LMFDB - Embedded newform 6044.2.a.a.1.1

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			1.1
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)

$f(q)$	$=$	$q-3.43683 q^{3} +1.31396 q^{5} +2.97589 q^{7} +8.81178 q^{9} +O(q^{10})$	$q-3.43683 q^{3} +1.31396 q^{5} +2.97589 q^{7} +8.81178 q^{9} -3.04814 q^{11} +4.12747 q^{13} -4.51587 q^{15} +1.31007 q^{17} +1.92060 q^{19} -10.2276 q^{21} +2.38694 q^{23} -3.27350 q^{25} -19.9741 q^{27} -0.127258 q^{29} -0.273738 q^{31} +10.4759 q^{33} +3.91021 q^{35} -10.5818 q^{37} -14.1854 q^{39} -5.48068 q^{41} -0.698763 q^{43} +11.5784 q^{45} -13.7050 q^{47} +1.85594 q^{49} -4.50248 q^{51} +3.84971 q^{53} -4.00514 q^{55} -6.60077 q^{57} -6.08417 q^{59} -12.5724 q^{61} +26.2229 q^{63} +5.42334 q^{65} -9.19524 q^{67} -8.20349 q^{69} +7.97288 q^{71} +9.98119 q^{73} +11.2505 q^{75} -9.07093 q^{77} -15.0822 q^{79} +42.2122 q^{81} -4.25697 q^{83} +1.72138 q^{85} +0.437365 q^{87} +10.6184 q^{89} +12.2829 q^{91} +0.940792 q^{93} +2.52360 q^{95} +15.6409 q^{97} -26.8595 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.43683	−1.98425	−0.992127	−	0.125239i	$-0.960030\pi$
$3$	−3.43683	−1.98425	−0.992127	+	0.125239i	$0.960030\pi$
$4$	0	0
$5$	1.31396	0.587622	0.293811	−	0.955863i	$-0.405076\pi$
$5$	1.31396	0.587622	0.293811	+	0.955863i	$0.405076\pi$
$6$	0	0
$7$	2.97589	1.12478	0.562391	−	0.826871i	$-0.309882\pi$
$7$	2.97589	1.12478	0.562391	+	0.826871i	$0.309882\pi$
$8$	0	0
$9$	8.81178	2.93726
$10$	0	0
$11$	−3.04814	−0.919048	−0.459524	−	0.888165i	$-0.651980\pi$
$11$	−3.04814	−0.919048	−0.459524	+	0.888165i	$0.651980\pi$
$12$	0	0
$13$	4.12747	1.14475	0.572377	−	0.819991i	$-0.306021\pi$
$13$	4.12747	1.14475	0.572377	+	0.819991i	$0.306021\pi$
$14$	0	0
$15$	−4.51587	−1.16599
$16$	0	0
$17$	1.31007	0.317738	0.158869	−	0.987300i	$-0.449215\pi$
$17$	1.31007	0.317738	0.158869	+	0.987300i	$0.449215\pi$
$18$	0	0
$19$	1.92060	0.440616	0.220308	−	0.975430i	$-0.429294\pi$
$19$	1.92060	0.440616	0.220308	+	0.975430i	$0.429294\pi$
$20$	0	0
$21$	−10.2276	−2.23185
$22$	0	0
$23$	2.38694	0.497711	0.248855	−	0.968541i	$-0.419946\pi$
$23$	2.38694	0.497711	0.248855	+	0.968541i	$0.419946\pi$
$24$	0	0
$25$	−3.27350	−0.654700
$26$	0	0
$27$	−19.9741	−3.84402
$28$	0	0
$29$	−0.127258	−0.0236313	−0.0118156	−	0.999930i	$-0.503761\pi$
$29$	−0.127258	−0.0236313	−0.0118156	+	0.999930i	$0.503761\pi$
$30$	0	0
$31$	−0.273738	−0.0491649	−0.0245824	−	0.999698i	$-0.507826\pi$
$31$	−0.273738	−0.0491649	−0.0245824	+	0.999698i	$0.507826\pi$
$32$	0	0
$33$	10.4759	1.82362
$34$	0	0
$35$	3.91021	0.660947
$36$	0	0
$37$	−10.5818	−1.73964	−0.869820	−	0.493370i	$-0.835765\pi$
$37$	−10.5818	−1.73964	−0.869820	+	0.493370i	$0.835765\pi$
$38$	0	0
$39$	−14.1854	−2.27148
$40$	0	0
$41$	−5.48068	−0.855938	−0.427969	−	0.903793i	$-0.640771\pi$
$41$	−5.48068	−0.855938	−0.427969	+	0.903793i	$0.640771\pi$
$42$	0	0
$43$	−0.698763	−0.106560	−0.0532802	−	0.998580i	$-0.516968\pi$
$43$	−0.698763	−0.106560	−0.0532802	+	0.998580i	$0.516968\pi$
$44$	0	0
$45$	11.5784	1.72600
$46$	0	0
$47$	−13.7050	−1.99908	−0.999538	−	0.0303977i	$-0.990323\pi$
$47$	−13.7050	−1.99908	−0.999538	+	0.0303977i	$0.990323\pi$
$48$	0	0
$49$	1.85594	0.265134
$50$	0	0
$51$	−4.50248	−0.630473
$52$	0	0
$53$	3.84971	0.528799	0.264399	−	0.964413i	$-0.414826\pi$
$53$	3.84971	0.528799	0.264399	+	0.964413i	$0.414826\pi$
$54$	0	0
$55$	−4.00514	−0.540053
$56$	0	0
$57$	−6.60077	−0.874293
$58$	0	0
$59$	−6.08417	−0.792091	−0.396046	−	0.918231i	$-0.629618\pi$
$59$	−6.08417	−0.792091	−0.396046	+	0.918231i	$0.629618\pi$
$60$	0	0
$61$	−12.5724	−1.60974	−0.804868	−	0.593454i	$-0.797764\pi$
$61$	−12.5724	−1.60974	−0.804868	+	0.593454i	$0.797764\pi$
$62$	0	0
$63$	26.2229	3.30378
$64$	0	0
$65$	5.42334	0.672683
$66$	0	0
$67$	−9.19524	−1.12338	−0.561689	−	0.827349i	$-0.689848\pi$
$67$	−9.19524	−1.12338	−0.561689	+	0.827349i	$0.689848\pi$
$68$	0	0
$69$	−8.20349	−0.987585
$70$	0	0
$71$	7.97288	0.946207	0.473104	−	0.881007i	$-0.343134\pi$
$71$	7.97288	0.946207	0.473104	+	0.881007i	$0.343134\pi$
$72$	0	0
$73$	9.98119	1.16821	0.584105	−	0.811678i	$-0.301446\pi$
$73$	9.98119	1.16821	0.584105	+	0.811678i	$0.301446\pi$
$74$	0	0
$75$	11.2505	1.29909
$76$	0	0
$77$	−9.07093	−1.03373
$78$	0	0
$79$	−15.0822	−1.69688	−0.848439	−	0.529294i	$-0.822457\pi$
$79$	−15.0822	−1.69688	−0.848439	+	0.529294i	$0.822457\pi$
$80$	0	0
$81$	42.2122	4.69024
$82$	0	0
$83$	−4.25697	−0.467263	−0.233632	−	0.972325i	$-0.575061\pi$
