LMFDB - Embedded newform 1620.3.t.f.269.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(269,1620)]

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, 1, 3]))

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

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

chi := DirichletCharacter("1620.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.t (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:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.4
Character	$\chi$	$=$	1620.269
Dual form			1620.3.t.f.1349.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.58175 + 2.00188i) q^{5} +(9.28893 + 5.36297i) q^{7} +O(q^{10})$	$q+(-4.58175 + 2.00188i) q^{5} +(9.28893 + 5.36297i) q^{7} +(-10.5523 - 6.09235i) q^{11} +(20.2632 - 11.6990i) q^{13} -1.67925 q^{17} +0.234357 q^{19} +(9.93181 + 17.2024i) q^{23} +(16.9849 - 18.3443i) q^{25} +(-29.5094 - 17.0373i) q^{29} +(9.53439 + 16.5140i) q^{31} +(-53.2956 - 5.97645i) q^{35} -32.7402i q^{37} +(4.57102 - 2.63908i) q^{41} +(-45.8845 - 26.4914i) q^{43} +(26.8573 - 46.5182i) q^{47} +(33.0229 + 57.1973i) q^{49} +84.6464 q^{53} +(60.5441 + 6.78927i) q^{55} +(76.4925 - 44.1630i) q^{59} +(32.7982 - 56.8082i) q^{61} +(-69.4210 + 94.1663i) q^{65} +(-86.1488 + 49.7380i) q^{67} +36.8639i q^{71} +79.0196i q^{73} +(-65.3462 - 113.183i) q^{77} +(-17.6981 + 30.6540i) q^{79} +(-13.9232 + 24.1156i) q^{83} +(7.69392 - 3.36167i) q^{85} +152.513i q^{89} +250.965 q^{91} +(-1.07377 + 0.469156i) q^{95} +(84.0834 + 48.5456i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 48 q^{25} + 288 q^{49} - 72 q^{55} - 120 q^{61} - 480 q^{79} + 24 q^{85} + 96 q^{91}+O(q^{100}) $ 48 * q + 48 * q^25 + 288 * q^49 - 72 * q^55 - 120 * q^61 - 480 * q^79 + 24 * q^85 + 96 * q^91

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
$5$	−4.58175	+	2.00188i	−0.916351	+	0.400376i
$5$	−4.58175	+	2.00188i	−0.916351	+	0.400376i
$6$		0			0
$7$	9.28893	+	5.36297i	1.32699	+	0.766138i	0.984833	−	0.173504i	$-0.0555090\pi$
$7$	9.28893	+	5.36297i	1.32699	+	0.766138i	0.342158	+	0.939643i	$0.388842\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−10.5523	−	6.09235i	−0.959297	−	0.553850i	−0.0633403	−	0.997992i	$-0.520175\pi$
$11$	−10.5523	−	6.09235i	−0.959297	−	0.553850i	−0.895957	+	0.444142i	$0.853509\pi$
$12$		0			0
$13$	20.2632	−	11.6990i	1.55871	−	0.899920i	0.561326	−	0.827595i	$-0.310291\pi$
$13$	20.2632	−	11.6990i	1.55871	−	0.899920i	0.997381	−	0.0723256i	$-0.0230421\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.67925			−0.0987796			−0.0493898	−	0.998780i	$-0.515728\pi$
$17$	−1.67925			−0.0987796			−0.0493898	+	0.998780i	$0.515728\pi$
$18$		0			0
$19$	0.234357			0.0123346			0.00616730	−	0.999981i	$-0.498037\pi$
$19$	0.234357			0.0123346			0.00616730	+	0.999981i	$0.498037\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	9.93181	+	17.2024i	0.431818	+	0.747930i	0.997030	−	0.0770154i	$-0.0245391\pi$
$23$	9.93181	+	17.2024i	0.431818	+	0.747930i	−0.565212	+	0.824946i	$0.691206\pi$
$24$		0			0
$25$	16.9849	−	18.3443i	0.679397	−	0.733771i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−29.5094	−	17.0373i	−1.01757	−	0.587492i	−0.104168	−	0.994560i	$-0.533218\pi$
$29$	−29.5094	−	17.0373i	−1.01757	−	0.587492i	−0.913398	+	0.407068i	$0.866551\pi$
$30$		0			0
$31$	9.53439	+	16.5140i	0.307561	+	0.532711i	0.977828	−	0.209409i	$-0.0671539\pi$
$31$	9.53439	+	16.5140i	0.307561	+	0.532711i	−0.670267	+	0.742120i	$0.733821\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−53.2956	−	5.97645i	−1.52273	−	0.170756i
$36$		0			0
$37$		−	32.7402i		−	0.884871i	−0.896800	−	0.442436i	$-0.854114\pi$
$37$		−	32.7402i		−	0.884871i	0.896800	−	0.442436i	$-0.145886\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.57102	−	2.63908i	0.111488	−	0.0643679i	−0.443219	−	0.896413i	$-0.646164\pi$
$41$	4.57102	−	2.63908i	0.111488	−	0.0643679i	0.554707	+	0.832046i	$0.312830\pi$
$42$		0			0
$43$	−45.8845	−	26.4914i	−1.06708	−	0.616080i	−0.139699	−	0.990194i	$-0.544614\pi$
$43$	−45.8845	−	26.4914i	−1.06708	−	0.616080i	−0.927383	+	0.374114i	$0.877947\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	26.8573	−	46.5182i	0.571432	−	0.989749i	−0.424987	−	0.905199i	$-0.639721\pi$
$47$	26.8573	−	46.5182i	0.571432	−	0.989749i	0.996419	−	0.0845497i	$-0.0269452\pi$
$48$		0			0
$49$	33.0229	+	57.1973i	0.673936	+	1.16729i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	84.6464			1.59710			0.798551	−	0.601927i	$-0.205600\pi$
$53$	84.6464			1.59710			0.798551	+	0.601927i	$0.205600\pi$
$54$		0			0
$55$	60.5441	+	6.78927i	1.10080	+	0.123441i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	76.4925	−	44.1630i	1.29648	−	0.748525i	0.316688	−	0.948530i	$-0.397429\pi$
$59$	76.4925	−	44.1630i	1.29648	−	0.748525i	0.979795	+	0.200005i	$0.0640959\pi$
$60$		0			0
$61$	32.7982	−	56.8082i	0.537676	−	0.931282i	−0.461353	−	0.887217i	$-0.652636\pi$
$61$	32.7982	−	56.8082i	0.537676	−	0.931282i	0.999029	−	0.0440652i	$-0.0140309\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−69.4210	+	94.1663i	−1.06802	+	1.44871i
$66$		0			0
$67$	−86.1488	+	49.7380i	−1.28580	+	0.742359i	−0.977903	−	0.209060i	$-0.932960\pi$
$67$	−86.1488	+	49.7380i	−1.28580	+	0.742359i	−0.307900	+	0.951419i	$0.599626\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			36.8639i			0.519210i	0.965715	+	0.259605i	$0.0835924\pi$
$71$			36.8639i			0.519210i	−0.965715	+	0.259605i	$0.916408\pi$
$72$		0			0
$73$			79.0196i			1.08246i	0.840875	+	0.541230i	$0.182041\pi$
$73$			79.0196i			1.08246i	−0.840875	+	0.541230i	$0.817959\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−65.3462	−	113.183i	−0.848652	−	1.46991i
$78$		0			0
$79$	−17.6981	+	30.6540i	−0.224027	+	0.388025i	−0.956027	−	0.293279i	$-0.905254\pi$
$79$	−17.6981	+	30.6540i	−0.224027	+	0.388025i	0.732000	+	0.681304i	$0.238587\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−13.9232	+	24.1156i	−0.167749	+	0.290550i	−0.937628	−	0.347640i	$-0.886983\pi$
