LMFDB - Embedded newform 5520.2.a.cb.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.339102$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +0.845563 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +0.845563 q^{7} +1.00000 q^{9} +4.55883 q^{11} +5.23704 q^{13} -1.00000 q^{15} +4.72619 q^{17} +5.54591 q^{19} +0.845563 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +7.06968 q^{29} +1.83264 q^{31} +4.55883 q^{33} -0.845563 q^{35} -9.93738 q^{37} +5.23704 q^{39} +5.71327 q^{41} -4.78294 q^{43} -1.00000 q^{45} -5.54591 q^{47} -6.28502 q^{49} +4.72619 q^{51} -14.4962 q^{53} -4.55883 q^{55} +5.54591 q^{57} -5.06968 q^{59} -6.22411 q^{61} +0.845563 q^{63} -5.23704 q^{65} +7.85849 q^{67} -1.00000 q^{69} -3.71327 q^{71} +1.23704 q^{73} +1.00000 q^{75} +3.85478 q^{77} +13.7700 q^{79} +1.00000 q^{81} -7.50913 q^{83} -4.72619 q^{85} +7.06968 q^{87} +2.98708 q^{89} +4.42825 q^{91} +1.83264 q^{93} -5.54591 q^{95} -2.33472 q^{97} +4.55883 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9} - 2 q^{13} - 4 q^{15} + 2 q^{17} + 6 q^{19} - 4 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} + 6 q^{31} - 4 q^{37} - 2 q^{39} + 8 q^{41} + 20 q^{43} - 4 q^{45} - 6 q^{47} + 10 q^{49} + 2 q^{51} - 4 q^{53} + 6 q^{57} + 4 q^{59} - 4 q^{61} + 2 q^{65} + 26 q^{67} - 4 q^{69} - 18 q^{73} + 4 q^{75} + 6 q^{77} + 18 q^{79} + 4 q^{81} + 26 q^{83} - 2 q^{85} + 4 q^{87} + 14 q^{89} + 38 q^{91} + 6 q^{93} - 6 q^{95} - 12 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9 - 2 * q^13 - 4 * q^15 + 2 * q^17 + 6 * q^19 - 4 * q^23 + 4 * q^25 + 4 * q^27 + 4 * q^29 + 6 * q^31 - 4 * q^37 - 2 * q^39 + 8 * q^41 + 20 * q^43 - 4 * q^45 - 6 * q^47 + 10 * q^49 + 2 * q^51 - 4 * q^53 + 6 * q^57 + 4 * q^59 - 4 * q^61 + 2 * q^65 + 26 * q^67 - 4 * q^69 - 18 * q^73 + 4 * q^75 + 6 * q^77 + 18 * q^79 + 4 * q^81 + 26 * q^83 - 2 * q^85 + 4 * q^87 + 14 * q^89 + 38 * q^91 + 6 * q^93 - 6 * q^95 - 12 * 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
$7$	0.845563	0.319593	0.159796	−	0.987150i	$-0.448916\pi$
$7$	0.845563	0.319593	0.159796	+	0.987150i	$0.448916\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.55883	1.37454	0.687270	−	0.726402i	$-0.258809\pi$
$11$	4.55883	1.37454	0.687270	+	0.726402i	$0.258809\pi$
$12$	0	0
$13$	5.23704	1.45249	0.726246	−	0.687435i	$-0.241263\pi$
$13$	5.23704	1.45249	0.726246	+	0.687435i	$0.241263\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	4.72619	1.14627	0.573135	−	0.819461i	$-0.305727\pi$
$17$	4.72619	1.14627	0.573135	+	0.819461i	$0.305727\pi$
$18$	0	0
$19$	5.54591	1.27232	0.636159	−	0.771558i	$-0.280522\pi$
$19$	5.54591	1.27232	0.636159	+	0.771558i	$0.280522\pi$
$20$	0	0
$21$	0.845563	0.184517
$22$	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
$29$	7.06968	1.31281	0.656403	−	0.754411i	$-0.272077\pi$
$29$	7.06968	1.31281	0.656403	+	0.754411i	$0.272077\pi$
$30$	0	0
$31$	1.83264	0.329152	0.164576	−	0.986364i	$-0.447374\pi$
$31$	1.83264	0.329152	0.164576	+	0.986364i	$0.447374\pi$
$32$	0	0
$33$	4.55883	0.793591
$34$	0	0
$35$	−0.845563	−0.142926
$36$	0	0
$37$	−9.93738	−1.63370	−0.816848	−	0.576854i	$-0.804280\pi$
$37$	−9.93738	−1.63370	−0.816848	+	0.576854i	$0.804280\pi$
$38$	0	0
$39$	5.23704	0.838597
$40$	0	0
$41$	5.71327	0.892263	0.446131	−	0.894968i	$-0.352801\pi$
$41$	5.71327	0.892263	0.446131	+	0.894968i	$0.352801\pi$
$42$	0	0
$43$	−4.78294	−0.729392	−0.364696	−	0.931127i	$-0.618827\pi$
$43$	−4.78294	−0.729392	−0.364696	+	0.931127i	$0.618827\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−5.54591	−0.808954	−0.404477	−	0.914548i	$-0.632546\pi$
$47$	−5.54591	−0.808954	−0.404477	+	0.914548i	$0.632546\pi$
$48$	0	0
$49$	−6.28502	−0.897860
$50$	0	0
$51$	4.72619	0.661799
$52$	0	0
$53$	−14.4962	−1.99121	−0.995604	−	0.0936641i	$-0.970142\pi$
$53$	−14.4962	−1.99121	−0.995604	+	0.0936641i	$0.970142\pi$
$54$	0	0
$55$	−4.55883	−0.614713
$56$	0	0
$57$	5.54591	0.734573
$58$	0	0
$59$	−5.06968	−0.660015	−0.330008	−	0.943978i	$-0.607051\pi$
$59$	−5.06968	−0.660015	−0.330008	+	0.943978i	$0.607051\pi$
$60$	0	0
$61$	−6.22411	−0.796916	−0.398458	−	0.917187i	$-0.630455\pi$
$61$	−6.22411	−0.796916	−0.398458	+	0.917187i	$0.630455\pi$
$62$	0	0
$63$	0.845563	0.106531
$64$	0	0
$65$	−5.23704	−0.649574
$66$	0	0
$67$	7.85849	0.960067	0.480033	−	0.877250i	$-0.340625\pi$
$67$	7.85849	0.960067	0.480033	+	0.877250i	$0.340625\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−3.71327	−0.440684	−0.220342	−	0.975423i	$-0.570717\pi$
$71$	−3.71327	−0.440684	−0.220342	+	0.975423i	$0.570717\pi$
$72$	0	0
$73$	1.23704	0.144784	0.0723920	−	0.997376i	$-0.476937\pi$
