LMFDB - Embedded newform 8048.2.a.u.1.3

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			1.3
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)

$f(q)$	$=$	$q-3.12024 q^{3} -4.14994 q^{5} +3.36863 q^{7} +6.73592 q^{9} +O(q^{10})$	$q-3.12024 q^{3} -4.14994 q^{5} +3.36863 q^{7} +6.73592 q^{9} -2.17417 q^{11} -2.33650 q^{13} +12.9488 q^{15} +0.306294 q^{17} +7.75425 q^{19} -10.5110 q^{21} -1.14764 q^{23} +12.2220 q^{25} -11.6570 q^{27} +2.03862 q^{29} +2.13184 q^{31} +6.78395 q^{33} -13.9796 q^{35} +7.84768 q^{37} +7.29046 q^{39} +0.876006 q^{41} +5.12346 q^{43} -27.9536 q^{45} +3.55634 q^{47} +4.34770 q^{49} -0.955712 q^{51} +1.93178 q^{53} +9.02267 q^{55} -24.1951 q^{57} -10.5698 q^{59} +3.77497 q^{61} +22.6909 q^{63} +9.69634 q^{65} -4.79689 q^{67} +3.58092 q^{69} +9.50868 q^{71} +0.0400986 q^{73} -38.1355 q^{75} -7.32399 q^{77} -9.19028 q^{79} +16.1649 q^{81} +12.0203 q^{83} -1.27110 q^{85} -6.36100 q^{87} +8.41066 q^{89} -7.87083 q^{91} -6.65187 q^{93} -32.1796 q^{95} +12.6582 q^{97} -14.6451 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.12024	−1.80147	−0.900737	−	0.434365i	$-0.856973\pi$
$3$	−3.12024	−1.80147	−0.900737	+	0.434365i	$0.856973\pi$
$4$	0	0
$5$	−4.14994	−1.85591	−0.927954	−	0.372695i	$-0.878434\pi$
$5$	−4.14994	−1.85591	−0.927954	+	0.372695i	$0.878434\pi$
$6$	0	0
$7$	3.36863	1.27322	0.636612	−	0.771184i	$-0.280335\pi$
$7$	3.36863	1.27322	0.636612	+	0.771184i	$0.280335\pi$
$8$	0	0
$9$	6.73592	2.24531
$10$	0	0
$11$	−2.17417	−0.655538	−0.327769	−	0.944758i	$-0.606297\pi$
$11$	−2.17417	−0.655538	−0.327769	+	0.944758i	$0.606297\pi$
$12$	0	0
$13$	−2.33650	−0.648030	−0.324015	−	0.946052i	$-0.605033\pi$
$13$	−2.33650	−0.648030	−0.324015	+	0.946052i	$0.605033\pi$
$14$	0	0
$15$	12.9488	3.34337
$16$	0	0
$17$	0.306294	0.0742872	0.0371436	−	0.999310i	$-0.488174\pi$
$17$	0.306294	0.0742872	0.0371436	+	0.999310i	$0.488174\pi$
$18$	0	0
$19$	7.75425	1.77895	0.889473	−	0.456988i	$-0.151072\pi$
$19$	7.75425	1.77895	0.889473	+	0.456988i	$0.151072\pi$
$20$	0	0
$21$	−10.5110	−2.29368
$22$	0	0
$23$	−1.14764	−0.239300	−0.119650	−	0.992816i	$-0.538177\pi$
$23$	−1.14764	−0.239300	−0.119650	+	0.992816i	$0.538177\pi$
$24$	0	0
$25$	12.2220	2.44439
$26$	0	0
$27$	−11.6570	−2.24339
$28$	0	0
$29$	2.03862	0.378563	0.189281	−	0.981923i	$-0.439384\pi$
$29$	2.03862	0.378563	0.189281	+	0.981923i	$0.439384\pi$
$30$	0	0
$31$	2.13184	0.382890	0.191445	−	0.981503i	$-0.438683\pi$
$31$	2.13184	0.382890	0.191445	+	0.981503i	$0.438683\pi$
$32$	0	0
$33$	6.78395	1.18093
$34$	0	0
$35$	−13.9796	−2.36299
$36$	0	0
$37$	7.84768	1.29015	0.645075	−	0.764119i	$-0.276826\pi$
$37$	7.84768	1.29015	0.645075	+	0.764119i	$0.276826\pi$
$38$	0	0
$39$	7.29046	1.16741
$40$	0	0
$41$	0.876006	0.136809	0.0684046	−	0.997658i	$-0.478209\pi$
$41$	0.876006	0.136809	0.0684046	+	0.997658i	$0.478209\pi$
$42$	0	0
$43$	5.12346	0.781320	0.390660	−	0.920535i	$-0.372247\pi$
$43$	5.12346	0.781320	0.390660	+	0.920535i	$0.372247\pi$
$44$	0	0
$45$	−27.9536	−4.16708
$46$	0	0
$47$	3.55634	0.518746	0.259373	−	0.965777i	$-0.416484\pi$
$47$	3.55634	0.518746	0.259373	+	0.965777i	$0.416484\pi$
$48$	0	0
$49$	4.34770	0.621100
$50$	0	0
$51$	−0.955712	−0.133827
$52$	0	0
$53$	1.93178	0.265351	0.132675	−	0.991160i	$-0.457643\pi$
$53$	1.93178	0.265351	0.132675	+	0.991160i	$0.457643\pi$
$54$	0	0
$55$	9.02267	1.21662
$56$	0	0
$57$	−24.1951	−3.20472
$58$	0	0
$59$	−10.5698	−1.37607	−0.688037	−	0.725675i	$-0.741528\pi$
$59$	−10.5698	−1.37607	−0.688037	+	0.725675i	$0.741528\pi$
$60$	0	0
$61$	3.77497	0.483335	0.241667	−	0.970359i	$-0.422306\pi$
$61$	3.77497	0.483335	0.241667	+	0.970359i	$0.422306\pi$
$62$	0	0
$63$	22.6909	2.85878
$64$	0	0
$65$	9.69634	1.20268
$66$	0	0
$67$	−4.79689	−0.586034	−0.293017	−	0.956107i	$-0.594659\pi$
$67$	−4.79689	−0.586034	−0.293017	+	0.956107i	$0.594659\pi$
$68$	0	0
$69$	3.58092	0.431092
$70$	0	0
$71$	9.50868	1.12847	0.564236	−	0.825613i	$-0.309171\pi$
$71$	9.50868	1.12847	0.564236	+	0.825613i	$0.309171\pi$
$72$	0	0
$73$	0.0400986	0.00469319	0.00234659	−	0.999997i	$-0.499253\pi$
$73$	0.0400986	0.00469319	0.00234659	+	0.999997i	$0.499253\pi$
$74$	0	0
$75$	−38.1355	−4.40351
$76$	0	0
$77$	−7.32399	−0.834646
$78$	0	0
$79$	−9.19028	−1.03399	−0.516994	−	0.855989i	$-0.672949\pi$
$79$	−9.19028	−1.03399	−0.516994	+	0.855989i	$0.672949\pi$
$80$	0	0
$81$	16.1649	1.79610
$82$	0	0
$83$	12.0203	1.31940	0.659699	−	0.751530i	$-0.270684\pi$
