LMFDB - Embedded newform 480.2.d.b.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,2,Mod(49,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("480.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	480.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:	$3.83281929702$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.839056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 6x^{4} + 8x^{2} + 1 $ x^6 + 6x^4 + 8x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$2.02852i$ of defining polynomial
Character	$\chi$	$=$	480.49
Dual form			480.2.d.b.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +(-2.11491 - 0.726062i) q^{5} +4.05705i q^{7} +1.00000 q^{9} +0.985939i q^{11} +4.94567 q^{13} +(-2.11491 - 0.726062i) q^{15} +4.52323i q^{17} +2.60492i q^{19} +4.05705i q^{21} +3.53729i q^{23} +(3.94567 + 3.07111i) q^{25} +1.00000 q^{27} -7.59434i q^{29} +3.28415 q^{31} +0.985939i q^{33} +(2.94567 - 8.58028i) q^{35} -0.945668 q^{37} +4.94567 q^{39} +0.568295 q^{41} -8.45963 q^{43} +(-2.11491 - 0.726062i) q^{45} -2.60492i q^{47} -9.45963 q^{49} +4.52323i q^{51} +0.229815 q^{53} +(0.715853 - 2.08517i) q^{55} +2.60492i q^{57} -9.10003i q^{59} +11.0183i q^{61} +4.05705i q^{63} +(-10.4596 - 3.59086i) q^{65} +8.45963 q^{67} +3.53729i q^{69} -1.43171 q^{71} -11.9507i q^{73} +(3.94567 + 3.07111i) q^{75} -4.00000 q^{77} -3.28415 q^{79} +1.00000 q^{81} -9.89134 q^{83} +(3.28415 - 9.56622i) q^{85} -7.59434i q^{87} +12.3510 q^{89} +20.0648i q^{91} +3.28415 q^{93} +(1.89134 - 5.50917i) q^{95} -3.23797i q^{97} +0.985939i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 6 q^{9} + 8 q^{13} + 2 q^{25} + 6 q^{27} + 16 q^{31} - 4 q^{35} + 16 q^{37} + 8 q^{39} - 4 q^{41} - 6 q^{49} - 24 q^{53} + 8 q^{55} - 12 q^{65} - 16 q^{71} + 2 q^{75} - 24 q^{77} - 16 q^{79}+ \cdots - 32 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 + 6 * q^9 + 8 * q^13 + 2 * q^25 + 6 * q^27 + 16 * q^31 - 4 * q^35 + 16 * q^37 + 8 * q^39 - 4 * q^41 - 6 * q^49 - 24 * q^53 + 8 * q^55 - 12 * q^65 - 16 * q^71 + 2 * q^75 - 24 * q^77 - 16 * q^79 + 6 * q^81 - 16 * q^83 + 16 * q^85 - 20 * q^89 + 16 * q^93 - 32 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	−2.11491	−	0.726062i	−0.945815	−	0.324705i
$5$	−2.11491	−	0.726062i	−0.945815	−	0.324705i
$6$		0			0
$7$			4.05705i			1.53342i	0.641994	+	0.766710i	$0.278107\pi$
$7$			4.05705i			1.53342i	−0.641994	+	0.766710i	$0.721893\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			0.985939i			0.297272i	0.988892	+	0.148636i	$0.0474882\pi$
$11$			0.985939i			0.297272i	−0.988892	+	0.148636i	$0.952512\pi$
$12$		0			0
$13$	4.94567			1.37168			0.685841	−	0.727752i	$-0.259435\pi$
$13$	4.94567			1.37168			0.685841	+	0.727752i	$0.259435\pi$
$14$		0			0
$15$	−2.11491	−	0.726062i	−0.546067	−	0.187468i
$16$		0			0
$17$			4.52323i			1.09704i	0.836136	+	0.548522i	$0.184809\pi$
$17$			4.52323i			1.09704i	−0.836136	+	0.548522i	$0.815191\pi$
$18$		0			0
$19$			2.60492i			0.597610i	0.954314	+	0.298805i	$0.0965881\pi$
$19$			2.60492i			0.597610i	−0.954314	+	0.298805i	$0.903412\pi$
$20$		0			0
$21$			4.05705i			0.885320i
$22$		0			0
$23$			3.53729i			0.737577i	0.929513	+	0.368788i	$0.120227\pi$
$23$			3.53729i			0.737577i	−0.929513	+	0.368788i	$0.879773\pi$
$24$		0			0
$25$	3.94567	+	3.07111i	0.789134	+	0.614222i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		−	7.59434i		−	1.41023i	−0.709091	−	0.705117i	$-0.750895\pi$
$29$		−	7.59434i		−	1.41023i	0.709091	−	0.705117i	$-0.249105\pi$
$30$		0			0
$31$	3.28415			0.589850			0.294925	−	0.955520i	$-0.404705\pi$
$31$	3.28415			0.589850			0.294925	+	0.955520i	$0.404705\pi$
$32$		0			0
$33$			0.985939i			0.171630i
$34$		0			0
$35$	2.94567	−	8.58028i	0.497909	−	1.45033i
$36$		0			0
$37$	−0.945668			−0.155467			−0.0777334	−	0.996974i	$-0.524768\pi$
$37$	−0.945668			−0.155467			−0.0777334	+	0.996974i	$0.524768\pi$
$38$		0			0
$39$	4.94567			0.791941
$40$		0			0
$41$	0.568295			0.0887527			0.0443763	−	0.999015i	$-0.485870\pi$
$41$	0.568295			0.0887527			0.0443763	+	0.999015i	$0.485870\pi$
$42$		0			0
$43$	−8.45963			−1.29008			−0.645041	−	0.764148i	$-0.723160\pi$
$43$	−8.45963			−1.29008			−0.645041	+	0.764148i	$0.723160\pi$
$44$		0			0
$45$	−2.11491	−	0.726062i	−0.315272	−	0.108235i
$46$		0			0
$47$		−	2.60492i		−	0.379967i	−0.981787	−	0.189984i	$-0.939157\pi$
$47$		−	2.60492i		−	0.379967i	0.981787	−	0.189984i	$-0.0608435\pi$
$48$		0			0
$49$	−9.45963			−1.35138
$50$		0			0
$51$			4.52323i			0.633379i
$52$		0			0
$53$	0.229815			0.0315675			0.0157838	−	0.999875i	$-0.494976\pi$
$53$	0.229815			0.0315675			0.0157838	+	0.999875i	$0.494976\pi$
$54$		0			0
$55$	0.715853	−	2.08517i	0.0965256	−	0.281164i
$56$		0			0
$57$			2.60492i			0.345030i
$58$		0			0
$59$		−	9.10003i		−	1.18472i	−0.805672	−	0.592362i	$-0.798196\pi$
$59$		−	9.10003i		−	1.18472i	0.805672	−	0.592362i	$-0.201804\pi$
$60$		0			0
$61$			11.0183i			1.41075i	0.708832	+	0.705377i	$0.249222\pi$
$61$			11.0183i			1.41075i	−0.708832	+	0.705377i	$0.750778\pi$
$62$		0			0
$63$			4.05705i			0.511140i
$64$		0			0
$65$	−10.4596	−	3.59086i	−1.29736	−	0.445392i
$66$		0			0
$67$	8.45963			1.03351			0.516754	−	0.856134i	$-0.327140\pi$
$67$	8.45963			1.03351			0.516754	+	0.856134i	$0.327140\pi$
$68$		0			0
$69$			3.53729i			0.425840i
$70$		0			0
$71$	−1.43171			−0.169912			−0.0849561	−	0.996385i	$-0.527075\pi$
$71$	−1.43171			−0.169912			−0.0849561	+	0.996385i	$0.527075\pi$
$72$		0			0
$73$		−	11.9507i		−	1.39873i	−0.714767	−	0.699363i	$-0.753467\pi$
$73$		−	11.9507i		−	1.39873i	0.714767	−	0.699363i	$-0.246533\pi$
$74$		0			0
$75$	3.94567	+	3.07111i	0.455606	+	0.354621i
$76$		0			0
$77$	−4.00000			−0.455842
$78$		0			0
$79$	−3.28415			−0.369495			−0.184748	−	0.982786i	$-0.559147\pi$
