LMFDB - Embedded newform 1344.4.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.31662$ of defining polynomial
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +2.63325 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +2.63325 q^{5} -7.00000 q^{7} +9.00000 q^{9} -24.4327 q^{11} +83.0660 q^{13} -7.89975 q^{15} +110.232 q^{17} +26.7335 q^{19} +21.0000 q^{21} -50.1003 q^{23} -118.066 q^{25} -27.0000 q^{27} +16.7335 q^{29} -129.266 q^{31} +73.2982 q^{33} -18.4327 q^{35} -122.264 q^{37} -249.198 q^{39} -417.430 q^{41} +153.330 q^{43} +23.6992 q^{45} +175.662 q^{47} +49.0000 q^{49} -330.697 q^{51} +487.594 q^{53} -64.3375 q^{55} -80.2005 q^{57} +496.327 q^{59} -274.464 q^{61} -63.0000 q^{63} +218.734 q^{65} +396.201 q^{67} +150.301 q^{69} -844.824 q^{71} +895.931 q^{73} +354.198 q^{75} +171.029 q^{77} -1310.86 q^{79} +81.0000 q^{81} +1365.33 q^{83} +290.269 q^{85} -50.2005 q^{87} +683.890 q^{89} -581.462 q^{91} +387.799 q^{93} +70.3960 q^{95} +1430.73 q^{97} -219.895 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 8 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 8 * q^5 - 14 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} - 8 q^{5} - 14 q^{7} + 18 q^{9} + 44 q^{11} + 60 q^{13} + 24 q^{15} + 48 q^{17} + 80 q^{19} + 42 q^{21} - 140 q^{23} - 130 q^{25} - 54 q^{27} + 60 q^{29} - 232 q^{31} - 132 q^{33} + 56 q^{35} + 180 q^{37} - 180 q^{39} - 344 q^{41} - 224 q^{43} - 72 q^{45} - 312 q^{47} + 98 q^{49} - 144 q^{51} + 20 q^{53} - 792 q^{55} - 240 q^{57} + 64 q^{59} - 204 q^{61} - 126 q^{63} + 464 q^{65} + 872 q^{67} + 420 q^{69} - 164 q^{71} + 1500 q^{73} + 390 q^{75} - 308 q^{77} - 1640 q^{79} + 162 q^{81} + 2200 q^{83} + 952 q^{85} - 180 q^{87} - 264 q^{89} - 420 q^{91} + 696 q^{93} - 496 q^{95} + 2092 q^{97} + 396 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 8 * q^5 - 14 * q^7 + 18 * q^9 + 44 * q^11 + 60 * q^13 + 24 * q^15 + 48 * q^17 + 80 * q^19 + 42 * q^21 - 140 * q^23 - 130 * q^25 - 54 * q^27 + 60 * q^29 - 232 * q^31 - 132 * q^33 + 56 * q^35 + 180 * q^37 - 180 * q^39 - 344 * q^41 - 224 * q^43 - 72 * q^45 - 312 * q^47 + 98 * q^49 - 144 * q^51 + 20 * q^53 - 792 * q^55 - 240 * q^57 + 64 * q^59 - 204 * q^61 - 126 * q^63 + 464 * q^65 + 872 * q^67 + 420 * q^69 - 164 * q^71 + 1500 * q^73 + 390 * q^75 - 308 * q^77 - 1640 * q^79 + 162 * q^81 + 2200 * q^83 + 952 * q^85 - 180 * q^87 - 264 * q^89 - 420 * q^91 + 696 * q^93 - 496 * q^95 + 2092 * q^97 + 396 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	2.63325	0.235525	0.117763	−	0.993042i	$-0.462428\pi$
$5$	2.63325	0.235525	0.117763	+	0.993042i	$0.462428\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−24.4327	−0.669705	−0.334852	−	0.942271i	$-0.608686\pi$
$11$	−24.4327	−0.669705	−0.334852	+	0.942271i	$0.608686\pi$
$12$	0	0
$13$	83.0660	1.77218	0.886091	−	0.463512i	$-0.153411\pi$
$13$	83.0660	1.77218	0.886091	+	0.463512i	$0.153411\pi$
$14$	0	0
$15$	−7.89975	−0.135980
$16$	0	0
$17$	110.232	1.57266	0.786331	−	0.617806i	$-0.211978\pi$
$17$	110.232	1.57266	0.786331	+	0.617806i	$0.211978\pi$
$18$	0	0
$19$	26.7335	0.322794	0.161397	−	0.986890i	$-0.448400\pi$
$19$	26.7335	0.322794	0.161397	+	0.986890i	$0.448400\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	−50.1003	−0.454201	−0.227101	−	0.973871i	$-0.572925\pi$
$23$	−50.1003	−0.454201	−0.227101	+	0.973871i	$0.572925\pi$
$24$	0	0
$25$	−118.066	−0.944528
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	16.7335	0.107149	0.0535747	−	0.998564i	$-0.482938\pi$
$29$	16.7335	0.107149	0.0535747	+	0.998564i	$0.482938\pi$
$30$	0	0
$31$	−129.266	−0.748934	−0.374467	−	0.927240i	$-0.622174\pi$
$31$	−129.266	−0.748934	−0.374467	+	0.927240i	$0.622174\pi$
$32$	0	0
$33$	73.2982	0.386654
$34$	0	0
$35$	−18.4327	−0.0890201
$36$	0	0
$37$	−122.264	−0.543245	−0.271623	−	0.962404i	$-0.587560\pi$
$37$	−122.264	−0.543245	−0.271623	+	0.962404i	$0.587560\pi$
$38$	0	0
$39$	−249.198	−1.02317
$40$	0	0
$41$	−417.430	−1.59004	−0.795020	−	0.606583i	$-0.792540\pi$
$41$	−417.430	−1.59004	−0.795020	+	0.606583i	$0.792540\pi$
$42$	0	0
$43$	153.330	0.543781	0.271891	−	0.962328i	$-0.412351\pi$
$43$	153.330	0.543781	0.271891	+	0.962328i	$0.412351\pi$
$44$	0	0
$45$	23.6992	0.0785083
$46$	0	0
$47$	175.662	0.545170	0.272585	−	0.962132i	$-0.412121\pi$
$47$	175.662	0.545170	0.272585	+	0.962132i	$0.412121\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−330.697	−0.907977
$52$	0	0
$53$	487.594	1.26370	0.631851	−	0.775090i	$-0.282296\pi$
$53$	487.594	1.26370	0.631851	+	0.775090i	$0.282296\pi$
$54$	0	0
$55$	−64.3375	−0.157732
$56$	0	0
$57$	−80.2005	−0.186365
$58$	0	0
$59$	496.327	1.09519	0.547596	−	0.836743i	$-0.315543\pi$
$59$	496.327	1.09519	0.547596	+	0.836743i	$0.315543\pi$
$60$	0	0
$61$	−274.464	−0.576091	−0.288046	−	0.957617i	$-0.593006\pi$
