LMFDB - Embedded newform 5472.2.a.bs.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(1,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, 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("5472.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.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:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.69353$ of defining polynomial
Character	$\chi$	$=$	5472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.20608 q^{5} +1.74252 q^{7} +O(q^{10})$	$q+4.20608 q^{5} +1.74252 q^{7} -6.20608 q^{11} +4.82550 q^{13} +5.12957 q^{17} +1.00000 q^{19} -0.463560 q^{23} +12.6911 q^{25} -4.82550 q^{29} -1.63806 q^{31} +7.32919 q^{35} +3.63806 q^{37} +5.28906 q^{41} +1.08298 q^{43} +0.555087 q^{47} -3.96362 q^{49} -2.65955 q^{53} -26.1033 q^{55} +1.85061 q^{59} -9.32919 q^{61} +20.2964 q^{65} +4.72751 q^{67} +11.4850 q^{71} -0.00646614 q^{73} -10.8142 q^{77} +6.76117 q^{79} +11.8972 q^{83} +21.5754 q^{85} +7.31909 q^{89} +8.40853 q^{91} +4.20608 q^{95} -10.4122 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{5} - q^{7}+O(q^{10}) $ 4 * q - q^5 - q^7	$ 4 q - q^{5} - q^{7} - 7 q^{11} + 10 q^{13} - 5 q^{17} + 4 q^{19} + 8 q^{23} + 17 q^{25} - 10 q^{29} - 6 q^{31} - 5 q^{35} + 14 q^{37} + 2 q^{41} + 3 q^{43} + 3 q^{47} + 7 q^{49} - 4 q^{53} - 35 q^{55} - 20 q^{59} - 3 q^{61} + 12 q^{65} + 8 q^{67} + 30 q^{71} + 9 q^{73} + 7 q^{77} + 10 q^{79} - 4 q^{83} + 19 q^{85} + 16 q^{89} + 10 q^{91} - q^{95} - 6 q^{97}+O(q^{100}) $ 4 * q - q^5 - q^7 - 7 * q^11 + 10 * q^13 - 5 * q^17 + 4 * q^19 + 8 * q^23 + 17 * q^25 - 10 * q^29 - 6 * q^31 - 5 * q^35 + 14 * q^37 + 2 * q^41 + 3 * q^43 + 3 * q^47 + 7 * q^49 - 4 * q^53 - 35 * q^55 - 20 * q^59 - 3 * q^61 + 12 * q^65 + 8 * q^67 + 30 * q^71 + 9 * q^73 + 7 * q^77 + 10 * q^79 - 4 * q^83 + 19 * q^85 + 16 * q^89 + 10 * q^91 - q^95 - 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	4.20608	1.88102	0.940508	−	0.339770i	$-0.110349\pi$
$5$	4.20608	1.88102	0.940508	+	0.339770i	$0.110349\pi$
$6$	0	0
$7$	1.74252	0.658611	0.329306	−	0.944223i	$-0.393185\pi$
$7$	1.74252	0.658611	0.329306	+	0.944223i	$0.393185\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.20608	−1.87120	−0.935602	−	0.353056i	$-0.885142\pi$
$11$	−6.20608	−1.87120	−0.935602	+	0.353056i	$0.885142\pi$
$12$	0	0
$13$	4.82550	1.33835	0.669176	−	0.743104i	$-0.266647\pi$
$13$	4.82550	1.33835	0.669176	+	0.743104i	$0.266647\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.12957	1.24410	0.622052	−	0.782976i	$-0.286299\pi$
$17$	5.12957	1.24410	0.622052	+	0.782976i	$0.286299\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.463560	−0.0966590	−0.0483295	−	0.998831i	$-0.515390\pi$
$23$	−0.463560	−0.0966590	−0.0483295	+	0.998831i	$0.515390\pi$
$24$	0	0
$25$	12.6911	2.53822
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.82550	−0.896072	−0.448036	−	0.894015i	$-0.647876\pi$
$29$	−4.82550	−0.896072	−0.448036	+	0.894015i	$0.647876\pi$
$30$	0	0
$31$	−1.63806	−0.294205	−0.147102	−	0.989121i	$-0.546995\pi$
$31$	−1.63806	−0.294205	−0.147102	+	0.989121i	$0.546995\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	7.32919	1.23886
$36$	0	0
$37$	3.63806	0.598094	0.299047	−	0.954238i	$-0.403331\pi$
$37$	3.63806	0.598094	0.299047	+	0.954238i	$0.403331\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.28906	0.826012	0.413006	−	0.910728i	$-0.364479\pi$
$41$	5.28906	0.826012	0.413006	+	0.910728i	$0.364479\pi$
$42$	0	0
$43$	1.08298	0.165152	0.0825762	−	0.996585i	$-0.473685\pi$
$43$	1.08298	0.165152	0.0825762	+	0.996585i	$0.473685\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.555087	0.0809677	0.0404839	−	0.999180i	$-0.487110\pi$
$47$	0.555087	0.0809677	0.0404839	+	0.999180i	$0.487110\pi$
$48$	0	0
$49$	−3.96362	−0.566231
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.65955	−0.365317	−0.182658	−	0.983176i	$-0.558470\pi$
$53$	−2.65955	−0.365317	−0.182658	+	0.983176i	$0.558470\pi$
$54$	0	0
$55$	−26.1033	−3.51977
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.85061	0.240929	0.120465	−	0.992718i	$-0.461562\pi$
$59$	1.85061	0.240929	0.120465	+	0.992718i	$0.461562\pi$
$60$	0	0
$61$	−9.32919	−1.19448	−0.597240	−	0.802063i	$-0.703736\pi$
$61$	−9.32919	−1.19448	−0.597240	+	0.802063i	$0.703736\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	20.2964	2.51746
$66$	0	0
$67$	4.72751	0.577557	0.288778	−	0.957396i	$-0.406751\pi$
$67$	4.72751	0.577557	0.288778	+	0.957396i	$0.406751\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.4850	1.36302	0.681512	−	0.731807i	$-0.261323\pi$
