LMFDB - Embedded newform 448.4.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,4,Mod(1,448)] 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("448.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-8,0,14,0,-7] 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:	yes
Analytic conductor:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	448.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{3} +14.0000 q^{5} -7.00000 q^{7} +37.0000 q^{9} +28.0000 q^{11} -18.0000 q^{13} -112.000 q^{15} +74.0000 q^{17} -80.0000 q^{19} +56.0000 q^{21} -112.000 q^{23} +71.0000 q^{25} -80.0000 q^{27} -190.000 q^{29} +72.0000 q^{31} -224.000 q^{33} -98.0000 q^{35} +346.000 q^{37} +144.000 q^{39} +162.000 q^{41} +412.000 q^{43} +518.000 q^{45} +24.0000 q^{47} +49.0000 q^{49} -592.000 q^{51} -318.000 q^{53} +392.000 q^{55} +640.000 q^{57} +200.000 q^{59} +198.000 q^{61} -259.000 q^{63} -252.000 q^{65} +716.000 q^{67} +896.000 q^{69} +392.000 q^{71} +538.000 q^{73} -568.000 q^{75} -196.000 q^{77} +240.000 q^{79} -359.000 q^{81} +1072.00 q^{83} +1036.00 q^{85} +1520.00 q^{87} +810.000 q^{89} +126.000 q^{91} -576.000 q^{93} -1120.00 q^{95} +1354.00 q^{97} +1036.00 q^{99} +O(q^{100})$

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
$3$	−8.00000	−1.53960	−0.769800	−	0.638285i	$-0.779644\pi$
$3$	−8.00000	−1.53960	−0.769800	+	0.638285i	$0.779644\pi$
$4$	0	0
$5$	14.0000	1.25220	0.626099	−	0.779744i	$-0.284651\pi$
$5$	14.0000	1.25220	0.626099	+	0.779744i	$0.284651\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	28.0000	0.767483	0.383742	−	0.923440i	$-0.374635\pi$
$11$	28.0000	0.767483	0.383742	+	0.923440i	$0.374635\pi$
$12$	0	0
$13$	−18.0000	−0.384023	−0.192012	−	0.981393i	$-0.561501\pi$
$13$	−18.0000	−0.384023	−0.192012	+	0.981393i	$0.561501\pi$
$14$	0	0
$15$	−112.000	−1.92789
$16$	0	0
$17$	74.0000	1.05574	0.527872	−	0.849324i	$-0.322990\pi$
$17$	74.0000	1.05574	0.527872	+	0.849324i	$0.322990\pi$
$18$	0	0
$19$	−80.0000	−0.965961	−0.482980	−	0.875631i	$-0.660446\pi$
$19$	−80.0000	−0.965961	−0.482980	+	0.875631i	$0.660446\pi$
$20$	0	0
$21$	56.0000	0.581914
$22$	0	0
$23$	−112.000	−1.01537	−0.507687	−	0.861541i	$-0.669499\pi$
$23$	−112.000	−1.01537	−0.507687	+	0.861541i	$0.669499\pi$
$24$	0	0
$25$	71.0000	0.568000
$26$	0	0
$27$	−80.0000	−0.570222
$28$	0	0
$29$	−190.000	−1.21662	−0.608312	−	0.793698i	$-0.708153\pi$
$29$	−190.000	−1.21662	−0.608312	+	0.793698i	$0.708153\pi$
$30$	0	0
$31$	72.0000	0.417148	0.208574	−	0.978007i	$-0.433118\pi$
$31$	72.0000	0.417148	0.208574	+	0.978007i	$0.433118\pi$
$32$	0	0
$33$	−224.000	−1.18162
$34$	0	0
$35$	−98.0000	−0.473286
$36$	0	0
$37$	346.000	1.53735	0.768676	−	0.639638i	$-0.220916\pi$
$37$	346.000	1.53735	0.768676	+	0.639638i	$0.220916\pi$
$38$	0	0
$39$	144.000	0.591242
$40$	0	0
$41$	162.000	0.617077	0.308538	−	0.951212i	$-0.400160\pi$
$41$	162.000	0.617077	0.308538	+	0.951212i	$0.400160\pi$
$42$	0	0
$43$	412.000	1.46115	0.730575	−	0.682833i	$-0.239252\pi$
$43$	412.000	1.46115	0.730575	+	0.682833i	$0.239252\pi$
$44$	0	0
$45$	518.000	1.71598
$46$	0	0
$47$	24.0000	0.0744843	0.0372421	−	0.999306i	$-0.488143\pi$
$47$	24.0000	0.0744843	0.0372421	+	0.999306i	$0.488143\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−592.000	−1.62542
$52$	0	0
$53$	−318.000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−318.000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	392.000	0.961041
$56$	0	0
$57$	640.000	1.48719
$58$	0	0
$59$	200.000	0.441318	0.220659	−	0.975351i	$-0.429179\pi$
$59$	200.000	0.441318	0.220659	+	0.975351i	$0.429179\pi$
$60$	0	0
$61$	198.000	0.415595	0.207798	−	0.978172i	$-0.433370\pi$
$61$	198.000	0.415595	0.207798	+	0.978172i	$0.433370\pi$
$62$	0	0
$63$	−259.000	−0.517951
$64$	0	0
$65$	−252.000	−0.480873
$66$	0	0
$67$	716.000	1.30557	0.652786	−	0.757542i	$-0.273600\pi$
$67$	716.000	1.30557	0.652786	+	0.757542i	$0.273600\pi$
$68$	0	0
$69$	896.000	1.56327
$70$	0	0
$71$	392.000	0.655237	0.327619	−	0.944810i	$-0.393754\pi$
$71$	392.000	0.655237	0.327619	+	0.944810i	$0.393754\pi$
$72$	0	0
$73$	538.000	0.862577	0.431289	−	0.902214i	$-0.358059\pi$
$73$	538.000	0.862577	0.431289	+	0.902214i	$0.358059\pi$
$74$	0	0
$75$	−568.000	−0.874493
$76$	0	0
$77$	−196.000	−0.290081
$78$	0	0
$79$	240.000	0.341799	0.170899	−	0.985288i	$-0.445333\pi$
$79$	240.000	0.341799	0.170899	+	0.985288i	$0.445333\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	1072.00	1.41768	0.708839	−	0.705370i	$-0.249219\pi$
