LMFDB - Embedded newform 548.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [548,2,Mod(1,548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(548, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("548.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 548 = 2^{2} \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	548.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:	$4.37580203077$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1957.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} - x + 1 $ x^4 - 4*x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.693822$ of defining polynomial
Character	$\chi$	$=$	548.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.26608 q^{3} -1.69382 q^{5} +2.95990 q^{7} +7.66727 q^{9} +O(q^{10})$	$q-3.26608 q^{3} -1.69382 q^{5} +2.95990 q^{7} +7.66727 q^{9} +3.09087 q^{11} -6.18876 q^{13} +5.53216 q^{15} -2.13097 q^{17} -0.198886 q^{19} -9.66727 q^{21} +1.79172 q^{23} -2.13097 q^{25} -15.2437 q^{27} +0.158787 q^{29} -2.15879 q^{31} -10.0950 q^{33} -5.01355 q^{35} -11.7683 q^{37} +20.2130 q^{39} -8.77456 q^{41} -6.47541 q^{43} -12.9870 q^{45} +1.53288 q^{47} +1.76102 q^{49} +6.95990 q^{51} +10.0477 q^{53} -5.23538 q^{55} +0.649576 q^{57} +7.30960 q^{59} +9.68082 q^{61} +22.6944 q^{63} +10.4827 q^{65} -13.5015 q^{67} -5.85188 q^{69} -8.54045 q^{71} -4.06067 q^{73} +6.95990 q^{75} +9.14866 q^{77} +0.993701 q^{79} +26.7852 q^{81} +8.19291 q^{83} +3.60948 q^{85} -0.518610 q^{87} -17.2850 q^{89} -18.3181 q^{91} +7.05077 q^{93} +0.336877 q^{95} -11.5837 q^{97} +23.6985 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} - 4 q^{5} - q^{7} + q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 - 4 * q^5 - q^7 + q^9	$ 4 q - 3 q^{3} - 4 q^{5} - q^{7} + q^{9} - q^{11} - 14 q^{13} + 2 q^{15} - 8 q^{17} - 6 q^{19} - 9 q^{21} + 7 q^{23} - 8 q^{25} - 15 q^{27} - 7 q^{29} - q^{31} - 18 q^{33} - 6 q^{35} - 20 q^{37} + 17 q^{39} - 3 q^{41} - 7 q^{43} - 11 q^{45} + 7 q^{47} - 11 q^{49} + 15 q^{51} + 14 q^{53} + 7 q^{55} - 4 q^{57} + 2 q^{59} - 5 q^{61} + 33 q^{63} + 14 q^{65} - 16 q^{67} + 8 q^{69} + 4 q^{71} - 31 q^{73} + 15 q^{75} + 13 q^{77} + 9 q^{79} + 36 q^{81} + 13 q^{83} - 5 q^{85} + 4 q^{87} - 2 q^{89} - 3 q^{91} + 2 q^{93} + 22 q^{95} - 21 q^{97} + 28 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 - 4 * q^5 - q^7 + q^9 - q^11 - 14 * q^13 + 2 * q^15 - 8 * q^17 - 6 * q^19 - 9 * q^21 + 7 * q^23 - 8 * q^25 - 15 * q^27 - 7 * q^29 - q^31 - 18 * q^33 - 6 * q^35 - 20 * q^37 + 17 * q^39 - 3 * q^41 - 7 * q^43 - 11 * q^45 + 7 * q^47 - 11 * q^49 + 15 * q^51 + 14 * q^53 + 7 * q^55 - 4 * q^57 + 2 * q^59 - 5 * q^61 + 33 * q^63 + 14 * q^65 - 16 * q^67 + 8 * q^69 + 4 * q^71 - 31 * q^73 + 15 * q^75 + 13 * q^77 + 9 * q^79 + 36 * q^81 + 13 * q^83 - 5 * q^85 + 4 * q^87 - 2 * q^89 - 3 * q^91 + 2 * q^93 + 22 * q^95 - 21 * q^97 + 28 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.26608	−1.88567	−0.942836	−	0.333258i	$-0.891852\pi$
$3$	−3.26608	−1.88567	−0.942836	+	0.333258i	$0.891852\pi$
$4$	0	0
$5$	−1.69382	−0.757500	−0.378750	−	0.925499i	$-0.623646\pi$
$5$	−1.69382	−0.757500	−0.378750	+	0.925499i	$0.623646\pi$
$6$	0	0
$7$	2.95990	1.11874	0.559369	−	0.828919i	$-0.311044\pi$
$7$	2.95990	1.11874	0.559369	+	0.828919i	$0.311044\pi$
$8$	0	0
$9$	7.66727	2.55576
$10$	0	0
$11$	3.09087	0.931931	0.465966	−	0.884803i	$-0.345707\pi$
$11$	3.09087	0.931931	0.465966	+	0.884803i	$0.345707\pi$
$12$	0	0
$13$	−6.18876	−1.71645	−0.858227	−	0.513271i	$-0.828434\pi$
$13$	−6.18876	−1.71645	−0.858227	+	0.513271i	$0.828434\pi$
$14$	0	0
$15$	5.53216	1.42840
$16$	0	0
$17$	−2.13097	−0.516835	−0.258418	−	0.966033i	$-0.583201\pi$
$17$	−2.13097	−0.516835	−0.258418	+	0.966033i	$0.583201\pi$
$18$	0	0
$19$	−0.198886	−0.0456275	−0.0228137	−	0.999740i	$-0.507262\pi$
$19$	−0.198886	−0.0456275	−0.0228137	+	0.999740i	$0.507262\pi$
$20$	0	0
$21$	−9.66727	−2.10957
$22$	0	0
$23$	1.79172	0.373598	0.186799	−	0.982398i	$-0.440189\pi$
$23$	1.79172	0.373598	0.186799	+	0.982398i	$0.440189\pi$
$24$	0	0
$25$	−2.13097	−0.426193
$26$	0	0
$27$	−15.2437	−2.93365
$28$	0	0
$29$	0.158787	0.0294860	0.0147430	−	0.999891i	$-0.495307\pi$
$29$	0.158787	0.0294860	0.0147430	+	0.999891i	$0.495307\pi$
$30$	0	0
$31$	−2.15879	−0.387730	−0.193865	−	0.981028i	$-0.562102\pi$
$31$	−2.15879	−0.387730	−0.193865	+	0.981028i	$0.562102\pi$
$32$	0	0
$33$	−10.0950	−1.75732
$34$	0	0
$35$	−5.01355	−0.847444
$36$	0	0
$37$	−11.7683	−1.93469	−0.967345	−	0.253462i	$-0.918431\pi$
$37$	−11.7683	−1.93469	−0.967345	+	0.253462i	$0.918431\pi$
$38$	0	0
$39$	20.2130	3.23667
$40$	0	0
$41$	−8.77456	−1.37036	−0.685178	−	0.728375i	$-0.740276\pi$
$41$	−8.77456	−1.37036	−0.685178	+	0.728375i	$0.740276\pi$
$42$	0	0
$43$	−6.47541	−0.987491	−0.493745	−	0.869606i	$-0.664373\pi$
$43$	−6.47541	−0.987491	−0.493745	+	0.869606i	$0.664373\pi$
