LMFDB - Embedded newform 5520.2.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, 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("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -3.18953 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -3.18953 q^{7} +1.00000 q^{9} -3.36266 q^{13} -1.00000 q^{15} +5.18953 q^{17} +3.18953 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.18953 q^{29} -2.17313 q^{31} -3.18953 q^{35} +9.53579 q^{37} +3.36266 q^{39} -6.55220 q^{41} +1.01641 q^{43} +1.00000 q^{45} -6.37907 q^{47} +3.17313 q^{49} -5.18953 q^{51} -2.55220 q^{53} -8.55220 q^{59} -3.01641 q^{61} -3.18953 q^{63} -3.36266 q^{65} +7.53579 q^{67} +1.00000 q^{69} -13.9149 q^{71} +8.37907 q^{73} -1.00000 q^{75} +7.74173 q^{79} +1.00000 q^{81} +9.91486 q^{83} +5.18953 q^{85} +1.18953 q^{87} +10.3463 q^{89} +10.7253 q^{91} +2.17313 q^{93} +3.01641 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 4 q^{13} - 3 q^{15} + 7 q^{17} + q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 5 q^{29} - q^{31} - q^{35} + 9 q^{37} - 4 q^{39} + 3 q^{41} + 3 q^{45} - 2 q^{47} + 4 q^{49} - 7 q^{51} + 15 q^{53} - 3 q^{59} - 6 q^{61} - q^{63} + 4 q^{65} + 3 q^{67} + 3 q^{69} - 5 q^{71} + 8 q^{73} - 3 q^{75} - 8 q^{79} + 3 q^{81} - 7 q^{83} + 7 q^{85} - 5 q^{87} + 20 q^{89} + 4 q^{91} + q^{93} + 6 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 + 4 * q^13 - 3 * q^15 + 7 * q^17 + q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 + 5 * q^29 - q^31 - q^35 + 9 * q^37 - 4 * q^39 + 3 * q^41 + 3 * q^45 - 2 * q^47 + 4 * q^49 - 7 * q^51 + 15 * q^53 - 3 * q^59 - 6 * q^61 - q^63 + 4 * q^65 + 3 * q^67 + 3 * q^69 - 5 * q^71 + 8 * q^73 - 3 * q^75 - 8 * q^79 + 3 * q^81 - 7 * q^83 + 7 * q^85 - 5 * q^87 + 20 * q^89 + 4 * q^91 + q^93 + 6 * q^97

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
$7$	−3.18953	−1.20553	−0.602765	−	0.797919i	$-0.705934\pi$
$7$	−3.18953	−1.20553	−0.602765	+	0.797919i	$0.705934\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−3.36266	−0.932634	−0.466317	−	0.884618i	$-0.654419\pi$
$13$	−3.36266	−0.932634	−0.466317	+	0.884618i	$0.654419\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.18953	1.25865	0.629323	−	0.777143i	$-0.283332\pi$
$17$	5.18953	1.25865	0.629323	+	0.777143i	$0.283332\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	3.18953	0.696013
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.18953	−0.220891	−0.110445	−	0.993882i	$-0.535228\pi$
$29$	−1.18953	−0.220891	−0.110445	+	0.993882i	$0.535228\pi$
$30$	0	0
$31$	−2.17313	−0.390305	−0.195153	−	0.980773i	$-0.562520\pi$
$31$	−2.17313	−0.390305	−0.195153	+	0.980773i	$0.562520\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.18953	−0.539130
$36$	0	0
$37$	9.53579	1.56767	0.783837	−	0.620967i	$-0.213260\pi$
$37$	9.53579	1.56767	0.783837	+	0.620967i	$0.213260\pi$
$38$	0	0
$39$	3.36266	0.538457
$40$	0	0
$41$	−6.55220	−1.02328	−0.511640	−	0.859200i	$-0.670962\pi$
$41$	−6.55220	−1.02328	−0.511640	+	0.859200i	$0.670962\pi$
$42$	0	0
$43$	1.01641	0.155001	0.0775003	−	0.996992i	$-0.475306\pi$
$43$	1.01641	0.155001	0.0775003	+	0.996992i	$0.475306\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−6.37907	−0.930483	−0.465241	−	0.885184i	$-0.654032\pi$
$47$	−6.37907	−0.930483	−0.465241	+	0.885184i	$0.654032\pi$
$48$	0	0
$49$	3.17313	0.453304
$50$	0	0
$51$	−5.18953	−0.726680
$52$	0	0
$53$	−2.55220	−0.350571	−0.175285	−	0.984518i	$-0.556085\pi$
$53$	−2.55220	−0.350571	−0.175285	+	0.984518i	$0.556085\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.55220	−1.11340	−0.556700	−	0.830713i	$-0.687933\pi$
$59$	−8.55220	−1.11340	−0.556700	+	0.830713i	$0.687933\pi$
$60$	0	0
$61$	−3.01641	−0.386211	−0.193106	−	0.981178i	$-0.561856\pi$
$61$	−3.01641	−0.386211	−0.193106	+	0.981178i	$0.561856\pi$
$62$	0	0
$63$	−3.18953	−0.401844
$64$	0	0
$65$	−3.36266	−0.417087
$66$	0	0
$67$	7.53579	0.920643	0.460322	−	0.887752i	$-0.347734\pi$
$67$	7.53579	0.920643	0.460322	+	0.887752i	$0.347734\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−13.9149	−1.65139	−0.825695	−	0.564117i	$-0.809217\pi$
$71$	−13.9149	−1.65139	−0.825695	+	0.564117i	$0.809217\pi$
$72$	0	0
$73$	8.37907	0.980696	0.490348	−	0.871527i	$-0.336870\pi$
$73$	8.37907	0.980696	0.490348	+	0.871527i	$0.336870\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.74173	0.871013	0.435506	−	0.900186i	$-0.356569\pi$
