LMFDB - Embedded newform 1620.3.b.a.809.4

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			809.4
Character	$\chi$	$=$	1620.809
Dual form			1620.3.b.a.809.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.19322 + 2.72340i) q^{5} +3.43857i q^{7} +O(q^{10})$	$q+(-4.19322 + 2.72340i) q^{5} +3.43857i q^{7} -12.8217i q^{11} +1.79388i q^{13} +30.9523 q^{17} -19.2201 q^{19} -2.38401 q^{23} +(10.1662 - 22.8396i) q^{25} +35.8847i q^{29} -40.9076 q^{31} +(-9.36459 - 14.4187i) q^{35} -53.6302i q^{37} +2.48241i q^{41} -55.1497i q^{43} +57.0500 q^{47} +37.1763 q^{49} +19.4030 q^{53} +(34.9187 + 53.7643i) q^{55} +69.1469i q^{59} +17.6094 q^{61} +(-4.88544 - 7.52212i) q^{65} +51.1686i q^{67} +53.9194i q^{71} +42.7832i q^{73} +44.0883 q^{77} +88.8456 q^{79} +28.0944 q^{83} +(-129.790 + 84.2956i) q^{85} +68.3114i q^{89} -6.16836 q^{91} +(80.5940 - 52.3440i) q^{95} -156.100i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.19322	+	2.72340i	−0.838644	+	0.544680i
$5$	−4.19322	+	2.72340i	−0.838644	+	0.544680i
$6$		0			0
$7$			3.43857i			0.491224i	0.969368	+	0.245612i	$0.0789889\pi$
$7$			3.43857i			0.491224i	−0.969368	+	0.245612i	$0.921011\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	12.8217i		−	1.16561i	−0.812612	−	0.582806i	$-0.801955\pi$
$11$		−	12.8217i		−	1.16561i	0.812612	−	0.582806i	$-0.198045\pi$
$12$		0			0
$13$			1.79388i			0.137990i	0.997617	+	0.0689952i	$0.0219793\pi$
$13$			1.79388i			0.137990i	−0.997617	+	0.0689952i	$0.978021\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	30.9523			1.82073			0.910363	−	0.413811i	$-0.135803\pi$
$17$	30.9523			1.82073			0.910363	+	0.413811i	$0.135803\pi$
$18$		0			0
$19$	−19.2201			−1.01158			−0.505792	−	0.862656i	$-0.668800\pi$
$19$	−19.2201			−1.01158			−0.505792	+	0.862656i	$0.668800\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.38401			−0.103653			−0.0518263	−	0.998656i	$-0.516504\pi$
$23$	−2.38401			−0.103653			−0.0518263	+	0.998656i	$0.516504\pi$
$24$		0			0
$25$	10.1662	−	22.8396i	0.406647	−	0.913585i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			35.8847i			1.23740i	0.785626	+	0.618701i	$0.212341\pi$
$29$			35.8847i			1.23740i	−0.785626	+	0.618701i	$0.787659\pi$
$30$		0			0
$31$	−40.9076			−1.31960			−0.659800	−	0.751441i	$-0.729359\pi$
$31$	−40.9076			−1.31960			−0.659800	+	0.751441i	$0.729359\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.36459	−	14.4187i	−0.267560	−	0.411962i
$36$		0			0
$37$		−	53.6302i		−	1.44947i	−0.689030	−	0.724733i	$-0.741963\pi$
$37$		−	53.6302i		−	1.44947i	0.689030	−	0.724733i	$-0.258037\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.48241i			0.0605465i	0.999542	+	0.0302732i	$0.00963774\pi$
$41$			2.48241i			0.0605465i	−0.999542	+	0.0302732i	$0.990362\pi$
$42$		0			0
$43$		−	55.1497i		−	1.28255i	−0.767311	−	0.641276i	$-0.778405\pi$
$43$		−	55.1497i		−	1.28255i	0.767311	−	0.641276i	$-0.221595\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	57.0500			1.21383			0.606914	−	0.794767i	$-0.292407\pi$
$47$	57.0500			1.21383			0.606914	+	0.794767i	$0.292407\pi$
$48$		0			0
$49$	37.1763			0.758699
$50$		0			0
$51$		0			0
$52$		0			0
$53$	19.4030			0.366094			0.183047	−	0.983104i	$-0.441404\pi$
$53$	19.4030			0.366094			0.183047	+	0.983104i	$0.441404\pi$
$54$		0			0
$55$	34.9187	+	53.7643i	0.634886	+	0.977533i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			69.1469i			1.17198i	0.810318	+	0.585991i	$0.199295\pi$
$59$			69.1469i			1.17198i	−0.810318	+	0.585991i	$0.800705\pi$
$60$		0			0
$61$	17.6094			0.288679			0.144340	−	0.989528i	$-0.453894\pi$
$61$	17.6094			0.288679			0.144340	+	0.989528i	$0.453894\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.88544	−	7.52212i	−0.0751607	−	0.115725i
$66$		0			0
$67$			51.1686i			0.763711i	0.924222	+	0.381855i	$0.124715\pi$
$67$			51.1686i			0.763711i	−0.924222	+	0.381855i	$0.875285\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			53.9194i			0.759428i	0.925104	+	0.379714i	$0.123978\pi$
$71$			53.9194i			0.759428i	−0.925104	+	0.379714i	$0.876022\pi$
$72$		0			0
$73$			42.7832i			0.586072i	0.956101	+	0.293036i	$0.0946655\pi$
$73$			42.7832i			0.586072i	−0.956101	+	0.293036i	$0.905334\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	44.0883			0.572576
$78$		0			0
$79$	88.8456			1.12463			0.562314	−	0.826924i	$-0.309911\pi$
$79$	88.8456			1.12463			0.562314	+	0.826924i	$0.309911\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	28.0944			0.338487			0.169243	−	0.985574i	$-0.445868\pi$
$83$	28.0944			0.338487			0.169243	+	0.985574i	$0.445868\pi$
$84$		0			0
$85$	−129.790	+	84.2956i	−1.52694	+	0.991713i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			68.3114i			0.767544i	0.923428	+	0.383772i	$0.125375\pi$
$89$			68.3114i			0.767544i	−0.923428	+	0.383772i	$0.874625\pi$
$90$		0			0
$91$	−6.16836			−0.0677842
$92$		0			0
$93$		0			0
$94$		0			0
$95$	80.5940	−	52.3440i	0.848358	−	0.550989i
$96$		0			0
$97$		−	156.100i		−	1.60927i	−0.593767	−	0.804637i	$-0.702360\pi$
