LMFDB - Embedded newform 960.2.d.f.289.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [960,2,Mod(289,960)] 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("960.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$7.66563859404$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.5
Root			$-1.09445 + 0.895644i$ of defining polynomial
Character	$\chi$	$=$	960.289
Dual form			960.2.d.f.289.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +(0.456850 - 2.18890i) q^{5} +4.37780i q^{7} +1.00000 q^{9} -5.58258i q^{11} +4.37780 q^{13} +(0.456850 - 2.18890i) q^{15} -5.58258i q^{17} +4.00000i q^{19} +4.37780i q^{21} +(-4.58258 - 2.00000i) q^{25} +1.00000 q^{27} +2.55040i q^{29} +5.29150 q^{31} -5.58258i q^{33} +(9.58258 + 2.00000i) q^{35} +2.55040 q^{37} +4.37780 q^{39} +6.00000 q^{41} +11.1652 q^{43} +(0.456850 - 2.18890i) q^{45} +6.92820i q^{47} -12.1652 q^{49} -5.58258i q^{51} -7.84190 q^{53} +(-12.2197 - 2.55040i) q^{55} +4.00000i q^{57} -1.58258i q^{59} -10.5830i q^{61} +4.37780i q^{63} +(2.00000 - 9.58258i) q^{65} +3.16515 q^{67} -6.92820 q^{71} -12.0000i q^{73} +(-4.58258 - 2.00000i) q^{75} +24.4394 q^{77} -5.29150 q^{79} +1.00000 q^{81} -7.16515 q^{83} +(-12.2197 - 2.55040i) q^{85} +2.55040i q^{87} -2.00000 q^{89} +19.1652i q^{91} +5.29150 q^{93} +(8.75560 + 1.82740i) q^{95} +11.1652i q^{97} -5.58258i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 8 q^{9} + 8 q^{27} + 40 q^{35} + 48 q^{41} + 16 q^{43} - 24 q^{49} + 16 q^{65} - 48 q^{67} + 8 q^{81} + 16 q^{83} - 16 q^{89}+O(q^{100}) $ 8 * q + 8 * q^3 + 8 * q^9 + 8 * q^27 + 40 * q^35 + 48 * q^41 + 16 * q^43 - 24 * q^49 + 16 * q^65 - 48 * q^67 + 8 * q^81 + 16 * q^83 - 16 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	0.456850	−	2.18890i	0.204310	−	0.978906i
$5$	0.456850	−	2.18890i	0.204310	−	0.978906i
$6$		0			0
$7$			4.37780i			1.65465i	0.561721	+	0.827327i	$0.310140\pi$
$7$			4.37780i			1.65465i	−0.561721	+	0.827327i	$0.689860\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	5.58258i		−	1.68321i	−0.540094	−	0.841605i	$-0.681611\pi$
$11$		−	5.58258i		−	1.68321i	0.540094	−	0.841605i	$-0.318389\pi$
$12$		0			0
$13$	4.37780			1.21418			0.607092	−	0.794632i	$-0.292336\pi$
$13$	4.37780			1.21418			0.607092	+	0.794632i	$0.292336\pi$
$14$		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
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$			4.37780i			0.955315i
$22$		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
$27$	1.00000			0.192450
$28$		0			0
$29$			2.55040i			0.473598i	0.971559	+	0.236799i	$0.0760982\pi$
$29$			2.55040i			0.473598i	−0.971559	+	0.236799i	$0.923902\pi$
$30$		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
$35$	9.58258	+	2.00000i	1.61975	+	0.338062i
$36$		0			0
$37$	2.55040			0.419283			0.209642	−	0.977778i	$-0.432770\pi$
$37$	2.55040			0.419283			0.209642	+	0.977778i	$0.432770\pi$
$38$		0			0
$39$	4.37780			0.701009
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$	11.1652			1.70267			0.851335	−	0.524623i	$-0.175794\pi$
$43$	11.1652			1.70267			0.851335	+	0.524623i	$0.175794\pi$
$44$		0			0
$45$	0.456850	−	2.18890i	0.0681032	−	0.326302i
$46$		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
$51$		−	5.58258i		−	0.781717i
$52$		0			0
$53$	−7.84190			−1.07717			−0.538584	−	0.842572i	$-0.681041\pi$
$53$	−7.84190			−1.07717			−0.538584	+	0.842572i	$0.681041\pi$
$54$		0			0
$55$	−12.2197	−	2.55040i	−1.64770	−	0.343896i
$56$		0			0
$57$			4.00000i			0.529813i
$58$		0			0
$59$		−	1.58258i		−	0.206034i	−0.994680	−	0.103017i	$-0.967150\pi$
$59$		−	1.58258i		−	0.206034i	0.994680	−	0.103017i	$-0.0328496\pi$
$60$		0			0
$61$		−	10.5830i		−	1.35501i	−0.735516	−	0.677507i	$-0.763060\pi$
$61$		−	10.5830i		−	1.35501i	0.735516	−	0.677507i	$-0.236940\pi$
$62$		0			0
$63$			4.37780i			0.551551i
$64$		0			0
$65$	2.00000	−	9.58258i	0.248069	−	1.18857i
$66$		0			0
$67$	3.16515			0.386685			0.193342	−	0.981131i	$-0.438067\pi$
$67$	3.16515			0.386685			0.193342	+	0.981131i	$0.438067\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.92820			−0.822226			−0.411113	−	0.911584i	$-0.634860\pi$
$71$	−6.92820			−0.822226			−0.411113	+	0.911584i	$0.634860\pi$
$72$		0			0
$73$		−	12.0000i		−	1.40449i	−0.711934	−	0.702247i	$-0.752180\pi$
$73$		−	12.0000i		−	1.40449i	0.711934	−	0.702247i	$-0.247820\pi$
$74$		0			0
$75$	−4.58258	−	2.00000i	−0.529150	−	0.230940i
$76$		0			0
$77$	24.4394			2.78513
$78$		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
$83$	−7.16515			−0.786478			−0.393239	−	0.919436i	$-0.628645\pi$
$83$	−7.16515			−0.786478			−0.393239	+	0.919436i	$0.628645\pi$
$84$		0			0
$85$	−12.2197	−	2.55040i	−1.32541	−	0.276630i
$86$		0			0
$87$			2.55040i			0.273432i
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$			19.1652i			2.00905i
