LMFDB - Embedded newform 4020.2.f.a.401.11

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			401.11
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.a.401.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.30028 - 1.14423i) q^{3} -1.00000 q^{5} +3.38238i q^{7} +(0.381467 + 2.97565i) q^{9} +O(q^{10})$	\(q+(-1.30028 - 1.14423i) q^{3} -1.00000 q^{5} +3.38238i q^{7} +(0.381467 + 2.97565i) q^{9} -1.64322 q^{11} -5.60143i q^{13} +(1.30028 + 1.14423i) q^{15} -6.51058i q^{17} -0.200715 q^{19} +(3.87022 - 4.39804i) q^{21} +6.34593i q^{23} +1.00000 q^{25} +(2.90882 - 4.30567i) q^{27} +6.66051i q^{29} +4.74947i q^{31} +(2.13664 + 1.88022i) q^{33} -3.38238i q^{35} +8.17050 q^{37} +(-6.40934 + 7.28344i) q^{39} -7.39582 q^{41} +4.85695i q^{43} +(-0.381467 - 2.97565i) q^{45} +6.48161i q^{47} -4.44047 q^{49} +(-7.44961 + 8.46559i) q^{51} -2.27123 q^{53} +1.64322 q^{55} +(0.260986 + 0.229665i) q^{57} +2.26181i q^{59} -3.37575i q^{61} +(-10.0648 + 1.29027i) q^{63} +5.60143i q^{65} +(-0.760978 - 8.14990i) q^{67} +(7.26121 - 8.25150i) q^{69} -12.7332i q^{71} +6.58032 q^{73} +(-1.30028 - 1.14423i) q^{75} -5.55798i q^{77} -9.54473i q^{79} +(-8.70897 + 2.27023i) q^{81} -3.35806i q^{83} +6.51058i q^{85} +(7.62116 - 8.66054i) q^{87} -8.56116i q^{89} +18.9462 q^{91} +(5.43450 - 6.17565i) q^{93} +0.200715 q^{95} -0.600983i q^{97} +(-0.626833 - 4.88963i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) $ 46 * q - 46 * q^5 - 4 * q^9	$ 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) $ 46 * q - 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 - 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 - 4 * q^41 + 4 * q^45 - 62 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 - 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.30028	−	1.14423i	−0.750718	−	0.660623i
$3$	−1.30028	−	1.14423i	−0.750718	−	0.660623i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			3.38238i			1.27842i	0.769033	+	0.639209i	$0.220738\pi$
$7$			3.38238i			1.27842i	−0.769033	+	0.639209i	$0.779262\pi$
$8$		0			0
$9$	0.381467	+	2.97565i	0.127156	+	0.991883i
$10$		0			0
$11$	−1.64322			−0.495448			−0.247724	−	0.968831i	$-0.579683\pi$
$11$	−1.64322			−0.495448			−0.247724	+	0.968831i	$0.579683\pi$
$12$		0			0
$13$		−	5.60143i		−	1.55356i	−0.629773	−	0.776779i	$-0.716852\pi$
$13$		−	5.60143i		−	1.55356i	0.629773	−	0.776779i	$-0.283148\pi$
$14$		0			0
$15$	1.30028	+	1.14423i	0.335731	+	0.295439i
$16$		0			0
$17$		−	6.51058i		−	1.57905i	−0.613721	−	0.789523i	$-0.710328\pi$
$17$		−	6.51058i		−	1.57905i	0.613721	−	0.789523i	$-0.289672\pi$
$18$		0			0
$19$	−0.200715			−0.0460472			−0.0230236	−	0.999735i	$-0.507329\pi$
$19$	−0.200715			−0.0460472			−0.0230236	+	0.999735i	$0.507329\pi$
$20$		0			0
$21$	3.87022	−	4.39804i	0.844552	−	0.959732i
$22$		0			0
$23$			6.34593i			1.32322i	0.749849	+	0.661609i	$0.230126\pi$
$23$			6.34593i			1.32322i	−0.749849	+	0.661609i	$0.769874\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	2.90882	−	4.30567i	0.559802	−	0.828626i
$28$		0			0
$29$			6.66051i			1.23682i	0.785854	+	0.618412i	$0.212224\pi$
$29$			6.66051i			1.23682i	−0.785854	+	0.618412i	$0.787776\pi$
$30$		0			0
$31$			4.74947i			0.853030i	0.904480	+	0.426515i	$0.140259\pi$
$31$			4.74947i			0.853030i	−0.904480	+	0.426515i	$0.859741\pi$
$32$		0			0
$33$	2.13664	+	1.88022i	0.371942	+	0.327304i
$34$		0			0
$35$		−	3.38238i		−	0.571726i
$36$		0			0
$37$	8.17050			1.34322			0.671611	−	0.740904i	$-0.265603\pi$
$37$	8.17050			1.34322			0.671611	+	0.740904i	$0.265603\pi$
$38$		0			0
$39$	−6.40934	+	7.28344i	−1.02632	+	1.16628i
$40$		0			0
$41$	−7.39582			−1.15503			−0.577517	−	0.816379i	$-0.695978\pi$
$41$	−7.39582			−1.15503			−0.577517	+	0.816379i	$0.695978\pi$
$42$		0			0
$43$			4.85695i			0.740677i	0.928897	+	0.370339i	$0.120758\pi$
$43$			4.85695i			0.740677i	−0.928897	+	0.370339i	$0.879242\pi$
$44$		0			0
$45$	−0.381467	−	2.97565i	−0.0568658	−	0.443583i
$46$		0			0
$47$			6.48161i			0.945439i	0.881213	+	0.472720i	$0.156728\pi$
$47$			6.48161i			0.945439i	−0.881213	+	0.472720i	$0.843272\pi$
$48$		0			0
$49$	−4.44047			−0.634353
$50$		0			0
$51$	−7.44961	+	8.46559i	−1.04315	+	1.18542i
$52$		0			0
$53$	−2.27123			−0.311978			−0.155989	−	0.987759i	$-0.549856\pi$
$53$	−2.27123			−0.311978			−0.155989	+	0.987759i	$0.549856\pi$
$54$		0			0
$55$	1.64322			0.221571
$56$		0			0
$57$	0.260986	+	0.229665i	0.0345685	+	0.0304198i
$58$		0			0
$59$			2.26181i			0.294462i	0.989102	+	0.147231i	$0.0470361\pi$
$59$			2.26181i			0.294462i	−0.989102	+	0.147231i	$0.952964\pi$
$60$		0			0
$61$		−	3.37575i		−	0.432221i	−0.976369	−	0.216110i	$-0.930663\pi$
$61$		−	3.37575i		−	0.432221i	0.976369	−	0.216110i	$-0.0693371\pi$
$62$		0			0
$63$	−10.0648	+	1.29027i	−1.26804	+	0.162558i
$64$		0			0
$65$			5.60143i			0.694772i
$66$		0			0
$67$	−0.760978	−	8.14990i	−0.0929682	−	0.995669i
$67$	−0.760978	−	8.14990i	−0.0929682	−	0.995669i
$68$		0			0
$69$	7.26121	−	8.25150i	0.874147	−	0.993363i
$70$		0			0
$71$		−	12.7332i		−	1.51116i	−0.655059	−	0.755578i	$-0.727356\pi$
$71$		−	12.7332i		−	1.51116i	0.655059	−	0.755578i	$-0.272644\pi$
$72$		0			0
$73$	6.58032			0.770168			0.385084	−	0.922882i	$-0.374173\pi$
$73$	6.58032			0.770168			0.385084	+	0.922882i	$0.374173\pi$
$74$		0			0
$75$	−1.30028	−	1.14423i	−0.150144	−	0.132125i
$76$		0			0
$77$		−	5.55798i		−	0.633390i
$78$		0			0
$79$		−	9.54473i		−	1.07387i	−0.843625	−	0.536933i	$-0.819583\pi$
$79$		−	9.54473i		−	1.07387i	0.843625	−	0.536933i	$-0.180417\pi$
$80$		0			0
$81$	−8.70897	+	2.27023i	−0.967663	+	0.252247i
$82$		0			0
