LMFDB - Embedded newform 3840.2.f.i.769.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3840,2,Mod(769,3840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [8,0,0,0,0,0,0,0,-8,0,-8,0,0,0,0,0,0,0,-32,0,0,0,0,0,0,0,0,0, 0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 960)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.5
Root			$0.228425 - 1.39564i$ of defining polynomial
Character	$\chi$	$=$	3840.769
Dual form			3840.2.f.i.769.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-2.18890 - 0.456850i) q^{5} -4.37780i q^{7} -1.00000 q^{9} -5.58258 q^{11} +4.37780i q^{13} +(0.456850 - 2.18890i) q^{15} -5.58258i q^{17} -4.00000 q^{19} +4.37780 q^{21} +(4.58258 + 2.00000i) q^{25} -1.00000i q^{27} -2.55040 q^{29} +5.29150 q^{31} -5.58258i q^{33} +(-2.00000 + 9.58258i) q^{35} -2.55040i q^{37} -4.37780 q^{39} -6.00000 q^{41} -11.1652i q^{43} +(2.18890 + 0.456850i) q^{45} +6.92820i q^{47} -12.1652 q^{49} +5.58258 q^{51} +7.84190i q^{53} +(12.2197 + 2.55040i) q^{55} -4.00000i q^{57} -1.58258 q^{59} +10.5830 q^{61} +4.37780i q^{63} +(2.00000 - 9.58258i) q^{65} +3.16515i q^{67} +6.92820 q^{71} +12.0000i q^{73} +(-2.00000 + 4.58258i) q^{75} +24.4394i q^{77} -5.29150 q^{79} +1.00000 q^{81} -7.16515i q^{83} +(-2.55040 + 12.2197i) q^{85} -2.55040i q^{87} +2.00000 q^{89} +19.1652 q^{91} +5.29150i q^{93} +(8.75560 + 1.82740i) q^{95} +11.1652i q^{97} +5.58258 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} - 8 q^{11} - 32 q^{19} - 16 q^{35} - 48 q^{41} - 24 q^{49} + 8 q^{51} + 24 q^{59} + 16 q^{65} - 16 q^{75} + 8 q^{81} + 16 q^{89} + 80 q^{91} + 8 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 8 * q^11 - 32 * q^19 - 16 * q^35 - 48 * q^41 - 24 * q^49 + 8 * q^51 + 24 * q^59 + 16 * q^65 - 16 * q^75 + 8 * q^81 + 16 * q^89 + 80 * q^91 + 8 * q^99

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
$5$	−2.18890	−	0.456850i	−0.978906	−	0.204310i
$5$	−2.18890	−	0.456850i	−0.978906	−	0.204310i
$6$		0			0
$7$		−	4.37780i		−	1.65465i	−0.561721	−	0.827327i	$-0.689860\pi$
$7$		−	4.37780i		−	1.65465i	0.561721	−	0.827327i	$-0.310140\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−5.58258			−1.68321			−0.841605	−	0.540094i	$-0.818389\pi$
$11$	−5.58258			−1.68321			−0.841605	+	0.540094i	$0.818389\pi$
$12$		0			0
$13$			4.37780i			1.21418i	0.794632	+	0.607092i	$0.207664\pi$
$13$			4.37780i			1.21418i	−0.794632	+	0.607092i	$0.792336\pi$
$14$		0			0
$15$	0.456850	−	2.18890i	0.117958	−	0.565172i
$16$		0			0
$17$		−	5.58258i		−	1.35397i	−0.735995	−	0.676987i	$-0.763285\pi$
$17$		−	5.58258i		−	1.35397i	0.735995	−	0.676987i	$-0.236715\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	4.37780			0.955315
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	4.58258	+	2.00000i	0.916515	+	0.400000i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−2.55040			−0.473598			−0.236799	−	0.971559i	$-0.576098\pi$
$29$	−2.55040			−0.473598			−0.236799	+	0.971559i	$0.576098\pi$
$30$		0			0
$31$	5.29150			0.950382			0.475191	−	0.879883i	$-0.342379\pi$
$31$	5.29150			0.950382			0.475191	+	0.879883i	$0.342379\pi$
$32$		0			0
$33$		−	5.58258i		−	0.971802i
$34$		0			0
$35$	−2.00000	+	9.58258i	−0.338062	+	1.61975i
$36$		0			0
$37$		−	2.55040i		−	0.419283i	−0.977778	−	0.209642i	$-0.932770\pi$
$37$		−	2.55040i		−	0.419283i	0.977778	−	0.209642i	$-0.0672297\pi$
$38$		0			0
$39$	−4.37780			−0.701009
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$		−	11.1652i		−	1.70267i	−0.524623	−	0.851335i	$-0.675794\pi$
$43$		−	11.1652i		−	1.70267i	0.524623	−	0.851335i	$-0.324206\pi$
$44$		0			0
$45$	2.18890	+	0.456850i	0.326302	+	0.0681032i
$46$		0			0
$47$			6.92820i			1.01058i	0.862949	+	0.505291i	$0.168615\pi$
$47$			6.92820i			1.01058i	−0.862949	+	0.505291i	$0.831385\pi$
$48$		0			0
$49$	−12.1652			−1.73788
$50$		0			0
$51$	5.58258			0.781717
$52$		0			0
$53$			7.84190i			1.07717i	0.842572	+	0.538584i	$0.181041\pi$
$53$			7.84190i			1.07717i	−0.842572	+	0.538584i	$0.818959\pi$
$54$		0			0
$55$	12.2197	+	2.55040i	1.64770	+	0.343896i
$56$		0			0
$57$		−	4.00000i		−	0.529813i
$58$		0			0
$59$	−1.58258			−0.206034			−0.103017	−	0.994680i	$-0.532850\pi$
$59$	−1.58258			−0.206034			−0.103017	+	0.994680i	$0.532850\pi$
$60$		0			0
$61$	10.5830			1.35501			0.677507	−	0.735516i	$-0.263060\pi$
$61$	10.5830			1.35501			0.677507	+	0.735516i	$0.263060\pi$
$62$		0			0
$63$			4.37780i			0.551551i
$64$		0			0
$65$	2.00000	−	9.58258i	0.248069	−	1.18857i
$66$		0			0
$67$			3.16515i			0.386685i	0.981131	+	0.193342i	$0.0619328\pi$
$67$			3.16515i			0.386685i	−0.981131	+	0.193342i	$0.938067\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.92820			0.822226			0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820			0.822226			0.411113	+	0.911584i	$0.365140\pi$
$72$		0			0
$73$			12.0000i			1.40449i	0.711934	+	0.702247i	$0.247820\pi$
$73$			12.0000i			1.40449i	−0.711934	+	0.702247i	$0.752180\pi$
$74$		0			0
$75$	−2.00000	+	4.58258i	−0.230940	+	0.529150i
$76$		0			0
$77$			24.4394i			2.78513i
$78$		0			0
$79$	−5.29150			−0.595341			−0.297670	−	0.954669i	$-0.596210\pi$
$79$	−5.29150			−0.595341			−0.297670	+	0.954669i	$0.596210\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	7.16515i		−	0.786478i	−0.919436	−	0.393239i	$-0.871355\pi$
$83$		−	7.16515i		−	0.786478i	0.919436	−	0.393239i	$-0.128645\pi$
$84$		0			0
$85$	−2.55040	+	12.2197i	−0.276630	+	1.32541i
$86$		0			0
$87$		−	2.55040i		−	0.273432i
$88$		0			0
