LMFDB - Embedded newform 3840.2.f.l.769.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(769,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3840.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.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:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.180227832610816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 $ x^12 + x^10 - 8x^6 + 16x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.4
Root			$-1.37729 - 0.321037i$ of defining polynomial
Character	$\chi$	$=$	3840.769
Dual form			3840.2.f.l.769.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(0.726062 + 2.11491i) q^{5} -4.05705i q^{7} -1.00000 q^{9} +0.985939 q^{11} +4.94567i q^{13} +(2.11491 - 0.726062i) q^{15} -4.52323i q^{17} -2.60492 q^{19} -4.05705 q^{21} -3.53729i q^{23} +(-3.94567 + 3.07111i) q^{25} +1.00000i q^{27} -7.59434 q^{29} -3.28415 q^{31} -0.985939i q^{33} +(8.58028 - 2.94567i) q^{35} +0.945668i q^{37} +4.94567 q^{39} -0.568295 q^{41} -8.45963i q^{43} +(-0.726062 - 2.11491i) q^{45} -2.60492i q^{47} -9.45963 q^{49} -4.52323 q^{51} -0.229815i q^{53} +(0.715853 + 2.08517i) q^{55} +2.60492i q^{57} -9.10003 q^{59} +11.0183 q^{61} +4.05705i q^{63} +(-10.4596 + 3.59086i) q^{65} -8.45963i q^{67} -3.53729 q^{69} -1.43171 q^{71} -11.9507i q^{73} +(3.07111 + 3.94567i) q^{75} -4.00000i q^{77} +3.28415 q^{79} +1.00000 q^{81} +9.89134i q^{83} +(9.56622 - 3.28415i) q^{85} +7.59434i q^{87} -12.3510 q^{89} +20.0648 q^{91} +3.28415i q^{93} +(-1.89134 - 5.50917i) q^{95} +3.23797i q^{97} -0.985939 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9} - 4 q^{25} - 32 q^{31} + 16 q^{39} + 8 q^{41} - 12 q^{49} + 16 q^{55} - 24 q^{65} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89} + 64 q^{95}+O(q^{100}) $ 12 * q - 12 * q^9 - 4 * q^25 - 32 * q^31 + 16 * q^39 + 8 * q^41 - 12 * q^49 + 16 * q^55 - 24 * q^65 - 32 * q^71 + 32 * q^79 + 12 * q^81 + 40 * q^89 + 64 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	0.726062	+	2.11491i	0.324705	+	0.945815i
$5$	0.726062	+	2.11491i	0.324705	+	0.945815i
$6$		0			0
$7$		−	4.05705i		−	1.53342i	−0.641994	−	0.766710i	$-0.721893\pi$
$7$		−	4.05705i		−	1.53342i	0.641994	−	0.766710i	$-0.278107\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	0.985939			0.297272			0.148636	−	0.988892i	$-0.452512\pi$
$11$	0.985939			0.297272			0.148636	+	0.988892i	$0.452512\pi$
$12$		0			0
$13$			4.94567i			1.37168i	0.727752	+	0.685841i	$0.240565\pi$
$13$			4.94567i			1.37168i	−0.727752	+	0.685841i	$0.759435\pi$
$14$		0			0
$15$	2.11491	−	0.726062i	0.546067	−	0.187468i
$16$		0			0
$17$		−	4.52323i		−	1.09704i	−0.836136	−	0.548522i	$-0.815191\pi$
$17$		−	4.52323i		−	1.09704i	0.836136	−	0.548522i	$-0.184809\pi$
$18$		0			0
$19$	−2.60492			−0.597610			−0.298805	−	0.954314i	$-0.596588\pi$
$19$	−2.60492			−0.597610			−0.298805	+	0.954314i	$0.596588\pi$
$20$		0			0
$21$	−4.05705			−0.885320
$22$		0			0
$23$		−	3.53729i		−	0.737577i	−0.929513	−	0.368788i	$-0.879773\pi$
$23$		−	3.53729i		−	0.737577i	0.929513	−	0.368788i	$-0.120227\pi$
$24$		0			0
$25$	−3.94567	+	3.07111i	−0.789134	+	0.614222i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−7.59434			−1.41023			−0.705117	−	0.709091i	$-0.749105\pi$
$29$	−7.59434			−1.41023			−0.705117	+	0.709091i	$0.749105\pi$
$30$		0			0
$31$	−3.28415			−0.589850			−0.294925	−	0.955520i	$-0.595295\pi$
$31$	−3.28415			−0.589850			−0.294925	+	0.955520i	$0.595295\pi$
$32$		0			0
$33$		−	0.985939i		−	0.171630i
$34$		0			0
$35$	8.58028	−	2.94567i	1.45033	−	0.497909i
$36$		0			0
$37$			0.945668i			0.155467i	0.996974	+	0.0777334i	$0.0247683\pi$
$37$			0.945668i			0.155467i	−0.996974	+	0.0777334i	$0.975232\pi$
$38$		0			0
$39$	4.94567			0.791941
$40$		0			0
$41$	−0.568295			−0.0887527			−0.0443763	−	0.999015i	$-0.514130\pi$
$41$	−0.568295			−0.0887527			−0.0443763	+	0.999015i	$0.514130\pi$
$42$		0			0
$43$		−	8.45963i		−	1.29008i	−0.764148	−	0.645041i	$-0.776840\pi$
$43$		−	8.45963i		−	1.29008i	0.764148	−	0.645041i	$-0.223160\pi$
$44$		0			0
$45$	−0.726062	−	2.11491i	−0.108235	−	0.315272i
$46$		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
$51$	−4.52323			−0.633379
$52$		0			0
$53$		−	0.229815i		−	0.0315675i	−0.999875	−	0.0157838i	$-0.994976\pi$
$53$		−	0.229815i		−	0.0315675i	0.999875	−	0.0157838i	$-0.00502434\pi$
$54$		0			0
$55$	0.715853	+	2.08517i	0.0965256	+	0.281164i
$56$		0			0
$57$			2.60492i			0.345030i
$58$		0			0
$59$	−9.10003			−1.18472			−0.592362	−	0.805672i	$-0.701804\pi$
$59$	−9.10003			−1.18472			−0.592362	+	0.805672i	$0.701804\pi$
$60$		0			0
$61$	11.0183			1.41075			0.705377	−	0.708832i	$-0.250778\pi$
$61$	11.0183			1.41075			0.705377	+	0.708832i	$0.250778\pi$
$62$		0			0
$63$			4.05705i			0.511140i
$64$		0			0
$65$	−10.4596	+	3.59086i	−1.29736	+	0.445392i
$66$		0			0
$67$		−	8.45963i		−	1.03351i	−0.856134	−	0.516754i	$-0.827140\pi$
$67$		−	8.45963i		−	1.03351i	0.856134	−	0.516754i	$-0.172860\pi$
$68$		0			0
$69$	−3.53729			−0.425840
$70$		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
$75$	3.07111	+	3.94567i	0.354621	+	0.455606i
$76$		0			0
$77$		−	4.00000i		−	0.455842i
