LMFDB - Embedded newform 8040.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8040,2,Mod(1,8040)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8040.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:	$64.1997232251$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
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			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	8040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +2.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +2.56155 q^{7} +1.00000 q^{9} +1.43845 q^{11} +5.12311 q^{13} +1.00000 q^{15} +1.12311 q^{17} -1.12311 q^{19} +2.56155 q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +7.12311 q^{29} +7.12311 q^{31} +1.43845 q^{33} +2.56155 q^{35} -5.68466 q^{37} +5.12311 q^{39} +6.00000 q^{41} +2.87689 q^{43} +1.00000 q^{45} -5.12311 q^{47} -0.438447 q^{49} +1.12311 q^{51} +3.12311 q^{53} +1.43845 q^{55} -1.12311 q^{57} -12.2462 q^{59} +5.68466 q^{61} +2.56155 q^{63} +5.12311 q^{65} +1.00000 q^{67} -4.00000 q^{69} +7.43845 q^{71} -11.1231 q^{73} +1.00000 q^{75} +3.68466 q^{77} -2.00000 q^{79} +1.00000 q^{81} -9.43845 q^{83} +1.12311 q^{85} +7.12311 q^{87} -2.80776 q^{89} +13.1231 q^{91} +7.12311 q^{93} -1.12311 q^{95} +14.5616 q^{97} +1.43845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 2 q^{13} + 2 q^{15} - 6 q^{17} + 6 q^{19} + q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} + 6 q^{31} + 7 q^{33} + q^{35} + q^{37} + 2 q^{39} + 12 q^{41} + 14 q^{43} + 2 q^{45} - 2 q^{47} - 5 q^{49} - 6 q^{51} - 2 q^{53} + 7 q^{55} + 6 q^{57} - 8 q^{59} - q^{61} + q^{63} + 2 q^{65} + 2 q^{67} - 8 q^{69} + 19 q^{71} - 14 q^{73} + 2 q^{75} - 5 q^{77} - 4 q^{79} + 2 q^{81} - 23 q^{83} - 6 q^{85} + 6 q^{87} + 15 q^{89} + 18 q^{91} + 6 q^{93} + 6 q^{95} + 25 q^{97} + 7 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9 + 7 * q^11 + 2 * q^13 + 2 * q^15 - 6 * q^17 + 6 * q^19 + q^21 - 8 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 + 6 * q^31 + 7 * q^33 + q^35 + q^37 + 2 * q^39 + 12 * q^41 + 14 * q^43 + 2 * q^45 - 2 * q^47 - 5 * q^49 - 6 * q^51 - 2 * q^53 + 7 * q^55 + 6 * q^57 - 8 * q^59 - q^61 + q^63 + 2 * q^65 + 2 * q^67 - 8 * q^69 + 19 * q^71 - 14 * q^73 + 2 * q^75 - 5 * q^77 - 4 * q^79 + 2 * q^81 - 23 * q^83 - 6 * q^85 + 6 * q^87 + 15 * q^89 + 18 * q^91 + 6 * q^93 + 6 * q^95 + 25 * q^97 + 7 * 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
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.43845	0.433708	0.216854	−	0.976204i	$-0.430420\pi$
$11$	1.43845	0.433708	0.216854	+	0.976204i	$0.430420\pi$
$12$	0	0
$13$	5.12311	1.42089	0.710447	−	0.703751i	$-0.248493\pi$
$13$	5.12311	1.42089	0.710447	+	0.703751i	$0.248493\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	1.12311	0.272393	0.136197	−	0.990682i	$-0.456512\pi$
$17$	1.12311	0.272393	0.136197	+	0.990682i	$0.456512\pi$
$18$	0	0
$19$	−1.12311	−0.257658	−0.128829	−	0.991667i	$-0.541122\pi$
$19$	−1.12311	−0.257658	−0.128829	+	0.991667i	$0.541122\pi$
$20$	0	0
$21$	2.56155	0.558977
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.12311	1.32273	0.661364	−	0.750065i	$-0.269978\pi$
$29$	7.12311	1.32273	0.661364	+	0.750065i	$0.269978\pi$
$30$	0	0
$31$	7.12311	1.27935	0.639674	−	0.768647i	$-0.279069\pi$
$31$	7.12311	1.27935	0.639674	+	0.768647i	$0.279069\pi$
$32$	0	0
$33$	1.43845	0.250402
$34$	0	0
$35$	2.56155	0.432981
$36$	0	0
$37$	−5.68466	−0.934552	−0.467276	−	0.884111i	$-0.654765\pi$
$37$	−5.68466	−0.934552	−0.467276	+	0.884111i	$0.654765\pi$
$38$	0	0
$39$	5.12311	0.820353
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	2.87689	0.438722	0.219361	−	0.975644i	$-0.429603\pi$
$43$	2.87689	0.438722	0.219361	+	0.975644i	$0.429603\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−5.12311	−0.747282	−0.373641	−	0.927573i	$-0.621891\pi$
$47$	−5.12311	−0.747282	−0.373641	+	0.927573i	$0.621891\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	1.12311	0.157266
$52$	0	0
$53$	3.12311	0.428992	0.214496	−	0.976725i	$-0.431189\pi$
$53$	3.12311	0.428992	0.214496	+	0.976725i	$0.431189\pi$
$54$	0	0
$55$	1.43845	0.193960
$56$	0	0
$57$	−1.12311	−0.148759
$58$	0	0
$59$	−12.2462	−1.59432	−0.797160	−	0.603768i	$-0.793666\pi$
$59$	−12.2462	−1.59432	−0.797160	+	0.603768i	$0.793666\pi$
$60$	0	0
$61$	5.68466	0.727846	0.363923	−	0.931429i	$-0.381437\pi$
$61$	5.68466	0.727846	0.363923	+	0.931429i	$0.381437\pi$
$62$	0	0
$63$	2.56155	0.322725
$64$	0	0
$65$	5.12311	0.635443
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	7.43845	0.882781	0.441391	−	0.897315i	$-0.354485\pi$
