LMFDB - Embedded newform 672.2.j.d.239.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(239,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("672.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.2593100598870016.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 $ x^12 - 2x^10 + x^8 + 4x^6 + 4x^4 - 32x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.9
Root			$0.430469 + 1.34711i$ of defining polynomial
Character	$\chi$	$=$	672.239
Dual form			672.2.j.d.239.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.46962 - 0.916638i) q^{3} -1.83328 q^{5} +1.00000i q^{7} +(1.31955 - 2.69421i) q^{9} +O(q^{10})$	$q+(1.46962 - 0.916638i) q^{3} -1.83328 q^{5} +1.00000i q^{7} +(1.31955 - 2.69421i) q^{9} -5.57834i q^{13} +(-2.69421 + 1.68045i) q^{15} -6.08565i q^{17} +6.93923 q^{19} +(0.916638 + 1.46962i) q^{21} -0.697224 q^{23} -1.63910 q^{25} +(-0.530383 - 5.16901i) q^{27} +7.11030 q^{29} -1.83328i q^{35} -1.87847i q^{37} +(-5.11331 - 8.19802i) q^{39} +4.69120i q^{41} -1.36090 q^{43} +(-2.41910 + 4.93923i) q^{45} +3.44375 q^{47} -1.00000 q^{49} +(-5.57834 - 8.94358i) q^{51} -12.1713 q^{53} +(10.1980 - 6.36076i) q^{57} +8.94358i q^{59} +4.30013i q^{61} +(2.69421 + 1.31955i) q^{63} +10.2266i q^{65} -4.51757 q^{67} +(-1.02465 + 0.639102i) q^{69} +6.08565 q^{71} -6.00000 q^{73} +(-2.40885 + 1.50246i) q^{75} +11.7569i q^{79} +(-5.51757 - 7.11030i) q^{81} +0.438828i q^{83} +11.1567i q^{85} +(10.4494 - 6.51757i) q^{87} +2.64190i q^{89} +5.57834 q^{91} -12.7215 q^{95} +11.0351 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{9}+O(q^{10}) $ 12 * q + 4 * q^9	$ 12 q + 4 q^{9} + 48 q^{19} + 4 q^{25} - 24 q^{27} - 40 q^{43} - 12 q^{49} - 8 q^{51} + 40 q^{57} + 40 q^{67} - 72 q^{73} + 24 q^{75} + 28 q^{81} + 8 q^{91} - 56 q^{97}+O(q^{100}) $ 12 * q + 4 * q^9 + 48 * q^19 + 4 * q^25 - 24 * q^27 - 40 * q^43 - 12 * q^49 - 8 * q^51 + 40 * q^57 + 40 * q^67 - 72 * q^73 + 24 * q^75 + 28 * q^81 + 8 * q^91 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	1.46962	−	0.916638i	0.848484	−	0.529221i
$3$	1.46962	−	0.916638i	0.848484	−	0.529221i
$4$		0			0
$5$	−1.83328			−0.819866			−0.409933	−	0.912116i	$-0.634448\pi$
$5$	−1.83328			−0.819866			−0.409933	+	0.912116i	$0.634448\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.31955	−	2.69421i	0.439850	−	0.898071i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		−	5.57834i		−	1.54715i	−0.633703	−	0.773576i	$-0.718466\pi$
$13$		−	5.57834i		−	1.54715i	0.633703	−	0.773576i	$-0.281534\pi$
$14$		0			0
$15$	−2.69421	+	1.68045i	−0.695643	+	0.433890i
$16$		0			0
$17$		−	6.08565i		−	1.47599i	−0.674808	−	0.737994i	$-0.735773\pi$
$17$		−	6.08565i		−	1.47599i	0.674808	−	0.737994i	$-0.264227\pi$
$18$		0			0
$19$	6.93923			1.59197			0.795985	−	0.605317i	$-0.206953\pi$
$19$	6.93923			1.59197			0.795985	+	0.605317i	$0.206953\pi$
$20$		0			0
$21$	0.916638	+	1.46962i	0.200027	+	0.320697i
$22$		0			0
$23$	−0.697224			−0.145381			−0.0726906	−	0.997355i	$-0.523159\pi$
$23$	−0.697224			−0.145381			−0.0726906	+	0.997355i	$0.523159\pi$
$24$		0			0
$25$	−1.63910			−0.327820
$26$		0			0
$27$	−0.530383	−	5.16901i	−0.102072	−	0.994777i
$28$		0			0
$29$	7.11030			1.32035			0.660175	−	0.751112i	$-0.270482\pi$
$29$	7.11030			1.32035			0.660175	+	0.751112i	$0.270482\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1.83328i		−	0.309880i
$36$		0			0
$37$		−	1.87847i		−	0.308819i	−0.988007	−	0.154409i	$-0.950653\pi$
$37$		−	1.87847i		−	0.308819i	0.988007	−	0.154409i	$-0.0493474\pi$
$38$		0			0
$39$	−5.11331	−	8.19802i	−0.818785	−	1.31273i
$40$		0			0
$41$			4.69120i			0.732643i	0.930488	+	0.366321i	$0.119383\pi$
$41$			4.69120i			0.732643i	−0.930488	+	0.366321i	$0.880617\pi$
$42$		0			0
$43$	−1.36090			−0.207535			−0.103768	−	0.994602i	$-0.533090\pi$
$43$	−1.36090			−0.207535			−0.103768	+	0.994602i	$0.533090\pi$
$44$		0			0
$45$	−2.41910	+	4.93923i	−0.360618	+	0.736298i
$46$		0			0
$47$	3.44375			0.502323			0.251161	−	0.967945i	$-0.419187\pi$
$47$	3.44375			0.502323			0.251161	+	0.967945i	$0.419187\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−5.57834	−	8.94358i	−0.781123	−	1.25235i
$52$		0			0
$53$	−12.1713			−1.67186			−0.835928	−	0.548839i	$-0.815070\pi$
$53$	−12.1713			−1.67186			−0.835928	+	0.548839i	$0.815070\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	10.1980	−	6.36076i	1.35076	−	0.842504i
$58$		0			0
$59$			8.94358i			1.16435i	0.813062	+	0.582177i	$0.197799\pi$
$59$			8.94358i			1.16435i	−0.813062	+	0.582177i	$0.802201\pi$
$60$		0			0
$61$			4.30013i			0.550576i	0.961362	+	0.275288i	$0.0887732\pi$
$61$			4.30013i			0.550576i	−0.961362	+	0.275288i	$0.911227\pi$
$62$		0			0
$63$	2.69421	+	1.31955i	0.339439	+	0.166248i
$64$		0			0
$65$			10.2266i			1.26846i
$66$		0			0
$67$	−4.51757			−0.551909			−0.275955	−	0.961171i	$-0.588994\pi$
$67$	−4.51757			−0.551909			−0.275955	+	0.961171i	$0.588994\pi$
$68$		0			0
$69$	−1.02465	+	0.639102i	−0.123354	+	0.0769388i
$70$		0			0
$71$	6.08565			0.722234			0.361117	−	0.932521i	$-0.382396\pi$
$71$	6.08565			0.722234			0.361117	+	0.932521i	$0.382396\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$	−2.40885	+	1.50246i	−0.278150	+	0.173489i
$76$		0			0
$77$		0			0
$78$		0			0
$79$			11.7569i			1.32276i	0.750051	+	0.661380i	$0.230029\pi$
$79$			11.7569i			1.32276i	−0.750051	+	0.661380i	$0.769971\pi$
$80$		0			0
$81$	−5.51757	−	7.11030i	−0.613063	−	0.790034i
$82$		0			0
$83$			0.438828i			0.0481676i	0.999710	+	0.0240838i	$0.00766685\pi$
