LMFDB - Embedded newform 4020.2.f.b.401.3

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			401.3
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.b.401.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.68102 - 0.417348i) q^{3} +1.00000 q^{5} +2.42754i q^{7} +(2.65164 + 1.40314i) q^{9} +O(q^{10})$	\(q+(-1.68102 - 0.417348i) q^{3} +1.00000 q^{5} +2.42754i q^{7} +(2.65164 + 1.40314i) q^{9} -2.52837 q^{11} +0.420456i q^{13} +(-1.68102 - 0.417348i) q^{15} -1.53240i q^{17} -1.99037 q^{19} +(1.01313 - 4.08074i) q^{21} -2.90732i q^{23} +1.00000 q^{25} +(-3.87186 - 3.46536i) q^{27} -1.32422i q^{29} -4.77115i q^{31} +(4.25024 + 1.05521i) q^{33} +2.42754i q^{35} +9.45031 q^{37} +(0.175476 - 0.706795i) q^{39} +2.00106 q^{41} +2.63886i q^{43} +(2.65164 + 1.40314i) q^{45} -2.60954i q^{47} +1.10703 q^{49} +(-0.639542 + 2.57598i) q^{51} +1.37490 q^{53} -2.52837 q^{55} +(3.34585 + 0.830677i) q^{57} -14.2114i q^{59} +11.2815i q^{61} +(-3.40618 + 6.43698i) q^{63} +0.420456i q^{65} +(7.34375 - 3.61516i) q^{67} +(-1.21336 + 4.88726i) q^{69} -3.60277i q^{71} -1.92733 q^{73} +(-1.68102 - 0.417348i) q^{75} -6.13774i q^{77} +7.44491i q^{79} +(5.06241 + 7.44124i) q^{81} +13.4573i q^{83} -1.53240i q^{85} +(-0.552661 + 2.22604i) q^{87} +2.46697i q^{89} -1.02068 q^{91} +(-1.99123 + 8.02040i) q^{93} -1.99037 q^{95} +4.51661i q^{97} +(-6.70434 - 3.54766i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.68102	−	0.417348i	−0.970536	−	0.240956i
$3$	−1.68102	−	0.417348i	−0.970536	−	0.240956i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			2.42754i			0.917525i	0.888559	+	0.458763i	$0.151707\pi$
$7$			2.42754i			0.917525i	−0.888559	+	0.458763i	$0.848293\pi$
$8$		0			0
$9$	2.65164	+	1.40314i	0.883881	+	0.467712i
$10$		0			0
$11$	−2.52837			−0.762333			−0.381167	−	0.924506i	$-0.624478\pi$
$11$	−2.52837			−0.762333			−0.381167	+	0.924506i	$0.624478\pi$
$12$		0			0
$13$			0.420456i			0.116614i	0.998299	+	0.0583068i	$0.0185702\pi$
$13$			0.420456i			0.116614i	−0.998299	+	0.0583068i	$0.981430\pi$
$14$		0			0
$15$	−1.68102	−	0.417348i	−0.434037	−	0.107759i
$16$		0			0
$17$		−	1.53240i		−	0.371661i	−0.982582	−	0.185830i	$-0.940503\pi$
$17$		−	1.53240i		−	0.371661i	0.982582	−	0.185830i	$-0.0594975\pi$
$18$		0			0
$19$	−1.99037			−0.456623			−0.228311	−	0.973588i	$-0.573320\pi$
$19$	−1.99037			−0.456623			−0.228311	+	0.973588i	$0.573320\pi$
$20$		0			0
$21$	1.01313	−	4.08074i	0.221083	−	0.890492i
$22$		0			0
$23$		−	2.90732i		−	0.606219i	−0.952956	−	0.303109i	$-0.901975\pi$
$23$		−	2.90732i		−	0.606219i	0.952956	−	0.303109i	$-0.0980248\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−3.87186	−	3.46536i	−0.745140	−	0.666908i
$28$		0			0
$29$		−	1.32422i		−	0.245902i	−0.992413	−	0.122951i	$-0.960764\pi$
$29$		−	1.32422i		−	0.245902i	0.992413	−	0.122951i	$-0.0392358\pi$
$30$		0			0
$31$		−	4.77115i		−	0.856925i	−0.903560	−	0.428462i	$-0.859055\pi$
$31$		−	4.77115i		−	0.856925i	0.903560	−	0.428462i	$-0.140945\pi$
$32$		0			0
$33$	4.25024	+	1.05521i	0.739872	+	0.183689i
$34$		0			0
$35$			2.42754i			0.410330i
$36$		0			0
$37$	9.45031			1.55362			0.776810	−	0.629735i	$-0.216836\pi$
$37$	9.45031			1.55362			0.776810	+	0.629735i	$0.216836\pi$
$38$		0			0
$39$	0.175476	−	0.706795i	0.0280987	−	0.113178i
$40$		0			0
$41$	2.00106			0.312513			0.156256	−	0.987717i	$-0.450057\pi$
$41$	2.00106			0.312513			0.156256	+	0.987717i	$0.450057\pi$
$42$		0			0
$43$			2.63886i			0.402422i	0.979548	+	0.201211i	$0.0644877\pi$
$43$			2.63886i			0.402422i	−0.979548	+	0.201211i	$0.935512\pi$
$44$		0			0
$45$	2.65164	+	1.40314i	0.395283	+	0.209167i
$46$		0			0
$47$		−	2.60954i		−	0.380641i	−0.981722	−	0.190321i	$-0.939047\pi$
$47$		−	2.60954i		−	0.380641i	0.981722	−	0.190321i	$-0.0609528\pi$
$48$		0			0
$49$	1.10703			0.158147
$50$		0			0
$51$	−0.639542	+	2.57598i	−0.0895538	+	0.360710i
$52$		0			0
$53$	1.37490			0.188858			0.0944289	−	0.995532i	$-0.469898\pi$
$53$	1.37490			0.188858			0.0944289	+	0.995532i	$0.469898\pi$
$54$		0			0
$55$	−2.52837			−0.340926
$56$		0			0
$57$	3.34585	+	0.830677i	0.443169	+	0.110026i
$58$		0			0
$59$		−	14.2114i		−	1.85017i	−0.379758	−	0.925086i	$-0.623993\pi$
$59$		−	14.2114i		−	1.85017i	0.379758	−	0.925086i	$-0.376007\pi$
$60$		0			0
$61$			11.2815i			1.44444i	0.691662	+	0.722221i	$0.256879\pi$
$61$			11.2815i			1.44444i	−0.691662	+	0.722221i	$0.743121\pi$
$62$		0			0
$63$	−3.40618	+	6.43698i	−0.429138	+	0.810983i
$64$		0			0
$65$			0.420456i			0.0521512i
$66$		0			0
$67$	7.34375	−	3.61516i	0.897182	−	0.441662i
$67$	7.34375	−	3.61516i	0.897182	−	0.441662i
$68$		0			0
$69$	−1.21336	+	4.88726i	−0.146072	+	0.588357i
$70$		0			0
$71$		−	3.60277i		−	0.427570i	−0.976881	−	0.213785i	$-0.931421\pi$
$71$		−	3.60277i		−	0.427570i	0.976881	−	0.213785i	$-0.0685793\pi$
$72$		0			0
$73$	−1.92733			−0.225577			−0.112789	−	0.993619i	$-0.535978\pi$
$73$	−1.92733			−0.225577			−0.112789	+	0.993619i	$0.535978\pi$
$74$		0			0
$75$	−1.68102	−	0.417348i	−0.194107	−	0.0481911i
$76$		0			0
$77$		−	6.13774i		−	0.699460i
$78$		0			0
$79$			7.44491i			0.837618i	0.908074	+	0.418809i	$0.137552\pi$
$79$			7.44491i			0.837618i	−0.908074	+	0.418809i	$0.862448\pi$
$80$		0			0
$81$	5.06241	+	7.44124i	0.562490	+	0.826804i
$82$		0			0
$83$			13.4573i			1.47713i	0.674183	+	0.738565i	$0.264496\pi$
$83$			13.4573i			1.47713i	−0.674183	+	0.738565i	$0.735504\pi$
$84$		0			0
$85$		−	1.53240i		−	0.166212i
$86$		0			0
$87$	−0.552661	+	2.22604i	−0.0592515	+	0.238657i
$88$		0			0
$89$			2.46697i			0.261498i	0.991416	+	0.130749i	$0.0417382\pi$