$44$	0	0
$45$	−12.9870	−1.93599
$46$	0	0
$47$	1.53288	0.223594	0.111797	−	0.993731i	$-0.464339\pi$
$47$	1.53288	0.223594	0.111797	+	0.993731i	$0.464339\pi$
$48$	0	0
$49$	1.76102	0.251574
$50$	0	0
$51$	6.95990	0.974581
$52$	0	0
$53$	10.0477	1.38015	0.690077	−	0.723736i	$-0.257577\pi$
$53$	10.0477	1.38015	0.690077	+	0.723736i	$0.257577\pi$
$54$	0	0
$55$	−5.23538	−0.705938
$56$	0	0
$57$	0.649576	0.0860384
$58$	0	0
$59$	7.30960	0.951629	0.475814	−	0.879546i	$-0.342153\pi$
$59$	7.30960	0.951629	0.475814	+	0.879546i	$0.342153\pi$
$60$	0	0
$61$	9.68082	1.23950	0.619751	−	0.784799i	$-0.287234\pi$
$61$	9.68082	1.23950	0.619751	+	0.784799i	$0.287234\pi$
$62$	0	0
$63$	22.6944	2.85922
$64$	0	0
$65$	10.4827	1.30021
$66$	0	0
$67$	−13.5015	−1.64947	−0.824733	−	0.565523i	$-0.808675\pi$
$67$	−13.5015	−1.64947	−0.824733	+	0.565523i	$0.808675\pi$
$68$	0	0
$69$	−5.85188	−0.704484
$70$	0	0
$71$	−8.54045	−1.01357	−0.506783	−	0.862074i	$-0.669165\pi$
$71$	−8.54045	−1.01357	−0.506783	+	0.862074i	$0.669165\pi$
$72$	0	0
$73$	−4.06067	−0.475266	−0.237633	−	0.971355i	$-0.576372\pi$
$73$	−4.06067	−0.475266	−0.237633	+	0.971355i	$0.576372\pi$
$74$	0	0
$75$	6.95990	0.803660
$76$	0	0
$77$	9.14866	1.04259
$78$	0	0
$79$	0.993701	0.111800	0.0559000	−	0.998436i	$-0.482197\pi$
$79$	0.993701	0.111800	0.0559000	+	0.998436i	$0.482197\pi$
$80$	0	0
$81$	26.7852	2.97614
$82$	0	0
$83$	8.19291	0.899288	0.449644	−	0.893208i	$-0.351551\pi$
$83$	8.19291	0.899288	0.449644	+	0.893208i	$0.351551\pi$
$84$	0	0
$85$	3.60948	0.391503
$86$	0	0
$87$	−0.518610	−0.0556009
$88$	0	0
$89$	−17.2850	−1.83221	−0.916106	−	0.400937i	$-0.868685\pi$
$89$	−17.2850	−1.83221	−0.916106	+	0.400937i	$0.868685\pi$
$90$	0	0
$91$	−18.3181	−1.92026
$92$	0	0
$93$	7.05077	0.731131
$94$	0	0
$95$	0.336877	0.0345628
$96$	0	0
$97$	−11.5837	−1.17614	−0.588071	−	0.808809i	$-0.700112\pi$
$97$	−11.5837	−1.17614	−0.588071	+	0.808809i	$0.700112\pi$
$98$	0	0
$99$	23.6985	2.38179
$100$	0	0
$101$	−12.2106	−1.21500	−0.607500	−	0.794320i	$-0.707828\pi$
$101$	−12.2106	−1.21500	−0.607500	+	0.794320i	$0.707828\pi$
$102$	0	0
$103$	−6.16221	−0.607180	−0.303590	−	0.952803i	$-0.598185\pi$
$103$	−6.16221	−0.607180	−0.303590	+	0.952803i	$0.598185\pi$
$104$	0	0
$105$	16.3746	1.59800
$106$	0	0
$107$	−5.20933	−0.503605	−0.251803	−	0.967779i	$-0.581023\pi$
$107$	−5.20933	−0.503605	−0.251803	+	0.967779i	$0.581023\pi$
$108$	0	0
$109$	−8.95702	−0.857927	−0.428964	−	0.903322i	$-0.641121\pi$
$109$	−8.95702	−0.857927	−0.428964	+	0.903322i	$0.641121\pi$
$110$	0	0
$111$	38.4361	3.64819
$112$	0	0
$113$	18.6366	1.75318	0.876591	−	0.481237i	$-0.159812\pi$
$113$	18.6366	1.75318	0.876591	+	0.481237i	$0.159812\pi$
$114$	0	0
$115$	−3.03485	−0.283001
$116$	0	0
$117$	−47.4509	−4.38684
$118$	0	0
$119$	−6.30745	−0.578203
$120$	0	0
$121$	−1.44654	−0.131504
$122$	0	0
$123$	28.6584	2.58404
$124$	0	0
$125$	12.0786	1.08034
$126$	0	0
$127$	14.7209	1.30627	0.653135	−	0.757241i	$-0.273453\pi$
$127$	14.7209	1.30627	0.653135	+	0.757241i	$0.273453\pi$
$128$	0	0
$129$	21.1492	1.86208
$130$	0	0
$131$	−11.2161	−0.979953	−0.489977	−	0.871736i	$-0.662995\pi$
$131$	−11.2161	−0.979953	−0.489977	+	0.871736i	$0.662995\pi$
$132$	0	0
$133$	−0.588682	−0.0510452
$134$	0	0
$135$	25.8201	2.22224
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	19.4337	1.64835	0.824173	−	0.566339i	$-0.191641\pi$
$139$	19.4337	1.64835	0.824173	+	0.566339i	$0.191641\pi$
$140$	0	0
$141$	−5.00652	−0.421625
$142$	0	0
$143$	−19.1286	−1.59962
$144$	0	0
$145$	−0.268957	−0.0223356
$146$	0	0
$147$	−5.75162	−0.474385
$148$	0	0
$149$	8.10751	0.664193	0.332097	−	0.943245i	$-0.392244\pi$
$149$	8.10751	0.664193	0.332097	+	0.943245i	$0.392244\pi$
$150$	0	0
$151$	−16.2986	−1.32636	−0.663180	−	0.748460i	$-0.730794\pi$
$151$	−16.2986	−1.32636	−0.663180	+	0.748460i	$0.730794\pi$
$152$	0	0
$153$	−16.3387	−1.32090
$154$	0	0
$155$	3.65660	0.293705
$156$	0	0
$157$	1.86091	0.148516	0.0742582	−	0.997239i	$-0.476341\pi$
$157$	1.86091	0.148516	0.0742582	+	0.997239i	$0.476341\pi$
$158$	0	0
$159$	−32.8165	−2.60252
$160$	0	0
$161$	5.30330	0.417959
$162$	0	0
$163$	−14.6088	−1.14425	−0.572123	−	0.820168i	$-0.693880\pi$
$163$	−14.6088	−1.14425	−0.572123	+	0.820168i	$0.693880\pi$
$164$	0	0
$165$	17.0992	1.33117
$166$	0	0
$167$	−11.0385	−0.854184	−0.427092	−	0.904208i	$-0.640462\pi$
$167$	−11.0385	−0.854184	−0.427092	+	0.904208i	$0.640462\pi$
$168$	0	0
$169$	25.3007	1.94621
$170$	0	0
$171$	−1.52491	−0.116613
$172$	0	0
$173$	9.62645	0.731885	0.365943	−	0.930637i	$-0.380747\pi$
$173$	9.62645	0.731885	0.365943	+	0.930637i	$0.380747\pi$
$174$	0	0
$175$	−6.30745	−0.476798
$176$	0	0
$177$	−23.8737	−1.79446
$178$	0	0
$179$	12.2697	0.917082	0.458541	−	0.888673i	$-0.348372\pi$
$179$	12.2697	0.917082	0.458541	+	0.888673i	$0.348372\pi$
$180$	0	0
$181$	−0.601181	−0.0446855	−0.0223427	−	0.999750i	$-0.507113\pi$
