LMFDB - Embedded newform 1024.4.a.n.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,4,Mod(1,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1024.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1024.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:	$60.4179558459$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 36x^{8} + 405x^{6} - 1380x^{4} + 420x^{2} - 32 $ x^10 - 36x^8 + 405x^6 - 1380x^4 + 420x^2 - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 16)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.34476$ of defining polynomial
Character	$\chi$	$=$	1024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.43597 q^{3} +12.2748 q^{5} +1.63924 q^{7} +44.1656 q^{9} +O(q^{10})$	$q+8.43597 q^{3} +12.2748 q^{5} +1.63924 q^{7} +44.1656 q^{9} -25.7416 q^{11} -13.2187 q^{13} +103.550 q^{15} -53.6113 q^{17} +100.391 q^{19} +13.8286 q^{21} +25.1189 q^{23} +25.6706 q^{25} +144.809 q^{27} +256.105 q^{29} +132.684 q^{31} -217.155 q^{33} +20.1214 q^{35} +247.306 q^{37} -111.512 q^{39} +198.660 q^{41} +404.064 q^{43} +542.124 q^{45} -78.3629 q^{47} -340.313 q^{49} -452.264 q^{51} +743.559 q^{53} -315.973 q^{55} +846.894 q^{57} +65.8036 q^{59} -273.392 q^{61} +72.3982 q^{63} -162.256 q^{65} -399.066 q^{67} +211.903 q^{69} +727.536 q^{71} +106.065 q^{73} +216.556 q^{75} -42.1968 q^{77} +58.9970 q^{79} +29.1298 q^{81} -580.049 q^{83} -658.068 q^{85} +2160.50 q^{87} -768.959 q^{89} -21.6686 q^{91} +1119.32 q^{93} +1232.28 q^{95} -809.953 q^{97} -1136.89 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 28 q^{7} + 54 q^{9}+O(q^{10}) $ 10 * q + 28 * q^7 + 54 * q^9	$ 10 q + 28 q^{7} + 54 q^{9} + 124 q^{15} + 4 q^{17} + 276 q^{23} + 50 q^{25} + 368 q^{31} - 4 q^{33} + 732 q^{39} + 944 q^{47} - 94 q^{49} + 1380 q^{55} + 108 q^{57} + 2628 q^{63} - 492 q^{65} + 3468 q^{71} - 296 q^{73} + 4416 q^{79} - 482 q^{81} + 6036 q^{87} + 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q + 28 * q^7 + 54 * q^9 + 124 * q^15 + 4 * q^17 + 276 * q^23 + 50 * q^25 + 368 * q^31 - 4 * q^33 + 732 * q^39 + 944 * q^47 - 94 * q^49 + 1380 * q^55 + 108 * q^57 + 2628 * q^63 - 492 * q^65 + 3468 * q^71 - 296 * q^73 + 4416 * q^79 - 482 * q^81 + 6036 * q^87 + 88 * q^89 + 6900 * q^95 - 4 * q^97

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
$3$	8.43597	1.62350	0.811752	−	0.584003i	$-0.198514\pi$
$3$	8.43597	1.62350	0.811752	+	0.584003i	$0.198514\pi$
$4$	0	0
$5$	12.2748	1.09789	0.548945	−	0.835858i	$-0.315029\pi$
$5$	12.2748	1.09789	0.548945	+	0.835858i	$0.315029\pi$
$6$	0	0
$7$	1.63924	0.0885109	0.0442554	−	0.999020i	$-0.485908\pi$
$7$	1.63924	0.0885109	0.0442554	+	0.999020i	$0.485908\pi$
$8$	0	0
$9$	44.1656	1.63576
$10$	0	0
$11$	−25.7416	−0.705580	−0.352790	−	0.935702i	$-0.614767\pi$
$11$	−25.7416	−0.705580	−0.352790	+	0.935702i	$0.614767\pi$
$12$	0	0
$13$	−13.2187	−0.282015	−0.141008	−	0.990009i	$-0.545034\pi$
$13$	−13.2187	−0.282015	−0.141008	+	0.990009i	$0.545034\pi$
$14$	0	0
$15$	103.550	1.78243
$16$	0	0
$17$	−53.6113	−0.764862	−0.382431	−	0.923984i	$-0.624913\pi$
$17$	−53.6113	−0.764862	−0.382431	+	0.923984i	$0.624913\pi$
$18$	0	0
$19$	100.391	1.21217	0.606085	−	0.795400i	$-0.292739\pi$
$19$	100.391	1.21217	0.606085	+	0.795400i	$0.292739\pi$
$20$	0	0
$21$	13.8286	0.143698
$22$	0	0
$23$	25.1189	0.227724	0.113862	−	0.993497i	$-0.463678\pi$
$23$	25.1189	0.227724	0.113862	+	0.993497i	$0.463678\pi$
$24$	0	0
$25$	25.6706	0.205365
$26$	0	0
$27$	144.809	1.03216
$28$	0	0
$29$	256.105	1.63992	0.819958	−	0.572424i	$-0.193997\pi$
$29$	256.105	1.63992	0.819958	+	0.572424i	$0.193997\pi$
$30$	0	0
$31$	132.684	0.768733	0.384367	−	0.923180i	$-0.374420\pi$
$31$	132.684	0.768733	0.384367	+	0.923180i	$0.374420\pi$
$32$	0	0
$33$	−217.155	−1.14551
$34$	0	0
$35$	20.1214	0.0971753
$36$	0	0
$37$	247.306	1.09884	0.549418	−	0.835548i	$-0.314849\pi$
$37$	247.306	1.09884	0.549418	+	0.835548i	$0.314849\pi$
$38$	0	0
$39$	−111.512	−0.457853
$40$	0	0
$41$	198.660	0.756720	0.378360	−	0.925658i	$-0.376488\pi$
$41$	198.660	0.756720	0.378360	+	0.925658i	$0.376488\pi$
$42$	0	0
$43$	404.064	1.43300	0.716502	−	0.697585i	$-0.245742\pi$
$43$	404.064	1.43300	0.716502	+	0.697585i	$0.245742\pi$
$44$	0	0
$45$	542.124	1.79589
$46$	0	0
$47$	−78.3629	−0.243200	−0.121600	−	0.992579i	$-0.538803\pi$
$47$	−78.3629	−0.243200	−0.121600	+	0.992579i	$0.538803\pi$
$48$	0	0
$49$	−340.313	−0.992166
$50$	0	0
$51$	−452.264	−1.24176
$52$	0	0
$53$	743.559	1.92709	0.963544	−	0.267549i	$-0.0862138\pi$
$53$	743.559	1.92709	0.963544	+	0.267549i	$0.0862138\pi$
$54$	0	0
$55$	−315.973	−0.774650
$56$	0	0
$57$	846.894	1.96796
$58$	0	0
$59$	65.8036	0.145202	0.0726008	−	0.997361i	$-0.476870\pi$
$59$	65.8036	0.145202	0.0726008	+	0.997361i	$0.476870\pi$
$60$	0	0
$61$	−273.392	−0.573841	−0.286921	−	0.957954i	$-0.592632\pi$
$61$	−273.392	−0.573841	−0.286921	+	0.957954i	$0.592632\pi$
