LMFDB - Embedded newform 4020.2.f.b.401.10

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			401.10
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.b.401.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.24570 + 1.20343i) q^{3} +1.00000 q^{5} +4.23048i q^{7} +(0.103517 - 2.99821i) q^{9} +O(q^{10})$	\(q+(-1.24570 + 1.20343i) q^{3} +1.00000 q^{5} +4.23048i q^{7} +(0.103517 - 2.99821i) q^{9} -5.66147 q^{11} +6.93967i q^{13} +(-1.24570 + 1.20343i) q^{15} -4.78020i q^{17} +1.51768 q^{19} +(-5.09108 - 5.26989i) q^{21} +3.29071i q^{23} +1.00000 q^{25} +(3.47919 + 3.85944i) q^{27} +10.0596i q^{29} +6.69843i q^{31} +(7.05247 - 6.81318i) q^{33} +4.23048i q^{35} +3.64442 q^{37} +(-8.35140 - 8.64472i) q^{39} -4.89894 q^{41} -5.51885i q^{43} +(0.103517 - 2.99821i) q^{45} -7.87576i q^{47} -10.8970 q^{49} +(5.75263 + 5.95468i) q^{51} -7.35304 q^{53} -5.66147 q^{55} +(-1.89057 + 1.82642i) q^{57} -0.429940i q^{59} -8.83793i q^{61} +(12.6839 + 0.437925i) q^{63} +6.93967i q^{65} +(-8.12402 + 1.00015i) q^{67} +(-3.96013 - 4.09922i) q^{69} +10.0962i q^{71} +11.5301 q^{73} +(-1.24570 + 1.20343i) q^{75} -23.9507i q^{77} +4.49158i q^{79} +(-8.97857 - 0.620730i) q^{81} +14.7247i q^{83} -4.78020i q^{85} +(-12.1060 - 12.5312i) q^{87} -4.24849i q^{89} -29.3581 q^{91} +(-8.06108 - 8.34421i) q^{93} +1.51768 q^{95} -17.5432i q^{97} +(-0.586056 + 16.9743i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.24570	+	1.20343i	−0.719203	+	0.694800i
$3$	−1.24570	+	1.20343i	−0.719203	+	0.694800i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			4.23048i			1.59897i	0.600685	+	0.799486i	$0.294894\pi$
$7$			4.23048i			1.59897i	−0.600685	+	0.799486i	$0.705106\pi$
$8$		0			0
$9$	0.103517	−	2.99821i	0.0345055	−	0.999405i
$10$		0			0
$11$	−5.66147			−1.70700			−0.853499	−	0.521095i	$-0.825524\pi$
$11$	−5.66147			−1.70700			−0.853499	+	0.521095i	$0.825524\pi$
$12$		0			0
$13$			6.93967i			1.92472i	0.271780	+	0.962359i	$0.412388\pi$
$13$			6.93967i			1.92472i	−0.271780	+	0.962359i	$0.587612\pi$
$14$		0			0
$15$	−1.24570	+	1.20343i	−0.321637	+	0.310724i
$16$		0			0
$17$		−	4.78020i		−	1.15937i	−0.814841	−	0.579684i	$-0.803176\pi$
$17$		−	4.78020i		−	1.15937i	0.814841	−	0.579684i	$-0.196824\pi$
$18$		0			0
$19$	1.51768			0.348179			0.174090	−	0.984730i	$-0.444302\pi$
$19$	1.51768			0.348179			0.174090	+	0.984730i	$0.444302\pi$
$20$		0			0
$21$	−5.09108	−	5.26989i	−1.11097	−	1.14998i
$22$		0			0
$23$			3.29071i			0.686160i	0.939306	+	0.343080i	$0.111470\pi$
$23$			3.29071i			0.686160i	−0.939306	+	0.343080i	$0.888530\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	3.47919	+	3.85944i	0.669570	+	0.742749i
$28$		0			0
$29$			10.0596i			1.86802i	0.357243	+	0.934012i	$0.383717\pi$
$29$			10.0596i			1.86802i	−0.357243	+	0.934012i	$0.616283\pi$
$30$		0			0
$31$			6.69843i			1.20307i	0.798845	+	0.601537i	$0.205445\pi$
$31$			6.69843i			1.20307i	−0.798845	+	0.601537i	$0.794555\pi$
$32$		0			0
$33$	7.05247	−	6.81318i	1.22768	−	1.18602i
$34$		0			0
$35$			4.23048i			0.715082i
$36$		0			0
$37$	3.64442			0.599139			0.299570	−	0.954074i	$-0.403157\pi$
$37$	3.64442			0.599139			0.299570	+	0.954074i	$0.403157\pi$
$38$		0			0
$39$	−8.35140	−	8.64472i	−1.33729	−	1.38426i
$40$		0			0
$41$	−4.89894			−0.765087			−0.382543	−	0.923938i	$-0.624952\pi$
$41$	−4.89894			−0.765087			−0.382543	+	0.923938i	$0.624952\pi$
$42$		0			0
$43$		−	5.51885i		−	0.841616i	−0.907150	−	0.420808i	$-0.861747\pi$
$43$		−	5.51885i		−	0.841616i	0.907150	−	0.420808i	$-0.138253\pi$
$44$		0			0
$45$	0.103517	−	2.99821i	0.0154313	−	0.446947i
$46$		0			0
$47$		−	7.87576i		−	1.14880i	−0.818576	−	0.574399i	$-0.805236\pi$
$47$		−	7.87576i		−	1.14880i	0.818576	−	0.574399i	$-0.194764\pi$
$48$		0			0
$49$	−10.8970			−1.55671
$50$		0			0
$51$	5.75263	+	5.95468i	0.805530	+	0.833821i
$52$		0			0
$53$	−7.35304			−1.01002			−0.505009	−	0.863114i	$-0.668511\pi$
$53$	−7.35304			−1.01002			−0.505009	+	0.863114i	$0.668511\pi$
$54$		0			0
$55$	−5.66147			−0.763392
$56$		0			0
$57$	−1.89057	+	1.82642i	−0.250412	+	0.241915i
$58$		0			0
$59$		−	0.429940i		−	0.0559734i	−0.999608	−	0.0279867i	$-0.991090\pi$
$59$		−	0.429940i		−	0.0559734i	0.999608	−	0.0279867i	$-0.00890961\pi$
$60$		0			0
$61$		−	8.83793i		−	1.13158i	−0.824549	−	0.565790i	$-0.808571\pi$
$61$		−	8.83793i		−	1.13158i	0.824549	−	0.565790i	$-0.191429\pi$
$62$		0			0
$63$	12.6839	+	0.437925i	1.59802	+	0.0551734i
$64$		0			0
$65$			6.93967i			0.860760i
$66$		0			0
$67$	−8.12402	+	1.00015i	−0.992507	+	0.122188i
$67$	−8.12402	+	1.00015i	−0.992507	+	0.122188i
$68$		0			0
$69$	−3.96013	−	4.09922i	−0.476744	−	0.493488i
$70$		0			0
$71$			10.0962i			1.19820i	0.800674	+	0.599100i	$0.204475\pi$
$71$			10.0962i			1.19820i	−0.800674	+	0.599100i	$0.795525\pi$
$72$		0			0
$73$	11.5301			1.34950			0.674748	−	0.738048i	$-0.264252\pi$
$73$	11.5301			1.34950			0.674748	+	0.738048i	$0.264252\pi$
$74$		0			0
$75$	−1.24570	+	1.20343i	−0.143841	+	0.138960i
$76$		0			0
$77$		−	23.9507i		−	2.72944i
$78$		0			0
$79$			4.49158i			0.505342i	0.967552	+	0.252671i	$0.0813090\pi$
$79$			4.49158i			0.505342i	−0.967552	+	0.252671i	$0.918691\pi$
$80$		0			0
$81$	−8.97857	−	0.620730i	−0.997619	−	0.0689700i
$82$		0			0
$83$			14.7247i			1.61624i	0.589016	+	0.808122i	$0.299516\pi$
$83$			14.7247i			1.61624i	−0.589016	+	0.808122i	$0.700484\pi$
$84$		0			0
$85$		−	4.78020i		−	0.518486i
$86$		0			0
$87$	−12.1060	−	12.5312i	−1.29790	−	1.34349i
$88$		0			0
$89$		−	4.24849i		−	0.450339i	−0.974320	−	0.225170i	$-0.927706\pi$