$83$	−13.9232	+	24.1156i	−0.167749	+	0.290550i	0.769879	+	0.638190i	$0.220316\pi$
$84$		0			0
$85$	7.69392	−	3.36167i	0.0905167	−	0.0395490i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			152.513i			1.71362i	0.515628	+	0.856812i	$0.327559\pi$
$89$			152.513i			1.71362i	−0.515628	+	0.856812i	$0.672441\pi$
$90$		0			0
$91$	250.965			2.75785
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.07377	+	0.469156i	−0.0113028	+	0.00493848i
$96$		0			0
$97$	84.0834	+	48.5456i	0.866839	+	0.500470i	0.866296	−	0.499530i	$-0.166494\pi$
$97$	84.0834	+	48.5456i	0.866839	+	0.500470i	0.000542334		1.00000i	$0.499827\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−24.7954	−	14.3156i	−0.245499	−	0.141739i	0.372203	−	0.928152i	$-0.378603\pi$
$101$	−24.7954	−	14.3156i	−0.245499	−	0.141739i	−0.617702	+	0.786413i	$0.711936\pi$
$102$		0			0
$103$	37.5687	−	21.6903i	0.364745	−	0.210585i	−0.306415	−	0.951898i	$-0.599130\pi$
$103$	37.5687	−	21.6903i	0.364745	−	0.210585i	0.671160	+	0.741312i	$0.265796\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	161.010			1.50476			0.752381	−	0.658728i	$-0.228905\pi$
$107$	161.010			1.50476			0.752381	+	0.658728i	$0.228905\pi$
$108$		0			0
$109$	142.306			1.30556			0.652779	−	0.757548i	$-0.273603\pi$
$109$	142.306			1.30556			0.652779	+	0.757548i	$0.273603\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	42.0279	+	72.7944i	0.371928	+	0.644198i	0.989862	−	0.142032i	$-0.0453635\pi$
$113$	42.0279	+	72.7944i	0.371928	+	0.644198i	−0.617934	+	0.786230i	$0.712030\pi$
$114$		0			0
$115$	−79.9423	−	58.9348i	−0.695150	−	0.512477i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−15.5985	−	9.00578i	−0.131080	−	0.0756788i
$120$		0			0
$121$	13.7335	+	23.7872i	0.113500	+	0.196588i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−41.0977	+	118.051i	−0.328782	+	0.944406i
$126$		0			0
$127$		−	188.651i		−	1.48544i	−0.669602	−	0.742720i	$-0.733535\pi$
$127$		−	188.651i		−	1.48544i	0.669602	−	0.742720i	$-0.266465\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	91.8070	−	53.0048i	0.700817	−	0.404617i	−0.106835	−	0.994277i	$-0.534072\pi$
$131$	91.8070	−	53.0048i	0.700817	−	0.404617i	0.807651	+	0.589660i	$0.200738\pi$
$132$		0			0
$133$	2.17693	+	1.25685i	0.0163679	+	0.00945001i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	128.067	−	221.818i	0.934793	−	1.61911i	0.159790	−	0.987151i	$-0.448918\pi$
$137$	128.067	−	221.818i	0.934793	−	1.61911i	0.775003	−	0.631958i	$-0.217748\pi$
$138$		0			0
$139$	90.8442	+	157.347i	0.653555	+	1.13199i	0.982254	+	0.187556i	$0.0600567\pi$
$139$	90.8442	+	157.347i	0.653555	+	1.13199i	−0.328698	+	0.944435i	$0.606610\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−285.097			−1.99368
$144$		0			0
$145$	169.311	+	18.9862i	1.16766	+	0.130939i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	249.248	−	143.904i	1.67281	−	0.965797i	0.706754	−	0.707460i	$-0.250159\pi$
$149$	249.248	−	143.904i	1.67281	−	0.965797i	0.966055	−	0.258337i	$-0.0831745\pi$
$150$		0			0
$151$	7.57042	−	13.1124i	0.0501353	−	0.0868368i	−0.839869	−	0.542790i	$-0.817368\pi$
$151$	7.57042	−	13.1124i	0.0501353	−	0.0868368i	0.890004	+	0.455953i	$0.150701\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−76.7434	−	56.5766i	−0.495119	−	0.365010i
$156$		0			0
$157$	78.0606	−	45.0683i	0.497202	−	0.287059i	−0.230356	−	0.973107i	$-0.573989\pi$
$157$	78.0606	−	45.0683i	0.497202	−	0.287059i	0.727557	+	0.686047i	$0.240656\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			213.056i			1.32333i
$162$		0			0
$163$		−	131.354i		−	0.805851i	−0.915233	−	0.402925i	$-0.867993\pi$
$163$		−	131.354i		−	0.805851i	0.915233	−	0.402925i	$-0.132007\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	77.9700	+	135.048i	0.466886	+	0.808670i	0.999284	−	0.0378235i	$-0.0120425\pi$
$167$	77.9700	+	135.048i	0.466886	+	0.808670i	−0.532398	+	0.846494i	$0.678709\pi$
$168$		0			0
$169$	189.231	−	327.758i	1.11971	−	1.93940i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−67.1594	+	116.323i	−0.388205	+	0.672390i	−0.992208	−	0.124592i	$-0.960238\pi$
$173$	−67.1594	+	116.323i	−0.388205	+	0.672390i	0.604004	+	0.796982i	$0.293571\pi$
$174$		0			0
$175$	256.152	−	79.3090i	1.46372	−	0.453194i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	201.999i		−	1.12849i	−0.825608	−	0.564244i	$-0.809168\pi$
$179$		−	201.999i		−	1.12849i	0.825608	−	0.564244i	$-0.190832\pi$
$180$		0			0
$181$	6.27313			0.0346582			0.0173291	−	0.999850i	$-0.494484\pi$
$181$	6.27313			0.0346582			0.0173291	+	0.999850i	$0.494484\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	65.5421	+	150.008i	0.354282	+	0.810853i
$186$		0			0
$187$	17.7199	+	10.2306i	0.0947589	+	0.0547091i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−257.787	−	148.834i	−1.34967	−	0.779233i	−0.361469	−	0.932384i	$-0.617725\pi$
$191$	−257.787	−	148.834i	−1.34967	−	0.779233i	−0.988203	+	0.153151i	$0.951058\pi$
$192$		0			0
$193$	−48.2175	+	27.8384i	−0.249831	+	0.144240i	−0.619687	−	0.784849i	$-0.712740\pi$
$193$	−48.2175	+	27.8384i	−0.249831	+	0.144240i	0.369856	+	0.929089i	$0.379407\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−22.9424			−0.116459			−0.0582296	−	0.998303i	$-0.518546\pi$
$197$	−22.9424			−0.116459			−0.0582296	+	0.998303i	$0.518546\pi$
$198$		0			0
$199$	204.154			1.02590			0.512951	−	0.858418i	$-0.328552\pi$
$199$	204.154			1.02590			0.512951	+	0.858418i	$0.328552\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−182.741	−	316.516i	−0.900200	−	1.55919i
$204$		0			0
$205$	−15.6602	+	21.2423i	−0.0763911	+	0.103621i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.47300	−	1.42779i	−0.0118325	−	0.00683152i
$210$		0			0
$211$	178.897	+	309.859i	0.847854	+	1.46853i	0.883120	+	0.469148i	$0.155439\pi$