$71$	7.43845	0.882781	0.441391	+	0.897315i	$0.354485\pi$
$72$	0	0
$73$	−11.1231	−1.30186	−0.650931	−	0.759137i	$-0.725621\pi$
$73$	−11.1231	−1.30186	−0.650931	+	0.759137i	$0.725621\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	3.68466	0.419906
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.43845	−1.03600	−0.518002	−	0.855379i	$-0.673324\pi$
$83$	−9.43845	−1.03600	−0.518002	+	0.855379i	$0.673324\pi$
$84$	0	0
$85$	1.12311	0.121818
$86$	0	0
$87$	7.12311	0.763677
$88$	0	0
$89$	−2.80776	−0.297622	−0.148811	−	0.988866i	$-0.547545\pi$
$89$	−2.80776	−0.297622	−0.148811	+	0.988866i	$0.547545\pi$
$90$	0	0
$91$	13.1231	1.37568
$92$	0	0
$93$	7.12311	0.738632
$94$	0	0
$95$	−1.12311	−0.115228
$96$	0	0
$97$	14.5616	1.47850	0.739251	−	0.673430i	$-0.235180\pi$
$97$	14.5616	1.47850	0.739251	+	0.673430i	$0.235180\pi$
$98$	0	0
$99$	1.43845	0.144569
$100$	0	0
$101$	−6.80776	−0.677398	−0.338699	−	0.940895i	$-0.609987\pi$
$101$	−6.80776	−0.677398	−0.338699	+	0.940895i	$0.609987\pi$
$102$	0	0
$103$	2.87689	0.283469	0.141734	−	0.989905i	$-0.454732\pi$
$103$	2.87689	0.283469	0.141734	+	0.989905i	$0.454732\pi$
$104$	0	0
$105$	2.56155	0.249982
$106$	0	0
$107$	−2.24621	−0.217149	−0.108575	−	0.994088i	$-0.534629\pi$
$107$	−2.24621	−0.217149	−0.108575	+	0.994088i	$0.534629\pi$
$108$	0	0
$109$	12.5616	1.20318	0.601589	−	0.798806i	$-0.294534\pi$
$109$	12.5616	1.20318	0.601589	+	0.798806i	$0.294534\pi$
$110$	0	0
$111$	−5.68466	−0.539564
$112$	0	0
$113$	−0.561553	−0.0528264	−0.0264132	−	0.999651i	$-0.508409\pi$
$113$	−0.561553	−0.0528264	−0.0264132	+	0.999651i	$0.508409\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	5.12311	0.473631
$118$	0	0
$119$	2.87689	0.263724
$120$	0	0
$121$	−8.93087	−0.811897
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.12311	0.454602	0.227301	−	0.973825i	$-0.427010\pi$
$127$	5.12311	0.454602	0.227301	+	0.973825i	$0.427010\pi$
$128$	0	0
$129$	2.87689	0.253296
$130$	0	0
$131$	4.87689	0.426096	0.213048	−	0.977042i	$-0.431661\pi$
$131$	4.87689	0.426096	0.213048	+	0.977042i	$0.431661\pi$
$132$	0	0
$133$	−2.87689	−0.249458
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−4.56155	−0.389720	−0.194860	−	0.980831i	$-0.562425\pi$
$137$	−4.56155	−0.389720	−0.194860	+	0.980831i	$0.562425\pi$
$138$	0	0
$139$	9.36932	0.794695	0.397348	−	0.917668i	$-0.369931\pi$
$139$	9.36932	0.794695	0.397348	+	0.917668i	$0.369931\pi$
$140$	0	0
$141$	−5.12311	−0.431443
$142$	0	0
$143$	7.36932	0.616253
$144$	0	0
$145$	7.12311	0.591542
$146$	0	0
$147$	−0.438447	−0.0361625
$148$	0	0
$149$	−4.87689	−0.399531	−0.199765	−	0.979844i	$-0.564018\pi$
$149$	−4.87689	−0.399531	−0.199765	+	0.979844i	$0.564018\pi$
$150$	0	0
$151$	−10.5616	−0.859487	−0.429743	−	0.902951i	$-0.641396\pi$
$151$	−10.5616	−0.859487	−0.429743	+	0.902951i	$0.641396\pi$
$152$	0	0
$153$	1.12311	0.0907977
$154$	0	0
$155$	7.12311	0.572142
$156$	0	0
$157$	14.4924	1.15662	0.578311	−	0.815817i	$-0.303712\pi$
$157$	14.4924	1.15662	0.578311	+	0.815817i	$0.303712\pi$
$158$	0	0
$159$	3.12311	0.247678
$160$	0	0
$161$	−10.2462	−0.807515
$162$	0	0
$163$	−13.4384	−1.05258	−0.526290	−	0.850305i	$-0.676417\pi$
$163$	−13.4384	−1.05258	−0.526290	+	0.850305i	$0.676417\pi$
$164$	0	0
$165$	1.43845	0.111983
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.2462	1.01894
$170$	0	0
$171$	−1.12311	−0.0858860
$172$	0	0
$173$	−8.31534	−0.632204	−0.316102	−	0.948725i	$-0.602374\pi$
$173$	−8.31534	−0.632204	−0.316102	+	0.948725i	$0.602374\pi$
$174$	0	0
$175$	2.56155	0.193635
$176$	0	0
$177$	−12.2462	−0.920482
$178$	0	0
$179$	−2.24621	−0.167890	−0.0839449	−	0.996470i	$-0.526752\pi$
$179$	−2.24621	−0.167890	−0.0839449	+	0.996470i	$0.526752\pi$
$180$	0	0
$181$	−23.6155	−1.75533	−0.877664	−	0.479276i	$-0.840899\pi$
$181$	−23.6155	−1.75533	−0.877664	+	0.479276i	$0.840899\pi$
$182$	0	0
$183$	5.68466	0.420222
$184$	0	0
$185$	−5.68466	−0.417944
$186$	0	0
$187$	1.61553	0.118139
$188$	0	0
$189$	2.56155	0.186326
$190$	0	0
$191$	−5.12311	−0.370695	−0.185347	−	0.982673i	$-0.559341\pi$
$191$	−5.12311	−0.370695	−0.185347	+	0.982673i	$0.559341\pi$
$192$	0	0
$193$	16.8769	1.21483	0.607413	−	0.794386i	$-0.292207\pi$
$193$	16.8769	1.21483	0.607413	+	0.794386i	$0.292207\pi$
$194$	0	0
$195$	5.12311	0.366873
$196$	0	0
$197$	−16.2462	−1.15749	−0.578747	−	0.815507i	$-0.696458\pi$
$197$	−16.2462	−1.15749	−0.578747	+	0.815507i	$0.696458\pi$
$198$	0	0
$199$	−13.9309	−0.987533	−0.493767	−	0.869594i	$-0.664380\pi$
