LMFDB - Embedded newform 1440.4.d.d.1009.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,4,Mod(1009,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1440.1009");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1440.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$84.9627504083$
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:	$\Z[a_1, \ldots, a_{13}]$
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			1009.8
Root			$-2.15123 - 1.83636i$ of defining polynomial
Character	$\chi$	$=$	1440.1009
Dual form			1440.4.d.d.1009.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.63740 + 10.8648i) q^{5} -4.74133i q^{7} +O(q^{10})$	$q+(-2.63740 + 10.8648i) q^{5} -4.74133i q^{7} +31.8831i q^{11} -59.2167 q^{13} -80.9370i q^{17} +114.647i q^{19} -28.3687i q^{23} +(-111.088 - 57.3097i) q^{25} -146.340i q^{29} -77.6995 q^{31} +(51.5136 + 12.5048i) q^{35} -201.444 q^{37} +99.3433 q^{41} +210.291 q^{43} +33.3026i q^{47} +320.520 q^{49} -255.386 q^{53} +(-346.404 - 84.0885i) q^{55} -589.854i q^{59} +573.462i q^{61} +(156.178 - 643.378i) q^{65} +23.9755 q^{67} +470.655 q^{71} -676.417i q^{73} +151.168 q^{77} -1085.76 q^{79} +78.2562 q^{83} +(879.366 + 213.463i) q^{85} +606.174 q^{89} +280.766i q^{91} +(-1245.62 - 302.371i) q^{95} +151.583i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 24 q^{25} + 112 q^{31} - 232 q^{41} - 200 q^{49} - 392 q^{55} + 600 q^{65} + 2096 q^{71} - 2992 q^{79} + 208 q^{89} - 1064 q^{95}+O(q^{100}) $ 16 * q - 24 * q^25 + 112 * q^31 - 232 * q^41 - 200 * q^49 - 392 * q^55 + 600 * q^65 + 2096 * q^71 - 2992 * q^79 + 208 * q^89 - 1064 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		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	$-0.959143\pi$
$7$		−	4.74133i		−	0.256008i	0.991774	−	0.128004i	$-0.0408570\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			31.8831i			0.873921i	0.899481	+	0.436960i	$0.143945\pi$
$11$			31.8831i			0.873921i	−0.899481	+	0.436960i	$0.856055\pi$
$12$		0			0
$13$	−59.2167			−1.26337			−0.631683	−	0.775227i	$-0.717635\pi$
$13$	−59.2167			−1.26337			−0.631683	+	0.775227i	$0.717635\pi$
$14$		0			0
$15$		0			0
$16$		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
$21$		0			0
$22$		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
$27$		0			0
$28$		0			0
$29$		−	146.340i		−	0.937060i	−0.883448	−	0.468530i	$-0.844784\pi$
$29$		−	146.340i		−	0.937060i	0.883448	−	0.468530i	$-0.155216\pi$
$30$		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
$33$		0			0
$34$		0			0
$35$	51.5136	+	12.5048i	0.248783	+	0.0603912i
$36$		0			0
$37$	−201.444			−0.895060			−0.447530	−	0.894269i	$-0.647696\pi$
$37$	−201.444			−0.895060			−0.447530	+	0.894269i	$0.647696\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	99.3433			0.378410			0.189205	−	0.981938i	$-0.439409\pi$
$41$	99.3433			0.378410			0.189205	+	0.981938i	$0.439409\pi$
$42$		0			0
$43$	210.291			0.745794			0.372897	−	0.927873i	$-0.378365\pi$
$43$	210.291			0.745794			0.372897	+	0.927873i	$0.378365\pi$
$44$		0			0
$45$		0			0
$46$		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
$51$		0			0
$52$		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
$57$		0			0
$58$		0			0
$59$		−	589.854i		−	1.30157i	−0.759263	−	0.650783i	$-0.774441\pi$
$59$		−	589.854i		−	1.30157i	0.759263	−	0.650783i	$-0.225559\pi$
$60$		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
$63$		0			0
$64$		0			0
$65$	156.178	−	643.378i	0.298023	−	1.22771i
$66$		0			0
$67$	23.9755			0.0437176			0.0218588	−	0.999761i	$-0.493042\pi$
$67$	23.9755			0.0437176			0.0218588	+	0.999761i	$0.493042\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	470.655			0.786711			0.393355	−	0.919386i	$-0.371314\pi$
