LMFDB - Embedded newform 9680.2.a.cp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.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:	$77.2951891566$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2525.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 5 $ x^4 - 2x^3 - 4x^2 + 5*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.777484$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39552 q^{3} +1.00000 q^{5} +4.65351 q^{7} +2.73851 q^{9} +O(q^{10})$	$q-2.39552 q^{3} +1.00000 q^{5} +4.65351 q^{7} +2.73851 q^{9} -6.03548 q^{13} -2.39552 q^{15} -3.32106 q^{17} -2.73851 q^{19} -11.1476 q^{21} +1.97807 q^{23} +1.00000 q^{25} +0.626410 q^{27} +2.29348 q^{29} +6.95265 q^{31} +4.65351 q^{35} -3.83705 q^{37} +14.4581 q^{39} +3.79320 q^{41} -7.98463 q^{43} +2.73851 q^{45} +12.1590 q^{47} +14.6552 q^{49} +7.95566 q^{51} -11.5020 q^{53} +6.56014 q^{57} -7.09855 q^{59} -1.33811 q^{61} +12.7437 q^{63} -6.03548 q^{65} +7.18872 q^{67} -4.73851 q^{69} -6.74368 q^{71} -8.71658 q^{73} -2.39552 q^{75} -5.48569 q^{79} -9.71610 q^{81} +8.21414 q^{83} -3.32106 q^{85} -5.49406 q^{87} -11.4213 q^{89} -28.0862 q^{91} -16.6552 q^{93} -2.73851 q^{95} +6.54798 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 3 * q^7 + 4 * q^9	$ 4 q + 4 q^{5} + 3 q^{7} + 4 q^{9} - 13 q^{13} - 13 q^{17} - 4 q^{19} - 14 q^{21} + 5 q^{23} + 4 q^{25} + 15 q^{27} - 8 q^{29} + 9 q^{31} + 3 q^{35} - 3 q^{37} + 9 q^{39} - 3 q^{41} + 11 q^{43} + 4 q^{45} + 3 q^{47} - q^{49} - 19 q^{51} - 3 q^{53} - 15 q^{57} - 23 q^{59} - 4 q^{61} + 13 q^{63} - 13 q^{65} + q^{67} - 12 q^{69} + 11 q^{71} - 25 q^{73} + 10 q^{79} + 16 q^{81} + 21 q^{83} - 13 q^{85} - 7 q^{87} + 4 q^{89} - 32 q^{91} - 7 q^{93} - 4 q^{95} - 10 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 3 * q^7 + 4 * q^9 - 13 * q^13 - 13 * q^17 - 4 * q^19 - 14 * q^21 + 5 * q^23 + 4 * q^25 + 15 * q^27 - 8 * q^29 + 9 * q^31 + 3 * q^35 - 3 * q^37 + 9 * q^39 - 3 * q^41 + 11 * q^43 + 4 * q^45 + 3 * q^47 - q^49 - 19 * q^51 - 3 * q^53 - 15 * q^57 - 23 * q^59 - 4 * q^61 + 13 * q^63 - 13 * q^65 + q^67 - 12 * q^69 + 11 * q^71 - 25 * q^73 + 10 * q^79 + 16 * q^81 + 21 * q^83 - 13 * q^85 - 7 * q^87 + 4 * q^89 - 32 * q^91 - 7 * q^93 - 4 * q^95 - 10 * 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
$3$	−2.39552	−1.38305	−0.691527	−	0.722351i	$-0.743062\pi$
$3$	−2.39552	−1.38305	−0.691527	+	0.722351i	$0.743062\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.65351	1.75886	0.879432	−	0.476025i	$-0.157923\pi$
$7$	4.65351	1.75886	0.879432	+	0.476025i	$0.157923\pi$
$8$	0	0
$9$	2.73851	0.912836
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.03548	−1.67394	−0.836971	−	0.547248i	$-0.815675\pi$
$13$	−6.03548	−1.67394	−0.836971	+	0.547248i	$0.815675\pi$
$14$	0	0
$15$	−2.39552	−0.618520
$16$	0	0
$17$	−3.32106	−0.805476	−0.402738	−	0.915315i	$-0.631941\pi$
$17$	−3.32106	−0.805476	−0.402738	+	0.915315i	$0.631941\pi$
$18$	0	0
$19$	−2.73851	−0.628257	−0.314128	−	0.949381i	$-0.601712\pi$
$19$	−2.73851	−0.628257	−0.314128	+	0.949381i	$0.601712\pi$
$20$	0	0
$21$	−11.1476	−2.43260
$22$	0	0
$23$	1.97807	0.412457	0.206228	−	0.978504i	$-0.433881\pi$
$23$	1.97807	0.412457	0.206228	+	0.978504i	$0.433881\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.626410	0.120553
$28$	0	0
$29$	2.29348	0.425888	0.212944	−	0.977064i	$-0.431695\pi$
$29$	2.29348	0.425888	0.212944	+	0.977064i	$0.431695\pi$
$30$	0	0
$31$	6.95265	1.24873	0.624366	−	0.781132i	$-0.285357\pi$
$31$	6.95265	1.24873	0.624366	+	0.781132i	$0.285357\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.65351	0.786587
$36$	0	0
$37$	−3.83705	−0.630808	−0.315404	−	0.948958i	$-0.602140\pi$
$37$	−3.83705	−0.630808	−0.315404	+	0.948958i	$0.602140\pi$
$38$	0	0
$39$	14.4581	2.31515
$40$	0	0
$41$	3.79320	0.592398	0.296199	−	0.955126i	$-0.404281\pi$
$41$	3.79320	0.592398	0.296199	+	0.955126i	$0.404281\pi$
$42$	0	0
$43$	−7.98463	−1.21764	−0.608822	−	0.793307i	$-0.708358\pi$
$43$	−7.98463	−1.21764	−0.608822	+	0.793307i	$0.708358\pi$
$44$	0	0
$45$	2.73851	0.408233
$46$	0	0
$47$	12.1590	1.77357	0.886784	−	0.462184i	$-0.152934\pi$
$47$	12.1590	1.77357	0.886784	+	0.462184i	$0.152934\pi$
$48$	0	0
$49$	14.6552	2.09360
$50$	0	0
$51$	7.95566	1.11402
$52$	0	0
$53$	−11.5020	−1.57992	−0.789958	−	0.613161i	$-0.789898\pi$
$53$	−11.5020	−1.57992	−0.789958	+	0.613161i	$0.789898\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.56014	0.868912
$58$	0	0
$59$	−7.09855	−0.924152	−0.462076	−	0.886840i	$-0.652895\pi$
$59$	−7.09855	−0.924152	−0.462076	+	0.886840i	$0.652895\pi$
$60$	0	0
$61$	−1.33811	−0.171327	−0.0856637	−	0.996324i	$-0.527301\pi$
$61$	−1.33811	−0.171327	−0.0856637	+	0.996324i	$0.527301\pi$
$62$	0	0
$63$	12.7437	1.60555
$64$	0	0
$65$	−6.03548	−0.748609
$66$	0	0
$67$	7.18872	0.878241	0.439121	−	0.898428i	$-0.355290\pi$
