LMFDB - Embedded newform 8048.2.a.u.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.08803 q^{3} +3.67682 q^{5} -1.62501 q^{7} -1.81619 q^{9} +O(q^{10})$	$q+1.08803 q^{3} +3.67682 q^{5} -1.62501 q^{7} -1.81619 q^{9} +0.444097 q^{11} +6.96112 q^{13} +4.00050 q^{15} -2.64454 q^{17} -4.48749 q^{19} -1.76807 q^{21} +7.32387 q^{23} +8.51902 q^{25} -5.24017 q^{27} -6.81726 q^{29} -5.57222 q^{31} +0.483191 q^{33} -5.97488 q^{35} +1.88078 q^{37} +7.57392 q^{39} +2.69229 q^{41} +3.72008 q^{43} -6.67779 q^{45} +9.38780 q^{47} -4.35933 q^{49} -2.87735 q^{51} +5.46030 q^{53} +1.63286 q^{55} -4.88253 q^{57} +11.1052 q^{59} +11.2532 q^{61} +2.95133 q^{63} +25.5948 q^{65} +6.74986 q^{67} +7.96861 q^{69} +13.5015 q^{71} -0.377804 q^{73} +9.26897 q^{75} -0.721663 q^{77} +13.3256 q^{79} -0.252910 q^{81} +1.00812 q^{83} -9.72351 q^{85} -7.41740 q^{87} -6.79299 q^{89} -11.3119 q^{91} -6.06275 q^{93} -16.4997 q^{95} -14.5945 q^{97} -0.806562 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * 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
$3$	1.08803	0.628176	0.314088	−	0.949394i	$-0.398301\pi$
$3$	1.08803	0.628176	0.314088	+	0.949394i	$0.398301\pi$
$4$	0	0
$5$	3.67682	1.64433	0.822163	−	0.569253i	$-0.192767\pi$
$5$	3.67682	1.64433	0.822163	+	0.569253i	$0.192767\pi$
$6$	0	0
$7$	−1.62501	−0.614197	−0.307099	−	0.951678i	$-0.599358\pi$
$7$	−1.62501	−0.614197	−0.307099	+	0.951678i	$0.599358\pi$
$8$	0	0
$9$	−1.81619	−0.605395
$10$	0	0
$11$	0.444097	0.133900	0.0669501	−	0.997756i	$-0.478673\pi$
$11$	0.444097	0.133900	0.0669501	+	0.997756i	$0.478673\pi$
$12$	0	0
$13$	6.96112	1.93067	0.965333	−	0.261021i	$-0.0840593\pi$
$13$	6.96112	1.93067	0.965333	+	0.261021i	$0.0840593\pi$
$14$	0	0
$15$	4.00050	1.03292
$16$	0	0
$17$	−2.64454	−0.641396	−0.320698	−	0.947182i	$-0.603917\pi$
$17$	−2.64454	−0.641396	−0.320698	+	0.947182i	$0.603917\pi$
$18$	0	0
$19$	−4.48749	−1.02950	−0.514750	−	0.857340i	$-0.672115\pi$
$19$	−4.48749	−1.02950	−0.514750	+	0.857340i	$0.672115\pi$
$20$	0	0
$21$	−1.76807	−0.385824
$22$	0	0
$23$	7.32387	1.52713	0.763567	−	0.645729i	$-0.223446\pi$
$23$	7.32387	1.52713	0.763567	+	0.645729i	$0.223446\pi$
$24$	0	0
$25$	8.51902	1.70380
$26$	0	0
$27$	−5.24017	−1.00847
$28$	0	0
$29$	−6.81726	−1.26593	−0.632967	−	0.774179i	$-0.718163\pi$
$29$	−6.81726	−1.26593	−0.632967	+	0.774179i	$0.718163\pi$
$30$	0	0
$31$	−5.57222	−1.00080	−0.500400	−	0.865794i	$-0.666814\pi$
$31$	−5.57222	−1.00080	−0.500400	+	0.865794i	$0.666814\pi$
$32$	0	0
$33$	0.483191	0.0841128
$34$	0	0
$35$	−5.97488	−1.00994
$36$	0	0
$37$	1.88078	0.309199	0.154599	−	0.987977i	$-0.450591\pi$
$37$	1.88078	0.309199	0.154599	+	0.987977i	$0.450591\pi$
$38$	0	0
$39$	7.57392	1.21280
$40$	0	0
$41$	2.69229	0.420465	0.210233	−	0.977651i	$-0.432578\pi$
$41$	2.69229	0.420465	0.210233	+	0.977651i	$0.432578\pi$
$42$	0	0
$43$	3.72008	0.567308	0.283654	−	0.958927i	$-0.408453\pi$
$43$	3.72008	0.567308	0.283654	+	0.958927i	$0.408453\pi$
$44$	0	0
$45$	−6.67779	−0.995467
$46$	0	0
$47$	9.38780	1.36935	0.684676	−	0.728848i	$-0.259944\pi$
$47$	9.38780	1.36935	0.684676	+	0.728848i	$0.259944\pi$
$48$	0	0
$49$	−4.35933	−0.622762
$50$	0	0
$51$	−2.87735	−0.402909
$52$	0	0
$53$	5.46030	0.750029	0.375015	−	0.927019i	$-0.377638\pi$
$53$	5.46030	0.750029	0.375015	+	0.927019i	$0.377638\pi$
$54$	0	0
$55$	1.63286	0.220175
$56$	0	0
$57$	−4.88253	−0.646707
$58$	0	0
$59$	11.1052	1.44577	0.722886	−	0.690967i	$-0.242815\pi$
$59$	11.1052	1.44577	0.722886	+	0.690967i	$0.242815\pi$
$60$	0	0
$61$	11.2532	1.44083	0.720415	−	0.693544i	$-0.243952\pi$
$61$	11.2532	1.44083	0.720415	+	0.693544i	$0.243952\pi$
$62$	0	0
$63$	2.95133	0.371832
$64$	0	0
$65$	25.5948	3.17464
$66$	0	0
$67$	6.74986	0.824627	0.412313	−	0.911042i	$-0.364721\pi$
$67$	6.74986	0.824627	0.412313	+	0.911042i	$0.364721\pi$
$68$	0	0
$69$	7.96861	0.959308
$70$	0	0
$71$	13.5015	1.60233	0.801167	−	0.598441i	$-0.204213\pi$
$71$	13.5015	1.60233	0.801167	+	0.598441i	$0.204213\pi$
$72$	0	0
$73$	−0.377804	−0.0442186	−0.0221093	−	0.999756i	$-0.507038\pi$
$73$	−0.377804	−0.0442186	−0.0221093	+	0.999756i	$0.507038\pi$
$74$	0	0
$75$	9.26897	1.07029
$76$	0	0
$77$	−0.721663	−0.0822411
$78$	0	0
$79$	13.3256	1.49924	0.749622	−	0.661866i	$-0.230235\pi$
$79$	13.3256	1.49924	0.749622	+	0.661866i	$0.230235\pi$
$80$	0	0
$81$	−0.252910	−0.0281011
$82$	0	0
$83$	1.00812	0.110656	0.0553279	−	0.998468i	$-0.482380\pi$
$83$	1.00812	0.110656	0.0553279	+	0.998468i	$0.482380\pi$