$78$		0			0
$79$	3.28415			0.369495			0.184748	−	0.982786i	$-0.440853\pi$
$79$	3.28415			0.369495			0.184748	+	0.982786i	$0.440853\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			9.89134i			1.08572i	0.839825	+	0.542858i	$0.182658\pi$
$83$			9.89134i			1.08572i	−0.839825	+	0.542858i	$0.817342\pi$
$84$		0			0
$85$	9.56622	−	3.28415i	1.03760	−	0.356216i
$86$		0			0
$87$			7.59434i			0.814199i
$88$		0			0
$89$	−12.3510			−1.30920			−0.654600	−	0.755976i	$-0.727163\pi$
$89$	−12.3510			−1.30920			−0.654600	+	0.755976i	$0.727163\pi$
$90$		0			0
$91$	20.0648			2.10336
$92$		0			0
$93$			3.28415i			0.340550i
$94$		0			0
$95$	−1.89134	−	5.50917i	−0.194047	−	0.565229i
$96$		0			0
$97$			3.23797i			0.328766i	0.986397	+	0.164383i	$0.0525633\pi$
$97$			3.23797i			0.328766i	−0.986397	+	0.164383i	$0.947437\pi$
$98$		0			0
$99$	−0.985939			−0.0990906
$100$		0			0
$101$	−4.35637			−0.433475			−0.216738	−	0.976230i	$-0.569542\pi$
$101$	−4.35637			−0.433475			−0.216738	+	0.976230i	$0.569542\pi$
$102$		0			0
$103$			15.0754i			1.48542i	0.669612	+	0.742711i	$0.266460\pi$
$103$			15.0754i			1.48542i	−0.669612	+	0.742711i	$0.733540\pi$
$104$		0			0
$105$	−2.94567	−	8.58028i	−0.287468	−	0.837350i
$106$		0			0
$107$		−	4.00000i		−	0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$		−	4.00000i		−	0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$		0			0
$109$	4.17034			0.399446			0.199723	−	0.979852i	$-0.435996\pi$
$109$	4.17034			0.399446			0.199723	+	0.979852i	$0.435996\pi$
$110$		0			0
$111$	0.945668			0.0897588
$112$		0			0
$113$		−	1.28526i		−	0.120907i	−0.998171	−	0.0604537i	$-0.980745\pi$
$113$		−	1.28526i		−	0.120907i	0.998171	−	0.0604537i	$-0.0192548\pi$
$114$		0			0
$115$	7.48105	−	2.56829i	0.697611	−	0.239495i
$116$		0			0
$117$		−	4.94567i		−	0.457227i
$118$		0			0
$119$	−18.3510			−1.68223
$120$		0			0
$121$	−10.0279			−0.911630
$122$		0			0
$123$			0.568295i			0.0512414i
$124$		0			0
$125$	−9.35991	−	6.11491i	−0.837176	−	0.546934i
$126$		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
$131$	−3.89019			−0.339887			−0.169944	−	0.985454i	$-0.554359\pi$
$131$	−3.89019			−0.339887			−0.169944	+	0.985454i	$0.554359\pi$
$132$		0			0
$133$			10.5683i			0.916387i
$134$		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
$139$	−16.8612			−1.43015			−0.715076	−	0.699047i	$-0.753608\pi$
$139$	−16.8612			−1.43015			−0.715076	+	0.699047i	$0.753608\pi$
$140$		0			0
$141$	−2.60492			−0.219374
$142$		0			0
$143$			4.87613i			0.407762i
$144$		0			0
$145$	−5.51396	−	16.0613i	−0.457910	−	1.33382i
$146$		0			0
$147$			9.45963i			0.780217i
$148$		0			0
$149$	10.4986			0.860078			0.430039	−	0.902810i	$-0.358500\pi$
$149$	10.4986			0.860078			0.430039	+	0.902810i	$0.358500\pi$
$150$		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
$155$	−2.38449	−	6.94567i	−0.191527	−	0.557889i
$156$		0			0
$157$		−	8.94567i		−	0.713942i	−0.934115	−	0.356971i	$-0.883809\pi$
$157$		−	8.94567i		−	0.713942i	0.934115	−	0.356971i	$-0.116191\pi$
$158$		0			0
$159$	−0.229815			−0.0182255
$160$		0			0
$161$	−14.3510			−1.13101
$162$		0			0
$163$		−	15.7827i		−	1.23619i	−0.786102	−	0.618097i	$-0.787904\pi$
$163$		−	15.7827i		−	1.23619i	0.786102	−	0.618097i	$-0.212096\pi$
$164$		0			0
$165$	2.08517	−	0.715853i	0.162330	−	0.0557291i
$166$		0			0
$167$			5.50917i			0.426312i	0.977018	+	0.213156i	$0.0683743\pi$
$167$			5.50917i			0.426312i	−0.977018	+	0.213156i	$0.931626\pi$
$168$		0			0
$169$	−11.4596			−0.881510
$170$		0			0
$171$	2.60492			0.199203
$172$		0			0
$173$			10.3385i			0.786020i	0.919534	+	0.393010i	$0.128566\pi$
$173$			10.3385i			0.786020i	−0.919534	+	0.393010i	$0.871434\pi$
$174$		0			0
$175$	12.4596	+	16.0078i	0.941860	+	1.21007i
$176$		0			0
$177$			9.10003i			0.684000i
$178$		0			0
$179$	−16.1746			−1.20895			−0.604474	−	0.796625i	$-0.706617\pi$
$179$	−16.1746			−1.20895			−0.604474	+	0.796625i	$0.706617\pi$
$180$		0			0
$181$	−8.11409			−0.603116			−0.301558	−	0.953448i	$-0.597507\pi$
$181$	−8.11409			−0.603116			−0.301558	+	0.953448i	$0.597507\pi$
$182$		0			0
$183$		−	11.0183i		−	0.814499i
$184$		0			0
$185$	−2.00000	+	0.686614i	−0.147043	+	0.0504808i
$186$		0			0
$187$		−	4.45963i		−	0.326120i
$188$		0			0
$189$	4.05705			0.295107
$190$		0			0
$191$	−24.9193			−1.80309			−0.901547	−	0.432681i	$-0.857568\pi$
$191$	−24.9193			−1.80309			−0.901547	+	0.432681i	$0.857568\pi$
$192$		0			0
$193$			1.03951i			0.0748254i	0.999300	+	0.0374127i	$0.0119116\pi$
$193$			1.03951i			0.0748254i	−0.999300	+	0.0374127i	$0.988088\pi$
$194$		0			0
$195$	3.59086	+	10.4596i	0.257147	+	0.749030i
$196$		0			0
$197$		−	9.66152i		−	0.688355i	−0.938905	−	0.344177i	$-0.888158\pi$
$197$		−	9.66152i		−	0.688355i	0.938905	−	0.344177i	$-0.111842\pi$
$198$		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
$203$			30.8106i			2.16248i
$204$		0			0
$205$	−0.412617	−	1.20189i	−0.0288184	−	0.0839437i
$206$		0			0
$207$			3.53729i			0.245859i
$208$		0			0
$209$	−2.56829			−0.177653
$210$		0			0
$211$	6.44154			0.443454			0.221727	−	0.975109i	$-0.428831\pi$
$211$	6.44154			0.443454			0.221727	+	0.975109i	$0.428831\pi$
