LMFDB - Embedded newform 5520.2.a.bx.1.3

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +4.77846 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +4.77846 q^{7} +1.00000 q^{9} -3.30777 q^{11} +3.19051 q^{13} +1.00000 q^{15} -2.41205 q^{17} -2.24914 q^{19} -4.77846 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.96896 q^{29} +10.3354 q^{31} +3.30777 q^{33} -4.77846 q^{35} -11.2767 q^{37} -3.19051 q^{39} +1.02760 q^{41} -10.0552 q^{43} -1.00000 q^{45} -2.74742 q^{47} +15.8337 q^{49} +2.41205 q^{51} -5.58795 q^{53} +3.30777 q^{55} +2.24914 q^{57} +1.96896 q^{59} -8.24914 q^{61} +4.77846 q^{63} -3.19051 q^{65} +7.71982 q^{67} +1.00000 q^{69} -16.0242 q^{71} -7.92332 q^{73} -1.00000 q^{75} -15.8061 q^{77} +1.00000 q^{81} +12.5845 q^{83} +2.41205 q^{85} +7.96896 q^{87} -3.05863 q^{89} +15.2457 q^{91} -10.3354 q^{93} +2.24914 q^{95} -5.43965 q^{97} -3.30777 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{15} - 6 q^{17} + 2 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 6 q^{35} - 8 q^{37} - 14 q^{41} + 4 q^{43} - 3 q^{45} + 18 q^{47} + 5 q^{49} + 6 q^{51} - 18 q^{53} + 2 q^{55} - 2 q^{57} - 12 q^{59} - 16 q^{61} + 6 q^{63} + 14 q^{67} + 3 q^{69} + 4 q^{71} - 3 q^{75} - 22 q^{77} + 3 q^{81} + 4 q^{83} + 6 q^{85} + 6 q^{87} - 10 q^{89} + 2 q^{91} - 6 q^{93} - 2 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 - 2 * q^11 + 3 * q^15 - 6 * q^17 + 2 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 6 * q^29 + 6 * q^31 + 2 * q^33 - 6 * q^35 - 8 * q^37 - 14 * q^41 + 4 * q^43 - 3 * q^45 + 18 * q^47 + 5 * q^49 + 6 * q^51 - 18 * q^53 + 2 * q^55 - 2 * q^57 - 12 * q^59 - 16 * q^61 + 6 * q^63 + 14 * q^67 + 3 * q^69 + 4 * q^71 - 3 * q^75 - 22 * q^77 + 3 * q^81 + 4 * q^83 + 6 * q^85 + 6 * q^87 - 10 * q^89 + 2 * q^91 - 6 * q^93 - 2 * q^95 + 2 * q^97 - 2 * 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
$7$	4.77846	1.80609	0.903044	−	0.429549i	$-0.141327\pi$
$7$	4.77846	1.80609	0.903044	+	0.429549i	$0.141327\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.30777	−0.997331	−0.498666	−	0.866794i	$-0.666176\pi$
$11$	−3.30777	−0.997331	−0.498666	+	0.866794i	$0.666176\pi$
$12$	0	0
$13$	3.19051	0.884888	0.442444	−	0.896796i	$-0.354112\pi$
$13$	3.19051	0.884888	0.442444	+	0.896796i	$0.354112\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−2.41205	−0.585008	−0.292504	−	0.956264i	$-0.594488\pi$
$17$	−2.41205	−0.585008	−0.292504	+	0.956264i	$0.594488\pi$
$18$	0	0
$19$	−2.24914	−0.515988	−0.257994	−	0.966146i	$-0.583062\pi$
$19$	−2.24914	−0.515988	−0.257994	+	0.966146i	$0.583062\pi$
$20$	0	0
$21$	−4.77846	−1.04274
$22$	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
$29$	−7.96896	−1.47980	−0.739900	−	0.672717i	$-0.765127\pi$
$29$	−7.96896	−1.47980	−0.739900	+	0.672717i	$0.765127\pi$
$30$	0	0
$31$	10.3354	1.85629	0.928144	−	0.372222i	$-0.121404\pi$
$31$	10.3354	1.85629	0.928144	+	0.372222i	$0.121404\pi$
$32$	0	0
$33$	3.30777	0.575809
$34$	0	0
$35$	−4.77846	−0.807707
$36$	0	0
$37$	−11.2767	−1.85388	−0.926942	−	0.375204i	$-0.877573\pi$
$37$	−11.2767	−1.85388	−0.926942	+	0.375204i	$0.877573\pi$
$38$	0	0
$39$	−3.19051	−0.510890
$40$	0	0
$41$	1.02760	0.160484	0.0802419	−	0.996775i	$-0.474431\pi$
$41$	1.02760	0.160484	0.0802419	+	0.996775i	$0.474431\pi$
$42$	0	0
$43$	−10.0552	−1.53340	−0.766701	−	0.642004i	$-0.778103\pi$
$43$	−10.0552	−1.53340	−0.766701	+	0.642004i	$0.778103\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−2.74742	−0.400753	−0.200376	−	0.979719i	$-0.564216\pi$
$47$	−2.74742	−0.400753	−0.200376	+	0.979719i	$0.564216\pi$
$48$	0	0
$49$	15.8337	2.26195
$50$	0	0
$51$	2.41205	0.337755
$52$	0	0
$53$	−5.58795	−0.767564	−0.383782	−	0.923424i	$-0.625379\pi$
$53$	−5.58795	−0.767564	−0.383782	+	0.923424i	$0.625379\pi$
$54$	0	0
$55$	3.30777	0.446020
$56$	0	0
$57$	2.24914	0.297906
$58$	0	0
$59$	1.96896	0.256337	0.128169	−	0.991752i	$-0.459090\pi$
$59$	1.96896	0.256337	0.128169	+	0.991752i	$0.459090\pi$
$60$	0	0
$61$	−8.24914	−1.05619	−0.528097	−	0.849184i	$-0.677094\pi$
$61$	−8.24914	−1.05619	−0.528097	+	0.849184i	$0.677094\pi$
$62$	0	0
$63$	4.77846	0.602029
$64$	0	0
$65$	−3.19051	−0.395734
$66$	0	0
$67$	7.71982	0.943127	0.471563	−	0.881832i	$-0.343690\pi$
$67$	7.71982	0.943127	0.471563	+	0.881832i	$0.343690\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−16.0242	−1.90172	−0.950859	−	0.309624i	$-0.899797\pi$
$71$	−16.0242	−1.90172	−0.950859	+	0.309624i	$0.899797\pi$
$72$	0	0
$73$	−7.92332	−0.927355	−0.463677	−	0.886004i	$-0.653470\pi$