$83$		−	3.35806i		−	0.368595i	−0.982871	−	0.184297i	$-0.940999\pi$
$83$		−	3.35806i		−	0.368595i	0.982871	−	0.184297i	$-0.0590009\pi$
$84$		0			0
$85$			6.51058i			0.706171i
$86$		0			0
$87$	7.62116	−	8.66054i	0.817074	−	0.928507i
$88$		0			0
$89$		−	8.56116i		−	0.907481i	−0.891134	−	0.453741i	$-0.850089\pi$
$89$		−	8.56116i		−	0.907481i	0.891134	−	0.453741i	$-0.149911\pi$
$90$		0			0
$91$	18.9462			1.98610
$92$		0			0
$93$	5.43450	−	6.17565i	0.563531	−	0.640385i
$94$		0			0
$95$	0.200715			0.0205930
$96$		0			0
$97$		−	0.600983i		−	0.0610206i	−0.999534	−	0.0305103i	$-0.990287\pi$
$97$		−	0.600983i		−	0.0610206i	0.999534	−	0.0305103i	$-0.00971324\pi$
$98$		0			0
$99$	−0.626833	−	4.88963i	−0.0629991	−	0.491427i
$100$		0			0
$101$	−9.71076			−0.966256			−0.483128	−	0.875550i	$-0.660499\pi$
$101$	−9.71076			−0.966256			−0.483128	+	0.875550i	$0.660499\pi$
$102$		0			0
$103$	−2.21272			−0.218026			−0.109013	−	0.994040i	$-0.534769\pi$
$103$	−2.21272			−0.218026			−0.109013	+	0.994040i	$0.534769\pi$
$104$		0			0
$105$	−3.87022	+	4.39804i	−0.377695	+	0.429205i
$106$		0			0
$107$			4.58800i			0.443539i	0.975099	+	0.221770i	$0.0711833\pi$
$107$			4.58800i			0.443539i	−0.975099	+	0.221770i	$0.928817\pi$
$108$		0			0
$109$		−	4.54323i		−	0.435162i	−0.976042	−	0.217581i	$-0.930183\pi$
$109$		−	4.54323i		−	0.435162i	0.976042	−	0.217581i	$-0.0698167\pi$
$110$		0			0
$111$	−10.6240	−	9.34894i	−1.00838	−	0.887362i
$112$		0			0
$113$	−5.84587			−0.549933			−0.274967	−	0.961454i	$-0.588667\pi$
$113$	−5.84587			−0.549933			−0.274967	+	0.961454i	$0.588667\pi$
$114$		0			0
$115$		−	6.34593i		−	0.591761i
$116$		0			0
$117$	16.6679	−	2.13676i	1.54095	−	0.197544i
$118$		0			0
$119$	22.0212			2.01868
$120$		0			0
$121$	−8.29984			−0.754531
$122$		0			0
$123$	9.61666	+	8.46254i	0.867105	+	0.763041i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	17.4105			1.54493			0.772464	−	0.635058i	$-0.219024\pi$
$127$	17.4105			1.54493			0.772464	+	0.635058i	$0.219024\pi$
$128$		0			0
$129$	5.55747	−	6.31540i	0.489308	−	0.556040i
$130$		0			0
$131$		−	2.84207i		−	0.248313i	−0.992263	−	0.124157i	$-0.960377\pi$
$131$		−	2.84207i		−	0.248313i	0.992263	−	0.124157i	$-0.0396225\pi$
$132$		0			0
$133$		−	0.678895i		−	0.0588676i
$134$		0			0
$135$	−2.90882	+	4.30567i	−0.250351	+	0.370573i
$136$		0			0
$137$	−17.5858			−1.50245			−0.751227	−	0.660044i	$-0.770538\pi$
$137$	−17.5858			−1.50245			−0.751227	+	0.660044i	$0.770538\pi$
$138$		0			0
$139$		−	13.6292i		−	1.15602i	−0.816031	−	0.578008i	$-0.803830\pi$
$139$		−	13.6292i		−	1.15602i	0.816031	−	0.578008i	$-0.196170\pi$
$140$		0			0
$141$	7.41646	−	8.42792i	0.624579	−	0.709759i
$142$		0			0
$143$			9.20436i			0.769708i
$144$		0			0
$145$		−	6.66051i		−	0.553125i
$146$		0			0
$147$	5.77387	+	5.08093i	0.476221	+	0.419068i
$148$		0			0
$149$		−	23.2138i		−	1.90175i	−0.309581	−	0.950873i	$-0.600189\pi$
$149$		−	23.2138i		−	1.90175i	0.309581	−	0.950873i	$-0.399811\pi$
$150$		0			0
$151$	−17.7833			−1.44718			−0.723592	−	0.690228i	$-0.757510\pi$
$151$	−17.7833			−1.44718			−0.723592	+	0.690228i	$0.757510\pi$
$152$		0			0
$153$	19.3732	−	2.48357i	1.56623	−	0.200785i
$154$		0			0
$155$		−	4.74947i		−	0.381487i
$156$		0			0
$157$	7.75756			0.619120			0.309560	−	0.950880i	$-0.399818\pi$
$157$	7.75756			0.619120			0.309560	+	0.950880i	$0.399818\pi$
$158$		0			0
$159$	2.95324	+	2.59882i	0.234207	+	0.206099i
$160$		0			0
$161$	−21.4643			−1.69163
$162$		0			0
$163$	−1.58062			−0.123803			−0.0619017	−	0.998082i	$-0.519717\pi$
$163$	−1.58062			−0.123803			−0.0619017	+	0.998082i	$0.519717\pi$
$164$		0			0
$165$	−2.13664	−	1.88022i	−0.166338	−	0.146375i
$166$		0			0
$167$		−	17.4965i		−	1.35392i	−0.736022	−	0.676958i	$-0.763298\pi$
$167$		−	17.4965i		−	1.35392i	0.736022	−	0.676958i	$-0.236702\pi$
$168$		0			0
$169$	−18.3760			−1.41354
$170$		0			0
$171$	−0.0765663	−	0.597258i	−0.00585517	−	0.0456735i
$172$		0			0
$173$		−	15.2081i		−	1.15625i	−0.815949	−	0.578124i	$-0.803785\pi$
$173$		−	15.2081i		−	1.15625i	0.815949	−	0.578124i	$-0.196215\pi$
$174$		0			0
$175$			3.38238i			0.255684i
$176$		0			0
$177$	2.58803	−	2.94099i	0.194528	−	0.221058i
$178$		0			0
$179$	−2.75565			−0.205967			−0.102983	−	0.994683i	$-0.532839\pi$
$179$	−2.75565			−0.205967			−0.102983	+	0.994683i	$0.532839\pi$
$180$		0			0
$181$	13.6314			1.01322			0.506608	−	0.862176i	$-0.330899\pi$
$181$	13.6314			1.01322			0.506608	+	0.862176i	$0.330899\pi$
$182$		0			0
$183$	−3.86264	+	4.38943i	−0.285535	+	0.324476i
$184$		0			0
$185$	−8.17050			−0.600707
$186$		0			0
$187$			10.6983i			0.782336i
$188$		0			0
$189$	14.5634	+	9.83871i	1.05933	+	0.715661i
$190$		0			0
$191$	−2.27280			−0.164454			−0.0822270	−	0.996614i	$-0.526203\pi$
$191$	−2.27280			−0.164454			−0.0822270	+	0.996614i	$0.526203\pi$
$192$		0			0
$193$	−15.3493			−1.10487			−0.552434	−	0.833556i	$-0.686301\pi$
$193$	−15.3493			−1.10487			−0.552434	+	0.833556i	$0.686301\pi$
$194$		0			0
$195$	6.40934	−	7.28344i	0.458982	−	0.521578i
$196$		0			0
$197$	−2.25906			−0.160951			−0.0804755	−	0.996757i	$-0.525644\pi$
$197$	−2.25906			−0.160951			−0.0804755	+	0.996757i	$0.525644\pi$
$198$		0			0
$199$	23.8190			1.68848			0.844241	−	0.535964i	$-0.180052\pi$
$199$	23.8190			1.68848			0.844241	+	0.535964i	$0.180052\pi$
$200$		0			0
$201$	−8.33589	+	11.4679i	−0.587968	+	0.808884i
$202$		0			0
$203$	−22.5283			−1.58118
$204$		0			0
$205$	7.39582			0.516547
$206$		0			0
$207$	−18.8832	+	2.42076i	−1.31248	+	0.168255i
$208$		0			0
$209$	0.329819			0.0228140
$210$		0			0
$211$	−1.20526			−0.0829732			−0.0414866	−	0.999139i	$-0.513209\pi$