$83$			0.438828i			0.0481676i	−0.999710	+	0.0240838i	$0.992333\pi$
$84$		0			0
$85$			11.1567i			1.21011i
$86$		0			0
$87$	10.4494	−	6.51757i	1.12030	−	0.698757i
$88$		0			0
$89$			2.64190i			0.280041i	0.990149	+	0.140020i	$0.0447168\pi$
$89$			2.64190i			0.280041i	−0.990149	+	0.140020i	$0.955283\pi$
$90$		0			0
$91$	5.57834			0.584769
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−12.7215			−1.30520
$96$		0			0
$97$	11.0351			1.12045			0.560224	−	0.828341i	$-0.310715\pi$
$97$	11.0351			1.12045			0.560224	+	0.828341i	$0.310715\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.3380			1.02867			0.514336	−	0.857589i	$-0.328038\pi$
$101$	10.3380			1.02867			0.514336	+	0.857589i	$0.328038\pi$
$102$		0			0
$103$		−	11.7569i		−	1.15845i	−0.815169	−	0.579223i	$-0.803356\pi$
$103$		−	11.7569i		−	1.15845i	0.815169	−	0.579223i	$-0.196644\pi$
$104$		0			0
$105$	−1.68045	−	2.69421i	−0.163995	−	0.262928i
$106$		0			0
$107$			3.66655i			0.354459i	0.984170	+	0.177229i	$0.0567135\pi$
$107$			3.66655i			0.354459i	−0.984170	+	0.177229i	$0.943287\pi$
$108$		0			0
$109$		−	17.8785i		−	1.71245i	−0.516606	−	0.856223i	$-0.672805\pi$
$109$		−	17.8785i		−	1.71245i	0.516606	−	0.856223i	$-0.327195\pi$
$110$		0			0
$111$	−1.72188	−	2.76063i	−0.163433	−	0.262028i
$112$		0			0
$113$			5.38843i			0.506901i	0.967348	+	0.253450i	$0.0815654\pi$
$113$			5.38843i			0.506901i	−0.967348	+	0.253450i	$0.918435\pi$
$114$		0			0
$115$	1.27820			0.119193
$116$		0			0
$117$	−15.0292	−	7.36090i	−1.38945	−	0.680515i
$118$		0			0
$119$	6.08565			0.557871
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	4.30013	+	6.89427i	0.387730	+	0.621636i
$124$		0			0
$125$	12.1713			1.08863
$126$		0			0
$127$			8.39604i			0.745028i	0.928027	+	0.372514i	$0.121504\pi$
$127$			8.39604i			0.745028i	−0.928027	+	0.372514i	$0.878496\pi$
$128$		0			0
$129$	−2.00000	+	1.24745i	−0.176090	+	0.109832i
$130$		0			0
$131$			6.89427i			0.602355i	0.953568	+	0.301178i	$0.0973797\pi$
$131$			6.89427i			0.602355i	−0.953568	+	0.301178i	$0.902620\pi$
$132$		0			0
$133$			6.93923i			0.601708i
$134$		0			0
$135$	0.972337	+	9.47622i	0.0836855	+	0.815583i
$136$		0			0
$137$			19.5044i			1.66637i	0.552992	+	0.833187i	$0.313486\pi$
$137$			19.5044i			1.66637i	−0.552992	+	0.833187i	$0.686514\pi$
$138$		0			0
$139$	−12.2174			−1.03627			−0.518135	−	0.855299i	$-0.673373\pi$
$139$	−12.2174			−1.03627			−0.518135	+	0.855299i	$0.673373\pi$
$140$		0			0
$141$	5.06100	−	3.15667i	0.426213	−	0.265840i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−13.0351			−1.08251
$146$		0			0
$147$	−1.46962	+	0.916638i	−0.121212	+	0.0756030i
$148$		0			0
$149$	4.83820			0.396361			0.198180	−	0.980166i	$-0.436497\pi$
$149$	4.83820			0.396361			0.198180	+	0.980166i	$0.436497\pi$
$150$		0			0
$151$		−	15.1178i		−	1.23027i	−0.788421	−	0.615136i	$-0.789101\pi$
$151$		−	15.1178i		−	1.23027i	0.788421	−	0.615136i	$-0.210899\pi$
$152$		0			0
$153$	−16.3960	−	8.03033i	−1.32554	−	0.649213i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			9.33528i			0.745036i	0.928025	+	0.372518i	$0.121506\pi$
$157$			9.33528i			0.745036i	−0.928025	+	0.372518i	$0.878494\pi$
$158$		0			0
$159$	−17.8872	+	11.1567i	−1.41854	+	0.884782i
$160$		0			0
$161$		−	0.697224i		−	0.0549489i
$162$		0			0
$163$	−6.63910			−0.520015			−0.260007	−	0.965607i	$-0.583725\pi$
$163$	−6.63910			−0.520015			−0.260007	+	0.965607i	$0.583725\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	19.5044			1.50930			0.754648	−	0.656130i	$-0.227808\pi$
$167$	19.5044			1.50930			0.754648	+	0.656130i	$0.227808\pi$
$168$		0			0
$169$	−18.1178			−1.39368
$170$		0			0
$171$	9.15667	−	18.6958i	0.700228	−	1.42970i
$172$		0			0
$173$	11.2157			0.852712			0.426356	−	0.904555i	$-0.359797\pi$
$173$	11.2157			0.852712			0.426356	+	0.904555i	$0.359797\pi$
$174$		0			0
$175$		−	1.63910i		−	0.123904i
$176$		0			0
$177$	8.19802	+	13.1436i	0.616201	+	0.987936i
$178$		0			0
$179$			20.6761i			1.54540i	0.634771	+	0.772700i	$0.281094\pi$
$179$			20.6761i			1.54540i	−0.634771	+	0.772700i	$0.718906\pi$
$180$		0			0
$181$			4.30013i			0.319626i	0.987147	+	0.159813i	$0.0510892\pi$
$181$			4.30013i			0.319626i	−0.987147	+	0.159813i	$0.948911\pi$
$182$		0			0
$183$	3.94166	+	6.31955i	0.291376	+	0.467155i
$184$		0			0
$185$			3.44375i			0.253190i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	5.16901	−	0.530383i	0.375990	−	0.0385797i
$190$		0			0
$191$	−24.1956			−1.75073			−0.875366	−	0.483460i	$-0.839380\pi$
$191$	−24.1956			−1.75073			−0.875366	+	0.483460i	$0.839380\pi$
$192$		0			0
$193$	−6.39604			−0.460397			−0.230199	−	0.973144i	$-0.573938\pi$
$193$	−6.39604			−0.460397			−0.230199	+	0.973144i	$0.573938\pi$
$194$		0			0
$195$	9.37411	+	15.0292i	0.671294	+	1.07627i
$196$		0			0
$197$	−3.44375			−0.245357			−0.122679	−	0.992446i	$-0.539148\pi$
$197$	−3.44375			−0.245357			−0.122679	+	0.992446i	$0.539148\pi$
$198$		0			0
$199$			9.27820i			0.657714i	0.944380	+	0.328857i	$0.106663\pi$
$199$			9.27820i			0.657714i	−0.944380	+	0.328857i	$0.893337\pi$
$200$		0			0
$201$	−6.63910	+	4.14098i	−0.468286	+	0.292082i
$202$		0			0
$203$			7.11030i			0.499045i
$204$		0			0
$205$		−	8.60027i		−	0.600669i
$206$		0			0
$207$	−0.920022	+	1.87847i	−0.0639460	+	0.130563i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.760632			−0.0523641			−0.0261820	−	0.999657i	$-0.508335\pi$
$211$	−0.760632			−0.0523641			−0.0261820	+	0.999657i	$0.508335\pi$