$62$	0	0
$63$	72.3982	0.144783
$64$	0	0
$65$	−162.256	−0.309622
$66$	0	0
$67$	−399.066	−0.727667	−0.363834	−	0.931464i	$-0.618532\pi$
$67$	−399.066	−0.727667	−0.363834	+	0.931464i	$0.618532\pi$
$68$	0	0
$69$	211.903	0.369711
$70$	0	0
$71$	727.536	1.21609	0.608046	−	0.793901i	$-0.291953\pi$
$71$	727.536	1.21609	0.608046	+	0.793901i	$0.291953\pi$
$72$	0	0
$73$	106.065	0.170054	0.0850270	−	0.996379i	$-0.472902\pi$
$73$	106.065	0.170054	0.0850270	+	0.996379i	$0.472902\pi$
$74$	0	0
$75$	216.556	0.333410
$76$	0	0
$77$	−42.1968	−0.0624515
$78$	0	0
$79$	58.9970	0.0840213	0.0420107	−	0.999117i	$-0.486624\pi$
$79$	58.9970	0.0840213	0.0420107	+	0.999117i	$0.486624\pi$
$80$	0	0
$81$	29.1298	0.0399586
$82$	0	0
$83$	−580.049	−0.767092	−0.383546	−	0.923522i	$-0.625297\pi$
$83$	−580.049	−0.767092	−0.383546	+	0.923522i	$0.625297\pi$
$84$	0	0
$85$	−658.068	−0.839735
$86$	0	0
$87$	2160.50	2.66241
$88$	0	0
$89$	−768.959	−0.915837	−0.457918	−	0.888994i	$-0.651405\pi$
$89$	−768.959	−0.915837	−0.457918	+	0.888994i	$0.651405\pi$
$90$	0	0
$91$	−21.6686	−0.0249614
$92$	0	0
$93$	1119.32	1.24804
$94$	0	0
$95$	1232.28	1.33083
$96$	0	0
$97$	−809.953	−0.847817	−0.423908	−	0.905705i	$-0.639342\pi$
$97$	−809.953	−0.847817	−0.423908	+	0.905705i	$0.639342\pi$
$98$	0	0
$99$	−1136.89	−1.15416
$100$	0	0
$101$	−428.775	−0.422422	−0.211211	−	0.977440i	$-0.567741\pi$
$101$	−428.775	−0.422422	−0.211211	+	0.977440i	$0.567741\pi$
$102$	0	0
$103$	962.201	0.920471	0.460235	−	0.887797i	$-0.347765\pi$
$103$	962.201	0.920471	0.460235	+	0.887797i	$0.347765\pi$
$104$	0	0
$105$	169.743	0.157764
$106$	0	0
$107$	−1030.02	−0.930618	−0.465309	−	0.885148i	$-0.654057\pi$
$107$	−1030.02	−0.930618	−0.465309	+	0.885148i	$0.654057\pi$
$108$	0	0
$109$	−838.993	−0.737256	−0.368628	−	0.929577i	$-0.620172\pi$
$109$	−838.993	−0.737256	−0.368628	+	0.929577i	$0.620172\pi$
$110$	0	0
$111$	2086.27	1.78396
$112$	0	0
$113$	−351.938	−0.292987	−0.146493	−	0.989212i	$-0.546799\pi$
$113$	−351.938	−0.292987	−0.146493	+	0.989212i	$0.546799\pi$
$114$	0	0
$115$	308.330	0.250017
$116$	0	0
$117$	−583.810	−0.461310
$118$	0	0
$119$	−87.8821	−0.0676986
$120$	0	0
$121$	−668.370	−0.502157
$122$	0	0
$123$	1675.89	1.22854
$124$	0	0
$125$	−1219.25	−0.872423
$126$	0	0
$127$	−2365.81	−1.65301	−0.826504	−	0.562931i	$-0.809674\pi$
$127$	−2365.81	−1.65301	−0.826504	+	0.562931i	$0.809674\pi$
$128$	0	0
$129$	3408.67	2.32649
$130$	0	0
$131$	−570.570	−0.380541	−0.190271	−	0.981732i	$-0.560937\pi$
$131$	−570.570	−0.380541	−0.190271	+	0.981732i	$0.560937\pi$
$132$	0	0
$133$	164.565	0.107290
$134$	0	0
$135$	1777.50	1.13320
$136$	0	0
$137$	−856.850	−0.534348	−0.267174	−	0.963648i	$-0.586090\pi$
$137$	−856.850	−0.534348	−0.267174	+	0.963648i	$0.586090\pi$
$138$	0	0
$139$	2389.08	1.45783	0.728916	−	0.684603i	$-0.240024\pi$
$139$	2389.08	1.45783	0.728916	+	0.684603i	$0.240024\pi$
$140$	0	0
$141$	−661.067	−0.394836
$142$	0	0
$143$	340.269	0.198984
$144$	0	0
$145$	3143.64	1.80045
$146$	0	0
$147$	−2870.87	−1.61078
$148$	0	0
$149$	45.5671	0.0250537	0.0125269	−	0.999922i	$-0.496012\pi$
$149$	45.5671	0.0250537	0.0125269	+	0.999922i	$0.496012\pi$
$150$	0	0
$151$	1077.06	0.580460	0.290230	−	0.956957i	$-0.406268\pi$
$151$	1077.06	0.580460	0.290230	+	0.956957i	$0.406268\pi$
$152$	0	0
$153$	−2367.78	−1.25113
$154$	0	0
$155$	1628.67	0.843986
$156$	0	0
$157$	−2694.94	−1.36994	−0.684968	−	0.728573i	$-0.740184\pi$
$157$	−2694.94	−1.36994	−0.684968	+	0.728573i	$0.740184\pi$
$158$	0	0
$159$	6272.64	3.12863
$160$	0	0
$161$	41.1761	0.0201561
$162$	0	0
$163$	−1003.85	−0.482377	−0.241189	−	0.970478i	$-0.577537\pi$
$163$	−1003.85	−0.482377	−0.241189	+	0.970478i	$0.577537\pi$
$164$	0	0
$165$	−2665.54	−1.25765
$166$	0	0
$167$	−3460.66	−1.60356	−0.801778	−	0.597623i	$-0.796112\pi$
$167$	−3460.66	−1.60356	−0.801778	+	0.597623i	$0.796112\pi$
$168$	0	0
$169$	−2022.27	−0.920467
$170$	0	0
$171$	4433.82	1.98282
$172$	0	0
$173$	−1843.49	−0.810162	−0.405081	−	0.914281i	$-0.632757\pi$
$173$	−1843.49	−0.810162	−0.405081	+	0.914281i	$0.632757\pi$
$174$	0	0
$175$	42.0803	0.0181770
$176$	0	0
$177$	555.117	0.235735
$178$	0	0
$179$	−1530.67	−0.639149	−0.319575	−	0.947561i	$-0.603540\pi$
$179$	−1530.67	−0.639149	−0.319575	+	0.947561i	$0.603540\pi$
$180$	0	0
$181$	−4163.35	−1.70972	−0.854859	−	0.518860i	$-0.826357\pi$
$181$	−4163.35	−1.70972	−0.854859	+	0.518860i	$0.826357\pi$
$182$	0	0
$183$	−2306.33	−0.931633
$184$	0	0
$185$	3035.64	1.20640
$186$	0	0
$187$	1380.04	0.539672
$188$	0	0
$189$	237.377	0.0913577
$190$	0	0
$191$	−430.650	−0.163145	−0.0815726	−	0.996667i	$-0.525994\pi$
$191$	−430.650	−0.163145	−0.0815726	+	0.996667i	$0.525994\pi$
$192$	0	0
$193$	−2266.98	−0.845497	−0.422749	−	0.906247i	$-0.638935\pi$
$193$	−2266.98	−0.845497	−0.422749	+	0.906247i	$0.638935\pi$
$194$	0	0
$195$	−1368.79	−0.502672
