LMFDB - Embedded newform 1620.3.b.a.809.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(809,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("1620.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.b (of order $2$, degree $1$, 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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.24
Character	$\chi$	$=$	1620.809
Dual form			1620.3.b.a.809.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.99031 + 0.311180i) q^{5} -4.65609i q^{7} +O(q^{10})$	$q+(4.99031 + 0.311180i) q^{5} -4.65609i q^{7} +12.0275i q^{11} -8.46826i q^{13} -1.25172 q^{17} +16.4797 q^{19} +2.38818 q^{23} +(24.8063 + 3.10577i) q^{25} +23.9969i q^{29} -5.25415 q^{31} +(1.44888 - 23.2353i) q^{35} +25.1935i q^{37} -29.8425i q^{41} -25.5104i q^{43} +64.9783 q^{47} +27.3208 q^{49} +71.4673 q^{53} +(-3.74272 + 60.0210i) q^{55} +18.5016i q^{59} +19.1025 q^{61} +(2.63516 - 42.2592i) q^{65} -81.0903i q^{67} -21.9875i q^{71} +109.701i q^{73} +56.0012 q^{77} -134.193 q^{79} +15.6122 q^{83} +(-6.24645 - 0.389509i) q^{85} +79.4211i q^{89} -39.4290 q^{91} +(82.2389 + 5.12817i) q^{95} -130.304i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 48 q^{25} - 288 q^{49} - 36 q^{55} + 120 q^{61} + 480 q^{79} - 24 q^{85} + 48 q^{91}+O(q^{100}) $ 24 * q - 48 * q^25 - 288 * q^49 - 36 * q^55 + 120 * q^61 + 480 * q^79 - 24 * q^85 + 48 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	4.99031	+	0.311180i	0.998061	+	0.0622361i
$5$	4.99031	+	0.311180i	0.998061	+	0.0622361i
$6$		0			0
$7$		−	4.65609i		−	0.665156i	−0.943076	−	0.332578i	$-0.892082\pi$
$7$		−	4.65609i		−	0.665156i	0.943076	−	0.332578i	$-0.107918\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			12.0275i			1.09341i	0.837325	+	0.546705i	$0.184118\pi$
$11$			12.0275i			1.09341i	−0.837325	+	0.546705i	$0.815882\pi$
$12$		0			0
$13$		−	8.46826i		−	0.651405i	−0.945472	−	0.325702i	$-0.894399\pi$
$13$		−	8.46826i		−	0.651405i	0.945472	−	0.325702i	$-0.105601\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.25172			−0.0736304			−0.0368152	−	0.999322i	$-0.511721\pi$
$17$	−1.25172			−0.0736304			−0.0368152	+	0.999322i	$0.511721\pi$
$18$		0			0
$19$	16.4797			0.867354			0.433677	−	0.901068i	$-0.357216\pi$
$19$	16.4797			0.867354			0.433677	+	0.901068i	$0.357216\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.38818			0.103834			0.0519169	−	0.998651i	$-0.483467\pi$
$23$	2.38818			0.103834			0.0519169	+	0.998651i	$0.483467\pi$
$24$		0			0
$25$	24.8063	+	3.10577i	0.992253	+	0.124231i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			23.9969i			0.827481i	0.910395	+	0.413740i	$0.135778\pi$
$29$			23.9969i			0.827481i	−0.910395	+	0.413740i	$0.864222\pi$
$30$		0			0
$31$	−5.25415			−0.169489			−0.0847443	−	0.996403i	$-0.527007\pi$
$31$	−5.25415			−0.169489			−0.0847443	+	0.996403i	$0.527007\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.44888	−	23.2353i	0.0413967	−	0.663867i
$36$		0			0
$37$			25.1935i			0.680906i	0.940262	+	0.340453i	$0.110580\pi$
$37$			25.1935i			0.680906i	−0.940262	+	0.340453i	$0.889420\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	29.8425i		−	0.727866i	−0.931425	−	0.363933i	$-0.881434\pi$
$41$		−	29.8425i		−	0.727866i	0.931425	−	0.363933i	$-0.118566\pi$
$42$		0			0
$43$		−	25.5104i		−	0.593265i	−0.954992	−	0.296633i	$-0.904136\pi$
$43$		−	25.5104i		−	0.593265i	0.954992	−	0.296633i	$-0.0958637\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	64.9783			1.38252			0.691258	−	0.722608i	$-0.257057\pi$
$47$	64.9783			1.38252			0.691258	+	0.722608i	$0.257057\pi$
$48$		0			0
$49$	27.3208			0.557567
$50$		0			0
$51$		0			0
$52$		0			0
$53$	71.4673			1.34844			0.674219	−	0.738531i	$-0.264480\pi$
$53$	71.4673			1.34844			0.674219	+	0.738531i	$0.264480\pi$
$54$		0			0
$55$	−3.74272	+	60.0210i	−0.0680495	+	1.09129i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			18.5016i			0.313587i	0.987631	+	0.156794i	$0.0501157\pi$
$59$			18.5016i			0.313587i	−0.987631	+	0.156794i	$0.949884\pi$
$60$		0			0
$61$	19.1025			0.313155			0.156578	−	0.987666i	$-0.449954\pi$
$61$	19.1025			0.313155			0.156578	+	0.987666i	$0.449954\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.63516	−	42.2592i	0.0405409	−	0.650142i
$66$		0			0
$67$		−	81.0903i		−	1.21030i	−0.796110	−	0.605151i	$-0.793113\pi$
$67$		−	81.0903i		−	1.21030i	0.796110	−	0.605151i	$-0.206887\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	21.9875i		−	0.309683i	−0.987939	−	0.154842i	$-0.950513\pi$
$71$		−	21.9875i		−	0.309683i	0.987939	−	0.154842i	$-0.0494867\pi$
$72$		0			0
$73$			109.701i			1.50276i	0.659871	+	0.751379i	$0.270611\pi$
$73$			109.701i			1.50276i	−0.659871	+	0.751379i	$0.729389\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	56.0012			0.727288
$78$		0			0
$79$	−134.193			−1.69864			−0.849322	−	0.527875i	$-0.822989\pi$
$79$	−134.193			−1.69864			−0.849322	+	0.527875i	$0.822989\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	15.6122			0.188099			0.0940496	−	0.995568i	$-0.470019\pi$
$83$	15.6122			0.188099			0.0940496	+	0.995568i	$0.470019\pi$
$84$		0			0
