LMFDB - Embedded newform 756.4.t.d.269.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,4,Mod(269,756)] 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("756.269"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.t (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} - 214 x^{14} + 30952 x^{12} - 2415192 x^{10} + 136176800 x^{8} - 4497757024 x^{6} + \cdots + 7263930548224 $ x^16 - 214x^14 + 30952x^12 - 2415192x^10 + 136176800x^8 - 4497757024x^6 + 104934553152x^4 - 1026082799616*x^2 + 7263930548224
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.7
Root			$6.73954 + 3.89107i$ of defining polynomial
Character	$\chi$	$=$	756.269
Dual form			756.4.t.d.593.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.73954 + 11.6732i) q^{5} +(9.49487 + 15.9012i) q^{7} +(-3.36020 - 1.94001i) q^{11} -22.9512i q^{13} +(57.1882 - 99.0529i) q^{17} +(112.404 - 64.8964i) q^{19} +(145.313 - 83.8966i) q^{23} +(-28.3427 + 49.0910i) q^{25} +111.992i q^{29} +(13.4452 + 7.76260i) q^{31} +(-121.627 + 218.002i) q^{35} +(166.902 + 289.082i) q^{37} +289.289 q^{41} -38.3502 q^{43} +(-137.463 - 238.094i) q^{47} +(-162.695 + 301.959i) q^{49} +(-542.435 - 313.175i) q^{53} -52.2991i q^{55} +(-232.366 + 402.470i) q^{59} +(142.015 - 81.9922i) q^{61} +(267.915 - 154.681i) q^{65} +(-11.3096 + 19.5888i) q^{67} +740.773i q^{71} +(846.739 + 488.865i) q^{73} +(-1.05618 - 71.8512i) q^{77} +(463.725 + 803.195i) q^{79} -1152.68 q^{83} +1541.69 q^{85} +(-554.282 - 960.045i) q^{89} +(364.951 - 217.919i) q^{91} +(1515.10 + 874.743i) q^{95} +1076.97i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 50 q^{7} - 390 q^{19} - 284 q^{25} + 984 q^{31} - 182 q^{37} + 472 q^{43} + 130 q^{49} - 24 q^{61} - 1522 q^{67} + 4458 q^{73} + 434 q^{79} - 2616 q^{85} - 4560 q^{91}+O(q^{100}) $ 16 * q + 50 * q^7 - 390 * q^19 - 284 * q^25 + 984 * q^31 - 182 * q^37 + 472 * q^43 + 130 * q^49 - 24 * q^61 - 1522 * q^67 + 4458 * q^73 + 434 * q^79 - 2616 * q^85 - 4560 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	6.73954	+	11.6732i	0.602802	+	1.04408i	0.992395	+	0.123097i	$0.0392826\pi$
$5$	6.73954	+	11.6732i	0.602802	+	1.04408i	−0.389592	+	0.920987i	$0.627384\pi$
$6$		0			0
$7$	9.49487	+	15.9012i	0.512675	+	0.858583i
$7$	9.49487	+	15.9012i	0.512675	+	0.858583i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.36020	−	1.94001i	−0.0921034	−	0.0531759i	0.453241	−	0.891388i	$-0.350268\pi$
$11$	−3.36020	−	1.94001i	−0.0921034	−	0.0531759i	−0.545344	+	0.838212i	$0.683601\pi$
$12$		0			0
$13$		−	22.9512i		−	0.489656i	−0.969567	−	0.244828i	$-0.921269\pi$
$13$		−	22.9512i		−	0.489656i	0.969567	−	0.244828i	$-0.0787314\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	57.1882	−	99.0529i	0.815893	−	1.41317i	−0.0927924	−	0.995685i	$-0.529579\pi$
$17$	57.1882	−	99.0529i	0.815893	−	1.41317i	0.908685	−	0.417482i	$-0.137087\pi$
$18$		0			0
$19$	112.404	−	64.8964i	1.35722	−	0.783592i	0.367973	−	0.929836i	$-0.380052\pi$
$19$	112.404	−	64.8964i	1.35722	−	0.783592i	0.989249	+	0.146244i	$0.0467184\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	145.313	−	83.8966i	1.31739	−	0.760593i	0.334079	−	0.942545i	$-0.391575\pi$
$23$	145.313	−	83.8966i	1.31739	−	0.760593i	0.983307	+	0.181952i	$0.0582415\pi$
$24$		0			0
$25$	−28.3427	+	49.0910i	−0.226742	+	0.392728i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			111.992i			0.717117i	0.933507	+	0.358558i	$0.116732\pi$
$29$			111.992i			0.717117i	−0.933507	+	0.358558i	$0.883268\pi$
$30$		0			0
$31$	13.4452	+	7.76260i	0.0778979	+	0.0449743i	0.538443	−	0.842662i	$-0.319013\pi$
$31$	13.4452	+	7.76260i	0.0778979	+	0.0449743i	−0.460545	+	0.887636i	$0.652346\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−121.627	+	218.002i	−0.587391	+	1.05283i
$36$		0			0
$37$	166.902	+	289.082i	0.741580	+	1.28445i	0.951776	+	0.306795i	$0.0992565\pi$
$37$	166.902	+	289.082i	0.741580	+	1.28445i	−0.210196	+	0.977659i	$0.567410\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	289.289			1.10194			0.550969	−	0.834526i	$-0.314258\pi$
$41$	289.289			1.10194			0.550969	+	0.834526i	$0.314258\pi$
$42$		0			0
$43$	−38.3502			−0.136008			−0.0680041	−	0.997685i	$-0.521663\pi$
$43$	−38.3502			−0.136008			−0.0680041	+	0.997685i	$0.521663\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−137.463	−	238.094i	−0.426619	−	0.738926i	0.569951	−	0.821679i	$-0.306962\pi$
$47$	−137.463	−	238.094i	−0.426619	−	0.738926i	−0.996570	+	0.0827528i	$0.973629\pi$
$48$		0			0
$49$	−162.695	+	301.959i	−0.474329	+	0.880348i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−542.435	−	313.175i	−1.40583	−	0.811659i	−0.410851	−	0.911703i	$-0.634768\pi$
$53$	−542.435	−	313.175i	−1.40583	−	0.811659i	−0.994983	+	0.100044i	$0.968102\pi$
$54$		0			0
$55$		−	52.2991i		−	0.128218i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−232.366	+	402.470i	−0.512737	+	0.888087i	0.487154	+	0.873316i	$0.338035\pi$
$59$	−232.366	+	402.470i	−0.512737	+	0.888087i	−0.999891	+	0.0147706i	$0.995298\pi$
$60$		0			0
$61$	142.015	−	81.9922i	0.298084	−	0.172099i	−0.343498	−	0.939153i	$-0.611612\pi$
$61$	142.015	−	81.9922i	0.298084	−	0.172099i	0.641582	+	0.767055i	$0.278279\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	267.915	−	154.681i	0.511242	−	0.295166i
$66$		0			0
$67$	−11.3096	+	19.5888i	−0.0206222	+	0.0357186i	−0.876152	−	0.482034i	$-0.839898\pi$
$67$	−11.3096	+	19.5888i	−0.0206222	+	0.0357186i	0.855530	+	0.517753i	$0.173231\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			740.773i			1.23822i	0.785304	+	0.619110i	$0.212507\pi$
$71$			740.773i			1.23822i	−0.785304	+	0.619110i	$0.787493\pi$
$72$		0			0
$73$	846.739	+	488.865i	1.35758	+	0.783799i	0.989297	−	0.145916i	$-0.0466128\pi$
$73$	846.739	+	488.865i	1.35758	+	0.783799i	0.368282	+	0.929714i	$0.379946\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.05618	−	71.8512i	−0.00156316	−	0.106340i
