LMFDB - Embedded newform 160.6.d.a.81.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(81,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("160.81");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	160.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 $ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{90}\cdot 3^{4}\cdot 5^{12} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			81.14
Root			$3.72553 + 1.45618i$ of defining polynomial
Character	$\chi$	$=$	160.81
Dual form			160.6.d.a.81.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.8240i q^{3} +25.0000i q^{5} -163.706 q^{7} +125.841 q^{9} +321.520i q^{11} -128.246i q^{13} -270.600 q^{15} -2110.72 q^{17} -1454.37i q^{19} -1771.96i q^{21} +1231.18 q^{23} -625.000 q^{25} +3992.34i q^{27} -4073.19i q^{29} +3956.03 q^{31} -3480.14 q^{33} -4092.65i q^{35} -10656.6i q^{37} +1388.13 q^{39} -5907.19 q^{41} -16439.6i q^{43} +3146.02i q^{45} -23238.8 q^{47} +9992.68 q^{49} -22846.5i q^{51} -30634.0i q^{53} -8038.01 q^{55} +15742.1 q^{57} -25262.4i q^{59} +39115.5i q^{61} -20600.9 q^{63} +3206.14 q^{65} +20894.5i q^{67} +13326.3i q^{69} -13889.1 q^{71} -43451.2 q^{73} -6765.01i q^{75} -52634.8i q^{77} +12546.4 q^{79} -12633.8 q^{81} -6680.84i q^{83} -52768.0i q^{85} +44088.2 q^{87} -90400.9 q^{89} +20994.6i q^{91} +42820.2i q^{93} +36359.2 q^{95} +149616. q^{97} +40460.4i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63}+ \cdots + 147376 q^{97}+O(q^{100}) $ 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	10.8240i	0.694361i	0.937798	+	0.347180i	$0.112861\pi$
$3$	10.8240i	0.694361i	−0.937798	+	0.347180i	$0.887139\pi$
$4$	0	0
$5$	25.0000i	0.447214i
$5$	25.0000i	0.447214i
$6$	0	0
$7$	−163.706	−1.26276	−0.631378	−	0.775475i	$-0.717510\pi$
$7$	−163.706	−1.26276	−0.631378	+	0.775475i	$0.717510\pi$
$8$	0	0
$9$	125.841	0.517863
$10$	0	0
$11$	321.520i	0.801174i	0.916259	+	0.400587i	$0.131194\pi$
$11$	321.520i	0.801174i	−0.916259	+	0.400587i	$0.868806\pi$
$12$	0	0
$13$	− 128.246i	− 0.210467i	−0.994448	−	0.105233i	$-0.966441\pi$
$13$	− 128.246i	− 0.210467i	0.994448	−	0.105233i	$-0.0335590\pi$
$14$	0	0
$15$	−270.600	−0.310528
$16$	0	0
$17$	−2110.72	−1.77137	−0.885683	−	0.464290i	$-0.846310\pi$
$17$	−2110.72	−1.77137	−0.885683	+	0.464290i	$0.846310\pi$
$18$	0	0
$19$	− 1454.37i	− 0.924252i	−0.886814	−	0.462126i	$-0.847087\pi$
$19$	− 1454.37i	− 0.924252i	0.886814	−	0.462126i	$-0.152913\pi$
$20$	0	0
$21$	− 1771.96i	− 0.876809i
$22$	0	0
$23$	1231.18	0.485289	0.242645	−	0.970115i	$-0.421985\pi$
$23$	1231.18	0.485289	0.242645	+	0.970115i	$0.421985\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	3992.34i	1.05394i
$28$	0	0
$29$	− 4073.19i	− 0.899372i	−0.893187	−	0.449686i	$-0.851536\pi$
$29$	− 4073.19i	− 0.899372i	0.893187	−	0.449686i	$-0.148464\pi$
$30$	0	0
$31$	3956.03	0.739360	0.369680	−	0.929159i	$-0.379467\pi$
$31$	3956.03	0.739360	0.369680	+	0.929159i	$0.379467\pi$
$32$	0	0
$33$	−3480.14	−0.556304
$34$	0	0
$35$	− 4092.65i	− 0.564722i
$36$	0	0
$37$	− 10656.6i	− 1.27972i	−0.768490	−	0.639861i	$-0.778992\pi$
$37$	− 10656.6i	− 1.27972i	0.768490	−	0.639861i	$-0.221008\pi$
$38$	0	0
$39$	1388.13	0.146140
$40$	0	0
$41$	−5907.19	−0.548809	−0.274404	−	0.961614i	$-0.588481\pi$
$41$	−5907.19	−0.548809	−0.274404	+	0.961614i	$0.588481\pi$
$42$	0	0
$43$	− 16439.6i	− 1.35588i	−0.735119	−	0.677938i	$-0.762874\pi$
$43$	− 16439.6i	− 1.35588i	0.735119	−	0.677938i	$-0.237126\pi$
$44$	0	0
$45$	3146.02i	0.231595i
$46$	0	0
$47$	−23238.8	−1.53450	−0.767252	−	0.641345i	$-0.778377\pi$
$47$	−23238.8	−1.53450	−0.767252	+	0.641345i	$0.778377\pi$
$48$	0	0
$49$	9992.68	0.594555
$50$	0	0
$51$	− 22846.5i	− 1.22997i
$52$	0	0
$53$	− 30634.0i	− 1.49801i	−0.662567	−	0.749003i	$-0.730533\pi$
$53$	− 30634.0i	− 1.49801i	0.662567	−	0.749003i	$-0.269467\pi$
$54$	0	0
$55$	−8038.01	−0.358296
$56$	0	0
$57$	15742.1	0.641764
$58$	0	0
$59$	− 25262.4i	− 0.944810i	−0.881382	−	0.472405i	$-0.843386\pi$
$59$	− 25262.4i	− 0.944810i	0.881382	−	0.472405i	$-0.156614\pi$
$60$	0	0
$61$	39115.5i	1.34594i	0.739672	+	0.672968i	$0.234981\pi$
$61$	39115.5i	1.34594i	−0.739672	+	0.672968i	$0.765019\pi$
$62$	0	0
$63$	−20600.9	−0.653935
$64$	0	0
$65$	3206.14	0.0941237
$66$	0	0
$67$	20894.5i	0.568651i	0.958728	+	0.284325i	$0.0917696\pi$
$67$	20894.5i	0.568651i	−0.958728	+	0.284325i	$0.908230\pi$
$68$	0	0
$69$	13326.3i	0.336966i
$70$	0	0
$71$	−13889.1	−0.326984	−0.163492	−	0.986545i	$-0.552276\pi$
$71$	−13889.1	−0.326984	−0.163492	+	0.986545i	$0.552276\pi$
$72$	0	0
$73$	−43451.2	−0.954321	−0.477161	−	0.878816i	$-0.658334\pi$
$73$	−43451.2	−0.954321	−0.477161	+	0.878816i	$0.658334\pi$
$74$	0	0
$75$	− 6765.01i	− 0.138872i
$76$	0	0
$77$	− 52634.8i	− 1.01169i
$78$	0	0
$79$	12546.4	0.226179	0.113089	−	0.993585i	$-0.463925\pi$
$79$	12546.4	0.226179	0.113089	+	0.993585i	$0.463925\pi$
$80$	0	0
$81$	−12633.8	−0.213955