$199$	−13.9309	−0.987533	−0.493767	+	0.869594i	$0.664380\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	18.2462	1.28063
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−1.61553	−0.111748
$210$	0	0
$211$	2.24621	0.154636	0.0773178	−	0.997006i	$-0.475364\pi$
$211$	2.24621	0.154636	0.0773178	+	0.997006i	$0.475364\pi$
$212$	0	0
$213$	7.43845	0.509674
$214$	0	0
$215$	2.87689	0.196203
$216$	0	0
$217$	18.2462	1.23863
$218$	0	0
$219$	−11.1231	−0.751630
$220$	0	0
$221$	5.75379	0.387042
$222$	0	0
$223$	13.1231	0.878788	0.439394	−	0.898294i	$-0.355193\pi$
$223$	13.1231	0.878788	0.439394	+	0.898294i	$0.355193\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−14.2462	−0.945554	−0.472777	−	0.881182i	$-0.656748\pi$
$227$	−14.2462	−0.945554	−0.472777	+	0.881182i	$0.656748\pi$
$228$	0	0
$229$	−25.0540	−1.65561	−0.827807	−	0.561014i	$-0.810412\pi$
$229$	−25.0540	−1.65561	−0.827807	+	0.561014i	$0.810412\pi$
$230$	0	0
$231$	3.68466	0.242433
$232$	0	0
$233$	−4.56155	−0.298837	−0.149419	−	0.988774i	$-0.547740\pi$
$233$	−4.56155	−0.298837	−0.149419	+	0.988774i	$0.547740\pi$
$234$	0	0
$235$	−5.12311	−0.334195
$236$	0	0
$237$	−2.00000	−0.129914
$238$	0	0
$239$	−4.63068	−0.299534	−0.149767	−	0.988721i	$-0.547852\pi$
$239$	−4.63068	−0.299534	−0.149767	+	0.988721i	$0.547852\pi$
$240$	0	0
$241$	−2.80776	−0.180864	−0.0904320	−	0.995903i	$-0.528825\pi$
$241$	−2.80776	−0.180864	−0.0904320	+	0.995903i	$0.528825\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.438447	−0.0280114
$246$	0	0
$247$	−5.75379	−0.366105
$248$	0	0
$249$	−9.43845	−0.598137
$250$	0	0
$251$	12.8078	0.808419	0.404209	−	0.914666i	$-0.367547\pi$
$251$	12.8078	0.808419	0.404209	+	0.914666i	$0.367547\pi$
$252$	0	0
$253$	−5.75379	−0.361738
$254$	0	0
$255$	1.12311	0.0703316
$256$	0	0
$257$	0.630683	0.0393409	0.0196705	−	0.999807i	$-0.493738\pi$
$257$	0.630683	0.0393409	0.0196705	+	0.999807i	$0.493738\pi$
$258$	0	0
$259$	−14.5616	−0.904811
$260$	0	0
$261$	7.12311	0.440909
$262$	0	0
$263$	12.4924	0.770316	0.385158	−	0.922851i	$-0.374147\pi$
$263$	12.4924	0.770316	0.385158	+	0.922851i	$0.374147\pi$
$264$	0	0
$265$	3.12311	0.191851
$266$	0	0
$267$	−2.80776	−0.171832
$268$	0	0
$269$	13.3693	0.815142	0.407571	−	0.913174i	$-0.366376\pi$
$269$	13.3693	0.815142	0.407571	+	0.913174i	$0.366376\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	0	0
$273$	13.1231	0.794246
$274$	0	0
$275$	1.43845	0.0867416
$276$	0	0
$277$	5.19224	0.311971	0.155986	−	0.987759i	$-0.450145\pi$
$277$	5.19224	0.311971	0.155986	+	0.987759i	$0.450145\pi$
$278$	0	0
$279$	7.12311	0.426449
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	7.68466	0.456806	0.228403	−	0.973567i	$-0.426650\pi$
$283$	7.68466	0.456806	0.228403	+	0.973567i	$0.426650\pi$
$284$	0	0
$285$	−1.12311	−0.0665270
$286$	0	0
$287$	15.3693	0.907222
$288$	0	0
$289$	−15.7386	−0.925802
$290$	0	0
$291$	14.5616	0.853613
$292$	0	0
$293$	16.3153	0.953152	0.476576	−	0.879133i	$-0.341878\pi$
$293$	16.3153	0.953152	0.476576	+	0.879133i	$0.341878\pi$
$294$	0	0
$295$	−12.2462	−0.713002
$296$	0	0
$297$	1.43845	0.0834672
$298$	0	0
$299$	−20.4924	−1.18511
$300$	0	0
$301$	7.36932	0.424760
$302$	0	0
$303$	−6.80776	−0.391096
$304$	0	0
$305$	5.68466	0.325503
$306$	0	0
$307$	20.8078	1.18756	0.593781	−	0.804627i	$-0.297635\pi$
$307$	20.8078	1.18756	0.593781	+	0.804627i	$0.297635\pi$
$308$	0	0
$309$	2.87689	0.163661
$310$	0	0
$311$	−8.49242	−0.481561	−0.240781	−	0.970580i	$-0.577403\pi$
$311$	−8.49242	−0.481561	−0.240781	+	0.970580i	$0.577403\pi$
$312$	0	0
$313$	12.8078	0.723938	0.361969	−	0.932190i	$-0.382105\pi$
$313$	12.8078	0.723938	0.361969	+	0.932190i	$0.382105\pi$
$314$	0	0
$315$	2.56155	0.144327
$316$	0	0
$317$	−14.8769	−0.835570	−0.417785	−	0.908546i	$-0.637193\pi$
$317$	−14.8769	−0.835570	−0.417785	+	0.908546i	$0.637193\pi$
$318$	0	0
$319$	10.2462	0.573678
$320$	0	0
$321$	−2.24621	−0.125371
$322$	0	0
$323$	−1.26137	−0.0701843
$324$	0	0
$325$	5.12311	0.284179
$326$	0	0
$327$	12.5616	0.694655
$328$	0	0
$329$	−13.1231	−0.723500
$330$	0	0
$331$	−8.56155	−0.470586	−0.235293	−	0.971925i	$-0.575605\pi$
$331$	−8.56155	−0.470586	−0.235293	+	0.971925i	$0.575605\pi$
$332$	0	0
$333$	−5.68466	−0.311517
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	11.3693	0.619326	0.309663	−	0.950846i	$-0.399784\pi$
$337$	11.3693	0.619326	0.309663	+	0.950846i	$0.399784\pi$
$338$	0	0
$339$	−0.561553	−0.0304994
$340$	0	0
$341$	10.2462	0.554863
$342$	0	0
$343$	−19.0540	−1.02882
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−26.7386	−1.43541	−0.717703	−	0.696350i	$-0.754806\pi$
