LMFDB - Embedded newform 8048.2.a.n.1.2

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:	$5$
Coefficient field:	5.5.36497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 $ x^5 - 2x^4 - 3x^3 + 5*x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.55629$ of defining polynomial
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)

$f(q)$	$=$	$q-0.522539 q^{3} +0.813221 q^{5} +3.14341 q^{7} -2.72695 q^{9} +O(q^{10})$	$q-0.522539 q^{3} +0.813221 q^{5} +3.14341 q^{7} -2.72695 q^{9} +0.935416 q^{11} +6.27767 q^{13} -0.424940 q^{15} -4.52254 q^{17} -3.63511 q^{19} -1.64256 q^{21} +2.08440 q^{23} -4.33867 q^{25} +2.99256 q^{27} -7.17238 q^{29} +3.77203 q^{31} -0.488792 q^{33} +2.55629 q^{35} -8.05157 q^{37} -3.28033 q^{39} +0.104212 q^{41} -0.412877 q^{43} -2.21762 q^{45} -9.24902 q^{47} +2.88102 q^{49} +2.36320 q^{51} -4.95783 q^{53} +0.760700 q^{55} +1.89949 q^{57} -1.98123 q^{59} -11.8824 q^{61} -8.57193 q^{63} +5.10513 q^{65} -4.21544 q^{67} -1.08918 q^{69} +13.4960 q^{71} -11.3749 q^{73} +2.26713 q^{75} +2.94040 q^{77} -14.7420 q^{79} +6.61713 q^{81} +7.06816 q^{83} -3.67782 q^{85} +3.74785 q^{87} -7.76652 q^{89} +19.7333 q^{91} -1.97103 q^{93} -2.95615 q^{95} +11.1338 q^{97} -2.55084 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{5} - 3 q^{7} - 3 q^{9}+O(q^{10}) $ 5 * q - q^5 - 3 * q^7 - 3 * q^9	$ 5 q - q^{5} - 3 q^{7} - 3 q^{9} + 11 q^{11} + 5 q^{13} + 8 q^{15} - 20 q^{17} + 4 q^{19} - 4 q^{21} + 4 q^{23} - 6 q^{25} - 3 q^{27} - 6 q^{29} + 3 q^{31} - 7 q^{33} + 3 q^{35} - 10 q^{37} - 4 q^{39} - 6 q^{41} - 11 q^{43} - 17 q^{45} + 9 q^{47} - 4 q^{49} + 12 q^{51} - 22 q^{53} - 14 q^{55} + 10 q^{57} + 10 q^{59} - 11 q^{61} + 3 q^{63} - 12 q^{65} - 14 q^{67} - 3 q^{69} + 26 q^{71} - 7 q^{73} - 15 q^{75} - 26 q^{77} + 15 q^{79} - 7 q^{81} + 12 q^{83} + 12 q^{85} + 18 q^{87} - 5 q^{89} + 22 q^{91} - 21 q^{93} + 10 q^{95} + 6 q^{97} - 8 q^{99}+O(q^{100}) $ 5 * q - q^5 - 3 * q^7 - 3 * q^9 + 11 * q^11 + 5 * q^13 + 8 * q^15 - 20 * q^17 + 4 * q^19 - 4 * q^21 + 4 * q^23 - 6 * q^25 - 3 * q^27 - 6 * q^29 + 3 * q^31 - 7 * q^33 + 3 * q^35 - 10 * q^37 - 4 * q^39 - 6 * q^41 - 11 * q^43 - 17 * q^45 + 9 * q^47 - 4 * q^49 + 12 * q^51 - 22 * q^53 - 14 * q^55 + 10 * q^57 + 10 * q^59 - 11 * q^61 + 3 * q^63 - 12 * q^65 - 14 * q^67 - 3 * q^69 + 26 * q^71 - 7 * q^73 - 15 * q^75 - 26 * q^77 + 15 * q^79 - 7 * q^81 + 12 * q^83 + 12 * q^85 + 18 * q^87 - 5 * q^89 + 22 * q^91 - 21 * q^93 + 10 * q^95 + 6 * q^97 - 8 * 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
$3$	−0.522539	−0.301688	−0.150844	−	0.988558i	$-0.548199\pi$
$3$	−0.522539	−0.301688	−0.150844	+	0.988558i	$0.548199\pi$
$4$	0	0
$5$	0.813221	0.363683	0.181842	−	0.983328i	$-0.441794\pi$
$5$	0.813221	0.363683	0.181842	+	0.983328i	$0.441794\pi$
$6$	0	0
$7$	3.14341	1.18810	0.594049	−	0.804429i	$-0.297529\pi$
$7$	3.14341	1.18810	0.594049	+	0.804429i	$0.297529\pi$
$8$	0	0
$9$	−2.72695	−0.908984
$10$	0	0
$11$	0.935416	0.282039	0.141019	−	0.990007i	$-0.454962\pi$
$11$	0.935416	0.282039	0.141019	+	0.990007i	$0.454962\pi$
$12$	0	0
$13$	6.27767	1.74111	0.870556	−	0.492070i	$-0.163759\pi$
$13$	6.27767	1.74111	0.870556	+	0.492070i	$0.163759\pi$
$14$	0	0
$15$	−0.424940	−0.109719
$16$	0	0
$17$	−4.52254	−1.09688	−0.548438	−	0.836191i	$-0.684777\pi$
$17$	−4.52254	−1.09688	−0.548438	+	0.836191i	$0.684777\pi$
$18$	0	0
$19$	−3.63511	−0.833952	−0.416976	−	0.908917i	$-0.636910\pi$
$19$	−3.63511	−0.833952	−0.416976	+	0.908917i	$0.636910\pi$
$20$	0	0
$21$	−1.64256	−0.358435
$22$	0	0
$23$	2.08440	0.434627	0.217313	−	0.976102i	$-0.430271\pi$
$23$	2.08440	0.434627	0.217313	+	0.976102i	$0.430271\pi$
$24$	0	0
$25$	−4.33867	−0.867734
$26$	0	0
$27$	2.99256	0.575918
$28$	0	0
$29$	−7.17238	−1.33188	−0.665938	−	0.746007i	$-0.731969\pi$
$29$	−7.17238	−1.33188	−0.665938	+	0.746007i	$0.731969\pi$
$30$	0	0
$31$	3.77203	0.677477	0.338738	−	0.940881i	$-0.390000\pi$
$31$	3.77203	0.677477	0.338738	+	0.940881i	$0.390000\pi$
$32$	0	0
$33$	−0.488792	−0.0850877
$34$	0	0
$35$	2.55629	0.432091
$36$	0	0
$37$	−8.05157	−1.32367	−0.661835	−	0.749650i	$-0.730222\pi$
$37$	−8.05157	−1.32367	−0.661835	+	0.749650i	$0.730222\pi$
$38$	0	0
$39$	−3.28033	−0.525273
$40$	0	0
$41$	0.104212	0.0162751	0.00813755	−	0.999967i	$-0.497410\pi$
$41$	0.104212	0.0162751	0.00813755	+	0.999967i	$0.497410\pi$
$42$	0	0
$43$	−0.412877	−0.0629632	−0.0314816	−	0.999504i	$-0.510023\pi$
$43$	−0.412877	−0.0629632	−0.0314816	+	0.999504i	$0.510023\pi$
$44$	0	0
$45$	−2.21762	−0.330583
$46$	0	0
$47$	−9.24902	−1.34911	−0.674554	−	0.738225i	$-0.735664\pi$
$47$	−9.24902	−1.34911	−0.674554	+	0.738225i	$0.735664\pi$
$48$	0	0
$49$	2.88102	0.411575
$50$	0	0
$51$	2.36320	0.330915
$52$	0	0
$53$	−4.95783	−0.681011	−0.340505	−	0.940243i	$-0.610598\pi$