$73$	−7.92332	−0.927355	−0.463677	+	0.886004i	$0.653470\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−15.8061	−1.80127
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.5845	1.38133	0.690665	−	0.723175i	$-0.257318\pi$
$83$	12.5845	1.38133	0.690665	+	0.723175i	$0.257318\pi$
$84$	0	0
$85$	2.41205	0.261624
$86$	0	0
$87$	7.96896	0.854363
$88$	0	0
$89$	−3.05863	−0.324214	−0.162107	−	0.986773i	$-0.551829\pi$
$89$	−3.05863	−0.324214	−0.162107	+	0.986773i	$0.551829\pi$
$90$	0	0
$91$	15.2457	1.59818
$92$	0	0
$93$	−10.3354	−1.07173
$94$	0	0
$95$	2.24914	0.230757
$96$	0	0
$97$	−5.43965	−0.552313	−0.276156	−	0.961113i	$-0.589061\pi$
$97$	−5.43965	−0.552313	−0.276156	+	0.961113i	$0.589061\pi$
$98$	0	0
$99$	−3.30777	−0.332444
$100$	0	0
$101$	−8.52932	−0.848699	−0.424349	−	0.905499i	$-0.639497\pi$
$101$	−8.52932	−0.848699	−0.424349	+	0.905499i	$0.639497\pi$
$102$	0	0
$103$	9.32238	0.918562	0.459281	−	0.888291i	$-0.348107\pi$
$103$	9.32238	0.918562	0.459281	+	0.888291i	$0.348107\pi$
$104$	0	0
$105$	4.77846	0.466330
$106$	0	0
$107$	13.1449	1.27076	0.635381	−	0.772199i	$-0.280843\pi$
$107$	13.1449	1.27076	0.635381	+	0.772199i	$0.280843\pi$
$108$	0	0
$109$	10.0406	0.961714	0.480857	−	0.876799i	$-0.340326\pi$
$109$	10.0406	0.961714	0.480857	+	0.876799i	$0.340326\pi$
$110$	0	0
$111$	11.2767	1.07034
$112$	0	0
$113$	−3.70522	−0.348557	−0.174279	−	0.984696i	$-0.555759\pi$
$113$	−3.70522	−0.348557	−0.174279	+	0.984696i	$0.555759\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	3.19051	0.294963
$118$	0	0
$119$	−11.5259	−1.05658
$120$	0	0
$121$	−0.0586332	−0.00533029
$122$	0	0
$123$	−1.02760	−0.0926554
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.68879	−0.149856	−0.0749279	−	0.997189i	$-0.523873\pi$
$127$	−1.68879	−0.149856	−0.0749279	+	0.997189i	$0.523873\pi$
$128$	0	0
$129$	10.0552	0.885311
$130$	0	0
$131$	8.99656	0.786033	0.393017	−	0.919531i	$-0.371431\pi$
$131$	8.99656	0.786033	0.393017	+	0.919531i	$0.371431\pi$
$132$	0	0
$133$	−10.7474	−0.931920
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−19.5569	−1.67086	−0.835430	−	0.549597i	$-0.814781\pi$
$137$	−19.5569	−1.67086	−0.835430	+	0.549597i	$0.814781\pi$
$138$	0	0
$139$	15.3319	1.30044	0.650219	−	0.759747i	$-0.274677\pi$
$139$	15.3319	1.30044	0.650219	+	0.759747i	$0.274677\pi$
$140$	0	0
$141$	2.74742	0.231375
$142$	0	0
$143$	−10.5535	−0.882526
$144$	0	0
$145$	7.96896	0.661786
$146$	0	0
$147$	−15.8337	−1.30594
$148$	0	0
$149$	9.80605	0.803343	0.401672	−	0.915784i	$-0.368429\pi$
$149$	9.80605	0.803343	0.401672	+	0.915784i	$0.368429\pi$
$150$	0	0
$151$	−14.0552	−1.14380	−0.571898	−	0.820325i	$-0.693793\pi$
$151$	−14.0552	−1.14380	−0.571898	+	0.820325i	$0.693793\pi$
$152$	0	0
$153$	−2.41205	−0.195003
$154$	0	0
$155$	−10.3354	−0.830157
$156$	0	0
$157$	−2.28018	−0.181978	−0.0909889	−	0.995852i	$-0.529003\pi$
$157$	−2.28018	−0.181978	−0.0909889	+	0.995852i	$0.529003\pi$
$158$	0	0
$159$	5.58795	0.443153
$160$	0	0
$161$	−4.77846	−0.376595
$162$	0	0
$163$	−13.5569	−1.06186	−0.530930	−	0.847416i	$-0.678157\pi$
$163$	−13.5569	−1.06186	−0.530930	+	0.847416i	$0.678157\pi$
$164$	0	0
$165$	−3.30777	−0.257510
$166$	0	0
$167$	12.8647	0.995499	0.497750	−	0.867321i	$-0.334160\pi$
$167$	12.8647	0.995499	0.497750	+	0.867321i	$0.334160\pi$
$168$	0	0
$169$	−2.82066	−0.216974
$170$	0	0
$171$	−2.24914	−0.171996
$172$	0	0
$173$	−11.3224	−0.860825	−0.430412	−	0.902632i	$-0.641632\pi$
$173$	−11.3224	−0.860825	−0.430412	+	0.902632i	$0.641632\pi$
$174$	0	0
$175$	4.77846	0.361217
$176$	0	0
$177$	−1.96896	−0.147996
$178$	0	0
$179$	20.9345	1.56472	0.782359	−	0.622828i	$-0.214016\pi$
$179$	20.9345	1.56472	0.782359	+	0.622828i	$0.214016\pi$
$180$	0	0
$181$	1.93793	0.144045	0.0720226	−	0.997403i	$-0.477055\pi$
$181$	1.93793	0.144045	0.0720226	+	0.997403i	$0.477055\pi$
$182$	0	0
$183$	8.24914	0.609794
$184$	0	0
$185$	11.2767	0.829082
$186$	0	0
$187$	7.97852	0.583447
$188$	0	0
$189$	−4.77846	−0.347582
$190$	0	0
$191$	11.7440	0.849765	0.424882	−	0.905249i	$-0.360315\pi$
$191$	11.7440	0.849765	0.424882	+	0.905249i	$0.360315\pi$
$192$	0	0
$193$	3.29317	0.237047	0.118524	−	0.992951i	$-0.462184\pi$
$193$	3.29317	0.237047	0.118524	+	0.992951i	$0.462184\pi$
$194$	0	0
$195$	3.19051	0.228477
$196$	0	0
$197$	−6.73281	−0.479693	−0.239847	−	0.970811i	$-0.577097\pi$
$197$	−6.73281	−0.479693	−0.239847	+	0.970811i	$0.577097\pi$
$198$	0	0
$199$	−9.82066	−0.696168	−0.348084	−	0.937463i	$-0.613168\pi$
$199$	−9.82066	−0.696168	−0.348084	+	0.937463i	$0.613168\pi$
$200$	0	0
$201$	−7.71982	−0.544514
$202$	0	0
$203$	−38.0794	−2.67265
$204$	0	0