$211$	−1.20526			−0.0829732			−0.0414866	+	0.999139i	$0.513209\pi$
$212$		0			0
$213$	−14.5698	+	16.5568i	−0.998304	+	1.13445i
$214$		0			0
$215$		−	4.85695i		−	0.331241i
$216$		0			0
$217$	−16.0645			−1.09053
$218$		0			0
$219$	−8.55627	−	7.52941i	−0.578179	−	0.508790i
$220$		0			0
$221$	−36.4686			−2.45314
$222$		0			0
$223$	−24.6586			−1.65126			−0.825632	−	0.564209i	$-0.809181\pi$
$223$	−24.6586			−1.65126			−0.825632	+	0.564209i	$0.809181\pi$
$224$		0			0
$225$	0.381467	+	2.97565i	0.0254312	+	0.198377i
$226$		0			0
$227$		−	22.1163i		−	1.46791i	−0.679198	−	0.733955i	$-0.737672\pi$
$227$		−	22.1163i		−	1.46791i	0.679198	−	0.733955i	$-0.262328\pi$
$228$		0			0
$229$		−	12.8113i		−	0.846591i	−0.905992	−	0.423296i	$-0.860873\pi$
$229$		−	12.8113i		−	0.846591i	0.905992	−	0.423296i	$-0.139127\pi$
$230$		0			0
$231$	−6.35961	+	7.22694i	−0.418432	+	0.475498i
$232$		0			0
$233$	2.12589			0.139272			0.0696358	−	0.997572i	$-0.477816\pi$
$233$	2.12589			0.139272			0.0696358	+	0.997572i	$0.477816\pi$
$234$		0			0
$235$		−	6.48161i		−	0.422813i
$236$		0			0
$237$	−10.9214	+	12.4108i	−0.709420	+	0.806170i
$238$		0			0
$239$	−3.27146			−0.211613			−0.105806	−	0.994387i	$-0.533742\pi$
$239$	−3.27146			−0.211613			−0.105806	+	0.994387i	$0.533742\pi$
$240$		0			0
$241$	−30.2369			−1.94773			−0.973865	−	0.227130i	$-0.927066\pi$
$241$	−30.2369			−1.94773			−0.973865	+	0.227130i	$0.927066\pi$
$242$		0			0
$243$	13.9218	+	7.01314i	0.893082	+	0.449893i
$244$		0			0
$245$	4.44047			0.283691
$246$		0			0
$247$			1.12429i			0.0715370i
$248$		0			0
$249$	−3.84240	+	4.36642i	−0.243502	+	0.276711i
$250$		0			0
$251$	30.5392			1.92762			0.963809	−	0.266593i	$-0.0858980\pi$
$251$	30.5392			1.92762			0.963809	+	0.266593i	$0.0858980\pi$
$252$		0			0
$253$		−	10.4277i		−	0.655586i
$254$		0			0
$255$	7.44961	−	8.46559i	0.466513	−	0.530136i
$256$		0			0
$257$			18.0351i			1.12500i	0.826799	+	0.562498i	$0.190160\pi$
$257$			18.0351i			1.12500i	−0.826799	+	0.562498i	$0.809840\pi$
$258$		0			0
$259$			27.6357i			1.71720i
$260$		0			0
$261$	−19.8193	+	2.54077i	−1.22679	+	0.157269i
$262$		0			0
$263$			15.3556i			0.946868i	0.880829	+	0.473434i	$0.156986\pi$
$263$			15.3556i			0.946868i	−0.880829	+	0.473434i	$0.843014\pi$
$264$		0			0
$265$	2.27123			0.139521
$266$		0			0
$267$	−9.79595	+	11.1319i	−0.599503	+	0.681263i
$268$		0			0
$269$			4.51333i			0.275183i	0.990489	+	0.137591i	$0.0439360\pi$
$269$			4.51333i			0.275183i	−0.990489	+	0.137591i	$0.956064\pi$
$270$		0			0
$271$		−	20.3425i		−	1.23572i	−0.786288	−	0.617860i	$-0.788000\pi$
$271$		−	20.3425i		−	1.23572i	0.786288	−	0.617860i	$-0.212000\pi$
$272$		0			0
$273$	−24.6353	−	21.6788i	−1.49100	−	1.31206i
$274$		0			0
$275$	−1.64322			−0.0990897
$276$		0			0
$277$	17.1745			1.03191			0.515956	−	0.856615i	$-0.327437\pi$
$277$	17.1745			1.03191			0.515956	+	0.856615i	$0.327437\pi$
$278$		0			0
$279$	−14.1328	+	1.81177i	−0.846106	+	0.108468i
$280$		0			0
$281$	−3.10682			−0.185337			−0.0926686	−	0.995697i	$-0.529540\pi$
$281$	−3.10682			−0.185337			−0.0926686	+	0.995697i	$0.529540\pi$
$282$		0			0
$283$	−5.91507			−0.351614			−0.175807	−	0.984425i	$-0.556254\pi$
$283$	−5.91507			−0.351614			−0.175807	+	0.984425i	$0.556254\pi$
$284$		0			0
$285$	−0.260986	−	0.229665i	−0.0154595	−	0.0136042i
$286$		0			0
$287$		−	25.0155i		−	1.47662i
$288$		0			0
$289$	−25.3876			−1.49339
$290$		0			0
$291$	−0.687664	+	0.781448i	−0.0403116	+	0.0458093i
$292$		0			0
$293$			13.0092i			0.760004i	0.924986	+	0.380002i	$0.124077\pi$
$293$			13.0092i			0.760004i	−0.924986	+	0.380002i	$0.875923\pi$
$294$		0			0
$295$		−	2.26181i		−	0.131688i
$296$		0			0
$297$	−4.77981	+	7.07515i	−0.277353	+	0.410542i
$298$		0			0
$299$	35.5463			2.05569
$300$		0			0
$301$	−16.4280			−0.946895
$302$		0			0
$303$	12.6267	+	11.1114i	0.725386	+	0.638331i
$304$		0			0
$305$			3.37575i			0.193295i
$306$		0			0
$307$	−17.8574			−1.01918			−0.509589	−	0.860418i	$-0.670202\pi$
$307$	−17.8574			−1.01918			−0.509589	+	0.860418i	$0.670202\pi$
$308$		0			0
$309$	2.87716	+	2.53186i	0.163676	+	0.144033i
$310$		0			0
$311$	−7.08019			−0.401481			−0.200740	−	0.979644i	$-0.564335\pi$
$311$	−7.08019			−0.401481			−0.200740	+	0.979644i	$0.564335\pi$
$312$		0			0
$313$		−	28.6572i		−	1.61980i	−0.586567	−	0.809901i	$-0.699521\pi$
$313$		−	28.6572i		−	1.61980i	0.586567	−	0.809901i	$-0.300479\pi$
$314$		0			0
$315$	10.0648	−	1.29027i	0.567085	−	0.0726983i
$316$		0			0
$317$			6.50941i			0.365605i	0.983150	+	0.182802i	$0.0585169\pi$
$317$			6.50941i			0.365605i	−0.983150	+	0.182802i	$0.941483\pi$
$318$		0			0
$319$		−	10.9447i		−	0.612783i
$320$		0			0
$321$	5.24974	−	5.96570i	0.293012	−	0.332973i
$322$		0			0
$323$			1.30677i			0.0727107i
$324$		0			0
$325$		−	5.60143i		−	0.310712i
$326$		0			0
$327$	−5.19851	+	5.90748i	−0.287478	+	0.326684i
$328$		0			0
$329$	−21.9232			−1.20867
$330$		0			0
$331$		−	25.0674i		−	1.37783i	−0.724842	−	0.688915i	$-0.758087\pi$
$331$		−	25.0674i		−	1.37783i	0.724842	−	0.688915i	$-0.241913\pi$
$332$		0			0
$333$	3.11678	+	24.3125i	0.170798	+	1.33232i
$334$		0			0
$335$	0.760978	+	8.14990i	0.0415767	+	0.445277i
$336$		0			0
$337$		−	27.8993i		−	1.51977i	−0.650058	−	0.759885i	$-0.725255\pi$
$337$		−	27.8993i		−	1.51977i	0.650058	−	0.759885i	$-0.274745\pi$
$338$		0			0
$339$	7.60128	+	6.68903i	0.412845	+	0.363298i
$340$		0			0
$341$		−	7.80441i		−	0.422632i
$342$		0			0
$343$			8.65728i			0.467449i
$344$		0			0
$345$	−7.26121	+	8.25150i	−0.390930	+	0.444246i
$346$		0			0
$347$	−3.10852			−0.166874			−0.0834370	−	0.996513i	$-0.526590\pi$
$347$	−3.10852			−0.166874			−0.0834370	+	0.996513i	$0.526590\pi$
