LMFDB - Embedded newform 4020.2.f.b.401.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(401,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4020.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.f (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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.15
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.b.401.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.06681 - 1.36452i) q^{3} +1.00000 q^{5} -1.51153i q^{7} +(-0.723813 + 2.91137i) q^{9} +O(q^{10})$	\(q+(-1.06681 - 1.36452i) q^{3} +1.00000 q^{5} -1.51153i q^{7} +(-0.723813 + 2.91137i) q^{9} -2.91428 q^{11} -1.06747i q^{13} +(-1.06681 - 1.36452i) q^{15} -4.54062i q^{17} -8.24273 q^{19} +(-2.06250 + 1.61252i) q^{21} +1.28047i q^{23} +1.00000 q^{25} +(4.74479 - 2.11824i) q^{27} +7.49237i q^{29} +5.99822i q^{31} +(3.10900 + 3.97659i) q^{33} -1.51153i q^{35} -7.60740 q^{37} +(-1.45658 + 1.13879i) q^{39} +9.17275 q^{41} -3.43442i q^{43} +(-0.723813 + 2.91137i) q^{45} -2.02537i q^{47} +4.71529 q^{49} +(-6.19576 + 4.84400i) q^{51} +5.94923 q^{53} -2.91428 q^{55} +(8.79347 + 11.2473i) q^{57} +8.92625i q^{59} -7.65318i q^{61} +(4.40062 + 1.09406i) q^{63} -1.06747i q^{65} +(-2.57720 - 7.76904i) q^{67} +(1.74722 - 1.36602i) q^{69} +1.51803i q^{71} +4.97461 q^{73} +(-1.06681 - 1.36452i) q^{75} +4.40502i q^{77} +16.8186i q^{79} +(-7.95219 - 4.21458i) q^{81} +9.82948i q^{83} -4.54062i q^{85} +(10.2235 - 7.99297i) q^{87} +4.29002i q^{89} -1.61351 q^{91} +(8.18468 - 6.39899i) q^{93} -8.24273 q^{95} +16.1446i q^{97} +(2.10940 - 8.48457i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 46 q^{5} - 4 q^{9}+O(q^{10}) $ 46 * q + 46 * q^5 - 4 * q^9	$ 46 q + 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} + 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} + 4 q^{41} - 4 q^{45} - 62 q^{49} + 8 q^{51} + 8 q^{53} + 12 q^{57} + 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} + 8 q^{95}+O(q^{100}) $ 46 * q + 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 + 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 + 4 * q^41 - 4 * q^45 - 62 * q^49 + 8 * q^51 + 8 * q^53 + 12 * q^57 + 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−1.06681	−	1.36452i	−0.615926	−	0.787804i
$3$	−1.06681	−	1.36452i	−0.615926	−	0.787804i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	1.51153i		−	0.571304i	−0.958333	−	0.285652i	$-0.907790\pi$
$7$		−	1.51153i		−	0.571304i	0.958333	−	0.285652i	$-0.0922101\pi$
$8$		0			0
$9$	−0.723813	+	2.91137i	−0.241271	+	0.970458i
$10$		0			0
$11$	−2.91428			−0.878690			−0.439345	−	0.898318i	$-0.644789\pi$
$11$	−2.91428			−0.878690			−0.439345	+	0.898318i	$0.644789\pi$
$12$		0			0
$13$		−	1.06747i		−	0.296063i	−0.988983	−	0.148031i	$-0.952706\pi$
$13$		−	1.06747i		−	0.296063i	0.988983	−	0.148031i	$-0.0472937\pi$
$14$		0			0
$15$	−1.06681	−	1.36452i	−0.275450	−	0.352317i
$16$		0			0
$17$		−	4.54062i		−	1.10126i	−0.834748	−	0.550632i	$-0.814387\pi$
$17$		−	4.54062i		−	1.10126i	0.834748	−	0.550632i	$-0.185613\pi$
$18$		0			0
$19$	−8.24273			−1.89101			−0.945506	−	0.325604i	$-0.894432\pi$
$19$	−8.24273			−1.89101			−0.945506	+	0.325604i	$0.894432\pi$
$20$		0			0
$21$	−2.06250	+	1.61252i	−0.450075	+	0.351881i
$22$		0			0
$23$			1.28047i			0.266996i	0.991049	+	0.133498i	$0.0426210\pi$
$23$			1.28047i			0.266996i	−0.991049	+	0.133498i	$0.957379\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	4.74479	−	2.11824i	0.913136	−	0.407656i
$28$		0			0
$29$			7.49237i			1.39130i	0.718382	+	0.695649i	$0.244883\pi$
$29$			7.49237i			1.39130i	−0.718382	+	0.695649i	$0.755117\pi$
$30$		0			0
$31$			5.99822i			1.07731i	0.842526	+	0.538656i	$0.181068\pi$
$31$			5.99822i			1.07731i	−0.842526	+	0.538656i	$0.818932\pi$
$32$		0			0
$33$	3.10900	+	3.97659i	0.541208	+	0.692235i
$34$		0			0
$35$		−	1.51153i		−	0.255495i
$36$		0			0
$37$	−7.60740			−1.25065			−0.625325	−	0.780365i	$-0.715034\pi$
$37$	−7.60740			−1.25065			−0.625325	+	0.780365i	$0.715034\pi$
$38$		0			0
$39$	−1.45658	+	1.13879i	−0.233240	+	0.182353i
$40$		0			0
$41$	9.17275			1.43254			0.716271	−	0.697822i	$-0.245847\pi$
$41$	9.17275			1.43254			0.716271	+	0.697822i	$0.245847\pi$
$42$		0			0
$43$		−	3.43442i		−	0.523745i	−0.965103	−	0.261872i	$-0.915660\pi$
$43$		−	3.43442i		−	0.523745i	0.965103	−	0.261872i	$-0.0843399\pi$
$44$		0			0
$45$	−0.723813	+	2.91137i	−0.107900	+	0.434002i
$46$		0			0
$47$		−	2.02537i		−	0.295431i	−0.989030	−	0.147716i	$-0.952808\pi$
$47$		−	2.02537i		−	0.295431i	0.989030	−	0.147716i	$-0.0471920\pi$
$48$		0			0
$49$	4.71529			0.673612
$50$		0			0
$51$	−6.19576	+	4.84400i	−0.867580	+	0.678296i
$52$		0			0
$53$	5.94923			0.817189			0.408594	−	0.912716i	$-0.366019\pi$
$53$	5.94923			0.817189			0.408594	+	0.912716i	$0.366019\pi$
$54$		0			0
$55$	−2.91428			−0.392962
$56$		0			0
$57$	8.79347	+	11.2473i	1.16472	+	1.48975i
$58$		0			0
$59$			8.92625i			1.16210i	0.813868	+	0.581050i	$0.197358\pi$
$59$			8.92625i			1.16210i	−0.813868	+	0.581050i	$0.802642\pi$
$60$		0			0
$61$		−	7.65318i		−	0.979890i	−0.871753	−	0.489945i	$-0.837017\pi$
$61$		−	7.65318i		−	0.979890i	0.871753	−	0.489945i	$-0.162983\pi$
$62$		0			0
$63$	4.40062	+	1.09406i	0.554426	+	0.137839i
$64$		0			0
$65$		−	1.06747i		−	0.132403i
$66$		0			0
$67$	−2.57720	−	7.76904i	−0.314855	−	0.949140i
$67$	−2.57720	−	7.76904i	−0.314855	−	0.949140i
$68$		0			0
$69$	1.74722	−	1.36602i	0.210341	−	0.164450i
$70$		0			0
$71$			1.51803i			0.180158i	0.995935	+	0.0900788i	$0.0287119\pi$
$71$			1.51803i			0.180158i	−0.995935	+	0.0900788i	$0.971288\pi$
$72$		0			0
$73$	4.97461			0.582234			0.291117	−	0.956688i	$-0.405973\pi$