$83$	1072.00	1.41768	0.708839	+	0.705370i	$0.249219\pi$
$84$	0	0
$85$	1036.00	1.32200
$86$	0	0
$87$	1520.00	1.87312
$88$	0	0
$89$	810.000	0.964717	0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000	0.964717	0.482359	+	0.875974i	$0.339780\pi$
$90$	0	0
$91$	126.000	0.145147
$92$	0	0
$93$	−576.000	−0.642241
$94$	0	0
$95$	−1120.00	−1.20957
$96$	0	0
$97$	1354.00	1.41730	0.708649	−	0.705561i	$-0.249305\pi$
$97$	1354.00	1.41730	0.708649	+	0.705561i	$0.249305\pi$
$98$	0	0
$99$	1036.00	1.05174
$100$	0	0
$101$	1358.00	1.33788	0.668941	−	0.743316i	$-0.266748\pi$
$101$	1358.00	1.33788	0.668941	+	0.743316i	$0.266748\pi$
$102$	0	0
$103$	−832.000	−0.795916	−0.397958	−	0.917404i	$-0.630281\pi$
$103$	−832.000	−0.795916	−0.397958	+	0.917404i	$0.630281\pi$
$104$	0	0
$105$	784.000	0.728672
$106$	0	0
$107$	−444.000	−0.401150	−0.200575	−	0.979678i	$-0.564281\pi$
$107$	−444.000	−0.401150	−0.200575	+	0.979678i	$0.564281\pi$
$108$	0	0
$109$	−1870.00	−1.64324	−0.821622	−	0.570033i	$-0.806930\pi$
$109$	−1870.00	−1.64324	−0.821622	+	0.570033i	$0.806930\pi$
$110$	0	0
$111$	−2768.00	−2.36691
$112$	0	0
$113$	1378.00	1.14718	0.573590	−	0.819143i	$-0.305550\pi$
$113$	1378.00	1.14718	0.573590	+	0.819143i	$0.305550\pi$
$114$	0	0
$115$	−1568.00	−1.27145
$116$	0	0
$117$	−666.000	−0.526254
$118$	0	0
$119$	−518.000	−0.399033
$120$	0	0
$121$	−547.000	−0.410969
$122$	0	0
$123$	−1296.00	−0.950052
$124$	0	0
$125$	−756.000	−0.540950
$126$	0	0
$127$	1944.00	1.35828	0.679142	−	0.734007i	$-0.262352\pi$
$127$	1944.00	1.35828	0.679142	+	0.734007i	$0.262352\pi$
$128$	0	0
$129$	−3296.00	−2.24959
$130$	0	0
$131$	848.000	0.565573	0.282787	−	0.959183i	$-0.408741\pi$
$131$	848.000	0.565573	0.282787	+	0.959183i	$0.408741\pi$
$132$	0	0
$133$	560.000	0.365099
$134$	0	0
$135$	−1120.00	−0.714031
$136$	0	0
$137$	−2966.00	−1.84965	−0.924827	−	0.380389i	$-0.875790\pi$
$137$	−2966.00	−1.84965	−0.924827	+	0.380389i	$0.875790\pi$
$138$	0	0
$139$	−2800.00	−1.70858	−0.854291	−	0.519795i	$-0.826008\pi$
$139$	−2800.00	−1.70858	−0.854291	+	0.519795i	$0.826008\pi$
$140$	0	0
$141$	−192.000	−0.114676
$142$	0	0
$143$	−504.000	−0.294731
$144$	0	0
$145$	−2660.00	−1.52346
$146$	0	0
$147$	−392.000	−0.219943
$148$	0	0
$149$	−510.000	−0.280408	−0.140204	−	0.990123i	$-0.544776\pi$
$149$	−510.000	−0.280408	−0.140204	+	0.990123i	$0.544776\pi$
$150$	0	0
$151$	592.000	0.319048	0.159524	−	0.987194i	$-0.449004\pi$
$151$	592.000	0.319048	0.159524	+	0.987194i	$0.449004\pi$
$152$	0	0
$153$	2738.00	1.44676
$154$	0	0
$155$	1008.00	0.522352
$156$	0	0
$157$	2686.00	1.36539	0.682695	−	0.730704i	$-0.260808\pi$
$157$	2686.00	1.36539	0.682695	+	0.730704i	$0.260808\pi$
$158$	0	0
$159$	2544.00	1.26888
$160$	0	0
$161$	784.000	0.383776
$162$	0	0
$163$	1012.00	0.486294	0.243147	−	0.969989i	$-0.421820\pi$
$163$	1012.00	0.486294	0.243147	+	0.969989i	$0.421820\pi$
$164$	0	0
$165$	−3136.00	−1.47962
$166$	0	0
$167$	544.000	0.252072	0.126036	−	0.992026i	$-0.459775\pi$
$167$	544.000	0.252072	0.126036	+	0.992026i	$0.459775\pi$
$168$	0	0
$169$	−1873.00	−0.852526
$170$	0	0
$171$	−2960.00	−1.32372
$172$	0	0
$173$	−1858.00	−0.816538	−0.408269	−	0.912862i	$-0.633868\pi$
$173$	−1858.00	−0.816538	−0.408269	+	0.912862i	$0.633868\pi$
$174$	0	0
$175$	−497.000	−0.214684
$176$	0	0
$177$	−1600.00	−0.679454
$178$	0	0
$179$	300.000	0.125268	0.0626342	−	0.998037i	$-0.480050\pi$
$179$	300.000	0.125268	0.0626342	+	0.998037i	$0.480050\pi$
$180$	0	0
$181$	2358.00	0.968336	0.484168	−	0.874975i	$-0.339122\pi$
$181$	2358.00	0.968336	0.484168	+	0.874975i	$0.339122\pi$
$182$	0	0
$183$	−1584.00	−0.639851
$184$	0	0
$185$	4844.00	1.92507
$186$	0	0
$187$	2072.00	0.810265
$188$	0	0
$189$	560.000	0.215524
$190$	0	0
$191$	1392.00	0.527338	0.263669	−	0.964613i	$-0.415067\pi$
$191$	1392.00	0.527338	0.263669	+	0.964613i	$0.415067\pi$
$192$	0	0
$193$	1778.00	0.663126	0.331563	−	0.943433i	$-0.392424\pi$
$193$	1778.00	0.663126	0.331563	+	0.943433i	$0.392424\pi$
$194$	0	0
$195$	2016.00	0.740353
$196$	0	0
$197$	−1214.00	−0.439055	−0.219528	−	0.975606i	$-0.570452\pi$
$197$	−1214.00	−0.439055	−0.219528	+	0.975606i	$0.570452\pi$
$198$	0	0
$199$	1040.00	0.370471	0.185235	−	0.982694i	$-0.440695\pi$
$199$	1040.00	0.370471	0.185235	+	0.982694i	$0.440695\pi$
$200$	0	0
$201$	−5728.00	−2.01006
$202$	0	0
$203$	1330.00	0.459841
$204$	0	0
$205$	2268.00	0.772702
$206$	0	0
$207$	−4144.00	−1.39144
$208$	0	0
$209$	−2240.00	−0.741359
$210$	0	0
$211$	3868.00	1.26201	0.631005	−	0.775779i	$-0.282643\pi$
