LMFDB - Embedded newform 4024.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.80686 q^{3} -3.33080 q^{5} -2.85725 q^{7} +4.87847 q^{9} +O(q^{10})$	$q-2.80686 q^{3} -3.33080 q^{5} -2.85725 q^{7} +4.87847 q^{9} -6.16199 q^{11} -3.18285 q^{13} +9.34908 q^{15} -3.44720 q^{17} +6.59579 q^{19} +8.01992 q^{21} +9.22071 q^{23} +6.09421 q^{25} -5.27260 q^{27} +1.40217 q^{29} +0.702160 q^{31} +17.2959 q^{33} +9.51693 q^{35} -9.54930 q^{37} +8.93381 q^{39} -2.82571 q^{41} +8.28009 q^{43} -16.2492 q^{45} +10.2543 q^{47} +1.16390 q^{49} +9.67581 q^{51} -5.03097 q^{53} +20.5243 q^{55} -18.5135 q^{57} -3.57284 q^{59} +2.46580 q^{61} -13.9390 q^{63} +10.6014 q^{65} +0.271159 q^{67} -25.8813 q^{69} -6.18072 q^{71} +13.2210 q^{73} -17.1056 q^{75} +17.6064 q^{77} -7.67343 q^{79} +0.164052 q^{81} +1.85568 q^{83} +11.4819 q^{85} -3.93570 q^{87} -1.46966 q^{89} +9.09420 q^{91} -1.97087 q^{93} -21.9692 q^{95} -0.763796 q^{97} -30.0611 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.80686	−1.62054	−0.810271	−	0.586055i	$-0.800680\pi$
$3$	−2.80686	−1.62054	−0.810271	+	0.586055i	$0.800680\pi$
$4$	0	0
$5$	−3.33080	−1.48958	−0.744789	−	0.667300i	$-0.767450\pi$
$5$	−3.33080	−1.48958	−0.744789	+	0.667300i	$0.767450\pi$
$6$	0	0
$7$	−2.85725	−1.07994	−0.539970	−	0.841684i	$-0.681565\pi$
$7$	−2.85725	−1.07994	−0.539970	+	0.841684i	$0.681565\pi$
$8$	0	0
$9$	4.87847	1.62616
$10$	0	0
$11$	−6.16199	−1.85791	−0.928955	−	0.370192i	$-0.879292\pi$
$11$	−6.16199	−1.85791	−0.928955	+	0.370192i	$0.879292\pi$
$12$	0	0
$13$	−3.18285	−0.882763	−0.441381	−	0.897320i	$-0.645511\pi$
$13$	−3.18285	−0.882763	−0.441381	+	0.897320i	$0.645511\pi$
$14$	0	0
$15$	9.34908	2.41392
$16$	0	0
$17$	−3.44720	−0.836069	−0.418035	−	0.908431i	$-0.637281\pi$
$17$	−3.44720	−0.836069	−0.418035	+	0.908431i	$0.637281\pi$
$18$	0	0
$19$	6.59579	1.51318	0.756589	−	0.653890i	$-0.226864\pi$
$19$	6.59579	1.51318	0.756589	+	0.653890i	$0.226864\pi$
$20$	0	0
$21$	8.01992	1.75009
$22$	0	0
$23$	9.22071	1.92265	0.961326	−	0.275414i	$-0.0888149\pi$
$23$	9.22071	1.92265	0.961326	+	0.275414i	$0.0888149\pi$
$24$	0	0
$25$	6.09421	1.21884
$26$	0	0
$27$	−5.27260	−1.01471
$28$	0	0
$29$	1.40217	0.260377	0.130188	−	0.991489i	$-0.458442\pi$
$29$	1.40217	0.260377	0.130188	+	0.991489i	$0.458442\pi$
$30$	0	0
$31$	0.702160	0.126112	0.0630558	−	0.998010i	$-0.479915\pi$
$31$	0.702160	0.126112	0.0630558	+	0.998010i	$0.479915\pi$
$32$	0	0
$33$	17.2959	3.01082
$34$	0	0
$35$	9.51693	1.60866
$36$	0	0
$37$	−9.54930	−1.56989	−0.784947	−	0.619562i	$-0.787310\pi$
$37$	−9.54930	−1.56989	−0.784947	+	0.619562i	$0.787310\pi$
$38$	0	0
$39$	8.93381	1.43055
$40$	0	0
$41$	−2.82571	−0.441301	−0.220651	−	0.975353i	$-0.570818\pi$
$41$	−2.82571	−0.441301	−0.220651	+	0.975353i	$0.570818\pi$
$42$	0	0
$43$	8.28009	1.26270	0.631351	−	0.775497i	$-0.282501\pi$
$43$	8.28009	1.26270	0.631351	+	0.775497i	$0.282501\pi$
$44$	0	0
$45$	−16.2492	−2.42229
$46$	0	0
$47$	10.2543	1.49574	0.747872	−	0.663842i	$-0.231075\pi$
$47$	10.2543	1.49574	0.747872	+	0.663842i	$0.231075\pi$
$48$	0	0
$49$	1.16390	0.166272
$50$	0	0
$51$	9.67581	1.35488
$52$	0	0
$53$	−5.03097	−0.691057	−0.345529	−	0.938408i	$-0.612300\pi$
$53$	−5.03097	−0.691057	−0.345529	+	0.938408i	$0.612300\pi$
$54$	0	0
$55$	20.5243	2.76750
$56$	0	0
$57$	−18.5135	−2.45217
$58$	0	0
$59$	−3.57284	−0.465145	−0.232572	−	0.972579i	$-0.574714\pi$
$59$	−3.57284	−0.465145	−0.232572	+	0.972579i	$0.574714\pi$
$60$	0	0
$61$	2.46580	0.315713	0.157856	−	0.987462i	$-0.449542\pi$
$61$	2.46580	0.315713	0.157856	+	0.987462i	$0.449542\pi$
$62$	0	0
$63$	−13.9390	−1.75615
$64$	0	0
$65$	10.6014	1.31494
$66$	0	0
$67$	0.271159	0.0331273	0.0165636	−	0.999863i	$-0.494727\pi$
$67$	0.271159	0.0331273	0.0165636	+	0.999863i	$0.494727\pi$
$68$	0	0
$69$	−25.8813	−3.11574
$70$	0	0
$71$	−6.18072	−0.733517	−0.366759	−	0.930316i	$-0.619532\pi$
$71$	−6.18072	−0.733517	−0.366759	+	0.930316i	$0.619532\pi$
$72$	0	0
$73$	13.2210	1.54740	0.773699	−	0.633553i	$-0.218404\pi$
$73$	13.2210	1.54740	0.773699	+	0.633553i	$0.218404\pi$
$74$	0	0
$75$	−17.1056	−1.97518
$76$	0	0
$77$	17.6064	2.00643
$78$	0	0
$79$	−7.67343	−0.863329	−0.431664	−	0.902034i	$-0.642074\pi$
$79$	−7.67343	−0.863329	−0.431664	+	0.902034i	$0.642074\pi$
$80$	0	0
$81$	0.164052	0.0182280
$82$	0	0
$83$	1.85568	0.203688	0.101844	−	0.994800i	$-0.467526\pi$
$83$	1.85568	0.203688	0.101844	+	0.994800i	$0.467526\pi$
$84$	0	0
