LMFDB - Embedded newform 1620.3.o.g.1241.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(701,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 0]))

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

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

chi := DirichletCharacter("1620.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.o (of order $6$, 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:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1241.2
Character	$\chi$	$=$	1620.1241
Dual form			1620.3.o.g.701.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93649 + 1.11803i) q^{5} +(5.78429 - 10.0187i) q^{7} +O(q^{10})$	$q+(-1.93649 + 1.11803i) q^{5} +(5.78429 - 10.0187i) q^{7} +(-2.14482 - 1.23831i) q^{11} +(-2.00318 - 3.46961i) q^{13} +13.4803i q^{17} +34.7794 q^{19} +(7.48630 - 4.32222i) q^{23} +(2.50000 - 4.33013i) q^{25} +(-17.1307 - 9.89040i) q^{29} +(0.0722784 + 0.125190i) q^{31} +25.8681i q^{35} -12.6892 q^{37} +(-5.90139 + 3.40717i) q^{41} +(14.4568 - 25.0399i) q^{43} +(54.7773 + 31.6257i) q^{47} +(-42.4161 - 73.4669i) q^{49} -84.1419i q^{53} +5.53790 q^{55} +(-88.7205 + 51.2228i) q^{59} +(22.3604 - 38.7293i) q^{61} +(7.75827 + 4.47924i) q^{65} +(-14.3284 - 24.8175i) q^{67} -125.069i q^{71} -104.159 q^{73} +(-24.8125 + 14.3255i) q^{77} +(-10.4491 + 18.0984i) q^{79} +(75.3078 + 43.4790i) q^{83} +(-15.0715 - 26.1046i) q^{85} +104.707i q^{89} -46.3479 q^{91} +(-67.3500 + 38.8845i) q^{95} +(36.6995 - 63.5654i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7}+O(q^{10}) $ 32 * q - 8 * q^7	$ 32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100}) $ 32 * q - 8 * q^7 + 40 * q^13 + 112 * q^19 + 80 * q^25 + 64 * q^31 - 176 * q^37 - 128 * q^43 - 216 * q^49 - 8 * q^61 + 40 * q^67 + 112 * q^73 + 136 * q^79 - 784 * q^91 - 128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.93649	+	1.11803i	−0.387298	+	0.223607i
$5$	−1.93649	+	1.11803i	−0.387298	+	0.223607i
$6$		0			0
$7$	5.78429	−	10.0187i	0.826328	−	1.43124i	−0.0745726	−	0.997216i	$-0.523759\pi$
$7$	5.78429	−	10.0187i	0.826328	−	1.43124i	0.900900	−	0.434026i	$-0.142907\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.14482	−	1.23831i	−0.194984	−	0.112574i	0.399330	−	0.916807i	$-0.369243\pi$
$11$	−2.14482	−	1.23831i	−0.194984	−	0.112574i	−0.594314	+	0.804233i	$0.702576\pi$
$12$		0			0
$13$	−2.00318	−	3.46961i	−0.154091	−	0.266893i	0.778637	−	0.627475i	$-0.215911\pi$
$13$	−2.00318	−	3.46961i	−0.154091	−	0.266893i	−0.932728	+	0.360582i	$0.882578\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			13.4803i			0.792961i	0.918043	+	0.396481i	$0.129769\pi$
$17$			13.4803i			0.792961i	−0.918043	+	0.396481i	$0.870231\pi$
$18$		0			0
$19$	34.7794			1.83049			0.915247	−	0.402893i	$-0.131995\pi$
$19$	34.7794			1.83049			0.915247	+	0.402893i	$0.131995\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.48630	−	4.32222i	0.325491	−	0.187923i	−0.328346	−	0.944557i	$-0.606491\pi$
$23$	7.48630	−	4.32222i	0.325491	−	0.187923i	0.653838	+	0.756635i	$0.273158\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−17.1307	−	9.89040i	−0.590713	−	0.341048i	0.174666	−	0.984628i	$-0.444115\pi$
$29$	−17.1307	−	9.89040i	−0.590713	−	0.341048i	−0.765379	+	0.643579i	$0.777449\pi$
$30$		0			0
$31$	0.0722784	+	0.125190i	0.00233156	+	0.00403838i	0.867189	−	0.497979i	$-0.165925\pi$
$31$	0.0722784	+	0.125190i	0.00233156	+	0.00403838i	−0.864857	+	0.502018i	$0.832591\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			25.8681i			0.739090i
$36$		0			0
$37$	−12.6892			−0.342952			−0.171476	−	0.985188i	$-0.554854\pi$
$37$	−12.6892			−0.342952			−0.171476	+	0.985188i	$0.554854\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.90139	+	3.40717i	−0.143936	+	0.0831017i	−0.570239	−	0.821479i	$-0.693149\pi$
$41$	−5.90139	+	3.40717i	−0.143936	+	0.0831017i	0.426302	+	0.904581i	$0.359816\pi$
$42$		0			0
$43$	14.4568	−	25.0399i	0.336204	−	0.582323i	−0.647511	−	0.762056i	$-0.724190\pi$
$43$	14.4568	−	25.0399i	0.336204	−	0.582323i	0.983715	+	0.179733i	$0.0575235\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	54.7773	+	31.6257i	1.16547	+	0.672887i	0.952610	−	0.304195i	$-0.0983875\pi$
$47$	54.7773	+	31.6257i	1.16547	+	0.672887i	0.212864	+	0.977082i	$0.431721\pi$
$48$		0			0
$49$	−42.4161	−	73.4669i	−0.865635	−	1.49932i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	84.1419i		−	1.58758i	−0.608189	−	0.793792i	$-0.708104\pi$
$53$		−	84.1419i		−	1.58758i	0.608189	−	0.793792i	$-0.291896\pi$
$54$		0			0
$55$	5.53790			0.100689
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−88.7205	+	51.2228i	−1.50374	+	0.868184i	−0.503747	+	0.863851i	$0.668046\pi$
$59$	−88.7205	+	51.2228i	−1.50374	+	0.868184i	−0.999991	+	0.00433260i	$0.998621\pi$
$60$		0			0
$61$	22.3604	−	38.7293i	0.366564	−	0.634907i	−0.622462	−	0.782650i	$-0.713868\pi$
$61$	22.3604	−	38.7293i	0.366564	−	0.634907i	0.989026	+	0.147743i	$0.0472009\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.75827	+	4.47924i	0.119358	+	0.0689114i
$66$		0			0
$67$	−14.3284	−	24.8175i	−0.213857	−	0.370411i	0.739061	−	0.673638i	$-0.235269\pi$
$67$	−14.3284	−	24.8175i	−0.213857	−	0.370411i	−0.952918	+	0.303227i	$0.901936\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	125.069i		−	1.76153i	−0.473550	−	0.880767i	$-0.657028\pi$
$71$		−	125.069i		−	1.76153i	0.473550	−	0.880767i	$-0.342972\pi$
$72$		0			0
$73$	−104.159			−1.42684			−0.713420	−	0.700737i	$-0.752855\pi$
$73$	−104.159			−1.42684			−0.713420	+	0.700737i	$0.752855\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−24.8125	+	14.3255i	−0.322241	+	0.186046i
$78$		0			0
$79$	−10.4491	+	18.0984i	−0.132267	+	0.229094i	−0.924550	−	0.381060i	$-0.875559\pi$