$211$	3868.00	1.26201	0.631005	+	0.775779i	$0.282643\pi$
$212$	0	0
$213$	−3136.00	−1.00880
$214$	0	0
$215$	5768.00	1.82965
$216$	0	0
$217$	−504.000	−0.157667
$218$	0	0
$219$	−4304.00	−1.32802
$220$	0	0
$221$	−1332.00	−0.405430
$222$	0	0
$223$	3968.00	1.19156	0.595778	−	0.803149i	$-0.296844\pi$
$223$	3968.00	1.19156	0.595778	+	0.803149i	$0.296844\pi$
$224$	0	0
$225$	2627.00	0.778370
$226$	0	0
$227$	3936.00	1.15084	0.575422	−	0.817857i	$-0.304838\pi$
$227$	3936.00	1.15084	0.575422	+	0.817857i	$0.304838\pi$
$228$	0	0
$229$	−4810.00	−1.38801	−0.694004	−	0.719971i	$-0.744155\pi$
$229$	−4810.00	−1.38801	−0.694004	+	0.719971i	$0.744155\pi$
$230$	0	0
$231$	1568.00	0.446610
$232$	0	0
$233$	−2182.00	−0.613509	−0.306754	−	0.951789i	$-0.599243\pi$
$233$	−2182.00	−0.613509	−0.306754	+	0.951789i	$0.599243\pi$
$234$	0	0
$235$	336.000	0.0932690
$236$	0	0
$237$	−1920.00	−0.526234
$238$	0	0
$239$	−3000.00	−0.811941	−0.405970	−	0.913886i	$-0.633066\pi$
$239$	−3000.00	−0.811941	−0.405970	+	0.913886i	$0.633066\pi$
$240$	0	0
$241$	2042.00	0.545796	0.272898	−	0.962043i	$-0.412018\pi$
$241$	2042.00	0.545796	0.272898	+	0.962043i	$0.412018\pi$
$242$	0	0
$243$	5032.00	1.32841
$244$	0	0
$245$	686.000	0.178885
$246$	0	0
$247$	1440.00	0.370951
$248$	0	0
$249$	−8576.00	−2.18266
$250$	0	0
$251$	528.000	0.132777	0.0663886	−	0.997794i	$-0.478852\pi$
$251$	528.000	0.132777	0.0663886	+	0.997794i	$0.478852\pi$
$252$	0	0
$253$	−3136.00	−0.779283
$254$	0	0
$255$	−8288.00	−2.03535
$256$	0	0
$257$	5634.00	1.36747	0.683734	−	0.729731i	$-0.260355\pi$
$257$	5634.00	1.36747	0.683734	+	0.729731i	$0.260355\pi$
$258$	0	0
$259$	−2422.00	−0.581065
$260$	0	0
$261$	−7030.00	−1.66723
$262$	0	0
$263$	168.000	0.0393891	0.0196945	−	0.999806i	$-0.493731\pi$
$263$	168.000	0.0393891	0.0196945	+	0.999806i	$0.493731\pi$
$264$	0	0
$265$	−4452.00	−1.03202
$266$	0	0
$267$	−6480.00	−1.48528
$268$	0	0
$269$	1310.00	0.296922	0.148461	−	0.988918i	$-0.452568\pi$
$269$	1310.00	0.296922	0.148461	+	0.988918i	$0.452568\pi$
$270$	0	0
$271$	−2208.00	−0.494932	−0.247466	−	0.968897i	$-0.579598\pi$
$271$	−2208.00	−0.494932	−0.247466	+	0.968897i	$0.579598\pi$
$272$	0	0
$273$	−1008.00	−0.223469
$274$	0	0
$275$	1988.00	0.435931
$276$	0	0
$277$	−5294.00	−1.14832	−0.574162	−	0.818742i	$-0.694672\pi$
$277$	−5294.00	−1.14832	−0.574162	+	0.818742i	$0.694672\pi$
$278$	0	0
$279$	2664.00	0.571647
$280$	0	0
$281$	3242.00	0.688262	0.344131	−	0.938922i	$-0.388174\pi$
$281$	3242.00	0.688262	0.344131	+	0.938922i	$0.388174\pi$
$282$	0	0
$283$	1592.00	0.334398	0.167199	−	0.985923i	$-0.446528\pi$
$283$	1592.00	0.334398	0.167199	+	0.985923i	$0.446528\pi$
$284$	0	0
$285$	8960.00	1.86226
$286$	0	0
$287$	−1134.00	−0.233233
$288$	0	0
$289$	563.000	0.114594
$290$	0	0
$291$	−10832.0	−2.18207
$292$	0	0
$293$	5022.00	1.00133	0.500663	−	0.865642i	$-0.333090\pi$
$293$	5022.00	1.00133	0.500663	+	0.865642i	$0.333090\pi$
$294$	0	0
$295$	2800.00	0.552618
$296$	0	0
$297$	−2240.00	−0.437636
$298$	0	0
$299$	2016.00	0.389927
$300$	0	0
$301$	−2884.00	−0.552262
$302$	0	0
$303$	−10864.0	−2.05980
$304$	0	0
$305$	2772.00	0.520407
$306$	0	0
$307$	9536.00	1.77280	0.886398	−	0.462924i	$-0.153200\pi$
$307$	9536.00	1.77280	0.886398	+	0.462924i	$0.153200\pi$
$308$	0	0
$309$	6656.00	1.22539
$310$	0	0
$311$	−968.000	−0.176496	−0.0882480	−	0.996099i	$-0.528127\pi$
$311$	−968.000	−0.176496	−0.0882480	+	0.996099i	$0.528127\pi$
$312$	0	0
$313$	3058.00	0.552231	0.276116	−	0.961124i	$-0.410953\pi$
$313$	3058.00	0.552231	0.276116	+	0.961124i	$0.410953\pi$
$314$	0	0
$315$	−3626.00	−0.648578
$316$	0	0
$317$	4986.00	0.883412	0.441706	−	0.897160i	$-0.354373\pi$
$317$	4986.00	0.883412	0.441706	+	0.897160i	$0.354373\pi$
$318$	0	0
$319$	−5320.00	−0.933739
$320$	0	0
$321$	3552.00	0.617612
$322$	0	0
$323$	−5920.00	−1.01981
$324$	0	0
$325$	−1278.00	−0.218125
$326$	0	0
$327$	14960.0	2.52994
$328$	0	0
$329$	−168.000	−0.0281524
$330$	0	0
$331$	−8612.00	−1.43009	−0.715043	−	0.699081i	$-0.753593\pi$
$331$	−8612.00	−1.43009	−0.715043	+	0.699081i	$0.753593\pi$
$332$	0	0
$333$	12802.0	2.10674
$334$	0	0
$335$	10024.0	1.63483
$336$	0	0
$337$	−10206.0	−1.64972	−0.824861	−	0.565336i	$-0.808747\pi$
$337$	−10206.0	−1.64972	−0.824861	+	0.565336i	$0.808747\pi$
$338$	0	0
$339$	−11024.0	−1.76620
$340$	0	0
$341$	2016.00	0.320154
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	12544.0	1.95753
$346$	0	0
$347$	−2004.00	−0.310030	−0.155015	−	0.987912i	$-0.549543\pi$
$347$	−2004.00	−0.310030	−0.155015	+	0.987912i	$0.549543\pi$
