LMFDB - Embedded newform 160.4.f.a.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,4,Mod(49,160)]

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, 1]))

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

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

chi := DirichletCharacter("160.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	160.f (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:	$9.44030560092$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2x^{14} + 44x^{12} + 400x^{10} - 3200x^{8} + 25600x^{6} + 180224x^{4} - 524288x^{2} + 16777216 $ x^16 - 2x^14 + 44x^12 + 400x^10 - 3200x^8 + 25600x^6 + 180224x^4 - 524288*x^2 + 16777216
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$2.07496 + 1.92212i$ of defining polynomial
Character	$\chi$	$=$	160.49
Dual form			160.4.f.a.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.67494 q^{3} +(-11.1461 + 0.874848i) q^{5} -29.8250i q^{7} +17.5548 q^{9} +34.8520i q^{11} +20.6159 q^{13} +(74.3992 - 5.83956i) q^{15} +89.9765i q^{17} -0.475810i q^{19} +199.080i q^{21} +100.299i q^{23} +(123.469 - 19.5022i) q^{25} +63.0462 q^{27} -132.077i q^{29} +137.250 q^{31} -232.635i q^{33} +(26.0923 + 332.431i) q^{35} +57.1057 q^{37} -137.610 q^{39} +298.591 q^{41} -248.058 q^{43} +(-195.667 + 15.3578i) q^{45} +323.827i q^{47} -546.530 q^{49} -600.588i q^{51} -55.4295 q^{53} +(-30.4902 - 388.462i) q^{55} +3.17600i q^{57} -167.879i q^{59} +164.687i q^{61} -523.571i q^{63} +(-229.786 + 18.0358i) q^{65} +666.565 q^{67} -669.491i q^{69} -384.334 q^{71} +749.119i q^{73} +(-824.150 + 130.176i) q^{75} +1039.46 q^{77} -582.564 q^{79} -894.809 q^{81} +636.408 q^{83} +(-78.7158 - 1002.88i) q^{85} +881.607i q^{87} +759.913 q^{89} -614.869i q^{91} -916.133 q^{93} +(0.416261 + 5.30340i) q^{95} +1279.63i q^{97} +611.819i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 104 q^{9} + 56 q^{15} - 24 q^{25} + 112 q^{31} + 736 q^{39} + 232 q^{41} - 200 q^{49} - 392 q^{55} - 600 q^{65} - 2096 q^{71} - 2992 q^{79} - 728 q^{81} - 208 q^{89} + 1064 q^{95}+O(q^{100}) $ 16 * q + 104 * q^9 + 56 * q^15 - 24 * q^25 + 112 * q^31 + 736 * q^39 + 232 * q^41 - 200 * q^49 - 392 * q^55 - 600 * q^65 - 2096 * q^71 - 2992 * q^79 - 728 * q^81 - 208 * q^89 + 1064 * q^95

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
$3$	−6.67494			−1.28459			−0.642296	−	0.766457i	$-0.722018\pi$
$3$	−6.67494			−1.28459			−0.642296	+	0.766457i	$0.722018\pi$
$4$		0			0
$5$	−11.1461	+	0.874848i	−0.996934	+	0.0782488i
$5$	−11.1461	+	0.874848i	−0.996934	+	0.0782488i
$6$		0			0
$7$		−	29.8250i		−	1.61040i	−0.593005	−	0.805199i	$-0.702059\pi$
$7$		−	29.8250i		−	1.61040i	0.593005	−	0.805199i	$-0.297941\pi$
$8$		0			0
$9$	17.5548			0.650177
$10$		0			0
$11$			34.8520i			0.955297i	0.878551	+	0.477648i	$0.158511\pi$
$11$			34.8520i			0.955297i	−0.878551	+	0.477648i	$0.841489\pi$
$12$		0			0
$13$	20.6159			0.439833			0.219916	−	0.975519i	$-0.429422\pi$
$13$	20.6159			0.439833			0.219916	+	0.975519i	$0.429422\pi$
$14$		0			0
$15$	74.3992	−	5.83956i	1.28065	−	0.100518i
$16$		0			0
$17$			89.9765i			1.28368i	0.766840	+	0.641839i	$0.221828\pi$
$17$			89.9765i			1.28368i	−0.766840	+	0.641839i	$0.778172\pi$
$18$		0			0
$19$		−	0.475810i		−	0.00574517i	−0.999996	−	0.00287259i	$-0.999086\pi$
$19$		−	0.475810i		−	0.00574517i	0.999996	−	0.00287259i	$-0.000914374\pi$
$20$		0			0
$21$			199.080i			2.06870i
$22$		0			0
$23$			100.299i			0.909298i	0.890671	+	0.454649i	$0.150235\pi$
$23$			100.299i			0.909298i	−0.890671	+	0.454649i	$0.849765\pi$
$24$		0			0
$25$	123.469	−	19.5022i	0.987754	−	0.156018i
$26$		0			0
$27$	63.0462			0.449380
$28$		0			0
$29$		−	132.077i		−	0.845729i	−0.906193	−	0.422864i	$-0.861025\pi$
$29$		−	132.077i		−	0.845729i	0.906193	−	0.422864i	$-0.138975\pi$
$30$		0			0
$31$	137.250			0.795186			0.397593	−	0.917562i	$-0.369846\pi$
$31$	137.250			0.795186			0.397593	+	0.917562i	$0.369846\pi$
$32$		0			0
$33$		−	232.635i		−	1.22717i
$34$		0			0
$35$	26.0923	+	332.431i	0.126012	+	1.60546i
$36$		0			0
$37$	57.1057			0.253733			0.126866	−	0.991920i	$-0.459508\pi$
$37$	57.1057			0.253733			0.126866	+	0.991920i	$0.459508\pi$
$38$		0			0
$39$	−137.610			−0.565005
$40$		0			0
$41$	298.591			1.13737			0.568684	−	0.822556i	$-0.307453\pi$
$41$	298.591			1.13737			0.568684	+	0.822556i	$0.307453\pi$
$42$		0			0
$43$	−248.058			−0.879733			−0.439866	−	0.898063i	$-0.644974\pi$
$43$	−248.058			−0.879733			−0.439866	+	0.898063i	$0.644974\pi$
$44$		0			0
$45$	−195.667	+	15.3578i	−0.648184	+	0.0508756i
$46$		0			0
$47$			323.827i			1.00500i	0.864577	+	0.502501i	$0.167587\pi$
$47$			323.827i			1.00500i	−0.864577	+	0.502501i	$0.832413\pi$
$48$		0			0
$49$	−546.530			−1.59338
$50$		0			0
$51$		−	600.588i		−	1.64900i
$52$		0			0
$53$	−55.4295			−0.143657			−0.0718285	−	0.997417i	$-0.522883\pi$
$53$	−55.4295			−0.143657			−0.0718285	+	0.997417i	$0.522883\pi$
$54$		0			0
$55$	−30.4902	−	388.462i	−0.0747508	−	0.952368i
$56$		0			0
$57$			3.17600i			0.00738020i
$58$		0			0
$59$		−	167.879i		−	0.370440i	−0.982697	−	0.185220i	$-0.940700\pi$
$59$		−	167.879i		−	0.370440i	0.982697	−	0.185220i	$-0.0592998\pi$
$60$		0			0
$61$			164.687i			0.345672i	0.984951	+	0.172836i	$0.0552930\pi$
$61$			164.687i			0.345672i	−0.984951	+	0.172836i	$0.944707\pi$
$62$		0			0
$63$		−	523.571i		−	1.04704i
$64$		0			0
$65$	−229.786	+	18.0358i	−0.438484	+	0.0344164i
$66$		0			0
$67$	666.565			1.21543			0.607715	−	0.794155i	$-0.292086\pi$
$67$	666.565			1.21543			0.607715	+	0.794155i	$0.292086\pi$
$68$		0			0
$69$		−	669.491i		−	1.16808i
$70$		0			0
$71$	−384.334			−0.642424			−0.321212	−	0.947007i	$-0.604090\pi$
