LMFDB - Embedded newform 4020.2.f.b.401.24

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			401.24
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.b.401.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0844914 + 1.72999i) q^{3} +1.00000 q^{5} -1.02252i q^{7} +(-2.98572 - 0.292338i) q^{9} +O(q^{10})$	\(q+(-0.0844914 + 1.72999i) q^{3} +1.00000 q^{5} -1.02252i q^{7} +(-2.98572 - 0.292338i) q^{9} +5.03614 q^{11} +5.07170i q^{13} +(-0.0844914 + 1.72999i) q^{15} +3.80635i q^{17} -7.38561 q^{19} +(1.76895 + 0.0863945i) q^{21} -4.35385i q^{23} +1.00000 q^{25} +(0.758010 - 5.14057i) q^{27} +3.16752i q^{29} +4.85659i q^{31} +(-0.425510 + 8.71246i) q^{33} -1.02252i q^{35} +6.59186 q^{37} +(-8.77398 - 0.428515i) q^{39} -0.222957 q^{41} -2.43572i q^{43} +(-2.98572 - 0.292338i) q^{45} -0.209803i q^{47} +5.95445 q^{49} +(-6.58494 - 0.321604i) q^{51} -14.0405 q^{53} +5.03614 q^{55} +(0.624021 - 12.7770i) q^{57} +12.5080i q^{59} +8.58322i q^{61} +(-0.298923 + 3.05297i) q^{63} +5.07170i q^{65} +(4.63798 - 6.74456i) q^{67} +(7.53211 + 0.367863i) q^{69} -6.87699i q^{71} -16.1841 q^{73} +(-0.0844914 + 1.72999i) q^{75} -5.14957i q^{77} +2.75949i q^{79} +(8.82908 + 1.74568i) q^{81} +15.6526i q^{83} +3.80635i q^{85} +(-5.47978 - 0.267629i) q^{87} +3.59918i q^{89} +5.18593 q^{91} +(-8.40185 - 0.410340i) q^{93} -7.38561 q^{95} -0.794799i q^{97} +(-15.0365 - 1.47226i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.0844914	+	1.72999i	−0.0487812	+	0.998809i
$3$	−0.0844914	+	1.72999i	−0.0487812	+	0.998809i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	1.02252i		−	0.386478i	−0.981152	−	0.193239i	$-0.938101\pi$
$7$		−	1.02252i		−	0.386478i	0.981152	−	0.193239i	$-0.0618992\pi$
$8$		0			0
$9$	−2.98572	−	0.292338i	−0.995241	−	0.0974462i
$10$		0			0
$11$	5.03614			1.51845			0.759226	−	0.650827i	$-0.225578\pi$
$11$	5.03614			1.51845			0.759226	+	0.650827i	$0.225578\pi$
$12$		0			0
$13$			5.07170i			1.40664i	0.710875	+	0.703318i	$0.248299\pi$
$13$			5.07170i			1.40664i	−0.710875	+	0.703318i	$0.751701\pi$
$14$		0			0
$15$	−0.0844914	+	1.72999i	−0.0218156	+	0.446681i
$16$		0			0
$17$			3.80635i			0.923174i	0.887095	+	0.461587i	$0.152720\pi$
$17$			3.80635i			0.923174i	−0.887095	+	0.461587i	$0.847280\pi$
$18$		0			0
$19$	−7.38561			−1.69438			−0.847188	−	0.531293i	$-0.821706\pi$
$19$	−7.38561			−1.69438			−0.847188	+	0.531293i	$0.821706\pi$
$20$		0			0
$21$	1.76895	+	0.0863945i	0.386018	+	0.0188528i
$22$		0			0
$23$		−	4.35385i		−	0.907840i	−0.891042	−	0.453920i	$-0.850025\pi$
$23$		−	4.35385i		−	0.907840i	0.891042	−	0.453920i	$-0.149975\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	0.758010	−	5.14057i	0.145879	−	0.989302i
$28$		0			0
$29$			3.16752i			0.588195i	0.955775	+	0.294097i	$0.0950190\pi$
$29$			3.16752i			0.588195i	−0.955775	+	0.294097i	$0.904981\pi$
$30$		0			0
$31$			4.85659i			0.872270i	0.899881	+	0.436135i	$0.143653\pi$
$31$			4.85659i			0.872270i	−0.899881	+	0.436135i	$0.856347\pi$
$32$		0			0
$33$	−0.425510	+	8.71246i	−0.0740719	+	1.51664i
$34$		0			0
$35$		−	1.02252i		−	0.172838i
$36$		0			0
$37$	6.59186			1.08369			0.541847	−	0.840477i	$-0.317725\pi$
$37$	6.59186			1.08369			0.541847	+	0.840477i	$0.317725\pi$
$38$		0			0
$39$	−8.77398	−	0.428515i	−1.40496	−	0.0686173i
$40$		0			0
$41$	−0.222957			−0.0348201			−0.0174100	−	0.999848i	$-0.505542\pi$
$41$	−0.222957			−0.0348201			−0.0174100	+	0.999848i	$0.505542\pi$
$42$		0			0
$43$		−	2.43572i		−	0.371444i	−0.982602	−	0.185722i	$-0.940538\pi$
$43$		−	2.43572i		−	0.371444i	0.982602	−	0.185722i	$-0.0594623\pi$
$44$		0			0
$45$	−2.98572	−	0.292338i	−0.445085	−	0.0435792i
$46$		0			0
$47$		−	0.209803i		−	0.0306029i	−0.999883	−	0.0153015i	$-0.995129\pi$
$47$		−	0.209803i		−	0.0306029i	0.999883	−	0.0153015i	$-0.00487080\pi$
$48$		0			0
$49$	5.95445			0.850635
$50$		0			0
$51$	−6.58494	−	0.321604i	−0.922075	−	0.0450335i
$52$		0			0
$53$	−14.0405			−1.92861			−0.964305	−	0.264795i	$-0.914696\pi$
$53$	−14.0405			−1.92861			−0.964305	+	0.264795i	$0.914696\pi$
$54$		0			0
$55$	5.03614			0.679073
$56$		0			0
$57$	0.624021	−	12.7770i	0.0826536	−	1.69236i
$58$		0			0
$59$			12.5080i			1.62840i	0.580584	+	0.814201i	$0.302824\pi$
$59$			12.5080i			1.62840i	−0.580584	+	0.814201i	$0.697176\pi$
$60$		0			0
$61$			8.58322i			1.09897i	0.835504	+	0.549485i	$0.185176\pi$
$61$			8.58322i			1.09897i	−0.835504	+	0.549485i	$0.814824\pi$
$62$		0			0
$63$	−0.298923	+	3.05297i	−0.0376608	+	0.384638i
$64$		0			0
$65$			5.07170i			0.629067i
$66$		0			0
$67$	4.63798	−	6.74456i	0.566620	−	0.823979i
$67$	4.63798	−	6.74456i	0.566620	−	0.823979i
$68$		0			0
$69$	7.53211	+	0.367863i	0.906759	+	0.0442855i
$70$		0			0
$71$		−	6.87699i		−	0.816149i	−0.912949	−	0.408074i	$-0.866200\pi$
$71$		−	6.87699i		−	0.816149i	0.912949	−	0.408074i	$-0.133800\pi$
$72$		0			0
$73$	−16.1841			−1.89421			−0.947106	−	0.320922i	$-0.896007\pi$
$73$	−16.1841			−1.89421			−0.947106	+	0.320922i	$0.896007\pi$
$74$		0			0
$75$	−0.0844914	+	1.72999i	−0.00975623	+	0.199762i
$76$		0			0
$77$		−	5.14957i		−	0.586848i
$78$		0			0
$79$			2.75949i			0.310467i	0.987878	+	0.155234i	$0.0496131\pi$
$79$			2.75949i			0.310467i	−0.987878	+	0.155234i	$0.950387\pi$
$80$		0			0
$81$	8.82908	+	1.74568i	0.981008	+	0.193965i
$82$		0			0
$83$			15.6526i			1.71810i	0.511896	+	0.859048i	$0.328943\pi$
$83$			15.6526i			1.71810i	−0.511896	+	0.859048i	$0.671057\pi$
$84$		0			0
$85$			3.80635i			0.412856i
$86$		0			0
$87$	−5.47978	−	0.267629i	−0.587494	−	0.0286928i
$88$		0			0
