LMFDB - Embedded newform 1560.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,4,Mod(1,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1560.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1560.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:	$92.0429796090$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 112x^{2} - 28x + 1648 $ x^4 - x^3 - 112x^2 - 28x + 1648
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.89553$ of defining polynomial
Character	$\chi$	$=$	1560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} -27.9085 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -5.00000 q^{5} -27.9085 q^{7} +9.00000 q^{9} +33.8687 q^{11} -13.0000 q^{13} +15.0000 q^{15} -9.09757 q^{17} +12.6556 q^{19} +83.7256 q^{21} +47.3845 q^{23} +25.0000 q^{25} -27.0000 q^{27} -40.2765 q^{29} -71.6331 q^{31} -101.606 q^{33} +139.543 q^{35} +198.929 q^{37} +39.0000 q^{39} -9.14953 q^{41} +216.030 q^{43} -45.0000 q^{45} +498.302 q^{47} +435.886 q^{49} +27.2927 q^{51} +274.355 q^{53} -169.344 q^{55} -37.9668 q^{57} +372.775 q^{59} -494.715 q^{61} -251.177 q^{63} +65.0000 q^{65} -987.605 q^{67} -142.153 q^{69} +927.897 q^{71} -655.763 q^{73} -75.0000 q^{75} -945.226 q^{77} +664.001 q^{79} +81.0000 q^{81} -797.823 q^{83} +45.4878 q^{85} +120.830 q^{87} +1424.03 q^{89} +362.811 q^{91} +214.899 q^{93} -63.2780 q^{95} -1078.05 q^{97} +304.819 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9	$ 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9} + 35 q^{11} - 52 q^{13} + 60 q^{15} - 9 q^{17} + 32 q^{19} + 33 q^{21} - 149 q^{23} + 100 q^{25} - 108 q^{27} + 168 q^{29} + 280 q^{31} - 105 q^{33} + 55 q^{35} + 37 q^{37} + 156 q^{39} + 247 q^{41} + 132 q^{43} - 180 q^{45} + 217 q^{49} + 27 q^{51} + 247 q^{53} - 175 q^{55} - 96 q^{57} + 614 q^{59} + 719 q^{61} - 99 q^{63} + 260 q^{65} + 658 q^{67} + 447 q^{69} + 939 q^{71} - 1452 q^{73} - 300 q^{75} + 199 q^{77} + 289 q^{79} + 324 q^{81} - 118 q^{83} + 45 q^{85} - 504 q^{87} - 1319 q^{89} + 143 q^{91} - 840 q^{93} - 160 q^{95} - 993 q^{97} + 315 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9 + 35 * q^11 - 52 * q^13 + 60 * q^15 - 9 * q^17 + 32 * q^19 + 33 * q^21 - 149 * q^23 + 100 * q^25 - 108 * q^27 + 168 * q^29 + 280 * q^31 - 105 * q^33 + 55 * q^35 + 37 * q^37 + 156 * q^39 + 247 * q^41 + 132 * q^43 - 180 * q^45 + 217 * q^49 + 27 * q^51 + 247 * q^53 - 175 * q^55 - 96 * q^57 + 614 * q^59 + 719 * q^61 - 99 * q^63 + 260 * q^65 + 658 * q^67 + 447 * q^69 + 939 * q^71 - 1452 * q^73 - 300 * q^75 + 199 * q^77 + 289 * q^79 + 324 * q^81 - 118 * q^83 + 45 * q^85 - 504 * q^87 - 1319 * q^89 + 143 * q^91 - 840 * q^93 - 160 * q^95 - 993 * q^97 + 315 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−27.9085	−1.50692	−0.753459	−	0.657495i	$-0.771616\pi$
$7$	−27.9085	−1.50692	−0.753459	+	0.657495i	$0.771616\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	33.8687	0.928346	0.464173	−	0.885745i	$-0.346352\pi$
$11$	33.8687	0.928346	0.464173	+	0.885745i	$0.346352\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−9.09757	−0.129793	−0.0648966	−	0.997892i	$-0.520672\pi$
$17$	−9.09757	−0.129793	−0.0648966	+	0.997892i	$0.520672\pi$
$18$	0	0
$19$	12.6556	0.152810	0.0764051	−	0.997077i	$-0.475656\pi$
$19$	12.6556	0.152810	0.0764051	+	0.997077i	$0.475656\pi$
$20$	0	0
$21$	83.7256	0.870020
$22$	0	0
$23$	47.3845	0.429580	0.214790	−	0.976660i	$-0.431093\pi$
$23$	47.3845	0.429580	0.214790	+	0.976660i	$0.431093\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−40.2765	−0.257902	−0.128951	−	0.991651i	$-0.541161\pi$
$29$	−40.2765	−0.257902	−0.128951	+	0.991651i	$0.541161\pi$
$30$	0	0
$31$	−71.6331	−0.415022	−0.207511	−	0.978233i	$-0.566536\pi$
$31$	−71.6331	−0.415022	−0.207511	+	0.978233i	$0.566536\pi$
$32$	0	0
$33$	−101.606	−0.535981
$34$	0	0
$35$	139.543	0.673914
$36$	0	0
$37$	198.929	0.883883	0.441941	−	0.897044i	$-0.354290\pi$
$37$	198.929	0.883883	0.441941	+	0.897044i	$0.354290\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−9.14953	−0.0348516	−0.0174258	−	0.999848i	$-0.505547\pi$
$41$	−9.14953	−0.0348516	−0.0174258	+	0.999848i	$0.505547\pi$
$42$	0	0
$43$	216.030	0.766146	0.383073	−	0.923718i	$-0.374866\pi$
$43$	216.030	0.766146	0.383073	+	0.923718i	$0.374866\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	498.302	1.54649	0.773243	−	0.634110i	$-0.218633\pi$
$47$	498.302	1.54649	0.773243	+	0.634110i	$0.218633\pi$
$48$	0	0
$49$	435.886	1.27080
$50$	0	0
$51$	27.2927	0.0749361
$52$	0	0
$53$	274.355	0.711049	0.355525	−	0.934667i	$-0.384302\pi$
$53$	274.355	0.711049	0.355525	+	0.934667i	$0.384302\pi$
$54$	0	0
$55$	−169.344	−0.415169
$56$	0	0
$57$	−37.9668	−0.0882250
$58$	0	0
$59$	372.775	0.822561	0.411281	−	0.911509i	$-0.365082\pi$
$59$	372.775	0.822561	0.411281	+	0.911509i	$0.365082\pi$