$89$		−	4.24849i		−	0.450339i	0.974320	−	0.225170i	$-0.0722936\pi$
$90$		0			0
$91$	−29.3581			−3.07757
$92$		0			0
$93$	−8.06108	−	8.34421i	−0.835895	−	0.865254i
$94$		0			0
$95$	1.51768			0.155711
$96$		0			0
$97$		−	17.5432i		−	1.78124i	−0.454748	−	0.890620i	$-0.650271\pi$
$97$		−	17.5432i		−	1.78124i	0.454748	−	0.890620i	$-0.349729\pi$
$98$		0			0
$99$	−0.586056	+	16.9743i	−0.0589009	+	1.70598i
$100$		0			0
$101$	6.60071			0.656796			0.328398	−	0.944540i	$-0.393491\pi$
$101$	6.60071			0.656796			0.328398	+	0.944540i	$0.393491\pi$
$102$		0			0
$103$	7.38168			0.727338			0.363669	−	0.931528i	$-0.381524\pi$
$103$	7.38168			0.727338			0.363669	+	0.931528i	$0.381524\pi$
$104$		0			0
$105$	−5.09108	−	5.26989i	−0.496839	−	0.514289i
$106$		0			0
$107$			5.39495i			0.521549i	0.965400	+	0.260775i	$0.0839780\pi$
$107$			5.39495i			0.521549i	−0.965400	+	0.260775i	$0.916022\pi$
$108$		0			0
$109$		−	12.1979i		−	1.16834i	−0.811630	−	0.584172i	$-0.801420\pi$
$109$		−	12.1979i		−	1.16834i	0.811630	−	0.584172i	$-0.198580\pi$
$110$		0			0
$111$	−4.53984	+	4.38580i	−0.430903	+	0.416282i
$112$		0			0
$113$	−7.94248			−0.747166			−0.373583	−	0.927597i	$-0.621871\pi$
$113$	−7.94248			−0.747166			−0.373583	+	0.927597i	$0.621871\pi$
$114$		0			0
$115$			3.29071i			0.306860i
$116$		0			0
$117$	20.8066	+	0.718372i	1.92357	+	0.0664135i
$118$		0			0
$119$	20.2225			1.85380
$120$		0			0
$121$	21.0522			1.91384
$122$		0			0
$123$	6.10260	−	5.89553i	0.550252	−	0.531582i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	13.4767			1.19587			0.597933	−	0.801546i	$-0.295989\pi$
$127$	13.4767			1.19587			0.597933	+	0.801546i	$0.295989\pi$
$128$		0			0
$129$	6.64154	+	6.87480i	0.584755	+	0.605293i
$130$		0			0
$131$			7.31296i			0.638936i	0.947597	+	0.319468i	$0.103504\pi$
$131$			7.31296i			0.638936i	−0.947597	+	0.319468i	$0.896496\pi$
$132$		0			0
$133$			6.42051i			0.556729i
$134$		0			0
$135$	3.47919	+	3.85944i	0.299441	+	0.332167i
$136$		0			0
$137$	2.44990			0.209309			0.104655	−	0.994509i	$-0.466626\pi$
$137$	2.44990			0.209309			0.104655	+	0.994509i	$0.466626\pi$
$138$		0			0
$139$			7.09001i			0.601367i	0.953724	+	0.300683i	$0.0972147\pi$
$139$			7.09001i			0.601367i	−0.953724	+	0.300683i	$0.902785\pi$
$140$		0			0
$141$	9.47792	+	9.81080i	0.798185	+	0.826218i
$142$		0			0
$143$		−	39.2887i		−	3.28549i
$144$		0			0
$145$			10.0596i			0.835405i
$146$		0			0
$147$	13.5743	−	13.1137i	1.11959	−	1.08160i
$148$		0			0
$149$		−	3.04659i		−	0.249586i	−0.992183	−	0.124793i	$-0.960173\pi$
$149$		−	3.04659i		−	0.249586i	0.992183	−	0.124793i	$-0.0398267\pi$
$150$		0			0
$151$	−14.4134			−1.17295			−0.586473	−	0.809969i	$-0.699484\pi$
$151$	−14.4134			−1.17295			−0.586473	+	0.809969i	$0.699484\pi$
$152$		0			0
$153$	−14.3321	−	0.494830i	−1.15868	−	0.0400047i
$154$		0			0
$155$			6.69843i			0.538031i
$156$		0			0
$157$	−15.2776			−1.21928			−0.609642	−	0.792677i	$-0.708687\pi$
$157$	−15.2776			−1.21928			−0.609642	+	0.792677i	$0.708687\pi$
$158$		0			0
$159$	9.15965	−	8.84886i	0.726407	−	0.701760i
$160$		0			0
$161$	−13.9213			−1.09715
$162$		0			0
$163$	−1.20122			−0.0940865			−0.0470432	−	0.998893i	$-0.514980\pi$
$163$	−1.20122			−0.0940865			−0.0470432	+	0.998893i	$0.514980\pi$
$164$		0			0
$165$	7.05247	−	6.81318i	0.549034	−	0.530405i
$166$		0			0
$167$		−	20.9338i		−	1.61991i	−0.586494	−	0.809953i	$-0.699492\pi$
$167$		−	20.9338i		−	1.61991i	0.586494	−	0.809953i	$-0.300508\pi$
$168$		0			0
$169$	−35.1590			−2.70454
$170$		0			0
$171$	0.157105	−	4.55033i	0.0120141	−	0.347972i
$172$		0			0
$173$		−	9.10142i		−	0.691968i	−0.938240	−	0.345984i	$-0.887545\pi$
$173$		−	9.10142i		−	0.691968i	0.938240	−	0.345984i	$-0.112455\pi$
$174$		0			0
$175$			4.23048i			0.319794i
$176$		0			0
$177$	0.517402	+	0.535574i	0.0388903	+	0.0402562i
$178$		0			0
$179$	−8.72650			−0.652249			−0.326124	−	0.945327i	$-0.605743\pi$
$179$	−8.72650			−0.652249			−0.326124	+	0.945327i	$0.605743\pi$
$180$		0			0
$181$	−22.7761			−1.69294			−0.846469	−	0.532439i	$-0.821276\pi$
$181$	−22.7761			−1.69294			−0.846469	+	0.532439i	$0.821276\pi$
$182$		0			0
$183$	10.6358	+	11.0094i	0.786222	+	0.813836i
$184$		0			0
$185$	3.64442			0.267943
$186$		0			0
$187$			27.0630i			1.97904i
$188$		0			0
$189$	−16.3273	+	14.7186i	−1.18763	+	1.07062i
$190$		0			0
$191$	11.5212			0.833648			0.416824	−	0.908987i	$-0.363143\pi$
$191$	11.5212			0.833648			0.416824	+	0.908987i	$0.363143\pi$
$192$		0			0
$193$	10.2806			0.740015			0.370008	−	0.929029i	$-0.379355\pi$
$193$	10.2806			0.740015			0.370008	+	0.929029i	$0.379355\pi$
$194$		0			0
$195$	−8.35140	−	8.64472i	−0.598056	−	0.619061i
$196$		0			0
$197$	22.2185			1.58300			0.791502	−	0.611166i	$-0.209299\pi$
$197$	22.2185			1.58300			0.791502	+	0.611166i	$0.209299\pi$
$198$		0			0
$199$	0.0739130			0.00523955			0.00261978	−	0.999997i	$-0.499166\pi$
$199$	0.0739130			0.00523955			0.00261978	+	0.999997i	$0.499166\pi$
$200$		0			0
$201$	8.91644	−	11.0226i	0.628917	−	0.777472i
$202$		0			0
$203$	−42.5570			−2.98692
$204$		0			0
$205$	−4.89894			−0.342157
$206$		0			0
$207$	9.86625	+	0.340643i	0.685751	+	0.0236763i
$208$		0			0
$209$	−8.59229			−0.594341
$210$		0			0
$211$	12.4547			0.857420			0.428710	−	0.903442i	$-0.358968\pi$
$211$	12.4547			0.857420			0.428710	+	0.903442i	$0.358968\pi$
$212$		0			0
$213$	−12.1501	−	12.5768i	−0.832509	−	0.861749i
$214$		0			0
$215$		−	5.51885i		−	0.376382i
$216$		0			0
$217$	−28.3376			−1.92368
$218$		0			0