$83$	12.0203	1.31940	0.659699	+	0.751530i	$0.270684\pi$
$84$	0	0
$85$	−1.27110	−0.137870
$86$	0	0
$87$	−6.36100	−0.681971
$88$	0	0
$89$	8.41066	0.891528	0.445764	−	0.895150i	$-0.352932\pi$
$89$	8.41066	0.891528	0.445764	+	0.895150i	$0.352932\pi$
$90$	0	0
$91$	−7.87083	−0.825087
$92$	0	0
$93$	−6.65187	−0.689767
$94$	0	0
$95$	−32.1796	−3.30156
$96$	0	0
$97$	12.6582	1.28524	0.642622	−	0.766183i	$-0.277847\pi$
$97$	12.6582	1.28524	0.642622	+	0.766183i	$0.277847\pi$
$98$	0	0
$99$	−14.6451	−1.47188
$100$	0	0
$101$	−18.7113	−1.86184	−0.930922	−	0.365219i	$-0.880994\pi$
$101$	−18.7113	−1.86184	−0.930922	+	0.365219i	$0.880994\pi$
$102$	0	0
$103$	−20.1606	−1.98649	−0.993244	−	0.116047i	$-0.962978\pi$
$103$	−20.1606	−1.98649	−0.993244	+	0.116047i	$0.962978\pi$
$104$	0	0
$105$	43.6198	4.25686
$106$	0	0
$107$	14.0021	1.35364	0.676819	−	0.736149i	$-0.263358\pi$
$107$	14.0021	1.35364	0.676819	+	0.736149i	$0.263358\pi$
$108$	0	0
$109$	6.97140	0.667739	0.333870	−	0.942619i	$-0.391645\pi$
$109$	6.97140	0.667739	0.333870	+	0.942619i	$0.391645\pi$
$110$	0	0
$111$	−24.4867	−2.32417
$112$	0	0
$113$	−3.64403	−0.342802	−0.171401	−	0.985201i	$-0.554829\pi$
$113$	−3.64403	−0.342802	−0.171401	+	0.985201i	$0.554829\pi$
$114$	0	0
$115$	4.76263	0.444118
$116$	0	0
$117$	−15.7385	−1.45503
$118$	0	0
$119$	1.03179	0.0945843
$120$	0	0
$121$	−6.27298	−0.570271
$122$	0	0
$123$	−2.73335	−0.246458
$124$	0	0
$125$	−29.9707	−2.68066
$126$	0	0
$127$	−13.8654	−1.23036	−0.615179	−	0.788387i	$-0.710916\pi$
$127$	−13.8654	−1.23036	−0.615179	+	0.788387i	$0.710916\pi$
$128$	0	0
$129$	−15.9864	−1.40753
$130$	0	0
$131$	−9.03145	−0.789081	−0.394541	−	0.918878i	$-0.629096\pi$
$131$	−9.03145	−0.789081	−0.394541	+	0.918878i	$0.629096\pi$
$132$	0	0
$133$	26.1212	2.26500
$134$	0	0
$135$	48.3758	4.16352
$136$	0	0
$137$	−17.6765	−1.51021	−0.755104	−	0.655605i	$-0.772414\pi$
$137$	−17.6765	−1.51021	−0.755104	+	0.655605i	$0.772414\pi$
$138$	0	0
$139$	−1.34994	−0.114500	−0.0572502	−	0.998360i	$-0.518233\pi$
$139$	−1.34994	−0.114500	−0.0572502	+	0.998360i	$0.518233\pi$
$140$	0	0
$141$	−11.0967	−0.934507
$142$	0	0
$143$	5.07996	0.424808
$144$	0	0
$145$	−8.46015	−0.702577
$146$	0	0
$147$	−13.5659	−1.11889
$148$	0	0
$149$	−5.79044	−0.474371	−0.237186	−	0.971464i	$-0.576225\pi$
$149$	−5.79044	−0.474371	−0.237186	+	0.971464i	$0.576225\pi$
$150$	0	0
$151$	8.05592	0.655581	0.327791	−	0.944750i	$-0.393696\pi$
$151$	8.05592	0.655581	0.327791	+	0.944750i	$0.393696\pi$
$152$	0	0
$153$	2.06317	0.166798
$154$	0	0
$155$	−8.84702	−0.710609
$156$	0	0
$157$	11.5369	0.920745	0.460372	−	0.887726i	$-0.347716\pi$
$157$	11.5369	0.920745	0.460372	+	0.887726i	$0.347716\pi$
$158$	0	0
$159$	−6.02764	−0.478023
$160$	0	0
$161$	−3.86598	−0.304682
$162$	0	0
$163$	8.31020	0.650905	0.325452	−	0.945558i	$-0.394483\pi$
$163$	8.31020	0.650905	0.325452	+	0.945558i	$0.394483\pi$
$164$	0	0
$165$	−28.1529	−2.19170
$166$	0	0
$167$	−8.36990	−0.647683	−0.323841	−	0.946111i	$-0.604974\pi$
$167$	−8.36990	−0.647683	−0.323841	+	0.946111i	$0.604974\pi$
$168$	0	0
$169$	−7.54075	−0.580058
$170$	0	0
$171$	52.2320	3.99428
$172$	0	0
$173$	−7.01371	−0.533242	−0.266621	−	0.963801i	$-0.585907\pi$
$173$	−7.01371	−0.533242	−0.266621	+	0.963801i	$0.585907\pi$
$174$	0	0
$175$	41.1713	3.11226
$176$	0	0
$177$	32.9805	2.47896
$178$	0	0
$179$	−9.95348	−0.743958	−0.371979	−	0.928241i	$-0.621321\pi$
$179$	−9.95348	−0.743958	−0.371979	+	0.928241i	$0.621321\pi$
$180$	0	0
$181$	0.289269	0.0215012	0.0107506	−	0.999942i	$-0.496578\pi$
$181$	0.289269	0.0215012	0.0107506	+	0.999942i	$0.496578\pi$
$182$	0	0
$183$	−11.7788	−0.870715
$184$	0	0
$185$	−32.5674	−2.39440
$186$	0	0
$187$	−0.665936	−0.0486981
$188$	0	0
$189$	−39.2681	−2.85634
$190$	0	0
$191$	16.3497	1.18302	0.591512	−	0.806296i	$-0.298531\pi$
$191$	16.3497	1.18302	0.591512	+	0.806296i	$0.298531\pi$
$192$	0	0
$193$	11.1270	0.800938	0.400469	−	0.916310i	$-0.368847\pi$
$193$	11.1270	0.800938	0.400469	+	0.916310i	$0.368847\pi$
$194$	0	0
$195$	−30.2550	−2.16660
$196$	0	0
$197$	9.10776	0.648901	0.324451	−	0.945903i	$-0.394821\pi$
$197$	9.10776	0.648901	0.324451	+	0.945903i	$0.394821\pi$
$198$	0	0
$199$	−19.1909	−1.36040	−0.680202	−	0.733024i	$-0.738108\pi$
$199$	−19.1909	−1.36040	−0.680202	+	0.733024i	$0.738108\pi$
$200$	0	0
$201$	14.9675	1.05572
$202$	0	0
$203$	6.86737	0.481995
$204$	0	0
$205$	−3.63537	−0.253905
$206$	0	0
$207$	−7.73042	−0.537301
$208$	0	0
$209$	−16.8591	−1.16617