$92$		0			0
$93$	5.29150			0.548703
$94$		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
$99$		−	5.58258i		−	0.561070i
$100$		0			0
$101$		−	0.723000i		−	0.0719412i	−0.999353	−	0.0359706i	$-0.988548\pi$
$101$		−	0.723000i		−	0.0719412i	0.999353	−	0.0359706i	$-0.0114523\pi$
$102$		0			0
$103$		−	0.723000i		−	0.0712393i	−0.999365	−	0.0356197i	$-0.988660\pi$
$103$		−	0.723000i		−	0.0712393i	0.999365	−	0.0356197i	$-0.0113405\pi$
$104$		0			0
$105$	9.58258	+	2.00000i	0.935164	+	0.195180i
$106$		0			0
$107$	−4.00000			−0.386695			−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000			−0.386695			−0.193347	+	0.981130i	$0.561934\pi$
$108$		0			0
$109$		−	1.82740i		−	0.175033i	−0.996163	−	0.0875166i	$-0.972107\pi$
$109$		−	1.82740i		−	0.175033i	0.996163	−	0.0875166i	$-0.0278931\pi$
$110$		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
$117$	4.37780			0.404728
$118$		0			0
$119$	24.4394			2.24036
$120$		0			0
$121$	−20.1652			−1.83320
$122$		0			0
$123$	6.00000			0.541002
$124$		0			0
$125$	−6.47135	+	9.11710i	−0.578815	+	0.815459i
$126$		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
$131$		−	6.41742i		−	0.560693i	−0.959899	−	0.280346i	$-0.909551\pi$
$131$		−	6.41742i		−	0.560693i	0.959899	−	0.280346i	$-0.0904494\pi$
$132$		0			0
$133$	−17.5112			−1.51841
$134$		0			0
$135$	0.456850	−	2.18890i	0.0393194	−	0.188391i
$136$		0			0
$137$			10.4174i			0.890021i	0.895525	+	0.445010i	$0.146800\pi$
$137$			10.4174i			0.890021i	−0.895525	+	0.445010i	$0.853200\pi$
$138$		0			0
$139$			7.16515i			0.607740i	0.952713	+	0.303870i	$0.0982789\pi$
$139$			7.16515i			0.607740i	−0.952713	+	0.303870i	$0.901721\pi$
$140$		0			0
$141$			6.92820i			0.583460i
$142$		0			0
$143$		−	24.4394i		−	2.04373i
$144$		0			0
$145$	5.58258	+	1.16515i	0.463608	+	0.0967606i
$146$		0			0
$147$	−12.1652			−1.00336
$148$		0			0
$149$			14.9608i			1.22564i	0.790224	+	0.612819i	$0.209964\pi$
$149$			14.9608i			1.22564i	−0.790224	+	0.612819i	$0.790036\pi$
$150$		0			0
$151$	−15.8745			−1.29185			−0.645925	−	0.763401i	$-0.723528\pi$
$151$	−15.8745			−1.29185			−0.645925	+	0.763401i	$0.723528\pi$
$152$		0			0
$153$		−	5.58258i		−	0.451324i
$154$		0			0
$155$	2.41742	−	11.5826i	0.194172	−	0.930335i
$156$		0			0
$157$	16.4068			1.30941			0.654703	−	0.755886i	$-0.272794\pi$
$157$	16.4068			1.30941			0.654703	+	0.755886i	$0.272794\pi$
$158$		0			0
$159$	−7.84190			−0.621903
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$	−12.2197	−	2.55040i	−0.951303	−	0.198548i
$166$		0			0
$167$			12.0290i			0.930832i	0.885092	+	0.465416i	$0.154095\pi$
$167$			12.0290i			0.930832i	−0.885092	+	0.465416i	$0.845905\pi$
$168$		0			0
$169$	6.16515			0.474242
$170$		0			0
$171$			4.00000i			0.305888i
$172$		0			0
$173$	−6.01450			−0.457274			−0.228637	−	0.973512i	$-0.573427\pi$
$173$	−6.01450			−0.457274			−0.228637	+	0.973512i	$0.573427\pi$
$174$		0			0
$175$	8.75560	−	20.0616i	0.661861	−	1.51652i
$176$		0			0
$177$		−	1.58258i		−	0.118954i
$178$		0			0
$179$			1.58258i			0.118287i	0.998249	+	0.0591436i	$0.0188370\pi$
$179$			1.58258i			0.118287i	−0.998249	+	0.0591436i	$0.981163\pi$
$180$		0			0
$181$			1.82740i			0.135830i	0.997691	+	0.0679148i	$0.0216346\pi$
$181$			1.82740i			0.135830i	−0.997691	+	0.0679148i	$0.978365\pi$
$182$		0			0
$183$		−	10.5830i		−	0.782318i
$184$		0			0
$185$	1.16515	−	5.58258i	0.0856636	−	0.410439i
$186$		0			0
$187$	−31.1652			−2.27902
$188$		0			0
$189$			4.37780i			0.318438i
$190$		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
$195$	2.00000	−	9.58258i	0.143223	−	0.686222i
$196$		0			0
$197$	−22.0797			−1.57311			−0.786557	−	0.617518i	$-0.788138\pi$
$197$	−22.0797			−1.57311			−0.786557	+	0.617518i	$0.788138\pi$
$198$		0			0
$199$	−12.2197			−0.866232			−0.433116	−	0.901338i	$-0.642586\pi$
$199$	−12.2197			−0.866232			−0.433116	+	0.901338i	$0.642586\pi$
$200$		0			0
$201$	3.16515			0.223253
$202$		0			0
$203$	−11.1652			−0.783640
$204$		0			0
$205$	2.74110	−	13.1334i	0.191447	−	0.917277i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	22.3303			1.54462
$210$		0			0
$211$			7.16515i			0.493269i	0.969109	+	0.246635i	$0.0793248\pi$
$211$			7.16515i			0.493269i	−0.969109	+	0.246635i	$0.920675\pi$
$212$		0			0
$213$	−6.92820			−0.474713
$214$		0			0
$215$	5.10080	−	24.4394i	0.347872	−	1.66675i
$216$		0			0
$217$			23.1652i			1.57255i
$218$		0			0
$219$		−	12.0000i		−	0.810885i
$220$		0			0
$221$		−	24.4394i		−	1.64397i
$222$		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
$227$	−15.1652			−1.00655			−0.503273	−	0.864127i	$-0.667871\pi$
$227$	−15.1652			−1.00655			−0.503273	+	0.864127i	$0.667871\pi$
$228$		0			0
$229$		−	12.0290i		−	0.794899i	−0.917624	−	0.397450i	$-0.869895\pi$
