LMFDB - Embedded newform 448.3.g.b.351.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,3,Mod(351,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.351"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.1
Root			$-1.32288 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	448.351
Dual form			448.3.g.b.351.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.64575 q^{3} -7.64575i q^{5} +2.64575i q^{7} -6.29150 q^{9} -3.29150 q^{11} -4.35425i q^{13} +12.5830i q^{15} -12.5830 q^{17} +10.3542 q^{19} -4.35425i q^{21} +19.1660i q^{23} -33.4575 q^{25} +25.1660 q^{27} +52.4575i q^{29} +17.1660i q^{31} +5.41699 q^{33} +20.2288 q^{35} -28.4575i q^{37} +7.16601i q^{39} -50.9150 q^{41} -75.2915 q^{43} +48.1033i q^{45} +74.3320i q^{47} -7.00000 q^{49} +20.7085 q^{51} -8.91503i q^{53} +25.1660i q^{55} -17.0405 q^{57} -45.1882 q^{59} -86.4353i q^{61} -16.6458i q^{63} -33.2915 q^{65} -113.830 q^{67} -31.5425i q^{69} +67.7490i q^{71} -57.7490 q^{73} +55.0627 q^{75} -8.70850i q^{77} -62.5830i q^{79} +15.2065 q^{81} +119.727 q^{83} +96.2065i q^{85} -86.3320i q^{87} +51.6680 q^{89} +11.5203 q^{91} -28.2510i q^{93} -79.1660i q^{95} +164.996 q^{97} +20.7085 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{9} + 8 q^{11} - 8 q^{17} + 52 q^{19} - 28 q^{25} + 16 q^{27} + 64 q^{33} + 28 q^{35} + 8 q^{41} - 280 q^{43} - 28 q^{49} + 104 q^{51} + 80 q^{57} - 276 q^{59} - 112 q^{65} - 32 q^{67}+ \cdots + 104 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^9 + 8 * q^11 - 8 * q^17 + 52 * q^19 - 28 * q^25 + 16 * q^27 + 64 * q^33 + 28 * q^35 + 8 * q^41 - 280 * q^43 - 28 * q^49 + 104 * q^51 + 80 * q^57 - 276 * q^59 - 112 * q^65 - 32 * q^67 - 104 * q^73 + 252 * q^75 - 172 * q^81 + 172 * q^83 + 376 * q^89 - 28 * q^91 + 152 * q^97 + 104 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.64575	−0.548584	−0.274292	−	0.961646i	$-0.588443\pi$
$3$	−1.64575	−0.548584	−0.274292	+	0.961646i	$0.588443\pi$
$4$	0	0
$5$	− 7.64575i	− 1.52915i	−0.644535	−	0.764575i	$-0.722949\pi$
$5$	− 7.64575i	− 1.52915i	0.644535	−	0.764575i	$-0.277051\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−6.29150	−0.699056
$10$	0	0
$11$	−3.29150	−0.299228	−0.149614	−	0.988745i	$-0.547803\pi$
$11$	−3.29150	−0.299228	−0.149614	+	0.988745i	$0.547803\pi$
$12$	0	0
$13$	− 4.35425i	− 0.334942i	−0.985877	−	0.167471i	$-0.946440\pi$
$13$	− 4.35425i	− 0.334942i	0.985877	−	0.167471i	$-0.0535601\pi$
$14$	0	0
$15$	12.5830i	0.838867i
$16$	0	0
$17$	−12.5830	−0.740177	−0.370088	−	0.928997i	$-0.620673\pi$
$17$	−12.5830	−0.740177	−0.370088	+	0.928997i	$0.620673\pi$
$18$	0	0
$19$	10.3542	0.544960	0.272480	−	0.962161i	$-0.412156\pi$
$19$	10.3542	0.544960	0.272480	+	0.962161i	$0.412156\pi$
$20$	0	0
$21$	− 4.35425i	− 0.207345i
$22$	0	0
$23$	19.1660i	0.833305i	0.909066	+	0.416652i	$0.136797\pi$
$23$	19.1660i	0.833305i	−0.909066	+	0.416652i	$0.863203\pi$
$24$	0	0
$25$	−33.4575	−1.33830
$26$	0	0
$27$	25.1660	0.932074
$28$	0	0
$29$	52.4575i	1.80888i	0.426601	+	0.904440i	$0.359711\pi$
$29$	52.4575i	1.80888i	−0.426601	+	0.904440i	$0.640289\pi$
$30$	0	0
$31$	17.1660i	0.553742i	0.960907	+	0.276871i	$0.0892975\pi$
$31$	17.1660i	0.553742i	−0.960907	+	0.276871i	$0.910702\pi$
$32$	0	0
$33$	5.41699	0.164151
$34$	0	0
$35$	20.2288	0.577964
$36$	0	0
$37$	− 28.4575i	− 0.769122i	−0.923100	−	0.384561i	$-0.874353\pi$
$37$	− 28.4575i	− 0.769122i	0.923100	−	0.384561i	$-0.125647\pi$
$38$	0	0
$39$	7.16601i	0.183744i
$40$	0	0
$41$	−50.9150	−1.24183	−0.620915	−	0.783878i	$-0.713239\pi$
$41$	−50.9150	−1.24183	−0.620915	+	0.783878i	$0.713239\pi$
$42$	0	0
$43$	−75.2915	−1.75097	−0.875483	−	0.483250i	$-0.839456\pi$
$43$	−75.2915	−1.75097	−0.875483	+	0.483250i	$0.839456\pi$
$44$	0	0
$45$	48.1033i	1.06896i
$46$	0	0
$47$	74.3320i	1.58153i	0.612118	+	0.790766i	$0.290318\pi$
$47$	74.3320i	1.58153i	−0.612118	+	0.790766i	$0.709682\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	20.7085	0.406049
$52$	0	0
$53$	− 8.91503i	− 0.168208i	−0.996457	−	0.0841040i	$-0.973197\pi$
$53$	− 8.91503i	− 0.168208i	0.996457	−	0.0841040i	$-0.0268028\pi$
$54$	0	0
$55$	25.1660i	0.457564i
$56$	0	0
$57$	−17.0405	−0.298956
$58$	0	0
$59$	−45.1882	−0.765902	−0.382951	−	0.923769i	$-0.625092\pi$
$59$	−45.1882	−0.765902	−0.382951	+	0.923769i	$0.625092\pi$
$60$	0	0
$61$	− 86.4353i	− 1.41697i	−0.705725	−	0.708486i	$-0.749379\pi$
$61$	− 86.4353i	− 1.41697i	0.705725	−	0.708486i	$-0.250621\pi$
$62$	0	0
$63$	− 16.6458i	− 0.264218i
$64$	0	0
$65$	−33.2915	−0.512177
$66$	0	0
$67$	−113.830	−1.69896	−0.849478	−	0.527624i	$-0.823083\pi$
$67$	−113.830	−1.69896	−0.849478	+	0.527624i	$0.823083\pi$
$68$	0	0
$69$	− 31.5425i	− 0.457137i
$70$	0	0
$71$	67.7490i	0.954211i	0.878846	+	0.477106i	$0.158314\pi$
$71$	67.7490i	0.954211i	−0.878846	+	0.477106i	$0.841686\pi$
