LMFDB - Embedded newform 1280.4.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(1,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1280.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1280.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:	$75.5224448073$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 21x^{4} + 38x^{3} + 93x^{2} - 108x - 60 $ x^6 - 2x^5 - 21x^4 + 38x^3 + 93x^2 - 108*x - 60
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.78252$ of defining polynomial
Character	$\chi$	$=$	1280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.888401 q^{3} -5.00000 q^{5} -26.6173 q^{7} -26.2107 q^{9} +O(q^{10})$	$q+0.888401 q^{3} -5.00000 q^{5} -26.6173 q^{7} -26.2107 q^{9} -61.7277 q^{11} -45.0627 q^{13} -4.44201 q^{15} -71.8754 q^{17} +17.6319 q^{19} -23.6469 q^{21} -43.4131 q^{23} +25.0000 q^{25} -47.2725 q^{27} -267.633 q^{29} +50.2133 q^{31} -54.8390 q^{33} +133.087 q^{35} +75.0720 q^{37} -40.0338 q^{39} +221.685 q^{41} +188.998 q^{43} +131.054 q^{45} -384.142 q^{47} +365.481 q^{49} -63.8542 q^{51} -247.445 q^{53} +308.638 q^{55} +15.6642 q^{57} +518.596 q^{59} +62.0042 q^{61} +697.660 q^{63} +225.314 q^{65} +558.476 q^{67} -38.5683 q^{69} -313.194 q^{71} +263.926 q^{73} +22.2100 q^{75} +1643.03 q^{77} -732.940 q^{79} +665.693 q^{81} -717.705 q^{83} +359.377 q^{85} -237.766 q^{87} -1634.69 q^{89} +1199.45 q^{91} +44.6096 q^{93} -88.1597 q^{95} -1367.12 q^{97} +1617.93 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 30 q^{5} - 14 q^{7} + 54 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 30 * q^5 - 14 * q^7 + 54 * q^9	$ 6 q + 6 q^{3} - 30 q^{5} - 14 q^{7} + 54 q^{9} + 44 q^{11} - 30 q^{15} + 152 q^{19} - 4 q^{21} - 302 q^{23} + 150 q^{25} + 216 q^{27} - 132 q^{31} - 116 q^{33} + 70 q^{35} + 68 q^{37} - 300 q^{39} - 20 q^{41} + 602 q^{43} - 270 q^{45} - 470 q^{47} + 654 q^{49} + 612 q^{51} - 528 q^{53} - 220 q^{55} + 340 q^{57} + 472 q^{59} + 476 q^{61} - 650 q^{63} + 1206 q^{67} + 980 q^{69} + 796 q^{71} - 216 q^{73} + 150 q^{75} - 412 q^{77} + 1008 q^{79} + 1254 q^{81} + 1778 q^{83} + 984 q^{87} + 212 q^{89} + 3652 q^{91} + 1392 q^{93} - 760 q^{95} - 792 q^{97} + 5516 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 - 30 * q^5 - 14 * q^7 + 54 * q^9 + 44 * q^11 - 30 * q^15 + 152 * q^19 - 4 * q^21 - 302 * q^23 + 150 * q^25 + 216 * q^27 - 132 * q^31 - 116 * q^33 + 70 * q^35 + 68 * q^37 - 300 * q^39 - 20 * q^41 + 602 * q^43 - 270 * q^45 - 470 * q^47 + 654 * q^49 + 612 * q^51 - 528 * q^53 - 220 * q^55 + 340 * q^57 + 472 * q^59 + 476 * q^61 - 650 * q^63 + 1206 * q^67 + 980 * q^69 + 796 * q^71 - 216 * q^73 + 150 * q^75 - 412 * q^77 + 1008 * q^79 + 1254 * q^81 + 1778 * q^83 + 984 * q^87 + 212 * q^89 + 3652 * q^91 + 1392 * q^93 - 760 * q^95 - 792 * q^97 + 5516 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.888401	0.170973	0.0854865	−	0.996339i	$-0.472756\pi$
$3$	0.888401	0.170973	0.0854865	+	0.996339i	$0.472756\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−26.6173	−1.43720	−0.718600	−	0.695424i	$-0.755217\pi$
$7$	−26.6173	−1.43720	−0.718600	+	0.695424i	$0.755217\pi$
$8$	0	0
$9$	−26.2107	−0.970768
$10$	0	0
$11$	−61.7277	−1.69196	−0.845982	−	0.533212i	$-0.820985\pi$
$11$	−61.7277	−1.69196	−0.845982	+	0.533212i	$0.820985\pi$
$12$	0	0
$13$	−45.0627	−0.961396	−0.480698	−	0.876886i	$-0.659617\pi$
$13$	−45.0627	−0.961396	−0.480698	+	0.876886i	$0.659617\pi$
$14$	0	0
$15$	−4.44201	−0.0764614
$16$	0	0
$17$	−71.8754	−1.02543	−0.512716	−	0.858558i	$-0.671361\pi$
$17$	−71.8754	−1.02543	−0.512716	+	0.858558i	$0.671361\pi$
$18$	0	0
$19$	17.6319	0.212897	0.106449	−	0.994318i	$-0.466052\pi$
$19$	17.6319	0.212897	0.106449	+	0.994318i	$0.466052\pi$
$20$	0	0
$21$	−23.6469	−0.245722
$22$	0	0
$23$	−43.4131	−0.393577	−0.196788	−	0.980446i	$-0.563051\pi$
$23$	−43.4131	−0.393577	−0.196788	+	0.980446i	$0.563051\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−47.2725	−0.336948
$28$	0	0
$29$	−267.633	−1.71373	−0.856866	−	0.515540i	$-0.827591\pi$
$29$	−267.633	−1.71373	−0.856866	+	0.515540i	$0.827591\pi$
$30$	0	0
$31$	50.2133	0.290922	0.145461	−	0.989364i	$-0.453533\pi$
$31$	50.2133	0.290922	0.145461	+	0.989364i	$0.453533\pi$
$32$	0	0
$33$	−54.8390	−0.289280
$34$	0	0
$35$	133.087	0.642735
$36$	0	0
$37$	75.0720	0.333561	0.166781	−	0.985994i	$-0.446663\pi$
$37$	75.0720	0.333561	0.166781	+	0.985994i	$0.446663\pi$
$38$	0	0
$39$	−40.0338	−0.164373
$40$	0	0
$41$	221.685	0.844425	0.422213	−	0.906497i	$-0.361254\pi$
$41$	221.685	0.844425	0.422213	+	0.906497i	$0.361254\pi$
$42$	0	0
$43$	188.998	0.670277	0.335139	−	0.942169i	$-0.391217\pi$
$43$	188.998	0.670277	0.335139	+	0.942169i	$0.391217\pi$
$44$	0	0
$45$	131.054	0.434141
$46$	0	0
$47$	−384.142	−1.19219	−0.596094	−	0.802914i	$-0.703282\pi$
$47$	−384.142	−1.19219	−0.596094	+	0.802914i	$0.703282\pi$
$48$	0	0
$49$	365.481	1.06554
$50$	0	0
$51$	−63.8542	−0.175321
$52$	0	0
$53$	−247.445	−0.641306	−0.320653	−	0.947197i	$-0.603902\pi$
$53$	−247.445	−0.641306	−0.320653	+	0.947197i	$0.603902\pi$
