LMFDB - Embedded newform 1176.3.d.f.785.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1176,3,Mod(785,1176)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1176.785"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1176.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,8,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$32.0436790888$
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} - 4 x^{14} - 4 x^{13} - 33 x^{12} + 220 x^{11} + 516 x^{10} - 2136 x^{9} - 3744 x^{8} + \cdots + 43046721 $ x^16 - 4x^14 - 4x^13 - 33x^12 + 220x^11 + 516x^10 - 2136x^9 - 3744x^8 - 19224x^7 + 41796x^6 + 160380x^5 - 216513x^4 - 236196x^3 - 2125764*x^2 + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			785.7
Root			$0.0276734 + 2.99987i$ of defining polynomial
Character	$\chi$	$=$	1176.785
Dual form			1176.3.d.f.785.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0276734 - 2.99987i) q^{3} +0.703530i q^{5} +(-8.99847 + 0.166033i) q^{9} -2.23488i q^{11} +15.6713 q^{13} +(2.11050 - 0.0194691i) q^{15} +28.4346i q^{17} -16.9061 q^{19} +28.2927i q^{23} +24.5050 q^{25} +(0.747096 + 26.9897i) q^{27} -24.5065i q^{29} -16.5096 q^{31} +(-6.70435 + 0.0618467i) q^{33} +52.2959 q^{37} +(-0.433678 - 47.0120i) q^{39} -5.66937i q^{41} +15.5745 q^{43} +(-0.116809 - 6.33070i) q^{45} +19.1839i q^{47} +(85.3001 - 0.786881i) q^{51} +53.5467i q^{53} +1.57231 q^{55} +(0.467848 + 50.7161i) q^{57} +51.4628i q^{59} +48.2547 q^{61} +11.0252i q^{65} +18.3292 q^{67} +(84.8744 - 0.782954i) q^{69} -76.4062i q^{71} -50.2757 q^{73} +(-0.678137 - 73.5120i) q^{75} +67.2013 q^{79} +(80.9449 - 2.98809i) q^{81} +43.2847i q^{83} -20.0046 q^{85} +(-73.5163 + 0.678177i) q^{87} +105.873i q^{89} +(0.456876 + 49.5266i) q^{93} -11.8939i q^{95} +153.611 q^{97} +(0.371064 + 20.1105i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{9} - 4 q^{13} + 8 q^{15} - 16 q^{19} - 76 q^{25} - 12 q^{27} + 36 q^{31} - 32 q^{33} - 28 q^{37} - 4 q^{39} - 120 q^{43} - 28 q^{45} - 36 q^{51} - 28 q^{55} - 88 q^{57} + 152 q^{61} + 88 q^{67}+ \cdots + 368 q^{99}+O(q^{100}) $ 16 * q + 8 * q^9 - 4 * q^13 + 8 * q^15 - 16 * q^19 - 76 * q^25 - 12 * q^27 + 36 * q^31 - 32 * q^33 - 28 * q^37 - 4 * q^39 - 120 * q^43 - 28 * q^45 - 36 * q^51 - 28 * q^55 - 88 * q^57 + 152 * q^61 + 88 * q^67 - 112 * q^69 + 156 * q^73 + 312 * q^75 - 84 * q^79 + 164 * q^81 + 152 * q^85 + 176 * q^87 - 8 * q^93 - 20 * q^97 + 368 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	−0.0276734	−	2.99987i	−0.00922446	−	0.999957i
$3$	−0.0276734	−	2.99987i	−0.00922446	−	0.999957i
$4$		0			0
$5$			0.703530i			0.140706i	0.997522	+	0.0703530i	$0.0224126\pi$
$5$			0.703530i			0.140706i	−0.997522	+	0.0703530i	$0.977587\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−8.99847	+	0.166033i	−0.999830	+	0.0184481i
$10$		0			0
$11$		−	2.23488i		−	0.203171i	−0.994827	−	0.101585i	$-0.967608\pi$
$11$		−	2.23488i		−	0.203171i	0.994827	−	0.101585i	$-0.0323915\pi$
$12$		0			0
$13$	15.6713			1.20549			0.602743	−	0.797935i	$-0.294074\pi$
$13$	15.6713			1.20549			0.602743	+	0.797935i	$0.294074\pi$
$14$		0			0
$15$	2.11050	−	0.0194691i	0.140700	−	0.00129794i
$16$		0			0
$17$			28.4346i			1.67262i	0.548255	+	0.836311i	$0.315292\pi$
$17$			28.4346i			1.67262i	−0.548255	+	0.836311i	$0.684708\pi$
$18$		0			0
$19$	−16.9061			−0.889793			−0.444897	−	0.895582i	$-0.646760\pi$
$19$	−16.9061			−0.889793			−0.444897	+	0.895582i	$0.646760\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			28.2927i			1.23012i	0.788482	+	0.615058i	$0.210867\pi$
$23$			28.2927i			1.23012i	−0.788482	+	0.615058i	$0.789133\pi$
$24$		0			0
$25$	24.5050			0.980202
$26$		0			0
$27$	0.747096	+	26.9897i	0.0276702	+	0.999617i
$28$		0			0
$29$		−	24.5065i		−	0.845051i	−0.906351	−	0.422526i	$-0.861144\pi$
$29$		−	24.5065i		−	0.845051i	0.906351	−	0.422526i	$-0.138856\pi$
$30$		0			0
$31$	−16.5096			−0.532567			−0.266283	−	0.963895i	$-0.585796\pi$
$31$	−16.5096			−0.532567			−0.266283	+	0.963895i	$0.585796\pi$
$32$		0			0
$33$	−6.70435	+	0.0618467i	−0.203162	+	0.00187414i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	52.2959			1.41340			0.706702	−	0.707512i	$-0.250182\pi$
$37$	52.2959			1.41340			0.706702	+	0.707512i	$0.250182\pi$
$38$		0			0
$39$	−0.433678	−	47.0120i	−0.0111200	−	1.20543i
$40$		0			0
$41$		−	5.66937i		−	0.138277i	−0.997607	−	0.0691386i	$-0.977975\pi$
$41$		−	5.66937i		−	0.138277i	0.997607	−	0.0691386i	$-0.0220251\pi$
$42$		0			0
$43$	15.5745			0.362198			0.181099	−	0.983465i	$-0.442035\pi$
$43$	15.5745			0.362198			0.181099	+	0.983465i	$0.442035\pi$
$44$		0			0
$45$	−0.116809	−	6.33070i	−0.00259576	−	0.140682i
$46$		0			0
$47$			19.1839i			0.408169i	0.978953	+	0.204084i	$0.0654217\pi$
$47$			19.1839i			0.408169i	−0.978953	+	0.204084i	$0.934578\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	85.3001	−	0.786881i	1.67255	−	0.0154290i
$52$		0			0
$53$			53.5467i			1.01031i	0.863027	+	0.505157i	$0.168566\pi$
$53$			53.5467i			1.01031i	−0.863027	+	0.505157i	$0.831434\pi$
$54$		0			0
$55$	1.57231			0.0285874
$56$		0			0
$57$	0.467848	+	50.7161i	0.00820786	+	0.889756i
$58$		0			0
$59$			51.4628i			0.872252i	0.899886	+	0.436126i	$0.143650\pi$
$59$			51.4628i			0.872252i	−0.899886	+	0.436126i	$0.856350\pi$
$60$		0			0
$61$	48.2547			0.791060			0.395530	−	0.918453i	$-0.370561\pi$
$61$	48.2547			0.791060			0.395530	+	0.918453i	$0.370561\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			11.0252i			0.169619i
$66$		0			0
$67$	18.3292			0.273570			0.136785	−	0.990601i	$-0.456323\pi$
$67$	18.3292			0.273570			0.136785	+	0.990601i	$0.456323\pi$
$68$		0			0
$69$	84.8744	−	0.782954i	1.23006	−	0.0113472i