$83$	−4.25697	−0.467263	−0.233632	+	0.972325i	$0.575061\pi$
$84$	0	0
$85$	1.72138	0.186710
$86$	0	0
$87$	0.437365	0.0468904
$88$	0	0
$89$	10.6184	1.12555	0.562774	−	0.826611i	$-0.309734\pi$
$89$	10.6184	1.12555	0.562774	+	0.826611i	$0.309734\pi$
$90$	0	0
$91$	12.2829	1.28760
$92$	0	0
$93$	0.940792	0.0975556
$94$	0	0
$95$	2.52360	0.258916
$96$	0	0
$97$	15.6409	1.58809	0.794046	−	0.607857i	$-0.207971\pi$
$97$	15.6409	1.58809	0.794046	+	0.607857i	$0.207971\pi$
$98$	0	0
$99$	−26.8595	−2.69948
$100$	0	0
$101$	2.37551	0.236372	0.118186	−	0.992991i	$-0.462292\pi$
$101$	2.37551	0.236372	0.118186	+	0.992991i	$0.462292\pi$
$102$	0	0
$103$	−17.3325	−1.70782	−0.853909	−	0.520423i	$-0.825774\pi$
$103$	−17.3325	−1.70782	−0.853909	+	0.520423i	$0.825774\pi$
$104$	0	0
$105$	−13.4387	−1.31149
$106$	0	0
$107$	−10.0669	−0.973200	−0.486600	−	0.873625i	$-0.661763\pi$
$107$	−10.0669	−0.973200	−0.486600	+	0.873625i	$0.661763\pi$
$108$	0	0
$109$	−5.62668	−0.538938	−0.269469	−	0.963009i	$-0.586848\pi$
$109$	−5.62668	−0.538938	−0.269469	+	0.963009i	$0.586848\pi$
$110$	0	0
$111$	36.3679	3.45188
$112$	0	0
$113$	−10.5469	−0.992166	−0.496083	−	0.868275i	$-0.665229\pi$
$113$	−10.5469	−0.992166	−0.496083	+	0.868275i	$0.665229\pi$
$114$	0	0
$115$	3.13635	0.292466
$116$	0	0
$117$	36.3704	3.36244
$118$	0	0
$119$	3.89862	0.357386
$120$	0	0
$121$	−1.70886	−0.155351
$122$	0	0
$123$	18.8361	1.69840
$124$	0	0
$125$	−10.8711	−0.972339
$126$	0	0
$127$	−10.1798	−0.903311	−0.451655	−	0.892192i	$-0.649166\pi$
$127$	−10.1798	−0.903311	−0.451655	+	0.892192i	$0.649166\pi$
$128$	0	0
$129$	2.40153	0.211443
$130$	0	0
$131$	−10.4177	−0.910196	−0.455098	−	0.890441i	$-0.650396\pi$
$131$	−10.4177	−0.910196	−0.455098	+	0.890441i	$0.650396\pi$
$132$	0	0
$133$	5.71550	0.495597
$134$	0	0
$135$	−26.2452	−2.25883
$136$	0	0
$137$	13.5230	1.15535	0.577676	−	0.816266i	$-0.303960\pi$
$137$	13.5230	1.15535	0.577676	+	0.816266i	$0.303960\pi$
$138$	0	0
$139$	0.612591	0.0519593	0.0259797	−	0.999662i	$-0.491729\pi$
$139$	0.612591	0.0519593	0.0259797	+	0.999662i	$0.491729\pi$
$140$	0	0
$141$	47.1016	3.96667
$142$	0	0
$143$	−12.5811	−1.05208
$144$	0	0
$145$	−0.167213	−0.0138863
$146$	0	0
$147$	−6.37854	−0.526093
$148$	0	0
$149$	15.4821	1.26834	0.634171	−	0.773193i	$-0.281342\pi$
$149$	15.4821	1.26834	0.634171	+	0.773193i	$0.281342\pi$
$150$	0	0
$151$	18.2485	1.48504	0.742521	−	0.669822i	$-0.233630\pi$
$151$	18.2485	1.48504	0.742521	+	0.669822i	$0.233630\pi$
$152$	0	0
$153$	11.5440	0.933280
$154$	0	0
$155$	−0.359682	−0.0288904
$156$	0	0
$157$	−20.1637	−1.60924	−0.804620	−	0.593790i	$-0.797631\pi$
$157$	−20.1637	−1.60924	−0.804620	+	0.593790i	$0.797631\pi$
$158$	0	0
$159$	−13.2308	−1.04927
$160$	0	0
$161$	7.10327	0.559816
$162$	0	0
$163$	5.45585	0.427335	0.213668	−	0.976906i	$-0.431459\pi$
$163$	5.45585	0.427335	0.213668	+	0.976906i	$0.431459\pi$
$164$	0	0
$165$	13.7650	1.07160
$166$	0	0
$167$	13.8449	1.07135	0.535675	−	0.844424i	$-0.320057\pi$
$167$	13.8449	1.07135	0.535675	+	0.844424i	$0.320057\pi$
$168$	0	0
$169$	4.03600	0.310461
$170$	0	0
$171$	16.9239	1.29420
$172$	0	0
$173$	21.4674	1.63213	0.816067	−	0.577957i	$-0.196150\pi$
$173$	21.4674	1.63213	0.816067	+	0.577957i	$0.196150\pi$
$174$	0	0
$175$	−9.74159	−0.736395
$176$	0	0
$177$	20.9102	1.57171
$178$	0	0
$179$	16.3962	1.22551	0.612754	−	0.790274i	$-0.290062\pi$
$179$	16.3962	1.22551	0.612754	+	0.790274i	$0.290062\pi$
$180$	0	0
$181$	−25.4204	−1.88949	−0.944743	−	0.327812i	$-0.893689\pi$
$181$	−25.4204	−1.88949	−0.944743	+	0.327812i	$0.893689\pi$
$182$	0	0
$183$	43.2093	3.19412
$184$	0	0
$185$	−13.9041	−1.02225
$186$	0	0
$187$	−3.99327	−0.292017
$188$	0	0
$189$	−59.4408	−4.32368
$190$	0	0
$191$	−17.4317	−1.26131	−0.630656	−	0.776063i	$-0.717214\pi$
$191$	−17.4317	−1.26131	−0.630656	+	0.776063i	$0.717214\pi$
$192$	0	0
$193$	−1.88543	−0.135716	−0.0678581	−	0.997695i	$-0.521617\pi$
$193$	−1.88543	−0.135716	−0.0678581	+	0.997695i	$0.521617\pi$
$194$	0	0
$195$	−18.6391	−1.33477
$196$	0	0
$197$	−8.72400	−0.621559	−0.310779	−	0.950482i	$-0.600590\pi$
$197$	−8.72400	−0.621559	−0.310779	+	0.950482i	$0.600590\pi$
$198$	0	0
$199$	−0.450808	−0.0319569	−0.0159785	−	0.999872i	$-0.505086\pi$
$199$	−0.450808	−0.0319569	−0.0159785	+	0.999872i	$0.505086\pi$
$200$	0	0
$201$	31.6025	2.22907
$202$	0	0
$203$	−0.378707	−0.0265800
$204$	0	0
$205$	−7.20141	−0.502968
$206$	0	0
$207$	21.0332	1.46191