$53$	−4.95783	−0.681011	−0.340505	+	0.940243i	$0.610598\pi$
$54$	0	0
$55$	0.760700	0.102573
$56$	0	0
$57$	1.89949	0.251594
$58$	0	0
$59$	−1.98123	−0.257934	−0.128967	−	0.991649i	$-0.541166\pi$
$59$	−1.98123	−0.257934	−0.128967	+	0.991649i	$0.541166\pi$
$60$	0	0
$61$	−11.8824	−1.52139	−0.760694	−	0.649110i	$-0.775141\pi$
$61$	−11.8824	−1.52139	−0.760694	+	0.649110i	$0.775141\pi$
$62$	0	0
$63$	−8.57193	−1.07996
$64$	0	0
$65$	5.10513	0.633214
$66$	0	0
$67$	−4.21544	−0.514997	−0.257499	−	0.966279i	$-0.582898\pi$
$67$	−4.21544	−0.514997	−0.257499	+	0.966279i	$0.582898\pi$
$68$	0	0
$69$	−1.08918	−0.131122
$70$	0	0
$71$	13.4960	1.60168	0.800841	−	0.598877i	$-0.204386\pi$
$71$	13.4960	1.60168	0.800841	+	0.598877i	$0.204386\pi$
$72$	0	0
$73$	−11.3749	−1.33133	−0.665667	−	0.746249i	$-0.731853\pi$
$73$	−11.3749	−1.33133	−0.665667	+	0.746249i	$0.731853\pi$
$74$	0	0
$75$	2.26713	0.261785
$76$	0	0
$77$	2.94040	0.335089
$78$	0	0
$79$	−14.7420	−1.65860	−0.829300	−	0.558804i	$-0.811260\pi$
$79$	−14.7420	−1.65860	−0.829300	+	0.558804i	$0.811260\pi$
$80$	0	0
$81$	6.61713	0.735237
$82$	0	0
$83$	7.06816	0.775832	0.387916	−	0.921695i	$-0.373195\pi$
$83$	7.06816	0.775832	0.387916	+	0.921695i	$0.373195\pi$
$84$	0	0
$85$	−3.67782	−0.398916
$86$	0	0
$87$	3.74785	0.401811
$88$	0	0
$89$	−7.76652	−0.823250	−0.411625	−	0.911353i	$-0.635039\pi$
$89$	−7.76652	−0.823250	−0.411625	+	0.911353i	$0.635039\pi$
$90$	0	0
$91$	19.7333	2.06861
$92$	0	0
$93$	−1.97103	−0.204387
$94$	0	0
$95$	−2.95615	−0.303295
$96$	0	0
$97$	11.1338	1.13046	0.565232	−	0.824932i	$-0.308786\pi$
$97$	11.1338	1.13046	0.565232	+	0.824932i	$0.308786\pi$
$98$	0	0
$99$	−2.55084	−0.256369
$100$	0	0
$101$	−9.86429	−0.981534	−0.490767	−	0.871291i	$-0.663283\pi$
$101$	−9.86429	−0.981534	−0.490767	+	0.871291i	$0.663283\pi$
$102$	0	0
$103$	7.33892	0.723126	0.361563	−	0.932348i	$-0.382243\pi$
$103$	7.33892	0.723126	0.361563	+	0.932348i	$0.382243\pi$
$104$	0	0
$105$	−1.33576	−0.130357
$106$	0	0
$107$	−1.78748	−0.172802	−0.0864009	−	0.996260i	$-0.527537\pi$
$107$	−1.78748	−0.172802	−0.0864009	+	0.996260i	$0.527537\pi$
$108$	0	0
$109$	1.54188	0.147686	0.0738428	−	0.997270i	$-0.476474\pi$
$109$	1.54188	0.147686	0.0738428	+	0.997270i	$0.476474\pi$
$110$	0	0
$111$	4.20726	0.399336
$112$	0	0
$113$	15.5871	1.46631	0.733155	−	0.680061i	$-0.238047\pi$
$113$	15.5871	1.46631	0.733155	+	0.680061i	$0.238047\pi$
$114$	0	0
$115$	1.69508	0.158067
$116$	0	0
$117$	−17.1189	−1.58264
$118$	0	0
$119$	−14.2162	−1.30320
$120$	0	0
$121$	−10.1250	−0.920454
$122$	0	0
$123$	−0.0544546	−0.00491001
$124$	0	0
$125$	−7.59440	−0.679264
$126$	0	0
$127$	−9.83548	−0.872757	−0.436379	−	0.899763i	$-0.643739\pi$
$127$	−9.83548	−0.872757	−0.436379	+	0.899763i	$0.643739\pi$
$128$	0	0
$129$	0.215744	0.0189952
$130$	0	0
$131$	10.0639	0.879291	0.439645	−	0.898171i	$-0.355104\pi$
$131$	10.0639	0.879291	0.439645	+	0.898171i	$0.355104\pi$
$132$	0	0
$133$	−11.4266	−0.990816
$134$	0	0
$135$	2.43361	0.209452
$136$	0	0
$137$	−2.95002	−0.252037	−0.126019	−	0.992028i	$-0.540220\pi$
$137$	−2.95002	−0.252037	−0.126019	+	0.992028i	$0.540220\pi$
$138$	0	0
$139$	−5.88956	−0.499546	−0.249773	−	0.968304i	$-0.580356\pi$
$139$	−5.88956	−0.499546	−0.249773	+	0.968304i	$0.580356\pi$
$140$	0	0
$141$	4.83298	0.407010
$142$	0	0
$143$	5.87223	0.491061
$144$	0	0
$145$	−5.83273	−0.484382
$146$	0	0
$147$	−1.50545	−0.124167
$148$	0	0
$149$	−1.57392	−0.128941	−0.0644703	−	0.997920i	$-0.520536\pi$
$149$	−1.57392	−0.128941	−0.0644703	+	0.997920i	$0.520536\pi$
$150$	0	0
$151$	−18.2020	−1.48125	−0.740627	−	0.671916i	$-0.765471\pi$
$151$	−18.2020	−1.48125	−0.740627	+	0.671916i	$0.765471\pi$
$152$	0	0
$153$	12.3328	0.997044
$154$	0	0
$155$	3.06750	0.246387
$156$	0	0
$157$	−6.36868	−0.508276	−0.254138	−	0.967168i	$-0.581792\pi$
$157$	−6.36868	−0.508276	−0.254138	+	0.967168i	$0.581792\pi$
$158$	0	0
$159$	2.59066	0.205453
$160$	0	0
$161$	6.55212	0.516379
$162$	0	0
$163$	−19.6461	−1.53880	−0.769400	−	0.638767i	$-0.779445\pi$
$163$	−19.6461	−1.53880	−0.769400	+	0.638767i	$0.779445\pi$
$164$	0	0
$165$	−0.397496	−0.0309450
$166$	0	0
$167$	14.1270	1.09318	0.546591	−	0.837400i	$-0.315925\pi$
$167$	14.1270	1.09318	0.546591	+	0.837400i	$0.315925\pi$
$168$	0	0
$169$	26.4091	2.03147
$170$	0	0
$171$	9.91278	0.758049
$172$	0	0
$173$	8.56018	0.650818	0.325409	−	0.945573i	$-0.394498\pi$
$173$	8.56018	0.650818	0.325409	+	0.945573i	$0.394498\pi$
$174$	0	0
$175$	−13.6382	−1.03095
$176$	0	0
$177$	1.03527	0.0778156
$178$	0	0
$179$	6.04290	0.451667	0.225834	−	0.974166i	$-0.427489\pi$
$179$	6.04290	0.451667	0.225834	+	0.974166i	$0.427489\pi$
$180$	0	0
$181$	16.5698	1.23162	0.615810	−	0.787895i	$-0.288829\pi$
$181$	16.5698	1.23162	0.615810	+	0.787895i	$0.288829\pi$
$182$	0	0
$183$	6.20903	0.458985
$184$	0	0
$185$	−6.54771	−0.481397
$186$	0	0