$82$	0	0
$83$	− 6680.84i	− 0.106448i	−0.998583	−	0.0532238i	$-0.983050\pi$
$83$	− 6680.84i	− 0.106448i	0.998583	−	0.0532238i	$-0.0169497\pi$
$84$	0	0
$85$	− 52768.0i	− 0.792179i
$86$	0	0
$87$	44088.2	0.624488
$88$	0	0
$89$	−90400.9	−1.20976	−0.604878	−	0.796318i	$-0.706778\pi$
$89$	−90400.9	−1.20976	−0.604878	+	0.796318i	$0.706778\pi$
$90$	0	0
$91$	20994.6i	0.265769i
$92$	0	0
$93$	42820.2i	0.513382i
$94$	0	0
$95$	36359.2	0.413338
$96$	0	0
$97$	149616.	1.61454	0.807269	−	0.590184i	$-0.200945\pi$
$97$	149616.	1.61454	0.807269	+	0.590184i	$0.200945\pi$
$98$	0	0
$99$	40460.4i	0.414898i
$100$	0	0
$101$	114822.i	1.12001i	0.828491	+	0.560003i	$0.189200\pi$
$101$	114822.i	1.12001i	−0.828491	+	0.560003i	$0.810800\pi$
$102$	0	0
$103$	−38586.9	−0.358382	−0.179191	−	0.983814i	$-0.557348\pi$
$103$	−38586.9	−0.358382	−0.179191	+	0.983814i	$0.557348\pi$
$104$	0	0
$105$	44298.9	0.392121
$106$	0	0
$107$	189459.i	1.59976i	0.600158	+	0.799881i	$0.295104\pi$
$107$	189459.i	1.59976i	−0.600158	+	0.799881i	$0.704896\pi$
$108$	0	0
$109$	24392.6i	0.196649i	0.995154	+	0.0983245i	$0.0313483\pi$
$109$	24392.6i	0.196649i	−0.995154	+	0.0983245i	$0.968652\pi$
$110$	0	0
$111$	115348.	0.888589
$112$	0	0
$113$	−52918.6	−0.389863	−0.194932	−	0.980817i	$-0.562448\pi$
$113$	−52918.6	−0.389863	−0.194932	+	0.980817i	$0.562448\pi$
$114$	0	0
$115$	30779.4i	0.217028i
$116$	0	0
$117$	− 16138.5i	− 0.108993i
$118$	0	0
$119$	345538.	2.23681
$120$	0	0
$121$	57675.7	0.358120
$122$	0	0
$123$	− 63939.5i	− 0.381071i
$124$	0	0
$125$	− 15625.0i	− 0.0894427i
$126$	0	0
$127$	−293650.	−1.61555	−0.807777	−	0.589489i	$-0.799329\pi$
$127$	−293650.	−1.61555	−0.807777	+	0.589489i	$0.799329\pi$
$128$	0	0
$129$	177942.	0.941467
$130$	0	0
$131$	317738.i	1.61767i	0.588032	+	0.808837i	$0.299903\pi$
$131$	317738.i	1.61767i	−0.588032	+	0.808837i	$0.700097\pi$
$132$	0	0
$133$	238089.i	1.16711i
$134$	0	0
$135$	−99808.4	−0.471338
$136$	0	0
$137$	−162285.	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$137$	−162285.	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$138$	0	0
$139$	− 306759.i	− 1.34667i	−0.739338	−	0.673334i	$-0.764861\pi$
$139$	− 306759.i	− 1.34667i	0.739338	−	0.673334i	$-0.235139\pi$
$140$	0	0
$141$	− 251537.i	− 1.06550i
$142$	0	0
$143$	41233.5	0.168621
$144$	0	0
$145$	101830.	0.402211
$146$	0	0
$147$	108161.i	0.412835i
$148$	0	0
$149$	368107.i	1.35834i	0.733981	+	0.679170i	$0.237660\pi$
$149$	368107.i	1.35834i	−0.733981	+	0.679170i	$0.762340\pi$
$150$	0	0
$151$	−336822.	−1.20215	−0.601075	−	0.799193i	$-0.705261\pi$
$151$	−336822.	−1.20215	−0.601075	+	0.799193i	$0.705261\pi$
$152$	0	0
$153$	−265615.	−0.917326
$154$	0	0
$155$	98900.9i	0.330652i
$156$	0	0
$157$	271253.i	0.878264i	0.898423	+	0.439132i	$0.144714\pi$
$157$	271253.i	0.878264i	−0.898423	+	0.439132i	$0.855286\pi$
$158$	0	0
$159$	331582.	1.04016
$160$	0	0
$161$	−201551.	−0.612802
$162$	0	0
$163$	121354.i	0.357756i	0.983871	+	0.178878i	$0.0572467\pi$
$163$	121354.i	0.357756i	−0.983871	+	0.178878i	$0.942753\pi$
$164$	0	0
$165$	− 87003.5i	− 0.248787i
$166$	0	0
$167$	−56095.4	−0.155645	−0.0778227	−	0.996967i	$-0.524797\pi$
$167$	−56095.4	−0.155645	−0.0778227	+	0.996967i	$0.524797\pi$
$168$	0	0
$169$	354846.	0.955704
$170$	0	0
$171$	− 183019.i	− 0.478636i
$172$	0	0
$173$	167666.i	0.425921i	0.977061	+	0.212960i	$0.0683105\pi$
$173$	167666.i	0.425921i	−0.977061	+	0.212960i	$0.931689\pi$
$174$	0	0
$175$	102316.	0.252551
$176$	0	0
$177$	273440.	0.656039
$178$	0	0
$179$	− 535921.i	− 1.25017i	−0.780558	−	0.625083i	$-0.785065\pi$
$179$	− 535921.i	− 1.25017i	0.780558	−	0.625083i	$-0.214935\pi$
$180$	0	0
$181$	− 113446.i	− 0.257391i	−0.991684	−	0.128696i	$-0.958921\pi$
$181$	− 113446.i	− 0.257391i	0.991684	−	0.128696i	$-0.0410790\pi$
$182$	0	0
$183$	−423387.	−0.934565
$184$	0	0
$185$	266416.	0.572309
$186$	0	0
$187$	− 678640.i	− 1.41917i
$188$	0	0
$189$	− 653570.i	− 1.33088i
$190$	0	0
$191$	−391808.	−0.777124	−0.388562	−	0.921423i	$-0.627028\pi$
$191$	−391808.	−0.777124	−0.388562	+	0.921423i	$0.627028\pi$
$192$	0	0
$193$	−261513.	−0.505359	−0.252680	−	0.967550i	$-0.581312\pi$
$193$	−261513.	−0.505359	−0.252680	+	0.967550i	$0.581312\pi$
$194$	0	0
$195$	34703.3i	0.0653558i
$196$	0	0
$197$	− 898947.i	− 1.65032i	−0.564898	−	0.825161i	$-0.691084\pi$
$197$	− 898947.i	− 1.65032i	0.564898	−	0.825161i	$-0.308916\pi$
$198$	0	0
$199$	83719.3	0.149862	0.0749312	−	0.997189i	$-0.476126\pi$
$199$	83719.3	0.149862	0.0749312	+	0.997189i	$0.476126\pi$
$200$	0	0
$201$	−226163.	−0.394849
$202$	0	0
$203$	666805.i	1.13569i
$204$	0	0
$205$	− 147680.i	− 0.245435i
$206$	0	0
$207$	154932.	0.251313
$208$	0	0
$209$	467609.	0.740487
$210$	0	0
$211$	− 553641.i	− 0.856095i	−0.903756	−	0.428047i	$-0.859202\pi$
$211$	− 553641.i	− 0.856095i	0.903756	−	0.428047i	$-0.140798\pi$
$212$	0	0
$213$	− 150335.i	− 0.227045i
$214$	0	0
$215$	410990.	0.606366
$216$	0	0