$212$		0			0
$213$			1.43171i			0.0980988i
$214$		0			0
$215$	17.8913	−	6.14222i	1.22018	−	0.418896i
$216$		0			0
$217$			13.3239i			0.904488i
$218$		0			0
$219$	−11.9507			−0.807554
$220$		0			0
$221$	22.3704			1.50480
$222$		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
$227$			7.02792i			0.466460i	0.972422	+	0.233230i	$0.0749295\pi$
$227$			7.02792i			0.466460i	−0.972422	+	0.233230i	$0.925071\pi$
$228$		0			0
$229$	4.17034			0.275584			0.137792	−	0.990461i	$-0.455999\pi$
$229$	4.17034			0.275584			0.137792	+	0.990461i	$0.455999\pi$
$230$		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
$235$	5.50917	−	1.89134i	0.359379	−	0.123377i
$236$		0			0
$237$		−	3.28415i		−	0.213328i
$238$		0			0
$239$	8.91926			0.576939			0.288469	−	0.957489i	$-0.406854\pi$
$239$	8.91926			0.576939			0.288469	+	0.957489i	$0.406854\pi$
$240$		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
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−6.86828	−	20.0062i	−0.438798	−	1.27815i
$246$		0			0
$247$		−	12.8831i		−	0.819731i
$248$		0			0
$249$	9.89134			0.626838
$250$		0			0
$251$	4.22391			0.266611			0.133305	−	0.991075i	$-0.457441\pi$
$251$	4.22391			0.266611			0.133305	+	0.991075i	$0.457441\pi$
$252$		0			0
$253$		−	3.48755i		−	0.219261i
$254$		0			0
$255$	−3.28415	−	9.56622i	−0.205661	−	0.599060i
$256$		0			0
$257$		−	24.6952i		−	1.54044i	−0.637777	−	0.770221i	$-0.720146\pi$
$257$		−	24.6952i		−	1.54044i	0.637777	−	0.770221i	$-0.279854\pi$
$258$		0			0
$259$	3.83662			0.238396
$260$		0			0
$261$	7.59434			0.470078
$262$		0			0
$263$		−	14.6628i		−	0.904145i	−0.891981	−	0.452073i	$-0.850685\pi$
$263$		−	14.6628i		−	0.904145i	0.891981	−	0.452073i	$-0.149315\pi$
$264$		0			0
$265$	0.486038	−	0.166860i	0.0298571	−	0.0102501i
$266$		0			0
$267$			12.3510i			0.755867i
$268$		0			0
$269$	11.5381			0.703490			0.351745	−	0.936096i	$-0.385588\pi$
$269$	11.5381			0.703490			0.351745	+	0.936096i	$0.385588\pi$
$270$		0			0
$271$	5.63511			0.342309			0.171154	−	0.985244i	$-0.445250\pi$
$271$	5.63511			0.342309			0.171154	+	0.985244i	$0.445250\pi$
$272$		0			0
$273$		−	20.0648i		−	1.21438i
$274$		0			0
$275$	−3.89019	+	3.02792i	−0.234587	+	0.182591i
$276$		0			0
$277$		−	17.4053i		−	1.04578i	−0.852399	−	0.522892i	$-0.824853\pi$
$277$		−	17.4053i		−	1.04578i	0.852399	−	0.522892i	$-0.175147\pi$
$278$		0			0
$279$	3.28415			0.196617
$280$		0			0
$281$	−21.7827			−1.29945			−0.649723	−	0.760171i	$-0.725115\pi$
$281$	−21.7827			−1.29945			−0.649723	+	0.760171i	$0.725115\pi$
$282$		0			0
$283$		−	21.5962i		−	1.28376i	−0.766804	−	0.641881i	$-0.778154\pi$
$283$		−	21.5962i		−	1.28376i	0.766804	−	0.641881i	$-0.221846\pi$
$284$		0			0
$285$	−5.50917	+	1.89134i	−0.326335	+	0.112033i
$286$		0			0
$287$			2.30560i			0.136095i
$288$		0			0
$289$	−3.45963			−0.203508
$290$		0			0
$291$	3.23797			0.189813
$292$		0			0
$293$			32.0125i			1.87019i	0.354398	+	0.935095i	$0.384686\pi$
$293$			32.0125i			1.87019i	−0.354398	+	0.935095i	$0.615314\pi$
$294$		0			0
$295$	−6.60719	−	19.2457i	−0.384685	−	1.12053i
$296$		0			0
$297$			0.985939i			0.0572100i
$298$		0			0
$299$	17.4943			1.01172
$300$		0			0
$301$	−34.3211			−1.97824
$302$		0			0
$303$			4.35637i			0.250267i
$304$		0			0
$305$	8.00000	+	23.3028i	0.458079	+	1.33431i
$306$		0			0
$307$		−	1.13659i		−	0.0648686i	−0.999474	−	0.0324343i	$-0.989674\pi$
$307$		−	1.13659i		−	0.0648686i	0.999474	−	0.0324343i	$-0.0103260\pi$
$308$		0			0
$309$	15.0754			0.857609
$310$		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
$315$	−8.58028	+	2.94567i	−0.483444	+	0.165970i
$316$		0			0
$317$			1.66152i			0.0933203i	0.998911	+	0.0466601i	$0.0148578\pi$
$317$			1.66152i			0.0933203i	−0.998911	+	0.0466601i	$0.985142\pi$
$318$		0			0
$319$	−7.48755			−0.419223
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$			11.7827i			0.655605i
$324$		0			0
$325$	−15.1887	−	19.5140i	−0.842516	−	1.08244i
$326$		0			0
$327$		−	4.17034i		−	0.230620i
$328$		0			0
$329$	−10.5683			−0.582649
$330$		0			0
$331$	−25.9077			−1.42402			−0.712008	−	0.702171i	$-0.752214\pi$
$331$	−25.9077			−1.42402			−0.712008	+	0.702171i	$0.752214\pi$
$332$		0			0
$333$		−	0.945668i		−	0.0518223i
$334$		0			0
$335$	17.8913	−	6.14222i	0.977508	−	0.335585i
$336$		0			0
$337$		−	8.00696i		−	0.436167i	−0.975930	−	0.218083i	$-0.930020\pi$
$337$		−	8.00696i		−	0.436167i	0.975930	−	0.218083i	$-0.0699805\pi$
$338$		0			0
$339$	−1.28526			−0.0698060
$340$		0			0
$341$	−3.23797			−0.175346
$342$		0			0
$343$			9.97884i			0.538806i
$344$		0			0
$345$	−2.56829	−	7.48105i	−0.138272	−	0.402766i
$346$		0			0
$347$			23.0279i			1.23620i	0.786098	+	0.618102i	$0.212098\pi$
$347$			23.0279i			1.23620i	−0.786098	+	0.618102i	$0.787902\pi$
$348$		0			0
$349$	−21.4380			−1.14755			−0.573776	−	0.819012i	$-0.694522\pi$
$349$	−21.4380			−1.14755			−0.573776	+	0.819012i	$0.694522\pi$
$350$		0			0
$351$	−4.94567			−0.263980
$352$		0			0
$353$		−	4.52323i		−	0.240747i	−0.992729	−	0.120374i	$-0.961591\pi$
$353$		−	4.52323i		−	0.240747i	0.992729	−	0.120374i	$-0.0384093\pi$
$354$		0			0
$355$	−1.03951	−	3.02792i	−0.0551713	−	0.160706i
$356$		0			0