$97$		−	156.100i		−	1.60927i	0.593767	−	0.804637i	$-0.297640\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			69.9654i			0.692727i	0.938100	+	0.346364i	$0.112584\pi$
$101$			69.9654i			0.692727i	−0.938100	+	0.346364i	$0.887416\pi$
$102$		0			0
$103$			192.148i			1.86551i	0.360505	+	0.932757i	$0.382604\pi$
$103$			192.148i			1.86551i	−0.360505	+	0.932757i	$0.617396\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	78.5826			0.734417			0.367209	−	0.930139i	$-0.380314\pi$
$107$	78.5826			0.734417			0.367209	+	0.930139i	$0.380314\pi$
$108$		0			0
$109$	3.54619			0.0325339			0.0162669	−	0.999868i	$-0.494822\pi$
$109$	3.54619			0.0325339			0.0162669	+	0.999868i	$0.494822\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	212.485			1.88040			0.940201	−	0.340620i	$-0.110637\pi$
$113$	212.485			1.88040			0.940201	+	0.340620i	$0.110637\pi$
$114$		0			0
$115$	9.99668	−	6.49262i	0.0869277	−	0.0564576i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			106.432i			0.894384i
$120$		0			0
$121$	−43.3967			−0.358650
$122$		0			0
$123$		0			0
$124$		0			0
$125$	19.5725	+	123.458i	0.156580	+	0.987665i
$126$		0			0
$127$			183.486i			1.44477i	0.691489	+	0.722387i	$0.256955\pi$
$127$			183.486i			1.44477i	−0.691489	+	0.722387i	$0.743045\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			60.6738i			0.463159i	0.972816	+	0.231579i	$0.0743893\pi$
$131$			60.6738i			0.463159i	−0.972816	+	0.231579i	$0.925611\pi$
$132$		0			0
$133$		−	66.0895i		−	0.496914i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.4737			−0.0910489			−0.0455244	−	0.998963i	$-0.514496\pi$
$137$	−12.4737			−0.0910489			−0.0455244	+	0.998963i	$0.514496\pi$
$138$		0			0
$139$	−73.6800			−0.530072			−0.265036	−	0.964238i	$-0.585384\pi$
$139$	−73.6800			−0.530072			−0.265036	+	0.964238i	$0.585384\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	23.0006			0.160843
$144$		0			0
$145$	−97.7283	−	150.472i	−0.673989	−	1.03774i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	4.64593i		−	0.0311808i	−0.999878	−	0.0155904i	$-0.995037\pi$
$149$		−	4.64593i		−	0.0311808i	0.999878	−	0.0155904i	$-0.00496277\pi$
$150$		0			0
$151$	237.020			1.56967			0.784835	−	0.619705i	$-0.212748\pi$
$151$	237.020			1.56967			0.784835	+	0.619705i	$0.212748\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	171.535	−	111.408i	1.10667	−	0.718760i
$156$		0			0
$157$			157.598i			1.00381i	0.864923	+	0.501905i	$0.167367\pi$
$157$			157.598i			1.00381i	−0.864923	+	0.501905i	$0.832633\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	8.19758i		−	0.0509166i
$162$		0			0
$163$		−	82.9389i		−	0.508828i	−0.967095	−	0.254414i	$-0.918117\pi$
$163$		−	82.9389i		−	0.508828i	0.967095	−	0.254414i	$-0.0818826\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−90.4344			−0.541524			−0.270762	−	0.962646i	$-0.587276\pi$
$167$	−90.4344			−0.541524			−0.270762	+	0.962646i	$0.587276\pi$
$168$		0			0
$169$	165.782			0.980959
$170$		0			0
$171$		0			0
$172$		0			0
$173$	259.811			1.50180			0.750898	−	0.660418i	$-0.229621\pi$
$173$	259.811			1.50180			0.750898	+	0.660418i	$0.229621\pi$
$174$		0			0
$175$	78.5356	+	34.9570i	0.448775	+	0.199755i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	228.604i		−	1.27712i	−0.769573	−	0.638559i	$-0.779531\pi$
$179$		−	228.604i		−	1.27712i	0.769573	−	0.638559i	$-0.220469\pi$
$180$		0			0
$181$	30.8901			0.170664			0.0853318	−	0.996353i	$-0.472805\pi$
$181$	30.8901			0.170664			0.0853318	+	0.996353i	$0.472805\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	146.057	+	224.883i	0.789496	+	1.21559i
$186$		0			0
$187$		−	396.862i		−	2.12226i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			2.62942i			0.0137666i	0.999976	+	0.00688330i	$0.00219104\pi$
$191$			2.62942i			0.0137666i	−0.999976	+	0.00688330i	$0.997809\pi$
$192$		0			0
$193$			21.8326i			0.113122i	0.998399	+	0.0565610i	$0.0180136\pi$
$193$			21.8326i			0.113122i	−0.998399	+	0.0565610i	$0.981986\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	68.8358			0.349421			0.174710	−	0.984620i	$-0.444101\pi$
$197$	68.8358			0.349421			0.174710	+	0.984620i	$0.444101\pi$
$198$		0			0
$199$	116.185			0.583846			0.291923	−	0.956442i	$-0.405705\pi$
$199$	116.185			0.583846			0.291923	+	0.956442i	$0.405705\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−123.392			−0.607841
$204$		0			0
$205$	−6.76059	−	10.4093i	−0.0329785	−	0.0507769i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			246.435i			1.17911i
$210$		0			0
$211$	−224.613			−1.06452			−0.532258	−	0.846582i	$-0.678656\pi$
$211$	−224.613			−1.06452			−0.532258	+	0.846582i	$0.678656\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	150.195	+	231.255i	0.698580	+	1.07560i
$216$		0			0
$217$		−	140.664i		−	0.648219i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			55.5247i			0.251243i
$222$		0			0
$223$		−	253.749i		−	1.13789i	−0.822377	−	0.568944i	$-0.807352\pi$
