LMFDB - Embedded newform 6044.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, 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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$1$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.31051 q^{3} +0.0245297 q^{5} -4.08379 q^{7} +7.95946 q^{9} +O(q^{10})$	$q-3.31051 q^{3} +0.0245297 q^{5} -4.08379 q^{7} +7.95946 q^{9} -5.14741 q^{11} -2.56189 q^{13} -0.0812057 q^{15} +3.31301 q^{17} -4.52227 q^{19} +13.5194 q^{21} +3.46001 q^{23} -4.99940 q^{25} -16.4183 q^{27} +3.20315 q^{29} +7.64495 q^{31} +17.0405 q^{33} -0.100174 q^{35} +8.23230 q^{37} +8.48116 q^{39} -2.54639 q^{41} -3.73243 q^{43} +0.195243 q^{45} +8.43977 q^{47} +9.67733 q^{49} -10.9678 q^{51} -4.36150 q^{53} -0.126264 q^{55} +14.9710 q^{57} +4.35220 q^{59} -12.9772 q^{61} -32.5048 q^{63} -0.0628423 q^{65} +11.6291 q^{67} -11.4544 q^{69} -4.00098 q^{71} +7.01673 q^{73} +16.5505 q^{75} +21.0209 q^{77} -3.62527 q^{79} +30.4747 q^{81} -5.72854 q^{83} +0.0812671 q^{85} -10.6041 q^{87} -14.0316 q^{89} +10.4622 q^{91} -25.3087 q^{93} -0.110930 q^{95} +17.5332 q^{97} -40.9706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * 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
$3$	−3.31051	−1.91132	−0.955661	−	0.294468i	$-0.904858\pi$
$3$	−3.31051	−1.91132	−0.955661	+	0.294468i	$0.904858\pi$
$4$	0	0
$5$	0.0245297	0.0109700	0.00548500	−	0.999985i	$-0.498254\pi$
$5$	0.0245297	0.0109700	0.00548500	+	0.999985i	$0.498254\pi$
$6$	0	0
$7$	−4.08379	−1.54353	−0.771764	−	0.635910i	$-0.780625\pi$
$7$	−4.08379	−1.54353	−0.771764	+	0.635910i	$0.780625\pi$
$8$	0	0
$9$	7.95946	2.65315
$10$	0	0
$11$	−5.14741	−1.55200	−0.776001	−	0.630731i	$-0.782755\pi$
$11$	−5.14741	−1.55200	−0.776001	+	0.630731i	$0.782755\pi$
$12$	0	0
$13$	−2.56189	−0.710540	−0.355270	−	0.934764i	$-0.615611\pi$
$13$	−2.56189	−0.710540	−0.355270	+	0.934764i	$0.615611\pi$
$14$	0	0
$15$	−0.0812057	−0.0209672
$16$	0	0
$17$	3.31301	0.803524	0.401762	−	0.915744i	$-0.368398\pi$
$17$	3.31301	0.803524	0.401762	+	0.915744i	$0.368398\pi$
$18$	0	0
$19$	−4.52227	−1.03748	−0.518740	−	0.854932i	$-0.673599\pi$
$19$	−4.52227	−1.03748	−0.518740	+	0.854932i	$0.673599\pi$
$20$	0	0
$21$	13.5194	2.95018
$22$	0	0
$23$	3.46001	0.721462	0.360731	−	0.932670i	$-0.382527\pi$
$23$	3.46001	0.721462	0.360731	+	0.932670i	$0.382527\pi$
$24$	0	0
$25$	−4.99940	−0.999880
$26$	0	0
$27$	−16.4183	−3.15971
$28$	0	0
$29$	3.20315	0.594810	0.297405	−	0.954751i	$-0.403879\pi$
$29$	3.20315	0.594810	0.297405	+	0.954751i	$0.403879\pi$
$30$	0	0
$31$	7.64495	1.37307	0.686537	−	0.727095i	$-0.259130\pi$
$31$	7.64495	1.37307	0.686537	+	0.727095i	$0.259130\pi$
$32$	0	0
$33$	17.0405	2.96638
$34$	0	0
$35$	−0.100174	−0.0169325
$36$	0	0
$37$	8.23230	1.35338	0.676691	−	0.736268i	$-0.263413\pi$
$37$	8.23230	1.35338	0.676691	+	0.736268i	$0.263413\pi$
$38$	0	0
$39$	8.48116	1.35807
$40$	0	0
$41$	−2.54639	−0.397679	−0.198839	−	0.980032i	$-0.563717\pi$
$41$	−2.54639	−0.397679	−0.198839	+	0.980032i	$0.563717\pi$
$42$	0	0
$43$	−3.73243	−0.569190	−0.284595	−	0.958648i	$-0.591859\pi$
$43$	−3.73243	−0.569190	−0.284595	+	0.958648i	$0.591859\pi$
$44$	0	0
$45$	0.195243	0.0291051
$46$	0	0
$47$	8.43977	1.23107	0.615533	−	0.788111i	$-0.288941\pi$
$47$	8.43977	1.23107	0.615533	+	0.788111i	$0.288941\pi$
$48$	0	0
$49$	9.67733	1.38248
$50$	0	0
$51$	−10.9678	−1.53579
$52$	0	0
$53$	−4.36150	−0.599098	−0.299549	−	0.954081i	$-0.596836\pi$
$53$	−4.36150	−0.599098	−0.299549	+	0.954081i	$0.596836\pi$
$54$	0	0
$55$	−0.126264	−0.0170255
$56$	0	0
$57$	14.9710	1.98296
$58$	0	0
$59$	4.35220	0.566608	0.283304	−	0.959030i	$-0.408569\pi$
$59$	4.35220	0.566608	0.283304	+	0.959030i	$0.408569\pi$
$60$	0	0
$61$	−12.9772	−1.66157	−0.830783	−	0.556597i	$-0.812107\pi$
$61$	−12.9772	−1.66157	−0.830783	+	0.556597i	$0.812107\pi$
$62$	0	0
$63$	−32.5048	−4.09522
$64$	0	0
$65$	−0.0628423	−0.00779463
$66$	0	0
$67$	11.6291	1.42072	0.710362	−	0.703836i	$-0.248531\pi$
$67$	11.6291	1.42072	0.710362	+	0.703836i	$0.248531\pi$
$68$	0	0
$69$	−11.4544	−1.37895
$70$	0	0
$71$	−4.00098	−0.474829	−0.237414	−	0.971408i	$-0.576300\pi$
$71$	−4.00098	−0.474829	−0.237414	+	0.971408i	$0.576300\pi$
$72$	0	0
$73$	7.01673	0.821246	0.410623	−	0.911805i	$-0.365311\pi$
$73$	7.01673	0.821246	0.410623	+	0.911805i	$0.365311\pi$
$74$	0	0
$75$	16.5505	1.91109
$76$	0	0
$77$	21.0209	2.39556
$78$	0	0
$79$	−3.62527	−0.407875	−0.203937	−	0.978984i	$-0.565374\pi$
