LMFDB - Embedded newform 3840.2.f.l.769.5

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			769.5
Root			$-0.806504 + 1.16170i$ of defining polynomial
Character	$\chi$	$=$	3840.769
Dual form			3840.2.f.l.769.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(1.23992 - 1.86081i) q^{5} -0.746175i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(1.23992 - 1.86081i) q^{5} -0.746175i q^{7} -1.00000 q^{9} +5.36068 q^{11} +2.92520i q^{13} +(-1.86081 - 1.23992i) q^{15} +2.13466i q^{17} +1.73367 q^{19} -0.746175 q^{21} +7.49534i q^{23} +(-1.92520 - 4.61450i) q^{25} +1.00000i q^{27} +6.74916 q^{29} +2.64681 q^{31} -5.36068i q^{33} +(-1.38849 - 0.925197i) q^{35} -1.07480i q^{37} +2.92520 q^{39} +11.2936 q^{41} +7.44322i q^{43} +(-1.23992 + 1.86081i) q^{45} +1.73367i q^{47} +6.44322 q^{49} +2.13466 q^{51} +7.72161i q^{53} +(6.64681 - 9.97518i) q^{55} -1.73367i q^{57} -6.85302 q^{59} +6.45203 q^{61} +0.746175i q^{63} +(5.44322 + 3.62701i) q^{65} +7.44322i q^{67} +7.49534 q^{69} -13.2936 q^{71} -0.690358i q^{73} +(-4.61450 + 1.92520i) q^{75} -4.00000i q^{77} -2.64681 q^{79} +1.00000 q^{81} +5.85039i q^{83} +(3.97219 + 2.64681i) q^{85} -6.74916i q^{87} +7.59283 q^{89} +2.18271 q^{91} -2.64681i q^{93} +(2.14961 - 3.22601i) q^{95} -14.1887i q^{97} -5.36068 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 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

Display 100 coefficients 10 coefficients

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
$5$	1.23992	−	1.86081i	0.554509	−	0.832178i
$5$	1.23992	−	1.86081i	0.554509	−	0.832178i
$6$		0			0
$7$		−	0.746175i		−	0.282028i	−0.990008	−	0.141014i	$-0.954964\pi$
$7$		−	0.746175i		−	0.282028i	0.990008	−	0.141014i	$-0.0450362\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	5.36068			1.61630			0.808152	−	0.588974i	$-0.200468\pi$
$11$	5.36068			1.61630			0.808152	+	0.588974i	$0.200468\pi$
$12$		0			0
$13$			2.92520i			0.811304i	0.914028	+	0.405652i	$0.132955\pi$
$13$			2.92520i			0.811304i	−0.914028	+	0.405652i	$0.867045\pi$
$14$		0			0
$15$	−1.86081	−	1.23992i	−0.480458	−	0.320146i
$16$		0			0
$17$			2.13466i			0.517731i	0.965913	+	0.258866i	$0.0833487\pi$
$17$			2.13466i			0.517731i	−0.965913	+	0.258866i	$0.916651\pi$
$18$		0			0
$19$	1.73367			0.397730			0.198865	−	0.980027i	$-0.436274\pi$
$19$	1.73367			0.397730			0.198865	+	0.980027i	$0.436274\pi$
$20$		0			0
$21$	−0.746175			−0.162829
$22$		0			0
$23$			7.49534i			1.56289i	0.623977	+	0.781443i	$0.285516\pi$
$23$			7.49534i			1.56289i	−0.623977	+	0.781443i	$0.714484\pi$
$24$		0			0
$25$	−1.92520	−	4.61450i	−0.385039	−	0.922900i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	6.74916			1.25329			0.626644	−	0.779306i	$-0.284428\pi$
$29$	6.74916			1.25329			0.626644	+	0.779306i	$0.284428\pi$
$30$		0			0
$31$	2.64681			0.475381			0.237690	−	0.971341i	$-0.423610\pi$
$31$	2.64681			0.475381			0.237690	+	0.971341i	$0.423610\pi$
$32$		0			0
$33$		−	5.36068i		−	0.933174i
$34$		0			0
$35$	−1.38849	−	0.925197i	−0.234697	−	0.156387i
$36$		0			0
$37$		−	1.07480i		−	0.176697i	−0.996090	−	0.0883483i	$-0.971841\pi$
$37$		−	1.07480i		−	0.176697i	0.996090	−	0.0883483i	$-0.0281588\pi$
$38$		0			0
$39$	2.92520			0.468406
$40$		0			0
$41$	11.2936			1.76377			0.881883	−	0.471468i	$-0.156276\pi$
$41$	11.2936			1.76377			0.881883	+	0.471468i	$0.156276\pi$
$42$		0			0
$43$			7.44322i			1.13508i	0.823346	+	0.567540i	$0.192105\pi$
$43$			7.44322i			1.13508i	−0.823346	+	0.567540i	$0.807895\pi$
$44$		0			0
$45$	−1.23992	+	1.86081i	−0.184836	+	0.277393i
$46$		0			0
$47$			1.73367i			0.252881i	0.991974	+	0.126441i	$0.0403553\pi$
$47$			1.73367i			0.252881i	−0.991974	+	0.126441i	$0.959645\pi$
$48$		0			0
$49$	6.44322			0.920460
$50$		0			0
$51$	2.13466			0.298912
$52$		0			0
$53$			7.72161i			1.06064i	0.847796	+	0.530322i	$0.177929\pi$
$53$			7.72161i			1.06064i	−0.847796	+	0.530322i	$0.822071\pi$
$54$		0			0
$55$	6.64681	−	9.97518i	0.896255	−	1.34505i
$56$		0			0
$57$		−	1.73367i		−	0.229630i
$58$		0			0
$59$	−6.85302			−0.892188			−0.446094	−	0.894986i	$-0.647185\pi$
$59$	−6.85302			−0.892188			−0.446094	+	0.894986i	$0.647185\pi$
$60$		0			0
$61$	6.45203			0.826098			0.413049	−	0.910709i	$-0.364464\pi$
$61$	6.45203			0.826098			0.413049	+	0.910709i	$0.364464\pi$
$62$		0			0
$63$			0.746175i			0.0940092i
$64$		0			0
$65$	5.44322	+	3.62701i	0.675149	+	0.449875i
$66$		0			0
$67$			7.44322i			0.909334i	0.890661	+	0.454667i	$0.150242\pi$
$67$			7.44322i			0.909334i	−0.890661	+	0.454667i	$0.849758\pi$
$68$		0			0
$69$	7.49534			0.902332
$70$		0			0
$71$	−13.2936			−1.57766			−0.788831	−	0.614610i	$-0.789313\pi$
$71$	−13.2936			−1.57766			−0.788831	+	0.614610i	$0.789313\pi$
$72$		0			0
$73$		−	0.690358i		−	0.0808003i	−0.999184	−	0.0404002i	$-0.987137\pi$
$73$		−	0.690358i		−	0.0808003i	0.999184	−	0.0404002i	$-0.0128633\pi$
$74$		0			0
$75$	−4.61450	+	1.92520i	−0.532837	+	0.222303i
$76$		0			0
$77$		−	4.00000i		−	0.455842i
$78$		0			0
$79$	−2.64681			−0.297789			−0.148895	−	0.988853i	$-0.547572\pi$
$79$	−2.64681			−0.297789			−0.148895	+	0.988853i	$0.547572\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			5.85039i			0.642164i	0.947051	+	0.321082i	$0.104047\pi$
$83$			5.85039i			0.642164i	−0.947051	+	0.321082i	$0.895953\pi$
$84$		0			0
$85$	3.97219	+	2.64681i	0.430844	+	0.287087i
$86$		0			0
$87$		−	6.74916i		−	0.723586i
$88$		0			0
$89$	7.59283			0.804838			0.402419	−	0.915456i	$-0.368169\pi$
$89$	7.59283			0.804838			0.402419	+	0.915456i	$0.368169\pi$
$90$		0			0
$91$	2.18271			0.228810
$92$		0			0
$93$		−	2.64681i		−	0.274461i
$94$		0			0
$95$	2.14961	−	3.22601i	0.220545	−	0.330982i