$73$	4.97461			0.582234			0.291117	+	0.956688i	$0.405973\pi$
$74$		0			0
$75$	−1.06681	−	1.36452i	−0.123185	−	0.157561i
$76$		0			0
$77$			4.40502i			0.501999i
$78$		0			0
$79$			16.8186i			1.89225i	0.323808	+	0.946123i	$0.395037\pi$
$79$			16.8186i			1.89225i	−0.323808	+	0.946123i	$0.604963\pi$
$80$		0			0
$81$	−7.95219	−	4.21458i	−0.883577	−	0.468287i
$82$		0			0
$83$			9.82948i			1.07893i	0.842009	+	0.539463i	$0.181373\pi$
$83$			9.82948i			1.07893i	−0.842009	+	0.539463i	$0.818627\pi$
$84$		0			0
$85$		−	4.54062i		−	0.492500i
$86$		0			0
$87$	10.2235	−	7.99297i	1.09607	−	0.856937i
$88$		0			0
$89$			4.29002i			0.454742i	0.973808	+	0.227371i	$0.0730129\pi$
$89$			4.29002i			0.454742i	−0.973808	+	0.227371i	$0.926987\pi$
$90$		0			0
$91$	−1.61351			−0.169142
$92$		0			0
$93$	8.18468	−	6.39899i	0.848711	−	0.663544i
$94$		0			0
$95$	−8.24273			−0.845687
$96$		0			0
$97$			16.1446i			1.63924i	0.572910	+	0.819619i	$0.305815\pi$
$97$			16.1446i			1.63924i	−0.572910	+	0.819619i	$0.694185\pi$
$98$		0			0
$99$	2.10940	−	8.48457i	0.212002	−	0.852731i
$100$		0			0
$101$	−4.47308			−0.445088			−0.222544	−	0.974923i	$-0.571436\pi$
$101$	−4.47308			−0.445088			−0.222544	+	0.974923i	$0.571436\pi$
$102$		0			0
$103$	−6.60791			−0.651097			−0.325548	−	0.945525i	$-0.605549\pi$
$103$	−6.60791			−0.651097			−0.325548	+	0.945525i	$0.605549\pi$
$104$		0			0
$105$	−2.06250	+	1.61252i	−0.201280	+	0.157366i
$106$		0			0
$107$		−	10.2247i		−	0.988458i	−0.869332	−	0.494229i	$-0.835450\pi$
$107$		−	10.2247i		−	0.988458i	0.869332	−	0.494229i	$-0.164550\pi$
$108$		0			0
$109$			8.64732i			0.828263i	0.910217	+	0.414132i	$0.135915\pi$
$109$			8.64732i			0.828263i	−0.910217	+	0.414132i	$0.864085\pi$
$110$		0			0
$111$	8.11569	+	10.3804i	0.770307	+	0.985267i
$112$		0			0
$113$	−13.6310			−1.28230			−0.641150	−	0.767416i	$-0.721542\pi$
$113$	−13.6310			−1.28230			−0.641150	+	0.767416i	$0.721542\pi$
$114$		0			0
$115$			1.28047i			0.119404i
$116$		0			0
$117$	3.10780	+	0.772649i	0.287317	+	0.0714314i
$118$		0			0
$119$	−6.86328			−0.629156
$120$		0			0
$121$	−2.50695			−0.227904
$122$		0			0
$123$	−9.78562	−	12.5164i	−0.882340	−	1.12856i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−3.89073			−0.345246			−0.172623	−	0.984988i	$-0.555224\pi$
$127$	−3.89073			−0.345246			−0.172623	+	0.984988i	$0.555224\pi$
$128$		0			0
$129$	−4.68633	+	3.66389i	−0.412608	+	0.322588i
$130$		0			0
$131$			3.92563i			0.342984i	0.985186	+	0.171492i	$0.0548587\pi$
$131$			3.92563i			0.342984i	−0.985186	+	0.171492i	$0.945141\pi$
$132$		0			0
$133$			12.4591i			1.08034i
$134$		0			0
$135$	4.74479	−	2.11824i	0.408367	−	0.182309i
$136$		0			0
$137$	13.8677			1.18480			0.592399	−	0.805645i	$-0.298181\pi$
$137$	13.8677			1.18480			0.592399	+	0.805645i	$0.298181\pi$
$138$		0			0
$139$			1.40851i			0.119468i	0.998214	+	0.0597341i	$0.0190253\pi$
$139$			1.40851i			0.119468i	−0.998214	+	0.0597341i	$0.980975\pi$
$140$		0			0
$141$	−2.76366	+	2.16070i	−0.232742	+	0.181964i
$142$		0			0
$143$			3.11091i			0.260148i
$144$		0			0
$145$			7.49237i			0.622208i
$146$		0			0
$147$	−5.03034	−	6.43409i	−0.414895	−	0.530675i
$148$		0			0
$149$		−	13.2459i		−	1.08515i	−0.840007	−	0.542575i	$-0.817449\pi$
$149$		−	13.2459i		−	1.08515i	0.840007	−	0.542575i	$-0.182551\pi$
$150$		0			0
$151$	−16.9992			−1.38337			−0.691686	−	0.722198i	$-0.743132\pi$
$151$	−16.9992			−1.38337			−0.691686	+	0.722198i	$0.743132\pi$
$152$		0			0
$153$	13.2195	+	3.28656i	1.06873	+	0.265703i
$154$		0			0
$155$			5.99822i			0.481789i
$156$		0			0
$157$	7.32679			0.584741			0.292371	−	0.956305i	$-0.405556\pi$
$157$	7.32679			0.584741			0.292371	+	0.956305i	$0.405556\pi$
$158$		0			0
$159$	−6.34672	−	8.11782i	−0.503328	−	0.643785i
$160$		0			0
$161$	1.93546			0.152536
$162$		0			0
$163$	−16.7119			−1.30897			−0.654487	−	0.756073i	$-0.727115\pi$
$163$	−16.7119			−1.30897			−0.654487	+	0.756073i	$0.727115\pi$
$164$		0			0
$165$	3.10900	+	3.97659i	0.242035	+	0.309577i
$166$		0			0
$167$			13.2730i			1.02709i	0.858062	+	0.513547i	$0.171669\pi$
$167$			13.2730i			1.02709i	−0.858062	+	0.513547i	$0.828331\pi$
$168$		0			0
$169$	11.8605			0.912347
$170$		0			0
$171$	5.96620	−	23.9977i	0.456246	−	1.83515i
$172$		0			0
$173$			14.4002i			1.09483i	0.836862	+	0.547413i	$0.184387\pi$
$173$			14.4002i			1.09483i	−0.836862	+	0.547413i	$0.815613\pi$
$174$		0			0
$175$		−	1.51153i		−	0.114261i
$176$		0			0
$177$	12.1800	−	9.52266i	0.915507	−	0.715767i
$178$		0			0
$179$	18.4474			1.37883			0.689413	−	0.724368i	$-0.257868\pi$
$179$	18.4474			1.37883			0.689413	+	0.724368i	$0.257868\pi$
$180$		0			0
$181$	−4.38149			−0.325673			−0.162837	−	0.986653i	$-0.552064\pi$
$181$	−4.38149			−0.325673			−0.162837	+	0.986653i	$0.552064\pi$
$182$		0			0
$183$	−10.4429	+	8.16453i	−0.771961	+	0.603539i
$184$		0			0
$185$	−7.60740			−0.559308
$186$		0			0
$187$			13.2327i			0.967669i
$188$		0			0
$189$	−3.20178	−	7.17188i	−0.232895	−	0.521678i
$190$		0			0
$191$	22.0174			1.59313			0.796563	−	0.604556i	$-0.206649\pi$
$191$	22.0174			1.59313			0.796563	+	0.604556i	$0.206649\pi$
$192$		0			0
$193$	14.8853			1.07146			0.535732	−	0.844388i	$-0.320036\pi$
$193$	14.8853			1.07146			0.535732	+	0.844388i	$0.320036\pi$
$194$		0			0
$195$	−1.45658	+	1.13879i	−0.104308	+	0.0815507i
$196$		0			0
$197$	13.0057			0.926619			0.463309	−	0.886197i	$-0.346662\pi$
$197$	13.0057			0.926619			0.463309	+	0.886197i	$0.346662\pi$
$198$		0			0
$199$	−9.59551			−0.680208			−0.340104	−	0.940388i	$-0.610462\pi$
$199$	−9.59551			−0.680208			−0.340104	+	0.940388i	$0.610462\pi$
$200$		0			0
$201$	−7.85160	+	11.8048i	−0.553809	+	0.832644i
$202$		0			0
$203$	11.3249			0.794854
$204$		0			0