$61$	−274.464	−0.576091	−0.288046	+	0.957617i	$0.593006\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	218.734	0.417393
$66$	0	0
$67$	396.201	0.722442	0.361221	−	0.932480i	$-0.382360\pi$
$67$	396.201	0.722442	0.361221	+	0.932480i	$0.382360\pi$
$68$	0	0
$69$	150.301	0.262233
$70$	0	0
$71$	−844.824	−1.41214	−0.706071	−	0.708141i	$-0.749534\pi$
$71$	−844.824	−1.41214	−0.706071	+	0.708141i	$0.749534\pi$
$72$	0	0
$73$	895.931	1.43645	0.718225	−	0.695811i	$-0.244955\pi$
$73$	895.931	1.43645	0.718225	+	0.695811i	$0.244955\pi$
$74$	0	0
$75$	354.198	0.545323
$76$	0	0
$77$	171.029	0.253125
$78$	0	0
$79$	−1310.86	−1.86688	−0.933439	−	0.358737i	$-0.883208\pi$
$79$	−1310.86	−1.86688	−0.933439	+	0.358737i	$0.883208\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1365.33	1.80560	0.902798	−	0.430065i	$-0.141509\pi$
$83$	1365.33	1.80560	0.902798	+	0.430065i	$0.141509\pi$
$84$	0	0
$85$	290.269	0.370401
$86$	0	0
$87$	−50.2005	−0.0618627
$88$	0	0
$89$	683.890	0.814519	0.407259	−	0.913313i	$-0.366485\pi$
$89$	683.890	0.814519	0.407259	+	0.913313i	$0.366485\pi$
$90$	0	0
$91$	−581.462	−0.669822
$92$	0	0
$93$	387.799	0.432397
$94$	0	0
$95$	70.3960	0.0760261
$96$	0	0
$97$	1430.73	1.49761	0.748807	−	0.662789i	$-0.230627\pi$
$97$	1430.73	1.49761	0.748807	+	0.662789i	$0.230627\pi$
$98$	0	0
$99$	−219.895	−0.223235
$100$	0	0
$101$	−329.293	−0.324415	−0.162207	−	0.986757i	$-0.551861\pi$
$101$	−329.293	−0.324415	−0.162207	+	0.986757i	$0.551861\pi$
$102$	0	0
$103$	−40.7385	−0.0389717	−0.0194859	−	0.999810i	$-0.506203\pi$
$103$	−40.7385	−0.0389717	−0.0194859	+	0.999810i	$0.506203\pi$
$104$	0	0
$105$	55.2982	0.0513958
$106$	0	0
$107$	−1282.42	−1.15865	−0.579327	−	0.815095i	$-0.696685\pi$
$107$	−1282.42	−1.15865	−0.579327	+	0.815095i	$0.696685\pi$
$108$	0	0
$109$	−128.132	−0.112595	−0.0562973	−	0.998414i	$-0.517929\pi$
$109$	−128.132	−0.112595	−0.0562973	+	0.998414i	$0.517929\pi$
$110$	0	0
$111$	366.792	0.313643
$112$	0	0
$113$	−570.117	−0.474620	−0.237310	−	0.971434i	$-0.576266\pi$
$113$	−570.117	−0.474620	−0.237310	+	0.971434i	$0.576266\pi$
$114$	0	0
$115$	−131.926	−0.106976
$116$	0	0
$117$	747.594	0.590727
$118$	0	0
$119$	−771.626	−0.594410
$120$	0	0
$121$	−734.041	−0.551496
$122$	0	0
$123$	1252.29	0.918010
$124$	0	0
$125$	−640.053	−0.457985
$126$	0	0
$127$	43.1245	0.0301313	0.0150657	−	0.999887i	$-0.495204\pi$
$127$	43.1245	0.0301313	0.0150657	+	0.999887i	$0.495204\pi$
$128$	0	0
$129$	−459.990	−0.313952
$130$	0	0
$131$	796.802	0.531427	0.265713	−	0.964052i	$-0.414393\pi$
$131$	796.802	0.531427	0.265713	+	0.964052i	$0.414393\pi$
$132$	0	0
$133$	−187.135	−0.122005
$134$	0	0
$135$	−71.0977	−0.0453268
$136$	0	0
$137$	706.464	0.440564	0.220282	−	0.975436i	$-0.429302\pi$
$137$	706.464	0.440564	0.220282	+	0.975436i	$0.429302\pi$
$138$	0	0
$139$	1965.46	1.19934	0.599669	−	0.800248i	$-0.295299\pi$
$139$	1965.46	1.19934	0.599669	+	0.800248i	$0.295299\pi$
$140$	0	0
$141$	−526.987	−0.314754
$142$	0	0
$143$	−2029.53	−1.18684
$144$	0	0
$145$	44.0635	0.0252364
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	2118.54	1.16481	0.582407	−	0.812897i	$-0.302111\pi$
$149$	2118.54	1.16481	0.582407	+	0.812897i	$0.302111\pi$
$150$	0	0
$151$	−192.127	−0.103544	−0.0517718	−	0.998659i	$-0.516487\pi$
$151$	−192.127	−0.103544	−0.0517718	+	0.998659i	$0.516487\pi$
$152$	0	0
$153$	992.090	0.524221
$154$	0	0
$155$	−340.391	−0.176393
$156$	0	0
$157$	2991.63	1.52075	0.760375	−	0.649484i	$-0.225015\pi$
$157$	2991.63	1.52075	0.760375	+	0.649484i	$0.225015\pi$
$158$	0	0
$159$	−1462.78	−0.729598
$160$	0	0
$161$	350.702	0.171672
$162$	0	0
$163$	3083.76	1.48183	0.740916	−	0.671598i	$-0.234392\pi$
$163$	3083.76	1.48183	0.740916	+	0.671598i	$0.234392\pi$
$164$	0	0
$165$	193.013	0.0910667
$166$	0	0
$167$	0.591457	0.000274062 0	0.000137031	−	1.00000i	$-0.499956\pi$
$167$	0.591457	0.000274062 0	0.000137031		1.00000i	$0.499956\pi$
$168$	0	0
$169$	4702.96	2.14063
$170$	0	0
$171$	240.602	0.107598
$172$	0	0
$173$	3212.70	1.41189	0.705946	−	0.708266i	$-0.250522\pi$
$173$	3212.70	1.41189	0.705946	+	0.708266i	$0.250522\pi$
$174$	0	0
$175$	826.462	0.356998
$176$	0	0
$177$	−1488.98	−0.632309
$178$	0	0
$179$	624.179	0.260633	0.130317	−	0.991472i	$-0.458401\pi$
$179$	624.179	0.260633	0.130317	+	0.991472i	$0.458401\pi$
$180$	0	0
$181$	−1658.25	−0.680978	−0.340489	−	0.940248i	$-0.610593\pi$
$181$	−1658.25	−0.680978	−0.340489	+	0.940248i	$0.610593\pi$
$182$	0	0
$183$	823.393	0.332607
$184$	0	0
$185$	−321.952	−0.127948
$186$	0	0
$187$	−2693.28	−1.05322
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	3904.22	1.47906	0.739528	−	0.673126i	$-0.235049\pi$
$191$	3904.22	1.47906	0.739528	+	0.673126i	$0.235049\pi$
$192$	0	0
$193$	2236.53	0.834141	0.417070	−	0.908874i	$-0.363057\pi$
$193$	2236.53	0.834141	0.417070	+	0.908874i	$0.363057\pi$