$84$	0	0
$85$	−9.72351	−1.05466
$86$	0	0
$87$	−7.41740	−0.795228
$88$	0	0
$89$	−6.79299	−0.720055	−0.360028	−	0.932942i	$-0.617233\pi$
$89$	−6.79299	−0.720055	−0.360028	+	0.932942i	$0.617233\pi$
$90$	0	0
$91$	−11.3119	−1.18581
$92$	0	0
$93$	−6.06275	−0.628678
$94$	0	0
$95$	−16.4997	−1.69283
$96$	0	0
$97$	−14.5945	−1.48185	−0.740924	−	0.671589i	$-0.765612\pi$
$97$	−14.5945	−1.48185	−0.740924	+	0.671589i	$0.765612\pi$
$98$	0	0
$99$	−0.806562	−0.0810625
$100$	0	0
$101$	14.9813	1.49070	0.745350	−	0.666674i	$-0.232283\pi$
$101$	14.9813	1.49070	0.745350	+	0.666674i	$0.232283\pi$
$102$	0	0
$103$	0.348097	0.0342990	0.0171495	−	0.999853i	$-0.494541\pi$
$103$	0.348097	0.0342990	0.0171495	+	0.999853i	$0.494541\pi$
$104$	0	0
$105$	−6.50087	−0.634420
$106$	0	0
$107$	−9.93506	−0.960458	−0.480229	−	0.877143i	$-0.659447\pi$
$107$	−9.93506	−0.960458	−0.480229	+	0.877143i	$0.659447\pi$
$108$	0	0
$109$	9.50257	0.910181	0.455091	−	0.890445i	$-0.349607\pi$
$109$	9.50257	0.910181	0.455091	+	0.890445i	$0.349607\pi$
$110$	0	0
$111$	2.04635	0.194231
$112$	0	0
$113$	−9.80733	−0.922596	−0.461298	−	0.887245i	$-0.652616\pi$
$113$	−9.80733	−0.922596	−0.461298	+	0.887245i	$0.652616\pi$
$114$	0	0
$115$	26.9286	2.51110
$116$	0	0
$117$	−12.6427	−1.16882
$118$	0	0
$119$	4.29742	0.393943
$120$	0	0
$121$	−10.8028	−0.982071
$122$	0	0
$123$	2.92930	0.264126
$124$	0	0
$125$	12.9388	1.15728
$126$	0	0
$127$	6.46825	0.573964	0.286982	−	0.957936i	$-0.407348\pi$
$127$	6.46825	0.573964	0.286982	+	0.957936i	$0.407348\pi$
$128$	0	0
$129$	4.04757	0.356369
$130$	0	0
$131$	−10.3312	−0.902645	−0.451322	−	0.892361i	$-0.649048\pi$
$131$	−10.3312	−0.902645	−0.451322	+	0.892361i	$0.649048\pi$
$132$	0	0
$133$	7.29222	0.632316
$134$	0	0
$135$	−19.2672	−1.65825
$136$	0	0
$137$	6.01949	0.514280	0.257140	−	0.966374i	$-0.417220\pi$
$137$	6.01949	0.514280	0.257140	+	0.966374i	$0.417220\pi$
$138$	0	0
$139$	−12.0786	−1.02450	−0.512248	−	0.858837i	$-0.671187\pi$
$139$	−12.0786	−1.02450	−0.512248	+	0.858837i	$0.671187\pi$
$140$	0	0
$141$	10.2142	0.860194
$142$	0	0
$143$	3.09141	0.258516
$144$	0	0
$145$	−25.0659	−2.08161
$146$	0	0
$147$	−4.74309	−0.391204
$148$	0	0
$149$	3.60830	0.295604	0.147802	−	0.989017i	$-0.452780\pi$
$149$	3.60830	0.295604	0.147802	+	0.989017i	$0.452780\pi$
$150$	0	0
$151$	−14.3421	−1.16715	−0.583573	−	0.812061i	$-0.698346\pi$
$151$	−14.3421	−1.16715	−0.583573	+	0.812061i	$0.698346\pi$
$152$	0	0
$153$	4.80298	0.388298
$154$	0	0
$155$	−20.4881	−1.64564
$156$	0	0
$157$	19.7274	1.57442	0.787211	−	0.616684i	$-0.211524\pi$
$157$	19.7274	1.57442	0.787211	+	0.616684i	$0.211524\pi$
$158$	0	0
$159$	5.94098	0.471150
$160$	0	0
$161$	−11.9014	−0.937961
$162$	0	0
$163$	6.34095	0.496661	0.248331	−	0.968675i	$-0.420118\pi$
$163$	6.34095	0.496661	0.248331	+	0.968675i	$0.420118\pi$
$164$	0	0
$165$	1.77661	0.138309
$166$	0	0
$167$	0.899922	0.0696381	0.0348190	−	0.999394i	$-0.488915\pi$
$167$	0.899922	0.0696381	0.0348190	+	0.999394i	$0.488915\pi$
$168$	0	0
$169$	35.4571	2.72747
$170$	0	0
$171$	8.15011	0.623254
$172$	0	0
$173$	20.7113	1.57465	0.787327	−	0.616536i	$-0.211464\pi$
$173$	20.7113	1.57465	0.787327	+	0.616536i	$0.211464\pi$
$174$	0	0
$175$	−13.8435	−1.04647
$176$	0	0
$177$	12.0828	0.908199
$178$	0	0
$179$	−0.727433	−0.0543709	−0.0271855	−	0.999630i	$-0.508654\pi$
$179$	−0.727433	−0.0543709	−0.0271855	+	0.999630i	$0.508654\pi$
$180$	0	0
$181$	17.7066	1.31612	0.658059	−	0.752966i	$-0.271378\pi$
$181$	17.7066	1.31612	0.658059	+	0.752966i	$0.271378\pi$
$182$	0	0
$183$	12.2439	0.905094
$184$	0	0
$185$	6.91531	0.508424
$186$	0	0
$187$	−1.17443	−0.0858830
$188$	0	0
$189$	8.51534	0.619400
$190$	0	0
$191$	−14.3592	−1.03899	−0.519497	−	0.854473i	$-0.673881\pi$
$191$	−14.3592	−1.03899	−0.519497	+	0.854473i	$0.673881\pi$
$192$	0	0
$193$	5.92814	0.426717	0.213359	−	0.976974i	$-0.431560\pi$
$193$	5.92814	0.426717	0.213359	+	0.976974i	$0.431560\pi$
$194$	0	0
$195$	27.8479	1.99423
$196$	0	0
$197$	9.89369	0.704896	0.352448	−	0.935831i	$-0.385349\pi$
$197$	9.89369	0.704896	0.352448	+	0.935831i	$0.385349\pi$
$198$	0	0
$199$	−4.63808	−0.328785	−0.164393	−	0.986395i	$-0.552566\pi$
$199$	−4.63808	−0.328785	−0.164393	+	0.986395i	$0.552566\pi$
$200$	0	0
$201$	7.34406	0.518010
$202$	0	0
$203$	11.0781	0.777533
$204$	0	0
$205$	9.89908	0.691381
$206$	0	0
$207$	−13.3015	−0.924520
$208$	0	0
$209$	−1.99288	−0.137850
$210$	0	0
$211$	21.2753	1.46465	0.732326	−	0.680955i	$-0.238435\pi$
