LMFDB - Embedded newform 8048.2.a.v.1.13

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:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q-0.682343 q^{3} +3.69511 q^{5} +3.46637 q^{7} -2.53441 q^{9} +O(q^{10})$	$q-0.682343 q^{3} +3.69511 q^{5} +3.46637 q^{7} -2.53441 q^{9} -1.14497 q^{11} -0.915952 q^{13} -2.52133 q^{15} -7.09797 q^{17} +3.54009 q^{19} -2.36525 q^{21} -4.97901 q^{23} +8.65381 q^{25} +3.77636 q^{27} -4.41622 q^{29} -10.1478 q^{31} +0.781263 q^{33} +12.8086 q^{35} +2.60522 q^{37} +0.624993 q^{39} -9.41578 q^{41} -0.391661 q^{43} -9.36491 q^{45} +4.45651 q^{47} +5.01570 q^{49} +4.84325 q^{51} -1.57279 q^{53} -4.23079 q^{55} -2.41555 q^{57} +3.21174 q^{59} -5.61162 q^{61} -8.78519 q^{63} -3.38454 q^{65} +0.0216322 q^{67} +3.39739 q^{69} +1.49981 q^{71} -12.8005 q^{73} -5.90486 q^{75} -3.96889 q^{77} +6.15151 q^{79} +5.02645 q^{81} +0.0862293 q^{83} -26.2278 q^{85} +3.01338 q^{87} -16.8773 q^{89} -3.17503 q^{91} +6.92426 q^{93} +13.0810 q^{95} -10.8940 q^{97} +2.90182 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	−0.682343	−0.393951	−0.196975	−	0.980408i	$-0.563112\pi$
$3$	−0.682343	−0.393951	−0.196975	+	0.980408i	$0.563112\pi$
$4$	0	0
$5$	3.69511	1.65250	0.826251	−	0.563302i	$-0.190469\pi$
$5$	3.69511	1.65250	0.826251	+	0.563302i	$0.190469\pi$
$6$	0	0
$7$	3.46637	1.31016	0.655082	−	0.755558i	$-0.272634\pi$
$7$	3.46637	1.31016	0.655082	+	0.755558i	$0.272634\pi$
$8$	0	0
$9$	−2.53441	−0.844803
$10$	0	0
$11$	−1.14497	−0.345222	−0.172611	−	0.984990i	$-0.555220\pi$
$11$	−1.14497	−0.345222	−0.172611	+	0.984990i	$0.555220\pi$
$12$	0	0
$13$	−0.915952	−0.254039	−0.127020	−	0.991900i	$-0.540541\pi$
$13$	−0.915952	−0.254039	−0.127020	+	0.991900i	$0.540541\pi$
$14$	0	0
$15$	−2.52133	−0.651004
$16$	0	0
$17$	−7.09797	−1.72151	−0.860756	−	0.509018i	$-0.830009\pi$
$17$	−7.09797	−1.72151	−0.860756	+	0.509018i	$0.830009\pi$
$18$	0	0
$19$	3.54009	0.812152	0.406076	−	0.913839i	$-0.366897\pi$
$19$	3.54009	0.812152	0.406076	+	0.913839i	$0.366897\pi$
$20$	0	0
$21$	−2.36525	−0.516140
$22$	0	0
$23$	−4.97901	−1.03820	−0.519098	−	0.854715i	$-0.673732\pi$
$23$	−4.97901	−1.03820	−0.519098	+	0.854715i	$0.673732\pi$
$24$	0	0
$25$	8.65381	1.73076
$26$	0	0
$27$	3.77636	0.726761
$28$	0	0
$29$	−4.41622	−0.820071	−0.410036	−	0.912069i	$-0.634484\pi$
$29$	−4.41622	−0.820071	−0.410036	+	0.912069i	$0.634484\pi$
$30$	0	0
$31$	−10.1478	−1.82259	−0.911297	−	0.411749i	$-0.864918\pi$
$31$	−10.1478	−1.82259	−0.911297	+	0.411749i	$0.864918\pi$
$32$	0	0
$33$	0.781263	0.136000
$34$	0	0
$35$	12.8086	2.16505
$36$	0	0
$37$	2.60522	0.428296	0.214148	−	0.976801i	$-0.431303\pi$
$37$	2.60522	0.428296	0.214148	+	0.976801i	$0.431303\pi$
$38$	0	0
$39$	0.624993	0.100079
$40$	0	0
$41$	−9.41578	−1.47050	−0.735249	−	0.677797i	$-0.762935\pi$
$41$	−9.41578	−1.47050	−0.735249	+	0.677797i	$0.762935\pi$
$42$	0	0
$43$	−0.391661	−0.0597278	−0.0298639	−	0.999554i	$-0.509507\pi$
$43$	−0.391661	−0.0597278	−0.0298639	+	0.999554i	$0.509507\pi$
$44$	0	0
$45$	−9.36491	−1.39604
$46$	0	0
$47$	4.45651	0.650048	0.325024	−	0.945706i	$-0.394628\pi$
$47$	4.45651	0.650048	0.325024	+	0.945706i	$0.394628\pi$
$48$	0	0
$49$	5.01570	0.716529
$50$	0	0
$51$	4.84325	0.678191
$52$	0	0
$53$	−1.57279	−0.216039	−0.108020	−	0.994149i	$-0.534451\pi$
$53$	−1.57279	−0.216039	−0.108020	+	0.994149i	$0.534451\pi$
$54$	0	0
$55$	−4.23079	−0.570480
$56$	0	0
$57$	−2.41555	−0.319948
$58$	0	0
$59$	3.21174	0.418133	0.209066	−	0.977901i	$-0.432958\pi$
$59$	3.21174	0.418133	0.209066	+	0.977901i	$0.432958\pi$
$60$	0	0
$61$	−5.61162	−0.718495	−0.359247	−	0.933242i	$-0.616967\pi$
$61$	−5.61162	−0.718495	−0.359247	+	0.933242i	$0.616967\pi$
$62$	0	0
$63$	−8.78519	−1.10683
$64$	0	0
$65$	−3.38454	−0.419800
$66$	0	0
$67$	0.0216322	0.00264280	0.00132140	−	0.999999i	$-0.499579\pi$
$67$	0.0216322	0.00264280	0.00132140	+	0.999999i	$0.499579\pi$
$68$	0	0
$69$	3.39739	0.408998
$70$	0	0
$71$	1.49981	0.177994	0.0889972	−	0.996032i	$-0.471634\pi$
$71$	1.49981	0.177994	0.0889972	+	0.996032i	$0.471634\pi$
$72$	0	0
$73$	−12.8005	−1.49819	−0.749095	−	0.662463i	$-0.769511\pi$
$73$	−12.8005	−1.49819	−0.749095	+	0.662463i	$0.769511\pi$
$74$	0	0
$75$	−5.90486	−0.681835
$76$	0	0
$77$	−3.96889	−0.452297
$78$	0	0
$79$	6.15151	0.692099	0.346049	−	0.938216i	$-0.387523\pi$
$79$	6.15151	0.692099	0.346049	+	0.938216i	$0.387523\pi$
$80$	0	0
$81$	5.02645	0.558495
$82$	0	0
$83$	0.0862293	0.00946490	0.00473245	−	0.999989i	$-0.498494\pi$