$71$	11.4850	1.36302	0.681512	+	0.731807i	$0.261323\pi$
$72$	0	0
$73$	−0.00646614	−0.000756804 0	−0.000378402	−	1.00000i	$-0.500120\pi$
$73$	−0.00646614	−0.000756804 0	−0.000378402		1.00000i	$0.500120\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.8142	−1.23240
$78$	0	0
$79$	6.76117	0.760691	0.380345	−	0.924844i	$-0.375805\pi$
$79$	6.76117	0.760691	0.380345	+	0.924844i	$0.375805\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.8972	1.30589	0.652944	−	0.757406i	$-0.273534\pi$
$83$	11.8972	1.30589	0.652944	+	0.757406i	$0.273534\pi$
$84$	0	0
$85$	21.5754	2.34018
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.31909	0.775822	0.387911	−	0.921697i	$-0.373197\pi$
$89$	7.31909	0.775822	0.387911	+	0.921697i	$0.373197\pi$
$90$	0	0
$91$	8.40853	0.881454
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.20608	0.431535
$96$	0	0
$97$	−10.4122	−1.05720	−0.528598	−	0.848873i	$-0.677282\pi$
$97$	−10.4122	−1.05720	−0.528598	+	0.848873i	$0.677282\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.246211	−0.0244989	−0.0122495	−	0.999925i	$-0.503899\pi$
$101$	−0.246211	−0.0244989	−0.0122495	+	0.999925i	$0.503899\pi$
$102$	0	0
$103$	0.710942	0.0700512	0.0350256	−	0.999386i	$-0.488849\pi$
$103$	0.710942	0.0700512	0.0350256	+	0.999386i	$0.488849\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.6976	1.13085	0.565424	−	0.824800i	$-0.308712\pi$
$107$	11.6976	1.13085	0.565424	+	0.824800i	$0.308712\pi$
$108$	0	0
$109$	−2.99145	−0.286529	−0.143264	−	0.989684i	$-0.545760\pi$
$109$	−2.99145	−0.286529	−0.143264	+	0.989684i	$0.545760\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.41216	−0.603206	−0.301603	−	0.953434i	$-0.597522\pi$
$113$	−6.41216	−0.603206	−0.301603	+	0.953434i	$0.597522\pi$
$114$	0	0
$115$	−1.94977	−0.181817
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.93839	0.819381
$120$	0	0
$121$	27.5155	2.50140
$122$	0	0
$123$	0	0
$124$	0	0
$125$	32.3495	2.89343
$126$	0	0
$127$	−18.1434	−1.60997	−0.804984	−	0.593297i	$-0.797826\pi$
$127$	−18.1434	−1.60997	−0.804984	+	0.593297i	$0.797826\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.4652	−1.08909	−0.544546	−	0.838731i	$-0.683298\pi$
$131$	−12.4652	−1.08909	−0.544546	+	0.838731i	$0.683298\pi$
$132$	0	0
$133$	1.74252	0.151096
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.41863	−0.719252	−0.359626	−	0.933097i	$-0.617096\pi$
$137$	−8.41863	−0.719252	−0.359626	+	0.933097i	$0.617096\pi$
$138$	0	0
$139$	11.3292	0.960929	0.480465	−	0.877014i	$-0.340468\pi$
$139$	11.3292	0.960929	0.480465	+	0.877014i	$0.340468\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−29.9474	−2.50433
$144$	0	0
$145$	−20.2964	−1.68553
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.321808	0.0263635	0.0131818	−	0.999913i	$-0.495804\pi$
$149$	0.321808	0.0263635	0.0131818	+	0.999913i	$0.495804\pi$
$150$	0	0
$151$	−2.95715	−0.240650	−0.120325	−	0.992735i	$-0.538394\pi$
$151$	−2.95715	−0.240650	−0.120325	+	0.992735i	$0.538394\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.88983	−0.553404
$156$	0	0
$157$	6.52789	0.520982	0.260491	−	0.965476i	$-0.416116\pi$
$157$	6.52789	0.520982	0.260491	+	0.965476i	$0.416116\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.807764	−0.0636607
$162$	0	0
$163$	2.05023	0.160586	0.0802931	−	0.996771i	$-0.474414\pi$
$163$	2.05023	0.160586	0.0802931	+	0.996771i	$0.474414\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.2462	0.792876	0.396438	−	0.918062i	$-0.370246\pi$
$167$	10.2462	0.792876	0.396438	+	0.918062i	$0.370246\pi$
$168$	0	0
$169$	10.2854	0.791187
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.05023	−0.612047	−0.306024	−	0.952024i	$-0.598999\pi$
$173$	−8.05023	−0.612047	−0.306024	+	0.952024i	$0.598999\pi$
$174$	0	0
$175$	22.1146	1.67170
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.75379	−0.131084	−0.0655422	−	0.997850i	$-0.520878\pi$
$179$	−1.75379	−0.131084	−0.0655422	+	0.997850i	$0.520878\pi$
$180$	0	0
$181$	−11.0203	−0.819133	−0.409567	−	0.912280i	$-0.634320\pi$
$181$	−11.0203	−0.819133	−0.409567	+	0.912280i	$0.634320\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	15.3020	1.12502
$186$	0	0
$187$	−31.8345	−2.32797
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.2276	0.957113	0.478556	−	0.878057i	$-0.341160\pi$
$191$	13.2276	0.957113	0.478556	+	0.878057i	$0.341160\pi$
$192$	0	0
$193$	1.47211	0.105965	0.0529824	−	0.998595i	$-0.483127\pi$
$193$	1.47211	0.105965	0.0529824	+	0.998595i	$0.483127\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.84698	−0.559074	−0.279537	−	0.960135i	$-0.590181\pi$
