LMFDB - Embedded newform 45.3.g.a.28.1

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			28.1
Root			$-1.58114 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	45.28
Dual form			45.3.g.a.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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	−0.981587	−	0.191017i	$-0.938821\pi$
$2$	−1.58114	+	1.58114i	−0.790569	+	0.790569i	0.191017	+	0.981587i	$0.438821\pi$
$3$		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	$-0.244738\pi$
$11$	15.8114			1.43740			0.718699	+	0.695321i	$0.244738\pi$
$12$		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	−0.607955	−	0.793971i	$-0.708010\pi$
$17$	3.16228	−	3.16228i	0.186016	−	0.186016i	0.793971	+	0.607955i	$0.208010\pi$
$18$		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	0.772502	−	0.635012i	$-0.219005\pi$
$23$	3.16228	+	3.16228i	0.137490	+	0.137490i	−0.635012	+	0.772502i	$0.719005\pi$
$24$		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	$0.304823\pi$
$29$			47.4342i			1.63566i	−0.575459	+	0.817830i	$0.695177\pi$
$30$		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	$-0.626021\pi$
$41$	−31.6228			−0.771287			−0.385644	+	0.922648i	$0.626021\pi$
$42$		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	−0.118305	−	0.992977i	$-0.537746\pi$
$47$	41.1096	−	41.1096i	0.874673	−	0.874673i	0.992977	+	0.118305i	$0.0377460\pi$
$48$		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	0.426950	−	0.904275i	$-0.359588\pi$
$53$	−25.2982	−	25.2982i	−0.477325	−	0.477325i	−0.904275	+	0.426950i	$0.859588\pi$
$54$		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	$-0.868321\pi$
$59$		−	47.4342i		−	0.803969i	0.915646	−	0.401984i	$-0.131679\pi$
$60$		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	$-0.353065\pi$
$71$	63.2456			0.890782			0.445391	+	0.895336i	$0.353065\pi$
$72$		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	0.304740	−	0.952436i	$-0.401430\pi$
$83$	−53.7587	−	53.7587i	−0.647695	−	0.647695i	−0.952436	+	0.304740i	$0.901430\pi$
$84$		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	$-0.475059\pi$
$101$	15.8114			0.156548			0.0782742	+	0.996932i	$0.475059\pi$
$102$		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	−0.368215	−	0.929741i	$-0.620031\pi$
$107$	60.0833	−	60.0833i	0.561526	−	0.561526i	0.929741	+	0.368215i	$0.120031\pi$
$108$		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	$0.0114895\pi$
$113$	117.004	+	117.004i	1.03544	+	1.03544i	0.0360874	+	0.999349i	$0.488511\pi$
$114$		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	$-0.731073\pi$
$131$	−173.925			−1.32767			−0.663837	+	0.747877i	$0.731073\pi$
$132$		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	−0.762454	−	0.647042i	$-0.776006\pi$
$137$	−15.8114	+	15.8114i	−0.115412	+	0.115412i	0.647042	+	0.762454i	$0.276006\pi$
$138$		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	$-0.949117\pi$
$149$		−	47.4342i		−	0.318350i	0.987250	−	0.159175i	$-0.0508834\pi$
$150$		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	−0.994521	−	0.104539i	$-0.966663\pi$
$167$	−148.627	+	148.627i	−0.889982	+	0.889982i	0.104539	+	0.994521i	$0.466663\pi$
$168$		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	0.310731	−	0.950498i	$-0.399426\pi$
$173$	−110.680	−	110.680i	−0.639767	−	0.639767i	−0.950498	+	0.310731i	$0.899426\pi$
$174$		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	$-0.869880\pi$
$179$		−	142.302i		−	0.794986i	0.917605	−	0.397493i	$-0.130120\pi$
$180$		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	$-0.364163\pi$
$191$	158.114			0.827821			0.413911	+	0.910317i	$0.364163\pi$
$192$		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	−0.233868	−	0.972268i	$-0.575138\pi$
$197$	145.465	−	145.465i	0.738400	−	0.738400i	0.972268	+	0.233868i	$0.0751385\pi$
