0
$577$	−35.0000	+	35.0000i	−0.0606586	+	0.0606586i	−0.736785	−	0.676127i	$-0.763657\pi$
$577$	−35.0000	+	35.0000i	−0.0606586	+	0.0606586i	0.676127	+	0.736785i	$0.263657\pi$
$578$	425.326	+	425.326i	0.735859	+	0.735859i
$579$		0			0
$580$	−75.0000	+	225.000i	−0.129310	+	0.387931i
$581$	−537.587			−0.925279
$582$		0			0
$583$	−400.000	−	400.000i	−0.686106	−	0.686106i
$584$			521.776i			0.893452i
$585$		0			0
$586$	−640.000			−1.09215
$587$	−154.952	+	154.952i	−0.263972	+	0.263972i	−0.826666	−	0.562694i	$-0.809765\pi$
$587$	−154.952	+	154.952i	−0.263972	+	0.263972i	0.562694	+	0.826666i	$0.309765\pi$
$588$		0			0
$589$		−	144.000i		−	0.244482i
$590$	474.342	−	237.171i	0.803969	−	0.401984i
$591$		0			0
$592$	190.000	−	190.000i	0.320946	−	0.320946i
$593$	167.601	+	167.601i	0.282632	+	0.282632i	0.834158	−	0.551526i	$-0.185954\pi$
$593$	167.601	+	167.601i	0.282632	+	0.282632i	−0.551526	+	0.834158i	$0.685954\pi$
$594$		0			0
$595$	−150.000	−	50.0000i	−0.252101	−	0.0840336i
$596$	47.4342			0.0795875
$597$		0			0
$598$	−100.000	−	100.000i	−0.167224	−	0.167224i
$599$			1043.55i			1.74216i	0.491144	+	0.871078i	$0.336579\pi$
$599$			1043.55i			1.74216i	−0.491144	+	0.871078i	$0.663421\pi$
$600$		0			0
$601$	−382.000			−0.635607			−0.317804	−	0.948157i	$-0.602945\pi$
$601$	−382.000			−0.635607			−0.317804	+	0.948157i	$0.602945\pi$
$602$	−158.114	+	158.114i	−0.262648	+	0.262648i
$603$		0			0
$604$			22.0000i			0.0364238i
$605$	203.967	−	611.901i	0.337135	−	1.01141i
$606$		0			0
$607$	655.000	−	655.000i	1.07908	−	1.07908i	0.0824851	−	0.996592i	$-0.473714\pi$
$607$	655.000	−	655.000i	1.07908	−	1.07908i	0.996592	−	0.0824851i	$-0.0262857\pi$
$608$	−199.223	−	199.223i	−0.327670	−	0.327670i
$609$		0			0
$610$	290.000	+	580.000i	0.475410	+	0.950820i
$611$	−822.192			−1.34565
$612$		0			0
$613$	−620.000	−	620.000i	−1.01142	−	1.01142i	−0.999934	−	0.0114852i	$-0.996344\pi$
$613$	−620.000	−	620.000i	−1.01142	−	1.01142i	−0.0114852	−	0.999934i	$-0.503656\pi$
$614$		−	600.833i		−	0.978555i
$615$		0			0
$616$	750.000			1.21753
$617$	34.7851	−	34.7851i	0.0563777	−	0.0563777i	−0.678356	−	0.734734i	$-0.737307\pi$
$617$	34.7851	−	34.7851i	0.0563777	−	0.0563777i	0.734734	+	0.678356i	$0.237307\pi$
$618$		0			0
$619$			258.000i			0.416801i	0.978044	+	0.208401i	$0.0668258\pi$
$619$			258.000i			0.416801i	−0.978044	+	0.208401i	$0.933174\pi$
$620$	−37.9473	−	12.6491i	−0.0612054	−	0.0204018i
$621$		0			0
$622$	−400.000	+	400.000i	−0.643087	+	0.643087i
$623$		0			0
$624$		0			0