$211$	178.897	+	309.859i	0.847854	+	1.46853i	−0.0352657	+	0.999378i	$0.511228\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	263.264	+	29.5219i	1.22449	+	0.137311i
$216$		0			0
$217$			204.531i			0.942537i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−34.0270	+	19.6455i	−0.153968	+	0.0888937i
$222$		0			0
$223$	211.390	+	122.046i	0.947937	+	0.547292i	0.892440	−	0.451167i	$-0.148992\pi$
$223$	211.390	+	122.046i	0.947937	+	0.547292i	0.0554978	+	0.998459i	$0.482325\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−112.604	+	195.036i	−0.496053	+	0.859189i	−0.999990	−	0.00455151i	$-0.998551\pi$
$227$	−112.604	+	195.036i	−0.496053	+	0.859189i	0.503937	+	0.863741i	$0.331885\pi$
$228$		0			0
$229$	22.5813	+	39.1119i	0.0986082	+	0.170794i	0.911109	−	0.412166i	$-0.135228\pi$
$229$	22.5813	+	39.1119i	0.0986082	+	0.170794i	−0.812501	+	0.582960i	$0.801894\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	55.4732			0.238083			0.119041	−	0.992889i	$-0.462018\pi$
$233$	55.4732			0.238083			0.119041	+	0.992889i	$0.462018\pi$
$234$		0			0
$235$	−29.9296	+	266.900i	−0.127360	+	1.13575i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.5094	+	6.06761i	−0.0439724	+	0.0253875i	−0.521825	−	0.853053i	$-0.674749\pi$
$239$	−10.5094	+	6.06761i	−0.0439724	+	0.0253875i	0.477853	+	0.878440i	$0.341415\pi$
$240$		0			0
$241$	−26.4497	+	45.8123i	−0.109750	+	0.190092i	−0.915669	−	0.401933i	$-0.868338\pi$
$241$	−26.4497	+	45.8123i	−0.109750	+	0.190092i	0.805919	+	0.592026i	$0.201672\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−265.805	−	195.956i	−1.08492	−	0.799820i
$246$		0			0
$247$	4.74883	−	2.74174i	0.0192260	−	0.0111002i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			44.3567i			0.176720i	0.996089	+	0.0883599i	$0.0281626\pi$
$251$			44.3567i			0.176720i	−0.996089	+	0.0883599i	$0.971837\pi$
$252$		0			0
$253$		−	242.032i		−	0.956649i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−85.6501	−	148.350i	−0.333269	−	0.577239i	0.649882	−	0.760035i	$-0.274818\pi$
$257$	−85.6501	−	148.350i	−0.333269	−	0.577239i	−0.983151	+	0.182796i	$0.941485\pi$
$258$		0			0
$259$	175.585	−	304.122i	0.677934	−	1.17422i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−61.5235	+	106.562i	−0.233929	+	0.405178i	−0.958961	−	0.283538i	$-0.908492\pi$
$263$	−61.5235	+	106.562i	−0.233929	+	0.405178i	0.725032	+	0.688716i	$0.241825\pi$
$264$		0			0
$265$	−387.829	+	169.452i	−1.46351	+	0.639442i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	128.164i		−	0.476446i	−0.971210	−	0.238223i	$-0.923435\pi$
$269$		−	128.164i		−	0.476446i	0.971210	−	0.238223i	$-0.0765649\pi$
$270$		0			0
$271$	295.707			1.09117			0.545586	−	0.838055i	$-0.316307\pi$
$271$	295.707			1.09117			0.545586	+	0.838055i	$0.316307\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−290.989	+	90.0953i	−1.05814	+	0.327619i
$276$		0			0
$277$	−420.027	−	242.503i	−1.51634	−	0.875462i	−0.999816	−	0.0191923i	$-0.993891\pi$
$277$	−420.027	−	242.503i	−1.51634	−	0.875462i	−0.516529	−	0.856270i	$-0.672776\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−217.686	−	125.681i	−0.774684	−	0.447264i	0.0598587	−	0.998207i	$-0.480935\pi$
$281$	−217.686	−	125.681i	−0.774684	−	0.447264i	−0.834543	+	0.550943i	$0.814268\pi$
$282$		0			0
$283$	−239.790	+	138.443i	−0.847314	+	0.489197i	−0.859744	−	0.510726i	$-0.829377\pi$
$283$	−239.790	+	138.443i	−0.847314	+	0.489197i	0.0124298	+	0.999923i	$0.496043\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	56.6133			0.197259
$288$		0			0
$289$	−286.180			−0.990243
$290$		0			0
$291$		0			0
$292$		0			0
$293$	87.0957	+	150.854i	0.297255	+	0.514861i	0.975507	−	0.219969i	$-0.0705955\pi$
$293$	87.0957	+	150.854i	0.297255	+	0.514861i	−0.678252	+	0.734829i	$0.737262\pi$
$294$		0			0
$295$	−262.061	+	355.473i	−0.888341	+	1.20499i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	402.500	+	232.384i	1.34615	+	0.777203i
$300$		0			0
$301$	−284.146	−	492.155i	−0.944005	−	1.63507i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−36.5501	+	325.939i	−0.119836	+	1.06865i
$306$		0			0
$307$			343.344i			1.11838i	0.829038	+	0.559192i	$0.188888\pi$
$307$			343.344i			1.11838i	−0.829038	+	0.559192i	$0.811112\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−199.491	+	115.176i	−0.641449	+	0.370341i	−0.785172	−	0.619277i	$-0.787426\pi$
$311$	−199.491	+	115.176i	−0.641449	+	0.370341i	0.143724	+	0.989618i	$0.454092\pi$
$312$		0			0
$313$	−347.732	−	200.763i	−1.11097	−	0.641416i	−0.171886	−	0.985117i	$-0.554986\pi$
$313$	−347.732	−	200.763i	−1.11097	−	0.641416i	−0.939079	+	0.343701i	$0.888319\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−208.287	+	360.764i	−0.657057	+	1.13806i	0.324316	+	0.945949i	$0.394866\pi$
$317$	−208.287	+	360.764i	−0.657057	+	1.13806i	−0.981374	+	0.192108i	$0.938468\pi$
$318$		0			0
$319$	207.594	+	359.563i	0.650765	+	1.12716i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.393545			−0.00121841
$324$		0			0
$325$	129.560	−	570.420i	0.398647	−	1.75514i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	498.951	−	288.070i	1.51657	−	0.875592i
$330$		0			0
$331$	−100.674	+	174.373i	−0.304151	+	0.526806i	−0.977072	−	0.212909i	$-0.931706\pi$
$331$	−100.674	+	174.373i	−0.304151	+	0.526806i	0.672921	+	0.739715i	$0.265040\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	295.143	−	400.347i	0.881024	−	1.19507i
$336$		0			0
$337$	38.8060	−	22.4046i	0.115151	−	0.0664826i	−0.441318	−	0.897351i	$-0.645489\pi$
$337$	38.8060	−	22.4046i	0.115151	−	0.0664826i	0.556469	+	0.830868i	$0.312156\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	232.347i		−	0.681371i
$342$		0			0
$343$			182.832i			0.533037i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	227.065	+	393.288i	0.654365	+	1.13339i	0.982053	+	0.188607i	$0.0603973\pi$