$73$	1.23704	0.144784	0.0723920	+	0.997376i	$0.476937\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	3.85478	0.439293
$78$	0	0
$79$	13.7700	1.54925	0.774624	−	0.632422i	$-0.217939\pi$
$79$	13.7700	1.54925	0.774624	+	0.632422i	$0.217939\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.50913	−0.824235	−0.412117	−	0.911131i	$-0.635211\pi$
$83$	−7.50913	−0.824235	−0.412117	+	0.911131i	$0.635211\pi$
$84$	0	0
$85$	−4.72619	−0.512627
$86$	0	0
$87$	7.06968	0.757949
$88$	0	0
$89$	2.98708	0.316630	0.158315	−	0.987389i	$-0.449394\pi$
$89$	2.98708	0.316630	0.158315	+	0.987389i	$0.449394\pi$
$90$	0	0
$91$	4.42825	0.464206
$92$	0	0
$93$	1.83264	0.190036
$94$	0	0
$95$	−5.54591	−0.568998
$96$	0	0
$97$	−2.33472	−0.237055	−0.118527	−	0.992951i	$-0.537817\pi$
$97$	−2.33472	−0.237055	−0.118527	+	0.992951i	$0.537817\pi$
$98$	0	0
$99$	4.55883	0.458180
$100$	0	0
$101$	−12.1873	−1.21269	−0.606343	−	0.795203i	$-0.707364\pi$
$101$	−12.1873	−1.21269	−0.606343	+	0.795203i	$0.707364\pi$
$102$	0	0
$103$	18.1047	1.78391	0.891956	−	0.452121i	$-0.149333\pi$
$103$	18.1047	1.78391	0.891956	+	0.452121i	$0.149333\pi$
$104$	0	0
$105$	−0.845563	−0.0825185
$106$	0	0
$107$	6.39147	0.617887	0.308943	−	0.951080i	$-0.400025\pi$
$107$	6.39147	0.617887	0.308943	+	0.951080i	$0.400025\pi$
$108$	0	0
$109$	6.55883	0.628222	0.314111	−	0.949386i	$-0.398294\pi$
$109$	6.55883	0.628222	0.314111	+	0.949386i	$0.398294\pi$
$110$	0	0
$111$	−9.93738	−0.943214
$112$	0	0
$113$	−14.0826	−1.32478	−0.662390	−	0.749159i	$-0.730458\pi$
$113$	−14.0826	−1.32478	−0.662390	+	0.749159i	$0.730458\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	5.23704	0.484164
$118$	0	0
$119$	3.99629	0.366340
$120$	0	0
$121$	9.78294	0.889358
$122$	0	0
$123$	5.71327	0.515148
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.2629	1.17689	0.588445	−	0.808537i	$-0.299740\pi$
$127$	13.2629	1.17689	0.588445	+	0.808537i	$0.299740\pi$
$128$	0	0
$129$	−4.78294	−0.421115
$130$	0	0
$131$	7.38225	0.644991	0.322495	−	0.946571i	$-0.395478\pi$
$131$	7.38225	0.644991	0.322495	+	0.946571i	$0.395478\pi$
$132$	0	0
$133$	4.68942	0.406624
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	10.1306	0.865514	0.432757	−	0.901511i	$-0.357541\pi$
$137$	10.1306	0.865514	0.432757	+	0.901511i	$0.357541\pi$
$138$	0	0
$139$	−11.7332	−0.995201	−0.497600	−	0.867406i	$-0.665785\pi$
$139$	−11.7332	−0.995201	−0.497600	+	0.867406i	$0.665785\pi$
$140$	0	0
$141$	−5.54591	−0.467050
$142$	0	0
$143$	23.8748	1.99651
$144$	0	0
$145$	−7.06968	−0.587105
$146$	0	0
$147$	−6.28502	−0.518380
$148$	0	0
$149$	−10.5934	−0.867849	−0.433924	−	0.900949i	$-0.642871\pi$
$149$	−10.5934	−0.867849	−0.433924	+	0.900949i	$0.642871\pi$
$150$	0	0
$151$	−8.16520	−0.664474	−0.332237	−	0.943196i	$-0.607803\pi$
$151$	−8.16520	−0.664474	−0.332237	+	0.943196i	$0.607803\pi$
$152$	0	0
$153$	4.72619	0.382090
$154$	0	0
$155$	−1.83264	−0.147201
$156$	0	0
$157$	−10.6068	−0.846516	−0.423258	−	0.906009i	$-0.639114\pi$
$157$	−10.6068	−0.846516	−0.423258	+	0.906009i	$0.639114\pi$
$158$	0	0
$159$	−14.4962	−1.14962
$160$	0	0
$161$	−0.845563	−0.0666397
$162$	0	0
$163$	15.6653	1.22700	0.613500	−	0.789695i	$-0.289761\pi$
$163$	15.6653	1.22700	0.613500	+	0.789695i	$0.289761\pi$
$164$	0	0
$165$	−4.55883	−0.354905
$166$	0	0
$167$	2.16365	0.167429	0.0837143	−	0.996490i	$-0.473322\pi$
$167$	2.16365	0.167429	0.0837143	+	0.996490i	$0.473322\pi$
$168$	0	0
$169$	14.4265	1.10973
$170$	0	0
$171$	5.54591	0.424106
$172$	0	0
$173$	−8.30887	−0.631712	−0.315856	−	0.948807i	$-0.602292\pi$
$173$	−8.30887	−0.631712	−0.315856	+	0.948807i	$0.602292\pi$
$174$	0	0
$175$	0.845563	0.0639186
$176$	0	0
$177$	−5.06968	−0.381060
$178$	0	0
$179$	10.7571	0.804023	0.402012	−	0.915635i	$-0.368311\pi$
$179$	10.7571	0.804023	0.402012	+	0.915635i	$0.368311\pi$
$180$	0	0
$181$	19.2482	1.43071	0.715356	−	0.698761i	$-0.246265\pi$
$181$	19.2482	1.43071	0.715356	+	0.698761i	$0.246265\pi$
$182$	0	0
$183$	−6.22411	−0.460100
$184$	0	0
$185$	9.93738	0.730611
$186$	0	0
$187$	21.5459	1.57559
$188$	0	0
$189$	0.845563	0.0615057
$190$	0	0
$191$	−1.78465	−0.129133	−0.0645665	−	0.997913i	$-0.520566\pi$
$191$	−1.78465	−0.129133	−0.0645665	+	0.997913i	$0.520566\pi$
$192$	0	0
$193$	−11.4524	−0.824361	−0.412180	−	0.911102i	$-0.635233\pi$
$193$	−11.4524	−0.824361	−0.412180	+	0.911102i	$0.635233\pi$
$194$	0	0
$195$	−5.23704	−0.375032
$196$	0	0
$197$	2.23874	0.159504	0.0797520	−	0.996815i	$-0.474587\pi$
$197$	2.23874	0.159504	0.0797520	+	0.996815i	$0.474587\pi$
$198$	0	0
$199$	−18.5530	−1.31518	−0.657592	−	0.753374i	$-0.728425\pi$
$199$	−18.5530	−1.31518	−0.657592	+	0.753374i	$0.728425\pi$