$194$	0	0
$195$	−656.201	−0.240982
$196$	0	0
$197$	−1276.25	−0.461568	−0.230784	−	0.973005i	$-0.574129\pi$
$197$	−1276.25	−0.461568	−0.230784	+	0.973005i	$0.574129\pi$
$198$	0	0
$199$	4069.15	1.44952	0.724760	−	0.689001i	$-0.241951\pi$
$199$	4069.15	1.44952	0.724760	+	0.689001i	$0.241951\pi$
$200$	0	0
$201$	−1188.60	−0.417102
$202$	0	0
$203$	−117.135	−0.0404987
$204$	0	0
$205$	−1099.20	−0.374494
$206$	0	0
$207$	−450.902	−0.151400
$208$	0	0
$209$	−653.173	−0.216177
$210$	0	0
$211$	−2093.57	−0.683069	−0.341535	−	0.939869i	$-0.610947\pi$
$211$	−2093.57	−0.683069	−0.341535	+	0.939869i	$0.610947\pi$
$212$	0	0
$213$	2534.47	0.815301
$214$	0	0
$215$	403.756	0.128074
$216$	0	0
$217$	904.865	0.283070
$218$	0	0
$219$	−2687.79	−0.829335
$220$	0	0
$221$	9156.55	2.78704
$222$	0	0
$223$	−6481.00	−1.94619	−0.973093	−	0.230412i	$-0.925993\pi$
$223$	−6481.00	−1.94619	−0.973093	+	0.230412i	$0.925993\pi$
$224$	0	0
$225$	−1062.59	−0.314843
$226$	0	0
$227$	3001.49	0.877603	0.438802	−	0.898584i	$-0.355403\pi$
$227$	3001.49	0.877603	0.438802	+	0.898584i	$0.355403\pi$
$228$	0	0
$229$	−3843.71	−1.10917	−0.554584	−	0.832128i	$-0.687123\pi$
$229$	−3843.71	−1.10917	−0.554584	+	0.832128i	$0.687123\pi$
$230$	0	0
$231$	−513.088	−0.146142
$232$	0	0
$233$	3378.85	0.950024	0.475012	−	0.879979i	$-0.342444\pi$
$233$	3378.85	0.950024	0.475012	+	0.879979i	$0.342444\pi$
$234$	0	0
$235$	462.563	0.128401
$236$	0	0
$237$	3932.58	1.07784
$238$	0	0
$239$	4653.65	1.25950	0.629748	−	0.776800i	$-0.283158\pi$
$239$	4653.65	1.25950	0.629748	+	0.776800i	$0.283158\pi$
$240$	0	0
$241$	5108.74	1.36549	0.682745	−	0.730657i	$-0.260786\pi$
$241$	5108.74	1.36549	0.682745	+	0.730657i	$0.260786\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	129.029	0.0336464
$246$	0	0
$247$	2220.64	0.572050
$248$	0	0
$249$	−4095.99	−1.04246
$250$	0	0
$251$	2.20553	0.000554628 0	0.000277314	−	1.00000i	$-0.499912\pi$
$251$	2.20553	0.000554628 0	0.000277314		1.00000i	$0.499912\pi$
$252$	0	0
$253$	1224.09	0.304181
$254$	0	0
$255$	−870.807	−0.213851
$256$	0	0
$257$	1348.65	0.327341	0.163670	−	0.986515i	$-0.447667\pi$
$257$	1348.65	0.327341	0.163670	+	0.986515i	$0.447667\pi$
$258$	0	0
$259$	855.848	0.205327
$260$	0	0
$261$	150.602	0.0357165
$262$	0	0
$263$	2645.77	0.620324	0.310162	−	0.950684i	$-0.399617\pi$
$263$	2645.77	0.620324	0.310162	+	0.950684i	$0.399617\pi$
$264$	0	0
$265$	1283.96	0.297633
$266$	0	0
$267$	−2051.67	−0.470263
$268$	0	0
$269$	−3378.90	−0.765856	−0.382928	−	0.923778i	$-0.625084\pi$
$269$	−3378.90	−0.765856	−0.382928	+	0.923778i	$0.625084\pi$
$270$	0	0
$271$	2875.38	0.644527	0.322263	−	0.946650i	$-0.395556\pi$
$271$	2875.38	0.644527	0.322263	+	0.946650i	$0.395556\pi$
$272$	0	0
$273$	1744.39	0.386722
$274$	0	0
$275$	2884.68	0.632555
$276$	0	0
$277$	3125.88	0.678037	0.339018	−	0.940780i	$-0.389905\pi$
$277$	3125.88	0.678037	0.339018	+	0.940780i	$0.389905\pi$
$278$	0	0
$279$	−1163.40	−0.249645
$280$	0	0
$281$	−3185.38	−0.676241	−0.338121	−	0.941103i	$-0.609791\pi$
$281$	−3185.38	−0.676241	−0.338121	+	0.941103i	$0.609791\pi$
$282$	0	0
$283$	−2516.34	−0.528554	−0.264277	−	0.964447i	$-0.585133\pi$
$283$	−2516.34	−0.528554	−0.264277	+	0.964447i	$0.585133\pi$
$284$	0	0
$285$	−211.188	−0.0438937
$286$	0	0
$287$	2922.01	0.600979
$288$	0	0
$289$	7238.15	1.47326
$290$	0	0
$291$	−4292.19	−0.864647
$292$	0	0
$293$	−978.409	−0.195083	−0.0975415	−	0.995231i	$-0.531098\pi$
$293$	−978.409	−0.195083	−0.0975415	+	0.995231i	$0.531098\pi$
$294$	0	0
$295$	1306.95	0.257945
$296$	0	0
$297$	659.684	0.128885
$298$	0	0
$299$	−4161.63	−0.804927
$300$	0	0
$301$	−1073.31	−0.205530
$302$	0	0
$303$	987.880	0.187301
$304$	0	0
$305$	−722.734	−0.135684
$306$	0	0
$307$	−10331.0	−1.92059	−0.960297	−	0.278981i	$-0.910003\pi$
$307$	−10331.0	−1.92059	−0.960297	+	0.278981i	$0.910003\pi$
$308$	0	0
$309$	122.216	0.0225003
$310$	0	0
$311$	9222.57	1.68156	0.840778	−	0.541380i	$-0.182098\pi$
$311$	9222.57	1.68156	0.840778	+	0.541380i	$0.182098\pi$
$312$	0	0
$313$	−5812.84	−1.04972	−0.524858	−	0.851190i	$-0.675882\pi$
$313$	−5812.84	−1.04972	−0.524858	+	0.851190i	$0.675882\pi$
$314$	0	0
$315$	−165.895	−0.0296734
$316$	0	0
$317$	5280.27	0.935550	0.467775	−	0.883847i	$-0.345056\pi$
$317$	5280.27	0.935550	0.467775	+	0.883847i	$0.345056\pi$
$318$	0	0
$319$	−408.845	−0.0717585
$320$	0	0
$321$	3847.25	0.668949
$322$	0	0
$323$	2946.89	0.507646
$324$	0	0
$325$	−9807.27	−1.67388
$326$	0	0
$327$	384.396	0.0650066
$328$	0	0
$329$	−1229.64	−0.206055
$330$	0	0
$331$	−7462.67	−1.23923	−0.619615	−	0.784906i	$-0.712711\pi$
$331$	−7462.67	−1.23923	−0.619615	+	0.784906i	$0.712711\pi$
$332$	0	0
$333$	−1100.38	−0.181082
$334$	0	0
$335$	1043.29	0.170153
$336$	0	0
$337$	6866.68	1.10995	0.554973	−	0.831869i	$-0.312729\pi$
$337$	6866.68	1.10995	0.554973	+	0.831869i	$0.312729\pi$