$347$	227.065	+	393.288i	0.654365	+	1.13339i	−0.327688	+	0.944786i	$0.606269\pi$
$348$		0			0
$349$	29.2232	−	50.6161i	0.0837341	−	0.145032i	−0.821117	−	0.570760i	$-0.806649\pi$
$349$	29.2232	−	50.6161i	0.0837341	−	0.145032i	0.904851	+	0.425728i	$0.139982\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	280.574	−	485.968i	0.794827	−	1.37668i	−0.128122	−	0.991758i	$-0.540895\pi$
$353$	280.574	−	485.968i	0.794827	−	1.37668i	0.922949	−	0.384922i	$-0.125772\pi$
$354$		0			0
$355$	−73.7972	−	168.901i	−0.207880	−	0.475779i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	139.320i		−	0.388079i	−0.980994	−	0.194040i	$-0.937841\pi$
$359$		−	139.320i		−	0.388079i	0.980994	−	0.194040i	$-0.0621590\pi$
$360$		0			0
$361$	−360.945			−0.999848
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−158.188	−	362.048i	−0.433391	−	0.991913i
$366$		0			0
$367$	294.621	+	170.100i	0.802782	+	0.463486i	0.844443	−	0.535645i	$-0.179932\pi$
$367$	294.621	+	170.100i	0.802782	+	0.463486i	−0.0416610	+	0.999132i	$0.513265\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	786.275	+	453.956i	2.11934	+	1.22360i
$372$		0			0
$373$	335.661	−	193.794i	0.899897	−	0.519556i	0.0227299	−	0.999742i	$-0.492764\pi$
$373$	335.661	−	193.794i	0.899897	−	0.519556i	0.877167	+	0.480186i	$0.159431\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−797.273			−2.11478
$378$		0			0
$379$	−546.876			−1.44295			−0.721473	−	0.692443i	$-0.756535\pi$
$379$	−546.876			−1.44295			−0.721473	+	0.692443i	$0.756535\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.8567	+	24.0005i	0.0361794	+	0.0626646i	0.883548	−	0.468340i	$-0.155148\pi$
$383$	13.8567	+	24.0005i	0.0361794	+	0.0626646i	−0.847369	+	0.531005i	$0.821815\pi$
$384$		0			0
$385$	525.979	+	387.761i	1.36618	+	1.00717i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	418.857	+	241.827i	1.07675	+	0.621664i	0.930019	−	0.367512i	$-0.119790\pi$
$389$	418.857	+	241.827i	1.07675	+	0.621664i	0.146735	+	0.989176i	$0.453123\pi$
$390$		0			0
$391$	−16.6780	−	28.8872i	−0.0426548	−	0.0738802i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	19.7226	−	175.879i	0.0499307	−	0.445262i
$396$		0			0
$397$		−	347.456i		−	0.875204i	−0.899169	−	0.437602i	$-0.855828\pi$
$397$		−	347.456i		−	0.875204i	0.899169	−	0.437602i	$-0.144172\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−240.121	+	138.634i	−0.598806	+	0.345721i	−0.768572	−	0.639763i	$-0.779032\pi$
$401$	−240.121	+	138.634i	−0.598806	+	0.345721i	0.169765	+	0.985484i	$0.445699\pi$
$402$		0			0
$403$	386.394	+	223.085i	0.958795	+	0.553561i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−199.465	+	345.484i	−0.490086	+	0.848854i
$408$		0			0
$409$	−263.838	−	456.981i	−0.645081	−	1.11731i	−0.984283	−	0.176599i	$-0.943490\pi$
$409$	−263.838	−	456.981i	−0.645081	−	1.11731i	0.339202	−	0.940713i	$-0.389843\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	947.378			2.29389
$414$		0			0
$415$	15.5159	−	138.364i	0.0373876	−	0.333408i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	551.429	−	318.368i	1.31606	−	0.759827i	0.332967	−	0.942938i	$-0.391950\pi$
$419$	551.429	−	318.368i	1.31606	−	0.759827i	0.983092	+	0.183111i	$0.0586167\pi$
$420$		0			0
$421$	−15.3331	+	26.5578i	−0.0364207	+	0.0630825i	−0.883661	−	0.468127i	$-0.844929\pi$
$421$	−15.3331	+	26.5578i	−0.0364207	+	0.0630825i	0.847240	+	0.531210i	$0.178262\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−28.5220	+	30.8046i	−0.0671106	+	0.0724815i
$426$		0			0
$427$	609.321	−	351.792i	1.42698	−	0.823868i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			381.687i			0.885584i	0.896624	+	0.442792i	$0.146012\pi$
$431$			381.687i			0.885584i	−0.896624	+	0.442792i	$0.853988\pi$
$432$		0			0
$433$			691.143i			1.59617i	0.602543	+	0.798086i	$0.294154\pi$
$433$			691.143i			1.59617i	−0.602543	+	0.798086i	$0.705846\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.32759	+	4.03151i	0.00532630	+	0.00922542i
$438$		0			0
$439$	−245.061	+	424.458i	−0.558225	+	0.966874i	0.439420	+	0.898282i	$0.355184\pi$
$439$	−245.061	+	424.458i	−0.558225	+	0.966874i	−0.997645	+	0.0685920i	$0.978149\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	287.938	−	498.724i	0.649974	−	1.12579i	−0.333155	−	0.942872i	$-0.608113\pi$
$443$	287.938	−	498.724i	0.649974	−	1.12579i	0.983129	−	0.182916i	$-0.0585535\pi$
$444$		0			0
$445$	−305.312	−	698.775i	−0.686095	−	1.57028i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	515.232i		−	1.14751i	−0.819027	−	0.573755i	$-0.805486\pi$
$449$		−	515.232i		−	1.14751i	0.819027	−	0.573755i	$-0.194514\pi$
$450$		0			0
$451$	−64.3129			−0.142601
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1149.86	+	502.402i	−2.52716	+	1.10418i
$456$		0			0
$457$	−83.8428	−	48.4067i	−0.183463	−	0.105923i	0.405455	−	0.914115i	$-0.367113\pi$
$457$	−83.8428	−	48.4067i	−0.183463	−	0.105923i	−0.588919	+	0.808192i	$0.700446\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−539.356	−	311.398i	−1.16997	−	0.675483i	−0.216297	−	0.976328i	$-0.569398\pi$
$461$	−539.356	−	311.398i	−1.16997	−	0.675483i	−0.953673	+	0.300845i	$0.902731\pi$
$462$		0			0
$463$	87.0283	−	50.2458i	0.187966	−	0.108522i	−0.403064	−	0.915172i	$-0.632055\pi$
$463$	87.0283	−	50.2458i	0.187966	−	0.108522i	0.591030	+	0.806650i	$0.298721\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−224.688			−0.481131			−0.240566	−	0.970633i	$-0.577333\pi$
$467$	−224.688			−0.481131			−0.240566	+	0.970633i	$0.577333\pi$
$468$		0			0
$469$	−1066.97			−2.27500
$470$		0			0
$471$		0			0
$472$		0			0
$473$	322.790	+	559.090i	0.682432	+	1.18201i
$474$		0			0
$475$	3.98055	−	4.29911i	0.00838010	−	0.00905077i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	605.459	+	349.562i	1.26401	+	0.729774i	0.973847	−	0.227205i	$-0.0729586\pi$
$479$	605.459	+	349.562i	1.26401	+	0.729774i	0.290159	+	0.956979i	$0.406292\pi$
$480$		0			0