$219$	−14.3630	+	13.8757i	−0.970562	+	0.937630i
$220$		0			0
$221$	33.1730			2.23146
$222$		0			0
$223$	−17.1631			−1.14933			−0.574663	−	0.818390i	$-0.694867\pi$
$223$	−17.1631			−1.14933			−0.574663	+	0.818390i	$0.694867\pi$
$224$		0			0
$225$	0.103517	−	2.99821i	0.00690111	−	0.199881i
$226$		0			0
$227$			11.3287i			0.751912i	0.926638	+	0.375956i	$0.122686\pi$
$227$			11.3287i			0.751912i	−0.926638	+	0.375956i	$0.877314\pi$
$228$		0			0
$229$		−	0.701935i		−	0.0463852i	−0.999731	−	0.0231926i	$-0.992617\pi$
$229$		−	0.701935i		−	0.0463852i	0.999731	−	0.0231926i	$-0.00738309\pi$
$230$		0			0
$231$	28.8230	+	29.8353i	1.89641	+	1.96302i
$232$		0			0
$233$	−17.1010			−1.12033			−0.560163	−	0.828383i	$-0.689261\pi$
$233$	−17.1010			−1.12033			−0.560163	+	0.828383i	$0.689261\pi$
$234$		0			0
$235$		−	7.87576i		−	0.513758i
$236$		0			0
$237$	−5.40530	−	5.59514i	−0.351112	−	0.363443i
$238$		0			0
$239$	−7.43160			−0.480710			−0.240355	−	0.970685i	$-0.577264\pi$
$239$	−7.43160			−0.480710			−0.240355	+	0.970685i	$0.577264\pi$
$240$		0			0
$241$	21.0275			1.35450			0.677249	−	0.735754i	$-0.263172\pi$
$241$	21.0275			1.35450			0.677249	+	0.735754i	$0.263172\pi$
$242$		0			0
$243$	11.9316	−	10.0318i	0.765411	−	0.643542i
$244$		0			0
$245$	−10.8970			−0.696181
$246$		0			0
$247$			10.5322i			0.670148i
$248$		0			0
$249$	−17.7201	−	18.3425i	−1.12297	−	1.16241i
$250$		0			0
$251$	−8.77827			−0.554079			−0.277040	−	0.960859i	$-0.589353\pi$
$251$	−8.77827			−0.554079			−0.277040	+	0.960859i	$0.589353\pi$
$252$		0			0
$253$		−	18.6302i		−	1.17127i
$254$		0			0
$255$	5.75263	+	5.95468i	0.360244	+	0.372896i
$256$		0			0
$257$		−	5.52818i		−	0.344838i	−0.985024	−	0.172419i	$-0.944842\pi$
$257$		−	5.52818i		−	0.344838i	0.985024	−	0.172419i	$-0.0551584\pi$
$258$		0			0
$259$			15.4177i			0.958006i
$260$		0			0
$261$	30.1609	+	1.04134i	1.86691	+	0.0644572i
$262$		0			0
$263$		−	17.0364i		−	1.05051i	−0.850944	−	0.525256i	$-0.823970\pi$
$263$		−	17.0364i		−	1.05051i	0.850944	−	0.525256i	$-0.176030\pi$
$264$		0			0
$265$	−7.35304			−0.451693
$266$		0			0
$267$	5.11276	+	5.29233i	0.312896	+	0.323885i
$268$		0			0
$269$		−	29.2706i		−	1.78466i	−0.451385	−	0.892329i	$-0.649070\pi$
$269$		−	29.2706i		−	1.78466i	0.451385	−	0.892329i	$-0.350930\pi$
$270$		0			0
$271$		−	23.9544i		−	1.45513i	−0.686040	−	0.727563i	$-0.740653\pi$
$271$		−	23.9544i		−	1.45513i	0.686040	−	0.727563i	$-0.259347\pi$
$272$		0			0
$273$	36.5713	−	35.3304i	2.21340	−	2.13830i
$274$		0			0
$275$	−5.66147			−0.341399
$276$		0			0
$277$	2.35565			0.141537			0.0707685	−	0.997493i	$-0.477455\pi$
$277$	2.35565			0.141537			0.0707685	+	0.997493i	$0.477455\pi$
$278$		0			0
$279$	20.0833	+	0.693399i	1.20236	+	0.0415127i
$280$		0			0
$281$	5.31664			0.317164			0.158582	−	0.987346i	$-0.449308\pi$
$281$	5.31664			0.317164			0.158582	+	0.987346i	$0.449308\pi$
$282$		0			0
$283$	−9.11084			−0.541583			−0.270792	−	0.962638i	$-0.587285\pi$
$283$	−9.11084			−0.541583			−0.270792	+	0.962638i	$0.587285\pi$
$284$		0			0
$285$	−1.89057	+	1.82642i	−0.111988	+	0.108188i
$286$		0			0
$287$		−	20.7249i		−	1.22335i
$288$		0			0
$289$	−5.85032			−0.344136
$290$		0			0
$291$	21.1120	+	21.8535i	1.23761	+	1.28107i
$292$		0			0
$293$			8.29393i			0.484537i	0.970209	+	0.242268i	$0.0778914\pi$
$293$			8.29393i			0.484537i	−0.970209	+	0.242268i	$0.922109\pi$
$294$		0			0
$295$		−	0.429940i		−	0.0250321i
$296$		0			0
$297$	−19.6973	−	21.8501i	−1.14295	−	1.26787i
$298$		0			0
$299$	−22.8364			−1.32067
$300$		0			0
$301$	23.3474			1.34572
$302$		0			0
$303$	−8.22248	+	7.94349i	−0.472369	+	0.456342i
$304$		0			0
$305$		−	8.83793i		−	0.506058i
$306$		0			0
$307$	3.80162			0.216970			0.108485	−	0.994098i	$-0.465400\pi$
$307$	3.80162			0.216970			0.108485	+	0.994098i	$0.465400\pi$
$308$		0			0
$309$	−9.19533	+	8.88333i	−0.523104	+	0.505355i
$310$		0			0
$311$	−31.4015			−1.78061			−0.890307	−	0.455362i	$-0.849510\pi$
$311$	−31.4015			−1.78061			−0.890307	+	0.455362i	$0.849510\pi$
$312$		0			0
$313$			6.79150i			0.383878i	0.981407	+	0.191939i	$0.0614776\pi$
$313$			6.79150i			0.383878i	−0.981407	+	0.191939i	$0.938522\pi$
$314$		0			0
$315$	12.6839	+	0.437925i	0.714656	+	0.0246743i
$316$		0			0
$317$			10.4589i			0.587430i	0.955893	+	0.293715i	$0.0948917\pi$
$317$			10.4589i			0.587430i	−0.955893	+	0.293715i	$0.905108\pi$
$318$		0			0
$319$		−	56.9522i		−	3.18871i
$320$		0			0
$321$	−6.49244	−	6.72047i	−0.362373	−	0.375100i
$322$		0			0
$323$		−	7.25481i		−	0.403668i
$324$		0			0
$325$			6.93967i			0.384944i
$326$		0			0
$327$	14.6793	+	15.1948i	0.811765	+	0.840276i
$328$		0			0
$329$	33.3182			1.83689
$330$		0			0
$331$		−	13.9267i		−	0.765479i	−0.923856	−	0.382739i	$-0.874981\pi$
$331$		−	13.9267i		−	0.765479i	0.923856	−	0.382739i	$-0.125019\pi$
$332$		0			0
$333$	0.377258	−	10.9268i	0.0206736	−	0.598782i
$334$		0			0
$335$	−8.12402	+	1.00015i	−0.443863	+	0.0546443i
$336$		0			0
$337$		−	9.79449i		−	0.533540i	−0.963760	−	0.266770i	$-0.914044\pi$
$337$		−	9.79449i		−	0.533540i	0.963760	−	0.266770i	$-0.0859564\pi$
$338$		0			0
$339$	9.89391	−	9.55821i	0.537364	−	0.519131i
$340$		0			0
$341$		−	37.9229i		−	2.05364i
$342$		0			0
$343$		−	16.4860i		−	0.890161i
$344$		0			0
$345$	−3.96013	−	4.09922i	−0.213206	−	0.220695i
$346$		0			0
$347$	20.4815			1.09951			0.549753	−	0.835327i	$-0.314722\pi$
$347$	20.4815			1.09951			0.549753	+	0.835327i	$0.314722\pi$
$348$		0			0
$349$	19.5855			1.04839			0.524193	−	0.851599i	$-0.324367\pi$
