LMFDB - Embedded newform 1680.3.l.a.1121.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,3,Mod(1121,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.1121"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,-8,0,0,0,0,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$45.7766844125$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 46x^{14} + 823x^{12} + 7252x^{10} + 32831x^{8} + 71486x^{6} + 60809x^{4} + 15680x^{2} + 576 $ x^16 + 46x^14 + 823x^12 + 7252x^10 + 32831x^8 + 71486x^6 + 60809x^4 + 15680*x^2 + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.15
Root			$3.57278i$ of defining polynomial
Character	$\chi$	$=$	1680.1121
Dual form			1680.3.l.a.1121.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.99481 - 0.176471i) q^{3} -2.23607i q^{5} +2.64575 q^{7} +(8.93772 - 1.05699i) q^{9} -12.0685i q^{11} -12.7142 q^{13} +(-0.394601 - 6.69659i) q^{15} -22.2556i q^{17} +28.9289 q^{19} +(7.92351 - 0.466899i) q^{21} +21.9101i q^{23} -5.00000 q^{25} +(26.5802 - 4.74274i) q^{27} +11.4122i q^{29} +4.93977 q^{31} +(-2.12974 - 36.1428i) q^{33} -5.91608i q^{35} -36.7045 q^{37} +(-38.0765 + 2.24369i) q^{39} -57.6874i q^{41} -57.7493 q^{43} +(-2.36351 - 19.9853i) q^{45} -15.6773i q^{47} +7.00000 q^{49} +(-3.92747 - 66.6513i) q^{51} -91.2304i q^{53} -26.9860 q^{55} +(86.6363 - 5.10511i) q^{57} -34.3101i q^{59} +71.1225 q^{61} +(23.6470 - 2.79654i) q^{63} +28.4298i q^{65} -20.1716 q^{67} +(3.86649 + 65.6164i) q^{69} +69.7718i q^{71} -96.8693 q^{73} +(-14.9740 + 0.882355i) q^{75} -31.9302i q^{77} +84.2348 q^{79} +(78.7655 - 18.8942i) q^{81} +20.4474i q^{83} -49.7651 q^{85} +(2.01393 + 34.1774i) q^{87} -1.65255i q^{89} -33.6386 q^{91} +(14.7936 - 0.871726i) q^{93} -64.6869i q^{95} +29.7125 q^{97} +(-12.7563 - 107.865i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{3} + 22 q^{9} - 10 q^{15} + 16 q^{19} - 14 q^{21} - 80 q^{25} + 148 q^{27} + 72 q^{31} - 4 q^{33} - 40 q^{37} - 90 q^{39} - 280 q^{43} + 40 q^{45} + 112 q^{49} - 38 q^{51} - 80 q^{55} - 36 q^{57}+ \cdots + 166 q^{99}+O(q^{100}) $ 16 * q - 8 * q^3 + 22 * q^9 - 10 * q^15 + 16 * q^19 - 14 * q^21 - 80 * q^25 + 148 * q^27 + 72 * q^31 - 4 * q^33 - 40 * q^37 - 90 * q^39 - 280 * q^43 + 40 * q^45 + 112 * q^49 - 38 * q^51 - 80 * q^55 - 36 * q^57 - 56 * q^61 + 56 * q^63 + 120 * q^67 + 60 * q^69 - 208 * q^73 + 40 * q^75 + 204 * q^79 + 458 * q^81 + 100 * q^85 + 324 * q^87 + 28 * q^91 + 108 * q^93 + 728 * q^97 + 166 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$	2.99481	−	0.176471i	0.998268	−	0.0588237i
$3$	2.99481	−	0.176471i	0.998268	−	0.0588237i
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$	2.64575			0.377964
$7$	2.64575			0.377964
$8$		0			0
$9$	8.93772	−	1.05699i	0.993080	−	0.117444i
$10$		0			0
$11$		−	12.0685i		−	1.09714i	−0.836106	−	0.548568i	$-0.815173\pi$
$11$		−	12.0685i		−	1.09714i	0.836106	−	0.548568i	$-0.184827\pi$
$12$		0			0
$13$	−12.7142			−0.978014			−0.489007	−	0.872280i	$-0.662641\pi$
$13$	−12.7142			−0.978014			−0.489007	+	0.872280i	$0.662641\pi$
$14$		0			0
$15$	−0.394601	−	6.69659i	−0.0263068	−	0.446439i
$16$		0			0
$17$		−	22.2556i		−	1.30915i	−0.755995	−	0.654577i	$-0.772847\pi$
$17$		−	22.2556i		−	1.30915i	0.755995	−	0.654577i	$-0.227153\pi$
$18$		0			0
$19$	28.9289			1.52257			0.761286	−	0.648417i	$-0.224568\pi$
$19$	28.9289			1.52257			0.761286	+	0.648417i	$0.224568\pi$
$20$		0			0
$21$	7.92351	−	0.466899i	0.377310	−	0.0222333i
$22$		0			0
$23$			21.9101i			0.952612i	0.879280	+	0.476306i	$0.158025\pi$
$23$			21.9101i			0.952612i	−0.879280	+	0.476306i	$0.841975\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	26.5802	−	4.74274i	0.984451	−	0.175657i
$28$		0			0
$29$			11.4122i			0.393525i	0.980451	+	0.196763i	$0.0630428\pi$
$29$			11.4122i			0.393525i	−0.980451	+	0.196763i	$0.936957\pi$
$30$		0			0
$31$	4.93977			0.159347			0.0796737	−	0.996821i	$-0.474612\pi$
$31$	4.93977			0.159347			0.0796737	+	0.996821i	$0.474612\pi$
$32$		0			0
$33$	−2.12974	−	36.1428i	−0.0645376	−	1.09524i
$34$		0			0
$35$		−	5.91608i		−	0.169031i
$36$		0			0
$37$	−36.7045			−0.992014			−0.496007	−	0.868319i	$-0.665201\pi$
$37$	−36.7045			−0.992014			−0.496007	+	0.868319i	$0.665201\pi$
$38$		0			0
$39$	−38.0765	+	2.24369i	−0.976321	+	0.0575304i
$40$		0			0
$41$		−	57.6874i		−	1.40701i	−0.710691	−	0.703504i	$-0.751618\pi$
$41$		−	57.6874i		−	1.40701i	0.710691	−	0.703504i	$-0.248382\pi$
$42$		0			0
$43$	−57.7493			−1.34301			−0.671503	−	0.741002i	$-0.734351\pi$
$43$	−57.7493			−1.34301			−0.671503	+	0.741002i	$0.734351\pi$
$44$		0			0
$45$	−2.36351	−	19.9853i	−0.0525224	−	0.444119i
$46$		0			0
$47$		−	15.6773i		−	0.333561i	−0.985994	−	0.166780i	$-0.946663\pi$
$47$		−	15.6773i		−	0.333561i	0.985994	−	0.166780i	$-0.0533370\pi$
$48$		0			0
$49$	7.00000			0.142857
$50$		0			0
$51$	−3.92747	−	66.6513i	−0.0770093	−	1.30689i
$52$		0			0
$53$		−	91.2304i		−	1.72133i	−0.509173	−	0.860664i	$-0.670049\pi$
$53$		−	91.2304i		−	1.72133i	0.509173	−	0.860664i	$-0.329951\pi$
$54$		0			0
$55$	−26.9860			−0.490654
$56$		0			0
$57$	86.6363	−	5.10511i	1.51993	−	0.0895633i
$58$		0			0
$59$		−	34.3101i		−	0.581527i	−0.956795	−	0.290763i	$-0.906091\pi$
$59$		−	34.3101i		−	0.581527i	0.956795	−	0.290763i	$-0.0939093\pi$
$60$		0			0
$61$	71.1225			1.16594			0.582971	−	0.812493i	$-0.301890\pi$
$61$	71.1225			1.16594			0.582971	+	0.812493i	$0.301890\pi$
$62$		0			0
$63$	23.6470	−	2.79654i	0.375349	−	0.0443895i
$64$		0			0
$65$			28.4298i			0.437381i
$66$		0			0
$67$	−20.1716			−0.301069			−0.150534	−	0.988605i	$-0.548099\pi$
$67$	−20.1716			−0.301069			−0.150534	+	0.988605i	$0.548099\pi$
$68$		0			0
$69$	3.86649	+	65.6164i	0.0560362	+	0.950963i
$70$		0			0
$71$			69.7718i			0.982702i	0.870962	+	0.491351i	$0.163497\pi$