$229$		−	12.0290i		−	0.794899i	0.917624	−	0.397450i	$-0.130105\pi$
$230$		0			0
$231$	24.4394			1.60800
$232$		0			0
$233$		−	10.4174i		−	0.682468i	−0.939978	−	0.341234i	$-0.889155\pi$
$233$		−	10.4174i		−	0.682468i	0.939978	−	0.341234i	$-0.110845\pi$
$234$		0			0
$235$	15.1652	+	3.16515i	0.989265	+	0.206472i
$236$		0			0
$237$	−5.29150			−0.343720
$238$		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
$243$	1.00000			0.0641500
$244$		0			0
$245$	−5.55765	+	26.6283i	−0.355065	+	1.70122i
$246$		0			0
$247$			17.5112i			1.11421i
$248$		0			0
$249$	−7.16515			−0.454073
$250$		0			0
$251$		−	2.41742i		−	0.152586i	−0.997085	−	0.0762932i	$-0.975691\pi$
$251$		−	2.41742i		−	0.152586i	0.997085	−	0.0762932i	$-0.0243085\pi$
$252$		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
$259$			11.1652i			0.693769i
$260$		0			0
$261$			2.55040i			0.157866i
$262$		0			0
$263$		−	1.82740i		−	0.112682i	−0.998412	−	0.0563412i	$-0.982057\pi$
$263$		−	1.82740i		−	0.112682i	0.998412	−	0.0563412i	$-0.0179435\pi$
$264$		0			0
$265$	−3.58258	+	17.1652i	−0.220076	+	1.05445i
$266$		0			0
$267$	−2.00000			−0.122398
$268$		0			0
$269$		−	6.20520i		−	0.378338i	−0.981945	−	0.189169i	$-0.939421\pi$
$269$		−	6.20520i		−	0.378338i	0.981945	−	0.189169i	$-0.0605794\pi$
$270$		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
$275$	−11.1652	+	25.5826i	−0.673284	+	1.54269i
$276$		0			0
$277$	9.47860			0.569514			0.284757	−	0.958600i	$-0.408087\pi$
$277$	9.47860			0.569514			0.284757	+	0.958600i	$0.408087\pi$
$278$		0			0
$279$	5.29150			0.316794
$280$		0			0
$281$	32.3303			1.92866			0.964332	−	0.264695i	$-0.0852714\pi$
$281$	32.3303			1.92866			0.964332	+	0.264695i	$0.0852714\pi$
$282$		0			0
$283$	−11.1652			−0.663699			−0.331850	−	0.943332i	$-0.607673\pi$
$283$	−11.1652			−0.663699			−0.331850	+	0.943332i	$0.607673\pi$
$284$		0			0
$285$	8.75560	+	1.82740i	0.518637	+	0.108246i
$286$		0			0
$287$			26.2668i			1.55048i
$288$		0			0
$289$	−14.1652			−0.833244
$290$		0			0
$291$			11.1652i			0.654513i
$292$		0			0
$293$	0.913701			0.0533790			0.0266895	−	0.999644i	$-0.491503\pi$
$293$	0.913701			0.0533790			0.0266895	+	0.999644i	$0.491503\pi$
$294$		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
$301$			48.8788i			2.81733i
$302$		0			0
$303$		−	0.723000i		−	0.0415353i
$304$		0			0
$305$	−23.1652	−	4.83485i	−1.32643	−	0.276843i
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$		0			0
$309$		−	0.723000i		−	0.0411300i
$310$		0			0
$311$	3.65480			0.207245			0.103622	−	0.994617i	$-0.466957\pi$
$311$	3.65480			0.207245			0.103622	+	0.994617i	$0.466957\pi$
$312$		0			0
$313$		−	0.834849i		−	0.0471884i	−0.999722	−	0.0235942i	$-0.992489\pi$
$313$		−	0.834849i		−	0.0471884i	0.999722	−	0.0235942i	$-0.00751097\pi$
$314$		0			0
$315$	9.58258	+	2.00000i	0.539917	+	0.112687i
$316$		0			0
$317$	34.1087			1.91574			0.957868	−	0.287208i	$-0.0927272\pi$
$317$	34.1087			1.91574			0.957868	+	0.287208i	$0.0927272\pi$
$318$		0			0
$319$	14.2378			0.797164
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$	22.3303			1.24249
$324$		0			0
$325$	−20.0616	−	8.75560i	−1.11282	−	0.485674i
$326$		0			0
$327$		−	1.82740i		−	0.101056i
$328$		0			0
$329$	−30.3303			−1.67216
$330$		0			0
$331$		−	10.3303i		−	0.567805i	−0.958853	−	0.283902i	$-0.908371\pi$
$331$		−	10.3303i		−	0.567805i	0.958853	−	0.283902i	$-0.0916292\pi$
$332$		0			0
$333$	2.55040			0.139761
$334$		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
$339$			13.5826i			0.737704i
$340$		0			0
$341$		−	29.5402i		−	1.59969i
$342$		0			0
$343$		−	22.6120i		−	1.22093i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.834849			−0.0448170			−0.0224085	−	0.999749i	$-0.507133\pi$
$347$	−0.834849			−0.0448170			−0.0224085	+	0.999749i	$0.507133\pi$
$348$		0			0
$349$		−	24.4394i		−	1.30821i	−0.756403	−	0.654106i	$-0.773045\pi$
$349$		−	24.4394i		−	1.30821i	0.756403	−	0.654106i	$-0.226955\pi$
$350$		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
$355$	−3.16515	+	15.1652i	−0.167989	+	0.804883i
$356$		0			0
$357$	24.4394			1.29347
$358$		0			0
$359$	−6.92820			−0.365657			−0.182828	−	0.983145i	$-0.558525\pi$
$359$	−6.92820			−0.365657			−0.182828	+	0.983145i	$0.558525\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	−20.1652			−1.05840
$364$		0			0
$365$	−26.2668	−	5.48220i	−1.37487	−	0.286952i
$366$		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
$371$		−	34.3303i		−	1.78234i
$372$		0			0
$373$	−11.3060			−0.585403			−0.292701	−	0.956204i	$-0.594554\pi$
$373$	−11.3060			−0.585403			−0.292701	+	0.956204i	$0.594554\pi$
$374$		0			0
$375$	−6.47135	+	9.11710i	−0.334179	+	0.470805i
$376$		0			0
$377$			11.1652i			0.575035i
$378$		0			0
$379$		−	23.1652i		−	1.18991i	−0.803758	−	0.594957i	$-0.797169\pi$