$625$	175.000	+	600.000i	0.280000	+	0.960000i
$626$	458.530			0.732476
$627$		0			0
$628$	200.000	+	200.000i	0.318471	+	0.318471i
$629$			63.2456i			0.100549i
$630$		0			0
$631$	812.000			1.28685			0.643423	−	0.765511i	$-0.277514\pi$
$631$	812.000			1.28685			0.643423	+	0.765511i	$0.277514\pi$
$632$	56.9210	−	56.9210i	0.0900649	−	0.0900649i
$633$		0			0
$634$		−	50.0000i		−	0.0788644i
$635$	−173.925	−	347.851i	−0.273898	−	0.547796i
$636$		0			0
$637$	10.0000	−	10.0000i	0.0156986	−	0.0156986i
$638$	1185.85	+	1185.85i	1.85871	+	1.85871i
$639$		0			0
$640$	690.000	−	345.000i	1.07812	−	0.539062i
$641$	−442.719			−0.690669			−0.345335	−	0.938480i	$-0.612235\pi$
$641$	−442.719			−0.690669			−0.345335	+	0.938480i	$0.612235\pi$
$642$		0			0
$643$	820.000	+	820.000i	1.27527	+	1.27527i	0.943283	+	0.331989i	$0.107720\pi$
$643$	820.000	+	820.000i	1.27527	+	1.27527i	0.331989	+	0.943283i	$0.392280\pi$
$644$		−	31.6228i		−	0.0491037i
$645$		0			0
$646$	180.000			0.278638
$647$	679.890	−	679.890i	1.05083	−	1.05083i	0.0521974	−	0.998637i	$-0.483378\pi$
$647$	679.890	−	679.890i	1.05083	−	1.05083i	0.998637	−	0.0521974i	$-0.0166225\pi$
$648$		0			0
$649$		−	750.000i		−	1.15562i
$650$	−632.456	−	474.342i	−0.973009	−	0.729756i
$651$		0			0
$652$	100.000	−	100.000i	0.153374	−	0.153374i
$653$	−515.451	−	515.451i	−0.789359	−	0.789359i	0.192030	−	0.981389i	$-0.438493\pi$
$653$	−515.451	−	515.451i	−0.789359	−	0.789359i	−0.981389	+	0.192030i	$0.938493\pi$
$654$		0			0
$655$	275.000	−	825.000i	0.419847	−	1.25954i
$656$	600.833			0.915904
$657$		0			0
$658$	−650.000	−	650.000i	−0.987842	−	0.987842i
$659$		−	901.249i		−	1.36760i	−0.729669	−	0.683801i	$-0.760326\pi$
$659$		−	901.249i		−	1.36760i	0.729669	−	0.683801i	$-0.239674\pi$
$660$		0			0
$661$	−802.000			−1.21331			−0.606657	−	0.794964i	$-0.707490\pi$
$661$	−802.000			−1.21331			−0.606657	+	0.794964i	$0.707490\pi$
$662$	762.109	−	762.109i	1.15122	−	1.15122i
$663$		0			0
$664$			510.000i			0.768072i
$665$	569.210	−	284.605i	0.855955	−	0.427977i
$666$		0			0
$667$	−150.000	+	150.000i	−0.224888	+	0.224888i
$668$	−148.627	−	148.627i	−0.222496	−	0.222496i
$669$		0			0
$670$	−1050.00	−	350.000i	−1.56716	−	0.522388i
$671$	917.061			1.36671
$672$		0			0
$673$	775.000	+	775.000i	1.15156	+	1.15156i	0.986240	+	0.165320i	$0.0528658\pi$
$673$	775.000	+	775.000i	1.15156	+	1.15156i	0.165320	+	0.986240i	$0.447134\pi$
$674$			490.153i			0.727230i
$675$		0			0
$676$	31.0000			0.0458580
$677$	−496.478	+	496.478i	−0.733349	+	0.733349i	−0.971282	−	0.237932i	$-0.923530\pi$