$79$	−3.62527	−0.407875	−0.203937	+	0.978984i	$0.565374\pi$
$80$	0	0
$81$	30.4747	3.38607
$82$	0	0
$83$	−5.72854	−0.628789	−0.314395	−	0.949292i	$-0.601802\pi$
$83$	−5.72854	−0.628789	−0.314395	+	0.949292i	$0.601802\pi$
$84$	0	0
$85$	0.0812671	0.00881466
$86$	0	0
$87$	−10.6041	−1.13687
$88$	0	0
$89$	−14.0316	−1.48734	−0.743672	−	0.668544i	$-0.766918\pi$
$89$	−14.0316	−1.48734	−0.743672	+	0.668544i	$0.766918\pi$
$90$	0	0
$91$	10.4622	1.09674
$92$	0	0
$93$	−25.3087	−2.62439
$94$	0	0
$95$	−0.110930	−0.0113812
$96$	0	0
$97$	17.5332	1.78023	0.890114	−	0.455737i	$-0.150624\pi$
$97$	17.5332	1.78023	0.890114	+	0.455737i	$0.150624\pi$
$98$	0	0
$99$	−40.9706	−4.11770
$100$	0	0
$101$	5.32179	0.529538	0.264769	−	0.964312i	$-0.414704\pi$
$101$	5.32179	0.529538	0.264769	+	0.964312i	$0.414704\pi$
$102$	0	0
$103$	15.0111	1.47909	0.739544	−	0.673109i	$-0.235041\pi$
$103$	15.0111	1.47909	0.739544	+	0.673109i	$0.235041\pi$
$104$	0	0
$105$	0.331627	0.0323635
$106$	0	0
$107$	−17.9603	−1.73628	−0.868142	−	0.496316i	$-0.834686\pi$
$107$	−17.9603	−1.73628	−0.868142	+	0.496316i	$0.834686\pi$
$108$	0	0
$109$	7.82579	0.749575	0.374787	−	0.927111i	$-0.377716\pi$
$109$	7.82579	0.749575	0.374787	+	0.927111i	$0.377716\pi$
$110$	0	0
$111$	−27.2531	−2.58675
$112$	0	0
$113$	19.4919	1.83364	0.916822	−	0.399295i	$-0.130745\pi$
$113$	19.4919	1.83364	0.916822	+	0.399295i	$0.130745\pi$
$114$	0	0
$115$	0.0848729	0.00791443
$116$	0	0
$117$	−20.3913	−1.88517
$118$	0	0
$119$	−13.5296	−1.24026
$120$	0	0
$121$	15.4958	1.40871
$122$	0	0
$123$	8.42984	0.760093
$124$	0	0
$125$	−0.245282	−0.0219387
$126$	0	0
$127$	3.38965	0.300783	0.150391	−	0.988627i	$-0.451947\pi$
$127$	3.38965	0.300783	0.150391	+	0.988627i	$0.451947\pi$
$128$	0	0
$129$	12.3562	1.08791
$130$	0	0
$131$	7.33129	0.640537	0.320269	−	0.947327i	$-0.396227\pi$
$131$	7.33129	0.640537	0.320269	+	0.947327i	$0.396227\pi$
$132$	0	0
$133$	18.4680	1.60138
$134$	0	0
$135$	−0.402737	−0.0346620
$136$	0	0
$137$	−6.36728	−0.543993	−0.271997	−	0.962298i	$-0.587684\pi$
$137$	−6.36728	−0.543993	−0.271997	+	0.962298i	$0.587684\pi$
$138$	0	0
$139$	14.1365	1.19904	0.599520	−	0.800360i	$-0.295358\pi$
$139$	14.1365	1.19904	0.599520	+	0.800360i	$0.295358\pi$
$140$	0	0
$141$	−27.9399	−2.35297
$142$	0	0
$143$	13.1871	1.10276
$144$	0	0
$145$	0.0785723	0.00652507
$146$	0	0
$147$	−32.0369	−2.64236
$148$	0	0
$149$	11.1650	0.914671	0.457336	−	0.889294i	$-0.348804\pi$
$149$	11.1650	0.914671	0.457336	+	0.889294i	$0.348804\pi$
$150$	0	0
$151$	7.73453	0.629427	0.314714	−	0.949187i	$-0.398091\pi$
$151$	7.73453	0.629427	0.314714	+	0.949187i	$0.398091\pi$
$152$	0	0
$153$	26.3698	2.13187
$154$	0	0
$155$	0.187528	0.0150626
$156$	0	0
$157$	0.559386	0.0446439	0.0223219	−	0.999751i	$-0.492894\pi$
$157$	0.559386	0.0446439	0.0223219	+	0.999751i	$0.492894\pi$
$158$	0	0
$159$	14.4388	1.14507
$160$	0	0
$161$	−14.1299	−1.11360
$162$	0	0
$163$	−6.19618	−0.485322	−0.242661	−	0.970111i	$-0.578020\pi$
$163$	−6.19618	−0.485322	−0.242661	+	0.970111i	$0.578020\pi$
$164$	0	0
$165$	0.417999	0.0325412
$166$	0	0
$167$	14.6295	1.13207	0.566033	−	0.824382i	$-0.308477\pi$
$167$	14.6295	1.13207	0.566033	+	0.824382i	$0.308477\pi$
$168$	0	0
$169$	−6.43672	−0.495132
$170$	0	0
$171$	−35.9948	−2.75259
$172$	0	0
$173$	4.50351	0.342395	0.171198	−	0.985237i	$-0.445236\pi$
$173$	4.50351	0.342395	0.171198	+	0.985237i	$0.445236\pi$
$174$	0	0
$175$	20.4165	1.54334
$176$	0	0
$177$	−14.4080	−1.08297
$178$	0	0
$179$	−15.6372	−1.16878	−0.584391	−	0.811472i	$-0.698667\pi$
$179$	−15.6372	−1.16878	−0.584391	+	0.811472i	$0.698667\pi$
$180$	0	0
$181$	14.8975	1.10732	0.553662	−	0.832741i	$-0.313230\pi$
$181$	14.8975	1.10732	0.553662	+	0.832741i	$0.313230\pi$
$182$	0	0
$183$	42.9613	3.17579
$184$	0	0
$185$	0.201936	0.0148466
$186$	0	0
$187$	−17.0534	−1.24707
$188$	0	0
$189$	67.0490	4.87710
$190$	0	0
$191$	−24.2245	−1.75282	−0.876412	−	0.481562i	$-0.840070\pi$
$191$	−24.2245	−1.75282	−0.876412	+	0.481562i	$0.840070\pi$
$192$	0	0
$193$	−17.2831	−1.24407	−0.622033	−	0.782991i	$-0.713693\pi$
$193$	−17.2831	−1.24407	−0.622033	+	0.782991i	$0.713693\pi$
$194$	0	0
$195$	0.208040	0.0148981
$196$	0	0
$197$	8.00826	0.570565	0.285282	−	0.958444i	$-0.407913\pi$
$197$	8.00826	0.570565	0.285282	+	0.958444i	$0.407913\pi$
$198$	0	0
$199$	−17.7052	−1.25509	−0.627543	−	0.778582i	$-0.715939\pi$
$199$	−17.7052	−1.25509	−0.627543	+	0.778582i	$0.715939\pi$
$200$	0	0
$201$	−38.4983	−2.71546
$202$	0	0
$203$	−13.0810	−0.918106