$85$	11.4819	1.24539
$86$	0	0
$87$	−3.93570	−0.421952
$88$	0	0
$89$	−1.46966	−0.155784	−0.0778920	−	0.996962i	$-0.524819\pi$
$89$	−1.46966	−0.155784	−0.0778920	+	0.996962i	$0.524819\pi$
$90$	0	0
$91$	9.09420	0.953331
$92$	0	0
$93$	−1.97087	−0.204369
$94$	0	0
$95$	−21.9692	−2.25400
$96$	0	0
$97$	−0.763796	−0.0775518	−0.0387759	−	0.999248i	$-0.512346\pi$
$97$	−0.763796	−0.0775518	−0.0387759	+	0.999248i	$0.512346\pi$
$98$	0	0
$99$	−30.0611	−3.02125
$100$	0	0
$101$	4.20829	0.418741	0.209370	−	0.977836i	$-0.432859\pi$
$101$	4.20829	0.418741	0.209370	+	0.977836i	$0.432859\pi$
$102$	0	0
$103$	15.5406	1.53126	0.765632	−	0.643279i	$-0.222427\pi$
$103$	15.5406	1.53126	0.765632	+	0.643279i	$0.222427\pi$
$104$	0	0
$105$	−26.7127	−2.60689
$106$	0	0
$107$	8.77861	0.848660	0.424330	−	0.905508i	$-0.360510\pi$
$107$	8.77861	0.848660	0.424330	+	0.905508i	$0.360510\pi$
$108$	0	0
$109$	−10.4923	−1.00498	−0.502492	−	0.864582i	$-0.667583\pi$
$109$	−10.4923	−1.00498	−0.502492	+	0.864582i	$0.667583\pi$
$110$	0	0
$111$	26.8036	2.54408
$112$	0	0
$113$	−9.15796	−0.861509	−0.430754	−	0.902469i	$-0.641752\pi$
$113$	−9.15796	−0.861509	−0.430754	+	0.902469i	$0.641752\pi$
$114$	0	0
$115$	−30.7123	−2.86394
$116$	0	0
$117$	−15.5274	−1.43551
$118$	0	0
$119$	9.84953	0.902905
$120$	0	0
$121$	26.9701	2.45183
$122$	0	0
$123$	7.93137	0.715147
$124$	0	0
$125$	−3.64458	−0.325981
$126$	0	0
$127$	9.14813	0.811765	0.405883	−	0.913925i	$-0.366964\pi$
$127$	9.14813	0.811765	0.405883	+	0.913925i	$0.366964\pi$
$128$	0	0
$129$	−23.2411	−2.04626
$130$	0	0
$131$	−8.38571	−0.732662	−0.366331	−	0.930485i	$-0.619386\pi$
$131$	−8.38571	−0.732662	−0.366331	+	0.930485i	$0.619386\pi$
$132$	0	0
$133$	−18.8459	−1.63414
$134$	0	0
$135$	17.5620	1.51149
$136$	0	0
$137$	18.8496	1.61043	0.805216	−	0.592981i	$-0.202049\pi$
$137$	18.8496	1.61043	0.805216	+	0.592981i	$0.202049\pi$
$138$	0	0
$139$	−10.0650	−0.853706	−0.426853	−	0.904321i	$-0.640378\pi$
$139$	−10.0650	−0.853706	−0.426853	+	0.904321i	$0.640378\pi$
$140$	0	0
$141$	−28.7824	−2.42392
$142$	0	0
$143$	19.6127	1.64009
$144$	0	0
$145$	−4.67035	−0.387852
$146$	0	0
$147$	−3.26692	−0.269451
$148$	0	0
$149$	−18.5789	−1.52205	−0.761023	−	0.648725i	$-0.775303\pi$
$149$	−18.5789	−1.52205	−0.761023	+	0.648725i	$0.775303\pi$
$150$	0	0
$151$	−6.78604	−0.552240	−0.276120	−	0.961123i	$-0.589049\pi$
$151$	−6.78604	−0.552240	−0.276120	+	0.961123i	$0.589049\pi$
$152$	0	0
$153$	−16.8171	−1.35958
$154$	0	0
$155$	−2.33875	−0.187853
$156$	0	0
$157$	19.7470	1.57598	0.787990	−	0.615688i	$-0.211122\pi$
$157$	19.7470	1.57598	0.787990	+	0.615688i	$0.211122\pi$
$158$	0	0
$159$	14.1212	1.11989
$160$	0	0
$161$	−26.3459	−2.07635
$162$	0	0
$163$	11.4905	0.900005	0.450002	−	0.893027i	$-0.351423\pi$
$163$	11.4905	0.900005	0.450002	+	0.893027i	$0.351423\pi$
$164$	0	0
$165$	−57.6090	−4.48485
$166$	0	0
$167$	−0.914891	−0.0707964	−0.0353982	−	0.999373i	$-0.511270\pi$
$167$	−0.914891	−0.0707964	−0.0353982	+	0.999373i	$0.511270\pi$
$168$	0	0
$169$	−2.86949	−0.220730
$170$	0	0
$171$	32.1774	2.46067
$172$	0	0
$173$	24.8348	1.88815	0.944076	−	0.329727i	$-0.106957\pi$
$173$	24.8348	1.88815	0.944076	+	0.329727i	$0.106957\pi$
$174$	0	0
$175$	−17.4127	−1.31628
$176$	0	0
$177$	10.0285	0.753787
$178$	0	0
$179$	−11.6568	−0.871274	−0.435637	−	0.900123i	$-0.643477\pi$
$179$	−11.6568	−0.871274	−0.435637	+	0.900123i	$0.643477\pi$
$180$	0	0
$181$	17.2535	1.28244	0.641221	−	0.767356i	$-0.278428\pi$
$181$	17.2535	1.28244	0.641221	+	0.767356i	$0.278428\pi$
$182$	0	0
$183$	−6.92115	−0.511626
$184$	0	0
$185$	31.8068	2.33848
$186$	0	0
$187$	21.2416	1.55334
$188$	0	0
$189$	15.0652	1.09583
$190$	0	0
$191$	−24.5481	−1.77624	−0.888119	−	0.459614i	$-0.847988\pi$
$191$	−24.5481	−1.77624	−0.888119	+	0.459614i	$0.847988\pi$
$192$	0	0
$193$	−18.1606	−1.30723	−0.653616	−	0.756826i	$-0.726749\pi$
$193$	−18.1606	−1.30723	−0.653616	+	0.756826i	$0.726749\pi$
$194$	0	0
$195$	−29.7567	−2.13092
$196$	0	0
$197$	−15.6761	−1.11688	−0.558439	−	0.829546i	$-0.688600\pi$
$197$	−15.6761	−1.11688	−0.558439	+	0.829546i	$0.688600\pi$
$198$	0	0
$199$	9.24717	0.655514	0.327757	−	0.944762i	$-0.393707\pi$
$199$	9.24717	0.655514	0.327757	+	0.944762i	$0.393707\pi$
$200$	0	0
$201$	−0.761104	−0.0536842
$202$	0	0
$203$	−4.00636	−0.281192
$204$	0	0
$205$	9.41185	0.657352
$206$	0	0
$207$	44.9830	3.12653
$208$	0	0
$209$	−40.6432	−2.81135
$210$	0	0