$357$			18.3510i			0.971236i
$358$		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
$363$			10.0279i			0.526330i
$364$		0			0
$365$	25.2747	−	8.67696i	1.32294	−	0.454173i
$366$		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
$371$	−0.932371			−0.0484063
$372$		0			0
$373$			30.0823i			1.55760i	0.627272	+	0.778800i	$0.284171\pi$
$373$			30.0823i			1.55760i	−0.627272	+	0.778800i	$0.715829\pi$
$374$		0			0
$375$	−6.11491	+	9.35991i	−0.315772	+	0.483344i
$376$		0			0
$377$		−	37.5591i		−	1.93439i
$378$		0			0
$379$	33.6881			1.73044			0.865220	−	0.501392i	$-0.167179\pi$
$379$	33.6881			1.73044			0.865220	+	0.501392i	$0.167179\pi$
$380$		0			0
$381$	1.15280			0.0590597
$382$		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
$387$			8.45963i			0.430027i
$388$		0			0
$389$	−16.6408			−0.843722			−0.421861	−	0.906660i	$-0.638623\pi$
$389$	−16.6408			−0.843722			−0.421861	+	0.906660i	$0.638623\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$			3.89019i			0.196234i
$394$		0			0
$395$	2.38449	+	6.94567i	0.119977	+	0.349474i
$396$		0			0
$397$			20.4332i			1.02551i	0.858534	+	0.512757i	$0.171376\pi$
$397$			20.4332i			1.02551i	−0.858534	+	0.512757i	$0.828624\pi$
$398$		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
$403$		−	16.2423i		−	0.809087i
$404$		0			0
$405$	0.726062	+	2.11491i	0.0360783	+	0.105091i
$406$		0			0
$407$			0.932371i			0.0462159i
$408$		0			0
$409$	19.8913			0.983563			0.491782	−	0.870719i	$-0.336346\pi$
$409$	19.8913			0.983563			0.491782	+	0.870719i	$0.336346\pi$
$410$		0			0
$411$	−17.5135			−0.863875
$412$		0			0
$413$			36.9193i			1.81668i
$414$		0			0
$415$	−20.9193	+	7.18172i	−1.02689	+	0.352537i
$416$		0			0
$417$			16.8612i			0.825698i
$418$		0			0
$419$	0.387288			0.0189203			0.00946013	−	0.999955i	$-0.496989\pi$
$419$	0.387288			0.0189203			0.00946013	+	0.999955i	$0.496989\pi$
$420$		0			0
$421$	12.0578			0.587664			0.293832	−	0.955857i	$-0.405069\pi$
$421$	12.0578			0.587664			0.293832	+	0.955857i	$0.405069\pi$
$422$		0			0
$423$			2.60492i			0.126656i
$424$		0			0
$425$	13.8913	+	17.8472i	0.673829	+	0.865715i
$426$		0			0
$427$		−	44.7019i		−	2.16328i
$428$		0			0
$429$	4.87613			0.235422
$430$		0			0
$431$	40.4068			1.94633			0.973164	−	0.230113i	$-0.0739096\pi$
$431$	40.4068			1.94633			0.973164	+	0.230113i	$0.0739096\pi$
$432$		0			0
$433$			36.1859i			1.73898i	0.493949	+	0.869491i	$0.335553\pi$
$433$			36.1859i			1.73898i	−0.493949	+	0.869491i	$0.664447\pi$
$434$		0			0
$435$	−16.0613	+	5.51396i	−0.770082	+	0.264374i
$436$		0			0
$437$			9.21438i			0.440783i
$438$		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
$443$			7.02792i			0.333907i	0.985965	+	0.166953i	$0.0533929\pi$
$443$			7.02792i			0.333907i	−0.985965	+	0.166953i	$0.946607\pi$
$444$		0			0
$445$	−8.96757	−	26.1212i	−0.425103	−	1.23826i
$446$		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
$451$	−0.560304			−0.0263837
$452$		0			0
$453$			4.71585i			0.221570i
$454$		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
$459$	4.52323			0.211126
$460$		0			0
$461$	−41.0902			−1.91376			−0.956881	−	0.290479i	$-0.906185\pi$
$461$	−41.0902			−1.91376			−0.956881	+	0.290479i	$0.906185\pi$
$462$		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
$467$		−	1.89134i		−	0.0875206i	−0.999042	−	0.0437603i	$-0.986066\pi$
$467$		−	1.89134i		−	0.0875206i	0.999042	−	0.0437603i	$-0.0139338\pi$
$468$		0			0
$469$	−34.3211			−1.58480
$470$		0			0
$471$	−8.94567			−0.412195
$472$		0			0
$473$		−	8.34068i		−	0.383505i
$474$		0			0
$475$	10.2782	−	8.00000i	0.471594	−	0.367065i
$476$		0			0
$477$			0.229815i			0.0105225i
$478$		0			0
$479$	−31.4876			−1.43870			−0.719352	−	0.694646i	$-0.755561\pi$
$479$	−31.4876			−1.43870			−0.719352	+	0.694646i	$0.755561\pi$
$480$		0			0
$481$	−4.67696			−0.213251
$482$		0			0
$483$			14.3510i			0.652992i
$484$		0			0
$485$	−6.84800	+	2.35097i	−0.310952	+	0.106752i
$486$		0			0
$487$			12.9964i			0.588922i	0.955664	+	0.294461i	$0.0951401\pi$
$487$			12.9964i			0.588922i	−0.955664	+	0.294461i	$0.904860\pi$
$488$		0			0
$489$	−15.7827			−0.713717
$490$		0			0
$491$	−14.9085			−0.672812			−0.336406	−	0.941717i	$-0.609212\pi$
$491$	−14.9085			−0.672812			−0.336406	+	0.941717i	$0.609212\pi$
$492$		0			0
$493$			34.3510i			1.54709i
$494$		0			0
$495$	−0.715853	−	2.08517i	−0.0321752	−	0.0937214i
$496$		0			0
$497$			5.80850i			0.260547i
$498$		0			0
$499$	35.6599			1.59636			0.798179	−	0.602420i	$-0.205797\pi$
$499$	35.6599			1.59636			0.798179	+	0.602420i	$0.205797\pi$
$500$		0			0
$501$	5.50917			0.246132
$502$		0			0
$503$			25.3090i			1.12847i	0.825613	+	0.564237i	$0.190830\pi$
$503$			25.3090i			1.12847i	−0.825613	+	0.564237i	$0.809170\pi$
$504$		0			0
$505$	−3.16300	−	9.21332i	−0.140751	−	0.409988i
$506$		0			0
$507$			11.4596i			0.508940i
$508$		0			0
$509$	13.7366			0.608862			0.304431	−	0.952534i	$-0.401534\pi$
$509$	13.7366			0.608862			0.304431	+	0.952534i	$0.401534\pi$
$510$		0			0
$511$	−48.4846			−2.14483
$512$		0			0
$513$		−	2.60492i		−	0.115010i
$514$		0			0
$515$	−31.8831	+	10.9457i	−1.40494	+	0.482324i
$516$		0			0