$71$	−384.334			−0.642424			−0.321212	+	0.947007i	$0.604090\pi$
$72$		0			0
$73$			749.119i			1.20107i	0.799600	+	0.600533i	$0.205045\pi$
$73$			749.119i			1.20107i	−0.799600	+	0.600533i	$0.794955\pi$
$74$		0			0
$75$	−824.150	+	130.176i	−1.26886	+	0.200419i
$76$		0			0
$77$	1039.46			1.53841
$78$		0			0
$79$	−582.564			−0.829666			−0.414833	−	0.909898i	$-0.636160\pi$
$79$	−582.564			−0.829666			−0.414833	+	0.909898i	$0.636160\pi$
$80$		0			0
$81$	−894.809			−1.22745
$82$		0			0
$83$	636.408			0.841625			0.420813	−	0.907148i	$-0.361745\pi$
$83$	636.408			0.841625			0.420813	+	0.907148i	$0.361745\pi$
$84$		0			0
$85$	−78.7158	−	1002.88i	−0.100446	−	1.27974i
$86$		0			0
$87$			881.607i			1.08642i
$88$		0			0
$89$	759.913			0.905064			0.452532	−	0.891748i	$-0.350521\pi$
$89$	759.913			0.905064			0.452532	+	0.891748i	$0.350521\pi$
$90$		0			0
$91$		−	614.869i		−	0.708305i
$92$		0			0
$93$	−916.133			−1.02149
$94$		0			0
$95$	0.416261	+	5.30340i	0.000449553	+	0.00572756i
$96$		0			0
$97$			1279.63i			1.33945i	0.742611	+	0.669723i	$0.233587\pi$
$97$			1279.63i			1.33945i	−0.742611	+	0.669723i	$0.766413\pi$
$98$		0			0
$99$			611.819i			0.621112i
$100$		0			0
$101$			806.217i			0.794273i	0.917760	+	0.397137i	$0.129996\pi$
$101$			806.217i			0.794273i	−0.917760	+	0.397137i	$0.870004\pi$
$102$		0			0
$103$		−	497.704i		−	0.476119i	−0.971251	−	0.238059i	$-0.923489\pi$
$103$		−	497.704i		−	0.476119i	0.971251	−	0.238059i	$-0.0765113\pi$
$104$		0			0
$105$	−174.165	−	2218.96i	−0.161874	−	2.06236i
$106$		0			0
$107$	111.714			0.100933			0.0504666	−	0.998726i	$-0.483929\pi$
$107$	111.714			0.100933			0.0504666	+	0.998726i	$0.483929\pi$
$108$		0			0
$109$			1098.19i			0.965022i	0.875890	+	0.482511i	$0.160275\pi$
$109$			1098.19i			0.965022i	−0.875890	+	0.482511i	$0.839725\pi$
$110$		0			0
$111$	−381.177			−0.325943
$112$		0			0
$113$		−	2118.96i		−	1.76403i	−0.471221	−	0.882015i	$-0.656187\pi$
$113$		−	2118.96i		−	1.76403i	0.471221	−	0.882015i	$-0.343813\pi$
$114$		0			0
$115$	−87.7466	−	1117.94i	−0.0711514	−	0.906510i
$116$		0			0
$117$	361.908			0.285969
$118$		0			0
$119$	2683.55			2.06723
$120$		0			0
$121$	116.341			0.0874084
$122$		0			0
$123$	−1993.08			−1.46105
$124$		0			0
$125$	−1359.13	+	325.390i	−0.972517	+	0.232830i
$126$		0			0
$127$			567.970i			0.396844i	0.980117	+	0.198422i	$0.0635816\pi$
$127$			567.970i			0.396844i	−0.980117	+	0.198422i	$0.936418\pi$
$128$		0			0
$129$	1655.77			1.13010
$130$		0			0
$131$			2238.84i			1.49319i	0.665278	+	0.746596i	$0.268313\pi$
$131$			2238.84i			1.49319i	−0.665278	+	0.746596i	$0.731687\pi$
$132$		0			0
$133$	−14.1910			−0.00925201
$134$		0			0
$135$	−702.717	+	55.1559i	−0.448002	+	0.0351634i
$136$		0			0
$137$			1428.64i			0.890925i	0.895301	+	0.445462i	$0.146961\pi$
$137$			1428.64i			0.890925i	−0.895301	+	0.445462i	$0.853039\pi$
$138$		0			0
$139$			1467.79i			0.895659i	0.894119	+	0.447829i	$0.147803\pi$
$139$			1467.79i			0.895659i	−0.894119	+	0.447829i	$0.852197\pi$
$140$		0			0
$141$		−	2161.53i		−	1.29102i
$142$		0			0
$143$			718.505i			0.420171i
$144$		0			0
$145$	115.548	+	1472.14i	0.0661772	+	0.843136i
$146$		0			0
$147$	3648.05			2.04684
$148$		0			0
$149$			2881.15i			1.58411i	0.610449	+	0.792056i	$0.290989\pi$
$149$			2881.15i			1.58411i	−0.610449	+	0.792056i	$0.709011\pi$
$150$		0			0
$151$	949.291			0.511604			0.255802	−	0.966729i	$-0.417660\pi$
$151$	949.291			0.511604			0.255802	+	0.966729i	$0.417660\pi$
$152$		0			0
$153$			1579.52i			0.834617i
$154$		0			0
$155$	−1529.79	+	120.073i	−0.792748	+	0.0622223i
$156$		0			0
$157$	3159.30			1.60599			0.802993	−	0.595988i	$-0.203239\pi$
$157$	3159.30			1.60599			0.802993	+	0.595988i	$0.203239\pi$
$158$		0			0
$159$	369.988			0.184541
$160$		0			0
$161$	2991.42			1.46433
$162$		0			0
$163$	1361.75			0.654360			0.327180	−	0.944962i	$-0.393902\pi$
$163$	1361.75			0.654360			0.327180	+	0.944962i	$0.393902\pi$
$164$		0			0
$165$	203.520	+	2592.96i	0.0960243	+	1.22340i
$166$		0			0
$167$			641.323i			0.297168i	0.988900	+	0.148584i	$0.0474716\pi$
$167$			641.323i			0.297168i	−0.988900	+	0.148584i	$0.952528\pi$
$168$		0			0
$169$	−1771.98			−0.806547
$170$		0			0
$171$		−	8.35274i		−	0.00373538i
$172$		0			0
$173$	−57.9969			−0.0254880			−0.0127440	−	0.999919i	$-0.504057\pi$
$173$	−57.9969			−0.0254880			−0.0127440	+	0.999919i	$0.504057\pi$
$174$		0			0
$175$	−581.653	−	3682.47i	−0.251251	−	1.59068i
$176$		0			0
$177$			1120.58i			0.475864i
$178$		0			0
$179$		−	1608.75i		−	0.671753i	−0.941906	−	0.335876i	$-0.890968\pi$
$179$		−	1608.75i		−	0.671753i	0.941906	−	0.335876i	$-0.109032\pi$
$180$		0			0
$181$		−	2665.60i		−	1.09465i	−0.836919	−	0.547327i	$-0.815646\pi$
$181$		−	2665.60i		−	1.09465i	0.836919	−	0.547327i	$-0.184354\pi$
$182$		0			0
$183$		−	1099.27i		−	0.444047i
$184$		0			0
$185$	−636.504	+	49.9588i	−0.252955	+	0.0198543i
$186$		0			0
$187$	−3135.86			−1.22629
$188$		0			0
$189$		−	1880.35i		−	0.723680i
$190$		0			0
$191$	2879.60			1.09089			0.545446	−	0.838146i	$-0.316360\pi$
$191$	2879.60			1.09089			0.545446	+	0.838146i	$0.316360\pi$
$192$		0			0
$193$		−	2401.35i		−	0.895611i	−0.894131	−	0.447805i	$-0.852206\pi$
$193$		−	2401.35i		−	0.895611i	0.894131	−	0.447805i	$-0.147794\pi$
$194$		0			0
$195$	1533.81	−	120.388i	0.563273	−	0.0442110i
$196$		0			0
$197$	−4454.03			−1.61085			−0.805423	−	0.592700i	$-0.798062\pi$
$197$	−4454.03			−1.61085			−0.805423	+	0.592700i	$0.798062\pi$
$198$		0			0
$199$	−4903.81			−1.74684			−0.873422	−	0.486964i	$-0.838104\pi$
$199$	−4903.81			−1.74684			−0.873422	+	0.486964i	$0.838104\pi$
$200$		0			0
$201$	−4449.28			−1.56133
$202$		0			0
