LMFDB - Embedded newform 4020.2.f.b.401.14

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			401.14
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.b.401.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.10897 + 1.33049i) q^{3} +1.00000 q^{5} -2.36440i q^{7} +(-0.540386 - 2.95093i) q^{9} +O(q^{10})$	\(q+(-1.10897 + 1.33049i) q^{3} +1.00000 q^{5} -2.36440i q^{7} +(-0.540386 - 2.95093i) q^{9} -0.927662 q^{11} -1.43383i q^{13} +(-1.10897 + 1.33049i) q^{15} -5.72886i q^{17} -1.85976 q^{19} +(3.14580 + 2.62204i) q^{21} +7.68356i q^{23} +1.00000 q^{25} +(4.52544 + 2.55351i) q^{27} -7.94643i q^{29} +0.732407i q^{31} +(1.02875 - 1.23424i) q^{33} -2.36440i q^{35} -1.54199 q^{37} +(1.90769 + 1.59007i) q^{39} -6.32704 q^{41} +5.59280i q^{43} +(-0.540386 - 2.95093i) q^{45} -9.43194i q^{47} +1.40963 q^{49} +(7.62216 + 6.35311i) q^{51} -12.1843 q^{53} -0.927662 q^{55} +(2.06241 - 2.47438i) q^{57} +6.47969i q^{59} +13.3752i q^{61} +(-6.97716 + 1.27769i) q^{63} -1.43383i q^{65} +(-7.02590 - 4.19961i) q^{67} +(-10.2229 - 8.52081i) q^{69} -5.24005i q^{71} -3.58783 q^{73} +(-1.10897 + 1.33049i) q^{75} +2.19336i q^{77} +12.6561i q^{79} +(-8.41597 + 3.18928i) q^{81} +4.64638i q^{83} -5.72886i q^{85} +(10.5726 + 8.81232i) q^{87} -1.75065i q^{89} -3.39015 q^{91} +(-0.974457 - 0.812214i) q^{93} -1.85976 q^{95} +12.2273i q^{97} +(0.501296 + 2.73746i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.10897	+	1.33049i	−0.640262	+	0.768156i
$3$	−1.10897	+	1.33049i	−0.640262	+	0.768156i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	2.36440i		−	0.893658i	−0.894619	−	0.446829i	$-0.852553\pi$
$7$		−	2.36440i		−	0.893658i	0.894619	−	0.446829i	$-0.147447\pi$
$8$		0			0
$9$	−0.540386	−	2.95093i	−0.180129	−	0.983643i
$10$		0			0
$11$	−0.927662			−0.279701			−0.139850	−	0.990173i	$-0.544662\pi$
$11$	−0.927662			−0.279701			−0.139850	+	0.990173i	$0.544662\pi$
$12$		0			0
$13$		−	1.43383i		−	0.397674i	−0.980033	−	0.198837i	$-0.936284\pi$
$13$		−	1.43383i		−	0.397674i	0.980033	−	0.198837i	$-0.0637164\pi$
$14$		0			0
$15$	−1.10897	+	1.33049i	−0.286334	+	0.343530i
$16$		0			0
$17$		−	5.72886i		−	1.38945i	−0.719275	−	0.694726i	$-0.755526\pi$
$17$		−	5.72886i		−	1.38945i	0.719275	−	0.694726i	$-0.244474\pi$
$18$		0			0
$19$	−1.85976			−0.426658			−0.213329	−	0.976980i	$-0.568431\pi$
$19$	−1.85976			−0.426658			−0.213329	+	0.976980i	$0.568431\pi$
$20$		0			0
$21$	3.14580	+	2.62204i	0.686469	+	0.572175i
$22$		0			0
$23$			7.68356i			1.60213i	0.598575	+	0.801066i	$0.295734\pi$
$23$			7.68356i			1.60213i	−0.598575	+	0.801066i	$0.704266\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	4.52544	+	2.55351i	0.870921	+	0.491422i
$28$		0			0
$29$		−	7.94643i		−	1.47561i	−0.675012	−	0.737807i	$-0.735861\pi$
$29$		−	7.94643i		−	1.47561i	0.675012	−	0.737807i	$-0.264139\pi$
$30$		0			0
$31$			0.732407i			0.131544i	0.997835	+	0.0657720i	$0.0209510\pi$
$31$			0.732407i			0.131544i	−0.997835	+	0.0657720i	$0.979049\pi$
$32$		0			0
$33$	1.02875	−	1.23424i	0.179082	−	0.214854i
$34$		0			0
$35$		−	2.36440i		−	0.399656i
$36$		0			0
$37$	−1.54199			−0.253502			−0.126751	−	0.991935i	$-0.540455\pi$
$37$	−1.54199			−0.253502			−0.126751	+	0.991935i	$0.540455\pi$
$38$		0			0
$39$	1.90769	+	1.59007i	0.305476	+	0.254615i
$40$		0			0
$41$	−6.32704			−0.988117			−0.494059	−	0.869429i	$-0.664487\pi$
$41$	−6.32704			−0.988117			−0.494059	+	0.869429i	$0.664487\pi$
$42$		0			0
$43$			5.59280i			0.852894i	0.904513	+	0.426447i	$0.140235\pi$
$43$			5.59280i			0.852894i	−0.904513	+	0.426447i	$0.859765\pi$
$44$		0			0
$45$	−0.540386	−	2.95093i	−0.0805560	−	0.439899i
$46$		0			0
$47$		−	9.43194i		−	1.37579i	−0.725810	−	0.687895i	$-0.758535\pi$
$47$		−	9.43194i		−	1.37579i	0.725810	−	0.687895i	$-0.241465\pi$
$48$		0			0
$49$	1.40963			0.201376
$50$		0			0
$51$	7.62216	+	6.35311i	1.06732	+	0.889613i
$52$		0			0
$53$	−12.1843			−1.67364			−0.836822	−	0.547475i	$-0.815589\pi$
$53$	−12.1843			−1.67364			−0.836822	+	0.547475i	$0.815589\pi$
$54$		0			0
$55$	−0.927662			−0.125086
$56$		0			0
$57$	2.06241	−	2.47438i	0.273173	−	0.327740i
$58$		0			0
$59$			6.47969i			0.843584i	0.906693	+	0.421792i	$0.138599\pi$
$59$			6.47969i			0.843584i	−0.906693	+	0.421792i	$0.861401\pi$
$60$		0			0
$61$			13.3752i			1.71251i	0.516549	+	0.856257i	$0.327216\pi$
$61$			13.3752i			1.71251i	−0.516549	+	0.856257i	$0.672784\pi$
$62$		0			0
$63$	−6.97716	+	1.27769i	−0.879040	+	0.160973i
$64$		0			0
$65$		−	1.43383i		−	0.177845i
$66$		0			0
$67$	−7.02590	−	4.19961i	−0.858350	−	0.513064i
$67$	−7.02590	−	4.19961i	−0.858350	−	0.513064i
$68$		0			0
$69$	−10.2229	−	8.52081i	−1.23069	−	1.02578i
$70$		0			0
$71$		−	5.24005i		−	0.621880i	−0.950430	−	0.310940i	$-0.899356\pi$
$71$		−	5.24005i		−	0.621880i	0.950430	−	0.310940i	$-0.100644\pi$
$72$		0			0
$73$	−3.58783			−0.419924			−0.209962	−	0.977710i	$-0.567334\pi$
$73$	−3.58783			−0.419924			−0.209962	+	0.977710i	$0.567334\pi$
$74$		0			0
$75$	−1.10897	+	1.33049i	−0.128052	+	0.153631i
$76$		0			0
$77$			2.19336i			0.249957i
$78$		0			0
$79$			12.6561i			1.42392i	0.702221	+	0.711959i	$0.252192\pi$
$79$			12.6561i			1.42392i	−0.702221	+	0.711959i	$0.747808\pi$
$80$		0			0
$81$	−8.41597	+	3.18928i	−0.935107	+	0.354365i
$82$		0			0
$83$			4.64638i			0.510007i	0.966940	+	0.255003i	$0.0820766\pi$
$83$			4.64638i			0.510007i	−0.966940	+	0.255003i	$0.917923\pi$
$84$		0			0
$85$		−	5.72886i		−	0.621382i
$86$		0			0
$87$	10.5726	+	8.81232i	1.13350	+	0.944780i
$88$		0			0
$89$		−	1.75065i		−	0.185569i	−0.995686	−	0.0927843i	$-0.970423\pi$
