LMFDB - Embedded newform 1344.4.a.q.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,4,Mod(1,1344)] 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("1344.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,-6,0,7,0,9,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1344.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -6.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} -36.0000 q^{11} -62.0000 q^{13} -18.0000 q^{15} +114.000 q^{17} +76.0000 q^{19} +21.0000 q^{21} -24.0000 q^{23} -89.0000 q^{25} +27.0000 q^{27} -54.0000 q^{29} -112.000 q^{31} -108.000 q^{33} -42.0000 q^{35} +178.000 q^{37} -186.000 q^{39} +378.000 q^{41} +172.000 q^{43} -54.0000 q^{45} -192.000 q^{47} +49.0000 q^{49} +342.000 q^{51} +402.000 q^{53} +216.000 q^{55} +228.000 q^{57} -396.000 q^{59} -254.000 q^{61} +63.0000 q^{63} +372.000 q^{65} +1012.00 q^{67} -72.0000 q^{69} +840.000 q^{71} +890.000 q^{73} -267.000 q^{75} -252.000 q^{77} +80.0000 q^{79} +81.0000 q^{81} +108.000 q^{83} -684.000 q^{85} -162.000 q^{87} -1638.00 q^{89} -434.000 q^{91} -336.000 q^{93} -456.000 q^{95} +1010.00 q^{97} -324.000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−6.00000	−0.536656	−0.268328	−	0.963328i	$-0.586471\pi$
$5$	−6.00000	−0.536656	−0.268328	+	0.963328i	$0.586471\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$11$	−36.0000	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$12$	0	0
$13$	−62.0000	−1.32275	−0.661373	−	0.750057i	$-0.730026\pi$
$13$	−62.0000	−1.32275	−0.661373	+	0.750057i	$0.730026\pi$
$14$	0	0
$15$	−18.0000	−0.309839
$16$	0	0
$17$	114.000	1.62642	0.813208	−	0.581974i	$-0.197719\pi$
$17$	114.000	1.62642	0.813208	+	0.581974i	$0.197719\pi$
$18$	0	0
$19$	76.0000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	76.0000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	−24.0000	−0.217580	−0.108790	−	0.994065i	$-0.534698\pi$
$23$	−24.0000	−0.217580	−0.108790	+	0.994065i	$0.534698\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−54.0000	−0.345778	−0.172889	−	0.984941i	$-0.555310\pi$
$29$	−54.0000	−0.345778	−0.172889	+	0.984941i	$0.555310\pi$
$30$	0	0
$31$	−112.000	−0.648897	−0.324448	−	0.945903i	$-0.605179\pi$
$31$	−112.000	−0.648897	−0.324448	+	0.945903i	$0.605179\pi$
$32$	0	0
$33$	−108.000	−0.569709
$34$	0	0
$35$	−42.0000	−0.202837
$36$	0	0
$37$	178.000	0.790892	0.395446	−	0.918489i	$-0.370590\pi$
$37$	178.000	0.790892	0.395446	+	0.918489i	$0.370590\pi$
$38$	0	0
$39$	−186.000	−0.763688
$40$	0	0
$41$	378.000	1.43985	0.719923	−	0.694054i	$-0.244177\pi$
$41$	378.000	1.43985	0.719923	+	0.694054i	$0.244177\pi$
$42$	0	0
$43$	172.000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	172.000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−54.0000	−0.178885
$46$	0	0
$47$	−192.000	−0.595874	−0.297937	−	0.954586i	$-0.596299\pi$
$47$	−192.000	−0.595874	−0.297937	+	0.954586i	$0.596299\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	342.000	0.939011
$52$	0	0
$53$	402.000	1.04187	0.520933	−	0.853597i	$-0.325584\pi$
$53$	402.000	1.04187	0.520933	+	0.853597i	$0.325584\pi$
$54$	0	0
$55$	216.000	0.529553
$56$	0	0
$57$	228.000	0.529813
$58$	0	0
$59$	−396.000	−0.873810	−0.436905	−	0.899508i	$-0.643925\pi$
$59$	−396.000	−0.873810	−0.436905	+	0.899508i	$0.643925\pi$
$60$	0	0
$61$	−254.000	−0.533137	−0.266569	−	0.963816i	$-0.585890\pi$
$61$	−254.000	−0.533137	−0.266569	+	0.963816i	$0.585890\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	372.000	0.709860
$66$	0	0
$67$	1012.00	1.84531	0.922653	−	0.385632i	$-0.126016\pi$
$67$	1012.00	1.84531	0.922653	+	0.385632i	$0.126016\pi$
$68$	0	0
$69$	−72.0000	−0.125620
$70$	0	0
$71$	840.000	1.40408	0.702040	−	0.712138i	$-0.252273\pi$
$71$	840.000	1.40408	0.702040	+	0.712138i	$0.252273\pi$
$72$	0	0
$73$	890.000	1.42694	0.713470	−	0.700686i	$-0.247122\pi$
$73$	890.000	1.42694	0.713470	+	0.700686i	$0.247122\pi$
$74$	0	0
$75$	−267.000	−0.411073
$76$	0	0
$77$	−252.000	−0.372962
$78$	0	0
$79$	80.0000	0.113933	0.0569665	−	0.998376i	$-0.481857\pi$
$79$	80.0000	0.113933	0.0569665	+	0.998376i	$0.481857\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	108.000	0.142826	0.0714129	−	0.997447i	$-0.477249\pi$
$83$	108.000	0.142826	0.0714129	+	0.997447i	$0.477249\pi$
$84$	0	0
$85$	−684.000	−0.872826
$86$	0	0
$87$	−162.000	−0.199635
$88$	0	0
$89$	−1638.00	−1.95087	−0.975436	−	0.220282i	$-0.929302\pi$
$89$	−1638.00	−1.95087	−0.975436	+	0.220282i	$0.929302\pi$
$90$	0	0