$83$	0.0862293	0.00946490	0.00473245	+	0.999989i	$0.498494\pi$
$84$	0	0
$85$	−26.2278	−2.84480
$86$	0	0
$87$	3.01338	0.323068
$88$	0	0
$89$	−16.8773	−1.78900	−0.894498	−	0.447073i	$-0.852466\pi$
$89$	−16.8773	−1.78900	−0.894498	+	0.447073i	$0.852466\pi$
$90$	0	0
$91$	−3.17503	−0.332833
$92$	0	0
$93$	6.92426	0.718012
$94$	0	0
$95$	13.0810	1.34208
$96$	0	0
$97$	−10.8940	−1.10612	−0.553059	−	0.833142i	$-0.686540\pi$
$97$	−10.8940	−1.10612	−0.553059	+	0.833142i	$0.686540\pi$
$98$	0	0
$99$	2.90182	0.291644
$100$	0	0
$101$	15.4041	1.53277	0.766384	−	0.642382i	$-0.222054\pi$
$101$	15.4041	1.53277	0.766384	+	0.642382i	$0.222054\pi$
$102$	0	0
$103$	7.32004	0.721265	0.360632	−	0.932708i	$-0.382561\pi$
$103$	7.32004	0.721265	0.360632	+	0.932708i	$0.382561\pi$
$104$	0	0
$105$	−8.73985	−0.852922
$106$	0	0
$107$	0.797681	0.0771147	0.0385574	−	0.999256i	$-0.487724\pi$
$107$	0.797681	0.0771147	0.0385574	+	0.999256i	$0.487724\pi$
$108$	0	0
$109$	−16.1247	−1.54446	−0.772232	−	0.635341i	$-0.780860\pi$
$109$	−16.1247	−1.54446	−0.772232	+	0.635341i	$0.780860\pi$
$110$	0	0
$111$	−1.77765	−0.168727
$112$	0	0
$113$	−7.24664	−0.681707	−0.340853	−	0.940116i	$-0.610716\pi$
$113$	−7.24664	−0.681707	−0.340853	+	0.940116i	$0.610716\pi$
$114$	0	0
$115$	−18.3980	−1.71562
$116$	0	0
$117$	2.32140	0.214613
$118$	0	0
$119$	−24.6042	−2.25546
$120$	0	0
$121$	−9.68904	−0.880822
$122$	0	0
$123$	6.42479	0.579304
$124$	0	0
$125$	13.5012	1.20758
$126$	0	0
$127$	−20.1778	−1.79049	−0.895246	−	0.445572i	$-0.853000\pi$
$127$	−20.1778	−1.79049	−0.895246	+	0.445572i	$0.853000\pi$
$128$	0	0
$129$	0.267247	0.0235298
$130$	0	0
$131$	2.17787	0.190281	0.0951407	−	0.995464i	$-0.469670\pi$
$131$	2.17787	0.190281	0.0951407	+	0.995464i	$0.469670\pi$
$132$	0	0
$133$	12.2712	1.06405
$134$	0	0
$135$	13.9541	1.20097
$136$	0	0
$137$	9.58889	0.819235	0.409617	−	0.912257i	$-0.365662\pi$
$137$	9.58889	0.819235	0.409617	+	0.912257i	$0.365662\pi$
$138$	0	0
$139$	−13.0832	−1.10970	−0.554851	−	0.831950i	$-0.687225\pi$
$139$	−13.0832	−1.10970	−0.554851	+	0.831950i	$0.687225\pi$
$140$	0	0
$141$	−3.04086	−0.256087
$142$	0	0
$143$	1.04874	0.0876999
$144$	0	0
$145$	−16.3184	−1.35517
$146$	0	0
$147$	−3.42243	−0.282277
$148$	0	0
$149$	−7.71463	−0.632007	−0.316004	−	0.948758i	$-0.602341\pi$
$149$	−7.71463	−0.632007	−0.316004	+	0.948758i	$0.602341\pi$
$150$	0	0
$151$	6.19163	0.503867	0.251934	−	0.967744i	$-0.418934\pi$
$151$	6.19163	0.503867	0.251934	+	0.967744i	$0.418934\pi$
$152$	0	0
$153$	17.9892	1.45434
$154$	0	0
$155$	−37.4971	−3.01184
$156$	0	0
$157$	−16.6597	−1.32959	−0.664793	−	0.747027i	$-0.731480\pi$
$157$	−16.6597	−1.32959	−0.664793	+	0.747027i	$0.731480\pi$
$158$	0	0
$159$	1.07318	0.0851088
$160$	0	0
$161$	−17.2591	−1.36021
$162$	0	0
$163$	−1.01575	−0.0795601	−0.0397800	−	0.999208i	$-0.512666\pi$
$163$	−1.01575	−0.0795601	−0.0397800	+	0.999208i	$0.512666\pi$
$164$	0	0
$165$	2.88685	0.224741
$166$	0	0
$167$	18.5419	1.43482	0.717409	−	0.696652i	$-0.245328\pi$
$167$	18.5419	1.43482	0.717409	+	0.696652i	$0.245328\pi$
$168$	0	0
$169$	−12.1610	−0.935464
$170$	0	0
$171$	−8.97203	−0.686108
$172$	0	0
$173$	19.5050	1.48294	0.741470	−	0.670986i	$-0.234129\pi$
$173$	19.5050	1.48294	0.741470	+	0.670986i	$0.234129\pi$
$174$	0	0
$175$	29.9973	2.26758
$176$	0	0
$177$	−2.19151	−0.164724
$178$	0	0
$179$	20.3190	1.51872	0.759358	−	0.650673i	$-0.225513\pi$
$179$	20.3190	1.51872	0.759358	+	0.650673i	$0.225513\pi$
$180$	0	0
$181$	11.4088	0.848011	0.424005	−	0.905660i	$-0.360624\pi$
$181$	11.4088	0.848011	0.424005	+	0.905660i	$0.360624\pi$
$182$	0	0
$183$	3.82905	0.283051
$184$	0	0
$185$	9.62656	0.707759
$186$	0	0
$187$	8.12697	0.594303
$188$	0	0
$189$	13.0903	0.952176
$190$	0	0
$191$	−14.3105	−1.03547	−0.517734	−	0.855541i	$-0.673224\pi$
$191$	−14.3105	−1.03547	−0.517734	+	0.855541i	$0.673224\pi$
$192$	0	0
$193$	4.59672	0.330879	0.165440	−	0.986220i	$-0.447096\pi$
$193$	4.59672	0.330879	0.165440	+	0.986220i	$0.447096\pi$
$194$	0	0
$195$	2.30942	0.165381
$196$	0	0
$197$	18.2817	1.30251	0.651257	−	0.758857i	$-0.274242\pi$
$197$	18.2817	1.30251	0.651257	+	0.758857i	$0.274242\pi$
$198$	0	0
$199$	12.0533	0.854436	0.427218	−	0.904149i	$-0.359494\pi$
$199$	12.0533	0.854436	0.427218	+	0.904149i	$0.359494\pi$
$200$	0	0
$201$	−0.0147606	−0.00104113
$202$	0	0
$203$	−15.3082	−1.07443
$204$	0	0
$205$	−34.7923	−2.43000
$206$	0	0
$207$	12.6188	0.877071
$208$	0	0