$210$	0	0
$211$	18.5781	1.27897	0.639484	−	0.768804i	$-0.279148\pi$
$211$	18.5781	1.27897	0.639484	+	0.768804i	$0.279148\pi$
$212$	0	0
$213$	−29.6694	−2.03291
$214$	0	0
$215$	−21.2620	−1.45006
$216$	0	0
$217$	7.18140	0.487505
$218$	0	0
$219$	−0.125117	−0.00845465
$220$	0	0
$221$	−0.715657	−0.0481403
$222$	0	0
$223$	13.9735	0.935738	0.467869	−	0.883798i	$-0.345022\pi$
$223$	13.9735	0.935738	0.467869	+	0.883798i	$0.345022\pi$
$224$	0	0
$225$	82.3262	5.48842
$226$	0	0
$227$	−3.68637	−0.244673	−0.122336	−	0.992489i	$-0.539039\pi$
$227$	−3.68637	−0.244673	−0.122336	+	0.992489i	$0.539039\pi$
$228$	0	0
$229$	1.63242	0.107873	0.0539365	−	0.998544i	$-0.482823\pi$
$229$	1.63242	0.107873	0.0539365	+	0.998544i	$0.482823\pi$
$230$	0	0
$231$	22.8526	1.50359
$232$	0	0
$233$	9.13109	0.598198	0.299099	−	0.954222i	$-0.403314\pi$
$233$	9.13109	0.598198	0.299099	+	0.954222i	$0.403314\pi$
$234$	0	0
$235$	−14.7586	−0.962744
$236$	0	0
$237$	28.6759	1.86270
$238$	0	0
$239$	12.6751	0.819881	0.409941	−	0.912112i	$-0.365549\pi$
$239$	12.6751	0.819881	0.409941	+	0.912112i	$0.365549\pi$
$240$	0	0
$241$	8.06927	0.519788	0.259894	−	0.965637i	$-0.416312\pi$
$241$	8.06927	0.519788	0.259894	+	0.965637i	$0.416312\pi$
$242$	0	0
$243$	−15.4674	−0.992235
$244$	0	0
$245$	−18.0427	−1.15270
$246$	0	0
$247$	−18.1178	−1.15281
$248$	0	0
$249$	−37.5062	−2.37686
$250$	0	0
$251$	−0.982198	−0.0619958	−0.0309979	−	0.999519i	$-0.509869\pi$
$251$	−0.982198	−0.0619958	−0.0309979	+	0.999519i	$0.509869\pi$
$252$	0	0
$253$	2.49517	0.156870
$254$	0	0
$255$	3.96615	0.248370
$256$	0	0
$257$	3.94561	0.246121	0.123060	−	0.992399i	$-0.460729\pi$
$257$	3.94561	0.246121	0.123060	+	0.992399i	$0.460729\pi$
$258$	0	0
$259$	26.4360	1.64265
$260$	0	0
$261$	13.7320	0.849990
$262$	0	0
$263$	7.86634	0.485059	0.242530	−	0.970144i	$-0.422023\pi$
$263$	7.86634	0.485059	0.242530	+	0.970144i	$0.422023\pi$
$264$	0	0
$265$	−8.01678	−0.492467
$266$	0	0
$267$	−26.2433	−1.60606
$268$	0	0
$269$	−19.7669	−1.20521	−0.602604	−	0.798040i	$-0.705870\pi$
$269$	−19.7669	−1.20521	−0.602604	+	0.798040i	$0.705870\pi$
$270$	0	0
$271$	−1.07635	−0.0653839	−0.0326919	−	0.999465i	$-0.510408\pi$
$271$	−1.07635	−0.0653839	−0.0326919	+	0.999465i	$0.510408\pi$
$272$	0	0
$273$	24.5589	1.48637
$274$	0	0
$275$	−26.5727	−1.60239
$276$	0	0
$277$	−29.0455	−1.74518	−0.872588	−	0.488457i	$-0.837560\pi$
$277$	−29.0455	−1.74518	−0.872588	+	0.488457i	$0.837560\pi$
$278$	0	0
$279$	14.3599	0.859707
$280$	0	0
$281$	−21.5140	−1.28342	−0.641708	−	0.766949i	$-0.721774\pi$
$281$	−21.5140	−1.28342	−0.641708	+	0.766949i	$0.721774\pi$
$282$	0	0
$283$	32.3270	1.92164	0.960821	−	0.277169i	$-0.0893962\pi$
$283$	32.3270	1.92164	0.960821	+	0.277169i	$0.0893962\pi$
$284$	0	0
$285$	100.408	5.94767
$286$	0	0
$287$	2.95094	0.174189
$288$	0	0
$289$	−16.9062	−0.994481
$290$	0	0
$291$	−39.4966	−2.31533
$292$	0	0
$293$	6.07444	0.354873	0.177436	−	0.984132i	$-0.443220\pi$
$293$	6.07444	0.354873	0.177436	+	0.984132i	$0.443220\pi$
$294$	0	0
$295$	43.8641	2.55387
$296$	0	0
$297$	25.3443	1.47063
$298$	0	0
$299$	2.68147	0.155073
$300$	0	0
$301$	17.2591	0.994796
$302$	0	0
$303$	58.3838	3.35406
$304$	0	0
$305$	−15.6659	−0.897025
$306$	0	0
$307$	−16.2055	−0.924897	−0.462448	−	0.886646i	$-0.653029\pi$
$307$	−16.2055	−0.924897	−0.462448	+	0.886646i	$0.653029\pi$
$308$	0	0
$309$	62.9061	3.57861
$310$	0	0
$311$	−1.63460	−0.0926896	−0.0463448	−	0.998926i	$-0.514757\pi$
$311$	−1.63460	−0.0926896	−0.0463448	+	0.998926i	$0.514757\pi$
$312$	0	0
$313$	3.74362	0.211602	0.105801	−	0.994387i	$-0.466259\pi$
$313$	3.74362	0.211602	0.105801	+	0.994387i	$0.466259\pi$
$314$	0	0
$315$	−94.1656	−5.30563
$316$	0	0
$317$	7.35534	0.413117	0.206559	−	0.978434i	$-0.433774\pi$
$317$	7.35534	0.413117	0.206559	+	0.978434i	$0.433774\pi$
$318$	0	0
$319$	−4.43232	−0.248162
$320$	0	0
$321$	−43.6901	−2.43854
$322$	0	0
$323$	2.37508	0.132153
$324$	0	0
$325$	−28.5567	−1.58404
$326$	0	0
$327$	−21.7525	−1.20291
$328$	0	0
$329$	11.9800	0.660479
$330$	0	0
$331$	10.7493	0.590837	0.295418	−	0.955368i	$-0.404541\pi$
$331$	10.7493	0.590837	0.295418	+	0.955368i	$0.404541\pi$
$332$	0	0
$333$	52.8614	2.89679
$334$	0	0
$335$	19.9068	1.08762
$336$	0	0
$337$	29.6080	1.61285	0.806424	−	0.591337i	$-0.201400\pi$
$337$	29.6080	1.61285	0.806424	+	0.591337i	$0.201400\pi$
$338$	0	0
$339$	11.3703	0.617548
$340$	0	0
$341$	−4.63500	−0.250999
$342$	0	0
$343$	−8.93464	−0.482425