$217$	−647627.	−0.933632
$218$	0	0
$219$	− 470316.i	− 0.662643i
$220$	0	0
$221$	270691.i	0.372814i
$222$	0	0
$223$	−279400.	−0.376239	−0.188120	−	0.982146i	$-0.560239\pi$
$223$	−279400.	−0.376239	−0.188120	+	0.982146i	$0.560239\pi$
$224$	0	0
$225$	−78650.5	−0.103573
$226$	0	0
$227$	593068.i	0.763905i	0.924182	+	0.381953i	$0.124748\pi$
$227$	593068.i	0.763905i	−0.924182	+	0.381953i	$0.875252\pi$
$228$	0	0
$229$	− 927873.i	− 1.16923i	−0.811311	−	0.584615i	$-0.801246\pi$
$229$	− 927873.i	− 1.16923i	0.811311	−	0.584615i	$-0.198754\pi$
$230$	0	0
$231$	569720.	0.702476
$232$	0	0
$233$	−1.09279e6	−1.31871	−0.659353	−	0.751833i	$-0.729170\pi$
$233$	−1.09279e6	−1.31871	−0.659353	+	0.751833i	$0.729170\pi$
$234$	0	0
$235$	− 580969.i	− 0.686251i
$236$	0	0
$237$	135803.i	0.157050i
$238$	0	0
$239$	797967.	0.903630	0.451815	−	0.892112i	$-0.350777\pi$
$239$	797967.	0.903630	0.451815	+	0.892112i	$0.350777\pi$
$240$	0	0
$241$	−1.61861e6	−1.79515	−0.897573	−	0.440865i	$-0.854672\pi$
$241$	−1.61861e6	−1.79515	−0.897573	+	0.440865i	$0.854672\pi$
$242$	0	0
$243$	833389.i	0.905383i
$244$	0	0
$245$	249817.i	0.265893i
$246$	0	0
$247$	−186516.	−0.194525
$248$	0	0
$249$	72313.5	0.0739130
$250$	0	0
$251$	− 449688.i	− 0.450533i	−0.974297	−	0.225266i	$-0.927675\pi$
$251$	− 449688.i	− 0.450533i	0.974297	−	0.225266i	$-0.0723253\pi$
$252$	0	0
$253$	395848.i	0.388801i
$254$	0	0
$255$	571162.	0.550058
$256$	0	0
$257$	396434.	0.374402	0.187201	−	0.982322i	$-0.440058\pi$
$257$	396434.	0.374402	0.187201	+	0.982322i	$0.440058\pi$
$258$	0	0
$259$	1.74456e6i	1.61598i
$260$	0	0
$261$	− 512573.i	− 0.465752i
$262$	0	0
$263$	−1.93423e6	−1.72432	−0.862160	−	0.506635i	$-0.830889\pi$
$263$	−1.93423e6	−1.72432	−0.862160	+	0.506635i	$0.830889\pi$
$264$	0	0
$265$	765849.	0.669928
$266$	0	0
$267$	− 978500.i	− 0.840007i
$268$	0	0
$269$	− 670016.i	− 0.564553i	−0.959333	−	0.282276i	$-0.908910\pi$
$269$	− 670016.i	− 0.564553i	0.959333	−	0.282276i	$-0.0910895\pi$
$270$	0	0
$271$	−2.08940e6	−1.72822	−0.864109	−	0.503305i	$-0.832117\pi$
$271$	−2.08940e6	−1.72822	−0.864109	+	0.503305i	$0.832117\pi$
$272$	0	0
$273$	−227246.	−0.184539
$274$	0	0
$275$	− 200950.i	− 0.160235i
$276$	0	0
$277$	828834.i	0.649035i	0.945880	+	0.324517i	$0.105202\pi$
$277$	828834.i	0.649035i	−0.945880	+	0.324517i	$0.894798\pi$
$278$	0	0
$279$	497830.	0.382887
$280$	0	0
$281$	2.39932e6	1.81268	0.906342	−	0.422545i	$-0.138863\pi$
$281$	2.39932e6	1.81268	0.906342	+	0.422545i	$0.138863\pi$
$282$	0	0
$283$	− 2.00868e6i	− 1.49089i	−0.666568	−	0.745444i	$-0.732237\pi$
$283$	− 2.00868e6i	− 1.49089i	0.666568	−	0.745444i	$-0.267763\pi$
$284$	0	0
$285$	393553.i	0.287006i
$286$	0	0
$287$	967042.	0.693012
$288$	0	0
$289$	3.03529e6	2.13774
$290$	0	0
$291$	1.61944e6i	1.12107i
$292$	0	0
$293$	1.74203e6i	1.18546i	0.805402	+	0.592729i	$0.201950\pi$
$293$	1.74203e6i	1.18546i	−0.805402	+	0.592729i	$0.798050\pi$
$294$	0	0
$295$	631560.	0.422532
$296$	0	0
$297$	−1.28362e6	−0.844393
$298$	0	0
$299$	− 157893.i	− 0.102137i
$300$	0	0
$301$	2.69126e6i	1.71214i
$302$	0	0
$303$	−1.24283e6	−0.777688
$304$	0	0
$305$	−977887.	−0.601921
$306$	0	0
$307$	978690.i	0.592651i	0.955087	+	0.296326i	$0.0957614\pi$
$307$	978690.i	0.592651i	−0.955087	+	0.296326i	$0.904239\pi$
$308$	0	0
$309$	− 417665.i	− 0.248847i
$310$	0	0
$311$	1.57652e6	0.924268	0.462134	−	0.886810i	$-0.347084\pi$
$311$	1.57652e6	0.924268	0.462134	+	0.886810i	$0.347084\pi$
$312$	0	0
$313$	1.60962e6	0.928670	0.464335	−	0.885660i	$-0.346293\pi$
$313$	1.60962e6	0.928670	0.464335	+	0.885660i	$0.346293\pi$
$314$	0	0
$315$	− 515022.i	− 0.292449i
$316$	0	0
$317$	1.50670e6i	0.842130i	0.907030	+	0.421065i	$0.138344\pi$
$317$	1.50670e6i	0.842130i	−0.907030	+	0.421065i	$0.861656\pi$
$318$	0	0
$319$	1.30961e6	0.720553
$320$	0	0
$321$	−2.05070e6	−1.11081
$322$	0	0
$323$	3.06977e6i	1.63719i
$324$	0	0
$325$	80153.5i	0.0420934i
$326$	0	0
$327$	−264026.	−0.136545
$328$	0	0
$329$	3.80433e6	1.93771
$330$	0	0
$331$	− 394453.i	− 0.197891i	−0.995093	−	0.0989454i	$-0.968453\pi$
$331$	− 394453.i	− 0.197891i	0.995093	−	0.0989454i	$-0.0315469\pi$
$332$	0	0
$333$	− 1.34104e6i	− 0.662721i
$334$	0	0
$335$	−522363.	−0.254308
$336$	0	0
$337$	−1.36879e6	−0.656543	−0.328272	−	0.944583i	$-0.606466\pi$
$337$	−1.36879e6	−0.656543	−0.328272	+	0.944583i	$0.606466\pi$
$338$	0	0
$339$	− 572791.i	− 0.270706i
$340$	0	0
$341$	1.27195e6i	0.592356i
$342$	0	0
$343$	1.11555e6	0.511979
$344$	0	0
$345$	−333157.	−0.150696
$346$	0	0
$347$	569376.i	0.253849i	0.991912	+	0.126925i	$0.0405106\pi$
$347$	569376.i	0.253849i	−0.991912	+	0.126925i	$0.959489\pi$
$348$	0	0
$349$	− 2.20053e6i	− 0.967081i	−0.875322	−	0.483541i	$-0.839351\pi$
$349$	− 2.20053e6i	− 0.967081i	0.875322	−	0.483541i	$-0.160649\pi$
$350$	0	0
$351$	511999.	0.221820
$352$	0	0
$353$	−2.51557e6	−1.07448	−0.537242	−	0.843428i	$-0.680534\pi$