$517$		−	2.56829i		−	0.112953i
$518$		0			0
$519$	10.3385			0.453809
$520$		0			0
$521$	30.9193			1.35460			0.677299	−	0.735708i	$-0.263150\pi$
$521$	30.9193			1.35460			0.677299	+	0.735708i	$0.263150\pi$
$522$		0			0
$523$		−	21.8385i		−	0.954932i	−0.878650	−	0.477466i	$-0.841555\pi$
$523$		−	21.8385i		−	0.954932i	0.878650	−	0.477466i	$-0.158445\pi$
$524$		0			0
$525$	16.0078	−	12.4596i	0.698636	−	0.543783i
$526$		0			0
$527$			14.8550i			0.647092i
$528$		0			0
$529$	10.4876			0.455981
$530$		0			0
$531$	9.10003			0.394908
$532$		0			0
$533$		−	2.81060i		−	0.121740i
$534$		0			0
$535$	8.45963	−	2.90425i	0.365742	−	0.125562i
$536$		0			0
$537$			16.1746i			0.697986i
$538$		0			0
$539$	−9.32662			−0.401726
$540$		0			0
$541$	24.3423			1.04656			0.523278	−	0.852162i	$-0.324709\pi$
$541$	24.3423			1.04656			0.523278	+	0.852162i	$0.324709\pi$
$542$		0			0
$543$			8.11409i			0.348209i
$544$		0			0
$545$	3.02792	+	8.81988i	0.129702	+	0.377802i
$546$		0			0
$547$		−	33.3789i		−	1.42718i	−0.700564	−	0.713589i	$-0.747068\pi$
$547$		−	33.3789i		−	1.42718i	0.700564	−	0.713589i	$-0.252932\pi$
$548$		0			0
$549$	−11.0183			−0.470251
$550$		0			0
$551$	19.7827			0.842770
$552$		0			0
$553$		−	13.3239i		−	0.566592i
$554$		0			0
$555$	0.686614	+	2.00000i	0.0291451	+	0.0848953i
$556$		0			0
$557$		−	30.5808i		−	1.29575i	−0.761747	−	0.647875i	$-0.775658\pi$
$557$		−	30.5808i		−	1.29575i	0.761747	−	0.647875i	$-0.224342\pi$
$558$		0			0
$559$	41.8385			1.76958
$560$		0			0
$561$	−4.45963			−0.188286
$562$		0			0
$563$			7.02792i			0.296192i	0.988973	+	0.148096i	$0.0473144\pi$
$563$			7.02792i			0.296192i	−0.988973	+	0.148096i	$0.952686\pi$
$564$		0			0
$565$	2.71821	−	0.933181i	0.114356	−	0.0392592i
$566$		0			0
$567$		−	4.05705i		−	0.170380i
$568$		0			0
$569$	24.3510			1.02085			0.510423	−	0.859924i	$-0.329489\pi$
$569$	24.3510			1.02085			0.510423	+	0.859924i	$0.329489\pi$
$570$		0			0
$571$	−20.0992			−0.841125			−0.420563	−	0.907263i	$-0.638167\pi$
$571$	−20.0992			−0.841125			−0.420563	+	0.907263i	$0.638167\pi$
$572$		0			0
$573$			24.9193i			1.04102i
$574$		0			0
$575$	10.8634	+	13.9570i	0.453036	+	0.582047i
$576$		0			0
$577$			4.76899i			0.198536i	0.995061	+	0.0992678i	$0.0316501\pi$
$577$			4.76899i			0.198536i	−0.995061	+	0.0992678i	$0.968350\pi$
$578$		0			0
$579$	1.03951			0.0432004
$580$		0			0
$581$	40.1296			1.66486
$582$		0			0
$583$		−	0.226584i		−	0.00938413i
$584$		0			0
$585$	10.4596	−	3.59086i	0.432452	−	0.148464i
$586$		0			0
$587$		−	8.21733i		−	0.339165i	−0.985516	−	0.169583i	$-0.945758\pi$
$587$		−	8.21733i		−	0.339165i	0.985516	−	0.169583i	$-0.0542420\pi$
$588$		0			0
$589$	8.55495			0.352501
$590$		0			0
$591$	−9.66152			−0.397422
$592$		0			0
$593$		−	7.76120i		−	0.318714i	−0.987221	−	0.159357i	$-0.949058\pi$
$593$		−	7.76120i		−	0.318714i	0.987221	−	0.159357i	$-0.0509421\pi$
$594$		0			0
$595$	−13.3239	−	38.8106i	−0.546228	−	1.59108i
$596$		0			0
$597$		−	23.0668i		−	0.944062i
$598$		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	$-0.220497\pi$
$601$	37.7299			1.53903			0.769516	+	0.638627i	$0.220497\pi$
$602$		0			0
$603$			8.45963i			0.344503i
$604$		0			0
$605$	−7.28090	−	21.2081i	−0.296010	−	0.862233i
$606$		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
$611$	12.8831			0.521194
$612$		0			0
$613$			0.703366i			0.0284087i	0.999899	+	0.0142044i	$0.00452154\pi$
$613$			0.703366i			0.0284087i	−0.999899	+	0.0142044i	$0.995478\pi$
$614$		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
$619$	39.4966			1.58750			0.793751	−	0.608243i	$-0.208126\pi$
$619$	39.4966			1.58750			0.793751	+	0.608243i	$0.208126\pi$
$620$		0			0
$621$	3.53729			0.141947
$622$		0			0
$623$			50.1084i			2.00755i
$624$		0			0
$625$	6.13659	−	24.2351i	0.245464	−	0.969406i
$626$		0			0
$627$			2.56829i			0.102568i
$628$		0			0
$629$	4.27748			0.170554
$630$		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
$635$	−2.43806	+	0.837003i	−0.0967516	+	0.0332155i
$636$		0			0
$637$		−	46.7842i		−	1.85366i
$638$		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
$643$		−	1.13659i		−	0.0448227i	−0.999749	−	0.0224113i	$-0.992866\pi$
$643$		−	1.13659i		−	0.0448227i	0.999749	−	0.0224113i	$-0.00713435\pi$
$644$		0			0
$645$	−6.14222	−	17.8913i	−0.241850	−	0.704471i
$646$		0			0
$647$			6.54868i			0.257455i	0.991680	+	0.128728i	$0.0410893\pi$
$647$			6.54868i			0.257455i	−0.991680	+	0.128728i	$0.958911\pi$
$648$		0			0
$649$	−8.97208			−0.352185
$650$		0			0
$651$	13.3239			0.522206
$652$		0			0
$653$			8.74226i			0.342111i	0.985261	+	0.171056i	$0.0547178\pi$
$653$			8.74226i			0.342111i	−0.985261	+	0.171056i	$0.945282\pi$
$654$		0			0
$655$	−2.82452	−	8.22739i	−0.110363	−	0.321471i
$656$		0			0
$657$			11.9507i			0.466242i
$658$		0			0
$659$	35.5336			1.38419			0.692097	−	0.721804i	$-0.256687\pi$
$659$	35.5336			1.38419			0.692097	+	0.721804i	$0.256687\pi$
$660$		0			0
$661$	23.3028			0.906373			0.453186	−	0.891416i	$-0.350287\pi$
$661$	23.3028			0.906373			0.453186	+	0.891416i	$0.350287\pi$
$662$		0			0
$663$		−	22.3704i		−	0.868795i
$664$		0			0
$665$	−22.3510	+	7.67324i	−0.866733	+	0.297555i