$72$	0	0
$73$	−57.7490	−0.791082	−0.395541	−	0.918448i	$-0.629443\pi$
$73$	−57.7490	−0.791082	−0.395541	+	0.918448i	$0.629443\pi$
$74$	0	0
$75$	55.0627	0.734170
$76$	0	0
$77$	− 8.70850i	− 0.113097i
$78$	0	0
$79$	− 62.5830i	− 0.792190i	−0.918210	−	0.396095i	$-0.870365\pi$
$79$	− 62.5830i	− 0.792190i	0.918210	−	0.396095i	$-0.129635\pi$
$80$	0	0
$81$	15.2065	0.187735
$82$	0	0
$83$	119.727	1.44249	0.721246	−	0.692679i	$-0.243570\pi$
$83$	119.727	1.44249	0.721246	+	0.692679i	$0.243570\pi$
$84$	0	0
$85$	96.2065i	1.13184i
$86$	0	0
$87$	− 86.3320i	− 0.992322i
$88$	0	0
$89$	51.6680	0.580539	0.290270	−	0.956945i	$-0.406255\pi$
$89$	51.6680	0.580539	0.290270	+	0.956945i	$0.406255\pi$
$90$	0	0
$91$	11.5203	0.126596
$92$	0	0
$93$	− 28.2510i	− 0.303774i
$94$	0	0
$95$	− 79.1660i	− 0.833326i
$96$	0	0
$97$	164.996	1.70099	0.850495	−	0.525983i	$-0.176302\pi$
$97$	164.996	1.70099	0.850495	+	0.525983i	$0.176302\pi$
$98$	0	0
$99$	20.7085	0.209177
$100$	0	0
$101$	85.4758i	0.846295i	0.906061	+	0.423148i	$0.139075\pi$
$101$	85.4758i	0.846295i	−0.906061	+	0.423148i	$0.860925\pi$
$102$	0	0
$103$	60.5020i	0.587398i	0.955898	+	0.293699i	$0.0948863\pi$
$103$	60.5020i	0.587398i	−0.955898	+	0.293699i	$0.905114\pi$
$104$	0	0
$105$	−33.2915	−0.317062
$106$	0	0
$107$	41.4170	0.387075	0.193537	−	0.981093i	$-0.438004\pi$
$107$	41.4170	0.387075	0.193537	+	0.981093i	$0.438004\pi$
$108$	0	0
$109$	− 87.7046i	− 0.804629i	−0.915502	−	0.402314i	$-0.868206\pi$
$109$	− 87.7046i	− 0.804629i	0.915502	−	0.402314i	$-0.131794\pi$
$110$	0	0
$111$	46.8340i	0.421928i
$112$	0	0
$113$	126.790	1.12203	0.561016	−	0.827805i	$-0.310411\pi$
$113$	126.790	1.12203	0.561016	+	0.827805i	$0.310411\pi$
$114$	0	0
$115$	146.539	1.27425
$116$	0	0
$117$	27.3948i	0.234143i
$118$	0	0
$119$	− 33.2915i	− 0.279761i
$120$	0	0
$121$	−110.166	−0.910463
$122$	0	0
$123$	83.7935	0.681248
$124$	0	0
$125$	64.6640i	0.517312i
$126$	0	0
$127$	− 102.162i	− 0.804426i	−0.915546	−	0.402213i	$-0.868241\pi$
$127$	− 102.162i	− 0.804426i	0.915546	−	0.402213i	$-0.131759\pi$
$128$	0	0
$129$	123.911	0.960551
$130$	0	0
$131$	−171.018	−1.30548	−0.652742	−	0.757581i	$-0.726381\pi$
$131$	−171.018	−1.30548	−0.652742	+	0.757581i	$0.726381\pi$
$132$	0	0
$133$	27.3948i	0.205976i
$134$	0	0
$135$	− 192.413i	− 1.42528i
$136$	0	0
$137$	−164.332	−1.19950	−0.599752	−	0.800186i	$-0.704734\pi$
$137$	−164.332	−1.19950	−0.599752	+	0.800186i	$0.704734\pi$
$138$	0	0
$139$	−177.395	−1.27622	−0.638111	−	0.769945i	$-0.720284\pi$
$139$	−177.395	−1.27622	−0.638111	+	0.769945i	$0.720284\pi$
$140$	0	0
$141$	− 122.332i	− 0.867603i
$142$	0	0
$143$	14.3320i	0.100224i
$144$	0	0
$145$	401.077	2.76605
$146$	0	0
$147$	11.5203	0.0783691
$148$	0	0
$149$	− 20.1621i	− 0.135316i	−0.997709	−	0.0676580i	$-0.978447\pi$
$149$	− 20.1621i	− 0.135316i	0.997709	−	0.0676580i	$-0.0215527\pi$
$150$	0	0
$151$	− 205.498i	− 1.36091i	−0.732788	−	0.680457i	$-0.761781\pi$
$151$	− 205.498i	− 1.36091i	0.732788	−	0.680457i	$-0.238219\pi$
$152$	0	0
$153$	79.1660	0.517425
$154$	0	0
$155$	131.247	0.846755
$156$	0	0
$157$	− 159.601i	− 1.01657i	−0.861189	−	0.508284i	$-0.830280\pi$
$157$	− 159.601i	− 1.01657i	0.861189	−	0.508284i	$-0.169720\pi$
$158$	0	0
$159$	14.6719i	0.0922762i
$160$	0	0
$161$	−50.7085	−0.314960
$162$	0	0
$163$	−252.207	−1.54728	−0.773640	−	0.633626i	$-0.781566\pi$
$163$	−252.207	−1.54728	−0.773640	+	0.633626i	$0.781566\pi$
$164$	0	0
$165$	− 41.4170i	− 0.251012i
$166$	0	0
$167$	− 37.1660i	− 0.222551i	−0.993790	−	0.111275i	$-0.964506\pi$
$167$	− 37.1660i	− 0.222551i	0.993790	−	0.111275i	$-0.0354936\pi$
$168$	0	0
$169$	150.041	0.887814
$170$	0	0
$171$	−65.1438	−0.380958
$172$	0	0
$173$	93.5647i	0.540837i	0.962743	+	0.270418i	$0.0871620\pi$
$173$	93.5647i	0.540837i	−0.962743	+	0.270418i	$0.912838\pi$
$174$	0	0
$175$	− 88.5203i	− 0.505830i
$176$	0	0
$177$	74.3686	0.420162
$178$	0	0
$179$	−198.583	−1.10940	−0.554701	−	0.832050i	$-0.687167\pi$
$179$	−198.583	−1.10940	−0.554701	+	0.832050i	$0.687167\pi$
$180$	0	0
$181$	173.314i	0.957534i	0.877942	+	0.478767i	$0.158916\pi$
$181$	173.314i	0.957534i	−0.877942	+	0.478767i	$0.841084\pi$
$182$	0	0
$183$	142.251i	0.777328i
$184$	0	0
$185$	−217.579	−1.17610
$186$	0	0
$187$	41.4170	0.221481
$188$	0	0
$189$	66.5830i	0.352291i
$190$	0	0
$191$	− 104.502i	− 0.547131i	−0.961853	−	0.273565i	$-0.911797\pi$
$191$	− 104.502i	− 0.547131i	0.961853	−	0.273565i	$-0.0882030\pi$
$192$	0	0
$193$	−11.1294	−0.0576654	−0.0288327	−	0.999584i	$-0.509179\pi$
$193$	−11.1294	−0.0576654	−0.0288327	+	0.999584i	$0.509179\pi$
$194$	0	0
$195$	54.7895	0.280972
$196$	0	0
$197$	− 299.247i	− 1.51902i	−0.650495	−	0.759510i	$-0.725439\pi$
$197$	− 299.247i	− 1.51902i	0.650495	−	0.759510i	$-0.274561\pi$
$198$	0	0
$199$	− 304.162i	− 1.52845i	−0.644948	−	0.764226i	$-0.723121\pi$