$349$	19.5855			1.04839			0.524193	+	0.851599i	$0.324367\pi$
$350$		0			0
$351$	−26.7832	+	24.1444i	−1.42958	+	1.28873i
$352$		0			0
$353$	−3.29759			−0.175513			−0.0877564	−	0.996142i	$-0.527970\pi$
$353$	−3.29759			−0.175513			−0.0877564	+	0.996142i	$0.527970\pi$
$354$		0			0
$355$			10.0962i			0.535851i
$356$		0			0
$357$	−25.1911	+	24.3364i	−1.33326	+	1.28802i
$358$		0			0
$359$		−	5.99278i		−	0.316287i	−0.987416	−	0.158144i	$-0.949449\pi$
$359$		−	5.99278i		−	0.316287i	0.987416	−	0.158144i	$-0.0505509\pi$
$360$		0			0
$361$	−16.6967			−0.878771
$362$		0			0
$363$	−26.2247	+	25.3349i	−1.37644	+	1.32974i
$364$		0			0
$365$	11.5301			0.603513
$366$		0			0
$367$			17.3056i			0.903347i	0.892183	+	0.451673i	$0.149173\pi$
$367$			17.3056i			0.903347i	−0.892183	+	0.451673i	$0.850827\pi$
$368$		0			0
$369$	−0.507122	+	14.6881i	−0.0263997	+	0.764631i
$370$		0			0
$371$		−	31.1069i		−	1.61499i
$372$		0			0
$373$			1.46118i			0.0756569i	0.999284	+	0.0378284i	$0.0120440\pi$
$373$			1.46118i			0.0756569i	−0.999284	+	0.0378284i	$0.987956\pi$
$374$		0			0
$375$	−1.24570	+	1.20343i	−0.0643275	+	0.0621448i
$376$		0			0
$377$	−69.8104			−3.59542
$378$		0			0
$379$		−	30.4666i		−	1.56496i	−0.622673	−	0.782482i	$-0.713953\pi$
$379$		−	30.4666i		−	1.56496i	0.622673	−	0.782482i	$-0.286047\pi$
$380$		0			0
$381$	−16.7879	+	16.2183i	−0.860070	+	0.830887i
$382$		0			0
$383$	15.6622			0.800303			0.400152	−	0.916449i	$-0.368957\pi$
$383$	15.6622			0.800303			0.400152	+	0.916449i	$0.368957\pi$
$384$		0			0
$385$		−	23.9507i		−	1.22064i
$386$		0			0
$387$	−16.5467	−	0.571292i	−0.841115	−	0.0290404i
$388$		0			0
$389$			16.2578i			0.824305i	0.911115	+	0.412153i	$0.135223\pi$
$389$			16.2578i			0.824305i	−0.911115	+	0.412153i	$0.864777\pi$
$390$		0			0
$391$	15.7302			0.795513
$392$		0			0
$393$	−8.80063	−	9.10972i	−0.443933	−	0.459525i
$394$		0			0
$395$			4.49158i			0.225996i
$396$		0			0
$397$	11.1400			0.559103			0.279552	−	0.960131i	$-0.409814\pi$
$397$	11.1400			0.559103			0.279552	+	0.960131i	$0.409814\pi$
$398$		0			0
$399$	−7.72663	−	7.99800i	−0.386815	−	0.400401i
$400$		0			0
$401$	−0.580797			−0.0290036			−0.0145018	−	0.999895i	$-0.504616\pi$
$401$	−0.580797			−0.0290036			−0.0145018	+	0.999895i	$0.504616\pi$
$402$		0			0
$403$	−46.4849			−2.31558
$404$		0			0
$405$	−8.97857	−	0.620730i	−0.446149	−	0.0308443i
$406$		0			0
$407$	−20.6328			−1.02273
$408$		0			0
$409$			17.2824i			0.854560i	0.904119	+	0.427280i	$0.140528\pi$
$409$			17.2824i			0.854560i	−0.904119	+	0.427280i	$0.859472\pi$
$410$		0			0
$411$	−3.05183	+	2.94828i	−0.150536	+	0.145428i
$412$		0			0
$413$	1.81885			0.0894999
$414$		0			0
$415$			14.7247i			0.722806i
$416$		0			0
$417$	−8.53232	−	8.83199i	−0.417830	−	0.432505i
$418$		0			0
$419$			8.36548i			0.408680i	0.978900	+	0.204340i	$0.0655049\pi$
$419$			8.36548i			0.408680i	−0.978900	+	0.204340i	$0.934495\pi$
$420$		0			0
$421$	−6.51840			−0.317687			−0.158844	−	0.987304i	$-0.550777\pi$
$421$	−6.51840			−0.317687			−0.158844	+	0.987304i	$0.550777\pi$
$422$		0			0
$423$	−23.6132	−	0.815272i	−1.14811	−	0.0396399i
$424$		0			0
$425$		−	4.78020i		−	0.231874i
$426$		0			0
$427$	37.3887			1.80936
$428$		0			0
$429$	47.2812	+	48.9418i	2.28276	+	2.36293i
$430$		0			0
$431$			28.2764i			1.36203i	0.732270	+	0.681014i	$0.238461\pi$
$431$			28.2764i			1.36203i	−0.732270	+	0.681014i	$0.761539\pi$
$432$		0			0
$433$		−	29.8043i		−	1.43231i	−0.697944	−	0.716153i	$-0.745901\pi$
$433$		−	29.8043i		−	1.43231i	0.697944	−	0.716153i	$-0.254099\pi$
$434$		0			0
$435$	−12.1060	−	12.5312i	−0.580440	−	0.600826i
$436$		0			0
$437$			4.99424i			0.238907i
$438$		0			0
$439$	−24.3874			−1.16395			−0.581975	−	0.813207i	$-0.697720\pi$
$439$	−24.3874			−1.16395			−0.581975	+	0.813207i	$0.697720\pi$
$440$		0			0
$441$	−1.12802	+	32.6714i	−0.0537151	+	1.55578i
$442$		0			0
$443$	−26.9390			−1.27991			−0.639956	−	0.768412i	$-0.721047\pi$
$443$	−26.9390			−1.27991			−0.639956	+	0.768412i	$0.721047\pi$
$444$		0			0
$445$		−	4.24849i		−	0.201398i
$446$		0			0
$447$	3.66635	+	3.79512i	0.173412	+	0.179503i
$448$		0			0
$449$			4.87231i			0.229939i	0.993369	+	0.114969i	$0.0366770\pi$
$449$			4.87231i			0.229939i	−0.993369	+	0.114969i	$0.963323\pi$
$450$		0			0
$451$	27.7352			1.30600
$452$		0			0
$453$	17.9547	−	17.3455i	0.843586	−	0.814963i
$454$		0			0
$455$	−29.3581			−1.37633
$456$		0			0
$457$	−23.0359			−1.07757			−0.538786	−	0.842443i	$-0.681117\pi$
$457$	−23.0359			−1.07757			−0.538786	+	0.842443i	$0.681117\pi$
$458$		0			0
$459$	18.4489	−	16.6312i	0.861120	−	0.776279i
$460$		0			0
$461$			41.5386i			1.93465i	0.253548	+	0.967323i	$0.418402\pi$
$461$			41.5386i			1.93465i	−0.253548	+	0.967323i	$0.581598\pi$
$462$		0			0
$463$		−	22.6958i		−	1.05476i	−0.849629	−	0.527381i	$-0.823174\pi$
$463$		−	22.6958i		−	1.05476i	0.849629	−	0.527381i	$-0.176826\pi$
$464$		0			0
$465$	−8.06108	−	8.34421i	−0.373824	−	0.386953i
$466$		0			0
$467$			10.0719i			0.466073i	0.972468	+	0.233037i	$0.0748662\pi$
$467$			10.0719i			0.466073i	−0.972468	+	0.233037i	$0.925134\pi$
$468$		0			0
$469$	−4.23113	−	34.3685i	−0.195376	−	1.58699i
$470$		0			0
$471$	19.0312	−	18.3855i	0.876913	−	0.847159i
$472$		0			0
$473$			31.2448i			1.43664i
$474$		0			0
$475$	1.51768			0.0696359
$476$		0			0
$477$	−0.761162	+	22.0460i	−0.0348512	+	1.00942i
$478$		0			0
$479$			0.248641i			0.0113607i	0.999984	+	0.00568035i	$0.00180812\pi$
$479$			0.248641i			0.0113607i	−0.999984	+	0.00568035i	$0.998192\pi$
$480$		0			0