$71$	470.655			0.786711			0.393355	+	0.919386i	$0.371314\pi$
$72$		0			0
$73$		−	676.417i		−	1.08450i	−0.840217	−	0.542251i	$-0.817572\pi$
$73$		−	676.417i		−	1.08450i	0.840217	−	0.542251i	$-0.182428\pi$
$74$		0			0
$75$		0			0
$76$		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
$81$		0			0
$82$		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
$87$		0			0
$88$		0			0
$89$	606.174			0.721959			0.360979	−	0.932574i	$-0.382442\pi$
$89$	606.174			0.721959			0.360979	+	0.932574i	$0.382442\pi$
$90$		0			0
$91$			280.766i			0.323431i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1245.62	−	302.371i	−1.34524	−	0.326554i
$96$		0			0
$97$			151.583i			0.158670i	0.996848	+	0.0793349i	$0.0252796\pi$
$97$			151.583i			0.158670i	−0.996848	+	0.0793349i	$0.974720\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			64.6577i			0.0636999i	0.999493	+	0.0318499i	$0.0101399\pi$
$101$			64.6577i			0.0636999i	−0.999493	+	0.0318499i	$0.989860\pi$
$102$		0			0
$103$			1294.08i			1.23796i	0.785409	+	0.618978i	$0.212453\pi$
$103$			1294.08i			1.23796i	−0.785409	+	0.618978i	$0.787547\pi$
$104$		0			0
$105$		0			0
$106$		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
$111$		0			0
$112$		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
$117$		0			0
$118$		0			0
$119$	−383.749			−0.295615
$120$		0			0
$121$	314.466			0.236263
$122$		0			0
$123$		0			0
$124$		0			0
$125$	915.643	−	1055.80i	0.655181	−	0.755472i
$126$		0			0
$127$		−	1916.02i		−	1.33874i	−0.742931	−	0.669368i	$-0.766565\pi$
$127$		−	1916.02i		−	1.33874i	0.742931	−	0.669368i	$-0.233435\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	126.455i		−	0.0843394i	−0.999110	−	0.0421697i	$-0.986573\pi$
$131$		−	126.455i		−	0.0843394i	0.999110	−	0.0421697i	$-0.0134270\pi$
$132$		0			0
$133$	543.581			0.354395
$134$		0			0
$135$		0			0
$136$		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
$141$		0			0
$142$		0			0
$143$		−	1888.01i		−	1.10408i
$144$		0			0
$145$	1589.96	+	385.958i	0.910615	+	0.221049i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	1074.70i		−	0.590893i	−0.955359	−	0.295446i	$-0.904532\pi$
$149$		−	1074.70i		−	0.590893i	0.955359	−	0.295446i	$-0.0954684\pi$
$150$		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
$153$		0			0
$154$		0			0
$155$	204.925	−	844.191i	0.106193	−	0.437465i
$156$		0			0
$157$	1126.04			0.572407			0.286204	−	0.958169i	$-0.407607\pi$
$157$	1126.04			0.572407			0.286204	+	0.958169i	$0.407607\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−134.505			−0.0658416
$162$		0			0
$163$	−309.194			−0.148576			−0.0742882	−	0.997237i	$-0.523668\pi$
$163$	−309.194			−0.148576			−0.0742882	+	0.997237i	$0.523668\pi$
$164$		0			0
$165$		0			0
$166$		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
$171$		0			0
$172$		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
$177$		0			0
$178$		0			0
$179$			3074.45i			1.28377i	0.766799	+	0.641887i	$0.221848\pi$
$179$			3074.45i			1.28377i	−0.766799	+	0.641887i	$0.778152\pi$
$180$		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
$183$		0			0
$184$		0			0
$185$	531.288	−	2188.65i	0.211141	−	0.869799i
$186$		0			0
$187$	2580.53			1.00913
$188$		0			0
$189$		0			0
$190$		0			0
$191$	195.880			0.0742061			0.0371030	−	0.999311i	$-0.488187\pi$
$191$	195.880			0.0742061			0.0371030	+	0.999311i	$0.488187\pi$
$192$		0			0
$193$		−	2873.89i		−	1.07185i	−0.844265	−	0.535926i	$-0.819963\pi$
$193$		−	2873.89i		−	1.07185i	0.844265	−	0.535926i	$-0.180037\pi$
$194$		0			0
$195$		0			0
$196$		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
$201$		0			0
$202$		0			0
$203$	−693.848			−0.239895
$204$		0			0