$79$	7.74173	0.871013	0.435506	+	0.900186i	$0.356569\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.91486	1.08830	0.544148	−	0.838989i	$-0.316853\pi$
$83$	9.91486	1.08830	0.544148	+	0.838989i	$0.316853\pi$
$84$	0	0
$85$	5.18953	0.562884
$86$	0	0
$87$	1.18953	0.127531
$88$	0	0
$89$	10.3463	1.09670	0.548350	−	0.836249i	$-0.315256\pi$
$89$	10.3463	1.09670	0.548350	+	0.836249i	$0.315256\pi$
$90$	0	0
$91$	10.7253	1.12432
$92$	0	0
$93$	2.17313	0.225343
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.01641	0.306270	0.153135	−	0.988205i	$-0.451063\pi$
$97$	3.01641	0.306270	0.153135	+	0.988205i	$0.451063\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.18953	−0.118363	−0.0591815	−	0.998247i	$-0.518849\pi$
$101$	−1.18953	−0.118363	−0.0591815	+	0.998247i	$0.518849\pi$
$102$	0	0
$103$	11.7417	1.15695	0.578473	−	0.815701i	$-0.303649\pi$
$103$	11.7417	1.15695	0.578473	+	0.815701i	$0.303649\pi$
$104$	0	0
$105$	3.18953	0.311267
$106$	0	0
$107$	7.18953	0.695038	0.347519	−	0.937673i	$-0.387024\pi$
$107$	7.18953	0.695038	0.347519	+	0.937673i	$0.387024\pi$
$108$	0	0
$109$	−10.0880	−0.966254	−0.483127	−	0.875550i	$-0.660499\pi$
$109$	−10.0880	−0.966254	−0.483127	+	0.875550i	$0.660499\pi$
$110$	0	0
$111$	−9.53579	−0.905097
$112$	0	0
$113$	15.9149	1.49714	0.748572	−	0.663054i	$-0.230740\pi$
$113$	15.9149	1.49714	0.748572	+	0.663054i	$0.230740\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−3.36266	−0.310878
$118$	0	0
$119$	−16.5522	−1.51734
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	6.55220	0.590792
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.36266	−0.120917	−0.0604583	−	0.998171i	$-0.519256\pi$
$127$	−1.36266	−0.120917	−0.0604583	+	0.998171i	$0.519256\pi$
$128$	0	0
$129$	−1.01641	−0.0894896
$130$	0	0
$131$	10.7253	0.937076	0.468538	−	0.883443i	$-0.344781\pi$
$131$	10.7253	0.937076	0.468538	+	0.883443i	$0.344781\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	3.01641	0.257709	0.128855	−	0.991664i	$-0.458870\pi$
$137$	3.01641	0.257709	0.128855	+	0.991664i	$0.458870\pi$
$138$	0	0
$139$	8.20594	0.696019	0.348009	−	0.937491i	$-0.386858\pi$
$139$	8.20594	0.696019	0.348009	+	0.937491i	$0.386858\pi$
$140$	0	0
$141$	6.37907	0.537214
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.18953	−0.0987854
$146$	0	0
$147$	−3.17313	−0.261715
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	5.18953	0.419549
$154$	0	0
$155$	−2.17313	−0.174550
$156$	0	0
$157$	−7.56860	−0.604040	−0.302020	−	0.953302i	$-0.597661\pi$
$157$	−7.56860	−0.604040	−0.302020	+	0.953302i	$0.597661\pi$
$158$	0	0
$159$	2.55220	0.202402
$160$	0	0
$161$	3.18953	0.251370
$162$	0	0
$163$	−5.10439	−0.399807	−0.199903	−	0.979816i	$-0.564063\pi$
$163$	−5.10439	−0.399807	−0.199903	+	0.979816i	$0.564063\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.62093	0.125431	0.0627157	−	0.998031i	$-0.480024\pi$
$167$	1.62093	0.125431	0.0627157	+	0.998031i	$0.480024\pi$
$168$	0	0
$169$	−1.69251	−0.130193
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.62093	0.275294	0.137647	−	0.990481i	$-0.456046\pi$
$173$	3.62093	0.275294	0.137647	+	0.990481i	$0.456046\pi$
$174$	0	0
$175$	−3.18953	−0.241106
$176$	0	0
$177$	8.55220	0.642822
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	16.1208	1.19825	0.599125	−	0.800656i	$-0.295515\pi$
$181$	16.1208	1.19825	0.599125	+	0.800656i	$0.295515\pi$
$182$	0	0
$183$	3.01641	0.222979
$184$	0	0
$185$	9.53579	0.701085
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.18953	0.232004
$190$	0	0
$191$	17.1044	1.23763	0.618815	−	0.785537i	$-0.287613\pi$
$191$	17.1044	1.23763	0.618815	+	0.785537i	$0.287613\pi$
$192$	0	0
$193$	11.1044	0.799312	0.399656	−	0.916665i	$-0.369130\pi$
$193$	11.1044	0.799312	0.399656	+	0.916665i	$0.369130\pi$
$194$	0	0
$195$	3.36266	0.240805
$196$	0	0
$197$	25.4835	1.81562	0.907811	−	0.419380i	$-0.137753\pi$
$197$	25.4835	1.81562	0.907811	+	0.419380i	$0.137753\pi$
$198$	0	0
$199$	1.36266	0.0965965	0.0482982	−	0.998833i	$-0.484620\pi$
$199$	1.36266	0.0965965	0.0482982	+	0.998833i	$0.484620\pi$
$200$	0	0
$201$	−7.53579	−0.531534
$202$	0	0
$203$	3.79406	0.266291
$204$	0	0
$205$	−6.55220	−0.457625
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−10.9313	−0.752539	−0.376270	−	0.926510i	$-0.622793\pi$
$211$	−10.9313	−0.752539	−0.376270	+	0.926510i	$0.622793\pi$
$212$	0	0
$213$	13.9149	0.953430
$214$	0	0
$215$	1.01641	0.0693184
$216$	0	0
$217$	6.93126	0.470525
$218$	0	0
$219$	−8.37907	−0.566205
