LMFDB - Embedded newform 4020.2.g.b.1609.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1609,4020)]

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, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4020.1609");

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.g (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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.15
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.b.1609.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-1.71974 + 1.42915i) q^{5} +1.37984i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-1.71974 + 1.42915i) q^{5} +1.37984i q^{7} -1.00000 q^{9} +3.21525 q^{11} -1.24439i q^{13} +(-1.42915 - 1.71974i) q^{15} -5.00989i q^{17} -0.139104 q^{19} -1.37984 q^{21} +1.01041i q^{23} +(0.915034 - 4.91556i) q^{25} -1.00000i q^{27} +8.55341 q^{29} +1.37797 q^{31} +3.21525i q^{33} +(-1.97200 - 2.37297i) q^{35} +5.04284i q^{37} +1.24439 q^{39} -0.647462 q^{41} -9.78726i q^{43} +(1.71974 - 1.42915i) q^{45} -3.18464i q^{47} +5.09605 q^{49} +5.00989 q^{51} -2.68009i q^{53} +(-5.52940 + 4.59509i) q^{55} -0.139104i q^{57} +9.87278 q^{59} +12.9260 q^{61} -1.37984i q^{63} +(1.77842 + 2.14003i) q^{65} -1.00000i q^{67} -1.01041 q^{69} -15.7434 q^{71} -16.0722i q^{73} +(4.91556 + 0.915034i) q^{75} +4.43652i q^{77} +1.22583 q^{79} +1.00000 q^{81} +3.38678i q^{83} +(7.15990 + 8.61572i) q^{85} +8.55341i q^{87} +2.10216 q^{89} +1.71705 q^{91} +1.37797i q^{93} +(0.239224 - 0.198802i) q^{95} -8.14957i q^{97} -3.21525 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) $ 24 * q - 4 * q^5 - 24 * q^9	$ 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) $ 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−1.71974	+	1.42915i	−0.769093	+	0.639137i
$5$	−1.71974	+	1.42915i	−0.769093	+	0.639137i
$6$		0			0
$7$			1.37984i			0.521530i	0.965402	+	0.260765i	$0.0839748\pi$
$7$			1.37984i			0.521530i	−0.965402	+	0.260765i	$0.916025\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	3.21525			0.969434			0.484717	−	0.874671i	$-0.338923\pi$
$11$	3.21525			0.969434			0.484717	+	0.874671i	$0.338923\pi$
$12$		0			0
$13$		−	1.24439i		−	0.345131i	−0.984998	−	0.172565i	$-0.944794\pi$
$13$		−	1.24439i		−	0.345131i	0.984998	−	0.172565i	$-0.0552056\pi$
$14$		0			0
$15$	−1.42915	−	1.71974i	−0.369006	−	0.444036i
$16$		0			0
$17$		−	5.00989i		−	1.21508i	−0.794291	−	0.607538i	$-0.792157\pi$
$17$		−	5.00989i		−	1.21508i	0.794291	−	0.607538i	$-0.207843\pi$
$18$		0			0
$19$	−0.139104			−0.0319127			−0.0159564	−	0.999873i	$-0.505079\pi$
$19$	−0.139104			−0.0319127			−0.0159564	+	0.999873i	$0.505079\pi$
$20$		0			0
$21$	−1.37984			−0.301105
$22$		0			0
$23$			1.01041i			0.210685i	0.994436	+	0.105342i	$0.0335938\pi$
$23$			1.01041i			0.210685i	−0.994436	+	0.105342i	$0.966406\pi$
$24$		0			0
$25$	0.915034	−	4.91556i	0.183007	−	0.983112i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	8.55341			1.58833			0.794164	−	0.607703i	$-0.207909\pi$
$29$	8.55341			1.58833			0.794164	+	0.607703i	$0.207909\pi$
$30$		0			0
$31$	1.37797			0.247490			0.123745	−	0.992314i	$-0.460509\pi$
$31$	1.37797			0.247490			0.123745	+	0.992314i	$0.460509\pi$
$32$		0			0
$33$			3.21525i			0.559703i
$34$		0			0
$35$	−1.97200	−	2.37297i	−0.333329	−	0.401105i
$36$		0			0
$37$			5.04284i			0.829038i	0.910041	+	0.414519i	$0.136050\pi$
$37$			5.04284i			0.829038i	−0.910041	+	0.414519i	$0.863950\pi$
$38$		0			0
$39$	1.24439			0.199261
$40$		0			0
$41$	−0.647462			−0.101117			−0.0505583	−	0.998721i	$-0.516100\pi$
$41$	−0.647462			−0.101117			−0.0505583	+	0.998721i	$0.516100\pi$
$42$		0			0
$43$		−	9.78726i		−	1.49254i	−0.665642	−	0.746272i	$-0.731842\pi$
$43$		−	9.78726i		−	1.49254i	0.665642	−	0.746272i	$-0.268158\pi$
$44$		0			0
$45$	1.71974	−	1.42915i	0.256364	−	0.213046i
$46$		0			0
$47$		−	3.18464i		−	0.464528i	−0.972653	−	0.232264i	$-0.925387\pi$
$47$		−	3.18464i		−	0.464528i	0.972653	−	0.232264i	$-0.0746134\pi$
$48$		0			0
$49$	5.09605			0.728007
$50$		0			0
$51$	5.00989			0.701524
$52$		0			0
$53$		−	2.68009i		−	0.368138i	−0.982913	−	0.184069i	$-0.941073\pi$
$53$		−	2.68009i		−	0.368138i	0.982913	−	0.184069i	$-0.0589271\pi$
$54$		0			0
$55$	−5.52940	+	4.59509i	−0.745584	+	0.619601i
$56$		0			0
$57$		−	0.139104i		−	0.0184248i
$58$		0			0
$59$	9.87278			1.28533			0.642664	−	0.766149i	$-0.277829\pi$
$59$	9.87278			1.28533			0.642664	+	0.766149i	$0.277829\pi$
$60$		0			0
$61$	12.9260			1.65501			0.827503	−	0.561462i	$-0.189761\pi$
$61$	12.9260			1.65501			0.827503	+	0.561462i	$0.189761\pi$
$62$		0			0
$63$		−	1.37984i		−	0.173843i
$64$		0			0
$65$	1.77842	+	2.14003i	0.220586	+	0.265438i
$66$		0			0
$67$		−	1.00000i		−	0.122169i
$67$		−	1.00000i		−	0.122169i
$68$		0			0
$69$	−1.01041			−0.121639
$70$		0			0
$71$	−15.7434			−1.86840			−0.934201	−	0.356747i	$-0.883886\pi$
$71$	−15.7434			−1.86840			−0.934201	+	0.356747i	$0.883886\pi$
$72$		0			0
$73$		−	16.0722i		−	1.88111i	−0.339640	−	0.940556i	$-0.610305\pi$
$73$		−	16.0722i		−	1.88111i	0.339640	−	0.940556i	$-0.389695\pi$
$74$		0			0
$75$	4.91556	+	0.915034i	0.567600	+	0.105659i
$76$		0			0
$77$			4.43652i			0.505589i
$78$		0			0
$79$	1.22583			0.137917			0.0689585	−	0.997620i	$-0.478032\pi$
$79$	1.22583			0.137917			0.0689585	+	0.997620i	$0.478032\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			3.38678i			0.371747i	0.982574	+	0.185874i	$0.0595115\pi$