$85$	−6.24645	−	0.389509i	−0.0734876	−	0.00458246i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			79.4211i			0.892372i	0.894940	+	0.446186i	$0.147218\pi$
$89$			79.4211i			0.892372i	−0.894940	+	0.446186i	$0.852782\pi$
$90$		0			0
$91$	−39.4290			−0.433286
$92$		0			0
$93$		0			0
$94$		0			0
$95$	82.2389	+	5.12817i	0.865672	+	0.0539807i
$96$		0			0
$97$		−	130.304i		−	1.34334i	−0.740852	−	0.671668i	$-0.765578\pi$
$97$		−	130.304i		−	1.34334i	0.740852	−	0.671668i	$-0.234422\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	129.939i		−	1.28652i	−0.765647	−	0.643260i	$-0.777581\pi$
$101$		−	129.939i		−	1.28652i	0.765647	−	0.643260i	$-0.222419\pi$
$102$		0			0
$103$		−	127.533i		−	1.23818i	−0.785319	−	0.619091i	$-0.787501\pi$
$103$		−	127.533i		−	1.23818i	0.785319	−	0.619091i	$-0.212499\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	28.0983			0.262601			0.131301	−	0.991343i	$-0.458085\pi$
$107$	28.0983			0.262601			0.131301	+	0.991343i	$0.458085\pi$
$108$		0			0
$109$	−14.6527			−0.134429			−0.0672144	−	0.997739i	$-0.521411\pi$
$109$	−14.6527			−0.134429			−0.0672144	+	0.997739i	$0.521411\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	133.085			1.17774			0.588870	−	0.808228i	$-0.299573\pi$
$113$	133.085			1.17774			0.588870	+	0.808228i	$0.299573\pi$
$114$		0			0
$115$	11.9177	+	0.743154i	0.103633	+	0.00646221i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			5.82811i			0.0489757i
$120$		0			0
$121$	−23.6610			−0.195545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	122.825	+	23.2180i	0.982598	+	0.185744i
$126$		0			0
$127$			219.646i			1.72950i	0.502206	+	0.864748i	$0.332522\pi$
$127$			219.646i			1.72950i	−0.502206	+	0.864748i	$0.667478\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			152.004i			1.16034i	0.814496	+	0.580170i	$0.197014\pi$
$131$			152.004i			1.16034i	−0.814496	+	0.580170i	$0.802986\pi$
$132$		0			0
$133$		−	76.7311i		−	0.576926i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	201.697			1.47224			0.736120	−	0.676851i	$-0.236656\pi$
$137$	201.697			1.47224			0.736120	+	0.676851i	$0.236656\pi$
$138$		0			0
$139$	−166.356			−1.19680			−0.598401	−	0.801196i	$-0.704197\pi$
$139$	−166.356			−1.19680			−0.598401	+	0.801196i	$0.704197\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	101.852			0.712252
$144$		0			0
$145$	−7.46737	+	119.752i	−0.0514991	+	0.825876i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	188.301i		−	1.26377i	−0.775063	−	0.631884i	$-0.782282\pi$
$149$		−	188.301i		−	1.26377i	0.775063	−	0.631884i	$-0.217718\pi$
$150$		0			0
$151$	52.6932			0.348961			0.174481	−	0.984661i	$-0.444175\pi$
$151$	52.6932			0.348961			0.174481	+	0.984661i	$0.444175\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−26.2198	−	1.63499i	−0.169160	−	0.0105483i
$156$		0			0
$157$		−	160.058i		−	1.01948i	−0.860329	−	0.509739i	$-0.829742\pi$
$157$		−	160.058i		−	1.01948i	0.860329	−	0.509739i	$-0.170258\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	11.1196i		−	0.0690657i
$162$		0			0
$163$			300.512i			1.84363i	0.387630	+	0.921815i	$0.373294\pi$
$163$			300.512i			1.84363i	−0.387630	+	0.921815i	$0.626706\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.85288			−0.0410352			−0.0205176	−	0.999789i	$-0.506531\pi$
$167$	−6.85288			−0.0410352			−0.0205176	+	0.999789i	$0.506531\pi$
$168$		0			0
$169$	97.2885			0.575672
$170$		0			0
$171$		0			0
$172$		0			0
$173$	208.309			1.20410			0.602049	−	0.798459i	$-0.294351\pi$
$173$	208.309			1.20410			0.602049	+	0.798459i	$0.294351\pi$
$174$		0			0
$175$	14.4608	−	115.501i	0.0826329	−	0.660003i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	23.2298i		−	0.129776i	−0.997893	−	0.0648878i	$-0.979331\pi$
$179$		−	23.2298i		−	0.129776i	0.997893	−	0.0648878i	$-0.0206690\pi$
$180$		0			0
$181$	259.043			1.43118			0.715588	−	0.698522i	$-0.246159\pi$
$181$	259.043			1.43118			0.715588	+	0.698522i	$0.246159\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.83973	+	125.723i	−0.0423769	+	0.679586i
$186$		0			0
$187$		−	15.0550i		−	0.0805082i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			169.995i			0.890029i	0.895523	+	0.445014i	$0.146801\pi$
$191$			169.995i			0.890029i	−0.895523	+	0.445014i	$0.853199\pi$
$192$		0			0
$193$			79.9172i			0.414079i	0.978333	+	0.207039i	$0.0663828\pi$
$193$			79.9172i			0.414079i	−0.978333	+	0.207039i	$0.933617\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	211.199			1.07208			0.536039	−	0.844193i	$-0.319920\pi$
$197$	211.199			1.07208			0.536039	+	0.844193i	$0.319920\pi$
$198$		0			0
$199$	−175.943			−0.884133			−0.442067	−	0.896982i	$-0.645755\pi$
$199$	−175.943			−0.884133			−0.442067	+	0.896982i	$0.645755\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	111.732			0.550404
$204$		0			0
$205$	9.28640	−	148.923i	0.0452995	−	0.726455i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			198.210i			0.948373i
$210$		0			0
$211$	225.758			1.06994			0.534971	−	0.844870i	$-0.320322\pi$
$211$	225.758			1.06994			0.534971	+	0.844870i	$0.320322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.93834	−	127.305i	0.0369225	−	0.592115i