$181$	−0.601181	−0.0446855	−0.0223427	+	0.999750i	$0.507113\pi$
$182$	0	0
$183$	−31.6183	−2.33729
$184$	0	0
$185$	19.9333	1.46553
$186$	0	0
$187$	−6.58653	−0.481655
$188$	0	0
$189$	−45.1198	−3.28198
$190$	0	0
$191$	−14.2890	−1.03392	−0.516959	−	0.856010i	$-0.672936\pi$
$191$	−14.2890	−1.03392	−0.516959	+	0.856010i	$0.672936\pi$
$192$	0	0
$193$	−8.86593	−0.638184	−0.319092	−	0.947724i	$-0.603378\pi$
$193$	−8.86593	−0.638184	−0.319092	+	0.947724i	$0.603378\pi$
$194$	0	0
$195$	−34.2372	−2.45178
$196$	0	0
$197$	−25.7287	−1.83309	−0.916547	−	0.399926i	$-0.869036\pi$
$197$	−25.7287	−1.83309	−0.916547	+	0.399926i	$0.869036\pi$
$198$	0	0
$199$	23.2614	1.64895	0.824477	−	0.565895i	$-0.191469\pi$
$199$	23.2614	1.64895	0.824477	+	0.565895i	$0.191469\pi$
$200$	0	0
$201$	44.0968	3.11035
$202$	0	0
$203$	0.469993	0.0329871
$204$	0	0
$205$	14.8626	1.03805
$206$	0	0
$207$	13.7376	0.954827
$208$	0	0
$209$	−0.614729	−0.0425217
$210$	0	0
$211$	3.97633	0.273742	0.136871	−	0.990589i	$-0.456295\pi$
$211$	3.97633	0.273742	0.136871	+	0.990589i	$0.456295\pi$
$212$	0	0
$213$	27.8938	1.91125
$214$	0	0
$215$	10.9682	0.748025
$216$	0	0
$217$	−6.38980	−0.433768
$218$	0	0
$219$	13.2625	0.896195
$220$	0	0
$221$	13.1880	0.887123
$222$	0	0
$223$	18.6564	1.24933	0.624663	−	0.780894i	$-0.285236\pi$
$223$	18.6564	1.24933	0.624663	+	0.780894i	$0.285236\pi$
$224$	0	0
$225$	−16.3387	−1.08925
$226$	0	0
$227$	−7.62536	−0.506113	−0.253056	−	0.967452i	$-0.581436\pi$
$227$	−7.62536	−0.506113	−0.253056	+	0.967452i	$0.581436\pi$
$228$	0	0
$229$	−3.29783	−0.217926	−0.108963	−	0.994046i	$-0.534753\pi$
$229$	−3.29783	−0.217926	−0.108963	+	0.994046i	$0.534753\pi$
$230$	0	0
$231$	−29.8802	−1.96598
$232$	0	0
$233$	−19.1666	−1.25564	−0.627822	−	0.778357i	$-0.716053\pi$
$233$	−19.1666	−1.25564	−0.627822	+	0.778357i	$0.716053\pi$
$234$	0	0
$235$	−2.59643	−0.169373
$236$	0	0
$237$	−3.24551	−0.210818
$238$	0	0
$239$	19.0036	1.22924	0.614622	−	0.788822i	$-0.289309\pi$
$239$	19.0036	1.22924	0.614622	+	0.788822i	$0.289309\pi$
$240$	0	0
$241$	23.3546	1.50440	0.752201	−	0.658933i	$-0.228992\pi$
$241$	23.3546	1.50440	0.752201	+	0.658933i	$0.228992\pi$
$242$	0	0
$243$	−41.7517	−2.67837
$244$	0	0
$245$	−2.98285	−0.190567
$246$	0	0
$247$	1.23085	0.0783174
$248$	0	0
$249$	−26.7587	−1.69576
$250$	0	0
$251$	−13.7686	−0.869065	−0.434533	−	0.900656i	$-0.643086\pi$
$251$	−13.7686	−0.869065	−0.434533	+	0.900656i	$0.643086\pi$
$252$	0	0
$253$	5.53795	0.348168
$254$	0	0
$255$	−11.7888	−0.738246
$256$	0	0
$257$	−9.94221	−0.620178	−0.310089	−	0.950708i	$-0.600359\pi$
$257$	−9.94221	−0.620178	−0.310089	+	0.950708i	$0.600359\pi$
$258$	0	0
$259$	−34.8329	−2.16441
$260$	0	0
$261$	1.21746	0.0753590
$262$	0	0
$263$	4.35910	0.268793	0.134397	−	0.990928i	$-0.457090\pi$
$263$	4.35910	0.268793	0.134397	+	0.990928i	$0.457090\pi$
$264$	0	0
$265$	−17.0190	−1.04547
$266$	0	0
$267$	56.4543	3.45495
$268$	0	0
$269$	25.1465	1.53321	0.766606	−	0.642118i	$-0.221944\pi$
$269$	25.1465	1.53321	0.766606	+	0.642118i	$0.221944\pi$
$270$	0	0
$271$	−13.8333	−0.840314	−0.420157	−	0.907452i	$-0.638025\pi$
$271$	−13.8333	−0.840314	−0.420157	+	0.907452i	$0.638025\pi$
$272$	0	0
$273$	59.8284	3.62098
$274$	0	0
$275$	−6.58653	−0.397183
$276$	0	0
$277$	3.38581	0.203434	0.101717	−	0.994813i	$-0.467566\pi$
$277$	3.38581	0.203434	0.101717	+	0.994813i	$0.467566\pi$
$278$	0	0
$279$	−16.5520	−0.990943
$280$	0	0
$281$	18.7570	1.11895	0.559475	−	0.828847i	$-0.311003\pi$
$281$	18.7570	1.11895	0.559475	+	0.828847i	$0.311003\pi$
$282$	0	0
$283$	16.0130	0.951873	0.475937	−	0.879480i	$-0.342109\pi$
$283$	16.0130	0.951873	0.475937	+	0.879480i	$0.342109\pi$
$284$	0	0
$285$	−1.10027	−0.0651741
$286$	0	0
$287$	−25.9718	−1.53307
$288$	0	0
$289$	−12.4590	−0.732882
$290$	0	0
$291$	37.8331	2.21782
$292$	0	0
$293$	7.26051	0.424163	0.212082	−	0.977252i	$-0.431976\pi$
$293$	7.26051	0.424163	0.212082	+	0.977252i	$0.431976\pi$
$294$	0	0
$295$	−12.3812	−0.720859
$296$	0	0
$297$	−47.1162	−2.73396
$298$	0	0
$299$	−11.0885	−0.641264
$300$	0	0
$301$	−19.1666	−1.10474
$302$	0	0
$303$	39.8808	2.29109
$304$	0	0
$305$	−16.3976	−0.938923
$306$	0	0
$307$	10.5936	0.604608	0.302304	−	0.953212i	$-0.402244\pi$
$307$	10.5936	0.604608	0.302304	+	0.953212i	$0.402244\pi$
$308$	0	0
$309$	20.1263	1.14494
$310$	0	0
$311$	2.72060	0.154271	0.0771354	−	0.997021i	$-0.475423\pi$
$311$	2.72060	0.154271	0.0771354	+	0.997021i	$0.475423\pi$
$312$	0	0
$313$	−13.8321	−0.781838	−0.390919	−	0.920425i	$-0.627843\pi$
$313$	−13.8321	−0.781838	−0.390919	+	0.920425i	$0.627843\pi$
$314$	0	0
$315$	−38.4402	−2.16586
$316$	0	0
$317$	32.3474	1.81681	0.908404	−	0.418093i	$-0.137301\pi$
$317$	32.3474	1.81681	0.908404	+	0.418093i	$0.137301\pi$
$318$	0	0
$319$	0.490789	0.0274789
$320$	0	0
$321$	17.0141	0.949634
$322$	0	0
$323$	0.423818	0.0235819
$324$	0	0
$325$	13.1880	0.731540
