LMFDB - Embedded newform 8024.2.a.x.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.63019 q^{3} +0.886839 q^{5} -4.44237 q^{7} -0.342474 q^{9} +O(q^{10})$	$q-1.63019 q^{3} +0.886839 q^{5} -4.44237 q^{7} -0.342474 q^{9} -1.70705 q^{11} -3.51934 q^{13} -1.44572 q^{15} -1.00000 q^{17} +3.26112 q^{19} +7.24192 q^{21} +4.44303 q^{23} -4.21352 q^{25} +5.44887 q^{27} +2.22206 q^{29} +6.17257 q^{31} +2.78282 q^{33} -3.93967 q^{35} -3.96451 q^{37} +5.73721 q^{39} +0.720474 q^{41} +10.1809 q^{43} -0.303719 q^{45} +4.13250 q^{47} +12.7347 q^{49} +1.63019 q^{51} +5.79616 q^{53} -1.51388 q^{55} -5.31626 q^{57} +1.00000 q^{59} +0.334786 q^{61} +1.52140 q^{63} -3.12109 q^{65} +5.32282 q^{67} -7.24299 q^{69} +10.6524 q^{71} +3.22136 q^{73} +6.86884 q^{75} +7.58335 q^{77} -11.9751 q^{79} -7.85529 q^{81} -12.6494 q^{83} -0.886839 q^{85} -3.62238 q^{87} -2.45717 q^{89} +15.6342 q^{91} -10.0625 q^{93} +2.89209 q^{95} -12.7605 q^{97} +0.584620 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * 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
$3$	−1.63019	−0.941192	−0.470596	−	0.882349i	$-0.655961\pi$
$3$	−1.63019	−0.941192	−0.470596	+	0.882349i	$0.655961\pi$
$4$	0	0
$5$	0.886839	0.396607	0.198303	−	0.980141i	$-0.436457\pi$
$5$	0.886839	0.396607	0.198303	+	0.980141i	$0.436457\pi$
$6$	0	0
$7$	−4.44237	−1.67906	−0.839530	−	0.543314i	$-0.817169\pi$
$7$	−4.44237	−1.67906	−0.839530	+	0.543314i	$0.817169\pi$
$8$	0	0
$9$	−0.342474	−0.114158
$10$	0	0
$11$	−1.70705	−0.514695	−0.257347	−	0.966319i	$-0.582848\pi$
$11$	−1.70705	−0.514695	−0.257347	+	0.966319i	$0.582848\pi$
$12$	0	0
$13$	−3.51934	−0.976091	−0.488045	−	0.872818i	$-0.662290\pi$
$13$	−3.51934	−0.976091	−0.488045	+	0.872818i	$0.662290\pi$
$14$	0	0
$15$	−1.44572	−0.373283
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	3.26112	0.748153	0.374077	−	0.927398i	$-0.377960\pi$
$19$	3.26112	0.748153	0.374077	+	0.927398i	$0.377960\pi$
$20$	0	0
$21$	7.24192	1.58032
$22$	0	0
$23$	4.44303	0.926435	0.463217	−	0.886245i	$-0.346695\pi$
$23$	4.44303	0.926435	0.463217	+	0.886245i	$0.346695\pi$
$24$	0	0
$25$	−4.21352	−0.842703
$26$	0	0
$27$	5.44887	1.04864
$28$	0	0
$29$	2.22206	0.412626	0.206313	−	0.978486i	$-0.433854\pi$
$29$	2.22206	0.412626	0.206313	+	0.978486i	$0.433854\pi$
$30$	0	0
$31$	6.17257	1.10863	0.554313	−	0.832308i	$-0.312981\pi$
$31$	6.17257	1.10863	0.554313	+	0.832308i	$0.312981\pi$
$32$	0	0
$33$	2.78282	0.484427
$34$	0	0
$35$	−3.93967	−0.665926
$36$	0	0
$37$	−3.96451	−0.651761	−0.325880	−	0.945411i	$-0.605661\pi$
$37$	−3.96451	−0.651761	−0.325880	+	0.945411i	$0.605661\pi$
$38$	0	0
$39$	5.73721	0.918689
$40$	0	0
$41$	0.720474	0.112519	0.0562596	−	0.998416i	$-0.482083\pi$
$41$	0.720474	0.112519	0.0562596	+	0.998416i	$0.482083\pi$
$42$	0	0
$43$	10.1809	1.55257	0.776284	−	0.630383i	$-0.217102\pi$
$43$	10.1809	1.55257	0.776284	+	0.630383i	$0.217102\pi$
$44$	0	0
$45$	−0.303719	−0.0452758
$46$	0	0
$47$	4.13250	0.602787	0.301394	−	0.953500i	$-0.402548\pi$
$47$	4.13250	0.602787	0.301394	+	0.953500i	$0.402548\pi$
$48$	0	0
$49$	12.7347	1.81924
$50$	0	0
$51$	1.63019	0.228273
$52$	0	0
$53$	5.79616	0.796163	0.398082	−	0.917350i	$-0.369676\pi$
$53$	5.79616	0.796163	0.398082	+	0.917350i	$0.369676\pi$
$54$	0	0
$55$	−1.51388	−0.204131
$56$	0	0
$57$	−5.31626	−0.704155
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	0.334786	0.0428650	0.0214325	−	0.999770i	$-0.493177\pi$
$61$	0.334786	0.0428650	0.0214325	+	0.999770i	$0.493177\pi$
$62$	0	0
$63$	1.52140	0.191678
$64$	0	0
$65$	−3.12109	−0.387124
$66$	0	0
$67$	5.32282	0.650285	0.325143	−	0.945665i	$-0.394588\pi$
$67$	5.32282	0.650285	0.325143	+	0.945665i	$0.394588\pi$
$68$	0	0
$69$	−7.24299	−0.871953
$70$	0	0
$71$	10.6524	1.26421	0.632103	−	0.774884i	$-0.282192\pi$
$71$	10.6524	1.26421	0.632103	+	0.774884i	$0.282192\pi$
$72$	0	0
$73$	3.22136	0.377032	0.188516	−	0.982070i	$-0.439632\pi$
$73$	3.22136	0.377032	0.188516	+	0.982070i	$0.439632\pi$
$74$	0	0
$75$	6.86884	0.793145
$76$	0	0
$77$	7.58335	0.864203
$78$	0	0
$79$	−11.9751	−1.34730	−0.673651	−	0.739049i	$-0.735275\pi$
$79$	−11.9751	−1.34730	−0.673651	+	0.739049i	$0.735275\pi$
$80$	0	0
$81$	−7.85529	−0.872810
$82$	0	0
$83$	−12.6494	−1.38845	−0.694226	−	0.719757i	$-0.744253\pi$
$83$	−12.6494	−1.38845	−0.694226	+	0.719757i	$0.744253\pi$
$84$	0	0
$85$	−0.886839	−0.0961912
$86$	0	0
$87$	−3.62238	−0.388360
$88$	0	0
$89$	−2.45717	−0.260460	−0.130230	−	0.991484i	$-0.541572\pi$