$67$	7.18872	0.878241	0.439121	+	0.898428i	$0.355290\pi$
$68$	0	0
$69$	−4.73851	−0.570449
$70$	0	0
$71$	−6.74368	−0.800328	−0.400164	−	0.916444i	$-0.631047\pi$
$71$	−6.74368	−0.800328	−0.400164	+	0.916444i	$0.631047\pi$
$72$	0	0
$73$	−8.71658	−1.02020	−0.510099	−	0.860116i	$-0.670391\pi$
$73$	−8.71658	−1.02020	−0.510099	+	0.860116i	$0.670391\pi$
$74$	0	0
$75$	−2.39552	−0.276611
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.48569	−0.617188	−0.308594	−	0.951194i	$-0.599858\pi$
$79$	−5.48569	−0.617188	−0.308594	+	0.951194i	$0.599858\pi$
$80$	0	0
$81$	−9.71610	−1.07957
$82$	0	0
$83$	8.21414	0.901619	0.450809	−	0.892620i	$-0.351135\pi$
$83$	8.21414	0.901619	0.450809	+	0.892620i	$0.351135\pi$
$84$	0	0
$85$	−3.32106	−0.360220
$86$	0	0
$87$	−5.49406	−0.589025
$88$	0	0
$89$	−11.4213	−1.21065	−0.605327	−	0.795977i	$-0.706958\pi$
$89$	−11.4213	−1.21065	−0.605327	+	0.795977i	$0.706958\pi$
$90$	0	0
$91$	−28.0862	−2.94423
$92$	0	0
$93$	−16.6552	−1.72706
$94$	0	0
$95$	−2.73851	−0.280965
$96$	0	0
$97$	6.54798	0.664846	0.332423	−	0.943130i	$-0.392134\pi$
$97$	6.54798	0.664846	0.332423	+	0.943130i	$0.392134\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.272884	−0.0271529	−0.0135765	−	0.999908i	$-0.504322\pi$
$101$	−0.272884	−0.0271529	−0.0135765	+	0.999908i	$0.504322\pi$
$102$	0	0
$103$	−8.19221	−0.807203	−0.403601	−	0.914935i	$-0.632242\pi$
$103$	−8.19221	−0.807203	−0.403601	+	0.914935i	$0.632242\pi$
$104$	0	0
$105$	−11.1476	−1.08789
$106$	0	0
$107$	−6.58773	−0.636860	−0.318430	−	0.947946i	$-0.603156\pi$
$107$	−6.58773	−0.636860	−0.318430	+	0.947946i	$0.603156\pi$
$108$	0	0
$109$	3.06258	0.293342	0.146671	−	0.989185i	$-0.453144\pi$
$109$	3.06258	0.293342	0.146671	+	0.989185i	$0.453144\pi$
$110$	0	0
$111$	9.19173	0.872441
$112$	0	0
$113$	−5.78586	−0.544288	−0.272144	−	0.962257i	$-0.587733\pi$
$113$	−5.78586	−0.544288	−0.272144	+	0.962257i	$0.587733\pi$
$114$	0	0
$115$	1.97807	0.184456
$116$	0	0
$117$	−16.5282	−1.52803
$118$	0	0
$119$	−15.4546	−1.41672
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−9.08667	−0.819318
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.75340	−0.155589	−0.0777943	−	0.996969i	$-0.524788\pi$
$127$	−1.75340	−0.155589	−0.0777943	+	0.996969i	$0.524788\pi$
$128$	0	0
$129$	19.1273	1.68407
$130$	0	0
$131$	12.7047	1.11002	0.555008	−	0.831845i	$-0.312715\pi$
$131$	12.7047	1.11002	0.555008	+	0.831845i	$0.312715\pi$
$132$	0	0
$133$	−12.7437	−1.10502
$134$	0	0
$135$	0.626410	0.0539128
$136$	0	0
$137$	−1.73635	−0.148346	−0.0741731	−	0.997245i	$-0.523632\pi$
$137$	−1.73635	−0.148346	−0.0741731	+	0.997245i	$0.523632\pi$
$138$	0	0
$139$	−6.46998	−0.548776	−0.274388	−	0.961619i	$-0.588475\pi$
$139$	−6.46998	−0.548776	−0.274388	+	0.961619i	$0.588475\pi$
$140$	0	0
$141$	−29.1270	−2.45294
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.29348	0.190463
$146$	0	0
$147$	−35.1068	−2.89556
$148$	0	0
$149$	1.30836	0.107185	0.0535927	−	0.998563i	$-0.482933\pi$
$149$	1.30836	0.107185	0.0535927	+	0.998563i	$0.482933\pi$
$150$	0	0
$151$	20.0578	1.63228	0.816138	−	0.577857i	$-0.196111\pi$
$151$	20.0578	1.63228	0.816138	+	0.577857i	$0.196111\pi$
$152$	0	0
$153$	−9.09475	−0.735267
$154$	0	0
$155$	6.95265	0.558450
$156$	0	0
$157$	17.6614	1.40953	0.704767	−	0.709439i	$-0.251052\pi$
$157$	17.6614	1.40953	0.704767	+	0.709439i	$0.251052\pi$
$158$	0	0
$159$	27.5532	2.18511
$160$	0	0
$161$	9.20499	0.725455
$162$	0	0
$163$	−4.62425	−0.362199	−0.181100	−	0.983465i	$-0.557966\pi$
$163$	−4.62425	−0.362199	−0.181100	+	0.983465i	$0.557966\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.4278	−1.73552	−0.867758	−	0.496988i	$-0.834439\pi$
$167$	−22.4278	−1.73552	−0.867758	+	0.496988i	$0.834439\pi$
$168$	0	0
$169$	23.4270	1.80208
$170$	0	0
$171$	−7.49942	−0.573495
$172$	0	0
$173$	−7.11042	−0.540595	−0.270298	−	0.962777i	$-0.587122\pi$
$173$	−7.11042	−0.540595	−0.270298	+	0.962777i	$0.587122\pi$
$174$	0	0
$175$	4.65351	0.351773
$176$	0	0
$177$	17.0047	1.27815
$178$	0	0
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	0	0
$181$	16.9841	1.26242	0.631208	−	0.775613i	$-0.282559\pi$
$181$	16.9841	1.26242	0.631208	+	0.775613i	$0.282559\pi$
$182$	0	0
$183$	3.20547	0.236955
$184$	0	0
$185$	−3.83705	−0.282106
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.91501	0.212036
$190$	0	0
$191$	−4.59045	−0.332153	−0.166077	−	0.986113i	$-0.553110\pi$
$191$	−4.59045	−0.332153	−0.166077	+	0.986113i	$0.553110\pi$
$192$	0	0
$193$	−8.75388	−0.630118	−0.315059	−	0.949072i	$-0.602024\pi$
$193$	−8.75388	−0.630118	−0.315059	+	0.949072i	$0.602024\pi$
$194$	0	0
$195$	14.4581	1.03537
$196$	0	0
$197$	13.4511	0.958352	0.479176	−	0.877719i	$-0.340936\pi$
$197$	13.4511	0.958352	0.479176	+	0.877719i	$0.340936\pi$
$198$	0	0