$326$	0	0
$327$	29.2543	1.61777
$328$	0	0
$329$	4.53719	0.250143
$330$	0	0
$331$	6.04242	0.332121	0.166061	−	0.986116i	$-0.446895\pi$
$331$	6.04242	0.332121	0.166061	+	0.986116i	$0.446895\pi$
$332$	0	0
$333$	−90.2305	−4.94460
$334$	0	0
$335$	22.8691	1.24947
$336$	0	0
$337$	−3.72794	−0.203074	−0.101537	−	0.994832i	$-0.532376\pi$
$337$	−3.72794	−0.203074	−0.101537	+	0.994832i	$0.532376\pi$
$338$	0	0
$339$	−60.8685	−3.30592
$340$	0	0
$341$	−6.67252	−0.361337
$342$	0	0
$343$	−15.5069	−0.837293
$344$	0	0
$345$	9.91205	0.533647
$346$	0	0
$347$	21.4584	1.15195	0.575974	−	0.817468i	$-0.304623\pi$
$347$	21.4584	1.15195	0.575974	+	0.817468i	$0.304623\pi$
$348$	0	0
$349$	−12.5345	−0.670958	−0.335479	−	0.942048i	$-0.608898\pi$
$349$	−12.5345	−0.670958	−0.335479	+	0.942048i	$0.608898\pi$
$350$	0	0
$351$	94.3394	5.03547
$352$	0	0
$353$	10.0811	0.536561	0.268280	−	0.963341i	$-0.413545\pi$
$353$	10.0811	0.536561	0.268280	+	0.963341i	$0.413545\pi$
$354$	0	0
$355$	14.4660	0.767776
$356$	0	0
$357$	20.6006	1.09030
$358$	0	0
$359$	−14.0846	−0.743355	−0.371678	−	0.928362i	$-0.621217\pi$
$359$	−14.0846	−0.743355	−0.371678	+	0.928362i	$0.621217\pi$
$360$	0	0
$361$	−18.9604	−0.997918
$362$	0	0
$363$	4.72452	0.247973
$364$	0	0
$365$	6.87806	0.360014
$366$	0	0
$367$	30.7370	1.60446	0.802229	−	0.597016i	$-0.203647\pi$
$367$	30.7370	1.60446	0.802229	+	0.597016i	$0.203647\pi$
$368$	0	0
$369$	−67.2770	−3.50230
$370$	0	0
$371$	29.7401	1.54403
$372$	0	0
$373$	−33.5863	−1.73904	−0.869518	−	0.493902i	$-0.835570\pi$
$373$	−33.5863	−1.73904	−0.869518	+	0.493902i	$0.835570\pi$
$374$	0	0
$375$	−39.4496	−2.03717
$376$	0	0
$377$	−0.982694	−0.0506113
$378$	0	0
$379$	−27.2200	−1.39820	−0.699099	−	0.715025i	$-0.746415\pi$
$379$	−27.2200	−1.39820	−0.699099	+	0.715025i	$0.746415\pi$
$380$	0	0
$381$	−48.0797	−2.46320
$382$	0	0
$383$	28.0457	1.43307	0.716535	−	0.697551i	$-0.245727\pi$
$383$	28.0457	1.43307	0.716535	+	0.697551i	$0.245727\pi$
$384$	0	0
$385$	−15.4962	−0.789760
$386$	0	0
$387$	−49.6487	−2.52379
$388$	0	0
$389$	18.6418	0.945178	0.472589	−	0.881283i	$-0.343320\pi$
$389$	18.6418	0.945178	0.472589	+	0.881283i	$0.343320\pi$
$390$	0	0
$391$	−3.81808	−0.193089
$392$	0	0
$393$	36.6326	1.84787
$394$	0	0
$395$	−1.68315	−0.0846886
$396$	0	0
$397$	−17.4158	−0.874073	−0.437037	−	0.899444i	$-0.643972\pi$
$397$	−17.4158	−0.874073	−0.437037	+	0.899444i	$0.643972\pi$
$398$	0	0
$399$	1.92268	0.0962544
$400$	0	0
$401$	−12.8907	−0.643731	−0.321865	−	0.946785i	$-0.604310\pi$
$401$	−12.8907	−0.643731	−0.321865	+	0.946785i	$0.604310\pi$
$402$	0	0
$403$	13.3602	0.665520
$404$	0	0
$405$	−45.3694	−2.25443
$406$	0	0
$407$	−36.3741	−1.80300
$408$	0	0
$409$	2.76317	0.136630	0.0683149	−	0.997664i	$-0.478238\pi$
$409$	2.76317	0.136630	0.0683149	+	0.997664i	$0.478238\pi$
$410$	0	0
$411$	−3.26608	−0.161104
$412$	0	0
$413$	21.6357	1.06462
$414$	0	0
$415$	−13.8773	−0.681211
$416$	0	0
$417$	−63.4720	−3.10824
$418$	0	0
$419$	18.3869	0.898260	0.449130	−	0.893466i	$-0.351734\pi$
$419$	18.3869	0.898260	0.449130	+	0.893466i	$0.351734\pi$
$420$	0	0
$421$	9.36435	0.456390	0.228195	−	0.973615i	$-0.426718\pi$
$421$	9.36435	0.456390	0.228195	+	0.973615i	$0.426718\pi$
$422$	0	0
$423$	11.7530	0.571452
$424$	0	0
$425$	4.54101	0.220272
$426$	0	0
$427$	28.6543	1.38668
$428$	0	0
$429$	62.4756	3.01635
$430$	0	0
$431$	8.49709	0.409290	0.204645	−	0.978836i	$-0.434396\pi$
$431$	8.49709	0.409290	0.204645	+	0.978836i	$0.434396\pi$
$432$	0	0
$433$	−6.07627	−0.292007	−0.146003	−	0.989284i	$-0.546641\pi$
$433$	−6.07627	−0.292007	−0.146003	+	0.989284i	$0.546641\pi$
$434$	0	0
$435$	0.878434	0.0421177
$436$	0	0
$437$	−0.356346	−0.0170464
$438$	0	0
$439$	−14.9801	−0.714963	−0.357481	−	0.933920i	$-0.616364\pi$
$439$	−14.9801	−0.714963	−0.357481	+	0.933920i	$0.616364\pi$
$440$	0	0
$441$	13.5022	0.642961
$442$	0	0
$443$	−21.4666	−1.01991	−0.509953	−	0.860202i	$-0.670337\pi$
$443$	−21.4666	−1.01991	−0.509953	+	0.860202i	$0.670337\pi$
$444$	0	0
$445$	29.2778	1.38790
$446$	0	0
$447$	−26.4798	−1.25245
$448$	0	0
$449$	30.4536	1.43719	0.718596	−	0.695428i	$-0.244785\pi$
$449$	30.4536	1.43719	0.718596	+	0.695428i	$0.244785\pi$
$450$	0	0
$451$	−27.1210	−1.27708
$452$	0	0
$453$	53.2325	2.50108
$454$	0	0
$455$	31.0276	1.45460
$456$	0	0
$457$	−23.9678	−1.12117	−0.560584	−	0.828097i	$-0.689423\pi$
$457$	−23.9678	−1.12117	−0.560584	+	0.828097i	$0.689423\pi$
$458$	0	0
$459$	32.4837	1.51621
$460$	0	0
$461$	−26.9386	−1.25465	−0.627327	−	0.778756i	$-0.715851\pi$
$461$	−26.9386	−1.25465	−0.627327	+	0.778756i	$0.715851\pi$
$462$	0	0
$463$	−1.67977	−0.0780656	−0.0390328	−	0.999238i	$-0.512428\pi$
$463$	−1.67977	−0.0780656	−0.0390328	+	0.999238i	$0.512428\pi$
$464$	0	0
$465$	−11.9427	−0.553832
$466$	0	0
$467$	12.5180	0.579266	0.289633	−	0.957138i	$-0.406467\pi$
$467$	12.5180	0.579266	0.289633	+	0.957138i	$0.406467\pi$