$200$	0	0
$201$	7.85849	0.554295
$202$	0	0
$203$	5.97786	0.419563
$204$	0	0
$205$	−5.71327	−0.399032
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	25.2829	1.74885
$210$	0	0
$211$	25.0204	1.72248	0.861239	−	0.508201i	$-0.169689\pi$
$211$	25.0204	1.72248	0.861239	+	0.508201i	$0.169689\pi$
$212$	0	0
$213$	−3.71327	−0.254429
$214$	0	0
$215$	4.78294	0.326194
$216$	0	0
$217$	1.54961	0.105195
$218$	0	0
$219$	1.23704	0.0835911
$220$	0	0
$221$	24.7512	1.66495
$222$	0	0
$223$	7.71697	0.516767	0.258383	−	0.966042i	$-0.416810\pi$
$223$	7.71697	0.516767	0.258383	+	0.966042i	$0.416810\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	14.4395	0.958381	0.479190	−	0.877711i	$-0.340930\pi$
$227$	14.4395	0.958381	0.479190	+	0.877711i	$0.340930\pi$
$228$	0	0
$229$	15.8660	1.04845	0.524227	−	0.851578i	$-0.324354\pi$
$229$	15.8660	1.04845	0.524227	+	0.851578i	$0.324354\pi$
$230$	0	0
$231$	3.85478	0.253626
$232$	0	0
$233$	−8.36057	−0.547719	−0.273859	−	0.961770i	$-0.588300\pi$
$233$	−8.36057	−0.547719	−0.273859	+	0.961770i	$0.588300\pi$
$234$	0	0
$235$	5.54591	0.361775
$236$	0	0
$237$	13.7700	0.894459
$238$	0	0
$239$	−11.6173	−0.751460	−0.375730	−	0.926729i	$-0.622608\pi$
$239$	−11.6173	−0.751460	−0.375730	+	0.926729i	$0.622608\pi$
$240$	0	0
$241$	−25.0675	−1.61474	−0.807370	−	0.590045i	$-0.799110\pi$
$241$	−25.0675	−1.61474	−0.807370	+	0.590045i	$0.799110\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.28502	0.401535
$246$	0	0
$247$	29.0441	1.84803
$248$	0	0
$249$	−7.50913	−0.475872
$250$	0	0
$251$	13.4870	0.851291	0.425646	−	0.904890i	$-0.360047\pi$
$251$	13.4870	0.851291	0.425646	+	0.904890i	$0.360047\pi$
$252$	0	0
$253$	−4.55883	−0.286611
$254$	0	0
$255$	−4.72619	−0.295966
$256$	0	0
$257$	−1.85478	−0.115698	−0.0578490	−	0.998325i	$-0.518424\pi$
$257$	−1.85478	−0.115698	−0.0578490	+	0.998325i	$0.518424\pi$
$258$	0	0
$259$	−8.40269	−0.522117
$260$	0	0
$261$	7.06968	0.437602
$262$	0	0
$263$	−16.2132	−0.999748	−0.499874	−	0.866098i	$-0.666620\pi$
$263$	−16.2132	−0.999748	−0.499874	+	0.866098i	$0.666620\pi$
$264$	0	0
$265$	14.4962	0.890495
$266$	0	0
$267$	2.98708	0.182806
$268$	0	0
$269$	−8.80508	−0.536855	−0.268428	−	0.963300i	$-0.586504\pi$
$269$	−8.80508	−0.536855	−0.268428	+	0.963300i	$0.586504\pi$
$270$	0	0
$271$	−3.23333	−0.196411	−0.0982054	−	0.995166i	$-0.531310\pi$
$271$	−3.23333	−0.196411	−0.0982054	+	0.995166i	$0.531310\pi$
$272$	0	0
$273$	4.42825	0.268010
$274$	0	0
$275$	4.55883	0.274908
$276$	0	0
$277$	2.61775	0.157285	0.0786426	−	0.996903i	$-0.474941\pi$
$277$	2.61775	0.157285	0.0786426	+	0.996903i	$0.474941\pi$
$278$	0	0
$279$	1.83264	0.109717
$280$	0	0
$281$	−14.6031	−0.871149	−0.435574	−	0.900153i	$-0.643455\pi$
$281$	−14.6031	−0.871149	−0.435574	+	0.900153i	$0.643455\pi$
$282$	0	0
$283$	22.7808	1.35418	0.677088	−	0.735902i	$-0.263241\pi$
$283$	22.7808	1.35418	0.677088	+	0.735902i	$0.263241\pi$
$284$	0	0
$285$	−5.54591	−0.328511
$286$	0	0
$287$	4.83093	0.285161
$288$	0	0
$289$	5.33688	0.313934
$290$	0	0
$291$	−2.33472	−0.136864
$292$	0	0
$293$	−25.5880	−1.49487	−0.747434	−	0.664336i	$-0.768715\pi$
$293$	−25.5880	−1.49487	−0.747434	+	0.664336i	$0.768715\pi$
$294$	0	0
$295$	5.06968	0.295168
$296$	0	0
$297$	4.55883	0.264530
$298$	0	0
$299$	−5.23704	−0.302866
$300$	0	0
$301$	−4.04428	−0.233109
$302$	0	0
$303$	−12.1873	−0.700144
$304$	0	0
$305$	6.22411	0.356392
$306$	0	0
$307$	−12.4507	−0.710597	−0.355299	−	0.934753i	$-0.615621\pi$
$307$	−12.4507	−0.710597	−0.355299	+	0.934753i	$0.615621\pi$
$308$	0	0
$309$	18.1047	1.02994
$310$	0	0
$311$	−10.9048	−0.618352	−0.309176	−	0.951005i	$-0.600053\pi$
$311$	−10.9048	−0.618352	−0.309176	+	0.951005i	$0.600053\pi$
$312$	0	0
$313$	−30.7549	−1.73837	−0.869186	−	0.494486i	$-0.835356\pi$
$313$	−30.7549	−1.73837	−0.869186	+	0.494486i	$0.835356\pi$
$314$	0	0
$315$	−0.845563	−0.0476421
$316$	0	0
$317$	24.9983	1.40404	0.702022	−	0.712155i	$-0.252281\pi$
$317$	24.9983	1.40404	0.702022	+	0.712155i	$0.252281\pi$
$318$	0	0
$319$	32.2295	1.80450
$320$	0	0
$321$	6.39147	0.356737
$322$	0	0
$323$	26.2110	1.45842
$324$	0	0
$325$	5.23704	0.290498
$326$	0	0
$327$	6.55883	0.362704
$328$	0	0
$329$	−4.68942	−0.258536
$330$	0	0
$331$	9.59390	0.527328	0.263664	−	0.964615i	$-0.415069\pi$
$331$	9.59390	0.527328	0.263664	+	0.964615i	$0.415069\pi$
$332$	0	0
$333$	−9.93738	−0.544565
$334$	0	0
$335$	−7.85849	−0.429355
$336$	0	0
$337$	3.97415	0.216486	0.108243	−	0.994124i	$-0.465478\pi$
$337$	3.97415	0.216486	0.108243	+	0.994124i	$0.465478\pi$
$338$	0	0
$339$	−14.0826	−0.764862
$340$	0	0
$341$	8.35470	0.452432
$342$	0	0
$343$	−11.2333	−0.606543
