LMFDB - Embedded newform 160.4.f.a.49.15

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			49.15
Root			$2.15123 + 1.83636i$ of defining polynomial
Character	$\chi$	$=$	160.49
Dual form			160.4.f.a.49.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.63004 q^{3} +(-2.63740 - 10.8648i) q^{5} +4.74133i q^{7} +47.4777 q^{9} +O(q^{10})$	$q+8.63004 q^{3} +(-2.63740 - 10.8648i) q^{5} +4.74133i q^{7} +47.4777 q^{9} -31.8831i q^{11} +59.2167 q^{13} +(-22.7609 - 93.7638i) q^{15} -80.9370i q^{17} +114.647i q^{19} +40.9179i q^{21} -28.3687i q^{23} +(-111.088 + 57.3097i) q^{25} +176.723 q^{27} +146.340i q^{29} -77.6995 q^{31} -275.153i q^{33} +(51.5136 - 12.5048i) q^{35} +201.444 q^{37} +511.042 q^{39} -99.3433 q^{41} -210.291 q^{43} +(-125.217 - 515.836i) q^{45} +33.3026i q^{47} +320.520 q^{49} -698.490i q^{51} -255.386 q^{53} +(-346.404 + 84.0885i) q^{55} +989.413i q^{57} +589.854i q^{59} +573.462i q^{61} +225.107i q^{63} +(-156.178 - 643.378i) q^{65} -23.9755 q^{67} -244.823i q^{69} -470.655 q^{71} +676.417i q^{73} +(-958.697 + 494.585i) q^{75} +151.168 q^{77} -1085.76 q^{79} +243.231 q^{81} +78.2562 q^{83} +(-879.366 + 213.463i) q^{85} +1262.92i q^{87} -606.174 q^{89} +280.766i q^{91} -670.550 q^{93} +(1245.62 - 302.371i) q^{95} -151.583i q^{97} -1513.74i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 104 q^{9}+O(q^{10}) $ 16 * q + 104 * q^9	$ 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

Display 100 coefficients 10 coefficients

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
$3$	8.63004			1.66085			0.830426	−	0.557128i	$-0.188097\pi$
$3$	8.63004			1.66085			0.830426	+	0.557128i	$0.188097\pi$
$4$		0			0
$5$	−2.63740	−	10.8648i	−0.235896	−	0.971778i
$5$	−2.63740	−	10.8648i	−0.235896	−	0.971778i
$6$		0			0
$7$			4.74133i			0.256008i	0.991774	+	0.128004i	$0.0408570\pi$
$7$			4.74133i			0.256008i	−0.991774	+	0.128004i	$0.959143\pi$
$8$		0			0
$9$	47.4777			1.75843
$10$		0			0
$11$		−	31.8831i		−	0.873921i	−0.899481	−	0.436960i	$-0.856055\pi$
$11$		−	31.8831i		−	0.873921i	0.899481	−	0.436960i	$-0.143945\pi$
$12$		0			0
$13$	59.2167			1.26337			0.631683	−	0.775227i	$-0.282365\pi$
$13$	59.2167			1.26337			0.631683	+	0.775227i	$0.282365\pi$
$14$		0			0
$15$	−22.7609	−	93.7638i	−0.391789	−	1.61398i
$16$		0			0
$17$		−	80.9370i		−	1.15471i	−0.816492	−	0.577356i	$-0.804084\pi$
$17$		−	80.9370i		−	1.15471i	0.816492	−	0.577356i	$-0.195916\pi$
$18$		0			0
$19$			114.647i			1.38431i	0.721748	+	0.692156i	$0.243339\pi$
$19$			114.647i			1.38431i	−0.721748	+	0.692156i	$0.756661\pi$
$20$		0			0
$21$			40.9179i			0.425191i
$22$		0			0
$23$		−	28.3687i		−	0.257186i	−0.991697	−	0.128593i	$-0.958954\pi$
$23$		−	28.3687i		−	0.257186i	0.991697	−	0.128593i	$-0.0410461\pi$
$24$		0			0
$25$	−111.088	+	57.3097i	−0.888706	+	0.458477i
$26$		0			0
$27$	176.723			1.25964
$28$		0			0
$29$			146.340i			0.937060i	0.883448	+	0.468530i	$0.155216\pi$
$29$			146.340i			0.937060i	−0.883448	+	0.468530i	$0.844784\pi$
$30$		0			0
$31$	−77.6995			−0.450169			−0.225085	−	0.974339i	$-0.572266\pi$
$31$	−77.6995			−0.450169			−0.225085	+	0.974339i	$0.572266\pi$
$32$		0			0
$33$		−	275.153i		−	1.45145i
$34$		0			0
$35$	51.5136	−	12.5048i	0.248783	−	0.0603912i
$36$		0			0
$37$	201.444			0.895060			0.447530	−	0.894269i	$-0.352304\pi$
$37$	201.444			0.895060			0.447530	+	0.894269i	$0.352304\pi$
$38$		0			0
$39$	511.042			2.09826
$40$		0			0
$41$	−99.3433			−0.378410			−0.189205	−	0.981938i	$-0.560591\pi$
$41$	−99.3433			−0.378410			−0.189205	+	0.981938i	$0.560591\pi$
$42$		0			0
$43$	−210.291			−0.745794			−0.372897	−	0.927873i	$-0.621635\pi$
$43$	−210.291			−0.745794			−0.372897	+	0.927873i	$0.621635\pi$
$44$		0			0
$45$	−125.217	−	515.836i	−0.414807	−	1.70881i
$46$		0			0
$47$			33.3026i			0.103355i	0.998664	+	0.0516776i	$0.0164568\pi$
$47$			33.3026i			0.103355i	−0.998664	+	0.0516776i	$0.983543\pi$
$48$		0			0
$49$	320.520			0.934460
$50$		0			0
$51$		−	698.490i		−	1.91781i
$52$		0			0
$53$	−255.386			−0.661886			−0.330943	−	0.943651i	$-0.607367\pi$
$53$	−255.386			−0.661886			−0.330943	+	0.943651i	$0.607367\pi$
$54$		0			0
$55$	−346.404	+	84.0885i	−0.849257	+	0.206154i
$56$		0			0
$57$			989.413i			2.29914i
$58$		0			0
$59$			589.854i			1.30157i	0.759263	+	0.650783i	$0.225559\pi$
$59$			589.854i			1.30157i	−0.759263	+	0.650783i	$0.774441\pi$
$60$		0			0
$61$			573.462i			1.20368i	0.798618	+	0.601838i	$0.205565\pi$
$61$			573.462i			1.20368i	−0.798618	+	0.601838i	$0.794435\pi$
$62$		0			0
$63$			225.107i			0.450172i
$64$		0			0
$65$	−156.178	−	643.378i	−0.298023	−	1.22771i
$66$		0			0
$67$	−23.9755			−0.0437176			−0.0218588	−	0.999761i	$-0.506958\pi$
$67$	−23.9755			−0.0437176			−0.0218588	+	0.999761i	$0.506958\pi$
$68$		0			0
$69$		−	244.823i		−	0.427148i
$70$		0			0
$71$	−470.655			−0.786711			−0.393355	−	0.919386i	$-0.628686\pi$
$71$	−470.655			−0.786711			−0.393355	+	0.919386i	$0.628686\pi$
$72$		0			0
$73$			676.417i			1.08450i	0.840217	+	0.542251i	$0.182428\pi$
$73$			676.417i			1.08450i	−0.840217	+	0.542251i	$0.817572\pi$
$74$		0			0
$75$	−958.697	+	494.585i	−1.47601	+	0.761463i
$76$		0			0
$77$	151.168			0.223730
$78$		0			0
$79$	−1085.76			−1.54630			−0.773152	−	0.634221i	$-0.781321\pi$
$79$	−1085.76			−1.54630			−0.773152	+	0.634221i	$0.781321\pi$
$80$		0			0
$81$	243.231			0.333650
$82$		0			0
$83$	78.2562			0.103491			0.0517454	−	0.998660i	$-0.483522\pi$
$83$	78.2562			0.103491			0.0517454	+	0.998660i	$0.483522\pi$
$84$		0			0
$85$	−879.366	+	213.463i	−1.12212	+	0.272392i
$86$		0			0