$208$	0	0
$209$	−5.85425	−0.404947
$210$	0	0
$211$	16.3506	1.12562	0.562812	−	0.826585i	$-0.309720\pi$
$211$	16.3506	1.12562	0.562812	+	0.826585i	$0.309720\pi$
$212$	0	0
$213$	−27.4014	−1.87751
$214$	0	0
$215$	−0.918149	−0.0626172
$216$	0	0
$217$	−0.814616	−0.0552998
$218$	0	0
$219$	−34.3036	−2.31802
$220$	0	0
$221$	5.40727	0.363732
$222$	0	0
$223$	−7.38285	−0.494392	−0.247196	−	0.968965i	$-0.579509\pi$
$223$	−7.38285	−0.494392	−0.247196	+	0.968965i	$0.579509\pi$
$224$	0	0
$225$	−28.8454	−1.92302
$226$	0	0
$227$	14.4615	0.959843	0.479922	−	0.877311i	$-0.340665\pi$
$227$	14.4615	0.959843	0.479922	+	0.877311i	$0.340665\pi$
$228$	0	0
$229$	−20.7858	−1.37356	−0.686782	−	0.726863i	$-0.740977\pi$
$229$	−20.7858	−1.37356	−0.686782	+	0.726863i	$0.740977\pi$
$230$	0	0
$231$	31.1752	2.05118
$232$	0	0
$233$	16.8030	1.10080	0.550400	−	0.834901i	$-0.314475\pi$
$233$	16.8030	1.10080	0.550400	+	0.834901i	$0.314475\pi$
$234$	0	0
$235$	−18.0078	−1.17470
$236$	0	0
$237$	51.8348	3.36703
$238$	0	0
$239$	−9.38777	−0.607244	−0.303622	−	0.952792i	$-0.598196\pi$
$239$	−9.38777	−0.607244	−0.303622	+	0.952792i	$0.598196\pi$
$240$	0	0
$241$	14.8491	0.956516	0.478258	−	0.878219i	$-0.341268\pi$
$241$	14.8491	0.956516	0.478258	+	0.878219i	$0.341268\pi$
$242$	0	0
$243$	−85.1536	−5.46261
$244$	0	0
$245$	2.43864	0.155799
$246$	0	0
$247$	7.92722	0.504397
$248$	0	0
$249$	14.6305	0.927168
$250$	0	0
$251$	−0.542070	−0.0342152	−0.0171076	−	0.999854i	$-0.505446\pi$
$251$	−0.542070	−0.0342152	−0.0171076	+	0.999854i	$0.505446\pi$
$252$	0	0
$253$	−7.27571	−0.457420
$254$	0	0
$255$	−5.91609	−0.370480
$256$	0	0
$257$	5.97182	0.372512	0.186256	−	0.982501i	$-0.440365\pi$
$257$	5.97182	0.372512	0.186256	+	0.982501i	$0.440365\pi$
$258$	0	0
$259$	−31.4903	−1.95671
$260$	0	0
$261$	−1.12137	−0.0694112
$262$	0	0
$263$	−3.67681	−0.226722	−0.113361	−	0.993554i	$-0.536162\pi$
$263$	−3.67681	−0.226722	−0.113361	+	0.993554i	$0.536162\pi$
$264$	0	0
$265$	5.05838	0.310734
$266$	0	0
$267$	−36.4936	−2.23337
$268$	0	0
$269$	4.60618	0.280844	0.140422	−	0.990092i	$-0.455154\pi$
$269$	4.60618	0.280844	0.140422	+	0.990092i	$0.455154\pi$
$270$	0	0
$271$	−31.7365	−1.92786	−0.963929	−	0.266161i	$-0.914245\pi$
$271$	−31.7365	−1.92786	−0.963929	+	0.266161i	$0.914245\pi$
$272$	0	0
$273$	−42.2142	−2.55492
$274$	0	0
$275$	9.97808	0.601701
$276$	0	0
$277$	19.4328	1.16760	0.583800	−	0.811897i	$-0.301565\pi$
$277$	19.4328	1.16760	0.583800	+	0.811897i	$0.301565\pi$
$278$	0	0
$279$	−2.41212	−0.144410
$280$	0	0
$281$	0.238157	0.0142072	0.00710362	−	0.999975i	$-0.497739\pi$
$281$	0.238157	0.0142072	0.00710362	+	0.999975i	$0.497739\pi$
$282$	0	0
$283$	−20.8132	−1.23721	−0.618607	−	0.785701i	$-0.712303\pi$
$283$	−20.8132	−1.23721	−0.618607	+	0.785701i	$0.712303\pi$
$284$	0	0
$285$	−8.67317	−0.513754
$286$	0	0
$287$	−16.3099	−0.962744
$288$	0	0
$289$	−15.2837	−0.899042
$290$	0	0
$291$	−53.7551	−3.15118
$292$	0	0
$293$	23.7369	1.38672	0.693361	−	0.720590i	$-0.256129\pi$
$293$	23.7369	1.38672	0.693361	+	0.720590i	$0.256129\pi$
$294$	0	0
$295$	−7.99437	−0.465450
$296$	0	0
$297$	60.8838	3.53283
$298$	0	0
$299$	9.85201	0.569757
$300$	0	0
$301$	−2.07944	−0.119857
$302$	0	0
$303$	−8.16423	−0.469023
$304$	0	0
$305$	−16.5197	−0.945917
$306$	0	0
$307$	0.529978	0.0302474	0.0151237	−	0.999886i	$-0.495186\pi$
$307$	0.529978	0.0302474	0.0151237	+	0.999886i	$0.495186\pi$
$308$	0	0
$309$	59.5687	3.38874
$310$	0	0
$311$	19.7427	1.11951	0.559754	−	0.828659i	$-0.310896\pi$
$311$	19.7427	1.11951	0.559754	+	0.828659i	$0.310896\pi$
$312$	0	0
$313$	−4.05191	−0.229028	−0.114514	−	0.993422i	$-0.536531\pi$
$313$	−4.05191	−0.229028	−0.114514	+	0.993422i	$0.536531\pi$
$314$	0	0
$315$	34.4560	1.94137
$316$	0	0
$317$	−10.0232	−0.562959	−0.281480	−	0.959567i	$-0.590825\pi$
$317$	−10.0232	−0.562959	−0.281480	+	0.959567i	$0.590825\pi$
$318$	0	0
$319$	0.387901	0.0217183
$320$	0	0
$321$	34.5981	1.93108
$322$	0	0
$323$	2.51612	0.140000
$324$	0	0
$325$	−13.5113	−0.749470
$326$	0	0
$327$	19.3379	1.06939
$328$	0	0
$329$	−40.7845	−2.24852
$330$	0	0
$331$	17.7200	0.973979	0.486989	−	0.873408i	$-0.338095\pi$
$331$	17.7200	0.973979	0.486989	+	0.873408i	$0.338095\pi$
$332$	0	0
$333$	−93.2446	−5.10977
$334$	0	0
$335$	−12.0822	−0.660122
$336$	0	0
$337$	−2.36241	−0.128689	−0.0643444	−	0.997928i	$-0.520496\pi$
$337$	−2.36241	−0.128689	−0.0643444	+	0.997928i	$0.520496\pi$
$338$	0	0
$339$	36.2478	1.96871
$340$	0	0