$348$	0	0
$349$	−1330.00	−0.203992	−0.101996	−	0.994785i	$-0.532523\pi$
$349$	−1330.00	−0.203992	−0.101996	+	0.994785i	$0.532523\pi$
$350$	0	0
$351$	1440.00	0.218979
$352$	0	0
$353$	978.000	0.147461	0.0737304	−	0.997278i	$-0.476510\pi$
$353$	978.000	0.147461	0.0737304	+	0.997278i	$0.476510\pi$
$354$	0	0
$355$	5488.00	0.820487
$356$	0	0
$357$	4144.00	0.614352
$358$	0	0
$359$	−9680.00	−1.42309	−0.711547	−	0.702638i	$-0.752005\pi$
$359$	−9680.00	−1.42309	−0.711547	+	0.702638i	$0.752005\pi$
$360$	0	0
$361$	−459.000	−0.0669194
$362$	0	0
$363$	4376.00	0.632728
$364$	0	0
$365$	7532.00	1.08012
$366$	0	0
$367$	−8656.00	−1.23117	−0.615585	−	0.788070i	$-0.711080\pi$
$367$	−8656.00	−1.23117	−0.615585	+	0.788070i	$0.711080\pi$
$368$	0	0
$369$	5994.00	0.845624
$370$	0	0
$371$	2226.00	0.311504
$372$	0	0
$373$	−5278.00	−0.732666	−0.366333	−	0.930484i	$-0.619387\pi$
$373$	−5278.00	−0.732666	−0.366333	+	0.930484i	$0.619387\pi$
$374$	0	0
$375$	6048.00	0.832846
$376$	0	0
$377$	3420.00	0.467212
$378$	0	0
$379$	−6340.00	−0.859272	−0.429636	−	0.903002i	$-0.641358\pi$
$379$	−6340.00	−0.859272	−0.429636	+	0.903002i	$0.641358\pi$
$380$	0	0
$381$	−15552.0	−2.09122
$382$	0	0
$383$	−6232.00	−0.831437	−0.415718	−	0.909493i	$-0.636470\pi$
$383$	−6232.00	−0.831437	−0.415718	+	0.909493i	$0.636470\pi$
$384$	0	0
$385$	−2744.00	−0.363239
$386$	0	0
$387$	15244.0	2.00232
$388$	0	0
$389$	14810.0	1.93033	0.965163	−	0.261649i	$-0.0842664\pi$
$389$	14810.0	1.93033	0.965163	+	0.261649i	$0.0842664\pi$
$390$	0	0
$391$	−8288.00	−1.07197
$392$	0	0
$393$	−6784.00	−0.870757
$394$	0	0
$395$	3360.00	0.428000
$396$	0	0
$397$	−5154.00	−0.651566	−0.325783	−	0.945445i	$-0.605628\pi$
$397$	−5154.00	−0.651566	−0.325783	+	0.945445i	$0.605628\pi$
$398$	0	0
$399$	−4480.00	−0.562107
$400$	0	0
$401$	3282.00	0.408716	0.204358	−	0.978896i	$-0.434489\pi$
$401$	3282.00	0.408716	0.204358	+	0.978896i	$0.434489\pi$
$402$	0	0
$403$	−1296.00	−0.160194
$404$	0	0
$405$	−5026.00	−0.616652
$406$	0	0
$407$	9688.00	1.17989
$408$	0	0
$409$	5810.00	0.702411	0.351205	−	0.936298i	$-0.385772\pi$
$409$	5810.00	0.702411	0.351205	+	0.936298i	$0.385772\pi$
$410$	0	0
$411$	23728.0	2.84773
$412$	0	0
$413$	−1400.00	−0.166803
$414$	0	0
$415$	15008.0	1.77521
$416$	0	0
$417$	22400.0	2.63053
$418$	0	0
$419$	−13560.0	−1.58102	−0.790512	−	0.612446i	$-0.790186\pi$
$419$	−13560.0	−1.58102	−0.790512	+	0.612446i	$0.790186\pi$
$420$	0	0
$421$	738.000	0.0854345	0.0427172	−	0.999087i	$-0.486399\pi$
$421$	738.000	0.0854345	0.0427172	+	0.999087i	$0.486399\pi$
$422$	0	0
$423$	888.000	0.102071
$424$	0	0
$425$	5254.00	0.599662
$426$	0	0
$427$	−1386.00	−0.157080
$428$	0	0
$429$	4032.00	0.453769
$430$	0	0
$431$	1272.00	0.142158	0.0710790	−	0.997471i	$-0.477356\pi$
$431$	1272.00	0.142158	0.0710790	+	0.997471i	$0.477356\pi$
$432$	0	0
$433$	−5062.00	−0.561811	−0.280906	−	0.959735i	$-0.590635\pi$
$433$	−5062.00	−0.561811	−0.280906	+	0.959735i	$0.590635\pi$
$434$	0	0
$435$	21280.0	2.34551
$436$	0	0
$437$	8960.00	0.980812
$438$	0	0
$439$	5640.00	0.613172	0.306586	−	0.951843i	$-0.400813\pi$
$439$	5640.00	0.613172	0.306586	+	0.951843i	$0.400813\pi$
$440$	0	0
$441$	1813.00	0.195767
$442$	0	0
$443$	−13388.0	−1.43585	−0.717927	−	0.696119i	$-0.754909\pi$
$443$	−13388.0	−1.43585	−0.717927	+	0.696119i	$0.754909\pi$
$444$	0	0
$445$	11340.0	1.20802
$446$	0	0
$447$	4080.00	0.431717
$448$	0	0
$449$	−3230.00	−0.339495	−0.169747	−	0.985488i	$-0.554295\pi$
$449$	−3230.00	−0.339495	−0.169747	+	0.985488i	$0.554295\pi$
$450$	0	0
$451$	4536.00	0.473596
$452$	0	0
$453$	−4736.00	−0.491207
$454$	0	0
$455$	1764.00	0.181753
$456$	0	0
$457$	−10646.0	−1.08971	−0.544857	−	0.838529i	$-0.683416\pi$
$457$	−10646.0	−1.08971	−0.544857	+	0.838529i	$0.683416\pi$
$458$	0	0
$459$	−5920.00	−0.602009
$460$	0	0
$461$	−7282.00	−0.735698	−0.367849	−	0.929886i	$-0.619906\pi$
$461$	−7282.00	−0.735698	−0.367849	+	0.929886i	$0.619906\pi$
$462$	0	0
$463$	12688.0	1.27357	0.636783	−	0.771043i	$-0.280265\pi$
$463$	12688.0	1.27357	0.636783	+	0.771043i	$0.280265\pi$
$464$	0	0
$465$	−8064.00	−0.804213
$466$	0	0
$467$	2816.00	0.279034	0.139517	−	0.990220i	$-0.455445\pi$
$467$	2816.00	0.279034	0.139517	+	0.990220i	$0.455445\pi$
$468$	0	0
$469$	−5012.00	−0.493460
$470$	0	0
$471$	−21488.0	−2.10215
$472$	0	0
$473$	11536.0	1.12141
$474$	0	0
$475$	−5680.00	−0.548666
$476$	0	0
$477$	−11766.0	−1.12941
$478$	0	0
$479$	−3160.00	−0.301428	−0.150714	−	0.988577i	$-0.548157\pi$