$211$	11.9897	0.825403	0.412702	−	0.910866i	$-0.364585\pi$
$211$	11.9897	0.825403	0.412702	+	0.910866i	$0.364585\pi$
$212$	0	0
$213$	17.3484	1.18870
$214$	0	0
$215$	−27.5793	−1.88089
$216$	0	0
$217$	−2.00625	−0.136193
$218$	0	0
$219$	−37.1094	−2.50762
$220$	0	0
$221$	10.9719	0.738051
$222$	0	0
$223$	−8.80822	−0.589842	−0.294921	−	0.955522i	$-0.595293\pi$
$223$	−8.80822	−0.589842	−0.294921	+	0.955522i	$0.595293\pi$
$224$	0	0
$225$	29.7304	1.98203
$226$	0	0
$227$	−12.3011	−0.816454	−0.408227	−	0.912881i	$-0.633853\pi$
$227$	−12.3011	−0.816454	−0.408227	+	0.912881i	$0.633853\pi$
$228$	0	0
$229$	3.49454	0.230926	0.115463	−	0.993312i	$-0.463165\pi$
$229$	3.49454	0.230926	0.115463	+	0.993312i	$0.463165\pi$
$230$	0	0
$231$	−49.4187	−3.25151
$232$	0	0
$233$	−8.16901	−0.535170	−0.267585	−	0.963534i	$-0.586226\pi$
$233$	−8.16901	−0.535170	−0.267585	+	0.963534i	$0.586226\pi$
$234$	0	0
$235$	−34.1550	−2.22803
$236$	0	0
$237$	21.5383	1.39906
$238$	0	0
$239$	−16.2842	−1.05334	−0.526668	−	0.850071i	$-0.676559\pi$
$239$	−16.2842	−1.05334	−0.526668	+	0.850071i	$0.676559\pi$
$240$	0	0
$241$	17.2260	1.10963	0.554813	−	0.831975i	$-0.312790\pi$
$241$	17.2260	1.10963	0.554813	+	0.831975i	$0.312790\pi$
$242$	0	0
$243$	15.3573	0.985173
$244$	0	0
$245$	−3.87673	−0.247675
$246$	0	0
$247$	−20.9934	−1.33578
$248$	0	0
$249$	−5.20864	−0.330084
$250$	0	0
$251$	−6.70328	−0.423107	−0.211553	−	0.977366i	$-0.567852\pi$
$251$	−6.70328	−0.423107	−0.211553	+	0.977366i	$0.567852\pi$
$252$	0	0
$253$	−56.8180	−3.57211
$254$	0	0
$255$	−32.2282	−2.01821
$256$	0	0
$257$	22.3504	1.39418	0.697091	−	0.716983i	$-0.254477\pi$
$257$	22.3504	1.39418	0.697091	+	0.716983i	$0.254477\pi$
$258$	0	0
$259$	27.2848	1.69539
$260$	0	0
$261$	6.84045	0.423414
$262$	0	0
$263$	−13.2208	−0.815229	−0.407614	−	0.913154i	$-0.633639\pi$
$263$	−13.2208	−0.815229	−0.407614	+	0.913154i	$0.633639\pi$
$264$	0	0
$265$	16.7572	1.02938
$266$	0	0
$267$	4.12514	0.252455
$268$	0	0
$269$	−28.2269	−1.72102	−0.860511	−	0.509431i	$-0.829856\pi$
$269$	−28.2269	−1.72102	−0.860511	+	0.509431i	$0.829856\pi$
$270$	0	0
$271$	21.3618	1.29764	0.648818	−	0.760944i	$-0.275264\pi$
$271$	21.3618	1.29764	0.648818	+	0.760944i	$0.275264\pi$
$272$	0	0
$273$	−25.5262	−1.54491
$274$	0	0
$275$	−37.5524	−2.26450
$276$	0	0
$277$	−26.8538	−1.61349	−0.806744	−	0.590902i	$-0.798772\pi$
$277$	−26.8538	−1.61349	−0.806744	+	0.590902i	$0.798772\pi$
$278$	0	0
$279$	3.42547	0.205077
$280$	0	0
$281$	25.1142	1.49819	0.749093	−	0.662465i	$-0.230490\pi$
$281$	25.1142	1.49819	0.749093	+	0.662465i	$0.230490\pi$
$282$	0	0
$283$	25.7681	1.53175	0.765877	−	0.642987i	$-0.222305\pi$
$283$	25.7681	1.53175	0.765877	+	0.642987i	$0.222305\pi$
$284$	0	0
$285$	61.6646	3.65270
$286$	0	0
$287$	8.07376	0.476579
$288$	0	0
$289$	−5.11681	−0.300989
$290$	0	0
$291$	2.14387	0.125676
$292$	0	0
$293$	18.2273	1.06485	0.532424	−	0.846478i	$-0.321281\pi$
$293$	18.2273	1.06485	0.532424	+	0.846478i	$0.321281\pi$
$294$	0	0
$295$	11.9004	0.692869
$296$	0	0
$297$	32.4897	1.88524
$298$	0	0
$299$	−29.3481	−1.69724
$300$	0	0
$301$	−23.6583	−1.36364
$302$	0	0
$303$	−11.8121	−0.678587
$304$	0	0
$305$	−8.21306	−0.470279
$306$	0	0
$307$	−4.85756	−0.277236	−0.138618	−	0.990346i	$-0.544266\pi$
$307$	−4.85756	−0.277236	−0.138618	+	0.990346i	$0.544266\pi$
$308$	0	0
$309$	−43.6204	−2.48148
$310$	0	0
$311$	−17.1372	−0.971763	−0.485882	−	0.874025i	$-0.661501\pi$
$311$	−17.1372	−0.971763	−0.485882	+	0.874025i	$0.661501\pi$
$312$	0	0
$313$	7.12125	0.402517	0.201258	−	0.979538i	$-0.435497\pi$
$313$	7.12125	0.402517	0.201258	+	0.979538i	$0.435497\pi$
$314$	0	0
$315$	46.4281	2.61593
$316$	0	0
$317$	−30.5055	−1.71336	−0.856680	−	0.515848i	$-0.827477\pi$
$317$	−30.5055	−1.71336	−0.856680	+	0.515848i	$0.827477\pi$
$318$	0	0
$319$	−8.64017	−0.483757
$320$	0	0
$321$	−24.6403	−1.37529
$322$	0	0
$323$	−22.7370	−1.26512
$324$	0	0
$325$	−19.3969	−1.07595
$326$	0	0
$327$	29.4505	1.62862
$328$	0	0
$329$	−29.2992	−1.61532
$330$	0	0
$331$	−7.25577	−0.398813	−0.199407	−	0.979917i	$-0.563901\pi$
$331$	−7.25577	−0.398813	−0.199407	+	0.979917i	$0.563901\pi$
$332$	0	0
$333$	−46.5860	−2.55289
$334$	0	0
$335$	−0.903174	−0.0493457
$336$	0	0
$337$	1.67495	0.0912405	0.0456203	−	0.998959i	$-0.485474\pi$
$337$	1.67495	0.0912405	0.0456203	+	0.998959i	$0.485474\pi$
$338$	0	0
$339$	25.7051	1.39611
$340$	0	0
$341$	−4.32670	−0.234304
$342$	0	0
$343$	16.6752	0.900377
$344$	0	0