$198$		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	$0.571680\pi$
$227$	−271.956	+	271.956i	−1.19804	+	1.19804i	−0.974752	+	0.223292i	$0.928320\pi$
$228$		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	0.428515	−	0.903535i	$-0.359037\pi$
$233$	−110.680	−	110.680i	−0.475020	−	0.475020i	−0.903535	+	0.428515i	$0.859037\pi$
$234$		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	$0.291941\pi$
$239$			379.473i			1.58775i	−0.608078	+	0.793877i	$0.708059\pi$
$240$		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	$-0.757895\pi$
$251$	−363.662			−1.44885			−0.724426	+	0.689352i	$0.757895\pi$
$252$		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	$-0.430216\pi$
$257$	306.741	−	306.741i	1.19354	−	1.19354i	0.976065	−	0.217480i	$-0.0697836\pi$
$258$		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	$0.0315945\pi$
$263$	287.767	+	287.767i	1.09417	+	1.09417i	0.0990940	+	0.995078i	$0.468406\pi$
$264$		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	$0.0852078\pi$
$269$			142.302i			0.529006i	−0.964385	+	0.264503i	$0.914792\pi$
$270$		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	$-0.409221\pi$
$281$	158.114			0.562683			0.281341	+	0.959608i	$0.409221\pi$
$282$		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	0.962394	−	0.271658i	$-0.0875718\pi$
$293$	202.386	+	202.386i	0.690736	+	0.690736i	−0.271658	+	0.962394i	$0.587572\pi$
$294$		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	$-0.366671\pi$
$311$	252.982			0.813448			0.406724	+	0.913551i	$0.366671\pi$
$312$		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	−0.731606	−	0.681728i	$-0.761229\pi$
$317$	−15.8114	+	15.8114i	−0.0498782	+	0.0498782i	0.681728	+	0.731606i	$0.261229\pi$
$318$		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	−0.429284	−	0.903170i	$-0.641234\pi$
$347$	164.438	−	164.438i	0.473886	−	0.473886i	0.903170	+	0.429284i	$0.141234\pi$
$348$		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	0.313537	−	0.949576i	$-0.398486\pi$
$353$	−224.522	−	224.522i	−0.636039	−	0.636039i	−0.949576	+	0.313537i	$0.898486\pi$
$354$		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	$0.129736\pi$
$359$			284.605i			0.792772i	−0.918084	+	0.396386i	$0.870264\pi$
$360$		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	0.236737	−	0.971574i	$-0.423922\pi$
$383$	−281.443	−	281.443i	−0.734837	−	0.734837i	−0.971574	+	0.236737i	$0.923922\pi$
$384$		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	$-0.766010\pi$
$389$		−	521.776i		−	1.34133i	0.741762	−	0.670663i	$-0.233990\pi$
$390$		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	$-0.397848\pi$
$401$	252.982			0.630878			0.315439	+	0.948946i	$0.397848\pi$
$402$		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	$-0.736782\pi$
$419$		−	616.644i		−	1.47170i	0.677142	−	0.735852i	$-0.263218\pi$
$420$		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	$-0.582667\pi$
$431$	−221.359			−0.513595			−0.256797	+	0.966465i	$0.582667\pi$
$432$		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	0.450301	−	0.892877i	$-0.351317\pi$
$443$	−196.061	−	196.061i	−0.442576	−	0.442576i	−0.892877	+	0.450301i	$0.851317\pi$
$444$		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	$0.102653\pi$
$449$			284.605i			0.633864i	−0.948448	+	0.316932i	$0.897347\pi$
$450$		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	$-0.863075\pi$
$461$	−838.004			−1.81780			−0.908898	+	0.417019i	$0.863075\pi$
$462$		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	−0.570649	−	0.821194i	$-0.693308\pi$
$467$	117.004	−	117.004i	0.250544	−	0.250544i	0.821194	+	0.570649i	$0.193308\pi$
$468$		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	$-0.870360\pi$
$479$		−	379.473i		−	0.792220i	0.918203	−	0.396110i	$-0.129640\pi$
$480$		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	$-0.654642\pi$