$54$	0	0
$55$	308.638	0.756669
$56$	0	0
$57$	15.6642	0.0363997
$58$	0	0
$59$	518.596	1.14433	0.572165	−	0.820139i	$-0.306104\pi$
$59$	518.596	1.14433	0.572165	+	0.820139i	$0.306104\pi$
$60$	0	0
$61$	62.0042	0.130145	0.0650723	−	0.997881i	$-0.479272\pi$
$61$	62.0042	0.130145	0.0650723	+	0.997881i	$0.479272\pi$
$62$	0	0
$63$	697.660	1.39519
$64$	0	0
$65$	225.314	0.429949
$66$	0	0
$67$	558.476	1.01834	0.509169	−	0.860666i	$-0.329953\pi$
$67$	558.476	1.01834	0.509169	+	0.860666i	$0.329953\pi$
$68$	0	0
$69$	−38.5683	−0.0672910
$70$	0	0
$71$	−313.194	−0.523512	−0.261756	−	0.965134i	$-0.584301\pi$
$71$	−313.194	−0.523512	−0.261756	+	0.965134i	$0.584301\pi$
$72$	0	0
$73$	263.926	0.423153	0.211576	−	0.977361i	$-0.432140\pi$
$73$	263.926	0.423153	0.211576	+	0.977361i	$0.432140\pi$
$74$	0	0
$75$	22.2100	0.0341946
$76$	0	0
$77$	1643.03	2.43169
$78$	0	0
$79$	−732.940	−1.04383	−0.521913	−	0.852999i	$-0.674781\pi$
$79$	−732.940	−1.04383	−0.521913	+	0.852999i	$0.674781\pi$
$80$	0	0
$81$	665.693	0.913159
$82$	0	0
$83$	−717.705	−0.949137	−0.474569	−	0.880218i	$-0.657396\pi$
$83$	−717.705	−0.949137	−0.474569	+	0.880218i	$0.657396\pi$
$84$	0	0
$85$	359.377	0.458587
$86$	0	0
$87$	−237.766	−0.293002
$88$	0	0
$89$	−1634.69	−1.94693	−0.973463	−	0.228845i	$-0.926505\pi$
$89$	−1634.69	−1.94693	−0.973463	+	0.228845i	$0.926505\pi$
$90$	0	0
$91$	1199.45	1.38172
$92$	0	0
$93$	44.6096	0.0497398
$94$	0	0
$95$	−88.1597	−0.0952105
$96$	0	0
$97$	−1367.12	−1.43103	−0.715516	−	0.698596i	$-0.753809\pi$
$97$	−1367.12	−1.43103	−0.715516	+	0.698596i	$0.753809\pi$
$98$	0	0
$99$	1617.93	1.64250
$100$	0	0
$101$	58.3234	0.0574594	0.0287297	−	0.999587i	$-0.490854\pi$
$101$	58.3234	0.0574594	0.0287297	+	0.999587i	$0.490854\pi$
$102$	0	0
$103$	9.69032	0.00927005	0.00463503	−	0.999989i	$-0.498525\pi$
$103$	9.69032	0.00927005	0.00463503	+	0.999989i	$0.498525\pi$
$104$	0	0
$105$	118.234	0.109890
$106$	0	0
$107$	439.045	0.396674	0.198337	−	0.980134i	$-0.436446\pi$
$107$	439.045	0.396674	0.198337	+	0.980134i	$0.436446\pi$
$108$	0	0
$109$	−1616.51	−1.42049	−0.710245	−	0.703955i	$-0.751416\pi$
$109$	−1616.51	−1.42049	−0.710245	+	0.703955i	$0.751416\pi$
$110$	0	0
$111$	66.6941	0.0570299
$112$	0	0
$113$	−1281.25	−1.06663	−0.533316	−	0.845916i	$-0.679054\pi$
$113$	−1281.25	−1.06663	−0.533316	+	0.845916i	$0.679054\pi$
$114$	0	0
$115$	217.066	0.176013
$116$	0	0
$117$	1181.13	0.933293
$118$	0	0
$119$	1913.13	1.47375
$120$	0	0
$121$	2479.31	1.86274
$122$	0	0
$123$	196.946	0.144374
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	235.326	0.164423	0.0822117	−	0.996615i	$-0.473802\pi$
$127$	235.326	0.164423	0.0822117	+	0.996615i	$0.473802\pi$
$128$	0	0
$129$	167.906	0.114599
$130$	0	0
$131$	993.121	0.662362	0.331181	−	0.943567i	$-0.392553\pi$
$131$	993.121	0.662362	0.331181	+	0.943567i	$0.392553\pi$
$132$	0	0
$133$	−469.315	−0.305976
$134$	0	0
$135$	236.363	0.150688
$136$	0	0
$137$	−2070.09	−1.29095	−0.645474	−	0.763782i	$-0.723340\pi$
$137$	−2070.09	−1.29095	−0.645474	+	0.763782i	$0.723340\pi$
$138$	0	0
$139$	−288.330	−0.175941	−0.0879707	−	0.996123i	$-0.528038\pi$
$139$	−288.330	−0.175941	−0.0879707	+	0.996123i	$0.528038\pi$
$140$	0	0
$141$	−341.272	−0.203832
$142$	0	0
$143$	2781.62	1.62665
$144$	0	0
$145$	1338.17	0.766404
$146$	0	0
$147$	324.694	0.182179
$148$	0	0
$149$	−900.665	−0.495204	−0.247602	−	0.968862i	$-0.579642\pi$
$149$	−900.665	−0.495204	−0.247602	+	0.968862i	$0.579642\pi$
$150$	0	0
$151$	−2005.71	−1.08094	−0.540472	−	0.841362i	$-0.681754\pi$
$151$	−2005.71	−1.08094	−0.540472	+	0.841362i	$0.681754\pi$
$152$	0	0
$153$	1883.91	0.995457
$154$	0	0
$155$	−251.067	−0.130104
$156$	0	0
$157$	−3098.13	−1.57489	−0.787445	−	0.616385i	$-0.788597\pi$
$157$	−3098.13	−1.57489	−0.787445	+	0.616385i	$0.788597\pi$
$158$	0	0
$159$	−219.831	−0.109646
$160$	0	0
$161$	1155.54	0.565648
$162$	0	0
$163$	3566.57	1.71383	0.856917	−	0.515454i	$-0.172377\pi$
$163$	3566.57	1.71383	0.856917	+	0.515454i	$0.172377\pi$
$164$	0	0
$165$	274.195	0.129370
$166$	0	0
$167$	−1326.87	−0.614830	−0.307415	−	0.951576i	$-0.599464\pi$
$167$	−1326.87	−0.614830	−0.307415	+	0.951576i	$0.599464\pi$
$168$	0	0
$169$	−166.353	−0.0757180
$170$	0	0
$171$	−462.146	−0.206674
$172$	0	0
$173$	1035.22	0.454952	0.227476	−	0.973784i	$-0.426953\pi$
$173$	1035.22	0.454952	0.227476	+	0.973784i	$0.426953\pi$
$174$	0	0
$175$	−665.433	−0.287440
$176$	0	0
$177$	460.721	0.195649
$178$	0	0
$179$	1811.28	0.756319	0.378160	−	0.925740i	$-0.376557\pi$
$179$	1811.28	0.756319	0.378160	+	0.925740i	$0.376557\pi$
$180$	0	0
$181$	−2286.18	−0.938842	−0.469421	−	0.882975i	$-0.655537\pi$
$181$	−2286.18	−0.938842	−0.469421	+	0.882975i	$0.655537\pi$
$182$	0	0
$183$	55.0846	0.0222512
$184$	0	0
$185$	−375.360	−0.149173
$186$	0	0
$187$	4436.70	1.73499
$188$	0	0
$189$	1258.27	0.484262
$190$	0	0
$191$	−2392.90	−0.906513	−0.453256	−	0.891380i	$-0.649738\pi$
$191$	−2392.90	−0.906513	−0.453256	+	0.891380i	$0.649738\pi$
$192$	0	0
$193$	−638.414	−0.238104	−0.119052	−	0.992888i	$-0.537985\pi$