$79$	−10.4491	+	18.0984i	−0.132267	+	0.229094i	0.792283	+	0.610154i	$0.208892\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	75.3078	+	43.4790i	0.907323	+	0.523843i	0.879569	−	0.475771i	$-0.157831\pi$
$83$	75.3078	+	43.4790i	0.907323	+	0.523843i	0.0277543	+	0.999615i	$0.491164\pi$
$84$		0			0
$85$	−15.0715	−	26.1046i	−0.177312	−	0.307113i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			104.707i			1.17649i	0.808684	+	0.588243i	$0.200180\pi$
$89$			104.707i			1.17649i	−0.808684	+	0.588243i	$0.799820\pi$
$90$		0			0
$91$	−46.3479			−0.509317
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−67.3500	+	38.8845i	−0.708947	+	0.409311i
$96$		0			0
$97$	36.6995	−	63.5654i	0.378345	−	0.655313i	−0.612476	−	0.790489i	$-0.709827\pi$
$97$	36.6995	−	63.5654i	0.378345	−	0.655313i	0.990822	+	0.135176i	$0.0431599\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	43.7169	+	25.2400i	0.432841	+	0.249901i	0.700556	−	0.713597i	$-0.252935\pi$
$101$	43.7169	+	25.2400i	0.432841	+	0.249901i	−0.267715	+	0.963498i	$0.586269\pi$
$102$		0			0
$103$	2.46131	+	4.26312i	0.0238963	+	0.0413895i	0.877726	−	0.479162i	$-0.159060\pi$
$103$	2.46131	+	4.26312i	0.0238963	+	0.0413895i	−0.853830	+	0.520552i	$0.825726\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	201.164i		−	1.88004i	−0.341118	−	0.940020i	$-0.610806\pi$
$107$		−	201.164i		−	1.88004i	0.341118	−	0.940020i	$-0.389194\pi$
$108$		0			0
$109$	60.8959			0.558678			0.279339	−	0.960193i	$-0.409885\pi$
$109$	60.8959			0.558678			0.279339	+	0.960193i	$0.409885\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	126.138	−	72.8256i	1.11626	−	0.644475i	0.175818	−	0.984423i	$-0.443743\pi$
$113$	126.138	−	72.8256i	1.11626	−	0.644475i	0.940444	+	0.339948i	$0.110409\pi$
$114$		0			0
$115$	−9.66477	+	16.7399i	−0.0840415	+	0.145564i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	135.055	+	77.9743i	1.13492	+	0.655246i
$120$		0			0
$121$	−57.4332	−	99.4772i	−0.474654	−	0.822125i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	96.6822			0.761277			0.380638	−	0.924724i	$-0.375704\pi$
$127$	96.6822			0.761277			0.380638	+	0.924724i	$0.375704\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	19.8722	−	11.4732i	0.151696	−	0.0875817i	−0.422231	−	0.906488i	$-0.638753\pi$
$131$	19.8722	−	11.4732i	0.151696	−	0.0875817i	0.573927	+	0.818907i	$0.305419\pi$
$132$		0			0
$133$	201.174	−	348.444i	1.51259	−	2.61988i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−152.705	−	88.1640i	−1.11463	−	0.643533i	−0.174607	−	0.984638i	$-0.555865\pi$
$137$	−152.705	−	88.1640i	−1.11463	−	0.643533i	−0.940025	+	0.341105i	$0.889199\pi$
$138$		0			0
$139$	−74.5089	−	129.053i	−0.536035	−	0.928440i	−0.999112	−	0.0421222i	$-0.986588\pi$
$139$	−74.5089	−	129.053i	−0.536035	−	0.928440i	0.463077	−	0.886318i	$-0.346745\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			9.92224i			0.0693863i
$144$		0			0
$145$	44.2312			0.305043
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−236.222	+	136.383i	−1.58539	+	0.915323i	−0.591332	+	0.806428i	$0.701398\pi$
$149$	−236.222	+	136.383i	−1.58539	+	0.915323i	−0.994053	+	0.108895i	$0.965269\pi$
$150$		0			0
$151$	114.497	−	198.315i	0.758262	−	1.31335i	−0.185475	−	0.982649i	$-0.559382\pi$
$151$	114.497	−	198.315i	0.758262	−	1.31335i	0.943736	−	0.330699i	$-0.107284\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.279933	−	0.161619i	−0.00180602	−	0.00104271i
$156$		0			0
$157$	−102.521	−	177.572i	−0.653003	−	1.13103i	−0.982390	−	0.186839i	$-0.940176\pi$
$157$	−102.521	−	177.572i	−0.653003	−	1.13103i	0.329388	−	0.944195i	$-0.393158\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	100.004i		−	0.621142i
$162$		0			0
$163$	6.71334			0.0411861			0.0205931	−	0.999788i	$-0.493445\pi$
$163$	6.71334			0.0411861			0.0205931	+	0.999788i	$0.493445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	37.0809	−	21.4087i	0.222041	−	0.128196i	−0.384854	−	0.922978i	$-0.625748\pi$
$167$	37.0809	−	21.4087i	0.222041	−	0.128196i	0.606895	+	0.794782i	$0.292415\pi$
$168$		0			0
$169$	76.4746	−	132.458i	0.452512	−	0.783774i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	170.982	+	98.7166i	0.988336	+	0.570616i	0.904777	−	0.425887i	$-0.140038\pi$
$173$	170.982	+	98.7166i	0.988336	+	0.570616i	0.0835596	+	0.996503i	$0.473371\pi$
$174$		0			0
$175$	−28.9215	−	50.0935i	−0.165266	−	0.286248i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			43.8415i			0.244925i	0.992473	+	0.122462i	$0.0390791\pi$
$179$			43.8415i			0.244925i	−0.992473	+	0.122462i	$0.960921\pi$
$180$		0			0
$181$	286.982			1.58554			0.792768	−	0.609523i	$-0.208639\pi$
$181$	286.982			1.58554			0.792768	+	0.609523i	$0.208639\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	24.5726	−	14.1870i	0.132825	−	0.0766864i
$186$		0			0
$187$	16.6929	−	28.9129i	0.0892667	−	0.154615i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−117.528	−	67.8549i	−0.615330	−	0.355261i	0.159718	−	0.987163i	$-0.448941\pi$
$191$	−117.528	−	67.8549i	−0.615330	−	0.355261i	−0.775049	+	0.631901i	$0.782275\pi$
$192$		0			0
$193$	64.2378	+	111.263i	0.332838	+	0.576493i	0.983067	−	0.183245i	$-0.0586603\pi$
$193$	64.2378	+	111.263i	0.332838	+	0.576493i	−0.650229	+	0.759738i	$0.725327\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	39.8188i		−	0.202126i	−0.994880	−	0.101063i	$-0.967776\pi$
$197$		−	39.8188i		−	0.202126i	0.994880	−	0.101063i	$-0.0322244\pi$
$198$		0			0
$199$	−64.0049			−0.321632			−0.160816	−	0.986984i	$-0.551413\pi$
$199$	−64.0049			−0.321632			−0.160816	+	0.986984i	$0.551413\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−198.178	+	114.418i	−0.976245	+	0.563636i
$204$		0			0