$345$	86.2052	4.64113
$346$	0	0
$347$	13.7470	0.737979	0.368989	−	0.929434i	$-0.379704\pi$
$347$	13.7470	0.737979	0.368989	+	0.929434i	$0.379704\pi$
$348$	0	0
$349$	−18.1465	−0.971360	−0.485680	−	0.874137i	$-0.661428\pi$
$349$	−18.1465	−0.971360	−0.485680	+	0.874137i	$0.661428\pi$
$350$	0	0
$351$	16.7819	0.895750
$352$	0	0
$353$	−22.2901	−1.18638	−0.593191	−	0.805061i	$-0.702132\pi$
$353$	−22.2901	−1.18638	−0.593191	+	0.805061i	$0.702132\pi$
$354$	0	0
$355$	20.5867	1.09263
$356$	0	0
$357$	−27.6463	−1.46320
$358$	0	0
$359$	18.1349	0.957125	0.478563	−	0.878053i	$-0.341158\pi$
$359$	18.1349	0.957125	0.478563	+	0.878053i	$0.341158\pi$
$360$	0	0
$361$	24.5045	1.28971
$362$	0	0
$363$	−75.7014	−3.97329
$364$	0	0
$365$	−44.0364	−2.30497
$366$	0	0
$367$	−31.0147	−1.61896	−0.809478	−	0.587150i	$-0.800250\pi$
$367$	−31.0147	−1.61896	−0.809478	+	0.587150i	$0.800250\pi$
$368$	0	0
$369$	−13.7851	−0.717625
$370$	0	0
$371$	14.3748	0.746301
$372$	0	0
$373$	25.7975	1.33574	0.667871	−	0.744277i	$-0.267206\pi$
$373$	25.7975	1.33574	0.667871	+	0.744277i	$0.267206\pi$
$374$	0	0
$375$	10.2298	0.528265
$376$	0	0
$377$	−4.46290	−0.229851
$378$	0	0
$379$	7.02969	0.361091	0.180545	−	0.983567i	$-0.442214\pi$
$379$	7.02969	0.361091	0.180545	+	0.983567i	$0.442214\pi$
$380$	0	0
$381$	−25.6775	−1.31550
$382$	0	0
$383$	4.24162	0.216737	0.108368	−	0.994111i	$-0.465437\pi$
$383$	4.24162	0.216737	0.108368	+	0.994111i	$0.465437\pi$
$384$	0	0
$385$	−58.6433	−2.98874
$386$	0	0
$387$	40.3942	2.05335
$388$	0	0
$389$	3.56789	0.180899	0.0904497	−	0.995901i	$-0.471170\pi$
$389$	3.56789	0.180899	0.0904497	+	0.995901i	$0.471170\pi$
$390$	0	0
$391$	−31.7856	−1.60747
$392$	0	0
$393$	23.5375	1.18731
$394$	0	0
$395$	25.5586	1.28599
$396$	0	0
$397$	−16.3759	−0.821883	−0.410941	−	0.911662i	$-0.634800\pi$
$397$	−16.3759	−0.821883	−0.410941	+	0.911662i	$0.634800\pi$
$398$	0	0
$399$	52.8977	2.64820
$400$	0	0
$401$	6.06998	0.303120	0.151560	−	0.988448i	$-0.451570\pi$
$401$	6.06998	0.303120	0.151560	+	0.988448i	$0.451570\pi$
$402$	0	0
$403$	−2.23487	−0.111327
$404$	0	0
$405$	−0.546425	−0.0271521
$406$	0	0
$407$	58.8427	2.91672
$408$	0	0
$409$	30.2379	1.49517	0.747584	−	0.664167i	$-0.231214\pi$
$409$	30.2379	1.49517	0.747584	+	0.664167i	$0.231214\pi$
$410$	0	0
$411$	−52.9083	−2.60977
$412$	0	0
$413$	10.2085	0.502329
$414$	0	0
$415$	−6.18090	−0.303408
$416$	0	0
$417$	28.2512	1.38347
$418$	0	0
$419$	2.93225	0.143250	0.0716249	−	0.997432i	$-0.477182\pi$
$419$	2.93225	0.143250	0.0716249	+	0.997432i	$0.477182\pi$
$420$	0	0
$421$	6.43693	0.313717	0.156858	−	0.987621i	$-0.449863\pi$
$421$	6.43693	0.313717	0.156858	+	0.987621i	$0.449863\pi$
$422$	0	0
$423$	50.0253	2.43231
$424$	0	0
$425$	−21.0080	−1.01904
$426$	0	0
$427$	−7.04541	−0.340951
$428$	0	0
$429$	−55.0500	−2.65784
$430$	0	0
$431$	34.4701	1.66037	0.830184	−	0.557490i	$-0.188235\pi$
$431$	34.4701	1.66037	0.830184	+	0.557490i	$0.188235\pi$
$432$	0	0
$433$	4.07058	0.195620	0.0978098	−	0.995205i	$-0.468816\pi$
$433$	4.07058	0.195620	0.0978098	+	0.995205i	$0.468816\pi$
$434$	0	0
$435$	13.1090	0.628530
$436$	0	0
$437$	60.8179	2.90932
$438$	0	0
$439$	9.95459	0.475107	0.237553	−	0.971374i	$-0.423655\pi$
$439$	9.95459	0.475107	0.237553	+	0.971374i	$0.423655\pi$
$440$	0	0
$441$	5.67807	0.270384
$442$	0	0
$443$	17.5544	0.834036	0.417018	−	0.908898i	$-0.363075\pi$
$443$	17.5544	0.834036	0.417018	+	0.908898i	$0.363075\pi$
$444$	0	0
$445$	4.89515	0.232052
$446$	0	0
$447$	52.1485	2.46654
$448$	0	0
$449$	27.8811	1.31579	0.657896	−	0.753109i	$-0.271447\pi$
$449$	27.8811	1.31579	0.657896	+	0.753109i	$0.271447\pi$
$450$	0	0
$451$	17.4120	0.819898
$452$	0	0
$453$	19.0475	0.894928
$454$	0	0
$455$	−30.2909	−1.42006
$456$	0	0
$457$	−21.1029	−0.987152	−0.493576	−	0.869703i	$-0.664311\pi$
$457$	−21.1029	−0.987152	−0.493576	+	0.869703i	$0.664311\pi$
$458$	0	0
$459$	18.1757	0.848370
$460$	0	0
$461$	−30.4681	−1.41904	−0.709520	−	0.704685i	$-0.751088\pi$
$461$	−30.4681	−1.41904	−0.709520	+	0.704685i	$0.751088\pi$
$462$	0	0
$463$	31.6310	1.47002	0.735008	−	0.678058i	$-0.237178\pi$
$463$	31.6310	1.47002	0.735008	+	0.678058i	$0.237178\pi$
$464$	0	0
$465$	6.56455	0.304424
$466$	0	0
$467$	−2.78847	−0.129035	−0.0645174	−	0.997917i	$-0.520551\pi$
$467$	−2.78847	−0.129035	−0.0645174	+	0.997917i	$0.520551\pi$
$468$	0	0
$469$	−0.774769	−0.0357755
$470$	0	0
$471$	−55.4270	−2.55394
$472$	0	0
$473$	−51.0219	−2.34599
