LMFDB - Embedded newform 6028.2.a.d.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.05741 q^{3} -1.81288 q^{5} +4.15919 q^{7} +1.23295 q^{9} +O(q^{10})$	$q+2.05741 q^{3} -1.81288 q^{5} +4.15919 q^{7} +1.23295 q^{9} -1.00000 q^{11} +1.63656 q^{13} -3.72985 q^{15} -7.92852 q^{17} +1.08787 q^{19} +8.55718 q^{21} -5.24909 q^{23} -1.71346 q^{25} -3.63555 q^{27} -1.03069 q^{29} -4.77126 q^{31} -2.05741 q^{33} -7.54013 q^{35} -4.73209 q^{37} +3.36708 q^{39} -5.23766 q^{41} -2.38368 q^{43} -2.23519 q^{45} +0.857313 q^{47} +10.2989 q^{49} -16.3122 q^{51} -7.45270 q^{53} +1.81288 q^{55} +2.23821 q^{57} +4.62076 q^{59} -11.6905 q^{61} +5.12807 q^{63} -2.96689 q^{65} -2.36070 q^{67} -10.7995 q^{69} -13.1687 q^{71} +6.55889 q^{73} -3.52529 q^{75} -4.15919 q^{77} +13.7218 q^{79} -11.1787 q^{81} +15.2923 q^{83} +14.3735 q^{85} -2.12055 q^{87} +18.6978 q^{89} +6.80676 q^{91} -9.81646 q^{93} -1.97219 q^{95} -1.28250 q^{97} -1.23295 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * 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
$3$	2.05741	1.18785	0.593924	−	0.804521i	$-0.297578\pi$
$3$	2.05741	1.18785	0.593924	+	0.804521i	$0.297578\pi$
$4$	0	0
$5$	−1.81288	−0.810746	−0.405373	−	0.914151i	$-0.632858\pi$
$5$	−1.81288	−0.810746	−0.405373	+	0.914151i	$0.632858\pi$
$6$	0	0
$7$	4.15919	1.57203	0.786014	−	0.618209i	$-0.212141\pi$
$7$	4.15919	1.57203	0.786014	+	0.618209i	$0.212141\pi$
$8$	0	0
$9$	1.23295	0.410983
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.63656	0.453899	0.226950	−	0.973906i	$-0.427125\pi$
$13$	1.63656	0.453899	0.226950	+	0.973906i	$0.427125\pi$
$14$	0	0
$15$	−3.72985	−0.963043
$16$	0	0
$17$	−7.92852	−1.92295	−0.961474	−	0.274895i	$-0.911357\pi$
$17$	−7.92852	−1.92295	−0.961474	+	0.274895i	$0.911357\pi$
$18$	0	0
$19$	1.08787	0.249576	0.124788	−	0.992183i	$-0.460175\pi$
$19$	1.08787	0.249576	0.124788	+	0.992183i	$0.460175\pi$
$20$	0	0
$21$	8.55718	1.86733
$22$	0	0
$23$	−5.24909	−1.09451	−0.547256	−	0.836966i	$-0.684327\pi$
$23$	−5.24909	−1.09451	−0.547256	+	0.836966i	$0.684327\pi$
$24$	0	0
$25$	−1.71346	−0.342692
$26$	0	0
$27$	−3.63555	−0.699663
$28$	0	0
$29$	−1.03069	−0.191394	−0.0956969	−	0.995411i	$-0.530508\pi$
$29$	−1.03069	−0.191394	−0.0956969	+	0.995411i	$0.530508\pi$
$30$	0	0
$31$	−4.77126	−0.856944	−0.428472	−	0.903555i	$-0.640948\pi$
$31$	−4.77126	−0.856944	−0.428472	+	0.903555i	$0.640948\pi$
$32$	0	0
$33$	−2.05741	−0.358150
$34$	0	0
$35$	−7.54013	−1.27451
$36$	0	0
$37$	−4.73209	−0.777951	−0.388975	−	0.921248i	$-0.627171\pi$
$37$	−4.73209	−0.777951	−0.388975	+	0.921248i	$0.627171\pi$
$38$	0	0
$39$	3.36708	0.539164
$40$	0	0
$41$	−5.23766	−0.817985	−0.408993	−	0.912538i	$-0.634120\pi$
$41$	−5.23766	−0.817985	−0.408993	+	0.912538i	$0.634120\pi$
$42$	0	0
$43$	−2.38368	−0.363507	−0.181754	−	0.983344i	$-0.558177\pi$
$43$	−2.38368	−0.363507	−0.181754	+	0.983344i	$0.558177\pi$
$44$	0	0
$45$	−2.23519	−0.333203
$46$	0	0
$47$	0.857313	0.125052	0.0625260	−	0.998043i	$-0.480084\pi$
$47$	0.857313	0.125052	0.0625260	+	0.998043i	$0.480084\pi$
$48$	0	0
$49$	10.2989	1.47127
$50$	0	0
$51$	−16.3122	−2.28417
$52$	0	0
$53$	−7.45270	−1.02371	−0.511853	−	0.859073i	$-0.671041\pi$
$53$	−7.45270	−1.02371	−0.511853	+	0.859073i	$0.671041\pi$
$54$	0	0
$55$	1.81288	0.244449
$56$	0	0
$57$	2.23821	0.296458
$58$	0	0
$59$	4.62076	0.601571	0.300786	−	0.953692i	$-0.402751\pi$
$59$	4.62076	0.601571	0.300786	+	0.953692i	$0.402751\pi$
$60$	0	0
$61$	−11.6905	−1.49681	−0.748406	−	0.663241i	$-0.769180\pi$
$61$	−11.6905	−1.49681	−0.748406	+	0.663241i	$0.769180\pi$
$62$	0	0
$63$	5.12807	0.646077
$64$	0	0
$65$	−2.96689	−0.367997
$66$	0	0
$67$	−2.36070	−0.288406	−0.144203	−	0.989548i	$-0.546062\pi$
$67$	−2.36070	−0.288406	−0.144203	+	0.989548i	$0.546062\pi$
$68$	0	0
$69$	−10.7995	−1.30011
$70$	0	0
$71$	−13.1687	−1.56284	−0.781419	−	0.624006i	$-0.785504\pi$
$71$	−13.1687	−1.56284	−0.781419	+	0.624006i	$0.785504\pi$
$72$	0	0
$73$	6.55889	0.767660	0.383830	−	0.923404i	$-0.374605\pi$
$73$	6.55889	0.767660	0.383830	+	0.923404i	$0.374605\pi$
$74$	0	0
$75$	−3.52529	−0.407066
$76$	0	0
$77$	−4.15919	−0.473984
$78$	0	0
$79$	13.7218	1.54383	0.771914	−	0.635727i	$-0.219300\pi$
$79$	13.7218	1.54383	0.771914	+	0.635727i	$0.219300\pi$
$80$	0	0
$81$	−11.1787	−1.24208
$82$	0	0
$83$	15.2923	1.67854	0.839272	−	0.543712i	$-0.182982\pi$
$83$	15.2923	1.67854	0.839272	+	0.543712i	$0.182982\pi$