$91$	−434.000	−0.499951
$92$	0	0
$93$	−336.000	−0.374641
$94$	0	0
$95$	−456.000	−0.492470
$96$	0	0
$97$	1010.00	1.05722	0.528608	−	0.848866i	$-0.322714\pi$
$97$	1010.00	1.05722	0.528608	+	0.848866i	$0.322714\pi$
$98$	0	0
$99$	−324.000	−0.328921
$100$	0	0
$101$	−6.00000	−0.00591111	−0.00295556	−	0.999996i	$-0.500941\pi$
$101$	−6.00000	−0.00591111	−0.00295556	+	0.999996i	$0.500941\pi$
$102$	0	0
$103$	−472.000	−0.451530	−0.225765	−	0.974182i	$-0.572488\pi$
$103$	−472.000	−0.451530	−0.225765	+	0.974182i	$0.572488\pi$
$104$	0	0
$105$	−126.000	−0.117108
$106$	0	0
$107$	972.000	0.878194	0.439097	−	0.898440i	$-0.355298\pi$
$107$	972.000	0.878194	0.439097	+	0.898440i	$0.355298\pi$
$108$	0	0
$109$	1786.00	1.56943	0.784715	−	0.619857i	$-0.212810\pi$
$109$	1786.00	1.56943	0.784715	+	0.619857i	$0.212810\pi$
$110$	0	0
$111$	534.000	0.456622
$112$	0	0
$113$	−2286.00	−1.90309	−0.951543	−	0.307515i	$-0.900503\pi$
$113$	−2286.00	−1.90309	−0.951543	+	0.307515i	$0.900503\pi$
$114$	0	0
$115$	144.000	0.116766
$116$	0	0
$117$	−558.000	−0.440916
$118$	0	0
$119$	798.000	0.614727
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	1134.00	0.831295
$124$	0	0
$125$	1284.00	0.918756
$126$	0	0
$127$	1328.00	0.927881	0.463941	−	0.885866i	$-0.346435\pi$
$127$	1328.00	0.927881	0.463941	+	0.885866i	$0.346435\pi$
$128$	0	0
$129$	516.000	0.352180
$130$	0	0
$131$	1212.00	0.808343	0.404171	−	0.914683i	$-0.367560\pi$
$131$	1212.00	0.808343	0.404171	+	0.914683i	$0.367560\pi$
$132$	0	0
$133$	532.000	0.346844
$134$	0	0
$135$	−162.000	−0.103280
$136$	0	0
$137$	−1254.00	−0.782018	−0.391009	−	0.920387i	$-0.627874\pi$
$137$	−1254.00	−0.782018	−0.391009	+	0.920387i	$0.627874\pi$
$138$	0	0
$139$	340.000	0.207471	0.103735	−	0.994605i	$-0.466921\pi$
$139$	340.000	0.207471	0.103735	+	0.994605i	$0.466921\pi$
$140$	0	0
$141$	−576.000	−0.344028
$142$	0	0
$143$	2232.00	1.30524
$144$	0	0
$145$	324.000	0.185564
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−1038.00	−0.570713	−0.285357	−	0.958421i	$-0.592112\pi$
$149$	−1038.00	−0.570713	−0.285357	+	0.958421i	$0.592112\pi$
$150$	0	0
$151$	2936.00	1.58231	0.791153	−	0.611618i	$-0.209481\pi$
$151$	2936.00	1.58231	0.791153	+	0.611618i	$0.209481\pi$
$152$	0	0
$153$	1026.00	0.542138
$154$	0	0
$155$	672.000	0.348234
$156$	0	0
$157$	1330.00	0.676086	0.338043	−	0.941131i	$-0.390235\pi$
$157$	1330.00	0.676086	0.338043	+	0.941131i	$0.390235\pi$
$158$	0	0
$159$	1206.00	0.601522
$160$	0	0
$161$	−168.000	−0.0822376
$162$	0	0
$163$	3364.00	1.61650	0.808248	−	0.588842i	$-0.200416\pi$
$163$	3364.00	1.61650	0.808248	+	0.588842i	$0.200416\pi$
$164$	0	0
$165$	648.000	0.305738
$166$	0	0
$167$	3048.00	1.41234	0.706172	−	0.708041i	$-0.250421\pi$
$167$	3048.00	1.41234	0.706172	+	0.708041i	$0.250421\pi$
$168$	0	0
$169$	1647.00	0.749659
$170$	0	0
$171$	684.000	0.305888
$172$	0	0
$173$	2706.00	1.18921	0.594605	−	0.804018i	$-0.297308\pi$
$173$	2706.00	1.18921	0.594605	+	0.804018i	$0.297308\pi$
$174$	0	0
$175$	−623.000	−0.269111
$176$	0	0
$177$	−1188.00	−0.504495
$178$	0	0
$179$	−4716.00	−1.96922	−0.984610	−	0.174766i	$-0.944083\pi$
$179$	−4716.00	−1.96922	−0.984610	+	0.174766i	$0.944083\pi$
$180$	0	0
$181$	−1910.00	−0.784360	−0.392180	−	0.919888i	$-0.628279\pi$
$181$	−1910.00	−0.784360	−0.392180	+	0.919888i	$0.628279\pi$
$182$	0	0
$183$	−762.000	−0.307807
$184$	0	0
$185$	−1068.00	−0.424437
$186$	0	0
$187$	−4104.00	−1.60489
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	4080.00	1.54565	0.772823	−	0.634621i	$-0.218844\pi$
$191$	4080.00	1.54565	0.772823	+	0.634621i	$0.218844\pi$
$192$	0	0
$193$	−2686.00	−1.00177	−0.500887	−	0.865512i	$-0.666993\pi$
$193$	−2686.00	−1.00177	−0.500887	+	0.865512i	$0.666993\pi$
$194$	0	0
$195$	1116.00	0.409838
$196$	0	0
$197$	−510.000	−0.184447	−0.0922233	−	0.995738i	$-0.529397\pi$
$197$	−510.000	−0.184447	−0.0922233	+	0.995738i	$0.529397\pi$
$198$	0	0
$199$	1352.00	0.481612	0.240806	−	0.970573i	$-0.422588\pi$
$199$	1352.00	0.481612	0.240806	+	0.970573i	$0.422588\pi$
$200$	0	0
$201$	3036.00	1.06539
$202$	0	0
$203$	−378.000	−0.130692
$204$	0	0
$205$	−2268.00	−0.772702
$206$	0	0
$207$	−216.000	−0.0725268
$208$	0	0
$209$	−2736.00	−0.905517
$210$	0	0
$211$	3364.00	1.09757	0.548785	−	0.835963i	$-0.315091\pi$
$211$	3364.00	1.09757	0.548785	+	0.835963i	$0.315091\pi$
$212$	0	0
$213$	2520.00	0.810646
$214$	0	0
$215$	−1032.00	−0.327357
$216$	0	0
$217$	−784.000	−0.245260
$218$	0	0
$219$	2670.00	0.823844