$468$	0	0
$469$	−39.9630	−1.84532
$470$	0	0
$471$	−6.07786	−0.280053
$472$	0	0
$473$	−20.0146	−0.920274
$474$	0	0
$475$	0.423818	0.0194461
$476$	0	0
$477$	77.0382	3.52734
$478$	0	0
$479$	22.3908	1.02306	0.511531	−	0.859265i	$-0.329079\pi$
$479$	22.3908	1.02306	0.511531	+	0.859265i	$0.329079\pi$
$480$	0	0
$481$	72.8310	3.32081
$482$	0	0
$483$	−17.3210	−0.788133
$484$	0	0
$485$	19.6207	0.890928
$486$	0	0
$487$	18.3400	0.831065	0.415533	−	0.909578i	$-0.363595\pi$
$487$	18.3400	0.831065	0.415533	+	0.909578i	$0.363595\pi$
$488$	0	0
$489$	47.7133	2.15767
$490$	0	0
$491$	−38.9328	−1.75701	−0.878507	−	0.477730i	$-0.841460\pi$
$491$	−38.9328	−1.75701	−0.878507	+	0.477730i	$0.841460\pi$
$492$	0	0
$493$	−0.338369	−0.0152394
$494$	0	0
$495$	−40.1411	−1.80421
$496$	0	0
$497$	−25.2789	−1.13391
$498$	0	0
$499$	−6.67180	−0.298671	−0.149335	−	0.988787i	$-0.547713\pi$
$499$	−6.67180	−0.298671	−0.149335	+	0.988787i	$0.547713\pi$
$500$	0	0
$501$	36.0526	1.61071
$502$	0	0
$503$	16.5775	0.739152	0.369576	−	0.929200i	$-0.379503\pi$
$503$	16.5775	0.739152	0.369576	+	0.929200i	$0.379503\pi$
$504$	0	0
$505$	20.6826	0.920363
$506$	0	0
$507$	−82.6342	−3.66991
$508$	0	0
$509$	9.88075	0.437957	0.218978	−	0.975730i	$-0.429728\pi$
$509$	9.88075	0.437957	0.218978	+	0.975730i	$0.429728\pi$
$510$	0	0
$511$	−12.0192	−0.531698
$512$	0	0
$513$	3.03175	0.133855
$514$	0	0
$515$	10.4377	0.459939
$516$	0	0
$517$	4.73794	0.208374
$518$	0	0
$519$	−31.4407	−1.38009
$520$	0	0
$521$	10.8548	0.475556	0.237778	−	0.971320i	$-0.423581\pi$
$521$	10.8548	0.475556	0.237778	+	0.971320i	$0.423581\pi$
$522$	0	0
$523$	18.4348	0.806098	0.403049	−	0.915178i	$-0.367950\pi$
$523$	18.4348	0.806098	0.403049	+	0.915178i	$0.367950\pi$
$524$	0	0
$525$	20.6006	0.899085
$526$	0	0
$527$	4.60030	0.200392
$528$	0	0
$529$	−19.7898	−0.860424
$530$	0	0
$531$	56.0447	2.43213
$532$	0	0
$533$	54.3037	2.35215
$534$	0	0
$535$	8.82368	0.381481
$536$	0	0
$537$	−40.0739	−1.72932
$538$	0	0
$539$	5.44307	0.234449
$540$	0	0
$541$	25.6738	1.10380	0.551901	−	0.833909i	$-0.313903\pi$
$541$	25.6738	1.10380	0.551901	+	0.833909i	$0.313903\pi$
$542$	0	0
$543$	1.96351	0.0842621
$544$	0	0
$545$	15.1716	0.649880
$546$	0	0
$547$	20.6704	0.883801	0.441901	−	0.897064i	$-0.354304\pi$
$547$	20.6704	0.883801	0.441901	+	0.897064i	$0.354304\pi$
$548$	0	0
$549$	74.2255	3.16787
$550$	0	0
$551$	−0.0315804	−0.00134537
$552$	0	0
$553$	2.94126	0.125075
$554$	0	0
$555$	−65.1039	−2.76351
$556$	0	0
$557$	0.853713	0.0361730	0.0180865	−	0.999836i	$-0.494243\pi$
$557$	0.853713	0.0361730	0.0180865	+	0.999836i	$0.494243\pi$
$558$	0	0
$559$	40.0748	1.69498
$560$	0	0
$561$	21.5121	0.908243
$562$	0	0
$563$	−29.9370	−1.26169	−0.630847	−	0.775907i	$-0.717292\pi$
$563$	−29.9370	−1.26169	−0.630847	+	0.775907i	$0.717292\pi$
$564$	0	0
$565$	−31.5670	−1.32804
$566$	0	0
$567$	79.2816	3.32952
$568$	0	0
$569$	3.84394	0.161146	0.0805731	−	0.996749i	$-0.474325\pi$
$569$	3.84394	0.161146	0.0805731	+	0.996749i	$0.474325\pi$
$570$	0	0
$571$	32.9640	1.37950	0.689751	−	0.724046i	$-0.257720\pi$
$571$	32.9640	1.37950	0.689751	+	0.724046i	$0.257720\pi$
$572$	0	0
$573$	46.6691	1.94963
$574$	0	0
$575$	−3.81808	−0.159225
$576$	0	0
$577$	−4.54279	−0.189119	−0.0945594	−	0.995519i	$-0.530144\pi$
$577$	−4.54279	−0.189119	−0.0945594	+	0.995519i	$0.530144\pi$
$578$	0	0
$579$	28.9568	1.20341
$580$	0	0
$581$	24.2502	1.00607
$582$	0	0
$583$	31.0560	1.28621
$584$	0	0
$585$	80.3734	3.32303
$586$	0	0
$587$	−45.5656	−1.88069	−0.940347	−	0.340216i	$-0.889500\pi$
$587$	−45.5656	−1.88069	−0.940347	+	0.340216i	$0.889500\pi$
$588$	0	0
$589$	0.429352	0.0176911
$590$	0	0
$591$	84.0320	3.45661
$592$	0	0
$593$	−34.0456	−1.39808	−0.699042	−	0.715081i	$-0.746390\pi$
$593$	−34.0456	−1.39808	−0.699042	+	0.715081i	$0.746390\pi$
$594$	0	0
$595$	10.6837	0.437989
$596$	0	0
$597$	−75.9735	−3.10939
$598$	0	0
$599$	−9.09163	−0.371474	−0.185737	−	0.982599i	$-0.559467\pi$
$599$	−9.09163	−0.371474	−0.185737	+	0.982599i	$0.559467\pi$
$600$	0	0
$601$	28.3494	1.15640	0.578198	−	0.815896i	$-0.303756\pi$
$601$	28.3494	1.15640	0.578198	+	0.815896i	$0.303756\pi$
$602$	0	0
$603$	−103.519	−4.21563
$604$	0	0
$605$	2.45019	0.0996142
$606$	0	0
$607$	30.1456	1.22357	0.611786	−	0.791023i	$-0.290451\pi$
$607$	30.1456	1.22357	0.611786	+	0.791023i	$0.290451\pi$
$608$	0	0
$609$	−1.53504	−0.0622028
$610$	0	0
$611$	−9.48665	−0.383789
$612$	0	0
$613$	−28.3420	−1.14472	−0.572360	−	0.820002i	$-0.693972\pi$
$613$	−28.3420	−1.14472	−0.572360	+	0.820002i	$0.693972\pi$
$614$	0	0
$615$	−48.5423	−1.95741
$616$	0	0
$617$	3.24696	0.130718	0.0653588	−	0.997862i	$-0.479181\pi$
$617$	3.24696	0.130718	0.0653588	+	0.997862i	$0.479181\pi$
$618$	0	0
$619$	−4.92473	−0.197942	−0.0989709	−	0.995090i	$-0.531555\pi$
$619$	−4.92473	−0.197942	−0.0989709	+	0.995090i	$0.531555\pi$
$620$	0	0