$96$		0			0
$97$		−	14.1887i		−	1.44064i	−0.693641	−	0.720321i	$-0.743994\pi$
$97$		−	14.1887i		−	1.44064i	0.693641	−	0.720321i	$-0.256006\pi$
$98$		0			0
$99$	−5.36068			−0.538768
$100$		0			0
$101$	−7.43952			−0.740260			−0.370130	−	0.928980i	$-0.620687\pi$
$101$	−7.43952			−0.740260			−0.370130	+	0.928980i	$0.620687\pi$
$102$		0			0
$103$			7.19820i			0.709260i	0.935007	+	0.354630i	$0.115393\pi$
$103$			7.19820i			0.709260i	−0.935007	+	0.354630i	$0.884607\pi$
$104$		0			0
$105$	−0.925197	+	1.38849i	−0.0902900	+	0.135502i
$106$		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
$109$	−19.9504			−1.91090			−0.955449	−	0.295158i	$-0.904628\pi$
$109$	−19.9504			−1.91090			−0.955449	+	0.295158i	$0.904628\pi$
$110$		0			0
$111$	−1.07480			−0.102016
$112$		0			0
$113$		−	12.0540i		−	1.13395i	−0.823736	−	0.566973i	$-0.808114\pi$
$113$		−	12.0540i		−	1.13395i	0.823736	−	0.566973i	$-0.191886\pi$
$114$		0			0
$115$	13.9474	+	9.29362i	1.30060	+	0.866634i
$116$		0			0
$117$		−	2.92520i		−	0.270435i
$118$		0			0
$119$	1.59283			0.146014
$120$		0			0
$121$	17.7368			1.61244
$122$		0			0
$123$		−	11.2936i		−	1.01831i
$124$		0			0
$125$	−10.9738	−	2.13919i	−0.981525	−	0.191335i
$126$		0			0
$127$		−	4.21351i		−	0.373888i	−0.982371	−	0.186944i	$-0.940142\pi$
$127$		−	4.21351i		−	0.373888i	0.982371	−	0.186944i	$-0.0598583\pi$
$128$		0			0
$129$	7.44322			0.655339
$130$		0			0
$131$	−10.3204			−0.901694			−0.450847	−	0.892601i	$-0.648878\pi$
$131$	−10.3204			−0.901694			−0.450847	+	0.892601i	$0.648878\pi$
$132$		0			0
$133$		−	1.29362i		−	0.112171i
$134$		0			0
$135$	1.86081	+	1.23992i	0.160153	+	0.106715i
$136$		0			0
$137$		−	15.0387i		−	1.28484i	−0.766351	−	0.642422i	$-0.777930\pi$
$137$		−	15.0387i		−	1.28484i	0.766351	−	0.642422i	$-0.222070\pi$
$138$		0			0
$139$	9.47032			0.803262			0.401631	−	0.915802i	$-0.368443\pi$
$139$	9.47032			0.803262			0.401631	+	0.915802i	$0.368443\pi$
$140$		0			0
$141$	1.73367			0.146001
$142$		0			0
$143$			15.6810i			1.31131i
$144$		0			0
$145$	8.36842	−	12.5589i	0.694959	−	1.04296i
$146$		0			0
$147$		−	6.44322i		−	0.531428i
$148$		0			0
$149$	−1.78948			−0.146600			−0.0733000	−	0.997310i	$-0.523353\pi$
$149$	−1.78948			−0.146600			−0.0733000	+	0.997310i	$0.523353\pi$
$150$		0			0
$151$	−10.6468			−0.866425			−0.433212	−	0.901292i	$-0.642620\pi$
$151$	−10.6468			−0.866425			−0.433212	+	0.901292i	$0.642620\pi$
$152$		0			0
$153$		−	2.13466i		−	0.172577i
$154$		0			0
$155$	3.28183	−	4.92520i	0.263603	−	0.395601i
$156$		0			0
$157$		−	6.92520i		−	0.552691i	−0.961058	−	0.276345i	$-0.910877\pi$
$157$		−	6.92520i		−	0.552691i	0.961058	−	0.276345i	$-0.0891234\pi$
$158$		0			0
$159$	7.72161			0.612364
$160$		0			0
$161$	5.59283			0.440777
$162$		0			0
$163$		−	7.70079i		−	0.603172i	−0.953439	−	0.301586i	$-0.902484\pi$
$163$		−	7.70079i		−	0.603172i	0.953439	−	0.301586i	$-0.0975161\pi$
$164$		0			0
$165$	−9.97518	−	6.64681i	−0.776566	−	0.517453i
$166$		0			0
$167$			3.22601i			0.249637i	0.992180	+	0.124818i	$0.0398348\pi$
$167$			3.22601i			0.249637i	−0.992180	+	0.124818i	$0.960165\pi$
$168$		0			0
$169$	4.44322			0.341786
$170$		0			0
$171$	−1.73367			−0.132577
$172$		0			0
$173$			6.42799i			0.488711i	0.969686	+	0.244356i	$0.0785764\pi$
$173$			6.42799i			0.488711i	−0.969686	+	0.244356i	$0.921424\pi$
$174$		0			0
$175$	−3.44322	+	1.43653i	−0.260283	+	0.108592i
$176$		0			0
$177$			6.85302i			0.515105i
$178$		0			0
$179$	8.13765			0.608236			0.304118	−	0.952634i	$-0.401638\pi$
$179$	8.13765			0.608236			0.304118	+	0.952634i	$0.401638\pi$
$180$		0			0
$181$	−1.49235			−0.110925			−0.0554627	−	0.998461i	$-0.517663\pi$
$181$	−1.49235			−0.110925			−0.0554627	+	0.998461i	$0.517663\pi$
$182$		0			0
$183$		−	6.45203i		−	0.476948i
$184$		0			0
$185$	−2.00000	−	1.33267i	−0.147043	−	0.0979798i
$186$		0			0
$187$			11.4432i			0.836811i
$188$		0			0
$189$	0.746175			0.0542762
$190$		0			0
$191$	6.88645			0.498286			0.249143	−	0.968467i	$-0.419851\pi$
$191$	6.88645			0.498286			0.249143	+	0.968467i	$0.419851\pi$
$192$		0			0
$193$			16.4830i			1.18647i	0.805028	+	0.593237i	$0.202150\pi$
$193$			16.4830i			1.18647i	−0.805028	+	0.593237i	$0.797850\pi$
$194$		0			0
$195$	3.62701	−	5.44322i	0.259736	−	0.389797i
$196$		0			0
$197$		−	13.5720i		−	0.966965i	−0.875354	−	0.483483i	$-0.839372\pi$
$197$		−	13.5720i		−	0.966965i	0.875354	−	0.483483i	$-0.160628\pi$
$198$		0			0
$199$	9.05398			0.641820			0.320910	−	0.947110i	$-0.396011\pi$
$199$	9.05398			0.641820			0.320910	+	0.947110i	$0.396011\pi$
$200$		0			0
$201$	7.44322			0.525005
$202$		0			0
$203$		−	5.03605i		−	0.353462i
$204$		0			0
$205$	14.0032	−	21.0152i	0.978025	−	1.46777i
$206$		0			0
$207$		−	7.49534i		−	0.520962i
$208$		0			0
$209$	9.29362			0.642853
$210$		0			0
$211$	−2.53566			−0.174562			−0.0872809	−	0.996184i	$-0.527818\pi$
$211$	−2.53566			−0.174562			−0.0872809	+	0.996184i	$0.527818\pi$
$212$		0			0
$213$			13.2936i			0.910864i
$214$		0			0
$215$	13.8504	+	9.22900i	0.944589	+	0.629413i
$216$		0			0
$217$		−	1.97498i		−	0.134070i
$218$		0			0
$219$	−0.690358			−0.0466501
$220$		0			0
$221$	−6.24430			−0.420037
$222$		0			0
$223$		−	12.1579i		−	0.814152i	−0.913394	−	0.407076i	$-0.866548\pi$
$223$		−	12.1579i		−	0.814152i	0.913394	−	0.407076i	$-0.133452\pi$
$224$		0			0
$225$	1.92520	+	4.61450i	0.128346	+	0.307633i
$226$		0			0
$227$		−	20.7368i		−	1.37635i	−0.725544	−	0.688176i	$-0.758412\pi$
$227$		−	20.7368i		−	1.37635i	0.725544	−	0.688176i	$-0.241588\pi$
$228$		0			0
$229$	−19.9504			−1.31836			−0.659178	−	0.751987i	$-0.729096\pi$
$229$	−19.9504			−1.31836			−0.659178	+	0.751987i	$0.729096\pi$