$71$			69.7718i			0.982702i	−0.870962	+	0.491351i	$0.836503\pi$
$72$		0			0
$73$	−96.8693			−1.32698			−0.663489	−	0.748186i	$-0.730925\pi$
$73$	−96.8693			−1.32698			−0.663489	+	0.748186i	$0.730925\pi$
$74$		0			0
$75$	−14.9740	+	0.882355i	−0.199654	+	0.0117647i
$76$		0			0
$77$		−	31.9302i		−	0.414679i
$78$		0			0
$79$	84.2348			1.06626			0.533132	−	0.846032i	$-0.321015\pi$
$79$	84.2348			1.06626			0.533132	+	0.846032i	$0.321015\pi$
$80$		0			0
$81$	78.7655	−	18.8942i	0.972414	−	0.233262i
$82$		0			0
$83$			20.4474i			0.246354i	0.992385	+	0.123177i	$0.0393083\pi$
$83$			20.4474i			0.246354i	−0.992385	+	0.123177i	$0.960692\pi$
$84$		0			0
$85$	−49.7651			−0.585472
$86$		0			0
$87$	2.01393	+	34.1774i	0.0231486	+	0.392844i
$88$		0			0
$89$		−	1.65255i		−	0.0185680i	−0.999957	−	0.00928398i	$-0.997045\pi$
$89$		−	1.65255i		−	0.0185680i	0.999957	−	0.00928398i	$-0.00295523\pi$
$90$		0			0
$91$	−33.6386			−0.369655
$92$		0			0
$93$	14.7936	−	0.871726i	0.159071	−	0.00937340i
$94$		0			0
$95$		−	64.6869i		−	0.680915i
$96$		0			0
$97$	29.7125			0.306314			0.153157	−	0.988202i	$-0.451056\pi$
$97$	29.7125			0.306314			0.153157	+	0.988202i	$0.451056\pi$
$98$		0			0
$99$	−12.7563	−	107.865i	−0.128852	−	1.08954i
$100$		0			0
$101$		−	33.7684i		−	0.334341i	−0.985928	−	0.167170i	$-0.946537\pi$
$101$		−	33.7684i		−	0.334341i	0.985928	−	0.167170i	$-0.0534630\pi$
$102$		0			0
$103$	120.360			1.16855			0.584274	−	0.811557i	$-0.301379\pi$
$103$	120.360			1.16855			0.584274	+	0.811557i	$0.301379\pi$
$104$		0			0
$105$	−1.04402	−	17.7175i	−0.00994302	−	0.168738i
$106$		0			0
$107$		−	174.545i		−	1.63126i	−0.578570	−	0.815632i	$-0.696389\pi$
$107$		−	174.545i		−	1.63126i	0.578570	−	0.815632i	$-0.303611\pi$
$108$		0			0
$109$	25.2313			0.231480			0.115740	−	0.993280i	$-0.463076\pi$
$109$	25.2313			0.231480			0.115740	+	0.993280i	$0.463076\pi$
$110$		0			0
$111$	−109.923	+	6.47728i	−0.990296	+	0.0583539i
$112$		0			0
$113$			126.424i			1.11880i	0.828898	+	0.559400i	$0.188969\pi$
$113$			126.424i			1.11880i	−0.828898	+	0.559400i	$0.811031\pi$
$114$		0			0
$115$	48.9924			0.426021
$116$		0			0
$117$	−113.636	+	13.4388i	−0.971246	+	0.114862i
$118$		0			0
$119$		−	58.8829i		−	0.494814i
$120$		0			0
$121$	−24.6487			−0.203708
$122$		0			0
$123$	−10.1801	−	172.762i	−0.0827654	−	1.40457i
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	7.33093			0.0577238			0.0288619	−	0.999583i	$-0.490812\pi$
$127$	7.33093			0.0577238			0.0288619	+	0.999583i	$0.490812\pi$
$128$		0			0
$129$	−172.948	+	10.1911i	−1.34068	+	0.0790006i
$130$		0			0
$131$			96.8346i			0.739195i	0.929192	+	0.369598i	$0.120505\pi$
$131$			96.8346i			0.739195i	−0.929192	+	0.369598i	$0.879495\pi$
$132$		0			0
$133$	76.5386			0.575478
$134$		0			0
$135$	−10.6051	−	59.4351i	−0.0785561	−	0.440260i
$136$		0			0
$137$		−	228.866i		−	1.67055i	−0.549830	−	0.835277i	$-0.685307\pi$
$137$		−	228.866i		−	1.67055i	0.549830	−	0.835277i	$-0.314693\pi$
$138$		0			0
$139$	−162.799			−1.17121			−0.585606	−	0.810596i	$-0.699144\pi$
$139$	−162.799			−1.17121			−0.585606	+	0.810596i	$0.699144\pi$
$140$		0			0
$141$	−2.76660	−	46.9506i	−0.0196213	−	0.332983i
$142$		0			0
$143$			153.441i			1.07301i
$144$		0			0
$145$	25.5185			0.175990
$146$		0			0
$147$	20.9636	−	1.23530i	0.142610	−	0.00840338i
$148$		0			0
$149$			1.91716i			0.0128668i	0.999979	+	0.00643341i	$0.00204783\pi$
$149$			1.91716i			0.0128668i	−0.999979	+	0.00643341i	$0.997952\pi$
$150$		0			0
$151$	−3.34803			−0.0221724			−0.0110862	−	0.999939i	$-0.503529\pi$
$151$	−3.34803			−0.0221724			−0.0110862	+	0.999939i	$0.503529\pi$
$152$		0			0
$153$	−23.5240	−	198.915i	−0.153752	−	1.30009i
$154$		0			0
$155$		−	11.0457i		−	0.0712623i
$156$		0			0
$157$	3.98288			0.0253687			0.0126843	−	0.999920i	$-0.495962\pi$
$157$	3.98288			0.0253687			0.0126843	+	0.999920i	$0.495962\pi$
$158$		0			0
$159$	−16.0995	−	273.217i	−0.101255	−	1.71835i
$160$		0			0
$161$			57.9686i			0.360054i
$162$		0			0
$163$	238.081			1.46062			0.730310	−	0.683116i	$-0.239376\pi$
$163$	238.081			1.46062			0.730310	+	0.683116i	$0.239376\pi$
$164$		0			0
$165$	−80.8178	+	4.76225i	−0.489805	+	0.0288621i
$166$		0			0
$167$		−	93.0174i		−	0.556990i	−0.960438	−	0.278495i	$-0.910164\pi$
$167$		−	93.0174i		−	0.556990i	0.960438	−	0.278495i	$-0.0898356\pi$
$168$		0			0
$169$	−7.34951			−0.0434882
$170$		0			0
$171$	258.558	−	30.5776i	1.51203	−	0.178816i
$172$		0			0
$173$		−	20.2396i		−	0.116992i	−0.998288	−	0.0584959i	$-0.981370\pi$
$173$		−	20.2396i		−	0.116992i	0.998288	−	0.0584959i	$-0.0186305\pi$
$174$		0			0
$175$	−13.2288			−0.0755929
$176$		0			0
$177$	−6.05474	−	102.752i	−0.0342076	−	0.580520i
$178$		0			0
$179$			115.119i			0.643122i	0.946889	+	0.321561i	$0.104207\pi$
$179$			115.119i			0.643122i	−0.946889	+	0.321561i	$0.895793\pi$
$180$		0			0
$181$	−35.1309			−0.194093			−0.0970467	−	0.995280i	$-0.530940\pi$
$181$	−35.1309			−0.194093			−0.0970467	+	0.995280i	$0.530940\pi$
$182$		0			0
$183$	212.998	−	12.5511i	1.16392	−	0.0685850i
$184$		0			0
$185$			82.0738i			0.443642i
$186$		0			0
$187$	−268.592			−1.43632
$188$		0			0
$189$	70.3246	−	12.5481i	0.372088	−	0.0663921i
$190$		0			0
$191$		−	128.664i		−	0.673631i	−0.941571	−	0.336816i	$-0.890650\pi$
$191$		−	128.664i		−	0.673631i	0.941571	−	0.336816i	$-0.109350\pi$
$192$		0			0
$193$	149.261			0.773373			0.386686	−	0.922211i	$-0.373620\pi$
$193$	149.261			0.773373			0.386686	+	0.922211i	$0.373620\pi$
$194$		0			0
$195$	5.01703	+	85.1417i	0.0257284	+	0.436624i
$196$		0			0
$197$			304.740i			1.54691i	0.633854	+	0.773453i	$0.281472\pi$
$197$			304.740i			1.54691i	−0.633854	+	0.773453i	$0.718528\pi$
$198$		0			0
$199$	98.5989			0.495472			0.247736	−	0.968828i	$-0.420313\pi$