$621$	−27.3123	−1.09601
$622$	0	0
$623$	−51.1620	−2.04976
$624$	0	0
$625$	−9.80416	−0.392166
$626$	0	0
$627$	2.00775	0.0801819
$628$	0	0
$629$	25.0778	0.999916
$630$	0	0
$631$	7.15641	0.284892	0.142446	−	0.989803i	$-0.454503\pi$
$631$	7.15641	0.284892	0.142446	+	0.989803i	$0.454503\pi$
$632$	0	0
$633$	−12.9870	−0.516187
$634$	0	0
$635$	−24.9346	−0.989500
$636$	0	0
$637$	−10.8985	−0.431814
$638$	0	0
$639$	−65.4820	−2.59043
$640$	0	0
$641$	−32.4953	−1.28349	−0.641744	−	0.766919i	$-0.721789\pi$
$641$	−32.4953	−1.28349	−0.641744	+	0.766919i	$0.721789\pi$
$642$	0	0
$643$	−23.4249	−0.923787	−0.461893	−	0.886936i	$-0.652830\pi$
$643$	−23.4249	−0.923787	−0.461893	+	0.886936i	$0.652830\pi$
$644$	0	0
$645$	−35.8230	−1.41053
$646$	0	0
$647$	15.2261	0.598602	0.299301	−	0.954159i	$-0.403247\pi$
$647$	15.2261	0.598602	0.299301	+	0.954159i	$0.403247\pi$
$648$	0	0
$649$	22.5930	0.886853
$650$	0	0
$651$	20.8696	0.817943
$652$	0	0
$653$	−6.38697	−0.249942	−0.124971	−	0.992160i	$-0.539884\pi$
$653$	−6.38697	−0.249942	−0.124971	+	0.992160i	$0.539884\pi$
$654$	0	0
$655$	18.9980	0.742315
$656$	0	0
$657$	−31.1343	−1.21466
$658$	0	0
$659$	−19.6348	−0.764862	−0.382431	−	0.923984i	$-0.624913\pi$
$659$	−19.6348	−0.764862	−0.382431	+	0.923984i	$0.624913\pi$
$660$	0	0
$661$	−22.7884	−0.886367	−0.443184	−	0.896431i	$-0.646151\pi$
$661$	−22.7884	−0.886367	−0.443184	+	0.896431i	$0.646151\pi$
$662$	0	0
$663$	−43.0732	−1.67282
$664$	0	0
$665$	0.997122	0.0386667
$666$	0	0
$667$	0.284501	0.0110159
$668$	0	0
$669$	−60.9333	−2.35582
$670$	0	0
$671$	29.9221	1.15513
$672$	0	0
$673$	−8.60423	−0.331669	−0.165834	−	0.986154i	$-0.553032\pi$
$673$	−8.60423	−0.331669	−0.165834	+	0.986154i	$0.553032\pi$
$674$	0	0
$675$	32.4837	1.25030
$676$	0	0
$677$	20.0430	0.770314	0.385157	−	0.922851i	$-0.374147\pi$
$677$	20.0430	0.770314	0.385157	+	0.922851i	$0.374147\pi$
$678$	0	0
$679$	−34.2865	−1.31579
$680$	0	0
$681$	24.9050	0.954363
$682$	0	0
$683$	12.3666	0.473194	0.236597	−	0.971608i	$-0.423968\pi$
$683$	12.3666	0.473194	0.236597	+	0.971608i	$0.423968\pi$
$684$	0	0
$685$	−1.69382	−0.0647176
$686$	0	0
$687$	10.7710	0.410938
$688$	0	0
$689$	−62.1826	−2.36897
$690$	0	0
$691$	−18.9841	−0.722190	−0.361095	−	0.932529i	$-0.617597\pi$
$691$	−18.9841	−0.722190	−0.361095	+	0.932529i	$0.617597\pi$
$692$	0	0
$693$	70.1453	2.66460
$694$	0	0
$695$	−32.9172	−1.24862
$696$	0	0
$697$	18.6983	0.708248
$698$	0	0
$699$	62.5996	2.36773
$700$	0	0
$701$	−19.5026	−0.736604	−0.368302	−	0.929706i	$-0.620061\pi$
$701$	−19.5026	−0.736604	−0.368302	+	0.929706i	$0.620061\pi$
$702$	0	0
$703$	2.34054	0.0882750
$704$	0	0
$705$	8.48016	0.319381
$706$	0	0
$707$	−36.1422	−1.35927
$708$	0	0
$709$	30.2717	1.13688	0.568438	−	0.822726i	$-0.307548\pi$
$709$	30.2717	1.13688	0.568438	+	0.822726i	$0.307548\pi$
$710$	0	0
$711$	7.61897	0.285734
$712$	0	0
$713$	−3.86793	−0.144855
$714$	0	0
$715$	32.4005	1.21171
$716$	0	0
$717$	−62.0674	−2.31795
$718$	0	0
$719$	−24.6863	−0.920643	−0.460322	−	0.887752i	$-0.652266\pi$
$719$	−24.6863	−0.920643	−0.460322	+	0.887752i	$0.652266\pi$
$720$	0	0
$721$	−18.2395	−0.679275
$722$	0	0
$723$	−76.2780	−2.83681
$724$	0	0
$725$	−0.338369	−0.0125667
$726$	0	0
$727$	−9.79624	−0.363322	−0.181661	−	0.983361i	$-0.558147\pi$
$727$	−9.79624	−0.363322	−0.181661	+	0.983361i	$0.558147\pi$
$728$	0	0
$729$	56.0085	2.07439
$730$	0	0
$731$	13.7989	0.510370
$732$	0	0
$733$	32.6532	1.20607	0.603037	−	0.797713i	$-0.293957\pi$
$733$	32.6532	1.20607	0.603037	+	0.797713i	$0.293957\pi$
$734$	0	0
$735$	9.74222	0.359347
$736$	0	0
$737$	−41.7312	−1.53719
$738$	0	0
$739$	0.240797	0.00885788	0.00442894	−	0.999990i	$-0.498590\pi$
$739$	0.240797	0.00885788	0.00442894	+	0.999990i	$0.498590\pi$
$740$	0	0
$741$	−4.02007	−0.147681
$742$	0	0
$743$	−5.51805	−0.202438	−0.101219	−	0.994864i	$-0.532274\pi$
$743$	−5.51805	−0.202438	−0.101219	+	0.994864i	$0.532274\pi$
$744$	0	0
$745$	−13.7327	−0.503127
$746$	0	0
$747$	62.8172	2.29836
$748$	0	0
$749$	−15.4191	−0.563402
$750$	0	0
$751$	−27.1532	−0.990833	−0.495417	−	0.868656i	$-0.664985\pi$
$751$	−27.1532	−0.990833	−0.495417	+	0.868656i	$0.664985\pi$
$752$	0	0
$753$	44.9693	1.63877
$754$	0	0
$755$	27.6069	1.00472
$756$	0	0
$757$	42.5719	1.54730	0.773651	−	0.633612i	$-0.218428\pi$
$757$	42.5719	1.54730	0.773651	+	0.633612i	$0.218428\pi$
$758$	0	0
$759$	−18.0874	−0.656531
$760$	0	0
$761$	−16.2602	−0.589430	−0.294715	−	0.955585i	$-0.595225\pi$
$761$	−16.2602	−0.589430	−0.294715	+	0.955585i	$0.595225\pi$
$762$	0	0
$763$	−26.5119	−0.959795
$764$	0	0
$765$	27.6748	1.00059
$766$	0	0
$767$	−45.2373	−1.63343
$768$	0	0
$769$	−23.9488	−0.863615	−0.431807	−	0.901966i	$-0.642124\pi$
$769$	−23.9488	−0.863615	−0.431807	+	0.901966i	$0.642124\pi$
$770$	0	0
$771$	32.4720	1.16945
$772$	0	0
$773$	−26.6256	−0.957655	−0.478827	−	0.877909i	$-0.658938\pi$