$216$		0			0
$217$			24.4638i			0.112736i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			10.5999i			0.0479632i
$222$		0			0
$223$			4.71078i			0.0211246i	0.999944	+	0.0105623i	$0.00336214\pi$
$223$			4.71078i			0.0211246i	−0.999944	+	0.0105623i	$0.996638\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.49794			−0.0374359			−0.0187179	−	0.999825i	$-0.505958\pi$
$227$	−8.49794			−0.0374359			−0.0187179	+	0.999825i	$0.505958\pi$
$228$		0			0
$229$	−35.5759			−0.155353			−0.0776766	−	0.996979i	$-0.524750\pi$
$229$	−35.5759			−0.155353			−0.0776766	+	0.996979i	$0.524750\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−110.611			−0.474726			−0.237363	−	0.971421i	$-0.576283\pi$
$233$	−110.611			−0.474726			−0.237363	+	0.971421i	$0.576283\pi$
$234$		0			0
$235$	324.261	+	20.2200i	1.37984	+	0.0860424i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	444.033i		−	1.85788i	−0.370234	−	0.928939i	$-0.620722\pi$
$239$		−	444.033i		−	1.85788i	0.370234	−	0.928939i	$-0.379278\pi$
$240$		0			0
$241$	−167.993			−0.697065			−0.348533	−	0.937297i	$-0.613320\pi$
$241$	−167.993			−0.697065			−0.348533	+	0.937297i	$0.613320\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	136.339	+	8.50170i	0.556487	+	0.0347008i
$246$		0			0
$247$		−	139.555i		−	0.564998i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	54.9639i		−	0.218980i	−0.993988	−	0.109490i	$-0.965078\pi$
$251$		−	54.9639i		−	0.218980i	0.993988	−	0.109490i	$-0.0349217\pi$
$252$		0			0
$253$			28.7238i			0.113533i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	282.402			1.09884			0.549419	−	0.835547i	$-0.314849\pi$
$257$	282.402			1.09884			0.549419	+	0.835547i	$0.314849\pi$
$258$		0			0
$259$	117.303			0.452909
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−115.831			−0.440421			−0.220211	−	0.975452i	$-0.570674\pi$
$263$	−115.831			−0.440421			−0.220211	+	0.975452i	$0.570674\pi$
$264$		0			0
$265$	356.644	+	22.2392i	1.34582	+	0.0839215i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	97.7513i		−	0.363388i	−0.983355	−	0.181694i	$-0.941842\pi$
$269$		−	97.7513i		−	0.363388i	0.983355	−	0.181694i	$-0.0581580\pi$
$270$		0			0
$271$	−110.971			−0.409486			−0.204743	−	0.978816i	$-0.565636\pi$
$271$	−110.971			−0.409486			−0.204743	+	0.978816i	$0.565636\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−37.3547	+	298.358i	−0.135835	+	1.08494i
$276$		0			0
$277$		−	190.229i		−	0.686746i	−0.939199	−	0.343373i	$-0.888430\pi$
$277$		−	190.229i		−	0.686746i	0.939199	−	0.343373i	$-0.111570\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	66.1684i		−	0.235475i	−0.993045	−	0.117737i	$-0.962436\pi$
$281$		−	66.1684i		−	0.235475i	0.993045	−	0.117737i	$-0.0375641\pi$
$282$		0			0
$283$		−	346.751i		−	1.22527i	−0.790366	−	0.612635i	$-0.790110\pi$
$283$		−	346.751i		−	1.22527i	0.790366	−	0.612635i	$-0.209890\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−138.949			−0.484144
$288$		0			0
$289$	−287.433			−0.994579
$290$		0			0
$291$		0			0
$292$		0			0
$293$	23.0494			0.0786669			0.0393334	−	0.999226i	$-0.487477\pi$
$293$	23.0494			0.0786669			0.0393334	+	0.999226i	$0.487477\pi$
$294$		0			0
$295$	−5.75735	+	92.3289i	−0.0195164	+	0.312979i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	20.2237i		−	0.0676379i
$300$		0			0
$301$	−118.779			−0.394614
$302$		0			0
$303$		0			0
$304$		0			0
$305$	95.3272	+	5.94432i	0.312548	+	0.0194896i
$306$		0			0
$307$			169.526i			0.552200i	0.961129	+	0.276100i	$0.0890422\pi$
$307$			169.526i			0.552200i	−0.961129	+	0.276100i	$0.910958\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	73.1261i		−	0.235132i	−0.993065	−	0.117566i	$-0.962491\pi$
$311$		−	73.1261i		−	0.235132i	0.993065	−	0.117566i	$-0.0375092\pi$
$312$		0			0
$313$		−	5.41919i		−	0.0173137i	−0.999963	−	0.00865685i	$-0.997244\pi$
$313$		−	5.41919i		−	0.0173137i	0.999963	−	0.00865685i	$-0.00275560\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−339.171			−1.06994			−0.534970	−	0.844871i	$-0.679677\pi$
$317$	−339.171			−1.06994			−0.534970	+	0.844871i	$0.679677\pi$
$318$		0			0
$319$	−288.623			−0.904775
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−20.6279			−0.0638636
$324$		0			0
$325$	26.3005	−	210.067i	0.0809246	−	0.646359i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	302.545i		−	0.919589i
$330$		0			0
$331$	255.702			0.772513			0.386256	−	0.922391i	$-0.373768\pi$
$331$	255.702			0.772513			0.386256	+	0.922391i	$0.373768\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	25.2337	−	404.665i	0.0753245	−	1.20796i
$336$		0			0
$337$			621.566i			1.84441i	0.386701	+	0.922205i	$0.373615\pi$
$337$			621.566i			1.84441i	−0.386701	+	0.922205i	$0.626385\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	63.1943i		−	0.185321i
$342$		0			0
$343$		−	355.357i		−	1.03603i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−347.716			−1.00206			−0.501032	−	0.865429i	$-0.667046\pi$
$347$	−347.716			−1.00206			−0.501032	+	0.865429i	$0.667046\pi$
$348$		0			0
$349$	−560.751			−1.60674			−0.803368	−	0.595482i	$-0.796961\pi$