$220$	0	0
$221$	−7068.00	−2.15134
$222$	0	0
$223$	−4768.00	−1.43179	−0.715894	−	0.698209i	$-0.753981\pi$
$223$	−4768.00	−1.43179	−0.715894	+	0.698209i	$0.753981\pi$
$224$	0	0
$225$	−801.000	−0.237333
$226$	0	0
$227$	−420.000	−0.122803	−0.0614017	−	0.998113i	$-0.519557\pi$
$227$	−420.000	−0.122803	−0.0614017	+	0.998113i	$0.519557\pi$
$228$	0	0
$229$	1882.00	0.543083	0.271542	−	0.962427i	$-0.412467\pi$
$229$	1882.00	0.543083	0.271542	+	0.962427i	$0.412467\pi$
$230$	0	0
$231$	−756.000	−0.215330
$232$	0	0
$233$	5082.00	1.42890	0.714448	−	0.699688i	$-0.246678\pi$
$233$	5082.00	1.42890	0.714448	+	0.699688i	$0.246678\pi$
$234$	0	0
$235$	1152.00	0.319780
$236$	0	0
$237$	240.000	0.0657792
$238$	0	0
$239$	−5424.00	−1.46799	−0.733995	−	0.679155i	$-0.762346\pi$
$239$	−5424.00	−1.46799	−0.733995	+	0.679155i	$0.762346\pi$
$240$	0	0
$241$	−2590.00	−0.692268	−0.346134	−	0.938185i	$-0.612506\pi$
$241$	−2590.00	−0.692268	−0.346134	+	0.938185i	$0.612506\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−294.000	−0.0766652
$246$	0	0
$247$	−4712.00	−1.21384
$248$	0	0
$249$	324.000	0.0824605
$250$	0	0
$251$	4932.00	1.24026	0.620130	−	0.784499i	$-0.287080\pi$
$251$	4932.00	1.24026	0.620130	+	0.784499i	$0.287080\pi$
$252$	0	0
$253$	864.000	0.214700
$254$	0	0
$255$	−2052.00	−0.503926
$256$	0	0
$257$	−3438.00	−0.834461	−0.417231	−	0.908801i	$-0.636999\pi$
$257$	−3438.00	−0.834461	−0.417231	+	0.908801i	$0.636999\pi$
$258$	0	0
$259$	1246.00	0.298929
$260$	0	0
$261$	−486.000	−0.115259
$262$	0	0
$263$	6120.00	1.43489	0.717444	−	0.696617i	$-0.245312\pi$
$263$	6120.00	1.43489	0.717444	+	0.696617i	$0.245312\pi$
$264$	0	0
$265$	−2412.00	−0.559124
$266$	0	0
$267$	−4914.00	−1.12634
$268$	0	0
$269$	18.0000	0.00407985	0.00203992	−	0.999998i	$-0.499351\pi$
$269$	18.0000	0.00407985	0.00203992	+	0.999998i	$0.499351\pi$
$270$	0	0
$271$	6896.00	1.54576	0.772882	−	0.634549i	$-0.218814\pi$
$271$	6896.00	1.54576	0.772882	+	0.634549i	$0.218814\pi$
$272$	0	0
$273$	−1302.00	−0.288647
$274$	0	0
$275$	3204.00	0.702576
$276$	0	0
$277$	−6254.00	−1.35656	−0.678279	−	0.734805i	$-0.737274\pi$
$277$	−6254.00	−1.35656	−0.678279	+	0.734805i	$0.737274\pi$
$278$	0	0
$279$	−1008.00	−0.216299
$280$	0	0
$281$	1194.00	0.253481	0.126740	−	0.991936i	$-0.459549\pi$
$281$	1194.00	0.253481	0.126740	+	0.991936i	$0.459549\pi$
$282$	0	0
$283$	7156.00	1.50311	0.751555	−	0.659671i	$-0.229304\pi$
$283$	7156.00	1.50311	0.751555	+	0.659671i	$0.229304\pi$
$284$	0	0
$285$	−1368.00	−0.284327
$286$	0	0
$287$	2646.00	0.544211
$288$	0	0
$289$	8083.00	1.64523
$290$	0	0
$291$	3030.00	0.610384
$292$	0	0
$293$	3738.00	0.745312	0.372656	−	0.927970i	$-0.378447\pi$
$293$	3738.00	0.745312	0.372656	+	0.927970i	$0.378447\pi$
$294$	0	0
$295$	2376.00	0.468936
$296$	0	0
$297$	−972.000	−0.189903
$298$	0	0
$299$	1488.00	0.287804
$300$	0	0
$301$	1204.00	0.230556
$302$	0	0
$303$	−18.0000	−0.00341278
$304$	0	0
$305$	1524.00	0.286111
$306$	0	0
$307$	844.000	0.156904	0.0784522	−	0.996918i	$-0.475002\pi$
$307$	844.000	0.156904	0.0784522	+	0.996918i	$0.475002\pi$
$308$	0	0
$309$	−1416.00	−0.260691
$310$	0	0
$311$	6312.00	1.15087	0.575435	−	0.817847i	$-0.304833\pi$
$311$	6312.00	1.15087	0.575435	+	0.817847i	$0.304833\pi$
$312$	0	0
$313$	8282.00	1.49561	0.747806	−	0.663918i	$-0.231108\pi$
$313$	8282.00	1.49561	0.747806	+	0.663918i	$0.231108\pi$
$314$	0	0
$315$	−378.000	−0.0676123
$316$	0	0
$317$	−9318.00	−1.65095	−0.825475	−	0.564439i	$-0.809093\pi$
$317$	−9318.00	−1.65095	−0.825475	+	0.564439i	$0.809093\pi$
$318$	0	0
$319$	1944.00	0.341201
$320$	0	0
$321$	2916.00	0.507026
$322$	0	0
$323$	8664.00	1.49250
$324$	0	0
$325$	5518.00	0.941796
$326$	0	0
$327$	5358.00	0.906110
$328$	0	0
$329$	−1344.00	−0.225219
$330$	0	0
$331$	−1652.00	−0.274327	−0.137163	−	0.990548i	$-0.543799\pi$
$331$	−1652.00	−0.274327	−0.137163	+	0.990548i	$0.543799\pi$
$332$	0	0
$333$	1602.00	0.263631
$334$	0	0
$335$	−6072.00	−0.990295
$336$	0	0
$337$	−1294.00	−0.209165	−0.104583	−	0.994516i	$-0.533351\pi$
$337$	−1294.00	−0.209165	−0.104583	+	0.994516i	$0.533351\pi$
$338$	0	0
$339$	−6858.00	−1.09875
$340$	0	0
$341$	4032.00	0.640308
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	432.000	0.0674148
$346$	0	0
$347$	−3636.00	−0.562509	−0.281255	−	0.959633i	$-0.590751\pi$
$347$	−3636.00	−0.562509	−0.281255	+	0.959633i	$0.590751\pi$
$348$	0	0
$349$	−10478.0	−1.60709	−0.803545	−	0.595244i	$-0.797055\pi$
$349$	−10478.0	−1.60709	−0.803545	+	0.595244i	$0.797055\pi$