$89$	2.00000			0.212000			0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000			0.212000			0.106000	+	0.994366i	$0.466196\pi$
$90$		0			0
$91$	19.1652			2.00905
$92$		0			0
$93$			5.29150i			0.548703i
$94$		0			0
$95$	8.75560	+	1.82740i	0.898306	+	0.187487i
$96$		0			0
$97$			11.1652i			1.13365i	0.823838	+	0.566825i	$0.191828\pi$
$97$			11.1652i			1.13365i	−0.823838	+	0.566825i	$0.808172\pi$
$98$		0			0
$99$	5.58258			0.561070
$100$		0			0
$101$	−0.723000			−0.0719412			−0.0359706	−	0.999353i	$-0.511452\pi$
$101$	−0.723000			−0.0719412			−0.0359706	+	0.999353i	$0.511452\pi$
$102$		0			0
$103$			0.723000i			0.0712393i	0.999365	+	0.0356197i	$0.0113405\pi$
$103$			0.723000i			0.0712393i	−0.999365	+	0.0356197i	$0.988660\pi$
$104$		0			0
$105$	−9.58258	−	2.00000i	−0.935164	−	0.195180i
$106$		0			0
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$	1.82740			0.175033			0.0875166	−	0.996163i	$-0.472107\pi$
$109$	1.82740			0.175033			0.0875166	+	0.996163i	$0.472107\pi$
$110$		0			0
$111$	2.55040			0.242073
$112$		0			0
$113$			13.5826i			1.27774i	0.769314	+	0.638871i	$0.220598\pi$
$113$			13.5826i			1.27774i	−0.769314	+	0.638871i	$0.779402\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	4.37780i		−	0.404728i
$118$		0			0
$119$	−24.4394			−2.24036
$120$		0			0
$121$	20.1652			1.83320
$122$		0			0
$123$		−	6.00000i		−	0.541002i
$124$		0			0
$125$	−9.11710	−	6.47135i	−0.815459	−	0.578815i
$126$		0			0
$127$			9.47860i			0.841090i	0.907272	+	0.420545i	$0.138161\pi$
$127$			9.47860i			0.841090i	−0.907272	+	0.420545i	$0.861839\pi$
$128$		0			0
$129$	11.1652			0.983037
$130$		0			0
$131$	6.41742			0.560693			0.280346	−	0.959899i	$-0.409551\pi$
$131$	6.41742			0.560693			0.280346	+	0.959899i	$0.409551\pi$
$132$		0			0
$133$			17.5112i			1.51841i
$134$		0			0
$135$	−0.456850	+	2.18890i	−0.0393194	+	0.188391i
$136$		0			0
$137$		−	10.4174i		−	0.890021i	−0.895525	−	0.445010i	$-0.853200\pi$
$137$		−	10.4174i		−	0.890021i	0.895525	−	0.445010i	$-0.146800\pi$
$138$		0			0
$139$	7.16515			0.607740			0.303870	−	0.952713i	$-0.401721\pi$
$139$	7.16515			0.607740			0.303870	+	0.952713i	$0.401721\pi$
$140$		0			0
$141$	−6.92820			−0.583460
$142$		0			0
$143$		−	24.4394i		−	2.04373i
$144$		0			0
$145$	5.58258	+	1.16515i	0.463608	+	0.0967606i
$146$		0			0
$147$		−	12.1652i		−	1.00336i
$148$		0			0
$149$	14.9608			1.22564			0.612819	−	0.790224i	$-0.290036\pi$
$149$	14.9608			1.22564			0.612819	+	0.790224i	$0.290036\pi$
$150$		0			0
$151$	15.8745			1.29185			0.645925	−	0.763401i	$-0.276472\pi$
$151$	15.8745			1.29185			0.645925	+	0.763401i	$0.276472\pi$
$152$		0			0
$153$			5.58258i			0.451324i
$154$		0			0
$155$	−11.5826	−	2.41742i	−0.930335	−	0.194172i
$156$		0			0
$157$			16.4068i			1.30941i	0.755886	+	0.654703i	$0.227206\pi$
$157$			16.4068i			1.30941i	−0.755886	+	0.654703i	$0.772794\pi$
$158$		0			0
$159$	−7.84190			−0.621903
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$	−2.55040	+	12.2197i	−0.198548	+	0.951303i
$166$		0			0
$167$		−	12.0290i		−	0.930832i	−0.885092	−	0.465416i	$-0.845905\pi$
$167$		−	12.0290i		−	0.930832i	0.885092	−	0.465416i	$-0.154095\pi$
$168$		0			0
$169$	−6.16515			−0.474242
$170$		0			0
$171$	4.00000			0.305888
$172$		0			0
$173$		−	6.01450i		−	0.457274i	−0.973512	−	0.228637i	$-0.926573\pi$
$173$		−	6.01450i		−	0.457274i	0.973512	−	0.228637i	$-0.0734269\pi$
$174$		0			0
$175$	8.75560	−	20.0616i	0.661861	−	1.51652i
$176$		0			0
$177$		−	1.58258i		−	0.118954i
$178$		0			0
$179$	−1.58258			−0.118287			−0.0591436	−	0.998249i	$-0.518837\pi$
$179$	−1.58258			−0.118287			−0.0591436	+	0.998249i	$0.518837\pi$
$180$		0			0
$181$	1.82740			0.135830			0.0679148	−	0.997691i	$-0.478365\pi$
$181$	1.82740			0.135830			0.0679148	+	0.997691i	$0.478365\pi$
$182$		0			0
$183$			10.5830i			0.782318i
$184$		0			0
$185$	−1.16515	+	5.58258i	−0.0856636	+	0.410439i
$186$		0			0
$187$			31.1652i			2.27902i
$188$		0			0
$189$	−4.37780			−0.318438
$190$		0			0
$191$	−13.8564			−1.00261			−0.501307	−	0.865269i	$-0.667147\pi$
$191$	−13.8564			−1.00261			−0.501307	+	0.865269i	$0.667147\pi$
$192$		0			0
$193$			15.1652i			1.09161i	0.837912	+	0.545806i	$0.183776\pi$
$193$			15.1652i			1.09161i	−0.837912	+	0.545806i	$0.816224\pi$
$194$		0			0
$195$	9.58258	+	2.00000i	0.686222	+	0.143223i
$196$		0			0
$197$			22.0797i			1.57311i	0.617518	+	0.786557i	$0.288138\pi$
$197$			22.0797i			1.57311i	−0.617518	+	0.786557i	$0.711862\pi$
$198$		0			0
$199$	12.2197			0.866232			0.433116	−	0.901338i	$-0.357414\pi$
$199$	12.2197			0.866232			0.433116	+	0.901338i	$0.357414\pi$
$200$		0			0
$201$	−3.16515			−0.223253
$202$		0			0
$203$			11.1652i			0.783640i
$204$		0			0
$205$	13.1334	+	2.74110i	0.917277	+	0.191447i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	22.3303			1.54462
$210$		0			0
$211$	−7.16515			−0.493269			−0.246635	−	0.969109i	$-0.579325\pi$
$211$	−7.16515			−0.493269			−0.246635	+	0.969109i	$0.579325\pi$
$212$		0			0
$213$			6.92820i			0.474713i
$214$		0			0
$215$	−5.10080	+	24.4394i	−0.347872	+	1.66675i
$216$		0			0
$217$		−	23.1652i		−	1.57255i
$218$		0			0
$219$	−12.0000			−0.810885
$220$		0			0
$221$	24.4394			1.64397
$222$		0			0
$223$			4.37780i			0.293159i	0.989199	+	0.146580i	$0.0468265\pi$
$223$			4.37780i			0.293159i	−0.989199	+	0.146580i	$0.953174\pi$
$224$		0			0
$225$	−4.58258	−	2.00000i	−0.305505	−	0.133333i
$226$		0			0
$227$		−	15.1652i		−	1.00655i	−0.864127	−	0.503273i	$-0.832129\pi$
$227$		−	15.1652i		−	1.00655i	0.864127	−	0.503273i	$-0.167871\pi$