$83$			3.38678i			0.371747i	−0.982574	+	0.185874i	$0.940488\pi$
$84$		0			0
$85$	7.15990	+	8.61572i	0.776600	+	0.934506i
$86$		0			0
$87$			8.55341i			0.917022i
$88$		0			0
$89$	2.10216			0.222829			0.111414	−	0.993774i	$-0.464462\pi$
$89$	2.10216			0.222829			0.111414	+	0.993774i	$0.464462\pi$
$90$		0			0
$91$	1.71705			0.179996
$92$		0			0
$93$			1.37797i			0.142889i
$94$		0			0
$95$	0.239224	−	0.198802i	0.0245439	−	0.0203966i
$96$		0			0
$97$		−	8.14957i		−	0.827464i	−0.910399	−	0.413732i	$-0.864225\pi$
$97$		−	8.14957i		−	0.827464i	0.910399	−	0.413732i	$-0.135775\pi$
$98$		0			0
$99$	−3.21525			−0.323145
$100$		0			0
$101$	−5.04519			−0.502015			−0.251008	−	0.967985i	$-0.580762\pi$
$101$	−5.04519			−0.502015			−0.251008	+	0.967985i	$0.580762\pi$
$102$		0			0
$103$			16.5579i			1.63150i	0.578406	+	0.815749i	$0.303675\pi$
$103$			16.5579i			1.63150i	−0.578406	+	0.815749i	$0.696325\pi$
$104$		0			0
$105$	2.37297	−	1.97200i	0.231578	−	0.192448i
$106$		0			0
$107$			10.9770i			1.06118i	0.847627	+	0.530592i	$0.178030\pi$
$107$			10.9770i			1.06118i	−0.847627	+	0.530592i	$0.821970\pi$
$108$		0			0
$109$	4.71636			0.451745			0.225873	−	0.974157i	$-0.427477\pi$
$109$	4.71636			0.451745			0.225873	+	0.974157i	$0.427477\pi$
$110$		0			0
$111$	−5.04284			−0.478646
$112$		0			0
$113$			9.35021i			0.879594i	0.898097	+	0.439797i	$0.144950\pi$
$113$			9.35021i			0.879594i	−0.898097	+	0.439797i	$0.855050\pi$
$114$		0			0
$115$	−1.44403	−	1.73764i	−0.134656	−	0.162036i
$116$		0			0
$117$			1.24439i			0.115044i
$118$		0			0
$119$	6.91283			0.633698
$120$		0			0
$121$	−0.662180			−0.0601982
$122$		0			0
$123$		−	0.647462i		−	0.0583796i
$124$		0			0
$125$	5.45147	+	9.76122i	0.487594	+	0.873070i
$126$		0			0
$127$		−	4.63486i		−	0.411277i	−0.978628	−	0.205639i	$-0.934073\pi$
$127$		−	4.63486i		−	0.411277i	0.978628	−	0.205639i	$-0.0659271\pi$
$128$		0			0
$129$	9.78726			0.861720
$130$		0			0
$131$	−19.2600			−1.68275			−0.841377	−	0.540448i	$-0.818255\pi$
$131$	−19.2600			−1.68275			−0.841377	+	0.540448i	$0.818255\pi$
$132$		0			0
$133$		−	0.191942i		−	0.0166435i
$134$		0			0
$135$	1.42915	+	1.71974i	0.123002	+	0.148012i
$136$		0			0
$137$			11.3789i			0.972164i	0.873913	+	0.486082i	$0.161574\pi$
$137$			11.3789i			0.972164i	−0.873913	+	0.486082i	$0.838426\pi$
$138$		0			0
$139$	3.21532			0.272720			0.136360	−	0.990659i	$-0.456460\pi$
$139$	3.21532			0.272720			0.136360	+	0.990659i	$0.456460\pi$
$140$		0			0
$141$	3.18464			0.268195
$142$		0			0
$143$		−	4.00101i		−	0.334582i
$144$		0			0
$145$	−14.7097	+	12.2241i	−1.22157	+	1.01516i
$146$		0			0
$147$			5.09605i			0.420315i
$148$		0			0
$149$	−8.57555			−0.702537			−0.351268	−	0.936275i	$-0.614250\pi$
$149$	−8.57555			−0.702537			−0.351268	+	0.936275i	$0.614250\pi$
$150$		0			0
$151$	13.5884			1.10581			0.552903	−	0.833245i	$-0.313520\pi$
$151$	13.5884			1.10581			0.552903	+	0.833245i	$0.313520\pi$
$152$		0			0
$153$			5.00989i			0.405025i
$154$		0			0
$155$	−2.36975	+	1.96933i	−0.190343	+	0.158180i
$156$		0			0
$157$			3.87121i			0.308957i	0.987996	+	0.154478i	$0.0493697\pi$
$157$			3.87121i			0.308957i	−0.987996	+	0.154478i	$0.950630\pi$
$158$		0			0
$159$	2.68009			0.212545
$160$		0			0
$161$	−1.39420			−0.109878
$162$		0			0
$163$		−	0.361794i		−	0.0283379i	−0.999900	−	0.0141690i	$-0.995490\pi$
$163$		−	0.361794i		−	0.0283379i	0.999900	−	0.0141690i	$-0.00451027\pi$
$164$		0			0
$165$	−4.59509	−	5.52940i	−0.357727	−	0.430463i
$166$		0			0
$167$		−	1.09311i		−	0.0845877i	−0.999105	−	0.0422939i	$-0.986533\pi$
$167$		−	1.09311i		−	0.0845877i	0.999105	−	0.0422939i	$-0.0134666\pi$
$168$		0			0
$169$	11.4515			0.880885
$170$		0			0
$171$	0.139104			0.0106376
$172$		0			0
$173$			16.2610i			1.23630i	0.786061	+	0.618149i	$0.212117\pi$
$173$			16.2610i			1.23630i	−0.786061	+	0.618149i	$0.787883\pi$
$174$		0			0
$175$	6.78268	+	1.26260i	0.512722	+	0.0954436i
$176$		0			0
$177$			9.87278i			0.742084i
$178$		0			0
$179$	21.1958			1.58425			0.792126	−	0.610358i	$-0.208974\pi$
$179$	21.1958			1.58425			0.792126	+	0.610358i	$0.208974\pi$
$180$		0			0
$181$	2.79730			0.207922			0.103961	−	0.994581i	$-0.466848\pi$
$181$	2.79730			0.207922			0.103961	+	0.994581i	$0.466848\pi$
$182$		0			0
$183$			12.9260i			0.955518i
$184$		0			0
$185$	−7.20700	−	8.67240i	−0.529869	−	0.637607i
$186$		0			0
$187$		−	16.1080i		−	1.17794i
$188$		0			0
$189$	1.37984			0.100368
$190$		0			0
$191$	9.31561			0.674054			0.337027	−	0.941495i	$-0.390579\pi$
$191$	9.31561			0.674054			0.337027	+	0.941495i	$0.390579\pi$
$192$		0			0
$193$			18.5754i			1.33708i	0.743675	+	0.668542i	$0.233081\pi$
$193$			18.5754i			1.33708i	−0.743675	+	0.668542i	$0.766919\pi$
$194$		0			0
$195$	−2.14003	+	1.77842i	−0.153250	+	0.127355i
$196$		0			0
$197$			1.70902i			0.121763i	0.998145	+	0.0608813i	$0.0193911\pi$
$197$			1.70902i			0.121763i	−0.998145	+	0.0608813i	$0.980609\pi$
$198$		0			0
$199$	17.4929			1.24004			0.620018	−	0.784587i	$-0.287125\pi$
$199$	17.4929			1.24004			0.620018	+	0.784587i	$0.287125\pi$
$200$		0			0
$201$	1.00000			0.0705346
$202$		0			0
$203$			11.8023i			0.828361i
$204$		0			0
$205$	1.11347	−	0.925323i	0.0777680	−	0.0646273i
$206$		0			0
$207$		−	1.01041i		−	0.0702282i
$208$		0			0
$209$	−0.447255			−0.0309373
$210$		0			0
$211$	4.86128			0.334664			0.167332	−	0.985901i	$-0.446485\pi$