$205$	7.61867	−	13.1959i	0.0371642	−	0.0643703i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−74.5956	−	43.0678i	−0.356917	−	0.206066i
$210$		0			0
$211$	−78.8984	−	136.656i	−0.373926	−	0.647659i	0.616240	−	0.787559i	$-0.288655\pi$
$211$	−78.8984	−	136.656i	−0.373926	−	0.647659i	−0.990166	+	0.139900i	$0.955322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			64.6527i			0.300710i
$216$		0			0
$217$	1.67232			0.00770654
$218$		0			0
$219$		0			0
$220$		0			0
$221$	46.7715	−	27.0035i	0.211636	−	0.122188i
$222$		0			0
$223$	−25.5374	+	44.2321i	−0.114518	+	0.198350i	−0.917587	−	0.397535i	$-0.869866\pi$
$223$	−25.5374	+	44.2321i	−0.114518	+	0.198350i	0.803069	+	0.595886i	$0.203199\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	190.295	+	109.867i	0.838305	+	0.483996i	0.856688	−	0.515835i	$-0.172518\pi$
$227$	190.295	+	109.867i	0.838305	+	0.483996i	−0.0183826	+	0.999831i	$0.505852\pi$
$228$		0			0
$229$	−16.4448	−	28.4832i	−0.0718113	−	0.124381i	0.827884	−	0.560900i	$-0.189545\pi$
$229$	−16.4448	−	28.4832i	−0.0718113	−	0.124381i	−0.899695	+	0.436519i	$0.856211\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			201.245i			0.863712i	0.901943	+	0.431856i	$0.142141\pi$
$233$			201.245i			0.863712i	−0.901943	+	0.431856i	$0.857859\pi$
$234$		0			0
$235$	−141.434			−0.601848
$236$		0			0
$237$		0			0
$238$		0			0
$239$	88.6564	−	51.1858i	0.370947	−	0.214166i	−0.302925	−	0.953014i	$-0.597963\pi$
$239$	88.6564	−	51.1858i	0.370947	−	0.214166i	0.673872	+	0.738848i	$0.264630\pi$
$240$		0			0
$241$	−168.043	+	291.058i	−0.697273	+	1.20771i	0.272136	+	0.962259i	$0.412270\pi$
$241$	−168.043	+	291.058i	−0.697273	+	1.20771i	−0.969409	+	0.245453i	$0.921063\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	164.277	+	94.8453i	0.670518	+	0.387124i
$246$		0			0
$247$	−69.6693	−	120.671i	−0.282062	−	0.488546i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	276.108i		−	1.10003i	−0.835154	−	0.550017i	$-0.814621\pi$
$251$		−	276.108i		−	1.10003i	0.835154	−	0.550017i	$-0.185379\pi$
$252$		0			0
$253$	−21.4090			−0.0846207
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−306.351	+	176.872i	−1.19203	+	0.688218i	−0.958766	−	0.284195i	$-0.908274\pi$
$257$	−306.351	+	176.872i	−1.19203	+	0.688218i	−0.233263	+	0.972414i	$0.574940\pi$
$258$		0			0
$259$	−73.3982	+	127.129i	−0.283391	+	0.490847i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	197.068	+	113.777i	0.749307	+	0.432613i	0.825444	−	0.564485i	$-0.190925\pi$
$263$	197.068	+	113.777i	0.749307	+	0.432613i	−0.0761362	+	0.997097i	$0.524258\pi$
$264$		0			0
$265$	94.0736	+	162.940i	0.354995	+	0.614869i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	514.993i		−	1.91447i	−0.289306	−	0.957237i	$-0.593425\pi$
$269$		−	514.993i		−	1.91447i	0.289306	−	0.957237i	$-0.406575\pi$
$270$		0			0
$271$	−343.018			−1.26575			−0.632874	−	0.774255i	$-0.718125\pi$
$271$	−343.018			−1.26575			−0.632874	+	0.774255i	$0.718125\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.7241	+	6.19156i	−0.0389967	+	0.0225148i
$276$		0			0
$277$	97.4246	−	168.744i	0.351713	−	0.609185i	−0.634837	−	0.772646i	$-0.718933\pi$
$277$	97.4246	−	168.744i	0.351713	−	0.609185i	0.986550	+	0.163461i	$0.0522659\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	19.9148	+	11.4978i	0.0708711	+	0.0409175i	0.535017	−	0.844841i	$-0.320305\pi$
$281$	19.9148	+	11.4978i	0.0708711	+	0.0409175i	−0.464146	+	0.885759i	$0.653639\pi$
$282$		0			0
$283$	−108.744	−	188.351i	−0.384256	−	0.665551i	0.607410	−	0.794389i	$-0.292209\pi$
$283$	−108.744	−	188.351i	−0.384256	−	0.665551i	−0.991666	+	0.128838i	$0.958875\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			78.8323i			0.274677i
$288$		0			0
$289$	107.280			0.371212
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−214.466	+	123.822i	−0.731967	+	0.422602i	−0.819142	−	0.573591i	$-0.805550\pi$
$293$	−214.466	+	123.822i	−0.731967	+	0.422602i	0.0871740	+	0.996193i	$0.472216\pi$
$294$		0			0
$295$	114.538	−	198.385i	0.388263	−	0.672492i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−29.9928	−	17.3163i	−0.100310	−	0.0579142i
$300$		0			0
$301$	−167.244	−	289.676i	−0.555629	−	0.962378i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			99.9987i			0.327865i
$306$		0			0
$307$	−262.586			−0.855329			−0.427665	−	0.903937i	$-0.640664\pi$
$307$	−262.586			−0.855329			−0.427665	+	0.903937i	$0.640664\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−374.706	+	216.336i	−1.20484	+	0.695616i	−0.961628	−	0.274357i	$-0.911535\pi$
$311$	−374.706	+	216.336i	−1.20484	+	0.695616i	−0.243214	+	0.969973i	$0.578202\pi$
$312$		0			0
$313$	−81.0013	+	140.298i	−0.258790	+	0.448237i	−0.965918	−	0.258848i	$-0.916657\pi$
$313$	−81.0013	+	140.298i	−0.258790	+	0.448237i	0.707128	+	0.707086i	$0.249990\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−154.242	−	89.0518i	−0.486569	−	0.280921i	0.236581	−	0.971612i	$-0.423973\pi$
$317$	−154.242	−	89.0518i	−0.486569	−	0.280921i	−0.723150	+	0.690691i	$0.757306\pi$
$318$		0			0
$319$	24.4948	+	42.4263i	0.0767863	+	0.132998i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			468.838i			1.45151i
$324$		0			0
$325$	−20.0318			−0.0616362
$326$		0			0
$327$		0			0
$328$		0			0
$329$	633.696	−	365.864i	1.92613	−	1.11205i
$330$		0			0
$331$	−220.201	+	381.399i	−0.665259	+	1.15226i	0.313956	+	0.949438i	$0.398346\pi$
$331$	−220.201	+	381.399i	−0.665259	+	1.15226i	−0.979215	+	0.202825i	$0.934988\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	55.4937	+	32.0393i	0.165653	+	0.0956397i
$336$		0			0
$337$	272.957	+	472.775i	0.809961	+	1.40289i	0.912890	+	0.408205i	$0.133845\pi$
$337$	272.957	+	472.775i	0.809961	+	1.40289i	−0.102930	+	0.994689i	$0.532822\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	0.358013i		−	0.00104989i