$220$	0	0
$221$	−17.4506	−1.17386
$222$	0	0
$223$	−4.08798	−0.273752	−0.136876	−	0.990588i	$-0.543706\pi$
$223$	−4.08798	−0.273752	−0.136876	+	0.990588i	$0.543706\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	17.7745	1.17974	0.589869	−	0.807499i	$-0.299179\pi$
$227$	17.7745	1.17974	0.589869	+	0.807499i	$0.299179\pi$
$228$	0	0
$229$	20.0552	1.32528	0.662641	−	0.748937i	$-0.269435\pi$
$229$	20.0552	1.32528	0.662641	+	0.748937i	$0.269435\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.65375	0.108340	0.0541702	−	0.998532i	$-0.482749\pi$
$233$	1.65375	0.108340	0.0541702	+	0.998532i	$0.482749\pi$
$234$	0	0
$235$	−6.37907	−0.416125
$236$	0	0
$237$	−7.74173	−0.502879
$238$	0	0
$239$	−19.6014	−1.26791	−0.633955	−	0.773370i	$-0.718570\pi$
$239$	−19.6014	−1.26791	−0.633955	+	0.773370i	$0.718570\pi$
$240$	0	0
$241$	−10.3463	−0.666461	−0.333230	−	0.942845i	$-0.608139\pi$
$241$	−10.3463	−0.666461	−0.333230	+	0.942845i	$0.608139\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.17313	0.202724
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−9.91486	−0.628329
$250$	0	0
$251$	−2.31344	−0.146023	−0.0730116	−	0.997331i	$-0.523261\pi$
$251$	−2.31344	−0.146023	−0.0730116	+	0.997331i	$0.523261\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−5.18953	−0.324981
$256$	0	0
$257$	18.7581	1.17010	0.585050	−	0.810997i	$-0.301075\pi$
$257$	18.7581	1.17010	0.585050	+	0.810997i	$0.301075\pi$
$258$	0	0
$259$	−30.4147	−1.88988
$260$	0	0
$261$	−1.18953	−0.0736303
$262$	0	0
$263$	5.82687	0.359300	0.179650	−	0.983731i	$-0.442503\pi$
$263$	5.82687	0.359300	0.179650	+	0.983731i	$0.442503\pi$
$264$	0	0
$265$	−2.55220	−0.156780
$266$	0	0
$267$	−10.3463	−0.633180
$268$	0	0
$269$	8.43140	0.514071	0.257036	−	0.966402i	$-0.417254\pi$
$269$	8.43140	0.514071	0.257036	+	0.966402i	$0.417254\pi$
$270$	0	0
$271$	20.8984	1.26949	0.634745	−	0.772722i	$-0.281105\pi$
$271$	20.8984	1.26949	0.634745	+	0.772722i	$0.281105\pi$
$272$	0	0
$273$	−10.7253	−0.649126
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.4999	−1.35189	−0.675943	−	0.736954i	$-0.736263\pi$
$277$	−22.4999	−1.35189	−0.675943	+	0.736954i	$0.736263\pi$
$278$	0	0
$279$	−2.17313	−0.130102
$280$	0	0
$281$	−2.41188	−0.143881	−0.0719404	−	0.997409i	$-0.522919\pi$
$281$	−2.41188	−0.143881	−0.0719404	+	0.997409i	$0.522919\pi$
$282$	0	0
$283$	19.8820	1.18186	0.590932	−	0.806721i	$-0.298760\pi$
$283$	19.8820	1.18186	0.590932	+	0.806721i	$0.298760\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.8984	1.23360
$288$	0	0
$289$	9.93126	0.584192
$290$	0	0
$291$	−3.01641	−0.176825
$292$	0	0
$293$	27.3103	1.59549	0.797743	−	0.602997i	$-0.206027\pi$
$293$	27.3103	1.59549	0.797743	+	0.602997i	$0.206027\pi$
$294$	0	0
$295$	−8.55220	−0.497928
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.36266	0.194468
$300$	0	0
$301$	−3.24186	−0.186858
$302$	0	0
$303$	1.18953	0.0683369
$304$	0	0
$305$	−3.01641	−0.172719
$306$	0	0
$307$	14.7253	0.840419	0.420209	−	0.907427i	$-0.361957\pi$
$307$	14.7253	0.840419	0.420209	+	0.907427i	$0.361957\pi$
$308$	0	0
$309$	−11.7417	−0.667964
$310$	0	0
$311$	13.7745	0.781083	0.390541	−	0.920585i	$-0.372288\pi$
$311$	13.7745	0.781083	0.390541	+	0.920585i	$0.372288\pi$
$312$	0	0
$313$	−7.56860	−0.427803	−0.213901	−	0.976855i	$-0.568617\pi$
$313$	−7.56860	−0.427803	−0.213901	+	0.976855i	$0.568617\pi$
$314$	0	0
$315$	−3.18953	−0.179710
$316$	0	0
$317$	−24.2088	−1.35970	−0.679850	−	0.733351i	$-0.737955\pi$
$317$	−24.2088	−1.35970	−0.679850	+	0.733351i	$0.737955\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−7.18953	−0.401281
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−3.36266	−0.186527
$326$	0	0
$327$	10.0880	0.557867
$328$	0	0
$329$	20.3463	1.12173
$330$	0	0
$331$	14.5850	0.801665	0.400832	−	0.916151i	$-0.368721\pi$
$331$	14.5850	0.801665	0.400832	+	0.916151i	$0.368721\pi$
$332$	0	0
$333$	9.53579	0.522558
$334$	0	0
$335$	7.53579	0.411724
$336$	0	0
$337$	−30.4999	−1.66143	−0.830717	−	0.556695i	$-0.812069\pi$
$337$	−30.4999	−1.66143	−0.830717	+	0.556695i	$0.812069\pi$
$338$	0	0
$339$	−15.9149	−0.864376
$340$	0	0
$341$	0	0
$342$	0	0
$343$	12.2059	0.659059
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	12.9313	0.692195	0.346097	−	0.938199i	$-0.387507\pi$
$349$	12.9313	0.692195	0.346097	+	0.938199i	$0.387507\pi$
$350$	0	0
$351$	3.36266	0.179486
$352$	0	0
$353$	−17.4835	−0.930551	−0.465275	−	0.885166i	$-0.654045\pi$