$230$		0			0
$231$	−4.00000			−0.263181
$232$		0			0
$233$		−	13.3386i		−	0.873844i	−0.899499	−	0.436922i	$-0.856069\pi$
$233$		−	13.3386i		−	0.873844i	0.899499	−	0.436922i	$-0.143931\pi$
$234$		0			0
$235$	3.22601	+	2.14961i	0.210442	+	0.140225i
$236$		0			0
$237$			2.64681i			0.171929i
$238$		0			0
$239$	−22.8864			−1.48040			−0.740201	−	0.672386i	$-0.765269\pi$
$239$	−22.8864			−1.48040			−0.740201	+	0.672386i	$0.765269\pi$
$240$		0			0
$241$	3.59283			0.231435			0.115717	−	0.993282i	$-0.463083\pi$
$241$	3.59283			0.231435			0.115717	+	0.993282i	$0.463083\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	7.98908	−	11.9896i	0.510404	−	0.765987i
$246$		0			0
$247$			5.07131i			0.322680i
$248$		0			0
$249$	5.85039			0.370754
$250$		0			0
$251$	−8.82801			−0.557219			−0.278609	−	0.960404i	$-0.589873\pi$
$251$	−8.82801			−0.557219			−0.278609	+	0.960404i	$0.589873\pi$
$252$		0			0
$253$			40.1801i			2.52610i
$254$		0			0
$255$	2.64681	−	3.97219i	0.165750	−	0.248748i
$256$		0			0
$257$		−	22.2927i		−	1.39058i	−0.718728	−	0.695291i	$-0.755275\pi$
$257$		−	22.2927i		−	1.39058i	0.718728	−	0.695291i	$-0.244725\pi$
$258$		0			0
$259$	−0.801991			−0.0498333
$260$		0			0
$261$	−6.74916			−0.417763
$262$		0			0
$263$		−	21.2014i		−	1.30733i	−0.756783	−	0.653667i	$-0.773230\pi$
$263$		−	21.2014i		−	1.30733i	0.756783	−	0.653667i	$-0.226770\pi$
$264$		0			0
$265$	14.3684	+	9.57418i	0.882645	+	0.588137i
$266$		0			0
$267$		−	7.59283i		−	0.464674i
$268$		0			0
$269$	14.6935			0.895881			0.447940	−	0.894063i	$-0.352158\pi$
$269$	14.6935			0.895881			0.447940	+	0.894063i	$0.352158\pi$
$270$		0			0
$271$	−20.2396			−1.22947			−0.614735	−	0.788734i	$-0.710737\pi$
$271$	−20.2396			−1.22947			−0.614735	+	0.788734i	$0.710737\pi$
$272$		0			0
$273$		−	2.18271i		−	0.132103i
$274$		0			0
$275$	−10.3204	−	24.7368i	−0.622341	−	1.49169i
$276$		0			0
$277$			0.518027i			0.0311252i	0.999879	+	0.0155626i	$0.00495393\pi$
$277$			0.518027i			0.0311252i	−0.999879	+	0.0155626i	$0.995046\pi$
$278$		0			0
$279$	−2.64681			−0.158460
$280$		0			0
$281$	−13.7008			−0.817320			−0.408660	−	0.912687i	$-0.634004\pi$
$281$	−13.7008			−0.817320			−0.408660	+	0.912687i	$0.634004\pi$
$282$		0			0
$283$			18.0305i			1.07180i	0.844282	+	0.535900i	$0.180027\pi$
$283$			18.0305i			1.07180i	−0.844282	+	0.535900i	$0.819973\pi$
$284$		0			0
$285$	−3.22601	−	2.14961i	−0.191093	−	0.127332i
$286$		0			0
$287$		−	8.42701i		−	0.497431i
$288$		0			0
$289$	12.4432			0.731954
$290$		0			0
$291$	−14.1887			−0.831755
$292$		0			0
$293$			15.9792i			0.933513i	0.884386	+	0.466757i	$0.154578\pi$
$293$			15.9792i			0.933513i	−0.884386	+	0.466757i	$0.845422\pi$
$294$		0			0
$295$	−8.49720	+	12.7521i	−0.494726	+	0.742459i
$296$		0			0
$297$			5.36068i			0.311058i
$298$		0			0
$299$	−21.9253			−1.26797
$300$		0			0
$301$	5.55394			0.320124
$302$		0			0
$303$			7.43952i			0.427389i
$304$		0			0
$305$	8.00000	−	12.0060i	0.458079	−	0.687460i
$306$		0			0
$307$			22.5872i			1.28912i	0.764553	+	0.644561i	$0.222960\pi$
$307$			22.5872i			1.28912i	−0.764553	+	0.644561i	$0.777040\pi$
$308$		0			0
$309$	7.19820			0.409492
$310$		0			0
$311$	18.5872			1.05399			0.526993	−	0.849870i	$-0.323320\pi$
$311$	18.5872			1.05399			0.526993	+	0.849870i	$0.323320\pi$
$312$		0			0
$313$		−	29.3871i		−	1.66106i	−0.556977	−	0.830528i	$-0.688039\pi$
$313$		−	29.3871i		−	1.66106i	0.556977	−	0.830528i	$-0.311961\pi$
$314$		0			0
$315$	1.38849	+	0.925197i	0.0782323	+	0.0521289i
$316$		0			0
$317$			5.57201i			0.312955i	0.987682	+	0.156478i	$0.0500139\pi$
$317$			5.57201i			0.312955i	−0.987682	+	0.156478i	$0.949986\pi$
$318$		0			0
$319$	36.1801			2.02569
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$			3.70079i			0.205917i
$324$		0			0
$325$	13.4983	−	5.63158i	0.748752	−	0.312384i
$326$		0			0
$327$			19.9504i			1.10326i
$328$		0			0
$329$	1.29362			0.0713194
$330$		0			0
$331$	13.7396			0.755199			0.377599	−	0.925969i	$-0.376750\pi$
$331$	13.7396			0.755199			0.377599	+	0.925969i	$0.376750\pi$
$332$		0			0
$333$			1.07480i			0.0588989i
$334$		0			0
$335$	13.8504	+	9.22900i	0.756728	+	0.504234i
$336$		0			0
$337$			20.7523i			1.13045i	0.824936	+	0.565226i	$0.191211\pi$
$337$			20.7523i			1.13045i	−0.824936	+	0.565226i	$0.808789\pi$
$338$		0			0
$339$	−12.0540			−0.654685
$340$		0			0
$341$	14.1887			0.768360
$342$		0			0
$343$		−	10.0310i		−	0.541623i
$344$		0			0
$345$	9.29362	−	13.9474i	0.500352	−	0.750901i
$346$		0			0
$347$		−	4.73684i		−	0.254287i	−0.991884	−	0.127143i	$-0.959419\pi$
$347$		−	4.73684i		−	0.254287i	0.991884	−	0.127143i	$-0.0405809\pi$
$348$		0			0
$349$	0.482632			0.0258347			0.0129174	−	0.999917i	$-0.495888\pi$
$349$	0.482632			0.0258347			0.0129174	+	0.999917i	$0.495888\pi$
$350$		0			0
$351$	−2.92520			−0.156135
$352$		0			0
$353$			2.13466i			0.113617i	0.998385	+	0.0568083i	$0.0180924\pi$
$353$			2.13466i			0.113617i	−0.998385	+	0.0568083i	$0.981908\pi$
$354$		0			0
$355$	−16.4830	+	24.7368i	−0.874828	+	1.31290i
$356$		0			0
$357$		−	1.59283i		−	0.0843015i
$358$		0			0
$359$	−9.59283			−0.506290			−0.253145	−	0.967428i	$-0.581465\pi$
$359$	−9.59283			−0.506290			−0.253145	+	0.967428i	$0.581465\pi$
$360$		0			0
$361$	−15.9944			−0.841811
$362$		0			0
$363$		−	17.7368i		−	0.930943i
$364$		0			0
$365$	−1.28462	−	0.855989i	−0.0672402	−	0.0448045i
$366$		0			0
$367$			34.0832i			1.77913i	0.456809	+	0.889565i	$0.348992\pi$
$367$			34.0832i			1.77913i	−0.456809	+	0.889565i	$0.651008\pi$
$368$		0			0
$369$	−11.2936			−0.587922
$370$		0			0
$371$	5.76167			0.299131
$372$		0			0