$211$	21.2753	1.46465	0.732326	+	0.680955i	$0.238435\pi$
$212$	0	0
$213$	14.6901	1.00655
$214$	0	0
$215$	13.6781	0.932838
$216$	0	0
$217$	9.05493	0.614689
$218$	0	0
$219$	−0.411063	−0.0277771
$220$	0	0
$221$	−18.4090	−1.23832
$222$	0	0
$223$	−6.33262	−0.424064	−0.212032	−	0.977263i	$-0.568008\pi$
$223$	−6.33262	−0.424064	−0.212032	+	0.977263i	$0.568008\pi$
$224$	0	0
$225$	−15.4721	−1.03148
$226$	0	0
$227$	−6.86948	−0.455943	−0.227972	−	0.973668i	$-0.573209\pi$
$227$	−6.86948	−0.455943	−0.227972	+	0.973668i	$0.573209\pi$
$228$	0	0
$229$	−2.10060	−0.138811	−0.0694057	−	0.997589i	$-0.522110\pi$
$229$	−2.10060	−0.138811	−0.0694057	+	0.997589i	$0.522110\pi$
$230$	0	0
$231$	−0.785192	−0.0516619
$232$	0	0
$233$	−11.5359	−0.755741	−0.377871	−	0.925858i	$-0.623344\pi$
$233$	−11.5359	−0.755741	−0.377871	+	0.925858i	$0.623344\pi$
$234$	0	0
$235$	34.5173	2.25166
$236$	0	0
$237$	14.4987	0.941789
$238$	0	0
$239$	−0.509199	−0.0329373	−0.0164687	−	0.999864i	$-0.505242\pi$
$239$	−0.509199	−0.0329373	−0.0164687	+	0.999864i	$0.505242\pi$
$240$	0	0
$241$	−21.4263	−1.38019	−0.690095	−	0.723718i	$-0.742431\pi$
$241$	−21.4263	−1.38019	−0.690095	+	0.723718i	$0.742431\pi$
$242$	0	0
$243$	15.4453	0.990818
$244$	0	0
$245$	−16.0285	−1.02402
$246$	0	0
$247$	−31.2379	−1.98762
$248$	0	0
$249$	1.09687	0.0695112
$250$	0	0
$251$	0.188092	0.0118723	0.00593613	−	0.999982i	$-0.498110\pi$
$251$	0.188092	0.0118723	0.00593613	+	0.999982i	$0.498110\pi$
$252$	0	0
$253$	3.25251	0.204483
$254$	0	0
$255$	−10.5795	−0.662514
$256$	0	0
$257$	−17.1373	−1.06900	−0.534498	−	0.845170i	$-0.679499\pi$
$257$	−17.1373	−1.06900	−0.534498	+	0.845170i	$0.679499\pi$
$258$	0	0
$259$	−3.05630	−0.189909
$260$	0	0
$261$	12.3814	0.766390
$262$	0	0
$263$	−9.83295	−0.606326	−0.303163	−	0.952939i	$-0.598043\pi$
$263$	−9.83295	−0.606326	−0.303163	+	0.952939i	$0.598043\pi$
$264$	0	0
$265$	20.0765	1.23329
$266$	0	0
$267$	−7.39099	−0.452321
$268$	0	0
$269$	−5.98584	−0.364963	−0.182481	−	0.983209i	$-0.558413\pi$
$269$	−5.98584	−0.364963	−0.182481	+	0.983209i	$0.558413\pi$
$270$	0	0
$271$	−3.21445	−0.195264	−0.0976320	−	0.995223i	$-0.531127\pi$
$271$	−3.21445	−0.195264	−0.0976320	+	0.995223i	$0.531127\pi$
$272$	0	0
$273$	−12.3077	−0.744897
$274$	0	0
$275$	3.78327	0.228140
$276$	0	0
$277$	−8.08036	−0.485502	−0.242751	−	0.970089i	$-0.578050\pi$
$277$	−8.08036	−0.485502	−0.242751	+	0.970089i	$0.578050\pi$
$278$	0	0
$279$	10.1202	0.605880
$280$	0	0
$281$	−0.364131	−0.0217222	−0.0108611	−	0.999941i	$-0.503457\pi$
$281$	−0.364131	−0.0217222	−0.0108611	+	0.999941i	$0.503457\pi$
$282$	0	0
$283$	−18.4654	−1.09765	−0.548826	−	0.835936i	$-0.684925\pi$
$283$	−18.4654	−1.09765	−0.548826	+	0.835936i	$0.684925\pi$
$284$	0	0
$285$	−17.9522	−1.06340
$286$	0	0
$287$	−4.37501	−0.258249
$288$	0	0
$289$	−10.0064	−0.588611
$290$	0	0
$291$	−15.8793	−0.930861
$292$	0	0
$293$	−24.6728	−1.44140	−0.720699	−	0.693248i	$-0.756179\pi$
$293$	−24.6728	−1.44140	−0.720699	+	0.693248i	$0.756179\pi$
$294$	0	0
$295$	40.8318	2.37732
$296$	0	0
$297$	−2.32714	−0.135034
$298$	0	0
$299$	50.9823	2.94838
$300$	0	0
$301$	−6.04518	−0.348439
$302$	0	0
$303$	16.3002	0.936421
$304$	0	0
$305$	41.3761	2.36919
$306$	0	0
$307$	−30.4706	−1.73905	−0.869526	−	0.493888i	$-0.835575\pi$
$307$	−30.4706	−1.73905	−0.869526	+	0.493888i	$0.835575\pi$
$308$	0	0
$309$	0.378741	0.0215458
$310$	0	0
$311$	−12.1823	−0.690794	−0.345397	−	0.938457i	$-0.612256\pi$
$311$	−12.1823	−0.690794	−0.345397	+	0.938457i	$0.612256\pi$
$312$	0	0
$313$	−31.0316	−1.75401	−0.877004	−	0.480482i	$-0.840462\pi$
$313$	−31.0316	−1.75401	−0.877004	+	0.480482i	$0.840462\pi$
$314$	0	0
$315$	10.8515	0.611413
$316$	0	0
$317$	24.4624	1.37394	0.686972	−	0.726683i	$-0.258939\pi$
$317$	24.4624	1.37394	0.686972	+	0.726683i	$0.258939\pi$
$318$	0	0
$319$	−3.02752	−0.169509
$320$	0	0
$321$	−10.8097	−0.603337
$322$	0	0
$323$	11.8673	0.660317
$324$	0	0
$325$	59.3019	3.28948
$326$	0	0
$327$	10.3391	0.571754
$328$	0	0
$329$	−15.2553	−0.841052
$330$	0	0
$331$	16.5187	0.907949	0.453975	−	0.891015i	$-0.350006\pi$
$331$	16.5187	0.907949	0.453975	+	0.891015i	$0.350006\pi$
$332$	0	0
$333$	−3.41585	−0.187188
$334$	0	0
$335$	24.8180	1.35595
$336$	0	0
$337$	31.4723	1.71441	0.857203	−	0.514979i	$-0.172200\pi$
$337$	31.4723	1.71441	0.857203	+	0.514979i	$0.172200\pi$
$338$	0	0
$339$	−10.6707	−0.579552
$340$	0	0
$341$	−2.47460	−0.134007
$342$	0	0
$343$	18.4591	0.996696
$344$	0	0
