LMFDB - Embedded newform 552.4.m.a.137.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [552,4,Mod(137,552)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("552.137"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	552.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$32.5690543232$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			137.5
Character	$\chi$	$=$	552.137
Dual form			552.4.m.a.137.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-5.11929 - 0.890422i) q^{3} -20.0544 q^{5} +33.3044i q^{7} +(25.4143 + 9.11666i) q^{9} +57.9527 q^{11} -50.6877 q^{13} +(102.664 + 17.8569i) q^{15} +59.5814 q^{17} +71.6126i q^{19} +(29.6550 - 170.495i) q^{21} +(24.4434 + 107.562i) q^{23} +277.178 q^{25} +(-121.986 - 69.3003i) q^{27} +34.8835i q^{29} -5.52541 q^{31} +(-296.677 - 51.6024i) q^{33} -667.899i q^{35} +144.241i q^{37} +(259.485 + 45.1334i) q^{39} +430.206i q^{41} -264.484i q^{43} +(-509.668 - 182.829i) q^{45} +215.856i q^{47} -766.183 q^{49} +(-305.014 - 53.0526i) q^{51} -62.8922 q^{53} -1162.21 q^{55} +(63.7654 - 366.606i) q^{57} +151.374i q^{59} +549.877i q^{61} +(-303.625 + 846.408i) q^{63} +1016.51 q^{65} -526.152i q^{67} +(-29.3574 - 572.405i) q^{69} -221.978i q^{71} -803.576 q^{73} +(-1418.96 - 246.806i) q^{75} +1930.08i q^{77} +973.309i q^{79} +(562.773 + 463.387i) q^{81} -244.352 q^{83} -1194.87 q^{85} +(31.0610 - 178.579i) q^{87} +966.847 q^{89} -1688.12i q^{91} +(28.2862 + 4.91994i) q^{93} -1436.15i q^{95} -1068.16i q^{97} +(1472.83 + 528.336i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 28 q^{9} + 1872 q^{25} - 444 q^{27} - 216 q^{31} - 68 q^{39} - 4200 q^{49} - 576 q^{55} + 1376 q^{69} - 384 q^{73} - 3592 q^{75} - 252 q^{81} + 3480 q^{85} - 412 q^{87} + 780 q^{93}+O(q^{100}) $ 72 * q + 28 * q^9 + 1872 * q^25 - 444 * q^27 - 216 * q^31 - 68 * q^39 - 4200 * q^49 - 576 * q^55 + 1376 * q^69 - 384 * q^73 - 3592 * q^75 - 252 * q^81 + 3480 * q^85 - 412 * q^87 + 780 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$.

$n$	$97$	$185$	$277$	$415$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−5.11929	−	0.890422i	−0.985208	−	0.171362i
$3$	−5.11929	−	0.890422i	−0.985208	−	0.171362i
$4$		0			0
$5$	−20.0544			−1.79372			−0.896859	−	0.442316i	$-0.854157\pi$
$5$	−20.0544			−1.79372			−0.896859	+	0.442316i	$0.854157\pi$
$6$		0			0
$7$			33.3044i			1.79827i	0.437673	+	0.899134i	$0.355803\pi$
$7$			33.3044i			1.79827i	−0.437673	+	0.899134i	$0.644197\pi$
$8$		0			0
$9$	25.4143	+	9.11666i	0.941270	+	0.337654i
$10$		0			0
$11$	57.9527			1.58849			0.794246	−	0.607596i	$-0.207866\pi$
$11$	57.9527			1.58849			0.794246	+	0.607596i	$0.207866\pi$
$12$		0			0
$13$	−50.6877			−1.08140			−0.540701	−	0.841215i	$-0.681841\pi$
$13$	−50.6877			−1.08140			−0.540701	+	0.841215i	$0.681841\pi$
$14$		0			0
$15$	102.664	+	17.8569i	1.76719	+	0.307375i
$16$		0			0
$17$	59.5814			0.850036			0.425018	−	0.905185i	$-0.360268\pi$
$17$	59.5814			0.850036			0.425018	+	0.905185i	$0.360268\pi$
$18$		0			0
$19$			71.6126i			0.864687i	0.901709	+	0.432344i	$0.142313\pi$
$19$			71.6126i			0.864687i	−0.901709	+	0.432344i	$0.857687\pi$
$20$		0			0
$21$	29.6550	−	170.495i	0.308154	−	1.77167i
$22$		0			0
$23$	24.4434	+	107.562i	0.221600	+	0.975138i
$23$	24.4434	+	107.562i	0.221600	+	0.975138i
$24$		0			0
$25$	277.178			2.21743
$26$		0			0
$27$	−121.986	−	69.3003i	−0.869486	−	0.493957i
$28$		0			0
$29$			34.8835i			0.223369i	0.993744	+	0.111684i	$0.0356246\pi$
$29$			34.8835i			0.223369i	−0.993744	+	0.111684i	$0.964375\pi$
$30$		0			0
$31$	−5.52541			−0.0320127			−0.0160063	−	0.999872i	$-0.505095\pi$
$31$	−5.52541			−0.0320127			−0.0160063	+	0.999872i	$0.505095\pi$
$32$		0			0
$33$	−296.677	−	51.6024i	−1.56500	−	0.272207i
$34$		0			0
$35$		−	667.899i		−	3.22559i
$36$		0			0
$37$			144.241i			0.640893i	0.947267	+	0.320446i	$0.103833\pi$
$37$			144.241i			0.640893i	−0.947267	+	0.320446i	$0.896167\pi$
$38$		0			0
$39$	259.485	+	45.1334i	1.06541	+	0.185311i
$40$		0			0
$41$			430.206i			1.63870i	0.573290	+	0.819352i	$0.305667\pi$
$41$			430.206i			1.63870i	−0.573290	+	0.819352i	$0.694333\pi$
$42$		0			0
$43$		−	264.484i		−	0.937985i	−0.883202	−	0.468993i	$-0.844617\pi$
$43$		−	264.484i		−	0.937985i	0.883202	−	0.468993i	$-0.155383\pi$
$44$		0			0
$45$	−509.668	−	182.829i	−1.68837	−	0.605657i
$46$		0			0
$47$			215.856i			0.669910i	0.942234	+	0.334955i	$0.108721\pi$
$47$			215.856i			0.669910i	−0.942234	+	0.334955i	$0.891279\pi$
$48$		0			0
$49$	−766.183			−2.23377
$50$		0			0
$51$	−305.014	−	53.0526i	−0.837462	−	0.145664i
$52$		0			0
$53$	−62.8922			−0.162998			−0.0814991	−	0.996673i	$-0.525971\pi$
$53$	−62.8922			−0.162998			−0.0814991	+	0.996673i	$0.525971\pi$
$54$		0			0
$55$	−1162.21			−2.84931
$56$		0			0
$57$	63.7654	−	366.606i	0.148174	−	0.851897i
$58$		0			0
$59$			151.374i			0.334022i	0.985955	+	0.167011i	$0.0534115\pi$
$59$			151.374i			0.334022i	−0.985955	+	0.167011i	$0.946589\pi$
$60$		0			0
$61$			549.877i			1.15417i	0.816683	+	0.577087i	$0.195811\pi$
$61$			549.877i			1.15417i	−0.816683	+	0.577087i	$0.804189\pi$
$62$		0			0
$63$	−303.625	+	846.408i	−0.607193	+	1.69266i
$64$		0			0
$65$	1016.51			1.93973
$66$		0			0
$67$		−	526.152i		−	0.959399i	−0.877433	−	0.479699i	$-0.840746\pi$
$67$		−	526.152i		−	0.959399i	0.877433	−	0.479699i	$-0.159254\pi$
$68$		0			0
$69$	−29.3574	−	572.405i	−0.0512205	−	0.998687i
$70$		0			0
$71$		−	221.978i		−	0.371042i	−0.982640	−	0.185521i	$-0.940603\pi$
$71$		−	221.978i		−	0.371042i	0.982640	−	0.185521i	$-0.0593973\pi$
$72$		0			0
$73$	−803.576			−1.28838			−0.644188	−	0.764867i	$-0.722805\pi$
$73$	−803.576			−1.28838			−0.644188	+	0.764867i	$0.722805\pi$