$223$		−	253.749i		−	1.13789i	0.822377	−	0.568944i	$-0.192648\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	55.5992			0.244930			0.122465	−	0.992473i	$-0.460920\pi$
$227$	55.5992			0.244930			0.122465	+	0.992473i	$0.460920\pi$
$228$		0			0
$229$	−222.782			−0.972845			−0.486423	−	0.873724i	$-0.661698\pi$
$229$	−222.782			−0.972845			−0.486423	+	0.873724i	$0.661698\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	399.458			1.71441			0.857207	−	0.514972i	$-0.172198\pi$
$233$	399.458			1.71441			0.857207	+	0.514972i	$0.172198\pi$
$234$		0			0
$235$	−239.223	+	155.370i	−1.01797	+	0.661149i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	20.2025i		−	0.0845294i	−0.999106	−	0.0422647i	$-0.986543\pi$
$239$		−	20.2025i		−	0.0845294i	0.999106	−	0.0422647i	$-0.0134573\pi$
$240$		0			0
$241$	−336.598			−1.39667			−0.698336	−	0.715770i	$-0.746076\pi$
$241$	−336.598			−1.39667			−0.698336	+	0.715770i	$0.746076\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−155.888	+	101.246i	−0.636278	+	0.413249i
$246$		0			0
$247$		−	34.4784i		−	0.139589i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			280.569i			1.11781i	0.829233	+	0.558903i	$0.188778\pi$
$251$			280.569i			1.11781i	−0.829233	+	0.558903i	$0.811222\pi$
$252$		0			0
$253$			30.5671i			0.120819i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	164.293			0.639271			0.319636	−	0.947541i	$-0.396440\pi$
$257$	164.293			0.639271			0.319636	+	0.947541i	$0.396440\pi$
$258$		0			0
$259$	184.411			0.712012
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−294.049			−1.11806			−0.559028	−	0.829149i	$-0.688826\pi$
$263$	−294.049			−1.11806			−0.559028	+	0.829149i	$0.688826\pi$
$264$		0			0
$265$	−81.3609	+	52.8421i	−0.307022	+	0.199404i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	373.214i		−	1.38741i	−0.720259	−	0.693706i	$-0.755977\pi$
$269$		−	373.214i		−	1.38741i	0.720259	−	0.693706i	$-0.244023\pi$
$270$		0			0
$271$	−388.909			−1.43509			−0.717543	−	0.696514i	$-0.754733\pi$
$271$	−388.909			−1.43509			−0.717543	+	0.696514i	$0.754733\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−292.844	−	130.348i	−1.06489	−	0.473992i
$276$		0			0
$277$			371.465i			1.34103i	0.741897	+	0.670514i	$0.233926\pi$
$277$			371.465i			1.34103i	−0.741897	+	0.670514i	$0.766074\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	271.037i		−	0.964545i	−0.876021	−	0.482273i	$-0.839812\pi$
$281$		−	271.037i		−	0.964545i	0.876021	−	0.482273i	$-0.160188\pi$
$282$		0			0
$283$			33.3411i			0.117813i	0.998264	+	0.0589065i	$0.0187614\pi$
$283$			33.3411i			0.117813i	−0.998264	+	0.0589065i	$0.981239\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.53591			−0.0297419
$288$		0			0
$289$	669.047			2.31504
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−119.022			−0.406218			−0.203109	−	0.979156i	$-0.565105\pi$
$293$	−119.022			−0.406218			−0.203109	+	0.979156i	$0.565105\pi$
$294$		0			0
$295$	−188.315	−	289.948i	−0.638356	−	0.982876i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	4.27662i		−	0.0143031i
$300$		0			0
$301$	189.636			0.630020
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−73.8402	+	47.9575i	−0.242099	+	0.157238i
$306$		0			0
$307$		−	63.9942i		−	0.208450i	−0.994554	−	0.104225i	$-0.966764\pi$
$307$		−	63.9942i		−	0.208450i	0.994554	−	0.104225i	$-0.0332362\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			106.616i			0.342816i	0.985200	+	0.171408i	$0.0548317\pi$
$311$			106.616i			0.342816i	−0.985200	+	0.171408i	$0.945168\pi$
$312$		0			0
$313$			218.725i			0.698801i	0.936973	+	0.349401i	$0.113615\pi$
$313$			218.725i			0.698801i	−0.936973	+	0.349401i	$0.886385\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	352.689			1.11258			0.556292	−	0.830987i	$-0.312224\pi$
$317$	352.689			1.11258			0.556292	+	0.830987i	$0.312224\pi$
$318$		0			0
$319$	460.103			1.44233
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−594.906			−1.84182
$324$		0			0
$325$	40.9715	+	18.2369i	0.126066	+	0.0561134i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			196.170i			0.596261i
$330$		0			0
$331$	60.7020			0.183390			0.0916949	−	0.995787i	$-0.470772\pi$
$331$	60.7020			0.183390			0.0916949	+	0.995787i	$0.470772\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−139.353	−	214.561i	−0.415978	−	0.640481i
$336$		0			0
$337$			22.1238i			0.0656492i	0.999461	+	0.0328246i	$0.0104503\pi$
$337$			22.1238i			0.0656492i	−0.999461	+	0.0328246i	$0.989550\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			524.506i			1.53814i
$342$		0			0
$343$			296.323i			0.863915i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	349.943			1.00848			0.504241	−	0.863563i	$-0.331772\pi$
$347$	349.943			1.00848			0.504241	+	0.863563i	$0.331772\pi$
$348$		0			0
$349$	281.407			0.806323			0.403161	−	0.915129i	$-0.367911\pi$
$349$	281.407			0.806323			0.403161	+	0.915129i	$0.367911\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	159.986			0.453219			0.226609	−	0.973986i	$-0.427236\pi$
$353$	159.986			0.453219			0.226609	+	0.973986i	$0.427236\pi$
$354$		0			0
