LMFDB - Embedded newform 8048.2.a.n.1.4

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			1.4
Root			$2.31801$ 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)

$f(q)$	$=$	$q+1.12789 q^{3} +2.61447 q^{5} -0.504123 q^{7} -1.72786 q^{9} +O(q^{10})$	$q+1.12789 q^{3} +2.61447 q^{5} -0.504123 q^{7} -1.72786 q^{9} -0.941783 q^{11} -4.19532 q^{13} +2.94884 q^{15} -2.87211 q^{17} +5.76392 q^{19} -0.568596 q^{21} +2.15926 q^{23} +1.83545 q^{25} -5.33252 q^{27} -4.90044 q^{29} -0.527925 q^{31} -1.06223 q^{33} -1.31801 q^{35} +3.99590 q^{37} -4.73187 q^{39} -1.62326 q^{41} -0.186109 q^{43} -4.51744 q^{45} -5.70727 q^{47} -6.74586 q^{49} -3.23943 q^{51} -1.93376 q^{53} -2.46226 q^{55} +6.50108 q^{57} +5.26685 q^{59} -4.08879 q^{61} +0.871053 q^{63} -10.9685 q^{65} -10.7702 q^{67} +2.43542 q^{69} +3.25283 q^{71} -7.02884 q^{73} +2.07019 q^{75} +0.474774 q^{77} +15.8779 q^{79} -0.830923 q^{81} +6.52369 q^{83} -7.50904 q^{85} -5.52716 q^{87} +8.10631 q^{89} +2.11496 q^{91} -0.595442 q^{93} +15.0696 q^{95} +3.41610 q^{97} +1.62727 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.12789	0.651189	0.325594	−	0.945510i	$-0.394436\pi$
$3$	1.12789	0.651189	0.325594	+	0.945510i	$0.394436\pi$
$4$	0	0
$5$	2.61447	1.16923	0.584613	−	0.811312i	$-0.301246\pi$
$5$	2.61447	1.16923	0.584613	+	0.811312i	$0.301246\pi$
$6$	0	0
$7$	−0.504123	−0.190541	−0.0952703	−	0.995451i	$-0.530372\pi$
$7$	−0.504123	−0.190541	−0.0952703	+	0.995451i	$0.530372\pi$
$8$	0	0
$9$	−1.72786	−0.575953
$10$	0	0
$11$	−0.941783	−0.283958	−0.141979	−	0.989870i	$-0.545347\pi$
$11$	−0.941783	−0.283958	−0.141979	+	0.989870i	$0.545347\pi$
$12$	0	0
$13$	−4.19532	−1.16357	−0.581787	−	0.813341i	$-0.697646\pi$
$13$	−4.19532	−1.16357	−0.581787	+	0.813341i	$0.697646\pi$
$14$	0	0
$15$	2.94884	0.761387
$16$	0	0
$17$	−2.87211	−0.696588	−0.348294	−	0.937385i	$-0.613239\pi$
$17$	−2.87211	−0.696588	−0.348294	+	0.937385i	$0.613239\pi$
$18$	0	0
$19$	5.76392	1.32233	0.661167	−	0.750239i	$-0.270062\pi$
$19$	5.76392	1.32233	0.661167	+	0.750239i	$0.270062\pi$
$20$	0	0
$21$	−0.568596	−0.124078
$22$	0	0
$23$	2.15926	0.450238	0.225119	−	0.974331i	$-0.427723\pi$
$23$	2.15926	0.450238	0.225119	+	0.974331i	$0.427723\pi$
$24$	0	0
$25$	1.83545	0.367090
$26$	0	0
$27$	−5.33252	−1.02624
$28$	0	0
$29$	−4.90044	−0.909988	−0.454994	−	0.890495i	$-0.650359\pi$
$29$	−4.90044	−0.909988	−0.454994	+	0.890495i	$0.650359\pi$
$30$	0	0
$31$	−0.527925	−0.0948181	−0.0474090	−	0.998876i	$-0.515096\pi$
$31$	−0.527925	−0.0948181	−0.0474090	+	0.998876i	$0.515096\pi$
$32$	0	0
$33$	−1.06223	−0.184910
$34$	0	0
$35$	−1.31801	−0.222785
$36$	0	0
$37$	3.99590	0.656922	0.328461	−	0.944518i	$-0.393470\pi$
$37$	3.99590	0.656922	0.328461	+	0.944518i	$0.393470\pi$
$38$	0	0
$39$	−4.73187	−0.757706
$40$	0	0
$41$	−1.62326	−0.253510	−0.126755	−	0.991934i	$-0.540456\pi$
$41$	−1.62326	−0.253510	−0.126755	+	0.991934i	$0.540456\pi$
$42$	0	0
$43$	−0.186109	−0.0283814	−0.0141907	−	0.999899i	$-0.504517\pi$
$43$	−0.186109	−0.0283814	−0.0141907	+	0.999899i	$0.504517\pi$
$44$	0	0
$45$	−4.51744	−0.673419
$46$	0	0
$47$	−5.70727	−0.832491	−0.416245	−	0.909252i	$-0.636654\pi$
$47$	−5.70727	−0.832491	−0.416245	+	0.909252i	$0.636654\pi$
$48$	0	0
$49$	−6.74586	−0.963694
$50$	0	0
$51$	−3.23943	−0.453611
$52$	0	0
$53$	−1.93376	−0.265622	−0.132811	−	0.991141i	$-0.542400\pi$
$53$	−1.93376	−0.265622	−0.132811	+	0.991141i	$0.542400\pi$
$54$	0	0
$55$	−2.46226	−0.332011
$56$	0	0
$57$	6.50108	0.861089
$58$	0	0
$59$	5.26685	0.685686	0.342843	−	0.939393i	$-0.388610\pi$
$59$	5.26685	0.685686	0.342843	+	0.939393i	$0.388610\pi$
$60$	0	0
$61$	−4.08879	−0.523516	−0.261758	−	0.965134i	$-0.584302\pi$
$61$	−4.08879	−0.523516	−0.261758	+	0.965134i	$0.584302\pi$
$62$	0	0
$63$	0.871053	0.109742
$64$	0	0
$65$	−10.9685	−1.36048
$66$	0	0
$67$	−10.7702	−1.31579	−0.657893	−	0.753111i	$-0.728552\pi$
$67$	−10.7702	−1.31579	−0.657893	+	0.753111i	$0.728552\pi$
$68$	0	0
$69$	2.43542	0.293190
$70$	0	0
$71$	3.25283	0.386039	0.193020	−	0.981195i	$-0.438172\pi$
$71$	3.25283	0.386039	0.193020	+	0.981195i	$0.438172\pi$
$72$	0	0
$73$	−7.02884	−0.822664	−0.411332	−	0.911486i	$-0.634936\pi$
$73$	−7.02884	−0.822664	−0.411332	+	0.911486i	$0.634936\pi$
$74$	0	0
$75$	2.07019	0.239045
$76$	0	0
$77$	0.474774	0.0541055
$78$	0	0
$79$	15.8779	1.78640	0.893202	−	0.449656i	$-0.148454\pi$
$79$	15.8779	1.78640	0.893202	+	0.449656i	$0.148454\pi$
$80$	0	0
$81$	−0.830923	−0.0923248
