LMFDB - Embedded newform 624.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,4,Mod(1,624)] 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("624.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,4,0,-4] 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:	$36.8171918436$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +4.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} -2.00000 q^{11} -13.0000 q^{13} -12.0000 q^{15} -6.00000 q^{17} +36.0000 q^{19} +12.0000 q^{21} +20.0000 q^{23} -109.000 q^{25} -27.0000 q^{27} -14.0000 q^{29} +152.000 q^{31} +6.00000 q^{33} -16.0000 q^{35} -258.000 q^{37} +39.0000 q^{39} +84.0000 q^{41} +188.000 q^{43} +36.0000 q^{45} -254.000 q^{47} -327.000 q^{49} +18.0000 q^{51} +366.000 q^{53} -8.00000 q^{55} -108.000 q^{57} -550.000 q^{59} -14.0000 q^{61} -36.0000 q^{63} -52.0000 q^{65} -448.000 q^{67} -60.0000 q^{69} -926.000 q^{71} +254.000 q^{73} +327.000 q^{75} +8.00000 q^{77} -1328.00 q^{79} +81.0000 q^{81} -186.000 q^{83} -24.0000 q^{85} +42.0000 q^{87} -336.000 q^{89} +52.0000 q^{91} -456.000 q^{93} +144.000 q^{95} +614.000 q^{97} -18.0000 q^{99} +O(q^{100})$

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
$5$	4.00000	0.357771	0.178885	−	0.983870i	$-0.442751\pi$
$5$	4.00000	0.357771	0.178885	+	0.983870i	$0.442751\pi$
$6$	0	0
$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$7$	−4.00000	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.0548202	−0.0274101	−	0.999624i	$-0.508726\pi$
$11$	−2.00000	−0.0548202	−0.0274101	+	0.999624i	$0.508726\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−12.0000	−0.206559
$16$	0	0
$17$	−6.00000	−0.0856008	−0.0428004	−	0.999084i	$-0.513628\pi$
$17$	−6.00000	−0.0856008	−0.0428004	+	0.999084i	$0.513628\pi$
$18$	0	0
$19$	36.0000	0.434682	0.217341	−	0.976096i	$-0.430262\pi$
$19$	36.0000	0.434682	0.217341	+	0.976096i	$0.430262\pi$
$20$	0	0
$21$	12.0000	0.124696
$22$	0	0
$23$	20.0000	0.181317	0.0906584	−	0.995882i	$-0.471103\pi$
$23$	20.0000	0.181317	0.0906584	+	0.995882i	$0.471103\pi$
$24$	0	0
$25$	−109.000	−0.872000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−14.0000	−0.0896460	−0.0448230	−	0.998995i	$-0.514272\pi$
$29$	−14.0000	−0.0896460	−0.0448230	+	0.998995i	$0.514272\pi$
$30$	0	0
$31$	152.000	0.880645	0.440323	−	0.897840i	$-0.354864\pi$
$31$	152.000	0.880645	0.440323	+	0.897840i	$0.354864\pi$
$32$	0	0
$33$	6.00000	0.0316505
$34$	0	0
$35$	−16.0000	−0.0772712
$36$	0	0
$37$	−258.000	−1.14635	−0.573175	−	0.819433i	$-0.694288\pi$
$37$	−258.000	−1.14635	−0.573175	+	0.819433i	$0.694288\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	84.0000	0.319966	0.159983	−	0.987120i	$-0.448856\pi$