$89$	−2.45717	−0.260460	−0.130230	+	0.991484i	$0.541572\pi$
$90$	0	0
$91$	15.6342	1.63891
$92$	0	0
$93$	−10.0625	−1.04343
$94$	0	0
$95$	2.89209	0.296722
$96$	0	0
$97$	−12.7605	−1.29564	−0.647818	−	0.761795i	$-0.724318\pi$
$97$	−12.7605	−1.29564	−0.647818	+	0.761795i	$0.724318\pi$
$98$	0	0
$99$	0.584620	0.0587565
$100$	0	0
$101$	−17.5015	−1.74146	−0.870732	−	0.491757i	$-0.836355\pi$
$101$	−17.5015	−1.74146	−0.870732	+	0.491757i	$0.836355\pi$
$102$	0	0
$103$	11.6965	1.15249	0.576243	−	0.817278i	$-0.304518\pi$
$103$	11.6965	1.15249	0.576243	+	0.817278i	$0.304518\pi$
$104$	0	0
$105$	6.42242	0.626764
$106$	0	0
$107$	−11.0043	−1.06383	−0.531915	−	0.846798i	$-0.678527\pi$
$107$	−11.0043	−1.06383	−0.531915	+	0.846798i	$0.678527\pi$
$108$	0	0
$109$	−5.39661	−0.516901	−0.258451	−	0.966024i	$-0.583212\pi$
$109$	−5.39661	−0.516901	−0.258451	+	0.966024i	$0.583212\pi$
$110$	0	0
$111$	6.46291	0.613432
$112$	0	0
$113$	17.3284	1.63012	0.815061	−	0.579376i	$-0.196704\pi$
$113$	17.3284	1.63012	0.815061	+	0.579376i	$0.196704\pi$
$114$	0	0
$115$	3.94025	0.367430
$116$	0	0
$117$	1.20528	0.111429
$118$	0	0
$119$	4.44237	0.407232
$120$	0	0
$121$	−8.08598	−0.735089
$122$	0	0
$123$	−1.17451	−0.105902
$124$	0	0
$125$	−8.17091	−0.730828
$126$	0	0
$127$	6.62380	0.587767	0.293884	−	0.955841i	$-0.405052\pi$
$127$	6.62380	0.587767	0.293884	+	0.955841i	$0.405052\pi$
$128$	0	0
$129$	−16.5968	−1.46126
$130$	0	0
$131$	9.56733	0.835901	0.417951	−	0.908470i	$-0.362749\pi$
$131$	9.56733	0.835901	0.417951	+	0.908470i	$0.362749\pi$
$132$	0	0
$133$	−14.4871	−1.25619
$134$	0	0
$135$	4.83228	0.415896
$136$	0	0
$137$	−3.36512	−0.287502	−0.143751	−	0.989614i	$-0.545916\pi$
$137$	−3.36512	−0.287502	−0.143751	+	0.989614i	$0.545916\pi$
$138$	0	0
$139$	−9.59881	−0.814161	−0.407080	−	0.913392i	$-0.633453\pi$
$139$	−9.59881	−0.814161	−0.407080	+	0.913392i	$0.633453\pi$
$140$	0	0
$141$	−6.73677	−0.567339
$142$	0	0
$143$	6.00770	0.502389
$144$	0	0
$145$	1.97061	0.163650
$146$	0	0
$147$	−20.7600	−1.71225
$148$	0	0
$149$	−21.2234	−1.73869	−0.869344	−	0.494207i	$-0.835458\pi$
$149$	−21.2234	−1.73869	−0.869344	+	0.494207i	$0.835458\pi$
$150$	0	0
$151$	−9.52358	−0.775018	−0.387509	−	0.921866i	$-0.626664\pi$
$151$	−9.52358	−0.775018	−0.387509	+	0.921866i	$0.626664\pi$
$152$	0	0
$153$	0.342474	0.0276874
$154$	0	0
$155$	5.47408	0.439688
$156$	0	0
$157$	13.9040	1.10966	0.554831	−	0.831963i	$-0.312783\pi$
$157$	13.9040	1.10966	0.554831	+	0.831963i	$0.312783\pi$
$158$	0	0
$159$	−9.44885	−0.749343
$160$	0	0
$161$	−19.7376	−1.55554
$162$	0	0
$163$	1.85633	0.145399	0.0726995	−	0.997354i	$-0.476839\pi$
$163$	1.85633	0.145399	0.0726995	+	0.997354i	$0.476839\pi$
$164$	0	0
$165$	2.46791	0.192127
$166$	0	0
$167$	12.0787	0.934679	0.467340	−	0.884078i	$-0.345213\pi$
$167$	12.0787	0.934679	0.467340	+	0.884078i	$0.345213\pi$
$168$	0	0
$169$	−0.614211	−0.0472470
$170$	0	0
$171$	−1.11685	−0.0854076
$172$	0	0
$173$	22.2688	1.69306	0.846532	−	0.532338i	$-0.178686\pi$
$173$	22.2688	1.69306	0.846532	+	0.532338i	$0.178686\pi$
$174$	0	0
$175$	18.7180	1.41495
$176$	0	0
$177$	−1.63019	−0.122533
$178$	0	0
$179$	17.1065	1.27860	0.639300	−	0.768957i	$-0.279224\pi$
$179$	17.1065	1.27860	0.639300	+	0.768957i	$0.279224\pi$
$180$	0	0
$181$	−11.5104	−0.855560	−0.427780	−	0.903883i	$-0.640704\pi$
$181$	−11.5104	−0.855560	−0.427780	+	0.903883i	$0.640704\pi$
$182$	0	0
$183$	−0.545766	−0.0403442
$184$	0	0
$185$	−3.51588	−0.258493
$186$	0	0
$187$	1.70705	0.124832
$188$	0	0
$189$	−24.2059	−1.76072
$190$	0	0
$191$	−18.1416	−1.31268	−0.656340	−	0.754465i	$-0.727896\pi$
$191$	−18.1416	−1.31268	−0.656340	+	0.754465i	$0.727896\pi$
$192$	0	0
$193$	10.7323	0.772531	0.386266	−	0.922388i	$-0.373765\pi$
$193$	10.7323	0.772531	0.386266	+	0.922388i	$0.373765\pi$
$194$	0	0
$195$	5.08798	0.364358
$196$	0	0
$197$	−21.3891	−1.52391	−0.761955	−	0.647630i	$-0.775760\pi$
$197$	−21.3891	−1.52391	−0.761955	+	0.647630i	$0.775760\pi$
$198$	0	0
$199$	−24.5820	−1.74258	−0.871288	−	0.490773i	$-0.836715\pi$
$199$	−24.5820	−1.74258	−0.871288	+	0.490773i	$0.836715\pi$
$200$	0	0
$201$	−8.67721	−0.612043
$202$	0	0
$203$	−9.87121	−0.692823
$204$	0	0
$205$	0.638945	0.0446258
$206$	0	0
$207$	−1.52162	−0.105760
$208$	0	0
$209$	−5.56690	−0.385071
$210$	0	0
$211$	11.2041	0.771321	0.385661	−	0.922641i	$-0.373974\pi$
$211$	11.2041	0.771321	0.385661	+	0.922641i	$0.373974\pi$
$212$	0	0
$213$	−17.3654	−1.18986