$79$	−3.28415			−0.369495			−0.184748	+	0.982786i	$0.559147\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−9.89134			−1.08572			−0.542858	−	0.839825i	$-0.682658\pi$
$83$	−9.89134			−1.08572			−0.542858	+	0.839825i	$0.682658\pi$
$84$		0			0
$85$	3.28415	−	9.56622i	0.356216	−	1.03760i
$86$		0			0
$87$		−	7.59434i		−	0.814199i
$88$		0			0
$89$	12.3510			1.30920			0.654600	−	0.755976i	$-0.272837\pi$
$89$	12.3510			1.30920			0.654600	+	0.755976i	$0.272837\pi$
$90$		0			0
$91$			20.0648i			2.10336i
$92$		0			0
$93$	3.28415			0.340550
$94$		0			0
$95$	1.89134	−	5.50917i	0.194047	−	0.565229i
$96$		0			0
$97$		−	3.23797i		−	0.328766i	−0.986397	−	0.164383i	$-0.947437\pi$
$97$		−	3.23797i		−	0.328766i	0.986397	−	0.164383i	$-0.0525633\pi$
$98$		0			0
$99$			0.985939i			0.0990906i
$100$		0			0
$101$			4.35637i			0.433475i	0.976230	+	0.216738i	$0.0695416\pi$
$101$			4.35637i			0.433475i	−0.976230	+	0.216738i	$0.930458\pi$
$102$		0			0
$103$		−	15.0754i		−	1.48542i	−0.669612	−	0.742711i	$-0.733540\pi$
$103$		−	15.0754i		−	1.48542i	0.669612	−	0.742711i	$-0.266460\pi$
$104$		0			0
$105$	2.94567	−	8.58028i	0.287468	−	0.837350i
$106$		0			0
$107$	−4.00000			−0.386695			−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000			−0.386695			−0.193347	+	0.981130i	$0.561934\pi$
$108$		0			0
$109$			4.17034i			0.399446i	0.979852	+	0.199723i	$0.0640042\pi$
$109$			4.17034i			0.399446i	−0.979852	+	0.199723i	$0.935996\pi$
$110$		0			0
$111$	−0.945668			−0.0897588
$112$		0			0
$113$			1.28526i			0.120907i	0.998171	+	0.0604537i	$0.0192548\pi$
$113$			1.28526i			0.120907i	−0.998171	+	0.0604537i	$0.980745\pi$
$114$		0			0
$115$	2.56829	−	7.48105i	0.239495	−	0.697611i
$116$		0			0
$117$	4.94567			0.457227
$118$		0			0
$119$	−18.3510			−1.68223
$120$		0			0
$121$	10.0279			0.911630
$122$		0			0
$123$	0.568295			0.0512414
$124$		0			0
$125$	−6.11491	−	9.35991i	−0.546934	−	0.837176i
$126$		0			0
$127$			1.15280i			0.102294i	0.998691	+	0.0511472i	$0.0162878\pi$
$127$			1.15280i			0.102294i	−0.998691	+	0.0511472i	$0.983712\pi$
$128$		0			0
$129$	−8.45963			−0.744829
$130$		0			0
$131$			3.89019i			0.339887i	0.985454	+	0.169944i	$0.0543586\pi$
$131$			3.89019i			0.339887i	−0.985454	+	0.169944i	$0.945641\pi$
$132$		0			0
$133$	−10.5683			−0.916387
$134$		0			0
$135$	−2.11491	−	0.726062i	−0.182022	−	0.0624895i
$136$		0			0
$137$		−	17.5135i		−	1.49628i	−0.663544	−	0.748138i	$-0.730948\pi$
$137$		−	17.5135i		−	1.49628i	0.663544	−	0.748138i	$-0.269052\pi$
$138$		0			0
$139$		−	16.8612i		−	1.43015i	−0.699047	−	0.715076i	$-0.746392\pi$
$139$		−	16.8612i		−	1.43015i	0.699047	−	0.715076i	$-0.253608\pi$
$140$		0			0
$141$		−	2.60492i		−	0.219374i
$142$		0			0
$143$			4.87613i			0.407762i
$144$		0			0
$145$	−5.51396	+	16.0613i	−0.457910	+	1.33382i
$146$		0			0
$147$	−9.45963			−0.780217
$148$		0			0
$149$		−	10.4986i		−	0.860078i	−0.902810	−	0.430039i	$-0.858500\pi$
$149$		−	10.4986i		−	0.860078i	0.902810	−	0.430039i	$-0.141500\pi$
$150$		0			0
$151$	−4.71585			−0.383771			−0.191885	−	0.981417i	$-0.561460\pi$
$151$	−4.71585			−0.383771			−0.191885	+	0.981417i	$0.561460\pi$
$152$		0			0
$153$			4.52323i			0.365682i
$154$		0			0
$155$	−6.94567	−	2.38449i	−0.557889	−	0.191527i
$156$		0			0
$157$	−8.94567			−0.713942			−0.356971	−	0.934115i	$-0.616191\pi$
$157$	−8.94567			−0.713942			−0.356971	+	0.934115i	$0.616191\pi$
$158$		0			0
$159$	0.229815			0.0182255
$160$		0			0
$161$	−14.3510			−1.13101
$162$		0			0
$163$	15.7827			1.23619			0.618097	−	0.786102i	$-0.287904\pi$
$163$	15.7827			1.23619			0.618097	+	0.786102i	$0.287904\pi$
$164$		0			0
$165$	0.715853	−	2.08517i	0.0557291	−	0.162330i
$166$		0			0
$167$		−	5.50917i		−	0.426312i	−0.977018	−	0.213156i	$-0.931626\pi$
$167$		−	5.50917i		−	0.426312i	0.977018	−	0.213156i	$-0.0683743\pi$
$168$		0			0
$169$	11.4596			0.881510
$170$		0			0
$171$			2.60492i			0.199203i
$172$		0			0
$173$	10.3385			0.786020			0.393010	−	0.919534i	$-0.371434\pi$
$173$	10.3385			0.786020			0.393010	+	0.919534i	$0.371434\pi$
$174$		0			0
$175$	−12.4596	+	16.0078i	−0.941860	+	1.21007i
$176$		0			0
$177$		−	9.10003i		−	0.684000i
$178$		0			0
$179$			16.1746i			1.20895i	0.796625	+	0.604474i	$0.206617\pi$
$179$			16.1746i			1.20895i	−0.796625	+	0.604474i	$0.793383\pi$
$180$		0			0
$181$			8.11409i			0.603116i	0.953448	+	0.301558i	$0.0975067\pi$
$181$			8.11409i			0.603116i	−0.953448	+	0.301558i	$0.902493\pi$
$182$		0			0
$183$			11.0183i			0.814499i
$184$		0			0
$185$	2.00000	+	0.686614i	0.147043	+	0.0504808i
$186$		0			0
$187$	−4.45963			−0.326120
$188$		0			0
$189$			4.05705i			0.295107i
$190$		0			0
$191$	24.9193			1.80309			0.901547	−	0.432681i	$-0.142432\pi$
$191$	24.9193			1.80309			0.901547	+	0.432681i	$0.142432\pi$
$192$		0			0
$193$		−	1.03951i		−	0.0748254i	−0.999300	−	0.0374127i	$-0.988088\pi$
$193$		−	1.03951i		−	0.0748254i	0.999300	−	0.0374127i	$-0.0119116\pi$
$194$		0			0
$195$	−10.4596	−	3.59086i	−0.749030	−	0.257147i
$196$		0			0
$197$	9.66152			0.688355			0.344177	−	0.938905i	$-0.388158\pi$
$197$	9.66152			0.688355			0.344177	+	0.938905i	$0.388158\pi$
$198$		0			0
$199$	23.0668			1.63516			0.817582	−	0.575813i	$-0.195314\pi$
$199$	23.0668			1.63516			0.817582	+	0.575813i	$0.195314\pi$
$200$		0			0
$201$	8.45963			0.596696
$202$		0			0
$203$	30.8106			2.16248
$204$		0			0
$205$	−1.20189	−	0.412617i	−0.0839437	−	0.0288184i
$206$		0			0
$207$			3.53729i			0.245859i
$208$		0			0
$209$	−2.56829			−0.177653
$210$		0			0
$211$		−	6.44154i		−	0.443454i	−0.975109	−	0.221727i	$-0.928831\pi$
$211$		−	6.44154i		−	0.443454i	0.975109	−	0.221727i	$-0.0711694\pi$
$212$		0			0
$213$	−1.43171			−0.0980988
$214$		0			0
$215$	17.8913	+	6.14222i	1.22018	+	0.418896i
$216$		0			0
$217$			13.3239i			0.904488i
