LMFDB - Embedded newform 4020.2.a.c.1.1

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:	4.4.98117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 4x + 1 $ x^4 - x^3 - 8x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.60457$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.60457 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -2.60457 q^{7} +1.00000 q^{9} +0.988510 q^{11} -3.16772 q^{13} -1.00000 q^{15} +3.38835 q^{17} +3.78378 q^{19} -2.60457 q^{21} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.60457 q^{29} -1.55166 q^{31} +0.988510 q^{33} +2.60457 q^{35} -2.20914 q^{37} -3.16772 q^{39} -6.78378 q^{41} -5.39984 q^{43} -1.00000 q^{45} +12.8181 q^{47} -0.216218 q^{49} +3.38835 q^{51} +0.551664 q^{53} -0.988510 q^{55} +3.78378 q^{57} -15.2506 q^{59} -4.21622 q^{61} -2.60457 q^{63} +3.16772 q^{65} -1.00000 q^{67} -3.00000 q^{69} -15.1606 q^{71} +9.36537 q^{73} +1.00000 q^{75} -2.57464 q^{77} -6.16772 q^{79} +1.00000 q^{81} +10.0573 q^{83} -3.38835 q^{85} +1.60457 q^{87} -13.2894 q^{89} +8.25056 q^{91} -1.55166 q^{93} -3.78378 q^{95} -18.8710 q^{97} +0.988510 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} - 5 q^{11} - 5 q^{13} - 4 q^{15} - 8 q^{17} + 5 q^{19} + q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 5 q^{29} - q^{31} - 5 q^{33} - q^{35} + 14 q^{37} - 5 q^{39} - 17 q^{41} - 9 q^{43} - 4 q^{45} - 7 q^{47} - 11 q^{49} - 8 q^{51} - 3 q^{53} + 5 q^{55} + 5 q^{57} - 23 q^{59} - 27 q^{61} + q^{63} + 5 q^{65} - 4 q^{67} - 12 q^{69} - 20 q^{71} - 2 q^{73} + 4 q^{75} - 23 q^{77} - 17 q^{79} + 4 q^{81} + 10 q^{83} + 8 q^{85} - 5 q^{87} - 18 q^{89} - 5 q^{91} - q^{93} - 5 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 - 5 * q^11 - 5 * q^13 - 4 * q^15 - 8 * q^17 + 5 * q^19 + q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 5 * q^29 - q^31 - 5 * q^33 - q^35 + 14 * q^37 - 5 * q^39 - 17 * q^41 - 9 * q^43 - 4 * q^45 - 7 * q^47 - 11 * q^49 - 8 * q^51 - 3 * q^53 + 5 * q^55 + 5 * q^57 - 23 * q^59 - 27 * q^61 + q^63 + 5 * q^65 - 4 * q^67 - 12 * q^69 - 20 * q^71 - 2 * q^73 + 4 * q^75 - 23 * q^77 - 17 * q^79 + 4 * q^81 + 10 * q^83 + 8 * q^85 - 5 * q^87 - 18 * q^89 - 5 * q^91 - q^93 - 5 * 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
$7$	−2.60457	−0.984435	−0.492217	−	0.870472i	$-0.663813\pi$
$7$	−2.60457	−0.984435	−0.492217	+	0.870472i	$0.663813\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.988510	0.298047	0.149024	−	0.988834i	$-0.452387\pi$
$11$	0.988510	0.298047	0.149024	+	0.988834i	$0.452387\pi$
$12$	0	0
$13$	−3.16772	−0.878568	−0.439284	−	0.898348i	$-0.644768\pi$
$13$	−3.16772	−0.878568	−0.439284	+	0.898348i	$0.644768\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	3.38835	0.821796	0.410898	−	0.911681i	$-0.365215\pi$
$17$	3.38835	0.821796	0.410898	+	0.911681i	$0.365215\pi$
$18$	0	0
$19$	3.78378	0.868059	0.434030	−	0.900899i	$-0.357091\pi$
$19$	3.78378	0.868059	0.434030	+	0.900899i	$0.357091\pi$
$20$	0	0
$21$	−2.60457	−0.568364
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.60457	0.297961	0.148981	−	0.988840i	$-0.452401\pi$
$29$	1.60457	0.297961	0.148981	+	0.988840i	$0.452401\pi$
$30$	0	0
$31$	−1.55166	−0.278687	−0.139344	−	0.990244i	$-0.544499\pi$
$31$	−1.55166	−0.278687	−0.139344	+	0.990244i	$0.544499\pi$
$32$	0	0
$33$	0.988510	0.172078
$34$	0	0
$35$	2.60457	0.440253
$36$	0	0
$37$	−2.20914	−0.363180	−0.181590	−	0.983374i	$-0.558124\pi$
$37$	−2.20914	−0.363180	−0.181590	+	0.983374i	$0.558124\pi$
$38$	0	0
$39$	−3.16772	−0.507242
$40$	0	0
$41$	−6.78378	−1.05945	−0.529724	−	0.848170i	$-0.677705\pi$
$41$	−6.78378	−1.05945	−0.529724	+	0.848170i	$0.677705\pi$
$42$	0	0
$43$	−5.39984	−0.823468	−0.411734	−	0.911304i	$-0.635077\pi$
$43$	−5.39984	−0.823468	−0.411734	+	0.911304i	$0.635077\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	12.8181	1.86971	0.934857	−	0.355023i	$-0.115527\pi$
$47$	12.8181	1.86971	0.934857	+	0.355023i	$0.115527\pi$
$48$	0	0
$49$	−0.216218	−0.0308882
$50$	0	0
$51$	3.38835	0.474464
$52$	0	0
$53$	0.551664	0.0757768	0.0378884	−	0.999282i	$-0.487937\pi$
$53$	0.551664	0.0757768	0.0378884	+	0.999282i	$0.487937\pi$
$54$	0	0
$55$	−0.988510	−0.133291
$56$	0	0
$57$	3.78378	0.501174
$58$	0	0
$59$	−15.2506	−1.98545	−0.992727	−	0.120391i	$-0.961585\pi$
$59$	−15.2506	−1.98545	−0.992727	+	0.120391i	$0.961585\pi$
$60$	0	0
$61$	−4.21622	−0.539831	−0.269916	−	0.962884i	$-0.586996\pi$
$61$	−4.21622	−0.539831	−0.269916	+	0.962884i	$0.586996\pi$
$62$	0	0
$63$	−2.60457	−0.328145
$64$	0	0
$65$	3.16772	0.392908
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−15.1606	−1.79924	−0.899619	−	0.436676i	$-0.856156\pi$