$197$	−7.84698	−0.559074	−0.279537	+	0.960135i	$0.590181\pi$
$198$	0	0
$199$	−11.8057	−0.836882	−0.418441	−	0.908244i	$-0.637423\pi$
$199$	−11.8057	−0.836882	−0.418441	+	0.908244i	$0.637423\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.40853	−0.590163
$204$	0	0
$205$	22.2462	1.55374
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.20608	−0.429284
$210$	0	0
$211$	−14.5745	−1.00335	−0.501674	−	0.865056i	$-0.667282\pi$
$211$	−14.5745	−1.00335	−0.501674	+	0.865056i	$0.667282\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.55509	0.310654
$216$	0	0
$217$	−2.85436	−0.193767
$218$	0	0
$219$	0	0
$220$	0	0
$221$	24.7527	1.66505
$222$	0	0
$223$	26.6713	1.78604	0.893021	−	0.450014i	$-0.148581\pi$
$223$	26.6713	1.78604	0.893021	+	0.450014i	$0.148581\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.01656	0.266589	0.133294	−	0.991076i	$-0.457444\pi$
$227$	4.01656	0.266589	0.133294	+	0.991076i	$0.457444\pi$
$228$	0	0
$229$	23.1834	1.53200	0.766002	−	0.642838i	$-0.222243\pi$
$229$	23.1834	1.53200	0.766002	+	0.642838i	$0.222243\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.2692	−1.45891	−0.729453	−	0.684031i	$-0.760225\pi$
$233$	−22.2692	−1.45891	−0.729453	+	0.684031i	$0.760225\pi$
$234$	0	0
$235$	2.33474	0.152302
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.0818	1.49304	0.746519	−	0.665364i	$-0.231724\pi$
$239$	23.0818	1.49304	0.746519	+	0.665364i	$0.231724\pi$
$240$	0	0
$241$	−0.565184	−0.0364067	−0.0182033	−	0.999834i	$-0.505795\pi$
$241$	−0.565184	−0.0364067	−0.0182033	+	0.999834i	$0.505795\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−16.6713	−1.06509
$246$	0	0
$247$	4.82550	0.307039
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.8441	−1.00007	−0.500037	−	0.866004i	$-0.666680\pi$
$251$	−15.8441	−1.00007	−0.500037	+	0.866004i	$0.666680\pi$
$252$	0	0
$253$	2.87689	0.180869
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.0632	−0.752479	−0.376240	−	0.926522i	$-0.622783\pi$
$257$	−12.0632	−0.752479	−0.376240	+	0.926522i	$0.622783\pi$
$258$	0	0
$259$	6.33940	0.393911
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−23.2135	−1.43140	−0.715702	−	0.698406i	$-0.753893\pi$
$263$	−23.2135	−1.43140	−0.715702	+	0.698406i	$0.753893\pi$
$264$	0	0
$265$	−11.1863	−0.687167
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−26.2964	−1.60332	−0.801661	−	0.597779i	$-0.796050\pi$
$269$	−26.2964	−1.60332	−0.801661	+	0.597779i	$0.796050\pi$
$270$	0	0
$271$	12.9560	0.787020	0.393510	−	0.919320i	$-0.371261\pi$
$271$	12.9560	0.787020	0.393510	+	0.919320i	$0.371261\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−78.7622	−4.74954
$276$	0	0
$277$	−10.8645	−0.652782	−0.326391	−	0.945235i	$-0.605833\pi$
$277$	−10.8645	−0.652782	−0.326391	+	0.945235i	$0.605833\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.8470	0.945352	0.472676	−	0.881236i	$-0.343288\pi$
$281$	15.8470	0.945352	0.472676	+	0.881236i	$0.343288\pi$
$282$	0	0
$283$	−27.6740	−1.64505	−0.822525	−	0.568729i	$-0.807435\pi$
$283$	−27.6740	−1.64505	−0.822525	+	0.568729i	$0.807435\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.21630	0.544021
$288$	0	0
$289$	9.31251	0.547795
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.3179	1.24541	0.622703	−	0.782458i	$-0.286034\pi$
$293$	21.3179	1.24541	0.622703	+	0.782458i	$0.286034\pi$
$294$	0	0
$295$	7.78382	0.453192
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.23691	−0.129364
$300$	0	0
$301$	1.88711	0.108771
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−39.2393	−2.24684
$306$	0	0
$307$	−26.6584	−1.52147	−0.760737	−	0.649060i	$-0.775162\pi$
$307$	−26.6584	−1.52147	−0.760737	+	0.649060i	$0.775162\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.0690	−1.13801	−0.569004	−	0.822335i	$-0.692671\pi$
$311$	−20.0690	−1.13801	−0.569004	+	0.822335i	$0.692671\pi$
$312$	0	0
$313$	−11.1397	−0.629651	−0.314826	−	0.949150i	$-0.601946\pi$
$313$	−11.1397	−0.629651	−0.314826	+	0.949150i	$0.601946\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33.1349	−1.86104	−0.930520	−	0.366242i	$-0.880644\pi$
$317$	−33.1349	−1.86104	−0.930520	+	0.366242i	$0.880644\pi$
$318$	0	0
$319$	29.9474	1.67673
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.12957	0.285417
$324$	0	0
$325$	61.2410	3.39704
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.967250	0.0533262
$330$	0	0
$331$	4.30241	0.236482	0.118241	−	0.992985i	$-0.462274\pi$
$331$	4.30241	0.236482	0.118241	+	0.992985i	$0.462274\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	19.8843	1.08639
$336$	0	0
$337$	31.3393	1.70716	0.853580	−	0.520962i	$-0.174427\pi$