$204$	0	0
$205$	−0.0624620	−0.00436254
$206$	0	0
$207$	27.5398	1.91415
$208$	0	0
$209$	23.2780	1.61017
$210$	0	0
$211$	−5.29730	−0.364681	−0.182341	−	0.983235i	$-0.558367\pi$
$211$	−5.29730	−0.364681	−0.182341	+	0.983235i	$0.558367\pi$
$212$	0	0
$213$	13.2453	0.907551
$214$	0	0
$215$	−0.0915553	−0.00624402
$216$	0	0
$217$	−31.2204	−2.11938
$218$	0	0
$219$	−23.2290	−1.56967
$220$	0	0
$221$	−8.48758	−0.570936
$222$	0	0
$223$	1.25482	0.0840291	0.0420146	−	0.999117i	$-0.486622\pi$
$223$	1.25482	0.0840291	0.0420146	+	0.999117i	$0.486622\pi$
$224$	0	0
$225$	−39.7925	−2.65283
$226$	0	0
$227$	−12.2898	−0.815702	−0.407851	−	0.913048i	$-0.633722\pi$
$227$	−12.2898	−0.815702	−0.407851	+	0.913048i	$0.633722\pi$
$228$	0	0
$229$	−19.5489	−1.29183	−0.645915	−	0.763409i	$-0.723524\pi$
$229$	−19.5489	−1.29183	−0.645915	+	0.763409i	$0.723524\pi$
$230$	0	0
$231$	−69.5900	−4.57868
$232$	0	0
$233$	−17.3373	−1.13581	−0.567903	−	0.823095i	$-0.692245\pi$
$233$	−17.3373	−1.13581	−0.567903	+	0.823095i	$0.692245\pi$
$234$	0	0
$235$	0.207025	0.0135048
$236$	0	0
$237$	12.0015	0.779580
$238$	0	0
$239$	4.53451	0.293313	0.146656	−	0.989187i	$-0.453149\pi$
$239$	4.53451	0.293313	0.146656	+	0.989187i	$0.453149\pi$
$240$	0	0
$241$	−23.6712	−1.52480	−0.762399	−	0.647107i	$-0.775979\pi$
$241$	−23.6712	−1.52480	−0.762399	+	0.647107i	$0.775979\pi$
$242$	0	0
$243$	−51.6316	−3.31217
$244$	0	0
$245$	0.237382	0.0151658
$246$	0	0
$247$	11.5856	0.737172
$248$	0	0
$249$	18.9644	1.20182
$250$	0	0
$251$	29.2340	1.84524	0.922618	−	0.385716i	$-0.126045\pi$
$251$	29.2340	1.84524	0.922618	+	0.385716i	$0.126045\pi$
$252$	0	0
$253$	−17.8101	−1.11971
$254$	0	0
$255$	−0.269036	−0.0168477
$256$	0	0
$257$	−5.08038	−0.316906	−0.158453	−	0.987367i	$-0.550651\pi$
$257$	−5.08038	−0.316906	−0.158453	+	0.987367i	$0.550651\pi$
$258$	0	0
$259$	−33.6190	−2.08898
$260$	0	0
$261$	25.4954	1.57812
$262$	0	0
$263$	17.5062	1.07948	0.539738	−	0.841833i	$-0.318523\pi$
$263$	17.5062	1.07948	0.539738	+	0.841833i	$0.318523\pi$
$264$	0	0
$265$	−0.106986	−0.00657210
$266$	0	0
$267$	46.4517	2.84280
$268$	0	0
$269$	−10.5726	−0.644624	−0.322312	−	0.946633i	$-0.604460\pi$
$269$	−10.5726	−0.644624	−0.322312	+	0.946633i	$0.604460\pi$
$270$	0	0
$271$	24.7009	1.50047	0.750237	−	0.661169i	$-0.229940\pi$
$271$	24.7009	1.50047	0.750237	+	0.661169i	$0.229940\pi$
$272$	0	0
$273$	−34.6353	−2.09622
$274$	0	0
$275$	25.7339	1.55182
$276$	0	0
$277$	2.92622	0.175820	0.0879099	−	0.996128i	$-0.471981\pi$
$277$	2.92622	0.175820	0.0879099	+	0.996128i	$0.471981\pi$
$278$	0	0
$279$	60.8497	3.64298
$280$	0	0
$281$	−18.2509	−1.08876	−0.544378	−	0.838840i	$-0.683234\pi$
$281$	−18.2509	−1.08876	−0.544378	+	0.838840i	$0.683234\pi$
$282$	0	0
$283$	2.26554	0.134672	0.0673361	−	0.997730i	$-0.478550\pi$
$283$	2.26554	0.134672	0.0673361	+	0.997730i	$0.478550\pi$
$284$	0	0
$285$	0.367234	0.0217531
$286$	0	0
$287$	10.3989	0.613828
$288$	0	0
$289$	−6.02394	−0.354349
$290$	0	0
$291$	−58.0439	−3.40259
$292$	0	0
$293$	−13.0432	−0.761992	−0.380996	−	0.924577i	$-0.624419\pi$
$293$	−13.0432	−0.761992	−0.380996	+	0.924577i	$0.624419\pi$
$294$	0	0
$295$	0.106758	0.00621570
$296$	0	0
$297$	84.5119	4.90388
$298$	0	0
$299$	−8.86416	−0.512628
$300$	0	0
$301$	15.2425	0.878560
$302$	0	0
$303$	−17.6178	−1.01212
$304$	0	0
$305$	−0.318327	−0.0182274
$306$	0	0
$307$	13.8040	0.787837	0.393919	−	0.919145i	$-0.371119\pi$
$307$	13.8040	0.787837	0.393919	+	0.919145i	$0.371119\pi$
$308$	0	0
$309$	−49.6944	−2.82701
$310$	0	0
$311$	6.64222	0.376646	0.188323	−	0.982107i	$-0.439695\pi$
$311$	6.64222	0.376646	0.188323	+	0.982107i	$0.439695\pi$
$312$	0	0
$313$	2.50076	0.141351	0.0706755	−	0.997499i	$-0.477485\pi$
$313$	2.50076	0.141351	0.0706755	+	0.997499i	$0.477485\pi$
$314$	0	0
$315$	−0.797331	−0.0449245
$316$	0	0
$317$	−30.5894	−1.71807	−0.859035	−	0.511916i	$-0.828936\pi$
$317$	−30.5894	−1.71807	−0.859035	+	0.511916i	$0.828936\pi$
$318$	0	0
$319$	−16.4879	−0.923147
$320$	0	0
$321$	59.4576	3.31860
$322$	0	0
$323$	−14.9823	−0.833640
$324$	0	0
$325$	12.8079	0.710455
$326$	0	0
$327$	−25.9073	−1.43268
$328$	0	0
$329$	−34.4662	−1.90018
$330$	0	0
$331$	−36.1939	−1.98939	−0.994697	−	0.102847i	$-0.967205\pi$
$331$	−36.1939	−1.98939	−0.994697	+	0.102847i	$0.967205\pi$
$332$	0	0
$333$	65.5247	3.59073
$334$	0	0
$335$	0.285259	0.0155854
$336$	0	0
$337$	28.9170	1.57521	0.787604	−	0.616181i	$-0.211321\pi$
$337$	28.9170	1.57521	0.787604	+	0.616181i	$0.211321\pi$
$338$	0	0