$211$	4.86128			0.334664			0.167332	+	0.985901i	$0.446485\pi$
$212$		0			0
$213$		−	15.7434i		−	1.07872i
$214$		0			0
$215$	13.9875	+	16.8316i	0.953940	+	1.14790i
$216$		0			0
$217$			1.90137i			0.129074i
$218$		0			0
$219$	16.0722			1.08606
$220$		0			0
$221$	−6.23424			−0.419360
$222$		0			0
$223$		−	0.732877i		−	0.0490771i	−0.999699	−	0.0245385i	$-0.992188\pi$
$223$		−	0.732877i		−	0.0490771i	0.999699	−	0.0245385i	$-0.00781164\pi$
$224$		0			0
$225$	−0.915034	+	4.91556i	−0.0610023	+	0.327704i
$226$		0			0
$227$			4.36928i			0.289999i	0.989432	+	0.145000i	$0.0463181\pi$
$227$			4.36928i			0.289999i	−0.989432	+	0.145000i	$0.953682\pi$
$228$		0			0
$229$	20.5903			1.36065			0.680323	−	0.732912i	$-0.261840\pi$
$229$	20.5903			1.36065			0.680323	+	0.732912i	$0.261840\pi$
$230$		0			0
$231$	−4.43652			−0.291902
$232$		0			0
$233$			1.59737i			0.104647i	0.998630	+	0.0523237i	$0.0166628\pi$
$233$			1.59737i			0.104647i	−0.998630	+	0.0523237i	$0.983337\pi$
$234$		0			0
$235$	4.55135	+	5.47677i	0.296897	+	0.357265i
$236$		0			0
$237$			1.22583i			0.0796264i
$238$		0			0
$239$	−3.07220			−0.198724			−0.0993619	−	0.995051i	$-0.531680\pi$
$239$	−3.07220			−0.198724			−0.0993619	+	0.995051i	$0.531680\pi$
$240$		0			0
$241$	27.3067			1.75898			0.879490	−	0.475917i	$-0.157884\pi$
$241$	27.3067			1.75898			0.879490	+	0.475917i	$0.157884\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−8.76389	+	7.28304i	−0.559904	+	0.465296i
$246$		0			0
$247$			0.173100i			0.0110141i
$248$		0			0
$249$	−3.38678			−0.214629
$250$		0			0
$251$	−1.06807			−0.0674161			−0.0337080	−	0.999432i	$-0.510732\pi$
$251$	−1.06807			−0.0674161			−0.0337080	+	0.999432i	$0.510732\pi$
$252$		0			0
$253$			3.24871i			0.204245i
$254$		0			0
$255$	−8.61572	+	7.15990i	−0.539537	+	0.448370i
$256$		0			0
$257$			9.76072i			0.608857i	0.952535	+	0.304429i	$0.0984655\pi$
$257$			9.76072i			0.608857i	−0.952535	+	0.304429i	$0.901535\pi$
$258$		0			0
$259$	−6.95831			−0.432368
$260$		0			0
$261$	−8.55341			−0.529443
$262$		0			0
$263$			7.09365i			0.437413i	0.975791	+	0.218707i	$0.0701838\pi$
$263$			7.09365i			0.437413i	−0.975791	+	0.218707i	$0.929816\pi$
$264$		0			0
$265$	3.83026	+	4.60906i	0.235291	+	0.283133i
$266$		0			0
$267$			2.10216i			0.128650i
$268$		0			0
$269$	−24.4888			−1.49311			−0.746555	−	0.665324i	$-0.768293\pi$
$269$	−24.4888			−1.49311			−0.746555	+	0.665324i	$0.768293\pi$
$270$		0			0
$271$	−13.6330			−0.828148			−0.414074	−	0.910243i	$-0.635894\pi$
$271$	−13.6330			−0.828148			−0.414074	+	0.910243i	$0.635894\pi$
$272$		0			0
$273$			1.71705i			0.103921i
$274$		0			0
$275$	2.94206	−	15.8047i	0.177413	−	0.953062i
$276$		0			0
$277$		−	25.9812i		−	1.56106i	−0.625120	−	0.780528i	$-0.714950\pi$
$277$		−	25.9812i		−	1.56106i	0.625120	−	0.780528i	$-0.285050\pi$
$278$		0			0
$279$	−1.37797			−0.0824967
$280$		0			0
$281$	−3.41129			−0.203501			−0.101750	−	0.994810i	$-0.532444\pi$
$281$	−3.41129			−0.203501			−0.101750	+	0.994810i	$0.532444\pi$
$282$		0			0
$283$		−	16.1212i		−	0.958304i	−0.877732	−	0.479152i	$-0.840944\pi$
$283$		−	16.1212i		−	0.958304i	0.877732	−	0.479152i	$-0.159056\pi$
$284$		0			0
$285$	0.198802	+	0.239224i	0.0117760	+	0.0141704i
$286$		0			0
$287$		−	0.893392i		−	0.0527353i
$288$		0			0
$289$	−8.09896			−0.476409
$290$		0			0
$291$	8.14957			0.477736
$292$		0			0
$293$		−	32.8616i		−	1.91979i	−0.280353	−	0.959897i	$-0.590452\pi$
$293$		−	32.8616i		−	1.91979i	0.280353	−	0.959897i	$-0.409548\pi$
$294$		0			0
$295$	−16.9787	+	14.1097i	−0.988536	+	0.821501i
$296$		0			0
$297$		−	3.21525i		−	0.186568i
$298$		0			0
$299$	1.25734			0.0727137
$300$		0			0
$301$	13.5048			0.778406
$302$		0			0
$303$		−	5.04519i		−	0.289839i
$304$		0			0
$305$	−22.2294	+	18.4733i	−1.27285	+	1.05778i
$306$		0			0
$307$			23.5637i			1.34485i	0.740164	+	0.672427i	$0.234748\pi$
$307$			23.5637i			1.34485i	−0.740164	+	0.672427i	$0.765252\pi$
$308$		0			0
$309$	−16.5579			−0.941946
$310$		0			0
$311$	23.8330			1.35145			0.675723	−	0.737155i	$-0.263831\pi$
$311$	23.8330			1.35145			0.675723	+	0.737155i	$0.263831\pi$
$312$		0			0
$313$		−	19.8108i		−	1.11977i	−0.828570	−	0.559885i	$-0.810845\pi$
$313$		−	19.8108i		−	1.11977i	0.828570	−	0.559885i	$-0.189155\pi$
$314$		0			0
$315$	1.97200	+	2.37297i	0.111110	+	0.133702i
$316$		0			0
$317$			7.70559i			0.432789i	0.976306	+	0.216394i	$0.0694297\pi$
$317$			7.70559i			0.432789i	−0.976306	+	0.216394i	$0.930570\pi$
$318$		0			0
$319$	27.5013			1.53978
$320$		0			0
$321$	−10.9770			−0.612675
$322$		0			0
$323$			0.696897i			0.0387764i
$324$		0			0
$325$	−6.11686	−	1.13866i	−0.339302	−	0.0631613i
$326$		0			0
$327$			4.71636i			0.260815i
$328$		0			0
$329$	4.39429			0.242265
$330$		0			0
$331$	−15.9273			−0.875444			−0.437722	−	0.899110i	$-0.644215\pi$
$331$	−15.9273			−0.875444			−0.437722	+	0.899110i	$0.644215\pi$
$332$		0			0
$333$		−	5.04284i		−	0.276346i
$334$		0			0
$335$	1.42915	+	1.71974i	0.0780831	+	0.0939596i
$336$		0			0
$337$		−	34.3453i		−	1.87091i	−0.353451	−	0.935453i	$-0.614992\pi$
$337$		−	34.3453i		−	1.87091i	0.353451	−	0.935453i	$-0.385008\pi$
$338$		0			0
$339$	−9.35021			−0.507834
$340$		0			0
$341$	4.43051			0.239925
$342$		0			0
$343$			16.6906i			0.901207i
$344$		0			0
$345$	1.73764	−	1.44403i	0.0935515	−	0.0777439i
$346$		0			0