$89$		−	1.75065i		−	0.185569i	0.995686	−	0.0927843i	$-0.0295767\pi$
$90$		0			0
$91$	−3.39015			−0.355384
$92$		0			0
$93$	−0.974457	−	0.812214i	−0.101046	−	0.0842227i
$94$		0			0
$95$	−1.85976			−0.190807
$96$		0			0
$97$			12.2273i			1.24149i	0.784011	+	0.620747i	$0.213171\pi$
$97$			12.2273i			1.24149i	−0.784011	+	0.620747i	$0.786829\pi$
$98$		0			0
$99$	0.501296	+	2.73746i	0.0503821	+	0.275125i
$100$		0			0
$101$	−13.8443			−1.37756			−0.688779	−	0.724971i	$-0.741853\pi$
$101$	−13.8443			−1.37756			−0.688779	+	0.724971i	$0.741853\pi$
$102$		0			0
$103$	−1.49968			−0.147768			−0.0738838	−	0.997267i	$-0.523539\pi$
$103$	−1.49968			−0.147768			−0.0738838	+	0.997267i	$0.523539\pi$
$104$		0			0
$105$	3.14580	+	2.62204i	0.306998	+	0.255885i
$106$		0			0
$107$		−	2.04145i		−	0.197355i	−0.995119	−	0.0986774i	$-0.968539\pi$
$107$		−	2.04145i		−	0.197355i	0.995119	−	0.0986774i	$-0.0314612\pi$
$108$		0			0
$109$			4.71192i			0.451321i	0.974206	+	0.225660i	$0.0724539\pi$
$109$			4.71192i			0.451321i	−0.974206	+	0.225660i	$0.927546\pi$
$110$		0			0
$111$	1.71002	−	2.05160i	0.162308	−	0.194729i
$112$		0			0
$113$	7.16322			0.673859			0.336929	−	0.941530i	$-0.390612\pi$
$113$	7.16322			0.673859			0.336929	+	0.941530i	$0.390612\pi$
$114$		0			0
$115$			7.68356i			0.716496i
$116$		0			0
$117$	−4.23114	+	0.774824i	−0.391169	+	0.0716325i
$118$		0			0
$119$	−13.5453			−1.24169
$120$		0			0
$121$	−10.1394			−0.921768
$122$		0			0
$123$	7.01647	−	8.41804i	0.632654	−	0.759029i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−2.01144			−0.178486			−0.0892432	−	0.996010i	$-0.528445\pi$
$127$	−2.01144			−0.178486			−0.0892432	+	0.996010i	$0.528445\pi$
$128$		0			0
$129$	−7.44114	−	6.20222i	−0.655156	−	0.546075i
$130$		0			0
$131$			0.515995i			0.0450827i	0.999746	+	0.0225414i	$0.00717575\pi$
$131$			0.515995i			0.0450827i	−0.999746	+	0.0225414i	$0.992824\pi$
$132$		0			0
$133$			4.39720i			0.381286i
$134$		0			0
$135$	4.52544	+	2.55351i	0.389488	+	0.219771i
$136$		0			0
$137$	−15.1340			−1.29298			−0.646491	−	0.762921i	$-0.723764\pi$
$137$	−15.1340			−1.29298			−0.646491	+	0.762921i	$0.723764\pi$
$138$		0			0
$139$		−	6.23922i		−	0.529204i	−0.964358	−	0.264602i	$-0.914759\pi$
$139$		−	6.23922i		−	0.529204i	0.964358	−	0.264602i	$-0.0852405\pi$
$140$		0			0
$141$	12.5491	+	10.4597i	1.05682	+	0.880866i
$142$		0			0
$143$			1.33011i			0.111230i
$144$		0			0
$145$		−	7.94643i		−	0.659915i
$146$		0			0
$147$	−1.56323	+	1.87550i	−0.128933	+	0.154688i
$148$		0			0
$149$		−	4.39766i		−	0.360270i	−0.983642	−	0.180135i	$-0.942347\pi$
$149$		−	4.39766i		−	0.360270i	0.983642	−	0.180135i	$-0.0576535\pi$
$150$		0			0
$151$	14.9994			1.22063			0.610316	−	0.792158i	$-0.291042\pi$
$151$	14.9994			1.22063			0.610316	+	0.792158i	$0.291042\pi$
$152$		0			0
$153$	−16.9054	+	3.09580i	−1.36672	+	0.250280i
$154$		0			0
$155$			0.732407i			0.0588283i
$156$		0			0
$157$	0.102891			0.00821162			0.00410581	−	0.999992i	$-0.498693\pi$
$157$	0.102891			0.00821162			0.00410581	+	0.999992i	$0.498693\pi$
$158$		0			0
$159$	13.5120	−	16.2111i	1.07157	−	1.28562i
$160$		0			0
$161$	18.1670			1.43176
$162$		0			0
$163$	−2.04062			−0.159834			−0.0799170	−	0.996802i	$-0.525466\pi$
$163$	−2.04062			−0.159834			−0.0799170	+	0.996802i	$0.525466\pi$
$164$		0			0
$165$	1.02875	−	1.23424i	0.0800878	−	0.0960855i
$166$		0			0
$167$		−	23.8616i		−	1.84646i	−0.384244	−	0.923232i	$-0.625538\pi$
$167$		−	23.8616i		−	1.84646i	0.384244	−	0.923232i	$-0.374462\pi$
$168$		0			0
$169$	10.9441			0.841856
$170$		0			0
$171$	1.00499	+	5.48801i	0.0768533	+	0.419679i
$172$		0			0
$173$			3.84748i			0.292519i	0.989246	+	0.146259i	$0.0467234\pi$
$173$			3.84748i			0.292519i	−0.989246	+	0.146259i	$0.953277\pi$
$174$		0			0
$175$		−	2.36440i		−	0.178732i
$176$		0			0
$177$	−8.62114	−	7.18576i	−0.648005	−	0.540115i
$178$		0			0
$179$	−4.57671			−0.342079			−0.171040	−	0.985264i	$-0.554713\pi$
$179$	−4.57671			−0.342079			−0.171040	+	0.985264i	$0.554713\pi$
$180$		0			0
$181$	15.0237			1.11671			0.558353	−	0.829603i	$-0.311433\pi$
$181$	15.0237			1.11671			0.558353	+	0.829603i	$0.311433\pi$
$182$		0			0
$183$	−17.7955	−	14.8326i	−1.31548	−	1.09646i
$184$		0			0
$185$	−1.54199			−0.113369
$186$		0			0
$187$			5.31444i			0.388630i
$188$		0			0
$189$	6.03750	−	10.6999i	0.439163	−	0.778306i
$190$		0			0
$191$	12.3225			0.891623			0.445812	−	0.895127i	$-0.352915\pi$
$191$	12.3225			0.891623			0.445812	+	0.895127i	$0.352915\pi$
$192$		0			0
$193$	−25.9166			−1.86552			−0.932759	−	0.360502i	$-0.882606\pi$
$193$	−25.9166			−1.86552			−0.932759	+	0.360502i	$0.882606\pi$
$194$		0			0
$195$	1.90769	+	1.59007i	0.136613	+	0.113867i
$196$		0			0
$197$	−21.3265			−1.51945			−0.759725	−	0.650244i	$-0.774666\pi$
$197$	−21.3265			−1.51945			−0.759725	+	0.650244i	$0.774666\pi$
$198$		0			0
$199$	−3.62204			−0.256759			−0.128380	−	0.991725i	$-0.540978\pi$
$199$	−3.62204			−0.256759			−0.128380	+	0.991725i	$0.540978\pi$
$200$		0			0
$201$	13.3790	−	4.69064i	0.943683	−	0.330852i
$202$		0			0
$203$	−18.7885			−1.31869
$204$		0			0
$205$	−6.32704			−0.441899
$206$		0			0
$207$	22.6736	−	4.15209i	1.57593	−	0.288590i
$208$		0			0
$209$	1.72523			0.119336
$210$		0			0
$211$	−23.8349			−1.64086			−0.820431	−	0.571746i	$-0.806266\pi$
$211$	−23.8349			−1.64086			−0.820431	+	0.571746i	$0.806266\pi$
$212$		0			0
$213$	6.97182	+	5.81104i	0.477701	+	0.398166i
$214$		0			0
$215$			5.59280i			0.381426i
$216$		0			0
$217$	1.73170			0.117555
$218$		0			0