$347$	−26.7386	−1.43541	−0.717703	+	0.696350i	$0.754806\pi$
$348$	0	0
$349$	22.4924	1.20399	0.601996	−	0.798499i	$-0.294372\pi$
$349$	22.4924	1.20399	0.601996	+	0.798499i	$0.294372\pi$
$350$	0	0
$351$	5.12311	0.273451
$352$	0	0
$353$	−4.24621	−0.226003	−0.113002	−	0.993595i	$-0.536046\pi$
$353$	−4.24621	−0.226003	−0.113002	+	0.993595i	$0.536046\pi$
$354$	0	0
$355$	7.43845	0.394792
$356$	0	0
$357$	2.87689	0.152261
$358$	0	0
$359$	21.0540	1.11119	0.555593	−	0.831454i	$-0.312491\pi$
$359$	21.0540	1.11119	0.555593	+	0.831454i	$0.312491\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	−8.93087	−0.468749
$364$	0	0
$365$	−11.1231	−0.582210
$366$	0	0
$367$	4.49242	0.234503	0.117251	−	0.993102i	$-0.462592\pi$
$367$	4.49242	0.234503	0.117251	+	0.993102i	$0.462592\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	25.1231	1.30083	0.650413	−	0.759581i	$-0.274596\pi$
$373$	25.1231	1.30083	0.650413	+	0.759581i	$0.274596\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	36.4924	1.87946
$378$	0	0
$379$	−27.6155	−1.41851	−0.709257	−	0.704950i	$-0.750970\pi$
$379$	−27.6155	−1.41851	−0.709257	+	0.704950i	$0.750970\pi$
$380$	0	0
$381$	5.12311	0.262465
$382$	0	0
$383$	25.4384	1.29984	0.649922	−	0.760001i	$-0.274802\pi$
$383$	25.4384	1.29984	0.649922	+	0.760001i	$0.274802\pi$
$384$	0	0
$385$	3.68466	0.187788
$386$	0	0
$387$	2.87689	0.146241
$388$	0	0
$389$	5.36932	0.272235	0.136118	−	0.990693i	$-0.456538\pi$
$389$	5.36932	0.272235	0.136118	+	0.990693i	$0.456538\pi$
$390$	0	0
$391$	−4.49242	−0.227192
$392$	0	0
$393$	4.87689	0.246007
$394$	0	0
$395$	−2.00000	−0.100631
$396$	0	0
$397$	9.19224	0.461345	0.230673	−	0.973031i	$-0.425907\pi$
$397$	9.19224	0.461345	0.230673	+	0.973031i	$0.425907\pi$
$398$	0	0
$399$	−2.87689	−0.144025
$400$	0	0
$401$	25.8617	1.29147	0.645737	−	0.763560i	$-0.276550\pi$
$401$	25.8617	1.29147	0.645737	+	0.763560i	$0.276550\pi$
$402$	0	0
$403$	36.4924	1.81782
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−8.17708	−0.405323
$408$	0	0
$409$	30.4924	1.50775	0.753877	−	0.657016i	$-0.228182\pi$
$409$	30.4924	1.50775	0.753877	+	0.657016i	$0.228182\pi$
$410$	0	0
$411$	−4.56155	−0.225005
$412$	0	0
$413$	−31.3693	−1.54358
$414$	0	0
$415$	−9.43845	−0.463315
$416$	0	0
$417$	9.36932	0.458817
$418$	0	0
$419$	−11.1231	−0.543399	−0.271700	−	0.962382i	$-0.587586\pi$
$419$	−11.1231	−0.543399	−0.271700	+	0.962382i	$0.587586\pi$
$420$	0	0
$421$	33.3693	1.62632	0.813160	−	0.582040i	$-0.197745\pi$
$421$	33.3693	1.62632	0.813160	+	0.582040i	$0.197745\pi$
$422$	0	0
$423$	−5.12311	−0.249094
$424$	0	0
$425$	1.12311	0.0544786
$426$	0	0
$427$	14.5616	0.704683
$428$	0	0
$429$	7.36932	0.355794
$430$	0	0
$431$	−11.4384	−0.550971	−0.275485	−	0.961305i	$-0.588839\pi$
$431$	−11.4384	−0.550971	−0.275485	+	0.961305i	$0.588839\pi$
$432$	0	0
$433$	−5.12311	−0.246201	−0.123100	−	0.992394i	$-0.539284\pi$
$433$	−5.12311	−0.246201	−0.123100	+	0.992394i	$0.539284\pi$
$434$	0	0
$435$	7.12311	0.341527
$436$	0	0
$437$	4.49242	0.214902
$438$	0	0
$439$	−22.5616	−1.07680	−0.538402	−	0.842688i	$-0.680972\pi$
$439$	−22.5616	−1.07680	−0.538402	+	0.842688i	$0.680972\pi$
$440$	0	0
$441$	−0.438447	−0.0208784
$442$	0	0
$443$	−30.2462	−1.43704	−0.718520	−	0.695506i	$-0.755180\pi$
$443$	−30.2462	−1.43704	−0.718520	+	0.695506i	$0.755180\pi$
$444$	0	0
$445$	−2.80776	−0.133101
$446$	0	0
$447$	−4.87689	−0.230669
$448$	0	0
$449$	10.1771	0.480286	0.240143	−	0.970738i	$-0.422806\pi$
$449$	10.1771	0.480286	0.240143	+	0.970738i	$0.422806\pi$
$450$	0	0
$451$	8.63068	0.406403
$452$	0	0
$453$	−10.5616	−0.496225
$454$	0	0
$455$	13.1231	0.615221
$456$	0	0
$457$	−8.87689	−0.415244	−0.207622	−	0.978209i	$-0.566572\pi$
$457$	−8.87689	−0.415244	−0.207622	+	0.978209i	$0.566572\pi$
$458$	0	0
$459$	1.12311	0.0524221
$460$	0	0
$461$	31.6155	1.47248	0.736241	−	0.676719i	$-0.236599\pi$
$461$	31.6155	1.47248	0.736241	+	0.676719i	$0.236599\pi$
$462$	0	0
$463$	−30.7386	−1.42855	−0.714273	−	0.699867i	$-0.753242\pi$
$463$	−30.7386	−1.42855	−0.714273	+	0.699867i	$0.753242\pi$
$464$	0	0
$465$	7.12311	0.330326
$466$	0	0
$467$	−23.0540	−1.06681	−0.533405	−	0.845860i	$-0.679088\pi$
$467$	−23.0540	−1.06681	−0.533405	+	0.845860i	$0.679088\pi$
$468$	0	0
$469$	2.56155	0.118282
$470$	0	0
$471$	14.4924	0.667776
$472$	0	0
$473$	4.13826	0.190277
$474$	0	0
$475$	−1.12311	−0.0515316
$476$	0	0
$477$	3.12311	0.142997
$478$	0	0
$479$	6.80776	0.311055	0.155527	−	0.987832i	$-0.450292\pi$