$373$			4.33796i			0.224611i	0.993674	+	0.112306i	$0.0358236\pi$
$373$			4.33796i			0.224611i	−0.993674	+	0.112306i	$0.964176\pi$
$374$		0			0
$375$	−2.13919	+	10.9738i	−0.110468	+	0.566684i
$376$		0			0
$377$			19.7426i			1.01680i
$378$		0			0
$379$	6.90107			0.354484			0.177242	−	0.984167i	$-0.443282\pi$
$379$	6.90107			0.354484			0.177242	+	0.984167i	$0.443282\pi$
$380$		0			0
$381$	−4.21351			−0.215864
$382$		0			0
$383$		−	22.3744i		−	1.14328i	−0.820506	−	0.571639i	$-0.806308\pi$
$383$		−	22.3744i		−	1.14328i	0.820506	−	0.571639i	$-0.193692\pi$
$384$		0			0
$385$	−7.44322	−	4.95968i	−0.379342	−	0.252769i
$386$		0			0
$387$		−	7.44322i		−	0.378360i
$388$		0			0
$389$	11.0185			0.558659			0.279330	−	0.960195i	$-0.409888\pi$
$389$	11.0185			0.558659			0.279330	+	0.960195i	$0.409888\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$			10.3204i			0.520593i
$394$		0			0
$395$	−3.28183	+	4.92520i	−0.165127	+	0.247814i
$396$		0			0
$397$		−	25.2549i		−	1.26751i	−0.773536	−	0.633753i	$-0.781514\pi$
$397$		−	25.2549i		−	1.26751i	0.773536	−	0.633753i	$-0.218486\pi$
$398$		0			0
$399$	−1.29362			−0.0647619
$400$		0			0
$401$	7.29362			0.364226			0.182113	−	0.983278i	$-0.441706\pi$
$401$	7.29362			0.364226			0.182113	+	0.983278i	$0.441706\pi$
$402$		0			0
$403$			7.74244i			0.385678i
$404$		0			0
$405$	1.23992	−	1.86081i	0.0616121	−	0.0924642i
$406$		0			0
$407$		−	5.76167i		−	0.285595i
$408$		0			0
$409$	15.8504			0.783752			0.391876	−	0.920018i	$-0.371826\pi$
$409$	15.8504			0.783752			0.391876	+	0.920018i	$0.371826\pi$
$410$		0			0
$411$	−15.0387			−0.741805
$412$		0			0
$413$			5.11355i			0.251622i
$414$		0			0
$415$	10.8864	+	7.25402i	0.534395	+	0.356086i
$416$		0			0
$417$		−	9.47032i		−	0.463763i
$418$		0			0
$419$	−8.02602			−0.392097			−0.196048	−	0.980594i	$-0.562811\pi$
$419$	−8.02602			−0.392097			−0.196048	+	0.980594i	$0.562811\pi$
$420$		0			0
$421$	22.9351			1.11779			0.558893	−	0.829240i	$-0.311226\pi$
$421$	22.9351			1.11779			0.558893	+	0.829240i	$0.311226\pi$
$422$		0			0
$423$		−	1.73367i		−	0.0842937i
$424$		0			0
$425$	9.85039	−	4.10964i	0.477814	−	0.199347i
$426$		0			0
$427$		−	4.81434i		−	0.232982i
$428$		0			0
$429$	15.6810			0.757087
$430$		0			0
$431$	−35.0665			−1.68909			−0.844547	−	0.535481i	$-0.820130\pi$
$431$	−35.0665			−1.68909			−0.844547	+	0.535481i	$0.820130\pi$
$432$		0			0
$433$		−	17.0773i		−	0.820682i	−0.911932	−	0.410341i	$-0.865410\pi$
$433$		−	17.0773i		−	0.820682i	0.911932	−	0.410341i	$-0.134590\pi$
$434$		0			0
$435$	−12.5589	−	8.36842i	−0.602152	−	0.401235i
$436$		0			0
$437$			12.9944i			0.621607i
$438$		0			0
$439$	8.53885			0.407537			0.203769	−	0.979019i	$-0.434681\pi$
$439$	8.53885			0.407537			0.203769	+	0.979019i	$0.434681\pi$
$440$		0			0
$441$	−6.44322			−0.306820
$442$		0			0
$443$		−	20.7368i		−	0.985237i	−0.870245	−	0.492619i	$-0.836040\pi$
$443$		−	20.7368i		−	0.985237i	0.870245	−	0.492619i	$-0.163960\pi$
$444$		0			0
$445$	9.41450	−	14.1288i	0.446290	−	0.669769i
$446$		0			0
$447$			1.78948i			0.0846396i
$448$		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
$451$	60.5414			2.85078
$452$		0			0
$453$			10.6468i			0.500231i
$454$		0			0
$455$	2.70638	−	4.06160i	0.126877	−	0.190411i
$456$		0			0
$457$		−	1.28462i		−	0.0600921i	−0.999549	−	0.0300461i	$-0.990435\pi$
$457$		−	1.28462i		−	0.0600921i	0.999549	−	0.0300461i	$-0.00956540\pi$
$458$		0			0
$459$	−2.13466			−0.0996374
$460$		0			0
$461$	−15.7033			−0.731374			−0.365687	−	0.930738i	$-0.619166\pi$
$461$	−15.7033			−0.731374			−0.365687	+	0.930738i	$0.619166\pi$
$462$		0			0
$463$		−	18.7215i		−	0.870064i	−0.900415	−	0.435032i	$-0.856737\pi$
$463$		−	18.7215i		−	0.870064i	0.900415	−	0.435032i	$-0.143263\pi$
$464$		0			0
$465$	−4.92520	−	3.28183i	−0.228401	−	0.152191i
$466$		0			0
$467$			2.14961i			0.0994719i	0.998762	+	0.0497360i	$0.0158380\pi$
$467$			2.14961i			0.0994719i	−0.998762	+	0.0497360i	$0.984162\pi$
$468$		0			0
$469$	5.55394			0.256457
$470$		0			0
$471$	−6.92520			−0.319096
$472$		0			0
$473$			39.9007i			1.83464i
$474$		0			0
$475$	−3.33765	−	8.00000i	−0.153142	−	0.367065i
$476$		0			0
$477$		−	7.72161i		−	0.353548i
$478$		0			0
$479$	12.1801			0.556521			0.278261	−	0.960506i	$-0.410242\pi$
$479$	12.1801			0.556521			0.278261	+	0.960506i	$0.410242\pi$
$480$		0			0
$481$	3.14401			0.143355
$482$		0			0
$483$		−	5.59283i		−	0.254483i
$484$		0			0
$485$	−26.4024	−	17.5928i	−1.19887	−	0.798849i
$486$		0			0
$487$		−	25.7678i		−	1.16765i	−0.811879	−	0.583826i	$-0.801555\pi$
$487$		−	25.7678i		−	1.16765i	0.811879	−	0.583826i	$-0.198445\pi$
$488$		0			0
$489$	−7.70079			−0.348242
$490$		0			0
$491$	−16.7724			−0.756927			−0.378464	−	0.925616i	$-0.623547\pi$
$491$	−16.7724			−0.756927			−0.378464	+	0.925616i	$0.623547\pi$
$492$		0			0
$493$			14.4072i			0.648866i
$494$		0			0
$495$	−6.64681	+	9.97518i	−0.298752	+	0.448351i
$496$		0			0
$497$			9.91936i			0.444944i
$498$		0			0
$499$	17.6224			0.788888			0.394444	−	0.918920i	$-0.370937\pi$
$499$	17.6224			0.788888			0.394444	+	0.918920i	$0.370937\pi$
$500$		0			0
$501$	3.22601			0.144128
$502$		0			0
$503$		−	27.1263i		−	1.20950i	−0.796414	−	0.604752i	$-0.793272\pi$
$503$		−	27.1263i		−	1.20950i	0.796414	−	0.604752i	$-0.206728\pi$
$504$		0			0
$505$	−9.22441	+	13.8435i	−0.410481	+	0.616028i
$506$		0			0
$507$		−	4.44322i		−	0.197330i
$508$		0			0
$509$	−15.9782			−0.708220			−0.354110	−	0.935204i	$-0.615216\pi$
$509$	−15.9782			−0.708220			−0.354110	+	0.935204i	$0.615216\pi$
$510$		0			0
$511$	−0.515128			−0.0227879
$512$		0			0
$513$			1.73367i			0.0765432i
$514$		0			0
$515$	13.3945	+	8.92520i	0.590230	+	0.393291i