$773$	−26.6256	−0.957655	−0.478827	+	0.877909i	$0.658938\pi$
$774$	0	0
$775$	4.60030	0.165248
$776$	0	0
$777$	113.767	4.08137
$778$	0	0
$779$	1.74513	0.0625259
$780$	0	0
$781$	−26.3974	−0.944573
$782$	0	0
$783$	−2.42050	−0.0865014
$784$	0	0
$785$	−3.15204	−0.112501
$786$	0	0
$787$	−34.2797	−1.22194	−0.610969	−	0.791654i	$-0.709220\pi$
$787$	−34.2797	−1.22194	−0.610969	+	0.791654i	$0.709220\pi$
$788$	0	0
$789$	−14.2372	−0.506856
$790$	0	0
$791$	55.1624	1.96135
$792$	0	0
$793$	−59.9123	−2.12755
$794$	0	0
$795$	55.5853	1.97141
$796$	0	0
$797$	−13.6760	−0.484428	−0.242214	−	0.970223i	$-0.577874\pi$
$797$	−13.6760	−0.484428	−0.242214	+	0.970223i	$0.577874\pi$
$798$	0	0
$799$	−3.26652	−0.115561
$800$	0	0
$801$	−132.529	−4.68269
$802$	0	0
$803$	−12.5510	−0.442915
$804$	0	0
$805$	−8.98285	−0.316604
$806$	0	0
$807$	−82.1306	−2.89113
$808$	0	0
$809$	30.1021	1.05833	0.529167	−	0.848517i	$-0.322504\pi$
$809$	30.1021	1.05833	0.529167	+	0.848517i	$0.322504\pi$
$810$	0	0
$811$	−1.23914	−0.0435120	−0.0217560	−	0.999763i	$-0.506926\pi$
$811$	−1.23914	−0.0435120	−0.0217560	+	0.999763i	$0.506926\pi$
$812$	0	0
$813$	45.1807	1.58456
$814$	0	0
$815$	24.7446	0.866767
$816$	0	0
$817$	1.28787	0.0450567
$818$	0	0
$819$	−140.450	−4.90772
$820$	0	0
$821$	−2.63625	−0.0920058	−0.0460029	−	0.998941i	$-0.514648\pi$
$821$	−2.63625	−0.0920058	−0.0460029	+	0.998941i	$0.514648\pi$
$822$	0	0
$823$	2.38555	0.0831550	0.0415775	−	0.999135i	$-0.486762\pi$
$823$	2.38555	0.0831550	0.0415775	+	0.999135i	$0.486762\pi$
$824$	0	0
$825$	21.5121	0.748956
$826$	0	0
$827$	39.0021	1.35624	0.678119	−	0.734953i	$-0.262796\pi$
$827$	39.0021	1.35624	0.678119	+	0.734953i	$0.262796\pi$
$828$	0	0
$829$	5.62484	0.195359	0.0976793	−	0.995218i	$-0.468858\pi$
$829$	5.62484	0.195359	0.0976793	+	0.995218i	$0.468858\pi$
$830$	0	0
$831$	−11.0583	−0.383609
$832$	0	0
$833$	−3.75266	−0.130022
$834$	0	0
$835$	18.6972	0.647045
$836$	0	0
$837$	32.9078	1.13746
$838$	0	0
$839$	−39.9612	−1.37961	−0.689807	−	0.723993i	$-0.742305\pi$
$839$	−39.9612	−1.37961	−0.689807	+	0.723993i	$0.742305\pi$
$840$	0	0
$841$	−28.9748	−0.999131
$842$	0	0
$843$	−61.2620	−2.10997
$844$	0	0
$845$	−42.8550	−1.47426
$846$	0	0
$847$	−4.28162	−0.147118
$848$	0	0
$849$	−52.2997	−1.79492
$850$	0	0
$851$	−21.0854	−0.722797
$852$	0	0
$853$	−12.4525	−0.426366	−0.213183	−	0.977012i	$-0.568383\pi$
$853$	−12.4525	−0.426366	−0.213183	+	0.977012i	$0.568383\pi$
$854$	0	0
$855$	2.58293	0.0883342
$856$	0	0
$857$	6.61435	0.225942	0.112971	−	0.993598i	$-0.463963\pi$
$857$	6.61435	0.225942	0.112971	+	0.993598i	$0.463963\pi$
$858$	0	0
$859$	−15.6756	−0.534843	−0.267422	−	0.963580i	$-0.586172\pi$
$859$	−15.6756	−0.534843	−0.267422	+	0.963580i	$0.586172\pi$
$860$	0	0
$861$	84.8261	2.89087
$862$	0	0
$863$	12.6707	0.431315	0.215658	−	0.976469i	$-0.430810\pi$
$863$	12.6707	0.431315	0.215658	+	0.976469i	$0.430810\pi$
$864$	0	0
$865$	−16.3055	−0.554403
$866$	0	0
$867$	40.6920	1.38197
$868$	0	0
$869$	3.07140	0.104190
$870$	0	0
$871$	83.5573	2.83123
$872$	0	0
$873$	−88.8150	−3.00593
$874$	0	0
$875$	35.7514	1.20862
$876$	0	0
$877$	−8.76462	−0.295960	−0.147980	−	0.988990i	$-0.547277\pi$
$877$	−8.76462	−0.295960	−0.147980	+	0.988990i	$0.547277\pi$
$878$	0	0
$879$	−23.7134	−0.799833
$880$	0	0
$881$	−31.5682	−1.06356	−0.531779	−	0.846883i	$-0.678476\pi$
$881$	−31.5682	−1.06356	−0.531779	+	0.846883i	$0.678476\pi$
$882$	0	0
$883$	17.3490	0.583839	0.291920	−	0.956443i	$-0.405706\pi$
$883$	17.3490	0.583839	0.291920	+	0.956443i	$0.405706\pi$
$884$	0	0
$885$	40.4379	1.35930
$886$	0	0
$887$	29.5161	0.991052	0.495526	−	0.868593i	$-0.334975\pi$
$887$	29.5161	0.991052	0.495526	+	0.868593i	$0.334975\pi$
$888$	0	0
$889$	43.5725	1.46137
$890$	0	0
$891$	82.7896	2.77356
$892$	0	0
$893$	−0.304869	−0.0102020
$894$	0	0
$895$	−20.7827	−0.694690
$896$	0	0
$897$	36.2159	1.20921
$898$	0	0
$899$	−0.342787	−0.0114326
$900$	0	0
$901$	−21.4112	−0.713312
$902$	0	0
$903$	62.5996	2.08318
$904$	0	0
$905$	1.01829	0.0338493
$906$	0	0
$907$	23.6966	0.786832	0.393416	−	0.919361i	$-0.371293\pi$
$907$	23.6966	0.786832	0.393416	+	0.919361i	$0.371293\pi$
$908$	0	0
$909$	−93.6220	−3.10525
$910$	0	0
$911$	−34.7136	−1.15011	−0.575056	−	0.818114i	$-0.695020\pi$
$911$	−34.7136	−1.15011	−0.575056	+	0.818114i	$0.695020\pi$
$912$	0	0
$913$	25.3232	0.838075
$914$	0	0
$915$	53.5558	1.77050
$916$	0	0
$917$	−33.1985	−1.09631
$918$	0	0
$919$	19.0733	0.629170	0.314585	−	0.949229i	$-0.398135\pi$
$919$	19.0733	0.629170	0.314585	+	0.949229i	$0.398135\pi$
$920$	0	0
$921$	−34.5995	−1.14009
$922$	0	0
$923$	52.8548	1.73974
$924$	0	0
$925$	25.0778	0.824552
$926$	0	0
$927$	−47.2473	−1.55181
$928$	0	0
$929$	39.9604	1.31106	0.655530	−	0.755170i	$-0.272445\pi$
$929$	39.9604	1.31106	0.655530	+	0.755170i	$0.272445\pi$
$930$	0	0
$931$	−0.350241	−0.0114787
$932$	0	0
$933$	−8.88568	−0.290904
$934$	0	0
$935$	11.1564	0.364854