$199$	98.5989			0.495472			0.247736	+	0.968828i	$0.420313\pi$
$200$		0			0
$201$	−60.4101	+	3.55971i	−0.300548	+	0.0177100i
$202$		0			0
$203$			30.1939i			0.148739i
$204$		0			0
$205$	−128.993			−0.629233
$206$		0			0
$207$	23.1588	+	195.826i	0.111878	+	0.946020i
$208$		0			0
$209$		−	349.128i		−	1.67047i
$210$		0			0
$211$	−255.428			−1.21056			−0.605280	−	0.796013i	$-0.706939\pi$
$211$	−255.428			−1.21056			−0.605280	+	0.796013i	$0.706939\pi$
$212$		0			0
$213$	12.3127	+	208.953i	0.0578061	+	0.981000i
$214$		0			0
$215$			129.131i			0.600611i
$216$		0			0
$217$	13.0694			0.0602276
$218$		0			0
$219$	−290.105	+	17.0946i	−1.32468	+	0.0780577i
$220$		0			0
$221$			282.962i			1.28037i
$222$		0			0
$223$	30.8285			0.138244			0.0691222	−	0.997608i	$-0.477980\pi$
$223$	30.8285			0.138244			0.0691222	+	0.997608i	$0.477980\pi$
$224$		0			0
$225$	−44.6886	+	5.28496i	−0.198616	+	0.0234887i
$226$		0			0
$227$			123.793i			0.545343i	0.962107	+	0.272672i	$0.0879072\pi$
$227$			123.793i			0.545343i	−0.962107	+	0.272672i	$0.912093\pi$
$228$		0			0
$229$	−38.8512			−0.169656			−0.0848279	−	0.996396i	$-0.527034\pi$
$229$	−38.8512			−0.169656			−0.0848279	+	0.996396i	$0.527034\pi$
$230$		0			0
$231$	−5.63476	−	95.6249i	−0.0243929	−	0.413960i
$232$		0			0
$233$		−	194.201i		−	0.833479i	−0.909026	−	0.416740i	$-0.863173\pi$
$233$		−	194.201i		−	0.833479i	0.909026	−	0.416740i	$-0.136827\pi$
$234$		0			0
$235$	−35.0556			−0.149173
$236$		0			0
$237$	252.267	−	14.8650i	1.06442	−	0.0627215i
$238$		0			0
$239$			228.989i			0.958115i	0.877784	+	0.479057i	$0.159021\pi$
$239$			228.989i			0.958115i	−0.877784	+	0.479057i	$0.840979\pi$
$240$		0			0
$241$	407.123			1.68931			0.844654	−	0.535313i	$-0.179806\pi$
$241$	407.123			1.68931			0.844654	+	0.535313i	$0.179806\pi$
$242$		0			0
$243$	232.553	−	70.4843i	0.957009	−	0.290059i
$244$		0			0
$245$		−	15.6525i		−	0.0638877i
$246$		0			0
$247$	−367.807			−1.48910
$248$		0			0
$249$	3.60838	+	61.2360i	0.0144915	+	0.245928i
$250$		0			0
$251$		−	164.458i		−	0.655211i	−0.944815	−	0.327605i	$-0.893758\pi$
$251$		−	164.458i		−	0.655211i	0.944815	−	0.327605i	$-0.106242\pi$
$252$		0			0
$253$	264.422			1.04515
$254$		0			0
$255$	−149.037	+	8.78210i	−0.584458	+	0.0344396i
$256$		0			0
$257$			384.159i			1.49478i	0.664385	+	0.747390i	$0.268693\pi$
$257$			384.159i			1.49478i	−0.664385	+	0.747390i	$0.731307\pi$
$258$		0			0
$259$	−97.1110			−0.374946
$260$		0			0
$261$	12.0626	+	101.999i	0.0462170	+	0.390802i
$262$		0			0
$263$			198.192i			0.753582i	0.926298	+	0.376791i	$0.122972\pi$
$263$			198.192i			0.753582i	−0.926298	+	0.376791i	$0.877028\pi$
$264$		0			0
$265$	−203.997			−0.769802
$266$		0			0
$267$	−0.291627	−	4.94906i	−0.00109224	−	0.0185358i
$268$		0			0
$269$		−	14.3502i		−	0.0533465i	−0.999644	−	0.0266732i	$-0.991509\pi$
$269$		−	14.3502i		−	0.0533465i	0.999644	−	0.0266732i	$-0.00849136\pi$
$270$		0			0
$271$	26.2455			0.0968468			0.0484234	−	0.998827i	$-0.484580\pi$
$271$	26.2455			0.0968468			0.0484234	+	0.998827i	$0.484580\pi$
$272$		0			0
$273$	−100.741	+	5.93623i	−0.369015	+	0.0217444i
$274$		0			0
$275$			60.3425i			0.219427i
$276$		0			0
$277$	21.2740			0.0768014			0.0384007	−	0.999262i	$-0.487774\pi$
$277$	21.2740			0.0768014			0.0384007	+	0.999262i	$0.487774\pi$
$278$		0			0
$279$	44.1503	−	5.22130i	0.158245	−	0.0187143i
$280$		0			0
$281$		−	322.063i		−	1.14613i	−0.819509	−	0.573066i	$-0.805754\pi$
$281$		−	322.063i		−	1.14613i	0.819509	−	0.573066i	$-0.194246\pi$
$282$		0			0
$283$	259.309			0.916288			0.458144	−	0.888878i	$-0.348514\pi$
$283$	259.309			0.916288			0.458144	+	0.888878i	$0.348514\pi$
$284$		0			0
$285$	−11.4154	−	193.725i	−0.0400539	−	0.679736i
$286$		0			0
$287$		−	152.626i		−	0.531799i
$288$		0			0
$289$	−206.313			−0.713886
$290$		0			0
$291$	88.9830	−	5.24339i	0.305784	−	0.0180185i
$292$		0			0
$293$		−	301.196i		−	1.02797i	−0.857798	−	0.513986i	$-0.828168\pi$
$293$		−	301.196i		−	1.02797i	0.857798	−	0.513986i	$-0.171832\pi$
$294$		0			0
$295$	−76.7197			−0.260067
$296$		0			0
$297$	−57.2377	−	320.783i	−0.192720	−	1.08008i
$298$		0			0
$299$		−	278.569i		−	0.931668i
$300$		0			0
$301$	−152.790			−0.507609
$302$		0			0
$303$	−5.95915	−	101.130i	−0.0196672	−	0.333762i
$304$		0			0
$305$		−	159.035i		−	0.521425i
$306$		0			0
$307$	262.697			0.855690			0.427845	−	0.903852i	$-0.359273\pi$
$307$	262.697			0.855690			0.427845	+	0.903852i	$0.359273\pi$
$308$		0			0
$309$	360.456	−	21.2401i	1.16652	−	0.0687383i
$310$		0			0
$311$			248.536i			0.799151i	0.916700	+	0.399575i	$0.130842\pi$
$311$			248.536i			0.799151i	−0.916700	+	0.399575i	$0.869158\pi$
$312$		0			0
$313$	426.733			1.36336			0.681682	−	0.731648i	$-0.261249\pi$
$313$	426.733			1.36336			0.681682	+	0.731648i	$0.261249\pi$
$314$		0			0
$315$	−6.25325	−	52.8762i	−0.0198516	−	0.167861i
$316$		0			0
$317$			107.379i			0.338735i	0.985553	+	0.169367i	$0.0541725\pi$
$317$			107.379i			0.338735i	−0.985553	+	0.169367i	$0.945828\pi$
$318$		0			0
$319$	137.728			0.431751
$320$		0			0
$321$	−30.8022	−	522.729i	−0.0959570	−	1.62844i
$322$		0			0
$323$		−	643.830i		−	1.99328i
$324$		0			0
$325$	63.5709			0.195603
$326$		0			0
$327$	75.5630	−	4.45260i	0.231079	−	0.0136165i
$328$		0			0
$329$		−	41.4784i		−	0.126074i
$330$		0			0
$331$	451.938			1.36537			0.682687	−	0.730711i	$-0.260811\pi$
$331$	451.938			1.36537			0.682687	+	0.730711i	$0.260811\pi$
$332$		0			0
$333$	−328.054	+	38.7964i	−0.985148	+	0.116506i
$334$		0			0
$335$			45.1051i			0.134642i
$336$		0			0
$337$	−638.072			−1.89339			−0.946695	−	0.322133i	$-0.895600\pi$
$337$	−638.072			−1.89339			−0.946695	+	0.322133i	$0.895600\pi$
$338$		0			0
$339$	22.3103	+	378.617i	0.0658120	+	1.11686i
$340$		0			0
$341$		−	59.6156i		−	0.174826i
$342$		0			0