$89$			2.46697i			0.261498i	−0.991416	+	0.130749i	$0.958262\pi$
$90$		0			0
$91$	−1.02068			−0.106996
$92$		0			0
$93$	−1.99123	+	8.02040i	−0.206481	+	0.831676i
$94$		0			0
$95$	−1.99037			−0.204208
$96$		0			0
$97$			4.51661i			0.458593i	0.973357	+	0.229296i	$0.0736425\pi$
$97$			4.51661i			0.458593i	−0.973357	+	0.229296i	$0.926357\pi$
$98$		0			0
$99$	−6.70434	−	3.54766i	−0.673812	−	0.356553i
$100$		0			0
$101$	−7.70241			−0.766419			−0.383209	−	0.923662i	$-0.625181\pi$
$101$	−7.70241			−0.766419			−0.383209	+	0.923662i	$0.625181\pi$
$102$		0			0
$103$	−0.858145			−0.0845556			−0.0422778	−	0.999106i	$-0.513461\pi$
$103$	−0.858145			−0.0845556			−0.0422778	+	0.999106i	$0.513461\pi$
$104$		0			0
$105$	1.01313	−	4.08074i	0.0988713	−	0.398240i
$106$		0			0
$107$			7.38436i			0.713873i	0.934129	+	0.356937i	$0.116179\pi$
$107$			7.38436i			0.713873i	−0.934129	+	0.356937i	$0.883821\pi$
$108$		0			0
$109$			8.74050i			0.837188i	0.908173	+	0.418594i	$0.137477\pi$
$109$			8.74050i			0.837188i	−0.908173	+	0.418594i	$0.862523\pi$
$110$		0			0
$111$	−15.8861	−	3.94406i	−1.50785	−	0.374354i
$112$		0			0
$113$	8.82442			0.830132			0.415066	−	0.909791i	$-0.363758\pi$
$113$	8.82442			0.830132			0.415066	+	0.909791i	$0.363758\pi$
$114$		0			0
$115$		−	2.90732i		−	0.271109i
$116$		0			0
$117$	−0.589958	+	1.11490i	−0.0545417	+	0.103073i
$118$		0			0
$119$	3.71996			0.341008
$120$		0			0
$121$	−4.60733			−0.418848
$122$		0			0
$123$	−3.36381	−	0.835137i	−0.303305	−	0.0753018i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	11.6506			1.03383			0.516913	−	0.856038i	$-0.327081\pi$
$127$	11.6506			1.03383			0.516913	+	0.856038i	$0.327081\pi$
$128$		0			0
$129$	1.10132	−	4.43597i	0.0969659	−	0.390565i
$130$		0			0
$131$			6.95159i			0.607363i	0.952774	+	0.303681i	$0.0982158\pi$
$131$			6.95159i			0.607363i	−0.952774	+	0.303681i	$0.901784\pi$
$132$		0			0
$133$		−	4.83172i		−	0.418963i
$134$		0			0
$135$	−3.87186	−	3.46536i	−0.333237	−	0.298250i
$136$		0			0
$137$	15.3836			1.31431			0.657154	−	0.753757i	$-0.271760\pi$
$137$	15.3836			1.31431			0.657154	+	0.753757i	$0.271760\pi$
$138$		0			0
$139$			13.5800i			1.15184i	0.817504	+	0.575922i	$0.195357\pi$
$139$			13.5800i			1.15184i	−0.817504	+	0.575922i	$0.804643\pi$
$140$		0			0
$141$	−1.08909	+	4.38669i	−0.0917177	+	0.369426i
$142$		0			0
$143$		−	1.06307i		−	0.0888985i
$144$		0			0
$145$		−	1.32422i		−	0.109971i
$146$		0			0
$147$	−1.86094	−	0.462016i	−0.153487	−	0.0381065i
$148$		0			0
$149$		−	19.4796i		−	1.59583i	−0.602767	−	0.797917i	$-0.705935\pi$
$149$		−	19.4796i		−	1.59583i	0.602767	−	0.797917i	$-0.294065\pi$
$150$		0			0
$151$	9.96107			0.810621			0.405310	−	0.914179i	$-0.367163\pi$
$151$	9.96107			0.810621			0.405310	+	0.914179i	$0.367163\pi$
$152$		0			0
$153$	2.15016	−	4.06337i	0.173830	−	0.328504i
$154$		0			0
$155$		−	4.77115i		−	0.383228i
$156$		0			0
$157$	8.59295			0.685792			0.342896	−	0.939373i	$-0.388592\pi$
$157$	8.59295			0.685792			0.342896	+	0.939373i	$0.388592\pi$
$158$		0			0
$159$	−2.31124	−	0.573813i	−0.183293	−	0.0455063i
$160$		0			0
$161$	7.05766			0.556221
$162$		0			0
$163$	10.2882			0.805832			0.402916	−	0.915237i	$-0.367997\pi$
$163$	10.2882			0.805832			0.402916	+	0.915237i	$0.367997\pi$
$164$		0			0
$165$	4.25024	+	1.05521i	0.330881	+	0.0821480i
$166$		0			0
$167$		−	0.722509i		−	0.0559094i	−0.999609	−	0.0279547i	$-0.991101\pi$
$167$		−	0.722509i		−	0.0559094i	0.999609	−	0.0279547i	$-0.00889942\pi$
$168$		0			0
$169$	12.8232			0.986401
$170$		0			0
$171$	−5.27776	−	2.79277i	−0.403600	−	0.213568i
$172$		0			0
$173$			9.18108i			0.698024i	0.937118	+	0.349012i	$0.113483\pi$
$173$			9.18108i			0.698024i	−0.937118	+	0.349012i	$0.886517\pi$
$174$		0			0
$175$			2.42754i			0.183505i
$176$		0			0
$177$	−5.93111	+	23.8897i	−0.445809	+	1.79566i
$178$		0			0
$179$	−6.77495			−0.506384			−0.253192	−	0.967416i	$-0.581480\pi$
$179$	−6.77495			−0.506384			−0.253192	+	0.967416i	$0.581480\pi$
$180$		0			0
$181$	−7.55595			−0.561630			−0.280815	−	0.959762i	$-0.590605\pi$
$181$	−7.55595			−0.561630			−0.280815	+	0.959762i	$0.590605\pi$
$182$		0			0
$183$	4.70829	−	18.9643i	0.348047	−	1.40188i
$184$		0			0
$185$	9.45031			0.694800
$186$		0			0
$187$			3.87447i			0.283329i
$188$		0			0
$189$	8.41230	−	9.39911i	0.611905	−	0.683685i
$190$		0			0
$191$	−11.9026			−0.861240			−0.430620	−	0.902533i	$-0.641705\pi$
$191$	−11.9026			−0.861240			−0.430620	+	0.902533i	$0.641705\pi$
$192$		0			0
$193$	13.2551			0.954121			0.477061	−	0.878870i	$-0.341702\pi$
$193$	13.2551			0.954121			0.477061	+	0.878870i	$0.341702\pi$
$194$		0			0
$195$	0.175476	−	0.706795i	0.0125661	−	0.0506146i
$196$		0			0
$197$	5.58564			0.397961			0.198980	−	0.980003i	$-0.436237\pi$
$197$	5.58564			0.397961			0.198980	+	0.980003i	$0.436237\pi$
$198$		0			0
$199$	16.0996			1.14127			0.570634	−	0.821205i	$-0.306698\pi$
$199$	16.0996			1.14127			0.570634	+	0.821205i	$0.306698\pi$
$200$		0			0
$201$	−13.8537	+	3.01225i	−0.977168	+	0.212468i
$202$		0			0
$203$	3.21461			0.225621
$204$		0			0
$205$	2.00106			0.139760
$206$		0			0
$207$	4.07937	−	7.70918i	0.283536	−	0.535825i
$208$		0			0
$209$	5.03241			0.348099
$210$		0			0
$211$	−22.0275			−1.51644			−0.758218	−	0.652001i	$-0.773930\pi$
$211$	−22.0275			−1.51644			−0.758218	+	0.652001i	$0.773930\pi$
$212$		0			0
$213$	−1.50361	+	6.05632i	−0.103026	+	0.414972i
$214$		0			0
$215$			2.63886i			0.179969i
$216$		0			0
$217$	11.5822			0.786250
$218$		0			0
$219$	3.23988	+	0.804368i	0.218931	+	0.0543541i