$341$	0.834392	0.0451849
$342$	0	0
$343$	−15.3082	−0.826564
$344$	0	0
$345$	−10.7791	−0.580327
$346$	0	0
$347$	−25.1626	−1.35080	−0.675399	−	0.737452i	$-0.736029\pi$
$347$	−25.1626	−1.35080	−0.675399	+	0.737452i	$0.736029\pi$
$348$	0	0
$349$	−15.4417	−0.826577	−0.413288	−	0.910600i	$-0.635620\pi$
$349$	−15.4417	−0.826577	−0.413288	+	0.910600i	$0.635620\pi$
$350$	0	0
$351$	−82.4424	−4.40045
$352$	0	0
$353$	−32.1171	−1.70942	−0.854710	−	0.519105i	$-0.826265\pi$
$353$	−32.1171	−1.70942	−0.854710	+	0.519105i	$0.826265\pi$
$354$	0	0
$355$	10.4761	0.556012
$356$	0	0
$357$	−13.3989	−0.709145
$358$	0	0
$359$	−30.3047	−1.59942	−0.799711	−	0.600385i	$-0.795014\pi$
$359$	−30.3047	−1.59942	−0.799711	+	0.600385i	$0.795014\pi$
$360$	0	0
$361$	−15.3113	−0.805858
$362$	0	0
$363$	5.87305	0.308255
$364$	0	0
$365$	13.1149	0.686466
$366$	0	0
$367$	−6.95465	−0.363030	−0.181515	−	0.983388i	$-0.558100\pi$
$367$	−6.95465	−0.363030	−0.181515	+	0.983388i	$0.558100\pi$
$368$	0	0
$369$	−48.2946	−2.51411
$370$	0	0
$371$	11.4563	0.594783
$372$	0	0
$373$	−13.3785	−0.692713	−0.346356	−	0.938103i	$-0.612581\pi$
$373$	−13.3785	−0.692713	−0.346356	+	0.938103i	$0.612581\pi$
$374$	0	0
$375$	37.3620	1.92937
$376$	0	0
$377$	−0.525255	−0.0270520
$378$	0	0
$379$	13.0746	0.671596	0.335798	−	0.941934i	$-0.390994\pi$
$379$	13.0746	0.671596	0.335798	+	0.941934i	$0.390994\pi$
$380$	0	0
$381$	34.9862	1.79240
$382$	0	0
$383$	19.5105	0.996939	0.498470	−	0.866907i	$-0.333896\pi$
$383$	19.5105	0.996939	0.498470	+	0.866907i	$0.333896\pi$
$384$	0	0
$385$	−11.9189	−0.607442
$386$	0	0
$387$	−6.15735	−0.312996
$388$	0	0
$389$	−13.0781	−0.663087	−0.331544	−	0.943440i	$-0.607569\pi$
$389$	−13.0781	−0.663087	−0.331544	+	0.943440i	$0.607569\pi$
$390$	0	0
$391$	3.12705	0.158142
$392$	0	0
$393$	35.8037	1.80606
$394$	0	0
$395$	−19.8174	−0.997123
$396$	0	0
$397$	−23.3656	−1.17268	−0.586342	−	0.810064i	$-0.699432\pi$
$397$	−23.3656	−1.17268	−0.586342	+	0.810064i	$0.699432\pi$
$398$	0	0
$399$	−19.6432	−0.983389
$400$	0	0
$401$	−2.79033	−0.139343	−0.0696713	−	0.997570i	$-0.522195\pi$
$401$	−2.79033	−0.139343	−0.0696713	+	0.997570i	$0.522195\pi$
$402$	0	0
$403$	−1.12985	−0.0562817
$404$	0	0
$405$	55.4652	2.75609
$406$	0	0
$407$	32.2548	1.59881
$408$	0	0
$409$	−10.7319	−0.530658	−0.265329	−	0.964158i	$-0.585481\pi$
$409$	−10.7319	−0.530658	−0.265329	+	0.964158i	$0.585481\pi$
$410$	0	0
$411$	−46.4763	−2.29251
$412$	0	0
$413$	−18.1058	−0.890930
$414$	0	0
$415$	−5.59350	−0.274574
$416$	0	0
$417$	−2.10537	−0.103100
$418$	0	0
$419$	8.14851	0.398081	0.199040	−	0.979991i	$-0.436218\pi$
$419$	8.14851	0.398081	0.199040	+	0.979991i	$0.436218\pi$
$420$	0	0
$421$	−9.55862	−0.465858	−0.232929	−	0.972494i	$-0.574831\pi$
$421$	−9.55862	−0.465858	−0.232929	+	0.972494i	$0.574831\pi$
$422$	0	0
$423$	−120.765	−5.87181
$424$	0	0
$425$	−4.28851	−0.208023
$426$	0	0
$427$	−37.4142	−1.81060
$428$	0	0
$429$	43.2390	2.08760
$430$	0	0
$431$	−4.05762	−0.195448	−0.0977242	−	0.995214i	$-0.531156\pi$
$431$	−4.05762	−0.195448	−0.0977242	+	0.995214i	$0.531156\pi$
$432$	0	0
$433$	13.9993	0.672763	0.336381	−	0.941726i	$-0.390797\pi$
$433$	13.9993	0.672763	0.336381	+	0.941726i	$0.390797\pi$
$434$	0	0
$435$	0.574681	0.0275539
$436$	0	0
$437$	4.58435	0.219299
$438$	0	0
$439$	39.1866	1.87027	0.935137	−	0.354288i	$-0.115277\pi$
$439$	39.1866	1.87027	0.935137	+	0.354288i	$0.115277\pi$
$440$	0	0
$441$	16.3541	0.778768
$442$	0	0
$443$	38.0814	1.80930	0.904651	−	0.426153i	$-0.140131\pi$
$443$	38.0814	1.80930	0.904651	+	0.426153i	$0.140131\pi$
$444$	0	0
$445$	13.9522	0.661397
$446$	0	0
$447$	−53.2093	−2.51671
$448$	0	0
$449$	29.9507	1.41346	0.706729	−	0.707484i	$-0.250170\pi$
$449$	29.9507	1.41346	0.706729	+	0.707484i	$0.250170\pi$
$450$	0	0
$451$	16.7059	0.786648
$452$	0	0
$453$	−62.7170	−2.94670
$454$	0	0
$455$	16.1393	0.756622
$456$	0	0
$457$	15.5239	0.726178	0.363089	−	0.931755i	$-0.381722\pi$
$457$	15.5239	0.726178	0.363089	+	0.931755i	$0.381722\pi$
$458$	0	0
$459$	−26.1674	−1.22139
$460$	0	0
$461$	−4.80886	−0.223971	−0.111985	−	0.993710i	$-0.535721\pi$
$461$	−4.80886	−0.223971	−0.111985	+	0.993710i	$0.535721\pi$
$462$	0	0
$463$	8.76396	0.407296	0.203648	−	0.979044i	$-0.434720\pi$
$463$	8.76396	0.407296	0.203648	+	0.979044i	$0.434720\pi$
$464$	0	0
$465$	1.23617	0.0573258
$466$	0	0
$467$	−34.6178	−1.60192	−0.800961	−	0.598716i	$-0.795678\pi$
$467$	−34.6178	−1.60192	−0.800961	+	0.598716i	$0.795678\pi$
