LMFDB - Embedded newform 448.4.a.k.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 = [1,0,2,0,12,0,7] 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:	$1$
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)

$f(q)$

$=$

$q+2.00000 q^{3} +12.0000 q^{5} +7.00000 q^{7} -23.0000 q^{9} -48.0000 q^{11} -56.0000 q^{13} +24.0000 q^{15} -114.000 q^{17} -2.00000 q^{19} +14.0000 q^{21} -120.000 q^{23} +19.0000 q^{25} -100.000 q^{27} +54.0000 q^{29} +236.000 q^{31} -96.0000 q^{33} +84.0000 q^{35} -146.000 q^{37} -112.000 q^{39} +126.000 q^{41} +376.000 q^{43} -276.000 q^{45} -12.0000 q^{47} +49.0000 q^{49} -228.000 q^{51} -174.000 q^{53} -576.000 q^{55} -4.00000 q^{57} -138.000 q^{59} -380.000 q^{61} -161.000 q^{63} -672.000 q^{65} +484.000 q^{67} -240.000 q^{69} +576.000 q^{71} -1150.00 q^{73} +38.0000 q^{75} -336.000 q^{77} +776.000 q^{79} +421.000 q^{81} -378.000 q^{83} -1368.00 q^{85} +108.000 q^{87} -390.000 q^{89} -392.000 q^{91} +472.000 q^{93} -24.0000 q^{95} -1330.00 q^{97} +1104.00 q^{99} +O(q^{100})$

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
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	0	0
$5$	12.0000	1.07331	0.536656	−	0.843801i	$-0.319687\pi$
$5$	12.0000	1.07331	0.536656	+	0.843801i	$0.319687\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−23.0000	−0.851852
$10$	0	0
$11$	−48.0000	−1.31569	−0.657843	−	0.753155i	$-0.728531\pi$
$11$	−48.0000	−1.31569	−0.657843	+	0.753155i	$0.728531\pi$
$12$	0	0
$13$	−56.0000	−1.19474	−0.597369	−	0.801966i	$-0.703787\pi$
$13$	−56.0000	−1.19474	−0.597369	+	0.801966i	$0.703787\pi$
$14$	0	0
$15$	24.0000	0.413118
$16$	0	0
$17$	−114.000	−1.62642	−0.813208	−	0.581974i	$-0.802281\pi$
$17$	−114.000	−1.62642	−0.813208	+	0.581974i	$0.802281\pi$
$18$	0	0
$19$	−2.00000	−0.0241490	−0.0120745	−	0.999927i	$-0.503844\pi$
$19$	−2.00000	−0.0241490	−0.0120745	+	0.999927i	$0.503844\pi$
$20$	0	0
$21$	14.0000	0.145479
$22$	0	0
$23$	−120.000	−1.08790	−0.543951	−	0.839117i	$-0.683072\pi$
$23$	−120.000	−1.08790	−0.543951	+	0.839117i	$0.683072\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	0	0
$27$	−100.000	−0.712778
$28$	0	0
$29$	54.0000	0.345778	0.172889	−	0.984941i	$-0.444690\pi$
$29$	54.0000	0.345778	0.172889	+	0.984941i	$0.444690\pi$
$30$	0	0
$31$	236.000	1.36732	0.683659	−	0.729802i	$-0.260388\pi$
$31$	236.000	1.36732	0.683659	+	0.729802i	$0.260388\pi$
$32$	0	0
$33$	−96.0000	−0.506408
$34$	0	0
$35$	84.0000	0.405674
$36$	0	0
$37$	−146.000	−0.648710	−0.324355	−	0.945936i	$-0.605147\pi$
$37$	−146.000	−0.648710	−0.324355	+	0.945936i	$0.605147\pi$
$38$	0	0
$39$	−112.000	−0.459855
$40$	0	0
$41$	126.000	0.479949	0.239974	−	0.970779i	$-0.422861\pi$
$41$	126.000	0.479949	0.239974	+	0.970779i	$0.422861\pi$
$42$	0	0
$43$	376.000	1.33348	0.666738	−	0.745292i	$-0.267690\pi$
$43$	376.000	1.33348	0.666738	+	0.745292i	$0.267690\pi$
$44$	0	0
$45$	−276.000	−0.914303
$46$	0	0
$47$	−12.0000	−0.0372421	−0.0186211	−	0.999827i	$-0.505928\pi$
$47$	−12.0000	−0.0372421	−0.0186211	+	0.999827i	$0.505928\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−228.000	−0.626008
$52$	0	0
$53$	−174.000	−0.450957	−0.225479	−	0.974248i	$-0.572395\pi$
$53$	−174.000	−0.450957	−0.225479	+	0.974248i	$0.572395\pi$
$54$	0	0
$55$	−576.000	−1.41214
$56$	0	0
$57$	−4.00000	−0.00929496
$58$	0	0
$59$	−138.000	−0.304510	−0.152255	−	0.988341i	$-0.548653\pi$
$59$	−138.000	−0.304510	−0.152255	+	0.988341i	$0.548653\pi$
$60$	0	0
$61$	−380.000	−0.797607	−0.398803	−	0.917036i	$-0.630574\pi$
$61$	−380.000	−0.797607	−0.398803	+	0.917036i	$0.630574\pi$
$62$	0	0
$63$	−161.000	−0.321970
$64$	0	0
$65$	−672.000	−1.28233
$66$	0	0
$67$	484.000	0.882537	0.441269	−	0.897375i	$-0.354529\pi$
$67$	484.000	0.882537	0.441269	+	0.897375i	$0.354529\pi$
$68$	0	0
$69$	−240.000	−0.418733
$70$	0	0
$71$	576.000	0.962798	0.481399	−	0.876502i	$-0.340129\pi$
$71$	576.000	0.962798	0.481399	+	0.876502i	$0.340129\pi$
$72$	0	0
$73$	−1150.00	−1.84380	−0.921899	−	0.387429i	$-0.873363\pi$
$73$	−1150.00	−1.84380	−0.921899	+	0.387429i	$0.873363\pi$
$74$	0	0
$75$	38.0000	0.0585048
$76$	0	0
$77$	−336.000	−0.497283
$78$	0	0
$79$	776.000	1.10515	0.552575	−	0.833463i	$-0.313645\pi$
$79$	776.000	1.10515	0.552575	+	0.833463i	$0.313645\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	−378.000	−0.499890	−0.249945	−	0.968260i	$-0.580413\pi$