$218$		0			0
$219$		−	11.9507i		−	0.807554i
$220$		0			0
$221$			22.3704i			1.50480i
$222$		0			0
$223$		−	17.9796i		−	1.20401i	−0.798494	−	0.602003i	$-0.794370\pi$
$223$		−	17.9796i		−	1.20401i	0.798494	−	0.602003i	$-0.205630\pi$
$224$		0			0
$225$	3.94567	+	3.07111i	0.263045	+	0.204741i
$226$		0			0
$227$	−7.02792			−0.466460			−0.233230	−	0.972422i	$-0.574929\pi$
$227$	−7.02792			−0.466460			−0.233230	+	0.972422i	$0.574929\pi$
$228$		0			0
$229$		−	4.17034i		−	0.275584i	−0.990461	−	0.137792i	$-0.955999\pi$
$229$		−	4.17034i		−	0.275584i	0.990461	−	0.137792i	$-0.0440005\pi$
$230$		0			0
$231$	−4.00000			−0.263181
$232$		0			0
$233$			23.9894i			1.57160i	0.618483	+	0.785799i	$0.287748\pi$
$233$			23.9894i			1.57160i	−0.618483	+	0.785799i	$0.712252\pi$
$234$		0			0
$235$	−1.89134	+	5.50917i	−0.123377	+	0.359379i
$236$		0			0
$237$	−3.28415			−0.213328
$238$		0			0
$239$	−8.91926			−0.576939			−0.288469	−	0.957489i	$-0.593146\pi$
$239$	−8.91926			−0.576939			−0.288469	+	0.957489i	$0.593146\pi$
$240$		0			0
$241$	−16.3510			−1.05326			−0.526629	−	0.850095i	$-0.676544\pi$
$241$	−16.3510			−1.05326			−0.526629	+	0.850095i	$0.676544\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	20.0062	+	6.86828i	1.27815	+	0.438798i
$246$		0			0
$247$			12.8831i			0.819731i
$248$		0			0
$249$	−9.89134			−0.626838
$250$		0			0
$251$			4.22391i			0.266611i	0.991075	+	0.133305i	$0.0425591\pi$
$251$			4.22391i			0.266611i	−0.991075	+	0.133305i	$0.957441\pi$
$252$		0			0
$253$	−3.48755			−0.219261
$254$		0			0
$255$	3.28415	−	9.56622i	0.205661	−	0.599060i
$256$		0			0
$257$			24.6952i			1.54044i	0.637777	+	0.770221i	$0.279854\pi$
$257$			24.6952i			1.54044i	−0.637777	+	0.770221i	$0.720146\pi$
$258$		0			0
$259$		−	3.83662i		−	0.238396i
$260$		0			0
$261$		−	7.59434i		−	0.470078i
$262$		0			0
$263$			14.6628i			0.904145i	0.891981	+	0.452073i	$0.149315\pi$
$263$			14.6628i			0.904145i	−0.891981	+	0.452073i	$0.850685\pi$
$264$		0			0
$265$	−0.486038	−	0.166860i	−0.0298571	−	0.0102501i
$266$		0			0
$267$	12.3510			0.755867
$268$		0			0
$269$			11.5381i			0.703490i	0.936096	+	0.351745i	$0.114412\pi$
$269$			11.5381i			0.703490i	−0.936096	+	0.351745i	$0.885588\pi$
$270$		0			0
$271$	−5.63511			−0.342309			−0.171154	−	0.985244i	$-0.554750\pi$
$271$	−5.63511			−0.342309			−0.171154	+	0.985244i	$0.554750\pi$
$272$		0			0
$273$			20.0648i			1.21438i
$274$		0			0
$275$	−3.02792	+	3.89019i	−0.182591	+	0.234587i
$276$		0			0
$277$	17.4053			1.04578			0.522892	−	0.852399i	$-0.324853\pi$
$277$	17.4053			1.04578			0.522892	+	0.852399i	$0.324853\pi$
$278$		0			0
$279$	3.28415			0.196617
$280$		0			0
$281$	21.7827			1.29945			0.649723	−	0.760171i	$-0.274885\pi$
$281$	21.7827			1.29945			0.649723	+	0.760171i	$0.274885\pi$
$282$		0			0
$283$	−21.5962			−1.28376			−0.641881	−	0.766804i	$-0.721846\pi$
$283$	−21.5962			−1.28376			−0.641881	+	0.766804i	$0.721846\pi$
$284$		0			0
$285$	1.89134	−	5.50917i	0.112033	−	0.326335i
$286$		0			0
$287$			2.30560i			0.136095i
$288$		0			0
$289$	−3.45963			−0.203508
$290$		0			0
$291$		−	3.23797i		−	0.189813i
$292$		0			0
$293$	−32.0125			−1.87019			−0.935095	−	0.354398i	$-0.884686\pi$
$293$	−32.0125			−1.87019			−0.935095	+	0.354398i	$0.884686\pi$
$294$		0			0
$295$	−6.60719	+	19.2457i	−0.384685	+	1.12053i
$296$		0			0
$297$			0.985939i			0.0572100i
$298$		0			0
$299$			17.4943i			1.01172i
$300$		0			0
$301$		−	34.3211i		−	1.97824i
$302$		0			0
$303$			4.35637i			0.250267i
$304$		0			0
$305$	8.00000	−	23.3028i	0.458079	−	1.33431i
$306$		0			0
$307$	1.13659			0.0648686			0.0324343	−	0.999474i	$-0.489674\pi$
$307$	1.13659			0.0648686			0.0324343	+	0.999474i	$0.489674\pi$
$308$		0			0
$309$		−	15.0754i		−	0.857609i
$310$		0			0
$311$	−5.13659			−0.291269			−0.145635	−	0.989338i	$-0.546522\pi$
$311$	−5.13659			−0.291269			−0.145635	+	0.989338i	$0.546522\pi$
$312$		0			0
$313$		−	23.0762i		−	1.30434i	−0.758071	−	0.652172i	$-0.773858\pi$
$313$		−	23.0762i		−	1.30434i	0.758071	−	0.652172i	$-0.226142\pi$
$314$		0			0
$315$	2.94567	−	8.58028i	0.165970	−	0.483444i
$316$		0			0
$317$	1.66152			0.0933203			0.0466601	−	0.998911i	$-0.485142\pi$
$317$	1.66152			0.0933203			0.0466601	+	0.998911i	$0.485142\pi$
$318$		0			0
$319$	7.48755			0.419223
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$	−11.7827			−0.655605
$324$		0			0
$325$	19.5140	+	15.1887i	1.08244	+	0.842516i
$326$		0			0
$327$			4.17034i			0.230620i
$328$		0			0
$329$	10.5683			0.582649
$330$		0			0
$331$		−	25.9077i		−	1.42402i	−0.702171	−	0.712008i	$-0.747786\pi$
$331$		−	25.9077i		−	1.42402i	0.702171	−	0.712008i	$-0.252214\pi$
$332$		0			0
$333$	−0.945668			−0.0518223
$334$		0			0
$335$	−17.8913	−	6.14222i	−0.977508	−	0.335585i
$336$		0			0
$337$			8.00696i			0.436167i	0.975930	+	0.218083i	$0.0699805\pi$
$337$			8.00696i			0.436167i	−0.975930	+	0.218083i	$0.930020\pi$
$338$		0			0
$339$			1.28526i			0.0698060i
$340$		0			0
$341$			3.23797i			0.175346i
$342$		0			0
$343$		−	9.97884i		−	0.538806i
$344$		0			0
$345$	2.56829	−	7.48105i	0.138272	−	0.402766i
$346$		0			0
$347$	23.0279			1.23620			0.618102	−	0.786098i	$-0.287902\pi$
$347$	23.0279			1.23620			0.618102	+	0.786098i	$0.287902\pi$
$348$		0			0
$349$		−	21.4380i		−	1.14755i	−0.819012	−	0.573776i	$-0.805478\pi$
$349$		−	21.4380i		−	1.14755i	0.819012	−	0.573776i	$-0.194522\pi$
$350$		0			0
$351$	4.94567			0.263980
$352$		0			0
$353$			4.52323i			0.240747i	0.992729	+	0.120374i	$0.0384093\pi$
$353$			4.52323i			0.240747i	−0.992729	+	0.120374i	$0.961591\pi$
$354$		0			0
$355$	3.02792	+	1.03951i	0.160706	+	0.0551713i
$356$		0			0
$357$	−18.3510			−0.971236
$358$		0			0
$359$	10.3510			0.546303			0.273152	−	0.961971i	$-0.411934\pi$
$359$	10.3510			0.546303			0.273152	+	0.961971i	$0.411934\pi$