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	−13.1177	−0.704193	−0.352097	−	0.935964i	$-0.614531\pi$
$347$	−13.1177	−0.704193	−0.352097	+	0.935964i	$0.614531\pi$
$348$	0	0
$349$	9.80679	0.524946	0.262473	−	0.964939i	$-0.415462\pi$
$349$	9.80679	0.524946	0.262473	+	0.964939i	$0.415462\pi$
$350$	0	0
$351$	5.23704	0.279532
$352$	0	0
$353$	−8.55007	−0.455074	−0.227537	−	0.973769i	$-0.573067\pi$
$353$	−8.55007	−0.455074	−0.227537	+	0.973769i	$0.573067\pi$
$354$	0	0
$355$	3.71327	0.197080
$356$	0	0
$357$	3.99629	0.211506
$358$	0	0
$359$	−28.3990	−1.49884	−0.749420	−	0.662094i	$-0.769668\pi$
$359$	−28.3990	−1.49884	−0.749420	+	0.662094i	$0.769668\pi$
$360$	0	0
$361$	11.7571	0.618795
$362$	0	0
$363$	9.78294	0.513471
$364$	0	0
$365$	−1.23704	−0.0647494
$366$	0	0
$367$	−24.3121	−1.26908	−0.634539	−	0.772890i	$-0.718810\pi$
$367$	−24.3121	−1.26908	−0.634539	+	0.772890i	$0.718810\pi$
$368$	0	0
$369$	5.71327	0.297421
$370$	0	0
$371$	−12.2575	−0.636376
$372$	0	0
$373$	−20.3435	−1.05335	−0.526673	−	0.850068i	$-0.676561\pi$
$373$	−20.3435	−1.05335	−0.526673	+	0.850068i	$0.676561\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	37.0241	1.90684
$378$	0	0
$379$	−7.33056	−0.376546	−0.188273	−	0.982117i	$-0.560289\pi$
$379$	−7.33056	−0.376546	−0.188273	+	0.982117i	$0.560289\pi$
$380$	0	0
$381$	13.2629	0.679478
$382$	0	0
$383$	−28.2575	−1.44389	−0.721945	−	0.691951i	$-0.756751\pi$
$383$	−28.2575	−1.44389	−0.721945	+	0.691951i	$0.756751\pi$
$384$	0	0
$385$	−3.85478	−0.196458
$386$	0	0
$387$	−4.78294	−0.243131
$388$	0	0
$389$	23.9966	1.21667	0.608337	−	0.793678i	$-0.291837\pi$
$389$	23.9966	1.21667	0.608337	+	0.793678i	$0.291837\pi$
$390$	0	0
$391$	−4.72619	−0.239014
$392$	0	0
$393$	7.38225	0.372385
$394$	0	0
$395$	−13.7700	−0.692845
$396$	0	0
$397$	−3.02169	−0.151654	−0.0758271	−	0.997121i	$-0.524160\pi$
$397$	−3.02169	−0.151654	−0.0758271	+	0.997121i	$0.524160\pi$
$398$	0	0
$399$	4.68942	0.234765
$400$	0	0
$401$	−29.6263	−1.47947	−0.739735	−	0.672899i	$-0.765049\pi$
$401$	−29.6263	−1.47947	−0.739735	+	0.672899i	$0.765049\pi$
$402$	0	0
$403$	9.59760	0.478091
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−45.3028	−2.24558
$408$	0	0
$409$	−7.42437	−0.367112	−0.183556	−	0.983009i	$-0.558761\pi$
$409$	−7.42437	−0.367112	−0.183556	+	0.983009i	$0.558761\pi$
$410$	0	0
$411$	10.1306	0.499705
$412$	0	0
$413$	−4.28673	−0.210936
$414$	0	0
$415$	7.50913	0.368609
$416$	0	0
$417$	−11.7332	−0.574580
$418$	0	0
$419$	2.20568	0.107754	0.0538772	−	0.998548i	$-0.482842\pi$
$419$	2.20568	0.107754	0.0538772	+	0.998548i	$0.482842\pi$
$420$	0	0
$421$	34.0989	1.66188	0.830939	−	0.556364i	$-0.187804\pi$
$421$	34.0989	1.66188	0.830939	+	0.556364i	$0.187804\pi$
$422$	0	0
$423$	−5.54591	−0.269651
$424$	0	0
$425$	4.72619	0.229254
$426$	0	0
$427$	−5.26288	−0.254689
$428$	0	0
$429$	23.8748	1.15268
$430$	0	0
$431$	−4.16952	−0.200839	−0.100419	−	0.994945i	$-0.532018\pi$
$431$	−4.16952	−0.200839	−0.100419	+	0.994945i	$0.532018\pi$
$432$	0	0
$433$	7.08965	0.340707	0.170353	−	0.985383i	$-0.445509\pi$
$433$	7.08965	0.340707	0.170353	+	0.985383i	$0.445509\pi$
$434$	0	0
$435$	−7.06968	−0.338965
$436$	0	0
$437$	−5.54591	−0.265297
$438$	0	0
$439$	36.7537	1.75416	0.877079	−	0.480347i	$-0.159489\pi$
$439$	36.7537	1.75416	0.877079	+	0.480347i	$0.159489\pi$
$440$	0	0
$441$	−6.28502	−0.299287
$442$	0	0
$443$	−23.3505	−1.10942	−0.554709	−	0.832045i	$-0.687170\pi$
$443$	−23.3505	−1.10942	−0.554709	+	0.832045i	$0.687170\pi$
$444$	0	0
$445$	−2.98708	−0.141601
$446$	0	0
$447$	−10.5934	−0.501053
$448$	0	0
$449$	30.0437	1.41785	0.708924	−	0.705285i	$-0.249181\pi$
$449$	30.0437	1.41785	0.708924	+	0.705285i	$0.249181\pi$
$450$	0	0
$451$	26.0458	1.22645
$452$	0	0
$453$	−8.16520	−0.383634
$454$	0	0
$455$	−4.42825	−0.207599
$456$	0	0
$457$	19.5496	0.914492	0.457246	−	0.889340i	$-0.348836\pi$
$457$	19.5496	0.914492	0.457246	+	0.889340i	$0.348836\pi$
$458$	0	0
$459$	4.72619	0.220600
$460$	0	0
$461$	−22.6577	−1.05527	−0.527637	−	0.849470i	$-0.676922\pi$
$461$	−22.6577	−1.05527	−0.527637	+	0.849470i	$0.676922\pi$
$462$	0	0
$463$	21.7037	1.00866	0.504328	−	0.863512i	$-0.331740\pi$
$463$	21.7037	1.00866	0.504328	+	0.863512i	$0.331740\pi$
$464$	0	0
$465$	−1.83264	−0.0849867
$466$	0	0
$467$	−25.2003	−1.16613	−0.583065	−	0.812426i	$-0.698147\pi$
$467$	−25.2003	−1.16613	−0.583065	+	0.812426i	$0.698147\pi$
$468$	0	0
$469$	6.64485	0.306831
$470$	0	0
$471$	−10.6068	−0.488736
$472$	0	0
$473$	−21.8046	−1.00258
$474$	0	0
$475$	5.54591	0.254464
$476$	0	0
$477$	−14.4962	−0.663736
$478$	0	0
$479$	19.1635	0.875602	0.437801	−	0.899072i	$-0.355757\pi$