$212$		0			0
$213$	8.94358	−	5.57834i	0.612804	−	0.382221i
$214$		0			0
$215$	2.49490			0.170151
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−8.81770	+	5.49983i	−0.595845	+	0.371644i
$220$		0			0
$221$	−33.9478			−2.28358
$222$		0			0
$223$			29.0351i			1.94434i	0.234283	+	0.972168i	$0.424726\pi$
$223$			29.0351i			1.94434i	−0.234283	+	0.972168i	$0.575274\pi$
$224$		0			0
$225$	−2.16288	+	4.41609i	−0.144192	+	0.294406i
$226$		0			0
$227$			3.88258i			0.257696i	0.991664	+	0.128848i	$0.0411279\pi$
$227$			3.88258i			0.257696i	−0.991664	+	0.128848i	$0.958872\pi$
$228$		0			0
$229$		−	3.02193i		−	0.199695i	−0.995003	−	0.0998474i	$-0.968165\pi$
$229$		−	3.02193i		−	0.199695i	0.995003	−	0.0998474i	$-0.0318355\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	12.1713i		−	0.797368i	−0.917088	−	0.398684i	$-0.869467\pi$
$233$		−	12.1713i		−	0.797368i	0.917088	−	0.398684i	$-0.130533\pi$
$234$		0			0
$235$	−6.31335			−0.411837
$236$		0			0
$237$	10.7769	+	17.2782i	0.700032	+	1.12234i
$238$		0			0
$239$	22.2509			1.43929			0.719647	−	0.694341i	$-0.244304\pi$
$239$	22.2509			1.43929			0.719647	+	0.694341i	$0.244304\pi$
$240$		0			0
$241$	15.0351			0.968499			0.484249	−	0.874930i	$-0.339093\pi$
$241$	15.0351			0.968499			0.484249	+	0.874930i	$0.339093\pi$
$242$		0			0
$243$	−14.6263	−	5.39181i	−0.938277	−	0.345885i
$244$		0			0
$245$	1.83328			0.117124
$246$		0			0
$247$		−	38.7094i		−	2.46302i
$248$		0			0
$249$	0.402246	+	0.644909i	0.0254913	+	0.0408694i
$250$		0			0
$251$		−	23.3870i		−	1.47617i	−0.674706	−	0.738087i	$-0.735730\pi$
$251$		−	23.3870i		−	1.47617i	0.674706	−	0.738087i	$-0.264270\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	10.2266	+	16.3960i	0.640416	+	1.02676i
$256$		0			0
$257$		−	20.3063i		−	1.26667i	−0.773878	−	0.633335i	$-0.781686\pi$
$257$		−	20.3063i		−	1.26667i	0.773878	−	0.633335i	$-0.218314\pi$
$258$		0			0
$259$	1.87847			0.116722
$260$		0			0
$261$	9.38241	−	19.1567i	0.580756	−	1.18577i
$262$		0			0
$263$	−18.2570			−1.12577			−0.562886	−	0.826535i	$-0.690309\pi$
$263$	−18.2570			−1.12577			−0.562886	+	0.826535i	$0.690309\pi$
$264$		0			0
$265$	22.3133			1.37070
$266$		0			0
$267$	2.42166	+	3.88258i	0.148203	+	0.237610i
$268$		0			0
$269$	−0.216029			−0.0131715			−0.00658577	−	0.999978i	$-0.502096\pi$
$269$	−0.216029			−0.0131715			−0.00658577	+	0.999978i	$0.502096\pi$
$270$		0			0
$271$		−	10.5564i		−	0.641256i	−0.947205	−	0.320628i	$-0.896106\pi$
$271$		−	10.5564i		−	0.641256i	0.947205	−	0.320628i	$-0.103894\pi$
$272$		0			0
$273$	8.19802	−	5.11331i	0.496167	−	0.309472i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			6.72180i			0.403874i	0.979399	+	0.201937i	$0.0647236\pi$
$277$			6.72180i			0.403874i	−0.979399	+	0.201937i	$0.935276\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4.83820i		−	0.288623i	−0.989532	−	0.144311i	$-0.953903\pi$
$281$		−	4.83820i		−	0.288623i	0.989532	−	0.144311i	$-0.0460967\pi$
$282$		0			0
$283$	−2.33897			−0.139037			−0.0695186	−	0.997581i	$-0.522146\pi$
$283$	−2.33897			−0.139037			−0.0695186	+	0.997581i	$0.522146\pi$
$284$		0			0
$285$	−18.6958	+	11.6610i	−1.10744	+	0.690740i
$286$		0			0
$287$	−4.69120			−0.276913
$288$		0			0
$289$	−20.0351			−1.17854
$290$		0			0
$291$	16.2174	−	10.1152i	0.950683	−	0.592965i
$292$		0			0
$293$	11.2157			0.655227			0.327614	−	0.944812i	$-0.393756\pi$
$293$	11.2157			0.655227			0.327614	+	0.944812i	$0.393756\pi$
$294$		0			0
$295$		−	16.3960i		−	0.954614i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			3.88935i			0.224927i
$300$		0			0
$301$		−	1.36090i		−	0.0784409i
$302$		0			0
$303$	15.1929	−	9.47622i	0.872812	−	0.544395i
$304$		0			0
$305$		−	7.88333i		−	0.451398i
$306$		0			0
$307$	1.73870			0.0992330			0.0496165	−	0.998768i	$-0.484200\pi$
$307$	1.73870			0.0992330			0.0496165	+	0.998768i	$0.484200\pi$
$308$		0			0
$309$	−10.7769	−	17.2782i	−0.613074	−	0.982923i
$310$		0			0
$311$	10.7769			0.611099			0.305550	−	0.952176i	$-0.401160\pi$
$311$	10.7769			0.611099			0.305550	+	0.952176i	$0.401160\pi$
$312$		0			0
$313$	16.4787			0.931433			0.465717	−	0.884934i	$-0.345797\pi$
$313$	16.4787			0.931433			0.465717	+	0.884934i	$0.345797\pi$
$314$		0			0
$315$	−4.93923	−	2.41910i	−0.278294	−	0.136301i
$316$		0			0
$317$	−33.7250			−1.89419			−0.947093	−	0.320960i	$-0.895994\pi$
$317$	−33.7250			−1.89419			−0.947093	+	0.320960i	$0.895994\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	3.36090	+	5.38843i	0.187587	+	0.300753i
$322$		0			0
$323$		−	42.2298i		−	2.34973i
$324$		0			0
$325$			9.14346i			0.507188i
$326$		0			0
$327$	−16.3881	−	26.2745i	−0.906263	−	1.45298i
$328$		0			0
$329$			3.44375i			0.189860i
$330$		0			0
$331$	23.6742			1.30125			0.650627	−	0.759397i	$-0.274506\pi$
$331$	23.6742			1.30125			0.650627	+	0.759397i	$0.274506\pi$
$332$		0			0
$333$	−5.06100	−	2.47874i	−0.277341	−	0.135834i
$334$		0			0
$335$	8.28195			0.452491
$336$		0			0
$337$	13.1178			0.714574			0.357287	−	0.933995i	$-0.383702\pi$
$337$	13.1178			0.714574			0.357287	+	0.933995i	$0.383702\pi$
$338$		0			0
$339$	4.93923	+	7.91893i	0.268262	+	0.430097i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	1.87847	−	1.17165i	0.101133	−	0.0630795i
$346$		0			0
$347$		−	28.8868i		−	1.55073i	−0.631516	−	0.775363i	$-0.717567\pi$
$347$		−	28.8868i		−	1.55073i	0.631516	−	0.775363i	$-0.282433\pi$
$348$		0			0
$349$			8.05707i			0.431285i	0.976472	+	0.215643i	$0.0691846\pi$