$379$		−	23.1652i		−	1.18991i	0.803758	−	0.594957i	$-0.202831\pi$
$380$		0			0
$381$			9.47860i			0.485604i
$382$		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
$387$	11.1652			0.567557
$388$		0			0
$389$		−	8.03260i		−	0.407269i	−0.979047	−	0.203635i	$-0.934725\pi$
$389$		−	8.03260i		−	0.407269i	0.979047	−	0.203635i	$-0.0652755\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		−	6.41742i		−	0.323716i
$394$		0			0
$395$	−2.41742	+	11.5826i	−0.121634	+	0.582783i
$396$		0			0
$397$	−2.55040			−0.128001			−0.0640005	−	0.997950i	$-0.520386\pi$
$397$	−2.55040			−0.128001			−0.0640005	+	0.997950i	$0.520386\pi$
$398$		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
$403$	23.1652			1.15394
$404$		0			0
$405$	0.456850	−	2.18890i	0.0227011	−	0.108767i
$406$		0			0
$407$		−	14.2378i		−	0.705742i
$408$		0			0
$409$	37.1652			1.83770			0.918849	−	0.394609i	$-0.129120\pi$
$409$	37.1652			1.83770			0.918849	+	0.394609i	$0.129120\pi$
$410$		0			0
$411$			10.4174i			0.513854i
$412$		0			0
$413$	6.92820			0.340915
$414$		0			0
$415$	−3.27340	+	15.6838i	−0.160685	+	0.769888i
$416$		0			0
$417$			7.16515i			0.350879i
$418$		0			0
$419$		−	16.7477i		−	0.818180i	−0.912494	−	0.409090i	$-0.865846\pi$
$419$		−	16.7477i		−	0.818180i	0.912494	−	0.409090i	$-0.134154\pi$
$420$		0			0
$421$			29.5402i			1.43970i	0.694129	+	0.719851i	$0.255790\pi$
$421$			29.5402i			1.43970i	−0.694129	+	0.719851i	$0.744210\pi$
$422$		0			0
$423$			6.92820i			0.336861i
$424$		0			0
$425$	−11.1652	+	25.5826i	−0.541589	+	1.24094i
$426$		0			0
$427$	46.3303			2.24208
$428$		0			0
$429$		−	24.4394i		−	1.17995i
$430$		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
$435$	5.58258	+	1.16515i	0.267664	+	0.0558647i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−29.7309			−1.41898			−0.709490	−	0.704716i	$-0.751074\pi$
$439$	−29.7309			−1.41898			−0.709490	+	0.704716i	$0.751074\pi$
$440$		0			0
$441$	−12.1652			−0.579293
$442$		0			0
$443$	31.1652			1.48070			0.740351	−	0.672221i	$-0.234660\pi$
$443$	31.1652			1.48070			0.740351	+	0.672221i	$0.234660\pi$
$444$		0			0
$445$	−0.913701	+	4.37780i	−0.0433136	+	0.207528i
$446$		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
$451$		−	33.4955i		−	1.57724i
$452$		0			0
$453$	−15.8745			−0.745849
$454$		0			0
$455$	41.9506	+	8.75560i	1.96668	+	0.410469i
$456$		0			0
$457$			4.83485i			0.226165i	0.993586	+	0.113082i	$0.0360724\pi$
$457$			4.83485i			0.226165i	−0.993586	+	0.113082i	$0.963928\pi$
$458$		0			0
$459$		−	5.58258i		−	0.260572i
$460$		0			0
$461$			1.10440i			0.0514371i	0.999669	+	0.0257185i	$0.00818737\pi$
$461$			1.10440i			0.0514371i	−0.999669	+	0.0257185i	$0.991813\pi$
$462$		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
$467$	−0.834849			−0.0386322			−0.0193161	−	0.999813i	$-0.506149\pi$
$467$	−0.834849			−0.0386322			−0.0193161	+	0.999813i	$0.506149\pi$
$468$		0			0
$469$			13.8564i			0.639829i
$470$		0			0
$471$	16.4068			0.755986
$472$		0			0
$473$		−	62.3303i		−	2.86595i
$474$		0			0
$475$	8.00000	−	18.3303i	0.367065	−	0.841052i
$476$		0			0
$477$	−7.84190			−0.359056
$478$		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
$485$	24.4394	+	5.10080i	1.10974	+	0.231615i
$486$		0			0
$487$			16.7882i			0.760746i	0.924833	+	0.380373i	$0.124204\pi$
$487$			16.7882i			0.760746i	−0.924833	+	0.380373i	$0.875796\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$			12.7477i			0.575297i	0.957736	+	0.287648i	$0.0928735\pi$
$491$			12.7477i			0.575297i	−0.957736	+	0.287648i	$0.907127\pi$
$492$		0			0
$493$	14.2378			0.641239
$494$		0			0
$495$	−12.2197	−	2.55040i	−0.549235	−	0.114632i
$496$		0			0
$497$		−	30.3303i		−	1.36050i
$498$		0			0
$499$		−	18.3303i		−	0.820577i	−0.911956	−	0.410289i	$-0.865428\pi$
$499$		−	18.3303i		−	0.820577i	0.911956	−	0.410289i	$-0.134572\pi$
$500$		0			0
$501$			12.0290i			0.537416i
$502$		0			0
$503$			28.0942i			1.25266i	0.779558	+	0.626330i	$0.215444\pi$
$503$			28.0942i			1.25266i	−0.779558	+	0.626330i	$0.784556\pi$
$504$		0			0
$505$	−1.58258	−	0.330303i	−0.0704237	−	0.0146983i
$506$		0			0
$507$	6.16515			0.273804
$508$		0			0
$509$			39.4002i			1.74638i	0.487376	+	0.873192i	$0.337954\pi$
$509$			39.4002i			1.74638i	−0.487376	+	0.873192i	$0.662046\pi$
$510$		0			0
$511$	52.5336			2.32395
$512$		0			0
$513$			4.00000i			0.176604i
$514$		0			0
$515$	−1.58258	−	0.330303i	−0.0697366	−	0.0145549i
$516$		0			0
$517$	38.6772			1.70102
$518$		0			0
$519$	−6.01450			−0.264007
$520$		0			0
$521$	−34.6606			−1.51851			−0.759254	−	0.650794i	$-0.774436\pi$
$521$	−34.6606			−1.51851			−0.759254	+	0.650794i	$0.774436\pi$
$522$		0			0
$523$	−18.3303			−0.801528			−0.400764	−	0.916181i	$-0.631255\pi$
$523$	−18.3303			−0.801528			−0.400764	+	0.916181i	$0.631255\pi$
$524$		0			0