$337$	31.3393	1.70716	0.853580	+	0.520962i	$0.174427\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.1660	0.550517
$342$	0	0
$343$	−19.1043	−1.03154
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.6782	1.16375	0.581873	−	0.813280i	$-0.302320\pi$
$347$	21.6782	1.16375	0.581873	+	0.813280i	$0.302320\pi$
$348$	0	0
$349$	4.50486	0.241140	0.120570	−	0.992705i	$-0.461528\pi$
$349$	4.50486	0.241140	0.120570	+	0.992705i	$0.461528\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.4158	1.19307	0.596536	−	0.802586i	$-0.296543\pi$
$353$	22.4158	1.19307	0.596536	+	0.802586i	$0.296543\pi$
$354$	0	0
$355$	48.3070	2.56387
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.4940	−0.606629	−0.303314	−	0.952891i	$-0.598093\pi$
$359$	−11.4940	−0.606629	−0.303314	+	0.952891i	$0.598093\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.0271971	−0.00142356
$366$	0	0
$367$	−19.7944	−1.03326	−0.516630	−	0.856209i	$-0.672814\pi$
$367$	−19.7944	−1.03326	−0.516630	+	0.856209i	$0.672814\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.63431	−0.240602
$372$	0	0
$373$	−16.6199	−0.860546	−0.430273	−	0.902699i	$-0.641583\pi$
$373$	−16.6199	−0.860546	−0.430273	+	0.902699i	$0.641583\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−23.2854	−1.19926
$378$	0	0
$379$	16.3956	0.842185	0.421093	−	0.907018i	$-0.361647\pi$
$379$	16.3956	0.842185	0.421093	+	0.907018i	$0.361647\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.6915	1.56826	0.784131	−	0.620595i	$-0.213109\pi$
$383$	30.6915	1.56826	0.784131	+	0.620595i	$0.213109\pi$
$384$	0	0
$385$	−45.4855	−2.31816
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.24905	0.114031	0.0570156	−	0.998373i	$-0.481842\pi$
$389$	2.24905	0.114031	0.0570156	+	0.998373i	$0.481842\pi$
$390$	0	0
$391$	−2.37787	−0.120254
$392$	0	0
$393$	0	0
$394$	0	0
$395$	28.4380	1.43087
$396$	0	0
$397$	−4.12582	−0.207069	−0.103535	−	0.994626i	$-0.533015\pi$
$397$	−4.12582	−0.207069	−0.103535	+	0.994626i	$0.533015\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.3992	1.11856	0.559282	−	0.828977i	$-0.311077\pi$
$401$	22.3992	1.11856	0.559282	+	0.828977i	$0.311077\pi$
$402$	0	0
$403$	−7.90447	−0.393750
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−22.5781	−1.11916
$408$	0	0
$409$	13.1992	0.652658	0.326329	−	0.945256i	$-0.394188\pi$
$409$	13.1992	0.652658	0.326329	+	0.945256i	$0.394188\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.22473	0.158679
$414$	0	0
$415$	50.0406	2.45640
$416$	0	0
$417$	0	0
$418$	0	0
$419$	31.5984	1.54368	0.771842	−	0.635814i	$-0.219336\pi$
$419$	31.5984	1.54368	0.771842	+	0.635814i	$0.219336\pi$
$420$	0	0
$421$	−25.8587	−1.26028	−0.630139	−	0.776482i	$-0.717002\pi$
$421$	−25.8587	−1.26028	−0.630139	+	0.776482i	$0.717002\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	65.1000	3.15782
$426$	0	0
$427$	−16.2563	−0.786698
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.8972	1.53643	0.768217	−	0.640189i	$-0.221144\pi$
$431$	31.8972	1.53643	0.768217	+	0.640189i	$0.221144\pi$
$432$	0	0
$433$	23.1006	1.11014	0.555071	−	0.831803i	$-0.312691\pi$
$433$	23.1006	1.11014	0.555071	+	0.831803i	$0.312691\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.463560	−0.0221751
$438$	0	0
$439$	15.7944	0.753826	0.376913	−	0.926249i	$-0.376986\pi$
$439$	15.7944	0.753826	0.376913	+	0.926249i	$0.376986\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−40.5584	−1.92699	−0.963494	−	0.267729i	$-0.913727\pi$
$443$	−40.5584	−1.92699	−0.963494	+	0.267729i	$0.913727\pi$
$444$	0	0
$445$	30.7847	1.45933
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.4026	1.43479	0.717393	−	0.696669i	$-0.245335\pi$
$449$	30.4026	1.43479	0.717393	+	0.696669i	$0.245335\pi$
$450$	0	0
$451$	−32.8243	−1.54564
$452$	0	0
$453$	0	0
$454$	0	0
$455$	35.3670	1.65803
$456$	0	0
$457$	1.33516	0.0624561	0.0312280	−	0.999512i	$-0.490058\pi$
$457$	1.33516	0.0624561	0.0312280	+	0.999512i	$0.490058\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.0604	1.07403	0.537016	−	0.843572i	$-0.319552\pi$
$461$	23.0604	1.07403	0.537016	+	0.843572i	$0.319552\pi$
$462$	0	0
$463$	9.29916	0.432168	0.216084	−	0.976375i	$-0.430671\pi$
$463$	9.29916	0.432168	0.216084	+	0.976375i	$0.430671\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.8344	0.871553	0.435777	−	0.900055i	$-0.356474\pi$
$467$	18.8344	0.871553	0.435777	+	0.900055i	$0.356474\pi$
$468$	0	0
$469$	8.23778	0.380385
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.72104	−0.309034
$474$	0	0
$475$	12.6911	0.582309
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.9701	−0.501236	−0.250618	−	0.968086i	$-0.580634\pi$