$203$	−3939.20			−1.36196
$204$		0			0
$205$	−3328.11	+	261.222i	−1.13388	+	0.0889977i
$206$		0			0
$207$			1760.73i			0.591205i
$208$		0			0
$209$	16.5829			0.00548834
$210$		0			0
$211$			1226.59i			0.400200i	0.979776	+	0.200100i	$0.0641266\pi$
$211$			1226.59i			0.400200i	−0.979776	+	0.200100i	$0.935873\pi$
$212$		0			0
$213$	2565.41			0.825253
$214$		0			0
$215$	2764.87	−	217.013i	0.877035	−	0.0688380i
$216$		0			0
$217$		−	4093.47i		−	1.28057i
$218$		0			0
$219$		−	5000.32i		−	1.54288i
$220$		0			0
$221$			1854.95i			0.564603i
$222$		0			0
$223$			1542.77i			0.463281i	0.972801	+	0.231641i	$0.0744093\pi$
$223$			1542.77i			0.463281i	−0.972801	+	0.231641i	$0.925591\pi$
$224$		0			0
$225$	2167.48	−	342.357i	0.642215	−	0.101439i
$226$		0			0
$227$	3851.34			1.12609			0.563045	−	0.826426i	$-0.309630\pi$
$227$	3851.34			1.12609			0.563045	+	0.826426i	$0.309630\pi$
$228$		0			0
$229$		−	5232.61i		−	1.50996i	−0.655748	−	0.754980i	$-0.727647\pi$
$229$		−	5232.61i		−	1.50996i	0.655748	−	0.754980i	$-0.272353\pi$
$230$		0			0
$231$	−6938.32			−1.97623
$232$		0			0
$233$		−	181.419i		−	0.0510094i	−0.999675	−	0.0255047i	$-0.991881\pi$
$233$		−	181.419i		−	0.0510094i	0.999675	−	0.0255047i	$-0.00811927\pi$
$234$		0			0
$235$	−283.300	−	3609.40i	−0.0786402	−	1.00192i
$236$		0			0
$237$	3888.58			1.06578
$238$		0			0
$239$	−2507.75			−0.678716			−0.339358	−	0.940657i	$-0.610210\pi$
$239$	−2507.75			−0.678716			−0.339358	+	0.940657i	$0.610210\pi$
$240$		0			0
$241$	−1089.46			−0.291196			−0.145598	−	0.989344i	$-0.546511\pi$
$241$	−1089.46			−0.291196			−0.145598	+	0.989344i	$0.546511\pi$
$242$		0			0
$243$	4270.54			1.12739
$244$		0			0
$245$	6091.65	−	478.130i	1.58850	−	0.124680i
$246$		0			0
$247$		−	9.80925i		−	0.00252691i
$248$		0			0
$249$	−4247.99			−1.08115
$250$		0			0
$251$			5380.02i			1.35292i	0.736478	+	0.676461i	$0.236487\pi$
$251$			5380.02i			1.35292i	−0.736478	+	0.676461i	$0.763513\pi$
$252$		0			0
$253$	−3495.63			−0.868649
$254$		0			0
$255$	525.423	+	6694.18i	0.129032	+	1.64395i
$256$		0			0
$257$			1377.98i			0.334458i	0.985918	+	0.167229i	$0.0534819\pi$
$257$			1377.98i			0.334458i	−0.985918	+	0.167229i	$0.946518\pi$
$258$		0			0
$259$		−	1703.18i		−	0.408611i
$260$		0			0
$261$		−	2318.59i		−	0.549873i
$262$		0			0
$263$			411.340i			0.0964422i	0.998837	+	0.0482211i	$0.0153552\pi$
$263$			411.340i			0.0964422i	−0.998837	+	0.0482211i	$0.984645\pi$
$264$		0			0
$265$	617.820	−	48.4924i	0.143217	−	0.0112410i
$266$		0			0
$267$	−5072.37			−1.16264
$268$		0			0
$269$			407.204i			0.0922961i	0.998935	+	0.0461481i	$0.0146946\pi$
$269$			407.204i			0.0922961i	−0.998935	+	0.0461481i	$0.985305\pi$
$270$		0			0
$271$	1329.04			0.297909			0.148954	−	0.988844i	$-0.452409\pi$
$271$	1329.04			0.297909			0.148954	+	0.988844i	$0.452409\pi$
$272$		0			0
$273$			4104.21i			0.909884i
$274$		0			0
$275$	679.691	+	4303.15i	0.149043	+	0.943598i
$276$		0			0
$277$	−4411.34			−0.956866			−0.478433	−	0.878124i	$-0.658795\pi$
$277$	−4411.34			−0.956866			−0.478433	+	0.878124i	$0.658795\pi$
$278$		0			0
$279$	2409.39			0.517012
$280$		0			0
$281$	5134.23			1.08997			0.544986	−	0.838445i	$-0.316535\pi$
$281$	5134.23			1.08997			0.544986	+	0.838445i	$0.316535\pi$
$282$		0			0
$283$	3093.54			0.649794			0.324897	−	0.945749i	$-0.394670\pi$
$283$	3093.54			0.649794			0.324897	+	0.945749i	$0.394670\pi$
$284$		0			0
$285$	−2.77852	−	35.3999i	−0.000577492	−	0.00735757i
$286$		0			0
$287$		−	8905.47i		−	1.83161i
$288$		0			0
$289$	−3182.77			−0.647827
$290$		0			0
$291$		−	8541.42i		−	1.72064i
$292$		0			0
$293$	−671.319			−0.133853			−0.0669264	−	0.997758i	$-0.521319\pi$
$293$	−671.319			−0.133853			−0.0669264	+	0.997758i	$0.521319\pi$
$294$		0			0
$295$	146.868	+	1871.19i	0.0289865	+	0.369304i
$296$		0			0
$297$			2197.28i			0.429291i
$298$		0			0
$299$			2067.76i			0.399939i
$300$		0			0
$301$			7398.33i			1.41672i
$302$		0			0
$303$		−	5381.45i		−	1.02032i
$304$		0			0
$305$	−144.076	−	1835.61i	−0.0270484	−	0.344612i
$306$		0			0
$307$	−5810.78			−1.08026			−0.540128	−	0.841583i	$-0.681624\pi$
$307$	−5810.78			−1.08026			−0.540128	+	0.841583i	$0.681624\pi$
$308$		0			0
$309$			3322.14i			0.611619i
$310$		0			0
$311$	7223.30			1.31703			0.658514	−	0.752568i	$-0.271185\pi$
$311$	7223.30			1.31703			0.658514	+	0.752568i	$0.271185\pi$
$312$		0			0
$313$			1804.40i			0.325849i	0.986638	+	0.162925i	$0.0520928\pi$
$313$			1804.40i			0.325849i	−0.986638	+	0.162925i	$0.947907\pi$
$314$		0			0
$315$	458.045	+	5835.75i	0.0819299	+	1.04383i
$316$		0			0
$317$	1593.07			0.282259			0.141129	−	0.989991i	$-0.454927\pi$
$317$	1593.07			0.282259			0.141129	+	0.989991i	$0.454927\pi$
$318$		0			0
$319$	4603.15			0.807922
$320$		0			0
$321$	−745.687			−0.129658
$322$		0			0
$323$	42.8117			0.00737495
$324$		0			0
$325$	2545.43	−	402.056i	0.434446	−	0.0686217i
$326$		0			0
$327$		−	7330.34i		−	1.23966i
$328$		0			0
$329$	9658.14			1.61845
$330$		0			0
$331$		−	6213.17i		−	1.03174i	−0.856666	−	0.515871i	$-0.827468\pi$
$331$		−	6213.17i		−	1.03174i	0.856666	−	0.515871i	$-0.172532\pi$
$332$		0			0
$333$	1002.48			0.164971
$334$		0			0
$335$	−7429.57	+	583.143i	−1.21170	+	0.0951059i
$336$		0			0
$337$		−	11379.1i		−	1.83935i	−0.392686	−	0.919673i	$-0.628454\pi$
$337$		−	11379.1i		−	1.83935i	0.392686	−	0.919673i	$-0.371546\pi$
$338$		0			0
$339$			14144.0i			2.26606i
$340$		0			0
$341$			4783.42i			0.759638i
$342$		0			0
$343$			6070.27i			0.955579i
$344$		0			0
$345$	585.703	+	7462.19i	0.0914006	+	1.16450i
$346$		0			0
$347$	−2528.79			−0.391217			−0.195609	−	0.980682i	$-0.562668\pi$
$347$	−2528.79			−0.391217			−0.195609	+	0.980682i	$0.562668\pi$