$205$	−1.02760	−0.0717705
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	7.43965	0.514611
$210$	0	0
$211$	−27.0974	−1.86546	−0.932731	−	0.360573i	$-0.882581\pi$
$211$	−27.0974	−1.86546	−0.932731	+	0.360573i	$0.882581\pi$
$212$	0	0
$213$	16.0242	1.09796
$214$	0	0
$215$	10.0552	0.685759
$216$	0	0
$217$	49.3871	3.35262
$218$	0	0
$219$	7.92332	0.535408
$220$	0	0
$221$	−7.69566	−0.517666
$222$	0	0
$223$	−20.9053	−1.39992	−0.699960	−	0.714182i	$-0.746799\pi$
$223$	−20.9053	−1.39992	−0.699960	+	0.714182i	$0.746799\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	29.0518	1.91979	0.959897	−	0.280353i	$-0.0904514\pi$
$229$	29.0518	1.91979	0.959897	+	0.280353i	$0.0904514\pi$
$230$	0	0
$231$	15.8061	1.03996
$232$	0	0
$233$	−20.5535	−1.34650	−0.673252	−	0.739414i	$-0.735103\pi$
$233$	−20.5535	−1.34650	−0.673252	+	0.739414i	$0.735103\pi$
$234$	0	0
$235$	2.74742	0.179222
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.0862	−1.29927	−0.649635	−	0.760246i	$-0.725078\pi$
$239$	−20.0862	−1.29927	−0.649635	+	0.760246i	$0.725078\pi$
$240$	0	0
$241$	−0.574960	−0.0370364	−0.0185182	−	0.999829i	$-0.505895\pi$
$241$	−0.574960	−0.0370364	−0.0185182	+	0.999829i	$0.505895\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−15.8337	−1.01157
$246$	0	0
$247$	−7.17590	−0.456592
$248$	0	0
$249$	−12.5845	−0.797511
$250$	0	0
$251$	−5.32238	−0.335946	−0.167973	−	0.985792i	$-0.553722\pi$
$251$	−5.32238	−0.335946	−0.167973	+	0.985792i	$0.553722\pi$
$252$	0	0
$253$	3.30777	0.207958
$254$	0	0
$255$	−2.41205	−0.151048
$256$	0	0
$257$	17.9164	1.11760	0.558799	−	0.829303i	$-0.311262\pi$
$257$	17.9164	1.11760	0.558799	+	0.829303i	$0.311262\pi$
$258$	0	0
$259$	−53.8854	−3.34828
$260$	0	0
$261$	−7.96896	−0.493267
$262$	0	0
$263$	18.9655	1.16946	0.584732	−	0.811226i	$-0.301200\pi$
$263$	18.9655	1.16946	0.584732	+	0.811226i	$0.301200\pi$
$264$	0	0
$265$	5.58795	0.343265
$266$	0	0
$267$	3.05863	0.187185
$268$	0	0
$269$	−13.7604	−0.838987	−0.419494	−	0.907758i	$-0.637792\pi$
$269$	−13.7604	−0.838987	−0.419494	+	0.907758i	$0.637792\pi$
$270$	0	0
$271$	−19.1595	−1.16386	−0.581928	−	0.813241i	$-0.697701\pi$
$271$	−19.1595	−1.16386	−0.581928	+	0.813241i	$0.697701\pi$
$272$	0	0
$273$	−15.2457	−0.922712
$274$	0	0
$275$	−3.30777	−0.199466
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	10.3354	0.618762
$280$	0	0
$281$	−5.63359	−0.336072	−0.168036	−	0.985781i	$-0.553742\pi$
$281$	−5.63359	−0.336072	−0.168036	+	0.985781i	$0.553742\pi$
$282$	0	0
$283$	−12.0456	−0.716039	−0.358020	−	0.933714i	$-0.616548\pi$
$283$	−12.0456	−0.716039	−0.358020	+	0.933714i	$0.616548\pi$
$284$	0	0
$285$	−2.24914	−0.133228
$286$	0	0
$287$	4.91033	0.289848
$288$	0	0
$289$	−11.1820	−0.657766
$290$	0	0
$291$	5.43965	0.318878
$292$	0	0
$293$	−3.73443	−0.218168	−0.109084	−	0.994033i	$-0.534792\pi$
$293$	−3.73443	−0.218168	−0.109084	+	0.994033i	$0.534792\pi$
$294$	0	0
$295$	−1.96896	−0.114638
$296$	0	0
$297$	3.30777	0.191936
$298$	0	0
$299$	−3.19051	−0.184512
$300$	0	0
$301$	−48.0483	−2.76946
$302$	0	0
$303$	8.52932	0.489996
$304$	0	0
$305$	8.24914	0.472344
$306$	0	0
$307$	−20.6302	−1.17743	−0.588713	−	0.808342i	$-0.700365\pi$
$307$	−20.6302	−1.17743	−0.588713	+	0.808342i	$0.700365\pi$
$308$	0	0
$309$	−9.32238	−0.530332
$310$	0	0
$311$	−19.1138	−1.08385	−0.541923	−	0.840428i	$-0.682304\pi$
$311$	−19.1138	−1.08385	−0.541923	+	0.840428i	$0.682304\pi$
$312$	0	0
$313$	−2.84053	−0.160556	−0.0802781	−	0.996773i	$-0.525581\pi$
$313$	−2.84053	−0.160556	−0.0802781	+	0.996773i	$0.525581\pi$
$314$	0	0
$315$	−4.77846	−0.269236
$316$	0	0
$317$	−18.9560	−1.06467	−0.532337	−	0.846533i	$-0.678686\pi$
$317$	−18.9560	−1.06467	−0.532337	+	0.846533i	$0.678686\pi$
$318$	0	0
$319$	26.3595	1.47585
$320$	0	0
$321$	−13.1449	−0.733675
$322$	0	0
$323$	5.42504	0.301857
$324$	0	0
$325$	3.19051	0.176978
$326$	0	0
$327$	−10.0406	−0.555246
$328$	0	0
$329$	−13.1284	−0.723794
$330$	0	0
$331$	−30.5078	−1.67686	−0.838431	−	0.545008i	$-0.816527\pi$
$331$	−30.5078	−1.67686	−0.838431	+	0.545008i	$0.816527\pi$
$332$	0	0
$333$	−11.2767	−0.617961
$334$	0	0
$335$	−7.71982	−0.421779
$336$	0	0
$337$	−0.117266	−0.00638790	−0.00319395	−	0.999995i	$-0.501017\pi$
$337$	−0.117266	−0.00638790	−0.00319395	+	0.999995i	$0.501017\pi$
$338$	0	0
$339$	3.70522	0.201240
$340$	0	0
$341$	−34.1871	−1.85133
$342$	0	0
$343$	42.2112	2.27919
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−21.3845	−1.14798	−0.573989	−	0.818863i	$-0.694605\pi$
$347$	−21.3845	−1.14798	−0.573989	+	0.818863i	$0.694605\pi$
$348$	0	0