$479$	−10.9701	−0.501236	−0.250618	+	0.968086i	$0.580634\pi$
$480$	0	0
$481$	17.5555	0.800460
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−43.7944	−1.98860
$486$	0	0
$487$	19.3895	0.878623	0.439311	−	0.898335i	$-0.355223\pi$
$487$	19.3895	0.878623	0.439311	+	0.898335i	$0.355223\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−33.1562	−1.49632	−0.748160	−	0.663518i	$-0.769062\pi$
$491$	−33.1562	−1.49632	−0.748160	+	0.663518i	$0.769062\pi$
$492$	0	0
$493$	−24.7527	−1.11481
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.0129	0.897703
$498$	0	0
$499$	−25.8814	−1.15861	−0.579306	−	0.815110i	$-0.696676\pi$
$499$	−25.8814	−1.15861	−0.579306	+	0.815110i	$0.696676\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.3178	−0.460048	−0.230024	−	0.973185i	$-0.573880\pi$
$503$	−10.3178	−0.460048	−0.230024	+	0.973185i	$0.573880\pi$
$504$	0	0
$505$	−1.03558	−0.0460829
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.21618	0.275527	0.137764	−	0.990465i	$-0.456009\pi$
$509$	6.21618	0.275527	0.137764	+	0.990465i	$0.456009\pi$
$510$	0	0
$511$	−0.0112674	−0.000498440 0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.99028	0.131767
$516$	0	0
$517$	−3.44491	−0.151507
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22.8917	−1.00290	−0.501451	−	0.865186i	$-0.667200\pi$
$521$	−22.8917	−1.00290	−0.501451	+	0.865186i	$0.667200\pi$
$522$	0	0
$523$	9.02628	0.394692	0.197346	−	0.980334i	$-0.436768\pi$
$523$	9.02628	0.394692	0.197346	+	0.980334i	$0.436768\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.40256	−0.366021
$528$	0	0
$529$	−22.7851	−0.990657
$530$	0	0
$531$	0	0
$532$	0	0
$533$	25.5223	1.10550
$534$	0	0
$535$	49.2010	2.12715
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24.5985	1.05953
$540$	0	0
$541$	−21.2937	−0.915489	−0.457744	−	0.889084i	$-0.651342\pi$
$541$	−21.2937	−0.915489	−0.457744	+	0.889084i	$0.651342\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.5823	−0.538966
$546$	0	0
$547$	3.60803	0.154268	0.0771341	−	0.997021i	$-0.475423\pi$
$547$	3.60803	0.154268	0.0771341	+	0.997021i	$0.475423\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.82550	−0.205573
$552$	0	0
$553$	11.7815	0.501000
$554$	0	0
$555$	0	0
$556$	0	0
$557$	34.8645	1.47725	0.738627	−	0.674114i	$-0.235474\pi$
$557$	34.8645	1.47725	0.738627	+	0.674114i	$0.235474\pi$
$558$	0	0
$559$	5.22590	0.221032
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−41.3167	−1.74129	−0.870647	−	0.491909i	$-0.836299\pi$
$563$	−41.3167	−1.74129	−0.870647	+	0.491909i	$0.836299\pi$
$564$	0	0
$565$	−26.9701	−1.13464
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.0835	−1.72231	−0.861154	−	0.508344i	$-0.830258\pi$
$569$	−41.0835	−1.72231	−0.861154	+	0.508344i	$0.830258\pi$
$570$	0	0
$571$	−36.8243	−1.54105	−0.770525	−	0.637410i	$-0.780006\pi$
$571$	−36.8243	−1.54105	−0.770525	+	0.637410i	$0.780006\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.88310	−0.245342
$576$	0	0
$577$	15.7004	0.653617	0.326809	−	0.945091i	$-0.394027\pi$
$577$	15.7004	0.653617	0.326809	+	0.945091i	$0.394027\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	20.7311	0.860072
$582$	0	0
$583$	16.5054	0.683582
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.3099	−1.41612	−0.708060	−	0.706152i	$-0.750429\pi$
$587$	−34.3099	−1.41612	−0.708060	+	0.706152i	$0.750429\pi$
$588$	0	0
$589$	−1.63806	−0.0674952
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.11584	−0.292213	−0.146106	−	0.989269i	$-0.546674\pi$
$593$	−7.11584	−0.292213	−0.146106	+	0.989269i	$0.546674\pi$
$594$	0	0
$595$	37.5956	1.54127
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.8340	0.565244	0.282622	−	0.959231i	$-0.408796\pi$
$599$	13.8340	0.565244	0.282622	+	0.959231i	$0.408796\pi$
$600$	0	0
$601$	39.2140	1.59957	0.799785	−	0.600286i	$-0.204947\pi$
$601$	39.2140	1.59957	0.799785	+	0.600286i	$0.204947\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	115.732	4.70518
$606$	0	0
$607$	35.1733	1.42764	0.713821	−	0.700328i	$-0.246963\pi$
$607$	35.1733	1.42764	0.713821	+	0.700328i	$0.246963\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.67857	0.108363
$612$	0	0
$613$	6.24905	0.252397	0.126198	−	0.992005i	$-0.459722\pi$
$613$	6.24905	0.252397	0.126198	+	0.992005i	$0.459722\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.0378	1.69238	0.846189	−	0.532883i	$-0.178891\pi$
$617$	42.0378	1.69238	0.846189	+	0.532883i	$0.178891\pi$
$618$	0	0
$619$	31.0633	1.24854	0.624269	−	0.781209i	$-0.285397\pi$