$348$		0			0
$349$			110.562i			0.0169577i	0.999964	+	0.00847885i	$0.00269893\pi$
$349$			110.562i			0.0169577i	−0.999964	+	0.00847885i	$0.997301\pi$
$350$		0			0
$351$	1299.76			0.197652
$352$		0			0
$353$			7118.00i			1.07324i	0.843825	+	0.536619i	$0.180299\pi$
$353$			7118.00i			1.07324i	−0.843825	+	0.536619i	$0.819701\pi$
$354$		0			0
$355$	4283.81	−	336.234i	0.640454	−	0.0502689i
$356$		0			0
$357$	−17912.5			−2.65555
$358$		0			0
$359$	3118.04			0.458396			0.229198	−	0.973380i	$-0.426390\pi$
$359$	3118.04			0.458396			0.229198	+	0.973380i	$0.426390\pi$
$360$		0			0
$361$	6858.77			0.999967
$362$		0			0
$363$	−776.566			−0.112284
$364$		0			0
$365$	−655.365	−	8349.73i	−0.0939819	−	1.19738i
$366$		0			0
$367$			11007.8i			1.56567i	0.622230	+	0.782835i	$0.286227\pi$
$367$			11007.8i			1.56567i	−0.622230	+	0.782835i	$0.713773\pi$
$368$		0			0
$369$	5241.70			0.739491
$370$		0			0
$371$			1653.18i			0.231345i
$372$		0			0
$373$	12872.3			1.78687			0.893436	−	0.449190i	$-0.148287\pi$
$373$	12872.3			1.78687			0.893436	+	0.449190i	$0.148287\pi$
$374$		0			0
$375$	9072.14	−	2171.96i	1.24929	−	0.299092i
$376$		0			0
$377$		−	2722.89i		−	0.371979i
$378$		0			0
$379$			6010.29i			0.814586i	0.913298	+	0.407293i	$0.133527\pi$
$379$			6010.29i			0.814586i	−0.913298	+	0.407293i	$0.866473\pi$
$380$		0			0
$381$		−	3791.16i		−	0.509783i
$382$		0			0
$383$			4012.72i			0.535354i	0.963509	+	0.267677i	$0.0862559\pi$
$383$			4012.72i			0.535354i	−0.963509	+	0.267677i	$0.913744\pi$
$384$		0			0
$385$	−11585.9	+	909.369i	−1.53369	+	0.120379i
$386$		0			0
$387$	−4354.61			−0.571982
$388$		0			0
$389$			9488.02i			1.23666i	0.785918	+	0.618331i	$0.212191\pi$
$389$			9488.02i			1.23666i	−0.785918	+	0.618331i	$0.787809\pi$
$390$		0			0
$391$	−9024.58			−1.16724
$392$		0			0
$393$		−	14944.1i		−	1.91814i
$394$		0			0
$395$	6493.30	−	509.655i	0.827122	−	0.0649204i
$396$		0			0
$397$	4950.44			0.625832			0.312916	−	0.949781i	$-0.398694\pi$
$397$	4950.44			0.625832			0.312916	+	0.949781i	$0.398694\pi$
$398$		0			0
$399$	94.7242			0.0118851
$400$		0			0
$401$	−8799.56			−1.09583			−0.547917	−	0.836533i	$-0.684579\pi$
$401$	−8799.56			−1.09583			−0.547917	+	0.836533i	$0.684579\pi$
$402$		0			0
$403$	2829.53			0.349749
$404$		0			0
$405$	9973.59	−	782.822i	1.22368	−	0.0960462i
$406$		0			0
$407$			1990.25i			0.242390i
$408$		0			0
$409$	7179.75			0.868009			0.434005	−	0.900911i	$-0.357100\pi$
$409$	7179.75			0.868009			0.434005	+	0.900911i	$0.357100\pi$
$410$		0			0
$411$		−	9536.06i		−	1.14447i
$412$		0			0
$413$	−5006.98			−0.596556
$414$		0			0
$415$	−7093.45	+	556.761i	−0.839045	+	0.0658562i
$416$		0			0
$417$		−	9797.42i		−	1.15056i
$418$		0			0
$419$		−	13557.2i		−	1.58070i	−0.612657	−	0.790349i	$-0.709899\pi$
$419$		−	13557.2i		−	1.58070i	0.612657	−	0.790349i	$-0.290101\pi$
$420$		0			0
$421$			7428.36i			0.859943i	0.902843	+	0.429971i	$0.141476\pi$
$421$			7428.36i			0.859943i	−0.902843	+	0.429971i	$0.858524\pi$
$422$		0			0
$423$			5684.72i			0.653429i
$424$		0			0
$425$	1754.74	+	11109.3i	0.200276	+	1.26796i
$426$		0			0
$427$	4911.78			0.556669
$428$		0			0
$429$		−	4795.97i		−	0.539748i
$430$		0			0
$431$	1591.77			0.177895			0.0889475	−	0.996036i	$-0.471650\pi$
$431$	1591.77			0.177895			0.0889475	+	0.996036i	$0.471650\pi$
$432$		0			0
$433$			10811.6i			1.19994i	0.800023	+	0.599970i	$0.204821\pi$
$433$			10811.6i			1.19994i	−0.800023	+	0.599970i	$0.795179\pi$
$434$		0			0
$435$	−771.273	−	9826.45i	−0.0850108	−	1.08309i
$436$		0			0
$437$	47.7234			0.00522407
$438$		0			0
$439$	4891.74			0.531823			0.265911	−	0.963997i	$-0.414327\pi$
$439$	4891.74			0.531823			0.265911	+	0.963997i	$0.414327\pi$
$440$		0			0
$441$	−9594.21			−1.03598
$442$		0			0
$443$	2116.86			0.227032			0.113516	−	0.993536i	$-0.463789\pi$
$443$	2116.86			0.227032			0.113516	+	0.993536i	$0.463789\pi$
$444$		0			0
$445$	−8470.04	+	664.809i	−0.902289	+	0.0708201i
$446$		0			0
$447$		−	19231.5i		−	2.03494i
$448$		0			0
$449$	−13253.6			−1.39305			−0.696523	−	0.717535i	$-0.745270\pi$
$449$	−13253.6			−1.39305			−0.696523	+	0.717535i	$0.745270\pi$
$450$		0			0
$451$			10406.5i			1.08652i
$452$		0			0
$453$	−6336.46			−0.657202
$454$		0			0
$455$	537.917	+	6853.37i	0.0554240	+	0.706134i
$456$		0			0
$457$			3969.19i			0.406282i	0.979150	+	0.203141i	$0.0651150\pi$
$457$			3969.19i			0.406282i	−0.979150	+	0.203141i	$0.934885\pi$
$458$		0			0
$459$			5672.68i			0.576858i
$460$		0			0
$461$		−	7068.99i		−	0.714177i	−0.934071	−	0.357089i	$-0.883769\pi$
$461$		−	7068.99i		−	0.714177i	0.934071	−	0.357089i	$-0.116231\pi$
$462$		0			0
$463$		−	178.581i		−	0.0179251i	−0.999960	−	0.00896257i	$-0.997147\pi$
$463$		−	178.581i		−	0.0179251i	0.999960	−	0.00896257i	$-0.00285291\pi$
$464$		0			0
$465$	10211.3	−	801.477i	1.01836	−	0.0799303i
$466$		0			0
$467$	−11576.2			−1.14707			−0.573536	−	0.819180i	$-0.694429\pi$
$467$	−11576.2			−1.14707			−0.573536	+	0.819180i	$0.694429\pi$
$468$		0			0
$469$		−	19880.3i		−	1.95733i
$470$		0			0
$471$	−21088.2			−2.06304
$472$		0			0
$473$		−	8645.31i		−	0.840406i
$474$		0			0
$475$	−9.27935	−	58.7479i	−0.000896349	−	0.00567482i
$476$		0			0
$477$	−973.053			−0.0934025
$478$		0			0
$479$	−12742.0			−1.21544			−0.607720	−	0.794152i	$-0.707916\pi$
$479$	−12742.0			−1.21544			−0.607720	+	0.794152i	$0.707916\pi$
$480$		0			0
$481$	1177.29			0.111600
$482$		0			0
$483$	−19967.6			−1.88107
$484$		0			0
$485$	−1119.48	−	14262.8i	−0.104810	−	1.33534i
$486$		0			0
$487$		−	13309.0i		−	1.23838i	−0.785243	−	0.619188i	$-0.787462\pi$
$487$		−	13309.0i		−	1.23838i	0.785243	−	0.619188i	$-0.212538\pi$
$488$		0			0