$209$	−4.05330	−0.280372
$210$	0	0
$211$	−24.0136	−1.65317	−0.826583	−	0.562815i	$-0.809718\pi$
$211$	−24.0136	−1.65317	−0.826583	+	0.562815i	$0.809718\pi$
$212$	0	0
$213$	−1.02338	−0.0701210
$214$	0	0
$215$	−1.44723	−0.0987002
$216$	0	0
$217$	−35.1759	−2.38790
$218$	0	0
$219$	8.73435	0.590213
$220$	0	0
$221$	6.50140	0.437332
$222$	0	0
$223$	−1.04220	−0.0697911	−0.0348955	−	0.999391i	$-0.511110\pi$
$223$	−1.04220	−0.0697911	−0.0348955	+	0.999391i	$0.511110\pi$
$224$	0	0
$225$	−21.9323	−1.46215
$226$	0	0
$227$	−7.91737	−0.525495	−0.262747	−	0.964865i	$-0.584629\pi$
$227$	−7.91737	−0.525495	−0.262747	+	0.964865i	$0.584629\pi$
$228$	0	0
$229$	3.95775	0.261535	0.130768	−	0.991413i	$-0.458256\pi$
$229$	3.95775	0.261535	0.130768	+	0.991413i	$0.458256\pi$
$230$	0	0
$231$	2.70814	0.178183
$232$	0	0
$233$	16.8280	1.10244	0.551221	−	0.834359i	$-0.314162\pi$
$233$	16.8280	1.10244	0.551221	+	0.834359i	$0.314162\pi$
$234$	0	0
$235$	16.4673	1.07421
$236$	0	0
$237$	−4.19743	−0.272653
$238$	0	0
$239$	12.8700	0.832494	0.416247	−	0.909252i	$-0.363345\pi$
$239$	12.8700	0.832494	0.416247	+	0.909252i	$0.363345\pi$
$240$	0	0
$241$	2.95315	0.190229	0.0951145	−	0.995466i	$-0.469678\pi$
$241$	2.95315	0.190229	0.0951145	+	0.995466i	$0.469678\pi$
$242$	0	0
$243$	−14.7589	−0.946781
$244$	0	0
$245$	18.5336	1.18407
$246$	0	0
$247$	−3.24255	−0.206318
$248$	0	0
$249$	−0.0588380	−0.00372870
$250$	0	0
$251$	−23.7939	−1.50186	−0.750930	−	0.660381i	$-0.770395\pi$
$251$	−23.7939	−1.50186	−0.750930	+	0.660381i	$0.770395\pi$
$252$	0	0
$253$	5.70082	0.358408
$254$	0	0
$255$	17.8963	1.12071
$256$	0	0
$257$	18.8437	1.17544	0.587718	−	0.809066i	$-0.300026\pi$
$257$	18.8437	1.17544	0.587718	+	0.809066i	$0.300026\pi$
$258$	0	0
$259$	9.03065	0.561137
$260$	0	0
$261$	11.1925	0.692799
$262$	0	0
$263$	14.3719	0.886209	0.443105	−	0.896470i	$-0.353877\pi$
$263$	14.3719	0.886209	0.443105	+	0.896470i	$0.353877\pi$
$264$	0	0
$265$	−5.81162	−0.357005
$266$	0	0
$267$	11.5161	0.704776
$268$	0	0
$269$	26.5833	1.62082	0.810408	−	0.585867i	$-0.199246\pi$
$269$	26.5833	1.62082	0.810408	+	0.585867i	$0.199246\pi$
$270$	0	0
$271$	4.11918	0.250222	0.125111	−	0.992143i	$-0.460071\pi$
$271$	4.11918	0.250222	0.125111	+	0.992143i	$0.460071\pi$
$272$	0	0
$273$	2.16646	0.131120
$274$	0	0
$275$	−9.90836	−0.597497
$276$	0	0
$277$	−26.1074	−1.56864	−0.784320	−	0.620357i	$-0.786988\pi$
$277$	−26.1074	−1.56864	−0.784320	+	0.620357i	$0.786988\pi$
$278$	0	0
$279$	25.7186	1.53973
$280$	0	0
$281$	3.97747	0.237276	0.118638	−	0.992938i	$-0.462147\pi$
$281$	3.97747	0.237276	0.118638	+	0.992938i	$0.462147\pi$
$282$	0	0
$283$	−10.5153	−0.625073	−0.312536	−	0.949906i	$-0.601179\pi$
$283$	−10.5153	−0.625073	−0.312536	+	0.949906i	$0.601179\pi$
$284$	0	0
$285$	−8.92572	−0.528714
$286$	0	0
$287$	−32.6386	−1.92659
$288$	0	0
$289$	33.3812	1.96360
$290$	0	0
$291$	7.43345	0.435756
$292$	0	0
$293$	24.9505	1.45762	0.728811	−	0.684715i	$-0.240073\pi$
$293$	24.9505	1.45762	0.728811	+	0.684715i	$0.240073\pi$
$294$	0	0
$295$	11.8677	0.690965
$296$	0	0
$297$	−4.32383	−0.250894
$298$	0	0
$299$	4.56053	0.263743
$300$	0	0
$301$	−1.35764	−0.0782532
$302$	0	0
$303$	−10.5109	−0.603835
$304$	0	0
$305$	−20.7355	−1.18731
$306$	0	0
$307$	−0.621755	−0.0354855	−0.0177427	−	0.999843i	$-0.505648\pi$
$307$	−0.621755	−0.0354855	−0.0177427	+	0.999843i	$0.505648\pi$
$308$	0	0
$309$	−4.99477	−0.284143
$310$	0	0
$311$	20.8523	1.18242	0.591212	−	0.806516i	$-0.298650\pi$
$311$	20.8523	1.18242	0.591212	+	0.806516i	$0.298650\pi$
$312$	0	0
$313$	25.4816	1.44030	0.720152	−	0.693817i	$-0.244072\pi$
$313$	25.4816	1.44030	0.720152	+	0.693817i	$0.244072\pi$
$314$	0	0
$315$	−32.4622	−1.82904
$316$	0	0
$317$	−6.05887	−0.340300	−0.170150	−	0.985418i	$-0.554425\pi$
$317$	−6.05887	−0.340300	−0.170150	+	0.985418i	$0.554425\pi$
$318$	0	0
$319$	5.05644	0.283107
$320$	0	0
$321$	−0.544292	−0.0303794
$322$	0	0
$323$	−25.1274	−1.39813
$324$	0	0
$325$	−7.92647	−0.439682
$326$	0	0
$327$	11.0026	0.608442
$328$	0	0
$329$	15.4479	0.851670
$330$	0	0
$331$	−26.6371	−1.46411	−0.732055	−	0.681246i	$-0.761438\pi$
$331$	−26.6371	−1.46411	−0.732055	+	0.681246i	$0.761438\pi$
$332$	0	0
$333$	−6.60269	−0.361825
$334$	0	0
$335$	0.0799334	0.00436723
$336$	0	0
$337$	3.62374	0.197398	0.0986988	−	0.995117i	$-0.468532\pi$
$337$	3.62374	0.197398	0.0986988	+	0.995117i	$0.468532\pi$
$338$	0	0
$339$	4.94469	0.268559
$340$	0	0
$341$	11.6189	0.629199
$342$	0	0