$74$		0			0
$75$	−1418.96	−	246.806i	−2.18463	−	0.379982i
$76$		0			0
$77$			1930.08i			2.85653i
$78$		0			0
$79$			973.309i			1.38615i	0.720866	+	0.693075i	$0.243745\pi$
$79$			973.309i			1.38615i	−0.720866	+	0.693075i	$0.756255\pi$
$80$		0			0
$81$	562.773	+	463.387i	0.771979	+	0.635648i
$82$		0			0
$83$	−244.352			−0.323146			−0.161573	−	0.986861i	$-0.551657\pi$
$83$	−244.352			−0.323146			−0.161573	+	0.986861i	$0.551657\pi$
$84$		0			0
$85$	−1194.87			−1.52473
$86$		0			0
$87$	31.0610	−	178.579i	0.0382769	−	0.220065i
$88$		0			0
$89$	966.847			1.15152			0.575762	−	0.817617i	$-0.304706\pi$
$89$	966.847			1.15152			0.575762	+	0.817617i	$0.304706\pi$
$90$		0			0
$91$		−	1688.12i		−	1.94465i
$92$		0			0
$93$	28.2862	+	4.91994i	0.0315391	+	0.00548575i
$94$		0			0
$95$		−	1436.15i		−	1.55101i
$96$		0			0
$97$		−	1068.16i		−	1.11810i	−0.829134	−	0.559050i	$-0.811166\pi$
$97$		−	1068.16i		−	1.11810i	0.829134	−	0.559050i	$-0.188834\pi$
$98$		0			0
$99$	1472.83	+	528.336i	1.49520	+	0.536361i
$100$		0			0
$101$		−	1173.08i		−	1.15570i	−0.816144	−	0.577849i	$-0.803892\pi$
$101$		−	1173.08i		−	1.15570i	0.816144	−	0.577849i	$-0.196108\pi$
$102$		0			0
$103$		−	1661.92i		−	1.58984i	−0.606715	−	0.794920i	$-0.707513\pi$
$103$		−	1661.92i		−	1.58984i	0.606715	−	0.794920i	$-0.292487\pi$
$104$		0			0
$105$	−594.712	+	3419.17i	−0.552743	+	3.17788i
$106$		0			0
$107$	162.335			0.146668			0.0733342	−	0.997307i	$-0.476636\pi$
$107$	162.335			0.146668			0.0733342	+	0.997307i	$0.476636\pi$
$108$		0			0
$109$			1447.94i			1.27236i	0.771539	+	0.636181i	$0.219487\pi$
$109$			1447.94i			1.27236i	−0.771539	+	0.636181i	$0.780513\pi$
$110$		0			0
$111$	128.435	−	738.411i	0.109825	−	0.631413i
$112$		0			0
$113$	−304.959			−0.253877			−0.126939	−	0.991911i	$-0.540515\pi$
$113$	−304.959			−0.253877			−0.126939	+	0.991911i	$0.540515\pi$
$114$		0			0
$115$	−490.197	−	2157.08i	−0.397488	−	1.74912i
$116$		0			0
$117$	−1288.19	−	462.102i	−1.01789	−	0.365140i
$118$		0			0
$119$			1984.32i			1.52859i
$120$		0			0
$121$	2027.52			1.52331
$122$		0			0
$123$	383.065	−	2202.35i	0.280811	−	1.61447i
$124$		0			0
$125$	−3051.84			−2.18372
$126$		0			0
$127$	−2713.71			−1.89608			−0.948041	−	0.318148i	$-0.896939\pi$
$127$	−2713.71			−1.89608			−0.948041	+	0.318148i	$0.896939\pi$
$128$		0			0
$129$	−235.502	+	1353.97i	−0.160735	+	0.924111i
$130$		0			0
$131$		−	291.584i		−	0.194472i	−0.995261	−	0.0972360i	$-0.969000\pi$
$131$		−	291.584i		−	0.194472i	0.995261	−	0.0972360i	$-0.0310002\pi$
$132$		0			0
$133$	−2385.01			−1.55494
$134$		0			0
$135$	2446.34	+	1389.77i	1.55961	+	0.886021i
$136$		0			0
$137$	−198.109			−0.123545			−0.0617724	−	0.998090i	$-0.519675\pi$
$137$	−198.109			−0.123545			−0.0617724	+	0.998090i	$0.519675\pi$
$138$		0			0
$139$	−1531.61			−0.934598			−0.467299	−	0.884099i	$-0.654773\pi$
$139$	−1531.61			−0.934598			−0.467299	+	0.884099i	$0.654773\pi$
$140$		0			0
$141$	192.203	−	1105.03i	0.114797	−	0.660001i
$142$		0			0
$143$	−2937.49			−1.71780
$144$		0			0
$145$		−	699.566i		−	0.400661i
$146$		0			0
$147$	3922.31	+	682.226i	2.20073	+	0.382783i
$148$		0			0
$149$	662.731			0.364383			0.182192	−	0.983263i	$-0.441681\pi$
$149$	662.731			0.364383			0.182192	+	0.983263i	$0.441681\pi$
$150$		0			0
$151$	3233.77			1.74278			0.871392	−	0.490588i	$-0.163218\pi$
$151$	3233.77			1.74278			0.871392	+	0.490588i	$0.163218\pi$
$152$		0			0
$153$	1514.22	+	543.183i	0.800113	+	0.287018i
$154$		0			0
$155$	110.809			0.0574217
$156$		0			0
$157$		−	165.429i		−	0.0840934i	−0.999116	−	0.0420467i	$-0.986612\pi$
$157$		−	165.429i		−	0.0840934i	0.999116	−	0.0420467i	$-0.0133878\pi$
$158$		0			0
$159$	321.964	+	56.0006i	0.160587	+	0.0279317i
$160$		0			0
$161$	−3582.28	+	814.071i	−1.75356	+	0.398496i
$162$		0			0
$163$	−1171.01			−0.562704			−0.281352	−	0.959605i	$-0.590783\pi$
$163$	−1171.01			−0.562704			−0.281352	+	0.959605i	$0.590783\pi$
$164$		0			0
$165$	5949.68	+	1034.85i	2.80716	+	0.488263i
$166$		0			0
$167$		−	2748.16i		−	1.27341i	−0.771108	−	0.636705i	$-0.780297\pi$
$167$		−	2748.16i		−	1.27341i	0.771108	−	0.636705i	$-0.219703\pi$
$168$		0			0
$169$	372.238			0.169430
$170$		0			0
$171$	−652.868	+	1819.98i	−0.291965	+	0.813904i
$172$		0			0
$173$			160.394i			0.0704885i	0.999379	+	0.0352442i	$0.0112209\pi$
$173$			160.394i			0.0704885i	−0.999379	+	0.0352442i	$0.988779\pi$
$174$		0			0
$175$			9231.26i			3.98753i
$176$		0			0
$177$	134.787	−	774.930i	0.0572385	−	0.329081i
$178$		0			0
$179$		−	1324.21i		−	0.552938i	−0.961023	−	0.276469i	$-0.910836\pi$
$179$		−	1324.21i		−	0.552938i	0.961023	−	0.276469i	$-0.0891643\pi$
$180$		0			0
$181$			2965.18i			1.21768i	0.793293	+	0.608841i	$0.208365\pi$
$181$			2965.18i			1.21768i	−0.793293	+	0.608841i	$0.791635\pi$
$182$		0			0
$183$	489.623	−	2814.98i	0.197781	−	1.13710i
$184$		0			0
$185$		−	2892.66i		−	1.14958i
$186$		0			0
$187$	3452.90			1.35027
$188$		0			0
$189$	2308.00	−	4062.65i	0.888268	−	1.56357i
$190$		0			0
$191$	4189.76			1.58723			0.793614	−	0.608422i	$-0.208197\pi$
$191$	4189.76			1.58723			0.793614	+	0.608422i	$0.208197\pi$
$192$		0			0
$193$	−1493.28			−0.556937			−0.278468	−	0.960445i	$-0.589827\pi$
$193$	−1493.28			−0.556937			−0.278468	+	0.960445i	$0.589827\pi$
$194$		0			0
$195$	−5203.81	−	905.123i	−1.91104	−	0.332396i
$196$		0			0
$197$		−	4483.73i		−	1.62159i	−0.585333	−	0.810793i	$-0.699036\pi$
$197$		−	4483.73i		−	1.62159i	0.585333	−	0.810793i	$-0.300964\pi$
$198$		0			0
$199$			2192.98i			0.781187i	0.920563	+	0.390594i	$0.127730\pi$
$199$			2192.98i			0.781187i	−0.920563	+	0.390594i	$0.872270\pi$
$200$		0			0
$201$	−468.497	+	2693.53i	−0.164404	+	0.945207i