$89$			3.59918i			0.381512i	0.981637	+	0.190756i	$0.0610940\pi$
$89$			3.59918i			0.381512i	−0.981637	+	0.190756i	$0.938906\pi$
$90$		0			0
$91$	5.18593			0.543633
$92$		0			0
$93$	−8.40185	−	0.410340i	−0.871231	−	0.0425503i
$94$		0			0
$95$	−7.38561			−0.757748
$96$		0			0
$97$		−	0.794799i		−	0.0806996i	−0.999186	−	0.0403498i	$-0.987153\pi$
$97$		−	0.794799i		−	0.0806996i	0.999186	−	0.0403498i	$-0.0128472\pi$
$98$		0			0
$99$	−15.0365	−	1.47226i	−1.51123	−	0.147967i
$100$		0			0
$101$	5.39557			0.536879			0.268440	−	0.963297i	$-0.413492\pi$
$101$	5.39557			0.536879			0.268440	+	0.963297i	$0.413492\pi$
$102$		0			0
$103$	14.3994			1.41882			0.709409	−	0.704797i	$-0.248962\pi$
$103$	14.3994			1.41882			0.709409	+	0.704797i	$0.248962\pi$
$104$		0			0
$105$	1.76895	+	0.0863945i	0.172632	+	0.00843124i
$106$		0			0
$107$			6.28415i			0.607512i	0.952750	+	0.303756i	$0.0982407\pi$
$107$			6.28415i			0.607512i	−0.952750	+	0.303756i	$0.901759\pi$
$108$		0			0
$109$		−	14.4703i		−	1.38600i	−0.720937	−	0.693001i	$-0.756288\pi$
$109$		−	14.4703i		−	1.38600i	0.720937	−	0.693001i	$-0.243712\pi$
$110$		0			0
$111$	−0.556956	+	11.4038i	−0.0528639	+	1.08240i
$112$		0			0
$113$	−16.5460			−1.55651			−0.778257	−	0.627946i	$-0.783896\pi$
$113$	−16.5460			−1.55651			−0.778257	+	0.627946i	$0.783896\pi$
$114$		0			0
$115$		−	4.35385i		−	0.405998i
$116$		0			0
$117$	1.48265	−	15.1427i	0.137071	−	1.39994i
$118$		0			0
$119$	3.89208			0.356786
$120$		0			0
$121$	14.3627			1.30570
$122$		0			0
$123$	0.0188380	−	0.385713i	0.00169856	−	0.0347786i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	0.0559039			0.00496067			0.00248034	−	0.999997i	$-0.499210\pi$
$127$	0.0559039			0.00496067			0.00248034	+	0.999997i	$0.499210\pi$
$128$		0			0
$129$	4.21377	+	0.205797i	0.371001	+	0.0181195i
$130$		0			0
$131$			11.6652i			1.01920i	0.860412	+	0.509599i	$0.170206\pi$
$131$			11.6652i			1.01920i	−0.860412	+	0.509599i	$0.829794\pi$
$132$		0			0
$133$			7.55197i			0.654839i
$134$		0			0
$135$	0.758010	−	5.14057i	0.0652391	−	0.442429i
$136$		0			0
$137$	11.3166			0.966846			0.483423	−	0.875387i	$-0.339393\pi$
$137$	11.3166			0.966846			0.483423	+	0.875387i	$0.339393\pi$
$138$		0			0
$139$			4.40701i			0.373798i	0.982379	+	0.186899i	$0.0598437\pi$
$139$			4.40701i			0.373798i	−0.982379	+	0.186899i	$0.940156\pi$
$140$		0			0
$141$	0.362957	+	0.0177266i	0.0305665	+	0.00149285i
$142$		0			0
$143$			25.5418i			2.13591i
$144$		0			0
$145$			3.16752i			0.263049i
$146$		0			0
$147$	−0.503100	+	10.3011i	−0.0414950	+	0.849622i
$148$		0			0
$149$			21.9536i			1.79851i	0.437425	+	0.899255i	$0.355890\pi$
$149$			21.9536i			1.79851i	−0.437425	+	0.899255i	$0.644110\pi$
$150$		0			0
$151$	−0.311270			−0.0253308			−0.0126654	−	0.999920i	$-0.504032\pi$
$151$	−0.311270			−0.0253308			−0.0126654	+	0.999920i	$0.504032\pi$
$152$		0			0
$153$	1.11274	−	11.3647i	0.0899598	−	0.918781i
$154$		0			0
$155$			4.85659i			0.390091i
$156$		0			0
$157$	−18.2395			−1.45567			−0.727833	−	0.685754i	$-0.759473\pi$
$157$	−18.2395			−1.45567			−0.727833	+	0.685754i	$0.759473\pi$
$158$		0			0
$159$	1.18630	−	24.2899i	0.0940798	−	1.92631i
$160$		0			0
$161$	−4.45191			−0.350860
$162$		0			0
$163$	−16.1768			−1.26706			−0.633532	−	0.773716i	$-0.718396\pi$
$163$	−16.1768			−1.26706			−0.633532	+	0.773716i	$0.718396\pi$
$164$		0			0
$165$	−0.425510	+	8.71246i	−0.0331259	+	0.678264i
$166$		0			0
$167$			19.8346i			1.53484i	0.641143	+	0.767422i	$0.278461\pi$
$167$			19.8346i			1.53484i	−0.641143	+	0.767422i	$0.721539\pi$
$168$		0			0
$169$	−12.7221			−0.978626
$170$		0			0
$171$	22.0514	+	2.15910i	1.68631	+	0.165110i
$172$		0			0
$173$		−	4.05348i		−	0.308181i	−0.988057	−	0.154090i	$-0.950755\pi$
$173$		−	4.05348i		−	0.308181i	0.988057	−	0.154090i	$-0.0492447\pi$
$174$		0			0
$175$		−	1.02252i		−	0.0772955i
$176$		0			0
$177$	−21.6387	−	1.05682i	−1.62646	−	0.0794353i
$178$		0			0
$179$	16.6434			1.24399			0.621994	−	0.783022i	$-0.286323\pi$
$179$	16.6434			1.24399			0.621994	+	0.783022i	$0.286323\pi$
$180$		0			0
$181$	−6.72887			−0.500153			−0.250077	−	0.968226i	$-0.580456\pi$
$181$	−6.72887			−0.500153			−0.250077	+	0.968226i	$0.580456\pi$
$182$		0			0
$183$	−14.8489	−	0.725209i	−1.09766	−	0.0536090i
$184$		0			0
$185$	6.59186			0.484643
$186$		0			0
$187$			19.1693i			1.40180i
$188$		0			0
$189$	−5.25635	−	0.775084i	−0.382343	−	0.0563790i
$190$		0			0
$191$	7.32069			0.529706			0.264853	−	0.964289i	$-0.414676\pi$
$191$	7.32069			0.529706			0.264853	+	0.964289i	$0.414676\pi$
$192$		0			0
$193$	14.7105			1.05889			0.529443	−	0.848345i	$-0.322401\pi$
$193$	14.7105			1.05889			0.529443	+	0.848345i	$0.322401\pi$
$194$		0			0
$195$	−8.77398	−	0.428515i	−0.628318	−	0.0306866i
$196$		0			0
$197$	−20.4668			−1.45820			−0.729100	−	0.684407i	$-0.760061\pi$
$197$	−20.4668			−1.45820			−0.729100	+	0.684407i	$0.760061\pi$
$198$		0			0
$199$	13.8416			0.981205			0.490602	−	0.871384i	$-0.336777\pi$
$199$	13.8416			0.981205			0.490602	+	0.871384i	$0.336777\pi$
$200$		0			0
$201$	11.2761	+	8.59352i	0.795358	+	0.606140i
$202$		0			0
$203$	3.23887			0.227324
$204$		0			0
$205$	−0.222957			−0.0155720
$206$		0			0
$207$	−1.27280	+	12.9994i	−0.0884655	+	0.903520i
$208$		0			0
$209$	−37.1950			−2.57283
$210$		0			0
$211$	−13.7619			−0.947406			−0.473703	−	0.880685i	$-0.657083\pi$
$211$	−13.7619			−0.947406			−0.473703	+	0.880685i	$0.657083\pi$
$212$		0			0
$213$	11.8971	+	0.581047i	0.815177	+	0.0398127i
$214$		0			0
$215$		−	2.43572i		−	0.166115i
$216$		0			0
$217$	4.96598			0.337113
$218$		0			0