$481$			25.2911i			1.15317i
$482$		0			0
$483$	17.3417	−	16.7533i	0.789074	−	0.762300i
$484$		0			0
$485$		−	17.5432i		−	0.796595i
$486$		0			0
$487$			1.98824i			0.0900958i	0.998985	+	0.0450479i	$0.0143441\pi$
$487$			1.98824i			0.0900958i	−0.998985	+	0.0450479i	$0.985656\pi$
$488$		0			0
$489$	1.49635	−	1.44558i	0.0676673	−	0.0653713i
$490$		0			0
$491$			24.9540i			1.12616i	0.826402	+	0.563080i	$0.190384\pi$
$491$			24.9540i			1.12616i	−0.826402	+	0.563080i	$0.809616\pi$
$492$		0			0
$493$	48.0870			2.16573
$494$		0			0
$495$	−0.586056	+	16.9743i	−0.0263413	+	0.762938i
$496$		0			0
$497$	−42.7118			−1.91589
$498$		0			0
$499$		−	12.7569i		−	0.571076i	−0.958367	−	0.285538i	$-0.907828\pi$
$499$		−	12.7569i		−	0.571076i	0.958367	−	0.285538i	$-0.0921723\pi$
$500$		0			0
$501$	25.1924	+	26.0772i	1.12551	+	1.16504i
$502$		0			0
$503$	−19.0680			−0.850200			−0.425100	−	0.905146i	$-0.639761\pi$
$503$	−19.0680			−0.850200			−0.425100	+	0.905146i	$0.639761\pi$
$504$		0			0
$505$	6.60071			0.293728
$506$		0			0
$507$	43.7975	−	42.3114i	1.94511	−	1.87912i
$508$		0			0
$509$			7.05340i			0.312637i	0.987707	+	0.156318i	$0.0499626\pi$
$509$			7.05340i			0.312637i	−0.987707	+	0.156318i	$0.950037\pi$
$510$		0			0
$511$			48.7779i			2.15781i
$512$		0			0
$513$	5.28029	+	5.85739i	0.233130	+	0.258610i
$514$		0			0
$515$	7.38168			0.325276
$516$		0			0
$517$			44.5884i			1.96099i
$518$		0			0
$519$	10.9529	+	11.3376i	0.480780	+	0.497666i
$520$		0			0
$521$	8.93986			0.391662			0.195831	−	0.980638i	$-0.437260\pi$
$521$	8.93986			0.391662			0.195831	+	0.980638i	$0.437260\pi$
$522$		0			0
$523$	−14.5291			−0.635311			−0.317656	−	0.948206i	$-0.602896\pi$
$523$	−14.5291			−0.635311			−0.317656	+	0.948206i	$0.602896\pi$
$524$		0			0
$525$	−5.09108	−	5.26989i	−0.222193	−	0.229997i
$526$		0			0
$527$	32.0198			1.39481
$528$		0			0
$529$	12.1712			0.529184
$530$		0			0
$531$	−1.28905	−	0.0445059i	−0.0559401	−	0.00193139i
$532$		0			0
$533$		−	33.9971i		−	1.47258i
$534$		0			0
$535$			5.39495i			0.233244i
$536$		0			0
$537$	10.8706	−	10.5017i	0.469099	−	0.453183i
$538$		0			0
$539$	61.6928			2.65730
$540$		0			0
$541$			16.0223i			0.688854i	0.938813	+	0.344427i	$0.111927\pi$
$541$			16.0223i			0.688854i	−0.938813	+	0.344427i	$0.888073\pi$
$542$		0			0
$543$	28.3721	−	27.4095i	1.21757	−	1.17625i
$544$		0			0
$545$		−	12.1979i		−	0.522499i
$546$		0			0
$547$			35.6165i			1.52285i	0.648252	+	0.761426i	$0.275500\pi$
$547$			35.6165i			1.52285i	−0.648252	+	0.761426i	$0.724500\pi$
$548$		0			0
$549$	−26.4980	−	0.914872i	−1.13091	−	0.0390458i
$550$		0			0
$551$			15.2673i			0.650407i
$552$		0			0
$553$	−19.0015			−0.808027
$554$		0			0
$555$	−4.53984	+	4.38580i	−0.192706	+	0.186167i
$556$		0			0
$557$		−	15.6127i		−	0.661531i	−0.943713	−	0.330765i	$-0.892693\pi$
$557$		−	15.6127i		−	0.661531i	0.943713	−	0.330765i	$-0.107307\pi$
$558$		0			0
$559$	38.2990			1.61987
$560$		0			0
$561$	−32.5684	−	33.7122i	−1.37504	−	1.42333i
$562$		0			0
$563$	−5.91899			−0.249456			−0.124728	−	0.992191i	$-0.539806\pi$
$563$	−5.91899			−0.249456			−0.124728	+	0.992191i	$0.539806\pi$
$564$		0			0
$565$	−7.94248			−0.334143
$566$		0			0
$567$	2.62599	−	37.9837i	0.110281	−	1.59516i
$568$		0			0
$569$		−	3.20204i		−	0.134236i	−0.997745	−	0.0671182i	$-0.978620\pi$
$569$		−	3.20204i		−	0.134236i	0.997745	−	0.0671182i	$-0.0213805\pi$
$570$		0			0
$571$	18.5496			0.776277			0.388138	−	0.921601i	$-0.373118\pi$
$571$	18.5496			0.776277			0.388138	+	0.921601i	$0.373118\pi$
$572$		0			0
$573$	−14.3520	+	13.8650i	−0.599562	+	0.579219i
$574$		0			0
$575$			3.29071i			0.137232i
$576$		0			0
$577$			31.8111i			1.32431i	0.749365	+	0.662157i	$0.230359\pi$
$577$			31.8111i			1.32431i	−0.749365	+	0.662157i	$0.769641\pi$
$578$		0			0
$579$	−12.8065	+	12.3720i	−0.532221	+	0.514163i
$580$		0			0
$581$	−62.2924			−2.58433
$582$		0			0
$583$	41.6290			1.72410
$584$		0			0
$585$	20.8066	+	0.718372i	0.860248	+	0.0297010i
$586$		0			0
$587$	33.8833			1.39851			0.699256	−	0.714871i	$-0.253515\pi$
$587$	33.8833			1.39851			0.699256	+	0.714871i	$0.253515\pi$
$588$		0			0
$589$			10.1661i			0.418885i
$590$		0			0
$591$	−27.6775	+	26.7384i	−1.13850	+	1.09987i
$592$		0			0
$593$	−4.04590			−0.166145			−0.0830727	−	0.996543i	$-0.526473\pi$
$593$	−4.04590			−0.166145			−0.0830727	+	0.996543i	$0.526473\pi$
$594$		0			0
$595$	20.2225			0.829043
$596$		0			0
$597$	−0.0920731	+	0.0889490i	−0.00376830	+	0.00364044i
$598$		0			0
$599$	−21.4285			−0.875547			−0.437773	−	0.899085i	$-0.644233\pi$
$599$	−21.4285			−0.875547			−0.437773	+	0.899085i	$0.644233\pi$
$600$		0			0
$601$	26.7482			1.09108			0.545540	−	0.838085i	$-0.316324\pi$
$601$	26.7482			1.09108			0.545540	+	0.838085i	$0.316324\pi$
$602$		0			0
$603$	2.15771	+	24.4611i	0.0878686	+	0.996132i
$604$		0			0
$605$	21.0522			0.855895
$606$		0			0
$607$	19.5752			0.794534			0.397267	−	0.917703i	$-0.369959\pi$
$607$	19.5752			0.794534			0.397267	+	0.917703i	$0.369959\pi$
$608$		0			0
$609$	53.0131	−	51.2143i	2.14820	−	2.07531i
$610$		0			0
$611$	54.6552			2.21111
$612$		0			0
$613$	−45.8941			−1.85364			−0.926822	−	0.375501i	$-0.877471\pi$
$613$	−45.8941			−1.85364			−0.926822	+	0.375501i	$0.877471\pi$
$614$		0			0
$615$	6.10260	−	5.89553i	0.246080	−	0.237731i
$616$		0			0
$617$			46.6578i			1.87837i	0.343407	+	0.939187i	$0.388419\pi$
$617$			46.6578i			1.87837i	−0.343407	+	0.939187i	$0.611581\pi$
$618$		0			0
$619$	−16.3383			−0.656692			−0.328346	−	0.944557i	$-0.606491\pi$