$193$	−638.414	−0.238104	−0.119052	+	0.992888i	$0.537985\pi$
$194$	0	0
$195$	200.169	0.0735097
$196$	0	0
$197$	−654.303	−0.236635	−0.118318	−	0.992976i	$-0.537750\pi$
$197$	−654.303	−0.236635	−0.118318	+	0.992976i	$0.537750\pi$
$198$	0	0
$199$	1637.06	0.583155	0.291578	−	0.956547i	$-0.405820\pi$
$199$	1637.06	0.583155	0.291578	+	0.956547i	$0.405820\pi$
$200$	0	0
$201$	496.151	0.174108
$202$	0	0
$203$	7123.67	2.46297
$204$	0	0
$205$	−1108.43	−0.377638
$206$	0	0
$207$	1137.89	0.382072
$208$	0	0
$209$	−1088.38	−0.360214
$210$	0	0
$211$	2769.26	0.903524	0.451762	−	0.892139i	$-0.350796\pi$
$211$	2769.26	0.903524	0.451762	+	0.892139i	$0.350796\pi$
$212$	0	0
$213$	−278.242	−0.0895063
$214$	0	0
$215$	−944.990	−0.299757
$216$	0	0
$217$	−1336.54	−0.418113
$218$	0	0
$219$	234.472	0.0723477
$220$	0	0
$221$	3238.90	0.985846
$222$	0	0
$223$	−4312.57	−1.29503	−0.647514	−	0.762054i	$-0.724191\pi$
$223$	−4312.57	−1.29503	−0.647514	+	0.762054i	$0.724191\pi$
$224$	0	0
$225$	−655.269	−0.194154
$226$	0	0
$227$	−575.097	−0.168152	−0.0840761	−	0.996459i	$-0.526794\pi$
$227$	−575.097	−0.168152	−0.0840761	+	0.996459i	$0.526794\pi$
$228$	0	0
$229$	2396.42	0.691529	0.345765	−	0.938321i	$-0.387620\pi$
$229$	2396.42	0.691529	0.345765	+	0.938321i	$0.387620\pi$
$230$	0	0
$231$	1459.67	0.415753
$232$	0	0
$233$	−3307.22	−0.929885	−0.464943	−	0.885341i	$-0.653925\pi$
$233$	−3307.22	−0.929885	−0.464943	+	0.885341i	$0.653925\pi$
$234$	0	0
$235$	1920.71	0.533163
$236$	0	0
$237$	−651.145	−0.178466
$238$	0	0
$239$	−1534.33	−0.415262	−0.207631	−	0.978207i	$-0.566575\pi$
$239$	−1534.33	−0.415262	−0.207631	+	0.978207i	$0.566575\pi$
$240$	0	0
$241$	−461.143	−0.123257	−0.0616283	−	0.998099i	$-0.519629\pi$
$241$	−461.143	−0.123257	−0.0616283	+	0.998099i	$0.519629\pi$
$242$	0	0
$243$	1867.76	0.493074
$244$	0	0
$245$	−1827.41	−0.476526
$246$	0	0
$247$	−794.543	−0.204678
$248$	0	0
$249$	−637.611	−0.162277
$250$	0	0
$251$	6200.39	1.55922	0.779612	−	0.626263i	$-0.215416\pi$
$251$	6200.39	1.55922	0.779612	+	0.626263i	$0.215416\pi$
$252$	0	0
$253$	2679.79	0.665917
$254$	0	0
$255$	319.271	0.0784060
$256$	0	0
$257$	−2381.71	−0.578082	−0.289041	−	0.957317i	$-0.593336\pi$
$257$	−2381.71	−0.578082	−0.289041	+	0.957317i	$0.593336\pi$
$258$	0	0
$259$	−1998.22	−0.479394
$260$	0	0
$261$	7014.86	1.66364
$262$	0	0
$263$	420.996	0.0987061	0.0493531	−	0.998781i	$-0.484284\pi$
$263$	420.996	0.0987061	0.0493531	+	0.998781i	$0.484284\pi$
$264$	0	0
$265$	1237.23	0.286801
$266$	0	0
$267$	−1452.26	−0.332872
$268$	0	0
$269$	6748.92	1.52970	0.764849	−	0.644209i	$-0.222813\pi$
$269$	6748.92	1.52970	0.764849	+	0.644209i	$0.222813\pi$
$270$	0	0
$271$	5718.47	1.28182	0.640908	−	0.767618i	$-0.278558\pi$
$271$	5718.47	1.28182	0.640908	+	0.767618i	$0.278558\pi$
$272$	0	0
$273$	1065.59	0.236236
$274$	0	0
$275$	−1543.19	−0.338393
$276$	0	0
$277$	−6245.97	−1.35481	−0.677407	−	0.735608i	$-0.736897\pi$
$277$	−6245.97	−1.35481	−0.677407	+	0.735608i	$0.736897\pi$
$278$	0	0
$279$	−1316.13	−0.282418
$280$	0	0
$281$	−2883.17	−0.612084	−0.306042	−	0.952018i	$-0.599005\pi$
$281$	−2883.17	−0.612084	−0.306042	+	0.952018i	$0.599005\pi$
$282$	0	0
$283$	−7520.83	−1.57974	−0.789871	−	0.613273i	$-0.789852\pi$
$283$	−7520.83	−1.57974	−0.789871	+	0.613273i	$0.789852\pi$
$284$	0	0
$285$	−78.3212	−0.0162784
$286$	0	0
$287$	−5900.67	−1.21361
$288$	0	0
$289$	253.072	0.0515107
$290$	0	0
$291$	−1214.55	−0.244668
$292$	0	0
$293$	6762.89	1.34844	0.674219	−	0.738531i	$-0.264480\pi$
$293$	6762.89	1.34844	0.674219	+	0.738531i	$0.264480\pi$
$294$	0	0
$295$	−2592.98	−0.511760
$296$	0	0
$297$	2918.02	0.570104
$298$	0	0
$299$	1956.31	0.378383
$300$	0	0
$301$	−5030.62	−0.963323
$302$	0	0
$303$	51.8146	0.00982400
$304$	0	0
$305$	−310.021	−0.0582024
$306$	0	0
$307$	10085.5	1.87495	0.937473	−	0.348058i	$-0.113159\pi$
$307$	10085.5	1.87495	0.937473	+	0.348058i	$0.113159\pi$
$308$	0	0
$309$	8.60889	0.00158493
$310$	0	0
$311$	7683.50	1.40094	0.700469	−	0.713683i	$-0.252974\pi$
$311$	7683.50	1.40094	0.700469	+	0.713683i	$0.252974\pi$
$312$	0	0
$313$	−1253.17	−0.226304	−0.113152	−	0.993578i	$-0.536095\pi$
$313$	−1253.17	−0.226304	−0.113152	+	0.993578i	$0.536095\pi$
$314$	0	0
$315$	−3488.30	−0.623947
$316$	0	0
$317$	−958.616	−0.169846	−0.0849231	−	0.996388i	$-0.527064\pi$
$317$	−958.616	−0.169846	−0.0849231	+	0.996388i	$0.527064\pi$
$318$	0	0
$319$	16520.4	2.89957
$320$	0	0
$321$	390.048	0.0678204
$322$	0	0
$323$	−1267.30	−0.218312
$324$	0	0
$325$	−1126.57	−0.192279
$326$	0	0
$327$	−1436.11	−0.242865
$328$	0	0
$329$	10224.8	1.71341
$330$	0	0
$331$	−4252.70	−0.706192	−0.353096	−	0.935587i	$-0.614871\pi$
$331$	−4252.70	−0.706192	−0.353096	+	0.935587i	$0.614871\pi$
$332$	0	0
$333$	−1967.69	−0.323811
$334$	0	0
$335$	−2792.38	−0.455415
$336$	0	0
$337$	7662.86	1.23864	0.619321	−	0.785138i	$-0.287408\pi$
$337$	7662.86	1.23864	0.619321	+	0.785138i	$0.287408\pi$
$338$	0	0
$339$	−1138.26	−0.182365
$340$	0	0
$341$	−3099.55	−0.492229
$342$	0	0
$343$	−598.394	−0.0941990
$344$	0	0
$345$	192.841	0.0300934