$345$	29.2992	1.57741
$346$	0	0
$347$	−4.69804	−0.252204	−0.126102	−	0.992017i	$-0.540247\pi$
$347$	−4.69804	−0.252204	−0.126102	+	0.992017i	$0.540247\pi$
$348$	0	0
$349$	0.337268	0.0180536	0.00902678	−	0.999959i	$-0.497127\pi$
$349$	0.337268	0.0180536	0.00902678	+	0.999959i	$0.497127\pi$
$350$	0	0
$351$	−36.4774	−1.94702
$352$	0	0
$353$	28.8116	1.53349	0.766743	−	0.641954i	$-0.221876\pi$
$353$	28.8116	1.53349	0.766743	+	0.641954i	$0.221876\pi$
$354$	0	0
$355$	49.6426	2.63476
$356$	0	0
$357$	4.67573	0.247466
$358$	0	0
$359$	23.5557	1.24322	0.621612	−	0.783326i	$-0.286478\pi$
$359$	23.5557	1.24322	0.621612	+	0.783326i	$0.286478\pi$
$360$	0	0
$361$	1.13753	0.0598701
$362$	0	0
$363$	−11.7538	−0.616913
$364$	0	0
$365$	−1.38912	−0.0727098
$366$	0	0
$367$	−32.4009	−1.69132	−0.845658	−	0.533726i	$-0.820791\pi$
$367$	−32.4009	−1.69132	−0.845658	+	0.533726i	$0.820791\pi$
$368$	0	0
$369$	−4.88970	−0.254548
$370$	0	0
$371$	−8.87305	−0.460666
$372$	0	0
$373$	−19.0971	−0.988810	−0.494405	−	0.869232i	$-0.664614\pi$
$373$	−19.0971	−0.988810	−0.494405	+	0.869232i	$0.664614\pi$
$374$	0	0
$375$	14.0779	0.726978
$376$	0	0
$377$	−47.4557	−2.44409
$378$	0	0
$379$	20.0690	1.03087	0.515437	−	0.856928i	$-0.327630\pi$
$379$	20.0690	1.03087	0.515437	+	0.856928i	$0.327630\pi$
$380$	0	0
$381$	7.03766	0.360550
$382$	0	0
$383$	−23.9798	−1.22531	−0.612656	−	0.790350i	$-0.709899\pi$
$383$	−23.9798	−1.22531	−0.612656	+	0.790350i	$0.709899\pi$
$384$	0	0
$385$	−2.65343	−0.135231
$386$	0	0
$387$	−6.75637	−0.343445
$388$	0	0
$389$	5.94704	0.301527	0.150764	−	0.988570i	$-0.451827\pi$
$389$	5.94704	0.301527	0.150764	+	0.988570i	$0.451827\pi$
$390$	0	0
$391$	−19.3683	−0.979497
$392$	0	0
$393$	−11.2407	−0.567019
$394$	0	0
$395$	48.9958	2.46524
$396$	0	0
$397$	−34.7059	−1.74184	−0.870920	−	0.491425i	$-0.836476\pi$
$397$	−34.7059	−1.74184	−0.870920	+	0.491425i	$0.836476\pi$
$398$	0	0
$399$	7.93417	0.397205
$400$	0	0
$401$	−2.22066	−0.110895	−0.0554474	−	0.998462i	$-0.517658\pi$
$401$	−2.22066	−0.110895	−0.0554474	+	0.998462i	$0.517658\pi$
$402$	0	0
$403$	−38.7889	−1.93221
$404$	0	0
$405$	−0.929904	−0.0462073
$406$	0	0
$407$	0.835250	0.0414018
$408$	0	0
$409$	−22.2378	−1.09959	−0.549795	−	0.835300i	$-0.685294\pi$
$409$	−22.2378	−1.09959	−0.549795	+	0.835300i	$0.685294\pi$
$410$	0	0
$411$	6.54940	0.323058
$412$	0	0
$413$	−18.0461	−0.887989
$414$	0	0
$415$	3.70668	0.181954
$416$	0	0
$417$	−13.1419	−0.643564
$418$	0	0
$419$	13.8531	0.676767	0.338384	−	0.941008i	$-0.390120\pi$
$419$	13.8531	0.676767	0.338384	+	0.941008i	$0.390120\pi$
$420$	0	0
$421$	9.06617	0.441858	0.220929	−	0.975290i	$-0.429091\pi$
$421$	9.06617	0.441858	0.220929	+	0.975290i	$0.429091\pi$
$422$	0	0
$423$	−17.0500	−0.828999
$424$	0	0
$425$	−22.5289	−1.09281
$426$	0	0
$427$	−18.2867	−0.884953
$428$	0	0
$429$	3.36355	0.162394
$430$	0	0
$431$	6.69719	0.322592	0.161296	−	0.986906i	$-0.448433\pi$
$431$	6.69719	0.322592	0.161296	+	0.986906i	$0.448433\pi$
$432$	0	0
$433$	16.8557	0.810034	0.405017	−	0.914309i	$-0.367266\pi$
$433$	16.8557	0.810034	0.405017	+	0.914309i	$0.367266\pi$
$434$	0	0
$435$	−27.2725	−1.30761
$436$	0	0
$437$	−32.8658	−1.57218
$438$	0	0
$439$	−13.3161	−0.635545	−0.317772	−	0.948167i	$-0.602935\pi$
$439$	−13.3161	−0.635545	−0.317772	+	0.948167i	$0.602935\pi$
$440$	0	0
$441$	7.91736	0.377017
$442$	0	0
$443$	37.0767	1.76157	0.880784	−	0.473519i	$-0.157016\pi$
$443$	37.0767	1.76157	0.880784	+	0.473519i	$0.157016\pi$
$444$	0	0
$445$	−24.9766	−1.18400
$446$	0	0
$447$	3.92595	0.185691
$448$	0	0
$449$	7.08211	0.334225	0.167113	−	0.985938i	$-0.446556\pi$
$449$	7.08211	0.334225	0.167113	+	0.985938i	$0.446556\pi$
$450$	0	0
$451$	1.19564	0.0563004
$452$	0	0
$453$	−15.6047	−0.733172
$454$	0	0
$455$	−41.5919	−1.94986
$456$	0	0
$457$	4.25876	0.199216	0.0996082	−	0.995027i	$-0.468241\pi$
$457$	4.25876	0.199216	0.0996082	+	0.995027i	$0.468241\pi$
$458$	0	0
$459$	13.8578	0.646829
$460$	0	0
$461$	18.7753	0.874452	0.437226	−	0.899352i	$-0.355961\pi$
$461$	18.7753	0.874452	0.437226	+	0.899352i	$0.355961\pi$
$462$	0	0
$463$	9.28290	0.431413	0.215706	−	0.976458i	$-0.430795\pi$
$463$	9.28290	0.431413	0.215706	+	0.976458i	$0.430795\pi$
$464$	0	0
$465$	−22.2917	−1.03375
$466$	0	0
$467$	−8.54771	−0.395541	−0.197770	−	0.980248i	$-0.563370\pi$
$467$	−8.54771	−0.395541	−0.197770	+	0.980248i	$0.563370\pi$
$468$	0	0
$469$	−10.9686	−0.506483
$470$	0	0
$471$	21.4641	0.989013
$472$	0	0