$338$	0	0
$339$	1710.35	0.274022
$340$	0	0
$341$	3158.34	0.501564
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	395.779	0.0617625
$346$	0	0
$347$	6347.14	0.981937	0.490969	−	0.871177i	$-0.336643\pi$
$347$	6347.14	0.981937	0.490969	+	0.871177i	$0.336643\pi$
$348$	0	0
$349$	−6712.06	−1.02948	−0.514740	−	0.857347i	$-0.672111\pi$
$349$	−6712.06	−1.02948	−0.514740	+	0.857347i	$0.672111\pi$
$350$	0	0
$351$	−2242.78	−0.341057
$352$	0	0
$353$	4834.01	0.728862	0.364431	−	0.931230i	$-0.381263\pi$
$353$	4834.01	0.728862	0.364431	+	0.931230i	$0.381263\pi$
$354$	0	0
$355$	−2224.63	−0.332595
$356$	0	0
$357$	2314.88	0.343183
$358$	0	0
$359$	2504.35	0.368174	0.184087	−	0.982910i	$-0.441067\pi$
$359$	2504.35	0.368174	0.184087	+	0.982910i	$0.441067\pi$
$360$	0	0
$361$	−6144.32	−0.895804
$362$	0	0
$363$	2202.12	0.318406
$364$	0	0
$365$	2359.21	0.338320
$366$	0	0
$367$	−10447.9	−1.48604	−0.743019	−	0.669270i	$-0.766607\pi$
$367$	−10447.9	−1.48604	−0.743019	+	0.669270i	$0.766607\pi$
$368$	0	0
$369$	−3756.87	−0.530013
$370$	0	0
$371$	−3413.16	−0.477634
$372$	0	0
$373$	2205.68	0.306181	0.153091	−	0.988212i	$-0.451077\pi$
$373$	2205.68	0.306181	0.153091	+	0.988212i	$0.451077\pi$
$374$	0	0
$375$	1920.16	0.264418
$376$	0	0
$377$	1389.98	0.189888
$378$	0	0
$379$	3786.31	0.513165	0.256582	−	0.966522i	$-0.417404\pi$
$379$	3786.31	0.513165	0.256582	+	0.966522i	$0.417404\pi$
$380$	0	0
$381$	−129.373	−0.0173963
$382$	0	0
$383$	1139.11	0.151973	0.0759865	−	0.997109i	$-0.475789\pi$
$383$	1139.11	0.151973	0.0759865	+	0.997109i	$0.475789\pi$
$384$	0	0
$385$	450.363	0.0596172
$386$	0	0
$387$	1379.97	0.181260
$388$	0	0
$389$	6105.58	0.795797	0.397898	−	0.917429i	$-0.369740\pi$
$389$	6105.58	0.795797	0.397898	+	0.917429i	$0.369740\pi$
$390$	0	0
$391$	−5522.66	−0.714305
$392$	0	0
$393$	−2390.41	−0.306819
$394$	0	0
$395$	−3451.82	−0.439696
$396$	0	0
$397$	−3444.26	−0.435422	−0.217711	−	0.976013i	$-0.569859\pi$
$397$	−3444.26	−0.435422	−0.217711	+	0.976013i	$0.569859\pi$
$398$	0	0
$399$	561.404	0.0704394
$400$	0	0
$401$	−4128.09	−0.514082	−0.257041	−	0.966400i	$-0.582748\pi$
$401$	−4128.09	−0.514082	−0.257041	+	0.966400i	$0.582748\pi$
$402$	0	0
$403$	−10737.7	−1.32725
$404$	0	0
$405$	213.293	0.0261694
$406$	0	0
$407$	2987.25	0.363814
$408$	0	0
$409$	10637.8	1.28607	0.643036	−	0.765836i	$-0.277675\pi$
$409$	10637.8	1.28607	0.643036	+	0.765836i	$0.277675\pi$
$410$	0	0
$411$	−2119.39	−0.254360
$412$	0	0
$413$	−3474.29	−0.413944
$414$	0	0
$415$	3595.25	0.425263
$416$	0	0
$417$	−5896.37	−0.692438
$418$	0	0
$419$	−5226.38	−0.609368	−0.304684	−	0.952453i	$-0.598551\pi$
$419$	−5226.38	−0.609368	−0.304684	+	0.952453i	$0.598551\pi$
$420$	0	0
$421$	−8271.22	−0.957517	−0.478759	−	0.877947i	$-0.658913\pi$
$421$	−8271.22	−0.957517	−0.478759	+	0.877947i	$0.658913\pi$
$422$	0	0
$423$	1580.96	0.181723
$424$	0	0
$425$	−13014.7	−1.48542
$426$	0	0
$427$	1921.25	0.217742
$428$	0	0
$429$	6088.59	0.685221
$430$	0	0
$431$	−8341.80	−0.932275	−0.466137	−	0.884712i	$-0.654355\pi$
$431$	−8341.80	−0.932275	−0.466137	+	0.884712i	$0.654355\pi$
$432$	0	0
$433$	12866.3	1.42798	0.713991	−	0.700155i	$-0.246886\pi$
$433$	12866.3	1.42798	0.713991	+	0.700155i	$0.246886\pi$
$434$	0	0
$435$	−132.190	−0.0145702
$436$	0	0
$437$	−1339.36	−0.146613
$438$	0	0
$439$	−479.599	−0.0521413	−0.0260706	−	0.999660i	$-0.508299\pi$
$439$	−479.599	−0.0521413	−0.0260706	+	0.999660i	$0.508299\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−3894.58	−0.417691	−0.208845	−	0.977949i	$-0.566971\pi$
$443$	−3894.58	−0.417691	−0.208845	+	0.977949i	$0.566971\pi$
$444$	0	0
$445$	1800.85	0.191840
$446$	0	0
$447$	−6355.61	−0.672506
$448$	0	0
$449$	−5428.50	−0.570572	−0.285286	−	0.958442i	$-0.592089\pi$
$449$	−5428.50	−0.570572	−0.285286	+	0.958442i	$0.592089\pi$
$450$	0	0
$451$	10199.0	1.06486
$452$	0	0
$453$	576.381	0.0597809
$454$	0	0
$455$	−1531.13	−0.157760
$456$	0	0
$457$	−9577.50	−0.980343	−0.490171	−	0.871626i	$-0.663066\pi$
$457$	−9577.50	−0.980343	−0.490171	+	0.871626i	$0.663066\pi$
$458$	0	0
$459$	−2976.27	−0.302659
$460$	0	0
$461$	−11198.0	−1.13133	−0.565664	−	0.824636i	$-0.691380\pi$
$461$	−11198.0	−1.13133	−0.565664	+	0.824636i	$0.691380\pi$
$462$	0	0
$463$	17391.1	1.74564	0.872821	−	0.488041i	$-0.162288\pi$
$463$	17391.1	1.74564	0.872821	+	0.488041i	$0.162288\pi$
$464$	0	0
$465$	1021.17	0.101840
$466$	0	0
$467$	10757.5	1.06595	0.532973	−	0.846132i	$-0.321075\pi$
$467$	10757.5	1.06595	0.532973	+	0.846132i	$0.321075\pi$
$468$	0	0
$469$	−2773.40	−0.273057
$470$	0	0
$471$	−8974.88	−0.878006
$472$	0	0
$473$	−3746.27	−0.364173
$474$	0	0
$475$	−3156.32	−0.304888
$476$	0	0
$477$	4388.35	0.421234
$478$	0	0
$479$	2742.63	0.261615	0.130808	−	0.991408i	$-0.458243\pi$
$479$	2742.63	0.261615	0.130808	+	0.991408i	$0.458243\pi$
$480$	0	0