$346$	0	0
$347$	−3626.75	−0.561078	−0.280539	−	0.959843i	$-0.590513\pi$
$347$	−3626.75	−0.561078	−0.280539	+	0.959843i	$0.590513\pi$
$348$	0	0
$349$	−8867.03	−1.36000	−0.680002	−	0.733210i	$-0.738021\pi$
$349$	−8867.03	−1.36000	−0.680002	+	0.733210i	$0.738021\pi$
$350$	0	0
$351$	2130.23	0.323940
$352$	0	0
$353$	8775.59	1.32317	0.661583	−	0.749872i	$-0.269885\pi$
$353$	8775.59	1.32317	0.661583	+	0.749872i	$0.269885\pi$
$354$	0	0
$355$	1565.97	0.234121
$356$	0	0
$357$	1699.63	0.251971
$358$	0	0
$359$	−7668.51	−1.12738	−0.563688	−	0.825988i	$-0.690618\pi$
$359$	−7668.51	−1.12738	−0.563688	+	0.825988i	$0.690618\pi$
$360$	0	0
$361$	−6548.11	−0.954675
$362$	0	0
$363$	2202.62	0.318478
$364$	0	0
$365$	−1319.63	−0.189240
$366$	0	0
$367$	−8276.28	−1.17716	−0.588581	−	0.808438i	$-0.700313\pi$
$367$	−8276.28	−1.17716	−0.588581	+	0.808438i	$0.700313\pi$
$368$	0	0
$369$	−5810.54	−0.819741
$370$	0	0
$371$	6586.33	0.921685
$372$	0	0
$373$	270.669	0.0375730	0.0187865	−	0.999824i	$-0.494020\pi$
$373$	270.669	0.0375730	0.0187865	+	0.999824i	$0.494020\pi$
$374$	0	0
$375$	−111.050	−0.0152923
$376$	0	0
$377$	12060.3	1.64757
$378$	0	0
$379$	8455.66	1.14601	0.573006	−	0.819551i	$-0.305777\pi$
$379$	8455.66	1.14601	0.573006	+	0.819551i	$0.305777\pi$
$380$	0	0
$381$	209.064	0.0281120
$382$	0	0
$383$	−6527.25	−0.870827	−0.435414	−	0.900231i	$-0.643398\pi$
$383$	−6527.25	−0.870827	−0.435414	+	0.900231i	$0.643398\pi$
$384$	0	0
$385$	−8215.13	−1.08748
$386$	0	0
$387$	−4953.78	−0.650684
$388$	0	0
$389$	−683.173	−0.0890443	−0.0445222	−	0.999008i	$-0.514177\pi$
$389$	−683.173	−0.0890443	−0.0445222	+	0.999008i	$0.514177\pi$
$390$	0	0
$391$	3120.34	0.403586
$392$	0	0
$393$	882.290	0.113246
$394$	0	0
$395$	3664.70	0.466813
$396$	0	0
$397$	2579.34	0.326079	0.163040	−	0.986620i	$-0.447870\pi$
$397$	2579.34	0.326079	0.163040	+	0.986620i	$0.447870\pi$
$398$	0	0
$399$	−416.940	−0.0523136
$400$	0	0
$401$	−8344.02	−1.03910	−0.519552	−	0.854439i	$-0.673901\pi$
$401$	−8344.02	−1.03910	−0.519552	+	0.854439i	$0.673901\pi$
$402$	0	0
$403$	−2262.75	−0.279691
$404$	0	0
$405$	−3328.47	−0.408377
$406$	0	0
$407$	−4634.02	−0.564373
$408$	0	0
$409$	−1939.68	−0.234501	−0.117251	−	0.993102i	$-0.537408\pi$
$409$	−1939.68	−0.234501	−0.117251	+	0.993102i	$0.537408\pi$
$410$	0	0
$411$	−1839.07	−0.220717
$412$	0	0
$413$	−13803.6	−1.64463
$414$	0	0
$415$	3588.53	0.424467
$416$	0	0
$417$	−256.153	−0.0300812
$418$	0	0
$419$	−4046.49	−0.471799	−0.235900	−	0.971777i	$-0.575804\pi$
$419$	−4046.49	−0.471799	−0.235900	+	0.971777i	$0.575804\pi$
$420$	0	0
$421$	−3305.28	−0.382636	−0.191318	−	0.981528i	$-0.561276\pi$
$421$	−3305.28	−0.382636	−0.191318	+	0.981528i	$0.561276\pi$
$422$	0	0
$423$	10068.6	1.15734
$424$	0	0
$425$	−1796.88	−0.205086
$426$	0	0
$427$	−1650.38	−0.187044
$428$	0	0
$429$	2471.19	0.278113
$430$	0	0
$431$	−2953.12	−0.330038	−0.165019	−	0.986290i	$-0.552769\pi$
$431$	−2953.12	−0.330038	−0.165019	+	0.986290i	$0.552769\pi$
$432$	0	0
$433$	1380.70	0.153239	0.0766193	−	0.997060i	$-0.475587\pi$
$433$	1380.70	0.153239	0.0766193	+	0.997060i	$0.475587\pi$
$434$	0	0
$435$	1188.83	0.131034
$436$	0	0
$437$	−765.458	−0.0837914
$438$	0	0
$439$	14233.4	1.54743	0.773717	−	0.633532i	$-0.218395\pi$
$439$	14233.4	1.54743	0.773717	+	0.633532i	$0.218395\pi$
$440$	0	0
$441$	−9579.54	−1.03440
$442$	0	0
$443$	−8541.25	−0.916042	−0.458021	−	0.888941i	$-0.651442\pi$
$443$	−8541.25	−0.916042	−0.458021	+	0.888941i	$0.651442\pi$
$444$	0	0
$445$	8173.43	0.870692
$446$	0	0
$447$	−800.152	−0.0846664
$448$	0	0
$449$	9994.76	1.05052	0.525258	−	0.850943i	$-0.323969\pi$
$449$	9994.76	1.05052	0.525258	+	0.850943i	$0.323969\pi$
$450$	0	0
$451$	−13684.1	−1.42874
$452$	0	0
$453$	−1781.88	−0.184812
$454$	0	0
$455$	−5997.24	−0.617923
$456$	0	0
$457$	−6435.58	−0.658739	−0.329370	−	0.944201i	$-0.606836\pi$
$457$	−6435.58	−0.658739	−0.329370	+	0.944201i	$0.606836\pi$
$458$	0	0
$459$	3397.73	0.345517
$460$	0	0
$461$	7851.78	0.793262	0.396631	−	0.917978i	$-0.370179\pi$
$461$	7851.78	0.793262	0.396631	+	0.917978i	$0.370179\pi$
$462$	0	0
$463$	7073.58	0.710016	0.355008	−	0.934863i	$-0.384478\pi$
$463$	7073.58	0.710016	0.355008	+	0.934863i	$0.384478\pi$
$464$	0	0
$465$	−223.048	−0.0222443
$466$	0	0
$467$	−15753.8	−1.56102	−0.780511	−	0.625142i	$-0.785041\pi$
$467$	−15753.8	−1.56102	−0.780511	+	0.625142i	$0.785041\pi$
$468$	0	0
$469$	−14865.1	−1.46356
$470$	0	0
$471$	−2752.38	−0.269264
$472$	0	0
$473$	−11666.4	−1.13408
$474$	0	0
$475$	440.799	0.0425794
$476$	0	0
$477$	6485.73	0.622560
$478$	0	0
$479$	14747.9	1.40678	0.703391	−	0.710803i	$-0.251668\pi$
$479$	14747.9	1.40678	0.703391	+	0.710803i	$0.251668\pi$
$480$	0	0
$481$	−3382.95	−0.320684
$482$	0	0
$483$	1026.58	0.0967106
$484$	0	0
$485$	6835.61	0.639977
$486$	0	0
$487$	−5631.77	−0.524024	−0.262012	−	0.965065i	$-0.584386\pi$
$487$	−5631.77	−0.524024	−0.262012	+	0.965065i	$0.584386\pi$
$488$	0	0
$489$	3168.54	0.293019
$490$	0	0
$491$	−18259.0	−1.67824	−0.839121	−	0.543945i	$-0.816930\pi$