$84$	0	0
$85$	14.3735	1.55902
$86$	0	0
$87$	−2.12055	−0.227347
$88$	0	0
$89$	18.6978	1.98196	0.990979	−	0.134019i	$-0.0427883\pi$
$89$	18.6978	1.98196	0.990979	+	0.134019i	$0.0427883\pi$
$90$	0	0
$91$	6.80676	0.713543
$92$	0	0
$93$	−9.81646	−1.01792
$94$	0	0
$95$	−1.97219	−0.202342
$96$	0	0
$97$	−1.28250	−0.130218	−0.0651092	−	0.997878i	$-0.520740\pi$
$97$	−1.28250	−0.130218	−0.0651092	+	0.997878i	$0.520740\pi$
$98$	0	0
$99$	−1.23295	−0.123916
$100$	0	0
$101$	1.91447	0.190497	0.0952485	−	0.995454i	$-0.469635\pi$
$101$	1.91447	0.190497	0.0952485	+	0.995454i	$0.469635\pi$
$102$	0	0
$103$	−7.15585	−0.705087	−0.352544	−	0.935795i	$-0.614683\pi$
$103$	−7.15585	−0.705087	−0.352544	+	0.935795i	$0.614683\pi$
$104$	0	0
$105$	−15.5132	−1.51393
$106$	0	0
$107$	−10.9370	−1.05732	−0.528661	−	0.848833i	$-0.677306\pi$
$107$	−10.9370	−1.05732	−0.528661	+	0.848833i	$0.677306\pi$
$108$	0	0
$109$	−11.0417	−1.05760	−0.528801	−	0.848746i	$-0.677358\pi$
$109$	−11.0417	−1.05760	−0.528801	+	0.848746i	$0.677358\pi$
$110$	0	0
$111$	−9.73586	−0.924087
$112$	0	0
$113$	−20.1460	−1.89518	−0.947589	−	0.319493i	$-0.896487\pi$
$113$	−20.1460	−1.89518	−0.947589	+	0.319493i	$0.896487\pi$
$114$	0	0
$115$	9.51598	0.887370
$116$	0	0
$117$	2.01779	0.186545
$118$	0	0
$119$	−32.9763	−3.02293
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−10.7760	−0.971642
$124$	0	0
$125$	12.1707	1.08858
$126$	0	0
$127$	6.58555	0.584373	0.292186	−	0.956361i	$-0.405617\pi$
$127$	6.58555	0.584373	0.292186	+	0.956361i	$0.405617\pi$
$128$	0	0
$129$	−4.90421	−0.431791
$130$	0	0
$131$	10.3420	0.903587	0.451794	−	0.892122i	$-0.350784\pi$
$131$	10.3420	0.903587	0.451794	+	0.892122i	$0.350784\pi$
$132$	0	0
$133$	4.52468	0.392340
$134$	0	0
$135$	6.59083	0.567249
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	16.7807	1.42332	0.711658	−	0.702526i	$-0.247944\pi$
$139$	16.7807	1.42332	0.711658	+	0.702526i	$0.247944\pi$
$140$	0	0
$141$	1.76385	0.148543
$142$	0	0
$143$	−1.63656	−0.136856
$144$	0	0
$145$	1.86851	0.155172
$146$	0	0
$147$	21.1891	1.74765
$148$	0	0
$149$	−12.2276	−1.00172	−0.500860	−	0.865528i	$-0.666983\pi$
$149$	−12.2276	−1.00172	−0.500860	+	0.865528i	$0.666983\pi$
$150$	0	0
$151$	21.1390	1.72027	0.860135	−	0.510066i	$-0.170379\pi$
$151$	21.1390	1.72027	0.860135	+	0.510066i	$0.170379\pi$
$152$	0	0
$153$	−9.77546	−0.790299
$154$	0	0
$155$	8.64974	0.694764
$156$	0	0
$157$	16.6281	1.32707	0.663534	−	0.748146i	$-0.269056\pi$
$157$	16.6281	1.32707	0.663534	+	0.748146i	$0.269056\pi$
$158$	0	0
$159$	−15.3333	−1.21601
$160$	0	0
$161$	−21.8320	−1.72060
$162$	0	0
$163$	−20.2603	−1.58691	−0.793455	−	0.608629i	$-0.791720\pi$
$163$	−20.2603	−1.58691	−0.793455	+	0.608629i	$0.791720\pi$
$164$	0	0
$165$	3.72985	0.290368
$166$	0	0
$167$	18.5819	1.43791	0.718957	−	0.695055i	$-0.244620\pi$
$167$	18.5819	1.43791	0.718957	+	0.695055i	$0.244620\pi$
$168$	0	0
$169$	−10.3217	−0.793975
$170$	0	0
$171$	1.34129	0.102571
$172$	0	0
$173$	5.59944	0.425717	0.212859	−	0.977083i	$-0.431723\pi$
$173$	5.59944	0.425717	0.212859	+	0.977083i	$0.431723\pi$
$174$	0	0
$175$	−7.12661	−0.538721
$176$	0	0
$177$	9.50681	0.714575
$178$	0	0
$179$	20.0625	1.49954	0.749772	−	0.661696i	$-0.230163\pi$
$179$	20.0625	1.49954	0.749772	+	0.661696i	$0.230163\pi$
$180$	0	0
$181$	−3.94183	−0.292994	−0.146497	−	0.989211i	$-0.546800\pi$
$181$	−3.94183	−0.292994	−0.146497	+	0.989211i	$0.546800\pi$
$182$	0	0
$183$	−24.0521	−1.77798
$184$	0	0
$185$	8.57872	0.630720
$186$	0	0
$187$	7.92852	0.579791
$188$	0	0
$189$	−15.1210	−1.09989
$190$	0	0
$191$	4.44080	0.321325	0.160662	−	0.987009i	$-0.448637\pi$
$191$	4.44080	0.321325	0.160662	+	0.987009i	$0.448637\pi$
$192$	0	0
$193$	−15.2892	−1.10054	−0.550272	−	0.834986i	$-0.685476\pi$
$193$	−15.2892	−1.10054	−0.550272	+	0.834986i	$0.685476\pi$
$194$	0	0
$195$	−6.10411	−0.437124
$196$	0	0
$197$	−22.9530	−1.63533	−0.817666	−	0.575693i	$-0.804732\pi$
$197$	−22.9530	−1.63533	−0.817666	+	0.575693i	$0.804732\pi$
$198$	0	0
$199$	22.2288	1.57576	0.787880	−	0.615829i	$-0.211179\pi$
$199$	22.2288	1.57576	0.787880	+	0.615829i	$0.211179\pi$
$200$	0	0
$201$	−4.85694	−0.342582
$202$	0	0
$203$	−4.28683	−0.300876
$204$	0	0
$205$	9.49526	0.663178
$206$	0	0
$207$	−6.47186	−0.449825
$208$	0	0
$209$	−1.08787	−0.0752498
$210$	0	0
$211$	1.23839	0.0852545	0.0426272	−	0.999091i	$-0.486427\pi$
$211$	1.23839	0.0852545	0.0426272	+	0.999091i	$0.486427\pi$