$219$	1.36742	−	27.9984i	0.0924018	−	1.89196i
$220$		0			0
$221$	−19.3046			−1.29857
$222$		0			0
$223$	7.73704			0.518111			0.259055	−	0.965862i	$-0.416589\pi$
$223$	7.73704			0.518111			0.259055	+	0.965862i	$0.416589\pi$
$224$		0			0
$225$	−2.98572	−	0.292338i	−0.199048	−	0.0194892i
$226$		0			0
$227$		−	11.9698i		−	0.794463i	−0.917718	−	0.397231i	$-0.869971\pi$
$227$		−	11.9698i		−	0.794463i	0.917718	−	0.397231i	$-0.130029\pi$
$228$		0			0
$229$			15.3185i			1.01227i	0.862454	+	0.506136i	$0.168927\pi$
$229$			15.3185i			1.01227i	−0.862454	+	0.506136i	$0.831073\pi$
$230$		0			0
$231$	8.90870	+	0.435095i	0.586149	+	0.0286271i
$232$		0			0
$233$	−5.48218			−0.359150			−0.179575	−	0.983744i	$-0.557472\pi$
$233$	−5.48218			−0.359150			−0.179575	+	0.983744i	$0.557472\pi$
$234$		0			0
$235$		−	0.209803i		−	0.0136860i
$236$		0			0
$237$	−4.77390	−	0.233154i	−0.310098	−	0.0151450i
$238$		0			0
$239$	−11.4914			−0.743314			−0.371657	−	0.928370i	$-0.621210\pi$
$239$	−11.4914			−0.743314			−0.371657	+	0.928370i	$0.621210\pi$
$240$		0			0
$241$	−8.87565			−0.571731			−0.285866	−	0.958270i	$-0.592281\pi$
$241$	−8.87565			−0.571731			−0.285866	+	0.958270i	$0.592281\pi$
$242$		0			0
$243$	−3.76599	+	15.1267i	−0.241589	+	0.970379i
$244$		0			0
$245$	5.95445			0.380416
$246$		0			0
$247$		−	37.4576i		−	2.38337i
$248$		0			0
$249$	−27.0788	−	1.32251i	−1.71605	−	0.0838107i
$250$		0			0
$251$	−1.37120			−0.0865493			−0.0432746	−	0.999063i	$-0.513779\pi$
$251$	−1.37120			−0.0865493			−0.0432746	+	0.999063i	$0.513779\pi$
$252$		0			0
$253$		−	21.9266i		−	1.37851i
$254$		0			0
$255$	−6.58494	−	0.321604i	−0.412365	−	0.0201396i
$256$		0			0
$257$		−	19.0264i		−	1.18683i	−0.804895	−	0.593417i	$-0.797779\pi$
$257$		−	19.0264i		−	1.18683i	0.804895	−	0.593417i	$-0.202221\pi$
$258$		0			0
$259$		−	6.74033i		−	0.418824i
$260$		0			0
$261$	0.925989	−	9.45735i	0.0573173	−	0.585395i
$262$		0			0
$263$		−	25.6694i		−	1.58284i	−0.611270	−	0.791422i	$-0.709341\pi$
$263$		−	25.6694i		−	1.58284i	0.611270	−	0.791422i	$-0.290659\pi$
$264$		0			0
$265$	−14.0405			−0.862500
$266$		0			0
$267$	−6.22654	−	0.304100i	−0.381058	−	0.0186106i
$268$		0			0
$269$		−	14.9952i		−	0.914274i	−0.889396	−	0.457137i	$-0.848875\pi$
$269$		−	14.9952i		−	0.914274i	0.889396	−	0.457137i	$-0.151125\pi$
$270$		0			0
$271$		−	18.6314i		−	1.13177i	−0.824483	−	0.565887i	$-0.808534\pi$
$271$		−	18.6314i		−	1.13177i	0.824483	−	0.565887i	$-0.191466\pi$
$272$		0			0
$273$	−0.438167	+	8.97161i	−0.0265191	+	0.542986i
$274$		0			0
$275$	5.03614			0.303690
$276$		0			0
$277$	−16.9552			−1.01874			−0.509370	−	0.860548i	$-0.670122\pi$
$277$	−16.9552			−1.01874			−0.509370	+	0.860548i	$0.670122\pi$
$278$		0			0
$279$	1.41977	−	14.5004i	0.0849993	−	0.868118i
$280$		0			0
$281$	−6.82588			−0.407198			−0.203599	−	0.979054i	$-0.565264\pi$
$281$	−6.82588			−0.407198			−0.203599	+	0.979054i	$0.565264\pi$
$282$		0			0
$283$	−6.49627			−0.386163			−0.193081	−	0.981183i	$-0.561848\pi$
$283$	−6.49627			−0.386163			−0.193081	+	0.981183i	$0.561848\pi$
$284$		0			0
$285$	0.624021	−	12.7770i	0.0369638	−	0.756846i
$286$		0			0
$287$			0.227979i			0.0134572i
$288$		0			0
$289$	2.51173			0.147749
$290$		0			0
$291$	1.37499	+	0.0671537i	0.0806036	+	0.00393662i
$292$		0			0
$293$			20.4202i			1.19296i	0.802627	+	0.596481i	$0.203435\pi$
$293$			20.4202i			1.19296i	−0.802627	+	0.596481i	$0.796565\pi$
$294$		0			0
$295$			12.5080i			0.728243i
$296$		0			0
$297$	3.81744	−	25.8886i	0.221511	−	1.50221i
$298$		0			0
$299$	22.0814			1.27700
$300$		0			0
$301$	−2.49058			−0.143555
$302$		0			0
$303$	−0.455879	+	9.33427i	−0.0261896	+	0.536240i
$304$		0			0
$305$			8.58322i			0.491474i
$306$		0			0
$307$	4.83495			0.275945			0.137973	−	0.990436i	$-0.455941\pi$
$307$	4.83495			0.275945			0.137973	+	0.990436i	$0.455941\pi$
$308$		0			0
$309$	−1.21663	+	24.9108i	−0.0692115	+	1.41713i
$310$		0			0
$311$	−12.4652			−0.706837			−0.353419	−	0.935465i	$-0.614981\pi$
$311$	−12.4652			−0.706837			−0.353419	+	0.935465i	$0.614981\pi$
$312$		0			0
$313$		−	1.24688i		−	0.0704780i	−0.999379	−	0.0352390i	$-0.988781\pi$
$313$		−	1.24688i		−	0.0704780i	0.999379	−	0.0352390i	$-0.0112192\pi$
$314$		0			0
$315$	−0.298923	+	3.05297i	−0.0168424	+	0.172015i
$316$		0			0
$317$		−	5.44180i		−	0.305642i	−0.988254	−	0.152821i	$-0.951164\pi$
$317$		−	5.44180i		−	0.305642i	0.988254	−	0.152821i	$-0.0488357\pi$
$318$		0			0
$319$			15.9521i			0.893145i
$320$		0			0
$321$	−10.8715	−	0.530957i	−0.606788	−	0.0296351i
$322$		0			0
$323$		−	28.1122i		−	1.56420i
$324$		0			0
$325$			5.07170i			0.281327i
$326$		0			0
$327$	25.0334	+	1.22261i	1.38435	+	0.0676108i
$328$		0			0
$329$	−0.214529			−0.0118273
$330$		0			0
$331$		−	19.6678i		−	1.08104i	−0.841331	−	0.540520i	$-0.818228\pi$
$331$		−	19.6678i		−	1.08104i	0.841331	−	0.540520i	$-0.181772\pi$
$332$		0			0
$333$	−19.6815	−	1.92705i	−1.07854	−	0.105602i
$334$		0			0
$335$	4.63798	−	6.74456i	0.253400	−	0.368495i
$336$		0			0
$337$			20.5957i			1.12192i	0.827844	+	0.560959i	$0.189567\pi$
$337$			20.5957i			1.12192i	−0.827844	+	0.560959i	$0.810433\pi$
$338$		0			0
$339$	1.39799	−	28.6244i	0.0759286	−	1.55466i
$340$		0			0
$341$			24.4585i			1.32450i
$342$		0			0
$343$		−	13.2462i		−	0.715229i
$344$		0			0
$345$	7.53211	+	0.367863i	0.405515	+	0.0198051i
$346$		0			0
$347$	−8.42236			−0.452136			−0.226068	−	0.974112i	$-0.572587\pi$
$347$	−8.42236			−0.452136			−0.226068	+	0.974112i	$0.572587\pi$
$348$		0			0
$349$	26.9726			1.44381			0.721904	−	0.691993i	$-0.243267\pi$