$83$	−378.000	−0.499890	−0.249945	+	0.968260i	$0.580413\pi$
$84$	0	0
$85$	−1368.00	−1.74565
$86$	0	0
$87$	108.000	0.133090
$88$	0	0
$89$	−390.000	−0.464493	−0.232247	−	0.972657i	$-0.574608\pi$
$89$	−390.000	−0.464493	−0.232247	+	0.972657i	$0.574608\pi$
$90$	0	0
$91$	−392.000	−0.451569
$92$	0	0
$93$	472.000	0.526281
$94$	0	0
$95$	−24.0000	−0.0259195
$96$	0	0
$97$	−1330.00	−1.39218	−0.696088	−	0.717957i	$-0.745078\pi$
$97$	−1330.00	−1.39218	−0.696088	+	0.717957i	$0.745078\pi$
$98$	0	0
$99$	1104.00	1.12077
$100$	0	0
$101$	1500.00	1.47778	0.738889	−	0.673827i	$-0.235351\pi$
$101$	1500.00	1.47778	0.738889	+	0.673827i	$0.235351\pi$
$102$	0	0
$103$	380.000	0.363520	0.181760	−	0.983343i	$-0.441821\pi$
$103$	380.000	0.363520	0.181760	+	0.983343i	$0.441821\pi$
$104$	0	0
$105$	168.000	0.156144
$106$	0	0
$107$	−636.000	−0.574621	−0.287310	−	0.957838i	$-0.592761\pi$
$107$	−636.000	−0.574621	−0.287310	+	0.957838i	$0.592761\pi$
$108$	0	0
$109$	−146.000	−0.128296	−0.0641480	−	0.997940i	$-0.520433\pi$
$109$	−146.000	−0.128296	−0.0641480	+	0.997940i	$0.520433\pi$
$110$	0	0
$111$	−292.000	−0.249688
$112$	0	0
$113$	198.000	0.164834	0.0824171	−	0.996598i	$-0.473736\pi$
$113$	198.000	0.164834	0.0824171	+	0.996598i	$0.473736\pi$
$114$	0	0
$115$	−1440.00	−1.16766
$116$	0	0
$117$	1288.00	1.01774
$118$	0	0
$119$	−798.000	−0.614727
$120$	0	0
$121$	973.000	0.731029
$122$	0	0
$123$	252.000	0.184732
$124$	0	0
$125$	−1272.00	−0.910169
$126$	0	0
$127$	−376.000	−0.262713	−0.131357	−	0.991335i	$-0.541933\pi$
$127$	−376.000	−0.262713	−0.131357	+	0.991335i	$0.541933\pi$
$128$	0	0
$129$	752.000	0.513255
$130$	0	0
$131$	−2130.00	−1.42060	−0.710301	−	0.703898i	$-0.751441\pi$
$131$	−2130.00	−1.42060	−0.710301	+	0.703898i	$0.751441\pi$
$132$	0	0
$133$	−14.0000	−0.00912747
$134$	0	0
$135$	−1200.00	−0.765034
$136$	0	0
$137$	−78.0000	−0.0486423	−0.0243211	−	0.999704i	$-0.507742\pi$
$137$	−78.0000	−0.0486423	−0.0243211	+	0.999704i	$0.507742\pi$
$138$	0	0
$139$	2338.00	1.42667	0.713333	−	0.700825i	$-0.247185\pi$
$139$	2338.00	1.42667	0.713333	+	0.700825i	$0.247185\pi$
$140$	0	0
$141$	−24.0000	−0.0143345
$142$	0	0
$143$	2688.00	1.57190
$144$	0	0
$145$	648.000	0.371127
$146$	0	0
$147$	98.0000	0.0549857
$148$	0	0
$149$	1002.00	0.550920	0.275460	−	0.961313i	$-0.411170\pi$
$149$	1002.00	0.550920	0.275460	+	0.961313i	$0.411170\pi$
$150$	0	0
$151$	−2752.00	−1.48314	−0.741571	−	0.670874i	$-0.765919\pi$
$151$	−2752.00	−1.48314	−0.741571	+	0.670874i	$0.765919\pi$
$152$	0	0
$153$	2622.00	1.38546
$154$	0	0
$155$	2832.00	1.46756
$156$	0	0
$157$	520.000	0.264335	0.132167	−	0.991227i	$-0.457806\pi$
$157$	520.000	0.264335	0.132167	+	0.991227i	$0.457806\pi$
$158$	0	0
$159$	−348.000	−0.173574
$160$	0	0
$161$	−840.000	−0.411188
$162$	0	0
$163$	−1280.00	−0.615076	−0.307538	−	0.951536i	$-0.599505\pi$
$163$	−1280.00	−0.615076	−0.307538	+	0.951536i	$0.599505\pi$
$164$	0	0
$165$	−1152.00	−0.543534
$166$	0	0
$167$	1764.00	0.817380	0.408690	−	0.912673i	$-0.365986\pi$
$167$	1764.00	0.817380	0.408690	+	0.912673i	$0.365986\pi$
$168$	0	0
$169$	939.000	0.427401
$170$	0	0
$171$	46.0000	0.0205714
$172$	0	0
$173$	768.000	0.337514	0.168757	−	0.985658i	$-0.446025\pi$
$173$	768.000	0.337514	0.168757	+	0.985658i	$0.446025\pi$
$174$	0	0
$175$	133.000	0.0574506
$176$	0	0
$177$	−276.000	−0.117206
$178$	0	0
$179$	−1812.00	−0.756621	−0.378311	−	0.925679i	$-0.623495\pi$
$179$	−1812.00	−0.756621	−0.378311	+	0.925679i	$0.623495\pi$
$180$	0	0
$181$	448.000	0.183976	0.0919878	−	0.995760i	$-0.470678\pi$
$181$	448.000	0.183976	0.0919878	+	0.995760i	$0.470678\pi$
$182$	0	0
$183$	−760.000	−0.306999
$184$	0	0
$185$	−1752.00	−0.696268
$186$	0	0
$187$	5472.00	2.13985
$188$	0	0
$189$	−700.000	−0.269405
$190$	0	0
$191$	−2136.00	−0.809191	−0.404596	−	0.914496i	$-0.632588\pi$
$191$	−2136.00	−0.809191	−0.404596	+	0.914496i	$0.632588\pi$
$192$	0	0
$193$	4430.00	1.65222	0.826110	−	0.563509i	$-0.190549\pi$
$193$	4430.00	1.65222	0.826110	+	0.563509i	$0.190549\pi$
$194$	0	0
$195$	−1344.00	−0.493568
$196$	0	0
$197$	−198.000	−0.0716087	−0.0358044	−	0.999359i	$-0.511399\pi$
$197$	−198.000	−0.0716087	−0.0358044	+	0.999359i	$0.511399\pi$
$198$	0	0
$199$	−2284.00	−0.813610	−0.406805	−	0.913515i	$-0.633357\pi$
$199$	−2284.00	−0.813610	−0.406805	+	0.913515i	$0.633357\pi$