$350$	0	0
$351$	−1674.00	−0.254563
$352$	0	0
$353$	−7566.00	−1.14079	−0.570393	−	0.821372i	$-0.693209\pi$
$353$	−7566.00	−1.14079	−0.570393	+	0.821372i	$0.693209\pi$
$354$	0	0
$355$	−5040.00	−0.753508
$356$	0	0
$357$	2394.00	0.354913
$358$	0	0
$359$	−8040.00	−1.18199	−0.590996	−	0.806675i	$-0.701265\pi$
$359$	−8040.00	−1.18199	−0.590996	+	0.806675i	$0.701265\pi$
$360$	0	0
$361$	−1083.00	−0.157895
$362$	0	0
$363$	−105.000	−0.0151820
$364$	0	0
$365$	−5340.00	−0.765776
$366$	0	0
$367$	7568.00	1.07642	0.538210	−	0.842811i	$-0.319101\pi$
$367$	7568.00	1.07642	0.538210	+	0.842811i	$0.319101\pi$
$368$	0	0
$369$	3402.00	0.479949
$370$	0	0
$371$	2814.00	0.393789
$372$	0	0
$373$	13522.0	1.87706	0.938529	−	0.345200i	$-0.112189\pi$
$373$	13522.0	1.87706	0.938529	+	0.345200i	$0.112189\pi$
$374$	0	0
$375$	3852.00	0.530444
$376$	0	0
$377$	3348.00	0.457376
$378$	0	0
$379$	−2468.00	−0.334492	−0.167246	−	0.985915i	$-0.553487\pi$
$379$	−2468.00	−0.334492	−0.167246	+	0.985915i	$0.553487\pi$
$380$	0	0
$381$	3984.00	0.535713
$382$	0	0
$383$	−12336.0	−1.64580	−0.822898	−	0.568189i	$-0.807644\pi$
$383$	−12336.0	−1.64580	−0.822898	+	0.568189i	$0.807644\pi$
$384$	0	0
$385$	1512.00	0.200152
$386$	0	0
$387$	1548.00	0.203331
$388$	0	0
$389$	3762.00	0.490337	0.245168	−	0.969481i	$-0.421157\pi$
$389$	3762.00	0.490337	0.245168	+	0.969481i	$0.421157\pi$
$390$	0	0
$391$	−2736.00	−0.353876
$392$	0	0
$393$	3636.00	0.466697
$394$	0	0
$395$	−480.000	−0.0611428
$396$	0	0
$397$	8770.00	1.10870	0.554350	−	0.832284i	$-0.312967\pi$
$397$	8770.00	1.10870	0.554350	+	0.832284i	$0.312967\pi$
$398$	0	0
$399$	1596.00	0.200250
$400$	0	0
$401$	6642.00	0.827146	0.413573	−	0.910471i	$-0.364281\pi$
$401$	6642.00	0.827146	0.413573	+	0.910471i	$0.364281\pi$
$402$	0	0
$403$	6944.00	0.858326
$404$	0	0
$405$	−486.000	−0.0596285
$406$	0	0
$407$	−6408.00	−0.780424
$408$	0	0
$409$	−1510.00	−0.182554	−0.0912771	−	0.995826i	$-0.529095\pi$
$409$	−1510.00	−0.182554	−0.0912771	+	0.995826i	$0.529095\pi$
$410$	0	0
$411$	−3762.00	−0.451498
$412$	0	0
$413$	−2772.00	−0.330269
$414$	0	0
$415$	−648.000	−0.0766484
$416$	0	0
$417$	1020.00	0.119783
$418$	0	0
$419$	1260.00	0.146909	0.0734547	−	0.997299i	$-0.476598\pi$
$419$	1260.00	0.146909	0.0734547	+	0.997299i	$0.476598\pi$
$420$	0	0
$421$	−3998.00	−0.462828	−0.231414	−	0.972855i	$-0.574335\pi$
$421$	−3998.00	−0.462828	−0.231414	+	0.972855i	$0.574335\pi$
$422$	0	0
$423$	−1728.00	−0.198625
$424$	0	0
$425$	−10146.0	−1.15801
$426$	0	0
$427$	−1778.00	−0.201507
$428$	0	0
$429$	6696.00	0.753580
$430$	0	0
$431$	−2736.00	−0.305774	−0.152887	−	0.988244i	$-0.548857\pi$
$431$	−2736.00	−0.305774	−0.152887	+	0.988244i	$0.548857\pi$
$432$	0	0
$433$	2690.00	0.298552	0.149276	−	0.988796i	$-0.452306\pi$
$433$	2690.00	0.298552	0.149276	+	0.988796i	$0.452306\pi$
$434$	0	0
$435$	972.000	0.107135
$436$	0	0
$437$	−1824.00	−0.199665
$438$	0	0
$439$	−1240.00	−0.134811	−0.0674054	−	0.997726i	$-0.521472\pi$
$439$	−1240.00	−0.134811	−0.0674054	+	0.997726i	$0.521472\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	3900.00	0.418272	0.209136	−	0.977887i	$-0.432935\pi$
$443$	3900.00	0.418272	0.209136	+	0.977887i	$0.432935\pi$
$444$	0	0
$445$	9828.00	1.04695
$446$	0	0
$447$	−3114.00	−0.329501
$448$	0	0
$449$	−10878.0	−1.14335	−0.571675	−	0.820480i	$-0.693706\pi$
$449$	−10878.0	−1.14335	−0.571675	+	0.820480i	$0.693706\pi$
$450$	0	0
$451$	−13608.0	−1.42079
$452$	0	0
$453$	8808.00	0.913545
$454$	0	0
$455$	2604.00	0.268302
$456$	0	0
$457$	2330.00	0.238496	0.119248	−	0.992864i	$-0.461952\pi$
$457$	2330.00	0.238496	0.119248	+	0.992864i	$0.461952\pi$
$458$	0	0
$459$	3078.00	0.313004
$460$	0	0
$461$	−15150.0	−1.53060	−0.765299	−	0.643675i	$-0.777409\pi$
$461$	−15150.0	−1.53060	−0.765299	+	0.643675i	$0.777409\pi$
$462$	0	0
$463$	−2992.00	−0.300324	−0.150162	−	0.988661i	$-0.547980\pi$
$463$	−2992.00	−0.300324	−0.150162	+	0.988661i	$0.547980\pi$
$464$	0	0
$465$	2016.00	0.201053
$466$	0	0
$467$	−8724.00	−0.864451	−0.432225	−	0.901766i	$-0.642272\pi$
$467$	−8724.00	−0.864451	−0.432225	+	0.901766i	$0.642272\pi$
$468$	0	0
$469$	7084.00	0.697460
$470$	0	0
$471$	3990.00	0.390339
$472$	0	0
$473$	−6192.00	−0.601921
$474$	0	0
$475$	−6764.00	−0.653376
$476$	0	0
$477$	3618.00	0.347289
$478$	0	0
$479$	9744.00	0.929467	0.464734	−	0.885451i	$-0.346150\pi$
$479$	9744.00	0.929467	0.464734	+	0.885451i	$0.346150\pi$
$480$	0	0
$481$	−11036.0	−1.04615
$482$	0	0