$348$		0			0
$349$	−12.0666			−0.645909			−0.322954	−	0.946415i	$-0.604676\pi$
$349$	−12.0666			−0.645909			−0.322954	+	0.946415i	$0.604676\pi$
$350$		0			0
$351$	−24.1179	−	16.2935i	−1.28732	−	0.869685i
$352$		0			0
$353$	33.8530			1.80181			0.900906	−	0.434014i	$-0.142903\pi$
$353$	33.8530			1.80181			0.900906	+	0.434014i	$0.142903\pi$
$354$		0			0
$355$			12.7332i			0.675810i
$356$		0			0
$357$	−28.6338	−	25.1974i	−1.51546	−	1.33359i
$358$		0			0
$359$			6.99805i			0.369343i	0.982800	+	0.184671i	$0.0591221\pi$
$359$			6.99805i			0.369343i	−0.982800	+	0.184671i	$0.940878\pi$
$360$		0			0
$361$	−18.9597			−0.997880
$362$		0			0
$363$	10.7921	+	9.49694i	0.566440	+	0.498460i
$364$		0			0
$365$	−6.58032			−0.344429
$366$		0			0
$367$		−	12.8119i		−	0.668773i	−0.942436	−	0.334387i	$-0.891471\pi$
$367$		−	12.8119i		−	0.668773i	0.942436	−	0.334387i	$-0.108529\pi$
$368$		0			0
$369$	−2.82127	−	22.0074i	−0.146869	−	1.14566i
$370$		0			0
$371$		−	7.68216i		−	0.398838i
$372$		0			0
$373$			7.33120i			0.379595i	0.981823	+	0.189798i	$0.0607832\pi$
$373$			7.33120i			0.379595i	−0.981823	+	0.189798i	$0.939217\pi$
$374$		0			0
$375$	1.30028	+	1.14423i	0.0671463	+	0.0590879i
$376$		0			0
$377$	37.3084			1.92148
$378$		0			0
$379$			24.7249i			1.27003i	0.772498	+	0.635017i	$0.219007\pi$
$379$			24.7249i			1.27003i	−0.772498	+	0.635017i	$0.780993\pi$
$380$		0			0
$381$	−22.6385	−	19.9216i	−1.15981	−	1.02061i
$382$		0			0
$383$	22.5672			1.15313			0.576566	−	0.817051i	$-0.304392\pi$
$383$	22.5672			1.15313			0.576566	+	0.817051i	$0.304392\pi$
$384$		0			0
$385$			5.55798i			0.283261i
$386$		0			0
$387$	−14.4526	+	1.85277i	−0.734665	+	0.0941814i
$388$		0			0
$389$			5.37921i			0.272737i	0.990658	+	0.136368i	$0.0435431\pi$
$389$			5.37921i			0.272737i	−0.990658	+	0.136368i	$0.956457\pi$
$390$		0			0
$391$	41.3156			2.08942
$392$		0			0
$393$	−3.25199	+	3.69550i	−0.164041	+	0.186413i
$394$		0			0
$395$			9.54473i			0.480247i
$396$		0			0
$397$	−2.58412			−0.129693			−0.0648467	−	0.997895i	$-0.520656\pi$
$397$	−2.58412			−0.129693			−0.0648467	+	0.997895i	$0.520656\pi$
$398$		0			0
$399$	−0.776813	+	0.882755i	−0.0388893	+	0.0441930i
$400$		0			0
$401$	27.4915			1.37286			0.686429	−	0.727197i	$-0.259177\pi$
$401$	27.4915			1.37286			0.686429	+	0.727197i	$0.259177\pi$
$402$		0			0
$403$	26.6038			1.32523
$404$		0			0
$405$	8.70897	−	2.27023i	0.432752	−	0.112808i
$406$		0			0
$407$	−13.4259			−0.665497
$408$		0			0
$409$			35.1794i			1.73951i	0.493485	+	0.869754i	$0.335723\pi$
$409$			35.1794i			1.73951i	−0.493485	+	0.869754i	$0.664277\pi$
$410$		0			0
$411$	22.8665	+	20.1222i	1.12792	+	0.992555i
$412$		0			0
$413$	−7.65029			−0.376446
$414$		0			0
$415$			3.35806i			0.164841i
$416$		0			0
$417$	−15.5950	+	17.7218i	−0.763690	+	0.867842i
$418$		0			0
$419$		−	22.1375i		−	1.08149i	−0.841187	−	0.540744i	$-0.818143\pi$
$419$		−	22.1375i		−	1.08149i	0.841187	−	0.540744i	$-0.181857\pi$
$420$		0			0
$421$	31.3294			1.52690			0.763450	−	0.645867i	$-0.223504\pi$
$421$	31.3294			1.52690			0.763450	+	0.645867i	$0.223504\pi$
$422$		0			0
$423$	−19.2870	+	2.47252i	−0.937765	+	0.120218i
$424$		0			0
$425$		−	6.51058i		−	0.315809i
$426$		0			0
$427$	11.4181			0.552559
$428$		0			0
$429$	10.5319	−	11.9683i	0.508486	−	0.577834i
$430$		0			0
$431$		−	28.9743i		−	1.39564i	−0.716271	−	0.697822i	$-0.754153\pi$
$431$		−	28.9743i		−	1.39564i	0.716271	−	0.697822i	$-0.245847\pi$
$432$		0			0
$433$		−	30.7583i		−	1.47815i	−0.673623	−	0.739075i	$-0.735263\pi$
$433$		−	30.7583i		−	1.47815i	0.673623	−	0.739075i	$-0.264737\pi$
$434$		0			0
$435$	−7.62116	+	8.66054i	−0.365407	+	0.415241i
$436$		0			0
$437$		−	1.27372i		−	0.0609305i
$438$		0			0
$439$	−24.4994			−1.16929			−0.584646	−	0.811288i	$-0.698767\pi$
$439$	−24.4994			−1.16929			−0.584646	+	0.811288i	$0.698767\pi$
$440$		0			0
$441$	−1.69390	−	13.2133i	−0.0806617	−	0.629204i
$442$		0			0
$443$	16.2216			0.770710			0.385355	−	0.922768i	$-0.374079\pi$
$443$	16.2216			0.770710			0.385355	+	0.922768i	$0.374079\pi$
$444$		0			0
$445$			8.56116i			0.405838i
$446$		0			0
$447$	−26.5619	+	30.1844i	−1.25634	+	1.42768i
$448$		0			0
$449$		−	22.1378i		−	1.04475i	−0.852716	−	0.522374i	$-0.825046\pi$
$449$		−	22.1378i		−	1.04475i	0.852716	−	0.522374i	$-0.174954\pi$
$450$		0			0
$451$	12.1529			0.572260
$452$		0			0
$453$	23.1233	+	20.3482i	1.08643	+	0.956042i
$454$		0			0
$455$	−18.9462			−0.888209
$456$		0			0
$457$	13.9497			0.652541			0.326271	−	0.945276i	$-0.394208\pi$
$457$	13.9497			0.652541			0.326271	+	0.945276i	$0.394208\pi$
$458$		0			0
$459$	−28.0324	−	18.9381i	−1.30844	−	0.883953i
$460$		0			0
$461$		−	37.0288i		−	1.72460i	−0.506394	−	0.862302i	$-0.669022\pi$
$461$		−	37.0288i		−	1.72460i	0.506394	−	0.862302i	$-0.330978\pi$
$462$		0			0
$463$			29.1664i			1.35548i	0.735303	+	0.677738i	$0.237040\pi$
$463$			29.1664i			1.35548i	−0.735303	+	0.677738i	$0.762960\pi$
$464$		0			0
$465$	−5.43450	+	6.17565i	−0.252019	+	0.286389i
$466$		0			0
$467$			23.1347i			1.07054i	0.844679	+	0.535272i	$0.179791\pi$
$467$			23.1347i			1.07054i	−0.844679	+	0.535272i	$0.820209\pi$
$468$		0			0
$469$	27.5660	−	2.57391i	1.27288	−	0.118852i
$470$		0			0
$471$	−10.0870	−	8.87644i	−0.464785	−	0.409005i
$472$		0			0
$473$		−	7.98101i		−	0.366967i
$474$		0			0
$475$	−0.200715			−0.00920945
$476$		0			0
$477$	−0.866401	−	6.75839i	−0.0396698	−	0.309445i
$478$		0			0
$479$		−	35.1711i		−	1.60701i	−0.595299	−	0.803505i	$-0.702966\pi$
$479$		−	35.1711i		−	1.60701i	0.595299	−	0.803505i	$-0.297034\pi$
$480$		0			0
$481$		−	45.7665i		−	2.08677i