$349$	26.9726			1.44381			0.721904	+	0.691993i	$0.243267\pi$
$350$		0			0
$351$	26.0714	+	3.84440i	1.39159	+	0.205199i
$352$		0			0
$353$	−7.94737			−0.422996			−0.211498	−	0.977378i	$-0.567834\pi$
$353$	−7.94737			−0.422996			−0.211498	+	0.977378i	$0.567834\pi$
$354$		0			0
$355$		−	6.87699i		−	0.364993i
$356$		0			0
$357$	−0.328847	+	6.73325i	−0.0174044	+	0.356362i
$358$		0			0
$359$			9.50959i			0.501897i	0.968000	+	0.250948i	$0.0807425\pi$
$359$			9.50959i			0.501897i	−0.968000	+	0.250948i	$0.919258\pi$
$360$		0			0
$361$	35.5473			1.87091
$362$		0			0
$363$	−1.21352	+	24.8473i	−0.0636934	+	1.30414i
$364$		0			0
$365$	−16.1841			−0.847117
$366$		0			0
$367$		−	1.38608i		−	0.0723529i	−0.999345	−	0.0361765i	$-0.988482\pi$
$367$		−	1.38608i		−	0.0723529i	0.999345	−	0.0361765i	$-0.0115178\pi$
$368$		0			0
$369$	0.665688	+	0.0651790i	0.0346543	+	0.00339308i
$370$		0			0
$371$			14.3567i			0.745364i
$372$		0			0
$373$			29.7932i			1.54263i	0.636451	+	0.771317i	$0.280402\pi$
$373$			29.7932i			1.54263i	−0.636451	+	0.771317i	$0.719598\pi$
$374$		0			0
$375$	−0.0844914	+	1.72999i	−0.00436312	+	0.0893362i
$376$		0			0
$377$	−16.0647			−0.827376
$378$		0			0
$379$			4.19417i			0.215440i	0.994181	+	0.107720i	$0.0343550\pi$
$379$			4.19417i			0.215440i	−0.994181	+	0.107720i	$0.965645\pi$
$380$		0			0
$381$	−0.00472340	+	0.0967132i	−0.000241987	+	0.00495477i
$382$		0			0
$383$	27.8936			1.42530			0.712649	−	0.701521i	$-0.247495\pi$
$383$	27.8936			1.42530			0.712649	+	0.701521i	$0.247495\pi$
$384$		0			0
$385$		−	5.14957i		−	0.262446i
$386$		0			0
$387$	−0.712054	+	7.27238i	−0.0361958	+	0.369676i
$388$		0			0
$389$		−	5.11261i		−	0.259219i	−0.991565	−	0.129610i	$-0.958628\pi$
$389$		−	5.11261i		−	0.259219i	0.991565	−	0.129610i	$-0.0413725\pi$
$390$		0			0
$391$	16.5723			0.838095
$392$		0			0
$393$	−20.1807	−	0.985613i	−1.01798	−	0.0497176i
$394$		0			0
$395$			2.75949i			0.138845i
$396$		0			0
$397$	−2.10591			−0.105693			−0.0528464	−	0.998603i	$-0.516829\pi$
$397$	−2.10591			−0.105693			−0.0528464	+	0.998603i	$0.516829\pi$
$398$		0			0
$399$	−13.0648	−	0.638077i	−0.654059	−	0.0319438i
$400$		0			0
$401$	−24.4194			−1.21944			−0.609722	−	0.792615i	$-0.708719\pi$
$401$	−24.4194			−1.21944			−0.609722	+	0.792615i	$0.708719\pi$
$402$		0			0
$403$	−24.6312			−1.22697
$404$		0			0
$405$	8.82908	+	1.74568i	0.438720	+	0.0867437i
$406$		0			0
$407$	33.1975			1.64554
$408$		0			0
$409$		−	36.2184i		−	1.79088i	−0.445178	−	0.895442i	$-0.646860\pi$
$409$		−	36.2184i		−	1.79088i	0.445178	−	0.895442i	$-0.353140\pi$
$410$		0			0
$411$	−0.956159	+	19.5777i	−0.0471639	+	0.965695i
$412$		0			0
$413$	12.7897			0.629341
$414$		0			0
$415$			15.6526i			0.768355i
$416$		0			0
$417$	−7.62408	−	0.372355i	−0.373353	−	0.0182343i
$418$		0			0
$419$			17.8459i			0.871829i	0.899988	+	0.435914i	$0.143575\pi$
$419$			17.8459i			0.871829i	−0.899988	+	0.435914i	$0.856425\pi$
$420$		0			0
$421$	17.0365			0.830311			0.415155	−	0.909751i	$-0.363727\pi$
$421$	17.0365			0.830311			0.415155	+	0.909751i	$0.363727\pi$
$422$		0			0
$423$	−0.0613335	+	0.626414i	−0.00298214	+	0.0304573i
$424$		0			0
$425$			3.80635i			0.184635i
$426$		0			0
$427$	8.77655			0.424727
$428$		0			0
$429$	−44.1870	−	2.15806i	−2.13337	−	0.104192i
$430$		0			0
$431$		−	37.6272i		−	1.81244i	−0.422809	−	0.906219i	$-0.638956\pi$
$431$		−	37.6272i		−	1.81244i	0.422809	−	0.906219i	$-0.361044\pi$
$432$		0			0
$433$		−	12.7843i		−	0.614376i	−0.951649	−	0.307188i	$-0.900612\pi$
$433$		−	12.7843i		−	0.614376i	0.951649	−	0.307188i	$-0.0993880\pi$
$434$		0			0
$435$	−5.47978	−	0.267629i	−0.262735	−	0.0128318i
$436$		0			0
$437$			32.1558i			1.53822i
$438$		0			0
$439$	−11.4580			−0.546861			−0.273431	−	0.961892i	$-0.588158\pi$
$439$	−11.4580			−0.546861			−0.273431	+	0.961892i	$0.588158\pi$
$440$		0			0
$441$	−17.7783	−	1.74071i	−0.846587	−	0.0828911i
$442$		0			0
$443$	29.2071			1.38767			0.693837	−	0.720132i	$-0.255919\pi$
$443$	29.2071			1.38767			0.693837	+	0.720132i	$0.255919\pi$
$444$		0			0
$445$			3.59918i			0.170617i
$446$		0			0
$447$	−37.9795	−	1.85489i	−1.79637	−	0.0877334i
$448$		0			0
$449$		−	8.52127i		−	0.402144i	−0.979577	−	0.201072i	$-0.935558\pi$
$449$		−	8.52127i		−	0.402144i	0.979577	−	0.201072i	$-0.0644424\pi$
$450$		0			0
$451$	−1.12284			−0.0528726
$452$		0			0
$453$	0.0262996	−	0.538493i	0.00123566	−	0.0253006i
$454$		0			0
$455$	5.18593			0.243120
$456$		0			0
$457$	−30.3282			−1.41869			−0.709347	−	0.704859i	$-0.751010\pi$
$457$	−30.3282			−1.41869			−0.709347	+	0.704859i	$0.751010\pi$
$458$		0			0
$459$	19.5668	+	2.88525i	0.913299	+	0.134672i
$460$		0			0
$461$			16.1416i			0.751788i	0.926663	+	0.375894i	$0.122664\pi$
$461$			16.1416i			0.751788i	−0.926663	+	0.375894i	$0.877336\pi$
$462$		0			0
$463$			25.3051i			1.17603i	0.808851	+	0.588013i	$0.200090\pi$
$463$			25.3051i			1.17603i	−0.808851	+	0.588013i	$0.799910\pi$
$464$		0			0
$465$	−8.40185	−	0.410340i	−0.389626	−	0.0190291i
$466$		0			0
$467$			32.4500i			1.50160i	0.660527	+	0.750802i	$0.270333\pi$
$467$			32.4500i			1.50160i	−0.660527	+	0.750802i	$0.729667\pi$
$468$		0			0
$469$	−6.89648	−	4.74245i	−0.318450	−	0.218986i
$470$		0			0
$471$	1.54108	−	31.5540i	0.0710091	−	1.45393i
$472$		0			0
$473$		−	12.2666i		−	0.564020i
$474$		0			0
$475$	−7.38561			−0.338875
$476$		0			0
$477$	41.9210	+	4.10457i	1.91943	+	0.187936i
$478$		0			0
$479$			2.50482i			0.114448i	0.998361	+	0.0572240i	$0.0182249\pi$
$479$			2.50482i			0.114448i	−0.998361	+	0.0572240i	$0.981775\pi$
$480$		0			0