$187$	−4.23046	−0.309362
$188$	0	0
$189$	9.40684	0.684247
$190$	0	0
$191$	−0.168957	−0.0122253	−0.00611263	−	0.999981i	$-0.501946\pi$
$191$	−0.168957	−0.0122253	−0.00611263	+	0.999981i	$0.501946\pi$
$192$	0	0
$193$	5.62723	0.405057	0.202528	−	0.979276i	$-0.435084\pi$
$193$	5.62723	0.405057	0.202528	+	0.979276i	$0.435084\pi$
$194$	0	0
$195$	−2.66763	−0.191033
$196$	0	0
$197$	15.9688	1.13773	0.568864	−	0.822431i	$-0.307383\pi$
$197$	15.9688	1.13773	0.568864	+	0.822431i	$0.307383\pi$
$198$	0	0
$199$	−17.5515	−1.24420	−0.622098	−	0.782940i	$-0.713719\pi$
$199$	−17.5515	−1.24420	−0.622098	+	0.782940i	$0.713719\pi$
$200$	0	0
$201$	2.20273	0.155369
$202$	0	0
$203$	−22.5457	−1.58240
$204$	0	0
$205$	0.0847470	0.00591899
$206$	0	0
$207$	−5.68405	−0.395069
$208$	0	0
$209$	−3.40034	−0.235207
$210$	0	0
$211$	20.5889	1.41740	0.708700	−	0.705510i	$-0.249282\pi$
$211$	20.5889	1.41740	0.708700	+	0.705510i	$0.249282\pi$
$212$	0	0
$213$	−7.05220	−0.483209
$214$	0	0
$215$	−0.335760	−0.0228987
$216$	0	0
$217$	11.8570	0.804908
$218$	0	0
$219$	5.94384	0.401648
$220$	0	0
$221$	−28.3910	−1.90979
$222$	0	0
$223$	4.38001	0.293307	0.146654	−	0.989188i	$-0.453150\pi$
$223$	4.38001	0.293307	0.146654	+	0.989188i	$0.453150\pi$
$224$	0	0
$225$	11.8314	0.788757
$226$	0	0
$227$	25.8731	1.71726	0.858628	−	0.512599i	$-0.171317\pi$
$227$	25.8731	1.71726	0.858628	+	0.512599i	$0.171317\pi$
$228$	0	0
$229$	−14.6201	−0.966123	−0.483061	−	0.875586i	$-0.660475\pi$
$229$	−14.6201	−0.966123	−0.483061	+	0.875586i	$0.660475\pi$
$230$	0	0
$231$	−1.53647	−0.101092
$232$	0	0
$233$	−17.8827	−1.17154	−0.585768	−	0.810478i	$-0.699207\pi$
$233$	−17.8827	−1.17154	−0.585768	+	0.810478i	$0.699207\pi$
$234$	0	0
$235$	−7.52150	−0.490648
$236$	0	0
$237$	7.70325	0.500380
$238$	0	0
$239$	2.21652	0.143375	0.0716873	−	0.997427i	$-0.477162\pi$
$239$	2.21652	0.143375	0.0716873	+	0.997427i	$0.477162\pi$
$240$	0	0
$241$	−2.40173	−0.154709	−0.0773546	−	0.997004i	$-0.524647\pi$
$241$	−2.40173	−0.154709	−0.0773546	+	0.997004i	$0.524647\pi$
$242$	0	0
$243$	−12.4354	−0.797730
$244$	0	0
$245$	2.34291	0.149683
$246$	0	0
$247$	−22.8200	−1.45200
$248$	0	0
$249$	−3.69339	−0.234059
$250$	0	0
$251$	2.82570	0.178357	0.0891783	−	0.996016i	$-0.471576\pi$
$251$	2.82570	0.178357	0.0891783	+	0.996016i	$0.471576\pi$
$252$	0	0
$253$	1.94978	0.122582
$254$	0	0
$255$	1.92181	0.120348
$256$	0	0
$257$	12.8910	0.804117	0.402058	−	0.915614i	$-0.368295\pi$
$257$	12.8910	0.804117	0.402058	+	0.915614i	$0.368295\pi$
$258$	0	0
$259$	−25.3094	−1.57265
$260$	0	0
$261$	19.5587	1.21065
$262$	0	0
$263$	19.4774	1.20103	0.600515	−	0.799613i	$-0.294962\pi$
$263$	19.4774	1.20103	0.600515	+	0.799613i	$0.294962\pi$
$264$	0	0
$265$	−4.03181	−0.247672
$266$	0	0
$267$	4.05831	0.248365
$268$	0	0
$269$	−30.6652	−1.86969	−0.934845	−	0.355055i	$-0.884462\pi$
$269$	−30.6652	−1.86969	−0.934845	+	0.355055i	$0.884462\pi$
$270$	0	0
$271$	−7.27064	−0.441660	−0.220830	−	0.975312i	$-0.570877\pi$
$271$	−7.27064	−0.441660	−0.220830	+	0.975312i	$0.570877\pi$
$272$	0	0
$273$	−10.3114	−0.624075
$274$	0	0
$275$	−4.05846	−0.244735
$276$	0	0
$277$	13.8174	0.830209	0.415104	−	0.909774i	$-0.363745\pi$
$277$	13.8174	0.830209	0.415104	+	0.909774i	$0.363745\pi$
$278$	0	0
$279$	−10.2862	−0.615816
$280$	0	0
$281$	10.7396	0.640672	0.320336	−	0.947304i	$-0.396204\pi$
$281$	10.7396	0.640672	0.320336	+	0.947304i	$0.396204\pi$
$282$	0	0
$283$	−18.4761	−1.09829	−0.549145	−	0.835727i	$-0.685046\pi$
$283$	−18.4761	−1.09829	−0.549145	+	0.835727i	$0.685046\pi$
$284$	0	0
$285$	1.54470	0.0915004
$286$	0	0
$287$	0.327580	0.0193364
$288$	0	0
$289$	3.45336	0.203139
$290$	0	0
$291$	−5.81784	−0.341048
$292$	0	0
$293$	22.6917	1.32567	0.662833	−	0.748767i	$-0.269354\pi$
$293$	22.6917	1.32567	0.662833	+	0.748767i	$0.269354\pi$
$294$	0	0
$295$	−1.61118	−0.0938062
$296$	0	0
$297$	2.79929	0.162431
$298$	0	0
$299$	13.0852	0.756734
$300$	0	0
$301$	−1.29784	−0.0748064
$302$	0	0
$303$	5.15448	0.296117
$304$	0	0
$305$	−9.66304	−0.553304
$306$	0	0
$307$	−16.6848	−0.952251	−0.476126	−	0.879377i	$-0.657959\pi$
$307$	−16.6848	−0.952251	−0.476126	+	0.879377i	$0.657959\pi$
$308$	0	0
$309$	−3.83488	−0.218158
$310$	0	0
$311$	−18.9279	−1.07330	−0.536652	−	0.843804i	$-0.680311\pi$
$311$	−18.9279	−1.07330	−0.536652	+	0.843804i	$0.680311\pi$
$312$	0	0
$313$	18.6899	1.05642	0.528208	−	0.849115i	$-0.322864\pi$
$313$	18.6899	1.05642	0.528208	+	0.849115i	$0.322864\pi$
$314$	0	0
$315$	−6.97087	−0.392764
$316$	0	0
$317$	31.8168	1.78701	0.893504	−	0.449054i	$-0.148239\pi$
$317$	31.8168	1.78701	0.893504	+	0.449054i	$0.148239\pi$
$318$	0	0
$319$	−6.70916	−0.375641
$320$	0	0
$321$	0.934026	0.0521323
$322$	0	0
$323$	16.4399	0.914743
$324$	0	0
$325$	−27.2367	−1.51082
$326$	0	0
$327$	−0.805694	−0.0445550
$328$	0	0