$60$	0	0
$61$	−494.715	−1.03839	−0.519195	−	0.854656i	$-0.673768\pi$
$61$	−494.715	−1.03839	−0.519195	+	0.854656i	$0.673768\pi$
$62$	0	0
$63$	−251.177	−0.502306
$64$	0	0
$65$	65.0000	0.124035
$66$	0	0
$67$	−987.605	−1.80082	−0.900412	−	0.435038i	$-0.856735\pi$
$67$	−987.605	−1.80082	−0.900412	+	0.435038i	$0.856735\pi$
$68$	0	0
$69$	−142.153	−0.248018
$70$	0	0
$71$	927.897	1.55100	0.775501	−	0.631347i	$-0.217498\pi$
$71$	927.897	1.55100	0.775501	+	0.631347i	$0.217498\pi$
$72$	0	0
$73$	−655.763	−1.05139	−0.525693	−	0.850674i	$-0.676194\pi$
$73$	−655.763	−1.05139	−0.525693	+	0.850674i	$0.676194\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−945.226	−1.39894
$78$	0	0
$79$	664.001	0.945645	0.472823	−	0.881158i	$-0.343235\pi$
$79$	664.001	0.945645	0.472823	+	0.881158i	$0.343235\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−797.823	−1.05509	−0.527545	−	0.849527i	$-0.676887\pi$
$83$	−797.823	−1.05509	−0.527545	+	0.849527i	$0.676887\pi$
$84$	0	0
$85$	45.4878	0.0580453
$86$	0	0
$87$	120.830	0.148900
$88$	0	0
$89$	1424.03	1.69604	0.848019	−	0.529967i	$-0.177796\pi$
$89$	1424.03	1.69604	0.848019	+	0.529967i	$0.177796\pi$
$90$	0	0
$91$	362.811	0.417944
$92$	0	0
$93$	214.899	0.239613
$94$	0	0
$95$	−63.2780	−0.0683388
$96$	0	0
$97$	−1078.05	−1.12845	−0.564223	−	0.825622i	$-0.690824\pi$
$97$	−1078.05	−1.12845	−0.564223	+	0.825622i	$0.690824\pi$
$98$	0	0
$99$	304.819	0.309449
$100$	0	0
$101$	−1033.96	−1.01864	−0.509321	−	0.860577i	$-0.670103\pi$
$101$	−1033.96	−1.01864	−0.509321	+	0.860577i	$0.670103\pi$
$102$	0	0
$103$	869.166	0.831471	0.415735	−	0.909486i	$-0.363524\pi$
$103$	869.166	0.831471	0.415735	+	0.909486i	$0.363524\pi$
$104$	0	0
$105$	−418.628	−0.389085
$106$	0	0
$107$	−354.582	−0.320362	−0.160181	−	0.987088i	$-0.551208\pi$
$107$	−354.582	−0.320362	−0.160181	+	0.987088i	$0.551208\pi$
$108$	0	0
$109$	−66.0402	−0.0580322	−0.0290161	−	0.999579i	$-0.509237\pi$
$109$	−66.0402	−0.0580322	−0.0290161	+	0.999579i	$0.509237\pi$
$110$	0	0
$111$	−596.786	−0.510310
$112$	0	0
$113$	−2123.52	−1.76782	−0.883910	−	0.467657i	$-0.845098\pi$
$113$	−2123.52	−1.76782	−0.883910	+	0.467657i	$0.845098\pi$
$114$	0	0
$115$	−236.922	−0.192114
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	253.900	0.195588
$120$	0	0
$121$	−183.909	−0.138173
$122$	0	0
$123$	27.4486	0.0201216
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1560.31	−1.09020	−0.545098	−	0.838372i	$-0.683508\pi$
$127$	−1560.31	−1.09020	−0.545098	+	0.838372i	$0.683508\pi$
$128$	0	0
$129$	−648.090	−0.442334
$130$	0	0
$131$	1985.41	1.32417	0.662084	−	0.749430i	$-0.269672\pi$
$131$	1985.41	1.32417	0.662084	+	0.749430i	$0.269672\pi$
$132$	0	0
$133$	−353.199	−0.230273
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−1656.02	−1.03273	−0.516364	−	0.856369i	$-0.672715\pi$
$137$	−1656.02	−1.03273	−0.516364	+	0.856369i	$0.672715\pi$
$138$	0	0
$139$	−1358.43	−0.828927	−0.414463	−	0.910066i	$-0.636031\pi$
$139$	−1358.43	−0.828927	−0.414463	+	0.910066i	$0.636031\pi$
$140$	0	0
$141$	−1494.91	−0.892864
$142$	0	0
$143$	−440.294	−0.257477
$144$	0	0
$145$	201.383	0.115337
$146$	0	0
$147$	−1307.66	−0.733699
$148$	0	0
$149$	−879.294	−0.483454	−0.241727	−	0.970344i	$-0.577714\pi$
$149$	−879.294	−0.483454	−0.241727	+	0.970344i	$0.577714\pi$
$150$	0	0
$151$	−1139.51	−0.614120	−0.307060	−	0.951690i	$-0.599345\pi$
$151$	−1139.51	−0.614120	−0.307060	+	0.951690i	$0.599345\pi$
$152$	0	0
$153$	−81.8781	−0.0432644
$154$	0	0
$155$	358.165	0.185603
$156$	0	0
$157$	933.193	0.474375	0.237188	−	0.971464i	$-0.423774\pi$
$157$	933.193	0.474375	0.237188	+	0.971464i	$0.423774\pi$
$158$	0	0
$159$	−823.066	−0.410524
$160$	0	0
$161$	−1322.43	−0.647342
$162$	0	0
$163$	−1479.84	−0.711103	−0.355552	−	0.934657i	$-0.615707\pi$
$163$	−1479.84	−0.711103	−0.355552	+	0.934657i	$0.615707\pi$
$164$	0	0
$165$	508.031	0.239698
$166$	0	0
$167$	1530.61	0.709234	0.354617	−	0.935012i	$-0.384611\pi$
$167$	1530.61	0.709234	0.354617	+	0.935012i	$0.384611\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	113.900	0.0509367
$172$	0	0
$173$	1108.98	0.487365	0.243683	−	0.969855i	$-0.421644\pi$
$173$	1108.98	0.487365	0.243683	+	0.969855i	$0.421644\pi$
$174$	0	0
$175$	−697.713	−0.301384
$176$	0	0
$177$	−1118.32	−0.474906
$178$	0	0
$179$	−3002.06	−1.25354	−0.626772	−	0.779203i	$-0.715624\pi$
$179$	−3002.06	−1.25354	−0.626772	+	0.779203i	$0.715624\pi$
$180$	0	0
$181$	3832.02	1.57366	0.786828	−	0.617173i	$-0.211722\pi$
$181$	3832.02	1.57366	0.786828	+	0.617173i	$0.211722\pi$
$182$	0	0
$183$	1484.15	0.599515
$184$	0	0
$185$	−994.643	−0.395284
$186$	0	0
$187$	−308.123	−0.120493
$188$	0	0
$189$	753.530	0.290007
$190$	0	0
$191$	−2158.44	−0.817694	−0.408847	−	0.912603i	$-0.634069\pi$
$191$	−2158.44	−0.817694	−0.408847	+	0.912603i	$0.634069\pi$