$481$	−10156.0	−0.962729
$482$	0	0
$483$	−1052.11	−0.0991148
$484$	0	0
$485$	3767.47	0.352725
$486$	0	0
$487$	−16970.3	−1.57905	−0.789526	−	0.613718i	$-0.789673\pi$
$487$	−16970.3	−1.57905	−0.789526	+	0.613718i	$0.789673\pi$
$488$	0	0
$489$	−9251.28	−0.855536
$490$	0	0
$491$	18794.0	1.72742	0.863709	−	0.503991i	$-0.168136\pi$
$491$	18794.0	1.72742	0.863709	+	0.503991i	$0.168136\pi$
$492$	0	0
$493$	1844.57	0.168510
$494$	0	0
$495$	−579.038	−0.0525774
$496$	0	0
$497$	5913.77	0.533740
$498$	0	0
$499$	62.5430	0.00561084	0.00280542	−	0.999996i	$-0.499107\pi$
$499$	62.5430	0.00561084	0.00280542	+	0.999996i	$0.499107\pi$
$500$	0	0
$501$	−1.77437	−0.000158230 0
$502$	0	0
$503$	5222.59	0.462950	0.231475	−	0.972841i	$-0.425645\pi$
$503$	5222.59	0.462950	0.231475	+	0.972841i	$0.425645\pi$
$504$	0	0
$505$	−867.111	−0.0764078
$506$	0	0
$507$	−14108.9	−1.23589
$508$	0	0
$509$	4387.65	0.382081	0.191040	−	0.981582i	$-0.438814\pi$
$509$	4387.65	0.382081	0.191040	+	0.981582i	$0.438814\pi$
$510$	0	0
$511$	−6271.52	−0.542927
$512$	0	0
$513$	−721.805	−0.0621217
$514$	0	0
$515$	−107.275	−0.00917881
$516$	0	0
$517$	−4291.92	−0.365103
$518$	0	0
$519$	−9638.11	−0.815156
$520$	0	0
$521$	−15313.8	−1.28773	−0.643865	−	0.765139i	$-0.722670\pi$
$521$	−15313.8	−1.28773	−0.643865	+	0.765139i	$0.722670\pi$
$522$	0	0
$523$	3114.32	0.260381	0.130191	−	0.991489i	$-0.458441\pi$
$523$	3114.32	0.260381	0.130191	+	0.991489i	$0.458441\pi$
$524$	0	0
$525$	−2479.39	−0.206113
$526$	0	0
$527$	−14249.3	−1.17782
$528$	0	0
$529$	−9656.96	−0.793701
$530$	0	0
$531$	4466.95	0.365064
$532$	0	0
$533$	−34674.3	−2.81784
$534$	0	0
$535$	−3376.93	−0.272892
$536$	0	0
$537$	−1872.54	−0.150477
$538$	0	0
$539$	−1197.20	−0.0956721
$540$	0	0
$541$	5285.36	0.420028	0.210014	−	0.977698i	$-0.432649\pi$
$541$	5285.36	0.420028	0.210014	+	0.977698i	$0.432649\pi$
$542$	0	0
$543$	4974.76	0.393163
$544$	0	0
$545$	−337.404	−0.0265189
$546$	0	0
$547$	1010.34	0.0789741	0.0394871	−	0.999220i	$-0.487428\pi$
$547$	1010.34	0.0789741	0.0394871	+	0.999220i	$0.487428\pi$
$548$	0	0
$549$	−2470.18	−0.192030
$550$	0	0
$551$	447.345	0.0345872
$552$	0	0
$553$	9176.02	0.705613
$554$	0	0
$555$	965.855	0.0738707
$556$	0	0
$557$	21461.8	1.63262	0.816308	−	0.577616i	$-0.196017\pi$
$557$	21461.8	1.63262	0.816308	+	0.577616i	$0.196017\pi$
$558$	0	0
$559$	12736.5	0.963680
$560$	0	0
$561$	8079.83	0.608076
$562$	0	0
$563$	17791.0	1.33180	0.665900	−	0.746041i	$-0.268048\pi$
$563$	17791.0	1.33180	0.665900	+	0.746041i	$0.268048\pi$
$564$	0	0
$565$	−1501.26	−0.111785
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−5817.45	−0.428612	−0.214306	−	0.976767i	$-0.568749\pi$
$569$	−5817.45	−0.428612	−0.214306	+	0.976767i	$0.568749\pi$
$570$	0	0
$571$	13391.6	0.981476	0.490738	−	0.871307i	$-0.336727\pi$
$571$	13391.6	0.981476	0.490738	+	0.871307i	$0.336727\pi$
$572$	0	0
$573$	−11712.7	−0.853933
$574$	0	0
$575$	5915.14	0.429006
$576$	0	0
$577$	−8391.88	−0.605474	−0.302737	−	0.953074i	$-0.597900\pi$
$577$	−8391.88	−0.605474	−0.302737	+	0.953074i	$0.597900\pi$
$578$	0	0
$579$	−6709.60	−0.481591
$580$	0	0
$581$	−9557.31	−0.682451
$582$	0	0
$583$	−11913.3	−0.846307
$584$	0	0
$585$	1968.60	0.139131
$586$	0	0
$587$	23828.4	1.67547	0.837736	−	0.546075i	$-0.183879\pi$
$587$	23828.4	1.67547	0.837736	+	0.546075i	$0.183879\pi$
$588$	0	0
$589$	−3455.75	−0.241751
$590$	0	0
$591$	3828.75	0.266487
$592$	0	0
$593$	−4159.17	−0.288021	−0.144011	−	0.989576i	$-0.546000\pi$
$593$	−4159.17	−0.288021	−0.144011	+	0.989576i	$0.546000\pi$
$594$	0	0
$595$	−2031.88	−0.139998
$596$	0	0
$597$	−12207.5	−0.836881
$598$	0	0
$599$	27461.8	1.87322	0.936610	−	0.350374i	$-0.113945\pi$
$599$	27461.8	1.87322	0.936610	+	0.350374i	$0.113945\pi$
$600$	0	0
$601$	−1290.98	−0.0876209	−0.0438104	−	0.999040i	$-0.513950\pi$
$601$	−1290.98	−0.0876209	−0.0438104	+	0.999040i	$0.513950\pi$
$602$	0	0
$603$	3565.80	0.240814
$604$	0	0
$605$	−1932.91	−0.129891
$606$	0	0
$607$	−21953.6	−1.46799	−0.733994	−	0.679156i	$-0.762346\pi$
$607$	−21953.6	−1.46799	−0.733994	+	0.679156i	$0.762346\pi$
$608$	0	0
$609$	351.404	0.0233819
$610$	0	0
$611$	14591.6	0.966141
$612$	0	0
$613$	12068.0	0.795141	0.397570	−	0.917572i	$-0.369853\pi$
$613$	12068.0	0.795141	0.397570	+	0.917572i	$0.369853\pi$
$614$	0	0
$615$	3297.59	0.216214
$616$	0	0
$617$	10838.3	0.707184	0.353592	−	0.935400i	$-0.384960\pi$
$617$	10838.3	0.707184	0.353592	+	0.935400i	$0.384960\pi$
$618$	0	0
$619$	7521.36	0.488383	0.244191	−	0.969727i	$-0.421477\pi$
$619$	7521.36	0.488383	0.244191	+	0.969727i	$0.421477\pi$
$620$	0	0
$621$	1352.71	0.0874110
$622$	0	0
$623$	−4787.23	−0.307859
$624$	0	0
$625$	13072.8	0.836661
$626$	0	0
$627$	1959.52	0.124810
$628$	0	0
$629$	−13477.4	−0.854341
$630$	0	0
$631$	−11401.1	−0.719287	−0.359643	−	0.933090i	$-0.617102\pi$