$199$	− 304.162i	− 1.52845i	0.644948	−	0.764226i	$-0.276879\pi$
$200$	0	0
$201$	187.336	0.932020
$202$	0	0
$203$	−138.790	−0.683692
$204$	0	0
$205$	389.284i	1.89894i
$206$	0	0
$207$	− 120.583i	− 0.582527i
$208$	0	0
$209$	−34.0810	−0.163067
$210$	0	0
$211$	341.077	1.61648	0.808240	−	0.588854i	$-0.200421\pi$
$211$	341.077	1.61648	0.808240	+	0.588854i	$0.200421\pi$
$212$	0	0
$213$	− 111.498i	− 0.523465i
$214$	0	0
$215$	575.660i	2.67749i
$216$	0	0
$217$	−45.4170	−0.209295
$218$	0	0
$219$	95.0405	0.433975
$220$	0	0
$221$	54.7895i	0.247916i
$222$	0	0
$223$	61.0039i	0.273560i	0.990601	+	0.136780i	$0.0436754\pi$
$223$	61.0039i	0.273560i	−0.990601	+	0.136780i	$0.956325\pi$
$224$	0	0
$225$	210.498	0.935547
$226$	0	0
$227$	147.771	0.650975	0.325487	−	0.945546i	$-0.394472\pi$
$227$	147.771	0.650975	0.325487	+	0.945546i	$0.394472\pi$
$228$	0	0
$229$	164.265i	0.717316i	0.933469	+	0.358658i	$0.116766\pi$
$229$	164.265i	0.717316i	−0.933469	+	0.358658i	$0.883234\pi$
$230$	0	0
$231$	14.3320i	0.0620434i
$232$	0	0
$233$	−341.660	−1.46635	−0.733176	−	0.680039i	$-0.761963\pi$
$233$	−341.660	−1.46635	−0.733176	+	0.680039i	$0.761963\pi$
$234$	0	0
$235$	568.324	2.41840
$236$	0	0
$237$	102.996i	0.434583i
$238$	0	0
$239$	333.158i	1.39397i	0.717087	+	0.696984i	$0.245475\pi$
$239$	333.158i	1.39397i	−0.717087	+	0.696984i	$0.754525\pi$
$240$	0	0
$241$	−183.166	−0.760025	−0.380012	−	0.924981i	$-0.624080\pi$
$241$	−183.166	−0.760025	−0.380012	+	0.924981i	$0.624080\pi$
$242$	0	0
$243$	−251.520	−1.03506
$244$	0	0
$245$	53.5203i	0.218450i
$246$	0	0
$247$	− 45.0850i	− 0.182530i
$248$	0	0
$249$	−197.041	−0.791327
$250$	0	0
$251$	−263.180	−1.04853	−0.524264	−	0.851556i	$-0.675659\pi$
$251$	−263.180	−1.04853	−0.524264	+	0.851556i	$0.675659\pi$
$252$	0	0
$253$	− 63.0850i	− 0.249348i
$254$	0	0
$255$	− 158.332i	− 0.620910i
$256$	0	0
$257$	189.838	0.738669	0.369334	−	0.929297i	$-0.379586\pi$
$257$	189.838	0.738669	0.369334	+	0.929297i	$0.379586\pi$
$258$	0	0
$259$	75.2915	0.290701
$260$	0	0
$261$	− 330.037i	− 1.26451i
$262$	0	0
$263$	− 178.421i	− 0.678407i	−0.940713	−	0.339203i	$-0.889843\pi$
$263$	− 178.421i	− 0.678407i	0.940713	−	0.339203i	$-0.110157\pi$
$264$	0	0
$265$	−68.1621	−0.257215
$266$	0	0
$267$	−85.0326	−0.318474
$268$	0	0
$269$	− 122.848i	− 0.456685i	−0.973581	−	0.228343i	$-0.926669\pi$
$269$	− 122.848i	− 0.456685i	0.973581	−	0.228343i	$-0.0733306\pi$
$270$	0	0
$271$	537.992i	1.98521i	0.121386	+	0.992605i	$0.461266\pi$
$271$	537.992i	1.98521i	−0.121386	+	0.992605i	$0.538734\pi$
$272$	0	0
$273$	−18.9595	−0.0694487
$274$	0	0
$275$	110.125	0.400456
$276$	0	0
$277$	20.5751i	0.0742785i	0.999310	+	0.0371392i	$0.0118245\pi$
$277$	20.5751i	0.0742785i	−0.999310	+	0.0371392i	$0.988175\pi$
$278$	0	0
$279$	− 108.000i	− 0.387097i
$280$	0	0
$281$	−216.656	−0.771018	−0.385509	−	0.922704i	$-0.625974\pi$
$281$	−216.656	−0.771018	−0.385509	+	0.922704i	$0.625974\pi$
$282$	0	0
$283$	−10.5608	−0.0373172	−0.0186586	−	0.999826i	$-0.505940\pi$
$283$	−10.5608	−0.0373172	−0.0186586	+	0.999826i	$0.505940\pi$
$284$	0	0
$285$	130.288i	0.457149i
$286$	0	0
$287$	− 134.708i	− 0.469368i
$288$	0	0
$289$	−130.668	−0.452138
$290$	0	0
$291$	−271.542	−0.933136
$292$	0	0
$293$	− 393.844i	− 1.34418i	−0.740470	−	0.672089i	$-0.765397\pi$
$293$	− 393.844i	− 1.34418i	0.740470	−	0.672089i	$-0.234603\pi$
$294$	0	0
$295$	345.498i	1.17118i
$296$	0	0
$297$	−82.8340	−0.278902
$298$	0	0
$299$	83.4536	0.279109
$300$	0	0
$301$	− 199.203i	− 0.661803i
$302$	0	0
$303$	− 140.672i	− 0.464264i
$304$	0	0
$305$	−660.863	−2.16676
$306$	0	0
$307$	−86.6784	−0.282340	−0.141170	−	0.989985i	$-0.545086\pi$
$307$	−86.6784	−0.282340	−0.141170	+	0.989985i	$0.545086\pi$
$308$	0	0
$309$	− 99.5712i	− 0.322237i
$310$	0	0
$311$	459.984i	1.47905i	0.673130	+	0.739525i	$0.264950\pi$
$311$	459.984i	1.47905i	−0.673130	+	0.739525i	$0.735050\pi$
$312$	0	0
$313$	239.919	0.766514	0.383257	−	0.923642i	$-0.374802\pi$
$313$	239.919	0.766514	0.383257	+	0.923642i	$0.374802\pi$
$314$	0	0
$315$	−127.269	−0.404029
$316$	0	0
$317$	− 156.340i	− 0.493186i	−0.969119	−	0.246593i	$-0.920689\pi$
$317$	− 156.340i	− 0.493186i	0.969119	−	0.246593i	$-0.0793110\pi$
$318$	0	0
$319$	− 172.664i	− 0.541267i
$320$	0	0
$321$	−68.1621	−0.212343
$322$	0	0
$323$	−130.288	−0.403367
$324$	0	0
$325$	145.682i	0.448253i
$326$	0	0
$327$	144.340i	0.441406i
$328$	0	0
$329$	−196.664	−0.597763
$330$	0	0
$331$	−206.125	−0.622736	−0.311368	−	0.950289i	$-0.600787\pi$
$331$	−206.125	−0.622736	−0.311368	+	0.950289i	$0.600787\pi$
$332$	0	0
$333$	179.041i	0.537659i
$334$	0	0
$335$	870.316i	2.59796i
$336$	0	0
$337$	273.373	0.811194	0.405597	−	0.914052i	$-0.367064\pi$
$337$	273.373	0.811194	0.405597	+	0.914052i	$0.367064\pi$
$338$	0	0
$339$	−208.664	−0.615528
$340$	0	0