$214$	0	0
$215$	9.02880	0.615759
$216$	0	0
$217$	−27.4209	−1.86145
$218$	0	0
$219$	−5.25144	−0.354859
$220$	0	0
$221$	3.51934	0.236737
$222$	0	0
$223$	7.00259	0.468928	0.234464	−	0.972125i	$-0.424666\pi$
$223$	7.00259	0.468928	0.234464	+	0.972125i	$0.424666\pi$
$224$	0	0
$225$	1.44302	0.0962013
$226$	0	0
$227$	−27.6510	−1.83526	−0.917632	−	0.397431i	$-0.869902\pi$
$227$	−27.6510	−1.83526	−0.917632	+	0.397431i	$0.869902\pi$
$228$	0	0
$229$	−9.83751	−0.650081	−0.325040	−	0.945700i	$-0.605378\pi$
$229$	−9.83751	−0.650081	−0.325040	+	0.945700i	$0.605378\pi$
$230$	0	0
$231$	−12.3623	−0.813381
$232$	0	0
$233$	−18.3237	−1.20042	−0.600212	−	0.799841i	$-0.704917\pi$
$233$	−18.3237	−1.20042	−0.600212	+	0.799841i	$0.704917\pi$
$234$	0	0
$235$	3.66487	0.239069
$236$	0	0
$237$	19.5217	1.26807
$238$	0	0
$239$	−2.85850	−0.184901	−0.0924504	−	0.995717i	$-0.529470\pi$
$239$	−2.85850	−0.184901	−0.0924504	+	0.995717i	$0.529470\pi$
$240$	0	0
$241$	4.88246	0.314507	0.157253	−	0.987558i	$-0.449736\pi$
$241$	4.88246	0.314507	0.157253	+	0.987558i	$0.449736\pi$
$242$	0	0
$243$	−3.54099	−0.227155
$244$	0	0
$245$	11.2936	0.721523
$246$	0	0
$247$	−11.4770	−0.730265
$248$	0	0
$249$	20.6209	1.30680
$250$	0	0
$251$	−25.0569	−1.58158	−0.790788	−	0.612090i	$-0.790329\pi$
$251$	−25.0569	−1.58158	−0.790788	+	0.612090i	$0.790329\pi$
$252$	0	0
$253$	−7.58447	−0.476831
$254$	0	0
$255$	1.44572	0.0905344
$256$	0	0
$257$	−2.17102	−0.135425	−0.0677123	−	0.997705i	$-0.521570\pi$
$257$	−2.17102	−0.135425	−0.0677123	+	0.997705i	$0.521570\pi$
$258$	0	0
$259$	17.6118	1.09435
$260$	0	0
$261$	−0.760996	−0.0471045
$262$	0	0
$263$	−16.2735	−1.00347	−0.501733	−	0.865023i	$-0.667304\pi$
$263$	−16.2735	−1.00347	−0.501733	+	0.865023i	$0.667304\pi$
$264$	0	0
$265$	5.14026	0.315764
$266$	0	0
$267$	4.00566	0.245143
$268$	0	0
$269$	6.63144	0.404326	0.202163	−	0.979352i	$-0.435203\pi$
$269$	6.63144	0.404326	0.202163	+	0.979352i	$0.435203\pi$
$270$	0	0
$271$	24.9281	1.51427	0.757137	−	0.653256i	$-0.226598\pi$
$271$	24.9281	1.51427	0.757137	+	0.653256i	$0.226598\pi$
$272$	0	0
$273$	−25.4868	−1.54253
$274$	0	0
$275$	7.19268	0.433735
$276$	0	0
$277$	32.7397	1.96714	0.983568	−	0.180540i	$-0.0577846\pi$
$277$	32.7397	1.96714	0.983568	+	0.180540i	$0.0577846\pi$
$278$	0	0
$279$	−2.11394	−0.126558
$280$	0	0
$281$	15.8090	0.943086	0.471543	−	0.881843i	$-0.343697\pi$
$281$	15.8090	0.943086	0.471543	+	0.881843i	$0.343697\pi$
$282$	0	0
$283$	26.4977	1.57512	0.787562	−	0.616235i	$-0.211343\pi$
$283$	26.4977	1.57512	0.787562	+	0.616235i	$0.211343\pi$
$284$	0	0
$285$	−4.71467	−0.279273
$286$	0	0
$287$	−3.20062	−0.188926
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	20.8021	1.21944
$292$	0	0
$293$	−1.85774	−0.108530	−0.0542652	−	0.998527i	$-0.517282\pi$
$293$	−1.85774	−0.108530	−0.0542652	+	0.998527i	$0.517282\pi$
$294$	0	0
$295$	0.886839	0.0516338
$296$	0	0
$297$	−9.30150	−0.539728
$298$	0	0
$299$	−15.6365	−0.904284
$300$	0	0
$301$	−45.2272	−2.60685
$302$	0	0
$303$	28.5308	1.63905
$304$	0	0
$305$	0.296901	0.0170005
$306$	0	0
$307$	7.38232	0.421331	0.210666	−	0.977558i	$-0.432437\pi$
$307$	7.38232	0.421331	0.210666	+	0.977558i	$0.432437\pi$
$308$	0	0
$309$	−19.0675	−1.08471
$310$	0	0
$311$	−17.6558	−1.00117	−0.500585	−	0.865687i	$-0.666882\pi$
$311$	−17.6558	−1.00117	−0.500585	+	0.865687i	$0.666882\pi$
$312$	0	0
$313$	−12.5261	−0.708020	−0.354010	−	0.935242i	$-0.615182\pi$
$313$	−12.5261	−0.708020	−0.354010	+	0.935242i	$0.615182\pi$
$314$	0	0
$315$	1.34923	0.0760208
$316$	0	0
$317$	21.4035	1.20214	0.601069	−	0.799197i	$-0.294742\pi$
$317$	21.4035	1.20214	0.601069	+	0.799197i	$0.294742\pi$
$318$	0	0
$319$	−3.79316	−0.212376
$320$	0	0
$321$	17.9392	1.00127
$322$	0	0
$323$	−3.26112	−0.181454
$324$	0	0
$325$	14.8288	0.822555
$326$	0	0
$327$	8.79750	0.486503
$328$	0	0
$329$	−18.3581	−1.01212
$330$	0	0
$331$	−12.5984	−0.692468	−0.346234	−	0.938148i	$-0.612540\pi$
$331$	−12.5984	−0.692468	−0.346234	+	0.938148i	$0.612540\pi$
$332$	0	0
$333$	1.35774	0.0744037
$334$	0	0
$335$	4.72048	0.257907
$336$	0	0
$337$	10.1330	0.551979	0.275989	−	0.961161i	$-0.410995\pi$
$337$	10.1330	0.551979	0.275989	+	0.961161i	$0.410995\pi$
$338$	0	0
$339$	−28.2487	−1.53426
$340$	0	0
$341$	−10.5369	−0.570604
$342$	0	0
$343$	−25.4756	−1.37555
$344$	0	0
$345$	−6.42336	−0.345822
$346$	0	0
$347$	17.0649	0.916094	0.458047	−	0.888928i	$-0.348549\pi$