$349$	3.10428	0.166168	0.0830841	−	0.996543i	$-0.473523\pi$
$349$	3.10428	0.166168	0.0830841	+	0.996543i	$0.473523\pi$
$350$	0	0
$351$	−3.19051	−0.170297
$352$	0	0
$353$	25.9164	1.37939	0.689697	−	0.724098i	$-0.257744\pi$
$353$	25.9164	1.37939	0.689697	+	0.724098i	$0.257744\pi$
$354$	0	0
$355$	16.0242	0.850474
$356$	0	0
$357$	11.5259	0.610014
$358$	0	0
$359$	2.01461	0.106327	0.0531635	−	0.998586i	$-0.483070\pi$
$359$	2.01461	0.106327	0.0531635	+	0.998586i	$0.483070\pi$
$360$	0	0
$361$	−13.9414	−0.733756
$362$	0	0
$363$	0.0586332	0.00307744
$364$	0	0
$365$	7.92332	0.414726
$366$	0	0
$367$	12.7785	0.667030	0.333515	−	0.942745i	$-0.391765\pi$
$367$	12.7785	0.667030	0.333515	+	0.942745i	$0.391765\pi$
$368$	0	0
$369$	1.02760	0.0534946
$370$	0	0
$371$	−26.7018	−1.38629
$372$	0	0
$373$	14.9345	0.773279	0.386639	−	0.922231i	$-0.373636\pi$
$373$	14.9345	0.773279	0.386639	+	0.922231i	$0.373636\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−25.4250	−1.30946
$378$	0	0
$379$	13.9931	0.718779	0.359389	−	0.933188i	$-0.382985\pi$
$379$	13.9931	0.718779	0.359389	+	0.933188i	$0.382985\pi$
$380$	0	0
$381$	1.68879	0.0865193
$382$	0	0
$383$	18.1414	0.926984	0.463492	−	0.886101i	$-0.346596\pi$
$383$	18.1414	0.926984	0.463492	+	0.886101i	$0.346596\pi$
$384$	0	0
$385$	15.8061	0.805551
$386$	0	0
$387$	−10.0552	−0.511134
$388$	0	0
$389$	−0.850080	−0.0431008	−0.0215504	−	0.999768i	$-0.506860\pi$
$389$	−0.850080	−0.0431008	−0.0215504	+	0.999768i	$0.506860\pi$
$390$	0	0
$391$	2.41205	0.121983
$392$	0	0
$393$	−8.99656	−0.453817
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.67762	0.234763	0.117381	−	0.993087i	$-0.462550\pi$
$397$	4.67762	0.234763	0.117381	+	0.993087i	$0.462550\pi$
$398$	0	0
$399$	10.7474	0.538044
$400$	0	0
$401$	31.4948	1.57278	0.786389	−	0.617732i	$-0.211948\pi$
$401$	31.4948	1.57278	0.786389	+	0.617732i	$0.211948\pi$
$402$	0	0
$403$	32.9751	1.64261
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	37.3009	1.84894
$408$	0	0
$409$	−32.6803	−1.61594	−0.807968	−	0.589226i	$-0.799433\pi$
$409$	−32.6803	−1.61594	−0.807968	+	0.589226i	$0.799433\pi$
$410$	0	0
$411$	19.5569	0.964671
$412$	0	0
$413$	9.40861	0.462968
$414$	0	0
$415$	−12.5845	−0.617749
$416$	0	0
$417$	−15.3319	−0.750808
$418$	0	0
$419$	−0.926759	−0.0452751	−0.0226376	−	0.999744i	$-0.507206\pi$
$419$	−0.926759	−0.0452751	−0.0226376	+	0.999744i	$0.507206\pi$
$420$	0	0
$421$	−25.4182	−1.23881	−0.619403	−	0.785073i	$-0.712625\pi$
$421$	−25.4182	−1.23881	−0.619403	+	0.785073i	$0.712625\pi$
$422$	0	0
$423$	−2.74742	−0.133584
$424$	0	0
$425$	−2.41205	−0.117002
$426$	0	0
$427$	−39.4182	−1.90758
$428$	0	0
$429$	10.5535	0.509527
$430$	0	0
$431$	−31.6121	−1.52270	−0.761351	−	0.648340i	$-0.775464\pi$
$431$	−31.6121	−1.52270	−0.761351	+	0.648340i	$0.775464\pi$
$432$	0	0
$433$	−27.1855	−1.30645	−0.653225	−	0.757164i	$-0.726584\pi$
$433$	−27.1855	−1.30645	−0.653225	+	0.757164i	$0.726584\pi$
$434$	0	0
$435$	−7.96896	−0.382083
$436$	0	0
$437$	2.24914	0.107591
$438$	0	0
$439$	−17.5569	−0.837946	−0.418973	−	0.907999i	$-0.637610\pi$
$439$	−17.5569	−0.837946	−0.418973	+	0.907999i	$0.637610\pi$
$440$	0	0
$441$	15.8337	0.753983
$442$	0	0
$443$	27.9785	1.32930	0.664650	−	0.747155i	$-0.268581\pi$
$443$	27.9785	1.32930	0.664650	+	0.747155i	$0.268581\pi$
$444$	0	0
$445$	3.05863	0.144993
$446$	0	0
$447$	−9.80605	−0.463810
$448$	0	0
$449$	−9.43459	−0.445246	−0.222623	−	0.974905i	$-0.571462\pi$
$449$	−9.43459	−0.445246	−0.222623	+	0.974905i	$0.571462\pi$
$450$	0	0
$451$	−3.39906	−0.160056
$452$	0	0
$453$	14.0552	0.660371
$454$	0	0
$455$	−15.2457	−0.714730
$456$	0	0
$457$	12.1629	0.568957	0.284478	−	0.958682i	$-0.408180\pi$
$457$	12.1629	0.568957	0.284478	+	0.958682i	$0.408180\pi$
$458$	0	0
$459$	2.41205	0.112585
$460$	0	0
$461$	−14.7328	−0.686176	−0.343088	−	0.939303i	$-0.611473\pi$
$461$	−14.7328	−0.686176	−0.343088	+	0.939303i	$0.611473\pi$
$462$	0	0
$463$	−20.2423	−0.940738	−0.470369	−	0.882470i	$-0.655879\pi$
$463$	−20.2423	−0.940738	−0.470369	+	0.882470i	$0.655879\pi$
$464$	0	0
$465$	10.3354	0.479291
$466$	0	0
$467$	−34.5484	−1.59871	−0.799355	−	0.600859i	$-0.794825\pi$
$467$	−34.5484	−1.59871	−0.799355	+	0.600859i	$0.794825\pi$
$468$	0	0
$469$	36.8888	1.70337
$470$	0	0
$471$	2.28018	0.105065
$472$	0	0
$473$	33.2603	1.52931
$474$	0	0
$475$	−2.24914	−0.103198
$476$	0	0
$477$	−5.58795	−0.255855
$478$	0	0
$479$	−0.0214836	−0.000981610 0	−0.000490805	−	1.00000i	$-0.500156\pi$
$479$	−0.0214836	−0.000981610 0	−0.000490805		1.00000i	$0.500156\pi$
$480$	0	0
$481$	−35.9785	−1.64048
$482$	0	0