$473$	1.65208	0.0759626
$474$	0	0
$475$	−38.2290	−1.75407
$476$	0	0
$477$	−9.91691	−0.454064
$478$	0	0
$479$	−0.720515	−0.0329212	−0.0164606	−	0.999865i	$-0.505240\pi$
$479$	−0.720515	−0.0329212	−0.0164606	+	0.999865i	$0.505240\pi$
$480$	0	0
$481$	13.0924	0.596960
$482$	0	0
$483$	−12.9491	−0.589204
$484$	0	0
$485$	−53.6614	−2.43664
$486$	0	0
$487$	−40.4584	−1.83334	−0.916672	−	0.399641i	$-0.869135\pi$
$487$	−40.4584	−1.83334	−0.916672	+	0.399641i	$0.869135\pi$
$488$	0	0
$489$	6.89915	0.311991
$490$	0	0
$491$	15.7811	0.712190	0.356095	−	0.934450i	$-0.384108\pi$
$491$	15.7811	0.712190	0.356095	+	0.934450i	$0.384108\pi$
$492$	0	0
$493$	18.0285	0.811964
$494$	0	0
$495$	−2.96559	−0.133293
$496$	0	0
$497$	−21.9401	−0.984149
$498$	0	0
$499$	40.8096	1.82689	0.913444	−	0.406965i	$-0.133413\pi$
$499$	40.8096	1.82689	0.913444	+	0.406965i	$0.133413\pi$
$500$	0	0
$501$	0.979144	0.0437449
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	55.0837	2.45119
$506$	0	0
$507$	38.5785	1.71333
$508$	0	0
$509$	15.5394	0.688773	0.344387	−	0.938828i	$-0.388087\pi$
$509$	15.5394	0.688773	0.344387	+	0.938828i	$0.388087\pi$
$510$	0	0
$511$	0.613937	0.0271590
$512$	0	0
$513$	23.5152	1.03822
$514$	0	0
$515$	1.27989	0.0563987
$516$	0	0
$517$	4.16909	0.183356
$518$	0	0
$519$	22.5346	0.989159
$520$	0	0
$521$	0.0128309	0.000562132 0	0.000281066	−	1.00000i	$-0.499911\pi$
$521$	0.0128309	0.000562132 0	0.000281066		1.00000i	$0.499911\pi$
$522$	0	0
$523$	−21.2701	−0.930078	−0.465039	−	0.885290i	$-0.653960\pi$
$523$	−21.2701	−0.930078	−0.465039	+	0.885290i	$0.653960\pi$
$524$	0	0
$525$	−15.0622	−0.657368
$526$	0	0
$527$	14.7360	0.641909
$528$	0	0
$529$	30.6391	1.33214
$530$	0	0
$531$	−20.1691	−0.875264
$532$	0	0
$533$	18.7413	0.811778
$534$	0	0
$535$	−36.5295	−1.57931
$536$	0	0
$537$	−0.791471	−0.0341545
$538$	0	0
$539$	−1.93596	−0.0833879
$540$	0	0
$541$	−29.2729	−1.25854	−0.629270	−	0.777187i	$-0.716646\pi$
$541$	−29.2729	−1.25854	−0.629270	+	0.777187i	$0.716646\pi$
$542$	0	0
$543$	19.2653	0.826753
$544$	0	0
$545$	34.9393	1.49663
$546$	0	0
$547$	22.3283	0.954690	0.477345	−	0.878716i	$-0.341599\pi$
$547$	22.3283	0.954690	0.477345	+	0.878716i	$0.341599\pi$
$548$	0	0
$549$	−20.4380	−0.872271
$550$	0	0
$551$	30.5924	1.30328
$552$	0	0
$553$	−21.6542	−0.920832
$554$	0	0
$555$	7.52408	0.319379
$556$	0	0
$557$	−36.2502	−1.53597	−0.767985	−	0.640468i	$-0.778740\pi$
$557$	−36.2502	−1.53597	−0.767985	+	0.640468i	$0.778740\pi$
$558$	0	0
$559$	25.8959	1.09528
$560$	0	0
$561$	−1.27782	−0.0539496
$562$	0	0
$563$	−18.9489	−0.798602	−0.399301	−	0.916820i	$-0.630747\pi$
$563$	−18.9489	−0.798602	−0.399301	+	0.916820i	$0.630747\pi$
$564$	0	0
$565$	−36.0598	−1.51705
$566$	0	0
$567$	0.410981	0.0172596
$568$	0	0
$569$	24.2317	1.01584	0.507922	−	0.861403i	$-0.330414\pi$
$569$	24.2317	1.01584	0.507922	+	0.861403i	$0.330414\pi$
$570$	0	0
$571$	−43.1512	−1.80582	−0.902911	−	0.429827i	$-0.858574\pi$
$571$	−43.1512	−1.80582	−0.902911	+	0.429827i	$0.858574\pi$
$572$	0	0
$573$	−15.6232	−0.652670
$574$	0	0
$575$	62.3923	2.60194
$576$	0	0
$577$	21.8954	0.911518	0.455759	−	0.890103i	$-0.349368\pi$
$577$	21.8954	0.911518	0.455759	+	0.890103i	$0.349368\pi$
$578$	0	0
$579$	6.45001	0.268053
$580$	0	0
$581$	−1.63821	−0.0679644
$582$	0	0
$583$	2.42490	0.100429
$584$	0	0
$585$	−46.4849	−1.92191
$586$	0	0
$587$	−24.1518	−0.996850	−0.498425	−	0.866933i	$-0.666088\pi$
$587$	−24.1518	−0.996850	−0.498425	+	0.866933i	$0.666088\pi$
$588$	0	0
$589$	25.0053	1.03032
$590$	0	0
$591$	10.7647	0.442799
$592$	0	0
$593$	24.4219	1.00289	0.501444	−	0.865190i	$-0.332802\pi$
$593$	24.4219	1.00289	0.501444	+	0.865190i	$0.332802\pi$
$594$	0	0
$595$	15.8008	0.647771
$596$	0	0
$597$	−5.04638	−0.206535
$598$	0	0
$599$	24.1038	0.984854	0.492427	−	0.870354i	$-0.336110\pi$
$599$	24.1038	0.984854	0.492427	+	0.870354i	$0.336110\pi$
$600$	0	0
$601$	−16.4944	−0.672821	−0.336410	−	0.941715i	$-0.609213\pi$
$601$	−16.4944	−0.672821	−0.336410	+	0.941715i	$0.609213\pi$
$602$	0	0
$603$	−12.2590	−0.499225
$604$	0	0
$605$	−39.7199	−1.61484
$606$	0	0
$607$	−38.2300	−1.55171	−0.775854	−	0.630912i	$-0.782681\pi$
$607$	−38.2300	−1.55171	−0.775854	+	0.630912i	$0.782681\pi$
$608$	0	0
$609$	12.0534	0.488427
$610$	0	0
$611$	65.3496	2.64376
$612$	0	0
$613$	21.5346	0.869773	0.434886	−	0.900485i	$-0.356789\pi$
$613$	21.5346	0.869773	0.434886	+	0.900485i	$0.356789\pi$
$614$	0	0
$615$	10.7705	0.434309
$616$	0	0