$481$	−383.027	−	663.422i	−0.796314	−	1.37926i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−482.432	−	54.0988i	−0.994705	−	0.111544i
$486$		0			0
$487$			222.666i			0.457220i	0.973518	+	0.228610i	$0.0734181\pi$
$487$			222.666i			0.457220i	−0.973518	+	0.228610i	$0.926582\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−789.824	+	456.005i	−1.60860	+	0.928728i	−0.618920	+	0.785454i	$0.712430\pi$
$491$	−789.824	+	456.005i	−1.60860	+	0.928728i	−0.989683	+	0.143274i	$0.954237\pi$
$492$		0			0
$493$	49.5537	+	28.6099i	0.100515	+	0.0580322i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−197.700	+	342.427i	−0.397787	+	0.688987i
$498$		0			0
$499$	392.875	+	680.479i	0.787324	+	1.36369i	0.927601	+	0.373573i	$0.121868\pi$
$499$	392.875	+	680.479i	0.787324	+	1.36369i	−0.140276	+	0.990112i	$0.544799\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−288.196			−0.572955			−0.286477	−	0.958087i	$-0.592484\pi$
$503$	−288.196			−0.572955			−0.286477	+	0.958087i	$0.592484\pi$
$504$		0			0
$505$	142.265	+	15.9532i	0.281712	+	0.0315905i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	412.825	−	238.345i	0.811052	−	0.468261i	−0.0362693	−	0.999342i	$-0.511547\pi$
$509$	412.825	−	238.345i	0.811052	−	0.468261i	0.847321	+	0.531081i	$0.178214\pi$
$510$		0			0
$511$	−423.779	+	734.007i	−0.829314	+	1.43641i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−128.709	+	174.588i	−0.249921	+	0.339005i
$516$		0			0
$517$	−566.811	+	327.248i	−1.09635	+	0.632975i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	337.377i		−	0.647557i	−0.946133	−	0.323778i	$-0.895047\pi$
$521$		−	337.377i		−	0.647557i	0.946133	−	0.323778i	$-0.104953\pi$
$522$		0			0
$523$		−	820.192i		−	1.56824i	−0.620607	−	0.784122i	$-0.713114\pi$
$523$		−	820.192i		−	1.56824i	0.620607	−	0.784122i	$-0.286886\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−16.0106	−	27.7312i	−0.0303807	−	0.0526210i
$528$		0			0
$529$	67.2184	−	116.426i	0.127067	−	0.220086i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	61.7490	−	106.952i	0.115852	−	0.200661i
$534$		0			0
$535$	−737.706	+	322.322i	−1.37889	+	0.602471i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	804.748i		−	1.49304i
$540$		0			0
$541$	−770.684			−1.42456			−0.712278	−	0.701898i	$-0.752336\pi$
$541$	−770.684			−1.42456			−0.712278	+	0.701898i	$0.752336\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−652.011	+	284.880i	−1.19635	+	0.522715i
$546$		0			0
$547$	−750.002	−	433.014i	−1.37112	−	0.791616i	−0.380050	−	0.924966i	$-0.624093\pi$
$547$	−750.002	−	433.014i	−1.37112	−	0.791616i	−0.991069	+	0.133350i	$0.957427\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.91575	−	3.99281i	−0.0125513	−	0.00724647i
$552$		0			0
$553$	−328.793	+	189.829i	−0.594562	+	0.343271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	245.790			0.441275			0.220637	−	0.975356i	$-0.429186\pi$
$557$	245.790			0.441275			0.220637	+	0.975356i	$0.429186\pi$
$558$		0			0
$559$	−1239.69			−2.21769
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−169.294	−	293.226i	−0.300700	−	0.520829i	0.675594	−	0.737274i	$-0.263887\pi$
$563$	−169.294	−	293.226i	−0.300700	−	0.520829i	−0.976295	+	0.216445i	$0.930554\pi$
$564$		0			0
$565$	−338.287	−	249.391i	−0.598738	−	0.441400i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−265.584	−	153.335i	−0.466755	−	0.269481i	0.248125	−	0.968728i	$-0.420186\pi$
$569$	−265.584	−	153.335i	−0.466755	−	0.269481i	−0.714880	+	0.699247i	$0.753519\pi$
$570$		0			0
$571$	−9.09849	−	15.7590i	−0.0159343	−	0.0275990i	0.857948	−	0.513736i	$-0.171739\pi$
$571$	−9.09849	−	15.7590i	−0.0159343	−	0.0275990i	−0.873883	+	0.486137i	$0.838406\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	484.256	+	109.990i	0.842185	+	0.191287i
$576$		0			0
$577$			766.613i			1.32862i	0.747458	+	0.664309i	$0.231274\pi$
$577$			766.613i			1.32862i	−0.747458	+	0.664309i	$0.768726\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−258.663	+	149.339i	−0.445203	+	0.257038i
$582$		0			0
$583$	−893.211	−	515.696i	−1.53209	−	0.884555i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	424.596	−	735.422i	0.723332	−	1.25285i	−0.236324	−	0.971674i	$-0.575943\pi$
$587$	424.596	−	735.422i	0.723332	−	1.25285i	0.959657	−	0.281174i	$-0.0907238\pi$
$588$		0			0
$589$	2.23445	+	3.87019i	0.00379364	+	0.00657078i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	91.0522			0.153545			0.0767725	−	0.997049i	$-0.475538\pi$
$593$	91.0522			0.153545			0.0767725	+	0.997049i	$0.475538\pi$
$594$		0			0
$595$	89.4968	+	10.0360i	0.150415	+	0.0168672i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	49.1183	−	28.3585i	0.0820005	−	0.0473430i	−0.458439	−	0.888726i	$-0.651591\pi$
$599$	49.1183	−	28.3585i	0.0820005	−	0.0473430i	0.540440	+	0.841383i	$0.318258\pi$
$600$		0			0
$601$	107.324	−	185.890i	0.178575	−	0.309301i	−0.762818	−	0.646614i	$-0.776185\pi$
$601$	107.324	−	185.890i	0.178575	−	0.309301i	0.941393	+	0.337313i	$0.109518\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−110.543	−	81.4941i	−0.182716	−	0.134701i
$606$		0			0
$607$	−235.162	+	135.771i	−0.387417	+	0.223675i	−0.681040	−	0.732246i	$-0.738472\pi$
$607$	−235.162	+	135.771i	−0.387417	+	0.223675i	0.293623	+	0.955921i	$0.405139\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1256.81i		−	2.05697i
$612$		0			0
$613$			576.529i			0.940504i	0.882532	+	0.470252i	$0.155837\pi$
$613$			576.529i			0.940504i	−0.882532	+	0.470252i	$0.844163\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−171.771	−	297.517i	−0.278398	−	0.482199i	0.692589	−	0.721332i	$-0.256470\pi$
$617$	−171.771	−	297.517i	−0.278398	−	0.482199i	−0.970987	+	0.239134i	$0.923137\pi$
$618$		0			0
$619$	196.673	−	340.648i	0.317727	−	0.550319i	−0.662286	−	0.749251i	$-0.730414\pi$