$70$		0			0
$71$		−	76.4062i		−	1.07614i	−0.842899	−	0.538072i	$-0.819153\pi$
$71$		−	76.4062i		−	1.07614i	0.842899	−	0.538072i	$-0.180847\pi$
$72$		0			0
$73$	−50.2757			−0.688708			−0.344354	−	0.938840i	$-0.611902\pi$
$73$	−50.2757			−0.688708			−0.344354	+	0.938840i	$0.611902\pi$
$74$		0			0
$75$	−0.678137	−	73.5120i	−0.00904183	−	0.980160i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	67.2013			0.850649			0.425325	−	0.905041i	$-0.360160\pi$
$79$	67.2013			0.850649			0.425325	+	0.905041i	$0.360160\pi$
$80$		0			0
$81$	80.9449	−	2.98809i	0.999319	−	0.0368900i
$82$		0			0
$83$			43.2847i			0.521502i	0.965406	+	0.260751i	$0.0839702\pi$
$83$			43.2847i			0.521502i	−0.965406	+	0.260751i	$0.916030\pi$
$84$		0			0
$85$	−20.0046			−0.235348
$86$		0			0
$87$	−73.5163	+	0.678177i	−0.845015	+	0.00779514i
$88$		0			0
$89$			105.873i			1.18958i	0.803879	+	0.594792i	$0.202766\pi$
$89$			105.873i			1.18958i	−0.803879	+	0.594792i	$0.797234\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.456876	+	49.5266i	0.00491264	+	0.532544i
$94$		0			0
$95$		−	11.8939i		−	0.125199i
$96$		0			0
$97$	153.611			1.58362			0.791811	−	0.610766i	$-0.209138\pi$
$97$	153.611			1.58362			0.791811	+	0.610766i	$0.209138\pi$
$98$		0			0
$99$	0.371064	+	20.1105i	0.00374812	+	0.203136i
$100$		0			0
$101$		−	160.218i		−	1.58631i	−0.609017	−	0.793157i	$-0.708436\pi$
$101$		−	160.218i		−	1.58631i	0.609017	−	0.793157i	$-0.291564\pi$
$102$		0			0
$103$	−179.204			−1.73985			−0.869923	−	0.493188i	$-0.835831\pi$
$103$	−179.204			−1.73985			−0.869923	+	0.493188i	$0.835831\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	193.523i		−	1.80863i	−0.426869	−	0.904314i	$-0.640383\pi$
$107$		−	193.523i		−	1.80863i	0.426869	−	0.904314i	$-0.359617\pi$
$108$		0			0
$109$	117.022			1.07359			0.536797	−	0.843711i	$-0.319634\pi$
$109$	117.022			1.07359			0.536797	+	0.843711i	$0.319634\pi$
$110$		0			0
$111$	−1.44720	−	156.881i	−0.0130379	−	1.41334i
$112$		0			0
$113$			117.473i			1.03958i	0.854294	+	0.519790i	$0.173990\pi$
$113$			117.473i			1.03958i	−0.854294	+	0.519790i	$0.826010\pi$
$114$		0			0
$115$	−19.9048			−0.173085
$116$		0			0
$117$	−141.018	+	2.60196i	−1.20528	+	0.0222390i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	116.005			0.958722
$122$		0			0
$123$	−17.0074	+	0.156891i	−0.138271	+	0.00127553i
$124$		0			0
$125$			34.8283i			0.278626i
$126$		0			0
$127$	118.965			0.936732			0.468366	−	0.883534i	$-0.344843\pi$
$127$	118.965			0.936732			0.468366	+	0.883534i	$0.344843\pi$
$128$		0			0
$129$	−0.431000	−	46.7216i	−0.00334108	−	0.362183i
$130$		0			0
$131$		−	188.489i		−	1.43885i	−0.694573	−	0.719423i	$-0.744406\pi$
$131$		−	188.489i		−	1.43885i	0.694573	−	0.719423i	$-0.255594\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−18.9880	+	0.525605i	−0.140652	+	0.00389337i
$136$		0			0
$137$			147.116i			1.07384i	0.843634	+	0.536919i	$0.180412\pi$
$137$			147.116i			1.07384i	−0.843634	+	0.536919i	$0.819588\pi$
$138$		0			0
$139$	208.158			1.49754			0.748768	−	0.662832i	$-0.230646\pi$
$139$	208.158			1.49754			0.748768	+	0.662832i	$0.230646\pi$
$140$		0			0
$141$	57.5493	−	0.530884i	0.408151	−	0.00376514i
$142$		0			0
$143$		−	35.0235i		−	0.244920i
$144$		0			0
$145$	17.2411			0.118904
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	56.0952i		−	0.376478i	−0.982123	−	0.188239i	$-0.939722\pi$
$149$		−	56.0952i		−	0.376478i	0.982123	−	0.188239i	$-0.0602779\pi$
$150$		0			0
$151$	178.264			1.18056			0.590279	−	0.807199i	$-0.299018\pi$
$151$	178.264			1.18056			0.590279	+	0.807199i	$0.299018\pi$
$152$		0			0
$153$	−4.72108	−	255.868i	−0.0308568	−	1.67234i
$154$		0			0
$155$		−	11.6150i		−	0.0749354i
$156$		0			0
$157$	−2.41273			−0.0153677			−0.00768385	−	0.999970i	$-0.502446\pi$
$157$	−2.41273			−0.0153677			−0.00768385	+	0.999970i	$0.502446\pi$
$158$		0			0
$159$	160.633	−	1.48182i	1.01027	−	0.00931961i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	180.353			1.10646			0.553231	−	0.833028i	$-0.313395\pi$
$163$	180.353			1.10646			0.553231	+	0.833028i	$0.313395\pi$
$164$		0			0
$165$	−0.0435110	−	4.71672i	−0.000263703	−	0.0285862i
$166$		0			0
$167$			260.953i			1.56259i	0.624161	+	0.781296i	$0.285441\pi$
$167$			260.953i			1.56259i	−0.624161	+	0.781296i	$0.714559\pi$
$168$		0			0
$169$	76.5903			0.453197
$170$		0			0
$171$	152.129	−	2.80697i	0.889642	−	0.0164150i
$172$		0			0
$173$			201.213i			1.16308i	0.813517	+	0.581540i	$0.197550\pi$
$173$			201.213i			1.16308i	−0.813517	+	0.581540i	$0.802450\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	154.382	−	1.42415i	0.872215	−	0.00804605i
$178$		0			0
$179$		−	165.241i		−	0.923133i	−0.887106	−	0.461566i	$-0.847288\pi$
$179$		−	165.241i		−	0.923133i	0.887106	−	0.461566i	$-0.152712\pi$
$180$		0			0
$181$	−173.350			−0.957734			−0.478867	−	0.877888i	$-0.658952\pi$
$181$	−173.350			−0.957734			−0.478867	+	0.877888i	$0.658952\pi$
$182$		0			0
$183$	−1.33537	−	144.758i	−0.00729710	−	0.791027i
$184$		0			0
$185$			36.7918i			0.198874i
$186$		0			0
$187$	63.5479			0.339828
$188$		0			0
$189$		0			0
$190$		0			0
$191$			133.531i			0.699116i	0.936915	+	0.349558i	$0.113668\pi$
$191$			133.531i			0.699116i	−0.936915	+	0.349558i	$0.886332\pi$
$192$		0			0
$193$	−233.409			−1.20937			−0.604686	−	0.796464i	$-0.706702\pi$
$193$	−233.409			−1.20937			−0.604686	+	0.796464i	$0.706702\pi$
$194$		0			0
$195$	33.0743	−	0.305106i	0.169612	−	0.00156465i
$196$		0			0
$197$			183.668i			0.932324i	0.884699	+	0.466162i	$0.154364\pi$
$197$			183.668i			0.932324i	−0.884699	+	0.466162i	$0.845636\pi$
$198$		0			0
$199$	−210.198			−1.05627			−0.528135	−	0.849160i	$-0.677109\pi$