$343$	−6.87830	−0.371393
$344$	0	0
$345$	12.5537	0.675870
$346$	0	0
$347$	36.0019	1.93268	0.966340	−	0.257267i	$-0.0828220\pi$
$347$	36.0019	1.93268	0.966340	+	0.257267i	$0.0828220\pi$
$348$	0	0
$349$	15.2420	0.815885	0.407943	−	0.913008i	$-0.366246\pi$
$349$	15.2420	0.815885	0.407943	+	0.913008i	$0.366246\pi$
$350$	0	0
$351$	−3.45897	−0.184626
$352$	0	0
$353$	−24.5906	−1.30883	−0.654414	−	0.756137i	$-0.727084\pi$
$353$	−24.5906	−1.30883	−0.654414	+	0.756137i	$0.727084\pi$
$354$	0	0
$355$	5.54195	0.294136
$356$	0	0
$357$	16.7885	0.888541
$358$	0	0
$359$	17.6695	0.932562	0.466281	−	0.884637i	$-0.345593\pi$
$359$	17.6695	0.932562	0.466281	+	0.884637i	$0.345593\pi$
$360$	0	0
$361$	−6.46778	−0.340410
$362$	0	0
$363$	6.61125	0.347000
$364$	0	0
$365$	−47.2993	−2.47576
$366$	0	0
$367$	−19.1835	−1.00137	−0.500684	−	0.865630i	$-0.666918\pi$
$367$	−19.1835	−1.00137	−0.500684	+	0.865630i	$0.666918\pi$
$368$	0	0
$369$	23.8634	1.24228
$370$	0	0
$371$	−5.45187	−0.283047
$372$	0	0
$373$	−14.3175	−0.741330	−0.370665	−	0.928767i	$-0.620870\pi$
$373$	−14.3175	−0.741330	−0.370665	+	0.928767i	$0.620870\pi$
$374$	0	0
$375$	−9.21245	−0.475729
$376$	0	0
$377$	4.04505	0.208330
$378$	0	0
$379$	30.1133	1.54681	0.773407	−	0.633910i	$-0.218551\pi$
$379$	30.1133	1.54681	0.773407	+	0.633910i	$0.218551\pi$
$380$	0	0
$381$	13.7682	0.705366
$382$	0	0
$383$	−13.2583	−0.677466	−0.338733	−	0.940882i	$-0.609998\pi$
$383$	−13.2583	−0.677466	−0.338733	+	0.940882i	$0.609998\pi$
$384$	0	0
$385$	−14.6655	−0.747422
$386$	0	0
$387$	0.992629	0.0504582
$388$	0	0
$389$	7.00355	0.355094	0.177547	−	0.984112i	$-0.443184\pi$
$389$	7.00355	0.355094	0.177547	+	0.984112i	$0.443184\pi$
$390$	0	0
$391$	35.3409	1.78727
$392$	0	0
$393$	−1.48605	−0.0749615
$394$	0	0
$395$	22.7305	1.14369
$396$	0	0
$397$	10.4033	0.522128	0.261064	−	0.965321i	$-0.415927\pi$
$397$	10.4033	0.522128	0.261064	+	0.965321i	$0.415927\pi$
$398$	0	0
$399$	−8.37319	−0.419184
$400$	0	0
$401$	−12.4393	−0.621190	−0.310595	−	0.950542i	$-0.600528\pi$
$401$	−12.4393	−0.621190	−0.310595	+	0.950542i	$0.600528\pi$
$402$	0	0
$403$	9.29487	0.463011
$404$	0	0
$405$	18.5733	0.922913
$406$	0	0
$407$	−2.98290	−0.147857
$408$	0	0
$409$	2.06701	0.102207	0.0511035	−	0.998693i	$-0.483726\pi$
$409$	2.06701	0.102207	0.0511035	+	0.998693i	$0.483726\pi$
$410$	0	0
$411$	−6.54291	−0.322738
$412$	0	0
$413$	11.1331	0.547822
$414$	0	0
$415$	0.318627	0.0156408
$416$	0	0
$417$	8.92722	0.437168
$418$	0	0
$419$	2.23093	0.108988	0.0544940	−	0.998514i	$-0.482645\pi$
$419$	2.23093	0.108988	0.0544940	+	0.998514i	$0.482645\pi$
$420$	0	0
$421$	−3.43410	−0.167368	−0.0836840	−	0.996492i	$-0.526669\pi$
$421$	−3.43410	−0.167368	−0.0836840	+	0.996492i	$0.526669\pi$
$422$	0	0
$423$	−11.2946	−0.549163
$424$	0	0
$425$	−61.4245	−2.97953
$426$	0	0
$427$	−19.4519	−0.941346
$428$	0	0
$429$	−0.715599	−0.0345494
$430$	0	0
$431$	−28.0604	−1.35162	−0.675811	−	0.737075i	$-0.736207\pi$
$431$	−28.0604	−1.35162	−0.675811	+	0.737075i	$0.736207\pi$
$432$	0	0
$433$	−10.0537	−0.483152	−0.241576	−	0.970382i	$-0.577664\pi$
$433$	−10.0537	−0.483152	−0.241576	+	0.970382i	$0.577664\pi$
$434$	0	0
$435$	11.1347	0.533870
$436$	0	0
$437$	−17.6261	−0.843172
$438$	0	0
$439$	−2.16976	−0.103557	−0.0517786	−	0.998659i	$-0.516489\pi$
$439$	−2.16976	−0.103557	−0.0517786	+	0.998659i	$0.516489\pi$
$440$	0	0
$441$	−12.7118	−0.605326
$442$	0	0
$443$	19.8482	0.943019	0.471509	−	0.881861i	$-0.343709\pi$
$443$	19.8482	0.943019	0.471509	+	0.881861i	$0.343709\pi$
$444$	0	0
$445$	−62.3636	−2.95632
$446$	0	0
$447$	5.26402	0.248980
$448$	0	0
$449$	−37.5743	−1.77324	−0.886621	−	0.462497i	$-0.846954\pi$
$449$	−37.5743	−1.77324	−0.886621	+	0.462497i	$0.846954\pi$
$450$	0	0
$451$	10.7808	0.507648
$452$	0	0
$453$	−4.22481	−0.198499
$454$	0	0
$455$	−11.7321	−0.550007
$456$	0	0
$457$	−16.7147	−0.781882	−0.390941	−	0.920416i	$-0.627850\pi$
$457$	−16.7147	−0.781882	−0.390941	+	0.920416i	$0.627850\pi$
$458$	0	0
$459$	−26.8045	−1.25113
$460$	0	0
$461$	−36.8100	−1.71441	−0.857205	−	0.514975i	$-0.827801\pi$
$461$	−36.8100	−1.71441	−0.857205	+	0.514975i	$0.827801\pi$
$462$	0	0
$463$	−5.00847	−0.232764	−0.116382	−	0.993205i	$-0.537130\pi$
$463$	−5.00847	−0.232764	−0.116382	+	0.993205i	$0.537130\pi$
$464$	0	0
$465$	25.5859	1.18652
$466$	0	0
$467$	−32.7816	−1.51695	−0.758475	−	0.651702i	$-0.774055\pi$
$467$	−32.7816	−1.51695	−0.758475	+	0.651702i	$0.774055\pi$
$468$	0	0
$469$	0.0749853	0.00346250