$341$	− 56.5020i	− 0.165695i
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−241.166	−0.699032
$346$	0	0
$347$	174.450	0.502737	0.251368	−	0.967891i	$-0.419119\pi$
$347$	174.450	0.502737	0.251368	+	0.967891i	$0.419119\pi$
$348$	0	0
$349$	− 102.140i	− 0.292664i	−0.989236	−	0.146332i	$-0.953253\pi$
$349$	− 102.140i	− 0.292664i	0.989236	−	0.146332i	$-0.0467468\pi$
$350$	0	0
$351$	− 109.579i	− 0.312191i
$352$	0	0
$353$	−294.073	−0.833068	−0.416534	−	0.909120i	$-0.636755\pi$
$353$	−294.073	−0.833068	−0.416534	+	0.909120i	$0.636755\pi$
$354$	0	0
$355$	517.992	1.45913
$356$	0	0
$357$	54.7895i	0.153472i
$358$	0	0
$359$	218.162i	0.607694i	0.952721	+	0.303847i	$0.0982711\pi$
$359$	218.162i	0.607694i	−0.952721	+	0.303847i	$0.901729\pi$
$360$	0	0
$361$	−253.790	−0.703018
$362$	0	0
$363$	181.306	0.499465
$364$	0	0
$365$	441.535i	1.20968i
$366$	0	0
$367$	− 129.166i	− 0.351951i	−0.984395	−	0.175975i	$-0.943692\pi$
$367$	− 129.166i	− 0.351951i	0.984395	−	0.175975i	$-0.0563079\pi$
$368$	0	0
$369$	320.332	0.868108
$370$	0	0
$371$	23.5869	0.0635767
$372$	0	0
$373$	565.166i	1.51519i	0.652725	+	0.757595i	$0.273626\pi$
$373$	565.166i	1.51519i	−0.652725	+	0.757595i	$0.726374\pi$
$374$	0	0
$375$	− 106.421i	− 0.283789i
$376$	0	0
$377$	228.413	0.605870
$378$	0	0
$379$	−557.624	−1.47130	−0.735651	−	0.677361i	$-0.763124\pi$
$379$	−557.624	−1.47130	−0.735651	+	0.677361i	$0.763124\pi$
$380$	0	0
$381$	168.133i	0.441295i
$382$	0	0
$383$	141.668i	0.369890i	0.982749	+	0.184945i	$0.0592107\pi$
$383$	141.668i	0.369890i	−0.982749	+	0.184945i	$0.940789\pi$
$384$	0	0
$385$	−66.5830	−0.172943
$386$	0	0
$387$	473.697	1.22402
$388$	0	0
$389$	616.944i	1.58597i	0.609239	+	0.792987i	$0.291475\pi$
$389$	616.944i	1.58597i	−0.609239	+	0.792987i	$0.708525\pi$
$390$	0	0
$391$	− 241.166i	− 0.616793i
$392$	0	0
$393$	281.454	0.716167
$394$	0	0
$395$	−478.494	−1.21138
$396$	0	0
$397$	− 641.387i	− 1.61558i	−0.589468	−	0.807792i	$-0.700662\pi$
$397$	− 641.387i	− 1.61558i	0.589468	−	0.807792i	$-0.299338\pi$
$398$	0	0
$399$	− 45.0850i	− 0.112995i
$400$	0	0
$401$	−651.025	−1.62350	−0.811752	−	0.584003i	$-0.801486\pi$
$401$	−651.025	−1.62350	−0.811752	+	0.584003i	$0.801486\pi$
$402$	0	0
$403$	74.7451	0.185472
$404$	0	0
$405$	− 116.265i	− 0.287075i
$406$	0	0
$407$	93.6680i	0.230142i
$408$	0	0
$409$	13.1581	0.0321715	0.0160857	−	0.999871i	$-0.494880\pi$
$409$	13.1581	0.0321715	0.0160857	+	0.999871i	$0.494880\pi$
$410$	0	0
$411$	270.450	0.658028
$412$	0	0
$413$	− 119.557i	− 0.289484i
$414$	0	0
$415$	− 915.401i	− 2.20579i
$416$	0	0
$417$	291.948	0.700114
$418$	0	0
$419$	415.269	0.991096	0.495548	−	0.868581i	$-0.334967\pi$
$419$	415.269	0.991096	0.495548	+	0.868581i	$0.334967\pi$
$420$	0	0
$421$	487.409i	1.15774i	0.815419	+	0.578871i	$0.196506\pi$
$421$	487.409i	1.15774i	−0.815419	+	0.578871i	$0.803494\pi$
$422$	0	0
$423$	− 467.660i	− 1.10558i
$424$	0	0
$425$	420.996	0.990579
$426$	0	0
$427$	228.686	0.535565
$428$	0	0
$429$	− 23.5869i	− 0.0549812i
$430$	0	0
$431$	476.818i	1.10631i	0.833079	+	0.553153i	$0.186576\pi$
$431$	476.818i	1.10631i	−0.833079	+	0.553153i	$0.813424\pi$
$432$	0	0
$433$	416.907	0.962834	0.481417	−	0.876492i	$-0.340122\pi$
$433$	416.907	0.962834	0.481417	+	0.876492i	$0.340122\pi$
$434$	0	0
$435$	−660.073	−1.51741
$436$	0	0
$437$	198.450i	0.454118i
$438$	0	0
$439$	202.170i	0.460524i	0.973129	+	0.230262i	$0.0739583\pi$
$439$	202.170i	0.460524i	−0.973129	+	0.230262i	$0.926042\pi$
$440$	0	0
$441$	44.0405	0.0998651
$442$	0	0
$443$	358.494	0.809242	0.404621	−	0.914485i	$-0.367403\pi$
$443$	358.494	0.809242	0.404621	+	0.914485i	$0.367403\pi$
$444$	0	0
$445$	− 395.041i	− 0.887732i
$446$	0	0
$447$	33.1818i	0.0742321i
$448$	0	0
$449$	−420.656	−0.936873	−0.468437	−	0.883497i	$-0.655183\pi$
$449$	−420.656	−0.936873	−0.468437	+	0.883497i	$0.655183\pi$
$450$	0	0
$451$	167.587	0.371590
$452$	0	0
$453$	338.199i	0.746575i
$454$	0	0
$455$	− 88.0810i	− 0.193585i
$456$	0	0
$457$	331.778	0.725991	0.362995	−	0.931791i	$-0.381754\pi$
$457$	331.778	0.725991	0.362995	+	0.931791i	$0.381754\pi$
$458$	0	0
$459$	−316.664	−0.689900
$460$	0	0
$461$	− 210.480i	− 0.456572i	−0.973594	−	0.228286i	$-0.926688\pi$
$461$	− 210.480i	− 0.456572i	0.973594	−	0.228286i	$-0.0733121\pi$
$462$	0	0
$463$	36.6640i	0.0791880i	0.999216	+	0.0395940i	$0.0126065\pi$
$463$	36.6640i	0.0791880i	−0.999216	+	0.0395940i	$0.987394\pi$
$464$	0	0
$465$	−216.000	−0.464516
$466$	0	0
$467$	−45.4679	−0.0973617	−0.0486809	−	0.998814i	$-0.515502\pi$
$467$	−45.4679	−0.0973617	−0.0486809	+	0.998814i	$0.515502\pi$
$468$	0	0
$469$	− 301.166i	− 0.642145i
$470$	0	0
$471$	262.664i	0.557673i
$472$	0	0
$473$	247.822	0.523937
$474$	0	0
$475$	−346.427	−0.729321
$476$	0	0
$477$	56.0889i	0.117587i
$478$	0	0