$617$	8.20984	0.330516	0.165258	−	0.986250i	$-0.447154\pi$
$617$	8.20984	0.330516	0.165258	+	0.986250i	$0.447154\pi$
$618$	0	0
$619$	−21.7969	−0.876091	−0.438046	−	0.898953i	$-0.644329\pi$
$619$	−21.7969	−0.876091	−0.438046	+	0.898953i	$0.644329\pi$
$620$	0	0
$621$	−38.3783	−1.54007
$622$	0	0
$623$	11.0387	0.442256
$624$	0	0
$625$	4.97865	0.199146
$626$	0	0
$627$	−2.16831	−0.0865941
$628$	0	0
$629$	−4.97381	−0.198319
$630$	0	0
$631$	−0.416993	−0.0166002	−0.00830012	−	0.999966i	$-0.502642\pi$
$631$	−0.416993	−0.0166002	−0.00830012	+	0.999966i	$0.502642\pi$
$632$	0	0
$633$	23.1482	0.920058
$634$	0	0
$635$	23.7826	0.943784
$636$	0	0
$637$	−30.3458	−1.20235
$638$	0	0
$639$	−24.5212	−0.970045
$640$	0	0
$641$	3.93254	0.155326	0.0776629	−	0.996980i	$-0.475254\pi$
$641$	3.93254	0.155326	0.0776629	+	0.996980i	$0.475254\pi$
$642$	0	0
$643$	26.3585	1.03948	0.519739	−	0.854325i	$-0.326029\pi$
$643$	26.3585	1.03948	0.519739	+	0.854325i	$0.326029\pi$
$644$	0	0
$645$	14.8822	0.585986
$646$	0	0
$647$	−0.284750	−0.0111947	−0.00559734	−	0.999984i	$-0.501782\pi$
$647$	−0.284750	−0.0111947	−0.00559734	+	0.999984i	$0.501782\pi$
$648$	0	0
$649$	4.93178	0.193589
$650$	0	0
$651$	9.85205	0.386132
$652$	0	0
$653$	−21.6597	−0.847609	−0.423805	−	0.905754i	$-0.639306\pi$
$653$	−21.6597	−0.847609	−0.423805	+	0.905754i	$0.639306\pi$
$654$	0	0
$655$	−37.9861	−1.48424
$656$	0	0
$657$	0.686163	0.0267698
$658$	0	0
$659$	−20.3180	−0.791476	−0.395738	−	0.918363i	$-0.629511\pi$
$659$	−20.3180	−0.791476	−0.395738	+	0.918363i	$0.629511\pi$
$660$	0	0
$661$	−40.4018	−1.57145	−0.785724	−	0.618578i	$-0.787709\pi$
$661$	−40.4018	−1.57145	−0.785724	+	0.618578i	$0.787709\pi$
$662$	0	0
$663$	−20.0295	−0.777883
$664$	0	0
$665$	26.8122	1.03973
$666$	0	0
$667$	−49.9287	−1.93325
$668$	0	0
$669$	−6.89010	−0.266387
$670$	0	0
$671$	4.99752	0.192927
$672$	0	0
$673$	3.82061	0.147274	0.0736369	−	0.997285i	$-0.476539\pi$
$673$	3.82061	0.147274	0.0736369	+	0.997285i	$0.476539\pi$
$674$	0	0
$675$	−44.6411	−1.71824
$676$	0	0
$677$	38.4091	1.47618	0.738090	−	0.674702i	$-0.235728\pi$
$677$	38.4091	1.47618	0.738090	+	0.674702i	$0.235728\pi$
$678$	0	0
$679$	23.7163	0.910147
$680$	0	0
$681$	−7.47422	−0.286413
$682$	0	0
$683$	−2.05708	−0.0787118	−0.0393559	−	0.999225i	$-0.512531\pi$
$683$	−2.05708	−0.0787118	−0.0393559	+	0.999225i	$0.512531\pi$
$684$	0	0
$685$	22.1326	0.845644
$686$	0	0
$687$	−2.28552	−0.0871979
$688$	0	0
$689$	38.0097	1.44806
$690$	0	0
$691$	46.0370	1.75133	0.875665	−	0.482919i	$-0.160423\pi$
$691$	46.0370	1.75133	0.875665	+	0.482919i	$0.160423\pi$
$692$	0	0
$693$	1.31067	0.0497884
$694$	0	0
$695$	−44.4110	−1.68461
$696$	0	0
$697$	−7.11988	−0.269685
$698$	0	0
$699$	−12.5514	−0.474738
$700$	0	0
$701$	8.97682	0.339050	0.169525	−	0.985526i	$-0.445777\pi$
$701$	8.97682	0.339050	0.169525	+	0.985526i	$0.445777\pi$
$702$	0	0
$703$	−8.43999	−0.318320
$704$	0	0
$705$	37.5559	1.41444
$706$	0	0
$707$	−24.3449	−0.915583
$708$	0	0
$709$	12.9810	0.487513	0.243756	−	0.969837i	$-0.421620\pi$
$709$	12.9810	0.487513	0.243756	+	0.969837i	$0.421620\pi$
$710$	0	0
$711$	−24.2017	−0.907635
$712$	0	0
$713$	−40.8102	−1.52836
$714$	0	0
$715$	11.3666	0.425085
$716$	0	0
$717$	−0.554025	−0.0206904
$718$	0	0
$719$	33.0346	1.23198	0.615991	−	0.787754i	$-0.288756\pi$
$719$	33.0346	1.23198	0.615991	+	0.787754i	$0.288756\pi$
$720$	0	0
$721$	−0.565662	−0.0210664
$722$	0	0
$723$	−23.3125	−0.867002
$724$	0	0
$725$	−58.0764	−2.15690
$726$	0	0
$727$	−17.1741	−0.636952	−0.318476	−	0.947931i	$-0.603171\pi$
$727$	−17.1741	−0.636952	−0.318476	+	0.947931i	$0.603171\pi$
$728$	0	0
$729$	17.5637	0.650509
$730$	0	0
$731$	−9.83792	−0.363869
$732$	0	0
$733$	10.1322	0.374242	0.187121	−	0.982337i	$-0.440084\pi$
$733$	10.1322	0.374242	0.187121	+	0.982337i	$0.440084\pi$
$734$	0	0
$735$	−17.4395	−0.643266
$736$	0	0
$737$	2.99759	0.110418
$738$	0	0
$739$	29.3839	1.08091	0.540453	−	0.841374i	$-0.318253\pi$
$739$	29.3839	1.08091	0.540453	+	0.841374i	$0.318253\pi$
$740$	0	0
$741$	−33.9878	−1.24857
$742$	0	0
$743$	−8.73127	−0.320319	−0.160160	−	0.987091i	$-0.551201\pi$
$743$	−8.73127	−0.320319	−0.160160	+	0.987091i	$0.551201\pi$
$744$	0	0
$745$	13.2671	0.486069
$746$	0	0
$747$	−1.83094	−0.0669905
$748$	0	0
$749$	16.1446	0.589911
$750$	0	0
$751$	−18.2247	−0.665030	−0.332515	−	0.943098i	$-0.607897\pi$
$751$	−18.2247	−0.665030	−0.332515	+	0.943098i	$0.607897\pi$
$752$	0	0
$753$	0.204650	0.00745786
$754$	0	0
$755$	−52.7335	−1.91917