$353$	−2.51557e6	−1.07448	−0.537242	+	0.843428i	$0.680534\pi$
$354$	0	0
$355$	− 347226.i	− 0.146232i
$356$	0	0
$357$	3.74011e6i	1.55315i
$358$	0	0
$359$	794808.	0.325481	0.162741	−	0.986669i	$-0.447967\pi$
$359$	794808.	0.325481	0.162741	+	0.986669i	$0.447967\pi$
$360$	0	0
$361$	360911.	0.145758
$362$	0	0
$363$	624282.i	0.248665i
$364$	0	0
$365$	− 1.08628e6i	− 0.426785i
$366$	0	0
$367$	874302.	0.338841	0.169421	−	0.985544i	$-0.445810\pi$
$367$	874302.	0.338841	0.169421	+	0.985544i	$0.445810\pi$
$368$	0	0
$369$	−743365.	−0.284208
$370$	0	0
$371$	5.01497e6i	1.89162i
$372$	0	0
$373$	− 4.86205e6i	− 1.80945i	−0.425994	−	0.904726i	$-0.640076\pi$
$373$	− 4.86205e6i	− 1.80945i	0.425994	−	0.904726i	$-0.359924\pi$
$374$	0	0
$375$	169125.	0.0621055
$376$	0	0
$377$	−522368.	−0.189288
$378$	0	0
$379$	− 1.24150e6i	− 0.443964i	−0.975051	−	0.221982i	$-0.928747\pi$
$379$	− 1.24150e6i	− 0.443964i	0.975051	−	0.221982i	$-0.0712527\pi$
$380$	0	0
$381$	− 3.17848e6i	− 1.12178i
$382$	0	0
$383$	30969.9	0.0107880	0.00539402	−	0.999985i	$-0.498283\pi$
$383$	30969.9	0.0107880	0.00539402	+	0.999985i	$0.498283\pi$
$384$	0	0
$385$	1.31587e6	0.452440
$386$	0	0
$387$	− 2.06877e6i	− 0.702158i
$388$	0	0
$389$	4.27860e6i	1.43360i	0.697279	+	0.716800i	$0.254394\pi$
$389$	4.27860e6i	1.43360i	−0.697279	+	0.716800i	$0.745606\pi$
$390$	0	0
$391$	−2.59867e6	−0.859626
$392$	0	0
$393$	−3.43920e6	−1.12325
$394$	0	0
$395$	313660.i	0.101150i
$396$	0	0
$397$	− 2.31119e6i	− 0.735968i	−0.929832	−	0.367984i	$-0.880048\pi$
$397$	− 2.31119e6i	− 0.735968i	0.929832	−	0.367984i	$-0.119952\pi$
$398$	0	0
$399$	−2.57708e6	−0.810392
$400$	0	0
$401$	996347.	0.309421	0.154710	−	0.987960i	$-0.450556\pi$
$401$	996347.	0.309421	0.154710	+	0.987960i	$0.450556\pi$
$402$	0	0
$403$	− 507344.i	− 0.155611i
$404$	0	0
$405$	− 315845.i	− 0.0956834i
$406$	0	0
$407$	3.42633e6	1.02528
$408$	0	0
$409$	−5.17948e6	−1.53101	−0.765505	−	0.643430i	$-0.777511\pi$
$409$	−5.17948e6	−1.53101	−0.765505	+	0.643430i	$0.777511\pi$
$410$	0	0
$411$	− 1.75658e6i	− 0.512936i
$412$	0	0
$413$	4.13561e6i	1.19306i
$414$	0	0
$415$	167021.	0.0476048
$416$	0	0
$417$	3.32037e6	0.935074
$418$	0	0
$419$	3.57698e6i	0.995363i	0.867360	+	0.497682i	$0.165815\pi$
$419$	3.57698e6i	0.995363i	−0.867360	+	0.497682i	$0.834185\pi$
$420$	0	0
$421$	− 2.15848e6i	− 0.593531i	−0.954950	−	0.296765i	$-0.904092\pi$
$421$	− 2.15848e6i	− 0.593531i	0.954950	−	0.296765i	$-0.0959080\pi$
$422$	0	0
$423$	−2.92438e6	−0.794664
$424$	0	0
$425$	1.31920e6	0.354273
$426$	0	0
$427$	− 6.40344e6i	− 1.69959i
$428$	0	0
$429$	446312.i	0.117084i
$430$	0	0
$431$	1.80890e6	0.469052	0.234526	−	0.972110i	$-0.424646\pi$
$431$	1.80890e6	0.469052	0.234526	+	0.972110i	$0.424646\pi$
$432$	0	0
$433$	1.28911e6	0.330423	0.165211	−	0.986258i	$-0.447169\pi$
$433$	1.28911e6	0.330423	0.165211	+	0.986258i	$0.447169\pi$
$434$	0	0
$435$	1.10221e6i	0.279280i
$436$	0	0
$437$	− 1.79058e6i	− 0.448530i
$438$	0	0
$439$	−1.89694e6	−0.469777	−0.234889	−	0.972022i	$-0.575473\pi$
$439$	−1.89694e6	−0.469777	−0.234889	+	0.972022i	$0.575473\pi$
$440$	0	0
$441$	1.25749e6	0.307898
$442$	0	0
$443$	− 6.01979e6i	− 1.45738i	−0.684845	−	0.728689i	$-0.740130\pi$
$443$	− 6.01979e6i	− 1.45738i	0.684845	−	0.728689i	$-0.259870\pi$
$444$	0	0
$445$	− 2.26002e6i	− 0.541019i
$446$	0	0
$447$	−3.98440e6	−0.943178
$448$	0	0
$449$	−2.40081e6	−0.562007	−0.281003	−	0.959707i	$-0.590667\pi$
$449$	−2.40081e6	−0.562007	−0.281003	+	0.959707i	$0.590667\pi$
$450$	0	0
$451$	− 1.89928e6i	− 0.439691i
$452$	0	0
$453$	− 3.64577e6i	− 0.834726i
$454$	0	0
$455$	−524864.	−0.118855
$456$	0	0
$457$	−858952.	−0.192388	−0.0961941	−	0.995363i	$-0.530667\pi$
$457$	−858952.	−0.192388	−0.0961941	+	0.995363i	$0.530667\pi$
$458$	0	0
$459$	− 8.42671e6i	− 1.86692i
$460$	0	0
$461$	− 2.33481e6i	− 0.511680i	−0.966719	−	0.255840i	$-0.917648\pi$
$461$	− 2.33481e6i	− 0.511680i	0.966719	−	0.255840i	$-0.0823521\pi$
$462$	0	0
$463$	1.35195e6	0.293096	0.146548	−	0.989204i	$-0.453184\pi$
$463$	1.35195e6	0.293096	0.146548	+	0.989204i	$0.453184\pi$
$464$	0	0
$465$	−1.07050e6	−0.229592
$466$	0	0
$467$	6.44013e6i	1.36648i	0.730195	+	0.683239i	$0.239429\pi$
$467$	6.44013e6i	1.36648i	−0.730195	+	0.683239i	$0.760571\pi$
$468$	0	0
$469$	− 3.42056e6i	− 0.718068i
$470$	0	0
$471$	−2.93604e6	−0.609832
$472$	0	0
$473$	5.28566e6	1.08629
$474$	0	0
$475$	908980.i	0.184850i
$476$	0	0
$477$	− 3.85500e6i	− 0.775762i
$478$	0	0
$479$	1.85358e6	0.369124	0.184562	−	0.982821i	$-0.440913\pi$
$479$	1.85358e6	0.369124	0.184562	+	0.982821i	$0.440913\pi$
$480$	0	0
$481$	−1.36667e6	−0.269339
$482$	0	0
$483$	− 2.18159e6i	− 0.425506i
$484$	0	0
$485$	3.74040e6i	0.722043i
$486$	0	0
$487$	−2.51668e6	−0.480846	−0.240423	−	0.970668i	$-0.577286\pi$
$487$	−2.51668e6	−0.480846	−0.240423	+	0.970668i	$0.577286\pi$
$488$	0	0
$489$	−1.31354e6	−0.248412
$490$	0	0
$491$	− 5.39115e6i	− 1.00920i	−0.863353	−	0.504601i	$-0.831640\pi$