$192$	0	0
$193$	−3146.52	−1.17353	−0.586766	−	0.809756i	$-0.699599\pi$
$193$	−3146.52	−1.17353	−0.586766	+	0.809756i	$0.699599\pi$
$194$	0	0
$195$	−195.000	−0.0716115
$196$	0	0
$197$	−2463.68	−0.891016	−0.445508	−	0.895278i	$-0.646977\pi$
$197$	−2463.68	−0.891016	−0.445508	+	0.895278i	$0.646977\pi$
$198$	0	0
$199$	4312.14	1.53608	0.768039	−	0.640404i	$-0.221233\pi$
$199$	4312.14	1.53608	0.768039	+	0.640404i	$0.221233\pi$
$200$	0	0
$201$	2962.82	1.03971
$202$	0	0
$203$	1124.06	0.388637
$204$	0	0
$205$	45.7477	0.0155861
$206$	0	0
$207$	426.460	0.143193
$208$	0	0
$209$	428.629	0.141861
$210$	0	0
$211$	−1869.98	−0.610116	−0.305058	−	0.952334i	$-0.598676\pi$
$211$	−1869.98	−0.610116	−0.305058	+	0.952334i	$0.598676\pi$
$212$	0	0
$213$	−2783.69	−0.895471
$214$	0	0
$215$	−1080.15	−0.342631
$216$	0	0
$217$	1999.17	0.625404
$218$	0	0
$219$	1967.29	0.607018
$220$	0	0
$221$	118.268	0.0359982
$222$	0	0
$223$	3502.51	1.05177	0.525886	−	0.850555i	$-0.323734\pi$
$223$	3502.51	1.05177	0.525886	+	0.850555i	$0.323734\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−2471.96	−0.722775	−0.361387	−	0.932416i	$-0.617697\pi$
$227$	−2471.96	−0.722775	−0.361387	+	0.932416i	$0.617697\pi$
$228$	0	0
$229$	4620.12	1.33322	0.666608	−	0.745409i	$-0.267746\pi$
$229$	4620.12	1.33322	0.666608	+	0.745409i	$0.267746\pi$
$230$	0	0
$231$	2835.68	0.807680
$232$	0	0
$233$	−6770.60	−1.90368	−0.951838	−	0.306600i	$-0.900809\pi$
$233$	−6770.60	−1.90368	−0.951838	+	0.306600i	$0.900809\pi$
$234$	0	0
$235$	−2491.51	−0.691610
$236$	0	0
$237$	−1992.00	−0.545968
$238$	0	0
$239$	1273.99	0.344801	0.172401	−	0.985027i	$-0.444848\pi$
$239$	1273.99	0.344801	0.172401	+	0.985027i	$0.444848\pi$
$240$	0	0
$241$	−294.684	−0.0787645	−0.0393823	−	0.999224i	$-0.512539\pi$
$241$	−294.684	−0.0787645	−0.0393823	+	0.999224i	$0.512539\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−2179.43	−0.568321
$246$	0	0
$247$	−164.523	−0.0423819
$248$	0	0
$249$	2393.47	0.609157
$250$	0	0
$251$	−3551.21	−0.893029	−0.446515	−	0.894776i	$-0.647335\pi$
$251$	−3551.21	−0.893029	−0.446515	+	0.894776i	$0.647335\pi$
$252$	0	0
$253$	1604.85	0.398799
$254$	0	0
$255$	−136.464	−0.0335125
$256$	0	0
$257$	−2738.23	−0.664614	−0.332307	−	0.943171i	$-0.607827\pi$
$257$	−2738.23	−0.664614	−0.332307	+	0.943171i	$0.607827\pi$
$258$	0	0
$259$	−5551.80	−1.33194
$260$	0	0
$261$	−362.489	−0.0859674
$262$	0	0
$263$	−1259.35	−0.295267	−0.147633	−	0.989042i	$-0.547166\pi$
$263$	−1259.35	−0.295267	−0.147633	+	0.989042i	$0.547166\pi$
$264$	0	0
$265$	−1371.78	−0.317991
$266$	0	0
$267$	−4272.10	−0.979207
$268$	0	0
$269$	−1897.23	−0.430022	−0.215011	−	0.976612i	$-0.568979\pi$
$269$	−1897.23	−0.430022	−0.215011	+	0.976612i	$0.568979\pi$
$270$	0	0
$271$	−3001.63	−0.672827	−0.336413	−	0.941714i	$-0.609214\pi$
$271$	−3001.63	−0.672827	−0.336413	+	0.941714i	$0.609214\pi$
$272$	0	0
$273$	−1088.43	−0.241300
$274$	0	0
$275$	846.718	0.185669
$276$	0	0
$277$	−186.791	−0.0405168	−0.0202584	−	0.999795i	$-0.506449\pi$
$277$	−186.791	−0.0405168	−0.0202584	+	0.999795i	$0.506449\pi$
$278$	0	0
$279$	−644.698	−0.138341
$280$	0	0
$281$	−5284.49	−1.12187	−0.560936	−	0.827859i	$-0.689559\pi$
$281$	−5284.49	−1.12187	−0.560936	+	0.827859i	$0.689559\pi$
$282$	0	0
$283$	2579.86	0.541896	0.270948	−	0.962594i	$-0.412663\pi$
$283$	2579.86	0.541896	0.270948	+	0.962594i	$0.412663\pi$
$284$	0	0
$285$	189.834	0.0394554
$286$	0	0
$287$	255.350	0.0525186
$288$	0	0
$289$	−4830.23	−0.983154
$290$	0	0
$291$	3234.15	0.651509
$292$	0	0
$293$	−6341.90	−1.26450	−0.632248	−	0.774766i	$-0.717868\pi$
$293$	−6341.90	−1.26450	−0.632248	+	0.774766i	$0.717868\pi$
$294$	0	0
$295$	−1863.87	−0.367861
$296$	0	0
$297$	−914.456	−0.178660
$298$	0	0
$299$	−615.998	−0.119144
$300$	0	0
$301$	−6029.08	−1.15452
$302$	0	0
$303$	3101.88	0.588113
$304$	0	0
$305$	2473.58	0.464382
$306$	0	0
$307$	1045.81	0.194421	0.0972106	−	0.995264i	$-0.469008\pi$
$307$	1045.81	0.194421	0.0972106	+	0.995264i	$0.469008\pi$
$308$	0	0
$309$	−2607.50	−0.480050
$310$	0	0
$311$	4101.16	0.747766	0.373883	−	0.927476i	$-0.378026\pi$
$311$	4101.16	0.747766	0.373883	+	0.927476i	$0.378026\pi$
$312$	0	0
$313$	387.677	0.0700089	0.0350045	−	0.999387i	$-0.488855\pi$
$313$	387.677	0.0700089	0.0350045	+	0.999387i	$0.488855\pi$
$314$	0	0
$315$	1255.88	0.224638
$316$	0	0
$317$	−1587.69	−0.281305	−0.140653	−	0.990059i	$-0.544920\pi$
$317$	−1587.69	−0.281305	−0.140653	+	0.990059i	$0.544920\pi$
$318$	0	0
$319$	−1364.11	−0.239422
$320$	0	0
$321$	1063.75	0.184961
$322$	0	0
$323$	−115.135	−0.0198337
$324$	0	0
$325$	−325.000	−0.0554700
$326$	0	0
$327$	198.121	0.0335049
$328$	0	0
$329$	−13906.9	−2.33043
$330$	0	0
$331$	2593.37	0.430648	0.215324	−	0.976543i	$-0.430919\pi$
$331$	2593.37	0.430648	0.215324	+	0.976543i	$0.430919\pi$