$199$	−210.198			−1.05627			−0.528135	+	0.849160i	$0.677109\pi$
$200$		0			0
$201$	−0.507231	−	54.9853i	−0.00252354	−	0.273559i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	3.98857			0.0194564
$206$		0			0
$207$	−4.69752	−	254.591i	−0.0226933	−	1.22991i
$208$		0			0
$209$			37.7830i			0.180780i
$210$		0			0
$211$	−345.747			−1.63861			−0.819306	−	0.573356i	$-0.805641\pi$
$211$	−345.747			−1.63861			−0.819306	+	0.573356i	$0.805641\pi$
$212$		0			0
$213$	−229.209	+	2.11442i	−1.07610	+	0.00992684i
$214$		0			0
$215$			10.9572i			0.0509635i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	1.39130	+	150.821i	0.00635296	+	0.688679i
$220$		0			0
$221$			445.607i			2.01632i
$222$		0			0
$223$	−142.876			−0.640699			−0.320350	−	0.947299i	$-0.603800\pi$
$223$	−142.876			−0.640699			−0.320350	+	0.947299i	$0.603800\pi$
$224$		0			0
$225$	−220.508	+	4.06865i	−0.980035	+	0.0180829i
$226$		0			0
$227$			115.091i			0.507010i	0.967334	+	0.253505i	$0.0815835\pi$
$227$			115.091i			0.507010i	−0.967334	+	0.253505i	$0.918417\pi$
$228$		0			0
$229$	17.3186			0.0756269			0.0378135	−	0.999285i	$-0.487961\pi$
$229$	17.3186			0.0756269			0.0378135	+	0.999285i	$0.487961\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	192.079i		−	0.824374i	−0.911099	−	0.412187i	$-0.864765\pi$
$233$		−	192.079i		−	0.824374i	0.911099	−	0.412187i	$-0.135235\pi$
$234$		0			0
$235$	−13.4965			−0.0574318
$236$		0			0
$237$	−1.85969	−	201.595i	−0.00784678	−	0.850613i
$238$		0			0
$239$		−	115.397i		−	0.482835i	−0.970421	−	0.241417i	$-0.922388\pi$
$239$		−	115.397i		−	0.482835i	0.970421	−	0.241417i	$-0.0776123\pi$
$240$		0			0
$241$	139.920			0.580579			0.290290	−	0.956939i	$-0.406248\pi$
$241$	139.920			0.580579			0.290290	+	0.956939i	$0.406248\pi$
$242$		0			0
$243$	−11.2039	−	242.742i	−0.0461066	−	0.998937i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−264.941			−1.07263
$248$		0			0
$249$	129.849	−	1.19783i	0.521480	−	0.00481058i
$250$		0			0
$251$			271.600i			1.08207i	0.841000	+	0.541036i	$0.181968\pi$
$251$			271.600i			1.08207i	−0.841000	+	0.541036i	$0.818032\pi$
$252$		0			0
$253$	63.2307			0.249924
$254$		0			0
$255$	0.553594	+	60.0112i	0.00217096	+	0.235338i
$256$		0			0
$257$			182.055i			0.708387i	0.935172	+	0.354193i	$0.115245\pi$
$257$			182.055i			0.708387i	−0.935172	+	0.354193i	$0.884755\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	4.06889	+	220.521i	0.0155896	+	0.844907i
$262$		0			0
$263$		−	97.2622i		−	0.369818i	−0.982756	−	0.184909i	$-0.940801\pi$
$263$		−	97.2622i		−	0.369818i	0.982756	−	0.184909i	$-0.0591991\pi$
$264$		0			0
$265$	−37.6717			−0.142157
$266$		0			0
$267$	317.606	−	2.92987i	1.18953	−	0.0109733i
$268$		0			0
$269$			288.053i			1.07083i	0.844589	+	0.535415i	$0.179845\pi$
$269$			288.053i			1.07083i	−0.844589	+	0.535415i	$0.820155\pi$
$270$		0			0
$271$	79.0737			0.291785			0.145892	−	0.989300i	$-0.453395\pi$
$271$	79.0737			0.291785			0.145892	+	0.989300i	$0.453395\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	54.7658i		−	0.199148i
$276$		0			0
$277$	−173.329			−0.625737			−0.312868	−	0.949796i	$-0.601290\pi$
$277$	−173.329			−0.625737			−0.312868	+	0.949796i	$0.601290\pi$
$278$		0			0
$279$	148.561	−	2.74114i	0.532476	−	0.00982486i
$280$		0			0
$281$		−	440.721i		−	1.56840i	−0.620506	−	0.784201i	$-0.713073\pi$
$281$		−	440.721i		−	1.56840i	0.620506	−	0.784201i	$-0.286927\pi$
$282$		0			0
$283$	278.458			0.983952			0.491976	−	0.870609i	$-0.336275\pi$
$283$	278.458			0.983952			0.491976	+	0.870609i	$0.336275\pi$
$284$		0			0
$285$	−35.6803	+	0.329145i	−0.125194	+	0.00115490i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−519.525			−1.79766
$290$		0			0
$291$	−4.25095	−	460.814i	−0.0146081	−	1.58355i
$292$		0			0
$293$			433.492i			1.47949i	0.672885	+	0.739747i	$0.265055\pi$
$293$			433.492i			1.47949i	−0.672885	+	0.739747i	$0.734945\pi$
$294$		0			0
$295$	−36.2057			−0.122731
$296$		0			0
$297$	60.3187	−	1.66967i	0.203093	−	0.00562179i
$298$		0			0
$299$			443.384i			1.48289i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−480.633	+	4.43377i	−1.58625	+	0.0146329i
$304$		0			0
$305$			33.9486i			0.111307i
$306$		0			0
$307$	66.8130			0.217632			0.108816	−	0.994062i	$-0.465294\pi$
$307$	66.8130			0.217632			0.108816	+	0.994062i	$0.465294\pi$
$308$		0			0
$309$	4.95918	+	537.589i	0.0160491	+	1.73977i
$310$		0			0
$311$			242.194i			0.778760i	0.921077	+	0.389380i	$0.127311\pi$
$311$			242.194i			0.778760i	−0.921077	+	0.389380i	$0.872689\pi$
$312$		0			0
$313$	−159.099			−0.508303			−0.254151	−	0.967164i	$-0.581796\pi$
$313$	−159.099			−0.508303			−0.254151	+	0.967164i	$0.581796\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			598.180i			1.88700i	0.331369	+	0.943501i	$0.392490\pi$
$317$			598.180i			1.88700i	−0.331369	+	0.943501i	$0.607510\pi$
$318$		0			0
$319$	−54.7690			−0.171690
$320$		0			0
$321$	−580.545	+	5.35544i	−1.80855	+	0.0166836i
$322$		0			0
$323$		−	480.717i		−	1.48829i
$324$		0			0
$325$	384.026			1.18162
$326$		0			0
$327$	−3.23839	−	351.050i	−0.00990333	−	1.07355i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−21.6968			−0.0655491			−0.0327746	−	0.999463i	$-0.510434\pi$
$331$	−21.6968			−0.0655491			−0.0327746	+	0.999463i	$0.510434\pi$
$332$		0			0
$333$	−470.583	+	8.68286i	−1.41316	+	0.0260746i
$334$		0			0
$335$			12.8952i			0.0384930i
$336$		0			0
$337$	574.634			1.70514			0.852572	−	0.522610i	$-0.175042\pi$
$337$	574.634			1.70514			0.852572	+	0.522610i	$0.175042\pi$
$338$		0			0
$339$	352.403	−	3.25086i	1.03954	−	0.00958957i
$340$		0			0