$78$		0			0
$79$	463.725	+	803.195i	0.660419	+	1.14388i	0.980506	+	0.196491i	$0.0629548\pi$
$79$	463.725	+	803.195i	0.660419	+	1.14388i	−0.320086	+	0.947388i	$0.603712\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1152.68			−1.52438			−0.762188	−	0.647356i	$-0.775875\pi$
$83$	−1152.68			−1.52438			−0.762188	+	0.647356i	$0.775875\pi$
$84$		0			0
$85$	1541.69			1.96729
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−554.282	−	960.045i	−0.660155	−	1.14342i	−0.980575	−	0.196146i	$-0.937157\pi$
$89$	−554.282	−	960.045i	−0.660155	−	1.14342i	0.320420	−	0.947276i	$-0.396176\pi$
$90$		0			0
$91$	364.951	−	217.919i	0.420410	−	0.251034i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1515.10	+	874.743i	1.63627	+	0.944703i
$96$		0			0
$97$			1076.97i			1.12732i	0.826007	+	0.563659i	$0.190607\pi$
$97$			1076.97i			1.12732i	−0.826007	+	0.563659i	$0.809393\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−417.629	+	723.355i	−0.411442	+	0.712638i	−0.995048	−	0.0993989i	$-0.968308\pi$
$101$	−417.629	+	723.355i	−0.411442	+	0.712638i	0.583606	+	0.812037i	$0.301641\pi$
$102$		0			0
$103$	488.227	−	281.878i	0.467053	−	0.269653i	−0.247952	−	0.968772i	$-0.579758\pi$
$103$	488.227	−	281.878i	0.467053	−	0.269653i	0.715005	+	0.699119i	$0.246424\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	112.506	−	64.9552i	0.101648	−	0.0586865i	−0.448314	−	0.893876i	$-0.647975\pi$
$107$	112.506	−	64.9552i	0.101648	−	0.0586865i	0.549962	+	0.835190i	$0.314642\pi$
$108$		0			0
$109$	39.5821	−	68.5582i	0.0347824	−	0.0602449i	−0.848110	−	0.529820i	$-0.822260\pi$
$109$	39.5821	−	68.5582i	0.0347824	−	0.0602449i	0.882893	+	0.469575i	$0.155593\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1143.03i		−	0.951571i	−0.879561	−	0.475786i	$-0.842164\pi$
$113$		−	1143.03i		−	0.951571i	0.879561	−	0.475786i	$-0.157836\pi$
$114$		0			0
$115$	1958.69	+	1130.85i	1.58825	+	0.916975i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2118.05	−	31.1345i	1.63161	−	0.0239840i
$120$		0			0
$121$	−657.973	−	1139.64i	−0.494345	−	0.856230i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	920.818			0.658884
$126$		0			0
$127$	−435.975			−0.304618			−0.152309	−	0.988333i	$-0.548671\pi$
$127$	−435.975			−0.304618			−0.152309	+	0.988333i	$0.548671\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−505.534	−	875.611i	−0.337166	−	0.583989i	0.646733	−	0.762717i	$-0.276135\pi$
$131$	−505.534	−	875.611i	−0.337166	−	0.583989i	−0.983898	+	0.178728i	$0.942802\pi$
$132$		0			0
$133$	2099.19	+	1171.17i	1.36859	+	0.763559i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−889.720	−	513.680i	−0.554846	−	0.320340i	0.196228	−	0.980558i	$-0.437131\pi$
$137$	−889.720	−	513.680i	−0.554846	−	0.320340i	−0.751074	+	0.660218i	$0.770464\pi$
$138$		0			0
$139$			2093.41i			1.27741i	0.769450	+	0.638707i	$0.220530\pi$
$139$			2093.41i			1.27741i	−0.769450	+	0.638707i	$0.779470\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−44.5256	+	77.1206i	−0.0260379	+	0.0450989i
$144$		0			0
$145$	−1307.31	+	754.774i	−0.748730	+	0.432280i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2188.48	−	1263.52i	1.20327	−	0.694709i	0.241991	−	0.970279i	$-0.422200\pi$
$149$	2188.48	−	1263.52i	1.20327	−	0.694709i	0.961281	+	0.275569i	$0.0888664\pi$
$150$		0			0
$151$	−631.489	+	1093.77i	−0.340330	+	0.589469i	−0.984494	−	0.175419i	$-0.943872\pi$
$151$	−631.489	+	1093.77i	−0.340330	+	0.589469i	0.644164	+	0.764887i	$0.277205\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			209.265i			0.108443i
$156$		0			0
$157$	866.760	+	500.424i	0.440605	+	0.254384i	0.703854	−	0.710344i	$-0.251461\pi$
$157$	866.760	+	500.424i	0.440605	+	0.254384i	−0.263249	+	0.964728i	$0.584794\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2713.78	+	1514.06i	1.32842	+	0.741148i
$162$		0			0
$163$	699.008	+	1210.72i	0.335893	+	0.581784i	0.983656	−	0.180058i	$-0.0576286\pi$
$163$	699.008	+	1210.72i	0.335893	+	0.581784i	−0.647763	+	0.761842i	$0.724295\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4131.29			−1.91430			−0.957152	−	0.289586i	$-0.906482\pi$
$167$	−4131.29			−1.91430			−0.957152	+	0.289586i	$0.906482\pi$
$168$		0			0
$169$	1670.24			0.760237
$170$		0			0
$171$		0			0
$172$		0			0
$173$	376.529	+	652.167i	0.165474	+	0.286609i	0.936823	−	0.349803i	$-0.113751\pi$
$173$	376.529	+	652.167i	0.165474	+	0.286609i	−0.771350	+	0.636412i	$0.780418\pi$
$174$		0			0
$175$	−1049.71	+	15.4304i	−0.453434	+	0.00666529i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1468.90	−	848.071i	−0.613357	−	0.354122i	0.160921	−	0.986967i	$-0.448553\pi$
$179$	−1468.90	−	848.071i	−0.613357	−	0.354122i	−0.774278	+	0.632846i	$0.781887\pi$
$180$		0			0
$181$		−	2704.01i		−	1.11043i	−0.831708	−	0.555213i	$-0.812637\pi$
$181$		−	2704.01i		−	1.11043i	0.831708	−	0.555213i	$-0.187363\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2249.68	+	3896.56i	−0.894052	+	1.54854i
$186$		0			0
$187$	−384.327	+	221.891i	−0.150293	+	0.0867717i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2561.33	−	1478.78i	0.970320	−	0.560215i	0.0709865	−	0.997477i	$-0.477385\pi$
$191$	2561.33	−	1478.78i	0.970320	−	0.560215i	0.899334	+	0.437263i	$0.144052\pi$
$192$		0			0
$193$	2209.84	−	3827.55i	0.824184	−	1.42753i	−0.0783576	−	0.996925i	$-0.524968\pi$
$193$	2209.84	−	3827.55i	0.824184	−	1.42753i	0.902541	−	0.430603i	$-0.141699\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	136.187i		−	0.0492534i	−0.999697	−	0.0246267i	$-0.992160\pi$
$197$		−	136.187i		−	0.0492534i	0.999697	−	0.0246267i	$-0.00783971\pi$
$198$		0			0
$199$	−881.994	−	509.219i	−0.314185	−	0.181395i	0.334612	−	0.942356i	$-0.391395\pi$
