LMFDB - Embedded newform 4020.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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, 0, 0]))

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

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

chi := DirichletCharacter("4020.1");

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.a (trivial)

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:	yes
Analytic conductor:	$32.0998616126$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.209057$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +2.57433 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +2.57433 q^{7} +1.00000 q^{9} -5.06316 q^{11} -1.30211 q^{13} +1.00000 q^{15} -4.52498 q^{17} -0.372829 q^{19} +2.57433 q^{21} -6.09464 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.59667 q^{29} -8.38006 q^{31} -5.06316 q^{33} +2.57433 q^{35} -8.09464 q^{37} -1.30211 q^{39} -5.44512 q^{41} +9.10495 q^{43} +1.00000 q^{45} -6.61048 q^{47} -0.372829 q^{49} -4.52498 q^{51} +0.143988 q^{53} -5.06316 q^{55} -0.372829 q^{57} -9.34102 q^{59} +6.21628 q^{61} +2.57433 q^{63} -1.30211 q^{65} +1.00000 q^{67} -6.09464 q^{69} +13.6671 q^{71} +3.93204 q^{73} +1.00000 q^{75} -13.0342 q^{77} +4.42843 q^{79} +1.00000 q^{81} +0.605807 q^{83} -4.52498 q^{85} -4.59667 q^{87} -7.85122 q^{89} -3.35207 q^{91} -8.38006 q^{93} -0.372829 q^{95} -6.62270 q^{97} -5.06316 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 5 q^{11} - 9 q^{13} + 4 q^{15} - 8 q^{17} - 7 q^{19} - 3 q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 9 q^{29} - 9 q^{31} - 5 q^{33} - 3 q^{35} - 20 q^{37} - 9 q^{39} - 11 q^{41} + q^{43} + 4 q^{45} - 5 q^{47} - 7 q^{49} - 8 q^{51} - 15 q^{53} - 5 q^{55} - 7 q^{57} + 7 q^{59} - 7 q^{61} - 3 q^{63} - 9 q^{65} + 4 q^{67} - 12 q^{69} - 26 q^{73} + 4 q^{75} - 15 q^{77} - 9 q^{79} + 4 q^{81} - 8 q^{83} - 8 q^{85} - 9 q^{87} + 10 q^{89} + 3 q^{91} - 9 q^{93} - 7 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 5 * q^11 - 9 * q^13 + 4 * q^15 - 8 * q^17 - 7 * q^19 - 3 * q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 9 * q^29 - 9 * q^31 - 5 * q^33 - 3 * q^35 - 20 * q^37 - 9 * q^39 - 11 * q^41 + q^43 + 4 * q^45 - 5 * q^47 - 7 * q^49 - 8 * q^51 - 15 * q^53 - 5 * q^55 - 7 * q^57 + 7 * q^59 - 7 * q^61 - 3 * q^63 - 9 * q^65 + 4 * q^67 - 12 * q^69 - 26 * q^73 + 4 * q^75 - 15 * q^77 - 9 * q^79 + 4 * q^81 - 8 * q^83 - 8 * q^85 - 9 * q^87 + 10 * q^89 + 3 * q^91 - 9 * q^93 - 7 * q^95 - 11 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.57433	0.973005	0.486502	−	0.873679i	$-0.338272\pi$
$7$	2.57433	0.973005	0.486502	+	0.873679i	$0.338272\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.06316	−1.52660	−0.763300	−	0.646044i	$-0.776422\pi$
$11$	−5.06316	−1.52660	−0.763300	+	0.646044i	$0.776422\pi$
$12$	0	0
$13$	−1.30211	−0.361141	−0.180571	−	0.983562i	$-0.557794\pi$
$13$	−1.30211	−0.361141	−0.180571	+	0.983562i	$0.557794\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.52498	−1.09747	−0.548734	−	0.835997i	$-0.684890\pi$
$17$	−4.52498	−1.09747	−0.548734	+	0.835997i	$0.684890\pi$
$18$	0	0
$19$	−0.372829	−0.0855329	−0.0427664	−	0.999085i	$-0.513617\pi$
$19$	−0.372829	−0.0855329	−0.0427664	+	0.999085i	$0.513617\pi$
$20$	0	0
$21$	2.57433	0.561765
$22$	0	0
$23$	−6.09464	−1.27082	−0.635410	−	0.772175i	$-0.719169\pi$
$23$	−6.09464	−1.27082	−0.635410	+	0.772175i	$0.719169\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.59667	−0.853580	−0.426790	−	0.904351i	$-0.640356\pi$
$29$	−4.59667	−0.853580	−0.426790	+	0.904351i	$0.640356\pi$
$30$	0	0
$31$	−8.38006	−1.50510	−0.752551	−	0.658534i	$-0.771177\pi$
$31$	−8.38006	−1.50510	−0.752551	+	0.658534i	$0.771177\pi$
$32$	0	0
$33$	−5.06316	−0.881383
$34$	0	0
$35$	2.57433	0.435141
$36$	0	0
$37$	−8.09464	−1.33075	−0.665375	−	0.746509i	$-0.731728\pi$
$37$	−8.09464	−1.33075	−0.665375	+	0.746509i	$0.731728\pi$
$38$	0	0
$39$	−1.30211	−0.208505
$40$	0	0
$41$	−5.44512	−0.850386	−0.425193	−	0.905103i	$-0.639794\pi$
$41$	−5.44512	−0.850386	−0.425193	+	0.905103i	$0.639794\pi$
$42$	0	0
$43$	9.10495	1.38849	0.694246	−	0.719738i	$-0.255738\pi$
$43$	9.10495	1.38849	0.694246	+	0.719738i	$0.255738\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−6.61048	−0.964237	−0.482119	−	0.876106i	$-0.660133\pi$
$47$	−6.61048	−0.964237	−0.482119	+	0.876106i	$0.660133\pi$
$48$	0	0
$49$	−0.372829	−0.0532613
$50$	0	0
$51$	−4.52498	−0.633624
$52$	0	0
$53$	0.143988	0.0197783	0.00988913	−	0.999951i	$-0.496852\pi$
$53$	0.143988	0.0197783	0.00988913	+	0.999951i	$0.496852\pi$
$54$	0	0
$55$	−5.06316	−0.682716
$56$	0	0
$57$	−0.372829	−0.0493824
$58$	0	0
$59$	−9.34102	−1.21610	−0.608049	−	0.793900i	$-0.708047\pi$