$219$	3.97879	−	4.77356i	0.268862	−	0.322567i
$220$		0			0
$221$	−8.21422			−0.552548
$222$		0			0
$223$	−0.470089			−0.0314795			−0.0157397	−	0.999876i	$-0.505010\pi$
$223$	−0.470089			−0.0314795			−0.0157397	+	0.999876i	$0.505010\pi$
$224$		0			0
$225$	−0.540386	−	2.95093i	−0.0360258	−	0.196729i
$226$		0			0
$227$			2.53155i			0.168025i	0.996465	+	0.0840125i	$0.0267736\pi$
$227$			2.53155i			0.168025i	−0.996465	+	0.0840125i	$0.973226\pi$
$228$		0			0
$229$			7.18554i			0.474834i	0.971408	+	0.237417i	$0.0763007\pi$
$229$			7.18554i			0.474834i	−0.971408	+	0.237417i	$0.923699\pi$
$230$		0			0
$231$	−2.91823	−	2.43236i	−0.192006	−	0.160038i
$232$		0			0
$233$	3.01455			0.197489			0.0987447	−	0.995113i	$-0.468517\pi$
$233$	3.01455			0.197489			0.0987447	+	0.995113i	$0.468517\pi$
$234$		0			0
$235$		−	9.43194i		−	0.615272i
$236$		0			0
$237$	−16.8387	−	14.0352i	−1.09379	−	0.911681i
$238$		0			0
$239$	−11.7654			−0.761039			−0.380520	−	0.924773i	$-0.624255\pi$
$239$	−11.7654			−0.761039			−0.380520	+	0.924773i	$0.624255\pi$
$240$		0			0
$241$	−14.4500			−0.930805			−0.465403	−	0.885099i	$-0.654090\pi$
$241$	−14.4500			−0.930805			−0.465403	+	0.885099i	$0.654090\pi$
$242$		0			0
$243$	5.08973	−	14.7341i	0.326506	−	0.945195i
$244$		0			0
$245$	1.40963			0.0900581
$246$		0			0
$247$			2.66658i			0.169670i
$248$		0			0
$249$	−6.18195	−	5.15268i	−0.391765	−	0.326538i
$250$		0			0
$251$	−5.68981			−0.359138			−0.179569	−	0.983745i	$-0.557470\pi$
$251$	−5.68981			−0.359138			−0.179569	+	0.983745i	$0.557470\pi$
$252$		0			0
$253$		−	7.12774i		−	0.448117i
$254$		0			0
$255$	7.62216	+	6.35311i	0.477318	+	0.397847i
$256$		0			0
$257$		−	20.6129i		−	1.28580i	−0.765951	−	0.642898i	$-0.777732\pi$
$257$		−	20.6129i		−	1.28580i	0.765951	−	0.642898i	$-0.222268\pi$
$258$		0			0
$259$			3.64588i			0.226544i
$260$		0			0
$261$	−23.4493	+	4.29414i	−1.45148	+	0.265801i
$262$		0			0
$263$			7.16878i			0.442046i	0.975269	+	0.221023i	$0.0709396\pi$
$263$			7.16878i			0.442046i	−0.975269	+	0.221023i	$0.929060\pi$
$264$		0			0
$265$	−12.1843			−0.748476
$266$		0			0
$267$	2.32922	+	1.94141i	0.142546	+	0.118813i
$268$		0			0
$269$		−	6.66430i		−	0.406330i	−0.979145	−	0.203165i	$-0.934877\pi$
$269$		−	6.66430i		−	0.406330i	0.979145	−	0.203165i	$-0.0651228\pi$
$270$		0			0
$271$			6.01096i			0.365140i	0.983193	+	0.182570i	$0.0584416\pi$
$271$			6.01096i			0.365140i	−0.983193	+	0.182570i	$0.941558\pi$
$272$		0			0
$273$	3.75956	−	4.51055i	0.227539	−	0.272991i
$274$		0			0
$275$	−0.927662			−0.0559401
$276$		0			0
$277$	0.613955			0.0368890			0.0184445	−	0.999830i	$-0.494129\pi$
$277$	0.613955			0.0368890			0.0184445	+	0.999830i	$0.494129\pi$
$278$		0			0
$279$	2.16128	−	0.395782i	0.129392	−	0.0236949i
$280$		0			0
$281$	28.2661			1.68621			0.843107	−	0.537746i	$-0.180724\pi$
$281$	28.2661			1.68621			0.843107	+	0.537746i	$0.180724\pi$
$282$		0			0
$283$	11.1321			0.661733			0.330867	−	0.943678i	$-0.392659\pi$
$283$	11.1321			0.661733			0.330867	+	0.943678i	$0.392659\pi$
$284$		0			0
$285$	2.06241	−	2.47438i	0.122167	−	0.146570i
$286$		0			0
$287$			14.9596i			0.883039i
$288$		0			0
$289$	−15.8198			−0.930576
$290$		0			0
$291$	−16.2683	−	13.5597i	−0.953662	−	0.794882i
$292$		0			0
$293$		−	14.8685i		−	0.868626i	−0.900762	−	0.434313i	$-0.856991\pi$
$293$		−	14.8685i		−	0.868626i	0.900762	−	0.434313i	$-0.143009\pi$
$294$		0			0
$295$			6.47969i			0.377262i
$296$		0			0
$297$	−4.19808	−	2.36879i	−0.243597	−	0.137451i
$298$		0			0
$299$	11.0169			0.637126
$300$		0			0
$301$	13.2236			0.762195
$302$		0			0
$303$	15.3529	−	18.4196i	0.881998	−	1.05818i
$304$		0			0
$305$			13.3752i			0.765860i
$306$		0			0
$307$	−15.2248			−0.868927			−0.434463	−	0.900690i	$-0.643062\pi$
$307$	−15.2248			−0.868927			−0.434463	+	0.900690i	$0.643062\pi$
$308$		0			0
$309$	1.66309	−	1.99530i	0.0946100	−	0.113509i
$310$		0			0
$311$	11.9664			0.678554			0.339277	−	0.940687i	$-0.389818\pi$
$311$	11.9664			0.678554			0.339277	+	0.940687i	$0.389818\pi$
$312$		0			0
$313$		−	25.3121i		−	1.43072i	−0.698754	−	0.715362i	$-0.746262\pi$
$313$		−	25.3121i		−	1.43072i	0.698754	−	0.715362i	$-0.253738\pi$
$314$		0			0
$315$	−6.97716	+	1.27769i	−0.393119	+	0.0719895i
$316$		0			0
$317$		−	1.23948i		−	0.0696163i	−0.999394	−	0.0348082i	$-0.988918\pi$
$317$		−	1.23948i		−	0.0696163i	0.999394	−	0.0348082i	$-0.0110820\pi$
$318$		0			0
$319$			7.37159i			0.412730i
$320$		0			0
$321$	2.71613	+	2.26390i	0.151599	+	0.126359i
$322$		0			0
$323$			10.6543i			0.592820i
$324$		0			0
$325$		−	1.43383i		−	0.0795347i
$326$		0			0
$327$	−6.26915	−	5.22537i	−0.346685	−	0.288963i
$328$		0			0
$329$	−22.3008			−1.22949
$330$		0			0
$331$			22.6045i			1.24246i	0.783630	+	0.621228i	$0.213366\pi$
$331$			22.6045i			1.24246i	−0.783630	+	0.621228i	$0.786634\pi$
$332$		0			0
$333$	0.833271	+	4.55030i	0.0456629	+	0.249355i
$334$		0			0
$335$	−7.02590	−	4.19961i	−0.383866	−	0.229449i
$336$		0			0
$337$		−	14.0097i		−	0.763156i	−0.924337	−	0.381578i	$-0.875381\pi$
$337$		−	14.0097i		−	0.763156i	0.924337	−	0.381578i	$-0.124619\pi$
$338$		0			0
$339$	−7.94377	+	9.53056i	−0.431446	+	0.517629i
$340$		0			0
$341$		−	0.679425i		−	0.0367930i
$342$		0			0
$343$		−	19.8837i		−	1.07362i
$344$		0			0
$345$	−10.2229	−	8.52081i	−0.550381	−	0.458745i
$346$		0			0
$347$	16.9440			0.909599			0.454800	−	0.890594i	$-0.349711\pi$
$347$	16.9440			0.909599			0.454800	+	0.890594i	$0.349711\pi$
$348$		0			0
$349$	8.01945			0.429271			0.214636	−	0.976694i	$-0.431144\pi$