$341$			36.8969i			0.108202i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	0.550832	+	59.7117i	0.00159661	+	0.173077i
$346$		0			0
$347$			419.932i			1.21018i	0.796157	+	0.605090i	$0.206863\pi$
$347$			419.932i			1.21018i	−0.796157	+	0.605090i	$0.793137\pi$
$348$		0			0
$349$	−187.113			−0.536141			−0.268070	−	0.963399i	$-0.586386\pi$
$349$	−187.113			−0.536141			−0.268070	+	0.963399i	$0.586386\pi$
$350$		0			0
$351$	11.7080	+	422.964i	0.0333561	+	1.20502i
$352$		0			0
$353$		−	157.632i		−	0.446550i	−0.974755	−	0.223275i	$-0.928325\pi$
$353$		−	157.632i		−	0.446550i	0.974755	−	0.223275i	$-0.0716749\pi$
$354$		0			0
$355$	53.7541			0.151420
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	567.641i		−	1.58117i	−0.612350	−	0.790587i	$-0.709776\pi$
$359$		−	567.641i		−	1.58117i	0.612350	−	0.790587i	$-0.290224\pi$
$360$		0			0
$361$	−75.1846			−0.208268
$362$		0			0
$363$	−3.21026	−	348.001i	−0.00884369	−	0.958681i
$364$		0			0
$365$		−	35.3705i		−	0.0969054i
$366$		0			0
$367$	−254.477			−0.693397			−0.346698	−	0.937977i	$-0.612697\pi$
$367$	−254.477			−0.693397			−0.346698	+	0.937977i	$0.612697\pi$
$368$		0			0
$369$	0.941303	+	51.0156i	0.00255096	+	0.138254i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−142.323			−0.381564			−0.190782	−	0.981632i	$-0.561102\pi$
$373$	−142.323			−0.381564			−0.190782	+	0.981632i	$0.561102\pi$
$374$		0			0
$375$	104.480	−	0.963817i	0.278615	−	0.00257018i
$376$		0			0
$377$		−	384.049i		−	1.01870i
$378$		0			0
$379$	−493.492			−1.30209			−0.651045	−	0.759039i	$-0.725669\pi$
$379$	−493.492			−1.30209			−0.651045	+	0.759039i	$0.725669\pi$
$380$		0			0
$381$	−3.29216	−	356.880i	−0.00864085	−	0.936693i
$382$		0			0
$383$		−	427.921i		−	1.11729i	−0.829408	−	0.558643i	$-0.811322\pi$
$383$		−	427.921i		−	1.11729i	0.829408	−	0.558643i	$-0.188678\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−140.147	+	2.58589i	−0.362137	+	0.00668188i
$388$		0			0
$389$			211.271i			0.543112i	0.962423	+	0.271556i	$0.0875383\pi$
$389$			211.271i			0.543112i	−0.962423	+	0.271556i	$0.912462\pi$
$390$		0			0
$391$	−804.490			−2.05752
$392$		0			0
$393$	−565.442	+	5.21612i	−1.43878	+	0.0132726i
$394$		0			0
$395$			47.2781i			0.119691i
$396$		0			0
$397$	451.543			1.13739			0.568694	−	0.822549i	$-0.307449\pi$
$397$	451.543			1.13739			0.568694	+	0.822549i	$0.307449\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	735.067i		−	1.83308i	−0.399938	−	0.916542i	$-0.630969\pi$
$401$		−	735.067i		−	1.83308i	0.399938	−	0.916542i	$-0.369031\pi$
$402$		0			0
$403$	−258.727			−0.642002
$404$		0			0
$405$	2.10221	+	56.9472i	0.00519064	+	0.140610i
$406$		0			0
$407$		−	116.875i		−	0.287162i
$408$		0			0
$409$	−258.371			−0.631713			−0.315856	−	0.948807i	$-0.602292\pi$
$409$	−258.371			−0.631713			−0.315856	+	0.948807i	$0.602292\pi$
$410$		0			0
$411$	441.329	−	4.07119i	1.07379	−	0.00990558i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−30.4521			−0.0733786
$416$		0			0
$417$	−5.76042	−	624.446i	−0.0138140	−	1.49747i
$418$		0			0
$419$			20.6559i			0.0492982i	0.999696	+	0.0246491i	$0.00784685\pi$
$419$			20.6559i			0.0492982i	−0.999696	+	0.0246491i	$0.992153\pi$
$420$		0			0
$421$	−416.377			−0.989019			−0.494509	−	0.869172i	$-0.664652\pi$
$421$	−416.377			−0.989019			−0.494509	+	0.869172i	$0.664652\pi$
$422$		0			0
$423$	−3.18517	−	172.626i	−0.00752995	−	0.408099i
$424$		0			0
$425$			696.791i			1.63951i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−105.066	+	0.969219i	−0.244909	+	0.00225925i
$430$		0			0
$431$			366.099i			0.849418i	0.905330	+	0.424709i	$0.139624\pi$
$431$			366.099i			0.849418i	−0.905330	+	0.424709i	$0.860376\pi$
$432$		0			0
$433$	196.017			0.452694			0.226347	−	0.974047i	$-0.427322\pi$
$433$	196.017			0.452694			0.226347	+	0.974047i	$0.427322\pi$
$434$		0			0
$435$	−0.477118	−	51.7210i	−0.00109682	−	0.118899i
$436$		0			0
$437$		−	478.318i		−	1.09455i
$438$		0			0
$439$	−50.9385			−0.116033			−0.0580165	−	0.998316i	$-0.518478\pi$
$439$	−50.9385			−0.116033			−0.0580165	+	0.998316i	$0.518478\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			509.210i			1.14946i	0.818344	+	0.574729i	$0.194892\pi$
$443$			509.210i			1.14946i	−0.818344	+	0.574729i	$0.805108\pi$
$444$		0			0
$445$	−74.4849			−0.167382
$446$		0			0
$447$	−168.278	+	1.55234i	−0.376462	+	0.00347281i
$448$		0			0
$449$			614.241i			1.36802i	0.729473	+	0.684010i	$0.239765\pi$
$449$			614.241i			1.36802i	−0.729473	+	0.684010i	$0.760235\pi$
$450$		0			0
$451$	−12.6704			−0.0280939
$452$		0			0
$453$	−4.93318	−	534.770i	−0.0108900	−	1.18051i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−314.514			−0.688215			−0.344107	−	0.938930i	$-0.611818\pi$
$457$	−314.514			−0.688215			−0.344107	+	0.938930i	$0.611818\pi$
$458$		0			0
$459$	−767.440	+	21.2434i	−1.67198	+	0.0462819i
$460$		0			0
$461$			92.8694i			0.201452i	0.994914	+	0.100726i	$0.0321166\pi$
$461$			92.8694i			0.201452i	−0.994914	+	0.100726i	$0.967883\pi$
$462$		0			0
$463$	−66.6851			−0.144028			−0.0720141	−	0.997404i	$-0.522943\pi$
$463$	−66.6851			−0.144028			−0.0720141	+	0.997404i	$0.522943\pi$
$464$		0			0
$465$	−34.8435	+	0.321426i	−0.0749322	+	0.000691238i
$466$		0			0
$467$			120.568i			0.258175i	0.991633	+	0.129088i	$0.0412048\pi$
$467$			120.568i			0.258175i	−0.991633	+	0.129088i	$0.958795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.0667683	+	7.23787i	0.000141759	+	0.0153670i
$472$		0			0
$473$		−	34.8072i		−	0.0735882i
$474$		0			0
$475$	−414.284			−0.872177
$476$		0			0
$477$	−8.89053	−	481.838i	−0.0186384	−	1.01014i
$478$		0			0
$479$			44.0038i			0.0918659i	0.998945	+	0.0459330i	$0.0146261\pi$