$756$	0	0
$757$	−21.5058	−0.781640	−0.390820	−	0.920467i	$-0.627809\pi$
$757$	−21.5058	−0.781640	−0.390820	+	0.920467i	$0.627809\pi$
$758$	0	0
$759$	3.53883	0.128451
$760$	0	0
$761$	−34.3434	−1.24495	−0.622474	−	0.782641i	$-0.713872\pi$
$761$	−34.3434	−1.24495	−0.622474	+	0.782641i	$0.713872\pi$
$762$	0	0
$763$	−15.4418	−0.559031
$764$	0	0
$765$	17.6597	0.638488
$766$	0	0
$767$	77.3045	2.79130
$768$	0	0
$769$	−20.2341	−0.729660	−0.364830	−	0.931074i	$-0.618873\pi$
$769$	−20.2341	−0.729660	−0.364830	+	0.931074i	$0.618873\pi$
$770$	0	0
$771$	−18.6459	−0.671517
$772$	0	0
$773$	21.5030	0.773409	0.386704	−	0.922204i	$-0.373613\pi$
$773$	21.5030	0.773409	0.386704	+	0.922204i	$0.373613\pi$
$774$	0	0
$775$	−47.4699	−1.70517
$776$	0	0
$777$	−3.32535	−0.119296
$778$	0	0
$779$	−12.0816	−0.432869
$780$	0	0
$781$	5.99597	0.214553
$782$	0	0
$783$	35.7236	1.27666
$784$	0	0
$785$	72.5343	2.58886
$786$	0	0
$787$	−9.43366	−0.336273	−0.168137	−	0.985764i	$-0.553775\pi$
$787$	−9.43366	−0.336273	−0.168137	+	0.985764i	$0.553775\pi$
$788$	0	0
$789$	−10.6986	−0.380879
$790$	0	0
$791$	15.9370	0.566656
$792$	0	0
$793$	78.3351	2.78176
$794$	0	0
$795$	21.8439	0.774724
$796$	0	0
$797$	−15.0608	−0.533482	−0.266741	−	0.963768i	$-0.585947\pi$
$797$	−15.0608	−0.533482	−0.266741	+	0.963768i	$0.585947\pi$
$798$	0	0
$799$	−24.8265	−0.878297
$800$	0	0
$801$	12.3373	0.435918
$802$	0	0
$803$	−0.167782	−0.00592088
$804$	0	0
$805$	−43.7593	−1.54231
$806$	0	0
$807$	−6.51278	−0.229261
$808$	0	0
$809$	−11.1181	−0.390890	−0.195445	−	0.980715i	$-0.562615\pi$
$809$	−11.1181	−0.390890	−0.195445	+	0.980715i	$0.562615\pi$
$810$	0	0
$811$	−13.3948	−0.470356	−0.235178	−	0.971952i	$-0.575567\pi$
$811$	−13.3948	−0.470356	−0.235178	+	0.971952i	$0.575567\pi$
$812$	0	0
$813$	−3.49743	−0.122660
$814$	0	0
$815$	23.3145	0.816673
$816$	0	0
$817$	−16.6938	−0.584043
$818$	0	0
$819$	20.5445	0.717884
$820$	0	0
$821$	17.8389	0.622583	0.311292	−	0.950314i	$-0.399238\pi$
$821$	17.8389	0.622583	0.311292	+	0.950314i	$0.399238\pi$
$822$	0	0
$823$	−13.8929	−0.484278	−0.242139	−	0.970242i	$-0.577849\pi$
$823$	−13.8929	−0.484278	−0.242139	+	0.970242i	$0.577849\pi$
$824$	0	0
$825$	4.11632	0.143312
$826$	0	0
$827$	−34.8154	−1.21065	−0.605325	−	0.795978i	$-0.706957\pi$
$827$	−34.8154	−1.21065	−0.605325	+	0.795978i	$0.706957\pi$
$828$	0	0
$829$	28.7227	0.997581	0.498791	−	0.866723i	$-0.333778\pi$
$829$	28.7227	0.997581	0.498791	+	0.866723i	$0.333778\pi$
$830$	0	0
$831$	−8.79170	−0.304981
$832$	0	0
$833$	11.5284	0.399437
$834$	0	0
$835$	3.30885	0.114508
$836$	0	0
$837$	29.1993	1.00928
$838$	0	0
$839$	−31.7200	−1.09510	−0.547549	−	0.836774i	$-0.684439\pi$
$839$	−31.7200	−1.09510	−0.547549	+	0.836774i	$0.684439\pi$
$840$	0	0
$841$	17.4750	0.602587
$842$	0	0
$843$	−0.396186	−0.0136454
$844$	0	0
$845$	130.370	4.48485
$846$	0	0
$847$	17.5547	0.603185
$848$	0	0
$849$	−20.0909	−0.689518
$850$	0	0
$851$	13.7746	0.472188
$852$	0	0
$853$	−29.2937	−1.00300	−0.501498	−	0.865159i	$-0.667217\pi$
$853$	−29.2937	−1.00300	−0.501498	+	0.865159i	$0.667217\pi$
$854$	0	0
$855$	29.9665	1.02483
$856$	0	0
$857$	−23.3315	−0.796988	−0.398494	−	0.917171i	$-0.630467\pi$
$857$	−23.3315	−0.796988	−0.398494	+	0.917171i	$0.630467\pi$
$858$	0	0
$859$	26.4734	0.903261	0.451630	−	0.892205i	$-0.350843\pi$
$859$	26.4734	0.903261	0.451630	+	0.892205i	$0.350843\pi$
$860$	0	0
$861$	−4.76015	−0.162225
$862$	0	0
$863$	−13.0103	−0.442875	−0.221438	−	0.975175i	$-0.571075\pi$
$863$	−13.0103	−0.442875	−0.221438	+	0.975175i	$0.571075\pi$
$864$	0	0
$865$	76.1519	2.58924
$866$	0	0
$867$	−10.8873	−0.369751
$868$	0	0
$869$	5.91784	0.200749
$870$	0	0
$871$	46.9865	1.59208
$872$	0	0
$873$	26.5064	0.897104
$874$	0	0
$875$	−21.0258	−0.710800
$876$	0	0
$877$	−33.7884	−1.14095	−0.570476	−	0.821314i	$-0.693241\pi$
$877$	−33.7884	−1.14095	−0.570476	+	0.821314i	$0.693241\pi$
$878$	0	0
$879$	−26.8447	−0.905451
$880$	0	0
$881$	−30.3070	−1.02107	−0.510535	−	0.859857i	$-0.670553\pi$
$881$	−30.3070	−1.02107	−0.510535	+	0.859857i	$0.670553\pi$
$882$	0	0
$883$	16.8677	0.567645	0.283822	−	0.958877i	$-0.408397\pi$
$883$	16.8677	0.567645	0.283822	+	0.958877i	$0.408397\pi$
$884$	0	0
$885$	44.4263	1.49337
$886$	0	0
$887$	7.70638	0.258755	0.129377	−	0.991595i	$-0.458702\pi$
$887$	7.70638	0.258755	0.129377	+	0.991595i	$0.458702\pi$
$888$	0	0
$889$	−10.5110	−0.352527
$890$	0	0
$891$	−0.112316	−0.00376274
$892$	0	0
$893$	−42.1276	−1.40975
$894$	0	0