$202$		0			0
$203$	−1161.77			−0.401677
$204$		0			0
$205$		−	8627.52i		−	2.93938i
$206$		0			0
$207$	−359.393	+	2956.45i	−0.120674	+	0.992692i
$208$		0			0
$209$			4150.15i			1.37355i
$210$		0			0
$211$	−1078.73			−0.351956			−0.175978	−	0.984394i	$-0.556309\pi$
$211$	−1078.73			−0.351956			−0.175978	+	0.984394i	$0.556309\pi$
$212$		0			0
$213$	−197.654	+	1136.37i	−0.0635824	+	0.365554i
$214$		0			0
$215$			5304.06i			1.68248i
$216$		0			0
$217$		−	184.020i		−	0.0575673i
$218$		0			0
$219$	4113.74	+	715.522i	1.26932	+	0.220778i
$220$		0			0
$221$	−3020.04			−0.919230
$222$		0			0
$223$	−2649.88			−0.795737			−0.397868	−	0.917442i	$-0.630250\pi$
$223$	−2649.88			−0.795737			−0.397868	+	0.917442i	$0.630250\pi$
$224$		0			0
$225$	7044.29	+	2526.94i	2.08720	+	0.748723i
$226$		0			0
$227$	3454.65			1.01010			0.505051	−	0.863090i	$-0.331474\pi$
$227$	3454.65			1.01010			0.505051	+	0.863090i	$0.331474\pi$
$228$		0			0
$229$			952.474i			0.274853i	0.990512	+	0.137426i	$0.0438830\pi$
$229$			952.474i			0.274853i	−0.990512	+	0.137426i	$0.956117\pi$
$230$		0			0
$231$	1718.59	−	9880.65i	0.489501	−	2.81428i
$232$		0			0
$233$			4803.31i			1.35054i	0.737572	+	0.675269i	$0.235972\pi$
$233$			4803.31i			1.35054i	−0.737572	+	0.675269i	$0.764028\pi$
$234$		0			0
$235$		−	4328.85i		−	1.20163i
$236$		0			0
$237$	866.656	−	4982.65i	0.237533	−	1.36565i
$238$		0			0
$239$		−	4589.78i		−	1.24221i	−0.783727	−	0.621105i	$-0.786684\pi$
$239$		−	4589.78i		−	1.24221i	0.783727	−	0.621105i	$-0.213316\pi$
$240$		0			0
$241$		−	1558.16i		−	0.416473i	−0.978078	−	0.208237i	$-0.933228\pi$
$241$		−	1558.16i		−	0.416473i	0.978078	−	0.208237i	$-0.0667724\pi$
$242$		0			0
$243$	−2468.39	−	2873.32i	−0.651635	−	0.758533i
$244$		0			0
$245$	15365.3			4.00675
$246$		0			0
$247$		−	3629.87i		−	0.935075i
$248$		0			0
$249$	1250.91	+	217.577i	0.318367	+	0.0553750i
$250$		0			0
$251$	406.202			0.102148			0.0510742	−	0.998695i	$-0.483735\pi$
$251$	406.202			0.102148			0.0510742	+	0.998695i	$0.483735\pi$
$252$		0			0
$253$	1416.56	+	6233.50i	0.352009	+	1.54900i
$254$		0			0
$255$	6116.88	+	1063.94i	1.50217	+	0.261280i
$256$		0			0
$257$			5017.55i			1.21784i	0.793230	+	0.608922i	$0.208398\pi$
$257$			5017.55i			1.21784i	−0.793230	+	0.608922i	$0.791602\pi$
$258$		0			0
$259$	−4803.85			−1.15250
$260$		0			0
$261$	−318.021	+	886.539i	−0.0754214	+	0.210250i
$262$		0			0
$263$	534.553			0.125331			0.0626653	−	0.998035i	$-0.480040\pi$
$263$	534.553			0.125331			0.0626653	+	0.998035i	$0.480040\pi$
$264$		0			0
$265$	1261.26			0.292373
$266$		0			0
$267$	−4949.57	−	860.902i	−1.13449	−	0.197327i
$268$		0			0
$269$			862.225i			0.195430i	0.995214	+	0.0977152i	$0.0311534\pi$
$269$			862.225i			0.195430i	−0.995214	+	0.0977152i	$0.968847\pi$
$270$		0			0
$271$	−1434.10			−0.321458			−0.160729	−	0.986999i	$-0.551385\pi$
$271$	−1434.10			−0.321458			−0.160729	+	0.986999i	$0.551385\pi$
$272$		0			0
$273$	−1503.14	+	8641.99i	−0.333239	+	1.91589i
$274$		0			0
$275$	16063.3			3.52237
$276$		0			0
$277$	203.521			0.0441458			0.0220729	−	0.999756i	$-0.492973\pi$
$277$	203.521			0.0441458			0.0220729	+	0.999756i	$0.492973\pi$
$278$		0			0
$279$	−140.424	−	50.3733i	−0.0301326	−	0.0108092i
$280$		0			0
$281$	−1807.02			−0.383622			−0.191811	−	0.981432i	$-0.561436\pi$
$281$	−1807.02			−0.383622			−0.191811	+	0.981432i	$0.561436\pi$
$282$		0			0
$283$			4149.51i			0.871600i	0.900044	+	0.435800i	$0.143535\pi$
$283$			4149.51i			0.871600i	−0.900044	+	0.435800i	$0.856465\pi$
$284$		0			0
$285$	−1278.78	+	7352.06i	−0.265783	+	1.52806i
$286$		0			0
$287$	−14327.8			−2.94683
$288$		0			0
$289$	−1363.06			−0.277439
$290$		0			0
$291$	−951.118	+	5468.25i	−0.191600	+	1.10156i
$292$		0			0
$293$	−9833.94			−1.96077			−0.980384	−	0.197099i	$-0.936848\pi$
$293$	−9833.94			−1.96077			−0.980384	+	0.197099i	$0.936848\pi$
$294$		0			0
$295$		−	3035.72i		−	0.599141i
$296$		0			0
$297$	−7069.40	−	4016.14i	−1.38117	−	0.784647i
$298$		0			0
$299$	−1238.98	−	5452.05i	−0.239638	−	1.05452i
$300$		0			0
$301$	8808.47			1.68675
$302$		0			0
$303$	−1044.53	+	6005.32i	−0.198043	+	1.13860i
$304$		0			0
$305$		−	11027.5i		−	2.07026i
$306$		0			0
$307$	6517.24			1.21159			0.605796	−	0.795620i	$-0.292855\pi$
$307$	6517.24			1.21159			0.605796	+	0.795620i	$0.292855\pi$
$308$		0			0
$309$	−1479.81	+	8507.83i	−0.272438	+	1.56632i
$310$		0			0
$311$			1323.87i			0.241381i	0.992690	+	0.120691i	$0.0385109\pi$
$311$			1323.87i			0.241381i	−0.992690	+	0.120691i	$0.961489\pi$
$312$		0			0
$313$			5770.09i			1.04200i	0.853558	+	0.520998i	$0.174440\pi$
$313$			5770.09i			1.04200i	−0.853558	+	0.520998i	$0.825560\pi$
$314$		0			0
$315$	6089.01	−	16974.2i	1.08913	−	3.03615i
$316$		0			0
$317$		−	7870.38i		−	1.39446i	−0.716846	−	0.697231i	$-0.754415\pi$
$317$		−	7870.38i		−	1.39446i	0.716846	−	0.697231i	$-0.245585\pi$
$318$		0			0
$319$			2021.59i			0.354820i
$320$		0			0
$321$	−831.041	−	144.547i	−0.144499	−	0.0251334i
$322$		0			0
$323$			4266.78i			0.735015i
$324$		0			0
$325$	−14049.5			−2.39793
$326$		0			0
$327$	1289.28	−	7412.43i	0.218034	−	1.25354i
$328$		0			0
$329$	−7188.94			−1.20468
$330$		0			0
$331$	−2918.55			−0.484647			−0.242324	−	0.970195i	$-0.577910\pi$
$331$	−2918.55			−0.484647			−0.242324	+	0.970195i	$0.577910\pi$
$332$		0			0
$333$	−1314.99	+	3665.78i	−0.216400	+	0.603253i
$334$		0			0
$335$			10551.7i			1.72089i
$336$		0			0
$337$		−	1579.31i		−	0.255283i	−0.991820	−	0.127642i	$-0.959259\pi$
$337$		−	1579.31i		−	0.255283i	0.991820	−	0.127642i	$-0.0407407\pi$
$338$		0			0
$339$	1561.17	+	271.542i	0.250122	+	0.0435048i
$340$		0			0
$341$	−320.213			−0.0508518
$342$		0			0