$87$			1262.92i			1.55632i
$88$		0			0
$89$	−606.174			−0.721959			−0.360979	−	0.932574i	$-0.617558\pi$
$89$	−606.174			−0.721959			−0.360979	+	0.932574i	$0.617558\pi$
$90$		0			0
$91$			280.766i			0.323431i
$92$		0			0
$93$	−670.550			−0.747665
$94$		0			0
$95$	1245.62	−	302.371i	1.34524	−	0.326554i
$96$		0			0
$97$		−	151.583i		−	0.158670i	−0.996848	−	0.0793349i	$-0.974720\pi$
$97$		−	151.583i		−	0.158670i	0.996848	−	0.0793349i	$-0.0252796\pi$
$98$		0			0
$99$		−	1513.74i		−	1.53673i
$100$		0			0
$101$		−	64.6577i		−	0.0636999i	−0.999493	−	0.0318499i	$-0.989860\pi$
$101$		−	64.6577i		−	0.0636999i	0.999493	−	0.0318499i	$-0.0101399\pi$
$102$		0			0
$103$		−	1294.08i		−	1.23796i	−0.785409	−	0.618978i	$-0.787547\pi$
$103$		−	1294.08i		−	1.23796i	0.785409	−	0.618978i	$-0.212453\pi$
$104$		0			0
$105$	444.565	−	107.917i	0.413191	−	0.100301i
$106$		0			0
$107$	1624.59			1.46780			0.733902	−	0.679256i	$-0.237697\pi$
$107$	1624.59			1.46780			0.733902	+	0.679256i	$0.237697\pi$
$108$		0			0
$109$		−	1247.60i		−	1.09632i	−0.836375	−	0.548158i	$-0.815329\pi$
$109$		−	1247.60i		−	1.09632i	0.836375	−	0.548158i	$-0.184671\pi$
$110$		0			0
$111$	1738.47			1.48656
$112$		0			0
$113$		−	261.173i		−	0.217426i	−0.994073	−	0.108713i	$-0.965327\pi$
$113$		−	261.173i		−	0.217426i	0.994073	−	0.108713i	$-0.0346729\pi$
$114$		0			0
$115$	−308.220	+	74.8195i	−0.249928	+	0.0606692i
$116$		0			0
$117$	2811.47			2.22154
$118$		0			0
$119$	383.749			0.295615
$120$		0			0
$121$	314.466			0.236263
$122$		0			0
$123$	−857.337			−0.628483
$124$		0			0
$125$	915.643	+	1055.80i	0.655181	+	0.755472i
$126$		0			0
$127$			1916.02i			1.33874i	0.742931	+	0.669368i	$0.233435\pi$
$127$			1916.02i			1.33874i	−0.742931	+	0.669368i	$0.766565\pi$
$128$		0			0
$129$	−1814.82			−1.23865
$130$		0			0
$131$			126.455i			0.0843394i	0.999110	+	0.0421697i	$0.0134270\pi$
$131$			126.455i			0.0843394i	−0.999110	+	0.0421697i	$0.986573\pi$
$132$		0			0
$133$	−543.581			−0.354395
$134$		0			0
$135$	−466.089	−	1920.06i	−0.297145	−	1.22409i
$136$		0			0
$137$		−	491.778i		−	0.306682i	−0.988173	−	0.153341i	$-0.950997\pi$
$137$		−	491.778i		−	0.306682i	0.988173	−	0.153341i	$-0.0490033\pi$
$138$		0			0
$139$			393.926i			0.240377i	0.992751	+	0.120188i	$0.0383498\pi$
$139$			393.926i			0.240377i	−0.992751	+	0.120188i	$0.961650\pi$
$140$		0			0
$141$			287.403i			0.171658i
$142$		0			0
$143$		−	1888.01i		−	1.10408i
$144$		0			0
$145$	1589.96	−	385.958i	0.910615	−	0.221049i
$146$		0			0
$147$	2766.10			1.55200
$148$		0			0
$149$			1074.70i			0.590893i	0.955359	+	0.295446i	$0.0954684\pi$
$149$			1074.70i			0.590893i	−0.955359	+	0.295446i	$0.904532\pi$
$150$		0			0
$151$	−1130.15			−0.609075			−0.304537	−	0.952500i	$-0.598502\pi$
$151$	−1130.15			−0.609075			−0.304537	+	0.952500i	$0.598502\pi$
$152$		0			0
$153$		−	3842.70i		−	2.03048i
$154$		0			0
$155$	204.925	+	844.191i	0.106193	+	0.437465i
$156$		0			0
$157$	−1126.04			−0.572407			−0.286204	−	0.958169i	$-0.592393\pi$
$157$	−1126.04			−0.572407			−0.286204	+	0.958169i	$0.592393\pi$
$158$		0			0
$159$	−2203.99			−1.09930
$160$		0			0
$161$	134.505			0.0658416
$162$		0			0
$163$	309.194			0.148576			0.0742882	−	0.997237i	$-0.476332\pi$
$163$	309.194			0.148576			0.0742882	+	0.997237i	$0.476332\pi$
$164$		0			0
$165$	−2989.48	+	725.688i	−1.41049	+	0.342392i
$166$		0			0
$167$		−	2556.08i		−	1.18440i	−0.805790	−	0.592201i	$-0.798259\pi$
$167$		−	2556.08i		−	1.18440i	0.805790	−	0.592201i	$-0.201741\pi$
$168$		0			0
$169$	1309.61			0.596092
$170$		0			0
$171$			5443.19i			2.43422i
$172$		0			0
$173$	3229.54			1.41929			0.709645	−	0.704559i	$-0.248855\pi$
$173$	3229.54			1.41929			0.709645	+	0.704559i	$0.248855\pi$
$174$		0			0
$175$	−271.724	−	526.706i	−0.117374	−	0.227515i
$176$		0			0
$177$			5090.47i			2.16171i
$178$		0			0
$179$		−	3074.45i		−	1.28377i	−0.766799	−	0.641887i	$-0.778152\pi$
$179$		−	3074.45i		−	1.28377i	0.766799	−	0.641887i	$-0.221848\pi$
$180$		0			0
$181$		−	3277.05i		−	1.34575i	−0.739755	−	0.672876i	$-0.765059\pi$
$181$		−	3277.05i		−	1.34575i	0.739755	−	0.672876i	$-0.234941\pi$
$182$		0			0
$183$			4949.00i			1.99913i
$184$		0			0
$185$	−531.288	−	2188.65i	−0.211141	−	0.869799i
$186$		0			0
$187$	−2580.53			−1.00913
$188$		0			0
$189$			837.902i			0.322478i
$190$		0			0
$191$	−195.880			−0.0742061			−0.0371030	−	0.999311i	$-0.511813\pi$
$191$	−195.880			−0.0742061			−0.0371030	+	0.999311i	$0.511813\pi$
$192$		0			0
$193$			2873.89i			1.07185i	0.844265	+	0.535926i	$0.180037\pi$
$193$			2873.89i			1.07185i	−0.844265	+	0.535926i	$0.819963\pi$
$194$		0			0
$195$	−1347.82	−	5552.38i	−0.494972	−	2.03905i
$196$		0			0
$197$	4109.31			1.48617			0.743087	−	0.669195i	$-0.233361\pi$
$197$	4109.31			1.48617			0.743087	+	0.669195i	$0.233361\pi$
$198$		0			0
$199$	2649.19			0.943700			0.471850	−	0.881679i	$-0.343586\pi$
$199$	2649.19			0.943700			0.471850	+	0.881679i	$0.343586\pi$
$200$		0			0
$201$	−206.910			−0.0726085
$202$		0			0
$203$	−693.848			−0.239895
$204$		0			0
$205$	262.008	+	1079.35i	0.0892655	+	0.367731i
$206$		0			0
$207$		−	1346.88i		−	0.452244i
$208$		0			0
$209$	3655.32			1.20978
$210$		0			0
$211$		−	4482.21i		−	1.46241i	−0.682160	−	0.731203i	$-0.738959\pi$
$211$		−	4482.21i		−	1.46241i	0.682160	−	0.731203i	$-0.261041\pi$
$212$		0			0
$213$	−4061.77			−1.30661
$214$		0			0
$215$	554.622	+	2284.78i	0.175930	+	0.724746i
$216$		0			0
$217$		−	368.399i		−	0.115247i
$218$		0			0
$219$			5837.51i			1.80120i
$220$		0			0
$221$		−	4792.82i		−	1.45882i
$222$		0			0