$491$	− 5.39115e6i	− 1.00920i	0.863353	−	0.504601i	$-0.168360\pi$
$492$	0	0
$493$	8.59736e6i	1.59312i
$494$	0	0
$495$	−1.01151e6	−0.185548
$496$	0	0
$497$	2.27372e6	0.412902
$498$	0	0
$499$	3.18358e6i	0.572353i	0.958177	+	0.286177i	$0.0923844\pi$
$499$	3.18358e6i	0.572353i	−0.958177	+	0.286177i	$0.907616\pi$
$500$	0	0
$501$	− 607177.i	− 0.108074i
$502$	0	0
$503$	8.99291e6	1.58482	0.792410	−	0.609989i	$-0.208826\pi$
$503$	8.99291e6	1.58482	0.792410	+	0.609989i	$0.208826\pi$
$504$	0	0
$505$	−2.87054e6	−0.500882
$506$	0	0
$507$	3.84086e6i	0.663603i
$508$	0	0
$509$	5.35388e6i	0.915956i	0.888964	+	0.457978i	$0.151426\pi$
$509$	5.35388e6i	0.915956i	−0.888964	+	0.457978i	$0.848574\pi$
$510$	0	0
$511$	7.11322e6	1.20508
$512$	0	0
$513$	5.80633e6	0.974111
$514$	0	0
$515$	− 964672.i	− 0.160273i
$516$	0	0
$517$	− 7.47173e6i	− 1.22941i
$518$	0	0
$519$	−1.81481e6	−0.295743
$520$	0	0
$521$	−2.62401e6	−0.423518	−0.211759	−	0.977322i	$-0.567919\pi$
$521$	−2.62401e6	−0.423518	−0.211759	+	0.977322i	$0.567919\pi$
$522$	0	0
$523$	− 1.31228e6i	− 0.209784i	−0.994484	−	0.104892i	$-0.966550\pi$
$523$	− 1.31228e6i	− 0.209784i	0.994484	−	0.104892i	$-0.0334497\pi$
$524$	0	0
$525$	1.10747e6i	0.175362i
$526$	0	0
$527$	−8.35008e6	−1.30968
$528$	0	0
$529$	−4.92055e6	−0.764494
$530$	0	0
$531$	− 3.17904e6i	− 0.489282i
$532$	0	0
$533$	757570.i	0.115506i
$534$	0	0
$535$	−4.73647e6	−0.715435
$536$	0	0
$537$	5.80081e6	0.868067
$538$	0	0
$539$	3.21285e6i	0.476342i
$540$	0	0
$541$	6.84935e6i	1.00613i	0.864247	+	0.503067i	$0.167795\pi$
$541$	6.84935e6i	1.00613i	−0.864247	+	0.503067i	$0.832205\pi$
$542$	0	0
$543$	1.22794e6	0.178722
$544$	0	0
$545$	−609815.	−0.0879441
$546$	0	0
$547$	6.39084e6i	0.913251i	0.889659	+	0.456625i	$0.150942\pi$
$547$	6.39084e6i	0.913251i	−0.889659	+	0.456625i	$0.849058\pi$
$548$	0	0
$549$	4.92232e6i	0.697010i
$550$	0	0
$551$	−5.92391e6	−0.831246
$552$	0	0
$553$	−2.05392e6	−0.285609
$554$	0	0
$555$	2.88369e6i	0.397389i
$556$	0	0
$557$	463389.i	0.0632861i	0.999499	+	0.0316430i	$0.0100740\pi$
$557$	463389.i	0.0632861i	−0.999499	+	0.0316430i	$0.989926\pi$
$558$	0	0
$559$	−2.10830e6	−0.285367
$560$	0	0
$561$	7.34561e6	0.985418
$562$	0	0
$563$	1.07609e7i	1.43080i	0.698715	+	0.715400i	$0.253756\pi$
$563$	1.07609e7i	1.43080i	−0.698715	+	0.715400i	$0.746244\pi$
$564$	0	0
$565$	− 1.32296e6i	− 0.174352i
$566$	0	0
$567$	2.06823e6	0.270173
$568$	0	0
$569$	−1.04253e7	−1.34992	−0.674961	−	0.737853i	$-0.735840\pi$
$569$	−1.04253e7	−1.34992	−0.674961	+	0.737853i	$0.735840\pi$
$570$	0	0
$571$	− 1.58675e6i	− 0.203666i	−0.994802	−	0.101833i	$-0.967529\pi$
$571$	− 1.58675e6i	− 0.203666i	0.994802	−	0.101833i	$-0.0324707\pi$
$572$	0	0
$573$	− 4.24094e6i	− 0.539605i
$574$	0	0
$575$	−769485.	−0.0970579
$576$	0	0
$577$	−2.79056e6	−0.348941	−0.174471	−	0.984662i	$-0.555821\pi$
$577$	−2.79056e6	−0.348941	−0.174471	+	0.984662i	$0.555821\pi$
$578$	0	0
$579$	− 2.83062e6i	− 0.350901i
$580$	0	0
$581$	1.09369e6i	0.134417i
$582$	0	0
$583$	9.84944e6	1.20016
$584$	0	0
$585$	403463.	0.0487432
$586$	0	0
$587$	1.94272e6i	0.232710i	0.993208	+	0.116355i	$0.0371210\pi$
$587$	1.94272e6i	0.232710i	−0.993208	+	0.116355i	$0.962879\pi$
$588$	0	0
$589$	− 5.75353e6i	− 0.683355i
$590$	0	0
$591$	9.73022e6	1.14592
$592$	0	0
$593$	8.14862e6	0.951584	0.475792	−	0.879558i	$-0.342161\pi$
$593$	8.14862e6	0.951584	0.475792	+	0.879558i	$0.342161\pi$
$594$	0	0
$595$	8.63845e6i	1.00033i
$596$	0	0
$597$	906179.i	0.104059i
$598$	0	0
$599$	−1.49677e6	−0.170447	−0.0852234	−	0.996362i	$-0.527160\pi$
$599$	−1.49677e6	−0.170447	−0.0852234	+	0.996362i	$0.527160\pi$
$600$	0	0
$601$	−9.04082e6	−1.02099	−0.510495	−	0.859881i	$-0.670538\pi$
$601$	−9.04082e6	−1.02099	−0.510495	+	0.859881i	$0.670538\pi$
$602$	0	0
$603$	2.62938e6i	0.294483i
$604$	0	0
$605$	1.44189e6i	0.160156i
$606$	0	0
$607$	3.63200e6	0.400105	0.200052	−	0.979785i	$-0.435889\pi$
$607$	3.63200e6	0.400105	0.200052	+	0.979785i	$0.435889\pi$
$608$	0	0
$609$	−7.21751e6	−0.788577
$610$	0	0
$611$	2.98027e6i	0.322962i
$612$	0	0
$613$	− 1.89937e6i	− 0.204154i	−0.994776	−	0.102077i	$-0.967451\pi$
$613$	− 1.89937e6i	− 0.204154i	0.994776	−	0.102077i	$-0.0325488\pi$
$614$	0	0
$615$	1.59849e6	0.170420
$616$	0	0
$617$	5.96746e6	0.631069	0.315534	−	0.948914i	$-0.397816\pi$
$617$	5.96746e6	0.631069	0.315534	+	0.948914i	$0.397816\pi$
$618$	0	0
$619$	1.46307e7i	1.53475i	0.641196	+	0.767377i	$0.278439\pi$
$619$	1.46307e7i	1.53475i	−0.641196	+	0.767377i	$0.721561\pi$
$620$	0	0
$621$	4.91527e6i	0.511468i
$622$	0	0
$623$	1.47992e7	1.52763
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	5.06141e6i	0.514165i
$628$	0	0
$629$	2.24932e7i	2.26686i
$630$	0	0
$631$	1.47747e7	1.47722	0.738609	−	0.674134i	$-0.235483\pi$
$631$	1.47747e7	1.47722	0.738609	+	0.674134i	$0.235483\pi$
$632$	0	0
$633$	5.99261e6	0.594438
$634$	0	0
$635$	− 7.34126e6i	− 0.722497i