$205$	9.17275			0.640652
$206$		0			0
$207$	−3.72793	−	0.926821i	−0.259109	−	0.0644185i
$208$		0			0
$209$	24.0217			1.66161
$210$		0			0
$211$	11.1278			0.766068			0.383034	−	0.923734i	$-0.374879\pi$
$211$	11.1278			0.766068			0.383034	+	0.923734i	$0.374879\pi$
$212$		0			0
$213$	2.07138	−	1.61946i	0.141929	−	0.110964i
$214$		0			0
$215$		−	3.43442i		−	0.234226i
$216$		0			0
$217$	9.06647			0.615472
$218$		0			0
$219$	−5.30698	−	6.78793i	−0.358613	−	0.458686i
$220$		0			0
$221$	−4.84698			−0.326043
$222$		0			0
$223$	17.2132			1.15268			0.576342	−	0.817209i	$-0.304480\pi$
$223$	17.2132			1.15268			0.576342	+	0.817209i	$0.304480\pi$
$224$		0			0
$225$	−0.723813	+	2.91137i	−0.0482542	+	0.194092i
$226$		0			0
$227$			23.7558i			1.57673i	0.615207	+	0.788365i	$0.289072\pi$
$227$			23.7558i			1.57673i	−0.615207	+	0.788365i	$0.710928\pi$
$228$		0			0
$229$		−	3.12837i		−	0.206728i	−0.994644	−	0.103364i	$-0.967039\pi$
$229$		−	3.12837i		−	0.206728i	0.994644	−	0.103364i	$-0.0329607\pi$
$230$		0			0
$231$	6.01072	−	4.69934i	0.395477	−	0.309194i
$232$		0			0
$233$	24.9797			1.63648			0.818238	−	0.574879i	$-0.194951\pi$
$233$	24.9797			1.63648			0.818238	+	0.574879i	$0.194951\pi$
$234$		0			0
$235$		−	2.02537i		−	0.132121i
$236$		0			0
$237$	22.9493	−	17.9424i	1.49072	−	1.16548i
$238$		0			0
$239$	10.8930			0.704612			0.352306	−	0.935885i	$-0.385398\pi$
$239$	10.8930			0.704612			0.352306	+	0.935885i	$0.385398\pi$
$240$		0			0
$241$	−12.4204			−0.800069			−0.400034	−	0.916500i	$-0.631002\pi$
$241$	−12.4204			−0.800069			−0.400034	+	0.916500i	$0.631002\pi$
$242$		0			0
$243$	2.73265	+	15.3471i	0.175299	+	0.984515i
$244$		0			0
$245$	4.71529			0.301249
$246$		0			0
$247$			8.79887i			0.559859i
$248$		0			0
$249$	13.4125	−	10.4862i	0.849982	−	0.664538i
$250$		0			0
$251$	−22.2285			−1.40305			−0.701525	−	0.712645i	$-0.747497\pi$
$251$	−22.2285			−1.40305			−0.701525	+	0.712645i	$0.747497\pi$
$252$		0			0
$253$		−	3.73165i		−	0.234607i
$254$		0			0
$255$	−6.19576	+	4.84400i	−0.387993	+	0.303343i
$256$		0			0
$257$			0.556494i			0.0347132i	0.999849	+	0.0173566i	$0.00552505\pi$
$257$			0.556494i			0.0347132i	−0.999849	+	0.0173566i	$0.994475\pi$
$258$		0			0
$259$			11.4988i			0.714501i
$260$		0			0
$261$	−21.8131	−	5.42308i	−1.35020	−	0.335680i
$262$		0			0
$263$			14.1074i			0.869901i	0.900454	+	0.434951i	$0.143234\pi$
$263$			14.1074i			0.869901i	−0.900454	+	0.434951i	$0.856766\pi$
$264$		0			0
$265$	5.94923			0.365458
$266$		0			0
$267$	5.85381	−	4.57666i	0.358247	−	0.280087i
$268$		0			0
$269$		−	9.90184i		−	0.603726i	−0.953351	−	0.301863i	$-0.902392\pi$
$269$		−	9.90184i		−	0.603726i	0.953351	−	0.301863i	$-0.0976084\pi$
$270$		0			0
$271$		−	5.07509i		−	0.308290i	−0.988048	−	0.154145i	$-0.950738\pi$
$271$		−	5.07509i		−	0.308290i	0.988048	−	0.154145i	$-0.0492623\pi$
$272$		0			0
$273$	1.72132	+	2.20166i	0.104179	+	0.133251i
$274$		0			0
$275$	−2.91428			−0.175738
$276$		0			0
$277$	−1.17216			−0.0704283			−0.0352141	−	0.999380i	$-0.511211\pi$
$277$	−1.17216			−0.0704283			−0.0352141	+	0.999380i	$0.511211\pi$
$278$		0			0
$279$	−17.4631	−	4.34159i	−1.04549	−	0.259924i
$280$		0			0
$281$	−22.8128			−1.36090			−0.680449	−	0.732795i	$-0.738215\pi$
$281$	−22.8128			−1.36090			−0.680449	+	0.732795i	$0.738215\pi$
$282$		0			0
$283$	19.8426			1.17952			0.589760	−	0.807579i	$-0.299223\pi$
$283$	19.8426			1.17952			0.589760	+	0.807579i	$0.299223\pi$
$284$		0			0
$285$	8.79347	+	11.2473i	0.520880	+	0.666235i
$286$		0			0
$287$		−	13.8649i		−	0.818417i
$288$		0			0
$289$	−3.61727			−0.212780
$290$		0			0
$291$	22.0296	−	17.2233i	1.29140	−	1.00965i
$292$		0			0
$293$			17.4402i			1.01887i	0.860509	+	0.509435i	$0.170145\pi$
$293$			17.4402i			1.01887i	−0.860509	+	0.509435i	$0.829855\pi$
$294$		0			0
$295$			8.92625i			0.519707i
$296$		0			0
$297$	−13.8277	+	6.17316i	−0.802363	+	0.358203i
$298$		0			0
$299$	1.36686			0.0790478
$300$		0			0
$301$	−5.19122			−0.299217
$302$		0			0
$303$	4.77194	+	6.10359i	0.274141	+	0.350642i
$304$		0			0
$305$		−	7.65318i		−	0.438220i
$306$		0			0
$307$	−6.55544			−0.374139			−0.187069	−	0.982347i	$-0.559899\pi$
$307$	−6.55544			−0.374139			−0.187069	+	0.982347i	$0.559899\pi$
$308$		0			0
$309$	7.04941	+	9.01660i	0.401027	+	0.512937i
$310$		0			0
$311$	−27.6982			−1.57062			−0.785311	−	0.619102i	$-0.787497\pi$
$311$	−27.6982			−1.57062			−0.785311	+	0.619102i	$0.787497\pi$
$312$		0			0
$313$			10.8663i			0.614199i	0.951677	+	0.307100i	$0.0993584\pi$
$313$			10.8663i			0.614199i	−0.951677	+	0.307100i	$0.900642\pi$
$314$		0			0
$315$	4.40062	+	1.09406i	0.247947	+	0.0616435i
$316$		0			0
$317$			26.3852i			1.48194i	0.671538	+	0.740970i	$0.265634\pi$
$317$			26.3852i			1.48194i	−0.671538	+	0.740970i	$0.734366\pi$
$318$		0			0
$319$		−	21.8349i		−	1.22252i
$320$		0			0
$321$	−13.9518	+	10.9079i	−0.778712	+	0.608817i
$322$		0			0
$323$			37.4272i			2.08250i
$324$		0			0
$325$		−	1.06747i		−	0.0592126i
$326$		0			0
$327$	11.7994	−	9.22509i	0.652509	−	0.510149i
$328$		0			0
$329$	−3.06141			−0.168781
$330$		0			0
$331$		−	5.49210i		−	0.301873i	−0.988543	−	0.150937i	$-0.951771\pi$
$331$		−	5.49210i		−	0.301873i	0.988543	−	0.150937i	$-0.0482289\pi$
$332$		0			0
$333$	5.50634	−	22.1480i	0.301745	−	1.21370i
$334$		0			0
$335$	−2.57720	−	7.76904i	−0.140807	−	0.424468i
$336$		0			0
$337$			12.4996i			0.680896i	0.940263	+	0.340448i	$0.110579\pi$
$337$			12.4996i			0.680896i	−0.940263	+	0.340448i	$0.889421\pi$
$338$		0			0
$339$	14.5418	+	18.5998i	0.789801	+	1.01020i
$340$		0			0
$341$		−	17.4805i		−	0.946623i
$342$		0			0
$343$		−	17.7080i		−	0.956141i
$344$		0			0
$345$	1.74722	−	1.36602i	0.0940673	−	0.0735443i