$342$		0			0
$343$	−414.528			−1.20854
$344$		0			0
$345$		0			0
$346$		0			0
$347$	311.300	−	179.729i	0.897119	−	0.517952i	0.0208549	−	0.999783i	$-0.493361\pi$
$347$	311.300	−	179.729i	0.897119	−	0.517952i	0.876265	+	0.481830i	$0.160028\pi$
$348$		0			0
$349$	223.688	−	387.439i	0.640940	−	1.11014i	−0.344283	−	0.938866i	$-0.611878\pi$
$349$	223.688	−	387.439i	0.640940	−	1.11014i	0.985223	−	0.171275i	$-0.0547886\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	161.481	+	93.2310i	0.457453	+	0.264111i	0.710973	−	0.703220i	$-0.248255\pi$
$353$	161.481	+	93.2310i	0.457453	+	0.264111i	−0.253520	+	0.967330i	$0.581588\pi$
$354$		0			0
$355$	139.831	+	242.195i	0.393891	+	0.682239i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	404.037i		−	1.12545i	−0.826643	−	0.562726i	$-0.809753\pi$
$359$		−	404.037i		−	1.12545i	0.826643	−	0.562726i	$-0.190247\pi$
$360$		0			0
$361$	848.606			2.35071
$362$		0			0
$363$		0			0
$364$		0			0
$365$	201.704	−	116.454i	0.552612	−	0.319051i
$366$		0			0
$367$	−316.232	+	547.730i	−0.861668	+	1.49245i	0.00864919	+	0.999963i	$0.497247\pi$
$367$	−316.232	+	547.730i	−0.861668	+	1.49245i	−0.870318	+	0.492491i	$0.836086\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−842.992	−	486.702i	−2.27222	−	1.31186i
$372$		0			0
$373$	168.398	+	291.673i	0.451468	+	0.781965i	0.998477	−	0.0551610i	$-0.0175672\pi$
$373$	168.398	+	291.673i	0.451468	+	0.781965i	−0.547010	+	0.837126i	$0.684234\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			79.2489i			0.210209i
$378$		0			0
$379$	−225.416			−0.594765			−0.297382	−	0.954758i	$-0.596114\pi$
$379$	−225.416			−0.594765			−0.297382	+	0.954758i	$0.596114\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−199.383	+	115.114i	−0.520581	+	0.300558i	−0.737173	−	0.675705i	$-0.763840\pi$
$383$	−199.383	+	115.114i	−0.520581	+	0.300558i	0.216591	+	0.976262i	$0.430506\pi$
$384$		0			0
$385$	32.0329	−	55.4825i	0.0832022	−	0.144111i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−439.906	−	253.980i	−1.13086	−	0.652904i	−0.186711	−	0.982415i	$-0.559783\pi$
$389$	−439.906	−	253.980i	−1.13086	−	0.652904i	−0.944152	+	0.329511i	$0.893116\pi$
$390$		0			0
$391$	58.2650	+	100.918i	0.149015	+	0.258102i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	46.7299i		−	0.118303i
$396$		0			0
$397$	576.342			1.45174			0.725871	−	0.687831i	$-0.241437\pi$
$397$	576.342			1.45174			0.725871	+	0.687831i	$0.241437\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	545.792	−	315.113i	1.36108	−	0.785819i	0.371311	−	0.928509i	$-0.378908\pi$
$401$	545.792	−	315.113i	1.36108	−	0.785819i	0.989767	+	0.142690i	$0.0455751\pi$
$402$		0			0
$403$	0.289573	−	0.501555i	0.000718544	−	0.00124455i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	27.2161	+	15.7132i	0.0668701	+	0.0386074i
$408$		0			0
$409$	343.648	+	595.215i	0.840214	+	1.45529i	0.889713	+	0.456520i	$0.150904\pi$
$409$	343.648	+	595.215i	0.840214	+	1.45529i	−0.0494989	+	0.998774i	$0.515762\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1185.15i			2.86962i
$414$		0			0
$415$	−194.444			−0.468540
$416$		0			0
$417$		0			0
$418$		0			0
$419$	419.091	−	241.962i	1.00022	−	0.577475i	0.0919040	−	0.995768i	$-0.470705\pi$
$419$	419.091	−	241.962i	1.00022	−	0.577475i	0.908312	+	0.418293i	$0.137371\pi$
$420$		0			0
$421$	99.6032	−	172.518i	0.236587	−	0.409781i	−0.723146	−	0.690696i	$-0.757304\pi$
$421$	99.6032	−	172.518i	0.236587	−	0.409781i	0.959733	+	0.280915i	$0.0906378\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	58.3716	+	33.7009i	0.137345	+	0.0792961i
$426$		0			0
$427$	−258.678	−	448.044i	−0.605803	−	1.04928i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	204.291i		−	0.473994i	−0.971510	−	0.236997i	$-0.923837\pi$
$431$		−	204.291i		−	0.473994i	0.971510	−	0.236997i	$-0.0761631\pi$
$432$		0			0
$433$	452.463			1.04495			0.522475	−	0.852655i	$-0.325009\pi$
$433$	452.463			1.04495			0.522475	+	0.852655i	$0.325009\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	260.369	−	150.324i	0.595810	−	0.343991i
$438$		0			0
$439$	−81.1094	+	140.486i	−0.184760	+	0.320013i	−0.943495	−	0.331385i	$-0.892484\pi$
$439$	−81.1094	+	140.486i	−0.184760	+	0.320013i	0.758736	+	0.651398i	$0.225817\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	435.494	+	251.432i	0.983056	+	0.567568i	0.903191	−	0.429238i	$-0.141218\pi$
$443$	435.494	+	251.432i	0.983056	+	0.567568i	0.0798645	+	0.996806i	$0.474551\pi$
$444$		0			0
$445$	−117.066	−	202.765i	−0.263070	−	0.455651i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			356.208i			0.793336i	0.917962	+	0.396668i	$0.129834\pi$
$449$			356.208i			0.793336i	−0.917962	+	0.396668i	$0.870166\pi$
$450$		0			0
$451$	16.8766			0.0374203
$452$		0			0
$453$		0			0
$454$		0			0
$455$	89.7523	−	51.8185i	0.197258	−	0.113887i
$456$		0			0
$457$	−95.0244	+	164.587i	−0.207931	+	0.360147i	−0.951063	−	0.308998i	$-0.900006\pi$
$457$	−95.0244	+	164.587i	−0.207931	+	0.360147i	0.743132	+	0.669145i	$0.233340\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−33.7914	−	19.5094i	−0.0733001	−	0.0423198i	0.462902	−	0.886409i	$-0.346808\pi$
$461$	−33.7914	−	19.5094i	−0.0733001	−	0.0423198i	−0.536202	+	0.844090i	$0.680142\pi$
$462$		0			0
$463$	375.130	+	649.745i	0.810217	+	1.40334i	0.912712	+	0.408603i	$0.133984\pi$
$463$	375.130	+	649.745i	0.810217	+	1.40334i	−0.102495	+	0.994733i	$0.532683\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	622.444i		−	1.33286i	−0.745569	−	0.666428i	$-0.767822\pi$
$467$		−	622.444i		−	1.33286i	0.745569	−	0.666428i	$-0.232178\pi$
$468$		0			0
$469$	−331.519			−0.706864
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−62.0144	+	35.8040i	−0.131109	+	0.0756956i