$353$	−17.4835	−0.930551	−0.465275	+	0.885166i	$0.654045\pi$
$354$	0	0
$355$	−13.9149	−0.738524
$356$	0	0
$357$	16.5522	0.876035
$358$	0	0
$359$	1.10439	0.0582875	0.0291438	−	0.999575i	$-0.490722\pi$
$359$	1.10439	0.0582875	0.0291438	+	0.999575i	$0.490722\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	8.37907	0.438580
$366$	0	0
$367$	−9.56860	−0.499477	−0.249738	−	0.968313i	$-0.580345\pi$
$367$	−9.56860	−0.499477	−0.249738	+	0.968313i	$0.580345\pi$
$368$	0	0
$369$	−6.55220	−0.341094
$370$	0	0
$371$	8.14031	0.422624
$372$	0	0
$373$	−18.4342	−0.954489	−0.477244	−	0.878771i	$-0.658364\pi$
$373$	−18.4342	−0.954489	−0.477244	+	0.878771i	$0.658364\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	18.0328	0.926283	0.463142	−	0.886284i	$-0.346722\pi$
$379$	18.0328	0.926283	0.463142	+	0.886284i	$0.346722\pi$
$380$	0	0
$381$	1.36266	0.0698113
$382$	0	0
$383$	5.82687	0.297739	0.148870	−	0.988857i	$-0.452437\pi$
$383$	5.82687	0.297739	0.148870	+	0.988857i	$0.452437\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.01641	0.0516669
$388$	0	0
$389$	3.27468	0.166033	0.0830164	−	0.996548i	$-0.473545\pi$
$389$	3.27468	0.166033	0.0830164	+	0.996548i	$0.473545\pi$
$390$	0	0
$391$	−5.18953	−0.262446
$392$	0	0
$393$	−10.7253	−0.541021
$394$	0	0
$395$	7.74173	0.389529
$396$	0	0
$397$	10.0880	0.506301	0.253151	−	0.967427i	$-0.418533\pi$
$397$	10.0880	0.506301	0.253151	+	0.967427i	$0.418533\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	34.9341	1.74453	0.872263	−	0.489037i	$-0.162652\pi$
$401$	34.9341	1.74453	0.872263	+	0.489037i	$0.162652\pi$
$402$	0	0
$403$	7.30749	0.364012
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	21.2775	1.05211	0.526053	−	0.850452i	$-0.323671\pi$
$409$	21.2775	1.05211	0.526053	+	0.850452i	$0.323671\pi$
$410$	0	0
$411$	−3.01641	−0.148788
$412$	0	0
$413$	27.2775	1.34224
$414$	0	0
$415$	9.91486	0.486701
$416$	0	0
$417$	−8.20594	−0.401847
$418$	0	0
$419$	−23.4835	−1.14724	−0.573621	−	0.819121i	$-0.694462\pi$
$419$	−23.4835	−1.14724	−0.573621	+	0.819121i	$0.694462\pi$
$420$	0	0
$421$	18.8461	0.918504	0.459252	−	0.888306i	$-0.348118\pi$
$421$	18.8461	0.918504	0.459252	+	0.888306i	$0.348118\pi$
$422$	0	0
$423$	−6.37907	−0.310161
$424$	0	0
$425$	5.18953	0.251729
$426$	0	0
$427$	9.62093	0.465590
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.03281	−0.0979172	−0.0489586	−	0.998801i	$-0.515590\pi$
$431$	−2.03281	−0.0979172	−0.0489586	+	0.998801i	$0.515590\pi$
$432$	0	0
$433$	−0.497025	−0.0238855	−0.0119427	−	0.999929i	$-0.503802\pi$
$433$	−0.497025	−0.0238855	−0.0119427	+	0.999929i	$0.503802\pi$
$434$	0	0
$435$	1.18953	0.0570338
$436$	0	0
$437$	0	0
$438$	0	0
$439$	19.4178	0.926763	0.463381	−	0.886159i	$-0.346636\pi$
$439$	19.4178	0.926763	0.463381	+	0.886159i	$0.346636\pi$
$440$	0	0
$441$	3.17313	0.151101
$442$	0	0
$443$	26.3791	1.25331	0.626654	−	0.779298i	$-0.284424\pi$
$443$	26.3791	1.25331	0.626654	+	0.779298i	$0.284424\pi$
$444$	0	0
$445$	10.3463	0.490460
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	5.10155	0.240757	0.120379	−	0.992728i	$-0.461589\pi$
$449$	5.10155	0.240757	0.120379	+	0.992728i	$0.461589\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.7253	0.502811
$456$	0	0
$457$	−16.2611	−0.760663	−0.380331	−	0.924850i	$-0.624190\pi$
$457$	−16.2611	−0.760663	−0.380331	+	0.924850i	$0.624190\pi$
$458$	0	0
$459$	−5.18953	−0.242227
$460$	0	0
$461$	−27.7745	−1.29359	−0.646795	−	0.762664i	$-0.723891\pi$
$461$	−27.7745	−1.29359	−0.646795	+	0.762664i	$0.723891\pi$
$462$	0	0
$463$	31.7417	1.47516	0.737582	−	0.675258i	$-0.235968\pi$
$463$	31.7417	1.47516	0.737582	+	0.675258i	$0.235968\pi$
$464$	0	0
$465$	2.17313	0.100776
$466$	0	0
$467$	1.73889	0.0804662	0.0402331	−	0.999190i	$-0.487190\pi$
$467$	1.73889	0.0804662	0.0402331	+	0.999190i	$0.487190\pi$
$468$	0	0
$469$	−24.0357	−1.10986
$470$	0	0
$471$	7.56860	0.348743
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.55220	−0.116857
$478$	0	0
$479$	−37.8625	−1.72998	−0.864992	−	0.501787i	$-0.832676\pi$
$479$	−37.8625	−1.72998	−0.864992	+	0.501787i	$0.832676\pi$
$480$	0	0
$481$	−32.0656	−1.46207
$482$	0	0
$483$	−3.18953	−0.145129
$484$	0	0
$485$	3.01641	0.136968
$486$	0	0
$487$	−18.0552	−0.818158	−0.409079	−	0.912499i	$-0.634150\pi$
$487$	−18.0552	−0.818158	−0.409079	+	0.912499i	$0.634150\pi$
$488$	0	0
$489$	5.10439	0.230829
$490$	0	0
$491$	−4.89845	−0.221064	−0.110532	−	0.993873i	$-0.535255\pi$