$199$	−4.55665	−0.323012	−0.161506	−	0.986872i	$-0.551635\pi$
$199$	−4.55665	−0.323012	−0.161506	+	0.986872i	$0.551635\pi$
$200$	0	0
$201$	−17.2207	−1.21465
$202$	0	0
$203$	10.6727	0.749078
$204$	0	0
$205$	3.79320	0.264928
$206$	0	0
$207$	5.41697	0.376505
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0.206025	0.0141833	0.00709166	−	0.999975i	$-0.497743\pi$
$211$	0.206025	0.0141833	0.00709166	+	0.999975i	$0.497743\pi$
$212$	0	0
$213$	16.1546	1.10690
$214$	0	0
$215$	−7.98463	−0.544547
$216$	0	0
$217$	32.3542	2.19635
$218$	0	0
$219$	20.8807	1.41099
$220$	0	0
$221$	20.0442	1.34832
$222$	0	0
$223$	−23.9806	−1.60586	−0.802929	−	0.596075i	$-0.796726\pi$
$223$	−23.9806	−1.60586	−0.802929	+	0.596075i	$0.796726\pi$
$224$	0	0
$225$	2.73851	0.182567
$226$	0	0
$227$	−23.4113	−1.55386	−0.776932	−	0.629585i	$-0.783225\pi$
$227$	−23.4113	−1.55386	−0.776932	+	0.629585i	$0.783225\pi$
$228$	0	0
$229$	−19.9592	−1.31894	−0.659471	−	0.751730i	$-0.729220\pi$
$229$	−19.9592	−1.31894	−0.659471	+	0.751730i	$0.729220\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.30695	−0.0856213	−0.0428106	−	0.999083i	$-0.513631\pi$
$233$	−1.30695	−0.0856213	−0.0428106	+	0.999083i	$0.513631\pi$
$234$	0	0
$235$	12.1590	0.793164
$236$	0	0
$237$	13.1411	0.853604
$238$	0	0
$239$	12.1243	0.784257	0.392128	−	0.919911i	$-0.371739\pi$
$239$	12.1243	0.784257	0.392128	+	0.919911i	$0.371739\pi$
$240$	0	0
$241$	−18.3095	−1.17942	−0.589709	−	0.807616i	$-0.700758\pi$
$241$	−18.3095	−1.17942	−0.589709	+	0.807616i	$0.700758\pi$
$242$	0	0
$243$	21.3959	1.37255
$244$	0	0
$245$	14.6552	0.936286
$246$	0	0
$247$	16.5282	1.05166
$248$	0	0
$249$	−19.6771	−1.24699
$250$	0	0
$251$	26.7897	1.69095	0.845475	−	0.534014i	$-0.179317\pi$
$251$	26.7897	1.69095	0.845475	+	0.534014i	$0.179317\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	7.95566	0.498203
$256$	0	0
$257$	−7.20227	−0.449265	−0.224633	−	0.974444i	$-0.572118\pi$
$257$	−7.20227	−0.449265	−0.224633	+	0.974444i	$0.572118\pi$
$258$	0	0
$259$	−17.8558	−1.10950
$260$	0	0
$261$	6.28070	0.388766
$262$	0	0
$263$	−11.7870	−0.726816	−0.363408	−	0.931630i	$-0.618387\pi$
$263$	−11.7870	−0.726816	−0.363408	+	0.931630i	$0.618387\pi$
$264$	0	0
$265$	−11.5020	−0.706560
$266$	0	0
$267$	27.3599	1.67440
$268$	0	0
$269$	14.7994	0.902336	0.451168	−	0.892439i	$-0.351008\pi$
$269$	14.7994	0.902336	0.451168	+	0.892439i	$0.351008\pi$
$270$	0	0
$271$	−23.2838	−1.41439	−0.707194	−	0.707020i	$-0.750039\pi$
$271$	−23.2838	−1.41439	−0.707194	+	0.707020i	$0.750039\pi$
$272$	0	0
$273$	67.2810	4.07203
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−15.4870	−0.930525	−0.465263	−	0.885173i	$-0.654040\pi$
$277$	−15.4870	−0.930525	−0.465263	+	0.885173i	$0.654040\pi$
$278$	0	0
$279$	19.0399	1.13989
$280$	0	0
$281$	−15.0218	−0.896125	−0.448062	−	0.894002i	$-0.647886\pi$
$281$	−15.0218	−0.896125	−0.448062	+	0.894002i	$0.647886\pi$
$282$	0	0
$283$	14.4830	0.860923	0.430462	−	0.902609i	$-0.358351\pi$
$283$	14.4830	0.860923	0.430462	+	0.902609i	$0.358351\pi$
$284$	0	0
$285$	6.56014	0.388589
$286$	0	0
$287$	17.6517	1.04195
$288$	0	0
$289$	−5.97055	−0.351209
$290$	0	0
$291$	−15.6858	−0.919518
$292$	0	0
$293$	−16.6763	−0.974240	−0.487120	−	0.873335i	$-0.661953\pi$
$293$	−16.6763	−0.974240	−0.487120	+	0.873335i	$0.661953\pi$
$294$	0	0
$295$	−7.09855	−0.413293
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−11.9386	−0.690428
$300$	0	0
$301$	−37.1566	−2.14167
$302$	0	0
$303$	0.653698	0.0375540
$304$	0	0
$305$	−1.33811	−0.0766200
$306$	0	0
$307$	−0.537575	−0.0306810	−0.0153405	−	0.999882i	$-0.504883\pi$
$307$	−0.537575	−0.0306810	−0.0153405	+	0.999882i	$0.504883\pi$
$308$	0	0
$309$	19.6246	1.11640
$310$	0	0
$311$	−12.4946	−0.708505	−0.354252	−	0.935150i	$-0.615265\pi$
$311$	−12.4946	−0.708505	−0.354252	+	0.935150i	$0.615265\pi$
$312$	0	0
$313$	18.8371	1.06474	0.532369	−	0.846513i	$-0.321302\pi$
$313$	18.8371	1.06474	0.532369	+	0.846513i	$0.321302\pi$
$314$	0	0
$315$	12.7437	0.718025
$316$	0	0
$317$	−7.06090	−0.396580	−0.198290	−	0.980143i	$-0.563539\pi$
$317$	−7.06090	−0.396580	−0.198290	+	0.980143i	$0.563539\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	15.7810	0.880811
$322$	0	0
$323$	9.09475	0.506046
$324$	0	0
$325$	−6.03548	−0.334788
$326$	0	0
$327$	−7.33648	−0.405708
$328$	0	0
$329$	56.5819	3.11946
$330$	0	0
$331$	−7.83433	−0.430614	−0.215307	−	0.976546i	$-0.569075\pi$
$331$	−7.83433	−0.430614	−0.215307	+	0.976546i	$0.569075\pi$
$332$	0	0
$333$	−10.5078	−0.575824
$334$	0	0
$335$	7.18872	0.392761
$336$	0	0
$337$	7.30949	0.398173	0.199087	−	0.979982i	$-0.436203\pi$
$337$	7.30949	0.398173	0.199087	+	0.979982i	$0.436203\pi$
$338$	0	0
$339$	13.8601	0.752779
$340$	0	0
$341$	0	0
$342$	0	0
$343$	35.6236	1.92349
$344$	0	0
$345$	−4.73851	−0.255113
$346$	0	0
$347$	17.5746	0.943454	0.471727	−	0.881745i	$-0.343631\pi$