$200$	0	0
$201$	968.000	0.339689
$202$	0	0
$203$	378.000	0.130692
$204$	0	0
$205$	1512.00	0.515135
$206$	0	0
$207$	2760.00	0.926731
$208$	0	0
$209$	96.0000	0.0317725
$210$	0	0
$211$	−4412.00	−1.43950	−0.719750	−	0.694233i	$-0.755744\pi$
$211$	−4412.00	−1.43950	−0.719750	+	0.694233i	$0.755744\pi$
$212$	0	0
$213$	1152.00	0.370581
$214$	0	0
$215$	4512.00	1.43124
$216$	0	0
$217$	1652.00	0.516798
$218$	0	0
$219$	−2300.00	−0.709679
$220$	0	0
$221$	6384.00	1.94314
$222$	0	0
$223$	2072.00	0.622204	0.311102	−	0.950377i	$-0.399302\pi$
$223$	2072.00	0.622204	0.311102	+	0.950377i	$0.399302\pi$
$224$	0	0
$225$	−437.000	−0.129481
$226$	0	0
$227$	366.000	0.107014	0.0535072	−	0.998567i	$-0.482960\pi$
$227$	366.000	0.107014	0.0535072	+	0.998567i	$0.482960\pi$
$228$	0	0
$229$	376.000	0.108501	0.0542506	−	0.998527i	$-0.482723\pi$
$229$	376.000	0.108501	0.0542506	+	0.998527i	$0.482723\pi$
$230$	0	0
$231$	−672.000	−0.191404
$232$	0	0
$233$	−2262.00	−0.636002	−0.318001	−	0.948090i	$-0.603012\pi$
$233$	−2262.00	−0.636002	−0.318001	+	0.948090i	$0.603012\pi$
$234$	0	0
$235$	−144.000	−0.0399724
$236$	0	0
$237$	1552.00	0.425372
$238$	0	0
$239$	2592.00	0.701517	0.350758	−	0.936466i	$-0.385924\pi$
$239$	2592.00	0.701517	0.350758	+	0.936466i	$0.385924\pi$
$240$	0	0
$241$	110.000	0.0294013	0.0147007	−	0.999892i	$-0.495320\pi$
$241$	110.000	0.0294013	0.0147007	+	0.999892i	$0.495320\pi$
$242$	0	0
$243$	3542.00	0.935059
$244$	0	0
$245$	588.000	0.153330
$246$	0	0
$247$	112.000	0.0288518
$248$	0	0
$249$	−756.000	−0.192408
$250$	0	0
$251$	1890.00	0.475282	0.237641	−	0.971353i	$-0.423626\pi$
$251$	1890.00	0.475282	0.237641	+	0.971353i	$0.423626\pi$
$252$	0	0
$253$	5760.00	1.43134
$254$	0	0
$255$	−2736.00	−0.671902
$256$	0	0
$257$	2130.00	0.516987	0.258494	−	0.966013i	$-0.416774\pi$
$257$	2130.00	0.516987	0.258494	+	0.966013i	$0.416774\pi$
$258$	0	0
$259$	−1022.00	−0.245189
$260$	0	0
$261$	−1242.00	−0.294551
$262$	0	0
$263$	−4992.00	−1.17042	−0.585209	−	0.810883i	$-0.698988\pi$
$263$	−4992.00	−1.17042	−0.585209	+	0.810883i	$0.698988\pi$
$264$	0	0
$265$	−2088.00	−0.484018
$266$	0	0
$267$	−780.000	−0.178784
$268$	0	0
$269$	−6816.00	−1.54490	−0.772451	−	0.635074i	$-0.780970\pi$
$269$	−6816.00	−1.54490	−0.772451	+	0.635074i	$0.780970\pi$
$270$	0	0
$271$	8192.00	1.83627	0.918134	−	0.396270i	$-0.129696\pi$
$271$	8192.00	1.83627	0.918134	+	0.396270i	$0.129696\pi$
$272$	0	0
$273$	−784.000	−0.173809
$274$	0	0
$275$	−912.000	−0.199984
$276$	0	0
$277$	−2414.00	−0.523622	−0.261811	−	0.965119i	$-0.584320\pi$
$277$	−2414.00	−0.523622	−0.261811	+	0.965119i	$0.584320\pi$
$278$	0	0
$279$	−5428.00	−1.16475
$280$	0	0
$281$	1962.00	0.416524	0.208262	−	0.978073i	$-0.433219\pi$
$281$	1962.00	0.416524	0.208262	+	0.978073i	$0.433219\pi$
$282$	0	0
$283$	−5402.00	−1.13468	−0.567342	−	0.823482i	$-0.692028\pi$
$283$	−5402.00	−1.13468	−0.567342	+	0.823482i	$0.692028\pi$
$284$	0	0
$285$	−48.0000	−0.00997640
$286$	0	0
$287$	882.000	0.181404
$288$	0	0
$289$	8083.00	1.64523
$290$	0	0
$291$	−2660.00	−0.535849
$292$	0	0
$293$	4788.00	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$293$	4788.00	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$294$	0	0
$295$	−1656.00	−0.326834
$296$	0	0
$297$	4800.00	0.937792
$298$	0	0
$299$	6720.00	1.29976
$300$	0	0
$301$	2632.00	0.504007
$302$	0	0
$303$	3000.00	0.568797
$304$	0	0
$305$	−4560.00	−0.856081
$306$	0	0
$307$	574.000	0.106710	0.0533549	−	0.998576i	$-0.483009\pi$
$307$	574.000	0.106710	0.0533549	+	0.998576i	$0.483009\pi$
$308$	0	0
$309$	760.000	0.139919
$310$	0	0
$311$	−8808.00	−1.60597	−0.802984	−	0.596001i	$-0.796755\pi$
$311$	−8808.00	−1.60597	−0.802984	+	0.596001i	$0.796755\pi$
$312$	0	0
$313$	−2770.00	−0.500223	−0.250111	−	0.968217i	$-0.580467\pi$
$313$	−2770.00	−0.500223	−0.250111	+	0.968217i	$0.580467\pi$
$314$	0	0
$315$	−1932.00	−0.345574
$316$	0	0
$317$	−7566.00	−1.34053	−0.670266	−	0.742121i	$-0.733820\pi$
$317$	−7566.00	−1.34053	−0.670266	+	0.742121i	$0.733820\pi$
$318$	0	0
$319$	−2592.00	−0.454935
$320$	0	0
$321$	−1272.00	−0.221172
$322$	0	0
$323$	228.000	0.0392763
$324$	0	0
$325$	−1064.00	−0.181600
$326$	0	0
$327$	−292.000	−0.0493812
$328$	0	0
$329$	−84.0000	−0.0140762
$330$	0	0
$331$	11320.0	1.87977	0.939884	−	0.341493i	$-0.110932\pi$