$349$	8.01945			0.429271			0.214636	+	0.976694i	$0.431144\pi$
$350$		0			0
$351$	3.66130	−	6.48873i	0.195426	−	0.346343i
$352$		0			0
$353$	−24.6483			−1.31190			−0.655949	−	0.754806i	$-0.727731\pi$
$353$	−24.6483			−1.31190			−0.655949	+	0.754806i	$0.727731\pi$
$354$		0			0
$355$		−	5.24005i		−	0.278113i
$356$		0			0
$357$	15.0213	−	18.0218i	0.795010	−	0.953815i
$358$		0			0
$359$			13.8474i			0.730838i	0.930843	+	0.365419i	$0.119074\pi$
$359$			13.8474i			0.730838i	−0.930843	+	0.365419i	$0.880926\pi$
$360$		0			0
$361$	−15.5413			−0.817963
$362$		0			0
$363$	11.2443	−	13.4904i	0.590173	−	0.708062i
$364$		0			0
$365$	−3.58783			−0.187796
$366$		0			0
$367$		−	32.0553i		−	1.67327i	−0.547758	−	0.836637i	$-0.684518\pi$
$367$		−	32.0553i		−	1.67327i	0.547758	−	0.836637i	$-0.315482\pi$
$368$		0			0
$369$	3.41904	+	18.6706i	0.177988	+	0.971955i
$370$		0			0
$371$			28.8085i			1.49566i
$372$		0			0
$373$		−	18.3802i		−	0.951692i	−0.879529	−	0.475846i	$-0.842142\pi$
$373$		−	18.3802i		−	0.951692i	0.879529	−	0.475846i	$-0.157858\pi$
$374$		0			0
$375$	−1.10897	+	1.33049i	−0.0572668	+	0.0687060i
$376$		0			0
$377$	−11.3938			−0.586813
$378$		0			0
$379$		−	0.0787586i		−	0.00404556i	−0.999998	−	0.00202278i	$-0.999356\pi$
$379$		−	0.0787586i		−	0.00404556i	0.999998	−	0.00202278i	$-0.000643871\pi$
$380$		0			0
$381$	2.23062	−	2.67619i	0.114278	−	0.137105i
$382$		0			0
$383$	−35.1704			−1.79712			−0.898562	−	0.438846i	$-0.855387\pi$
$383$	−35.1704			−1.79712			−0.898562	+	0.438846i	$0.855387\pi$
$384$		0			0
$385$			2.19336i			0.111784i
$386$		0			0
$387$	16.5039	−	3.02227i	0.838943	−	0.153631i
$388$		0			0
$389$		−	30.1117i		−	1.52672i	−0.645973	−	0.763361i	$-0.723548\pi$
$389$		−	30.1117i		−	1.52672i	0.645973	−	0.763361i	$-0.276452\pi$
$390$		0			0
$391$	44.0180			2.22609
$392$		0			0
$393$	−0.686525	−	0.572222i	−0.0346306	−	0.0288648i
$394$		0			0
$395$			12.6561i			0.636796i
$396$		0			0
$397$	−24.0289			−1.20597			−0.602987	−	0.797751i	$-0.706023\pi$
$397$	−24.0289			−1.20597			−0.602987	+	0.797751i	$0.706023\pi$
$398$		0			0
$399$	−5.85042	−	4.87635i	−0.292887	−	0.244123i
$400$		0			0
$401$	−7.40188			−0.369632			−0.184816	−	0.982773i	$-0.559169\pi$
$401$	−7.40188			−0.369632			−0.184816	+	0.982773i	$0.559169\pi$
$402$		0			0
$403$	1.05015			0.0523116
$404$		0			0
$405$	−8.41597	+	3.18928i	−0.418193	+	0.158477i
$406$		0			0
$407$	1.43045			0.0709045
$408$		0			0
$409$		−	15.4090i		−	0.761927i	−0.924590	−	0.380963i	$-0.875592\pi$
$409$		−	15.4090i		−	0.761927i	0.924590	−	0.380963i	$-0.124408\pi$
$410$		0			0
$411$	16.7831	−	20.1355i	0.827848	−	0.993213i
$412$		0			0
$413$	15.3206			0.753875
$414$		0			0
$415$			4.64638i			0.228082i
$416$		0			0
$417$	8.30119	+	6.91908i	0.406511	+	0.338829i
$418$		0			0
$419$		−	2.44364i		−	0.119380i	−0.998217	−	0.0596898i	$-0.980989\pi$
$419$		−	2.44364i		−	0.119380i	0.998217	−	0.0596898i	$-0.0190112\pi$
$420$		0			0
$421$	−12.7433			−0.621070			−0.310535	−	0.950562i	$-0.600508\pi$
$421$	−12.7433			−0.621070			−0.310535	+	0.950562i	$0.600508\pi$
$422$		0			0
$423$	−27.8330	+	5.09689i	−1.35329	+	0.247819i
$424$		0			0
$425$		−	5.72886i		−	0.277890i
$426$		0			0
$427$	31.6242			1.53040
$428$		0			0
$429$	−1.76970	−	1.47505i	−0.0854417	−	0.0712161i
$430$		0			0
$431$			11.7002i			0.563577i	0.959476	+	0.281789i	$0.0909277\pi$
$431$			11.7002i			0.563577i	−0.959476	+	0.281789i	$0.909072\pi$
$432$		0			0
$433$		−	13.1099i		−	0.630020i	−0.949088	−	0.315010i	$-0.897992\pi$
$433$		−	13.1099i		−	0.630020i	0.949088	−	0.315010i	$-0.102008\pi$
$434$		0			0
$435$	10.5726	+	8.81232i	0.506918	+	0.422518i
$436$		0			0
$437$		−	14.2896i		−	0.683562i
$438$		0			0
$439$	−11.6037			−0.553812			−0.276906	−	0.960897i	$-0.589309\pi$
$439$	−11.6037			−0.553812			−0.276906	+	0.960897i	$0.589309\pi$
$440$		0			0
$441$	−0.761746	−	4.15972i	−0.0362736	−	0.198082i
$442$		0			0
$443$	2.06052			0.0978983			0.0489492	−	0.998801i	$-0.484413\pi$
$443$	2.06052			0.0978983			0.0489492	+	0.998801i	$0.484413\pi$
$444$		0			0
$445$		−	1.75065i		−	0.0829888i
$446$		0			0
$447$	5.85102	+	4.87686i	0.276744	+	0.230667i
$448$		0			0
$449$		−	12.5225i		−	0.590972i	−0.955347	−	0.295486i	$-0.904518\pi$
$449$		−	12.5225i		−	0.590972i	0.955347	−	0.295486i	$-0.0954816\pi$
$450$		0			0
$451$	5.86935			0.276377
$452$		0			0
$453$	−16.6338	+	19.9565i	−0.781525	+	0.937637i
$454$		0			0
$455$	−3.39015			−0.158933
$456$		0			0
$457$	14.6075			0.683310			0.341655	−	0.939825i	$-0.389013\pi$
$457$	14.6075			0.683310			0.341655	+	0.939825i	$0.389013\pi$
$458$		0			0
$459$	14.6287	−	25.9256i	0.682808	−	1.21010i
$460$		0			0
$461$			21.8169i			1.01612i	0.861323	+	0.508058i	$0.169636\pi$
$461$			21.8169i			1.01612i	−0.861323	+	0.508058i	$0.830364\pi$
$462$		0			0
$463$		−	9.64976i		−	0.448462i	−0.974536	−	0.224231i	$-0.928013\pi$
$463$		−	9.64976i		−	0.448462i	0.974536	−	0.224231i	$-0.0719870\pi$
$464$		0			0
$465$	−0.974457	−	0.812214i	−0.0451893	−	0.0376655i
$466$		0			0
$467$			0.493069i			0.0228165i	0.999935	+	0.0114083i	$0.00363144\pi$
$467$			0.493069i			0.0228165i	−0.999935	+	0.0114083i	$0.996369\pi$
$468$		0			0
$469$	−9.92954	+	16.6120i	−0.458504	+	0.767071i
$470$		0			0
$471$	−0.114103	+	0.136895i	−0.00525759	+	0.00630781i
$472$		0			0
$473$		−	5.18822i		−	0.238555i
$474$		0			0
$475$	−1.85976			−0.0853315
$476$		0			0
$477$	6.58424	+	35.9550i	0.301471	+	1.64627i
$478$		0			0
$479$			32.9905i			1.50737i	0.657234	+	0.753687i	$0.271726\pi$
$479$			32.9905i			1.50737i	−0.657234	+	0.753687i	$0.728274\pi$