$474$		0			0
$475$	86.9485	−	150.599i	0.183049	−	0.317051i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	288.814	+	166.747i	0.602952	+	0.348114i	0.770202	−	0.637800i	$-0.220155\pi$
$479$	288.814	+	166.747i	0.602952	+	0.348114i	−0.167250	+	0.985914i	$0.553489\pi$
$480$		0			0
$481$	25.4188	+	44.0266i	0.0528457	+	0.0915314i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			164.125i			0.338402i
$486$		0			0
$487$	390.299			0.801435			0.400718	−	0.916202i	$-0.368761\pi$
$487$	390.299			0.801435			0.400718	+	0.916202i	$0.368761\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−657.499	+	379.607i	−1.33910	+	0.773130i	−0.986674	−	0.162709i	$-0.947977\pi$
$491$	−657.499	+	379.607i	−1.33910	+	0.773130i	−0.352427	+	0.935839i	$0.614644\pi$
$492$		0			0
$493$	133.326	−	230.927i	0.270438	−	0.468413i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1253.03	−	723.435i	−2.52118	−	1.45560i
$498$		0			0
$499$	227.809	+	394.577i	0.456531	+	0.790735i	0.998775	−	0.0494863i	$-0.0157584\pi$
$499$	227.809	+	394.577i	0.456531	+	0.790735i	−0.542244	+	0.840221i	$0.682425\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			999.141i			1.98636i	0.116572	+	0.993182i	$0.462809\pi$
$503$			999.141i			1.98636i	−0.116572	+	0.993182i	$0.537191\pi$
$504$		0			0
$505$	−112.877			−0.223518
$506$		0			0
$507$		0			0
$508$		0			0
$509$	148.656	−	85.8268i	0.292056	−	0.168618i	−0.346813	−	0.937934i	$-0.612736\pi$
$509$	148.656	−	85.8268i	0.292056	−	0.168618i	0.638869	+	0.769316i	$0.279403\pi$
$510$		0			0
$511$	−602.488	+	1043.54i	−1.17904	+	2.04215i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.53263	−	5.50367i	−0.0185100	−	0.0106867i
$516$		0			0
$517$	−78.3250	−	135.663i	−0.151499	−	0.262404i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	479.287i		−	0.919936i	−0.887936	−	0.459968i	$-0.847861\pi$
$521$		−	479.287i		−	0.919936i	0.887936	−	0.459968i	$-0.152139\pi$
$522$		0			0
$523$	−833.620			−1.59392			−0.796960	−	0.604032i	$-0.793560\pi$
$523$	−833.620			−1.59392			−0.796960	+	0.604032i	$0.793560\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.68760	+	0.974338i	−0.00320228	+	0.00184884i
$528$		0			0
$529$	−227.137	+	393.413i	−0.429370	+	0.743691i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	23.6431	+	13.6503i	0.0443585	+	0.0256104i
$534$		0			0
$535$	224.909	+	389.553i	0.420390	+	0.728137i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			210.098i			0.389792i
$540$		0			0
$541$	−119.053			−0.220060			−0.110030	−	0.993928i	$-0.535095\pi$
$541$	−119.053			−0.220060			−0.110030	+	0.993928i	$0.535095\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−117.924	+	68.0837i	−0.216375	+	0.124924i
$546$		0			0
$547$	−461.621	+	799.551i	−0.843914	+	1.46170i	0.0426471	+	0.999090i	$0.486421\pi$
$547$	−461.621	+	799.551i	−0.843914	+	1.46170i	−0.886561	+	0.462612i	$0.846912\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−595.795	−	343.982i	−1.08130	−	0.624287i
$552$		0			0
$553$	120.882	+	209.373i	0.218592	+	0.378613i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			80.8321i			0.145121i	0.997364	+	0.0725603i	$0.0231170\pi$
$557$			80.8321i			0.145121i	−0.997364	+	0.0725603i	$0.976883\pi$
$558$		0			0
$559$	−115.838			−0.207224
$560$		0			0
$561$		0			0
$562$		0			0
$563$	389.616	−	224.945i	0.692035	−	0.399547i	−0.112339	−	0.993670i	$-0.535834\pi$
$563$	389.616	−	224.945i	0.692035	−	0.399547i	0.804374	+	0.594123i	$0.202501\pi$
$564$		0			0
$565$	−162.843	+	282.052i	−0.288218	+	0.499208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	544.886	+	314.590i	0.957621	+	0.552883i	0.895440	−	0.445182i	$-0.146861\pi$
$569$	544.886	+	314.590i	0.957621	+	0.552883i	0.0621807	+	0.998065i	$0.480194\pi$
$570$		0			0
$571$	55.3534	+	95.8749i	0.0969412	+	0.167907i	0.910417	−	0.413692i	$-0.135761\pi$
$571$	55.3534	+	95.8749i	0.0969412	+	0.167907i	−0.813476	+	0.581599i	$0.802427\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	43.2222i		−	0.0751690i
$576$		0			0
$577$	−382.867			−0.663547			−0.331774	−	0.943359i	$-0.607647\pi$
$577$	−382.867			−0.663547			−0.331774	+	0.943359i	$0.607647\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	871.205	−	502.991i	1.49949	−	0.865733i
$582$		0			0
$583$	−104.194	+	180.469i	−0.178720	+	0.309553i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−153.533	−	88.6421i	−0.261555	−	0.151009i	0.363489	−	0.931599i	$-0.381585\pi$
$587$	−153.533	−	88.6421i	−0.261555	−	0.151009i	−0.625044	+	0.780590i	$0.714919\pi$
$588$		0			0
$589$	2.51380	+	4.35403i	0.00426791	+	0.00739224i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			869.475i			1.46623i	0.680104	+	0.733115i	$0.261935\pi$
$593$			869.475i			1.46623i	−0.680104	+	0.733115i	$0.738065\pi$
$594$		0			0
$595$	−348.711			−0.586070
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−749.960	+	432.990i	−1.25202	+	0.722854i	−0.971510	−	0.236997i	$-0.923837\pi$
$599$	−749.960	+	432.990i	−1.25202	+	0.722854i	−0.280510	+	0.959851i	$0.590504\pi$
$600$		0			0
$601$	−393.103	+	680.874i	−0.654081	+	1.13290i	0.328042	+	0.944663i	$0.393611\pi$
$601$	−393.103	+	680.874i	−0.654081	+	1.13290i	−0.982123	+	0.188239i	$0.939722\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	222.438	+	128.424i	0.367666	+	0.212272i
$606$		0			0
$607$	331.980	+	575.007i	0.546920	+	0.947293i	0.998483	+	0.0550541i	$0.0175331\pi$
$607$	331.980	+	575.007i	0.546920	+	0.947293i	−0.451563	+	0.892239i	$0.649134\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	253.407i		−	0.414742i
$612$		0			0
$613$	−709.117			−1.15680			−0.578399	−	0.815754i	$-0.696322\pi$
$613$	−709.117			−1.15680			−0.578399	+	0.815754i	$0.696322\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−176.205	+	101.732i	−0.285583	+	0.164882i	−0.635948	−	0.771732i	$-0.719391\pi$