$331$	11320.0	1.87977	0.939884	+	0.341493i	$0.110932\pi$
$332$	0	0
$333$	3358.00	0.552604
$334$	0	0
$335$	5808.00	0.947239
$336$	0	0
$337$	−4786.00	−0.773620	−0.386810	−	0.922159i	$-0.626423\pi$
$337$	−4786.00	−0.773620	−0.386810	+	0.922159i	$0.626423\pi$
$338$	0	0
$339$	396.000	0.0634447
$340$	0	0
$341$	−11328.0	−1.79896
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−2880.00	−0.449432
$346$	0	0
$347$	−12648.0	−1.95672	−0.978358	−	0.206921i	$-0.933656\pi$
$347$	−12648.0	−1.95672	−0.978358	+	0.206921i	$0.933656\pi$
$348$	0	0
$349$	−9632.00	−1.47733	−0.738666	−	0.674071i	$-0.764544\pi$
$349$	−9632.00	−1.47733	−0.738666	+	0.674071i	$0.764544\pi$
$350$	0	0
$351$	5600.00	0.851584
$352$	0	0
$353$	−3390.00	−0.511137	−0.255569	−	0.966791i	$-0.582263\pi$
$353$	−3390.00	−0.511137	−0.255569	+	0.966791i	$0.582263\pi$
$354$	0	0
$355$	6912.00	1.03338
$356$	0	0
$357$	−1596.00	−0.236609
$358$	0	0
$359$	−10704.0	−1.57364	−0.786818	−	0.617185i	$-0.788273\pi$
$359$	−10704.0	−1.57364	−0.786818	+	0.617185i	$0.788273\pi$
$360$	0	0
$361$	−6855.00	−0.999417
$362$	0	0
$363$	1946.00	0.281373
$364$	0	0
$365$	−13800.0	−1.97897
$366$	0	0
$367$	−8584.00	−1.22093	−0.610465	−	0.792043i	$-0.709017\pi$
$367$	−8584.00	−1.22093	−0.610465	+	0.792043i	$0.709017\pi$
$368$	0	0
$369$	−2898.00	−0.408845
$370$	0	0
$371$	−1218.00	−0.170446
$372$	0	0
$373$	2122.00	0.294566	0.147283	−	0.989094i	$-0.452947\pi$
$373$	2122.00	0.294566	0.147283	+	0.989094i	$0.452947\pi$
$374$	0	0
$375$	−2544.00	−0.350324
$376$	0	0
$377$	−3024.00	−0.413114
$378$	0	0
$379$	4912.00	0.665732	0.332866	−	0.942974i	$-0.391984\pi$
$379$	4912.00	0.665732	0.332866	+	0.942974i	$0.391984\pi$
$380$	0	0
$381$	−752.000	−0.101118
$382$	0	0
$383$	9060.00	1.20873	0.604366	−	0.796707i	$-0.293426\pi$
$383$	9060.00	1.20873	0.604366	+	0.796707i	$0.293426\pi$
$384$	0	0
$385$	−4032.00	−0.533740
$386$	0	0
$387$	−8648.00	−1.13592
$388$	0	0
$389$	−8994.00	−1.17227	−0.586136	−	0.810213i	$-0.699352\pi$
$389$	−8994.00	−1.17227	−0.586136	+	0.810213i	$0.699352\pi$
$390$	0	0
$391$	13680.0	1.76938
$392$	0	0
$393$	−4260.00	−0.546790
$394$	0	0
$395$	9312.00	1.18617
$396$	0	0
$397$	12976.0	1.64042	0.820210	−	0.572062i	$-0.193857\pi$
$397$	12976.0	1.64042	0.820210	+	0.572062i	$0.193857\pi$
$398$	0	0
$399$	−28.0000	−0.00351317
$400$	0	0
$401$	−3522.00	−0.438604	−0.219302	−	0.975657i	$-0.570378\pi$
$401$	−3522.00	−0.438604	−0.219302	+	0.975657i	$0.570378\pi$
$402$	0	0
$403$	−13216.0	−1.63359
$404$	0	0
$405$	5052.00	0.619842
$406$	0	0
$407$	7008.00	0.853498
$408$	0	0
$409$	12710.0	1.53660	0.768300	−	0.640090i	$-0.221103\pi$
$409$	12710.0	1.53660	0.768300	+	0.640090i	$0.221103\pi$
$410$	0	0
$411$	−156.000	−0.0187224
$412$	0	0
$413$	−966.000	−0.115094
$414$	0	0
$415$	−4536.00	−0.536539
$416$	0	0
$417$	4676.00	0.549124
$418$	0	0
$419$	−1638.00	−0.190982	−0.0954911	−	0.995430i	$-0.530442\pi$
$419$	−1638.00	−0.190982	−0.0954911	+	0.995430i	$0.530442\pi$
$420$	0	0
$421$	12850.0	1.48758	0.743789	−	0.668414i	$-0.233027\pi$
$421$	12850.0	1.48758	0.743789	+	0.668414i	$0.233027\pi$
$422$	0	0
$423$	276.000	0.0317248
$424$	0	0
$425$	−2166.00	−0.247215
$426$	0	0
$427$	−2660.00	−0.301467
$428$	0	0
$429$	5376.00	0.605025
$430$	0	0
$431$	−8016.00	−0.895863	−0.447932	−	0.894068i	$-0.647839\pi$
$431$	−8016.00	−0.895863	−0.447932	+	0.894068i	$0.647839\pi$
$432$	0	0
$433$	2198.00	0.243947	0.121974	−	0.992533i	$-0.461078\pi$
$433$	2198.00	0.243947	0.121974	+	0.992533i	$0.461078\pi$
$434$	0	0
$435$	1296.00	0.142847
$436$	0	0
$437$	240.000	0.0262718
$438$	0	0
$439$	−376.000	−0.0408781	−0.0204391	−	0.999791i	$-0.506506\pi$
$439$	−376.000	−0.0408781	−0.0204391	+	0.999791i	$0.506506\pi$
$440$	0	0
$441$	−1127.00	−0.121693
$442$	0	0
$443$	−7188.00	−0.770908	−0.385454	−	0.922727i	$-0.625955\pi$
$443$	−7188.00	−0.770908	−0.385454	+	0.922727i	$0.625955\pi$
$444$	0	0
$445$	−4680.00	−0.498547
$446$	0	0
$447$	2004.00	0.212049
$448$	0	0
$449$	−14670.0	−1.54192	−0.770958	−	0.636886i	$-0.780222\pi$
$449$	−14670.0	−1.54192	−0.770958	+	0.636886i	$0.780222\pi$
$450$	0	0
$451$	−6048.00	−0.631462
$452$	0	0
$453$	−5504.00	−0.570862
$454$	0	0
$455$	−4704.00	−0.484675
$456$	0	0
$457$	−5146.00	−0.526739	−0.263370	−	0.964695i	$-0.584834\pi$
$457$	−5146.00	−0.526739	−0.263370	+	0.964695i	$0.584834\pi$
$458$	0	0
$459$	11400.0	1.15927