$491$	−18259.0	−1.67824	−0.839121	+	0.543945i	$0.816930\pi$
$492$	0	0
$493$	19236.2	1.75731
$494$	0	0
$495$	−8089.64	−0.734550
$496$	0	0
$497$	8336.39	0.752391
$498$	0	0
$499$	729.599	0.0654536	0.0327268	−	0.999464i	$-0.489581\pi$
$499$	729.599	0.0654536	0.0327268	+	0.999464i	$0.489581\pi$
$500$	0	0
$501$	−1178.80	−0.105119
$502$	0	0
$503$	−1551.81	−0.137558	−0.0687790	−	0.997632i	$-0.521910\pi$
$503$	−1551.81	−0.137558	−0.0687790	+	0.997632i	$0.521910\pi$
$504$	0	0
$505$	−291.617	−0.0256966
$506$	0	0
$507$	−147.788	−0.0129457
$508$	0	0
$509$	−6145.59	−0.535164	−0.267582	−	0.963535i	$-0.586225\pi$
$509$	−6145.59	−0.535164	−0.267582	+	0.963535i	$0.586225\pi$
$510$	0	0
$511$	−7024.99	−0.608155
$512$	0	0
$513$	−833.506	−0.0717353
$514$	0	0
$515$	−48.4516	−0.00414569
$516$	0	0
$517$	23712.2	2.01714
$518$	0	0
$519$	919.695	0.0777845
$520$	0	0
$521$	−588.432	−0.0494812	−0.0247406	−	0.999694i	$-0.507876\pi$
$521$	−588.432	−0.0494812	−0.0247406	+	0.999694i	$0.507876\pi$
$522$	0	0
$523$	866.147	0.0724168	0.0362084	−	0.999344i	$-0.488472\pi$
$523$	866.147	0.0724168	0.0362084	+	0.999344i	$0.488472\pi$
$524$	0	0
$525$	−591.172	−0.0491445
$526$	0	0
$527$	−3609.10	−0.298321
$528$	0	0
$529$	−10282.3	−0.845097
$530$	0	0
$531$	−13592.8	−1.11088
$532$	0	0
$533$	−9989.74	−0.811827
$534$	0	0
$535$	−2195.22	−0.177398
$536$	0	0
$537$	1609.14	0.129310
$538$	0	0
$539$	−22560.3	−1.80286
$540$	0	0
$541$	−10380.3	−0.824926	−0.412463	−	0.910974i	$-0.635331\pi$
$541$	−10380.3	−0.824926	−0.412463	+	0.910974i	$0.635331\pi$
$542$	0	0
$543$	−2031.05	−0.160517
$544$	0	0
$545$	8082.54	0.635262
$546$	0	0
$547$	−1208.43	−0.0944584	−0.0472292	−	0.998884i	$-0.515039\pi$
$547$	−1208.43	−0.0944584	−0.0472292	+	0.998884i	$0.515039\pi$
$548$	0	0
$549$	−1625.18	−0.126340
$550$	0	0
$551$	−4718.89	−0.364849
$552$	0	0
$553$	19508.9	1.50019
$554$	0	0
$555$	−333.470	−0.0255046
$556$	0	0
$557$	10284.9	0.782379	0.391190	−	0.920310i	$-0.372064\pi$
$557$	10284.9	0.782379	0.391190	+	0.920310i	$0.372064\pi$
$558$	0	0
$559$	−8516.76	−0.644402
$560$	0	0
$561$	3941.57	0.296637
$562$	0	0
$563$	−18420.7	−1.37894	−0.689469	−	0.724316i	$-0.742156\pi$
$563$	−18420.7	−1.37894	−0.689469	+	0.724316i	$0.742156\pi$
$564$	0	0
$565$	6406.23	0.477013
$566$	0	0
$567$	−17719.0	−1.31239
$568$	0	0
$569$	13111.3	0.966002	0.483001	−	0.875620i	$-0.339547\pi$
$569$	13111.3	0.966002	0.483001	+	0.875620i	$0.339547\pi$
$570$	0	0
$571$	12226.6	0.896088	0.448044	−	0.894012i	$-0.352121\pi$
$571$	12226.6	0.896088	0.448044	+	0.894012i	$0.352121\pi$
$572$	0	0
$573$	−2125.85	−0.154989
$574$	0	0
$575$	−1085.33	−0.0787153
$576$	0	0
$577$	3294.12	0.237671	0.118836	−	0.992914i	$-0.462084\pi$
$577$	3294.12	0.237671	0.118836	+	0.992914i	$0.462084\pi$
$578$	0	0
$579$	−567.168	−0.0407093
$580$	0	0
$581$	19103.4	1.36410
$582$	0	0
$583$	15274.2	1.08507
$584$	0	0
$585$	−5905.64	−0.417381
$586$	0	0
$587$	−16607.6	−1.16775	−0.583874	−	0.811844i	$-0.698464\pi$
$587$	−16607.6	−1.16775	−0.583874	+	0.811844i	$0.698464\pi$
$588$	0	0
$589$	885.358	0.0619364
$590$	0	0
$591$	−581.284	−0.0404582
$592$	0	0
$593$	−15288.4	−1.05872	−0.529359	−	0.848398i	$-0.677567\pi$
$593$	−15288.4	−1.05872	−0.529359	+	0.848398i	$0.677567\pi$
$594$	0	0
$595$	−9565.65	−0.659081
$596$	0	0
$597$	1454.36	0.0997038
$598$	0	0
$599$	23543.0	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$599$	23543.0	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$600$	0	0
$601$	15689.7	1.06488	0.532442	−	0.846467i	$-0.321275\pi$
$601$	15689.7	1.06488	0.532442	+	0.846467i	$0.321275\pi$
$602$	0	0
$603$	−14638.1	−0.988571
$604$	0	0
$605$	−12396.5	−0.833043
$606$	0	0
$607$	−26462.9	−1.76952	−0.884759	−	0.466048i	$-0.845677\pi$
$607$	−26462.9	−1.76952	−0.884759	+	0.466048i	$0.845677\pi$
$608$	0	0
$609$	6328.68	0.421102
$610$	0	0
$611$	17310.5	1.14617
$612$	0	0
$613$	18256.6	1.20290	0.601448	−	0.798912i	$-0.294591\pi$
$613$	18256.6	1.20290	0.601448	+	0.798912i	$0.294591\pi$
$614$	0	0
$615$	−984.728	−0.0645659
$616$	0	0
$617$	−393.579	−0.0256805	−0.0128403	−	0.999918i	$-0.504087\pi$
$617$	−393.579	−0.0256805	−0.0128403	+	0.999918i	$0.504087\pi$
$618$	0	0
$619$	−23039.6	−1.49602	−0.748012	−	0.663685i	$-0.768991\pi$
$619$	−23039.6	−1.49602	−0.748012	+	0.663685i	$0.768991\pi$
$620$	0	0
$621$	2052.25	0.132615
$622$	0	0
$623$	43511.0	2.79812
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−966.918	−0.0615869
$628$	0	0
$629$	−5395.83	−0.342044
$630$	0	0
$631$	−28960.4	−1.82709	−0.913547	−	0.406734i	$-0.866668\pi$
$631$	−28960.4	−1.82709	−0.913547	+	0.406734i	$0.866668\pi$
$632$	0	0
$633$	2460.21	0.154478
$634$	0	0
$635$	−1176.63	−0.0735324
$636$	0	0
$637$	−16469.6	−1.02441
$638$	0	0
$639$	8209.05	0.508208
$640$	0	0
$641$	−7160.30	−0.441209	−0.220604	−	0.975363i	$-0.570803\pi$
$641$	−7160.30	−0.441209	−0.220604	+	0.975363i	$0.570803\pi$
$642$	0	0
$643$	1843.33	0.113054	0.0565270	−	0.998401i	$-0.481997\pi$
$643$	1843.33	0.113054	0.0565270	+	0.998401i	$0.481997\pi$
$644$	0	0
$645$	−839.531	−0.0512504
$646$	0	0