$470$	0	0
$471$	11.3676	0.523792
$472$	0	0
$473$	0.448441	0.0206193
$474$	0	0
$475$	30.6352	1.40564
$476$	0	0
$477$	3.98609	0.182511
$478$	0	0
$479$	23.0131	1.05150	0.525748	−	0.850640i	$-0.323785\pi$
$479$	23.0131	1.05150	0.525748	+	0.850640i	$0.323785\pi$
$480$	0	0
$481$	−2.38626	−0.108804
$482$	0	0
$483$	11.7766	0.535854
$484$	0	0
$485$	−40.2545	−1.82786
$486$	0	0
$487$	11.9167	0.539999	0.270000	−	0.962860i	$-0.412976\pi$
$487$	11.9167	0.539999	0.270000	+	0.962860i	$0.412976\pi$
$488$	0	0
$489$	0.693093	0.0313427
$490$	0	0
$491$	5.07035	0.228822	0.114411	−	0.993434i	$-0.463502\pi$
$491$	5.07035	0.228822	0.114411	+	0.993434i	$0.463502\pi$
$492$	0	0
$493$	31.3462	1.41176
$494$	0	0
$495$	10.7225	0.481943
$496$	0	0
$497$	5.19888	0.233202
$498$	0	0
$499$	−11.0312	−0.493822	−0.246911	−	0.969038i	$-0.579416\pi$
$499$	−11.0312	−0.493822	−0.246911	+	0.969038i	$0.579416\pi$
$500$	0	0
$501$	−12.6520	−0.565248
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	56.9199	2.53290
$506$	0	0
$507$	8.29799	0.368527
$508$	0	0
$509$	1.01308	0.0449039	0.0224520	−	0.999748i	$-0.492853\pi$
$509$	1.01308	0.0449039	0.0224520	+	0.999748i	$0.492853\pi$
$510$	0	0
$511$	−44.3714	−1.96287
$512$	0	0
$513$	13.3687	0.590240
$514$	0	0
$515$	27.0483	1.19189
$516$	0	0
$517$	−5.10257	−0.224411
$518$	0	0
$519$	−13.3091	−0.584205
$520$	0	0
$521$	27.5158	1.20549	0.602744	−	0.797934i	$-0.294074\pi$
$521$	27.5158	1.20549	0.602744	+	0.797934i	$0.294074\pi$
$522$	0	0
$523$	−21.7823	−0.952474	−0.476237	−	0.879317i	$-0.658000\pi$
$523$	−21.7823	−0.952474	−0.476237	+	0.879317i	$0.658000\pi$
$524$	0	0
$525$	−20.4684	−0.893315
$526$	0	0
$527$	72.0286	3.13762
$528$	0	0
$529$	1.79055	0.0778501
$530$	0	0
$531$	−8.13986	−0.353240
$532$	0	0
$533$	8.62440	0.373564
$534$	0	0
$535$	2.94752	0.127432
$536$	0	0
$537$	−13.8646	−0.598300
$538$	0	0
$539$	−5.74284	−0.247361
$540$	0	0
$541$	−39.5644	−1.70101	−0.850503	−	0.525970i	$-0.823702\pi$
$541$	−39.5644	−1.70101	−0.850503	+	0.525970i	$0.823702\pi$
$542$	0	0
$543$	−7.78472	−0.334074
$544$	0	0
$545$	−59.5824	−2.55223
$546$	0	0
$547$	4.64052	0.198414	0.0992072	−	0.995067i	$-0.468369\pi$
$547$	4.64052	0.198414	0.0992072	+	0.995067i	$0.468369\pi$
$548$	0	0
$549$	14.2221	0.606986
$550$	0	0
$551$	−15.6338	−0.666022
$552$	0	0
$553$	21.3234	0.906762
$554$	0	0
$555$	−6.56861	−0.278822
$556$	0	0
$557$	38.9567	1.65065	0.825325	−	0.564658i	$-0.190992\pi$
$557$	38.9567	1.65065	0.825325	+	0.564658i	$0.190992\pi$
$558$	0	0
$559$	0.358743	0.0151732
$560$	0	0
$561$	−5.54538	−0.234126
$562$	0	0
$563$	2.24123	0.0944567	0.0472284	−	0.998884i	$-0.484961\pi$
$563$	2.24123	0.0944567	0.0472284	+	0.998884i	$0.484961\pi$
$564$	0	0
$565$	−26.7771	−1.12652
$566$	0	0
$567$	17.4235	0.731720
$568$	0	0
$569$	−21.7992	−0.913869	−0.456935	−	0.889500i	$-0.651053\pi$
$569$	−21.7992	−0.913869	−0.456935	+	0.889500i	$0.651053\pi$
$570$	0	0
$571$	26.8351	1.12301	0.561507	−	0.827472i	$-0.310222\pi$
$571$	26.8351	1.12301	0.561507	+	0.827472i	$0.310222\pi$
$572$	0	0
$573$	9.76464	0.407924
$574$	0	0
$575$	−43.0874	−1.79687
$576$	0	0
$577$	−7.50651	−0.312500	−0.156250	−	0.987718i	$-0.549941\pi$
$577$	−7.50651	−0.312500	−0.156250	+	0.987718i	$0.549941\pi$
$578$	0	0
$579$	−3.13654	−0.130350
$580$	0	0
$581$	0.298903	0.0124006
$582$	0	0
$583$	1.80080	0.0745814
$584$	0	0
$585$	8.57781	0.354649
$586$	0	0
$587$	16.1847	0.668015	0.334008	−	0.942570i	$-0.391599\pi$
$587$	16.1847	0.668015	0.334008	+	0.942570i	$0.391599\pi$
$588$	0	0
$589$	−35.9240	−1.48022
$590$	0	0
$591$	−12.4744	−0.513126
$592$	0	0
$593$	10.1623	0.417315	0.208657	−	0.977989i	$-0.433091\pi$
$593$	10.1623	0.417315	0.208657	+	0.977989i	$0.433091\pi$
$594$	0	0
$595$	−90.9151	−3.72715
$596$	0	0
$597$	−8.22448	−0.336606
$598$	0	0
$599$	37.0050	1.51198	0.755991	−	0.654582i	$-0.227155\pi$
$599$	37.0050	1.51198	0.755991	+	0.654582i	$0.227155\pi$
$600$	0	0
$601$	−17.0209	−0.694295	−0.347148	−	0.937811i	$-0.612850\pi$
$601$	−17.0209	−0.694295	−0.347148	+	0.937811i	$0.612850\pi$
$602$	0	0
$603$	−0.0548249	−0.00223264
$604$	0	0
$605$	−35.8020	−1.45556
$606$	0	0
$607$	−33.6693	−1.36659	−0.683297	−	0.730140i	$-0.739455\pi$
$607$	−33.6693	−1.36659	−0.683297	+	0.730140i	$0.739455\pi$
$608$	0	0
$609$	10.4455	0.423272
$610$	0	0
$611$	−4.08195	−0.165138
$612$	0	0
$613$	13.3329	0.538510	0.269255	−	0.963069i	$-0.413222\pi$
$613$	13.3329	0.538510	0.269255	+	0.963069i	$0.413222\pi$
$614$	0	0
$615$	23.7403	0.957300