$228$		0			0
$229$	−12.0290			−0.794899			−0.397450	−	0.917624i	$-0.630105\pi$
$229$	−12.0290			−0.794899			−0.397450	+	0.917624i	$0.630105\pi$
$230$		0			0
$231$	−24.4394			−1.60800
$232$		0			0
$233$			10.4174i			0.682468i	0.939978	+	0.341234i	$0.110845\pi$
$233$			10.4174i			0.682468i	−0.939978	+	0.341234i	$0.889155\pi$
$234$		0			0
$235$	3.16515	−	15.1652i	0.206472	−	0.989265i
$236$		0			0
$237$		−	5.29150i		−	0.343720i
$238$		0			0
$239$	−21.1660			−1.36912			−0.684558	−	0.728959i	$-0.740005\pi$
$239$	−21.1660			−1.36912			−0.684558	+	0.728959i	$0.740005\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	26.6283	+	5.55765i	1.70122	+	0.355065i
$246$		0			0
$247$		−	17.5112i		−	1.11421i
$248$		0			0
$249$	7.16515			0.454073
$250$		0			0
$251$	−2.41742			−0.152586			−0.0762932	−	0.997085i	$-0.524309\pi$
$251$	−2.41742			−0.152586			−0.0762932	+	0.997085i	$0.524309\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−12.2197	−	2.55040i	−0.765228	−	0.159712i
$256$		0			0
$257$			24.7477i			1.54372i	0.635792	+	0.771860i	$0.280674\pi$
$257$			24.7477i			1.54372i	−0.635792	+	0.771860i	$0.719326\pi$
$258$		0			0
$259$	−11.1652			−0.693769
$260$		0			0
$261$	2.55040			0.157866
$262$		0			0
$263$			1.82740i			0.112682i	0.998412	+	0.0563412i	$0.0179435\pi$
$263$			1.82740i			0.112682i	−0.998412	+	0.0563412i	$0.982057\pi$
$264$		0			0
$265$	3.58258	−	17.1652i	0.220076	−	1.05445i
$266$		0			0
$267$			2.00000i			0.122398i
$268$		0			0
$269$	6.20520			0.378338			0.189169	−	0.981945i	$-0.439421\pi$
$269$	6.20520			0.378338			0.189169	+	0.981945i	$0.439421\pi$
$270$		0			0
$271$	15.4931			0.941139			0.470570	−	0.882363i	$-0.344048\pi$
$271$	15.4931			0.941139			0.470570	+	0.882363i	$0.344048\pi$
$272$		0			0
$273$			19.1652i			1.15993i
$274$		0			0
$275$	−25.5826	−	11.1652i	−1.54269	−	0.673284i
$276$		0			0
$277$		−	9.47860i		−	0.569514i	−0.958600	−	0.284757i	$-0.908087\pi$
$277$		−	9.47860i		−	0.569514i	0.958600	−	0.284757i	$-0.0919129\pi$
$278$		0			0
$279$	−5.29150			−0.316794
$280$		0			0
$281$	−32.3303			−1.92866			−0.964332	−	0.264695i	$-0.914729\pi$
$281$	−32.3303			−1.92866			−0.964332	+	0.264695i	$0.914729\pi$
$282$		0			0
$283$			11.1652i			0.663699i	0.943332	+	0.331850i	$0.107673\pi$
$283$			11.1652i			0.663699i	−0.943332	+	0.331850i	$0.892327\pi$
$284$		0			0
$285$	−1.82740	+	8.75560i	−0.108246	+	0.518637i
$286$		0			0
$287$			26.2668i			1.55048i
$288$		0			0
$289$	−14.1652			−0.833244
$290$		0			0
$291$	−11.1652			−0.654513
$292$		0			0
$293$		−	0.913701i		−	0.0533790i	−0.999644	−	0.0266895i	$-0.991503\pi$
$293$		−	0.913701i		−	0.0533790i	0.999644	−	0.0266895i	$-0.00849653\pi$
$294$		0			0
$295$	3.46410	+	0.723000i	0.201688	+	0.0420947i
$296$		0			0
$297$			5.58258i			0.323934i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−48.8788			−2.81733
$302$		0			0
$303$		−	0.723000i		−	0.0415353i
$304$		0			0
$305$	−23.1652	−	4.83485i	−1.32643	−	0.276843i
$306$		0			0
$307$		−	4.00000i		−	0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$		−	4.00000i		−	0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$		0			0
$309$	−0.723000			−0.0411300
$310$		0			0
$311$	−3.65480			−0.207245			−0.103622	−	0.994617i	$-0.533043\pi$
$311$	−3.65480			−0.207245			−0.103622	+	0.994617i	$0.533043\pi$
$312$		0			0
$313$			0.834849i			0.0471884i	0.999722	+	0.0235942i	$0.00751097\pi$
$313$			0.834849i			0.0471884i	−0.999722	+	0.0235942i	$0.992489\pi$
$314$		0			0
$315$	2.00000	−	9.58258i	0.112687	−	0.539917i
$316$		0			0
$317$			34.1087i			1.91574i	0.287208	+	0.957868i	$0.407273\pi$
$317$			34.1087i			1.91574i	−0.287208	+	0.957868i	$0.592727\pi$
$318$		0			0
$319$	14.2378			0.797164
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$			22.3303i			1.24249i
$324$		0			0
$325$	−8.75560	+	20.0616i	−0.485674	+	1.11282i
$326$		0			0
$327$			1.82740i			0.101056i
$328$		0			0
$329$	30.3303			1.67216
$330$		0			0
$331$	−10.3303			−0.567805			−0.283902	−	0.958853i	$-0.591629\pi$
$331$	−10.3303			−0.567805			−0.283902	+	0.958853i	$0.591629\pi$
$332$		0			0
$333$			2.55040i			0.139761i
$334$		0			0
$335$	1.44600	−	6.92820i	0.0790034	−	0.378528i
$336$		0			0
$337$		−	26.3303i		−	1.43430i	−0.696917	−	0.717151i	$-0.745446\pi$
$337$		−	26.3303i		−	1.43430i	0.696917	−	0.717151i	$-0.254554\pi$
$338$		0			0
$339$	−13.5826			−0.737704
$340$		0			0
$341$	−29.5402			−1.59969
$342$		0			0
$343$			22.6120i			1.22093i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			0.834849i			0.0448170i	0.999749	+	0.0224085i	$0.00713345\pi$
$347$			0.834849i			0.0448170i	−0.999749	+	0.0224085i	$0.992867\pi$
$348$		0			0
$349$	24.4394			1.30821			0.654106	−	0.756403i	$-0.273045\pi$
$349$	24.4394			1.30821			0.654106	+	0.756403i	$0.273045\pi$
$350$		0			0
$351$	4.37780			0.233670
$352$		0			0
$353$		−	11.9129i		−	0.634059i	−0.948416	−	0.317029i	$-0.897315\pi$
$353$		−	11.9129i		−	0.634059i	0.948416	−	0.317029i	$-0.102685\pi$
$354$		0			0
$355$	−15.1652	−	3.16515i	−0.804883	−	0.167989i
$356$		0			0
$357$		−	24.4394i		−	1.29347i
$358$		0			0
$359$	6.92820			0.365657			0.182828	−	0.983145i	$-0.441475\pi$
$359$	6.92820			0.365657			0.182828	+	0.983145i	$0.441475\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$			20.1652i			1.05840i
$364$		0			0
$365$	5.48220	−	26.2668i	0.286952	−	1.37487i
$366$		0			0
$367$		−	11.6874i		−	0.610078i	−0.952340	−	0.305039i	$-0.901331\pi$
$367$		−	11.6874i		−	0.610078i	0.952340	−	0.305039i	$-0.0986694\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$	34.3303			1.78234