$677$	−496.478	+	496.478i	−0.733349	+	0.733349i	0.237932	+	0.971282i	$0.423530\pi$
$678$		0			0
$679$		−	50.0000i		−	0.0736377i
$680$	−47.4342	+	142.302i	−0.0697561	+	0.209268i
$681$		0			0
$682$	−200.000	+	200.000i	−0.293255	+	0.293255i
$683$	−60.0833	−	60.0833i	−0.0879697	−	0.0879697i	0.661753	−	0.749722i	$-0.269813\pi$
$683$	−60.0833	−	60.0833i	−0.0879697	−	0.0879697i	−0.749722	+	0.661753i	$0.769813\pi$
$684$		0			0
$685$	−50.0000	−	100.000i	−0.0729927	−	0.145985i
$686$	−758.947			−1.10634
$687$		0			0
$688$	190.000	+	190.000i	0.276163	+	0.276163i
$689$			505.964i			0.734346i
$690$		0			0
$691$	−562.000			−0.813314			−0.406657	−	0.913581i	$-0.633306\pi$
$691$	−562.000			−0.813314			−0.406657	+	0.913581i	$0.633306\pi$
$692$	110.680	−	110.680i	0.159942	−	0.159942i
$693$		0			0
$694$			520.000i			0.749280i
$695$	483.828	+	161.276i	0.696156	+	0.232052i
$696$		0			0
$697$	−100.000	+	100.000i	−0.143472	+	0.143472i
$698$	−502.802	−	502.802i	−0.720347	−	0.720347i
$699$		0			0
$700$	−25.0000	−	175.000i	−0.0357143	−	0.250000i
$701$	363.662			0.518776			0.259388	−	0.965773i	$-0.416479\pi$
$701$	363.662			0.518776			0.259388	+	0.965773i	$0.416479\pi$
$702$		0			0
$703$	−180.000	−	180.000i	−0.256046	−	0.256046i
$704$		−	648.267i		−	0.920834i
$705$		0			0
$706$	710.000			1.00567
$707$	79.0569	−	79.0569i	0.111820	−	0.111820i
$708$		0			0
$709$		−	498.000i		−	0.702398i	−0.936301	−	0.351199i	$-0.885774\pi$
$709$		−	498.000i		−	0.702398i	0.936301	−	0.351199i	$-0.114226\pi$
$710$	316.228	+	632.456i	0.445391	+	0.890782i
$711$		0			0
$712$		0			0
$713$	−25.2982	−	25.2982i	−0.0354814	−	0.0354814i
$714$		0			0
$715$	−1000.00	+	500.000i	−1.39860	+	0.699301i
$716$	142.302			0.198747
$717$		0			0
$718$	−450.000	−	450.000i	−0.626741	−	0.626741i
$719$		−	569.210i		−	0.791669i	−0.918322	−	0.395834i	$-0.870455\pi$
$719$		−	569.210i		−	0.791669i	0.918322	−	0.395834i	$-0.129545\pi$
$720$		0			0
$721$	350.000			0.485437
$722$	58.5021	−	58.5021i	0.0810279	−	0.0810279i
$723$		0			0
$724$		−	218.000i		−	0.301105i
$725$	−711.512	+	948.683i	−0.981397	+	1.30853i
$726$		0			0
$727$	−905.000	+	905.000i	−1.24484	+	1.24484i	−0.286873	+	0.957969i	$0.592616\pi$
$727$	−905.000	+	905.000i	−1.24484	+	1.24484i	−0.957969	+	0.286873i	$0.907384\pi$
$728$	−474.342	−	474.342i	−0.651568	−	0.651568i
$729$		0			0
$730$	275.000	−	825.000i	0.376712	−	1.13014i
$731$	−63.2456			−0.0865192
$732$		0			0
$733$	550.000	+	550.000i	0.750341	+	0.750341i	0.974543	−	0.224202i	$-0.0719774\pi$