$82$	0	0
$83$	6.52369	0.716068	0.358034	−	0.933709i	$-0.383447\pi$
$83$	6.52369	0.716068	0.358034	+	0.933709i	$0.383447\pi$
$84$	0	0
$85$	−7.50904	−0.814469
$86$	0	0
$87$	−5.52716	−0.592574
$88$	0	0
$89$	8.10631	0.859267	0.429634	−	0.903003i	$-0.358643\pi$
$89$	8.10631	0.859267	0.429634	+	0.903003i	$0.358643\pi$
$90$	0	0
$91$	2.11496	0.221708
$92$	0	0
$93$	−0.595442	−0.0617445
$94$	0	0
$95$	15.0696	1.54611
$96$	0	0
$97$	3.41610	0.346853	0.173426	−	0.984847i	$-0.444516\pi$
$97$	3.41610	0.346853	0.173426	+	0.984847i	$0.444516\pi$
$98$	0	0
$99$	1.62727	0.163547
$100$	0	0
$101$	−18.8747	−1.87810	−0.939049	−	0.343783i	$-0.888292\pi$
$101$	−18.8747	−1.87810	−0.939049	+	0.343783i	$0.888292\pi$
$102$	0	0
$103$	−17.0847	−1.68340	−0.841701	−	0.539944i	$-0.818446\pi$
$103$	−17.0847	−1.68340	−0.841701	+	0.539944i	$0.818446\pi$
$104$	0	0
$105$	−1.48658	−0.145075
$106$	0	0
$107$	11.0922	1.07232	0.536162	−	0.844115i	$-0.319874\pi$
$107$	11.0922	1.07232	0.536162	+	0.844115i	$0.319874\pi$
$108$	0	0
$109$	−2.80311	−0.268489	−0.134245	−	0.990948i	$-0.542861\pi$
$109$	−2.80311	−0.268489	−0.134245	+	0.990948i	$0.542861\pi$
$110$	0	0
$111$	4.50695	0.427780
$112$	0	0
$113$	−15.6040	−1.46790	−0.733952	−	0.679202i	$-0.762326\pi$
$113$	−15.6040	−1.46790	−0.733952	+	0.679202i	$0.762326\pi$
$114$	0	0
$115$	5.64533	0.526429
$116$	0	0
$117$	7.24893	0.670164
$118$	0	0
$119$	1.44790	0.132728
$120$	0	0
$121$	−10.1130	−0.919368
$122$	0	0
$123$	−1.83086	−0.165083
$124$	0	0
$125$	−8.27362	−0.740015
$126$	0	0
$127$	−2.61723	−0.232241	−0.116121	−	0.993235i	$-0.537046\pi$
$127$	−2.61723	−0.232241	−0.116121	+	0.993235i	$0.537046\pi$
$128$	0	0
$129$	−0.209911	−0.0184817
$130$	0	0
$131$	−12.1109	−1.05814	−0.529069	−	0.848579i	$-0.677459\pi$
$131$	−12.1109	−1.05814	−0.529069	+	0.848579i	$0.677459\pi$
$132$	0	0
$133$	−2.90572	−0.251958
$134$	0	0
$135$	−13.9417	−1.19991
$136$	0	0
$137$	−4.55455	−0.389121	−0.194561	−	0.980890i	$-0.562328\pi$
$137$	−4.55455	−0.389121	−0.194561	+	0.980890i	$0.562328\pi$
$138$	0	0
$139$	−13.4014	−1.13669	−0.568346	−	0.822790i	$-0.692416\pi$
$139$	−13.4014	−1.13669	−0.568346	+	0.822790i	$0.692416\pi$
$140$	0	0
$141$	−6.43718	−0.542109
$142$	0	0
$143$	3.95108	0.330406
$144$	0	0
$145$	−12.8120	−1.06398
$146$	0	0
$147$	−7.60860	−0.627547
$148$	0	0
$149$	−0.152207	−0.0124693	−0.00623465	−	0.999981i	$-0.501985\pi$
$149$	−0.152207	−0.0124693	−0.00623465	+	0.999981i	$0.501985\pi$
$150$	0	0
$151$	−9.93979	−0.808889	−0.404444	−	0.914563i	$-0.632535\pi$
$151$	−9.93979	−0.808889	−0.404444	+	0.914563i	$0.632535\pi$
$152$	0	0
$153$	4.96260	0.401202
$154$	0	0
$155$	−1.38024	−0.110864
$156$	0	0
$157$	7.48026	0.596990	0.298495	−	0.954411i	$-0.403515\pi$
$157$	7.48026	0.596990	0.298495	+	0.954411i	$0.403515\pi$
$158$	0	0
$159$	−2.18107	−0.172970
$160$	0	0
$161$	−1.08853	−0.0857885
$162$	0	0
$163$	−13.5736	−1.06317	−0.531585	−	0.847005i	$-0.678403\pi$
$163$	−13.5736	−1.06317	−0.531585	+	0.847005i	$0.678403\pi$
$164$	0	0
$165$	−2.77717	−0.216202
$166$	0	0
$167$	1.98752	0.153799	0.0768995	−	0.997039i	$-0.475498\pi$
$167$	1.98752	0.153799	0.0768995	+	0.997039i	$0.475498\pi$
$168$	0	0
$169$	4.60073	0.353903
$170$	0	0
$171$	−9.95924	−0.761602
$172$	0	0
$173$	−4.71619	−0.358565	−0.179283	−	0.983798i	$-0.557378\pi$
$173$	−4.71619	−0.358565	−0.179283	+	0.983798i	$0.557378\pi$
$174$	0	0
$175$	−0.925292	−0.0699455
$176$	0	0
$177$	5.94044	0.446511
$178$	0	0
$179$	6.99696	0.522977	0.261489	−	0.965207i	$-0.415787\pi$
$179$	6.99696	0.522977	0.261489	+	0.965207i	$0.415787\pi$
$180$	0	0
$181$	11.3817	0.845994	0.422997	−	0.906131i	$-0.360978\pi$
$181$	11.3817	0.845994	0.422997	+	0.906131i	$0.360978\pi$
$182$	0	0
$183$	−4.61171	−0.340908
$184$	0	0
$185$	10.4472	0.768091
$186$	0	0
$187$	2.70490	0.197802
$188$	0	0
$189$	2.68824	0.195541
$190$	0	0
$191$	−8.19131	−0.592702	−0.296351	−	0.955079i	$-0.595770\pi$
$191$	−8.19131	−0.592702	−0.296351	+	0.955079i	$0.595770\pi$
$192$	0	0
$193$	18.4288	1.32653	0.663267	−	0.748383i	$-0.269170\pi$
$193$	18.4288	1.32653	0.663267	+	0.748383i	$0.269170\pi$
$194$	0	0
$195$	−12.3713	−0.885929
$196$	0	0
$197$	−15.1466	−1.07915	−0.539576	−	0.841937i	$-0.681415\pi$
$197$	−15.1466	−1.07915	−0.539576	+	0.841937i	$0.681415\pi$
$198$	0	0
$199$	2.87212	0.203599	0.101800	−	0.994805i	$-0.467540\pi$
$199$	2.87212	0.203599	0.101800	+	0.994805i	$0.467540\pi$
$200$	0	0
$201$	−12.1476	−0.856825
$202$	0	0
$203$	2.47042	0.173390