$41$	84.0000	0.319966	0.159983	+	0.987120i	$0.448856\pi$
$42$	0	0
$43$	188.000	0.666738	0.333369	−	0.942796i	$-0.391815\pi$
$43$	188.000	0.666738	0.333369	+	0.942796i	$0.391815\pi$
$44$	0	0
$45$	36.0000	0.119257
$46$	0	0
$47$	−254.000	−0.788292	−0.394146	−	0.919048i	$-0.628960\pi$
$47$	−254.000	−0.788292	−0.394146	+	0.919048i	$0.628960\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	18.0000	0.0494217
$52$	0	0
$53$	366.000	0.948565	0.474283	−	0.880373i	$-0.342707\pi$
$53$	366.000	0.948565	0.474283	+	0.880373i	$0.342707\pi$
$54$	0	0
$55$	−8.00000	−0.0196131
$56$	0	0
$57$	−108.000	−0.250964
$58$	0	0
$59$	−550.000	−1.21363	−0.606813	−	0.794845i	$-0.707552\pi$
$59$	−550.000	−1.21363	−0.606813	+	0.794845i	$0.707552\pi$
$60$	0	0
$61$	−14.0000	−0.0293855	−0.0146928	−	0.999892i	$-0.504677\pi$
$61$	−14.0000	−0.0293855	−0.0146928	+	0.999892i	$0.504677\pi$
$62$	0	0
$63$	−36.0000	−0.0719932
$64$	0	0
$65$	−52.0000	−0.0992278
$66$	0	0
$67$	−448.000	−0.816894	−0.408447	−	0.912782i	$-0.633930\pi$
$67$	−448.000	−0.816894	−0.408447	+	0.912782i	$0.633930\pi$
$68$	0	0
$69$	−60.0000	−0.104683
$70$	0	0
$71$	−926.000	−1.54783	−0.773915	−	0.633289i	$-0.781704\pi$
$71$	−926.000	−1.54783	−0.773915	+	0.633289i	$0.781704\pi$
$72$	0	0
$73$	254.000	0.407239	0.203620	−	0.979050i	$-0.434729\pi$
$73$	254.000	0.407239	0.203620	+	0.979050i	$0.434729\pi$
$74$	0	0
$75$	327.000	0.503449
$76$	0	0
$77$	8.00000	0.0118401
$78$	0	0
$79$	−1328.00	−1.89129	−0.945644	−	0.325205i	$-0.894567\pi$
$79$	−1328.00	−1.89129	−0.945644	+	0.325205i	$0.894567\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−186.000	−0.245978	−0.122989	−	0.992408i	$-0.539248\pi$
$83$	−186.000	−0.245978	−0.122989	+	0.992408i	$0.539248\pi$
$84$	0	0
$85$	−24.0000	−0.0306255
$86$	0	0
$87$	42.0000	0.0517572
$88$	0	0
$89$	−336.000	−0.400179	−0.200089	−	0.979778i	$-0.564123\pi$
$89$	−336.000	−0.400179	−0.200089	+	0.979778i	$0.564123\pi$
$90$	0	0
$91$	52.0000	0.0599020
$92$	0	0
$93$	−456.000	−0.508441
$94$	0	0
$95$	144.000	0.155517
$96$	0	0
$97$	614.000	0.642704	0.321352	−	0.946960i	$-0.395863\pi$
$97$	614.000	0.642704	0.321352	+	0.946960i	$0.395863\pi$
$98$	0	0
$99$	−18.0000	−0.0182734
$100$	0	0
$101$	−1606.00	−1.58221	−0.791104	−	0.611682i	$-0.790493\pi$
$101$	−1606.00	−1.58221	−0.791104	+	0.611682i	$0.790493\pi$
$102$	0	0
$103$	−208.000	−0.198979	−0.0994896	−	0.995039i	$-0.531721\pi$
$103$	−208.000	−0.198979	−0.0994896	+	0.995039i	$0.531721\pi$
$104$	0	0
$105$	48.0000	0.0446126
$106$	0	0
$107$	248.000	0.224066	0.112033	−	0.993704i	$-0.464264\pi$
$107$	248.000	0.224066	0.112033	+	0.993704i	$0.464264\pi$
$108$	0	0