$489$	−9089.62			−0.840586
$490$		0			0
$491$			216.689i			0.0199166i	0.999950	+	0.00995830i	$0.00316988\pi$
$491$			216.689i			0.0199166i	−0.999950	+	0.00995830i	$0.996830\pi$
$492$		0			0
$493$	11883.9			1.08564
$494$		0			0
$495$	−535.248	−	6819.37i	−0.0486013	−	0.619208i
$496$		0			0
$497$			11462.8i			1.03456i
$498$		0			0
$499$		−	10800.4i		−	0.968921i	−0.874813	−	0.484460i	$-0.839016\pi$
$499$		−	10800.4i		−	0.968921i	0.874813	−	0.484460i	$-0.160984\pi$
$500$		0			0
$501$		−	4280.79i		−	0.381740i
$502$		0			0
$503$		−	16697.6i		−	1.48013i	−0.672533	−	0.740067i	$-0.734794\pi$
$503$		−	16697.6i		−	1.48013i	0.672533	−	0.740067i	$-0.265206\pi$
$504$		0			0
$505$	−705.317	−	8986.14i	−0.0621509	−	0.791838i
$506$		0			0
$507$	11827.9			1.03608
$508$		0			0
$509$			13066.1i			1.13781i	0.822402	+	0.568906i	$0.192633\pi$
$509$			13066.1i			1.13781i	−0.822402	+	0.568906i	$0.807367\pi$
$510$		0			0
$511$	22342.5			1.93419
$512$		0			0
$513$		−	29.9980i		−	0.00258176i
$514$		0			0
$515$	435.415	+	5547.44i	0.0372557	+	0.474659i
$516$		0			0
$517$	−11286.0			−0.960075
$518$		0			0
$519$	387.126			0.0327417
$520$		0			0
$521$	−269.920			−0.0226975			−0.0113488	−	0.999936i	$-0.503613\pi$
$521$	−269.920			−0.0226975			−0.0113488	+	0.999936i	$0.503613\pi$
$522$		0			0
$523$	19287.7			1.61260			0.806300	−	0.591506i	$-0.201467\pi$
$523$	19287.7			1.61260			0.806300	+	0.591506i	$0.201467\pi$
$524$		0			0
$525$	3882.50	+	24580.2i	0.322755	+	2.04337i
$526$		0			0
$527$			12349.2i			1.02076i
$528$		0			0
$529$	2107.06			0.173178
$530$		0			0
$531$		−	2947.08i		−	0.240852i
$532$		0			0
$533$	6155.72			0.500251
$534$		0			0
$535$	−1245.18	+	97.7332i	−0.100624	+	0.00789790i
$536$		0			0
$537$			10738.3i			0.862928i
$538$		0			0
$539$		−	19047.6i		−	1.52215i
$540$		0			0
$541$		−	8722.31i		−	0.693164i	−0.938020	−	0.346582i	$-0.887342\pi$
$541$		−	8722.31i		−	0.693164i	0.938020	−	0.346582i	$-0.112658\pi$
$542$		0			0
$543$			17792.7i			1.40618i
$544$		0			0
$545$	−960.748	−	12240.5i	−0.0755118	−	0.962063i
$546$		0			0
$547$	−8718.86			−0.681520			−0.340760	−	0.940150i	$-0.610684\pi$
$547$	−8718.86			−0.681520			−0.340760	+	0.940150i	$0.610684\pi$
$548$		0			0
$549$			2891.04i			0.224748i
$550$		0			0
$551$	−62.8437			−0.00485886
$552$		0			0
$553$			17375.0i			1.33609i
$554$		0			0
$555$	4248.62	−	333.472i	0.324944	−	0.0255047i
$556$		0			0
$557$	11758.7			0.894489			0.447245	−	0.894412i	$-0.352405\pi$
$557$	11758.7			0.894489			0.447245	+	0.894412i	$0.352405\pi$
$558$		0			0
$559$	−5113.94			−0.386935
$560$		0			0
$561$	20931.7			1.57529
$562$		0			0
$563$	5260.52			0.393791			0.196896	−	0.980424i	$-0.436914\pi$
$563$	5260.52			0.393791			0.196896	+	0.980424i	$0.436914\pi$
$564$		0			0
$565$	1853.77	+	23618.1i	0.138033	+	1.75862i
$566$		0			0
$567$			26687.7i			1.97668i
$568$		0			0
$569$	−19030.4			−1.40210			−0.701050	−	0.713112i	$-0.747285\pi$
$569$	−19030.4			−1.40210			−0.701050	+	0.713112i	$0.747285\pi$
$570$		0			0
$571$			23227.4i			1.70234i	0.524887	+	0.851172i	$0.324108\pi$
$571$			23227.4i			1.70234i	−0.524887	+	0.851172i	$0.675892\pi$
$572$		0			0
$573$	−19221.1			−1.40135
$574$		0			0
$575$	1956.06	+	12383.9i	0.141867	+	0.898163i
$576$		0			0
$577$		−	526.632i		−	0.0379965i	−0.999820	−	0.0189983i	$-0.993952\pi$
$577$		−	526.632i		−	0.0379965i	0.999820	−	0.0189983i	$-0.00604770\pi$
$578$		0			0
$579$			16028.9i			1.15049i
$580$		0			0
$581$		−	18980.9i		−	1.35535i
$582$		0			0
$583$		−	1931.83i		−	0.137235i
$584$		0			0
$585$	−4033.85	+	316.614i	−0.285092	+	0.0223767i
$586$		0			0
$587$	14220.5			0.999902			0.499951	−	0.866054i	$-0.333351\pi$
$587$	14220.5			0.999902			0.499951	+	0.866054i	$0.333351\pi$
$588$		0			0
$589$		−	65.3047i		−	0.00456848i
$590$		0			0
$591$	29730.4			2.06928
$592$		0			0
$593$		−	10794.3i		−	0.747500i	−0.927529	−	0.373750i	$-0.878072\pi$
$593$		−	10794.3i		−	0.747500i	0.927529	−	0.373750i	$-0.121928\pi$
$594$		0			0
$595$	−29911.0	+	2347.70i	−2.06089	+	0.161758i
$596$		0			0
$597$	32732.6			2.24398
$598$		0			0
$599$	−10036.9			−0.684634			−0.342317	−	0.939584i	$-0.611212\pi$
$599$	−10036.9			−0.684634			−0.342317	+	0.939584i	$0.611212\pi$
$600$		0			0
$601$	8251.09			0.560015			0.280008	−	0.959998i	$-0.409663\pi$
$601$	8251.09			0.560015			0.280008	+	0.959998i	$0.409663\pi$
$602$		0			0
$603$	11701.4			0.790245
$604$		0			0
$605$	−1296.74	+	101.780i	−0.0871404	+	0.00683960i
$606$		0			0
$607$		−	8119.05i		−	0.542903i	−0.962452	−	0.271451i	$-0.912496\pi$
$607$		−	8119.05i		−	0.542903i	0.962452	−	0.271451i	$-0.0875036\pi$
$608$		0			0
$609$	26293.9			1.74956
$610$		0			0
$611$			6675.99i			0.442032i
$612$		0			0
$613$	−20527.3			−1.35251			−0.676255	−	0.736668i	$-0.736398\pi$
$613$	−20527.3			−1.35251			−0.676255	+	0.736668i	$0.736398\pi$
$614$		0			0
$615$	22214.9	−	1743.64i	1.45657	−	0.114326i
$616$		0			0
$617$		−	27545.7i		−	1.79732i	−0.438643	−	0.898661i	$-0.644541\pi$
$617$		−	27545.7i		−	1.79732i	0.438643	−	0.898661i	$-0.355459\pi$
$618$		0			0
$619$			1777.17i			0.115396i	0.998334	+	0.0576982i	$0.0183761\pi$
$619$			1777.17i			0.115396i	−0.998334	+	0.0576982i	$0.981624\pi$
$620$		0			0
$621$			6323.49i			0.408620i
$622$		0			0
$623$		−	22664.4i		−	1.45751i
$624$		0			0
$625$	14864.3	−	4815.85i	0.951317	−	0.308214i
$626$		0			0
$627$	−110.690			−0.00705028
$628$		0			0
$629$			5138.17i			0.325711i
$630$		0			0
$631$	−16484.4			−1.03999			−0.519995	−	0.854169i	$-0.674066\pi$
$631$	−16484.4			−1.03999			−0.519995	+	0.854169i	$0.674066\pi$
$632$		0			0
$633$		−	8187.43i		−	0.514093i
$634$		0			0
$635$	−496.887	−	6330.63i	−0.0310526	−	0.395627i
$636$		0			0