$619$	196.673	−	340.648i	0.317727	−	0.550319i	0.980013	+	0.198931i	$0.0637471\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−817.920	+	1416.68i	−1.31287	+	2.27396i
$624$		0			0
$625$	−48.0240	−	623.152i	−0.0768383	−	0.997044i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			54.9791i			0.0874072i
$630$		0			0
$631$	−224.300			−0.355467			−0.177734	−	0.984079i	$-0.556877\pi$
$631$	−224.300			−0.355467			−0.177734	+	0.984079i	$0.556877\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	377.657	+	864.352i	0.594735	+	1.36118i
$636$		0			0
$637$	1338.30	+	772.667i	2.10094	+	1.21298i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	227.262	+	131.210i	0.354544	+	0.204696i	0.666685	−	0.745340i	$-0.267713\pi$
$641$	227.262	+	131.210i	0.354544	+	0.204696i	−0.312141	+	0.950036i	$0.601046\pi$
$642$		0			0
$643$	−282.643	+	163.184i	−0.439569	+	0.253785i	−0.703415	−	0.710779i	$-0.748342\pi$
$643$	−282.643	+	163.184i	−0.439569	+	0.253785i	0.263846	+	0.964565i	$0.415009\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	438.462			0.677685			0.338842	−	0.940843i	$-0.389965\pi$
$647$	438.462			0.677685			0.338842	+	0.940843i	$0.389965\pi$
$648$		0			0
$649$	−1076.23			−1.65828
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−439.932	−	761.984i	−0.673709	−	1.16690i	−0.976845	−	0.213950i	$-0.931367\pi$
$653$	−439.932	−	761.984i	−0.673709	−	1.16690i	0.303136	−	0.952947i	$-0.401966\pi$
$654$		0			0
$655$	−314.528	+	426.642i	−0.480195	+	0.651361i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	888.871	+	513.190i	1.34882	+	0.778741i	0.988082	−	0.153927i	$-0.0491921\pi$
$659$	888.871	+	513.190i	1.34882	+	0.778741i	0.360736	+	0.932668i	$0.382525\pi$
$660$		0			0
$661$	−253.494	−	439.065i	−0.383501	−	0.664243i	0.608059	−	0.793892i	$-0.291948\pi$
$661$	−253.494	−	439.065i	−0.383501	−	0.664243i	−0.991560	+	0.129649i	$0.958615\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12.4902	−	1.40063i	−0.0187823	−	0.00210620i
$666$		0			0
$667$		−	676.843i		−	1.01476i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−692.191	+	399.637i	−1.03158	+	0.595584i
$672$		0			0
$673$	454.331	+	262.308i	0.675084	+	0.389760i	0.798000	−	0.602657i	$-0.205891\pi$
$673$	454.331	+	262.308i	0.675084	+	0.389760i	−0.122916	+	0.992417i	$0.539225\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	22.1441	−	38.3547i	0.0327091	−	0.0566539i	−0.849207	−	0.528059i	$-0.822920\pi$
$677$	22.1441	−	38.3547i	0.0327091	−	0.0566539i	0.881917	+	0.471406i	$0.156253\pi$
$678$		0			0
$679$	520.697	+	901.873i	0.766858	+	1.32824i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1020.38			−1.49396			−0.746982	−	0.664845i	$-0.768498\pi$
$683$	−1020.38			−1.49396			−0.746982	+	0.664845i	$0.768498\pi$
$684$		0			0
$685$	−142.716	+	1272.69i	−0.208345	+	1.85794i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1715.21	−	990.275i	2.48941	−	1.43726i
$690$		0			0
$691$	−72.5955	+	125.739i	−0.105059	+	0.181967i	−0.913762	−	0.406250i	$-0.866836\pi$
$691$	−72.5955	+	125.739i	−0.105059	+	0.181967i	0.808704	+	0.588216i	$0.200170\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−731.215	−	539.065i	−1.05211	−	0.775633i
$696$		0			0
$697$	−7.67590	+	4.43169i	−0.0110128	+	0.00635823i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1136.43i			1.62116i	0.585629	+	0.810579i	$0.300848\pi$
$701$			1136.43i			1.62116i	−0.585629	+	0.810579i	$0.699152\pi$
$702$		0			0
$703$		−	7.67292i		−	0.0109145i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−153.549	−	265.954i	−0.217183	−	0.376172i
$708$		0			0
$709$	402.678	−	697.459i	0.567953	−	0.983723i	−0.428816	−	0.903392i	$-0.641069\pi$
$709$	402.678	−	697.459i	0.567953	−	0.983723i	0.996768	−	0.0803307i	$-0.0255976\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−189.387	+	328.029i	−0.265620	+	0.460068i
$714$		0			0
$715$	1306.24	−	570.730i	1.82691	−	0.798224i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1266.26i		−	1.76115i	−0.473910	−	0.880573i	$-0.657158\pi$
$719$		−	1266.26i		−	1.76115i	0.473910	−	0.880573i	$-0.342842\pi$
$720$		0			0
$721$	465.298			0.645351
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−813.751	+	251.951i	−1.12242	+	0.347519i
$726$		0			0
$727$	−211.476	−	122.095i	−0.290888	−	0.167944i	0.347454	−	0.937697i	$-0.387046\pi$
$727$	−211.476	−	122.095i	−0.290888	−	0.167944i	−0.638342	+	0.769753i	$0.720380\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	77.0517	+	44.4858i	0.105406	+	0.0608561i
$732$		0			0
$733$	569.850	−	329.003i	0.777421	−	0.448844i	−0.0580943	−	0.998311i	$-0.518502\pi$
$733$	569.850	−	329.003i	0.777421	−	0.448844i	0.835516	+	0.549467i	$0.185169\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1212.09			1.64462
$738$		0			0
$739$	178.113			0.241019			0.120509	−	0.992712i	$-0.461547\pi$
$739$	178.113			0.241019			0.120509	+	0.992712i	$0.461547\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	206.395	+	357.486i	0.277786	+	0.481139i	0.970834	−	0.239752i	$-0.0770661\pi$
$743$	206.395	+	357.486i	0.277786	+	0.481139i	−0.693048	+	0.720891i	$0.743733\pi$
$744$		0			0
$745$	−853.917	+	1158.30i	−1.14620	+	1.55476i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1495.61	+	863.489i	1.99681	+	1.15286i
$750$		0			0
$751$	−199.806	−	346.075i	−0.266054	−	0.460819i	0.701785	−	0.712388i	$-0.252387\pi$
$751$	−199.806	−	346.075i	−0.266054	−	0.460819i	−0.967839	+	0.251570i	$0.919053\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.43642	+	75.2327i	−0.0111741	+	0.0996460i
$756$		0			0
$757$		−	1032.23i		−	1.36358i	−0.731546	−	0.681792i	$-0.761201\pi$
$757$		−	1032.23i		−	1.36358i	0.731546	−	0.681792i	$-0.238799\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−248.216	+	143.308i	−0.326171	+	0.188315i	−0.654140	−	0.756373i	$-0.726969\pi$
$761$	−248.216	+	143.308i	−0.326171	+	0.188315i	0.327969	+	0.944689i	$0.393636\pi$
$762$		0			0