$355$	−146.844	−	226.096i	−0.413646	−	0.636890i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	643.967i		−	1.79378i	−0.442254	−	0.896890i	$-0.645821\pi$
$359$		−	643.967i		−	1.79378i	0.442254	−	0.896890i	$-0.354179\pi$
$360$		0			0
$361$	8.41153			0.0233006
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−116.516	−	179.399i	−0.319222	−	0.491505i
$366$		0			0
$367$			619.982i			1.68933i	0.535299	+	0.844663i	$0.320199\pi$
$367$			619.982i			1.68933i	−0.535299	+	0.844663i	$0.679801\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			66.7184i			0.179834i
$372$		0			0
$373$		−	143.026i		−	0.383447i	−0.981449	−	0.191724i	$-0.938592\pi$
$373$		−	143.026i		−	0.383447i	0.981449	−	0.191724i	$-0.0614077\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−64.3726			−0.170750
$378$		0			0
$379$	−57.1486			−0.150788			−0.0753940	−	0.997154i	$-0.524021\pi$
$379$	−57.1486			−0.150788			−0.0753940	+	0.997154i	$0.524021\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	290.908			0.759550			0.379775	−	0.925079i	$-0.376001\pi$
$383$	290.908			0.759550			0.379775	+	0.925079i	$0.376001\pi$
$384$		0			0
$385$	−184.872	+	120.070i	−0.480187	+	0.311871i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			654.106i			1.68151i	0.541419	+	0.840753i	$0.317887\pi$
$389$			654.106i			1.68151i	−0.541419	+	0.840753i	$0.682113\pi$
$390$		0			0
$391$	−73.7907			−0.188723
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−372.549	+	241.962i	−0.943162	+	0.612563i
$396$		0			0
$397$		−	229.991i		−	0.579323i	−0.957129	−	0.289661i	$-0.906457\pi$
$397$		−	229.991i		−	0.579323i	0.957129	−	0.289661i	$-0.0935427\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	510.947i		−	1.27418i	−0.770788	−	0.637091i	$-0.780137\pi$
$401$		−	510.947i		−	1.27418i	0.770788	−	0.637091i	$-0.219863\pi$
$402$		0			0
$403$		−	73.3832i		−	0.182092i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−687.632			−1.68951
$408$		0			0
$409$	599.265			1.46520			0.732598	−	0.680662i	$-0.238308\pi$
$409$	599.265			1.46520			0.732598	+	0.680662i	$0.238308\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−237.766			−0.575705
$414$		0			0
$415$	−117.806	+	76.5123i	−0.283870	+	0.184367i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	606.996i		−	1.44868i	−0.689444	−	0.724339i	$-0.742145\pi$
$419$		−	606.996i		−	1.44868i	0.689444	−	0.724339i	$-0.257855\pi$
$420$		0			0
$421$	−452.155			−1.07400			−0.537001	−	0.843582i	$-0.680443\pi$
$421$	−452.155			−1.07400			−0.537001	+	0.843582i	$0.680443\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	314.667	−	706.940i	0.740392	−	1.66339i
$426$		0			0
$427$			60.5512i			0.141806i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	684.701i		−	1.58863i	−0.607505	−	0.794316i	$-0.707829\pi$
$431$		−	684.701i		−	1.58863i	0.607505	−	0.794316i	$-0.292171\pi$
$432$		0			0
$433$		−	458.058i		−	1.05787i	−0.848662	−	0.528936i	$-0.822591\pi$
$433$		−	458.058i		−	1.05787i	0.848662	−	0.528936i	$-0.177409\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	45.8209			0.104853
$438$		0			0
$439$	12.1452			0.0276655			0.0138328	−	0.999904i	$-0.495597\pi$
$439$	12.1452			0.0276655			0.0138328	+	0.999904i	$0.495597\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−474.449			−1.07099			−0.535495	−	0.844538i	$-0.679875\pi$
$443$	−474.449			−1.07099			−0.535495	+	0.844538i	$0.679875\pi$
$444$		0			0
$445$	−186.039	−	286.445i	−0.418066	−	0.643696i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	269.662i		−	0.600583i	−0.953847	−	0.300291i	$-0.902916\pi$
$449$		−	269.662i		−	0.600583i	0.953847	−	0.300291i	$-0.0970840\pi$
$450$		0			0
$451$	31.8287			0.0705737
$452$		0			0
$453$		0			0
$454$		0			0
$455$	25.8653	−	16.7989i	0.0568468	−	0.0369207i
$456$		0			0
$457$			558.523i			1.22215i	0.791573	+	0.611075i	$0.209263\pi$
$457$			558.523i			1.22215i	−0.791573	+	0.611075i	$0.790737\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	700.113i		−	1.51868i	−0.650692	−	0.759341i	$-0.725521\pi$
$461$		−	700.113i		−	1.51868i	0.650692	−	0.759341i	$-0.274479\pi$
$462$		0			0
$463$		−	802.069i		−	1.73233i	−0.499758	−	0.866165i	$-0.666578\pi$
$463$		−	802.069i		−	1.73233i	0.499758	−	0.866165i	$-0.333422\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−868.960			−1.86073			−0.930364	−	0.366636i	$-0.880509\pi$
$467$	−868.960			−1.86073			−0.930364	+	0.366636i	$0.880509\pi$
$468$		0			0
$469$	−175.947			−0.375153
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−707.115			−1.49496
$474$		0			0
$475$	−195.395	+	438.980i	−0.411357	+	0.924168i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			705.287i			1.47242i	0.676755	+	0.736208i	$0.263386\pi$
$479$			705.287i			1.47242i	−0.676755	+	0.736208i	$0.736614\pi$
$480$		0			0
$481$	96.2060			0.200013
$482$		0			0
$483$		0			0
$484$		0			0
$485$	425.122	+	654.560i	0.876540	+	1.34961i
$486$		0			0
$487$			715.753i			1.46972i	0.678220	+	0.734859i	$0.262752\pi$