$223$			1605.92i			0.482244i	0.970495	+	0.241122i	$0.0775153\pi$
$223$			1605.92i			0.482244i	−0.970495	+	0.241122i	$0.922485\pi$
$224$		0			0
$225$	−5274.21	+	2720.93i	−1.56273	+	0.806201i
$226$		0			0
$227$	1611.34			0.471139			0.235570	−	0.971857i	$-0.424304\pi$
$227$	1611.34			0.471139			0.235570	+	0.971857i	$0.424304\pi$
$228$		0			0
$229$			2221.77i			0.641129i	0.947227	+	0.320565i	$0.103873\pi$
$229$			2221.77i			0.641129i	−0.947227	+	0.320565i	$0.896127\pi$
$230$		0			0
$231$	1304.59			0.371583
$232$		0			0
$233$		−	424.194i		−	0.119270i	−0.998220	−	0.0596350i	$-0.981006\pi$
$233$		−	424.194i		−	0.119270i	0.998220	−	0.0596350i	$-0.0189937\pi$
$234$		0			0
$235$	361.827	−	87.8323i	0.100438	−	0.0243811i
$236$		0			0
$237$	−9370.19			−2.56818
$238$		0			0
$239$	4772.32			1.29161			0.645807	−	0.763501i	$-0.276521\pi$
$239$	4772.32			1.29161			0.645807	+	0.763501i	$0.276521\pi$
$240$		0			0
$241$	−5675.88			−1.51708			−0.758539	−	0.651628i	$-0.774086\pi$
$241$	−5675.88			−1.51708			−0.758539	+	0.651628i	$0.774086\pi$
$242$		0			0
$243$	−2672.43			−0.705500
$244$		0			0
$245$	−845.338	−	3482.39i	−0.220435	−	0.908088i
$246$		0			0
$247$			6789.04i			1.74889i
$248$		0			0
$249$	675.354			0.171883
$250$		0			0
$251$			334.026i			0.0839982i	0.999118	+	0.0419991i	$0.0133727\pi$
$251$			334.026i			0.0839982i	−0.999118	+	0.0419991i	$0.986627\pi$
$252$		0			0
$253$	−904.483			−0.224760
$254$		0			0
$255$	−7588.96	+	1842.20i	−1.86368	+	0.452403i
$256$		0			0
$257$		−	3516.77i		−	0.853580i	−0.904351	−	0.426790i	$-0.859644\pi$
$257$		−	3516.77i		−	0.853580i	0.904351	−	0.426790i	$-0.140356\pi$
$258$		0			0
$259$			955.112i			0.229142i
$260$		0			0
$261$			6947.90i			1.64776i
$262$		0			0
$263$		−	7550.95i		−	1.77039i	−0.465224	−	0.885193i	$-0.654026\pi$
$263$		−	7550.95i		−	1.77039i	0.465224	−	0.885193i	$-0.345974\pi$
$264$		0			0
$265$	673.554	+	2774.72i	0.156136	+	0.643206i
$266$		0			0
$267$	−5231.31			−1.19907
$268$		0			0
$269$			5979.76i			1.35536i	0.735356	+	0.677681i	$0.237015\pi$
$269$			5979.76i			1.35536i	−0.735356	+	0.677681i	$0.762985\pi$
$270$		0			0
$271$	−2543.95			−0.570237			−0.285118	−	0.958492i	$-0.592033\pi$
$271$	−2543.95			−0.570237			−0.285118	+	0.958492i	$0.592033\pi$
$272$		0			0
$273$			2423.02i			0.537171i
$274$		0			0
$275$	1827.21	+	3541.84i	0.400673	+	0.776659i
$276$		0			0
$277$	5858.01			1.27066			0.635332	−	0.772239i	$-0.280863\pi$
$277$	5858.01			1.27066			0.635332	+	0.772239i	$0.280863\pi$
$278$		0			0
$279$	−3688.99			−0.791592
$280$		0			0
$281$	998.531			0.211983			0.105992	−	0.994367i	$-0.466198\pi$
$281$	998.531			0.211983			0.105992	+	0.994367i	$0.466198\pi$
$282$		0			0
$283$	2850.81			0.598809			0.299404	−	0.954126i	$-0.403212\pi$
$283$	2850.81			0.598809			0.299404	+	0.954126i	$0.403212\pi$
$284$		0			0
$285$	10749.8	−	2609.48i	2.23425	−	0.542358i
$286$		0			0
$287$		−	471.019i		−	0.0968759i
$288$		0			0
$289$	−1637.80			−0.333361
$290$		0			0
$291$		−	1308.17i		−	0.263527i
$292$		0			0
$293$	−75.5914			−0.0150720			−0.00753601	−	0.999972i	$-0.502399\pi$
$293$	−75.5914			−0.0150720			−0.00753601	+	0.999972i	$0.502399\pi$
$294$		0			0
$295$	6408.65	−	1555.68i	1.26483	−	0.307035i
$296$		0			0
$297$		−	5634.48i		−	1.10083i
$298$		0			0
$299$		−	1679.90i		−	0.324920i
$300$		0			0
$301$		−	997.060i		−	0.190929i
$302$		0			0
$303$		−	557.999i		−	0.105796i
$304$		0			0
$305$	6230.55	−	1512.45i	1.16971	−	0.283942i
$306$		0			0
$307$	−3156.88			−0.586881			−0.293441	−	0.955977i	$-0.594800\pi$
$307$	−3156.88			−0.586881			−0.293441	+	0.955977i	$0.594800\pi$
$308$		0			0
$309$		−	11168.0i		−	2.05606i
$310$		0			0
$311$	−5028.64			−0.916874			−0.458437	−	0.888727i	$-0.651591\pi$
$311$	−5028.64			−0.916874			−0.458437	+	0.888727i	$0.651591\pi$
$312$		0			0
$313$			8961.07i			1.61824i	0.587642	+	0.809121i	$0.300056\pi$
$313$			8961.07i			1.61824i	−0.587642	+	0.809121i	$0.699944\pi$
$314$		0			0
$315$	2445.75	−	593.697i	0.437467	−	0.106194i
$316$		0			0
$317$	−8441.61			−1.49567			−0.747836	−	0.663884i	$-0.768907\pi$
$317$	−8441.61			−1.49567			−0.747836	+	0.663884i	$0.768907\pi$
$318$		0			0
$319$	4665.79			0.818916
$320$		0			0
$321$	14020.3			2.43780
$322$		0			0
$323$	9279.23			1.59848
$324$		0			0
$325$	−6578.28	+	3393.69i	−1.12276	+	0.579224i
$326$		0			0
$327$		−	10766.8i		−	1.82082i
$328$		0			0
$329$	−157.899			−0.0264597
$330$		0			0
$331$		−	3295.83i		−	0.547297i	−0.961830	−	0.273649i	$-0.911769\pi$
$331$		−	3295.83i		−	0.547297i	0.961830	−	0.273649i	$-0.0882305\pi$
$332$		0			0
$333$	9564.09			1.57390
$334$		0			0
$335$	63.2330	+	260.490i	0.0103128	+	0.0424838i
$336$		0			0
$337$		−	12162.4i		−	1.96596i	−0.183709	−	0.982981i	$-0.558811\pi$
$337$		−	12162.4i		−	1.96596i	0.183709	−	0.982981i	$-0.441189\pi$
$338$		0			0
$339$		−	2253.93i		−	0.361112i
$340$		0			0
$341$			2477.30i			0.393412i
$342$		0			0
$343$			3145.96i			0.495236i
$344$		0			0
$345$	−2659.96	+	645.696i	−0.415093	+	0.100763i
$346$		0			0
$347$	2870.18			0.444032			0.222016	−	0.975043i	$-0.428736\pi$
$347$	2870.18			0.444032			0.222016	+	0.975043i	$0.428736\pi$
$348$		0			0
$349$		−	4290.16i		−	0.658015i	−0.944327	−	0.329007i	$-0.893286\pi$
$349$		−	4290.16i		−	0.658015i	0.944327	−	0.329007i	$-0.106714\pi$
$350$		0			0
$351$	10464.9			1.59139
$352$		0			0
$353$		−	11246.6i		−	1.69574i	−0.530201	−	0.847872i	$-0.677883\pi$
$353$		−	11246.6i		−	1.69574i	0.530201	−	0.847872i	$-0.322117\pi$
$354$		0			0
$355$	1241.30	+	5113.58i	0.185582	+	0.764509i
$356$		0			0
$357$	3311.77			0.490973