$647$	−28494.4	−1.73142	−0.865712	−	0.500542i	$-0.833134\pi$
$647$	−28494.4	−1.73142	−0.865712	+	0.500542i	$0.833134\pi$
$648$	0	0
$649$	−32011.7	−1.93616
$650$	0	0
$651$	−1187.39	−0.0714860
$652$	0	0
$653$	−15898.2	−0.952746	−0.476373	−	0.879243i	$-0.658049\pi$
$653$	−15898.2	−0.952746	−0.476373	+	0.879243i	$0.658049\pi$
$654$	0	0
$655$	−4965.61	−0.296217
$656$	0	0
$657$	−6917.69	−0.410783
$658$	0	0
$659$	−6729.26	−0.397776	−0.198888	−	0.980022i	$-0.563733\pi$
$659$	−6729.26	−0.397776	−0.198888	+	0.980022i	$0.563733\pi$
$660$	0	0
$661$	−24516.3	−1.44262	−0.721311	−	0.692611i	$-0.756460\pi$
$661$	−24516.3	−1.44262	−0.721311	+	0.692611i	$0.756460\pi$
$662$	0	0
$663$	2877.44	0.168553
$664$	0	0
$665$	2346.58	0.136837
$666$	0	0
$667$	11618.8	0.674485
$668$	0	0
$669$	−3831.29	−0.221415
$670$	0	0
$671$	−3827.37	−0.220200
$672$	0	0
$673$	−8192.84	−0.469258	−0.234629	−	0.972085i	$-0.575388\pi$
$673$	−8192.84	−0.469258	−0.234629	+	0.972085i	$0.575388\pi$
$674$	0	0
$675$	−1181.81	−0.0673896
$676$	0	0
$677$	−5219.11	−0.296287	−0.148144	−	0.988966i	$-0.547330\pi$
$677$	−5219.11	−0.296287	−0.148144	+	0.988966i	$0.547330\pi$
$678$	0	0
$679$	36389.1	2.05668
$680$	0	0
$681$	−510.917	−0.0287495
$682$	0	0
$683$	−7903.11	−0.442759	−0.221379	−	0.975188i	$-0.571056\pi$
$683$	−7903.11	−0.442759	−0.221379	+	0.975188i	$0.571056\pi$
$684$	0	0
$685$	10350.5	0.577330
$686$	0	0
$687$	2128.99	0.118233
$688$	0	0
$689$	11150.6	0.616549
$690$	0	0
$691$	−13969.2	−0.769051	−0.384526	−	0.923114i	$-0.625635\pi$
$691$	−13969.2	−0.769051	−0.384526	+	0.923114i	$0.625635\pi$
$692$	0	0
$693$	−43064.9	−2.36061
$694$	0	0
$695$	1441.65	0.0786834
$696$	0	0
$697$	−15933.7	−0.865900
$698$	0	0
$699$	−2938.14	−0.158985
$700$	0	0
$701$	−1929.28	−0.103948	−0.0519742	−	0.998648i	$-0.516551\pi$
$701$	−1929.28	−0.103948	−0.0519742	+	0.998648i	$0.516551\pi$
$702$	0	0
$703$	1323.67	0.0710142
$704$	0	0
$705$	1706.36	0.0911564
$706$	0	0
$707$	−1552.41	−0.0825806
$708$	0	0
$709$	33077.6	1.75213	0.876063	−	0.482197i	$-0.160161\pi$
$709$	33077.6	1.75213	0.876063	+	0.482197i	$0.160161\pi$
$710$	0	0
$711$	19210.9	1.01331
$712$	0	0
$713$	−2179.92	−0.114500
$714$	0	0
$715$	−13908.1	−0.727458
$716$	0	0
$717$	−1363.10	−0.0709985
$718$	0	0
$719$	−5643.22	−0.292707	−0.146354	−	0.989232i	$-0.546754\pi$
$719$	−5643.22	−0.292707	−0.146354	+	0.989232i	$0.546754\pi$
$720$	0	0
$721$	−257.930	−0.0133229
$722$	0	0
$723$	−409.680	−0.0210735
$724$	0	0
$725$	−6690.83	−0.342746
$726$	0	0
$727$	−17112.0	−0.872969	−0.436484	−	0.899712i	$-0.643777\pi$
$727$	−17112.0	−0.872969	−0.436484	+	0.899712i	$0.643777\pi$
$728$	0	0
$729$	−16314.4	−0.828857
$730$	0	0
$731$	−13584.3	−0.687324
$732$	0	0
$733$	20314.2	1.02363	0.511817	−	0.859095i	$-0.328973\pi$
$733$	20314.2	1.02363	0.511817	+	0.859095i	$0.328973\pi$
$734$	0	0
$735$	−1623.47	−0.0814730
$736$	0	0
$737$	−34473.4	−1.72299
$738$	0	0
$739$	5845.97	0.290998	0.145499	−	0.989358i	$-0.453521\pi$
$739$	5845.97	0.290998	0.145499	+	0.989358i	$0.453521\pi$
$740$	0	0
$741$	−705.873	−0.0349945
$742$	0	0
$743$	−4964.00	−0.245103	−0.122551	−	0.992462i	$-0.539108\pi$
$743$	−4964.00	−0.245103	−0.122551	+	0.992462i	$0.539108\pi$
$744$	0	0
$745$	4503.32	0.221462
$746$	0	0
$747$	18811.6	0.921393
$748$	0	0
$749$	−11686.2	−0.570099
$750$	0	0
$751$	30765.7	1.49488	0.747441	−	0.664329i	$-0.231282\pi$
$751$	30765.7	1.49488	0.747441	+	0.664329i	$0.231282\pi$
$752$	0	0
$753$	5508.44	0.266585
$754$	0	0
$755$	10028.6	0.483413
$756$	0	0
$757$	−34346.8	−1.64908	−0.824542	−	0.565801i	$-0.808567\pi$
$757$	−34346.8	−1.64908	−0.824542	+	0.565801i	$0.808567\pi$
$758$	0	0
$759$	2380.73	0.113854
$760$	0	0
$761$	12083.9	0.575610	0.287805	−	0.957689i	$-0.407074\pi$
$761$	12083.9	0.575610	0.287805	+	0.957689i	$0.407074\pi$
$762$	0	0
$763$	43027.1	2.04153
$764$	0	0
$765$	−9419.54	−0.445182
$766$	0	0
$767$	−23369.3	−1.10015
$768$	0	0
$769$	9054.84	0.424611	0.212305	−	0.977203i	$-0.431903\pi$
$769$	9054.84	0.424611	0.212305	+	0.977203i	$0.431903\pi$
$770$	0	0
$771$	−2115.92	−0.0988364
$772$	0	0
$773$	13846.2	0.644260	0.322130	−	0.946695i	$-0.395601\pi$
$773$	13846.2	0.644260	0.322130	+	0.946695i	$0.395601\pi$
$774$	0	0
$775$	1255.33	0.0581844
$776$	0	0
$777$	−1775.22	−0.0819634
$778$	0	0
$779$	3908.74	0.179776
$780$	0	0
$781$	19332.8	0.885762
$782$	0	0
$783$	12651.7	0.577438
$784$	0	0
$785$	15490.7	0.704312
$786$	0	0
$787$	−9393.21	−0.425454	−0.212727	−	0.977112i	$-0.568234\pi$
$787$	−9393.21	−0.425454	−0.212727	+	0.977112i	$0.568234\pi$
$788$	0	0
$789$	374.013	0.0168761
$790$	0	0
$791$	34103.3	1.53296
$792$	0	0
$793$	−2794.08	−0.125120
$794$	0	0
$795$	1099.15	0.0490352
$796$	0	0
$797$	−7921.92	−0.352081	−0.176041	−	0.984383i	$-0.556329\pi$
$797$	−7921.92	−0.352081	−0.176041	+	0.984383i	$0.556329\pi$
$798$	0	0
$799$	27610.4	1.22251
$800$	0	0
$801$	42846.3	1.89001
$802$	0	0
$803$	−16291.5	−0.715959
$804$	0	0
$805$	−5777.71	−0.252966
$806$	0	0
$807$	5995.75	0.261537
$808$	0	0
$809$	−8042.02	−0.349496	−0.174748	−	0.984613i	$-0.555911\pi$
$809$	−8042.02	−0.349496	−0.174748	+	0.984613i	$0.555911\pi$