$637$	−11267.2			−0.700821
$638$		0			0
$639$	−6746.91			−0.417689
$640$		0			0
$641$	−24.7805			−0.00152694			−0.000763470	−	1.00000i	$-0.500243\pi$
$641$	−24.7805			−0.00152694			−0.000763470		1.00000i	$0.500243\pi$
$642$		0			0
$643$	−29107.1			−1.78518			−0.892590	−	0.450870i	$-0.851114\pi$
$643$	−29107.1			−1.78518			−0.892590	+	0.450870i	$0.851114\pi$
$644$		0			0
$645$	−18455.3	+	1448.55i	−1.12663	+	0.0884288i
$646$		0			0
$647$			13419.6i			0.815421i	0.913111	+	0.407711i	$0.133673\pi$
$647$			13419.6i			0.815421i	−0.913111	+	0.407711i	$0.866327\pi$
$648$		0			0
$649$	5850.91			0.353880
$650$		0			0
$651$			27323.6i			1.64500i
$652$		0			0
$653$	−1926.26			−0.115437			−0.0577186	−	0.998333i	$-0.518383\pi$
$653$	−1926.26			−0.115437			−0.0577186	+	0.998333i	$0.518383\pi$
$654$		0			0
$655$	−1958.64	−	24954.2i	−0.116840	−	1.48861i
$656$		0			0
$657$			13150.6i			0.780905i
$658$		0			0
$659$		−	13158.1i		−	0.777797i	−0.921281	−	0.388899i	$-0.872856\pi$
$659$		−	13158.1i		−	0.777797i	0.921281	−	0.388899i	$-0.127144\pi$
$660$		0			0
$661$			27392.1i			1.61184i	0.592023	+	0.805921i	$0.298330\pi$
$661$			27392.1i			1.61184i	−0.592023	+	0.805921i	$0.701670\pi$
$662$		0			0
$663$		−	12381.7i		−	0.725285i
$664$		0			0
$665$	158.174	−	12.4150i	0.00922364	−	0.000723959i
$666$		0			0
$667$	13247.3			0.769019
$668$		0			0
$669$		−	10297.9i		−	0.595128i
$670$		0			0
$671$	−5739.65			−0.330219
$672$		0			0
$673$			5569.75i			0.319017i	0.987197	+	0.159508i	$0.0509909\pi$
$673$			5569.75i			0.319017i	−0.987197	+	0.159508i	$0.949009\pi$
$674$		0			0
$675$	7784.27	−	1229.54i	0.443877	−	0.0701112i
$676$		0			0
$677$	−4130.23			−0.234472			−0.117236	−	0.993104i	$-0.537403\pi$
$677$	−4130.23			−0.234472			−0.117236	+	0.993104i	$0.537403\pi$
$678$		0			0
$679$	38164.8			2.15704
$680$		0			0
$681$	−25707.5			−1.44657
$682$		0			0
$683$	−22875.3			−1.28155			−0.640776	−	0.767728i	$-0.721387\pi$
$683$	−22875.3			−1.28155			−0.640776	+	0.767728i	$0.721387\pi$
$684$		0			0
$685$	−1249.84	−	15923.7i	−0.0697138	−	0.888193i
$686$		0			0
$687$			34927.3i			1.93968i
$688$		0			0
$689$	−1142.73			−0.0631851
$690$		0			0
$691$		−	19910.4i		−	1.09613i	−0.836435	−	0.548066i	$-0.815364\pi$
$691$		−	19910.4i		−	1.09613i	0.836435	−	0.548066i	$-0.184636\pi$
$692$		0			0
$693$	18247.5			1.00024
$694$		0			0
$695$	−1284.10	−	16360.1i	−0.0700842	−	0.892912i
$696$		0			0
$697$			26866.2i			1.46001i
$698$		0			0
$699$			1210.96i			0.0655262i
$700$		0			0
$701$		−	3428.44i		−	0.184722i	−0.995726	−	0.0923612i	$-0.970559\pi$
$701$		−	3428.44i		−	0.184722i	0.995726	−	0.0923612i	$-0.0294414\pi$
$702$		0			0
$703$		−	27.1715i		−	0.00145774i
$704$		0			0
$705$	1891.01	+	24092.5i	0.101021	+	1.28706i
$706$		0			0
$707$	24045.4			1.27910
$708$		0			0
$709$			15608.3i			0.826771i	0.910556	+	0.413385i	$0.135654\pi$
$709$			15608.3i			0.826771i	−0.910556	+	0.413385i	$0.864346\pi$
$710$		0			0
$711$	−10226.8			−0.539430
$712$		0			0
$713$			13766.0i			0.723061i
$714$		0			0
$715$	−628.583	−	8008.50i	−0.0328778	−	0.418882i
$716$		0			0
$717$	16739.1			0.871873
$718$		0			0
$719$	−8282.41			−0.429599			−0.214800	−	0.976658i	$-0.568910\pi$
$719$	−8282.41			−0.429599			−0.214800	+	0.976658i	$0.568910\pi$
$720$		0			0
$721$	−14844.0			−0.766741
$722$		0			0
$723$	7272.08			0.374068
$724$		0			0
$725$	−2575.80	−	16307.5i	−0.131949	−	0.835372i
$726$		0			0
$727$		−	20590.4i		−	1.05042i	−0.850973	−	0.525210i	$-0.823987\pi$
$727$		−	20590.4i		−	1.05042i	0.850973	−	0.525210i	$-0.176013\pi$
$728$		0			0
$729$	−4345.77			−0.220788
$730$		0			0
$731$		−	22319.4i		−	1.12929i
$732$		0			0
$733$	24257.3			1.22232			0.611162	−	0.791506i	$-0.290703\pi$
$733$	24257.3			1.22232			0.611162	+	0.791506i	$0.290703\pi$
$734$		0			0
$735$	−40661.4	+	3191.49i	−2.04057	+	0.160163i
$736$		0			0
$737$			23231.1i			1.16110i
$738$		0			0
$739$			25426.8i			1.26569i	0.774280	+	0.632843i	$0.218112\pi$
$739$			25426.8i			1.26569i	−0.774280	+	0.632843i	$0.781888\pi$
$740$		0			0
$741$			65.4761i			0.00324605i
$742$		0			0
$743$			22658.4i			1.11878i	0.828904	+	0.559391i	$0.188965\pi$
$743$			22658.4i			1.11878i	−0.828904	+	0.559391i	$0.811035\pi$
$744$		0			0
$745$	−2520.56	−	32113.4i	−0.123955	−	1.57925i
$746$		0			0
$747$	11172.0			0.547205
$748$		0			0
$749$		−	3331.88i		−	0.162542i
$750$		0			0
$751$	10999.0			0.534431			0.267215	−	0.963637i	$-0.413896\pi$
$751$	10999.0			0.534431			0.267215	+	0.963637i	$0.413896\pi$
$752$		0			0
$753$		−	35911.3i		−	1.73795i
$754$		0			0
$755$	−10580.9	+	830.485i	−0.510035	+	0.0400324i
$756$		0			0
$757$	30279.9			1.45382			0.726909	−	0.686734i	$-0.240956\pi$
$757$	30279.9			1.45382			0.726909	+	0.686734i	$0.240956\pi$
$758$		0			0
$759$	23333.1			1.11586
$760$		0			0
$761$	4374.09			0.208358			0.104179	−	0.994559i	$-0.466778\pi$
$761$	4374.09			0.208358			0.104179	+	0.994559i	$0.466778\pi$
$762$		0			0
$763$	32753.4			1.55407
$764$		0			0
$765$	−1381.84	−	17605.4i	−0.0653078	−	0.832058i
$766$		0			0
$767$		−	3460.97i		−	0.162932i
$768$		0			0
$769$	−12982.1			−0.608774			−0.304387	−	0.952548i	$-0.598452\pi$
$769$	−12982.1			−0.608774			−0.304387	+	0.952548i	$0.598452\pi$
$770$		0			0
$771$		−	9197.90i		−	0.429642i
$772$		0			0
$773$	−34056.3			−1.58463			−0.792316	−	0.610112i	$-0.791125\pi$
$773$	−34056.3			−1.58463			−0.792316	+	0.610112i	$0.791125\pi$
$774$		0			0
$775$	16946.1	−	2676.67i	0.785448	−	0.124063i
$776$		0			0
$777$			11368.6i			0.524898i
$778$		0			0
$779$		−	142.073i		−	0.00653437i
$780$		0			0
$781$		−	13394.8i		−	0.613706i
$782$		0			0
$783$		−	8326.97i		−	0.380053i
$784$		0			0
$785$	−35213.8	+	2763.91i	−1.60106	+	0.125667i