$204$	0	0
$205$	−4.24395	−0.296411
$206$	0	0
$207$	−3.73090	−0.259316
$208$	0	0
$209$	−5.42836	−0.375487
$210$	0	0
$211$	−10.4536	−0.719652	−0.359826	−	0.933019i	$-0.617164\pi$
$211$	−10.4536	−0.719652	−0.359826	+	0.933019i	$0.617164\pi$
$212$	0	0
$213$	3.66884	0.251385
$214$	0	0
$215$	−0.486577	−0.0331843
$216$	0	0
$217$	0.266139	0.0180667
$218$	0	0
$219$	−7.92777	−0.535709
$220$	0	0
$221$	12.0494	0.810532
$222$	0	0
$223$	−10.5459	−0.706207	−0.353104	−	0.935584i	$-0.614874\pi$
$223$	−10.5459	−0.706207	−0.353104	+	0.935584i	$0.614874\pi$
$224$	0	0
$225$	−3.17140	−0.211426
$226$	0	0
$227$	−24.3890	−1.61875	−0.809377	−	0.587289i	$-0.800195\pi$
$227$	−24.3890	−1.61875	−0.809377	+	0.587289i	$0.800195\pi$
$228$	0	0
$229$	−18.7112	−1.23647	−0.618234	−	0.785994i	$-0.712152\pi$
$229$	−18.7112	−1.23647	−0.618234	+	0.785994i	$0.712152\pi$
$230$	0	0
$231$	0.535494	0.0352329
$232$	0	0
$233$	−3.10870	−0.203658	−0.101829	−	0.994802i	$-0.532469\pi$
$233$	−3.10870	−0.203658	−0.101829	+	0.994802i	$0.532469\pi$
$234$	0	0
$235$	−14.9215	−0.973370
$236$	0	0
$237$	17.9086	1.16329
$238$	0	0
$239$	2.29767	0.148624	0.0743121	−	0.997235i	$-0.476324\pi$
$239$	2.29767	0.148624	0.0743121	+	0.997235i	$0.476324\pi$
$240$	0	0
$241$	9.02423	0.581302	0.290651	−	0.956829i	$-0.406128\pi$
$241$	9.02423	0.581302	0.290651	+	0.956829i	$0.406128\pi$
$242$	0	0
$243$	15.0604	0.966122
$244$	0	0
$245$	−17.6368	−1.12678
$246$	0	0
$247$	−24.1815	−1.53863
$248$	0	0
$249$	7.35802	0.466296
$250$	0	0
$251$	23.6327	1.49168	0.745841	−	0.666124i	$-0.232048\pi$
$251$	23.6327	1.49168	0.745841	+	0.666124i	$0.232048\pi$
$252$	0	0
$253$	−2.03356	−0.127849
$254$	0	0
$255$	−8.46938	−0.530373
$256$	0	0
$257$	−13.2000	−0.823392	−0.411696	−	0.911321i	$-0.635063\pi$
$257$	−13.2000	−0.823392	−0.411696	+	0.911321i	$0.635063\pi$
$258$	0	0
$259$	−2.01443	−0.125170
$260$	0	0
$261$	8.46726	0.524110
$262$	0	0
$263$	−1.59776	−0.0985223	−0.0492611	−	0.998786i	$-0.515687\pi$
$263$	−1.59776	−0.0985223	−0.0492611	+	0.998786i	$0.515687\pi$
$264$	0	0
$265$	−5.05575	−0.310572
$266$	0	0
$267$	9.14305	0.559545
$268$	0	0
$269$	12.5307	0.764013	0.382006	−	0.924160i	$-0.375233\pi$
$269$	12.5307	0.764013	0.382006	+	0.924160i	$0.375233\pi$
$270$	0	0
$271$	21.9043	1.33059	0.665295	−	0.746581i	$-0.268306\pi$
$271$	21.9043	1.33059	0.665295	+	0.746581i	$0.268306\pi$
$272$	0	0
$273$	2.38544	0.144374
$274$	0	0
$275$	−1.72859	−0.104238
$276$	0	0
$277$	−18.0061	−1.08188	−0.540942	−	0.841060i	$-0.681932\pi$
$277$	−18.0061	−1.08188	−0.540942	+	0.841060i	$0.681932\pi$
$278$	0	0
$279$	0.912180	0.0546108
$280$	0	0
$281$	16.4701	0.982522	0.491261	−	0.871013i	$-0.336536\pi$
$281$	16.4701	0.982522	0.491261	+	0.871013i	$0.336536\pi$
$282$	0	0
$283$	1.85554	0.110300	0.0551502	−	0.998478i	$-0.482436\pi$
$283$	1.85554	0.110300	0.0551502	+	0.998478i	$0.482436\pi$
$284$	0	0
$285$	16.9969	1.00681
$286$	0	0
$287$	0.818320	0.0483039
$288$	0	0
$289$	−8.75100	−0.514764
$290$	0	0
$291$	3.85299	0.225867
$292$	0	0
$293$	3.37578	0.197215	0.0986076	−	0.995126i	$-0.468561\pi$
$293$	3.37578	0.197215	0.0986076	+	0.995126i	$0.468561\pi$
$294$	0	0
$295$	13.7700	0.801722
$296$	0	0
$297$	5.02207	0.291410
$298$	0	0
$299$	−9.05881	−0.523884
$300$	0	0
$301$	0.0938219	0.00540781
$302$	0	0
$303$	−21.2886	−1.22300
$304$	0	0
$305$	−10.6900	−0.612108
$306$	0	0
$307$	−12.6616	−0.722637	−0.361318	−	0.932443i	$-0.617673\pi$
$307$	−12.6616	−0.722637	−0.361318	+	0.932443i	$0.617673\pi$
$308$	0	0
$309$	−19.2697	−1.09621
$310$	0	0
$311$	9.98185	0.566019	0.283009	−	0.959117i	$-0.408667\pi$
$311$	9.98185	0.566019	0.283009	+	0.959117i	$0.408667\pi$
$312$	0	0
$313$	−0.774696	−0.0437884	−0.0218942	−	0.999760i	$-0.506970\pi$
$313$	−0.774696	−0.0437884	−0.0218942	+	0.999760i	$0.506970\pi$
$314$	0	0
$315$	2.27734	0.128314
$316$	0	0
$317$	8.47072	0.475763	0.237882	−	0.971294i	$-0.423547\pi$
$317$	8.47072	0.475763	0.237882	+	0.971294i	$0.423547\pi$
$318$	0	0
$319$	4.61515	0.258399
$320$	0	0
$321$	12.5108	0.698285
$322$	0	0
$323$	−16.5546	−0.921122
$324$	0	0
$325$	−7.70030	−0.427136
$326$	0	0
$327$	−3.16160	−0.174837
$328$	0	0
$329$	2.87716	0.158623
$330$	0	0
$331$	4.35192	0.239203	0.119602	−	0.992822i	$-0.461838\pi$
$331$	4.35192	0.239203	0.119602	+	0.992822i	$0.461838\pi$
$332$	0	0
$333$	−6.90436	−0.378356
$334$	0	0
$335$	−28.1583	−1.53845
$336$	0	0
$337$	25.0488	1.36449	0.682246	−	0.731122i	$-0.261003\pi$
$337$	25.0488	1.36449	0.682246	+	0.731122i	$0.261003\pi$