$347$	17.0649	0.916094	0.458047	+	0.888928i	$0.348549\pi$
$348$	0	0
$349$	10.1605	0.543877	0.271939	−	0.962315i	$-0.412335\pi$
$349$	10.1605	0.543877	0.271939	+	0.962315i	$0.412335\pi$
$350$	0	0
$351$	−19.1765	−1.02356
$352$	0	0
$353$	−2.99358	−0.159332	−0.0796661	−	0.996822i	$-0.525385\pi$
$353$	−2.99358	−0.159332	−0.0796661	+	0.996822i	$0.525385\pi$
$354$	0	0
$355$	9.44696	0.501392
$356$	0	0
$357$	−7.24192	−0.383283
$358$	0	0
$359$	9.07364	0.478888	0.239444	−	0.970910i	$-0.423035\pi$
$359$	9.07364	0.478888	0.239444	+	0.970910i	$0.423035\pi$
$360$	0	0
$361$	−8.36507	−0.440267
$362$	0	0
$363$	13.1817	0.691860
$364$	0	0
$365$	2.85683	0.149533
$366$	0	0
$367$	23.3879	1.22084	0.610420	−	0.792078i	$-0.291001\pi$
$367$	23.3879	1.22084	0.610420	+	0.792078i	$0.291001\pi$
$368$	0	0
$369$	−0.246744	−0.0128450
$370$	0	0
$371$	−25.7487	−1.33681
$372$	0	0
$373$	−7.01317	−0.363128	−0.181564	−	0.983379i	$-0.558116\pi$
$373$	−7.01317	−0.363128	−0.181564	+	0.983379i	$0.558116\pi$
$374$	0	0
$375$	13.3201	0.687850
$376$	0	0
$377$	−7.82018	−0.402760
$378$	0	0
$379$	−7.13990	−0.366752	−0.183376	−	0.983043i	$-0.558703\pi$
$379$	−7.13990	−0.366752	−0.183376	+	0.983043i	$0.558703\pi$
$380$	0	0
$381$	−10.7981	−0.553202
$382$	0	0
$383$	30.6022	1.56370	0.781849	−	0.623468i	$-0.214277\pi$
$383$	30.6022	1.56370	0.781849	+	0.623468i	$0.214277\pi$
$384$	0	0
$385$	6.72522	0.342749
$386$	0	0
$387$	−3.48668	−0.177238
$388$	0	0
$389$	9.15691	0.464274	0.232137	−	0.972683i	$-0.425428\pi$
$389$	9.15691	0.464274	0.232137	+	0.972683i	$0.425428\pi$
$390$	0	0
$391$	−4.44303	−0.224693
$392$	0	0
$393$	−15.5966	−0.786743
$394$	0	0
$395$	−10.6200	−0.534349
$396$	0	0
$397$	38.5579	1.93516	0.967582	−	0.252557i	$-0.0812717\pi$
$397$	38.5579	1.93516	0.967582	+	0.252557i	$0.0812717\pi$
$398$	0	0
$399$	23.6168	1.18232
$400$	0	0
$401$	10.3074	0.514726	0.257363	−	0.966315i	$-0.417146\pi$
$401$	10.3074	0.514726	0.257363	+	0.966315i	$0.417146\pi$
$402$	0	0
$403$	−21.7234	−1.08212
$404$	0	0
$405$	−6.96638	−0.346162
$406$	0	0
$407$	6.76761	0.335458
$408$	0	0
$409$	12.3572	0.611022	0.305511	−	0.952188i	$-0.401173\pi$
$409$	12.3572	0.611022	0.305511	+	0.952188i	$0.401173\pi$
$410$	0	0
$411$	5.48579	0.270594
$412$	0	0
$413$	−4.44237	−0.218595
$414$	0	0
$415$	−11.2180	−0.550669
$416$	0	0
$417$	15.6479	0.766281
$418$	0	0
$419$	3.16628	0.154683	0.0773415	−	0.997005i	$-0.475357\pi$
$419$	3.16628	0.154683	0.0773415	+	0.997005i	$0.475357\pi$
$420$	0	0
$421$	17.2123	0.838878	0.419439	−	0.907784i	$-0.362227\pi$
$421$	17.2123	0.838878	0.419439	+	0.907784i	$0.362227\pi$
$422$	0	0
$423$	−1.41527	−0.0688130
$424$	0	0
$425$	4.21352	0.204386
$426$	0	0
$427$	−1.48724	−0.0719728
$428$	0	0
$429$	−9.79370	−0.472844
$430$	0	0
$431$	−1.83662	−0.0884669	−0.0442334	−	0.999021i	$-0.514085\pi$
$431$	−1.83662	−0.0884669	−0.0442334	+	0.999021i	$0.514085\pi$
$432$	0	0
$433$	24.8735	1.19535	0.597673	−	0.801740i	$-0.296092\pi$
$433$	24.8735	1.19535	0.597673	+	0.801740i	$0.296092\pi$
$434$	0	0
$435$	−3.21247	−0.154026
$436$	0	0
$437$	14.4893	0.693115
$438$	0	0
$439$	−38.1269	−1.81970	−0.909849	−	0.414939i	$-0.863803\pi$
$439$	−38.1269	−1.81970	−0.909849	+	0.414939i	$0.863803\pi$
$440$	0	0
$441$	−4.36130	−0.207681
$442$	0	0
$443$	10.7122	0.508950	0.254475	−	0.967079i	$-0.418097\pi$
$443$	10.7122	0.508950	0.254475	+	0.967079i	$0.418097\pi$
$444$	0	0
$445$	−2.17912	−0.103300
$446$	0	0
$447$	34.5982	1.63644
$448$	0	0
$449$	−13.5415	−0.639062	−0.319531	−	0.947576i	$-0.603525\pi$
$449$	−13.5415	−0.639062	−0.319531	+	0.947576i	$0.603525\pi$
$450$	0	0
$451$	−1.22989	−0.0579130
$452$	0	0
$453$	15.5253	0.729441
$454$	0	0
$455$	13.8651	0.650004
$456$	0	0
$457$	−38.0639	−1.78056	−0.890278	−	0.455418i	$-0.849490\pi$
$457$	−38.0639	−1.78056	−0.890278	+	0.455418i	$0.849490\pi$
$458$	0	0
$459$	−5.44887	−0.254332
$460$	0	0
$461$	−19.1640	−0.892555	−0.446278	−	0.894895i	$-0.647251\pi$
$461$	−19.1640	−0.892555	−0.446278	+	0.894895i	$0.647251\pi$
$462$	0	0
$463$	−2.89598	−0.134587	−0.0672937	−	0.997733i	$-0.521436\pi$
$463$	−2.89598	−0.134587	−0.0672937	+	0.997733i	$0.521436\pi$
$464$	0	0
$465$	−8.92380	−0.413831
$466$	0	0
$467$	−30.9611	−1.43271	−0.716356	−	0.697735i	$-0.754191\pi$
$467$	−30.9611	−1.43271	−0.716356	+	0.697735i	$0.754191\pi$
$468$	0	0
$469$	−23.6459	−1.09187
$470$	0	0
$471$	−22.6662	−1.04440
$472$	0	0
$473$	−17.3793	−0.799099
$474$	0	0
$475$	−13.7408	−0.630471
$476$	0	0
$477$	−1.98503	−0.0908884