$483$	−504.000	−0.0474799
$484$	0	0
$485$	−6060.00	−0.567362
$486$	0	0
$487$	4136.00	0.384846	0.192423	−	0.981312i	$-0.438365\pi$
$487$	4136.00	0.384846	0.192423	+	0.981312i	$0.438365\pi$
$488$	0	0
$489$	10092.0	0.933284
$490$	0	0
$491$	−16212.0	−1.49010	−0.745048	−	0.667011i	$-0.767574\pi$
$491$	−16212.0	−1.49010	−0.745048	+	0.667011i	$0.767574\pi$
$492$	0	0
$493$	−6156.00	−0.562378
$494$	0	0
$495$	1944.00	0.176518
$496$	0	0
$497$	5880.00	0.530692
$498$	0	0
$499$	−2396.00	−0.214949	−0.107475	−	0.994208i	$-0.534276\pi$
$499$	−2396.00	−0.214949	−0.107475	+	0.994208i	$0.534276\pi$
$500$	0	0
$501$	9144.00	0.815417
$502$	0	0
$503$	13128.0	1.16371	0.581857	−	0.813291i	$-0.302326\pi$
$503$	13128.0	1.16371	0.581857	+	0.813291i	$0.302326\pi$
$504$	0	0
$505$	36.0000	0.00317224
$506$	0	0
$507$	4941.00	0.432816
$508$	0	0
$509$	−12798.0	−1.11446	−0.557231	−	0.830357i	$-0.688136\pi$
$509$	−12798.0	−1.11446	−0.557231	+	0.830357i	$0.688136\pi$
$510$	0	0
$511$	6230.00	0.539333
$512$	0	0
$513$	2052.00	0.176604
$514$	0	0
$515$	2832.00	0.242316
$516$	0	0
$517$	6912.00	0.587987
$518$	0	0
$519$	8118.00	0.686591
$520$	0	0
$521$	7386.00	0.621087	0.310544	−	0.950559i	$-0.399489\pi$
$521$	7386.00	0.621087	0.310544	+	0.950559i	$0.399489\pi$
$522$	0	0
$523$	−5180.00	−0.433089	−0.216545	−	0.976273i	$-0.569479\pi$
$523$	−5180.00	−0.433089	−0.216545	+	0.976273i	$0.569479\pi$
$524$	0	0
$525$	−1869.00	−0.155371
$526$	0	0
$527$	−12768.0	−1.05538
$528$	0	0
$529$	−11591.0	−0.952659
$530$	0	0
$531$	−3564.00	−0.291270
$532$	0	0
$533$	−23436.0	−1.90455
$534$	0	0
$535$	−5832.00	−0.471288
$536$	0	0
$537$	−14148.0	−1.13693
$538$	0	0
$539$	−1764.00	−0.140966
$540$	0	0
$541$	−4070.00	−0.323444	−0.161722	−	0.986836i	$-0.551705\pi$
$541$	−4070.00	−0.323444	−0.161722	+	0.986836i	$0.551705\pi$
$542$	0	0
$543$	−5730.00	−0.452851
$544$	0	0
$545$	−10716.0	−0.842244
$546$	0	0
$547$	−14780.0	−1.15530	−0.577648	−	0.816286i	$-0.696029\pi$
$547$	−14780.0	−1.15530	−0.577648	+	0.816286i	$0.696029\pi$
$548$	0	0
$549$	−2286.00	−0.177712
$550$	0	0
$551$	−4104.00	−0.317307
$552$	0	0
$553$	560.000	0.0430626
$554$	0	0
$555$	−3204.00	−0.245049
$556$	0	0
$557$	6858.00	0.521693	0.260846	−	0.965380i	$-0.415998\pi$
$557$	6858.00	0.521693	0.260846	+	0.965380i	$0.415998\pi$
$558$	0	0
$559$	−10664.0	−0.806868
$560$	0	0
$561$	−12312.0	−0.926583
$562$	0	0
$563$	−6660.00	−0.498553	−0.249277	−	0.968432i	$-0.580193\pi$
$563$	−6660.00	−0.498553	−0.249277	+	0.968432i	$0.580193\pi$
$564$	0	0
$565$	13716.0	1.02130
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	−150.000	−0.0110515	−0.00552577	−	0.999985i	$-0.501759\pi$
$569$	−150.000	−0.0110515	−0.00552577	+	0.999985i	$0.501759\pi$
$570$	0	0
$571$	8188.00	0.600100	0.300050	−	0.953923i	$-0.402997\pi$
$571$	8188.00	0.600100	0.300050	+	0.953923i	$0.402997\pi$
$572$	0	0
$573$	12240.0	0.892379
$574$	0	0
$575$	2136.00	0.154917
$576$	0	0
$577$	−5854.00	−0.422366	−0.211183	−	0.977447i	$-0.567732\pi$
$577$	−5854.00	−0.422366	−0.211183	+	0.977447i	$0.567732\pi$
$578$	0	0
$579$	−8058.00	−0.578375
$580$	0	0
$581$	756.000	0.0539831
$582$	0	0
$583$	−14472.0	−1.02808
$584$	0	0
$585$	3348.00	0.236620
$586$	0	0
$587$	−17580.0	−1.23612	−0.618062	−	0.786130i	$-0.712082\pi$
$587$	−17580.0	−1.23612	−0.618062	+	0.786130i	$0.712082\pi$
$588$	0	0
$589$	−8512.00	−0.595468
$590$	0	0
$591$	−1530.00	−0.106490
$592$	0	0
$593$	17154.0	1.18791	0.593955	−	0.804498i	$-0.297566\pi$
$593$	17154.0	1.18791	0.593955	+	0.804498i	$0.297566\pi$
$594$	0	0
$595$	−4788.00	−0.329897
$596$	0	0
$597$	4056.00	0.278059
$598$	0	0
$599$	−18120.0	−1.23600	−0.617999	−	0.786179i	$-0.712057\pi$
$599$	−18120.0	−1.23600	−0.617999	+	0.786179i	$0.712057\pi$
$600$	0	0
$601$	17546.0	1.19088	0.595438	−	0.803401i	$-0.296979\pi$
$601$	17546.0	1.19088	0.595438	+	0.803401i	$0.296979\pi$
$602$	0	0
$603$	9108.00	0.615102
$604$	0	0
$605$	210.000	0.0141119
$606$	0	0
$607$	−14560.0	−0.973595	−0.486798	−	0.873515i	$-0.661835\pi$
$607$	−14560.0	−0.973595	−0.486798	+	0.873515i	$0.661835\pi$
$608$	0	0
$609$	−1134.00	−0.0754548
$610$	0	0
$611$	11904.0	0.788190
$612$	0	0
$613$	4498.00	0.296366	0.148183	−	0.988960i	$-0.452657\pi$
$613$	4498.00	0.296366	0.148183	+	0.988960i	$0.452657\pi$
$614$	0	0
$615$	−6804.00	−0.446120
$616$	0	0
$617$	−5478.00	−0.357433	−0.178716	−	0.983901i	$-0.557194\pi$
$617$	−5478.00	−0.357433	−0.178716	+	0.983901i	$0.557194\pi$
$618$	0	0
$619$	−6044.00	−0.392454	−0.196227	−	0.980559i	$-0.562869\pi$