$349$	−560.751			−1.60674			−0.803368	+	0.595482i	$0.796961\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	265.523			0.752189			0.376094	−	0.926581i	$-0.377267\pi$
$353$	265.523			0.752189			0.376094	+	0.926581i	$0.377267\pi$
$354$		0			0
$355$	6.84208	−	109.724i	0.0192735	−	0.309083i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			502.476i			1.39965i	0.714313	+	0.699827i	$0.246739\pi$
$359$			502.476i			1.39965i	−0.714313	+	0.699827i	$0.753261\pi$
$360$		0			0
$361$	−89.4187			−0.247697
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−34.1369	+	547.443i	−0.0935257	+	1.49984i
$366$		0			0
$367$			658.120i			1.79324i	0.442798	+	0.896622i	$0.353986\pi$
$367$			658.120i			1.79324i	−0.442798	+	0.896622i	$0.646014\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	332.758i		−	0.896922i
$372$		0			0
$373$			638.191i			1.71097i	0.517829	+	0.855484i	$0.326740\pi$
$373$			638.191i			1.71097i	−0.517829	+	0.855484i	$0.673260\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	203.212			0.539025
$378$		0			0
$379$	−126.043			−0.332567			−0.166284	−	0.986078i	$-0.553177\pi$
$379$	−126.043			−0.332567			−0.166284	+	0.986078i	$0.553177\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−639.275			−1.66912			−0.834562	−	0.550914i	$-0.814279\pi$
$383$	−639.275			−1.66912			−0.834562	+	0.550914i	$0.814279\pi$
$384$		0			0
$385$	279.463	+	17.4265i	0.725878	+	0.0452636i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			281.326i			0.723202i	0.932333	+	0.361601i	$0.117770\pi$
$389$			281.326i			0.723202i	−0.932333	+	0.361601i	$0.882230\pi$
$390$		0			0
$391$	−2.98932			−0.00764532
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−669.664	−	41.7582i	−1.69535	−	0.105717i
$396$		0			0
$397$		−	408.350i		−	1.02859i	−0.857613	−	0.514295i	$-0.828054\pi$
$397$		−	408.350i		−	1.02859i	0.857613	−	0.514295i	$-0.171946\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			594.709i			1.48307i	0.670917	+	0.741533i	$0.265901\pi$
$401$			594.709i			1.48307i	−0.670917	+	0.741533i	$0.734099\pi$
$402$		0			0
$403$			44.4935i			0.110406i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−303.015			−0.744509
$408$		0			0
$409$	−524.955			−1.28351			−0.641754	−	0.766910i	$-0.721793\pi$
$409$	−524.955			−1.28351			−0.641754	+	0.766910i	$0.721793\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	86.1454			0.208584
$414$		0			0
$415$	77.9099	+	4.85822i	0.187735	+	0.0117066i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	354.853i		−	0.846904i	−0.905919	−	0.423452i	$-0.860818\pi$
$419$		−	354.853i		−	0.846904i	0.905919	−	0.423452i	$-0.139182\pi$
$420$		0			0
$421$	18.8951			0.0448814			0.0224407	−	0.999748i	$-0.492856\pi$
$421$	18.8951			0.0448814			0.0224407	+	0.999748i	$0.492856\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−31.0505	−	3.88754i	−0.0730600	−	0.00914716i
$426$		0			0
$427$		−	88.9429i		−	0.208297i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			294.610i			0.683549i	0.939782	+	0.341774i	$0.111028\pi$
$431$			294.610i			0.683549i	−0.939782	+	0.341774i	$0.888972\pi$
$432$		0			0
$433$		−	541.170i		−	1.24982i	−0.780698	−	0.624908i	$-0.785136\pi$
$433$		−	541.170i		−	1.24982i	0.780698	−	0.624908i	$-0.214864\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	39.3565			0.0900607
$438$		0			0
$439$	−532.252			−1.21242			−0.606209	−	0.795305i	$-0.707311\pi$
$439$	−532.252			−1.21242			−0.606209	+	0.795305i	$0.707311\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−740.961			−1.67260			−0.836299	−	0.548274i	$-0.815285\pi$
$443$	−740.961			−1.67260			−0.836299	+	0.548274i	$0.815285\pi$
$444$		0			0
$445$	−24.7143	+	396.336i	−0.0555377	+	0.890642i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			575.484i			1.28170i	0.767665	+	0.640851i	$0.221418\pi$
$449$			575.484i			1.28170i	−0.767665	+	0.640851i	$0.778582\pi$
$450$		0			0
$451$	358.931			0.795855
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−196.763	−	12.2695i	−0.432446	−	0.0269660i
$456$		0			0
$457$			165.883i			0.362983i	0.983393	+	0.181492i	$0.0580925\pi$
$457$			165.883i			0.362983i	−0.983393	+	0.181492i	$0.941908\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	356.860i		−	0.774100i	−0.922059	−	0.387050i	$-0.873494\pi$
$461$		−	356.860i		−	0.774100i	0.922059	−	0.387050i	$-0.126506\pi$
$462$		0			0
$463$		−	393.449i		−	0.849782i	−0.905244	−	0.424891i	$-0.860312\pi$
$463$		−	393.449i		−	0.849782i	0.905244	−	0.424891i	$-0.139688\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−413.760			−0.885997			−0.442998	−	0.896522i	$-0.646085\pi$
$467$	−413.760			−0.885997			−0.442998	+	0.896522i	$0.646085\pi$
$468$		0			0
$469$	−377.564			−0.805040
$470$		0			0
$471$		0			0
$472$		0			0
$473$	306.827			0.648682
$474$		0			0
$475$	408.802	+	51.1822i	0.860635	+	0.107752i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			879.375i			1.83586i	0.396748	+	0.917928i	$0.370139\pi$
$479$			879.375i			1.83586i	−0.396748	+	0.917928i	$0.629861\pi$
$480$		0			0
$481$	213.345			0.443545
$482$		0			0
$483$		0			0
$484$		0			0