$347$		−	18.4834i		−	0.992242i	−0.868253	−	0.496121i	$-0.834757\pi$
$347$		−	18.4834i		−	0.992242i	0.868253	−	0.496121i	$-0.165243\pi$
$348$		0			0
$349$	17.4357			0.933312			0.466656	−	0.884439i	$-0.345459\pi$
$349$	17.4357			0.933312			0.466656	+	0.884439i	$0.345459\pi$
$350$		0			0
$351$	−1.24439			−0.0664205
$352$		0			0
$353$		−	18.0673i		−	0.961627i	−0.876823	−	0.480813i	$-0.840341\pi$
$353$		−	18.0673i		−	0.961627i	0.876823	−	0.480813i	$-0.159659\pi$
$354$		0			0
$355$	27.0747	−	22.4998i	1.43697	−	1.19417i
$356$		0			0
$357$			6.91283i			0.365866i
$358$		0			0
$359$	19.7760			1.04374			0.521868	−	0.853026i	$-0.325235\pi$
$359$	19.7760			1.04374			0.521868	+	0.853026i	$0.325235\pi$
$360$		0			0
$361$	−18.9806			−0.998982
$362$		0			0
$363$		−	0.662180i		−	0.0347554i
$364$		0			0
$365$	22.9697	+	27.6401i	1.20229	+	1.44675i
$366$		0			0
$367$			13.9318i			0.727233i	0.931549	+	0.363616i	$0.118458\pi$
$367$			13.9318i			0.727233i	−0.931549	+	0.363616i	$0.881542\pi$
$368$		0			0
$369$	0.647462			0.0337055
$370$		0			0
$371$	3.69809			0.191995
$372$		0			0
$373$		−	26.3171i		−	1.36265i	−0.731983	−	0.681323i	$-0.761405\pi$
$373$		−	26.3171i		−	1.36265i	0.731983	−	0.681323i	$-0.238595\pi$
$374$		0			0
$375$	−9.76122	+	5.45147i	−0.504067	+	0.281513i
$376$		0			0
$377$		−	10.6438i		−	0.548181i
$378$		0			0
$379$	−24.0224			−1.23395			−0.616975	−	0.786983i	$-0.711642\pi$
$379$	−24.0224			−1.23395			−0.616975	+	0.786983i	$0.711642\pi$
$380$		0			0
$381$	4.63486			0.237451
$382$		0			0
$383$			29.2552i			1.49487i	0.664334	+	0.747436i	$0.268715\pi$
$383$			29.2552i			1.49487i	−0.664334	+	0.747436i	$0.731285\pi$
$384$		0			0
$385$	−6.34048	−	7.62968i	−0.323141	−	0.388845i
$386$		0			0
$387$			9.78726i			0.497514i
$388$		0			0
$389$	−8.43519			−0.427681			−0.213841	−	0.976869i	$-0.568597\pi$
$389$	−8.43519			−0.427681			−0.213841	+	0.976869i	$0.568597\pi$
$390$		0			0
$391$	5.06203			0.255998
$392$		0			0
$393$		−	19.2600i		−	0.971539i
$394$		0			0
$395$	−2.10812	+	1.75191i	−0.106071	+	0.0881479i
$396$		0			0
$397$		−	19.0271i		−	0.954943i	−0.878647	−	0.477471i	$-0.841553\pi$
$397$		−	19.0271i		−	0.954943i	0.878647	−	0.477471i	$-0.158447\pi$
$398$		0			0
$399$	0.191942			0.00960910
$400$		0			0
$401$	−33.8977			−1.69277			−0.846386	−	0.532570i	$-0.821226\pi$
$401$	−33.8977			−1.69277			−0.846386	+	0.532570i	$0.821226\pi$
$402$		0			0
$403$		−	1.71472i		−	0.0854165i
$404$		0			0
$405$	−1.71974	+	1.42915i	−0.0854547	+	0.0710153i
$406$		0			0
$407$			16.2140i			0.803698i
$408$		0			0
$409$	−5.06819			−0.250606			−0.125303	−	0.992119i	$-0.539990\pi$
$409$	−5.06819			−0.250606			−0.125303	+	0.992119i	$0.539990\pi$
$410$		0			0
$411$	−11.3789			−0.561279
$412$		0			0
$413$			13.6228i			0.670337i
$414$		0			0
$415$	−4.84023	−	5.82439i	−0.237598	−	0.285908i
$416$		0			0
$417$			3.21532i			0.157455i
$418$		0			0
$419$	−7.14074			−0.348848			−0.174424	−	0.984671i	$-0.555806\pi$
$419$	−7.14074			−0.348848			−0.174424	+	0.984671i	$0.555806\pi$
$420$		0			0
$421$	26.6859			1.30059			0.650295	−	0.759681i	$-0.274645\pi$
$421$	26.6859			1.30059			0.650295	+	0.759681i	$0.274645\pi$
$422$		0			0
$423$			3.18464i			0.154843i
$424$		0			0
$425$	−24.6264	−	4.58422i	−1.19456	−	0.222367i
$426$		0			0
$427$			17.8358i			0.863135i
$428$		0			0
$429$	4.00101			0.193171
$430$		0			0
$431$	25.8374			1.24454			0.622272	−	0.782801i	$-0.286210\pi$
$431$	25.8374			1.24454			0.622272	+	0.782801i	$0.286210\pi$
$432$		0			0
$433$			15.2078i			0.730838i	0.930843	+	0.365419i	$0.119074\pi$
$433$			15.2078i			0.730838i	−0.930843	+	0.365419i	$0.880926\pi$
$434$		0			0
$435$	−12.2241	−	14.7097i	−0.586103	−	0.705275i
$436$		0			0
$437$		−	0.140552i		−	0.00672352i
$438$		0			0
$439$	−23.5804			−1.12543			−0.562715	−	0.826651i	$-0.690243\pi$
$439$	−23.5804			−1.12543			−0.562715	+	0.826651i	$0.690243\pi$
$440$		0			0
$441$	−5.09605			−0.242669
$442$		0			0
$443$			17.7399i			0.842849i	0.906864	+	0.421424i	$0.138470\pi$
$443$			17.7399i			0.842849i	−0.906864	+	0.421424i	$0.861530\pi$
$444$		0			0
$445$	−3.61518	+	3.00432i	−0.171376	+	0.142418i
$446$		0			0
$447$		−	8.57555i		−	0.405610i
$448$		0			0
$449$	11.2690			0.531815			0.265907	−	0.963999i	$-0.414328\pi$
$449$	11.2690			0.531815			0.265907	+	0.963999i	$0.414328\pi$
$450$		0			0
$451$	−2.08175			−0.0980258
$452$		0			0
$453$			13.5884i			0.638438i
$454$		0			0
$455$	−2.95289	+	2.45393i	−0.138434	+	0.115042i
$456$		0			0
$457$			7.40159i			0.346232i	0.984901	+	0.173116i	$0.0553835\pi$
$457$			7.40159i			0.346232i	−0.984901	+	0.173116i	$0.944617\pi$
$458$		0			0
$459$	−5.00989			−0.233841
$460$		0			0
$461$	−14.4254			−0.671858			−0.335929	−	0.941887i	$-0.609050\pi$
$461$	−14.4254			−0.671858			−0.335929	+	0.941887i	$0.609050\pi$
$462$		0			0
$463$			15.6194i			0.725893i	0.931810	+	0.362947i	$0.118229\pi$
$463$			15.6194i			0.725893i	−0.931810	+	0.362947i	$0.881771\pi$
$464$		0			0
$465$	−1.96933	−	2.36975i	−0.0913254	−	0.109894i
$466$		0			0
$467$			16.9400i			0.783888i	0.919989	+	0.391944i	$0.128197\pi$
$467$			16.9400i			0.783888i	−0.919989	+	0.391944i	$0.871803\pi$
$468$		0			0
$469$	1.37984			0.0637150
$470$		0			0
$471$	−3.87121			−0.178376
$472$		0			0
$473$		−	31.4685i		−	1.44692i
$474$		0			0
$475$	−0.127285	+	0.683776i	−0.00584025	+	0.0313738i
$476$		0			0
$477$			2.68009i			0.122713i
$478$		0			0