$339$	−64.5281	−3.50469
$340$	0	0
$341$	−39.3517	−2.13101
$342$	0	0
$343$	−10.9337	−0.590362
$344$	0	0
$345$	−0.280972	−0.0151270
$346$	0	0
$347$	26.5897	1.42741	0.713705	−	0.700446i	$-0.247016\pi$
$347$	26.5897	1.42741	0.713705	+	0.700446i	$0.247016\pi$
$348$	0	0
$349$	−7.36884	−0.394445	−0.197223	−	0.980359i	$-0.563192\pi$
$349$	−7.36884	−0.394445	−0.197223	+	0.980359i	$0.563192\pi$
$350$	0	0
$351$	42.0620	2.24510
$352$	0	0
$353$	5.69115	0.302909	0.151455	−	0.988464i	$-0.451604\pi$
$353$	5.69115	0.302909	0.151455	+	0.988464i	$0.451604\pi$
$354$	0	0
$355$	−0.0981427	−0.00520887
$356$	0	0
$357$	44.7900	2.37054
$358$	0	0
$359$	−5.09161	−0.268725	−0.134362	−	0.990932i	$-0.542899\pi$
$359$	−5.09161	−0.268725	−0.134362	+	0.990932i	$0.542899\pi$
$360$	0	0
$361$	1.45093	0.0763649
$362$	0	0
$363$	−51.2990	−2.69250
$364$	0	0
$365$	0.172118	0.00900908
$366$	0	0
$367$	−21.7827	−1.13705	−0.568525	−	0.822666i	$-0.692486\pi$
$367$	−21.7827	−1.13705	−0.568525	+	0.822666i	$0.692486\pi$
$368$	0	0
$369$	−20.2679	−1.05510
$370$	0	0
$371$	17.8114	0.924723
$372$	0	0
$373$	−22.7441	−1.17764	−0.588822	−	0.808263i	$-0.700408\pi$
$373$	−22.7441	−1.17764	−0.588822	+	0.808263i	$0.700408\pi$
$374$	0	0
$375$	0.812008	0.0419319
$376$	0	0
$377$	−8.20612	−0.422637
$378$	0	0
$379$	−7.38241	−0.379209	−0.189604	−	0.981861i	$-0.560721\pi$
$379$	−7.38241	−0.379209	−0.189604	+	0.981861i	$0.560721\pi$
$380$	0	0
$381$	−11.2215	−0.574893
$382$	0	0
$383$	−10.8097	−0.552349	−0.276174	−	0.961108i	$-0.589067\pi$
$383$	−10.8097	−0.552349	−0.276174	+	0.961108i	$0.589067\pi$
$384$	0	0
$385$	0.515637	0.0262793
$386$	0	0
$387$	−29.7081	−1.51015
$388$	0	0
$389$	12.0514	0.611030	0.305515	−	0.952187i	$-0.401171\pi$
$389$	12.0514	0.611030	0.305515	+	0.952187i	$0.401171\pi$
$390$	0	0
$391$	11.4631	0.579712
$392$	0	0
$393$	−24.2703	−1.22427
$394$	0	0
$395$	−0.0889267	−0.00447439
$396$	0	0
$397$	24.4195	1.22558	0.612791	−	0.790245i	$-0.290047\pi$
$397$	24.4195	1.22558	0.612791	+	0.790245i	$0.290047\pi$
$398$	0	0
$399$	−61.1385	−3.06075
$400$	0	0
$401$	1.82917	0.0913446	0.0456723	−	0.998956i	$-0.485457\pi$
$401$	1.82917	0.0913446	0.0456723	+	0.998956i	$0.485457\pi$
$402$	0	0
$403$	−19.5855	−0.975624
$404$	0	0
$405$	0.747533	0.0371452
$406$	0	0
$407$	−42.3750	−2.10045
$408$	0	0
$409$	−24.6309	−1.21792	−0.608960	−	0.793201i	$-0.708413\pi$
$409$	−24.6309	−1.21792	−0.608960	+	0.793201i	$0.708413\pi$
$410$	0	0
$411$	21.0789	1.03975
$412$	0	0
$413$	−17.7735	−0.874575
$414$	0	0
$415$	−0.140519	−0.00689782
$416$	0	0
$417$	−46.7989	−2.29175
$418$	0	0
$419$	−5.21610	−0.254823	−0.127412	−	0.991850i	$-0.540667\pi$
$419$	−5.21610	−0.254823	−0.127412	+	0.991850i	$0.540667\pi$
$420$	0	0
$421$	−21.1837	−1.03243	−0.516214	−	0.856459i	$-0.672659\pi$
$421$	−21.1837	−1.03243	−0.516214	+	0.856459i	$0.672659\pi$
$422$	0	0
$423$	67.1760	3.26621
$424$	0	0
$425$	−16.5631	−0.803427
$426$	0	0
$427$	52.9963	2.56467
$428$	0	0
$429$	−43.6560	−2.10773
$430$	0	0
$431$	33.3927	1.60847	0.804235	−	0.594311i	$-0.202575\pi$
$431$	33.3927	1.60847	0.804235	+	0.594311i	$0.202575\pi$
$432$	0	0
$433$	20.5782	0.988924	0.494462	−	0.869199i	$-0.335365\pi$
$433$	20.5782	0.988924	0.494462	+	0.869199i	$0.335365\pi$
$434$	0	0
$435$	−0.260114	−0.0124715
$436$	0	0
$437$	−15.6471	−0.748502
$438$	0	0
$439$	−6.75142	−0.322228	−0.161114	−	0.986936i	$-0.551509\pi$
$439$	−6.75142	−0.322228	−0.161114	+	0.986936i	$0.551509\pi$
$440$	0	0
$441$	77.0264	3.66792
$442$	0	0
$443$	−15.9338	−0.757035	−0.378518	−	0.925594i	$-0.623566\pi$
$443$	−15.9338	−0.757035	−0.378518	+	0.925594i	$0.623566\pi$
$444$	0	0
$445$	−0.344190	−0.0163162
$446$	0	0
$447$	−36.9618	−1.74823
$448$	0	0
$449$	−12.6834	−0.598567	−0.299284	−	0.954164i	$-0.596748\pi$
$449$	−12.6834	−0.598567	−0.299284	+	0.954164i	$0.596748\pi$
$450$	0	0
$451$	13.1073	0.617199
$452$	0	0
$453$	−25.6052	−1.20304
$454$	0	0
$455$	0.256635	0.0120312
$456$	0	0
$457$	27.4515	1.28413	0.642063	−	0.766652i	$-0.278079\pi$
$457$	27.4515	1.28413	0.642063	+	0.766652i	$0.278079\pi$
$458$	0	0
$459$	−54.3942	−2.53890
$460$	0	0
$461$	29.4668	1.37241	0.686203	−	0.727410i	$-0.259276\pi$
$461$	29.4668	1.37241	0.686203	+	0.727410i	$0.259276\pi$
$462$	0	0
$463$	−30.4898	−1.41698	−0.708491	−	0.705719i	$-0.750624\pi$
$463$	−30.4898	−1.41698	−0.708491	+	0.705719i	$0.750624\pi$
$464$	0	0
$465$	−0.620813	−0.0287895
$466$	0	0
$467$	−8.23685	−0.381156	−0.190578	−	0.981672i	$-0.561036\pi$