$616$	0	0
$617$	32.0158	1.28891	0.644454	−	0.764643i	$-0.277085\pi$
$617$	32.0158	1.28891	0.644454	+	0.764643i	$0.277085\pi$
$618$	0	0
$619$	−32.7902	−1.31795	−0.658974	−	0.752166i	$-0.729009\pi$
$619$	−32.7902	−1.31795	−0.658974	+	0.752166i	$0.729009\pi$
$620$	0	0
$621$	−18.8026	−0.754520
$622$	0	0
$623$	−58.5031	−2.34388
$624$	0	0
$625$	6.61935	0.264774
$626$	0	0
$627$	2.76574	0.110453
$628$	0	0
$629$	−18.4918	−0.737316
$630$	0	0
$631$	−37.7022	−1.50090	−0.750450	−	0.660927i	$-0.770163\pi$
$631$	−37.7022	−1.50090	−0.750450	+	0.660927i	$0.770163\pi$
$632$	0	0
$633$	16.3855	0.651266
$634$	0	0
$635$	−74.5592	−2.95879
$636$	0	0
$637$	−4.59414	−0.182027
$638$	0	0
$639$	−3.80113	−0.150370
$640$	0	0
$641$	−28.6119	−1.13010	−0.565051	−	0.825056i	$-0.691144\pi$
$641$	−28.6119	−1.13010	−0.565051	+	0.825056i	$0.691144\pi$
$642$	0	0
$643$	20.3342	0.801904	0.400952	−	0.916099i	$-0.368679\pi$
$643$	20.3342	0.801904	0.400952	+	0.916099i	$0.368679\pi$
$644$	0	0
$645$	0.987506	0.0388830
$646$	0	0
$647$	−49.6208	−1.95080	−0.975398	−	0.220452i	$-0.929247\pi$
$647$	−49.6208	−1.95080	−0.975398	+	0.220452i	$0.929247\pi$
$648$	0	0
$649$	−3.67735	−0.144349
$650$	0	0
$651$	24.0020	0.940714
$652$	0	0
$653$	−19.0540	−0.745642	−0.372821	−	0.927903i	$-0.621609\pi$
$653$	−19.0540	−0.745642	−0.372821	+	0.927903i	$0.621609\pi$
$654$	0	0
$655$	8.04746	0.314440
$656$	0	0
$657$	32.4418	1.26567
$658$	0	0
$659$	−39.9543	−1.55640	−0.778198	−	0.628019i	$-0.783866\pi$
$659$	−39.9543	−1.55640	−0.778198	+	0.628019i	$0.783866\pi$
$660$	0	0
$661$	39.7928	1.54776	0.773880	−	0.633332i	$-0.218313\pi$
$661$	39.7928	1.54776	0.773880	+	0.633332i	$0.218313\pi$
$662$	0	0
$663$	−4.43618	−0.172287
$664$	0	0
$665$	45.3435	1.75835
$666$	0	0
$667$	21.9884	0.851395
$668$	0	0
$669$	0.711140	0.0274943
$670$	0	0
$671$	6.42514	0.248040
$672$	0	0
$673$	28.7641	1.10877	0.554387	−	0.832259i	$-0.312953\pi$
$673$	28.7641	1.10877	0.554387	+	0.832259i	$0.312953\pi$
$674$	0	0
$675$	32.6799	1.25785
$676$	0	0
$677$	−37.2623	−1.43211	−0.716053	−	0.698046i	$-0.754053\pi$
$677$	−37.2623	−1.43211	−0.716053	+	0.698046i	$0.754053\pi$
$678$	0	0
$679$	−37.7626	−1.44920
$680$	0	0
$681$	5.40236	0.207019
$682$	0	0
$683$	−27.9895	−1.07099	−0.535493	−	0.844539i	$-0.679874\pi$
$683$	−27.9895	−1.07099	−0.535493	+	0.844539i	$0.679874\pi$
$684$	0	0
$685$	35.4320	1.35379
$686$	0	0
$687$	−2.70054	−0.103032
$688$	0	0
$689$	1.44060	0.0548825
$690$	0	0
$691$	41.2535	1.56936	0.784679	−	0.619902i	$-0.212828\pi$
$691$	41.2535	1.56936	0.784679	+	0.619902i	$0.212828\pi$
$692$	0	0
$693$	10.0588	0.382102
$694$	0	0
$695$	−48.3438	−1.83378
$696$	0	0
$697$	66.8329	2.53148
$698$	0	0
$699$	−11.4825	−0.434308
$700$	0	0
$701$	25.1272	0.949041	0.474520	−	0.880245i	$-0.342622\pi$
$701$	25.1272	0.949041	0.474520	+	0.880245i	$0.342622\pi$
$702$	0	0
$703$	9.22271	0.347841
$704$	0	0
$705$	−11.2363	−0.423184
$706$	0	0
$707$	53.3964	2.00818
$708$	0	0
$709$	−15.7683	−0.592191	−0.296096	−	0.955158i	$-0.595685\pi$
$709$	−15.7683	−0.592191	−0.296096	+	0.955158i	$0.595685\pi$
$710$	0	0
$711$	−15.5904	−0.584687
$712$	0	0
$713$	50.5259	1.89221
$714$	0	0
$715$	3.87520	0.144924
$716$	0	0
$717$	−8.78178	−0.327962
$718$	0	0
$719$	41.9447	1.56427	0.782137	−	0.623106i	$-0.214129\pi$
$719$	41.9447	1.56427	0.782137	+	0.623106i	$0.214129\pi$
$720$	0	0
$721$	25.3739	0.944975
$722$	0	0
$723$	−2.01506	−0.0749409
$724$	0	0
$725$	−38.2171	−1.41935
$726$	0	0
$727$	−28.4589	−1.05548	−0.527742	−	0.849405i	$-0.676961\pi$
$727$	−28.4589	−1.05548	−0.527742	+	0.849405i	$0.676961\pi$
$728$	0	0
$729$	−5.00877	−0.185510
$730$	0	0
$731$	2.78000	0.102822
$732$	0	0
$733$	−20.6873	−0.764102	−0.382051	−	0.924141i	$-0.624782\pi$
$733$	−20.6873	−0.764102	−0.382051	+	0.924141i	$0.624782\pi$
$734$	0	0
$735$	−12.6462	−0.466463
$736$	0	0
$737$	−0.0247683	−0.000912351 0
$738$	0	0
$739$	−40.7104	−1.49756	−0.748778	−	0.662820i	$-0.769359\pi$
$739$	−40.7104	−1.49756	−0.748778	+	0.662820i	$0.769359\pi$
$740$	0	0
$741$	2.21253	0.0812793
$742$	0	0
$743$	26.9510	0.988735	0.494368	−	0.869253i	$-0.335400\pi$
$743$	26.9510	0.988735	0.494368	+	0.869253i	$0.335400\pi$
$744$	0	0
$745$	−28.5064	−1.04439
$746$	0	0
$747$	−0.218540	−0.00799597
$748$	0	0
$749$	2.76506	0.101033
$750$	0	0
$751$	19.2460	0.702298	0.351149	−	0.936320i	$-0.385791\pi$
$751$	19.2460	0.702298	0.351149	+	0.936320i	$0.385791\pi$
$752$	0	0
$753$	16.2356	0.591659
$754$	0	0
$755$	22.8787	0.832642