$347$	17.5746	0.943454	0.471727	+	0.881745i	$0.343631\pi$
$348$	0	0
$349$	−21.3794	−1.14441	−0.572206	−	0.820110i	$-0.693912\pi$
$349$	−21.3794	−1.14441	−0.572206	+	0.820110i	$0.693912\pi$
$350$	0	0
$351$	−3.78068	−0.201798
$352$	0	0
$353$	7.59779	0.404389	0.202195	−	0.979345i	$-0.435193\pi$
$353$	7.59779	0.404389	0.202195	+	0.979345i	$0.435193\pi$
$354$	0	0
$355$	−6.74368	−0.357918
$356$	0	0
$357$	37.0218	1.95940
$358$	0	0
$359$	34.6662	1.82961	0.914805	−	0.403896i	$-0.132344\pi$
$359$	34.6662	1.82961	0.914805	+	0.403896i	$0.132344\pi$
$360$	0	0
$361$	−11.5006	−0.605293
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.71658	−0.456247
$366$	0	0
$367$	15.0303	0.784575	0.392288	−	0.919843i	$-0.371684\pi$
$367$	15.0303	0.784575	0.392288	+	0.919843i	$0.371684\pi$
$368$	0	0
$369$	10.3877	0.540762
$370$	0	0
$371$	−53.5245	−2.77886
$372$	0	0
$373$	−13.5055	−0.699286	−0.349643	−	0.936883i	$-0.613697\pi$
$373$	−13.5055	−0.699286	−0.349643	+	0.936883i	$0.613697\pi$
$374$	0	0
$375$	−2.39552	−0.123704
$376$	0	0
$377$	−13.8422	−0.712911
$378$	0	0
$379$	23.1937	1.19138	0.595689	−	0.803215i	$-0.296879\pi$
$379$	23.1937	1.19138	0.595689	+	0.803215i	$0.296879\pi$
$380$	0	0
$381$	4.20029	0.215187
$382$	0	0
$383$	−35.1392	−1.79553	−0.897765	−	0.440475i	$-0.854810\pi$
$383$	−35.1392	−1.79553	−0.897765	+	0.440475i	$0.854810\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−21.8660	−1.11151
$388$	0	0
$389$	−35.7603	−1.81312	−0.906560	−	0.422078i	$-0.861301\pi$
$389$	−35.7603	−1.81312	−0.906560	+	0.422078i	$0.861301\pi$
$390$	0	0
$391$	−6.56930	−0.332224
$392$	0	0
$393$	−30.4344	−1.53521
$394$	0	0
$395$	−5.48569	−0.276015
$396$	0	0
$397$	−3.62347	−0.181857	−0.0909284	−	0.995857i	$-0.528983\pi$
$397$	−3.62347	−0.181857	−0.0909284	+	0.995857i	$0.528983\pi$
$398$	0	0
$399$	30.5277	1.52830
$400$	0	0
$401$	1.05301	0.0525848	0.0262924	−	0.999654i	$-0.491630\pi$
$401$	1.05301	0.0525848	0.0262924	+	0.999654i	$0.491630\pi$
$402$	0	0
$403$	−41.9626	−2.09030
$404$	0	0
$405$	−9.71610	−0.482797
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−11.4892	−0.568106	−0.284053	−	0.958809i	$-0.591679\pi$
$409$	−11.4892	−0.568106	−0.284053	+	0.958809i	$0.591679\pi$
$410$	0	0
$411$	4.15945	0.205171
$412$	0	0
$413$	−33.0332	−1.62546
$414$	0	0
$415$	8.21414	0.403216
$416$	0	0
$417$	15.4989	0.758987
$418$	0	0
$419$	−9.12829	−0.445946	−0.222973	−	0.974825i	$-0.571576\pi$
$419$	−9.12829	−0.445946	−0.222973	+	0.974825i	$0.571576\pi$
$420$	0	0
$421$	−25.1610	−1.22627	−0.613136	−	0.789977i	$-0.710092\pi$
$421$	−25.1610	−1.22627	−0.613136	+	0.789977i	$0.710092\pi$
$422$	0	0
$423$	33.2974	1.61898
$424$	0	0
$425$	−3.32106	−0.161095
$426$	0	0
$427$	−6.22691	−0.301342
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.8014	0.761128	0.380564	−	0.924754i	$-0.375730\pi$
$431$	15.8014	0.761128	0.380564	+	0.924754i	$0.375730\pi$
$432$	0	0
$433$	−2.68676	−0.129117	−0.0645586	−	0.997914i	$-0.520564\pi$
$433$	−2.68676	−0.129117	−0.0645586	+	0.997914i	$0.520564\pi$
$434$	0	0
$435$	−5.49406	−0.263420
$436$	0	0
$437$	−5.41697	−0.259129
$438$	0	0
$439$	18.0888	0.863333	0.431666	−	0.902033i	$-0.357926\pi$
$439$	18.0888	0.863333	0.431666	+	0.902033i	$0.357926\pi$
$440$	0	0
$441$	40.1334	1.91111
$442$	0	0
$443$	32.6959	1.55343	0.776713	−	0.629854i	$-0.216885\pi$
$443$	32.6959	1.55343	0.776713	+	0.629854i	$0.216885\pi$
$444$	0	0
$445$	−11.4213	−0.541421
$446$	0	0
$447$	−3.13421	−0.148243
$448$	0	0
$449$	9.06419	0.427765	0.213883	−	0.976859i	$-0.431389\pi$
$449$	9.06419	0.427765	0.213883	+	0.976859i	$0.431389\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−48.0487	−2.25753
$454$	0	0
$455$	−28.0862	−1.31670
$456$	0	0
$457$	15.7502	0.736763	0.368382	−	0.929675i	$-0.379912\pi$
$457$	15.7502	0.736763	0.368382	+	0.929675i	$0.379912\pi$
$458$	0	0
$459$	−2.08035	−0.0971022
$460$	0	0
$461$	−33.2263	−1.54750	−0.773751	−	0.633489i	$-0.781622\pi$
$461$	−33.2263	−1.54750	−0.773751	+	0.633489i	$0.781622\pi$
$462$	0	0
$463$	−6.68368	−0.310617	−0.155309	−	0.987866i	$-0.549637\pi$
$463$	−6.68368	−0.310617	−0.155309	+	0.987866i	$0.549637\pi$
$464$	0	0
$465$	−16.6552	−0.772366
$466$	0	0
$467$	−21.3345	−0.987242	−0.493621	−	0.869677i	$-0.664327\pi$
$467$	−21.3345	−0.987242	−0.493621	+	0.869677i	$0.664327\pi$
$468$	0	0
$469$	33.4528	1.54471
$470$	0	0
$471$	−42.3082	−1.94946
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.73851	−0.125651
$476$	0	0
$477$	−31.4982	−1.44220
$478$	0	0
$479$	−39.2657	−1.79409	−0.897047	−	0.441935i	$-0.854292\pi$
$479$	−39.2657	−1.79409	−0.897047	+	0.441935i	$0.854292\pi$
$480$	0	0
$481$	23.1585	1.05593
$482$	0	0
$483$	−22.0507	−1.00334
$484$	0	0
$485$	6.54798	0.297328
$486$	0	0
$487$	−12.7297	−0.576836	−0.288418	−	0.957505i	$-0.593129\pi$
$487$	−12.7297	−0.576836	−0.288418	+	0.957505i	$0.593129\pi$