$71$	−15.1606	−1.79924	−0.899619	+	0.436676i	$0.856156\pi$
$72$	0	0
$73$	9.36537	1.09613	0.548067	−	0.836434i	$-0.315364\pi$
$73$	9.36537	1.09613	0.548067	+	0.836434i	$0.315364\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.57464	−0.293408
$78$	0	0
$79$	−6.16772	−0.693923	−0.346962	−	0.937879i	$-0.612787\pi$
$79$	−6.16772	−0.693923	−0.346962	+	0.937879i	$0.612787\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.0573	1.10393	0.551967	−	0.833866i	$-0.313877\pi$
$83$	10.0573	1.10393	0.551967	+	0.833866i	$0.313877\pi$
$84$	0	0
$85$	−3.38835	−0.367518
$86$	0	0
$87$	1.60457	0.172028
$88$	0	0
$89$	−13.2894	−1.40868	−0.704339	−	0.709864i	$-0.748756\pi$
$89$	−13.2894	−1.40868	−0.704339	+	0.709864i	$0.748756\pi$
$90$	0	0
$91$	8.25056	0.864893
$92$	0	0
$93$	−1.55166	−0.160900
$94$	0	0
$95$	−3.78378	−0.388208
$96$	0	0
$97$	−18.8710	−1.91606	−0.958031	−	0.286664i	$-0.907454\pi$
$97$	−18.8710	−1.91606	−0.958031	+	0.286664i	$0.907454\pi$
$98$	0	0
$99$	0.988510	0.0993490
$100$	0	0
$101$	−9.31247	−0.926625	−0.463313	−	0.886195i	$-0.653339\pi$
$101$	−9.31247	−0.926625	−0.463313	+	0.886195i	$0.653339\pi$
$102$	0	0
$103$	1.80930	0.178275	0.0891377	−	0.996019i	$-0.471589\pi$
$103$	1.80930	0.178275	0.0891377	+	0.996019i	$0.471589\pi$
$104$	0	0
$105$	2.60457	0.254180
$106$	0	0
$107$	−8.66897	−0.838061	−0.419030	−	0.907972i	$-0.637630\pi$
$107$	−8.66897	−0.838061	−0.419030	+	0.907972i	$0.637630\pi$
$108$	0	0
$109$	4.05732	0.388621	0.194310	−	0.980940i	$-0.437753\pi$
$109$	4.05732	0.388621	0.194310	+	0.980940i	$0.437753\pi$
$110$	0	0
$111$	−2.20914	−0.209682
$112$	0	0
$113$	6.24614	0.587588	0.293794	−	0.955869i	$-0.405082\pi$
$113$	6.24614	0.587588	0.293794	+	0.955869i	$0.405082\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	−3.16772	−0.292856
$118$	0	0
$119$	−8.82520	−0.809005
$120$	0	0
$121$	−10.0228	−0.911168
$122$	0	0
$123$	−6.78378	−0.611673
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.79527	−0.248040	−0.124020	−	0.992280i	$-0.539579\pi$
$127$	−2.79527	−0.248040	−0.124020	+	0.992280i	$0.539579\pi$
$128$	0	0
$129$	−5.39984	−0.475430
$130$	0	0
$131$	13.9354	1.21754	0.608772	−	0.793345i	$-0.291663\pi$
$131$	13.9354	1.21754	0.608772	+	0.793345i	$0.291663\pi$
$132$	0	0
$133$	−9.85512	−0.854548
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−0.862203	−0.0736630	−0.0368315	−	0.999321i	$-0.511726\pi$
$137$	−0.862203	−0.0736630	−0.0368315	+	0.999321i	$0.511726\pi$
$138$	0	0
$139$	−15.7907	−1.33935	−0.669676	−	0.742653i	$-0.733567\pi$
$139$	−15.7907	−1.33935	−0.669676	+	0.742653i	$0.733567\pi$
$140$	0	0
$141$	12.8181	1.07948
$142$	0	0
$143$	−3.13133	−0.261855
$144$	0	0
$145$	−1.60457	−0.133252
$146$	0	0
$147$	−0.216218	−0.0178333
$148$	0	0
$149$	−13.7608	−1.12733	−0.563664	−	0.826004i	$-0.690609\pi$
$149$	−13.7608	−1.12733	−0.563664	+	0.826004i	$0.690609\pi$
$150$	0	0
$151$	−9.93101	−0.808174	−0.404087	−	0.914721i	$-0.632411\pi$
$151$	−9.93101	−0.808174	−0.404087	+	0.914721i	$0.632411\pi$
$152$	0	0
$153$	3.38835	0.273932
$154$	0	0
$155$	1.55166	0.124633
$156$	0	0
$157$	−2.39543	−0.191176	−0.0955881	−	0.995421i	$-0.530473\pi$
$157$	−2.39543	−0.191176	−0.0955881	+	0.995421i	$0.530473\pi$
$158$	0	0
$159$	0.551664	0.0437498
$160$	0	0
$161$	7.81371	0.615807
$162$	0	0
$163$	23.1906	1.81643	0.908213	−	0.418509i	$-0.137447\pi$
$163$	23.1906	1.81643	0.908213	+	0.418509i	$0.137447\pi$
$164$	0	0
$165$	−0.988510	−0.0769554
$166$	0	0
$167$	−13.7723	−1.06573	−0.532866	−	0.846200i	$-0.678885\pi$
$167$	−13.7723	−1.06573	−0.532866	+	0.846200i	$0.678885\pi$
$168$	0	0
$169$	−2.96553	−0.228118
$170$	0	0
$171$	3.78378	0.289353
$172$	0	0
$173$	−15.0987	−1.14794	−0.573968	−	0.818878i	$-0.694597\pi$
$173$	−15.0987	−1.14794	−0.573968	+	0.818878i	$0.694597\pi$
$174$	0	0
$175$	−2.60457	−0.196887
$176$	0	0
$177$	−15.2506	−1.14630
$178$	0	0
$179$	23.0432	1.72233	0.861163	−	0.508328i	$-0.169736\pi$
$179$	23.0432	1.72233	0.861163	+	0.508328i	$0.169736\pi$
$180$	0	0
$181$	5.06632	0.376577	0.188288	−	0.982114i	$-0.439706\pi$
$181$	5.06632	0.376577	0.188288	+	0.982114i	$0.439706\pi$
$182$	0	0
$183$	−4.21622	−0.311672
$184$	0	0
$185$	2.20914	0.162419
$186$	0	0
$187$	3.34942	0.244934
$188$	0	0
$189$	−2.60457	−0.189455
$190$	0	0
$191$	−24.1191	−1.74520	−0.872598	−	0.488439i	$-0.837567\pi$
$191$	−24.1191	−1.74520	−0.872598	+	0.488439i	$0.837567\pi$
$192$	0	0
$193$	22.3926	1.61186	0.805928	−	0.592013i	$-0.201667\pi$
$193$	22.3926	1.61186	0.805928	+	0.592013i	$0.201667\pi$
$194$	0	0
$195$	3.16772	0.226845
$196$	0	0
$197$	−13.6594	−0.973192	−0.486596	−	0.873627i	$-0.661762\pi$
$197$	−13.6594	−0.973192	−0.486596	+	0.873627i	$0.661762\pi$
$198$	0	0