$338$	0	0
$339$	−17.5997	−0.955882
$340$	0	0
$341$	0.497190	0.0269244
$342$	0	0
$343$	6.92960	0.374163
$344$	0	0
$345$	6.36732	0.342805
$346$	0	0
$347$	1.12276	0.0602729	0.0301365	−	0.999546i	$-0.490406\pi$
$347$	1.12276	0.0602729	0.0301365	+	0.999546i	$0.490406\pi$
$348$	0	0
$349$	24.2198	1.29646	0.648228	−	0.761447i	$-0.275511\pi$
$349$	24.2198	1.29646	0.648228	+	0.761447i	$0.275511\pi$
$350$	0	0
$351$	22.3716	1.19411
$352$	0	0
$353$	23.6325	1.25783	0.628916	−	0.777473i	$-0.283499\pi$
$353$	23.6325	1.25783	0.628916	+	0.777473i	$0.283499\pi$
$354$	0	0
$355$	8.50441	0.451367
$356$	0	0
$357$	1.63307	0.0864312
$358$	0	0
$359$	−4.37103	−0.230694	−0.115347	−	0.993325i	$-0.536798\pi$
$359$	−4.37103	−0.230694	−0.115347	+	0.993325i	$0.536798\pi$
$360$	0	0
$361$	14.2228	0.748567
$362$	0	0
$363$	−11.4064	−0.598682
$364$	0	0
$365$	−18.3767	−0.961880
$366$	0	0
$367$	36.2181	1.89057	0.945285	−	0.326246i	$-0.105784\pi$
$367$	36.2181	1.89057	0.945285	+	0.326246i	$0.105784\pi$
$368$	0	0
$369$	2.80476	0.146010
$370$	0	0
$371$	0.974852	0.0506118
$372$	0	0
$373$	3.00453	0.155568	0.0777842	−	0.996970i	$-0.475215\pi$
$373$	3.00453	0.155568	0.0777842	+	0.996970i	$0.475215\pi$
$374$	0	0
$375$	−9.33175	−0.481890
$376$	0	0
$377$	20.5589	1.05884
$378$	0	0
$379$	−16.1682	−0.830504	−0.415252	−	0.909706i	$-0.636307\pi$
$379$	−16.1682	−0.830504	−0.415252	+	0.909706i	$0.636307\pi$
$380$	0	0
$381$	−2.95195	−0.151233
$382$	0	0
$383$	26.1690	1.33718	0.668588	−	0.743633i	$-0.266899\pi$
$383$	26.1690	1.33718	0.668588	+	0.743633i	$0.266899\pi$
$384$	0	0
$385$	1.24128	0.0632616
$386$	0	0
$387$	0.321571	0.0163464
$388$	0	0
$389$	−17.4666	−0.885590	−0.442795	−	0.896623i	$-0.646013\pi$
$389$	−17.4666	−0.885590	−0.442795	+	0.896623i	$0.646013\pi$
$390$	0	0
$391$	−6.20164	−0.313630
$392$	0	0
$393$	−13.6598	−0.689047
$394$	0	0
$395$	41.5123	2.08871
$396$	0	0
$397$	16.7142	0.838862	0.419431	−	0.907787i	$-0.362230\pi$
$397$	16.7142	0.838862	0.419431	+	0.907787i	$0.362230\pi$
$398$	0	0
$399$	−3.27734	−0.164072
$400$	0	0
$401$	−28.4130	−1.41888	−0.709439	−	0.704767i	$-0.751052\pi$
$401$	−28.4130	−1.41888	−0.709439	+	0.704767i	$0.751052\pi$
$402$	0	0
$403$	2.21481	0.110328
$404$	0	0
$405$	−2.17242	−0.107949
$406$	0	0
$407$	−3.76327	−0.186538
$408$	0	0
$409$	9.17703	0.453775	0.226887	−	0.973921i	$-0.427145\pi$
$409$	9.17703	0.453775	0.226887	+	0.973921i	$0.427145\pi$
$410$	0	0
$411$	−5.13704	−0.253392
$412$	0	0
$413$	−2.65514	−0.130651
$414$	0	0
$415$	17.0560	0.837246
$416$	0	0
$417$	−15.1153	−0.740200
$418$	0	0
$419$	−33.3908	−1.63125	−0.815623	−	0.578584i	$-0.803605\pi$
$419$	−33.3908	−1.63125	−0.815623	+	0.578584i	$0.803605\pi$
$420$	0	0
$421$	23.9385	1.16669	0.583345	−	0.812224i	$-0.301743\pi$
$421$	23.9385	1.16669	0.583345	+	0.812224i	$0.301743\pi$
$422$	0	0
$423$	9.86136	0.479476
$424$	0	0
$425$	−5.27161	−0.255710
$426$	0	0
$427$	2.06125	0.0997510
$428$	0	0
$429$	4.45640	0.215157
$430$	0	0
$431$	16.6090	0.800026	0.400013	−	0.916509i	$-0.369006\pi$
$431$	16.6090	0.800026	0.400013	+	0.916509i	$0.369006\pi$
$432$	0	0
$433$	3.81464	0.183320	0.0916600	−	0.995790i	$-0.470783\pi$
$433$	3.81464	0.183320	0.0916600	+	0.995790i	$0.470783\pi$
$434$	0	0
$435$	−14.4506	−0.692853
$436$	0	0
$437$	12.4458	0.595364
$438$	0	0
$439$	14.6668	0.700008	0.350004	−	0.936748i	$-0.386180\pi$
$439$	14.6668	0.700008	0.350004	+	0.936748i	$0.386180\pi$
$440$	0	0
$441$	11.6559	0.555043
$442$	0	0
$443$	1.47171	0.0699230	0.0349615	−	0.999389i	$-0.488869\pi$
$443$	1.47171	0.0699230	0.0349615	+	0.999389i	$0.488869\pi$
$444$	0	0
$445$	21.1937	1.00468
$446$	0	0
$447$	−0.171673	−0.00811986
$448$	0	0
$449$	−5.26148	−0.248305	−0.124152	−	0.992263i	$-0.539621\pi$
$449$	−5.26148	−0.248305	−0.124152	+	0.992263i	$0.539621\pi$
$450$	0	0
$451$	1.52875	0.0719862
$452$	0	0
$453$	−11.2110	−0.526739
$454$	0	0
$455$	5.52949	0.259227
$456$	0	0
$457$	11.3116	0.529135	0.264567	−	0.964367i	$-0.414771\pi$
$457$	11.3116	0.529135	0.264567	+	0.964367i	$0.414771\pi$
$458$	0	0
$459$	15.3156	0.714869
$460$	0	0
$461$	−18.7953	−0.875386	−0.437693	−	0.899125i	$-0.644204\pi$
$461$	−18.7953	−0.875386	−0.437693	+	0.899125i	$0.644204\pi$
$462$	0	0
$463$	−37.9662	−1.76444	−0.882220	−	0.470837i	$-0.843952\pi$
$463$	−37.9662	−1.76444	−0.882220	+	0.470837i	$0.843952\pi$
$464$	0	0
$465$	−1.55676	−0.0721932
$466$	0	0
$467$	−27.0190	−1.25029	−0.625145	−	0.780508i	$-0.714960\pi$