$491$	−4.89845	−0.221064	−0.110532	+	0.993873i	$0.535255\pi$
$492$	0	0
$493$	−6.17313	−0.278024
$494$	0	0
$495$	0	0
$496$	0	0
$497$	44.3819	1.99080
$498$	0	0
$499$	43.1072	1.92974	0.964872	−	0.262719i	$-0.0846193\pi$
$499$	43.1072	1.92974	0.964872	+	0.262719i	$0.0846193\pi$
$500$	0	0
$501$	−1.62093	−0.0724179
$502$	0	0
$503$	16.5522	0.738026	0.369013	−	0.929424i	$-0.379696\pi$
$503$	16.5522	0.738026	0.369013	+	0.929424i	$0.379696\pi$
$504$	0	0
$505$	−1.18953	−0.0529336
$506$	0	0
$507$	1.69251	0.0751670
$508$	0	0
$509$	−2.95078	−0.130791	−0.0653955	−	0.997859i	$-0.520831\pi$
$509$	−2.95078	−0.130791	−0.0653955	+	0.997859i	$0.520831\pi$
$510$	0	0
$511$	−26.7253	−1.18226
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.7417	0.517402
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−3.62093	−0.158941
$520$	0	0
$521$	−0.379068	−0.0166073	−0.00830364	−	0.999966i	$-0.502643\pi$
$521$	−0.379068	−0.0166073	−0.00830364	+	0.999966i	$0.502643\pi$
$522$	0	0
$523$	14.9836	0.655187	0.327593	−	0.944819i	$-0.393762\pi$
$523$	14.9836	0.655187	0.327593	+	0.944819i	$0.393762\pi$
$524$	0	0
$525$	3.18953	0.139203
$526$	0	0
$527$	−11.2775	−0.491256
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−8.55220	−0.371134
$532$	0	0
$533$	22.0328	0.954347
$534$	0	0
$535$	7.18953	0.310831
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10.0656	0.432755	0.216378	−	0.976310i	$-0.430576\pi$
$541$	10.0656	0.432755	0.216378	+	0.976310i	$0.430576\pi$
$542$	0	0
$543$	−16.1208	−0.691810
$544$	0	0
$545$	−10.0880	−0.432122
$546$	0	0
$547$	−17.4506	−0.746136	−0.373068	−	0.927804i	$-0.621694\pi$
$547$	−17.4506	−0.746136	−0.373068	+	0.927804i	$0.621694\pi$
$548$	0	0
$549$	−3.01641	−0.128737
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−24.6925	−1.05003
$554$	0	0
$555$	−9.53579	−0.404772
$556$	0	0
$557$	12.9313	0.547915	0.273958	−	0.961742i	$-0.411667\pi$
$557$	12.9313	0.547915	0.273958	+	0.961742i	$0.411667\pi$
$558$	0	0
$559$	−3.41783	−0.144559
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.8105	1.04564	0.522818	−	0.852444i	$-0.324881\pi$
$563$	24.8105	1.04564	0.522818	+	0.852444i	$0.324881\pi$
$564$	0	0
$565$	15.9149	0.669543
$566$	0	0
$567$	−3.18953	−0.133948
$568$	0	0
$569$	32.0328	1.34289	0.671443	−	0.741056i	$-0.265675\pi$
$569$	32.0328	1.34289	0.671443	+	0.741056i	$0.265675\pi$
$570$	0	0
$571$	0.280628	0.0117439	0.00587195	−	0.999983i	$-0.498131\pi$
$571$	0.280628	0.0117439	0.00587195	+	0.999983i	$0.498131\pi$
$572$	0	0
$573$	−17.1044	−0.714546
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−33.1372	−1.37952	−0.689760	−	0.724038i	$-0.742284\pi$
$577$	−33.1372	−1.37952	−0.689760	+	0.724038i	$0.742284\pi$
$578$	0	0
$579$	−11.1044	−0.461483
$580$	0	0
$581$	−31.6238	−1.31198
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−3.36266	−0.139029
$586$	0	0
$587$	41.8625	1.72785	0.863926	−	0.503619i	$-0.167999\pi$
$587$	41.8625	1.72785	0.863926	+	0.503619i	$0.167999\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−25.4835	−1.04825
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−16.5522	−0.678574
$596$	0	0
$597$	−1.36266	−0.0557700
$598$	0	0
$599$	−36.4342	−1.48866	−0.744331	−	0.667811i	$-0.767232\pi$
$599$	−36.4342	−1.48866	−0.744331	+	0.667811i	$0.767232\pi$
$600$	0	0
$601$	−40.5850	−1.65550	−0.827749	−	0.561099i	$-0.810379\pi$
$601$	−40.5850	−1.65550	−0.827749	+	0.561099i	$0.810379\pi$
$602$	0	0
$603$	7.53579	0.306881
$604$	0	0
$605$	−11.0000	−0.447214
$606$	0	0
$607$	9.36266	0.380019	0.190009	−	0.981782i	$-0.439148\pi$
$607$	9.36266	0.380019	0.190009	+	0.981782i	$0.439148\pi$
$608$	0	0
$609$	−3.79406	−0.153743
$610$	0	0
$611$	21.4506	0.867800
$612$	0	0
$613$	−13.6761	−0.552373	−0.276186	−	0.961104i	$-0.589071\pi$
$613$	−13.6761	−0.552373	−0.276186	+	0.961104i	$0.589071\pi$
$614$	0	0
$615$	6.55220	0.264210
$616$	0	0
$617$	−40.6730	−1.63743	−0.818717	−	0.574198i	$-0.805314\pi$
$617$	−40.6730	−1.63743	−0.818717	+	0.574198i	$0.805314\pi$
$618$	0	0
$619$	20.7581	0.834340	0.417170	−	0.908828i	$-0.363022\pi$
$619$	20.7581	0.834340	0.417170	+	0.908828i	$0.363022\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−32.9997	−1.32211
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	49.4863	1.97315
$630$	0	0
$631$	14.8133	0.589708	0.294854	−	0.955542i	$-0.404729\pi$
$631$	14.8133	0.589708	0.294854	+	0.955542i	$0.404729\pi$
$632$	0	0
$633$	10.9313	0.434479
$634$	0	0
$635$	−1.36266	−0.0540756