$199$	16.5816	1.17544	0.587719	−	0.809065i	$-0.300026\pi$
$199$	16.5816	1.17544	0.587719	+	0.809065i	$0.300026\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	−4.17921	−0.293323
$204$	0	0
$205$	6.78378	0.473800
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	3.74031	0.258722
$210$	0	0
$211$	−17.7077	−1.21905	−0.609525	−	0.792767i	$-0.708640\pi$
$211$	−17.7077	−1.21905	−0.609525	+	0.792767i	$0.708640\pi$
$212$	0	0
$213$	−15.1606	−1.03879
$214$	0	0
$215$	5.39984	0.368266
$216$	0	0
$217$	4.04142	0.274349
$218$	0	0
$219$	9.36537	0.632853
$220$	0	0
$221$	−10.7334	−0.722004
$222$	0	0
$223$	14.0987	0.944121	0.472061	−	0.881566i	$-0.343510\pi$
$223$	14.0987	0.944121	0.472061	+	0.881566i	$0.343510\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−9.91070	−0.657796	−0.328898	−	0.944365i	$-0.606677\pi$
$227$	−9.91070	−0.657796	−0.328898	+	0.944365i	$0.606677\pi$
$228$	0	0
$229$	−10.5586	−0.697729	−0.348865	−	0.937173i	$-0.613433\pi$
$229$	−10.5586	−0.697729	−0.348865	+	0.937173i	$0.613433\pi$
$230$	0	0
$231$	−2.57464	−0.169399
$232$	0	0
$233$	15.3998	1.00888	0.504439	−	0.863448i	$-0.331699\pi$
$233$	15.3998	1.00888	0.504439	+	0.863448i	$0.331699\pi$
$234$	0	0
$235$	−12.8181	−0.836162
$236$	0	0
$237$	−6.16772	−0.400637
$238$	0	0
$239$	−22.2021	−1.43613	−0.718066	−	0.695975i	$-0.754972\pi$
$239$	−22.2021	−1.43613	−0.718066	+	0.695975i	$0.754972\pi$
$240$	0	0
$241$	24.9812	1.60918	0.804592	−	0.593828i	$-0.202384\pi$
$241$	24.9812	1.60918	0.804592	+	0.593828i	$0.202384\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.216218	0.0138136
$246$	0	0
$247$	−11.9860	−0.762649
$248$	0	0
$249$	10.0573	0.637357
$250$	0	0
$251$	−19.8639	−1.25380	−0.626901	−	0.779099i	$-0.715677\pi$
$251$	−19.8639	−1.25380	−0.626901	+	0.779099i	$0.715677\pi$
$252$	0	0
$253$	−2.96553	−0.186441
$254$	0	0
$255$	−3.38835	−0.212187
$256$	0	0
$257$	−1.66207	−0.103677	−0.0518385	−	0.998655i	$-0.516508\pi$
$257$	−1.66207	−0.103677	−0.0518385	+	0.998655i	$0.516508\pi$
$258$	0	0
$259$	5.75386	0.357527
$260$	0	0
$261$	1.60457	0.0993204
$262$	0	0
$263$	0.0159013	0.000980515 0	0.000490257	−	1.00000i	$-0.499844\pi$
$263$	0.0159013	0.000980515 0	0.000490257		1.00000i	$0.499844\pi$
$264$	0	0
$265$	−0.551664	−0.0338884
$266$	0	0
$267$	−13.2894	−0.813300
$268$	0	0
$269$	−30.0298	−1.83095	−0.915474	−	0.402376i	$-0.868184\pi$
$269$	−30.0298	−1.83095	−0.915474	+	0.402376i	$0.868184\pi$
$270$	0	0
$271$	20.1720	1.22536	0.612681	−	0.790330i	$-0.290091\pi$
$271$	20.1720	1.22536	0.612681	+	0.790330i	$0.290091\pi$
$272$	0	0
$273$	8.25056	0.499346
$274$	0	0
$275$	0.988510	0.0596094
$276$	0	0
$277$	19.6753	1.18217	0.591087	−	0.806608i	$-0.298699\pi$
$277$	19.6753	1.18217	0.591087	+	0.806608i	$0.298699\pi$
$278$	0	0
$279$	−1.55166	−0.0928957
$280$	0	0
$281$	16.1147	0.961322	0.480661	−	0.876907i	$-0.340397\pi$
$281$	16.1147	0.961322	0.480661	+	0.876907i	$0.340397\pi$
$282$	0	0
$283$	4.12190	0.245021	0.122511	−	0.992467i	$-0.460905\pi$
$283$	4.12190	0.245021	0.122511	+	0.992467i	$0.460905\pi$
$284$	0	0
$285$	−3.78378	−0.224132
$286$	0	0
$287$	17.6688	1.04296
$288$	0	0
$289$	−5.51907	−0.324651
$290$	0	0
$291$	−18.8710	−1.10624
$292$	0	0
$293$	2.55415	0.149215	0.0746075	−	0.997213i	$-0.476230\pi$
$293$	2.55415	0.149215	0.0746075	+	0.997213i	$0.476230\pi$
$294$	0	0
$295$	15.2506	0.887922
$296$	0	0
$297$	0.988510	0.0573592
$298$	0	0
$299$	9.50317	0.549582
$300$	0	0
$301$	14.0643	0.810651
$302$	0	0
$303$	−9.31247	−0.534987
$304$	0	0
$305$	4.21622	0.241420
$306$	0	0
$307$	21.4367	1.22346	0.611729	−	0.791067i	$-0.290474\pi$
$307$	21.4367	1.22346	0.611729	+	0.791067i	$0.290474\pi$
$308$	0	0
$309$	1.80930	0.102927
$310$	0	0
$311$	8.69181	0.492867	0.246434	−	0.969160i	$-0.420741\pi$
$311$	8.69181	0.492867	0.246434	+	0.969160i	$0.420741\pi$
$312$	0	0
$313$	4.11923	0.232833	0.116416	−	0.993201i	$-0.462859\pi$
$313$	4.11923	0.232833	0.116416	+	0.993201i	$0.462859\pi$
$314$	0	0
$315$	2.60457	0.146751
$316$	0	0
$317$	−10.5032	−0.589917	−0.294958	−	0.955510i	$-0.595306\pi$
$317$	−10.5032	−0.589917	−0.294958	+	0.955510i	$0.595306\pi$
$318$	0	0
$319$	1.58613	0.0888064
$320$	0	0
$321$	−8.66897	−0.483854
$322$	0	0
$323$	12.8208	0.713368
$324$	0	0
$325$	−3.16772	−0.175714
$326$	0	0
$327$	4.05732	0.224370
$328$	0	0
$329$	−33.3857	−1.84061
$330$	0	0
$331$	35.0457	1.92629	0.963143	−	0.268991i	$-0.0866900\pi$
$331$	35.0457	1.92629	0.963143	+	0.268991i	$0.0866900\pi$
$332$	0	0
$333$	−2.20914	−0.121060
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	−21.1580	−1.15255	−0.576274	−	0.817256i	$-0.695494\pi$
$337$	−21.1580	−1.15255	−0.576274	+	0.817256i	$0.695494\pi$
$338$	0	0
$339$	6.24614	0.339244