$488$	0	0
$489$	11.0775	0.500941
$490$	0	0
$491$	5.40592	0.243966	0.121983	−	0.992532i	$-0.461075\pi$
$491$	5.40592	0.243966	0.121983	+	0.992532i	$0.461075\pi$
$492$	0	0
$493$	−7.61678	−0.343042
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−31.3818	−1.40767
$498$	0	0
$499$	4.76471	0.213298	0.106649	−	0.994297i	$-0.465988\pi$
$499$	4.76471	0.213298	0.106649	+	0.994297i	$0.465988\pi$
$500$	0	0
$501$	53.7262	2.40031
$502$	0	0
$503$	−16.4005	−0.731261	−0.365631	−	0.930760i	$-0.619147\pi$
$503$	−16.4005	−0.731261	−0.365631	+	0.930760i	$0.619147\pi$
$504$	0	0
$505$	−0.272884	−0.0121432
$506$	0	0
$507$	−56.1199	−2.49237
$508$	0	0
$509$	41.0717	1.82047	0.910236	−	0.414090i	$-0.135900\pi$
$509$	41.0717	1.82047	0.910236	+	0.414090i	$0.135900\pi$
$510$	0	0
$511$	−40.5627	−1.79439
$512$	0	0
$513$	−1.71543	−0.0757380
$514$	0	0
$515$	−8.19221	−0.360992
$516$	0	0
$517$	0	0
$518$	0	0
$519$	17.0331	0.747672
$520$	0	0
$521$	39.3976	1.72604	0.863021	−	0.505169i	$-0.168570\pi$
$521$	39.3976	1.72604	0.863021	+	0.505169i	$0.168570\pi$
$522$	0	0
$523$	16.7391	0.731948	0.365974	−	0.930625i	$-0.380736\pi$
$523$	16.7391	0.731948	0.365974	+	0.930625i	$0.380736\pi$
$524$	0	0
$525$	−11.1476	−0.486520
$526$	0	0
$527$	−23.0902	−1.00582
$528$	0	0
$529$	−19.0872	−0.829880
$530$	0	0
$531$	−19.4394	−0.843599
$532$	0	0
$533$	−22.8938	−0.991639
$534$	0	0
$535$	−6.58773	−0.284812
$536$	0	0
$537$	11.9776	0.516871
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.98943	−0.386486	−0.193243	−	0.981151i	$-0.561901\pi$
$541$	−8.98943	−0.386486	−0.193243	+	0.981151i	$0.561901\pi$
$542$	0	0
$543$	−40.6857	−1.74599
$544$	0	0
$545$	3.06258	0.131187
$546$	0	0
$547$	−20.9605	−0.896206	−0.448103	−	0.893982i	$-0.647900\pi$
$547$	−20.9605	−0.896206	−0.448103	+	0.893982i	$0.647900\pi$
$548$	0	0
$549$	−3.66442	−0.156394
$550$	0	0
$551$	−6.28070	−0.267567
$552$	0	0
$553$	−25.5277	−1.08555
$554$	0	0
$555$	9.19173	0.390167
$556$	0	0
$557$	−15.2334	−0.645461	−0.322730	−	0.946491i	$-0.604601\pi$
$557$	−15.2334	−0.645461	−0.322730	+	0.946491i	$0.604601\pi$
$558$	0	0
$559$	48.1911	2.03827
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15.2106	−0.641052	−0.320526	−	0.947240i	$-0.603860\pi$
$563$	−15.2106	−0.641052	−0.320526	+	0.947240i	$0.603860\pi$
$564$	0	0
$565$	−5.78586	−0.243413
$566$	0	0
$567$	−45.2140	−1.89881
$568$	0	0
$569$	38.4882	1.61351	0.806755	−	0.590886i	$-0.201222\pi$
$569$	38.4882	1.61351	0.806755	+	0.590886i	$0.201222\pi$
$570$	0	0
$571$	−17.3549	−0.726282	−0.363141	−	0.931734i	$-0.618296\pi$
$571$	−17.3549	−0.726282	−0.363141	+	0.931734i	$0.618296\pi$
$572$	0	0
$573$	10.9965	0.459386
$574$	0	0
$575$	1.97807	0.0824913
$576$	0	0
$577$	4.98019	0.207328	0.103664	−	0.994612i	$-0.466943\pi$
$577$	4.98019	0.207328	0.103664	+	0.994612i	$0.466943\pi$
$578$	0	0
$579$	20.9701	0.871486
$580$	0	0
$581$	38.2246	1.58582
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−16.5282	−0.683357
$586$	0	0
$587$	39.1518	1.61597	0.807985	−	0.589203i	$-0.200558\pi$
$587$	39.1518	1.61597	0.807985	+	0.589203i	$0.200558\pi$
$588$	0	0
$589$	−19.0399	−0.784525
$590$	0	0
$591$	−32.2224	−1.32545
$592$	0	0
$593$	36.2454	1.48842	0.744211	−	0.667945i	$-0.232826\pi$
$593$	36.2454	1.48842	0.744211	+	0.667945i	$0.232826\pi$
$594$	0	0
$595$	−15.4546	−0.633577
$596$	0	0
$597$	10.9155	0.446743
$598$	0	0
$599$	−12.0593	−0.492732	−0.246366	−	0.969177i	$-0.579236\pi$
$599$	−12.0593	−0.492732	−0.246366	+	0.969177i	$0.579236\pi$
$600$	0	0
$601$	−27.8585	−1.13637	−0.568186	−	0.822900i	$-0.692355\pi$
$601$	−27.8585	−1.13637	−0.568186	+	0.822900i	$0.692355\pi$
$602$	0	0
$603$	19.6864	0.801690
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−41.5181	−1.68517	−0.842584	−	0.538565i	$-0.818966\pi$
$607$	−41.5181	−1.68517	−0.842584	+	0.538565i	$0.818966\pi$
$608$	0	0
$609$	−25.5667	−1.03602
$610$	0	0
$611$	−73.3852	−2.96885
$612$	0	0
$613$	−43.4438	−1.75468	−0.877339	−	0.479870i	$-0.840684\pi$
$613$	−43.4438	−1.75468	−0.877339	+	0.479870i	$0.840684\pi$
$614$	0	0
$615$	−9.08667	−0.366410
$616$	0	0
$617$	−18.9692	−0.763672	−0.381836	−	0.924230i	$-0.624708\pi$
$617$	−18.9692	−0.763672	−0.381836	+	0.924230i	$0.624708\pi$
$618$	0	0
$619$	−2.80636	−0.112797	−0.0563985	−	0.998408i	$-0.517962\pi$
$619$	−2.80636	−0.112797	−0.0563985	+	0.998408i	$0.517962\pi$
$620$	0	0
$621$	1.23908	0.0497227
$622$	0	0
$623$	−53.1491	−2.12937
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.7431	0.508100
$630$	0	0
$631$	23.6412	0.941140	0.470570	−	0.882363i	$-0.344048\pi$
$631$	23.6412	0.941140	0.470570	+	0.882363i	$0.344048\pi$
$632$	0	0
$633$	−0.493536	−0.0196163
$634$	0	0
$635$	−1.75340	−0.0695814
$636$	0	0
$637$	−88.4511	−3.50456
$638$	0	0
$639$	−18.4676	−0.730568
$640$	0	0
$641$	−5.43527	−0.214680	−0.107340	−	0.994222i	$-0.534233\pi$