$360$		0			0
$361$	12.2144			0.642862
$362$		0			0
$363$	10.0279			0.526330
$364$		0			0
$365$	−8.67696	+	25.2747i	−0.454173	+	1.32294i
$366$		0			0
$367$			0.485359i			0.0253355i	0.999920	+	0.0126678i	$0.00403238\pi$
$367$			0.485359i			0.0253355i	−0.999920	+	0.0126678i	$0.995968\pi$
$368$		0			0
$369$	0.568295			0.0295842
$370$		0			0
$371$			0.932371i			0.0484063i
$372$		0			0
$373$	−30.0823			−1.55760			−0.778800	−	0.627272i	$-0.784171\pi$
$373$	−30.0823			−1.55760			−0.778800	+	0.627272i	$0.784171\pi$
$374$		0			0
$375$	−6.11491	−	9.35991i	−0.315772	−	0.483344i
$376$		0			0
$377$		−	37.5591i		−	1.93439i
$378$		0			0
$379$			33.6881i			1.73044i	0.501392	+	0.865220i	$0.332821\pi$
$379$			33.6881i			1.73044i	−0.501392	+	0.865220i	$0.667179\pi$
$380$		0			0
$381$			1.15280i			0.0590597i
$382$		0			0
$383$		−	5.17545i		−	0.264453i	−0.991220	−	0.132227i	$-0.957787\pi$
$383$		−	5.17545i		−	0.264453i	0.991220	−	0.132227i	$-0.0422127\pi$
$384$		0			0
$385$	8.45963	+	2.90425i	0.431143	+	0.148014i
$386$		0			0
$387$	−8.45963			−0.430027
$388$		0			0
$389$			16.6408i			0.843722i	0.906660	+	0.421861i	$0.138623\pi$
$389$			16.6408i			0.843722i	−0.906660	+	0.421861i	$0.861377\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$			3.89019i			0.196234i
$394$		0			0
$395$	6.94567	+	2.38449i	0.349474	+	0.119977i
$396$		0			0
$397$	20.4332			1.02551			0.512757	−	0.858534i	$-0.328624\pi$
$397$	20.4332			1.02551			0.512757	+	0.858534i	$0.328624\pi$
$398$		0			0
$399$	−10.5683			−0.529076
$400$		0			0
$401$	−4.56829			−0.228130			−0.114065	−	0.993473i	$-0.536387\pi$
$401$	−4.56829			−0.228130			−0.114065	+	0.993473i	$0.536387\pi$
$402$		0			0
$403$	16.2423			0.809087
$404$		0			0
$405$	−2.11491	−	0.726062i	−0.105091	−	0.0360783i
$406$		0			0
$407$		−	0.932371i		−	0.0462159i
$408$		0			0
$409$	−19.8913			−0.983563			−0.491782	−	0.870719i	$-0.663654\pi$
$409$	−19.8913			−0.983563			−0.491782	+	0.870719i	$0.663654\pi$
$410$		0			0
$411$		−	17.5135i		−	0.863875i
$412$		0			0
$413$	36.9193			1.81668
$414$		0			0
$415$	20.9193	+	7.18172i	1.02689	+	0.352537i
$416$		0			0
$417$		−	16.8612i		−	0.825698i
$418$		0			0
$419$		−	0.387288i		−	0.0189203i	−0.999955	−	0.00946013i	$-0.996989\pi$
$419$		−	0.387288i		−	0.0189203i	0.999955	−	0.00946013i	$-0.00301130\pi$
$420$		0			0
$421$		−	12.0578i		−	0.587664i	−0.955857	−	0.293832i	$-0.905069\pi$
$421$		−	12.0578i		−	0.587664i	0.955857	−	0.293832i	$-0.0949306\pi$
$422$		0			0
$423$		−	2.60492i		−	0.126656i
$424$		0			0
$425$	−13.8913	+	17.8472i	−0.673829	+	0.865715i
$426$		0			0
$427$	−44.7019			−2.16328
$428$		0			0
$429$			4.87613i			0.235422i
$430$		0			0
$431$	−40.4068			−1.94633			−0.973164	−	0.230113i	$-0.926090\pi$
$431$	−40.4068			−1.94633			−0.973164	+	0.230113i	$0.926090\pi$
$432$		0			0
$433$		−	36.1859i		−	1.73898i	−0.493949	−	0.869491i	$-0.664447\pi$
$433$		−	36.1859i		−	1.73898i	0.493949	−	0.869491i	$-0.335553\pi$
$434$		0			0
$435$	−5.51396	+	16.0613i	−0.264374	+	0.770082i
$436$		0			0
$437$	−9.21438			−0.440783
$438$		0			0
$439$	−25.4178			−1.21312			−0.606562	−	0.795036i	$-0.707452\pi$
$439$	−25.4178			−1.21312			−0.606562	+	0.795036i	$0.707452\pi$
$440$		0			0
$441$	−9.45963			−0.450459
$442$		0			0
$443$	7.02792			0.333907			0.166953	−	0.985965i	$-0.446607\pi$
$443$	7.02792			0.333907			0.166953	+	0.985965i	$0.446607\pi$
$444$		0			0
$445$	−26.1212	−	8.96757i	−1.23826	−	0.425103i
$446$		0			0
$447$		−	10.4986i		−	0.496566i
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$		0			0
$451$			0.560304i			0.0263837i
$452$		0			0
$453$	−4.71585			−0.221570
$454$		0			0
$455$	14.5683	−	42.4352i	0.682972	−	1.98939i
$456$		0			0
$457$			25.2747i			1.18230i	0.806562	+	0.591149i	$0.201326\pi$
$457$			25.2747i			1.18230i	−0.806562	+	0.591149i	$0.798674\pi$
$458$		0			0
$459$			4.52323i			0.211126i
$460$		0			0
$461$		−	41.0902i		−	1.91376i	−0.290479	−	0.956881i	$-0.593815\pi$
$461$		−	41.0902i		−	1.91376i	0.290479	−	0.956881i	$-0.406185\pi$
$462$		0			0
$463$		−	13.2106i		−	0.613951i	−0.951717	−	0.306975i	$-0.900683\pi$
$463$		−	13.2106i		−	0.613951i	0.951717	−	0.306975i	$-0.0993169\pi$
$464$		0			0
$465$	−6.94567	−	2.38449i	−0.322098	−	0.110578i
$466$		0			0
$467$	1.89134			0.0875206			0.0437603	−	0.999042i	$-0.486066\pi$
$467$	1.89134			0.0875206			0.0437603	+	0.999042i	$0.486066\pi$
$468$		0			0
$469$			34.3211i			1.58480i
$470$		0			0
$471$	−8.94567			−0.412195
$472$		0			0
$473$		−	8.34068i		−	0.383505i
$474$		0			0
$475$	−8.00000	+	10.2782i	−0.367065	+	0.471594i
$476$		0			0
$477$	0.229815			0.0105225
$478$		0			0
$479$	31.4876			1.43870			0.719352	−	0.694646i	$-0.244439\pi$
$479$	31.4876			1.43870			0.719352	+	0.694646i	$0.244439\pi$
$480$		0			0
$481$	−4.67696			−0.213251
$482$		0			0
$483$	−14.3510			−0.652992
$484$		0			0
$485$	−2.35097	+	6.84800i	−0.106752	+	0.310952i
$486$		0			0
$487$		−	12.9964i		−	0.588922i	−0.955664	−	0.294461i	$-0.904860\pi$
$487$		−	12.9964i		−	0.588922i	0.955664	−	0.294461i	$-0.0951401\pi$
$488$		0			0
$489$	15.7827			0.713717
$490$		0			0
$491$		−	14.9085i		−	0.672812i	−0.941717	−	0.336406i	$-0.890788\pi$
$491$		−	14.9085i		−	0.672812i	0.941717	−	0.336406i	$-0.109212\pi$
$492$		0			0
$493$	34.3510			1.54709
$494$		0			0
$495$	0.715853	−	2.08517i	0.0321752	−	0.0937214i
$496$		0			0
$497$		−	5.80850i		−	0.260547i
$498$		0			0
$499$		−	35.6599i		−	1.59636i	−0.602420	−	0.798179i	$-0.705797\pi$
$499$		−	35.6599i		−	1.59636i	0.602420	−	0.798179i	$-0.294203\pi$
$500$		0			0
$501$		−	5.50917i		−	0.246132i
$502$		0			0
$503$		−	25.3090i		−	1.12847i	−0.825613	−	0.564237i	$-0.809170\pi$
$503$		−	25.3090i		−	1.12847i	0.825613	−	0.564237i	$-0.190830\pi$
$504$		0			0
$505$	3.16300	−	9.21332i	0.140751	−	0.409988i
