LMFDB - Embedded newform 4020.2.f.a.401.26

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			401.26
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.a.401.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.293264 + 1.70704i) q^{3} -1.00000 q^{5} +1.30750i q^{7} +(-2.82799 + 1.00123i) q^{9} +O(q^{10})$	\(q+(0.293264 + 1.70704i) q^{3} -1.00000 q^{5} +1.30750i q^{7} +(-2.82799 + 1.00123i) q^{9} -0.0724072 q^{11} +4.54252i q^{13} +(-0.293264 - 1.70704i) q^{15} +5.16393i q^{17} +5.55603 q^{19} +(-2.23195 + 0.383442i) q^{21} +9.12827i q^{23} +1.00000 q^{25} +(-2.53849 - 4.53388i) q^{27} +9.23488i q^{29} -10.8188i q^{31} +(-0.0212345 - 0.123602i) q^{33} -1.30750i q^{35} -0.716347 q^{37} +(-7.75428 + 1.33216i) q^{39} +7.26752 q^{41} -3.55439i q^{43} +(2.82799 - 1.00123i) q^{45} -5.42151i q^{47} +5.29045 q^{49} +(-8.81505 + 1.51440i) q^{51} -2.69163 q^{53} +0.0724072 q^{55} +(1.62938 + 9.48438i) q^{57} -6.20881i q^{59} -11.4692i q^{61} +(-1.30910 - 3.69759i) q^{63} -4.54252i q^{65} +(-0.243839 + 8.18172i) q^{67} +(-15.5824 + 2.67700i) q^{69} +5.55100i q^{71} -6.69308 q^{73} +(0.293264 + 1.70704i) q^{75} -0.0946722i q^{77} +8.46695i q^{79} +(6.99508 - 5.66294i) q^{81} +9.48531i q^{83} -5.16393i q^{85} +(-15.7643 + 2.70826i) q^{87} -6.51193i q^{89} -5.93933 q^{91} +(18.4682 - 3.17278i) q^{93} -5.55603 q^{95} +12.3494i q^{97} +(0.204767 - 0.0724963i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.293264	+	1.70704i	0.169316	+	0.985562i
$3$	0.293264	+	1.70704i	0.169316	+	0.985562i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			1.30750i			0.494187i	0.968992	+	0.247094i	$0.0794755\pi$
$7$			1.30750i			0.494187i	−0.968992	+	0.247094i	$0.920524\pi$
$8$		0			0
$9$	−2.82799	+	1.00123i	−0.942664	+	0.333743i
$10$		0			0
$11$	−0.0724072			−0.0218316			−0.0109158	−	0.999940i	$-0.503475\pi$
$11$	−0.0724072			−0.0218316			−0.0109158	+	0.999940i	$0.503475\pi$
$12$		0			0
$13$			4.54252i			1.25987i	0.776648	+	0.629934i	$0.216918\pi$
$13$			4.54252i			1.25987i	−0.776648	+	0.629934i	$0.783082\pi$
$14$		0			0
$15$	−0.293264	−	1.70704i	−0.0757205	−	0.440757i
$16$		0			0
$17$			5.16393i			1.25244i	0.779648	+	0.626218i	$0.215398\pi$
$17$			5.16393i			1.25244i	−0.779648	+	0.626218i	$0.784602\pi$
$18$		0			0
$19$	5.55603			1.27464			0.637320	−	0.770599i	$-0.280043\pi$
$19$	5.55603			1.27464			0.637320	+	0.770599i	$0.280043\pi$
$20$		0			0
$21$	−2.23195	+	0.383442i	−0.487052	+	0.0836739i
$22$		0			0
$23$			9.12827i			1.90338i	0.307066	+	0.951688i	$0.400653\pi$
$23$			9.12827i			1.90338i	−0.307066	+	0.951688i	$0.599347\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−2.53849	−	4.53388i	−0.488533	−	0.872546i
$28$		0			0
$29$			9.23488i			1.71487i	0.514589	+	0.857437i	$0.327944\pi$
$29$			9.23488i			1.71487i	−0.514589	+	0.857437i	$0.672056\pi$
$30$		0			0
$31$		−	10.8188i		−	1.94312i	−0.236795	−	0.971560i	$-0.576097\pi$
$31$		−	10.8188i		−	1.94312i	0.236795	−	0.971560i	$-0.423903\pi$
$32$		0			0
$33$	−0.0212345	−	0.123602i	−0.00369644	−	0.0215164i
$34$		0			0
$35$		−	1.30750i		−	0.221007i
$36$		0			0
$37$	−0.716347			−0.117767			−0.0588834	−	0.998265i	$-0.518754\pi$
$37$	−0.716347			−0.117767			−0.0588834	+	0.998265i	$0.518754\pi$
$38$		0			0
$39$	−7.75428	+	1.33216i	−1.24168	+	0.213316i
$40$		0			0
$41$	7.26752			1.13500			0.567498	−	0.823375i	$-0.307912\pi$
$41$	7.26752			1.13500			0.567498	+	0.823375i	$0.307912\pi$
$42$		0			0
$43$		−	3.55439i		−	0.542039i	−0.962574	−	0.271019i	$-0.912639\pi$
$43$		−	3.55439i		−	0.542039i	0.962574	−	0.271019i	$-0.0873608\pi$
$44$		0			0
$45$	2.82799	−	1.00123i	0.421572	−	0.149254i
$46$		0			0
$47$		−	5.42151i		−	0.790808i	−0.918507	−	0.395404i	$-0.870605\pi$
$47$		−	5.42151i		−	0.790808i	0.918507	−	0.395404i	$-0.129395\pi$
$48$		0			0
$49$	5.29045			0.755779
$50$		0			0
$51$	−8.81505	+	1.51440i	−1.23435	+	0.212058i
$52$		0			0
$53$	−2.69163			−0.369724			−0.184862	−	0.982764i	$-0.559184\pi$
$53$	−2.69163			−0.369724			−0.184862	+	0.982764i	$0.559184\pi$
$54$		0			0
$55$	0.0724072			0.00976339
$56$		0			0
$57$	1.62938	+	9.48438i	0.215817	+	1.25624i
$58$		0			0
$59$		−	6.20881i		−	0.808319i	−0.914689	−	0.404159i	$-0.867564\pi$
$59$		−	6.20881i		−	0.808319i	0.914689	−	0.404159i	$-0.132436\pi$
$60$		0			0
$61$		−	11.4692i		−	1.46848i	−0.678891	−	0.734239i	$-0.737539\pi$
$61$		−	11.4692i		−	1.46848i	0.678891	−	0.734239i	$-0.262461\pi$
$62$		0			0
$63$	−1.30910	−	3.69759i	−0.164932	−	0.465853i
$64$		0			0
$65$		−	4.54252i		−	0.563430i
$66$		0			0
$67$	−0.243839	+	8.18172i	−0.0297896	+	0.999556i
$67$	−0.243839	+	8.18172i	−0.0297896	+	0.999556i
$68$		0			0
$69$	−15.5824	+	2.67700i	−1.87590	+	0.322273i
$70$		0			0
$71$			5.55100i			0.658782i	0.944194	+	0.329391i	$0.106843\pi$
$71$			5.55100i			0.658782i	−0.944194	+	0.329391i	$0.893157\pi$
$72$		0			0
$73$	−6.69308			−0.783365			−0.391683	−	0.920100i	$-0.628107\pi$
$73$	−6.69308			−0.783365			−0.391683	+	0.920100i	$0.628107\pi$
$74$		0			0
$75$	0.293264	+	1.70704i	0.0338632	+	0.197112i
$76$		0			0
$77$		−	0.0946722i		−	0.0107889i
$78$		0			0
$79$			8.46695i			0.952607i	0.879281	+	0.476303i	$0.158024\pi$
$79$			8.46695i			0.952607i	−0.879281	+	0.476303i	$0.841976\pi$
$80$		0			0
$81$	6.99508	−	5.66294i	0.777231	−	0.629215i
$82$		0			0
$83$			9.48531i			1.04115i	0.853817	+	0.520574i	$0.174282\pi$
$83$			9.48531i			1.04115i	−0.853817	+	0.520574i	$0.825718\pi$
$84$		0			0
$85$		−	5.16393i		−	0.560107i
$86$		0			0
$87$	−15.7643	+	2.70826i	−1.69011	+	0.290356i
$88$		0			0