$343$		−	14093.8i		−	2.21865i
$344$		0			0
$345$	588.744	+	11479.2i	0.0918751	+	1.79136i
$346$		0			0
$347$			9345.12i			1.44574i	0.690983	+	0.722871i	$0.257178\pi$
$347$			9345.12i			1.44574i	−0.690983	+	0.722871i	$0.742822\pi$
$348$		0			0
$349$	2015.77			0.309174			0.154587	−	0.987979i	$-0.450595\pi$
$349$	2015.77			0.309174			0.154587	+	0.987979i	$0.450595\pi$
$350$		0			0
$351$	6183.16	+	3512.67i	0.940264	+	0.534166i
$352$		0			0
$353$			515.899i			0.0777862i	0.999243	+	0.0388931i	$0.0123832\pi$
$353$			515.899i			0.0777862i	−0.999243	+	0.0388931i	$0.987617\pi$
$354$		0			0
$355$			4451.64i			0.665545i
$356$		0			0
$357$	1766.88	−	10158.3i	0.261942	−	1.50598i
$358$		0			0
$359$	−3438.96			−0.505575			−0.252787	−	0.967522i	$-0.581347\pi$
$359$	−3438.96			−0.505575			−0.252787	+	0.967522i	$0.581347\pi$
$360$		0			0
$361$	1730.63			0.252316
$362$		0			0
$363$	−10379.5	−	1805.35i	−1.50077	−	0.261037i
$364$		0			0
$365$	16115.2			2.31098
$366$		0			0
$367$		−	4236.03i		−	0.602504i	−0.953545	−	0.301252i	$-0.902595\pi$
$367$		−	4236.03i		−	0.602504i	0.953545	−	0.301252i	$-0.0974046\pi$
$368$		0			0
$369$	−3922.04	+	10933.4i	−0.553315	+	1.54246i
$370$		0			0
$371$		−	2094.59i		−	0.293115i
$372$		0			0
$373$			8604.37i			1.19442i	0.802086	+	0.597208i	$0.203723\pi$
$373$			8604.37i			1.19442i	−0.802086	+	0.597208i	$0.796277\pi$
$374$		0			0
$375$	15623.3	+	2717.43i	2.15142	+	0.374207i
$376$		0			0
$377$		−	1768.16i		−	0.241551i
$378$		0			0
$379$			3190.97i			0.432478i	0.976340	+	0.216239i	$0.0693790\pi$
$379$			3190.97i			0.432478i	−0.976340	+	0.216239i	$0.930621\pi$
$380$		0			0
$381$	13892.3	+	2416.34i	1.86804	+	0.324916i
$382$		0			0
$383$	−5211.50			−0.695288			−0.347644	−	0.937627i	$-0.613018\pi$
$383$	−5211.50			−0.695288			−0.347644	+	0.937627i	$0.613018\pi$
$384$		0			0
$385$		−	38706.6i		−	5.12382i
$386$		0			0
$387$	2411.21	−	6721.67i	0.316715	−	0.882898i
$388$		0			0
$389$	6735.90			0.877953			0.438976	−	0.898499i	$-0.355341\pi$
$389$	6735.90			0.877953			0.438976	+	0.898499i	$0.355341\pi$
$390$		0			0
$391$	1456.37	+	6408.68i	0.188368	+	0.828902i
$392$		0			0
$393$	−259.633	+	1492.70i	−0.0333251	+	0.191595i
$394$		0			0
$395$		−	19519.1i		−	2.48636i
$396$		0			0
$397$	−8708.38			−1.10091			−0.550455	−	0.834865i	$-0.685546\pi$
$397$	−8708.38			−1.10091			−0.550455	+	0.834865i	$0.685546\pi$
$398$		0			0
$399$	12209.6	+	2123.67i	1.53194	+	0.266457i
$400$		0			0
$401$	4057.38			0.505276			0.252638	−	0.967561i	$-0.418702\pi$
$401$	4057.38			0.505276			0.252638	+	0.967561i	$0.418702\pi$
$402$		0			0
$403$	280.070			0.0346185
$404$		0			0
$405$	−11286.1	−	9292.94i	−1.38471	−	1.14017i
$406$		0			0
$407$			8359.15i			1.01805i
$408$		0			0
$409$	6583.44			0.795918			0.397959	−	0.917403i	$-0.369719\pi$
$409$	6583.44			0.795918			0.397959	+	0.917403i	$0.369719\pi$
$410$		0			0
$411$	1014.18	+	176.401i	0.121717	+	0.0211708i
$412$		0			0
$413$	−5041.43			−0.600660
$414$		0			0
$415$	4900.34			0.579634
$416$		0			0
$417$	7840.74	+	1363.78i	0.920773	+	0.160154i
$418$		0			0
$419$	−4410.29			−0.514217			−0.257108	−	0.966383i	$-0.582770\pi$
$419$	−4410.29			−0.514217			−0.257108	+	0.966383i	$0.582770\pi$
$420$		0			0
$421$		−	23.2747i		−	0.00269439i	−0.999999	−	0.00134720i	$-0.999571\pi$
$421$		−	23.2747i		−	0.00269439i	0.999999	−	0.00134720i	$-0.000428826\pi$
$422$		0			0
$423$	−1967.88	+	5485.82i	−0.226198	+	0.630566i
$424$		0			0
$425$	16514.7			1.88489
$426$		0			0
$427$	−18313.3			−2.07551
$428$		0			0
$429$	15037.9	+	2615.60i	1.69239	+	0.294365i
$430$		0			0
$431$	−3734.23			−0.417335			−0.208667	−	0.977987i	$-0.566913\pi$
$431$	−3734.23			−0.417335			−0.208667	+	0.977987i	$0.566913\pi$
$432$		0			0
$433$		−	5952.10i		−	0.660600i	−0.943876	−	0.330300i	$-0.892850\pi$
$433$		−	5952.10i		−	0.660600i	0.943876	−	0.330300i	$-0.107150\pi$
$434$		0			0
$435$	−622.909	+	3581.28i	−0.0686580	+	0.394734i
$436$		0			0
$437$	−7702.77	+	1750.45i	−0.843189	+	0.191614i
$438$		0			0
$439$	−14010.4			−1.52319			−0.761594	−	0.648054i	$-0.775583\pi$
$439$	−14010.4			−1.52319			−0.761594	+	0.648054i	$0.775583\pi$
$440$		0			0
$441$	−19472.0	−	6985.03i	−2.10258	−	0.754241i
$442$		0			0
$443$			840.098i			0.0900999i	0.998985	+	0.0450500i	$0.0143447\pi$
$443$			840.098i			0.0900999i	−0.998985	+	0.0450500i	$0.985655\pi$
$444$		0			0
$445$	−19389.5			−2.06551
$446$		0			0
$447$	−3392.72	−	590.111i	−0.358993	−	0.0624413i
$448$		0			0
$449$		−	12556.3i		−	1.31975i	−0.751375	−	0.659875i	$-0.770609\pi$
$449$		−	12556.3i		−	1.31975i	0.751375	−	0.659875i	$-0.229391\pi$
$450$		0			0
$451$			24931.6i			2.60307i
$452$		0			0
$453$	−16554.6	−	2879.42i	−1.71700	−	0.298647i
$454$		0			0
$455$			33854.2i			3.48816i
$456$		0			0
$457$			5646.01i			0.577920i	0.957341	+	0.288960i	$0.0933095\pi$
$457$			5646.01i			0.577920i	−0.957341	+	0.288960i	$0.906691\pi$
$458$		0			0
$459$	−7268.07	−	4129.01i	−0.739094	−	0.419881i
$460$		0			0
$461$		−	12972.6i		−	1.31062i	−0.755359	−	0.655311i	$-0.772538\pi$
$461$		−	12972.6i		−	1.31062i	0.755359	−	0.655311i	$-0.227462\pi$
$462$		0			0
$463$	14034.2			1.40869			0.704346	−	0.709857i	$-0.251240\pi$
$463$	14034.2			1.40869			0.704346	+	0.709857i	$0.251240\pi$
$464$		0			0
$465$	−567.262	−	98.6665i	−0.0565723	−	0.00983989i
$466$		0			0
$467$	2862.18			0.283610			0.141805	−	0.989895i	$-0.454709\pi$
$467$	2862.18			0.283610			0.141805	+	0.989895i	$0.454709\pi$
$468$		0			0
$469$	17523.2			1.72526
$470$		0			0
$471$	−147.302	+	846.879i	−0.0144104	+	0.0828495i
$472$		0			0
$473$		−	15327.6i		−	1.48998i
$474$		0			0
$475$			19849.5i			1.91738i
$476$		0			0
$477$	−1598.36	−	573.367i	−0.153425	−	0.0550370i
$478$		0			0