$506$		0			0
$507$	11.4596			0.508940
$508$		0			0
$509$			13.7366i			0.608862i	0.952534	+	0.304431i	$0.0984663\pi$
$509$			13.7366i			0.608862i	−0.952534	+	0.304431i	$0.901534\pi$
$510$		0			0
$511$	48.4846			2.14483
$512$		0			0
$513$			2.60492i			0.115010i
$514$		0			0
$515$	−10.9457	+	31.8831i	−0.482324	+	1.40494i
$516$		0			0
$517$	2.56829			0.112953
$518$		0			0
$519$	10.3385			0.453809
$520$		0			0
$521$	−30.9193			−1.35460			−0.677299	−	0.735708i	$-0.736850\pi$
$521$	−30.9193			−1.35460			−0.677299	+	0.735708i	$0.736850\pi$
$522$		0			0
$523$	−21.8385			−0.954932			−0.477466	−	0.878650i	$-0.658445\pi$
$523$	−21.8385			−0.954932			−0.477466	+	0.878650i	$0.658445\pi$
$524$		0			0
$525$	−12.4596	+	16.0078i	−0.543783	+	0.698636i
$526$		0			0
$527$			14.8550i			0.647092i
$528$		0			0
$529$	10.4876			0.455981
$530$		0			0
$531$		−	9.10003i		−	0.394908i
$532$		0			0
$533$	2.81060			0.121740
$534$		0			0
$535$	8.45963	+	2.90425i	0.365742	+	0.125562i
$536$		0			0
$537$			16.1746i			0.697986i
$538$		0			0
$539$		−	9.32662i		−	0.401726i
$540$		0			0
$541$			24.3423i			1.04656i	0.852162	+	0.523278i	$0.175291\pi$
$541$			24.3423i			1.04656i	−0.852162	+	0.523278i	$0.824709\pi$
$542$		0			0
$543$			8.11409i			0.348209i
$544$		0			0
$545$	3.02792	−	8.81988i	0.129702	−	0.377802i
$546$		0			0
$547$	33.3789			1.42718			0.713589	−	0.700564i	$-0.247068\pi$
$547$	33.3789			1.42718			0.713589	+	0.700564i	$0.247068\pi$
$548$		0			0
$549$			11.0183i			0.470251i
$550$		0			0
$551$	19.7827			0.842770
$552$		0			0
$553$		−	13.3239i		−	0.566592i
$554$		0			0
$555$	2.00000	+	0.686614i	0.0848953	+	0.0291451i
$556$		0			0
$557$	−30.5808			−1.29575			−0.647875	−	0.761747i	$-0.724342\pi$
$557$	−30.5808			−1.29575			−0.647875	+	0.761747i	$0.724342\pi$
$558$		0			0
$559$	−41.8385			−1.76958
$560$		0			0
$561$	−4.45963			−0.188286
$562$		0			0
$563$	−7.02792			−0.296192			−0.148096	−	0.988973i	$-0.547314\pi$
$563$	−7.02792			−0.296192			−0.148096	+	0.988973i	$0.547314\pi$
$564$		0			0
$565$	0.933181	−	2.71821i	0.0392592	−	0.114356i
$566$		0			0
$567$			4.05705i			0.170380i
$568$		0			0
$569$	−24.3510			−1.02085			−0.510423	−	0.859924i	$-0.670511\pi$
$569$	−24.3510			−1.02085			−0.510423	+	0.859924i	$0.670511\pi$
$570$		0			0
$571$		−	20.0992i		−	0.841125i	−0.907263	−	0.420563i	$-0.861833\pi$
$571$		−	20.0992i		−	0.841125i	0.907263	−	0.420563i	$-0.138167\pi$
$572$		0			0
$573$	24.9193			1.04102
$574$		0			0
$575$	−10.8634	+	13.9570i	−0.453036	+	0.582047i
$576$		0			0
$577$		−	4.76899i		−	0.198536i	−0.995061	−	0.0992678i	$-0.968350\pi$
$577$		−	4.76899i		−	0.198536i	0.995061	−	0.0992678i	$-0.0316501\pi$
$578$		0			0
$579$		−	1.03951i		−	0.0432004i
$580$		0			0
$581$		−	40.1296i		−	1.66486i
$582$		0			0
$583$			0.226584i			0.00938413i
$584$		0			0
$585$	−10.4596	−	3.59086i	−0.432452	−	0.148464i
$586$		0			0
$587$	−8.21733			−0.339165			−0.169583	−	0.985516i	$-0.554242\pi$
$587$	−8.21733			−0.339165			−0.169583	+	0.985516i	$0.554242\pi$
$588$		0			0
$589$			8.55495i			0.352501i
$590$		0			0
$591$	9.66152			0.397422
$592$		0			0
$593$			7.76120i			0.318714i	0.987221	+	0.159357i	$0.0509421\pi$
$593$			7.76120i			0.318714i	−0.987221	+	0.159357i	$0.949058\pi$
$594$		0			0
$595$	38.8106	+	13.3239i	1.59108	+	0.546228i
$596$		0			0
$597$	23.0668			0.944062
$598$		0			0
$599$	−5.64903			−0.230813			−0.115407	−	0.993318i	$-0.536817\pi$
$599$	−5.64903			−0.230813			−0.115407	+	0.993318i	$0.536817\pi$
$600$		0			0
$601$	−37.7299			−1.53903			−0.769516	−	0.638627i	$-0.779503\pi$
$601$	−37.7299			−1.53903			−0.769516	+	0.638627i	$0.779503\pi$
$602$		0			0
$603$	8.45963			0.344503
$604$		0			0
$605$	−21.2081	−	7.28090i	−0.862233	−	0.296010i
$606$		0			0
$607$			0.113292i			0.00459837i	0.999997	+	0.00229919i	$0.000731854\pi$
$607$			0.113292i			0.00459837i	−0.999997	+	0.00229919i	$0.999268\pi$
$608$		0			0
$609$	30.8106			1.24851
$610$		0			0
$611$		−	12.8831i		−	0.521194i
$612$		0			0
$613$	−0.703366			−0.0284087			−0.0142044	−	0.999899i	$-0.504522\pi$
$613$	−0.703366			−0.0284087			−0.0142044	+	0.999899i	$0.504522\pi$
$614$		0			0
$615$	−1.20189	−	0.412617i	−0.0484649	−	0.0166383i
$616$		0			0
$617$			24.4809i			0.985564i	0.870153	+	0.492782i	$0.164020\pi$
$617$			24.4809i			0.985564i	−0.870153	+	0.492782i	$0.835980\pi$
$618$		0			0
$619$			39.4966i			1.58750i	0.608243	+	0.793751i	$0.291874\pi$
$619$			39.4966i			1.58750i	−0.608243	+	0.793751i	$0.708126\pi$
$620$		0			0
$621$			3.53729i			0.141947i
$622$		0			0
$623$			50.1084i			2.00755i
$624$		0			0
$625$	6.13659	+	24.2351i	0.245464	+	0.969406i
$626$		0			0
$627$	−2.56829			−0.102568
$628$		0			0
$629$		−	4.27748i		−	0.170554i
$630$		0			0
$631$	−17.3400			−0.690294			−0.345147	−	0.938549i	$-0.612171\pi$
$631$	−17.3400			−0.690294			−0.345147	+	0.938549i	$0.612171\pi$
$632$		0			0
$633$		−	6.44154i		−	0.256028i
$634$		0			0
$635$	0.837003	−	2.43806i	0.0332155	−	0.0967516i
$636$		0			0
$637$	−46.7842			−1.85366
$638$		0			0
$639$	−1.43171			−0.0566374
$640$		0			0
$641$	38.7019			1.52863			0.764317	−	0.644840i	$-0.223076\pi$
$641$	38.7019			1.52863			0.764317	+	0.644840i	$0.223076\pi$
$642$		0			0
$643$	1.13659			0.0448227			0.0224113	−	0.999749i	$-0.492866\pi$
$643$	1.13659			0.0448227			0.0224113	+	0.999749i	$0.492866\pi$
$644$		0			0
$645$	17.8913	+	6.14222i	0.704471	+	0.241850i
$646$		0			0
$647$		−	6.54868i		−	0.257455i	−0.991680	−	0.128728i	$-0.958911\pi$
$647$		−	6.54868i		−	0.257455i	0.991680	−	0.128728i	$-0.0410893\pi$
$648$		0			0
$649$	8.97208			0.352185
$650$		0			0
$651$			13.3239i			0.522206i
$652$		0			0
$653$	8.74226			0.342111			0.171056	−	0.985261i	$-0.445282\pi$
$653$	8.74226			0.342111			0.171056	+	0.985261i	$0.445282\pi$
$654$		0			0