$199$	−881.994	−	509.219i	−0.314185	−	0.181395i	−0.648798	+	0.760961i	$0.724728\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1780.80	+	1063.35i	−0.615704	+	0.367648i
$204$		0			0
$205$	1949.68	+	3376.94i	0.664250	+	1.15052i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−503.599			−0.166673
$210$		0			0
$211$	−4421.97			−1.44275			−0.721377	−	0.692543i	$-0.756490\pi$
$211$	−4421.97			−1.44275			−0.721377	+	0.692543i	$0.756490\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−258.463	−	447.671i	−0.0819861	−	0.142004i
$216$		0			0
$217$	4.22613	+	287.500i	0.00132207	+	0.0899390i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2273.38	−	1312.54i	−0.691965	−	0.399506i
$222$		0			0
$223$			2676.81i			0.803823i	0.915679	+	0.401911i	$0.131654\pi$
$223$			2676.81i			0.803823i	−0.915679	+	0.401911i	$0.868346\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	917.491	−	1589.14i	0.268264	−	0.464648i	−0.700149	−	0.713997i	$-0.746883\pi$
$227$	917.491	−	1589.14i	0.268264	−	0.464648i	0.968414	+	0.249349i	$0.0802166\pi$
$228$		0			0
$229$	2248.39	−	1298.11i	0.648813	−	0.374592i	−0.139189	−	0.990266i	$-0.544449\pi$
$229$	2248.39	−	1298.11i	0.648813	−	0.374592i	0.788001	+	0.615674i	$0.211116\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−700.293	+	404.314i	−0.196900	+	0.113680i	−0.595209	−	0.803571i	$-0.702931\pi$
$233$	−700.293	+	404.314i	−0.196900	+	0.113680i	0.398309	+	0.917251i	$0.369597\pi$
$234$		0			0
$235$	1852.88	−	3209.28i	0.514334	−	0.890853i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1514.16i			0.409803i	0.978783	+	0.204901i	$0.0656874\pi$
$239$			1514.16i			0.409803i	−0.978783	+	0.204901i	$0.934313\pi$
$240$		0			0
$241$	2377.48	+	1372.64i	0.635465	+	0.366886i	0.782866	−	0.622191i	$-0.213757\pi$
$241$	2377.48	+	1372.64i	0.635465	+	0.366886i	−0.147400	+	0.989077i	$0.547091\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4621.32	+	135.892i	−1.20508	+	0.0354361i
$246$		0			0
$247$	−1489.45	−	2579.81i	−0.383690	−	0.664571i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2572.17			0.646829			0.323415	−	0.946257i	$-0.395169\pi$
$251$	2572.17			0.646829			0.323415	+	0.946257i	$0.395169\pi$
$252$		0			0
$253$	−651.041			−0.161781
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2593.23	−	4491.60i	−0.629420	−	1.09019i	−0.987668	−	0.156561i	$-0.949959\pi$
$257$	−2593.23	−	4491.60i	−0.629420	−	1.09019i	0.358248	−	0.933626i	$-0.383374\pi$
$258$		0			0
$259$	−3012.04	+	5398.73i	−0.722621	+	1.29522i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2388.44	+	1378.97i	0.559990	+	0.323310i	0.753142	−	0.657859i	$-0.228538\pi$
$263$	2388.44	+	1378.97i	0.559990	+	0.323310i	−0.193151	+	0.981169i	$0.561871\pi$
$264$		0			0
$265$		−	8442.62i		−	1.95708i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−624.369	+	1081.44i	−0.141518	+	0.245117i	−0.928069	−	0.372409i	$-0.878532\pi$
$269$	−624.369	+	1081.44i	−0.141518	+	0.245117i	0.786550	+	0.617526i	$0.211865\pi$
$270$		0			0
$271$	−304.361	+	175.723i	−0.0682237	+	0.0393890i	−0.533724	−	0.845659i	$-0.679208\pi$
$271$	−304.361	+	175.723i	−0.0682237	+	0.0393890i	0.465500	+	0.885048i	$0.345874\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	190.474	−	109.970i	0.0417673	−	0.0241144i
$276$		0			0
$277$	−942.774	+	1632.93i	−0.204498	+	0.354200i	−0.949973	−	0.312333i	$-0.898889\pi$
$277$	−942.774	+	1632.93i	−0.204498	+	0.354200i	0.745475	+	0.666534i	$0.232223\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			8690.49i			1.84495i	0.386057	+	0.922475i	$0.373837\pi$
$281$			8690.49i			1.84495i	−0.386057	+	0.922475i	$0.626163\pi$
$282$		0			0
$283$	−7757.78	−	4478.95i	−1.62951	−	0.940799i	−0.984238	−	0.176848i	$-0.943410\pi$
$283$	−7757.78	−	4478.95i	−1.62951	−	0.940799i	−0.645274	−	0.763951i	$-0.723257\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2746.77	+	4600.04i	0.564935	+	0.946104i
$288$		0			0
$289$	−4084.48	−	7074.53i	−0.831362	−	1.43996i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7151.22			1.42587			0.712933	−	0.701232i	$-0.247366\pi$
$293$	7151.22			1.42587			0.712933	+	0.701232i	$0.247366\pi$
$294$		0			0
$295$	−6264.16			−1.23632
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1925.53	−	3335.11i	−0.372429	−	0.645065i
$300$		0			0
$301$	−364.131	−	609.814i	−0.0697280	−	0.116774i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1914.22	+	1105.18i	0.359371	+	0.207483i
$306$		0			0
$307$		−	263.121i		−	0.0489157i	−0.999701	−	0.0244578i	$-0.992214\pi$
$307$		−	263.121i		−	0.0489157i	0.999701	−	0.0244578i	$-0.00778595\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3283.90	−	5687.88i	0.598755	−	1.03707i	−0.394250	−	0.919003i	$-0.628996\pi$
$311$	3283.90	−	5687.88i	0.598755	−	1.03707i	0.993005	−	0.118071i	$-0.0376711\pi$
$312$		0			0
$313$	−1003.18	+	579.188i	−0.181161	+	0.104593i	−0.587838	−	0.808979i	$-0.700021\pi$
$313$	−1003.18	+	579.188i	−0.181161	+	0.104593i	0.406677	+	0.913572i	$0.366687\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1434.46	−	828.187i	0.254156	−	0.146737i	−0.367510	−	0.930020i	$-0.619790\pi$
$317$	1434.46	−	828.187i	0.254156	−	0.146737i	0.621666	+	0.783283i	$0.286456\pi$
$318$		0			0
$319$	217.266	−	376.315i	0.0381333	−	0.0660489i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	14845.2i		−	2.55731i
$324$		0			0
$325$	1126.70	+	650.499i	0.192301	+	0.111025i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2480.77	−	4446.50i	0.415712	−	0.745117i
$330$		0			0
$331$	−2701.35	−	4678.87i	−0.448578	−	0.776960i	0.549716	−	0.835352i	$-0.314736\pi$
$331$	−2701.35	−	4678.87i	−0.448578	−	0.776960i	−0.998294	+	0.0583918i	$0.981403\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−304.885			−0.0497243
$336$		0			0
$337$	−9033.79			−1.46024			−0.730122	−	0.683317i	$-0.760537\pi$