$344$	0	0
$345$	−14.8606	−0.800067
$346$	0	0
$347$	9.49018	0.509460	0.254730	−	0.967012i	$-0.418013\pi$
$347$	9.49018	0.509460	0.254730	+	0.967012i	$0.418013\pi$
$348$	0	0
$349$	9.32926	0.499384	0.249692	−	0.968325i	$-0.419671\pi$
$349$	9.32926	0.499384	0.249692	+	0.968325i	$0.419671\pi$
$350$	0	0
$351$	27.2366	1.45378
$352$	0	0
$353$	15.2284	0.810527	0.405264	−	0.914200i	$-0.367180\pi$
$353$	15.2284	0.810527	0.405264	+	0.914200i	$0.367180\pi$
$354$	0	0
$355$	−39.4604	−2.09434
$356$	0	0
$357$	−3.21945	−0.170391
$358$	0	0
$359$	6.87738	0.362974	0.181487	−	0.983393i	$-0.441909\pi$
$359$	6.87738	0.362974	0.181487	+	0.983393i	$0.441909\pi$
$360$	0	0
$361$	41.1283	2.16465
$362$	0	0
$363$	19.5732	1.02733
$364$	0	0
$365$	−0.166407	−0.00871012
$366$	0	0
$367$	−15.0529	−0.785755	−0.392877	−	0.919591i	$-0.628520\pi$
$367$	−15.0529	−0.785755	−0.392877	+	0.919591i	$0.628520\pi$
$368$	0	0
$369$	5.90071	0.307179
$370$	0	0
$371$	6.50747	0.337851
$372$	0	0
$373$	30.9037	1.60013	0.800067	−	0.599911i	$-0.204797\pi$
$373$	30.9037	1.60013	0.800067	+	0.599911i	$0.204797\pi$
$374$	0	0
$375$	93.5159	4.82914
$376$	0	0
$377$	−4.76325	−0.245320
$378$	0	0
$379$	18.1402	0.931800	0.465900	−	0.884837i	$-0.345731\pi$
$379$	18.1402	0.931800	0.465900	+	0.884837i	$0.345731\pi$
$380$	0	0
$381$	43.2635	2.21646
$382$	0	0
$383$	7.73708	0.395346	0.197673	−	0.980268i	$-0.436662\pi$
$383$	7.73708	0.395346	0.197673	+	0.980268i	$0.436662\pi$
$384$	0	0
$385$	30.3941	1.54903
$386$	0	0
$387$	34.5112	1.75430
$388$	0	0
$389$	20.1338	1.02082	0.510411	−	0.859930i	$-0.329493\pi$
$389$	20.1338	1.02082	0.510411	+	0.859930i	$0.329493\pi$
$390$	0	0
$391$	−0.351516	−0.0177769
$392$	0	0
$393$	28.1803	1.42151
$394$	0	0
$395$	38.1391	1.91899
$396$	0	0
$397$	−5.90728	−0.296478	−0.148239	−	0.988952i	$-0.547360\pi$
$397$	−5.90728	−0.296478	−0.148239	+	0.988952i	$0.547360\pi$
$398$	0	0
$399$	−81.5046	−4.08033
$400$	0	0
$401$	26.2279	1.30976	0.654880	−	0.755733i	$-0.272719\pi$
$401$	26.2279	1.30976	0.654880	+	0.755733i	$0.272719\pi$
$402$	0	0
$403$	−4.98106	−0.248124
$404$	0	0
$405$	−67.0832	−3.33339
$406$	0	0
$407$	−17.0622	−0.845742
$408$	0	0
$409$	−22.0811	−1.09184	−0.545920	−	0.837837i	$-0.683820\pi$
$409$	−22.0811	−1.09184	−0.545920	+	0.837837i	$0.683820\pi$
$410$	0	0
$411$	55.1551	2.72060
$412$	0	0
$413$	−35.6059	−1.75205
$414$	0	0
$415$	−49.8834	−2.44868
$416$	0	0
$417$	4.21214	0.206269
$418$	0	0
$419$	−26.6871	−1.30375	−0.651875	−	0.758327i	$-0.726017\pi$
$419$	−26.6871	−1.30375	−0.651875	+	0.758327i	$0.726017\pi$
$420$	0	0
$421$	−15.4487	−0.752922	−0.376461	−	0.926433i	$-0.622859\pi$
$421$	−15.4487	−0.752922	−0.376461	+	0.926433i	$0.622859\pi$
$422$	0	0
$423$	23.9552	1.16474
$424$	0	0
$425$	3.74352	0.181587
$426$	0	0
$427$	12.7165	0.615394
$428$	0	0
$429$	−15.8507	−0.765280
$430$	0	0
$431$	−4.21132	−0.202852	−0.101426	−	0.994843i	$-0.532341\pi$
$431$	−4.21132	−0.202852	−0.101426	+	0.994843i	$0.532341\pi$
$432$	0	0
$433$	−34.6530	−1.66532	−0.832659	−	0.553786i	$-0.813183\pi$
$433$	−34.6530	−1.66532	−0.832659	+	0.553786i	$0.813183\pi$
$434$	0	0
$435$	26.3977	1.26567
$436$	0	0
$437$	−8.89909	−0.425701
$438$	0	0
$439$	−30.1295	−1.43800	−0.719001	−	0.695009i	$-0.755400\pi$
$439$	−30.1295	−1.43800	−0.719001	+	0.695009i	$0.755400\pi$
$440$	0	0
$441$	29.2858	1.39456
$442$	0	0
$443$	18.7775	0.892147	0.446073	−	0.894996i	$-0.352822\pi$
$443$	18.7775	0.892147	0.446073	+	0.894996i	$0.352822\pi$
$444$	0	0
$445$	−34.9037	−1.65459
$446$	0	0
$447$	18.0676	0.854568
$448$	0	0
$449$	−6.47494	−0.305571	−0.152786	−	0.988259i	$-0.548824\pi$
$449$	−6.47494	−0.305571	−0.152786	+	0.988259i	$0.548824\pi$
$450$	0	0
$451$	−1.90459	−0.0896835
$452$	0	0
$453$	−25.1364	−1.18101
$454$	0	0
$455$	32.6634	1.53129
$456$	0	0
$457$	38.6925	1.80996	0.904979	−	0.425455i	$-0.139886\pi$
$457$	38.6925	1.80996	0.904979	+	0.425455i	$0.139886\pi$
$458$	0	0
$459$	−3.57047	−0.166655
$460$	0	0
$461$	11.4734	0.534368	0.267184	−	0.963645i	$-0.413907\pi$
$461$	11.4734	0.534368	0.267184	+	0.963645i	$0.413907\pi$
$462$	0	0
$463$	16.5097	0.767272	0.383636	−	0.923484i	$-0.374672\pi$
$463$	16.5097	0.767272	0.383636	+	0.923484i	$0.374672\pi$
$464$	0	0
$465$	27.6048	1.28014
$466$	0	0
$467$	−16.8267	−0.778649	−0.389325	−	0.921101i	$-0.627292\pi$
$467$	−16.8267	−0.778649	−0.389325	+	0.921101i	$0.627292\pi$
$468$	0	0
$469$	−16.1590	−0.746152
$470$	0	0
$471$	−35.9979	−1.65870
$472$	0	0
$473$	−11.1393	−0.512185