$372$		0			0
$373$			11.3060i			0.585403i	0.956204	+	0.292701i	$0.0945542\pi$
$373$			11.3060i			0.585403i	−0.956204	+	0.292701i	$0.905446\pi$
$374$		0			0
$375$	6.47135	−	9.11710i	0.334179	−	0.470805i
$376$		0			0
$377$		−	11.1652i		−	0.575035i
$378$		0			0
$379$	−23.1652			−1.18991			−0.594957	−	0.803758i	$-0.702831\pi$
$379$	−23.1652			−1.18991			−0.594957	+	0.803758i	$0.702831\pi$
$380$		0			0
$381$	−9.47860			−0.485604
$382$		0			0
$383$			29.9216i			1.52892i	0.644669	+	0.764462i	$0.276995\pi$
$383$			29.9216i			1.52892i	−0.644669	+	0.764462i	$0.723005\pi$
$384$		0			0
$385$	11.1652	−	53.4955i	0.569029	−	2.72638i
$386$		0			0
$387$			11.1652i			0.567557i
$388$		0			0
$389$	−8.03260			−0.407269			−0.203635	−	0.979047i	$-0.565275\pi$
$389$	−8.03260			−0.407269			−0.203635	+	0.979047i	$0.565275\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			6.41742i			0.323716i
$394$		0			0
$395$	11.5826	+	2.41742i	0.582783	+	0.121634i
$396$		0			0
$397$		−	2.55040i		−	0.128001i	−0.997950	−	0.0640005i	$-0.979614\pi$
$397$		−	2.55040i		−	0.128001i	0.997950	−	0.0640005i	$-0.0203859\pi$
$398$		0			0
$399$	−17.5112			−0.876657
$400$		0			0
$401$	22.0000			1.09863			0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000			1.09863			0.549314	+	0.835616i	$0.314889\pi$
$402$		0			0
$403$			23.1652i			1.15394i
$404$		0			0
$405$	−2.18890	−	0.456850i	−0.108767	−	0.0227011i
$406$		0			0
$407$			14.2378i			0.705742i
$408$		0			0
$409$	−37.1652			−1.83770			−0.918849	−	0.394609i	$-0.870880\pi$
$409$	−37.1652			−1.83770			−0.918849	+	0.394609i	$0.870880\pi$
$410$		0			0
$411$	10.4174			0.513854
$412$		0			0
$413$			6.92820i			0.340915i
$414$		0			0
$415$	−3.27340	+	15.6838i	−0.160685	+	0.769888i
$416$		0			0
$417$			7.16515i			0.350879i
$418$		0			0
$419$	16.7477			0.818180			0.409090	−	0.912494i	$-0.365846\pi$
$419$	16.7477			0.818180			0.409090	+	0.912494i	$0.365846\pi$
$420$		0			0
$421$	29.5402			1.43970			0.719851	−	0.694129i	$-0.244210\pi$
$421$	29.5402			1.43970			0.719851	+	0.694129i	$0.244210\pi$
$422$		0			0
$423$		−	6.92820i		−	0.336861i
$424$		0			0
$425$	11.1652	−	25.5826i	0.541589	−	1.24094i
$426$		0			0
$427$		−	46.3303i		−	2.24208i
$428$		0			0
$429$	24.4394			1.17995
$430$		0			0
$431$	−38.2958			−1.84464			−0.922322	−	0.386421i	$-0.873711\pi$
$431$	−38.2958			−1.84464			−0.922322	+	0.386421i	$0.873711\pi$
$432$		0			0
$433$			1.66970i			0.0802405i	0.999195	+	0.0401203i	$0.0127741\pi$
$433$			1.66970i			0.0802405i	−0.999195	+	0.0401203i	$0.987226\pi$
$434$		0			0
$435$	−1.16515	+	5.58258i	−0.0558647	+	0.267664i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	29.7309			1.41898			0.709490	−	0.704716i	$-0.248926\pi$
$439$	29.7309			1.41898			0.709490	+	0.704716i	$0.248926\pi$
$440$		0			0
$441$	12.1652			0.579293
$442$		0			0
$443$		−	31.1652i		−	1.48070i	−0.672221	−	0.740351i	$-0.734660\pi$
$443$		−	31.1652i		−	1.48070i	0.672221	−	0.740351i	$-0.265340\pi$
$444$		0			0
$445$	−4.37780	−	0.913701i	−0.207528	−	0.0433136i
$446$		0			0
$447$			14.9608i			0.707622i
$448$		0			0
$449$	−28.3303			−1.33699			−0.668495	−	0.743717i	$-0.733061\pi$
$449$	−28.3303			−1.33699			−0.668495	+	0.743717i	$0.733061\pi$
$450$		0			0
$451$	33.4955			1.57724
$452$		0			0
$453$			15.8745i			0.745849i
$454$		0			0
$455$	−41.9506	−	8.75560i	−1.96668	−	0.410469i
$456$		0			0
$457$		−	4.83485i		−	0.226165i	−0.993586	−	0.113082i	$-0.963928\pi$
$457$		−	4.83485i		−	0.226165i	0.993586	−	0.113082i	$-0.0360724\pi$
$458$		0			0
$459$	−5.58258			−0.260572
$460$		0			0
$461$	−1.10440			−0.0514371			−0.0257185	−	0.999669i	$-0.508187\pi$
$461$	−1.10440			−0.0514371			−0.0257185	+	0.999669i	$0.508187\pi$
$462$		0			0
$463$		−	29.1986i		−	1.35697i	−0.734612	−	0.678487i	$-0.762636\pi$
$463$		−	29.1986i		−	1.35697i	0.734612	−	0.678487i	$-0.237364\pi$
$464$		0			0
$465$	2.41742	−	11.5826i	0.112105	−	0.537129i
$466$		0			0
$467$		−	0.834849i		−	0.0386322i	−0.999813	−	0.0193161i	$-0.993851\pi$
$467$		−	0.834849i		−	0.0386322i	0.999813	−	0.0193161i	$-0.00614889\pi$
$468$		0			0
$469$	13.8564			0.639829
$470$		0			0
$471$	−16.4068			−0.755986
$472$		0			0
$473$			62.3303i			2.86595i
$474$		0			0
$475$	−18.3303	−	8.00000i	−0.841052	−	0.367065i
$476$		0			0
$477$		−	7.84190i		−	0.359056i
$478$		0			0
$479$	31.7490			1.45065			0.725325	−	0.688407i	$-0.241690\pi$
$479$	31.7490			1.45065			0.725325	+	0.688407i	$0.241690\pi$
$480$		0			0
$481$	11.1652			0.509087
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.10080	−	24.4394i	0.231615	−	1.10974i
$486$		0			0
$487$		−	16.7882i		−	0.760746i	−0.924833	−	0.380373i	$-0.875796\pi$
$487$		−	16.7882i		−	0.760746i	0.924833	−	0.380373i	$-0.124204\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	12.7477			0.575297			0.287648	−	0.957736i	$-0.407127\pi$
$491$	12.7477			0.575297			0.287648	+	0.957736i	$0.407127\pi$
$492$		0			0
$493$			14.2378i			0.641239i
$494$		0			0
$495$	−12.2197	−	2.55040i	−0.549235	−	0.114632i
$496$		0			0
$497$		−	30.3303i		−	1.36050i
$498$		0			0
$499$	18.3303			0.820577			0.410289	−	0.911956i	$-0.365428\pi$
$499$	18.3303			0.820577			0.410289	+	0.911956i	$0.365428\pi$
$500$		0			0
$501$	12.0290			0.537416
$502$		0			0
$503$		−	28.0942i		−	1.25266i	−0.779558	−	0.626330i	$-0.784556\pi$
$503$		−	28.0942i		−	1.25266i	0.779558	−	0.626330i	$-0.215444\pi$
$504$		0			0
$505$	1.58258	+	0.330303i	0.0704237	+	0.0146983i
$506$		0			0