$478$	0	0
$479$	21.5665	0.985401	0.492700	−	0.870199i	$-0.336010\pi$
$479$	21.5665	0.985401	0.492700	+	0.870199i	$0.336010\pi$
$480$	0	0
$481$	13.9525	0.636178
$482$	0	0
$483$	32.1760	1.46406
$484$	0	0
$485$	−11.3165	−0.513858
$486$	0	0
$487$	3.25825	0.147646	0.0738228	−	0.997271i	$-0.476480\pi$
$487$	3.25825	0.147646	0.0738228	+	0.997271i	$0.476480\pi$
$488$	0	0
$489$	−3.02618	−0.136848
$490$	0	0
$491$	−18.5111	−0.835394	−0.417697	−	0.908586i	$-0.637163\pi$
$491$	−18.5111	−0.835394	−0.417697	+	0.908586i	$0.637163\pi$
$492$	0	0
$493$	−2.22206	−0.100076
$494$	0	0
$495$	0.518464	0.0233032
$496$	0	0
$497$	−47.3219	−2.12268
$498$	0	0
$499$	13.2944	0.595141	0.297570	−	0.954700i	$-0.403824\pi$
$499$	13.2944	0.595141	0.297570	+	0.954700i	$0.403824\pi$
$500$	0	0
$501$	−19.6906	−0.879712
$502$	0	0
$503$	−11.7714	−0.524863	−0.262431	−	0.964951i	$-0.584524\pi$
$503$	−11.7714	−0.524863	−0.262431	+	0.964951i	$0.584524\pi$
$504$	0	0
$505$	−15.5210	−0.690676
$506$	0	0
$507$	1.00128	0.0444685
$508$	0	0
$509$	−21.9599	−0.973354	−0.486677	−	0.873582i	$-0.661791\pi$
$509$	−21.9599	−0.973354	−0.486677	+	0.873582i	$0.661791\pi$
$510$	0	0
$511$	−14.3105	−0.633059
$512$	0	0
$513$	17.7695	0.784540
$514$	0	0
$515$	10.3729	0.457084
$516$	0	0
$517$	−7.05439	−0.310252
$518$	0	0
$519$	−36.3024	−1.59350
$520$	0	0
$521$	10.7272	0.469966	0.234983	−	0.971999i	$-0.424497\pi$
$521$	10.7272	0.469966	0.234983	+	0.971999i	$0.424497\pi$
$522$	0	0
$523$	−7.71141	−0.337196	−0.168598	−	0.985685i	$-0.553924\pi$
$523$	−7.71141	−0.337196	−0.168598	+	0.985685i	$0.553924\pi$
$524$	0	0
$525$	−30.5140	−1.33174
$526$	0	0
$527$	−6.17257	−0.268881
$528$	0	0
$529$	−3.25952	−0.141718
$530$	0	0
$531$	−0.342474	−0.0148621
$532$	0	0
$533$	−2.53560	−0.109829
$534$	0	0
$535$	−9.75908	−0.421922
$536$	0	0
$537$	−27.8869	−1.20341
$538$	0	0
$539$	−21.7387	−0.936354
$540$	0	0
$541$	−13.9196	−0.598452	−0.299226	−	0.954182i	$-0.596728\pi$
$541$	−13.9196	−0.598452	−0.299226	+	0.954182i	$0.596728\pi$
$542$	0	0
$543$	18.7641	0.805246
$544$	0	0
$545$	−4.78592	−0.205006
$546$	0	0
$547$	0.0672567	0.00287569	0.00143784	−	0.999999i	$-0.499542\pi$
$547$	0.0672567	0.00287569	0.00143784	+	0.999999i	$0.499542\pi$
$548$	0	0
$549$	−0.114656	−0.00489338
$550$	0	0
$551$	7.24640	0.308707
$552$	0	0
$553$	53.1978	2.26220
$554$	0	0
$555$	5.73156	0.243291
$556$	0	0
$557$	−14.2035	−0.601821	−0.300910	−	0.953652i	$-0.597290\pi$
$557$	−14.2035	−0.601821	−0.300910	+	0.953652i	$0.597290\pi$
$558$	0	0
$559$	−35.8300	−1.51545
$560$	0	0
$561$	−2.78282	−0.117491
$562$	0	0
$563$	14.7398	0.621210	0.310605	−	0.950539i	$-0.399468\pi$
$563$	14.7398	0.621210	0.310605	+	0.950539i	$0.399468\pi$
$564$	0	0
$565$	15.3675	0.646517
$566$	0	0
$567$	34.8961	1.46550
$568$	0	0
$569$	−8.56560	−0.359088	−0.179544	−	0.983750i	$-0.557462\pi$
$569$	−8.56560	−0.359088	−0.179544	+	0.983750i	$0.557462\pi$
$570$	0	0
$571$	−20.1110	−0.841621	−0.420811	−	0.907149i	$-0.638254\pi$
$571$	−20.1110	−0.841621	−0.420811	+	0.907149i	$0.638254\pi$
$572$	0	0
$573$	29.5743	1.23548
$574$	0	0
$575$	−18.7208	−0.780710
$576$	0	0
$577$	−16.8904	−0.703157	−0.351578	−	0.936158i	$-0.614355\pi$
$577$	−16.8904	−0.703157	−0.351578	+	0.936158i	$0.614355\pi$
$578$	0	0
$579$	−17.4958	−0.727100
$580$	0	0
$581$	56.1933	2.33129
$582$	0	0
$583$	−9.89433	−0.409781
$584$	0	0
$585$	1.06889	0.0441933
$586$	0	0
$587$	−32.6326	−1.34689	−0.673446	−	0.739237i	$-0.735187\pi$
$587$	−32.6326	−1.34689	−0.673446	+	0.739237i	$0.735187\pi$
$588$	0	0
$589$	20.1295	0.829422
$590$	0	0
$591$	34.8683	1.43429
$592$	0	0
$593$	−34.7082	−1.42529	−0.712647	−	0.701523i	$-0.752504\pi$
$593$	−34.7082	−1.42529	−0.712647	+	0.701523i	$0.752504\pi$
$594$	0	0
$595$	3.93967	0.161511
$596$	0	0
$597$	40.0735	1.64010
$598$	0	0
$599$	−5.54149	−0.226419	−0.113209	−	0.993571i	$-0.536113\pi$
$599$	−5.54149	−0.226419	−0.113209	+	0.993571i	$0.536113\pi$
$600$	0	0
$601$	−20.1000	−0.819898	−0.409949	−	0.912109i	$-0.634453\pi$
$601$	−20.1000	−0.819898	−0.409949	+	0.912109i	$0.634453\pi$
$602$	0	0
$603$	−1.82293	−0.0742353
$604$	0	0
$605$	−7.17096	−0.291541
$606$	0	0
$607$	−10.7073	−0.434597	−0.217299	−	0.976105i	$-0.569725\pi$
$607$	−10.7073	−0.434597	−0.217299	+	0.976105i	$0.569725\pi$
$608$	0	0
$609$	16.0920	0.652079
$610$	0	0
$611$	−14.5437	−0.588375
$612$	0	0
$613$	−26.7353	−1.07983	−0.539915	−	0.841720i	$-0.681544\pi$
$613$	−26.7353	−1.07983	−0.539915	+	0.841720i	$0.681544\pi$