$810$	0	0
$811$	28973.7	1.25451	0.627253	−	0.778816i	$-0.284179\pi$
$811$	28973.7	1.25451	0.627253	+	0.778816i	$0.284179\pi$
$812$	0	0
$813$	5080.29	0.219156
$814$	0	0
$815$	−17832.8	−0.766450
$816$	0	0
$817$	3332.40	0.142700
$818$	0	0
$819$	−31438.4	−1.34133
$820$	0	0
$821$	−11126.0	−0.472960	−0.236480	−	0.971636i	$-0.575994\pi$
$821$	−11126.0	−0.472960	−0.236480	+	0.971636i	$0.575994\pi$
$822$	0	0
$823$	−3077.71	−0.130355	−0.0651775	−	0.997874i	$-0.520761\pi$
$823$	−3077.71	−0.130355	−0.0651775	+	0.997874i	$0.520761\pi$
$824$	0	0
$825$	−1370.97	−0.0578560
$826$	0	0
$827$	−37902.1	−1.59369	−0.796847	−	0.604181i	$-0.793501\pi$
$827$	−37902.1	−1.59369	−0.796847	+	0.604181i	$0.793501\pi$
$828$	0	0
$829$	38596.7	1.61703	0.808516	−	0.588474i	$-0.200271\pi$
$829$	38596.7	1.61703	0.808516	+	0.588474i	$0.200271\pi$
$830$	0	0
$831$	−5548.92	−0.231637
$832$	0	0
$833$	−26269.1	−1.09264
$834$	0	0
$835$	6634.37	0.274960
$836$	0	0
$837$	−2373.71	−0.0980255
$838$	0	0
$839$	7585.24	0.312123	0.156062	−	0.987747i	$-0.450120\pi$
$839$	7585.24	0.312123	0.156062	+	0.987747i	$0.450120\pi$
$840$	0	0
$841$	47238.4	1.93687
$842$	0	0
$843$	−2561.41	−0.104650
$844$	0	0
$845$	831.763	0.0338621
$846$	0	0
$847$	−65992.5	−2.67713
$848$	0	0
$849$	−6681.52	−0.270093
$850$	0	0
$851$	−3259.11	−0.131282
$852$	0	0
$853$	37228.6	1.49435	0.747177	−	0.664625i	$-0.231409\pi$
$853$	37228.6	1.49435	0.747177	+	0.664625i	$0.231409\pi$
$854$	0	0
$855$	2310.73	0.0924273
$856$	0	0
$857$	−7510.28	−0.299354	−0.149677	−	0.988735i	$-0.547823\pi$
$857$	−7510.28	−0.299354	−0.149677	+	0.988735i	$0.547823\pi$
$858$	0	0
$859$	19383.6	0.769918	0.384959	−	0.922934i	$-0.374216\pi$
$859$	19383.6	0.769918	0.384959	+	0.922934i	$0.374216\pi$
$860$	0	0
$861$	−5242.16	−0.207494
$862$	0	0
$863$	7835.20	0.309054	0.154527	−	0.987989i	$-0.450615\pi$
$863$	7835.20	0.309054	0.154527	+	0.987989i	$0.450615\pi$
$864$	0	0
$865$	−5176.12	−0.203461
$866$	0	0
$867$	224.830	0.00880694
$868$	0	0
$869$	45242.7	1.76611
$870$	0	0
$871$	−25166.4	−0.979026
$872$	0	0
$873$	35833.3	1.38920
$874$	0	0
$875$	3327.16	0.128547
$876$	0	0
$877$	−8353.53	−0.321640	−0.160820	−	0.986984i	$-0.551414\pi$
$877$	−8353.53	−0.321640	−0.160820	+	0.986984i	$0.551414\pi$
$878$	0	0
$879$	6008.16	0.230546
$880$	0	0
$881$	17964.9	0.687008	0.343504	−	0.939151i	$-0.388386\pi$
$881$	17964.9	0.687008	0.343504	+	0.939151i	$0.388386\pi$
$882$	0	0
$883$	34012.6	1.29628	0.648140	−	0.761521i	$-0.275547\pi$
$883$	34012.6	1.29628	0.648140	+	0.761521i	$0.275547\pi$
$884$	0	0
$885$	−2303.61	−0.0874971
$886$	0	0
$887$	−32352.3	−1.22467	−0.612335	−	0.790598i	$-0.709770\pi$
$887$	−32352.3	−1.22467	−0.612335	+	0.790598i	$0.709770\pi$
$888$	0	0
$889$	−6263.74	−0.236309
$890$	0	0
$891$	−41091.7	−1.54503
$892$	0	0
$893$	−6773.17	−0.253814
$894$	0	0
$895$	−9056.38	−0.338236
$896$	0	0
$897$	1737.99	0.0646933
$898$	0	0
$899$	−13438.7	−0.498562
$900$	0	0
$901$	17785.2	0.657616
$902$	0	0
$903$	−4469.21	−0.164702
$904$	0	0
$905$	11430.9	0.419863
$906$	0	0
$907$	−15523.0	−0.568284	−0.284142	−	0.958782i	$-0.591709\pi$
$907$	−15523.0	−0.568284	−0.284142	+	0.958782i	$0.591709\pi$
$908$	0	0
$909$	−1528.70	−0.0557797
$910$	0	0
$911$	8000.66	0.290970	0.145485	−	0.989360i	$-0.453526\pi$
$911$	8000.66	0.290970	0.145485	+	0.989360i	$0.453526\pi$
$912$	0	0
$913$	44302.3	1.60591
$914$	0	0
$915$	−275.423	−0.00995104
$916$	0	0
$917$	−26434.2	−0.951946
$918$	0	0
$919$	23631.8	0.848251	0.424125	−	0.905603i	$-0.360582\pi$
$919$	23631.8	0.848251	0.424125	+	0.905603i	$0.360582\pi$
$920$	0	0
$921$	8959.95	0.320565
$922$	0	0
$923$	14113.4	0.503302
$924$	0	0
$925$	1876.80	0.0667122
$926$	0	0
$927$	−253.991	−0.00899908
$928$	0	0
$929$	−7564.75	−0.267160	−0.133580	−	0.991038i	$-0.542647\pi$
$929$	−7564.75	−0.267160	−0.133580	+	0.991038i	$0.542647\pi$
$930$	0	0
$931$	6444.15	0.226851
$932$	0	0
$933$	6826.03	0.239522
$934$	0	0
$935$	−22183.5	−0.775913
$936$	0	0
$937$	−6226.13	−0.217074	−0.108537	−	0.994092i	$-0.534617\pi$
$937$	−6226.13	−0.217074	−0.108537	+	0.994092i	$0.534617\pi$
$938$	0	0
$939$	−1113.32	−0.0386919
$940$	0	0
$941$	14012.2	0.485426	0.242713	−	0.970098i	$-0.421963\pi$
$941$	14012.2	0.485426	0.242713	+	0.970098i	$0.421963\pi$
$942$	0	0
$943$	−9624.05	−0.332346
$944$	0	0
$945$	−6291.33	−0.216568
$946$	0	0
$947$	39575.5	1.35801	0.679003	−	0.734135i	$-0.262412\pi$
$947$	39575.5	1.35801	0.679003	+	0.734135i	$0.262412\pi$
$948$	0	0
$949$	−11893.2	−0.406817
$950$	0	0
$951$	−851.636	−0.0290391
$952$	0	0
$953$	−36806.3	−1.25107	−0.625537	−	0.780194i	$-0.715120\pi$
$953$	−36806.3	−1.25107	−0.625537	+	0.780194i	$0.715120\pi$
$954$	0	0
$955$	11964.5	0.405405
$956$	0	0
$957$	14676.7	0.495748
$958$	0	0
$959$	55100.3	1.85535
$960$	0	0
$961$	−27269.6	−0.915364
$962$	0	0
$963$	−11507.7	−0.385078
$964$	0	0
$965$	3192.07	0.106483
$966$	0	0
$967$	−33422.7	−1.11148	−0.555740	−	0.831356i	$-0.687565\pi$
$967$	−33422.7	−1.11148	−0.555740	+	0.831356i	$0.687565\pi$
$968$	0	0
$969$	−1125.87	−0.0373254