$485$	40.5479	−	650.255i	0.0836040	−	1.34073i
$486$		0			0
$487$		−	492.034i		−	1.01034i	−0.863021	−	0.505169i	$-0.831430\pi$
$487$		−	492.034i		−	1.01034i	0.863021	−	0.505169i	$-0.168570\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			446.789i			0.909958i	0.890502	+	0.454979i	$0.150353\pi$
$491$			446.789i			0.909958i	−0.890502	+	0.454979i	$0.849647\pi$
$492$		0			0
$493$		−	30.0374i		−	0.0609277i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−102.376			−0.205988
$498$		0			0
$499$	144.988			0.290557			0.145278	−	0.989391i	$-0.453592\pi$
$499$	144.988			0.290557			0.145278	+	0.989391i	$0.453592\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−542.902			−1.07933			−0.539664	−	0.841881i	$-0.681449\pi$
$503$	−542.902			−1.07933			−0.539664	+	0.841881i	$0.681449\pi$
$504$		0			0
$505$	40.4343	−	648.434i	0.0800680	−	1.28403i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	795.577i		−	1.56302i	−0.623894	−	0.781509i	$-0.714450\pi$
$509$		−	795.577i		−	1.56302i	0.623894	−	0.781509i	$-0.285550\pi$
$510$		0			0
$511$	510.779			0.999568
$512$		0			0
$513$		0			0
$514$		0			0
$515$	39.6857	−	636.428i	0.0770596	−	1.23578i
$516$		0			0
$517$			781.526i			1.51166i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	857.089i		−	1.64508i	−0.568705	−	0.822542i	$-0.692555\pi$
$521$		−	857.089i		−	1.64508i	0.568705	−	0.822542i	$-0.307445\pi$
$522$		0			0
$523$		−	616.196i		−	1.17819i	−0.808062	−	0.589097i	$-0.799483\pi$
$523$		−	616.196i		−	1.17819i	0.808062	−	0.589097i	$-0.200517\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.57670			0.0124795
$528$		0			0
$529$	−523.297			−0.989219
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−252.714			−0.474135
$534$		0			0
$535$	140.219	+	8.74364i	0.262092	+	0.0163433i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			328.601i			0.609650i
$540$		0			0
$541$	145.542			0.269025			0.134512	−	0.990912i	$-0.457053\pi$
$541$	145.542			0.269025			0.134512	+	0.990912i	$0.457053\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−73.1217	−	4.55964i	−0.134168	−	0.00836632i
$546$		0			0
$547$			60.0207i			0.109727i	0.998494	+	0.0548635i	$0.0174724\pi$
$547$			60.0207i			0.109727i	−0.998494	+	0.0548635i	$0.982528\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			395.463i			0.717718i
$552$		0			0
$553$			624.815i			1.12986i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−557.225			−1.00040			−0.500202	−	0.865909i	$-0.666741\pi$
$557$	−557.225			−1.00040			−0.500202	+	0.865909i	$0.666741\pi$
$558$		0			0
$559$	−216.029			−0.386456
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−978.859			−1.73865			−0.869324	−	0.494243i	$-0.835445\pi$
$563$	−978.859			−1.73865			−0.869324	+	0.494243i	$0.835445\pi$
$564$		0			0
$565$	664.134	+	41.4133i	1.17546	+	0.0732979i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			784.582i			1.37888i	0.724343	+	0.689440i	$0.242143\pi$
$569$			784.582i			1.37888i	−0.724343	+	0.689440i	$0.757857\pi$
$570$		0			0
$571$	738.813			1.29389			0.646947	−	0.762535i	$-0.276046\pi$
$571$	738.813			1.29389			0.646947	+	0.762535i	$0.276046\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	59.2419	+	7.41713i	0.103029	+	0.0128994i
$576$		0			0
$577$		−	707.354i		−	1.22592i	−0.790115	−	0.612959i	$-0.789979\pi$
$577$		−	707.354i		−	1.22592i	0.790115	−	0.612959i	$-0.210021\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	72.6920i		−	0.125115i
$582$		0			0
$583$			859.573i			1.47440i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−913.998			−1.55707			−0.778533	−	0.627603i	$-0.784036\pi$
$587$	−913.998			−1.55707			−0.778533	+	0.627603i	$0.784036\pi$
$588$		0			0
$589$	−86.5869			−0.147007
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−97.8573			−0.165021			−0.0825103	−	0.996590i	$-0.526294\pi$
$593$	−97.8573			−0.165021			−0.0825103	+	0.996590i	$0.526294\pi$
$594$		0			0
$595$	−1.81359	+	29.0840i	−0.00304805	+	0.0488807i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1144.87i		−	1.91130i	−0.294501	−	0.955651i	$-0.595153\pi$
$599$		−	1144.87i		−	1.91130i	0.294501	−	0.955651i	$-0.404847\pi$
$600$		0			0
$601$	310.828			0.517185			0.258593	−	0.965986i	$-0.416741\pi$
$601$	310.828			0.517185			0.258593	+	0.965986i	$0.416741\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−118.075	−	7.36282i	−0.195166	−	0.0121700i
$606$		0			0
$607$		−	255.687i		−	0.421230i	−0.977569	−	0.210615i	$-0.932453\pi$
$607$		−	255.687i		−	0.421230i	0.977569	−	0.210615i	$-0.0675467\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	550.253i		−	0.900578i
$612$		0			0
$613$		−	423.661i		−	0.691127i	−0.938395	−	0.345564i	$-0.887688\pi$
$613$		−	423.661i		−	0.691127i	0.938395	−	0.345564i	$-0.112312\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−787.746			−1.27674			−0.638368	−	0.769731i	$-0.720390\pi$
$617$	−787.746			−1.27674			−0.638368	+	0.769731i	$0.720390\pi$
$618$		0			0
$619$	−594.956			−0.961156			−0.480578	−	0.876952i	$-0.659573\pi$
$619$	−594.956			−0.961156			−0.480578	+	0.876952i	$0.659573\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	369.792			0.593566