$460$	0	0
$461$	1512.00	0.152757	0.0763784	−	0.997079i	$-0.475664\pi$
$461$	1512.00	0.152757	0.0763784	+	0.997079i	$0.475664\pi$
$462$	0	0
$463$	7184.00	0.721099	0.360549	−	0.932740i	$-0.382589\pi$
$463$	7184.00	0.721099	0.360549	+	0.932740i	$0.382589\pi$
$464$	0	0
$465$	5664.00	0.564864
$466$	0	0
$467$	16518.0	1.63675	0.818375	−	0.574685i	$-0.194875\pi$
$467$	16518.0	1.63675	0.818375	+	0.574685i	$0.194875\pi$
$468$	0	0
$469$	3388.00	0.333568
$470$	0	0
$471$	1040.00	0.101742
$472$	0	0
$473$	−18048.0	−1.75444
$474$	0	0
$475$	−38.0000	−0.00367065
$476$	0	0
$477$	4002.00	0.384149
$478$	0	0
$479$	10092.0	0.962662	0.481331	−	0.876539i	$-0.340153\pi$
$479$	10092.0	0.962662	0.481331	+	0.876539i	$0.340153\pi$
$480$	0	0
$481$	8176.00	0.775038
$482$	0	0
$483$	−1680.00	−0.158266
$484$	0	0
$485$	−15960.0	−1.49424
$486$	0	0
$487$	7832.00	0.728751	0.364376	−	0.931252i	$-0.381282\pi$
$487$	7832.00	0.728751	0.364376	+	0.931252i	$0.381282\pi$
$488$	0	0
$489$	−2560.00	−0.236743
$490$	0	0
$491$	6732.00	0.618759	0.309380	−	0.950939i	$-0.399879\pi$
$491$	6732.00	0.618759	0.309380	+	0.950939i	$0.399879\pi$
$492$	0	0
$493$	−6156.00	−0.562378
$494$	0	0
$495$	13248.0	1.20294
$496$	0	0
$497$	4032.00	0.363903
$498$	0	0
$499$	−18668.0	−1.67474	−0.837369	−	0.546638i	$-0.815907\pi$
$499$	−18668.0	−1.67474	−0.837369	+	0.546638i	$0.815907\pi$
$500$	0	0
$501$	3528.00	0.314610
$502$	0	0
$503$	−6048.00	−0.536117	−0.268059	−	0.963403i	$-0.586382\pi$
$503$	−6048.00	−0.536117	−0.268059	+	0.963403i	$0.586382\pi$
$504$	0	0
$505$	18000.0	1.58612
$506$	0	0
$507$	1878.00	0.164507
$508$	0	0
$509$	−11328.0	−0.986453	−0.493227	−	0.869901i	$-0.664183\pi$
$509$	−11328.0	−0.986453	−0.493227	+	0.869901i	$0.664183\pi$
$510$	0	0
$511$	−8050.00	−0.696890
$512$	0	0
$513$	200.000	0.0172129
$514$	0	0
$515$	4560.00	0.390170
$516$	0	0
$517$	576.000	0.0489989
$518$	0	0
$519$	1536.00	0.129909
$520$	0	0
$521$	−4146.00	−0.348636	−0.174318	−	0.984689i	$-0.555772\pi$
$521$	−4146.00	−0.348636	−0.174318	+	0.984689i	$0.555772\pi$
$522$	0	0
$523$	1006.00	0.0841096	0.0420548	−	0.999115i	$-0.486610\pi$
$523$	1006.00	0.0841096	0.0420548	+	0.999115i	$0.486610\pi$
$524$	0	0
$525$	266.000	0.0221127
$526$	0	0
$527$	−26904.0	−2.22383
$528$	0	0
$529$	2233.00	0.183529
$530$	0	0
$531$	3174.00	0.259397
$532$	0	0
$533$	−7056.00	−0.573413
$534$	0	0
$535$	−7632.00	−0.616748
$536$	0	0
$537$	−3624.00	−0.291224
$538$	0	0
$539$	−2352.00	−0.187955
$540$	0	0
$541$	14722.0	1.16996	0.584980	−	0.811048i	$-0.301102\pi$
$541$	14722.0	1.16996	0.584980	+	0.811048i	$0.301102\pi$
$542$	0	0
$543$	896.000	0.0708122
$544$	0	0
$545$	−1752.00	−0.137702
$546$	0	0
$547$	13480.0	1.05368	0.526840	−	0.849964i	$-0.323377\pi$
$547$	13480.0	1.05368	0.526840	+	0.849964i	$0.323377\pi$
$548$	0	0
$549$	8740.00	0.679443
$550$	0	0
$551$	−108.000	−0.00835019
$552$	0	0
$553$	5432.00	0.417707
$554$	0	0
$555$	−3504.00	−0.267994
$556$	0	0
$557$	−6222.00	−0.473312	−0.236656	−	0.971594i	$-0.576051\pi$
$557$	−6222.00	−0.473312	−0.236656	+	0.971594i	$0.576051\pi$
$558$	0	0
$559$	−21056.0	−1.59316
$560$	0	0
$561$	10944.0	0.823629
$562$	0	0
$563$	−4926.00	−0.368750	−0.184375	−	0.982856i	$-0.559026\pi$
$563$	−4926.00	−0.368750	−0.184375	+	0.982856i	$0.559026\pi$
$564$	0	0
$565$	2376.00	0.176919
$566$	0	0
$567$	2947.00	0.218276
$568$	0	0
$569$	22182.0	1.63430	0.817151	−	0.576424i	$-0.195552\pi$
$569$	22182.0	1.63430	0.817151	+	0.576424i	$0.195552\pi$
$570$	0	0
$571$	−3296.00	−0.241564	−0.120782	−	0.992679i	$-0.538540\pi$
$571$	−3296.00	−0.241564	−0.120782	+	0.992679i	$0.538540\pi$
$572$	0	0
$573$	−4272.00	−0.311458
$574$	0	0
$575$	−2280.00	−0.165361
$576$	0	0
$577$	−24334.0	−1.75570	−0.877849	−	0.478938i	$-0.841022\pi$
$577$	−24334.0	−1.75570	−0.877849	+	0.478938i	$0.841022\pi$
$578$	0	0
$579$	8860.00	0.635940
$580$	0	0
$581$	−2646.00	−0.188941
$582$	0	0
$583$	8352.00	0.593318
$584$	0	0
$585$	15456.0	1.09235
$586$	0	0
$587$	−1638.00	−0.115175	−0.0575873	−	0.998340i	$-0.518341\pi$
$587$	−1638.00	−0.115175	−0.0575873	+	0.998340i	$0.518341\pi$
$588$	0	0
$589$	−472.000	−0.0330194
$590$	0	0
$591$	−396.000	−0.0275622
$592$	0	0
$593$	−7446.00	−0.515633	−0.257817	−	0.966194i	$-0.583003\pi$
$593$	−7446.00	−0.515633	−0.257817	+	0.966194i	$0.583003\pi$
$594$	0	0
$595$	−9576.00	−0.659794
$596$	0	0
$597$	−4568.00	−0.313159