$479$	−3160.00	−0.301428	−0.150714	+	0.988577i	$0.548157\pi$
$480$	0	0
$481$	−6228.00	−0.590379
$482$	0	0
$483$	−6272.00	−0.590861
$484$	0	0
$485$	18956.0	1.77474
$486$	0	0
$487$	−14176.0	−1.31905	−0.659523	−	0.751684i	$-0.729242\pi$
$487$	−14176.0	−1.31905	−0.659523	+	0.751684i	$0.729242\pi$
$488$	0	0
$489$	−8096.00	−0.748699
$490$	0	0
$491$	11268.0	1.03568	0.517839	−	0.855478i	$-0.326737\pi$
$491$	11268.0	1.03568	0.517839	+	0.855478i	$0.326737\pi$
$492$	0	0
$493$	−14060.0	−1.28444
$494$	0	0
$495$	14504.0	1.31698
$496$	0	0
$497$	−2744.00	−0.247656
$498$	0	0
$499$	4460.00	0.400114	0.200057	−	0.979784i	$-0.435887\pi$
$499$	4460.00	0.400114	0.200057	+	0.979784i	$0.435887\pi$
$500$	0	0
$501$	−4352.00	−0.388090
$502$	0	0
$503$	−1512.00	−0.134029	−0.0670147	−	0.997752i	$-0.521347\pi$
$503$	−1512.00	−0.134029	−0.0670147	+	0.997752i	$0.521347\pi$
$504$	0	0
$505$	19012.0	1.67529
$506$	0	0
$507$	14984.0	1.31255
$508$	0	0
$509$	11790.0	1.02668	0.513342	−	0.858184i	$-0.328407\pi$
$509$	11790.0	1.02668	0.513342	+	0.858184i	$0.328407\pi$
$510$	0	0
$511$	−3766.00	−0.326024
$512$	0	0
$513$	6400.00	0.550813
$514$	0	0
$515$	−11648.0	−0.996645
$516$	0	0
$517$	672.000	0.0571654
$518$	0	0
$519$	14864.0	1.25714
$520$	0	0
$521$	1362.00	0.114530	0.0572652	−	0.998359i	$-0.481762\pi$
$521$	1362.00	0.114530	0.0572652	+	0.998359i	$0.481762\pi$
$522$	0	0
$523$	−6968.00	−0.582580	−0.291290	−	0.956635i	$-0.594084\pi$
$523$	−6968.00	−0.582580	−0.291290	+	0.956635i	$0.594084\pi$
$524$	0	0
$525$	3976.00	0.330527
$526$	0	0
$527$	5328.00	0.440401
$528$	0	0
$529$	377.000	0.0309855
$530$	0	0
$531$	7400.00	0.604770
$532$	0	0
$533$	−2916.00	−0.236972
$534$	0	0
$535$	−6216.00	−0.502320
$536$	0	0
$537$	−2400.00	−0.192863
$538$	0	0
$539$	1372.00	0.109640
$540$	0	0
$541$	−7062.00	−0.561218	−0.280609	−	0.959822i	$-0.590536\pi$
$541$	−7062.00	−0.561218	−0.280609	+	0.959822i	$0.590536\pi$
$542$	0	0
$543$	−18864.0	−1.49085
$544$	0	0
$545$	−26180.0	−2.05767
$546$	0	0
$547$	8196.00	0.640650	0.320325	−	0.947308i	$-0.396208\pi$
$547$	8196.00	0.640650	0.320325	+	0.947308i	$0.396208\pi$
$548$	0	0
$549$	7326.00	0.569519
$550$	0	0
$551$	15200.0	1.17521
$552$	0	0
$553$	−1680.00	−0.129188
$554$	0	0
$555$	−38752.0	−2.96384
$556$	0	0
$557$	7466.00	0.567944	0.283972	−	0.958833i	$-0.408348\pi$
$557$	7466.00	0.567944	0.283972	+	0.958833i	$0.408348\pi$
$558$	0	0
$559$	−7416.00	−0.561115
$560$	0	0
$561$	−16576.0	−1.24749
$562$	0	0
$563$	−24968.0	−1.86905	−0.934526	−	0.355896i	$-0.884176\pi$
$563$	−24968.0	−1.86905	−0.934526	+	0.355896i	$0.884176\pi$
$564$	0	0
$565$	19292.0	1.43650
$566$	0	0
$567$	2513.00	0.186131
$568$	0	0
$569$	14250.0	1.04990	0.524948	−	0.851134i	$-0.324085\pi$
$569$	14250.0	1.04990	0.524948	+	0.851134i	$0.324085\pi$
$570$	0	0
$571$	−6372.00	−0.467005	−0.233503	−	0.972356i	$-0.575019\pi$
$571$	−6372.00	−0.467005	−0.233503	+	0.972356i	$0.575019\pi$
$572$	0	0
$573$	−11136.0	−0.811890
$574$	0	0
$575$	−7952.00	−0.576733
$576$	0	0
$577$	−8366.00	−0.603607	−0.301803	−	0.953370i	$-0.597589\pi$
$577$	−8366.00	−0.603607	−0.301803	+	0.953370i	$0.597589\pi$
$578$	0	0
$579$	−14224.0	−1.02095
$580$	0	0
$581$	−7504.00	−0.535832
$582$	0	0
$583$	−8904.00	−0.632532
$584$	0	0
$585$	−9324.00	−0.658974
$586$	0	0
$587$	−20384.0	−1.43328	−0.716642	−	0.697441i	$-0.754322\pi$
$587$	−20384.0	−1.43328	−0.716642	+	0.697441i	$0.754322\pi$
$588$	0	0
$589$	−5760.00	−0.402948
$590$	0	0
$591$	9712.00	0.675970
$592$	0	0
$593$	9378.00	0.649424	0.324712	−	0.945813i	$-0.394733\pi$
$593$	9378.00	0.649424	0.324712	+	0.945813i	$0.394733\pi$
$594$	0	0
$595$	−7252.00	−0.499669
$596$	0	0
$597$	−8320.00	−0.570377
$598$	0	0
$599$	−9000.00	−0.613907	−0.306953	−	0.951725i	$-0.599310\pi$
$599$	−9000.00	−0.613907	−0.306953	+	0.951725i	$0.599310\pi$
$600$	0	0
$601$	7562.00	0.513245	0.256623	−	0.966512i	$-0.417390\pi$
$601$	7562.00	0.513245	0.256623	+	0.966512i	$0.417390\pi$
$602$	0	0
$603$	26492.0	1.78912
$604$	0	0
$605$	−7658.00	−0.514615
$606$	0	0
$607$	−2976.00	−0.198999	−0.0994993	−	0.995038i	$-0.531724\pi$
$607$	−2976.00	−0.198999	−0.0994993	+	0.995038i	$0.531724\pi$
$608$	0	0
$609$	−10640.0	−0.707971
$610$	0	0
$611$	−432.000	−0.0286037
$612$	0	0
$613$	−4278.00	−0.281871	−0.140935	−	0.990019i	$-0.545011\pi$
$613$	−4278.00	−0.281871	−0.140935	+	0.990019i	$0.545011\pi$
$614$	0	0
$615$	−18144.0	−1.18965
$616$	0	0
$617$	18794.0	1.22629	0.613143	−	0.789972i	$-0.289905\pi$
$617$	18794.0	1.22629	0.613143	+	0.789972i	$0.289905\pi$
$618$	0	0