$733$	550.000	+	550.000i	0.750341	+	0.750341i	−0.224202	+	0.974543i	$0.571977\pi$
$734$			585.021i			0.797032i
$735$		0			0
$736$	−70.0000			−0.0951087
$737$	−1106.80	+	1106.80i	−1.50176	+	1.50176i
$738$		0			0
$739$		−	198.000i		−	0.267930i	−0.990986	−	0.133965i	$-0.957229\pi$
$739$		−	198.000i		−	0.267930i	0.990986	−	0.133965i	$-0.0427709\pi$
$740$	−63.2456	+	31.6228i	−0.0854670	+	0.0427335i
$741$		0			0
$742$	−400.000	+	400.000i	−0.539084	+	0.539084i
$743$	964.495	+	964.495i	1.29811	+	1.29811i	0.929640	+	0.368468i	$0.120118\pi$
$743$	964.495	+	964.495i	1.29811	+	1.29811i	0.368468	+	0.929640i	$0.379882\pi$
$744$		0			0
$745$	225.000	+	75.0000i	0.302013	+	0.100671i
$746$	316.228			0.423898
$747$		0			0
$748$	−50.0000	−	50.0000i	−0.0668449	−	0.0668449i
$749$		−	600.833i		−	0.802180i
$750$		0			0
$751$	728.000			0.969374			0.484687	−	0.874688i	$-0.338933\pi$
$751$	728.000			0.969374			0.484687	+	0.874688i	$0.338933\pi$
$752$	−781.083	+	781.083i	−1.03867	+	1.03867i
$753$		0			0
$754$		−	1500.00i		−	1.98939i
$755$	−34.7851	+	104.355i	−0.0460729	+	0.138219i
$756$		0			0
$757$	−170.000	+	170.000i	−0.224571	+	0.224571i	−0.810420	−	0.585849i	$-0.800761\pi$
$757$	−170.000	+	170.000i	−0.224571	+	0.224571i	0.585849	+	0.810420i	$0.300761\pi$
$758$	−882.275	−	882.275i	−1.16395	−	1.16395i
$759$		0			0
$760$	−270.000	−	540.000i	−0.355263	−	0.710526i
$761$	31.6228			0.0415542			0.0207771	−	0.999784i	$-0.493386\pi$
$761$	31.6228			0.0415542			0.0207771	+	0.999784i	$0.493386\pi$
$762$		0			0
$763$	810.000	+	810.000i	1.06160	+	1.06160i
$764$			158.114i			0.206955i
$765$		0			0
$766$	890.000			1.16188
$767$	−474.342	+	474.342i	−0.618438	+	0.618438i
$768$		0			0
$769$			318.000i			0.413524i	0.978391	+	0.206762i	$0.0662926\pi$
$769$			318.000i			0.413524i	−0.978391	+	0.206762i	$0.933707\pi$
$770$	−1185.85	−	395.285i	−1.54007	−	0.513357i
$771$		0			0
$772$	−125.000	+	125.000i	−0.161917	+	0.161917i
$773$	452.206	+	452.206i	0.585001	+	0.585001i	0.936273	−	0.351272i	$-0.114251\pi$
$773$	452.206	+	452.206i	0.585001	+	0.585001i	−0.351272	+	0.936273i	$0.614251\pi$
$774$		0			0
$775$	−160.000	−	120.000i	−0.206452	−	0.154839i
$776$	−47.4342			−0.0611265
$777$		0			0
$778$	825.000	+	825.000i	1.06041	+	1.06041i
$779$		−	569.210i		−	0.730693i
$780$		0			0
$781$	1000.00			1.28041
$782$	31.6228	−	31.6228i	0.0404383	−	0.0404383i