$467$	−8.23685	−0.381156	−0.190578	+	0.981672i	$0.561036\pi$
$468$	0	0
$469$	−47.4909	−2.19293
$470$	0	0
$471$	−1.85185	−0.0853289
$472$	0	0
$473$	19.2123	0.883384
$474$	0	0
$475$	22.6086	1.03736
$476$	0	0
$477$	−34.7152	−1.58950
$478$	0	0
$479$	−18.7992	−0.858957	−0.429478	−	0.903077i	$-0.641303\pi$
$479$	−18.7992	−0.858957	−0.429478	+	0.903077i	$0.641303\pi$
$480$	0	0
$481$	−21.0902	−0.961632
$482$	0	0
$483$	46.7773	2.12844
$484$	0	0
$485$	0.430084	0.0195291
$486$	0	0
$487$	0.258705	0.0117230	0.00586152	−	0.999983i	$-0.498134\pi$
$487$	0.258705	0.0117230	0.00586152	+	0.999983i	$0.498134\pi$
$488$	0	0
$489$	20.5125	0.927608
$490$	0	0
$491$	32.0782	1.44767	0.723835	−	0.689973i	$-0.242378\pi$
$491$	32.0782	1.44767	0.723835	+	0.689973i	$0.242378\pi$
$492$	0	0
$493$	10.6121	0.477944
$494$	0	0
$495$	−1.00500	−0.0451712
$496$	0	0
$497$	16.3392	0.732911
$498$	0	0
$499$	−11.3916	−0.509959	−0.254979	−	0.966946i	$-0.582069\pi$
$499$	−11.3916	−0.509959	−0.254979	+	0.966946i	$0.582069\pi$
$500$	0	0
$501$	−48.4311	−2.16374
$502$	0	0
$503$	−4.53562	−0.202233	−0.101117	−	0.994875i	$-0.532242\pi$
$503$	−4.53562	−0.202233	−0.101117	+	0.994875i	$0.532242\pi$
$504$	0	0
$505$	0.130542	0.00580904
$506$	0	0
$507$	21.3088	0.946358
$508$	0	0
$509$	−8.75560	−0.388085	−0.194043	−	0.980993i	$-0.562160\pi$
$509$	−8.75560	−0.388085	−0.194043	+	0.980993i	$0.562160\pi$
$510$	0	0
$511$	−28.6549	−1.26762
$512$	0	0
$513$	74.2482	3.27814
$514$	0	0
$515$	0.368217	0.0162256
$516$	0	0
$517$	−43.4429	−1.91062
$518$	0	0
$519$	−14.9089	−0.654428
$520$	0	0
$521$	−17.7098	−0.775882	−0.387941	−	0.921684i	$-0.626814\pi$
$521$	−17.7098	−0.775882	−0.387941	+	0.921684i	$0.626814\pi$
$522$	0	0
$523$	−41.8685	−1.83078	−0.915390	−	0.402568i	$-0.868118\pi$
$523$	−41.8685	−1.83078	−0.915390	+	0.402568i	$0.868118\pi$
$524$	0	0
$525$	−67.5889	−2.94982
$526$	0	0
$527$	25.3278	1.10330
$528$	0	0
$529$	−11.0283	−0.479493
$530$	0	0
$531$	34.6412	1.50330
$532$	0	0
$533$	6.52356	0.282567
$534$	0	0
$535$	−0.440559	−0.0190470
$536$	0	0
$537$	51.7672	2.23392
$538$	0	0
$539$	−49.8132	−2.14561
$540$	0	0
$541$	−26.0805	−1.12129	−0.560644	−	0.828057i	$-0.689446\pi$
$541$	−26.0805	−1.12129	−0.560644	+	0.828057i	$0.689446\pi$
$542$	0	0
$543$	−49.3184	−2.11645
$544$	0	0
$545$	0.191964	0.00822284
$546$	0	0
$547$	−4.99389	−0.213523	−0.106762	−	0.994285i	$-0.534048\pi$
$547$	−4.99389	−0.213523	−0.106762	+	0.994285i	$0.534048\pi$
$548$	0	0
$549$	−103.292	−4.40839
$550$	0	0
$551$	−14.4855	−0.617104
$552$	0	0
$553$	14.8048	0.629566
$554$	0	0
$555$	−0.668509	−0.0283766
$556$	0	0
$557$	32.9197	1.39485	0.697425	−	0.716658i	$-0.254329\pi$
$557$	32.9197	1.39485	0.697425	+	0.716658i	$0.254329\pi$
$558$	0	0
$559$	9.56207	0.404433
$560$	0	0
$561$	56.4555	2.38355
$562$	0	0
$563$	−1.87959	−0.0792152	−0.0396076	−	0.999215i	$-0.512611\pi$
$563$	−1.87959	−0.0792152	−0.0396076	+	0.999215i	$0.512611\pi$
$564$	0	0
$565$	0.478130	0.0201151
$566$	0	0
$567$	−124.452	−5.22650
$568$	0	0
$569$	−6.19629	−0.259762	−0.129881	−	0.991530i	$-0.541459\pi$
$569$	−6.19629	−0.259762	−0.129881	+	0.991530i	$0.541459\pi$
$570$	0	0
$571$	12.4571	0.521313	0.260656	−	0.965432i	$-0.416061\pi$
$571$	12.4571	0.521313	0.260656	+	0.965432i	$0.416061\pi$
$572$	0	0
$573$	80.1954	3.35021
$574$	0	0
$575$	−17.2980	−0.721375
$576$	0	0
$577$	28.7401	1.19647	0.598233	−	0.801323i	$-0.295870\pi$
$577$	28.7401	1.19647	0.598233	+	0.801323i	$0.295870\pi$
$578$	0	0
$579$	57.2159	2.37781
$580$	0	0
$581$	23.3942	0.970553
$582$	0	0
$583$	22.4504	0.929801
$584$	0	0
$585$	−0.500191	−0.0206804
$586$	0	0
$587$	−3.02021	−0.124657	−0.0623287	−	0.998056i	$-0.519853\pi$
$587$	−3.02021	−0.124657	−0.0623287	+	0.998056i	$0.519853\pi$
$588$	0	0
$589$	−34.5725	−1.42454
$590$	0	0
$591$	−26.5114	−1.09053
$592$	0	0
$593$	−46.5429	−1.91129	−0.955644	−	0.294524i	$-0.904839\pi$
$593$	−46.5429	−1.91129	−0.955644	+	0.294524i	$0.904839\pi$
$594$	0	0
$595$	−0.331878	−0.0136057
$596$	0	0
$597$	58.6131	2.39887
$598$	0	0
$599$	−21.2922	−0.869977	−0.434988	−	0.900436i	$-0.643248\pi$
$599$	−21.2922	−0.869977	−0.434988	+	0.900436i	$0.643248\pi$
$600$	0	0
$601$	−43.7276	−1.78368	−0.891842	−	0.452347i	$-0.850587\pi$
$601$	−43.7276	−1.78368	−0.891842	+	0.452347i	$0.850587\pi$
$602$	0	0
$603$	92.5616	3.76940
$604$	0	0
$605$	0.380107	0.0154536
$606$	0	0
$607$	−6.32390	−0.256679	−0.128340	−	0.991730i	$-0.540965\pi$
$607$	−6.32390	−0.256679	−0.128340	+	0.991730i	$0.540965\pi$
$608$	0	0
$609$	43.3047	1.75480
$610$	0	0