$636$	0	0
$637$	− 1.28152e6i	− 0.125134i
$638$	0	0
$639$	−1.74781e6	−0.169333
$640$	0	0
$641$	1.81017e7	1.74010	0.870050	−	0.492964i	$-0.164087\pi$
$641$	1.81017e7	1.74010	0.870050	+	0.492964i	$0.164087\pi$
$642$	0	0
$643$	− 7.21849e6i	− 0.688524i	−0.938874	−	0.344262i	$-0.888129\pi$
$643$	− 7.21849e6i	− 0.688524i	0.938874	−	0.344262i	$-0.111871\pi$
$644$	0	0
$645$	4.44856e6i	0.421037i
$646$	0	0
$647$	−2.44672e6	−0.229786	−0.114893	−	0.993378i	$-0.536653\pi$
$647$	−2.44672e6	−0.229786	−0.114893	+	0.993378i	$0.536653\pi$
$648$	0	0
$649$	8.12237e6	0.756957
$650$	0	0
$651$	− 7.00992e6i	− 0.648277i
$652$	0	0
$653$	− 5.92231e6i	− 0.543511i	−0.962366	−	0.271755i	$-0.912396\pi$
$653$	− 5.92231e6i	− 0.543511i	0.962366	−	0.271755i	$-0.0876041\pi$
$654$	0	0
$655$	−7.94346e6	−0.723446
$656$	0	0
$657$	−5.46793e6	−0.494208
$658$	0	0
$659$	2.68146e6i	0.240523i	0.992742	+	0.120262i	$0.0383734\pi$
$659$	2.68146e6i	0.240523i	−0.992742	+	0.120262i	$0.961627\pi$
$660$	0	0
$661$	− 1.14098e7i	− 1.01572i	−0.861440	−	0.507859i	$-0.830437\pi$
$661$	− 1.14098e7i	− 1.01572i	0.861440	−	0.507859i	$-0.169563\pi$
$662$	0	0
$663$	−2.92996e6	−0.258868
$664$	0	0
$665$	−5.95223e6	−0.521946
$666$	0	0
$667$	− 5.01481e6i	− 0.436456i
$668$	0	0
$669$	− 3.02423e6i	− 0.261246i
$670$	0	0
$671$	−1.25764e7	−1.07833
$672$	0	0
$673$	−5.13646e6	−0.437146	−0.218573	−	0.975821i	$-0.570140\pi$
$673$	−5.13646e6	−0.437146	−0.218573	+	0.975821i	$0.570140\pi$
$674$	0	0
$675$	− 2.49521e6i	− 0.210789i
$676$	0	0
$677$	4.91407e6i	0.412068i	0.978545	+	0.206034i	$0.0660558\pi$
$677$	4.91407e6i	0.412068i	−0.978545	+	0.206034i	$0.933944\pi$
$678$	0	0
$679$	−2.44930e7	−2.03877
$680$	0	0
$681$	−6.41937e6	−0.530426
$682$	0	0
$683$	− 4.30841e6i	− 0.353399i	−0.984265	−	0.176700i	$-0.943458\pi$
$683$	− 4.30841e6i	− 0.353399i	0.984265	−	0.176700i	$-0.0565421\pi$
$684$	0	0
$685$	− 4.05713e6i	− 0.330364i
$686$	0	0
$687$	1.00433e7	0.811868
$688$	0	0
$689$	−3.92867e6	−0.315281
$690$	0	0
$691$	− 2.17672e6i	− 0.173423i	−0.996233	−	0.0867116i	$-0.972364\pi$
$691$	− 2.17672e6i	− 0.173423i	0.996233	−	0.0867116i	$-0.0276359\pi$
$692$	0	0
$693$	− 6.62361e6i	− 0.523916i
$694$	0	0
$695$	7.66898e6	0.602249
$696$	0	0
$697$	1.24684e7	0.972142
$698$	0	0
$699$	− 1.18284e7i	− 0.915658i
$700$	0	0
$701$	− 1.31991e6i	− 0.101450i	−0.998713	−	0.0507248i	$-0.983847\pi$
$701$	− 1.31991e6i	− 0.101450i	0.998713	−	0.0507248i	$-0.0161531\pi$
$702$	0	0
$703$	−1.54987e7	−1.18279
$704$	0	0
$705$	6.28842e6	0.476506
$706$	0	0
$707$	− 1.87970e7i	− 1.41429i
$708$	0	0
$709$	1.33410e7i	0.996721i	0.866970	+	0.498360i	$0.166064\pi$
$709$	1.33410e7i	0.996721i	−0.866970	+	0.498360i	$0.833936\pi$
$710$	0	0
$711$	1.57885e6	0.117130
$712$	0	0
$713$	4.87058e6	0.358803
$714$	0	0
$715$	1.03084e6i	0.0754094i
$716$	0	0
$717$	8.63721e6i	0.627445i
$718$	0	0
$719$	−2.25289e7	−1.62524	−0.812620	−	0.582794i	$-0.801960\pi$
$719$	−2.25289e7	−1.62524	−0.812620	+	0.582794i	$0.801960\pi$
$720$	0	0
$721$	6.31691e6	0.452550
$722$	0	0
$723$	− 1.75199e7i	− 1.24648i
$724$	0	0
$725$	2.54574e6i	0.179874i
$726$	0	0
$727$	1.49088e7	1.04618	0.523091	−	0.852277i	$-0.324779\pi$
$727$	1.49088e7	1.04618	0.523091	+	0.852277i	$0.324779\pi$
$728$	0	0
$729$	−1.20906e7	−0.842617
$730$	0	0
$731$	3.46994e7i	2.40175i
$732$	0	0
$733$	7.98329e6i	0.548810i	0.961614	+	0.274405i	$0.0884809\pi$
$733$	7.98329e6i	0.548810i	−0.961614	+	0.274405i	$0.911519\pi$
$734$	0	0
$735$	−2.70402e6	−0.184626
$736$	0	0
$737$	−6.71802e6	−0.455588
$738$	0	0
$739$	3.81018e6i	0.256646i	0.991732	+	0.128323i	$0.0409594\pi$
$739$	3.81018e6i	0.256646i	−0.991732	+	0.128323i	$0.959041\pi$
$740$	0	0
$741$	− 2.01885e6i	− 0.135070i
$742$	0	0
$743$	−8.99457e6	−0.597734	−0.298867	−	0.954295i	$-0.596609\pi$
$743$	−8.99457e6	−0.597734	−0.298867	+	0.954295i	$0.596609\pi$
$744$	0	0
$745$	−9.20268e6	−0.607468
$746$	0	0
$747$	− 840722.i	− 0.0551253i
$748$	0	0
$749$	− 3.10156e7i	− 2.02011i
$750$	0	0
$751$	−1.18325e7	−0.765559	−0.382779	−	0.923840i	$-0.625033\pi$
$751$	−1.18325e7	−0.765559	−0.382779	+	0.923840i	$0.625033\pi$
$752$	0	0
$753$	4.86742e6	0.312832
$754$	0	0
$755$	− 8.42056e6i	− 0.537618i
$756$	0	0
$757$	− 5.14086e6i	− 0.326059i	−0.986621	−	0.163030i	$-0.947873\pi$
$757$	− 5.14086e6i	− 0.326059i	0.986621	−	0.163030i	$-0.0521266\pi$
$758$	0	0
$759$	−4.28467e6	−0.269968
$760$	0	0
$761$	8.21979e6	0.514516	0.257258	−	0.966343i	$-0.417181\pi$
$761$	8.21979e6	0.514516	0.257258	+	0.966343i	$0.417181\pi$
$762$	0	0
$763$	− 3.99322e6i	− 0.248320i
$764$	0	0
$765$	− 6.64037e6i	− 0.410241i
$766$	0	0
$767$	−3.23979e6	−0.198851
$768$	0	0
$769$	−1.11390e7	−0.679251	−0.339626	−	0.940561i	$-0.610300\pi$
$769$	−1.11390e7	−0.679251	−0.339626	+	0.940561i	$0.610300\pi$
$770$	0	0
$771$	4.29101e6i	0.259970i
$772$	0	0
$773$	− 5.63671e6i	− 0.339294i	−0.985505	−	0.169647i	$-0.945737\pi$
$773$	− 5.63671e6i	− 0.339294i	0.985505	−	0.169647i	$-0.0542628\pi$
