LMFDB - Embedded newform 5120.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5120,2,Mod(1,5120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5120, 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("5120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5120 = 2^{10} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5120.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:	$40.8834058349$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	5120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.93185 q^{3} +1.00000 q^{5} +4.34607 q^{7} +5.59575 q^{9} +O(q^{10})$	$q+2.93185 q^{3} +1.00000 q^{5} +4.34607 q^{7} +5.59575 q^{9} +3.76733 q^{11} -0.682163 q^{13} +2.93185 q^{15} -0.353113 q^{17} -7.63103 q^{19} +12.7420 q^{21} +6.62863 q^{23} +1.00000 q^{25} +7.61037 q^{27} +4.84909 q^{29} +0.635674 q^{31} +11.0452 q^{33} +4.34607 q^{35} -4.42883 q^{37} -2.00000 q^{39} -2.33245 q^{41} -10.7350 q^{43} +5.59575 q^{45} -5.41057 q^{47} +11.8883 q^{49} -1.03528 q^{51} +6.35311 q^{53} +3.76733 q^{55} -22.3730 q^{57} -13.3877 q^{59} -8.79191 q^{61} +24.3195 q^{63} -0.682163 q^{65} +0.512994 q^{67} +19.4341 q^{69} -7.32780 q^{71} -10.3682 q^{73} +2.93185 q^{75} +16.3730 q^{77} +8.13630 q^{79} +5.52520 q^{81} +11.9065 q^{83} -0.353113 q^{85} +14.2168 q^{87} -4.55583 q^{89} -2.96472 q^{91} +1.86370 q^{93} -7.63103 q^{95} +2.17549 q^{97} +21.0810 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7	$ 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{17} - 4 q^{19} + 16 q^{21} + 16 q^{23} + 4 q^{25} + 4 q^{27} + 8 q^{29} + 12 q^{33} + 4 q^{35} - 8 q^{37} - 8 q^{39} - 4 q^{41} + 4 q^{47} + 20 q^{53} + 4 q^{55} - 28 q^{57} + 12 q^{59} + 8 q^{61} + 24 q^{63} - 4 q^{65} - 24 q^{67} + 16 q^{69} - 12 q^{73} + 4 q^{75} + 4 q^{77} + 48 q^{79} + 8 q^{81} + 16 q^{83} + 4 q^{85} + 36 q^{87} + 24 q^{89} - 16 q^{91} - 8 q^{93} - 4 q^{95} - 12 q^{97} + 28 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^17 - 4 * q^19 + 16 * q^21 + 16 * q^23 + 4 * q^25 + 4 * q^27 + 8 * q^29 + 12 * q^33 + 4 * q^35 - 8 * q^37 - 8 * q^39 - 4 * q^41 + 4 * q^47 + 20 * q^53 + 4 * q^55 - 28 * q^57 + 12 * q^59 + 8 * q^61 + 24 * q^63 - 4 * q^65 - 24 * q^67 + 16 * q^69 - 12 * q^73 + 4 * q^75 + 4 * q^77 + 48 * q^79 + 8 * q^81 + 16 * q^83 + 4 * q^85 + 36 * q^87 + 24 * q^89 - 16 * q^91 - 8 * q^93 - 4 * q^95 - 12 * 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
$3$	2.93185	1.69271	0.846353	−	0.532623i	$-0.178794\pi$
$3$	2.93185	1.69271	0.846353	+	0.532623i	$0.178794\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.34607	1.64266	0.821329	−	0.570455i	$-0.193233\pi$
$7$	4.34607	1.64266	0.821329	+	0.570455i	$0.193233\pi$
$8$	0	0
$9$	5.59575	1.86525
$10$	0	0
$11$	3.76733	1.13589	0.567946	−	0.823066i	$-0.307738\pi$
$11$	3.76733	1.13589	0.567946	+	0.823066i	$0.307738\pi$
$12$	0	0
$13$	−0.682163	−0.189198	−0.0945990	−	0.995515i	$-0.530157\pi$
$13$	−0.682163	−0.189198	−0.0945990	+	0.995515i	$0.530157\pi$
$14$	0	0
$15$	2.93185	0.757001
$16$	0	0
$17$	−0.353113	−0.0856426	−0.0428213	−	0.999083i	$-0.513635\pi$
$17$	−0.353113	−0.0856426	−0.0428213	+	0.999083i	$0.513635\pi$
$18$	0	0
$19$	−7.63103	−1.75068	−0.875339	−	0.483509i	$-0.839362\pi$
$19$	−7.63103	−1.75068	−0.875339	+	0.483509i	$0.839362\pi$
$20$	0	0
$21$	12.7420	2.78054
$22$	0	0
$23$	6.62863	1.38216	0.691082	−	0.722776i	$-0.257134\pi$
$23$	6.62863	1.38216	0.691082	+	0.722776i	$0.257134\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	7.61037	1.46462
$28$	0	0
$29$	4.84909	0.900454	0.450227	−	0.892914i	$-0.351343\pi$
$29$	4.84909	0.900454	0.450227	+	0.892914i	$0.351343\pi$
$30$	0	0
$31$	0.635674	0.114171	0.0570853	−	0.998369i	$-0.481819\pi$
$31$	0.635674	0.114171	0.0570853	+	0.998369i	$0.481819\pi$
$32$	0	0
$33$	11.0452	1.92273
$34$	0	0
$35$	4.34607	0.734619
$36$	0	0
$37$	−4.42883	−0.728094	−0.364047	−	0.931380i	$-0.618605\pi$
$37$	−4.42883	−0.728094	−0.364047	+	0.931380i	$0.618605\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−2.33245	−0.364267	−0.182134	−	0.983274i	$-0.558300\pi$
$41$	−2.33245	−0.364267	−0.182134	+	0.983274i	$0.558300\pi$
$42$	0	0
$43$	−10.7350	−1.63707	−0.818534	−	0.574458i	$-0.805213\pi$
$43$	−10.7350	−1.63707	−0.818534	+	0.574458i	$0.805213\pi$
$44$	0	0
$45$	5.59575	0.834166
$46$	0	0
$47$	−5.41057	−0.789212	−0.394606	−	0.918850i	$-0.629119\pi$
$47$	−5.41057	−0.789212	−0.394606	+	0.918850i	$0.629119\pi$
$48$	0	0
$49$	11.8883	1.69833
$50$	0	0
$51$	−1.03528	−0.144968
$52$	0	0
$53$	6.35311	0.872667	0.436334	−	0.899785i	$-0.356277\pi$
$53$	6.35311	0.872667	0.436334	+	0.899785i	$0.356277\pi$
$54$	0	0
$55$	3.76733	0.507986
$56$	0	0
$57$	−22.3730	−2.96338
$58$	0	0
$59$	−13.3877	−1.74292	−0.871462	−	0.490462i	$-0.836828\pi$
$59$	−13.3877	−1.74292	−0.871462	+	0.490462i	$0.836828\pi$
$60$	0	0