$467$	−27.0190	−1.25029	−0.625145	+	0.780508i	$0.714960\pi$
$468$	0	0
$469$	5.42949	0.250711
$470$	0	0
$471$	8.43693	0.388753
$472$	0	0
$473$	0.175275	0.00805913
$474$	0	0
$475$	10.5794	0.485415
$476$	0	0
$477$	3.34126	0.152986
$478$	0	0
$479$	5.99935	0.274117	0.137059	−	0.990563i	$-0.456235\pi$
$479$	5.99935	0.274117	0.137059	+	0.990563i	$0.456235\pi$
$480$	0	0
$481$	−16.7641	−0.764377
$482$	0	0
$483$	−1.22775	−0.0558645
$484$	0	0
$485$	8.93129	0.405549
$486$	0	0
$487$	−0.963000	−0.0436377	−0.0218188	−	0.999762i	$-0.506946\pi$
$487$	−0.963000	−0.0436377	−0.0218188	+	0.999762i	$0.506946\pi$
$488$	0	0
$489$	−15.3096	−0.692324
$490$	0	0
$491$	25.7659	1.16280	0.581399	−	0.813619i	$-0.302506\pi$
$491$	25.7659	1.16280	0.581399	+	0.813619i	$0.302506\pi$
$492$	0	0
$493$	14.0746	0.633887
$494$	0	0
$495$	4.25444	0.191223
$496$	0	0
$497$	−1.63982	−0.0735562
$498$	0	0
$499$	14.8570	0.665089	0.332545	−	0.943088i	$-0.392093\pi$
$499$	14.8570	0.665089	0.332545	+	0.943088i	$0.392093\pi$
$500$	0	0
$501$	2.24171	0.100152
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−49.3472	−2.19592
$506$	0	0
$507$	5.18913	0.230457
$508$	0	0
$509$	−29.1194	−1.29069	−0.645347	−	0.763890i	$-0.723287\pi$
$509$	−29.1194	−1.29069	−0.645347	+	0.763890i	$0.723287\pi$
$510$	0	0
$511$	3.54340	0.156751
$512$	0	0
$513$	−30.7362	−1.35704
$514$	0	0
$515$	−44.6673	−1.96828
$516$	0	0
$517$	5.37501	0.236393
$518$	0	0
$519$	−5.31936	−0.233494
$520$	0	0
$521$	40.4159	1.77065	0.885325	−	0.464972i	$-0.153936\pi$
$521$	40.4159	1.77065	0.885325	+	0.464972i	$0.153936\pi$
$522$	0	0
$523$	−7.90850	−0.345815	−0.172907	−	0.984938i	$-0.555316\pi$
$523$	−7.90850	−0.345815	−0.172907	+	0.984938i	$0.555316\pi$
$524$	0	0
$525$	−1.04363	−0.0455477
$526$	0	0
$527$	1.51626	0.0660492
$528$	0	0
$529$	−18.3376	−0.797286
$530$	0	0
$531$	−9.10038	−0.394923
$532$	0	0
$533$	6.81008	0.294977
$534$	0	0
$535$	29.0002	1.25379
$536$	0	0
$537$	7.89181	0.340557
$538$	0	0
$539$	6.35313	0.273649
$540$	0	0
$541$	38.0902	1.63762	0.818812	−	0.574062i	$-0.194633\pi$
$541$	38.0902	1.63762	0.818812	+	0.574062i	$0.194633\pi$
$542$	0	0
$543$	12.8373	0.550902
$544$	0	0
$545$	−7.32864	−0.313925
$546$	0	0
$547$	11.0313	0.471665	0.235832	−	0.971794i	$-0.424218\pi$
$547$	11.0313	0.471665	0.235832	+	0.971794i	$0.424218\pi$
$548$	0	0
$549$	7.06485	0.301521
$550$	0	0
$551$	−28.2457	−1.20331
$552$	0	0
$553$	−8.00441	−0.340382
$554$	0	0
$555$	11.7833	0.500172
$556$	0	0
$557$	1.78310	0.0755526	0.0377763	−	0.999286i	$-0.487973\pi$
$557$	1.78310	0.0755526	0.0377763	+	0.999286i	$0.487973\pi$
$558$	0	0
$559$	0.780789	0.0330238
$560$	0	0
$561$	3.05084	0.128806
$562$	0	0
$563$	9.68353	0.408112	0.204056	−	0.978959i	$-0.434587\pi$
$563$	9.68353	0.408112	0.204056	+	0.978959i	$0.434587\pi$
$564$	0	0
$565$	−40.7962	−1.71631
$566$	0	0
$567$	0.418887	0.0175916
$568$	0	0
$569$	17.1719	0.719885	0.359943	−	0.932974i	$-0.382796\pi$
$569$	17.1719	0.719885	0.359943	+	0.932974i	$0.382796\pi$
$570$	0	0
$571$	21.5870	0.903387	0.451693	−	0.892173i	$-0.350820\pi$
$571$	21.5870	0.903387	0.451693	+	0.892173i	$0.350820\pi$
$572$	0	0
$573$	−9.23891	−0.385961
$574$	0	0
$575$	3.96322	0.165278
$576$	0	0
$577$	−30.6592	−1.27636	−0.638179	−	0.769888i	$-0.720312\pi$
$577$	−30.6592	−1.27636	−0.638179	+	0.769888i	$0.720312\pi$
$578$	0	0
$579$	20.7857	0.863823
$580$	0	0
$581$	−3.28874	−0.136440
$582$	0	0
$583$	1.82118	0.0754256
$584$	0	0
$585$	18.9521	0.783573
$586$	0	0
$587$	−38.9682	−1.60839	−0.804195	−	0.594365i	$-0.797403\pi$
$587$	−38.9682	−1.60839	−0.804195	+	0.594365i	$0.797403\pi$
$588$	0	0
$589$	−3.04292	−0.125381
$590$	0	0
$591$	−17.0837	−0.702731
$592$	0	0
$593$	−28.6248	−1.17548	−0.587739	−	0.809051i	$-0.699982\pi$
$593$	−28.6248	−1.17548	−0.587739	+	0.809051i	$0.699982\pi$
$594$	0	0
$595$	3.78548	0.155189
$596$	0	0
$597$	3.23944	0.132581
$598$	0	0
$599$	1.52866	0.0624594	0.0312297	−	0.999512i	$-0.490058\pi$
$599$	1.52866	0.0624594	0.0312297	+	0.999512i	$0.490058\pi$
$600$	0	0
$601$	18.4340	0.751939	0.375970	−	0.926632i	$-0.377310\pi$
$601$	18.4340	0.751939	0.375970	+	0.926632i	$0.377310\pi$
$602$	0	0
$603$	18.6093	0.757831
$604$	0	0
$605$	−26.4402	−1.07495
$606$	0	0
$607$	−7.99251	−0.324406	−0.162203	−	0.986757i	$-0.551860\pi$
$607$	−7.99251	−0.324406	−0.162203	+	0.986757i	$0.551860\pi$
$608$	0	0
$609$	2.78637	0.112909
$610$	0	0
$611$	23.9438	0.968664
$612$	0	0
$613$	28.2168	1.13967	0.569833	−	0.821760i	$-0.307008\pi$