$614$	0	0
$615$	−1.04160	−0.0420015
$616$	0	0
$617$	5.33093	0.214615	0.107308	−	0.994226i	$-0.465777\pi$
$617$	5.33093	0.214615	0.107308	+	0.994226i	$0.465777\pi$
$618$	0	0
$619$	−7.65187	−0.307554	−0.153777	−	0.988106i	$-0.549144\pi$
$619$	−7.65187	−0.307554	−0.153777	+	0.988106i	$0.549144\pi$
$620$	0	0
$621$	24.2095	0.971493
$622$	0	0
$623$	10.9157	0.437327
$624$	0	0
$625$	13.8213	0.552852
$626$	0	0
$627$	9.07512	0.362425
$628$	0	0
$629$	3.96451	0.158075
$630$	0	0
$631$	−15.0160	−0.597778	−0.298889	−	0.954288i	$-0.596616\pi$
$631$	−15.0160	−0.597778	−0.298889	+	0.954288i	$0.596616\pi$
$632$	0	0
$633$	−18.2648	−0.725961
$634$	0	0
$635$	5.87425	0.233112
$636$	0	0
$637$	−44.8177	−1.77574
$638$	0	0
$639$	−3.64817	−0.144319
$640$	0	0
$641$	−14.1622	−0.559373	−0.279686	−	0.960091i	$-0.590230\pi$
$641$	−14.1622	−0.559373	−0.279686	+	0.960091i	$0.590230\pi$
$642$	0	0
$643$	−5.07287	−0.200054	−0.100027	−	0.994985i	$-0.531893\pi$
$643$	−5.07287	−0.200054	−0.100027	+	0.994985i	$0.531893\pi$
$644$	0	0
$645$	−14.7187	−0.579547
$646$	0	0
$647$	15.3300	0.602685	0.301343	−	0.953516i	$-0.402565\pi$
$647$	15.3300	0.602685	0.301343	+	0.953516i	$0.402565\pi$
$648$	0	0
$649$	−1.70705	−0.0670076
$650$	0	0
$651$	44.7013	1.75198
$652$	0	0
$653$	0.522185	0.0204347	0.0102173	−	0.999948i	$-0.496748\pi$
$653$	0.522185	0.0204347	0.0102173	+	0.999948i	$0.496748\pi$
$654$	0	0
$655$	8.48468	0.331524
$656$	0	0
$657$	−1.10323	−0.0430412
$658$	0	0
$659$	−45.2064	−1.76099	−0.880495	−	0.474056i	$-0.842789\pi$
$659$	−45.2064	−1.76099	−0.880495	+	0.474056i	$0.842789\pi$
$660$	0	0
$661$	34.0679	1.32509	0.662545	−	0.749022i	$-0.269476\pi$
$661$	34.0679	1.32509	0.662545	+	0.749022i	$0.269476\pi$
$662$	0	0
$663$	−5.73721	−0.222815
$664$	0	0
$665$	−12.8478	−0.498215
$666$	0	0
$667$	9.87266	0.382271
$668$	0	0
$669$	−11.4156	−0.441351
$670$	0	0
$671$	−0.571497	−0.0220624
$672$	0	0
$673$	0.0515325	0.00198643	0.000993215	−	1.00000i	$-0.499684\pi$
$673$	0.0515325	0.00198643	0.000993215		1.00000i	$0.499684\pi$
$674$	0	0
$675$	−22.9589	−0.883689
$676$	0	0
$677$	−42.1155	−1.61863	−0.809315	−	0.587375i	$-0.800161\pi$
$677$	−42.1155	−1.61863	−0.809315	+	0.587375i	$0.800161\pi$
$678$	0	0
$679$	56.6871	2.17545
$680$	0	0
$681$	45.0765	1.72734
$682$	0	0
$683$	−37.1034	−1.41972	−0.709861	−	0.704341i	$-0.751242\pi$
$683$	−37.1034	−1.41972	−0.709861	+	0.704341i	$0.751242\pi$
$684$	0	0
$685$	−2.98432	−0.114025
$686$	0	0
$687$	16.0370	0.611851
$688$	0	0
$689$	−20.3987	−0.777128
$690$	0	0
$691$	24.7795	0.942658	0.471329	−	0.881957i	$-0.343774\pi$
$691$	24.7795	0.942658	0.471329	+	0.881957i	$0.343774\pi$
$692$	0	0
$693$	−2.59710	−0.0986557
$694$	0	0
$695$	−8.51260	−0.322901
$696$	0	0
$697$	−0.720474	−0.0272899
$698$	0	0
$699$	29.8711	1.12983
$700$	0	0
$701$	38.1468	1.44078	0.720392	−	0.693567i	$-0.243962\pi$
$701$	38.1468	1.44078	0.720392	+	0.693567i	$0.243962\pi$
$702$	0	0
$703$	−12.9287	−0.487617
$704$	0	0
$705$	−5.97443	−0.225010
$706$	0	0
$707$	77.7482	2.92402
$708$	0	0
$709$	25.9314	0.973874	0.486937	−	0.873437i	$-0.338114\pi$
$709$	25.9314	0.973874	0.486937	+	0.873437i	$0.338114\pi$
$710$	0	0
$711$	4.10116	0.153805
$712$	0	0
$713$	27.4249	1.02707
$714$	0	0
$715$	5.32786	0.199251
$716$	0	0
$717$	4.65990	0.174027
$718$	0	0
$719$	−28.9582	−1.07996	−0.539980	−	0.841678i	$-0.681568\pi$
$719$	−28.9582	−1.07996	−0.539980	+	0.841678i	$0.681568\pi$
$720$	0	0
$721$	−51.9600	−1.93509
$722$	0	0
$723$	−7.95935	−0.296011
$724$	0	0
$725$	−9.36267	−0.347721
$726$	0	0
$727$	−43.8257	−1.62540	−0.812702	−	0.582680i	$-0.802004\pi$
$727$	−43.8257	−1.62540	−0.812702	+	0.582680i	$0.802004\pi$
$728$	0	0
$729$	29.3384	1.08661
$730$	0	0
$731$	−10.1809	−0.376553
$732$	0	0
$733$	−25.7353	−0.950554	−0.475277	−	0.879836i	$-0.657652\pi$
$733$	−25.7353	−0.950554	−0.475277	+	0.879836i	$0.657652\pi$
$734$	0	0
$735$	−18.4108	−0.679091
$736$	0	0
$737$	−9.08631	−0.334699
$738$	0	0
$739$	25.3341	0.931931	0.465966	−	0.884803i	$-0.345707\pi$
$739$	25.3341	0.931931	0.465966	+	0.884803i	$0.345707\pi$
$740$	0	0
$741$	18.7097	0.687320
$742$	0	0
$743$	3.10066	0.113752	0.0568760	−	0.998381i	$-0.481886\pi$
$743$	3.10066	0.113752	0.0568760	+	0.998381i	$0.481886\pi$
$744$	0	0
$745$	−18.8217	−0.689575
$746$	0	0
$747$	4.33209	0.158503
$748$	0	0
$749$	48.8854	1.78623
$750$	0	0
$751$	−29.8965	−1.09094	−0.545469	−	0.838131i	$-0.683649\pi$
$751$	−29.8965	−1.09094	−0.545469	+	0.838131i	$0.683649\pi$