$89$		−	6.51193i		−	0.690263i	−0.938554	−	0.345132i	$-0.887834\pi$
$89$		−	6.51193i		−	0.690263i	0.938554	−	0.345132i	$-0.112166\pi$
$90$		0			0
$91$	−5.93933			−0.622611
$92$		0			0
$93$	18.4682	−	3.17278i	1.91506	−	0.329002i
$94$		0			0
$95$	−5.55603			−0.570036
$96$		0			0
$97$			12.3494i			1.25389i	0.779063	+	0.626946i	$0.215695\pi$
$97$			12.3494i			1.25389i	−0.779063	+	0.626946i	$0.784305\pi$
$98$		0			0
$99$	0.204767	−	0.0724963i	0.0205799	−	0.00728615i
$100$		0			0
$101$	−8.94926			−0.890485			−0.445242	−	0.895410i	$-0.646883\pi$
$101$	−8.94926			−0.890485			−0.445242	+	0.895410i	$0.646883\pi$
$102$		0			0
$103$	−5.98662			−0.589879			−0.294940	−	0.955516i	$-0.595300\pi$
$103$	−5.98662			−0.589879			−0.294940	+	0.955516i	$0.595300\pi$
$104$		0			0
$105$	2.23195	−	0.383442i	0.217816	−	0.0374201i
$106$		0			0
$107$			10.9432i			1.05791i	0.848648	+	0.528957i	$0.177417\pi$
$107$			10.9432i			1.05791i	−0.848648	+	0.528957i	$0.822583\pi$
$108$		0			0
$109$		−	4.12295i		−	0.394907i	−0.980312	−	0.197453i	$-0.936733\pi$
$109$		−	4.12295i		−	0.394907i	0.980312	−	0.197453i	$-0.0632671\pi$
$110$		0			0
$111$	−0.210079	−	1.22284i	−0.0199398	−	0.116066i
$112$		0			0
$113$	−18.9022			−1.77817			−0.889084	−	0.457745i	$-0.848657\pi$
$113$	−18.9022			−1.77817			−0.889084	+	0.457745i	$0.848657\pi$
$114$		0			0
$115$		−	9.12827i		−	0.851216i
$116$		0			0
$117$	−4.54811	−	12.8462i	−0.420473	−	1.18763i
$118$		0			0
$119$	−6.75182			−0.618938
$120$		0			0
$121$	−10.9948			−0.999523
$122$		0			0
$123$	2.13130	+	12.4060i	0.192173	+	1.11861i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−17.1369			−1.52066			−0.760329	−	0.649538i	$-0.774962\pi$
$127$	−17.1369			−1.52066			−0.760329	+	0.649538i	$0.774962\pi$
$128$		0			0
$129$	6.06749	−	1.04237i	0.534213	−	0.0917760i
$130$		0			0
$131$			5.88155i			0.513874i	0.966428	+	0.256937i	$0.0827132\pi$
$131$			5.88155i			0.513874i	−0.966428	+	0.256937i	$0.917287\pi$
$132$		0			0
$133$			7.26449i			0.629911i
$134$		0			0
$135$	2.53849	+	4.53388i	0.218479	+	0.390214i
$136$		0			0
$137$	−10.0223			−0.856265			−0.428132	−	0.903716i	$-0.640828\pi$
$137$	−10.0223			−0.856265			−0.428132	+	0.903716i	$0.640828\pi$
$138$		0			0
$139$		−	16.0089i		−	1.35785i	−0.734206	−	0.678927i	$-0.762445\pi$
$139$		−	16.0089i		−	1.35785i	0.734206	−	0.678927i	$-0.237555\pi$
$140$		0			0
$141$	9.25475	−	1.58994i	0.779391	−	0.133897i
$142$		0			0
$143$		−	0.328911i		−	0.0275050i
$144$		0			0
$145$		−	9.23488i		−	0.766915i
$146$		0			0
$147$	1.55150	+	9.03103i	0.127966	+	0.744867i
$148$		0			0
$149$			3.50939i			0.287501i	0.989614	+	0.143750i	$0.0459162\pi$
$149$			3.50939i			0.287501i	−0.989614	+	0.143750i	$0.954084\pi$
$150$		0			0
$151$	−4.85995			−0.395497			−0.197749	−	0.980253i	$-0.563363\pi$
$151$	−4.85995			−0.395497			−0.197749	+	0.980253i	$0.563363\pi$
$152$		0			0
$153$	−5.17028	−	14.6035i	−0.417992	−	1.18063i
$154$		0			0
$155$			10.8188i			0.868989i
$156$		0			0
$157$	−7.99694			−0.638225			−0.319113	−	0.947717i	$-0.603385\pi$
$157$	−7.99694			−0.638225			−0.319113	+	0.947717i	$0.603385\pi$
$158$		0			0
$159$	−0.789359	−	4.59473i	−0.0626003	−	0.364386i
$160$		0			0
$161$	−11.9352			−0.940624
$162$		0			0
$163$	5.01155			0.392535			0.196268	−	0.980550i	$-0.437118\pi$
$163$	5.01155			0.392535			0.196268	+	0.980550i	$0.437118\pi$
$164$		0			0
$165$	0.0212345	+	0.123602i	0.00165310	+	0.00962242i
$166$		0			0
$167$		−	3.60915i		−	0.279285i	−0.990202	−	0.139642i	$-0.955405\pi$
$167$		−	3.60915i		−	0.279285i	0.990202	−	0.139642i	$-0.0445953\pi$
$168$		0			0
$169$	−7.63450			−0.587269
$170$		0			0
$171$	−15.7124	+	5.56286i	−1.20156	+	0.425402i
$172$		0			0
$173$		−	15.1821i		−	1.15427i	−0.816648	−	0.577136i	$-0.804170\pi$
$173$		−	15.1821i		−	1.15427i	0.816648	−	0.577136i	$-0.195830\pi$
$174$		0			0
$175$			1.30750i			0.0988375i
$176$		0			0
$177$	10.5987	−	1.82082i	0.796648	−	0.136861i
$178$		0			0
$179$	13.4800			1.00754			0.503772	−	0.863837i	$-0.331945\pi$
$179$	13.4800			1.00754			0.503772	+	0.863837i	$0.331945\pi$
$180$		0			0
$181$	12.8237			0.953181			0.476590	−	0.879126i	$-0.341873\pi$
$181$	12.8237			0.953181			0.476590	+	0.879126i	$0.341873\pi$
$182$		0			0
$183$	19.5784	−	3.36350i	1.44728	−	0.248637i
$184$		0			0
$185$	0.716347			0.0526669
$186$		0			0
$187$		−	0.373906i		−	0.0273427i
$188$		0			0
$189$	5.92803	−	3.31907i	0.431201	−	0.241427i
$190$		0			0
$191$	14.4808			1.04779			0.523897	−	0.851782i	$-0.324478\pi$
$191$	14.4808			1.04779			0.523897	+	0.851782i	$0.324478\pi$
$192$		0			0
$193$	12.3760			0.890844			0.445422	−	0.895321i	$-0.353054\pi$
$193$	12.3760			0.890844			0.445422	+	0.895321i	$0.353054\pi$
$194$		0			0
$195$	7.75428	−	1.33216i	0.555295	−	0.0953979i
$196$		0			0
$197$	−23.7952			−1.69534			−0.847668	−	0.530527i	$-0.821994\pi$
$197$	−23.7952			−1.69534			−0.847668	+	0.530527i	$0.821994\pi$
$198$		0			0
$199$	13.1980			0.935581			0.467790	−	0.883839i	$-0.345050\pi$
$199$	13.1980			0.935581			0.467790	+	0.883839i	$0.345050\pi$
$200$		0			0
$201$	−14.0381	+	1.98316i	−0.990168	+	0.139882i
$202$		0			0
$203$	−12.0746			−0.847469
$204$		0			0
$205$	−7.26752			−0.507585
$206$		0			0
$207$	−9.13950	−	25.8147i	−0.635239	−	1.79424i
$208$		0			0
$209$	−0.402297			−0.0278274
$210$		0			0
$211$	−10.5217			−0.724344			−0.362172	−	0.932111i	$-0.617965\pi$
$211$	−10.5217			−0.724344			−0.362172	+	0.932111i	$0.617965\pi$
$212$		0			0
$213$	−9.47579	+	1.62791i	−0.649270	+	0.111542i
$214$		0			0
$215$			3.55439i			0.242407i
$216$		0			0
$217$	14.1456			0.960265
$218$		0			0