$61$	−8.79191	−1.12569	−0.562844	−	0.826563i	$-0.690293\pi$
$61$	−8.79191	−1.12569	−0.562844	+	0.826563i	$0.690293\pi$
$62$	0	0
$63$	24.3195	3.06397
$64$	0	0
$65$	−0.682163	−0.0846119
$66$	0	0
$67$	0.512994	0.0626722	0.0313361	−	0.999509i	$-0.490024\pi$
$67$	0.512994	0.0626722	0.0313361	+	0.999509i	$0.490024\pi$
$68$	0	0
$69$	19.4341	2.33960
$70$	0	0
$71$	−7.32780	−0.869650	−0.434825	−	0.900515i	$-0.643190\pi$
$71$	−7.32780	−0.869650	−0.434825	+	0.900515i	$0.643190\pi$
$72$	0	0
$73$	−10.3682	−1.21351	−0.606755	−	0.794889i	$-0.707529\pi$
$73$	−10.3682	−1.21351	−0.606755	+	0.794889i	$0.707529\pi$
$74$	0	0
$75$	2.93185	0.338541
$76$	0	0
$77$	16.3730	1.86588
$78$	0	0
$79$	8.13630	0.915405	0.457702	−	0.889105i	$-0.348673\pi$
$79$	8.13630	0.915405	0.457702	+	0.889105i	$0.348673\pi$
$80$	0	0
$81$	5.52520	0.613911
$82$	0	0
$83$	11.9065	1.30691	0.653456	−	0.756964i	$-0.273318\pi$
$83$	11.9065	1.30691	0.653456	+	0.756964i	$0.273318\pi$
$84$	0	0
$85$	−0.353113	−0.0383005
$86$	0	0
$87$	14.2168	1.52420
$88$	0	0
$89$	−4.55583	−0.482917	−0.241459	−	0.970411i	$-0.577626\pi$
$89$	−4.55583	−0.482917	−0.241459	+	0.970411i	$0.577626\pi$
$90$	0	0
$91$	−2.96472	−0.310788
$92$	0	0
$93$	1.86370	0.193257
$94$	0	0
$95$	−7.63103	−0.782927
$96$	0	0
$97$	2.17549	0.220887	0.110444	−	0.993882i	$-0.464773\pi$
$97$	2.17549	0.220887	0.110444	+	0.993882i	$0.464773\pi$
$98$	0	0
$99$	21.0810	2.11872
$100$	0	0
$101$	5.70674	0.567842	0.283921	−	0.958848i	$-0.408365\pi$
$101$	5.70674	0.567842	0.283921	+	0.958848i	$0.408365\pi$
$102$	0	0
$103$	−2.92116	−0.287830	−0.143915	−	0.989590i	$-0.545969\pi$
$103$	−2.92116	−0.287830	−0.143915	+	0.989590i	$0.545969\pi$
$104$	0	0
$105$	12.7420	1.24349
$106$	0	0
$107$	−3.78950	−0.366345	−0.183173	−	0.983081i	$-0.558637\pi$
$107$	−3.78950	−0.366345	−0.183173	+	0.983081i	$0.558637\pi$
$108$	0	0
$109$	11.0504	1.05844	0.529218	−	0.848486i	$-0.322485\pi$
$109$	11.0504	1.05844	0.529218	+	0.848486i	$0.322485\pi$
$110$	0	0
$111$	−12.9847	−1.23245
$112$	0	0
$113$	14.3923	1.35391	0.676957	−	0.736022i	$-0.263298\pi$
$113$	14.3923	1.35391	0.676957	+	0.736022i	$0.263298\pi$
$114$	0	0
$115$	6.62863	0.618123
$116$	0	0
$117$	−3.81722	−0.352902
$118$	0	0
$119$	−1.53465	−0.140681
$120$	0	0
$121$	3.19275	0.290250
$122$	0	0
$123$	−6.83839	−0.616597
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.6534	−1.83269	−0.916345	−	0.400389i	$-0.868875\pi$
$127$	−20.6534	−1.83269	−0.916345	+	0.400389i	$0.868875\pi$
$128$	0	0
$129$	−31.4733	−2.77107
$130$	0	0
$131$	−2.93890	−0.256773	−0.128386	−	0.991724i	$-0.540980\pi$
$131$	−2.93890	−0.256773	−0.128386	+	0.991724i	$0.540980\pi$
$132$	0	0
$133$	−33.1650	−2.87577
$134$	0	0
$135$	7.61037	0.654996
$136$	0	0
$137$	−14.4195	−1.23194	−0.615972	−	0.787768i	$-0.711237\pi$
$137$	−14.4195	−1.23194	−0.615972	+	0.787768i	$0.711237\pi$
$138$	0	0
$139$	1.39695	0.118488	0.0592438	−	0.998244i	$-0.481131\pi$
$139$	1.39695	0.118488	0.0592438	+	0.998244i	$0.481131\pi$
$140$	0	0
$141$	−15.8630	−1.33590
$142$	0	0
$143$	−2.56993	−0.214908
$144$	0	0
$145$	4.84909	0.402695
$146$	0	0
$147$	34.8547	2.87477
$148$	0	0
$149$	−1.52985	−0.125330	−0.0626649	−	0.998035i	$-0.519960\pi$
$149$	−1.52985	−0.125330	−0.0626649	+	0.998035i	$0.519960\pi$
$150$	0	0
$151$	12.2220	0.994610	0.497305	−	0.867576i	$-0.334323\pi$
$151$	12.2220	0.994610	0.497305	+	0.867576i	$0.334323\pi$
$152$	0	0
$153$	−1.97594	−0.159745
$154$	0	0
$155$	0.635674	0.0510586
$156$	0	0
$157$	−0.404410	−0.0322754	−0.0161377	−	0.999870i	$-0.505137\pi$
$157$	−0.404410	−0.0322754	−0.0161377	+	0.999870i	$0.505137\pi$
$158$	0	0
$159$	18.6264	1.47717
$160$	0	0
$161$	28.8084	2.27042
$162$	0	0
$163$	1.63327	0.127928	0.0639638	−	0.997952i	$-0.479626\pi$
$163$	1.63327	0.127928	0.0639638	+	0.997952i	$0.479626\pi$
$164$	0	0
$165$	11.0452	0.859871
$166$	0	0
$167$	9.52280	0.736896	0.368448	−	0.929648i	$-0.379889\pi$
$167$	9.52280	0.736896	0.368448	+	0.929648i	$0.379889\pi$
$168$	0	0
$169$	−12.5347	−0.964204
$170$	0	0
$171$	−42.7014	−3.26546
$172$	0	0
$173$	−3.86370	−0.293752	−0.146876	−	0.989155i	$-0.546922\pi$
$173$	−3.86370	−0.293752	−0.146876	+	0.989155i	$0.546922\pi$
$174$	0	0
$175$	4.34607	0.328532
$176$	0	0
$177$	−39.2506	−2.95026
$178$	0	0
$179$	−25.1244	−1.87788	−0.938941	−	0.344077i	$-0.888192\pi$
$179$	−25.1244	−1.87788	−0.938941	+	0.344077i	$0.888192\pi$
$180$	0	0
$181$	−5.31444	−0.395019	−0.197510	−	0.980301i	$-0.563285\pi$
$181$	−5.31444	−0.395019	−0.197510	+	0.980301i	$0.563285\pi$
$182$	0	0
$183$	−25.7766	−1.90546
$184$	0	0
$185$	−4.42883	−0.325614
$186$	0	0
$187$	−1.33029	−0.0972807
$188$	0	0
$189$	33.0751	2.40586
$190$	0	0
$191$	14.4702	1.04702	0.523512	−	0.852018i	$-0.324622\pi$
$191$	14.4702	1.04702	0.523512	+	0.852018i	$0.324622\pi$
$192$	0	0
$193$	4.41886	0.318076	0.159038	−	0.987272i	$-0.449161\pi$