$786$		0			0
$787$	−11962.9			−0.541842			−0.270921	−	0.962602i	$-0.587328\pi$
$787$	−11962.9			−0.541842			−0.270921	+	0.962602i	$0.587328\pi$
$788$		0			0
$789$		−	2745.67i		−	0.123889i
$790$		0			0
$791$	−63198.1			−2.84079
$792$		0			0
$793$			3395.17i			0.152038i
$794$		0			0
$795$	−4123.91	+	323.684i	−0.183975	+	0.0144401i
$796$		0			0
$797$	−17898.9			−0.795498			−0.397749	−	0.917494i	$-0.630209\pi$
$797$	−17898.9			−0.795498			−0.397749	+	0.917494i	$0.630209\pi$
$798$		0			0
$799$	−29136.9			−1.29010
$800$		0			0
$801$	13340.1			0.588452
$802$		0			0
$803$	−26108.3			−1.14737
$804$		0			0
$805$	−33342.6	+	2617.04i	−1.45984	+	0.114582i
$806$		0			0
$807$		−	2718.06i		−	0.118563i
$808$		0			0
$809$	27690.6			1.20340			0.601700	−	0.798722i	$-0.294490\pi$
$809$	27690.6			1.20340			0.601700	+	0.798722i	$0.294490\pi$
$810$		0			0
$811$		−	44013.9i		−	1.90572i	−0.303412	−	0.952859i	$-0.598126\pi$
$811$		−	44013.9i		−	1.90572i	0.303412	−	0.952859i	$-0.401874\pi$
$812$		0			0
$813$	−8871.24			−0.382691
$814$		0			0
$815$	−15178.2	+	1191.33i	−0.652354	+	0.0512029i
$816$		0			0
$817$			118.028i			0.00505422i
$818$		0			0
$819$		−	10793.9i		−	0.460524i
$820$		0			0
$821$		−	30978.8i		−	1.31689i	−0.752628	−	0.658446i	$-0.771214\pi$
$821$		−	30978.8i		−	1.31689i	0.752628	−	0.658446i	$-0.228786\pi$
$822$		0			0
$823$			41684.8i			1.76554i	0.469804	+	0.882771i	$0.344325\pi$
$823$			41684.8i			1.76554i	−0.469804	+	0.882771i	$0.655675\pi$
$824$		0			0
$825$	−4536.89	−	28723.2i	−0.191460	−	1.21214i
$826$		0			0
$827$	−11988.7			−0.504096			−0.252048	−	0.967715i	$-0.581104\pi$
$827$	−11988.7			−0.504096			−0.252048	+	0.967715i	$0.581104\pi$
$828$		0			0
$829$			2023.64i			0.0847814i	0.999101	+	0.0423907i	$0.0134974\pi$
$829$			2023.64i			0.0847814i	−0.999101	+	0.0423907i	$0.986503\pi$
$830$		0			0
$831$	29445.4			1.22918
$832$		0			0
$833$		−	49174.8i		−	2.04539i
$834$		0			0
$835$	−561.061	−	7148.23i	−0.0232531	−	0.296257i
$836$		0			0
$837$	8653.07			0.357340
$838$		0			0
$839$	10706.3			0.440553			0.220276	−	0.975438i	$-0.429304\pi$
$839$	10706.3			0.440553			0.220276	+	0.975438i	$0.429304\pi$
$840$		0			0
$841$	6944.60			0.284743
$842$		0			0
$843$	−34270.6			−1.40017
$844$		0			0
$845$	19750.6	−	1550.22i	0.804074	−	0.0631114i
$846$		0			0
$847$		−	3469.86i		−	0.140762i
$848$		0			0
$849$	−20649.2			−0.834721
$850$		0			0
$851$			5727.66i			0.230719i
$852$		0			0
$853$	29816.5			1.19683			0.598416	−	0.801186i	$-0.295797\pi$
$853$	29816.5			1.19683			0.598416	+	0.801186i	$0.295797\pi$
$854$		0			0
$855$	7.30738	+	93.1001i	0.000292289	+	0.00372393i
$856$		0			0
$857$		−	6572.14i		−	0.261960i	−0.991385	−	0.130980i	$-0.958188\pi$
$857$		−	6572.14i		−	0.261960i	0.991385	−	0.130980i	$-0.0418124\pi$
$858$		0			0
$859$			45723.7i			1.81615i	0.418809	+	0.908075i	$0.362448\pi$
$859$			45723.7i			1.81615i	−0.418809	+	0.908075i	$0.637552\pi$
$860$		0			0
$861$			59443.5i			2.35288i
$862$		0			0
$863$		−	13343.7i		−	0.526331i	−0.964751	−	0.263166i	$-0.915233\pi$
$863$		−	13343.7i		−	0.526331i	0.964751	−	0.263166i	$-0.0847665\pi$
$864$		0			0
$865$	646.437	−	50.7385i	0.0254098	−	0.00199440i
$866$		0			0
$867$	21244.8			0.832193
$868$		0			0
$869$		−	20303.5i		−	0.792577i
$870$		0			0
$871$	13741.8			0.534586
$872$		0			0
$873$			22463.6i			0.870878i
$874$		0			0
$875$	9704.74	+	40536.2i	0.374949	+	1.56614i
$876$		0			0
$877$	−41636.8			−1.60317			−0.801583	−	0.597884i	$-0.796008\pi$
$877$	−41636.8			−1.60317			−0.801583	+	0.597884i	$0.796008\pi$
$878$		0			0
$879$	4481.01			0.171946
$880$		0			0
$881$	−31220.6			−1.19392			−0.596962	−	0.802269i	$-0.703626\pi$
$881$	−31220.6			−1.19392			−0.596962	+	0.802269i	$0.703626\pi$
$882$		0			0
$883$	35199.4			1.34151			0.670755	−	0.741679i	$-0.265970\pi$
$883$	35199.4			1.34151			0.670755	+	0.741679i	$0.265970\pi$
$884$		0			0
$885$	−980.338	−	12490.1i	−0.0372358	−	0.474405i
$886$		0			0
$887$			12744.8i			0.482444i	0.970470	+	0.241222i	$0.0775482\pi$
$887$			12744.8i			0.482444i	−0.970470	+	0.241222i	$0.922452\pi$
$888$		0			0
$889$	16939.7			0.639076
$890$		0			0
$891$		−	31185.8i		−	1.17258i
$892$		0			0
$893$	154.080			0.00577391
$894$		0			0
$895$	1407.41	+	17931.2i	0.0525638	+	0.669693i
$896$		0			0
$897$		−	13802.2i		−	0.513758i
$898$		0			0
$899$		−	18127.6i		−	0.672512i
$900$		0			0
$901$		−	4987.35i		−	0.184409i
$902$		0			0
$903$		−	49383.4i		−	1.81991i
$904$		0			0
$905$	2331.99	+	29710.9i	0.0856553	+	1.09130i
$906$		0			0
$907$	16588.8			0.607299			0.303650	−	0.952784i	$-0.401795\pi$
$907$	16588.8			0.607299			0.303650	+	0.952784i	$0.401795\pi$
$908$		0			0
$909$			14153.0i			0.516418i
$910$		0			0
$911$	46114.7			1.67711			0.838555	−	0.544816i	$-0.183401\pi$
$911$	46114.7			1.67711			0.838555	+	0.544816i	$0.183401\pi$
$912$		0			0
$913$			22180.1i			0.804002i
$914$		0			0
$915$	961.697	+	12252.6i	0.0347461	+	0.442686i
$916$		0			0
$917$	66773.3			2.40463
$918$		0			0
$919$	−47522.6			−1.70580			−0.852898	−	0.522077i	$-0.825157\pi$
$919$	−47522.6			−1.70580			−0.852898	+	0.522077i	$0.825157\pi$
$920$		0			0
$921$	38786.6			1.38769
$922$		0			0
$923$	−7923.40			−0.282559
$924$		0			0
$925$	7050.80	−	1113.69i	0.250626	−	0.0395868i
$926$		0			0
$927$		−	8737.09i		−	0.309562i
$928$		0			0
$929$	13494.3			0.476570			0.238285	−	0.971195i	$-0.423415\pi$
$929$	13494.3			0.476570			0.238285	+	0.971195i	$0.423415\pi$
$930$		0			0
$931$			260.044i			0.00915425i
$932$		0			0
$933$	−48215.1			−1.69184
$934$		0			0
$935$	34952.5	−	2743.40i	1.22253	−	0.0959559i
$936$		0			0
$937$		−	12453.1i		−	0.434179i	−0.976152	−	0.217089i	$-0.930344\pi$