$346$		0			0
$347$	−11.5310			−0.619016			−0.309508	−	0.950897i	$-0.600164\pi$
$347$	−11.5310			−0.619016			−0.309508	+	0.950897i	$0.600164\pi$
$348$		0			0
$349$	−15.4774			−0.828485			−0.414242	−	0.910167i	$-0.635953\pi$
$349$	−15.4774			−0.828485			−0.414242	+	0.910167i	$0.635953\pi$
$350$		0			0
$351$	−2.26116	−	5.06493i	−0.120692	−	0.270346i
$352$		0			0
$353$	−4.14277			−0.220498			−0.110249	−	0.993904i	$-0.535165\pi$
$353$	−4.14277			−0.220498			−0.110249	+	0.993904i	$0.535165\pi$
$354$		0			0
$355$			1.51803i			0.0805689i
$356$		0			0
$357$	7.32184	+	9.36506i	0.387513	+	0.495651i
$358$		0			0
$359$		−	33.2259i		−	1.75360i	−0.480859	−	0.876798i	$-0.659675\pi$
$359$		−	33.2259i		−	1.75360i	0.480859	−	0.876798i	$-0.340325\pi$
$360$		0			0
$361$	48.9426			2.57593
$362$		0			0
$363$	2.67445	+	3.42077i	0.140372	+	0.179544i
$364$		0			0
$365$	4.97461			0.260383
$366$		0			0
$367$		−	24.5728i		−	1.28269i	−0.767252	−	0.641345i	$-0.778377\pi$
$367$		−	24.5728i		−	1.28269i	0.767252	−	0.641345i	$-0.221623\pi$
$368$		0			0
$369$	−6.63935	+	26.7053i	−0.345631	+	1.39022i
$370$		0			0
$371$		−	8.99242i		−	0.466863i
$372$		0			0
$373$			15.9856i			0.827701i	0.910345	+	0.413850i	$0.135816\pi$
$373$			15.9856i			0.827701i	−0.910345	+	0.413850i	$0.864184\pi$
$374$		0			0
$375$	−1.06681	−	1.36452i	−0.0550901	−	0.0704634i
$376$		0			0
$377$	7.99788			0.411912
$378$		0			0
$379$			38.4640i			1.97576i	0.155204	+	0.987882i	$0.450396\pi$
$379$			38.4640i			1.97576i	−0.155204	+	0.987882i	$0.549604\pi$
$380$		0			0
$381$	4.15069	+	5.30897i	0.212646	+	0.271987i
$382$		0			0
$383$	−11.2036			−0.572477			−0.286239	−	0.958158i	$-0.592405\pi$
$383$	−11.2036			−0.572477			−0.286239	+	0.958158i	$0.592405\pi$
$384$		0			0
$385$			4.40502i			0.224501i
$386$		0			0
$387$	9.99889	+	2.48588i	0.508272	+	0.126364i
$388$		0			0
$389$		−	9.48315i		−	0.480815i	−0.970672	−	0.240407i	$-0.922719\pi$
$389$		−	9.48315i		−	0.480815i	0.970672	−	0.240407i	$-0.0772810\pi$
$390$		0			0
$391$	5.81413			0.294033
$392$		0			0
$393$	5.35659	−	4.18792i	0.270204	−	0.211253i
$394$		0			0
$395$			16.8186i			0.846238i
$396$		0			0
$397$	−33.6795			−1.69033			−0.845164	−	0.534508i	$-0.820497\pi$
$397$	−33.6795			−1.69033			−0.845164	+	0.534508i	$0.820497\pi$
$398$		0			0
$399$	17.0007	−	13.2916i	0.851098	−	0.665411i
$400$		0			0
$401$	17.0538			0.851624			0.425812	−	0.904812i	$-0.359988\pi$
$401$	17.0538			0.851624			0.425812	+	0.904812i	$0.359988\pi$
$402$		0			0
$403$	6.40292			0.318952
$404$		0			0
$405$	−7.95219	−	4.21458i	−0.395147	−	0.209424i
$406$		0			0
$407$	22.1701			1.09893
$408$		0			0
$409$			15.4762i			0.765251i	0.923904	+	0.382625i	$0.124980\pi$
$409$			15.4762i			0.765251i	−0.923904	+	0.382625i	$0.875020\pi$
$410$		0			0
$411$	−14.7943	−	18.9227i	−0.729747	−	0.933389i
$412$		0			0
$413$	13.4923			0.663911
$414$		0			0
$415$			9.82948i			0.482510i
$416$		0			0
$417$	1.92194	−	1.50262i	0.0941176	−	0.0735836i
$418$		0			0
$419$			17.4402i			0.852010i	0.904721	+	0.426005i	$0.140079\pi$
$419$			17.4402i			0.852010i	−0.904721	+	0.426005i	$0.859921\pi$
$420$		0			0
$421$	−19.4314			−0.947026			−0.473513	−	0.880787i	$-0.657014\pi$
$421$	−19.4314			−0.947026			−0.473513	+	0.880787i	$0.657014\pi$
$422$		0			0
$423$	5.89662	+	1.46599i	0.286703	+	0.0712790i
$424$		0			0
$425$		−	4.54062i		−	0.220253i
$426$		0			0
$427$	−11.5680			−0.559814
$428$		0			0
$429$	4.24489	−	3.31877i	0.204945	−	0.160232i
$430$		0			0
$431$			36.5313i			1.75965i	0.475295	+	0.879826i	$0.342341\pi$
$431$			36.5313i			1.75965i	−0.475295	+	0.879826i	$0.657659\pi$
$432$		0			0
$433$			20.0649i			0.964256i	0.876101	+	0.482128i	$0.160136\pi$
$433$			20.0649i			0.964256i	−0.876101	+	0.482128i	$0.839864\pi$
$434$		0			0
$435$	10.2235	−	7.99297i	0.490178	−	0.383234i
$436$		0			0
$437$		−	10.5546i		−	0.504894i
$438$		0			0
$439$	16.4468			0.784963			0.392482	−	0.919760i	$-0.371617\pi$
$439$	16.4468			0.784963			0.392482	+	0.919760i	$0.371617\pi$
$440$		0			0
$441$	−3.41298	+	13.7280i	−0.162523	+	0.653712i
$442$		0			0
$443$	−29.2521			−1.38981			−0.694904	−	0.719103i	$-0.744553\pi$
$443$	−29.2521			−1.38981			−0.694904	+	0.719103i	$0.744553\pi$
$444$		0			0
$445$			4.29002i			0.203367i
$446$		0			0
$447$	−18.0743	+	14.1310i	−0.854885	+	0.668372i
$448$		0			0
$449$			35.8537i			1.69204i	0.533149	+	0.846021i	$0.321009\pi$
$449$			35.8537i			1.69204i	−0.533149	+	0.846021i	$0.678991\pi$
$450$		0			0
$451$	−26.7320			−1.25876
$452$		0			0
$453$	18.1350	+	23.1956i	0.852054	+	1.08983i
$454$		0			0
$455$	−1.61351			−0.0756425
$456$		0			0
$457$	−7.09402			−0.331844			−0.165922	−	0.986139i	$-0.553060\pi$
$457$	−7.09402			−0.331844			−0.165922	+	0.986139i	$0.553060\pi$
$458$		0			0
$459$	−9.61814	−	21.5443i	−0.448936	−	1.00560i
$460$		0			0
$461$		−	1.21893i		−	0.0567714i	−0.999597	−	0.0283857i	$-0.990963\pi$
$461$		−	1.21893i		−	0.0567714i	0.999597	−	0.0283857i	$-0.00903666\pi$
$462$		0			0
$463$		−	21.7611i		−	1.01132i	−0.862732	−	0.505661i	$-0.831249\pi$
$463$		−	21.7611i		−	1.01132i	0.862732	−	0.505661i	$-0.168751\pi$
$464$		0			0
$465$	8.18468	−	6.39899i	0.379555	−	0.296746i
$466$		0			0
$467$		−	1.15563i		−	0.0534760i	−0.999642	−	0.0267380i	$-0.991488\pi$
$467$		−	1.15563i		−	0.0534760i	0.999642	−	0.0267380i	$-0.00851199\pi$
$468$		0			0
$469$	−11.7431	+	3.89550i	−0.542247	+	0.179878i
$470$		0			0
$471$	−7.81632	−	9.99752i	−0.360157	−	0.460662i
$472$		0			0
$473$			10.0089i			0.460209i
$474$		0			0
$475$	−8.24273			−0.378203
$476$		0			0
$477$	−4.30613	+	17.3204i	−0.197164	+	0.793047i
$478$		0			0
$479$			21.1004i			0.964104i	0.876143	+	0.482052i	$0.160108\pi$