$196$	0	0
$197$	−1469.95	−0.531621	−0.265810	−	0.964025i	$-0.585640\pi$
$197$	−1469.95	−0.531621	−0.265810	+	0.964025i	$0.585640\pi$
$198$	0	0
$199$	4989.44	1.77735	0.888674	−	0.458540i	$-0.151627\pi$
$199$	4989.44	1.77735	0.888674	+	0.458540i	$0.151627\pi$
$200$	0	0
$201$	−3366.51	−1.18137
$202$	0	0
$203$	419.819	0.145150
$204$	0	0
$205$	2438.51	0.830796
$206$	0	0
$207$	1109.39	0.372503
$208$	0	0
$209$	−2584.22	−0.855283
$210$	0	0
$211$	3749.79	1.22344	0.611720	−	0.791074i	$-0.290478\pi$
$211$	3749.79	1.22344	0.611720	+	0.791074i	$0.290478\pi$
$212$	0	0
$213$	6137.47	1.97433
$214$	0	0
$215$	4959.80	1.57328
$216$	0	0
$217$	217.501	0.0680413
$218$	0	0
$219$	894.760	0.276083
$220$	0	0
$221$	708.670	0.215703
$222$	0	0
$223$	3690.85	1.10833	0.554165	−	0.832407i	$-0.313038\pi$
$223$	3690.85	1.10833	0.554165	+	0.832407i	$0.313038\pi$
$224$	0	0
$225$	1133.76	0.335928
$226$	0	0
$227$	2418.90	0.707259	0.353630	−	0.935385i	$-0.384947\pi$
$227$	2418.90	0.707259	0.353630	+	0.935385i	$0.384947\pi$
$228$	0	0
$229$	−129.827	−0.0374637	−0.0187318	−	0.999825i	$-0.505963\pi$
$229$	−129.827	−0.0374637	−0.0187318	+	0.999825i	$0.505963\pi$
$230$	0	0
$231$	−355.971	−0.101390
$232$	0	0
$233$	4259.71	1.19769	0.598847	−	0.800863i	$-0.295626\pi$
$233$	4259.71	1.19769	0.598847	+	0.800863i	$0.295626\pi$
$234$	0	0
$235$	−961.889	−0.267007
$236$	0	0
$237$	497.697	0.136409
$238$	0	0
$239$	5053.12	1.36761	0.683806	−	0.729664i	$-0.260324\pi$
$239$	5053.12	1.36761	0.683806	+	0.729664i	$0.260324\pi$
$240$	0	0
$241$	−48.8379	−0.0130536	−0.00652681	−	0.999979i	$-0.502078\pi$
$241$	−48.8379	−0.0130536	−0.00652681	+	0.999979i	$0.502078\pi$
$242$	0	0
$243$	−3664.09	−0.967291
$244$	0	0
$245$	−4177.27	−1.08929
$246$	0	0
$247$	−1327.03	−0.341850
$248$	0	0
$249$	−4893.28	−1.24538
$250$	0	0
$251$	−3683.69	−0.926345	−0.463172	−	0.886268i	$-0.653289\pi$
$251$	−3683.69	−0.926345	−0.463172	+	0.886268i	$0.653289\pi$
$252$	0	0
$253$	−646.602	−0.160678
$254$	0	0
$255$	−5551.44	−1.36331
$256$	0	0
$257$	739.054	0.179381	0.0896905	−	0.995970i	$-0.471412\pi$
$257$	739.054	0.179381	0.0896905	+	0.995970i	$0.471412\pi$
$258$	0	0
$259$	405.396	0.0972589
$260$	0	0
$261$	11311.0	2.68251
$262$	0	0
$263$	2448.30	0.574025	0.287012	−	0.957927i	$-0.407338\pi$
$263$	2448.30	0.574025	0.287012	+	0.957927i	$0.407338\pi$
$264$	0	0
$265$	9127.03	2.11573
$266$	0	0
$267$	−6486.91	−1.48686
$268$	0	0
$269$	−1173.73	−0.266035	−0.133018	−	0.991114i	$-0.542467\pi$
$269$	−1173.73	−0.266035	−0.133018	+	0.991114i	$0.542467\pi$
$270$	0	0
$271$	−1404.85	−0.314902	−0.157451	−	0.987527i	$-0.550328\pi$
$271$	−1404.85	−0.314902	−0.157451	+	0.987527i	$0.550328\pi$
$272$	0	0
$273$	−182.796	−0.0405249
$274$	0	0
$275$	−660.801	−0.144901
$276$	0	0
$277$	3175.88	0.688881	0.344440	−	0.938808i	$-0.388069\pi$
$277$	3175.88	0.688881	0.344440	+	0.938808i	$0.388069\pi$
$278$	0	0
$279$	5860.07	1.25747
$280$	0	0
$281$	6045.97	1.28353	0.641766	−	0.766900i	$-0.278202\pi$
$281$	6045.97	1.28353	0.641766	+	0.766900i	$0.278202\pi$
$282$	0	0
$283$	3478.45	0.730644	0.365322	−	0.930881i	$-0.380959\pi$
$283$	3478.45	0.730644	0.365322	+	0.930881i	$0.380959\pi$
$284$	0	0
$285$	10395.4	2.16061
$286$	0	0
$287$	325.653	0.0669780
$288$	0	0
$289$	−2038.82	−0.414986
$290$	0	0
$291$	−6832.74	−1.37643
$292$	0	0
$293$	−2619.82	−0.522360	−0.261180	−	0.965290i	$-0.584112\pi$
$293$	−2619.82	−0.522360	−0.261180	+	0.965290i	$0.584112\pi$
$294$	0	0
$295$	807.725	0.159415
$296$	0	0
$297$	−3727.60	−0.728275
$298$	0	0
$299$	−332.039	−0.0642217
$300$	0	0
$301$	662.360	0.126837
$302$	0	0
$303$	−3617.13	−0.685804
$304$	0	0
$305$	−3355.83	−0.630015
$306$	0	0
$307$	2980.24	0.554044	0.277022	−	0.960864i	$-0.410653\pi$
$307$	2980.24	0.554044	0.277022	+	0.960864i	$0.410653\pi$
$308$	0	0
$309$	8117.10	1.49439
$310$	0	0
$311$	−5294.90	−0.965422	−0.482711	−	0.875780i	$-0.660348\pi$
$311$	−5294.90	−0.965422	−0.482711	+	0.875780i	$0.660348\pi$
$312$	0	0
$313$	4005.87	0.723403	0.361702	−	0.932294i	$-0.382196\pi$
$313$	4005.87	0.723403	0.361702	+	0.932294i	$0.382196\pi$
$314$	0	0
$315$	888.673	0.158956
$316$	0	0
$317$	−1144.44	−0.202770	−0.101385	−	0.994847i	$-0.532327\pi$
$317$	−1144.44	−0.202770	−0.101385	+	0.994847i	$0.532327\pi$
$318$	0	0
$319$	−6592.56	−1.15709
$320$	0	0
$321$	−8689.25	−1.51086
$322$	0	0
$323$	−5382.08	−0.927143
$324$	0	0
$325$	−339.331	−0.0579159
$326$	0	0
$327$	−7077.72	−1.19694
$328$	0	0
$329$	−128.456	−0.0215259
$330$	0	0
$331$	5981.64	0.993295	0.496648	−	0.867952i	$-0.334564\pi$
$331$	5981.64	0.993295	0.496648	+	0.867952i	$0.334564\pi$
$332$	0	0
$333$	10922.4	1.79744
$334$	0	0
$335$	−4898.46	−0.798899
$336$	0	0
$337$	−10002.6	−1.61684	−0.808419	−	0.588607i	$-0.799677\pi$
$337$	−10002.6	−1.61684	−0.808419	+	0.588607i	$0.799677\pi$
$338$	0	0
$339$	−2968.94	−0.475665
$340$	0	0
$341$	−3415.50	−0.542403