$343$	18.5203			0.0539949
$344$		0			0
$345$	146.723	−	8.64575i	0.425283	−	0.0250601i
$346$		0			0
$347$		−	129.949i		−	0.374493i	−0.982313	−	0.187246i	$-0.940044\pi$
$347$		−	129.949i		−	0.374493i	0.982313	−	0.187246i	$-0.0599563\pi$
$348$		0			0
$349$	−15.1067			−0.0432856			−0.0216428	−	0.999766i	$-0.506890\pi$
$349$	−15.1067			−0.0432856			−0.0216428	+	0.999766i	$0.506890\pi$
$350$		0			0
$351$	−337.945	+	60.3000i	−0.962807	+	0.171795i
$352$		0			0
$353$			688.966i			1.95174i	0.218343	+	0.975872i	$0.429935\pi$
$353$			688.966i			1.95174i	−0.218343	+	0.975872i	$0.570065\pi$
$354$		0			0
$355$	156.015			0.439478
$356$		0			0
$357$	−10.3911	−	176.343i	−0.0291068	−	0.493957i
$358$		0			0
$359$			596.806i			1.66241i	0.555965	+	0.831206i	$0.312349\pi$
$359$			596.806i			1.66241i	−0.555965	+	0.831206i	$0.687651\pi$
$360$		0			0
$361$	475.879			1.31822
$362$		0			0
$363$	−73.8180	+	4.34978i	−0.203355	+	0.0119829i
$364$		0			0
$365$			216.606i			0.593442i
$366$		0			0
$367$	−653.268			−1.78002			−0.890011	−	0.455939i	$-0.849304\pi$
$367$	−653.268			−1.78002			−0.890011	+	0.455939i	$0.849304\pi$
$368$		0			0
$369$	−60.9751	−	515.593i	−0.165244	−	1.39727i
$370$		0			0
$371$		−	241.373i		−	0.650601i
$372$		0			0
$373$	−106.838			−0.286430			−0.143215	−	0.989692i	$-0.545744\pi$
$373$	−106.838			−0.286430			−0.143215	+	0.989692i	$0.545744\pi$
$374$		0			0
$375$	1.97301	+	33.4829i	0.00526135	+	0.0892878i
$376$		0			0
$377$		−	145.097i		−	0.384873i
$378$		0			0
$379$	152.839			0.403270			0.201635	−	0.979461i	$-0.435375\pi$
$379$	152.839			0.403270			0.201635	+	0.979461i	$0.435375\pi$
$380$		0			0
$381$	21.9547	−	1.29370i	0.0576239	−	0.00339553i
$382$		0			0
$383$			201.860i			0.527050i	0.964653	+	0.263525i	$0.0848851\pi$
$383$			201.860i			0.527050i	−0.964653	+	0.263525i	$0.915115\pi$
$384$		0			0
$385$	−71.3982			−0.185450
$386$		0			0
$387$	−516.147	+	61.0406i	−1.33371	+	0.157728i
$388$		0			0
$389$		−	414.977i		−	1.06678i	−0.845870	−	0.533389i	$-0.820918\pi$
$389$		−	414.977i		−	1.06678i	0.845870	−	0.533389i	$-0.179082\pi$
$390$		0			0
$391$	487.623			1.24712
$392$		0			0
$393$	17.0885	+	290.001i	0.0434822	+	0.737915i
$394$		0			0
$395$		−	188.355i		−	0.476847i
$396$		0			0
$397$	−410.015			−1.03278			−0.516391	−	0.856353i	$-0.672725\pi$
$397$	−410.015			−1.03278			−0.516391	+	0.856353i	$0.672725\pi$
$398$		0			0
$399$	229.218	−	13.5068i	0.574481	−	0.0338517i
$400$		0			0
$401$			347.186i			0.865801i	0.901442	+	0.432901i	$0.142510\pi$
$401$			347.186i			0.865801i	−0.901442	+	0.432901i	$0.857490\pi$
$402$		0			0
$403$	−62.8051			−0.155844
$404$		0			0
$405$	−42.2487	−	176.125i	−0.104318	−	0.434877i
$406$		0			0
$407$			442.968i			1.08837i
$408$		0			0
$409$	−103.523			−0.253112			−0.126556	−	0.991959i	$-0.540392\pi$
$409$	−103.523			−0.253112			−0.126556	+	0.991959i	$0.540392\pi$
$410$		0			0
$411$	−40.3882	−	685.409i	−0.0982681	−	1.66766i
$412$		0			0
$413$		−	90.7760i		−	0.219797i
$414$		0			0
$415$	45.7218			0.110173
$416$		0			0
$417$	−487.550	+	28.7292i	−1.16918	+	0.0688951i
$418$		0			0
$419$			85.1144i			0.203137i	0.994829	+	0.101569i	$0.0323861\pi$
$419$			85.1144i			0.203137i	−0.994829	+	0.101569i	$0.967614\pi$
$420$		0			0
$421$	−489.724			−1.16324			−0.581620	−	0.813461i	$-0.697581\pi$
$421$	−489.724			−1.16324			−0.581620	+	0.813461i	$0.697581\pi$
$422$		0			0
$423$	−16.5708	−	140.120i	−0.0391746	−	0.331252i
$424$		0			0
$425$			111.278i			0.261831i
$426$		0			0
$427$	188.172			0.440685
$428$		0			0
$429$	27.0779	+	459.526i	0.0631187	+	1.07116i
$430$		0			0
$431$			577.191i			1.33919i	0.742727	+	0.669595i	$0.233532\pi$
$431$			577.191i			1.33919i	−0.742727	+	0.669595i	$0.766468\pi$
$432$		0			0
$433$	94.5352			0.218326			0.109163	−	0.994024i	$-0.465183\pi$
$433$	94.5352			0.218326			0.109163	+	0.994024i	$0.465183\pi$
$434$		0			0
$435$	76.4230	−	4.50328i	0.175685	−	0.0103524i
$436$		0			0
$437$			633.834i			1.45042i
$438$		0			0
$439$	−159.446			−0.363202			−0.181601	−	0.983372i	$-0.558128\pi$
$439$	−159.446			−0.363202			−0.181601	+	0.983372i	$0.558128\pi$
$440$		0			0
$441$	62.5640	−	7.39895i	0.141869	−	0.0167777i
$442$		0			0
$443$		−	129.501i		−	0.292326i	−0.989261	−	0.146163i	$-0.953308\pi$
$443$		−	129.501i		−	0.292326i	0.989261	−	0.146163i	$-0.0466924\pi$
$444$		0			0
$445$	−3.69521			−0.00830385
$446$		0			0
$447$	0.338323	+	5.74151i	0.000756874	+	0.0128445i
$448$		0			0
$449$			306.203i			0.681967i	0.940069	+	0.340984i	$0.110760\pi$
$449$			306.203i			0.681967i	−0.940069	+	0.340984i	$0.889240\pi$
$450$		0			0
$451$	−696.200			−1.54368
$452$		0			0
$453$	−10.0267	+	0.590830i	−0.0221340	+	0.00130426i
$454$		0			0
$455$			75.2181i			0.165315i
$456$		0			0
$457$	−143.189			−0.313325			−0.156662	−	0.987652i	$-0.550073\pi$
$457$	−143.189			−0.313325			−0.156662	+	0.987652i	$0.550073\pi$
$458$		0			0
$459$	−105.553	−	591.559i	−0.229962	−	1.28880i
$460$		0			0
$461$			743.362i			1.61250i	0.591575	+	0.806250i	$0.298506\pi$
$461$			743.362i			1.61250i	−0.591575	+	0.806250i	$0.701494\pi$
$462$		0			0
$463$	577.246			1.24675			0.623375	−	0.781923i	$-0.285761\pi$
$463$	577.246			1.24675			0.623375	+	0.781923i	$0.285761\pi$
$464$		0			0
$465$	−1.94924	−	33.0796i	−0.00419191	−	0.0711389i
$466$		0			0
$467$		−	338.396i		−	0.724617i	−0.932058	−	0.362308i	$-0.881989\pi$
$467$		−	338.396i		−	0.724617i	0.932058	−	0.362308i	$-0.118011\pi$
$468$		0			0
$469$	−53.3691			−0.113793
$470$		0			0
$471$	11.9280	−	0.702863i	0.0253247	−	0.00149228i
$472$		0			0
$473$			696.947i			1.47346i
$474$		0			0
$475$	−144.644			−0.304514
$476$		0			0
$477$	−96.4299	−	815.392i	−0.202159	−	1.70942i
$478$		0			0
$479$			239.609i			0.500228i	0.968216	+	0.250114i	$0.0804681\pi$
$479$			239.609i			0.500228i	−0.968216	+	0.250114i	$0.919532\pi$