$483$	4.77846	0.217427
$484$	0	0
$485$	5.43965	0.247002
$486$	0	0
$487$	14.8387	0.672406	0.336203	−	0.941790i	$-0.390857\pi$
$487$	14.8387	0.672406	0.336203	+	0.941790i	$0.390857\pi$
$488$	0	0
$489$	13.5569	0.613065
$490$	0	0
$491$	4.81904	0.217480	0.108740	−	0.994070i	$-0.465318\pi$
$491$	4.81904	0.217480	0.108740	+	0.994070i	$0.465318\pi$
$492$	0	0
$493$	19.2215	0.865695
$494$	0	0
$495$	3.30777	0.148673
$496$	0	0
$497$	−76.5708	−3.43467
$498$	0	0
$499$	−40.0647	−1.79354	−0.896772	−	0.442493i	$-0.854094\pi$
$499$	−40.0647	−1.79354	−0.896772	+	0.442493i	$0.854094\pi$
$500$	0	0
$501$	−12.8647	−0.574752
$502$	0	0
$503$	33.1449	1.47786	0.738928	−	0.673784i	$-0.235332\pi$
$503$	33.1449	1.47786	0.738928	+	0.673784i	$0.235332\pi$
$504$	0	0
$505$	8.52932	0.379550
$506$	0	0
$507$	2.82066	0.125270
$508$	0	0
$509$	34.9053	1.54715	0.773575	−	0.633705i	$-0.218467\pi$
$509$	34.9053	1.54715	0.773575	+	0.633705i	$0.218467\pi$
$510$	0	0
$511$	−37.8613	−1.67488
$512$	0	0
$513$	2.24914	0.0993020
$514$	0	0
$515$	−9.32238	−0.410793
$516$	0	0
$517$	9.08785	0.399683
$518$	0	0
$519$	11.3224	0.496997
$520$	0	0
$521$	−15.0180	−0.657953	−0.328976	−	0.944338i	$-0.606704\pi$
$521$	−15.0180	−0.657953	−0.328976	+	0.944338i	$0.606704\pi$
$522$	0	0
$523$	3.00344	0.131331	0.0656656	−	0.997842i	$-0.479083\pi$
$523$	3.00344	0.131331	0.0656656	+	0.997842i	$0.479083\pi$
$524$	0	0
$525$	−4.77846	−0.208549
$526$	0	0
$527$	−24.9294	−1.08594
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.96896	0.0854458
$532$	0	0
$533$	3.27856	0.142010
$534$	0	0
$535$	−13.1449	−0.568302
$536$	0	0
$537$	−20.9345	−0.903390
$538$	0	0
$539$	−52.3741	−2.25591
$540$	0	0
$541$	5.90871	0.254035	0.127018	−	0.991900i	$-0.459459\pi$
$541$	5.90871	0.254035	0.127018	+	0.991900i	$0.459459\pi$
$542$	0	0
$543$	−1.93793	−0.0831645
$544$	0	0
$545$	−10.0406	−0.430092
$546$	0	0
$547$	−39.1138	−1.67239	−0.836193	−	0.548435i	$-0.815224\pi$
$547$	−39.1138	−1.67239	−0.836193	+	0.548435i	$0.815224\pi$
$548$	0	0
$549$	−8.24914	−0.352065
$550$	0	0
$551$	17.9233	0.763559
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−11.2767	−0.478671
$556$	0	0
$557$	−4.59139	−0.194543	−0.0972717	−	0.995258i	$-0.531012\pi$
$557$	−4.59139	−0.194543	−0.0972717	+	0.995258i	$0.531012\pi$
$558$	0	0
$559$	−32.0812	−1.35689
$560$	0	0
$561$	−7.97852	−0.336853
$562$	0	0
$563$	12.3208	0.519258	0.259629	−	0.965708i	$-0.416400\pi$
$563$	12.3208	0.519258	0.259629	+	0.965708i	$0.416400\pi$
$564$	0	0
$565$	3.70522	0.155880
$566$	0	0
$567$	4.77846	0.200676
$568$	0	0
$569$	25.1430	1.05405	0.527026	−	0.849849i	$-0.323307\pi$
$569$	25.1430	1.05405	0.527026	+	0.849849i	$0.323307\pi$
$570$	0	0
$571$	17.2197	0.720623	0.360311	−	0.932832i	$-0.382670\pi$
$571$	17.2197	0.720623	0.360311	+	0.932832i	$0.382670\pi$
$572$	0	0
$573$	−11.7440	−0.490612
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	32.7880	1.36498	0.682491	−	0.730894i	$-0.260896\pi$
$577$	32.7880	1.36498	0.682491	+	0.730894i	$0.260896\pi$
$578$	0	0
$579$	−3.29317	−0.136859
$580$	0	0
$581$	60.1346	2.49480
$582$	0	0
$583$	18.4837	0.765516
$584$	0	0
$585$	−3.19051	−0.131911
$586$	0	0
$587$	29.9018	1.23418	0.617090	−	0.786892i	$-0.288311\pi$
$587$	29.9018	1.23418	0.617090	+	0.786892i	$0.288311\pi$
$588$	0	0
$589$	−23.2457	−0.957822
$590$	0	0
$591$	6.73281	0.276951
$592$	0	0
$593$	0.656135	0.0269442	0.0134721	−	0.999909i	$-0.495712\pi$
$593$	0.656135	0.0269442	0.0134721	+	0.999909i	$0.495712\pi$
$594$	0	0
$595$	11.5259	0.472515
$596$	0	0
$597$	9.82066	0.401933
$598$	0	0
$599$	−41.2603	−1.68585	−0.842925	−	0.538031i	$-0.819168\pi$
$599$	−41.2603	−1.68585	−0.842925	+	0.538031i	$0.819168\pi$
$600$	0	0
$601$	−12.1008	−0.493604	−0.246802	−	0.969066i	$-0.579380\pi$
$601$	−12.1008	−0.493604	−0.246802	+	0.969066i	$0.579380\pi$
$602$	0	0
$603$	7.71982	0.314376
$604$	0	0
$605$	0.0586332	0.00238378
$606$	0	0
$607$	37.1475	1.50777	0.753886	−	0.657005i	$-0.228177\pi$
$607$	37.1475	1.50777	0.753886	+	0.657005i	$0.228177\pi$
$608$	0	0
$609$	38.0794	1.54305
$610$	0	0
$611$	−8.76567	−0.354621
$612$	0	0
$613$	−31.3415	−1.26587	−0.632935	−	0.774205i	$-0.718150\pi$
$613$	−31.3415	−1.26587	−0.632935	+	0.774205i	$0.718150\pi$
$614$	0	0
$615$	1.02760	0.0414367
$616$	0	0
$617$	1.26213	0.0508115	0.0254057	−	0.999677i	$-0.491912\pi$
$617$	1.26213	0.0508115	0.0254057	+	0.999677i	$0.491912\pi$
$618$	0	0
$619$	31.1138	1.25057	0.625285	−	0.780396i	$-0.284983\pi$
$619$	31.1138	1.25057	0.625285	+	0.780396i	$0.284983\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−14.6155	−0.585560