$59$	−9.34102	−1.21610	−0.608049	+	0.793900i	$0.708047\pi$
$60$	0	0
$61$	6.21628	0.795914	0.397957	−	0.917404i	$-0.369719\pi$
$61$	6.21628	0.795914	0.397957	+	0.917404i	$0.369719\pi$
$62$	0	0
$63$	2.57433	0.324335
$64$	0	0
$65$	−1.30211	−0.161507
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−6.09464	−0.733708
$70$	0	0
$71$	13.6671	1.62198	0.810991	−	0.585059i	$-0.198929\pi$
$71$	13.6671	1.62198	0.810991	+	0.585059i	$0.198929\pi$
$72$	0	0
$73$	3.93204	0.460211	0.230105	−	0.973166i	$-0.426093\pi$
$73$	3.93204	0.460211	0.230105	+	0.973166i	$0.426093\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−13.0342	−1.48539
$78$	0	0
$79$	4.42843	0.498237	0.249119	−	0.968473i	$-0.419859\pi$
$79$	4.42843	0.498237	0.249119	+	0.968473i	$0.419859\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.605807	0.0664959	0.0332480	−	0.999447i	$-0.489415\pi$
$83$	0.605807	0.0664959	0.0332480	+	0.999447i	$0.489415\pi$
$84$	0	0
$85$	−4.52498	−0.490803
$86$	0	0
$87$	−4.59667	−0.492815
$88$	0	0
$89$	−7.85122	−0.832227	−0.416114	−	0.909313i	$-0.636608\pi$
$89$	−7.85122	−0.832227	−0.416114	+	0.909313i	$0.636608\pi$
$90$	0	0
$91$	−3.35207	−0.351392
$92$	0	0
$93$	−8.38006	−0.868971
$94$	0	0
$95$	−0.372829	−0.0382515
$96$	0	0
$97$	−6.62270	−0.672434	−0.336217	−	0.941785i	$-0.609148\pi$
$97$	−6.62270	−0.672434	−0.336217	+	0.941785i	$0.609148\pi$
$98$	0	0
$99$	−5.06316	−0.508867
$100$	0	0
$101$	−17.0572	−1.69725	−0.848627	−	0.528992i	$-0.822570\pi$
$101$	−17.0572	−1.69725	−0.848627	+	0.528992i	$0.822570\pi$
$102$	0	0
$103$	−6.74979	−0.665077	−0.332539	−	0.943090i	$-0.607905\pi$
$103$	−6.74979	−0.665077	−0.332539	+	0.943090i	$0.607905\pi$
$104$	0	0
$105$	2.57433	0.251229
$106$	0	0
$107$	9.12185	0.881843	0.440921	−	0.897546i	$-0.354652\pi$
$107$	9.12185	0.881843	0.440921	+	0.897546i	$0.354652\pi$
$108$	0	0
$109$	14.1096	1.35146	0.675728	−	0.737151i	$-0.263829\pi$
$109$	14.1096	1.35146	0.675728	+	0.737151i	$0.263829\pi$
$110$	0	0
$111$	−8.09464	−0.768309
$112$	0	0
$113$	18.5441	1.74448	0.872241	−	0.489076i	$-0.162666\pi$
$113$	18.5441	1.74448	0.872241	+	0.489076i	$0.162666\pi$
$114$	0	0
$115$	−6.09464	−0.568328
$116$	0	0
$117$	−1.30211	−0.120380
$118$	0	0
$119$	−11.6488	−1.06784
$120$	0	0
$121$	14.6356	1.33051
$122$	0	0
$123$	−5.44512	−0.490970
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	20.2326	1.79536	0.897679	−	0.440651i	$-0.145252\pi$
$127$	20.2326	1.79536	0.897679	+	0.440651i	$0.145252\pi$
$128$	0	0
$129$	9.10495	0.801646
$130$	0	0
$131$	1.75170	0.153047	0.0765236	−	0.997068i	$-0.475618\pi$
$131$	1.75170	0.153047	0.0765236	+	0.997068i	$0.475618\pi$
$132$	0	0
$133$	−0.959785	−0.0832239
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	5.29923	0.452744	0.226372	−	0.974041i	$-0.427314\pi$
$137$	5.29923	0.452744	0.226372	+	0.974041i	$0.427314\pi$
$138$	0	0
$139$	−17.9663	−1.52388	−0.761940	−	0.647648i	$-0.775753\pi$
$139$	−17.9663	−1.52388	−0.761940	+	0.647648i	$0.775753\pi$
$140$	0	0
$141$	−6.61048	−0.556703
$142$	0	0
$143$	6.59281	0.551318
$144$	0	0
$145$	−4.59667	−0.381733
$146$	0	0
$147$	−0.372829	−0.0307504
$148$	0	0
$149$	17.5604	1.43860	0.719302	−	0.694698i	$-0.244462\pi$
$149$	17.5604	1.43860	0.719302	+	0.694698i	$0.244462\pi$
$150$	0	0
$151$	11.2511	0.915598	0.457799	−	0.889056i	$-0.348638\pi$
$151$	11.2511	0.915598	0.457799	+	0.889056i	$0.348638\pi$
$152$	0	0
$153$	−4.52498	−0.365823
$154$	0	0
$155$	−8.38006	−0.673102
$156$	0	0
$157$	−11.1411	−0.889157	−0.444578	−	0.895740i	$-0.646646\pi$
$157$	−11.1411	−0.889157	−0.444578	+	0.895740i	$0.646646\pi$
$158$	0	0
$159$	0.143988	0.0114190
$160$	0	0
$161$	−15.6896	−1.23651
$162$	0	0
$163$	−17.0059	−1.33201	−0.666003	−	0.745949i	$-0.731996\pi$
$163$	−17.0059	−1.33201	−0.666003	+	0.745949i	$0.731996\pi$
$164$	0	0
$165$	−5.06316	−0.394166
$166$	0	0
$167$	15.2571	1.18063	0.590315	−	0.807173i	$-0.299004\pi$
$167$	15.2571	1.18063	0.590315	+	0.807173i	$0.299004\pi$
$168$	0	0
$169$	−11.3045	−0.869577
$170$	0	0
$171$	−0.372829	−0.0285110
$172$	0	0
$173$	−17.9396	−1.36392	−0.681961	−	0.731388i	$-0.738873\pi$
$173$	−17.9396	−1.36392	−0.681961	+	0.731388i	$0.738873\pi$
$174$	0	0
$175$	2.57433	0.194601
$176$	0	0
$177$	−9.34102	−0.702114
$178$	0	0
$179$	22.9072	1.71216	0.856082	−	0.516840i	$-0.172892\pi$
$179$	22.9072	1.71216	0.856082	+	0.516840i	$0.172892\pi$
$180$	0	0
$181$	25.9439	1.92840	0.964199	−	0.265180i	$-0.0854313\pi$
$181$	25.9439	1.92840	0.964199	+	0.265180i	$0.0854313\pi$
$182$	0	0
$183$	6.21628	0.459521
$184$	0	0
$185$	−8.09464	−0.595129
$186$	0	0
$187$	22.9107	1.67539
$188$	0	0
$189$	2.57433	0.187255
$190$	0	0
$191$	2.55454	0.184840	0.0924202	−	0.995720i	$-0.470540\pi$
$191$	2.55454	0.184840	0.0924202	+	0.995720i	$0.470540\pi$
$192$	0	0