$507$		−	6.16515i		−	0.273804i
$508$		0			0
$509$	−39.4002			−1.74638			−0.873192	−	0.487376i	$-0.837954\pi$
$509$	−39.4002			−1.74638			−0.873192	+	0.487376i	$0.837954\pi$
$510$		0			0
$511$	52.5336			2.32395
$512$		0			0
$513$			4.00000i			0.176604i
$514$		0			0
$515$	0.330303	−	1.58258i	0.0145549	−	0.0697366i
$516$		0			0
$517$		−	38.6772i		−	1.70102i
$518$		0			0
$519$	6.01450			0.264007
$520$		0			0
$521$	34.6606			1.51851			0.759254	−	0.650794i	$-0.225564\pi$
$521$	34.6606			1.51851			0.759254	+	0.650794i	$0.225564\pi$
$522$		0			0
$523$			18.3303i			0.801528i	0.916181	+	0.400764i	$0.131255\pi$
$523$			18.3303i			0.801528i	−0.916181	+	0.400764i	$0.868745\pi$
$524$		0			0
$525$	20.0616	+	8.75560i	0.875560	+	0.382126i
$526$		0			0
$527$		−	29.5402i		−	1.28679i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	1.58258			0.0686779
$532$		0			0
$533$		−	26.2668i		−	1.13774i
$534$		0			0
$535$	1.82740	−	8.75560i	0.0790054	−	0.378538i
$536$		0			0
$537$		−	1.58258i		−	0.0682932i
$538$		0			0
$539$	67.9129			2.92521
$540$		0			0
$541$	36.8498			1.58430			0.792149	−	0.610328i	$-0.208962\pi$
$541$	36.8498			1.58430			0.792149	+	0.610328i	$0.208962\pi$
$542$		0			0
$543$			1.82740i			0.0784213i
$544$		0			0
$545$	−4.00000	−	0.834849i	−0.171341	−	0.0357610i
$546$		0			0
$547$		−	25.4955i		−	1.09011i	−0.838401	−	0.545053i	$-0.816509\pi$
$547$		−	25.4955i		−	1.09011i	0.838401	−	0.545053i	$-0.183491\pi$
$548$		0			0
$549$	−10.5830			−0.451672
$550$		0			0
$551$	10.2016			0.434603
$552$		0			0
$553$			23.1652i			0.985082i
$554$		0			0
$555$	−5.58258	−	1.16515i	−0.236967	−	0.0494579i
$556$		0			0
$557$			4.18710i			0.177413i	0.996058	+	0.0887066i	$0.0282733\pi$
$557$			4.18710i			0.177413i	−0.996058	+	0.0887066i	$0.971727\pi$
$558$		0			0
$559$	48.8788			2.06735
$560$		0			0
$561$	−31.1652			−1.31579
$562$		0			0
$563$			29.4955i			1.24309i	0.783380	+	0.621543i	$0.213494\pi$
$563$			29.4955i			1.24309i	−0.783380	+	0.621543i	$0.786506\pi$
$564$		0			0
$565$	6.20520	−	29.7309i	0.261055	−	1.25079i
$566$		0			0
$567$		−	4.37780i		−	0.183850i
$568$		0			0
$569$	−36.3303			−1.52305			−0.761523	−	0.648138i	$-0.775548\pi$
$569$	−36.3303			−1.52305			−0.761523	+	0.648138i	$0.775548\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$		−	13.8564i		−	0.578860i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	15.1652i		−	0.631334i	−0.948870	−	0.315667i	$-0.897772\pi$
$577$		−	15.1652i		−	0.631334i	0.948870	−	0.315667i	$-0.102228\pi$
$578$		0			0
$579$	−15.1652			−0.630242
$580$		0			0
$581$	−31.3676			−1.30135
$582$		0			0
$583$		−	43.7780i		−	1.81310i
$584$		0			0
$585$	−2.00000	+	9.58258i	−0.0826898	+	0.396191i
$586$		0			0
$587$		−	5.66970i		−	0.234013i	−0.993131	−	0.117007i	$-0.962670\pi$
$587$		−	5.66970i		−	0.234013i	0.993131	−	0.117007i	$-0.0373299\pi$
$588$		0			0
$589$	−21.1660			−0.872130
$590$		0			0
$591$	−22.0797			−0.908238
$592$		0			0
$593$			35.9129i			1.47477i	0.675475	+	0.737383i	$0.263938\pi$
$593$			35.9129i			1.47477i	−0.675475	+	0.737383i	$0.736062\pi$
$594$		0			0
$595$	53.4955	+	11.1652i	2.19310	+	0.457727i
$596$		0			0
$597$			12.2197i			0.500119i
$598$		0			0
$599$	14.2378			0.581741			0.290871	−	0.956762i	$-0.406055\pi$
$599$	14.2378			0.581741			0.290871	+	0.956762i	$0.406055\pi$
$600$		0			0
$601$	−13.1652			−0.537018			−0.268509	−	0.963277i	$-0.586531\pi$
$601$	−13.1652			−0.537018			−0.268509	+	0.963277i	$0.586531\pi$
$602$		0			0
$603$		−	3.16515i		−	0.128895i
$604$		0			0
$605$	−44.1395	−	9.21245i	−1.79453	−	0.374540i
$606$		0			0
$607$			35.7454i			1.45086i	0.688295	+	0.725431i	$0.258359\pi$
$607$			35.7454i			1.45086i	−0.688295	+	0.725431i	$0.741641\pi$
$608$		0			0
$609$	−11.1652			−0.452435
$610$		0			0
$611$	−30.3303			−1.22703
$612$		0			0
$613$		−	32.4720i		−	1.31153i	−0.754964	−	0.655766i	$-0.772346\pi$
$613$		−	32.4720i		−	1.31153i	0.754964	−	0.655766i	$-0.227654\pi$
$614$		0			0
$615$	−2.74110	+	13.1334i	−0.110532	+	0.529590i
$616$		0			0
$617$			13.5826i			0.546814i	0.961898	+	0.273407i	$0.0881506\pi$
$617$			13.5826i			0.546814i	−0.961898	+	0.273407i	$0.911849\pi$
$618$		0			0
$619$	−45.4955			−1.82862			−0.914308	−	0.405019i	$-0.867265\pi$
$619$	−45.4955			−1.82862			−0.914308	+	0.405019i	$0.867265\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	8.75560i		−	0.350786i
$624$		0			0
$625$	17.0000	+	18.3303i	0.680000	+	0.733212i
$626$		0			0
$627$			22.3303i			0.891786i
$628$		0			0
$629$	−14.2378			−0.567699
$630$		0			0
$631$	−12.2197			−0.486459			−0.243229	−	0.969969i	$-0.578207\pi$
$631$	−12.2197			−0.486459			−0.243229	+	0.969969i	$0.578207\pi$
$632$		0			0
$633$		−	7.16515i		−	0.284789i
$634$		0			0
$635$	4.33030	−	20.7477i	0.171843	−	0.823348i
$636$		0			0
$637$		−	53.2566i		−	2.11010i
$638$		0			0
$639$	−6.92820			−0.274075
$640$		0			0
$641$	−4.33030			−0.171037			−0.0855183	−	0.996337i	$-0.527255\pi$
$641$	−4.33030			−0.171037			−0.0855183	+	0.996337i	$0.527255\pi$
$642$		0			0
$643$		−	18.3303i		−	0.722877i	−0.932396	−	0.361438i	$-0.882286\pi$
$643$		−	18.3303i		−	0.722877i	0.932396	−	0.361438i	$-0.117714\pi$
$644$		0			0
$645$	−24.4394	−	5.10080i	−0.962301	−	0.200844i
$646$		0			0
$647$		−	17.8926i		−	0.703431i	−0.936107	−	0.351716i	$-0.885598\pi$
$647$		−	17.8926i		−	0.703431i	0.936107	−	0.351716i	$-0.114402\pi$
$648$		0			0
$649$	8.83485			0.346798
$650$		0			0
$651$	23.1652			0.907914
$652$		0			0
$653$		−	22.0797i		−	0.864046i	−0.901863	−	0.432023i	$-0.857800\pi$