$774$	0	0
$775$	−2.47252e6	−0.147872
$776$	0	0
$777$	−1.88831e7	−1.12207
$778$	0	0
$779$	8.59123e6i	0.507238i
$780$	0	0
$781$	− 4.46561e6i	− 0.261971i
$782$	0	0
$783$	1.62615e7	0.947888
$784$	0	0
$785$	−6.78132e6	−0.392771
$786$	0	0
$787$	− 3.07197e7i	− 1.76799i	−0.467495	−	0.883996i	$-0.654843\pi$
$787$	− 3.07197e7i	− 1.76799i	0.467495	−	0.883996i	$-0.345157\pi$
$788$	0	0
$789$	− 2.09361e7i	− 1.19730i
$790$	0	0
$791$	8.66309e6	0.492302
$792$	0	0
$793$	5.01639e6	0.283275
$794$	0	0
$795$	8.28956e6i	0.465172i
$796$	0	0
$797$	2.22339e7i	1.23985i	0.784660	+	0.619926i	$0.212837\pi$
$797$	2.22339e7i	1.23985i	−0.784660	+	0.619926i	$0.787163\pi$
$798$	0	0
$799$	4.90505e7	2.71817
$800$	0	0
$801$	−1.13761e7	−0.626488
$802$	0	0
$803$	− 1.39704e7i	− 0.764577i
$804$	0	0
$805$	− 5.03878e6i	− 0.274054i
$806$	0	0
$807$	7.25226e6	0.392003
$808$	0	0
$809$	1.51247e7	0.812484	0.406242	−	0.913765i	$-0.366839\pi$
$809$	1.51247e7	0.812484	0.406242	+	0.913765i	$0.366839\pi$
$810$	0	0
$811$	− 1.35982e7i	− 0.725990i	−0.931791	−	0.362995i	$-0.881754\pi$
$811$	− 1.35982e7i	− 0.725990i	0.931791	−	0.362995i	$-0.118246\pi$
$812$	0	0
$813$	− 2.26157e7i	− 1.20001i
$814$	0	0
$815$	−3.03386e6	−0.159993
$816$	0	0
$817$	−2.39092e7	−1.25317
$818$	0	0
$819$	2.64197e6i	0.137632i
$820$	0	0
$821$	− 2.56951e7i	− 1.33043i	−0.746651	−	0.665216i	$-0.768340\pi$
$821$	− 2.56951e7i	− 1.33043i	0.746651	−	0.665216i	$-0.231660\pi$
$822$	0	0
$823$	−8.73184e6	−0.449372	−0.224686	−	0.974431i	$-0.572136\pi$
$823$	−8.73184e6	−0.449372	−0.224686	+	0.974431i	$0.572136\pi$
$824$	0	0
$825$	2.17509e6	0.111261
$826$	0	0
$827$	− 1.46565e6i	− 0.0745191i	−0.999306	−	0.0372596i	$-0.988137\pi$
$827$	− 1.46565e6i	− 0.0745191i	0.999306	−	0.0372596i	$-0.0118628\pi$
$828$	0	0
$829$	− 4.49887e6i	− 0.227362i	−0.993517	−	0.113681i	$-0.963736\pi$
$829$	− 4.49887e6i	− 0.227362i	0.993517	−	0.113681i	$-0.0362641\pi$
$830$	0	0
$831$	−8.97130e6	−0.450664
$832$	0	0
$833$	−2.10918e7	−1.05317
$834$	0	0
$835$	− 1.40239e6i	− 0.0696067i
$836$	0	0
$837$	1.57938e7i	0.779244i
$838$	0	0
$839$	1.43801e7	0.705273	0.352636	−	0.935760i	$-0.385285\pi$
$839$	1.43801e7	0.705273	0.352636	+	0.935760i	$0.385285\pi$
$840$	0	0
$841$	3.92030e6	0.191130
$842$	0	0
$843$	2.59702e7i	1.25866i
$844$	0	0
$845$	8.87115e6i	0.427404i
$846$	0	0
$847$	−9.44185e6	−0.452219
$848$	0	0
$849$	2.17420e7	1.03521
$850$	0	0
$851$	− 1.31202e7i	− 0.621036i
$852$	0	0
$853$	− 2.62365e7i	− 1.23462i	−0.786720	−	0.617310i	$-0.788222\pi$
$853$	− 2.62365e7i	− 1.23462i	0.786720	−	0.617310i	$-0.211778\pi$
$854$	0	0
$855$	4.57547e6	0.214053
$856$	0	0
$857$	−6.34066e6	−0.294905	−0.147453	−	0.989069i	$-0.547107\pi$
$857$	−6.34066e6	−0.294905	−0.147453	+	0.989069i	$0.547107\pi$
$858$	0	0
$859$	1.09488e7i	0.506273i	0.967431	+	0.253137i	$0.0814622\pi$
$859$	1.09488e7i	0.506273i	−0.967431	+	0.253137i	$0.918538\pi$
$860$	0	0
$861$	1.04673e7i	0.481200i
$862$	0	0
$863$	3.28604e6	0.150192	0.0750958	−	0.997176i	$-0.476074\pi$
$863$	3.28604e6	0.150192	0.0750958	+	0.997176i	$0.476074\pi$
$864$	0	0
$865$	−4.19164e6	−0.190478
$866$	0	0
$867$	3.28540e7i	1.48436i
$868$	0	0
$869$	4.03393e6i	0.181209i
$870$	0	0
$871$	2.67963e6	0.119682
$872$	0	0
$873$	1.88278e7	0.836110
$874$	0	0
$875$	2.55791e6i	0.112944i
$876$	0	0
$877$	7.75326e6i	0.340397i	0.985410	+	0.170198i	$0.0544409\pi$
$877$	7.75326e6i	0.340397i	−0.985410	+	0.170198i	$0.945559\pi$
$878$	0	0
$879$	−1.88557e7	−0.823136
$880$	0	0
$881$	−449453.	−0.0195094	−0.00975471	−	0.999952i	$-0.503105\pi$
$881$	−449453.	−0.0195094	−0.00975471	+	0.999952i	$0.503105\pi$
$882$	0	0
$883$	2.42001e7i	1.04452i	0.852787	+	0.522259i	$0.174911\pi$
$883$	2.42001e7i	1.04452i	−0.852787	+	0.522259i	$0.825089\pi$
$884$	0	0
$885$	6.83601e6i	0.293389i
$886$	0	0
$887$	−8.80204e6	−0.375642	−0.187821	−	0.982203i	$-0.560143\pi$
$887$	−8.80204e6	−0.375642	−0.187821	+	0.982203i	$0.560143\pi$
$888$	0	0
$889$	4.80724e7	2.04005
$890$	0	0
$891$	− 4.06203e6i	− 0.171415i
$892$	0	0
$893$	3.37977e7i	1.41827i
$894$	0	0
$895$	1.33980e7	0.559091
$896$	0	0
$897$	1.70903e6	0.0709202
$898$	0	0
$899$	− 1.61137e7i	− 0.664959i
$900$	0	0
$901$	6.46597e7i	2.65352i
$902$	0	0
$903$	−2.91302e7	−1.18884
$904$	0	0
$905$	2.83616e6	0.115109
$906$	0	0
$907$	1.35035e7i	0.545040i	0.962150	+	0.272520i	$0.0878571\pi$
$907$	1.35035e7i	0.545040i	−0.962150	+	0.272520i	$0.912143\pi$
$908$	0	0
$909$	1.44492e7i	0.580010i
$910$	0	0
$911$	1.74802e7	0.697831	0.348915	−	0.937154i	$-0.386550\pi$
$911$	1.74802e7	0.697831	0.348915	+	0.937154i	$0.386550\pi$
$912$	0	0
$913$	2.14803e6	0.0852830
$914$	0	0
$915$	− 1.05847e7i	− 0.417950i
$916$	0	0
$917$	− 5.20157e7i	− 2.04273i
$918$	0	0
$919$	−3.84416e7	−1.50146	−0.750728	−	0.660612i	$-0.770297\pi$
$919$	−3.84416e7	−1.50146	−0.750728	+	0.660612i	$0.770297\pi$
$920$	0	0
$921$	−1.05934e7	−0.411514
$922$	0	0
$923$	1.78121e6i	0.0688194i