$193$	14.4032	1.03677	0.518383	−	0.855149i	$-0.326534\pi$
$193$	14.4032	1.03677	0.518383	+	0.855149i	$0.326534\pi$
$194$	0	0
$195$	−1.30211	−0.0932463
$196$	0	0
$197$	17.4775	1.24522	0.622608	−	0.782534i	$-0.286073\pi$
$197$	17.4775	1.24522	0.622608	+	0.782534i	$0.286073\pi$
$198$	0	0
$199$	−7.74744	−0.549202	−0.274601	−	0.961558i	$-0.588546\pi$
$199$	−7.74744	−0.549202	−0.274601	+	0.961558i	$0.588546\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−11.8333	−0.830538
$204$	0	0
$205$	−5.44512	−0.380304
$206$	0	0
$207$	−6.09464	−0.423607
$208$	0	0
$209$	1.88769	0.130574
$210$	0	0
$211$	−14.5573	−1.00217	−0.501083	−	0.865399i	$-0.667065\pi$
$211$	−14.5573	−1.00217	−0.501083	+	0.865399i	$0.667065\pi$
$212$	0	0
$213$	13.6671	0.936451
$214$	0	0
$215$	9.10495	0.620953
$216$	0	0
$217$	−21.5730	−1.46447
$218$	0	0
$219$	3.93204	0.265703
$220$	0	0
$221$	5.89203	0.396341
$222$	0	0
$223$	22.3743	1.49829	0.749146	−	0.662404i	$-0.230464\pi$
$223$	22.3743	1.49829	0.749146	+	0.662404i	$0.230464\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	21.1483	1.40366	0.701832	−	0.712343i	$-0.252366\pi$
$227$	21.1483	1.40366	0.701832	+	0.712343i	$0.252366\pi$
$228$	0	0
$229$	−4.25243	−0.281009	−0.140504	−	0.990080i	$-0.544872\pi$
$229$	−4.25243	−0.281009	−0.140504	+	0.990080i	$0.544872\pi$
$230$	0	0
$231$	−13.0342	−0.857590
$232$	0	0
$233$	−24.8578	−1.62849	−0.814246	−	0.580521i	$-0.802849\pi$
$233$	−24.8578	−1.62849	−0.814246	+	0.580521i	$0.802849\pi$
$234$	0	0
$235$	−6.61048	−0.431220
$236$	0	0
$237$	4.42843	0.287658
$238$	0	0
$239$	−12.0138	−0.777111	−0.388556	−	0.921425i	$-0.627026\pi$
$239$	−12.0138	−0.777111	−0.388556	+	0.921425i	$0.627026\pi$
$240$	0	0
$241$	−26.9245	−1.73436	−0.867179	−	0.497996i	$-0.834069\pi$
$241$	−26.9245	−1.73436	−0.867179	+	0.497996i	$0.834069\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.372829	−0.0238192
$246$	0	0
$247$	0.485466	0.0308895
$248$	0	0
$249$	0.605807	0.0383914
$250$	0	0
$251$	−7.24776	−0.457475	−0.228737	−	0.973488i	$-0.573460\pi$
$251$	−7.24776	−0.457475	−0.228737	+	0.973488i	$0.573460\pi$
$252$	0	0
$253$	30.8581	1.94003
$254$	0	0
$255$	−4.52498	−0.283365
$256$	0	0
$257$	17.5200	1.09287	0.546433	−	0.837503i	$-0.315985\pi$
$257$	17.5200	1.09287	0.546433	+	0.837503i	$0.315985\pi$
$258$	0	0
$259$	−20.8383	−1.29483
$260$	0	0
$261$	−4.59667	−0.284527
$262$	0	0
$263$	−24.4730	−1.50907	−0.754534	−	0.656260i	$-0.772137\pi$
$263$	−24.4730	−1.50907	−0.754534	+	0.656260i	$0.772137\pi$
$264$	0	0
$265$	0.143988	0.00884511
$266$	0	0
$267$	−7.85122	−0.480487
$268$	0	0
$269$	−7.73115	−0.471376	−0.235688	−	0.971829i	$-0.575734\pi$
$269$	−7.73115	−0.471376	−0.235688	+	0.971829i	$0.575734\pi$
$270$	0	0
$271$	−26.2956	−1.59734	−0.798672	−	0.601766i	$-0.794464\pi$
$271$	−26.2956	−1.59734	−0.798672	+	0.601766i	$0.794464\pi$
$272$	0	0
$273$	−3.35207	−0.202876
$274$	0	0
$275$	−5.06316	−0.305320
$276$	0	0
$277$	−5.81860	−0.349606	−0.174803	−	0.984603i	$-0.555929\pi$
$277$	−5.81860	−0.349606	−0.174803	+	0.984603i	$0.555929\pi$
$278$	0	0
$279$	−8.38006	−0.501701
$280$	0	0
$281$	5.82887	0.347722	0.173861	−	0.984770i	$-0.444376\pi$
$281$	5.82887	0.347722	0.173861	+	0.984770i	$0.444376\pi$
$282$	0	0
$283$	18.2550	1.08515	0.542573	−	0.840009i	$-0.317450\pi$
$283$	18.2550	1.08515	0.542573	+	0.840009i	$0.317450\pi$
$284$	0	0
$285$	−0.372829	−0.0220845
$286$	0	0
$287$	−14.0175	−0.827429
$288$	0	0
$289$	3.47542	0.204437
$290$	0	0
$291$	−6.62270	−0.388230
$292$	0	0
$293$	−17.3911	−1.01600	−0.508000	−	0.861357i	$-0.669615\pi$
$293$	−17.3911	−1.01600	−0.508000	+	0.861357i	$0.669615\pi$
$294$	0	0
$295$	−9.34102	−0.543855
$296$	0	0
$297$	−5.06316	−0.293794
$298$	0	0
$299$	7.93591	0.458945
$300$	0	0
$301$	23.4391	1.35101
$302$	0	0
$303$	−17.0572	−0.979909
$304$	0	0
$305$	6.21628	0.355943
$306$	0	0
$307$	−3.06223	−0.174770	−0.0873852	−	0.996175i	$-0.527851\pi$
$307$	−3.06223	−0.174770	−0.0873852	+	0.996175i	$0.527851\pi$
$308$	0	0
$309$	−6.74979	−0.383982
$310$	0	0
$311$	−22.6209	−1.28271	−0.641356	−	0.767244i	$-0.721628\pi$
$311$	−22.6209	−1.28271	−0.641356	+	0.767244i	$0.721628\pi$
$312$	0	0
$313$	−7.48810	−0.423252	−0.211626	−	0.977351i	$-0.567876\pi$
$313$	−7.48810	−0.423252	−0.211626	+	0.977351i	$0.567876\pi$
$314$	0	0
$315$	2.57433	0.145047
$316$	0	0
$317$	−34.6415	−1.94566	−0.972830	−	0.231520i	$-0.925630\pi$
$317$	−34.6415	−1.94566	−0.972830	+	0.231520i	$0.925630\pi$
$318$	0	0
$319$	23.2737	1.30308
$320$	0	0
$321$	9.12185	0.509132
$322$	0	0
$323$	1.68704	0.0938696
$324$	0	0
$325$	−1.30211	−0.0722283
$326$	0	0
$327$	14.1096	0.780264
$328$	0	0
$329$	−17.0175	−0.938208
$330$	0	0
$331$	18.4794	1.01572	0.507859	−	0.861440i	$-0.330437\pi$
$331$	18.4794	1.01572	0.507859	+	0.861440i	$0.330437\pi$