$479$	6.80776	0.311055	0.155527	+	0.987832i	$0.450292\pi$
$480$	0	0
$481$	−29.1231	−1.32790
$482$	0	0
$483$	−10.2462	−0.466219
$484$	0	0
$485$	14.5616	0.661206
$486$	0	0
$487$	−35.5464	−1.61076	−0.805381	−	0.592758i	$-0.798039\pi$
$487$	−35.5464	−1.61076	−0.805381	+	0.592758i	$0.798039\pi$
$488$	0	0
$489$	−13.4384	−0.607708
$490$	0	0
$491$	13.3693	0.603349	0.301674	−	0.953411i	$-0.402454\pi$
$491$	13.3693	0.603349	0.301674	+	0.953411i	$0.402454\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	0	0
$495$	1.43845	0.0646534
$496$	0	0
$497$	19.0540	0.854688
$498$	0	0
$499$	−1.68466	−0.0754157	−0.0377078	−	0.999289i	$-0.512006\pi$
$499$	−1.68466	−0.0754157	−0.0377078	+	0.999289i	$0.512006\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.9309	−0.977849	−0.488925	−	0.872326i	$-0.662611\pi$
$503$	−21.9309	−0.977849	−0.488925	+	0.872326i	$0.662611\pi$
$504$	0	0
$505$	−6.80776	−0.302942
$506$	0	0
$507$	13.2462	0.588285
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	−28.4924	−1.26043
$512$	0	0
$513$	−1.12311	−0.0495863
$514$	0	0
$515$	2.87689	0.126771
$516$	0	0
$517$	−7.36932	−0.324102
$518$	0	0
$519$	−8.31534	−0.365003
$520$	0	0
$521$	−8.87689	−0.388904	−0.194452	−	0.980912i	$-0.562293\pi$
$521$	−8.87689	−0.388904	−0.194452	+	0.980912i	$0.562293\pi$
$522$	0	0
$523$	7.68466	0.336027	0.168013	−	0.985785i	$-0.446265\pi$
$523$	7.68466	0.336027	0.168013	+	0.985785i	$0.446265\pi$
$524$	0	0
$525$	2.56155	0.111795
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−12.2462	−0.531440
$532$	0	0
$533$	30.7386	1.33144
$534$	0	0
$535$	−2.24621	−0.0971122
$536$	0	0
$537$	−2.24621	−0.0969312
$538$	0	0
$539$	−0.630683	−0.0271654
$540$	0	0
$541$	32.2462	1.38637	0.693186	−	0.720758i	$-0.256206\pi$
$541$	32.2462	1.38637	0.693186	+	0.720758i	$0.256206\pi$
$542$	0	0
$543$	−23.6155	−1.01344
$544$	0	0
$545$	12.5616	0.538078
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	5.68466	0.242615
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	−5.12311	−0.217857
$554$	0	0
$555$	−5.68466	−0.241300
$556$	0	0
$557$	−3.36932	−0.142763	−0.0713813	−	0.997449i	$-0.522741\pi$
$557$	−3.36932	−0.142763	−0.0713813	+	0.997449i	$0.522741\pi$
$558$	0	0
$559$	14.7386	0.623378
$560$	0	0
$561$	1.61553	0.0682077
$562$	0	0
$563$	−22.8769	−0.964146	−0.482073	−	0.876131i	$-0.660116\pi$
$563$	−22.8769	−0.964146	−0.482073	+	0.876131i	$0.660116\pi$
$564$	0	0
$565$	−0.561553	−0.0236247
$566$	0	0
$567$	2.56155	0.107575
$568$	0	0
$569$	−33.6847	−1.41213	−0.706067	−	0.708145i	$-0.749532\pi$
$569$	−33.6847	−1.41213	−0.706067	+	0.708145i	$0.749532\pi$
$570$	0	0
$571$	28.4924	1.19237	0.596185	−	0.802847i	$-0.296682\pi$
$571$	28.4924	1.19237	0.596185	+	0.802847i	$0.296682\pi$
$572$	0	0
$573$	−5.12311	−0.214021
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	5.12311	0.213278	0.106639	−	0.994298i	$-0.465991\pi$
$577$	5.12311	0.213278	0.106639	+	0.994298i	$0.465991\pi$
$578$	0	0
$579$	16.8769	0.701380
$580$	0	0
$581$	−24.1771	−1.00303
$582$	0	0
$583$	4.49242	0.186057
$584$	0	0
$585$	5.12311	0.211814
$586$	0	0
$587$	−25.1231	−1.03694	−0.518471	−	0.855095i	$-0.673498\pi$
$587$	−25.1231	−1.03694	−0.518471	+	0.855095i	$0.673498\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−16.2462	−0.668280
$592$	0	0
$593$	−12.0691	−0.495620	−0.247810	−	0.968809i	$-0.579711\pi$
$593$	−12.0691	−0.495620	−0.247810	+	0.968809i	$0.579711\pi$
$594$	0	0
$595$	2.87689	0.117941
$596$	0	0
$597$	−13.9309	−0.570153
$598$	0	0
$599$	30.8769	1.26160	0.630798	−	0.775947i	$-0.282728\pi$
$599$	30.8769	1.26160	0.630798	+	0.775947i	$0.282728\pi$
$600$	0	0
$601$	10.9460	0.446498	0.223249	−	0.974761i	$-0.428334\pi$
$601$	10.9460	0.446498	0.223249	+	0.974761i	$0.428334\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	−8.93087	−0.363091
$606$	0	0
$607$	15.3693	0.623821	0.311911	−	0.950111i	$-0.399031\pi$
$607$	15.3693	0.623821	0.311911	+	0.950111i	$0.399031\pi$
$608$	0	0
$609$	18.2462	0.739374
$610$	0	0
$611$	−26.2462	−1.06181
$612$	0	0
$613$	−37.2311	−1.50375	−0.751874	−	0.659307i	$-0.770850\pi$
$613$	−37.2311	−1.50375	−0.751874	+	0.659307i	$0.770850\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	17.7538	0.714741	0.357370	−	0.933963i	$-0.383673\pi$
$617$	17.7538	0.714741	0.357370	+	0.933963i	$0.383673\pi$
$618$	0	0
$619$	0.630683	0.0253493	0.0126746	−	0.999920i	$-0.495965\pi$
$619$	0.630683	0.0253493	0.0126746	+	0.999920i	$0.495965\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−7.19224	−0.288151
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.61553	−0.0645180