$220$		0			0
$221$	0.644306			0.0433407
$222$		0			0
$223$	−2.09341			−0.140185			−0.0700926	−	0.997540i	$-0.522329\pi$
$223$	−2.09341			−0.140185			−0.0700926	+	0.997540i	$0.522329\pi$
$224$		0			0
$225$	2.65164	+	1.40314i	0.176776	+	0.0935425i
$226$		0			0
$227$			0.0320395i			0.00212653i	0.999999	+	0.00106327i	$0.000338448\pi$
$227$			0.0320395i			0.00212653i	−0.999999	+	0.00106327i	$0.999662\pi$
$228$		0			0
$229$		−	3.71101i		−	0.245230i	−0.992454	−	0.122615i	$-0.960872\pi$
$229$		−	3.71101i		−	0.245230i	0.992454	−	0.122615i	$-0.0391281\pi$
$230$		0			0
$231$	−2.56157	+	10.3176i	−0.168539	+	0.678851i
$232$		0			0
$233$	18.0674			1.18364			0.591819	−	0.806071i	$-0.298410\pi$
$233$	18.0674			1.18364			0.591819	+	0.806071i	$0.298410\pi$
$234$		0			0
$235$		−	2.60954i		−	0.170228i
$236$		0			0
$237$	3.10711	−	12.5150i	0.201829	−	0.812938i
$238$		0			0
$239$	5.57206			0.360427			0.180213	−	0.983628i	$-0.442321\pi$
$239$	5.57206			0.360427			0.180213	+	0.983628i	$0.442321\pi$
$240$		0			0
$241$	21.3553			1.37562			0.687809	−	0.725892i	$-0.258573\pi$
$241$	21.3553			1.37562			0.687809	+	0.725892i	$0.258573\pi$
$242$		0			0
$243$	−5.40442	−	14.6216i	−0.346694	−	0.937978i
$244$		0			0
$245$	1.10703			0.0707255
$246$		0			0
$247$		−	0.836865i		−	0.0532485i
$248$		0			0
$249$	5.61637	−	22.6219i	0.355923	−	1.43361i
$250$		0			0
$251$	10.0803			0.636265			0.318133	−	0.948046i	$-0.396944\pi$
$251$	10.0803			0.636265			0.318133	+	0.948046i	$0.396944\pi$
$252$		0			0
$253$			7.35080i			0.462141i
$254$		0			0
$255$	−0.639542	+	2.57598i	−0.0400497	+	0.161314i
$256$		0			0
$257$			7.37780i			0.460214i	0.973165	+	0.230107i	$0.0739077\pi$
$257$			7.37780i			0.460214i	−0.973165	+	0.230107i	$0.926092\pi$
$258$		0			0
$259$			22.9410i			1.42549i
$260$		0			0
$261$	1.85807	−	3.51136i	0.115011	−	0.217348i
$262$		0			0
$263$			8.74055i			0.538965i	0.963005	+	0.269483i	$0.0868527\pi$
$263$			8.74055i			0.538965i	−0.963005	+	0.269483i	$0.913147\pi$
$264$		0			0
$265$	1.37490			0.0844597
$266$		0			0
$267$	1.02958	−	4.14702i	0.0630094	−	0.253793i
$268$		0			0
$269$			1.21198i			0.0738958i	0.999317	+	0.0369479i	$0.0117636\pi$
$269$			1.21198i			0.0738958i	−0.999317	+	0.0369479i	$0.988236\pi$
$270$		0			0
$271$			0.452403i			0.0274816i	0.999906	+	0.0137408i	$0.00437396\pi$
$271$			0.452403i			0.0274816i	−0.999906	+	0.0137408i	$0.995626\pi$
$272$		0			0
$273$	1.71578	+	0.425977i	0.103843	+	0.0257813i
$274$		0			0
$275$	−2.52837			−0.152467
$276$		0			0
$277$	13.7483			0.826053			0.413027	−	0.910719i	$-0.364472\pi$
$277$	13.7483			0.826053			0.413027	+	0.910719i	$0.364472\pi$
$278$		0			0
$279$	6.69459	−	12.6514i	0.400794	−	0.757419i
$280$		0			0
$281$	−11.6935			−0.697574			−0.348787	−	0.937202i	$-0.613406\pi$
$281$	−11.6935			−0.697574			−0.348787	+	0.937202i	$0.613406\pi$
$282$		0			0
$283$	13.4059			0.796901			0.398450	−	0.917190i	$-0.369548\pi$
$283$	13.4059			0.796901			0.398450	+	0.917190i	$0.369548\pi$
$284$		0			0
$285$	3.34585	+	0.830677i	0.198191	+	0.0492051i
$286$		0			0
$287$			4.85766i			0.286738i
$288$		0			0
$289$	14.6518			0.861868
$290$		0			0
$291$	1.88500	−	7.59251i	0.110501	−	0.445081i
$292$		0			0
$293$			16.2764i			0.950879i	0.879748	+	0.475440i	$0.157711\pi$
$293$			16.2764i			0.950879i	−0.879748	+	0.475440i	$0.842289\pi$
$294$		0			0
$295$		−	14.2114i		−	0.827422i
$296$		0			0
$297$	9.78951	+	8.76171i	0.568045	+	0.508406i
$298$		0			0
$299$	1.22240			0.0706934
$300$		0			0
$301$	−6.40594			−0.369232
$302$		0			0
$303$	12.9479	+	3.21458i	0.743837	+	0.184673i
$304$		0			0
$305$			11.2815i			0.645974i
$306$		0			0
$307$	−2.07309			−0.118318			−0.0591589	−	0.998249i	$-0.518842\pi$
$307$	−2.07309			−0.118318			−0.0591589	+	0.998249i	$0.518842\pi$
$308$		0			0
$309$	1.44256	+	0.358145i	0.0820642	+	0.0203742i
$310$		0			0
$311$	22.0659			1.25124			0.625622	−	0.780126i	$-0.284845\pi$
$311$	22.0659			1.25124			0.625622	+	0.780126i	$0.284845\pi$
$312$		0			0
$313$		−	13.5417i		−	0.765424i	−0.923868	−	0.382712i	$-0.874990\pi$
$313$		−	13.5417i		−	0.765424i	0.923868	−	0.382712i	$-0.125010\pi$
$314$		0			0
$315$	−3.40618	+	6.43698i	−0.191916	+	0.362683i
$316$		0			0
$317$		−	20.4476i		−	1.14845i	−0.818696	−	0.574227i	$-0.805303\pi$
$317$		−	20.4476i		−	1.14845i	0.818696	−	0.574227i	$-0.194697\pi$
$318$		0			0
$319$			3.34813i			0.187459i
$320$		0			0
$321$	3.08185	−	12.4132i	0.172012	−	0.692840i
$322$		0			0
$323$			3.05004i			0.169709i
$324$		0			0
$325$			0.420456i			0.0233227i
$326$		0			0
$327$	3.64783	−	14.6929i	0.201725	−	0.812522i
$328$		0			0
$329$	6.33478			0.349248
$330$		0			0
$331$		−	24.5476i		−	1.34926i	−0.738156	−	0.674630i	$-0.764303\pi$
$331$		−	24.5476i		−	1.34926i	0.738156	−	0.674630i	$-0.235697\pi$
$332$		0			0
$333$	25.0588	+	13.2601i	1.37322	+	0.726648i
$334$		0			0
$335$	7.34375	−	3.61516i	0.401232	−	0.197517i
$336$		0			0
$337$		−	10.0872i		−	0.549484i	−0.961518	−	0.274742i	$-0.911408\pi$
$337$		−	10.0872i		−	0.549484i	0.961518	−	0.274742i	$-0.0885924\pi$
$338$		0			0
$339$	−14.8340	−	3.68285i	−0.805673	−	0.200025i
$340$		0			0
$341$			12.0633i			0.653262i
$342$		0			0
$343$			19.6802i			1.06263i
$344$		0			0
$345$	−1.21336	+	4.88726i	−0.0653254	+	0.263121i
$346$		0			0
$347$	3.82450			0.205310			0.102655	−	0.994717i	$-0.467266\pi$
$347$	3.82450			0.205310			0.102655	+	0.994717i	$0.467266\pi$
$348$		0			0
$349$	3.49170			0.186907			0.0934533	−	0.995624i	$-0.470209\pi$
$349$	3.49170			0.186907			0.0934533	+	0.995624i	$0.470209\pi$