$636$	0	0
$637$	−10.6702	−0.422767
$638$	0	0
$639$	−13.9149	−0.550463
$640$	0	0
$641$	6.41188	0.253254	0.126627	−	0.991950i	$-0.459585\pi$
$641$	6.41188	0.253254	0.126627	+	0.991950i	$0.459585\pi$
$642$	0	0
$643$	13.5030	0.532505	0.266253	−	0.963903i	$-0.414214\pi$
$643$	13.5030	0.532505	0.266253	+	0.963903i	$0.414214\pi$
$644$	0	0
$645$	−1.01641	−0.0400210
$646$	0	0
$647$	−13.4506	−0.528799	−0.264400	−	0.964413i	$-0.585174\pi$
$647$	−13.4506	−0.528799	−0.264400	+	0.964413i	$0.585174\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−6.93126	−0.271658
$652$	0	0
$653$	−0.961236	−0.0376161	−0.0188080	−	0.999823i	$-0.505987\pi$
$653$	−0.961236	−0.0376161	−0.0188080	+	0.999823i	$0.505987\pi$
$654$	0	0
$655$	10.7253	0.419073
$656$	0	0
$657$	8.37907	0.326899
$658$	0	0
$659$	33.5163	1.30561	0.652804	−	0.757527i	$-0.273592\pi$
$659$	33.5163	1.30561	0.652804	+	0.757527i	$0.273592\pi$
$660$	0	0
$661$	13.6761	0.531939	0.265969	−	0.963981i	$-0.414308\pi$
$661$	13.6761	0.531939	0.265969	+	0.963981i	$0.414308\pi$
$662$	0	0
$663$	17.4506	0.677727
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.18953	0.0460589
$668$	0	0
$669$	4.08798	0.158051
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−41.4178	−1.59654	−0.798270	−	0.602300i	$-0.794251\pi$
$673$	−41.4178	−1.59654	−0.798270	+	0.602300i	$0.794251\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−41.6238	−1.59973	−0.799866	−	0.600179i	$-0.795096\pi$
$677$	−41.6238	−1.59973	−0.799866	+	0.600179i	$0.795096\pi$
$678$	0	0
$679$	−9.62093	−0.369217
$680$	0	0
$681$	−17.7745	−0.681122
$682$	0	0
$683$	36.4119	1.39326	0.696631	−	0.717430i	$-0.254682\pi$
$683$	36.4119	1.39326	0.696631	+	0.717430i	$0.254682\pi$
$684$	0	0
$685$	3.01641	0.115251
$686$	0	0
$687$	−20.0552	−0.765152
$688$	0	0
$689$	8.58217	0.326955
$690$	0	0
$691$	−18.6597	−0.709848	−0.354924	−	0.934895i	$-0.615493\pi$
$691$	−18.6597	−0.709848	−0.354924	+	0.934895i	$0.615493\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.20594	0.311269
$696$	0	0
$697$	−34.0028	−1.28795
$698$	0	0
$699$	−1.65375	−0.0625504
$700$	0	0
$701$	24.2088	0.914353	0.457177	−	0.889376i	$-0.348861\pi$
$701$	24.2088	0.914353	0.457177	+	0.889376i	$0.348861\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	6.37907	0.240250
$706$	0	0
$707$	3.79406	0.142690
$708$	0	0
$709$	4.98359	0.187163	0.0935814	−	0.995612i	$-0.470168\pi$
$709$	4.98359	0.187163	0.0935814	+	0.995612i	$0.470168\pi$
$710$	0	0
$711$	7.74173	0.290338
$712$	0	0
$713$	2.17313	0.0813843
$714$	0	0
$715$	0	0
$716$	0	0
$717$	19.6014	0.732028
$718$	0	0
$719$	24.1236	0.899660	0.449830	−	0.893114i	$-0.351485\pi$
$719$	24.1236	0.899660	0.449830	+	0.893114i	$0.351485\pi$
$720$	0	0
$721$	−37.4506	−1.39473
$722$	0	0
$723$	10.3463	0.384781
$724$	0	0
$725$	−1.18953	−0.0441782
$726$	0	0
$727$	0.640179	0.0237429	0.0118715	−	0.999930i	$-0.496221\pi$
$727$	0.640179	0.0237429	0.0118715	+	0.999930i	$0.496221\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.27468	0.195091
$732$	0	0
$733$	4.26111	0.157388	0.0786939	−	0.996899i	$-0.474925\pi$
$733$	4.26111	0.157388	0.0786939	+	0.996899i	$0.474925\pi$
$734$	0	0
$735$	−3.17313	−0.117043
$736$	0	0
$737$	0	0
$738$	0	0
$739$	41.3103	1.51963	0.759813	−	0.650142i	$-0.225291\pi$
$739$	41.3103	1.51963	0.759813	+	0.650142i	$0.225291\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.4506	1.08044	0.540220	−	0.841524i	$-0.318341\pi$
$743$	29.4506	1.08044	0.540220	+	0.841524i	$0.318341\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	9.91486	0.362766
$748$	0	0
$749$	−22.9313	−0.837890
$750$	0	0
$751$	−45.1924	−1.64909	−0.824547	−	0.565794i	$-0.808570\pi$
$751$	−45.1924	−1.64909	−0.824547	+	0.565794i	$0.808570\pi$
$752$	0	0
$753$	2.31344	0.0843065
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−34.9864	−1.27160	−0.635802	−	0.771852i	$-0.719330\pi$
$757$	−34.9864	−1.27160	−0.635802	+	0.771852i	$0.719330\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.5522	−0.817516	−0.408758	−	0.912643i	$-0.634038\pi$
$761$	−22.5522	−0.817516	−0.408758	+	0.912643i	$0.634038\pi$
$762$	0	0
$763$	32.1760	1.16485
$764$	0	0
$765$	5.18953	0.187628
$766$	0	0
$767$	28.7581	1.03840
$768$	0	0
$769$	32.9669	1.18882	0.594409	−	0.804163i	$-0.297386\pi$
$769$	32.9669	1.18882	0.594409	+	0.804163i	$0.297386\pi$
$770$	0	0
$771$	−18.7581	−0.675558
$772$	0	0
$773$	−1.48346	−0.0533563	−0.0266781	−	0.999644i	$-0.508493\pi$