$924$	0	0
$925$	6.66040e6i	0.255945i
$926$	0	0
$927$	−4.85580e6	−0.185593
$928$	0	0
$929$	−694053.	−0.0263848	−0.0131924	−	0.999913i	$-0.504199\pi$
$929$	−694053.	−0.0263848	−0.0131924	+	0.999913i	$0.504199\pi$
$930$	0	0
$931$	− 1.45330e7i	− 0.549518i
$932$	0	0
$933$	1.70643e7i	0.641776i
$934$	0	0
$935$	1.69660e7	0.634673
$936$	0	0
$937$	4.08849e7	1.52130	0.760648	−	0.649165i	$-0.224881\pi$
$937$	4.08849e7	1.52130	0.760648	+	0.649165i	$0.224881\pi$
$938$	0	0
$939$	1.74225e7i	0.644832i
$940$	0	0
$941$	− 1.21324e7i	− 0.446655i	−0.974743	−	0.223328i	$-0.928308\pi$
$941$	− 1.21324e7i	− 0.446655i	0.974743	−	0.223328i	$-0.0716920\pi$
$942$	0	0
$943$	−7.27279e6	−0.266331
$944$	0	0
$945$	1.63392e7	0.595186
$946$	0	0
$947$	− 3.38948e7i	− 1.22817i	−0.789241	−	0.614084i	$-0.789526\pi$
$947$	− 3.38948e7i	− 1.22817i	0.789241	−	0.614084i	$-0.210474\pi$
$948$	0	0
$949$	5.57242e6i	0.200853i
$950$	0	0
$951$	−1.63086e7	−0.584742
$952$	0	0
$953$	−5.18152e7	−1.84810	−0.924049	−	0.382274i	$-0.875141\pi$
$953$	−5.18152e7	−1.84810	−0.924049	+	0.382274i	$0.875141\pi$
$954$	0	0
$955$	− 9.79521e6i	− 0.347541i
$956$	0	0
$957$	1.41753e7i	0.500324i
$958$	0	0
$959$	2.65671e7	0.932819
$960$	0	0
$961$	−1.29789e7	−0.453347
$962$	0	0
$963$	2.38416e7i	0.828458i
$964$	0	0
$965$	− 6.53783e6i	− 0.226003i
$966$	0	0
$967$	4.41440e7	1.51812	0.759058	−	0.651023i	$-0.225660\pi$
$967$	4.41440e7	1.51812	0.759058	+	0.651023i	$0.225660\pi$
$968$	0	0
$969$	−3.32272e7	−1.13680
$970$	0	0
$971$	2.25369e7i	0.767091i	0.923522	+	0.383546i	$0.125297\pi$
$971$	2.25369e7i	0.767091i	−0.923522	+	0.383546i	$0.874703\pi$
$972$	0	0
$973$	5.02184e7i	1.70052i
$974$	0	0
$975$	−867582.	−0.0292280
$976$	0	0
$977$	−508955.	−0.0170586	−0.00852929	−	0.999964i	$-0.502715\pi$
$977$	−508955.	−0.0170586	−0.00852929	+	0.999964i	$0.502715\pi$
$978$	0	0
$979$	− 2.90657e7i	− 0.969224i
$980$	0	0
$981$	3.06958e6i	0.101837i
$982$	0	0
$983$	1.38312e7	0.456536	0.228268	−	0.973598i	$-0.426694\pi$
$983$	1.38312e7	0.456536	0.228268	+	0.973598i	$0.426694\pi$
$984$	0	0
$985$	2.24737e7	0.738046
$986$	0	0
$987$	4.11781e7i	1.34547i
$988$	0	0
$989$	− 2.02400e7i	− 0.657992i
$990$	0	0
$991$	−1.13368e7	−0.366696	−0.183348	−	0.983048i	$-0.558693\pi$
$991$	−1.13368e7	−0.366696	−0.183348	+	0.983048i	$0.558693\pi$
$992$	0	0
$993$	4.26957e6	0.137408
$994$	0	0
$995$	2.09298e6i	0.0670205i
$996$	0	0
$997$	3.66390e7i	1.16736i	0.811983	+	0.583681i	$0.198388\pi$
$997$	3.66390e7i	1.16736i	−0.811983	+	0.583681i	$0.801612\pi$
$998$	0	0
$999$	4.25449e7	1.34876

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.d.a.81.14		20
4.3	odd	2		40.6.d.a.21.18	yes	20
5.2	odd	4		800.6.f.c.49.13		20
5.3	odd	4		800.6.f.b.49.8		20
5.4	even	2		800.6.d.c.401.7		20
8.3	odd	2		40.6.d.a.21.17	✓	20
8.5	even	2	inner	160.6.d.a.81.7		20
12.11	even	2		360.6.k.b.181.3		20
20.3	even	4		200.6.f.c.149.13		20
20.7	even	4		200.6.f.b.149.8		20
20.19	odd	2		200.6.d.b.101.3		20
24.11	even	2		360.6.k.b.181.4		20
40.3	even	4		200.6.f.b.149.7		20
40.13	odd	4		800.6.f.c.49.14		20
40.19	odd	2		200.6.d.b.101.4		20
40.27	even	4		200.6.f.c.149.14		20
40.29	even	2		800.6.d.c.401.14		20
40.37	odd	4		800.6.f.b.49.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.17	✓	20	8.3	odd	2
40.6.d.a.21.18	yes	20	4.3	odd	2
160.6.d.a.81.7		20	8.5	even	2	inner
160.6.d.a.81.14		20	1.1	even	1	trivial
200.6.d.b.101.3		20	20.19	odd	2
200.6.d.b.101.4		20	40.19	odd	2
200.6.f.b.149.7		20	40.3	even	4
200.6.f.b.149.8		20	20.7	even	4
200.6.f.c.149.13		20	20.3	even	4
200.6.f.c.149.14		20	40.27	even	4
360.6.k.b.181.3		20	12.11	even	2
360.6.k.b.181.4		20	24.11	even	2
800.6.d.c.401.7		20	5.4	even	2
800.6.d.c.401.14		20	40.29	even	2
800.6.f.b.49.7		20	40.37	odd	4
800.6.f.b.49.8		20	5.3	odd	4
800.6.f.c.49.13		20	5.2	odd	4
800.6.f.c.49.14		20	40.13	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	160.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.8240i q^{3} +25.0000i q^{5} -163.706 q^{7} +125.841 q^{9} +321.520i q^{11} -128.246i q^{13} -270.600 q^{15} -2110.72 q^{17} -1454.37i q^{19} -1771.96i q^{21} +1231.18 q^{23} -625.000 q^{25} +3992.34i q^{27} -4073.19i q^{29} +3956.03 q^{31} -3480.14 q^{33} -4092.65i q^{35} -10656.6i q^{37} +1388.13 q^{39} -5907.19 q^{41} -16439.6i q^{43} +3146.02i q^{45} -23238.8 q^{47} +9992.68 q^{49} -22846.5i q^{51} -30634.0i q^{53} -8038.01 q^{55} +15742.1 q^{57} -25262.4i q^{59} +39115.5i q^{61} -20600.9 q^{63} +3206.14 q^{65} +20894.5i q^{67} +13326.3i q^{69} -13889.1 q^{71} -43451.2 q^{73} -6765.01i q^{75} -52634.8i q^{77} +12546.4 q^{79} -12633.8 q^{81} -6680.84i q^{83} -52768.0i q^{85} +44088.2 q^{87} -90400.9 q^{89} +20994.6i q^{91} +42820.2i q^{93} +36359.2 q^{95} +149616. q^{97} +40460.4i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63}+ \cdots + 147376 q^{97}+O(q^{100}) \) 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more