$219$	−1.96284	−	11.4254i	−0.132636	−	0.772055i
$220$		0			0
$221$	−23.4572			−1.57791
$222$		0			0
$223$	10.8726			0.728081			0.364041	−	0.931383i	$-0.381397\pi$
$223$	10.8726			0.728081			0.364041	+	0.931383i	$0.381397\pi$
$224$		0			0
$225$	−2.82799	+	1.00123i	−0.188533	+	0.0667486i
$226$		0			0
$227$		−	2.16075i		−	0.143414i	−0.997426	−	0.0717070i	$-0.977155\pi$
$227$		−	2.16075i		−	0.143414i	0.997426	−	0.0717070i	$-0.0228447\pi$
$228$		0			0
$229$			0.939414i			0.0620782i	0.999518	+	0.0310391i	$0.00988164\pi$
$229$			0.939414i			0.0620782i	−0.999518	+	0.0310391i	$0.990118\pi$
$230$		0			0
$231$	0.161610	−	0.0277640i	0.0106331	−	0.00182674i
$232$		0			0
$233$	0.440087			0.0288311			0.0144155	−	0.999896i	$-0.495411\pi$
$233$	0.440087			0.0288311			0.0144155	+	0.999896i	$0.495411\pi$
$234$		0			0
$235$			5.42151i			0.353660i
$236$		0			0
$237$	−14.4535	+	2.48305i	−0.938853	+	0.161292i
$238$		0			0
$239$	5.91155			0.382386			0.191193	−	0.981552i	$-0.438764\pi$
$239$	5.91155			0.382386			0.191193	+	0.981552i	$0.438764\pi$
$240$		0			0
$241$	22.7799			1.46738			0.733690	−	0.679484i	$-0.237796\pi$
$241$	22.7799			1.46738			0.733690	+	0.679484i	$0.237796\pi$
$242$		0			0
$243$	11.7183	+	10.2802i	0.751728	+	0.659473i
$244$		0			0
$245$	−5.29045			−0.337995
$246$		0			0
$247$			25.2384i			1.60588i
$248$		0			0
$249$	−16.1918	+	2.78170i	−1.02612	+	0.176283i
$250$		0			0
$251$	−0.460541			−0.0290691			−0.0145345	−	0.999894i	$-0.504627\pi$
$251$	−0.460541			−0.0290691			−0.0145345	+	0.999894i	$0.504627\pi$
$252$		0			0
$253$		−	0.660953i		−	0.0415538i
$254$		0			0
$255$	8.81505	−	1.51440i	0.552020	−	0.0948351i
$256$		0			0
$257$		−	17.8791i		−	1.11527i	−0.830088	−	0.557633i	$-0.811710\pi$
$257$		−	17.8791i		−	1.11527i	0.830088	−	0.557633i	$-0.188290\pi$
$258$		0			0
$259$		−	0.936621i		−	0.0581988i
$260$		0			0
$261$	−9.24623	−	26.1162i	−0.572327	−	1.61655i
$262$		0			0
$263$		−	1.37436i		−	0.0847467i	−0.999102	−	0.0423733i	$-0.986508\pi$
$263$		−	1.37436i		−	0.0847467i	0.999102	−	0.0423733i	$-0.0134919\pi$
$264$		0			0
$265$	2.69163			0.165346
$266$		0			0
$267$	11.1161	−	1.90972i	0.680297	−	0.116873i
$268$		0			0
$269$			26.4771i			1.61434i	0.590322	+	0.807168i	$0.299001\pi$
$269$			26.4771i			1.61434i	−0.590322	+	0.807168i	$0.700999\pi$
$270$		0			0
$271$		−	16.8902i		−	1.02601i	−0.858386	−	0.513004i	$-0.828533\pi$
$271$		−	16.8902i		−	1.02601i	0.858386	−	0.513004i	$-0.171467\pi$
$272$		0			0
$273$	−1.74179	−	10.1387i	−0.105418	−	0.613622i
$274$		0			0
$275$	−0.0724072			−0.00436632
$276$		0			0
$277$	−1.42007			−0.0853236			−0.0426618	−	0.999090i	$-0.513584\pi$
$277$	−1.42007			−0.0853236			−0.0426618	+	0.999090i	$0.513584\pi$
$278$		0			0
$279$	10.8321	+	30.5956i	0.648503	+	1.83171i
$280$		0			0
$281$	2.52630			0.150706			0.0753532	−	0.997157i	$-0.475992\pi$
$281$	2.52630			0.150706			0.0753532	+	0.997157i	$0.475992\pi$
$282$		0			0
$283$	22.6658			1.34734			0.673670	−	0.739033i	$-0.264717\pi$
$283$	22.6658			1.34734			0.673670	+	0.739033i	$0.264717\pi$
$284$		0			0
$285$	−1.62938	−	9.48438i	−0.0965164	−	0.561806i
$286$		0			0
$287$			9.50226i			0.560900i
$288$		0			0
$289$	−9.66615			−0.568597
$290$		0			0
$291$	−21.0810	+	3.62164i	−1.23579	+	0.212304i
$292$		0			0
$293$		−	9.65665i		−	0.564148i	−0.959393	−	0.282074i	$-0.908978\pi$
$293$		−	9.65665i		−	0.564148i	0.959393	−	0.282074i	$-0.0910223\pi$
$294$		0			0
$295$			6.20881i			0.361491i
$296$		0			0
$297$	0.183805	+	0.328286i	0.0106655	+	0.0190491i
$298$		0			0
$299$	−41.4654			−2.39800
$300$		0			0
$301$	4.64735			0.267869
$302$		0			0
$303$	−2.62450	−	15.2768i	−0.150774	−	0.877628i
$304$		0			0
$305$			11.4692i			0.656723i
$306$		0			0
$307$	−26.6120			−1.51883			−0.759414	−	0.650608i	$-0.774514\pi$
$307$	−26.6120			−1.51883			−0.759414	+	0.650608i	$0.774514\pi$
$308$		0			0
$309$	−1.75566	−	10.2194i	−0.0998761	−	0.581362i
$310$		0			0
$311$	14.2984			0.810786			0.405393	−	0.914142i	$-0.367135\pi$
$311$	14.2984			0.810786			0.405393	+	0.914142i	$0.367135\pi$
$312$		0			0
$313$		−	15.0265i		−	0.849347i	−0.905347	−	0.424673i	$-0.860389\pi$
$313$		−	15.0265i		−	0.849347i	0.905347	−	0.424673i	$-0.139611\pi$
$314$		0			0
$315$	1.30910	+	3.69759i	0.0737597	+	0.208336i
$316$		0			0
$317$		−	18.2899i		−	1.02726i	−0.858011	−	0.513631i	$-0.828300\pi$
$317$		−	18.2899i		−	1.02726i	0.858011	−	0.513631i	$-0.171700\pi$
$318$		0			0
$319$		−	0.668672i		−	0.0374384i
$320$		0			0
$321$	−18.6804	+	3.20924i	−1.04264	+	0.179122i
$322$		0			0
$323$			28.6909i			1.59641i
$324$		0			0
$325$			4.54252i			0.251974i
$326$		0			0
$327$	7.03805	−	1.20911i	0.389205	−	0.0668642i
$328$		0			0
$329$	7.08861			0.390807
$330$		0			0
$331$		−	3.27913i		−	0.180237i	−0.995931	−	0.0901187i	$-0.971275\pi$
$331$		−	3.27913i		−	0.180237i	0.995931	−	0.0901187i	$-0.0287246\pi$
$332$		0			0
$333$	2.02582	−	0.717228i	0.111014	−	0.0393038i
$334$		0			0
$335$	0.243839	−	8.18172i	0.0133223	−	0.447015i
$336$		0			0
$337$		−	27.5435i		−	1.50039i	−0.661217	−	0.750194i	$-0.729960\pi$
$337$		−	27.5435i		−	1.50039i	0.661217	−	0.750194i	$-0.270040\pi$
$338$		0			0
$339$	−5.54333	−	32.2668i	−0.301073	−	1.75249i
$340$		0			0
$341$			0.783362i			0.0424214i
$342$		0			0
$343$			16.0697i			0.867684i
$344$		0			0
$345$	15.5824	−	2.67700i	0.838926	−	0.144125i
$346$		0			0
$347$	−1.31660			−0.0706789			−0.0353394	−	0.999375i	$-0.511251\pi$
$347$	−1.31660			−0.0706789			−0.0353394	+	0.999375i	$0.511251\pi$
$348$		0			0
$349$	20.7662			1.11159			0.555794	−	0.831320i	$-0.312415\pi$