$340$	0	0
$341$	−1.53384	−0.0830618
$342$	0	0
$343$	18.7951	1.01484
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	4.95404	0.265947	0.132973	−	0.991120i	$-0.457548\pi$
$347$	4.95404	0.265947	0.132973	+	0.991120i	$0.457548\pi$
$348$	0	0
$349$	4.16313	0.222847	0.111424	−	0.993773i	$-0.464459\pi$
$349$	4.16313	0.222847	0.111424	+	0.993773i	$0.464459\pi$
$350$	0	0
$351$	−3.16772	−0.169081
$352$	0	0
$353$	13.7723	0.733025	0.366513	−	0.930413i	$-0.380552\pi$
$353$	13.7723	0.733025	0.366513	+	0.930413i	$0.380552\pi$
$354$	0	0
$355$	15.1606	0.804643
$356$	0	0
$357$	−8.82520	−0.467079
$358$	0	0
$359$	−25.9398	−1.36905	−0.684526	−	0.728988i	$-0.739991\pi$
$359$	−25.9398	−1.36905	−0.684526	+	0.728988i	$0.739991\pi$
$360$	0	0
$361$	−4.68299	−0.246473
$362$	0	0
$363$	−10.0228	−0.526063
$364$	0	0
$365$	−9.36537	−0.490206
$366$	0	0
$367$	−8.38394	−0.437638	−0.218819	−	0.975765i	$-0.570220\pi$
$367$	−8.38394	−0.437638	−0.218819	+	0.975765i	$0.570220\pi$
$368$	0	0
$369$	−6.78378	−0.353150
$370$	0	0
$371$	−1.43685	−0.0745974
$372$	0	0
$373$	−13.3487	−0.691168	−0.345584	−	0.938388i	$-0.612319\pi$
$373$	−13.3487	−0.691168	−0.345584	+	0.938388i	$0.612319\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−5.08283	−0.261779
$378$	0	0
$379$	5.33357	0.273967	0.136984	−	0.990573i	$-0.456259\pi$
$379$	5.33357	0.273967	0.136984	+	0.990573i	$0.456259\pi$
$380$	0	0
$381$	−2.79527	−0.143206
$382$	0	0
$383$	−9.20206	−0.470203	−0.235102	−	0.971971i	$-0.575542\pi$
$383$	−9.20206	−0.470203	−0.235102	+	0.971971i	$0.575542\pi$
$384$	0	0
$385$	2.57464	0.131216
$386$	0	0
$387$	−5.39984	−0.274489
$388$	0	0
$389$	0.630084	0.0319465	0.0159733	−	0.999872i	$-0.494915\pi$
$389$	0.630084	0.0319465	0.0159733	+	0.999872i	$0.494915\pi$
$390$	0	0
$391$	−10.1651	−0.514069
$392$	0	0
$393$	13.9354	0.702949
$394$	0	0
$395$	6.16772	0.310332
$396$	0	0
$397$	−38.2222	−1.91832	−0.959160	−	0.282865i	$-0.908715\pi$
$397$	−38.2222	−1.91832	−0.959160	+	0.282865i	$0.908715\pi$
$398$	0	0
$399$	−9.85512	−0.493373
$400$	0	0
$401$	10.6320	0.530935	0.265467	−	0.964120i	$-0.414474\pi$
$401$	10.6320	0.530935	0.265467	+	0.964120i	$0.414474\pi$
$402$	0	0
$403$	4.91524	0.244846
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−2.18376	−0.108245
$408$	0	0
$409$	8.61859	0.426162	0.213081	−	0.977035i	$-0.431650\pi$
$409$	8.61859	0.426162	0.213081	+	0.977035i	$0.431650\pi$
$410$	0	0
$411$	−0.862203	−0.0425293
$412$	0	0
$413$	39.7211	1.95455
$414$	0	0
$415$	−10.0573	−0.493694
$416$	0	0
$417$	−15.7907	−0.773275
$418$	0	0
$419$	36.5304	1.78463	0.892313	−	0.451417i	$-0.149081\pi$
$419$	36.5304	1.78463	0.892313	+	0.451417i	$0.149081\pi$
$420$	0	0
$421$	3.43436	0.167381	0.0836903	−	0.996492i	$-0.473329\pi$
$421$	3.43436	0.167381	0.0836903	+	0.996492i	$0.473329\pi$
$422$	0	0
$423$	12.8181	0.623238
$424$	0	0
$425$	3.38835	0.164359
$426$	0	0
$427$	10.9814	0.531429
$428$	0	0
$429$	−3.13133	−0.151182
$430$	0	0
$431$	−2.96801	−0.142964	−0.0714821	−	0.997442i	$-0.522773\pi$
$431$	−2.96801	−0.142964	−0.0714821	+	0.997442i	$0.522773\pi$
$432$	0	0
$433$	−39.5673	−1.90148	−0.950740	−	0.309988i	$-0.899675\pi$
$433$	−39.5673	−1.90148	−0.950740	+	0.309988i	$0.899675\pi$
$434$	0	0
$435$	−1.60457	−0.0769332
$436$	0	0
$437$	−11.3513	−0.543009
$438$	0	0
$439$	33.8455	1.61536	0.807679	−	0.589622i	$-0.200723\pi$
$439$	33.8455	1.61536	0.807679	+	0.589622i	$0.200723\pi$
$440$	0	0
$441$	−0.216218	−0.0102961
$442$	0	0
$443$	4.22958	0.200954	0.100477	−	0.994939i	$-0.467963\pi$
$443$	4.22958	0.200954	0.100477	+	0.994939i	$0.467963\pi$
$444$	0	0
$445$	13.2894	0.629980
$446$	0	0
$447$	−13.7608	−0.650864
$448$	0	0
$449$	−3.25069	−0.153409	−0.0767047	−	0.997054i	$-0.524440\pi$
$449$	−3.25069	−0.153409	−0.0767047	+	0.997054i	$0.524440\pi$
$450$	0	0
$451$	−6.70584	−0.315766
$452$	0	0
$453$	−9.93101	−0.466600
$454$	0	0
$455$	−8.25056	−0.386792
$456$	0	0
$457$	32.6611	1.52782	0.763912	−	0.645320i	$-0.223276\pi$
$457$	32.6611	1.52782	0.763912	+	0.645320i	$0.223276\pi$
$458$	0	0
$459$	3.38835	0.158155
$460$	0	0
$461$	29.0660	1.35374	0.676869	−	0.736103i	$-0.263336\pi$
$461$	29.0660	1.35374	0.676869	+	0.736103i	$0.263336\pi$
$462$	0	0
$463$	8.29398	0.385454	0.192727	−	0.981252i	$-0.438267\pi$
$463$	8.29398	0.385454	0.192727	+	0.981252i	$0.438267\pi$
$464$	0	0
$465$	1.55166	0.0719567
$466$	0	0
$467$	12.1951	0.564323	0.282161	−	0.959367i	$-0.408949\pi$
$467$	12.1951	0.564323	0.282161	+	0.959367i	$0.408949\pi$
$468$	0	0
$469$	2.60457	0.120268
$470$	0	0
$471$	−2.39543	−0.110376
$472$	0	0
$473$	−5.33780	−0.245432
$474$	0	0
$475$	3.78378	0.173612
$476$	0	0
$477$	0.551664	0.0252589
$478$	0	0