$598$	0	0
$599$	−6504.00	−0.443650	−0.221825	−	0.975087i	$-0.571201\pi$
$599$	−6504.00	−0.443650	−0.221825	+	0.975087i	$0.571201\pi$
$600$	0	0
$601$	16058.0	1.08988	0.544941	−	0.838474i	$-0.316552\pi$
$601$	16058.0	1.08988	0.544941	+	0.838474i	$0.316552\pi$
$602$	0	0
$603$	−11132.0	−0.751791
$604$	0	0
$605$	11676.0	0.784623
$606$	0	0
$607$	10208.0	0.682586	0.341293	−	0.939957i	$-0.389135\pi$
$607$	10208.0	0.682586	0.341293	+	0.939957i	$0.389135\pi$
$608$	0	0
$609$	756.000	0.0503032
$610$	0	0
$611$	672.000	0.0444946
$612$	0	0
$613$	14974.0	0.986614	0.493307	−	0.869855i	$-0.335788\pi$
$613$	14974.0	0.986614	0.493307	+	0.869855i	$0.335788\pi$
$614$	0	0
$615$	3024.00	0.198276
$616$	0	0
$617$	7254.00	0.473314	0.236657	−	0.971593i	$-0.423948\pi$
$617$	7254.00	0.473314	0.236657	+	0.971593i	$0.423948\pi$
$618$	0	0
$619$	−12458.0	−0.808933	−0.404466	−	0.914553i	$-0.632543\pi$
$619$	−12458.0	−0.808933	−0.404466	+	0.914553i	$0.632543\pi$
$620$	0	0
$621$	12000.0	0.775432
$622$	0	0
$623$	−2730.00	−0.175562
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	192.000	0.0122293
$628$	0	0
$629$	16644.0	1.05507
$630$	0	0
$631$	28352.0	1.78871	0.894354	−	0.447359i	$-0.147635\pi$
$631$	28352.0	1.78871	0.894354	+	0.447359i	$0.147635\pi$
$632$	0	0
$633$	−8824.00	−0.554064
$634$	0	0
$635$	−4512.00	−0.281974
$636$	0	0
$637$	−2744.00	−0.170677
$638$	0	0
$639$	−13248.0	−0.820161
$640$	0	0
$641$	27390.0	1.68774	0.843869	−	0.536549i	$-0.180272\pi$
$641$	27390.0	1.68774	0.843869	+	0.536549i	$0.180272\pi$
$642$	0	0
$643$	21490.0	1.31801	0.659007	−	0.752137i	$-0.270977\pi$
$643$	21490.0	1.31801	0.659007	+	0.752137i	$0.270977\pi$
$644$	0	0
$645$	9024.00	0.550883
$646$	0	0
$647$	17652.0	1.07260	0.536300	−	0.844028i	$-0.319822\pi$
$647$	17652.0	1.07260	0.536300	+	0.844028i	$0.319822\pi$
$648$	0	0
$649$	6624.00	0.400639
$650$	0	0
$651$	3304.00	0.198915
$652$	0	0
$653$	4782.00	0.286576	0.143288	−	0.989681i	$-0.454232\pi$
$653$	4782.00	0.286576	0.143288	+	0.989681i	$0.454232\pi$
$654$	0	0
$655$	−25560.0	−1.52475
$656$	0	0
$657$	26450.0	1.57064
$658$	0	0
$659$	27144.0	1.60452	0.802261	−	0.596973i	$-0.203630\pi$
$659$	27144.0	1.60452	0.802261	+	0.596973i	$0.203630\pi$
$660$	0	0
$661$	11860.0	0.697883	0.348941	−	0.937145i	$-0.386541\pi$
$661$	11860.0	0.697883	0.348941	+	0.937145i	$0.386541\pi$
$662$	0	0
$663$	12768.0	0.747916
$664$	0	0
$665$	−168.000	−0.00979663
$666$	0	0
$667$	−6480.00	−0.376172
$668$	0	0
$669$	4144.00	0.239486
$670$	0	0
$671$	18240.0	1.04940
$672$	0	0
$673$	5546.00	0.317656	0.158828	−	0.987306i	$-0.449228\pi$
$673$	5546.00	0.317656	0.158828	+	0.987306i	$0.449228\pi$
$674$	0	0
$675$	−1900.00	−0.108342
$676$	0	0
$677$	14880.0	0.844734	0.422367	−	0.906425i	$-0.361199\pi$
$677$	14880.0	0.844734	0.422367	+	0.906425i	$0.361199\pi$
$678$	0	0
$679$	−9310.00	−0.526193
$680$	0	0
$681$	732.000	0.0411899
$682$	0	0
$683$	−20964.0	−1.17447	−0.587237	−	0.809415i	$-0.699784\pi$
$683$	−20964.0	−1.17447	−0.587237	+	0.809415i	$0.699784\pi$
$684$	0	0
$685$	−936.000	−0.0522084
$686$	0	0
$687$	752.000	0.0417621
$688$	0	0
$689$	9744.00	0.538776
$690$	0	0
$691$	−13106.0	−0.721528	−0.360764	−	0.932657i	$-0.617484\pi$
$691$	−13106.0	−0.721528	−0.360764	+	0.932657i	$0.617484\pi$
$692$	0	0
$693$	7728.00	0.423611
$694$	0	0
$695$	28056.0	1.53126
$696$	0	0
$697$	−14364.0	−0.780596
$698$	0	0
$699$	−4524.00	−0.244797
$700$	0	0
$701$	4590.00	0.247307	0.123653	−	0.992325i	$-0.460539\pi$
$701$	4590.00	0.247307	0.123653	+	0.992325i	$0.460539\pi$
$702$	0	0
$703$	292.000	0.0156657
$704$	0	0
$705$	−288.000	−0.0153854
$706$	0	0
$707$	10500.0	0.558548
$708$	0	0
$709$	862.000	0.0456602	0.0228301	−	0.999739i	$-0.492732\pi$
$709$	862.000	0.0456602	0.0228301	+	0.999739i	$0.492732\pi$
$710$	0	0
$711$	−17848.0	−0.941424
$712$	0	0
$713$	−28320.0	−1.48751
$714$	0	0
$715$	32256.0	1.68714
$716$	0	0
$717$	5184.00	0.270014
$718$	0	0
$719$	−3540.00	−0.183616	−0.0918079	−	0.995777i	$-0.529265\pi$
$719$	−3540.00	−0.183616	−0.0918079	+	0.995777i	$0.529265\pi$
$720$	0	0
$721$	2660.00	0.137397
$722$	0	0
$723$	220.000	0.0113166
$724$	0	0
$725$	1026.00	0.0525582
$726$	0	0
$727$	−4228.00	−0.215692	−0.107846	−	0.994168i	$-0.534395\pi$
$727$	−4228.00	−0.215692	−0.107846	+	0.994168i	$0.534395\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	0	0