$482$		0			0
$483$	27.9097	+	24.5602i	1.26993	+	1.11753i
$484$		0			0
$485$			0.600983i			0.0272892i
$486$		0			0
$487$			36.3041i			1.64509i	0.568697	+	0.822547i	$0.307448\pi$
$487$			36.3041i			1.64509i	−0.568697	+	0.822547i	$0.692552\pi$
$488$		0			0
$489$	2.05525	+	1.80859i	0.0929415	+	0.0817874i
$490$		0			0
$491$		−	42.2424i		−	1.90637i	−0.302386	−	0.953186i	$-0.597783\pi$
$491$		−	42.2424i		−	1.90637i	0.302386	−	0.953186i	$-0.402217\pi$
$492$		0			0
$493$	43.3637			1.95300
$494$		0			0
$495$	0.626833	+	4.88963i	0.0281741	+	0.219773i
$496$		0			0
$497$	43.0686			1.93189
$498$		0			0
$499$			31.1477i			1.39436i	0.716894	+	0.697182i	$0.245563\pi$
$499$			31.1477i			1.39436i	−0.716894	+	0.697182i	$0.754437\pi$
$500$		0			0
$501$	−20.0200	+	22.7503i	−0.894427	+	1.01641i
$502$		0			0
$503$	−10.9780			−0.489485			−0.244743	−	0.969588i	$-0.578704\pi$
$503$	−10.9780			−0.489485			−0.244743	+	0.969588i	$0.578704\pi$
$504$		0			0
$505$	9.71076			0.432123
$506$		0			0
$507$	23.8940	+	21.0264i	1.06117	+	0.933817i
$508$		0			0
$509$			27.2886i			1.20955i	0.796397	+	0.604774i	$0.206736\pi$
$509$			27.2886i			1.20955i	−0.796397	+	0.604774i	$0.793264\pi$
$510$		0			0
$511$			22.2571i			0.984596i
$512$		0			0
$513$	−0.583844	+	0.864214i	−0.0257773	+	0.0381560i
$514$		0			0
$515$	2.21272			0.0975040
$516$		0			0
$517$		−	10.6507i		−	0.468416i
$518$		0			0
$519$	−17.4016	+	19.7748i	−0.763844	+	0.868017i
$520$		0			0
$521$	−12.5995			−0.551995			−0.275998	−	0.961158i	$-0.589008\pi$
$521$	−12.5995			−0.551995			−0.275998	+	0.961158i	$0.589008\pi$
$522$		0			0
$523$	19.7879			0.865265			0.432632	−	0.901570i	$-0.357585\pi$
$523$	19.7879			0.865265			0.432632	+	0.901570i	$0.357585\pi$
$524$		0			0
$525$	3.87022	−	4.39804i	0.168910	−	0.191946i
$526$		0			0
$527$	30.9218			1.34697
$528$		0			0
$529$	−17.2708			−0.750904
$530$		0			0
$531$	−6.73035	+	0.862806i	−0.292072	+	0.0374426i
$532$		0			0
$533$			41.4272i			1.79441i
$534$		0			0
$535$		−	4.58800i		−	0.198357i
$536$		0			0
$537$	3.58312	+	3.15310i	0.154623	+	0.136066i
$538$		0			0
$539$	7.29666			0.314289
$540$		0			0
$541$		−	9.01068i		−	0.387399i	−0.981061	−	0.193700i	$-0.937951\pi$
$541$		−	9.01068i		−	0.387399i	0.981061	−	0.193700i	$-0.0620488\pi$
$542$		0			0
$543$	−17.7247	−	15.5975i	−0.760640	−	0.669354i
$544$		0			0
$545$			4.54323i			0.194611i
$546$		0			0
$547$			2.03235i			0.0868970i	0.999056	+	0.0434485i	$0.0138344\pi$
$547$			2.03235i			0.0868970i	−0.999056	+	0.0434485i	$0.986166\pi$
$548$		0			0
$549$	10.0450	−	1.28774i	0.428712	−	0.0549594i
$550$		0			0
$551$		−	1.33687i		−	0.0569524i
$552$		0			0
$553$	32.2839			1.37285
$554$		0			0
$555$	10.6240	+	9.34894i	0.450962	+	0.396841i
$556$		0			0
$557$		−	4.61146i		−	0.195394i	−0.995216	−	0.0976969i	$-0.968852\pi$
$557$		−	4.61146i		−	0.195394i	0.995216	−	0.0976969i	$-0.0311476\pi$
$558$		0			0
$559$	27.2058			1.15068
$560$		0			0
$561$	12.2413	−	13.9108i	0.516829	−	0.587314i
$562$		0			0
$563$	−18.4062			−0.775728			−0.387864	−	0.921717i	$-0.626787\pi$
$563$	−18.4062			−0.775728			−0.387864	+	0.921717i	$0.626787\pi$
$564$		0			0
$565$	5.84587			0.245938
$566$		0			0
$567$	−7.67876	−	29.4570i	−0.322478	−	1.23708i
$568$		0			0
$569$		−	2.10365i		−	0.0881894i	−0.999027	−	0.0440947i	$-0.985960\pi$
$569$		−	2.10365i		−	0.0881894i	0.999027	−	0.0440947i	$-0.0140403\pi$
$570$		0			0
$571$	−14.6904			−0.614776			−0.307388	−	0.951584i	$-0.599455\pi$
$571$	−14.6904			−0.614776			−0.307388	+	0.951584i	$0.599455\pi$
$572$		0			0
$573$	2.95528	+	2.60061i	0.123459	+	0.108642i
$574$		0			0
$575$			6.34593i			0.264643i
$576$		0			0
$577$		−	25.9530i		−	1.08044i	−0.841524	−	0.540219i	$-0.818341\pi$
$577$		−	25.9530i		−	1.08044i	0.841524	−	0.540219i	$-0.181659\pi$
$578$		0			0
$579$	19.9585	+	17.5632i	0.829445	+	0.729901i
$580$		0			0
$581$	11.3582			0.471218
$582$		0			0
$583$	3.73212			0.154569
$584$		0			0
$585$	−16.6679	+	2.13676i	−0.689132	+	0.0883443i
$586$		0			0
$587$	2.44530			0.100928			0.0504641	−	0.998726i	$-0.483930\pi$
$587$	2.44530			0.100928			0.0504641	+	0.998726i	$0.483930\pi$
$588$		0			0
$589$		−	0.953291i		−	0.0392797i
$590$		0			0
$591$	2.93741	+	2.58488i	0.120829	+	0.106328i
$592$		0			0
$593$	−0.197091			−0.00809356			−0.00404678	−	0.999992i	$-0.501288\pi$
$593$	−0.197091			−0.00809356			−0.00404678	+	0.999992i	$0.501288\pi$
$594$		0			0
$595$	−22.0212			−0.902782
$596$		0			0
$597$	−30.9714	−	27.2544i	−1.26757	−	1.11545i
$598$		0			0
$599$	−29.9565			−1.22399			−0.611994	−	0.790863i	$-0.709632\pi$
$599$	−29.9565			−1.22399			−0.611994	+	0.790863i	$0.709632\pi$
$600$		0			0
$601$	−41.8495			−1.70708			−0.853539	−	0.521028i	$-0.825549\pi$
$601$	−41.8495			−1.70708			−0.853539	+	0.521028i	$0.825549\pi$
$602$		0			0
$603$	23.9610	−	5.37332i	0.975766	−	0.218819i
$604$		0			0
$605$	8.29984			0.337436
$606$		0			0
$607$	−21.5128			−0.873177			−0.436588	−	0.899661i	$-0.643813\pi$
$607$	−21.5128			−0.873177			−0.436588	+	0.899661i	$0.643813\pi$
$608$		0			0
$609$	29.2932	+	25.7776i	1.18702	+	1.04456i
$610$		0			0
$611$	36.3063			1.46879
$612$		0			0
$613$	−6.40089			−0.258530			−0.129265	−	0.991610i	$-0.541262\pi$
$613$	−6.40089			−0.258530			−0.129265	+	0.991610i	$0.541262\pi$
$614$		0			0
$615$	−9.61666	−	8.46254i	−0.387781	−	0.341242i
$616$		0			0
$617$		−	29.0749i		−	1.17051i	−0.810849	−	0.585256i	$-0.800994\pi$
$617$		−	29.0749i		−	1.17051i	0.810849	−	0.585256i	$-0.199006\pi$
$618$		0			0
$619$	−35.0758			−1.40982			−0.704908	−	0.709299i	$-0.749012\pi$
$619$	−35.0758			−1.40982			−0.704908	+	0.709299i	$0.749012\pi$
$620$		0			0