$613$	28.2168	1.13967	0.569833	+	0.821760i	$0.307008\pi$
$614$	0	0
$615$	−4.78672	−0.193019
$616$	0	0
$617$	−4.08652	−0.164517	−0.0822585	−	0.996611i	$-0.526213\pi$
$617$	−4.08652	−0.164517	−0.0822585	+	0.996611i	$0.526213\pi$
$618$	0	0
$619$	−18.8799	−0.758848	−0.379424	−	0.925223i	$-0.623878\pi$
$619$	−18.8799	−0.758848	−0.379424	+	0.925223i	$0.623878\pi$
$620$	0	0
$621$	−11.5143	−0.462053
$622$	0	0
$623$	−4.08658	−0.163725
$624$	0	0
$625$	−30.8084	−1.23233
$626$	0	0
$627$	−6.12260	−0.244513
$628$	0	0
$629$	−11.4767	−0.457604
$630$	0	0
$631$	3.09383	0.123164	0.0615818	−	0.998102i	$-0.480386\pi$
$631$	3.09383	0.123164	0.0615818	+	0.998102i	$0.480386\pi$
$632$	0	0
$633$	−11.7905	−0.468629
$634$	0	0
$635$	−6.84266	−0.271543
$636$	0	0
$637$	28.3011	1.12133
$638$	0	0
$639$	−5.62043	−0.222341
$640$	0	0
$641$	−26.2410	−1.03646	−0.518228	−	0.855242i	$-0.673408\pi$
$641$	−26.2410	−1.03646	−0.518228	+	0.855242i	$0.673408\pi$
$642$	0	0
$643$	−23.6424	−0.932364	−0.466182	−	0.884689i	$-0.654371\pi$
$643$	−23.6424	−0.932364	−0.466182	+	0.884689i	$0.654371\pi$
$644$	0	0
$645$	−0.548806	−0.0216092
$646$	0	0
$647$	20.5495	0.807886	0.403943	−	0.914784i	$-0.367639\pi$
$647$	20.5495	0.807886	0.403943	+	0.914784i	$0.367639\pi$
$648$	0	0
$649$	−4.96023	−0.194706
$650$	0	0
$651$	0.300176	0.0117648
$652$	0	0
$653$	−6.90175	−0.270087	−0.135043	−	0.990840i	$-0.543117\pi$
$653$	−6.90175	−0.270087	−0.135043	+	0.990840i	$0.543117\pi$
$654$	0	0
$655$	−31.6637	−1.23720
$656$	0	0
$657$	12.1448	0.473816
$658$	0	0
$659$	32.1598	1.25277	0.626385	−	0.779514i	$-0.284534\pi$
$659$	32.1598	1.25277	0.626385	+	0.779514i	$0.284534\pi$
$660$	0	0
$661$	−24.2868	−0.944648	−0.472324	−	0.881425i	$-0.656585\pi$
$661$	−24.2868	−0.944648	−0.472324	+	0.881425i	$0.656585\pi$
$662$	0	0
$663$	13.5904	0.527809
$664$	0	0
$665$	−7.59692	−0.294596
$666$	0	0
$667$	−10.5813	−0.409711
$668$	0	0
$669$	−11.8947	−0.459874
$670$	0	0
$671$	3.85075	0.148657
$672$	0	0
$673$	−21.2095	−0.817565	−0.408783	−	0.912632i	$-0.634047\pi$
$673$	−21.2095	−0.817565	−0.408783	+	0.912632i	$0.634047\pi$
$674$	0	0
$675$	−9.78756	−0.376723
$676$	0	0
$677$	1.61274	0.0619828	0.0309914	−	0.999520i	$-0.490134\pi$
$677$	1.61274	0.0619828	0.0309914	+	0.999520i	$0.490134\pi$
$678$	0	0
$679$	−1.72213	−0.0660895
$680$	0	0
$681$	−27.5082	−1.05411
$682$	0	0
$683$	19.7241	0.754722	0.377361	−	0.926066i	$-0.376832\pi$
$683$	19.7241	0.754722	0.377361	+	0.926066i	$0.376832\pi$
$684$	0	0
$685$	−11.9077	−0.454971
$686$	0	0
$687$	−21.1042	−0.805174
$688$	0	0
$689$	8.11274	0.309071
$690$	0	0
$691$	−8.88226	−0.337897	−0.168949	−	0.985625i	$-0.554037\pi$
$691$	−8.88226	−0.337897	−0.168949	+	0.985625i	$0.554037\pi$
$692$	0	0
$693$	−0.820343	−0.0311623
$694$	0	0
$695$	−35.0375	−1.32905
$696$	0	0
$697$	4.66217	0.176592
$698$	0	0
$699$	−3.50628	−0.132620
$700$	0	0
$701$	−18.3901	−0.694586	−0.347293	−	0.937757i	$-0.612899\pi$
$701$	−18.3901	−0.694586	−0.347293	+	0.937757i	$0.612899\pi$
$702$	0	0
$703$	23.0321	0.868670
$704$	0	0
$705$	−16.8298	−0.633848
$706$	0	0
$707$	9.51514	0.357854
$708$	0	0
$709$	30.9027	1.16058	0.580288	−	0.814411i	$-0.302940\pi$
$709$	30.9027	1.16058	0.580288	+	0.814411i	$0.302940\pi$
$710$	0	0
$711$	−27.4348	−1.02888
$712$	0	0
$713$	−1.13993	−0.0426907
$714$	0	0
$715$	10.3300	0.386320
$716$	0	0
$717$	2.59153	0.0967824
$718$	0	0
$719$	−17.1226	−0.638565	−0.319283	−	0.947660i	$-0.603442\pi$
$719$	−17.1226	−0.638565	−0.319283	+	0.947660i	$0.603442\pi$
$720$	0	0
$721$	8.61277	0.320756
$722$	0	0
$723$	10.1784	0.378537
$724$	0	0
$725$	−8.99450	−0.334047
$726$	0	0
$727$	19.2445	0.713740	0.356870	−	0.934154i	$-0.383844\pi$
$727$	19.2445	0.713740	0.356870	+	0.934154i	$0.383844\pi$
$728$	0	0
$729$	19.4792	0.721453
$730$	0	0
$731$	0.534526	0.0197702
$732$	0	0
$733$	−18.6074	−0.687278	−0.343639	−	0.939102i	$-0.611660\pi$
$733$	−18.6074	−0.687278	−0.343639	+	0.939102i	$0.611660\pi$
$734$	0	0
$735$	−19.8925	−0.733744
$736$	0	0
$737$	10.1432	0.373628
$738$	0	0
$739$	−24.8671	−0.914752	−0.457376	−	0.889273i	$-0.651211\pi$
$739$	−24.8671	−0.914752	−0.457376	+	0.889273i	$0.651211\pi$
$740$	0	0
$741$	−27.2741	−1.00194
$742$	0	0
$743$	−25.7228	−0.943678	−0.471839	−	0.881685i	$-0.656410\pi$
$743$	−25.7228	−0.943678	−0.471839	+	0.881685i	$0.656410\pi$
$744$	0	0
$745$	−0.397941	−0.0145794
$746$	0	0
$747$	−11.2720	−0.412422
$748$	0	0
$749$	−5.59183	−0.204321
$750$	0	0
$751$	−0.538830	−0.0196622	−0.00983109	−	0.999952i	$-0.503129\pi$