$731$	−42864.0	−2.16879
$732$	0	0
$733$	−5420.00	−0.273114	−0.136557	−	0.990632i	$-0.543604\pi$
$733$	−5420.00	−0.273114	−0.136557	+	0.990632i	$0.543604\pi$
$734$	0	0
$735$	1176.00	0.0590169
$736$	0	0
$737$	−23232.0	−1.16114
$738$	0	0
$739$	−1280.00	−0.0637152	−0.0318576	−	0.999492i	$-0.510142\pi$
$739$	−1280.00	−0.0637152	−0.0318576	+	0.999492i	$0.510142\pi$
$740$	0	0
$741$	224.000	0.0111051
$742$	0	0
$743$	−35712.0	−1.76332	−0.881660	−	0.471886i	$-0.843573\pi$
$743$	−35712.0	−1.76332	−0.881660	+	0.471886i	$0.843573\pi$
$744$	0	0
$745$	12024.0	0.591309
$746$	0	0
$747$	8694.00	0.425832
$748$	0	0
$749$	−4452.00	−0.217186
$750$	0	0
$751$	24464.0	1.18869	0.594344	−	0.804211i	$-0.297412\pi$
$751$	24464.0	1.18869	0.594344	+	0.804211i	$0.297412\pi$
$752$	0	0
$753$	3780.00	0.182936
$754$	0	0
$755$	−33024.0	−1.59188
$756$	0	0
$757$	−30242.0	−1.45200	−0.726000	−	0.687695i	$-0.758623\pi$
$757$	−30242.0	−1.45200	−0.726000	+	0.687695i	$0.758623\pi$
$758$	0	0
$759$	11520.0	0.550922
$760$	0	0
$761$	−2154.00	−0.102605	−0.0513025	−	0.998683i	$-0.516337\pi$
$761$	−2154.00	−0.102605	−0.0513025	+	0.998683i	$0.516337\pi$
$762$	0	0
$763$	−1022.00	−0.0484913
$764$	0	0
$765$	31464.0	1.48704
$766$	0	0
$767$	7728.00	0.363810
$768$	0	0
$769$	10262.0	0.481219	0.240609	−	0.970622i	$-0.422653\pi$
$769$	10262.0	0.481219	0.240609	+	0.970622i	$0.422653\pi$
$770$	0	0
$771$	4260.00	0.198989
$772$	0	0
$773$	−9084.00	−0.422676	−0.211338	−	0.977413i	$-0.567782\pi$
$773$	−9084.00	−0.422676	−0.211338	+	0.977413i	$0.567782\pi$
$774$	0	0
$775$	4484.00	0.207832
$776$	0	0
$777$	−2044.00	−0.0943733
$778$	0	0
$779$	−252.000	−0.0115903
$780$	0	0
$781$	−27648.0	−1.26674
$782$	0	0
$783$	−5400.00	−0.246463
$784$	0	0
$785$	6240.00	0.283714
$786$	0	0
$787$	19798.0	0.896725	0.448362	−	0.893852i	$-0.352007\pi$
$787$	19798.0	0.896725	0.448362	+	0.893852i	$0.352007\pi$
$788$	0	0
$789$	−9984.00	−0.450494
$790$	0	0
$791$	1386.00	0.0623015
$792$	0	0
$793$	21280.0	0.952932
$794$	0	0
$795$	−4176.00	−0.186299
$796$	0	0
$797$	−30240.0	−1.34398	−0.671992	−	0.740558i	$-0.734561\pi$
$797$	−30240.0	−1.34398	−0.671992	+	0.740558i	$0.734561\pi$
$798$	0	0
$799$	1368.00	0.0605712
$800$	0	0
$801$	8970.00	0.395680
$802$	0	0
$803$	55200.0	2.42586
$804$	0	0
$805$	−10080.0	−0.441333
$806$	0	0
$807$	−13632.0	−0.594633
$808$	0	0
$809$	−2346.00	−0.101954	−0.0509771	−	0.998700i	$-0.516234\pi$
$809$	−2346.00	−0.101954	−0.0509771	+	0.998700i	$0.516234\pi$
$810$	0	0
$811$	29806.0	1.29054	0.645271	−	0.763953i	$-0.276744\pi$
$811$	29806.0	1.29054	0.645271	+	0.763953i	$0.276744\pi$
$812$	0	0
$813$	16384.0	0.706780
$814$	0	0
$815$	−15360.0	−0.660169
$816$	0	0
$817$	−752.000	−0.0322021
$818$	0	0
$819$	9016.00	0.384670
$820$	0	0
$821$	1506.00	0.0640192	0.0320096	−	0.999488i	$-0.489809\pi$
$821$	1506.00	0.0640192	0.0320096	+	0.999488i	$0.489809\pi$
$822$	0	0
$823$	−20392.0	−0.863694	−0.431847	−	0.901947i	$-0.642138\pi$
$823$	−20392.0	−0.863694	−0.431847	+	0.901947i	$0.642138\pi$
$824$	0	0
$825$	−1824.00	−0.0769740
$826$	0	0
$827$	−36108.0	−1.51826	−0.759128	−	0.650941i	$-0.774374\pi$
$827$	−36108.0	−1.51826	−0.759128	+	0.650941i	$0.774374\pi$
$828$	0	0
$829$	13876.0	0.581343	0.290672	−	0.956823i	$-0.406121\pi$
$829$	13876.0	0.581343	0.290672	+	0.956823i	$0.406121\pi$
$830$	0	0
$831$	−4828.00	−0.201542
$832$	0	0
$833$	−5586.00	−0.232345
$834$	0	0
$835$	21168.0	0.877304
$836$	0	0
$837$	−23600.0	−0.974594
$838$	0	0
$839$	23436.0	0.964363	0.482182	−	0.876071i	$-0.339845\pi$
$839$	23436.0	0.964363	0.482182	+	0.876071i	$0.339845\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	0	0
$843$	3924.00	0.160320
$844$	0	0
$845$	11268.0	0.458735
$846$	0	0
$847$	6811.00	0.276303
$848$	0	0
$849$	−10804.0	−0.436740
$850$	0	0
$851$	17520.0	0.705732
$852$	0	0
$853$	−8120.00	−0.325936	−0.162968	−	0.986631i	$-0.552107\pi$
$853$	−8120.00	−0.325936	−0.162968	+	0.986631i	$0.552107\pi$
$854$	0	0
$855$	552.000	0.0220795
$856$	0	0
$857$	−50010.0	−1.99336	−0.996680	−	0.0814218i	$-0.974054\pi$
$857$	−50010.0	−1.99336	−0.996680	+	0.0814218i	$0.974054\pi$
$858$	0	0
$859$	−34526.0	−1.37138	−0.685688	−	0.727896i	$-0.740499\pi$
$859$	−34526.0	−1.37138	−0.685688	+	0.727896i	$0.740499\pi$
$860$	0	0
$861$	1764.00	0.0698223
$862$	0	0
$863$	−17256.0	−0.680650	−0.340325	−	0.940308i	$-0.610537\pi$