$479$	0.400450	0.0182970	0.00914851	−	0.999958i	$-0.497088\pi$
$479$	0.400450	0.0182970	0.00914851	+	0.999958i	$0.497088\pi$
$480$	0	0
$481$	6.99794	0.319079
$482$	0	0
$483$	7.81371	0.355536
$484$	0	0
$485$	18.8710	0.856889
$486$	0	0
$487$	6.95151	0.315003	0.157501	−	0.987519i	$-0.449656\pi$
$487$	6.95151	0.315003	0.157501	+	0.987519i	$0.449656\pi$
$488$	0	0
$489$	23.1906	1.04871
$490$	0	0
$491$	−26.1994	−1.18236	−0.591181	−	0.806539i	$-0.701338\pi$
$491$	−26.1994	−1.18236	−0.591181	+	0.806539i	$0.701338\pi$
$492$	0	0
$493$	5.43685	0.244863
$494$	0	0
$495$	−0.988510	−0.0444302
$496$	0	0
$497$	39.4870	1.77123
$498$	0	0
$499$	30.1287	1.34874	0.674372	−	0.738392i	$-0.264414\pi$
$499$	30.1287	1.34874	0.674372	+	0.738392i	$0.264414\pi$
$500$	0	0
$501$	−13.7723	−0.615301
$502$	0	0
$503$	−43.5922	−1.94368	−0.971839	−	0.235645i	$-0.924280\pi$
$503$	−43.5922	−1.94368	−0.971839	+	0.235645i	$0.924280\pi$
$504$	0	0
$505$	9.31247	0.414399
$506$	0	0
$507$	−2.96553	−0.131704
$508$	0	0
$509$	−3.51273	−0.155699	−0.0778496	−	0.996965i	$-0.524805\pi$
$509$	−3.51273	−0.155699	−0.0778496	+	0.996965i	$0.524805\pi$
$510$	0	0
$511$	−24.3928	−1.07907
$512$	0	0
$513$	3.78378	0.167058
$514$	0	0
$515$	−1.80930	−0.0797272
$516$	0	0
$517$	12.6708	0.557263
$518$	0	0
$519$	−15.0987	−0.662761
$520$	0	0
$521$	29.5374	1.29406	0.647028	−	0.762466i	$-0.276012\pi$
$521$	29.5374	1.29406	0.647028	+	0.762466i	$0.276012\pi$
$522$	0	0
$523$	16.0529	0.701945	0.350972	−	0.936386i	$-0.385851\pi$
$523$	16.0529	0.701945	0.350972	+	0.936386i	$0.385851\pi$
$524$	0	0
$525$	−2.60457	−0.113673
$526$	0	0
$527$	−5.25758	−0.229024
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−15.2506	−0.661818
$532$	0	0
$533$	21.4891	0.930798
$534$	0	0
$535$	8.66897	0.374792
$536$	0	0
$537$	23.0432	0.994386
$538$	0	0
$539$	−0.213733	−0.00920614
$540$	0	0
$541$	−24.9877	−1.07431	−0.537153	−	0.843485i	$-0.680500\pi$
$541$	−24.9877	−1.07431	−0.537153	+	0.843485i	$0.680500\pi$
$542$	0	0
$543$	5.06632	0.217417
$544$	0	0
$545$	−4.05732	−0.173796
$546$	0	0
$547$	−22.5101	−0.962463	−0.481232	−	0.876594i	$-0.659810\pi$
$547$	−22.5101	−0.962463	−0.481232	+	0.876594i	$0.659810\pi$
$548$	0	0
$549$	−4.21622	−0.179944
$550$	0	0
$551$	6.07134	0.258648
$552$	0	0
$553$	16.0643	0.683122
$554$	0	0
$555$	2.20914	0.0937727
$556$	0	0
$557$	−7.92472	−0.335781	−0.167891	−	0.985806i	$-0.553696\pi$
$557$	−7.92472	−0.335781	−0.167891	+	0.985806i	$0.553696\pi$
$558$	0	0
$559$	17.1052	0.723473
$560$	0	0
$561$	3.34942	0.141413
$562$	0	0
$563$	−38.3210	−1.61504	−0.807518	−	0.589843i	$-0.799190\pi$
$563$	−38.3210	−1.61504	−0.807518	+	0.589843i	$0.799190\pi$
$564$	0	0
$565$	−6.24614	−0.262777
$566$	0	0
$567$	−2.60457	−0.109382
$568$	0	0
$569$	−12.5632	−0.526675	−0.263337	−	0.964704i	$-0.584823\pi$
$569$	−12.5632	−0.526675	−0.263337	+	0.964704i	$0.584823\pi$
$570$	0	0
$571$	25.6435	1.07315	0.536573	−	0.843854i	$-0.319719\pi$
$571$	25.6435	1.07315	0.536573	+	0.843854i	$0.319719\pi$
$572$	0	0
$573$	−24.1191	−1.00759
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	−13.3010	−0.553727	−0.276863	−	0.960909i	$-0.589295\pi$
$577$	−13.3010	−0.553727	−0.276863	+	0.960909i	$0.589295\pi$
$578$	0	0
$579$	22.3926	0.930606
$580$	0	0
$581$	−26.1950	−1.08675
$582$	0	0
$583$	0.545325	0.0225851
$584$	0	0
$585$	3.16772	0.130969
$586$	0	0
$587$	−5.69121	−0.234901	−0.117451	−	0.993079i	$-0.537472\pi$
$587$	−5.69121	−0.234901	−0.117451	+	0.993079i	$0.537472\pi$
$588$	0	0
$589$	−5.87116	−0.241917
$590$	0	0
$591$	−13.6594	−0.561873
$592$	0	0
$593$	1.26471	0.0519355	0.0259677	−	0.999663i	$-0.491733\pi$
$593$	1.26471	0.0519355	0.0259677	+	0.999663i	$0.491733\pi$
$594$	0	0
$595$	8.82520	0.361798
$596$	0	0
$597$	16.5816	0.678639
$598$	0	0
$599$	14.4597	0.590807	0.295404	−	0.955373i	$-0.404546\pi$
$599$	14.4597	0.590807	0.295404	+	0.955373i	$0.404546\pi$
$600$	0	0
$601$	−30.6268	−1.24929	−0.624647	−	0.780907i	$-0.714757\pi$
$601$	−30.6268	−1.24929	−0.624647	+	0.780907i	$0.714757\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	10.0228	0.407487
$606$	0	0
$607$	−20.2301	−0.821113	−0.410556	−	0.911835i	$-0.634666\pi$
$607$	−20.2301	−0.821113	−0.410556	+	0.911835i	$0.634666\pi$
$608$	0	0
$609$	−4.17921	−0.169350
$610$	0	0
$611$	−40.6043	−1.64267
$612$	0	0
$613$	18.6230	0.752174	0.376087	−	0.926584i	$-0.377269\pi$
$613$	18.6230	0.752174	0.376087	+	0.926584i	$0.377269\pi$
$614$	0	0
$615$	6.78378	0.273549
$616$	0	0
$617$	−24.6071	−0.990645	−0.495322	−	0.868709i	$-0.664950\pi$
$617$	−24.6071	−0.990645	−0.495322	+	0.868709i	$0.664950\pi$
$618$	0	0
$619$	−14.7925	−0.594560	−0.297280	−	0.954790i	$-0.596079\pi$