$628$	0	0
$629$	−6.38447	−0.254566
$630$	0	0
$631$	37.8617	1.50725	0.753626	−	0.657303i	$-0.228303\pi$
$631$	37.8617	1.50725	0.753626	+	0.657303i	$0.228303\pi$
$632$	0	0
$633$	2.24621	0.0892789
$634$	0	0
$635$	5.12311	0.203304
$636$	0	0
$637$	−2.24621	−0.0889981
$638$	0	0
$639$	7.43845	0.294260
$640$	0	0
$641$	3.75379	0.148266	0.0741329	−	0.997248i	$-0.476381\pi$
$641$	3.75379	0.148266	0.0741329	+	0.997248i	$0.476381\pi$
$642$	0	0
$643$	−22.2462	−0.877305	−0.438652	−	0.898657i	$-0.644544\pi$
$643$	−22.2462	−0.877305	−0.438652	+	0.898657i	$0.644544\pi$
$644$	0	0
$645$	2.87689	0.113278
$646$	0	0
$647$	−24.8078	−0.975294	−0.487647	−	0.873041i	$-0.662145\pi$
$647$	−24.8078	−0.975294	−0.487647	+	0.873041i	$0.662145\pi$
$648$	0	0
$649$	−17.6155	−0.691470
$650$	0	0
$651$	18.2462	0.715125
$652$	0	0
$653$	11.6155	0.454551	0.227275	−	0.973831i	$-0.427018\pi$
$653$	11.6155	0.454551	0.227275	+	0.973831i	$0.427018\pi$
$654$	0	0
$655$	4.87689	0.190556
$656$	0	0
$657$	−11.1231	−0.433954
$658$	0	0
$659$	33.3693	1.29988	0.649942	−	0.759984i	$-0.274793\pi$
$659$	33.3693	1.29988	0.649942	+	0.759984i	$0.274793\pi$
$660$	0	0
$661$	12.2462	0.476322	0.238161	−	0.971226i	$-0.423455\pi$
$661$	12.2462	0.476322	0.238161	+	0.971226i	$0.423455\pi$
$662$	0	0
$663$	5.75379	0.223459
$664$	0	0
$665$	−2.87689	−0.111561
$666$	0	0
$667$	−28.4924	−1.10323
$668$	0	0
$669$	13.1231	0.507369
$670$	0	0
$671$	8.17708	0.315673
$672$	0	0
$673$	35.3693	1.36339	0.681693	−	0.731638i	$-0.261244\pi$
$673$	35.3693	1.36339	0.681693	+	0.731638i	$0.261244\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−19.7538	−0.759200	−0.379600	−	0.925151i	$-0.623938\pi$
$677$	−19.7538	−0.759200	−0.379600	+	0.925151i	$0.623938\pi$
$678$	0	0
$679$	37.3002	1.43145
$680$	0	0
$681$	−14.2462	−0.545916
$682$	0	0
$683$	29.6155	1.13321	0.566603	−	0.823991i	$-0.308257\pi$
$683$	29.6155	1.13321	0.566603	+	0.823991i	$0.308257\pi$
$684$	0	0
$685$	−4.56155	−0.174288
$686$	0	0
$687$	−25.0540	−0.955869
$688$	0	0
$689$	16.0000	0.609551
$690$	0	0
$691$	18.8769	0.718111	0.359055	−	0.933316i	$-0.383099\pi$
$691$	18.8769	0.718111	0.359055	+	0.933316i	$0.383099\pi$
$692$	0	0
$693$	3.68466	0.139969
$694$	0	0
$695$	9.36932	0.355398
$696$	0	0
$697$	6.73863	0.255244
$698$	0	0
$699$	−4.56155	−0.172534
$700$	0	0
$701$	12.5616	0.474443	0.237222	−	0.971456i	$-0.423763\pi$
$701$	12.5616	0.474443	0.237222	+	0.971456i	$0.423763\pi$
$702$	0	0
$703$	6.38447	0.240795
$704$	0	0
$705$	−5.12311	−0.192947
$706$	0	0
$707$	−17.4384	−0.655840
$708$	0	0
$709$	7.12311	0.267514	0.133757	−	0.991014i	$-0.457296\pi$
$709$	7.12311	0.267514	0.133757	+	0.991014i	$0.457296\pi$
$710$	0	0
$711$	−2.00000	−0.0750059
$712$	0	0
$713$	−28.4924	−1.06705
$714$	0	0
$715$	7.36932	0.275597
$716$	0	0
$717$	−4.63068	−0.172936
$718$	0	0
$719$	−45.3693	−1.69199	−0.845995	−	0.533191i	$-0.820993\pi$
$719$	−45.3693	−1.69199	−0.845995	+	0.533191i	$0.820993\pi$
$720$	0	0
$721$	7.36932	0.274448
$722$	0	0
$723$	−2.80776	−0.104422
$724$	0	0
$725$	7.12311	0.264546
$726$	0	0
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.23106	0.119505
$732$	0	0
$733$	9.61553	0.355158	0.177579	−	0.984107i	$-0.443174\pi$
$733$	9.61553	0.355158	0.177579	+	0.984107i	$0.443174\pi$
$734$	0	0
$735$	−0.438447	−0.0161724
$736$	0	0
$737$	1.43845	0.0529859
$738$	0	0
$739$	1.36932	0.0503711	0.0251856	−	0.999683i	$-0.491982\pi$
$739$	1.36932	0.0503711	0.0251856	+	0.999683i	$0.491982\pi$
$740$	0	0
$741$	−5.75379	−0.211371
$742$	0	0
$743$	5.12311	0.187949	0.0939743	−	0.995575i	$-0.470043\pi$
$743$	5.12311	0.187949	0.0939743	+	0.995575i	$0.470043\pi$
$744$	0	0
$745$	−4.87689	−0.178676
$746$	0	0
$747$	−9.43845	−0.345335
$748$	0	0
$749$	−5.75379	−0.210239
$750$	0	0
$751$	−24.9848	−0.911710	−0.455855	−	0.890054i	$-0.650666\pi$
$751$	−24.9848	−0.911710	−0.455855	+	0.890054i	$0.650666\pi$
$752$	0	0
$753$	12.8078	0.466741
$754$	0	0
$755$	−10.5616	−0.384374
$756$	0	0
$757$	31.3693	1.14014	0.570069	−	0.821597i	$-0.306917\pi$
$757$	31.3693	1.14014	0.570069	+	0.821597i	$0.306917\pi$
$758$	0	0
$759$	−5.75379	−0.208849
$760$	0	0
$761$	−17.0540	−0.618206	−0.309103	−	0.951029i	$-0.600029\pi$
$761$	−17.0540	−0.618206	−0.309103	+	0.951029i	$0.600029\pi$
$762$	0	0
$763$	32.1771	1.16489
$764$	0	0
$765$	1.12311	0.0406060
$766$	0	0
$767$	−62.7386	−2.26536
$768$	0	0
$769$	39.6155	1.42857	0.714286	−	0.699854i	$-0.246751\pi$
$769$	39.6155	1.42857	0.714286	+	0.699854i	$0.246751\pi$
$770$	0	0
$771$	0.630683	0.0227135
$772$	0	0