$342$	0	0
$343$	−1120.12	−0.176328
$344$	0	0
$345$	2601.06	0.405903
$346$	0	0
$347$	−9064.38	−1.40231	−0.701155	−	0.713009i	$-0.747332\pi$
$347$	−9064.38	−1.40231	−0.701155	+	0.713009i	$0.747332\pi$
$348$	0	0
$349$	7782.74	1.19370	0.596849	−	0.802354i	$-0.296419\pi$
$349$	7782.74	1.19370	0.596849	+	0.802354i	$0.296419\pi$
$350$	0	0
$351$	−1914.18	−0.291086
$352$	0	0
$353$	−1411.35	−0.212800	−0.106400	−	0.994323i	$-0.533932\pi$
$353$	−1411.35	−0.212800	−0.106400	+	0.994323i	$0.533932\pi$
$354$	0	0
$355$	8930.35	1.33514
$356$	0	0
$357$	−741.371	−0.109909
$358$	0	0
$359$	−2160.73	−0.317658	−0.158829	−	0.987306i	$-0.550772\pi$
$359$	−2160.73	−0.317658	−0.158829	+	0.987306i	$0.550772\pi$
$360$	0	0
$361$	3219.31	0.469355
$362$	0	0
$363$	−5638.35	−0.815253
$364$	0	0
$365$	1301.92	0.186701
$366$	0	0
$367$	−10757.7	−1.53010	−0.765052	−	0.643969i	$-0.777287\pi$
$367$	−10757.7	−1.53010	−0.765052	+	0.643969i	$0.777287\pi$
$368$	0	0
$369$	8773.95	1.23782
$370$	0	0
$371$	1218.87	0.170568
$372$	0	0
$373$	−1989.79	−0.276213	−0.138107	−	0.990417i	$-0.544102\pi$
$373$	−1989.79	−0.276213	−0.138107	+	0.990417i	$0.544102\pi$
$374$	0	0
$375$	−10285.5	−1.41638
$376$	0	0
$377$	−3385.37	−0.462481
$378$	0	0
$379$	−1622.04	−0.219838	−0.109919	−	0.993941i	$-0.535059\pi$
$379$	−1622.04	−0.219838	−0.109919	+	0.993941i	$0.535059\pi$
$380$	0	0
$381$	−19957.9	−2.68366
$382$	0	0
$383$	9042.17	1.20635	0.603176	−	0.797608i	$-0.293902\pi$
$383$	9042.17	1.20635	0.603176	+	0.797608i	$0.293902\pi$
$384$	0	0
$385$	−517.957	−0.0685650
$386$	0	0
$387$	17845.7	2.34406
$388$	0	0
$389$	3642.08	0.474706	0.237353	−	0.971423i	$-0.423720\pi$
$389$	3642.08	0.474706	0.237353	+	0.971423i	$0.423720\pi$
$390$	0	0
$391$	−1346.66	−0.174178
$392$	0	0
$393$	−4813.31	−0.617810
$394$	0	0
$395$	724.176	0.0922462
$396$	0	0
$397$	−10071.3	−1.27320	−0.636602	−	0.771192i	$-0.719661\pi$
$397$	−10071.3	−1.27320	−0.636602	+	0.771192i	$0.719661\pi$
$398$	0	0
$399$	1388.27	0.174186
$400$	0	0
$401$	3025.14	0.376729	0.188365	−	0.982099i	$-0.439681\pi$
$401$	3025.14	0.376729	0.188365	+	0.982099i	$0.439681\pi$
$402$	0	0
$403$	−1753.90	−0.216794
$404$	0	0
$405$	357.562	0.0438702
$406$	0	0
$407$	−6366.06	−0.775317
$408$	0	0
$409$	9440.21	1.14129	0.570646	−	0.821196i	$-0.306693\pi$
$409$	9440.21	1.14129	0.570646	+	0.821196i	$0.306693\pi$
$410$	0	0
$411$	−7228.36	−0.867515
$412$	0	0
$413$	107.868	0.0128519
$414$	0	0
$415$	−7119.98	−0.842183
$416$	0	0
$417$	20154.2	2.36680
$418$	0	0
$419$	4604.25	0.536831	0.268416	−	0.963303i	$-0.413500\pi$
$419$	4604.25	0.536831	0.268416	+	0.963303i	$0.413500\pi$
$420$	0	0
$421$	−13347.4	−1.54516	−0.772580	−	0.634917i	$-0.781034\pi$
$421$	−13347.4	−1.54516	−0.772580	+	0.634917i	$0.781034\pi$
$422$	0	0
$423$	−3460.95	−0.397818
$424$	0	0
$425$	−1376.23	−0.157076
$426$	0	0
$427$	−448.157	−0.0507912
$428$	0	0
$429$	2870.50	0.323052
$430$	0	0
$431$	10617.7	1.18663	0.593314	−	0.804971i	$-0.297819\pi$
$431$	10617.7	1.18663	0.593314	+	0.804971i	$0.297819\pi$
$432$	0	0
$433$	−706.479	−0.0784093	−0.0392046	−	0.999231i	$-0.512482\pi$
$433$	−706.479	−0.0784093	−0.0392046	+	0.999231i	$0.512482\pi$
$434$	0	0
$435$	26519.7	2.92303
$436$	0	0
$437$	2521.71	0.276041
$438$	0	0
$439$	13611.8	1.47985	0.739926	−	0.672688i	$-0.234860\pi$
$439$	13611.8	1.47985	0.739926	+	0.672688i	$0.234860\pi$
$440$	0	0
$441$	−15030.1	−1.62295
$442$	0	0
$443$	4422.21	0.474279	0.237139	−	0.971476i	$-0.423790\pi$
$443$	4422.21	0.474279	0.237139	+	0.971476i	$0.423790\pi$
$444$	0	0
$445$	−9438.81	−1.00549
$446$	0	0
$447$	384.403	0.0406748
$448$	0	0
$449$	−5231.76	−0.549893	−0.274947	−	0.961460i	$-0.588660\pi$
$449$	−5231.76	−0.549893	−0.274947	+	0.961460i	$0.588660\pi$
$450$	0	0
$451$	−5113.83	−0.533927
$452$	0	0
$453$	9086.01	0.942379
$454$	0	0
$455$	−265.978	−0.0274049
$456$	0	0
$457$	−6833.10	−0.699429	−0.349715	−	0.936856i	$-0.613721\pi$
$457$	−6833.10	−0.699429	−0.349715	+	0.936856i	$0.613721\pi$
$458$	0	0
$459$	−7763.38	−0.789463
$460$	0	0
$461$	−8451.19	−0.853821	−0.426910	−	0.904294i	$-0.640398\pi$
$461$	−8451.19	−0.853821	−0.426910	+	0.904294i	$0.640398\pi$
$462$	0	0
$463$	4273.38	0.428943	0.214472	−	0.976730i	$-0.431197\pi$
$463$	4273.38	0.428943	0.214472	+	0.976730i	$0.431197\pi$
$464$	0	0
$465$	13739.4	1.37021
$466$	0	0
$467$	17317.9	1.71601	0.858004	−	0.513643i	$-0.171704\pi$
$467$	17317.9	1.71601	0.858004	+	0.513643i	$0.171704\pi$
$468$	0	0
$469$	−654.167	−0.0644065
$470$	0	0
$471$	−22734.5	−2.22410
$472$	0	0
$473$	−10401.3	−1.01110
$474$	0	0
$475$	2577.09	0.248937
$476$	0	0
$477$	32839.7	3.15226
$478$	0	0
$479$	−4067.97	−0.388038	−0.194019	−	0.980998i	$-0.562152\pi$
$479$	−4067.97	−0.388038	−0.194019	+	0.980998i	$0.562152\pi$
$480$	0	0
$481$	−3269.06	−0.309888
$482$	0	0
$483$	347.360	0.0327235
$484$	0	0
$485$	−9942.00	−0.930811
$486$	0	0