$621$	27.3235	+	18.4591i	1.09645	+	0.740740i
$622$		0			0
$623$	28.9571			1.16014
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−0.428857	−	0.377389i	−0.0171269	−	0.0150715i
$628$		0			0
$629$		−	53.1947i		−	2.12101i
$630$		0			0
$631$			31.6271i			1.25906i	0.776978	+	0.629528i	$0.216752\pi$
$631$			31.6271i			1.25906i	−0.776978	+	0.629528i	$0.783248\pi$
$632$		0			0
$633$	1.56717	+	1.37909i	0.0622895	+	0.0548140i
$634$		0			0
$635$	−17.4105			−0.690913
$636$		0			0
$637$			24.8730i			0.985504i
$638$		0			0
$639$	37.8896	−	4.85731i	1.49889	−	0.192152i
$640$		0			0
$641$	−12.0966			−0.477789			−0.238894	−	0.971046i	$-0.576785\pi$
$641$	−12.0966			−0.477789			−0.238894	+	0.971046i	$0.576785\pi$
$642$		0			0
$643$	40.9178			1.61364			0.806821	−	0.590796i	$-0.201186\pi$
$643$	40.9178			1.61364			0.806821	+	0.590796i	$0.201186\pi$
$644$		0			0
$645$	−5.55747	+	6.31540i	−0.218825	+	0.248669i
$646$		0			0
$647$	29.2904			1.15152			0.575762	−	0.817618i	$-0.304706\pi$
$647$	29.2904			1.15152			0.575762	+	0.817618i	$0.304706\pi$
$648$		0			0
$649$		−	3.71664i		−	0.145891i
$650$		0			0
$651$	20.8884	+	18.3815i	0.818680	+	0.720428i
$652$		0			0
$653$	−27.4780			−1.07530			−0.537648	−	0.843169i	$-0.680687\pi$
$653$	−27.4780			−1.07530			−0.537648	+	0.843169i	$0.680687\pi$
$654$		0			0
$655$			2.84207i			0.111049i
$656$		0			0
$657$	2.51018	+	19.5807i	0.0979313	+	0.763916i
$658$		0			0
$659$			23.9762i			0.933980i	0.884263	+	0.466990i	$0.154662\pi$
$659$			23.9762i			0.933980i	−0.884263	+	0.466990i	$0.845338\pi$
$660$		0			0
$661$			8.03559i			0.312548i	0.987714	+	0.156274i	$0.0499483\pi$
$661$			8.03559i			0.312548i	−0.987714	+	0.156274i	$0.950052\pi$
$662$		0			0
$663$	47.4194	+	41.7285i	1.84162	+	1.62060i
$664$		0			0
$665$			0.678895i			0.0263264i
$666$		0			0
$667$	−42.2671			−1.63659
$668$		0			0
$669$	32.0632	+	28.2152i	1.23963	+	1.09086i
$670$		0			0
$671$			5.54709i			0.214143i
$672$		0			0
$673$		−	2.62739i		−	0.101279i	−0.998717	−	0.0506393i	$-0.983874\pi$
$673$		−	2.62739i		−	0.101279i	0.998717	−	0.0506393i	$-0.0161259\pi$
$674$		0			0
$675$	2.90882	−	4.30567i	0.111960	−	0.165725i
$676$		0			0
$677$	1.98247			0.0761925			0.0380963	−	0.999274i	$-0.487871\pi$
$677$	1.98247			0.0761925			0.0380963	+	0.999274i	$0.487871\pi$
$678$		0			0
$679$	2.03275			0.0780099
$680$		0			0
$681$	−25.3062	+	28.7574i	−0.969735	+	1.10199i
$682$		0			0
$683$	25.2868			0.967573			0.483786	−	0.875186i	$-0.339261\pi$
$683$	25.2868			0.967573			0.483786	+	0.875186i	$0.339261\pi$
$684$		0			0
$685$	17.5858			0.671918
$686$		0			0
$687$	−14.6590	+	16.6582i	−0.559277	+	0.635552i
$688$		0			0
$689$			12.7222i			0.484675i
$690$		0			0
$691$	−28.7226			−1.09266			−0.546329	−	0.837570i	$-0.683975\pi$
$691$	−28.7226			−1.09266			−0.546329	+	0.837570i	$0.683975\pi$
$692$		0			0
$693$	16.5386	−	2.12019i	0.628249	−	0.0805392i
$694$		0			0
$695$			13.6292i			0.516986i
$696$		0			0
$697$			48.1511i			1.82385i
$698$		0			0
$699$	−2.76425	−	2.43251i	−0.104554	−	0.0920059i
$700$		0			0
$701$	−48.6306			−1.83675			−0.918375	−	0.395711i	$-0.870498\pi$
$701$	−48.6306			−1.83675			−0.918375	+	0.395711i	$0.870498\pi$
$702$		0			0
$703$	−1.63994			−0.0618517
$704$		0			0
$705$	−7.41646	+	8.42792i	−0.279320	+	0.317414i
$706$		0			0
$707$		−	32.8454i		−	1.23528i
$708$		0			0
$709$	−12.8857			−0.483931			−0.241966	−	0.970285i	$-0.577792\pi$
$709$	−12.8857			−0.483931			−0.241966	+	0.970285i	$0.577792\pi$
$710$		0			0
$711$	28.4017	−	3.64100i	1.06515	−	0.136548i
$712$		0			0
$713$	−30.1398			−1.12874
$714$		0			0
$715$		−	9.20436i		−	0.344224i
$716$		0			0
$717$	4.25382	+	3.74330i	0.158862	+	0.139796i
$718$		0			0
$719$			21.9257i			0.817690i	0.912604	+	0.408845i	$0.134068\pi$
$719$			21.9257i			0.817690i	−0.912604	+	0.408845i	$0.865932\pi$
$720$		0			0
$721$		−	7.48425i		−	0.278728i
$722$		0			0
$723$	39.3165	+	34.5980i	1.46220	+	1.28671i
$724$		0			0
$725$			6.66051i			0.247365i
$726$		0			0
$727$		−	22.4712i		−	0.833410i	−0.909042	−	0.416705i	$-0.863185\pi$
$727$		−	22.4712i		−	0.833410i	0.909042	−	0.416705i	$-0.136815\pi$
$728$		0			0
$729$	−10.0776	−	25.0488i	−0.373244	−	0.927733i
$730$		0			0
$731$	31.6215			1.16956
$732$		0			0
$733$		−	5.43248i		−	0.200653i	−0.994955	−	0.100327i	$-0.968011\pi$
$733$		−	5.43248i		−	0.200653i	0.994955	−	0.100327i	$-0.0319887\pi$
$734$		0			0
$735$	−5.77387	−	5.08093i	−0.212972	−	0.187413i
$736$		0			0
$737$	1.25045	+	13.3921i	0.0460610	+	0.493303i
$738$		0			0
$739$		−	4.14510i		−	0.152480i	−0.997089	−	0.0762400i	$-0.975708\pi$
$739$		−	4.14510i		−	0.152480i	0.997089	−	0.0762400i	$-0.0242915\pi$
$740$		0			0
$741$	1.28645	−	1.46190i	0.0472590	−	0.0537042i
$742$		0			0
$743$		−	6.15641i		−	0.225857i	−0.993603	−	0.112928i	$-0.963977\pi$
$743$		−	6.15641i		−	0.225857i	0.993603	−	0.112928i	$-0.0360231\pi$
$744$		0			0
$745$			23.2138i			0.850487i
$746$		0			0
$747$	9.99240	−	1.28099i	0.365603	−	0.0468690i
$748$		0			0
$749$	−15.5184			−0.567028
$750$		0			0
$751$	−26.9437			−0.983191			−0.491595	−	0.870824i	$-0.663586\pi$
$751$	−26.9437			−0.983191			−0.491595	+	0.870824i	$0.663586\pi$
$752$		0			0
$753$	−39.7096	−	34.9439i	−1.44710	−	1.27343i
$754$		0			0
$755$	17.7833			0.647200
$756$		0			0
$757$		−	22.5096i		−	0.818127i	−0.912506	−	0.409063i	$-0.865856\pi$
$757$		−	22.5096i		−	0.818127i	0.912506	−	0.409063i	$-0.134144\pi$
$758$		0			0
$759$	−11.9317	+	13.5590i	−0.433095	+	0.492160i
$760$		0			0
$761$			3.12182i			0.113166i	0.998398	+	0.0565829i	$0.0180205\pi$
$761$			3.12182i			0.113166i	−0.998398	+	0.0565829i	$0.981979\pi$
$762$		0			0