$655$	2.82452	−	8.22739i	0.110363	−	0.321471i
$656$		0			0
$657$		−	11.9507i		−	0.466242i
$658$		0			0
$659$		−	35.5336i		−	1.38419i	−0.721804	−	0.692097i	$-0.756687\pi$
$659$		−	35.5336i		−	1.38419i	0.721804	−	0.692097i	$-0.243313\pi$
$660$		0			0
$661$		−	23.3028i		−	0.906373i	−0.891416	−	0.453186i	$-0.850287\pi$
$661$		−	23.3028i		−	0.906373i	0.891416	−	0.453186i	$-0.149713\pi$
$662$		0			0
$663$			22.3704i			0.868795i
$664$		0			0
$665$	22.3510	+	7.67324i	0.866733	+	0.297555i
$666$		0			0
$667$	26.8634			1.04016
$668$		0			0
$669$		−	17.9796i		−	0.695133i
$670$		0			0
$671$	−10.8634			−0.419377
$672$		0			0
$673$			14.3634i			0.553670i	0.960917	+	0.276835i	$0.0892856\pi$
$673$			14.3634i			0.553670i	−0.960917	+	0.276835i	$0.910714\pi$
$674$		0			0
$675$	3.94567	+	3.07111i	0.151869	+	0.118207i
$676$		0			0
$677$	24.3076			0.934217			0.467109	−	0.884200i	$-0.345296\pi$
$677$	24.3076			0.934217			0.467109	+	0.884200i	$0.345296\pi$
$678$		0			0
$679$	13.1366			0.504136
$680$		0			0
$681$	−7.02792			−0.269311
$682$		0			0
$683$	−38.5933			−1.47673			−0.738365	−	0.674401i	$-0.764402\pi$
$683$	−38.5933			−1.47673			−0.738365	+	0.674401i	$0.764402\pi$
$684$		0			0
$685$	−12.7159	+	37.0393i	−0.485848	+	1.41520i
$686$		0			0
$687$		−	4.17034i		−	0.159108i
$688$		0			0
$689$	1.13659			0.0433006
$690$		0			0
$691$		−	13.4090i		−	0.510102i	−0.966928	−	0.255051i	$-0.917908\pi$
$691$		−	13.4090i		−	0.510102i	0.966928	−	0.255051i	$-0.0820923\pi$
$692$		0			0
$693$	−4.00000			−0.151947
$694$		0			0
$695$	−12.2423	+	35.6599i	−0.464377	+	1.35266i
$696$		0			0
$697$			2.57053i			0.0973657i
$698$		0			0
$699$			23.9894i			0.907362i
$700$		0			0
$701$		−	15.6013i		−	0.589253i	−0.955613	−	0.294626i	$-0.904805\pi$
$701$		−	15.6013i		−	0.589253i	0.955613	−	0.294626i	$-0.0951952\pi$
$702$		0			0
$703$		−	2.46339i		−	0.0929086i
$704$		0			0
$705$	−1.89134	+	5.50917i	−0.0712318	+	0.207487i
$706$		0			0
$707$	−17.6740			−0.664699
$708$		0			0
$709$		−	15.7873i		−	0.592906i	−0.955047	−	0.296453i	$-0.904196\pi$
$709$		−	15.7873i		−	0.592906i	0.955047	−	0.296453i	$-0.0958038\pi$
$710$		0			0
$711$	−3.28415			−0.123165
$712$		0			0
$713$			11.6170i			0.435060i
$714$		0			0
$715$	3.54037	−	10.3126i	0.132402	−	0.385668i
$716$		0			0
$717$	−8.91926			−0.333096
$718$		0			0
$719$	22.5683			0.841655			0.420828	−	0.907141i	$-0.361740\pi$
$719$	22.5683			0.841655			0.420828	+	0.907141i	$0.361740\pi$
$720$		0			0
$721$	61.1616			2.27778
$722$		0			0
$723$	−16.3510			−0.608099
$724$		0			0
$725$	23.3230	−	29.9647i	0.866196	−	1.11286i
$726$		0			0
$727$		−	2.79096i		−	0.103511i	−0.998660	−	0.0517554i	$-0.983518\pi$
$727$		−	2.79096i		−	0.103511i	0.998660	−	0.0517554i	$-0.0164816\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	38.2649i		−	1.41528i
$732$		0			0
$733$	16.4860			0.608926			0.304463	−	0.952524i	$-0.401523\pi$
$733$	16.4860			0.608926			0.304463	+	0.952524i	$0.401523\pi$
$734$		0			0
$735$	20.0062	+	6.86828i	0.737941	+	0.253340i
$736$		0			0
$737$			8.34068i			0.307233i
$738$		0			0
$739$			14.8894i			0.547714i	0.961770	+	0.273857i	$0.0882995\pi$
$739$			14.8894i			0.547714i	−0.961770	+	0.273857i	$0.911701\pi$
$740$		0			0
$741$			12.8831i			0.473272i
$742$		0			0
$743$			41.4301i			1.51992i	0.649968	+	0.759961i	$0.274782\pi$
$743$			41.4301i			1.51992i	−0.649968	+	0.759961i	$0.725218\pi$
$744$		0			0
$745$	−7.62263	+	22.2035i	−0.279271	+	0.813475i
$746$		0			0
$747$	−9.89134			−0.361905
$748$		0			0
$749$		−	16.2282i		−	0.592965i
$750$		0			0
$751$	−30.5544			−1.11494			−0.557472	−	0.830195i	$-0.688229\pi$
$751$	−30.5544			−1.11494			−0.557472	+	0.830195i	$0.688229\pi$
$752$		0			0
$753$			4.22391i			0.153928i
$754$		0			0
$755$	9.97359	+	3.42400i	0.362976	+	0.124612i
$756$		0			0
$757$	−0.433223			−0.0157457			−0.00787287	−	0.999969i	$-0.502506\pi$
$757$	−0.433223			−0.0157457			−0.00787287	+	0.999969i	$0.502506\pi$
$758$		0			0
$759$	−3.48755			−0.126590
$760$		0			0
$761$	−14.9193			−0.540823			−0.270411	−	0.962745i	$-0.587160\pi$
$761$	−14.9193			−0.540823			−0.270411	+	0.962745i	$0.587160\pi$
$762$		0			0
$763$	−16.9193			−0.612518
$764$		0			0
$765$	3.28415	−	9.56622i	0.118739	−	0.345867i
$766$		0			0
$767$		−	45.0057i		−	1.62506i
$768$		0			0
$769$	31.3789			1.13155			0.565776	−	0.824559i	$-0.308577\pi$
$769$	31.3789			1.13155			0.565776	+	0.824559i	$0.308577\pi$
$770$		0			0
$771$			24.6952i			0.889375i
$772$		0			0
$773$	−13.4192			−0.482656			−0.241328	−	0.970444i	$-0.577583\pi$
$773$	−13.4192			−0.482656			−0.241328	+	0.970444i	$0.577583\pi$
$774$		0			0
$775$	12.9582	+	10.0860i	0.465471	+	0.362299i
$776$		0			0
$777$		−	3.83662i		−	0.137638i
$778$		0			0
$779$			1.48036i			0.0530395i
$780$		0			0
$781$		−	1.41157i		−	0.0505101i
$782$		0			0
$783$		−	7.59434i		−	0.271400i
$784$		0			0
$785$	18.9193	+	6.49511i	0.675257	+	0.231820i
$786$		0			0
$787$	36.4846			1.30054			0.650268	−	0.759705i	$-0.274657\pi$
$787$	36.4846			1.30054			0.650268	+	0.759705i	$0.274657\pi$
$788$		0			0
$789$			14.6628i			0.522009i
$790$		0			0
$791$	−5.21438			−0.185402
$792$		0			0
$793$			54.4931i			1.93511i
$794$		0			0
$795$	−0.486038	−	0.166860i	−0.0172380	−	0.00591791i
$796$		0			0
$797$	26.0683			0.923388			0.461694	−	0.887039i	$-0.347242\pi$
$797$	26.0683			0.923388			0.461694	+	0.887039i	$0.347242\pi$
$798$		0			0
$799$	11.7827			0.416841
$800$		0			0
$801$	12.3510			0.436400
$802$		0			0
$803$	11.7827			0.415801
$804$		0			0
$805$	30.3510	+	10.4197i	1.06973	+	0.367246i
$806$		0			0
$807$			11.5381i			0.406160i
$808$		0			0
$809$	−35.6212			−1.25237			−0.626187	−	0.779673i	$-0.715385\pi$
$809$	−35.6212			−1.25237			−0.626187	+	0.779673i	$0.715385\pi$
$810$		0			0
$811$			43.8935i			1.54131i	0.637253	+	0.770654i	$0.280071\pi$