$349$			8.05707i			0.431285i	−0.976472	+	0.215643i	$0.930815\pi$
$350$		0			0
$351$	−28.8345	+	2.95865i	−1.53907	+	0.157921i
$352$		0			0
$353$		−	2.64190i		−	0.140614i	−0.997525	−	0.0703070i	$-0.977602\pi$
$353$		−	2.64190i		−	0.140614i	0.997525	−	0.0703070i	$-0.0223979\pi$
$354$		0			0
$355$	−11.1567			−0.592135
$356$		0			0
$357$	8.94358	−	5.57834i	0.473344	−	0.295237i
$358$		0			0
$359$	−6.19028			−0.326711			−0.163355	−	0.986567i	$-0.552232\pi$
$359$	−6.19028			−0.326711			−0.163355	+	0.986567i	$0.552232\pi$
$360$		0			0
$361$	29.1530			1.53437
$362$		0			0
$363$	16.1658	−	10.0830i	0.848484	−	0.529221i
$364$		0			0
$365$	10.9997			0.575748
$366$		0			0
$367$			10.4787i			0.546986i	0.961874	+	0.273493i	$0.0881790\pi$
$367$			10.4787i			0.546986i	−0.961874	+	0.273493i	$0.911821\pi$
$368$		0			0
$369$	12.6391	+	6.19028i	0.657965	+	0.322253i
$370$		0			0
$371$		−	12.1713i		−	0.631902i
$372$		0			0
$373$			26.4787i			1.37102i	0.728065	+	0.685508i	$0.240420\pi$
$373$			26.4787i			1.37102i	−0.728065	+	0.685508i	$0.759580\pi$
$374$		0			0
$375$	17.8872	−	11.1567i	0.923689	−	0.576128i
$376$		0			0
$377$		−	39.6637i		−	2.04278i
$378$		0			0
$379$	20.9524			1.07625			0.538127	−	0.842863i	$-0.319132\pi$
$379$	20.9524			1.07625			0.538127	+	0.842863i	$0.319132\pi$
$380$		0			0
$381$	7.69613	+	12.3390i	0.394285	+	0.632144i
$382$		0			0
$383$	−8.72755			−0.445957			−0.222978	−	0.974823i	$-0.571578\pi$
$383$	−8.72755			−0.445957			−0.222978	+	0.974823i	$0.571578\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.79577	+	3.66655i	−0.0912844	+	0.186381i
$388$		0			0
$389$	7.11030			0.360507			0.180253	−	0.983620i	$-0.442308\pi$
$389$	7.11030			0.360507			0.180253	+	0.983620i	$0.442308\pi$
$390$		0			0
$391$			4.24306i			0.214581i
$392$		0			0
$393$	6.31955	+	10.1319i	0.318779	+	0.511089i
$394$		0			0
$395$		−	21.5537i		−	1.08448i
$396$		0			0
$397$		−	19.2137i		−	0.964310i	−0.876086	−	0.482155i	$-0.839854\pi$
$397$		−	19.2137i		−	0.964310i	0.876086	−	0.482155i	$-0.160146\pi$
$398$		0			0
$399$	6.36076	+	10.1980i	0.318436	+	0.510540i
$400$		0			0
$401$		−	17.5597i		−	0.876891i	−0.898758	−	0.438445i	$-0.855529\pi$
$401$		−	17.5597i		−	0.876891i	0.898758	−	0.438445i	$-0.144471\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	10.1152	+	13.0351i	0.502630	+	0.647721i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−7.20053			−0.356043			−0.178022	−	0.984027i	$-0.556970\pi$
$409$	−7.20053			−0.356043			−0.178022	+	0.984027i	$0.556970\pi$
$410$		0			0
$411$	17.8785	+	28.6640i	0.881880	+	1.41389i
$412$		0			0
$413$	−8.94358			−0.440085
$414$		0			0
$415$		−	0.804492i		−	0.0394910i
$416$		0			0
$417$	−17.9550	+	11.1990i	−0.879258	+	0.548416i
$418$		0			0
$419$			13.7818i			0.673284i	0.941633	+	0.336642i	$0.109291\pi$
$419$			13.7818i			0.673284i	−0.941633	+	0.336642i	$0.890709\pi$
$420$		0			0
$421$		−	13.0351i		−	0.635294i	−0.948209	−	0.317647i	$-0.897107\pi$
$421$		−	13.0351i		−	0.635294i	0.948209	−	0.317647i	$-0.102893\pi$
$422$		0			0
$423$	4.54421	−	9.27820i	0.220947	−	0.451122i
$424$		0			0
$425$			9.97500i			0.483859i
$426$		0			0
$427$	−4.30013			−0.208098
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−30.5329			−1.47072			−0.735359	−	0.677678i	$-0.762986\pi$
$431$	−30.5329			−1.47072			−0.735359	+	0.677678i	$0.762986\pi$
$432$		0			0
$433$	1.75694			0.0844331			0.0422166	−	0.999108i	$-0.486558\pi$
$433$	1.75694			0.0844331			0.0422166	+	0.999108i	$0.486558\pi$
$434$		0			0
$435$	−19.1567	+	11.9485i	−0.918492	+	0.572887i
$436$		0			0
$437$	−4.83820			−0.231442
$438$		0			0
$439$			24.7921i			1.18326i	0.806209	+	0.591631i	$0.201516\pi$
$439$			24.7921i			1.18326i	−0.806209	+	0.591631i	$0.798484\pi$
$440$		0			0
$441$	−1.31955	+	2.69421i	−0.0628358	+	0.128296i
$442$		0			0
$443$			17.4416i			0.828673i	0.910124	+	0.414337i	$0.135986\pi$
$443$			17.4416i			0.828673i	−0.910124	+	0.414337i	$0.864014\pi$
$444$		0			0
$445$		−	4.84333i		−	0.229596i
$446$		0			0
$447$	7.11030	−	4.43488i	0.336306	−	0.209762i
$448$		0			0
$449$		−	6.88750i		−	0.325041i	−0.986705	−	0.162521i	$-0.948038\pi$
$449$		−	6.88750i		−	0.325041i	0.986705	−	0.162521i	$-0.0519624\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−13.8576	−	22.2174i	−0.651086	−	1.04387i
$454$		0			0
$455$	−10.2266			−0.479432
$456$		0			0
$457$	−32.9524			−1.54145			−0.770725	−	0.637168i	$-0.780106\pi$
$457$	−32.9524			−1.54145			−0.770725	+	0.637168i	$0.780106\pi$
$458$		0			0
$459$	−31.4568	+	3.22772i	−1.46828	+	0.150657i
$460$		0			0
$461$	5.49983			0.256152			0.128076	−	0.991764i	$-0.459120\pi$
$461$	5.49983			0.256152			0.128076	+	0.991764i	$0.459120\pi$
$462$		0			0
$463$			5.44359i			0.252985i	0.991968	+	0.126493i	$0.0403720\pi$
$463$			5.44359i			0.252985i	−0.991968	+	0.126493i	$0.959628\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	38.7792i		−	1.79449i	−0.441534	−	0.897245i	$-0.645566\pi$
$467$		−	38.7792i		−	1.79449i	0.441534	−	0.897245i	$-0.354434\pi$
$468$		0			0
$469$		−	4.51757i		−	0.208602i
$470$		0			0
$471$	8.55707	+	13.7193i	0.394289	+	0.632151i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−11.3741			−0.521880
$476$		0			0
$477$	−16.0607	+	32.7921i	−0.735367	+	1.50145i
$478$		0			0
$479$	−14.2206			−0.649756			−0.324878	−	0.945756i	$-0.605323\pi$
$479$	−14.2206			−0.649756			−0.324878	+	0.945756i	$0.605323\pi$
$480$		0			0
$481$	−10.4787			−0.477789
$482$		0			0
$483$	−0.639102	−	1.02465i	−0.0290801	−	0.0466233i
$484$		0			0