$329$	−29.0735	−1.60287
$330$	0	0
$331$	13.4665	0.740184	0.370092	−	0.928995i	$-0.379326\pi$
$331$	13.4665	0.740184	0.370092	+	0.928995i	$0.379326\pi$
$332$	0	0
$333$	21.9563	1.20320
$334$	0	0
$335$	−3.42808	−0.187296
$336$	0	0
$337$	−27.9427	−1.52214	−0.761068	−	0.648672i	$-0.775325\pi$
$337$	−27.9427	−1.52214	−0.761068	+	0.648672i	$0.775325\pi$
$338$	0	0
$339$	−8.14487	−0.442369
$340$	0	0
$341$	3.52842	0.191075
$342$	0	0
$343$	−12.9476	−0.699106
$344$	0	0
$345$	−0.885744	−0.0476868
$346$	0	0
$347$	−3.49880	−0.187825	−0.0939127	−	0.995580i	$-0.529937\pi$
$347$	−3.49880	−0.187825	−0.0939127	+	0.995580i	$0.529937\pi$
$348$	0	0
$349$	−2.41464	−0.129253	−0.0646263	−	0.997910i	$-0.520586\pi$
$349$	−2.41464	−0.129253	−0.0646263	+	0.997910i	$0.520586\pi$
$350$	0	0
$351$	18.7863	1.00274
$352$	0	0
$353$	−6.00091	−0.319396	−0.159698	−	0.987166i	$-0.551052\pi$
$353$	−6.00091	−0.319396	−0.159698	+	0.987166i	$0.551052\pi$
$354$	0	0
$355$	10.9752	0.582505
$356$	0	0
$357$	7.42852	0.393159
$358$	0	0
$359$	−10.6230	−0.560663	−0.280332	−	0.959903i	$-0.590444\pi$
$359$	−10.6230	−0.560663	−0.280332	+	0.959903i	$0.590444\pi$
$360$	0	0
$361$	−5.78595	−0.304524
$362$	0	0
$363$	5.29071	0.277690
$364$	0	0
$365$	−9.25032	−0.484184
$366$	0	0
$367$	−23.3767	−1.22025	−0.610127	−	0.792304i	$-0.708881\pi$
$367$	−23.3767	−1.22025	−0.610127	+	0.792304i	$0.708881\pi$
$368$	0	0
$369$	−0.284180	−0.0147938
$370$	0	0
$371$	−15.5845	−0.809107
$372$	0	0
$373$	4.27846	0.221530	0.110765	−	0.993847i	$-0.464670\pi$
$373$	4.27846	0.221530	0.110765	+	0.993847i	$0.464670\pi$
$374$	0	0
$375$	3.96837	0.204926
$376$	0	0
$377$	−45.0258	−2.31895
$378$	0	0
$379$	−13.4590	−0.691343	−0.345671	−	0.938356i	$-0.612349\pi$
$379$	−13.4590	−0.691343	−0.345671	+	0.938356i	$0.612349\pi$
$380$	0	0
$381$	5.13942	0.263301
$382$	0	0
$383$	−12.2710	−0.627020	−0.313510	−	0.949585i	$-0.601505\pi$
$383$	−12.2710	−0.627020	−0.313510	+	0.949585i	$0.601505\pi$
$384$	0	0
$385$	2.39119	0.121866
$386$	0	0
$387$	1.12590	0.0572325
$388$	0	0
$389$	24.3325	1.23371	0.616854	−	0.787078i	$-0.288407\pi$
$389$	24.3325	1.23371	0.616854	+	0.787078i	$0.288407\pi$
$390$	0	0
$391$	−9.42677	−0.476732
$392$	0	0
$393$	−5.25880	−0.265272
$394$	0	0
$395$	−11.9885	−0.603205
$396$	0	0
$397$	13.5772	0.681420	0.340710	−	0.940168i	$-0.389333\pi$
$397$	13.5772	0.681420	0.340710	+	0.940168i	$0.389333\pi$
$398$	0	0
$399$	5.97087	0.298918
$400$	0	0
$401$	−11.3626	−0.567421	−0.283711	−	0.958910i	$-0.591566\pi$
$401$	−11.3626	−0.567421	−0.283711	+	0.958910i	$0.591566\pi$
$402$	0	0
$403$	23.6796	1.17956
$404$	0	0
$405$	5.38119	0.267393
$406$	0	0
$407$	−7.53157	−0.373326
$408$	0	0
$409$	−24.8484	−1.22867	−0.614337	−	0.789044i	$-0.710576\pi$
$409$	−24.8484	−1.22867	−0.614337	+	0.789044i	$0.710576\pi$
$410$	0	0
$411$	1.54150	0.0760366
$412$	0	0
$413$	−6.22781	−0.306450
$414$	0	0
$415$	5.74798	0.282157
$416$	0	0
$417$	3.07753	0.150707
$418$	0	0
$419$	10.4331	0.509689	0.254844	−	0.966982i	$-0.417976\pi$
$419$	10.4331	0.509689	0.254844	+	0.966982i	$0.417976\pi$
$420$	0	0
$421$	3.33127	0.162356	0.0811782	−	0.996700i	$-0.474132\pi$
$421$	3.33127	0.162356	0.0811782	+	0.996700i	$0.474132\pi$
$422$	0	0
$423$	25.2216	1.22632
$424$	0	0
$425$	19.6218	0.951798
$426$	0	0
$427$	−37.3513	−1.80756
$428$	0	0
$429$	−3.06847	−0.148147
$430$	0	0
$431$	31.1223	1.49911	0.749554	−	0.661943i	$-0.230268\pi$
$431$	31.1223	1.49911	0.749554	+	0.661943i	$0.230268\pi$
$432$	0	0
$433$	−33.6809	−1.61860	−0.809301	−	0.587395i	$-0.800154\pi$
$433$	−33.6809	−1.61860	−0.809301	+	0.587395i	$0.800154\pi$
$434$	0	0
$435$	3.04783	0.146132
$436$	0	0
$437$	−7.57702	−0.362458
$438$	0	0
$439$	−9.72891	−0.464336	−0.232168	−	0.972676i	$-0.574582\pi$
$439$	−9.72891	−0.464336	−0.232168	+	0.972676i	$0.574582\pi$
$440$	0	0
$441$	−7.85642	−0.374115
$442$	0	0
$443$	−36.9189	−1.75407	−0.877035	−	0.480427i	$-0.840482\pi$
$443$	−36.9189	−1.75407	−0.877035	+	0.480427i	$0.840482\pi$
$444$	0	0
$445$	−6.31590	−0.299402
$446$	0	0
$447$	0.822436	0.0388999
$448$	0	0
$449$	−32.4703	−1.53237	−0.766185	−	0.642620i	$-0.777847\pi$
$449$	−32.4703	−1.53237	−0.766185	+	0.642620i	$0.777847\pi$
$450$	0	0
$451$	0.0974812	0.00459021
$452$	0	0
$453$	9.51123	0.446877
$454$	0	0
$455$	16.0475	0.752319
$456$	0	0
$457$	9.95036	0.465458	0.232729	−	0.972542i	$-0.425234\pi$
$457$	9.95036	0.465458	0.232729	+	0.972542i	$0.425234\pi$
$458$	0	0
$459$	−13.5340	−0.631711
$460$	0	0
$461$	−1.65594	−0.0771246	−0.0385623	−	0.999256i	$-0.512278\pi$
$461$	−1.65594	−0.0771246	−0.0385623	+	0.999256i	$0.512278\pi$
$462$	0	0
$463$	−36.4531	−1.69412	−0.847059	−	0.531498i	$-0.821629\pi$
$463$	−36.4531	−1.69412	−0.847059	+	0.531498i	$0.821629\pi$
$464$	0	0
$465$	−1.60289	−0.0743321
$466$	0	0
$467$	−3.67241	−0.169939	−0.0849694	−	0.996384i	$-0.527079\pi$