$474$	0	0
$475$	40.1961	1.84432
$476$	0	0
$477$	−24.5435	−1.12377
$478$	0	0
$479$	−0.186867	−0.00853819	−0.00426910	−	0.999991i	$-0.501359\pi$
$479$	−0.186867	−0.00853819	−0.00426910	+	0.999991i	$0.501359\pi$
$480$	0	0
$481$	30.3939	1.38584
$482$	0	0
$483$	73.9493	3.36481
$484$	0	0
$485$	2.54405	0.115519
$486$	0	0
$487$	−15.6009	−0.706943	−0.353472	−	0.935445i	$-0.614999\pi$
$487$	−15.6009	−0.706943	−0.353472	+	0.935445i	$0.614999\pi$
$488$	0	0
$489$	−32.2522	−1.45850
$490$	0	0
$491$	35.6367	1.60826	0.804130	−	0.594453i	$-0.202631\pi$
$491$	35.6367	1.60826	0.804130	+	0.594453i	$0.202631\pi$
$492$	0	0
$493$	−4.83357	−0.217693
$494$	0	0
$495$	100.127	4.50039
$496$	0	0
$497$	17.6599	0.792155
$498$	0	0
$499$	25.9141	1.16007	0.580037	−	0.814590i	$-0.303038\pi$
$499$	25.9141	1.16007	0.580037	+	0.814590i	$0.303038\pi$
$500$	0	0
$501$	2.56797	0.114728
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−14.0170	−0.623747
$506$	0	0
$507$	8.05426	0.357702
$508$	0	0
$509$	43.1469	1.91245	0.956227	−	0.292627i	$-0.0945293\pi$
$509$	43.1469	1.91245	0.956227	+	0.292627i	$0.0945293\pi$
$510$	0	0
$511$	−37.7757	−1.67110
$512$	0	0
$513$	−34.7770	−1.53544
$514$	0	0
$515$	−51.7627	−2.28094
$516$	0	0
$517$	−63.1870	−2.77896
$518$	0	0
$519$	−69.7078	−3.05983
$520$	0	0
$521$	−22.9789	−1.00673	−0.503363	−	0.864075i	$-0.667904\pi$
$521$	−22.9789	−1.00673	−0.503363	+	0.864075i	$0.667904\pi$
$522$	0	0
$523$	−2.68716	−0.117501	−0.0587506	−	0.998273i	$-0.518712\pi$
$523$	−2.68716	−0.117501	−0.0587506	+	0.998273i	$0.518712\pi$
$524$	0	0
$525$	48.8750	2.13308
$526$	0	0
$527$	−2.42049	−0.105438
$528$	0	0
$529$	62.0215	2.69659
$530$	0	0
$531$	−17.4300	−0.756398
$532$	0	0
$533$	8.99379	0.389564
$534$	0	0
$535$	−29.2398	−1.26415
$536$	0	0
$537$	32.7192	1.41194
$538$	0	0
$539$	−7.17197	−0.308919
$540$	0	0
$541$	13.7406	0.590754	0.295377	−	0.955381i	$-0.404555\pi$
$541$	13.7406	0.590754	0.295377	+	0.955381i	$0.404555\pi$
$542$	0	0
$543$	−48.4281	−2.07825
$544$	0	0
$545$	34.9478	1.49700
$546$	0	0
$547$	38.1975	1.63321	0.816604	−	0.577199i	$-0.195854\pi$
$547$	38.1975	1.63321	0.816604	+	0.577199i	$0.195854\pi$
$548$	0	0
$549$	12.0293	0.513398
$550$	0	0
$551$	9.24844	0.393997
$552$	0	0
$553$	21.9250	0.932344
$554$	0	0
$555$	−89.2772	−3.78961
$556$	0	0
$557$	−34.0835	−1.44416	−0.722082	−	0.691807i	$-0.756815\pi$
$557$	−34.0835	−1.44416	−0.722082	+	0.691807i	$0.756815\pi$
$558$	0	0
$559$	−26.3543	−1.11467
$560$	0	0
$561$	−59.6223	−2.51725
$562$	0	0
$563$	19.5491	0.823896	0.411948	−	0.911207i	$-0.364849\pi$
$563$	19.5491	0.823896	0.411948	+	0.911207i	$0.364849\pi$
$564$	0	0
$565$	30.5033	1.28328
$566$	0	0
$567$	−0.468739	−0.0196852
$568$	0	0
$569$	16.6019	0.695987	0.347994	−	0.937497i	$-0.386863\pi$
$569$	16.6019	0.695987	0.347994	+	0.937497i	$0.386863\pi$
$570$	0	0
$571$	13.2433	0.554213	0.277107	−	0.960839i	$-0.410624\pi$
$571$	13.2433	0.554213	0.277107	+	0.960839i	$0.410624\pi$
$572$	0	0
$573$	68.9031	2.87847
$574$	0	0
$575$	56.1929	2.34341
$576$	0	0
$577$	−24.5290	−1.02116	−0.510578	−	0.859831i	$-0.670569\pi$
$577$	−24.5290	−1.02116	−0.510578	+	0.859831i	$0.670569\pi$
$578$	0	0
$579$	50.9744	2.11842
$580$	0	0
$581$	−5.30216	−0.219971
$582$	0	0
$583$	31.0008	1.28392
$584$	0	0
$585$	51.7187	2.13830
$586$	0	0
$587$	−28.2759	−1.16707	−0.583535	−	0.812088i	$-0.698331\pi$
$587$	−28.2759	−1.16707	−0.583535	+	0.812088i	$0.698331\pi$
$588$	0	0
$589$	4.63130	0.190829
$590$	0	0
$591$	44.0007	1.80995
$592$	0	0
$593$	−15.9323	−0.654260	−0.327130	−	0.944979i	$-0.606082\pi$
$593$	−15.9323	−0.654260	−0.327130	+	0.944979i	$0.606082\pi$
$594$	0	0
$595$	−32.8068	−1.34495
$596$	0	0
$597$	−25.9555	−1.06229
$598$	0	0
$599$	−40.9491	−1.67313	−0.836567	−	0.547865i	$-0.815441\pi$
$599$	−40.9491	−1.67313	−0.836567	+	0.547865i	$0.815441\pi$
$600$	0	0
$601$	−39.2764	−1.60212	−0.801058	−	0.598587i	$-0.795729\pi$
$601$	−39.2764	−1.60212	−0.801058	+	0.598587i	$0.795729\pi$
$602$	0	0
$603$	1.32284	0.0538701
$604$	0	0
$605$	−89.8320	−3.65219
$606$	0	0
$607$	−4.80079	−0.194858	−0.0974290	−	0.995242i	$-0.531062\pi$
$607$	−4.80079	−0.194858	−0.0974290	+	0.995242i	$0.531062\pi$
$608$	0	0
$609$	11.2453	0.455683
$610$	0	0
$611$	−32.6379	−1.32039
$612$	0	0
$613$	8.56017	0.345742	0.172871	−	0.984944i	$-0.444696\pi$
$613$	8.56017	0.345742	0.172871	+	0.984944i	$0.444696\pi$
$614$	0	0
$615$	−26.4178	−1.06527
$616$	0	0