$481$			33.4319i			1.52436i
$482$		0			0
$483$	0.376149	−	7.70176i	0.0171154	−	0.350442i
$484$		0			0
$485$		−	0.794799i		−	0.0360900i
$486$		0			0
$487$			2.86807i			0.129965i	0.997886	+	0.0649823i	$0.0206991\pi$
$487$			2.86807i			0.129965i	−0.997886	+	0.0649823i	$0.979301\pi$
$488$		0			0
$489$	1.36680	−	27.9857i	0.0618089	−	1.26556i
$490$		0			0
$491$			5.04444i			0.227652i	0.993501	+	0.113826i	$0.0363107\pi$
$491$			5.04444i			0.227652i	−0.993501	+	0.113826i	$0.963689\pi$
$492$		0			0
$493$	−12.0567			−0.543006
$494$		0			0
$495$	−15.0365	−	1.47226i	−0.675841	−	0.0661730i
$496$		0			0
$497$	−7.03189			−0.315423
$498$		0			0
$499$			19.5820i			0.876611i	0.898826	+	0.438306i	$0.144421\pi$
$499$			19.5820i			0.876611i	−0.898826	+	0.438306i	$0.855579\pi$
$500$		0			0
$501$	−34.3136	−	1.67585i	−1.53302	−	0.0748714i
$502$		0			0
$503$	−5.42635			−0.241949			−0.120975	−	0.992656i	$-0.538602\pi$
$503$	−5.42635			−0.241949			−0.120975	+	0.992656i	$0.538602\pi$
$504$		0			0
$505$	5.39557			0.240100
$506$		0			0
$507$	1.07491	−	22.0091i	0.0477385	−	0.977461i
$508$		0			0
$509$			37.3945i			1.65748i	0.559632	+	0.828741i	$0.310943\pi$
$509$			37.3945i			1.65748i	−0.559632	+	0.828741i	$0.689057\pi$
$510$		0			0
$511$			16.5487i			0.732070i
$512$		0			0
$513$	−5.59837	+	37.9662i	−0.247174	+	1.67625i
$514$		0			0
$515$	14.3994			0.634514
$516$		0			0
$517$		−	1.05660i		−	0.0464691i
$518$		0			0
$519$	7.01248	+	0.342485i	0.307814	+	0.0150334i
$520$		0			0
$521$	29.4984			1.29235			0.646174	−	0.763190i	$-0.276368\pi$
$521$	29.4984			1.29235			0.646174	+	0.763190i	$0.276368\pi$
$522$		0			0
$523$	−16.3490			−0.714891			−0.357445	−	0.933934i	$-0.616352\pi$
$523$	−16.3490			−0.714891			−0.357445	+	0.933934i	$0.616352\pi$
$524$		0			0
$525$	1.76895	+	0.0863945i	0.0772035	+	0.00377057i
$526$		0			0
$527$	−18.4859			−0.805257
$528$		0			0
$529$	4.04400			0.175826
$530$		0			0
$531$	3.65657	−	37.3454i	0.158681	−	1.62065i
$532$		0			0
$533$		−	1.13077i		−	0.0489792i
$534$		0			0
$535$			6.28415i			0.271688i
$536$		0			0
$537$	−1.40623	+	28.7929i	−0.0606832	+	1.24251i
$538$		0			0
$539$	29.9874			1.29165
$540$		0			0
$541$		−	17.5378i		−	0.754011i	−0.926211	−	0.377006i	$-0.876954\pi$
$541$		−	17.5378i		−	0.754011i	0.926211	−	0.377006i	$-0.123046\pi$
$542$		0			0
$543$	0.568532	−	11.6409i	0.0243981	−	0.499558i
$544$		0			0
$545$		−	14.4703i		−	0.619839i
$546$		0			0
$547$		−	30.1742i		−	1.29016i	−0.764116	−	0.645078i	$-0.776825\pi$
$547$		−	30.1742i		−	1.29016i	0.764116	−	0.645078i	$-0.223175\pi$
$548$		0			0
$549$	2.50921	−	25.6271i	0.107090	−	1.09374i
$550$		0			0
$551$		−	23.3941i		−	0.996623i
$552$		0			0
$553$	2.82165			0.119989
$554$		0			0
$555$	−0.556956	+	11.4038i	−0.0236414	+	0.484066i
$556$		0			0
$557$			37.7591i			1.59990i	0.600064	+	0.799952i	$0.295142\pi$
$557$			37.7591i			1.59990i	−0.600064	+	0.799952i	$0.704858\pi$
$558$		0			0
$559$	12.3532			0.522486
$560$		0			0
$561$	−33.1626	−	1.61964i	−1.40013	−	0.0683812i
$562$		0			0
$563$	32.5130			1.37026			0.685131	−	0.728420i	$-0.259745\pi$
$563$	32.5130			1.37026			0.685131	+	0.728420i	$0.259745\pi$
$564$		0			0
$565$	−16.5460			−0.696094
$566$		0			0
$567$	1.78500	−	9.02794i	0.0749631	−	0.379138i
$568$		0			0
$569$			10.5680i			0.443033i	0.975157	+	0.221516i	$0.0711006\pi$
$569$			10.5680i			0.443033i	−0.975157	+	0.221516i	$0.928899\pi$
$570$		0			0
$571$	21.2408			0.888902			0.444451	−	0.895803i	$-0.353399\pi$
$571$	21.2408			0.888902			0.444451	+	0.895803i	$0.353399\pi$
$572$		0			0
$573$	−0.618536	+	12.6647i	−0.0258397	+	0.529076i
$574$		0			0
$575$		−	4.35385i		−	0.181568i
$576$		0			0
$577$		−	39.0108i		−	1.62404i	−0.583630	−	0.812020i	$-0.698368\pi$
$577$		−	39.0108i		−	1.62404i	0.583630	−	0.812020i	$-0.301632\pi$
$578$		0			0
$579$	−1.24291	+	25.4490i	−0.0516537	+	1.05763i
$580$		0			0
$581$	16.0051			0.664005
$582$		0			0
$583$	−70.7098			−2.92850
$584$		0			0
$585$	1.48265	−	15.1427i	0.0613002	−	0.626073i
$586$		0			0
$587$	40.4758			1.67062			0.835308	−	0.549782i	$-0.185289\pi$
$587$	40.4758			1.67062			0.835308	+	0.549782i	$0.185289\pi$
$588$		0			0
$589$		−	35.8689i		−	1.47795i
$590$		0			0
$591$	1.72927	−	35.4074i	0.0711327	−	1.45646i
$592$		0			0
$593$	−16.0511			−0.659138			−0.329569	−	0.944132i	$-0.606903\pi$
$593$	−16.0511			−0.659138			−0.329569	+	0.944132i	$0.606903\pi$
$594$		0			0
$595$	3.89208			0.159560
$596$		0			0
$597$	−1.16950	+	23.9458i	−0.0478643	+	0.980037i
$598$		0			0
$599$	42.8316			1.75005			0.875026	−	0.484076i	$-0.160844\pi$
$599$	42.8316			1.75005			0.875026	+	0.484076i	$0.160844\pi$
$600$		0			0
$601$	36.0818			1.47181			0.735903	−	0.677087i	$-0.236758\pi$
$601$	36.0818			1.47181			0.735903	+	0.677087i	$0.236758\pi$
$602$		0			0
$603$	−15.8194	+	18.7815i	−0.644217	+	0.764843i
$604$		0			0
$605$	14.3627			0.583926
$606$		0			0
$607$	6.12800			0.248728			0.124364	−	0.992237i	$-0.460311\pi$
$607$	6.12800			0.248728			0.124364	+	0.992237i	$0.460311\pi$
$608$		0			0
$609$	−0.273657	+	5.60321i	−0.0110891	+	0.227053i
$610$		0			0
$611$	1.06406			0.0430472
$612$		0			0
$613$	9.82991			0.397026			0.198513	−	0.980098i	$-0.436389\pi$
$613$	9.82991			0.397026			0.198513	+	0.980098i	$0.436389\pi$
$614$		0			0
$615$	0.0188380	−	0.385713i	0.000759620	−	0.0155535i
$616$		0			0
$617$		−	16.0977i		−	0.648069i	−0.946045	−	0.324034i	$-0.894961\pi$
$617$		−	16.0977i		−	0.648069i	0.946045	−	0.324034i	$-0.105039\pi$
$618$		0			0
$619$	33.2120			1.33490			0.667451	−	0.744653i	$-0.267385\pi$