$641$	−5.43527	−0.214680	−0.107340	+	0.994222i	$0.534233\pi$
$642$	0	0
$643$	−14.5956	−0.575595	−0.287798	−	0.957691i	$-0.592923\pi$
$643$	−14.5956	−0.575595	−0.287798	+	0.957691i	$0.592923\pi$
$644$	0	0
$645$	19.1273	0.753138
$646$	0	0
$647$	11.4578	0.450453	0.225226	−	0.974306i	$-0.427688\pi$
$647$	11.4578	0.450453	0.225226	+	0.974306i	$0.427688\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−77.5052	−3.03767
$652$	0	0
$653$	−7.08857	−0.277397	−0.138699	−	0.990335i	$-0.544292\pi$
$653$	−7.08857	−0.277397	−0.138699	+	0.990335i	$0.544292\pi$
$654$	0	0
$655$	12.7047	0.496414
$656$	0	0
$657$	−23.8704	−0.931274
$658$	0	0
$659$	−10.2209	−0.398150	−0.199075	−	0.979984i	$-0.563794\pi$
$659$	−10.2209	−0.398150	−0.199075	+	0.979984i	$0.563794\pi$
$660$	0	0
$661$	−15.7641	−0.613151	−0.306575	−	0.951846i	$-0.599183\pi$
$661$	−15.7641	−0.613151	−0.306575	+	0.951846i	$0.599183\pi$
$662$	0	0
$663$	−48.0162	−1.86480
$664$	0	0
$665$	−12.7437	−0.494179
$666$	0	0
$667$	4.53666	0.175660
$668$	0	0
$669$	57.4459	2.22099
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−11.8914	−0.458380	−0.229190	−	0.973382i	$-0.573608\pi$
$673$	−11.8914	−0.458380	−0.229190	+	0.973382i	$0.573608\pi$
$674$	0	0
$675$	0.626410	0.0241105
$676$	0	0
$677$	−20.6643	−0.794195	−0.397098	−	0.917776i	$-0.629983\pi$
$677$	−20.6643	−0.794195	−0.397098	+	0.917776i	$0.629983\pi$
$678$	0	0
$679$	30.4711	1.16937
$680$	0	0
$681$	56.0822	2.14908
$682$	0	0
$683$	22.8239	0.873331	0.436666	−	0.899624i	$-0.356159\pi$
$683$	22.8239	0.873331	0.436666	+	0.899624i	$0.356159\pi$
$684$	0	0
$685$	−1.73635	−0.0663424
$686$	0	0
$687$	47.8126	1.82417
$688$	0	0
$689$	69.4198	2.64469
$690$	0	0
$691$	40.9712	1.55862	0.779309	−	0.626640i	$-0.215570\pi$
$691$	40.9712	1.55862	0.779309	+	0.626640i	$0.215570\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.46998	−0.245420
$696$	0	0
$697$	−12.5974	−0.477162
$698$	0	0
$699$	3.13083	0.118419
$700$	0	0
$701$	33.3585	1.25993	0.629967	−	0.776622i	$-0.283068\pi$
$701$	33.3585	1.25993	0.629967	+	0.776622i	$0.283068\pi$
$702$	0	0
$703$	10.5078	0.396309
$704$	0	0
$705$	−29.1270	−1.09699
$706$	0	0
$707$	−1.26987	−0.0477583
$708$	0	0
$709$	−16.0315	−0.602074	−0.301037	−	0.953612i	$-0.597333\pi$
$709$	−16.0315	−0.602074	−0.301037	+	0.953612i	$0.597333\pi$
$710$	0	0
$711$	−15.0226	−0.563391
$712$	0	0
$713$	13.7528	0.515048
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−29.0440	−1.08467
$718$	0	0
$719$	−13.2094	−0.492626	−0.246313	−	0.969190i	$-0.579219\pi$
$719$	−13.2094	−0.492626	−0.246313	+	0.969190i	$0.579219\pi$
$720$	0	0
$721$	−38.1226	−1.41976
$722$	0	0
$723$	43.8607	1.63120
$724$	0	0
$725$	2.29348	0.0851776
$726$	0	0
$727$	−21.9855	−0.815395	−0.407698	−	0.913117i	$-0.633668\pi$
$727$	−21.9855	−0.815395	−0.407698	+	0.913117i	$0.633668\pi$
$728$	0	0
$729$	−22.1059	−0.818736
$730$	0	0
$731$	26.5175	0.980783
$732$	0	0
$733$	−33.7989	−1.24839	−0.624196	−	0.781268i	$-0.714573\pi$
$733$	−33.7989	−1.24839	−0.624196	+	0.781268i	$0.714573\pi$
$734$	0	0
$735$	−35.1068	−1.29493
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−14.7537	−0.542725	−0.271363	−	0.962477i	$-0.587474\pi$
$739$	−14.7537	−0.542725	−0.271363	+	0.962477i	$0.587474\pi$
$740$	0	0
$741$	−39.5936	−1.45451
$742$	0	0
$743$	−24.3494	−0.893292	−0.446646	−	0.894711i	$-0.647382\pi$
$743$	−24.3494	−0.893292	−0.446646	+	0.894711i	$0.647382\pi$
$744$	0	0
$745$	1.30836	0.0479347
$746$	0	0
$747$	22.4945	0.823030
$748$	0	0
$749$	−30.6561	−1.12015
$750$	0	0
$751$	37.6302	1.37315	0.686573	−	0.727061i	$-0.259114\pi$
$751$	37.6302	1.37315	0.686573	+	0.727061i	$0.259114\pi$
$752$	0	0
$753$	−64.1752	−2.33867
$754$	0	0
$755$	20.0578	0.729976
$756$	0	0
$757$	−43.9401	−1.59703	−0.798515	−	0.601975i	$-0.794381\pi$
$757$	−43.9401	−1.59703	−0.798515	+	0.601975i	$0.794381\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41.2126	1.49396	0.746978	−	0.664849i	$-0.231504\pi$
$761$	41.2126	1.49396	0.746978	+	0.664849i	$0.231504\pi$
$762$	0	0
$763$	14.2518	0.515949
$764$	0	0
$765$	−9.09475	−0.328821
$766$	0	0
$767$	42.8431	1.54698
$768$	0	0
$769$	−21.4283	−0.772724	−0.386362	−	0.922347i	$-0.626268\pi$
$769$	−21.4283	−0.772724	−0.386362	+	0.922347i	$0.626268\pi$
$770$	0	0
$771$	17.2532	0.621358
$772$	0	0
$773$	5.79976	0.208603	0.104301	−	0.994546i	$-0.466739\pi$
$773$	5.79976	0.208603	0.104301	+	0.994546i	$0.466739\pi$
$774$	0	0
$775$	6.95265	0.249746
$776$	0	0
$777$	42.7739	1.53450
$778$	0	0
$779$	−10.3877	−0.372178
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.43666	0.0513419
$784$	0	0
$785$	17.6614	0.630363
$786$	0	0
$787$	−25.2120	−0.898710	−0.449355	−	0.893353i	$-0.648346\pi$
$787$	−25.2120	−0.898710	−0.449355	+	0.893353i	$0.648346\pi$
$788$	0	0
$789$	28.2359	1.00523
$790$	0	0
$791$	−26.9246	−0.957328