$467$	−3.67241	−0.169939	−0.0849694	+	0.996384i	$0.527079\pi$
$468$	0	0
$469$	−13.2508	−0.611867
$470$	0	0
$471$	3.32788	0.153341
$472$	0	0
$473$	−0.386212	−0.0177580
$474$	0	0
$475$	15.7716	0.723649
$476$	0	0
$477$	13.5198	0.619028
$478$	0	0
$479$	19.1667	0.875748	0.437874	−	0.899036i	$-0.355732\pi$
$479$	19.1667	0.875748	0.437874	+	0.899036i	$0.355732\pi$
$480$	0	0
$481$	−50.5451	−2.30466
$482$	0	0
$483$	−3.42374	−0.155785
$484$	0	0
$485$	9.05423	0.411131
$486$	0	0
$487$	36.7939	1.66729	0.833646	−	0.552298i	$-0.186249\pi$
$487$	36.7939	1.66729	0.833646	+	0.552298i	$0.186249\pi$
$488$	0	0
$489$	10.2659	0.464238
$490$	0	0
$491$	−13.9423	−0.629205	−0.314603	−	0.949223i	$-0.601871\pi$
$491$	−13.9423	−0.629205	−0.314603	+	0.949223i	$0.601871\pi$
$492$	0	0
$493$	32.4373	1.46090
$494$	0	0
$495$	−2.07439	−0.0932371
$496$	0	0
$497$	42.4235	1.90295
$498$	0	0
$499$	12.0514	0.539496	0.269748	−	0.962931i	$-0.413060\pi$
$499$	12.0514	0.539496	0.269748	+	0.962931i	$0.413060\pi$
$500$	0	0
$501$	−7.38192	−0.329800
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−8.02185	−0.356968
$506$	0	0
$507$	−13.7998	−0.612871
$508$	0	0
$509$	−13.3438	−0.591452	−0.295726	−	0.955273i	$-0.595562\pi$
$509$	−13.3438	−0.591452	−0.295726	+	0.955273i	$0.595562\pi$
$510$	0	0
$511$	−35.7560	−1.58175
$512$	0	0
$513$	−10.8783	−0.480288
$514$	0	0
$515$	5.96817	0.262989
$516$	0	0
$517$	−8.65169	−0.380501
$518$	0	0
$519$	−4.47303	−0.196344
$520$	0	0
$521$	−17.4349	−0.763835	−0.381918	−	0.924196i	$-0.624736\pi$
$521$	−17.4349	−0.763835	−0.381918	+	0.924196i	$0.624736\pi$
$522$	0	0
$523$	40.2946	1.76196	0.880981	−	0.473152i	$-0.156884\pi$
$523$	40.2946	1.76196	0.880981	+	0.473152i	$0.156884\pi$
$524$	0	0
$525$	7.12651	0.311026
$526$	0	0
$527$	−17.0592	−0.743109
$528$	0	0
$529$	−18.6553	−0.811099
$530$	0	0
$531$	5.40271	0.234458
$532$	0	0
$533$	0.654205	0.0283368
$534$	0	0
$535$	−1.45361	−0.0628452
$536$	0	0
$537$	−3.15765	−0.136263
$538$	0	0
$539$	2.69496	0.116080
$540$	0	0
$541$	−24.7221	−1.06288	−0.531442	−	0.847094i	$-0.678350\pi$
$541$	−24.7221	−1.06288	−0.531442	+	0.847094i	$0.678350\pi$
$542$	0	0
$543$	−8.65835	−0.371565
$544$	0	0
$545$	1.25389	0.0537108
$546$	0	0
$547$	32.4623	1.38799	0.693994	−	0.719980i	$-0.255849\pi$
$547$	32.4623	1.38799	0.693994	+	0.719980i	$0.255849\pi$
$548$	0	0
$549$	32.4028	1.38292
$550$	0	0
$551$	26.0724	1.11072
$552$	0	0
$553$	−46.3400	−1.97058
$554$	0	0
$555$	3.42143	0.145232
$556$	0	0
$557$	−38.2242	−1.61961	−0.809806	−	0.586698i	$-0.800427\pi$
$557$	−38.2242	−1.61961	−0.809806	+	0.586698i	$0.800427\pi$
$558$	0	0
$559$	−2.59190	−0.109626
$560$	0	0
$561$	2.21058	0.0933308
$562$	0	0
$563$	8.20606	0.345844	0.172922	−	0.984936i	$-0.444679\pi$
$563$	8.20606	0.345844	0.172922	+	0.984936i	$0.444679\pi$
$564$	0	0
$565$	12.6758	0.533273
$566$	0	0
$567$	20.8003	0.873533
$568$	0	0
$569$	9.53803	0.399855	0.199928	−	0.979811i	$-0.435929\pi$
$569$	9.53803	0.399855	0.199928	+	0.979811i	$0.435929\pi$
$570$	0	0
$571$	32.6514	1.36642	0.683210	−	0.730222i	$-0.260583\pi$
$571$	32.6514	1.36642	0.683210	+	0.730222i	$0.260583\pi$
$572$	0	0
$573$	0.0882865	0.00368822
$574$	0	0
$575$	−9.04352	−0.377141
$576$	0	0
$577$	18.3529	0.764042	0.382021	−	0.924154i	$-0.375228\pi$
$577$	18.3529	0.764042	0.382021	+	0.924154i	$0.375228\pi$
$578$	0	0
$579$	−2.94045	−0.122201
$580$	0	0
$581$	22.2181	0.921764
$582$	0	0
$583$	−4.63764	−0.192071
$584$	0	0
$585$	−13.9215	−0.575581
$586$	0	0
$587$	14.9281	0.616150	0.308075	−	0.951362i	$-0.400315\pi$
$587$	14.9281	0.616150	0.308075	+	0.951362i	$0.400315\pi$
$588$	0	0
$589$	−13.7118	−0.564983
$590$	0	0
$591$	−8.34432	−0.343239
$592$	0	0
$593$	24.3426	0.999630	0.499815	−	0.866132i	$-0.333401\pi$
$593$	24.3426	0.999630	0.499815	+	0.866132i	$0.333401\pi$
$594$	0	0
$595$	−11.5609	−0.473951
$596$	0	0
$597$	9.17136	0.375359
$598$	0	0
$599$	18.2919	0.747386	0.373693	−	0.927552i	$-0.378091\pi$
$599$	18.2919	0.747386	0.373693	+	0.927552i	$0.378091\pi$
$600$	0	0
$601$	−26.1164	−1.06531	−0.532655	−	0.846333i	$-0.678806\pi$
$601$	−26.1164	−1.06531	−0.532655	+	0.846333i	$0.678806\pi$
$602$	0	0
$603$	11.4953	0.468125
$604$	0	0
$605$	−8.23386	−0.334754
$606$	0	0
$607$	12.8673	0.522269	0.261135	−	0.965302i	$-0.415903\pi$
$607$	12.8673	0.522269	0.261135	+	0.965302i	$0.415903\pi$
$608$	0	0
$609$	11.7810	0.477391
$610$	0	0
$611$	−58.0623	−2.34895
$612$	0	0
$613$	−10.5919	−0.427804	−0.213902	−	0.976855i	$-0.568617\pi$
$613$	−10.5919	−0.427804	−0.213902	+	0.976855i	$0.568617\pi$
$614$	0	0
$615$	−0.0442836	−0.00178569
$616$	0	0
$617$	−2.94095	−0.118398	−0.0591990	−	0.998246i	$-0.518855\pi$
$617$	−2.94095	−0.118398	−0.0591990	+	0.998246i	$0.518855\pi$
$618$	0	0
$619$	−10.1326	−0.407263	−0.203632	−	0.979048i	$-0.565275\pi$
$619$	−10.1326	−0.407263	−0.203632	+	0.979048i	$0.565275\pi$