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.43965	−0.297111
$628$	0	0
$629$	27.2001	1.08454
$630$	0	0
$631$	43.2648	1.72234	0.861172	−	0.508313i	$-0.169731\pi$
$631$	43.2648	1.72234	0.861172	+	0.508313i	$0.169731\pi$
$632$	0	0
$633$	27.0974	1.07702
$634$	0	0
$635$	1.68879	0.0670175
$636$	0	0
$637$	50.5174	2.00157
$638$	0	0
$639$	−16.0242	−0.633906
$640$	0	0
$641$	−8.01461	−0.316558	−0.158279	−	0.987394i	$-0.550595\pi$
$641$	−8.01461	−0.316558	−0.158279	+	0.987394i	$0.550595\pi$
$642$	0	0
$643$	−27.0061	−1.06502	−0.532509	−	0.846425i	$-0.678751\pi$
$643$	−27.0061	−1.06502	−0.532509	+	0.846425i	$0.678751\pi$
$644$	0	0
$645$	−10.0552	−0.395923
$646$	0	0
$647$	−14.6854	−0.577341	−0.288670	−	0.957429i	$-0.593213\pi$
$647$	−14.6854	−0.577341	−0.288670	+	0.957429i	$0.593213\pi$
$648$	0	0
$649$	−6.51289	−0.255653
$650$	0	0
$651$	−49.3871	−1.93563
$652$	0	0
$653$	11.4871	0.449525	0.224763	−	0.974414i	$-0.427839\pi$
$653$	11.4871	0.449525	0.224763	+	0.974414i	$0.427839\pi$
$654$	0	0
$655$	−8.99656	−0.351525
$656$	0	0
$657$	−7.92332	−0.309118
$658$	0	0
$659$	18.9199	0.737014	0.368507	−	0.929625i	$-0.379869\pi$
$659$	18.9199	0.737014	0.368507	+	0.929625i	$0.379869\pi$
$660$	0	0
$661$	−47.3707	−1.84251	−0.921253	−	0.388963i	$-0.872833\pi$
$661$	−47.3707	−1.84251	−0.921253	+	0.388963i	$0.872833\pi$
$662$	0	0
$663$	7.69566	0.298875
$664$	0	0
$665$	10.7474	0.416767
$666$	0	0
$667$	7.96896	0.308560
$668$	0	0
$669$	20.9053	0.808245
$670$	0	0
$671$	27.2863	1.05338
$672$	0	0
$673$	38.2130	1.47300	0.736502	−	0.676435i	$-0.236476\pi$
$673$	38.2130	1.47300	0.736502	+	0.676435i	$0.236476\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	47.8156	1.83770	0.918852	−	0.394603i	$-0.129118\pi$
$677$	47.8156	1.83770	0.918852	+	0.394603i	$0.129118\pi$
$678$	0	0
$679$	−25.9931	−0.997525
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	26.2491	1.00440	0.502198	−	0.864753i	$-0.332525\pi$
$683$	26.2491	1.00440	0.502198	+	0.864753i	$0.332525\pi$
$684$	0	0
$685$	19.5569	0.747231
$686$	0	0
$687$	−29.0518	−1.10839
$688$	0	0
$689$	−17.8284	−0.679208
$690$	0	0
$691$	−18.0552	−0.686852	−0.343426	−	0.939180i	$-0.611587\pi$
$691$	−18.0552	−0.686852	−0.343426	+	0.939180i	$0.611587\pi$
$692$	0	0
$693$	−15.8061	−0.600422
$694$	0	0
$695$	−15.3319	−0.581573
$696$	0	0
$697$	−2.47862	−0.0938843
$698$	0	0
$699$	20.5535	0.777404
$700$	0	0
$701$	0.0766789	0.00289612	0.00144806	−	0.999999i	$-0.499539\pi$
$701$	0.0766789	0.00289612	0.00144806	+	0.999999i	$0.499539\pi$
$702$	0	0
$703$	25.3630	0.956582
$704$	0	0
$705$	−2.74742	−0.103474
$706$	0	0
$707$	−40.7570	−1.53282
$708$	0	0
$709$	−5.48024	−0.205815	−0.102907	−	0.994691i	$-0.532814\pi$
$709$	−5.48024	−0.205815	−0.102907	+	0.994691i	$0.532814\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10.3354	−0.387063
$714$	0	0
$715$	10.5535	0.394678
$716$	0	0
$717$	20.0862	0.750134
$718$	0	0
$719$	17.5811	0.655663	0.327832	−	0.944736i	$-0.393682\pi$
$719$	17.5811	0.655663	0.327832	+	0.944736i	$0.393682\pi$
$720$	0	0
$721$	44.5466	1.65900
$722$	0	0
$723$	0.574960	0.0213830
$724$	0	0
$725$	−7.96896	−0.295960
$726$	0	0
$727$	28.2993	1.04956	0.524781	−	0.851237i	$-0.324147\pi$
$727$	28.2993	1.04956	0.524781	+	0.851237i	$0.324147\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.2536	0.897053
$732$	0	0
$733$	13.9836	0.516495	0.258248	−	0.966079i	$-0.416855\pi$
$733$	13.9836	0.516495	0.258248	+	0.966079i	$0.416855\pi$
$734$	0	0
$735$	15.8337	0.584033
$736$	0	0
$737$	−25.5354	−0.940610
$738$	0	0
$739$	12.9509	0.476407	0.238204	−	0.971215i	$-0.423441\pi$
$739$	12.9509	0.476407	0.238204	+	0.971215i	$0.423441\pi$
$740$	0	0
$741$	7.17590	0.263613
$742$	0	0
$743$	−1.64820	−0.0604666	−0.0302333	−	0.999543i	$-0.509625\pi$
$743$	−1.64820	−0.0604666	−0.0302333	+	0.999543i	$0.509625\pi$
$744$	0	0
$745$	−9.80605	−0.359266
$746$	0	0
$747$	12.5845	0.460443
$748$	0	0
$749$	62.8122	2.29511
$750$	0	0
$751$	34.6233	1.26342	0.631711	−	0.775204i	$-0.282353\pi$
$751$	34.6233	1.26342	0.631711	+	0.775204i	$0.282353\pi$
$752$	0	0
$753$	5.32238	0.193958
$754$	0	0
$755$	14.0552	0.511521
$756$	0	0
$757$	15.9183	0.578559	0.289280	−	0.957245i	$-0.406584\pi$
$757$	15.9183	0.578559	0.289280	+	0.957245i	$0.406584\pi$
$758$	0	0
$759$	−3.30777	−0.120065
$760$	0	0
$761$	1.58795	0.0575631	0.0287816	−	0.999586i	$-0.490837\pi$
$761$	1.58795	0.0575631	0.0287816	+	0.999586i	$0.490837\pi$
$762$	0	0
$763$	47.9785	1.73694
$764$	0	0
$765$	2.41205	0.0872079
$766$	0	0
$767$	6.28200	0.226830
$768$	0	0
$769$	50.6302	1.82577	0.912885	−	0.408217i	$-0.133849\pi$