$350$		0			0
$351$	1.45703	−	1.62795i	0.0777706	−	0.0868935i
$352$		0			0
$353$	30.7386			1.63605			0.818026	−	0.575182i	$-0.195069\pi$
$353$	30.7386			1.63605			0.818026	+	0.575182i	$0.195069\pi$
$354$		0			0
$355$		−	3.60277i		−	0.191215i
$356$		0			0
$357$	−6.25332	−	1.55252i	−0.330961	−	0.0821678i
$358$		0			0
$359$			8.84544i			0.466845i	0.972375	+	0.233422i	$0.0749924\pi$
$359$			8.84544i			0.466845i	−0.972375	+	0.233422i	$0.925008\pi$
$360$		0			0
$361$	−15.0384			−0.791495
$362$		0			0
$363$	7.74500	+	1.92286i	0.406507	+	0.100924i
$364$		0			0
$365$	−1.92733			−0.100881
$366$		0			0
$367$			10.7003i			0.558551i	0.960211	+	0.279275i	$0.0900942\pi$
$367$			10.7003i			0.558551i	−0.960211	+	0.279275i	$0.909906\pi$
$368$		0			0
$369$	5.30609	+	2.80776i	0.276224	+	0.146166i
$370$		0			0
$371$			3.33764i			0.173282i
$372$		0			0
$373$			13.8227i			0.715713i	0.933777	+	0.357857i	$0.116492\pi$
$373$			13.8227i			0.715713i	−0.933777	+	0.357857i	$0.883508\pi$
$374$		0			0
$375$	−1.68102	−	0.417348i	−0.0868074	−	0.0215517i
$376$		0			0
$377$	0.556778			0.0286755
$378$		0			0
$379$		−	15.5521i		−	0.798859i	−0.916764	−	0.399429i	$-0.869208\pi$
$379$		−	15.5521i		−	0.798859i	0.916764	−	0.399429i	$-0.130792\pi$
$380$		0			0
$381$	−19.5849	−	4.86236i	−1.00336	−	0.249106i
$382$		0			0
$383$	18.2968			0.934920			0.467460	−	0.884014i	$-0.345169\pi$
$383$	18.2968			0.934920			0.467460	+	0.884014i	$0.345169\pi$
$384$		0			0
$385$		−	6.13774i		−	0.312808i
$386$		0			0
$387$	−3.70268	+	6.99731i	−0.188218	+	0.355693i
$388$		0			0
$389$			0.443282i			0.0224753i	0.999937	+	0.0112376i	$0.00357713\pi$
$389$			0.443282i			0.0224753i	−0.999937	+	0.0112376i	$0.996423\pi$
$390$		0			0
$391$	−4.45517			−0.225308
$392$		0			0
$393$	2.90123	−	11.6857i	0.146348	−	0.589468i
$394$		0			0
$395$			7.44491i			0.374594i
$396$		0			0
$397$	−10.8994			−0.547023			−0.273512	−	0.961869i	$-0.588185\pi$
$397$	−10.8994			−0.547023			−0.273512	+	0.961869i	$0.588185\pi$
$398$		0			0
$399$	−2.01651	+	8.12221i	−0.100952	+	0.406619i
$400$		0			0
$401$	−25.4840			−1.27261			−0.636304	−	0.771438i	$-0.719538\pi$
$401$	−25.4840			−1.27261			−0.636304	+	0.771438i	$0.719538\pi$
$402$		0			0
$403$	2.00606			0.0999291
$404$		0			0
$405$	5.06241	+	7.44124i	0.251553	+	0.369758i
$406$		0			0
$407$	−23.8939			−1.18438
$408$		0			0
$409$		−	10.5766i		−	0.522979i	−0.965206	−	0.261489i	$-0.915786\pi$
$409$		−	10.5766i		−	0.522979i	0.965206	−	0.261489i	$-0.0842137\pi$
$410$		0			0
$411$	−25.8601	−	6.42030i	−1.27558	−	0.316690i
$412$		0			0
$413$	34.4989			1.69758
$414$		0			0
$415$			13.4573i			0.660592i
$416$		0			0
$417$	5.66760	−	22.8283i	0.277544	−	1.11791i
$418$		0			0
$419$			39.1762i			1.91389i	0.290277	+	0.956943i	$0.406252\pi$
$419$			39.1762i			1.91389i	−0.290277	+	0.956943i	$0.593748\pi$
$420$		0			0
$421$	−29.4203			−1.43386			−0.716929	−	0.697146i	$-0.754453\pi$
$421$	−29.4203			−1.43386			−0.716929	+	0.697146i	$0.754453\pi$
$422$		0			0
$423$	3.66155	−	6.91958i	0.178031	−	0.336441i
$424$		0			0
$425$		−	1.53240i		−	0.0743321i
$426$		0			0
$427$	−27.3862			−1.32531
$428$		0			0
$429$	−0.443670	+	1.78704i	−0.0214206	+	0.0862792i
$430$		0			0
$431$		−	22.5435i		−	1.08588i	−0.839771	−	0.542941i	$-0.817311\pi$
$431$		−	22.5435i		−	1.08588i	0.839771	−	0.542941i	$-0.182689\pi$
$432$		0			0
$433$		−	36.4052i		−	1.74952i	−0.484553	−	0.874762i	$-0.661018\pi$
$433$		−	36.4052i		−	1.74952i	0.484553	−	0.874762i	$-0.338982\pi$
$434$		0			0
$435$	−0.552661	+	2.22604i	−0.0264981	+	0.106730i
$436$		0			0
$437$			5.78666i			0.276813i
$438$		0			0
$439$	5.22089			0.249180			0.124590	−	0.992208i	$-0.460239\pi$
$439$	5.22089			0.249180			0.124590	+	0.992208i	$0.460239\pi$
$440$		0			0
$441$	2.93545	+	1.55332i	0.139783	+	0.0739674i
$442$		0			0
$443$	23.1367			1.09926			0.549628	−	0.835410i	$-0.314770\pi$
$443$	23.1367			1.09926			0.549628	+	0.835410i	$0.314770\pi$
$444$		0			0
$445$			2.46697i			0.116945i
$446$		0			0
$447$	−8.12978	+	32.7456i	−0.384525	+	1.54881i
$448$		0			0
$449$			15.5127i			0.732091i	0.930597	+	0.366045i	$0.119288\pi$
$449$			15.5127i			0.732091i	−0.930597	+	0.366045i	$0.880712\pi$
$450$		0			0
$451$	−5.05942			−0.238239
$452$		0			0
$453$	−16.7447	−	4.15723i	−0.786737	−	0.195324i
$454$		0			0
$455$	−1.02068			−0.0478501
$456$		0			0
$457$	19.0892			0.892955			0.446477	−	0.894795i	$-0.352678\pi$
$457$	19.0892			0.892955			0.446477	+	0.894795i	$0.352678\pi$
$458$		0			0
$459$	−5.31030	+	5.93322i	−0.247863	+	0.276939i
$460$		0			0
$461$			7.79717i			0.363150i	0.983377	+	0.181575i	$0.0581195\pi$
$461$			7.79717i			0.363150i	−0.983377	+	0.181575i	$0.941880\pi$
$462$		0			0
$463$			13.4365i			0.624449i	0.950008	+	0.312224i	$0.101074\pi$
$463$			13.4365i			0.624449i	−0.950008	+	0.312224i	$0.898926\pi$
$464$		0			0
$465$	−1.99123	+	8.02040i	−0.0923411	+	0.371937i
$466$		0			0
$467$		−	16.4705i		−	0.762163i	−0.924542	−	0.381081i	$-0.875552\pi$
$467$		−	16.4705i		−	0.762163i	0.924542	−	0.381081i	$-0.124448\pi$
$468$		0			0
$469$	8.77596	+	17.8273i	0.405236	+	0.823187i
$470$		0			0
$471$	−14.4449	−	3.58625i	−0.665586	−	0.165246i
$472$		0			0
$473$		−	6.67202i		−	0.306780i
$474$		0			0
$475$	−1.99037			−0.0913246
$476$		0			0
$477$	3.64576	+	1.92918i	0.166928	+	0.0883311i
$478$		0			0
$479$			13.6228i			0.622443i	0.950337	+	0.311221i	$0.100738\pi$
$479$			13.6228i			0.622443i	−0.950337	+	0.311221i	$0.899262\pi$
$480$		0			0
$481$			3.97344i			0.181173i
$482$		0			0
$483$	−11.8640	−	2.94550i	−0.539833	−	0.134025i
$484$		0			0