$337$	−9033.79			−1.46024			−0.730122	+	0.683317i	$0.760537\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−30.1191	−	52.1677i	−0.00478310	−	0.00828458i
$342$		0			0
$343$	−6346.27	+	280.024i	−0.999028	+	0.0440813i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1876.21	+	1083.23i	0.290260	+	0.167582i	0.638059	−	0.769987i	$-0.279737\pi$
$347$	1876.21	+	1083.23i	0.290260	+	0.167582i	−0.347799	+	0.937569i	$0.613071\pi$
$348$		0			0
$349$			5910.25i			0.906499i	0.891384	+	0.453250i	$0.149735\pi$
$349$			5910.25i			0.906499i	−0.891384	+	0.453250i	$0.850265\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1201.00	−	2080.20i	0.181085	−	0.313648i	−0.761165	−	0.648558i	$-0.775373\pi$
$353$	1201.00	−	2080.20i	0.181085	−	0.313648i	0.942250	+	0.334910i	$0.108706\pi$
$354$		0			0
$355$	−8647.21	+	4992.47i	−1.29281	+	0.746402i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7559.87	+	4364.69i	−1.11141	+	0.641670i	−0.939193	−	0.343390i	$-0.888425\pi$
$359$	−7559.87	+	4364.69i	−1.11141	+	0.641670i	−0.172213	+	0.985060i	$0.555092\pi$
$360$		0			0
$361$	4993.59	−	8649.15i	0.728034	−	1.26099i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			13178.9i			1.88990i
$366$		0			0
$367$	10060.7	+	5808.55i	1.43096	+	0.826168i	0.997194	−	0.0748556i	$-0.0238496\pi$
$367$	10060.7	+	5808.55i	1.43096	+	0.826168i	0.433770	+	0.901023i	$0.357183\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−170.499	−	11598.9i	−0.0238595	−	1.62314i
$372$		0			0
$373$	−4242.65	−	7348.48i	−0.588944	−	1.02008i	−0.994371	−	0.105953i	$-0.966211\pi$
$373$	−4242.65	−	7348.48i	−0.588944	−	1.02008i	0.405427	−	0.914127i	$-0.367123\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2570.35			0.351140
$378$		0			0
$379$	−7285.37			−0.987399			−0.493699	−	0.869633i	$-0.664356\pi$
$379$	−7285.37			−0.987399			−0.493699	+	0.869633i	$0.664356\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7291.06	+	12628.5i	0.972731	+	1.68482i	0.687229	+	0.726441i	$0.258827\pi$
$383$	7291.06	+	12628.5i	0.972731	+	1.68482i	0.285502	+	0.958378i	$0.407840\pi$
$384$		0			0
$385$	831.617	−	496.573i	0.110086	−	0.0657343i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1257.64	+	726.099i	0.163920	+	0.0946393i	0.579716	−	0.814819i	$-0.303164\pi$
$389$	1257.64	+	726.099i	0.163920	+	0.0946393i	−0.415796	+	0.909458i	$0.636497\pi$
$390$		0			0
$391$		−	19191.6i		−	2.48225i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6250.58	+	10826.3i	−0.796205	+	1.37907i
$396$		0			0
$397$	−1501.67	+	866.988i	−0.189840	+	0.109604i	−0.591908	−	0.806006i	$-0.701625\pi$
$397$	−1501.67	+	866.988i	−0.189840	+	0.109604i	0.402068	+	0.915610i	$0.368292\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3683.42	+	2126.62i	−0.458706	+	0.264834i	−0.711500	−	0.702686i	$-0.751984\pi$
$401$	−3683.42	+	2126.62i	−0.458706	+	0.264834i	0.252794	+	0.967520i	$0.418650\pi$
$402$		0			0
$403$	178.161	−	308.584i	0.0220219	−	0.0381431i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1295.16i		−	0.157737i
$408$		0			0
$409$	7407.36	+	4276.64i	0.895527	+	0.517033i	0.875746	−	0.482772i	$-0.160370\pi$
$409$	7407.36	+	4276.64i	0.895527	+	0.517033i	0.0197804	+	0.999804i	$0.493703\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8606.03	+	126.505i	−1.02536	+	0.0150724i
$414$		0			0
$415$	−7768.53	−	13455.5i	−0.918897	−	1.59158i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11821.3			−1.37830			−0.689151	−	0.724618i	$-0.742016\pi$
$419$	−11821.3			−1.37830			−0.689151	+	0.724618i	$0.742016\pi$
$420$		0			0
$421$	9584.58			1.10956			0.554779	−	0.831998i	$-0.312803\pi$
$421$	9584.58			1.10956			0.554779	+	0.831998i	$0.312803\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3241.73	+	5614.85i	0.369993	+	0.640848i
$426$		0			0
$427$	2652.18	+	1479.69i	0.300581	+	0.167699i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7118.04	−	4109.60i	−0.795508	−	0.459287i	0.0463903	−	0.998923i	$-0.485228\pi$
$431$	−7118.04	−	4109.60i	−0.795508	−	0.459287i	−0.841898	+	0.539637i	$0.818562\pi$
$432$		0			0
$433$		−	183.154i		−	0.0203276i	−0.999948	−	0.0101638i	$-0.996765\pi$
$433$		−	183.154i		−	0.0203276i	0.999948	−	0.0101638i	$-0.00323529\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10889.2	−	18860.6i	1.19199	−	2.06459i
$438$		0			0
$439$	−11916.7	+	6880.10i	−1.29556	+	0.747993i	−0.979634	−	0.200790i	$-0.935649\pi$
$439$	−11916.7	+	6880.10i	−1.29556	+	0.747993i	−0.315928	+	0.948783i	$0.602316\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5187.15	+	2994.80i	−0.556318	+	0.321191i	−0.751666	−	0.659543i	$-0.770750\pi$
$443$	−5187.15	+	2994.80i	−0.556318	+	0.321191i	0.195348	+	0.980734i	$0.437416\pi$
$444$		0			0
$445$	7471.21	−	12940.5i	0.795886	−	1.37852i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	16572.5i		−	1.74188i	−0.491385	−	0.870942i	$-0.663509\pi$
$449$		−	16572.5i		−	1.74188i	0.491385	−	0.870942i	$-0.336491\pi$
$450$		0			0
$451$	−972.069	−	561.225i	−0.101492	−	0.0585965i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5003.42	+	2791.49i	0.515525	+	0.287619i
$456$		0			0
$457$	5588.89	+	9680.24i	0.572073	+	0.990859i	0.996353	+	0.0853283i	$0.0271939\pi$
$457$	5588.89	+	9680.24i	0.572073	+	0.990859i	−0.424280	+	0.905531i	$0.639473\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12231.1			−1.23570			−0.617852	−	0.786295i	$-0.711997\pi$
$461$	−12231.1			−1.23570			−0.617852	+	0.786295i	$0.711997\pi$
$462$		0			0
$463$	−13290.3			−1.33402			−0.667011	−	0.745048i	$-0.732427\pi$
$463$	−13290.3			−1.33402			−0.667011	+	0.745048i	$0.732427\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6213.58	+	10762.2i	0.615697	+	1.06642i	0.990262	+	0.139217i	$0.0444586\pi$
$467$	6213.58	+	10762.2i	0.615697	+	1.06642i	−0.374565	+	0.927201i	$0.622208\pi$
$468$		0			0