$332$	0	0
$333$	1790.36	0.294628
$334$	0	0
$335$	4938.03	0.805353
$336$	0	0
$337$	−3597.89	−0.581571	−0.290786	−	0.956788i	$-0.593917\pi$
$337$	−3597.89	−0.581571	−0.290786	+	0.956788i	$0.593917\pi$
$338$	0	0
$339$	6370.55	1.02065
$340$	0	0
$341$	−2426.12	−0.385284
$342$	0	0
$343$	−2592.30	−0.408079
$344$	0	0
$345$	710.767	0.110917
$346$	0	0
$347$	−2583.61	−0.399699	−0.199849	−	0.979827i	$-0.564045\pi$
$347$	−2583.61	−0.399699	−0.199849	+	0.979827i	$0.564045\pi$
$348$	0	0
$349$	6563.25	1.00666	0.503328	−	0.864095i	$-0.332109\pi$
$349$	6563.25	1.00666	0.503328	+	0.864095i	$0.332109\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	−8342.25	−1.25783	−0.628914	−	0.777475i	$-0.716500\pi$
$353$	−8342.25	−1.25783	−0.628914	+	0.777475i	$0.716500\pi$
$354$	0	0
$355$	−4639.48	−0.693629
$356$	0	0
$357$	−761.699	−0.112923
$358$	0	0
$359$	−3719.91	−0.546878	−0.273439	−	0.961889i	$-0.588161\pi$
$359$	−3719.91	−0.546878	−0.273439	+	0.961889i	$0.588161\pi$
$360$	0	0
$361$	−6698.84	−0.976649
$362$	0	0
$363$	551.726	0.0797744
$364$	0	0
$365$	3278.81	0.470194
$366$	0	0
$367$	6821.06	0.970181	0.485091	−	0.874464i	$-0.338787\pi$
$367$	6821.06	0.970181	0.485091	+	0.874464i	$0.338787\pi$
$368$	0	0
$369$	−82.3458	−0.0116172
$370$	0	0
$371$	−7656.85	−1.07149
$372$	0	0
$373$	−8613.76	−1.19572	−0.597860	−	0.801600i	$-0.703982\pi$
$373$	−8613.76	−1.19572	−0.597860	+	0.801600i	$0.703982\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	523.595	0.0715292
$378$	0	0
$379$	11666.0	1.58111	0.790556	−	0.612390i	$-0.209792\pi$
$379$	11666.0	1.58111	0.790556	+	0.612390i	$0.209792\pi$
$380$	0	0
$381$	4680.92	0.629425
$382$	0	0
$383$	−5197.39	−0.693405	−0.346702	−	0.937975i	$-0.612699\pi$
$383$	−5197.39	−0.693405	−0.346702	+	0.937975i	$0.612699\pi$
$384$	0	0
$385$	4726.13	0.625626
$386$	0	0
$387$	1944.27	0.255382
$388$	0	0
$389$	79.8475	0.0104073	0.00520364	−	0.999986i	$-0.498344\pi$
$389$	79.8475	0.0104073	0.00520364	+	0.999986i	$0.498344\pi$
$390$	0	0
$391$	−431.083	−0.0557566
$392$	0	0
$393$	−5956.22	−0.764508
$394$	0	0
$395$	−3320.01	−0.422905
$396$	0	0
$397$	983.486	0.124332	0.0621659	−	0.998066i	$-0.480199\pi$
$397$	983.486	0.124332	0.0621659	+	0.998066i	$0.480199\pi$
$398$	0	0
$399$	1059.60	0.132948
$400$	0	0
$401$	6944.01	0.864757	0.432378	−	0.901692i	$-0.357675\pi$
$401$	6944.01	0.864757	0.432378	+	0.901692i	$0.357675\pi$
$402$	0	0
$403$	931.230	0.115106
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	6737.46	0.820549
$408$	0	0
$409$	638.585	0.0772030	0.0386015	−	0.999255i	$-0.487710\pi$
$409$	638.585	0.0772030	0.0386015	+	0.999255i	$0.487710\pi$
$410$	0	0
$411$	4968.07	0.596245
$412$	0	0
$413$	−10403.6	−1.23953
$414$	0	0
$415$	3989.12	0.471851
$416$	0	0
$417$	4075.30	0.478581
$418$	0	0
$419$	141.915	0.0165465	0.00827327	−	0.999966i	$-0.497367\pi$
$419$	141.915	0.0165465	0.00827327	+	0.999966i	$0.497367\pi$
$420$	0	0
$421$	−14145.3	−1.63753	−0.818765	−	0.574129i	$-0.805341\pi$
$421$	−14145.3	−1.63753	−0.818765	+	0.574129i	$0.805341\pi$
$422$	0	0
$423$	4484.72	0.515495
$424$	0	0
$425$	−227.439	−0.0259586
$426$	0	0
$427$	13806.8	1.56477
$428$	0	0
$429$	1320.88	0.148654
$430$	0	0
$431$	14042.2	1.56935	0.784673	−	0.619910i	$-0.212831\pi$
$431$	14042.2	1.56935	0.784673	+	0.619910i	$0.212831\pi$
$432$	0	0
$433$	−5333.35	−0.591927	−0.295964	−	0.955199i	$-0.595641\pi$
$433$	−5333.35	−0.591927	−0.295964	+	0.955199i	$0.595641\pi$
$434$	0	0
$435$	−604.148	−0.0665900
$436$	0	0
$437$	599.679	0.0656442
$438$	0	0
$439$	−16325.3	−1.77487	−0.887433	−	0.460938i	$-0.847513\pi$
$439$	−16325.3	−1.77487	−0.887433	+	0.460938i	$0.847513\pi$
$440$	0	0
$441$	3922.97	0.423601
$442$	0	0
$443$	5377.29	0.576711	0.288355	−	0.957523i	$-0.406892\pi$
$443$	5377.29	0.576711	0.288355	+	0.957523i	$0.406892\pi$
$444$	0	0
$445$	−7120.17	−0.758491
$446$	0	0
$447$	2637.88	0.279122
$448$	0	0
$449$	5756.01	0.604996	0.302498	−	0.953150i	$-0.402180\pi$
$449$	5756.01	0.604996	0.302498	+	0.953150i	$0.402180\pi$
$450$	0	0
$451$	−309.883	−0.0323544
$452$	0	0
$453$	3418.54	0.354562
$454$	0	0
$455$	−1814.05	−0.186910
$456$	0	0
$457$	8111.75	0.830310	0.415155	−	0.909751i	$-0.363727\pi$
$457$	8111.75	0.830310	0.415155	+	0.909751i	$0.363727\pi$
$458$	0	0
$459$	245.634	0.0249787
$460$	0	0
$461$	8956.85	0.904907	0.452454	−	0.891788i	$-0.350549\pi$
$461$	8956.85	0.904907	0.452454	+	0.891788i	$0.350549\pi$
$462$	0	0
$463$	1626.66	0.163277	0.0816385	−	0.996662i	$-0.473985\pi$
$463$	1626.66	0.163277	0.0816385	+	0.996662i	$0.473985\pi$
$464$	0	0
$465$	−1074.50	−0.107158
$466$	0	0
$467$	10170.2	1.00775	0.503875	−	0.863777i	$-0.331907\pi$
$467$	10170.2	1.00775	0.503875	+	0.863777i	$0.331907\pi$
$468$	0	0
$469$	27562.6	2.71370
$470$	0	0
$471$	−2799.58	−0.273881
$472$	0	0
$473$	7316.66	0.711248
$474$	0	0