$487$	−16174.3	−1.50499	−0.752493	−	0.658600i	$-0.771149\pi$
$487$	−16174.3	−1.50499	−0.752493	+	0.658600i	$0.771149\pi$
$488$	0	0
$489$	−8468.44	−0.783141
$490$	0	0
$491$	−19228.6	−1.76736	−0.883680	−	0.468091i	$-0.844942\pi$
$491$	−19228.6	−1.76736	−0.883680	+	0.468091i	$0.844942\pi$
$492$	0	0
$493$	−13730.1	−1.25431
$494$	0	0
$495$	−13955.1	−1.26714
$496$	0	0
$497$	1192.61	0.107637
$498$	0	0
$499$	−20713.6	−1.85825	−0.929127	−	0.369762i	$-0.879439\pi$
$499$	−20713.6	−1.85825	−0.929127	+	0.369762i	$0.879439\pi$
$500$	0	0
$501$	−29194.0	−2.60338
$502$	0	0
$503$	9828.84	0.871265	0.435632	−	0.900125i	$-0.356525\pi$
$503$	9828.84	0.871265	0.435632	+	0.900125i	$0.356525\pi$
$504$	0	0
$505$	−5263.12	−0.463774
$506$	0	0
$507$	−17059.8	−1.49438
$508$	0	0
$509$	−19029.8	−1.65714	−0.828568	−	0.559889i	$-0.810844\pi$
$509$	−19029.8	−1.65714	−0.828568	+	0.559889i	$0.810844\pi$
$510$	0	0
$511$	173.866	0.0150516
$512$	0	0
$513$	14537.5	1.25116
$514$	0	0
$515$	11810.8	1.01058
$516$	0	0
$517$	2017.19	0.171597
$518$	0	0
$519$	−15551.6	−1.31530
$520$	0	0
$521$	10607.1	0.891950	0.445975	−	0.895045i	$-0.352857\pi$
$521$	10607.1	0.891950	0.445975	+	0.895045i	$0.352857\pi$
$522$	0	0
$523$	−5519.88	−0.461506	−0.230753	−	0.973012i	$-0.574119\pi$
$523$	−5519.88	−0.461506	−0.230753	+	0.973012i	$0.574119\pi$
$524$	0	0
$525$	354.989	0.0295104
$526$	0	0
$527$	−7113.36	−0.587975
$528$	0	0
$529$	−11536.0	−0.948142
$530$	0	0
$531$	2906.25	0.237515
$532$	0	0
$533$	−2626.02	−0.213407
$534$	0	0
$535$	−12643.3	−1.02172
$536$	0	0
$537$	−12912.7	−1.03766
$538$	0	0
$539$	8760.20	0.700053
$540$	0	0
$541$	−13481.4	−1.07137	−0.535683	−	0.844419i	$-0.679946\pi$
$541$	−13481.4	−1.07137	−0.535683	+	0.844419i	$0.679946\pi$
$542$	0	0
$543$	−35121.9	−2.77573
$544$	0	0
$545$	−10298.5	−0.809427
$546$	0	0
$547$	1743.55	0.136287	0.0681434	−	0.997676i	$-0.478292\pi$
$547$	1743.55	0.136287	0.0681434	+	0.997676i	$0.478292\pi$
$548$	0	0
$549$	−12074.5	−0.938668
$550$	0	0
$551$	25710.6	1.98786
$552$	0	0
$553$	96.7105	0.00743680
$554$	0	0
$555$	25608.5	1.95860
$556$	0	0
$557$	−4086.47	−0.310861	−0.155430	−	0.987847i	$-0.549676\pi$
$557$	−4086.47	−0.310861	−0.155430	+	0.987847i	$0.549676\pi$
$558$	0	0
$559$	−5341.19	−0.404129
$560$	0	0
$561$	11642.0	0.876159
$562$	0	0
$563$	99.1014	0.00741852	0.00370926	−	0.999993i	$-0.498819\pi$
$563$	99.1014	0.00741852	0.00370926	+	0.999993i	$0.498819\pi$
$564$	0	0
$565$	−4319.96	−0.321668
$566$	0	0
$567$	47.7509	0.00353677
$568$	0	0
$569$	8915.23	0.656847	0.328423	−	0.944531i	$-0.393483\pi$
$569$	8915.23	0.656847	0.328423	+	0.944531i	$0.393483\pi$
$570$	0	0
$571$	6995.13	0.512674	0.256337	−	0.966587i	$-0.417484\pi$
$571$	6995.13	0.512674	0.256337	+	0.966587i	$0.417484\pi$
$572$	0	0
$573$	−3632.95	−0.264867
$574$	0	0
$575$	644.818	0.0467665
$576$	0	0
$577$	17911.5	1.29232	0.646159	−	0.763203i	$-0.276374\pi$
$577$	17911.5	1.29232	0.646159	+	0.763203i	$0.276374\pi$
$578$	0	0
$579$	−19124.2	−1.37267
$580$	0	0
$581$	−950.842	−0.0678960
$582$	0	0
$583$	−19140.4	−1.35972
$584$	0	0
$585$	−7166.15	−0.506468
$586$	0	0
$587$	−11229.2	−0.789573	−0.394787	−	0.918773i	$-0.629181\pi$
$587$	−11229.2	−0.789573	−0.394787	+	0.918773i	$0.629181\pi$
$588$	0	0
$589$	13320.2	0.931835
$590$	0	0
$591$	−12400.4	−0.863088
$592$	0	0
$593$	7006.26	0.485181	0.242591	−	0.970129i	$-0.422003\pi$
$593$	7006.26	0.485181	0.242591	+	0.970129i	$0.422003\pi$
$594$	0	0
$595$	−1078.73	−0.0743257
$596$	0	0
$597$	42090.8	2.88553
$598$	0	0
$599$	−8502.74	−0.579987	−0.289994	−	0.957029i	$-0.593653\pi$
$599$	−8502.74	−0.579987	−0.289994	+	0.957029i	$0.593653\pi$
$600$	0	0
$601$	11936.2	0.810127	0.405063	−	0.914289i	$-0.367249\pi$
$601$	11936.2	0.810127	0.405063	+	0.914289i	$0.367249\pi$
$602$	0	0
$603$	−17625.0	−1.19029
$604$	0	0
$605$	−8204.11	−0.551313
$606$	0	0
$607$	−3850.00	−0.257441	−0.128721	−	0.991681i	$-0.541087\pi$
$607$	−3850.00	−0.257441	−0.128721	+	0.991681i	$0.541087\pi$
$608$	0	0
$609$	3541.58	0.235652
$610$	0	0
$611$	1035.85	0.0685861
$612$	0	0
$613$	−8938.34	−0.588933	−0.294467	−	0.955662i	$-0.595142\pi$
$613$	−8938.34	−0.588933	−0.294467	+	0.955662i	$0.595142\pi$
$614$	0	0
$615$	20571.2	1.34880
$616$	0	0
$617$	−2585.09	−0.168674	−0.0843370	−	0.996437i	$-0.526877\pi$
$617$	−2585.09	−0.168674	−0.0843370	+	0.996437i	$0.526877\pi$
$618$	0	0
$619$	−10359.1	−0.672648	−0.336324	−	0.941746i	$-0.609184\pi$
$619$	−10359.1	−0.672648	−0.336324	+	0.941746i	$0.609184\pi$
$620$	0	0
$621$	3637.44	0.235049
$622$	0	0
$623$	−1260.51	−0.0810615
$624$	0	0
$625$	−18174.8	−1.16319
$626$	0	0
$627$	−21800.4	−1.38855
$628$	0	0
$629$	−13258.4	−0.840458
$630$	0	0
$631$	14411.5	0.909210	0.454605	−	0.890693i	$-0.349780\pi$
$631$	14411.5	0.909210	0.454605	+	0.890693i	$0.349780\pi$
$632$	0	0
$633$	31633.1	1.98626
$634$	0	0
$635$	−29039.9	−1.81482
$636$	0	0
$637$	4498.48	0.279806