$619$	31.0633	1.24854	0.624269	+	0.781209i	$0.285397\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.7537	0.510965
$624$	0	0
$625$	72.6090	2.90436
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.6617	0.744091
$630$	0	0
$631$	−40.0378	−1.59388	−0.796940	−	0.604059i	$-0.793549\pi$
$631$	−40.0378	−1.59388	−0.796940	+	0.604059i	$0.793549\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−76.3127	−3.02838
$636$	0	0
$637$	−19.1264	−0.757817
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.3619	0.804248	0.402124	−	0.915585i	$-0.368272\pi$
$641$	20.3619	0.804248	0.402124	+	0.915585i	$0.368272\pi$
$642$	0	0
$643$	13.7041	0.540435	0.270218	−	0.962799i	$-0.412904\pi$
$643$	13.7041	0.540435	0.270218	+	0.962799i	$0.412904\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.9588	−1.05986	−0.529930	−	0.848041i	$-0.677782\pi$
$647$	−26.9588	−1.05986	−0.529930	+	0.848041i	$0.677782\pi$
$648$	0	0
$649$	−11.4850	−0.450827
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.7842	−1.20468	−0.602339	−	0.798240i	$-0.705765\pi$
$653$	−30.7842	−1.20468	−0.602339	+	0.798240i	$0.705765\pi$
$654$	0	0
$655$	−52.4298	−2.04860
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.57854	−0.295218	−0.147609	−	0.989046i	$-0.547158\pi$
$659$	−7.57854	−0.295218	−0.147609	+	0.989046i	$0.547158\pi$
$660$	0	0
$661$	47.8191	1.85995	0.929974	−	0.367626i	$-0.119829\pi$
$661$	47.8191	1.85995	0.929974	+	0.367626i	$0.119829\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.32919	0.284214
$666$	0	0
$667$	2.23691	0.0866135
$668$	0	0
$669$	0	0
$670$	0	0
$671$	57.8977	2.23512
$672$	0	0
$673$	12.2389	0.471777	0.235888	−	0.971780i	$-0.424200\pi$
$673$	12.2389	0.471777	0.235888	+	0.971780i	$0.424200\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23.3609	−0.897832	−0.448916	−	0.893574i	$-0.648190\pi$
$677$	−23.3609	−0.897832	−0.448916	+	0.893574i	$0.648190\pi$
$678$	0	0
$679$	−18.1434	−0.696280
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.7239	−0.946033	−0.473016	−	0.881054i	$-0.656835\pi$
$683$	−24.7239	−0.946033	−0.473016	+	0.881054i	$0.656835\pi$
$684$	0	0
$685$	−35.4094	−1.35293
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.8336	−0.488922
$690$	0	0
$691$	6.78837	0.258242	0.129121	−	0.991629i	$-0.458785\pi$
$691$	6.78837	0.258242	0.129121	+	0.991629i	$0.458785\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	47.6515	1.80752
$696$	0	0
$697$	27.1306	1.02764
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−26.9175	−1.01666	−0.508330	−	0.861162i	$-0.669737\pi$
$701$	−26.9175	−1.01666	−0.508330	+	0.861162i	$0.669737\pi$
$702$	0	0
$703$	3.63806	0.137212
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.429028	−0.0161353
$708$	0	0
$709$	−21.5984	−0.811146	−0.405573	−	0.914063i	$-0.632928\pi$
$709$	−21.5984	−0.811146	−0.405573	+	0.914063i	$0.632928\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0.759341	0.0284376
$714$	0	0
$715$	−125.961	−4.71069
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22.2575	0.830064	0.415032	−	0.909807i	$-0.363770\pi$
$719$	22.2575	0.830064	0.415032	+	0.909807i	$0.363770\pi$
$720$	0	0
$721$	1.23883	0.0461365
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−61.2410	−2.27443
$726$	0	0
$727$	−51.9118	−1.92530	−0.962651	−	0.270745i	$-0.912730\pi$
$727$	−51.9118	−1.92530	−0.962651	+	0.270745i	$0.912730\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.55520	0.205467
$732$	0	0
$733$	20.0575	0.740840	0.370420	−	0.928864i	$-0.379214\pi$
$733$	20.0575	0.740840	0.370420	+	0.928864i	$0.379214\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−29.3393	−1.08073
$738$	0	0
$739$	7.36465	0.270913	0.135457	−	0.990783i	$-0.456750\pi$
$739$	7.36465	0.270913	0.135457	+	0.990783i	$0.456750\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.85436	0.104716	0.0523581	−	0.998628i	$-0.483326\pi$
$743$	2.85436	0.104716	0.0523581	+	0.998628i	$0.483326\pi$
$744$	0	0
$745$	1.35355	0.0495902
$746$	0	0
$747$	0	0
$748$	0	0
$749$	20.3833	0.744790
$750$	0	0
$751$	−22.3725	−0.816385	−0.408193	−	0.912896i	$-0.633841\pi$
$751$	−22.3725	−0.816385	−0.408193	+	0.912896i	$0.633841\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.4380	−0.452666
$756$	0	0
$757$	6.07326	0.220736	0.110368	−	0.993891i	$-0.464797\pi$
$757$	6.07326	0.220736	0.110368	+	0.993891i	$0.464797\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.1166	−0.475478	−0.237739	−	0.971329i	$-0.576406\pi$
$761$	−13.1166	−0.475478	−0.237739	+	0.971329i	$0.576406\pi$
$762$	0	0
$763$	−5.21267	−0.188711
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.93012	0.322448
$768$	0	0