$620$	0	0
$621$	6.23768	0.250309
$622$	0	0
$623$	−24.4134	−0.978101
$624$	0	0
$625$	15.5174	0.620697
$626$	0	0
$627$	1.77681	0.0709591
$628$	0	0
$629$	36.4135	1.45190
$630$	0	0
$631$	−20.3719	−0.810993	−0.405497	−	0.914097i	$-0.632901\pi$
$631$	−20.3719	−0.810993	−0.405497	+	0.914097i	$0.632901\pi$
$632$	0	0
$633$	−10.7585	−0.427613
$634$	0	0
$635$	−7.99842	−0.317407
$636$	0	0
$637$	18.0861	0.716598
$638$	0	0
$639$	−36.8030	−1.45590
$640$	0	0
$641$	−15.2866	−0.603783	−0.301892	−	0.953342i	$-0.597618\pi$
$641$	−15.2866	−0.603783	−0.301892	+	0.953342i	$0.597618\pi$
$642$	0	0
$643$	16.8979	0.666390	0.333195	−	0.942858i	$-0.391873\pi$
$643$	16.8979	0.666390	0.333195	+	0.942858i	$0.391873\pi$
$644$	0	0
$645$	0.175448	0.00690826
$646$	0	0
$647$	−6.90398	−0.271423	−0.135712	−	0.990748i	$-0.543332\pi$
$647$	−6.90398	−0.271423	−0.135712	+	0.990748i	$0.543332\pi$
$648$	0	0
$649$	−1.85327	−0.0727473
$650$	0	0
$651$	−6.19577	−0.242831
$652$	0	0
$653$	27.7005	1.08400	0.542002	−	0.840377i	$-0.317667\pi$
$653$	27.7005	1.08400	0.542002	+	0.840377i	$0.317667\pi$
$654$	0	0
$655$	8.18421	0.319783
$656$	0	0
$657$	31.0189	1.21016
$658$	0	0
$659$	−19.9962	−0.778942	−0.389471	−	0.921039i	$-0.627342\pi$
$659$	−19.9962	−0.778942	−0.389471	+	0.921039i	$0.627342\pi$
$660$	0	0
$661$	−6.81293	−0.264992	−0.132496	−	0.991184i	$-0.542299\pi$
$661$	−6.81293	−0.264992	−0.132496	+	0.991184i	$0.542299\pi$
$662$	0	0
$663$	14.8354	0.576160
$664$	0	0
$665$	−9.29239	−0.360343
$666$	0	0
$667$	−14.9501	−0.578869
$668$	0	0
$669$	−2.28873	−0.0884873
$670$	0	0
$671$	−11.1150	−0.429090
$672$	0	0
$673$	1.57260	0.0606194	0.0303097	−	0.999541i	$-0.490351\pi$
$673$	1.57260	0.0606194	0.0303097	+	0.999541i	$0.490351\pi$
$674$	0	0
$675$	−12.9837	−0.499744
$676$	0	0
$677$	−38.0167	−1.46110	−0.730551	−	0.682858i	$-0.760737\pi$
$677$	−38.0167	−1.46110	−0.730551	+	0.682858i	$0.760737\pi$
$678$	0	0
$679$	34.9981	1.34310
$680$	0	0
$681$	−13.5197	−0.518076
$682$	0	0
$683$	38.6081	1.47730	0.738649	−	0.674090i	$-0.235464\pi$
$683$	38.6081	1.47730	0.738649	+	0.674090i	$0.235464\pi$
$684$	0	0
$685$	−2.39902	−0.0916617
$686$	0	0
$687$	7.63957	0.291468
$688$	0	0
$689$	−31.1236	−1.18572
$690$	0	0
$691$	37.9253	1.44275	0.721374	−	0.692546i	$-0.243511\pi$
$691$	37.9253	1.44275	0.721374	+	0.692546i	$0.243511\pi$
$692$	0	0
$693$	−8.01832	−0.304591
$694$	0	0
$695$	−4.78951	−0.181677
$696$	0	0
$697$	−0.471301	−0.0178518
$698$	0	0
$699$	9.34443	0.353439
$700$	0	0
$701$	35.7661	1.35087	0.675433	−	0.737421i	$-0.263957\pi$
$701$	35.7661	1.35087	0.675433	+	0.737421i	$0.263957\pi$
$702$	0	0
$703$	29.2684	1.10388
$704$	0	0
$705$	3.93028	0.148023
$706$	0	0
$707$	−31.0075	−1.16616
$708$	0	0
$709$	21.2634	0.798564	0.399282	−	0.916828i	$-0.369260\pi$
$709$	21.2634	0.798564	0.399282	+	0.916828i	$0.369260\pi$
$710$	0	0
$711$	40.2006	1.50764
$712$	0	0
$713$	7.86241	0.294450
$714$	0	0
$715$	4.77542	0.178591
$716$	0	0
$717$	−1.15822	−0.0432544
$718$	0	0
$719$	−45.4811	−1.69616	−0.848079	−	0.529870i	$-0.822241\pi$
$719$	−45.4811	−1.69616	−0.848079	+	0.529870i	$0.822241\pi$
$720$	0	0
$721$	23.0692	0.859143
$722$	0	0
$723$	1.25500	0.0466740
$724$	0	0
$725$	31.1186	1.15572
$726$	0	0
$727$	−43.4589	−1.61180	−0.805901	−	0.592051i	$-0.798319\pi$
$727$	−43.4589	−1.61180	−0.805901	+	0.592051i	$0.798319\pi$
$728$	0	0
$729$	−13.3534	−0.494571
$730$	0	0
$731$	1.86725	0.0690628
$732$	0	0
$733$	−7.44886	−0.275130	−0.137565	−	0.990493i	$-0.543928\pi$
$733$	−7.44886	−0.275130	−0.137565	+	0.990493i	$0.543928\pi$
$734$	0	0
$735$	−1.22426	−0.0451576
$736$	0	0
$737$	−3.94319	−0.145249
$738$	0	0
$739$	−4.31285	−0.158651	−0.0793253	−	0.996849i	$-0.525277\pi$
$739$	−4.31285	−0.158651	−0.0793253	+	0.996849i	$0.525277\pi$
$740$	0	0
$741$	11.9244	0.438052
$742$	0	0
$743$	−37.3328	−1.36961	−0.684805	−	0.728727i	$-0.740112\pi$
$743$	−37.3328	−1.36961	−0.684805	+	0.728727i	$0.740112\pi$
$744$	0	0
$745$	−1.27995	−0.0468936
$746$	0	0
$747$	−19.2745	−0.705219
$748$	0	0
$749$	−5.61877	−0.205305
$750$	0	0
$751$	−5.53928	−0.202131	−0.101066	−	0.994880i	$-0.532225\pi$
$751$	−5.53928	−0.202131	−0.101066	+	0.994880i	$0.532225\pi$
$752$	0	0
$753$	−1.47654	−0.0538081
$754$	0	0
$755$	−14.8022	−0.538708
$756$	0	0
$757$	−25.5143	−0.927333	−0.463666	−	0.886010i	$-0.653466\pi$
$757$	−25.5143	−0.927333	−0.463666	+	0.886010i	$0.653466\pi$
$758$	0	0
$759$	−1.01884	−0.0369814
$760$	0	0
$761$	−31.5088	−1.14219	−0.571097	−	0.820883i	$-0.693482\pi$
$761$	−31.5088	−1.14219	−0.571097	+	0.820883i	$0.693482\pi$
$762$	0	0
$763$	4.84677	0.175465
$764$	0	0
$765$	10.0293	0.362608
$766$	0	0
$767$	−12.4375	−0.449091
$768$	0	0
$769$	−3.95793	−0.142727	−0.0713633	−	0.997450i	$-0.522735\pi$
$769$	−3.95793	−0.142727	−0.0713633	+	0.997450i	$0.522735\pi$
$770$	0	0