$474$	0	0
$475$	94.7722	4.34845
$476$	0	0
$477$	13.0123	0.595795
$478$	0	0
$479$	31.4121	1.43525	0.717627	−	0.696428i	$-0.245228\pi$
$479$	31.4121	1.43525	0.717627	+	0.696428i	$0.245228\pi$
$480$	0	0
$481$	−18.3361	−0.836056
$482$	0	0
$483$	12.0628	0.548877
$484$	0	0
$485$	−52.5307	−2.38529
$486$	0	0
$487$	37.8951	1.71719	0.858594	−	0.512655i	$-0.171338\pi$
$487$	37.8951	1.71719	0.858594	+	0.512655i	$0.171338\pi$
$488$	0	0
$489$	−25.9298	−1.17259
$490$	0	0
$491$	8.40738	0.379420	0.189710	−	0.981840i	$-0.439245\pi$
$491$	8.40738	0.379420	0.189710	+	0.981840i	$0.439245\pi$
$492$	0	0
$493$	0.624418	0.0281224
$494$	0	0
$495$	60.7760	2.73168
$496$	0	0
$497$	32.0313	1.43680
$498$	0	0
$499$	−41.7088	−1.86714	−0.933571	−	0.358392i	$-0.883325\pi$
$499$	−41.7088	−1.86714	−0.933571	+	0.358392i	$0.883325\pi$
$500$	0	0
$501$	26.1161	1.16678
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	77.6507	3.45541
$506$	0	0
$507$	23.5290	1.04496
$508$	0	0
$509$	5.75657	0.255156	0.127578	−	0.991829i	$-0.459280\pi$
$509$	5.75657	0.255156	0.127578	+	0.991829i	$0.459280\pi$
$510$	0	0
$511$	0.135078	0.00597548
$512$	0	0
$513$	−90.3912	−3.99087
$514$	0	0
$515$	83.6654	3.68674
$516$	0	0
$517$	−7.73210	−0.340057
$518$	0	0
$519$	21.8845	0.960622
$520$	0	0
$521$	13.0000	0.569541	0.284770	−	0.958596i	$-0.408083\pi$
$521$	13.0000	0.569541	0.284770	+	0.958596i	$0.408083\pi$
$522$	0	0
$523$	22.2404	0.972504	0.486252	−	0.873819i	$-0.338364\pi$
$523$	22.2404	0.972504	0.486252	+	0.873819i	$0.338364\pi$
$524$	0	0
$525$	−128.465	−5.60666
$526$	0	0
$527$	0.652971	0.0284439
$528$	0	0
$529$	−21.6829	−0.942736
$530$	0	0
$531$	−71.1976	−3.08971
$532$	0	0
$533$	−2.04679	−0.0886564
$534$	0	0
$535$	−58.1080	−2.51223
$536$	0	0
$537$	31.0573	1.34022
$538$	0	0
$539$	−9.45264	−0.407154
$540$	0	0
$541$	−8.53197	−0.366818	−0.183409	−	0.983037i	$-0.558713\pi$
$541$	−8.53197	−0.366818	−0.183409	+	0.983037i	$0.558713\pi$
$542$	0	0
$543$	−0.902591	−0.0387339
$544$	0	0
$545$	−28.9309	−1.23926
$546$	0	0
$547$	−21.7708	−0.930851	−0.465426	−	0.885087i	$-0.654099\pi$
$547$	−21.7708	−0.930851	−0.465426	+	0.885087i	$0.654099\pi$
$548$	0	0
$549$	25.4279	1.08524
$550$	0	0
$551$	15.8080	0.673443
$552$	0	0
$553$	−30.9587	−1.31650
$554$	0	0
$555$	101.618	4.31345
$556$	0	0
$557$	23.6155	1.00062	0.500311	−	0.865846i	$-0.333219\pi$
$557$	23.6155	1.00062	0.500311	+	0.865846i	$0.333219\pi$
$558$	0	0
$559$	−11.9710	−0.506319
$560$	0	0
$561$	2.07788	0.0877283
$562$	0	0
$563$	−18.6326	−0.785270	−0.392635	−	0.919694i	$-0.628436\pi$
$563$	−18.6326	−0.785270	−0.392635	+	0.919694i	$0.628436\pi$
$564$	0	0
$565$	15.1225	0.636209
$566$	0	0
$567$	54.4536	2.28684
$568$	0	0
$569$	−27.7565	−1.16361	−0.581807	−	0.813327i	$-0.697654\pi$
$569$	−27.7565	−1.16361	−0.581807	+	0.813327i	$0.697654\pi$
$570$	0	0
$571$	46.8643	1.96121	0.980605	−	0.195995i	$-0.0627936\pi$
$571$	46.8643	1.96121	0.980605	+	0.195995i	$0.0627936\pi$
$572$	0	0
$573$	−51.0151	−2.13119
$574$	0	0
$575$	−14.0264	−0.584942
$576$	0	0
$577$	26.5352	1.10468	0.552338	−	0.833620i	$-0.313736\pi$
$577$	26.5352	1.10468	0.552338	+	0.833620i	$0.313736\pi$
$578$	0	0
$579$	−34.7189	−1.44287
$580$	0	0
$581$	40.4919	1.67989
$582$	0	0
$583$	−4.20003	−0.173948
$584$	0	0
$585$	65.3138	2.70039
$586$	0	0
$587$	−4.31579	−0.178132	−0.0890659	−	0.996026i	$-0.528388\pi$
$587$	−4.31579	−0.178132	−0.0890659	+	0.996026i	$0.528388\pi$
$588$	0	0
$589$	16.5308	0.681142
$590$	0	0
$591$	−28.4184	−1.16898
$592$	0	0
$593$	14.5019	0.595520	0.297760	−	0.954641i	$-0.403761\pi$
$593$	14.5019	0.595520	0.297760	+	0.954641i	$0.403761\pi$
$594$	0	0
$595$	−4.28187	−0.175540
$596$	0	0
$597$	59.8802	2.45073
$598$	0	0
$599$	10.1663	0.415385	0.207692	−	0.978194i	$-0.433405\pi$
$599$	10.1663	0.415385	0.207692	+	0.978194i	$0.433405\pi$
$600$	0	0
$601$	30.7418	1.25398	0.626992	−	0.779025i	$-0.284286\pi$
$601$	30.7418	1.25398	0.626992	+	0.779025i	$0.284286\pi$
$602$	0	0
$603$	−32.3115	−1.31583
$604$	0	0
$605$	26.0325	1.05837
$606$	0	0
$607$	−19.4514	−0.789508	−0.394754	−	0.918787i	$-0.629170\pi$
$607$	−19.4514	−0.789508	−0.394754	+	0.918787i	$0.629170\pi$
$608$	0	0
$609$	−21.4279	−0.868301
$610$	0	0
$611$	−8.30940	−0.336163
$612$	0	0
$613$	−16.3747	−0.661370	−0.330685	−	0.943741i	$-0.607280\pi$
$613$	−16.3747	−0.661370	−0.330685	+	0.943741i	$0.607280\pi$
$614$	0	0
$615$	11.3432	0.457404
$616$	0	0
$617$	15.8833	0.639439	0.319720	−	0.947512i	$-0.396411\pi$