$479$	− 155.320i	− 0.324259i	−0.986769	−	0.162130i	$-0.948164\pi$
$479$	− 155.320i	− 0.324259i	0.986769	−	0.162130i	$-0.0518363\pi$
$480$	0	0
$481$	−123.911	−0.257611
$482$	0	0
$483$	83.4536	0.172782
$484$	0	0
$485$	− 1261.52i	− 2.60107i
$486$	0	0
$487$	− 799.814i	− 1.64233i	−0.570691	−	0.821165i	$-0.693325\pi$
$487$	− 799.814i	− 1.64233i	0.570691	−	0.821165i	$-0.306675\pi$
$488$	0	0
$489$	415.069	0.848812
$490$	0	0
$491$	48.4131	0.0986009	0.0493005	−	0.998784i	$-0.484301\pi$
$491$	48.4131	0.0986009	0.0493005	+	0.998784i	$0.484301\pi$
$492$	0	0
$493$	− 660.073i	− 1.33889i
$494$	0	0
$495$	− 158.332i	− 0.319863i
$496$	0	0
$497$	−179.247	−0.360658
$498$	0	0
$499$	−748.397	−1.49979	−0.749897	−	0.661554i	$-0.769897\pi$
$499$	−748.397	−1.49979	−0.749897	+	0.661554i	$0.769897\pi$
$500$	0	0
$501$	61.1660i	0.122088i
$502$	0	0
$503$	− 564.000i	− 1.12127i	−0.828062	−	0.560636i	$-0.810557\pi$
$503$	− 564.000i	− 1.12127i	0.828062	−	0.560636i	$-0.189443\pi$
$504$	0	0
$505$	653.527	1.29411
$506$	0	0
$507$	−246.929	−0.487040
$508$	0	0
$509$	20.6052i	0.0404818i	0.999795	+	0.0202409i	$0.00644332\pi$
$509$	20.6052i	0.0404818i	−0.999795	+	0.0202409i	$0.993557\pi$
$510$	0	0
$511$	− 152.790i	− 0.299001i
$512$	0	0
$513$	260.575	0.507944
$514$	0	0
$515$	462.583	0.898219
$516$	0	0
$517$	− 244.664i	− 0.473238i
$518$	0	0
$519$	− 153.984i	− 0.296694i
$520$	0	0
$521$	717.158	1.37650	0.688252	−	0.725472i	$-0.258379\pi$
$521$	717.158	1.37650	0.688252	+	0.725472i	$0.258379\pi$
$522$	0	0
$523$	133.233	0.254747	0.127374	−	0.991855i	$-0.459345\pi$
$523$	133.233	0.254747	0.127374	+	0.991855i	$0.459345\pi$
$524$	0	0
$525$	145.682i	0.277490i
$526$	0	0
$527$	− 216.000i	− 0.409867i
$528$	0	0
$529$	161.664	0.305603
$530$	0	0
$531$	284.302	0.535409
$532$	0	0
$533$	221.697i	0.415941i
$534$	0	0
$535$	− 316.664i	− 0.591895i
$536$	0	0
$537$	326.818	0.608600
$538$	0	0
$539$	23.0405	0.0427468
$540$	0	0
$541$	454.907i	0.840863i	0.907324	+	0.420432i	$0.138121\pi$
$541$	454.907i	0.840863i	−0.907324	+	0.420432i	$0.861879\pi$
$542$	0	0
$543$	− 285.231i	− 0.525288i
$544$	0	0
$545$	−670.567	−1.23040
$546$	0	0
$547$	77.2105	0.141153	0.0705763	−	0.997506i	$-0.477516\pi$
$547$	77.2105	0.141153	0.0705763	+	0.997506i	$0.477516\pi$
$548$	0	0
$549$	543.808i	0.990542i
$550$	0	0
$551$	543.158i	0.985768i
$552$	0	0
$553$	165.579	0.299420
$554$	0	0
$555$	358.081	0.645191
$556$	0	0
$557$	10.4209i	0.0187090i	0.999956	+	0.00935452i	$0.00297768\pi$
$557$	10.4209i	0.0187090i	−0.999956	+	0.00935452i	$0.997022\pi$
$558$	0	0
$559$	327.838i	0.586472i
$560$	0	0
$561$	−68.1621	−0.121501
$562$	0	0
$563$	−116.775	−0.207416	−0.103708	−	0.994608i	$-0.533071\pi$
$563$	−116.775	−0.207416	−0.103708	+	0.994608i	$0.533071\pi$
$564$	0	0
$565$	− 969.401i	− 1.71575i
$566$	0	0
$567$	40.2327i	0.0709571i
$568$	0	0
$569$	884.855	1.55511	0.777553	−	0.628818i	$-0.216461\pi$
$569$	884.855	1.55511	0.777553	+	0.628818i	$0.216461\pi$
$570$	0	0
$571$	−475.616	−0.832952	−0.416476	−	0.909147i	$-0.636735\pi$
$571$	−475.616	−0.832952	−0.416476	+	0.909147i	$0.636735\pi$
$572$	0	0
$573$	171.984i	0.300147i
$574$	0	0
$575$	− 641.247i	− 1.11521i
$576$	0	0
$577$	409.571	0.709829	0.354914	−	0.934899i	$-0.384510\pi$
$577$	409.571	0.709829	0.354914	+	0.934899i	$0.384510\pi$
$578$	0	0
$579$	18.3163	0.0316343
$580$	0	0
$581$	316.767i	0.545211i
$582$	0	0
$583$	29.3438i	0.0503325i
$584$	0	0
$585$	209.454	0.358040
$586$	0	0
$587$	515.593	0.878353	0.439177	−	0.898401i	$-0.355270\pi$
$587$	515.593	0.878353	0.439177	+	0.898401i	$0.355270\pi$
$588$	0	0
$589$	177.741i	0.301768i
$590$	0	0
$591$	492.486i	0.833310i
$592$	0	0
$593$	−218.162	−0.367896	−0.183948	−	0.982936i	$-0.558888\pi$
$593$	−218.162	−0.367896	−0.183948	+	0.982936i	$0.558888\pi$
$594$	0	0
$595$	−254.539	−0.427796
$596$	0	0
$597$	500.575i	0.838484i
$598$	0	0
$599$	− 633.328i	− 1.05731i	−0.848837	−	0.528654i	$-0.822697\pi$
$599$	− 633.328i	− 1.05731i	0.848837	−	0.528654i	$-0.177303\pi$
$600$	0	0
$601$	−226.826	−0.377415	−0.188707	−	0.982033i	$-0.560430\pi$
$601$	−226.826	−0.377415	−0.188707	+	0.982033i	$0.560430\pi$
$602$	0	0
$603$	716.162	1.18767
$604$	0	0
$605$	842.302i	1.39223i
$606$	0	0
$607$	− 1117.81i	− 1.84154i	−0.390107	−	0.920770i	$-0.627562\pi$
$607$	− 1117.81i	− 1.84154i	0.390107	−	0.920770i	$-0.372438\pi$
$608$	0	0
$609$	228.413	0.375062
$610$	0	0
$611$	323.660	0.529722
$612$	0	0
$613$	− 69.1947i	− 0.112879i	−0.998406	−	0.0564394i	$-0.982025\pi$
$613$	− 69.1947i	− 0.112879i	0.998406	−	0.0564394i	$-0.0179748\pi$
$614$	0	0
$615$	− 640.664i	− 1.04173i
$616$	0	0
$617$	94.7738	0.153604	0.0768021	−	0.997046i	$-0.475529\pi$
$617$	94.7738	0.153604	0.0768021	+	0.997046i	$0.475529\pi$
$618$	0	0
$619$	1148.99	1.85621	0.928106	−	0.372317i	$-0.121437\pi$
$619$	1148.99	1.85621	0.928106	+	0.372317i	$0.121437\pi$