$205$	−262.008	+	1079.35i	−0.0892655	+	0.367731i
$206$		0			0
$207$		0			0
$208$		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
$213$		0			0
$214$		0			0
$215$	−554.622	+	2284.78i	−0.175930	+	0.724746i
$216$		0			0
$217$			368.399i			0.115247i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			4792.82i			1.45882i
$222$		0			0
$223$		−	1605.92i		−	0.482244i	−0.970495	−	0.241122i	$-0.922485\pi$
$223$		−	1605.92i		−	0.482244i	0.970495	−	0.241122i	$-0.0775153\pi$
$224$		0			0
$225$		0			0
$226$		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
$231$		0			0
$232$		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
$237$		0			0
$238$		0			0
$239$	−4772.32			−1.29161			−0.645807	−	0.763501i	$-0.723479\pi$
$239$	−4772.32			−1.29161			−0.645807	+	0.763501i	$0.723479\pi$
$240$		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
$243$		0			0
$244$		0			0
$245$	−845.338	+	3482.39i	−0.220435	+	0.908088i
$246$		0			0
$247$		−	6789.04i		−	1.74889i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	334.026i		−	0.0839982i	−0.999118	−	0.0419991i	$-0.986627\pi$
$251$		−	334.026i		−	0.0839982i	0.999118	−	0.0419991i	$-0.0133727\pi$
$252$		0			0
$253$	904.483			0.224760
$254$		0			0
$255$		0			0
$256$		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
$261$		0			0
$262$		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
$267$		0			0
$268$		0			0
$269$		−	5979.76i		−	1.35536i	−0.735356	−	0.677681i	$-0.762985\pi$
$269$		−	5979.76i		−	1.35536i	0.735356	−	0.677681i	$-0.237015\pi$
$270$		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
$273$		0			0
$274$		0			0
$275$	1827.21	−	3541.84i	0.400673	−	0.776659i
$276$		0			0
$277$	−5858.01			−1.27066			−0.635332	−	0.772239i	$-0.719137\pi$
$277$	−5858.01			−1.27066			−0.635332	+	0.772239i	$0.719137\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−998.531			−0.211983			−0.105992	−	0.994367i	$-0.533802\pi$
$281$	−998.531			−0.211983			−0.105992	+	0.994367i	$0.533802\pi$
$282$		0			0
$283$	−2850.81			−0.598809			−0.299404	−	0.954126i	$-0.596788\pi$
$283$	−2850.81			−0.598809			−0.299404	+	0.954126i	$0.596788\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	471.019i		−	0.0968759i
$288$		0			0
$289$	−1637.80			−0.333361
$290$		0			0
$291$		0			0
$292$		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
$297$		0			0
$298$		0			0
$299$			1679.90i			0.324920i
$300$		0			0
$301$		−	997.060i		−	0.190929i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6230.55	−	1512.45i	−1.16971	−	0.283942i
$306$		0			0
$307$	3156.88			0.586881			0.293441	−	0.955977i	$-0.405200\pi$
$307$	3156.88			0.586881			0.293441	+	0.955977i	$0.405200\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5028.64			0.916874			0.458437	−	0.888727i	$-0.348409\pi$
$311$	5028.64			0.916874			0.458437	+	0.888727i	$0.348409\pi$
$312$		0			0
$313$		−	8961.07i		−	1.61824i	−0.587642	−	0.809121i	$-0.699944\pi$
$313$		−	8961.07i		−	1.61824i	0.587642	−	0.809121i	$-0.300056\pi$
$314$		0			0
$315$		0			0
$316$		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
$321$		0			0
$322$		0			0
$323$	9279.23			1.59848
$324$		0			0
$325$	6578.28	+	3393.69i	1.12276	+	0.579224i
$326$		0			0
$327$		0			0
$328$		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
$333$		0			0
$334$		0			0
$335$	−63.2330	+	260.490i	−0.0103128	+	0.0424838i
$336$		0			0
$337$			12162.4i			1.96596i	0.183709	+	0.982981i	$0.441189\pi$
$337$			12162.4i			1.96596i	−0.183709	+	0.982981i	$0.558811\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	2477.30i		−	0.393412i
$342$		0			0
$343$		−	3145.96i		−	0.495236i
$344$		0			0
$345$		0			0
$346$		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
$351$		0			0