$479$	−14342.1			−1.36808			−0.684038	−	0.729446i	$-0.739778\pi$
$479$	−14342.1			−1.36808			−0.684038	+	0.729446i	$0.739778\pi$
$480$		0			0
$481$		−	7311.22i		−	0.693063i
$482$		0			0
$483$	19063.6	−	977.730i	1.79591	−	0.0921082i
$484$		0			0
$485$			21421.4i			2.00556i
$486$		0			0
$487$	1338.32			0.124528			0.0622641	−	0.998060i	$-0.480168\pi$
$487$	1338.32			0.124528			0.0622641	+	0.998060i	$0.480168\pi$
$488$		0			0
$489$	5994.75	+	1042.69i	0.554380	+	0.0964259i
$490$		0			0
$491$			1238.33i			0.113819i	0.998379	+	0.0569095i	$0.0181246\pi$
$491$			1238.33i			0.113819i	−0.998379	+	0.0569095i	$0.981875\pi$
$492$		0			0
$493$			2078.40i			0.189871i
$494$		0			0
$495$	−29536.7	−	10595.4i	−2.68197	−	0.962080i
$496$		0			0
$497$	7392.85			0.667233
$498$		0			0
$499$	−16838.6			−1.51062			−0.755312	−	0.655365i	$-0.772515\pi$
$499$	−16838.6			−1.51062			−0.755312	+	0.655365i	$0.772515\pi$
$500$		0			0
$501$	−2447.03	+	14068.7i	−0.218214	+	1.25457i
$502$		0			0
$503$	3495.77			0.309878			0.154939	−	0.987924i	$-0.450482\pi$
$503$	3495.77			0.309878			0.154939	+	0.987924i	$0.450482\pi$
$504$		0			0
$505$			23525.3i			2.07300i
$506$		0			0
$507$	−1905.59	−	331.449i	−0.166924	−	0.0290338i
$508$		0			0
$509$		−	7029.51i		−	0.612137i	−0.952010	−	0.306068i	$-0.900986\pi$
$509$		−	7029.51i		−	0.612137i	0.952010	−	0.306068i	$-0.0990136\pi$
$510$		0			0
$511$		−	26762.6i		−	2.31685i
$512$		0			0
$513$	4962.78	−	8735.70i	0.427119	−	0.751834i
$514$		0			0
$515$			33328.7i			2.85172i
$516$		0			0
$517$			12509.4i			1.06415i
$518$		0			0
$519$	142.818	−	821.102i	0.0120790	−	0.0694458i
$520$		0			0
$521$	−8244.43			−0.693272			−0.346636	−	0.938000i	$-0.612676\pi$
$521$	−8244.43			−0.693272			−0.346636	+	0.938000i	$0.612676\pi$
$522$		0			0
$523$		−	13656.2i		−	1.14176i	−0.821032	−	0.570882i	$-0.806601\pi$
$523$		−	13656.2i		−	1.14176i	0.821032	−	0.570882i	$-0.193399\pi$
$524$		0			0
$525$	8219.72	−	47257.5i	0.683310	−	3.92855i
$526$		0			0
$527$	−329.211			−0.0272119
$528$		0			0
$529$	−10972.0	+	5258.34i	−0.901787	+	0.432180i
$530$		0			0
$531$	−1380.03	+	3847.07i	−0.112784	+	0.314405i
$532$		0			0
$533$		−	21806.1i		−	1.77210i
$534$		0			0
$535$	−3255.53			−0.263082
$536$		0			0
$537$	−1179.10	+	6779.00i	−0.0947525	+	0.544759i
$538$		0			0
$539$	−44402.4			−3.54832
$540$		0			0
$541$	−21151.7			−1.68093			−0.840464	−	0.541868i	$-0.817717\pi$
$541$	−21151.7			−1.68093			−0.840464	+	0.541868i	$0.817717\pi$
$542$		0			0
$543$	2640.26	−	15179.6i	0.208664	−	1.19967i
$544$		0			0
$545$		−	29037.6i		−	2.28226i
$546$		0			0
$547$	13822.1			1.08042			0.540211	−	0.841530i	$-0.318344\pi$
$547$	13822.1			1.08042			0.540211	+	0.841530i	$0.318344\pi$
$548$		0			0
$549$	−5013.05	+	13974.7i	−0.389711	+	1.08639i
$550$		0			0
$551$	−2498.10			−0.193144
$552$		0			0
$553$	−32415.5			−2.49267
$554$		0			0
$555$	−2575.69	+	14808.4i	−0.196994	+	1.13258i
$556$		0			0
$557$	−17563.3			−1.33606			−0.668028	−	0.744136i	$-0.732861\pi$
$557$	−17563.3			−1.33606			−0.668028	+	0.744136i	$0.732861\pi$
$558$		0			0
$559$			13406.1i			1.01434i
$560$		0			0
$561$	−17676.4	−	3074.54i	−1.33030	−	0.231386i
$562$		0			0
$563$	9217.05			0.689968			0.344984	−	0.938608i	$-0.387884\pi$
$563$	9217.05			0.689968			0.344984	+	0.938608i	$0.387884\pi$
$564$		0			0
$565$	6115.76			0.455384
$566$		0			0
$567$	−15432.8	+	18742.8i	−1.14306	+	1.38823i
$568$		0			0
$569$	1380.04			0.101677			0.0508387	−	0.998707i	$-0.483811\pi$
$569$	1380.04			0.101677			0.0508387	+	0.998707i	$0.483811\pi$
$570$		0			0
$571$		−	9745.55i		−	0.714253i	−0.934056	−	0.357127i	$-0.883756\pi$
$571$		−	9745.55i		−	0.714253i	0.934056	−	0.357127i	$-0.116244\pi$
$572$		0			0
$573$	−21448.6	−	3730.66i	−1.56375	−	0.271990i
$574$		0			0
$575$	6775.17	+	29813.8i	0.491381	+	2.16230i
$576$		0			0
$577$	−6481.97			−0.467674			−0.233837	−	0.972276i	$-0.575128\pi$
$577$	−6481.97			−0.467674			−0.233837	+	0.972276i	$0.575128\pi$
$578$		0			0
$579$	7644.54	+	1329.65i	0.548698	+	0.0954377i
$580$		0			0
$581$		−	8138.01i		−	0.581104i
$582$		0			0
$583$	−3644.78			−0.258921
$584$		0			0
$585$	25833.9	+	9267.17i	1.82581	+	0.654958i
$586$		0			0
$587$		−	18352.4i		−	1.29043i	−0.764000	−	0.645216i	$-0.776767\pi$
$587$		−	18352.4i		−	1.29043i	0.764000	−	0.645216i	$-0.223233\pi$
$588$		0			0
$589$		−	395.689i		−	0.0276809i
$590$		0			0
$591$	−3992.41	+	22953.5i	−0.277878	+	1.59760i
$592$		0			0
$593$			5951.20i			0.412119i	0.978539	+	0.206059i	$0.0660640\pi$
$593$			5951.20i			0.412119i	−0.978539	+	0.206059i	$0.933936\pi$
$594$		0			0
$595$		−	39794.4i		−	2.74186i
$596$		0			0
$597$	1952.68	−	11226.5i	0.133866	−	0.769632i
$598$		0			0
$599$		−	16582.5i		−	1.13112i	−0.824707	−	0.565561i	$-0.808660\pi$
$599$		−	16582.5i		−	1.13112i	0.824707	−	0.565561i	$-0.191340\pi$
$600$		0			0
$601$	−1178.04			−0.0799554			−0.0399777	−	0.999201i	$-0.512729\pi$
$601$	−1178.04			−0.0799554			−0.0399777	+	0.999201i	$0.512729\pi$
$602$		0			0
$603$	4796.75	−	13371.8i	0.323945	−	0.903053i
$604$		0			0
$605$	−40660.7			−2.73238
$606$		0			0
$607$	18457.2			1.23419			0.617097	−	0.786887i	$-0.288309\pi$
$607$	18457.2			1.23419			0.617097	+	0.786887i	$0.288309\pi$
$608$		0			0
$609$	5947.45	+	1034.47i	0.395735	+	0.0688321i
$610$		0			0
$611$		−	10941.2i		−	0.724442i
$612$		0			0
$613$			8799.63i			0.579794i	0.957058	+	0.289897i	$0.0936211\pi$
$613$			8799.63i			0.579794i	−0.957058	+	0.289897i	$0.906379\pi$
$614$		0			0
$615$	−7682.13	+	44166.8i	−0.503697	+	2.89590i
$616$		0			0
$617$	8358.12			0.545357			0.272679	−	0.962105i	$-0.412090\pi$
$617$	8358.12			0.545357			0.272679	+	0.962105i	$0.412090\pi$
$618$		0			0