$193$	4.41886	0.318076	0.159038	+	0.987272i	$0.449161\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−12.7537	−0.908667	−0.454333	−	0.890832i	$-0.650123\pi$
$197$	−12.7537	−0.908667	−0.454333	+	0.890832i	$0.650123\pi$
$198$	0	0
$199$	−0.585057	−0.0414736	−0.0207368	−	0.999785i	$-0.506601\pi$
$199$	−0.585057	−0.0414736	−0.0207368	+	0.999785i	$0.506601\pi$
$200$	0	0
$201$	1.50402	0.106086
$202$	0	0
$203$	21.0745	1.47914
$204$	0	0
$205$	−2.33245	−0.162905
$206$	0	0
$207$	37.0922	2.57808
$208$	0	0
$209$	−28.7486	−1.98858
$210$	0	0
$211$	−0.383066	−0.0263714	−0.0131857	−	0.999913i	$-0.504197\pi$
$211$	−0.383066	−0.0263714	−0.0131857	+	0.999913i	$0.504197\pi$
$212$	0	0
$213$	−21.4840	−1.47206
$214$	0	0
$215$	−10.7350	−0.732119
$216$	0	0
$217$	2.76268	0.187543
$218$	0	0
$219$	−30.3981	−2.05412
$220$	0	0
$221$	0.240881	0.0162034
$222$	0	0
$223$	−0.964475	−0.0645860	−0.0322930	−	0.999478i	$-0.510281\pi$
$223$	−0.964475	−0.0645860	−0.0322930	+	0.999478i	$0.510281\pi$
$224$	0	0
$225$	5.59575	0.373050
$226$	0	0
$227$	9.63911	0.639770	0.319885	−	0.947456i	$-0.396356\pi$
$227$	9.63911	0.639770	0.319885	+	0.947456i	$0.396356\pi$
$228$	0	0
$229$	−3.58454	−0.236873	−0.118437	−	0.992962i	$-0.537788\pi$
$229$	−3.58454	−0.236873	−0.118437	+	0.992962i	$0.537788\pi$
$230$	0	0
$231$	48.0034	3.15839
$232$	0	0
$233$	−10.8233	−0.709056	−0.354528	−	0.935045i	$-0.615358\pi$
$233$	−10.8233	−0.709056	−0.354528	+	0.935045i	$0.615358\pi$
$234$	0	0
$235$	−5.41057	−0.352946
$236$	0	0
$237$	23.8544	1.54951
$238$	0	0
$239$	21.1137	1.36573	0.682865	−	0.730545i	$-0.260734\pi$
$239$	21.1137	1.36573	0.682865	+	0.730545i	$0.260734\pi$
$240$	0	0
$241$	18.9296	1.21936	0.609682	−	0.792646i	$-0.291297\pi$
$241$	18.9296	1.21936	0.609682	+	0.792646i	$0.291297\pi$
$242$	0	0
$243$	−6.63203	−0.425445
$244$	0	0
$245$	11.8883	0.759515
$246$	0	0
$247$	5.20560	0.331225
$248$	0	0
$249$	34.9082	2.21222
$250$	0	0
$251$	−16.4454	−1.03802	−0.519011	−	0.854768i	$-0.673700\pi$
$251$	−16.4454	−1.03802	−0.519011	+	0.854768i	$0.673700\pi$
$252$	0	0
$253$	24.9722	1.56999
$254$	0	0
$255$	−1.03528	−0.0648315
$256$	0	0
$257$	−6.49258	−0.404996	−0.202498	−	0.979283i	$-0.564906\pi$
$257$	−6.49258	−0.404996	−0.202498	+	0.979283i	$0.564906\pi$
$258$	0	0
$259$	−19.2480	−1.19601
$260$	0	0
$261$	27.1343	1.67957
$262$	0	0
$263$	−2.66102	−0.164085	−0.0820427	−	0.996629i	$-0.526144\pi$
$263$	−2.66102	−0.164085	−0.0820427	+	0.996629i	$0.526144\pi$
$264$	0	0
$265$	6.35311	0.390269
$266$	0	0
$267$	−13.3570	−0.817437
$268$	0	0
$269$	−17.3185	−1.05593	−0.527964	−	0.849267i	$-0.677045\pi$
$269$	−17.3185	−1.05593	−0.527964	+	0.849267i	$0.677045\pi$
$270$	0	0
$271$	25.0280	1.52034	0.760171	−	0.649723i	$-0.225115\pi$
$271$	25.0280	1.52034	0.760171	+	0.649723i	$0.225115\pi$
$272$	0	0
$273$	−8.69213	−0.526072
$274$	0	0
$275$	3.76733	0.227178
$276$	0	0
$277$	−0.450219	−0.0270510	−0.0135255	−	0.999909i	$-0.504305\pi$
$277$	−0.450219	−0.0270510	−0.0135255	+	0.999909i	$0.504305\pi$
$278$	0	0
$279$	3.55708	0.212957
$280$	0	0
$281$	7.90843	0.471777	0.235889	−	0.971780i	$-0.424200\pi$
$281$	7.90843	0.471777	0.235889	+	0.971780i	$0.424200\pi$
$282$	0	0
$283$	16.9026	1.00476	0.502378	−	0.864648i	$-0.332458\pi$
$283$	16.9026	1.00476	0.502378	+	0.864648i	$0.332458\pi$
$284$	0	0
$285$	−22.3730	−1.32527
$286$	0	0
$287$	−10.1370	−0.598367
$288$	0	0
$289$	−16.8753	−0.992665
$290$	0	0
$291$	6.37821	0.373897
$292$	0	0
$293$	25.3205	1.47924	0.739620	−	0.673025i	$-0.235005\pi$
$293$	25.3205	1.47924	0.739620	+	0.673025i	$0.235005\pi$
$294$	0	0
$295$	−13.3877	−0.779460
$296$	0	0
$297$	28.6707	1.66364
$298$	0	0
$299$	−4.52180	−0.261503
$300$	0	0
$301$	−46.6549	−2.68914
$302$	0	0
$303$	16.7313	0.961189
$304$	0	0
$305$	−8.79191	−0.503423
$306$	0	0
$307$	−1.35071	−0.0770891	−0.0385445	−	0.999257i	$-0.512272\pi$
$307$	−1.35071	−0.0770891	−0.0385445	+	0.999257i	$0.512272\pi$
$308$	0	0
$309$	−8.56439	−0.487211
$310$	0	0
$311$	13.3550	0.757295	0.378647	−	0.925541i	$-0.376389\pi$
$311$	13.3550	0.757295	0.378647	+	0.925541i	$0.376389\pi$
$312$	0	0
$313$	21.1069	1.19303	0.596515	−	0.802602i	$-0.296552\pi$
$313$	21.1069	1.19303	0.596515	+	0.802602i	$0.296552\pi$
$314$	0	0
$315$	24.3195	1.37025
$316$	0	0
$317$	−29.8352	−1.67571	−0.837855	−	0.545893i	$-0.816190\pi$
$317$	−29.8352	−1.67571	−0.837855	+	0.545893i	$0.816190\pi$
$318$	0	0
$319$	18.2681	1.02282
$320$	0	0
$321$	−11.1103	−0.620114
$322$	0	0
$323$	2.69462	0.149933
$324$	0	0
$325$	−0.682163	−0.0378396
$326$	0	0
$327$	32.3981	1.79162
$328$	0	0
$329$	−23.5147	−1.29641
$330$	0	0
$331$	30.7963	1.69272	0.846360	−	0.532611i	$-0.178789\pi$
$331$	30.7963	1.69272	0.846360	+	0.532611i	$0.178789\pi$
$332$	0	0
$333$	−24.7826	−1.35808
$334$	0	0
$335$	0.512994	0.0280279
$336$	0	0
$337$	19.0260	1.03641	0.518206	−	0.855256i	$-0.326600\pi$
$337$	19.0260	1.03641	0.518206	+	0.855256i	$0.326600\pi$