$352$		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
$357$		0			0
$358$		0			0
$359$	10864.7			1.59726			0.798632	−	0.601820i	$-0.205558\pi$
$359$	10864.7			1.59726			0.798632	+	0.601820i	$0.205558\pi$
$360$		0			0
$361$	−6285.05			−0.916322
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7349.15	+	1783.98i	1.05390	+	0.255830i
$366$		0			0
$367$			11131.3i			1.58323i	0.611018	+	0.791617i	$0.290760\pi$
$367$			11131.3i			1.58323i	−0.611018	+	0.791617i	$0.709240\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1210.87i			0.169448i
$372$		0			0
$373$	−3357.27			−0.466040			−0.233020	−	0.972472i	$-0.574861\pi$
$373$	−3357.27			−0.466040			−0.233020	+	0.972472i	$0.574861\pi$
$374$		0			0
$375$		0			0
$376$		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
$381$		0			0
$382$		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
$387$		0			0
$388$		0			0
$389$		−	5583.44i		−	0.727741i	−0.931449	−	0.363871i	$-0.881455\pi$
$389$		−	5583.44i		−	0.727741i	0.931449	−	0.363871i	$-0.118545\pi$
$390$		0			0
$391$	−2296.08			−0.296976
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2863.59	−	11796.6i	0.364767	−	1.50266i
$396$		0			0
$397$	12509.3			1.58142			0.790708	−	0.612194i	$-0.209713\pi$
$397$	12509.3			1.58142			0.790708	+	0.612194i	$0.209713\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12573.4			1.56579			0.782897	−	0.622151i	$-0.213741\pi$
$401$	12573.4			1.56579			0.782897	+	0.622151i	$0.213741\pi$
$402$		0			0
$403$	4601.11			0.568728
$404$		0			0
$405$		0			0
$406$		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
$411$		0			0
$412$		0			0
$413$	−2796.69			−0.333211
$414$		0			0
$415$	−206.393	+	850.239i	−0.0244131	+	0.100570i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1832.65i			0.213677i	0.994276	+	0.106839i	$0.0340728\pi$
$419$			1832.65i			0.213677i	−0.994276	+	0.106839i	$0.965927\pi$
$420$		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
$423$		0			0
$424$		0			0
$425$	−4638.47	+	8991.15i	−0.529409	+	1.02620i
$426$		0			0
$427$	2718.97			0.308150
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6563.74			−0.733559			−0.366780	−	0.930308i	$-0.619540\pi$
$431$	−6563.74			−0.733559			−0.366780	+	0.930308i	$0.619540\pi$
$432$		0			0
$433$			6985.19i			0.775258i	0.921816	+	0.387629i	$0.126706\pi$
$433$			6985.19i			0.775258i	−0.921816	+	0.387629i	$0.873294\pi$
$434$		0			0
$435$		0			0
$436$		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
$441$		0			0
$442$		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
$447$		0			0
$448$		0			0
$449$	−2927.44			−0.307694			−0.153847	−	0.988095i	$-0.549166\pi$
$449$	−2927.44			−0.307694			−0.153847	+	0.988095i	$0.549166\pi$
$450$		0			0
$451$			3167.38i			0.330700i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3050.47	−	740.491i	−0.314303	−	0.0762961i
$456$		0			0
$457$		−	2998.66i		−	0.306940i	−0.988153	−	0.153470i	$-0.950955\pi$
$457$		−	2998.66i		−	0.306940i	0.988153	−	0.153470i	$-0.0490448\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			13198.4i			1.33343i	0.745314	+	0.666714i	$0.232300\pi$
$461$			13198.4i			1.33343i	−0.745314	+	0.666714i	$0.767700\pi$
$462$		0			0
$463$		−	4324.69i		−	0.434094i	−0.976161	−	0.217047i	$-0.930358\pi$
$463$		−	4324.69i		−	0.434094i	0.976161	−	0.217047i	$-0.0696425\pi$
$464$		0			0
$465$		0			0
$466$		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
$471$		0			0
$472$		0			0
$473$			6704.75i			0.651765i
$474$		0			0
$475$	6570.41	−	12736.0i	0.634676	−	1.23025i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5114.22			0.487838			0.243919	−	0.969796i	$-0.421567\pi$
$479$	5114.22			0.487838			0.243919	+	0.969796i	$0.421567\pi$
$480$		0			0
$481$	11928.8			1.13079