$763$	15.3669			0.556320
$764$		0			0
$765$	−19.3732	+	2.48357i	−0.700439	+	0.0897938i
$766$		0			0
$767$	12.6694			0.457464
$768$		0			0
$769$		−	3.72947i		−	0.134488i	−0.997737	−	0.0672440i	$-0.978579\pi$
$769$		−	3.72947i		−	0.134488i	0.997737	−	0.0672440i	$-0.0214206\pi$
$770$		0			0
$771$	20.6363	−	23.4507i	0.743198	−	0.844555i
$772$		0			0
$773$		−	23.7794i		−	0.855287i	−0.903948	−	0.427643i	$-0.859344\pi$
$773$		−	23.7794i		−	0.855287i	0.903948	−	0.427643i	$-0.140656\pi$
$774$		0			0
$775$			4.74947i			0.170606i
$776$		0			0
$777$	31.6217	−	35.9342i	1.13442	−	1.28913i
$778$		0			0
$779$	1.48445			0.0531861
$780$		0			0
$781$			20.9234i			0.748700i
$782$		0			0
$783$	28.6779	+	19.3742i	1.02487	+	0.692377i
$784$		0			0
$785$	−7.75756			−0.276879
$786$		0			0
$787$			29.6169i			1.05573i	0.849328	+	0.527865i	$0.177007\pi$
$787$			29.6169i			1.05573i	−0.849328	+	0.527865i	$0.822993\pi$
$788$		0			0
$789$	17.5704	−	19.9666i	0.625523	−	0.710831i
$790$		0			0
$791$		−	19.7729i		−	0.703045i
$792$		0			0
$793$	−18.9090			−0.671480
$794$		0			0
$795$	−2.95324	−	2.59882i	−0.104741	−	0.0921705i
$796$		0			0
$797$			4.99560i			0.176953i	0.996078	+	0.0884766i	$0.0281998\pi$
$797$			4.99560i			0.176953i	−0.996078	+	0.0884766i	$0.971800\pi$
$798$		0			0
$799$	42.1990			1.49289
$800$		0			0
$801$	25.4750	−	3.26580i	0.900115	−	0.115392i
$802$		0			0
$803$	−10.8129			−0.381578
$804$		0			0
$805$	21.4643			0.756518
$806$		0			0
$807$	5.16430	−	5.86860i	0.181792	−	0.206585i
$808$		0			0
$809$	−37.3885			−1.31451			−0.657255	−	0.753668i	$-0.728283\pi$
$809$	−37.3885			−1.31451			−0.657255	+	0.753668i	$0.728283\pi$
$810$		0			0
$811$			24.2616i			0.851942i	0.904737	+	0.425971i	$0.140067\pi$
$811$			24.2616i			0.851942i	−0.904737	+	0.425971i	$0.859933\pi$
$812$		0			0
$813$	−23.2765	+	26.4510i	−0.816344	+	0.927677i
$814$		0			0
$815$	1.58062			0.0553666
$816$		0			0
$817$		−	0.974863i		−	0.0341061i
$818$		0			0
$819$	7.22734	+	56.3771i	0.252544	+	1.96997i
$820$		0			0
$821$		−	28.0790i		−	0.979966i	−0.871732	−	0.489983i	$-0.837003\pi$
$821$		−	28.0790i		−	0.979966i	0.871732	−	0.489983i	$-0.162997\pi$
$822$		0			0
$823$	−28.8295			−1.00493			−0.502466	−	0.864597i	$-0.667574\pi$
$823$	−28.8295			−1.00493			−0.502466	+	0.864597i	$0.667574\pi$
$824$		0			0
$825$	2.13664	+	1.88022i	0.0743884	+	0.0654609i
$826$		0			0
$827$		−	26.2258i		−	0.911959i	−0.889990	−	0.455980i	$-0.849289\pi$
$827$		−	26.2258i		−	0.911959i	0.889990	−	0.455980i	$-0.150711\pi$
$828$		0			0
$829$	−30.0230			−1.04274			−0.521371	−	0.853330i	$-0.674579\pi$
$829$	−30.0230			−1.04274			−0.521371	+	0.853330i	$0.674579\pi$
$830$		0			0
$831$	−22.3316	−	19.6516i	−0.774676	−	0.681705i
$832$		0			0
$833$			28.9100i			1.00167i
$834$		0			0
$835$			17.4965i			0.605490i
$836$		0			0
$837$	20.4497	+	13.8153i	0.706843	+	0.477528i
$838$		0			0
$839$		−	20.3764i		−	0.703473i	−0.936099	−	0.351736i	$-0.885591\pi$
$839$		−	20.3764i		−	0.703473i	0.936099	−	0.351736i	$-0.114409\pi$
$840$		0			0
$841$	−15.3623			−0.529736
$842$		0			0
$843$	4.03974	+	3.55492i	0.139136	+	0.122438i
$844$		0			0
$845$	18.3760			0.632155
$846$		0			0
$847$		−	28.0732i		−	0.964606i
$848$		0			0
$849$	7.69126	+	6.76821i	0.263963	+	0.232284i
$850$		0			0
$851$			51.8494i			1.77737i
$852$		0			0
$853$	44.6724			1.52955			0.764776	−	0.644296i	$-0.222850\pi$
$853$	44.6724			1.52955			0.764776	+	0.644296i	$0.222850\pi$
$854$		0			0
$855$	0.0765663	+	0.597258i	0.00261851	+	0.0204258i
$856$		0			0
$857$	8.52585			0.291237			0.145619	−	0.989341i	$-0.453483\pi$
$857$	8.52585			0.291237			0.145619	+	0.989341i	$0.453483\pi$
$858$		0			0
$859$	−17.3101			−0.590614			−0.295307	−	0.955402i	$-0.595422\pi$
$859$	−17.3101			−0.590614			−0.295307	+	0.955402i	$0.595422\pi$
$860$		0			0
$861$	−28.6235	+	32.5272i	−0.975486	+	1.10852i
$862$		0			0
$863$		−	10.3106i		−	0.350976i	−0.984482	−	0.175488i	$-0.943850\pi$
$863$		−	10.3106i		−	0.350976i	0.984482	−	0.175488i	$-0.0561503\pi$
$864$		0			0
$865$			15.2081i			0.517090i
$866$		0			0
$867$	33.0110	+	29.0493i	1.12111	+	0.986566i
$868$		0			0
$869$			15.6840i			0.532045i
$870$		0			0
$871$	−45.6511	+	4.26257i	−1.54683	+	0.144432i
$872$		0			0
$873$	1.78831	−	0.229256i	0.0605253	−	0.00775912i
$874$		0			0
$875$		−	3.38238i		−	0.114345i
$876$		0			0
$877$	18.9445			0.639710			0.319855	−	0.947466i	$-0.396366\pi$
$877$	18.9445			0.639710			0.319855	+	0.947466i	$0.396366\pi$
$878$		0			0
$879$	14.8855	−	16.9156i	0.502076	−	0.570549i
$880$		0			0
$881$		−	11.1742i		−	0.376469i	−0.982124	−	0.188235i	$-0.939723\pi$
$881$		−	11.1742i		−	0.376469i	0.982124	−	0.188235i	$-0.0602765\pi$
$882$		0			0
$883$		−	43.8751i		−	1.47651i	−0.674520	−	0.738257i	$-0.735649\pi$
$883$		−	43.8751i		−	1.47651i	0.674520	−	0.738257i	$-0.264351\pi$
$884$		0			0
$885$	−2.58803	+	2.94099i	−0.0869958	+	0.0988603i
$886$		0			0
$887$		−	28.1567i		−	0.945411i	−0.881221	−	0.472705i	$-0.843277\pi$
$887$		−	28.1567i		−	0.945411i	0.881221	−	0.472705i	$-0.156723\pi$
$888$		0			0
$889$			58.8887i			1.97506i
$890$		0			0
$891$	14.3107	−	3.73047i	0.479427	−	0.124976i
$892$		0			0
$893$		−	1.30096i		−	0.0435349i
$894$		0			0
$895$	2.75565			0.0921111
$896$		0			0
$897$	−46.2202	−	40.6732i	−1.54325	−	1.35804i
$898$		0			0
$899$	−31.6339			−1.05505
$900$		0			0
$901$			14.7870i			0.492627i
$902$		0			0
$903$	21.3611	+	18.7975i	0.710852	+	0.625540i
$904$		0			0
$905$	−13.6314			−0.453124
$906$		0			0
$907$	16.2712			0.540276			0.270138	−	0.962822i	$-0.412931\pi$
$907$	16.2712			0.540276			0.270138	+	0.962822i	$0.412931\pi$
$908$		0			0