$487$			715.753i			1.46972i	−0.678220	+	0.734859i	$0.737248\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	101.390i		−	0.206498i	−0.994656	−	0.103249i	$-0.967076\pi$
$491$		−	101.390i		−	0.206498i	0.994656	−	0.103249i	$-0.0329238\pi$
$492$		0			0
$493$			1110.71i			2.25297i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−185.405			−0.373049
$498$		0			0
$499$	937.926			1.87961			0.939805	−	0.341711i	$-0.111006\pi$
$499$	937.926			1.87961			0.939805	+	0.341711i	$0.111006\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	766.791			1.52444			0.762218	−	0.647321i	$-0.224111\pi$
$503$	766.791			1.52444			0.762218	+	0.647321i	$0.224111\pi$
$504$		0			0
$505$	−190.544	−	293.380i	−0.377315	−	0.580951i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			747.402i			1.46837i	0.678948	+	0.734186i	$0.262436\pi$
$509$			747.402i			1.46837i	−0.678948	+	0.734186i	$0.737564\pi$
$510$		0			0
$511$	−147.113			−0.287892
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−523.296	−	805.719i	−1.01611	−	1.56450i
$516$		0			0
$517$		−	731.479i		−	1.41485i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			39.7011i			0.0762018i	0.999274	+	0.0381009i	$0.0121308\pi$
$521$			39.7011i			0.0762018i	−0.999274	+	0.0381009i	$0.987869\pi$
$522$		0			0
$523$			135.722i			0.259506i	0.991546	+	0.129753i	$0.0414184\pi$
$523$			135.722i			0.259506i	−0.991546	+	0.129753i	$0.958582\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1266.19			−2.40263
$528$		0			0
$529$	−523.316			−0.989256
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.45313			−0.00835484
$534$		0			0
$535$	−329.514	+	214.012i	−0.615914	+	0.400022i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	476.664i		−	0.884349i
$540$		0			0
$541$	840.659			1.55390			0.776949	−	0.629563i	$-0.216766\pi$
$541$	840.659			1.55390			0.776949	+	0.629563i	$0.216766\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−14.8700	+	9.65771i	−0.0272843	+	0.0177206i
$546$		0			0
$547$		−	945.998i		−	1.72943i	−0.502263	−	0.864715i	$-0.667499\pi$
$547$		−	945.998i		−	1.72943i	0.502263	−	0.864715i	$-0.332501\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	689.706i		−	1.25174i
$552$		0			0
$553$			305.501i			0.552444i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	78.6518			0.141206			0.0706031	−	0.997504i	$-0.477508\pi$
$557$	78.6518			0.141206			0.0706031	+	0.997504i	$0.477508\pi$
$558$		0			0
$559$	98.9317			0.176980
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−278.855			−0.495302			−0.247651	−	0.968849i	$-0.579659\pi$
$563$	−278.855			−0.495302			−0.247651	+	0.968849i	$0.579659\pi$
$564$		0			0
$565$	−890.998	+	578.683i	−1.57699	+	1.02422i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			582.055i			1.02294i	0.859300	+	0.511472i	$0.170900\pi$
$569$			582.055i			1.02294i	−0.859300	+	0.511472i	$0.829100\pi$
$570$		0			0
$571$	−307.250			−0.538092			−0.269046	−	0.963127i	$-0.586708\pi$
$571$	−307.250			−0.538092			−0.269046	+	0.963127i	$0.586708\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.2363	+	54.4500i	−0.0421500	+	0.0946956i
$576$		0			0
$577$			24.6333i			0.0426921i	0.999772	+	0.0213460i	$0.00679517\pi$
$577$			24.6333i			0.0426921i	−0.999772	+	0.0213460i	$0.993205\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			96.6044i			0.166273i
$582$		0			0
$583$		−	248.780i		−	0.426723i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.4524			−0.0348423			−0.0174211	−	0.999848i	$-0.505546\pi$
$587$	−20.4524			−0.0348423			−0.0174211	+	0.999848i	$0.505546\pi$
$588$		0			0
$589$	786.248			1.33489
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−676.813			−1.14134			−0.570669	−	0.821180i	$-0.693316\pi$
$593$	−676.813			−1.14134			−0.570669	+	0.821180i	$0.693316\pi$
$594$		0			0
$595$	−289.856	−	446.291i	−0.487153	−	0.750069i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	54.2796i		−	0.0906170i	−0.998973	−	0.0453085i	$-0.985573\pi$
$599$		−	54.2796i		−	0.0906170i	0.998973	−	0.0453085i	$-0.0144271\pi$
$600$		0			0
$601$	−285.301			−0.474711			−0.237355	−	0.971423i	$-0.576281\pi$
$601$	−285.301			−0.474711			−0.237355	+	0.971423i	$0.576281\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	181.972	−	118.187i	0.300780	−	0.195350i
$606$		0			0
$607$		−	982.351i		−	1.61837i	−0.587554	−	0.809185i	$-0.699909\pi$
$607$		−	982.351i		−	1.61837i	0.587554	−	0.809185i	$-0.300091\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			102.341i			0.167497i
$612$		0			0
$613$		−	850.701i		−	1.38777i	−0.720087	−	0.693883i	$-0.755898\pi$
$613$		−	850.701i		−	1.38777i	0.720087	−	0.693883i	$-0.244102\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−128.853			−0.208838			−0.104419	−	0.994533i	$-0.533298\pi$
$617$	−128.853			−0.208838			−0.104419	+	0.994533i	$0.533298\pi$
$618$		0			0
$619$	905.661			1.46310			0.731552	−	0.681786i	$-0.238797\pi$
$619$	905.661			1.46310			0.731552	+	0.681786i	$0.238797\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−234.893			−0.377036
$624$		0			0