$332$	0	0
$333$	−8.09464	−0.443583
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	11.4422	0.623298	0.311649	−	0.950197i	$-0.399119\pi$
$337$	11.4422	0.623298	0.311649	+	0.950197i	$0.399119\pi$
$338$	0	0
$339$	18.5441	1.00718
$340$	0	0
$341$	42.4296	2.29769
$342$	0	0
$343$	−18.9801	−1.02483
$344$	0	0
$345$	−6.09464	−0.328124
$346$	0	0
$347$	0.563613	0.0302563	0.0151282	−	0.999886i	$-0.495184\pi$
$347$	0.563613	0.0302563	0.0151282	+	0.999886i	$0.495184\pi$
$348$	0	0
$349$	29.8015	1.59524	0.797619	−	0.603162i	$-0.206093\pi$
$349$	29.8015	1.59524	0.797619	+	0.603162i	$0.206093\pi$
$350$	0	0
$351$	−1.30211	−0.0695017
$352$	0	0
$353$	−16.4473	−0.875400	−0.437700	−	0.899121i	$-0.644207\pi$
$353$	−16.4473	−0.875400	−0.437700	+	0.899121i	$0.644207\pi$
$354$	0	0
$355$	13.6671	0.725372
$356$	0	0
$357$	−11.6488	−0.616519
$358$	0	0
$359$	−6.99963	−0.369426	−0.184713	−	0.982792i	$-0.559136\pi$
$359$	−6.99963	−0.369426	−0.184713	+	0.982792i	$0.559136\pi$
$360$	0	0
$361$	−18.8610	−0.992684
$362$	0	0
$363$	14.6356	0.768169
$364$	0	0
$365$	3.93204	0.205813
$366$	0	0
$367$	16.0983	0.840326	0.420163	−	0.907449i	$-0.361973\pi$
$367$	16.0983	0.840326	0.420163	+	0.907449i	$0.361973\pi$
$368$	0	0
$369$	−5.44512	−0.283462
$370$	0	0
$371$	0.370672	0.0192443
$372$	0	0
$373$	−28.4151	−1.47128	−0.735639	−	0.677374i	$-0.763118\pi$
$373$	−28.4151	−1.47128	−0.735639	+	0.677374i	$0.763118\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	5.98539	0.308263
$378$	0	0
$379$	−23.8629	−1.22576	−0.612878	−	0.790178i	$-0.709988\pi$
$379$	−23.8629	−1.22576	−0.612878	+	0.790178i	$0.709988\pi$
$380$	0	0
$381$	20.2326	1.03655
$382$	0	0
$383$	−18.3688	−0.938600	−0.469300	−	0.883039i	$-0.655494\pi$
$383$	−18.3688	−0.938600	−0.469300	+	0.883039i	$0.655494\pi$
$384$	0	0
$385$	−13.0342	−0.664286
$386$	0	0
$387$	9.10495	0.462831
$388$	0	0
$389$	−23.4868	−1.19083	−0.595414	−	0.803419i	$-0.703012\pi$
$389$	−23.4868	−1.19083	−0.595414	+	0.803419i	$0.703012\pi$
$390$	0	0
$391$	27.5781	1.39468
$392$	0	0
$393$	1.75170	0.0883618
$394$	0	0
$395$	4.42843	0.222819
$396$	0	0
$397$	−6.88229	−0.345412	−0.172706	−	0.984973i	$-0.555251\pi$
$397$	−6.88229	−0.345412	−0.172706	+	0.984973i	$0.555251\pi$
$398$	0	0
$399$	−0.959785	−0.0480494
$400$	0	0
$401$	−32.7534	−1.63562	−0.817812	−	0.575485i	$-0.804813\pi$
$401$	−32.7534	−1.63562	−0.817812	+	0.575485i	$0.804813\pi$
$402$	0	0
$403$	10.9118	0.543555
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	40.9844	2.03152
$408$	0	0
$409$	−2.99066	−0.147879	−0.0739393	−	0.997263i	$-0.523557\pi$
$409$	−2.99066	−0.147879	−0.0739393	+	0.997263i	$0.523557\pi$
$410$	0	0
$411$	5.29923	0.261392
$412$	0	0
$413$	−24.0469	−1.18327
$414$	0	0
$415$	0.605807	0.0297379
$416$	0	0
$417$	−17.9663	−0.879813
$418$	0	0
$419$	13.3733	0.653329	0.326664	−	0.945140i	$-0.394075\pi$
$419$	13.3733	0.653329	0.326664	+	0.945140i	$0.394075\pi$
$420$	0	0
$421$	26.6216	1.29746	0.648729	−	0.761020i	$-0.275301\pi$
$421$	26.6216	1.29746	0.648729	+	0.761020i	$0.275301\pi$
$422$	0	0
$423$	−6.61048	−0.321412
$424$	0	0
$425$	−4.52498	−0.219494
$426$	0	0
$427$	16.0028	0.774428
$428$	0	0
$429$	6.59281	0.318304
$430$	0	0
$431$	31.3697	1.51102	0.755512	−	0.655135i	$-0.227388\pi$
$431$	31.3697	1.51102	0.755512	+	0.655135i	$0.227388\pi$
$432$	0	0
$433$	−26.4646	−1.27181	−0.635904	−	0.771768i	$-0.719373\pi$
$433$	−26.4646	−1.27181	−0.635904	+	0.771768i	$0.719373\pi$
$434$	0	0
$435$	−4.59667	−0.220393
$436$	0	0
$437$	2.27226	0.108697
$438$	0	0
$439$	1.53687	0.0733510	0.0366755	−	0.999327i	$-0.488323\pi$
$439$	1.53687	0.0733510	0.0366755	+	0.999327i	$0.488323\pi$
$440$	0	0
$441$	−0.372829	−0.0177538
$442$	0	0
$443$	−5.95781	−0.283064	−0.141532	−	0.989934i	$-0.545203\pi$
$443$	−5.95781	−0.283064	−0.141532	+	0.989934i	$0.545203\pi$
$444$	0	0
$445$	−7.85122	−0.372183
$446$	0	0
$447$	17.5604	0.830578
$448$	0	0
$449$	4.83558	0.228205	0.114103	−	0.993469i	$-0.463601\pi$
$449$	4.83558	0.228205	0.114103	+	0.993469i	$0.463601\pi$
$450$	0	0
$451$	27.5695	1.29820
$452$	0	0
$453$	11.2511	0.528621
$454$	0	0
$455$	−3.35207	−0.157147
$456$	0	0
$457$	10.6204	0.496801	0.248401	−	0.968657i	$-0.420095\pi$
$457$	10.6204	0.496801	0.248401	+	0.968657i	$0.420095\pi$
$458$	0	0
$459$	−4.52498	−0.211208
$460$	0	0
$461$	3.10784	0.144747	0.0723733	−	0.997378i	$-0.476943\pi$
$461$	3.10784	0.144747	0.0723733	+	0.997378i	$0.476943\pi$
$462$	0	0
$463$	−22.5563	−1.04828	−0.524141	−	0.851632i	$-0.675613\pi$
$463$	−22.5563	−1.04828	−0.524141	+	0.851632i	$0.675613\pi$
$464$	0	0
$465$	−8.38006	−0.388616
$466$	0	0
$467$	−25.7110	−1.18976	−0.594882	−	0.803813i	$-0.702801\pi$
$467$	−25.7110	−1.18976	−0.594882	+	0.803813i	$0.702801\pi$
$468$	0	0