$617$	−21.4421	−0.863228	−0.431614	−	0.902058i	$-0.642056\pi$
$617$	−21.4421	−0.863228	−0.431614	+	0.902058i	$0.642056\pi$
$618$	0	0
$619$	11.7723	0.473167	0.236584	−	0.971611i	$-0.423972\pi$
$619$	11.7723	0.473167	0.236584	+	0.971611i	$0.423972\pi$
$620$	0	0
$621$	−48.6171	−1.95094
$622$	0	0
$623$	4.19920	0.168238
$624$	0	0
$625$	−18.3317	−0.733268
$626$	0	0
$627$	114.080	4.55591
$628$	0	0
$629$	32.9184	1.31254
$630$	0	0
$631$	−36.7998	−1.46498	−0.732489	−	0.680779i	$-0.761641\pi$
$631$	−36.7998	−1.46498	−0.732489	+	0.680779i	$0.761641\pi$
$632$	0	0
$633$	−33.6534	−1.33760
$634$	0	0
$635$	−30.4706	−1.20919
$636$	0	0
$637$	−3.70453	−0.146779
$638$	0	0
$639$	−30.1525	−1.19281
$640$	0	0
$641$	35.1857	1.38975	0.694877	−	0.719129i	$-0.255459\pi$
$641$	35.1857	1.38975	0.694877	+	0.719129i	$0.255459\pi$
$642$	0	0
$643$	22.3621	0.881874	0.440937	−	0.897538i	$-0.354646\pi$
$643$	22.3621	0.881874	0.440937	+	0.897538i	$0.354646\pi$
$644$	0	0
$645$	77.4113	3.04807
$646$	0	0
$647$	−12.3698	−0.486308	−0.243154	−	0.969988i	$-0.578182\pi$
$647$	−12.3698	−0.486308	−0.243154	+	0.969988i	$0.578182\pi$
$648$	0	0
$649$	22.0158	0.864197
$650$	0	0
$651$	5.63126	0.220707
$652$	0	0
$653$	4.19397	0.164123	0.0820613	−	0.996627i	$-0.473850\pi$
$653$	4.19397	0.164123	0.0820613	+	0.996627i	$0.473850\pi$
$654$	0	0
$655$	27.9311	1.09136
$656$	0	0
$657$	64.4981	2.51631
$658$	0	0
$659$	−18.2209	−0.709784	−0.354892	−	0.934907i	$-0.615482\pi$
$659$	−18.2209	−0.709784	−0.354892	+	0.934907i	$0.615482\pi$
$660$	0	0
$661$	−21.2704	−0.827324	−0.413662	−	0.910430i	$-0.635751\pi$
$661$	−21.2704	−0.827324	−0.413662	+	0.910430i	$0.635751\pi$
$662$	0	0
$663$	−30.7966	−1.19604
$664$	0	0
$665$	62.7717	2.43418
$666$	0	0
$667$	12.9290	0.500614
$668$	0	0
$669$	24.7235	0.955864
$670$	0	0
$671$	−15.1942	−0.586566
$672$	0	0
$673$	−15.5656	−0.600011	−0.300005	−	0.953938i	$-0.596988\pi$
$673$	−15.5656	−0.600011	−0.300005	+	0.953938i	$0.596988\pi$
$674$	0	0
$675$	−32.1323	−1.23677
$676$	0	0
$677$	−5.63298	−0.216493	−0.108247	−	0.994124i	$-0.534524\pi$
$677$	−5.63298	−0.216493	−0.108247	+	0.994124i	$0.534524\pi$
$678$	0	0
$679$	2.18236	0.0837513
$680$	0	0
$681$	34.5275	1.32310
$682$	0	0
$683$	−3.67984	−0.140805	−0.0704026	−	0.997519i	$-0.522428\pi$
$683$	−3.67984	−0.140805	−0.0704026	+	0.997519i	$0.522428\pi$
$684$	0	0
$685$	−62.7843	−2.39886
$686$	0	0
$687$	−9.80868	−0.374225
$688$	0	0
$689$	16.0128	0.610040
$690$	0	0
$691$	−6.37801	−0.242631	−0.121316	−	0.992614i	$-0.538711\pi$
$691$	−6.37801	−0.242631	−0.121316	+	0.992614i	$0.538711\pi$
$692$	0	0
$693$	85.8922	3.26277
$694$	0	0
$695$	33.5246	1.27166
$696$	0	0
$697$	9.74078	0.368958
$698$	0	0
$699$	22.9293	0.867265
$700$	0	0
$701$	24.3301	0.918936	0.459468	−	0.888194i	$-0.348040\pi$
$701$	24.3301	0.918936	0.459468	+	0.888194i	$0.348040\pi$
$702$	0	0
$703$	−62.9852	−2.37553
$704$	0	0
$705$	95.8684	3.61061
$706$	0	0
$707$	−12.0242	−0.452215
$708$	0	0
$709$	−18.8738	−0.708822	−0.354411	−	0.935090i	$-0.615319\pi$
$709$	−18.8738	−0.708822	−0.354411	+	0.935090i	$0.615319\pi$
$710$	0	0
$711$	−37.4346	−1.40391
$712$	0	0
$713$	6.47441	0.242469
$714$	0	0
$715$	−65.3258	−2.44305
$716$	0	0
$717$	45.7074	1.70697
$718$	0	0
$719$	45.6619	1.70290	0.851450	−	0.524435i	$-0.175724\pi$
$719$	45.6619	1.70290	0.851450	+	0.524435i	$0.175724\pi$
$720$	0	0
$721$	−44.4035	−1.65367
$722$	0	0
$723$	−48.3511	−1.79820
$724$	0	0
$725$	8.54513	0.317358
$726$	0	0
$727$	38.1174	1.41370	0.706849	−	0.707365i	$-0.250116\pi$
$727$	38.1174	1.41370	0.706849	+	0.707365i	$0.250116\pi$
$728$	0	0
$729$	−43.5981	−1.61474
$730$	0	0
$731$	−28.5431	−1.05571
$732$	0	0
$733$	−3.54071	−0.130779	−0.0653895	−	0.997860i	$-0.520829\pi$
$733$	−3.54071	−0.130779	−0.0653895	+	0.997860i	$0.520829\pi$
$734$	0	0
$735$	10.8814	0.401368
$736$	0	0
$737$	−1.67088	−0.0615475
$738$	0	0
$739$	−19.2057	−0.706495	−0.353247	−	0.935530i	$-0.614923\pi$
$739$	−19.2057	−0.706495	−0.353247	+	0.935530i	$0.614923\pi$
$740$	0	0
$741$	58.9255	2.16468
$742$	0	0
$743$	−6.96467	−0.255509	−0.127755	−	0.991806i	$-0.540777\pi$
$743$	−6.96467	−0.255509	−0.127755	+	0.991806i	$0.540777\pi$
$744$	0	0
$745$	61.8827	2.26721
$746$	0	0
$747$	9.05289	0.331228
$748$	0	0
$749$	−25.0827	−0.916503
$750$	0	0
$751$	−12.2850	−0.448288	−0.224144	−	0.974556i	$-0.571959\pi$
$751$	−12.2850	−0.448288	−0.224144	+	0.974556i	$0.571959\pi$
$752$	0	0
$753$	18.8152	0.685663
$754$	0	0