$480$		0			0
$481$	466.668			0.970203
$482$		0			0
$483$	10.2298	+	173.605i	0.0211797	+	0.359430i
$484$		0			0
$485$		−	66.4391i		−	0.136988i
$486$		0			0
$487$	95.6658			0.196439			0.0982195	−	0.995165i	$-0.468685\pi$
$487$	95.6658			0.196439			0.0982195	+	0.995165i	$0.468685\pi$
$488$		0			0
$489$	713.007	−	42.0144i	1.45809	−	0.0859191i
$490$		0			0
$491$			352.470i			0.717861i	0.933364	+	0.358930i	$0.116858\pi$
$491$			352.470i			0.717861i	−0.933364	+	0.358930i	$0.883142\pi$
$492$		0			0
$493$	253.986			0.515185
$494$		0			0
$495$	−241.193	+	28.5240i	−0.487259	+	0.0576242i
$496$		0			0
$497$			184.599i			0.371426i
$498$		0			0
$499$	317.440			0.636152			0.318076	−	0.948065i	$-0.396963\pi$
$499$	317.440			0.636152			0.318076	+	0.948065i	$0.396963\pi$
$500$		0			0
$501$	−16.4149	−	278.569i	−0.0327642	−	0.556026i
$502$		0			0
$503$			244.292i			0.485670i	0.970068	+	0.242835i	$0.0780773\pi$
$503$			244.292i			0.485670i	−0.970068	+	0.242835i	$0.921923\pi$
$504$		0			0
$505$	−75.5085			−0.149522
$506$		0			0
$507$	−22.0104	+	1.29698i	−0.0434129	+	0.00255814i
$508$		0			0
$509$		−	4.58079i		−	0.00899959i	−0.999990	−	0.00449979i	$-0.998568\pi$
$509$		−	4.58079i		−	0.00899959i	0.999990	−	0.00449979i	$-0.00143233\pi$
$510$		0			0
$511$	−256.292			−0.501550
$512$		0			0
$513$	768.935	−	137.202i	1.49890	−	0.267450i
$514$		0			0
$515$		−	269.134i		−	0.522590i
$516$		0			0
$517$	−189.202			−0.365961
$518$		0			0
$519$	−3.57170	−	60.6136i	−0.00688189	−	0.116789i
$520$		0			0
$521$			813.583i			1.56158i	0.624794	+	0.780789i	$0.285183\pi$
$521$			813.583i			1.56158i	−0.624794	+	0.780789i	$0.714817\pi$
$522$		0			0
$523$	6.71882			0.0128467			0.00642335	−	0.999979i	$-0.497955\pi$
$523$	6.71882			0.0128467			0.00642335	+	0.999979i	$0.497955\pi$
$524$		0			0
$525$	−39.6175	+	2.33449i	−0.0754620	+	0.00444665i
$526$		0			0
$527$		−	109.938i		−	0.208610i
$528$		0			0
$529$	48.9484			0.0925301
$530$		0			0
$531$	−36.2655	−	306.654i	−0.0682966	−	0.577502i
$532$		0			0
$533$			733.448i			1.37607i
$534$		0			0
$535$	−390.295			−0.729524
$536$		0			0
$537$	20.3151	+	344.758i	0.0378308	+	0.642008i
$538$		0			0
$539$		−	84.4795i		−	0.156734i
$540$		0			0
$541$	18.1305			0.0335129			0.0167565	−	0.999860i	$-0.494666\pi$
$541$	18.1305			0.0335129			0.0167565	+	0.999860i	$0.494666\pi$
$542$		0			0
$543$	−105.210	+	6.19959i	−0.193757	+	0.0114173i
$544$		0			0
$545$		−	56.4190i		−	0.103521i
$546$		0			0
$547$	−111.345			−0.203556			−0.101778	−	0.994807i	$-0.532453\pi$
$547$	−111.345			−0.203556			−0.101778	+	0.994807i	$0.532453\pi$
$548$		0			0
$549$	635.673	−	75.1760i	1.15787	−	0.136933i
$550$		0			0
$551$			330.143i			0.599170i
$552$		0			0
$553$	222.864			0.403010
$554$		0			0
$555$	14.4836	+	245.795i	0.0260967	+	0.442874i
$556$		0			0
$557$		−	100.916i		−	0.181177i	−0.995888	−	0.0905885i	$-0.971125\pi$
$557$		−	100.916i		−	0.181177i	0.995888	−	0.0905885i	$-0.0288748\pi$
$558$		0			0
$559$	734.235			1.31348
$560$		0			0
$561$	−804.381	+	47.3987i	−1.43383	+	0.0844897i
$562$		0			0
$563$			942.043i			1.67326i	0.547772	+	0.836628i	$0.315476\pi$
$563$			942.043i			1.67326i	−0.547772	+	0.836628i	$0.684524\pi$
$564$		0			0
$565$	282.694			0.500343
$566$		0			0
$567$	208.394	−	49.9894i	0.367538	−	0.0881647i
$568$		0			0
$569$			1113.81i			1.95749i	0.205086	+	0.978744i	$0.434253\pi$
$569$			1113.81i			1.95749i	−0.205086	+	0.978744i	$0.565747\pi$
$570$		0			0
$571$	489.370			0.857041			0.428520	−	0.903532i	$-0.359035\pi$
$571$	489.370			0.857041			0.428520	+	0.903532i	$0.359035\pi$
$572$		0			0
$573$	−22.7054	−	385.322i	−0.0396255	−	0.672465i
$574$		0			0
$575$		−	109.550i		−	0.190522i
$576$		0			0
$577$	176.145			0.305277			0.152638	−	0.988282i	$-0.451223\pi$
$577$	176.145			0.305277			0.152638	+	0.988282i	$0.451223\pi$
$578$		0			0
$579$	447.008	−	26.3402i	0.772034	−	0.0454926i
$580$		0			0
$581$			54.0988i			0.0931132i
$582$		0			0
$583$	−1101.01			−1.88853
$584$		0			0
$585$	30.0501	+	254.097i	0.0513677	+	0.434354i
$586$		0			0
$587$			120.248i			0.204852i	0.994741	+	0.102426i	$0.0326604\pi$
$587$			120.248i			0.204852i	−0.994741	+	0.102426i	$0.967340\pi$
$588$		0			0
$589$	142.902			0.242618
$590$		0			0
$591$	53.7779	+	912.638i	0.0909947	+	1.54423i
$592$		0			0
$593$		−	808.407i		−	1.36325i	−0.731702	−	0.681625i	$-0.761274\pi$
$593$		−	808.407i		−	1.36325i	0.731702	−	0.681625i	$-0.238726\pi$
$594$		0			0
$595$	−131.666			−0.221288
$596$		0			0
$597$	295.285	−	17.3999i	0.494614	−	0.0291455i
$598$		0			0
$599$			679.382i			1.13419i	0.823651	+	0.567097i	$0.191934\pi$
$599$			679.382i			1.13419i	−0.823651	+	0.567097i	$0.808066\pi$
$600$		0			0
$601$	349.965			0.582305			0.291153	−	0.956677i	$-0.405961\pi$
$601$	349.965			0.582305			0.291153	+	0.956677i	$0.405961\pi$
$602$		0			0
$603$	−180.288	+	21.3213i	−0.298985	+	0.0353586i
$604$		0			0
$605$			55.1161i			0.0911011i
$606$		0			0
$607$	−463.847			−0.764163			−0.382082	−	0.924129i	$-0.624793\pi$
$607$	−463.847			−0.764163			−0.382082	+	0.924129i	$0.624793\pi$
$608$		0			0
$609$	5.32835	+	90.4249i	0.00874935	+	0.148481i
$610$		0			0
$611$			199.325i			0.326227i
$612$		0			0
$613$	−137.987			−0.225102			−0.112551	−	0.993646i	$-0.535902\pi$
$613$	−137.987			−0.225102			−0.112551	+	0.993646i	$0.535902\pi$
$614$		0			0
$615$	−386.308	+	22.7635i	−0.628144	+	0.0370138i
$616$		0			0
$617$			90.3121i			0.146373i	0.997318	+	0.0731864i	$0.0233168\pi$
$617$			90.3121i			0.146373i	−0.997318	+	0.0731864i	$0.976683\pi$
$618$		0			0
$619$	558.874			0.902866			0.451433	−	0.892305i	$-0.350913\pi$
$619$	558.874			0.902866			0.451433	+	0.892305i	$0.350913\pi$
$620$		0			0
$621$	103.914	+	582.374i	0.167333	+	0.937800i
$622$		0			0