$349$	20.7662			1.11159			0.555794	+	0.831320i	$0.312415\pi$
$350$		0			0
$351$	20.5952	−	11.5311i	1.09929	−	0.615487i
$352$		0			0
$353$	26.1494			1.39179			0.695897	−	0.718141i	$-0.255007\pi$
$353$	26.1494			1.39179			0.695897	+	0.718141i	$0.255007\pi$
$354$		0			0
$355$		−	5.55100i		−	0.294616i
$356$		0			0
$357$	−1.98007	−	11.5256i	−0.104796	−	0.610002i
$358$		0			0
$359$			35.3265i			1.86446i	0.361863	+	0.932231i	$0.382141\pi$
$359$			35.3265i			1.86446i	−0.361863	+	0.932231i	$0.617859\pi$
$360$		0			0
$361$	11.8694			0.624708
$362$		0			0
$363$	−3.22437	−	18.7685i	−0.169236	−	0.985092i
$364$		0			0
$365$	6.69308			0.350332
$366$		0			0
$367$		−	15.4816i		−	0.808131i	−0.914730	−	0.404066i	$-0.867597\pi$
$367$		−	15.4816i		−	0.808131i	0.914730	−	0.404066i	$-0.132403\pi$
$368$		0			0
$369$	−20.5525	+	7.27645i	−1.06992	+	0.378797i
$370$		0			0
$371$		−	3.51930i		−	0.182713i
$372$		0			0
$373$			33.1977i			1.71891i	0.511211	+	0.859455i	$0.329197\pi$
$373$			33.1977i			1.71891i	−0.511211	+	0.859455i	$0.670803\pi$
$374$		0			0
$375$	−0.293264	−	1.70704i	−0.0151441	−	0.0881513i
$376$		0			0
$377$	−41.9496			−2.16052
$378$		0			0
$379$			4.50990i			0.231658i	0.993269	+	0.115829i	$0.0369524\pi$
$379$			4.50990i			0.231658i	−0.993269	+	0.115829i	$0.963048\pi$
$380$		0			0
$381$	−5.02565	−	29.2535i	−0.257472	−	1.49870i
$382$		0			0
$383$	−27.9411			−1.42773			−0.713863	−	0.700285i	$-0.753056\pi$
$383$	−27.9411			−1.42773			−0.713863	+	0.700285i	$0.753056\pi$
$384$		0			0
$385$			0.0946722i			0.00482494i
$386$		0			0
$387$	3.55876	+	10.0518i	0.180902	+	0.510961i
$388$		0			0
$389$		−	2.48529i		−	0.126009i	−0.998013	−	0.0630045i	$-0.979932\pi$
$389$		−	2.48529i		−	0.126009i	0.998013	−	0.0630045i	$-0.0200682\pi$
$390$		0			0
$391$	−47.1377			−2.38386
$392$		0			0
$393$	−10.0401	+	1.72485i	−0.506454	+	0.0870072i
$394$		0			0
$395$		−	8.46695i		−	0.426019i
$396$		0			0
$397$	4.05299			0.203414			0.101707	−	0.994814i	$-0.467570\pi$
$397$	4.05299			0.203414			0.101707	+	0.994814i	$0.467570\pi$
$398$		0			0
$399$	−12.4008	+	2.13041i	−0.620816	+	0.106654i
$400$		0			0
$401$	27.5484			1.37570			0.687851	−	0.725851i	$-0.258554\pi$
$401$	27.5484			1.37570			0.687851	+	0.725851i	$0.258554\pi$
$402$		0			0
$403$	49.1448			2.44808
$404$		0			0
$405$	−6.99508	+	5.66294i	−0.347588	+	0.281394i
$406$		0			0
$407$	0.0518687			0.00257104
$408$		0			0
$409$			4.83676i			0.239162i	0.992824	+	0.119581i	$0.0381552\pi$
$409$			4.83676i			0.239162i	−0.992824	+	0.119581i	$0.961845\pi$
$410$		0			0
$411$	−2.93919	−	17.1085i	−0.144979	−	0.843902i
$412$		0			0
$413$	8.11800			0.399461
$414$		0			0
$415$		−	9.48531i		−	0.465615i
$416$		0			0
$417$	27.3278	−	4.69483i	1.33825	−	0.229907i
$418$		0			0
$419$			16.5904i			0.810493i	0.914208	+	0.405246i	$0.132814\pi$
$419$			16.5904i			0.810493i	−0.914208	+	0.405246i	$0.867186\pi$
$420$		0			0
$421$	−11.1409			−0.542973			−0.271486	−	0.962442i	$-0.587515\pi$
$421$	−11.1409			−0.542973			−0.271486	+	0.962442i	$0.587515\pi$
$422$		0			0
$423$	5.42818	+	15.3320i	0.263927	+	0.745467i
$424$		0			0
$425$			5.16393i			0.250487i
$426$		0			0
$427$	14.9959			0.725703
$428$		0			0
$429$	0.561466	−	0.0964580i	0.0271078	−	0.00465703i
$430$		0			0
$431$		−	12.3187i		−	0.593373i	−0.954975	−	0.296686i	$-0.904118\pi$
$431$		−	12.3187i		−	0.593373i	0.954975	−	0.296686i	$-0.0958816\pi$
$432$		0			0
$433$		−	10.2701i		−	0.493548i	−0.969073	−	0.246774i	$-0.920629\pi$
$433$		−	10.2701i		−	0.493548i	0.969073	−	0.246774i	$-0.0793706\pi$
$434$		0			0
$435$	15.7643	−	2.70826i	0.755842	−	0.129851i
$436$		0			0
$437$			50.7169i			2.42612i
$438$		0			0
$439$	−25.1870			−1.20211			−0.601054	−	0.799209i	$-0.705252\pi$
$439$	−25.1870			−1.20211			−0.601054	+	0.799209i	$0.705252\pi$
$440$		0			0
$441$	−14.9614	+	5.29696i	−0.712446	+	0.252236i
$442$		0			0
$443$	22.0748			1.04880			0.524402	−	0.851471i	$-0.324289\pi$
$443$	22.0748			1.04880			0.524402	+	0.851471i	$0.324289\pi$
$444$		0			0
$445$			6.51193i			0.308695i
$446$		0			0
$447$	−5.99069	+	1.02918i	−0.283350	+	0.0486785i
$448$		0			0
$449$		−	30.3019i		−	1.43004i	−0.699106	−	0.715018i	$-0.746418\pi$
$449$		−	30.3019i		−	1.43004i	0.699106	−	0.715018i	$-0.253582\pi$
$450$		0			0
$451$	−0.526221			−0.0247788
$452$		0			0
$453$	−1.42525	−	8.29614i	−0.0669641	−	0.389787i
$454$		0			0
$455$	5.93933			0.278440
$456$		0			0
$457$	24.6028			1.15087			0.575434	−	0.817848i	$-0.304833\pi$
$457$	24.6028			1.15087			0.575434	+	0.817848i	$0.304833\pi$
$458$		0			0
$459$	23.4126	−	13.1086i	1.09281	−	0.611856i
$460$		0			0
$461$		−	16.5490i		−	0.770762i	−0.922758	−	0.385381i	$-0.874070\pi$
$461$		−	16.5490i		−	0.770762i	0.922758	−	0.385381i	$-0.125930\pi$
$462$		0			0
$463$			1.65161i			0.0767571i	0.999263	+	0.0383785i	$0.0122193\pi$
$463$			1.65161i			0.0767571i	−0.999263	+	0.0383785i	$0.987781\pi$
$464$		0			0
$465$	−18.4682	+	3.17278i	−0.856443	+	0.147134i
$466$		0			0
$467$			36.0952i			1.67028i	0.550034	+	0.835142i	$0.314615\pi$
$467$			36.0952i			1.67028i	−0.550034	+	0.835142i	$0.685385\pi$
$468$		0			0
$469$	−10.6976	−	0.318818i	−0.493968	−	0.0147217i
$470$		0			0
$471$	−2.34522	−	13.6511i	−0.108062	−	0.629011i
$472$		0			0
$473$			0.257363i			0.0118336i
$474$		0			0
$475$	5.55603			0.254928
$476$		0			0
$477$	7.61191	−	2.69494i	0.348525	−	0.123393i
$478$		0			0
$479$		−	4.28687i		−	0.195872i	−0.995193	−	0.0979360i	$-0.968776\pi$
$479$		−	4.28687i		−	0.195872i	0.995193	−	0.0979360i	$-0.0312240\pi$
$480$		0			0
$481$		−	3.25402i		−	0.148371i