$895$	−2.67464	−0.0894035
$896$	0	0
$897$	55.4704	1.85210
$898$	0	0
$899$	37.9873	1.26695
$900$	0	0
$901$	−14.4400	−0.481066
$902$	0	0
$903$	−6.57736	−0.218881
$904$	0	0
$905$	65.1039	2.16413
$906$	0	0
$907$	12.9574	0.430244	0.215122	−	0.976587i	$-0.430985\pi$
$907$	12.9574	0.430244	0.215122	+	0.976587i	$0.430985\pi$
$908$	0	0
$909$	−27.2089	−0.902462
$910$	0	0
$911$	16.1166	0.533965	0.266983	−	0.963701i	$-0.413973\pi$
$911$	16.1166	0.533965	0.266983	+	0.963701i	$0.413973\pi$
$912$	0	0
$913$	0.447703	0.0148168
$914$	0	0
$915$	45.0186	1.48827
$916$	0	0
$917$	16.7884	0.554402
$918$	0	0
$919$	42.9283	1.41607	0.708037	−	0.706176i	$-0.249581\pi$
$919$	42.9283	1.41607	0.708037	+	0.706176i	$0.249581\pi$
$920$	0	0
$921$	−33.1530	−1.09243
$922$	0	0
$923$	93.9855	3.09357
$924$	0	0
$925$	16.0224	0.526815
$926$	0	0
$927$	−0.632209	−0.0207645
$928$	0	0
$929$	−46.6902	−1.53186	−0.765928	−	0.642926i	$-0.777720\pi$
$929$	−46.6902	−1.53186	−0.765928	+	0.642926i	$0.777720\pi$
$930$	0	0
$931$	19.5624	0.641133
$932$	0	0
$933$	−13.2547	−0.433940
$934$	0	0
$935$	−4.31818	−0.141220
$936$	0	0
$937$	−32.5383	−1.06298	−0.531490	−	0.847064i	$-0.678368\pi$
$937$	−32.5383	−1.06298	−0.531490	+	0.847064i	$0.678368\pi$
$938$	0	0
$939$	−33.7634	−1.10183
$940$	0	0
$941$	16.3652	0.533492	0.266746	−	0.963767i	$-0.414052\pi$
$941$	16.3652	0.533492	0.266746	+	0.963767i	$0.414052\pi$
$942$	0	0
$943$	19.7180	0.642107
$944$	0	0
$945$	31.3094	1.01849
$946$	0	0
$947$	36.4764	1.18532	0.592661	−	0.805452i	$-0.298077\pi$
$947$	36.4764	1.18532	0.592661	+	0.805452i	$0.298077\pi$
$948$	0	0
$949$	−2.62994	−0.0853714
$950$	0	0
$951$	26.6159	0.863079
$952$	0	0
$953$	−12.5154	−0.405412	−0.202706	−	0.979240i	$-0.564974\pi$
$953$	−12.5154	−0.405412	−0.202706	+	0.979240i	$0.564974\pi$
$954$	0	0
$955$	−52.7961	−1.70844
$956$	0	0
$957$	−3.29404	−0.106481
$958$	0	0
$959$	−9.78175	−0.315869
$960$	0	0
$961$	0.0496228	0.00160074
$962$	0	0
$963$	18.0439	0.581457
$964$	0	0
$965$	21.7967	0.701661
$966$	0	0
$967$	21.7860	0.700589	0.350295	−	0.936640i	$-0.386081\pi$
$967$	21.7860	0.700589	0.350295	+	0.936640i	$0.386081\pi$
$968$	0	0
$969$	12.9121	0.414795
$970$	0	0
$971$	−23.2256	−0.745345	−0.372673	−	0.927963i	$-0.621559\pi$
$971$	−23.2256	−0.745345	−0.372673	+	0.927963i	$0.621559\pi$
$972$	0	0
$973$	19.6279	0.629243
$974$	0	0
$975$	64.5224	2.06637
$976$	0	0
$977$	−54.9006	−1.75643	−0.878213	−	0.478270i	$-0.841264\pi$
$977$	−54.9006	−1.75643	−0.878213	+	0.478270i	$0.841264\pi$
$978$	0	0
$979$	−3.01674	−0.0964155
$980$	0	0
$981$	−17.2584	−0.551020
$982$	0	0
$983$	−29.8038	−0.950594	−0.475297	−	0.879825i	$-0.657659\pi$
$983$	−29.8038	−0.950594	−0.475297	+	0.879825i	$0.657659\pi$
$984$	0	0
$985$	36.3774	1.15908
$986$	0	0
$987$	−16.5983	−0.528328
$988$	0	0
$989$	27.2454	0.866354
$990$	0	0
$991$	45.0849	1.43217	0.716084	−	0.698014i	$-0.245933\pi$
$991$	45.0849	1.43217	0.716084	+	0.698014i	$0.245933\pi$
$992$	0	0
$993$	17.9729	0.570351
$994$	0	0
$995$	−17.0534	−0.540630
$996$	0	0
$997$	36.4935	1.15576	0.577881	−	0.816121i	$-0.303880\pi$
$997$	36.4935	1.15576	0.577881	+	0.816121i	$0.303880\pi$
$998$	0	0
$999$	−9.85562	−0.311818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.u.1.18		26
4.3	odd	2		503.2.a.f.1.20	✓	26
12.11	even	2		4527.2.a.o.1.7		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.20	✓	26	4.3	odd	2
4527.2.a.o.1.7		26	12.11	even	2
8048.2.a.u.1.18		26	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+1.08803 q^{3} +3.67682 q^{5} -1.62501 q^{7} -1.81619 q^{9} +O(q^{10})\)	\(q+1.08803 q^{3} +3.67682 q^{5} -1.62501 q^{7} -1.81619 q^{9} +0.444097 q^{11} +6.96112 q^{13} +4.00050 q^{15} -2.64454 q^{17} -4.48749 q^{19} -1.76807 q^{21} +7.32387 q^{23} +8.51902 q^{25} -5.24017 q^{27} -6.81726 q^{29} -5.57222 q^{31} +0.483191 q^{33} -5.97488 q^{35} +1.88078 q^{37} +7.57392 q^{39} +2.69229 q^{41} +3.72008 q^{43} -6.67779 q^{45} +9.38780 q^{47} -4.35933 q^{49} -2.87735 q^{51} +5.46030 q^{53} +1.63286 q^{55} -4.88253 q^{57} +11.1052 q^{59} +11.2532 q^{61} +2.95133 q^{63} +25.5948 q^{65} +6.74986 q^{67} +7.96861 q^{69} +13.5015 q^{71} -0.377804 q^{73} +9.26897 q^{75} -0.721663 q^{77} +13.3256 q^{79} -0.252910 q^{81} +1.00812 q^{83} -9.72351 q^{85} -7.41740 q^{87} -6.79299 q^{89} -11.3119 q^{91} -6.06275 q^{93} -16.4997 q^{95} -14.5945 q^{97} -0.806562 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

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