$358$		0			0
$359$	−10864.7			−1.59726			−0.798632	−	0.601820i	$-0.794442\pi$
$359$	−10864.7			−1.59726			−0.798632	+	0.601820i	$0.794442\pi$
$360$		0			0
$361$	−6285.05			−0.916322
$362$		0			0
$363$	2713.85			0.392397
$364$		0			0
$365$	7349.15	−	1783.98i	1.05390	−	0.255830i
$366$		0			0
$367$		−	11131.3i		−	1.58323i	−0.611018	−	0.791617i	$-0.709240\pi$
$367$		−	11131.3i		−	1.58323i	0.611018	−	0.791617i	$-0.290760\pi$
$368$		0			0
$369$	−4716.59			−0.665408
$370$		0			0
$371$		−	1210.87i		−	0.169448i
$372$		0			0
$373$	3357.27			0.466040			0.233020	−	0.972472i	$-0.425139\pi$
$373$	3357.27			0.466040			0.233020	+	0.972472i	$0.425139\pi$
$374$		0			0
$375$	7902.04	+	9111.64i	1.08816	+	1.25473i
$376$		0			0
$377$			8665.80i			1.18385i
$378$		0			0
$379$			2203.08i			0.298587i	0.988793	+	0.149293i	$0.0476999\pi$
$379$			2203.08i			0.298587i	−0.988793	+	0.149293i	$0.952300\pi$
$380$		0			0
$381$			16535.4i			2.22344i
$382$		0			0
$383$			8022.25i			1.07028i	0.844763	+	0.535141i	$0.179741\pi$
$383$			8022.25i			1.07028i	−0.844763	+	0.535141i	$0.820259\pi$
$384$		0			0
$385$	−398.691	−	1642.42i	−0.0527771	−	0.217416i
$386$		0			0
$387$	−9984.14			−1.31143
$388$		0			0
$389$			5583.44i			0.727741i	0.931449	+	0.363871i	$0.118545\pi$
$389$			5583.44i			0.727741i	−0.931449	+	0.363871i	$0.881455\pi$
$390$		0			0
$391$	−2296.08			−0.296976
$392$		0			0
$393$			1091.32i			0.140075i
$394$		0			0
$395$	2863.59	+	11796.6i	0.364767	+	1.50266i
$396$		0			0
$397$	−12509.3			−1.58142			−0.790708	−	0.612194i	$-0.790287\pi$
$397$	−12509.3			−1.58142			−0.790708	+	0.612194i	$0.790287\pi$
$398$		0			0
$399$	−4691.13			−0.588597
$400$		0			0
$401$	−12573.4			−1.56579			−0.782897	−	0.622151i	$-0.786259\pi$
$401$	−12573.4			−1.56579			−0.782897	+	0.622151i	$0.786259\pi$
$402$		0			0
$403$	−4601.11			−0.568728
$404$		0			0
$405$	−641.496	−	2642.66i	−0.0787067	−	0.324234i
$406$		0			0
$407$		−	6422.67i		−	0.782211i
$408$		0			0
$409$	−9675.87			−1.16978			−0.584891	−	0.811112i	$-0.698863\pi$
$409$	−9675.87			−1.16978			−0.584891	+	0.811112i	$0.698863\pi$
$410$		0			0
$411$		−	4244.07i		−	0.509354i
$412$		0			0
$413$	−2796.69			−0.333211
$414$		0			0
$415$	−206.393	−	850.239i	−0.0244131	−	0.100570i
$416$		0			0
$417$			3399.60i			0.399230i
$418$		0			0
$419$		−	1832.65i		−	0.213677i	−0.994276	−	0.106839i	$-0.965927\pi$
$419$		−	1832.65i		−	0.213677i	0.994276	−	0.106839i	$-0.0340728\pi$
$420$		0			0
$421$			1917.39i			0.221967i	0.993822	+	0.110983i	$0.0354000\pi$
$421$			1917.39i			0.221967i	−0.993822	+	0.110983i	$0.964600\pi$
$422$		0			0
$423$			1581.13i			0.181743i
$424$		0			0
$425$	4638.47	+	8991.15i	0.529409	+	1.02620i
$426$		0			0
$427$	−2718.97			−0.308150
$428$		0			0
$429$		−	16293.6i		−	1.83372i
$430$		0			0
$431$	6563.74			0.733559			0.366780	−	0.930308i	$-0.380460\pi$
$431$	6563.74			0.733559			0.366780	+	0.930308i	$0.380460\pi$
$432$		0			0
$433$		−	6985.19i		−	0.775258i	−0.921816	−	0.387629i	$-0.873294\pi$
$433$		−	6985.19i		−	0.775258i	0.921816	−	0.387629i	$-0.126706\pi$
$434$		0			0
$435$	13721.4	−	3330.84i	1.51240	−	0.367129i
$436$		0			0
$437$	3252.40			0.356026
$438$		0			0
$439$	1340.81			0.145771			0.0728855	−	0.997340i	$-0.476779\pi$
$439$	1340.81			0.145771			0.0728855	+	0.997340i	$0.476779\pi$
$440$		0			0
$441$	15217.5			1.64318
$442$		0			0
$443$	−14680.3			−1.57445			−0.787225	−	0.616666i	$-0.788483\pi$
$443$	−14680.3			−1.57445			−0.787225	+	0.616666i	$0.788483\pi$
$444$		0			0
$445$	1598.72	+	6585.97i	0.170307	+	0.701584i
$446$		0			0
$447$			9274.73i			0.981386i
$448$		0			0
$449$	2927.44			0.307694			0.153847	−	0.988095i	$-0.450834\pi$
$449$	2927.44			0.307694			0.153847	+	0.988095i	$0.450834\pi$
$450$		0			0
$451$			3167.38i			0.330700i
$452$		0			0
$453$	−9753.24			−1.01158
$454$		0			0
$455$	3050.47	−	740.491i	0.314303	−	0.0762961i
$456$		0			0
$457$			2998.66i			0.306940i	0.988153	+	0.153470i	$0.0490448\pi$
$457$			2998.66i			0.306940i	−0.988153	+	0.153470i	$0.950955\pi$
$458$		0			0
$459$		−	14303.4i		−	1.45453i
$460$		0			0
$461$		−	13198.4i		−	1.33343i	−0.745314	−	0.666714i	$-0.767700\pi$
$461$		−	13198.4i		−	1.33343i	0.745314	−	0.666714i	$-0.232300\pi$
$462$		0			0
$463$			4324.69i			0.434094i	0.976161	+	0.217047i	$0.0696425\pi$
$463$			4324.69i			0.434094i	−0.976161	+	0.217047i	$0.930358\pi$
$464$		0			0
$465$	1768.51	+	7285.40i	0.176371	+	0.726564i
$466$		0			0
$467$	−7778.87			−0.770799			−0.385400	−	0.922750i	$-0.625936\pi$
$467$	−7778.87			−0.770799			−0.385400	+	0.922750i	$0.625936\pi$
$468$		0			0
$469$		−	113.676i		−	0.0111920i
$470$		0			0
$471$	−9717.79			−0.950684
$472$		0			0
$473$			6704.75i			0.651765i
$474$		0			0
$475$	−6570.41	−	12736.0i	−0.634676	−	1.23025i
$476$		0			0
$477$	−12125.1			−1.16388
$478$		0			0
$479$	−5114.22			−0.487838			−0.243919	−	0.969796i	$-0.578433\pi$
$479$	−5114.22			−0.487838			−0.243919	+	0.969796i	$0.578433\pi$
$480$		0			0
$481$	11928.8			1.13079
$482$		0			0
$483$	1160.79			0.109353
$484$		0			0
$485$	−1646.93	+	399.786i	−0.154192	+	0.0374296i
$486$		0			0
$487$			10457.3i			0.973030i	0.873672	+	0.486515i	$0.161732\pi$
$487$			10457.3i			0.973030i	−0.873672	+	0.486515i	$0.838268\pi$
$488$		0			0
$489$	2668.36			0.246764
$490$		0			0
$491$		−	3777.66i		−	0.347217i	−0.984815	−	0.173608i	$-0.944457\pi$
$491$		−	3777.66i		−	0.347217i	0.984815	−	0.173608i	$-0.0555427\pi$
$492$		0			0
$493$	11844.4			1.08204
$494$		0			0
$495$	−16446.5	+	3992.33i	−1.49336	+	0.362508i
$496$		0			0
$497$		−	2231.53i		−	0.201404i
$498$		0			0