$482$		0			0
$483$	−3.50016	−	20.3739i	−0.159263	−	0.927043i
$484$		0			0
$485$		−	12.3494i		−	0.560758i
$486$		0			0
$487$			32.6270i			1.47847i	0.673448	+	0.739235i	$0.264813\pi$
$487$			32.6270i			1.47847i	−0.673448	+	0.739235i	$0.735187\pi$
$488$		0			0
$489$	1.46971	+	8.55494i	0.0664626	+	0.386868i
$490$		0			0
$491$		−	8.30493i		−	0.374796i	−0.982284	−	0.187398i	$-0.939995\pi$
$491$		−	8.30493i		−	0.374796i	0.982284	−	0.187398i	$-0.0600055\pi$
$492$		0			0
$493$	−47.6882			−2.14777
$494$		0			0
$495$	−0.204767	+	0.0724963i	−0.00920360	+	0.00325846i
$496$		0			0
$497$	−7.25791			−0.325562
$498$		0			0
$499$			17.7454i			0.794395i	0.917733	+	0.397197i	$0.130017\pi$
$499$			17.7454i			0.794395i	−0.917733	+	0.397197i	$0.869983\pi$
$500$		0			0
$501$	6.16098	−	1.05844i	0.275252	−	0.0472874i
$502$		0			0
$503$	7.72456			0.344421			0.172210	−	0.985060i	$-0.444909\pi$
$503$	7.72456			0.344421			0.172210	+	0.985060i	$0.444909\pi$
$504$		0			0
$505$	8.94926			0.398237
$506$		0			0
$507$	−2.23893	−	13.0324i	−0.0994342	−	0.578790i
$508$		0			0
$509$		−	1.91659i		−	0.0849515i	−0.999097	−	0.0424757i	$-0.986475\pi$
$509$		−	1.91659i		−	0.0849515i	0.999097	−	0.0424757i	$-0.0135245\pi$
$510$		0			0
$511$		−	8.75118i		−	0.387129i
$512$		0			0
$513$	−14.1039	−	25.1904i	−0.622704	−	1.11218i
$514$		0			0
$515$	5.98662			0.263802
$516$		0			0
$517$			0.392557i			0.0172646i
$518$		0			0
$519$	25.9165	−	4.45236i	1.13761	−	0.195437i
$520$		0			0
$521$	−1.26414			−0.0553828			−0.0276914	−	0.999617i	$-0.508816\pi$
$521$	−1.26414			−0.0553828			−0.0276914	+	0.999617i	$0.508816\pi$
$522$		0			0
$523$	22.7024			0.992705			0.496353	−	0.868121i	$-0.334672\pi$
$523$	22.7024			0.992705			0.496353	+	0.868121i	$0.334672\pi$
$524$		0			0
$525$	−2.23195	+	0.383442i	−0.0974104	+	0.0167348i
$526$		0			0
$527$	55.8677			2.43363
$528$		0			0
$529$	−60.3254			−2.62284
$530$		0			0
$531$	6.21645	+	17.5585i	0.269771	+	0.761973i
$532$		0			0
$533$			33.0129i			1.42995i
$534$		0			0
$535$		−	10.9432i		−	0.473114i
$536$		0			0
$537$	3.95321	+	23.0110i	0.170594	+	0.992997i
$538$		0			0
$539$	−0.383067			−0.0164999
$540$		0			0
$541$			26.5991i			1.14359i	0.820398	+	0.571793i	$0.193752\pi$
$541$			26.5991i			1.14359i	−0.820398	+	0.571793i	$0.806248\pi$
$542$		0			0
$543$	3.76074	+	21.8907i	0.161389	+	0.939418i
$544$		0			0
$545$			4.12295i			0.176608i
$546$		0			0
$547$		−	22.9422i		−	0.980938i	−0.871459	−	0.490469i	$-0.836825\pi$
$547$		−	22.9422i		−	0.980938i	0.871459	−	0.490469i	$-0.163175\pi$
$548$		0			0
$549$	11.4833	+	32.4348i	0.490095	+	1.38428i
$550$		0			0
$551$			51.3092i			2.18585i
$552$		0			0
$553$	−11.0705			−0.470766
$554$		0			0
$555$	0.210079	+	1.22284i	0.00891736	+	0.0519065i
$556$		0			0
$557$			9.52407i			0.403548i	0.979432	+	0.201774i	$0.0646706\pi$
$557$			9.52407i			0.403548i	−0.979432	+	0.201774i	$0.935329\pi$
$558$		0			0
$559$	16.1459			0.682898
$560$		0			0
$561$	0.638273	−	0.109653i	0.0269479	−	0.00462956i
$562$		0			0
$563$	16.3106			0.687409			0.343704	−	0.939078i	$-0.388318\pi$
$563$	16.3106			0.687409			0.343704	+	0.939078i	$0.388318\pi$
$564$		0			0
$565$	18.9022			0.795221
$566$		0			0
$567$	7.40427	+	9.14604i	0.310950	+	0.384098i
$568$		0			0
$569$		−	24.0937i		−	1.01006i	−0.863102	−	0.505030i	$-0.831481\pi$
$569$		−	24.0937i		−	1.01006i	0.863102	−	0.505030i	$-0.168519\pi$
$570$		0			0
$571$	−11.7402			−0.491312			−0.245656	−	0.969357i	$-0.579003\pi$
$571$	−11.7402			−0.491312			−0.245656	+	0.969357i	$0.579003\pi$
$572$		0			0
$573$	4.24670	+	24.7193i	0.177408	+	1.03267i
$574$		0			0
$575$			9.12827i			0.380675i
$576$		0			0
$577$			23.4877i			0.977807i	0.872338	+	0.488904i	$0.162603\pi$
$577$			23.4877i			0.977807i	−0.872338	+	0.488904i	$0.837397\pi$
$578$		0			0
$579$	3.62944	+	21.1264i	0.150834	+	0.877982i
$580$		0			0
$581$	−12.4020			−0.514522
$582$		0			0
$583$	0.194894			0.00807167
$584$		0			0
$585$	4.54811	+	12.8462i	0.188041	+	0.531126i
$586$		0			0
$587$	3.92594			0.162041			0.0810205	−	0.996712i	$-0.474182\pi$
$587$	3.92594			0.162041			0.0810205	+	0.996712i	$0.474182\pi$
$588$		0			0
$589$		−	60.1097i		−	2.47678i
$590$		0			0
$591$	−6.97827	−	40.6194i	−0.287048	−	1.67086i
$592$		0			0
$593$	−42.9777			−1.76488			−0.882441	−	0.470423i	$-0.844101\pi$
$593$	−42.9777			−1.76488			−0.882441	+	0.470423i	$0.844101\pi$
$594$		0			0
$595$	6.75182			0.276798
$596$		0			0
$597$	3.87050	+	22.5295i	0.158409	+	0.922073i
$598$		0			0
$599$	−19.8502			−0.811059			−0.405529	−	0.914082i	$-0.632913\pi$
$599$	−19.8502			−0.811059			−0.405529	+	0.914082i	$0.632913\pi$
$600$		0			0
$601$	−31.9544			−1.30345			−0.651724	−	0.758456i	$-0.725954\pi$
$601$	−31.9544			−1.30345			−0.651724	+	0.758456i	$0.725954\pi$
$602$		0			0
$603$	−7.50221	−	23.3820i	−0.305513	−	0.952188i
$604$		0			0
$605$	10.9948			0.447000
$606$		0			0
$607$	20.8282			0.845392			0.422696	−	0.906272i	$-0.361084\pi$
$607$	20.8282			0.845392			0.422696	+	0.906272i	$0.361084\pi$
$608$		0			0
$609$	−3.54104	−	20.6118i	−0.143490	−	0.835233i
$610$		0			0
$611$	24.6273			0.996315
$612$		0			0
$613$	36.4468			1.47207			0.736037	−	0.676941i	$-0.236695\pi$
$613$	36.4468			1.47207			0.736037	+	0.676941i	$0.236695\pi$
$614$		0			0
$615$	−2.13130	−	12.4060i	−0.0859425	−	0.500257i
$616$		0			0
$617$			2.66431i			0.107261i	0.998561	+	0.0536306i	$0.0170794\pi$
$617$			2.66431i			0.107261i	−0.998561	+	0.0536306i	$0.982921\pi$
$618$		0			0
$619$	23.2794			0.935680			0.467840	−	0.883813i	$-0.345032\pi$