$773$	−1.48346	−0.0533563	−0.0266781	+	0.999644i	$0.508493\pi$
$774$	0	0
$775$	−2.17313	−0.0780610
$776$	0	0
$777$	30.4147	1.09112
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.18953	0.0425105
$784$	0	0
$785$	−7.56860	−0.270135
$786$	0	0
$787$	−49.4639	−1.76320	−0.881600	−	0.471998i	$-0.843533\pi$
$787$	−49.4639	−1.76320	−0.881600	+	0.471998i	$0.843533\pi$
$788$	0	0
$789$	−5.82687	−0.207442
$790$	0	0
$791$	−50.7610	−1.80485
$792$	0	0
$793$	10.1432	0.360194
$794$	0	0
$795$	2.55220	0.0905170
$796$	0	0
$797$	−14.7225	−0.521497	−0.260749	−	0.965407i	$-0.583969\pi$
$797$	−14.7225	−0.521497	−0.260749	+	0.965407i	$0.583969\pi$
$798$	0	0
$799$	−33.1044	−1.17115
$800$	0	0
$801$	10.3463	0.365567
$802$	0	0
$803$	0	0
$804$	0	0
$805$	3.18953	0.112416
$806$	0	0
$807$	−8.43140	−0.296799
$808$	0	0
$809$	21.7941	0.766238	0.383119	−	0.923699i	$-0.374850\pi$
$809$	21.7941	0.766238	0.383119	+	0.923699i	$0.374850\pi$
$810$	0	0
$811$	24.6178	0.864449	0.432224	−	0.901766i	$-0.357729\pi$
$811$	24.6178	0.864449	0.432224	+	0.901766i	$0.357729\pi$
$812$	0	0
$813$	−20.8984	−0.732941
$814$	0	0
$815$	−5.10439	−0.178799
$816$	0	0
$817$	0	0
$818$	0	0
$819$	10.7253	0.374773
$820$	0	0
$821$	37.9177	1.32334	0.661668	−	0.749797i	$-0.269849\pi$
$821$	37.9177	1.32334	0.661668	+	0.749797i	$0.269849\pi$
$822$	0	0
$823$	−16.4342	−0.572862	−0.286431	−	0.958101i	$-0.592469\pi$
$823$	−16.4342	−0.572862	−0.286431	+	0.958101i	$0.592469\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.7058	−0.859105	−0.429553	−	0.903042i	$-0.641329\pi$
$827$	−24.7058	−0.859105	−0.429553	+	0.903042i	$0.641329\pi$
$828$	0	0
$829$	34.1013	1.18439	0.592193	−	0.805796i	$-0.298262\pi$
$829$	34.1013	1.18439	0.592193	+	0.805796i	$0.298262\pi$
$830$	0	0
$831$	22.4999	0.780512
$832$	0	0
$833$	16.4671	0.570550
$834$	0	0
$835$	1.62093	0.0560947
$836$	0	0
$837$	2.17313	0.0751143
$838$	0	0
$839$	26.6207	0.919047	0.459524	−	0.888166i	$-0.348020\pi$
$839$	26.6207	0.919047	0.459524	+	0.888166i	$0.348020\pi$
$840$	0	0
$841$	−27.5850	−0.951207
$842$	0	0
$843$	2.41188	0.0830696
$844$	0	0
$845$	−1.69251	−0.0582241
$846$	0	0
$847$	35.0849	1.20553
$848$	0	0
$849$	−19.8820	−0.682350
$850$	0	0
$851$	−9.53579	−0.326883
$852$	0	0
$853$	−19.0820	−0.653356	−0.326678	−	0.945136i	$-0.605929\pi$
$853$	−19.0820	−0.653356	−0.326678	+	0.945136i	$0.605929\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.54935	−0.155403	−0.0777015	−	0.996977i	$-0.524758\pi$
$857$	−4.54935	−0.155403	−0.0777015	+	0.996977i	$0.524758\pi$
$858$	0	0
$859$	−43.1072	−1.47080	−0.735400	−	0.677633i	$-0.763006\pi$
$859$	−43.1072	−1.47080	−0.735400	+	0.677633i	$0.763006\pi$
$860$	0	0
$861$	−20.8984	−0.712217
$862$	0	0
$863$	−9.10439	−0.309917	−0.154959	−	0.987921i	$-0.549524\pi$
$863$	−9.10439	−0.309917	−0.154959	+	0.987921i	$0.549524\pi$
$864$	0	0
$865$	3.62093	0.123115
$866$	0	0
$867$	−9.93126	−0.337283
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−25.3403	−0.858623
$872$	0	0
$873$	3.01641	0.102090
$874$	0	0
$875$	−3.18953	−0.107826
$876$	0	0
$877$	17.1596	0.579437	0.289719	−	0.957112i	$-0.406438\pi$
$877$	17.1596	0.579437	0.289719	+	0.957112i	$0.406438\pi$
$878$	0	0
$879$	−27.3103	−0.921155
$880$	0	0
$881$	−54.4176	−1.83337	−0.916687	−	0.399606i	$-0.869147\pi$
$881$	−54.4176	−1.83337	−0.916687	+	0.399606i	$0.869147\pi$
$882$	0	0
$883$	18.3791	0.618505	0.309252	−	0.950980i	$-0.399921\pi$
$883$	18.3791	0.618505	0.309252	+	0.950980i	$0.399921\pi$
$884$	0	0
$885$	8.55220	0.287479
$886$	0	0
$887$	39.7641	1.33515	0.667574	−	0.744544i	$-0.267333\pi$
$887$	39.7641	1.33515	0.667574	+	0.744544i	$0.267333\pi$
$888$	0	0
$889$	4.34625	0.145769
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.36266	−0.112276
$898$	0	0
$899$	2.58501	0.0862149
$900$	0	0
$901$	−13.2447	−0.441245
$902$	0	0
$903$	3.24186	0.107882
$904$	0	0
$905$	16.1208	0.535873
$906$	0	0
$907$	15.0192	0.498706	0.249353	−	0.968413i	$-0.419782\pi$
$907$	15.0192	0.498706	0.249353	+	0.968413i	$0.419782\pi$
$908$	0	0
$909$	−1.18953	−0.0394544
$910$	0	0
$911$	−19.2419	−0.637511	−0.318756	−	0.947837i	$-0.603265\pi$
$911$	−19.2419	−0.637511	−0.318756	+	0.947837i	$0.603265\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.01641	0.0997193
$916$	0	0
$917$	−34.2088	−1.12967
$918$	0	0
$919$	−14.2255	−0.469255	−0.234627	−	0.972085i	$-0.575387\pi$
$919$	−14.2255	−0.469255	−0.234627	+	0.972085i	$0.575387\pi$
$920$	0	0
$921$	−14.7253	−0.485216
$922$	0	0