$631$	−11401.1	−0.719287	−0.359643	+	0.933090i	$0.617102\pi$
$632$	0	0
$633$	6280.72	0.394370
$634$	0	0
$635$	113.557	0.00709668
$636$	0	0
$637$	4070.23	0.253169
$638$	0	0
$639$	−7603.41	−0.470714
$640$	0	0
$641$	−6995.08	−0.431028	−0.215514	−	0.976501i	$-0.569143\pi$
$641$	−6995.08	−0.431028	−0.215514	+	0.976501i	$0.569143\pi$
$642$	0	0
$643$	964.431	0.0591500	0.0295750	−	0.999563i	$-0.490585\pi$
$643$	964.431	0.0591500	0.0295750	+	0.999563i	$0.490585\pi$
$644$	0	0
$645$	−1211.27	−0.0739436
$646$	0	0
$647$	19435.6	1.18098	0.590489	−	0.807046i	$-0.298935\pi$
$647$	19435.6	1.18098	0.590489	+	0.807046i	$0.298935\pi$
$648$	0	0
$649$	−12126.6	−0.733455
$650$	0	0
$651$	−2714.60	−0.163431
$652$	0	0
$653$	−4134.87	−0.247795	−0.123897	−	0.992295i	$-0.539539\pi$
$653$	−4134.87	−0.247795	−0.123897	+	0.992295i	$0.539539\pi$
$654$	0	0
$655$	2098.18	0.125164
$656$	0	0
$657$	8063.38	0.478817
$658$	0	0
$659$	16958.3	1.00243	0.501214	−	0.865323i	$-0.332887\pi$
$659$	16958.3	1.00243	0.501214	+	0.865323i	$0.332887\pi$
$660$	0	0
$661$	−25074.5	−1.47547	−0.737736	−	0.675090i	$-0.764105\pi$
$661$	−25074.5	−1.47547	−0.737736	+	0.675090i	$0.764105\pi$
$662$	0	0
$663$	−27469.7	−1.60910
$664$	0	0
$665$	−492.772	−0.0287351
$666$	0	0
$667$	−838.353	−0.0486674
$668$	0	0
$669$	19443.0	1.12363
$670$	0	0
$671$	6705.92	0.385811
$672$	0	0
$673$	−17078.4	−0.978195	−0.489098	−	0.872229i	$-0.662674\pi$
$673$	−17078.4	−0.978195	−0.489098	+	0.872229i	$0.662674\pi$
$674$	0	0
$675$	3187.78	0.181774
$676$	0	0
$677$	−16960.2	−0.962829	−0.481414	−	0.876493i	$-0.659877\pi$
$677$	−16960.2	−0.962829	−0.481414	+	0.876493i	$0.659877\pi$
$678$	0	0
$679$	−10015.1	−0.566045
$680$	0	0
$681$	−9004.47	−0.506684
$682$	0	0
$683$	5365.07	0.300569	0.150285	−	0.988643i	$-0.451981\pi$
$683$	5365.07	0.300569	0.150285	+	0.988643i	$0.451981\pi$
$684$	0	0
$685$	1860.30	0.103764
$686$	0	0
$687$	11531.1	0.640379
$688$	0	0
$689$	40502.5	2.23951
$690$	0	0
$691$	−17463.3	−0.961411	−0.480706	−	0.876882i	$-0.659619\pi$
$691$	−17463.3	−0.961411	−0.480706	+	0.876882i	$0.659619\pi$
$692$	0	0
$693$	1539.26	0.0843748
$694$	0	0
$695$	5175.54	0.282474
$696$	0	0
$697$	−46014.3	−2.50060
$698$	0	0
$699$	−10136.5	−0.548496
$700$	0	0
$701$	3312.57	0.178479	0.0892396	−	0.996010i	$-0.471556\pi$
$701$	3312.57	0.178479	0.0892396	+	0.996010i	$0.471556\pi$
$702$	0	0
$703$	−3268.54	−0.175356
$704$	0	0
$705$	−1387.69	−0.0741325
$706$	0	0
$707$	2305.05	0.122617
$708$	0	0
$709$	−1722.66	−0.0912494	−0.0456247	−	0.998959i	$-0.514528\pi$
$709$	−1722.66	−0.0912494	−0.0456247	+	0.998959i	$0.514528\pi$
$710$	0	0
$711$	−11797.7	−0.622293
$712$	0	0
$713$	6476.28	0.340167
$714$	0	0
$715$	−5344.26	−0.279530
$716$	0	0
$717$	−13960.9	−0.727170
$718$	0	0
$719$	−16424.6	−0.851927	−0.425964	−	0.904740i	$-0.640065\pi$
$719$	−16424.6	−0.851927	−0.425964	+	0.904740i	$0.640065\pi$
$720$	0	0
$721$	285.170	0.0147299
$722$	0	0
$723$	−15326.2	−0.788366
$724$	0	0
$725$	−1975.66	−0.101206
$726$	0	0
$727$	−3615.78	−0.184459	−0.0922297	−	0.995738i	$-0.529399\pi$
$727$	−3615.78	−0.184459	−0.0922297	+	0.995738i	$0.529399\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	16901.9	0.855184
$732$	0	0
$733$	12093.0	0.609368	0.304684	−	0.952453i	$-0.401449\pi$
$733$	12093.0	0.609368	0.304684	+	0.952453i	$0.401449\pi$
$734$	0	0
$735$	−387.088	−0.0194258
$736$	0	0
$737$	−9680.27	−0.483823
$738$	0	0
$739$	18840.3	0.937824	0.468912	−	0.883245i	$-0.344646\pi$
$739$	18840.3	0.937824	0.468912	+	0.883245i	$0.344646\pi$
$740$	0	0
$741$	−6661.93	−0.330273
$742$	0	0
$743$	−5559.28	−0.274495	−0.137248	−	0.990537i	$-0.543826\pi$
$743$	−5559.28	−0.274495	−0.137248	+	0.990537i	$0.543826\pi$
$744$	0	0
$745$	5578.64	0.274343
$746$	0	0
$747$	12288.0	0.601865
$748$	0	0
$749$	8976.92	0.437930
$750$	0	0
$751$	−30552.5	−1.48452	−0.742260	−	0.670112i	$-0.766246\pi$
$751$	−30552.5	−1.48452	−0.742260	+	0.670112i	$0.766246\pi$
$752$	0	0
$753$	−6.61658	−0.000320215 0
$754$	0	0
$755$	−505.918	−0.0243871
$756$	0	0
$757$	−20415.9	−0.980223	−0.490112	−	0.871660i	$-0.663044\pi$
$757$	−20415.9	−0.980223	−0.490112	+	0.871660i	$0.663044\pi$
$758$	0	0
$759$	−3672.26	−0.175619
$760$	0	0
$761$	−37041.9	−1.76448	−0.882238	−	0.470804i	$-0.843964\pi$
$761$	−37041.9	−1.76448	−0.882238	+	0.470804i	$0.843964\pi$
$762$	0	0
$763$	896.924	0.0425568
$764$	0	0
$765$	2612.42	0.123467
$766$	0	0
$767$	41227.9	1.94088
$768$	0	0
$769$	5163.09	0.242114	0.121057	−	0.992646i	$-0.461372\pi$
$769$	5163.09	0.242114	0.121057	+	0.992646i	$0.461372\pi$
$770$	0	0
$771$	−4045.95	−0.188990
$772$	0	0
$773$	−6634.38	−0.308696	−0.154348	−	0.988017i	$-0.549328\pi$
$773$	−6634.38	−0.308696	−0.154348	+	0.988017i	$0.549328\pi$
$774$	0	0
$775$	15262.0	0.707389
$776$	0	0
$777$	−2567.54	−0.118546
$778$	0	0
$779$	−11159.4	−0.513255
$780$	0	0