$611$	−21.6218	−0.874723
$612$	0	0
$613$	11.9047	0.480827	0.240414	−	0.970671i	$-0.422717\pi$
$613$	11.9047	0.480827	0.240414	+	0.970671i	$0.422717\pi$
$614$	0	0
$615$	0.206781	0.00833822
$616$	0	0
$617$	−34.1388	−1.37438	−0.687189	−	0.726479i	$-0.741156\pi$
$617$	−34.1388	−1.37438	−0.687189	+	0.726479i	$0.741156\pi$
$618$	0	0
$619$	38.1884	1.53492	0.767462	−	0.641095i	$-0.221520\pi$
$619$	38.1884	1.53492	0.767462	+	0.641095i	$0.221520\pi$
$620$	0	0
$621$	−56.8076	−2.27961
$622$	0	0
$623$	57.3020	2.29576
$624$	0	0
$625$	24.9910	0.999639
$626$	0	0
$627$	−77.0619	−3.07756
$628$	0	0
$629$	27.2737	1.08747
$630$	0	0
$631$	−4.18018	−0.166410	−0.0832051	−	0.996532i	$-0.526516\pi$
$631$	−4.18018	−0.166410	−0.0832051	+	0.996532i	$0.526516\pi$
$632$	0	0
$633$	17.5367	0.697023
$634$	0	0
$635$	0.0831470	0.00329959
$636$	0	0
$637$	−24.7923	−0.982305
$638$	0	0
$639$	−31.8456	−1.25979
$640$	0	0
$641$	15.9355	0.629415	0.314708	−	0.949189i	$-0.398094\pi$
$641$	15.9355	0.629415	0.314708	+	0.949189i	$0.398094\pi$
$642$	0	0
$643$	32.8024	1.29360	0.646800	−	0.762660i	$-0.276107\pi$
$643$	32.8024	1.29360	0.646800	+	0.762660i	$0.276107\pi$
$644$	0	0
$645$	0.303094	0.0119343
$646$	0	0
$647$	10.5392	0.414339	0.207170	−	0.978305i	$-0.433575\pi$
$647$	10.5392	0.414339	0.207170	+	0.978305i	$0.433575\pi$
$648$	0	0
$649$	−22.4026	−0.879378
$650$	0	0
$651$	103.355	4.05081
$652$	0	0
$653$	32.0684	1.25493	0.627467	−	0.778643i	$-0.284092\pi$
$653$	32.0684	1.25493	0.627467	+	0.778643i	$0.284092\pi$
$654$	0	0
$655$	0.179834	0.00702670
$656$	0	0
$657$	55.8494	2.17889
$658$	0	0
$659$	−11.1432	−0.434077	−0.217038	−	0.976163i	$-0.569640\pi$
$659$	−11.1432	−0.434077	−0.217038	+	0.976163i	$0.569640\pi$
$660$	0	0
$661$	−20.2323	−0.786945	−0.393473	−	0.919336i	$-0.628726\pi$
$661$	−20.2323	−0.786945	−0.393473	+	0.919336i	$0.628726\pi$
$662$	0	0
$663$	28.0982	1.09124
$664$	0	0
$665$	0.453014	0.0175671
$666$	0	0
$667$	11.0829	0.429133
$668$	0	0
$669$	−4.15410	−0.160607
$670$	0	0
$671$	66.7992	2.57875
$672$	0	0
$673$	−41.9584	−1.61738	−0.808689	−	0.588237i	$-0.799822\pi$
$673$	−41.9584	−1.61738	−0.808689	+	0.588237i	$0.799822\pi$
$674$	0	0
$675$	82.0818	3.15933
$676$	0	0
$677$	30.0630	1.15542	0.577708	−	0.816244i	$-0.303947\pi$
$677$	30.0630	1.15542	0.577708	+	0.816244i	$0.303947\pi$
$678$	0	0
$679$	−71.6020	−2.74783
$680$	0	0
$681$	40.6855	1.55907
$682$	0	0
$683$	−46.8392	−1.79225	−0.896126	−	0.443799i	$-0.853630\pi$
$683$	−46.8392	−1.79225	−0.896126	+	0.443799i	$0.853630\pi$
$684$	0	0
$685$	−0.156187	−0.00596761
$686$	0	0
$687$	64.7169	2.46911
$688$	0	0
$689$	11.1737	0.425683
$690$	0	0
$691$	31.5282	1.19939	0.599695	−	0.800229i	$-0.295289\pi$
$691$	31.5282	1.19939	0.599695	+	0.800229i	$0.295289\pi$
$692$	0	0
$693$	167.315	6.35578
$694$	0	0
$695$	0.346763	0.0131535
$696$	0	0
$697$	−8.43622	−0.319544
$698$	0	0
$699$	57.3954	2.17089
$700$	0	0
$701$	7.70959	0.291187	0.145594	−	0.989344i	$-0.453491\pi$
$701$	7.70959	0.291187	0.145594	+	0.989344i	$0.453491\pi$
$702$	0	0
$703$	−37.2287	−1.40411
$704$	0	0
$705$	−0.685357	−0.0258120
$706$	0	0
$707$	−21.7331	−0.817357
$708$	0	0
$709$	48.1301	1.80756	0.903781	−	0.427994i	$-0.140780\pi$
$709$	48.1301	1.80756	0.903781	+	0.427994i	$0.140780\pi$
$710$	0	0
$711$	−28.8552	−1.08215
$712$	0	0
$713$	26.4516	0.990620
$714$	0	0
$715$	0.323475	0.0120973
$716$	0	0
$717$	−15.0115	−0.560616
$718$	0	0
$719$	43.8696	1.63606	0.818030	−	0.575175i	$-0.195066\pi$
$719$	43.8696	1.63606	0.818030	+	0.575175i	$0.195066\pi$
$720$	0	0
$721$	−61.3022	−2.28301
$722$	0	0
$723$	78.3638	2.91438
$724$	0	0
$725$	−16.0138	−0.594739
$726$	0	0
$727$	46.4387	1.72232	0.861158	−	0.508338i	$-0.169740\pi$
$727$	46.4387	1.72232	0.861158	+	0.508338i	$0.169740\pi$
$728$	0	0
$729$	79.5028	2.94455
$730$	0	0
$731$	−12.3656	−0.457358
$732$	0	0
$733$	23.9792	0.885693	0.442847	−	0.896597i	$-0.353969\pi$
$733$	23.9792	0.885693	0.442847	+	0.896597i	$0.353969\pi$
$734$	0	0
$735$	−0.785854	−0.0289867
$736$	0	0
$737$	−59.8599	−2.20497
$738$	0	0
$739$	−45.2879	−1.66594	−0.832971	−	0.553317i	$-0.813362\pi$
$739$	−45.2879	−1.66594	−0.832971	+	0.553317i	$0.813362\pi$
$740$	0	0
$741$	−38.3541	−1.40897
$742$	0	0
$743$	7.39508	0.271299	0.135650	−	0.990757i	$-0.456688\pi$
$743$	7.39508	0.271299	0.135650	+	0.990757i	$0.456688\pi$
$744$	0	0
$745$	0.273874	0.0100339
$746$	0	0
$747$	−45.5961	−1.66827
$748$	0	0
$749$	73.3459	2.68000
$750$	0	0
$751$	17.4833	0.637975	0.318988	−	0.947759i	$-0.396657\pi$