$936$	0	0
$937$	4.27884	0.139784	0.0698919	−	0.997555i	$-0.477735\pi$
$937$	4.27884	0.139784	0.0698919	+	0.997555i	$0.477735\pi$
$938$	0	0
$939$	45.1768	1.47429
$940$	0	0
$941$	28.9493	0.943719	0.471860	−	0.881674i	$-0.343583\pi$
$941$	28.9493	0.943719	0.471860	+	0.881674i	$0.343583\pi$
$942$	0	0
$943$	−15.7215	−0.511963
$944$	0	0
$945$	76.4249	2.48610
$946$	0	0
$947$	22.1568	0.720000	0.360000	−	0.932952i	$-0.382777\pi$
$947$	22.1568	0.720000	0.360000	+	0.932952i	$0.382777\pi$
$948$	0	0
$949$	25.1305	0.815771
$950$	0	0
$951$	−105.649	−3.42590
$952$	0	0
$953$	39.8286	1.29018	0.645088	−	0.764108i	$-0.276821\pi$
$953$	39.8286	1.29018	0.645088	+	0.764108i	$0.276821\pi$
$954$	0	0
$955$	24.2031	0.783193
$956$	0	0
$957$	−1.60296	−0.0518162
$958$	0	0
$959$	2.95990	0.0955802
$960$	0	0
$961$	−26.3396	−0.849666
$962$	0	0
$963$	−39.9414	−1.28709
$964$	0	0
$965$	15.0173	0.483425
$966$	0	0
$967$	29.6628	0.953891	0.476946	−	0.878933i	$-0.341744\pi$
$967$	29.6628	0.953891	0.476946	+	0.878933i	$0.341744\pi$
$968$	0	0
$969$	−1.38422	−0.0444677
$970$	0	0
$971$	−3.95225	−0.126834	−0.0634168	−	0.997987i	$-0.520200\pi$
$971$	−3.95225	−0.126834	−0.0634168	+	0.997987i	$0.520200\pi$
$972$	0	0
$973$	57.5218	1.84407
$974$	0	0
$975$	−43.0732	−1.37944
$976$	0	0
$977$	51.6391	1.65208	0.826041	−	0.563610i	$-0.190588\pi$
$977$	51.6391	1.65208	0.826041	+	0.563610i	$0.190588\pi$
$978$	0	0
$979$	−53.4258	−1.70749
$980$	0	0
$981$	−68.6759	−2.19265
$982$	0	0
$983$	−48.3895	−1.54339	−0.771693	−	0.635995i	$-0.780590\pi$
$983$	−48.3895	−1.54339	−0.771693	+	0.635995i	$0.780590\pi$
$984$	0	0
$985$	43.5799	1.38857
$986$	0	0
$987$	−14.8188	−0.471688
$988$	0	0
$989$	−11.6021	−0.368925
$990$	0	0
$991$	31.4758	0.999861	0.499931	−	0.866065i	$-0.333359\pi$
$991$	31.4758	0.999861	0.499931	+	0.866065i	$0.333359\pi$
$992$	0	0
$993$	−19.7350	−0.626272
$994$	0	0
$995$	−39.4006	−1.24908
$996$	0	0
$997$	1.78906	0.0566601	0.0283300	−	0.999599i	$-0.490981\pi$
$997$	1.78906	0.0566601	0.0283300	+	0.999599i	$0.490981\pi$
$998$	0	0
$999$	179.392	5.67570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	548.2.a.a.1.1	✓	4
3.2	odd	2		4932.2.a.e.1.3		4
4.3	odd	2		2192.2.a.i.1.4		4
8.3	odd	2		8768.2.a.s.1.1		4
8.5	even	2		8768.2.a.u.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
548.2.a.a.1.1	✓	4	1.1	even	1	trivial
2192.2.a.i.1.4		4	4.3	odd	2
4932.2.a.e.1.3		4	3.2	odd	2
8768.2.a.s.1.1		4	8.3	odd	2
8768.2.a.u.1.4		4	8.5	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:	\( N \)	\(=\)	\( 548 = 2^{2} \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	548.a (trivial)

\(f(q)\)	\(=\)	\(q-3.26608 q^{3} -1.69382 q^{5} +2.95990 q^{7} +7.66727 q^{9} +O(q^{10})\)	\(q-3.26608 q^{3} -1.69382 q^{5} +2.95990 q^{7} +7.66727 q^{9} +3.09087 q^{11} -6.18876 q^{13} +5.53216 q^{15} -2.13097 q^{17} -0.198886 q^{19} -9.66727 q^{21} +1.79172 q^{23} -2.13097 q^{25} -15.2437 q^{27} +0.158787 q^{29} -2.15879 q^{31} -10.0950 q^{33} -5.01355 q^{35} -11.7683 q^{37} +20.2130 q^{39} -8.77456 q^{41} -6.47541 q^{43} -12.9870 q^{45} +1.53288 q^{47} +1.76102 q^{49} +6.95990 q^{51} +10.0477 q^{53} -5.23538 q^{55} +0.649576 q^{57} +7.30960 q^{59} +9.68082 q^{61} +22.6944 q^{63} +10.4827 q^{65} -13.5015 q^{67} -5.85188 q^{69} -8.54045 q^{71} -4.06067 q^{73} +6.95990 q^{75} +9.14866 q^{77} +0.993701 q^{79} +26.7852 q^{81} +8.19291 q^{83} +3.60948 q^{85} -0.518610 q^{87} -17.2850 q^{89} -18.3181 q^{91} +7.05077 q^{93} +0.336877 q^{95} -11.5837 q^{97} +23.6985 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} - 4 q^{5} - q^{7} + q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 - 4 * q^5 - q^7 + q^9	\( 4 q - 3 q^{3} - 4 q^{5} - q^{7} + q^{9} - q^{11} - 14 q^{13} + 2 q^{15} - 8 q^{17} - 6 q^{19} - 9 q^{21} + 7 q^{23} - 8 q^{25} - 15 q^{27} - 7 q^{29} - q^{31} - 18 q^{33} - 6 q^{35} - 20 q^{37} + 17 q^{39} - 3 q^{41} - 7 q^{43} - 11 q^{45} + 7 q^{47} - 11 q^{49} + 15 q^{51} + 14 q^{53} + 7 q^{55} - 4 q^{57} + 2 q^{59} - 5 q^{61} + 33 q^{63} + 14 q^{65} - 16 q^{67} + 8 q^{69} + 4 q^{71} - 31 q^{73} + 15 q^{75} + 13 q^{77} + 9 q^{79} + 36 q^{81} + 13 q^{83} - 5 q^{85} + 4 q^{87} - 2 q^{89} - 3 q^{91} + 2 q^{93} + 22 q^{95} - 21 q^{97} + 28 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 - 4 * q^5 - q^7 + q^9 - q^11 - 14 * q^13 + 2 * q^15 - 8 * q^17 - 6 * q^19 - 9 * q^21 + 7 * q^23 - 8 * q^25 - 15 * q^27 - 7 * q^29 - q^31 - 18 * q^33 - 6 * q^35 - 20 * q^37 + 17 * q^39 - 3 * q^41 - 7 * q^43 - 11 * q^45 + 7 * q^47 - 11 * q^49 + 15 * q^51 + 14 * q^53 + 7 * q^55 - 4 * q^57 + 2 * q^59 - 5 * q^61 + 33 * q^63 + 14 * q^65 - 16 * q^67 + 8 * q^69 + 4 * q^71 - 31 * q^73 + 15 * q^75 + 13 * q^77 + 9 * q^79 + 36 * q^81 + 13 * q^83 - 5 * q^85 + 4 * q^87 - 2 * q^89 - 3 * q^91 + 2 * q^93 + 22 * q^95 - 21 * q^97 + 28 * q^99

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