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1646.93	−	399.786i	−0.154192	−	0.0374296i
$486$		0			0
$487$		−	10457.3i		−	0.973030i	−0.873672	−	0.486515i	$-0.838268\pi$
$487$		−	10457.3i		−	0.973030i	0.873672	−	0.486515i	$-0.161732\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			3777.66i			0.347217i	0.984815	+	0.173608i	$0.0555427\pi$
$491$			3777.66i			0.347217i	−0.984815	+	0.173608i	$0.944457\pi$
$492$		0			0
$493$	−11844.4			−1.08204
$494$		0			0
$495$		0			0
$496$		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
$501$		0			0
$502$		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
$507$		0			0
$508$		0			0
$509$		−	19760.1i		−	1.72073i	−0.509680	−	0.860364i	$-0.670236\pi$
$509$		−	19760.1i		−	1.72073i	0.509680	−	0.860364i	$-0.329764\pi$
$510$		0			0
$511$	−3207.11			−0.277641
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−14059.9	−	3413.00i	−1.20302	−	0.292029i
$516$		0			0
$517$	−1061.79			−0.0903242
$518$		0			0
$519$		0			0
$520$		0			0
$521$	16342.3			1.37422			0.687109	−	0.726554i	$-0.258880\pi$
$521$	16342.3			1.37422			0.687109	+	0.726554i	$0.258880\pi$
$522$		0			0
$523$	−22862.3			−1.91147			−0.955735	−	0.294230i	$-0.904937\pi$
$523$	−22862.3			−1.91147			−0.955735	+	0.294230i	$0.904937\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			6288.77i			0.519816i
$528$		0			0
$529$	11362.2			0.933855
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5882.78			−0.478070
$534$		0			0
$535$	−4284.69	+	17650.9i	−0.346249	+	1.42638i
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		0			0
$545$	13554.9	+	3290.42i	1.06538	+	0.258616i
$546$		0			0
$547$	1515.34			0.118449			0.0592243	−	0.998245i	$-0.481137\pi$
$547$	1515.34			0.118449			0.0592243	+	0.998245i	$0.481137\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	16777.6			1.29718
$552$		0			0
$553$			5147.96i			0.395865i
$554$		0			0
$555$		0			0
$556$		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
$561$		0			0
$562$		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
$567$		0			0
$568$		0			0
$569$	−14779.9			−1.08893			−0.544467	−	0.838782i	$-0.683268\pi$
$569$	−14779.9			−1.08893			−0.544467	+	0.838782i	$0.683268\pi$
$570$		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
$573$		0			0
$574$		0			0
$575$	−1625.80	+	3151.43i	−0.117914	+	0.228563i
$576$		0			0
$577$		−	9935.61i		−	0.716854i	−0.933558	−	0.358427i	$-0.883313\pi$
$577$		−	9935.61i		−	0.716854i	0.933558	−	0.358427i	$-0.116687\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	371.038i		−	0.0264944i
$582$		0			0
$583$		−	8142.51i		−	0.578436i
$584$		0			0
$585$		0			0
$586$		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
$591$		0			0
$592$		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
$597$		0			0
$598$		0			0
$599$	−28945.9			−1.97445			−0.987225	−	0.159330i	$-0.949067\pi$
$599$	−28945.9			−1.97445			−0.987225	+	0.159330i	$0.949067\pi$
$600$		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
$603$		0			0
$604$		0			0
$605$	−829.371	+	3416.61i	−0.0557334	+	0.229595i
$606$		0			0
$607$			18635.9i			1.24614i	0.782165	+	0.623072i	$0.214115\pi$
$607$			18635.9i			1.24614i	−0.782165	+	0.623072i	$0.785885\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1972.07i		−	0.130575i
$612$		0			0
$613$	−8792.08			−0.579296			−0.289648	−	0.957133i	$-0.593538\pi$
$613$	−8792.08			−0.579296			−0.289648	+	0.957133i	$0.593538\pi$
$614$		0			0
$615$		0			0
$616$		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
$621$		0			0
$622$		0			0
$623$		−	2874.07i		−	0.184827i
$624$		0			0
$625$	9056.20	+	12732.9i	0.579597	+	0.814903i
$626$		0			0
$627$		0			0
$628$		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