$469$	−418.867	+	6.15717i	−0.0412399	+	0.000606209i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	128.864	+	74.3998i	0.0125268	+	0.00723236i
$474$		0			0
$475$			7357.35i			0.710692i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2655.35	+	4599.21i	−0.253291	+	0.438712i	−0.964430	−	0.264339i	$-0.914846\pi$
$479$	−2655.35	+	4599.21i	−0.253291	+	0.438712i	0.711139	+	0.703051i	$0.248180\pi$
$480$		0			0
$481$	6634.78	−	3830.59i	0.628940	−	0.363119i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12571.7	+	7258.29i	−1.17702	+	0.679551i
$486$		0			0
$487$	−1409.76	+	2441.78i	−0.131176	+	0.227203i	−0.924130	−	0.382078i	$-0.875209\pi$
$487$	−1409.76	+	2441.78i	−0.131176	+	0.227203i	0.792954	+	0.609281i	$0.208542\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	11622.3i		−	1.06824i	−0.845407	−	0.534122i	$-0.820642\pi$
$491$		−	11622.3i		−	1.06824i	0.845407	−	0.534122i	$-0.179358\pi$
$492$		0			0
$493$	11093.1	+	6404.62i	1.01341	+	0.585090i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11779.2	+	7033.55i	−1.06311	+	0.634804i
$498$		0			0
$499$	1565.67	+	2711.82i	0.140459	+	0.243282i	0.927670	−	0.373402i	$-0.121809\pi$
$499$	1565.67	+	2711.82i	0.140459	+	0.243282i	−0.787211	+	0.616684i	$0.788476\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	18250.0			1.61774			0.808872	−	0.587984i	$-0.200078\pi$
$503$	18250.0			1.61774			0.808872	+	0.587984i	$0.200078\pi$
$504$		0			0
$505$	−11258.5			−0.992073
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4710.36	+	8158.58i	0.410182	+	0.710457i	0.994909	−	0.100773i	$-0.0321317\pi$
$509$	4710.36	+	8158.58i	0.410182	+	0.710457i	−0.584727	+	0.811230i	$0.698798\pi$
$510$		0			0
$511$	266.148	+	18105.8i	0.0230405	+	1.56743i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6580.85	+	3799.46i	0.563081	+	0.325095i
$516$		0			0
$517$			1066.72i			0.0907434i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4266.12	−	7389.13i	0.358737	−	0.621351i	−0.629013	−	0.777395i	$-0.716541\pi$
$521$	4266.12	−	7389.13i	0.358737	−	0.621351i	0.987750	+	0.156044i	$0.0498742\pi$
$522$		0			0
$523$	−5291.29	+	3054.93i	−0.442394	+	0.255416i	−0.704613	−	0.709592i	$-0.748879\pi$
$523$	−5291.29	+	3054.93i	−0.442394	+	0.255416i	0.262219	+	0.965008i	$0.415546\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1537.82	−	887.859i	0.127113	−	0.0733885i
$528$		0			0
$529$	7993.77	−	13845.6i	0.657004	−	1.13796i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	6639.55i		−	0.539570i
$534$		0			0
$535$	1516.47	+	875.536i	0.122547	+	0.0707527i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1132.49	−	699.013i	0.0905006	−	0.0558601i
$540$		0			0
$541$	−3917.88	−	6785.97i	−0.311355	−	0.539282i	0.667301	−	0.744788i	$-0.267449\pi$
$541$	−3917.88	−	6785.97i	−0.311355	−	0.539282i	−0.978656	+	0.205506i	$0.934116\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1067.06			0.0838676
$546$		0			0
$547$	−16124.4			−1.26039			−0.630193	−	0.776438i	$-0.717024\pi$
$547$	−16124.4			−1.26039			−0.630193	+	0.776438i	$0.717024\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7267.88	+	12588.3i	0.561927	+	0.973287i
$552$		0			0
$553$	−8368.74	+	15000.0i	−0.643535	+	1.15346i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	10054.8	+	5805.12i	0.764872	+	0.441599i	0.831042	−	0.556209i	$-0.187745\pi$
$557$	10054.8	+	5805.12i	0.764872	+	0.441599i	−0.0661703	+	0.997808i	$0.521078\pi$
$558$		0			0
$559$			880.185i			0.0665972i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5721.06	+	9909.16i	−0.428266	+	0.741778i	−0.996719	−	0.0809371i	$-0.974209\pi$
$563$	−5721.06	+	9909.16i	−0.428266	+	0.741778i	0.568453	+	0.822716i	$0.307542\pi$
$564$		0			0
$565$	13342.9	−	7703.52i	0.993521	−	0.573609i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	22603.5	−	13050.1i	1.66536	−	0.961495i	0.695267	−	0.718751i	$-0.255286\pi$
$569$	22603.5	−	13050.1i	1.66536	−	0.961495i	0.970091	−	0.242743i	$-0.0780473\pi$
$570$		0			0
$571$	2595.16	−	4494.95i	0.190200	−	0.329436i	−0.755116	−	0.655591i	$-0.772420\pi$
$571$	2595.16	−	4494.95i	0.190200	−	0.329436i	0.945316	+	0.326155i	$0.105753\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			9511.42i			0.689832i
$576$		0			0
$577$	19142.6	+	11052.0i	1.38114	+	0.797399i	0.992294	−	0.123905i	$-0.0395419\pi$
$577$	19142.6	+	11052.0i	1.38114	+	0.797399i	0.388842	+	0.921304i	$0.372875\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10944.6	−	18329.0i	−0.781509	−	1.30880i
$582$		0			0
$583$	1215.13	+	2104.66i	0.0863214	+	0.149513i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4437.47			−0.312017			−0.156009	−	0.987756i	$-0.549863\pi$
$587$	−4437.47			−0.312017			−0.156009	+	0.987756i	$0.549863\pi$
$588$		0			0
$589$	2015.06			0.140966
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13896.3	−	24069.1i	−0.962315	−	1.66678i	−0.716661	−	0.697421i	$-0.754331\pi$
$593$	−13896.3	−	24069.1i	−0.962315	−	1.66678i	−0.245654	−	0.969358i	$-0.579003\pi$
$594$		0			0
$595$	14638.1	+	24514.6i	1.00858	+	1.68908i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6526.51	−	3768.08i	−0.445185	−	0.257028i	0.260609	−	0.965444i	$-0.416077\pi$
$599$	−6526.51	−	3768.08i	−0.445185	−	0.257028i	−0.705795	+	0.708417i	$0.749410\pi$
$600$		0			0
$601$		−	348.179i		−	0.0236315i	−0.999930	−	0.0118157i	$-0.996239\pi$
$601$		−	348.179i		−	0.0236315i	0.999930	−	0.0118157i	$-0.00376115\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8868.86	−	15361.3i	0.595984	−	1.03228i
$606$		0			0
$607$	12737.0	−	7353.70i	0.851694	−	0.491726i	−0.00952801	−	0.999955i	$-0.503033\pi$
$607$	12737.0	−	7353.70i	0.851694	−	0.491726i	0.861222	+	0.508229i	$0.169700\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5464.54	+	3154.95i	−0.361819	+	0.208896i
$612$		0			0
$613$	8696.67	−	15063.1i	0.573010	−	0.992483i	−0.423245	−	0.906015i	$-0.639109\pi$