$751$	17.4833	0.637975	0.318988	+	0.947759i	$0.396657\pi$
$752$	0	0
$753$	−96.7795	−3.52684
$754$	0	0
$755$	0.189726	0.00690482
$756$	0	0
$757$	10.2996	0.374345	0.187172	−	0.982327i	$-0.440068\pi$
$757$	10.2996	0.374345	0.187172	+	0.982327i	$0.440068\pi$
$758$	0	0
$759$	58.9604	2.14013
$760$	0	0
$761$	30.0987	1.09108	0.545539	−	0.838085i	$-0.316325\pi$
$761$	30.0987	1.09108	0.545539	+	0.838085i	$0.316325\pi$
$762$	0	0
$763$	−31.9589	−1.15699
$764$	0	0
$765$	0.646843	0.0233866
$766$	0	0
$767$	−11.1499	−0.402598
$768$	0	0
$769$	17.4769	0.630235	0.315117	−	0.949053i	$-0.397956\pi$
$769$	17.4769	0.630235	0.315117	+	0.949053i	$0.397956\pi$
$770$	0	0
$771$	16.8186	0.605709
$772$	0	0
$773$	5.59250	0.201148	0.100574	−	0.994930i	$-0.467932\pi$
$773$	5.59250	0.201148	0.100574	+	0.994930i	$0.467932\pi$
$774$	0	0
$775$	−38.2201	−1.37291
$776$	0	0
$777$	111.296	3.99272
$778$	0	0
$779$	11.5155	0.412584
$780$	0	0
$781$	20.5947	0.736936
$782$	0	0
$783$	−52.5904	−1.87943
$784$	0	0
$785$	0.0137216	0.000489743 0
$786$	0	0
$787$	−26.2154	−0.934478	−0.467239	−	0.884131i	$-0.654751\pi$
$787$	−26.2154	−0.934478	−0.467239	+	0.884131i	$0.654751\pi$
$788$	0	0
$789$	−57.9543	−2.06323
$790$	0	0
$791$	−79.6008	−2.83028
$792$	0	0
$793$	33.2463	1.18061
$794$	0	0
$795$	0.354178	0.0125614
$796$	0	0
$797$	−22.6017	−0.800595	−0.400297	−	0.916385i	$-0.631093\pi$
$797$	−22.6017	−0.800595	−0.400297	+	0.916385i	$0.631093\pi$
$798$	0	0
$799$	27.9611	0.989191
$800$	0	0
$801$	−111.684	−3.94615
$802$	0	0
$803$	−36.1180	−1.27458
$804$	0	0
$805$	−0.346603	−0.0122161
$806$	0	0
$807$	35.0008	1.23209
$808$	0	0
$809$	−20.1002	−0.706685	−0.353343	−	0.935494i	$-0.614955\pi$
$809$	−20.1002	−0.706685	−0.353343	+	0.935494i	$0.614955\pi$
$810$	0	0
$811$	1.06814	0.0375075	0.0187537	−	0.999824i	$-0.494030\pi$
$811$	1.06814	0.0375075	0.0187537	+	0.999824i	$0.494030\pi$
$812$	0	0
$813$	−81.7726	−2.86789
$814$	0	0
$815$	−0.151990	−0.00532399
$816$	0	0
$817$	16.8791	0.590523
$818$	0	0
$819$	83.2736	2.90982
$820$	0	0
$821$	15.7392	0.549301	0.274650	−	0.961544i	$-0.411438\pi$
$821$	15.7392	0.549301	0.274650	+	0.961544i	$0.411438\pi$
$822$	0	0
$823$	−38.2042	−1.33171	−0.665857	−	0.746080i	$-0.731934\pi$
$823$	−38.2042	−1.33171	−0.665857	+	0.746080i	$0.731934\pi$
$824$	0	0
$825$	−85.1924	−2.96602
$826$	0	0
$827$	12.8015	0.445152	0.222576	−	0.974915i	$-0.428553\pi$
$827$	12.8015	0.445152	0.222576	+	0.974915i	$0.428553\pi$
$828$	0	0
$829$	10.2710	0.356727	0.178363	−	0.983965i	$-0.442920\pi$
$829$	10.2710	0.356727	0.178363	+	0.983965i	$0.442920\pi$
$830$	0	0
$831$	−9.68729	−0.336048
$832$	0	0
$833$	32.0611	1.11085
$834$	0	0
$835$	0.358857	0.0124188
$836$	0	0
$837$	−125.517	−4.33852
$838$	0	0
$839$	1.80700	0.0623847	0.0311923	−	0.999513i	$-0.490070\pi$
$839$	1.80700	0.0623847	0.0311923	+	0.999513i	$0.490070\pi$
$840$	0	0
$841$	−18.7398	−0.646201
$842$	0	0
$843$	60.4197	2.08096
$844$	0	0
$845$	−0.157891	−0.00543160
$846$	0	0
$847$	−63.2817	−2.17438
$848$	0	0
$849$	−7.50008	−0.257402
$850$	0	0
$851$	28.4838	0.976412
$852$	0	0
$853$	−47.0022	−1.60932	−0.804662	−	0.593734i	$-0.797653\pi$
$853$	−47.0022	−1.60932	−0.804662	+	0.593734i	$0.797653\pi$
$854$	0	0
$855$	−0.882942	−0.0301960
$856$	0	0
$857$	−17.5209	−0.598504	−0.299252	−	0.954174i	$-0.596737\pi$
$857$	−17.5209	−0.598504	−0.299252	+	0.954174i	$0.596737\pi$
$858$	0	0
$859$	−41.3863	−1.41208	−0.706041	−	0.708171i	$-0.749521\pi$
$859$	−41.3863	−1.41208	−0.706041	+	0.708171i	$0.749521\pi$
$860$	0	0
$861$	−34.4257	−1.17322
$862$	0	0
$863$	17.8428	0.607377	0.303689	−	0.952771i	$-0.401782\pi$
$863$	17.8428	0.607377	0.303689	+	0.952771i	$0.401782\pi$
$864$	0	0
$865$	0.110470	0.00375608
$866$	0	0
$867$	19.9423	0.677276
$868$	0	0
$869$	18.6607	0.633023
$870$	0	0
$871$	−29.7926	−1.00948
$872$	0	0
$873$	139.555	4.72322
$874$	0	0
$875$	1.00168	0.0338630
$876$	0	0
$877$	−35.0414	−1.18326	−0.591631	−	0.806209i	$-0.701516\pi$
$877$	−35.0414	−1.18326	−0.591631	+	0.806209i	$0.701516\pi$
$878$	0	0
$879$	43.1796	1.45641
$880$	0	0
$881$	−55.1615	−1.85844	−0.929219	−	0.369528i	$-0.879519\pi$
$881$	−55.1615	−1.85844	−0.929219	+	0.369528i	$0.879519\pi$
$882$	0	0
$883$	−7.42446	−0.249853	−0.124927	−	0.992166i	$-0.539870\pi$
$883$	−7.42446	−0.249853	−0.124927	+	0.992166i	$0.539870\pi$
$884$	0	0
$885$	−0.353423	−0.0118802
$886$	0	0
$887$	29.8435	1.00205	0.501024	−	0.865433i	$-0.332957\pi$
$887$	29.8435	1.00205	0.501024	+	0.865433i	$0.332957\pi$