$617$	−176.205	+	101.732i	−0.285583	+	0.164882i	0.350365	+	0.936613i	$0.386058\pi$
$618$		0			0
$619$	−546.645	+	946.816i	−0.883109	+	1.52959i	−0.0352434	+	0.999379i	$0.511221\pi$
$619$	−546.645	+	946.816i	−0.883109	+	1.52959i	−0.847866	+	0.530211i	$0.822113\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1049.03	+	605.658i	1.68384	+	0.972163i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	171.055i		−	0.271948i
$630$		0			0
$631$	992.298			1.57258			0.786290	−	0.617858i	$-0.211999\pi$
$631$	992.298			1.57258			0.786290	+	0.617858i	$0.211999\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−187.224	+	108.094i	−0.294841	+	0.170227i
$636$		0			0
$637$	−169.934	+	294.334i	−0.266772	+	0.462063i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1021.02	−	589.487i	−1.59286	−	0.919637i	−0.992814	−	0.119671i	$-0.961816\pi$
$641$	−1021.02	−	589.487i	−1.59286	−	0.919637i	−0.600045	−	0.799967i	$-0.704850\pi$
$642$		0			0
$643$	380.476	+	659.003i	0.591719	+	1.02489i	0.994001	+	0.109372i	$0.0348840\pi$
$643$	380.476	+	659.003i	0.591719	+	1.02489i	−0.402282	+	0.915516i	$0.631783\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	445.973i		−	0.689293i	−0.938732	−	0.344647i	$-0.887999\pi$
$647$		−	445.973i		−	0.689293i	0.938732	−	0.344647i	$-0.112001\pi$
$648$		0			0
$649$	253.720			0.390939
$650$		0			0
$651$		0			0
$652$		0			0
$653$	273.062	−	157.653i	0.418166	−	0.241428i	−0.276127	−	0.961121i	$-0.589051\pi$
$653$	273.062	−	157.653i	0.418166	−	0.241428i	0.694292	+	0.719693i	$0.255718\pi$
$654$		0			0
$655$	−25.6549	+	44.4355i	−0.0391677	+	0.0678405i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−534.867	−	308.806i	−0.811635	−	0.468597i	0.0358887	−	0.999356i	$-0.488574\pi$
$659$	−534.867	−	308.806i	−0.811635	−	0.468597i	−0.847523	+	0.530758i	$0.821907\pi$
$660$		0			0
$661$	−390.512	−	676.387i	−0.590790	−	1.02328i	−0.994126	−	0.108226i	$-0.965483\pi$
$661$	−390.512	−	676.387i	−0.590790	−	1.02328i	0.403336	−	0.915052i	$-0.367851\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			899.679i			1.35290i
$666$		0			0
$667$	−170.994			−0.256363
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−95.9180	+	55.3783i	−0.142948	+	0.0825310i
$672$		0			0
$673$	441.469	−	764.646i	0.655971	−	1.13618i	−0.325678	−	0.945481i	$-0.605593\pi$
$673$	441.469	−	764.646i	0.655971	−	1.13618i	0.981649	−	0.190695i	$-0.0610741\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	447.860	+	258.572i	0.661537	+	0.381938i	0.792862	−	0.609401i	$-0.208590\pi$
$677$	447.860	+	258.572i	0.661537	+	0.381938i	−0.131326	+	0.991339i	$0.541923\pi$
$678$		0			0
$679$	−424.561	−	735.362i	−0.625274	−	1.08301i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	663.535i		−	0.971501i	−0.874097	−	0.485751i	$-0.838546\pi$
$683$		−	663.535i		−	0.971501i	0.874097	−	0.485751i	$-0.161454\pi$
$684$		0			0
$685$	394.282			0.575594
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−291.939	+	168.551i	−0.423715	+	0.244632i
$690$		0			0
$691$	183.906	−	318.535i	0.266145	−	0.460977i	−0.701718	−	0.712455i	$-0.747583\pi$
$691$	183.906	−	318.535i	0.266145	−	0.460977i	0.967863	+	0.251478i	$0.0809167\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	288.572	+	166.607i	0.415211	+	0.239722i
$696$		0			0
$697$	−45.9298	−	79.5528i	−0.0658965	−	0.114136i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	579.637i		−	0.826872i	−0.910533	−	0.413436i	$-0.864329\pi$
$701$		−	579.637i		−	0.826872i	0.910533	−	0.413436i	$-0.135671\pi$
$702$		0			0
$703$	−441.324			−0.627772
$704$		0			0
$705$		0			0
$706$		0			0
$707$	505.743	−	291.991i	0.715337	−	0.413000i
$708$		0			0
$709$	107.770	−	186.662i	0.152002	−	0.263276i	−0.779961	−	0.625828i	$-0.784761\pi$
$709$	107.770	−	186.662i	0.152002	−	0.263276i	0.931963	+	0.362552i	$0.118095\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.08220	+	0.624806i	0.00151781	+	0.000876306i
$714$		0			0
$715$	−11.0934	−	19.2143i	−0.0155153	−	0.0268732i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			621.418i			0.864281i	0.901806	+	0.432141i	$0.142242\pi$
$719$			621.418i			0.864281i	−0.901806	+	0.432141i	$0.857758\pi$
$720$		0			0
$721$	56.9479			0.0789845
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−85.6534	+	49.4520i	−0.118143	+	0.0682097i
$726$		0			0
$727$	307.517	−	532.635i	0.422994	−	0.732648i	−0.573236	−	0.819390i	$-0.694312\pi$
$727$	307.517	−	532.635i	0.422994	−	0.732648i	0.996231	+	0.0867423i	$0.0276457\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	337.546	+	194.882i	0.461759	+	0.266597i
$732$		0			0
$733$	406.464	+	704.016i	0.554521	+	0.960458i	0.997941	+	0.0641443i	$0.0204318\pi$
$733$	406.464	+	704.016i	0.554521	+	0.960458i	−0.443420	+	0.896314i	$0.646235\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			70.9722i			0.0962988i
$738$		0			0
$739$	577.062			0.780869			0.390434	−	0.920631i	$-0.372325\pi$
$739$	577.062			0.780869			0.390434	+	0.920631i	$0.372325\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	174.558	−	100.781i	0.234937	−	0.135641i	−0.377911	−	0.925842i	$-0.623357\pi$
$743$	174.558	−	100.781i	0.234937	−	0.135641i	0.612848	+	0.790201i	$0.290024\pi$
$744$		0			0
$745$	304.962	−	528.210i	0.409345	−	0.709006i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2015.40	−	1163.59i	−2.69079	−	1.55353i
$750$		0			0
$751$	365.075	+	632.329i	0.486119	+	0.841982i	0.999873	−	0.0159554i	$-0.00507898\pi$
$751$	365.075	+	632.329i	0.486119	+	0.841982i	−0.513754	+	0.857937i	$0.671746\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			512.048i			0.678210i
$756$		0			0
$757$	1123.40			1.48401			0.742006	−	0.670394i	$-0.233875\pi$