$619$	−18040.0	−1.17139	−0.585694	−	0.810532i	$-0.699178\pi$
$619$	−18040.0	−1.17139	−0.585694	+	0.810532i	$0.699178\pi$
$620$	0	0
$621$	8960.00	0.578989
$622$	0	0
$623$	−5670.00	−0.364629
$624$	0	0
$625$	−19459.0	−1.24538
$626$	0	0
$627$	17920.0	1.14140
$628$	0	0
$629$	25604.0	1.62305
$630$	0	0
$631$	−21688.0	−1.36828	−0.684141	−	0.729350i	$-0.739823\pi$
$631$	−21688.0	−1.36828	−0.684141	+	0.729350i	$0.739823\pi$
$632$	0	0
$633$	−30944.0	−1.94299
$634$	0	0
$635$	27216.0	1.70084
$636$	0	0
$637$	−882.000	−0.0548605
$638$	0	0
$639$	14504.0	0.897918
$640$	0	0
$641$	−10558.0	−0.650571	−0.325285	−	0.945616i	$-0.605460\pi$
$641$	−10558.0	−0.650571	−0.325285	+	0.945616i	$0.605460\pi$
$642$	0	0
$643$	26152.0	1.60394	0.801971	−	0.597363i	$-0.203785\pi$
$643$	26152.0	1.60394	0.801971	+	0.597363i	$0.203785\pi$
$644$	0	0
$645$	−46144.0	−2.81693
$646$	0	0
$647$	25584.0	1.55458	0.777288	−	0.629145i	$-0.216595\pi$
$647$	25584.0	1.55458	0.777288	+	0.629145i	$0.216595\pi$
$648$	0	0
$649$	5600.00	0.338705
$650$	0	0
$651$	4032.00	0.242744
$652$	0	0
$653$	−15198.0	−0.910787	−0.455393	−	0.890290i	$-0.650501\pi$
$653$	−15198.0	−0.910787	−0.455393	+	0.890290i	$0.650501\pi$
$654$	0	0
$655$	11872.0	0.708210
$656$	0	0
$657$	19906.0	1.18205
$658$	0	0
$659$	6100.00	0.360580	0.180290	−	0.983613i	$-0.442296\pi$
$659$	6100.00	0.360580	0.180290	+	0.983613i	$0.442296\pi$
$660$	0	0
$661$	2318.00	0.136399	0.0681995	−	0.997672i	$-0.478275\pi$
$661$	2318.00	0.136399	0.0681995	+	0.997672i	$0.478275\pi$
$662$	0	0
$663$	10656.0	0.624200
$664$	0	0
$665$	7840.00	0.457176
$666$	0	0
$667$	21280.0	1.23533
$668$	0	0
$669$	−31744.0	−1.83452
$670$	0	0
$671$	5544.00	0.318962
$672$	0	0
$673$	−10222.0	−0.585482	−0.292741	−	0.956192i	$-0.594567\pi$
$673$	−10222.0	−0.585482	−0.292741	+	0.956192i	$0.594567\pi$
$674$	0	0
$675$	−5680.00	−0.323886
$676$	0	0
$677$	−25434.0	−1.44388	−0.721941	−	0.691955i	$-0.756750\pi$
$677$	−25434.0	−1.44388	−0.721941	+	0.691955i	$0.756750\pi$
$678$	0	0
$679$	−9478.00	−0.535688
$680$	0	0
$681$	−31488.0	−1.77184
$682$	0	0
$683$	8532.00	0.477991	0.238996	−	0.971021i	$-0.423182\pi$
$683$	8532.00	0.477991	0.238996	+	0.971021i	$0.423182\pi$
$684$	0	0
$685$	−41524.0	−2.31613
$686$	0	0
$687$	38480.0	2.13698
$688$	0	0
$689$	5724.00	0.316498
$690$	0	0
$691$	−20672.0	−1.13806	−0.569030	−	0.822317i	$-0.692681\pi$
$691$	−20672.0	−1.13806	−0.569030	+	0.822317i	$0.692681\pi$
$692$	0	0
$693$	−7252.00	−0.397519
$694$	0	0
$695$	−39200.0	−2.13948
$696$	0	0
$697$	11988.0	0.651475
$698$	0	0
$699$	17456.0	0.944559
$700$	0	0
$701$	21458.0	1.15614	0.578072	−	0.815985i	$-0.303805\pi$
$701$	21458.0	1.15614	0.578072	+	0.815985i	$0.303805\pi$
$702$	0	0
$703$	−27680.0	−1.48502
$704$	0	0
$705$	−2688.00	−0.143597
$706$	0	0
$707$	−9506.00	−0.505672
$708$	0	0
$709$	9850.00	0.521755	0.260878	−	0.965372i	$-0.415988\pi$
$709$	9850.00	0.521755	0.260878	+	0.965372i	$0.415988\pi$
$710$	0	0
$711$	8880.00	0.468391
$712$	0	0
$713$	−8064.00	−0.423561
$714$	0	0
$715$	−7056.00	−0.369062
$716$	0	0
$717$	24000.0	1.25006
$718$	0	0
$719$	−18840.0	−0.977209	−0.488605	−	0.872505i	$-0.662494\pi$
$719$	−18840.0	−0.977209	−0.488605	+	0.872505i	$0.662494\pi$
$720$	0	0
$721$	5824.00	0.300828
$722$	0	0
$723$	−16336.0	−0.840308
$724$	0	0
$725$	−13490.0	−0.691043
$726$	0	0
$727$	37504.0	1.91327	0.956634	−	0.291291i	$-0.0940849\pi$
$727$	37504.0	1.91327	0.956634	+	0.291291i	$0.0940849\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	30488.0	1.54260
$732$	0	0
$733$	−13338.0	−0.672101	−0.336051	−	0.941844i	$-0.609091\pi$
$733$	−13338.0	−0.672101	−0.336051	+	0.941844i	$0.609091\pi$
$734$	0	0
$735$	−5488.00	−0.275412
$736$	0	0
$737$	20048.0	1.00200
$738$	0	0
$739$	−17100.0	−0.851196	−0.425598	−	0.904912i	$-0.639936\pi$
$739$	−17100.0	−0.851196	−0.425598	+	0.904912i	$0.639936\pi$
$740$	0	0
$741$	−11520.0	−0.571117
$742$	0	0
$743$	−19632.0	−0.969352	−0.484676	−	0.874694i	$-0.661062\pi$
$743$	−19632.0	−0.969352	−0.484676	+	0.874694i	$0.661062\pi$
$744$	0	0
$745$	−7140.00	−0.351127
$746$	0	0
$747$	39664.0	1.94274
$748$	0	0
$749$	3108.00	0.151621
$750$	0	0
$751$	33912.0	1.64776	0.823879	−	0.566766i	$-0.191805\pi$
$751$	33912.0	1.64776	0.823879	+	0.566766i	$0.191805\pi$
$752$	0	0
$753$	−4224.00	−0.204424
$754$	0	0
$755$	8288.00	0.399512
$756$	0	0
$757$	31386.0	1.50693	0.753463	−	0.657490i	$-0.228382\pi$
$757$	31386.0	1.50693	0.753463	+	0.657490i	$0.228382\pi$
$758$	0	0
$759$	25088.0	1.19978
$760$	0	0
$761$	−34558.0	−1.64616	−0.823079	−	0.567927i	$-0.807746\pi$