$479$	−3.32582			−0.151961			−0.0759803	−	0.997109i	$-0.524209\pi$
$479$	−3.32582			−0.151961			−0.0759803	+	0.997109i	$0.524209\pi$
$480$		0			0
$481$	6.27525			0.286127
$482$		0			0
$483$		−	1.39420i		−	0.0634383i
$484$		0			0
$485$	11.6470	+	14.0152i	0.528863	+	0.636396i
$486$		0			0
$487$			23.4222i			1.06136i	0.847572	+	0.530680i	$0.178063\pi$
$487$			23.4222i			1.06136i	−0.847572	+	0.530680i	$0.821937\pi$
$488$		0			0
$489$	0.361794			0.0163609
$490$		0			0
$491$	33.5478			1.51399			0.756996	−	0.653419i	$-0.226666\pi$
$491$	33.5478			1.51399			0.756996	+	0.653419i	$0.226666\pi$
$492$		0			0
$493$		−	42.8516i		−	1.92994i
$494$		0			0
$495$	5.52940	−	4.59509i	0.248528	−	0.206534i
$496$		0			0
$497$		−	21.7234i		−	0.974428i
$498$		0			0
$499$	39.9695			1.78928			0.894640	−	0.446787i	$-0.147432\pi$
$499$	39.9695			1.78928			0.894640	+	0.446787i	$0.147432\pi$
$500$		0			0
$501$	1.09311			0.0488367
$502$		0			0
$503$		−	8.72475i		−	0.389017i	−0.980901	−	0.194509i	$-0.937689\pi$
$503$		−	8.72475i		−	0.389017i	0.980901	−	0.194509i	$-0.0623113\pi$
$504$		0			0
$505$	8.67643	−	7.21036i	0.386096	−	0.320857i
$506$		0			0
$507$			11.4515i			0.508579i
$508$		0			0
$509$	19.0080			0.842514			0.421257	−	0.906941i	$-0.361589\pi$
$509$	19.0080			0.842514			0.421257	+	0.906941i	$0.361589\pi$
$510$		0			0
$511$	22.1771			0.981056
$512$		0			0
$513$			0.139104i			0.00614161i
$514$		0			0
$515$	−23.6638	−	28.4753i	−1.04275	−	1.25477i
$516$		0			0
$517$		−	10.2394i		−	0.450329i
$518$		0			0
$519$	−16.2610			−0.713777
$520$		0			0
$521$	−29.0434			−1.27242			−0.636208	−	0.771518i	$-0.719498\pi$
$521$	−29.0434			−1.27242			−0.636208	+	0.771518i	$0.719498\pi$
$522$		0			0
$523$		−	37.3180i		−	1.63180i	−0.578190	−	0.815902i	$-0.696241\pi$
$523$		−	37.3180i		−	1.63180i	0.578190	−	0.815902i	$-0.303759\pi$
$524$		0			0
$525$	−1.26260	+	6.78268i	−0.0551044	+	0.296020i
$526$		0			0
$527$		−	6.90346i		−	0.300719i
$528$		0			0
$529$	21.9791			0.955612
$530$		0			0
$531$	−9.87278			−0.428442
$532$		0			0
$533$			0.805693i			0.0348984i
$534$		0			0
$535$	−15.6878	−	18.8776i	−0.678242	−	0.816149i
$536$		0			0
$537$			21.1958i			0.914668i
$538$		0			0
$539$	16.3851			0.705754
$540$		0			0
$541$	−9.14688			−0.393255			−0.196627	−	0.980478i	$-0.562999\pi$
$541$	−9.14688			−0.393255			−0.196627	+	0.980478i	$0.562999\pi$
$542$		0			0
$543$			2.79730i			0.120044i
$544$		0			0
$545$	−8.11093	+	6.74041i	−0.347434	+	0.288727i
$546$		0			0
$547$			14.7131i			0.629088i	0.949243	+	0.314544i	$0.101852\pi$
$547$			14.7131i			0.629088i	−0.949243	+	0.314544i	$0.898148\pi$
$548$		0			0
$549$	−12.9260			−0.551668
$550$		0			0
$551$	−1.18982			−0.0506879
$552$		0			0
$553$			1.69145i			0.0719278i
$554$		0			0
$555$	8.67240	−	7.20700i	0.368123	−	0.305920i
$556$		0			0
$557$			0.112803i			0.00477962i	0.999997	+	0.00238981i	$0.000760701\pi$
$557$			0.112803i			0.00477962i	−0.999997	+	0.00238981i	$0.999239\pi$
$558$		0			0
$559$	−12.1791			−0.515123
$560$		0			0
$561$	16.1080			0.680081
$562$		0			0
$563$		−	42.3205i		−	1.78359i	−0.452435	−	0.891797i	$-0.649445\pi$
$563$		−	42.3205i		−	1.78359i	0.452435	−	0.891797i	$-0.350555\pi$
$564$		0			0
$565$	−13.3629	−	16.0800i	−0.562181	−	0.676489i
$566$		0			0
$567$			1.37984i			0.0579478i
$568$		0			0
$569$	3.67609			0.154110			0.0770548	−	0.997027i	$-0.475448\pi$
$569$	3.67609			0.154110			0.0770548	+	0.997027i	$0.475448\pi$
$570$		0			0
$571$	−27.1635			−1.13676			−0.568378	−	0.822768i	$-0.692429\pi$
$571$	−27.1635			−1.13676			−0.568378	+	0.822768i	$0.692429\pi$
$572$		0			0
$573$			9.31561i			0.389165i
$574$		0			0
$575$	4.96672	+	0.924558i	0.207126	+	0.0385567i
$576$		0			0
$577$		−	17.3776i		−	0.723439i	−0.932287	−	0.361719i	$-0.882190\pi$
$577$		−	17.3776i		−	0.723439i	0.932287	−	0.361719i	$-0.117810\pi$
$578$		0			0
$579$	−18.5754			−0.771966
$580$		0			0
$581$	−4.67321			−0.193877
$582$		0			0
$583$		−	8.61715i		−	0.356886i
$584$		0			0
$585$	−1.77842	−	2.14003i	−0.0735287	−	0.0884792i
$586$		0			0
$587$			16.3397i			0.674413i	0.941431	+	0.337206i	$0.109482\pi$
$587$			16.3397i			0.674413i	−0.941431	+	0.337206i	$0.890518\pi$
$588$		0			0
$589$	−0.191681			−0.00789809
$590$		0			0
$591$	−1.70902			−0.0702997
$592$		0			0
$593$		−	42.0886i		−	1.72837i	−0.503171	−	0.864187i	$-0.667833\pi$
$593$		−	42.0886i		−	1.72837i	0.503171	−	0.864187i	$-0.332167\pi$
$594$		0			0
$595$	−11.8883	+	9.87951i	−0.487373	+	0.405020i
$596$		0			0
$597$			17.4929i			0.715935i
$598$		0			0
$599$	45.2137			1.84738			0.923690	−	0.383140i	$-0.125157\pi$
$599$	45.2137			1.84738			0.923690	+	0.383140i	$0.125157\pi$
$600$		0			0
$601$	−30.2577			−1.23424			−0.617120	−	0.786869i	$-0.711700\pi$
$601$	−30.2577			−1.23424			−0.617120	+	0.786869i	$0.711700\pi$
$602$		0			0
$603$			1.00000i			0.0407231i
$604$		0			0
$605$	1.13878	−	0.946357i	0.0462980	−	0.0384749i
$606$		0			0
$607$			9.53779i			0.387127i	0.981088	+	0.193563i	$0.0620045\pi$
$607$			9.53779i			0.387127i	−0.981088	+	0.193563i	$0.937995\pi$
$608$		0			0
$609$	−11.8023			−0.478254
$610$		0			0
$611$	−3.96293			−0.160323
$612$		0			0
$613$		−	8.34523i		−	0.337061i	−0.985696	−	0.168530i	$-0.946098\pi$
$613$		−	8.34523i		−	0.337061i	0.985696	−	0.168530i	$-0.0539021\pi$
$614$		0			0
$615$	0.925323	+	1.11347i	0.0373126	+	0.0448994i
$616$		0			0
$617$		−	1.24011i		−	0.0499250i	−0.999688	−	0.0249625i	$-0.992053\pi$