$624$		0			0
$625$	605.708	+	154.086i	0.969133	+	0.246537i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	31.5351i		−	0.0501353i
$630$		0			0
$631$	265.641			0.420984			0.210492	−	0.977596i	$-0.432493\pi$
$631$	265.641			0.420984			0.210492	+	0.977596i	$0.432493\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−68.3495	+	1096.10i	−0.107637	+	1.72614i
$636$		0			0
$637$		−	231.360i		−	0.363202i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	539.960i		−	0.842371i	−0.906975	−	0.421185i	$-0.861614\pi$
$641$		−	539.960i		−	0.842371i	0.906975	−	0.421185i	$-0.138386\pi$
$642$		0			0
$643$			359.162i			0.558573i	0.960208	+	0.279286i	$0.0900979\pi$
$643$			359.162i			0.558573i	−0.960208	+	0.279286i	$0.909902\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−210.526			−0.325387			−0.162694	−	0.986677i	$-0.552018\pi$
$647$	−210.526			−0.325387			−0.162694	+	0.986677i	$0.552018\pi$
$648$		0			0
$649$	−222.529			−0.342879
$650$		0			0
$651$		0			0
$652$		0			0
$653$	153.489			0.235052			0.117526	−	0.993070i	$-0.462504\pi$
$653$	153.489			0.235052			0.117526	+	0.993070i	$0.462504\pi$
$654$		0			0
$655$	−47.3008	+	758.549i	−0.0722150	+	1.15809i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			567.982i			0.861884i	0.902380	+	0.430942i	$0.141819\pi$
$659$			567.982i			0.861884i	−0.902380	+	0.430942i	$0.858181\pi$
$660$		0			0
$661$	2.85938			0.00432584			0.00216292	−	0.999998i	$-0.499312\pi$
$661$	2.85938			0.00432584			0.00216292	+	0.999998i	$0.499312\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	23.8772	−	382.912i	0.0359056	−	0.575807i
$666$		0			0
$667$			57.3090i			0.0859205i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			229.755i			0.342407i
$672$		0			0
$673$		−	259.229i		−	0.385184i	−0.981279	−	0.192592i	$-0.938311\pi$
$673$		−	259.229i		−	0.385184i	0.981279	−	0.192592i	$-0.0616894\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1174.34			1.73462			0.867308	−	0.497771i	$-0.165848\pi$
$677$	1174.34			1.73462			0.867308	+	0.497771i	$0.165848\pi$
$678$		0			0
$679$	−606.706			−0.893528
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−118.941			−0.174145			−0.0870725	−	0.996202i	$-0.527751\pi$
$683$	−118.941			−0.174145			−0.0870725	+	0.996202i	$0.527751\pi$
$684$		0			0
$685$	1006.53	+	62.7641i	1.46939	+	0.0916264i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	605.203i		−	0.878379i
$690$		0			0
$691$	−49.7749			−0.0720332			−0.0360166	−	0.999351i	$-0.511467\pi$
$691$	−49.7749			−0.0720332			−0.0360166	+	0.999351i	$0.511467\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−830.166	−	51.7666i	−1.19448	−	0.0744843i
$696$		0			0
$697$			37.3543i			0.0535930i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1120.97i			1.59910i	0.600600	+	0.799550i	$0.294929\pi$
$701$			1120.97i			1.59910i	−0.600600	+	0.799550i	$0.705071\pi$
$702$		0			0
$703$			415.182i			0.590586i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−605.006			−0.855737
$708$		0			0
$709$	−277.784			−0.391797			−0.195899	−	0.980624i	$-0.562762\pi$
$709$	−277.784			−0.391797			−0.195899	+	0.980624i	$0.562762\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.5478			−0.0175987
$714$		0			0
$715$	508.273	+	31.6944i	0.710872	+	0.0443278i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			450.600i			0.626704i	0.949637	+	0.313352i	$0.101452\pi$
$719$			450.600i			0.626704i	−0.949637	+	0.313352i	$0.898548\pi$
$720$		0			0
$721$	−593.804			−0.823584
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−74.5290	+	595.276i	−0.102799	+	0.821070i
$726$		0			0
$727$			467.492i			0.643042i	0.946902	+	0.321521i	$0.104194\pi$
$727$			467.492i			0.643042i	−0.946902	+	0.321521i	$0.895806\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			31.9318i			0.0436823i
$732$		0			0
$733$			695.638i			0.949029i	0.880248	+	0.474514i	$0.157376\pi$
$733$			695.638i			0.949029i	−0.880248	+	0.474514i	$0.842624\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	975.314			1.32336
$738$		0			0
$739$	−521.831			−0.706132			−0.353066	−	0.935598i	$-0.614861\pi$
$739$	−521.831			−0.706132			−0.353066	+	0.935598i	$0.614861\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	941.305			1.26690			0.633449	−	0.773784i	$-0.281639\pi$
$743$	941.305			1.26690			0.633449	+	0.773784i	$0.281639\pi$
$744$		0			0
$745$	58.5957	−	939.682i	0.0786520	−	1.26132i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	130.828i		−	0.174671i
$750$		0			0
$751$	−719.465			−0.958010			−0.479005	−	0.877812i	$-0.659002\pi$
$751$	−719.465			−0.958010			−0.479005	+	0.877812i	$0.659002\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	262.955	+	16.3971i	0.348285	+	0.0217180i
$756$		0			0
$757$		−	578.171i		−	0.763767i	−0.924211	−	0.381883i	$-0.875276\pi$
$757$		−	578.171i		−	0.763767i	0.924211	−	0.381883i	$-0.124724\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			368.553i			0.484300i	0.970239	+	0.242150i	$0.0778527\pi$
$761$			368.553i			0.484300i	−0.970239	+	0.242150i	$0.922147\pi$
$762$		0			0
$763$			68.2245i			0.0894161i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	156.677			0.204272