$761$	−34558.0	−1.64616	−0.823079	+	0.567927i	$0.807746\pi$
$762$	0	0
$763$	13090.0	0.621088
$764$	0	0
$765$	38332.0	1.81163
$766$	0	0
$767$	−3600.00	−0.169476
$768$	0	0
$769$	39130.0	1.83493	0.917467	−	0.397812i	$-0.130231\pi$
$769$	39130.0	1.83493	0.917467	+	0.397812i	$0.130231\pi$
$770$	0	0
$771$	−45072.0	−2.10535
$772$	0	0
$773$	25982.0	1.20894	0.604468	−	0.796629i	$-0.293386\pi$
$773$	25982.0	1.20894	0.604468	+	0.796629i	$0.293386\pi$
$774$	0	0
$775$	5112.00	0.236940
$776$	0	0
$777$	19376.0	0.894608
$778$	0	0
$779$	−12960.0	−0.596072
$780$	0	0
$781$	10976.0	0.502884
$782$	0	0
$783$	15200.0	0.693747
$784$	0	0
$785$	37604.0	1.70974
$786$	0	0
$787$	−35424.0	−1.60448	−0.802242	−	0.596999i	$-0.796360\pi$
$787$	−35424.0	−1.60448	−0.802242	+	0.596999i	$0.796360\pi$
$788$	0	0
$789$	−1344.00	−0.0606434
$790$	0	0
$791$	−9646.00	−0.433593
$792$	0	0
$793$	−3564.00	−0.159598
$794$	0	0
$795$	35616.0	1.58889
$796$	0	0
$797$	30606.0	1.36025	0.680126	−	0.733096i	$-0.261925\pi$
$797$	30606.0	1.36025	0.680126	+	0.733096i	$0.261925\pi$
$798$	0	0
$799$	1776.00	0.0786362
$800$	0	0
$801$	29970.0	1.32202
$802$	0	0
$803$	15064.0	0.662014
$804$	0	0
$805$	10976.0	0.480563
$806$	0	0
$807$	−10480.0	−0.457142
$808$	0	0
$809$	16810.0	0.730542	0.365271	−	0.930901i	$-0.380976\pi$
$809$	16810.0	0.730542	0.365271	+	0.930901i	$0.380976\pi$
$810$	0	0
$811$	9368.00	0.405616	0.202808	−	0.979218i	$-0.434993\pi$
$811$	9368.00	0.405616	0.202808	+	0.979218i	$0.434993\pi$
$812$	0	0
$813$	17664.0	0.761997
$814$	0	0
$815$	14168.0	0.608937
$816$	0	0
$817$	−32960.0	−1.41141
$818$	0	0
$819$	4662.00	0.198905
$820$	0	0
$821$	−34382.0	−1.46156	−0.730780	−	0.682614i	$-0.760843\pi$
$821$	−34382.0	−1.46156	−0.730780	+	0.682614i	$0.760843\pi$
$822$	0	0
$823$	−4472.00	−0.189410	−0.0947048	−	0.995505i	$-0.530191\pi$
$823$	−4472.00	−0.189410	−0.0947048	+	0.995505i	$0.530191\pi$
$824$	0	0
$825$	−15904.0	−0.671159
$826$	0	0
$827$	1716.00	0.0721538	0.0360769	−	0.999349i	$-0.488514\pi$
$827$	1716.00	0.0721538	0.0360769	+	0.999349i	$0.488514\pi$
$828$	0	0
$829$	7910.00	0.331394	0.165697	−	0.986177i	$-0.447013\pi$
$829$	7910.00	0.331394	0.165697	+	0.986177i	$0.447013\pi$
$830$	0	0
$831$	42352.0	1.76796
$832$	0	0
$833$	3626.00	0.150820
$834$	0	0
$835$	7616.00	0.315644
$836$	0	0
$837$	−5760.00	−0.237867
$838$	0	0
$839$	−19360.0	−0.796641	−0.398320	−	0.917246i	$-0.630407\pi$
$839$	−19360.0	−0.796641	−0.398320	+	0.917246i	$0.630407\pi$
$840$	0	0
$841$	11711.0	0.480175
$842$	0	0
$843$	−25936.0	−1.05965
$844$	0	0
$845$	−26222.0	−1.06753
$846$	0	0
$847$	3829.00	0.155332
$848$	0	0
$849$	−12736.0	−0.514839
$850$	0	0
$851$	−38752.0	−1.56099
$852$	0	0
$853$	−698.000	−0.0280177	−0.0140088	−	0.999902i	$-0.504459\pi$
$853$	−698.000	−0.0280177	−0.0140088	+	0.999902i	$0.504459\pi$
$854$	0	0
$855$	−41440.0	−1.65757
$856$	0	0
$857$	−23406.0	−0.932945	−0.466472	−	0.884536i	$-0.654475\pi$
$857$	−23406.0	−0.932945	−0.466472	+	0.884536i	$0.654475\pi$
$858$	0	0
$859$	−7280.00	−0.289162	−0.144581	−	0.989493i	$-0.546183\pi$
$859$	−7280.00	−0.289162	−0.144581	+	0.989493i	$0.546183\pi$
$860$	0	0
$861$	9072.00	0.359086
$862$	0	0
$863$	9808.00	0.386869	0.193435	−	0.981113i	$-0.438037\pi$
$863$	9808.00	0.386869	0.193435	+	0.981113i	$0.438037\pi$
$864$	0	0
$865$	−26012.0	−1.02247
$866$	0	0
$867$	−4504.00	−0.176429
$868$	0	0
$869$	6720.00	0.262325
$870$	0	0
$871$	−12888.0	−0.501370
$872$	0	0
$873$	50098.0	1.94222
$874$	0	0
$875$	5292.00	0.204460
$876$	0	0
$877$	8066.00	0.310570	0.155285	−	0.987870i	$-0.450370\pi$
$877$	8066.00	0.310570	0.155285	+	0.987870i	$0.450370\pi$
$878$	0	0
$879$	−40176.0	−1.54164
$880$	0	0
$881$	25842.0	0.988240	0.494120	−	0.869394i	$-0.335490\pi$
$881$	25842.0	0.988240	0.494120	+	0.869394i	$0.335490\pi$
$882$	0	0
$883$	5692.00	0.216932	0.108466	−	0.994100i	$-0.465406\pi$
$883$	5692.00	0.216932	0.108466	+	0.994100i	$0.465406\pi$
$884$	0	0
$885$	−22400.0	−0.850811
$886$	0	0
$887$	−13536.0	−0.512395	−0.256198	−	0.966624i	$-0.582470\pi$
$887$	−13536.0	−0.512395	−0.256198	+	0.966624i	$0.582470\pi$
$888$	0	0
$889$	−13608.0	−0.513383
$890$	0	0
$891$	−10052.0	−0.377951
$892$	0	0
$893$	−1920.00	−0.0719489
$894$	0	0
$895$	4200.00	0.156861
$896$	0	0
$897$	−16128.0	−0.600332
$898$	0	0
$899$	−13680.0	−0.507512
$900$	0	0
$901$	−23532.0	−0.870105
$902$	0	0
$903$	23072.0	0.850264
$904$	0	0
$905$	33012.0	1.21255
$906$	0	0
$907$	−17004.0	−0.622501	−0.311251	−	0.950328i	$-0.600748\pi$
$907$	−17004.0	−0.622501	−0.311251	+	0.950328i	$0.600748\pi$