$863$	−17256.0	−0.680650	−0.340325	+	0.940308i	$0.610537\pi$
$864$	0	0
$865$	9216.00	0.362258
$866$	0	0
$867$	16166.0	0.633248
$868$	0	0
$869$	−37248.0	−1.45403
$870$	0	0
$871$	−27104.0	−1.05440
$872$	0	0
$873$	30590.0	1.18593
$874$	0	0
$875$	−8904.00	−0.344012
$876$	0	0
$877$	−8714.00	−0.335520	−0.167760	−	0.985828i	$-0.553653\pi$
$877$	−8714.00	−0.335520	−0.167760	+	0.985828i	$0.553653\pi$
$878$	0	0
$879$	9576.00	0.367452
$880$	0	0
$881$	−22806.0	−0.872138	−0.436069	−	0.899913i	$-0.643630\pi$
$881$	−22806.0	−0.872138	−0.436069	+	0.899913i	$0.643630\pi$
$882$	0	0
$883$	−40196.0	−1.53194	−0.765970	−	0.642876i	$-0.777741\pi$
$883$	−40196.0	−1.53194	−0.765970	+	0.642876i	$0.777741\pi$
$884$	0	0
$885$	−3312.00	−0.125798
$886$	0	0
$887$	40812.0	1.54491	0.772454	−	0.635071i	$-0.219029\pi$
$887$	40812.0	1.54491	0.772454	+	0.635071i	$0.219029\pi$
$888$	0	0
$889$	−2632.00	−0.0992963
$890$	0	0
$891$	−20208.0	−0.759813
$892$	0	0
$893$	24.0000	0.000899361 0
$894$	0	0
$895$	−21744.0	−0.812091
$896$	0	0
$897$	13440.0	0.500277
$898$	0	0
$899$	12744.0	0.472788
$900$	0	0
$901$	19836.0	0.733444
$902$	0	0
$903$	5264.00	0.193992
$904$	0	0
$905$	5376.00	0.197463
$906$	0	0
$907$	13588.0	0.497444	0.248722	−	0.968575i	$-0.419989\pi$
$907$	13588.0	0.497444	0.248722	+	0.968575i	$0.419989\pi$
$908$	0	0
$909$	−34500.0	−1.25885
$910$	0	0
$911$	−47304.0	−1.72036	−0.860182	−	0.509987i	$-0.829650\pi$
$911$	−47304.0	−1.72036	−0.860182	+	0.509987i	$0.829650\pi$
$912$	0	0
$913$	18144.0	0.657699
$914$	0	0
$915$	−9120.00	−0.329506
$916$	0	0
$917$	−14910.0	−0.536937
$918$	0	0
$919$	1784.00	0.0640356	0.0320178	−	0.999487i	$-0.489807\pi$
$919$	1784.00	0.0640356	0.0320178	+	0.999487i	$0.489807\pi$
$920$	0	0
$921$	1148.00	0.0410726
$922$	0	0
$923$	−32256.0	−1.15029
$924$	0	0
$925$	−2774.00	−0.0986038
$926$	0	0
$927$	−8740.00	−0.309665
$928$	0	0
$929$	−35922.0	−1.26864	−0.634318	−	0.773072i	$-0.718719\pi$
$929$	−35922.0	−1.26864	−0.634318	+	0.773072i	$0.718719\pi$
$930$	0	0
$931$	−98.0000	−0.00344986
$932$	0	0
$933$	−17616.0	−0.618137
$934$	0	0
$935$	65664.0	2.29673
$936$	0	0
$937$	−26782.0	−0.933756	−0.466878	−	0.884322i	$-0.654621\pi$
$937$	−26782.0	−0.933756	−0.466878	+	0.884322i	$0.654621\pi$
$938$	0	0
$939$	−5540.00	−0.192536
$940$	0	0
$941$	−4044.00	−0.140096	−0.0700482	−	0.997544i	$-0.522315\pi$
$941$	−4044.00	−0.140096	−0.0700482	+	0.997544i	$0.522315\pi$
$942$	0	0
$943$	−15120.0	−0.522137
$944$	0	0
$945$	−8400.00	−0.289156
$946$	0	0
$947$	2136.00	0.0732953	0.0366477	−	0.999328i	$-0.488332\pi$
$947$	2136.00	0.0732953	0.0366477	+	0.999328i	$0.488332\pi$
$948$	0	0
$949$	64400.0	2.20286
$950$	0	0
$951$	−15132.0	−0.515971
$952$	0	0
$953$	−15174.0	−0.515776	−0.257888	−	0.966175i	$-0.583026\pi$
$953$	−15174.0	−0.515776	−0.257888	+	0.966175i	$0.583026\pi$
$954$	0	0
$955$	−25632.0	−0.868515
$956$	0	0
$957$	−5184.00	−0.175104
$958$	0	0
$959$	−546.000	−0.0183850
$960$	0	0
$961$	25905.0	0.869558
$962$	0	0
$963$	14628.0	0.489492
$964$	0	0
$965$	53160.0	1.77335
$966$	0	0
$967$	25832.0	0.859050	0.429525	−	0.903055i	$-0.358681\pi$
$967$	25832.0	0.859050	0.429525	+	0.903055i	$0.358681\pi$
$968$	0	0
$969$	456.000	0.0151175
$970$	0	0
$971$	37686.0	1.24552	0.622761	−	0.782412i	$-0.286011\pi$
$971$	37686.0	1.24552	0.622761	+	0.782412i	$0.286011\pi$
$972$	0	0
$973$	16366.0	0.539229
$974$	0	0
$975$	−2128.00	−0.0698980
$976$	0	0
$977$	−54006.0	−1.76848	−0.884240	−	0.467033i	$-0.845323\pi$
$977$	−54006.0	−1.76848	−0.884240	+	0.467033i	$0.845323\pi$
$978$	0	0
$979$	18720.0	0.611127
$980$	0	0
$981$	3358.00	0.109289
$982$	0	0
$983$	33276.0	1.07969	0.539847	−	0.841763i	$-0.318482\pi$
$983$	33276.0	1.07969	0.539847	+	0.841763i	$0.318482\pi$
$984$	0	0
$985$	−2376.00	−0.0768585
$986$	0	0
$987$	−168.000	−0.00541793
$988$	0	0
$989$	−45120.0	−1.45069
$990$	0	0
$991$	−3760.00	−0.120525	−0.0602625	−	0.998183i	$-0.519194\pi$
$991$	−3760.00	−0.120525	−0.0602625	+	0.998183i	$0.519194\pi$
$992$	0	0
$993$	22640.0	0.723523
$994$	0	0
$995$	−27408.0	−0.873258
$996$	0	0
$997$	−36524.0	−1.16021	−0.580104	−	0.814543i	$-0.696988\pi$
$997$	−36524.0	−1.16021	−0.580104	+	0.814543i	$0.696988\pi$
$998$	0	0
$999$	14600.0	0.462386

Twists

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

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