$516$		0			0
$517$			9.29362i			0.408733i
$518$		0			0
$519$	6.42799			0.282158
$520$		0			0
$521$	−0.886447			−0.0388359			−0.0194180	−	0.999811i	$-0.506181\pi$
$521$	−0.886447			−0.0388359			−0.0194180	+	0.999811i	$0.506181\pi$
$522$		0			0
$523$			41.7729i			1.82660i	0.407286	+	0.913301i	$0.366475\pi$
$523$			41.7729i			1.82660i	−0.407286	+	0.913301i	$0.633525\pi$
$524$		0			0
$525$	1.43653	+	3.44322i	0.0626954	+	0.150275i
$526$		0			0
$527$			5.65004i			0.246120i
$528$		0			0
$529$	−33.1801			−1.44261
$530$		0			0
$531$	6.85302			0.297396
$532$		0			0
$533$			33.0361i			1.43095i
$534$		0			0
$535$	−7.44322	−	4.95968i	−0.321799	−	0.214426i
$536$		0			0
$537$		−	8.13765i		−	0.351165i
$538$		0			0
$539$	34.5400			1.48774
$540$		0			0
$541$	4.47705			0.192483			0.0962417	−	0.995358i	$-0.469318\pi$
$541$	4.47705			0.192483			0.0962417	+	0.995358i	$0.469318\pi$
$542$		0			0
$543$			1.49235i			0.0640428i
$544$		0			0
$545$	−24.7368	+	37.1237i	−1.05961	+	1.59021i
$546$		0			0
$547$			14.3297i			0.612692i	0.951920	+	0.306346i	$0.0991065\pi$
$547$			14.3297i			0.612692i	−0.951920	+	0.306346i	$0.900893\pi$
$548$		0			0
$549$	−6.45203			−0.275366
$550$		0			0
$551$	11.7008			0.498470
$552$		0			0
$553$			1.97498i			0.0839848i
$554$		0			0
$555$	−1.33267	+	2.00000i	−0.0565687	+	0.0848953i
$556$		0			0
$557$		−	2.68556i		−	0.113791i	−0.998380	−	0.0568954i	$-0.981880\pi$
$557$		−	2.68556i		−	0.113791i	0.998380	−	0.0568954i	$-0.0181201\pi$
$558$		0			0
$559$	−21.7729			−0.920895
$560$		0			0
$561$	11.4432			0.483133
$562$		0			0
$563$		−	20.7368i		−	0.873954i	−0.899473	−	0.436977i	$-0.856049\pi$
$563$		−	20.7368i		−	0.873954i	0.899473	−	0.436977i	$-0.143951\pi$
$564$		0			0
$565$	−22.4302	−	14.9460i	−0.943645	−	0.628784i
$566$		0			0
$567$		−	0.746175i		−	0.0313364i
$568$		0			0
$569$	4.40717			0.184758			0.0923791	−	0.995724i	$-0.470553\pi$
$569$	4.40717			0.184758			0.0923791	+	0.995724i	$0.470553\pi$
$570$		0			0
$571$	23.6590			0.990098			0.495049	−	0.868865i	$-0.335150\pi$
$571$	23.6590			0.990098			0.495049	+	0.868865i	$0.335150\pi$
$572$		0			0
$573$		−	6.88645i		−	0.287685i
$574$		0			0
$575$	34.5872	−	14.4300i	1.44239	−	0.601772i
$576$		0			0
$577$		−	6.56366i		−	0.273249i	−0.990623	−	0.136624i	$-0.956375\pi$
$577$		−	6.56366i		−	0.273249i	0.990623	−	0.136624i	$-0.0436253\pi$
$578$		0			0
$579$	16.4830			0.685011
$580$		0			0
$581$	4.36542			0.181108
$582$		0			0
$583$			41.3931i			1.71433i
$584$		0			0
$585$	−5.44322	−	3.62701i	−0.225050	−	0.149958i
$586$		0			0
$587$		−	16.2992i		−	0.672741i	−0.941730	−	0.336370i	$-0.890801\pi$
$587$		−	16.2992i		−	0.672741i	0.941730	−	0.336370i	$-0.109199\pi$
$588$		0			0
$589$	4.58868			0.189073
$590$		0			0
$591$	−13.5720			−0.558278
$592$		0			0
$593$			16.3233i			0.670319i	0.942161	+	0.335160i	$0.108790\pi$
$593$			16.3233i			0.670319i	−0.942161	+	0.335160i	$0.891210\pi$
$594$		0			0
$595$	1.97498	−	2.96395i	0.0809663	−	0.121510i
$596$		0			0
$597$		−	9.05398i		−	0.370555i
$598$		0			0
$599$	−25.5928			−1.04569			−0.522847	−	0.852426i	$-0.675130\pi$
$599$	−25.5928			−1.04569			−0.522847	+	0.852426i	$0.675130\pi$
$600$		0			0
$601$	−29.9225			−1.22056			−0.610282	−	0.792184i	$-0.708944\pi$
$601$	−29.9225			−1.22056			−0.610282	+	0.792184i	$0.708944\pi$
$602$		0			0
$603$		−	7.44322i		−	0.303111i
$604$		0			0
$605$	21.9923	−	33.0048i	0.894113	−	1.34184i
$606$		0			0
$607$		−	20.6965i		−	0.840046i	−0.907513	−	0.420023i	$-0.862022\pi$
$607$		−	20.6965i		−	0.840046i	0.907513	−	0.420023i	$-0.137978\pi$
$608$		0			0
$609$	−5.03605			−0.204071
$610$		0			0
$611$	−5.07131			−0.205163
$612$		0			0
$613$			22.6676i			0.915537i	0.889071	+	0.457769i	$0.151351\pi$
$613$			22.6676i			0.915537i	−0.889071	+	0.457769i	$0.848649\pi$
$614$		0			0
$615$	−21.0152	−	14.0032i	−0.847416	−	0.564663i
$616$		0			0
$617$		−	22.1966i		−	0.893603i	−0.894633	−	0.446802i	$-0.852563\pi$
$617$		−	22.1966i		−	0.893603i	0.894633	−	0.446802i	$-0.147437\pi$
$618$		0			0
$619$	16.8204			0.676070			0.338035	−	0.941133i	$-0.390238\pi$
$619$	16.8204			0.676070			0.338035	+	0.941133i	$0.390238\pi$
$620$		0			0
$621$	−7.49534			−0.300777
$622$		0			0
$623$		−	5.66558i		−	0.226987i
$624$		0			0
$625$	−17.5872	+	17.7676i	−0.703489	+	0.710706i
$626$		0			0
$627$		−	9.29362i		−	0.371151i
$628$		0			0
$629$	2.29434			0.0914813
$630$		0			0
$631$	44.1205			1.75641			0.878204	−	0.478285i	$-0.158742\pi$
$631$	44.1205			1.75641			0.878204	+	0.478285i	$0.158742\pi$
$632$		0			0
$633$			2.53566i			0.100783i
$634$		0			0
$635$	−7.84052	−	5.22441i	−0.311141	−	0.207324i
$636$		0			0
$637$			18.8477i			0.746773i
$638$		0			0
$639$	13.2936			0.525887
$640$		0			0
$641$	−1.18566			−0.0468307			−0.0234154	−	0.999726i	$-0.507454\pi$
$641$	−1.18566			−0.0468307			−0.0234154	+	0.999726i	$0.507454\pi$
$642$		0			0
$643$			22.5872i			0.890754i	0.895343	+	0.445377i	$0.146930\pi$
$643$			22.5872i			0.890754i	−0.895343	+	0.445377i	$0.853070\pi$
$644$		0			0
$645$	9.22900	−	13.8504i	0.363392	−	0.545359i
$646$		0			0
$647$			19.7090i			0.774842i	0.921903	+	0.387421i	$0.126634\pi$
$647$			19.7090i			0.774842i	−0.921903	+	0.387421i	$0.873366\pi$
$648$		0			0
$649$	−36.7368			−1.44205
$650$		0			0
$651$	−1.97498			−0.0774056
$652$		0			0
$653$			44.4585i			1.73979i	0.493234	+	0.869897i	$0.335815\pi$
$653$			44.4585i			1.73979i	−0.493234	+	0.869897i	$0.664185\pi$
$654$		0			0
$655$	−12.7964	+	19.2042i	−0.499997	+	0.750369i
$656$		0			0
$657$			0.690358i			0.0269334i
$658$		0			0
$659$	−41.5863			−1.61997			−0.809987	−	0.586448i	$-0.800526\pi$
$659$	−41.5863			−1.61997			−0.809987	+	0.586448i	$0.800526\pi$