$623$		−	4.37223i		−	0.00701803i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−61.6110	−	1045.57i	−0.0982631	−	1.66758i
$628$		0			0
$629$			816.882i			1.29870i
$630$		0			0
$631$	699.239			1.10814			0.554072	−	0.832469i	$-0.313073\pi$
$631$	699.239			1.10814			0.554072	+	0.832469i	$0.313073\pi$
$632$		0			0
$633$	−764.958	+	45.0757i	−1.20846	+	0.0712096i
$634$		0			0
$635$		−	16.3925i		−	0.0258149i
$636$		0			0
$637$	−88.9993			−0.139716
$638$		0			0
$639$	73.7483	+	623.601i	0.115412	+	0.975901i
$640$		0			0
$641$		−	773.823i		−	1.20721i	−0.797283	−	0.603606i	$-0.793730\pi$
$641$		−	773.823i		−	1.20721i	0.797283	−	0.603606i	$-0.206270\pi$
$642$		0			0
$643$	−93.1342			−0.144843			−0.0724216	−	0.997374i	$-0.523073\pi$
$643$	−93.1342			−0.144843			−0.0724216	+	0.997374i	$0.523073\pi$
$644$		0			0
$645$	22.7879	+	386.723i	0.0353301	+	0.599571i
$646$		0			0
$647$			1130.06i			1.74662i	0.487164	+	0.873311i	$0.338031\pi$
$647$			1130.06i			1.74662i	−0.487164	+	0.873311i	$0.661969\pi$
$648$		0			0
$649$	−414.071			−0.638014
$650$		0			0
$651$	39.1403	−	2.30637i	0.0601234	−	0.00354281i
$652$		0			0
$653$		−	482.760i		−	0.739296i	−0.929172	−	0.369648i	$-0.879478\pi$
$653$		−	482.760i		−	0.739296i	0.929172	−	0.369648i	$-0.120522\pi$
$654$		0			0
$655$	216.529			0.330578
$656$		0			0
$657$	−865.791	+	102.390i	−1.31779	+	0.155845i
$658$		0			0
$659$		−	39.2039i		−	0.0594901i	−0.999558	−	0.0297450i	$-0.990530\pi$
$659$		−	39.2039i		−	0.0594901i	0.999558	−	0.0297450i	$-0.00946953\pi$
$660$		0			0
$661$	−638.260			−0.965598			−0.482799	−	0.875731i	$-0.660380\pi$
$661$	−638.260			−0.965598			−0.482799	+	0.875731i	$0.660380\pi$
$662$		0			0
$663$	49.9346	+	847.417i	0.0753162	+	1.27815i
$664$		0			0
$665$		−	171.145i		−	0.257362i
$666$		0			0
$667$	−250.043			−0.374877
$668$		0			0
$669$	92.3253	−	5.44033i	0.138005	−	0.00813204i
$670$		0			0
$671$		−	858.342i		−	1.27920i
$672$		0			0
$673$	233.289			0.346641			0.173320	−	0.984866i	$-0.444550\pi$
$673$	233.289			0.346641			0.173320	+	0.984866i	$0.444550\pi$
$674$		0			0
$675$	−132.901	+	23.7137i	−0.196890	+	0.0351314i
$676$		0			0
$677$		−	550.920i		−	0.813766i	−0.913480	−	0.406883i	$-0.866616\pi$
$677$		−	550.920i		−	0.813766i	0.913480	−	0.406883i	$-0.133384\pi$
$678$		0			0
$679$	78.6118			0.115776
$680$		0			0
$681$	21.8459	+	370.736i	0.0320791	+	0.544399i
$682$		0			0
$683$			117.573i			0.172142i	0.996289	+	0.0860712i	$0.0274313\pi$
$683$			117.573i			0.172142i	−0.996289	+	0.0860712i	$0.972569\pi$
$684$		0			0
$685$	−511.760			−0.747094
$686$		0			0
$687$	−116.352	+	6.85611i	−0.169362	+	0.00997978i
$688$		0			0
$689$			1159.92i			1.68348i
$690$		0			0
$691$	−831.462			−1.20327			−0.601637	−	0.798770i	$-0.705485\pi$
$691$	−831.462			−1.20327			−0.601637	+	0.798770i	$0.705485\pi$
$692$		0			0
$693$	−33.7500	−	285.383i	−0.0487014	−	0.411809i
$694$		0			0
$695$			364.029i			0.523782i
$696$		0			0
$697$	−1283.87			−1.84199
$698$		0			0
$699$	−34.2708	−	581.593i	−0.0490283	−	0.832036i
$700$		0			0
$701$		−	1265.38i		−	1.80510i	−0.430583	−	0.902551i	$-0.641692\pi$
$701$		−	1265.38i		−	1.80510i	0.430583	−	0.902551i	$-0.358308\pi$
$702$		0			0
$703$	−1061.82			−1.51041
$704$		0			0
$705$	−104.985	+	6.18630i	−0.148914	+	0.00877489i
$706$		0			0
$707$		−	89.3429i		−	0.126369i
$708$		0			0
$709$	404.244			0.570161			0.285080	−	0.958504i	$-0.407980\pi$
$709$	404.244			0.570161			0.285080	+	0.958504i	$0.407980\pi$
$710$		0			0
$711$	752.867	−	89.0356i	1.05888	−	0.125226i
$712$		0			0
$713$			108.231i			0.151796i
$714$		0			0
$715$	343.105			0.479867
$716$		0			0
$717$	40.4100	+	685.779i	0.0563598	+	0.956456i
$718$		0			0
$719$			790.437i			1.09936i	0.835377	+	0.549678i	$0.185250\pi$
$719$			790.437i			1.09936i	−0.835377	+	0.549678i	$0.814750\pi$
$720$		0			0
$721$	318.444			0.441670
$722$		0			0
$723$	1219.25	−	71.8454i	1.68638	−	0.0993713i
$724$		0			0
$725$		−	57.0611i		−	0.0787050i
$726$		0			0
$727$	−951.432			−1.30871			−0.654355	−	0.756188i	$-0.727060\pi$
$727$	−951.432			−1.30871			−0.654355	+	0.756188i	$0.727060\pi$
$728$		0			0
$729$	684.013	−	252.126i	0.938289	−	0.345851i
$730$		0			0
$731$			1285.25i			1.75820i
$732$		0			0
$733$	920.746			1.25613			0.628067	−	0.778159i	$-0.283846\pi$
$733$	920.746			1.25613			0.628067	+	0.778159i	$0.283846\pi$
$734$		0			0
$735$	−2.76221	−	46.8761i	−0.00375811	−	0.0637770i
$736$		0			0
$737$			243.441i			0.330314i
$738$		0			0
$739$	312.768			0.423231			0.211615	−	0.977353i	$-0.432128\pi$
$739$	312.768			0.423231			0.211615	+	0.977353i	$0.432128\pi$
$740$		0			0
$741$	−1101.51	+	64.9073i	−1.48652	+	0.0875941i
$742$		0			0
$743$		−	1116.08i		−	1.50212i	−0.660234	−	0.751060i	$-0.729543\pi$
$743$		−	1116.08i		−	1.50212i	0.660234	−	0.751060i	$-0.270457\pi$
$744$		0			0
$745$	4.28689			0.00575422
$746$		0			0
$747$	21.6128	+	182.753i	0.0289328	+	0.244649i
$748$		0			0
$749$		−	461.804i		−	0.616560i
$750$		0			0
$751$	1419.71			1.89043			0.945215	−	0.326448i	$-0.105852\pi$
$751$	1419.71			1.89043			0.945215	+	0.326448i	$0.105852\pi$
$752$		0			0
$753$	−29.0221	−	492.519i	−0.0385419	−	0.654076i
$754$		0			0
$755$			7.48642i			0.00991578i
$756$		0			0
$757$	822.074			1.08596			0.542981	−	0.839745i	$-0.317295\pi$
$757$	822.074			1.08596			0.542981	+	0.839745i	$0.317295\pi$
$758$		0			0
$759$	791.892	−	46.6628i	1.04334	−	0.0614793i
$760$		0			0
$761$			1366.47i			1.79562i	0.440380	+	0.897812i	$0.354844\pi$
$761$			1366.47i			1.79562i	−0.440380	+	0.897812i	$0.645156\pi$
$762$		0			0
$763$	66.7559			0.0874913
$764$		0			0
$765$	−444.786	+	52.6014i	−0.581420	+	0.0687600i
$766$		0			0
$767$			436.225i			0.568742i
$768$		0			0
$769$	117.440			0.152718			0.0763591	−	0.997080i	$-0.475670\pi$
$769$	117.440			0.152718			0.0763591	+	0.997080i	$0.475670\pi$