$638$	0	0
$639$	32132.1	1.98924
$640$	0	0
$641$	−25724.0	−1.58508	−0.792542	−	0.609818i	$-0.791243\pi$
$641$	−25724.0	−1.58508	−0.792542	+	0.609818i	$0.791243\pi$
$642$	0	0
$643$	11081.4	0.679640	0.339820	−	0.940491i	$-0.389634\pi$
$643$	11081.4	0.679640	0.339820	+	0.940491i	$0.389634\pi$
$644$	0	0
$645$	41840.8	2.55423
$646$	0	0
$647$	1247.43	0.0757981	0.0378991	−	0.999282i	$-0.487933\pi$
$647$	1247.43	0.0757981	0.0378991	+	0.999282i	$0.487933\pi$
$648$	0	0
$649$	−1693.89	−0.102451
$650$	0	0
$651$	1834.84	0.110465
$652$	0	0
$653$	11741.1	0.703621	0.351811	−	0.936071i	$-0.385566\pi$
$653$	11741.1	0.703621	0.351811	+	0.936071i	$0.385566\pi$
$654$	0	0
$655$	−7003.63	−0.417793
$656$	0	0
$657$	4684.42	0.278168
$658$	0	0
$659$	2398.73	0.141792	0.0708961	−	0.997484i	$-0.477414\pi$
$659$	2398.73	0.141792	0.0708961	+	0.997484i	$0.477414\pi$
$660$	0	0
$661$	12428.5	0.731337	0.365669	−	0.930745i	$-0.380840\pi$
$661$	12428.5	0.731337	0.365669	+	0.930745i	$0.380840\pi$
$662$	0	0
$663$	5978.32	0.350194
$664$	0	0
$665$	2020.00	0.117793
$666$	0	0
$667$	6433.09	0.373449
$668$	0	0
$669$	31135.9	1.79938
$670$	0	0
$671$	7037.56	0.404891
$672$	0	0
$673$	−23869.3	−1.36716	−0.683578	−	0.729878i	$-0.739577\pi$
$673$	−23869.3	−1.36716	−0.683578	+	0.729878i	$0.739577\pi$
$674$	0	0
$675$	3717.32	0.211970
$676$	0	0
$677$	−8009.12	−0.454676	−0.227338	−	0.973816i	$-0.573002\pi$
$677$	−8009.12	−0.454676	−0.227338	+	0.973816i	$0.573002\pi$
$678$	0	0
$679$	−1327.71	−0.0750410
$680$	0	0
$681$	20405.8	1.14824
$682$	0	0
$683$	12944.0	0.725167	0.362583	−	0.931951i	$-0.381895\pi$
$683$	12944.0	0.725167	0.362583	+	0.931951i	$0.381895\pi$
$684$	0	0
$685$	−10517.7	−0.586655
$686$	0	0
$687$	−1095.21	−0.0608224
$688$	0	0
$689$	−9828.85	−0.543468
$690$	0	0
$691$	24123.5	1.32807	0.664037	−	0.747699i	$-0.268842\pi$
$691$	24123.5	1.32807	0.664037	+	0.747699i	$0.268842\pi$
$692$	0	0
$693$	−1863.65	−0.102156
$694$	0	0
$695$	29325.4	1.60054
$696$	0	0
$697$	−10650.4	−0.578787
$698$	0	0
$699$	35934.8	1.94446
$700$	0	0
$701$	10918.3	0.588274	0.294137	−	0.955763i	$-0.404968\pi$
$701$	10918.3	0.588274	0.294137	+	0.955763i	$0.404968\pi$
$702$	0	0
$703$	24827.3	1.33198
$704$	0	0
$705$	−8114.47	−0.433487
$706$	0	0
$707$	−702.866	−0.0373890
$708$	0	0
$709$	6473.79	0.342917	0.171459	−	0.985191i	$-0.445152\pi$
$709$	6473.79	0.342917	0.171459	+	0.985191i	$0.445152\pi$
$710$	0	0
$711$	2605.64	0.137439
$712$	0	0
$713$	3332.88	0.175059
$714$	0	0
$715$	4176.74	0.218463
$716$	0	0
$717$	42628.0	2.22032
$718$	0	0
$719$	30210.0	1.56696	0.783479	−	0.621418i	$-0.213443\pi$
$719$	30210.0	1.56696	0.783479	+	0.621418i	$0.213443\pi$
$720$	0	0
$721$	1577.28	0.0814717
$722$	0	0
$723$	−411.995	−0.0211926
$724$	0	0
$725$	6574.37	0.336781
$726$	0	0
$727$	−20721.3	−1.05710	−0.528549	−	0.848903i	$-0.677264\pi$
$727$	−20721.3	−1.05710	−0.528549	+	0.848903i	$0.677264\pi$
$728$	0	0
$729$	−31696.7	−1.61036
$730$	0	0
$731$	−21662.4	−1.09605
$732$	0	0
$733$	19629.0	0.989107	0.494553	−	0.869147i	$-0.335332\pi$
$733$	19629.0	0.989107	0.494553	+	0.869147i	$0.335332\pi$
$734$	0	0
$735$	−35239.3	−1.76847
$736$	0	0
$737$	10272.6	0.513428
$738$	0	0
$739$	−12436.5	−0.619058	−0.309529	−	0.950890i	$-0.600171\pi$
$739$	−12436.5	−0.619058	−0.309529	+	0.950890i	$0.600171\pi$
$740$	0	0
$741$	−11194.8	−0.554995
$742$	0	0
$743$	7669.27	0.378678	0.189339	−	0.981912i	$-0.439365\pi$
$743$	7669.27	0.378678	0.189339	+	0.981912i	$0.439365\pi$
$744$	0	0
$745$	559.327	0.0275062
$746$	0	0
$747$	−25618.2	−1.25478
$748$	0	0
$749$	−1688.46	−0.0823698
$750$	0	0
$751$	26531.8	1.28916	0.644580	−	0.764537i	$-0.277032\pi$
$751$	26531.8	1.28916	0.644580	+	0.764537i	$0.277032\pi$
$752$	0	0
$753$	−31075.5	−1.50392
$754$	0	0
$755$	13220.6	0.637282
$756$	0	0
$757$	112.316	0.00539258	0.00269629	−	0.999996i	$-0.499142\pi$
$757$	112.316	0.00539258	0.00269629	+	0.999996i	$0.499142\pi$
$758$	0	0
$759$	−5454.71	−0.260861
$760$	0	0
$761$	36991.3	1.76207	0.881033	−	0.473055i	$-0.156849\pi$
$761$	36991.3	1.76207	0.881033	+	0.473055i	$0.156849\pi$
$762$	0	0
$763$	−1375.31	−0.0652552
$764$	0	0
$765$	−29064.0	−1.37361
$766$	0	0
$767$	−869.835	−0.0409490
$768$	0	0
$769$	26637.0	1.24910	0.624548	−	0.780987i	$-0.285283\pi$
$769$	26637.0	1.24910	0.624548	+	0.780987i	$0.285283\pi$
$770$	0	0
$771$	6234.64	0.291226
$772$	0	0
$773$	27921.7	1.29919	0.649596	−	0.760280i	$-0.274938\pi$
$773$	27921.7	1.29919	0.649596	+	0.760280i	$0.274938\pi$
$774$	0	0
$775$	3406.07	0.157871
$776$	0	0
$777$	3419.91	0.157900
$778$	0	0
$779$	19943.7	0.917273
$780$	0	0
$781$	−18727.9	−0.858051
$782$	0	0
$783$	37086.2	1.69266
$784$	0	0
$785$	−33079.9	−1.50404
$786$	0	0
$787$	−40839.8	−1.84979	−0.924893	−	0.380229i	$-0.875845\pi$
$787$	−40839.8	−1.84979	−0.924893	+	0.380229i	$0.875845\pi$
$788$	0	0
$789$	20653.8	0.931931
$790$	0	0
$791$	−576.912	−0.0259325