$792$	0	0
$793$	8.07614	0.286792
$794$	0	0
$795$	27.5532	0.977210
$796$	0	0
$797$	18.4471	0.653428	0.326714	−	0.945123i	$-0.394059\pi$
$797$	18.4471	0.653428	0.326714	+	0.945123i	$0.394059\pi$
$798$	0	0
$799$	−40.3807	−1.42857
$800$	0	0
$801$	−31.2773	−1.10513
$802$	0	0
$803$	0	0
$804$	0	0
$805$	9.20499	0.324433
$806$	0	0
$807$	−35.4523	−1.24798
$808$	0	0
$809$	5.20841	0.183118	0.0915589	−	0.995800i	$-0.470815\pi$
$809$	5.20841	0.183118	0.0915589	+	0.995800i	$0.470815\pi$
$810$	0	0
$811$	7.79012	0.273548	0.136774	−	0.990602i	$-0.456327\pi$
$811$	7.79012	0.273548	0.136774	+	0.990602i	$0.456327\pi$
$812$	0	0
$813$	55.7767	1.95617
$814$	0	0
$815$	−4.62425	−0.161980
$816$	0	0
$817$	21.8660	0.764994
$818$	0	0
$819$	−76.9143	−2.68760
$820$	0	0
$821$	−9.56369	−0.333775	−0.166888	−	0.985976i	$-0.553372\pi$
$821$	−9.56369	−0.333775	−0.166888	+	0.985976i	$0.553372\pi$
$822$	0	0
$823$	13.4877	0.470152	0.235076	−	0.971977i	$-0.424466\pi$
$823$	13.4877	0.470152	0.235076	+	0.971977i	$0.424466\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.8316	−0.654839	−0.327420	−	0.944879i	$-0.606179\pi$
$827$	−18.8316	−0.654839	−0.327420	+	0.944879i	$0.606179\pi$
$828$	0	0
$829$	29.2659	1.01645	0.508223	−	0.861225i	$-0.330302\pi$
$829$	29.2659	1.01645	0.508223	+	0.861225i	$0.330302\pi$
$830$	0	0
$831$	37.0994	1.28697
$832$	0	0
$833$	−48.6708	−1.68634
$834$	0	0
$835$	−22.4278	−0.776146
$836$	0	0
$837$	4.35521	0.150538
$838$	0	0
$839$	−12.4646	−0.430326	−0.215163	−	0.976578i	$-0.569028\pi$
$839$	−12.4646	−0.430326	−0.215163	+	0.976578i	$0.569028\pi$
$840$	0	0
$841$	−23.7400	−0.818620
$842$	0	0
$843$	35.9850	1.23939
$844$	0	0
$845$	23.4270	0.805914
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−34.6942	−1.19070
$850$	0	0
$851$	−7.58997	−0.260181
$852$	0	0
$853$	34.3608	1.17649	0.588246	−	0.808682i	$-0.299819\pi$
$853$	34.3608	1.17649	0.588246	+	0.808682i	$0.299819\pi$
$854$	0	0
$855$	−7.49942	−0.256475
$856$	0	0
$857$	−14.9439	−0.510474	−0.255237	−	0.966879i	$-0.582153\pi$
$857$	−14.9439	−0.510474	−0.255237	+	0.966879i	$0.582153\pi$
$858$	0	0
$859$	−26.2500	−0.895640	−0.447820	−	0.894124i	$-0.647799\pi$
$859$	−26.2500	−0.895640	−0.447820	+	0.894124i	$0.647799\pi$
$860$	0	0
$861$	−42.2850	−1.44107
$862$	0	0
$863$	−20.0675	−0.683107	−0.341554	−	0.939862i	$-0.610953\pi$
$863$	−20.0675	−0.683107	−0.341554	+	0.939862i	$0.610953\pi$
$864$	0	0
$865$	−7.11042	−0.241761
$866$	0	0
$867$	14.3026	0.485741
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−43.3874	−1.47012
$872$	0	0
$873$	17.9317	0.606895
$874$	0	0
$875$	4.65351	0.157317
$876$	0	0
$877$	−36.2903	−1.22544	−0.612718	−	0.790302i	$-0.709924\pi$
$877$	−36.2903	−1.22544	−0.612718	+	0.790302i	$0.709924\pi$
$878$	0	0
$879$	39.9484	1.34743
$880$	0	0
$881$	26.3513	0.887796	0.443898	−	0.896077i	$-0.353595\pi$
$881$	26.3513	0.887796	0.443898	+	0.896077i	$0.353595\pi$
$882$	0	0
$883$	−54.0051	−1.81742	−0.908708	−	0.417432i	$-0.862930\pi$
$883$	−54.0051	−1.81742	−0.908708	+	0.417432i	$0.862930\pi$
$884$	0	0
$885$	17.0047	0.571607
$886$	0	0
$887$	12.7946	0.429601	0.214801	−	0.976658i	$-0.431090\pi$
$887$	12.7946	0.429601	0.214801	+	0.976658i	$0.431090\pi$
$888$	0	0
$889$	−8.15945	−0.273659
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−33.2974	−1.11426
$894$	0	0
$895$	−5.00000	−0.167132
$896$	0	0
$897$	28.5992	0.954898
$898$	0	0
$899$	15.9457	0.531820
$900$	0	0
$901$	38.1987	1.27258
$902$	0	0
$903$	89.0093	2.96204
$904$	0	0
$905$	16.9841	0.564570
$906$	0	0
$907$	−51.0096	−1.69375	−0.846873	−	0.531795i	$-0.821518\pi$
$907$	−51.0096	−1.69375	−0.846873	+	0.531795i	$0.821518\pi$
$908$	0	0
$909$	−0.747294	−0.0247862
$910$	0	0
$911$	1.54632	0.0512320	0.0256160	−	0.999672i	$-0.491845\pi$
$911$	1.54632	0.0512320	0.0256160	+	0.999672i	$0.491845\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.20547	0.105969
$916$	0	0
$917$	59.1215	1.95237
$918$	0	0
$919$	25.3314	0.835605	0.417802	−	0.908538i	$-0.362800\pi$
$919$	25.3314	0.835605	0.417802	+	0.908538i	$0.362800\pi$
$920$	0	0
$921$	1.28777	0.0424335
$922$	0	0
$923$	40.7014	1.33970
$924$	0	0
$925$	−3.83705	−0.126162
$926$	0	0
$927$	−22.4344	−0.736844
$928$	0	0
$929$	−54.7970	−1.79783	−0.898915	−	0.438122i	$-0.855644\pi$
$929$	−54.7970	−1.79783	−0.898915	+	0.438122i	$0.855644\pi$
$930$	0	0
$931$	−40.1334	−1.31532
$932$	0	0
$933$	29.9311	0.979900
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−32.3128	−1.05561	−0.527806	−	0.849365i	$-0.676985\pi$
$937$	−32.3128	−1.05561	−0.527806	+	0.849365i	$0.676985\pi$
$938$	0	0
$939$	−45.1247	−1.47259
$940$	0	0
$941$	19.5783	0.638234	0.319117	−	0.947715i	$-0.396614\pi$
$941$	19.5783	0.638234	0.319117	+	0.947715i	$0.396614\pi$
$942$	0	0
$943$	7.50322	0.244338
$944$	0	0
$945$	2.91501	0.0948252
$946$	0	0