$479$	19.1635	0.875602	0.437801	+	0.899072i	$0.355757\pi$
$480$	0	0
$481$	−52.0424	−2.37293
$482$	0	0
$483$	−0.845563	−0.0384745
$484$	0	0
$485$	2.33472	0.106014
$486$	0	0
$487$	−14.3805	−0.651645	−0.325822	−	0.945431i	$-0.605641\pi$
$487$	−14.3805	−0.651645	−0.325822	+	0.945431i	$0.605641\pi$
$488$	0	0
$489$	15.6653	0.708408
$490$	0	0
$491$	16.3267	0.736813	0.368407	−	0.929665i	$-0.379903\pi$
$491$	16.3267	0.736813	0.368407	+	0.929665i	$0.379903\pi$
$492$	0	0
$493$	33.4126	1.50483
$494$	0	0
$495$	−4.55883	−0.204904
$496$	0	0
$497$	−3.13980	−0.140839
$498$	0	0
$499$	−29.8726	−1.33728	−0.668641	−	0.743586i	$-0.733123\pi$
$499$	−29.8726	−1.33728	−0.668641	+	0.743586i	$0.733123\pi$
$500$	0	0
$501$	2.16365	0.0966649
$502$	0	0
$503$	−33.6831	−1.50186	−0.750928	−	0.660385i	$-0.770393\pi$
$503$	−33.6831	−1.50186	−0.750928	+	0.660385i	$0.770393\pi$
$504$	0	0
$505$	12.1873	0.542329
$506$	0	0
$507$	14.4265	0.640705
$508$	0	0
$509$	−29.9265	−1.32647	−0.663233	−	0.748413i	$-0.730816\pi$
$509$	−29.9265	−1.32647	−0.663233	+	0.748413i	$0.730816\pi$
$510$	0	0
$511$	1.04599	0.0462719
$512$	0	0
$513$	5.54591	0.244858
$514$	0	0
$515$	−18.1047	−0.797790
$516$	0	0
$517$	−25.2829	−1.11194
$518$	0	0
$519$	−8.30887	−0.364719
$520$	0	0
$521$	−8.19827	−0.359173	−0.179586	−	0.983742i	$-0.557476\pi$
$521$	−8.19827	−0.359173	−0.179586	+	0.983742i	$0.557476\pi$
$522$	0	0
$523$	8.15779	0.356715	0.178358	−	0.983966i	$-0.442922\pi$
$523$	8.15779	0.356715	0.178358	+	0.983966i	$0.442922\pi$
$524$	0	0
$525$	0.845563	0.0369034
$526$	0	0
$527$	8.66141	0.377297
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.06968	−0.220005
$532$	0	0
$533$	29.9206	1.29600
$534$	0	0
$535$	−6.39147	−0.276327
$536$	0	0
$537$	10.7571	0.464203
$538$	0	0
$539$	−28.6524	−1.23414
$540$	0	0
$541$	−43.4406	−1.86766	−0.933830	−	0.357718i	$-0.883555\pi$
$541$	−43.4406	−1.86766	−0.933830	+	0.357718i	$0.883555\pi$
$542$	0	0
$543$	19.2482	0.826021
$544$	0	0
$545$	−6.55883	−0.280949
$546$	0	0
$547$	37.5659	1.60620	0.803101	−	0.595843i	$-0.203182\pi$
$547$	37.5659	1.60620	0.803101	+	0.595843i	$0.203182\pi$
$548$	0	0
$549$	−6.22411	−0.265639
$550$	0	0
$551$	39.2078	1.67031
$552$	0	0
$553$	11.6434	0.495129
$554$	0	0
$555$	9.93738	0.421818
$556$	0	0
$557$	13.0438	0.552685	0.276342	−	0.961059i	$-0.410878\pi$
$557$	13.0438	0.552685	0.276342	+	0.961059i	$0.410878\pi$
$558$	0	0
$559$	−25.0484	−1.05944
$560$	0	0
$561$	21.5459	0.909669
$562$	0	0
$563$	4.25212	0.179206	0.0896028	−	0.995978i	$-0.471440\pi$
$563$	4.25212	0.179206	0.0896028	+	0.995978i	$0.471440\pi$
$564$	0	0
$565$	14.0826	0.592459
$566$	0	0
$567$	0.845563	0.0355103
$568$	0	0
$569$	45.8358	1.92154	0.960769	−	0.277350i	$-0.0894563\pi$
$569$	45.8358	1.92154	0.960769	+	0.277350i	$0.0894563\pi$
$570$	0	0
$571$	−22.7595	−0.952457	−0.476229	−	0.879321i	$-0.657997\pi$
$571$	−22.7595	−0.952457	−0.476229	+	0.879321i	$0.657997\pi$
$572$	0	0
$573$	−1.78465	−0.0745549
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	40.2310	1.67484	0.837419	−	0.546561i	$-0.184063\pi$
$577$	40.2310	1.67484	0.837419	+	0.546561i	$0.184063\pi$
$578$	0	0
$579$	−11.4524	−0.475945
$580$	0	0
$581$	−6.34945	−0.263420
$582$	0	0
$583$	−66.0858	−2.73699
$584$	0	0
$585$	−5.23704	−0.216525
$586$	0	0
$587$	42.7278	1.76357	0.881783	−	0.471655i	$-0.156343\pi$
$587$	42.7278	1.76357	0.881783	+	0.471655i	$0.156343\pi$
$588$	0	0
$589$	10.1637	0.418786
$590$	0	0
$591$	2.23874	0.0920897
$592$	0	0
$593$	22.1852	0.911036	0.455518	−	0.890227i	$-0.349454\pi$
$593$	22.1852	0.911036	0.455518	+	0.890227i	$0.349454\pi$
$594$	0	0
$595$	−3.99629	−0.163832
$596$	0	0
$597$	−18.5530	−0.759322
$598$	0	0
$599$	9.85632	0.402718	0.201359	−	0.979517i	$-0.435464\pi$
$599$	9.85632	0.402718	0.201359	+	0.979517i	$0.435464\pi$
$600$	0	0
$601$	12.1157	0.494208	0.247104	−	0.968989i	$-0.420521\pi$
$601$	12.1157	0.494208	0.247104	+	0.968989i	$0.420521\pi$
$602$	0	0
$603$	7.85849	0.320022
$604$	0	0
$605$	−9.78294	−0.397733
$606$	0	0
$607$	−14.4984	−0.588471	−0.294235	−	0.955733i	$-0.595065\pi$
$607$	−14.4984	−0.588471	−0.294235	+	0.955733i	$0.595065\pi$
$608$	0	0
$609$	5.97786	0.242235
$610$	0	0
$611$	−29.0441	−1.17500
$612$	0	0
$613$	−36.0312	−1.45529	−0.727643	−	0.685956i	$-0.759384\pi$
$613$	−36.0312	−1.45529	−0.727643	+	0.685956i	$0.759384\pi$
$614$	0	0
$615$	−5.71327	−0.230381
$616$	0	0
$617$	−44.0792	−1.77456	−0.887280	−	0.461230i	$-0.847408\pi$
$617$	−44.0792	−1.77456	−0.887280	+	0.461230i	$0.847408\pi$
$618$	0	0
$619$	−32.9998	−1.32638	−0.663188	−	0.748453i	$-0.730797\pi$
$619$	−32.9998	−1.32638	−0.663188	+	0.748453i	$0.730797\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	2.52576	0.101193