$619$			6442.47i			0.418327i	0.977881	+	0.209164i	$0.0670741\pi$
$619$			6442.47i			0.418327i	−0.977881	+	0.209164i	$0.932926\pi$
$620$		0			0
$621$	4472.32	−	14814.9i	0.288999	−	0.957330i
$622$		0			0
$623$			32200.3i			2.07075i
$624$		0			0
$625$	26555.6			1.69956
$626$		0			0
$627$	3695.38	−	21245.8i	0.235374	−	1.35323i
$628$		0			0
$629$			8594.06i			0.544782i
$630$		0			0
$631$		−	24060.7i		−	1.51797i	−0.651105	−	0.758987i	$-0.725694\pi$
$631$		−	24060.7i		−	1.51797i	0.651105	−	0.758987i	$-0.274306\pi$
$632$		0			0
$633$	5522.32	+	960.523i	0.346750	+	0.0603118i
$634$		0			0
$635$	54421.7			3.40104
$636$		0			0
$637$	38836.0			2.41560
$638$		0			0
$639$	2023.70	−	5641.42i	0.125284	−	0.349251i
$640$		0			0
$641$	−14564.2			−0.897430			−0.448715	−	0.893675i	$-0.648118\pi$
$641$	−14564.2			−0.897430			−0.448715	+	0.893675i	$0.648118\pi$
$642$		0			0
$643$		−	7514.39i		−	0.460869i	−0.973088	−	0.230435i	$-0.925985\pi$
$643$		−	7514.39i		−	0.460869i	0.973088	−	0.230435i	$-0.0740147\pi$
$644$		0			0
$645$	4722.85	−	27153.0i	0.288313	−	1.65760i
$646$		0			0
$647$		−	1735.28i		−	0.105442i	−0.998609	−	0.0527208i	$-0.983211\pi$
$647$		−	1735.28i		−	0.105442i	0.998609	−	0.0527208i	$-0.0167893\pi$
$648$		0			0
$649$			8772.56i			0.530590i
$650$		0			0
$651$	−163.856	+	942.054i	−0.00986484	+	0.0567158i
$652$		0			0
$653$			4420.65i			0.264921i	0.991188	+	0.132460i	$0.0422878\pi$
$653$			4420.65i			0.264921i	−0.991188	+	0.132460i	$0.957712\pi$
$654$		0			0
$655$			5847.54i			0.348828i
$656$		0			0
$657$	−20422.3	−	7325.93i	−1.21271	−	0.435026i
$658$		0			0
$659$	−3532.82			−0.208830			−0.104415	−	0.994534i	$-0.533297\pi$
$659$	−3532.82			−0.208830			−0.104415	+	0.994534i	$0.533297\pi$
$660$		0			0
$661$		−	9876.59i		−	0.581172i	−0.956849	−	0.290586i	$-0.906150\pi$
$661$		−	9876.59i		−	0.581172i	0.956849	−	0.290586i	$-0.0938503\pi$
$662$		0			0
$663$	15460.5	+	2689.11i	0.905633	+	0.157521i
$664$		0			0
$665$	47830.0			2.78912
$666$		0			0
$667$	−3752.12	+	852.669i	−0.217815	+	0.0494985i
$668$		0			0
$669$	13565.5	+	2359.51i	0.783967	+	0.136359i
$670$		0			0
$671$			31866.9i			1.83340i
$672$		0			0
$673$	2629.87			0.150630			0.0753151	−	0.997160i	$-0.476004\pi$
$673$	2629.87			0.150630			0.0753151	+	0.997160i	$0.476004\pi$
$674$		0			0
$675$	−33811.8	−	19208.5i	−1.92802	−	1.09531i
$676$		0			0
$677$	27298.3			1.54972			0.774859	−	0.632134i	$-0.217821\pi$
$677$	27298.3			1.54972			0.774859	+	0.632134i	$0.217821\pi$
$678$		0			0
$679$	35574.6			2.01064
$680$		0			0
$681$	−17685.4	−	3076.10i	−0.995161	−	0.173093i
$682$		0			0
$683$		−	11097.0i		−	0.621689i	−0.950461	−	0.310844i	$-0.899388\pi$
$683$		−	11097.0i		−	0.621689i	0.950461	−	0.310844i	$-0.100612\pi$
$684$		0			0
$685$	3972.96			0.221604
$686$		0			0
$687$	848.104	−	4875.99i	0.0470992	−	0.270787i
$688$		0			0
$689$	3187.86			0.176267
$690$		0			0
$691$	23414.6			1.28905			0.644525	−	0.764583i	$-0.277055\pi$
$691$	23414.6			1.28905			0.644525	+	0.764583i	$0.277055\pi$
$692$		0			0
$693$	−17595.9	+	49051.7i	−0.964521	+	2.68877i
$694$		0			0
$695$	30715.4			1.67641
$696$		0			0
$697$			25632.3i			1.39296i
$698$		0			0
$699$	4276.97	−	24589.5i	0.231431	−	1.33056i
$700$		0			0
$701$	25705.7			1.38501			0.692503	−	0.721415i	$-0.256508\pi$
$701$	25705.7			1.38501			0.692503	+	0.721415i	$0.256508\pi$
$702$		0			0
$703$	−10329.5			−0.554172
$704$		0			0
$705$	−3854.50	+	22160.6i	−0.205913	+	1.18386i
$706$		0			0
$707$	39068.6			2.07826
$708$		0			0
$709$			24217.6i			1.28281i	0.767204	+	0.641403i	$0.221648\pi$
$709$			24217.6i			1.28281i	−0.767204	+	0.641403i	$0.778352\pi$
$710$		0			0
$711$	−8873.33	+	24736.0i	−0.468039	+	1.30474i
$712$		0			0
$713$	−135.060	−	594.322i	−0.00709400	−	0.0312167i
$714$		0			0
$715$	58909.5			3.08125
$716$		0			0
$717$	−4086.84	+	23496.4i	−0.212867	+	1.22384i
$718$		0			0
$719$		−	12407.4i		−	0.643557i	−0.946815	−	0.321778i	$-0.895719\pi$
$719$		−	12407.4i		−	0.643557i	0.946815	−	0.321778i	$-0.104281\pi$
$720$		0			0
$721$	55349.1			2.85896
$722$		0			0
$723$	−1387.42	+	7976.69i	−0.0713676	+	0.410313i
$724$		0			0
$725$			9668.94i			0.495304i
$726$		0			0
$727$			30245.8i			1.54299i	0.636236	+	0.771494i	$0.280490\pi$
$727$			30245.8i			1.54299i	−0.636236	+	0.771494i	$0.719510\pi$
$728$		0			0
$729$	10077.9	+	16907.3i	0.512012	+	0.858978i
$730$		0			0
$731$		−	15758.3i		−	0.797321i
$732$		0			0
$733$			7723.66i			0.389195i	0.980883	+	0.194597i	$0.0623401\pi$
$733$			7723.66i			0.389195i	−0.980883	+	0.194597i	$0.937660\pi$
$734$		0			0
$735$	−78659.6	−	13681.6i	−3.94749	−	0.686604i
$736$		0			0
$737$		−	30492.0i		−	1.52400i
$738$		0			0
$739$	14577.4			0.725628			0.362814	−	0.931862i	$-0.381816\pi$
$739$	14577.4			0.725628			0.362814	+	0.931862i	$0.381816\pi$
$740$		0			0
$741$	−3232.12	+	18582.4i	−0.160236	+	0.921243i
$742$		0			0
$743$	20240.5			0.999395			0.499697	−	0.866200i	$-0.333445\pi$
$743$	20240.5			0.999395			0.499697	+	0.866200i	$0.333445\pi$
$744$		0			0
$745$	−13290.7			−0.653601
$746$		0			0
$747$	−6210.04	−	2227.68i	−0.304168	−	0.109112i
$748$		0			0
$749$			5406.47i			0.263749i
$750$		0			0
$751$		−	18003.6i		−	0.874779i	−0.899272	−	0.437390i	$-0.855903\pi$
$751$		−	18003.6i		−	0.874779i	0.899272	−	0.437390i	$-0.144097\pi$
$752$		0			0
$753$	−2079.47	−	361.692i	−0.100638	−	0.0175043i
$754$		0			0
$755$	−64851.2			−3.12606
$756$		0			0
$757$			11315.8i			0.543301i	0.962396	+	0.271650i	$0.0875695\pi$
$757$			11315.8i			0.543301i	−0.962396	+	0.271650i	$0.912431\pi$
$758$		0			0
$759$	−1701.34	−	33172.4i	−0.0813633	−	1.58641i
$760$		0			0
$761$			10.6496i			0.000507289i	1.00000		0.000253645i	$8.07376e-5\pi$