$769$	−44.5791	−1.60757	−0.803783	−	0.594923i	$-0.797182\pi$
$769$	−44.5791	−1.60757	−0.803783	+	0.594923i	$0.797182\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4.43920	0.159667	0.0798334	−	0.996808i	$-0.474561\pi$
$773$	4.43920	0.159667	0.0798334	+	0.996808i	$0.474561\pi$
$774$	0	0
$775$	−20.7889	−0.746758
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.28906	0.189500
$780$	0	0
$781$	−71.2771	−2.55050
$782$	0	0
$783$	0	0
$784$	0	0
$785$	27.4568	0.979977
$786$	0	0
$787$	0.315342	0.0112407	0.00562036	−	0.999984i	$-0.498211\pi$
$787$	0.315342	0.0112407	0.00562036	+	0.999984i	$0.498211\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11.1733	−0.397278
$792$	0	0
$793$	−45.0180	−1.59864
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.16712	−0.218451	−0.109225	−	0.994017i	$-0.534837\pi$
$797$	−6.16712	−0.218451	−0.109225	+	0.994017i	$0.534837\pi$
$798$	0	0
$799$	2.84736	0.100732
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.0401294	0.00141614
$804$	0	0
$805$	−3.39752	−0.119747
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−20.4057	−0.717426	−0.358713	−	0.933448i	$-0.616784\pi$
$809$	−20.4057	−0.717426	−0.358713	+	0.933448i	$0.616784\pi$
$810$	0	0
$811$	−36.5763	−1.28437	−0.642184	−	0.766551i	$-0.721971\pi$
$811$	−36.5763	−1.28437	−0.642184	+	0.766551i	$0.721971\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.62342	0.302065
$816$	0	0
$817$	1.08298	0.0378885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.6686	−0.686439	−0.343219	−	0.939255i	$-0.611517\pi$
$821$	−19.6686	−0.686439	−0.343219	+	0.939255i	$0.611517\pi$
$822$	0	0
$823$	−36.7328	−1.28042	−0.640212	−	0.768198i	$-0.721154\pi$
$823$	−36.7328	−1.28042	−0.640212	+	0.768198i	$0.721154\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−27.4920	−0.955991	−0.477995	−	0.878362i	$-0.658636\pi$
$827$	−27.4920	−0.955991	−0.477995	+	0.878362i	$0.658636\pi$
$828$	0	0
$829$	−12.1016	−0.420307	−0.210153	−	0.977668i	$-0.567396\pi$
$829$	−12.1016	−0.420307	−0.210153	+	0.977668i	$0.567396\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−20.3317	−0.704451
$834$	0	0
$835$	43.0964	1.49141
$836$	0	0
$837$	0	0
$838$	0	0
$839$	51.4731	1.77705	0.888524	−	0.458829i	$-0.151731\pi$
$839$	51.4731	1.77705	0.888524	+	0.458829i	$0.151731\pi$
$840$	0	0
$841$	−5.71457	−0.197054
$842$	0	0
$843$	0	0
$844$	0	0
$845$	43.2613	1.48824
$846$	0	0
$847$	47.9463	1.64745
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.68646	−0.0578112
$852$	0	0
$853$	−42.6187	−1.45924	−0.729619	−	0.683854i	$-0.760303\pi$
$853$	−42.6187	−1.45924	−0.729619	+	0.683854i	$0.760303\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.45501	−0.118021	−0.0590105	−	0.998257i	$-0.518795\pi$
$857$	−3.45501	−0.118021	−0.0590105	+	0.998257i	$0.518795\pi$
$858$	0	0
$859$	17.1161	0.583994	0.291997	−	0.956419i	$-0.405680\pi$
$859$	17.1161	0.583994	0.291997	+	0.956419i	$0.405680\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13.0130	0.442969	0.221485	−	0.975164i	$-0.428910\pi$
$863$	13.0130	0.442969	0.221485	+	0.975164i	$0.428910\pi$
$864$	0	0
$865$	−33.8599	−1.15127
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−41.9604	−1.42341
$870$	0	0
$871$	22.8126	0.772974
$872$	0	0
$873$	0	0
$874$	0	0
$875$	56.3697	1.90564
$876$	0	0
$877$	0.908097	0.0306642	0.0153321	−	0.999882i	$-0.495119\pi$
$877$	0.908097	0.0306642	0.0153321	+	0.999882i	$0.495119\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−29.1332	−0.981523	−0.490761	−	0.871294i	$-0.663281\pi$
$881$	−29.1332	−0.981523	−0.490761	+	0.871294i	$0.663281\pi$
$882$	0	0
$883$	−31.4780	−1.05932	−0.529660	−	0.848210i	$-0.677681\pi$
$883$	−31.4780	−1.05932	−0.529660	+	0.848210i	$0.677681\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.0933	−1.24547	−0.622736	−	0.782432i	$-0.713979\pi$
$887$	−37.0933	−1.24547	−0.622736	+	0.782432i	$0.713979\pi$
$888$	0	0
$889$	−31.6153	−1.06034
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.555087	0.0185753
$894$	0	0
$895$	−7.37658	−0.246572
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.90447	0.263629
$900$	0	0
$901$	−13.6423	−0.454492
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−46.3523	−1.54080
$906$	0	0
$907$	−24.0295	−0.797886	−0.398943	−	0.916976i	$-0.630623\pi$
$907$	−24.0295	−0.797886	−0.398943	+	0.916976i	$0.630623\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45.7644	1.51624	0.758121	−	0.652114i	$-0.226118\pi$
$911$	45.7644	1.51624	0.758121	+	0.652114i	$0.226118\pi$
$912$	0	0
$913$	−73.8350	−2.44358
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21.7209	−0.717288
$918$	0	0