$947$	22.1674	0.720345	0.360172	−	0.932886i	$-0.382718\pi$
$947$	22.1674	0.720345	0.360172	+	0.932886i	$0.382718\pi$
$948$	0	0
$949$	52.6087	1.70775
$950$	0	0
$951$	16.9145	0.548491
$952$	0	0
$953$	−52.4293	−1.69835	−0.849177	−	0.528109i	$-0.822901\pi$
$953$	−52.4293	−1.69835	−0.849177	+	0.528109i	$0.822901\pi$
$954$	0	0
$955$	−4.59045	−0.148543
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.08011	−0.260920
$960$	0	0
$961$	17.3393	0.559333
$962$	0	0
$963$	−18.0405	−0.581349
$964$	0	0
$965$	−8.75388	−0.281797
$966$	0	0
$967$	−9.72125	−0.312614	−0.156307	−	0.987709i	$-0.549959\pi$
$967$	−9.72125	−0.312614	−0.156307	+	0.987709i	$0.549959\pi$
$968$	0	0
$969$	−21.7866	−0.699888
$970$	0	0
$971$	47.2487	1.51628	0.758142	−	0.652090i	$-0.226108\pi$
$971$	47.2487	1.51628	0.758142	+	0.652090i	$0.226108\pi$
$972$	0	0
$973$	−30.1081	−0.965222
$974$	0	0
$975$	14.4581	0.463030
$976$	0	0
$977$	−41.8556	−1.33908	−0.669540	−	0.742776i	$-0.733509\pi$
$977$	−41.8556	−1.33908	−0.669540	+	0.742776i	$0.733509\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	8.38691	0.267773
$982$	0	0
$983$	35.9902	1.14791	0.573954	−	0.818887i	$-0.305409\pi$
$983$	35.9902	1.14791	0.573954	+	0.818887i	$0.305409\pi$
$984$	0	0
$985$	13.4511	0.428588
$986$	0	0
$987$	−135.543	−4.31438
$988$	0	0
$989$	−15.7942	−0.502226
$990$	0	0
$991$	−0.548830	−0.0174342	−0.00871709	−	0.999962i	$-0.502775\pi$
$991$	−0.548830	−0.0174342	−0.00871709	+	0.999962i	$0.502775\pi$
$992$	0	0
$993$	18.7673	0.595562
$994$	0	0
$995$	−4.55665	−0.144455
$996$	0	0
$997$	−9.25013	−0.292955	−0.146477	−	0.989214i	$-0.546794\pi$
$997$	−9.25013	−0.292955	−0.146477	+	0.989214i	$0.546794\pi$
$998$	0	0
$999$	−2.40357	−0.0760455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.cp.1.1		4
4.3	odd	2		2420.2.a.k.1.4		4
11.5	even	5		880.2.bo.c.641.1		8
11.9	even	5		880.2.bo.c.81.1		8
11.10	odd	2		9680.2.a.co.1.1		4
44.27	odd	10		220.2.m.b.201.2	yes	8
44.31	odd	10		220.2.m.b.81.2	✓	8
44.43	even	2		2420.2.a.l.1.4		4
132.71	even	10		1980.2.z.d.1081.1		8
132.119	even	10		1980.2.z.d.1621.1		8
220.27	even	20		1100.2.cb.b.949.4		16
220.119	odd	10		1100.2.n.b.301.1		8
220.159	odd	10		1100.2.n.b.201.1		8
220.163	even	20		1100.2.cb.b.1049.4		16
220.203	even	20		1100.2.cb.b.949.1		16
220.207	even	20		1100.2.cb.b.1049.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.m.b.81.2	✓	8	44.31	odd	10
220.2.m.b.201.2	yes	8	44.27	odd	10
880.2.bo.c.81.1		8	11.9	even	5
880.2.bo.c.641.1		8	11.5	even	5
1100.2.n.b.201.1		8	220.159	odd	10
1100.2.n.b.301.1		8	220.119	odd	10
1100.2.cb.b.949.1		16	220.203	even	20
1100.2.cb.b.949.4		16	220.27	even	20
1100.2.cb.b.1049.1		16	220.207	even	20
1100.2.cb.b.1049.4		16	220.163	even	20
1980.2.z.d.1081.1		8	132.71	even	10
1980.2.z.d.1621.1		8	132.119	even	10
2420.2.a.k.1.4		4	4.3	odd	2
2420.2.a.l.1.4		4	44.43	even	2
9680.2.a.co.1.1		4	11.10	odd	2
9680.2.a.cp.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39552 q^{3} +1.00000 q^{5} +4.65351 q^{7} +2.73851 q^{9} +O(q^{10})\)	\(q-2.39552 q^{3} +1.00000 q^{5} +4.65351 q^{7} +2.73851 q^{9} -6.03548 q^{13} -2.39552 q^{15} -3.32106 q^{17} -2.73851 q^{19} -11.1476 q^{21} +1.97807 q^{23} +1.00000 q^{25} +0.626410 q^{27} +2.29348 q^{29} +6.95265 q^{31} +4.65351 q^{35} -3.83705 q^{37} +14.4581 q^{39} +3.79320 q^{41} -7.98463 q^{43} +2.73851 q^{45} +12.1590 q^{47} +14.6552 q^{49} +7.95566 q^{51} -11.5020 q^{53} +6.56014 q^{57} -7.09855 q^{59} -1.33811 q^{61} +12.7437 q^{63} -6.03548 q^{65} +7.18872 q^{67} -4.73851 q^{69} -6.74368 q^{71} -8.71658 q^{73} -2.39552 q^{75} -5.48569 q^{79} -9.71610 q^{81} +8.21414 q^{83} -3.32106 q^{85} -5.49406 q^{87} -11.4213 q^{89} -28.0862 q^{91} -16.6552 q^{93} -2.73851 q^{95} +6.54798 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 3 * q^7 + 4 * q^9	\( 4 q + 4 q^{5} + 3 q^{7} + 4 q^{9} - 13 q^{13} - 13 q^{17} - 4 q^{19} - 14 q^{21} + 5 q^{23} + 4 q^{25} + 15 q^{27} - 8 q^{29} + 9 q^{31} + 3 q^{35} - 3 q^{37} + 9 q^{39} - 3 q^{41} + 11 q^{43} + 4 q^{45} + 3 q^{47} - q^{49} - 19 q^{51} - 3 q^{53} - 15 q^{57} - 23 q^{59} - 4 q^{61} + 13 q^{63} - 13 q^{65} + q^{67} - 12 q^{69} + 11 q^{71} - 25 q^{73} + 10 q^{79} + 16 q^{81} + 21 q^{83} - 13 q^{85} - 7 q^{87} + 4 q^{89} - 32 q^{91} - 7 q^{93} - 4 q^{95} - 10 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 3 * q^7 + 4 * q^9 - 13 * q^13 - 13 * q^17 - 4 * q^19 - 14 * q^21 + 5 * q^23 + 4 * q^25 + 15 * q^27 - 8 * q^29 + 9 * q^31 + 3 * q^35 - 3 * q^37 + 9 * q^39 - 3 * q^41 + 11 * q^43 + 4 * q^45 + 3 * q^47 - q^49 - 19 * q^51 - 3 * q^53 - 15 * q^57 - 23 * q^59 - 4 * q^61 + 13 * q^63 - 13 * q^65 + q^67 - 12 * q^69 + 11 * q^71 - 25 * q^73 + 10 * q^79 + 16 * q^81 + 21 * q^83 - 13 * q^85 - 7 * q^87 + 4 * q^89 - 32 * q^91 - 7 * q^93 - 4 * q^95 - 10 * q^97

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