$479$			44.0038i			0.0918659i	−0.998945	+	0.0459330i	$0.985374\pi$
$480$		0			0
$481$	819.546			1.70384
$482$		0			0
$483$		0			0
$484$		0			0
$485$			108.070i			0.222825i
$486$		0			0
$487$	−427.642			−0.878114			−0.439057	−	0.898459i	$-0.644687\pi$
$487$	−427.642			−0.878114			−0.439057	+	0.898459i	$0.644687\pi$
$488$		0			0
$489$	−4.99099	−	541.037i	−0.0102065	−	1.10642i
$490$		0			0
$491$			339.071i			0.690572i	0.938497	+	0.345286i	$0.112218\pi$
$491$			339.071i			0.690572i	−0.938497	+	0.345286i	$0.887782\pi$
$492$		0			0
$493$	696.831			1.41345
$494$		0			0
$495$	−14.1483	+	0.261055i	−0.0285825	+	0.000527384i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−180.103			−0.360928			−0.180464	−	0.983582i	$-0.557760\pi$
$499$	−180.103			−0.360928			−0.180464	+	0.983582i	$0.557760\pi$
$500$		0			0
$501$	782.825	−	7.22144i	1.56252	−	0.0144141i
$502$		0			0
$503$			310.294i			0.616886i	0.951243	+	0.308443i	$0.0998079\pi$
$503$			310.294i			0.616886i	−0.951243	+	0.308443i	$0.900192\pi$
$504$		0			0
$505$	112.718			0.223204
$506$		0			0
$507$	−2.11951	−	229.761i	−0.00418050	−	0.453178i
$508$		0			0
$509$		−	590.114i		−	1.15936i	−0.814844	−	0.579680i	$-0.803178\pi$
$509$		−	590.114i		−	1.15936i	0.814844	−	0.579680i	$-0.196822\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−12.6305	−	456.289i	−0.0246208	−	0.889453i
$514$		0			0
$515$		−	126.075i		−	0.244807i
$516$		0			0
$517$	42.8738			0.0829280
$518$		0			0
$519$	603.613	−	5.56824i	1.16303	−	0.0107288i
$520$		0			0
$521$			485.766i			0.932372i	0.884687	+	0.466186i	$0.154372\pi$
$521$			485.766i			0.932372i	−0.884687	+	0.466186i	$0.845628\pi$
$522$		0			0
$523$	−532.200			−1.01759			−0.508795	−	0.860888i	$-0.669909\pi$
$523$	−532.200			−1.01759			−0.508795	+	0.860888i	$0.669909\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	469.443i		−	0.890783i
$528$		0			0
$529$	−271.475			−0.513186
$530$		0			0
$531$	−8.54454	−	463.087i	−0.0160914	−	0.872103i
$532$		0			0
$533$		−	88.8465i		−	0.166691i
$534$		0			0
$535$	136.149			0.254485
$536$		0			0
$537$	−495.701	+	4.57277i	−0.923093	+	0.00851540i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−555.002			−1.02588			−0.512940	−	0.858424i	$-0.671444\pi$
$541$	−555.002			−1.02588			−0.512940	+	0.858424i	$0.671444\pi$
$542$		0			0
$543$	4.79718	+	520.027i	0.00883458	+	0.957693i
$544$		0			0
$545$			82.3284i			0.151061i
$546$		0			0
$547$	469.415			0.858163			0.429082	−	0.903266i	$-0.358837\pi$
$547$	469.415			0.858163			0.429082	+	0.903266i	$0.358837\pi$
$548$		0			0
$549$	−434.218	+	8.01188i	−0.790926	+	0.0145936i
$550$		0			0
$551$			414.308i			0.751921i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	110.371	−	1.01815i	0.198866	−	0.00183451i
$556$		0			0
$557$			434.470i			0.780018i	0.920811	+	0.390009i	$0.127528\pi$
$557$			434.470i			0.780018i	−0.920811	+	0.390009i	$0.872472\pi$
$558$		0			0
$559$	244.073			0.436625
$560$		0			0
$561$	−1.75858	−	190.635i	−0.00313473	−	0.339814i
$562$		0			0
$563$		−	869.419i		−	1.54426i	−0.635464	−	0.772131i	$-0.719191\pi$
$563$		−	869.419i		−	1.54426i	0.635464	−	0.772131i	$-0.280809\pi$
$564$		0			0
$565$	−82.6455			−0.146275
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	563.142i		−	0.989705i	−0.868977	−	0.494853i	$-0.835222\pi$
$569$		−	563.142i		−	0.989705i	0.868977	−	0.494853i	$-0.164778\pi$
$570$		0			0
$571$	832.936			1.45873			0.729366	−	0.684123i	$-0.239815\pi$
$571$	832.936			1.45873			0.729366	+	0.684123i	$0.239815\pi$
$572$		0			0
$573$	400.576	−	3.69526i	0.699086	−	0.00644897i
$574$		0			0
$575$			693.313i			1.20576i
$576$		0			0
$577$	363.951			0.630763			0.315382	−	0.948965i	$-0.397867\pi$
$577$	363.951			0.630763			0.315382	+	0.948965i	$0.397867\pi$
$578$		0			0
$579$	6.45921	+	700.197i	0.0111558	+	1.20932i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	119.670			0.205267
$584$		0			0
$585$	−1.83056	−	99.2104i	−0.00312916	−	0.169590i
$586$		0			0
$587$		−	126.612i		−	0.215693i	−0.994168	−	0.107846i	$-0.965605\pi$
$587$		−	126.612i		−	0.215693i	0.994168	−	0.107846i	$-0.0343955\pi$
$588$		0			0
$589$	279.112			0.473874
$590$		0			0
$591$	550.980	−	5.08271i	0.932284	−	0.00860019i
$592$		0			0
$593$		−	503.668i		−	0.849356i	−0.905344	−	0.424678i	$-0.860387\pi$
$593$		−	503.668i		−	0.849356i	0.905344	−	0.424678i	$-0.139613\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	5.81688	+	630.567i	0.00974352	+	1.05623i
$598$		0			0
$599$		−	310.266i		−	0.517973i	−0.965881	−	0.258986i	$-0.916612\pi$
$599$		−	310.266i		−	0.517973i	0.965881	−	0.258986i	$-0.0833885\pi$
$600$		0			0
$601$	270.793			0.450571			0.225285	−	0.974293i	$-0.427669\pi$
$601$	270.793			0.450571			0.225285	+	0.974293i	$0.427669\pi$
$602$		0			0
$603$	−164.935	+	3.04326i	−0.273524	+	0.00504686i
$604$		0			0
$605$			81.6133i			0.134898i
$606$		0			0
$607$	−1103.04			−1.81719			−0.908596	−	0.417676i	$-0.862845\pi$
$607$	−1103.04			−1.81719			−0.908596	+	0.417676i	$0.862845\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			300.637i			0.492042i
$612$		0			0
$613$	315.840			0.515236			0.257618	−	0.966247i	$-0.417062\pi$
$613$	315.840			0.515236			0.257618	+	0.966247i	$0.417062\pi$
$614$		0			0
$615$	−0.110377	−	11.9652i	−0.000179475	−	0.0194556i
$616$		0			0
$617$		−	896.128i		−	1.45240i	−0.687485	−	0.726198i	$-0.741286\pi$
$617$		−	896.128i		−	1.45240i	0.687485	−	0.726198i	$-0.258714\pi$
$618$		0			0
$619$	−574.118			−0.927493			−0.463746	−	0.885968i	$-0.653495\pi$
$619$	−574.118			−0.927493			−0.463746	+	0.885968i	$0.653495\pi$
$620$		0			0
$621$	−763.610	+	21.1374i	−1.22965	+	0.0340376i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	588.123			0.940997