$469$	2.57433	0.118871
$470$	0	0
$471$	−11.1411	−0.513355
$472$	0	0
$473$	−46.0998	−2.11967
$474$	0	0
$475$	−0.372829	−0.0171066
$476$	0	0
$477$	0.143988	0.00659275
$478$	0	0
$479$	35.1185	1.60460	0.802302	−	0.596918i	$-0.203608\pi$
$479$	35.1185	1.60460	0.802302	+	0.596918i	$0.203608\pi$
$480$	0	0
$481$	10.5401	0.480589
$482$	0	0
$483$	−15.6896	−0.713902
$484$	0	0
$485$	−6.62270	−0.300722
$486$	0	0
$487$	9.23237	0.418359	0.209179	−	0.977877i	$-0.432921\pi$
$487$	9.23237	0.418359	0.209179	+	0.977877i	$0.432921\pi$
$488$	0	0
$489$	−17.0059	−0.769034
$490$	0	0
$491$	10.0506	0.453577	0.226789	−	0.973944i	$-0.427177\pi$
$491$	10.0506	0.453577	0.226789	+	0.973944i	$0.427177\pi$
$492$	0	0
$493$	20.7998	0.936777
$494$	0	0
$495$	−5.06316	−0.227572
$496$	0	0
$497$	35.1835	1.57820
$498$	0	0
$499$	10.6487	0.476703	0.238352	−	0.971179i	$-0.423393\pi$
$499$	10.6487	0.476703	0.238352	+	0.971179i	$0.423393\pi$
$500$	0	0
$501$	15.2571	0.681637
$502$	0	0
$503$	−34.4118	−1.53435	−0.767174	−	0.641440i	$-0.778337\pi$
$503$	−34.4118	−1.53435	−0.767174	+	0.641440i	$0.778337\pi$
$504$	0	0
$505$	−17.0572	−0.759035
$506$	0	0
$507$	−11.3045	−0.502051
$508$	0	0
$509$	−8.28851	−0.367382	−0.183691	−	0.982984i	$-0.558805\pi$
$509$	−8.28851	−0.367382	−0.183691	+	0.982984i	$0.558805\pi$
$510$	0	0
$511$	10.1224	0.447788
$512$	0	0
$513$	−0.372829	−0.0164608
$514$	0	0
$515$	−6.74979	−0.297431
$516$	0	0
$517$	33.4699	1.47200
$518$	0	0
$519$	−17.9396	−0.787461
$520$	0	0
$521$	−2.78941	−0.122206	−0.0611030	−	0.998131i	$-0.519462\pi$
$521$	−2.78941	−0.122206	−0.0611030	+	0.998131i	$0.519462\pi$
$522$	0	0
$523$	−9.66957	−0.422821	−0.211410	−	0.977397i	$-0.567806\pi$
$523$	−9.66957	−0.422821	−0.211410	+	0.977397i	$0.567806\pi$
$524$	0	0
$525$	2.57433	0.112353
$526$	0	0
$527$	37.9196	1.65180
$528$	0	0
$529$	14.1446	0.614982
$530$	0	0
$531$	−9.34102	−0.405366
$532$	0	0
$533$	7.09017	0.307109
$534$	0	0
$535$	9.12185	0.394372
$536$	0	0
$537$	22.9072	0.988518
$538$	0	0
$539$	1.88769	0.0813087
$540$	0	0
$541$	10.9178	0.469395	0.234697	−	0.972069i	$-0.424590\pi$
$541$	10.9178	0.469395	0.234697	+	0.972069i	$0.424590\pi$
$542$	0	0
$543$	25.9439	1.11336
$544$	0	0
$545$	14.1096	0.604390
$546$	0	0
$547$	−22.3545	−0.955811	−0.477906	−	0.878411i	$-0.658604\pi$
$547$	−22.3545	−0.955811	−0.477906	+	0.878411i	$0.658604\pi$
$548$	0	0
$549$	6.21628	0.265305
$550$	0	0
$551$	1.71377	0.0730092
$552$	0	0
$553$	11.4002	0.484787
$554$	0	0
$555$	−8.09464	−0.343598
$556$	0	0
$557$	−7.73632	−0.327798	−0.163899	−	0.986477i	$-0.552407\pi$
$557$	−7.73632	−0.327798	−0.163899	+	0.986477i	$0.552407\pi$
$558$	0	0
$559$	−11.8557	−0.501442
$560$	0	0
$561$	22.9107	0.967290
$562$	0	0
$563$	−0.00564685	−0.000237986 0	−0.000118993	−	1.00000i	$-0.500038\pi$
$563$	−0.00564685	−0.000237986 0	−0.000118993		1.00000i	$0.500038\pi$
$564$	0	0
$565$	18.5441	0.780156
$566$	0	0
$567$	2.57433	0.108112
$568$	0	0
$569$	40.7994	1.71040	0.855200	−	0.518298i	$-0.173434\pi$
$569$	40.7994	1.71040	0.855200	+	0.518298i	$0.173434\pi$
$570$	0	0
$571$	24.2819	1.01617	0.508084	−	0.861308i	$-0.330354\pi$
$571$	24.2819	1.01617	0.508084	+	0.861308i	$0.330354\pi$
$572$	0	0
$573$	2.55454	0.106718
$574$	0	0
$575$	−6.09464	−0.254164
$576$	0	0
$577$	−32.8865	−1.36908	−0.684541	−	0.728974i	$-0.739997\pi$
$577$	−32.8865	−1.36908	−0.684541	+	0.728974i	$0.739997\pi$
$578$	0	0
$579$	14.4032	0.598577
$580$	0	0
$581$	1.55955	0.0647009
$582$	0	0
$583$	−0.729034	−0.0301935
$584$	0	0
$585$	−1.30211	−0.0538358
$586$	0	0
$587$	25.4277	1.04951	0.524756	−	0.851253i	$-0.324157\pi$
$587$	25.4277	1.04951	0.524756	+	0.851253i	$0.324157\pi$
$588$	0	0
$589$	3.12433	0.128736
$590$	0	0
$591$	17.4775	0.718926
$592$	0	0
$593$	−12.9267	−0.530837	−0.265419	−	0.964133i	$-0.585510\pi$
$593$	−12.9267	−0.530837	−0.265419	+	0.964133i	$0.585510\pi$
$594$	0	0
$595$	−11.6488	−0.477553
$596$	0	0
$597$	−7.74744	−0.317082
$598$	0	0
$599$	−6.82184	−0.278733	−0.139366	−	0.990241i	$-0.544507\pi$
$599$	−6.82184	−0.278733	−0.139366	+	0.990241i	$0.544507\pi$
$600$	0	0
$601$	−30.3127	−1.23648	−0.618240	−	0.785989i	$-0.712154\pi$
$601$	−30.3127	−1.23648	−0.618240	+	0.785989i	$0.712154\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	14.6356	0.595021
$606$	0	0
$607$	−15.4184	−0.625815	−0.312907	−	0.949784i	$-0.601303\pi$
$607$	−15.4184	−0.625815	−0.312907	+	0.949784i	$0.601303\pi$
$608$	0	0
$609$	−11.8333	−0.479511
$610$	0	0
$611$	8.60759	0.348226
$612$	0	0
$613$	7.28680	0.294311	0.147155	−	0.989113i	$-0.452988\pi$
$613$	7.28680	0.294311	0.147155	+	0.989113i	$0.452988\pi$
$614$	0	0
$615$	−5.44512	−0.219569
$616$	0	0
$617$	−12.6148	−0.507853	−0.253927	−	0.967223i	$-0.581722\pi$
$617$	−12.6148	−0.507853	−0.253927	+	0.967223i	$0.581722\pi$