$499$			18387.4i			1.64957i	0.565447	+	0.824785i	$0.308704\pi$
$499$			18387.4i			1.64957i	−0.565447	+	0.824785i	$0.691296\pi$
$500$		0			0
$501$		−	22059.1i		−	1.96712i
$502$		0			0
$503$			6909.13i			0.612451i	0.951959	+	0.306226i	$0.0990662\pi$
$503$			6909.13i			0.612451i	−0.951959	+	0.306226i	$0.900934\pi$
$504$		0			0
$505$	−702.494	+	170.528i	−0.0619021	+	0.0150265i
$506$		0			0
$507$	11302.0			0.990021
$508$		0			0
$509$			19760.1i			1.72073i	0.509680	+	0.860364i	$0.329764\pi$
$509$			19760.1i			1.72073i	−0.509680	+	0.860364i	$0.670236\pi$
$510$		0			0
$511$	−3207.11			−0.277641
$512$		0			0
$513$			20260.9i			1.74374i
$514$		0			0
$515$	−14059.9	+	3413.00i	−1.20302	+	0.292029i
$516$		0			0
$517$	1061.79			0.0903242
$518$		0			0
$519$	27871.1			2.35723
$520$		0			0
$521$	−16342.3			−1.37422			−0.687109	−	0.726554i	$-0.741120\pi$
$521$	−16342.3			−1.37422			−0.687109	+	0.726554i	$0.741120\pi$
$522$		0			0
$523$	22862.3			1.91147			0.955735	−	0.294230i	$-0.0950632\pi$
$523$	22862.3			1.91147			0.955735	+	0.294230i	$0.0950632\pi$
$524$		0			0
$525$	−2344.99	−	4545.49i	−0.194940	−	0.377870i
$526$		0			0
$527$			6288.77i			0.519816i
$528$		0			0
$529$	11362.2			0.933855
$530$		0			0
$531$			28004.9i			2.28872i
$532$		0			0
$533$	−5882.78			−0.478070
$534$		0			0
$535$	−4284.69	−	17650.9i	−0.346249	−	1.42638i
$536$		0			0
$537$		−	26532.7i		−	2.13216i
$538$		0			0
$539$		−	10219.2i		−	0.816644i
$540$		0			0
$541$			6981.75i			0.554841i	0.960749	+	0.277421i	$0.0894795\pi$
$541$			6981.75i			0.554841i	−0.960749	+	0.277421i	$0.910520\pi$
$542$		0			0
$543$		−	28281.1i		−	2.23510i
$544$		0			0
$545$	−13554.9	+	3290.42i	−1.06538	+	0.258616i
$546$		0			0
$547$	−1515.34			−0.118449			−0.0592243	−	0.998245i	$-0.518863\pi$
$547$	−1515.34			−0.118449			−0.0592243	+	0.998245i	$0.518863\pi$
$548$		0			0
$549$			27226.6i			2.11658i
$550$		0			0
$551$	−16777.6			−1.29718
$552$		0			0
$553$		−	5147.96i		−	0.395865i
$554$		0			0
$555$	−4585.04	−	18888.2i	−0.350674	−	1.44461i
$556$		0			0
$557$	9210.22			0.700628			0.350314	−	0.936632i	$-0.386075\pi$
$557$	9210.22			0.700628			0.350314	+	0.936632i	$0.386075\pi$
$558$		0			0
$559$	−12452.8			−0.942210
$560$		0			0
$561$	−22270.1			−1.67601
$562$		0			0
$563$	−13512.5			−1.01152			−0.505758	−	0.862675i	$-0.668787\pi$
$563$	−13512.5			−1.01152			−0.505758	+	0.862675i	$0.668787\pi$
$564$		0			0
$565$	−2837.60	+	688.817i	−0.211289	+	0.0512898i
$566$		0			0
$567$			1153.24i			0.0854169i
$568$		0			0
$569$	14779.9			1.08893			0.544467	−	0.838782i	$-0.316732\pi$
$569$	14779.9			1.08893			0.544467	+	0.838782i	$0.316732\pi$
$570$		0			0
$571$		−	10948.6i		−	0.802423i	−0.915985	−	0.401211i	$-0.868589\pi$
$571$		−	10948.6i		−	0.802423i	0.915985	−	0.401211i	$-0.131411\pi$
$572$		0			0
$573$	−1690.45			−0.123245
$574$		0			0
$575$	1625.80	+	3151.43i	0.117914	+	0.228563i
$576$		0			0
$577$			9935.61i			0.716854i	0.933558	+	0.358427i	$0.116687\pi$
$577$			9935.61i			0.716854i	−0.933558	+	0.358427i	$0.883313\pi$
$578$		0			0
$579$			24801.8i			1.78019i
$580$		0			0
$581$			371.038i			0.0264944i
$582$		0			0
$583$			8142.51i			0.578436i
$584$		0			0
$585$	−7414.96	−	30546.1i	−0.524053	−	2.15885i
$586$		0			0
$587$	802.589			0.0564334			0.0282167	−	0.999602i	$-0.491017\pi$
$587$	802.589			0.0564334			0.0282167	+	0.999602i	$0.491017\pi$
$588$		0			0
$589$		−	8908.05i		−	0.623175i
$590$		0			0
$591$	35463.5			2.46831
$592$		0			0
$593$			3331.26i			0.230689i	0.993326	+	0.115344i	$0.0367972\pi$
$593$			3331.26i			0.230689i	−0.993326	+	0.115344i	$0.963203\pi$
$594$		0			0
$595$	−1012.10	−	4169.36i	−0.0697344	−	0.287272i
$596$		0			0
$597$	22862.6			1.56735
$598$		0			0
$599$	28945.9			1.97445			0.987225	−	0.159330i	$-0.0509334\pi$
$599$	28945.9			1.97445			0.987225	+	0.159330i	$0.0509334\pi$
$600$		0			0
$601$	−1611.28			−0.109360			−0.0546801	−	0.998504i	$-0.517414\pi$
$601$	−1611.28			−0.109360			−0.0546801	+	0.998504i	$0.517414\pi$
$602$		0			0
$603$	−1138.30			−0.0768744
$604$		0			0
$605$	−829.371	−	3416.61i	−0.0557334	−	0.229595i
$606$		0			0
$607$		−	18635.9i		−	1.24614i	−0.782165	−	0.623072i	$-0.785885\pi$
$607$		−	18635.9i		−	1.24614i	0.782165	−	0.623072i	$-0.214115\pi$
$608$		0			0
$609$	−5987.94			−0.398429
$610$		0			0
$611$			1972.07i			0.130575i
$612$		0			0
$613$	8792.08			0.579296			0.289648	−	0.957133i	$-0.406462\pi$
$613$	8792.08			0.579296			0.289648	+	0.957133i	$0.406462\pi$
$614$		0			0
$615$	2261.14	+	9314.80i	0.148257	+	0.610747i
$616$		0			0
$617$			13584.0i			0.886340i	0.896438	+	0.443170i	$0.146146\pi$
$617$			13584.0i			0.886340i	−0.896438	+	0.443170i	$0.853854\pi$
$618$		0			0
$619$			1425.16i			0.0925394i	0.998929	+	0.0462697i	$0.0147334\pi$
$619$			1425.16i			0.0925394i	−0.998929	+	0.0462697i	$0.985267\pi$
$620$		0			0
$621$		−	5013.40i		−	0.323963i
$622$		0			0
$623$		−	2874.07i		−	0.184827i
$624$		0			0
$625$	9056.20	−	12732.9i	0.579597	−	0.814903i
$626$		0			0
$627$	31545.6			2.00927
$628$		0			0
$629$		−	16304.3i		−	1.03354i
$630$		0			0
$631$	−9225.47			−0.582029			−0.291014	−	0.956719i	$-0.593993\pi$
$631$	−9225.47			−0.582029			−0.291014	+	0.956719i	$0.593993\pi$
$632$		0			0
$633$		−	38681.6i		−	2.42884i
$634$		0			0
$635$	20817.2	−	5053.32i	1.30096	−	0.315803i
$636$		0			0
$637$	18980.1			1.18056
$638$		0			0
$639$	−22345.6			−1.38338
$640$		0			0
$641$	14596.9			0.899445			0.449723	−	0.893168i	$-0.351523\pi$
$641$	14596.9			0.899445			0.449723	+	0.893168i	$0.351523\pi$
$642$		0			0
$643$	23710.2			1.45418			0.727092	−	0.686540i	$-0.240871\pi$