$781$	20641.4	0.945718
$782$	0	0
$783$	−451.805	−0.0206209
$784$	0	0
$785$	7877.70	0.358175
$786$	0	0
$787$	−22692.6	−1.02783	−0.513915	−	0.857841i	$-0.671805\pi$
$787$	−22692.6	−1.02783	−0.513915	+	0.857841i	$0.671805\pi$
$788$	0	0
$789$	−7937.32	−0.358145
$790$	0	0
$791$	3990.82	0.179390
$792$	0	0
$793$	−22798.7	−1.02094
$794$	0	0
$795$	−3851.87	−0.171839
$796$	0	0
$797$	−24145.0	−1.07310	−0.536549	−	0.843869i	$-0.680272\pi$
$797$	−24145.0	−1.07310	−0.536549	+	0.843869i	$0.680272\pi$
$798$	0	0
$799$	19363.7	0.857368
$800$	0	0
$801$	6155.01	0.271506
$802$	0	0
$803$	−21890.1	−0.961997
$804$	0	0
$805$	923.485	0.0404330
$806$	0	0
$807$	10136.7	0.442167
$808$	0	0
$809$	−41683.5	−1.81151	−0.905756	−	0.423800i	$-0.860696\pi$
$809$	−41683.5	−1.81151	−0.905756	+	0.423800i	$0.860696\pi$
$810$	0	0
$811$	−37899.1	−1.64096	−0.820480	−	0.571675i	$-0.806294\pi$
$811$	−37899.1	−1.64096	−0.820480	+	0.571675i	$0.806294\pi$
$812$	0	0
$813$	−8626.14	−0.372118
$814$	0	0
$815$	8120.31	0.349009
$816$	0	0
$817$	4099.05	0.175529
$818$	0	0
$819$	−5233.16	−0.223274
$820$	0	0
$821$	−8724.19	−0.370860	−0.185430	−	0.982657i	$-0.559368\pi$
$821$	−8724.19	−0.370860	−0.185430	+	0.982657i	$0.559368\pi$
$822$	0	0
$823$	12245.0	0.518633	0.259317	−	0.965792i	$-0.416503\pi$
$823$	12245.0	0.518633	0.259317	+	0.965792i	$0.416503\pi$
$824$	0	0
$825$	−8654.03	−0.365206
$826$	0	0
$827$	16038.0	0.674360	0.337180	−	0.941440i	$-0.390527\pi$
$827$	16038.0	0.674360	0.337180	+	0.941440i	$0.390527\pi$
$828$	0	0
$829$	−15359.7	−0.643504	−0.321752	−	0.946824i	$-0.604272\pi$
$829$	−15359.7	−0.643504	−0.321752	+	0.946824i	$0.604272\pi$
$830$	0	0
$831$	−9377.65	−0.391465
$832$	0	0
$833$	5401.38	0.224666
$834$	0	0
$835$	1.55745	6.45484e−5 0
$836$	0	0
$837$	3490.20	0.144132
$838$	0	0
$839$	−21290.9	−0.876095	−0.438047	−	0.898952i	$-0.644330\pi$
$839$	−21290.9	−0.876095	−0.438047	+	0.898952i	$0.644330\pi$
$840$	0	0
$841$	−24109.0	−0.988519
$842$	0	0
$843$	9556.14	0.390428
$844$	0	0
$845$	12384.1	0.504171
$846$	0	0
$847$	5138.29	0.208446
$848$	0	0
$849$	7549.01	0.305161
$850$	0	0
$851$	6125.46	0.246743
$852$	0	0
$853$	−31764.2	−1.27501	−0.637506	−	0.770446i	$-0.720034\pi$
$853$	−31764.2	−1.27501	−0.637506	+	0.770446i	$0.720034\pi$
$854$	0	0
$855$	633.564	0.0253420
$856$	0	0
$857$	25128.3	1.00159	0.500797	−	0.865565i	$-0.333040\pi$
$857$	25128.3	1.00159	0.500797	+	0.865565i	$0.333040\pi$
$858$	0	0
$859$	−37614.5	−1.49405	−0.747026	−	0.664795i	$-0.768519\pi$
$859$	−37614.5	−1.49405	−0.747026	+	0.664795i	$0.768519\pi$
$860$	0	0
$861$	−8766.03	−0.346975
$862$	0	0
$863$	16990.0	0.670158	0.335079	−	0.942190i	$-0.391237\pi$
$863$	16990.0	0.670158	0.335079	+	0.942190i	$0.391237\pi$
$864$	0	0
$865$	8459.85	0.332536
$866$	0	0
$867$	−21714.4	−0.850590
$868$	0	0
$869$	32027.9	1.25026
$870$	0	0
$871$	32910.8	1.28030
$872$	0	0
$873$	12876.6	0.499204
$874$	0	0
$875$	4480.37	0.173102
$876$	0	0
$877$	39804.7	1.53262	0.766312	−	0.642469i	$-0.222090\pi$
$877$	39804.7	1.53262	0.766312	+	0.642469i	$0.222090\pi$
$878$	0	0
$879$	2935.23	0.112631
$880$	0	0
$881$	42647.7	1.63092	0.815458	−	0.578816i	$-0.196485\pi$
$881$	42647.7	1.63092	0.815458	+	0.578816i	$0.196485\pi$
$882$	0	0
$883$	42853.2	1.63321	0.816605	−	0.577198i	$-0.195854\pi$
$883$	42853.2	1.63321	0.816605	+	0.577198i	$0.195854\pi$
$884$	0	0
$885$	−3920.86	−0.148925
$886$	0	0
$887$	−13409.9	−0.507622	−0.253811	−	0.967254i	$-0.581684\pi$
$887$	−13409.9	−0.507622	−0.253811	+	0.967254i	$0.581684\pi$
$888$	0	0
$889$	−301.871	−0.0113886
$890$	0	0
$891$	−1979.05	−0.0744116
$892$	0	0
$893$	4696.07	0.175978
$894$	0	0
$895$	1643.62	0.0613856
$896$	0	0
$897$	12484.9	0.464725
$898$	0	0
$899$	−2163.08	−0.0802478
$900$	0	0
$901$	53748.6	1.98737
$902$	0	0
$903$	3219.93	0.118663
$904$	0	0
$905$	−4366.60	−0.160387
$906$	0	0
$907$	10223.5	0.374272	0.187136	−	0.982334i	$-0.440079\pi$
$907$	10223.5	0.374272	0.187136	+	0.982334i	$0.440079\pi$
$908$	0	0
$909$	−2963.64	−0.108138
$910$	0	0
$911$	12069.3	0.438938	0.219469	−	0.975619i	$-0.429567\pi$
$911$	12069.3	0.438938	0.219469	+	0.975619i	$0.429567\pi$
$912$	0	0
$913$	−33358.8	−1.20922
$914$	0	0
$915$	2168.20	0.0783372
$916$	0	0
$917$	−5577.61	−0.200860
$918$	0	0
$919$	−20488.5	−0.735421	−0.367711	−	0.929940i	$-0.619858\pi$
$919$	−20488.5	−0.735421	−0.367711	+	0.929940i	$0.619858\pi$
$920$	0	0
$921$	30993.0	1.10885
$922$	0	0
$923$	−70176.1	−2.50257
$924$	0	0
$925$	14435.2	0.513110
$926$	0	0
$927$	−366.647	−0.0129906
$928$	0	0
$929$	31335.6	1.10666	0.553330	−	0.832962i	$-0.313357\pi$
$929$	31335.6	1.10666	0.553330	+	0.832962i	$0.313357\pi$
$930$	0	0
$931$	1309.94	0.0461134
$932$	0	0
$933$	−27667.7	−0.970847
$934$	0	0
$935$	−7092.07	−0.248059
$936$	0	0
$937$	−2075.56	−0.0723646	−0.0361823	−	0.999345i	$-0.511520\pi$
$937$	−2075.56	−0.0723646	−0.0361823	+	0.999345i	$0.511520\pi$