$619$	−16.3383			−0.656692			−0.328346	+	0.944557i	$0.606491\pi$
$620$		0			0
$621$	−12.7003	+	11.4490i	−0.509645	+	0.459432i
$622$		0			0
$623$	17.9732			0.720079
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	10.7034	−	10.3402i	0.427452	−	0.412948i
$628$		0			0
$629$		−	17.4211i		−	0.694623i
$630$		0			0
$631$			29.9649i			1.19289i	0.802656	+	0.596443i	$0.203420\pi$
$631$			29.9649i			1.19289i	−0.802656	+	0.596443i	$0.796580\pi$
$632$		0			0
$633$	−15.5148	+	14.9884i	−0.616659	+	0.595736i
$634$		0			0
$635$	13.4767			0.534807
$636$		0			0
$637$		−	75.6213i		−	2.99623i
$638$		0			0
$639$	30.2706	+	1.04513i	1.19749	+	0.0413445i
$640$		0			0
$641$	15.6269			0.617227			0.308614	−	0.951188i	$-0.400135\pi$
$641$	15.6269			0.617227			0.308614	+	0.951188i	$0.400135\pi$
$642$		0			0
$643$	45.5900			1.79790			0.898948	−	0.438056i	$-0.144333\pi$
$643$	45.5900			1.79790			0.898948	+	0.438056i	$0.144333\pi$
$644$		0			0
$645$	6.64154	+	6.87480i	0.261510	+	0.270695i
$646$		0			0
$647$	−27.5674			−1.08379			−0.541894	−	0.840447i	$-0.682292\pi$
$647$	−27.5674			−1.08379			−0.541894	+	0.840447i	$0.682292\pi$
$648$		0			0
$649$			2.43409i			0.0955465i
$650$		0			0
$651$	35.3000	−	34.1023i	1.38352	−	1.33657i
$652$		0			0
$653$	13.1167			0.513294			0.256647	−	0.966505i	$-0.417382\pi$
$653$	13.1167			0.513294			0.256647	+	0.966505i	$0.417382\pi$
$654$		0			0
$655$			7.31296i			0.285741i
$656$		0			0
$657$	1.19356	−	34.5697i	0.0465651	−	1.34869i
$658$		0			0
$659$			17.9830i			0.700520i	0.936653	+	0.350260i	$0.113907\pi$
$659$			17.9830i			0.700520i	−0.936653	+	0.350260i	$0.886093\pi$
$660$		0			0
$661$			41.8152i			1.62642i	0.581969	+	0.813211i	$0.302282\pi$
$661$			41.8152i			1.62642i	−0.581969	+	0.813211i	$0.697718\pi$
$662$		0			0
$663$	−41.3235	+	39.9214i	−1.60487	+	1.55042i
$664$		0			0
$665$			6.42051i			0.248977i
$666$		0			0
$667$	−33.1032			−1.28176
$668$		0			0
$669$	21.3800	−	20.6546i	0.826599	−	0.798552i
$670$		0			0
$671$			50.0356i			1.93160i
$672$		0			0
$673$		−	1.66433i		−	0.0641553i	−0.999485	−	0.0320777i	$-0.989788\pi$
$673$		−	1.66433i		−	0.0641553i	0.999485	−	0.0320777i	$-0.0102124\pi$
$674$		0			0
$675$	3.47919	+	3.85944i	0.133914	+	0.148550i
$676$		0			0
$677$	14.9445			0.574363			0.287182	−	0.957876i	$-0.407282\pi$
$677$	14.9445			0.574363			0.287182	+	0.957876i	$0.407282\pi$
$678$		0			0
$679$	74.2161			2.84815
$680$		0			0
$681$	−13.6333	−	14.1121i	−0.522428	−	0.540777i
$682$		0			0
$683$	−43.6934			−1.67188			−0.835940	−	0.548821i	$-0.815077\pi$
$683$	−43.6934			−1.67188			−0.835940	+	0.548821i	$0.815077\pi$
$684$		0			0
$685$	2.44990			0.0936059
$686$		0			0
$687$	0.844729	+	0.874397i	0.0322284	+	0.0333603i
$688$		0			0
$689$		−	51.0277i		−	1.94400i
$690$		0			0
$691$	−31.9451			−1.21525			−0.607625	−	0.794224i	$-0.707878\pi$
$691$	−31.9451			−1.21525			−0.607625	+	0.794224i	$0.707878\pi$
$692$		0			0
$693$	−71.8094	−	2.47930i	−2.72781	−	0.0941808i
$694$		0			0
$695$			7.09001i			0.268939i
$696$		0			0
$697$			23.4179i			0.887018i
$698$		0			0
$699$	21.3027	−	20.5799i	0.805741	−	0.778402i
$700$		0			0
$701$	7.57857			0.286239			0.143119	−	0.989705i	$-0.454287\pi$
$701$	7.57857			0.286239			0.143119	+	0.989705i	$0.454287\pi$
$702$		0			0
$703$	5.53106			0.208608
$704$		0			0
$705$	9.47792	+	9.81080i	0.356959	+	0.369496i
$706$		0			0
$707$			27.9242i			1.05020i
$708$		0			0
$709$	−12.5352			−0.470768			−0.235384	−	0.971902i	$-0.575635\pi$
$709$	−12.5352			−0.470768			−0.235384	+	0.971902i	$0.575635\pi$
$710$		0			0
$711$	13.4667	+	0.464953i	0.505041	+	0.0174371i
$712$		0			0
$713$	−22.0426			−0.825501
$714$		0			0
$715$		−	39.2887i		−	1.46932i
$716$		0			0
$717$	9.25751	−	8.94340i	0.345728	−	0.333997i
$718$		0			0
$719$		−	29.3832i		−	1.09581i	−0.836541	−	0.547904i	$-0.815426\pi$
$719$		−	29.3832i		−	1.09581i	0.836541	−	0.547904i	$-0.184574\pi$
$720$		0			0
$721$			31.2280i			1.16299i
$722$		0			0
$723$	−26.1938	+	25.3051i	−0.974159	+	0.941106i
$724$		0			0
$725$			10.0596i			0.373605i
$726$		0			0
$727$			15.2591i			0.565928i	0.959131	+	0.282964i	$0.0913176\pi$
$727$			15.2591i			0.565928i	−0.959131	+	0.282964i	$0.908682\pi$
$728$		0			0
$729$	−2.79051	+	26.8554i	−0.103352	+	0.994645i
$730$		0			0
$731$	−26.3812			−0.975744
$732$		0			0
$733$			3.26830i			0.120718i	0.998177	+	0.0603588i	$0.0192245\pi$
$733$			3.26830i			0.120718i	−0.998177	+	0.0603588i	$0.980776\pi$
$734$		0			0
$735$	13.5743	−	13.1137i	0.500696	−	0.483707i
$736$		0			0
$737$	45.9939	−	5.66234i	1.69421	−	0.208575i
$738$		0			0
$739$		−	10.1806i		−	0.374501i	−0.982312	−	0.187250i	$-0.940042\pi$
$739$		−	10.1806i		−	0.374501i	0.982312	−	0.187250i	$-0.0599575\pi$
$740$		0			0
$741$	−12.6748	−	13.1199i	−0.465619	−	0.481972i
$742$		0			0
$743$		−	23.2874i		−	0.854333i	−0.904173	−	0.427166i	$-0.859512\pi$
$743$		−	23.2874i		−	0.854333i	0.904173	−	0.427166i	$-0.140488\pi$
$744$		0			0
$745$		−	3.04659i		−	0.111618i
$746$		0			0
$747$	44.1477	+	1.52425i	1.61528	+	0.0557693i
$748$		0			0
$749$	−22.8232			−0.833942
$750$		0			0
$751$	3.62466			0.132266			0.0661329	−	0.997811i	$-0.478934\pi$
$751$	3.62466			0.132266			0.0661329	+	0.997811i	$0.478934\pi$
$752$		0			0
$753$	10.9350	−	10.5640i	0.398495	−	0.384974i
$754$		0			0
$755$	−14.4134			−0.524557
$756$		0			0
$757$			8.59542i			0.312406i	0.987725	+	0.156203i	$0.0499254\pi$
$757$			8.59542i			0.312406i	−0.987725	+	0.156203i	$0.950075\pi$
$758$		0			0
$759$	22.4202	+	23.2076i	0.813801	+	0.842383i