$485$			4.51661i			0.205089i
$486$		0			0
$487$		−	34.0846i		−	1.54452i	−0.635305	−	0.772261i	$-0.719126\pi$
$487$		−	34.0846i		−	1.54452i	0.635305	−	0.772261i	$-0.280874\pi$
$488$		0			0
$489$	−17.2946	−	4.29374i	−0.782089	−	0.194170i
$490$		0			0
$491$			37.7437i			1.70335i	0.524070	+	0.851675i	$0.324413\pi$
$491$			37.7437i			1.70335i	−0.524070	+	0.851675i	$0.675587\pi$
$492$		0			0
$493$	−2.02923			−0.0913920
$494$		0			0
$495$	−6.70434	−	3.54766i	−0.301338	−	0.159455i
$496$		0			0
$497$	8.74589			0.392307
$498$		0			0
$499$		−	39.2910i		−	1.75891i	−0.475986	−	0.879453i	$-0.657909\pi$
$499$		−	39.2910i		−	1.75891i	0.475986	−	0.879453i	$-0.342091\pi$
$500$		0			0
$501$	−0.301537	+	1.21455i	−0.0134717	+	0.0542621i
$502$		0			0
$503$	4.34751			0.193846			0.0969230	−	0.995292i	$-0.469100\pi$
$503$	4.34751			0.193846			0.0969230	+	0.995292i	$0.469100\pi$
$504$		0			0
$505$	−7.70241			−0.342753
$506$		0			0
$507$	−21.5561	−	5.35174i	−0.957338	−	0.237679i
$508$		0			0
$509$		−	28.7652i		−	1.27499i	−0.770453	−	0.637497i	$-0.779970\pi$
$509$		−	28.7652i		−	1.27499i	0.770453	−	0.637497i	$-0.220030\pi$
$510$		0			0
$511$		−	4.67868i		−	0.206973i
$512$		0			0
$513$	7.70645	+	6.89735i	0.340248	+	0.304525i
$514$		0			0
$515$	−0.858145			−0.0378144
$516$		0			0
$517$			6.59790i			0.290176i
$518$		0			0
$519$	3.83170	−	15.4336i	0.168193	−	0.677458i
$520$		0			0
$521$	−6.51176			−0.285286			−0.142643	−	0.989774i	$-0.545560\pi$
$521$	−6.51176			−0.285286			−0.142643	+	0.989774i	$0.545560\pi$
$522$		0			0
$523$	−30.0962			−1.31602			−0.658008	−	0.753011i	$-0.728601\pi$
$523$	−30.0962			−1.31602			−0.658008	+	0.753011i	$0.728601\pi$
$524$		0			0
$525$	1.01313	−	4.08074i	0.0442166	−	0.178098i
$526$		0			0
$527$	−7.31130			−0.318485
$528$		0			0
$529$	14.5475			0.632499
$530$		0			0
$531$	19.9406	−	37.6836i	0.865348	−	1.63533i
$532$		0			0
$533$			0.841358i			0.0364433i
$534$		0			0
$535$			7.38436i			0.319254i
$536$		0			0
$537$	11.3888	+	2.82751i	0.491464	+	0.122016i
$538$		0			0
$539$	−2.79899			−0.120561
$540$		0			0
$541$			38.8023i			1.66824i	0.551583	+	0.834120i	$0.314024\pi$
$541$			38.8023i			1.66824i	−0.551583	+	0.834120i	$0.685976\pi$
$542$		0			0
$543$	12.7017	+	3.15346i	0.545082	+	0.135328i
$544$		0			0
$545$			8.74050i			0.374402i
$546$		0			0
$547$			21.7743i			0.931003i	0.885047	+	0.465502i	$0.154126\pi$
$547$			21.7743i			0.931003i	−0.885047	+	0.465502i	$0.845874\pi$
$548$		0			0
$549$	−15.8294	+	29.9144i	−0.675584	+	1.27671i
$550$		0			0
$551$			2.63570i			0.112284i
$552$		0			0
$553$	−18.0728			−0.768535
$554$		0			0
$555$	−15.8861	−	3.94406i	−0.674329	−	0.167416i
$556$		0			0
$557$			31.9112i			1.35212i	0.736847	+	0.676060i	$0.236314\pi$
$557$			31.9112i			1.35212i	−0.736847	+	0.676060i	$0.763686\pi$
$558$		0			0
$559$	−1.10952			−0.0469279
$560$		0			0
$561$	1.61700	−	6.51305i	0.0682698	−	0.274981i
$562$		0			0
$563$	34.9040			1.47103			0.735513	−	0.677510i	$-0.236941\pi$
$563$	34.9040			1.47103			0.735513	+	0.677510i	$0.236941\pi$
$564$		0			0
$565$	8.82442			0.371246
$566$		0			0
$567$	−18.0639	+	12.2892i	−0.758614	+	0.516099i
$568$		0			0
$569$		−	7.51890i		−	0.315209i	−0.987502	−	0.157604i	$-0.949623\pi$
$569$		−	7.51890i		−	0.315209i	0.987502	−	0.157604i	$-0.0503771\pi$
$570$		0			0
$571$	−5.31988			−0.222630			−0.111315	−	0.993785i	$-0.535506\pi$
$571$	−5.31988			−0.222630			−0.111315	+	0.993785i	$0.535506\pi$
$572$		0			0
$573$	20.0084	+	4.96751i	0.835864	+	0.207521i
$574$		0			0
$575$		−	2.90732i		−	0.121244i
$576$		0			0
$577$			6.43497i			0.267891i	0.990989	+	0.133946i	$0.0427648\pi$
$577$			6.43497i			0.267891i	−0.990989	+	0.133946i	$0.957235\pi$
$578$		0			0
$579$	−22.2820	−	5.53197i	−0.926009	−	0.229901i
$580$		0			0
$581$	−32.6682			−1.35530
$582$		0			0
$583$	−3.47627			−0.143973
$584$		0			0
$585$	−0.589958	+	1.11490i	−0.0243918	+	0.0460954i
$586$		0			0
$587$	0.508051			0.0209695			0.0104848	−	0.999945i	$-0.496663\pi$
$587$	0.508051			0.0209695			0.0104848	+	0.999945i	$0.496663\pi$
$588$		0			0
$589$			9.49638i			0.391291i
$590$		0			0
$591$	−9.38957	−	2.33115i	−0.386235	−	0.0958909i
$592$		0			0
$593$	6.25928			0.257038			0.128519	−	0.991707i	$-0.458978\pi$
$593$	6.25928			0.257038			0.128519	+	0.991707i	$0.458978\pi$
$594$		0			0
$595$	3.71996			0.152503
$596$		0			0
$597$	−27.0636	−	6.71911i	−1.10764	−	0.274995i
$598$		0			0
$599$	18.9213			0.773105			0.386552	−	0.922267i	$-0.373666\pi$
$599$	18.9213			0.773105			0.386552	+	0.922267i	$0.373666\pi$
$600$		0			0
$601$	−17.9349			−0.731581			−0.365791	−	0.930697i	$-0.619201\pi$
$601$	−17.9349			−0.731581			−0.365791	+	0.930697i	$0.619201\pi$
$602$		0			0
$603$	24.5456	+	0.718183i	0.999572	+	0.0292467i
$604$		0			0
$605$	−4.60733			−0.187314
$606$		0			0
$607$	−29.7897			−1.20913			−0.604564	−	0.796556i	$-0.706653\pi$
$607$	−29.7897			−1.20913			−0.604564	+	0.796556i	$0.706653\pi$
$608$		0			0
$609$	−5.40381	−	1.34161i	−0.218974	−	0.0543647i
$610$		0			0
$611$	1.09720			0.0443880
$612$		0			0
$613$	−8.42785			−0.340398			−0.170199	−	0.985410i	$-0.554441\pi$
$613$	−8.42785			−0.340398			−0.170199	+	0.985410i	$0.554441\pi$
$614$		0			0
$615$	−3.36381	−	0.835137i	−0.135642	−	0.0336760i
$616$		0			0
$617$		−	39.1035i		−	1.57425i	−0.616795	−	0.787124i	$-0.711569\pi$
$617$		−	39.1035i		−	1.57425i	0.616795	−	0.787124i	$-0.288431\pi$
$618$		0			0
$619$	−22.0283			−0.885391			−0.442696	−	0.896672i	$-0.645978\pi$
$619$	−22.0283			−0.885391			−0.442696	+	0.896672i	$0.645978\pi$
$620$		0			0