$752$	0	0
$753$	40.8475	1.48857
$754$	0	0
$755$	−8.44589	−0.307377
$756$	0	0
$757$	45.1831	1.64221	0.821104	−	0.570779i	$-0.193359\pi$
$757$	45.1831	1.64221	0.821104	+	0.570779i	$0.193359\pi$
$758$	0	0
$759$	12.3641	0.448790
$760$	0	0
$761$	−4.32190	−0.156669	−0.0783344	−	0.996927i	$-0.524960\pi$
$761$	−4.32190	−0.156669	−0.0783344	+	0.996927i	$0.524960\pi$
$762$	0	0
$763$	23.9737	0.867907
$764$	0	0
$765$	0.303719	0.0109810
$766$	0	0
$767$	−3.51934	−0.127076
$768$	0	0
$769$	−43.5895	−1.57188	−0.785938	−	0.618305i	$-0.787820\pi$
$769$	−43.5895	−1.57188	−0.785938	+	0.618305i	$0.787820\pi$
$770$	0	0
$771$	3.53918	0.127461
$772$	0	0
$773$	30.9804	1.11429	0.557145	−	0.830416i	$-0.311897\pi$
$773$	30.9804	1.11429	0.557145	+	0.830416i	$0.311897\pi$
$774$	0	0
$775$	−26.0082	−0.934243
$776$	0	0
$777$	−28.7106	−1.02999
$778$	0	0
$779$	2.34956	0.0841815
$780$	0	0
$781$	−18.1842	−0.650680
$782$	0	0
$783$	12.1077	0.432694
$784$	0	0
$785$	12.3306	0.440099
$786$	0	0
$787$	10.7866	0.384501	0.192250	−	0.981346i	$-0.438421\pi$
$787$	10.7866	0.384501	0.192250	+	0.981346i	$0.438421\pi$
$788$	0	0
$789$	26.5289	0.944453
$790$	0	0
$791$	−76.9793	−2.73707
$792$	0	0
$793$	−1.17823	−0.0418401
$794$	0	0
$795$	−8.37961	−0.297194
$796$	0	0
$797$	−34.9536	−1.23812	−0.619060	−	0.785343i	$-0.712486\pi$
$797$	−34.9536	−1.23812	−0.619060	+	0.785343i	$0.712486\pi$
$798$	0	0
$799$	−4.13250	−0.146197
$800$	0	0
$801$	0.841517	0.0297335
$802$	0	0
$803$	−5.49903	−0.194056
$804$	0	0
$805$	−17.5041	−0.616937
$806$	0	0
$807$	−10.8105	−0.380549
$808$	0	0
$809$	−10.1002	−0.355105	−0.177552	−	0.984111i	$-0.556818\pi$
$809$	−10.1002	−0.355105	−0.177552	+	0.984111i	$0.556818\pi$
$810$	0	0
$811$	0.118042	0.00414502	0.00207251	−	0.999998i	$-0.499340\pi$
$811$	0.118042	0.00414502	0.00207251	+	0.999998i	$0.499340\pi$
$812$	0	0
$813$	−40.6376	−1.42522
$814$	0	0
$815$	1.64627	0.0576662
$816$	0	0
$817$	33.2011	1.16156
$818$	0	0
$819$	−5.35432	−0.187095
$820$	0	0
$821$	13.4916	0.470860	0.235430	−	0.971891i	$-0.424350\pi$
$821$	13.4916	0.470860	0.235430	+	0.971891i	$0.424350\pi$
$822$	0	0
$823$	−38.9186	−1.35662	−0.678308	−	0.734778i	$-0.737286\pi$
$823$	−38.9186	−1.35662	−0.678308	+	0.734778i	$0.737286\pi$
$824$	0	0
$825$	−11.7255	−0.408228
$826$	0	0
$827$	45.2296	1.57279	0.786394	−	0.617726i	$-0.211946\pi$
$827$	45.2296	1.57279	0.786394	+	0.617726i	$0.211946\pi$
$828$	0	0
$829$	9.07415	0.315158	0.157579	−	0.987506i	$-0.449631\pi$
$829$	9.07415	0.315158	0.157579	+	0.987506i	$0.449631\pi$
$830$	0	0
$831$	−53.3719	−1.85145
$832$	0	0
$833$	−12.7347	−0.441230
$834$	0	0
$835$	10.7119	0.370700
$836$	0	0
$837$	33.6335	1.16255
$838$	0	0
$839$	51.4725	1.77703	0.888514	−	0.458850i	$-0.151739\pi$
$839$	51.4725	1.77703	0.888514	+	0.458850i	$0.151739\pi$
$840$	0	0
$841$	−24.0625	−0.829740
$842$	0	0
$843$	−25.7717	−0.887625
$844$	0	0
$845$	−0.544706	−0.0187385
$846$	0	0
$847$	35.9209	1.23426
$848$	0	0
$849$	−43.1963	−1.48249
$850$	0	0
$851$	−17.6144	−0.603814
$852$	0	0
$853$	−32.0661	−1.09792	−0.548962	−	0.835847i	$-0.684977\pi$
$853$	−32.0661	−1.09792	−0.548962	+	0.835847i	$0.684977\pi$
$854$	0	0
$855$	−0.990466	−0.0338732
$856$	0	0
$857$	−36.6740	−1.25276	−0.626380	−	0.779518i	$-0.715464\pi$
$857$	−36.6740	−1.25276	−0.626380	+	0.779518i	$0.715464\pi$
$858$	0	0
$859$	−2.42421	−0.0827131	−0.0413565	−	0.999144i	$-0.513168\pi$
$859$	−2.42421	−0.0827131	−0.0413565	+	0.999144i	$0.513168\pi$
$860$	0	0
$861$	5.21762	0.177816
$862$	0	0
$863$	45.4193	1.54609	0.773046	−	0.634350i	$-0.218732\pi$
$863$	45.4193	1.54609	0.773046	+	0.634350i	$0.218732\pi$
$864$	0	0
$865$	19.7488	0.671480
$866$	0	0
$867$	−1.63019	−0.0553642
$868$	0	0
$869$	20.4421	0.693450
$870$	0	0
$871$	−18.7328	−0.634738
$872$	0	0
$873$	4.37015	0.147907
$874$	0	0
$875$	36.2982	1.22710
$876$	0	0
$877$	−47.8443	−1.61559	−0.807793	−	0.589466i	$-0.799338\pi$
$877$	−47.8443	−1.61559	−0.807793	+	0.589466i	$0.799338\pi$
$878$	0	0
$879$	3.02848	0.102148
$880$	0	0
$881$	25.2381	0.850293	0.425146	−	0.905125i	$-0.360223\pi$
$881$	25.2381	0.850293	0.425146	+	0.905125i	$0.360223\pi$
$882$	0	0
$883$	−48.8054	−1.64243	−0.821217	−	0.570617i	$-0.806704\pi$
$883$	−48.8054	−1.64243	−0.821217	+	0.570617i	$0.806704\pi$
$884$	0	0
$885$	−1.44572	−0.0485973
$886$	0	0
$887$	−33.7584	−1.13350	−0.566748	−	0.823891i	$-0.691799\pi$
$887$	−33.7584	−1.13350	−0.566748	+	0.823891i	$0.691799\pi$
$888$	0	0