$619$	23.2794			0.935680			0.467840	+	0.883813i	$0.345032\pi$
$620$		0			0
$621$	41.3865	−	23.1720i	1.66078	−	0.929862i
$622$		0			0
$623$	8.51433			0.341119
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−0.117979	−	0.686738i	−0.00471164	−	0.0274257i
$628$		0			0
$629$		−	3.69916i		−	0.147495i
$630$		0			0
$631$			16.9004i			0.672793i	0.941721	+	0.336396i	$0.109208\pi$
$631$			16.9004i			0.672793i	−0.941721	+	0.336396i	$0.890792\pi$
$632$		0			0
$633$	−3.08564	−	17.9610i	−0.122643	−	0.713885i
$634$		0			0
$635$	17.1369			0.680059
$636$		0			0
$637$			24.0320i			0.952182i
$638$		0			0
$639$	−5.55782	−	15.6982i	−0.219864	−	0.621010i
$640$		0			0
$641$	0.420799			0.0166206			0.00831029	−	0.999965i	$-0.497355\pi$
$641$	0.420799			0.0166206			0.00831029	+	0.999965i	$0.497355\pi$
$642$		0			0
$643$	17.9640			0.708433			0.354216	−	0.935164i	$-0.384748\pi$
$643$	17.9640			0.708433			0.354216	+	0.935164i	$0.384748\pi$
$644$		0			0
$645$	−6.06749	+	1.04237i	−0.238907	+	0.0410435i
$646$		0			0
$647$	−10.6289			−0.417866			−0.208933	−	0.977930i	$-0.566999\pi$
$647$	−10.6289			−0.417866			−0.208933	+	0.977930i	$0.566999\pi$
$648$		0			0
$649$			0.449563i			0.0176469i
$650$		0			0
$651$	4.14839	+	24.1471i	0.162588	+	0.946400i
$652$		0			0
$653$	−34.4632			−1.34865			−0.674325	−	0.738435i	$-0.735565\pi$
$653$	−34.4632			−1.34865			−0.674325	+	0.738435i	$0.735565\pi$
$654$		0			0
$655$		−	5.88155i		−	0.229811i
$656$		0			0
$657$	18.9280	−	6.70131i	0.738450	−	0.261443i
$658$		0			0
$659$		−	37.4538i		−	1.45899i	−0.683984	−	0.729497i	$-0.739754\pi$
$659$		−	37.4538i		−	1.45899i	0.683984	−	0.729497i	$-0.260246\pi$
$660$		0			0
$661$			36.2279i			1.40910i	0.709654	+	0.704550i	$0.248851\pi$
$661$			36.2279i			1.40910i	−0.709654	+	0.704550i	$0.751149\pi$
$662$		0			0
$663$	−6.87917	−	40.0425i	−0.267165	−	1.55512i
$664$		0			0
$665$		−	7.26449i		−	0.281705i
$666$		0			0
$667$	−84.2985			−3.26405
$668$		0			0
$669$	3.18854	+	18.5600i	0.123276	+	0.717569i
$670$		0			0
$671$			0.830452i			0.0320592i
$672$		0			0
$673$		−	29.1286i		−	1.12282i	−0.827537	−	0.561412i	$-0.810258\pi$
$673$		−	29.1286i		−	1.12282i	0.827537	−	0.561412i	$-0.189742\pi$
$674$		0			0
$675$	−2.53849	−	4.53388i	−0.0977066	−	0.174509i
$676$		0			0
$677$	17.5318			0.673801			0.336901	−	0.941540i	$-0.390621\pi$
$677$	17.5318			0.673801			0.336901	+	0.941540i	$0.390621\pi$
$678$		0			0
$679$	−16.1468			−0.619657
$680$		0			0
$681$	3.68849	−	0.633671i	0.141343	−	0.0242823i
$682$		0			0
$683$	38.9032			1.48859			0.744296	−	0.667850i	$-0.232785\pi$
$683$	38.9032			1.48859			0.744296	+	0.667850i	$0.232785\pi$
$684$		0			0
$685$	10.0223			0.382933
$686$		0			0
$687$	−1.60362	+	0.275497i	−0.0611819	+	0.0105108i
$688$		0			0
$689$		−	12.2268i		−	0.465804i
$690$		0			0
$691$	−10.6725			−0.406003			−0.203001	−	0.979178i	$-0.565070\pi$
$691$	−10.6725			−0.406003			−0.203001	+	0.979178i	$0.565070\pi$
$692$		0			0
$693$	0.0947886	+	0.267732i	0.00360072	+	0.0101703i
$694$		0			0
$695$			16.0089i			0.607251i
$696$		0			0
$697$			37.5289i			1.42151i
$698$		0			0
$699$	0.129062	+	0.751248i	0.00488157	+	0.0284148i
$700$		0			0
$701$	−37.8021			−1.42776			−0.713882	−	0.700266i	$-0.753065\pi$
$701$	−37.8021			−1.42776			−0.713882	+	0.700266i	$0.753065\pi$
$702$		0			0
$703$	−3.98004			−0.150110
$704$		0			0
$705$	−9.25475	+	1.58994i	−0.348554	+	0.0598804i
$706$		0			0
$707$		−	11.7011i		−	0.440066i
$708$		0			0
$709$	−16.3845			−0.615333			−0.307666	−	0.951494i	$-0.599548\pi$
$709$	−16.3845			−0.615333			−0.307666	+	0.951494i	$0.599548\pi$
$710$		0			0
$711$	−8.47736	−	23.9445i	−0.317926	−	0.897988i
$712$		0			0
$713$	98.7572			3.69849
$714$		0			0
$715$			0.328911i			0.0123006i
$716$		0			0
$717$	1.73365	+	10.0913i	0.0647442	+	0.376865i
$718$		0			0
$719$			39.1840i			1.46132i	0.682743	+	0.730658i	$0.260787\pi$
$719$			39.1840i			1.46132i	−0.682743	+	0.730658i	$0.739213\pi$
$720$		0			0
$721$		−	7.82749i		−	0.291511i
$722$		0			0
$723$	6.68052	+	38.8862i	0.248451	+	1.44619i
$724$		0			0
$725$			9.23488i			0.342975i
$726$		0			0
$727$		−	7.20961i		−	0.267390i	−0.991023	−	0.133695i	$-0.957316\pi$
$727$		−	7.20961i		−	0.267390i	0.991023	−	0.133695i	$-0.0426842\pi$
$728$		0			0
$729$	−14.1121	+	23.0184i	−0.522671	+	0.852534i
$730$		0			0
$731$	18.3546			0.678869
$732$		0			0
$733$			14.7221i			0.543775i	0.962329	+	0.271887i	$0.0876478\pi$
$733$			14.7221i			0.543775i	−0.962329	+	0.271887i	$0.912352\pi$
$734$		0			0
$735$	−1.55150	−	9.03103i	−0.0572280	−	0.333115i
$736$		0			0
$737$	0.0176557	−	0.592416i	0.000650355	−	0.0218219i
$738$		0			0
$739$			14.2892i			0.525638i	0.964845	+	0.262819i	$0.0846522\pi$
$739$			14.2892i			0.525638i	−0.964845	+	0.262819i	$0.915348\pi$
$740$		0			0
$741$	−43.0830	+	7.40151i	−1.58269	+	0.271901i
$742$		0			0
$743$			11.0662i			0.405980i	0.979181	+	0.202990i	$0.0650659\pi$
$743$			11.0662i			0.405980i	−0.979181	+	0.202990i	$0.934934\pi$
$744$		0			0
$745$		−	3.50939i		−	0.128574i
$746$		0			0
$747$	−9.49697	−	26.8244i	−0.347476	−	0.981453i
$748$		0			0
$749$	−14.3081			−0.522808
$750$		0			0
$751$	−9.40805			−0.343305			−0.171652	−	0.985158i	$-0.554911\pi$
$751$	−9.40805			−0.343305			−0.171652	+	0.985158i	$0.554911\pi$
$752$		0			0
$753$	−0.135060	−	0.786163i	−0.00492187	−	0.0286494i
$754$		0			0
$755$	4.85995			0.176872
$756$		0			0
$757$		−	27.4346i		−	0.997128i	−0.866853	−	0.498564i	$-0.833861\pi$
$757$		−	27.4346i		−	0.997128i	0.866853	−	0.498564i	$-0.166139\pi$
$758$		0			0
$759$	1.12828	−	0.193834i	0.0409538	−	0.00703573i