$653$		−	22.0797i		−	0.864046i	0.901863	−	0.432023i	$-0.142200\pi$
$654$		0			0
$655$	−14.0471	−	2.93180i	−0.548866	−	0.114555i
$656$		0			0
$657$		−	12.0000i		−	0.468165i
$658$		0			0
$659$	25.5826			0.996556			0.498278	−	0.867017i	$-0.333966\pi$
$659$	25.5826			0.996556			0.498278	+	0.867017i	$0.333966\pi$
$660$		0			0
$661$	−17.8926			−0.695942			−0.347971	−	0.937505i	$-0.613129\pi$
$661$	−17.8926			−0.695942			−0.347971	+	0.937505i	$0.613129\pi$
$662$		0			0
$663$			24.4394i			0.949148i
$664$		0			0
$665$	8.00000	−	38.3303i	0.310227	−	1.48639i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−4.37780			−0.169256
$670$		0			0
$671$	−59.0804			−2.28077
$672$		0			0
$673$			40.0000i			1.54189i	0.636904	+	0.770943i	$0.280215\pi$
$673$			40.0000i			1.54189i	−0.636904	+	0.770943i	$0.719785\pi$
$674$		0			0
$675$	2.00000	−	4.58258i	0.0769800	−	0.176383i
$676$		0			0
$677$			28.6265i			1.10021i	0.835097	+	0.550103i	$0.185412\pi$
$677$			28.6265i			1.10021i	−0.835097	+	0.550103i	$0.814588\pi$
$678$		0			0
$679$	48.8788			1.87580
$680$		0			0
$681$	15.1652			0.581130
$682$		0			0
$683$		−	5.49545i		−	0.210278i	−0.994458	−	0.105139i	$-0.966471\pi$
$683$		−	5.49545i		−	0.210278i	0.994458	−	0.105139i	$-0.0335287\pi$
$684$		0			0
$685$	−4.75920	+	22.8027i	−0.181840	+	0.871247i
$686$		0			0
$687$		−	12.0290i		−	0.458935i
$688$		0			0
$689$	−34.3303			−1.30788
$690$		0			0
$691$	−40.6606			−1.54680			−0.773401	−	0.633917i	$-0.781446\pi$
$691$	−40.6606			−1.54680			−0.773401	+	0.633917i	$0.781446\pi$
$692$		0			0
$693$		−	24.4394i		−	0.928376i
$694$		0			0
$695$	−15.6838	−	3.27340i	−0.594921	−	0.124167i
$696$		0			0
$697$			33.4955i			1.26873i
$698$		0			0
$699$	−10.4174			−0.394023
$700$		0			0
$701$	45.9470			1.73540			0.867698	−	0.497093i	$-0.165599\pi$
$701$	45.9470			1.73540			0.867698	+	0.497093i	$0.165599\pi$
$702$		0			0
$703$			10.2016i			0.384761i
$704$		0			0
$705$	15.1652	+	3.16515i	0.571153	+	0.119206i
$706$		0			0
$707$			3.16515i			0.119038i
$708$		0			0
$709$	15.6838			0.589018			0.294509	−	0.955649i	$-0.404844\pi$
$709$	15.6838			0.589018			0.294509	+	0.955649i	$0.404844\pi$
$710$		0			0
$711$	5.29150			0.198447
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−11.1652	+	53.4955i	−0.417553	+	2.00062i
$716$		0			0
$717$		−	21.1660i		−	0.790459i
$718$		0			0
$719$	30.9862			1.15559			0.577795	−	0.816182i	$-0.303913\pi$
$719$	30.9862			1.15559			0.577795	+	0.816182i	$0.303913\pi$
$720$		0			0
$721$	3.16515			0.117876
$722$		0			0
$723$		−	10.0000i		−	0.371904i
$724$		0			0
$725$	−11.6874	−	5.10080i	−0.434059	−	0.189439i
$726$		0			0
$727$			49.6018i			1.83963i	0.392353	+	0.919815i	$0.371661\pi$
$727$			49.6018i			1.83963i	−0.392353	+	0.919815i	$0.628339\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−62.3303			−2.30537
$732$		0			0
$733$		−	29.1986i		−	1.07848i	−0.842154	−	0.539238i	$-0.818712\pi$
$733$		−	29.1986i		−	1.07848i	0.842154	−	0.539238i	$-0.181288\pi$
$734$		0			0
$735$	−5.55765	+	26.6283i	−0.204997	+	0.982200i
$736$		0			0
$737$		−	17.6697i		−	0.650872i
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$		0			0
$741$	17.5112			0.643290
$742$		0			0
$743$			47.4328i			1.74014i	0.492927	+	0.870071i	$0.335927\pi$
$743$			47.4328i			1.74014i	−0.492927	+	0.870071i	$0.664073\pi$
$744$		0			0
$745$	−32.7477	−	6.83485i	−1.19978	−	0.250409i
$746$		0			0
$747$			7.16515i			0.262159i
$748$		0			0
$749$	17.5112			0.639846
$750$		0			0
$751$	33.0043			1.20434			0.602172	−	0.798366i	$-0.294302\pi$
$751$	33.0043			1.20434			0.602172	+	0.798366i	$0.294302\pi$
$752$		0			0
$753$		−	2.41742i		−	0.0880958i
$754$		0			0
$755$	−34.7477	−	7.25227i	−1.26460	−	0.263937i
$756$		0			0
$757$			18.2342i			0.662734i	0.943502	+	0.331367i	$0.107510\pi$
$757$			18.2342i			0.662734i	−0.943502	+	0.331367i	$0.892490\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	22.0000			0.797499			0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000			0.797499			0.398750	+	0.917060i	$0.369444\pi$
$762$		0			0
$763$		−	8.00000i		−	0.289619i
$764$		0			0
$765$	2.55040	−	12.2197i	0.0922099	−	0.441804i
$766$		0			0
$767$		−	6.92820i		−	0.250163i
$768$		0			0
$769$	23.4955			0.847268			0.423634	−	0.905834i	$-0.360754\pi$
$769$	23.4955			0.847268			0.423634	+	0.905834i	$0.360754\pi$
$770$		0			0
$771$	−24.7477			−0.891268
$772$		0			0
$773$		−	30.4539i		−	1.09535i	−0.836691	−	0.547676i	$-0.815513\pi$
$773$		−	30.4539i		−	1.09535i	0.836691	−	0.547676i	$-0.184487\pi$
$774$		0			0
$775$	24.2487	+	10.5830i	0.871039	+	0.380153i
$776$		0			0
$777$		−	11.1652i		−	0.400548i
$778$		0			0
$779$	24.0000			0.859889
$780$		0			0
$781$	−38.6772			−1.38398
$782$		0			0
$783$			2.55040i			0.0911439i
$784$		0			0
$785$	7.49545	−	35.9129i	0.267524	−	1.28179i
$786$		0			0
$787$			36.0000i			1.28326i	0.767014	+	0.641631i	$0.221742\pi$
$787$			36.0000i			1.28326i	−0.767014	+	0.641631i	$0.778258\pi$
$788$		0			0
$789$	−1.82740			−0.0650572
$790$		0			0
$791$	59.4618			2.11422
$792$		0			0
$793$			46.3303i			1.64524i
$794$		0			0
$795$	17.1652	+	3.58258i	0.608785	+	0.127061i
$796$		0			0
$797$		−	53.0659i		−	1.87969i	−0.341599	−	0.939846i	$-0.610969\pi$
$797$		−	53.0659i		−	1.87969i	0.341599	−	0.939846i	$-0.389031\pi$
$798$		0			0
$799$	38.6772			1.36830
$800$		0			0
$801$	−2.00000			−0.0706665
$802$		0			0
$803$		−	66.9909i		−	2.36406i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			6.20520i			0.218433i
$808$		0			0