$525$	8.75560	−	20.0616i	0.382126	−	0.875560i
$526$		0			0
$527$		−	29.5402i		−	1.28679i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		−	1.58258i		−	0.0686779i
$532$		0			0
$533$	26.2668			1.13774
$534$		0			0
$535$	−1.82740	+	8.75560i	−0.0790054	+	0.378538i
$536$		0			0
$537$			1.58258i			0.0682932i
$538$		0			0
$539$			67.9129i			2.92521i
$540$		0			0
$541$		−	36.8498i		−	1.58430i	−0.610328	−	0.792149i	$-0.708962\pi$
$541$		−	36.8498i		−	1.58430i	0.610328	−	0.792149i	$-0.291038\pi$
$542$		0			0
$543$			1.82740i			0.0784213i
$544$		0			0
$545$	−4.00000	−	0.834849i	−0.171341	−	0.0357610i
$546$		0			0
$547$	−25.4955			−1.09011			−0.545053	−	0.838401i	$-0.683491\pi$
$547$	−25.4955			−1.09011			−0.545053	+	0.838401i	$0.683491\pi$
$548$		0			0
$549$		−	10.5830i		−	0.451672i
$550$		0			0
$551$	−10.2016			−0.434603
$552$		0			0
$553$		−	23.1652i		−	0.985082i
$554$		0			0
$555$	1.16515	−	5.58258i	0.0494579	−	0.236967i
$556$		0			0
$557$	4.18710			0.177413			0.0887066	−	0.996058i	$-0.471727\pi$
$557$	4.18710			0.177413			0.0887066	+	0.996058i	$0.471727\pi$
$558$		0			0
$559$	48.8788			2.06735
$560$		0			0
$561$	−31.1652			−1.31579
$562$		0			0
$563$	29.4955			1.24309			0.621543	−	0.783380i	$-0.286506\pi$
$563$	29.4955			1.24309			0.621543	+	0.783380i	$0.286506\pi$
$564$		0			0
$565$	29.7309	+	6.20520i	1.25079	+	0.261055i
$566$		0			0
$567$			4.37780i			0.183850i
$568$		0			0
$569$	36.3303			1.52305			0.761523	−	0.648138i	$-0.224452\pi$
$569$	36.3303			1.52305			0.761523	+	0.648138i	$0.224452\pi$
$570$		0			0
$571$			20.0000i			0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$			20.0000i			0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$		0			0
$573$	−13.8564			−0.578860
$574$		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
$579$			15.1652i			0.630242i
$580$		0			0
$581$		−	31.3676i		−	1.30135i
$582$		0			0
$583$			43.7780i			1.81310i
$584$		0			0
$585$	2.00000	−	9.58258i	0.0826898	−	0.396191i
$586$		0			0
$587$	5.66970			0.234013			0.117007	−	0.993131i	$-0.462670\pi$
$587$	5.66970			0.234013			0.117007	+	0.993131i	$0.462670\pi$
$588$		0			0
$589$			21.1660i			0.872130i
$590$		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
$595$	11.1652	−	53.4955i	0.457727	−	2.19310i
$596$		0			0
$597$	−12.2197			−0.500119
$598$		0			0
$599$	−14.2378			−0.581741			−0.290871	−	0.956762i	$-0.593945\pi$
$599$	−14.2378			−0.581741			−0.290871	+	0.956762i	$0.593945\pi$
$600$		0			0
$601$	13.1652			0.537018			0.268509	−	0.963277i	$-0.413469\pi$
$601$	13.1652			0.537018			0.268509	+	0.963277i	$0.413469\pi$
$602$		0			0
$603$	3.16515			0.128895
$604$		0			0
$605$	−9.21245	+	44.1395i	−0.374540	+	1.79453i
$606$		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
$611$			30.3303i			1.22703i
$612$		0			0
$613$	32.4720			1.31153			0.655766	−	0.754964i	$-0.272346\pi$
$613$	32.4720			1.31153			0.655766	+	0.754964i	$0.272346\pi$
$614$		0			0
$615$	2.74110	−	13.1334i	0.110532	−	0.529590i
$616$		0			0
$617$		−	13.5826i		−	0.546814i	−0.961898	−	0.273407i	$-0.911849\pi$
$617$		−	13.5826i		−	0.546814i	0.961898	−	0.273407i	$-0.0881506\pi$
$618$		0			0
$619$		−	45.4955i		−	1.82862i	−0.405019	−	0.914308i	$-0.632735\pi$
$619$		−	45.4955i		−	1.82862i	0.405019	−	0.914308i	$-0.367265\pi$
$620$		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
$627$	22.3303			0.891786
$628$		0			0
$629$		−	14.2378i		−	0.567699i
$630$		0			0
$631$	12.2197			0.486459			0.243229	−	0.969969i	$-0.421793\pi$
$631$	12.2197			0.486459			0.243229	+	0.969969i	$0.421793\pi$
$632$		0			0
$633$			7.16515i			0.284789i
$634$		0			0
$635$	20.7477	+	4.33030i	0.823348	+	0.171843i
$636$		0			0
$637$	−53.2566			−2.11010
$638$		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
$643$	−18.3303			−0.722877			−0.361438	−	0.932396i	$-0.617714\pi$
$643$	−18.3303			−0.722877			−0.361438	+	0.932396i	$0.617714\pi$
$644$		0			0
$645$	5.10080	−	24.4394i	0.200844	−	0.962301i
$646$		0			0
$647$			17.8926i			0.703431i	0.936107	+	0.351716i	$0.114402\pi$
$647$			17.8926i			0.703431i	−0.936107	+	0.351716i	$0.885598\pi$
$648$		0			0
$649$	−8.83485			−0.346798
$650$		0			0
$651$			23.1652i			0.907914i
$652$		0			0
$653$	−22.0797			−0.864046			−0.432023	−	0.901863i	$-0.642200\pi$
$653$	−22.0797			−0.864046			−0.432023	+	0.901863i	$0.642200\pi$
$654$		0			0
$655$	−14.0471	−	2.93180i	−0.548866	−	0.114555i
$656$		0			0
$657$		−	12.0000i		−	0.468165i
$658$		0			0
$659$		−	25.5826i		−	0.996556i	−0.867017	−	0.498278i	$-0.833966\pi$
$659$		−	25.5826i		−	0.996556i	0.867017	−	0.498278i	$-0.166034\pi$
$660$		0			0
$661$		−	17.8926i		−	0.695942i	−0.937505	−	0.347971i	$-0.886871\pi$
$661$		−	17.8926i		−	0.695942i	0.937505	−	0.347971i	$-0.113129\pi$
$662$		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