$633$		0			0
$634$		0			0
$635$	20817.2	+	5053.32i	1.30096	+	0.315803i
$636$		0			0
$637$	−18980.1			−1.18056
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14596.9			−0.899445			−0.449723	−	0.893168i	$-0.648477\pi$
$641$	−14596.9			−0.899445			−0.449723	+	0.893168i	$0.648477\pi$
$642$		0			0
$643$	−23710.2			−1.45418			−0.727092	−	0.686540i	$-0.759129\pi$
$643$	−23710.2			−1.45418			−0.727092	+	0.686540i	$0.759129\pi$
$644$		0			0
$645$		0			0
$646$		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
$651$		0			0
$652$		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
$657$		0			0
$658$		0			0
$659$			26334.9i			1.55670i	0.627832	+	0.778349i	$0.283942\pi$
$659$			26334.9i			1.55670i	−0.627832	+	0.778349i	$0.716058\pi$
$660$		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
$663$		0			0
$664$		0			0
$665$	−1433.64	+	5905.91i	−0.0836003	+	0.344393i
$666$		0			0
$667$	−4151.49			−0.240999
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−18283.8			−1.05192
$672$		0			0
$673$			1622.72i			0.0929438i	0.998920	+	0.0464719i	$0.0147978\pi$
$673$			1622.72i			0.0929438i	−0.998920	+	0.0464719i	$0.985202\pi$
$674$		0			0
$675$		0			0
$676$		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
$681$		0			0
$682$		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
$687$		0			0
$688$		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
$693$		0			0
$694$		0			0
$695$	−4279.93	−	1038.94i	−0.233593	−	0.0567039i
$696$		0			0
$697$		−	8040.55i		−	0.436955i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			15260.7i			0.822237i	0.911582	+	0.411118i	$0.134862\pi$
$701$			15260.7i			0.822237i	−0.911582	+	0.411118i	$0.865138\pi$
$702$		0			0
$703$		−	23095.1i		−	1.23904i
$704$		0			0
$705$		0			0
$706$		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
$711$		0			0
$712$		0			0
$713$			2204.23i			0.115777i
$714$		0			0
$715$	20512.9	+	4979.44i	1.07292	+	0.260448i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17279.4			0.896264			0.448132	−	0.893967i	$-0.352089\pi$
$719$	17279.4			0.896264			0.448132	+	0.893967i	$0.352089\pi$
$720$		0			0
$721$	6135.65			0.316926
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8386.72	+	16256.7i	−0.429621	+	0.832771i
$726$		0			0
$727$			34013.1i			1.73518i	0.497280	+	0.867590i	$0.334332\pi$
$727$			34013.1i			1.73518i	−0.497280	+	0.867590i	$0.665668\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	17020.4i		−	0.861177i
$732$		0			0
$733$	−12918.8			−0.650976			−0.325488	−	0.945546i	$-0.605529\pi$
$733$	−12918.8			−0.650976			−0.325488	+	0.945546i	$0.605529\pi$
$734$		0			0
$735$		0			0
$736$		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
$741$		0			0
$742$		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
$747$		0			0
$748$		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
$753$		0			0
$754$		0			0
$755$	2980.65	−	12278.9i	0.143678	−	0.591885i
$756$		0			0
$757$	33113.0			1.58984			0.794921	−	0.606712i	$-0.207512\pi$
$757$	33113.0			1.58984			0.794921	+	0.606712i	$0.207512\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3327.92			−0.158524			−0.0792622	−	0.996854i	$-0.525256\pi$
$761$	−3327.92			−0.158524			−0.0792622	+	0.996854i	$0.525256\pi$
$762$		0			0
$763$	−5915.28			−0.280665
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		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
$777$		0			0
$778$		0			0
$779$			11389.5i			0.523838i
$780$		0			0
$781$			15006.0i			0.687523i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2969.82	+	12234.2i	−0.135029	+	0.556253i
$786$		0			0
$787$	−19256.9			−0.872217			−0.436109	−	0.899894i	$-0.643644\pi$