$617$		−	1.24011i		−	0.0499250i	0.999688	−	0.0249625i	$-0.00794663\pi$
$618$		0			0
$619$	6.78730			0.272805			0.136402	−	0.990654i	$-0.456446\pi$
$619$	6.78730			0.272805			0.136402	+	0.990654i	$0.456446\pi$
$620$		0			0
$621$	1.01041			0.0405463
$622$		0			0
$623$			2.90065i			0.116212i
$624$		0			0
$625$	−23.3254	−	8.99581i	−0.933017	−	0.359832i
$626$		0			0
$627$		−	0.447255i		−	0.0178617i
$628$		0			0
$629$	25.2641			1.00734
$630$		0			0
$631$	38.5565			1.53491			0.767456	−	0.641102i	$-0.221522\pi$
$631$	38.5565			1.53491			0.767456	+	0.641102i	$0.221522\pi$
$632$		0			0
$633$			4.86128i			0.193218i
$634$		0			0
$635$	6.62393	+	7.97077i	0.262863	+	0.316310i
$636$		0			0
$637$		−	6.34145i		−	0.251258i
$638$		0			0
$639$	15.7434			0.622801
$640$		0			0
$641$	−43.5577			−1.72043			−0.860213	−	0.509935i	$-0.829669\pi$
$641$	−43.5577			−1.72043			−0.860213	+	0.509935i	$0.829669\pi$
$642$		0			0
$643$		−	6.88593i		−	0.271554i	−0.990739	−	0.135777i	$-0.956647\pi$
$643$		−	6.88593i		−	0.271554i	0.990739	−	0.135777i	$-0.0433531\pi$
$644$		0			0
$645$	−16.8316	+	13.9875i	−0.662743	+	0.550758i
$646$		0			0
$647$		−	31.4769i		−	1.23749i	−0.785593	−	0.618743i	$-0.787642\pi$
$647$		−	31.4769i		−	1.23749i	0.785593	−	0.618743i	$-0.212358\pi$
$648$		0			0
$649$	31.7435			1.24604
$650$		0			0
$651$	−1.90137			−0.0745206
$652$		0			0
$653$			47.5114i			1.85927i	0.368487	+	0.929633i	$0.379876\pi$
$653$			47.5114i			1.85927i	−0.368487	+	0.929633i	$0.620124\pi$
$654$		0			0
$655$	33.1223	−	27.5255i	1.29419	−	1.07551i
$656$		0			0
$657$			16.0722i			0.627037i
$658$		0			0
$659$	16.3074			0.635246			0.317623	−	0.948217i	$-0.397115\pi$
$659$	16.3074			0.635246			0.317623	+	0.948217i	$0.397115\pi$
$660$		0			0
$661$	−40.7121			−1.58352			−0.791758	−	0.610834i	$-0.790834\pi$
$661$	−40.7121			−1.58352			−0.791758	+	0.610834i	$0.790834\pi$
$662$		0			0
$663$		−	6.23424i		−	0.242118i
$664$		0			0
$665$	0.274314	+	0.330090i	0.0106375	+	0.0128004i
$666$		0			0
$667$			8.64243i			0.334636i
$668$		0			0
$669$	0.732877			0.0283347
$670$		0			0
$671$	41.5603			1.60442
$672$		0			0
$673$			47.3035i			1.82341i	0.410840	+	0.911707i	$0.365235\pi$
$673$			47.3035i			1.82341i	−0.410840	+	0.911707i	$0.634765\pi$
$674$		0			0
$675$	−4.91556	−	0.915034i	−0.189200	−	0.0352197i
$676$		0			0
$677$		−	40.7091i		−	1.56458i	−0.622915	−	0.782290i	$-0.714052\pi$
$677$		−	40.7091i		−	1.56458i	0.622915	−	0.782290i	$-0.285948\pi$
$678$		0			0
$679$	11.2451			0.431547
$680$		0			0
$681$	−4.36928			−0.167431
$682$		0			0
$683$			17.5106i			0.670025i	0.942214	+	0.335013i	$0.108741\pi$
$683$			17.5106i			0.670025i	−0.942214	+	0.335013i	$0.891259\pi$
$684$		0			0
$685$	−16.2622	−	19.5688i	−0.621346	−	0.747684i
$686$		0			0
$687$			20.5903i			0.785569i
$688$		0			0
$689$	−3.33507			−0.127056
$690$		0			0
$691$	−12.5338			−0.476809			−0.238404	−	0.971166i	$-0.576624\pi$
$691$	−12.5338			−0.476809			−0.238404	+	0.971166i	$0.576624\pi$
$692$		0			0
$693$		−	4.43652i		−	0.168530i
$694$		0			0
$695$	−5.52952	+	4.59519i	−0.209747	+	0.174305i
$696$		0			0
$697$			3.24371i			0.122864i
$698$		0			0
$699$	−1.59737			−0.0604182
$700$		0			0
$701$	−2.49750			−0.0943292			−0.0471646	−	0.998887i	$-0.515019\pi$
$701$	−2.49750			−0.0943292			−0.0471646	+	0.998887i	$0.515019\pi$
$702$		0			0
$703$		−	0.701482i		−	0.0264569i
$704$		0			0
$705$	−5.47677	+	4.55135i	−0.206267	+	0.171414i
$706$		0			0
$707$		−	6.96155i		−	0.261816i
$708$		0			0
$709$	24.8977			0.935052			0.467526	−	0.883979i	$-0.345145\pi$
$709$	24.8977			0.935052			0.467526	+	0.883979i	$0.345145\pi$
$710$		0			0
$711$	−1.22583			−0.0459723
$712$		0			0
$713$			1.39231i			0.0521423i
$714$		0			0
$715$	5.71807	+	6.88072i	0.213844	+	0.257324i
$716$		0			0
$717$		−	3.07220i		−	0.114733i
$718$		0			0
$719$	−30.4342			−1.13500			−0.567502	−	0.823372i	$-0.692090\pi$
$719$	−30.4342			−1.13500			−0.567502	+	0.823372i	$0.692090\pi$
$720$		0			0
$721$	−22.8472			−0.850875
$722$		0			0
$723$			27.3067i			1.01555i
$724$		0			0
$725$	7.82667	−	42.0448i	0.290675	−	1.56150i
$726$		0			0
$727$			29.7716i			1.10417i	0.833789	+	0.552083i	$0.186167\pi$
$727$			29.7716i			1.10417i	−0.833789	+	0.552083i	$0.813833\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−49.0331			−1.81355
$732$		0			0
$733$			5.43162i			0.200621i	0.994956	+	0.100311i	$0.0319837\pi$
$733$			5.43162i			0.200621i	−0.994956	+	0.100311i	$0.968016\pi$
$734$		0			0
$735$	−7.28304	−	8.76389i	−0.268639	−	0.323261i
$736$		0			0
$737$		−	3.21525i		−	0.118435i
$738$		0			0
$739$	−32.2870			−1.18770			−0.593848	−	0.804577i	$-0.702392\pi$
$739$	−32.2870			−1.18770			−0.593848	+	0.804577i	$0.702392\pi$
$740$		0			0
$741$	−0.173100			−0.00635898
$742$		0			0
$743$			36.8619i			1.35233i	0.736749	+	0.676166i	$0.236360\pi$
$743$			36.8619i			1.35233i	−0.736749	+	0.676166i	$0.763640\pi$
$744$		0			0
$745$	14.7478	−	12.2558i	0.540316	−	0.449017i
$746$		0			0
$747$		−	3.38678i		−	0.123916i
$748$		0			0
$749$	−15.1464			−0.553439
$750$		0			0
$751$	23.1860			0.846070			0.423035	−	0.906113i	$-0.360965\pi$
$751$	23.1860			0.846070			0.423035	+	0.906113i	$0.360965\pi$
$752$		0			0
$753$		−	1.06807i		−	0.0389227i
$754$		0			0
$755$	−23.3685	+	19.4199i	−0.850468	+	0.706762i
$756$		0			0
$757$		−	16.4866i		−	0.599217i	−0.954062	−	0.299608i	$-0.903144\pi$