$792$	0	0
$793$	3613.88	0.161832
$794$	0	0
$795$	76995.4	3.43490
$796$	0	0
$797$	23555.1	1.04688	0.523440	−	0.852062i	$-0.324648\pi$
$797$	23555.1	1.04688	0.523440	+	0.852062i	$0.324648\pi$
$798$	0	0
$799$	4201.14	0.186015
$800$	0	0
$801$	−33961.5	−1.49809
$802$	0	0
$803$	−2730.28	−0.119987
$804$	0	0
$805$	505.428	0.0221292
$806$	0	0
$807$	−9901.55	−0.431910
$808$	0	0
$809$	34940.4	1.51847	0.759233	−	0.650819i	$-0.225574\pi$
$809$	34940.4	1.51847	0.759233	+	0.650819i	$0.225574\pi$
$810$	0	0
$811$	21451.0	0.928789	0.464395	−	0.885628i	$-0.346272\pi$
$811$	21451.0	0.928789	0.464395	+	0.885628i	$0.346272\pi$
$812$	0	0
$813$	−11851.3	−0.511244
$814$	0	0
$815$	−12322.0	−0.529597
$816$	0	0
$817$	40564.3	1.73705
$818$	0	0
$819$	−957.008	−0.0408310
$820$	0	0
$821$	−12318.6	−0.523656	−0.261828	−	0.965115i	$-0.584325\pi$
$821$	−12318.6	−0.523656	−0.261828	+	0.965115i	$0.584325\pi$
$822$	0	0
$823$	−24493.5	−1.03741	−0.518707	−	0.854952i	$-0.673586\pi$
$823$	−24493.5	−1.03741	−0.518707	+	0.854952i	$0.673586\pi$
$824$	0	0
$825$	−5574.50	−0.235248
$826$	0	0
$827$	37233.4	1.56558	0.782789	−	0.622287i	$-0.213796\pi$
$827$	37233.4	1.56558	0.782789	+	0.622287i	$0.213796\pi$
$828$	0	0
$829$	−12881.0	−0.539658	−0.269829	−	0.962908i	$-0.586967\pi$
$829$	−12881.0	−0.539658	−0.269829	+	0.962908i	$0.586967\pi$
$830$	0	0
$831$	26791.6	1.11840
$832$	0	0
$833$	18244.6	0.758870
$834$	0	0
$835$	−42478.9	−1.76053
$836$	0	0
$837$	19213.8	0.793459
$838$	0	0
$839$	1394.89	0.0573982	0.0286991	−	0.999588i	$-0.490864\pi$
$839$	1394.89	0.0573982	0.0286991	+	0.999588i	$0.490864\pi$
$840$	0	0
$841$	41200.9	1.68932
$842$	0	0
$843$	51003.7	2.08382
$844$	0	0
$845$	−24822.9	−1.01057
$846$	0	0
$847$	−1095.62	−0.0444463
$848$	0	0
$849$	29344.1	1.18620
$850$	0	0
$851$	6212.08	0.250232
$852$	0	0
$853$	−21615.8	−0.867658	−0.433829	−	0.900995i	$-0.642838\pi$
$853$	−21615.8	−0.867658	−0.433829	+	0.900995i	$0.642838\pi$
$854$	0	0
$855$	54424.2	2.17692
$856$	0	0
$857$	−2273.70	−0.0906277	−0.0453139	−	0.998973i	$-0.514429\pi$
$857$	−2273.70	−0.0906277	−0.0453139	+	0.998973i	$0.514429\pi$
$858$	0	0
$859$	−30652.1	−1.21750	−0.608752	−	0.793361i	$-0.708329\pi$
$859$	−30652.1	−1.21750	−0.608752	+	0.793361i	$0.708329\pi$
$860$	0	0
$861$	2747.20	0.108739
$862$	0	0
$863$	23721.7	0.935686	0.467843	−	0.883812i	$-0.345031\pi$
$863$	23721.7	0.935686	0.467843	+	0.883812i	$0.345031\pi$
$864$	0	0
$865$	−22628.5	−0.889469
$866$	0	0
$867$	−17199.5	−0.673731
$868$	0	0
$869$	−1518.68	−0.0592838
$870$	0	0
$871$	5275.12	0.205213
$872$	0	0
$873$	−35772.1	−1.38683
$874$	0	0
$875$	−1998.65	−0.0772189
$876$	0	0
$877$	31720.2	1.22134	0.610670	−	0.791885i	$-0.290900\pi$
$877$	31720.2	1.22134	0.610670	+	0.791885i	$0.290900\pi$
$878$	0	0
$879$	−22100.7	−0.848054
$880$	0	0
$881$	−24603.0	−0.940859	−0.470429	−	0.882438i	$-0.655901\pi$
$881$	−24603.0	−0.940859	−0.470429	+	0.882438i	$0.655901\pi$
$882$	0	0
$883$	−33215.2	−1.26589	−0.632945	−	0.774197i	$-0.718154\pi$
$883$	−33215.2	−1.26589	−0.632945	+	0.774197i	$0.718154\pi$
$884$	0	0
$885$	6813.95	0.258812
$886$	0	0
$887$	39722.9	1.50368	0.751841	−	0.659345i	$-0.229166\pi$
$887$	39722.9	1.50368	0.751841	+	0.659345i	$0.229166\pi$
$888$	0	0
$889$	−3878.15	−0.146309
$890$	0	0
$891$	−749.848	−0.0281940
$892$	0	0
$893$	−7866.92	−0.294800
$894$	0	0
$895$	−18788.7	−0.701716
$896$	0	0
$897$	−2801.07	−0.104264
$898$	0	0
$899$	33981.1	1.26066
$900$	0	0
$901$	−39863.2	−1.47396
$902$	0	0
$903$	5587.65	0.205920
$904$	0	0
$905$	−51104.2	−1.87708
$906$	0	0
$907$	−6456.50	−0.236367	−0.118183	−	0.992992i	$-0.537707\pi$
$907$	−6456.50	−0.236367	−0.118183	+	0.992992i	$0.537707\pi$
$908$	0	0
$909$	−18937.1	−0.690983
$910$	0	0
$911$	2013.95	0.0732438	0.0366219	−	0.999329i	$-0.488340\pi$
$911$	2013.95	0.0732438	0.0366219	+	0.999329i	$0.488340\pi$
$912$	0	0
$913$	14931.4	0.541245
$914$	0	0
$915$	−28309.7	−1.02283
$916$	0	0
$917$	−935.303	−0.0336820
$918$	0	0
$919$	−37746.5	−1.35489	−0.677443	−	0.735575i	$-0.736912\pi$
$919$	−37746.5	−1.35489	−0.677443	+	0.735575i	$0.736912\pi$
$920$	0	0
$921$	25141.2	0.899492
$922$	0	0
$923$	−9617.05	−0.342957
$924$	0	0
$925$	6348.50	0.225662
$926$	0	0
$927$	42496.2	1.50567
$928$	0	0
$929$	45643.5	1.61197	0.805983	−	0.591939i	$-0.201637\pi$
$929$	45643.5	1.61197	0.805983	+	0.591939i	$0.201637\pi$
$930$	0	0
$931$	−34164.3	−1.20267
$932$	0	0
$933$	−44667.6	−1.56737
$934$	0	0
$935$	16939.7	0.592501
$936$	0	0
$937$	−47317.5	−1.64973	−0.824864	−	0.565331i	$-0.808748\pi$
$937$	−47317.5	−1.64973	−0.824864	+	0.565331i	$0.808748\pi$
$938$	0	0
$939$	33793.4	1.17445
$940$	0	0
$941$	21318.9	0.738550	0.369275	−	0.929320i	$-0.379606\pi$
$941$	21318.9	0.738550	0.369275	+	0.929320i	$0.379606\pi$
$942$	0	0
$943$	4990.14	0.172324
$944$	0	0
$945$	2913.75	0.100301
$946$	0	0
$947$	20601.8	0.706937	0.353468	−	0.935446i	$-0.385002\pi$