$617$	15.8833	0.639439	0.319720	+	0.947512i	$0.396411\pi$
$618$	0	0
$619$	−28.1027	−1.12954	−0.564771	−	0.825248i	$-0.691036\pi$
$619$	−28.1027	−1.12954	−0.564771	+	0.825248i	$0.691036\pi$
$620$	0	0
$621$	13.3780	0.536842
$622$	0	0
$623$	28.3324	1.13512
$624$	0	0
$625$	63.2667	2.53067
$626$	0	0
$627$	52.6044	2.10082
$628$	0	0
$629$	2.40370	0.0958418
$630$	0	0
$631$	18.4235	0.733427	0.366714	−	0.930334i	$-0.380483\pi$
$631$	18.4235	0.733427	0.366714	+	0.930334i	$0.380483\pi$
$632$	0	0
$633$	−57.9682	−2.30403
$634$	0	0
$635$	57.5407	2.28343
$636$	0	0
$637$	−10.1584	−0.402491
$638$	0	0
$639$	64.0497	2.53377
$640$	0	0
$641$	9.76989	0.385887	0.192944	−	0.981210i	$-0.438197\pi$
$641$	9.76989	0.385887	0.192944	+	0.981210i	$0.438197\pi$
$642$	0	0
$643$	6.63862	0.261802	0.130901	−	0.991395i	$-0.458213\pi$
$643$	6.63862	0.261802	0.130901	+	0.991395i	$0.458213\pi$
$644$	0	0
$645$	66.3427	2.61224
$646$	0	0
$647$	7.53902	0.296389	0.148195	−	0.988958i	$-0.452654\pi$
$647$	7.53902	0.296389	0.148195	+	0.988958i	$0.452654\pi$
$648$	0	0
$649$	22.9806	0.902069
$650$	0	0
$651$	−22.4077	−0.878228
$652$	0	0
$653$	−0.257001	−0.0100572	−0.00502861	−	0.999987i	$-0.501601\pi$
$653$	−0.257001	−0.0100572	−0.00502861	+	0.999987i	$0.501601\pi$
$654$	0	0
$655$	37.4799	1.46446
$656$	0	0
$657$	0.270101	0.0105376
$658$	0	0
$659$	22.7843	0.887552	0.443776	−	0.896138i	$-0.353639\pi$
$659$	22.7843	0.887552	0.443776	+	0.896138i	$0.353639\pi$
$660$	0	0
$661$	−16.1234	−0.627128	−0.313564	−	0.949567i	$-0.601523\pi$
$661$	−16.1234	−0.627128	−0.313564	+	0.949567i	$0.601523\pi$
$662$	0	0
$663$	2.23303	0.0867235
$664$	0	0
$665$	−108.401	−4.20363
$666$	0	0
$667$	−2.33961	−0.0905899
$668$	0	0
$669$	−43.6009	−1.68571
$670$	0	0
$671$	−8.20743	−0.316844
$672$	0	0
$673$	37.3214	1.43863	0.719316	−	0.694683i	$-0.244455\pi$
$673$	37.3214	1.43863	0.719316	+	0.694683i	$0.244455\pi$
$674$	0	0
$675$	−142.471	−5.48373
$676$	0	0
$677$	15.2255	0.585162	0.292581	−	0.956241i	$-0.405486\pi$
$677$	15.2255	0.585162	0.292581	+	0.956241i	$0.405486\pi$
$678$	0	0
$679$	42.6408	1.63640
$680$	0	0
$681$	11.5024	0.440771
$682$	0	0
$683$	−26.3670	−1.00891	−0.504453	−	0.863439i	$-0.668306\pi$
$683$	−26.3670	−1.00891	−0.504453	+	0.863439i	$0.668306\pi$
$684$	0	0
$685$	73.3565	2.80281
$686$	0	0
$687$	−5.09354	−0.194330
$688$	0	0
$689$	−4.51362	−0.171955
$690$	0	0
$691$	33.0164	1.25600	0.628002	−	0.778212i	$-0.283873\pi$
$691$	33.0164	1.25600	0.628002	+	0.778212i	$0.283873\pi$
$692$	0	0
$693$	−49.3338	−1.87404
$694$	0	0
$695$	5.60216	0.212502
$696$	0	0
$697$	0.268316	0.0101632
$698$	0	0
$699$	−28.4912	−1.07764
$700$	0	0
$701$	37.8830	1.43082	0.715410	−	0.698705i	$-0.246240\pi$
$701$	37.8830	1.43082	0.715410	+	0.698705i	$0.246240\pi$
$702$	0	0
$703$	60.8529	2.29511
$704$	0	0
$705$	46.0504	1.73436
$706$	0	0
$707$	−63.0315	−2.37054
$708$	0	0
$709$	33.3141	1.25114	0.625568	−	0.780169i	$-0.284867\pi$
$709$	33.3141	1.25114	0.625568	+	0.780169i	$0.284867\pi$
$710$	0	0
$711$	−61.9050	−2.32162
$712$	0	0
$713$	−2.44659	−0.0916255
$714$	0	0
$715$	−21.0815	−0.788404
$716$	0	0
$717$	−39.5493	−1.47699
$718$	0	0
$719$	−43.2501	−1.61296	−0.806479	−	0.591263i	$-0.798630\pi$
$719$	−43.2501	−1.61296	−0.806479	+	0.591263i	$0.798630\pi$
$720$	0	0
$721$	−67.9138	−2.52924
$722$	0	0
$723$	−25.1781	−0.936384
$724$	0	0
$725$	24.9160	0.925356
$726$	0	0
$727$	−14.0124	−0.519692	−0.259846	−	0.965650i	$-0.583672\pi$
$727$	−14.0124	−0.519692	−0.259846	+	0.965650i	$0.583672\pi$
$728$	0	0
$729$	−0.232537	−0.00861247
$730$	0	0
$731$	1.56929	0.0580421
$732$	0	0
$733$	−10.8264	−0.399883	−0.199942	−	0.979808i	$-0.564075\pi$
$733$	−10.8264	−0.399883	−0.199942	+	0.979808i	$0.564075\pi$
$734$	0	0
$735$	56.2975	2.07657
$736$	0	0
$737$	10.4293	0.384167
$738$	0	0
$739$	−21.5206	−0.791647	−0.395823	−	0.918327i	$-0.629541\pi$
$739$	−21.5206	−0.791647	−0.395823	+	0.918327i	$0.629541\pi$
$740$	0	0
$741$	56.5320	2.07676
$742$	0	0
$743$	26.9138	0.987373	0.493687	−	0.869640i	$-0.335649\pi$
$743$	26.9138	0.987373	0.493687	+	0.869640i	$0.335649\pi$
$744$	0	0
$745$	24.0300	0.880390
$746$	0	0
$747$	80.9677	2.96245
$748$	0	0
$749$	47.1681	1.72348
$750$	0	0
$751$	−30.0424	−1.09626	−0.548132	−	0.836392i	$-0.684661\pi$
$751$	−30.0424	−1.09626	−0.548132	+	0.836392i	$0.684661\pi$
$752$	0	0
$753$	3.06470	0.111684
$754$	0	0
$755$	−33.4315	−1.21670
$756$	0	0
$757$	−10.9089	−0.396489	−0.198245	−	0.980153i	$-0.563524\pi$