$773$	−32.6695	−1.17504	−0.587520	−	0.809210i	$-0.699896\pi$
$773$	−32.6695	−1.17504	−0.587520	+	0.809210i	$0.699896\pi$
$774$	0	0
$775$	7.12311	0.255870
$776$	0	0
$777$	−14.5616	−0.522393
$778$	0	0
$779$	−6.73863	−0.241437
$780$	0	0
$781$	10.6998	0.382869
$782$	0	0
$783$	7.12311	0.254559
$784$	0	0
$785$	14.4924	0.517257
$786$	0	0
$787$	35.3693	1.26078	0.630390	−	0.776279i	$-0.282895\pi$
$787$	35.3693	1.26078	0.630390	+	0.776279i	$0.282895\pi$
$788$	0	0
$789$	12.4924	0.444742
$790$	0	0
$791$	−1.43845	−0.0511453
$792$	0	0
$793$	29.1231	1.03419
$794$	0	0
$795$	3.12311	0.110765
$796$	0	0
$797$	14.4233	0.510899	0.255450	−	0.966822i	$-0.417777\pi$
$797$	14.4233	0.510899	0.255450	+	0.966822i	$0.417777\pi$
$798$	0	0
$799$	−5.75379	−0.203554
$800$	0	0
$801$	−2.80776	−0.0992075
$802$	0	0
$803$	−16.0000	−0.564628
$804$	0	0
$805$	−10.2462	−0.361131
$806$	0	0
$807$	13.3693	0.470622
$808$	0	0
$809$	−14.0000	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$810$	0	0
$811$	34.6695	1.21741	0.608705	−	0.793396i	$-0.291689\pi$
$811$	34.6695	1.21741	0.608705	+	0.793396i	$0.291689\pi$
$812$	0	0
$813$	14.0000	0.491001
$814$	0	0
$815$	−13.4384	−0.470728
$816$	0	0
$817$	−3.23106	−0.113040
$818$	0	0
$819$	13.1231	0.458558
$820$	0	0
$821$	50.3542	1.75737	0.878686	−	0.477400i	$-0.158421\pi$
$821$	50.3542	1.75737	0.878686	+	0.477400i	$0.158421\pi$
$822$	0	0
$823$	−24.6307	−0.858572	−0.429286	−	0.903169i	$-0.641235\pi$
$823$	−24.6307	−0.858572	−0.429286	+	0.903169i	$0.641235\pi$
$824$	0	0
$825$	1.43845	0.0500803
$826$	0	0
$827$	−47.0540	−1.63623	−0.818114	−	0.575057i	$-0.804980\pi$
$827$	−47.0540	−1.63623	−0.818114	+	0.575057i	$0.804980\pi$
$828$	0	0
$829$	−49.8617	−1.73177	−0.865885	−	0.500243i	$-0.833244\pi$
$829$	−49.8617	−1.73177	−0.865885	+	0.500243i	$0.833244\pi$
$830$	0	0
$831$	5.19224	0.180117
$832$	0	0
$833$	−0.492423	−0.0170614
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.12311	0.246211
$838$	0	0
$839$	9.36932	0.323465	0.161732	−	0.986835i	$-0.448292\pi$
$839$	9.36932	0.323465	0.161732	+	0.986835i	$0.448292\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	0	0
$843$	2.00000	0.0688837
$844$	0	0
$845$	13.2462	0.455684
$846$	0	0
$847$	−22.8769	−0.786059
$848$	0	0
$849$	7.68466	0.263737
$850$	0	0
$851$	22.7386	0.779470
$852$	0	0
$853$	44.2462	1.51496	0.757481	−	0.652858i	$-0.226430\pi$
$853$	44.2462	1.51496	0.757481	+	0.652858i	$0.226430\pi$
$854$	0	0
$855$	−1.12311	−0.0384094
$856$	0	0
$857$	23.4384	0.800642	0.400321	−	0.916375i	$-0.368899\pi$
$857$	23.4384	0.800642	0.400321	+	0.916375i	$0.368899\pi$
$858$	0	0
$859$	17.7538	0.605751	0.302876	−	0.953030i	$-0.402053\pi$
$859$	17.7538	0.605751	0.302876	+	0.953030i	$0.402053\pi$
$860$	0	0
$861$	15.3693	0.523785
$862$	0	0
$863$	9.12311	0.310554	0.155277	−	0.987871i	$-0.450373\pi$
$863$	9.12311	0.310554	0.155277	+	0.987871i	$0.450373\pi$
$864$	0	0
$865$	−8.31534	−0.282730
$866$	0	0
$867$	−15.7386	−0.534512
$868$	0	0
$869$	−2.87689	−0.0975920
$870$	0	0
$871$	5.12311	0.173590
$872$	0	0
$873$	14.5616	0.492834
$874$	0	0
$875$	2.56155	0.0865963
$876$	0	0
$877$	−38.9848	−1.31642	−0.658212	−	0.752832i	$-0.728687\pi$
$877$	−38.9848	−1.31642	−0.658212	+	0.752832i	$0.728687\pi$
$878$	0	0
$879$	16.3153	0.550303
$880$	0	0
$881$	29.1922	0.983511	0.491756	−	0.870733i	$-0.336355\pi$
$881$	29.1922	0.983511	0.491756	+	0.870733i	$0.336355\pi$
$882$	0	0
$883$	−22.2462	−0.748645	−0.374322	−	0.927299i	$-0.622125\pi$
$883$	−22.2462	−0.748645	−0.374322	+	0.927299i	$0.622125\pi$
$884$	0	0
$885$	−12.2462	−0.411652
$886$	0	0
$887$	16.0000	0.537227	0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000	0.537227	0.268614	+	0.963248i	$0.413434\pi$
$888$	0	0
$889$	13.1231	0.440135
$890$	0	0
$891$	1.43845	0.0481898
$892$	0	0
$893$	5.75379	0.192543
$894$	0	0
$895$	−2.24621	−0.0750826
$896$	0	0
$897$	−20.4924	−0.684222
$898$	0	0
$899$	50.7386	1.69223
$900$	0	0
$901$	3.50758	0.116854
$902$	0	0
$903$	7.36932	0.245236
$904$	0	0
$905$	−23.6155	−0.785007
$906$	0	0
$907$	−3.82292	−0.126938	−0.0634690	−	0.997984i	$-0.520216\pi$
$907$	−3.82292	−0.126938	−0.0634690	+	0.997984i	$0.520216\pi$
$908$	0	0
$909$	−6.80776	−0.225799
$910$	0	0
$911$	9.19224	0.304552	0.152276	−	0.988338i	$-0.451340\pi$
$911$	9.19224	0.304552	0.152276	+	0.988338i	$0.451340\pi$
$912$	0	0
$913$	−13.5767	−0.449323
$914$	0	0
$915$	5.68466	0.187929
$916$	0	0
$917$	12.4924	0.412536
$918$	0	0
$919$	−30.9848	−1.02210	−0.511048	−	0.859552i	$-0.670743\pi$
$919$	−30.9848	−1.02210	−0.511048	+	0.859552i	$0.670743\pi$