$771$	−6.73604	−0.242593
$772$	0	0
$773$	−9.80022	−0.352489	−0.176245	−	0.984346i	$-0.556395\pi$
$773$	−9.80022	−0.352489	−0.176245	+	0.984346i	$0.556395\pi$
$774$	0	0
$775$	−16.3656	−0.587870
$776$	0	0
$777$	13.2251	0.474450
$778$	0	0
$779$	−0.378821	−0.0135727
$780$	0	0
$781$	12.6244	0.451736
$782$	0	0
$783$	−21.4637	−0.767052
$784$	0	0
$785$	−5.17914	−0.184852
$786$	0	0
$787$	6.53777	0.233046	0.116523	−	0.993188i	$-0.462825\pi$
$787$	6.53777	0.233046	0.116523	+	0.993188i	$0.462825\pi$
$788$	0	0
$789$	−10.1777	−0.362337
$790$	0	0
$791$	48.9966	1.74212
$792$	0	0
$793$	−74.5939	−2.64891
$794$	0	0
$795$	2.10678	0.0747198
$796$	0	0
$797$	30.9321	1.09567	0.547836	−	0.836586i	$-0.315452\pi$
$797$	30.9321	1.09567	0.547836	+	0.836586i	$0.315452\pi$
$798$	0	0
$799$	41.8291	1.47981
$800$	0	0
$801$	21.1789	0.748321
$802$	0	0
$803$	−10.6403	−0.375487
$804$	0	0
$805$	5.32832	0.187799
$806$	0	0
$807$	16.0238	0.564064
$808$	0	0
$809$	−32.4321	−1.14025	−0.570125	−	0.821558i	$-0.693105\pi$
$809$	−32.4321	−1.14025	−0.570125	+	0.821558i	$0.693105\pi$
$810$	0	0
$811$	12.3578	0.433942	0.216971	−	0.976178i	$-0.430382\pi$
$811$	12.3578	0.433942	0.216971	+	0.976178i	$0.430382\pi$
$812$	0	0
$813$	3.79919	0.133244
$814$	0	0
$815$	−15.9766	−0.559636
$816$	0	0
$817$	1.50085	0.0525083
$818$	0	0
$819$	−53.8117	−1.88033
$820$	0	0
$821$	−40.0699	−1.39845	−0.699225	−	0.714902i	$-0.746471\pi$
$821$	−40.0699	−1.39845	−0.699225	+	0.714902i	$0.746471\pi$
$822$	0	0
$823$	−27.5103	−0.958948	−0.479474	−	0.877556i	$-0.659173\pi$
$823$	−27.5103	−0.958948	−0.479474	+	0.877556i	$0.659173\pi$
$824$	0	0
$825$	2.12071	0.0738335
$826$	0	0
$827$	32.8789	1.14331	0.571656	−	0.820493i	$-0.306301\pi$
$827$	32.8789	1.14331	0.571656	+	0.820493i	$0.306301\pi$
$828$	0	0
$829$	−21.7701	−0.756108	−0.378054	−	0.925784i	$-0.623407\pi$
$829$	−21.7701	−0.756108	−0.378054	+	0.925784i	$0.623407\pi$
$830$	0	0
$831$	−7.22015	−0.250464
$832$	0	0
$833$	−13.0295	−0.451447
$834$	0	0
$835$	11.4884	0.397572
$836$	0	0
$837$	11.2880	0.390171
$838$	0	0
$839$	−15.5554	−0.537032	−0.268516	−	0.963275i	$-0.586533\pi$
$839$	−15.5554	−0.537032	−0.268516	+	0.963275i	$0.586533\pi$
$840$	0	0
$841$	22.4430	0.773895
$842$	0	0
$843$	−5.61187	−0.193283
$844$	0	0
$845$	21.4764	0.738812
$846$	0	0
$847$	−31.8270	−1.09359
$848$	0	0
$849$	9.65448	0.331341
$850$	0	0
$851$	−16.7827	−0.575303
$852$	0	0
$853$	−51.2275	−1.75400	−0.876998	−	0.480493i	$-0.840458\pi$
$853$	−51.2275	−1.75400	−0.876998	+	0.480493i	$0.840458\pi$
$854$	0	0
$855$	8.06128	0.275690
$856$	0	0
$857$	57.7319	1.97208	0.986042	−	0.166497i	$-0.0532456\pi$
$857$	57.7319	1.97208	0.986042	+	0.166497i	$0.0532456\pi$
$858$	0	0
$859$	−6.86388	−0.234192	−0.117096	−	0.993121i	$-0.537359\pi$
$859$	−6.86388	−0.234192	−0.117096	+	0.993121i	$0.537359\pi$
$860$	0	0
$861$	−0.171173	−0.00583357
$862$	0	0
$863$	−44.8679	−1.52732	−0.763661	−	0.645618i	$-0.776600\pi$
$863$	−44.8679	−1.52732	−0.763661	+	0.645618i	$0.776600\pi$
$864$	0	0
$865$	6.96131	0.236692
$866$	0	0
$867$	−1.80452	−0.0612846
$868$	0	0
$869$	−13.7899	−0.467789
$870$	0	0
$871$	−26.4631	−0.896668
$872$	0	0
$873$	−30.3613	−1.02757
$874$	0	0
$875$	−23.8723	−0.807032
$876$	0	0
$877$	44.6205	1.50673	0.753363	−	0.657605i	$-0.228430\pi$
$877$	44.6205	1.50673	0.753363	+	0.657605i	$0.228430\pi$
$878$	0	0
$879$	−11.8573	−0.399938
$880$	0	0
$881$	−11.9111	−0.401295	−0.200647	−	0.979664i	$-0.564305\pi$
$881$	−11.9111	−0.401295	−0.200647	+	0.979664i	$0.564305\pi$
$882$	0	0
$883$	−11.3860	−0.383170	−0.191585	−	0.981476i	$-0.561363\pi$
$883$	−11.3860	−0.383170	−0.191585	+	0.981476i	$0.561363\pi$
$884$	0	0
$885$	0.841902	0.0283002
$886$	0	0
$887$	32.5755	1.09378	0.546889	−	0.837205i	$-0.315812\pi$
$887$	32.5755	1.09378	0.546889	+	0.837205i	$0.315812\pi$
$888$	0	0
$889$	−30.9169	−1.03692
$890$	0	0
$891$	6.18977	0.207365
$892$	0	0
$893$	33.6212	1.12509
$894$	0	0
$895$	4.91421	0.164264
$896$	0	0
$897$	−6.83751	−0.228298
$898$	0	0
$899$	−27.0544	−0.902316
$900$	0	0
$901$	22.4220	0.746985
$902$	0	0
$903$	0.678173	0.0225682
$904$	0	0
$905$	13.4749	0.447920
$906$	0	0
$907$	−32.6995	−1.08577	−0.542885	−	0.839807i	$-0.682668\pi$
$907$	−32.6995	−1.08577	−0.542885	+	0.839807i	$0.682668\pi$
$908$	0	0
$909$	26.8995	0.892199
$910$	0	0
$911$	50.7353	1.68093	0.840467	−	0.541863i	$-0.182281\pi$
$911$	50.7353	1.68093	0.840467	+	0.541863i	$0.182281\pi$
$912$	0	0
$913$	6.61168	0.218815
$914$	0	0
$915$	5.04932	0.166925
$916$	0	0
$917$	31.6351	1.04468
$918$	0	0
$919$	−30.2301	−0.997199	−0.498599	−	0.866833i	$-0.666152\pi$
$919$	−30.2301	−0.997199	−0.498599	+	0.866833i	$0.666152\pi$
$920$	0	0
$921$	8.71846	0.287283
$922$	0	0
$923$	84.7235	2.78871
$924$	0	0
$925$	34.9331	1.14859
$926$	0	0
$927$	−20.0129	−0.657310
$928$	0	0