$760$		0			0
$761$			25.0899i			0.909508i	0.890617	+	0.454754i	$0.150273\pi$
$761$			25.0899i			0.909508i	−0.890617	+	0.454754i	$0.849727\pi$
$762$		0			0
$763$	51.6028			1.86815
$764$		0			0
$765$	−14.3321	−	0.494830i	−0.518177	−	0.0178906i
$766$		0			0
$767$	2.98364			0.107733
$768$		0			0
$769$			25.3050i			0.912521i	0.889846	+	0.456260i	$0.150811\pi$
$769$			25.3050i			0.912521i	−0.889846	+	0.456260i	$0.849189\pi$
$770$		0			0
$771$	6.65277	+	6.88643i	0.239594	+	0.248009i
$772$		0			0
$773$			40.3358i			1.45078i	0.688339	+	0.725389i	$0.258340\pi$
$773$			40.3358i			1.45078i	−0.688339	+	0.725389i	$0.741660\pi$
$774$		0			0
$775$			6.69843i			0.240615i
$776$		0			0
$777$	−18.5541	−	19.2057i	−0.665623	−	0.689001i
$778$		0			0
$779$	−7.43503			−0.266387
$780$		0			0
$781$		−	57.1594i		−	2.04532i
$782$		0			0
$783$	−38.8244	+	34.9993i	−1.38747	+	1.25077i
$784$		0			0
$785$	−15.2776			−0.545281
$786$		0			0
$787$			7.72221i			0.275267i	0.990483	+	0.137633i	$0.0439496\pi$
$787$			7.72221i			0.275267i	−0.990483	+	0.137633i	$0.956050\pi$
$788$		0			0
$789$	20.5021	+	21.2222i	0.729896	+	0.755531i
$790$		0			0
$791$		−	33.6005i		−	1.19470i
$792$		0			0
$793$	61.3323			2.17797
$794$		0			0
$795$	9.15965	−	8.84886i	0.324859	−	0.313837i
$796$		0			0
$797$			1.42894i			0.0506157i	0.999680	+	0.0253078i	$0.00805660\pi$
$797$			1.42894i			0.0506157i	−0.999680	+	0.0253078i	$0.991943\pi$
$798$		0			0
$799$	−37.6477			−1.33188
$800$		0			0
$801$	−12.7379	−	0.439790i	−0.450071	−	0.0155392i
$802$		0			0
$803$	−65.2773			−2.30359
$804$		0			0
$805$	−13.9213			−0.490660
$806$		0			0
$807$	35.2251	+	36.4622i	1.23998	+	1.28353i
$808$		0			0
$809$	−32.0754			−1.12771			−0.563856	−	0.825873i	$-0.690683\pi$
$809$	−32.0754			−1.12771			−0.563856	+	0.825873i	$0.690683\pi$
$810$		0			0
$811$			47.4618i			1.66661i	0.552814	+	0.833305i	$0.313554\pi$
$811$			47.4618i			1.66661i	−0.552814	+	0.833305i	$0.686446\pi$
$812$		0			0
$813$	28.8274	+	29.8399i	1.01102	+	1.04653i
$814$		0			0
$815$	−1.20122			−0.0420767
$816$		0			0
$817$		−	8.37584i		−	0.293033i
$818$		0			0
$819$	−3.03906	+	88.0220i	−0.106193	+	3.07574i
$820$		0			0
$821$			19.5761i			0.683211i	0.939843	+	0.341605i	$0.110971\pi$
$821$			19.5761i			0.683211i	−0.939843	+	0.341605i	$0.889029\pi$
$822$		0			0
$823$	48.3690			1.68604			0.843019	−	0.537883i	$-0.180776\pi$
$823$	48.3690			1.68604			0.843019	+	0.537883i	$0.180776\pi$
$824$		0			0
$825$	7.05247	−	6.81318i	0.245535	−	0.237204i
$826$		0			0
$827$			19.0316i			0.661793i	0.943667	+	0.330897i	$0.107351\pi$
$827$			19.0316i			0.661793i	−0.943667	+	0.330897i	$0.892649\pi$
$828$		0			0
$829$	−9.21999			−0.320224			−0.160112	−	0.987099i	$-0.551185\pi$
$829$	−9.21999			−0.320224			−0.160112	+	0.987099i	$0.551185\pi$
$830$		0			0
$831$	−2.93442	+	2.83485i	−0.101794	+	0.0983399i
$832$		0			0
$833$			52.0897i			1.80480i
$834$		0			0
$835$		−	20.9338i		−	0.724444i
$836$		0			0
$837$	−25.8522	+	23.3051i	−0.893582	+	0.805542i
$838$		0			0
$839$		−	5.54523i		−	0.191443i	−0.995408	−	0.0957213i	$-0.969484\pi$
$839$		−	5.54523i		−	0.191443i	0.995408	−	0.0957213i	$-0.0305158\pi$
$840$		0			0
$841$	−72.1958			−2.48951
$842$		0			0
$843$	−6.62292	+	6.39820i	−0.228105	+	0.220366i
$844$		0			0
$845$	−35.1590			−1.20951
$846$		0			0
$847$			89.0611i			3.06017i
$848$		0			0
$849$	11.3493	−	10.9643i	0.389508	−	0.376292i
$850$		0			0
$851$			11.9927i			0.411105i
$852$		0			0
$853$	−38.2932			−1.31113			−0.655567	−	0.755137i	$-0.727570\pi$
$853$	−38.2932			−1.31113			−0.655567	+	0.755137i	$0.727570\pi$
$854$		0			0
$855$	0.157105	−	4.55033i	0.00537288	−	0.155618i
$856$		0			0
$857$	5.05773			0.172769			0.0863844	−	0.996262i	$-0.472469\pi$
$857$	5.05773			0.172769			0.0863844	+	0.996262i	$0.472469\pi$
$858$		0			0
$859$	−0.398092			−0.0135827			−0.00679135	−	0.999977i	$-0.502162\pi$
$859$	−0.398092			−0.0135827			−0.00679135	+	0.999977i	$0.502162\pi$
$860$		0			0
$861$	24.9409	+	25.8169i	0.849985	+	0.879838i
$862$		0			0
$863$			26.3978i			0.898592i	0.893383	+	0.449296i	$0.148325\pi$
$863$			26.3978i			0.898592i	−0.893383	+	0.449296i	$0.851675\pi$
$864$		0			0
$865$		−	9.10142i		−	0.309458i
$866$		0			0
$867$	7.28771	−	7.04044i	0.247504	−	0.239106i
$868$		0			0
$869$		−	25.4289i		−	0.862617i
$870$		0			0
$871$	−6.94075	−	56.3780i	−0.235178	−	1.91030i
$872$		0			0
$873$	−52.5982	−	1.81601i	−1.78018	−	0.0614627i
$874$		0			0
$875$			4.23048i			0.143016i
$876$		0			0
$877$	33.7979			1.14127			0.570637	−	0.821202i	$-0.306696\pi$
$877$	33.7979			1.14127			0.570637	+	0.821202i	$0.306696\pi$
$878$		0			0
$879$	−9.98116	−	10.3317i	−0.336656	−	0.348480i
$880$		0			0
$881$		−	29.2800i		−	0.986467i	−0.869897	−	0.493233i	$-0.835815\pi$
$881$		−	29.2800i		−	0.986467i	0.869897	−	0.493233i	$-0.164185\pi$
$882$		0			0
$883$		−	46.0434i		−	1.54948i	−0.632278	−	0.774742i	$-0.717880\pi$
$883$		−	46.0434i		−	1.54948i	0.632278	−	0.774742i	$-0.282120\pi$
$884$		0			0
$885$	0.517402	+	0.535574i	0.0173923	+	0.0180031i
$886$		0			0
$887$		−	18.8197i		−	0.631903i	−0.948775	−	0.315952i	$-0.897676\pi$
$887$		−	18.8197i		−	0.631903i	0.948775	−	0.315952i	$-0.102324\pi$
$888$		0			0
$889$			57.0130i			1.91215i
$890$		0			0
$891$	50.8319	+	3.51424i	1.70293	+	0.117732i
$892$		0			0
$893$		−	11.9529i		−	0.399988i
$894$		0			0
$895$	−8.72650			−0.291695
$896$		0			0
$897$	28.4473	−	27.4820i	0.949826	−	0.917598i
$898$		0			0
$899$	−67.3836			−2.24737
$900$		0			0
$901$			35.1490i			1.17098i
$902$		0			0