$889$	−29.4254	−0.986896
$890$	0	0
$891$	13.4094	0.449231
$892$	0	0
$893$	13.4766	0.450977
$894$	0	0
$895$	15.1707	0.507101
$896$	0	0
$897$	25.4906	0.851105
$898$	0	0
$899$	13.7158	0.457447
$900$	0	0
$901$	−5.79616	−0.193098
$902$	0	0
$903$	73.7291	2.45355
$904$	0	0
$905$	−10.2079	−0.339321
$906$	0	0
$907$	56.6732	1.88180	0.940901	−	0.338682i	$-0.109981\pi$
$907$	56.6732	1.88180	0.940901	+	0.338682i	$0.109981\pi$
$908$	0	0
$909$	5.99381	0.198802
$910$	0	0
$911$	−6.71623	−0.222519	−0.111259	−	0.993791i	$-0.535488\pi$
$911$	−6.71623	−0.222519	−0.111259	+	0.993791i	$0.535488\pi$
$912$	0	0
$913$	21.5932	0.714629
$914$	0	0
$915$	−0.484006	−0.0160008
$916$	0	0
$917$	−42.5016	−1.40353
$918$	0	0
$919$	−15.8758	−0.523694	−0.261847	−	0.965109i	$-0.584332\pi$
$919$	−15.8758	−0.523694	−0.261847	+	0.965109i	$0.584332\pi$
$920$	0	0
$921$	−12.0346	−0.396554
$922$	0	0
$923$	−37.4894	−1.23398
$924$	0	0
$925$	16.7045	0.549241
$926$	0	0
$927$	−4.00573	−0.131565
$928$	0	0
$929$	−3.18483	−0.104491	−0.0522454	−	0.998634i	$-0.516638\pi$
$929$	−3.18483	−0.104491	−0.0522454	+	0.998634i	$0.516638\pi$
$930$	0	0
$931$	41.5294	1.36107
$932$	0	0
$933$	28.7824	0.942293
$934$	0	0
$935$	1.51388	0.0495091
$936$	0	0
$937$	26.4897	0.865382	0.432691	−	0.901542i	$-0.357564\pi$
$937$	26.4897	0.865382	0.432691	+	0.901542i	$0.357564\pi$
$938$	0	0
$939$	20.4200	0.666382
$940$	0	0
$941$	42.4654	1.38433	0.692166	−	0.721739i	$-0.256657\pi$
$941$	42.4654	1.38433	0.692166	+	0.721739i	$0.256657\pi$
$942$	0	0
$943$	3.20109	0.104242
$944$	0	0
$945$	−21.4668	−0.698314
$946$	0	0
$947$	−2.19557	−0.0713463	−0.0356732	−	0.999364i	$-0.511358\pi$
$947$	−2.19557	−0.0713463	−0.0356732	+	0.999364i	$0.511358\pi$
$948$	0	0
$949$	−11.3371	−0.368017
$950$	0	0
$951$	−34.8918	−1.13144
$952$	0	0
$953$	−29.0964	−0.942523	−0.471262	−	0.881993i	$-0.656201\pi$
$953$	−29.0964	−0.942523	−0.471262	+	0.881993i	$0.656201\pi$
$954$	0	0
$955$	−16.0887	−0.520617
$956$	0	0
$957$	6.18358	0.199887
$958$	0	0
$959$	14.9491	0.482732
$960$	0	0
$961$	7.10060	0.229052
$962$	0	0
$963$	3.76870	0.121445
$964$	0	0
$965$	9.51786	0.306391
$966$	0	0
$967$	−22.4269	−0.721201	−0.360600	−	0.932720i	$-0.617428\pi$
$967$	−22.4269	−0.721201	−0.360600	+	0.932720i	$0.617428\pi$
$968$	0	0
$969$	5.31626	0.170783
$970$	0	0
$971$	−44.9142	−1.44136	−0.720682	−	0.693266i	$-0.756171\pi$
$971$	−44.9142	−1.44136	−0.720682	+	0.693266i	$0.756171\pi$
$972$	0	0
$973$	42.6415	1.36702
$974$	0	0
$975$	−24.1738	−0.774182
$976$	0	0
$977$	−39.7756	−1.27253	−0.636267	−	0.771469i	$-0.719522\pi$
$977$	−39.7756	−1.27253	−0.636267	+	0.771469i	$0.719522\pi$
$978$	0	0
$979$	4.19451	0.134057
$980$	0	0
$981$	1.84820	0.0590084
$982$	0	0
$983$	59.2777	1.89067	0.945333	−	0.326106i	$-0.105737\pi$
$983$	59.2777	1.89067	0.945333	+	0.326106i	$0.105737\pi$
$984$	0	0
$985$	−18.9687	−0.604393
$986$	0	0
$987$	29.9273	0.952595
$988$	0	0
$989$	45.2339	1.43835
$990$	0	0
$991$	30.8693	0.980594	0.490297	−	0.871555i	$-0.336888\pi$
$991$	30.8693	0.980594	0.490297	+	0.871555i	$0.336888\pi$
$992$	0	0
$993$	20.5377	0.651746
$994$	0	0
$995$	−21.8003	−0.691117
$996$	0	0
$997$	0.760691	0.0240913	0.0120457	−	0.999927i	$-0.496166\pi$
$997$	0.760691	0.0240913	0.0120457	+	0.999927i	$0.496166\pi$
$998$	0	0
$999$	−21.6021	−0.683460

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.x.1.7	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.x.1.7	✓	22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.63019 q^{3} +0.886839 q^{5} -4.44237 q^{7} -0.342474 q^{9} +O(q^{10})\)	\(q-1.63019 q^{3} +0.886839 q^{5} -4.44237 q^{7} -0.342474 q^{9} -1.70705 q^{11} -3.51934 q^{13} -1.44572 q^{15} -1.00000 q^{17} +3.26112 q^{19} +7.24192 q^{21} +4.44303 q^{23} -4.21352 q^{25} +5.44887 q^{27} +2.22206 q^{29} +6.17257 q^{31} +2.78282 q^{33} -3.93967 q^{35} -3.96451 q^{37} +5.73721 q^{39} +0.720474 q^{41} +10.1809 q^{43} -0.303719 q^{45} +4.13250 q^{47} +12.7347 q^{49} +1.63019 q^{51} +5.79616 q^{53} -1.51388 q^{55} -5.31626 q^{57} +1.00000 q^{59} +0.334786 q^{61} +1.52140 q^{63} -3.12109 q^{65} +5.32282 q^{67} -7.24299 q^{69} +10.6524 q^{71} +3.22136 q^{73} +6.86884 q^{75} +7.58335 q^{77} -11.9751 q^{79} -7.85529 q^{81} -12.6494 q^{83} -0.886839 q^{85} -3.62238 q^{87} -2.45717 q^{89} +15.6342 q^{91} -10.0625 q^{93} +2.89209 q^{95} -12.7605 q^{97} +0.584620 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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