$757$		−	16.4866i		−	0.599217i	0.954062	−	0.299608i	$-0.0968560\pi$
$758$		0			0
$759$	−3.24871			−0.117921
$760$		0			0
$761$	−6.23550			−0.226037			−0.113018	−	0.993593i	$-0.536052\pi$
$761$	−6.23550			−0.226037			−0.113018	+	0.993593i	$0.536052\pi$
$762$		0			0
$763$			6.50781i			0.235599i
$764$		0			0
$765$	−7.15990	−	8.61572i	−0.258867	−	0.311502i
$766$		0			0
$767$		−	12.2856i		−	0.443606i
$768$		0			0
$769$	29.7662			1.07340			0.536699	−	0.843774i	$-0.319671\pi$
$769$	29.7662			1.07340			0.536699	+	0.843774i	$0.319671\pi$
$770$		0			0
$771$	−9.76072			−0.351524
$772$		0			0
$773$			9.47745i			0.340880i	0.985368	+	0.170440i	$0.0545189\pi$
$773$			9.47745i			0.340880i	−0.985368	+	0.170440i	$0.945481\pi$
$774$		0			0
$775$	1.26089	−	6.77348i	0.0452924	−	0.243310i
$776$		0			0
$777$		−	6.95831i		−	0.249628i
$778$		0			0
$779$	0.0900648			0.00322691
$780$		0			0
$781$	−50.6191			−1.81129
$782$		0			0
$783$		−	8.55341i		−	0.305674i
$784$		0			0
$785$	−5.53256	−	6.65750i	−0.197466	−	0.237616i
$786$		0			0
$787$		−	14.5181i		−	0.517514i	−0.965942	−	0.258757i	$-0.916687\pi$
$787$		−	14.5181i		−	0.517514i	0.965942	−	0.258757i	$-0.0833130\pi$
$788$		0			0
$789$	−7.09365			−0.252541
$790$		0			0
$791$	−12.9018			−0.458734
$792$		0			0
$793$		−	16.0850i		−	0.571194i
$794$		0			0
$795$	−4.60906	+	3.83026i	−0.163467	+	0.135845i
$796$		0			0
$797$		−	6.78303i		−	0.240267i	−0.992758	−	0.120134i	$-0.961668\pi$
$797$		−	6.78303i		−	0.240267i	0.992758	−	0.120134i	$-0.0383323\pi$
$798$		0			0
$799$	−15.9547			−0.564437
$800$		0			0
$801$	−2.10216			−0.0742763
$802$		0			0
$803$		−	51.6762i		−	1.82361i
$804$		0			0
$805$	2.39766	−	1.99253i	0.0845066	−	0.0702273i
$806$		0			0
$807$		−	24.4888i		−	0.862047i
$808$		0			0
$809$	−42.2675			−1.48605			−0.743024	−	0.669265i	$-0.766609\pi$
$809$	−42.2675			−1.48605			−0.743024	+	0.669265i	$0.766609\pi$
$810$		0			0
$811$	4.30209			0.151067			0.0755335	−	0.997143i	$-0.475934\pi$
$811$	4.30209			0.151067			0.0755335	+	0.997143i	$0.475934\pi$
$812$		0			0
$813$		−	13.6330i		−	0.478131i
$814$		0			0
$815$	0.517060	+	0.622193i	0.0181118	+	0.0217945i
$816$		0			0
$817$			1.36145i			0.0476311i
$818$		0			0
$819$	−1.71705			−0.0599987
$820$		0			0
$821$	−37.4813			−1.30811			−0.654053	−	0.756449i	$-0.726933\pi$
$821$	−37.4813			−1.30811			−0.654053	+	0.756449i	$0.726933\pi$
$822$		0			0
$823$		−	31.0654i		−	1.08287i	−0.840742	−	0.541435i	$-0.817881\pi$
$823$		−	31.0654i		−	1.08287i	0.840742	−	0.541435i	$-0.182119\pi$
$824$		0			0
$825$	15.8047	+	2.94206i	0.550250	+	0.102429i
$826$		0			0
$827$			6.35419i			0.220957i	0.993879	+	0.110478i	$0.0352383\pi$
$827$			6.35419i			0.220957i	−0.993879	+	0.110478i	$0.964762\pi$
$828$		0			0
$829$	7.20436			0.250218			0.125109	−	0.992143i	$-0.460072\pi$
$829$	7.20436			0.250218			0.125109	+	0.992143i	$0.460072\pi$
$830$		0			0
$831$	25.9812			0.901276
$832$		0			0
$833$		−	25.5306i		−	0.884583i
$834$		0			0
$835$	1.56223	+	1.87988i	0.0540632	+	0.0650558i
$836$		0			0
$837$		−	1.37797i		−	0.0476295i
$838$		0			0
$839$	−18.9943			−0.655755			−0.327877	−	0.944720i	$-0.606333\pi$
$839$	−18.9943			−0.655755			−0.327877	+	0.944720i	$0.606333\pi$
$840$		0			0
$841$	44.1609			1.52279
$842$		0			0
$843$		−	3.41129i		−	0.117491i
$844$		0			0
$845$	−19.6936	+	16.3660i	−0.677482	+	0.563006i
$846$		0			0
$847$		−	0.913701i		−	0.0313951i
$848$		0			0
$849$	16.1212			0.553277
$850$		0			0
$851$	−5.09533			−0.174666
$852$		0			0
$853$		−	31.4748i		−	1.07768i	−0.842409	−	0.538838i	$-0.818864\pi$
$853$		−	31.4748i		−	1.07768i	0.842409	−	0.538838i	$-0.181136\pi$
$854$		0			0
$855$	−0.239224	+	0.198802i	−0.00818129	+	0.00679888i
$856$		0			0
$857$			19.2795i			0.658576i	0.944230	+	0.329288i	$0.106809\pi$
$857$			19.2795i			0.658576i	−0.944230	+	0.329288i	$0.893191\pi$
$858$		0			0
$859$	38.1150			1.30047			0.650233	−	0.759735i	$-0.274671\pi$
$859$	38.1150			1.30047			0.650233	+	0.759735i	$0.274671\pi$
$860$		0			0
$861$	0.893392			0.0304467
$862$		0			0
$863$			35.9823i			1.22485i	0.790528	+	0.612425i	$0.209806\pi$
$863$			35.9823i			1.22485i	−0.790528	+	0.612425i	$0.790194\pi$
$864$		0			0
$865$	−23.2394	−	27.9647i	−0.790164	−	0.950827i
$866$		0			0
$867$		−	8.09896i		−	0.275055i
$868$		0			0
$869$	3.94136			0.133701
$870$		0			0
$871$	−1.24439			−0.0421645
$872$		0			0
$873$			8.14957i			0.275821i
$874$		0			0
$875$	−13.4689	+	7.52215i	−0.455332	+	0.254295i
$876$		0			0
$877$		−	20.2807i		−	0.684830i	−0.939549	−	0.342415i	$-0.888755\pi$
$877$		−	20.2807i		−	0.684830i	0.939549	−	0.342415i	$-0.111245\pi$
$878$		0			0
$879$	32.8616			1.10839
$880$		0			0
$881$	45.4240			1.53037			0.765187	−	0.643809i	$-0.222647\pi$
$881$	45.4240			1.53037			0.765187	+	0.643809i	$0.222647\pi$
$882$		0			0
$883$		−	8.80229i		−	0.296221i	−0.988971	−	0.148110i	$-0.952681\pi$
$883$		−	8.80229i		−	0.296221i	0.988971	−	0.148110i	$-0.0473191\pi$
$884$		0			0
$885$	−14.1097	−	16.9787i	−0.474294	−	0.570731i
$886$		0			0
$887$		−	3.08114i		−	0.103455i	−0.998661	−	0.0517273i	$-0.983527\pi$
$887$		−	3.08114i		−	0.103455i	0.998661	−	0.0517273i	$-0.0164726\pi$
$888$		0			0
$889$	6.39536			0.214493
$890$		0			0
$891$	3.21525			0.107715
$892$		0			0
$893$			0.442998i			0.0148244i
$894$		0			0
$895$	−36.4514	+	30.2921i	−1.21844	+	1.01255i
$896$		0			0
$897$			1.25734i			0.0419813i