$903$	−29.0837	+	28.0969i	−0.967846	+	0.935006i
$904$		0			0
$905$	−22.7761			−0.757105
$906$		0			0
$907$	44.9017			1.49094			0.745468	−	0.666541i	$-0.232226\pi$
$907$	44.9017			1.49094			0.745468	+	0.666541i	$0.232226\pi$
$908$		0			0
$909$	0.683284	−	19.7903i	0.0226631	−	0.656404i
$910$		0			0
$911$		−	45.2999i		−	1.50085i	−0.660954	−	0.750426i	$-0.729848\pi$
$911$		−	45.2999i		−	1.50085i	0.660954	−	0.750426i	$-0.270152\pi$
$912$		0			0
$913$		−	83.3633i		−	2.75892i
$914$		0			0
$915$	10.6358	+	11.0094i	0.351609	+	0.363958i
$916$		0			0
$917$	−30.9373			−1.02164
$918$		0			0
$919$			20.0194i			0.660378i	0.943915	+	0.330189i	$0.107113\pi$
$919$			20.0194i			0.660378i	−0.943915	+	0.330189i	$0.892887\pi$
$920$		0			0
$921$	−4.73567	+	4.57498i	−0.156045	+	0.150751i
$922$		0			0
$923$	−70.0644			−2.30620
$924$		0			0
$925$	3.64442			0.119828
$926$		0			0
$927$	0.764127	−	22.1319i	0.0250972	−	0.726905i
$928$		0			0
$929$	30.8648			1.01264			0.506321	−	0.862345i	$-0.331005\pi$
$929$	30.8648			1.01264			0.506321	+	0.862345i	$0.331005\pi$
$930$		0			0
$931$	−16.5381			−0.542014
$932$		0			0
$933$	39.1167	−	37.7894i	1.28062	−	1.23717i
$934$		0			0
$935$			27.0630i			0.885053i
$936$		0			0
$937$			7.38374i			0.241216i	0.992700	+	0.120608i	$0.0384845\pi$
$937$			7.38374i			0.241216i	−0.992700	+	0.120608i	$0.961516\pi$
$938$		0			0
$939$	−8.17309	−	8.46014i	−0.266719	−	0.276086i
$940$		0			0
$941$	2.06904			0.0674488			0.0337244	−	0.999431i	$-0.489263\pi$
$941$	2.06904			0.0674488			0.0337244	+	0.999431i	$0.489263\pi$
$942$		0			0
$943$		−	16.1210i		−	0.524972i
$944$		0			0
$945$	−16.3273	+	14.7186i	−0.531126	+	0.478797i
$946$		0			0
$947$		−	54.2637i		−	1.76333i	−0.471872	−	0.881667i	$-0.656422\pi$
$947$		−	54.2637i		−	1.76333i	0.471872	−	0.881667i	$-0.343578\pi$
$948$		0			0
$949$			80.0151i			2.59740i
$950$		0			0
$951$	−12.5865	−	13.0286i	−0.408147	−	0.422482i
$952$		0			0
$953$			39.3888i			1.27593i	0.770065	+	0.637965i	$0.220224\pi$
$953$			39.3888i			1.27593i	−0.770065	+	0.637965i	$0.779776\pi$
$954$		0			0
$955$	11.5212			0.372819
$956$		0			0
$957$	68.5379	+	70.9451i	2.21552	+	2.29333i
$958$		0			0
$959$			10.3643i			0.334679i
$960$		0			0
$961$	−13.8689			−0.447385
$962$		0			0
$963$	16.1752	+	0.558467i	0.521239	+	0.0179963i
$964$		0			0
$965$	10.2806			0.330945
$966$		0			0
$967$	−24.8180			−0.798094			−0.399047	−	0.916930i	$-0.630659\pi$
$967$	−24.8180			−0.798094			−0.399047	+	0.916930i	$0.630659\pi$
$968$		0			0
$969$	8.73065	+	9.03729i	0.280469	+	0.290320i
$970$		0			0
$971$			43.3431i			1.39095i	0.718552	+	0.695473i	$0.244805\pi$
$971$			43.3431i			1.39095i	−0.718552	+	0.695473i	$0.755195\pi$
$972$		0			0
$973$	−29.9941			−0.961568
$974$		0			0
$975$	−8.35140	−	8.64472i	−0.267459	−	0.276853i
$976$		0			0
$977$			40.3711i			1.29158i	0.763513	+	0.645792i	$0.223473\pi$
$977$			40.3711i			1.29158i	−0.763513	+	0.645792i	$0.776527\pi$
$978$		0			0
$979$			24.0527i			0.768728i
$980$		0			0
$981$	−36.5718	−	1.26268i	−1.16765	−	0.0403143i
$982$		0			0
$983$	26.8527			0.856469			0.428234	−	0.903668i	$-0.359136\pi$
$983$	26.8527			0.856469			0.428234	+	0.903668i	$0.359136\pi$
$984$		0			0
$985$	22.2185			0.707941
$986$		0			0
$987$	−41.5044	+	40.0961i	−1.32110	+	1.27627i
$988$		0			0
$989$	18.1609			0.577483
$990$		0			0
$991$		−	16.4082i		−	0.521225i	−0.965443	−	0.260612i	$-0.916076\pi$
$991$		−	16.4082i		−	0.521225i	0.965443	−	0.260612i	$-0.0839245\pi$
$992$		0			0
$993$	16.7598	+	17.3484i	0.531855	+	0.550535i
$994$		0			0
$995$	0.0739130			0.00234320
$996$		0			0
$997$	−46.1358			−1.46114			−0.730568	−	0.682840i	$-0.760745\pi$
$997$	−46.1358			−1.46114			−0.730568	+	0.682840i	$0.760745\pi$
$998$		0			0
$999$	12.6796	+	14.0654i	0.401166	+	0.445010i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.b.401.10	yes	46
3.2	odd	2		4020.2.f.a.401.38	yes	46
67.66	odd	2		4020.2.f.a.401.37	✓	46
201.200	even	2	inner	4020.2.f.b.401.9	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.37	✓	46	67.66	odd	2
4020.2.f.a.401.38	yes	46	3.2	odd	2
4020.2.f.b.401.9	yes	46	201.200	even	2	inner
4020.2.f.b.401.10	yes	46	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.24570 + 1.20343i) q^{3} +1.00000 q^{5} +4.23048i q^{7} +(0.103517 - 2.99821i) q^{9} +O(q^{10})\)	\(q+(-1.24570 + 1.20343i) q^{3} +1.00000 q^{5} +4.23048i q^{7} +(0.103517 - 2.99821i) q^{9} -5.66147 q^{11} +6.93967i q^{13} +(-1.24570 + 1.20343i) q^{15} -4.78020i q^{17} +1.51768 q^{19} +(-5.09108 - 5.26989i) q^{21} +3.29071i q^{23} +1.00000 q^{25} +(3.47919 + 3.85944i) q^{27} +10.0596i q^{29} +6.69843i q^{31} +(7.05247 - 6.81318i) q^{33} +4.23048i q^{35} +3.64442 q^{37} +(-8.35140 - 8.64472i) q^{39} -4.89894 q^{41} -5.51885i q^{43} +(0.103517 - 2.99821i) q^{45} -7.87576i q^{47} -10.8970 q^{49} +(5.75263 + 5.95468i) q^{51} -7.35304 q^{53} -5.66147 q^{55} +(-1.89057 + 1.82642i) q^{57} -0.429940i q^{59} -8.83793i q^{61} +(12.6839 + 0.437925i) q^{63} +6.93967i q^{65} +(-8.12402 + 1.00015i) q^{67} +(-3.96013 - 4.09922i) q^{69} +10.0962i q^{71} +11.5301 q^{73} +(-1.24570 + 1.20343i) q^{75} -23.9507i q^{77} +4.49158i q^{79} +(-8.97857 - 0.620730i) q^{81} +14.7247i q^{83} -4.78020i q^{85} +(-12.1060 - 12.5312i) q^{87} -4.24849i q^{89} -29.3581 q^{91} +(-8.06108 - 8.34421i) q^{93} +1.51768 q^{95} -17.5432i q^{97} +(-0.586056 + 16.9743i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 46 q^{5} - 4 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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