$212$	0	0
$213$	−27.0935	−1.85641
$214$	0	0
$215$	4.32132	0.294712
$216$	0	0
$217$	−19.8446	−1.34714
$218$	0	0
$219$	13.4944	0.911864
$220$	0	0
$221$	−12.9755	−0.872825
$222$	0	0
$223$	5.32282	0.356442	0.178221	−	0.983990i	$-0.442966\pi$
$223$	5.32282	0.356442	0.178221	+	0.983990i	$0.442966\pi$
$224$	0	0
$225$	−2.11261	−0.140840
$226$	0	0
$227$	−23.3456	−1.54950	−0.774749	−	0.632269i	$-0.782124\pi$
$227$	−23.3456	−1.54950	−0.774749	+	0.632269i	$0.782124\pi$
$228$	0	0
$229$	−17.4780	−1.15498	−0.577488	−	0.816399i	$-0.695967\pi$
$229$	−17.4780	−1.15498	−0.577488	+	0.816399i	$0.695967\pi$
$230$	0	0
$231$	−8.55718	−0.563021
$232$	0	0
$233$	−14.7264	−0.964761	−0.482380	−	0.875962i	$-0.660228\pi$
$233$	−14.7264	−0.964761	−0.482380	+	0.875962i	$0.660228\pi$
$234$	0	0
$235$	−1.55421	−0.101385
$236$	0	0
$237$	28.2315	1.83383
$238$	0	0
$239$	19.8487	1.28391	0.641953	−	0.766744i	$-0.278125\pi$
$239$	19.8487	1.28391	0.641953	+	0.766744i	$0.278125\pi$
$240$	0	0
$241$	22.5958	1.45553	0.727763	−	0.685829i	$-0.240560\pi$
$241$	22.5958	1.45553	0.727763	+	0.685829i	$0.240560\pi$
$242$	0	0
$243$	−12.0925	−0.775735
$244$	0	0
$245$	−18.6707	−1.19283
$246$	0	0
$247$	1.78037	0.113282
$248$	0	0
$249$	31.4625	1.99386
$250$	0	0
$251$	9.61381	0.606818	0.303409	−	0.952860i	$-0.401875\pi$
$251$	9.61381	0.606818	0.303409	+	0.952860i	$0.401875\pi$
$252$	0	0
$253$	5.24909	0.330008
$254$	0	0
$255$	29.5722	1.85188
$256$	0	0
$257$	9.32555	0.581712	0.290856	−	0.956767i	$-0.406060\pi$
$257$	9.32555	0.581712	0.290856	+	0.956767i	$0.406060\pi$
$258$	0	0
$259$	−19.6817	−1.22296
$260$	0	0
$261$	−1.27078	−0.0786596
$262$	0	0
$263$	−23.5222	−1.45044	−0.725219	−	0.688518i	$-0.758262\pi$
$263$	−23.5222	−1.45044	−0.725219	+	0.688518i	$0.758262\pi$
$264$	0	0
$265$	13.5109	0.829966
$266$	0	0
$267$	38.4690	2.35426
$268$	0	0
$269$	−14.7702	−0.900555	−0.450278	−	0.892889i	$-0.648675\pi$
$269$	−14.7702	−0.900555	−0.450278	+	0.892889i	$0.648675\pi$
$270$	0	0
$271$	26.5320	1.61170	0.805852	−	0.592117i	$-0.201708\pi$
$271$	26.5320	1.61170	0.805852	+	0.592117i	$0.201708\pi$
$272$	0	0
$273$	14.0043	0.847580
$274$	0	0
$275$	1.71346	0.103325
$276$	0	0
$277$	24.4953	1.47178	0.735891	−	0.677100i	$-0.236763\pi$
$277$	24.4953	1.47178	0.735891	+	0.677100i	$0.236763\pi$
$278$	0	0
$279$	−5.88272	−0.352189
$280$	0	0
$281$	−21.4217	−1.27791	−0.638955	−	0.769244i	$-0.720633\pi$
$281$	−21.4217	−1.27791	−0.638955	+	0.769244i	$0.720633\pi$
$282$	0	0
$283$	29.4791	1.75235	0.876175	−	0.481993i	$-0.160087\pi$
$283$	29.4791	1.75235	0.876175	+	0.481993i	$0.160087\pi$
$284$	0	0
$285$	−4.05761	−0.240352
$286$	0	0
$287$	−21.7844	−1.28590
$288$	0	0
$289$	45.8614	2.69773
$290$	0	0
$291$	−2.63864	−0.154680
$292$	0	0
$293$	−20.6837	−1.20836	−0.604178	−	0.796850i	$-0.706498\pi$
$293$	−20.6837	−1.20836	−0.604178	+	0.796850i	$0.706498\pi$
$294$	0	0
$295$	−8.37689	−0.487721
$296$	0	0
$297$	3.63555	0.210956
$298$	0	0
$299$	−8.59044	−0.496798
$300$	0	0
$301$	−9.91417	−0.571443
$302$	0	0
$303$	3.93886	0.226282
$304$	0	0
$305$	21.1934	1.21353
$306$	0	0
$307$	−22.6016	−1.28994	−0.644971	−	0.764207i	$-0.723131\pi$
$307$	−22.6016	−1.28994	−0.644971	+	0.764207i	$0.723131\pi$
$308$	0	0
$309$	−14.7225	−0.837537
$310$	0	0
$311$	−6.68131	−0.378863	−0.189431	−	0.981894i	$-0.560664\pi$
$311$	−6.68131	−0.378863	−0.189431	+	0.981894i	$0.560664\pi$
$312$	0	0
$313$	−16.7075	−0.944366	−0.472183	−	0.881500i	$-0.656534\pi$
$313$	−16.7075	−0.944366	−0.472183	+	0.881500i	$0.656534\pi$
$314$	0	0
$315$	−9.29659	−0.523804
$316$	0	0
$317$	−27.4825	−1.54357	−0.771784	−	0.635885i	$-0.780635\pi$
$317$	−27.4825	−1.54357	−0.771784	+	0.635885i	$0.780635\pi$
$318$	0	0
$319$	1.03069	0.0577074
$320$	0	0
$321$	−22.5020	−1.25594
$322$	0	0
$323$	−8.62523	−0.479921
$324$	0	0
$325$	−2.80417	−0.155548
$326$	0	0
$327$	−22.7173	−1.25627
$328$	0	0
$329$	3.56573	0.196585
$330$	0	0
$331$	−28.4092	−1.56151	−0.780756	−	0.624836i	$-0.785166\pi$
$331$	−28.4092	−1.56151	−0.780756	+	0.624836i	$0.785166\pi$
$332$	0	0
$333$	−5.83442	−0.319725
$334$	0	0
$335$	4.27968	0.233824
$336$	0	0
$337$	−6.27704	−0.341932	−0.170966	−	0.985277i	$-0.554689\pi$
$337$	−6.27704	−0.341932	−0.170966	+	0.985277i	$0.554689\pi$
$338$	0	0
$339$	−41.4487	−2.25118
$340$	0	0
$341$	4.77126	0.258378
$342$	0	0
$343$	13.7208	0.740852
$344$	0	0
$345$	19.5783	1.05406
$346$	0	0
$347$	15.5336	0.833890	0.416945	−	0.908932i	$-0.363101\pi$