$338$	0	0
$339$	42.1961	2.29178
$340$	0	0
$341$	2.39479	0.129685
$342$	0	0
$343$	21.2448	1.14711
$344$	0	0
$345$	19.4341	1.04630
$346$	0	0
$347$	2.72183	0.146116	0.0730579	−	0.997328i	$-0.476724\pi$
$347$	2.72183	0.146116	0.0730579	+	0.997328i	$0.476724\pi$
$348$	0	0
$349$	22.9980	1.23106	0.615528	−	0.788115i	$-0.288943\pi$
$349$	22.9980	1.23106	0.615528	+	0.788115i	$0.288943\pi$
$350$	0	0
$351$	−5.19151	−0.277102
$352$	0	0
$353$	−22.9058	−1.21915	−0.609576	−	0.792728i	$-0.708660\pi$
$353$	−22.9058	−1.21915	−0.609576	+	0.792728i	$0.708660\pi$
$354$	0	0
$355$	−7.32780	−0.388920
$356$	0	0
$357$	−4.49938	−0.238132
$358$	0	0
$359$	21.5687	1.13835	0.569176	−	0.822216i	$-0.307262\pi$
$359$	21.5687	1.13835	0.569176	+	0.822216i	$0.307262\pi$
$360$	0	0
$361$	39.2326	2.06487
$362$	0	0
$363$	9.36068	0.491308
$364$	0	0
$365$	−10.3682	−0.542699
$366$	0	0
$367$	4.32097	0.225553	0.112776	−	0.993620i	$-0.464026\pi$
$367$	4.32097	0.225553	0.112776	+	0.993620i	$0.464026\pi$
$368$	0	0
$369$	−13.0518	−0.679450
$370$	0	0
$371$	27.6110	1.43349
$372$	0	0
$373$	−28.3490	−1.46785	−0.733927	−	0.679228i	$-0.762315\pi$
$373$	−28.3490	−1.46785	−0.733927	+	0.679228i	$0.762315\pi$
$374$	0	0
$375$	2.93185	0.151400
$376$	0	0
$377$	−3.30787	−0.170364
$378$	0	0
$379$	17.6923	0.908792	0.454396	−	0.890800i	$-0.349855\pi$
$379$	17.6923	0.908792	0.454396	+	0.890800i	$0.349855\pi$
$380$	0	0
$381$	−60.5526	−3.10220
$382$	0	0
$383$	12.0645	0.616469	0.308234	−	0.951310i	$-0.400262\pi$
$383$	12.0645	0.616469	0.308234	+	0.951310i	$0.400262\pi$
$384$	0	0
$385$	16.3730	0.834448
$386$	0	0
$387$	−60.0703	−3.05354
$388$	0	0
$389$	−33.8320	−1.71535	−0.857675	−	0.514192i	$-0.828092\pi$
$389$	−33.8320	−1.71535	−0.857675	+	0.514192i	$0.828092\pi$
$390$	0	0
$391$	−2.34066	−0.118372
$392$	0	0
$393$	−8.61642	−0.434641
$394$	0	0
$395$	8.13630	0.409382
$396$	0	0
$397$	10.0664	0.505219	0.252610	−	0.967568i	$-0.418711\pi$
$397$	10.0664	0.505219	0.252610	+	0.967568i	$0.418711\pi$
$398$	0	0
$399$	−97.2347	−4.86783
$400$	0	0
$401$	30.9895	1.54754	0.773770	−	0.633467i	$-0.218369\pi$
$401$	30.9895	1.54754	0.773770	+	0.633467i	$0.218369\pi$
$402$	0	0
$403$	−0.433633	−0.0216008
$404$	0	0
$405$	5.52520	0.274549
$406$	0	0
$407$	−16.6848	−0.827036
$408$	0	0
$409$	−0.344554	−0.0170371	−0.00851855	−	0.999964i	$-0.502712\pi$
$409$	−0.344554	−0.0170371	−0.00851855	+	0.999964i	$0.502712\pi$
$410$	0	0
$411$	−42.2759	−2.08532
$412$	0	0
$413$	−58.1836	−2.86303
$414$	0	0
$415$	11.9065	0.584469
$416$	0	0
$417$	4.09565	0.200565
$418$	0	0
$419$	19.3463	0.945130	0.472565	−	0.881296i	$-0.343328\pi$
$419$	19.3463	0.945130	0.472565	+	0.881296i	$0.343328\pi$
$420$	0	0
$421$	−28.9529	−1.41108	−0.705540	−	0.708670i	$-0.749296\pi$
$421$	−28.9529	−1.41108	−0.705540	+	0.708670i	$0.749296\pi$
$422$	0	0
$423$	−30.2762	−1.47208
$424$	0	0
$425$	−0.353113	−0.0171285
$426$	0	0
$427$	−38.2102	−1.84912
$428$	0	0
$429$	−7.53465	−0.363777
$430$	0	0
$431$	−0.928203	−0.0447100	−0.0223550	−	0.999750i	$-0.507116\pi$
$431$	−0.928203	−0.0447100	−0.0223550	+	0.999750i	$0.507116\pi$
$432$	0	0
$433$	−4.01926	−0.193153	−0.0965766	−	0.995326i	$-0.530789\pi$
$433$	−4.01926	−0.193153	−0.0965766	+	0.995326i	$0.530789\pi$
$434$	0	0
$435$	14.2168	0.681644
$436$	0	0
$437$	−50.5832	−2.41972
$438$	0	0
$439$	37.4578	1.78776	0.893882	−	0.448301i	$-0.147971\pi$
$439$	37.4578	1.78776	0.893882	+	0.448301i	$0.147971\pi$
$440$	0	0
$441$	66.5239	3.16781
$442$	0	0
$443$	−14.6379	−0.695466	−0.347733	−	0.937594i	$-0.613048\pi$
$443$	−14.6379	−0.695466	−0.347733	+	0.937594i	$0.613048\pi$
$444$	0	0
$445$	−4.55583	−0.215967
$446$	0	0
$447$	−4.48528	−0.212147
$448$	0	0
$449$	9.78623	0.461841	0.230920	−	0.972973i	$-0.425826\pi$
$449$	9.78623	0.461841	0.230920	+	0.972973i	$0.425826\pi$
$450$	0	0
$451$	−8.78710	−0.413768
$452$	0	0
$453$	35.8330	1.68358
$454$	0	0
$455$	−2.96472	−0.138988
$456$	0	0
$457$	−0.907307	−0.0424420	−0.0212210	−	0.999775i	$-0.506755\pi$
$457$	−0.907307	−0.0424420	−0.0212210	+	0.999775i	$0.506755\pi$
$458$	0	0
$459$	−2.68732	−0.125433
$460$	0	0
$461$	11.3636	0.529255	0.264628	−	0.964351i	$-0.414751\pi$
$461$	11.3636	0.529255	0.264628	+	0.964351i	$0.414751\pi$
$462$	0	0
$463$	39.0533	1.81496	0.907481	−	0.420093i	$-0.138003\pi$
$463$	39.0533	1.81496	0.907481	+	0.420093i	$0.138003\pi$
$464$	0	0
$465$	1.86370	0.0864272
$466$	0	0
$467$	13.8918	0.642834	0.321417	−	0.946938i	$-0.395841\pi$
$467$	13.8918	0.642834	0.321417	+	0.946938i	$0.395841\pi$
$468$	0	0
$469$	2.22950	0.102949
$470$	0	0
$471$	−1.18567	−0.0546328
$472$	0	0
$473$	−40.4421	−1.85953
$474$	0	0
$475$	−7.63103	−0.350136
$476$	0	0
$477$	35.5505	1.62774
$478$	0	0
$479$	31.0479	1.41862	0.709308	−	0.704899i	$-0.249008\pi$
$479$	31.0479	1.41862	0.709308	+	0.704899i	$0.249008\pi$
$480$	0	0
$481$	3.02118	0.137754