$475$	316.390	0.0305620
$476$	0	0
$477$	2469.20	0.237016
$478$	0	0
$479$	−7585.43	−0.723564	−0.361782	−	0.932263i	$-0.617832\pi$
$479$	−7585.43	−0.723564	−0.361782	+	0.932263i	$0.617832\pi$
$480$	0	0
$481$	−2586.07	−0.245145
$482$	0	0
$483$	3967.29	0.373743
$484$	0	0
$485$	5390.25	0.504657
$486$	0	0
$487$	9087.89	0.845609	0.422804	−	0.906221i	$-0.361046\pi$
$487$	9087.89	0.845609	0.422804	+	0.906221i	$0.361046\pi$
$488$	0	0
$489$	4439.51	0.410556
$490$	0	0
$491$	−841.579	−0.0773522	−0.0386761	−	0.999252i	$-0.512314\pi$
$491$	−841.579	−0.0773522	−0.0386761	+	0.999252i	$0.512314\pi$
$492$	0	0
$493$	366.418	0.0334739
$494$	0	0
$495$	−1524.09	−0.138390
$496$	0	0
$497$	−25896.2	−2.33723
$498$	0	0
$499$	2869.68	0.257444	0.128722	−	0.991681i	$-0.458913\pi$
$499$	2869.68	0.257444	0.128722	+	0.991681i	$0.458913\pi$
$500$	0	0
$501$	−4591.83	−0.409476
$502$	0	0
$503$	−9301.78	−0.824545	−0.412272	−	0.911061i	$-0.635265\pi$
$503$	−9301.78	−0.824545	−0.412272	+	0.911061i	$0.635265\pi$
$504$	0	0
$505$	5169.79	0.455550
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−3708.23	−0.322916	−0.161458	−	0.986880i	$-0.551620\pi$
$509$	−3708.23	−0.322916	−0.161458	+	0.986880i	$0.551620\pi$
$510$	0	0
$511$	18301.4	1.58435
$512$	0	0
$513$	−341.701	−0.0294083
$514$	0	0
$515$	−4345.83	−0.371845
$516$	0	0
$517$	16876.9	1.43567
$518$	0	0
$519$	−3326.94	−0.281380
$520$	0	0
$521$	−3873.32	−0.325707	−0.162853	−	0.986650i	$-0.552070\pi$
$521$	−3873.32	−0.325707	−0.162853	+	0.986650i	$0.552070\pi$
$522$	0	0
$523$	−9375.36	−0.783854	−0.391927	−	0.919996i	$-0.628191\pi$
$523$	−9375.36	−0.783854	−0.391927	+	0.919996i	$0.628191\pi$
$524$	0	0
$525$	2093.14	0.174004
$526$	0	0
$527$	651.687	0.0538670
$528$	0	0
$529$	−9921.71	−0.815461
$530$	0	0
$531$	3354.97	0.274187
$532$	0	0
$533$	118.944	0.00966611
$534$	0	0
$535$	1772.91	0.143270
$536$	0	0
$537$	9006.18	0.723734
$538$	0	0
$539$	14762.9	1.17975
$540$	0	0
$541$	−2382.60	−0.189346	−0.0946728	−	0.995508i	$-0.530181\pi$
$541$	−2382.60	−0.189346	−0.0946728	+	0.995508i	$0.530181\pi$
$542$	0	0
$543$	−11496.1	−0.908551
$544$	0	0
$545$	330.201	0.0259528
$546$	0	0
$547$	11772.5	0.920213	0.460106	−	0.887864i	$-0.347811\pi$
$547$	11772.5	0.920213	0.460106	+	0.887864i	$0.347811\pi$
$548$	0	0
$549$	−4452.44	−0.346130
$550$	0	0
$551$	−509.724	−0.0394101
$552$	0	0
$553$	−18531.3	−1.42501
$554$	0	0
$555$	2983.93	0.228218
$556$	0	0
$557$	−8398.84	−0.638905	−0.319453	−	0.947602i	$-0.603499\pi$
$557$	−8398.84	−0.638905	−0.319453	+	0.947602i	$0.603499\pi$
$558$	0	0
$559$	−2808.39	−0.212491
$560$	0	0
$561$	924.369	0.0695667
$562$	0	0
$563$	−481.383	−0.0360353	−0.0180176	−	0.999838i	$-0.505736\pi$
$563$	−481.383	−0.0360353	−0.0180176	+	0.999838i	$0.505736\pi$
$564$	0	0
$565$	10617.6	0.790593
$566$	0	0
$567$	−2260.59	−0.167435
$568$	0	0
$569$	11200.1	0.825187	0.412594	−	0.910915i	$-0.364623\pi$
$569$	11200.1	0.825187	0.412594	+	0.910915i	$0.364623\pi$
$570$	0	0
$571$	−7995.96	−0.586026	−0.293013	−	0.956109i	$-0.594658\pi$
$571$	−7995.96	−0.586026	−0.293013	+	0.956109i	$0.594658\pi$
$572$	0	0
$573$	6475.33	0.472096
$574$	0	0
$575$	1184.61	0.0859160
$576$	0	0
$577$	8538.77	0.616073	0.308036	−	0.951375i	$-0.400328\pi$
$577$	8538.77	0.616073	0.308036	+	0.951375i	$0.400328\pi$
$578$	0	0
$579$	9439.57	0.677539
$580$	0	0
$581$	22266.1	1.58994
$582$	0	0
$583$	9292.07	0.660100
$584$	0	0
$585$	585.000	0.0413449
$586$	0	0
$587$	25795.0	1.81375	0.906875	−	0.421399i	$-0.138461\pi$
$587$	25795.0	1.81375	0.906875	+	0.421399i	$0.138461\pi$
$588$	0	0
$589$	−906.560	−0.0634196
$590$	0	0
$591$	7391.05	0.514428
$592$	0	0
$593$	1733.01	0.120010	0.0600051	−	0.998198i	$-0.480888\pi$
$593$	1733.01	0.120010	0.0600051	+	0.998198i	$0.480888\pi$
$594$	0	0
$595$	−1269.50	−0.0874695
$596$	0	0
$597$	−12936.4	−0.886855
$598$	0	0
$599$	−23206.4	−1.58295	−0.791476	−	0.611200i	$-0.790687\pi$
$599$	−23206.4	−1.58295	−0.791476	+	0.611200i	$0.790687\pi$
$600$	0	0
$601$	8125.27	0.551475	0.275738	−	0.961233i	$-0.411078\pi$
$601$	8125.27	0.551475	0.275738	+	0.961233i	$0.411078\pi$
$602$	0	0
$603$	−8888.45	−0.600275
$604$	0	0
$605$	919.544	0.0617930
$606$	0	0
$607$	−9000.82	−0.601865	−0.300932	−	0.953645i	$-0.597298\pi$
$607$	−9000.82	−0.601865	−0.300932	+	0.953645i	$0.597298\pi$
$608$	0	0
$609$	−3372.17	−0.224380
$610$	0	0
$611$	−6477.93	−0.428918
$612$	0	0
$613$	−9269.56	−0.610757	−0.305378	−	0.952231i	$-0.598783\pi$
$613$	−9269.56	−0.610757	−0.305378	+	0.952231i	$0.598783\pi$
$614$	0	0
$615$	−137.243	−0.00899865
$616$	0	0
$617$	22577.2	1.47314	0.736568	−	0.676364i	$-0.236445\pi$
$617$	22577.2	1.47314	0.736568	+	0.676364i	$0.236445\pi$
$618$	0	0
$619$	20438.1	1.32710	0.663550	−	0.748132i	$-0.269049\pi$
$619$	20438.1	1.32710	0.663550	+	0.748132i	$0.269049\pi$
$620$	0	0
$621$	−1279.38	−0.0826727