$756$	0	0
$757$	−20.0449	−0.728544	−0.364272	−	0.931293i	$-0.618682\pi$
$757$	−20.0449	−0.728544	−0.364272	+	0.931293i	$0.618682\pi$
$758$	0	0
$759$	−3.88991	−0.141195
$760$	0	0
$761$	11.9482	0.433120	0.216560	−	0.976269i	$-0.430516\pi$
$761$	11.9482	0.433120	0.216560	+	0.976269i	$0.430516\pi$
$762$	0	0
$763$	−55.8940	−2.02350
$764$	0	0
$765$	66.4719	2.40330
$766$	0	0
$767$	−2.94180	−0.106222
$768$	0	0
$769$	−11.7469	−0.423603	−0.211802	−	0.977313i	$-0.567933\pi$
$769$	−11.7469	−0.423603	−0.211802	+	0.977313i	$0.567933\pi$
$770$	0	0
$771$	−12.8578	−0.463064
$772$	0	0
$773$	29.6378	1.06600	0.532999	−	0.846116i	$-0.321065\pi$
$773$	29.6378	1.06600	0.532999	+	0.846116i	$0.321065\pi$
$774$	0	0
$775$	−87.8169	−3.15448
$776$	0	0
$777$	−6.16200	−0.221060
$778$	0	0
$779$	−33.3327	−1.19427
$780$	0	0
$781$	−1.71724	−0.0614475
$782$	0	0
$783$	−16.6772	−0.595996
$784$	0	0
$785$	−61.5593	−2.19714
$786$	0	0
$787$	−5.52019	−0.196774	−0.0983868	−	0.995148i	$-0.531368\pi$
$787$	−5.52019	−0.196774	−0.0983868	+	0.995148i	$0.531368\pi$
$788$	0	0
$789$	−9.80656	−0.349123
$790$	0	0
$791$	−25.1195	−0.893148
$792$	0	0
$793$	5.13998	0.182526
$794$	0	0
$795$	3.96552	0.140642
$796$	0	0
$797$	−7.83445	−0.277511	−0.138755	−	0.990327i	$-0.544310\pi$
$797$	−7.83445	−0.277511	−0.138755	+	0.990327i	$0.544310\pi$
$798$	0	0
$799$	−31.6322	−1.11907
$800$	0	0
$801$	42.7741	1.51135
$802$	0	0
$803$	14.6562	0.517207
$804$	0	0
$805$	−63.7741	−2.24774
$806$	0	0
$807$	−18.1389	−0.638521
$808$	0	0
$809$	0.983806	0.0345888	0.0172944	−	0.999850i	$-0.494495\pi$
$809$	0.983806	0.0345888	0.0172944	+	0.999850i	$0.494495\pi$
$810$	0	0
$811$	39.8528	1.39942	0.699710	−	0.714427i	$-0.253313\pi$
$811$	39.8528	1.39942	0.699710	+	0.714427i	$0.253313\pi$
$812$	0	0
$813$	−2.81069	−0.0985752
$814$	0	0
$815$	−3.75332	−0.131473
$816$	0	0
$817$	−1.38651	−0.0485080
$818$	0	0
$819$	8.04681	0.281178
$820$	0	0
$821$	2.19951	0.0767634	0.0383817	−	0.999263i	$-0.487780\pi$
$821$	2.19951	0.0767634	0.0383817	+	0.999263i	$0.487780\pi$
$822$	0	0
$823$	−4.01532	−0.139965	−0.0699825	−	0.997548i	$-0.522294\pi$
$823$	−4.01532	−0.139965	−0.0699825	+	0.997548i	$0.522294\pi$
$824$	0	0
$825$	6.76090	0.235384
$826$	0	0
$827$	9.44160	0.328317	0.164158	−	0.986434i	$-0.447509\pi$
$827$	9.44160	0.328317	0.164158	+	0.986434i	$0.447509\pi$
$828$	0	0
$829$	43.4870	1.51036	0.755182	−	0.655515i	$-0.227548\pi$
$829$	43.4870	1.51036	0.755182	+	0.655515i	$0.227548\pi$
$830$	0	0
$831$	17.8142	0.617967
$832$	0	0
$833$	−35.6013	−1.23351
$834$	0	0
$835$	68.5144	2.37104
$836$	0	0
$837$	−38.3217	−1.32459
$838$	0	0
$839$	−31.1893	−1.07677	−0.538387	−	0.842698i	$-0.680966\pi$
$839$	−31.1893	−1.07677	−0.538387	+	0.842698i	$0.680966\pi$
$840$	0	0
$841$	−9.49700	−0.327483
$842$	0	0
$843$	−2.71400	−0.0934751
$844$	0	0
$845$	−44.9363	−1.54586
$846$	0	0
$847$	−33.5858	−1.15402
$848$	0	0
$849$	7.17507	0.246248
$850$	0	0
$851$	−12.9714	−0.444655
$852$	0	0
$853$	−0.320054	−0.0109584	−0.00547922	−	0.999985i	$-0.501744\pi$
$853$	−0.320054	−0.0109584	−0.00547922	+	0.999985i	$0.501744\pi$
$854$	0	0
$855$	−33.1526	−1.13379
$856$	0	0
$857$	−28.8466	−0.985381	−0.492690	−	0.870205i	$-0.663987\pi$
$857$	−28.8466	−0.985381	−0.492690	+	0.870205i	$0.663987\pi$
$858$	0	0
$859$	−37.0112	−1.26281	−0.631403	−	0.775455i	$-0.717520\pi$
$859$	−37.0112	−1.26281	−0.631403	+	0.775455i	$0.717520\pi$
$860$	0	0
$861$	22.2707	0.758983
$862$	0	0
$863$	8.39353	0.285719	0.142859	−	0.989743i	$-0.454370\pi$
$863$	8.39353	0.285719	0.142859	+	0.989743i	$0.454370\pi$
$864$	0	0
$865$	72.0732	2.45056
$866$	0	0
$867$	−22.7774	−0.773562
$868$	0	0
$869$	−7.04330	−0.238927
$870$	0	0
$871$	−0.0198141	−0.000671375 0
$872$	0	0
$873$	27.6099	0.934452
$874$	0	0
$875$	46.8001	1.58213
$876$	0	0
$877$	29.5901	0.999186	0.499593	−	0.866260i	$-0.333483\pi$
$877$	29.5901	0.999186	0.499593	+	0.866260i	$0.333483\pi$
$878$	0	0
$879$	−17.0248	−0.574231
$880$	0	0
$881$	17.3917	0.585943	0.292971	−	0.956121i	$-0.405356\pi$
$881$	17.3917	0.585943	0.292971	+	0.956121i	$0.405356\pi$
$882$	0	0
$883$	11.4133	0.384090	0.192045	−	0.981386i	$-0.438488\pi$
$883$	11.4133	0.384090	0.192045	+	0.981386i	$0.438488\pi$
$884$	0	0
$885$	−8.09785	−0.272206
$886$	0	0
$887$	−41.5424	−1.39486	−0.697428	−	0.716654i	$-0.745672\pi$
$887$	−41.5424	−1.39486	−0.697428	+	0.716654i	$0.745672\pi$
$888$	0	0
$889$	−69.9438	−2.34584
$890$	0	0
$891$	−5.75514	−0.192805
$892$	0	0