$811$			43.8935i			1.54131i	−0.637253	+	0.770654i	$0.719929\pi$
$812$		0			0
$813$	−5.63511			−0.197632
$814$		0			0
$815$	−33.3789	−	11.4592i	−1.16921	−	0.401398i
$816$		0			0
$817$		−	22.0367i		−	0.770966i
$818$		0			0
$819$			20.0648i			0.701121i
$820$		0			0
$821$			28.8058i			1.00533i	0.864482	+	0.502665i	$0.167647\pi$
$821$			28.8058i			1.00533i	−0.864482	+	0.502665i	$0.832353\pi$
$822$		0			0
$823$			27.9585i			0.974571i	0.873243	+	0.487286i	$0.162013\pi$
$823$			27.9585i			0.974571i	−0.873243	+	0.487286i	$0.837987\pi$
$824$		0			0
$825$	−3.02792	+	3.89019i	−0.105419	+	0.135439i
$826$		0			0
$827$	14.8634			0.516851			0.258426	−	0.966031i	$-0.416796\pi$
$827$	14.8634			0.516851			0.258426	+	0.966031i	$0.416796\pi$
$828$		0			0
$829$			41.7678i			1.45065i	0.688404	+	0.725327i	$0.258312\pi$
$829$			41.7678i			1.45065i	−0.688404	+	0.725327i	$0.741688\pi$
$830$		0			0
$831$	17.4053			0.603783
$832$		0			0
$833$		−	42.7881i		−	1.48252i
$834$		0			0
$835$	−4.00000	+	11.6514i	−0.138426	+	0.403213i
$836$		0			0
$837$	3.28415			0.113517
$838$		0			0
$839$	−21.6490			−0.747408			−0.373704	−	0.927548i	$-0.621912\pi$
$839$	−21.6490			−0.747408			−0.373704	+	0.927548i	$0.621912\pi$
$840$		0			0
$841$	−28.6740			−0.988759
$842$		0			0
$843$	21.7827			0.750235
$844$		0			0
$845$	−24.2361	−	8.32040i	−0.833746	−	0.286231i
$846$		0			0
$847$			40.6838i			1.39791i
$848$		0			0
$849$	−21.5962			−0.741180
$850$		0			0
$851$		−	3.34510i		−	0.114669i
$852$		0			0
$853$	−49.1880			−1.68416			−0.842082	−	0.539350i	$-0.818670\pi$
$853$	−49.1880			−1.68416			−0.842082	+	0.539350i	$0.818670\pi$
$854$		0			0
$855$	1.89134	−	5.50917i	0.0646823	−	0.188410i
$856$		0			0
$857$		−	2.65849i		−	0.0908123i	−0.998969	−	0.0454062i	$-0.985542\pi$
$857$		−	2.65849i		−	0.0908123i	0.998969	−	0.0454062i	$-0.0144582\pi$
$858$		0			0
$859$		−	4.57680i		−	0.156158i	−0.996947	−	0.0780792i	$-0.975121\pi$
$859$		−	4.57680i		−	0.156158i	0.996947	−	0.0780792i	$-0.0248787\pi$
$860$		0			0
$861$			2.30560i			0.0785746i
$862$		0			0
$863$		−	34.0218i		−	1.15812i	−0.815287	−	0.579058i	$-0.803421\pi$
$863$		−	34.0218i		−	1.15812i	0.815287	−	0.579058i	$-0.196579\pi$
$864$		0			0
$865$	−21.8649	−	7.50638i	−0.743430	−	0.255224i
$866$		0			0
$867$	−3.45963			−0.117495
$868$		0			0
$869$		−	3.23797i		−	0.109841i
$870$		0			0
$871$	41.8385			1.41764
$872$		0			0
$873$		−	3.23797i		−	0.109589i
$874$		0			0
$875$	37.9736	−	24.8085i	1.28374	−	0.838679i
$876$		0			0
$877$	11.7563			0.396981			0.198490	−	0.980103i	$-0.436396\pi$
$877$	11.7563			0.396981			0.198490	+	0.980103i	$0.436396\pi$
$878$		0			0
$879$	−32.0125			−1.07975
$880$		0			0
$881$	13.2702			0.447085			0.223543	−	0.974694i	$-0.428238\pi$
$881$	13.2702			0.447085			0.223543	+	0.974694i	$0.428238\pi$
$882$		0			0
$883$	17.1366			0.576692			0.288346	−	0.957526i	$-0.406895\pi$
$883$	17.1366			0.576692			0.288346	+	0.957526i	$0.406895\pi$
$884$		0			0
$885$	−6.60719	+	19.2457i	−0.222098	+	0.646938i
$886$		0			0
$887$			32.0883i			1.07742i	0.842492	+	0.538709i	$0.181088\pi$
$887$			32.0883i			1.07742i	−0.842492	+	0.538709i	$0.818912\pi$
$888$		0			0
$889$	−4.67696			−0.156860
$890$		0			0
$891$			0.985939i			0.0330302i
$892$		0			0
$893$	6.78562			0.227072
$894$		0			0
$895$	11.7438	−	34.2078i	0.392551	−	1.14344i
$896$		0			0
$897$			17.4943i			0.584117i
$898$		0			0
$899$		−	24.9409i		−	0.831827i
$900$		0			0
$901$			1.03951i			0.0346310i
$902$		0			0
$903$		−	34.3211i		−	1.14214i
$904$		0			0
$905$	5.89134	−	17.1606i	0.195835	−	0.570436i
$906$		0			0
$907$	−40.4596			−1.34344			−0.671720	−	0.740805i	$-0.734444\pi$
$907$	−40.4596			−1.34344			−0.671720	+	0.740805i	$0.734444\pi$
$908$		0			0
$909$			4.35637i			0.144492i
$910$		0			0
$911$	−16.0000			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$		−	9.75225i		−	0.322752i
$914$		0			0
$915$	8.00000	−	23.3028i	0.264472	−	0.770366i
$916$		0			0
$917$	−15.7827			−0.521190
$918$		0			0
$919$	50.8495			1.67737			0.838685	−	0.544617i	$-0.183325\pi$
$919$	50.8495			1.67737			0.838685	+	0.544617i	$0.183325\pi$
$920$		0			0
$921$	1.13659			0.0374519
$922$		0			0
$923$	−7.08074			−0.233065
$924$		0			0
$925$	−3.73129	−	2.90425i	−0.122684	−	0.0954911i
$926$		0			0
$927$		−	15.0754i		−	0.495141i
$928$		0			0
$929$	−12.6461			−0.414904			−0.207452	−	0.978245i	$-0.566517\pi$
$929$	−12.6461			−0.414904			−0.207452	+	0.978245i	$0.566517\pi$
$930$		0			0
$931$		−	24.6416i		−	0.807596i
$932$		0			0
$933$	−5.13659			−0.168164
$934$		0			0
$935$	9.43171	+	3.23797i	0.308450	+	0.105893i
$936$		0			0
$937$		−	40.7971i		−	1.33278i	−0.745603	−	0.666391i	$-0.767838\pi$
$937$		−	40.7971i		−	1.33278i	0.745603	−	0.666391i	$-0.232162\pi$
$938$		0			0
$939$		−	23.0762i		−	0.753063i
$940$		0			0
$941$			14.1086i			0.459928i	0.973199	+	0.229964i	$0.0738608\pi$
$941$			14.1086i			0.459928i	−0.973199	+	0.229964i	$0.926139\pi$
$942$		0			0
$943$			2.01022i			0.0654619i
$944$		0			0
$945$	2.94567	−	8.58028i	0.0958226	−	0.279117i
$946$		0			0
$947$	−45.8385			−1.48955			−0.744776	−	0.667315i	$-0.767444\pi$
$947$	−45.8385			−1.48955			−0.744776	+	0.667315i	$0.767444\pi$
$948$		0			0
$949$		−	59.1043i		−	1.91861i
$950$		0			0
$951$	1.66152			0.0538785
$952$		0			0
$953$			21.9104i			0.709747i	0.934914	+	0.354873i	$0.115476\pi$
$953$			21.9104i			0.709747i	−0.934914	+	0.354873i	$0.884524\pi$
$954$		0			0
$955$	−52.7019	−	18.0929i	−1.70539	−	0.585473i
$956$		0			0
$957$	7.48755			0.242038
$958$		0			0
$959$	71.0529			2.29442
$960$		0			0
$961$	−20.2144			−0.652077
$962$		0			0
$963$	−4.00000			−0.128898
$964$		0			0
$965$	−0.754747	+	2.19846i	−0.0242962	+	0.0707710i
$966$		0			0
$967$		−	14.0359i		−	0.451364i	−0.974201	−	0.225682i	$-0.927539\pi$