$660$		0			0
$661$	−12.0060			−0.466978			−0.233489	−	0.972359i	$-0.575014\pi$
$661$	−12.0060			−0.466978			−0.233489	+	0.972359i	$0.575014\pi$
$662$		0			0
$663$			6.24430i			0.242509i
$664$		0			0
$665$	−2.40717	−	1.60398i	−0.0933461	−	0.0621997i
$666$		0			0
$667$			50.5872i			1.95875i
$668$		0			0
$669$	−12.1579			−0.470051
$670$		0			0
$671$	34.5872			1.33523
$672$		0			0
$673$		−	14.5080i		−	0.559244i	−0.960110	−	0.279622i	$-0.909791\pi$
$673$		−	14.5080i		−	0.559244i	0.960110	−	0.279622i	$-0.0902091\pi$
$674$		0			0
$675$	4.61450	−	1.92520i	0.177612	−	0.0741009i
$676$		0			0
$677$		−	43.8600i		−	1.68568i	−0.538166	−	0.842839i	$-0.680883\pi$
$677$		−	43.8600i		−	1.68568i	0.538166	−	0.842839i	$-0.319117\pi$
$678$		0			0
$679$	−10.5872			−0.406301
$680$		0			0
$681$	−20.7368			−0.794637
$682$		0			0
$683$			5.33527i			0.204148i	0.994777	+	0.102074i	$0.0325479\pi$
$683$			5.33527i			0.204148i	−0.994777	+	0.102074i	$0.967452\pi$
$684$		0			0
$685$	−27.9841	−	18.6468i	−1.06922	−	0.712458i
$686$		0			0
$687$			19.9504i			0.761153i
$688$		0			0
$689$	−22.5872			−0.860505
$690$		0			0
$691$	−39.7710			−1.51296			−0.756480	−	0.654016i	$-0.773083\pi$
$691$	−39.7710			−1.51296			−0.756480	+	0.654016i	$0.773083\pi$
$692$		0			0
$693$			4.00000i			0.151947i
$694$		0			0
$695$	11.7424	−	17.6224i	0.445416	−	0.668457i
$696$		0			0
$697$			24.1080i			0.913157i
$698$		0			0
$699$	−13.3386			−0.504514
$700$		0			0
$701$	27.5015			1.03872			0.519359	−	0.854556i	$-0.326171\pi$
$701$	27.5015			1.03872			0.519359	+	0.854556i	$0.326171\pi$
$702$		0			0
$703$		−	1.86335i		−	0.0702775i
$704$		0			0
$705$	2.14961	−	3.22601i	0.0809589	−	0.121499i
$706$		0			0
$707$			5.55118i			0.208774i
$708$		0			0
$709$	−0.111632			−0.00419244			−0.00209622	−	0.999998i	$-0.500667\pi$
$709$	−0.111632			−0.00419244			−0.00209622	+	0.999998i	$0.500667\pi$
$710$		0			0
$711$	2.64681			0.0992631
$712$		0			0
$713$			19.8387i			0.742966i
$714$		0			0
$715$	29.1794	+	19.4432i	1.09125	+	0.727135i
$716$		0			0
$717$			22.8864i			0.854710i
$718$		0			0
$719$	−10.7064			−0.399281			−0.199640	−	0.979869i	$-0.563977\pi$
$719$	−10.7064			−0.399281			−0.199640	+	0.979869i	$0.563977\pi$
$720$		0			0
$721$	5.37112			0.200031
$722$		0			0
$723$		−	3.59283i		−	0.133619i
$724$		0			0
$725$	−12.9935	−	31.1440i	−0.482565	−	1.15666i
$726$		0			0
$727$			25.6562i			0.951536i	0.879571	+	0.475768i	$0.157830\pi$
$727$			25.6562i			0.951536i	−0.879571	+	0.475768i	$0.842170\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−15.8888			−0.587667
$732$		0			0
$733$			30.3684i			1.12168i	0.827923	+	0.560842i	$0.189522\pi$
$733$			30.3684i			1.12168i	−0.827923	+	0.560842i	$0.810478\pi$
$734$		0			0
$735$	−11.9896	−	7.98908i	−0.442243	−	0.294682i
$736$		0			0
$737$			39.9007i			1.46976i
$738$		0			0
$739$	20.1917			0.742763			0.371381	−	0.928480i	$-0.378884\pi$
$739$	20.1917			0.742763			0.371381	+	0.928480i	$0.378884\pi$
$740$		0			0
$741$	5.07131			0.186299
$742$		0			0
$743$			46.3863i			1.70175i	0.525369	+	0.850875i	$0.323927\pi$
$743$			46.3863i			1.70175i	−0.525369	+	0.850875i	$0.676073\pi$
$744$		0			0
$745$	−2.21881	+	3.32988i	−0.0812911	+	0.121997i
$746$		0			0
$747$		−	5.85039i		−	0.214055i
$748$		0			0
$749$	−2.98470			−0.109059
$750$		0			0
$751$	−27.1261			−0.989845			−0.494922	−	0.868937i	$-0.664804\pi$
$751$	−27.1261			−0.989845			−0.494922	+	0.868937i	$0.664804\pi$
$752$		0			0
$753$			8.82801i			0.321710i
$754$		0			0
$755$	−13.2012	+	19.8116i	−0.480441	+	0.721020i
$756$		0			0
$757$		−	45.2549i		−	1.64482i	−0.568898	−	0.822408i	$-0.692630\pi$
$757$		−	45.2549i		−	1.64482i	0.568898	−	0.822408i	$-0.307370\pi$
$758$		0			0
$759$	40.1801			1.45844
$760$		0			0
$761$	−16.8864			−0.612133			−0.306067	−	0.952010i	$-0.599013\pi$
$761$	−16.8864			−0.612133			−0.306067	+	0.952010i	$0.599013\pi$
$762$		0			0
$763$			14.8864i			0.538926i
$764$		0			0
$765$	−3.97219	−	2.64681i	−0.143615	−	0.0956956i
$766$		0			0
$767$		−	20.0464i		−	0.723835i
$768$		0			0
$769$	−16.3297			−0.588863			−0.294431	−	0.955673i	$-0.595130\pi$
$769$	−16.3297			−0.588863			−0.294431	+	0.955673i	$0.595130\pi$
$770$		0			0
$771$	−22.2927			−0.802853
$772$		0			0
$773$			41.3144i			1.48598i	0.669304	+	0.742989i	$0.266592\pi$
$773$			41.3144i			1.48598i	−0.669304	+	0.742989i	$0.733408\pi$
$774$		0			0
$775$	−5.09563	−	12.2137i	−0.183040	−	0.438729i
$776$		0			0
$777$			0.801991i			0.0287713i
$778$		0			0
$779$	19.5794			0.701503
$780$		0			0
$781$	−71.2628			−2.54998
$782$		0			0
$783$			6.74916i			0.241195i
$784$		0			0
$785$	−12.8864	−	8.58669i	−0.459937	−	0.306472i
$786$		0			0
$787$			11.4849i			0.409391i	0.978826	+	0.204696i	$0.0656205\pi$
$787$			11.4849i			0.409391i	−0.978826	+	0.204696i	$0.934380\pi$
$788$		0			0
$789$	−21.2014			−0.754789
$790$		0			0
$791$	−8.99440			−0.319804
$792$		0			0
$793$			18.8735i			0.670216i
$794$		0			0
$795$	9.57418	−	14.3684i	0.339561	−	0.509595i
$796$		0			0
$797$		−	45.4945i		−	1.61150i	−0.592257	−	0.805749i	$-0.701763\pi$
$797$		−	45.4945i		−	1.61150i	0.592257	−	0.805749i	$-0.298237\pi$
$798$		0			0
$799$	−3.70079			−0.130924
$800$		0			0
$801$	−7.59283			−0.268279
$802$		0			0
$803$		−	3.70079i		−	0.130598i
$804$		0			0
$805$	6.93466	−	10.4072i	0.244415	−	0.366805i
$806$		0			0
$807$		−	14.6935i		−	0.517237i
$808$		0			0
$809$	−36.0721			−1.26823			−0.634114	−	0.773240i	$-0.718635\pi$
$809$	−36.0721			−1.26823			−0.634114	+	0.773240i	$0.718635\pi$
$810$		0			0
$811$	−44.5230			−1.56341			−0.781707	−	0.623646i	$-0.785651\pi$
$811$	−44.5230			−1.56341			−0.781707	+	0.623646i	$0.785651\pi$