$757$	1123.40			1.48401			0.742006	+	0.670394i	$0.233875\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1049.72	−	606.054i	1.37939	−	0.796392i	0.387305	−	0.921952i	$-0.373406\pi$
$761$	1049.72	−	606.054i	1.37939	−	0.796392i	0.992086	+	0.125559i	$0.0400726\pi$
$762$		0			0
$763$	352.240	−	610.097i	0.461651	−	0.799603i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	355.446	+	205.217i	0.463424	+	0.267558i
$768$		0			0
$769$	−183.214	−	317.336i	−0.238250	−	0.412661i	0.721962	−	0.691932i	$-0.243240\pi$
$769$	−183.214	−	317.336i	−0.238250	−	0.412661i	−0.960212	+	0.279271i	$0.909907\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			843.532i			1.09124i	0.838031	+	0.545622i	$0.183707\pi$
$773$			843.532i			1.09124i	−0.838031	+	0.545622i	$0.816293\pi$
$774$		0			0
$775$	0.722784			0.000932625
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−205.247	+	118.499i	−0.263475	+	0.152117i
$780$		0			0
$781$	−154.874	+	268.250i	−0.198303	+	0.343470i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	397.064	+	229.245i	0.505814	+	0.292032i
$786$		0			0
$787$	−13.6421	−	23.6288i	−0.0173343	−	0.0300238i	0.857228	−	0.514937i	$-0.172185\pi$
$787$	−13.6421	−	23.6288i	−0.0173343	−	0.0300238i	−0.874562	+	0.484913i	$0.838851\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1684.98i		−	2.13019i
$792$		0			0
$793$	−179.167			−0.225936
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−326.043	+	188.241i	−0.409088	+	0.236187i	−0.690398	−	0.723430i	$-0.742564\pi$
$797$	−326.043	+	188.241i	−0.409088	+	0.236187i	0.281310	+	0.959617i	$0.409231\pi$
$798$		0			0
$799$	−426.325	+	738.416i	−0.533573	+	0.924176i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	223.403	+	128.982i	0.278210	+	0.160625i
$804$		0			0
$805$	111.808	+	193.657i	0.138892	+	0.240567i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			586.697i			0.725212i	0.931942	+	0.362606i	$0.118113\pi$
$809$			586.697i			0.725212i	−0.931942	+	0.362606i	$0.881887\pi$
$810$		0			0
$811$	−552.276			−0.680981			−0.340491	−	0.940248i	$-0.610593\pi$
$811$	−552.276			−0.680981			−0.340491	+	0.940248i	$0.610593\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−13.0003	+	7.50574i	−0.0159513	+	0.00920950i
$816$		0			0
$817$	502.798	−	870.871i	0.615420	−	1.06594i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−859.255	−	496.091i	−1.04660	−	0.604252i	−0.124901	−	0.992169i	$-0.539861\pi$
$821$	−859.255	−	496.091i	−1.04660	−	0.604252i	−0.921694	+	0.387917i	$0.873195\pi$
$822$		0			0
$823$	584.890	+	1013.06i	0.710680	+	1.23093i	0.964602	+	0.263709i	$0.0849460\pi$
$823$	584.890	+	1013.06i	0.710680	+	1.23093i	−0.253922	+	0.967225i	$0.581721\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			511.184i			0.618118i	0.951043	+	0.309059i	$0.100014\pi$
$827$			511.184i			0.618118i	−0.951043	+	0.309059i	$0.899986\pi$
$828$		0			0
$829$	1383.72			1.66915			0.834575	−	0.550895i	$-0.185713\pi$
$829$	1383.72			1.66915			0.834575	+	0.550895i	$0.185713\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	990.358	−	571.784i	1.18891	−	0.686415i
$834$		0			0
$835$	−47.8712	+	82.9154i	−0.0573308	+	0.0992999i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	399.555	+	230.683i	0.476228	+	0.274950i	0.718843	−	0.695172i	$-0.244672\pi$
$839$	399.555	+	230.683i	0.476228	+	0.274950i	−0.242615	+	0.970123i	$0.578005\pi$
$840$		0			0
$841$	−224.860	−	389.469i	−0.267372	−	0.463102i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			342.005i			0.404739i
$846$		0			0
$847$	−1328.84			−1.56888
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−94.9954	+	54.8456i	−0.111628	+	0.0644484i
$852$		0			0
$853$	591.204	−	1023.99i	0.693087	−	1.20046i	−0.277734	−	0.960658i	$-0.589583\pi$
$853$	591.204	−	1023.99i	0.693087	−	1.20046i	0.970821	−	0.239804i	$-0.0770833\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1006.90	−	581.334i	−1.17491	−	0.678336i	−0.220080	−	0.975482i	$-0.570632\pi$
$857$	−1006.90	−	581.334i	−1.17491	−	0.678336i	−0.954832	+	0.297146i	$0.903965\pi$
$858$		0			0
$859$	−604.709	−	1047.39i	−0.703968	−	1.21931i	−0.967063	−	0.254539i	$-0.918076\pi$
$859$	−604.709	−	1047.39i	−0.703968	−	1.21931i	0.263094	−	0.964770i	$-0.415257\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	156.071i		−	0.180847i	−0.995903	−	0.0904235i	$-0.971178\pi$
$863$		−	156.071i		−	0.180847i	0.995903	−	0.0904235i	$-0.0288221\pi$
$864$		0			0
$865$	−441.474			−0.510375
$866$		0			0
$867$		0			0
$868$		0			0
$869$	44.8230	−	25.8786i	0.0515799	−	0.0297797i
$870$		0			0
$871$	−57.4047	+	99.4279i	−0.0659067	+	0.114154i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	112.012	+	64.6704i	0.128014	+	0.0739090i
$876$		0			0
$877$	85.0718	+	147.349i	0.0970032	+	0.168014i	0.910443	−	0.413635i	$-0.135741\pi$
$877$	85.0718	+	147.349i	0.0970032	+	0.168014i	−0.813440	+	0.581649i	$0.802408\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	746.459i		−	0.847286i	−0.905829	−	0.423643i	$-0.860751\pi$
$881$		−	746.459i		−	0.847286i	0.905829	−	0.423643i	$-0.139249\pi$
$882$		0			0
$883$	−897.185			−1.01606			−0.508032	−	0.861338i	$-0.669627\pi$
$883$	−897.185			−1.01606			−0.508032	+	0.861338i	$0.669627\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	525.929	−	303.645i	0.592930	−	0.342328i	−0.173325	−	0.984865i	$-0.555451\pi$
$887$	525.929	−	303.645i	0.592930	−	0.342328i	0.766255	+	0.642536i	$0.222118\pi$
$888$		0			0
$889$	559.238	−	968.629i	0.629064	−	1.08957i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1905.12	+	1099.92i	2.13339	+	1.23172i
$894$		0			0
$895$	−49.0163	−	84.8988i	−0.0547669	−	0.0948590i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	2.85945i		−	0.00318070i
$900$		0			0
$901$	1134.26			1.25889