$485$	−20.2305			−0.918618
$486$		0			0
$487$			33.6742i			1.52593i	0.646442	+	0.762963i	$0.276256\pi$
$487$			33.6742i			1.52593i	−0.646442	+	0.762963i	$0.723744\pi$
$488$		0			0
$489$	−9.75694	+	6.08565i	−0.441224	+	0.275203i
$490$		0			0
$491$			32.5534i			1.46911i	0.678548	+	0.734556i	$0.262610\pi$
$491$			32.5534i			1.46911i	−0.678548	+	0.734556i	$0.737390\pi$
$492$		0			0
$493$		−	43.2708i		−	1.94882i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			6.08565i			0.272979i
$498$		0			0
$499$	18.9963			0.850392			0.425196	−	0.905101i	$-0.360205\pi$
$499$	18.9963			0.850392			0.425196	+	0.905101i	$0.360205\pi$
$500$		0			0
$501$	28.6640	−	17.8785i	1.28061	−	0.798751i
$502$		0			0
$503$	2.49490			0.111242			0.0556211	−	0.998452i	$-0.482286\pi$
$503$	2.49490			0.111242			0.0556211	+	0.998452i	$0.482286\pi$
$504$		0			0
$505$	−18.9524			−0.843373
$506$		0			0
$507$	−26.6263	+	16.6075i	−1.18252	+	0.737565i
$508$		0			0
$509$	33.0634			1.46551			0.732754	−	0.680493i	$-0.238234\pi$
$509$	33.0634			1.46551			0.732754	+	0.680493i	$0.238234\pi$
$510$		0			0
$511$		−	6.00000i		−	0.265424i
$512$		0			0
$513$	−3.68045	−	35.8690i	−0.162496	−	1.58365i
$514$		0			0
$515$			21.5537i			0.949770i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	16.4828	−	10.2807i	0.723513	−	0.451273i
$520$		0			0
$521$		−	30.1343i		−	1.32021i	−0.751175	−	0.660103i	$-0.770513\pi$
$521$		−	30.1343i		−	1.32021i	0.751175	−	0.660103i	$-0.229487\pi$
$522$		0			0
$523$	7.37411			0.322447			0.161224	−	0.986918i	$-0.448456\pi$
$523$	7.37411			0.322447			0.161224	+	0.986918i	$0.448456\pi$
$524$		0			0
$525$	−1.50246	−	2.40885i	−0.0655728	−	0.105131i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−22.5139			−0.978864
$530$		0			0
$531$	24.0959	+	11.8015i	1.04567	+	0.512142i
$532$		0			0
$533$	26.1691			1.13351
$534$		0			0
$535$		−	6.72180i		−	0.290609i
$536$		0			0
$537$	18.9524	+	30.3859i	0.817858	+	1.31125i
$538$		0			0
$539$		0			0
$540$		0			0
$541$			0.792082i			0.0340543i	0.999855	+	0.0170271i	$0.00542017\pi$
$541$			0.792082i			0.0340543i	−0.999855	+	0.0170271i	$0.994580\pi$
$542$		0			0
$543$	3.94166	+	6.31955i	0.169153	+	0.271198i
$544$		0			0
$545$			32.7762i			1.40398i
$546$		0			0
$547$	9.36090			0.400243			0.200122	−	0.979771i	$-0.435866\pi$
$547$	9.36090			0.400243			0.200122	+	0.979771i	$0.435866\pi$
$548$		0			0
$549$	11.5855	+	5.67424i	0.494456	+	0.242171i
$550$		0			0
$551$	49.3401			2.10196
$552$		0			0
$553$	−11.7569			−0.499956
$554$		0			0
$555$	3.15667	+	5.06100i	0.133993	+	0.214827i
$556$		0			0
$557$	−1.60371			−0.0679513			−0.0339756	−	0.999423i	$-0.510817\pi$
$557$	−1.60371			−0.0679513			−0.0339756	+	0.999423i	$0.510817\pi$
$558$		0			0
$559$			7.59155i			0.321088i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	8.94358i		−	0.376927i	−0.982080	−	0.188463i	$-0.939649\pi$
$563$		−	8.94358i		−	0.376927i	0.982080	−	0.188463i	$-0.0603507\pi$
$564$		0			0
$565$		−	9.87847i		−	0.415590i
$566$		0			0
$567$	7.11030	−	5.51757i	0.298605	−	0.231716i
$568$		0			0
$569$			39.1134i			1.63972i	0.572564	+	0.819860i	$0.305949\pi$
$569$			39.1134i			1.63972i	−0.572564	+	0.819860i	$0.694051\pi$
$570$		0			0
$571$	−14.6391			−0.612627			−0.306314	−	0.951931i	$-0.599096\pi$
$571$	−14.6391			−0.612627			−0.306314	+	0.951931i	$0.599096\pi$
$572$		0			0
$573$	−35.5583	+	22.1786i	−1.48547	+	0.926524i
$574$		0			0
$575$	1.14282			0.0476589
$576$		0			0
$577$	28.2357			1.17547			0.587733	−	0.809055i	$-0.300020\pi$
$577$	28.2357			1.17547			0.587733	+	0.809055i	$0.300020\pi$
$578$		0			0
$579$	−9.39973	+	5.86285i	−0.390640	+	0.243652i
$580$		0			0
$581$	−0.438828			−0.0182056
$582$		0			0
$583$		0			0
$584$		0			0
$585$	27.5527	+	13.4946i	1.13916	+	0.557931i
$586$		0			0
$587$			3.45052i			0.142418i	0.997461	+	0.0712091i	$0.0226858\pi$
$587$			3.45052i			0.142418i	−0.997461	+	0.0712091i	$0.977314\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−5.06100	+	3.15667i	−0.208182	+	0.129848i
$592$		0			0
$593$			4.03635i			0.165753i	0.996560	+	0.0828764i	$0.0264107\pi$
$593$			4.03635i			0.165753i	−0.996560	+	0.0828764i	$0.973589\pi$
$594$		0			0
$595$	−11.1567			−0.457379
$596$		0			0
$597$	8.50475	+	13.6354i	0.348076	+	0.558060i
$598$		0			0
$599$	−9.23541			−0.377349			−0.188674	−	0.982040i	$-0.560419\pi$
$599$	−9.23541			−0.377349			−0.188674	+	0.982040i	$0.560419\pi$
$600$		0			0
$601$	−38.7921			−1.58236			−0.791181	−	0.611582i	$-0.790533\pi$
$601$	−38.7921			−1.58236			−0.791181	+	0.611582i	$0.790533\pi$
$602$		0			0
$603$	−5.96116	+	12.1713i	−0.242757	+	0.495654i
$604$		0			0
$605$	−20.1660			−0.819866
$606$		0			0
$607$			6.31335i			0.256251i	0.991758	+	0.128125i	$0.0408960\pi$
$607$			6.31335i			0.256251i	−0.991758	+	0.128125i	$0.959104\pi$
$608$		0			0
$609$	6.51757	+	10.4494i	0.264105	+	0.423432i
$610$		0			0
$611$		−	19.2104i		−	0.777170i
$612$		0			0
$613$		−	8.19182i		−	0.330864i	−0.986221	−	0.165432i	$-0.947098\pi$
$613$		−	8.19182i		−	0.330864i	0.986221	−	0.165432i	$-0.0529019\pi$
$614$		0			0
$615$	−7.88333	−	12.6391i	−0.317886	−	0.509658i
$616$		0			0
$617$		−	18.6602i		−	0.751231i	−0.926776	−	0.375615i	$-0.877431\pi$
$617$		−	18.6602i		−	0.751231i	0.926776	−	0.375615i	$-0.122569\pi$
$618$		0			0
$619$	−14.6962			−0.590689			−0.295345	−	0.955391i	$-0.595434\pi$
$619$	−14.6962			−0.590689			−0.295345	+	0.955391i	$0.595434\pi$
$620$		0			0
$621$	0.369795	+	3.60396i	0.0148394	+	0.144622i
$622$		0			0