$621$	−10.0749	+	11.2568i	−0.404292	+	0.451718i
$622$		0			0
$623$	−5.98867			−0.239931
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−8.45957	−	2.10026i	−0.337843	−	0.0838764i
$628$		0			0
$629$		−	14.4816i		−	0.577420i
$630$		0			0
$631$		−	15.4627i		−	0.615561i	−0.951457	−	0.307781i	$-0.900414\pi$
$631$		−	15.4627i		−	0.615561i	0.951457	−	0.307781i	$-0.0995863\pi$
$632$		0			0
$633$	37.0286	+	9.19313i	1.47176	+	0.365394i
$634$		0			0
$635$	11.6506			0.462341
$636$		0			0
$637$			0.465458i			0.0184421i
$638$		0			0
$639$	5.05518	−	9.55326i	0.199980	−	0.377921i
$640$		0			0
$641$	−13.2575			−0.523638			−0.261819	−	0.965117i	$-0.584322\pi$
$641$	−13.2575			−0.523638			−0.261819	+	0.965117i	$0.584322\pi$
$642$		0			0
$643$	−8.05881			−0.317809			−0.158904	−	0.987294i	$-0.550796\pi$
$643$	−8.05881			−0.317809			−0.158904	+	0.987294i	$0.550796\pi$
$644$		0			0
$645$	1.10132	−	4.43597i	0.0433645	−	0.174666i
$646$		0			0
$647$	−29.4596			−1.15818			−0.579088	−	0.815265i	$-0.696591\pi$
$647$	−29.4596			−1.15818			−0.579088	+	0.815265i	$0.696591\pi$
$648$		0			0
$649$			35.9318i			1.41045i
$650$		0			0
$651$	−19.4699	−	4.83380i	−0.763084	−	0.189452i
$652$		0			0
$653$	1.97875			0.0774344			0.0387172	−	0.999250i	$-0.487673\pi$
$653$	1.97875			0.0774344			0.0387172	+	0.999250i	$0.487673\pi$
$654$		0			0
$655$			6.95159i			0.271621i
$656$		0			0
$657$	−5.11060	−	2.70431i	−0.199383	−	0.105505i
$658$		0			0
$659$			14.6853i			0.572056i	0.958221	+	0.286028i	$0.0923351\pi$
$659$			14.6853i			0.572056i	−0.958221	+	0.286028i	$0.907665\pi$
$660$		0			0
$661$		−	30.5712i		−	1.18908i	−0.804065	−	0.594542i	$-0.797334\pi$
$661$		−	30.5712i		−	1.18908i	0.804065	−	0.594542i	$-0.202666\pi$
$662$		0			0
$663$	−1.08309	−	0.268899i	−0.0420637	−	0.0104432i
$664$		0			0
$665$		−	4.83172i		−	0.187366i
$666$		0			0
$667$	−3.84994			−0.149070
$668$		0			0
$669$	3.51906	+	0.873680i	0.136055	+	0.0337784i
$670$		0			0
$671$		−	28.5237i		−	1.10115i
$672$		0			0
$673$		−	20.2408i		−	0.780225i	−0.920767	−	0.390112i	$-0.872436\pi$
$673$		−	20.2408i		−	0.780225i	0.920767	−	0.390112i	$-0.127564\pi$
$674$		0			0
$675$	−3.87186	−	3.46536i	−0.149028	−	0.133382i
$676$		0			0
$677$	11.6761			0.448750			0.224375	−	0.974503i	$-0.427966\pi$
$677$	11.6761			0.448750			0.224375	+	0.974503i	$0.427966\pi$
$678$		0			0
$679$	−10.9643			−0.420771
$680$		0			0
$681$	0.0133716	−	0.0538589i	0.000512401	−	0.00206388i
$682$		0			0
$683$	16.5015			0.631412			0.315706	−	0.948857i	$-0.397759\pi$
$683$	16.5015			0.631412			0.315706	+	0.948857i	$0.397759\pi$
$684$		0			0
$685$	15.3836			0.587776
$686$		0			0
$687$	−1.54878	+	6.23827i	−0.0590896	+	0.238005i
$688$		0			0
$689$			0.578088i			0.0220234i
$690$		0			0
$691$	−15.5174			−0.590309			−0.295154	−	0.955450i	$-0.595371\pi$
$691$	−15.5174			−0.590309			−0.295154	+	0.955450i	$0.595371\pi$
$692$		0			0
$693$	8.61209	−	16.2751i	0.327146	−	0.618239i
$694$		0			0
$695$			13.5800i			0.515121i
$696$		0			0
$697$		−	3.06641i		−	0.116149i
$698$		0			0
$699$	−30.3717	−	7.54041i	−1.14876	−	0.285204i
$700$		0			0
$701$	−32.7546			−1.23712			−0.618562	−	0.785736i	$-0.712284\pi$
$701$	−32.7546			−1.23712			−0.618562	+	0.785736i	$0.712284\pi$
$702$		0			0
$703$	−18.8096			−0.709419
$704$		0			0
$705$	−1.08909	+	4.38669i	−0.0410174	+	0.165212i
$706$		0			0
$707$		−	18.6979i		−	0.703209i
$708$		0			0
$709$	10.6450			0.399780			0.199890	−	0.979818i	$-0.435941\pi$
$709$	10.6450			0.399780			0.199890	+	0.979818i	$0.435941\pi$
$710$		0			0
$711$	−10.4462	+	19.7412i	−0.391764	+	0.740354i
$712$		0			0
$713$	−13.8713			−0.519484
$714$		0			0
$715$		−	1.06307i		−	0.0397566i
$716$		0			0
$717$	−9.36673	−	2.32549i	−0.349807	−	0.0868469i
$718$		0			0
$719$			23.7738i			0.886613i	0.896370	+	0.443307i	$0.146195\pi$
$719$			23.7738i			0.886613i	−0.896370	+	0.443307i	$0.853805\pi$
$720$		0			0
$721$		−	2.08319i		−	0.0775819i
$722$		0			0
$723$	−35.8987	−	8.91260i	−1.33509	−	0.331463i
$724$		0			0
$725$		−	1.32422i		−	0.0491804i
$726$		0			0
$727$			12.3179i			0.456847i	0.973562	+	0.228424i	$0.0733571\pi$
$727$			12.3179i			0.456847i	−0.973562	+	0.228424i	$0.926643\pi$
$728$		0			0
$729$	2.98262	+	26.8348i	0.110467	+	0.993880i
$730$		0			0
$731$	4.04378			0.149564
$732$		0			0
$733$		−	43.5859i		−	1.60988i	−0.593355	−	0.804941i	$-0.702197\pi$
$733$		−	43.5859i		−	1.60988i	0.593355	−	0.804941i	$-0.297803\pi$
$734$		0			0
$735$	−1.86094	−	0.462016i	−0.0686417	−	0.0170417i
$736$		0			0
$737$	−18.5677	+	9.14047i	−0.683952	+	0.336694i
$738$		0			0
$739$			16.9495i			0.623498i	0.950165	+	0.311749i	$0.100915\pi$
$739$			16.9495i			0.623498i	−0.950165	+	0.311749i	$0.899085\pi$
$740$		0			0
$741$	−0.349264	+	1.40679i	−0.0128305	+	0.0516796i
$742$		0			0
$743$			19.6974i			0.722629i	0.932444	+	0.361315i	$0.117672\pi$
$743$			19.6974i			0.722629i	−0.932444	+	0.361315i	$0.882328\pi$
$744$		0			0
$745$		−	19.4796i		−	0.713679i
$746$		0			0
$747$	−18.8824	+	35.6839i	−0.690872	+	1.30561i
$748$		0			0
$749$	−17.9259			−0.654997
$750$		0			0
$751$	−23.2894			−0.849841			−0.424920	−	0.905231i	$-0.639698\pi$
$751$	−23.2894			−0.849841			−0.424920	+	0.905231i	$0.639698\pi$
$752$		0			0
$753$	−16.9452	−	4.20700i	−0.617518	−	0.153312i
$754$		0			0
$755$	9.96107			0.362521
$756$		0			0
$757$			11.9943i			0.435942i	0.975955	+	0.217971i	$0.0699438\pi$
$757$			11.9943i			0.435942i	−0.975955	+	0.217971i	$0.930056\pi$
$758$		0			0
$759$	3.06784	−	12.3568i	0.111356	−	0.448524i
$760$		0			0
$761$		−	39.6678i		−	1.43796i	−0.695033	−	0.718978i	$-0.744610\pi$