$619$	33.2120			1.33490			0.667451	+	0.744653i	$0.267385\pi$
$620$		0			0
$621$	−22.3812	−	3.30026i	−0.898128	−	0.132435i
$622$		0			0
$623$	3.68025			0.147446
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	3.14266	−	64.3469i	0.125506	−	2.56977i
$628$		0			0
$629$			25.0909i			1.00044i
$630$		0			0
$631$			14.5617i			0.579691i	0.957073	+	0.289845i	$0.0936039\pi$
$631$			14.5617i			0.579691i	−0.957073	+	0.289845i	$0.906396\pi$
$632$		0			0
$633$	1.16276	−	23.8079i	0.0462156	−	0.946278i
$634$		0			0
$635$	0.0559039			0.00221848
$636$		0			0
$637$			30.1992i			1.19653i
$638$		0			0
$639$	−2.01041	+	20.5328i	−0.0795305	+	0.812264i
$640$		0			0
$641$	38.4106			1.51713			0.758564	−	0.651598i	$-0.225901\pi$
$641$	38.4106			1.51713			0.758564	+	0.651598i	$0.225901\pi$
$642$		0			0
$643$	−2.77646			−0.109493			−0.0547464	−	0.998500i	$-0.517435\pi$
$643$	−2.77646			−0.109493			−0.0547464	+	0.998500i	$0.517435\pi$
$644$		0			0
$645$	4.21377	+	0.205797i	0.165917	+	0.00810327i
$646$		0			0
$647$	29.7944			1.17134			0.585669	−	0.810551i	$-0.300832\pi$
$647$	29.7944			1.17134			0.585669	+	0.810551i	$0.300832\pi$
$648$		0			0
$649$			62.9919i			2.47265i
$650$		0			0
$651$	−0.419583	+	8.59109i	−0.0164448	+	0.336711i
$652$		0			0
$653$	23.3478			0.913671			0.456836	−	0.889551i	$-0.348983\pi$
$653$	23.3478			0.913671			0.456836	+	0.889551i	$0.348983\pi$
$654$		0			0
$655$			11.6652i			0.455799i
$656$		0			0
$657$	48.3214	+	4.73125i	1.88520	+	0.184584i
$658$		0			0
$659$			24.1904i			0.942322i	0.882047	+	0.471161i	$0.156165\pi$
$659$			24.1904i			0.942322i	−0.882047	+	0.471161i	$0.843835\pi$
$660$		0			0
$661$			2.96021i			0.115139i	0.998342	+	0.0575694i	$0.0183350\pi$
$661$			2.96021i			0.115139i	−0.998342	+	0.0575694i	$0.981665\pi$
$662$		0			0
$663$	1.63108	−	33.3968i	0.0633458	−	1.29702i
$664$		0			0
$665$			7.55197i			0.292853i
$666$		0			0
$667$	13.7909			0.533987
$668$		0			0
$669$	−0.653714	+	13.3850i	−0.0252740	+	0.517494i
$670$		0			0
$671$			43.2263i			1.66873i
$672$		0			0
$673$		−	18.2250i		−	0.702521i	−0.936278	−	0.351261i	$-0.885753\pi$
$673$		−	18.2250i		−	0.702521i	0.936278	−	0.351261i	$-0.114247\pi$
$674$		0			0
$675$	0.758010	−	5.14057i	0.0291758	−	0.197860i
$676$		0			0
$677$	11.9254			0.458328			0.229164	−	0.973388i	$-0.426401\pi$
$677$	11.9254			0.458328			0.229164	+	0.973388i	$0.426401\pi$
$678$		0			0
$679$	−0.812701			−0.0311886
$680$		0			0
$681$	20.7076	+	1.01134i	0.793517	+	0.0387548i
$682$		0			0
$683$	2.23545			0.0855372			0.0427686	−	0.999085i	$-0.486382\pi$
$683$	2.23545			0.0855372			0.0427686	+	0.999085i	$0.486382\pi$
$684$		0			0
$685$	11.3166			0.432387
$686$		0			0
$687$	−26.5007	−	1.29428i	−1.01107	−	0.0493798i
$688$		0			0
$689$		−	71.2091i		−	2.71285i
$690$		0			0
$691$	8.32720			0.316782			0.158391	−	0.987377i	$-0.449369\pi$
$691$	8.32720			0.316782			0.158391	+	0.987377i	$0.449369\pi$
$692$		0			0
$693$	−1.50542	+	15.3752i	−0.0571861	+	0.584055i
$694$		0			0
$695$			4.40701i			0.167167i
$696$		0			0
$697$		−	0.848652i		−	0.0321450i
$698$		0			0
$699$	0.463197	−	9.48411i	0.0175197	−	0.358722i
$700$		0			0
$701$	4.81177			0.181738			0.0908691	−	0.995863i	$-0.471036\pi$
$701$	4.81177			0.181738			0.0908691	+	0.995863i	$0.471036\pi$
$702$		0			0
$703$	−48.6849			−1.83619
$704$		0			0
$705$	0.362957	+	0.0177266i	0.0136698	+	0.000667621i
$706$		0			0
$707$		−	5.51710i		−	0.207492i
$708$		0			0
$709$	30.0462			1.12841			0.564204	−	0.825635i	$-0.309183\pi$
$709$	30.0462			1.12841			0.564204	+	0.825635i	$0.309183\pi$
$710$		0			0
$711$	0.806707	−	8.23909i	0.0302539	−	0.308990i
$712$		0			0
$713$	21.1449			0.791882
$714$		0			0
$715$			25.5418i			0.955208i
$716$		0			0
$717$	0.970922	−	19.8799i	0.0362597	−	0.742429i
$718$		0			0
$719$		−	34.0242i		−	1.26889i	−0.772968	−	0.634445i	$-0.781229\pi$
$719$		−	34.0242i		−	1.26889i	0.772968	−	0.634445i	$-0.218771\pi$
$720$		0			0
$721$		−	14.7238i		−	0.548341i
$722$		0			0
$723$	0.749917	−	15.3548i	0.0278897	−	0.571050i
$724$		0			0
$725$			3.16752i			0.117639i
$726$		0			0
$727$		−	26.7459i		−	0.991949i	−0.868337	−	0.495974i	$-0.834811\pi$
$727$		−	26.7459i		−	0.991949i	0.868337	−	0.495974i	$-0.165189\pi$
$728$		0			0
$729$	−25.8508	−	7.79320i	−0.957439	−	0.288637i
$730$		0			0
$731$	9.27119			0.342907
$732$		0			0
$733$		−	21.3959i		−	0.790275i	−0.918622	−	0.395137i	$-0.870697\pi$
$733$		−	21.3959i		−	0.790275i	0.918622	−	0.395137i	$-0.129303\pi$
$734$		0			0
$735$	−0.503100	+	10.3011i	−0.0185571	+	0.379963i
$736$		0			0
$737$	23.3575	−	33.9665i	0.860385	−	1.25117i
$738$		0			0
$739$			26.0921i			0.959812i	0.877320	+	0.479906i	$0.159329\pi$
$739$			26.0921i			0.959812i	−0.877320	+	0.479906i	$0.840671\pi$
$740$		0			0
$741$	64.8013	+	3.16485i	2.38053	+	0.116264i
$742$		0			0
$743$			4.51665i			0.165700i	0.996562	+	0.0828500i	$0.0264022\pi$
$743$			4.51665i			0.165700i	−0.996562	+	0.0828500i	$0.973598\pi$
$744$		0			0
$745$			21.9536i			0.804318i
$746$		0			0
$747$	4.57585	−	46.7343i	0.167422	−	1.70992i
$748$		0			0
$749$	6.42569			0.234790
$750$		0			0
$751$	35.5260			1.29636			0.648181	−	0.761487i	$-0.275530\pi$
$751$	35.5260			1.29636			0.648181	+	0.761487i	$0.275530\pi$
$752$		0			0
$753$	0.115855	−	2.37216i	0.00422197	−	0.0864462i
$754$		0			0
$755$	−0.311270			−0.0113283
$756$		0			0
$757$		−	17.6395i		−	0.641119i	−0.947228	−	0.320560i	$-0.896129\pi$
$757$		−	17.6395i		−	0.641119i	0.947228	−	0.320560i	$-0.103871\pi$
$758$		0			0
$759$	37.9327	+	1.85261i	1.37687	+	0.0672454i
$760$		0			0
$761$		−	9.88549i		−	0.358349i	−0.983817	−	0.179174i	$-0.942657\pi$