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$	−0.834849			−0.0293155			−0.0146577	−	0.999893i	$-0.504666\pi$
$811$	−0.834849			−0.0293155			−0.0146577	+	0.999893i	$0.504666\pi$
$812$		0			0
$813$			15.4931i			0.543367i
$814$		0			0
$815$	−1.82740	+	8.75560i	−0.0640111	+	0.306695i
$816$		0			0
$817$			44.6606i			1.56248i
$818$		0			0
$819$	−19.1652			−0.669685
$820$		0			0
$821$	6.20520			0.216563			0.108282	−	0.994120i	$-0.465465\pi$
$821$	6.20520			0.216563			0.108282	+	0.994120i	$0.465465\pi$
$822$		0			0
$823$			9.47860i			0.330403i	0.986260	+	0.165202i	$0.0528275\pi$
$823$			9.47860i			0.330403i	−0.986260	+	0.165202i	$0.947172\pi$
$824$		0			0
$825$	11.1652	−	25.5826i	0.388721	−	0.890671i
$826$		0			0
$827$			28.0000i			0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$			28.0000i			0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$		0			0
$829$	−36.8498			−1.27985			−0.639924	−	0.768439i	$-0.721034\pi$
$829$	−36.8498			−1.27985			−0.639924	+	0.768439i	$0.721034\pi$
$830$		0			0
$831$	9.47860			0.328809
$832$		0			0
$833$			67.9129i			2.35304i
$834$		0			0
$835$	−5.49545	+	26.3303i	−0.190178	+	0.911198i
$836$		0			0
$837$		−	5.29150i		−	0.182901i
$838$		0			0
$839$	−55.8070			−1.92667			−0.963336	−	0.268297i	$-0.913539\pi$
$839$	−55.8070			−1.92667			−0.963336	+	0.268297i	$0.913539\pi$
$840$		0			0
$841$	−22.4955			−0.775705
$842$		0			0
$843$		−	32.3303i		−	1.11351i
$844$		0			0
$845$	13.4949	+	2.81655i	0.464239	+	0.0968923i
$846$		0			0
$847$		−	88.2790i		−	3.03330i
$848$		0			0
$849$	−11.1652			−0.383187
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	7.65120i		−	0.261972i	−0.991384	−	0.130986i	$-0.958186\pi$
$853$		−	7.65120i		−	0.261972i	0.991384	−	0.130986i	$-0.0418143\pi$
$854$		0			0
$855$	−8.75560	−	1.82740i	−0.299435	−	0.0624958i
$856$		0			0
$857$		−	29.5826i		−	1.01052i	−0.862967	−	0.505261i	$-0.831396\pi$
$857$		−	29.5826i		−	1.01052i	0.862967	−	0.505261i	$-0.168604\pi$
$858$		0			0
$859$	−24.8348			−0.847354			−0.423677	−	0.905813i	$-0.639261\pi$
$859$	−24.8348			−0.847354			−0.423677	+	0.905813i	$0.639261\pi$
$860$		0			0
$861$	−26.2668			−0.895171
$862$		0			0
$863$			50.7062i			1.72606i	0.505153	+	0.863030i	$0.331436\pi$
$863$			50.7062i			1.72606i	−0.505153	+	0.863030i	$0.668564\pi$
$864$		0			0
$865$	−2.74773	+	13.1652i	−0.0934255	+	0.447629i
$866$		0			0
$867$		−	14.1652i		−	0.481074i
$868$		0			0
$869$	29.5402			1.00208
$870$		0			0
$871$	−13.8564			−0.469506
$872$		0			0
$873$		−	11.1652i		−	0.377883i
$874$		0			0
$875$	−28.3303	+	39.9129i	−0.957739	+	1.34930i
$876$		0			0
$877$		−	17.8528i		−	0.602846i	−0.953490	−	0.301423i	$-0.902538\pi$
$877$		−	17.8528i		−	0.602846i	0.953490	−	0.301423i	$-0.0974617\pi$
$878$		0			0
$879$	0.913701			0.0308184
$880$		0			0
$881$	36.3303			1.22400			0.612000	−	0.790858i	$-0.290365\pi$
$881$	36.3303			1.22400			0.612000	+	0.790858i	$0.290365\pi$
$882$		0			0
$883$		−	34.3303i		−	1.15531i	−0.816282	−	0.577653i	$-0.803969\pi$
$883$		−	34.3303i		−	1.15531i	0.816282	−	0.577653i	$-0.196031\pi$
$884$		0			0
$885$	−0.723000	+	3.46410i	−0.0243034	+	0.116445i
$886$		0			0
$887$			47.0514i			1.57983i	0.613215	+	0.789916i	$0.289876\pi$
$887$			47.0514i			1.57983i	−0.613215	+	0.789916i	$0.710124\pi$
$888$		0			0
$889$	41.4955			1.39171
$890$		0			0
$891$	−5.58258			−0.187023
$892$		0			0
$893$		−	27.7128i		−	0.927374i
$894$		0			0
$895$	3.46410	+	0.723000i	0.115792	+	0.0241672i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−13.4955			−0.450099
$900$		0			0
$901$	43.7780			1.45846
$902$		0			0
$903$		−	48.8788i		−	1.62659i
$904$		0			0
$905$	−4.00000	−	0.834849i	−0.132964	−	0.0277513i
$906$		0			0
$907$			17.4955i			0.580927i	0.956886	+	0.290464i	$0.0938095\pi$
$907$			17.4955i			0.580927i	−0.956886	+	0.290464i	$0.906191\pi$
$908$		0			0
$909$	0.723000			0.0239804
$910$		0			0
$911$	−7.30960			−0.242178			−0.121089	−	0.992642i	$-0.538639\pi$
$911$	−7.30960			−0.242178			−0.121089	+	0.992642i	$0.538639\pi$
$912$		0			0
$913$			40.0000i			1.32381i
$914$		0			0
$915$	4.83485	−	23.1652i	0.159835	−	0.765816i
$916$		0			0
$917$		−	28.0942i		−	0.927753i
$918$		0			0
$919$	−11.8383			−0.390510			−0.195255	−	0.980753i	$-0.562553\pi$
$919$	−11.8383			−0.390510			−0.195255	+	0.980753i	$0.562553\pi$
$920$		0			0
$921$	4.00000			0.131804
$922$		0			0
$923$			30.3303i			0.998334i
$924$		0			0
$925$	5.10080	−	11.6874i	0.167713	−	0.384280i
$926$		0			0
$927$		−	0.723000i		−	0.0237464i
$928$		0			0
$929$	30.6606			1.00594			0.502971	−	0.864303i	$-0.332240\pi$
$929$	30.6606			1.00594			0.502971	+	0.864303i	$0.332240\pi$
$930$		0			0
$931$	48.6606			1.59479
$932$		0			0
$933$		−	3.65480i		−	0.119653i
$934$		0			0
$935$	14.2378	−	68.2174i	0.465626	−	2.23095i
$936$		0			0
$937$		−	14.3303i		−	0.468151i	−0.972218	−	0.234075i	$-0.924794\pi$
$937$		−	14.3303i		−	0.468151i	0.972218	−	0.234075i	$-0.0752062\pi$
$938$		0			0
$939$	−0.834849			−0.0272443
$940$		0			0
$941$	28.4358			0.926981			0.463491	−	0.886102i	$-0.346597\pi$
$941$	28.4358			0.926981			0.463491	+	0.886102i	$0.346597\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	9.58258	+	2.00000i	0.311721	+	0.0650600i
$946$		0			0
$947$			52.0000i			1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$			52.0000i			1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$		0			0
$949$	−52.5336			−1.70531
$950$		0			0
$951$	−34.1087			−1.10605
$952$		0			0
$953$		−	26.2432i		−	0.850100i	−0.905170	−	0.425050i	$-0.860257\pi$