$669$			4.37780i			0.169256i
$670$		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
$675$	−4.58258	−	2.00000i	−0.176383	−	0.0769800i
$676$		0			0
$677$	−28.6265			−1.10021			−0.550103	−	0.835097i	$-0.685412\pi$
$677$	−28.6265			−1.10021			−0.550103	+	0.835097i	$0.685412\pi$
$678$		0			0
$679$	−48.8788			−1.87580
$680$		0			0
$681$	−15.1652			−0.581130
$682$		0			0
$683$	5.49545			0.210278			0.105139	−	0.994458i	$-0.466471\pi$
$683$	5.49545			0.210278			0.105139	+	0.994458i	$0.466471\pi$
$684$		0			0
$685$	22.8027	+	4.75920i	0.871247	+	0.181840i
$686$		0			0
$687$		−	12.0290i		−	0.458935i
$688$		0			0
$689$	−34.3303			−1.30788
$690$		0			0
$691$			40.6606i			1.54680i	0.633917	+	0.773401i	$0.281446\pi$
$691$			40.6606i			1.54680i	−0.633917	+	0.773401i	$0.718554\pi$
$692$		0			0
$693$	24.4394			0.928376
$694$		0			0
$695$	15.6838	+	3.27340i	0.594921	+	0.124167i
$696$		0			0
$697$		−	33.4955i		−	1.26873i
$698$		0			0
$699$		−	10.4174i		−	0.394023i
$700$		0			0
$701$		−	45.9470i		−	1.73540i	−0.497093	−	0.867698i	$-0.665599\pi$
$701$		−	45.9470i		−	1.73540i	0.497093	−	0.867698i	$-0.334401\pi$
$702$		0			0
$703$			10.2016i			0.384761i
$704$		0			0
$705$	15.1652	+	3.16515i	0.571153	+	0.119206i
$706$		0			0
$707$	3.16515			0.119038
$708$		0			0
$709$			15.6838i			0.589018i	0.955649	+	0.294509i	$0.0951561\pi$
$709$			15.6838i			0.589018i	−0.955649	+	0.294509i	$0.904844\pi$
$710$		0			0
$711$	−5.29150			−0.198447
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−53.4955	−	11.1652i	−2.00062	−	0.417553i
$716$		0			0
$717$	−21.1660			−0.790459
$718$		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
$723$	−10.0000			−0.371904
$724$		0			0
$725$	5.10080	−	11.6874i	0.189439	−	0.434059i
$726$		0			0
$727$		−	49.6018i		−	1.83963i	−0.392353	−	0.919815i	$-0.628339\pi$
$727$		−	49.6018i		−	1.83963i	0.392353	−	0.919815i	$-0.371661\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	62.3303i		−	2.30537i
$732$		0			0
$733$	−29.1986			−1.07848			−0.539238	−	0.842154i	$-0.681288\pi$
$733$	−29.1986			−1.07848			−0.539238	+	0.842154i	$0.681288\pi$
$734$		0			0
$735$	−5.55765	+	26.6283i	−0.204997	+	0.982200i
$736$		0			0
$737$		−	17.6697i		−	0.650872i
$738$		0			0
$739$			20.0000i			0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$			20.0000i			0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$		0			0
$741$			17.5112i			0.643290i
$742$		0			0
$743$		−	47.4328i		−	1.74014i	−0.492927	−	0.870071i	$-0.664073\pi$
$743$		−	47.4328i		−	1.74014i	0.492927	−	0.870071i	$-0.335927\pi$
$744$		0			0
$745$	32.7477	+	6.83485i	1.19978	+	0.250409i
$746$		0			0
$747$	−7.16515			−0.262159
$748$		0			0
$749$		−	17.5112i		−	0.639846i
$750$		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
$755$	−7.25227	+	34.7477i	−0.263937	+	1.26460i
$756$		0			0
$757$	−18.2342			−0.662734			−0.331367	−	0.943502i	$-0.607510\pi$
$757$	−18.2342			−0.662734			−0.331367	+	0.943502i	$0.607510\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$	8.00000			0.289619
$764$		0			0
$765$	−12.2197	−	2.55040i	−0.441804	−	0.0922099i
$766$		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
$771$			24.7477i			0.891268i
$772$		0			0
$773$	30.4539			1.09535			0.547676	−	0.836691i	$-0.315513\pi$
$773$	30.4539			1.09535			0.547676	+	0.836691i	$0.315513\pi$
$774$		0			0
$775$	−24.2487	−	10.5830i	−0.871039	−	0.380153i
$776$		0			0
$777$			11.1652i			0.400548i
$778$		0			0
$779$			24.0000i			0.859889i
$780$		0			0
$781$			38.6772i			1.38398i
$782$		0			0
$783$			2.55040i			0.0911439i
$784$		0			0
$785$	7.49545	−	35.9129i	0.267524	−	1.28179i
$786$		0			0
$787$	36.0000			1.28326			0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000			1.28326			0.641631	+	0.767014i	$0.278258\pi$
$788$		0			0
$789$		−	1.82740i		−	0.0650572i
$790$		0			0
$791$	−59.4618			−2.11422
$792$		0			0
$793$		−	46.3303i		−	1.64524i
$794$		0			0
$795$	−3.58258	+	17.1652i	−0.127061	+	0.608785i
$796$		0			0
$797$	−53.0659			−1.87969			−0.939846	−	0.341599i	$-0.889031\pi$
$797$	−53.0659			−1.87969			−0.939846	+	0.341599i	$0.889031\pi$
$798$		0			0
$799$	38.6772			1.36830
$800$		0			0
$801$	−2.00000			−0.0706665
$802$		0			0
$803$	−66.9909			−2.36406
$804$		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	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$		−	0.834849i		−	0.0293155i	−0.999893	−	0.0146577i	$-0.995334\pi$
$811$		−	0.834849i		−	0.0293155i	0.999893	−	0.0146577i	$-0.00466587\pi$
$812$		0			0
$813$	15.4931			0.543367
$814$		0			0
$815$	−1.82740	+	8.75560i	−0.0640111	+	0.306695i
$816$		0			0
$817$			44.6606i			1.56248i
$818$		0			0
$819$			19.1652i			0.669685i
$820$		0			0
$821$			6.20520i			0.216563i	0.994120	+	0.108282i	$0.0345348\pi$
$821$			6.20520i			0.216563i	−0.994120	+	0.108282i	$0.965465\pi$