$619$	−6044.00	−0.392454	−0.196227	+	0.980559i	$0.562869\pi$
$620$	0	0
$621$	−648.000	−0.0418733
$622$	0	0
$623$	−11466.0	−0.737360
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	−8208.00	−0.522801
$628$	0	0
$629$	20292.0	1.28632
$630$	0	0
$631$	−15352.0	−0.968547	−0.484274	−	0.874917i	$-0.660916\pi$
$631$	−15352.0	−0.968547	−0.484274	+	0.874917i	$0.660916\pi$
$632$	0	0
$633$	10092.0	0.633682
$634$	0	0
$635$	−7968.00	−0.497953
$636$	0	0
$637$	−3038.00	−0.188964
$638$	0	0
$639$	7560.00	0.468027
$640$	0	0
$641$	−22398.0	−1.38014	−0.690068	−	0.723744i	$-0.742420\pi$
$641$	−22398.0	−1.38014	−0.690068	+	0.723744i	$0.742420\pi$
$642$	0	0
$643$	−3764.00	−0.230852	−0.115426	−	0.993316i	$-0.536823\pi$
$643$	−3764.00	−0.230852	−0.115426	+	0.993316i	$0.536823\pi$
$644$	0	0
$645$	−3096.00	−0.189000
$646$	0	0
$647$	−17688.0	−1.07479	−0.537393	−	0.843332i	$-0.680591\pi$
$647$	−17688.0	−1.07479	−0.537393	+	0.843332i	$0.680591\pi$
$648$	0	0
$649$	14256.0	0.862245
$650$	0	0
$651$	−2352.00	−0.141601
$652$	0	0
$653$	−19878.0	−1.19125	−0.595625	−	0.803263i	$-0.703096\pi$
$653$	−19878.0	−1.19125	−0.595625	+	0.803263i	$0.703096\pi$
$654$	0	0
$655$	−7272.00	−0.433802
$656$	0	0
$657$	8010.00	0.475647
$658$	0	0
$659$	20004.0	1.18247	0.591233	−	0.806501i	$-0.298641\pi$
$659$	20004.0	1.18247	0.591233	+	0.806501i	$0.298641\pi$
$660$	0	0
$661$	1306.00	0.0768495	0.0384247	−	0.999261i	$-0.487766\pi$
$661$	1306.00	0.0768495	0.0384247	+	0.999261i	$0.487766\pi$
$662$	0	0
$663$	−21204.0	−1.24207
$664$	0	0
$665$	−3192.00	−0.186136
$666$	0	0
$667$	1296.00	0.0752344
$668$	0	0
$669$	−14304.0	−0.826644
$670$	0	0
$671$	9144.00	0.526081
$672$	0	0
$673$	−13054.0	−0.747689	−0.373845	−	0.927491i	$-0.621961\pi$
$673$	−13054.0	−0.747689	−0.373845	+	0.927491i	$0.621961\pi$
$674$	0	0
$675$	−2403.00	−0.137024
$676$	0	0
$677$	−5046.00	−0.286460	−0.143230	−	0.989689i	$-0.545749\pi$
$677$	−5046.00	−0.286460	−0.143230	+	0.989689i	$0.545749\pi$
$678$	0	0
$679$	7070.00	0.399590
$680$	0	0
$681$	−1260.00	−0.0709006
$682$	0	0
$683$	−12468.0	−0.698499	−0.349249	−	0.937030i	$-0.613563\pi$
$683$	−12468.0	−0.698499	−0.349249	+	0.937030i	$0.613563\pi$
$684$	0	0
$685$	7524.00	0.419675
$686$	0	0
$687$	5646.00	0.313549
$688$	0	0
$689$	−24924.0	−1.37813
$690$	0	0
$691$	23212.0	1.27790	0.638948	−	0.769250i	$-0.279370\pi$
$691$	23212.0	1.27790	0.638948	+	0.769250i	$0.279370\pi$
$692$	0	0
$693$	−2268.00	−0.124321
$694$	0	0
$695$	−2040.00	−0.111340
$696$	0	0
$697$	43092.0	2.34179
$698$	0	0
$699$	15246.0	0.824974
$700$	0	0
$701$	−35958.0	−1.93740	−0.968698	−	0.248241i	$-0.920147\pi$
$701$	−35958.0	−1.93740	−0.968698	+	0.248241i	$0.920147\pi$
$702$	0	0
$703$	13528.0	0.725773
$704$	0	0
$705$	3456.00	0.184625
$706$	0	0
$707$	−42.0000	−0.00223419
$708$	0	0
$709$	−6446.00	−0.341445	−0.170723	−	0.985319i	$-0.554610\pi$
$709$	−6446.00	−0.341445	−0.170723	+	0.985319i	$0.554610\pi$
$710$	0	0
$711$	720.000	0.0379777
$712$	0	0
$713$	2688.00	0.141187
$714$	0	0
$715$	−13392.0	−0.700465
$716$	0	0
$717$	−16272.0	−0.847544
$718$	0	0
$719$	−4704.00	−0.243991	−0.121996	−	0.992531i	$-0.538929\pi$
$719$	−4704.00	−0.243991	−0.121996	+	0.992531i	$0.538929\pi$
$720$	0	0
$721$	−3304.00	−0.170662
$722$	0	0
$723$	−7770.00	−0.399681
$724$	0	0
$725$	4806.00	0.246194
$726$	0	0
$727$	−10600.0	−0.540760	−0.270380	−	0.962754i	$-0.587149\pi$
$727$	−10600.0	−0.540760	−0.270380	+	0.962754i	$0.587149\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	19608.0	0.992104
$732$	0	0
$733$	−12542.0	−0.631991	−0.315995	−	0.948761i	$-0.602338\pi$
$733$	−12542.0	−0.631991	−0.315995	+	0.948761i	$0.602338\pi$
$734$	0	0
$735$	−882.000	−0.0442627
$736$	0	0
$737$	−36432.0	−1.82088
$738$	0	0
$739$	−23324.0	−1.16101	−0.580506	−	0.814256i	$-0.697145\pi$
$739$	−23324.0	−1.16101	−0.580506	+	0.814256i	$0.697145\pi$
$740$	0	0
$741$	−14136.0	−0.700808
$742$	0	0
$743$	−6312.00	−0.311662	−0.155831	−	0.987784i	$-0.549806\pi$
$743$	−6312.00	−0.311662	−0.155831	+	0.987784i	$0.549806\pi$
$744$	0	0
$745$	6228.00	0.306277
$746$	0	0
$747$	972.000	0.0476086
$748$	0	0
$749$	6804.00	0.331926
$750$	0	0
$751$	35840.0	1.74144	0.870719	−	0.491781i	$-0.163654\pi$
$751$	35840.0	1.74144	0.870719	+	0.491781i	$0.163654\pi$
$752$	0	0
$753$	14796.0	0.716064
$754$	0	0
$755$	−17616.0	−0.849155
$756$	0	0
$757$	34594.0	1.66095	0.830476	−	0.557055i	$-0.188069\pi$
$757$	34594.0	1.66095	0.830476	+	0.557055i	$0.188069\pi$
$758$	0	0
$759$	2592.00	0.123957