$468$	0	0
$469$	−27.3641	−1.26355
$470$	0	0
$471$	69.2992	3.19314
$472$	0	0
$473$	2.12993	0.0979341
$474$	0	0
$475$	−6.28708	−0.288471
$476$	0	0
$477$	33.9228	1.55322
$478$	0	0
$479$	−9.37031	−0.428141	−0.214070	−	0.976818i	$-0.568672\pi$
$479$	−9.37031	−0.428141	−0.214070	+	0.976818i	$0.568672\pi$
$480$	0	0
$481$	−43.6761	−1.99146
$482$	0	0
$483$	−24.4127	−1.11082
$484$	0	0
$485$	20.5516	0.933199
$486$	0	0
$487$	16.7625	0.759583	0.379791	−	0.925072i	$-0.375996\pi$
$487$	16.7625	0.759583	0.379791	+	0.925072i	$0.375996\pi$
$488$	0	0
$489$	−18.7508	−0.847941
$490$	0	0
$491$	−39.9793	−1.80424	−0.902120	−	0.431486i	$-0.857989\pi$
$491$	−39.9793	−1.80424	−0.902120	+	0.431486i	$0.857989\pi$
$492$	0	0
$493$	−0.166717	−0.00750856
$494$	0	0
$495$	−35.2924	−1.58628
$496$	0	0
$497$	23.7265	1.06428
$498$	0	0
$499$	3.21802	0.144058	0.0720291	−	0.997403i	$-0.477053\pi$
$499$	3.21802	0.144058	0.0720291	+	0.997403i	$0.477053\pi$
$500$	0	0
$501$	−47.5825	−2.12583
$502$	0	0
$503$	44.1707	1.96947	0.984737	−	0.174050i	$-0.0556854\pi$
$503$	44.1707	1.96947	0.984737	+	0.174050i	$0.0556854\pi$
$504$	0	0
$505$	3.12134	0.138898
$506$	0	0
$507$	−13.8710	−0.616034
$508$	0	0
$509$	−8.99004	−0.398476	−0.199238	−	0.979951i	$-0.563847\pi$
$509$	−8.99004	−0.398476	−0.199238	+	0.979951i	$0.563847\pi$
$510$	0	0
$511$	29.7029	1.31398
$512$	0	0
$513$	−38.3622	−1.69373
$514$	0	0
$515$	−22.7742	−1.00355
$516$	0	0
$517$	41.7746	1.83725
$518$	0	0
$519$	−73.7797	−3.23857
$520$	0	0
$521$	−4.07009	−0.178314	−0.0891568	−	0.996018i	$-0.528417\pi$
$521$	−4.07009	−0.178314	−0.0891568	+	0.996018i	$0.528417\pi$
$522$	0	0
$523$	−8.38613	−0.366700	−0.183350	−	0.983048i	$-0.558694\pi$
$523$	−8.38613	−0.366700	−0.183350	+	0.983048i	$0.558694\pi$
$524$	0	0
$525$	33.4802	1.46119
$526$	0	0
$527$	−0.358616	−0.0156216
$528$	0	0
$529$	−17.3025	−0.752284
$530$	0	0
$531$	−53.6123	−2.32658
$532$	0	0
$533$	−22.6213	−0.979839
$534$	0	0
$535$	−13.2275	−0.571874
$536$	0	0
$537$	−56.3509	−2.43172
$538$	0	0
$539$	−5.65716	−0.243671
$540$	0	0
$541$	−33.3465	−1.43368	−0.716839	−	0.697239i	$-0.754412\pi$
$541$	−33.3465	−1.43368	−0.716839	+	0.697239i	$0.754412\pi$
$542$	0	0
$543$	87.3656	3.74922
$544$	0	0
$545$	−7.39325	−0.316692
$546$	0	0
$547$	−24.3956	−1.04308	−0.521541	−	0.853226i	$-0.674643\pi$
$547$	−24.3956	−1.04308	−0.521541	+	0.853226i	$0.674643\pi$
$548$	0	0
$549$	−110.786	−4.72821
$550$	0	0
$551$	−0.244412	−0.0104123
$552$	0	0
$553$	−44.8829	−1.90862
$554$	0	0
$555$	47.7860	2.02840
$556$	0	0
$557$	−30.1152	−1.27602	−0.638011	−	0.770028i	$-0.720242\pi$
$557$	−30.1152	−1.27602	−0.638011	+	0.770028i	$0.720242\pi$
$558$	0	0
$559$	−2.88412	−0.121985
$560$	0	0
$561$	13.7242	0.579435
$562$	0	0
$563$	17.6373	0.743322	0.371661	−	0.928369i	$-0.378788\pi$
$563$	17.6373	0.743322	0.371661	+	0.928369i	$0.378788\pi$
$564$	0	0
$565$	−13.8582	−0.583019
$566$	0	0
$567$	125.619	5.27550
$568$	0	0
$569$	20.5360	0.860913	0.430456	−	0.902611i	$-0.358353\pi$
$569$	20.5360	0.860913	0.430456	+	0.902611i	$0.358353\pi$
$570$	0	0
$571$	14.2908	0.598053	0.299026	−	0.954245i	$-0.403338\pi$
$571$	14.2908	0.598053	0.299026	+	0.954245i	$0.403338\pi$
$572$	0	0
$573$	59.9097	2.50276
$574$	0	0
$575$	−7.81364	−0.325851
$576$	0	0
$577$	22.3185	0.929130	0.464565	−	0.885539i	$-0.346211\pi$
$577$	22.3185	0.929130	0.464565	+	0.885539i	$0.346211\pi$
$578$	0	0
$579$	6.47989	0.269295
$580$	0	0
$581$	−12.6683	−0.525569
$582$	0	0
$583$	−11.7344	−0.485991
$584$	0	0
$585$	47.7893	1.97584
$586$	0	0
$587$	−6.25908	−0.258340	−0.129170	−	0.991622i	$-0.541231\pi$
$587$	−6.25908	−0.258340	−0.129170	+	0.991622i	$0.541231\pi$
$588$	0	0
$589$	−0.525742	−0.0216628
$590$	0	0
$591$	29.9829	1.23333
$592$	0	0
$593$	2.09671	0.0861014	0.0430507	−	0.999073i	$-0.486292\pi$
$593$	2.09671	0.0861014	0.0430507	+	0.999073i	$0.486292\pi$
$594$	0	0
$595$	5.12265	0.210008
$596$	0	0
$597$	1.54935	0.0634106
$598$	0	0
$599$	−11.8150	−0.482747	−0.241373	−	0.970432i	$-0.577598\pi$
$599$	−11.8150	−0.482747	−0.241373	+	0.970432i	$0.577598\pi$
$600$	0	0
$601$	34.9546	1.42583	0.712914	−	0.701251i	$-0.247375\pi$
$601$	34.9546	1.42583	0.712914	+	0.701251i	$0.247375\pi$
$602$	0	0
$603$	−81.0265	−3.29965
$604$	0	0
$605$	−2.24538	−0.0912876
$606$	0	0
$607$	16.5048	0.669908	0.334954	−	0.942235i	$-0.391279\pi$
$607$	16.5048	0.669908	0.334954	+	0.942235i	$0.391279\pi$
$608$	0	0
$609$	1.30155	0.0527415
$610$	0	0