$920$	0	0
$921$	20.8078	0.685639
$922$	0	0
$923$	38.1080	1.25434
$924$	0	0
$925$	−5.68466	−0.186910
$926$	0	0
$927$	2.87689	0.0944896
$928$	0	0
$929$	−25.8617	−0.848496	−0.424248	−	0.905546i	$-0.639462\pi$
$929$	−25.8617	−0.848496	−0.424248	+	0.905546i	$0.639462\pi$
$930$	0	0
$931$	0.492423	0.0161385
$932$	0	0
$933$	−8.49242	−0.278029
$934$	0	0
$935$	1.61553	0.0528334
$936$	0	0
$937$	23.6847	0.773744	0.386872	−	0.922133i	$-0.373556\pi$
$937$	23.6847	0.773744	0.386872	+	0.922133i	$0.373556\pi$
$938$	0	0
$939$	12.8078	0.417966
$940$	0	0
$941$	−43.4773	−1.41732	−0.708659	−	0.705551i	$-0.750700\pi$
$941$	−43.4773	−1.41732	−0.708659	+	0.705551i	$0.750700\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	2.56155	0.0833273
$946$	0	0
$947$	5.30019	0.172233	0.0861165	−	0.996285i	$-0.472554\pi$
$947$	5.30019	0.172233	0.0861165	+	0.996285i	$0.472554\pi$
$948$	0	0
$949$	−56.9848	−1.84981
$950$	0	0
$951$	−14.8769	−0.482416
$952$	0	0
$953$	36.3542	1.17763	0.588813	−	0.808269i	$-0.299595\pi$
$953$	36.3542	1.17763	0.588813	+	0.808269i	$0.299595\pi$
$954$	0	0
$955$	−5.12311	−0.165780
$956$	0	0
$957$	10.2462	0.331213
$958$	0	0
$959$	−11.6847	−0.377317
$960$	0	0
$961$	19.7386	0.636730
$962$	0	0
$963$	−2.24621	−0.0723831
$964$	0	0
$965$	16.8769	0.543286
$966$	0	0
$967$	−35.8617	−1.15324	−0.576618	−	0.817014i	$-0.695628\pi$
$967$	−35.8617	−1.15324	−0.576618	+	0.817014i	$0.695628\pi$
$968$	0	0
$969$	−1.26137	−0.0405209
$970$	0	0
$971$	−21.8617	−0.701577	−0.350788	−	0.936455i	$-0.614086\pi$
$971$	−21.8617	−0.701577	−0.350788	+	0.936455i	$0.614086\pi$
$972$	0	0
$973$	24.0000	0.769405
$974$	0	0
$975$	5.12311	0.164071
$976$	0	0
$977$	29.6155	0.947485	0.473742	−	0.880663i	$-0.342903\pi$
$977$	29.6155	0.947485	0.473742	+	0.880663i	$0.342903\pi$
$978$	0	0
$979$	−4.03882	−0.129081
$980$	0	0
$981$	12.5616	0.401060
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	−16.2462	−0.517647
$986$	0	0
$987$	−13.1231	−0.417713
$988$	0	0
$989$	−11.5076	−0.365920
$990$	0	0
$991$	10.4924	0.333303	0.166651	−	0.986016i	$-0.446705\pi$
$991$	10.4924	0.333303	0.166651	+	0.986016i	$0.446705\pi$
$992$	0	0
$993$	−8.56155	−0.271693
$994$	0	0
$995$	−13.9309	−0.441638
$996$	0	0
$997$	9.82292	0.311095	0.155547	−	0.987828i	$-0.450286\pi$
$997$	9.82292	0.311095	0.155547	+	0.987828i	$0.450286\pi$
$998$	0	0
$999$	−5.68466	−0.179855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.m.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.m.1.2	✓	2	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:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.56155 q^{7} +1.00000 q^{9} +1.43845 q^{11} +5.12311 q^{13} +1.00000 q^{15} +1.12311 q^{17} -1.12311 q^{19} +2.56155 q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +7.12311 q^{29} +7.12311 q^{31} +1.43845 q^{33} +2.56155 q^{35} -5.68466 q^{37} +5.12311 q^{39} +6.00000 q^{41} +2.87689 q^{43} +1.00000 q^{45} -5.12311 q^{47} -0.438447 q^{49} +1.12311 q^{51} +3.12311 q^{53} +1.43845 q^{55} -1.12311 q^{57} -12.2462 q^{59} +5.68466 q^{61} +2.56155 q^{63} +5.12311 q^{65} +1.00000 q^{67} -4.00000 q^{69} +7.43845 q^{71} -11.1231 q^{73} +1.00000 q^{75} +3.68466 q^{77} -2.00000 q^{79} +1.00000 q^{81} -9.43845 q^{83} +1.12311 q^{85} +7.12311 q^{87} -2.80776 q^{89} +13.1231 q^{91} +7.12311 q^{93} -1.12311 q^{95} +14.5616 q^{97} +1.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 2 q^{13} + 2 q^{15} - 6 q^{17} + 6 q^{19} + q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} + 6 q^{31} + 7 q^{33} + q^{35} + q^{37} + 2 q^{39} + 12 q^{41} + 14 q^{43} + 2 q^{45} - 2 q^{47} - 5 q^{49} - 6 q^{51} - 2 q^{53} + 7 q^{55} + 6 q^{57} - 8 q^{59} - q^{61} + q^{63} + 2 q^{65} + 2 q^{67} - 8 q^{69} + 19 q^{71} - 14 q^{73} + 2 q^{75} - 5 q^{77} - 4 q^{79} + 2 q^{81} - 23 q^{83} - 6 q^{85} + 6 q^{87} + 15 q^{89} + 18 q^{91} + 6 q^{93} + 6 q^{95} + 25 q^{97} + 7 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9 + 7 * q^11 + 2 * q^13 + 2 * q^15 - 6 * q^17 + 6 * q^19 + q^21 - 8 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 + 6 * q^31 + 7 * q^33 + q^35 + q^37 + 2 * q^39 + 12 * q^41 + 14 * q^43 + 2 * q^45 - 2 * q^47 - 5 * q^49 - 6 * q^51 - 2 * q^53 + 7 * q^55 + 6 * q^57 - 8 * q^59 - q^61 + q^63 + 2 * q^65 + 2 * q^67 - 8 * q^69 + 19 * q^71 - 14 * q^73 + 2 * q^75 - 5 * q^77 - 4 * q^79 + 2 * q^81 - 23 * q^83 - 6 * q^85 + 6 * q^87 + 15 * q^89 + 18 * q^91 + 6 * q^93 + 6 * q^95 + 25 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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