$755$	22.6029	0.822604
$756$	0	0
$757$	−13.5894	−0.493915	−0.246958	−	0.969026i	$-0.579431\pi$
$757$	−13.5894	−0.493915	−0.246958	+	0.969026i	$0.579431\pi$
$758$	0	0
$759$	159.480	5.78876
$760$	0	0
$761$	−39.8241	−1.44362	−0.721811	−	0.692090i	$-0.756690\pi$
$761$	−39.8241	−1.44362	−0.721811	+	0.692090i	$0.756690\pi$
$762$	0	0
$763$	29.9793	1.08532
$764$	0	0
$765$	56.0142	2.02520
$766$	0	0
$767$	11.3718	0.410612
$768$	0	0
$769$	−27.0475	−0.975358	−0.487679	−	0.873023i	$-0.662156\pi$
$769$	−27.0475	−0.975358	−0.487679	+	0.873023i	$0.662156\pi$
$770$	0	0
$771$	−62.7346	−2.25933
$772$	0	0
$773$	−15.0712	−0.542073	−0.271036	−	0.962569i	$-0.587366\pi$
$773$	−15.0712	−0.542073	−0.271036	+	0.962569i	$0.587366\pi$
$774$	0	0
$775$	4.27911	0.153710
$776$	0	0
$777$	−76.5846	−2.74746
$778$	0	0
$779$	−18.6378	−0.667768
$780$	0	0
$781$	38.0856	1.36281
$782$	0	0
$783$	−7.39310	−0.264208
$784$	0	0
$785$	−65.7732	−2.34755
$786$	0	0
$787$	−28.1114	−1.00206	−0.501032	−	0.865429i	$-0.667046\pi$
$787$	−28.1114	−1.00206	−0.501032	+	0.865429i	$0.667046\pi$
$788$	0	0
$789$	37.1089	1.32111
$790$	0	0
$791$	26.1666	0.930378
$792$	0	0
$793$	−7.84825	−0.278699
$794$	0	0
$795$	−47.0350	−1.66816
$796$	0	0
$797$	−13.8844	−0.491810	−0.245905	−	0.969294i	$-0.579085\pi$
$797$	−13.8844	−0.491810	−0.245905	+	0.969294i	$0.579085\pi$
$798$	0	0
$799$	−35.3487	−1.25055
$800$	0	0
$801$	−7.16971	−0.253329
$802$	0	0
$803$	−81.4675	−2.87493
$804$	0	0
$805$	87.7529	3.09288
$806$	0	0
$807$	79.2289	2.78899
$808$	0	0
$809$	35.5748	1.25074	0.625371	−	0.780327i	$-0.284948\pi$
$809$	35.5748	1.25074	0.625371	+	0.780327i	$0.284948\pi$
$810$	0	0
$811$	23.0995	0.811133	0.405567	−	0.914065i	$-0.367074\pi$
$811$	23.0995	0.811133	0.405567	+	0.914065i	$0.367074\pi$
$812$	0	0
$813$	−59.9595	−2.10287
$814$	0	0
$815$	−38.2725	−1.34063
$816$	0	0
$817$	54.6138	1.91069
$818$	0	0
$819$	44.3658	1.55027
$820$	0	0
$821$	−21.0963	−0.736265	−0.368132	−	0.929773i	$-0.620003\pi$
$821$	−21.0963	−0.736265	−0.368132	+	0.929773i	$0.620003\pi$
$822$	0	0
$823$	28.7224	1.00120	0.500599	−	0.865679i	$-0.333113\pi$
$823$	28.7224	1.00120	0.500599	+	0.865679i	$0.333113\pi$
$824$	0	0
$825$	105.404	3.66971
$826$	0	0
$827$	−27.4860	−0.955782	−0.477891	−	0.878419i	$-0.658599\pi$
$827$	−27.4860	−0.955782	−0.477891	+	0.878419i	$0.658599\pi$
$828$	0	0
$829$	25.5174	0.886255	0.443127	−	0.896459i	$-0.353869\pi$
$829$	25.5174	0.886255	0.443127	+	0.896459i	$0.353869\pi$
$830$	0	0
$831$	75.3748	2.61472
$832$	0	0
$833$	−4.01221	−0.139015
$834$	0	0
$835$	3.04731	0.105457
$836$	0	0
$837$	−3.70221	−0.127967
$838$	0	0
$839$	−15.1427	−0.522784	−0.261392	−	0.965233i	$-0.584181\pi$
$839$	−15.1427	−0.522784	−0.261392	+	0.965233i	$0.584181\pi$
$840$	0	0
$841$	−27.0339	−0.932204
$842$	0	0
$843$	−70.4920	−2.42787
$844$	0	0
$845$	9.55769	0.328795
$846$	0	0
$847$	−77.0605	−2.64783
$848$	0	0
$849$	−72.3275	−2.48227
$850$	0	0
$851$	−88.0513	−3.01836
$852$	0	0
$853$	11.9554	0.409346	0.204673	−	0.978830i	$-0.434387\pi$
$853$	11.9554	0.409346	0.204673	+	0.978830i	$0.434387\pi$
$854$	0	0
$855$	−107.176	−3.66535
$856$	0	0
$857$	9.53741	0.325792	0.162896	−	0.986643i	$-0.447917\pi$
$857$	9.53741	0.325792	0.162896	+	0.986643i	$0.447917\pi$
$858$	0	0
$859$	21.9007	0.747243	0.373621	−	0.927581i	$-0.378116\pi$
$859$	21.9007	0.747243	0.373621	+	0.927581i	$0.378116\pi$
$860$	0	0
$861$	−22.6619	−0.772317
$862$	0	0
$863$	−7.82525	−0.266375	−0.133187	−	0.991091i	$-0.542521\pi$
$863$	−7.82525	−0.266375	−0.133187	+	0.991091i	$0.542521\pi$
$864$	0	0
$865$	−82.7196	−2.81255
$866$	0	0
$867$	14.3622	0.487765
$868$	0	0
$869$	47.2836	1.60399
$870$	0	0
$871$	−0.863056	−0.0292435
$872$	0	0
$873$	−3.72616	−0.126111
$874$	0	0
$875$	10.4135	0.352040
$876$	0	0
$877$	22.9618	0.775365	0.387682	−	0.921793i	$-0.373276\pi$
$877$	22.9618	0.775365	0.387682	+	0.921793i	$0.373276\pi$
$878$	0	0
$879$	−51.1614	−1.72563
$880$	0	0
$881$	13.0701	0.440343	0.220171	−	0.975461i	$-0.429338\pi$
$881$	13.0701	0.440343	0.220171	+	0.975461i	$0.429338\pi$
$882$	0	0
$883$	5.64794	0.190068	0.0950341	−	0.995474i	$-0.469704\pi$
$883$	5.64794	0.190068	0.0950341	+	0.995474i	$0.469704\pi$
$884$	0	0
$885$	−33.4028	−1.12282
$886$	0	0
$887$	2.03708	0.0683985	0.0341992	−	0.999415i	$-0.489112\pi$
$887$	2.03708	0.0683985	0.0341992	+	0.999415i	$0.489112\pi$
$888$	0	0
$889$	−26.1385	−0.876658
$890$	0	0
$891$	−1.01089	−0.0338661
$892$	0	0
$893$	67.6353	2.26333