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	25.2829	1.00970
$628$	0	0
$629$	−46.9660	−1.87266
$630$	0	0
$631$	12.8200	0.510356	0.255178	−	0.966894i	$-0.417866\pi$
$631$	12.8200	0.510356	0.255178	+	0.966894i	$0.417866\pi$
$632$	0	0
$633$	25.0204	0.994473
$634$	0	0
$635$	−13.2629	−0.526321
$636$	0	0
$637$	−32.9149	−1.30414
$638$	0	0
$639$	−3.71327	−0.146895
$640$	0	0
$641$	38.8126	1.53301	0.766503	−	0.642241i	$-0.221995\pi$
$641$	38.8126	1.53301	0.766503	+	0.642241i	$0.221995\pi$
$642$	0	0
$643$	33.1339	1.30667	0.653337	−	0.757067i	$-0.273368\pi$
$643$	33.1339	1.30667	0.653337	+	0.757067i	$0.273368\pi$
$644$	0	0
$645$	4.78294	0.188328
$646$	0	0
$647$	−24.3547	−0.957482	−0.478741	−	0.877956i	$-0.658907\pi$
$647$	−24.3547	−0.957482	−0.478741	+	0.877956i	$0.658907\pi$
$648$	0	0
$649$	−23.1118	−0.907217
$650$	0	0
$651$	1.54961	0.0607341
$652$	0	0
$653$	12.5943	0.492855	0.246427	−	0.969161i	$-0.420743\pi$
$653$	12.5943	0.492855	0.246427	+	0.969161i	$0.420743\pi$
$654$	0	0
$655$	−7.38225	−0.288449
$656$	0	0
$657$	1.23704	0.0482613
$658$	0	0
$659$	41.9259	1.63320	0.816601	−	0.577202i	$-0.195856\pi$
$659$	41.9259	1.63320	0.816601	+	0.577202i	$0.195856\pi$
$660$	0	0
$661$	45.1664	1.75677	0.878384	−	0.477955i	$-0.158622\pi$
$661$	45.1664	1.75677	0.878384	+	0.477955i	$0.158622\pi$
$662$	0	0
$663$	24.7512	0.961258
$664$	0	0
$665$	−4.68942	−0.181848
$666$	0	0
$667$	−7.06968	−0.273739
$668$	0	0
$669$	7.71697	0.298355
$670$	0	0
$671$	−28.3747	−1.09539
$672$	0	0
$673$	−13.4022	−0.516618	−0.258309	−	0.966062i	$-0.583165\pi$
$673$	−13.4022	−0.516618	−0.258309	+	0.966062i	$0.583165\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−41.5662	−1.59752	−0.798759	−	0.601651i	$-0.794510\pi$
$677$	−41.5662	−1.59752	−0.798759	+	0.601651i	$0.794510\pi$
$678$	0	0
$679$	−1.97415	−0.0757611
$680$	0	0
$681$	14.4395	0.553321
$682$	0	0
$683$	24.1778	0.925136	0.462568	−	0.886584i	$-0.346928\pi$
$683$	24.1778	0.925136	0.462568	+	0.886584i	$0.346928\pi$
$684$	0	0
$685$	−10.1306	−0.387070
$686$	0	0
$687$	15.8660	0.605325
$688$	0	0
$689$	−75.9172	−2.89221
$690$	0	0
$691$	−4.99242	−0.189921	−0.0949603	−	0.995481i	$-0.530272\pi$
$691$	−4.99242	−0.189921	−0.0949603	+	0.995481i	$0.530272\pi$
$692$	0	0
$693$	3.85478	0.146431
$694$	0	0
$695$	11.7332	0.445067
$696$	0	0
$697$	27.0020	1.02277
$698$	0	0
$699$	−8.36057	−0.316226
$700$	0	0
$701$	28.0384	1.05900	0.529498	−	0.848311i	$-0.322380\pi$
$701$	28.0384	1.05900	0.529498	+	0.848311i	$0.322380\pi$
$702$	0	0
$703$	−55.1118	−2.07858
$704$	0	0
$705$	5.54591	0.208871
$706$	0	0
$707$	−10.3052	−0.387566
$708$	0	0
$709$	40.9594	1.53826	0.769130	−	0.639092i	$-0.220690\pi$
$709$	40.9594	1.53826	0.769130	+	0.639092i	$0.220690\pi$
$710$	0	0
$711$	13.7700	0.516416
$712$	0	0
$713$	−1.83264	−0.0686329
$714$	0	0
$715$	−23.8748	−0.892865
$716$	0	0
$717$	−11.6173	−0.433856
$718$	0	0
$719$	−26.8568	−1.00159	−0.500794	−	0.865566i	$-0.666959\pi$
$719$	−26.8568	−1.00159	−0.500794	+	0.865566i	$0.666959\pi$
$720$	0	0
$721$	15.3087	0.570126
$722$	0	0
$723$	−25.0675	−0.932271
$724$	0	0
$725$	7.06968	0.262561
$726$	0	0
$727$	34.9849	1.29752	0.648759	−	0.760994i	$-0.275288\pi$
$727$	34.9849	1.29752	0.648759	+	0.760994i	$0.275288\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−22.6051	−0.836080
$732$	0	0
$733$	28.2170	1.04222	0.521109	−	0.853490i	$-0.325518\pi$
$733$	28.2170	1.04222	0.521109	+	0.853490i	$0.325518\pi$
$734$	0	0
$735$	6.28502	0.231827
$736$	0	0
$737$	35.8255	1.31965
$738$	0	0
$739$	−30.8251	−1.13392	−0.566959	−	0.823746i	$-0.691880\pi$
$739$	−30.8251	−1.13392	−0.566959	+	0.823746i	$0.691880\pi$
$740$	0	0
$741$	29.0441	1.06696
$742$	0	0
$743$	−53.2093	−1.95206	−0.976030	−	0.217635i	$-0.930166\pi$
$743$	−53.2093	−1.95206	−0.976030	+	0.217635i	$0.930166\pi$
$744$	0	0
$745$	10.5934	0.388114
$746$	0	0
$747$	−7.50913	−0.274745
$748$	0	0
$749$	5.40439	0.197472
$750$	0	0
$751$	−25.0513	−0.914136	−0.457068	−	0.889432i	$-0.651100\pi$
$751$	−25.0513	−0.914136	−0.457068	+	0.889432i	$0.651100\pi$
$752$	0	0
$753$	13.4870	0.491493
$754$	0	0
$755$	8.16520	0.297162
$756$	0	0
$757$	2.72032	0.0988718	0.0494359	−	0.998777i	$-0.484258\pi$
$757$	2.72032	0.0988718	0.0494359	+	0.998777i	$0.484258\pi$
$758$	0	0
$759$	−4.55883	−0.165475
$760$	0	0
$761$	−11.9779	−0.434197	−0.217099	−	0.976150i	$-0.569659\pi$
$761$	−11.9779	−0.434197	−0.217099	+	0.976150i	$0.569659\pi$
$762$	0	0
$763$	5.54591	0.200775
$764$	0	0
$765$	−4.72619	−0.170876
$766$	0	0
$767$	−26.5501	−0.958667
$768$	0	0
$769$	14.2412	0.513551	0.256775	−	0.966471i	$-0.417340\pi$