$770$		0			0
$771$	67.7929	+	1150.48i	0.0879285	+	1.49219i
$772$		0			0
$773$			306.116i			0.396011i	0.980201	+	0.198005i	$0.0634464\pi$
$773$			306.116i			0.396011i	−0.980201	+	0.198005i	$0.936554\pi$
$774$		0			0
$775$	−24.6988			−0.0318695
$776$		0			0
$777$	−290.828	+	17.1373i	−0.374297	+	0.0220557i
$778$		0			0
$779$		−	1668.83i		−	2.14227i
$780$		0			0
$781$	842.041			1.07816
$782$		0			0
$783$	54.1252	+	303.339i	0.0691254	+	0.387406i
$784$		0			0
$785$		−	8.90599i		−	0.0113452i
$786$		0			0
$787$	279.568			0.355232			0.177616	−	0.984100i	$-0.443161\pi$
$787$	279.568			0.355232			0.177616	+	0.984100i	$0.443161\pi$
$788$		0			0
$789$	34.9752	+	593.546i	0.0443285	+	0.752277i
$790$		0			0
$791$			334.488i			0.422867i
$792$		0			0
$793$	−904.264			−1.14031
$794$		0			0
$795$	−610.933	+	35.9996i	−0.768469	+	0.0452826i
$796$		0			0
$797$		−	327.577i		−	0.411013i	−0.978656	−	0.205507i	$-0.934116\pi$
$797$		−	327.577i		−	0.411013i	0.978656	−	0.205507i	$-0.0658842\pi$
$798$		0			0
$799$	−348.909			−0.436682
$800$		0			0
$801$	−1.74673	−	14.7700i	−0.00218069	−	0.0184395i
$802$		0			0
$803$			1169.07i			1.45588i
$804$		0			0
$805$	129.622			0.161021
$806$		0			0
$807$	−2.53240	−	42.9761i	−0.00313804	−	0.0532541i
$808$		0			0
$809$			219.021i			0.270731i	0.990796	+	0.135365i	$0.0432208\pi$
$809$			219.021i			0.270731i	−0.990796	+	0.135365i	$0.956779\pi$
$810$		0			0
$811$	−57.9161			−0.0714132			−0.0357066	−	0.999362i	$-0.511368\pi$
$811$	−57.9161			−0.0714132			−0.0357066	+	0.999362i	$0.511368\pi$
$812$		0			0
$813$	78.6001	−	4.63157i	0.0966791	−	0.00569688i
$814$		0			0
$815$		−	532.366i		−	0.653209i
$816$		0			0
$817$	−1670.62			−2.04482
$818$		0			0
$819$	−300.652	+	35.5557i	−0.367096	+	0.0434136i
$820$		0			0
$821$		−	1385.60i		−	1.68770i	−0.536577	−	0.843851i	$-0.680283\pi$
$821$		−	1385.60i		−	1.68770i	0.536577	−	0.843851i	$-0.319717\pi$
$822$		0			0
$823$	−1414.41			−1.71861			−0.859303	−	0.511468i	$-0.829102\pi$
$823$	−1414.41			−1.71861			−0.859303	+	0.511468i	$0.829102\pi$
$824$		0			0
$825$	10.6487	+	180.714i	0.0129075	+	0.219047i
$826$		0			0
$827$			305.144i			0.368977i	0.982835	+	0.184488i	$0.0590628\pi$
$827$			305.144i			0.368977i	−0.982835	+	0.184488i	$0.940937\pi$
$828$		0			0
$829$	517.903			0.624732			0.312366	−	0.949962i	$-0.398878\pi$
$829$	517.903			0.624732			0.312366	+	0.949962i	$0.398878\pi$
$830$		0			0
$831$	63.7114	−	3.75424i	0.0766684	−	0.00451774i
$832$		0			0
$833$		−	155.789i		−	0.187022i
$834$		0			0
$835$	−207.993			−0.249094
$836$		0			0
$837$	131.300	−	23.4280i	0.156870	−	0.0279905i
$838$		0			0
$839$			3.40272i			0.00405569i	0.999998	+	0.00202784i	$0.000645483\pi$
$839$			3.40272i			0.00405569i	−0.999998	+	0.00202784i	$0.999355\pi$
$840$		0			0
$841$	710.761			0.845138
$842$		0			0
$843$	−56.8349	−	964.517i	−0.0674198	−	1.14415i
$844$		0			0
$845$			16.4340i			0.0194485i
$846$		0			0
$847$	−65.2143			−0.0769944
$848$		0			0
$849$	776.581	−	45.7606i	0.914701	−	0.0538994i
$850$		0			0
$851$		−	804.199i		−	0.945004i
$852$		0			0
$853$	639.114			0.749255			0.374627	−	0.927175i	$-0.377771\pi$
$853$	639.114			0.749255			0.374627	+	0.927175i	$0.377771\pi$
$854$		0			0
$855$	−68.3736	−	578.153i	−0.0799691	−	0.676202i
$856$		0			0
$857$			724.119i			0.844946i	0.906375	+	0.422473i	$0.138838\pi$
$857$			724.119i			0.844946i	−0.906375	+	0.422473i	$0.861162\pi$
$858$		0			0
$859$	952.638			1.10901			0.554504	−	0.832181i	$-0.312908\pi$
$859$	952.638			1.10901			0.554504	+	0.832181i	$0.312908\pi$
$860$		0			0
$861$	−26.9341	−	457.086i	−0.0312824	−	0.530878i
$862$		0			0
$863$			1609.84i			1.86540i	0.360659	+	0.932698i	$0.382552\pi$
$863$			1609.84i			1.86540i	−0.360659	+	0.932698i	$0.617448\pi$
$864$		0			0
$865$	−45.2571			−0.0523203
$866$		0			0
$867$	−617.868	+	36.4083i	−0.712650	+	0.0419934i
$868$		0			0
$869$		−	1016.59i		−	1.16984i
$870$		0			0
$871$	256.466			0.294450
$872$		0			0
$873$	265.562	−	31.4059i	0.304194	−	0.0359746i
$874$		0			0
$875$			29.5804i			0.0338062i
$876$		0			0
$877$	740.088			0.843886			0.421943	−	0.906622i	$-0.361348\pi$
$877$	740.088			0.843886			0.421943	+	0.906622i	$0.361348\pi$
$878$		0			0
$879$	−53.1524	−	902.023i	−0.0604691	−	1.02619i
$880$		0			0
$881$			81.0232i			0.0919673i	0.998942	+	0.0459836i	$0.0146422\pi$
$881$			81.0232i			0.0919673i	−0.998942	+	0.0459836i	$0.985358\pi$
$882$		0			0
$883$	−783.824			−0.887682			−0.443841	−	0.896105i	$-0.646385\pi$
$883$	−783.824			−0.887682			−0.443841	+	0.896105i	$0.646385\pi$
$884$		0			0
$885$	−229.761	+	13.5388i	−0.259616	+	0.0152981i
$886$		0			0
$887$			1320.12i			1.48830i	0.668012	+	0.744150i	$0.267145\pi$
$887$			1320.12i			1.48830i	−0.668012	+	0.744150i	$0.732855\pi$
$888$		0			0
$889$	19.3958			0.0218176
$890$		0			0
$891$	−228.025	−	950.582i	−0.255920	−	1.06687i
$892$		0			0
$893$		−	453.528i		−	0.507870i
$894$		0			0
$895$	257.413			0.287613
$896$		0			0
$897$	−49.1593	−	834.259i	−0.0548042	−	0.930055i
$898$		0			0
$899$			56.3738i			0.0627072i
$900$		0			0
$901$	−2030.39			−2.25349
$902$		0			0
$903$	−457.577	+	26.9631i	−0.506730	+	0.0298594i
$904$		0			0
$905$			78.5551i			0.0868012i
$906$		0			0
$907$	−1149.97			−1.26788			−0.633942	−	0.773381i	$-0.718564\pi$
$907$	−1149.97			−1.26788			−0.633942	+	0.773381i	$0.718564\pi$
$908$		0			0
$909$	−35.6930	−	301.813i	−0.0392662	−	0.332027i
$910$		0			0
$911$		−	1275.15i		−	1.39972i	−0.714280	−	0.699860i	$-0.753246\pi$
$911$		−	1275.15i		−	1.39972i	0.714280	−	0.699860i	$-0.246754\pi$
$912$		0			0
$913$	246.770			0.270284
$914$		0			0
$915$	−28.0650	−	476.278i	−0.0306722	−	0.520522i
$916$		0			0
$917$			256.200i			0.279390i
$918$		0			0
$919$	228.393			0.248523			0.124262	−	0.992249i	$-0.460344\pi$