$967$		−	14.0359i		−	0.451364i	0.974201	−	0.225682i	$-0.0724610\pi$
$968$		0			0
$969$	−11.7827			−0.378514
$970$		0			0
$971$			21.6494i			0.694762i	0.937724	+	0.347381i	$0.112929\pi$
$971$			21.6494i			0.694762i	−0.937724	+	0.347381i	$0.887071\pi$
$972$		0			0
$973$	68.4068			2.19302
$974$		0			0
$975$	19.5140	+	15.1887i	0.624947	+	0.486427i
$976$		0			0
$977$			6.60225i			0.211225i	0.994407	+	0.105612i	$0.0336802\pi$
$977$			6.60225i			0.211225i	−0.994407	+	0.105612i	$0.966320\pi$
$978$		0			0
$979$			12.1773i			0.389188i
$980$		0			0
$981$			4.17034i			0.133149i
$982$		0			0
$983$		−	53.8600i		−	1.71787i	−0.512087	−	0.858934i	$-0.671127\pi$
$983$		−	53.8600i		−	1.71787i	0.512087	−	0.858934i	$-0.328873\pi$
$984$		0			0
$985$	−20.4332	−	7.01486i	−0.651057	−	0.223512i
$986$		0			0
$987$	10.5683			0.336393
$988$		0			0
$989$		−	29.9242i		−	0.951534i
$990$		0			0
$991$	−29.7129			−0.943861			−0.471931	−	0.881636i	$-0.656443\pi$
$991$	−29.7129			−0.943861			−0.471931	+	0.881636i	$0.656443\pi$
$992$		0			0
$993$		−	25.9077i		−	0.822156i
$994$		0			0
$995$	−48.7842	−	16.7479i	−1.54656	−	0.530945i
$996$		0			0
$997$	16.6506			0.527328			0.263664	−	0.964615i	$-0.415069\pi$
$997$	16.6506			0.527328			0.263664	+	0.964615i	$0.415069\pi$
$998$		0			0
$999$	−0.945668			−0.0299196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.2.d.b.49.1		6
3.2	odd	2		1440.2.d.f.1009.6		6
4.3	odd	2		120.2.d.b.109.3	yes	6
5.2	odd	4		2400.2.k.f.1201.1		12
5.3	odd	4		2400.2.k.f.1201.12		12
5.4	even	2		480.2.d.a.49.5		6
8.3	odd	2		120.2.d.a.109.3	✓	6
8.5	even	2		480.2.d.a.49.6		6
12.11	even	2		360.2.d.e.109.4		6
15.2	even	4		7200.2.k.u.3601.1		12
15.8	even	4		7200.2.k.u.3601.11		12
15.14	odd	2		1440.2.d.e.1009.2		6
16.3	odd	4		3840.2.f.l.769.3		12
16.5	even	4		3840.2.f.m.769.4		12
16.11	odd	4		3840.2.f.l.769.10		12
16.13	even	4		3840.2.f.m.769.9		12
20.3	even	4		600.2.k.f.301.1		12
20.7	even	4		600.2.k.f.301.12		12
20.19	odd	2		120.2.d.a.109.4	yes	6
24.5	odd	2		1440.2.d.e.1009.1		6
24.11	even	2		360.2.d.f.109.4		6
40.3	even	4		600.2.k.f.301.2		12
40.13	odd	4		2400.2.k.f.1201.6		12
40.19	odd	2		120.2.d.b.109.4	yes	6
40.27	even	4		600.2.k.f.301.11		12
40.29	even	2	inner	480.2.d.b.49.2		6
40.37	odd	4		2400.2.k.f.1201.7		12
60.23	odd	4		1800.2.k.u.901.12		12
60.47	odd	4		1800.2.k.u.901.1		12
60.59	even	2		360.2.d.f.109.3		6
80.19	odd	4		3840.2.f.l.769.9		12
80.29	even	4		3840.2.f.m.769.3		12
80.59	odd	4		3840.2.f.l.769.4		12
80.69	even	4		3840.2.f.m.769.10		12
120.29	odd	2		1440.2.d.f.1009.5		6
120.53	even	4		7200.2.k.u.3601.12		12
120.59	even	2		360.2.d.e.109.3		6
120.77	even	4		7200.2.k.u.3601.2		12
120.83	odd	4		1800.2.k.u.901.11		12
120.107	odd	4		1800.2.k.u.901.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.d.a.109.3	✓	6	8.3	odd	2
120.2.d.a.109.4	yes	6	20.19	odd	2
120.2.d.b.109.3	yes	6	4.3	odd	2
120.2.d.b.109.4	yes	6	40.19	odd	2
360.2.d.e.109.3		6	120.59	even	2
360.2.d.e.109.4		6	12.11	even	2
360.2.d.f.109.3		6	60.59	even	2
360.2.d.f.109.4		6	24.11	even	2
480.2.d.a.49.5		6	5.4	even	2
480.2.d.a.49.6		6	8.5	even	2
480.2.d.b.49.1		6	1.1	even	1	trivial
480.2.d.b.49.2		6	40.29	even	2	inner
600.2.k.f.301.1		12	20.3	even	4
600.2.k.f.301.2		12	40.3	even	4
600.2.k.f.301.11		12	40.27	even	4
600.2.k.f.301.12		12	20.7	even	4
1440.2.d.e.1009.1		6	24.5	odd	2
1440.2.d.e.1009.2		6	15.14	odd	2
1440.2.d.f.1009.5		6	120.29	odd	2
1440.2.d.f.1009.6		6	3.2	odd	2
1800.2.k.u.901.1		12	60.47	odd	4
1800.2.k.u.901.2		12	120.107	odd	4
1800.2.k.u.901.11		12	120.83	odd	4
1800.2.k.u.901.12		12	60.23	odd	4
2400.2.k.f.1201.1		12	5.2	odd	4
2400.2.k.f.1201.6		12	40.13	odd	4
2400.2.k.f.1201.7		12	40.37	odd	4
2400.2.k.f.1201.12		12	5.3	odd	4
3840.2.f.l.769.3		12	16.3	odd	4
3840.2.f.l.769.4		12	80.59	odd	4
3840.2.f.l.769.9		12	80.19	odd	4
3840.2.f.l.769.10		12	16.11	odd	4
3840.2.f.m.769.3		12	80.29	even	4
3840.2.f.m.769.4		12	16.5	even	4
3840.2.f.m.769.9		12	16.13	even	4
3840.2.f.m.769.10		12	80.69	even	4
7200.2.k.u.3601.1		12	15.2	even	4
7200.2.k.u.3601.2		12	120.77	even	4
7200.2.k.u.3601.11		12	15.8	even	4
7200.2.k.u.3601.12		12	120.53	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +(-2.11491 - 0.726062i) q^{5} +4.05705i q^{7} +1.00000 q^{9} +0.985939i q^{11} +4.94567 q^{13} +(-2.11491 - 0.726062i) q^{15} +4.52323i q^{17} +2.60492i q^{19} +4.05705i q^{21} +3.53729i q^{23} +(3.94567 + 3.07111i) q^{25} +1.00000 q^{27} -7.59434i q^{29} +3.28415 q^{31} +0.985939i q^{33} +(2.94567 - 8.58028i) q^{35} -0.945668 q^{37} +4.94567 q^{39} +0.568295 q^{41} -8.45963 q^{43} +(-2.11491 - 0.726062i) q^{45} -2.60492i q^{47} -9.45963 q^{49} +4.52323i q^{51} +0.229815 q^{53} +(0.715853 - 2.08517i) q^{55} +2.60492i q^{57} -9.10003i q^{59} +11.0183i q^{61} +4.05705i q^{63} +(-10.4596 - 3.59086i) q^{65} +8.45963 q^{67} +3.53729i q^{69} -1.43171 q^{71} -11.9507i q^{73} +(3.94567 + 3.07111i) q^{75} -4.00000 q^{77} -3.28415 q^{79} +1.00000 q^{81} -9.89134 q^{83} +(3.28415 - 9.56622i) q^{85} -7.59434i q^{87} +12.3510 q^{89} +20.0648i q^{91} +3.28415 q^{93} +(1.89134 - 5.50917i) q^{95} -3.23797i q^{97} +0.985939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 6 q^{9} + 8 q^{13} + 2 q^{25} + 6 q^{27} + 16 q^{31} - 4 q^{35} + 16 q^{37} + 8 q^{39} - 4 q^{41} - 6 q^{49} - 24 q^{53} + 8 q^{55} - 12 q^{65} - 16 q^{71} + 2 q^{75} - 24 q^{77} - 16 q^{79}+ \cdots - 32 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 + 6 * q^9 + 8 * q^13 + 2 * q^25 + 6 * q^27 + 16 * q^31 - 4 * q^35 + 16 * q^37 + 8 * q^39 - 4 * q^41 - 6 * q^49 - 24 * q^53 + 8 * q^55 - 12 * q^65 - 16 * q^71 + 2 * q^75 - 24 * q^77 - 16 * q^79 + 6 * q^81 - 16 * q^83 + 16 * q^85 - 20 * q^89 + 16 * q^93 - 32 * q^95

Introduction

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