$622$	0	0
$623$	−39742.7	−2.55579
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−1285.89	−0.0819034
$628$	0	0
$629$	−1809.77	−0.114722
$630$	0	0
$631$	−14650.8	−0.924312	−0.462156	−	0.886799i	$-0.652924\pi$
$631$	−14650.8	−0.924312	−0.462156	+	0.886799i	$0.652924\pi$
$632$	0	0
$633$	5609.93	0.352251
$634$	0	0
$635$	7801.54	0.487550
$636$	0	0
$637$	−5666.51	−0.352457
$638$	0	0
$639$	8351.07	0.517000
$640$	0	0
$641$	−24943.1	−1.53696	−0.768482	−	0.639871i	$-0.778988\pi$
$641$	−24943.1	−1.53696	−0.768482	+	0.639871i	$0.778988\pi$
$642$	0	0
$643$	−9791.99	−0.600557	−0.300279	−	0.953852i	$-0.597080\pi$
$643$	−9791.99	−0.600557	−0.300279	+	0.953852i	$0.597080\pi$
$644$	0	0
$645$	3240.45	0.197818
$646$	0	0
$647$	−12786.4	−0.776950	−0.388475	−	0.921459i	$-0.626998\pi$
$647$	−12786.4	−0.776950	−0.388475	+	0.921459i	$0.626998\pi$
$648$	0	0
$649$	12625.4	0.763621
$650$	0	0
$651$	−5997.52	−0.361077
$652$	0	0
$653$	18037.8	1.08097	0.540486	−	0.841353i	$-0.318240\pi$
$653$	18037.8	1.08097	0.540486	+	0.841353i	$0.318240\pi$
$654$	0	0
$655$	−9927.04	−0.592185
$656$	0	0
$657$	−5901.86	−0.350462
$658$	0	0
$659$	16362.6	0.967220	0.483610	−	0.875283i	$-0.339325\pi$
$659$	16362.6	0.967220	0.483610	+	0.875283i	$0.339325\pi$
$660$	0	0
$661$	14078.3	0.828413	0.414207	−	0.910183i	$-0.364059\pi$
$661$	14078.3	0.828413	0.414207	+	0.910183i	$0.364059\pi$
$662$	0	0
$663$	−354.805	−0.0207835
$664$	0	0
$665$	1766.00	0.102981
$666$	0	0
$667$	−1908.48	−0.110790
$668$	0	0
$669$	−10507.5	−0.607241
$670$	0	0
$671$	−16755.4	−0.963986
$672$	0	0
$673$	17100.5	0.979456	0.489728	−	0.871875i	$-0.337096\pi$
$673$	17100.5	0.979456	0.489728	+	0.871875i	$0.337096\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−12886.5	−0.731563	−0.365781	−	0.930701i	$-0.619198\pi$
$677$	−12886.5	−0.731563	−0.365781	+	0.930701i	$0.619198\pi$
$678$	0	0
$679$	30086.8	1.70048
$680$	0	0
$681$	7415.88	0.417294
$682$	0	0
$683$	−15041.3	−0.842661	−0.421331	−	0.906907i	$-0.638437\pi$
$683$	−15041.3	−0.842661	−0.421331	+	0.906907i	$0.638437\pi$
$684$	0	0
$685$	8280.12	0.461850
$686$	0	0
$687$	−13860.4	−0.769732
$688$	0	0
$689$	−3566.62	−0.197210
$690$	0	0
$691$	7851.56	0.432254	0.216127	−	0.976365i	$-0.430657\pi$
$691$	7851.56	0.432254	0.216127	+	0.976365i	$0.430657\pi$
$692$	0	0
$693$	−8507.04	−0.466314
$694$	0	0
$695$	6792.17	0.370707
$696$	0	0
$697$	83.2385	0.00452351
$698$	0	0
$699$	20311.8	1.09909
$700$	0	0
$701$	−30214.3	−1.62793	−0.813964	−	0.580915i	$-0.802695\pi$
$701$	−30214.3	−1.62793	−0.813964	+	0.580915i	$0.802695\pi$
$702$	0	0
$703$	2517.56	0.135066
$704$	0	0
$705$	7474.53	0.399301
$706$	0	0
$707$	28856.3	1.53501
$708$	0	0
$709$	5795.96	0.307013	0.153506	−	0.988148i	$-0.450943\pi$
$709$	5795.96	0.307013	0.153506	+	0.988148i	$0.450943\pi$
$710$	0	0
$711$	5976.01	0.315215
$712$	0	0
$713$	−3394.30	−0.178285
$714$	0	0
$715$	2201.47	0.115147
$716$	0	0
$717$	−3821.97	−0.199071
$718$	0	0
$719$	24818.8	1.28732	0.643661	−	0.765311i	$-0.277415\pi$
$719$	24818.8	1.28732	0.643661	+	0.765311i	$0.277415\pi$
$720$	0	0
$721$	−24257.1	−1.25296
$722$	0	0
$723$	884.052	0.0454747
$724$	0	0
$725$	−1006.91	−0.0515804
$726$	0	0
$727$	25691.4	1.31065	0.655324	−	0.755348i	$-0.272532\pi$
$727$	25691.4	1.31065	0.655324	+	0.755348i	$0.272532\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1965.35	−0.0994405
$732$	0	0
$733$	15971.9	0.804822	0.402411	−	0.915459i	$-0.368172\pi$
$733$	15971.9	0.804822	0.402411	+	0.915459i	$0.368172\pi$
$734$	0	0
$735$	6538.28	0.328120
$736$	0	0
$737$	−33448.9	−1.67179
$738$	0	0
$739$	39293.1	1.95591	0.977957	−	0.208806i	$-0.0669579\pi$
$739$	39293.1	1.95591	0.977957	+	0.208806i	$0.0669579\pi$
$740$	0	0
$741$	493.568	0.0244692
$742$	0	0
$743$	−1163.02	−0.0574252	−0.0287126	−	0.999588i	$-0.509141\pi$
$743$	−1163.02	−0.0574252	−0.0287126	+	0.999588i	$0.509141\pi$
$744$	0	0
$745$	4396.47	0.216207
$746$	0	0
$747$	−7180.41	−0.351697
$748$	0	0
$749$	9895.87	0.482760
$750$	0	0
$751$	−14430.8	−0.701182	−0.350591	−	0.936529i	$-0.614019\pi$
$751$	−14430.8	−0.701182	−0.350591	+	0.936529i	$0.614019\pi$
$752$	0	0
$753$	10653.6	0.515591
$754$	0	0
$755$	5697.56	0.274643
$756$	0	0
$757$	9375.28	0.450133	0.225066	−	0.974343i	$-0.427740\pi$
$757$	9375.28	0.450133	0.225066	+	0.974343i	$0.427740\pi$
$758$	0	0
$759$	−4814.55	−0.230247
$760$	0	0
$761$	15587.9	0.742524	0.371262	−	0.928528i	$-0.378925\pi$
$761$	15587.9	0.742524	0.371262	+	0.928528i	$0.378925\pi$
$762$	0	0
$763$	1843.09	0.0874498
$764$	0	0
$765$	409.391	0.0193484
$766$	0	0
$767$	−4846.07	−0.228137
$768$	0	0
$769$	−38230.1	−1.79274	−0.896368	−	0.443311i	$-0.853804\pi$
$769$	−38230.1	−1.79274	−0.896368	+	0.443311i	$0.853804\pi$
$770$	0	0
$771$	8214.68	0.383715
$772$	0	0
$773$	−8638.60	−0.401952	−0.200976	−	0.979596i	$-0.564411\pi$