$751$	−0.538830	−0.0196622	−0.00983109	+	0.999952i	$0.503129\pi$
$752$	0	0
$753$	26.6551	0.971366
$754$	0	0
$755$	−25.9873	−0.945774
$756$	0	0
$757$	27.0333	0.982542	0.491271	−	0.871007i	$-0.336533\pi$
$757$	27.0333	0.982542	0.491271	+	0.871007i	$0.336533\pi$
$758$	0	0
$759$	−2.29363	−0.0832536
$760$	0	0
$761$	−44.8896	−1.62725	−0.813624	−	0.581391i	$-0.802509\pi$
$761$	−44.8896	−1.62725	−0.813624	+	0.581391i	$0.802509\pi$
$762$	0	0
$763$	1.41311	0.0511581
$764$	0	0
$765$	12.9746	0.469096
$766$	0	0
$767$	−22.0961	−0.797846
$768$	0	0
$769$	−7.34134	−0.264736	−0.132368	−	0.991201i	$-0.542258\pi$
$769$	−7.34134	−0.264736	−0.132368	+	0.991201i	$0.542258\pi$
$770$	0	0
$771$	−14.8882	−0.536184
$772$	0	0
$773$	26.1605	0.940929	0.470465	−	0.882419i	$-0.344086\pi$
$773$	26.1605	0.940929	0.470465	+	0.882419i	$0.344086\pi$
$774$	0	0
$775$	−0.968979	−0.0348067
$776$	0	0
$777$	−2.27205	−0.0815095
$778$	0	0
$779$	−9.35632	−0.335225
$780$	0	0
$781$	−3.06346	−0.109619
$782$	0	0
$783$	26.1316	0.933869
$784$	0	0
$785$	19.5569	0.698016
$786$	0	0
$787$	17.1339	0.610758	0.305379	−	0.952231i	$-0.401217\pi$
$787$	17.1339	0.610758	0.305379	+	0.952231i	$0.401217\pi$
$788$	0	0
$789$	−1.80210	−0.0641566
$790$	0	0
$791$	7.86635	0.279695
$792$	0	0
$793$	17.1538	0.609149
$794$	0	0
$795$	−5.70234	−0.202241
$796$	0	0
$797$	43.8016	1.55153	0.775765	−	0.631021i	$-0.217364\pi$
$797$	43.8016	1.55153	0.775765	+	0.631021i	$0.217364\pi$
$798$	0	0
$799$	16.3919	0.579904
$800$	0	0
$801$	−14.0066	−0.494898
$802$	0	0
$803$	6.61964	0.233602
$804$	0	0
$805$	−2.84594	−0.100306
$806$	0	0
$807$	14.1333	0.497516
$808$	0	0
$809$	7.20294	0.253242	0.126621	−	0.991951i	$-0.459587\pi$
$809$	7.20294	0.253242	0.126621	+	0.991951i	$0.459587\pi$
$810$	0	0
$811$	−2.42833	−0.0852703	−0.0426352	−	0.999091i	$-0.513575\pi$
$811$	−2.42833	−0.0852703	−0.0426352	+	0.999091i	$0.513575\pi$
$812$	0	0
$813$	24.7057	0.866465
$814$	0	0
$815$	−35.4878	−1.24309
$816$	0	0
$817$	−1.07272	−0.0375297
$818$	0	0
$819$	−3.65435	−0.127693
$820$	0	0
$821$	43.9322	1.53324	0.766622	−	0.642098i	$-0.221936\pi$
$821$	43.9322	1.53324	0.766622	+	0.642098i	$0.221936\pi$
$822$	0	0
$823$	34.9644	1.21878	0.609391	−	0.792869i	$-0.291414\pi$
$823$	34.9644	1.21878	0.609391	+	0.792869i	$0.291414\pi$
$824$	0	0
$825$	−1.94967	−0.0678787
$826$	0	0
$827$	−21.3408	−0.742092	−0.371046	−	0.928615i	$-0.621001\pi$
$827$	−21.3408	−0.742092	−0.371046	+	0.928615i	$0.621001\pi$
$828$	0	0
$829$	−12.1679	−0.422607	−0.211304	−	0.977420i	$-0.567771\pi$
$829$	−12.1679	−0.422607	−0.211304	+	0.977420i	$0.567771\pi$
$830$	0	0
$831$	−20.3090	−0.704511
$832$	0	0
$833$	19.3748	0.671298
$834$	0	0
$835$	5.19631	0.179826
$836$	0	0
$837$	2.81517	0.0973064
$838$	0	0
$839$	26.6511	0.920098	0.460049	−	0.887893i	$-0.347832\pi$
$839$	26.6511	0.920098	0.460049	+	0.887893i	$0.347832\pi$
$840$	0	0
$841$	−4.98574	−0.171922
$842$	0	0
$843$	18.5765	0.639807
$844$	0	0
$845$	12.0285	0.413792
$846$	0	0
$847$	5.09822	0.175177
$848$	0	0
$849$	2.09285	0.0718264
$850$	0	0
$851$	8.62820	0.295771
$852$	0	0
$853$	41.7610	1.42987	0.714935	−	0.699191i	$-0.246456\pi$
$853$	41.7610	1.42987	0.714935	+	0.699191i	$0.246456\pi$
$854$	0	0
$855$	−26.0381	−0.890485
$856$	0	0
$857$	28.8868	0.986753	0.493377	−	0.869816i	$-0.335762\pi$
$857$	28.8868	0.986753	0.493377	+	0.869816i	$0.335762\pi$
$858$	0	0
$859$	6.27850	0.214220	0.107110	−	0.994247i	$-0.465840\pi$
$859$	6.27850	0.214220	0.107110	+	0.994247i	$0.465840\pi$
$860$	0	0
$861$	0.922977	0.0314550
$862$	0	0
$863$	15.2552	0.519293	0.259647	−	0.965704i	$-0.416394\pi$
$863$	15.2552	0.519293	0.259647	+	0.965704i	$0.416394\pi$
$864$	0	0
$865$	−12.3303	−0.419244
$866$	0	0
$867$	−9.87018	−0.335209
$868$	0	0
$869$	−14.9535	−0.507264
$870$	0	0
$871$	45.1844	1.53101
$872$	0	0
$873$	−5.90254	−0.199771
$874$	0	0
$875$	4.17092	0.141003
$876$	0	0
$877$	−24.3589	−0.822543	−0.411272	−	0.911513i	$-0.634915\pi$
$877$	−24.3589	−0.822543	−0.411272	+	0.911513i	$0.634915\pi$
$878$	0	0
$879$	3.80752	0.128424
$880$	0	0
$881$	−12.4734	−0.420241	−0.210120	−	0.977676i	$-0.567386\pi$
$881$	−12.4734	−0.420241	−0.210120	+	0.977676i	$0.567386\pi$
$882$	0	0
$883$	−4.06361	−0.136751	−0.0683757	−	0.997660i	$-0.521782\pi$
$883$	−4.06361	−0.136751	−0.0683757	+	0.997660i	$0.521782\pi$
$884$	0	0
$885$	15.5311	0.522072
$886$	0	0
$887$	−3.96992	−0.133297	−0.0666484	−	0.997777i	$-0.521231\pi$
$887$	−3.96992	−0.133297	−0.0666484	+	0.997777i	$0.521231\pi$