$480$		0			0
$481$			2.21096i			0.100811i
$482$		0			0
$483$	−20.1466	+	24.1709i	−0.916701	+	1.09981i
$484$		0			0
$485$			12.2273i			0.555213i
$486$		0			0
$487$		−	2.92977i		−	0.132761i	−0.997794	−	0.0663803i	$-0.978855\pi$
$487$		−	2.92977i		−	0.132761i	0.997794	−	0.0663803i	$-0.0211451\pi$
$488$		0			0
$489$	2.26298	−	2.71502i	0.102336	−	0.122778i
$490$		0			0
$491$			29.4557i			1.32932i	0.747148	+	0.664658i	$0.231423\pi$
$491$			29.4557i			1.32932i	−0.747148	+	0.664658i	$0.768577\pi$
$492$		0			0
$493$	−45.5239			−2.05029
$494$		0			0
$495$	0.501296	+	2.73746i	0.0225316	+	0.123040i
$496$		0			0
$497$	−12.3896			−0.555748
$498$		0			0
$499$			27.3412i			1.22396i	0.790873	+	0.611980i	$0.209627\pi$
$499$			27.3412i			1.22396i	−0.790873	+	0.611980i	$0.790373\pi$
$500$		0			0
$501$	31.7475	+	26.4617i	1.41837	+	1.18222i
$502$		0			0
$503$	−5.03157			−0.224346			−0.112173	−	0.993689i	$-0.535781\pi$
$503$	−5.03157			−0.224346			−0.112173	+	0.993689i	$0.535781\pi$
$504$		0			0
$505$	−13.8443			−0.616063
$506$		0			0
$507$	−12.1367	+	14.5610i	−0.539008	+	0.646677i
$508$		0			0
$509$			19.6328i			0.870210i	0.900380	+	0.435105i	$0.143289\pi$
$509$			19.6328i			0.870210i	−0.900380	+	0.435105i	$0.856711\pi$
$510$		0			0
$511$			8.48306i			0.375268i
$512$		0			0
$513$	−8.41622	−	4.74890i	−0.371585	−	0.209669i
$514$		0			0
$515$	−1.49968			−0.0660837
$516$		0			0
$517$			8.74965i			0.384809i
$518$		0			0
$519$	−5.11902	−	4.26673i	−0.224700	−	0.187289i
$520$		0			0
$521$	23.8347			1.04422			0.522108	−	0.852880i	$-0.325146\pi$
$521$	23.8347			1.04422			0.522108	+	0.852880i	$0.325146\pi$
$522$		0			0
$523$	−10.8341			−0.473741			−0.236870	−	0.971541i	$-0.576122\pi$
$523$	−10.8341			−0.473741			−0.236870	+	0.971541i	$0.576122\pi$
$524$		0			0
$525$	3.14580	+	2.62204i	0.137294	+	0.114435i
$526$		0			0
$527$	4.19585			0.182774
$528$		0			0
$529$	−36.0371			−1.56683
$530$		0			0
$531$	19.1211	−	3.50154i	0.829786	−	0.151954i
$532$		0			0
$533$			9.07192i			0.392948i
$534$		0			0
$535$		−	2.04145i		−	0.0882597i
$536$		0			0
$537$	5.07542	−	6.08925i	0.219021	−	0.262771i
$538$		0			0
$539$	−1.30766			−0.0563250
$540$		0			0
$541$			19.4673i			0.836963i	0.908225	+	0.418481i	$0.137437\pi$
$541$			19.4673i			0.836963i	−0.908225	+	0.418481i	$0.862563\pi$
$542$		0			0
$543$	−16.6608	+	19.9889i	−0.714985	+	0.857805i
$544$		0			0
$545$			4.71192i			0.201837i
$546$		0			0
$547$			30.7159i			1.31332i	0.754187	+	0.656659i	$0.228031\pi$
$547$			30.7159i			1.31332i	−0.754187	+	0.656659i	$0.771969\pi$
$548$		0			0
$549$	39.4692	−	7.22776i	1.68450	−	0.308473i
$550$		0			0
$551$			14.7784i			0.629582i
$552$		0			0
$553$	29.9239			1.27250
$554$		0			0
$555$	1.71002	−	2.05160i	0.0725861	−	0.0870854i
$556$		0			0
$557$			34.6116i			1.46654i	0.679936	+	0.733271i	$0.262007\pi$
$557$			34.6116i			1.46654i	−0.679936	+	0.733271i	$0.737993\pi$
$558$		0			0
$559$	8.01914			0.339173
$560$		0			0
$561$	−7.07079	−	5.89354i	−0.298529	−	0.248825i
$562$		0			0
$563$	−23.8342			−1.00449			−0.502245	−	0.864725i	$-0.667492\pi$
$563$	−23.8342			−1.00449			−0.502245	+	0.864725i	$0.667492\pi$
$564$		0			0
$565$	7.16322			0.301359
$566$		0			0
$567$	7.54073	+	19.8987i	0.316681	+	0.835666i
$568$		0			0
$569$		−	43.2245i		−	1.81207i	−0.423206	−	0.906034i	$-0.639095\pi$
$569$		−	43.2245i		−	1.81207i	0.423206	−	0.906034i	$-0.360905\pi$
$570$		0			0
$571$	26.9041			1.12590			0.562950	−	0.826491i	$-0.309666\pi$
$571$	26.9041			1.12590			0.562950	+	0.826491i	$0.309666\pi$
$572$		0			0
$573$	−13.6652	+	16.3949i	−0.570873	+	0.684906i
$574$		0			0
$575$			7.68356i			0.320427i
$576$		0			0
$577$		−	19.0660i		−	0.793728i	−0.917877	−	0.396864i	$-0.870098\pi$
$577$		−	19.0660i		−	0.793728i	0.917877	−	0.396864i	$-0.129902\pi$
$578$		0			0
$579$	28.7406	−	34.4817i	1.19442	−	1.43301i
$580$		0			0
$581$	10.9859			0.455771
$582$		0			0
$583$	11.3029			0.468119
$584$		0			0
$585$	−4.23114	+	0.774824i	−0.174936	+	0.0320350i
$586$		0			0
$587$	−26.6631			−1.10051			−0.550253	−	0.834998i	$-0.685469\pi$
$587$	−26.6631			−1.10051			−0.550253	+	0.834998i	$0.685469\pi$
$588$		0			0
$589$		−	1.36210i		−	0.0561243i
$590$		0			0
$591$	23.6504	−	28.3746i	0.972846	−	1.16718i
$592$		0			0
$593$	−15.6336			−0.641996			−0.320998	−	0.947080i	$-0.604018\pi$
$593$	−15.6336			−0.641996			−0.320998	+	0.947080i	$0.604018\pi$
$594$		0			0
$595$	−13.5453			−0.555303
$596$		0			0
$597$	4.01672	−	4.81907i	0.164393	−	0.197231i
$598$		0			0
$599$	38.5453			1.57492			0.787459	−	0.616366i	$-0.211396\pi$
$599$	38.5453			1.57492			0.787459	+	0.616366i	$0.211396\pi$
$600$		0			0
$601$	37.2372			1.51894			0.759468	−	0.650544i	$-0.225459\pi$
$601$	37.2372			1.51894			0.759468	+	0.650544i	$0.225459\pi$
$602$		0			0
$603$	−8.59605	+	23.0023i	−0.350058	+	0.936728i
$604$		0			0
$605$	−10.1394			−0.412227
$606$		0			0
$607$	13.0663			0.530346			0.265173	−	0.964201i	$-0.414571\pi$
$607$	13.0663			0.530346			0.265173	+	0.964201i	$0.414571\pi$
$608$		0			0
$609$	20.8358	−	24.9978i	0.844310	−	1.01296i
$610$		0			0
$611$	−13.5238			−0.547116
$612$		0			0
$613$	41.4688			1.67491			0.837455	−	0.546507i	$-0.184043\pi$
$613$	41.4688			1.67491			0.837455	+	0.546507i	$0.184043\pi$
$614$		0			0
$615$	7.01647	−	8.41804i	0.282931	−	0.339448i
$616$		0			0
$617$		−	34.9926i		−	1.40875i	−0.709829	−	0.704374i	$-0.751228\pi$
$617$		−	34.9926i		−	1.40875i	0.709829	−	0.704374i	$-0.248772\pi$
$618$		0			0
$619$	−12.9371			−0.519986			−0.259993	−	0.965610i	$-0.583720\pi$