$479$			21.1004i			0.964104i	−0.876143	+	0.482052i	$0.839892\pi$
$480$		0			0
$481$			8.12068i			0.370271i
$482$		0			0
$483$	−2.06478	−	2.64097i	−0.0939508	−	0.120168i
$484$		0			0
$485$			16.1446i			0.733089i
$486$		0			0
$487$		−	26.6790i		−	1.20894i	−0.796627	−	0.604471i	$-0.793385\pi$
$487$		−	26.6790i		−	1.20894i	0.796627	−	0.604471i	$-0.206615\pi$
$488$		0			0
$489$	17.8285	+	22.8036i	0.806231	+	1.03121i
$490$		0			0
$491$		−	6.18111i		−	0.278949i	−0.990226	−	0.139475i	$-0.955459\pi$
$491$		−	6.18111i		−	0.278949i	0.990226	−	0.139475i	$-0.0445414\pi$
$492$		0			0
$493$	34.0200			1.53219
$494$		0			0
$495$	2.10940	−	8.48457i	0.0948103	−	0.381353i
$496$		0			0
$497$	2.29455			0.102925
$498$		0			0
$499$		−	33.9824i		−	1.52126i	−0.649184	−	0.760632i	$-0.724889\pi$
$499$		−	33.9824i		−	1.52126i	0.649184	−	0.760632i	$-0.275111\pi$
$500$		0			0
$501$	18.1112	−	14.1598i	0.809149	−	0.632613i
$502$		0			0
$503$	0.285664			0.0127371			0.00636857	−	0.999980i	$-0.497973\pi$
$503$	0.285664			0.0127371			0.00636857	+	0.999980i	$0.497973\pi$
$504$		0			0
$505$	−4.47308			−0.199049
$506$		0			0
$507$	−12.6530	−	16.1839i	−0.561938	−	0.718751i
$508$		0			0
$509$		−	9.12712i		−	0.404552i	−0.979329	−	0.202276i	$-0.935166\pi$
$509$		−	9.12712i		−	0.404552i	0.979329	−	0.202276i	$-0.0648339\pi$
$510$		0			0
$511$		−	7.51925i		−	0.332632i
$512$		0			0
$513$	−39.1101	+	17.4601i	−1.72675	+	0.770882i
$514$		0			0
$515$	−6.60791			−0.291179
$516$		0			0
$517$			5.90252i			0.259592i
$518$		0			0
$519$	19.6493	−	15.3623i	0.862509	−	0.674332i
$520$		0			0
$521$	−39.6229			−1.73591			−0.867956	−	0.496642i	$-0.834566\pi$
$521$	−39.6229			−1.73591			−0.867956	+	0.496642i	$0.834566\pi$
$522$		0			0
$523$	40.1286			1.75470			0.877352	−	0.479848i	$-0.159308\pi$
$523$	40.1286			1.75470			0.877352	+	0.479848i	$0.159308\pi$
$524$		0			0
$525$	−2.06250	+	1.61252i	−0.0900151	+	0.0703761i
$526$		0			0
$527$	27.2357			1.18640
$528$		0			0
$529$	21.3604			0.928713
$530$		0			0
$531$	−25.9877	−	6.46094i	−1.12777	−	0.280381i
$532$		0			0
$533$		−	9.79164i		−	0.424123i
$534$		0			0
$535$		−	10.2247i		−	0.442052i
$536$		0			0
$537$	−19.6800	−	25.1718i	−0.849255	−	1.08625i
$538$		0			0
$539$	−13.7417			−0.591896
$540$		0			0
$541$		−	17.9888i		−	0.773400i	−0.922206	−	0.386700i	$-0.873615\pi$
$541$		−	17.9888i		−	0.773400i	0.922206	−	0.386700i	$-0.126385\pi$
$542$		0			0
$543$	4.67424	+	5.97861i	0.200591	+	0.256567i
$544$		0			0
$545$			8.64732i			0.370411i
$546$		0			0
$547$		−	14.8043i		−	0.632987i	−0.948595	−	0.316494i	$-0.897494\pi$
$547$		−	14.8043i		−	0.632987i	0.948595	−	0.316494i	$-0.102506\pi$
$548$		0			0
$549$	22.2813	+	5.53947i	0.950941	+	0.236419i
$550$		0			0
$551$		−	61.7576i		−	2.63096i
$552$		0			0
$553$	25.4218			1.08105
$554$		0			0
$555$	8.11569	+	10.3804i	0.344492	+	0.440625i
$556$		0			0
$557$			9.28084i			0.393242i	0.980480	+	0.196621i	$0.0629969\pi$
$557$			9.28084i			0.393242i	−0.980480	+	0.196621i	$0.937003\pi$
$558$		0			0
$559$	−3.66614			−0.155061
$560$		0			0
$561$	18.0562	−	14.1168i	0.762333	−	0.596012i
$562$		0			0
$563$	1.59378			0.0671698			0.0335849	−	0.999436i	$-0.489308\pi$
$563$	1.59378			0.0671698			0.0335849	+	0.999436i	$0.489308\pi$
$564$		0			0
$565$	−13.6310			−0.573462
$566$		0			0
$567$	−6.37045	+	12.0200i	−0.267534	+	0.504790i
$568$		0			0
$569$			2.02023i			0.0846925i	0.999103	+	0.0423462i	$0.0134833\pi$
$569$			2.02023i			0.0846925i	−0.999103	+	0.0423462i	$0.986517\pi$
$570$		0			0
$571$	−23.4043			−0.979439			−0.489719	−	0.871880i	$-0.662901\pi$
$571$	−23.4043			−0.979439			−0.489719	+	0.871880i	$0.662901\pi$
$572$		0			0
$573$	−23.4885	−	30.0432i	−0.981247	−	1.25507i
$574$		0			0
$575$			1.28047i			0.0533993i
$576$		0			0
$577$		−	38.1037i		−	1.58628i	−0.609042	−	0.793138i	$-0.708446\pi$
$577$		−	38.1037i		−	1.58628i	0.609042	−	0.793138i	$-0.291554\pi$
$578$		0			0
$579$	−15.8798	−	20.3112i	−0.659943	−	0.844104i
$580$		0			0
$581$	14.8575			0.616394
$582$		0			0
$583$	−17.3377			−0.718056
$584$		0			0
$585$	3.10780	+	0.772649i	0.128492	+	0.0319451i
$586$		0			0
$587$	−25.3069			−1.04453			−0.522265	−	0.852783i	$-0.674913\pi$
$587$	−25.3069			−1.04453			−0.522265	+	0.852783i	$0.674913\pi$
$588$		0			0
$589$		−	49.4417i		−	2.03721i
$590$		0			0
$591$	−13.8747	−	17.7465i	−0.570728	−	0.729994i
$592$		0			0
$593$	−36.7619			−1.50963			−0.754815	−	0.655938i	$-0.772273\pi$
$593$	−36.7619			−1.50963			−0.754815	+	0.655938i	$0.772273\pi$
$594$		0			0
$595$	−6.86328			−0.281367
$596$		0			0
$597$	10.2366	+	13.0932i	0.418958	+	0.535871i
$598$		0			0
$599$	10.0515			0.410693			0.205346	−	0.978689i	$-0.434168\pi$
$599$	10.0515			0.410693			0.205346	+	0.978689i	$0.434168\pi$
$600$		0			0
$601$	−40.6328			−1.65745			−0.828723	−	0.559658i	$-0.810932\pi$
$601$	−40.6328			−1.65745			−0.828723	+	0.559658i	$0.810932\pi$
$602$		0			0
$603$	24.4840	−	1.87985i	0.997065	−	0.0765532i
$604$		0			0
$605$	−2.50695			−0.101922
$606$		0			0
$607$	−11.0531			−0.448631			−0.224316	−	0.974517i	$-0.572015\pi$
$607$	−11.0531			−0.448631			−0.224316	+	0.974517i	$0.572015\pi$
$608$		0			0
$609$	−12.0816	−	15.4530i	−0.489571	−	0.626189i
$610$		0			0
$611$	−2.16203			−0.0874662
$612$		0			0
$613$	−21.3136			−0.860846			−0.430423	−	0.902627i	$-0.641636\pi$
$613$	−21.3136			−0.860846			−0.430423	+	0.902627i	$0.641636\pi$
$614$		0			0
$615$	−9.78562	−	12.5164i	−0.394594	−	0.504709i
$616$		0			0
$617$			20.7346i			0.834743i	0.908736	+	0.417372i	$0.137049\pi$
$617$			20.7346i			0.834743i	−0.908736	+	0.417372i	$0.862951\pi$
$618$		0			0
$619$	48.0497			1.93128			0.965641	−	0.259881i	$-0.0836833\pi$
$619$	48.0497			1.93128			0.965641	+	0.259881i	$0.0836833\pi$