$773$	−8638.60	−0.401952	−0.200976	+	0.979596i	$0.564411\pi$
$774$	0	0
$775$	−1790.83	−0.0830044
$776$	0	0
$777$	16655.4	0.768995
$778$	0	0
$779$	−115.793	−0.00532569
$780$	0	0
$781$	31426.7	1.43987
$782$	0	0
$783$	1087.47	0.0496333
$784$	0	0
$785$	−4665.97	−0.212147
$786$	0	0
$787$	2257.85	0.102267	0.0511333	−	0.998692i	$-0.483717\pi$
$787$	2257.85	0.102267	0.0511333	+	0.998692i	$0.483717\pi$
$788$	0	0
$789$	3778.06	0.170472
$790$	0	0
$791$	59264.2	2.66396
$792$	0	0
$793$	6431.30	0.287998
$794$	0	0
$795$	4115.33	0.183592
$796$	0	0
$797$	−11665.5	−0.518462	−0.259231	−	0.965815i	$-0.583469\pi$
$797$	−11665.5	−0.518462	−0.259231	+	0.965815i	$0.583469\pi$
$798$	0	0
$799$	−4533.34	−0.200723
$800$	0	0
$801$	12816.3	0.565346
$802$	0	0
$803$	−22209.9	−0.976051
$804$	0	0
$805$	6612.15	0.289500
$806$	0	0
$807$	5691.68	0.248274
$808$	0	0
$809$	24785.1	1.07713	0.538564	−	0.842585i	$-0.318967\pi$
$809$	24785.1	1.07713	0.538564	+	0.842585i	$0.318967\pi$
$810$	0	0
$811$	−11625.5	−0.503363	−0.251681	−	0.967810i	$-0.580983\pi$
$811$	−11625.5	−0.503363	−0.251681	+	0.967810i	$0.580983\pi$
$812$	0	0
$813$	9004.89	0.388457
$814$	0	0
$815$	7399.19	0.318015
$816$	0	0
$817$	2733.99	0.117075
$818$	0	0
$819$	3265.30	0.139315
$820$	0	0
$821$	−31824.7	−1.35285	−0.676425	−	0.736511i	$-0.736472\pi$
$821$	−31824.7	−1.35285	−0.676425	+	0.736511i	$0.736472\pi$
$822$	0	0
$823$	−41778.5	−1.76951	−0.884755	−	0.466056i	$-0.845674\pi$
$823$	−41778.5	−1.76951	−0.884755	+	0.466056i	$0.845674\pi$
$824$	0	0
$825$	−2540.16	−0.107196
$826$	0	0
$827$	−35090.2	−1.47546	−0.737729	−	0.675096i	$-0.764102\pi$
$827$	−35090.2	−1.47546	−0.737729	+	0.675096i	$0.764102\pi$
$828$	0	0
$829$	27248.7	1.14160	0.570801	−	0.821089i	$-0.306633\pi$
$829$	27248.7	1.14160	0.570801	+	0.821089i	$0.306633\pi$
$830$	0	0
$831$	560.372	0.0233924
$832$	0	0
$833$	−3965.50	−0.164942
$834$	0	0
$835$	−7653.04	−0.317179
$836$	0	0
$837$	1934.09	0.0798710
$838$	0	0
$839$	−3425.50	−0.140955	−0.0704775	−	0.997513i	$-0.522452\pi$
$839$	−3425.50	−0.140955	−0.0704775	+	0.997513i	$0.522452\pi$
$840$	0	0
$841$	−22766.8	−0.933487
$842$	0	0
$843$	15853.5	0.647713
$844$	0	0
$845$	−845.000	−0.0344010
$846$	0	0
$847$	5132.62	0.208216
$848$	0	0
$849$	−7739.58	−0.312864
$850$	0	0
$851$	9426.12	0.379698
$852$	0	0
$853$	−5910.04	−0.237229	−0.118614	−	0.992940i	$-0.537845\pi$
$853$	−5910.04	−0.237229	−0.118614	+	0.992940i	$0.537845\pi$
$854$	0	0
$855$	−569.502	−0.0227796
$856$	0	0
$857$	−27211.0	−1.08461	−0.542304	−	0.840183i	$-0.682448\pi$
$857$	−27211.0	−1.08461	−0.542304	+	0.840183i	$0.682448\pi$
$858$	0	0
$859$	−6909.17	−0.274433	−0.137216	−	0.990541i	$-0.543816\pi$
$859$	−6909.17	−0.274433	−0.137216	+	0.990541i	$0.543816\pi$
$860$	0	0
$861$	−766.050	−0.0303216
$862$	0	0
$863$	22136.2	0.873148	0.436574	−	0.899668i	$-0.356192\pi$
$863$	22136.2	0.873148	0.436574	+	0.899668i	$0.356192\pi$
$864$	0	0
$865$	−5544.90	−0.217956
$866$	0	0
$867$	14490.7	0.567624
$868$	0	0
$869$	22488.9	0.877886
$870$	0	0
$871$	12838.9	0.499459
$872$	0	0
$873$	−9702.44	−0.376149
$874$	0	0
$875$	3488.57	0.134783
$876$	0	0
$877$	2319.08	0.0892929	0.0446464	−	0.999003i	$-0.485784\pi$
$877$	2319.08	0.0892929	0.0446464	+	0.999003i	$0.485784\pi$
$878$	0	0
$879$	19025.7	0.730058
$880$	0	0
$881$	−10839.3	−0.414512	−0.207256	−	0.978287i	$-0.566453\pi$
$881$	−10839.3	−0.414512	−0.207256	+	0.978287i	$0.566453\pi$
$882$	0	0
$883$	31352.0	1.19488	0.597441	−	0.801913i	$-0.296184\pi$
$883$	31352.0	1.19488	0.597441	+	0.801913i	$0.296184\pi$
$884$	0	0
$885$	5591.62	0.212384
$886$	0	0
$887$	−2757.05	−0.104366	−0.0521830	−	0.998638i	$-0.516618\pi$
$887$	−2757.05	−0.104366	−0.0521830	+	0.998638i	$0.516618\pi$
$888$	0	0
$889$	43545.9	1.64284
$890$	0	0
$891$	2743.37	0.103150
$892$	0	0
$893$	6306.31	0.236319
$894$	0	0
$895$	15010.3	0.560602
$896$	0	0
$897$	1847.99	0.0687879
$898$	0	0
$899$	2885.13	0.107035
$900$	0	0
$901$	−2495.97	−0.0922894
$902$	0	0
$903$	18087.2	0.666562
$904$	0	0
$905$	−19160.1	−0.703760
$906$	0	0
$907$	38763.3	1.41909	0.709544	−	0.704661i	$-0.248901\pi$
$907$	38763.3	1.41909	0.709544	+	0.704661i	$0.248901\pi$
$908$	0	0
$909$	−9305.63	−0.339547
$910$	0	0
$911$	−35865.5	−1.30437	−0.652183	−	0.758061i	$-0.726147\pi$
$911$	−35865.5	−1.30437	−0.652183	+	0.758061i	$0.726147\pi$
$912$	0	0
$913$	−27021.3	−0.979489
$914$	0	0
$915$	−7420.73	−0.268111
$916$	0	0
$917$	−55409.8	−1.99541
$918$	0	0
$919$	5294.07	0.190028	0.0950138	−	0.995476i	$-0.469710\pi$
$919$	5294.07	0.190028	0.0950138	+	0.995476i	$0.469710\pi$
$920$	0	0
$921$	−3137.42	−0.112249
$922$	0	0
$923$	−12062.7	−0.430170
$924$	0	0
$925$	4973.21	0.176777
$926$	0	0
$927$	7822.49	0.277157
$928$	0	0
$929$	−19177.5	−0.677281	−0.338640	−	0.940916i	$-0.609967\pi$
$929$	−19177.5	−0.677281	−0.338640	+	0.940916i	$0.609967\pi$