$938$	0	0
$939$	17438.5	0.606054
$940$	0	0
$941$	35604.9	1.23346	0.616730	−	0.787174i	$-0.288457\pi$
$941$	35604.9	1.23346	0.616730	+	0.787174i	$0.288457\pi$
$942$	0	0
$943$	20913.4	0.722198
$944$	0	0
$945$	497.684	0.0171319
$946$	0	0
$947$	−30879.9	−1.05962	−0.529810	−	0.848116i	$-0.677737\pi$
$947$	−30879.9	−1.05962	−0.529810	+	0.848116i	$0.677737\pi$
$948$	0	0
$949$	74421.4	2.54565
$950$	0	0
$951$	−15840.8	−0.540140
$952$	0	0
$953$	35653.5	1.21189	0.605944	−	0.795507i	$-0.292795\pi$
$953$	35653.5	1.21189	0.605944	+	0.795507i	$0.292795\pi$
$954$	0	0
$955$	10280.8	0.348355
$956$	0	0
$957$	1226.54	0.0414298
$958$	0	0
$959$	−4945.25	−0.166518
$960$	0	0
$961$	−13081.2	−0.439098
$962$	0	0
$963$	−11541.8	−0.386218
$964$	0	0
$965$	5889.35	0.196461
$966$	0	0
$967$	44294.9	1.47304	0.736520	−	0.676416i	$-0.236468\pi$
$967$	44294.9	1.47304	0.736520	+	0.676416i	$0.236468\pi$
$968$	0	0
$969$	−8840.68	−0.293089
$970$	0	0
$971$	37669.1	1.24496	0.622482	−	0.782634i	$-0.286124\pi$
$971$	37669.1	1.24496	0.622482	+	0.782634i	$0.286124\pi$
$972$	0	0
$973$	−13758.2	−0.453307
$974$	0	0
$975$	29421.8	0.966412
$976$	0	0
$977$	−41450.1	−1.35732	−0.678662	−	0.734451i	$-0.737440\pi$
$977$	−41450.1	−1.35732	−0.678662	+	0.734451i	$0.737440\pi$
$978$	0	0
$979$	−16709.3	−0.545487
$980$	0	0
$981$	−1153.19	−0.0375316
$982$	0	0
$983$	−41027.9	−1.33122	−0.665609	−	0.746301i	$-0.731828\pi$
$983$	−41027.9	−1.33122	−0.665609	+	0.746301i	$0.731828\pi$
$984$	0	0
$985$	−3360.68	−0.108711
$986$	0	0
$987$	3688.91	0.118966
$988$	0	0
$989$	−7681.87	−0.246986
$990$	0	0
$991$	−2049.50	−0.0656959	−0.0328479	−	0.999460i	$-0.510458\pi$
$991$	−2049.50	−0.0656959	−0.0328479	+	0.999460i	$0.510458\pi$
$992$	0	0
$993$	22388.0	0.715470
$994$	0	0
$995$	10715.1	0.341398
$996$	0	0
$997$	12662.9	0.402245	0.201122	−	0.979566i	$-0.435541\pi$
$997$	12662.9	0.402245	0.201122	+	0.979566i	$0.435541\pi$
$998$	0	0
$999$	3301.13	0.104548

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bf.1.2		2
4.3	odd	2		1344.4.a.bn.1.2		2
8.3	odd	2		672.4.a.g.1.1	✓	2
8.5	even	2		672.4.a.l.1.1	yes	2
24.5	odd	2		2016.4.a.k.1.2		2
24.11	even	2		2016.4.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.g.1.1	✓	2	8.3	odd	2
672.4.a.l.1.1	yes	2	8.5	even	2
1344.4.a.bf.1.2		2	1.1	even	1	trivial
1344.4.a.bn.1.2		2	4.3	odd	2
2016.4.a.k.1.2		2	24.5	odd	2
2016.4.a.l.1.2		2	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +2.63325 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +2.63325 q^{5} -7.00000 q^{7} +9.00000 q^{9} -24.4327 q^{11} +83.0660 q^{13} -7.89975 q^{15} +110.232 q^{17} +26.7335 q^{19} +21.0000 q^{21} -50.1003 q^{23} -118.066 q^{25} -27.0000 q^{27} +16.7335 q^{29} -129.266 q^{31} +73.2982 q^{33} -18.4327 q^{35} -122.264 q^{37} -249.198 q^{39} -417.430 q^{41} +153.330 q^{43} +23.6992 q^{45} +175.662 q^{47} +49.0000 q^{49} -330.697 q^{51} +487.594 q^{53} -64.3375 q^{55} -80.2005 q^{57} +496.327 q^{59} -274.464 q^{61} -63.0000 q^{63} +218.734 q^{65} +396.201 q^{67} +150.301 q^{69} -844.824 q^{71} +895.931 q^{73} +354.198 q^{75} +171.029 q^{77} -1310.86 q^{79} +81.0000 q^{81} +1365.33 q^{83} +290.269 q^{85} -50.2005 q^{87} +683.890 q^{89} -581.462 q^{91} +387.799 q^{93} +70.3960 q^{95} +1430.73 q^{97} -219.895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 8 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 8 * q^5 - 14 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} - 8 q^{5} - 14 q^{7} + 18 q^{9} + 44 q^{11} + 60 q^{13} + 24 q^{15} + 48 q^{17} + 80 q^{19} + 42 q^{21} - 140 q^{23} - 130 q^{25} - 54 q^{27} + 60 q^{29} - 232 q^{31} - 132 q^{33} + 56 q^{35} + 180 q^{37} - 180 q^{39} - 344 q^{41} - 224 q^{43} - 72 q^{45} - 312 q^{47} + 98 q^{49} - 144 q^{51} + 20 q^{53} - 792 q^{55} - 240 q^{57} + 64 q^{59} - 204 q^{61} - 126 q^{63} + 464 q^{65} + 872 q^{67} + 420 q^{69} - 164 q^{71} + 1500 q^{73} + 390 q^{75} - 308 q^{77} - 1640 q^{79} + 162 q^{81} + 2200 q^{83} + 952 q^{85} - 180 q^{87} - 264 q^{89} - 420 q^{91} + 696 q^{93} - 496 q^{95} + 2092 q^{97} + 396 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 8 * q^5 - 14 * q^7 + 18 * q^9 + 44 * q^11 + 60 * q^13 + 24 * q^15 + 48 * q^17 + 80 * q^19 + 42 * q^21 - 140 * q^23 - 130 * q^25 - 54 * q^27 + 60 * q^29 - 232 * q^31 - 132 * q^33 + 56 * q^35 + 180 * q^37 - 180 * q^39 - 344 * q^41 - 224 * q^43 - 72 * q^45 - 312 * q^47 + 98 * q^49 - 144 * q^51 + 20 * q^53 - 792 * q^55 - 240 * q^57 + 64 * q^59 - 204 * q^61 - 126 * q^63 + 464 * q^65 + 872 * q^67 + 420 * q^69 - 164 * q^71 + 1500 * q^73 + 390 * q^75 - 308 * q^77 - 1640 * q^79 + 162 * q^81 + 2200 * q^83 + 952 * q^85 - 180 * q^87 - 264 * q^89 - 420 * q^91 + 696 * q^93 - 496 * q^95 + 2092 * q^97 + 396 * q^99

Introduction

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