$619$	−12.9371			−0.519986			−0.259993	+	0.965610i	$0.583720\pi$
$620$		0			0
$621$	−19.6200	+	34.7715i	−0.787324	+	1.39533i
$622$		0			0
$623$	−4.13923			−0.165835
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−1.91322	+	2.29539i	−0.0764065	+	0.0916690i
$628$		0			0
$629$			8.83384i			0.352228i
$630$		0			0
$631$		−	8.93494i		−	0.355694i	−0.984058	−	0.177847i	$-0.943087\pi$
$631$		−	8.93494i		−	0.355694i	0.984058	−	0.177847i	$-0.0569133\pi$
$632$		0			0
$633$	26.4321	−	31.7120i	1.05058	−	1.26044i
$634$		0			0
$635$	−2.01144			−0.0798215
$636$		0			0
$637$		−	2.02118i		−	0.0800819i
$638$		0			0
$639$	−15.4630	+	2.83165i	−0.611708	+	0.112018i
$640$		0			0
$641$	19.5231			0.771114			0.385557	−	0.922684i	$-0.374009\pi$
$641$	19.5231			0.771114			0.385557	+	0.922684i	$0.374009\pi$
$642$		0			0
$643$	43.8713			1.73011			0.865057	−	0.501673i	$-0.167282\pi$
$643$	43.8713			1.73011			0.865057	+	0.501673i	$0.167282\pi$
$644$		0			0
$645$	−7.44114	−	6.20222i	−0.292995	−	0.244212i
$646$		0			0
$647$	−22.2539			−0.874891			−0.437445	−	0.899245i	$-0.644117\pi$
$647$	−22.2539			−0.874891			−0.437445	+	0.899245i	$0.644117\pi$
$648$		0			0
$649$		−	6.01096i		−	0.235951i
$650$		0			0
$651$	−1.92040	+	2.30400i	−0.0752663	+	0.0903009i
$652$		0			0
$653$	−19.4354			−0.760568			−0.380284	−	0.924870i	$-0.624174\pi$
$653$	−19.4354			−0.760568			−0.380284	+	0.924870i	$0.624174\pi$
$654$		0			0
$655$			0.515995i			0.0201616i
$656$		0			0
$657$	1.93882	+	10.5874i	0.0756404	+	0.413055i
$658$		0			0
$659$			10.8355i			0.422090i	0.977476	+	0.211045i	$0.0676866\pi$
$659$			10.8355i			0.422090i	−0.977476	+	0.211045i	$0.932313\pi$
$660$		0			0
$661$			3.63079i			0.141221i	0.997504	+	0.0706107i	$0.0224948\pi$
$661$			3.63079i			0.141221i	−0.997504	+	0.0706107i	$0.977505\pi$
$662$		0			0
$663$	9.10930	−	10.9289i	0.353776	−	0.424444i
$664$		0			0
$665$			4.39720i			0.170516i
$666$		0			0
$667$	61.0568			2.36413
$668$		0			0
$669$	0.521313	−	0.625447i	0.0201551	−	0.0241812i
$670$		0			0
$671$		−	12.4076i		−	0.478991i
$672$		0			0
$673$		−	34.6060i		−	1.33396i	−0.745074	−	0.666982i	$-0.767586\pi$
$673$		−	34.6060i		−	1.33396i	0.745074	−	0.666982i	$-0.232414\pi$
$674$		0			0
$675$	4.52544	+	2.55351i	0.174184	+	0.0982845i
$676$		0			0
$677$	0.107831			0.00414427			0.00207214	−	0.999998i	$-0.499340\pi$
$677$	0.107831			0.00414427			0.00207214	+	0.999998i	$0.499340\pi$
$678$		0			0
$679$	28.9102			1.10947
$680$		0			0
$681$	−3.36819	−	2.80741i	−0.129069	−	0.107580i
$682$		0			0
$683$	−43.6386			−1.66979			−0.834893	−	0.550413i	$-0.814470\pi$
$683$	−43.6386			−1.66979			−0.834893	+	0.550413i	$0.814470\pi$
$684$		0			0
$685$	−15.1340			−0.578239
$686$		0			0
$687$	−9.56026	−	7.96852i	−0.364747	−	0.304018i
$688$		0			0
$689$			17.4703i			0.665564i
$690$		0			0
$691$	45.2615			1.72183			0.860914	−	0.508750i	$-0.169892\pi$
$691$	45.2615			1.72183			0.860914	+	0.508750i	$0.169892\pi$
$692$		0			0
$693$	6.47245	−	1.18526i	0.245868	−	0.0450244i
$694$		0			0
$695$		−	6.23922i		−	0.236667i
$696$		0			0
$697$			36.2467i			1.37294i
$698$		0			0
$699$	−3.34303	+	4.01081i	−0.126445	+	0.151703i
$700$		0			0
$701$	−46.8990			−1.77135			−0.885676	−	0.464304i	$-0.846304\pi$
$701$	−46.8990			−1.77135			−0.885676	+	0.464304i	$0.846304\pi$
$702$		0			0
$703$	2.86773			0.108158
$704$		0			0
$705$	12.5491	+	10.4597i	0.472625	+	0.393935i
$706$		0			0
$707$			32.7334i			1.23107i
$708$		0			0
$709$	43.1131			1.61915			0.809573	−	0.587019i	$-0.199699\pi$
$709$	43.1131			1.61915			0.809573	+	0.587019i	$0.199699\pi$
$710$		0			0
$711$	37.3471	−	6.83916i	1.40063	−	0.256489i
$712$		0			0
$713$	−5.62749			−0.210751
$714$		0			0
$715$			1.33011i			0.0497434i
$716$		0			0
$717$	13.0474	−	15.6537i	0.487265	−	0.584597i
$718$		0			0
$719$			9.62238i			0.358854i	0.983771	+	0.179427i	$0.0574244\pi$
$719$			9.62238i			0.358854i	−0.983771	+	0.179427i	$0.942576\pi$
$720$		0			0
$721$			3.54583i			0.132054i
$722$		0			0
$723$	16.0245	−	19.2255i	0.595959	−	0.715004i
$724$		0			0
$725$		−	7.94643i		−	0.295123i
$726$		0			0
$727$			32.5959i			1.20892i	0.796637	+	0.604458i	$0.206610\pi$
$727$			32.5959i			1.20892i	−0.796637	+	0.604458i	$0.793390\pi$
$728$		0			0
$729$	13.9592	+	23.1115i	0.517008	+	0.855980i
$730$		0			0
$731$	32.0403			1.18505
$732$		0			0
$733$			37.9422i			1.40143i	0.713443	+	0.700713i	$0.247135\pi$
$733$			37.9422i			1.40143i	−0.713443	+	0.700713i	$0.752865\pi$
$734$		0			0
$735$	−1.56323	+	1.87550i	−0.0576608	+	0.0691787i
$736$		0			0
$737$	6.51766	+	3.89582i	0.240081	+	0.143504i
$738$		0			0
$739$		−	21.8380i		−	0.803323i	−0.915788	−	0.401662i	$-0.868433\pi$
$739$		−	21.8380i		−	0.803323i	0.915788	−	0.401662i	$-0.131567\pi$
$740$		0			0
$741$	−3.54785	−	2.95715i	−0.130333	−	0.108634i
$742$		0			0
$743$		−	14.1470i		−	0.519003i	−0.965743	−	0.259502i	$-0.916442\pi$
$743$		−	14.1470i		−	0.519003i	0.965743	−	0.259502i	$-0.0835583\pi$
$744$		0			0
$745$		−	4.39766i		−	0.161118i
$746$		0			0
$747$	13.7111	−	2.51084i	0.501665	−	0.0918669i
$748$		0			0
$749$	−4.82680			−0.176368
$750$		0			0
$751$	−35.0298			−1.27825			−0.639127	−	0.769101i	$-0.720704\pi$
$751$	−35.0298			−1.27825			−0.639127	+	0.769101i	$0.720704\pi$
$752$		0			0
$753$	6.30981	−	7.57022i	0.229942	−	0.275874i
$754$		0			0
$755$	14.9994			0.545883
$756$		0			0
$757$		−	23.2440i		−	0.844817i	−0.906406	−	0.422408i	$-0.861185\pi$
$757$		−	23.2440i		−	0.844817i	0.906406	−	0.422408i	$-0.138815\pi$
$758$		0			0
$759$	9.48336	+	7.90443i	0.344224	+	0.286913i
$760$		0			0