$768$		0			0
$769$	−1035.64			−1.34674			−0.673369	−	0.739307i	$-0.735153\pi$
$769$	−1035.64			−1.34674			−0.673369	+	0.739307i	$0.735153\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−778.625			−1.00728			−0.503638	−	0.863915i	$-0.668006\pi$
$773$	−778.625			−1.00728			−0.503638	+	0.863915i	$0.668006\pi$
$774$		0			0
$775$	−130.336	−	16.3182i	−0.168176	−	0.0210557i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	491.796i		−	0.631317i
$780$		0			0
$781$	264.455			0.338611
$782$		0			0
$783$		0			0
$784$		0			0
$785$	49.8069	−	798.738i	0.0634483	−	1.01750i
$786$		0			0
$787$		−	590.471i		−	0.750281i	−0.926968	−	0.375141i	$-0.877594\pi$
$787$		−	590.471i		−	0.750281i	0.926968	−	0.375141i	$-0.122406\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	619.655i		−	0.783381i
$792$		0			0
$793$		−	161.765i		−	0.203991i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1025.50			1.28670			0.643348	−	0.765574i	$-0.277545\pi$
$797$	1025.50			1.28670			0.643348	+	0.765574i	$0.277545\pi$
$798$		0			0
$799$	−81.3343			−0.101795
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1319.43			−1.64313
$804$		0			0
$805$	3.46019	−	55.4901i	0.00429838	−	0.0689318i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			769.002i			0.950559i	0.879835	+	0.475280i	$0.157653\pi$
$809$			769.002i			0.950559i	−0.879835	+	0.475280i	$0.842347\pi$
$810$		0			0
$811$	699.204			0.862150			0.431075	−	0.902316i	$-0.358134\pi$
$811$	699.204			0.862150			0.431075	+	0.902316i	$0.358134\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−93.5133	+	1499.65i	−0.114740	+	1.84006i
$816$		0			0
$817$		−	420.404i		−	0.514571i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	321.482i		−	0.391574i	−0.980646	−	0.195787i	$-0.937274\pi$
$821$		−	321.482i		−	0.391574i	0.980646	−	0.195787i	$-0.0627261\pi$
$822$		0			0
$823$		−	1312.47i		−	1.59474i	−0.603489	−	0.797371i	$-0.706223\pi$
$823$		−	1312.47i		−	1.59474i	0.603489	−	0.797371i	$-0.293777\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1407.89			−1.70241			−0.851205	−	0.524834i	$-0.824127\pi$
$827$	−1407.89			−1.70241			−0.851205	+	0.524834i	$0.824127\pi$
$828$		0			0
$829$	−1307.92			−1.57771			−0.788855	−	0.614579i	$-0.789326\pi$
$829$	−1307.92			−1.57771			−0.788855	+	0.614579i	$0.789326\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−34.1979			−0.0410539
$834$		0			0
$835$	−34.1980	−	2.13248i	−0.0409556	−	0.00255387i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	276.400i		−	0.329440i	−0.986340	−	0.164720i	$-0.947328\pi$
$839$		−	276.400i		−	0.329440i	0.986340	−	0.164720i	$-0.0526721\pi$
$840$		0			0
$841$	265.147			0.315276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	485.500	+	30.2743i	0.574556	+	0.0358275i
$846$		0			0
$847$			110.168i			0.130068i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			60.1666i			0.0707011i
$852$		0			0
$853$			198.741i			0.232991i	0.993191	+	0.116496i	$0.0371661\pi$
$853$			198.741i			0.232991i	−0.993191	+	0.116496i	$0.962834\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	125.199			0.146090			0.0730448	−	0.997329i	$-0.476728\pi$
$857$	125.199			0.146090			0.0730448	+	0.997329i	$0.476728\pi$
$858$		0			0
$859$	−1414.49			−1.64667			−0.823335	−	0.567556i	$-0.807889\pi$
$859$	−1414.49			−1.64667			−0.823335	+	0.567556i	$0.807889\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	753.685			0.873331			0.436665	−	0.899624i	$-0.356159\pi$
$863$	753.685			0.873331			0.436665	+	0.899624i	$0.356159\pi$
$864$		0			0
$865$	1039.53	+	64.8217i	1.20176	+	0.0749384i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1614.01i		−	1.85731i
$870$		0			0
$871$	−686.694			−0.788397
$872$		0			0
$873$		0			0
$874$		0			0
$875$	108.105	−	571.883i	0.123549	−	0.653581i
$876$		0			0
$877$			200.489i			0.228607i	0.993446	+	0.114304i	$0.0364637\pi$
$877$			200.489i			0.228607i	−0.993446	+	0.114304i	$0.963536\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			755.616i			0.857680i	0.903381	+	0.428840i	$0.141078\pi$
$881$			755.616i			0.857680i	−0.903381	+	0.428840i	$0.858922\pi$
$882$		0			0
$883$			1582.03i			1.79166i	0.444398	+	0.895829i	$0.353418\pi$
$883$			1582.03i			1.79166i	−0.444398	+	0.895829i	$0.646582\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.06620			−0.00120203			−0.000601014	−	1.00000i	$-0.500191\pi$
$887$	−1.06620			−0.00120203			−0.000601014		1.00000i	$0.500191\pi$
$888$		0			0
$889$	1022.69			1.15038
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1070.82			1.19913
$894$		0			0
$895$	7.22867	−	115.924i	0.00807673	−	0.129524i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	126.083i		−	0.140249i
$900$		0			0
$901$	−89.4567			−0.0992860
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1292.70	+	80.6091i	1.42840	+	0.0890708i
$906$		0			0
$907$		−	1191.62i		−	1.31380i	−0.753978	−	0.656900i	$-0.771867\pi$
$907$		−	1191.62i		−	1.31380i	0.753978	−	0.656900i	$-0.228133\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	178.918i		−	0.196397i	−0.995167	−	0.0981986i	$-0.968692\pi$