$757$	−10.9089	−0.396489	−0.198245	+	0.980153i	$0.563524\pi$
$758$	0	0
$759$	−7.78553	−0.282597
$760$	0	0
$761$	37.2722	1.35112	0.675558	−	0.737306i	$-0.263903\pi$
$761$	37.2722	1.35112	0.675558	+	0.737306i	$0.263903\pi$
$762$	0	0
$763$	23.4841	0.850182
$764$	0	0
$765$	−8.56204	−0.309561
$766$	0	0
$767$	24.6965	0.891737
$768$	0	0
$769$	−36.6992	−1.32341	−0.661703	−	0.749766i	$-0.730166\pi$
$769$	−36.6992	−1.32341	−0.661703	+	0.749766i	$0.730166\pi$
$770$	0	0
$771$	−12.3113	−0.443380
$772$	0	0
$773$	−26.5771	−0.955912	−0.477956	−	0.878384i	$-0.658622\pi$
$773$	−26.5771	−0.955912	−0.477956	+	0.878384i	$0.658622\pi$
$774$	0	0
$775$	26.0553	0.935935
$776$	0	0
$777$	−82.4867	−2.95919
$778$	0	0
$779$	6.79277	0.243376
$780$	0	0
$781$	−20.6735	−0.739756
$782$	0	0
$783$	−23.7642	−0.849263
$784$	0	0
$785$	−47.8774	−1.70882
$786$	0	0
$787$	33.7734	1.20389	0.601946	−	0.798537i	$-0.294392\pi$
$787$	33.7734	1.20389	0.601946	+	0.798537i	$0.294392\pi$
$788$	0	0
$789$	−24.5449	−0.873821
$790$	0	0
$791$	−12.2754	−0.436463
$792$	0	0
$793$	−8.82022	−0.313215
$794$	0	0
$795$	25.0143	0.887166
$796$	0	0
$797$	−40.5803	−1.43743	−0.718713	−	0.695306i	$-0.755269\pi$
$797$	−40.5803	−1.43743	−0.718713	+	0.695306i	$0.755269\pi$
$798$	0	0
$799$	1.08929	0.0385362
$800$	0	0
$801$	56.6536	2.00176
$802$	0	0
$803$	−0.0871813	−0.00307656
$804$	0	0
$805$	16.0436	0.565462
$806$	0	0
$807$	61.6775	2.17115
$808$	0	0
$809$	−27.2353	−0.957541	−0.478770	−	0.877940i	$-0.658917\pi$
$809$	−27.2353	−0.957541	−0.478770	+	0.877940i	$0.658917\pi$
$810$	0	0
$811$	17.3936	0.610771	0.305386	−	0.952229i	$-0.401215\pi$
$811$	17.3936	0.610771	0.305386	+	0.952229i	$0.401215\pi$
$812$	0	0
$813$	3.35849	0.117787
$814$	0	0
$815$	−34.4868	−1.20802
$816$	0	0
$817$	39.7286	1.38993
$818$	0	0
$819$	−53.0173	−1.85257
$820$	0	0
$821$	26.6441	0.929886	0.464943	−	0.885341i	$-0.346075\pi$
$821$	26.6441	0.929886	0.464943	+	0.885341i	$0.346075\pi$
$822$	0	0
$823$	−42.8316	−1.49302	−0.746508	−	0.665377i	$-0.768271\pi$
$823$	−42.8316	−1.49302	−0.746508	+	0.665377i	$0.768271\pi$
$824$	0	0
$825$	82.9132	2.88667
$826$	0	0
$827$	−19.0330	−0.661842	−0.330921	−	0.943659i	$-0.607359\pi$
$827$	−19.0330	−0.661842	−0.330921	+	0.943659i	$0.607359\pi$
$828$	0	0
$829$	22.4300	0.779027	0.389513	−	0.921021i	$-0.372643\pi$
$829$	22.4300	0.779027	0.389513	+	0.921021i	$0.372643\pi$
$830$	0	0
$831$	90.6291	3.14389
$832$	0	0
$833$	1.33167	0.0461398
$834$	0	0
$835$	34.7346	1.20204
$836$	0	0
$837$	−24.8509	−0.858972
$838$	0	0
$839$	−20.3437	−0.702343	−0.351172	−	0.936311i	$-0.614217\pi$
$839$	−20.3437	−0.702343	−0.351172	+	0.936311i	$0.614217\pi$
$840$	0	0
$841$	−24.8440	−0.856690
$842$	0	0
$843$	67.1289	2.31204
$844$	0	0
$845$	31.2936	1.07653
$846$	0	0
$847$	−21.1314	−0.726082
$848$	0	0
$849$	−100.868	−3.46179
$850$	0	0
$851$	−9.00632	−0.308733
$852$	0	0
$853$	27.1582	0.929879	0.464940	−	0.885342i	$-0.346076\pi$
$853$	27.1582	0.929879	0.464940	+	0.885342i	$0.346076\pi$
$854$	0	0
$855$	−216.759	−7.41302
$856$	0	0
$857$	−48.3982	−1.65325	−0.826625	−	0.562752i	$-0.809742\pi$
$857$	−48.3982	−1.65325	−0.826625	+	0.562752i	$0.809742\pi$
$858$	0	0
$859$	−13.4357	−0.458420	−0.229210	−	0.973377i	$-0.573614\pi$
$859$	−13.4357	−0.458420	−0.229210	+	0.973377i	$0.573614\pi$
$860$	0	0
$861$	−9.20767	−0.313796
$862$	0	0
$863$	−41.1809	−1.40182	−0.700908	−	0.713252i	$-0.747222\pi$
$863$	−41.1809	−1.40182	−0.700908	+	0.713252i	$0.747222\pi$
$864$	0	0
$865$	29.1064	0.989649
$866$	0	0
$867$	52.7514	1.79153
$868$	0	0
$869$	19.9813	0.677818
$870$	0	0
$871$	11.2080	0.379767
$872$	0	0
$873$	85.2646	2.88577
$874$	0	0
$875$	−100.960	−3.41308
$876$	0	0
$877$	43.9373	1.48366	0.741829	−	0.670589i	$-0.233959\pi$
$877$	43.9373	1.48366	0.741829	+	0.670589i	$0.233959\pi$
$878$	0	0
$879$	−18.9538	−0.639294
$880$	0	0
$881$	0.893078	0.0300886	0.0150443	−	0.999887i	$-0.495211\pi$
$881$	0.893078	0.0300886	0.0150443	+	0.999887i	$0.495211\pi$
$882$	0	0
$883$	4.04475	0.136117	0.0680583	−	0.997681i	$-0.478320\pi$
$883$	4.04475	0.136117	0.0680583	+	0.997681i	$0.478320\pi$
$884$	0	0
$885$	−136.867	−4.60073
$886$	0	0
$887$	−13.0998	−0.439849	−0.219924	−	0.975517i	$-0.570581\pi$
$887$	−13.0998	−0.439849	−0.219924	+	0.975517i	$0.570581\pi$
$888$	0	0
$889$	−46.7076	−1.56652
$890$	0	0
$891$	−35.1452	−1.17741
$892$	0	0
$893$	27.5767	0.922821
$894$	0	0
$895$	41.3063	1.38072
$896$	0	0
$897$	−8.36683	−0.279360
$898$	0	0
$899$	4.34602	0.144948