$347$	15.5336	0.833890	0.416945	+	0.908932i	$0.363101\pi$
$348$	0	0
$349$	−16.5944	−0.888280	−0.444140	−	0.895957i	$-0.646491\pi$
$349$	−16.5944	−0.888280	−0.444140	+	0.895957i	$0.646491\pi$
$350$	0	0
$351$	−5.94979	−0.317577
$352$	0	0
$353$	−6.05342	−0.322191	−0.161096	−	0.986939i	$-0.551503\pi$
$353$	−6.05342	−0.322191	−0.161096	+	0.986939i	$0.551503\pi$
$354$	0	0
$355$	23.8733	1.26706
$356$	0	0
$357$	−67.8458	−3.59078
$358$	0	0
$359$	9.22303	0.486773	0.243386	−	0.969929i	$-0.421742\pi$
$359$	9.22303	0.486773	0.243386	+	0.969929i	$0.421742\pi$
$360$	0	0
$361$	−17.8165	−0.937712
$362$	0	0
$363$	2.05741	0.107986
$364$	0	0
$365$	−11.8905	−0.622377
$366$	0	0
$367$	−6.93239	−0.361868	−0.180934	−	0.983495i	$-0.557912\pi$
$367$	−6.93239	−0.361868	−0.180934	+	0.983495i	$0.557912\pi$
$368$	0	0
$369$	−6.45777	−0.336178
$370$	0	0
$371$	−30.9972	−1.60930
$372$	0	0
$373$	6.86468	0.355440	0.177720	−	0.984081i	$-0.443128\pi$
$373$	6.86468	0.355440	0.177720	+	0.984081i	$0.443128\pi$
$374$	0	0
$375$	25.0402	1.29307
$376$	0	0
$377$	−1.68678	−0.0868736
$378$	0	0
$379$	5.43536	0.279196	0.139598	−	0.990208i	$-0.455419\pi$
$379$	5.43536	0.279196	0.139598	+	0.990208i	$0.455419\pi$
$380$	0	0
$381$	13.5492	0.694146
$382$	0	0
$383$	−8.18223	−0.418093	−0.209046	−	0.977906i	$-0.567036\pi$
$383$	−8.18223	−0.418093	−0.209046	+	0.977906i	$0.567036\pi$
$384$	0	0
$385$	7.54013	0.384281
$386$	0	0
$387$	−2.93895	−0.149395
$388$	0	0
$389$	11.3007	0.572969	0.286485	−	0.958085i	$-0.407513\pi$
$389$	11.3007	0.572969	0.286485	+	0.958085i	$0.407513\pi$
$390$	0	0
$391$	41.6175	2.10469
$392$	0	0
$393$	21.2778	1.07332
$394$	0	0
$395$	−24.8761	−1.25165
$396$	0	0
$397$	−32.1068	−1.61140	−0.805698	−	0.592327i	$-0.798209\pi$
$397$	−32.1068	−1.61140	−0.805698	+	0.592327i	$0.798209\pi$
$398$	0	0
$399$	9.30914	0.466040
$400$	0	0
$401$	35.4948	1.77252	0.886262	−	0.463184i	$-0.153293\pi$
$401$	35.4948	1.77252	0.886262	+	0.463184i	$0.153293\pi$
$402$	0	0
$403$	−7.80845	−0.388967
$404$	0	0
$405$	20.2656	1.00701
$406$	0	0
$407$	4.73209	0.234561
$408$	0	0
$409$	−23.6652	−1.17017	−0.585085	−	0.810972i	$-0.698939\pi$
$409$	−23.6652	−1.17017	−0.585085	+	0.810972i	$0.698939\pi$
$410$	0	0
$411$	2.05741	0.101485
$412$	0	0
$413$	19.2186	0.945687
$414$	0	0
$415$	−27.7231	−1.36087
$416$	0	0
$417$	34.5247	1.69068
$418$	0	0
$419$	−8.09951	−0.395687	−0.197844	−	0.980234i	$-0.563394\pi$
$419$	−8.09951	−0.395687	−0.197844	+	0.980234i	$0.563394\pi$
$420$	0	0
$421$	0.821646	0.0400446	0.0200223	−	0.999800i	$-0.493626\pi$
$421$	0.821646	0.0400446	0.0200223	+	0.999800i	$0.493626\pi$
$422$	0	0
$423$	1.05702	0.0513942
$424$	0	0
$425$	13.5852	0.658978
$426$	0	0
$427$	−48.6229	−2.35303
$428$	0	0
$429$	−3.36708	−0.162564
$430$	0	0
$431$	−26.6619	−1.28426	−0.642129	−	0.766597i	$-0.721949\pi$
$431$	−26.6619	−1.28426	−0.642129	+	0.766597i	$0.721949\pi$
$432$	0	0
$433$	−6.85950	−0.329646	−0.164823	−	0.986323i	$-0.552705\pi$
$433$	−6.85950	−0.329646	−0.164823	+	0.986323i	$0.552705\pi$
$434$	0	0
$435$	3.84431	0.184320
$436$	0	0
$437$	−5.71035	−0.273163
$438$	0	0
$439$	19.9754	0.953373	0.476686	−	0.879073i	$-0.341838\pi$
$439$	19.9754	0.953373	0.476686	+	0.879073i	$0.341838\pi$
$440$	0	0
$441$	12.6980	0.604667
$442$	0	0
$443$	12.3517	0.586845	0.293423	−	0.955983i	$-0.405206\pi$
$443$	12.3517	0.586845	0.293423	+	0.955983i	$0.405206\pi$
$444$	0	0
$445$	−33.8968	−1.60686
$446$	0	0
$447$	−25.1571	−1.18989
$448$	0	0
$449$	−35.5190	−1.67625	−0.838123	−	0.545482i	$-0.816347\pi$
$449$	−35.5190	−1.67625	−0.838123	+	0.545482i	$0.816347\pi$
$450$	0	0
$451$	5.23766	0.246632
$452$	0	0
$453$	43.4917	2.04342
$454$	0	0
$455$	−12.3399	−0.578501
$456$	0	0
$457$	−31.8176	−1.48837	−0.744183	−	0.667976i	$-0.767161\pi$
$457$	−31.8176	−1.48837	−0.744183	+	0.667976i	$0.767161\pi$
$458$	0	0
$459$	28.8246	1.34542
$460$	0	0
$461$	38.6088	1.79819	0.899097	−	0.437750i	$-0.144225\pi$
$461$	38.6088	1.79819	0.899097	+	0.437750i	$0.144225\pi$
$462$	0	0
$463$	20.3072	0.943755	0.471878	−	0.881664i	$-0.343576\pi$
$463$	20.3072	0.943755	0.471878	+	0.881664i	$0.343576\pi$
$464$	0	0
$465$	17.7961	0.825274
$466$	0	0
$467$	−14.7396	−0.682065	−0.341033	−	0.940051i	$-0.610777\pi$
$467$	−14.7396	−0.682065	−0.341033	+	0.940051i	$0.610777\pi$
$468$	0	0
$469$	−9.81863	−0.453382
$470$	0	0
$471$	34.2109	1.57636
$472$	0	0
$473$	2.38368	0.109602
$474$	0	0
$475$	−1.86403	−0.0855274