$620$	0	0
$621$	482.332i	0.776702i
$622$	0	0
$623$	136.701i	0.219423i
$624$	0	0
$625$	−342.033	−0.547252
$626$	0	0
$627$	56.0889	0.0894560
$628$	0	0
$629$	358.081i	0.569286i
$630$	0	0
$631$	775.320i	1.22872i	0.789027	+	0.614358i	$0.210585\pi$
$631$	775.320i	1.22872i	−0.789027	+	0.614358i	$0.789415\pi$
$632$	0	0
$633$	−561.328	−0.886774
$634$	0	0
$635$	−781.106	−1.23009
$636$	0	0
$637$	30.4797i	0.0478489i
$638$	0	0
$639$	− 426.243i	− 0.667047i
$640$	0	0
$641$	−981.681	−1.53148	−0.765742	−	0.643148i	$-0.777628\pi$
$641$	−981.681	−1.53148	−0.765742	+	0.643148i	$0.777628\pi$
$642$	0	0
$643$	−852.273	−1.32546	−0.662732	−	0.748857i	$-0.730603\pi$
$643$	−852.273	−1.32546	−0.662732	+	0.748857i	$0.730603\pi$
$644$	0	0
$645$	− 947.393i	− 1.46883i
$646$	0	0
$647$	131.174i	0.202742i	0.994849	+	0.101371i	$0.0323229\pi$
$647$	131.174i	0.202742i	−0.994849	+	0.101371i	$0.967677\pi$
$648$	0	0
$649$	148.737	0.229179
$650$	0	0
$651$	74.7451	0.114816
$652$	0	0
$653$	317.284i	0.485886i	0.970041	+	0.242943i	$0.0781128\pi$
$653$	317.284i	0.485886i	−0.970041	+	0.242943i	$0.921887\pi$
$654$	0	0
$655$	1307.56i	1.99628i
$656$	0	0
$657$	363.328	0.553011
$658$	0	0
$659$	844.677	1.28176	0.640878	−	0.767643i	$-0.278571\pi$
$659$	844.677	1.28176	0.640878	+	0.767643i	$0.278571\pi$
$660$	0	0
$661$	− 1034.44i	− 1.56496i	−0.622679	−	0.782478i	$-0.713956\pi$
$661$	− 1034.44i	− 1.56496i	0.622679	−	0.782478i	$-0.286044\pi$
$662$	0	0
$663$	− 90.1699i	− 0.136003i
$664$	0	0
$665$	209.454	0.314968
$666$	0	0
$667$	−1005.40	−1.50735
$668$	0	0
$669$	− 100.397i	− 0.150071i
$670$	0	0
$671$	284.502i	0.423997i
$672$	0	0
$673$	−164.996	−0.245165	−0.122583	−	0.992458i	$-0.539118\pi$
$673$	−164.996	−0.245165	−0.122583	+	0.992458i	$0.539118\pi$
$674$	0	0
$675$	−841.992	−1.24740
$676$	0	0
$677$	1017.98i	1.50366i	0.659357	+	0.751830i	$0.270829\pi$
$677$	1017.98i	1.50366i	−0.659357	+	0.751830i	$0.729171\pi$
$678$	0	0
$679$	436.539i	0.642914i
$680$	0	0
$681$	−243.195	−0.357114
$682$	0	0
$683$	−1050.39	−1.53791	−0.768953	−	0.639306i	$-0.779222\pi$
$683$	−1050.39	−1.53791	−0.768953	+	0.639306i	$0.779222\pi$
$684$	0	0
$685$	1256.44i	1.83422i
$686$	0	0
$687$	− 270.340i	− 0.393508i
$688$	0	0
$689$	−38.8182	−0.0563400
$690$	0	0
$691$	−487.003	−0.704779	−0.352390	−	0.935853i	$-0.614631\pi$
$691$	−487.003	−0.704779	−0.352390	+	0.935853i	$0.614631\pi$
$692$	0	0
$693$	54.7895i	0.0790614i
$694$	0	0
$695$	1356.32i	1.95153i
$696$	0	0
$697$	640.664	0.919174
$698$	0	0
$699$	562.288	0.804417
$700$	0	0
$701$	− 746.125i	− 1.06437i	−0.846627	−	0.532187i	$-0.821371\pi$
$701$	− 746.125i	− 1.06437i	0.846627	−	0.532187i	$-0.178629\pi$
$702$	0	0
$703$	− 294.656i	− 0.419141i
$704$	0	0
$705$	−935.320	−1.32670
$706$	0	0
$707$	−226.148	−0.319869
$708$	0	0
$709$	− 340.604i	− 0.480400i	−0.970723	−	0.240200i	$-0.922787\pi$
$709$	− 340.604i	− 0.480400i	0.970723	−	0.240200i	$-0.0772131\pi$
$710$	0	0
$711$	393.741i	0.553785i
$712$	0	0
$713$	−329.004	−0.461436
$714$	0	0
$715$	109.579	0.153257
$716$	0	0
$717$	− 548.295i	− 0.764708i
$718$	0	0
$719$	− 341.344i	− 0.474748i	−0.971418	−	0.237374i	$-0.923713\pi$
$719$	− 341.344i	− 0.474748i	0.971418	−	0.237374i	$-0.0762867\pi$
$720$	0	0
$721$	−160.073	−0.222015
$722$	0	0
$723$	301.446	0.416937
$724$	0	0
$725$	− 1755.10i	− 2.42082i
$726$	0	0
$727$	− 616.000i	− 0.847318i	−0.905822	−	0.423659i	$-0.860746\pi$
$727$	− 616.000i	− 0.847318i	0.905822	−	0.423659i	$-0.139254\pi$
$728$	0	0
$729$	277.081	0.380084
$730$	0	0
$731$	947.393	1.29602
$732$	0	0
$733$	211.160i	0.288076i	0.989572	+	0.144038i	$0.0460087\pi$
$733$	211.160i	0.288076i	−0.989572	+	0.144038i	$0.953991\pi$
$734$	0	0
$735$	− 88.0810i	− 0.119838i
$736$	0	0
$737$	374.672	0.508374
$738$	0	0
$739$	624.959	0.845683	0.422841	−	0.906204i	$-0.361033\pi$
$739$	624.959	0.845683	0.422841	+	0.906204i	$0.361033\pi$
$740$	0	0
$741$	74.1987i	0.100133i
$742$	0	0
$743$	153.158i	0.206135i	0.994674	+	0.103067i	$0.0328657\pi$
$743$	153.158i	0.206135i	−0.994674	+	0.103067i	$0.967134\pi$
$744$	0	0
$745$	−154.154	−0.206918
$746$	0	0
$747$	−753.261	−1.00838
$748$	0	0
$749$	109.579i	0.146300i
$750$	0	0
$751$	940.463i	1.25228i	0.779710	+	0.626140i	$0.215366\pi$
$751$	940.463i	1.25228i	−0.779710	+	0.626140i	$0.784634\pi$
$752$	0	0
$753$	433.129	0.575205
$754$	0	0
$755$	−1571.19	−2.08104
$756$	0	0
$757$	− 110.052i	− 0.145380i	−0.997355	−	0.0726898i	$-0.976842\pi$
$757$	− 110.052i	− 0.145380i	0.997355	−	0.0726898i	$-0.0231583\pi$
$758$	0	0
$759$	103.822i	0.136788i
$760$	0	0
$761$	−909.571	−1.19523	−0.597616	−	0.801783i	$-0.703885\pi$
$761$	−909.571	−1.19523	−0.597616	+	0.801783i	$0.703885\pi$
$762$	0	0
$763$	232.044	0.304121
$764$	0	0
$765$	− 605.284i	− 0.791220i
$766$	0	0
$767$	196.761i	0.256533i
$768$	0	0
$769$	−427.328	−0.555693	−0.277847	−	0.960625i	$-0.589621\pi$