$620$		0			0
$621$	2.71234	+	6.07556i	0.108843	+	0.243804i
$622$		0			0
$623$	6.48449			0.259795
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−25.6267	−	32.7780i	−1.02343	−	1.30903i
$628$		0			0
$629$			34.5424i			1.37729i
$630$		0			0
$631$			9.84900i			0.392082i	0.980596	+	0.196041i	$0.0628086\pi$
$631$			9.84900i			0.392082i	−0.980596	+	0.196041i	$0.937191\pi$
$632$		0			0
$633$	−11.8713	−	15.1840i	−0.471841	−	0.603512i
$634$		0			0
$635$	−3.89073			−0.154399
$636$		0			0
$637$		−	5.03343i		−	0.199432i
$638$		0			0
$639$	−4.41957	−	1.09877i	−0.174835	−	0.0434668i
$640$		0			0
$641$	3.42899			0.135437			0.0677185	−	0.997704i	$-0.478428\pi$
$641$	3.42899			0.135437			0.0677185	+	0.997704i	$0.478428\pi$
$642$		0			0
$643$	17.9699			0.708663			0.354332	−	0.935120i	$-0.384708\pi$
$643$	17.9699			0.708663			0.354332	+	0.935120i	$0.384708\pi$
$644$		0			0
$645$	−4.68633	+	3.66389i	−0.184524	+	0.144266i
$646$		0			0
$647$	13.2627			0.521409			0.260705	−	0.965419i	$-0.416045\pi$
$647$	13.2627			0.521409			0.260705	+	0.965419i	$0.416045\pi$
$648$		0			0
$649$		−	26.0136i		−	1.02112i
$650$		0			0
$651$	−9.67225	−	12.3714i	−0.379085	−	0.484872i
$652$		0			0
$653$	9.93967			0.388969			0.194485	−	0.980906i	$-0.437697\pi$
$653$	9.93967			0.388969			0.194485	+	0.980906i	$0.437697\pi$
$654$		0			0
$655$			3.92563i			0.153387i
$656$		0			0
$657$	−3.60068	+	14.4829i	−0.140476	+	0.565033i
$658$		0			0
$659$		−	18.3219i		−	0.713721i	−0.934158	−	0.356861i	$-0.883847\pi$
$659$		−	18.3219i		−	0.713721i	0.934158	−	0.356861i	$-0.116153\pi$
$660$		0			0
$661$			17.6464i			0.686364i	0.939269	+	0.343182i	$0.111505\pi$
$661$			17.6464i			0.686364i	−0.939269	+	0.343182i	$0.888495\pi$
$662$		0			0
$663$	5.17083	+	6.61379i	0.200818	+	0.256858i
$664$		0			0
$665$			12.4591i			0.483144i
$666$		0			0
$667$	−9.59376			−0.371472
$668$		0			0
$669$	−18.3633	−	23.4878i	−0.709968	−	0.908089i
$670$		0			0
$671$			22.3035i			0.861019i
$672$		0			0
$673$		−	1.83591i		−	0.0707693i	−0.999374	−	0.0353847i	$-0.988734\pi$
$673$		−	1.83591i		−	0.0707693i	0.999374	−	0.0353847i	$-0.0112656\pi$
$674$		0			0
$675$	4.74479	−	2.11824i	0.182627	−	0.0815311i
$676$		0			0
$677$	−36.5375			−1.40425			−0.702125	−	0.712054i	$-0.747765\pi$
$677$	−36.5375			−1.40425			−0.702125	+	0.712054i	$0.747765\pi$
$678$		0			0
$679$	24.4030			0.936502
$680$		0			0
$681$	32.4152	−	25.3431i	1.24215	−	0.971149i
$682$		0			0
$683$	−34.3267			−1.31347			−0.656737	−	0.754119i	$-0.728064\pi$
$683$	−34.3267			−1.31347			−0.656737	+	0.754119i	$0.728064\pi$
$684$		0			0
$685$	13.8677			0.529858
$686$		0			0
$687$	−4.26871	+	3.33739i	−0.162861	+	0.127329i
$688$		0			0
$689$		−	6.35062i		−	0.241939i
$690$		0			0
$691$	38.8164			1.47665			0.738323	−	0.674447i	$-0.235618\pi$
$691$	38.8164			1.47665			0.738323	+	0.674447i	$0.235618\pi$
$692$		0			0
$693$	−12.8247	−	3.18841i	−0.487168	−	0.121118i
$694$		0			0
$695$			1.40851i			0.0534278i
$696$		0			0
$697$		−	41.6500i		−	1.57761i
$698$		0			0
$699$	−26.6487	−	34.0853i	−1.00795	−	1.28922i
$700$		0			0
$701$	1.65858			0.0626439			0.0313219	−	0.999509i	$-0.490028\pi$
$701$	1.65858			0.0626439			0.0313219	+	0.999509i	$0.490028\pi$
$702$		0			0
$703$	62.7058			2.36499
$704$		0			0
$705$	−2.76366	+	2.16070i	−0.104085	+	0.0813766i
$706$		0			0
$707$			6.76118i			0.254280i
$708$		0			0
$709$	−43.8829			−1.64806			−0.824029	−	0.566547i	$-0.808279\pi$
$709$	−43.8829			−1.64806			−0.824029	+	0.566547i	$0.808279\pi$
$710$		0			0
$711$	−48.9654	−	12.1736i	−1.83634	−	0.456544i
$712$		0			0
$713$	−7.68054			−0.287639
$714$		0			0
$715$			3.11091i			0.116342i
$716$		0			0
$717$	−11.6208	−	14.8637i	−0.433988	−	0.555096i
$718$		0			0
$719$			15.8640i			0.591629i	0.955245	+	0.295814i	$0.0955910\pi$
$719$			15.8640i			0.591629i	−0.955245	+	0.295814i	$0.904409\pi$
$720$		0			0
$721$			9.98803i			0.371974i
$722$		0			0
$723$	13.2503	+	16.9479i	0.492783	+	0.630298i
$724$		0			0
$725$			7.49237i			0.278260i
$726$		0			0
$727$		−	40.1203i		−	1.48798i	−0.668192	−	0.743989i	$-0.732931\pi$
$727$		−	40.1203i		−	1.48798i	0.668192	−	0.743989i	$-0.267069\pi$
$728$		0			0
$729$	18.0261	−	20.1012i	0.667634	−	0.744490i
$730$		0			0
$731$	−15.5944			−0.576781
$732$		0			0
$733$		−	3.87613i		−	0.143168i	−0.997435	−	0.0715841i	$-0.977195\pi$
$733$		−	3.87613i		−	0.143168i	0.997435	−	0.0715841i	$-0.0228054\pi$
$734$		0			0
$735$	−5.03034	−	6.43409i	−0.185547	−	0.237325i
$736$		0			0
$737$	7.51068	+	22.6412i	0.276660	+	0.833999i
$738$		0			0
$739$			50.4875i			1.85721i	0.371067	+	0.928606i	$0.378992\pi$
$739$			50.4875i			1.85721i	−0.371067	+	0.928606i	$0.621008\pi$
$740$		0			0
$741$	12.0062	−	9.38677i	0.441059	−	0.344831i
$742$		0			0
$743$			15.2996i			0.561287i	0.959812	+	0.280643i	$0.0905479\pi$
$743$			15.2996i			0.561287i	−0.959812	+	0.280643i	$0.909452\pi$
$744$		0			0
$745$		−	13.2459i		−	0.485294i
$746$		0			0
$747$	−28.6173	−	7.11470i	−1.04705	−	0.260313i
$748$		0			0
$749$	−15.4549			−0.564710
$750$		0			0
$751$	−42.7689			−1.56066			−0.780330	−	0.625368i	$-0.784949\pi$
$751$	−42.7689			−1.56066			−0.780330	+	0.625368i	$0.784949\pi$
$752$		0			0
$753$	23.7137	+	30.3312i	0.864175	+	1.10533i
$754$		0			0
$755$	−16.9992			−0.618663
$756$		0			0
$757$			2.68965i			0.0977570i	0.998805	+	0.0488785i	$0.0155647\pi$
$757$			2.68965i			0.0977570i	−0.998805	+	0.0488785i	$0.984435\pi$
$758$		0			0
$759$	−5.09190	+	3.98098i	−0.184824	+	0.144500i
$760$		0			0
$761$		−	12.7997i		−	0.463987i	−0.972717	−	0.231994i	$-0.925475\pi$
$761$		−	12.7997i		−	0.463987i	0.972717	−	0.231994i	$-0.0745248\pi$
$762$		0			0
$763$	13.0707			0.473190
$764$		0			0