$643$	23710.2			1.45418			0.727092	+	0.686540i	$0.240871\pi$
$644$		0			0
$645$	4786.41	+	19717.7i	0.292193	+	1.20370i
$646$		0			0
$647$			15897.0i			0.965962i	0.875631	+	0.482981i	$0.160446\pi$
$647$			15897.0i			0.965962i	−0.875631	+	0.482981i	$0.839554\pi$
$648$		0			0
$649$	18806.4			1.13747
$650$		0			0
$651$		−	3179.30i		−	0.191408i
$652$		0			0
$653$	−31188.2			−1.86905			−0.934524	−	0.355900i	$-0.884174\pi$
$653$	−31188.2			−1.86905			−0.934524	+	0.355900i	$0.884174\pi$
$654$		0			0
$655$	1373.91	−	333.513i	0.0819592	−	0.0198953i
$656$		0			0
$657$			32114.7i			1.90702i
$658$		0			0
$659$		−	26334.9i		−	1.55670i	−0.627832	−	0.778349i	$-0.716058\pi$
$659$		−	26334.9i		−	1.55670i	0.627832	−	0.778349i	$-0.283942\pi$
$660$		0			0
$661$		−	20361.4i		−	1.19813i	−0.800700	−	0.599066i	$-0.795539\pi$
$661$		−	20361.4i		−	1.19813i	0.800700	−	0.599066i	$-0.204461\pi$
$662$		0			0
$663$		−	41362.3i		−	2.42289i
$664$		0			0
$665$	1433.64	+	5905.91i	0.0836003	+	0.344393i
$666$		0			0
$667$	4151.49			0.240999
$668$		0			0
$669$			13859.2i			0.800936i
$670$		0			0
$671$	18283.8			1.05192
$672$		0			0
$673$		−	1622.72i		−	0.0929438i	−0.998920	−	0.0464719i	$-0.985202\pi$
$673$		−	1622.72i		−	0.0929438i	0.998920	−	0.0464719i	$-0.0147978\pi$
$674$		0			0
$675$	−19631.9	+	10127.9i	−1.11945	+	0.577518i
$676$		0			0
$677$	−15796.0			−0.896733			−0.448367	−	0.893850i	$-0.647994\pi$
$677$	−15796.0			−0.896733			−0.448367	+	0.893850i	$0.647994\pi$
$678$		0			0
$679$	718.707			0.0406207
$680$		0			0
$681$	13906.0			0.782493
$682$		0			0
$683$	−15901.0			−0.890828			−0.445414	−	0.895325i	$-0.646943\pi$
$683$	−15901.0			−0.890828			−0.445414	+	0.895325i	$0.646943\pi$
$684$		0			0
$685$	−5343.08	+	1297.02i	−0.298027	+	0.0723451i
$686$		0			0
$687$			19174.0i			1.06482i
$688$		0			0
$689$	−15123.1			−0.836204
$690$		0			0
$691$			14869.9i			0.818638i	0.912391	+	0.409319i	$0.134234\pi$
$691$			14869.9i			0.818638i	−0.912391	+	0.409319i	$0.865766\pi$
$692$		0			0
$693$	7177.12			0.393414
$694$		0			0
$695$	4279.93	−	1038.94i	0.233593	−	0.0567039i
$696$		0			0
$697$			8040.55i			0.436955i
$698$		0			0
$699$		−	3660.81i		−	0.198090i
$700$		0			0
$701$		−	15260.7i		−	0.822237i	−0.911582	−	0.411118i	$-0.865138\pi$
$701$		−	15260.7i		−	0.822237i	0.911582	−	0.411118i	$-0.134862\pi$
$702$		0			0
$703$			23095.1i			1.23904i
$704$		0			0
$705$	3122.58	−	757.997i	0.166813	−	0.0404933i
$706$		0			0
$707$	306.563			0.0163076
$708$		0			0
$709$		−	1280.70i		−	0.0678387i	−0.999425	−	0.0339194i	$-0.989201\pi$
$709$		−	1280.70i		−	0.0678387i	0.999425	−	0.0339194i	$-0.0107989\pi$
$710$		0			0
$711$	−51549.5			−2.71907
$712$		0			0
$713$			2204.23i			0.115777i
$714$		0			0
$715$	−20512.9	+	4979.44i	−1.07292	+	0.260448i
$716$		0			0
$717$	41185.3			2.14518
$718$		0			0
$719$	−17279.4			−0.896264			−0.448132	−	0.893967i	$-0.647911\pi$
$719$	−17279.4			−0.896264			−0.448132	+	0.893967i	$0.647911\pi$
$720$		0			0
$721$	6135.65			0.316926
$722$		0			0
$723$	−48983.1			−2.51964
$724$		0			0
$725$	−8386.72	−	16256.7i	−0.429621	−	0.832771i
$726$		0			0
$727$		−	34013.1i		−	1.73518i	−0.497280	−	0.867590i	$-0.665668\pi$
$727$		−	34013.1i		−	1.73518i	0.497280	−	0.867590i	$-0.334332\pi$
$728$		0			0
$729$	−29630.4			−1.50538
$730$		0			0
$731$			17020.4i			0.861177i
$732$		0			0
$733$	12918.8			0.650976			0.325488	−	0.945546i	$-0.394471\pi$
$733$	12918.8			0.650976			0.325488	+	0.945546i	$0.394471\pi$
$734$		0			0
$735$	−7295.31	−	30053.2i	−0.366111	−	1.50820i
$736$		0			0
$737$			764.415i			0.0382057i
$738$		0			0
$739$			6230.77i			0.310152i	0.987903	+	0.155076i	$0.0495623\pi$
$739$			6230.77i			0.310152i	−0.987903	+	0.155076i	$0.950438\pi$
$740$		0			0
$741$			58589.7i			2.90465i
$742$		0			0
$743$			27113.4i			1.33875i	0.742924	+	0.669376i	$0.233439\pi$
$743$			27113.4i			1.33875i	−0.742924	+	0.669376i	$0.766561\pi$
$744$		0			0
$745$	11676.4	−	2834.42i	0.574217	−	0.139389i
$746$		0			0
$747$	3715.42			0.181981
$748$		0			0
$749$			7702.71i			0.375769i
$750$		0			0
$751$	17626.9			0.856479			0.428239	−	0.903665i	$-0.359134\pi$
$751$	17626.9			0.856479			0.428239	+	0.903665i	$0.359134\pi$
$752$		0			0
$753$			2882.66i			0.139509i
$754$		0			0
$755$	2980.65	+	12278.9i	0.143678	+	0.591885i
$756$		0			0
$757$	−33113.0			−1.58984			−0.794921	−	0.606712i	$-0.792488\pi$
$757$	−33113.0			−1.58984			−0.794921	+	0.606712i	$0.792488\pi$
$758$		0			0
$759$	−7805.73			−0.373294
$760$		0			0
$761$	3327.92			0.158524			0.0792622	−	0.996854i	$-0.474744\pi$
$761$	3327.92			0.158524			0.0792622	+	0.996854i	$0.474744\pi$
$762$		0			0
$763$	5915.28			0.280665
$764$		0			0
$765$	−41750.2	+	10134.7i	−1.97318	+	0.478983i
$766$		0			0
$767$			34929.2i			1.64435i
$768$		0			0
$769$	−7278.36			−0.341306			−0.170653	−	0.985331i	$-0.554588\pi$
$769$	−7278.36			−0.341306			−0.170653	+	0.985331i	$0.554588\pi$
$770$		0			0
$771$		−	30349.9i		−	1.41767i
$772$		0			0
$773$	−8158.35			−0.379606			−0.189803	−	0.981822i	$-0.560785\pi$
$773$	−8158.35			−0.379606			−0.189803	+	0.981822i	$0.560785\pi$
$774$		0			0
$775$	8631.50	−	4452.93i	0.400068	−	0.206392i
$776$		0			0
$777$			8242.66i			0.380571i
$778$		0			0
$779$		−	11389.5i		−	0.523838i
$780$		0			0
$781$			15006.0i			0.687523i
$782$		0			0
$783$			25861.7i			1.18036i
$784$		0			0
$785$	2969.82	+	12234.2i	0.135029	+	0.556253i
$786$		0			0
$787$	19256.9			0.872217			0.436109	−	0.899894i	$-0.356356\pi$
$787$	19256.9			0.872217			0.436109	+	0.899894i	$0.356356\pi$
$788$		0			0