$626$		0			0
$627$	113.344	−	1.04558i	0.180772	−	0.00166760i
$628$		0			0
$629$			1487.01i			2.36409i
$630$		0			0
$631$	−463.162			−0.734013			−0.367007	−	0.930218i	$-0.619617\pi$
$631$	−463.162			−0.734013			−0.367007	+	0.930218i	$0.619617\pi$
$632$		0			0
$633$	9.56799	+	1037.20i	0.0151153	+	1.63854i
$634$		0			0
$635$			83.6955i			0.131804i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	12.6860	+	687.539i	0.0198528	+	1.07596i
$640$		0			0
$641$		−	1149.12i		−	1.79270i	−0.443348	−	0.896349i	$-0.646209\pi$
$641$		−	1149.12i		−	1.79270i	0.443348	−	0.896349i	$-0.353791\pi$
$642$		0			0
$643$	309.583			0.481466			0.240733	−	0.970591i	$-0.422612\pi$
$643$	309.583			0.481466			0.240733	+	0.970591i	$0.422612\pi$
$644$		0			0
$645$	32.8701	−	0.303221i	0.0509613	−	0.000470111i
$646$		0			0
$647$		−	108.760i		−	0.168099i	−0.996462	−	0.0840495i	$-0.973215\pi$
$647$		−	108.760i		−	0.168099i	0.996462	−	0.0840495i	$-0.0267854\pi$
$648$		0			0
$649$	115.013			0.177216
$650$		0			0
$651$		0			0
$652$		0			0
$653$			635.241i			0.972804i	0.873735	+	0.486402i	$0.161691\pi$
$653$			635.241i			0.972804i	−0.873735	+	0.486402i	$0.838309\pi$
$654$		0			0
$655$	132.608			0.202454
$656$		0			0
$657$	452.404	−	8.34743i	0.688591	−	0.0127054i
$658$		0			0
$659$			96.5679i			0.146537i	0.997312	+	0.0732685i	$0.0233430\pi$
$659$			96.5679i			0.146537i	−0.997312	+	0.0732685i	$0.976657\pi$
$660$		0			0
$661$	−700.902			−1.06037			−0.530183	−	0.847883i	$-0.677877\pi$
$661$	−700.902			−1.06037			−0.530183	+	0.847883i	$0.677877\pi$
$662$		0			0
$663$	1336.77	−	12.3315i	2.01624	−	0.0185995i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	693.354			1.03951
$668$		0			0
$669$	3.95386	+	428.609i	0.00591010	+	0.640672i
$670$		0			0
$671$		−	107.843i		−	0.160720i
$672$		0			0
$673$	154.319			0.229300			0.114650	−	0.993406i	$-0.463425\pi$
$673$	154.319			0.229300			0.114650	+	0.993406i	$0.463425\pi$
$674$		0			0
$675$	18.3076	+	661.383i	0.0271224	+	0.979826i
$676$		0			0
$677$		−	260.436i		−	0.384691i	−0.981327	−	0.192346i	$-0.938391\pi$
$677$		−	260.436i		−	0.384691i	0.981327	−	0.192346i	$-0.0616095\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	345.259	−	3.18497i	0.506989	−	0.00467689i
$682$		0			0
$683$			304.735i			0.446172i	0.974799	+	0.223086i	$0.0716130\pi$
$683$			304.735i			0.446172i	−0.974799	+	0.223086i	$0.928387\pi$
$684$		0			0
$685$	−103.500			−0.151096
$686$		0			0
$687$	−0.479263	−	51.9535i	−0.000697618	−	0.0756237i
$688$		0			0
$689$			839.147i			1.21792i
$690$		0			0
$691$	−80.0809			−0.115891			−0.0579457	−	0.998320i	$-0.518455\pi$
$691$	−80.0809			−0.115891			−0.0579457	+	0.998320i	$0.518455\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			146.445i			0.210712i
$696$		0			0
$697$	161.206			0.231286
$698$		0			0
$699$	−576.213	+	5.31548i	−0.824339	+	0.00760440i
$700$		0			0
$701$		−	259.739i		−	0.370526i	−0.982689	−	0.185263i	$-0.940686\pi$
$701$		−	259.739i		−	0.370526i	0.982689	−	0.185263i	$-0.0593138\pi$
$702$		0			0
$703$	−884.119			−1.25764
$704$		0			0
$705$	0.373493	+	40.4877i	0.000529777	+	0.0574294i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	687.849			0.970167			0.485084	−	0.874468i	$-0.338789\pi$
$709$	687.849			0.970167			0.485084	+	0.874468i	$0.338789\pi$
$710$		0			0
$711$	−604.709	+	11.1576i	−0.850504	+	0.0156929i
$712$		0			0
$713$		−	467.100i		−	0.655119i
$714$		0			0
$715$	24.6401			0.0344617
$716$		0			0
$717$	−346.178	+	3.19344i	−0.482814	+	0.00445389i
$718$		0			0
$719$			1027.59i			1.42920i	0.699535	+	0.714598i	$0.253391\pi$
$719$			1027.59i			1.42920i	−0.699535	+	0.714598i	$0.746609\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−3.87205	−	419.741i	−0.00535553	−	0.580555i
$724$		0			0
$725$		−	600.532i		−	0.828321i
$726$		0			0
$727$	672.734			0.925356			0.462678	−	0.886526i	$-0.346889\pi$
$727$	672.734			0.925356			0.462678	+	0.886526i	$0.346889\pi$
$728$		0			0
$729$	−727.884	+	40.3278i	−0.998469	+	0.0553193i
$730$		0			0
$731$			442.855i			0.605821i
$732$		0			0
$733$	1372.23			1.87207			0.936037	−	0.351903i	$-0.114465\pi$
$733$	1372.23			1.87207			0.936037	+	0.351903i	$0.114465\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	40.9636i		−	0.0555815i
$738$		0			0
$739$	162.565			0.219979			0.109990	−	0.993933i	$-0.464918\pi$
$739$	162.565			0.219979			0.109990	+	0.993933i	$0.464918\pi$
$740$		0			0
$741$	7.33180	+	794.788i	0.00989447	+	1.07259i
$742$		0			0
$743$		−	452.101i		−	0.608481i	−0.952595	−	0.304240i	$-0.901597\pi$
$743$		−	452.101i		−	0.608481i	0.952595	−	0.304240i	$-0.0984026\pi$
$744$		0			0
$745$	39.4647			0.0529727
$746$		0			0
$747$	−7.18670	−	389.496i	−0.00962075	−	0.521414i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	827.374			1.10170			0.550848	−	0.834605i	$-0.314304\pi$
$751$	827.374			1.10170			0.550848	+	0.834605i	$0.314304\pi$
$752$		0			0
$753$	814.765	−	7.51609i	1.08203	−	0.00998152i
$754$		0			0
$755$			125.414i			0.166112i
$756$		0			0
$757$	102.463			0.135354			0.0676768	−	0.997707i	$-0.478441\pi$
$757$	102.463			0.135354			0.0676768	+	0.997707i	$0.478441\pi$
$758$		0			0
$759$	−1.74981	−	189.684i	−0.00230541	−	0.249913i
$760$		0			0
$761$		−	796.043i		−	1.04605i	−0.852318	−	0.523024i	$-0.824804\pi$
$761$		−	796.043i		−	1.04605i	0.852318	−	0.523024i	$-0.175196\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	180.011	−	3.32143i	0.235308	−	0.00434173i
$766$		0			0
$767$			806.491i			1.05149i
$768$		0			0
$769$	322.892			0.419885			0.209943	−	0.977714i	$-0.432672\pi$
$769$	322.892			0.419885			0.209943	+	0.977714i	$0.432672\pi$
$770$		0			0
$771$	546.143	−	5.03809i	0.708357	−	0.00653448i