$769$	−427.328	−0.555693	−0.277847	+	0.960625i	$0.589621\pi$
$770$	0	0
$771$	−312.426	−0.405222
$772$	0	0
$773$	589.342i	0.762409i	0.924491	+	0.381205i	$0.124491\pi$
$773$	589.342i	0.762409i	−0.924491	+	0.381205i	$0.875509\pi$
$774$	0	0
$775$	− 574.332i	− 0.741074i
$776$	0	0
$777$	−123.911	−0.159474
$778$	0	0
$779$	−527.187	−0.676748
$780$	0	0
$781$	− 222.996i	− 0.285526i
$782$	0	0
$783$	1320.15i	1.68601i
$784$	0	0
$785$	−1220.27	−1.55449
$786$	0	0
$787$	1126.16	1.43095	0.715477	−	0.698636i	$-0.246209\pi$
$787$	1126.16	1.43095	0.715477	+	0.698636i	$0.246209\pi$
$788$	0	0
$789$	293.636i	0.372163i
$790$	0	0
$791$	335.454i	0.424088i
$792$	0	0
$793$	−376.361	−0.474604
$794$	0	0
$795$	112.178	0.141104
$796$	0	0
$797$	− 536.339i	− 0.672947i	−0.941693	−	0.336473i	$-0.890766\pi$
$797$	− 536.339i	− 0.672947i	0.941693	−	0.336473i	$-0.109234\pi$
$798$	0	0
$799$	− 935.320i	− 1.17061i
$800$	0	0
$801$	−325.069	−0.405829
$802$	0	0
$803$	190.081	0.236714
$804$	0	0
$805$	387.705i	0.481621i
$806$	0	0
$807$	202.178i	0.250530i
$808$	0	0
$809$	164.782	0.203686	0.101843	−	0.994801i	$-0.467526\pi$
$809$	164.782	0.203686	0.101843	+	0.994801i	$0.467526\pi$
$810$	0	0
$811$	903.918	1.11457	0.557286	−	0.830321i	$-0.311843\pi$
$811$	903.918	1.11457	0.557286	+	0.830321i	$0.311843\pi$
$812$	0	0
$813$	− 885.401i	− 1.08905i
$814$	0	0
$815$	1928.31i	2.36602i
$816$	0	0
$817$	−779.587	−0.954207
$818$	0	0
$819$	−72.4797	−0.0884979
$820$	0	0
$821$	1206.32i	1.46933i	0.678432	+	0.734663i	$0.262660\pi$
$821$	1206.32i	1.46933i	−0.678432	+	0.734663i	$0.737340\pi$
$822$	0	0
$823$	1430.05i	1.73761i	0.495158	+	0.868803i	$0.335110\pi$
$823$	1430.05i	1.73761i	−0.495158	+	0.868803i	$0.664890\pi$
$824$	0	0
$825$	−181.239	−0.219684
$826$	0	0
$827$	1327.97	1.60577	0.802883	−	0.596137i	$-0.203298\pi$
$827$	1327.97	1.60577	0.802883	+	0.596137i	$0.203298\pi$
$828$	0	0
$829$	224.205i	0.270453i	0.990815	+	0.135226i	$0.0431761\pi$
$829$	224.205i	0.270453i	−0.990815	+	0.135226i	$0.956824\pi$
$830$	0	0
$831$	− 33.8615i	− 0.0407480i
$832$	0	0
$833$	88.0810	0.105740
$834$	0	0
$835$	−284.162	−0.340314
$836$	0	0
$837$	432.000i	0.516129i
$838$	0	0
$839$	− 1544.16i	− 1.84048i	−0.391355	−	0.920240i	$-0.627994\pi$
$839$	− 1544.16i	− 1.84048i	0.391355	−	0.920240i	$-0.372006\pi$
$840$	0	0
$841$	−1910.79	−2.27205
$842$	0	0
$843$	356.562	0.422968
$844$	0	0
$845$	− 1147.17i	− 1.35760i
$846$	0	0
$847$	− 291.472i	− 0.344123i
$848$	0	0
$849$	17.3804	0.0204716
$850$	0	0
$851$	545.417	0.640913
$852$	0	0
$853$	1094.24i	1.28282i	0.767200	+	0.641408i	$0.221649\pi$
$853$	1094.24i	1.28282i	−0.767200	+	0.641408i	$0.778351\pi$
$854$	0	0
$855$	498.073i	0.582542i
$856$	0	0
$857$	1504.64	1.75571	0.877853	−	0.478930i	$-0.158975\pi$
$857$	1504.64	1.75571	0.877853	+	0.478930i	$0.158975\pi$
$858$	0	0
$859$	7.40264	0.00861774	0.00430887	−	0.999991i	$-0.498628\pi$
$859$	7.40264	0.00861774	0.00430887	+	0.999991i	$0.498628\pi$
$860$	0	0
$861$	221.697i	0.257487i
$862$	0	0
$863$	− 248.648i	− 0.288121i	−0.989569	−	0.144060i	$-0.953984\pi$
$863$	− 248.648i	− 0.288121i	0.989569	−	0.144060i	$-0.0460160\pi$
$864$	0	0
$865$	715.373	0.827020
$866$	0	0
$867$	215.047	0.248036
$868$	0	0
$869$	205.992i	0.237045i
$870$	0	0
$871$	495.644i	0.569052i
$872$	0	0
$873$	−1038.07	−1.18909
$874$	0	0
$875$	−171.085	−0.195526
$876$	0	0
$877$	689.210i	0.785873i	0.919566	+	0.392936i	$0.128541\pi$
$877$	689.210i	0.785873i	−0.919566	+	0.392936i	$0.871459\pi$
$878$	0	0
$879$	648.170i	0.737395i
$880$	0	0
$881$	−1017.23	−1.15463	−0.577316	−	0.816521i	$-0.695900\pi$
$881$	−1017.23	−1.15463	−0.577316	+	0.816521i	$0.695900\pi$
$882$	0	0
$883$	1018.71	1.15370	0.576848	−	0.816852i	$-0.304282\pi$
$883$	1018.71	1.15370	0.576848	+	0.816852i	$0.304282\pi$
$884$	0	0
$885$	− 568.604i	− 0.642490i
$886$	0	0
$887$	− 616.324i	− 0.694841i	−0.937709	−	0.347421i	$-0.887058\pi$
$887$	− 616.324i	− 0.694841i	0.937709	−	0.347421i	$-0.112942\pi$
$888$	0	0
$889$	270.295	0.304044
$890$	0	0
$891$	−50.0523	−0.0561755
$892$	0	0
$893$	769.652i	0.861873i
$894$	0	0
$895$	1518.32i	1.69644i
$896$	0	0
$897$	−137.344	−0.153115
$898$	0	0
$899$	−900.486	−1.00165
$900$	0	0
$901$	112.178i	0.124504i
$902$	0	0
$903$	327.838i	0.363054i
$904$	0	0
$905$	1325.11	1.46421
$906$	0	0
$907$	−949.166	−1.04649	−0.523245	−	0.852182i	$-0.675279\pi$
$907$	−949.166	−1.04649	−0.523245	+	0.852182i	$0.675279\pi$
$908$	0	0
$909$	− 537.771i	− 0.591608i
$910$	0	0
$911$	− 1034.84i	− 1.13594i	−0.823049	−	0.567970i	$-0.807729\pi$
$911$	− 1034.84i	− 1.13594i	0.823049	−	0.567970i	$-0.192271\pi$
$912$	0	0
$913$	−394.081	−0.431633
$914$	0	0
$915$	1087.62	1.18865
$916$	0	0
$917$	− 452.472i	− 0.493426i
$918$	0	0
$919$	486.761i	0.529664i	0.964295	+	0.264832i	$0.0853164\pi$