$618$	0	0
$619$	9.50093	0.381875	0.190937	−	0.981602i	$-0.438847\pi$
$619$	9.50093	0.381875	0.190937	+	0.981602i	$0.438847\pi$
$620$	0	0
$621$	−6.09464	−0.244569
$622$	0	0
$623$	−20.2116	−0.809761
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.88769	0.0753872
$628$	0	0
$629$	36.6280	1.46046
$630$	0	0
$631$	−2.85319	−0.113584	−0.0567919	−	0.998386i	$-0.518087\pi$
$631$	−2.85319	−0.113584	−0.0567919	+	0.998386i	$0.518087\pi$
$632$	0	0
$633$	−14.5573	−0.578601
$634$	0	0
$635$	20.2326	0.802908
$636$	0	0
$637$	0.485466	0.0192349
$638$	0	0
$639$	13.6671	0.540660
$640$	0	0
$641$	−31.2601	−1.23470	−0.617351	−	0.786688i	$-0.711794\pi$
$641$	−31.2601	−1.23470	−0.617351	+	0.786688i	$0.711794\pi$
$642$	0	0
$643$	33.5770	1.32415	0.662074	−	0.749438i	$-0.269676\pi$
$643$	33.5770	1.32415	0.662074	+	0.749438i	$0.269676\pi$
$644$	0	0
$645$	9.10495	0.358507
$646$	0	0
$647$	7.96977	0.313324	0.156662	−	0.987652i	$-0.449927\pi$
$647$	7.96977	0.313324	0.156662	+	0.987652i	$0.449927\pi$
$648$	0	0
$649$	47.2951	1.85649
$650$	0	0
$651$	−21.5730	−0.845513
$652$	0	0
$653$	−40.6214	−1.58964	−0.794819	−	0.606846i	$-0.792434\pi$
$653$	−40.6214	−1.58964	−0.794819	+	0.606846i	$0.792434\pi$
$654$	0	0
$655$	1.75170	0.0684448
$656$	0	0
$657$	3.93204	0.153404
$658$	0	0
$659$	39.0131	1.51974	0.759868	−	0.650077i	$-0.225263\pi$
$659$	39.0131	1.51974	0.759868	+	0.650077i	$0.225263\pi$
$660$	0	0
$661$	42.4171	1.64983	0.824917	−	0.565253i	$-0.191222\pi$
$661$	42.4171	1.64983	0.824917	+	0.565253i	$0.191222\pi$
$662$	0	0
$663$	5.89203	0.228828
$664$	0	0
$665$	−0.959785	−0.0372189
$666$	0	0
$667$	28.0150	1.08475
$668$	0	0
$669$	22.3743	0.865040
$670$	0	0
$671$	−31.4740	−1.21504
$672$	0	0
$673$	−18.4187	−0.709988	−0.354994	−	0.934869i	$-0.615517\pi$
$673$	−18.4187	−0.709988	−0.354994	+	0.934869i	$0.615517\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−32.4174	−1.24590	−0.622951	−	0.782261i	$-0.714066\pi$
$677$	−32.4174	−1.24590	−0.622951	+	0.782261i	$0.714066\pi$
$678$	0	0
$679$	−17.0490	−0.654281
$680$	0	0
$681$	21.1483	0.810406
$682$	0	0
$683$	−16.8551	−0.644942	−0.322471	−	0.946579i	$-0.604513\pi$
$683$	−16.8551	−0.644942	−0.322471	+	0.946579i	$0.604513\pi$
$684$	0	0
$685$	5.29923	0.202473
$686$	0	0
$687$	−4.25243	−0.162240
$688$	0	0
$689$	−0.187489	−0.00714275
$690$	0	0
$691$	5.55016	0.211138	0.105569	−	0.994412i	$-0.466334\pi$
$691$	5.55016	0.211138	0.105569	+	0.994412i	$0.466334\pi$
$692$	0	0
$693$	−13.0342	−0.495130
$694$	0	0
$695$	−17.9663	−0.681500
$696$	0	0
$697$	24.6391	0.933271
$698$	0	0
$699$	−24.8578	−0.940210
$700$	0	0
$701$	−34.3510	−1.29742	−0.648710	−	0.761036i	$-0.724691\pi$
$701$	−34.3510	−1.29742	−0.648710	+	0.761036i	$0.724691\pi$
$702$	0	0
$703$	3.01792	0.113823
$704$	0	0
$705$	−6.61048	−0.248965
$706$	0	0
$707$	−43.9108	−1.65144
$708$	0	0
$709$	8.75631	0.328850	0.164425	−	0.986390i	$-0.447423\pi$
$709$	8.75631	0.328850	0.164425	+	0.986390i	$0.447423\pi$
$710$	0	0
$711$	4.42843	0.166079
$712$	0	0
$713$	51.0734	1.91271
$714$	0	0
$715$	6.59281	0.246557
$716$	0	0
$717$	−12.0138	−0.448665
$718$	0	0
$719$	−20.7754	−0.774793	−0.387396	−	0.921913i	$-0.626626\pi$
$719$	−20.7754	−0.774793	−0.387396	+	0.921913i	$0.626626\pi$
$720$	0	0
$721$	−17.3762	−0.647123
$722$	0	0
$723$	−26.9245	−1.00133
$724$	0	0
$725$	−4.59667	−0.170716
$726$	0	0
$727$	24.4698	0.907535	0.453768	−	0.891120i	$-0.350080\pi$
$727$	24.4698	0.907535	0.453768	+	0.891120i	$0.350080\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−41.1997	−1.52383
$732$	0	0
$733$	−47.3763	−1.74988	−0.874942	−	0.484228i	$-0.839101\pi$
$733$	−47.3763	−1.74988	−0.874942	+	0.484228i	$0.839101\pi$
$734$	0	0
$735$	−0.372829	−0.0137520
$736$	0	0
$737$	−5.06316	−0.186504
$738$	0	0
$739$	43.6466	1.60557	0.802783	−	0.596272i	$-0.203352\pi$
$739$	43.6466	1.60557	0.802783	+	0.596272i	$0.203352\pi$
$740$	0	0
$741$	0.485466	0.0178340
$742$	0	0
$743$	−17.7961	−0.652874	−0.326437	−	0.945219i	$-0.605848\pi$
$743$	−17.7961	−0.652874	−0.326437	+	0.945219i	$0.605848\pi$
$744$	0	0
$745$	17.5604	0.643363
$746$	0	0
$747$	0.605807	0.0221653
$748$	0	0
$749$	23.4826	0.858037
$750$	0	0
$751$	16.3267	0.595770	0.297885	−	0.954602i	$-0.403719\pi$
$751$	16.3267	0.595770	0.297885	+	0.954602i	$0.403719\pi$
$752$	0	0
$753$	−7.24776	−0.264123
$754$	0	0
$755$	11.2511	0.409468
$756$	0	0
$757$	51.2890	1.86413	0.932065	−	0.362292i	$-0.118006\pi$
$757$	51.2890	1.86413	0.932065	+	0.362292i	$0.118006\pi$
$758$	0	0
$759$	30.8581	1.12008
$760$	0	0
$761$	27.8266	1.00871	0.504357	−	0.863495i	$-0.331729\pi$
$761$	27.8266	1.00871	0.504357	+	0.863495i	$0.331729\pi$
$762$	0	0
$763$	36.3228	1.31497
$764$	0	0
$765$	−4.52498	−0.163601
$766$	0	0
$767$	12.1631	0.439183
$768$	0	0