$623$	−2.64190			−0.105845
$624$		0			0
$625$	−14.1178			−0.564714
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−11.4317			−0.455812
$630$		0			0
$631$		−	10.5564i		−	0.420244i	−0.977675	−	0.210122i	$-0.932614\pi$
$631$		−	10.5564i		−	0.420244i	0.977675	−	0.210122i	$-0.0673861\pi$
$632$		0			0
$633$	−1.11784	+	0.697224i	−0.0444301	+	0.0277122i
$634$		0			0
$635$		−	15.3923i		−	0.610823i
$636$		0			0
$637$			5.57834i			0.221022i
$638$		0			0
$639$	8.03033	−	16.3960i	0.317675	−	0.648617i
$640$		0			0
$641$		−	14.5616i		−	0.575148i	−0.957758	−	0.287574i	$-0.907151\pi$
$641$		−	14.5616i		−	0.575148i	0.957758	−	0.287574i	$-0.0928486\pi$
$642$		0			0
$643$	14.3390			0.565474			0.282737	−	0.959198i	$-0.408758\pi$
$643$	14.3390			0.565474			0.282737	+	0.959198i	$0.408758\pi$
$644$		0			0
$645$	3.66655	−	2.28692i	0.144370	−	0.0900474i
$646$		0			0
$647$	−34.3799			−1.35161			−0.675806	−	0.737080i	$-0.736204\pi$
$647$	−34.3799			−1.35161			−0.675806	+	0.737080i	$0.736204\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	15.8379			0.619783			0.309892	−	0.950772i	$-0.399707\pi$
$653$	15.8379			0.619783			0.309892	+	0.950772i	$0.399707\pi$
$654$		0			0
$655$		−	12.6391i		−	0.493851i
$656$		0			0
$657$	−7.91730	+	16.1653i	−0.308883	+	0.630668i
$658$		0			0
$659$			34.4646i			1.34255i	0.741208	+	0.671275i	$0.234253\pi$
$659$			34.4646i			1.34255i	−0.741208	+	0.671275i	$0.765747\pi$
$660$		0			0
$661$			36.4919i			1.41937i	0.704518	+	0.709686i	$0.251163\pi$
$661$			36.4919i			1.41937i	−0.704518	+	0.709686i	$0.748837\pi$
$662$		0			0
$663$	−49.8903	+	31.1178i	−1.93758	+	1.20852i
$664$		0			0
$665$		−	12.7215i		−	0.493320i
$666$		0			0
$667$	−4.95747			−0.191954
$668$		0			0
$669$	26.6147	+	42.6706i	1.02898	+	1.64974i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	27.0351			1.04213			0.521064	−	0.853518i	$-0.325535\pi$
$673$	27.0351			1.04213			0.521064	+	0.853518i	$0.325535\pi$
$674$		0			0
$675$	0.869351	+	8.47254i	0.0334613	+	0.326108i
$676$		0			0
$677$	12.8329			0.493209			0.246605	−	0.969116i	$-0.420685\pi$
$677$	12.8329			0.493209			0.246605	+	0.969116i	$0.420685\pi$
$678$		0			0
$679$			11.0351i			0.423490i
$680$		0			0
$681$	3.55892	+	5.70591i	0.136378	+	0.218651i
$682$		0			0
$683$		−	6.45545i		−	0.247011i	−0.992344	−	0.123505i	$-0.960586\pi$
$683$		−	6.45545i		−	0.247011i	0.992344	−	0.123505i	$-0.0394136\pi$
$684$		0			0
$685$		−	35.7569i		−	1.36620i
$686$		0			0
$687$	−2.77001	−	4.44108i	−0.105683	−	0.169438i
$688$		0			0
$689$			67.8956i			2.58662i
$690$		0			0
$691$	1.90409			0.0724351			0.0362175	−	0.999344i	$-0.488469\pi$
$691$	1.90409			0.0724351			0.0362175	+	0.999344i	$0.488469\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	22.3979			0.849602
$696$		0			0
$697$	28.5490			1.08137
$698$		0			0
$699$	−11.1567	−	17.8872i	−0.421984	−	0.676554i
$700$		0			0
$701$	−8.50475			−0.321220			−0.160610	−	0.987018i	$-0.551346\pi$
$701$	−8.50475			−0.321220			−0.160610	+	0.987018i	$0.551346\pi$
$702$		0			0
$703$		−	13.0351i		−	0.491630i
$704$		0			0
$705$	−9.27820	+	5.78705i	−0.349437	+	0.217953i
$706$		0			0
$707$			10.3380i			0.388801i
$708$		0			0
$709$			20.4349i			0.767448i	0.923448	+	0.383724i	$0.125359\pi$
$709$			20.4349i			0.767448i	−0.923448	+	0.383724i	$0.874641\pi$
$710$		0			0
$711$	31.6757	+	15.5139i	1.18793	+	0.581816i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	32.7004	−	20.3960i	1.22122	−	0.761704i
$718$		0			0
$719$	30.2813			1.12930			0.564650	−	0.825330i	$-0.309011\pi$
$719$	30.2813			1.12930			0.564650	+	0.825330i	$0.309011\pi$
$720$		0			0
$721$	11.7569			0.437851
$722$		0			0
$723$	22.0959	−	13.7818i	0.821756	−	0.512550i
$724$		0			0
$725$	−11.6545			−0.432838
$726$		0			0
$727$		−	34.4787i		−	1.27875i	−0.768897	−	0.639373i	$-0.779194\pi$
$727$		−	34.4787i		−	1.27875i	0.768897	−	0.639373i	$-0.220806\pi$
$728$		0			0
$729$	−26.4374	+	5.48311i	−0.979163	+	0.203078i
$730$		0			0
$731$			8.28195i			0.306319i
$732$		0			0
$733$			22.9707i			0.848442i	0.905559	+	0.424221i	$0.139452\pi$
$733$			22.9707i			0.848442i	−0.905559	+	0.424221i	$0.860548\pi$
$734$		0			0
$735$	2.69421	−	1.68045i	0.0993776	−	0.0619843i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−6.80449			−0.250307			−0.125154	−	0.992137i	$-0.539942\pi$
$739$	−6.80449			−0.250307			−0.125154	+	0.992137i	$0.539942\pi$
$740$		0			0
$741$	−35.4825	−	56.8880i	−1.30348	−	2.08983i
$742$		0			0
$743$	−32.8185			−1.20399			−0.601997	−	0.798498i	$-0.705628\pi$
$743$	−32.8185			−1.20399			−0.601997	+	0.798498i	$0.705628\pi$
$744$		0			0
$745$	−8.86975			−0.324963
$746$		0			0
$747$	1.18230	+	0.579055i	0.0432579	+	0.0211865i
$748$		0			0
$749$	−3.66655			−0.133973
$750$		0			0
$751$			14.3258i			0.522754i	0.965237	+	0.261377i	$0.0841766\pi$
$751$			14.3258i			0.522754i	−0.965237	+	0.261377i	$0.915823\pi$
$752$		0			0
$753$	−21.4374	−	34.3699i	−0.781222	−	1.25251i
$754$		0			0
$755$			27.7152i			1.00866i
$756$		0			0
$757$			36.4349i			1.32425i	0.749394	+	0.662124i	$0.230345\pi$
$757$			36.4349i			1.32425i	−0.749394	+	0.662124i	$0.769655\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	14.3676i		−	0.520825i	−0.965497	−	0.260413i	$-0.916141\pi$
$761$		−	14.3676i		−	0.520825i	0.965497	−	0.260413i	$-0.0838586\pi$
$762$		0			0
$763$	17.8785			0.647244
$764$		0			0
$765$	30.0585	+	14.7218i	1.08677	+	0.532268i
$766$		0			0
$767$	49.8903			1.80143
$768$		0			0
$769$	12.4787			0.449995			0.224997	−	0.974359i	$-0.427763\pi$