$929$	−5.70119	−0.187050	−0.0935250	−	0.995617i	$-0.529814\pi$
$929$	−5.70119	−0.187050	−0.0935250	+	0.995617i	$0.529814\pi$
$930$	0	0
$931$	−10.4729	−0.343234
$932$	0	0
$933$	9.89058	0.323803
$934$	0	0
$935$	−3.44030	−0.112510
$936$	0	0
$937$	30.1864	0.986147	0.493073	−	0.869988i	$-0.335873\pi$
$937$	30.1864	0.986147	0.493073	+	0.869988i	$0.335873\pi$
$938$	0	0
$939$	−9.76621	−0.318708
$940$	0	0
$941$	−30.2132	−0.984922	−0.492461	−	0.870334i	$-0.663903\pi$
$941$	−30.2132	−0.984922	−0.492461	+	0.870334i	$0.663903\pi$
$942$	0	0
$943$	0.217218	0.00707360
$944$	0	0
$945$	7.64984	0.248849
$946$	0	0
$947$	60.8276	1.97663	0.988316	−	0.152422i	$-0.0487073\pi$
$947$	60.8276	1.97663	0.988316	+	0.152422i	$0.0487073\pi$
$948$	0	0
$949$	−71.4080	−2.31800
$950$	0	0
$951$	−16.6255	−0.539119
$952$	0	0
$953$	−44.0330	−1.42637	−0.713185	−	0.700976i	$-0.752748\pi$
$953$	−44.0330	−1.42637	−0.713185	+	0.700976i	$0.752748\pi$
$954$	0	0
$955$	−0.137399	−0.00444613
$956$	0	0
$957$	3.50580	0.113326
$958$	0	0
$959$	−9.27312	−0.299445
$960$	0	0
$961$	−16.7718	−0.541025
$962$	0	0
$963$	4.87436	0.157074
$964$	0	0
$965$	4.57618	0.147313
$966$	0	0
$967$	50.4567	1.62258	0.811290	−	0.584644i	$-0.198766\pi$
$967$	50.4567	1.62258	0.811290	+	0.584644i	$0.198766\pi$
$968$	0	0
$969$	−8.59051	−0.275967
$970$	0	0
$971$	−42.8065	−1.37373	−0.686863	−	0.726787i	$-0.741013\pi$
$971$	−42.8065	−1.37373	−0.686863	+	0.726787i	$0.741013\pi$
$972$	0	0
$973$	−18.5133	−0.593509
$974$	0	0
$975$	14.2323	0.455797
$976$	0	0
$977$	21.1059	0.675238	0.337619	−	0.941283i	$-0.390379\pi$
$977$	21.1059	0.675238	0.337619	+	0.941283i	$0.390379\pi$
$978$	0	0
$979$	−7.26493	−0.232188
$980$	0	0
$981$	−4.20464	−0.134244
$982$	0	0
$983$	17.4280	0.555867	0.277933	−	0.960600i	$-0.410351\pi$
$983$	17.4280	0.555867	0.277933	+	0.960600i	$0.410351\pi$
$984$	0	0
$985$	12.9861	0.413773
$986$	0	0
$987$	15.1920	0.483568
$988$	0	0
$989$	−0.860600	−0.0273655
$990$	0	0
$991$	27.7294	0.880853	0.440426	−	0.897789i	$-0.354827\pi$
$991$	27.7294	0.880853	0.440426	+	0.897789i	$0.354827\pi$
$992$	0	0
$993$	−7.03676	−0.223305
$994$	0	0
$995$	−14.2733	−0.452493
$996$	0	0
$997$	−26.3758	−0.835331	−0.417665	−	0.908601i	$-0.637152\pi$
$997$	−26.3758	−0.835331	−0.417665	+	0.908601i	$0.637152\pi$
$998$	0	0
$999$	−24.0948	−0.762325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.n.1.2		5
4.3	odd	2		1006.2.a.g.1.4	✓	5
12.11	even	2		9054.2.a.bb.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.g.1.4	✓	5	4.3	odd	2
8048.2.a.n.1.2		5	1.1	even	1	trivial
9054.2.a.bb.1.2		5	12.11	even	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-0.522539 q^{3} +0.813221 q^{5} +3.14341 q^{7} -2.72695 q^{9} +O(q^{10})\)	\(q-0.522539 q^{3} +0.813221 q^{5} +3.14341 q^{7} -2.72695 q^{9} +0.935416 q^{11} +6.27767 q^{13} -0.424940 q^{15} -4.52254 q^{17} -3.63511 q^{19} -1.64256 q^{21} +2.08440 q^{23} -4.33867 q^{25} +2.99256 q^{27} -7.17238 q^{29} +3.77203 q^{31} -0.488792 q^{33} +2.55629 q^{35} -8.05157 q^{37} -3.28033 q^{39} +0.104212 q^{41} -0.412877 q^{43} -2.21762 q^{45} -9.24902 q^{47} +2.88102 q^{49} +2.36320 q^{51} -4.95783 q^{53} +0.760700 q^{55} +1.89949 q^{57} -1.98123 q^{59} -11.8824 q^{61} -8.57193 q^{63} +5.10513 q^{65} -4.21544 q^{67} -1.08918 q^{69} +13.4960 q^{71} -11.3749 q^{73} +2.26713 q^{75} +2.94040 q^{77} -14.7420 q^{79} +6.61713 q^{81} +7.06816 q^{83} -3.67782 q^{85} +3.74785 q^{87} -7.76652 q^{89} +19.7333 q^{91} -1.97103 q^{93} -2.95615 q^{95} +11.1338 q^{97} -2.55084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{5} - 3 q^{7} - 3 q^{9}+O(q^{10}) \) 5 * q - q^5 - 3 * q^7 - 3 * q^9	\( 5 q - q^{5} - 3 q^{7} - 3 q^{9} + 11 q^{11} + 5 q^{13} + 8 q^{15} - 20 q^{17} + 4 q^{19} - 4 q^{21} + 4 q^{23} - 6 q^{25} - 3 q^{27} - 6 q^{29} + 3 q^{31} - 7 q^{33} + 3 q^{35} - 10 q^{37} - 4 q^{39} - 6 q^{41} - 11 q^{43} - 17 q^{45} + 9 q^{47} - 4 q^{49} + 12 q^{51} - 22 q^{53} - 14 q^{55} + 10 q^{57} + 10 q^{59} - 11 q^{61} + 3 q^{63} - 12 q^{65} - 14 q^{67} - 3 q^{69} + 26 q^{71} - 7 q^{73} - 15 q^{75} - 26 q^{77} + 15 q^{79} - 7 q^{81} + 12 q^{83} + 12 q^{85} + 18 q^{87} - 5 q^{89} + 22 q^{91} - 21 q^{93} + 10 q^{95} + 6 q^{97} - 8 q^{99}+O(q^{100}) \) 5 * q - q^5 - 3 * q^7 - 3 * q^9 + 11 * q^11 + 5 * q^13 + 8 * q^15 - 20 * q^17 + 4 * q^19 - 4 * q^21 + 4 * q^23 - 6 * q^25 - 3 * q^27 - 6 * q^29 + 3 * q^31 - 7 * q^33 + 3 * q^35 - 10 * q^37 - 4 * q^39 - 6 * q^41 - 11 * q^43 - 17 * q^45 + 9 * q^47 - 4 * q^49 + 12 * q^51 - 22 * q^53 - 14 * q^55 + 10 * q^57 + 10 * q^59 - 11 * q^61 + 3 * q^63 - 12 * q^65 - 14 * q^67 - 3 * q^69 + 26 * q^71 - 7 * q^73 - 15 * q^75 - 26 * q^77 + 15 * q^79 - 7 * q^81 + 12 * q^83 + 12 * q^85 + 18 * q^87 - 5 * q^89 + 22 * q^91 - 21 * q^93 + 10 * q^95 + 6 * q^97 - 8 * q^99

Introduction

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