$970$	0	0
$971$	35405.1	1.17014	0.585069	−	0.810983i	$-0.301067\pi$
$971$	35405.1	1.17014	0.585069	+	0.810983i	$0.301067\pi$
$972$	0	0
$973$	7674.58	0.252863
$974$	0	0
$975$	−1000.84	−0.0328745
$976$	0	0
$977$	−45772.5	−1.49887	−0.749433	−	0.662080i	$-0.769674\pi$
$977$	−45772.5	−1.49887	−0.749433	+	0.662080i	$0.769674\pi$
$978$	0	0
$979$	100905.	3.29413
$980$	0	0
$981$	42369.9	1.37897
$982$	0	0
$983$	18452.9	0.598736	0.299368	−	0.954138i	$-0.403224\pi$
$983$	18452.9	0.598736	0.299368	+	0.954138i	$0.403224\pi$
$984$	0	0
$985$	3271.51	0.105826
$986$	0	0
$987$	9083.75	0.292947
$988$	0	0
$989$	−8205.00	−0.263806
$990$	0	0
$991$	34921.1	1.11938	0.559689	−	0.828703i	$-0.310920\pi$
$991$	34921.1	1.11938	0.559689	+	0.828703i	$0.310920\pi$
$992$	0	0
$993$	−3778.10	−0.120740
$994$	0	0
$995$	−8185.29	−0.260795
$996$	0	0
$997$	18316.5	0.581835	0.290917	−	0.956748i	$-0.406039\pi$
$997$	18316.5	0.581835	0.290917	+	0.956748i	$0.406039\pi$
$998$	0	0
$999$	−3548.84	−0.112393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.a.bc.1.3		6
4.3	odd	2		1280.4.a.ba.1.4		6
8.3	odd	2		1280.4.a.bd.1.3		6
8.5	even	2		1280.4.a.bb.1.4		6
16.3	odd	4		160.4.d.a.81.6		12
16.5	even	4		40.4.d.a.21.4	yes	12
16.11	odd	4		160.4.d.a.81.7		12
16.13	even	4		40.4.d.a.21.3	✓	12
48.5	odd	4		360.4.k.c.181.9		12
48.11	even	4		1440.4.k.c.721.7		12
48.29	odd	4		360.4.k.c.181.10		12
48.35	even	4		1440.4.k.c.721.1		12
80.3	even	4		800.4.f.c.49.6		12
80.13	odd	4		200.4.f.b.149.6		12
80.19	odd	4		800.4.d.d.401.7		12
80.27	even	4		800.4.f.c.49.5		12
80.29	even	4		200.4.d.b.101.10		12
80.37	odd	4		200.4.f.b.149.5		12
80.43	even	4		800.4.f.b.49.8		12
80.53	odd	4		200.4.f.c.149.8		12
80.59	odd	4		800.4.d.d.401.6		12
80.67	even	4		800.4.f.b.49.7		12
80.69	even	4		200.4.d.b.101.9		12
80.77	odd	4		200.4.f.c.149.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.d.a.21.3	✓	12	16.13	even	4
40.4.d.a.21.4	yes	12	16.5	even	4
160.4.d.a.81.6		12	16.3	odd	4
160.4.d.a.81.7		12	16.11	odd	4
200.4.d.b.101.9		12	80.69	even	4
200.4.d.b.101.10		12	80.29	even	4
200.4.f.b.149.5		12	80.37	odd	4
200.4.f.b.149.6		12	80.13	odd	4
200.4.f.c.149.7		12	80.77	odd	4
200.4.f.c.149.8		12	80.53	odd	4
360.4.k.c.181.9		12	48.5	odd	4
360.4.k.c.181.10		12	48.29	odd	4
800.4.d.d.401.6		12	80.59	odd	4
800.4.d.d.401.7		12	80.19	odd	4
800.4.f.b.49.7		12	80.67	even	4
800.4.f.b.49.8		12	80.43	even	4
800.4.f.c.49.5		12	80.27	even	4
800.4.f.c.49.6		12	80.3	even	4
1280.4.a.ba.1.4		6	4.3	odd	2
1280.4.a.bb.1.4		6	8.5	even	2
1280.4.a.bc.1.3		6	1.1	even	1	trivial
1280.4.a.bd.1.3		6	8.3	odd	2
1440.4.k.c.721.1		12	48.35	even	4
1440.4.k.c.721.7		12	48.11	even	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.a (trivial)

\(f(q)\)	\(=\)	\(q+0.888401 q^{3} -5.00000 q^{5} -26.6173 q^{7} -26.2107 q^{9} +O(q^{10})\)	\(q+0.888401 q^{3} -5.00000 q^{5} -26.6173 q^{7} -26.2107 q^{9} -61.7277 q^{11} -45.0627 q^{13} -4.44201 q^{15} -71.8754 q^{17} +17.6319 q^{19} -23.6469 q^{21} -43.4131 q^{23} +25.0000 q^{25} -47.2725 q^{27} -267.633 q^{29} +50.2133 q^{31} -54.8390 q^{33} +133.087 q^{35} +75.0720 q^{37} -40.0338 q^{39} +221.685 q^{41} +188.998 q^{43} +131.054 q^{45} -384.142 q^{47} +365.481 q^{49} -63.8542 q^{51} -247.445 q^{53} +308.638 q^{55} +15.6642 q^{57} +518.596 q^{59} +62.0042 q^{61} +697.660 q^{63} +225.314 q^{65} +558.476 q^{67} -38.5683 q^{69} -313.194 q^{71} +263.926 q^{73} +22.2100 q^{75} +1643.03 q^{77} -732.940 q^{79} +665.693 q^{81} -717.705 q^{83} +359.377 q^{85} -237.766 q^{87} -1634.69 q^{89} +1199.45 q^{91} +44.6096 q^{93} -88.1597 q^{95} -1367.12 q^{97} +1617.93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 30 q^{5} - 14 q^{7} + 54 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 30 * q^5 - 14 * q^7 + 54 * q^9	\( 6 q + 6 q^{3} - 30 q^{5} - 14 q^{7} + 54 q^{9} + 44 q^{11} - 30 q^{15} + 152 q^{19} - 4 q^{21} - 302 q^{23} + 150 q^{25} + 216 q^{27} - 132 q^{31} - 116 q^{33} + 70 q^{35} + 68 q^{37} - 300 q^{39} - 20 q^{41} + 602 q^{43} - 270 q^{45} - 470 q^{47} + 654 q^{49} + 612 q^{51} - 528 q^{53} - 220 q^{55} + 340 q^{57} + 472 q^{59} + 476 q^{61} - 650 q^{63} + 1206 q^{67} + 980 q^{69} + 796 q^{71} - 216 q^{73} + 150 q^{75} - 412 q^{77} + 1008 q^{79} + 1254 q^{81} + 1778 q^{83} + 984 q^{87} + 212 q^{89} + 3652 q^{91} + 1392 q^{93} - 760 q^{95} - 792 q^{97} + 5516 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 - 30 * q^5 - 14 * q^7 + 54 * q^9 + 44 * q^11 - 30 * q^15 + 152 * q^19 - 4 * q^21 - 302 * q^23 + 150 * q^25 + 216 * q^27 - 132 * q^31 - 116 * q^33 + 70 * q^35 + 68 * q^37 - 300 * q^39 - 20 * q^41 + 602 * q^43 - 270 * q^45 - 470 * q^47 + 654 * q^49 + 612 * q^51 - 528 * q^53 - 220 * q^55 + 340 * q^57 + 472 * q^59 + 476 * q^61 - 650 * q^63 + 1206 * q^67 + 980 * q^69 + 796 * q^71 - 216 * q^73 + 150 * q^75 - 412 * q^77 + 1008 * q^79 + 1254 * q^81 + 1778 * q^83 + 984 * q^87 + 212 * q^89 + 3652 * q^91 + 1392 * q^93 - 760 * q^95 - 792 * q^97 + 5516 * q^99

Introduction

L-functions

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