$761$			1.88806i			0.0684420i	0.999414	+	0.0342210i	$0.0108950\pi$
$761$			1.88806i			0.0684420i	−0.999414	+	0.0342210i	$0.989105\pi$
$762$		0			0
$763$	11.1409			0.403326
$764$		0			0
$765$	−16.9054	+	3.09580i	−0.611218	+	0.111929i
$766$		0			0
$767$	9.29080			0.335471
$768$		0			0
$769$			28.5174i			1.02837i	0.857681	+	0.514183i	$0.171905\pi$
$769$			28.5174i			1.02837i	−0.857681	+	0.514183i	$0.828095\pi$
$770$		0			0
$771$	27.4252	+	22.8590i	0.987693	+	0.823247i
$772$		0			0
$773$			15.3480i			0.552028i	0.961154	+	0.276014i	$0.0890136\pi$
$773$			15.3480i			0.552028i	−0.961154	+	0.276014i	$0.910986\pi$
$774$		0			0
$775$			0.732407i			0.0263088i
$776$		0			0
$777$	−4.85079	−	4.04315i	−0.174021	−	0.145047i
$778$		0			0
$779$	11.7668			0.421588
$780$		0			0
$781$			4.86100i			0.173940i
$782$		0			0
$783$	20.2912	−	35.9611i	0.725150	−	1.28514i
$784$		0			0
$785$	0.102891			0.00367235
$786$		0			0
$787$		−	8.41537i		−	0.299976i	−0.988688	−	0.149988i	$-0.952077\pi$
$787$		−	8.41537i		−	0.299976i	0.988688	−	0.149988i	$-0.0479235\pi$
$788$		0			0
$789$	−9.53797	−	7.94994i	−0.339561	−	0.283025i
$790$		0			0
$791$		−	16.9367i		−	0.602199i
$792$		0			0
$793$	19.1778			0.681022
$794$		0			0
$795$	13.5120	−	16.2111i	0.479221	−	0.574947i
$796$		0			0
$797$			38.2236i			1.35395i	0.736006	+	0.676975i	$0.236709\pi$
$797$			38.2236i			1.35395i	−0.736006	+	0.676975i	$0.763291\pi$
$798$		0			0
$799$	−54.0342			−1.91159
$800$		0			0
$801$	−5.16605	+	0.946028i	−0.182533	+	0.0334262i
$802$		0			0
$803$	3.32830			0.117453
$804$		0			0
$805$	18.1670			0.640302
$806$		0			0
$807$	8.86676	+	7.39049i	0.312125	+	0.260158i
$808$		0			0
$809$	20.8270			0.732238			0.366119	−	0.930568i	$-0.380686\pi$
$809$	20.8270			0.732238			0.366119	+	0.930568i	$0.380686\pi$
$810$		0			0
$811$		−	25.6751i		−	0.901576i	−0.892631	−	0.450788i	$-0.851143\pi$
$811$		−	25.6751i		−	0.901576i	0.892631	−	0.450788i	$-0.148857\pi$
$812$		0			0
$813$	−7.99750	−	6.66596i	−0.280485	−	0.233785i
$814$		0			0
$815$	−2.04062			−0.0714800
$816$		0			0
$817$		−	10.4012i		−	0.363893i
$818$		0			0
$819$	1.83199	+	10.0041i	0.0640149	+	0.349571i
$820$		0			0
$821$		−	22.6505i		−	0.790508i	−0.918572	−	0.395254i	$-0.870657\pi$
$821$		−	22.6505i		−	0.790508i	0.918572	−	0.395254i	$-0.129343\pi$
$822$		0			0
$823$	−9.56954			−0.333573			−0.166787	−	0.985993i	$-0.553339\pi$
$823$	−9.56954			−0.333573			−0.166787	+	0.985993i	$0.553339\pi$
$824$		0			0
$825$	1.02875	−	1.23424i	0.0358163	−	0.0429708i
$826$		0			0
$827$			45.7249i			1.59001i	0.606603	+	0.795005i	$0.292532\pi$
$827$			45.7249i			1.59001i	−0.606603	+	0.795005i	$0.707468\pi$
$828$		0			0
$829$	53.8467			1.87017			0.935087	−	0.354419i	$-0.115321\pi$
$829$	53.8467			1.87017			0.935087	+	0.354419i	$0.115321\pi$
$830$		0			0
$831$	−0.680855	+	0.816858i	−0.0236186	+	0.0283365i
$832$		0			0
$833$		−	8.07558i		−	0.279802i
$834$		0			0
$835$		−	23.8616i		−	0.825763i
$836$		0			0
$837$	−1.87020	+	3.31446i	−0.0646437	+	0.114565i
$838$		0			0
$839$			6.13562i			0.211825i	0.994375	+	0.105913i	$0.0337764\pi$
$839$			6.13562i			0.211825i	−0.994375	+	0.105913i	$0.966224\pi$
$840$		0			0
$841$	−34.1457			−1.17744
$842$		0			0
$843$	−31.3462	+	37.6077i	−1.07962	+	1.29528i
$844$		0			0
$845$	10.9441			0.376489
$846$		0			0
$847$			23.9737i			0.823745i
$848$		0			0
$849$	−12.3451	+	14.8111i	−0.423683	+	0.508315i
$850$		0			0
$851$		−	11.8480i		−	0.406143i
$852$		0			0
$853$	−31.8059			−1.08901			−0.544506	−	0.838757i	$-0.683283\pi$
$853$	−31.8059			−1.08901			−0.544506	+	0.838757i	$0.683283\pi$
$854$		0			0
$855$	1.00499	+	5.48801i	0.0343698	+	0.187686i
$856$		0			0
$857$	44.4079			1.51695			0.758473	−	0.651704i	$-0.225946\pi$
$857$	44.4079			1.51695			0.758473	+	0.651704i	$0.225946\pi$
$858$		0			0
$859$	−22.2117			−0.757853			−0.378926	−	0.925427i	$-0.623707\pi$
$859$	−22.2117			−0.757853			−0.378926	+	0.925427i	$0.623707\pi$
$860$		0			0
$861$	−19.9036	−	16.5897i	−0.678312	−	0.565376i
$862$		0			0
$863$			32.7364i			1.11436i	0.830392	+	0.557180i	$0.188117\pi$
$863$			32.7364i			1.11436i	−0.830392	+	0.557180i	$0.811883\pi$
$864$		0			0
$865$			3.84748i			0.130818i
$866$		0			0
$867$	17.5436	−	21.0480i	0.595813	−	0.714828i
$868$		0			0
$869$		−	11.7405i		−	0.398271i
$870$		0			0
$871$	−6.02154	+	10.0740i	−0.204032	+	0.341343i
$872$		0			0
$873$	36.0819	−	6.60747i	1.22119	−	0.223629i
$874$		0			0
$875$		−	2.36440i		−	0.0799312i
$876$		0			0
$877$	44.6251			1.50688			0.753441	−	0.657516i	$-0.228393\pi$
$877$	44.6251			1.50688			0.753441	+	0.657516i	$0.228393\pi$
$878$		0			0
$879$	19.7823	+	16.4886i	0.667240	+	0.556148i
$880$		0			0
$881$		−	27.8999i		−	0.939971i	−0.882674	−	0.469986i	$-0.844259\pi$
$881$		−	27.8999i		−	0.939971i	0.882674	−	0.469986i	$-0.155741\pi$
$882$		0			0
$883$		−	39.1780i		−	1.31845i	−0.751947	−	0.659223i	$-0.770885\pi$
$883$		−	39.1780i		−	1.31845i	0.751947	−	0.659223i	$-0.229115\pi$
$884$		0			0
$885$	−8.62114	−	7.18576i	−0.289796	−	0.241547i
$886$		0			0
$887$			42.3688i			1.42260i	0.702887	+	0.711302i	$0.251894\pi$
$887$			42.3688i			1.42260i	−0.702887	+	0.711302i	$0.748106\pi$
$888$		0			0
$889$			4.75584i			0.159506i
$890$		0			0
$891$	7.80717	−	2.95858i	0.261550	−	0.0991160i
$892$		0			0
$893$			17.5411i			0.586991i
$894$		0			0
$895$	−4.57671			−0.152983
$896$		0			0
$897$	−12.2174	+	14.6579i	−0.407928	+	0.489413i
$898$		0			0
$899$	5.82001			0.194108
$900$		0			0
$901$			69.8022i			2.32545i
$902$		0			0
$903$	−14.6645	+	17.5938i	−0.488005	+	0.585485i
$904$		0			0