$812$		0			0
$813$			20.2396i			0.709835i
$814$		0			0
$815$	−14.3297	−	9.54836i	−0.501946	−	0.334464i
$816$		0			0
$817$			12.9041i			0.451456i
$818$		0			0
$819$	−2.18271			−0.0762700
$820$		0			0
$821$	−34.1613			−1.19224			−0.596118	−	0.802897i	$-0.703291\pi$
$821$	−34.1613			−1.19224			−0.596118	+	0.802897i	$0.703291\pi$
$822$		0			0
$823$		−	2.12689i		−	0.0741388i	−0.999313	−	0.0370694i	$-0.988198\pi$
$823$		−	2.12689i		−	0.0741388i	0.999313	−	0.0370694i	$-0.0118023\pi$
$824$		0			0
$825$	−24.7368	+	10.3204i	−0.861226	+	0.359309i
$826$		0			0
$827$			38.5872i			1.34181i	0.741543	+	0.670905i	$0.234094\pi$
$827$			38.5872i			1.34181i	−0.741543	+	0.670905i	$0.765906\pi$
$828$		0			0
$829$	34.2351			1.18904			0.594518	−	0.804083i	$-0.297343\pi$
$829$	34.2351			1.18904			0.594518	+	0.804083i	$0.297343\pi$
$830$		0			0
$831$	0.518027			0.0179701
$832$		0			0
$833$			13.7541i			0.476551i
$834$		0			0
$835$	6.00299	+	4.00000i	0.207742	+	0.138426i
$836$		0			0
$837$			2.64681i			0.0914871i
$838$		0			0
$839$	−41.5928			−1.43594			−0.717972	−	0.696072i	$-0.754929\pi$
$839$	−41.5928			−1.43594			−0.717972	+	0.696072i	$0.754929\pi$
$840$		0			0
$841$	16.5512			0.570730
$842$		0			0
$843$			13.7008i			0.471880i
$844$		0			0
$845$	5.50924	−	8.26798i	0.189524	−	0.284427i
$846$		0			0
$847$		−	13.2348i		−	0.454752i
$848$		0			0
$849$	18.0305			0.618804
$850$		0			0
$851$	8.05601			0.276156
$852$		0			0
$853$			23.1828i			0.793763i	0.917870	+	0.396881i	$0.129908\pi$
$853$			23.1828i			0.793763i	−0.917870	+	0.396881i	$0.870092\pi$
$854$		0			0
$855$	−2.14961	+	3.22601i	−0.0735150	+	0.110327i
$856$		0			0
$857$		−	9.38868i		−	0.320711i	−0.987059	−	0.160356i	$-0.948736\pi$
$857$		−	9.38868i		−	0.320711i	0.987059	−	0.160356i	$-0.0512641\pi$
$858$		0			0
$859$	−8.98769			−0.306656			−0.153328	−	0.988175i	$-0.548999\pi$
$859$	−8.98769			−0.306656			−0.153328	+	0.988175i	$0.548999\pi$
$860$		0			0
$861$	−8.42701			−0.287192
$862$		0			0
$863$			12.2473i			0.416903i	0.978033	+	0.208451i	$0.0668423\pi$
$863$			12.2473i			0.416903i	−0.978033	+	0.208451i	$0.933158\pi$
$864$		0			0
$865$	11.9612	+	7.97020i	0.406695	+	0.270995i
$866$		0			0
$867$		−	12.4432i		−	0.422594i
$868$		0			0
$869$	−14.1887			−0.481318
$870$		0			0
$871$	−21.7729			−0.737746
$872$		0			0
$873$			14.1887i			0.480214i
$874$		0			0
$875$	−1.59621	+	8.18836i	−0.0539618	+	0.276817i
$876$		0			0
$877$		−	26.1109i		−	0.881701i	−0.897581	−	0.440850i	$-0.854677\pi$
$877$		−	26.1109i		−	0.881701i	0.897581	−	0.440850i	$-0.145323\pi$
$878$		0			0
$879$	15.9792			0.538964
$880$		0			0
$881$	−38.4793			−1.29640			−0.648200	−	0.761470i	$-0.724478\pi$
$881$	−38.4793			−1.29640			−0.648200	+	0.761470i	$0.724478\pi$
$882$		0			0
$883$			6.58723i			0.221678i	0.993838	+	0.110839i	$0.0353538\pi$
$883$			6.58723i			0.221678i	−0.993838	+	0.110839i	$0.964646\pi$
$884$		0			0
$885$	12.7521	+	8.49720i	0.428659	+	0.285630i
$886$		0			0
$887$		−	50.9595i		−	1.71105i	−0.517760	−	0.855526i	$-0.673234\pi$
$887$		−	50.9595i		−	1.71105i	0.517760	−	0.855526i	$-0.326766\pi$
$888$		0			0
$889$	−3.14401			−0.105447
$890$		0			0
$891$	5.36068			0.179589
$892$		0			0
$893$			3.00560i			0.100578i
$894$		0			0
$895$	10.0900	−	15.1426i	0.337273	−	0.506161i
$896$		0			0
$897$			21.9253i			0.732066i
$898$		0			0
$899$	17.8637			0.595789
$900$		0			0
$901$	−16.4830			−0.549129
$902$		0			0
$903$		−	5.55394i		−	0.184824i
$904$		0			0
$905$	−1.85039	+	2.77697i	−0.0615092	+	0.0923097i
$906$		0			0
$907$		−	24.5568i		−	0.815394i	−0.913117	−	0.407697i	$-0.866332\pi$
$907$		−	24.5568i		−	0.815394i	0.913117	−	0.407697i	$-0.133668\pi$
$908$		0			0
$909$	7.43952			0.246753
$910$		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
$913$			31.3621i			1.03793i
$914$		0			0
$915$	−12.0060	−	8.00000i	−0.396905	−	0.264472i
$916$		0			0
$917$			7.70079i			0.254302i
$918$		0			0
$919$	28.7548			0.948532			0.474266	−	0.880382i	$-0.342713\pi$
$919$	28.7548			0.948532			0.474266	+	0.880382i	$0.342713\pi$
$920$		0			0
$921$	22.5872			0.744275
$922$		0			0
$923$		−	38.8864i		−	1.27996i
$924$		0			0
$925$	−4.95968	+	2.06921i	−0.163073	+	0.0680351i
$926$		0			0
$927$		−	7.19820i		−	0.236420i
$928$		0			0
$929$	−28.2880			−0.928100			−0.464050	−	0.885809i	$-0.653604\pi$
$929$	−28.2880			−0.928100			−0.464050	+	0.885809i	$0.653604\pi$
$930$		0			0
$931$	11.1704			0.366095
$932$		0			0
$933$		−	18.5872i		−	0.608519i
$934$		0			0
$935$	21.2936	+	14.1887i	0.696376	+	0.464020i
$936$		0			0
$937$			33.9313i			1.10849i	0.832354	+	0.554244i	$0.186992\pi$
$937$			33.9313i			1.10849i	−0.832354	+	0.554244i	$0.813008\pi$
$938$		0			0
$939$	−29.3871			−0.959011
$940$		0			0
$941$	38.8016			1.26490			0.632448	−	0.774603i	$-0.282050\pi$
$941$	38.8016			1.26490			0.632448	+	0.774603i	$0.282050\pi$
$942$		0			0
$943$			84.6495i			2.75657i
$944$		0			0
$945$	0.925197	−	1.38849i	0.0300967	−	0.0451675i
$946$		0			0
$947$		−	17.7729i		−	0.577541i	−0.957398	−	0.288771i	$-0.906753\pi$
$947$		−	17.7729i		−	0.577541i	0.957398	−	0.288771i	$-0.0932465\pi$
$948$		0			0
$949$	2.01943			0.0655536
$950$		0			0
$951$	5.57201			0.180685
$952$		0			0
$953$		−	46.3047i		−	1.49996i	−0.661463	−	0.749978i	$-0.730064\pi$
$953$		−	46.3047i		−	1.49996i	0.661463	−	0.749978i	$-0.269936\pi$
$954$		0			0
$955$	8.53864	−	12.8143i	0.276304	−	0.414662i
$956$		0			0
$957$		−	36.1801i		−	1.16954i
$958$		0			0
$959$	−11.2215			−0.362361
$960$		0			0
$961$	−23.9944			−0.774013
$962$		0			0
$963$			4.00000i			0.128898i
$964$		0			0
$965$	30.6717	+	20.4376i	0.987357	+	0.657911i
$966$		0			0
$967$		−	9.28482i		−	0.298580i	−0.988793	−	0.149290i	$-0.952301\pi$