$482$	0	0
$483$	84.4621	3.84316
$484$	0	0
$485$	2.17549	0.0987838
$486$	0	0
$487$	−11.5809	−0.524780	−0.262390	−	0.964962i	$-0.584511\pi$
$487$	−11.5809	−0.524780	−0.262390	+	0.964962i	$0.584511\pi$
$488$	0	0
$489$	4.78851	0.216544
$490$	0	0
$491$	−7.82826	−0.353285	−0.176642	−	0.984275i	$-0.556524\pi$
$491$	−7.82826	−0.353285	−0.176642	+	0.984275i	$0.556524\pi$
$492$	0	0
$493$	−1.71228	−0.0771172
$494$	0	0
$495$	21.0810	0.947522
$496$	0	0
$497$	−31.8471	−1.42854
$498$	0	0
$499$	7.09513	0.317622	0.158811	−	0.987309i	$-0.449234\pi$
$499$	7.09513	0.317622	0.158811	+	0.987309i	$0.449234\pi$
$500$	0	0
$501$	27.9194	1.24735
$502$	0	0
$503$	−30.2548	−1.34899	−0.674497	−	0.738277i	$-0.735640\pi$
$503$	−30.2548	−1.34899	−0.674497	+	0.738277i	$0.735640\pi$
$504$	0	0
$505$	5.70674	0.253947
$506$	0	0
$507$	−36.7497	−1.63211
$508$	0	0
$509$	−21.5140	−0.953591	−0.476795	−	0.879014i	$-0.658202\pi$
$509$	−21.5140	−0.953591	−0.476795	+	0.879014i	$0.658202\pi$
$510$	0	0
$511$	−45.0611	−1.99338
$512$	0	0
$513$	−58.0749	−2.56407
$514$	0	0
$515$	−2.92116	−0.128721
$516$	0	0
$517$	−20.3834	−0.896460
$518$	0	0
$519$	−11.3278	−0.497235
$520$	0	0
$521$	13.2742	0.581552	0.290776	−	0.956791i	$-0.406087\pi$
$521$	13.2742	0.581552	0.290776	+	0.956791i	$0.406087\pi$
$522$	0	0
$523$	16.1891	0.707900	0.353950	−	0.935264i	$-0.384838\pi$
$523$	16.1891	0.707900	0.353950	+	0.935264i	$0.384838\pi$
$524$	0	0
$525$	12.7420	0.556107
$526$	0	0
$527$	−0.224465	−0.00977786
$528$	0	0
$529$	20.9387	0.910378
$530$	0	0
$531$	−74.9141	−3.25099
$532$	0	0
$533$	1.59111	0.0689186
$534$	0	0
$535$	−3.78950	−0.163834
$536$	0	0
$537$	−73.6609	−3.17870
$538$	0	0
$539$	44.7870	1.92911
$540$	0	0
$541$	32.7761	1.40915	0.704576	−	0.709628i	$-0.251137\pi$
$541$	32.7761	1.40915	0.704576	+	0.709628i	$0.251137\pi$
$542$	0	0
$543$	−15.5811	−0.668651
$544$	0	0
$545$	11.0504	0.473347
$546$	0	0
$547$	2.31508	0.0989857	0.0494929	−	0.998774i	$-0.484239\pi$
$547$	2.31508	0.0989857	0.0494929	+	0.998774i	$0.484239\pi$
$548$	0	0
$549$	−49.1973	−2.09969
$550$	0	0
$551$	−37.0036	−1.57640
$552$	0	0
$553$	35.3609	1.50370
$554$	0	0
$555$	−12.9847	−0.551168
$556$	0	0
$557$	−8.25651	−0.349839	−0.174920	−	0.984583i	$-0.555967\pi$
$557$	−8.25651	−0.349839	−0.174920	+	0.984583i	$0.555967\pi$
$558$	0	0
$559$	7.32300	0.309730
$560$	0	0
$561$	−3.90022	−0.164668
$562$	0	0
$563$	19.4040	0.817781	0.408890	−	0.912583i	$-0.365916\pi$
$563$	19.4040	0.817781	0.408890	+	0.912583i	$0.365916\pi$
$564$	0	0
$565$	14.3923	0.605489
$566$	0	0
$567$	24.0129	1.00845
$568$	0	0
$569$	7.05861	0.295912	0.147956	−	0.988994i	$-0.452731\pi$
$569$	7.05861	0.295912	0.147956	+	0.988994i	$0.452731\pi$
$570$	0	0
$571$	26.5875	1.11265	0.556326	−	0.830964i	$-0.312211\pi$
$571$	26.5875	1.11265	0.556326	+	0.830964i	$0.312211\pi$
$572$	0	0
$573$	42.4243	1.77230
$574$	0	0
$575$	6.62863	0.276433
$576$	0	0
$577$	−30.6181	−1.27465	−0.637325	−	0.770595i	$-0.719959\pi$
$577$	−30.6181	−1.27465	−0.637325	+	0.770595i	$0.719959\pi$
$578$	0	0
$579$	12.9554	0.538410
$580$	0	0
$581$	51.7466	2.14681
$582$	0	0
$583$	23.9343	0.991256
$584$	0	0
$585$	−3.81722	−0.157822
$586$	0	0
$587$	−31.1273	−1.28476	−0.642380	−	0.766386i	$-0.722053\pi$
$587$	−31.1273	−1.28476	−0.642380	+	0.766386i	$0.722053\pi$
$588$	0	0
$589$	−4.85085	−0.199876
$590$	0	0
$591$	−37.3921	−1.53811
$592$	0	0
$593$	15.6014	0.640674	0.320337	−	0.947304i	$-0.396204\pi$
$593$	15.6014	0.640674	0.320337	+	0.947304i	$0.396204\pi$
$594$	0	0
$595$	−1.53465	−0.0629147
$596$	0	0
$597$	−1.71530	−0.0702026
$598$	0	0
$599$	−47.5264	−1.94188	−0.970938	−	0.239331i	$-0.923072\pi$
$599$	−47.5264	−1.94188	−0.970938	+	0.239331i	$0.923072\pi$
$600$	0	0
$601$	−4.35238	−0.177537	−0.0887687	−	0.996052i	$-0.528293\pi$
$601$	−4.35238	−0.177537	−0.0887687	+	0.996052i	$0.528293\pi$
$602$	0	0
$603$	2.87059	0.116899
$604$	0	0
$605$	3.19275	0.129804
$606$	0	0
$607$	7.08152	0.287430	0.143715	−	0.989619i	$-0.454095\pi$
$607$	7.08152	0.287430	0.143715	+	0.989619i	$0.454095\pi$
$608$	0	0
$609$	61.7872	2.50374
$610$	0	0
$611$	3.69089	0.149317
$612$	0	0
$613$	−0.100128	−0.00404412	−0.00202206	−	0.999998i	$-0.500644\pi$
$613$	−0.100128	−0.00404412	−0.00202206	+	0.999998i	$0.500644\pi$
$614$	0	0
$615$	−6.83839	−0.275751
$616$	0	0
$617$	4.13217	0.166355	0.0831774	−	0.996535i	$-0.473493\pi$
$617$	4.13217	0.166355	0.0831774	+	0.996535i	$0.473493\pi$
$618$	0	0
$619$	−6.26670	−0.251880	−0.125940	−	0.992038i	$-0.540195\pi$
$619$	−6.26670	−0.251880	−0.125940	+	0.992038i	$0.540195\pi$
$620$	0	0
$621$	50.4463	2.02434
$622$	0	0
$623$	−19.8000	−0.793268
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−84.2866	−3.36608
$628$	0	0
$629$	1.56388	0.0623559
$630$	0	0
$631$	14.7165	0.585856	0.292928	−	0.956134i	$-0.405370\pi$
$631$	14.7165	0.585856	0.292928	+	0.956134i	$0.405370\pi$