$787$	−19256.9			−0.872217			−0.436109	+	0.899894i	$0.643644\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1238.31			−0.0556626
$792$		0			0
$793$		−	33958.5i		−	1.52068i
$794$		0			0
$795$		0			0
$796$		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
$801$		0			0
$802$		0			0
$803$	21566.3			0.947769
$804$		0			0
$805$	354.744	−	1461.37i	0.0155318	−	0.0639834i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	32295.7			1.40353			0.701766	−	0.712408i	$-0.252395\pi$
$809$	32295.7			1.40353			0.701766	+	0.712408i	$0.252395\pi$
$810$		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
$813$		0			0
$814$		0			0
$815$	815.468	−	3359.34i	0.0350486	−	0.144383i
$816$		0			0
$817$			24109.4i			1.03241i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	5586.10i		−	0.237462i	−0.992926	−	0.118731i	$-0.962117\pi$
$821$		−	5586.10i		−	0.237462i	0.992926	−	0.118731i	$-0.0378826\pi$
$822$		0			0
$823$			3121.31i			0.132202i	0.997813	+	0.0661008i	$0.0210559\pi$
$823$			3121.31i			0.132202i	−0.997813	+	0.0661008i	$0.978944\pi$
$824$		0			0
$825$		0			0
$826$		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
$831$		0			0
$832$		0			0
$833$		−	25941.9i		−	1.07903i
$834$		0			0
$835$	27771.3	+	6741.39i	1.15098	+	0.279396i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19119.3			0.786736			0.393368	−	0.919381i	$-0.371310\pi$
$839$	19119.3			0.786736			0.393368	+	0.919381i	$0.371310\pi$
$840$		0			0
$841$	2973.46			0.121918
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3453.97	+	14228.7i	−0.140616	+	0.579269i
$846$		0			0
$847$		−	1490.98i		−	0.0604850i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			5714.71i			0.230197i
$852$		0			0
$853$	−38038.6			−1.52686			−0.763432	−	0.645888i	$-0.776487\pi$
$853$	−38038.6			−1.52686			−0.763432	+	0.645888i	$0.776487\pi$
$854$		0			0
$855$		0			0
$856$		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
$861$		0			0
$862$		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
$867$		0			0
$868$		0			0
$869$		−	34617.6i		−	1.35135i
$870$		0			0
$871$	−1419.75			−0.0552313
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5005.91	−	4341.36i	−0.193407	−	0.167731i
$876$		0			0
$877$	−9266.02			−0.356775			−0.178387	−	0.983960i	$-0.557088\pi$
$877$	−9266.02			−0.356775			−0.178387	+	0.983960i	$0.557088\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−50175.7			−1.91880			−0.959400	−	0.282048i	$-0.908986\pi$
$881$	−50175.7			−1.91880			−0.959400	+	0.282048i	$0.908986\pi$
$882$		0			0
$883$	36118.4			1.37653			0.688267	−	0.725457i	$-0.258372\pi$
$883$	36118.4			1.37653			0.688267	+	0.725457i	$0.258372\pi$
$884$		0			0
$885$		0			0
$886$		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
$891$		0			0
$892$		0			0
$893$	−3818.06			−0.143076
$894$		0			0
$895$	−33403.4	−	8108.56i	−1.24754	−	0.302837i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			11370.6i			0.421836i
$900$		0			0
$901$			20670.2i			0.764288i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	35604.5	+	8642.88i	1.30777	+	0.317458i
$906$		0			0
$907$	−14784.3			−0.541241			−0.270620	−	0.962686i	$-0.587229\pi$
$907$	−14784.3			−0.541241			−0.270620	+	0.962686i	$0.587229\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	30678.1			1.11571			0.557854	−	0.829939i	$-0.311625\pi$
$911$	30678.1			1.11571			0.557854	+	0.829939i	$0.311625\pi$
$912$		0			0
$913$			2495.05i			0.0904427i
$914$		0			0
$915$		0			0
$916$		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
$921$		0			0
$922$		0			0
$923$	−27870.6			−0.993903
$924$		0			0
$925$	22378.1	+	11544.7i	0.795445	+	0.410364i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15277.5			−0.539548			−0.269774	−	0.962924i	$-0.586949\pi$