$772$		0			0
$773$		−	675.102i		−	0.873353i	−0.899619	−	0.436677i	$-0.856155\pi$
$773$		−	675.102i		−	0.873353i	0.899619	−	0.436677i	$-0.143845\pi$
$774$		0			0
$775$	−404.568			−0.522023
$776$		0			0
$777$		0			0
$778$		0			0
$779$			95.8467i			0.123038i
$780$		0			0
$781$	−170.759			−0.218641
$782$		0			0
$783$	661.422	−	18.3087i	0.844728	−	0.0233828i
$784$		0			0
$785$		−	1.69743i		−	0.00216233i
$786$		0			0
$787$	237.395			0.301645			0.150823	−	0.988561i	$-0.451808\pi$
$787$	237.395			0.301645			0.150823	+	0.988561i	$0.451808\pi$
$788$		0			0
$789$	−291.774	+	2.69157i	−0.369802	+	0.00341137i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	756.214			0.953612
$794$		0			0
$795$	1.04250	+	113.010i	0.00131133	+	0.142151i
$796$		0			0
$797$		−	129.415i		−	0.162377i	−0.996699	−	0.0811886i	$-0.974128\pi$
$797$		−	129.415i		−	0.162377i	0.996699	−	0.0811886i	$-0.0258716\pi$
$798$		0			0
$799$	−545.487			−0.682712
$800$		0			0
$801$	−17.5784	−	952.695i	−0.0219456	−	1.18938i
$802$		0			0
$803$			112.360i			0.139925i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	864.123	−	7.97141i	1.07078	−	0.00987783i
$808$		0			0
$809$		−	1421.49i		−	1.75709i	−0.477658	−	0.878546i	$-0.658514\pi$
$809$		−	1421.49i		−	1.75709i	0.477658	−	0.878546i	$-0.341486\pi$
$810$		0			0
$811$	945.453			1.16579			0.582894	−	0.812548i	$-0.301920\pi$
$811$	945.453			1.16579			0.582894	+	0.812548i	$0.301920\pi$
$812$		0			0
$813$	−2.18824	−	237.211i	−0.00269156	−	0.291773i
$814$		0			0
$815$			126.884i			0.155686i
$816$		0			0
$817$	−263.304			−0.322282
$818$		0			0
$819$		0			0
$820$		0			0
$821$			271.908i			0.331192i	0.986194	+	0.165596i	$0.0529547\pi$
$821$			271.908i			0.331192i	−0.986194	+	0.165596i	$0.947045\pi$
$822$		0			0
$823$	1018.68			1.23777			0.618883	−	0.785483i	$-0.287585\pi$
$823$	1018.68			1.23777			0.618883	+	0.785483i	$0.287585\pi$
$824$		0			0
$825$	−164.291	+	1.51556i	−0.199140	+	0.00183704i
$826$		0			0
$827$			1079.81i			1.30569i	0.757489	+	0.652847i	$0.226426\pi$
$827$			1079.81i			1.30569i	−0.757489	+	0.652847i	$0.773574\pi$
$828$		0			0
$829$	−71.2718			−0.0859732			−0.0429866	−	0.999076i	$-0.513687\pi$
$829$	−71.2718			−0.0859732			−0.0429866	+	0.999076i	$0.513687\pi$
$830$		0			0
$831$	4.79660	+	519.965i	0.00577208	+	0.625710i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−183.588			−0.219866
$836$		0			0
$837$	−12.3342	−	445.588i	−0.0147362	−	0.532363i
$838$		0			0
$839$		−	1358.71i		−	1.61944i	−0.586818	−	0.809719i	$-0.699619\pi$
$839$		−	1358.71i		−	1.61944i	0.586818	−	0.809719i	$-0.300381\pi$
$840$		0			0
$841$	240.432			0.285889
$842$		0			0
$843$	−1322.11	+	12.1962i	−1.56834	+	0.0144677i
$844$		0			0
$845$			53.8836i			0.0637676i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−7.70588	−	835.339i	−0.00907642	−	0.983910i
$850$		0			0
$851$			1479.59i			1.73865i
$852$		0			0
$853$	447.661			0.524808			0.262404	−	0.964958i	$-0.415485\pi$
$853$	447.661			0.524808			0.262404	+	0.964958i	$0.415485\pi$
$854$		0			0
$855$	1.97479	+	107.027i	0.00230969	+	0.125178i
$856$		0			0
$857$			500.641i			0.584178i	0.956391	+	0.292089i	$0.0943504\pi$
$857$			500.641i			0.584178i	−0.956391	+	0.292089i	$0.905650\pi$
$858$		0			0
$859$	−664.001			−0.772993			−0.386497	−	0.922291i	$-0.626315\pi$
$859$	−664.001			−0.772993			−0.386497	+	0.922291i	$0.626315\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1448.53i			1.67848i	0.543763	+	0.839239i	$0.316999\pi$
$863$			1448.53i			1.67848i	−0.543763	+	0.839239i	$0.683001\pi$
$864$		0			0
$865$	−141.559			−0.163653
$866$		0			0
$867$	14.3770	+	1558.51i	0.0165825	+	1.79759i
$868$		0			0
$869$		−	150.187i		−	0.172827i
$870$		0			0
$871$	287.243			0.329785
$872$		0			0
$873$	−1382.27	+	25.5046i	−1.58335	+	0.0292149i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	673.203			0.767620			0.383810	−	0.923412i	$-0.374612\pi$
$877$	673.203			0.767620			0.383810	+	0.923412i	$0.374612\pi$
$878$		0			0
$879$	1300.42	−	11.9962i	1.47943	−	0.0136475i
$880$		0			0
$881$		−	1215.64i		−	1.37984i	−0.723886	−	0.689920i	$-0.757646\pi$
$881$		−	1215.64i		−	1.37984i	0.723886	−	0.689920i	$-0.242354\pi$
$882$		0			0
$883$	325.546			0.368682			0.184341	−	0.982862i	$-0.440985\pi$
$883$	325.546			0.368682			0.184341	+	0.982862i	$0.440985\pi$
$884$		0			0
$885$	1.00193	+	108.612i	0.00113213	+	0.122726i
$886$		0			0
$887$		−	429.209i		−	0.483889i	−0.970290	−	0.241944i	$-0.922215\pi$
$887$		−	429.209i		−	0.483889i	0.970290	−	0.241944i	$-0.0777852\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−6.67802	−	180.902i	−0.00749497	−	0.203033i
$892$		0			0
$893$		−	324.325i		−	0.363186i
$894$		0			0
$895$	116.252			0.129890
$896$		0			0
$897$	1330.09	−	12.2699i	1.48283	−	0.0136788i
$898$		0			0
$899$			404.591i			0.450046i
$900$		0			0
$901$	−1522.58			−1.68987
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	121.957i		−	0.134759i
$906$		0			0
$907$	1306.12			1.44004			0.720020	−	0.693954i	$-0.244133\pi$
$907$	1306.12			1.44004			0.720020	+	0.693954i	$0.244133\pi$
$908$		0			0
$909$	26.6015	+	1441.71i	0.0292645	+	1.58604i
$910$		0			0
$911$		−	17.1374i		−	0.0188116i	−0.999956	−	0.00940580i	$-0.997006\pi$
$911$		−	17.1374i		−	0.0188116i	0.999956	−	0.00940580i	$-0.00299400\pi$
$912$		0			0
$913$	96.7361			0.105954
$914$		0			0
$915$	101.842	−	0.939473i	0.111302	−	0.00102675i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	54.4468			0.0592457			0.0296229	−	0.999561i	$-0.490569\pi$
$919$	54.4468			0.0592457			0.0296229	+	0.999561i	$0.490569\pi$
$920$		0			0
$921$	−1.84894	−	200.430i	−0.00200754	−	0.217623i
$922$		0			0