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−555.738	+	320.856i	−0.614076	+	0.354537i
$906$		0			0
$907$	−266.140	+	460.968i	−0.293429	+	0.508234i	−0.974618	−	0.223873i	$-0.928130\pi$
$907$	−266.140	+	460.968i	−0.293429	+	0.508234i	0.681189	+	0.732108i	$0.261463\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−51.8970	−	29.9628i	−0.0569671	−	0.0328900i	0.471246	−	0.882002i	$-0.343804\pi$
$911$	−51.8970	−	29.9628i	−0.0569671	−	0.0328900i	−0.528213	+	0.849112i	$0.677138\pi$
$912$		0			0
$913$	−107.681	−	186.509i	−0.117942	−	0.204282i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	265.457i		−	0.289485i
$918$		0			0
$919$	1066.55			1.16056			0.580278	−	0.814419i	$-0.302944\pi$
$919$	1066.55			1.16056			0.580278	+	0.814419i	$0.302944\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−433.940	+	250.535i	−0.470141	+	0.271436i
$924$		0			0
$925$	−31.7231	+	54.9460i	−0.0342952	+	0.0594010i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1568.95	+	905.836i	1.68886	+	0.975066i	0.955390	+	0.295348i	$0.0954356\pi$
$929$	1568.95	+	905.836i	1.68886	+	0.975066i	0.733474	+	0.679718i	$0.237898\pi$
$930$		0			0
$931$	−1475.21	−	2555.13i	−1.58454	−	2.74450i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			74.6528i			0.0798426i
$936$		0			0
$937$	545.004			0.581648			0.290824	−	0.956777i	$-0.406071\pi$
$937$	545.004			0.581648			0.290824	+	0.956777i	$0.406071\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−655.609	+	378.516i	−0.696716	+	0.402249i	−0.806123	−	0.591748i	$-0.798438\pi$
$941$	−655.609	+	378.516i	−0.696716	+	0.402249i	0.109407	+	0.993997i	$0.465105\pi$
$942$		0			0
$943$	−29.4531	+	51.0142i	−0.0312334	+	0.0540978i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−287.457	−	165.963i	−0.303545	−	0.175252i	0.340489	−	0.940248i	$-0.389407\pi$
$947$	−287.457	−	165.963i	−0.303545	−	0.175252i	−0.644034	+	0.764997i	$0.722741\pi$
$948$		0			0
$949$	208.650	+	361.392i	0.219862	+	0.380813i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1583.10i		−	1.66118i	−0.556887	−	0.830588i	$-0.688004\pi$
$953$		−	1583.10i		−	1.66118i	0.556887	−	0.830588i	$-0.311996\pi$
$954$		0			0
$955$	303.456			0.317755
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1766.58	+	1019.93i	−1.84210	+	1.06354i
$960$		0			0
$961$	480.490	−	832.232i	0.499989	−	0.866007i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−248.792	−	143.640i	−0.257816	−	0.148850i
$966$		0			0
$967$	−442.570	−	766.553i	−0.457673	−	0.792713i	0.541165	−	0.840917i	$-0.317984\pi$
$967$	−442.570	−	766.553i	−0.457673	−	0.792713i	−0.998838	+	0.0482038i	$0.984650\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1035.63i		−	1.06656i	−0.845939	−	0.533280i	$-0.820959\pi$
$971$		−	1035.63i		−	1.06656i	0.845939	−	0.533280i	$-0.179041\pi$
$972$		0			0
$973$	−1723.93			−1.77176
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−811.611	+	468.584i	−0.830717	+	0.479615i	−0.854098	−	0.520112i	$-0.825890\pi$
$977$	−811.611	+	468.584i	−0.830717	+	0.479615i	0.0233808	+	0.999727i	$0.492557\pi$
$978$		0			0
$979$	129.660	−	224.578i	0.132442	−	0.229396i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	885.944	+	511.500i	0.901266	+	0.520346i	0.877611	−	0.479374i	$-0.159136\pi$
$983$	885.944	+	511.500i	0.901266	+	0.520346i	0.0236551	+	0.999720i	$0.492470\pi$
$984$		0			0
$985$	44.5188	+	77.1088i	0.0451967	+	0.0782830i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	249.941i		−	0.252721i
$990$		0			0
$991$	−845.732			−0.853412			−0.426706	−	0.904390i	$-0.640326\pi$
$991$	−845.732			−0.853412			−0.426706	+	0.904390i	$0.640326\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	123.945	−	71.5596i	0.124568	−	0.0719192i
$996$		0			0
$997$	−413.561	+	716.308i	−0.414805	+	0.718464i	−0.995408	−	0.0957233i	$-0.969484\pi$
$997$	−413.561	+	716.308i	−0.414805	+	0.718464i	0.580603	+	0.814187i	$0.302817\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.o.g.1241.2		32
3.2	odd	2	inner	1620.3.o.g.1241.14		32
9.2	odd	6		1620.3.g.c.161.9	yes	16
9.4	even	3	inner	1620.3.o.g.701.14		32
9.5	odd	6	inner	1620.3.o.g.701.2		32
9.7	even	3		1620.3.g.c.161.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.g.c.161.1	✓	16	9.7	even	3
1620.3.g.c.161.9	yes	16	9.2	odd	6
1620.3.o.g.701.2		32	9.5	odd	6	inner
1620.3.o.g.701.14		32	9.4	even	3	inner
1620.3.o.g.1241.2		32	1.1	even	1	trivial
1620.3.o.g.1241.14		32	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.93649 + 1.11803i) q^{5} +(5.78429 - 10.0187i) q^{7} +O(q^{10})\)	\(q+(-1.93649 + 1.11803i) q^{5} +(5.78429 - 10.0187i) q^{7} +(-2.14482 - 1.23831i) q^{11} +(-2.00318 - 3.46961i) q^{13} +13.4803i q^{17} +34.7794 q^{19} +(7.48630 - 4.32222i) q^{23} +(2.50000 - 4.33013i) q^{25} +(-17.1307 - 9.89040i) q^{29} +(0.0722784 + 0.125190i) q^{31} +25.8681i q^{35} -12.6892 q^{37} +(-5.90139 + 3.40717i) q^{41} +(14.4568 - 25.0399i) q^{43} +(54.7773 + 31.6257i) q^{47} +(-42.4161 - 73.4669i) q^{49} -84.1419i q^{53} +5.53790 q^{55} +(-88.7205 + 51.2228i) q^{59} +(22.3604 - 38.7293i) q^{61} +(7.75827 + 4.47924i) q^{65} +(-14.3284 - 24.8175i) q^{67} -125.069i q^{71} -104.159 q^{73} +(-24.8125 + 14.3255i) q^{77} +(-10.4491 + 18.0984i) q^{79} +(75.3078 + 43.4790i) q^{83} +(-15.0715 - 26.1046i) q^{85} +104.707i q^{89} -46.3479 q^{91} +(-67.3500 + 38.8845i) q^{95} +(36.6995 - 63.5654i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7}+O(q^{10}) \) 32 * q - 8 * q^7	\( 32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100}) \) 32 * q - 8 * q^7 + 40 * q^13 + 112 * q^19 + 80 * q^25 + 64 * q^31 - 176 * q^37 - 128 * q^43 - 216 * q^49 - 8 * q^61 + 40 * q^67 + 112 * q^73 + 136 * q^79 - 784 * q^91 - 128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more