$769$	14.2412	0.513551	0.256775	+	0.966471i	$0.417340\pi$
$770$	0	0
$771$	−1.85478	−0.0667983
$772$	0	0
$773$	15.0660	0.541885	0.270943	−	0.962595i	$-0.412665\pi$
$773$	15.0660	0.541885	0.270943	+	0.962595i	$0.412665\pi$
$774$	0	0
$775$	1.83264	0.0658304
$776$	0	0
$777$	−8.40269	−0.301445
$778$	0	0
$779$	31.6853	1.13524
$780$	0	0
$781$	−16.9282	−0.605737
$782$	0	0
$783$	7.06968	0.252650
$784$	0	0
$785$	10.6068	0.378574
$786$	0	0
$787$	−4.79922	−0.171074	−0.0855368	−	0.996335i	$-0.527261\pi$
$787$	−4.79922	−0.171074	−0.0855368	+	0.996335i	$0.527261\pi$
$788$	0	0
$789$	−16.2132	−0.577205
$790$	0	0
$791$	−11.9077	−0.423390
$792$	0	0
$793$	−32.5959	−1.15751
$794$	0	0
$795$	14.4962	0.514128
$796$	0	0
$797$	51.6538	1.82967	0.914836	−	0.403825i	$-0.132320\pi$
$797$	51.6538	1.82967	0.914836	+	0.403825i	$0.132320\pi$
$798$	0	0
$799$	−26.2110	−0.927279
$800$	0	0
$801$	2.98708	0.105543
$802$	0	0
$803$	5.63943	0.199011
$804$	0	0
$805$	0.845563	0.0298022
$806$	0	0
$807$	−8.80508	−0.309954
$808$	0	0
$809$	−9.30500	−0.327146	−0.163573	−	0.986531i	$-0.552302\pi$
$809$	−9.30500	−0.327146	−0.163573	+	0.986531i	$0.552302\pi$
$810$	0	0
$811$	39.2041	1.37664	0.688320	−	0.725407i	$-0.258348\pi$
$811$	39.2041	1.37664	0.688320	+	0.725407i	$0.258348\pi$
$812$	0	0
$813$	−3.23333	−0.113398
$814$	0	0
$815$	−15.6653	−0.548731
$816$	0	0
$817$	−26.5258	−0.928019
$818$	0	0
$819$	4.42825	0.154735
$820$	0	0
$821$	−9.23948	−0.322460	−0.161230	−	0.986917i	$-0.551546\pi$
$821$	−9.23948	−0.322460	−0.161230	+	0.986917i	$0.551546\pi$
$822$	0	0
$823$	26.3931	0.920006	0.460003	−	0.887917i	$-0.347848\pi$
$823$	26.3931	0.920006	0.460003	+	0.887917i	$0.347848\pi$
$824$	0	0
$825$	4.55883	0.158718
$826$	0	0
$827$	16.7520	0.582525	0.291263	−	0.956643i	$-0.405925\pi$
$827$	16.7520	0.582525	0.291263	+	0.956643i	$0.405925\pi$
$828$	0	0
$829$	−32.1381	−1.11620	−0.558101	−	0.829773i	$-0.688470\pi$
$829$	−32.1381	−1.11620	−0.558101	+	0.829773i	$0.688470\pi$
$830$	0	0
$831$	2.61775	0.0908086
$832$	0	0
$833$	−29.7042	−1.02919
$834$	0	0
$835$	−2.16365	−0.0748763
$836$	0	0
$837$	1.83264	0.0633453
$838$	0	0
$839$	−15.8748	−0.548058	−0.274029	−	0.961721i	$-0.588356\pi$
$839$	−15.8748	−0.548058	−0.274029	+	0.961721i	$0.588356\pi$
$840$	0	0
$841$	20.9803	0.723459
$842$	0	0
$843$	−14.6031	−0.502958
$844$	0	0
$845$	−14.4265	−0.496288
$846$	0	0
$847$	8.27210	0.284233
$848$	0	0
$849$	22.7808	0.781834
$850$	0	0
$851$	9.93738	0.340649
$852$	0	0
$853$	−27.4481	−0.939804	−0.469902	−	0.882719i	$-0.655711\pi$
$853$	−27.4481	−0.939804	−0.469902	+	0.882719i	$0.655711\pi$
$854$	0	0
$855$	−5.54591	−0.189666
$856$	0	0
$857$	−49.3929	−1.68723	−0.843615	−	0.536948i	$-0.819577\pi$
$857$	−49.3929	−1.68723	−0.843615	+	0.536948i	$0.819577\pi$
$858$	0	0
$859$	18.5863	0.634157	0.317078	−	0.948399i	$-0.397298\pi$
$859$	18.5863	0.634157	0.317078	+	0.948399i	$0.397298\pi$
$860$	0	0
$861$	4.83093	0.164638
$862$	0	0
$863$	33.6619	1.14586	0.572932	−	0.819603i	$-0.305806\pi$
$863$	33.6619	1.14586	0.572932	+	0.819603i	$0.305806\pi$
$864$	0	0
$865$	8.30887	0.282510
$866$	0	0
$867$	5.33688	0.181250
$868$	0	0
$869$	62.7752	2.12950
$870$	0	0
$871$	41.1552	1.39449
$872$	0	0
$873$	−2.33472	−0.0790183
$874$	0	0
$875$	−0.845563	−0.0285853
$876$	0	0
$877$	29.2321	0.987097	0.493548	−	0.869718i	$-0.335700\pi$
$877$	29.2321	0.987097	0.493548	+	0.869718i	$0.335700\pi$
$878$	0	0
$879$	−25.5880	−0.863063
$880$	0	0
$881$	1.20242	0.0405107	0.0202553	−	0.999795i	$-0.493552\pi$
$881$	1.20242	0.0405107	0.0202553	+	0.999795i	$0.493552\pi$
$882$	0	0
$883$	2.82877	0.0951956	0.0475978	−	0.998867i	$-0.484843\pi$
$883$	2.82877	0.0951956	0.0475978	+	0.998867i	$0.484843\pi$
$884$	0	0
$885$	5.06968	0.170415
$886$	0	0
$887$	6.86048	0.230352	0.115176	−	0.993345i	$-0.463257\pi$
$887$	6.86048	0.230352	0.115176	+	0.993345i	$0.463257\pi$
$888$	0	0
$889$	11.2146	0.376126
$890$	0	0
$891$	4.55883	0.152727
$892$	0	0
$893$	−30.7571	−1.02925
$894$	0	0
$895$	−10.7571	−0.359570
$896$	0	0
$897$	−5.23704	−0.174860
$898$	0	0
$899$	12.9562	0.432113
$900$	0	0
$901$	−68.5119	−2.28246
$902$	0	0
$903$	−4.04428	−0.134585
$904$	0	0
$905$	−19.2482	−0.639833
$906$	0	0
$907$	15.1198	0.502046	0.251023	−	0.967981i	$-0.419233\pi$
$907$	15.1198	0.502046	0.251023	+	0.967981i	$0.419233\pi$
$908$	0	0
$909$	−12.1873	−0.404228
$910$	0	0
$911$	50.3931	1.66960	0.834799	−	0.550555i	$-0.185584\pi$
$911$	50.3931	1.66960	0.834799	+	0.550555i	$0.185584\pi$
$912$	0	0
$913$	−34.2329	−1.13294
$914$	0	0
$915$	6.22411	0.205763
$916$	0	0
$917$	6.24216	0.206134
$918$	0	0
$919$	40.0053	1.31965	0.659827	−	0.751417i	$-0.270629\pi$