$894$	0	0
$895$	38.8266	1.29783
$896$	0	0
$897$	82.3761	2.75046
$898$	0	0
$899$	0.984549	0.0328366
$900$	0	0
$901$	17.3428	0.577772
$902$	0	0
$903$	66.4057	2.20984
$904$	0	0
$905$	−57.4679	−1.91030
$906$	0	0
$907$	49.0974	1.63025	0.815127	−	0.579282i	$-0.196667\pi$
$907$	49.0974	1.63025	0.815127	+	0.579282i	$0.196667\pi$
$908$	0	0
$909$	20.5300	0.680938
$910$	0	0
$911$	−23.9150	−0.792340	−0.396170	−	0.918177i	$-0.629661\pi$
$911$	−23.9150	−0.792340	−0.396170	+	0.918177i	$0.629661\pi$
$912$	0	0
$913$	−11.4347	−0.378433
$914$	0	0
$915$	23.0529	0.762106
$916$	0	0
$917$	23.9601	0.791232
$918$	0	0
$919$	−27.0185	−0.891260	−0.445630	−	0.895217i	$-0.647020\pi$
$919$	−27.0185	−0.891260	−0.445630	+	0.895217i	$0.647020\pi$
$920$	0	0
$921$	13.6345	0.449272
$922$	0	0
$923$	19.6723	0.647522
$924$	0	0
$925$	−58.1954	−1.91345
$926$	0	0
$927$	75.8145	2.49007
$928$	0	0
$929$	17.8916	0.587003	0.293501	−	0.955959i	$-0.405179\pi$
$929$	17.8916	0.587003	0.293501	+	0.955959i	$0.405179\pi$
$930$	0	0
$931$	7.67687	0.251599
$932$	0	0
$933$	48.1018	1.57478
$934$	0	0
$935$	−70.7515	−2.31382
$936$	0	0
$937$	−18.2903	−0.597519	−0.298760	−	0.954328i	$-0.596573\pi$
$937$	−18.2903	−0.597519	−0.298760	+	0.954328i	$0.596573\pi$
$938$	0	0
$939$	−19.9884	−0.652295
$940$	0	0
$941$	16.2134	0.528543	0.264272	−	0.964448i	$-0.414869\pi$
$941$	16.2134	0.528543	0.264272	+	0.964448i	$0.414869\pi$
$942$	0	0
$943$	−26.0550	−0.848469
$944$	0	0
$945$	−50.1790	−1.63232
$946$	0	0
$947$	−6.25401	−0.203228	−0.101614	−	0.994824i	$-0.532401\pi$
$947$	−6.25401	−0.203228	−0.101614	+	0.994824i	$0.532401\pi$
$948$	0	0
$949$	−42.0803	−1.36599
$950$	0	0
$951$	85.6247	2.77657
$952$	0	0
$953$	−18.2420	−0.590916	−0.295458	−	0.955356i	$-0.595472\pi$
$953$	−18.2420	−0.590916	−0.295458	+	0.955356i	$0.595472\pi$
$954$	0	0
$955$	81.7647	2.64584
$956$	0	0
$957$	24.2518	0.783948
$958$	0	0
$959$	−53.8582	−1.73917
$960$	0	0
$961$	−30.5070	−0.984096
$962$	0	0
$963$	42.8262	1.38005
$964$	0	0
$965$	60.4894	1.94722
$966$	0	0
$967$	−6.35525	−0.204371	−0.102186	−	0.994765i	$-0.532584\pi$
$967$	−6.35525	−0.204371	−0.102186	+	0.994765i	$0.532584\pi$
$968$	0	0
$969$	63.8197	2.05018
$970$	0	0
$971$	−56.2895	−1.80642	−0.903208	−	0.429204i	$-0.858794\pi$
$971$	−56.2895	−1.80642	−0.903208	+	0.429204i	$0.858794\pi$
$972$	0	0
$973$	28.7584	0.921952
$974$	0	0
$975$	54.4445	1.74362
$976$	0	0
$977$	51.3434	1.64262	0.821310	−	0.570482i	$-0.193243\pi$
$977$	51.3434	1.64262	0.821310	+	0.570482i	$0.193243\pi$
$978$	0	0
$979$	9.05606	0.289433
$980$	0	0
$981$	−51.1865	−1.63426
$982$	0	0
$983$	−13.8340	−0.441235	−0.220617	−	0.975360i	$-0.570807\pi$
$983$	−13.8340	−0.441235	−0.220617	+	0.975360i	$0.570807\pi$
$984$	0	0
$985$	52.2140	1.66368
$986$	0	0
$987$	82.2387	2.61769
$988$	0	0
$989$	76.3484	2.42774
$990$	0	0
$991$	7.41841	0.235653	0.117827	−	0.993034i	$-0.462407\pi$
$991$	7.41841	0.235653	0.117827	+	0.993034i	$0.462407\pi$
$992$	0	0
$993$	20.3659	0.646293
$994$	0	0
$995$	−30.8004	−0.976439
$996$	0	0
$997$	−33.9777	−1.07608	−0.538042	−	0.842918i	$-0.680836\pi$
$997$	−33.9777	−1.07608	−0.538042	+	0.842918i	$0.680836\pi$
$998$	0	0
$999$	50.3496	1.59299

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.d.1.3	✓	28
4.3	odd	2		8048.2.a.v.1.26		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.3	✓	28	1.1	even	1	trivial
8048.2.a.v.1.26		28	4.3	odd	2

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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.80686 q^{3} -3.33080 q^{5} -2.85725 q^{7} +4.87847 q^{9} +O(q^{10})\)	\(q-2.80686 q^{3} -3.33080 q^{5} -2.85725 q^{7} +4.87847 q^{9} -6.16199 q^{11} -3.18285 q^{13} +9.34908 q^{15} -3.44720 q^{17} +6.59579 q^{19} +8.01992 q^{21} +9.22071 q^{23} +6.09421 q^{25} -5.27260 q^{27} +1.40217 q^{29} +0.702160 q^{31} +17.2959 q^{33} +9.51693 q^{35} -9.54930 q^{37} +8.93381 q^{39} -2.82571 q^{41} +8.28009 q^{43} -16.2492 q^{45} +10.2543 q^{47} +1.16390 q^{49} +9.67581 q^{51} -5.03097 q^{53} +20.5243 q^{55} -18.5135 q^{57} -3.57284 q^{59} +2.46580 q^{61} -13.9390 q^{63} +10.6014 q^{65} +0.271159 q^{67} -25.8813 q^{69} -6.18072 q^{71} +13.2210 q^{73} -17.1056 q^{75} +17.6064 q^{77} -7.67343 q^{79} +0.164052 q^{81} +1.85568 q^{83} +11.4819 q^{85} -3.93570 q^{87} -1.46966 q^{89} +9.09420 q^{91} -1.97087 q^{93} -21.9692 q^{95} -0.763796 q^{97} -30.0611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more