$789$		−	65165.0i		−	2.94035i
$790$		0			0
$791$	1238.31			0.0556626
$792$		0			0
$793$			33958.5i			1.52068i
$794$		0			0
$795$	5812.80	+	23946.0i	0.259319	+	1.06827i
$796$		0			0
$797$	9099.25			0.404407			0.202203	−	0.979344i	$-0.435190\pi$
$797$	9099.25			0.404407			0.202203	+	0.979344i	$0.435190\pi$
$798$		0			0
$799$	2695.42			0.119345
$800$		0			0
$801$	−28779.7			−1.26951
$802$		0			0
$803$	21566.3			0.947769
$804$		0			0
$805$	−354.744	−	1461.37i	−0.0155318	−	0.0639834i
$806$		0			0
$807$			51605.6i			2.25106i
$808$		0			0
$809$	−32295.7			−1.40353			−0.701766	−	0.712408i	$-0.747605\pi$
$809$	−32295.7			−1.40353			−0.701766	+	0.712408i	$0.747605\pi$
$810$		0			0
$811$		−	38607.7i		−	1.67164i	−0.549005	−	0.835819i	$-0.684993\pi$
$811$		−	38607.7i		−	1.67164i	0.549005	−	0.835819i	$-0.315007\pi$
$812$		0			0
$813$	−21954.4			−0.947079
$814$		0			0
$815$	−815.468	−	3359.34i	−0.0350486	−	0.144383i
$816$		0			0
$817$		−	24109.4i		−	1.03241i
$818$		0			0
$819$			13330.1i			0.568731i
$820$		0			0
$821$			5586.10i			0.237462i	0.992926	+	0.118731i	$0.0378826\pi$
$821$			5586.10i			0.237462i	−0.992926	+	0.118731i	$0.962117\pi$
$822$		0			0
$823$		−	3121.31i		−	0.132202i	−0.997813	−	0.0661008i	$-0.978944\pi$
$823$		−	3121.31i		−	0.132202i	0.997813	−	0.0661008i	$-0.0210559\pi$
$824$		0			0
$825$	15768.9	+	30566.3i	0.665458	+	1.28992i
$826$		0			0
$827$	−10702.5			−0.450017			−0.225008	−	0.974357i	$-0.572241\pi$
$827$	−10702.5			−0.450017			−0.225008	+	0.974357i	$0.572241\pi$
$828$		0			0
$829$		−	28818.9i		−	1.20739i	−0.797217	−	0.603693i	$-0.793695\pi$
$829$		−	28818.9i		−	1.20739i	0.797217	−	0.603693i	$-0.206305\pi$
$830$		0			0
$831$	50554.9			2.11038
$832$		0			0
$833$		−	25941.9i		−	1.07903i
$834$		0			0
$835$	−27771.3	+	6741.39i	−1.15098	+	0.279396i
$836$		0			0
$837$	−13731.3			−0.567052
$838$		0			0
$839$	−19119.3			−0.786736			−0.393368	−	0.919381i	$-0.628690\pi$
$839$	−19119.3			−0.786736			−0.393368	+	0.919381i	$0.628690\pi$
$840$		0			0
$841$	2973.46			0.121918
$842$		0			0
$843$	8617.36			0.352073
$844$		0			0
$845$	−3453.97	−	14228.7i	−0.140616	−	0.579269i
$846$		0			0
$847$			1490.98i			0.0604850i
$848$		0			0
$849$	24602.6			0.994533
$850$		0			0
$851$		−	5714.71i		−	0.230197i
$852$		0			0
$853$	38038.6			1.52686			0.763432	−	0.645888i	$-0.223513\pi$
$853$	38038.6			1.52686			0.763432	+	0.645888i	$0.223513\pi$
$854$		0			0
$855$	59139.3	−	14355.9i	2.36552	−	0.574223i
$856$		0			0
$857$			22594.3i			0.900589i	0.892880	+	0.450295i	$0.148681\pi$
$857$			22594.3i			0.900589i	−0.892880	+	0.450295i	$0.851319\pi$
$858$		0			0
$859$		−	34869.4i		−	1.38502i	−0.721410	−	0.692508i	$-0.756506\pi$
$859$		−	34869.4i		−	1.38502i	0.721410	−	0.692508i	$-0.243494\pi$
$860$		0			0
$861$		−	4064.91i		−	0.160897i
$862$		0			0
$863$		−	40314.4i		−	1.59017i	−0.606496	−	0.795087i	$-0.707425\pi$
$863$		−	40314.4i		−	1.59017i	0.606496	−	0.795087i	$-0.292575\pi$
$864$		0			0
$865$	−8517.58	−	35088.3i	−0.334805	−	1.37924i
$866$		0			0
$867$	−14134.3			−0.553664
$868$		0			0
$869$			34617.6i			1.35135i
$870$		0			0
$871$	−1419.75			−0.0552313
$872$		0			0
$873$		−	7196.83i		−	0.279010i
$874$		0			0
$875$	−5005.91	+	4341.36i	−0.193407	+	0.167731i
$876$		0			0
$877$	9266.02			0.356775			0.178387	−	0.983960i	$-0.442912\pi$
$877$	9266.02			0.356775			0.178387	+	0.983960i	$0.442912\pi$
$878$		0			0
$879$	−652.357			−0.0250324
$880$		0			0
$881$	50175.7			1.91880			0.959400	−	0.282048i	$-0.0910138\pi$
$881$	50175.7			1.91880			0.959400	+	0.282048i	$0.0910138\pi$
$882$		0			0
$883$	−36118.4			−1.37653			−0.688267	−	0.725457i	$-0.741628\pi$
$883$	−36118.4			−1.37653			−0.688267	+	0.725457i	$0.741628\pi$
$884$		0			0
$885$	55307.0	−	13425.6i	2.10070	−	0.509939i
$886$		0			0
$887$			5070.40i			0.191936i	0.995384	+	0.0959680i	$0.0305946\pi$
$887$			5070.40i			0.191936i	−0.995384	+	0.0959680i	$0.969405\pi$
$888$		0			0
$889$	−9084.49			−0.342727
$890$		0			0
$891$		−	7754.96i		−	0.291583i
$892$		0			0
$893$	−3818.06			−0.143076
$894$		0			0
$895$	−33403.4	+	8108.56i	−1.24754	+	0.302837i
$896$		0			0
$897$		−	14497.6i		−	0.539644i
$898$		0			0
$899$		−	11370.6i		−	0.421836i
$900$		0			0
$901$			20670.2i			0.764288i
$902$		0			0
$903$		−	8604.67i		−	0.317105i
$904$		0			0
$905$	−35604.5	+	8642.88i	−1.30777	+	0.317458i
$906$		0			0
$907$	14784.3			0.541241			0.270620	−	0.962686i	$-0.412771\pi$
$907$	14784.3			0.541241			0.270620	+	0.962686i	$0.412771\pi$
$908$		0			0
$909$		−	3069.80i		−	0.112012i
$910$		0			0
$911$	−30678.1			−1.11571			−0.557854	−	0.829939i	$-0.688375\pi$
$911$	−30678.1			−1.11571			−0.557854	+	0.829939i	$0.688375\pi$
$912$		0			0
$913$		−	2495.05i		−	0.0904427i
$914$		0			0
$915$	53770.0	−	13052.5i	1.94271	−	0.471587i
$916$		0			0
$917$	−599.567			−0.0215915
$918$		0			0
$919$	13617.4			0.488790			0.244395	−	0.969676i	$-0.421411\pi$
$919$	13617.4			0.488790			0.244395	+	0.969676i	$0.421411\pi$
$920$		0			0
$921$	−27244.0			−0.974724
$922$		0			0
$923$	−27870.6			−0.993903
$924$		0			0
$925$	−22378.1	+	11544.7i	−0.795445	+	0.410364i
$926$		0			0
$927$		−	61439.8i		−	2.17686i
$928$		0			0
$929$	15277.5			0.539548			0.269774	−	0.962924i	$-0.413051\pi$
$929$	15277.5			0.539548			0.269774	+	0.962924i	$0.413051\pi$
$930$		0			0
$931$			36746.8i			1.29358i
$932$		0			0
$933$	−43397.4			−1.52279
$934$		0			0
$935$	6805.87	+	28036.9i	0.238049	+	0.980648i
$936$		0			0
$937$			16196.6i			0.564697i	0.959312	+	0.282348i	$0.0911134\pi$
$937$			16196.6i			0.564697i	−0.959312	+	0.282348i	$0.908887\pi$
$938$		0			0