$911$		−	178.918i		−	0.196397i	0.995167	−	0.0981986i	$-0.0313080\pi$
$912$		0			0
$913$			187.776i			0.205670i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	707.747			0.771807
$918$		0			0
$919$	−1400.32			−1.52374			−0.761870	−	0.647729i	$-0.775719\pi$
$919$	−1400.32			−1.52374			−0.761870	+	0.647729i	$0.775719\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−186.196			−0.201729
$924$		0			0
$925$	−78.2453	+	624.959i	−0.0845895	+	0.675631i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			417.203i			0.449088i	0.974464	+	0.224544i	$0.0720893\pi$
$929$			417.203i			0.449088i	−0.974464	+	0.224544i	$0.927911\pi$
$930$		0			0
$931$	450.239			0.483608
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.68483	−	75.1292i	0.00501051	−	0.0803521i
$936$		0			0
$937$			1300.03i			1.38744i	0.720246	+	0.693719i	$0.244029\pi$
$937$			1300.03i			1.38744i	−0.720246	+	0.693719i	$0.755971\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1586.25i			1.68571i	0.538140	+	0.842856i	$0.319127\pi$
$941$			1586.25i			1.68571i	−0.538140	+	0.842856i	$0.680873\pi$
$942$		0			0
$943$		−	71.2692i		−	0.0755771i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1024.86			1.08222			0.541108	−	0.840953i	$-0.318005\pi$
$947$	1024.86			1.08222			0.541108	+	0.840953i	$0.318005\pi$
$948$		0			0
$949$	928.980			0.978904
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1740.92			1.82678			0.913388	−	0.407091i	$-0.133457\pi$
$953$	1740.92			1.82678			0.913388	+	0.407091i	$0.133457\pi$
$954$		0			0
$955$	−52.8993	+	848.330i	−0.0553919	+	0.888303i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	939.119i		−	0.979269i
$960$		0			0
$961$	−933.394			−0.971274
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−24.8687	+	398.811i	−0.0257706	+	0.413276i
$966$		0			0
$967$		−	1214.71i		−	1.25616i	−0.778147	−	0.628082i	$-0.783840\pi$
$967$		−	1214.71i		−	1.25616i	0.778147	−	0.628082i	$-0.216160\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1150.91i		−	1.18529i	−0.805465	−	0.592644i	$-0.798084\pi$
$971$		−	1150.91i		−	1.18529i	0.805465	−	0.592644i	$-0.201916\pi$
$972$		0			0
$973$			774.567i			0.796061i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−544.530			−0.557349			−0.278674	−	0.960386i	$-0.589895\pi$
$977$	−544.530			−0.557349			−0.278674	+	0.960386i	$0.589895\pi$
$978$		0			0
$979$	−955.238			−0.975728
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1439.85			1.46475			0.732375	−	0.680902i	$-0.238412\pi$
$983$	1439.85			1.46475			0.732375	+	0.680902i	$0.238412\pi$
$984$		0			0
$985$	1053.95	+	65.7211i	1.07000	+	0.0667219i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	60.9234i		−	0.0616010i
$990$		0			0
$991$	1104.64			1.11467			0.557334	−	0.830288i	$-0.311824\pi$
$991$	1104.64			1.11467			0.557334	+	0.830288i	$0.311824\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−878.007	−	54.7499i	−0.882419	−	0.0550250i
$996$		0			0
$997$		−	611.431i		−	0.613271i	−0.951827	−	0.306636i	$-0.900797\pi$
$997$		−	611.431i		−	0.613271i	0.951827	−	0.306636i	$-0.0992033\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.b.a.809.24	yes	24
3.2	odd	2	inner	1620.3.b.a.809.1	✓	24
5.4	even	2	inner	1620.3.b.a.809.2	yes	24
9.2	odd	6		1620.3.t.f.1349.18		48
9.4	even	3		1620.3.t.f.269.8		48
9.5	odd	6		1620.3.t.f.269.17		48
9.7	even	3		1620.3.t.f.1349.7		48
15.14	odd	2	inner	1620.3.b.a.809.23	yes	24
45.4	even	6		1620.3.t.f.269.18		48
45.14	odd	6		1620.3.t.f.269.7		48
45.29	odd	6		1620.3.t.f.1349.8		48
45.34	even	6		1620.3.t.f.1349.17		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.b.a.809.1	✓	24	3.2	odd	2	inner
1620.3.b.a.809.2	yes	24	5.4	even	2	inner
1620.3.b.a.809.23	yes	24	15.14	odd	2	inner
1620.3.b.a.809.24	yes	24	1.1	even	1	trivial
1620.3.t.f.269.7		48	45.14	odd	6
1620.3.t.f.269.8		48	9.4	even	3
1620.3.t.f.269.17		48	9.5	odd	6
1620.3.t.f.269.18		48	45.4	even	6
1620.3.t.f.1349.7		48	9.7	even	3
1620.3.t.f.1349.8		48	45.29	odd	6
1620.3.t.f.1349.17		48	45.34	even	6
1620.3.t.f.1349.18		48	9.2	odd	6

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(4.99031 + 0.311180i) q^{5} -4.65609i q^{7} +O(q^{10})\)	\(q+(4.99031 + 0.311180i) q^{5} -4.65609i q^{7} +12.0275i q^{11} -8.46826i q^{13} -1.25172 q^{17} +16.4797 q^{19} +2.38818 q^{23} +(24.8063 + 3.10577i) q^{25} +23.9969i q^{29} -5.25415 q^{31} +(1.44888 - 23.2353i) q^{35} +25.1935i q^{37} -29.8425i q^{41} -25.5104i q^{43} +64.9783 q^{47} +27.3208 q^{49} +71.4673 q^{53} +(-3.74272 + 60.0210i) q^{55} +18.5016i q^{59} +19.1025 q^{61} +(2.63516 - 42.2592i) q^{65} -81.0903i q^{67} -21.9875i q^{71} +109.701i q^{73} +56.0012 q^{77} -134.193 q^{79} +15.6122 q^{83} +(-6.24645 - 0.389509i) q^{85} +79.4211i q^{89} -39.4290 q^{91} +(82.2389 + 5.12817i) q^{95} -130.304i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 48 q^{25} - 288 q^{49} - 36 q^{55} + 120 q^{61} + 480 q^{79} - 24 q^{85} + 48 q^{91}+O(q^{100}) \) 24 * q - 48 * q^25 - 288 * q^49 - 36 * q^55 + 120 * q^61 + 480 * q^79 - 24 * q^85 + 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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