$760$	0	0
$761$	23946.0	1.14066	0.570330	−	0.821416i	$-0.306815\pi$
$761$	23946.0	1.14066	0.570330	+	0.821416i	$0.306815\pi$
$762$	0	0
$763$	12502.0	0.593188
$764$	0	0
$765$	−6156.00	−0.290942
$766$	0	0
$767$	24552.0	1.15583
$768$	0	0
$769$	18770.0	0.880187	0.440093	−	0.897952i	$-0.354945\pi$
$769$	18770.0	0.880187	0.440093	+	0.897952i	$0.354945\pi$
$770$	0	0
$771$	−10314.0	−0.481776
$772$	0	0
$773$	−30342.0	−1.41181	−0.705903	−	0.708309i	$-0.749459\pi$
$773$	−30342.0	−1.41181	−0.705903	+	0.708309i	$0.749459\pi$
$774$	0	0
$775$	9968.00	0.462014
$776$	0	0
$777$	3738.00	0.172587
$778$	0	0
$779$	28728.0	1.32129
$780$	0	0
$781$	−30240.0	−1.38550
$782$	0	0
$783$	−1458.00	−0.0665449
$784$	0	0
$785$	−7980.00	−0.362826
$786$	0	0
$787$	26188.0	1.18615	0.593076	−	0.805147i	$-0.297913\pi$
$787$	26188.0	1.18615	0.593076	+	0.805147i	$0.297913\pi$
$788$	0	0
$789$	18360.0	0.828433
$790$	0	0
$791$	−16002.0	−0.719299
$792$	0	0
$793$	15748.0	0.705205
$794$	0	0
$795$	−7236.00	−0.322811
$796$	0	0
$797$	34818.0	1.54745	0.773724	−	0.633522i	$-0.218391\pi$
$797$	34818.0	1.54745	0.773724	+	0.633522i	$0.218391\pi$
$798$	0	0
$799$	−21888.0	−0.969139
$800$	0	0
$801$	−14742.0	−0.650291
$802$	0	0
$803$	−32040.0	−1.40805
$804$	0	0
$805$	1008.00	0.0441333
$806$	0	0
$807$	54.0000	0.00235550
$808$	0	0
$809$	−21702.0	−0.943142	−0.471571	−	0.881828i	$-0.656313\pi$
$809$	−21702.0	−0.943142	−0.471571	+	0.881828i	$0.656313\pi$
$810$	0	0
$811$	20356.0	0.881376	0.440688	−	0.897660i	$-0.354735\pi$
$811$	20356.0	0.881376	0.440688	+	0.897660i	$0.354735\pi$
$812$	0	0
$813$	20688.0	0.892448
$814$	0	0
$815$	−20184.0	−0.867503
$816$	0	0
$817$	13072.0	0.559769
$818$	0	0
$819$	−3906.00	−0.166650
$820$	0	0
$821$	19890.0	0.845513	0.422756	−	0.906243i	$-0.361063\pi$
$821$	19890.0	0.845513	0.422756	+	0.906243i	$0.361063\pi$
$822$	0	0
$823$	4232.00	0.179245	0.0896223	−	0.995976i	$-0.471434\pi$
$823$	4232.00	0.179245	0.0896223	+	0.995976i	$0.471434\pi$
$824$	0	0
$825$	9612.00	0.405633
$826$	0	0
$827$	−9636.00	−0.405171	−0.202586	−	0.979265i	$-0.564934\pi$
$827$	−9636.00	−0.405171	−0.202586	+	0.979265i	$0.564934\pi$
$828$	0	0
$829$	−35294.0	−1.47866	−0.739331	−	0.673342i	$-0.764858\pi$
$829$	−35294.0	−1.47866	−0.739331	+	0.673342i	$0.764858\pi$
$830$	0	0
$831$	−18762.0	−0.783209
$832$	0	0
$833$	5586.00	0.232345
$834$	0	0
$835$	−18288.0	−0.757943
$836$	0	0
$837$	−3024.00	−0.124880
$838$	0	0
$839$	−3768.00	−0.155049	−0.0775243	−	0.996990i	$-0.524702\pi$
$839$	−3768.00	−0.155049	−0.0775243	+	0.996990i	$0.524702\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	0	0
$843$	3582.00	0.146347
$844$	0	0
$845$	−9882.00	−0.402309
$846$	0	0
$847$	−245.000	−0.00993896
$848$	0	0
$849$	21468.0	0.867821
$850$	0	0
$851$	−4272.00	−0.172083
$852$	0	0
$853$	39466.0	1.58416	0.792081	−	0.610416i	$-0.208998\pi$
$853$	39466.0	1.58416	0.792081	+	0.610416i	$0.208998\pi$
$854$	0	0
$855$	−4104.00	−0.164157
$856$	0	0
$857$	−34038.0	−1.35673	−0.678364	−	0.734726i	$-0.737311\pi$
$857$	−34038.0	−1.35673	−0.678364	+	0.734726i	$0.737311\pi$
$858$	0	0
$859$	3364.00	0.133618	0.0668092	−	0.997766i	$-0.478718\pi$
$859$	3364.00	0.133618	0.0668092	+	0.997766i	$0.478718\pi$
$860$	0	0
$861$	7938.00	0.314200
$862$	0	0
$863$	13104.0	0.516878	0.258439	−	0.966028i	$-0.416792\pi$
$863$	13104.0	0.516878	0.258439	+	0.966028i	$0.416792\pi$
$864$	0	0
$865$	−16236.0	−0.638197
$866$	0	0
$867$	24249.0	0.949872
$868$	0	0
$869$	−2880.00	−0.112425
$870$	0	0
$871$	−62744.0	−2.44087
$872$	0	0
$873$	9090.00	0.352405
$874$	0	0
$875$	8988.00	0.347257
$876$	0	0
$877$	40858.0	1.57318	0.786589	−	0.617477i	$-0.211845\pi$
$877$	40858.0	1.57318	0.786589	+	0.617477i	$0.211845\pi$
$878$	0	0
$879$	11214.0	0.430306
$880$	0	0
$881$	−37374.0	−1.42924	−0.714621	−	0.699512i	$-0.753401\pi$
$881$	−37374.0	−1.42924	−0.714621	+	0.699512i	$0.753401\pi$
$882$	0	0
$883$	−9788.00	−0.373038	−0.186519	−	0.982451i	$-0.559721\pi$
$883$	−9788.00	−0.373038	−0.186519	+	0.982451i	$0.559721\pi$
$884$	0	0
$885$	7128.00	0.270740
$886$	0	0
$887$	50424.0	1.90876	0.954381	−	0.298591i	$-0.0965166\pi$
$887$	50424.0	1.90876	0.954381	+	0.298591i	$0.0965166\pi$
$888$	0	0
$889$	9296.00	0.350706
$890$	0	0
$891$	−2916.00	−0.109640
$892$	0	0
$893$	−14592.0	−0.546811
$894$	0	0
$895$	28296.0	1.05679
$896$	0	0
$897$	4464.00	0.166163
$898$	0	0
$899$	6048.00	0.224374
$900$	0	0
$901$	45828.0	1.69451
$902$	0	0
$903$	3612.00	0.133112
$904$	0	0
$905$	11460.0	0.420932