$769$	50.6302	1.82577	0.912885	+	0.408217i	$0.133849\pi$
$770$	0	0
$771$	−17.9164	−0.645245
$772$	0	0
$773$	5.37758	0.193418	0.0967090	−	0.995313i	$-0.469168\pi$
$773$	5.37758	0.193418	0.0967090	+	0.995313i	$0.469168\pi$
$774$	0	0
$775$	10.3354	0.371257
$776$	0	0
$777$	53.8854	1.93313
$778$	0	0
$779$	−2.31121	−0.0828077
$780$	0	0
$781$	53.0043	1.89664
$782$	0	0
$783$	7.96896	0.284788
$784$	0	0
$785$	2.28018	0.0813830
$786$	0	0
$787$	−6.42666	−0.229086	−0.114543	−	0.993418i	$-0.536540\pi$
$787$	−6.42666	−0.229086	−0.114543	+	0.993418i	$0.536540\pi$
$788$	0	0
$789$	−18.9655	−0.675191
$790$	0	0
$791$	−17.7052	−0.629525
$792$	0	0
$793$	−26.3189	−0.934613
$794$	0	0
$795$	−5.58795	−0.198184
$796$	0	0
$797$	−14.2295	−0.504034	−0.252017	−	0.967723i	$-0.581094\pi$
$797$	−14.2295	−0.504034	−0.252017	+	0.967723i	$0.581094\pi$
$798$	0	0
$799$	6.62692	0.234444
$800$	0	0
$801$	−3.05863	−0.108071
$802$	0	0
$803$	26.2086	0.924880
$804$	0	0
$805$	4.77846	0.168418
$806$	0	0
$807$	13.7604	0.484389
$808$	0	0
$809$	−45.5551	−1.60163	−0.800816	−	0.598911i	$-0.795600\pi$
$809$	−45.5551	−1.60163	−0.800816	+	0.598911i	$0.795600\pi$
$810$	0	0
$811$	−17.1043	−0.600612	−0.300306	−	0.953843i	$-0.597089\pi$
$811$	−17.1043	−0.600612	−0.300306	+	0.953843i	$0.597089\pi$
$812$	0	0
$813$	19.1595	0.671952
$814$	0	0
$815$	13.5569	0.474878
$816$	0	0
$817$	22.6155	0.791218
$818$	0	0
$819$	15.2457	0.532728
$820$	0	0
$821$	−47.9018	−1.67179	−0.835893	−	0.548893i	$-0.815050\pi$
$821$	−47.9018	−1.67179	−0.835893	+	0.548893i	$0.815050\pi$
$822$	0	0
$823$	−19.5309	−0.680806	−0.340403	−	0.940280i	$-0.610563\pi$
$823$	−19.5309	−0.680806	−0.340403	+	0.940280i	$0.610563\pi$
$824$	0	0
$825$	3.30777	0.115162
$826$	0	0
$827$	19.5879	0.681140	0.340570	−	0.940219i	$-0.389380\pi$
$827$	19.5879	0.681140	0.340570	+	0.940219i	$0.389380\pi$
$828$	0	0
$829$	−30.1560	−1.04736	−0.523681	−	0.851914i	$-0.675442\pi$
$829$	−30.1560	−1.04736	−0.523681	+	0.851914i	$0.675442\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−38.1916	−1.32326
$834$	0	0
$835$	−12.8647	−0.445201
$836$	0	0
$837$	−10.3354	−0.357243
$838$	0	0
$839$	−21.3224	−0.736130	−0.368065	−	0.929800i	$-0.619980\pi$
$839$	−21.3224	−0.736130	−0.368065	+	0.929800i	$0.619980\pi$
$840$	0	0
$841$	34.5044	1.18981
$842$	0	0
$843$	5.63359	0.194031
$844$	0	0
$845$	2.82066	0.0970337
$846$	0	0
$847$	−0.280176	−0.00962696
$848$	0	0
$849$	12.0456	0.413405
$850$	0	0
$851$	11.2767	0.386562
$852$	0	0
$853$	46.3741	1.58782	0.793910	−	0.608035i	$-0.208042\pi$
$853$	46.3741	1.58782	0.793910	+	0.608035i	$0.208042\pi$
$854$	0	0
$855$	2.24914	0.0769190
$856$	0	0
$857$	−50.9966	−1.74201	−0.871005	−	0.491275i	$-0.836531\pi$
$857$	−50.9966	−1.74201	−0.871005	+	0.491275i	$0.836531\pi$
$858$	0	0
$859$	−12.6872	−0.432881	−0.216440	−	0.976296i	$-0.569445\pi$
$859$	−12.6872	−0.432881	−0.216440	+	0.976296i	$0.569445\pi$
$860$	0	0
$861$	−4.91033	−0.167344
$862$	0	0
$863$	46.5827	1.58569	0.792847	−	0.609421i	$-0.208598\pi$
$863$	46.5827	1.58569	0.792847	+	0.609421i	$0.208598\pi$
$864$	0	0
$865$	11.3224	0.384973
$866$	0	0
$867$	11.1820	0.379761
$868$	0	0
$869$	0	0
$870$	0	0
$871$	24.6302	0.834561
$872$	0	0
$873$	−5.43965	−0.184104
$874$	0	0
$875$	−4.77846	−0.161541
$876$	0	0
$877$	0.270624	0.00913833	0.00456916	−	0.999990i	$-0.498546\pi$
$877$	0.270624	0.00913833	0.00456916	+	0.999990i	$0.498546\pi$
$878$	0	0
$879$	3.73443	0.125959
$880$	0	0
$881$	−22.5389	−0.759354	−0.379677	−	0.925119i	$-0.623965\pi$
$881$	−22.5389	−0.759354	−0.379677	+	0.925119i	$0.623965\pi$
$882$	0	0
$883$	48.3043	1.62557	0.812785	−	0.582564i	$-0.197950\pi$
$883$	48.3043	1.62557	0.812785	+	0.582564i	$0.197950\pi$
$884$	0	0
$885$	1.96896	0.0661860
$886$	0	0
$887$	11.9379	0.400836	0.200418	−	0.979710i	$-0.435770\pi$
$887$	11.9379	0.400836	0.200418	+	0.979710i	$0.435770\pi$
$888$	0	0
$889$	−8.06980	−0.270653
$890$	0	0
$891$	−3.30777	−0.110815
$892$	0	0
$893$	6.17934	0.206784
$894$	0	0
$895$	−20.9345	−0.699763
$896$	0	0
$897$	3.19051	0.106528
$898$	0	0
$899$	−82.3622	−2.74693
$900$	0	0
$901$	13.4784	0.449031
$902$	0	0
$903$	48.0483	1.59895
$904$	0	0
$905$	−1.93793	−0.0644189
$906$	0	0
$907$	28.8268	0.957177	0.478589	−	0.878039i	$-0.341149\pi$
$907$	28.8268	0.957177	0.478589	+	0.878039i	$0.341149\pi$
$908$	0	0
$909$	−8.52932	−0.282900
$910$	0	0
$911$	−54.5726	−1.80807	−0.904035	−	0.427458i	$-0.859409\pi$
$911$	−54.5726	−1.80807	−0.904035	+	0.427458i	$0.859409\pi$
$912$	0	0
$913$	−41.6267	−1.37764
$914$	0	0
$915$	−8.24914	−0.272708
$916$	0	0
$917$	42.9897	1.41964