$761$		−	9.88549i		−	0.358349i	0.983817	−	0.179174i	$-0.0573426\pi$
$762$		0			0
$763$	−14.7962			−0.535659
$764$		0			0
$765$	1.11274	−	11.3647i	0.0402312	−	0.410891i
$766$		0			0
$767$	−63.4367			−2.29057
$768$		0			0
$769$		−	26.7560i		−	0.964847i	−0.875938	−	0.482424i	$-0.839757\pi$
$769$		−	26.7560i		−	0.964847i	0.875938	−	0.482424i	$-0.160243\pi$
$770$		0			0
$771$	32.9154	+	1.60757i	1.18542	+	0.0578951i
$772$		0			0
$773$			6.77249i			0.243589i	0.992555	+	0.121795i	$0.0388649\pi$
$773$			6.77249i			0.243589i	−0.992555	+	0.121795i	$0.961135\pi$
$774$		0			0
$775$			4.85659i			0.174454i
$776$		0			0
$777$	11.6607	+	0.569500i	0.418325	+	0.0204307i
$778$		0			0
$779$	1.64668			0.0589983
$780$		0			0
$781$		−	34.6335i		−	1.23928i
$782$		0			0
$783$	16.2829	+	2.40102i	0.581902	+	0.0858053i
$784$		0			0
$785$	−18.2395			−0.650994
$786$		0			0
$787$		−	30.0258i		−	1.07031i	−0.844755	−	0.535153i	$-0.820254\pi$
$787$		−	30.0258i		−	1.07031i	0.844755	−	0.535153i	$-0.179746\pi$
$788$		0			0
$789$	44.4078	+	2.16885i	1.58096	+	0.0772130i
$790$		0			0
$791$			16.9187i			0.601558i
$792$		0			0
$793$	−43.5315			−1.54585
$794$		0			0
$795$	1.18630	−	24.2899i	0.0420738	−	0.861473i
$796$		0			0
$797$			0.293260i			0.0103878i	0.999987	+	0.00519391i	$0.00165328\pi$
$797$			0.293260i			0.0103878i	−0.999987	+	0.00519391i	$0.998347\pi$
$798$		0			0
$799$	0.798583			0.0282518
$800$		0			0
$801$	1.05218	−	10.7461i	0.0371769	−	0.379696i
$802$		0			0
$803$	−81.5056			−2.87627
$804$		0			0
$805$	−4.45191			−0.156909
$806$		0			0
$807$	25.9415	+	1.26697i	0.913185	+	0.0445993i
$808$		0			0
$809$	42.1305			1.48123			0.740616	−	0.671929i	$-0.234534\pi$
$809$	42.1305			1.48123			0.740616	+	0.671929i	$0.234534\pi$
$810$		0			0
$811$		−	25.4881i		−	0.895007i	−0.894282	−	0.447504i	$-0.852313\pi$
$811$		−	25.4881i		−	0.895007i	0.894282	−	0.447504i	$-0.147687\pi$
$812$		0			0
$813$	32.2321	+	1.57419i	1.13043	+	0.0552093i
$814$		0			0
$815$	−16.1768			−0.566648
$816$		0			0
$817$			17.9893i			0.629365i
$818$		0			0
$819$	−15.4838	−	1.51605i	−0.541046	−	0.0529750i
$820$		0			0
$821$			6.39520i			0.223194i	0.993754	+	0.111597i	$0.0355966\pi$
$821$			6.39520i			0.223194i	−0.993754	+	0.111597i	$0.964403\pi$
$822$		0			0
$823$	0.818848			0.0285433			0.0142716	−	0.999898i	$-0.495457\pi$
$823$	0.818848			0.0285433			0.0142716	+	0.999898i	$0.495457\pi$
$824$		0			0
$825$	−0.425510	+	8.71246i	−0.0148144	+	0.303329i
$826$		0			0
$827$			5.69895i			0.198172i	0.995079	+	0.0990860i	$0.0315919\pi$
$827$			5.69895i			0.198172i	−0.995079	+	0.0990860i	$0.968408\pi$
$828$		0			0
$829$	−42.5625			−1.47826			−0.739128	−	0.673565i	$-0.764762\pi$
$829$	−42.5625			−1.47826			−0.739128	+	0.673565i	$0.764762\pi$
$830$		0			0
$831$	1.43257	−	29.3323i	0.0496953	−	1.01753i
$832$		0			0
$833$			22.6647i			0.785284i
$834$		0			0
$835$			19.8346i			0.686403i
$836$		0			0
$837$	24.9656	+	3.68135i	0.862939	+	0.127246i
$838$		0			0
$839$			12.1210i			0.418462i	0.977866	+	0.209231i	$0.0670961\pi$
$839$			12.1210i			0.418462i	−0.977866	+	0.209231i	$0.932904\pi$
$840$		0			0
$841$	18.9668			0.654027
$842$		0			0
$843$	0.576728	−	11.8087i	0.0198636	−	0.406713i
$844$		0			0
$845$	−12.7221			−0.437655
$846$		0			0
$847$		−	14.6862i		−	0.504623i
$848$		0			0
$849$	0.548879	−	11.2385i	0.0188375	−	0.385703i
$850$		0			0
$851$		−	28.7000i		−	0.983822i
$852$		0			0
$853$	33.0265			1.13081			0.565403	−	0.824815i	$-0.308721\pi$
$853$	33.0265			1.13081			0.565403	+	0.824815i	$0.308721\pi$
$854$		0			0
$855$	22.0514	+	2.15910i	0.754142	+	0.0738396i
$856$		0			0
$857$	−9.05030			−0.309152			−0.154576	−	0.987981i	$-0.549401\pi$
$857$	−9.05030			−0.309152			−0.154576	+	0.987981i	$0.549401\pi$
$858$		0			0
$859$	−21.6497			−0.738677			−0.369338	−	0.929295i	$-0.620416\pi$
$859$	−21.6497			−0.738677			−0.369338	+	0.929295i	$0.620416\pi$
$860$		0			0
$861$	−0.394401	−	0.0192623i	−0.0134412	−	0.000656457i
$862$		0			0
$863$			47.8519i			1.62890i	0.580234	+	0.814450i	$0.302961\pi$
$863$			47.8519i			1.62890i	−0.580234	+	0.814450i	$0.697039\pi$
$864$		0			0
$865$		−	4.05348i		−	0.137823i
$866$		0			0
$867$	−0.212220	+	4.34527i	−0.00720737	+	0.147573i
$868$		0			0
$869$			13.8972i			0.471430i
$870$		0			0
$871$	34.2064	+	23.5225i	1.15904	+	0.797028i
$872$		0			0
$873$	−0.232350	+	2.37305i	−0.00786387	+	0.0803156i
$874$		0			0
$875$		−	1.02252i		−	0.0345676i
$876$		0			0
$877$	−12.3907			−0.418405			−0.209202	−	0.977872i	$-0.567087\pi$
$877$	−12.3907			−0.418405			−0.209202	+	0.977872i	$0.567087\pi$
$878$		0			0
$879$	−35.3267	−	1.72533i	−1.19154	−	0.0581940i
$880$		0			0
$881$			40.8844i			1.37743i	0.725032	+	0.688716i	$0.241825\pi$
$881$			40.8844i			1.37743i	−0.725032	+	0.688716i	$0.758175\pi$
$882$		0			0
$883$			33.7450i			1.13561i	0.823164	+	0.567804i	$0.192207\pi$
$883$			33.7450i			1.13561i	−0.823164	+	0.567804i	$0.807793\pi$
$884$		0			0
$885$	−21.6387	−	1.05682i	−0.727376	−	0.0355245i
$886$		0			0
$887$			40.2185i			1.35041i	0.737632	+	0.675203i	$0.235944\pi$
$887$			40.2185i			1.35041i	−0.737632	+	0.675203i	$0.764056\pi$
$888$		0			0
$889$		−	0.0571631i		−	0.00191719i
$890$		0			0
$891$	44.4644	+	8.79150i	1.48961	+	0.294526i
$892$		0			0
$893$			1.54953i			0.0518529i
$894$		0			0
$895$	16.6434			0.556328
$896$		0			0
$897$	−1.86569	+	38.2006i	−0.0622936	+	1.27548i
$898$		0			0
$899$	−15.3834			−0.513064
$900$		0			0
$901$		−	53.4429i		−	1.78044i
$902$		0			0
$903$	0.210433	−	4.30868i	0.00700276	−	0.143384i
$904$		0			0