$763$	1321.87	+	763.182i	1.73246	+	1.00024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1033.32	−	1789.77i	1.34722	−	2.33346i
$768$		0			0
$769$	−342.621	−	593.437i	−0.445541	−	0.771700i	0.552549	−	0.833481i	$-0.313655\pi$
$769$	−342.621	−	593.437i	−0.445541	−	0.771700i	−0.998090	+	0.0617809i	$0.980322\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1155.89			−1.49533			−0.747663	−	0.664079i	$-0.768824\pi$
$773$	−1155.89			−1.49533			−0.747663	+	0.664079i	$0.768824\pi$
$774$		0			0
$775$	464.879	+	105.589i	0.599844	+	0.136243i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.07125	−	0.618488i	0.00137516	−	0.000793952i
$780$		0			0
$781$	224.588	−	388.998i	0.287565	−	0.498077i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−267.433	+	362.760i	−0.340679	+	0.462115i
$786$		0			0
$787$	8.38210	−	4.83941i	0.0106507	−	0.00614919i	−0.494665	−	0.869084i	$-0.664709\pi$
$787$	8.38210	−	4.83941i	0.0106507	−	0.00614919i	0.505316	+	0.862934i	$0.331376\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			901.577i			1.13979i
$792$		0			0
$793$		−	1534.82i		−	1.93546i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	757.597	+	1312.20i	0.950561	+	1.64642i	0.744215	+	0.667940i	$0.232824\pi$
$797$	757.597	+	1312.20i	0.950561	+	1.64642i	0.206346	+	0.978479i	$0.433843\pi$
$798$		0			0
$799$	−45.1002	+	78.1158i	−0.0564458	+	0.0977670i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	481.415	−	833.835i	0.599521	−	1.03840i
$804$		0			0
$805$	−426.513	−	976.170i	−0.529830	−	1.21263i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			628.159i			0.776464i	0.921562	+	0.388232i	$0.126914\pi$
$809$			628.159i			0.776464i	−0.921562	+	0.388232i	$0.873086\pi$
$810$		0			0
$811$	1027.82			1.26735			0.633675	−	0.773600i	$-0.281546\pi$
$811$	1027.82			1.26735			0.633675	+	0.773600i	$0.281546\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	262.955	+	601.830i	0.322644	+	0.738442i
$816$		0			0
$817$	−10.7534	−	6.20847i	−0.0131620	−	0.00759910i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−969.780	−	559.903i	−1.18122	−	0.681977i	−0.224922	−	0.974377i	$-0.572213\pi$
$821$	−969.780	−	559.903i	−1.18122	−	0.681977i	−0.956296	+	0.292400i	$0.905546\pi$
$822$		0			0
$823$	−352.273	+	203.385i	−0.428036	+	0.247127i	−0.698510	−	0.715601i	$-0.746153\pi$
$823$	−352.273	+	203.385i	−0.428036	+	0.247127i	0.270474	+	0.962727i	$0.412820\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−445.615			−0.538833			−0.269416	−	0.963024i	$-0.586831\pi$
$827$	−445.615			−0.538833			−0.269416	+	0.963024i	$0.586831\pi$
$828$		0			0
$829$	763.809			0.921362			0.460681	−	0.887566i	$-0.347605\pi$
$829$	763.809			0.921362			0.460681	+	0.887566i	$0.347605\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−55.4537	−	96.0487i	−0.0665711	−	0.115305i
$834$		0			0
$835$	−627.589	−	462.670i	−0.751604	−	0.554096i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−386.267	−	223.011i	−0.460389	−	0.265806i	0.251819	−	0.967774i	$-0.418971\pi$
$839$	−386.267	−	223.011i	−0.460389	−	0.265806i	−0.712208	+	0.701969i	$0.752305\pi$
$840$		0			0
$841$	160.036	+	277.191i	0.190293	+	0.329597i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−210.878	+	1880.53i	−0.249560	+	2.22548i
$846$		0			0
$847$			294.610i			0.347828i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	563.211	−	325.170i	0.661822	−	0.382103i
$852$		0			0
$853$	−922.445	−	532.574i	−1.08141	−	0.624354i	−0.150135	−	0.988665i	$-0.547971\pi$
$853$	−922.445	−	532.574i	−1.08141	−	0.624354i	−0.931277	+	0.364312i	$0.881304\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	386.183	−	668.889i	0.450622	−	0.780501i	−0.547802	−	0.836608i	$-0.684535\pi$
$857$	386.183	−	668.889i	0.450622	−	0.780501i	0.998425	+	0.0561069i	$0.0178688\pi$
$858$		0			0
$859$	−34.0802	−	59.0286i	−0.0396742	−	0.0687178i	0.845506	−	0.533965i	$-0.179299\pi$
$859$	−34.0802	−	59.0286i	−0.0396742	−	0.0687178i	−0.885181	+	0.465247i	$0.845965\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−572.939			−0.663892			−0.331946	−	0.943298i	$-0.607705\pi$
$863$	−572.939			−0.663892			−0.331946	+	0.943298i	$0.607705\pi$
$864$		0			0
$865$	74.8419	−	667.411i	0.0865224	−	0.771573i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	373.510	−	215.646i	0.429816	−	0.248154i
$870$		0			0
$871$	−1163.77	+	2015.70i	−1.33613	+	2.31424i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1014.86	+	876.160i	−1.15984	+	1.00133i
$876$		0			0
$877$	−268.988	+	155.300i	−0.306713	+	0.177081i	−0.645455	−	0.763799i	$-0.723332\pi$
$877$	−268.988	+	155.300i	−0.306713	+	0.177081i	0.338742	+	0.940879i	$0.389999\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	81.2933i		−	0.0922739i	−0.998935	−	0.0461369i	$-0.985309\pi$
$881$		−	81.2933i		−	0.0922739i	0.998935	−	0.0461369i	$-0.0146911\pi$
$882$		0			0
$883$			673.174i			0.762372i	0.924498	+	0.381186i	$0.124484\pi$
$883$			673.174i			0.762372i	−0.924498	+	0.381186i	$0.875516\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−871.614	−	1509.68i	−0.982654	−	1.70201i	−0.651931	−	0.758279i	$-0.726041\pi$
$887$	−871.614	−	1509.68i	−0.982654	−	1.70201i	−0.330723	−	0.943728i	$-0.607293\pi$
$888$		0			0
$889$	1011.73	−	1752.37i	1.13805	−	1.97117i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.29421	−	10.9019i	0.00704838	−	0.0122082i
$894$		0			0
$895$	404.379	+	925.511i	0.451820	+	1.03409i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	649.759i		−	0.722758i
$900$		0			0
$901$	−142.143			−0.157761
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−28.7419	+	12.5581i	−0.0317590	+	0.0138763i
$906$		0			0
$907$	629.070	+	363.194i	0.693573	+	0.400434i	0.804949	−	0.593344i	$-0.202193\pi$
$907$	629.070	+	363.194i	0.693573	+	0.400434i	−0.111376	+	0.993778i	$0.535526\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	991.195	+	572.267i	1.08803	+	0.628174i	0.933051	−	0.359743i	$-0.117136\pi$