$476$	0	0
$477$	−9.18879	−0.420726
$478$	0	0
$479$	−11.0004	−0.502620	−0.251310	−	0.967907i	$-0.580861\pi$
$479$	−11.0004	−0.502620	−0.251310	+	0.967907i	$0.580861\pi$
$480$	0	0
$481$	−7.74434	−0.353111
$482$	0	0
$483$	−44.9174	−2.04381
$484$	0	0
$485$	2.32503	0.105574
$486$	0	0
$487$	9.43318	0.427458	0.213729	−	0.976893i	$-0.431439\pi$
$487$	9.43318	0.427458	0.213729	+	0.976893i	$0.431439\pi$
$488$	0	0
$489$	−41.6838	−1.88501
$490$	0	0
$491$	24.0029	1.08324	0.541619	−	0.840624i	$-0.317812\pi$
$491$	24.0029	1.08324	0.541619	+	0.840624i	$0.317812\pi$
$492$	0	0
$493$	8.17182	0.368040
$494$	0	0
$495$	2.23519	0.100464
$496$	0	0
$497$	−54.7712	−2.45683
$498$	0	0
$499$	−1.74625	−0.0781731	−0.0390866	−	0.999236i	$-0.512445\pi$
$499$	−1.74625	−0.0781731	−0.0390866	+	0.999236i	$0.512445\pi$
$500$	0	0
$501$	38.2307	1.70802
$502$	0	0
$503$	−21.4524	−0.956515	−0.478257	−	0.878220i	$-0.658731\pi$
$503$	−21.4524	−0.956515	−0.478257	+	0.878220i	$0.658731\pi$
$504$	0	0
$505$	−3.47071	−0.154445
$506$	0	0
$507$	−21.2360	−0.943122
$508$	0	0
$509$	−15.3962	−0.682424	−0.341212	−	0.939986i	$-0.610837\pi$
$509$	−15.3962	−0.682424	−0.341212	+	0.939986i	$0.610837\pi$
$510$	0	0
$511$	27.2797	1.20678
$512$	0	0
$513$	−3.95503	−0.174619
$514$	0	0
$515$	12.9727	0.571646
$516$	0	0
$517$	−0.857313	−0.0377046
$518$	0	0
$519$	11.5204	0.505687
$520$	0	0
$521$	3.03071	0.132778	0.0663890	−	0.997794i	$-0.478852\pi$
$521$	3.03071	0.132778	0.0663890	+	0.997794i	$0.478852\pi$
$522$	0	0
$523$	17.8070	0.778647	0.389324	−	0.921101i	$-0.372709\pi$
$523$	17.8070	0.778647	0.389324	+	0.921101i	$0.372709\pi$
$524$	0	0
$525$	−14.6624	−0.639918
$526$	0	0
$527$	37.8291	1.64786
$528$	0	0
$529$	4.55296	0.197955
$530$	0	0
$531$	5.69716	0.247236
$532$	0	0
$533$	−8.57173	−0.371283
$534$	0	0
$535$	19.8275	0.857219
$536$	0	0
$537$	41.2769	1.78123
$538$	0	0
$539$	−10.2989	−0.443605
$540$	0	0
$541$	−18.2644	−0.785248	−0.392624	−	0.919699i	$-0.628433\pi$
$541$	−18.2644	−0.785248	−0.392624	+	0.919699i	$0.628433\pi$
$542$	0	0
$543$	−8.10998	−0.348033
$544$	0	0
$545$	20.0173	0.857446
$546$	0	0
$547$	31.1060	1.32999	0.664997	−	0.746846i	$-0.268433\pi$
$547$	31.1060	1.32999	0.664997	+	0.746846i	$0.268433\pi$
$548$	0	0
$549$	−14.4138	−0.615164
$550$	0	0
$551$	−1.12126	−0.0477672
$552$	0	0
$553$	57.0718	2.42694
$554$	0	0
$555$	17.6500	0.749200
$556$	0	0
$557$	15.6397	0.662674	0.331337	−	0.943513i	$-0.392500\pi$
$557$	15.6397	0.662674	0.331337	+	0.943513i	$0.392500\pi$
$558$	0	0
$559$	−3.90102	−0.164996
$560$	0	0
$561$	16.3122	0.688703
$562$	0	0
$563$	−17.3043	−0.729291	−0.364645	−	0.931146i	$-0.618810\pi$
$563$	−17.3043	−0.729291	−0.364645	+	0.931146i	$0.618810\pi$
$564$	0	0
$565$	36.5223	1.53651
$566$	0	0
$567$	−46.4943	−1.95258
$568$	0	0
$569$	−37.1514	−1.55747	−0.778735	−	0.627353i	$-0.784138\pi$
$569$	−37.1514	−1.55747	−0.778735	+	0.627353i	$0.784138\pi$
$570$	0	0
$571$	−21.8987	−0.916434	−0.458217	−	0.888840i	$-0.651512\pi$
$571$	−21.8987	−0.916434	−0.458217	+	0.888840i	$0.651512\pi$
$572$	0	0
$573$	9.13655	0.381685
$574$	0	0
$575$	8.99410	0.375080
$576$	0	0
$577$	23.6753	0.985615	0.492807	−	0.870138i	$-0.335971\pi$
$577$	23.6753	0.985615	0.492807	+	0.870138i	$0.335971\pi$
$578$	0	0
$579$	−31.4563	−1.30728
$580$	0	0
$581$	63.6035	2.63872
$582$	0	0
$583$	7.45270	0.308659
$584$	0	0
$585$	−3.65802	−0.151240
$586$	0	0
$587$	−16.2444	−0.670479	−0.335240	−	0.942133i	$-0.608817\pi$
$587$	−16.2444	−0.670479	−0.335240	+	0.942133i	$0.608817\pi$
$588$	0	0
$589$	−5.19054	−0.213872
$590$	0	0
$591$	−47.2238	−1.94253
$592$	0	0
$593$	−29.0887	−1.19453	−0.597264	−	0.802044i	$-0.703746\pi$
$593$	−29.0887	−1.19453	−0.597264	+	0.802044i	$0.703746\pi$
$594$	0	0
$595$	59.7821	2.45083
$596$	0	0
$597$	45.7339	1.87176
$598$	0	0
$599$	1.73739	0.0709877	0.0354938	−	0.999370i	$-0.488700\pi$
$599$	1.73739	0.0709877	0.0354938	+	0.999370i	$0.488700\pi$
$600$	0	0
$601$	23.4254	0.955543	0.477772	−	0.878484i	$-0.341445\pi$
$601$	23.4254	0.955543	0.477772	+	0.878484i	$0.341445\pi$
$602$	0	0
$603$	−2.91063	−0.118530
$604$	0	0
$605$	−1.81288	−0.0737041
$606$	0	0
$607$	−12.1274	−0.492238	−0.246119	−	0.969240i	$-0.579155\pi$
$607$	−12.1274	−0.492238	−0.246119	+	0.969240i	$0.579155\pi$
$608$	0	0
$609$	−8.81978	−0.357395
$610$	0	0
$611$	1.40304	0.0567610
$612$	0	0
$613$	1.57264	0.0635182	0.0317591	−	0.999496i	$-0.489889\pi$
$613$	1.57264	0.0635182	0.0317591	+	0.999496i	$0.489889\pi$