$619$	−14.7925	−0.594560	−0.297280	+	0.954790i	$0.596079\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	0	0
$623$	34.6133	1.38675
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.74031	0.149373
$628$	0	0
$629$	−7.48534	−0.298460
$630$	0	0
$631$	−5.86680	−0.233554	−0.116777	−	0.993158i	$-0.537256\pi$
$631$	−5.86680	−0.233554	−0.116777	+	0.993158i	$0.537256\pi$
$632$	0	0
$633$	−17.7077	−0.703818
$634$	0	0
$635$	2.79527	0.110927
$636$	0	0
$637$	0.684917	0.0271374
$638$	0	0
$639$	−15.1606	−0.599746
$640$	0	0
$641$	−35.2196	−1.39109	−0.695545	−	0.718483i	$-0.744837\pi$
$641$	−35.2196	−1.39109	−0.695545	+	0.718483i	$0.744837\pi$
$642$	0	0
$643$	19.4643	0.767597	0.383798	−	0.923417i	$-0.374616\pi$
$643$	19.4643	0.767597	0.383798	+	0.923417i	$0.374616\pi$
$644$	0	0
$645$	5.39984	0.212619
$646$	0	0
$647$	−33.5074	−1.31731	−0.658657	−	0.752443i	$-0.728875\pi$
$647$	−33.5074	−1.31731	−0.658657	+	0.752443i	$0.728875\pi$
$648$	0	0
$649$	−15.0753	−0.591758
$650$	0	0
$651$	4.04142	0.158396
$652$	0	0
$653$	32.0528	1.25432	0.627161	−	0.778890i	$-0.284217\pi$
$653$	32.0528	1.25432	0.627161	+	0.778890i	$0.284217\pi$
$654$	0	0
$655$	−13.9354	−0.544502
$656$	0	0
$657$	9.36537	0.365378
$658$	0	0
$659$	−15.1332	−0.589506	−0.294753	−	0.955573i	$-0.595237\pi$
$659$	−15.1332	−0.589506	−0.294753	+	0.955573i	$0.595237\pi$
$660$	0	0
$661$	−36.5904	−1.42320	−0.711601	−	0.702584i	$-0.752030\pi$
$661$	−36.5904	−1.42320	−0.711601	+	0.702584i	$0.752030\pi$
$662$	0	0
$663$	−10.7334	−0.416849
$664$	0	0
$665$	9.85512	0.382165
$666$	0	0
$667$	−4.81371	−0.186388
$668$	0	0
$669$	14.0987	0.545089
$670$	0	0
$671$	−4.16777	−0.160895
$672$	0	0
$673$	−43.5851	−1.68008	−0.840041	−	0.542523i	$-0.817469\pi$
$673$	−43.5851	−1.68008	−0.840041	+	0.542523i	$0.817469\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−23.4704	−0.902040	−0.451020	−	0.892514i	$-0.648940\pi$
$677$	−23.4704	−0.902040	−0.451020	+	0.892514i	$0.648940\pi$
$678$	0	0
$679$	49.1509	1.88624
$680$	0	0
$681$	−9.91070	−0.379779
$682$	0	0
$683$	−1.09813	−0.0420186	−0.0210093	−	0.999779i	$-0.506688\pi$
$683$	−1.09813	−0.0420186	−0.0210093	+	0.999779i	$0.506688\pi$
$684$	0	0
$685$	0.862203	0.0329431
$686$	0	0
$687$	−10.5586	−0.402834
$688$	0	0
$689$	−1.74752	−0.0665751
$690$	0	0
$691$	22.8024	0.867442	0.433721	−	0.901047i	$-0.357200\pi$
$691$	22.8024	0.867442	0.433721	+	0.901047i	$0.357200\pi$
$692$	0	0
$693$	−2.57464	−0.0978026
$694$	0	0
$695$	15.7907	0.598976
$696$	0	0
$697$	−22.9858	−0.870651
$698$	0	0
$699$	15.3998	0.582476
$700$	0	0
$701$	23.4131	0.884300	0.442150	−	0.896941i	$-0.354216\pi$
$701$	23.4131	0.884300	0.442150	+	0.896941i	$0.354216\pi$
$702$	0	0
$703$	−8.35890	−0.315262
$704$	0	0
$705$	−12.8181	−0.482758
$706$	0	0
$707$	24.2550	0.912202
$708$	0	0
$709$	−4.17541	−0.156811	−0.0784054	−	0.996922i	$-0.524983\pi$
$709$	−4.17541	−0.156811	−0.0784054	+	0.996922i	$0.524983\pi$
$710$	0	0
$711$	−6.16772	−0.231308
$712$	0	0
$713$	4.65499	0.174331
$714$	0	0
$715$	3.13133	0.117105
$716$	0	0
$717$	−22.2021	−0.829151
$718$	0	0
$719$	44.7511	1.66893	0.834466	−	0.551059i	$-0.185776\pi$
$719$	44.7511	1.66893	0.834466	+	0.551059i	$0.185776\pi$
$720$	0	0
$721$	−4.71244	−0.175500
$722$	0	0
$723$	24.9812	0.929063
$724$	0	0
$725$	1.60457	0.0595922
$726$	0	0
$727$	41.4156	1.53602	0.768010	−	0.640438i	$-0.221247\pi$
$727$	41.4156	1.53602	0.768010	+	0.640438i	$0.221247\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−18.2966	−0.676723
$732$	0	0
$733$	4.25763	0.157259	0.0786296	−	0.996904i	$-0.474946\pi$
$733$	4.25763	0.157259	0.0786296	+	0.996904i	$0.474946\pi$
$734$	0	0
$735$	0.216218	0.00797530
$736$	0	0
$737$	−0.988510	−0.0364122
$738$	0	0
$739$	27.1173	0.997526	0.498763	−	0.866738i	$-0.333788\pi$
$739$	27.1173	0.997526	0.498763	+	0.866738i	$0.333788\pi$
$740$	0	0
$741$	−11.9860	−0.440316
$742$	0	0
$743$	18.5700	0.681266	0.340633	−	0.940196i	$-0.389359\pi$
$743$	18.5700	0.681266	0.340633	+	0.940196i	$0.389359\pi$
$744$	0	0
$745$	13.7608	0.504157
$746$	0	0
$747$	10.0573	0.367978
$748$	0	0
$749$	22.5789	0.825016
$750$	0	0
$751$	17.1051	0.624173	0.312086	−	0.950054i	$-0.398972\pi$
$751$	17.1051	0.624173	0.312086	+	0.950054i	$0.398972\pi$
$752$	0	0
$753$	−19.8639	−0.723882
$754$	0	0
$755$	9.93101	0.361426
$756$	0	0
$757$	38.0254	1.38206	0.691028	−	0.722828i	$-0.257158\pi$
$757$	38.0254	1.38206	0.691028	+	0.722828i	$0.257158\pi$
$758$	0	0
$759$	−2.96553	−0.107642
$760$	0	0
$761$	−21.0088	−0.761569	−0.380785	−	0.924664i	$-0.624346\pi$
$761$	−21.0088	−0.761569	−0.380785	+	0.924664i	$0.624346\pi$
$762$	0	0
$763$	−10.5676	−0.382572
$764$	0	0
$765$	−3.38835	−0.122506
$766$	0	0
$767$	48.3095	1.74436
$768$	0	0