$939$			77334.4i			2.68766i
$940$		0			0
$941$			17503.4i			0.606371i	0.952932	+	0.303186i	$0.0980502\pi$
$941$			17503.4i			0.606371i	−0.952932	+	0.303186i	$0.901950\pi$
$942$		0			0
$943$			2818.24i			0.0973218i
$944$		0			0
$945$	9103.64	−	2209.88i	0.313377	−	0.0760713i
$946$		0			0
$947$	12098.6			0.415154			0.207577	−	0.978219i	$-0.433442\pi$
$947$	12098.6			0.415154			0.207577	+	0.978219i	$0.433442\pi$
$948$		0			0
$949$			40055.2i			1.37012i
$950$		0			0
$951$	−72851.4			−2.48409
$952$		0			0
$953$			9549.00i			0.324578i	0.986743	+	0.162289i	$0.0518876\pi$
$953$			9549.00i			0.324578i	−0.986743	+	0.162289i	$0.948112\pi$
$954$		0			0
$955$	516.613	+	2128.20i	0.0175049	+	0.0721118i
$956$		0			0
$957$	40266.0			1.36010
$958$		0			0
$959$	2331.68			0.0785130
$960$		0			0
$961$	−23753.8			−0.797348
$962$		0			0
$963$	77131.7			2.58103
$964$		0			0
$965$	31224.3	−	7579.60i	1.04160	−	0.252846i
$966$		0			0
$967$		−	27009.8i		−	0.898216i	−0.893477	−	0.449108i	$-0.851742\pi$
$967$		−	27009.8i		−	0.898216i	0.893477	−	0.449108i	$-0.148258\pi$
$968$		0			0
$969$	80080.1			2.65484
$970$		0			0
$971$			35347.4i			1.16823i	0.811671	+	0.584115i	$0.198558\pi$
$971$			35347.4i			1.16823i	−0.811671	+	0.584115i	$0.801442\pi$
$972$		0			0
$973$	−1867.73			−0.0615382
$974$		0			0
$975$	−56770.8	+	29287.7i	−1.86474	+	0.962006i
$976$		0			0
$977$		−	32592.0i		−	1.06726i	−0.845719	−	0.533629i	$-0.820828\pi$
$977$		−	32592.0i		−	1.06726i	0.845719	−	0.533629i	$-0.179172\pi$
$978$		0			0
$979$			19326.7i			0.630935i
$980$		0			0
$981$		−	59233.1i		−	1.92780i
$982$		0			0
$983$		−	45120.5i		−	1.46401i	−0.681300	−	0.732004i	$-0.738585\pi$
$983$		−	45120.5i		−	1.46401i	0.681300	−	0.732004i	$-0.261415\pi$
$984$		0			0
$985$	−10837.9	−	44646.9i	−0.350582	−	1.44423i
$986$		0			0
$987$	−1362.67			−0.0439456
$988$		0			0
$989$			5965.69i			0.191808i
$990$		0			0
$991$	24588.7			0.788179			0.394089	−	0.919072i	$-0.371060\pi$
$991$	24588.7			0.788179			0.394089	+	0.919072i	$0.371060\pi$
$992$		0			0
$993$		−	28443.2i		−	0.908980i
$994$		0			0
$995$	−6986.97	−	28783.0i	−0.222615	−	0.917067i
$996$		0			0
$997$	−42000.6			−1.33418			−0.667088	−	0.744979i	$-0.732459\pi$
$997$	−42000.6			−1.33418			−0.667088	+	0.744979i	$0.732459\pi$
$998$		0			0
$999$	35599.8			1.12746

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.4.f.a.49.15		16
3.2	odd	2		1440.4.d.d.1009.10		16
4.3	odd	2		40.4.f.a.29.13	yes	16
5.2	odd	4		800.4.d.e.401.1		16
5.3	odd	4		800.4.d.e.401.16		16
5.4	even	2	inner	160.4.f.a.49.1		16
8.3	odd	2		40.4.f.a.29.3	✓	16
8.5	even	2	inner	160.4.f.a.49.2		16
12.11	even	2		360.4.d.d.109.4		16
15.14	odd	2		1440.4.d.d.1009.8		16
20.3	even	4		200.4.d.e.101.5		16
20.7	even	4		200.4.d.e.101.12		16
20.19	odd	2		40.4.f.a.29.4	yes	16
24.5	odd	2		1440.4.d.d.1009.7		16
24.11	even	2		360.4.d.d.109.14		16
40.3	even	4		200.4.d.e.101.6		16
40.13	odd	4		800.4.d.e.401.2		16
40.19	odd	2		40.4.f.a.29.14	yes	16
40.27	even	4		200.4.d.e.101.11		16
40.29	even	2	inner	160.4.f.a.49.16		16
40.37	odd	4		800.4.d.e.401.15		16
60.59	even	2		360.4.d.d.109.13		16
120.29	odd	2		1440.4.d.d.1009.9		16
120.59	even	2		360.4.d.d.109.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.f.a.29.3	✓	16	8.3	odd	2
40.4.f.a.29.4	yes	16	20.19	odd	2
40.4.f.a.29.13	yes	16	4.3	odd	2
40.4.f.a.29.14	yes	16	40.19	odd	2
160.4.f.a.49.1		16	5.4	even	2	inner
160.4.f.a.49.2		16	8.5	even	2	inner
160.4.f.a.49.15		16	1.1	even	1	trivial
160.4.f.a.49.16		16	40.29	even	2	inner
200.4.d.e.101.5		16	20.3	even	4
200.4.d.e.101.6		16	40.3	even	4
200.4.d.e.101.11		16	40.27	even	4
200.4.d.e.101.12		16	20.7	even	4
360.4.d.d.109.3		16	120.59	even	2
360.4.d.d.109.4		16	12.11	even	2
360.4.d.d.109.13		16	60.59	even	2
360.4.d.d.109.14		16	24.11	even	2
800.4.d.e.401.1		16	5.2	odd	4
800.4.d.e.401.2		16	40.13	odd	4
800.4.d.e.401.15		16	40.37	odd	4
800.4.d.e.401.16		16	5.3	odd	4
1440.4.d.d.1009.7		16	24.5	odd	2
1440.4.d.d.1009.8		16	15.14	odd	2
1440.4.d.d.1009.9		16	120.29	odd	2
1440.4.d.d.1009.10		16	3.2	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)

\(f(q)\)	\(=\)	\(q+8.63004 q^{3} +(-2.63740 - 10.8648i) q^{5} +4.74133i q^{7} +47.4777 q^{9} +O(q^{10})\)	\(q+8.63004 q^{3} +(-2.63740 - 10.8648i) q^{5} +4.74133i q^{7} +47.4777 q^{9} -31.8831i q^{11} +59.2167 q^{13} +(-22.7609 - 93.7638i) q^{15} -80.9370i q^{17} +114.647i q^{19} +40.9179i q^{21} -28.3687i q^{23} +(-111.088 + 57.3097i) q^{25} +176.723 q^{27} +146.340i q^{29} -77.6995 q^{31} -275.153i q^{33} +(51.5136 - 12.5048i) q^{35} +201.444 q^{37} +511.042 q^{39} -99.3433 q^{41} -210.291 q^{43} +(-125.217 - 515.836i) q^{45} +33.3026i q^{47} +320.520 q^{49} -698.490i q^{51} -255.386 q^{53} +(-346.404 + 84.0885i) q^{55} +989.413i q^{57} +589.854i q^{59} +573.462i q^{61} +225.107i q^{63} +(-156.178 - 643.378i) q^{65} -23.9755 q^{67} -244.823i q^{69} -470.655 q^{71} +676.417i q^{73} +(-958.697 + 494.585i) q^{75} +151.168 q^{77} -1085.76 q^{79} +243.231 q^{81} +78.2562 q^{83} +(-879.366 + 213.463i) q^{85} +1262.92i q^{87} -606.174 q^{89} +280.766i q^{91} -670.550 q^{93} +(1245.62 - 302.371i) q^{95} -151.583i q^{97} -1513.74i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 104 q^{9}+O(q^{10}) \) 16 * q + 104 * q^9	\( 16 q + 104 q^{9} + 56 q^{15} - 24 q^{25} + 112 q^{31} + 736 q^{39} + 232 q^{41} - 200 q^{49} - 392 q^{55} - 600 q^{65} - 2096 q^{71} - 2992 q^{79} - 728 q^{81} - 208 q^{89} + 1064 q^{95}+O(q^{100}) \) 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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