$614$	0	0
$615$	19.5357	0.787754
$616$	0	0
$617$	3.22418	0.129801	0.0649003	−	0.997892i	$-0.479327\pi$
$617$	3.22418	0.129801	0.0649003	+	0.997892i	$0.479327\pi$
$618$	0	0
$619$	21.0614	0.846527	0.423264	−	0.906007i	$-0.360884\pi$
$619$	21.0614	0.846527	0.423264	+	0.906007i	$0.360884\pi$
$620$	0	0
$621$	19.0834	0.765789
$622$	0	0
$623$	77.7676	3.11569
$624$	0	0
$625$	−13.4968	−0.539871
$626$	0	0
$627$	−2.23821	−0.0893854
$628$	0	0
$629$	37.5185	1.49596
$630$	0	0
$631$	29.8534	1.18844	0.594222	−	0.804301i	$-0.297460\pi$
$631$	29.8534	1.18844	0.594222	+	0.804301i	$0.297460\pi$
$632$	0	0
$633$	2.54789	0.101269
$634$	0	0
$635$	−11.9388	−0.473778
$636$	0	0
$637$	16.8547	0.667809
$638$	0	0
$639$	−16.2363	−0.642300
$640$	0	0
$641$	−12.3620	−0.488270	−0.244135	−	0.969741i	$-0.578504\pi$
$641$	−12.3620	−0.488270	−0.244135	+	0.969741i	$0.578504\pi$
$642$	0	0
$643$	−25.8167	−1.01811	−0.509055	−	0.860734i	$-0.670005\pi$
$643$	−25.8167	−1.01811	−0.509055	+	0.860734i	$0.670005\pi$
$644$	0	0
$645$	8.89075	0.350073
$646$	0	0
$647$	2.74969	0.108102	0.0540508	−	0.998538i	$-0.482787\pi$
$647$	2.74969	0.108102	0.0540508	+	0.998538i	$0.482787\pi$
$648$	0	0
$649$	−4.62076	−0.181381
$650$	0	0
$651$	−40.8286	−1.60020
$652$	0	0
$653$	39.8254	1.55849	0.779245	−	0.626719i	$-0.215603\pi$
$653$	39.8254	1.55849	0.779245	+	0.626719i	$0.215603\pi$
$654$	0	0
$655$	−18.7489	−0.732579
$656$	0	0
$657$	8.08678	0.315495
$658$	0	0
$659$	36.2528	1.41221	0.706105	−	0.708107i	$-0.250451\pi$
$659$	36.2528	1.41221	0.706105	+	0.708107i	$0.250451\pi$
$660$	0	0
$661$	32.5904	1.26762	0.633809	−	0.773490i	$-0.281490\pi$
$661$	32.5904	1.26762	0.633809	+	0.773490i	$0.281490\pi$
$662$	0	0
$663$	−26.6959	−1.03678
$664$	0	0
$665$	−8.20271	−0.318088
$666$	0	0
$667$	5.41017	0.209483
$668$	0	0
$669$	10.9512	0.423399
$670$	0	0
$671$	11.6905	0.451306
$672$	0	0
$673$	6.00271	0.231387	0.115694	−	0.993285i	$-0.463091\pi$
$673$	6.00271	0.231387	0.115694	+	0.993285i	$0.463091\pi$
$674$	0	0
$675$	6.22937	0.239769
$676$	0	0
$677$	−20.7074	−0.795850	−0.397925	−	0.917418i	$-0.630269\pi$
$677$	−20.7074	−0.795850	−0.397925	+	0.917418i	$0.630269\pi$
$678$	0	0
$679$	−5.33418	−0.204707
$680$	0	0
$681$	−48.0314	−1.84057
$682$	0	0
$683$	−15.7279	−0.601810	−0.300905	−	0.953654i	$-0.597289\pi$
$683$	−15.7279	−0.601810	−0.300905	+	0.953654i	$0.597289\pi$
$684$	0	0
$685$	−1.81288	−0.0692667
$686$	0	0
$687$	−35.9594	−1.37194
$688$	0	0
$689$	−12.1968	−0.464660
$690$	0	0
$691$	44.9348	1.70940	0.854700	−	0.519123i	$-0.173741\pi$
$691$	44.9348	1.70940	0.854700	+	0.519123i	$0.173741\pi$
$692$	0	0
$693$	−5.12807	−0.194799
$694$	0	0
$695$	−30.4214	−1.15395
$696$	0	0
$697$	41.5269	1.57294
$698$	0	0
$699$	−30.2984	−1.14599
$700$	0	0
$701$	17.4656	0.659667	0.329834	−	0.944039i	$-0.393007\pi$
$701$	17.4656	0.659667	0.329834	+	0.944039i	$0.393007\pi$
$702$	0	0
$703$	−5.14792	−0.194157
$704$	0	0
$705$	−3.19765	−0.120430
$706$	0	0
$707$	7.96266	0.299467
$708$	0	0
$709$	−30.0553	−1.12875	−0.564375	−	0.825518i	$-0.690883\pi$
$709$	−30.0553	−1.12875	−0.564375	+	0.825518i	$0.690883\pi$
$710$	0	0
$711$	16.9183	0.634487
$712$	0	0
$713$	25.0448	0.937935
$714$	0	0
$715$	2.96689	0.110955
$716$	0	0
$717$	40.8370	1.52508
$718$	0	0
$719$	−23.3566	−0.871055	−0.435528	−	0.900175i	$-0.643438\pi$
$719$	−23.3566	−0.871055	−0.435528	+	0.900175i	$0.643438\pi$
$720$	0	0
$721$	−29.7626	−1.10842
$722$	0	0
$723$	46.4890	1.72894
$724$	0	0
$725$	1.76604	0.0655891
$726$	0	0
$727$	−20.2224	−0.750007	−0.375004	−	0.927023i	$-0.622359\pi$
$727$	−20.2224	−0.750007	−0.375004	+	0.927023i	$0.622359\pi$
$728$	0	0
$729$	8.65677	0.320621
$730$	0	0
$731$	18.8990	0.699005
$732$	0	0
$733$	9.80756	0.362250	0.181125	−	0.983460i	$-0.442026\pi$
$733$	9.80756	0.362250	0.181125	+	0.983460i	$0.442026\pi$
$734$	0	0
$735$	−38.4133	−1.41690
$736$	0	0
$737$	2.36070	0.0869576
$738$	0	0
$739$	31.2789	1.15061	0.575307	−	0.817938i	$-0.304883\pi$
$739$	31.2789	1.15061	0.575307	+	0.817938i	$0.304883\pi$
$740$	0	0
$741$	3.66295	0.134562
$742$	0	0
$743$	4.10643	0.150650	0.0753251	−	0.997159i	$-0.476001\pi$
$743$	4.10643	0.150650	0.0753251	+	0.997159i	$0.476001\pi$
$744$	0	0
$745$	22.1671	0.812140
$746$	0	0
$747$	18.8546	0.689853
$748$	0	0
$749$	−45.4892	−1.66214
$750$	0	0
$751$	−33.1588	−1.20998	−0.604991	−	0.796232i	$-0.706823\pi$
$751$	−33.1588	−1.20998	−0.604991	+	0.796232i	$0.706823\pi$