$947$	20601.8	0.706937	0.353468	+	0.935446i	$0.385002\pi$
$948$	0	0
$949$	−1402.03	−0.0479578
$950$	0	0
$951$	−9654.45	−0.329198
$952$	0	0
$953$	−42987.2	−1.46117	−0.730583	−	0.682824i	$-0.760752\pi$
$953$	−42987.2	−1.46117	−0.730583	+	0.682824i	$0.760752\pi$
$954$	0	0
$955$	−5286.14	−0.179116
$956$	0	0
$957$	−55614.6	−1.87854
$958$	0	0
$959$	−1404.59	−0.0472956
$960$	0	0
$961$	−12186.0	−0.409049
$962$	0	0
$963$	−45491.6	−1.52227
$964$	0	0
$965$	−27826.7	−0.928264
$966$	0	0
$967$	44030.7	1.46425	0.732126	−	0.681170i	$-0.238528\pi$
$967$	44030.7	1.46425	0.732126	+	0.681170i	$0.238528\pi$
$968$	0	0
$969$	−45403.1	−1.50522
$970$	0	0
$971$	−50487.1	−1.66860	−0.834298	−	0.551313i	$-0.814127\pi$
$971$	−50487.1	−1.66860	−0.834298	+	0.551313i	$0.814127\pi$
$972$	0	0
$973$	3916.28	0.129034
$974$	0	0
$975$	−2862.58	−0.0940267
$976$	0	0
$977$	49515.3	1.62143	0.810714	−	0.585442i	$-0.199079\pi$
$977$	49515.3	1.62143	0.810714	+	0.585442i	$0.199079\pi$
$978$	0	0
$979$	19794.2	0.646196
$980$	0	0
$981$	−37054.6	−1.20598
$982$	0	0
$983$	40046.2	1.29936	0.649682	−	0.760206i	$-0.274902\pi$
$983$	40046.2	1.29936	0.649682	+	0.760206i	$0.274902\pi$
$984$	0	0
$985$	−18043.3	−0.583662
$986$	0	0
$987$	−1083.65	−0.0349473
$988$	0	0
$989$	10149.7	0.326330
$990$	0	0
$991$	18673.2	0.598560	0.299280	−	0.954165i	$-0.403254\pi$
$991$	18673.2	0.598560	0.299280	+	0.954165i	$0.403254\pi$
$992$	0	0
$993$	50460.9	1.61262
$994$	0	0
$995$	61244.4	1.95133
$996$	0	0
$997$	31087.3	0.987508	0.493754	−	0.869602i	$-0.335624\pi$
$997$	31087.3	0.987508	0.493754	+	0.869602i	$0.335624\pi$
$998$	0	0
$999$	35812.1	1.13418

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.4.a.n.1.10		10
4.3	odd	2		1024.4.a.m.1.1		10
8.3	odd	2		1024.4.a.m.1.10		10
8.5	even	2	inner	1024.4.a.n.1.1		10
16.3	odd	4		1024.4.b.k.513.1		10
16.5	even	4		1024.4.b.j.513.1		10
16.11	odd	4		1024.4.b.k.513.10		10
16.13	even	4		1024.4.b.j.513.10		10
32.3	odd	8		128.4.e.a.33.1		10
32.5	even	8		16.4.e.a.5.2	✓	10
32.11	odd	8		128.4.e.a.97.1		10
32.13	even	8		16.4.e.a.13.2	yes	10
32.19	odd	8		64.4.e.a.17.5		10
32.21	even	8		128.4.e.b.97.5		10
32.27	odd	8		64.4.e.a.49.5		10
32.29	even	8		128.4.e.b.33.5		10
96.5	odd	8		144.4.k.a.37.4		10
96.59	even	8		576.4.k.a.433.1		10
96.77	odd	8		144.4.k.a.109.4		10
96.83	even	8		576.4.k.a.145.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.2	✓	10	32.5	even	8
16.4.e.a.13.2	yes	10	32.13	even	8
64.4.e.a.17.5		10	32.19	odd	8
64.4.e.a.49.5		10	32.27	odd	8
128.4.e.a.33.1		10	32.3	odd	8
128.4.e.a.97.1		10	32.11	odd	8
128.4.e.b.33.5		10	32.29	even	8
128.4.e.b.97.5		10	32.21	even	8
144.4.k.a.37.4		10	96.5	odd	8
144.4.k.a.109.4		10	96.77	odd	8
576.4.k.a.145.1		10	96.83	even	8
576.4.k.a.433.1		10	96.59	even	8
1024.4.a.m.1.1		10	4.3	odd	2
1024.4.a.m.1.10		10	8.3	odd	2
1024.4.a.n.1.1		10	8.5	even	2	inner
1024.4.a.n.1.10		10	1.1	even	1	trivial
1024.4.b.j.513.1		10	16.5	even	4
1024.4.b.j.513.10		10	16.13	even	4
1024.4.b.k.513.1		10	16.3	odd	4
1024.4.b.k.513.10		10	16.11	odd	4

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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

\(f(q)\)	\(=\)	\(q+8.43597 q^{3} +12.2748 q^{5} +1.63924 q^{7} +44.1656 q^{9} +O(q^{10})\)	\(q+8.43597 q^{3} +12.2748 q^{5} +1.63924 q^{7} +44.1656 q^{9} -25.7416 q^{11} -13.2187 q^{13} +103.550 q^{15} -53.6113 q^{17} +100.391 q^{19} +13.8286 q^{21} +25.1189 q^{23} +25.6706 q^{25} +144.809 q^{27} +256.105 q^{29} +132.684 q^{31} -217.155 q^{33} +20.1214 q^{35} +247.306 q^{37} -111.512 q^{39} +198.660 q^{41} +404.064 q^{43} +542.124 q^{45} -78.3629 q^{47} -340.313 q^{49} -452.264 q^{51} +743.559 q^{53} -315.973 q^{55} +846.894 q^{57} +65.8036 q^{59} -273.392 q^{61} +72.3982 q^{63} -162.256 q^{65} -399.066 q^{67} +211.903 q^{69} +727.536 q^{71} +106.065 q^{73} +216.556 q^{75} -42.1968 q^{77} +58.9970 q^{79} +29.1298 q^{81} -580.049 q^{83} -658.068 q^{85} +2160.50 q^{87} -768.959 q^{89} -21.6686 q^{91} +1119.32 q^{93} +1232.28 q^{95} -809.953 q^{97} -1136.89 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 28 q^{7} + 54 q^{9}+O(q^{10}) \) 10 * q + 28 * q^7 + 54 * q^9	\( 10 q + 28 q^{7} + 54 q^{9} + 124 q^{15} + 4 q^{17} + 276 q^{23} + 50 q^{25} + 368 q^{31} - 4 q^{33} + 732 q^{39} + 944 q^{47} - 94 q^{49} + 1380 q^{55} + 108 q^{57} + 2628 q^{63} - 492 q^{65} + 3468 q^{71} - 296 q^{73} + 4416 q^{79} - 482 q^{81} + 6036 q^{87} + 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q + 28 * q^7 + 54 * q^9 + 124 * q^15 + 4 * q^17 + 276 * q^23 + 50 * q^25 + 368 * q^31 - 4 * q^33 + 732 * q^39 + 944 * q^47 - 94 * q^49 + 1380 * q^55 + 108 * q^57 + 2628 * q^63 - 492 * q^65 + 3468 * q^71 - 296 * q^73 + 4416 * q^79 - 482 * q^81 + 6036 * q^87 + 88 * q^89 + 6900 * q^95 - 4 * q^97

Introduction

L-functions

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