$769$	−19.4869	−0.702716	−0.351358	−	0.936241i	$-0.614280\pi$
$769$	−19.4869	−0.702716	−0.351358	+	0.936241i	$0.614280\pi$
$770$	0	0
$771$	17.5200	0.630967
$772$	0	0
$773$	−12.8209	−0.461136	−0.230568	−	0.973056i	$-0.574058\pi$
$773$	−12.8209	−0.461136	−0.230568	+	0.973056i	$0.574058\pi$
$774$	0	0
$775$	−8.38006	−0.301020
$776$	0	0
$777$	−20.8383	−0.747568
$778$	0	0
$779$	2.03010	0.0727359
$780$	0	0
$781$	−69.1985	−2.47612
$782$	0	0
$783$	−4.59667	−0.164272
$784$	0	0
$785$	−11.1411	−0.397643
$786$	0	0
$787$	32.3697	1.15386	0.576928	−	0.816795i	$-0.304251\pi$
$787$	32.3697	1.15386	0.576928	+	0.816795i	$0.304251\pi$
$788$	0	0
$789$	−24.4730	−0.871261
$790$	0	0
$791$	47.7386	1.69739
$792$	0	0
$793$	−8.09431	−0.287437
$794$	0	0
$795$	0.143988	0.00510673
$796$	0	0
$797$	16.8676	0.597483	0.298741	−	0.954334i	$-0.403433\pi$
$797$	16.8676	0.597483	0.298741	+	0.954334i	$0.403433\pi$
$798$	0	0
$799$	29.9123	1.05822
$800$	0	0
$801$	−7.85122	−0.277409
$802$	0	0
$803$	−19.9086	−0.702558
$804$	0	0
$805$	−15.6896	−0.552986
$806$	0	0
$807$	−7.73115	−0.272149
$808$	0	0
$809$	27.3027	0.959912	0.479956	−	0.877292i	$-0.340653\pi$
$809$	27.3027	0.959912	0.479956	+	0.877292i	$0.340653\pi$
$810$	0	0
$811$	48.4544	1.70146	0.850732	−	0.525599i	$-0.176159\pi$
$811$	48.4544	1.70146	0.850732	+	0.525599i	$0.176159\pi$
$812$	0	0
$813$	−26.2956	−0.922227
$814$	0	0
$815$	−17.0059	−0.595691
$816$	0	0
$817$	−3.39459	−0.118762
$818$	0	0
$819$	−3.35207	−0.117131
$820$	0	0
$821$	−31.1014	−1.08545	−0.542723	−	0.839912i	$-0.682606\pi$
$821$	−31.1014	−1.08545	−0.542723	+	0.839912i	$0.682606\pi$
$822$	0	0
$823$	1.57026	0.0547359	0.0273679	−	0.999625i	$-0.491287\pi$
$823$	1.57026	0.0547359	0.0273679	+	0.999625i	$0.491287\pi$
$824$	0	0
$825$	−5.06316	−0.176277
$826$	0	0
$827$	−46.5237	−1.61779	−0.808894	−	0.587954i	$-0.799933\pi$
$827$	−46.5237	−1.61779	−0.808894	+	0.587954i	$0.799933\pi$
$828$	0	0
$829$	−25.3503	−0.880454	−0.440227	−	0.897887i	$-0.645102\pi$
$829$	−25.3503	−0.880454	−0.440227	+	0.897887i	$0.645102\pi$
$830$	0	0
$831$	−5.81860	−0.201845
$832$	0	0
$833$	1.68704	0.0584526
$834$	0	0
$835$	15.2571	0.527994
$836$	0	0
$837$	−8.38006	−0.289657
$838$	0	0
$839$	−55.3139	−1.90965	−0.954823	−	0.297174i	$-0.903956\pi$
$839$	−55.3139	−1.90965	−0.954823	+	0.297174i	$0.903956\pi$
$840$	0	0
$841$	−7.87063	−0.271401
$842$	0	0
$843$	5.82887	0.200757
$844$	0	0
$845$	−11.3045	−0.388887
$846$	0	0
$847$	37.6768	1.29459
$848$	0	0
$849$	18.2550	0.626510
$850$	0	0
$851$	49.3339	1.69114
$852$	0	0
$853$	−14.5989	−0.499856	−0.249928	−	0.968264i	$-0.580407\pi$
$853$	−14.5989	−0.499856	−0.249928	+	0.968264i	$0.580407\pi$
$854$	0	0
$855$	−0.372829	−0.0127505
$856$	0	0
$857$	−52.9944	−1.81025	−0.905127	−	0.425140i	$-0.860225\pi$
$857$	−52.9944	−1.81025	−0.905127	+	0.425140i	$0.860225\pi$
$858$	0	0
$859$	−20.5313	−0.700518	−0.350259	−	0.936653i	$-0.613906\pi$
$859$	−20.5313	−0.700518	−0.350259	+	0.936653i	$0.613906\pi$
$860$	0	0
$861$	−14.0175	−0.477717
$862$	0	0
$863$	48.4513	1.64930	0.824651	−	0.565642i	$-0.191372\pi$
$863$	48.4513	1.64930	0.824651	+	0.565642i	$0.191372\pi$
$864$	0	0
$865$	−17.9396	−0.609965
$866$	0	0
$867$	3.47542	0.118032
$868$	0	0
$869$	−22.4219	−0.760609
$870$	0	0
$871$	−1.30211	−0.0441204
$872$	0	0
$873$	−6.62270	−0.224145
$874$	0	0
$875$	2.57433	0.0870282
$876$	0	0
$877$	−24.5558	−0.829189	−0.414595	−	0.910006i	$-0.636077\pi$
$877$	−24.5558	−0.829189	−0.414595	+	0.910006i	$0.636077\pi$
$878$	0	0
$879$	−17.3911	−0.586587
$880$	0	0
$881$	30.4769	1.02679	0.513397	−	0.858151i	$-0.328387\pi$
$881$	30.4769	1.02679	0.513397	+	0.858151i	$0.328387\pi$
$882$	0	0
$883$	51.1608	1.72170	0.860849	−	0.508860i	$-0.169933\pi$
$883$	51.1608	1.72170	0.860849	+	0.508860i	$0.169933\pi$
$884$	0	0
$885$	−9.34102	−0.313995
$886$	0	0
$887$	16.1528	0.542359	0.271180	−	0.962529i	$-0.412586\pi$
$887$	16.1528	0.542359	0.271180	+	0.962529i	$0.412586\pi$
$888$	0	0
$889$	52.0855	1.74689
$890$	0	0
$891$	−5.06316	−0.169622
$892$	0	0
$893$	2.46458	0.0824740
$894$	0	0
$895$	22.9072	0.765703
$896$	0	0
$897$	7.93591	0.264972
$898$	0	0
$899$	38.5203	1.28473
$900$	0	0
$901$	−0.651542	−0.0217060
$902$	0	0
$903$	23.4391	0.780006
$904$	0	0
$905$	25.9439	0.862406
$906$	0	0
$907$	49.8039	1.65371	0.826855	−	0.562415i	$-0.190127\pi$
$907$	49.8039	1.65371	0.826855	+	0.562415i	$0.190127\pi$
$908$	0	0
$909$	−17.0572	−0.565751
$910$	0	0
$911$	−25.2333	−0.836016	−0.418008	−	0.908443i	$-0.637272\pi$
$911$	−25.2333	−0.836016	−0.418008	+	0.908443i	$0.637272\pi$
$912$	0	0
$913$	−3.06729	−0.101513
$914$	0	0
$915$	6.21628	0.205504
$916$	0	0
$917$	4.50946	0.148916
$918$	0	0
$919$	−37.1163	−1.22435	−0.612176	−	0.790721i	$-0.709706\pi$