$900$	0	0
$901$	0.591694	0.0197122
$902$	0	0
$903$	−53.8525	−1.79210
$904$	0	0
$905$	−1.20045	−0.0399043
$906$	0	0
$907$	−43.6653	−1.44988	−0.724941	−	0.688811i	$-0.758133\pi$
$907$	−43.6653	−1.44988	−0.724941	+	0.688811i	$0.758133\pi$
$908$	0	0
$909$	−126.038	−4.18041
$910$	0	0
$911$	43.5333	1.44232	0.721161	−	0.692768i	$-0.243609\pi$
$911$	43.5333	1.44232	0.721161	+	0.692768i	$0.243609\pi$
$912$	0	0
$913$	−26.1342	−0.864914
$914$	0	0
$915$	48.8813	1.61597
$916$	0	0
$917$	−30.4237	−1.00468
$918$	0	0
$919$	−36.9619	−1.21926	−0.609630	−	0.792686i	$-0.708682\pi$
$919$	−36.9619	−1.21926	−0.609630	+	0.792686i	$0.708682\pi$
$920$	0	0
$921$	50.5651	1.66618
$922$	0	0
$923$	−22.2171	−0.731284
$924$	0	0
$925$	95.9141	3.15364
$926$	0	0
$927$	−135.801	−4.46028
$928$	0	0
$929$	−16.5783	−0.543918	−0.271959	−	0.962309i	$-0.587671\pi$
$929$	−16.5783	−0.543918	−0.271959	+	0.962309i	$0.587671\pi$
$930$	0	0
$931$	33.7131	1.10490
$932$	0	0
$933$	5.10035	0.166978
$934$	0	0
$935$	2.76359	0.0903791
$936$	0	0
$937$	21.5450	0.703843	0.351922	−	0.936030i	$-0.385528\pi$
$937$	21.5450	0.703843	0.351922	+	0.936030i	$0.385528\pi$
$938$	0	0
$939$	−11.6810	−0.381196
$940$	0	0
$941$	−28.9555	−0.943923	−0.471962	−	0.881619i	$-0.656454\pi$
$941$	−28.9555	−0.943923	−0.471962	+	0.881619i	$0.656454\pi$
$942$	0	0
$943$	−1.00534	−0.0327384
$944$	0	0
$945$	162.960	5.30110
$946$	0	0
$947$	34.9450	1.13556	0.567779	−	0.823181i	$-0.307803\pi$
$947$	34.9450	1.13556	0.567779	+	0.823181i	$0.307803\pi$
$948$	0	0
$949$	−0.0936905	−0.00304132
$950$	0	0
$951$	−22.9505	−0.744220
$952$	0	0
$953$	−18.9514	−0.613895	−0.306947	−	0.951727i	$-0.599308\pi$
$953$	−18.9514	−0.613895	−0.306947	+	0.951727i	$0.599308\pi$
$954$	0	0
$955$	−67.8503	−2.19558
$956$	0	0
$957$	13.8299	0.447057
$958$	0	0
$959$	−59.5458	−1.92283
$960$	0	0
$961$	−26.4552	−0.853395
$962$	0	0
$963$	94.3173	3.03933
$964$	0	0
$965$	−46.1763	−1.48647
$966$	0	0
$967$	−45.5262	−1.46402	−0.732012	−	0.681292i	$-0.761419\pi$
$967$	−45.5262	−1.46402	−0.732012	+	0.681292i	$0.761419\pi$
$968$	0	0
$969$	−7.41083	−0.238070
$970$	0	0
$971$	42.6255	1.36792	0.683959	−	0.729521i	$-0.260257\pi$
$971$	42.6255	1.36792	0.683959	+	0.729521i	$0.260257\pi$
$972$	0	0
$973$	−4.54745	−0.145785
$974$	0	0
$975$	89.1038	2.85361
$976$	0	0
$977$	21.0234	0.672597	0.336298	−	0.941755i	$-0.390825\pi$
$977$	21.0234	0.672597	0.336298	+	0.941755i	$0.390825\pi$
$978$	0	0
$979$	−18.2862	−0.584430
$980$	0	0
$981$	46.9588	1.49928
$982$	0	0
$983$	18.6739	0.595604	0.297802	−	0.954628i	$-0.403747\pi$
$983$	18.6739	0.595604	0.297802	+	0.954628i	$0.403747\pi$
$984$	0	0
$985$	−37.7966	−1.20430
$986$	0	0
$987$	−37.3806	−1.18984
$988$	0	0
$989$	−5.87989	−0.186970
$990$	0	0
$991$	−10.9999	−0.349422	−0.174711	−	0.984620i	$-0.555899\pi$
$991$	−10.9999	−0.349422	−0.174711	+	0.984620i	$0.555899\pi$
$992$	0	0
$993$	−33.5405	−1.06438
$994$	0	0
$995$	79.6409	2.52479
$996$	0	0
$997$	−48.8203	−1.54615	−0.773077	−	0.634312i	$-0.781284\pi$
$997$	−48.8203	−1.54615	−0.773077	+	0.634312i	$0.781284\pi$
$998$	0	0
$999$	−91.4803	−2.89431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.u.1.3		26
4.3	odd	2		503.2.a.f.1.24	✓	26
12.11	even	2		4527.2.a.o.1.3		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.24	✓	26	4.3	odd	2
4527.2.a.o.1.3		26	12.11	even	2
8048.2.a.u.1.3		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)

\(f(q)\)	\(=\)	\(q-3.12024 q^{3} -4.14994 q^{5} +3.36863 q^{7} +6.73592 q^{9} +O(q^{10})\)	\(q-3.12024 q^{3} -4.14994 q^{5} +3.36863 q^{7} +6.73592 q^{9} -2.17417 q^{11} -2.33650 q^{13} +12.9488 q^{15} +0.306294 q^{17} +7.75425 q^{19} -10.5110 q^{21} -1.14764 q^{23} +12.2220 q^{25} -11.6570 q^{27} +2.03862 q^{29} +2.13184 q^{31} +6.78395 q^{33} -13.9796 q^{35} +7.84768 q^{37} +7.29046 q^{39} +0.876006 q^{41} +5.12346 q^{43} -27.9536 q^{45} +3.55634 q^{47} +4.34770 q^{49} -0.955712 q^{51} +1.93178 q^{53} +9.02267 q^{55} -24.1951 q^{57} -10.5698 q^{59} +3.77497 q^{61} +22.6909 q^{63} +9.69634 q^{65} -4.79689 q^{67} +3.58092 q^{69} +9.50868 q^{71} +0.0400986 q^{73} -38.1355 q^{75} -7.32399 q^{77} -9.19028 q^{79} +16.1649 q^{81} +12.0203 q^{83} -1.27110 q^{85} -6.36100 q^{87} +8.41066 q^{89} -7.87083 q^{91} -6.65187 q^{93} -32.1796 q^{95} +12.6582 q^{97} -14.6451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) 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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