$905$	−6.72887			−0.223675
$906$		0			0
$907$	49.4913			1.64333			0.821665	−	0.569970i	$-0.193045\pi$
$907$	49.4913			1.64333			0.821665	+	0.569970i	$0.193045\pi$
$908$		0			0
$909$	−16.1097	−	1.57733i	−0.534324	−	0.0523168i
$910$		0			0
$911$			31.6315i			1.04800i	0.851719	+	0.524000i	$0.175561\pi$
$911$			31.6315i			1.04800i	−0.851719	+	0.524000i	$0.824439\pi$
$912$		0			0
$913$			78.8286i			2.60885i
$914$		0			0
$915$	−14.8489	−	0.725209i	−0.490889	−	0.0239747i
$916$		0			0
$917$	11.9280			0.393897
$918$		0			0
$919$			47.9617i			1.58211i	0.611745	+	0.791055i	$0.290468\pi$
$919$			47.9617i			1.58211i	−0.611745	+	0.791055i	$0.709532\pi$
$920$		0			0
$921$	−0.408512	+	8.36441i	−0.0134609	+	0.275617i
$922$		0			0
$923$	34.8780			1.14802
$924$		0			0
$925$	6.59186			0.216739
$926$		0			0
$927$	−42.9927	−	4.20951i	−1.41206	−	0.138258i
$928$		0			0
$929$	11.6762			0.383085			0.191543	−	0.981484i	$-0.438651\pi$
$929$	11.6762			0.383085			0.191543	+	0.981484i	$0.438651\pi$
$930$		0			0
$931$	−43.9772			−1.44130
$932$		0			0
$933$	1.05320	−	21.5647i	0.0344803	−	0.705996i
$934$		0			0
$935$			19.1693i			0.626902i
$936$		0			0
$937$		−	54.7267i		−	1.78784i	−0.448223	−	0.893922i	$-0.647943\pi$
$937$		−	54.7267i		−	1.78784i	0.448223	−	0.893922i	$-0.352057\pi$
$938$		0			0
$939$	2.15709	+	0.105351i	0.0703941	+	0.00343800i
$940$		0			0
$941$	35.1663			1.14639			0.573195	−	0.819419i	$-0.305704\pi$
$941$	35.1663			1.14639			0.573195	+	0.819419i	$0.305704\pi$
$942$		0			0
$943$			0.970722i			0.0316111i
$944$		0			0
$945$	−5.25635	−	0.775084i	−0.170989	−	0.0252135i
$946$		0			0
$947$		−	42.1107i		−	1.36842i	−0.729287	−	0.684208i	$-0.760148\pi$
$947$		−	42.1107i		−	1.36842i	0.729287	−	0.684208i	$-0.239852\pi$
$948$		0			0
$949$		−	82.0811i		−	2.66447i
$950$		0			0
$951$	9.41425	+	0.459785i	0.305278	+	0.0149096i
$952$		0			0
$953$		−	2.98852i		−	0.0968077i	−0.998828	−	0.0484039i	$-0.984587\pi$
$953$		−	2.98852i		−	0.0968077i	0.998828	−	0.0484039i	$-0.0154134\pi$
$954$		0			0
$955$	7.32069			0.236892
$956$		0			0
$957$	−27.5969	−	1.34781i	−0.892082	−	0.0435687i
$958$		0			0
$959$		−	11.5715i		−	0.373664i
$960$		0			0
$961$	7.41351			0.239145
$962$		0			0
$963$	1.83710	−	18.7627i	0.0591997	−	0.604620i
$964$		0			0
$965$	14.7105			0.473548
$966$		0			0
$967$	−4.23990			−0.136346			−0.0681730	−	0.997674i	$-0.521717\pi$
$967$	−4.23990			−0.136346			−0.0681730	+	0.997674i	$0.521717\pi$
$968$		0			0
$969$	48.6338	+	2.37524i	1.56234	+	0.0763037i
$970$		0			0
$971$		−	60.5710i		−	1.94382i	−0.235362	−	0.971908i	$-0.575628\pi$
$971$		−	60.5710i		−	1.94382i	0.235362	−	0.971908i	$-0.424372\pi$
$972$		0			0
$973$	4.50627			0.144464
$974$		0			0
$975$	−8.77398	−	0.428515i	−0.280992	−	0.0137235i
$976$		0			0
$977$		−	39.8692i		−	1.27553i	−0.770231	−	0.637765i	$-0.779859\pi$
$977$		−	39.8692i		−	1.27553i	0.770231	−	0.637765i	$-0.220141\pi$
$978$		0			0
$979$			18.1260i			0.579308i
$980$		0			0
$981$	−4.23022	+	43.2042i	−0.135061	+	1.37941i
$982$		0			0
$983$	0.868902			0.0277137			0.0138568	−	0.999904i	$-0.495589\pi$
$983$	0.868902			0.0277137			0.0138568	+	0.999904i	$0.495589\pi$
$984$		0			0
$985$	−20.4668			−0.652127
$986$		0			0
$987$	0.0181258	−	0.371132i	0.000576952	−	0.0118133i
$988$		0			0
$989$	−10.6048			−0.337212
$990$		0			0
$991$			5.60398i			0.178016i	0.996031	+	0.0890081i	$0.0283697\pi$
$991$			5.60398i			0.178016i	−0.996031	+	0.0890081i	$0.971630\pi$
$992$		0			0
$993$	34.0251	+	1.66176i	1.07975	+	0.0527344i
$994$		0			0
$995$	13.8416			0.438808
$996$		0			0
$997$	−1.26247			−0.0399830			−0.0199915	−	0.999800i	$-0.506364\pi$
$997$	−1.26247			−0.0399830			−0.0199915	+	0.999800i	$0.506364\pi$
$998$		0			0
$999$	4.99670	−	33.8859i	0.158088	−	1.07210i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.b.401.24	yes	46
3.2	odd	2		4020.2.f.a.401.24	yes	46
67.66	odd	2		4020.2.f.a.401.23	✓	46
201.200	even	2	inner	4020.2.f.b.401.23	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.23	✓	46	67.66	odd	2
4020.2.f.a.401.24	yes	46	3.2	odd	2
4020.2.f.b.401.23	yes	46	201.200	even	2	inner
4020.2.f.b.401.24	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)

\(f(q)\)	\(=\)	\(q+(-0.0844914 + 1.72999i) q^{3} +1.00000 q^{5} -1.02252i q^{7} +(-2.98572 - 0.292338i) q^{9} +O(q^{10})\)	\(q+(-0.0844914 + 1.72999i) q^{3} +1.00000 q^{5} -1.02252i q^{7} +(-2.98572 - 0.292338i) q^{9} +5.03614 q^{11} +5.07170i q^{13} +(-0.0844914 + 1.72999i) q^{15} +3.80635i q^{17} -7.38561 q^{19} +(1.76895 + 0.0863945i) q^{21} -4.35385i q^{23} +1.00000 q^{25} +(0.758010 - 5.14057i) q^{27} +3.16752i q^{29} +4.85659i q^{31} +(-0.425510 + 8.71246i) q^{33} -1.02252i q^{35} +6.59186 q^{37} +(-8.77398 - 0.428515i) q^{39} -0.222957 q^{41} -2.43572i q^{43} +(-2.98572 - 0.292338i) q^{45} -0.209803i q^{47} +5.95445 q^{49} +(-6.58494 - 0.321604i) q^{51} -14.0405 q^{53} +5.03614 q^{55} +(0.624021 - 12.7770i) q^{57} +12.5080i q^{59} +8.58322i q^{61} +(-0.298923 + 3.05297i) q^{63} +5.07170i q^{65} +(4.63798 - 6.74456i) q^{67} +(7.53211 + 0.367863i) q^{69} -6.87699i q^{71} -16.1841 q^{73} +(-0.0844914 + 1.72999i) q^{75} -5.14957i q^{77} +2.75949i q^{79} +(8.82908 + 1.74568i) q^{81} +15.6526i q^{83} +3.80635i q^{85} +(-5.47978 - 0.267629i) q^{87} +3.59918i q^{89} +5.18593 q^{91} +(-8.40185 - 0.410340i) q^{93} -7.38561 q^{95} -0.794799i q^{97} +(-15.0365 - 1.47226i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 46 q^{5} - 4 q^{9}+O(q^{10}) \) 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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