$822$		0			0
$823$		−	9.47860i		−	0.330403i	−0.986260	−	0.165202i	$-0.947172\pi$
$823$		−	9.47860i		−	0.330403i	0.986260	−	0.165202i	$-0.0528275\pi$
$824$		0			0
$825$	−11.1652	+	25.5826i	−0.388721	+	0.890671i
$826$		0			0
$827$	−28.0000			−0.973655			−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000			−0.973655			−0.486828	+	0.873498i	$0.661846\pi$
$828$		0			0
$829$			36.8498i			1.27985i	0.768439	+	0.639924i	$0.221034\pi$
$829$			36.8498i			1.27985i	−0.768439	+	0.639924i	$0.778966\pi$
$830$		0			0
$831$	9.47860			0.328809
$832$		0			0
$833$			67.9129i			2.35304i
$834$		0			0
$835$	26.3303	+	5.49545i	0.911198	+	0.190178i
$836$		0			0
$837$	5.29150			0.182901
$838$		0			0
$839$	55.8070			1.92667			0.963336	−	0.268297i	$-0.0864608\pi$
$839$	55.8070			1.92667			0.963336	+	0.268297i	$0.0864608\pi$
$840$		0			0
$841$	22.4955			0.775705
$842$		0			0
$843$	32.3303			1.11351
$844$		0			0
$845$	2.81655	−	13.4949i	0.0968923	−	0.464239i
$846$		0			0
$847$		−	88.2790i		−	3.03330i
$848$		0			0
$849$	−11.1652			−0.383187
$850$		0			0
$851$		0			0
$852$		0			0
$853$	7.65120			0.261972			0.130986	−	0.991384i	$-0.458186\pi$
$853$	7.65120			0.261972			0.130986	+	0.991384i	$0.458186\pi$
$854$		0			0
$855$	8.75560	+	1.82740i	0.299435	+	0.0624958i
$856$		0			0
$857$			29.5826i			1.01052i	0.862967	+	0.505261i	$0.168604\pi$
$857$			29.5826i			1.01052i	−0.862967	+	0.505261i	$0.831396\pi$
$858$		0			0
$859$		−	24.8348i		−	0.847354i	−0.905813	−	0.423677i	$-0.860739\pi$
$859$		−	24.8348i		−	0.847354i	0.905813	−	0.423677i	$-0.139261\pi$
$860$		0			0
$861$			26.2668i			0.895171i
$862$		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
$867$	−14.1652			−0.481074
$868$		0			0
$869$			29.5402i			1.00208i
$870$		0			0
$871$	13.8564			0.469506
$872$		0			0
$873$			11.1652i			0.377883i
$874$		0			0
$875$	−39.9129	−	28.3303i	−1.34930	−	0.957739i
$876$		0			0
$877$	−17.8528			−0.602846			−0.301423	−	0.953490i	$-0.597462\pi$
$877$	−17.8528			−0.602846			−0.301423	+	0.953490i	$0.597462\pi$
$878$		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
$883$	−34.3303			−1.15531			−0.577653	−	0.816282i	$-0.696031\pi$
$883$	−34.3303			−1.15531			−0.577653	+	0.816282i	$0.696031\pi$
$884$		0			0
$885$	−3.46410	−	0.723000i	−0.116445	−	0.0243034i
$886$		0			0
$887$		−	47.0514i		−	1.57983i	−0.613215	−	0.789916i	$-0.710124\pi$
$887$		−	47.0514i		−	1.57983i	0.613215	−	0.789916i	$-0.289876\pi$
$888$		0			0
$889$	−41.4955			−1.39171
$890$		0			0
$891$		−	5.58258i		−	0.187023i
$892$		0			0
$893$	−27.7128			−0.927374
$894$		0			0
$895$	3.46410	+	0.723000i	0.115792	+	0.0241672i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			13.4955i			0.450099i
$900$		0			0
$901$			43.7780i			1.45846i
$902$		0			0
$903$			48.8788i			1.62659i
$904$		0			0
$905$	4.00000	+	0.834849i	0.132964	+	0.0277513i
$906$		0			0
$907$	−17.4955			−0.580927			−0.290464	−	0.956886i	$-0.593809\pi$
$907$	−17.4955			−0.580927			−0.290464	+	0.956886i	$0.593809\pi$
$908$		0			0
$909$		−	0.723000i		−	0.0239804i
$910$		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
$915$	−23.1652	−	4.83485i	−0.765816	−	0.159835i
$916$		0			0
$917$	28.0942			0.927753
$918$		0			0
$919$	11.8383			0.390510			0.195255	−	0.980753i	$-0.437447\pi$
$919$	11.8383			0.390510			0.195255	+	0.980753i	$0.437447\pi$
$920$		0			0
$921$	−4.00000			−0.131804
$922$		0			0
$923$	−30.3303			−0.998334
$924$		0			0
$925$	−11.6874	−	5.10080i	−0.384280	−	0.167713i
$926$		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
$931$		−	48.6606i		−	1.59479i
$932$		0			0
$933$	3.65480			0.119653
$934$		0			0
$935$	−14.2378	+	68.2174i	−0.465626	+	2.23095i
$936$		0			0
$937$			14.3303i			0.468151i	0.972218	+	0.234075i	$0.0752062\pi$
$937$			14.3303i			0.468151i	−0.972218	+	0.234075i	$0.924794\pi$
$938$		0			0
$939$		−	0.834849i		−	0.0272443i
$940$		0			0
$941$		−	28.4358i		−	0.926981i	−0.886102	−	0.463491i	$-0.846597\pi$
$941$		−	28.4358i		−	0.926981i	0.886102	−	0.463491i	$-0.153403\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	9.58258	+	2.00000i	0.311721	+	0.0650600i
$946$		0			0
$947$	52.0000			1.68977			0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000			1.68977			0.844886	+	0.534946i	$0.179668\pi$
$948$		0			0
$949$		−	52.5336i		−	1.70531i
$950$		0			0
$951$	34.1087			1.10605
$952$		0			0
$953$			26.2432i			0.850100i	0.905170	+	0.425050i	$0.139743\pi$
$953$			26.2432i			0.850100i	−0.905170	+	0.425050i	$0.860257\pi$
$954$		0			0
$955$	−6.33030	+	30.3303i	−0.204844	+	0.981466i
$956$		0			0
$957$	14.2378			0.460243
$958$		0			0
$959$	−45.6054			−1.47268
$960$		0			0
$961$	−3.00000			−0.0967742
$962$		0			0
$963$	−4.00000			−0.128898
$964$		0			0
$965$	33.1950	+	6.92820i	1.06859	+	0.223027i