$769$	22.1785	0.799776	0.399888	−	0.916564i	$-0.369049\pi$
$769$	22.1785	0.799776	0.399888	+	0.916564i	$0.369049\pi$
$770$	0	0
$771$	−1.66207	−0.0598580
$772$	0	0
$773$	17.2754	0.621353	0.310677	−	0.950516i	$-0.399444\pi$
$773$	17.2754	0.621353	0.310677	+	0.950516i	$0.399444\pi$
$774$	0	0
$775$	−1.55166	−0.0557374
$776$	0	0
$777$	5.75386	0.206418
$778$	0	0
$779$	−25.6684	−0.919664
$780$	0	0
$781$	−14.9865	−0.536257
$782$	0	0
$783$	1.60457	0.0573426
$784$	0	0
$785$	2.39543	0.0854966
$786$	0	0
$787$	−38.9512	−1.38846	−0.694231	−	0.719752i	$-0.744255\pi$
$787$	−38.9512	−1.38846	−0.694231	+	0.719752i	$0.744255\pi$
$788$	0	0
$789$	0.0159013	0.000566101 0
$790$	0	0
$791$	−16.2685	−0.578442
$792$	0	0
$793$	13.3558	0.474279
$794$	0	0
$795$	−0.551664	−0.0195655
$796$	0	0
$797$	11.1764	0.395889	0.197944	−	0.980213i	$-0.436573\pi$
$797$	11.1764	0.395889	0.197944	+	0.980213i	$0.436573\pi$
$798$	0	0
$799$	43.4323	1.53652
$800$	0	0
$801$	−13.2894	−0.469559
$802$	0	0
$803$	9.25777	0.326699
$804$	0	0
$805$	−7.81371	−0.275397
$806$	0	0
$807$	−30.0298	−1.05710
$808$	0	0
$809$	−54.4501	−1.91436	−0.957182	−	0.289486i	$-0.906516\pi$
$809$	−54.4501	−1.91436	−0.957182	+	0.289486i	$0.906516\pi$
$810$	0	0
$811$	39.3787	1.38277	0.691386	−	0.722486i	$-0.257000\pi$
$811$	39.3787	1.38277	0.691386	+	0.722486i	$0.257000\pi$
$812$	0	0
$813$	20.1720	0.707463
$814$	0	0
$815$	−23.1906	−0.812330
$816$	0	0
$817$	−20.4318	−0.714819
$818$	0	0
$819$	8.25056	0.288298
$820$	0	0
$821$	11.6186	0.405492	0.202746	−	0.979231i	$-0.435013\pi$
$821$	11.6186	0.405492	0.202746	+	0.979231i	$0.435013\pi$
$822$	0	0
$823$	47.1555	1.64374	0.821869	−	0.569677i	$-0.192932\pi$
$823$	47.1555	1.64374	0.821869	+	0.569677i	$0.192932\pi$
$824$	0	0
$825$	0.988510	0.0344155
$826$	0	0
$827$	2.00703	0.0697912	0.0348956	−	0.999391i	$-0.488890\pi$
$827$	2.00703	0.0697912	0.0348956	+	0.999391i	$0.488890\pi$
$828$	0	0
$829$	−36.7025	−1.27473	−0.637365	−	0.770562i	$-0.719976\pi$
$829$	−36.7025	−1.27473	−0.637365	+	0.770562i	$0.719976\pi$
$830$	0	0
$831$	19.6753	0.682529
$832$	0	0
$833$	−0.732621	−0.0253838
$834$	0	0
$835$	13.7723	0.476610
$836$	0	0
$837$	−1.55166	−0.0536333
$838$	0	0
$839$	−47.2592	−1.63157	−0.815785	−	0.578355i	$-0.803695\pi$
$839$	−47.2592	−1.63157	−0.815785	+	0.578355i	$0.803695\pi$
$840$	0	0
$841$	−26.4254	−0.911219
$842$	0	0
$843$	16.1147	0.555019
$844$	0	0
$845$	2.96553	0.102017
$846$	0	0
$847$	26.1052	0.896985
$848$	0	0
$849$	4.12190	0.141463
$850$	0	0
$851$	6.62742	0.227185
$852$	0	0
$853$	28.7345	0.983850	0.491925	−	0.870637i	$-0.336293\pi$
$853$	28.7345	0.983850	0.491925	+	0.870637i	$0.336293\pi$
$854$	0	0
$855$	−3.78378	−0.129403
$856$	0	0
$857$	−3.73782	−0.127682	−0.0638408	−	0.997960i	$-0.520335\pi$
$857$	−3.73782	−0.127682	−0.0638408	+	0.997960i	$0.520335\pi$
$858$	0	0
$859$	28.1607	0.960831	0.480415	−	0.877041i	$-0.340486\pi$
$859$	28.1607	0.960831	0.480415	+	0.877041i	$0.340486\pi$
$860$	0	0
$861$	17.6688	0.602152
$862$	0	0
$863$	−38.0405	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$863$	−38.0405	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$864$	0	0
$865$	15.0987	0.513372
$866$	0	0
$867$	−5.51907	−0.187437
$868$	0	0
$869$	−6.09686	−0.206822
$870$	0	0
$871$	3.16772	0.107334
$872$	0	0
$873$	−18.8710	−0.638687
$874$	0	0
$875$	2.60457	0.0880505
$876$	0	0
$877$	45.7281	1.54413	0.772064	−	0.635545i	$-0.219225\pi$
$877$	45.7281	1.54413	0.772064	+	0.635545i	$0.219225\pi$
$878$	0	0
$879$	2.55415	0.0861493
$880$	0	0
$881$	40.0732	1.35010	0.675051	−	0.737772i	$-0.264122\pi$
$881$	40.0732	1.35010	0.675051	+	0.737772i	$0.264122\pi$
$882$	0	0
$883$	−22.0335	−0.741488	−0.370744	−	0.928735i	$-0.620897\pi$
$883$	−22.0335	−0.741488	−0.370744	+	0.928735i	$0.620897\pi$
$884$	0	0
$885$	15.2506	0.512642
$886$	0	0
$887$	−0.0235876	−0.000791994 0	−0.000395997	−	1.00000i	$-0.500126\pi$
$887$	−0.0235876	−0.000791994 0	−0.000395997		1.00000i	$0.500126\pi$
$888$	0	0
$889$	7.28048	0.244180
$890$	0	0
$891$	0.988510	0.0331163
$892$	0	0
$893$	48.5010	1.62302
$894$	0	0
$895$	−23.0432	−0.770248
$896$	0	0
$897$	9.50317	0.317302
$898$	0	0
$899$	−2.48975	−0.0830379
$900$	0	0
$901$	1.86923	0.0622731
$902$	0	0
$903$	14.0643	0.468029
$904$	0	0
$905$	−5.06632	−0.168410
$906$	0	0
$907$	15.3330	0.509123	0.254561	−	0.967057i	$-0.418069\pi$
$907$	15.3330	0.509123	0.254561	+	0.967057i	$0.418069\pi$
$908$	0	0
$909$	−9.31247	−0.308875
$910$	0	0
$911$	48.5074	1.60712	0.803562	−	0.595221i	$-0.202936\pi$
$911$	48.5074	1.60712	0.803562	+	0.595221i	$0.202936\pi$
$912$	0	0
$913$	9.94176	0.329024
$914$	0	0
$915$	4.21622	0.139384
$916$	0	0
$917$	−36.2958	−1.19859
$918$	0	0
$919$	28.7794	0.949344	0.474672	−	0.880163i	$-0.342567\pi$