$919$	40.0053	1.31965	0.659827	+	0.751417i	$0.270629\pi$
$920$	0	0
$921$	−12.4507	−0.410264
$922$	0	0
$923$	−19.4465	−0.640090
$924$	0	0
$925$	−9.93738	−0.326739
$926$	0	0
$927$	18.1047	0.594638
$928$	0	0
$929$	39.0921	1.28257	0.641285	−	0.767303i	$-0.278402\pi$
$929$	39.0921	1.28257	0.641285	+	0.767303i	$0.278402\pi$
$930$	0	0
$931$	−34.8562	−1.14236
$932$	0	0
$933$	−10.9048	−0.357006
$934$	0	0
$935$	−21.5459	−0.704627
$936$	0	0
$937$	48.2310	1.57564	0.787819	−	0.615907i	$-0.211210\pi$
$937$	48.2310	1.57564	0.787819	+	0.615907i	$0.211210\pi$
$938$	0	0
$939$	−30.7549	−1.00365
$940$	0	0
$941$	10.7337	0.349909	0.174954	−	0.984577i	$-0.444022\pi$
$941$	10.7337	0.349909	0.174954	+	0.984577i	$0.444022\pi$
$942$	0	0
$943$	−5.71327	−0.186050
$944$	0	0
$945$	−0.845563	−0.0275062
$946$	0	0
$947$	12.9257	0.420029	0.210015	−	0.977698i	$-0.432649\pi$
$947$	12.9257	0.420029	0.210015	+	0.977698i	$0.432649\pi$
$948$	0	0
$949$	6.47840	0.210298
$950$	0	0
$951$	24.9983	0.810625
$952$	0	0
$953$	−53.4051	−1.72996	−0.864981	−	0.501805i	$-0.832670\pi$
$953$	−53.4051	−1.72996	−0.864981	+	0.501805i	$0.832670\pi$
$954$	0	0
$955$	1.78465	0.0577500
$956$	0	0
$957$	32.2295	1.04183
$958$	0	0
$959$	8.56605	0.276612
$960$	0	0
$961$	−27.6414	−0.891659
$962$	0	0
$963$	6.39147	0.205962
$964$	0	0
$965$	11.4524	0.368665
$966$	0	0
$967$	28.1852	0.906374	0.453187	−	0.891415i	$-0.350287\pi$
$967$	28.1852	0.906374	0.453187	+	0.891415i	$0.350287\pi$
$968$	0	0
$969$	26.2110	0.842019
$970$	0	0
$971$	−33.1922	−1.06519	−0.532595	−	0.846370i	$-0.678783\pi$
$971$	−33.1922	−1.06519	−0.532595	+	0.846370i	$0.678783\pi$
$972$	0	0
$973$	−9.92120	−0.318059
$974$	0	0
$975$	5.23704	0.167719
$976$	0	0
$977$	11.0750	0.354321	0.177161	−	0.984182i	$-0.443309\pi$
$977$	11.0750	0.354321	0.177161	+	0.984182i	$0.443309\pi$
$978$	0	0
$979$	13.6176	0.435220
$980$	0	0
$981$	6.55883	0.209407
$982$	0	0
$983$	11.8785	0.378864	0.189432	−	0.981894i	$-0.439335\pi$
$983$	11.8785	0.378864	0.189432	+	0.981894i	$0.439335\pi$
$984$	0	0
$985$	−2.23874	−0.0713323
$986$	0	0
$987$	−4.68942	−0.149266
$988$	0	0
$989$	4.78294	0.152089
$990$	0	0
$991$	33.8800	1.07623	0.538117	−	0.842870i	$-0.319136\pi$
$991$	33.8800	1.07623	0.538117	+	0.842870i	$0.319136\pi$
$992$	0	0
$993$	9.59390	0.304453
$994$	0	0
$995$	18.5530	0.588168
$996$	0	0
$997$	−9.33797	−0.295737	−0.147868	−	0.989007i	$-0.547241\pi$
$997$	−9.33797	−0.295737	−0.147868	+	0.989007i	$0.547241\pi$
$998$	0	0
$999$	−9.93738	−0.314405

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.cb.1.3		4
4.3	odd	2		2760.2.a.v.1.2	✓	4
12.11	even	2		8280.2.a.bq.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.v.1.2	✓	4	4.3	odd	2
5520.2.a.cb.1.3		4	1.1	even	1	trivial
8280.2.a.bq.1.2		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +0.845563 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +0.845563 q^{7} +1.00000 q^{9} +4.55883 q^{11} +5.23704 q^{13} -1.00000 q^{15} +4.72619 q^{17} +5.54591 q^{19} +0.845563 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +7.06968 q^{29} +1.83264 q^{31} +4.55883 q^{33} -0.845563 q^{35} -9.93738 q^{37} +5.23704 q^{39} +5.71327 q^{41} -4.78294 q^{43} -1.00000 q^{45} -5.54591 q^{47} -6.28502 q^{49} +4.72619 q^{51} -14.4962 q^{53} -4.55883 q^{55} +5.54591 q^{57} -5.06968 q^{59} -6.22411 q^{61} +0.845563 q^{63} -5.23704 q^{65} +7.85849 q^{67} -1.00000 q^{69} -3.71327 q^{71} +1.23704 q^{73} +1.00000 q^{75} +3.85478 q^{77} +13.7700 q^{79} +1.00000 q^{81} -7.50913 q^{83} -4.72619 q^{85} +7.06968 q^{87} +2.98708 q^{89} +4.42825 q^{91} +1.83264 q^{93} -5.54591 q^{95} -2.33472 q^{97} +4.55883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9} - 2 q^{13} - 4 q^{15} + 2 q^{17} + 6 q^{19} - 4 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} + 6 q^{31} - 4 q^{37} - 2 q^{39} + 8 q^{41} + 20 q^{43} - 4 q^{45} - 6 q^{47} + 10 q^{49} + 2 q^{51} - 4 q^{53} + 6 q^{57} + 4 q^{59} - 4 q^{61} + 2 q^{65} + 26 q^{67} - 4 q^{69} - 18 q^{73} + 4 q^{75} + 6 q^{77} + 18 q^{79} + 4 q^{81} + 26 q^{83} - 2 q^{85} + 4 q^{87} + 14 q^{89} + 38 q^{91} + 6 q^{93} - 6 q^{95} - 12 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9 - 2 * q^13 - 4 * q^15 + 2 * q^17 + 6 * q^19 - 4 * q^23 + 4 * q^25 + 4 * q^27 + 4 * q^29 + 6 * q^31 - 4 * q^37 - 2 * q^39 + 8 * q^41 + 20 * q^43 - 4 * q^45 - 6 * q^47 + 10 * q^49 + 2 * q^51 - 4 * q^53 + 6 * q^57 + 4 * q^59 - 4 * q^61 + 2 * q^65 + 26 * q^67 - 4 * q^69 - 18 * q^73 + 4 * q^75 + 6 * q^77 + 18 * q^79 + 4 * q^81 + 26 * q^83 - 2 * q^85 + 4 * q^87 + 14 * q^89 + 38 * q^91 + 6 * q^93 - 6 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more