$611$	−56.5668	−2.28845
$612$	0	0
$613$	−44.5280	−1.79847	−0.899234	−	0.437469i	$-0.855875\pi$
$613$	−44.5280	−1.79847	−0.899234	+	0.437469i	$0.855875\pi$
$614$	0	0
$615$	24.7500	0.998017
$616$	0	0
$617$	20.7397	0.834951	0.417475	−	0.908688i	$-0.362915\pi$
$617$	20.7397	0.834951	0.417475	+	0.908688i	$0.362915\pi$
$618$	0	0
$619$	30.5683	1.22865	0.614323	−	0.789055i	$-0.289429\pi$
$619$	30.5683	1.22865	0.614323	+	0.789055i	$0.289429\pi$
$620$	0	0
$621$	−47.6769	−1.91321
$622$	0	0
$623$	31.5992	1.26600
$624$	0	0
$625$	2.08330	0.0833322
$626$	0	0
$627$	20.1201	0.803518
$628$	0	0
$629$	−13.8629	−0.552750
$630$	0	0
$631$	−39.6401	−1.57805	−0.789025	−	0.614362i	$-0.789414\pi$
$631$	−39.6401	−1.57805	−0.789025	+	0.614362i	$0.789414\pi$
$632$	0	0
$633$	−56.1943	−2.23352
$634$	0	0
$635$	−13.3759	−0.530806
$636$	0	0
$637$	7.66033	0.303513
$638$	0	0
$639$	70.2553	2.77926
$640$	0	0
$641$	−12.0341	−0.475318	−0.237659	−	0.971349i	$-0.576380\pi$
$641$	−12.0341	−0.475318	−0.237659	+	0.971349i	$0.576380\pi$
$642$	0	0
$643$	45.7562	1.80445	0.902225	−	0.431266i	$-0.141933\pi$
$643$	45.7562	1.80445	0.902225	+	0.431266i	$0.141933\pi$
$644$	0	0
$645$	3.15552	0.124248
$646$	0	0
$647$	31.9438	1.25584	0.627919	−	0.778278i	$-0.283907\pi$
$647$	31.9438	1.25584	0.627919	+	0.778278i	$0.283907\pi$
$648$	0	0
$649$	18.5454	0.727970
$650$	0	0
$651$	2.79970	0.109729
$652$	0	0
$653$	13.2709	0.519330	0.259665	−	0.965699i	$-0.416388\pi$
$653$	13.2709	0.519330	0.259665	+	0.965699i	$0.416388\pi$
$654$	0	0
$655$	−13.6884	−0.534852
$656$	0	0
$657$	87.9521	3.43134
$658$	0	0
$659$	3.02075	0.117672	0.0588358	−	0.998268i	$-0.481261\pi$
$659$	3.02075	0.117672	0.0588358	+	0.998268i	$0.481261\pi$
$660$	0	0
$661$	−46.7998	−1.82030	−0.910151	−	0.414276i	$-0.864035\pi$
$661$	−46.7998	−1.82030	−0.910151	+	0.414276i	$0.864035\pi$
$662$	0	0
$663$	−18.5838	−0.721736
$664$	0	0
$665$	7.50996	0.291224
$666$	0	0
$667$	−0.303758	−0.0117615
$668$	0	0
$669$	25.3736	0.980999
$670$	0	0
$671$	38.3225	1.47942
$672$	0	0
$673$	6.26705	0.241577	0.120789	−	0.992678i	$-0.461458\pi$
$673$	6.26705	0.241577	0.120789	+	0.992678i	$0.461458\pi$
$674$	0	0
$675$	65.3852	2.51668
$676$	0	0
$677$	9.28662	0.356914	0.178457	−	0.983948i	$-0.442889\pi$
$677$	9.28662	0.356914	0.178457	+	0.983948i	$0.442889\pi$
$678$	0	0
$679$	46.5456	1.78626
$680$	0	0
$681$	−49.7017	−1.90457
$682$	0	0
$683$	8.49380	0.325006	0.162503	−	0.986708i	$-0.448043\pi$
$683$	8.49380	0.325006	0.162503	+	0.986708i	$0.448043\pi$
$684$	0	0
$685$	17.7688	0.678910
$686$	0	0
$687$	71.4372	2.72550
$688$	0	0
$689$	15.8896	0.605344
$690$	0	0
$691$	26.0759	0.991973	0.495987	−	0.868330i	$-0.334806\pi$
$691$	26.0759	0.991973	0.495987	+	0.868330i	$0.334806\pi$
$692$	0	0
$693$	−79.9311	−3.03633
$694$	0	0
$695$	0.804923	0.0305325
$696$	0	0
$697$	−7.18006	−0.271964
$698$	0	0
$699$	−57.7490	−2.18427
$700$	0	0
$701$	8.55904	0.323271	0.161635	−	0.986851i	$-0.448323\pi$
$701$	8.55904	0.323271	0.161635	+	0.986851i	$0.448323\pi$
$702$	0	0
$703$	−20.3234	−0.766513
$704$	0	0
$705$	61.8898	2.33091
$706$	0	0
$707$	7.06927	0.265867
$708$	0	0
$709$	−0.569928	−0.0214041	−0.0107021	−	0.999943i	$-0.503407\pi$
$709$	−0.569928	−0.0214041	−0.0107021	+	0.999943i	$0.503407\pi$
$710$	0	0
$711$	−132.901	−4.98417
$712$	0	0
$713$	−0.653397	−0.0244699
$714$	0	0
$715$	−16.5311	−0.618228
$716$	0	0
$717$	32.2642	1.20493
$718$	0	0
$719$	19.7249	0.735616	0.367808	−	0.929902i	$-0.380108\pi$
$719$	19.7249	0.735616	0.367808	+	0.929902i	$0.380108\pi$
$720$	0	0
$721$	−51.5795	−1.92092
$722$	0	0
$723$	−51.0339	−1.89797
$724$	0	0
$725$	0.416580	0.0154714
$726$	0	0
$727$	−1.06123	−0.0393587	−0.0196794	−	0.999806i	$-0.506265\pi$
$727$	−1.06123	−0.0393587	−0.0196794	+	0.999806i	$0.506265\pi$
$728$	0	0
$729$	166.022	6.14896
$730$	0	0
$731$	−0.915427	−0.0338583
$732$	0	0
$733$	39.0915	1.44388	0.721939	−	0.691957i	$-0.243251\pi$
$733$	39.0915	1.44388	0.721939	+	0.691957i	$0.243251\pi$
$734$	0	0
$735$	−8.38117	−0.309144
$736$	0	0
$737$	28.0284	1.03244
$738$	0	0
$739$	34.7905	1.27979	0.639895	−	0.768463i	$-0.278978\pi$
$739$	34.7905	1.27979	0.639895	+	0.768463i	$0.278978\pi$
$740$	0	0
$741$	−27.2445	−1.00085
$742$	0	0
$743$	52.3955	1.92220	0.961102	−	0.276194i	$-0.0890733\pi$
$743$	52.3955	1.92220	0.961102	+	0.276194i	$0.0890733\pi$
$744$	0	0
$745$	20.3429	0.745306
$746$	0	0
$747$	−37.5115	−1.37247
$748$	0	0
$749$	−29.9579	−1.09464
$750$	0	0