$906$	0	0
$907$	12412.0	0.454392	0.227196	−	0.973849i	$-0.427044\pi$
$907$	12412.0	0.454392	0.227196	+	0.973849i	$0.427044\pi$
$908$	0	0
$909$	−54.0000	−0.00197037
$910$	0	0
$911$	6576.00	0.239158	0.119579	−	0.992825i	$-0.461846\pi$
$911$	6576.00	0.239158	0.119579	+	0.992825i	$0.461846\pi$
$912$	0	0
$913$	−3888.00	−0.140935
$914$	0	0
$915$	4572.00	0.165187
$916$	0	0
$917$	8484.00	0.305525
$918$	0	0
$919$	8264.00	0.296631	0.148316	−	0.988940i	$-0.452615\pi$
$919$	8264.00	0.296631	0.148316	+	0.988940i	$0.452615\pi$
$920$	0	0
$921$	2532.00	0.0905887
$922$	0	0
$923$	−52080.0	−1.85724
$924$	0	0
$925$	−15842.0	−0.563115
$926$	0	0
$927$	−4248.00	−0.150510
$928$	0	0
$929$	39426.0	1.39238	0.696192	−	0.717855i	$-0.254876\pi$
$929$	39426.0	1.39238	0.696192	+	0.717855i	$0.254876\pi$
$930$	0	0
$931$	3724.00	0.131095
$932$	0	0
$933$	18936.0	0.664455
$934$	0	0
$935$	24624.0	0.861274
$936$	0	0
$937$	−4678.00	−0.163099	−0.0815494	−	0.996669i	$-0.525987\pi$
$937$	−4678.00	−0.163099	−0.0815494	+	0.996669i	$0.525987\pi$
$938$	0	0
$939$	24846.0	0.863492
$940$	0	0
$941$	17346.0	0.600918	0.300459	−	0.953795i	$-0.402860\pi$
$941$	17346.0	0.600918	0.300459	+	0.953795i	$0.402860\pi$
$942$	0	0
$943$	−9072.00	−0.313282
$944$	0	0
$945$	−1134.00	−0.0390360
$946$	0	0
$947$	−19452.0	−0.667482	−0.333741	−	0.942665i	$-0.608311\pi$
$947$	−19452.0	−0.667482	−0.333741	+	0.942665i	$0.608311\pi$
$948$	0	0
$949$	−55180.0	−1.88748
$950$	0	0
$951$	−27954.0	−0.953176
$952$	0	0
$953$	4458.00	0.151531	0.0757654	−	0.997126i	$-0.475860\pi$
$953$	4458.00	0.151531	0.0757654	+	0.997126i	$0.475860\pi$
$954$	0	0
$955$	−24480.0	−0.829481
$956$	0	0
$957$	5832.00	0.196992
$958$	0	0
$959$	−8778.00	−0.295575
$960$	0	0
$961$	−17247.0	−0.578933
$962$	0	0
$963$	8748.00	0.292731
$964$	0	0
$965$	16116.0	0.537609
$966$	0	0
$967$	52520.0	1.74657	0.873283	−	0.487213i	$-0.161987\pi$
$967$	52520.0	1.74657	0.873283	+	0.487213i	$0.161987\pi$
$968$	0	0
$969$	25992.0	0.861696
$970$	0	0
$971$	10404.0	0.343852	0.171926	−	0.985110i	$-0.445001\pi$
$971$	10404.0	0.343852	0.171926	+	0.985110i	$0.445001\pi$
$972$	0	0
$973$	2380.00	0.0784165
$974$	0	0
$975$	16554.0	0.543746
$976$	0	0
$977$	−7566.00	−0.247756	−0.123878	−	0.992297i	$-0.539533\pi$
$977$	−7566.00	−0.247756	−0.123878	+	0.992297i	$0.539533\pi$
$978$	0	0
$979$	58968.0	1.92505
$980$	0	0
$981$	16074.0	0.523143
$982$	0	0
$983$	44376.0	1.43985	0.719926	−	0.694051i	$-0.244176\pi$
$983$	44376.0	1.43985	0.719926	+	0.694051i	$0.244176\pi$
$984$	0	0
$985$	3060.00	0.0989845
$986$	0	0
$987$	−4032.00	−0.130030
$988$	0	0
$989$	−4128.00	−0.132723
$990$	0	0
$991$	−27328.0	−0.875986	−0.437993	−	0.898978i	$-0.644311\pi$
$991$	−27328.0	−0.875986	−0.437993	+	0.898978i	$0.644311\pi$
$992$	0	0
$993$	−4956.00	−0.158383
$994$	0	0
$995$	−8112.00	−0.258460
$996$	0	0
$997$	−2774.00	−0.0881178	−0.0440589	−	0.999029i	$-0.514029\pi$
$997$	−2774.00	−0.0881178	−0.0440589	+	0.999029i	$0.514029\pi$
$998$	0	0
$999$	4806.00	0.152207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.q.1.1		1
4.3	odd	2		1344.4.a.d.1.1		1
8.3	odd	2		336.4.a.k.1.1		1
8.5	even	2		84.4.a.a.1.1	✓	1
24.5	odd	2		252.4.a.b.1.1		1
24.11	even	2		1008.4.a.h.1.1		1
40.13	odd	4		2100.4.k.j.1849.1		2
40.29	even	2		2100.4.a.l.1.1		1
40.37	odd	4		2100.4.k.j.1849.2		2
56.5	odd	6		588.4.i.c.361.1		2
56.13	odd	2		588.4.a.d.1.1		1
56.27	even	2		2352.4.a.d.1.1		1
56.37	even	6		588.4.i.f.361.1		2
56.45	odd	6		588.4.i.c.373.1		2
56.53	even	6		588.4.i.f.373.1		2
168.5	even	6		1764.4.k.f.361.1		2
168.53	odd	6		1764.4.k.l.1549.1		2
168.101	even	6		1764.4.k.f.1549.1		2
168.125	even	2		1764.4.a.j.1.1		1
168.149	odd	6		1764.4.k.l.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.a.a.1.1	✓	1	8.5	even	2
252.4.a.b.1.1		1	24.5	odd	2
336.4.a.k.1.1		1	8.3	odd	2
588.4.a.d.1.1		1	56.13	odd	2
588.4.i.c.361.1		2	56.5	odd	6
588.4.i.c.373.1		2	56.45	odd	6
588.4.i.f.361.1		2	56.37	even	6
588.4.i.f.373.1		2	56.53	even	6
1008.4.a.h.1.1		1	24.11	even	2
1344.4.a.d.1.1		1	4.3	odd	2
1344.4.a.q.1.1		1	1.1	even	1	trivial
1764.4.a.j.1.1		1	168.125	even	2
1764.4.k.f.361.1		2	168.5	even	6
1764.4.k.f.1549.1		2	168.101	even	6
1764.4.k.l.361.1		2	168.149	odd	6
1764.4.k.l.1549.1		2	168.53	odd	6
2100.4.a.l.1.1		1	40.29	even	2
2100.4.k.j.1849.1		2	40.13	odd	4
2100.4.k.j.1849.2		2	40.37	odd	4
2352.4.a.d.1.1		1	56.27	even	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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