$613$	8696.67	−	15063.1i	0.573010	−	0.992483i	0.996255	−	0.0864672i	$-0.0275578\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			17411.5i			1.13608i	0.823000	+	0.568041i	$0.192298\pi$
$617$			17411.5i			1.13608i	−0.823000	+	0.568041i	$0.807702\pi$
$618$		0			0
$619$	7394.09	+	4268.98i	0.480119	+	0.277197i	0.720466	−	0.693490i	$-0.243928\pi$
$619$	7394.09	+	4268.98i	0.480119	+	0.277197i	−0.240347	+	0.970687i	$0.577261\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10003.0	−	17929.2i	0.643278	−	1.15300i
$624$		0			0
$625$	9748.72	+	16885.3i	0.623918	+	1.08066i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	38179.2			2.42020
$630$		0			0
$631$	15670.8			0.988662			0.494331	−	0.869274i	$-0.335413\pi$
$631$	15670.8			0.988662			0.494331	+	0.869274i	$0.335413\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2938.27	−	5089.23i	−0.183625	−	0.318047i
$636$		0			0
$637$	6930.33	+	3734.04i	0.431067	+	0.232258i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3120.23	+	1801.46i	0.192265	+	0.111004i	0.593042	−	0.805171i	$-0.297927\pi$
$641$	3120.23	+	1801.46i	0.192265	+	0.111004i	−0.400778	+	0.916175i	$0.631260\pi$
$642$		0			0
$643$			18859.3i			1.15667i	0.815800	+	0.578334i	$0.196297\pi$
$643$			18859.3i			1.15667i	−0.815800	+	0.578334i	$0.803703\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8923.20	−	15455.4i	0.542206	−	0.939128i	−0.456571	−	0.889687i	$-0.650923\pi$
$647$	8923.20	−	15455.4i	0.542206	−	0.939128i	0.998777	−	0.0494414i	$-0.0157441\pi$
$648$		0			0
$649$	1561.59	−	901.585i	0.0944497	−	0.0545305i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−16920.5	+	9769.08i	−1.01401	+	0.585442i	−0.912365	−	0.409378i	$-0.865746\pi$
$653$	−16920.5	+	9769.08i	−1.01401	+	0.585442i	−0.101650	+	0.994820i	$0.532412\pi$
$654$		0			0
$655$	6814.13	−	11802.4i	0.406489	−	0.704059i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			17003.2i			1.00509i	0.864552	+	0.502544i	$0.167602\pi$
$659$			17003.2i			1.00509i	−0.864552	+	0.502544i	$0.832398\pi$
$660$		0			0
$661$	13929.6	+	8042.28i	0.819668	+	0.473235i	0.850302	−	0.526295i	$-0.176419\pi$
$661$	13929.6	+	8042.28i	0.819668	+	0.473235i	−0.0306343	+	0.999531i	$0.509753\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	476.229	+	32397.4i	0.0277705	+	1.88920i
$666$		0			0
$667$	9395.74	+	16273.9i	0.545434	+	0.944720i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−636.262			−0.0366060
$672$		0			0
$673$	5276.73			0.302233			0.151117	−	0.988516i	$-0.451713\pi$
$673$	5276.73			0.302233			0.151117	+	0.988516i	$0.451713\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4172.65	−	7227.24i	−0.236880	−	0.410288i	0.722937	−	0.690914i	$-0.242792\pi$
$677$	−4172.65	−	7227.24i	−0.236880	−	0.410288i	−0.959817	+	0.280625i	$0.909458\pi$
$678$		0			0
$679$	−17125.1	+	10225.7i	−0.967897	+	0.577948i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11941.9	−	6894.65i	−0.669024	−	0.386261i	0.126682	−	0.991943i	$-0.459567\pi$
$683$	−11941.9	−	6894.65i	−0.669024	−	0.386261i	−0.795707	+	0.605682i	$0.792900\pi$
$684$		0			0
$685$		−	13847.9i		−	0.772408i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7187.75	+	12449.5i	−0.397433	+	0.688374i
$690$		0			0
$691$	−30742.0	+	17748.9i	−1.69244	+	0.977133i	−0.739908	+	0.672708i	$0.765131\pi$
$691$	−30742.0	+	17748.9i	−1.69244	+	0.977133i	−0.952536	+	0.304426i	$0.901535\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−24436.8	+	14108.6i	−1.33373	+	0.770028i
$696$		0			0
$697$	16543.9	−	28655.0i	0.899063	−	1.55722i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			13027.5i			0.701912i	0.936392	+	0.350956i	$0.114143\pi$
$701$			13027.5i			0.701912i	−0.936392	+	0.350956i	$0.885857\pi$
$702$		0			0
$703$	37520.8	+	21662.6i	2.01298	+	1.16219i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−15467.5	+	227.366i	−0.822795	+	0.0120947i
$708$		0			0
$709$	−2669.66	−	4623.99i	−0.141412	−	0.244933i	0.786616	−	0.617442i	$-0.211831\pi$
$709$	−2669.66	−	4623.99i	−0.141412	−	0.244933i	−0.928029	+	0.372509i	$0.878498\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2605.02			0.136829
$714$		0			0
$715$	−1200.33			−0.0627828
$716$		0			0
$717$		0			0
$718$		0			0
$719$	480.344	+	831.979i	0.0249149	+	0.0431538i	0.878214	−	0.478268i	$-0.158735\pi$
$719$	480.344	+	831.979i	0.0249149	+	0.0431538i	−0.853299	+	0.521422i	$0.825402\pi$
$720$		0			0
$721$	9117.85	+	5086.99i	0.470966	+	0.262759i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5497.79	−	3174.15i	−0.281632	−	0.162600i
$726$		0			0
$727$		−	10608.2i		−	0.541178i	−0.962695	−	0.270589i	$-0.912782\pi$
$727$		−	10608.2i		−	0.541178i	0.962695	−	0.270589i	$-0.0872184\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2193.18	+	3798.70i	−0.110968	+	0.192203i
$732$		0			0
$733$	−4227.26	+	2440.61i	−0.213011	+	0.122982i	−0.602710	−	0.797960i	$-0.705912\pi$
$733$	−4227.26	+	2440.61i	−0.213011	+	0.122982i	0.389699	+	0.920942i	$0.372579\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	76.0048	−	43.8814i	0.00379874	−	0.00219320i
$738$		0			0
$739$	726.391	−	1258.15i	0.0361579	−	0.0626274i	−0.847380	−	0.530987i	$-0.821821\pi$
$739$	726.391	−	1258.15i	0.0361579	−	0.0626274i	0.883538	+	0.468359i	$0.155155\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	21934.1i		−	1.08302i	−0.840695	−	0.541510i	$-0.817853\pi$
$743$		−	21934.1i		−	1.08302i	0.840695	−	0.541510i	$-0.182147\pi$
$744$		0			0
$745$	29498.7	+	17031.1i	1.45067	+	0.837545i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2101.09	+	1172.23i	0.102500	+	0.0571862i
$750$		0			0
$751$	−8380.16	−	14514.9i	−0.407186	−	0.705266i	0.587388	−	0.809306i	$-0.300156\pi$
$751$	−8380.16	−	14514.9i	−0.407186	−	0.705266i	−0.994573	+	0.104040i	$0.966823\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−17023.8			−0.820607