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	45.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.58114 - 1.58114i) q^{2} -1.00000i q^{4} +(1.58114 - 4.74342i) q^{5} +(-5.00000 + 5.00000i) q^{7} +(4.74342 + 4.74342i) q^{8} +O(q^{10})\)	\(q+(1.58114 - 1.58114i) q^{2} -1.00000i q^{4} +(1.58114 - 4.74342i) q^{5} +(-5.00000 + 5.00000i) q^{7} +(4.74342 + 4.74342i) q^{8} +(-5.00000 - 10.0000i) q^{10} -15.8114 q^{11} +(10.0000 + 10.0000i) q^{13} +15.8114i q^{14} +19.0000 q^{16} +(-3.16228 + 3.16228i) q^{17} -18.0000i q^{19} +(-4.74342 - 1.58114i) q^{20} +(-25.0000 + 25.0000i) q^{22} +(-3.16228 - 3.16228i) q^{23} +(-20.0000 - 15.0000i) q^{25} +31.6228 q^{26} +(5.00000 + 5.00000i) q^{28} -47.4342i q^{29} +8.00000 q^{31} +(11.0680 - 11.0680i) q^{32} +10.0000i q^{34} +(15.8114 + 31.6228i) q^{35} +(10.0000 - 10.0000i) q^{37} +(-28.4605 - 28.4605i) q^{38} +(30.0000 - 15.0000i) q^{40} +31.6228 q^{41} +(10.0000 + 10.0000i) q^{43} +15.8114i q^{44} -10.0000 q^{46} +(-41.1096 + 41.1096i) q^{47} -1.00000i q^{49} +(-55.3399 + 7.90569i) q^{50} +(10.0000 - 10.0000i) q^{52} +(25.2982 + 25.2982i) q^{53} +(-25.0000 + 75.0000i) q^{55} -47.4342 q^{56} +(-75.0000 - 75.0000i) q^{58} +47.4342i q^{59} -58.0000 q^{61} +(12.6491 - 12.6491i) q^{62} +41.0000i q^{64} +(63.2456 - 31.6228i) q^{65} +(70.0000 - 70.0000i) q^{67} +(3.16228 + 3.16228i) q^{68} +(75.0000 + 25.0000i) q^{70} -63.2456 q^{71} +(55.0000 + 55.0000i) q^{73} -31.6228i q^{74} -18.0000 q^{76} +(79.0569 - 79.0569i) q^{77} -12.0000i q^{79} +(30.0416 - 90.1249i) q^{80} +(50.0000 - 50.0000i) q^{82} +(53.7587 + 53.7587i) q^{83} +(10.0000 + 20.0000i) q^{85} +31.6228 q^{86} +(-75.0000 - 75.0000i) q^{88} -100.000 q^{91} +(-3.16228 + 3.16228i) q^{92} +130.000i q^{94} +(-85.3815 - 28.4605i) q^{95} +(-5.00000 + 5.00000i) q^{97} +(-1.58114 - 1.58114i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{7}+O(q^{10}) \) 4 * q - 20 * q^7	\( 4 q - 20 q^{7} - 20 q^{10} + 40 q^{13} + 76 q^{16} - 100 q^{22} - 80 q^{25} + 20 q^{28} + 32 q^{31} + 40 q^{37} + 120 q^{40} + 40 q^{43} - 40 q^{46} + 40 q^{52} - 100 q^{55} - 300 q^{58} - 232 q^{61} + 280 q^{67} + 300 q^{70} + 220 q^{73} - 72 q^{76} + 200 q^{82} + 40 q^{85} - 300 q^{88} - 400 q^{91} - 20 q^{97}+O(q^{100}) \) 4 * q - 20 * q^7 - 20 * q^10 + 40 * q^13 + 76 * q^16 - 100 * q^22 - 80 * q^25 + 20 * q^28 + 32 * q^31 + 40 * q^37 + 120 * q^40 + 40 * q^43 - 40 * q^46 + 40 * q^52 - 100 * q^55 - 300 * q^58 - 232 * q^61 + 280 * q^67 + 300 * q^70 + 220 * q^73 - 72 * q^76 + 200 * q^82 + 40 * q^85 - 300 * q^88 - 400 * q^91 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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