$769$	12.4787			0.449995			0.224997	+	0.974359i	$0.427763\pi$
$770$		0			0
$771$	−18.6135	−	29.8424i	−0.670348	−	1.07475i
$772$		0			0
$773$	−26.1759			−0.941481			−0.470740	−	0.882272i	$-0.656013\pi$
$773$	−26.1759			−0.941481			−0.470740	+	0.882272i	$0.656013\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2.76063	−	1.72188i	0.0990371	−	0.0617720i
$778$		0			0
$779$			32.5534i			1.16635i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−3.77118	−	36.7532i	−0.134771	−	1.31345i
$784$		0			0
$785$		−	17.1141i		−	0.610830i
$786$		0			0
$787$	−20.2174			−0.720674			−0.360337	−	0.932822i	$-0.617338\pi$
$787$	−20.2174			−0.720674			−0.360337	+	0.932822i	$0.617338\pi$
$788$		0			0
$789$	−26.8307	+	16.7350i	−0.955199	+	0.595782i
$790$		0			0
$791$	−5.38843			−0.191590
$792$		0			0
$793$	23.9876			0.851824
$794$		0			0
$795$	32.7921	−	20.4533i	1.16302	−	0.725402i
$796$		0			0
$797$	−24.1266			−0.854607			−0.427304	−	0.904108i	$-0.640536\pi$
$797$	−24.1266			−0.854607			−0.427304	+	0.904108i	$0.640536\pi$
$798$		0			0
$799$		−	20.9575i		−	0.741422i
$800$		0			0
$801$	7.11784	+	3.48612i	0.251496	+	0.123176i
$802$		0			0
$803$		0			0
$804$		0			0
$805$			1.27820i			0.0450507i
$806$		0			0
$807$	−0.317480	+	0.198021i	−0.0111758	+	0.00697065i
$808$		0			0
$809$			13.3764i			0.470289i	0.971960	+	0.235144i	$0.0755563\pi$
$809$			13.3764i			0.470289i	−0.971960	+	0.235144i	$0.924444\pi$
$810$		0			0
$811$	−26.1736			−0.919079			−0.459539	−	0.888157i	$-0.651985\pi$
$811$	−26.1736			−0.919079			−0.459539	+	0.888157i	$0.651985\pi$
$812$		0			0
$813$	−9.67640	−	15.5139i	−0.339366	−	0.544095i
$814$		0			0
$815$	12.1713			0.426342
$816$		0			0
$817$	−9.44359			−0.330390
$818$		0			0
$819$	7.36090	−	15.0292i	0.257211	−	0.525164i
$820$		0			0
$821$	−12.6169			−0.440333			−0.220166	−	0.975462i	$-0.570660\pi$
$821$	−12.6169			−0.440333			−0.220166	+	0.975462i	$0.570660\pi$
$822$		0			0
$823$		−	6.79947i		−	0.237015i	−0.992953	−	0.118507i	$-0.962189\pi$
$823$		−	6.79947i		−	0.237015i	0.992953	−	0.118507i	$-0.0378109\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	0.877655i		−	0.0305191i	−0.999884	−	0.0152595i	$-0.995143\pi$
$827$		−	0.877655i		−	0.0305191i	0.999884	−	0.0152595i	$-0.00485745\pi$
$828$		0			0
$829$		−	30.2563i		−	1.05084i	−0.850842	−	0.525422i	$-0.823908\pi$
$829$		−	30.2563i		−	1.05084i	0.850842	−	0.525422i	$-0.176092\pi$
$830$		0			0
$831$	6.16145	+	9.87847i	0.213738	+	0.342680i
$832$		0			0
$833$			6.08565i			0.210855i
$834$		0			0
$835$	−35.7569			−1.23742
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−11.5164			−0.397592			−0.198796	−	0.980041i	$-0.563703\pi$
$839$	−11.5164			−0.397592			−0.198796	+	0.980041i	$0.563703\pi$
$840$		0			0
$841$	21.5564			0.743324
$842$		0			0
$843$	−4.43488	−	7.11030i	−0.152745	−	0.244892i
$844$		0			0
$845$	33.2150			1.14263
$846$		0			0
$847$			11.0000i			0.377964i
$848$		0			0
$849$	−3.43739	+	2.14399i	−0.117971	+	0.0735814i
$850$		0			0
$851$			1.30971i			0.0448964i
$852$		0			0
$853$		−	5.57834i		−	0.190999i	−0.995429	−	0.0954993i	$-0.969555\pi$
$853$		−	5.57834i		−	0.190999i	0.995429	−	0.0954993i	$-0.0304448\pi$
$854$		0			0
$855$	−16.7867	+	34.2745i	−0.574093	+	1.17216i
$856$		0			0
$857$			31.7380i			1.08415i	0.840331	+	0.542074i	$0.182361\pi$
$857$			31.7380i			1.08415i	−0.840331	+	0.542074i	$0.817639\pi$
$858$		0			0
$859$	−40.8177			−1.39268			−0.696341	−	0.717711i	$-0.745190\pi$
$859$	−40.8177			−1.39268			−0.696341	+	0.717711i	$0.745190\pi$
$860$		0			0
$861$	−6.89427	+	4.30013i	−0.234956	+	0.146548i
$862$		0			0
$863$	12.3183			0.419320			0.209660	−	0.977774i	$-0.432764\pi$
$863$	12.3183			0.419320			0.209660	+	0.977774i	$0.432764\pi$
$864$		0			0
$865$	−20.5614			−0.699110
$866$		0			0
$867$	−29.4440	+	18.3650i	−0.999970	+	0.623707i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			25.2005i			0.853887i
$872$		0			0
$873$	14.5614	−	29.7310i	0.492830	−	1.00624i
$874$		0			0
$875$			12.1713i			0.411465i
$876$		0			0
$877$		−	27.9488i		−	0.943762i	−0.881662	−	0.471881i	$-0.843575\pi$
$877$		−	27.9488i		−	0.943762i	0.881662	−	0.471881i	$-0.156425\pi$
$878$		0			0
$879$	16.4828	−	10.2807i	0.555950	−	0.346760i
$880$		0			0
$881$		−	28.5882i		−	0.963161i	−0.876402	−	0.481581i	$-0.840063\pi$
$881$		−	28.5882i		−	0.963161i	0.876402	−	0.481581i	$-0.159937\pi$
$882$		0			0
$883$	1.96116			0.0659985			0.0329992	−	0.999455i	$-0.489494\pi$
$883$	1.96116			0.0659985			0.0329992	+	0.999455i	$0.489494\pi$
$884$		0			0
$885$	−15.0292	−	24.0959i	−0.505202	−	0.809975i
$886$		0			0
$887$	38.2692			1.28495			0.642477	−	0.766305i	$-0.277907\pi$
$887$	38.2692			1.28495			0.642477	+	0.766305i	$0.277907\pi$
$888$		0			0
$889$	−8.39604			−0.281594
$890$		0			0
$891$		0			0
$892$		0			0
$893$	23.8970			0.799683
$894$		0			0
$895$		−	37.9049i		−	1.26702i
$896$		0			0
$897$	3.56512	+	5.71586i	0.119036	+	0.190847i
$898$		0			0
$899$		0			0
$900$		0			0
$901$			74.0703i			2.46764i
$902$		0			0
$903$	−1.24745	−	2.00000i	−0.0415126	−	0.0665558i
$904$		0			0
$905$		−	7.88333i		−	0.262051i
$906$		0			0
$907$	49.3097			1.63730			0.818650	−	0.574293i	$-0.194723\pi$
$907$	49.3097			1.63730			0.818650	+	0.574293i	$0.194723\pi$
$908$		0			0
$909$	13.6416	−	27.8528i	0.452462	−	0.923821i
$910$		0			0
$911$	32.7338			1.08452			0.542259	−	0.840211i	$-0.317569\pi$
$911$	32.7338			1.08452			0.542259	+	0.840211i	$0.317569\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−7.22616	−	11.5855i	−0.238889	−	0.383004i