$953$		−	26.2432i		−	0.850100i	0.905170	−	0.425050i	$-0.139743\pi$
$954$		0			0
$955$	30.3303	+	6.33030i	0.981466	+	0.204844i
$956$		0			0
$957$			14.2378i			0.460243i
$958$		0			0
$959$	−45.6054			−1.47268
$960$		0			0
$961$	−3.00000			−0.0967742
$962$		0			0
$963$		−	4.00000i		−	0.128898i
$964$		0			0
$965$	6.92820	−	33.1950i	0.223027	−	1.06859i
$966$		0			0
$967$			26.9898i			0.867934i	0.900929	+	0.433967i	$0.142887\pi$
$967$			26.9898i			0.867934i	−0.900929	+	0.433967i	$0.857113\pi$
$968$		0			0
$969$	−22.3303			−0.717353
$970$		0			0
$971$	−4.08712			−0.131162			−0.0655810	−	0.997847i	$-0.520890\pi$
$971$	−4.08712			−0.131162			−0.0655810	+	0.997847i	$0.520890\pi$
$972$		0			0
$973$		−	31.3676i		−	1.00560i
$974$		0			0
$975$	−20.0616	−	8.75560i	−0.642486	−	0.280404i
$976$		0			0
$977$		−	7.25227i		−	0.232021i	−0.993248	−	0.116010i	$-0.962989\pi$
$977$		−	7.25227i		−	0.232021i	0.993248	−	0.116010i	$-0.0370106\pi$
$978$		0			0
$979$	−11.1652			−0.356840
$980$		0			0
$981$	−1.82740			−0.0583444
$982$		0			0
$983$			2.20880i			0.0704498i	0.999379	+	0.0352249i	$0.0112148\pi$
$983$			2.20880i			0.0704498i	−0.999379	+	0.0352249i	$0.988785\pi$
$984$		0			0
$985$	10.0871	−	48.3303i	0.321402	−	1.53993i
$986$		0			0
$987$			30.3303i			0.965424i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−11.8383			−0.376056			−0.188028	−	0.982164i	$-0.560210\pi$
$991$	−11.8383			−0.376056			−0.188028	+	0.982164i	$0.560210\pi$
$992$		0			0
$993$		−	10.3303i		−	0.327822i
$994$		0			0
$995$	−26.7477	−	5.58258i	−0.847960	−	0.176980i
$996$		0			0
$997$			12.7520i			0.403860i	0.979400	+	0.201930i	$0.0647214\pi$
$997$			12.7520i			0.403860i	−0.979400	+	0.201930i	$0.935279\pi$
$998$		0			0
$999$	−2.55040			−0.0806911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.f.i.769.5		8
4.3	odd	2		3840.2.f.k.769.1		8
5.4	even	2	inner	3840.2.f.i.769.1		8
8.3	odd	2	inner	3840.2.f.i.769.8		8
8.5	even	2		3840.2.f.k.769.4		8
16.3	odd	4		960.2.d.f.289.4	yes	8
16.5	even	4		960.2.d.f.289.5	yes	8
16.11	odd	4		960.2.d.e.289.5	yes	8
16.13	even	4		960.2.d.e.289.4	yes	8
20.19	odd	2		3840.2.f.k.769.5		8
40.19	odd	2	inner	3840.2.f.i.769.4		8
40.29	even	2		3840.2.f.k.769.8		8
48.5	odd	4		2880.2.d.i.289.4		8
48.11	even	4		2880.2.d.j.289.4		8
48.29	odd	4		2880.2.d.j.289.5		8
48.35	even	4		2880.2.d.i.289.5		8
80.3	even	4		4800.2.k.q.2401.5		8
80.13	odd	4		4800.2.k.q.2401.4		8
80.19	odd	4		960.2.d.e.289.6	yes	8
80.27	even	4		4800.2.k.r.2401.8		8
80.29	even	4		960.2.d.f.289.6	yes	8
80.37	odd	4		4800.2.k.r.2401.1		8
80.43	even	4		4800.2.k.q.2401.1		8
80.53	odd	4		4800.2.k.q.2401.8		8
80.59	odd	4		960.2.d.f.289.3	yes	8
80.67	even	4		4800.2.k.r.2401.4		8
80.69	even	4		960.2.d.e.289.3	✓	8
80.77	odd	4		4800.2.k.r.2401.5		8
240.29	odd	4		2880.2.d.i.289.3		8
240.59	even	4		2880.2.d.i.289.6		8
240.149	odd	4		2880.2.d.j.289.6		8
240.179	even	4		2880.2.d.j.289.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.2.d.e.289.3	✓	8	80.69	even	4
960.2.d.e.289.4	yes	8	16.13	even	4
960.2.d.e.289.5	yes	8	16.11	odd	4
960.2.d.e.289.6	yes	8	80.19	odd	4
960.2.d.f.289.3	yes	8	80.59	odd	4
960.2.d.f.289.4	yes	8	16.3	odd	4
960.2.d.f.289.5	yes	8	16.5	even	4
960.2.d.f.289.6	yes	8	80.29	even	4
2880.2.d.i.289.3		8	240.29	odd	4
2880.2.d.i.289.4		8	48.5	odd	4
2880.2.d.i.289.5		8	48.35	even	4
2880.2.d.i.289.6		8	240.59	even	4
2880.2.d.j.289.3		8	240.179	even	4
2880.2.d.j.289.4		8	48.11	even	4
2880.2.d.j.289.5		8	48.29	odd	4
2880.2.d.j.289.6		8	240.149	odd	4
3840.2.f.i.769.1		8	5.4	even	2	inner
3840.2.f.i.769.4		8	40.19	odd	2	inner
3840.2.f.i.769.5		8	1.1	even	1	trivial
3840.2.f.i.769.8		8	8.3	odd	2	inner
3840.2.f.k.769.1		8	4.3	odd	2
3840.2.f.k.769.4		8	8.5	even	2
3840.2.f.k.769.5		8	20.19	odd	2
3840.2.f.k.769.8		8	40.29	even	2
4800.2.k.q.2401.1		8	80.43	even	4
4800.2.k.q.2401.4		8	80.13	odd	4
4800.2.k.q.2401.5		8	80.3	even	4
4800.2.k.q.2401.8		8	80.53	odd	4
4800.2.k.r.2401.1		8	80.37	odd	4
4800.2.k.r.2401.4		8	80.67	even	4
4800.2.k.r.2401.5		8	80.77	odd	4
4800.2.k.r.2401.8		8	80.27	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-2.18890 - 0.456850i) q^{5} -4.37780i q^{7} -1.00000 q^{9} -5.58258 q^{11} +4.37780i q^{13} +(0.456850 - 2.18890i) q^{15} -5.58258i q^{17} -4.00000 q^{19} +4.37780 q^{21} +(4.58258 + 2.00000i) q^{25} -1.00000i q^{27} -2.55040 q^{29} +5.29150 q^{31} -5.58258i q^{33} +(-2.00000 + 9.58258i) q^{35} -2.55040i q^{37} -4.37780 q^{39} -6.00000 q^{41} -11.1652i q^{43} +(2.18890 + 0.456850i) q^{45} +6.92820i q^{47} -12.1652 q^{49} +5.58258 q^{51} +7.84190i q^{53} +(12.2197 + 2.55040i) q^{55} -4.00000i q^{57} -1.58258 q^{59} +10.5830 q^{61} +4.37780i q^{63} +(2.00000 - 9.58258i) q^{65} +3.16515i q^{67} +6.92820 q^{71} +12.0000i q^{73} +(-2.00000 + 4.58258i) q^{75} +24.4394i q^{77} -5.29150 q^{79} +1.00000 q^{81} -7.16515i q^{83} +(-2.55040 + 12.2197i) q^{85} -2.55040i q^{87} +2.00000 q^{89} +19.1652 q^{91} +5.29150i q^{93} +(8.75560 + 1.82740i) q^{95} +11.1652i q^{97} +5.58258 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 8 q^{11} - 32 q^{19} - 16 q^{35} - 48 q^{41} - 24 q^{49} + 8 q^{51} + 24 q^{59} + 16 q^{65} - 16 q^{75} + 8 q^{81} + 16 q^{89} + 80 q^{91} + 8 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 8 * q^11 - 32 * q^19 - 16 * q^35 - 48 * q^41 - 24 * q^49 + 8 * q^51 + 24 * q^59 + 16 * q^65 - 16 * q^75 + 8 * q^81 + 16 * q^89 + 80 * q^91 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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