$756$		0			0
$757$	−29239.3			−1.40386			−0.701930	−	0.712246i	$-0.747678\pi$
$757$	−29239.3			−1.40386			−0.701930	+	0.712246i	$0.747678\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13062.4	−	22624.7i	−0.622222	−	1.07772i	−0.989071	−	0.147440i	$-0.952897\pi$
$761$	−13062.4	−	22624.7i	−0.622222	−	1.07772i	0.366849	−	0.930280i	$-0.380437\pi$
$762$		0			0
$763$	1465.98	−	21.5493i	0.0695573	−	0.00102246i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9237.17	+	5333.09i	0.434857	+	0.251065i
$768$		0			0
$769$			22095.8i			1.03614i	0.855337	+	0.518072i	$0.173350\pi$
$769$			22095.8i			1.03614i	−0.855337	+	0.518072i	$0.826650\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−11406.9	+	19757.3i	−0.530760	+	0.919303i	0.468596	+	0.883413i	$0.344760\pi$
$773$	−11406.9	+	19757.3i	−0.530760	+	0.919303i	−0.999356	+	0.0358904i	$0.988573\pi$
$774$		0			0
$775$	−762.148	+	440.026i	−0.0353254	+	0.0203951i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	32517.3	−	18773.8i	1.49557	−	0.863470i
$780$		0			0
$781$	1437.11	−	2489.14i	0.0658435	−	0.114044i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			13490.5i			0.613372i
$786$		0			0
$787$	20936.1	+	12087.5i	0.948276	+	0.547487i	0.892545	−	0.450959i	$-0.148918\pi$
$787$	20936.1	+	12087.5i	0.948276	+	0.547487i	0.0557309	+	0.998446i	$0.482251\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	18175.6	−	10853.0i	0.817003	−	0.487847i
$792$		0			0
$793$	−1881.82	−	3259.41i	−0.0842691	−	0.145958i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−16204.7			−0.720201			−0.360101	−	0.932913i	$-0.617258\pi$
$797$	−16204.7			−0.720201			−0.360101	+	0.932913i	$0.617258\pi$
$798$		0			0
$799$	−31445.1			−1.39230
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1896.81	−	3285.36i	−0.0833584	−	0.144381i
$804$		0			0
$805$	615.658	+	41882.7i	0.0269554	+	1.83375i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−24980.6	−	14422.5i	−1.08562	−	0.626785i	−0.153216	−	0.988193i	$-0.548963\pi$
$809$	−24980.6	−	14422.5i	−1.08562	−	0.626785i	−0.932408	+	0.361408i	$0.882296\pi$
$810$		0			0
$811$			24978.8i			1.08154i	0.841172	+	0.540768i	$0.181866\pi$
$811$			24978.8i			1.08154i	−0.841172	+	0.540768i	$0.818134\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9421.98	+	16319.4i	−0.404954	+	0.701401i
$816$		0			0
$817$	−4310.71	+	2488.79i	−0.184593	+	0.106575i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8539.34	−	4930.19i	0.363002	−	0.209579i	−0.307395	−	0.951582i	$-0.599457\pi$
$821$	8539.34	−	4930.19i	0.363002	−	0.209579i	0.670397	+	0.742003i	$0.266124\pi$
$822$		0			0
$823$	−4660.50	+	8072.22i	−0.197393	+	0.341895i	−0.947683	−	0.319214i	$-0.896581\pi$
$823$	−4660.50	+	8072.22i	−0.197393	+	0.341895i	0.750289	+	0.661110i	$0.229914\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	27727.4i		−	1.16587i	−0.812518	−	0.582936i	$-0.801904\pi$
$827$		−	27727.4i		−	1.16587i	0.812518	−	0.582936i	$-0.198096\pi$
$828$		0			0
$829$	15400.4	+	8891.42i	0.645208	+	0.372511i	0.786618	−	0.617440i	$-0.211830\pi$
$829$	15400.4	+	8891.42i	0.645208	+	0.372511i	−0.141410	+	0.989951i	$0.545164\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	20605.7	+	33383.9i	0.857077	+	1.38858i
$834$		0			0
$835$	−27843.0	−	48225.4i	−1.15395	−	1.99869i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−37414.7			−1.53957			−0.769786	−	0.638302i	$-0.779637\pi$
$839$	−37414.7			−1.53957			−0.769786	+	0.638302i	$0.779637\pi$
$840$		0			0
$841$	11846.8			0.485744
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11256.7	+	19497.1i	0.458273	+	0.793752i
$846$		0			0
$847$	11874.3	−	21283.3i	0.481706	−	0.863403i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	48506.0	+	28004.9i	1.95389	+	1.12808i
$852$		0			0
$853$		−	45000.8i		−	1.80633i	−0.429297	−	0.903163i	$-0.641239\pi$
$853$		−	45000.8i		−	1.80633i	0.429297	−	0.903163i	$-0.358761\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3211.66	−	5562.76i	0.128014	−	0.221727i	−0.794893	−	0.606750i	$-0.792473\pi$
$857$	3211.66	−	5562.76i	0.128014	−	0.221727i	0.922907	+	0.385023i	$0.125806\pi$
$858$		0			0
$859$	9746.17	−	5626.95i	0.387119	−	0.223503i	−0.293792	−	0.955869i	$-0.594917\pi$
$859$	9746.17	−	5626.95i	0.387119	−	0.223503i	0.680911	+	0.732366i	$0.261584\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	27330.4	−	15779.2i	1.07803	−	0.622399i	0.147663	−	0.989038i	$-0.452825\pi$
$863$	27330.4	−	15779.2i	1.07803	−	0.622399i	0.930363	+	0.366639i	$0.119492\pi$
$864$		0			0
$865$	−5075.26	+	8790.61i	−0.199496	+	0.345537i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	3598.52i		−	0.140474i
$870$		0			0
$871$	449.586	+	259.568i	0.0174898	+	0.0100978i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8743.05	+	14642.1i	0.337793	+	0.565706i
$876$		0			0
$877$	7959.71	+	13786.6i	0.306477	+	0.530834i	0.977589	−	0.210522i	$-0.0675164\pi$
$877$	7959.71	+	13786.6i	0.306477	+	0.530834i	−0.671112	+	0.741356i	$0.734183\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−23711.9			−0.906783			−0.453392	−	0.891311i	$-0.649786\pi$
$881$	−23711.9			−0.906783			−0.453392	+	0.891311i	$0.649786\pi$
$882$		0			0
$883$	−10170.8			−0.387629			−0.193814	−	0.981038i	$-0.562086\pi$
$883$	−10170.8			−0.387629			−0.193814	+	0.981038i	$0.562086\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4088.38	−	7081.29i	−0.154763	−	0.268057i	0.778210	−	0.628004i	$-0.216128\pi$
$887$	−4088.38	−	7081.29i	−0.154763	−	0.268057i	−0.932973	+	0.359947i	$0.882795\pi$
$888$		0			0
$889$	−4139.52	−	6932.51i	−0.156170	−	0.261540i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−30902.8	−	17841.8i	−1.15803	−	0.668591i
$894$		0			0
$895$		−	22862.4i		−	0.853861i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−869.349	+	1505.76i	−0.0322519	+	0.0558619i