$905$	15.0237			0.499406
$906$		0			0
$907$	−30.8752			−1.02519			−0.512597	−	0.858630i	$-0.671316\pi$
$907$	−30.8752			−1.02519			−0.512597	+	0.858630i	$0.671316\pi$
$908$		0			0
$909$	7.48127	+	40.8535i	0.248138	+	1.35503i
$910$		0			0
$911$		−	2.10921i		−	0.0698812i	−0.999389	−	0.0349406i	$-0.988876\pi$
$911$		−	2.10921i		−	0.0698812i	0.999389	−	0.0349406i	$-0.0111242\pi$
$912$		0			0
$913$		−	4.31027i		−	0.142649i
$914$		0			0
$915$	−17.7955	−	14.8326i	−0.588300	−	0.490351i
$916$		0			0
$917$	1.22002			0.0402885
$918$		0			0
$919$			54.1287i			1.78554i	0.450511	+	0.892771i	$0.351242\pi$
$919$			54.1287i			1.78554i	−0.450511	+	0.892771i	$0.648758\pi$
$920$		0			0
$921$	16.8838	−	20.2564i	0.556341	−	0.667472i
$922$		0			0
$923$	−7.51336			−0.247305
$924$		0			0
$925$	−1.54199			−0.0507003
$926$		0			0
$927$	0.810405	+	4.42544i	0.0266172	+	0.145351i
$928$		0			0
$929$	−5.58865			−0.183358			−0.0916788	−	0.995789i	$-0.529223\pi$
$929$	−5.58865			−0.183358			−0.0916788	+	0.995789i	$0.529223\pi$
$930$		0			0
$931$	−2.62157			−0.0859186
$932$		0			0
$933$	−13.2704	+	15.9212i	−0.434452	+	0.521235i
$934$		0			0
$935$			5.31444i			0.173801i
$936$		0			0
$937$			16.2323i			0.530285i	0.964209	+	0.265142i	$0.0854190\pi$
$937$			16.2323i			0.530285i	−0.964209	+	0.265142i	$0.914581\pi$
$938$		0			0
$939$	33.6774	+	28.0703i	1.09902	+	0.916038i
$940$		0			0
$941$	23.8565			0.777700			0.388850	−	0.921301i	$-0.372873\pi$
$941$	23.8565			0.777700			0.388850	+	0.921301i	$0.372873\pi$
$942$		0			0
$943$		−	48.6142i		−	1.58310i
$944$		0			0
$945$	6.03750	−	10.6999i	0.196400	−	0.348069i
$946$		0			0
$947$		−	29.6998i		−	0.965114i	−0.875864	−	0.482557i	$-0.839708\pi$
$947$		−	29.6998i		−	0.965114i	0.875864	−	0.482557i	$-0.160292\pi$
$948$		0			0
$949$			5.14435i			0.166993i
$950$		0			0
$951$	1.64912	+	1.37455i	0.0534762	+	0.0445727i
$952$		0			0
$953$			29.4517i			0.954034i	0.878894	+	0.477017i	$0.158282\pi$
$953$			29.4517i			0.954034i	−0.878894	+	0.477017i	$0.841718\pi$
$954$		0			0
$955$	12.3225			0.398746
$956$		0			0
$957$	−9.80780	−	8.17485i	−0.317041	−	0.264255i
$958$		0			0
$959$			35.7827i			1.15548i
$960$		0			0
$961$	30.4636			0.982696
$962$		0			0
$963$	−6.02418	+	1.10317i	−0.194127	+	0.0355493i
$964$		0			0
$965$	−25.9166			−0.834285
$966$		0			0
$967$	−55.2842			−1.77782			−0.888910	−	0.458082i	$-0.848537\pi$
$967$	−55.2842			−1.77782			−0.888910	+	0.458082i	$0.848537\pi$
$968$		0			0
$969$	−14.1754	−	11.8152i	−0.455379	−	0.379560i
$970$		0			0
$971$			3.70363i			0.118855i	0.998233	+	0.0594275i	$0.0189275\pi$
$971$			3.70363i			0.118855i	−0.998233	+	0.0594275i	$0.981072\pi$
$972$		0			0
$973$	−14.7520			−0.472927
$974$		0			0
$975$	1.90769	+	1.59007i	0.0610951	+	0.0509231i
$976$		0			0
$977$		−	21.2234i		−	0.678995i	−0.940607	−	0.339498i	$-0.889743\pi$
$977$		−	21.2234i		−	0.678995i	0.940607	−	0.339498i	$-0.110257\pi$
$978$		0			0
$979$			1.62401i			0.0519036i
$980$		0			0
$981$	13.9046	−	2.54626i	0.443938	−	0.0812958i
$982$		0			0
$983$	25.7168			0.820238			0.410119	−	0.912032i	$-0.365487\pi$
$983$	25.7168			0.820238			0.410119	+	0.912032i	$0.365487\pi$
$984$		0			0
$985$	−21.3265			−0.679519
$986$		0			0
$987$	24.7309	−	29.6710i	0.787193	−	0.944437i
$988$		0			0
$989$	−42.9726			−1.36645
$990$		0			0
$991$		−	54.6611i		−	1.73637i	−0.496244	−	0.868183i	$-0.665288\pi$
$991$		−	54.6611i		−	1.73637i	0.496244	−	0.868183i	$-0.334712\pi$
$992$		0			0
$993$	−30.0750	−	25.0677i	−0.954401	−	0.795498i
$994$		0			0
$995$	−3.62204			−0.114826
$996$		0			0
$997$	−3.21334			−0.101768			−0.0508838	−	0.998705i	$-0.516204\pi$
$997$	−3.21334			−0.101768			−0.0508838	+	0.998705i	$0.516204\pi$
$998$		0			0
$999$	−6.97819	−	3.93748i	−0.220780	−	0.124576i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.b.401.14	yes	46
3.2	odd	2		4020.2.f.a.401.34	yes	46
67.66	odd	2		4020.2.f.a.401.33	✓	46
201.200	even	2	inner	4020.2.f.b.401.13	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.33	✓	46	67.66	odd	2
4020.2.f.a.401.34	yes	46	3.2	odd	2
4020.2.f.b.401.13	yes	46	201.200	even	2	inner
4020.2.f.b.401.14	yes	46	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(-1.10897 + 1.33049i) q^{3} +1.00000 q^{5} -2.36440i q^{7} +(-0.540386 - 2.95093i) q^{9} +O(q^{10})\)	\(q+(-1.10897 + 1.33049i) q^{3} +1.00000 q^{5} -2.36440i q^{7} +(-0.540386 - 2.95093i) q^{9} -0.927662 q^{11} -1.43383i q^{13} +(-1.10897 + 1.33049i) q^{15} -5.72886i q^{17} -1.85976 q^{19} +(3.14580 + 2.62204i) q^{21} +7.68356i q^{23} +1.00000 q^{25} +(4.52544 + 2.55351i) q^{27} -7.94643i q^{29} +0.732407i q^{31} +(1.02875 - 1.23424i) q^{33} -2.36440i q^{35} -1.54199 q^{37} +(1.90769 + 1.59007i) q^{39} -6.32704 q^{41} +5.59280i q^{43} +(-0.540386 - 2.95093i) q^{45} -9.43194i q^{47} +1.40963 q^{49} +(7.62216 + 6.35311i) q^{51} -12.1843 q^{53} -0.927662 q^{55} +(2.06241 - 2.47438i) q^{57} +6.47969i q^{59} +13.3752i q^{61} +(-6.97716 + 1.27769i) q^{63} -1.43383i q^{65} +(-7.02590 - 4.19961i) q^{67} +(-10.2229 - 8.52081i) q^{69} -5.24005i q^{71} -3.58783 q^{73} +(-1.10897 + 1.33049i) q^{75} +2.19336i q^{77} +12.6561i q^{79} +(-8.41597 + 3.18928i) q^{81} +4.64638i q^{83} -5.72886i q^{85} +(10.5726 + 8.81232i) q^{87} -1.75065i q^{89} -3.39015 q^{91} +(-0.974457 - 0.812214i) q^{93} -1.85976 q^{95} +12.2273i q^{97} +(0.501296 + 2.73746i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 46 q^{5} - 4 q^{9}+O(q^{10}) \) 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

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