$666$		0			0
$667$			26.8634i			1.04016i
$668$		0			0
$669$	−17.9796			−0.695133
$670$		0			0
$671$	10.8634			0.419377
$672$		0			0
$673$		−	14.3634i		−	0.553670i	−0.960917	−	0.276835i	$-0.910714\pi$
$673$		−	14.3634i		−	0.553670i	0.960917	−	0.276835i	$-0.0892856\pi$
$674$		0			0
$675$	−3.07111	−	3.94567i	−0.118207	−	0.151869i
$676$		0			0
$677$		−	24.3076i		−	0.934217i	−0.884200	−	0.467109i	$-0.845296\pi$
$677$		−	24.3076i		−	0.934217i	0.884200	−	0.467109i	$-0.154704\pi$
$678$		0			0
$679$	13.1366			0.504136
$680$		0			0
$681$	7.02792			0.269311
$682$		0			0
$683$		−	38.5933i		−	1.47673i	−0.674401	−	0.738365i	$-0.735598\pi$
$683$		−	38.5933i		−	1.47673i	0.674401	−	0.738365i	$-0.264402\pi$
$684$		0			0
$685$	37.0393	−	12.7159i	1.41520	−	0.485848i
$686$		0			0
$687$		−	4.17034i		−	0.159108i
$688$		0			0
$689$	1.13659			0.0433006
$690$		0			0
$691$	13.4090			0.510102			0.255051	−	0.966928i	$-0.417908\pi$
$691$	13.4090			0.510102			0.255051	+	0.966928i	$0.417908\pi$
$692$		0			0
$693$			4.00000i			0.151947i
$694$		0			0
$695$	−12.2423	−	35.6599i	−0.464377	−	1.35266i
$696$		0			0
$697$			2.57053i			0.0973657i
$698$		0			0
$699$	23.9894			0.907362
$700$		0			0
$701$	−15.6013			−0.589253			−0.294626	−	0.955613i	$-0.595195\pi$
$701$	−15.6013			−0.589253			−0.294626	+	0.955613i	$0.595195\pi$
$702$		0			0
$703$		−	2.46339i		−	0.0929086i
$704$		0			0
$705$	−1.89134	−	5.50917i	−0.0712318	−	0.207487i
$706$		0			0
$707$			17.6740i			0.664699i
$708$		0			0
$709$	15.7873			0.592906			0.296453	−	0.955047i	$-0.404196\pi$
$709$	15.7873			0.592906			0.296453	+	0.955047i	$0.404196\pi$
$710$		0			0
$711$	−3.28415			−0.123165
$712$		0			0
$713$			11.6170i			0.435060i
$714$		0			0
$715$	−10.3126	+	3.54037i	−0.385668	+	0.132402i
$716$		0			0
$717$		−	8.91926i		−	0.333096i
$718$		0			0
$719$	−22.5683			−0.841655			−0.420828	−	0.907141i	$-0.638260\pi$
$719$	−22.5683			−0.841655			−0.420828	+	0.907141i	$0.638260\pi$
$720$		0			0
$721$	61.1616			2.27778
$722$		0			0
$723$			16.3510i			0.608099i
$724$		0			0
$725$	29.9647	−	23.3230i	1.11286	−	0.866196i
$726$		0			0
$727$			2.79096i			0.103511i	0.998660	+	0.0517554i	$0.0164816\pi$
$727$			2.79096i			0.103511i	−0.998660	+	0.0517554i	$0.983518\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−38.2649			−1.41528
$732$		0			0
$733$			16.4860i			0.608926i	0.952524	+	0.304463i	$0.0984769\pi$
$733$			16.4860i			0.608926i	−0.952524	+	0.304463i	$0.901523\pi$
$734$		0			0
$735$	−20.0062	+	6.86828i	−0.737941	+	0.253340i
$736$		0			0
$737$		−	8.34068i		−	0.307233i
$738$		0			0
$739$	−14.8894			−0.547714			−0.273857	−	0.961770i	$-0.588299\pi$
$739$	−14.8894			−0.547714			−0.273857	+	0.961770i	$0.588299\pi$
$740$		0			0
$741$	−12.8831			−0.473272
$742$		0			0
$743$		−	41.4301i		−	1.51992i	−0.649968	−	0.759961i	$-0.725218\pi$
$743$		−	41.4301i		−	1.51992i	0.649968	−	0.759961i	$-0.274782\pi$
$744$		0			0
$745$	7.62263	+	22.2035i	0.279271	+	0.813475i
$746$		0			0
$747$		−	9.89134i		−	0.361905i
$748$		0			0
$749$	−16.2282			−0.592965
$750$		0			0
$751$	30.5544			1.11494			0.557472	−	0.830195i	$-0.311771\pi$
$751$	30.5544			1.11494			0.557472	+	0.830195i	$0.311771\pi$
$752$		0			0
$753$		−	4.22391i		−	0.153928i
$754$		0			0
$755$	−3.42400	−	9.97359i	−0.124612	−	0.362976i
$756$		0			0
$757$			0.433223i			0.0157457i	0.999969	+	0.00787287i	$0.00250604\pi$
$757$			0.433223i			0.0157457i	−0.999969	+	0.00787287i	$0.997494\pi$
$758$		0			0
$759$	−3.48755			−0.126590
$760$		0			0
$761$	14.9193			0.540823			0.270411	−	0.962745i	$-0.412840\pi$
$761$	14.9193			0.540823			0.270411	+	0.962745i	$0.412840\pi$
$762$		0			0
$763$		−	16.9193i		−	0.612518i
$764$		0			0
$765$	−9.56622	+	3.28415i	−0.345867	+	0.118739i
$766$		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
$771$	−24.6952			−0.889375
$772$		0			0
$773$			13.4192i			0.482656i	0.970444	+	0.241328i	$0.0775829\pi$
$773$			13.4192i			0.482656i	−0.970444	+	0.241328i	$0.922417\pi$
$774$		0			0
$775$	12.9582	−	10.0860i	0.465471	−	0.362299i
$776$		0			0
$777$		−	3.83662i		−	0.137638i
$778$		0			0
$779$	1.48036			0.0530395
$780$		0			0
$781$	−1.41157			−0.0505101
$782$		0			0
$783$		−	7.59434i		−	0.271400i
$784$		0			0
$785$	18.9193	−	6.49511i	0.675257	−	0.231820i
$786$		0			0
$787$		−	36.4846i		−	1.30054i	−0.759705	−	0.650268i	$-0.774657\pi$
$787$		−	36.4846i		−	1.30054i	0.759705	−	0.650268i	$-0.225343\pi$
$788$		0			0
$789$	−14.6628			−0.522009
$790$		0			0
$791$	−5.21438			−0.185402
$792$		0			0
$793$			54.4931i			1.93511i
$794$		0			0
$795$	−0.166860	−	0.486038i	−0.00591791	−	0.0172380i
$796$		0			0
$797$			26.0683i			0.923388i	0.887039	+	0.461694i	$0.152758\pi$
$797$			26.0683i			0.923388i	−0.887039	+	0.461694i	$0.847242\pi$
$798$		0			0
$799$	−11.7827			−0.416841
$800$		0			0
$801$	12.3510			0.436400
$802$		0			0
$803$		−	11.7827i		−	0.415801i
$804$		0			0
$805$	−10.4197	−	30.3510i	−0.367246	−	1.06973i
$806$		0			0
$807$		−	11.5381i		−	0.406160i
$808$		0			0
$809$	35.6212			1.25237			0.626187	−	0.779673i	$-0.284615\pi$
$809$	35.6212			1.25237			0.626187	+	0.779673i	$0.284615\pi$
$810$		0			0