$919$	−2.91536	−0.0961688	−0.0480844	−	0.998843i	$-0.515312\pi$
$919$	−2.91536	−0.0961688	−0.0480844	+	0.998843i	$0.515312\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	55.4210	1.82421
$924$	0	0
$925$	46.1711	1.51810
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.4417	0.473815	0.236908	−	0.971532i	$-0.423866\pi$
$929$	14.4417	0.473815	0.236908	+	0.971532i	$0.423866\pi$
$930$	0	0
$931$	−3.96362	−0.129902
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−133.899	−4.37896
$936$	0	0
$937$	39.3960	1.28701	0.643505	−	0.765442i	$-0.277479\pi$
$937$	39.3960	1.28701	0.643505	+	0.765442i	$0.277479\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−37.5954	−1.22558	−0.612788	−	0.790247i	$-0.709952\pi$
$941$	−37.5954	−1.22558	−0.612788	+	0.790247i	$0.709952\pi$
$942$	0	0
$943$	−2.45180	−0.0798415
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.3522	0.693854	0.346927	−	0.937892i	$-0.387225\pi$
$947$	21.3522	0.693854	0.346927	+	0.937892i	$0.387225\pi$
$948$	0	0
$949$	−0.0312023	−0.00101287
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−37.5726	−1.21709	−0.608547	−	0.793518i	$-0.708247\pi$
$953$	−37.5726	−1.21709	−0.608547	+	0.793518i	$0.708247\pi$
$954$	0	0
$955$	55.6362	1.80035
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14.6696	−0.473707
$960$	0	0
$961$	−28.3167	−0.913444
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.19182	0.199322
$966$	0	0
$967$	11.5676	0.371990	0.185995	−	0.982551i	$-0.440449\pi$
$967$	11.5676	0.371990	0.185995	+	0.982551i	$0.440449\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	60.2009	1.93194	0.965970	−	0.258656i	$-0.0832796\pi$
$971$	60.2009	1.93194	0.965970	+	0.258656i	$0.0832796\pi$
$972$	0	0
$973$	19.7414	0.632879
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41.2738	−1.32047	−0.660233	−	0.751061i	$-0.729542\pi$
$977$	−41.2738	−1.32047	−0.660233	+	0.751061i	$0.729542\pi$
$978$	0	0
$979$	−45.4229	−1.45172
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16.5610	−0.528214	−0.264107	−	0.964493i	$-0.585077\pi$
$983$	−16.5610	−0.528214	−0.264107	+	0.964493i	$0.585077\pi$
$984$	0	0
$985$	−33.0050	−1.05163
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.502025	−0.0159635
$990$	0	0
$991$	35.6243	1.13164	0.565821	−	0.824528i	$-0.308559\pi$
$991$	35.6243	1.13164	0.565821	+	0.824528i	$0.308559\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−49.6557	−1.57419
$996$	0	0
$997$	41.0563	1.30027	0.650133	−	0.759821i	$-0.274713\pi$
$997$	41.0563	1.30027	0.650133	+	0.759821i	$0.274713\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.a.bs.1.4		4
3.2	odd	2		608.2.a.j.1.3	yes	4
4.3	odd	2		5472.2.a.bt.1.4		4
12.11	even	2		608.2.a.i.1.1	✓	4
24.5	odd	2		1216.2.a.w.1.2		4
24.11	even	2		1216.2.a.x.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.2.a.i.1.1	✓	4	12.11	even	2
608.2.a.j.1.3	yes	4	3.2	odd	2
1216.2.a.w.1.2		4	24.5	odd	2
1216.2.a.x.1.4		4	24.11	even	2
5472.2.a.bs.1.4		4	1.1	even	1	trivial
5472.2.a.bt.1.4		4	4.3	odd	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:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.a (trivial)

\(f(q)\)	\(=\)	\(q+4.20608 q^{5} +1.74252 q^{7} +O(q^{10})\)	\(q+4.20608 q^{5} +1.74252 q^{7} -6.20608 q^{11} +4.82550 q^{13} +5.12957 q^{17} +1.00000 q^{19} -0.463560 q^{23} +12.6911 q^{25} -4.82550 q^{29} -1.63806 q^{31} +7.32919 q^{35} +3.63806 q^{37} +5.28906 q^{41} +1.08298 q^{43} +0.555087 q^{47} -3.96362 q^{49} -2.65955 q^{53} -26.1033 q^{55} +1.85061 q^{59} -9.32919 q^{61} +20.2964 q^{65} +4.72751 q^{67} +11.4850 q^{71} -0.00646614 q^{73} -10.8142 q^{77} +6.76117 q^{79} +11.8972 q^{83} +21.5754 q^{85} +7.31909 q^{89} +8.40853 q^{91} +4.20608 q^{95} -10.4122 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{5} - q^{7}+O(q^{10}) \) 4 * q - q^5 - q^7	\( 4 q - q^{5} - q^{7} - 7 q^{11} + 10 q^{13} - 5 q^{17} + 4 q^{19} + 8 q^{23} + 17 q^{25} - 10 q^{29} - 6 q^{31} - 5 q^{35} + 14 q^{37} + 2 q^{41} + 3 q^{43} + 3 q^{47} + 7 q^{49} - 4 q^{53} - 35 q^{55} - 20 q^{59} - 3 q^{61} + 12 q^{65} + 8 q^{67} + 30 q^{71} + 9 q^{73} + 7 q^{77} + 10 q^{79} - 4 q^{83} + 19 q^{85} + 16 q^{89} + 10 q^{91} - q^{95} - 6 q^{97}+O(q^{100}) \) 4 * q - q^5 - q^7 - 7 * q^11 + 10 * q^13 - 5 * q^17 + 4 * q^19 + 8 * q^23 + 17 * q^25 - 10 * q^29 - 6 * q^31 - 5 * q^35 + 14 * q^37 + 2 * q^41 + 3 * q^43 + 3 * q^47 + 7 * q^49 - 4 * q^53 - 35 * q^55 - 20 * q^59 - 3 * q^61 + 12 * q^65 + 8 * q^67 + 30 * q^71 + 9 * q^73 + 7 * q^77 + 10 * q^79 - 4 * q^83 + 19 * q^85 + 16 * q^89 + 10 * q^91 - q^95 - 6 * q^97

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