$967$		−	9.28482i		−	0.298580i	0.988793	−	0.149290i	$-0.0476987\pi$
$968$		0			0
$969$	3.70079			0.118886
$970$		0			0
$971$	20.9301			0.671678			0.335839	−	0.941919i	$-0.390980\pi$
$971$	20.9301			0.671678			0.335839	+	0.941919i	$0.390980\pi$
$972$		0			0
$973$		−	7.06651i		−	0.226542i
$974$		0			0
$975$	−5.63158	−	13.4983i	−0.180355	−	0.432292i
$976$		0			0
$977$		−	30.8314i		−	0.986383i	−0.869921	−	0.493192i	$-0.835830\pi$
$977$		−	30.8314i		−	0.986383i	0.869921	−	0.493192i	$-0.164170\pi$
$978$		0			0
$979$	40.7027			1.30086
$980$		0			0
$981$	19.9504			0.636966
$982$		0			0
$983$			31.3285i			0.999223i	0.866250	+	0.499612i	$0.166524\pi$
$983$			31.3285i			0.999223i	−0.866250	+	0.499612i	$0.833476\pi$
$984$		0			0
$985$	−25.2549	−	16.8282i	−0.804687	−	0.536191i
$986$		0			0
$987$		−	1.29362i		−	0.0411763i
$988$		0			0
$989$	−55.7895			−1.77400
$990$		0			0
$991$	31.3420			0.995611			0.497806	−	0.867289i	$-0.334139\pi$
$991$	31.3420			0.995611			0.497806	+	0.867289i	$0.334139\pi$
$992$		0			0
$993$		−	13.7396i		−	0.436014i
$994$		0			0
$995$	11.2262	−	16.8477i	0.355895	−	0.534108i
$996$		0			0
$997$			20.9557i			0.663672i	0.943337	+	0.331836i	$0.107668\pi$
$997$			20.9557i			0.663672i	−0.943337	+	0.331836i	$0.892332\pi$
$998$		0			0
$999$	1.07480			0.0340053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.f.l.769.5		12
4.3	odd	2		3840.2.f.m.769.11		12
5.4	even	2	inner	3840.2.f.l.769.11		12
8.3	odd	2		3840.2.f.m.769.2		12
8.5	even	2	inner	3840.2.f.l.769.8		12
16.3	odd	4		480.2.d.a.49.1		6
16.5	even	4		120.2.d.b.109.2	yes	6
16.11	odd	4		480.2.d.b.49.6		6
16.13	even	4		120.2.d.a.109.6	yes	6
20.19	odd	2		3840.2.f.m.769.5		12
40.19	odd	2		3840.2.f.m.769.8		12
40.29	even	2	inner	3840.2.f.l.769.2		12
48.5	odd	4		360.2.d.e.109.5		6
48.11	even	4		1440.2.d.f.1009.1		6
48.29	odd	4		360.2.d.f.109.1		6
48.35	even	4		1440.2.d.e.1009.6		6
80.3	even	4		2400.2.k.f.1201.3		12
80.13	odd	4		600.2.k.f.301.9		12
80.19	odd	4		480.2.d.b.49.5		6
80.27	even	4		2400.2.k.f.1201.4		12
80.29	even	4		120.2.d.b.109.1	yes	6
80.37	odd	4		600.2.k.f.301.3		12
80.43	even	4		2400.2.k.f.1201.9		12
80.53	odd	4		600.2.k.f.301.10		12
80.59	odd	4		480.2.d.a.49.2		6
80.67	even	4		2400.2.k.f.1201.10		12
80.69	even	4		120.2.d.a.109.5	✓	6
80.77	odd	4		600.2.k.f.301.4		12
240.29	odd	4		360.2.d.e.109.6		6
240.53	even	4		1800.2.k.u.901.3		12
240.59	even	4		1440.2.d.e.1009.5		6
240.77	even	4		1800.2.k.u.901.9		12
240.83	odd	4		7200.2.k.u.3601.5		12
240.107	odd	4		7200.2.k.u.3601.8		12
240.149	odd	4		360.2.d.f.109.2		6
240.173	even	4		1800.2.k.u.901.4		12
240.179	even	4		1440.2.d.f.1009.2		6
240.197	even	4		1800.2.k.u.901.10		12
240.203	odd	4		7200.2.k.u.3601.6		12
240.227	odd	4		7200.2.k.u.3601.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.d.a.109.5	✓	6	80.69	even	4
120.2.d.a.109.6	yes	6	16.13	even	4
120.2.d.b.109.1	yes	6	80.29	even	4
120.2.d.b.109.2	yes	6	16.5	even	4
360.2.d.e.109.5		6	48.5	odd	4
360.2.d.e.109.6		6	240.29	odd	4
360.2.d.f.109.1		6	48.29	odd	4
360.2.d.f.109.2		6	240.149	odd	4
480.2.d.a.49.1		6	16.3	odd	4
480.2.d.a.49.2		6	80.59	odd	4
480.2.d.b.49.5		6	80.19	odd	4
480.2.d.b.49.6		6	16.11	odd	4
600.2.k.f.301.3		12	80.37	odd	4
600.2.k.f.301.4		12	80.77	odd	4
600.2.k.f.301.9		12	80.13	odd	4
600.2.k.f.301.10		12	80.53	odd	4
1440.2.d.e.1009.5		6	240.59	even	4
1440.2.d.e.1009.6		6	48.35	even	4
1440.2.d.f.1009.1		6	48.11	even	4
1440.2.d.f.1009.2		6	240.179	even	4
1800.2.k.u.901.3		12	240.53	even	4
1800.2.k.u.901.4		12	240.173	even	4
1800.2.k.u.901.9		12	240.77	even	4
1800.2.k.u.901.10		12	240.197	even	4
2400.2.k.f.1201.3		12	80.3	even	4
2400.2.k.f.1201.4		12	80.27	even	4
2400.2.k.f.1201.9		12	80.43	even	4
2400.2.k.f.1201.10		12	80.67	even	4
3840.2.f.l.769.2		12	40.29	even	2	inner
3840.2.f.l.769.5		12	1.1	even	1	trivial
3840.2.f.l.769.8		12	8.5	even	2	inner
3840.2.f.l.769.11		12	5.4	even	2	inner
3840.2.f.m.769.2		12	8.3	odd	2
3840.2.f.m.769.5		12	20.19	odd	2
3840.2.f.m.769.8		12	40.19	odd	2
3840.2.f.m.769.11		12	4.3	odd	2
7200.2.k.u.3601.5		12	240.83	odd	4
7200.2.k.u.3601.6		12	240.203	odd	4
7200.2.k.u.3601.7		12	240.227	odd	4
7200.2.k.u.3601.8		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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(1.23992 - 1.86081i) q^{5} -0.746175i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(1.23992 - 1.86081i) q^{5} -0.746175i q^{7} -1.00000 q^{9} +5.36068 q^{11} +2.92520i q^{13} +(-1.86081 - 1.23992i) q^{15} +2.13466i q^{17} +1.73367 q^{19} -0.746175 q^{21} +7.49534i q^{23} +(-1.92520 - 4.61450i) q^{25} +1.00000i q^{27} +6.74916 q^{29} +2.64681 q^{31} -5.36068i q^{33} +(-1.38849 - 0.925197i) q^{35} -1.07480i q^{37} +2.92520 q^{39} +11.2936 q^{41} +7.44322i q^{43} +(-1.23992 + 1.86081i) q^{45} +1.73367i q^{47} +6.44322 q^{49} +2.13466 q^{51} +7.72161i q^{53} +(6.64681 - 9.97518i) q^{55} -1.73367i q^{57} -6.85302 q^{59} +6.45203 q^{61} +0.746175i q^{63} +(5.44322 + 3.62701i) q^{65} +7.44322i q^{67} +7.49534 q^{69} -13.2936 q^{71} -0.690358i q^{73} +(-4.61450 + 1.92520i) q^{75} -4.00000i q^{77} -2.64681 q^{79} +1.00000 q^{81} +5.85039i q^{83} +(3.97219 + 2.64681i) q^{85} -6.74916i q^{87} +7.59283 q^{89} +2.18271 q^{91} -2.64681i q^{93} +(2.14961 - 3.22601i) q^{95} -14.1887i q^{97} -5.36068 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} - 4 q^{25} - 32 q^{31} + 16 q^{39} + 8 q^{41} - 12 q^{49} + 16 q^{55} - 24 q^{65} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89} + 64 q^{95}+O(q^{100}) \) 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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