$765$	13.2195	+	3.28656i	0.477950	+	0.118826i
$766$		0			0
$767$	9.52851			0.344055
$768$		0			0
$769$			9.14385i			0.329736i	0.986316	+	0.164868i	$0.0527197\pi$
$769$			9.14385i			0.329736i	−0.986316	+	0.164868i	$0.947280\pi$
$770$		0			0
$771$	0.759346	−	0.593676i	0.0273472	−	0.0213807i
$772$		0			0
$773$			37.7233i			1.35681i	0.734687	+	0.678407i	$0.237329\pi$
$773$			37.7233i			1.35681i	−0.734687	+	0.678407i	$0.762671\pi$
$774$		0			0
$775$			5.99822i			0.215462i
$776$		0			0
$777$	15.6903	−	12.2671i	0.562887	−	0.440079i
$778$		0			0
$779$	−75.6085			−2.70896
$780$		0			0
$781$		−	4.42398i		−	0.158303i
$782$		0			0
$783$	15.8706	+	35.5497i	0.567171	+	1.27044i
$784$		0			0
$785$	7.32679			0.261504
$786$		0			0
$787$		−	12.6663i		−	0.451505i	−0.974185	−	0.225753i	$-0.927516\pi$
$787$		−	12.6663i		−	0.451505i	0.974185	−	0.225753i	$-0.0724841\pi$
$788$		0			0
$789$	19.2498	−	15.0500i	0.685312	−	0.535794i
$790$		0			0
$791$			20.6037i			0.732582i
$792$		0			0
$793$	−8.16954			−0.290109
$794$		0			0
$795$	−6.34672	−	8.11782i	−0.225095	−	0.287909i
$796$		0			0
$797$			9.12811i			0.323334i	0.986845	+	0.161667i	$0.0516871\pi$
$797$			9.12811i			0.323334i	−0.986845	+	0.161667i	$0.948313\pi$
$798$		0			0
$799$	−9.19646			−0.325347
$800$		0			0
$801$	−12.4899	−	3.10517i	−0.441308	−	0.109716i
$802$		0			0
$803$	−14.4974			−0.511603
$804$		0			0
$805$	1.93546			0.0682162
$806$		0			0
$807$	−13.5112	+	10.5634i	−0.475618	+	0.371850i
$808$		0			0
$809$	18.9508			0.666275			0.333138	−	0.942878i	$-0.391893\pi$
$809$	18.9508			0.666275			0.333138	+	0.942878i	$0.391893\pi$
$810$		0			0
$811$		−	38.8542i		−	1.36435i	−0.731187	−	0.682177i	$-0.761033\pi$
$811$		−	38.8542i		−	1.36435i	0.731187	−	0.682177i	$-0.238967\pi$
$812$		0			0
$813$	−6.92505	+	5.41418i	−0.242872	+	0.189884i
$814$		0			0
$815$	−16.7119			−0.585391
$816$		0			0
$817$			28.3090i			0.990408i
$818$		0			0
$819$	1.16788	−	4.69753i	0.0408090	−	0.164145i
$820$		0			0
$821$		−	25.9045i		−	0.904074i	−0.891999	−	0.452037i	$-0.850697\pi$
$821$		−	25.9045i		−	0.904074i	0.891999	−	0.452037i	$-0.149303\pi$
$822$		0			0
$823$	−14.4419			−0.503411			−0.251706	−	0.967804i	$-0.580992\pi$
$823$	−14.4419			−0.503411			−0.251706	+	0.967804i	$0.580992\pi$
$824$		0			0
$825$	3.10900	+	3.97659i	0.108242	+	0.138447i
$826$		0			0
$827$		−	24.8339i		−	0.863558i	−0.901979	−	0.431779i	$-0.857886\pi$
$827$		−	24.8339i		−	0.863558i	0.901979	−	0.431779i	$-0.142114\pi$
$828$		0			0
$829$	−27.6445			−0.960133			−0.480067	−	0.877232i	$-0.659388\pi$
$829$	−27.6445			−0.960133			−0.480067	+	0.877232i	$0.659388\pi$
$830$		0			0
$831$	1.25048	+	1.59943i	0.0433786	+	0.0554837i
$832$		0			0
$833$		−	21.4103i		−	0.741824i
$834$		0			0
$835$			13.2730i			0.459330i
$836$		0			0
$837$	12.7057	+	28.4603i	0.439172	+	0.983732i
$838$		0			0
$839$			32.2235i			1.11248i	0.831022	+	0.556239i	$0.187756\pi$
$839$			32.2235i			1.11248i	−0.831022	+	0.556239i	$0.812244\pi$
$840$		0			0
$841$	−27.1356			−0.935712
$842$		0			0
$843$	24.3370	+	31.1285i	0.838212	+	1.07212i
$844$		0			0
$845$	11.8605			0.408014
$846$		0			0
$847$			3.78932i			0.130203i
$848$		0			0
$849$	−21.1684	−	27.0755i	−0.726496	−	0.929230i
$850$		0			0
$851$		−	9.74105i		−	0.333919i
$852$		0			0
$853$	−35.2725			−1.20771			−0.603854	−	0.797095i	$-0.706369\pi$
$853$	−35.2725			−1.20771			−0.603854	+	0.797095i	$0.706369\pi$
$854$		0			0
$855$	5.96620	−	23.9977i	0.204040	−	0.820703i
$856$		0			0
$857$	−16.9568			−0.579232			−0.289616	−	0.957143i	$-0.593528\pi$
$857$	−16.9568			−0.579232			−0.289616	+	0.957143i	$0.593528\pi$
$858$		0			0
$859$	−48.5260			−1.65568			−0.827842	−	0.560961i	$-0.810432\pi$
$859$	−48.5260			−1.65568			−0.827842	+	0.560961i	$0.810432\pi$
$860$		0			0
$861$	−18.9188	+	14.7912i	−0.644752	+	0.504084i
$862$		0			0
$863$			12.8496i			0.437406i	0.975791	+	0.218703i	$0.0701826\pi$
$863$			12.8496i			0.437406i	−0.975791	+	0.218703i	$0.929817\pi$
$864$		0			0
$865$			14.4002i			0.489621i
$866$		0			0
$867$	3.85895	+	4.93582i	0.131057	+	0.167629i
$868$		0			0
$869$		−	49.0143i		−	1.66270i
$870$		0			0
$871$	−8.29322	+	2.75108i	−0.281005	+	0.0932168i
$872$		0			0
$873$	−47.0030	−	11.6857i	−1.59081	−	0.395500i
$874$		0			0
$875$		−	1.51153i		−	0.0510989i
$876$		0			0
$877$	−3.22864			−0.109023			−0.0545117	−	0.998513i	$-0.517360\pi$
$877$	−3.22864			−0.109023			−0.0545117	+	0.998513i	$0.517360\pi$
$878$		0			0
$879$	23.7975	−	18.6055i	0.802669	−	0.627548i
$880$		0			0
$881$			9.52920i			0.321047i	0.987032	+	0.160523i	$0.0513182\pi$
$881$			9.52920i			0.321047i	−0.987032	+	0.160523i	$0.948682\pi$
$882$		0			0
$883$		−	29.0233i		−	0.976713i	−0.872644	−	0.488357i	$-0.837597\pi$
$883$		−	29.0233i		−	0.976713i	0.872644	−	0.488357i	$-0.162403\pi$
$884$		0			0
$885$	12.1800	−	9.52266i	0.409427	−	0.320101i
$886$		0			0
$887$			0.185684i			0.00623466i	0.999995	+	0.00311733i	$0.000992278\pi$
$887$			0.185684i			0.00623466i	−0.999995	+	0.00311733i	$0.999008\pi$
$888$		0			0
$889$			5.88094i			0.197240i
$890$		0			0
$891$	23.1749	+	12.2825i	0.776390	+	0.411479i
$892$		0			0
$893$			16.6946i			0.558664i
$894$		0			0
$895$	18.4474			0.616630
$896$		0			0
$897$	−1.45819	−	1.86511i	−0.0486875	−	0.0622742i
$898$		0			0
$899$	−44.9409			−1.49886
$900$		0			0
$901$		−	27.0132i		−	0.899940i
$902$		0			0
$903$	5.53807	+	7.08351i	0.184296	+	0.235725i
$904$		0			0
$905$	−4.38149			−0.145646
$906$		0			0
$907$	16.0164			0.531816			0.265908	−	0.963998i	$-0.414328\pi$
$907$	16.0164			0.531816			0.265908	+	0.963998i	$0.414328\pi$
$908$		0			0
$909$	3.23767	−	13.0228i	0.107387	−	0.431939i
$910$		0			0
$911$		−	49.4993i		−	1.63999i	−0.572373	−	0.819993i	$-0.693977\pi$