$625$	−418.298	−	464.383i	−0.669277	−	0.743013i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1659.98i		−	2.63908i
$630$		0			0
$631$	−1083.23			−1.71669			−0.858345	−	0.513073i	$-0.828507\pi$
$631$	−1083.23			−1.71669			−0.858345	+	0.513073i	$0.828507\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−499.707	−	769.398i	−0.786940	−	1.21165i
$636$		0			0
$637$			66.6896i			0.104693i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1011.30i			1.57770i	0.614588	+	0.788848i	$0.289322\pi$
$641$			1011.30i			1.57770i	−0.614588	+	0.788848i	$0.710678\pi$
$642$		0			0
$643$		−	1137.24i		−	1.76864i	−0.466881	−	0.884320i	$-0.654622\pi$
$643$		−	1137.24i		−	1.76864i	0.466881	−	0.884320i	$-0.345378\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−273.788			−0.423166			−0.211583	−	0.977360i	$-0.567862\pi$
$647$	−273.788			−0.423166			−0.211583	+	0.977360i	$0.567862\pi$
$648$		0			0
$649$	886.583			1.36608
$650$		0			0
$651$		0			0
$652$		0			0
$653$	401.233			0.614445			0.307223	−	0.951638i	$-0.400600\pi$
$653$	401.233			0.614445			0.307223	+	0.951638i	$0.400600\pi$
$654$		0			0
$655$	−165.239	−	254.419i	−0.252274	−	0.388425i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			373.828i			0.567266i	0.958933	+	0.283633i	$0.0915398\pi$
$659$			373.828i			0.567266i	−0.958933	+	0.283633i	$0.908460\pi$
$660$		0			0
$661$	130.958			0.198120			0.0990602	−	0.995081i	$-0.468416\pi$
$661$	130.958			0.198120			0.0990602	+	0.995081i	$0.468416\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	179.988	+	277.128i	0.270659	+	0.416734i
$666$		0			0
$667$		−	85.5494i		−	0.128260i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	225.783i		−	0.336488i
$672$		0			0
$673$			696.868i			1.03546i	0.855543	+	0.517732i	$0.173224\pi$
$673$			696.868i			1.03546i	−0.855543	+	0.517732i	$0.826776\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−492.873			−0.728025			−0.364013	−	0.931394i	$-0.618594\pi$
$677$	−492.873			−0.728025			−0.364013	+	0.931394i	$0.618594\pi$
$678$		0			0
$679$	536.759			0.790513
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−628.686			−0.920477			−0.460238	−	0.887795i	$-0.652236\pi$
$683$	−628.686			−0.920477			−0.460238	+	0.887795i	$0.652236\pi$
$684$		0			0
$685$	52.3049	−	33.9709i	0.0763576	−	0.0495925i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			34.8065i			0.0505174i
$690$		0			0
$691$	−1234.63			−1.78672			−0.893362	−	0.449337i	$-0.851660\pi$
$691$	−1234.63			−1.78672			−0.893362	+	0.449337i	$0.851660\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	308.957	−	200.660i	0.444542	−	0.288720i
$696$		0			0
$697$			76.8363i			0.110239i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			834.094i			1.18986i	0.803777	+	0.594931i	$0.202821\pi$
$701$			834.094i			1.18986i	−0.803777	+	0.594931i	$0.797179\pi$
$702$		0			0
$703$			1030.78i			1.46626i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−240.581			−0.340284
$708$		0			0
$709$	102.204			0.144153			0.0720765	−	0.997399i	$-0.477037\pi$
$709$	102.204			0.144153			0.0720765	+	0.997399i	$0.477037\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	97.5242			0.136780
$714$		0			0
$715$	−96.4465	+	62.6398i	−0.134890	+	0.0876082i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	593.832i		−	0.825913i	−0.910751	−	0.412957i	$-0.864496\pi$
$719$		−	593.832i		−	0.825913i	0.910751	−	0.412957i	$-0.135504\pi$
$720$		0			0
$721$	−660.714			−0.916385
$722$		0			0
$723$		0			0
$724$		0			0
$725$	819.593	+	364.810i	1.13047	+	0.503186i
$726$		0			0
$727$			670.784i			0.922674i	0.887225	+	0.461337i	$0.152630\pi$
$727$			670.784i			0.922674i	−0.887225	+	0.461337i	$0.847370\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1707.01i		−	2.33517i
$732$		0			0
$733$			842.366i			1.14920i	0.818433	+	0.574602i	$0.194843\pi$
$733$			842.366i			1.14920i	−0.818433	+	0.574602i	$0.805157\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	656.070			0.890190
$738$		0			0
$739$	1423.99			1.92692			0.963460	−	0.267853i	$-0.0863141\pi$
$739$	1423.99			1.92692			0.963460	+	0.267853i	$0.0863141\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−551.291			−0.741980			−0.370990	−	0.928637i	$-0.620982\pi$
$743$	−551.291			−0.741980			−0.370990	+	0.928637i	$0.620982\pi$
$744$		0			0
$745$	12.6527	+	19.4814i	0.0169835	+	0.0261496i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			270.212i			0.360763i
$750$		0			0
$751$	285.173			0.379724			0.189862	−	0.981811i	$-0.439196\pi$
$751$	285.173			0.379724			0.189862	+	0.981811i	$0.439196\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−993.877	+	645.501i	−1.31639	+	0.854968i
$756$		0			0
$757$		−	787.273i		−	1.03999i	−0.854169	−	0.519995i	$-0.825934\pi$
$757$		−	787.273i		−	1.03999i	0.854169	−	0.519995i	$-0.174066\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	101.175i		−	0.132950i	−0.997788	−	0.0664752i	$-0.978825\pi$
$761$		−	101.175i		−	0.132950i	0.997788	−	0.0664752i	$-0.0211753\pi$
$762$		0			0
$763$			12.1938i			0.0159814i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−124.041			−0.161722