$760$		0			0
$761$			30.4421i			1.10353i	0.834001	+	0.551763i	$0.186045\pi$
$761$			30.4421i			1.10353i	−0.834001	+	0.551763i	$0.813955\pi$
$762$		0			0
$763$	5.39074			0.195158
$764$		0			0
$765$	5.17028	+	14.6035i	0.186932	+	0.527992i
$766$		0			0
$767$	28.2037			1.01838
$768$		0			0
$769$			46.4727i			1.67585i	0.545788	+	0.837923i	$0.316230\pi$
$769$			46.4727i			1.67585i	−0.545788	+	0.837923i	$0.683770\pi$
$770$		0			0
$771$	30.5203	−	5.24329i	1.09916	−	0.188833i
$772$		0			0
$773$			55.0272i			1.97919i	0.143877	+	0.989596i	$0.454043\pi$
$773$			55.0272i			1.97919i	−0.143877	+	0.989596i	$0.545957\pi$
$774$		0			0
$775$		−	10.8188i		−	0.388624i
$776$		0			0
$777$	1.59885	−	0.274678i	0.0573585	−	0.00985400i
$778$		0			0
$779$	40.3785			1.44671
$780$		0			0
$781$		−	0.401932i		−	0.0143823i
$782$		0			0
$783$	41.8698	−	23.4427i	1.49631	−	0.837772i
$784$		0			0
$785$	7.99694			0.285423
$786$		0			0
$787$			19.6177i			0.699294i	0.936882	+	0.349647i	$0.113698\pi$
$787$			19.6177i			0.699294i	−0.936882	+	0.349647i	$0.886302\pi$
$788$		0			0
$789$	2.34609	−	0.403051i	0.0835231	−	0.0143490i
$790$		0			0
$791$		−	24.7145i		−	0.878748i
$792$		0			0
$793$	52.0990			1.85009
$794$		0			0
$795$	0.789359	+	4.59473i	0.0279957	+	0.162958i
$796$		0			0
$797$			20.3165i			0.719647i	0.933020	+	0.359823i	$0.117163\pi$
$797$			20.3165i			0.719647i	−0.933020	+	0.359823i	$0.882837\pi$
$798$		0			0
$799$	27.9963			0.990437
$800$		0			0
$801$	6.51994	+	18.4157i	0.230371	+	0.650686i
$802$		0			0
$803$	0.484627			0.0171021
$804$		0			0
$805$	11.9352			0.420660
$806$		0			0
$807$	−45.1975	+	7.76478i	−1.59103	+	0.273333i
$808$		0			0
$809$	−7.67891			−0.269976			−0.134988	−	0.990847i	$-0.543100\pi$
$809$	−7.67891			−0.269976			−0.134988	+	0.990847i	$0.543100\pi$
$810$		0			0
$811$		−	24.9123i		−	0.874789i	−0.899270	−	0.437395i	$-0.855901\pi$
$811$		−	24.9123i		−	0.874789i	0.899270	−	0.437395i	$-0.144099\pi$
$812$		0			0
$813$	28.8323	−	4.95330i	1.01119	−	0.173720i
$814$		0			0
$815$	−5.01155			−0.175547
$816$		0			0
$817$		−	19.7483i		−	0.690905i
$818$		0			0
$819$	16.7964	−	5.94663i	0.586913	−	0.207792i
$820$		0			0
$821$			24.1757i			0.843737i	0.906657	+	0.421868i	$0.138626\pi$
$821$			24.1757i			0.843737i	−0.906657	+	0.421868i	$0.861374\pi$
$822$		0			0
$823$	32.8763			1.14600			0.572998	−	0.819557i	$-0.305780\pi$
$823$	32.8763			1.14600			0.572998	+	0.819557i	$0.305780\pi$
$824$		0			0
$825$	−0.0212345	−	0.123602i	−0.000739289	−	0.00430328i
$826$		0			0
$827$		−	2.54312i		−	0.0884328i	−0.999022	−	0.0442164i	$-0.985921\pi$
$827$		−	2.54312i		−	0.0884328i	0.999022	−	0.0442164i	$-0.0140791\pi$
$828$		0			0
$829$	−3.53812			−0.122884			−0.0614420	−	0.998111i	$-0.519570\pi$
$829$	−3.53812			−0.122884			−0.0614420	+	0.998111i	$0.519570\pi$
$830$		0			0
$831$	−0.416455	−	2.42412i	−0.0144467	−	0.0840917i
$832$		0			0
$833$			27.3195i			0.946565i
$834$		0			0
$835$			3.60915i			0.124900i
$836$		0			0
$837$	−49.0513	+	27.4635i	−1.69546	+	0.949278i
$838$		0			0
$839$			43.6909i			1.50838i	0.656657	+	0.754189i	$0.271970\pi$
$839$			43.6909i			1.50838i	−0.656657	+	0.754189i	$0.728030\pi$
$840$		0			0
$841$	−56.2830			−1.94079
$842$		0			0
$843$	0.740873	+	4.31250i	0.0255170	+	0.148530i
$844$		0			0
$845$	7.63450			0.262635
$846$		0			0
$847$		−	14.3756i		−	0.493952i
$848$		0			0
$849$	6.64706	+	38.6914i	0.228126	+	1.32789i
$850$		0			0
$851$		−	6.53901i		−	0.224154i
$852$		0			0
$853$	21.0415			0.720447			0.360224	−	0.932866i	$-0.382700\pi$
$853$	21.0415			0.720447			0.360224	+	0.932866i	$0.382700\pi$
$854$		0			0
$855$	15.7124	−	5.56286i	0.537353	−	0.190246i
$856$		0			0
$857$	35.4001			1.20925			0.604623	−	0.796512i	$-0.293324\pi$
$857$	35.4001			1.20925			0.604623	+	0.796512i	$0.293324\pi$
$858$		0			0
$859$	25.5677			0.872359			0.436180	−	0.899860i	$-0.356331\pi$
$859$	25.5677			0.872359			0.436180	+	0.899860i	$0.356331\pi$
$860$		0			0
$861$	−16.2208	+	2.78667i	−0.552802	+	0.0949695i
$862$		0			0
$863$			14.6848i			0.499875i	0.968262	+	0.249938i	$0.0804101\pi$
$863$			14.6848i			0.499875i	−0.968262	+	0.249938i	$0.919590\pi$
$864$		0			0
$865$			15.1821i			0.516206i
$866$		0			0
$867$	−2.83474	−	16.5005i	−0.0962727	−	0.560387i
$868$		0			0
$869$		−	0.613069i		−	0.0207969i
$870$		0			0
$871$	−37.1656	−	1.10764i	−1.25931	−	0.0375310i
$872$		0			0
$873$	−12.3646	−	34.9240i	−0.418478	−	1.18200i
$874$		0			0
$875$		−	1.30750i		−	0.0442015i
$876$		0			0
$877$	−48.0445			−1.62235			−0.811174	−	0.584805i	$-0.801171\pi$
$877$	−48.0445			−1.62235			−0.811174	+	0.584805i	$0.801171\pi$
$878$		0			0
$879$	16.4843	−	2.83195i	0.556002	−	0.0955193i
$880$		0			0
$881$			50.5078i			1.70165i	0.525448	+	0.850826i	$0.323898\pi$
$881$			50.5078i			1.70165i	−0.525448	+	0.850826i	$0.676102\pi$
$882$		0			0
$883$			34.9220i			1.17522i	0.809145	+	0.587610i	$0.199931\pi$
$883$			34.9220i			1.17522i	−0.809145	+	0.587610i	$0.800069\pi$
$884$		0			0
$885$	−10.5987	+	1.82082i	−0.356272	+	0.0612063i
$886$		0			0
$887$		−	9.50529i		−	0.319157i	−0.987185	−	0.159578i	$-0.948987\pi$
$887$		−	9.50529i		−	0.319157i	0.987185	−	0.159578i	$-0.0510134\pi$
$888$		0			0
$889$		−	22.4065i		−	0.751490i
$890$		0			0
$891$	−0.506494	+	0.410038i	−0.0169682	+	0.0137368i
$892$		0			0
$893$		−	30.1221i		−	1.00800i
$894$		0			0
$895$	−13.4800			−0.450587
$896$		0			0
$897$	−12.1603	−	70.7832i	−0.406021	−	2.36338i
$898$		0			0
$899$	99.9106			3.33220
$900$		0			0
$901$		−	13.8994i		−	0.463056i
$902$		0			0
$903$	1.36290	+	7.93322i	0.0453545	+	0.264001i