$632$	0	0
$633$	−1.12309	−0.0446389
$634$	0	0
$635$	−20.6534	−0.819604
$636$	0	0
$637$	−8.10974	−0.321320
$638$	0	0
$639$	−41.0046	−1.62212
$640$	0	0
$641$	40.5416	1.60130	0.800649	−	0.599134i	$-0.204488\pi$
$641$	40.5416	1.60130	0.800649	+	0.599134i	$0.204488\pi$
$642$	0	0
$643$	33.3671	1.31587	0.657935	−	0.753074i	$-0.271430\pi$
$643$	33.3671	1.31587	0.657935	+	0.753074i	$0.271430\pi$
$644$	0	0
$645$	−31.4733	−1.23926
$646$	0	0
$647$	−43.4154	−1.70683	−0.853417	−	0.521228i	$-0.825474\pi$
$647$	−43.4154	−1.70683	−0.853417	+	0.521228i	$0.825474\pi$
$648$	0	0
$649$	−50.4357	−1.97977
$650$	0	0
$651$	8.09978	0.317455
$652$	0	0
$653$	6.25334	0.244712	0.122356	−	0.992486i	$-0.460955\pi$
$653$	6.25334	0.244712	0.122356	+	0.992486i	$0.460955\pi$
$654$	0	0
$655$	−2.93890	−0.114832
$656$	0	0
$657$	−58.0181	−2.26350
$658$	0	0
$659$	−11.0951	−0.432205	−0.216102	−	0.976371i	$-0.569335\pi$
$659$	−11.0951	−0.432205	−0.216102	+	0.976371i	$0.569335\pi$
$660$	0	0
$661$	15.0985	0.587265	0.293632	−	0.955918i	$-0.405136\pi$
$661$	15.0985	0.587265	0.293632	+	0.955918i	$0.405136\pi$
$662$	0	0
$663$	0.706227	0.0274276
$664$	0	0
$665$	−33.1650	−1.28608
$666$	0	0
$667$	32.1428	1.24457
$668$	0	0
$669$	−2.82770	−0.109325
$670$	0	0
$671$	−33.1220	−1.27866
$672$	0	0
$673$	11.8323	0.456103	0.228052	−	0.973649i	$-0.426764\pi$
$673$	11.8323	0.456103	0.228052	+	0.973649i	$0.426764\pi$
$674$	0	0
$675$	7.61037	0.292923
$676$	0	0
$677$	−28.7089	−1.10337	−0.551686	−	0.834052i	$-0.686015\pi$
$677$	−28.7089	−1.10337	−0.551686	+	0.834052i	$0.686015\pi$
$678$	0	0
$679$	9.45481	0.362842
$680$	0	0
$681$	28.2604	1.08294
$682$	0	0
$683$	33.9254	1.29812	0.649061	−	0.760737i	$-0.275162\pi$
$683$	33.9254	1.29812	0.649061	+	0.760737i	$0.275162\pi$
$684$	0	0
$685$	−14.4195	−0.550942
$686$	0	0
$687$	−10.5093	−0.400957
$688$	0	0
$689$	−4.33386	−0.165107
$690$	0	0
$691$	−26.6278	−1.01297	−0.506485	−	0.862249i	$-0.669055\pi$
$691$	−26.6278	−1.01297	−0.506485	+	0.862249i	$0.669055\pi$
$692$	0	0
$693$	91.6196	3.48034
$694$	0	0
$695$	1.39695	0.0529893
$696$	0	0
$697$	0.823619	0.0311968
$698$	0	0
$699$	−31.7322	−1.20022
$700$	0	0
$701$	29.5478	1.11600	0.558002	−	0.829839i	$-0.311568\pi$
$701$	29.5478	1.11600	0.558002	+	0.829839i	$0.311568\pi$
$702$	0	0
$703$	33.7965	1.27466
$704$	0	0
$705$	−15.8630	−0.597434
$706$	0	0
$707$	24.8019	0.932770
$708$	0	0
$709$	−0.149664	−0.00562076	−0.00281038	−	0.999996i	$-0.500895\pi$
$709$	−0.149664	−0.00562076	−0.00281038	+	0.999996i	$0.500895\pi$
$710$	0	0
$711$	45.5287	1.70746
$712$	0	0
$713$	4.21365	0.157802
$714$	0	0
$715$	−2.56993	−0.0961099
$716$	0	0
$717$	61.9021	2.31178
$718$	0	0
$719$	−32.4142	−1.20885	−0.604423	−	0.796663i	$-0.706596\pi$
$719$	−32.4142	−1.20885	−0.604423	+	0.796663i	$0.706596\pi$
$720$	0	0
$721$	−12.6955	−0.472806
$722$	0	0
$723$	55.4988	2.06402
$724$	0	0
$725$	4.84909	0.180091
$726$	0	0
$727$	−49.3403	−1.82993	−0.914966	−	0.403531i	$-0.867783\pi$
$727$	−49.3403	−1.82993	−0.914966	+	0.403531i	$0.867783\pi$
$728$	0	0
$729$	−36.0197	−1.33406
$730$	0	0
$731$	3.79066	0.140203
$732$	0	0
$733$	−1.70017	−0.0627974	−0.0313987	−	0.999507i	$-0.509996\pi$
$733$	−1.70017	−0.0627974	−0.0313987	+	0.999507i	$0.509996\pi$
$734$	0	0
$735$	34.8547	1.28563
$736$	0	0
$737$	1.93262	0.0711888
$738$	0	0
$739$	−11.1055	−0.408521	−0.204260	−	0.978917i	$-0.565479\pi$
$739$	−11.1055	−0.408521	−0.204260	+	0.978917i	$0.565479\pi$
$740$	0	0
$741$	15.2621	0.560666
$742$	0	0
$743$	39.8689	1.46265	0.731325	−	0.682029i	$-0.238902\pi$
$743$	39.8689	1.46265	0.731325	+	0.682029i	$0.238902\pi$
$744$	0	0
$745$	−1.52985	−0.0560492
$746$	0	0
$747$	66.6261	2.43772
$748$	0	0
$749$	−16.4694	−0.601780
$750$	0	0
$751$	−41.4417	−1.51223	−0.756115	−	0.654438i	$-0.772905\pi$
$751$	−41.4417	−1.51223	−0.756115	+	0.654438i	$0.772905\pi$
$752$	0	0
$753$	−48.2154	−1.75707
$754$	0	0
$755$	12.2220	0.444803
$756$	0	0
$757$	24.4170	0.887453	0.443726	−	0.896162i	$-0.353656\pi$
$757$	24.4170	0.887453	0.443726	+	0.896162i	$0.353656\pi$
$758$	0	0
$759$	73.2148	2.65753
$760$	0	0
$761$	−34.9189	−1.26581	−0.632905	−	0.774230i	$-0.718138\pi$
$761$	−34.9189	−1.26581	−0.632905	+	0.774230i	$0.718138\pi$
$762$	0	0
$763$	48.0258	1.73865
$764$	0	0
$765$	−1.97594	−0.0714401
$766$	0	0
$767$	9.13256	0.329758
$768$	0	0
$769$	−20.8099	−0.750422	−0.375211	−	0.926939i	$-0.622430\pi$
$769$	−20.8099	−0.750422	−0.375211	+	0.926939i	$0.622430\pi$
$770$	0	0
$771$	−19.0353	−0.685539
$772$	0	0
$773$	−34.0289	−1.22393	−0.611967	−	0.790884i	$-0.709621\pi$
$773$	−34.0289	−1.22393	−0.611967	+	0.790884i	$0.709621\pi$
$774$	0	0
$775$	0.635674	0.0228341
$776$	0	0
$777$	−56.4322	−2.02449
$778$	0	0
$779$	17.7990	0.637715
$780$	0	0
$781$	−27.6062	−0.987829
$782$	0	0
$783$	36.9034	1.31882
$784$	0	0
$785$	−0.404410	−0.0144340
$786$	0	0