$888$	0	0
$889$	1.31940	0.0442514
$890$	0	0
$891$	0.782549	0.0262164
$892$	0	0
$893$	−32.8962	−1.10083
$894$	0	0
$895$	18.2933	0.611478
$896$	0	0
$897$	−10.2174	−0.341148
$898$	0	0
$899$	2.58706	0.0862833
$900$	0	0
$901$	5.55396	0.185029
$902$	0	0
$903$	0.105821	0.00352150
$904$	0	0
$905$	29.7571	0.989158
$906$	0	0
$907$	−17.3051	−0.574606	−0.287303	−	0.957840i	$-0.592759\pi$
$907$	−17.3051	−0.574606	−0.287303	+	0.957840i	$0.592759\pi$
$908$	0	0
$909$	32.6128	1.08170
$910$	0	0
$911$	−19.1207	−0.633499	−0.316749	−	0.948509i	$-0.602591\pi$
$911$	−19.1207	−0.633499	−0.316749	+	0.948509i	$0.602591\pi$
$912$	0	0
$913$	−6.14390	−0.203333
$914$	0	0
$915$	−12.0572	−0.398598
$916$	0	0
$917$	6.10540	0.201618
$918$	0	0
$919$	39.1556	1.29162	0.645812	−	0.763496i	$-0.276519\pi$
$919$	39.1556	1.29162	0.645812	+	0.763496i	$0.276519\pi$
$920$	0	0
$921$	−14.2809	−0.470573
$922$	0	0
$923$	−13.6467	−0.449185
$924$	0	0
$925$	7.33427	0.241149
$926$	0	0
$927$	29.5199	0.969561
$928$	0	0
$929$	32.7045	1.07300	0.536499	−	0.843901i	$-0.319746\pi$
$929$	32.7045	1.07300	0.536499	+	0.843901i	$0.319746\pi$
$930$	0	0
$931$	−38.8826	−1.27433
$932$	0	0
$933$	11.2584	0.368585
$934$	0	0
$935$	7.07188	0.231275
$936$	0	0
$937$	−42.7575	−1.39683	−0.698413	−	0.715695i	$-0.746110\pi$
$937$	−42.7575	−1.39683	−0.698413	+	0.715695i	$0.746110\pi$
$938$	0	0
$939$	−0.873773	−0.0285145
$940$	0	0
$941$	28.4079	0.926071	0.463035	−	0.886340i	$-0.346760\pi$
$941$	28.4079	0.926071	0.463035	+	0.886340i	$0.346760\pi$
$942$	0	0
$943$	−3.50504	−0.114140
$944$	0	0
$945$	7.02833	0.228631
$946$	0	0
$947$	−52.0042	−1.68991	−0.844955	−	0.534837i	$-0.820373\pi$
$947$	−52.0042	−1.68991	−0.844955	+	0.534837i	$0.820373\pi$
$948$	0	0
$949$	29.4883	0.957229
$950$	0	0
$951$	9.55406	0.309812
$952$	0	0
$953$	−41.0806	−1.33073	−0.665366	−	0.746518i	$-0.731724\pi$
$953$	−41.0806	−1.33073	−0.665366	+	0.746518i	$0.731724\pi$
$954$	0	0
$955$	−21.4159	−0.693003
$956$	0	0
$957$	5.20539	0.168266
$958$	0	0
$959$	2.29605	0.0741434
$960$	0	0
$961$	−30.7213	−0.991010
$962$	0	0
$963$	−19.1658	−0.617608
$964$	0	0
$965$	48.1815	1.55102
$966$	0	0
$967$	−43.4278	−1.39654	−0.698272	−	0.715833i	$-0.746047\pi$
$967$	−43.4278	−1.39654	−0.698272	+	0.715833i	$0.746047\pi$
$968$	0	0
$969$	−18.6718	−0.599825
$970$	0	0
$971$	52.7901	1.69411	0.847057	−	0.531502i	$-0.178372\pi$
$971$	52.7901	1.69411	0.847057	+	0.531502i	$0.178372\pi$
$972$	0	0
$973$	6.75595	0.216586
$974$	0	0
$975$	−8.68511	−0.278146
$976$	0	0
$977$	−46.0277	−1.47256	−0.736278	−	0.676680i	$-0.763418\pi$
$977$	−46.0277	−1.47256	−0.736278	+	0.676680i	$0.763418\pi$
$978$	0	0
$979$	−7.63439	−0.243996
$980$	0	0
$981$	4.84338	0.154637
$982$	0	0
$983$	14.7585	0.470722	0.235361	−	0.971908i	$-0.424373\pi$
$983$	14.7585	0.470722	0.235361	+	0.971908i	$0.424373\pi$
$984$	0	0
$985$	−39.6003	−1.26177
$986$	0	0
$987$	3.24513	0.103294
$988$	0	0
$989$	−0.401859	−0.0127784
$990$	0	0
$991$	−3.21252	−0.102049	−0.0510245	−	0.998697i	$-0.516249\pi$
$991$	−3.21252	−0.102049	−0.0510245	+	0.998697i	$0.516249\pi$
$992$	0	0
$993$	4.90850	0.155766
$994$	0	0
$995$	7.50907	0.238053
$996$	0	0
$997$	−14.2785	−0.452205	−0.226103	−	0.974103i	$-0.572599\pi$
$997$	−14.2785	−0.452205	−0.226103	+	0.974103i	$0.572599\pi$
$998$	0	0
$999$	−21.3082	−0.674162

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.12789 q^{3} +2.61447 q^{5} -0.504123 q^{7} -1.72786 q^{9} +O(q^{10})\)	\(q+1.12789 q^{3} +2.61447 q^{5} -0.504123 q^{7} -1.72786 q^{9} -0.941783 q^{11} -4.19532 q^{13} +2.94884 q^{15} -2.87211 q^{17} +5.76392 q^{19} -0.568596 q^{21} +2.15926 q^{23} +1.83545 q^{25} -5.33252 q^{27} -4.90044 q^{29} -0.527925 q^{31} -1.06223 q^{33} -1.31801 q^{35} +3.99590 q^{37} -4.73187 q^{39} -1.62326 q^{41} -0.186109 q^{43} -4.51744 q^{45} -5.70727 q^{47} -6.74586 q^{49} -3.23943 q^{51} -1.93376 q^{53} -2.46226 q^{55} +6.50108 q^{57} +5.26685 q^{59} -4.08879 q^{61} +0.871053 q^{63} -10.9685 q^{65} -10.7702 q^{67} +2.43542 q^{69} +3.25283 q^{71} -7.02884 q^{73} +2.07019 q^{75} +0.474774 q^{77} +15.8779 q^{79} -0.830923 q^{81} +6.52369 q^{83} -7.50904 q^{85} -5.52716 q^{87} +8.10631 q^{89} +2.11496 q^{91} -0.595442 q^{93} +15.0696 q^{95} +3.41610 q^{97} +1.62727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{5} - 3 q^{7} - 3 q^{9}+O(q^{10}) \) 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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