$923$		−	1197.39i		−	1.29728i
$924$		0			0
$925$	1281.51			1.38542
$926$		0			0
$927$	1612.56	−	29.7538i	1.73955	−	0.0320969i
$928$		0			0
$929$			130.414i			0.140381i	0.997534	+	0.0701906i	$0.0223608\pi$
$929$			130.414i			0.140381i	−0.997534	+	0.0701906i	$0.977639\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	726.552	−	6.70233i	0.778727	−	0.00718364i
$934$		0			0
$935$			44.7078i			0.0478159i
$936$		0			0
$937$	−741.896			−0.791778			−0.395889	−	0.918298i	$-0.629563\pi$
$937$	−741.896			−0.791778			−0.395889	+	0.918298i	$0.629563\pi$
$938$		0			0
$939$	4.40280	+	477.276i	0.00468882	+	0.508281i
$940$		0			0
$941$		−	445.721i		−	0.473667i	−0.971550	−	0.236834i	$-0.923890\pi$
$941$		−	445.721i		−	0.473667i	0.971550	−	0.236834i	$-0.0761096\pi$
$942$		0			0
$943$	160.402			0.170097
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1729.07i		−	1.82584i	−0.408144	−	0.912918i	$-0.633824\pi$
$947$		−	1729.07i		−	1.82584i	0.408144	−	0.912918i	$-0.366176\pi$
$948$		0			0
$949$	−787.886			−0.830228
$950$		0			0
$951$	1794.46	−	16.5537i	1.88692	−	0.0174066i
$952$		0			0
$953$		−	522.827i		−	0.548612i	−0.961643	−	0.274306i	$-0.911552\pi$
$953$		−	522.827i		−	0.548612i	0.961643	−	0.274306i	$-0.0884480\pi$
$954$		0			0
$955$	−93.9432			−0.0983699
$956$		0			0
$957$	1.51564	+	164.300i	0.00158375	+	0.171682i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−688.434			−0.716373
$962$		0			0
$963$	32.1313	+	1741.41i	0.0333658	+	1.80832i
$964$		0			0
$965$		−	164.210i		−	0.170166i
$966$		0			0
$967$	−103.535			−0.107069			−0.0535343	−	0.998566i	$-0.517049\pi$
$967$	−103.535			−0.107069			−0.0535343	+	0.998566i	$0.517049\pi$
$968$		0			0
$969$	−1442.09	+	13.3031i	−1.48822	+	0.0137287i
$970$		0			0
$971$		−	1046.48i		−	1.07773i	−0.842392	−	0.538865i	$-0.818853\pi$
$971$		−	1046.48i		−	1.07773i	0.842392	−	0.538865i	$-0.181147\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−10.6273	−	1152.03i	−0.0108998	−	1.18157i
$976$		0			0
$977$			84.6422i			0.0866348i	0.999061	+	0.0433174i	$0.0137927\pi$
$977$			84.6422i			0.0866348i	−0.999061	+	0.0433174i	$0.986207\pi$
$978$		0			0
$979$	236.614			0.241689
$980$		0			0
$981$	−1053.02	+	19.4295i	−1.07341	+	0.0198058i
$982$		0			0
$983$		−	1509.13i		−	1.53523i	−0.640912	−	0.767615i	$-0.721444\pi$
$983$		−	1509.13i		−	1.53523i	0.640912	−	0.767615i	$-0.278556\pi$
$984$		0			0
$985$	−129.216			−0.131184
$986$		0			0
$987$		0			0
$988$		0			0
$989$			440.645i			0.445546i
$990$		0			0
$991$	−1309.19			−1.32108			−0.660538	−	0.750792i	$-0.729672\pi$
$991$	−1309.19			−1.32108			−0.660538	+	0.750792i	$0.729672\pi$
$992$		0			0
$993$	0.600423	+	65.0875i	0.000604655	+	0.0655463i
$994$		0			0
$995$		−	147.881i		−	0.148624i
$996$		0			0
$997$	190.121			0.190693			0.0953467	−	0.995444i	$-0.469604\pi$
$997$	190.121			0.190693			0.0953467	+	0.995444i	$0.469604\pi$
$998$		0			0
$999$	39.0701	+	1411.45i	0.0391092	+	1.41286i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.3.d.f.785.7		16
3.2	odd	2	inner	1176.3.d.f.785.8		16
7.2	even	3		168.3.bf.a.137.14	yes	32
7.4	even	3		168.3.bf.a.65.3	✓	32
7.6	odd	2		1176.3.d.g.785.10		16
21.2	odd	6		168.3.bf.a.137.3	yes	32
21.11	odd	6		168.3.bf.a.65.14	yes	32
21.20	even	2		1176.3.d.g.785.9		16
28.11	odd	6		336.3.bn.h.65.14		32
28.23	odd	6		336.3.bn.h.305.3		32
84.11	even	6		336.3.bn.h.65.3		32
84.23	even	6		336.3.bn.h.305.14		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.bf.a.65.3	✓	32	7.4	even	3
168.3.bf.a.65.14	yes	32	21.11	odd	6
168.3.bf.a.137.3	yes	32	21.2	odd	6
168.3.bf.a.137.14	yes	32	7.2	even	3
336.3.bn.h.65.3		32	84.11	even	6
336.3.bn.h.65.14		32	28.11	odd	6
336.3.bn.h.305.3		32	28.23	odd	6
336.3.bn.h.305.14		32	84.23	even	6
1176.3.d.f.785.7		16	1.1	even	1	trivial
1176.3.d.f.785.8		16	3.2	odd	2	inner
1176.3.d.g.785.9		16	21.20	even	2
1176.3.d.g.785.10		16	7.6	odd	2

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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1176.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.0276734 - 2.99987i) q^{3} +0.703530i q^{5} +(-8.99847 + 0.166033i) q^{9} -2.23488i q^{11} +15.6713 q^{13} +(2.11050 - 0.0194691i) q^{15} +28.4346i q^{17} -16.9061 q^{19} +28.2927i q^{23} +24.5050 q^{25} +(0.747096 + 26.9897i) q^{27} -24.5065i q^{29} -16.5096 q^{31} +(-6.70435 + 0.0618467i) q^{33} +52.2959 q^{37} +(-0.433678 - 47.0120i) q^{39} -5.66937i q^{41} +15.5745 q^{43} +(-0.116809 - 6.33070i) q^{45} +19.1839i q^{47} +(85.3001 - 0.786881i) q^{51} +53.5467i q^{53} +1.57231 q^{55} +(0.467848 + 50.7161i) q^{57} +51.4628i q^{59} +48.2547 q^{61} +11.0252i q^{65} +18.3292 q^{67} +(84.8744 - 0.782954i) q^{69} -76.4062i q^{71} -50.2757 q^{73} +(-0.678137 - 73.5120i) q^{75} +67.2013 q^{79} +(80.9449 - 2.98809i) q^{81} +43.2847i q^{83} -20.0046 q^{85} +(-73.5163 + 0.678177i) q^{87} +105.873i q^{89} +(0.456876 + 49.5266i) q^{93} -11.8939i q^{95} +153.611 q^{97} +(0.371064 + 20.1105i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9} - 4 q^{13} + 8 q^{15} - 16 q^{19} - 76 q^{25} - 12 q^{27} + 36 q^{31} - 32 q^{33} - 28 q^{37} - 4 q^{39} - 120 q^{43} - 28 q^{45} - 36 q^{51} - 28 q^{55} - 88 q^{57} + 152 q^{61} + 88 q^{67}+ \cdots + 368 q^{99}+O(q^{100}) \) 16 * q + 8 * q^9 - 4 * q^13 + 8 * q^15 - 16 * q^19 - 76 * q^25 - 12 * q^27 + 36 * q^31 - 32 * q^33 - 28 * q^37 - 4 * q^39 - 120 * q^43 - 28 * q^45 - 36 * q^51 - 28 * q^55 - 88 * q^57 + 152 * q^61 + 88 * q^67 - 112 * q^69 + 156 * q^73 + 312 * q^75 - 84 * q^79 + 164 * q^81 + 152 * q^85 + 176 * q^87 - 8 * q^93 - 20 * q^97 + 368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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