$768$		0			0
$769$	−3.68997			−0.00479840			−0.00239920	−	0.999997i	$-0.500764\pi$
$769$	−3.68997			−0.00479840			−0.00239920	+	0.999997i	$0.500764\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	504.260			0.652341			0.326171	−	0.945311i	$-0.394242\pi$
$773$	504.260			0.652341			0.326171	+	0.945311i	$0.394242\pi$
$774$		0			0
$775$	−415.874	+	934.315i	−0.536611	+	1.20557i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	47.7120i		−	0.0612478i
$780$		0			0
$781$	691.340			0.885198
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−429.203	−	660.844i	−0.546756	−	0.841839i
$786$		0			0
$787$			1412.86i			1.79524i	0.440766	+	0.897622i	$0.354707\pi$
$787$			1412.86i			1.79524i	−0.440766	+	0.897622i	$0.645293\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			730.645i			0.923698i
$792$		0			0
$793$			31.5891i			0.0398350i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	482.767			0.605730			0.302865	−	0.953034i	$-0.402057\pi$
$797$	482.767			0.605730			0.302865	+	0.953034i	$0.402057\pi$
$798$		0			0
$799$	1765.83			2.21005
$800$		0			0
$801$		0			0
$802$		0			0
$803$	548.555			0.683132
$804$		0			0
$805$	22.3253	+	34.3742i	0.0277333	+	0.0427009i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	874.774i		−	1.08130i	−0.841247	−	0.540652i	$-0.818178\pi$
$809$		−	874.774i		−	1.08130i	0.841247	−	0.540652i	$-0.181822\pi$
$810$		0			0
$811$	−568.734			−0.701275			−0.350638	−	0.936511i	$-0.614035\pi$
$811$	−568.734			−0.701275			−0.350638	+	0.936511i	$0.614035\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	225.876	+	347.781i	0.277149	+	0.426725i
$816$		0			0
$817$			1059.98i			1.29741i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	148.710i		−	0.181133i	−0.995890	−	0.0905664i	$-0.971132\pi$
$821$		−	148.710i		−	0.181133i	0.995890	−	0.0905664i	$-0.0288677\pi$
$822$		0			0
$823$			505.561i			0.614290i	0.951663	+	0.307145i	$0.0993737\pi$
$823$			505.561i			0.614290i	−0.951663	+	0.307145i	$0.900626\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	980.267			1.18533			0.592664	−	0.805450i	$-0.298076\pi$
$827$	980.267			1.18533			0.592664	+	0.805450i	$0.298076\pi$
$828$		0			0
$829$	190.269			0.229516			0.114758	−	0.993393i	$-0.463391\pi$
$829$	190.269			0.229516			0.114758	+	0.993393i	$0.463391\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1150.69			1.38138
$834$		0			0
$835$	379.211	−	246.289i	0.454145	−	0.294957i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1179.94i			1.40636i	0.711011	+	0.703181i	$0.248238\pi$
$839$			1179.94i			1.40636i	−0.711011	+	0.703181i	$0.751762\pi$
$840$		0			0
$841$	−446.709			−0.531164
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−695.160	+	451.491i	−0.822675	+	0.534309i
$846$		0			0
$847$		−	149.222i		−	0.176177i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			127.855i			0.150241i
$852$		0			0
$853$		−	1120.35i		−	1.31342i	−0.754142	−	0.656711i	$-0.771947\pi$
$853$		−	1120.35i		−	1.31342i	0.754142	−	0.656711i	$-0.228053\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1300.72			1.51776			0.758882	−	0.651228i	$-0.225746\pi$
$857$	1300.72			1.51776			0.758882	+	0.651228i	$0.225746\pi$
$858$		0			0
$859$	−867.902			−1.01036			−0.505182	−	0.863013i	$-0.668575\pi$
$859$	−867.902			−1.01036			−0.505182	+	0.863013i	$0.668575\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−258.845			−0.299937			−0.149968	−	0.988691i	$-0.547917\pi$
$863$	−258.845			−0.299937			−0.149968	+	0.988691i	$0.547917\pi$
$864$		0			0
$865$	−1089.44	+	707.569i	−1.25947	+	0.817999i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1139.15i		−	1.31088i
$870$		0			0
$871$	−91.7902			−0.105385
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−424.519	+	67.3014i	−0.485165	+	0.0769159i
$876$		0			0
$877$		−	19.0346i		−	0.0217042i	−0.999941	−	0.0108521i	$-0.996546\pi$
$877$		−	19.0346i		−	0.0217042i	0.999941	−	0.0108521i	$-0.00345440\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1688.86i			1.91699i	0.285117	+	0.958493i	$0.407968\pi$
$881$			1688.86i			1.91699i	−0.285117	+	0.958493i	$0.592032\pi$
$882$		0			0
$883$			373.276i			0.422736i	0.977407	+	0.211368i	$0.0677918\pi$
$883$			373.276i			0.422736i	−0.977407	+	0.211368i	$0.932208\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1196.59			1.34903			0.674516	−	0.738260i	$-0.264352\pi$
$887$	1196.59			1.34903			0.674516	+	0.738260i	$0.264352\pi$
$888$		0			0
$889$	−630.930			−0.709707
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1096.50			−1.22789
$894$		0			0
$895$	622.581	+	958.587i	0.695621	+	1.07105i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1467.96i		−	1.63288i
$900$		0			0
$901$	600.567			0.666556
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−129.529	+	84.1262i	−0.143126	+	0.0929571i
$906$		0			0
$907$		−	877.709i		−	0.967706i	−0.875149	−	0.483853i	$-0.839237\pi$
$907$		−	877.709i		−	0.967706i	0.875149	−	0.483853i	$-0.160763\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			245.697i			0.269700i	0.990866	+	0.134850i	$0.0430553\pi$