$893$	15.7764	0.527938
$894$	0	0
$895$	75.0810	2.50968
$896$	0	0
$897$	−3.11185	−0.103902
$898$	0	0
$899$	44.8148	1.49466
$900$	0	0
$901$	11.1636	0.371914
$902$	0	0
$903$	0.926377	0.0308279
$904$	0	0
$905$	42.1568	1.40134
$906$	0	0
$907$	−27.9945	−0.929542	−0.464771	−	0.885431i	$-0.653863\pi$
$907$	−27.9945	−0.929542	−0.464771	+	0.885431i	$0.653863\pi$
$908$	0	0
$909$	−39.0404	−1.29489
$910$	0	0
$911$	−43.1485	−1.42957	−0.714787	−	0.699342i	$-0.753477\pi$
$911$	−43.1485	−1.42957	−0.714787	+	0.699342i	$0.753477\pi$
$912$	0	0
$913$	−0.0987301	−0.00326749
$914$	0	0
$915$	14.1487	0.467743
$916$	0	0
$917$	7.54930	0.249300
$918$	0	0
$919$	52.2680	1.72416	0.862082	−	0.506769i	$-0.169160\pi$
$919$	52.2680	1.72416	0.862082	+	0.506769i	$0.169160\pi$
$920$	0	0
$921$	0.424250	0.0139795
$922$	0	0
$923$	−1.37375	−0.0452176
$924$	0	0
$925$	22.5451	0.741277
$926$	0	0
$927$	−18.5520	−0.609326
$928$	0	0
$929$	−30.0632	−0.986341	−0.493171	−	0.869933i	$-0.664162\pi$
$929$	−30.0632	−0.986341	−0.493171	+	0.869933i	$0.664162\pi$
$930$	0	0
$931$	17.7560	0.581930
$932$	0	0
$933$	−14.2284	−0.465817
$934$	0	0
$935$	30.0300	0.982087
$936$	0	0
$937$	23.9857	0.783579	0.391789	−	0.920055i	$-0.371856\pi$
$937$	23.9857	0.783579	0.391789	+	0.920055i	$0.371856\pi$
$938$	0	0
$939$	−17.3872	−0.567408
$940$	0	0
$941$	−13.9444	−0.454573	−0.227287	−	0.973828i	$-0.572985\pi$
$941$	−13.9444	−0.454573	−0.227287	+	0.973828i	$0.572985\pi$
$942$	0	0
$943$	46.8813	1.52666
$944$	0	0
$945$	48.3699	1.57347
$946$	0	0
$947$	−26.9833	−0.876840	−0.438420	−	0.898770i	$-0.644462\pi$
$947$	−26.9833	−0.876840	−0.438420	+	0.898770i	$0.644462\pi$
$948$	0	0
$949$	11.7247	0.380599
$950$	0	0
$951$	4.13422	0.134061
$952$	0	0
$953$	−24.0311	−0.778444	−0.389222	−	0.921144i	$-0.627256\pi$
$953$	−24.0311	−0.778444	−0.389222	+	0.921144i	$0.627256\pi$
$954$	0	0
$955$	−52.8787	−1.71111
$956$	0	0
$957$	−3.45023	−0.111530
$958$	0	0
$959$	33.2386	1.07333
$960$	0	0
$961$	71.9773	2.32185
$962$	0	0
$963$	−2.02165	−0.0651467
$964$	0	0
$965$	16.9854	0.546778
$966$	0	0
$967$	49.3261	1.58622	0.793110	−	0.609079i	$-0.208461\pi$
$967$	49.3261	1.58622	0.793110	+	0.609079i	$0.208461\pi$
$968$	0	0
$969$	17.1455	0.550794
$970$	0	0
$971$	−46.6947	−1.49851	−0.749253	−	0.662284i	$-0.769587\pi$
$971$	−46.6947	−1.49851	−0.749253	+	0.662284i	$0.769587\pi$
$972$	0	0
$973$	−45.3511	−1.45389
$974$	0	0
$975$	5.40857	0.173213
$976$	0	0
$977$	−21.8052	−0.697610	−0.348805	−	0.937195i	$-0.613412\pi$
$977$	−21.8052	−0.697610	−0.348805	+	0.937195i	$0.613412\pi$
$978$	0	0
$979$	19.3241	0.617600
$980$	0	0
$981$	40.8665	1.30477
$982$	0	0
$983$	13.9597	0.445245	0.222623	−	0.974905i	$-0.428538\pi$
$983$	13.9597	0.445245	0.222623	+	0.974905i	$0.428538\pi$
$984$	0	0
$985$	67.5527	2.15241
$986$	0	0
$987$	−10.5408	−0.335516
$988$	0	0
$989$	1.95009	0.0620091
$990$	0	0
$991$	42.9004	1.36278	0.681389	−	0.731922i	$-0.261376\pi$
$991$	42.9004	1.36278	0.681389	+	0.731922i	$0.261376\pi$
$992$	0	0
$993$	18.1756	0.576787
$994$	0	0
$995$	44.5382	1.41196
$996$	0	0
$997$	−0.122631	−0.00388375	−0.00194187	−	0.999998i	$-0.500618\pi$
$997$	−0.122631	−0.00388375	−0.00194187	+	0.999998i	$0.500618\pi$
$998$	0	0
$999$	9.83826	0.311269

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.13		28
4.3	odd	2		4024.2.a.d.1.16	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.16	✓	28	4.3	odd	2
8048.2.a.v.1.13		28	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)

\(f(q)\)	\(=\)	\(q-0.682343 q^{3} +3.69511 q^{5} +3.46637 q^{7} -2.53441 q^{9} +O(q^{10})\)	\(q-0.682343 q^{3} +3.69511 q^{5} +3.46637 q^{7} -2.53441 q^{9} -1.14497 q^{11} -0.915952 q^{13} -2.52133 q^{15} -7.09797 q^{17} +3.54009 q^{19} -2.36525 q^{21} -4.97901 q^{23} +8.65381 q^{25} +3.77636 q^{27} -4.41622 q^{29} -10.1478 q^{31} +0.781263 q^{33} +12.8086 q^{35} +2.60522 q^{37} +0.624993 q^{39} -9.41578 q^{41} -0.391661 q^{43} -9.36491 q^{45} +4.45651 q^{47} +5.01570 q^{49} +4.84325 q^{51} -1.57279 q^{53} -4.23079 q^{55} -2.41555 q^{57} +3.21174 q^{59} -5.61162 q^{61} -8.78519 q^{63} -3.38454 q^{65} +0.0216322 q^{67} +3.39739 q^{69} +1.49981 q^{71} -12.8005 q^{73} -5.90486 q^{75} -3.96889 q^{77} +6.15151 q^{79} +5.02645 q^{81} +0.0862293 q^{83} -26.2278 q^{85} +3.01338 q^{87} -16.8773 q^{89} -3.17503 q^{91} +6.92426 q^{93} +13.0810 q^{95} -10.8940 q^{97} +2.90182 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

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