$923$	46.7909	1.54014
$924$	0	0
$925$	9.53579	0.313535
$926$	0	0
$927$	11.7417	0.385649
$928$	0	0
$929$	−23.6566	−0.776147	−0.388074	−	0.921628i	$-0.626859\pi$
$929$	−23.6566	−0.776147	−0.388074	+	0.921628i	$0.626859\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−13.7745	−0.450958
$934$	0	0
$935$	0	0
$936$	0	0
$937$	45.9177	1.50007	0.750033	−	0.661401i	$-0.230038\pi$
$937$	45.9177	1.50007	0.750033	+	0.661401i	$0.230038\pi$
$938$	0	0
$939$	7.56860	0.246992
$940$	0	0
$941$	−26.6925	−0.870151	−0.435075	−	0.900394i	$-0.643278\pi$
$941$	−26.6925	−0.870151	−0.435075	+	0.900394i	$0.643278\pi$
$942$	0	0
$943$	6.55220	0.213369
$944$	0	0
$945$	3.18953	0.103756
$946$	0	0
$947$	16.9341	0.550284	0.275142	−	0.961404i	$-0.411275\pi$
$947$	16.9341	0.550284	0.275142	+	0.961404i	$0.411275\pi$
$948$	0	0
$949$	−28.1760	−0.914631
$950$	0	0
$951$	24.2088	0.785024
$952$	0	0
$953$	−12.4671	−0.403847	−0.201924	−	0.979401i	$-0.564719\pi$
$953$	−12.4671	−0.403847	−0.201924	+	0.979401i	$0.564719\pi$
$954$	0	0
$955$	17.1044	0.553485
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.62093	−0.310676
$960$	0	0
$961$	−26.2775	−0.847662
$962$	0	0
$963$	7.18953	0.231679
$964$	0	0
$965$	11.1044	0.357463
$966$	0	0
$967$	−54.1208	−1.74041	−0.870204	−	0.492692i	$-0.836013\pi$
$967$	−54.1208	−1.74041	−0.870204	+	0.492692i	$0.836013\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30.9669	0.993776	0.496888	−	0.867815i	$-0.334476\pi$
$971$	30.9669	0.993776	0.496888	+	0.867815i	$0.334476\pi$
$972$	0	0
$973$	−26.1731	−0.839072
$974$	0	0
$975$	3.36266	0.107691
$976$	0	0
$977$	0.326738	0.0104533	0.00522664	−	0.999986i	$-0.498336\pi$
$977$	0.326738	0.0104533	0.00522664	+	0.999986i	$0.498336\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−10.0880	−0.322085
$982$	0	0
$983$	−42.2444	−1.34739	−0.673694	−	0.739010i	$-0.735293\pi$
$983$	−42.2444	−1.34739	−0.673694	+	0.739010i	$0.735293\pi$
$984$	0	0
$985$	25.4835	0.811971
$986$	0	0
$987$	−20.3463	−0.647628
$988$	0	0
$989$	−1.01641	−0.0323199
$990$	0	0
$991$	−24.9641	−0.793010	−0.396505	−	0.918033i	$-0.629777\pi$
$991$	−24.9641	−0.793010	−0.396505	+	0.918033i	$0.629777\pi$
$992$	0	0
$993$	−14.5850	−0.462841
$994$	0	0
$995$	1.36266	0.0431993
$996$	0	0
$997$	40.8789	1.29465	0.647324	−	0.762215i	$-0.275888\pi$
$997$	40.8789	1.29465	0.647324	+	0.762215i	$0.275888\pi$
$998$	0	0
$999$	−9.53579	−0.301699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.ca.1.1		3
4.3	odd	2		2760.2.a.t.1.3	✓	3
12.11	even	2		8280.2.a.bh.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.t.1.3	✓	3	4.3	odd	2
5520.2.a.ca.1.1		3	1.1	even	1	trivial
8280.2.a.bh.1.3		3	12.11	even	2

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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.18953 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.18953 q^{7} +1.00000 q^{9} -3.36266 q^{13} -1.00000 q^{15} +5.18953 q^{17} +3.18953 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.18953 q^{29} -2.17313 q^{31} -3.18953 q^{35} +9.53579 q^{37} +3.36266 q^{39} -6.55220 q^{41} +1.01641 q^{43} +1.00000 q^{45} -6.37907 q^{47} +3.17313 q^{49} -5.18953 q^{51} -2.55220 q^{53} -8.55220 q^{59} -3.01641 q^{61} -3.18953 q^{63} -3.36266 q^{65} +7.53579 q^{67} +1.00000 q^{69} -13.9149 q^{71} +8.37907 q^{73} -1.00000 q^{75} +7.74173 q^{79} +1.00000 q^{81} +9.91486 q^{83} +5.18953 q^{85} +1.18953 q^{87} +10.3463 q^{89} +10.7253 q^{91} +2.17313 q^{93} +3.01641 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 4 q^{13} - 3 q^{15} + 7 q^{17} + q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 5 q^{29} - q^{31} - q^{35} + 9 q^{37} - 4 q^{39} + 3 q^{41} + 3 q^{45} - 2 q^{47} + 4 q^{49} - 7 q^{51} + 15 q^{53} - 3 q^{59} - 6 q^{61} - q^{63} + 4 q^{65} + 3 q^{67} + 3 q^{69} - 5 q^{71} + 8 q^{73} - 3 q^{75} - 8 q^{79} + 3 q^{81} - 7 q^{83} + 7 q^{85} - 5 q^{87} + 20 q^{89} + 4 q^{91} + q^{93} + 6 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 + 4 * q^13 - 3 * q^15 + 7 * q^17 + q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 + 5 * q^29 - q^31 - q^35 + 9 * q^37 - 4 * q^39 + 3 * q^41 + 3 * q^45 - 2 * q^47 + 4 * q^49 - 7 * q^51 + 15 * q^53 - 3 * q^59 - 6 * q^61 - q^63 + 4 * q^65 + 3 * q^67 + 3 * q^69 - 5 * q^71 + 8 * q^73 - 3 * q^75 - 8 * q^79 + 3 * q^81 - 7 * q^83 + 7 * q^85 - 5 * q^87 + 20 * q^89 + 4 * q^91 + q^93 + 6 * q^97

Introduction

L-functions

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