$904$		0			0
$905$	−12.8237			−0.426275
$906$		0			0
$907$	−5.35309			−0.177746			−0.0888732	−	0.996043i	$-0.528327\pi$
$907$	−5.35309			−0.177746			−0.0888732	+	0.996043i	$0.528327\pi$
$908$		0			0
$909$	25.3084	−	8.96027i	0.839428	−	0.297193i
$910$		0			0
$911$			3.78624i			0.125444i	0.998031	+	0.0627219i	$0.0199781\pi$
$911$			3.78624i			0.125444i	−0.998031	+	0.0627219i	$0.980022\pi$
$912$		0			0
$913$		−	0.686805i		−	0.0227299i
$914$		0			0
$915$	−19.5784	+	3.36350i	−0.647241	+	0.111194i
$916$		0			0
$917$	−7.69011			−0.253950
$918$		0			0
$919$		−	36.7180i		−	1.21122i	−0.795763	−	0.605608i	$-0.792930\pi$
$919$		−	36.7180i		−	1.21122i	0.795763	−	0.605608i	$-0.207070\pi$
$920$		0			0
$921$	−7.80435	−	45.4278i	−0.257162	−	1.49690i
$922$		0			0
$923$	−25.2155			−0.829979
$924$		0			0
$925$	−0.716347			−0.0235533
$926$		0			0
$927$	16.9301	−	5.99398i	0.556058	−	0.196868i
$928$		0			0
$929$	−41.9179			−1.37528			−0.687641	−	0.726051i	$-0.741354\pi$
$929$	−41.9179			−1.37528			−0.687641	+	0.726051i	$0.741354\pi$
$930$		0			0
$931$	29.3939			0.963346
$932$		0			0
$933$	4.19320	+	24.4079i	0.137279	+	0.799080i
$934$		0			0
$935$			0.373906i			0.0122280i
$936$		0			0
$937$		−	15.1976i		−	0.496485i	−0.968698	−	0.248243i	$-0.920147\pi$
$937$		−	15.1976i		−	0.496485i	0.968698	−	0.248243i	$-0.0798530\pi$
$938$		0			0
$939$	25.6508	−	4.40673i	0.837084	−	0.143808i
$940$		0			0
$941$	11.5583			0.376788			0.188394	−	0.982093i	$-0.439672\pi$
$941$	11.5583			0.376788			0.188394	+	0.982093i	$0.439672\pi$
$942$		0			0
$943$			66.3399i			2.16032i
$944$		0			0
$945$	−5.92803	+	3.31907i	−0.192839	+	0.107969i
$946$		0			0
$947$		−	18.9368i		−	0.615365i	−0.951489	−	0.307682i	$-0.900447\pi$
$947$		−	18.9368i		−	0.615365i	0.951489	−	0.307682i	$-0.0995534\pi$
$948$		0			0
$949$		−	30.4034i		−	0.986938i
$950$		0			0
$951$	31.2216	−	5.36376i	1.01243	−	0.173932i
$952$		0			0
$953$			15.5479i			0.503644i	0.967774	+	0.251822i	$0.0810298\pi$
$953$			15.5479i			0.503644i	−0.967774	+	0.251822i	$0.918970\pi$
$954$		0			0
$955$	−14.4808			−0.468588
$956$		0			0
$957$	1.14145	−	0.196098i	0.0368979	−	0.00633893i
$958$		0			0
$959$		−	13.1041i		−	0.423155i
$960$		0			0
$961$	−86.0471			−2.77571
$962$		0			0
$963$	−10.9566	−	30.9471i	−0.353072	−	0.997258i
$964$		0			0
$965$	−12.3760			−0.398398
$966$		0			0
$967$	−28.9970			−0.932481			−0.466241	−	0.884658i	$-0.654392\pi$
$967$	−28.9970			−0.932481			−0.466241	+	0.884658i	$0.654392\pi$
$968$		0			0
$969$	−48.9766	+	8.41402i	−1.57336	+	0.270297i
$970$		0			0
$971$			37.8381i			1.21428i	0.794594	+	0.607141i	$0.207684\pi$
$971$			37.8381i			1.21428i	−0.794594	+	0.607141i	$0.792316\pi$
$972$		0			0
$973$	20.9315			0.671034
$974$		0			0
$975$	−7.75428	+	1.33216i	−0.248336	+	0.0426632i
$976$		0			0
$977$		−	52.7945i		−	1.68904i	−0.535520	−	0.844522i	$-0.679884\pi$
$977$		−	52.7945i		−	1.68904i	0.535520	−	0.844522i	$-0.320116\pi$
$978$		0			0
$979$			0.471511i			0.0150696i
$980$		0			0
$981$	4.12802	+	11.6597i	0.131798	+	0.372265i
$982$		0			0
$983$	−45.5811			−1.45381			−0.726906	−	0.686737i	$-0.759042\pi$
$983$	−45.5811			−1.45381			−0.726906	+	0.686737i	$0.759042\pi$
$984$		0			0
$985$	23.7952			0.758177
$986$		0			0
$987$	2.07883	+	12.1006i	0.0661700	+	0.385165i
$988$		0			0
$989$	32.4454			1.03170
$990$		0			0
$991$			16.1580i			0.513277i	0.966507	+	0.256638i	$0.0826149\pi$
$991$			16.1580i			0.513277i	−0.966507	+	0.256638i	$0.917385\pi$
$992$		0			0
$993$	5.59762	−	0.961652i	0.177635	−	0.0305171i
$994$		0			0
$995$	−13.1980			−0.418404
$996$		0			0
$997$	−1.18741			−0.0376057			−0.0188028	−	0.999823i	$-0.505985\pi$
$997$	−1.18741			−0.0376057			−0.0188028	+	0.999823i	$0.505985\pi$
$998$		0			0
$999$	1.81844	+	3.24783i	0.0575329	+	0.102757i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.a.401.26	yes	46
3.2	odd	2		4020.2.f.b.401.22	yes	46
67.66	odd	2		4020.2.f.b.401.21	yes	46
201.200	even	2	inner	4020.2.f.a.401.25	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.25	✓	46	201.200	even	2	inner
4020.2.f.a.401.26	yes	46	1.1	even	1	trivial
4020.2.f.b.401.21	yes	46	67.66	odd	2
4020.2.f.b.401.22	yes	46	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q+(0.293264 + 1.70704i) q^{3} -1.00000 q^{5} +1.30750i q^{7} +(-2.82799 + 1.00123i) q^{9} +O(q^{10})\)	\(q+(0.293264 + 1.70704i) q^{3} -1.00000 q^{5} +1.30750i q^{7} +(-2.82799 + 1.00123i) q^{9} -0.0724072 q^{11} +4.54252i q^{13} +(-0.293264 - 1.70704i) q^{15} +5.16393i q^{17} +5.55603 q^{19} +(-2.23195 + 0.383442i) q^{21} +9.12827i q^{23} +1.00000 q^{25} +(-2.53849 - 4.53388i) q^{27} +9.23488i q^{29} -10.8188i q^{31} +(-0.0212345 - 0.123602i) q^{33} -1.30750i q^{35} -0.716347 q^{37} +(-7.75428 + 1.33216i) q^{39} +7.26752 q^{41} -3.55439i q^{43} +(2.82799 - 1.00123i) q^{45} -5.42151i q^{47} +5.29045 q^{49} +(-8.81505 + 1.51440i) q^{51} -2.69163 q^{53} +0.0724072 q^{55} +(1.62938 + 9.48438i) q^{57} -6.20881i q^{59} -11.4692i q^{61} +(-1.30910 - 3.69759i) q^{63} -4.54252i q^{65} +(-0.243839 + 8.18172i) q^{67} +(-15.5824 + 2.67700i) q^{69} +5.55100i q^{71} -6.69308 q^{73} +(0.293264 + 1.70704i) q^{75} -0.0946722i q^{77} +8.46695i q^{79} +(6.99508 - 5.66294i) q^{81} +9.48531i q^{83} -5.16393i q^{85} +(-15.7643 + 2.70826i) q^{87} -6.51193i q^{89} -5.93933 q^{91} +(18.4682 - 3.17278i) q^{93} -5.55603 q^{95} +12.3494i q^{97} +(0.204767 - 0.0724963i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) \) 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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