$908$	0	0
$909$	50246.0	1.83339
$910$	0	0
$911$	−14568.0	−0.529813	−0.264906	−	0.964274i	$-0.585341\pi$
$911$	−14568.0	−0.529813	−0.264906	+	0.964274i	$0.585341\pi$
$912$	0	0
$913$	30016.0	1.08804
$914$	0	0
$915$	−22176.0	−0.801220
$916$	0	0
$917$	−5936.00	−0.213767
$918$	0	0
$919$	−1400.00	−0.0502522	−0.0251261	−	0.999684i	$-0.507999\pi$
$919$	−1400.00	−0.0502522	−0.0251261	+	0.999684i	$0.507999\pi$
$920$	0	0
$921$	−76288.0	−2.72940
$922$	0	0
$923$	−7056.00	−0.251626
$924$	0	0
$925$	24566.0	0.873216
$926$	0	0
$927$	−30784.0	−1.09070
$928$	0	0
$929$	−13830.0	−0.488426	−0.244213	−	0.969722i	$-0.578530\pi$
$929$	−13830.0	−0.488426	−0.244213	+	0.969722i	$0.578530\pi$
$930$	0	0
$931$	−3920.00	−0.137994
$932$	0	0
$933$	7744.00	0.271733
$934$	0	0
$935$	29008.0	1.01461
$936$	0	0
$937$	−24166.0	−0.842549	−0.421275	−	0.906933i	$-0.638417\pi$
$937$	−24166.0	−0.842549	−0.421275	+	0.906933i	$0.638417\pi$
$938$	0	0
$939$	−24464.0	−0.850216
$940$	0	0
$941$	10838.0	0.375461	0.187730	−	0.982221i	$-0.439887\pi$
$941$	10838.0	0.375461	0.187730	+	0.982221i	$0.439887\pi$
$942$	0	0
$943$	−18144.0	−0.626564
$944$	0	0
$945$	7840.00	0.269879
$946$	0	0
$947$	40916.0	1.40400	0.702002	−	0.712175i	$-0.252290\pi$
$947$	40916.0	1.40400	0.702002	+	0.712175i	$0.252290\pi$
$948$	0	0
$949$	−9684.00	−0.331250
$950$	0	0
$951$	−39888.0	−1.36010
$952$	0	0
$953$	56618.0	1.92449	0.962244	−	0.272189i	$-0.0877475\pi$
$953$	56618.0	1.92449	0.962244	+	0.272189i	$0.0877475\pi$
$954$	0	0
$955$	19488.0	0.660332
$956$	0	0
$957$	42560.0	1.43759
$958$	0	0
$959$	20762.0	0.699103
$960$	0	0
$961$	−24607.0	−0.825988
$962$	0	0
$963$	−16428.0	−0.549725
$964$	0	0
$965$	24892.0	0.830365
$966$	0	0
$967$	17504.0	0.582100	0.291050	−	0.956708i	$-0.405995\pi$
$967$	17504.0	0.582100	0.291050	+	0.956708i	$0.405995\pi$
$968$	0	0
$969$	47360.0	1.57010
$970$	0	0
$971$	−23112.0	−0.763851	−0.381926	−	0.924193i	$-0.624739\pi$
$971$	−23112.0	−0.763851	−0.381926	+	0.924193i	$0.624739\pi$
$972$	0	0
$973$	19600.0	0.645783
$974$	0	0
$975$	10224.0	0.335826
$976$	0	0
$977$	23874.0	0.781778	0.390889	−	0.920438i	$-0.372168\pi$
$977$	23874.0	0.781778	0.390889	+	0.920438i	$0.372168\pi$
$978$	0	0
$979$	22680.0	0.740404
$980$	0	0
$981$	−69190.0	−2.25185
$982$	0	0
$983$	−15312.0	−0.496823	−0.248411	−	0.968655i	$-0.579909\pi$
$983$	−15312.0	−0.496823	−0.248411	+	0.968655i	$0.579909\pi$
$984$	0	0
$985$	−16996.0	−0.549784
$986$	0	0
$987$	1344.00	0.0433435
$988$	0	0
$989$	−46144.0	−1.48361
$990$	0	0
$991$	−16528.0	−0.529797	−0.264899	−	0.964276i	$-0.585339\pi$
$991$	−16528.0	−0.529797	−0.264899	+	0.964276i	$0.585339\pi$
$992$	0	0
$993$	68896.0	2.20176
$994$	0	0
$995$	14560.0	0.463903
$996$	0	0
$997$	28606.0	0.908687	0.454344	−	0.890827i	$-0.349874\pi$
$997$	28606.0	0.908687	0.454344	+	0.890827i	$0.349874\pi$
$998$	0	0
$999$	−27680.0	−0.876633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.b.1.1		1
4.3	odd	2		448.4.a.o.1.1		1
8.3	odd	2		112.4.a.a.1.1		1
8.5	even	2		14.4.a.a.1.1	✓	1
24.5	odd	2		126.4.a.h.1.1		1
24.11	even	2		1008.4.a.s.1.1		1
40.13	odd	4		350.4.c.b.99.2		2
40.29	even	2		350.4.a.l.1.1		1
40.37	odd	4		350.4.c.b.99.1		2
56.5	odd	6		98.4.c.f.67.1		2
56.13	odd	2		98.4.a.a.1.1		1
56.27	even	2		784.4.a.s.1.1		1
56.37	even	6		98.4.c.d.67.1		2
56.45	odd	6		98.4.c.f.79.1		2
56.53	even	6		98.4.c.d.79.1		2
88.21	odd	2		1694.4.a.g.1.1		1
104.77	even	2		2366.4.a.h.1.1		1
168.5	even	6		882.4.g.k.361.1		2
168.53	odd	6		882.4.g.b.667.1		2
168.101	even	6		882.4.g.k.667.1		2
168.125	even	2		882.4.a.i.1.1		1
168.149	odd	6		882.4.g.b.361.1		2
280.69	odd	2		2450.4.a.bo.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.4.a.a.1.1	✓	1	8.5	even	2
98.4.a.a.1.1		1	56.13	odd	2
98.4.c.d.67.1		2	56.37	even	6
98.4.c.d.79.1		2	56.53	even	6
98.4.c.f.67.1		2	56.5	odd	6
98.4.c.f.79.1		2	56.45	odd	6
112.4.a.a.1.1		1	8.3	odd	2
126.4.a.h.1.1		1	24.5	odd	2
350.4.a.l.1.1		1	40.29	even	2
350.4.c.b.99.1		2	40.37	odd	4
350.4.c.b.99.2		2	40.13	odd	4
448.4.a.b.1.1		1	1.1	even	1	trivial
448.4.a.o.1.1		1	4.3	odd	2
784.4.a.s.1.1		1	56.27	even	2
882.4.a.i.1.1		1	168.125	even	2
882.4.g.b.361.1		2	168.149	odd	6
882.4.g.b.667.1		2	168.53	odd	6
882.4.g.k.361.1		2	168.5	even	6
882.4.g.k.667.1		2	168.101	even	6
1008.4.a.s.1.1		1	24.11	even	2
1694.4.a.g.1.1		1	88.21	odd	2
2366.4.a.h.1.1		1	104.77	even	2
2450.4.a.bo.1.1		1	280.69	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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