$811$	43.8935			1.54131			0.770654	−	0.637253i	$-0.219929\pi$
$811$	43.8935			1.54131			0.770654	+	0.637253i	$0.219929\pi$
$812$		0			0
$813$		−	5.63511i		−	0.197632i
$814$		0			0
$815$	33.3789	−	11.4592i	1.16921	−	0.401398i
$816$		0			0
$817$			22.0367i			0.770966i
$818$		0			0
$819$	−20.0648			−0.701121
$820$		0			0
$821$	−28.8058			−1.00533			−0.502665	−	0.864482i	$-0.667647\pi$
$821$	−28.8058			−1.00533			−0.502665	+	0.864482i	$0.667647\pi$
$822$		0			0
$823$		−	27.9585i		−	0.974571i	−0.873243	−	0.487286i	$-0.837987\pi$
$823$		−	27.9585i		−	0.974571i	0.873243	−	0.487286i	$-0.162013\pi$
$824$		0			0
$825$	3.02792	+	3.89019i	0.105419	+	0.135439i
$826$		0			0
$827$			14.8634i			0.516851i	0.966031	+	0.258426i	$0.0832037\pi$
$827$			14.8634i			0.516851i	−0.966031	+	0.258426i	$0.916796\pi$
$828$		0			0
$829$	41.7678			1.45065			0.725327	−	0.688404i	$-0.241688\pi$
$829$	41.7678			1.45065			0.725327	+	0.688404i	$0.241688\pi$
$830$		0			0
$831$	−17.4053			−0.603783
$832$		0			0
$833$			42.7881i			1.48252i
$834$		0			0
$835$	−11.6514	+	4.00000i	−0.403213	+	0.138426i
$836$		0			0
$837$		−	3.28415i		−	0.113517i
$838$		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
$843$			21.7827i			0.750235i
$844$		0			0
$845$	−8.32040	−	24.2361i	−0.286231	−	0.833746i
$846$		0			0
$847$			40.6838i			1.39791i
$848$		0			0
$849$	−21.5962			−0.741180
$850$		0			0
$851$	3.34510			0.114669
$852$		0			0
$853$			49.1880i			1.68416i	0.539350	+	0.842082i	$0.318670\pi$
$853$			49.1880i			1.68416i	−0.539350	+	0.842082i	$0.681330\pi$
$854$		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
$859$	−4.57680			−0.156158			−0.0780792	−	0.996947i	$-0.524879\pi$
$859$	−4.57680			−0.156158			−0.0780792	+	0.996947i	$0.524879\pi$
$860$		0			0
$861$	2.30560			0.0785746
$862$		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
$867$			3.45963i			0.117495i
$868$		0			0
$869$	3.23797			0.109841
$870$		0			0
$871$	41.8385			1.41764
$872$		0			0
$873$		−	3.23797i		−	0.109589i
$874$		0			0
$875$	−24.8085	+	37.9736i	−0.838679	+	1.28374i
$876$		0			0
$877$			11.7563i			0.396981i	0.980103	+	0.198490i	$0.0636039\pi$
$877$			11.7563i			0.396981i	−0.980103	+	0.198490i	$0.936396\pi$
$878$		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
$883$		−	17.1366i		−	0.576692i	−0.957526	−	0.288346i	$-0.906895\pi$
$883$		−	17.1366i		−	0.576692i	0.957526	−	0.288346i	$-0.0931054\pi$
$884$		0			0
$885$	−19.2457	+	6.60719i	−0.646938	+	0.222098i
$886$		0			0
$887$		−	32.0883i		−	1.07742i	−0.842492	−	0.538709i	$-0.818912\pi$
$887$		−	32.0883i		−	1.07742i	0.842492	−	0.538709i	$-0.181088\pi$
$888$		0			0
$889$	4.67696			0.156860
$890$		0			0
$891$	0.985939			0.0330302
$892$		0			0
$893$			6.78562i			0.227072i
$894$		0			0
$895$	−11.7438	−	34.2078i	−0.392551	−	1.14344i
$896$		0			0
$897$		−	17.4943i		−	0.584117i
$898$		0			0
$899$	24.9409			0.831827
$900$		0			0
$901$	−1.03951			−0.0346310
$902$		0			0
$903$			34.3211i			1.14214i
$904$		0			0
$905$	−5.89134	−	17.1606i	−0.195835	−	0.570436i
$906$		0			0
$907$		−	40.4596i		−	1.34344i	−0.740805	−	0.671720i	$-0.765556\pi$
$907$		−	40.4596i		−	1.34344i	0.740805	−	0.671720i	$-0.234444\pi$
$908$		0			0
$909$	4.35637			0.144492
$910$		0			0
$911$	16.0000			0.530104			0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000			0.530104			0.265052	+	0.964234i	$0.414611\pi$
$912$		0			0
$913$			9.75225i			0.322752i
$914$		0			0
$915$	23.3028	−	8.00000i	0.770366	−	0.264472i
$916$		0			0
$917$			15.7827i			0.521190i
$918$		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
$923$		−	7.08074i		−	0.233065i
$924$		0			0
$925$	−2.90425	−	3.73129i	−0.0954911	−	0.122684i
$926$		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
$931$	24.6416			0.807596
$932$		0			0
$933$			5.13659i			0.168164i
$934$		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
$939$	−23.0762			−0.753063
$940$		0			0
$941$	14.1086			0.459928			0.229964	−	0.973199i	$-0.426139\pi$
$941$	14.1086			0.459928			0.229964	+	0.973199i	$0.426139\pi$
$942$		0			0
$943$			2.01022i			0.0654619i
$944$		0			0
$945$	2.94567	+	8.58028i	0.0958226	+	0.279117i
$946$		0			0
$947$			45.8385i			1.48955i	0.667315	+	0.744776i	$0.267444\pi$
$947$			45.8385i			1.48955i	−0.667315	+	0.744776i	$0.732556\pi$
$948$		0			0
$949$	59.1043			1.91861
$950$		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
$955$	−18.0929	−	52.7019i	−0.585473	−	1.70539i
$956$		0			0
$957$			7.48755i			0.242038i
$958$		0			0
$959$	−71.0529			−2.29442
$960$		0			0
$961$	−20.2144			−0.652077
$962$		0			0
$963$			4.00000i			0.128898i
$964$		0			0
$965$	−2.19846	+	0.754747i	−0.0707710	+	0.0242962i
$966$		0			0
$967$			14.0359i			0.451364i	0.974201	+	0.225682i	$0.0724610\pi$
$967$			14.0359i			0.451364i	−0.974201	+	0.225682i	$0.927539\pi$
$968$		0			0
$969$	11.7827			0.378514
$970$		0			0
$971$	21.6494			0.694762			0.347381	−	0.937724i	$-0.387071\pi$
$971$	21.6494			0.694762			0.347381	+	0.937724i	$0.387071\pi$
$972$		0			0
$973$			68.4068i			2.19302i
$974$		0			0
$975$	−19.5140	+	15.1887i	−0.624947	+	0.486427i
$976$		0			0
$977$		−	6.60225i		−	0.211225i	−0.994407	−	0.105612i	$-0.966320\pi$