$911$			245.697i			0.269700i	−0.990866	+	0.134850i	$0.956945\pi$
$912$		0			0
$913$		−	360.219i		−	0.394544i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−208.631			−0.227515
$918$		0			0
$919$	66.4769			0.0723361			0.0361681	−	0.999346i	$-0.488485\pi$
$919$	66.4769			0.0723361			0.0361681	+	0.999346i	$0.488485\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−96.7247			−0.104794
$924$		0			0
$925$	−1224.90	−	545.214i	−1.32421	−	0.589421i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1362.54i			1.46667i	0.679867	+	0.733336i	$0.262038\pi$
$929$			1362.54i			1.46667i	−0.679867	+	0.733336i	$0.737962\pi$
$930$		0			0
$931$	−714.531			−0.767487
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1080.82	+	1664.13i	1.15595	+	1.77982i
$936$		0			0
$937$		−	750.141i		−	0.800578i	−0.916389	−	0.400289i	$-0.868910\pi$
$937$		−	750.141i		−	0.800578i	0.916389	−	0.400289i	$-0.131090\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			262.183i			0.278621i	0.990249	+	0.139311i	$0.0444886\pi$
$941$			262.183i			0.278621i	−0.990249	+	0.139311i	$0.955511\pi$
$942$		0			0
$943$		−	5.91808i		−	0.00627580i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	658.541			0.695397			0.347699	−	0.937606i	$-0.386963\pi$
$947$	658.541			0.695397			0.347699	+	0.937606i	$0.386963\pi$
$948$		0			0
$949$	−76.7478			−0.0808723
$950$		0			0
$951$		0			0
$952$		0			0
$953$	122.390			0.128426			0.0642131	−	0.997936i	$-0.479546\pi$
$953$	122.390			0.128426			0.0642131	+	0.997936i	$0.479546\pi$
$954$		0			0
$955$	−7.16096	−	11.0257i	−0.00749839	−	0.0115453i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	42.8916i		−	0.0447254i
$960$		0			0
$961$	712.432			0.741345
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−59.4588	−	91.5487i	−0.0616154	−	0.0948691i
$966$		0			0
$967$		−	1066.24i		−	1.10262i	−0.834299	−	0.551312i	$-0.814127\pi$
$967$		−	1066.24i		−	1.10262i	0.834299	−	0.551312i	$-0.185873\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1798.30i		−	1.85201i	−0.377508	−	0.926006i	$-0.623219\pi$
$971$		−	1798.30i		−	1.85201i	0.377508	−	0.926006i	$-0.376781\pi$
$972$		0			0
$973$		−	253.354i		−	0.260384i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	574.073			0.587587			0.293794	−	0.955869i	$-0.405082\pi$
$977$	574.073			0.587587			0.293794	+	0.955869i	$0.405082\pi$
$978$		0			0
$979$	875.870			0.894658
$980$		0			0
$981$		0			0
$982$		0			0
$983$	764.267			0.777484			0.388742	−	0.921347i	$-0.372910\pi$
$983$	764.267			0.777484			0.388742	+	0.921347i	$0.372910\pi$
$984$		0			0
$985$	−288.644	+	187.468i	−0.293039	+	0.190322i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			131.478i			0.132940i
$990$		0			0
$991$	398.586			0.402206			0.201103	−	0.979570i	$-0.435547\pi$
$991$	398.586			0.402206			0.201103	+	0.979570i	$0.435547\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−487.191	+	316.419i	−0.489639	+	0.318009i
$996$		0			0
$997$		−	1476.05i		−	1.48049i	−0.672337	−	0.740246i	$-0.734709\pi$
$997$		−	1476.05i		−	1.48049i	0.672337	−	0.740246i	$-0.265291\pi$
$998$		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.809.4	yes	24
3.2	odd	2	inner	1620.3.b.a.809.21	yes	24
5.4	even	2	inner	1620.3.b.a.809.22	yes	24
9.2	odd	6		1620.3.t.f.1349.13		48
9.4	even	3		1620.3.t.f.269.20		48
9.5	odd	6		1620.3.t.f.269.5		48
9.7	even	3		1620.3.t.f.1349.12		48
15.14	odd	2	inner	1620.3.b.a.809.3	✓	24
45.4	even	6		1620.3.t.f.269.13		48
45.14	odd	6		1620.3.t.f.269.12		48
45.29	odd	6		1620.3.t.f.1349.20		48
45.34	even	6		1620.3.t.f.1349.5		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.b.a.809.3	✓	24	15.14	odd	2	inner
1620.3.b.a.809.4	yes	24	1.1	even	1	trivial
1620.3.b.a.809.21	yes	24	3.2	odd	2	inner
1620.3.b.a.809.22	yes	24	5.4	even	2	inner
1620.3.t.f.269.5		48	9.5	odd	6
1620.3.t.f.269.12		48	45.14	odd	6
1620.3.t.f.269.13		48	45.4	even	6
1620.3.t.f.269.20		48	9.4	even	3
1620.3.t.f.1349.5		48	45.34	even	6
1620.3.t.f.1349.12		48	9.7	even	3
1620.3.t.f.1349.13		48	9.2	odd	6
1620.3.t.f.1349.20		48	45.29	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)

\(f(q)\)	\(=\)	\(q+(-4.19322 + 2.72340i) q^{5} +3.43857i q^{7} +O(q^{10})\)	\(q+(-4.19322 + 2.72340i) q^{5} +3.43857i q^{7} -12.8217i q^{11} +1.79388i q^{13} +30.9523 q^{17} -19.2201 q^{19} -2.38401 q^{23} +(10.1662 - 22.8396i) q^{25} +35.8847i q^{29} -40.9076 q^{31} +(-9.36459 - 14.4187i) q^{35} -53.6302i q^{37} +2.48241i q^{41} -55.1497i q^{43} +57.0500 q^{47} +37.1763 q^{49} +19.4030 q^{53} +(34.9187 + 53.7643i) q^{55} +69.1469i q^{59} +17.6094 q^{61} +(-4.88544 - 7.52212i) q^{65} +51.1686i q^{67} +53.9194i q^{71} +42.7832i q^{73} +44.0883 q^{77} +88.8456 q^{79} +28.0944 q^{83} +(-129.790 + 84.2956i) q^{85} +68.3114i q^{89} -6.16836 q^{91} +(80.5940 - 52.3440i) q^{95} -156.100i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 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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