$919$	228.393			0.248523			0.124262	+	0.992249i	$0.460344\pi$
$920$		0			0
$921$	786.726	−	46.3584i	0.854209	−	0.0503349i
$922$		0			0
$923$		−	887.092i		−	0.961096i
$924$		0			0
$925$	183.523			0.198403
$926$		0			0
$927$	1075.75	−	127.220i	1.16046	−	0.137239i
$928$		0			0
$929$		−	71.5997i		−	0.0770718i	−0.999257	−	0.0385359i	$-0.987731\pi$
$929$		−	71.5997i		−	0.0770718i	0.999257	−	0.0385359i	$-0.0122694\pi$
$930$		0			0
$931$	202.502			0.217510
$932$		0			0
$933$	43.8594	+	744.317i	0.0470090	+	0.797767i
$934$		0			0
$935$			600.590i			0.642342i
$936$		0			0
$937$	606.742			0.647536			0.323768	−	0.946136i	$-0.395050\pi$
$937$	606.742			0.647536			0.323768	+	0.946136i	$0.395050\pi$
$938$		0			0
$939$	1277.98	−	75.3060i	1.36100	−	0.0801981i
$940$		0			0
$941$		−	1043.93i		−	1.10938i	−0.832056	−	0.554692i	$-0.812836\pi$
$941$		−	1043.93i		−	1.10938i	0.832056	−	0.554692i	$-0.187164\pi$
$942$		0			0
$943$	1263.93			1.34033
$944$		0			0
$945$	−28.0584	−	157.251i	−0.0296914	−	0.166403i
$946$		0			0
$947$			1057.69i			1.11688i	0.829545	+	0.558440i	$0.188600\pi$
$947$			1057.69i			1.11688i	−0.829545	+	0.558440i	$0.811400\pi$
$948$		0			0
$949$	1231.61			1.29780
$950$		0			0
$951$	18.9493	+	321.579i	0.0199256	+	0.338148i
$952$		0			0
$953$		−	278.572i		−	0.292311i	−0.989262	−	0.146155i	$-0.953310\pi$
$953$		−	278.572i		−	0.292311i	0.989262	−	0.146155i	$-0.0466900\pi$
$954$		0			0
$955$	−287.700			−0.301257
$956$		0			0
$957$	412.470	−	24.3051i	0.431003	−	0.0253972i
$958$		0			0
$959$		−	605.522i		−	0.631410i
$960$		0			0
$961$	−936.599			−0.974608
$962$		0			0
$963$	−184.493	−	1560.04i	−0.191582	−	1.61998i
$964$		0			0
$965$		−	333.758i		−	0.345863i
$966$		0			0
$967$	−717.616			−0.742105			−0.371053	−	0.928612i	$-0.621003\pi$
$967$	−717.616			−0.742105			−0.371053	+	0.928612i	$0.621003\pi$
$968$		0			0
$969$	−113.617	−	1928.15i	−0.117252	−	1.98983i
$970$		0			0
$971$		−	1566.83i		−	1.61363i	−0.590806	−	0.806814i	$-0.701190\pi$
$971$		−	1566.83i		−	1.61363i	0.590806	−	0.806814i	$-0.298810\pi$
$972$		0			0
$973$	−430.725			−0.442677
$974$		0			0
$975$	190.383	−	11.2184i	0.195264	−	0.0115061i
$976$		0			0
$977$		−	727.568i		−	0.744696i	−0.928093	−	0.372348i	$-0.878553\pi$
$977$		−	727.568i		−	0.744696i	0.928093	−	0.372348i	$-0.121447\pi$
$978$		0			0
$979$	−19.9438			−0.0203716
$980$		0			0
$981$	225.511	−	26.6694i	0.229878	−	0.0271859i
$982$		0			0
$983$		−	686.673i		−	0.698549i	−0.937021	−	0.349274i	$-0.886428\pi$
$983$		−	686.673i		−	0.698549i	0.937021	−	0.349274i	$-0.113572\pi$
$984$		0			0
$985$	681.420			0.691797
$986$		0			0
$987$	−7.31973	−	124.220i	−0.00741614	−	0.125856i
$988$		0			0
$989$		−	1265.29i		−	1.27936i
$990$		0			0
$991$	−468.764			−0.473022			−0.236511	−	0.971629i	$-0.576004\pi$
$991$	−468.764			−0.473022			−0.236511	+	0.971629i	$0.576004\pi$
$992$		0			0
$993$	1353.47	−	79.7541i	1.36301	−	0.0803163i
$994$		0			0
$995$		−	220.474i		−	0.221582i
$996$		0			0
$997$	−1860.83			−1.86643			−0.933213	−	0.359323i	$-0.883008\pi$
$997$	−1860.83			−1.86643			−0.933213	+	0.359323i	$0.883008\pi$
$998$		0			0
$999$	−975.613	+	174.080i	−0.976589	+	0.174254i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.l.a.1121.15		16
3.2	odd	2	inner	1680.3.l.a.1121.16		16
4.3	odd	2		105.3.c.a.71.2	✓	16
12.11	even	2		105.3.c.a.71.15	yes	16
20.3	even	4		525.3.f.b.449.4		32
20.7	even	4		525.3.f.b.449.30		32
20.19	odd	2		525.3.c.b.176.15		16
60.23	odd	4		525.3.f.b.449.29		32
60.47	odd	4		525.3.f.b.449.3		32
60.59	even	2		525.3.c.b.176.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.c.a.71.2	✓	16	4.3	odd	2
105.3.c.a.71.15	yes	16	12.11	even	2
525.3.c.b.176.2		16	60.59	even	2
525.3.c.b.176.15		16	20.19	odd	2
525.3.f.b.449.3		32	60.47	odd	4
525.3.f.b.449.4		32	20.3	even	4
525.3.f.b.449.29		32	60.23	odd	4
525.3.f.b.449.30		32	20.7	even	4
1680.3.l.a.1121.15		16	1.1	even	1	trivial
1680.3.l.a.1121.16		16	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.99481 - 0.176471i) q^{3} -2.23607i q^{5} +2.64575 q^{7} +(8.93772 - 1.05699i) q^{9} -12.0685i q^{11} -12.7142 q^{13} +(-0.394601 - 6.69659i) q^{15} -22.2556i q^{17} +28.9289 q^{19} +(7.92351 - 0.466899i) q^{21} +21.9101i q^{23} -5.00000 q^{25} +(26.5802 - 4.74274i) q^{27} +11.4122i q^{29} +4.93977 q^{31} +(-2.12974 - 36.1428i) q^{33} -5.91608i q^{35} -36.7045 q^{37} +(-38.0765 + 2.24369i) q^{39} -57.6874i q^{41} -57.7493 q^{43} +(-2.36351 - 19.9853i) q^{45} -15.6773i q^{47} +7.00000 q^{49} +(-3.92747 - 66.6513i) q^{51} -91.2304i q^{53} -26.9860 q^{55} +(86.6363 - 5.10511i) q^{57} -34.3101i q^{59} +71.1225 q^{61} +(23.6470 - 2.79654i) q^{63} +28.4298i q^{65} -20.1716 q^{67} +(3.86649 + 65.6164i) q^{69} +69.7718i q^{71} -96.8693 q^{73} +(-14.9740 + 0.882355i) q^{75} -31.9302i q^{77} +84.2348 q^{79} +(78.7655 - 18.8942i) q^{81} +20.4474i q^{83} -49.7651 q^{85} +(2.01393 + 34.1774i) q^{87} -1.65255i q^{89} -33.6386 q^{91} +(14.7936 - 0.871726i) q^{93} -64.6869i q^{95} +29.7125 q^{97} +(-12.7563 - 107.865i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{3} + 22 q^{9} - 10 q^{15} + 16 q^{19} - 14 q^{21} - 80 q^{25} + 148 q^{27} + 72 q^{31} - 4 q^{33} - 40 q^{37} - 90 q^{39} - 280 q^{43} + 40 q^{45} + 112 q^{49} - 38 q^{51} - 80 q^{55} - 36 q^{57}+ \cdots + 166 q^{99}+O(q^{100}) \) 16 * q - 8 * q^3 + 22 * q^9 - 10 * q^15 + 16 * q^19 - 14 * q^21 - 80 * q^25 + 148 * q^27 + 72 * q^31 - 4 * q^33 - 40 * q^37 - 90 * q^39 - 280 * q^43 + 40 * q^45 + 112 * q^49 - 38 * q^51 - 80 * q^55 - 36 * q^57 - 56 * q^61 + 56 * q^63 + 120 * q^67 + 60 * q^69 - 208 * q^73 + 40 * q^75 + 204 * q^79 + 458 * q^81 + 100 * q^85 + 324 * q^87 + 28 * q^91 + 108 * q^93 + 728 * q^97 + 166 * q^99

Introduction

L-functions

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