$751$	22.2741	0.812793	0.406397	−	0.913697i	$-0.366785\pi$
$751$	22.2741	0.812793	0.406397	+	0.913697i	$0.366785\pi$
$752$	0	0
$753$	1.86300	0.0678915
$754$	0	0
$755$	23.9779	0.872644
$756$	0	0
$757$	−5.78288	−0.210182	−0.105091	−	0.994463i	$-0.533513\pi$
$757$	−5.78288	−0.210182	−0.105091	+	0.994463i	$0.533513\pi$
$758$	0	0
$759$	25.0054	0.907638
$760$	0	0
$761$	35.2058	1.27621	0.638105	−	0.769949i	$-0.279718\pi$
$761$	35.2058	1.27621	0.638105	+	0.769949i	$0.279718\pi$
$762$	0	0
$763$	−16.7444	−0.606188
$764$	0	0
$765$	15.1684	0.548416
$766$	0	0
$767$	−25.1122	−0.906749
$768$	0	0
$769$	−11.2409	−0.405359	−0.202679	−	0.979245i	$-0.564965\pi$
$769$	−11.2409	−0.405359	−0.202679	+	0.979245i	$0.564965\pi$
$770$	0	0
$771$	−20.5241	−0.739157
$772$	0	0
$773$	6.72473	0.241872	0.120936	−	0.992660i	$-0.461410\pi$
$773$	6.72473	0.241872	0.120936	+	0.992660i	$0.461410\pi$
$774$	0	0
$775$	0.896083	0.0321882
$776$	0	0
$777$	108.227	3.88262
$778$	0	0
$779$	−10.5262	−0.377140
$780$	0	0
$781$	−24.3024	−0.869610
$782$	0	0
$783$	2.54187	0.0908390
$784$	0	0
$785$	−26.4944	−0.945625
$786$	0	0
$787$	20.5198	0.731451	0.365726	−	0.930723i	$-0.380821\pi$
$787$	20.5198	0.731451	0.365726	+	0.930723i	$0.380821\pi$
$788$	0	0
$789$	12.6366	0.449873
$790$	0	0
$791$	−31.3864	−1.11597
$792$	0	0
$793$	−51.8924	−1.84275
$794$	0	0
$795$	−17.3848	−0.616575
$796$	0	0
$797$	37.8530	1.34082	0.670411	−	0.741990i	$-0.266118\pi$
$797$	37.8530	1.34082	0.670411	+	0.741990i	$0.266118\pi$
$798$	0	0
$799$	−17.9544	−0.635183
$800$	0	0
$801$	93.5670	3.30603
$802$	0	0
$803$	−30.4240	−1.07364
$804$	0	0
$805$	9.33344	0.328961
$806$	0	0
$807$	−15.8306	−0.557265
$808$	0	0
$809$	−8.84978	−0.311142	−0.155571	−	0.987825i	$-0.549722\pi$
$809$	−8.84978	−0.311142	−0.155571	+	0.987825i	$0.549722\pi$
$810$	0	0
$811$	8.72294	0.306304	0.153152	−	0.988203i	$-0.451058\pi$
$811$	8.72294	0.306304	0.153152	+	0.988203i	$0.451058\pi$
$812$	0	0
$813$	109.073	3.82536
$814$	0	0
$815$	7.16879	0.251112
$816$	0	0
$817$	−1.34204	−0.0469522
$818$	0	0
$819$	108.234	3.78201
$820$	0	0
$821$	20.7577	0.724450	0.362225	−	0.932091i	$-0.382017\pi$
$821$	20.7577	0.724450	0.362225	+	0.932091i	$0.382017\pi$
$822$	0	0
$823$	24.6422	0.858974	0.429487	−	0.903073i	$-0.358694\pi$
$823$	24.6422	0.858974	0.429487	+	0.903073i	$0.358694\pi$
$824$	0	0
$825$	−34.2929	−1.19393
$826$	0	0
$827$	3.82099	0.132869	0.0664345	−	0.997791i	$-0.478838\pi$
$827$	3.82099	0.132869	0.0664345	+	0.997791i	$0.478838\pi$
$828$	0	0
$829$	−23.0658	−0.801109	−0.400554	−	0.916273i	$-0.631182\pi$
$829$	−23.0658	−0.801109	−0.400554	+	0.916273i	$0.631182\pi$
$830$	0	0
$831$	−66.7870	−2.31682
$832$	0	0
$833$	2.43141	0.0842433
$834$	0	0
$835$	18.1917	0.629549
$836$	0	0
$837$	5.46768	0.188991
$838$	0	0
$839$	−9.41885	−0.325175	−0.162587	−	0.986694i	$-0.551984\pi$
$839$	−9.41885	−0.325175	−0.162587	+	0.986694i	$0.551984\pi$
$840$	0	0
$841$	−28.9838	−0.999442
$842$	0	0
$843$	−0.818504	−0.0281908
$844$	0	0
$845$	5.30315	0.182434
$846$	0	0
$847$	−5.08538	−0.174736
$848$	0	0
$849$	71.5312	2.45495
$850$	0	0
$851$	−25.2581	−0.865838
$852$	0	0
$853$	8.84052	0.302694	0.151347	−	0.988481i	$-0.451639\pi$
$853$	8.84052	0.302694	0.151347	+	0.988481i	$0.451639\pi$
$854$	0	0
$855$	22.2374	0.760503
$856$	0	0
$857$	34.6723	1.18438	0.592192	−	0.805797i	$-0.298263\pi$
$857$	34.6723	1.18438	0.592192	+	0.805797i	$0.298263\pi$
$858$	0	0
$859$	6.01068	0.205082	0.102541	−	0.994729i	$-0.467303\pi$
$859$	6.01068	0.205082	0.102541	+	0.994729i	$0.467303\pi$
$860$	0	0
$861$	56.0544	1.91033
$862$	0	0
$863$	−18.6487	−0.634808	−0.317404	−	0.948290i	$-0.602811\pi$
$863$	−18.6487	−0.634808	−0.317404	+	0.948290i	$0.602811\pi$
$864$	0	0
$865$	28.2073	0.959079
$866$	0	0
$867$	52.5275	1.78393
$868$	0	0
$869$	45.9725	1.55951
$870$	0	0
$871$	−37.9531	−1.28599
$872$	0	0
$873$	137.824	4.66464
$874$	0	0
$875$	−32.3512	−1.09367
$876$	0	0
$877$	−35.3222	−1.19275	−0.596373	−	0.802707i	$-0.703392\pi$
$877$	−35.3222	−1.19275	−0.596373	+	0.802707i	$0.703392\pi$
$878$	0	0
$879$	−81.5795	−2.75161
$880$	0	0
$881$	−40.3739	−1.36023	−0.680116	−	0.733104i	$-0.738071\pi$
$881$	−40.3739	−1.36023	−0.680116	+	0.733104i	$0.738071\pi$
$882$	0	0
$883$	22.5817	0.759933	0.379967	−	0.925000i	$-0.375935\pi$
$883$	22.5817	0.759933	0.379967	+	0.925000i	$0.375935\pi$
$884$	0	0
$885$	27.4753	0.923571
$886$	0	0
$887$	−12.8314	−0.430837	−0.215419	−	0.976522i	$-0.569112\pi$