$911$	991.195	+	572.267i	1.08803	+	0.628174i	0.154979	+	0.987918i	$0.450469\pi$
$912$		0			0
$913$	293.842	−	169.650i	0.321842	−	0.185816i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1137.05			1.23997
$918$		0			0
$919$	725.840			0.789815			0.394907	−	0.918721i	$-0.370777\pi$
$919$	725.840			0.789815			0.394907	+	0.918721i	$0.370777\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	431.270	+	746.981i	0.467248	+	0.809297i
$924$		0			0
$925$	−600.596	−	556.091i	−0.649293	−	0.601179i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	623.881	+	360.198i	0.671562	+	0.387727i	0.796668	−	0.604417i	$-0.206594\pi$
$929$	623.881	+	360.198i	0.671562	+	0.387727i	−0.125106	+	0.992143i	$0.539927\pi$
$930$		0			0
$931$	7.73915	+	13.4046i	0.00831273	+	0.0143981i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−101.669	−	11.4009i	−0.108737	−	0.0121935i
$936$		0			0
$937$		−	821.308i		−	0.876529i	−0.898846	−	0.438265i	$-0.855593\pi$
$937$		−	821.308i		−	0.876529i	0.898846	−	0.438265i	$-0.144407\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	150.992	−	87.1752i	0.160459	−	0.0926410i	−0.417620	−	0.908622i	$-0.637136\pi$
$941$	150.992	−	87.1752i	0.160459	−	0.0926410i	0.578079	+	0.815981i	$0.303802\pi$
$942$		0			0
$943$	90.7971	+	52.4217i	0.0962853	+	0.0555904i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−579.276	+	1003.34i	−0.611696	+	1.05949i	0.379258	+	0.925291i	$0.376179\pi$
$947$	−579.276	+	1003.34i	−0.611696	+	1.05949i	−0.990955	+	0.134198i	$0.957154\pi$
$948$		0			0
$949$	924.447	+	1601.19i	0.974127	+	1.68724i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	734.811			0.771051			0.385525	−	0.922697i	$-0.374020\pi$
$953$	734.811			0.771051			0.385525	+	0.922697i	$0.374020\pi$
$954$		0			0
$955$	1479.06	+	165.859i	1.54876	+	0.173674i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2379.21	−	1373.63i	2.48092	−	1.43236i
$960$		0			0
$961$	298.691	−	517.348i	0.310813	−	0.538343i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	165.191	−	224.074i	0.171183	−	0.232201i
$966$		0			0
$967$	−636.681	+	367.588i	−0.658408	+	0.380132i	−0.791670	−	0.610949i	$-0.790788\pi$
$967$	−636.681	+	367.588i	−0.658408	+	0.380132i	0.133262	+	0.991081i	$0.457455\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			630.994i			0.649839i	0.945742	+	0.324920i	$0.105337\pi$
$971$			630.994i			0.649839i	−0.945742	+	0.324920i	$0.894663\pi$
$972$		0			0
$973$			1948.78i			2.00286i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−563.994	−	976.866i	−0.577271	−	0.999863i	−0.995791	−	0.0916552i	$-0.970784\pi$
$977$	−563.994	−	976.866i	−0.577271	−	0.999863i	0.418520	−	0.908208i	$-0.362549\pi$
$978$		0			0
$979$	929.161	−	1609.35i	0.949092	−	1.64388i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−467.565	+	809.846i	−0.475651	+	0.823851i	−0.999611	−	0.0278915i	$-0.991121\pi$
$983$	−467.565	+	809.846i	−0.475651	+	0.823851i	0.523960	+	0.851743i	$0.324454\pi$
$984$		0			0
$985$	105.117	−	45.9281i	0.106717	−	0.0466275i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1052.43i		−	1.06414i
$990$		0			0
$991$	839.459			0.847083			0.423541	−	0.905877i	$-0.360787\pi$
$991$	839.459			0.847083			0.423541	+	0.905877i	$0.360787\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−935.385	+	408.693i	−0.940085	+	0.410747i
$996$		0			0
$997$	−216.451	−	124.968i	−0.217102	−	0.125344i	0.387506	−	0.921867i	$-0.373337\pi$
$997$	−216.451	−	124.968i	−0.217102	−	0.125344i	−0.604608	+	0.796523i	$0.706670\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.t.f.269.4		48
3.2	odd	2	inner	1620.3.t.f.269.21		48
5.4	even	2	inner	1620.3.t.f.269.6		48
9.2	odd	6		1620.3.b.a.809.12	yes	24
9.4	even	3	inner	1620.3.t.f.1349.19		48
9.5	odd	6	inner	1620.3.t.f.1349.6		48
9.7	even	3		1620.3.b.a.809.13	yes	24
15.14	odd	2	inner	1620.3.t.f.269.19		48
45.4	even	6	inner	1620.3.t.f.1349.21		48
45.14	odd	6	inner	1620.3.t.f.1349.4		48
45.29	odd	6		1620.3.b.a.809.14	yes	24
45.34	even	6		1620.3.b.a.809.11	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.b.a.809.11	✓	24	45.34	even	6
1620.3.b.a.809.12	yes	24	9.2	odd	6
1620.3.b.a.809.13	yes	24	9.7	even	3
1620.3.b.a.809.14	yes	24	45.29	odd	6
1620.3.t.f.269.4		48	1.1	even	1	trivial
1620.3.t.f.269.6		48	5.4	even	2	inner
1620.3.t.f.269.19		48	15.14	odd	2	inner
1620.3.t.f.269.21		48	3.2	odd	2	inner
1620.3.t.f.1349.4		48	45.14	odd	6	inner
1620.3.t.f.1349.6		48	9.5	odd	6	inner
1620.3.t.f.1349.19		48	9.4	even	3	inner
1620.3.t.f.1349.21		48	45.4	even	6	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.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.58175 + 2.00188i) q^{5} +(9.28893 + 5.36297i) q^{7} +O(q^{10})\)	\(q+(-4.58175 + 2.00188i) q^{5} +(9.28893 + 5.36297i) q^{7} +(-10.5523 - 6.09235i) q^{11} +(20.2632 - 11.6990i) q^{13} -1.67925 q^{17} +0.234357 q^{19} +(9.93181 + 17.2024i) q^{23} +(16.9849 - 18.3443i) q^{25} +(-29.5094 - 17.0373i) q^{29} +(9.53439 + 16.5140i) q^{31} +(-53.2956 - 5.97645i) q^{35} -32.7402i q^{37} +(4.57102 - 2.63908i) q^{41} +(-45.8845 - 26.4914i) q^{43} +(26.8573 - 46.5182i) q^{47} +(33.0229 + 57.1973i) q^{49} +84.6464 q^{53} +(60.5441 + 6.78927i) q^{55} +(76.4925 - 44.1630i) q^{59} +(32.7982 - 56.8082i) q^{61} +(-69.4210 + 94.1663i) q^{65} +(-86.1488 + 49.7380i) q^{67} +36.8639i q^{71} +79.0196i q^{73} +(-65.3462 - 113.183i) q^{77} +(-17.6981 + 30.6540i) q^{79} +(-13.9232 + 24.1156i) q^{83} +(7.69392 - 3.36167i) q^{85} +152.513i q^{89} +250.965 q^{91} +(-1.07377 + 0.469156i) q^{95} +(84.0834 + 48.5456i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 48 q^{25} + 288 q^{49} - 72 q^{55} - 120 q^{61} - 480 q^{79} + 24 q^{85} + 96 q^{91}+O(q^{100}) \) 48 * q + 48 * q^25 + 288 * q^49 - 72 * q^55 - 120 * q^61 - 480 * q^79 + 24 * q^85 + 96 * q^91

Introduction

L-functions

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