$900$		0			0
$901$	−62041.8	+	35819.8i	−2.29402	+	1.32445i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	31564.5	−	18223.8i	1.15938	−	0.669368i
$906$		0			0
$907$	21720.2	−	37620.6i	0.795159	−	1.37726i	−0.127580	−	0.991828i	$-0.540721\pi$
$907$	21720.2	−	37620.6i	0.795159	−	1.37726i	0.922738	−	0.385427i	$-0.125946\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	6894.34i		−	0.250735i	−0.992110	−	0.125368i	$-0.959989\pi$
$911$		−	6894.34i		−	0.250735i	0.992110	−	0.125368i	$-0.0400110\pi$
$912$		0			0
$913$	3873.23	+	2236.21i	0.140400	+	0.0810600i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9123.26	−	16352.4i	0.328546	−	0.588881i
$918$		0			0
$919$	−22803.2	−	39496.3i	−0.818508	−	1.41770i	−0.906782	−	0.421601i	$-0.861468\pi$
$919$	−22803.2	−	39496.3i	−0.818508	−	1.41770i	0.0882738	−	0.996096i	$-0.471865\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	17001.7			0.606301
$924$		0			0
$925$	−18921.8			−0.672588
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2502.59	+	4334.61i	0.0883825	+	0.153083i	0.906828	−	0.421502i	$-0.138497\pi$
$929$	2502.59	+	4334.61i	0.0883825	+	0.153083i	−0.818445	+	0.574585i	$0.805164\pi$
$930$		0			0
$931$	1308.54	+	44499.7i	0.0460640	+	1.56651i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5180.37	−	2990.89i	−0.181194	−	0.104612i
$936$		0			0
$937$			1553.37i			0.0541583i	0.999633	+	0.0270791i	$0.00862061\pi$
$937$			1553.37i			0.0541583i	−0.999633	+	0.0270791i	$0.991379\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−9229.76	+	15986.4i	−0.319747	+	0.553817i	−0.980435	−	0.196843i	$-0.936931\pi$
$941$	−9229.76	+	15986.4i	−0.319747	+	0.553817i	0.660688	+	0.750660i	$0.270264\pi$
$942$		0			0
$943$	42037.6	−	24270.4i	1.45168	−	0.838126i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	32928.5	−	19011.3i	1.12992	−	0.652359i	0.186005	−	0.982549i	$-0.440446\pi$
$947$	32928.5	−	19011.3i	1.12992	−	0.652359i	0.943915	+	0.330190i	$0.107113\pi$
$948$		0			0
$949$	11220.0	−	19433.7i	0.383791	−	0.664746i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			7237.56i			0.246010i	0.992406	+	0.123005i	$0.0392531\pi$
$953$			7237.56i			0.246010i	−0.992406	+	0.123005i	$0.960747\pi$
$954$		0			0
$955$	34524.3	+	19932.6i	1.16982	+	0.675398i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−279.658	−	19024.9i	−0.00941672	−	0.640612i
$960$		0			0
$961$	−14775.0	−	25591.0i	−0.495955	−	0.859019i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	59573.1			1.98728
$966$		0			0
$967$	−29380.5			−0.977057			−0.488528	−	0.872548i	$-0.662466\pi$
$967$	−29380.5			−0.977057			−0.488528	+	0.872548i	$0.662466\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6132.73	+	10622.2i	0.202686	+	0.351063i	0.949393	−	0.314090i	$-0.101699\pi$
$971$	6132.73	+	10622.2i	0.202686	+	0.351063i	−0.746707	+	0.665153i	$0.768366\pi$
$972$		0			0
$973$	−33287.6	+	19876.6i	−1.09677	+	0.654898i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11649.2	+	6725.69i	0.381466	+	0.220239i	0.678456	−	0.734641i	$-0.262649\pi$
$977$	11649.2	+	6725.69i	0.381466	+	0.220239i	−0.296990	+	0.954881i	$0.595983\pi$
$978$		0			0
$979$			4301.25i			0.140417i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7782.25	+	13479.2i	−0.252508	+	0.437356i	−0.964216	−	0.265119i	$-0.914589\pi$
$983$	−7782.25	+	13479.2i	−0.252508	+	0.437356i	0.711708	+	0.702476i	$0.247922\pi$
$984$		0			0
$985$	1589.74	−	917.836i	0.0514247	−	0.0296900i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−5572.79	+	3217.45i	−0.179175	+	0.103447i
$990$		0			0
$991$	23244.2	−	40260.1i	0.745081	−	1.29052i	−0.205076	−	0.978746i	$-0.565744\pi$
$991$	23244.2	−	40260.1i	0.745081	−	1.29052i	0.950157	−	0.311772i	$-0.100923\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	13727.6i		−	0.437381i
$996$		0			0
$997$	−7592.45	−	4383.50i	−0.241179	−	0.139245i	0.374540	−	0.927211i	$-0.377801\pi$
$997$	−7592.45	−	4383.50i	−0.241179	−	0.139245i	−0.615718	+	0.787966i	$0.711134\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.t.d.269.7	yes	16
3.2	odd	2	inner	756.4.t.d.269.2	✓	16
7.5	odd	6	inner	756.4.t.d.593.2	yes	16
21.5	even	6	inner	756.4.t.d.593.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.t.d.269.2	✓	16	3.2	odd	2	inner
756.4.t.d.269.7	yes	16	1.1	even	1	trivial
756.4.t.d.593.2	yes	16	7.5	odd	6	inner
756.4.t.d.593.7	yes	16	21.5	even	6	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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(6.73954 + 11.6732i) q^{5} +(9.49487 + 15.9012i) q^{7} +(-3.36020 - 1.94001i) q^{11} -22.9512i q^{13} +(57.1882 - 99.0529i) q^{17} +(112.404 - 64.8964i) q^{19} +(145.313 - 83.8966i) q^{23} +(-28.3427 + 49.0910i) q^{25} +111.992i q^{29} +(13.4452 + 7.76260i) q^{31} +(-121.627 + 218.002i) q^{35} +(166.902 + 289.082i) q^{37} +289.289 q^{41} -38.3502 q^{43} +(-137.463 - 238.094i) q^{47} +(-162.695 + 301.959i) q^{49} +(-542.435 - 313.175i) q^{53} -52.2991i q^{55} +(-232.366 + 402.470i) q^{59} +(142.015 - 81.9922i) q^{61} +(267.915 - 154.681i) q^{65} +(-11.3096 + 19.5888i) q^{67} +740.773i q^{71} +(846.739 + 488.865i) q^{73} +(-1.05618 - 71.8512i) q^{77} +(463.725 + 803.195i) q^{79} -1152.68 q^{83} +1541.69 q^{85} +(-554.282 - 960.045i) q^{89} +(364.951 - 217.919i) q^{91} +(1515.10 + 874.743i) q^{95} +1076.97i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 50 q^{7} - 390 q^{19} - 284 q^{25} + 984 q^{31} - 182 q^{37} + 472 q^{43} + 130 q^{49} - 24 q^{61} - 1522 q^{67} + 4458 q^{73} + 434 q^{79} - 2616 q^{85} - 4560 q^{91}+O(q^{100}) \) 16 * q + 50 * q^7 - 390 * q^19 - 284 * q^25 + 984 * q^31 - 182 * q^37 + 472 * q^43 + 130 * q^49 - 24 * q^61 - 1522 * q^67 + 4458 * q^73 + 434 * q^79 - 2616 * q^85 - 4560 * q^91

Introduction

L-functions

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