$930$	0	0
$931$	5516.39	0.194192
$932$	0	0
$933$	−12303.5	−0.431723
$934$	0	0
$935$	1540.62	0.0538861
$936$	0	0
$937$	−27310.9	−0.952197	−0.476098	−	0.879392i	$-0.657949\pi$
$937$	−27310.9	−0.952197	−0.476098	+	0.879392i	$0.657949\pi$
$938$	0	0
$939$	−1163.03	−0.0404197
$940$	0	0
$941$	2411.64	0.0835466	0.0417733	−	0.999127i	$-0.486699\pi$
$941$	2411.64	0.0835466	0.0417733	+	0.999127i	$0.486699\pi$
$942$	0	0
$943$	−433.546	−0.0149716
$944$	0	0
$945$	−3767.65	−0.129695
$946$	0	0
$947$	−12225.3	−0.419502	−0.209751	−	0.977755i	$-0.567265\pi$
$947$	−12225.3	−0.419502	−0.209751	+	0.977755i	$0.567265\pi$
$948$	0	0
$949$	8524.91	0.291602
$950$	0	0
$951$	4763.08	0.162412
$952$	0	0
$953$	6048.00	0.205576	0.102788	−	0.994703i	$-0.467224\pi$
$953$	6048.00	0.205576	0.102788	+	0.994703i	$0.467224\pi$
$954$	0	0
$955$	10792.2	0.365684
$956$	0	0
$957$	4092.34	0.138231
$958$	0	0
$959$	46217.2	1.55624
$960$	0	0
$961$	−24659.7	−0.827757
$962$	0	0
$963$	−3191.24	−0.106787
$964$	0	0
$965$	15732.6	0.524820
$966$	0	0
$967$	−45754.7	−1.52159	−0.760793	−	0.648995i	$-0.775190\pi$
$967$	−45754.7	−1.52159	−0.760793	+	0.648995i	$0.775190\pi$
$968$	0	0
$969$	345.406	0.0114510
$970$	0	0
$971$	9716.29	0.321123	0.160562	−	0.987026i	$-0.448670\pi$
$971$	9716.29	0.321123	0.160562	+	0.987026i	$0.448670\pi$
$972$	0	0
$973$	37911.9	1.24912
$974$	0	0
$975$	975.000	0.0320256
$976$	0	0
$977$	−59456.6	−1.94697	−0.973483	−	0.228758i	$-0.926533\pi$
$977$	−59456.6	−1.94697	−0.973483	+	0.228758i	$0.926533\pi$
$978$	0	0
$979$	48230.2	1.57451
$980$	0	0
$981$	−594.362	−0.0193441
$982$	0	0
$983$	−30970.0	−1.00487	−0.502437	−	0.864614i	$-0.667563\pi$
$983$	−30970.0	−1.00487	−0.502437	+	0.864614i	$0.667563\pi$
$984$	0	0
$985$	12318.4	0.398474
$986$	0	0
$987$	41720.6	1.34547
$988$	0	0
$989$	10236.5	0.329121
$990$	0	0
$991$	1027.75	0.0329439	0.0164720	−	0.999864i	$-0.494757\pi$
$991$	1027.75	0.0329439	0.0164720	+	0.999864i	$0.494757\pi$
$992$	0	0
$993$	−7780.11	−0.248635
$994$	0	0
$995$	−21560.7	−0.686955
$996$	0	0
$997$	39012.9	1.23927	0.619635	−	0.784890i	$-0.287281\pi$
$997$	39012.9	1.23927	0.619635	+	0.784890i	$0.287281\pi$
$998$	0	0
$999$	−5371.07	−0.170103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.4.a.l.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.a.l.1.1	✓	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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1560.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} -27.9085 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -5.00000 q^{5} -27.9085 q^{7} +9.00000 q^{9} +33.8687 q^{11} -13.0000 q^{13} +15.0000 q^{15} -9.09757 q^{17} +12.6556 q^{19} +83.7256 q^{21} +47.3845 q^{23} +25.0000 q^{25} -27.0000 q^{27} -40.2765 q^{29} -71.6331 q^{31} -101.606 q^{33} +139.543 q^{35} +198.929 q^{37} +39.0000 q^{39} -9.14953 q^{41} +216.030 q^{43} -45.0000 q^{45} +498.302 q^{47} +435.886 q^{49} +27.2927 q^{51} +274.355 q^{53} -169.344 q^{55} -37.9668 q^{57} +372.775 q^{59} -494.715 q^{61} -251.177 q^{63} +65.0000 q^{65} -987.605 q^{67} -142.153 q^{69} +927.897 q^{71} -655.763 q^{73} -75.0000 q^{75} -945.226 q^{77} +664.001 q^{79} +81.0000 q^{81} -797.823 q^{83} +45.4878 q^{85} +120.830 q^{87} +1424.03 q^{89} +362.811 q^{91} +214.899 q^{93} -63.2780 q^{95} -1078.05 q^{97} +304.819 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9	\( 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9} + 35 q^{11} - 52 q^{13} + 60 q^{15} - 9 q^{17} + 32 q^{19} + 33 q^{21} - 149 q^{23} + 100 q^{25} - 108 q^{27} + 168 q^{29} + 280 q^{31} - 105 q^{33} + 55 q^{35} + 37 q^{37} + 156 q^{39} + 247 q^{41} + 132 q^{43} - 180 q^{45} + 217 q^{49} + 27 q^{51} + 247 q^{53} - 175 q^{55} - 96 q^{57} + 614 q^{59} + 719 q^{61} - 99 q^{63} + 260 q^{65} + 658 q^{67} + 447 q^{69} + 939 q^{71} - 1452 q^{73} - 300 q^{75} + 199 q^{77} + 289 q^{79} + 324 q^{81} - 118 q^{83} + 45 q^{85} - 504 q^{87} - 1319 q^{89} + 143 q^{91} - 840 q^{93} - 160 q^{95} - 993 q^{97} + 315 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9 + 35 * q^11 - 52 * q^13 + 60 * q^15 - 9 * q^17 + 32 * q^19 + 33 * q^21 - 149 * q^23 + 100 * q^25 - 108 * q^27 + 168 * q^29 + 280 * q^31 - 105 * q^33 + 55 * q^35 + 37 * q^37 + 156 * q^39 + 247 * q^41 + 132 * q^43 - 180 * q^45 + 217 * q^49 + 27 * q^51 + 247 * q^53 - 175 * q^55 - 96 * q^57 + 614 * q^59 + 719 * q^61 - 99 * q^63 + 260 * q^65 + 658 * q^67 + 447 * q^69 + 939 * q^71 - 1452 * q^73 - 300 * q^75 + 199 * q^77 + 289 * q^79 + 324 * q^81 - 118 * q^83 + 45 * q^85 - 504 * q^87 - 1319 * q^89 + 143 * q^91 - 840 * q^93 - 160 * q^95 - 993 * q^97 + 315 * q^99

Introduction

L-functions

Modular forms

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Database

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