$977$		−	6.60225i		−	0.211225i	0.994407	−	0.105612i	$-0.0336802\pi$
$978$		0			0
$979$	−12.1773			−0.389188
$980$		0			0
$981$	−4.17034			−0.133149
$982$		0			0
$983$			53.8600i			1.71787i	0.512087	+	0.858934i	$0.328873\pi$
$983$			53.8600i			1.71787i	−0.512087	+	0.858934i	$0.671127\pi$
$984$		0			0
$985$	20.4332	−	7.01486i	0.651057	−	0.223512i
$986$		0			0
$987$			10.5683i			0.336393i
$988$		0			0
$989$	−29.9242			−0.951534
$990$		0			0
$991$	29.7129			0.943861			0.471931	−	0.881636i	$-0.343557\pi$
$991$	29.7129			0.943861			0.471931	+	0.881636i	$0.343557\pi$
$992$		0			0
$993$			25.9077i			0.822156i
$994$		0			0
$995$	16.7479	+	48.7842i	0.530945	+	1.54656i
$996$		0			0
$997$		−	16.6506i		−	0.527328i	−0.964615	−	0.263664i	$-0.915069\pi$
$997$		−	16.6506i		−	0.527328i	0.964615	−	0.263664i	$-0.0849311\pi$
$998$		0			0
$999$	−0.945668			−0.0299196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.f.l.769.4		12
4.3	odd	2		3840.2.f.m.769.10		12
5.4	even	2	inner	3840.2.f.l.769.10		12
8.3	odd	2		3840.2.f.m.769.3		12
8.5	even	2	inner	3840.2.f.l.769.9		12
16.3	odd	4		480.2.d.a.49.5		6
16.5	even	4		120.2.d.b.109.4	yes	6
16.11	odd	4		480.2.d.b.49.2		6
16.13	even	4		120.2.d.a.109.4	yes	6
20.19	odd	2		3840.2.f.m.769.4		12
40.19	odd	2		3840.2.f.m.769.9		12
40.29	even	2	inner	3840.2.f.l.769.3		12
48.5	odd	4		360.2.d.e.109.3		6
48.11	even	4		1440.2.d.f.1009.5		6
48.29	odd	4		360.2.d.f.109.3		6
48.35	even	4		1440.2.d.e.1009.2		6
80.3	even	4		2400.2.k.f.1201.1		12
80.13	odd	4		600.2.k.f.301.12		12
80.19	odd	4		480.2.d.b.49.1		6
80.27	even	4		2400.2.k.f.1201.6		12
80.29	even	4		120.2.d.b.109.3	yes	6
80.37	odd	4		600.2.k.f.301.2		12
80.43	even	4		2400.2.k.f.1201.7		12
80.53	odd	4		600.2.k.f.301.11		12
80.59	odd	4		480.2.d.a.49.6		6
80.67	even	4		2400.2.k.f.1201.12		12
80.69	even	4		120.2.d.a.109.3	✓	6
80.77	odd	4		600.2.k.f.301.1		12
240.29	odd	4		360.2.d.e.109.4		6
240.53	even	4		1800.2.k.u.901.2		12
240.59	even	4		1440.2.d.e.1009.1		6
240.77	even	4		1800.2.k.u.901.12		12
240.83	odd	4		7200.2.k.u.3601.1		12
240.107	odd	4		7200.2.k.u.3601.12		12
240.149	odd	4		360.2.d.f.109.4		6
240.173	even	4		1800.2.k.u.901.1		12
240.179	even	4		1440.2.d.f.1009.6		6
240.197	even	4		1800.2.k.u.901.11		12
240.203	odd	4		7200.2.k.u.3601.2		12
240.227	odd	4		7200.2.k.u.3601.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.d.a.109.3	✓	6	80.69	even	4
120.2.d.a.109.4	yes	6	16.13	even	4
120.2.d.b.109.3	yes	6	80.29	even	4
120.2.d.b.109.4	yes	6	16.5	even	4
360.2.d.e.109.3		6	48.5	odd	4
360.2.d.e.109.4		6	240.29	odd	4
360.2.d.f.109.3		6	48.29	odd	4
360.2.d.f.109.4		6	240.149	odd	4
480.2.d.a.49.5		6	16.3	odd	4
480.2.d.a.49.6		6	80.59	odd	4
480.2.d.b.49.1		6	80.19	odd	4
480.2.d.b.49.2		6	16.11	odd	4
600.2.k.f.301.1		12	80.77	odd	4
600.2.k.f.301.2		12	80.37	odd	4
600.2.k.f.301.11		12	80.53	odd	4
600.2.k.f.301.12		12	80.13	odd	4
1440.2.d.e.1009.1		6	240.59	even	4
1440.2.d.e.1009.2		6	48.35	even	4
1440.2.d.f.1009.5		6	48.11	even	4
1440.2.d.f.1009.6		6	240.179	even	4
1800.2.k.u.901.1		12	240.173	even	4
1800.2.k.u.901.2		12	240.53	even	4
1800.2.k.u.901.11		12	240.197	even	4
1800.2.k.u.901.12		12	240.77	even	4
2400.2.k.f.1201.1		12	80.3	even	4
2400.2.k.f.1201.6		12	80.27	even	4
2400.2.k.f.1201.7		12	80.43	even	4
2400.2.k.f.1201.12		12	80.67	even	4
3840.2.f.l.769.3		12	40.29	even	2	inner
3840.2.f.l.769.4		12	1.1	even	1	trivial
3840.2.f.l.769.9		12	8.5	even	2	inner
3840.2.f.l.769.10		12	5.4	even	2	inner
3840.2.f.m.769.3		12	8.3	odd	2
3840.2.f.m.769.4		12	20.19	odd	2
3840.2.f.m.769.9		12	40.19	odd	2
3840.2.f.m.769.10		12	4.3	odd	2
7200.2.k.u.3601.1		12	240.83	odd	4
7200.2.k.u.3601.2		12	240.203	odd	4
7200.2.k.u.3601.11		12	240.227	odd	4
7200.2.k.u.3601.12		12	240.107	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(0.726062 + 2.11491i) q^{5} -4.05705i q^{7} -1.00000 q^{9} +0.985939 q^{11} +4.94567i q^{13} +(2.11491 - 0.726062i) q^{15} -4.52323i q^{17} -2.60492 q^{19} -4.05705 q^{21} -3.53729i q^{23} +(-3.94567 + 3.07111i) q^{25} +1.00000i q^{27} -7.59434 q^{29} -3.28415 q^{31} -0.985939i q^{33} +(8.58028 - 2.94567i) q^{35} +0.945668i q^{37} +4.94567 q^{39} -0.568295 q^{41} -8.45963i q^{43} +(-0.726062 - 2.11491i) q^{45} -2.60492i q^{47} -9.45963 q^{49} -4.52323 q^{51} -0.229815i q^{53} +(0.715853 + 2.08517i) q^{55} +2.60492i q^{57} -9.10003 q^{59} +11.0183 q^{61} +4.05705i q^{63} +(-10.4596 + 3.59086i) q^{65} -8.45963i q^{67} -3.53729 q^{69} -1.43171 q^{71} -11.9507i q^{73} +(3.07111 + 3.94567i) q^{75} -4.00000i q^{77} +3.28415 q^{79} +1.00000 q^{81} +9.89134i q^{83} +(9.56622 - 3.28415i) q^{85} +7.59434i q^{87} -12.3510 q^{89} +20.0648 q^{91} +3.28415i q^{93} +(-1.89134 - 5.50917i) q^{95} +3.23797i q^{97} -0.985939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9} - 4 q^{25} - 32 q^{31} + 16 q^{39} + 8 q^{41} - 12 q^{49} + 16 q^{55} - 24 q^{65} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89} + 64 q^{95}+O(q^{100}) \) 12 * q - 12 * q^9 - 4 * q^25 - 32 * q^31 + 16 * q^39 + 8 * q^41 - 12 * q^49 + 16 * q^55 - 24 * q^65 - 32 * q^71 + 32 * q^79 + 12 * q^81 + 40 * q^89 + 64 * q^95

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