$909$	−3.70434	−	28.8958i	−0.122865	−	0.958413i
$910$		0			0
$911$			28.4563i			0.942799i	0.881920	+	0.471400i	$0.156251\pi$
$911$			28.4563i			0.942799i	−0.881920	+	0.471400i	$0.843749\pi$
$912$		0			0
$913$			5.51801i			0.182620i
$914$		0			0
$915$	3.86264	−	4.38943i	0.127695	−	0.145110i
$916$		0			0
$917$	9.61297			0.317448
$918$		0			0
$919$			48.9977i			1.61629i	0.588987	+	0.808143i	$0.299527\pi$
$919$			48.9977i			1.61629i	−0.588987	+	0.808143i	$0.700473\pi$
$920$		0			0
$921$	23.2197	+	20.4330i	0.765115	+	0.673291i
$922$		0			0
$923$	−71.3243			−2.34767
$924$		0			0
$925$	8.17050			0.268644
$926$		0			0
$927$	−0.844080	−	6.58427i	−0.0277232	−	0.216256i
$928$		0			0
$929$	54.9073			1.80145			0.900726	−	0.434388i	$-0.143035\pi$
$929$	54.9073			1.80145			0.900726	+	0.434388i	$0.143035\pi$
$930$		0			0
$931$	0.891271			0.0292102
$932$		0			0
$933$	9.20624	+	8.10138i	0.301399	+	0.265227i
$934$		0			0
$935$		−	10.6983i		−	0.349871i
$936$		0			0
$937$			51.1971i			1.67254i	0.548321	+	0.836268i	$0.315267\pi$
$937$			51.1971i			1.67254i	−0.548321	+	0.836268i	$0.684733\pi$
$938$		0			0
$939$	−32.7905	+	37.2625i	−1.07008	+	1.21601i
$940$		0			0
$941$	48.9819			1.59676			0.798382	−	0.602152i	$-0.205690\pi$
$941$	48.9819			1.59676			0.798382	+	0.602152i	$0.205690\pi$
$942$		0			0
$943$		−	46.9334i		−	1.52836i
$944$		0			0
$945$	−14.5634	−	9.83871i	−0.473747	−	0.320053i
$946$		0			0
$947$		−	18.0944i		−	0.587988i	−0.955807	−	0.293994i	$-0.905015\pi$
$947$		−	18.0944i		−	0.587988i	0.955807	−	0.293994i	$-0.0949845\pi$
$948$		0			0
$949$		−	36.8592i		−	1.19650i
$950$		0			0
$951$	7.44827	−	8.46407i	0.241527	−	0.274466i
$952$		0			0
$953$			16.9120i			0.547834i	0.961753	+	0.273917i	$0.0883194\pi$
$953$			16.9120i			0.547834i	−0.961753	+	0.273917i	$0.911681\pi$
$954$		0			0
$955$	2.27280			0.0735461
$956$		0			0
$957$	−12.5232	+	14.2311i	−0.404818	+	0.460027i
$958$		0			0
$959$		−	59.4817i		−	1.92077i
$960$		0			0
$961$	8.44253			0.272340
$962$		0			0
$963$	−13.6523	+	1.75017i	−0.439939	+	0.0563986i
$964$		0			0
$965$	15.3493			0.494112
$966$		0			0
$967$	17.7297			0.570149			0.285074	−	0.958505i	$-0.407982\pi$
$967$	17.7297			0.570149			0.285074	+	0.958505i	$0.407982\pi$
$968$		0			0
$969$	1.49525	−	1.69917i	0.0480344	−	0.0545853i
$970$		0			0
$971$		−	23.6952i		−	0.760416i	−0.924901	−	0.380208i	$-0.875852\pi$
$971$		−	23.6952i		−	0.760416i	0.924901	−	0.380208i	$-0.124148\pi$
$972$		0			0
$973$	46.0992			1.47787
$974$		0			0
$975$	−6.40934	+	7.28344i	−0.205263	+	0.233257i
$976$		0			0
$977$			16.9650i			0.542759i	0.962472	+	0.271380i	$0.0874799\pi$
$977$			16.9650i			0.542759i	−0.962472	+	0.271380i	$0.912520\pi$
$978$		0			0
$979$			14.0678i			0.449610i
$980$		0			0
$981$	13.5191	−	1.73309i	0.431630	−	0.0553334i
$982$		0			0
$983$	40.4888			1.29139			0.645696	−	0.763594i	$-0.276567\pi$
$983$	40.4888			1.29139			0.645696	+	0.763594i	$0.276567\pi$
$984$		0			0
$985$	2.25906			0.0719795
$986$		0			0
$987$	28.5064	+	25.0853i	0.907368	+	0.798473i
$988$		0			0
$989$	−30.8218			−0.980077
$990$		0			0
$991$		−	16.5220i		−	0.524837i	−0.964954	−	0.262419i	$-0.915480\pi$
$991$		−	16.5220i		−	0.524837i	0.964954	−	0.262419i	$-0.0845201\pi$
$992$		0			0
$993$	−28.6829	+	32.5947i	−0.910225	+	1.03436i
$994$		0			0
$995$	−23.8190			−0.755112
$996$		0			0
$997$	27.8527			0.882104			0.441052	−	0.897482i	$-0.354605\pi$
$997$	27.8527			0.882104			0.441052	+	0.897482i	$0.354605\pi$
$998$		0			0
$999$	23.7665	−	35.1795i	0.751938	−	1.11303i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.a.401.11	✓	46
3.2	odd	2		4020.2.f.b.401.35	yes	46
67.66	odd	2		4020.2.f.b.401.36	yes	46
201.200	even	2	inner	4020.2.f.a.401.12	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.11	✓	46	1.1	even	1	trivial
4020.2.f.a.401.12	yes	46	201.200	even	2	inner
4020.2.f.b.401.35	yes	46	3.2	odd	2
4020.2.f.b.401.36	yes	46	67.66	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)

\(f(q)\)	\(=\)	\(q+(-1.30028 - 1.14423i) q^{3} -1.00000 q^{5} +3.38238i q^{7} +(0.381467 + 2.97565i) q^{9} +O(q^{10})\)	\(q+(-1.30028 - 1.14423i) q^{3} -1.00000 q^{5} +3.38238i q^{7} +(0.381467 + 2.97565i) q^{9} -1.64322 q^{11} -5.60143i q^{13} +(1.30028 + 1.14423i) q^{15} -6.51058i q^{17} -0.200715 q^{19} +(3.87022 - 4.39804i) q^{21} +6.34593i q^{23} +1.00000 q^{25} +(2.90882 - 4.30567i) q^{27} +6.66051i q^{29} +4.74947i q^{31} +(2.13664 + 1.88022i) q^{33} -3.38238i q^{35} +8.17050 q^{37} +(-6.40934 + 7.28344i) q^{39} -7.39582 q^{41} +4.85695i q^{43} +(-0.381467 - 2.97565i) q^{45} +6.48161i q^{47} -4.44047 q^{49} +(-7.44961 + 8.46559i) q^{51} -2.27123 q^{53} +1.64322 q^{55} +(0.260986 + 0.229665i) q^{57} +2.26181i q^{59} -3.37575i q^{61} +(-10.0648 + 1.29027i) q^{63} +5.60143i q^{65} +(-0.760978 - 8.14990i) q^{67} +(7.26121 - 8.25150i) q^{69} -12.7332i q^{71} +6.58032 q^{73} +(-1.30028 - 1.14423i) q^{75} -5.55798i q^{77} -9.54473i q^{79} +(-8.70897 + 2.27023i) q^{81} -3.35806i q^{83} +6.51058i q^{85} +(7.62116 - 8.66054i) q^{87} -8.56116i q^{89} +18.9462 q^{91} +(5.43450 - 6.17565i) q^{93} +0.200715 q^{95} -0.600983i q^{97} +(-0.626833 - 4.88963i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) \) 46 * q - 46 * q^5 - 4 * q^9	\( 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) \) 46 * q - 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 - 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 - 4 * q^41 + 4 * q^45 - 62 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 - 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 - 8 * q^95

Introduction

L-functions

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