$491$	−458.530			−0.933870			−0.466935	+	0.884292i	$0.654642\pi$
$492$		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	0.996397	−	0.0848065i	$-0.0270272\pi$
$503$	458.530	+	458.530i	0.911591	+	0.911591i	−0.0848065	+	0.996397i	$0.527027\pi$
$504$		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	$0.0748468\pi$
$509$			237.171i			0.465954i	−0.972482	+	0.232977i	$0.925153\pi$
$510$		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	$-0.774167\pi$
$521$	−790.569			−1.51741			−0.758704	+	0.651436i	$0.774167\pi$
$522$		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	−0.896742	−	0.442555i	$-0.854072\pi$
$557$	−252.982	+	252.982i	−0.454187	+	0.454187i	0.442555	+	0.896742i	$0.354072\pi$
$558$		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	0.863620	−	0.504143i	$-0.168192\pi$
$563$	202.386	+	202.386i	0.359477	+	0.359477i	−0.504143	+	0.863620i	$0.668192\pi$
$564$		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	$0.0533202\pi$
$569$			189.737i			0.333456i	−0.986003	+	0.166728i	$0.946680\pi$
$570$		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	−0.562694	−	0.826666i	$-0.690235\pi$
$587$	154.952	−	154.952i	0.263972	−	0.263972i	0.826666	+	0.562694i	$0.190235\pi$
$588$		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	0.551526	−	0.834158i	$-0.314046\pi$
$593$	−167.601	−	167.601i	−0.282632	−	0.282632i	−0.834158	+	0.551526i	$0.814046\pi$
$594$		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	$-0.663421\pi$
$599$		−	1043.55i		−	1.74216i	0.491144	−	0.871078i	$-0.336579\pi$
$600$		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	−0.734734	−	0.678356i	$-0.762693\pi$
$617$	−34.7851	+	34.7851i	−0.0563777	+	0.0563777i	0.678356	+	0.734734i	$0.262693\pi$
$618$		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	$-0.387765\pi$
$641$	442.719			0.690669			0.345335	+	0.938480i	$0.387765\pi$
$642$		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	$0.516622\pi$
$647$	−679.890	+	679.890i	−1.05083	+	1.05083i	−0.998637	+	0.0521974i	$0.983378\pi$
$648$		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	0.981389	−	0.192030i	$-0.0615072\pi$
$653$	515.451	+	515.451i	0.789359	+	0.789359i	−0.192030	+	0.981389i	$0.561507\pi$
$654$		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	$0.239674\pi$
$659$			901.249i			1.36760i	−0.729669	+	0.683801i	$0.760326\pi$
$660$		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	−0.237932	−	0.971282i	$-0.576470\pi$
$677$	496.478	−	496.478i	0.733349	−	0.733349i	0.971282	+	0.237932i	$0.0764696\pi$
$678$		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	0.749722	−	0.661753i	$-0.230187\pi$
$683$	60.0833	+	60.0833i	0.0879697	+	0.0879697i	−0.661753	+	0.749722i	$0.730187\pi$
$684$		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	$-0.583521\pi$
$701$	−363.662			−0.518776			−0.259388	+	0.965773i	$0.583521\pi$
$702$		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	$0.129545\pi$
$719$			569.210i			0.791669i	−0.918322	+	0.395834i	$0.870455\pi$
$720$		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	$-0.879882\pi$
$743$	−964.495	−	964.495i	−1.29811	−	1.29811i	−0.368468	−	0.929640i	$-0.620118\pi$
$744$		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	$-0.506614\pi$
$761$	−31.6228			−0.0415542			−0.0207771	+	0.999784i	$0.506614\pi$
$762$		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	0.351272	−	0.936273i	$-0.385749\pi$
$773$	−452.206	−	452.206i	−0.585001	−	0.585001i	−0.936273	+	0.351272i	$0.885749\pi$
$774$		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
$783$		0			0
$784$		−	19.0000i

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)

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

Fields

Representations

Groups

Database

Properties

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