$761$			10.6496i			0.000507289i	−1.00000		0.000253645i	$0.999919\pi$
$762$		0			0
$763$	−48222.8			−2.28805
$764$		0			0
$765$	−30366.7	−	10893.2i	−1.43518	−	0.514830i
$766$		0			0
$767$		−	7672.81i		−	0.361212i
$768$		0			0
$769$		−	24642.3i		−	1.15556i	−0.816194	−	0.577778i	$-0.803920\pi$
$769$		−	24642.3i		−	1.15556i	0.816194	−	0.577778i	$-0.196080\pi$
$770$		0			0
$771$	4467.74	−	25686.3i	0.208692	−	1.19983i
$772$		0			0
$773$	7708.00			0.358651			0.179326	−	0.983790i	$-0.442608\pi$
$773$	7708.00			0.358651			0.179326	+	0.983790i	$0.442608\pi$
$774$		0			0
$775$	−1531.52			−0.0709857
$776$		0			0
$777$	24592.3	+	4277.45i	1.13545	+	0.197494i
$778$		0			0
$779$	−30808.2			−1.41697
$780$		0			0
$781$		−	12864.3i		−	0.589397i
$782$		0			0
$783$	2417.43	−	4255.28i	0.110335	−	0.194216i
$784$		0			0
$785$			3317.57i			0.150840i
$786$		0			0
$787$		−	22970.2i		−	1.04040i	−0.854043	−	0.520202i	$-0.825857\pi$
$787$		−	22970.2i		−	1.04040i	0.854043	−	0.520202i	$-0.174143\pi$
$788$		0			0
$789$	−2736.53	−	475.977i	−0.123477	−	0.0214769i
$790$		0			0
$791$		−	10156.5i		−	0.456539i
$792$		0			0
$793$		−	27872.0i		−	1.24813i
$794$		0			0
$795$	−6456.78	−	1123.06i	−0.288048	−	0.0501016i
$796$		0			0
$797$	−17743.4			−0.788588			−0.394294	−	0.918984i	$-0.629011\pi$
$797$	−17743.4			−0.788588			−0.394294	+	0.918984i	$0.629011\pi$
$798$		0			0
$799$			12861.0i			0.569447i
$800$		0			0
$801$	24571.7	+	8814.42i	1.08389	+	0.388817i
$802$		0			0
$803$	−46569.4			−2.04658
$804$		0			0
$805$	71840.4	−	16325.7i	3.14539	−	0.714789i
$806$		0			0
$807$	767.744	−	4413.98i	0.0334893	−	0.192540i
$808$		0			0
$809$		−	22418.5i		−	0.974281i	−0.873324	−	0.487141i	$-0.838040\pi$
$809$		−	22418.5i		−	0.974281i	0.873324	−	0.487141i	$-0.161960\pi$
$810$		0			0
$811$	18917.8			0.819105			0.409553	−	0.912286i	$-0.365685\pi$
$811$	18917.8			0.819105			0.409553	+	0.912286i	$0.365685\pi$
$812$		0			0
$813$	7341.56	+	1276.95i	0.316703	+	0.0550857i
$814$		0			0
$815$	23483.9			1.00933
$816$		0			0
$817$	18940.4			0.811064
$818$		0			0
$819$	15390.0	−	42902.4i	0.656619	−	1.83044i
$820$		0			0
$821$			39044.4i			1.65975i	0.557946	+	0.829877i	$0.311589\pi$
$821$			39044.4i			1.65975i	−0.557946	+	0.829877i	$0.688411\pi$
$822$		0			0
$823$	14917.5			0.631826			0.315913	−	0.948788i	$-0.397689\pi$
$823$	14917.5			0.631826			0.315913	+	0.948788i	$0.397689\pi$
$824$		0			0
$825$	−82232.5	−	14303.1i	−3.47026	−	0.603599i
$826$		0			0
$827$	42665.6			1.79399			0.896995	−	0.442042i	$-0.145746\pi$
$827$	42665.6			1.79399			0.896995	+	0.442042i	$0.145746\pi$
$828$		0			0
$829$	−171.879			−0.00720097			−0.00360049	−	0.999994i	$-0.501146\pi$
$829$	−171.879			−0.00720097			−0.00360049	+	0.999994i	$0.501146\pi$
$830$		0			0
$831$	−1041.88	−	181.219i	−0.0434928	−	0.00756490i
$832$		0			0
$833$	−45650.2			−1.89878
$834$		0			0
$835$			55112.8i			2.28414i
$836$		0			0
$837$	674.020	+	382.912i	0.0278346	+	0.0158129i
$838$		0			0
$839$	−2047.36			−0.0842465			−0.0421232	−	0.999112i	$-0.513412\pi$
$839$	−2047.36			−0.0842465			−0.0421232	+	0.999112i	$0.513412\pi$
$840$		0			0
$841$	23172.1			0.950106
$842$		0			0
$843$	9250.67	+	1609.01i	0.377948	+	0.0657382i
$844$		0			0
$845$	−7465.00			−0.303910
$846$		0			0
$847$			67525.4i			2.73931i
$848$		0			0
$849$	3694.82	−	21242.6i	0.149359	−	0.858708i
$850$		0			0
$851$	−15514.8	+	3525.73i	−0.624959	+	0.142022i
$852$		0			0
$853$	−25416.9			−1.02023			−0.510116	−	0.860106i	$-0.670397\pi$
$853$	−25416.9			−1.02023			−0.510116	+	0.860106i	$0.670397\pi$
$854$		0			0
$855$	13092.9	−	36498.7i	0.523704	−	1.45992i
$856$		0			0
$857$			21389.0i			0.852550i	0.904594	+	0.426275i	$0.140174\pi$
$857$			21389.0i			0.852550i	−0.904594	+	0.426275i	$0.859826\pi$
$858$		0			0
$859$	22121.7			0.878678			0.439339	−	0.898321i	$-0.355213\pi$
$859$	22121.7			0.878678			0.439339	+	0.898321i	$0.355213\pi$
$860$		0			0
$861$	73347.9	+	12757.7i	2.90324	+	0.504974i
$862$		0			0
$863$			15118.3i			0.596330i	0.954514	+	0.298165i	$0.0963745\pi$
$863$			15118.3i			0.596330i	−0.954514	+	0.298165i	$0.903626\pi$
$864$		0			0
$865$		−	3216.60i		−	0.126436i
$866$		0			0
$867$	6977.90	+	1213.70i	0.273335	+	0.0475425i
$868$		0			0
$869$			56405.9i			2.20189i
$870$		0			0
$871$			26669.4i			1.03750i
$872$		0			0
$873$	9738.10	−	27146.7i	0.377531	−	1.05243i
$874$		0			0
$875$		−	101640.i		−	3.92692i
$876$		0			0
$877$	5834.48			0.224648			0.112324	−	0.993672i	$-0.464171\pi$
$877$	5834.48			0.224648			0.112324	+	0.993672i	$0.464171\pi$
$878$		0			0
$879$	50342.8	+	8756.36i	1.93176	+	0.336001i
$880$		0			0
$881$	−41979.2			−1.60535			−0.802677	−	0.596414i	$-0.796592\pi$
$881$	−41979.2			−1.60535			−0.802677	+	0.596414i	$0.796592\pi$
$882$		0			0
$883$	−12504.3			−0.476560			−0.238280	−	0.971197i	$-0.576584\pi$
$883$	−12504.3			−0.476560			−0.238280	+	0.971197i	$0.576584\pi$
$884$		0			0
$885$	−2703.07	+	15540.7i	−0.102670	+	0.590278i
$886$		0			0
$887$			19431.7i			0.735572i	0.929911	+	0.367786i	$0.119884\pi$
$887$			19431.7i			0.735572i	−0.929911	+	0.367786i	$0.880116\pi$
$888$		0			0
$889$		−	90378.3i		−	3.40966i
$890$		0			0
$891$	32614.2	+	26854.6i	1.22628	+	1.00972i
$892$		0			0
$893$	−15458.0			−0.579263
$894$		0			0
$895$			26556.2i			0.991815i
$896$		0			0
$897$	1488.06	+	29013.8i	0.0553899	+	1.07998i
$898$		0			0
$899$		−	192.745i		−	0.00715063i
$900$		0			0
$901$	−3747.20			−0.138554
$902$		0			0
$903$	−45093.1	−	7843.25i	−1.66180	−	0.289044i
$904$		0			0
$905$		−	59464.9i		−	2.18418i
$906$		0			0
$907$			31295.0i			1.14568i	0.819666	+	0.572841i	$0.194159\pi$
$907$			31295.0i			1.14568i	−0.819666	+	0.572841i	$0.805841\pi$
$908$		0			0