$752$	0	0
$753$	19.7796	0.720808
$754$	0	0
$755$	−38.3226	−1.39470
$756$	0	0
$757$	35.9560	1.30684	0.653422	−	0.756994i	$-0.273333\pi$
$757$	35.9560	1.30684	0.653422	+	0.756994i	$0.273333\pi$
$758$	0	0
$759$	10.7995	0.391999
$760$	0	0
$761$	−32.1673	−1.16607	−0.583033	−	0.812449i	$-0.698134\pi$
$761$	−32.1673	−1.16607	−0.583033	+	0.812449i	$0.698134\pi$
$762$	0	0
$763$	−45.9246	−1.66258
$764$	0	0
$765$	17.7218	0.640731
$766$	0	0
$767$	7.56214	0.273053
$768$	0	0
$769$	−40.7788	−1.47052	−0.735261	−	0.677784i	$-0.762940\pi$
$769$	−40.7788	−1.47052	−0.735261	+	0.677784i	$0.762940\pi$
$770$	0	0
$771$	19.1865	0.690985
$772$	0	0
$773$	52.0621	1.87254	0.936271	−	0.351278i	$-0.114253\pi$
$773$	52.0621	1.87254	0.936271	+	0.351278i	$0.114253\pi$
$774$	0	0
$775$	8.17536	0.293668
$776$	0	0
$777$	−40.4934	−1.45269
$778$	0	0
$779$	−5.69792	−0.204149
$780$	0	0
$781$	13.1687	0.471214
$782$	0	0
$783$	3.74712	0.133911
$784$	0	0
$785$	−30.1448	−1.07592
$786$	0	0
$787$	−38.4552	−1.37078	−0.685389	−	0.728177i	$-0.740368\pi$
$787$	−38.4552	−1.37078	−0.685389	+	0.728177i	$0.740368\pi$
$788$	0	0
$789$	−48.3948	−1.72290
$790$	0	0
$791$	−83.7912	−2.97927
$792$	0	0
$793$	−19.1321	−0.679402
$794$	0	0
$795$	27.7974	0.985873
$796$	0	0
$797$	37.4561	1.32676	0.663382	−	0.748281i	$-0.269120\pi$
$797$	37.4561	1.32676	0.663382	+	0.748281i	$0.269120\pi$
$798$	0	0
$799$	−6.79722	−0.240468
$800$	0	0
$801$	23.0534	0.814551
$802$	0	0
$803$	−6.55889	−0.231458
$804$	0	0
$805$	39.5788	1.39497
$806$	0	0
$807$	−30.3884	−1.06972
$808$	0	0
$809$	16.5447	0.581679	0.290839	−	0.956772i	$-0.406065\pi$
$809$	16.5447	0.581679	0.290839	+	0.956772i	$0.406065\pi$
$810$	0	0
$811$	−40.0458	−1.40620	−0.703099	−	0.711092i	$-0.748201\pi$
$811$	−40.0458	−1.40620	−0.703099	+	0.711092i	$0.748201\pi$
$812$	0	0
$813$	54.5873	1.91446
$814$	0	0
$815$	36.7296	1.28658
$816$	0	0
$817$	−2.59314	−0.0907225
$818$	0	0
$819$	8.39239	0.293254
$820$	0	0
$821$	39.5663	1.38087	0.690437	−	0.723393i	$-0.257418\pi$
$821$	39.5663	1.38087	0.690437	+	0.723393i	$0.257418\pi$
$822$	0	0
$823$	−0.914000	−0.0318600	−0.0159300	−	0.999873i	$-0.505071\pi$
$823$	−0.914000	−0.0318600	−0.0159300	+	0.999873i	$0.505071\pi$
$824$	0	0
$825$	3.52529	0.122735
$826$	0	0
$827$	25.9947	0.903925	0.451963	−	0.892037i	$-0.350724\pi$
$827$	25.9947	0.903925	0.451963	+	0.892037i	$0.350724\pi$
$828$	0	0
$829$	−22.6417	−0.786380	−0.393190	−	0.919457i	$-0.628629\pi$
$829$	−22.6417	−0.786380	−0.393190	+	0.919457i	$0.628629\pi$
$830$	0	0
$831$	50.3970	1.74825
$832$	0	0
$833$	−81.6550	−2.82918
$834$	0	0
$835$	−33.6869	−1.16578
$836$	0	0
$837$	17.3462	0.599572
$838$	0	0
$839$	23.4439	0.809374	0.404687	−	0.914455i	$-0.367380\pi$
$839$	23.4439	0.809374	0.404687	+	0.914455i	$0.367380\pi$
$840$	0	0
$841$	−27.9377	−0.963368
$842$	0	0
$843$	−44.0732	−1.51796
$844$	0	0
$845$	18.7120	0.643712
$846$	0	0
$847$	4.15919	0.142912
$848$	0	0
$849$	60.6507	2.08153
$850$	0	0
$851$	24.8392	0.851476
$852$	0	0
$853$	20.6799	0.708067	0.354033	−	0.935233i	$-0.384810\pi$
$853$	20.6799	0.708067	0.354033	+	0.935233i	$0.384810\pi$
$854$	0	0
$855$	−2.43161	−0.0831592
$856$	0	0
$857$	−51.4399	−1.75716	−0.878578	−	0.477600i	$-0.841507\pi$
$857$	−51.4399	−1.75716	−0.878578	+	0.477600i	$0.841507\pi$
$858$	0	0
$859$	32.4924	1.10863	0.554314	−	0.832308i	$-0.312981\pi$
$859$	32.4924	1.10863	0.554314	+	0.832308i	$0.312981\pi$
$860$	0	0
$861$	−44.8196	−1.52745
$862$	0	0
$863$	43.2538	1.47238	0.736188	−	0.676778i	$-0.236624\pi$
$863$	43.2538	1.47238	0.736188	+	0.676778i	$0.236624\pi$
$864$	0	0
$865$	−10.1511	−0.345148
$866$	0	0
$867$	94.3559	3.20449
$868$	0	0
$869$	−13.7218	−0.465482
$870$	0	0
$871$	−3.86343	−0.130907
$872$	0	0
$873$	−1.58126	−0.0535175
$874$	0	0
$875$	50.6203	1.71128
$876$	0	0
$877$	−39.1510	−1.32203	−0.661017	−	0.750371i	$-0.729875\pi$
$877$	−39.1510	−1.32203	−0.661017	+	0.750371i	$0.729875\pi$
$878$	0	0
$879$	−42.5549	−1.43534
$880$	0	0
$881$	48.9650	1.64967	0.824836	−	0.565372i	$-0.191267\pi$
$881$	48.9650	1.64967	0.824836	+	0.565372i	$0.191267\pi$
$882$	0	0
$883$	−34.8611	−1.17317	−0.586584	−	0.809888i	$-0.699528\pi$
$883$	−34.8611	−1.17317	−0.586584	+	0.809888i	$0.699528\pi$
$884$	0	0
$885$	−17.2347	−0.579339
$886$	0	0
$887$	−4.02371	−0.135103	−0.0675515	−	0.997716i	$-0.521519\pi$
$887$	−4.02371	−0.135103	−0.0675515	+	0.997716i	$0.521519\pi$
$888$	0	0
$889$	27.3906	0.918650