$898$		0			0
$899$	11.7863			0.393096
$900$		0			0
$901$	−13.4269			−0.447316
$902$		0			0
$903$			13.5048i			0.449413i
$904$		0			0
$905$	−4.81064	+	3.99777i	−0.159911	+	0.132890i
$906$		0			0
$907$		−	59.6761i		−	1.98151i	−0.135659	−	0.990756i	$-0.543315\pi$
$907$		−	59.6761i		−	1.98151i	0.135659	−	0.990756i	$-0.456685\pi$
$908$		0			0
$909$	5.04519			0.167338
$910$		0			0
$911$	3.00185			0.0994557			0.0497279	−	0.998763i	$-0.484165\pi$
$911$	3.00185			0.0994557			0.0497279	+	0.998763i	$0.484165\pi$
$912$		0			0
$913$			10.8893i			0.360385i
$914$		0			0
$915$	−18.4733	−	22.2294i	−0.610707	−	0.734882i
$916$		0			0
$917$		−	26.5757i		−	0.877607i
$918$		0			0
$919$	25.0779			0.827244			0.413622	−	0.910449i	$-0.364264\pi$
$919$	25.0779			0.827244			0.413622	+	0.910449i	$0.364264\pi$
$920$		0			0
$921$	−23.5637			−0.776452
$922$		0			0
$923$			19.5909i			0.644843i
$924$		0			0
$925$	24.7884	+	4.61437i	0.815037	+	0.151720i
$926$		0			0
$927$		−	16.5579i		−	0.543833i
$928$		0			0
$929$	32.9759			1.08191			0.540953	−	0.841053i	$-0.318064\pi$
$929$	32.9759			1.08191			0.540953	+	0.841053i	$0.318064\pi$
$930$		0			0
$931$	−0.708883			−0.0232327
$932$		0			0
$933$			23.8330i			0.780258i
$934$		0			0
$935$	23.0209	+	27.7017i	0.752863	+	0.905942i
$936$		0			0
$937$		−	34.0273i		−	1.11162i	−0.831308	−	0.555812i	$-0.812407\pi$
$937$		−	34.0273i		−	1.11162i	0.831308	−	0.555812i	$-0.187593\pi$
$938$		0			0
$939$	19.8108			0.646500
$940$		0			0
$941$	−40.6448			−1.32498			−0.662492	−	0.749069i	$-0.730501\pi$
$941$	−40.6448			−1.32498			−0.662492	+	0.749069i	$0.730501\pi$
$942$		0			0
$943$		−	0.654200i		−	0.0213037i
$944$		0			0
$945$	−2.37297	+	1.97200i	−0.0771927	+	0.0641492i
$946$		0			0
$947$			40.2683i			1.30854i	0.756260	+	0.654272i	$0.227025\pi$
$947$			40.2683i			1.30854i	−0.756260	+	0.654272i	$0.772975\pi$
$948$		0			0
$949$	−20.0001			−0.649230
$950$		0			0
$951$	−7.70559			−0.249871
$952$		0			0
$953$			24.1148i			0.781155i	0.920570	+	0.390578i	$0.127725\pi$
$953$			24.1148i			0.781155i	−0.920570	+	0.390578i	$0.872275\pi$
$954$		0			0
$955$	−16.0205	+	13.3135i	−0.518410	+	0.430813i
$956$		0			0
$957$			27.5013i			0.888992i
$958$		0			0
$959$	−15.7010			−0.507012
$960$		0			0
$961$	−29.1012			−0.938749
$962$		0			0
$963$		−	10.9770i		−	0.353728i
$964$		0			0
$965$	−26.5471	−	31.9448i	−0.854580	−	1.02834i
$966$		0			0
$967$		−	10.5601i		−	0.339591i	−0.985479	−	0.169795i	$-0.945689\pi$
$967$		−	10.5601i		−	0.339591i	0.985479	−	0.169795i	$-0.0543107\pi$
$968$		0			0
$969$	−0.696897			−0.0223876
$970$		0			0
$971$	−7.52267			−0.241414			−0.120707	−	0.992688i	$-0.538516\pi$
$971$	−7.52267			−0.241414			−0.120707	+	0.992688i	$0.538516\pi$
$972$		0			0
$973$			4.43662i			0.142232i
$974$		0			0
$975$	1.13866	−	6.11686i	0.0364662	−	0.195896i
$976$		0			0
$977$			14.9046i			0.476841i	0.971162	+	0.238421i	$0.0766297\pi$
$977$			14.9046i			0.476841i	−0.971162	+	0.238421i	$0.923370\pi$
$978$		0			0
$979$	6.75898			0.216018
$980$		0			0
$981$	−4.71636			−0.150582
$982$		0			0
$983$			11.6522i			0.371648i	0.982583	+	0.185824i	$0.0594954\pi$
$983$			11.6522i			0.371648i	−0.982583	+	0.185824i	$0.940505\pi$
$984$		0			0
$985$	−2.44245	−	2.93907i	−0.0778230	−	0.0936467i
$986$		0			0
$987$			4.39429i			0.139872i
$988$		0			0
$989$	9.88912			0.314456
$990$		0			0
$991$	11.7682			0.373830			0.186915	−	0.982376i	$-0.440151\pi$
$991$	11.7682			0.373830			0.186915	+	0.982376i	$0.440151\pi$
$992$		0			0
$993$		−	15.9273i		−	0.505438i
$994$		0			0
$995$	−30.0832	+	25.0000i	−0.953703	+	0.792554i
$996$		0			0
$997$			34.8434i			1.10350i	0.834009	+	0.551751i	$0.186040\pi$
$997$			34.8434i			1.10350i	−0.834009	+	0.551751i	$0.813960\pi$
$998$		0			0
$999$	5.04284			0.159549

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.b.1609.15	yes	24
5.4	even	2	inner	4020.2.g.b.1609.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.b.1609.3	✓	24	5.4	even	2	inner
4020.2.g.b.1609.15	yes	24	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.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-1.71974 + 1.42915i) q^{5} +1.37984i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-1.71974 + 1.42915i) q^{5} +1.37984i q^{7} -1.00000 q^{9} +3.21525 q^{11} -1.24439i q^{13} +(-1.42915 - 1.71974i) q^{15} -5.00989i q^{17} -0.139104 q^{19} -1.37984 q^{21} +1.01041i q^{23} +(0.915034 - 4.91556i) q^{25} -1.00000i q^{27} +8.55341 q^{29} +1.37797 q^{31} +3.21525i q^{33} +(-1.97200 - 2.37297i) q^{35} +5.04284i q^{37} +1.24439 q^{39} -0.647462 q^{41} -9.78726i q^{43} +(1.71974 - 1.42915i) q^{45} -3.18464i q^{47} +5.09605 q^{49} +5.00989 q^{51} -2.68009i q^{53} +(-5.52940 + 4.59509i) q^{55} -0.139104i q^{57} +9.87278 q^{59} +12.9260 q^{61} -1.37984i q^{63} +(1.77842 + 2.14003i) q^{65} -1.00000i q^{67} -1.01041 q^{69} -15.7434 q^{71} -16.0722i q^{73} +(4.91556 + 0.915034i) q^{75} +4.43652i q^{77} +1.22583 q^{79} +1.00000 q^{81} +3.38678i q^{83} +(7.15990 + 8.61572i) q^{85} +8.55341i q^{87} +2.10216 q^{89} +1.71705 q^{91} +1.37797i q^{93} +(0.239224 - 0.198802i) q^{95} -8.14957i q^{97} -3.21525 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) \) 24 * q - 4 * q^5 - 24 * q^9	\( 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) \) 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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