$919$	−37.1163	−1.22435	−0.612176	+	0.790721i	$0.709706\pi$
$920$	0	0
$921$	−3.06223	−0.100904
$922$	0	0
$923$	−17.7961	−0.585764
$924$	0	0
$925$	−8.09464	−0.266150
$926$	0	0
$927$	−6.74979	−0.221692
$928$	0	0
$929$	17.2282	0.565240	0.282620	−	0.959232i	$-0.408796\pi$
$929$	17.2282	0.565240	0.282620	+	0.959232i	$0.408796\pi$
$930$	0	0
$931$	0.139002	0.00455559
$932$	0	0
$933$	−22.6209	−0.740574
$934$	0	0
$935$	22.9107	0.749259
$936$	0	0
$937$	−8.84065	−0.288811	−0.144406	−	0.989519i	$-0.546127\pi$
$937$	−8.84065	−0.288811	−0.144406	+	0.989519i	$0.546127\pi$
$938$	0	0
$939$	−7.48810	−0.244365
$940$	0	0
$941$	36.9406	1.20423	0.602114	−	0.798410i	$-0.294325\pi$
$941$	36.9406	1.20423	0.602114	+	0.798410i	$0.294325\pi$
$942$	0	0
$943$	33.1861	1.08069
$944$	0	0
$945$	2.57433	0.0837429
$946$	0	0
$947$	−27.3195	−0.887765	−0.443883	−	0.896085i	$-0.646399\pi$
$947$	−27.3195	−0.887765	−0.443883	+	0.896085i	$0.646399\pi$
$948$	0	0
$949$	−5.11997	−0.166201
$950$	0	0
$951$	−34.6415	−1.12333
$952$	0	0
$953$	−25.1996	−0.816296	−0.408148	−	0.912916i	$-0.633825\pi$
$953$	−25.1996	−0.816296	−0.408148	+	0.912916i	$0.633825\pi$
$954$	0	0
$955$	2.55454	0.0826631
$956$	0	0
$957$	23.2737	0.752331
$958$	0	0
$959$	13.6420	0.440522
$960$	0	0
$961$	39.2253	1.26533
$962$	0	0
$963$	9.12185	0.293948
$964$	0	0
$965$	14.4032	0.463656
$966$	0	0
$967$	−45.0809	−1.44970	−0.724852	−	0.688905i	$-0.758092\pi$
$967$	−45.0809	−1.44970	−0.724852	+	0.688905i	$0.758092\pi$
$968$	0	0
$969$	1.68704	0.0541957
$970$	0	0
$971$	−15.6302	−0.501598	−0.250799	−	0.968039i	$-0.580693\pi$
$971$	−15.6302	−0.501598	−0.250799	+	0.968039i	$0.580693\pi$
$972$	0	0
$973$	−46.2511	−1.48274
$974$	0	0
$975$	−1.30211	−0.0417010
$976$	0	0
$977$	43.2542	1.38382	0.691912	−	0.721981i	$-0.256768\pi$
$977$	43.2542	1.38382	0.691912	+	0.721981i	$0.256768\pi$
$978$	0	0
$979$	39.7520	1.27048
$980$	0	0
$981$	14.1096	0.450486
$982$	0	0
$983$	50.3250	1.60512	0.802559	−	0.596573i	$-0.203471\pi$
$983$	50.3250	1.60512	0.802559	+	0.596573i	$0.203471\pi$
$984$	0	0
$985$	17.4775	0.556878
$986$	0	0
$987$	−17.0175	−0.541674
$988$	0	0
$989$	−55.4914	−1.76452
$990$	0	0
$991$	44.3166	1.40776	0.703881	−	0.710318i	$-0.251449\pi$
$991$	44.3166	1.40776	0.703881	+	0.710318i	$0.251449\pi$
$992$	0	0
$993$	18.4794	0.586425
$994$	0	0
$995$	−7.74744	−0.245610
$996$	0	0
$997$	−35.7805	−1.13318	−0.566589	−	0.824000i	$-0.691737\pi$
$997$	−35.7805	−1.13318	−0.566589	+	0.824000i	$0.691737\pi$
$998$	0	0
$999$	−8.09464	−0.256103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.d.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.d.1.4	✓	4	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.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.57433 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.57433 q^{7} +1.00000 q^{9} -5.06316 q^{11} -1.30211 q^{13} +1.00000 q^{15} -4.52498 q^{17} -0.372829 q^{19} +2.57433 q^{21} -6.09464 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.59667 q^{29} -8.38006 q^{31} -5.06316 q^{33} +2.57433 q^{35} -8.09464 q^{37} -1.30211 q^{39} -5.44512 q^{41} +9.10495 q^{43} +1.00000 q^{45} -6.61048 q^{47} -0.372829 q^{49} -4.52498 q^{51} +0.143988 q^{53} -5.06316 q^{55} -0.372829 q^{57} -9.34102 q^{59} +6.21628 q^{61} +2.57433 q^{63} -1.30211 q^{65} +1.00000 q^{67} -6.09464 q^{69} +13.6671 q^{71} +3.93204 q^{73} +1.00000 q^{75} -13.0342 q^{77} +4.42843 q^{79} +1.00000 q^{81} +0.605807 q^{83} -4.52498 q^{85} -4.59667 q^{87} -7.85122 q^{89} -3.35207 q^{91} -8.38006 q^{93} -0.372829 q^{95} -6.62270 q^{97} -5.06316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 5 q^{11} - 9 q^{13} + 4 q^{15} - 8 q^{17} - 7 q^{19} - 3 q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 9 q^{29} - 9 q^{31} - 5 q^{33} - 3 q^{35} - 20 q^{37} - 9 q^{39} - 11 q^{41} + q^{43} + 4 q^{45} - 5 q^{47} - 7 q^{49} - 8 q^{51} - 15 q^{53} - 5 q^{55} - 7 q^{57} + 7 q^{59} - 7 q^{61} - 3 q^{63} - 9 q^{65} + 4 q^{67} - 12 q^{69} - 26 q^{73} + 4 q^{75} - 15 q^{77} - 9 q^{79} + 4 q^{81} - 8 q^{83} - 8 q^{85} - 9 q^{87} + 10 q^{89} + 3 q^{91} - 9 q^{93} - 7 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 5 * q^11 - 9 * q^13 + 4 * q^15 - 8 * q^17 - 7 * q^19 - 3 * q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 9 * q^29 - 9 * q^31 - 5 * q^33 - 3 * q^35 - 20 * q^37 - 9 * q^39 - 11 * q^41 + q^43 + 4 * q^45 - 5 * q^47 - 7 * q^49 - 8 * q^51 - 15 * q^53 - 5 * q^55 - 7 * q^57 + 7 * q^59 - 7 * q^61 - 3 * q^63 - 9 * q^65 + 4 * q^67 - 12 * q^69 - 26 * q^73 + 4 * q^75 - 15 * q^77 - 9 * q^79 + 4 * q^81 - 8 * q^83 - 8 * q^85 - 9 * q^87 + 10 * q^89 + 3 * q^91 - 9 * q^93 - 7 * q^95 - 11 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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