$919$	28.7794	0.949344	0.474672	+	0.880163i	$0.342567\pi$
$920$	0	0
$921$	21.4367	0.706364
$922$	0	0
$923$	48.0247	1.58075
$924$	0	0
$925$	−2.20914	−0.0726360
$926$	0	0
$927$	1.80930	0.0594251
$928$	0	0
$929$	23.9277	0.785044	0.392522	−	0.919743i	$-0.371603\pi$
$929$	23.9277	0.785044	0.392522	+	0.919743i	$0.371603\pi$
$930$	0	0
$931$	−0.818120	−0.0268128
$932$	0	0
$933$	8.69181	0.284557
$934$	0	0
$935$	−3.34942	−0.109538
$936$	0	0
$937$	33.3741	1.09028	0.545142	−	0.838344i	$-0.316476\pi$
$937$	33.3741	1.09028	0.545142	+	0.838344i	$0.316476\pi$
$938$	0	0
$939$	4.11923	0.134426
$940$	0	0
$941$	−31.1039	−1.01396	−0.506980	−	0.861958i	$-0.669238\pi$
$941$	−31.1039	−1.01396	−0.506980	+	0.861958i	$0.669238\pi$
$942$	0	0
$943$	20.3513	0.662731
$944$	0	0
$945$	2.60457	0.0847267
$946$	0	0
$947$	49.2268	1.59966	0.799829	−	0.600228i	$-0.204924\pi$
$947$	49.2268	1.59966	0.799829	+	0.600228i	$0.204924\pi$
$948$	0	0
$949$	−29.6669	−0.963029
$950$	0	0
$951$	−10.5032	−0.340589
$952$	0	0
$953$	−6.47324	−0.209689	−0.104844	−	0.994489i	$-0.533434\pi$
$953$	−6.47324	−0.209689	−0.104844	+	0.994489i	$0.533434\pi$
$954$	0	0
$955$	24.1191	0.780476
$956$	0	0
$957$	1.58613	0.0512724
$958$	0	0
$959$	2.24567	0.0725164
$960$	0	0
$961$	−28.5923	−0.922334
$962$	0	0
$963$	−8.66897	−0.279354
$964$	0	0
$965$	−22.3926	−0.720844
$966$	0	0
$967$	18.9860	0.610548	0.305274	−	0.952265i	$-0.401252\pi$
$967$	18.9860	0.610548	0.305274	+	0.952265i	$0.401252\pi$
$968$	0	0
$969$	12.8208	0.411863
$970$	0	0
$971$	7.21360	0.231495	0.115748	−	0.993279i	$-0.463074\pi$
$971$	7.21360	0.231495	0.115748	+	0.993279i	$0.463074\pi$
$972$	0	0
$973$	41.1281	1.31850
$974$	0	0
$975$	−3.16772	−0.101448
$976$	0	0
$977$	−9.17232	−0.293448	−0.146724	−	0.989177i	$-0.546873\pi$
$977$	−9.17232	−0.293448	−0.146724	+	0.989177i	$0.546873\pi$
$978$	0	0
$979$	−13.1367	−0.419852
$980$	0	0
$981$	4.05732	0.129540
$982$	0	0
$983$	47.2134	1.50587	0.752937	−	0.658092i	$-0.228636\pi$
$983$	47.2134	1.50587	0.752937	+	0.658092i	$0.228636\pi$
$984$	0	0
$985$	13.6594	0.435225
$986$	0	0
$987$	−33.3857	−1.06268
$988$	0	0
$989$	16.1995	0.515115
$990$	0	0
$991$	2.36545	0.0751411	0.0375706	−	0.999294i	$-0.488038\pi$
$991$	2.36545	0.0751411	0.0375706	+	0.999294i	$0.488038\pi$
$992$	0	0
$993$	35.0457	1.11214
$994$	0	0
$995$	−16.5816	−0.525672
$996$	0	0
$997$	−38.5745	−1.22167	−0.610834	−	0.791759i	$-0.709166\pi$
$997$	−38.5745	−1.22167	−0.610834	+	0.791759i	$0.709166\pi$
$998$	0	0
$999$	−2.20914	−0.0698941

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.c.1.1	✓	4	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.60457 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.60457 q^{7} +1.00000 q^{9} +0.988510 q^{11} -3.16772 q^{13} -1.00000 q^{15} +3.38835 q^{17} +3.78378 q^{19} -2.60457 q^{21} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.60457 q^{29} -1.55166 q^{31} +0.988510 q^{33} +2.60457 q^{35} -2.20914 q^{37} -3.16772 q^{39} -6.78378 q^{41} -5.39984 q^{43} -1.00000 q^{45} +12.8181 q^{47} -0.216218 q^{49} +3.38835 q^{51} +0.551664 q^{53} -0.988510 q^{55} +3.78378 q^{57} -15.2506 q^{59} -4.21622 q^{61} -2.60457 q^{63} +3.16772 q^{65} -1.00000 q^{67} -3.00000 q^{69} -15.1606 q^{71} +9.36537 q^{73} +1.00000 q^{75} -2.57464 q^{77} -6.16772 q^{79} +1.00000 q^{81} +10.0573 q^{83} -3.38835 q^{85} +1.60457 q^{87} -13.2894 q^{89} +8.25056 q^{91} -1.55166 q^{93} -3.78378 q^{95} -18.8710 q^{97} +0.988510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} - 5 q^{11} - 5 q^{13} - 4 q^{15} - 8 q^{17} + 5 q^{19} + q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 5 q^{29} - q^{31} - 5 q^{33} - q^{35} + 14 q^{37} - 5 q^{39} - 17 q^{41} - 9 q^{43} - 4 q^{45} - 7 q^{47} - 11 q^{49} - 8 q^{51} - 3 q^{53} + 5 q^{55} + 5 q^{57} - 23 q^{59} - 27 q^{61} + q^{63} + 5 q^{65} - 4 q^{67} - 12 q^{69} - 20 q^{71} - 2 q^{73} + 4 q^{75} - 23 q^{77} - 17 q^{79} + 4 q^{81} + 10 q^{83} + 8 q^{85} - 5 q^{87} - 18 q^{89} - 5 q^{91} - q^{93} - 5 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 - 5 * q^11 - 5 * q^13 - 4 * q^15 - 8 * q^17 + 5 * q^19 + q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 5 * q^29 - q^31 - 5 * q^33 - q^35 + 14 * q^37 - 5 * q^39 - 17 * q^41 - 9 * q^43 - 4 * q^45 - 7 * q^47 - 11 * q^49 - 8 * q^51 - 3 * q^53 + 5 * q^55 + 5 * q^57 - 23 * q^59 - 27 * q^61 + q^63 + 5 * q^65 - 4 * q^67 - 12 * q^69 - 20 * q^71 - 2 * q^73 + 4 * q^75 - 23 * q^77 - 17 * q^79 + 4 * q^81 + 10 * q^83 + 8 * q^85 - 5 * q^87 - 18 * q^89 - 5 * q^91 - q^93 - 5 * q^95 - 11 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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