$761$		−	39.6678i		−	1.43796i	0.695033	−	0.718978i	$-0.255390\pi$
$762$		0			0
$763$	−21.2180			−0.768142
$764$		0			0
$765$	2.15016	−	4.06337i	0.0777393	−	0.146911i
$766$		0			0
$767$	5.97529			0.215755
$768$		0			0
$769$			12.5436i			0.452333i	0.974089	+	0.226167i	$0.0726194\pi$
$769$			12.5436i			0.452333i	−0.974089	+	0.226167i	$0.927381\pi$
$770$		0			0
$771$	3.07911	−	12.4022i	0.110891	−	0.446655i
$772$		0			0
$773$			21.4721i			0.772296i	0.922437	+	0.386148i	$0.126195\pi$
$773$			21.4721i			0.772296i	−0.922437	+	0.386148i	$0.873805\pi$
$774$		0			0
$775$		−	4.77115i		−	0.171385i
$776$		0			0
$777$	9.57439	−	38.5643i	0.343479	−	1.38349i
$778$		0			0
$779$	−3.98285			−0.142701
$780$		0			0
$781$			9.10915i			0.325951i
$782$		0			0
$783$	−4.58890	+	5.12720i	−0.163994	+	0.183231i
$784$		0			0
$785$	8.59295			0.306695
$786$		0			0
$787$		−	36.5376i		−	1.30242i	−0.758896	−	0.651212i	$-0.774261\pi$
$787$		−	36.5376i		−	1.30242i	0.758896	−	0.651212i	$-0.225739\pi$
$788$		0			0
$789$	3.64785	−	14.6930i	0.129867	−	0.523085i
$790$		0			0
$791$			21.4217i			0.761667i
$792$		0			0
$793$	−4.74336			−0.168442
$794$		0			0
$795$	−2.31124	−	0.573813i	−0.0819712	−	0.0203511i
$796$		0			0
$797$		−	28.9887i		−	1.02683i	−0.858140	−	0.513416i	$-0.828380\pi$
$797$		−	28.9887i		−	1.02683i	0.858140	−	0.513416i	$-0.171620\pi$
$798$		0			0
$799$	−3.99886			−0.141469
$800$		0			0
$801$	−3.46149	+	6.54151i	−0.122306	+	0.231133i
$802$		0			0
$803$	4.87302			0.171965
$804$		0			0
$805$	7.05766			0.248750
$806$		0			0
$807$	0.505817	−	2.03736i	0.0178056	−	0.0717185i
$808$		0			0
$809$	21.8104			0.766812			0.383406	−	0.923580i	$-0.374751\pi$
$809$	21.8104			0.766812			0.383406	+	0.923580i	$0.374751\pi$
$810$		0			0
$811$			27.8274i			0.977153i	0.872521	+	0.488576i	$0.162484\pi$
$811$			27.8274i			0.977153i	−0.872521	+	0.488576i	$0.837516\pi$
$812$		0			0
$813$	0.188809	−	0.760498i	0.00662184	−	0.0266718i
$814$		0			0
$815$	10.2882			0.360379
$816$		0			0
$817$		−	5.25231i		−	0.183755i
$818$		0			0
$819$	−2.70647	−	1.43215i	−0.0945717	−	0.0500434i
$820$		0			0
$821$		−	44.4600i		−	1.55166i	−0.630940	−	0.775832i	$-0.717330\pi$
$821$		−	44.4600i		−	1.55166i	0.630940	−	0.775832i	$-0.282670\pi$
$822$		0			0
$823$	3.01482			0.105090			0.0525450	−	0.998619i	$-0.483267\pi$
$823$	3.01482			0.105090			0.0525450	+	0.998619i	$0.483267\pi$
$824$		0			0
$825$	4.25024	+	1.05521i	0.147974	+	0.0367377i
$826$		0			0
$827$		−	15.3860i		−	0.535024i	−0.963554	−	0.267512i	$-0.913798\pi$
$827$		−	15.3860i		−	0.535024i	0.963554	−	0.267512i	$-0.0862016\pi$
$828$		0			0
$829$	−11.3596			−0.394534			−0.197267	−	0.980350i	$-0.563207\pi$
$829$	−11.3596			−0.394534			−0.197267	+	0.980350i	$0.563207\pi$
$830$		0			0
$831$	−23.1111	−	5.73781i	−0.801715	−	0.199042i
$832$		0			0
$833$		−	1.69641i		−	0.0587771i
$834$		0			0
$835$		−	0.722509i		−	0.0250034i
$836$		0			0
$837$	−16.5337	+	18.4733i	−0.571490	+	0.638529i
$838$		0			0
$839$			23.8599i			0.823737i	0.911243	+	0.411868i	$0.135124\pi$
$839$			23.8599i			0.823737i	−0.911243	+	0.411868i	$0.864876\pi$
$840$		0			0
$841$	27.2464			0.939532
$842$		0			0
$843$	19.6569	+	4.88024i	0.677021	+	0.168084i
$844$		0			0
$845$	12.8232			0.441132
$846$		0			0
$847$		−	11.1845i		−	0.384303i
$848$		0			0
$849$	−22.5356	−	5.59494i	−0.773421	−	0.192018i
$850$		0			0
$851$		−	27.4751i		−	0.941834i
$852$		0			0
$853$	−0.360566			−0.0123456			−0.00617278	−	0.999981i	$-0.501965\pi$
$853$	−0.360566			−0.0123456			−0.00617278	+	0.999981i	$0.501965\pi$
$854$		0			0
$855$	−5.27776	−	2.79277i	−0.180495	−	0.0955106i
$856$		0			0
$857$	7.81250			0.266870			0.133435	−	0.991058i	$-0.457399\pi$
$857$	7.81250			0.266870			0.133435	+	0.991058i	$0.457399\pi$
$858$		0			0
$859$	−23.6578			−0.807194			−0.403597	−	0.914937i	$-0.632240\pi$
$859$	−23.6578			−0.807194			−0.403597	+	0.914937i	$0.632240\pi$
$860$		0			0
$861$	2.02733	−	8.16581i	0.0690913	−	0.278290i
$862$		0			0
$863$			5.00090i			0.170233i	0.996371	+	0.0851163i	$0.0271262\pi$
$863$			5.00090i			0.170233i	−0.996371	+	0.0851163i	$0.972874\pi$
$864$		0			0
$865$			9.18108i			0.312166i
$866$		0			0
$867$	−24.6299	−	6.11488i	−0.836474	−	0.207672i
$868$		0			0
$869$		−	18.8235i		−	0.638544i
$870$		0			0
$871$	1.52002	+	3.08773i	0.0515038	+	0.104624i
$872$		0			0
$873$	−6.33743	+	11.9764i	−0.214490	+	0.405341i
$874$		0			0
$875$			2.42754i			0.0820660i
$876$		0			0
$877$	−4.40316			−0.148684			−0.0743421	−	0.997233i	$-0.523686\pi$
$877$	−4.40316			−0.148684			−0.0743421	+	0.997233i	$0.523686\pi$
$878$		0			0
$879$	6.79293	−	27.3610i	0.229120	−	0.922863i
$880$		0			0
$881$			23.2289i			0.782601i	0.920263	+	0.391300i	$0.127975\pi$
$881$			23.2289i			0.782601i	−0.920263	+	0.391300i	$0.872025\pi$
$882$		0			0
$883$		−	23.0604i		−	0.776045i	−0.921650	−	0.388023i	$-0.873158\pi$
$883$		−	23.0604i		−	0.776045i	0.921650	−	0.388023i	$-0.126842\pi$
$884$		0			0
$885$	−5.93111	+	23.8897i	−0.199372	+	0.803043i
$886$		0			0
$887$			15.0386i			0.504945i	0.967604	+	0.252473i	$0.0812438\pi$
$887$			15.0386i			0.504945i	−0.967604	+	0.252473i	$0.918756\pi$
$888$		0			0
$889$			28.2824i			0.948561i
$890$		0			0
$891$	−12.7997	−	18.8142i	−0.428805	−	0.630300i
$892$		0			0
$893$			5.19397i			0.173810i
$894$		0			0
$895$	−6.77495			−0.226462
$896$		0			0
$897$	−2.05488	−	0.510167i	−0.0686105	−	0.0170340i
$898$		0			0
$899$	−6.31807			−0.210719
$900$		0			0
$901$		−	2.10690i		−	0.0701910i
$902$		0			0
$903$	10.7685	+	2.67351i	0.358353	+	0.0889687i