$966$		0			0
$967$		−	26.9898i		−	0.867934i	−0.900929	−	0.433967i	$-0.857113\pi$
$967$		−	26.9898i		−	0.867934i	0.900929	−	0.433967i	$-0.142887\pi$
$968$		0			0
$969$	22.3303			0.717353
$970$		0			0
$971$		−	4.08712i		−	0.131162i	−0.997847	−	0.0655810i	$-0.979110\pi$
$971$		−	4.08712i		−	0.131162i	0.997847	−	0.0655810i	$-0.0208901\pi$
$972$		0			0
$973$	−31.3676			−1.00560
$974$		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
$979$			11.1652i			0.356840i
$980$		0			0
$981$		−	1.82740i		−	0.0583444i
$982$		0			0
$983$		−	2.20880i		−	0.0704498i	−0.999379	−	0.0352249i	$-0.988785\pi$
$983$		−	2.20880i		−	0.0704498i	0.999379	−	0.0352249i	$-0.0112148\pi$
$984$		0			0
$985$	−10.0871	+	48.3303i	−0.321402	+	1.53993i
$986$		0			0
$987$	−30.3303			−0.965424
$988$		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
$995$	−5.58258	+	26.7477i	−0.176980	+	0.847960i
$996$		0			0
$997$	−12.7520			−0.403860			−0.201930	−	0.979400i	$-0.564721\pi$
$997$	−12.7520			−0.403860			−0.201930	+	0.979400i	$0.564721\pi$
$998$		0			0
$999$	2.55040			0.0806911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.d.f.289.5	yes	8
3.2	odd	2		2880.2.d.i.289.4		8
4.3	odd	2		960.2.d.e.289.5	yes	8
5.2	odd	4		4800.2.k.r.2401.1		8
5.3	odd	4		4800.2.k.q.2401.8		8
5.4	even	2		960.2.d.e.289.3	✓	8
8.3	odd	2	inner	960.2.d.f.289.4	yes	8
8.5	even	2		960.2.d.e.289.4	yes	8
12.11	even	2		2880.2.d.j.289.4		8
15.14	odd	2		2880.2.d.j.289.6		8
16.3	odd	4		3840.2.f.k.769.1		8
16.5	even	4		3840.2.f.k.769.4		8
16.11	odd	4		3840.2.f.i.769.8		8
16.13	even	4		3840.2.f.i.769.5		8
20.3	even	4		4800.2.k.q.2401.1		8
20.7	even	4		4800.2.k.r.2401.8		8
20.19	odd	2	inner	960.2.d.f.289.3	yes	8
24.5	odd	2		2880.2.d.j.289.5		8
24.11	even	2		2880.2.d.i.289.5		8
40.3	even	4		4800.2.k.q.2401.5		8
40.13	odd	4		4800.2.k.q.2401.4		8
40.19	odd	2		960.2.d.e.289.6	yes	8
40.27	even	4		4800.2.k.r.2401.4		8
40.29	even	2	inner	960.2.d.f.289.6	yes	8
40.37	odd	4		4800.2.k.r.2401.5		8
60.59	even	2		2880.2.d.i.289.6		8
80.19	odd	4		3840.2.f.k.769.5		8
80.29	even	4		3840.2.f.i.769.1		8
80.59	odd	4		3840.2.f.i.769.4		8
80.69	even	4		3840.2.f.k.769.8		8
120.29	odd	2		2880.2.d.i.289.3		8
120.59	even	2		2880.2.d.j.289.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.2.d.e.289.3	✓	8	5.4	even	2
960.2.d.e.289.4	yes	8	8.5	even	2
960.2.d.e.289.5	yes	8	4.3	odd	2
960.2.d.e.289.6	yes	8	40.19	odd	2
960.2.d.f.289.3	yes	8	20.19	odd	2	inner
960.2.d.f.289.4	yes	8	8.3	odd	2	inner
960.2.d.f.289.5	yes	8	1.1	even	1	trivial
960.2.d.f.289.6	yes	8	40.29	even	2	inner
2880.2.d.i.289.3		8	120.29	odd	2
2880.2.d.i.289.4		8	3.2	odd	2
2880.2.d.i.289.5		8	24.11	even	2
2880.2.d.i.289.6		8	60.59	even	2
2880.2.d.j.289.3		8	120.59	even	2
2880.2.d.j.289.4		8	12.11	even	2
2880.2.d.j.289.5		8	24.5	odd	2
2880.2.d.j.289.6		8	15.14	odd	2
3840.2.f.i.769.1		8	80.29	even	4
3840.2.f.i.769.4		8	80.59	odd	4
3840.2.f.i.769.5		8	16.13	even	4
3840.2.f.i.769.8		8	16.11	odd	4
3840.2.f.k.769.1		8	16.3	odd	4
3840.2.f.k.769.4		8	16.5	even	4
3840.2.f.k.769.5		8	80.19	odd	4
3840.2.f.k.769.8		8	80.69	even	4
4800.2.k.q.2401.1		8	20.3	even	4
4800.2.k.q.2401.4		8	40.13	odd	4
4800.2.k.q.2401.5		8	40.3	even	4
4800.2.k.q.2401.8		8	5.3	odd	4
4800.2.k.r.2401.1		8	5.2	odd	4
4800.2.k.r.2401.4		8	40.27	even	4
4800.2.k.r.2401.5		8	40.37	odd	4
4800.2.k.r.2401.8		8	20.7	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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +(0.456850 - 2.18890i) q^{5} +4.37780i q^{7} +1.00000 q^{9} -5.58258i q^{11} +4.37780 q^{13} +(0.456850 - 2.18890i) q^{15} -5.58258i q^{17} +4.00000i q^{19} +4.37780i q^{21} +(-4.58258 - 2.00000i) q^{25} +1.00000 q^{27} +2.55040i q^{29} +5.29150 q^{31} -5.58258i q^{33} +(9.58258 + 2.00000i) q^{35} +2.55040 q^{37} +4.37780 q^{39} +6.00000 q^{41} +11.1652 q^{43} +(0.456850 - 2.18890i) q^{45} +6.92820i q^{47} -12.1652 q^{49} -5.58258i q^{51} -7.84190 q^{53} +(-12.2197 - 2.55040i) q^{55} +4.00000i q^{57} -1.58258i q^{59} -10.5830i q^{61} +4.37780i q^{63} +(2.00000 - 9.58258i) q^{65} +3.16515 q^{67} -6.92820 q^{71} -12.0000i q^{73} +(-4.58258 - 2.00000i) q^{75} +24.4394 q^{77} -5.29150 q^{79} +1.00000 q^{81} -7.16515 q^{83} +(-12.2197 - 2.55040i) q^{85} +2.55040i q^{87} -2.00000 q^{89} +19.1652i q^{91} +5.29150 q^{93} +(8.75560 + 1.82740i) q^{95} +11.1652i q^{97} -5.58258i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 8 q^{9} + 8 q^{27} + 40 q^{35} + 48 q^{41} + 16 q^{43} - 24 q^{49} + 16 q^{65} - 48 q^{67} + 8 q^{81} + 16 q^{83} - 16 q^{89}+O(q^{100}) \) 8 * q + 8 * q^3 + 8 * q^9 + 8 * q^27 + 40 * q^35 + 48 * q^41 + 16 * q^43 - 24 * q^49 + 16 * q^65 - 48 * q^67 + 8 * q^81 + 16 * q^83 - 16 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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