$888$	0	0
$889$	−13.8426	−0.464266
$890$	0	0
$891$	−156.866	−5.25519
$892$	0	0
$893$	−38.1669	−1.27721
$894$	0	0
$895$	−0.383576	−0.0128215
$896$	0	0
$897$	29.3449	0.979797
$898$	0	0
$899$	24.4879	0.816718
$900$	0	0
$901$	−14.4497	−0.481389
$902$	0	0
$903$	−50.4603	−1.67921
$904$	0	0
$905$	0.365431	0.0121474
$906$	0	0
$907$	45.4175	1.50806	0.754032	−	0.656838i	$-0.228106\pi$
$907$	45.4175	1.50806	0.754032	+	0.656838i	$0.228106\pi$
$908$	0	0
$909$	42.3586	1.40495
$910$	0	0
$911$	0.170581	0.00565162	0.00282581	−	0.999996i	$-0.499101\pi$
$911$	0.170581	0.00565162	0.00282581	+	0.999996i	$0.499101\pi$
$912$	0	0
$913$	29.4872	0.975882
$914$	0	0
$915$	1.05383	0.0348384
$916$	0	0
$917$	−29.9394	−0.988687
$918$	0	0
$919$	−16.1467	−0.532631	−0.266316	−	0.963886i	$-0.585806\pi$
$919$	−16.1467	−0.532631	−0.266316	+	0.963886i	$0.585806\pi$
$920$	0	0
$921$	−45.6983	−1.50581
$922$	0	0
$923$	10.2501	0.337385
$924$	0	0
$925$	−41.1565	−1.35322
$926$	0	0
$927$	119.480	3.92425
$928$	0	0
$929$	25.5567	0.838489	0.419244	−	0.907873i	$-0.362295\pi$
$929$	25.5567	0.838489	0.419244	+	0.907873i	$0.362295\pi$
$930$	0	0
$931$	−43.7635	−1.43429
$932$	0	0
$933$	−21.9891	−0.719891
$934$	0	0
$935$	−0.418315	−0.0136804
$936$	0	0
$937$	31.7983	1.03880	0.519402	−	0.854530i	$-0.326155\pi$
$937$	31.7983	1.03880	0.519402	+	0.854530i	$0.326155\pi$
$938$	0	0
$939$	−8.27877	−0.270167
$940$	0	0
$941$	−21.4250	−0.698435	−0.349217	−	0.937042i	$-0.613553\pi$
$941$	−21.4250	−0.698435	−0.349217	+	0.937042i	$0.613553\pi$
$942$	0	0
$943$	−8.81052	−0.286910
$944$	0	0
$945$	1.64469	0.0535018
$946$	0	0
$947$	52.0222	1.69049	0.845247	−	0.534376i	$-0.179453\pi$
$947$	52.0222	1.69049	0.845247	+	0.534376i	$0.179453\pi$
$948$	0	0
$949$	−17.9761	−0.583529
$950$	0	0
$951$	101.266	3.28379
$952$	0	0
$953$	40.9692	1.32712	0.663561	−	0.748122i	$-0.269044\pi$
$953$	40.9692	1.32712	0.663561	+	0.748122i	$0.269044\pi$
$954$	0	0
$955$	−0.594219	−0.0192285
$956$	0	0
$957$	54.5834	1.76443
$958$	0	0
$959$	26.0026	0.839669
$960$	0	0
$961$	27.4452	0.885330
$962$	0	0
$963$	−142.954	−4.60663
$964$	0	0
$965$	−0.423949	−0.0136474
$966$	0	0
$967$	−18.9111	−0.608140	−0.304070	−	0.952650i	$-0.598346\pi$
$967$	−18.9111	−0.608140	−0.304070	+	0.952650i	$0.598346\pi$
$968$	0	0
$969$	49.5992	1.59335
$970$	0	0
$971$	36.4685	1.17033	0.585165	−	0.810914i	$-0.301030\pi$
$971$	36.4685	1.17033	0.585165	+	0.810914i	$0.301030\pi$
$972$	0	0
$973$	−57.7303	−1.85075
$974$	0	0
$975$	−42.4007	−1.35791
$976$	0	0
$977$	27.2805	0.872780	0.436390	−	0.899758i	$-0.356257\pi$
$977$	27.2805	0.872780	0.436390	+	0.899758i	$0.356257\pi$
$978$	0	0
$979$	72.2263	2.30836
$980$	0	0
$981$	62.2891	1.98874
$982$	0	0
$983$	−18.3123	−0.584072	−0.292036	−	0.956407i	$-0.594333\pi$
$983$	−18.3123	−0.584072	−0.292036	+	0.956407i	$0.594333\pi$
$984$	0	0
$985$	0.196440	0.00625910
$986$	0	0
$987$	114.101	3.63187
$988$	0	0
$989$	−12.9142	−0.410649
$990$	0	0
$991$	45.2500	1.43741	0.718707	−	0.695313i	$-0.244734\pi$
$991$	45.2500	1.43741	0.718707	+	0.695313i	$0.244734\pi$
$992$	0	0
$993$	119.820	3.80237
$994$	0	0
$995$	−0.434302	−0.0137683
$996$	0	0
$997$	−27.6749	−0.876473	−0.438237	−	0.898860i	$-0.644397\pi$
$997$	−27.6749	−0.876473	−0.438237	+	0.898860i	$0.644397\pi$
$998$	0	0
$999$	−135.161	−4.27629

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.2	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.2	✓	63	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-3.31051 q^{3} +0.0245297 q^{5} -4.08379 q^{7} +7.95946 q^{9} +O(q^{10})\)	\(q-3.31051 q^{3} +0.0245297 q^{5} -4.08379 q^{7} +7.95946 q^{9} -5.14741 q^{11} -2.56189 q^{13} -0.0812057 q^{15} +3.31301 q^{17} -4.52227 q^{19} +13.5194 q^{21} +3.46001 q^{23} -4.99940 q^{25} -16.4183 q^{27} +3.20315 q^{29} +7.64495 q^{31} +17.0405 q^{33} -0.100174 q^{35} +8.23230 q^{37} +8.48116 q^{39} -2.54639 q^{41} -3.73243 q^{43} +0.195243 q^{45} +8.43977 q^{47} +9.67733 q^{49} -10.9678 q^{51} -4.36150 q^{53} -0.126264 q^{55} +14.9710 q^{57} +4.35220 q^{59} -12.9772 q^{61} -32.5048 q^{63} -0.0628423 q^{65} +11.6291 q^{67} -11.4544 q^{69} -4.00098 q^{71} +7.01673 q^{73} +16.5505 q^{75} +21.0209 q^{77} -3.62527 q^{79} +30.4747 q^{81} -5.72854 q^{83} +0.0812671 q^{85} -10.6041 q^{87} -14.0316 q^{89} +10.4622 q^{91} -25.3087 q^{93} -0.110930 q^{95} +17.5332 q^{97} -40.9706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

Introduction

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