$904$		0			0
$905$	−7.55595			−0.251168
$906$		0			0
$907$	4.42199			0.146830			0.0734149	−	0.997301i	$-0.476610\pi$
$907$	4.42199			0.146830			0.0734149	+	0.997301i	$0.476610\pi$
$908$		0			0
$909$	−20.4240	−	10.8075i	−0.677423	−	0.358464i
$910$		0			0
$911$		−	12.1786i		−	0.403496i	−0.979437	−	0.201748i	$-0.935338\pi$
$911$		−	12.1786i		−	0.403496i	0.979437	−	0.201748i	$-0.0646623\pi$
$912$		0			0
$913$		−	34.0251i		−	1.12606i
$914$		0			0
$915$	4.70829	−	18.9643i	0.155651	−	0.626941i
$916$		0			0
$917$	−16.8753			−0.557271
$918$		0			0
$919$		−	56.8418i		−	1.87504i	−0.347933	−	0.937519i	$-0.613116\pi$
$919$		−	56.8418i		−	1.87504i	0.347933	−	0.937519i	$-0.386884\pi$
$920$		0			0
$921$	3.48491	+	0.865201i	0.114832	+	0.0285093i
$922$		0			0
$923$	1.51481			0.0498605
$924$		0			0
$925$	9.45031			0.310724
$926$		0			0
$927$	−2.27549	−	1.20410i	−0.0747370	−	0.0395477i
$928$		0			0
$929$	−4.21303			−0.138225			−0.0691124	−	0.997609i	$-0.522017\pi$
$929$	−4.21303			−0.138225			−0.0691124	+	0.997609i	$0.522017\pi$
$930$		0			0
$931$	−2.20340			−0.0722136
$932$		0			0
$933$	−37.0932	−	9.20916i	−1.21438	−	0.301494i
$934$		0			0
$935$			3.87447i			0.126709i
$936$		0			0
$937$		−	10.1901i		−	0.332896i	−0.986050	−	0.166448i	$-0.946770\pi$
$937$		−	10.1901i		−	0.332896i	0.986050	−	0.166448i	$-0.0532298\pi$
$938$		0			0
$939$	−5.65161	+	22.7639i	−0.184433	+	0.742872i
$940$		0			0
$941$	12.3534			0.402710			0.201355	−	0.979518i	$-0.435466\pi$
$941$	12.3534			0.402710			0.201355	+	0.979518i	$0.435466\pi$
$942$		0			0
$943$		−	5.81772i		−	0.189451i
$944$		0			0
$945$	8.41230	−	9.39911i	0.273652	−	0.305753i
$946$		0			0
$947$			45.1800i			1.46815i	0.679067	+	0.734076i	$0.262385\pi$
$947$			45.1800i			1.46815i	−0.679067	+	0.734076i	$0.737615\pi$
$948$		0			0
$949$		−	0.810359i		−	0.0263054i
$950$		0			0
$951$	−8.53377	+	34.3728i	−0.276727	+	1.11462i
$952$		0			0
$953$		−	5.16355i		−	0.167264i	−0.996497	−	0.0836320i	$-0.973348\pi$
$953$		−	5.16355i		−	0.167264i	0.996497	−	0.0836320i	$-0.0266520\pi$
$954$		0			0
$955$	−11.9026			−0.385158
$956$		0			0
$957$	1.39733	−	5.62826i	0.0451694	−	0.181936i
$958$		0			0
$959$			37.3443i			1.20591i
$960$		0			0
$961$	8.23608			0.265680
$962$		0			0
$963$	−10.3613	+	19.5807i	−0.333887	+	0.630979i
$964$		0			0
$965$	13.2551			0.426696
$966$		0			0
$967$	−10.9992			−0.353709			−0.176854	−	0.984237i	$-0.556592\pi$
$967$	−10.9992			−0.353709			−0.176854	+	0.984237i	$0.556592\pi$
$968$		0			0
$969$	1.27293	−	5.12717i	0.0408923	−	0.164708i
$970$		0			0
$971$			9.50261i			0.304953i	0.988307	+	0.152477i	$0.0487249\pi$
$971$			9.50261i			0.304953i	−0.988307	+	0.152477i	$0.951275\pi$
$972$		0			0
$973$	−32.9662			−1.05685
$974$		0			0
$975$	0.175476	−	0.706795i	0.00561975	−	0.0226356i
$976$		0			0
$977$			42.2127i			1.35051i	0.737586	+	0.675253i	$0.235965\pi$
$977$			42.2127i			1.35051i	−0.737586	+	0.675253i	$0.764035\pi$
$978$		0			0
$979$		−	6.23742i		−	0.199349i
$980$		0			0
$981$	−12.2641	+	23.1767i	−0.391563	+	0.739975i
$982$		0			0
$983$	19.4727			0.621084			0.310542	−	0.950560i	$-0.399490\pi$
$983$	19.4727			0.621084			0.310542	+	0.950560i	$0.399490\pi$
$984$		0			0
$985$	5.58564			0.177973
$986$		0			0
$987$	−10.6489	−	2.64381i	−0.338958	−	0.0841533i
$988$		0			0
$989$	7.67201			0.243956
$990$		0			0
$991$		−	30.0521i		−	0.954635i	−0.878731	−	0.477318i	$-0.841609\pi$
$991$		−	30.0521i		−	0.954635i	0.878731	−	0.477318i	$-0.158391\pi$
$992$		0			0
$993$	−10.2449	+	41.2650i	−0.325112	+	1.30951i
$994$		0			0
$995$	16.0996			0.510390
$996$		0			0
$997$	−48.3064			−1.52988			−0.764939	−	0.644103i	$-0.777231\pi$
$997$	−48.3064			−1.52988			−0.764939	+	0.644103i	$0.777231\pi$
$998$		0			0
$999$	−36.5903	−	32.7487i	−1.15767	−	1.03612i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.b.401.3	yes	46
3.2	odd	2		4020.2.f.a.401.43	✓	46
67.66	odd	2		4020.2.f.a.401.44	yes	46
201.200	even	2	inner	4020.2.f.b.401.4	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.43	✓	46	3.2	odd	2
4020.2.f.a.401.44	yes	46	67.66	odd	2
4020.2.f.b.401.3	yes	46	1.1	even	1	trivial
4020.2.f.b.401.4	yes	46	201.200	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.68102 - 0.417348i) q^{3} +1.00000 q^{5} +2.42754i q^{7} +(2.65164 + 1.40314i) q^{9} +O(q^{10})\)	\(q+(-1.68102 - 0.417348i) q^{3} +1.00000 q^{5} +2.42754i q^{7} +(2.65164 + 1.40314i) q^{9} -2.52837 q^{11} +0.420456i q^{13} +(-1.68102 - 0.417348i) q^{15} -1.53240i q^{17} -1.99037 q^{19} +(1.01313 - 4.08074i) q^{21} -2.90732i q^{23} +1.00000 q^{25} +(-3.87186 - 3.46536i) q^{27} -1.32422i q^{29} -4.77115i q^{31} +(4.25024 + 1.05521i) q^{33} +2.42754i q^{35} +9.45031 q^{37} +(0.175476 - 0.706795i) q^{39} +2.00106 q^{41} +2.63886i q^{43} +(2.65164 + 1.40314i) q^{45} -2.60954i q^{47} +1.10703 q^{49} +(-0.639542 + 2.57598i) q^{51} +1.37490 q^{53} -2.52837 q^{55} +(3.34585 + 0.830677i) q^{57} -14.2114i q^{59} +11.2815i q^{61} +(-3.40618 + 6.43698i) q^{63} +0.420456i q^{65} +(7.34375 - 3.61516i) q^{67} +(-1.21336 + 4.88726i) q^{69} -3.60277i q^{71} -1.92733 q^{73} +(-1.68102 - 0.417348i) q^{75} -6.13774i q^{77} +7.44491i q^{79} +(5.06241 + 7.44124i) q^{81} +13.4573i q^{83} -1.53240i q^{85} +(-0.552661 + 2.22604i) q^{87} +2.46697i q^{89} -1.02068 q^{91} +(-1.99123 + 8.02040i) q^{93} -1.99037 q^{95} +4.51661i q^{97} +(-6.70434 - 3.54766i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 46 q^{5} - 4 q^{9}+O(q^{10}) \) 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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