$919$	486.761i	0.529664i	−0.964295	+	0.264832i	$0.914684\pi$
$920$	0	0
$921$	142.651	0.154887
$922$	0	0
$923$	294.996	0.319606
$924$	0	0
$925$	952.118i	1.02932i
$926$	0	0
$927$	− 380.648i	− 0.410624i
$928$	0	0
$929$	−625.749	−0.673573	−0.336786	−	0.941581i	$-0.609340\pi$
$929$	−625.749	−0.673573	−0.336786	+	0.941581i	$0.609340\pi$
$930$	0	0
$931$	−72.4797	−0.0778515
$932$	0	0
$933$	− 757.020i	− 0.811382i
$934$	0	0
$935$	− 316.664i	− 0.338678i
$936$	0	0
$937$	528.405	0.563933	0.281966	−	0.959424i	$-0.409013\pi$
$937$	528.405	0.563933	0.281966	+	0.959424i	$0.409013\pi$
$938$	0	0
$939$	−394.847	−0.420497
$940$	0	0
$941$	− 293.800i	− 0.312221i	−0.987740	−	0.156110i	$-0.950104\pi$
$941$	− 293.800i	− 0.312221i	0.987740	−	0.156110i	$-0.0498956\pi$
$942$	0	0
$943$	− 975.838i	− 1.03482i
$944$	0	0
$945$	509.077	0.538706
$946$	0	0
$947$	234.376	0.247494	0.123747	−	0.992314i	$-0.460509\pi$
$947$	234.376	0.247494	0.123747	+	0.992314i	$0.460509\pi$
$948$	0	0
$949$	251.454i	0.264967i
$950$	0	0
$951$	257.297i	0.270554i
$952$	0	0
$953$	−17.5138	−0.0183775	−0.00918876	−	0.999958i	$-0.502925\pi$
$953$	−17.5138	−0.0183775	−0.00918876	+	0.999958i	$0.502925\pi$
$954$	0	0
$955$	−798.996	−0.836645
$956$	0	0
$957$	284.162i	0.296930i
$958$	0	0
$959$	− 434.782i	− 0.453370i
$960$	0	0
$961$	666.328	0.693369
$962$	0	0
$963$	−260.575	−0.270587
$964$	0	0
$965$	85.0928i	0.0881791i
$966$	0	0
$967$	− 482.146i	− 0.498600i	−0.968426	−	0.249300i	$-0.919799\pi$
$967$	− 482.146i	− 0.498600i	0.968426	−	0.249300i	$-0.0802005\pi$
$968$	0	0
$969$	214.421	0.221281
$970$	0	0
$971$	−784.524	−0.807955	−0.403977	−	0.914769i	$-0.632373\pi$
$971$	−784.524	−0.807955	−0.403977	+	0.914769i	$0.632373\pi$
$972$	0	0
$973$	− 469.342i	− 0.482366i
$974$	0	0
$975$	− 239.757i	− 0.245905i
$976$	0	0
$977$	1158.97	1.18626	0.593128	−	0.805108i	$-0.297893\pi$
$977$	1158.97	1.18626	0.593128	+	0.805108i	$0.297893\pi$
$978$	0	0
$979$	−170.065	−0.173713
$980$	0	0
$981$	551.793i	0.562481i
$982$	0	0
$983$	− 559.629i	− 0.569307i	−0.958630	−	0.284653i	$-0.908121\pi$
$983$	− 559.629i	− 0.569307i	0.958630	−	0.284653i	$-0.0918785\pi$
$984$	0	0
$985$	−2287.97	−2.32281
$986$	0	0
$987$	323.660	0.327923
$988$	0	0
$989$	− 1443.04i	− 1.45909i
$990$	0	0
$991$	641.093i	0.646915i	0.946243	+	0.323458i	$0.104845\pi$
$991$	641.093i	0.646915i	−0.946243	+	0.323458i	$0.895155\pi$
$992$	0	0
$993$	339.231	0.341623
$994$	0	0
$995$	−2325.55	−2.33723
$996$	0	0
$997$	1114.86i	1.11822i	0.829094	+	0.559109i	$0.188857\pi$
$997$	1114.86i	1.11822i	−0.829094	+	0.559109i	$0.811143\pi$
$998$	0	0
$999$	− 716.162i	− 0.716879i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.g.b.351.1	yes	4
4.3	odd	2		448.3.g.a.351.3	✓	4
8.3	odd	2	inner	448.3.g.b.351.2	yes	4
8.5	even	2		448.3.g.a.351.4	yes	4
16.3	odd	4		1792.3.d.a.1023.3		4
16.5	even	4		1792.3.d.c.1023.3		4
16.11	odd	4		1792.3.d.c.1023.2		4
16.13	even	4		1792.3.d.a.1023.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.3.g.a.351.3	✓	4	4.3	odd	2
448.3.g.a.351.4	yes	4	8.5	even	2
448.3.g.b.351.1	yes	4	1.1	even	1	trivial
448.3.g.b.351.2	yes	4	8.3	odd	2	inner
1792.3.d.a.1023.2		4	16.13	even	4
1792.3.d.a.1023.3		4	16.3	odd	4
1792.3.d.c.1023.2		4	16.11	odd	4
1792.3.d.c.1023.3		4	16.5	even	4

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.64575 q^{3} -7.64575i q^{5} +2.64575i q^{7} -6.29150 q^{9} -3.29150 q^{11} -4.35425i q^{13} +12.5830i q^{15} -12.5830 q^{17} +10.3542 q^{19} -4.35425i q^{21} +19.1660i q^{23} -33.4575 q^{25} +25.1660 q^{27} +52.4575i q^{29} +17.1660i q^{31} +5.41699 q^{33} +20.2288 q^{35} -28.4575i q^{37} +7.16601i q^{39} -50.9150 q^{41} -75.2915 q^{43} +48.1033i q^{45} +74.3320i q^{47} -7.00000 q^{49} +20.7085 q^{51} -8.91503i q^{53} +25.1660i q^{55} -17.0405 q^{57} -45.1882 q^{59} -86.4353i q^{61} -16.6458i q^{63} -33.2915 q^{65} -113.830 q^{67} -31.5425i q^{69} +67.7490i q^{71} -57.7490 q^{73} +55.0627 q^{75} -8.70850i q^{77} -62.5830i q^{79} +15.2065 q^{81} +119.727 q^{83} +96.2065i q^{85} -86.3320i q^{87} +51.6680 q^{89} +11.5203 q^{91} -28.2510i q^{93} -79.1660i q^{95} +164.996 q^{97} +20.7085 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{9} + 8 q^{11} - 8 q^{17} + 52 q^{19} - 28 q^{25} + 16 q^{27} + 64 q^{33} + 28 q^{35} + 8 q^{41} - 280 q^{43} - 28 q^{49} + 104 q^{51} + 80 q^{57} - 276 q^{59} - 112 q^{65} - 32 q^{67}+ \cdots + 104 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^9 + 8 * q^11 - 8 * q^17 + 52 * q^19 - 28 * q^25 + 16 * q^27 + 64 * q^33 + 28 * q^35 + 8 * q^41 - 280 * q^43 - 28 * q^49 + 104 * q^51 + 80 * q^57 - 276 * q^59 - 112 * q^65 - 32 * q^67 - 104 * q^73 + 252 * q^75 - 172 * q^81 + 172 * q^83 + 376 * q^89 - 28 * q^91 + 152 * q^97 + 104 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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