$916$		0			0
$917$	−6.89427			−0.227669
$918$		0			0
$919$			37.8272i			1.24781i	0.781502	+	0.623903i	$0.214454\pi$
$919$			37.8272i			1.24781i	−0.781502	+	0.623903i	$0.785546\pi$
$920$		0			0
$921$	2.55523	−	1.59376i	0.0841976	−	0.0525162i
$922$		0			0
$923$		−	33.9478i		−	1.11741i
$924$		0			0
$925$			3.07900i			0.101237i
$926$		0			0
$927$	−31.6757	−	15.5139i	−1.04037	−	0.509543i
$928$		0			0
$929$			0.801854i			0.0263080i	0.999913	+	0.0131540i	$0.00418716\pi$
$929$			0.801854i			0.0263080i	−0.999913	+	0.0131540i	$0.995813\pi$
$930$		0			0
$931$	−6.93923			−0.227424
$932$		0			0
$933$	15.8379	−	9.87847i	0.518508	−	0.323407i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−43.5842			−1.42383			−0.711916	−	0.702264i	$-0.752172\pi$
$937$	−43.5842			−1.42383			−0.711916	+	0.702264i	$0.752172\pi$
$938$		0			0
$939$	24.2174	−	15.1050i	0.790306	−	0.492934i
$940$		0			0
$941$	1.09368			0.0356531			0.0178265	−	0.999841i	$-0.494325\pi$
$941$	1.09368			0.0356531			0.0178265	+	0.999841i	$0.494325\pi$
$942$		0			0
$943$		−	3.27082i		−	0.106513i
$944$		0			0
$945$	−9.47622	+	0.972337i	−0.308262	+	0.0316301i
$946$		0			0
$947$		−	24.3426i		−	0.791028i	−0.918460	−	0.395514i	$-0.870566\pi$
$947$		−	24.3426i		−	0.791028i	0.918460	−	0.395514i	$-0.129434\pi$
$948$		0			0
$949$			33.4700i			1.08648i
$950$		0			0
$951$	−49.5629	+	30.9136i	−1.60719	+	1.00244i
$952$		0			0
$953$		−	11.8773i		−	0.384744i	−0.981322	−	0.192372i	$-0.938382\pi$
$953$		−	11.8773i		−	0.384744i	0.981322	−	0.192372i	$-0.0616180\pi$
$954$		0			0
$955$	44.3572			1.43537
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−19.5044			−0.629830
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	9.87847	+	4.83820i	0.318329	+	0.155909i
$964$		0			0
$965$	11.7257			0.377464
$966$		0			0
$967$			33.6742i			1.08289i	0.840736	+	0.541445i	$0.182123\pi$
$967$			33.6742i			1.08289i	−0.840736	+	0.541445i	$0.817877\pi$
$968$		0			0
$969$	−38.7094	−	62.0616i	−1.24352	−	1.99371i
$970$		0			0
$971$		−	4.32818i		−	0.138898i	−0.997586	−	0.0694489i	$-0.977876\pi$
$971$		−	4.32818i		−	0.138898i	0.997586	−	0.0694489i	$-0.0221241\pi$
$972$		0			0
$973$		−	12.2174i		−	0.391673i
$974$		0			0
$975$	8.38124	+	13.4374i	0.268414	+	0.430341i
$976$		0			0
$977$			1.60371i			0.0513072i	0.999671	+	0.0256536i	$0.00816668\pi$
$977$			1.60371i			0.0513072i	−0.999671	+	0.0256536i	$0.991833\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−48.1684	−	23.5915i	−1.53790	−	0.753220i
$982$		0			0
$983$	−50.2313			−1.60213			−0.801064	−	0.598579i	$-0.795732\pi$
$983$	−50.2313			−1.60213			−0.801064	+	0.598579i	$0.795732\pi$
$984$		0			0
$985$	6.31335			0.201160
$986$		0			0
$987$	3.15667	+	5.06100i	0.100478	+	0.161093i
$988$		0			0
$989$	0.948851			0.0301717
$990$		0			0
$991$			1.68665i			0.0535783i	0.999641	+	0.0267892i	$0.00852828\pi$
$991$			1.68665i			0.0535783i	−0.999641	+	0.0267892i	$0.991472\pi$
$992$		0			0
$993$	34.7921	−	21.7007i	1.10409	−	0.688651i
$994$		0			0
$995$		−	17.0095i		−	0.539237i
$996$		0			0
$997$		−	27.8140i		−	0.880879i	−0.897782	−	0.440439i	$-0.854823\pi$
$997$		−	27.8140i		−	0.880879i	0.897782	−	0.440439i	$-0.145177\pi$
$998$		0			0
$999$	−9.70983	+	0.996308i	−0.307206	+	0.0315218i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.j.d.239.9		12
3.2	odd	2	inner	672.2.j.d.239.12		12
4.3	odd	2		168.2.j.d.155.1	✓	12
8.3	odd	2	inner	672.2.j.d.239.10		12
8.5	even	2		168.2.j.d.155.11	yes	12
12.11	even	2		168.2.j.d.155.12	yes	12
24.5	odd	2		168.2.j.d.155.2	yes	12
24.11	even	2	inner	672.2.j.d.239.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.j.d.155.1	✓	12	4.3	odd	2
168.2.j.d.155.2	yes	12	24.5	odd	2
168.2.j.d.155.11	yes	12	8.5	even	2
168.2.j.d.155.12	yes	12	12.11	even	2
672.2.j.d.239.9		12	1.1	even	1	trivial
672.2.j.d.239.10		12	8.3	odd	2	inner
672.2.j.d.239.11		12	24.11	even	2	inner
672.2.j.d.239.12		12	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.46962 - 0.916638i) q^{3} -1.83328 q^{5} +1.00000i q^{7} +(1.31955 - 2.69421i) q^{9} +O(q^{10})\)	\(q+(1.46962 - 0.916638i) q^{3} -1.83328 q^{5} +1.00000i q^{7} +(1.31955 - 2.69421i) q^{9} -5.57834i q^{13} +(-2.69421 + 1.68045i) q^{15} -6.08565i q^{17} +6.93923 q^{19} +(0.916638 + 1.46962i) q^{21} -0.697224 q^{23} -1.63910 q^{25} +(-0.530383 - 5.16901i) q^{27} +7.11030 q^{29} -1.83328i q^{35} -1.87847i q^{37} +(-5.11331 - 8.19802i) q^{39} +4.69120i q^{41} -1.36090 q^{43} +(-2.41910 + 4.93923i) q^{45} +3.44375 q^{47} -1.00000 q^{49} +(-5.57834 - 8.94358i) q^{51} -12.1713 q^{53} +(10.1980 - 6.36076i) q^{57} +8.94358i q^{59} +4.30013i q^{61} +(2.69421 + 1.31955i) q^{63} +10.2266i q^{65} -4.51757 q^{67} +(-1.02465 + 0.639102i) q^{69} +6.08565 q^{71} -6.00000 q^{73} +(-2.40885 + 1.50246i) q^{75} +11.7569i q^{79} +(-5.51757 - 7.11030i) q^{81} +0.438828i q^{83} +11.1567i q^{85} +(10.4494 - 6.51757i) q^{87} +2.64190i q^{89} +5.57834 q^{91} -12.7215 q^{95} +11.0351 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{9}+O(q^{10}) \) 12 * q + 4 * q^9	\( 12 q + 4 q^{9} + 48 q^{19} + 4 q^{25} - 24 q^{27} - 40 q^{43} - 12 q^{49} - 8 q^{51} + 40 q^{57} + 40 q^{67} - 72 q^{73} + 24 q^{75} + 28 q^{81} + 8 q^{91} - 56 q^{97}+O(q^{100}) \) 12 * q + 4 * q^9 + 48 * q^19 + 4 * q^25 - 24 * q^27 - 40 * q^43 - 12 * q^49 - 8 * q^51 + 40 * q^57 + 40 * q^67 - 72 * q^73 + 24 * q^75 + 28 * q^81 + 8 * q^91 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more