$918$	0	0
$919$	−30.2277	−0.997118	−0.498559	−	0.866856i	$-0.666137\pi$
$919$	−30.2277	−0.997118	−0.498559	+	0.866856i	$0.666137\pi$
$920$	0	0
$921$	20.6302	0.679787
$922$	0	0
$923$	−51.1252	−1.68281
$924$	0	0
$925$	−11.2767	−0.370777
$926$	0	0
$927$	9.32238	0.306187
$928$	0	0
$929$	40.1706	1.31796	0.658978	−	0.752162i	$-0.270989\pi$
$929$	40.1706	1.31796	0.658978	+	0.752162i	$0.270989\pi$
$930$	0	0
$931$	−35.6121	−1.16714
$932$	0	0
$933$	19.1138	0.625759
$934$	0	0
$935$	−7.97852	−0.260925
$936$	0	0
$937$	−10.3258	−0.337330	−0.168665	−	0.985673i	$-0.553946\pi$
$937$	−10.3258	−0.337330	−0.168665	+	0.985673i	$0.553946\pi$
$938$	0	0
$939$	2.84053	0.0926971
$940$	0	0
$941$	−43.9096	−1.43141	−0.715706	−	0.698402i	$-0.753895\pi$
$941$	−43.9096	−1.43141	−0.715706	+	0.698402i	$0.753895\pi$
$942$	0	0
$943$	−1.02760	−0.0334632
$944$	0	0
$945$	4.77846	0.155443
$946$	0	0
$947$	58.4914	1.90072	0.950358	−	0.311160i	$-0.100717\pi$
$947$	58.4914	1.90072	0.950358	+	0.311160i	$0.100717\pi$
$948$	0	0
$949$	−25.2794	−0.820605
$950$	0	0
$951$	18.9560	0.614690
$952$	0	0
$953$	21.5309	0.697455	0.348728	−	0.937224i	$-0.386614\pi$
$953$	21.5309	0.697455	0.348728	+	0.937224i	$0.386614\pi$
$954$	0	0
$955$	−11.7440	−0.380026
$956$	0	0
$957$	−26.3595	−0.852083
$958$	0	0
$959$	−93.4519	−3.01772
$960$	0	0
$961$	75.8199	2.44580
$962$	0	0
$963$	13.1449	0.423587
$964$	0	0
$965$	−3.29317	−0.106011
$966$	0	0
$967$	14.9629	0.481173	0.240586	−	0.970628i	$-0.422660\pi$
$967$	14.9629	0.481173	0.240586	+	0.970628i	$0.422660\pi$
$968$	0	0
$969$	−5.42504	−0.174277
$970$	0	0
$971$	41.0777	1.31825	0.659124	−	0.752035i	$-0.270927\pi$
$971$	41.0777	1.31825	0.659124	+	0.752035i	$0.270927\pi$
$972$	0	0
$973$	73.2630	2.34870
$974$	0	0
$975$	−3.19051	−0.102178
$976$	0	0
$977$	5.76041	0.184292	0.0921459	−	0.995746i	$-0.470627\pi$
$977$	5.76041	0.184292	0.0921459	+	0.995746i	$0.470627\pi$
$978$	0	0
$979$	10.1173	0.323349
$980$	0	0
$981$	10.0406	0.320571
$982$	0	0
$983$	14.1223	0.450432	0.225216	−	0.974309i	$-0.427691\pi$
$983$	14.1223	0.450432	0.225216	+	0.974309i	$0.427691\pi$
$984$	0	0
$985$	6.73281	0.214525
$986$	0	0
$987$	13.1284	0.417883
$988$	0	0
$989$	10.0552	0.319737
$990$	0	0
$991$	−30.4458	−0.967142	−0.483571	−	0.875305i	$-0.660660\pi$
$991$	−30.4458	−0.967142	−0.483571	+	0.875305i	$0.660660\pi$
$992$	0	0
$993$	30.5078	0.968137
$994$	0	0
$995$	9.82066	0.311336
$996$	0	0
$997$	1.17590	0.0372411	0.0186206	−	0.999827i	$-0.494073\pi$
$997$	1.17590	0.0372411	0.0186206	+	0.999827i	$0.494073\pi$
$998$	0	0
$999$	11.2767	0.356780

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.r.1.1	✓	3	4.3	odd	2
5520.2.a.bx.1.3		3	1.1	even	1	trivial
8280.2.a.bm.1.1		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +4.77846 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +4.77846 q^{7} +1.00000 q^{9} -3.30777 q^{11} +3.19051 q^{13} +1.00000 q^{15} -2.41205 q^{17} -2.24914 q^{19} -4.77846 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.96896 q^{29} +10.3354 q^{31} +3.30777 q^{33} -4.77846 q^{35} -11.2767 q^{37} -3.19051 q^{39} +1.02760 q^{41} -10.0552 q^{43} -1.00000 q^{45} -2.74742 q^{47} +15.8337 q^{49} +2.41205 q^{51} -5.58795 q^{53} +3.30777 q^{55} +2.24914 q^{57} +1.96896 q^{59} -8.24914 q^{61} +4.77846 q^{63} -3.19051 q^{65} +7.71982 q^{67} +1.00000 q^{69} -16.0242 q^{71} -7.92332 q^{73} -1.00000 q^{75} -15.8061 q^{77} +1.00000 q^{81} +12.5845 q^{83} +2.41205 q^{85} +7.96896 q^{87} -3.05863 q^{89} +15.2457 q^{91} -10.3354 q^{93} +2.24914 q^{95} -5.43965 q^{97} -3.30777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{15} - 6 q^{17} + 2 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 6 q^{35} - 8 q^{37} - 14 q^{41} + 4 q^{43} - 3 q^{45} + 18 q^{47} + 5 q^{49} + 6 q^{51} - 18 q^{53} + 2 q^{55} - 2 q^{57} - 12 q^{59} - 16 q^{61} + 6 q^{63} + 14 q^{67} + 3 q^{69} + 4 q^{71} - 3 q^{75} - 22 q^{77} + 3 q^{81} + 4 q^{83} + 6 q^{85} + 6 q^{87} - 10 q^{89} + 2 q^{91} - 6 q^{93} - 2 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 - 2 * q^11 + 3 * q^15 - 6 * q^17 + 2 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 6 * q^29 + 6 * q^31 + 2 * q^33 - 6 * q^35 - 8 * q^37 - 14 * q^41 + 4 * q^43 - 3 * q^45 + 18 * q^47 + 5 * q^49 + 6 * q^51 - 18 * q^53 + 2 * q^55 - 2 * q^57 - 12 * q^59 - 16 * q^61 + 6 * q^63 + 14 * q^67 + 3 * q^69 + 4 * q^71 - 3 * q^75 - 22 * q^77 + 3 * q^81 + 4 * q^83 + 6 * q^85 + 6 * q^87 - 10 * q^89 + 2 * q^91 - 6 * q^93 - 2 * q^95 + 2 * q^97 - 2 * q^99

Introduction

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