$787$	−12.1923	−0.434608	−0.217304	−	0.976104i	$-0.569726\pi$
$787$	−12.1923	−0.434608	−0.217304	+	0.976104i	$0.569726\pi$
$788$	0	0
$789$	−7.80171	−0.277748
$790$	0	0
$791$	62.5499	2.22402
$792$	0	0
$793$	5.99751	0.212978
$794$	0	0
$795$	18.6264	0.660610
$796$	0	0
$797$	26.2002	0.928060	0.464030	−	0.885820i	$-0.346403\pi$
$797$	26.2002	0.928060	0.464030	+	0.885820i	$0.346403\pi$
$798$	0	0
$799$	1.91054	0.0675902
$800$	0	0
$801$	−25.4933	−0.900762
$802$	0	0
$803$	−39.0606	−1.37842
$804$	0	0
$805$	28.8084	1.01536
$806$	0	0
$807$	−50.7753	−1.78738
$808$	0	0
$809$	29.0318	1.02070	0.510352	−	0.859965i	$-0.329515\pi$
$809$	29.0318	1.02070	0.510352	+	0.859965i	$0.329515\pi$
$810$	0	0
$811$	−46.9364	−1.64816	−0.824080	−	0.566473i	$-0.808307\pi$
$811$	−46.9364	−1.64816	−0.824080	+	0.566473i	$0.808307\pi$
$812$	0	0
$813$	73.3783	2.57349
$814$	0	0
$815$	1.63327	0.0572110
$816$	0	0
$817$	81.9189	2.86598
$818$	0	0
$819$	−16.5899	−0.579697
$820$	0	0
$821$	−34.5733	−1.20662	−0.603308	−	0.797508i	$-0.706151\pi$
$821$	−34.5733	−1.20662	−0.603308	+	0.797508i	$0.706151\pi$
$822$	0	0
$823$	7.59713	0.264819	0.132410	−	0.991195i	$-0.457729\pi$
$823$	7.59713	0.264819	0.132410	+	0.991195i	$0.457729\pi$
$824$	0	0
$825$	11.0452	0.384546
$826$	0	0
$827$	−10.9486	−0.380721	−0.190360	−	0.981714i	$-0.560966\pi$
$827$	−10.9486	−0.380721	−0.190360	+	0.981714i	$0.560966\pi$
$828$	0	0
$829$	−22.4170	−0.778576	−0.389288	−	0.921116i	$-0.627279\pi$
$829$	−22.4170	−0.778576	−0.389288	+	0.921116i	$0.627279\pi$
$830$	0	0
$831$	−1.31997	−0.0457894
$832$	0	0
$833$	−4.19791	−0.145449
$834$	0	0
$835$	9.52280	0.329550
$836$	0	0
$837$	4.83772	0.167216
$838$	0	0
$839$	0.821130	0.0283486	0.0141743	−	0.999900i	$-0.495488\pi$
$839$	0.821130	0.0283486	0.0141743	+	0.999900i	$0.495488\pi$
$840$	0	0
$841$	−5.48631	−0.189183
$842$	0	0
$843$	23.1863	0.798580
$844$	0	0
$845$	−12.5347	−0.431205
$846$	0	0
$847$	13.8759	0.476782
$848$	0	0
$849$	49.5560	1.70076
$850$	0	0
$851$	−29.3570	−1.00635
$852$	0	0
$853$	−18.5998	−0.636846	−0.318423	−	0.947949i	$-0.603153\pi$
$853$	−18.5998	−0.636846	−0.318423	+	0.947949i	$0.603153\pi$
$854$	0	0
$855$	−42.7014	−1.46036
$856$	0	0
$857$	28.1630	0.962031	0.481015	−	0.876712i	$-0.340268\pi$
$857$	28.1630	0.962031	0.481015	+	0.876712i	$0.340268\pi$
$858$	0	0
$859$	41.0869	1.40187	0.700933	−	0.713227i	$-0.252767\pi$
$859$	41.0869	1.40187	0.700933	+	0.713227i	$0.252767\pi$
$860$	0	0
$861$	−29.7201	−1.01286
$862$	0	0
$863$	40.4434	1.37671	0.688355	−	0.725374i	$-0.258333\pi$
$863$	40.4434	1.37671	0.688355	+	0.725374i	$0.258333\pi$
$864$	0	0
$865$	−3.86370	−0.131370
$866$	0	0
$867$	−49.4759	−1.68029
$868$	0	0
$869$	30.6521	1.03980
$870$	0	0
$871$	−0.349945	−0.0118574
$872$	0	0
$873$	12.1735	0.412010
$874$	0	0
$875$	4.34607	0.146924
$876$	0	0
$877$	7.00021	0.236380	0.118190	−	0.992991i	$-0.462291\pi$
$877$	7.00021	0.236380	0.118190	+	0.992991i	$0.462291\pi$
$878$	0	0
$879$	74.2360	2.50392
$880$	0	0
$881$	13.6262	0.459079	0.229540	−	0.973299i	$-0.426278\pi$
$881$	13.6262	0.459079	0.229540	+	0.973299i	$0.426278\pi$
$882$	0	0
$883$	−26.7576	−0.900464	−0.450232	−	0.892912i	$-0.648659\pi$
$883$	−26.7576	−0.900464	−0.450232	+	0.892912i	$0.648659\pi$
$884$	0	0
$885$	−39.2506	−1.31940
$886$	0	0
$887$	−9.21197	−0.309308	−0.154654	−	0.987969i	$-0.549426\pi$
$887$	−9.21197	−0.309308	−0.154654	+	0.987969i	$0.549426\pi$
$888$	0	0
$889$	−89.7609	−3.01048
$890$	0	0
$891$	20.8152	0.697337
$892$	0	0
$893$	41.2882	1.38166
$894$	0	0
$895$	−25.1244	−0.839815
$896$	0	0
$897$	−13.2573	−0.442647
$898$	0	0
$899$	3.08244	0.102805
$900$	0	0
$901$	−2.24337	−0.0747375
$902$	0	0
$903$	−136.785	−4.55193
$904$	0	0
$905$	−5.31444	−0.176658
$906$	0	0
$907$	−21.0097	−0.697616	−0.348808	−	0.937194i	$-0.613413\pi$
$907$	−21.0097	−0.697616	−0.348808	+	0.937194i	$0.613413\pi$
$908$	0	0
$909$	31.9335	1.05917
$910$	0	0
$911$	−37.0308	−1.22689	−0.613443	−	0.789739i	$-0.710216\pi$
$911$	−37.0308	−1.22689	−0.613443	+	0.789739i	$0.710216\pi$
$912$	0	0
$913$	44.8558	1.48451
$914$	0	0
$915$	−25.7766	−0.852147
$916$	0	0
$917$	−12.7727	−0.421790
$918$	0	0
$919$	−11.8760	−0.391754	−0.195877	−	0.980629i	$-0.562755\pi$
$919$	−11.8760	−0.391754	−0.195877	+	0.980629i	$0.562755\pi$
$920$	0	0
$921$	−3.96008	−0.130489
$922$	0	0
$923$	4.99876	0.164536
$924$	0	0
$925$	−4.42883	−0.145619
$926$	0	0
$927$	−16.3461	−0.536875
$928$	0	0
$929$	4.07018	0.133538	0.0667691	−	0.997768i	$-0.478731\pi$
$929$	4.07018	0.133538	0.0667691	+	0.997768i	$0.478731\pi$
$930$	0	0
$931$	−90.7198	−2.97322
$932$	0	0
$933$	39.1550	1.28188
$934$	0	0
$935$	−1.33029	−0.0435053
$936$	0	0
$937$	−51.6936	−1.68875	−0.844377	−	0.535749i	$-0.820029\pi$
$937$	−51.6936	−1.68875	−0.844377	+	0.535749i	$0.820029\pi$
$938$	0	0
$939$	61.8822	2.01945
$940$	0	0
$941$	−13.3694	−0.435829	−0.217914	−	0.975968i	$-0.569925\pi$