$909$	10694.5	−	29812.9i	0.390226	−	1.08782i
$910$		0			0
$911$	26675.6			0.970144			0.485072	−	0.874474i	$-0.338793\pi$
$911$	26675.6			0.970144			0.485072	+	0.874474i	$0.338793\pi$
$912$		0			0
$913$	−14160.9			−0.513316
$914$		0			0
$915$	−9819.09	+	56452.8i	−0.354764	+	2.03964i
$916$		0			0
$917$	9711.04			0.349713
$918$		0			0
$919$			8871.58i			0.318440i	0.987243	+	0.159220i	$0.0508979\pi$
$919$			8871.58i			0.318440i	−0.987243	+	0.159220i	$0.949102\pi$
$920$		0			0
$921$	−33363.6	−	5803.09i	−1.19367	−	0.207620i
$922$		0			0
$923$			11251.6i			0.401245i
$924$		0			0
$925$			39980.4i			1.42113i
$926$		0			0
$927$	15151.1	−	42236.4i	0.536816	−	1.49647i
$928$		0			0
$929$			37634.8i			1.32912i	0.747233	+	0.664562i	$0.231382\pi$
$929$			37634.8i			1.32912i	−0.747233	+	0.664562i	$0.768618\pi$
$930$		0			0
$931$		−	54868.3i		−	1.93151i
$932$		0			0
$933$	1178.80	−	6777.26i	0.0413635	−	0.237811i
$934$		0			0
$935$	−69245.9			−2.42201
$936$		0			0
$937$			27696.6i			0.965644i	0.875719	+	0.482822i	$0.160388\pi$
$937$			27696.6i			0.965644i	−0.875719	+	0.482822i	$0.839612\pi$
$938$		0			0
$939$	5137.81	−	29538.8i	0.178558	−	1.02658i
$940$		0			0
$941$	11405.8			0.395133			0.197566	−	0.980290i	$-0.436696\pi$
$941$	11405.8			0.395133			0.197566	+	0.980290i	$0.436696\pi$
$942$		0			0
$943$	−46273.7	+	10515.7i	−1.59796	+	0.363137i
$944$		0			0
$945$	−46285.6	+	81474.0i	−1.59330	+	2.80460i
$946$		0			0
$947$			10982.8i			0.376868i	0.982086	+	0.188434i	$0.0603412\pi$
$947$			10982.8i			0.376868i	−0.982086	+	0.188434i	$0.939659\pi$
$948$		0			0
$949$	40731.4			1.39325
$950$		0			0
$951$	−7007.96	+	40290.8i	−0.238958	+	1.37384i
$952$		0			0
$953$	150.697			0.00512229			0.00256115	−	0.999997i	$-0.499185\pi$
$953$	150.697			0.00512229			0.00256115	+	0.999997i	$0.499185\pi$
$954$		0			0
$955$	−84023.1			−2.84704
$956$		0			0
$957$	1800.07	−	10349.1i	0.0608025	−	0.349571i
$958$		0			0
$959$		−	6597.91i		−	0.222167i
$960$		0			0
$961$	−29760.5			−0.998975
$962$		0			0
$963$	4125.63	+	1479.95i	0.138055	+	0.0495232i
$964$		0			0
$965$	29946.8			0.998988
$966$		0			0
$967$	−35798.2			−1.19048			−0.595239	−	0.803549i	$-0.702943\pi$
$967$	−35798.2			−1.19048			−0.595239	+	0.803549i	$0.702943\pi$
$968$		0			0
$969$	3799.23	−	21842.9i	0.125954	−	0.724143i
$970$		0			0
$971$	49555.9			1.63782			0.818911	−	0.573920i	$-0.194578\pi$
$971$	49555.9			1.63782			0.818911	+	0.573920i	$0.194578\pi$
$972$		0			0
$973$		−	51009.2i		−	1.68066i
$974$		0			0
$975$	71923.6	+	12510.0i	2.36246	+	0.410914i
$976$		0			0
$977$	−42576.7			−1.39422			−0.697109	−	0.716965i	$-0.745531\pi$
$977$	−42576.7			−1.39422			−0.697109	+	0.716965i	$0.745531\pi$
$978$		0			0
$979$	56031.5			1.82919
$980$		0			0
$981$	−13200.4	+	36798.4i	−0.429619	+	1.19764i
$982$		0			0
$983$	33552.1			1.08865			0.544327	−	0.838873i	$-0.316785\pi$
$983$	33552.1			1.08865			0.544327	+	0.838873i	$0.316785\pi$
$984$		0			0
$985$			89918.5i			2.90867i
$986$		0			0
$987$	36802.3	+	6401.19i	1.18686	+	0.206436i
$988$		0			0
$989$	28448.3	−	6464.87i	0.914665	−	0.207857i
$990$		0			0
$991$	37162.8			1.19124			0.595618	−	0.803268i	$-0.296907\pi$
$991$	37162.8			1.19124			0.595618	+	0.803268i	$0.296907\pi$
$992$		0			0
$993$	14940.9	+	2598.74i	0.477478	+	0.0830500i
$994$		0			0
$995$		−	43978.9i		−	1.40123i
$996$		0			0
$997$	−48124.2			−1.52869			−0.764347	−	0.644805i	$-0.776939\pi$
$997$	−48124.2			−1.52869			−0.764347	+	0.644805i	$0.776939\pi$
$998$		0			0
$999$	9995.93	−	17595.3i	0.316574	−	0.557247i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.4.m.a.137.5	✓	72
3.2	odd	2	inner	552.4.m.a.137.8	yes	72
23.22	odd	2	inner	552.4.m.a.137.6	yes	72
69.68	even	2	inner	552.4.m.a.137.7	yes	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.4.m.a.137.5	✓	72	1.1	even	1	trivial
552.4.m.a.137.6	yes	72	23.22	odd	2	inner
552.4.m.a.137.7	yes	72	69.68	even	2	inner
552.4.m.a.137.8	yes	72	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:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	552.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.11929 - 0.890422i) q^{3} -20.0544 q^{5} +33.3044i q^{7} +(25.4143 + 9.11666i) q^{9} +57.9527 q^{11} -50.6877 q^{13} +(102.664 + 17.8569i) q^{15} +59.5814 q^{17} +71.6126i q^{19} +(29.6550 - 170.495i) q^{21} +(24.4434 + 107.562i) q^{23} +277.178 q^{25} +(-121.986 - 69.3003i) q^{27} +34.8835i q^{29} -5.52541 q^{31} +(-296.677 - 51.6024i) q^{33} -667.899i q^{35} +144.241i q^{37} +(259.485 + 45.1334i) q^{39} +430.206i q^{41} -264.484i q^{43} +(-509.668 - 182.829i) q^{45} +215.856i q^{47} -766.183 q^{49} +(-305.014 - 53.0526i) q^{51} -62.8922 q^{53} -1162.21 q^{55} +(63.7654 - 366.606i) q^{57} +151.374i q^{59} +549.877i q^{61} +(-303.625 + 846.408i) q^{63} +1016.51 q^{65} -526.152i q^{67} +(-29.3574 - 572.405i) q^{69} -221.978i q^{71} -803.576 q^{73} +(-1418.96 - 246.806i) q^{75} +1930.08i q^{77} +973.309i q^{79} +(562.773 + 463.387i) q^{81} -244.352 q^{83} -1194.87 q^{85} +(31.0610 - 178.579i) q^{87} +966.847 q^{89} -1688.12i q^{91} +(28.2862 + 4.91994i) q^{93} -1436.15i q^{95} -1068.16i q^{97} +(1472.83 + 528.336i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 28 q^{9} + 1872 q^{25} - 444 q^{27} - 216 q^{31} - 68 q^{39} - 4200 q^{49} - 576 q^{55} + 1376 q^{69} - 384 q^{73} - 3592 q^{75} - 252 q^{81} + 3480 q^{85} - 412 q^{87} + 780 q^{93}+O(q^{100}) \) 72 * q + 28 * q^9 + 1872 * q^25 - 444 * q^27 - 216 * q^31 - 68 * q^39 - 4200 * q^49 - 576 * q^55 + 1376 * q^69 - 384 * q^73 - 3592 * q^75 - 252 * q^81 + 3480 * q^85 - 412 * q^87 + 780 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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