$911$		−	49.4993i		−	1.63999i	0.572373	−	0.819993i	$-0.306023\pi$
$912$		0			0
$913$		−	28.6459i		−	0.948041i
$914$		0			0
$915$	−10.4429	+	8.16453i	−0.345232	+	0.269911i
$916$		0			0
$917$	5.93370			0.195948
$918$		0			0
$919$			24.1757i			0.797481i	0.917064	+	0.398741i	$0.130553\pi$
$919$			24.1757i			0.797481i	−0.917064	+	0.398741i	$0.869447\pi$
$920$		0			0
$921$	6.99344	+	8.94502i	0.230442	+	0.294748i
$922$		0			0
$923$	1.62046			0.0533380
$924$		0			0
$925$	−7.60740			−0.250130
$926$		0			0
$927$	4.78289	−	19.2381i	0.157091	−	0.631862i
$928$		0			0
$929$	10.4860			0.344034			0.172017	−	0.985094i	$-0.444972\pi$
$929$	10.4860			0.344034			0.172017	+	0.985094i	$0.444972\pi$
$930$		0			0
$931$	−38.8668			−1.27381
$932$		0			0
$933$	29.5489	+	37.7947i	0.967386	+	1.23734i
$934$		0			0
$935$			13.2327i			0.432755i
$936$		0			0
$937$			16.4531i			0.537498i	0.963210	+	0.268749i	$0.0866102\pi$
$937$			16.4531i			0.537498i	−0.963210	+	0.268749i	$0.913390\pi$
$938$		0			0
$939$	14.8272	−	11.5923i	0.483869	−	0.378301i
$940$		0			0
$941$	39.0311			1.27238			0.636188	−	0.771534i	$-0.280510\pi$
$941$	39.0311			1.27238			0.636188	+	0.771534i	$0.280510\pi$
$942$		0			0
$943$			11.7454i			0.382484i
$944$		0			0
$945$	−3.20178	−	7.17188i	−0.104154	−	0.233301i
$946$		0			0
$947$		−	59.9435i		−	1.94790i	−0.226759	−	0.973951i	$-0.572813\pi$
$947$		−	59.9435i		−	1.94790i	0.226759	−	0.973951i	$-0.427187\pi$
$948$		0			0
$949$		−	5.31024i		−	0.172378i
$950$		0			0
$951$	36.0031	−	28.1481i	1.16748	−	0.912766i
$952$		0			0
$953$		−	13.5516i		−	0.438980i	−0.975615	−	0.219490i	$-0.929561\pi$
$953$		−	13.5516i		−	0.438980i	0.975615	−	0.219490i	$-0.0704393\pi$
$954$		0			0
$955$	22.0174			0.712467
$956$		0			0
$957$	−29.7941	+	23.2938i	−0.963106	+	0.752981i
$958$		0			0
$959$		−	20.9614i		−	0.676879i
$960$		0			0
$961$	−4.97866			−0.160602
$962$		0			0
$963$	29.7679	+	7.40076i	0.959257	+	0.238486i
$964$		0			0
$965$	14.8853			0.479173
$966$		0			0
$967$	−6.79753			−0.218594			−0.109297	−	0.994009i	$-0.534860\pi$
$967$	−6.79753			−0.218594			−0.109297	+	0.994009i	$0.534860\pi$
$968$		0			0
$969$	51.0700	−	39.9278i	1.64060	−	1.28267i
$970$		0			0
$971$			14.4116i			0.462491i	0.972895	+	0.231246i	$0.0742801\pi$
$971$			14.4116i			0.462491i	−0.972895	+	0.231246i	$0.925720\pi$
$972$		0			0
$973$	2.12900			0.0682527
$974$		0			0
$975$	−1.45658	+	1.13879i	−0.0466479	+	0.0364706i
$976$		0			0
$977$		−	41.4363i		−	1.32567i	−0.748767	−	0.662833i	$-0.769354\pi$
$977$		−	41.4363i		−	1.32567i	0.748767	−	0.662833i	$-0.230646\pi$
$978$		0			0
$979$		−	12.5023i		−	0.399577i
$980$		0			0
$981$	−25.1756	−	6.25904i	−0.803794	−	0.199836i
$982$		0			0
$983$	−14.1655			−0.451810			−0.225905	−	0.974149i	$-0.572534\pi$
$983$	−14.1655			−0.451810			−0.225905	+	0.974149i	$0.572534\pi$
$984$		0			0
$985$	13.0057			0.414396
$986$		0			0
$987$	3.26595	+	4.17734i	0.103956	+	0.132966i
$988$		0			0
$989$	4.39767			0.139838
$990$		0			0
$991$			32.2256i			1.02368i	0.859081	+	0.511840i	$0.171036\pi$
$991$			32.2256i			1.02368i	−0.859081	+	0.511840i	$0.828964\pi$
$992$		0			0
$993$	−7.49407	+	5.85906i	−0.237817	+	0.185932i
$994$		0			0
$995$	−9.59551			−0.304198
$996$		0			0
$997$	60.9906			1.93159			0.965797	−	0.259301i	$-0.0834921\pi$
$997$	60.9906			1.93159			0.965797	+	0.259301i	$0.0834921\pi$
$998$		0			0
$999$	−36.0956	+	16.1143i	−1.14201	+	0.509834i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.b.401.15	yes	46
3.2	odd	2		4020.2.f.a.401.31	✓	46
67.66	odd	2		4020.2.f.a.401.32	yes	46
201.200	even	2	inner	4020.2.f.b.401.16	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.31	✓	46	3.2	odd	2
4020.2.f.a.401.32	yes	46	67.66	odd	2
4020.2.f.b.401.15	yes	46	1.1	even	1	trivial
4020.2.f.b.401.16	yes	46	201.200	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.06681 - 1.36452i) q^{3} +1.00000 q^{5} -1.51153i q^{7} +(-0.723813 + 2.91137i) q^{9} +O(q^{10})\)	\(q+(-1.06681 - 1.36452i) q^{3} +1.00000 q^{5} -1.51153i q^{7} +(-0.723813 + 2.91137i) q^{9} -2.91428 q^{11} -1.06747i q^{13} +(-1.06681 - 1.36452i) q^{15} -4.54062i q^{17} -8.24273 q^{19} +(-2.06250 + 1.61252i) q^{21} +1.28047i q^{23} +1.00000 q^{25} +(4.74479 - 2.11824i) q^{27} +7.49237i q^{29} +5.99822i q^{31} +(3.10900 + 3.97659i) q^{33} -1.51153i q^{35} -7.60740 q^{37} +(-1.45658 + 1.13879i) q^{39} +9.17275 q^{41} -3.43442i q^{43} +(-0.723813 + 2.91137i) q^{45} -2.02537i q^{47} +4.71529 q^{49} +(-6.19576 + 4.84400i) q^{51} +5.94923 q^{53} -2.91428 q^{55} +(8.79347 + 11.2473i) q^{57} +8.92625i q^{59} -7.65318i q^{61} +(4.40062 + 1.09406i) q^{63} -1.06747i q^{65} +(-2.57720 - 7.76904i) q^{67} +(1.74722 - 1.36602i) q^{69} +1.51803i q^{71} +4.97461 q^{73} +(-1.06681 - 1.36452i) q^{75} +4.40502i q^{77} +16.8186i q^{79} +(-7.95219 - 4.21458i) q^{81} +9.82948i q^{83} -4.54062i q^{85} +(10.2235 - 7.99297i) q^{87} +4.29002i q^{89} -1.61351 q^{91} +(8.18468 - 6.39899i) q^{93} -8.24273 q^{95} +16.1446i q^{97} +(2.10940 - 8.48457i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 46 q^{5} - 4 q^{9}+O(q^{10}) \) 46 * q + 46 * q^5 - 4 * q^9	\( 46 q + 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} + 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} + 4 q^{41} - 4 q^{45} - 62 q^{49} + 8 q^{51} + 8 q^{53} + 12 q^{57} + 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} + 8 q^{95}+O(q^{100}) \) 46 * q + 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 + 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 + 4 * q^41 - 4 * q^45 - 62 * q^49 + 8 * q^51 + 8 * q^53 + 12 * q^57 + 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

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