$941$	−13.3694	−0.435829	−0.217914	+	0.975968i	$0.569925\pi$
$942$	0	0
$943$	−15.4609	−0.503477
$944$	0	0
$945$	33.0751	1.07593
$946$	0	0
$947$	7.25126	0.235634	0.117817	−	0.993035i	$-0.462410\pi$
$947$	7.25126	0.235634	0.117817	+	0.993035i	$0.462410\pi$
$948$	0	0
$949$	7.07283	0.229594
$950$	0	0
$951$	−87.4723	−2.83648
$952$	0	0
$953$	18.4127	0.596447	0.298224	−	0.954496i	$-0.403606\pi$
$953$	18.4127	0.596447	0.298224	+	0.954496i	$0.403606\pi$
$954$	0	0
$955$	14.4702	0.468243
$956$	0	0
$957$	53.5594	1.73133
$958$	0	0
$959$	−62.6682	−2.02366
$960$	0	0
$961$	−30.5959	−0.986965
$962$	0	0
$963$	−21.2051	−0.683326
$964$	0	0
$965$	4.41886	0.142248
$966$	0	0
$967$	45.4253	1.46078	0.730390	−	0.683030i	$-0.239338\pi$
$967$	45.4253	1.46078	0.730390	+	0.683030i	$0.239338\pi$
$968$	0	0
$969$	7.90022	0.253792
$970$	0	0
$971$	−24.5345	−0.787349	−0.393675	−	0.919250i	$-0.628796\pi$
$971$	−24.5345	−0.787349	−0.393675	+	0.919250i	$0.628796\pi$
$972$	0	0
$973$	6.07123	0.194635
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	6.98733	0.223544	0.111772	−	0.993734i	$-0.464347\pi$
$977$	6.98733	0.223544	0.111772	+	0.993734i	$0.464347\pi$
$978$	0	0
$979$	−17.1633	−0.548542
$980$	0	0
$981$	61.8353	1.97425
$982$	0	0
$983$	−29.1408	−0.929447	−0.464723	−	0.885456i	$-0.653846\pi$
$983$	−29.1408	−0.929447	−0.464723	+	0.885456i	$0.653846\pi$
$984$	0	0
$985$	−12.7537	−0.406368
$986$	0	0
$987$	−68.9415	−2.19443
$988$	0	0
$989$	−71.1581	−2.26270
$990$	0	0
$991$	47.6299	1.51302	0.756508	−	0.653985i	$-0.226904\pi$
$991$	47.6299	1.51302	0.756508	+	0.653985i	$0.226904\pi$
$992$	0	0
$993$	90.2903	2.86528
$994$	0	0
$995$	−0.585057	−0.0185476
$996$	0	0
$997$	44.7382	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$997$	44.7382	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$998$	0	0
$999$	−33.7050	−1.06638

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5120.2.a.r.1.4		4
4.3	odd	2		5120.2.a.b.1.1		4
8.3	odd	2		5120.2.a.q.1.4		4
8.5	even	2		5120.2.a.a.1.1		4
32.3	odd	8		1280.2.l.d.321.1	yes	8
32.5	even	8		1280.2.l.a.961.1	yes	8
32.11	odd	8		1280.2.l.d.961.1	yes	8
32.13	even	8		1280.2.l.a.321.1	✓	8
32.19	odd	8		1280.2.l.f.321.4	yes	8
32.21	even	8		1280.2.l.g.961.4	yes	8
32.27	odd	8		1280.2.l.f.961.4	yes	8
32.29	even	8		1280.2.l.g.321.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1280.2.l.a.321.1	✓	8	32.13	even	8
1280.2.l.a.961.1	yes	8	32.5	even	8
1280.2.l.d.321.1	yes	8	32.3	odd	8
1280.2.l.d.961.1	yes	8	32.11	odd	8
1280.2.l.f.321.4	yes	8	32.19	odd	8
1280.2.l.f.961.4	yes	8	32.27	odd	8
1280.2.l.g.321.4	yes	8	32.29	even	8
1280.2.l.g.961.4	yes	8	32.21	even	8
5120.2.a.a.1.1		4	8.5	even	2
5120.2.a.b.1.1		4	4.3	odd	2
5120.2.a.q.1.4		4	8.3	odd	2
5120.2.a.r.1.4		4	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:	\( N \)	\(=\)	\( 5120 = 2^{10} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5120.a (trivial)

\(f(q)\)	\(=\)	\(q+2.93185 q^{3} +1.00000 q^{5} +4.34607 q^{7} +5.59575 q^{9} +O(q^{10})\)	\(q+2.93185 q^{3} +1.00000 q^{5} +4.34607 q^{7} +5.59575 q^{9} +3.76733 q^{11} -0.682163 q^{13} +2.93185 q^{15} -0.353113 q^{17} -7.63103 q^{19} +12.7420 q^{21} +6.62863 q^{23} +1.00000 q^{25} +7.61037 q^{27} +4.84909 q^{29} +0.635674 q^{31} +11.0452 q^{33} +4.34607 q^{35} -4.42883 q^{37} -2.00000 q^{39} -2.33245 q^{41} -10.7350 q^{43} +5.59575 q^{45} -5.41057 q^{47} +11.8883 q^{49} -1.03528 q^{51} +6.35311 q^{53} +3.76733 q^{55} -22.3730 q^{57} -13.3877 q^{59} -8.79191 q^{61} +24.3195 q^{63} -0.682163 q^{65} +0.512994 q^{67} +19.4341 q^{69} -7.32780 q^{71} -10.3682 q^{73} +2.93185 q^{75} +16.3730 q^{77} +8.13630 q^{79} +5.52520 q^{81} +11.9065 q^{83} -0.353113 q^{85} +14.2168 q^{87} -4.55583 q^{89} -2.96472 q^{91} +1.86370 q^{93} -7.63103 q^{95} +2.17549 q^{97} +21.0810 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7	\( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{17} - 4 q^{19} + 16 q^{21} + 16 q^{23} + 4 q^{25} + 4 q^{27} + 8 q^{29} + 12 q^{33} + 4 q^{35} - 8 q^{37} - 8 q^{39} - 4 q^{41} + 4 q^{47} + 20 q^{53} + 4 q^{55} - 28 q^{57} + 12 q^{59} + 8 q^{61} + 24 q^{63} - 4 q^{65} - 24 q^{67} + 16 q^{69} - 12 q^{73} + 4 q^{75} + 4 q^{77} + 48 q^{79} + 8 q^{81} + 16 q^{83} + 4 q^{85} + 36 q^{87} + 24 q^{89} - 16 q^{91} - 8 q^{93} - 4 q^{95} - 12 q^{97} + 28 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^17 - 4 * q^19 + 16 * q^21 + 16 * q^23 + 4 * q^25 + 4 * q^27 + 8 * q^29 + 12 * q^33 + 4 * q^35 - 8 * q^37 - 8 * q^39 - 4 * q^41 + 4 * q^47 + 20 * q^53 + 4 * q^55 - 28 * q^57 + 12 * q^59 + 8 * q^61 + 24 * q^63 - 4 * q^65 - 24 * q^67 + 16 * q^69 - 12 * q^73 + 4 * q^75 + 4 * q^77 + 48 * q^79 + 8 * q^81 + 16 * q^83 + 4 * q^85 + 36 * q^87 + 24 * q^89 - 16 * q^91 - 8 * q^93 - 4 * q^95 - 12 * q^97 + 28 * q^99

Introduction

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