$937$		−	12453.1i		−	0.434179i	0.976152	−	0.217089i	$-0.0696563\pi$
$938$		0			0
$939$		−	12044.3i		−	0.418584i
$940$		0			0
$941$			40466.5i			1.40188i	0.713220	+	0.700940i	$0.247236\pi$
$941$			40466.5i			1.40188i	−0.713220	+	0.700940i	$0.752764\pi$
$942$		0			0
$943$			29948.5i			1.03421i
$944$		0			0
$945$	1645.02	+	20958.5i	0.0566271	+	0.721461i
$946$		0			0
$947$	−2645.96			−0.0907943			−0.0453971	−	0.998969i	$-0.514455\pi$
$947$	−2645.96			−0.0907943			−0.0453971	+	0.998969i	$0.514455\pi$
$948$		0			0
$949$			15443.8i			0.528268i
$950$		0			0
$951$	−10633.7			−0.362587
$952$		0			0
$953$			40045.8i			1.36119i	0.732662	+	0.680593i	$0.238278\pi$
$953$			40045.8i			1.36119i	−0.732662	+	0.680593i	$0.761722\pi$
$954$		0			0
$955$	−32096.1	+	2519.21i	−1.08755	+	0.0853609i
$956$		0			0
$957$	−30725.8			−1.03785
$958$		0			0
$959$	42609.1			1.43474
$960$		0			0
$961$	−10953.5			−0.367679
$962$		0			0
$963$	1961.12			0.0656244
$964$		0			0
$965$	2100.82	+	26765.6i	0.0700805	+	0.892865i
$966$		0			0
$967$			11049.0i			0.367439i	0.982979	+	0.183719i	$0.0588137\pi$
$967$			11049.0i			0.367439i	−0.982979	+	0.183719i	$0.941186\pi$
$968$		0			0
$969$	−285.765			−0.00947380
$970$		0			0
$971$			26708.5i			0.882717i	0.897331	+	0.441358i	$0.145503\pi$
$971$			26708.5i			0.882717i	−0.897331	+	0.441358i	$0.854497\pi$
$972$		0			0
$973$	43776.9			1.44237
$974$		0			0
$975$	−16990.6	+	2683.70i	−0.558087	+	0.0881509i
$976$		0			0
$977$			11102.6i			0.363566i	0.983339	+	0.181783i	$0.0581869\pi$
$977$			11102.6i			0.363566i	−0.983339	+	0.181783i	$0.941813\pi$
$978$		0			0
$979$			26484.5i			0.864604i
$980$		0			0
$981$			19278.5i			0.627435i
$982$		0			0
$983$		−	33544.2i		−	1.08840i	−0.838957	−	0.544198i	$-0.816834\pi$
$983$		−	33544.2i		−	1.08840i	0.838957	−	0.544198i	$-0.183166\pi$
$984$		0			0
$985$	49644.9	−	3896.60i	1.60591	−	0.126047i
$986$		0			0
$987$	−64467.5			−2.07905
$988$		0			0
$989$		−	24880.0i		−	0.799939i
$990$		0			0
$991$	−5538.20			−0.177524			−0.0887622	−	0.996053i	$-0.528291\pi$
$991$	−5538.20			−0.177524			−0.0887622	+	0.996053i	$0.528291\pi$
$992$		0			0
$993$			41472.5i			1.32537i
$994$		0			0
$995$	54658.2	−	4290.09i	1.74149	−	0.136688i
$996$		0			0
$997$	−24192.6			−0.768492			−0.384246	−	0.923231i	$-0.625539\pi$
$997$	−24192.6			−0.768492			−0.384246	+	0.923231i	$0.625539\pi$
$998$		0			0
$999$	3600.30			0.114022

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.4.f.a.49.4		16
3.2	odd	2		1440.4.d.d.1009.15		16
4.3	odd	2		40.4.f.a.29.11	yes	16
5.2	odd	4		800.4.d.e.401.14		16
5.3	odd	4		800.4.d.e.401.3		16
5.4	even	2	inner	160.4.f.a.49.14		16
8.3	odd	2		40.4.f.a.29.5	✓	16
8.5	even	2	inner	160.4.f.a.49.13		16
12.11	even	2		360.4.d.d.109.6		16
15.14	odd	2		1440.4.d.d.1009.1		16
20.3	even	4		200.4.d.e.101.3		16
20.7	even	4		200.4.d.e.101.14		16
20.19	odd	2		40.4.f.a.29.6	yes	16
24.5	odd	2		1440.4.d.d.1009.2		16
24.11	even	2		360.4.d.d.109.12		16
40.3	even	4		200.4.d.e.101.4		16
40.13	odd	4		800.4.d.e.401.13		16
40.19	odd	2		40.4.f.a.29.12	yes	16
40.27	even	4		200.4.d.e.101.13		16
40.29	even	2	inner	160.4.f.a.49.3		16
40.37	odd	4		800.4.d.e.401.4		16
60.59	even	2		360.4.d.d.109.11		16
120.29	odd	2		1440.4.d.d.1009.16		16
120.59	even	2		360.4.d.d.109.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.f.a.29.5	✓	16	8.3	odd	2
40.4.f.a.29.6	yes	16	20.19	odd	2
40.4.f.a.29.11	yes	16	4.3	odd	2
40.4.f.a.29.12	yes	16	40.19	odd	2
160.4.f.a.49.3		16	40.29	even	2	inner
160.4.f.a.49.4		16	1.1	even	1	trivial
160.4.f.a.49.13		16	8.5	even	2	inner
160.4.f.a.49.14		16	5.4	even	2	inner
200.4.d.e.101.3		16	20.3	even	4
200.4.d.e.101.4		16	40.3	even	4
200.4.d.e.101.13		16	40.27	even	4
200.4.d.e.101.14		16	20.7	even	4
360.4.d.d.109.5		16	120.59	even	2
360.4.d.d.109.6		16	12.11	even	2
360.4.d.d.109.11		16	60.59	even	2
360.4.d.d.109.12		16	24.11	even	2
800.4.d.e.401.3		16	5.3	odd	4
800.4.d.e.401.4		16	40.37	odd	4
800.4.d.e.401.13		16	40.13	odd	4
800.4.d.e.401.14		16	5.2	odd	4
1440.4.d.d.1009.1		16	15.14	odd	2
1440.4.d.d.1009.2		16	24.5	odd	2
1440.4.d.d.1009.15		16	3.2	odd	2
1440.4.d.d.1009.16		16	120.29	odd	2

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.67494 q^{3} +(-11.1461 + 0.874848i) q^{5} -29.8250i q^{7} +17.5548 q^{9} +34.8520i q^{11} +20.6159 q^{13} +(74.3992 - 5.83956i) q^{15} +89.9765i q^{17} -0.475810i q^{19} +199.080i q^{21} +100.299i q^{23} +(123.469 - 19.5022i) q^{25} +63.0462 q^{27} -132.077i q^{29} +137.250 q^{31} -232.635i q^{33} +(26.0923 + 332.431i) q^{35} +57.1057 q^{37} -137.610 q^{39} +298.591 q^{41} -248.058 q^{43} +(-195.667 + 15.3578i) q^{45} +323.827i q^{47} -546.530 q^{49} -600.588i q^{51} -55.4295 q^{53} +(-30.4902 - 388.462i) q^{55} +3.17600i q^{57} -167.879i q^{59} +164.687i q^{61} -523.571i q^{63} +(-229.786 + 18.0358i) q^{65} +666.565 q^{67} -669.491i q^{69} -384.334 q^{71} +749.119i q^{73} +(-824.150 + 130.176i) q^{75} +1039.46 q^{77} -582.564 q^{79} -894.809 q^{81} +636.408 q^{83} +(-78.7158 - 1002.88i) q^{85} +881.607i q^{87} +759.913 q^{89} -614.869i q^{91} -916.133 q^{93} +(0.416261 + 5.30340i) q^{95} +1279.63i q^{97} +611.819i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 104 q^{9} + 56 q^{15} - 24 q^{25} + 112 q^{31} + 736 q^{39} + 232 q^{41} - 200 q^{49} - 392 q^{55} - 600 q^{65} - 2096 q^{71} - 2992 q^{79} - 728 q^{81} - 208 q^{89} + 1064 q^{95}+O(q^{100}) \) 16 * q + 104 * q^9 + 56 * q^15 - 24 * q^25 + 112 * q^31 + 736 * q^39 + 232 * q^41 - 200 * q^49 - 392 * q^55 - 600 * q^65 - 2096 * q^71 - 2992 * q^79 - 728 * q^81 - 208 * q^89 + 1064 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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