$929$	−15277.5			−0.539548			−0.269774	+	0.962924i	$0.586949\pi$
$930$		0			0
$931$			36746.8i			1.29358i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6805.87	+	28036.9i	−0.238049	+	0.980648i
$936$		0			0
$937$		−	16196.6i		−	0.564697i	−0.959312	−	0.282348i	$-0.908887\pi$
$937$		−	16196.6i		−	0.564697i	0.959312	−	0.282348i	$-0.0911134\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	17503.4i		−	0.606371i	−0.952932	−	0.303186i	$-0.901950\pi$
$941$		−	17503.4i		−	0.606371i	0.952932	−	0.303186i	$-0.0980502\pi$
$942$		0			0
$943$		−	2818.24i		−	0.0973218i
$944$		0			0
$945$		0			0
$946$		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
$951$		0			0
$952$		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
$957$		0			0
$958$		0			0
$959$	−2331.68			−0.0785130
$960$		0			0
$961$	−23753.8			−0.797348
$962$		0			0
$963$		0			0
$964$		0			0
$965$	31224.3	+	7579.60i	1.04160	+	0.252846i
$966$		0			0
$967$			27009.8i			0.898216i	0.893477	+	0.449108i	$0.148258\pi$
$967$			27009.8i			0.898216i	−0.893477	+	0.449108i	$0.851742\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	35347.4i		−	1.16823i	−0.811671	−	0.584115i	$-0.801442\pi$
$971$		−	35347.4i		−	1.16823i	0.811671	−	0.584115i	$-0.198558\pi$
$972$		0			0
$973$	1867.73			0.0615382
$974$		0			0
$975$		0			0
$976$		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
$981$		0			0
$982$		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
$987$		0			0
$988$		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
$993$		0			0
$994$		0			0
$995$	−6986.97	+	28783.0i	−0.222615	+	0.917067i
$996$		0			0
$997$	42000.6			1.33418			0.667088	−	0.744979i	$-0.267541\pi$
$997$	42000.6			1.33418			0.667088	+	0.744979i	$0.267541\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.f.a.29.3	✓	16	120.59	even	2
40.4.f.a.29.4	yes	16	12.11	even	2
40.4.f.a.29.13	yes	16	60.59	even	2
40.4.f.a.29.14	yes	16	24.11	even	2
160.4.f.a.49.1		16	3.2	odd	2
160.4.f.a.49.2		16	120.29	odd	2
160.4.f.a.49.15		16	15.14	odd	2
160.4.f.a.49.16		16	24.5	odd	2
200.4.d.e.101.5		16	60.47	odd	4
200.4.d.e.101.6		16	120.107	odd	4
200.4.d.e.101.11		16	120.83	odd	4
200.4.d.e.101.12		16	60.23	odd	4
360.4.d.d.109.3		16	8.3	odd	2
360.4.d.d.109.4		16	20.19	odd	2
360.4.d.d.109.13		16	4.3	odd	2
360.4.d.d.109.14		16	40.19	odd	2
800.4.d.e.401.1		16	15.8	even	4
800.4.d.e.401.2		16	120.77	even	4
800.4.d.e.401.15		16	120.53	even	4
800.4.d.e.401.16		16	15.2	even	4
1440.4.d.d.1009.7		16	40.29	even	2	inner
1440.4.d.d.1009.8		16	1.1	even	1	trivial
1440.4.d.d.1009.9		16	8.5	even	2	inner
1440.4.d.d.1009.10		16	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.63740 + 10.8648i) q^{5} -4.74133i q^{7} +O(q^{10})\)	\(q+(-2.63740 + 10.8648i) q^{5} -4.74133i q^{7} +31.8831i q^{11} -59.2167 q^{13} -80.9370i q^{17} +114.647i q^{19} -28.3687i q^{23} +(-111.088 - 57.3097i) q^{25} -146.340i q^{29} -77.6995 q^{31} +(51.5136 + 12.5048i) q^{35} -201.444 q^{37} +99.3433 q^{41} +210.291 q^{43} +33.3026i q^{47} +320.520 q^{49} -255.386 q^{53} +(-346.404 - 84.0885i) q^{55} -589.854i q^{59} +573.462i q^{61} +(156.178 - 643.378i) q^{65} +23.9755 q^{67} +470.655 q^{71} -676.417i q^{73} +151.168 q^{77} -1085.76 q^{79} +78.2562 q^{83} +(879.366 + 213.463i) q^{85} +606.174 q^{89} +280.766i q^{91} +(-1245.62 - 302.371i) q^{95} +151.583i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 24 q^{25} + 112 q^{31} - 232 q^{41} - 200 q^{49} - 392 q^{55} + 600 q^{65} + 2096 q^{71} - 2992 q^{79} + 208 q^{89} - 1064 q^{95}+O(q^{100}) \) 16 * q - 24 * q^25 + 112 * q^31 - 232 * q^41 - 200 * q^49 - 392 * q^55 + 600 * q^65 + 2096 * q^71 - 2992 * q^79 + 208 * q^89 - 1064 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more