$890$	0	0
$891$	11.1787	0.374500
$892$	0	0
$893$	0.932649	0.0312099
$894$	0	0
$895$	−36.3710	−1.21575
$896$	0	0
$897$	−17.6741	−0.590121
$898$	0	0
$899$	4.91768	0.164014
$900$	0	0
$901$	59.0889	1.96854
$902$	0	0
$903$	−20.3975	−0.678788
$904$	0	0
$905$	7.14608	0.237544
$906$	0	0
$907$	−52.6551	−1.74838	−0.874192	−	0.485580i	$-0.838609\pi$
$907$	−52.6551	−1.74838	−0.874192	+	0.485580i	$0.838609\pi$
$908$	0	0
$909$	2.36045	0.0782910
$910$	0	0
$911$	−9.93607	−0.329197	−0.164598	−	0.986361i	$-0.552633\pi$
$911$	−9.93607	−0.329197	−0.164598	+	0.986361i	$0.552633\pi$
$912$	0	0
$913$	−15.2923	−0.506100
$914$	0	0
$915$	43.6037	1.44149
$916$	0	0
$917$	43.0145	1.42046
$918$	0	0
$919$	18.1403	0.598394	0.299197	−	0.954191i	$-0.403281\pi$
$919$	18.1403	0.598394	0.299197	+	0.954191i	$0.403281\pi$
$920$	0	0
$921$	−46.5009	−1.53226
$922$	0	0
$923$	−21.5514	−0.709371
$924$	0	0
$925$	8.10824	0.266597
$926$	0	0
$927$	−8.82280	−0.289779
$928$	0	0
$929$	−10.6857	−0.350586	−0.175293	−	0.984516i	$-0.556087\pi$
$929$	−10.6857	−0.350586	−0.175293	+	0.984516i	$0.556087\pi$
$930$	0	0
$931$	11.2039	0.367193
$932$	0	0
$933$	−13.7462	−0.450031
$934$	0	0
$935$	−14.3735	−0.470063
$936$	0	0
$937$	−27.8079	−0.908443	−0.454221	−	0.890889i	$-0.650083\pi$
$937$	−27.8079	−0.908443	−0.454221	+	0.890889i	$0.650083\pi$
$938$	0	0
$939$	−34.3743	−1.12176
$940$	0	0
$941$	−34.7991	−1.13442	−0.567209	−	0.823574i	$-0.691977\pi$
$941$	−34.7991	−1.13442	−0.567209	+	0.823574i	$0.691977\pi$
$942$	0	0
$943$	27.4930	0.895294
$944$	0	0
$945$	27.4126	0.891730
$946$	0	0
$947$	1.98595	0.0645347	0.0322673	−	0.999479i	$-0.489727\pi$
$947$	1.98595	0.0645347	0.0322673	+	0.999479i	$0.489727\pi$
$948$	0	0
$949$	10.7340	0.348441
$950$	0	0
$951$	−56.5428	−1.83352
$952$	0	0
$953$	−46.9184	−1.51984	−0.759918	−	0.650019i	$-0.774761\pi$
$953$	−46.9184	−1.51984	−0.759918	+	0.650019i	$0.774761\pi$
$954$	0	0
$955$	−8.05064	−0.260513
$956$	0	0
$957$	2.12055	0.0685476
$958$	0	0
$959$	4.15919	0.134307
$960$	0	0
$961$	−8.23504	−0.265646
$962$	0	0
$963$	−13.4848	−0.434541
$964$	0	0
$965$	27.7176	0.892261
$966$	0	0
$967$	−32.5575	−1.04698	−0.523489	−	0.852033i	$-0.675370\pi$
$967$	−32.5575	−1.04698	−0.523489	+	0.852033i	$0.675370\pi$
$968$	0	0
$969$	−17.7457	−0.570073
$970$	0	0
$971$	36.3913	1.16785	0.583927	−	0.811806i	$-0.301515\pi$
$971$	36.3913	1.16785	0.583927	+	0.811806i	$0.301515\pi$
$972$	0	0
$973$	69.7940	2.23749
$974$	0	0
$975$	−5.76934	−0.184767
$976$	0	0
$977$	57.5771	1.84205	0.921027	−	0.389499i	$-0.127352\pi$
$977$	57.5771	1.84205	0.921027	+	0.389499i	$0.127352\pi$
$978$	0	0
$979$	−18.6978	−0.597583
$980$	0	0
$981$	−13.6138	−0.434657
$982$	0	0
$983$	−15.0795	−0.480962	−0.240481	−	0.970654i	$-0.577305\pi$
$983$	−15.0795	−0.480962	−0.240481	+	0.970654i	$0.577305\pi$
$984$	0	0
$985$	41.6111	1.32584
$986$	0	0
$987$	7.33618	0.233513
$988$	0	0
$989$	12.5121	0.397863
$990$	0	0
$991$	4.97159	0.157928	0.0789639	−	0.996877i	$-0.474839\pi$
$991$	4.97159	0.157928	0.0789639	+	0.996877i	$0.474839\pi$
$992$	0	0
$993$	−58.4495	−1.85484
$994$	0	0
$995$	−40.2982	−1.27754
$996$	0	0
$997$	38.8404	1.23009	0.615044	−	0.788493i	$-0.289138\pi$
$997$	38.8404	1.23009	0.615044	+	0.788493i	$0.289138\pi$
$998$	0	0
$999$	17.2038	0.544303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.d.1.22	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.d.1.22	✓	27	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q+2.05741 q^{3} -1.81288 q^{5} +4.15919 q^{7} +1.23295 q^{9} +O(q^{10})\)	\(q+2.05741 q^{3} -1.81288 q^{5} +4.15919 q^{7} +1.23295 q^{9} -1.00000 q^{11} +1.63656 q^{13} -3.72985 q^{15} -7.92852 q^{17} +1.08787 q^{19} +8.55718 q^{21} -5.24909 q^{23} -1.71346 q^{25} -3.63555 q^{27} -1.03069 q^{29} -4.77126 q^{31} -2.05741 q^{33} -7.54013 q^{35} -4.73209 q^{37} +3.36708 q^{39} -5.23766 q^{41} -2.38368 q^{43} -2.23519 q^{45} +0.857313 q^{47} +10.2989 q^{49} -16.3122 q^{51} -7.45270 q^{53} +1.81288 q^{55} +2.23821 q^{57} +4.62076 q^{59} -11.6905 q^{61} +5.12807 q^{63} -2.96689 q^{65} -2.36070 q^{67} -10.7995 q^{69} -13.1687 q^{71} +6.55889 q^{73} -3.52529 q^{75} -4.15919 q^{77} +13.7218 q^{79} -11.1787 q^{81} +15.2923 q^{83} +14.3735 q^{85} -2.12055 q^{87} +18.6978 q^{89} +6.80676 q^{91} -9.81646 q^{93} -1.97219 q^{95} -1.28250 q^{97} -1.23295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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