$887$	−12.8314	−0.430837	−0.215419	+	0.976522i	$0.569112\pi$
$888$	0	0
$889$	−30.2940	−1.01603
$890$	0	0
$891$	−128.668	−4.31055
$892$	0	0
$893$	−26.3218	−0.880824
$894$	0	0
$895$	21.5440	0.720136
$896$	0	0
$897$	−33.8597	−1.13054
$898$	0	0
$899$	0.0348355	0.00116183
$900$	0	0
$901$	5.04338	0.168019
$902$	0	0
$903$	7.14669	0.237827
$904$	0	0
$905$	−33.4015	−1.11030
$906$	0	0
$907$	−31.6544	−1.05107	−0.525534	−	0.850773i	$-0.676134\pi$
$907$	−31.6544	−1.05107	−0.525534	+	0.850773i	$0.676134\pi$
$908$	0	0
$909$	20.9325	0.694287
$910$	0	0
$911$	−14.4132	−0.477531	−0.238765	−	0.971077i	$-0.576743\pi$
$911$	−14.4132	−0.477531	−0.238765	+	0.971077i	$0.576743\pi$
$912$	0	0
$913$	12.9758	0.429437
$914$	0	0
$915$	56.7755	1.87694
$916$	0	0
$917$	−31.0019	−1.02377
$918$	0	0
$919$	−23.0964	−0.761879	−0.380939	−	0.924600i	$-0.624399\pi$
$919$	−23.0964	−0.761879	−0.380939	+	0.924600i	$0.624399\pi$
$920$	0	0
$921$	−1.82144	−0.0600185
$922$	0	0
$923$	32.9078	1.08317
$924$	0	0
$925$	34.6396	1.13894
$926$	0	0
$927$	−152.730	−5.01631
$928$	0	0
$929$	−21.4077	−0.702364	−0.351182	−	0.936307i	$-0.614220\pi$
$929$	−21.4077	−0.702364	−0.351182	+	0.936307i	$0.614220\pi$
$930$	0	0
$931$	3.56452	0.116822
$932$	0	0
$933$	−67.8524	−2.22139
$934$	0	0
$935$	−5.24701	−0.171595
$936$	0	0
$937$	26.8922	0.878531	0.439265	−	0.898357i	$-0.355239\pi$
$937$	26.8922	0.878531	0.439265	+	0.898357i	$0.355239\pi$
$938$	0	0
$939$	13.9257	0.454449
$940$	0	0
$941$	−39.4907	−1.28736	−0.643680	−	0.765295i	$-0.722593\pi$
$941$	−39.4907	−1.28736	−0.643680	+	0.765295i	$0.722593\pi$
$942$	0	0
$943$	−13.0820	−0.426010
$944$	0	0
$945$	−78.1030	−2.54069
$946$	0	0
$947$	−1.52937	−0.0496980	−0.0248490	−	0.999691i	$-0.507910\pi$
$947$	−1.52937	−0.0496980	−0.0248490	+	0.999691i	$0.507910\pi$
$948$	0	0
$949$	41.1970	1.33731
$950$	0	0
$951$	34.4480	1.11705
$952$	0	0
$953$	−37.0571	−1.20040	−0.600199	−	0.799851i	$-0.704912\pi$
$953$	−37.0571	−1.20040	−0.600199	+	0.799851i	$0.704912\pi$
$954$	0	0
$955$	−22.9046	−0.741175
$956$	0	0
$957$	−1.33315	−0.0430945
$958$	0	0
$959$	40.2431	1.29952
$960$	0	0
$961$	−30.9251	−0.997583
$962$	0	0
$963$	−88.7070	−2.85854
$964$	0	0
$965$	−2.47738	−0.0797498
$966$	0	0
$967$	34.0872	1.09617	0.548085	−	0.836423i	$-0.315357\pi$
$967$	34.0872	1.09617	0.548085	+	0.836423i	$0.315357\pi$
$968$	0	0
$969$	−8.64746	−0.277796
$970$	0	0
$971$	−44.7175	−1.43505	−0.717527	−	0.696531i	$-0.754726\pi$
$971$	−44.7175	−1.43505	−0.717527	+	0.696531i	$0.754726\pi$
$972$	0	0
$973$	1.82301	0.0584429
$974$	0	0
$975$	46.4359	1.48714
$976$	0	0
$977$	−31.8270	−1.01824	−0.509118	−	0.860696i	$-0.670028\pi$
$977$	−31.8270	−1.01824	−0.509118	+	0.860696i	$0.670028\pi$
$978$	0	0
$979$	−32.3663	−1.03443
$980$	0	0
$981$	−49.5811	−1.58300
$982$	0	0
$983$	−41.7358	−1.33117	−0.665583	−	0.746324i	$-0.731817\pi$
$983$	−41.7358	−1.33117	−0.665583	+	0.746324i	$0.731817\pi$
$984$	0	0
$985$	−11.4630	−0.365242
$986$	0	0
$987$	140.169	4.46164
$988$	0	0
$989$	−1.66790	−0.0530362
$990$	0	0
$991$	9.86565	0.313393	0.156696	−	0.987647i	$-0.449916\pi$
$991$	9.86565	0.313393	0.156696	+	0.987647i	$0.449916\pi$
$992$	0	0
$993$	−60.9005	−1.93262
$994$	0	0
$995$	−0.592345	−0.0187786
$996$	0	0
$997$	32.6133	1.03287	0.516437	−	0.856325i	$-0.327258\pi$
$997$	32.6133	1.03287	0.516437	+	0.856325i	$0.327258\pi$
$998$	0	0
$999$	211.362	6.68720

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-3.43683 q^{3} +1.31396 q^{5} +2.97589 q^{7} +8.81178 q^{9} +O(q^{10})\)	\(q-3.43683 q^{3} +1.31396 q^{5} +2.97589 q^{7} +8.81178 q^{9} -3.04814 q^{11} +4.12747 q^{13} -4.51587 q^{15} +1.31007 q^{17} +1.92060 q^{19} -10.2276 q^{21} +2.38694 q^{23} -3.27350 q^{25} -19.9741 q^{27} -0.127258 q^{29} -0.273738 q^{31} +10.4759 q^{33} +3.91021 q^{35} -10.5818 q^{37} -14.1854 q^{39} -5.48068 q^{41} -0.698763 q^{43} +11.5784 q^{45} -13.7050 q^{47} +1.85594 q^{49} -4.50248 q^{51} +3.84971 q^{53} -4.00514 q^{55} -6.60077 q^{57} -6.08417 q^{59} -12.5724 q^{61} +26.2229 q^{63} +5.42334 q^{65} -9.19524 q^{67} -8.20349 q^{69} +7.97288 q^{71} +9.98119 q^{73} +11.2505 q^{75} -9.07093 q^{77} -15.0822 q^{79} +42.2122 q^{81} -4.25697 q^{83} +1.72138 q^{85} +0.437365 q^{87} +10.6184 q^{89} +12.2829 q^{91} +0.940792 q^{93} +2.52360 q^{95} +15.6409 q^{97} -26.8595 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 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

L-functions

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