$109$	−542.000	−0.476277	−0.238138	−	0.971231i	$-0.576537\pi$
$109$	−542.000	−0.476277	−0.238138	+	0.971231i	$0.576537\pi$
$110$	0	0
$111$	774.000	0.661845
$112$	0	0
$113$	−2042.00	−1.69996	−0.849979	−	0.526817i	$-0.823385\pi$
$113$	−2042.00	−1.69996	−0.849979	+	0.526817i	$0.823385\pi$
$114$	0	0
$115$	80.0000	0.0648699
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	24.0000	0.0184880
$120$	0	0
$121$	−1327.00	−0.996995
$122$	0	0
$123$	−252.000	−0.184732
$124$	0	0
$125$	−936.000	−0.669747
$126$	0	0
$127$	488.000	0.340968	0.170484	−	0.985360i	$-0.445467\pi$
$127$	488.000	0.340968	0.170484	+	0.985360i	$0.445467\pi$
$128$	0	0
$129$	−564.000	−0.384941
$130$	0	0
$131$	−1744.00	−1.16316	−0.581580	−	0.813489i	$-0.697565\pi$
$131$	−1744.00	−1.16316	−0.581580	+	0.813489i	$0.697565\pi$
$132$	0	0
$133$	−144.000	−0.0938826
$134$	0	0
$135$	−108.000	−0.0688530
$136$	0	0
$137$	−828.000	−0.516356	−0.258178	−	0.966097i	$-0.583122\pi$
$137$	−828.000	−0.516356	−0.258178	+	0.966097i	$0.583122\pi$
$138$	0	0
$139$	404.000	0.246524	0.123262	−	0.992374i	$-0.460664\pi$
$139$	404.000	0.246524	0.123262	+	0.992374i	$0.460664\pi$
$140$	0	0
$141$	762.000	0.455120
$142$	0	0
$143$	26.0000	0.0152044
$144$	0	0
$145$	−56.0000	−0.0320727
$146$	0	0
$147$	981.000	0.550418
$148$	0	0
$149$	2928.00	1.60987	0.804937	−	0.593361i	$-0.202199\pi$
$149$	2928.00	1.60987	0.804937	+	0.593361i	$0.202199\pi$
$150$	0	0
$151$	−1944.00	−1.04769	−0.523843	−	0.851815i	$-0.675502\pi$
$151$	−1944.00	−1.04769	−0.523843	+	0.851815i	$0.675502\pi$
$152$	0	0
$153$	−54.0000	−0.0285336
$154$	0	0
$155$	608.000	0.315069
$156$	0	0
$157$	3590.00	1.82492	0.912462	−	0.409161i	$-0.134178\pi$
$157$	3590.00	1.82492	0.912462	+	0.409161i	$0.134178\pi$
$158$	0	0
$159$	−1098.00	−0.547654
$160$	0	0
$161$	−80.0000	−0.0391608
$162$	0	0
$163$	2284.00	1.09753	0.548763	−	0.835978i	$-0.315099\pi$
$163$	2284.00	1.09753	0.548763	+	0.835978i	$0.315099\pi$
$164$	0	0
$165$	24.0000	0.0113236
$166$	0	0
$167$	−3174.00	−1.47073	−0.735364	−	0.677673i	$-0.762989\pi$
$167$	−3174.00	−1.47073	−0.735364	+	0.677673i	$0.762989\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	324.000	0.144894
$172$	0	0
$173$	−1358.00	−0.596802	−0.298401	−	0.954441i	$-0.596453\pi$
$173$	−1358.00	−0.596802	−0.298401	+	0.954441i	$0.596453\pi$
$174$	0	0
$175$	436.000	0.188334
$176$	0	0
$177$	1650.00	0.700687
$178$	0	0
$179$	−708.000	−0.295634	−0.147817	−	0.989015i	$-0.547225\pi$
$179$	−708.000	−0.295634	−0.147817	+	0.989015i	$0.547225\pi$
$180$	0	0
$181$	−546.000	−0.224220	−0.112110	−	0.993696i	$-0.535761\pi$
$181$	−546.000	−0.224220	−0.112110	+	0.993696i	$0.535761\pi$
$182$	0	0
$183$	42.0000	0.0169657
$184$	0	0
$185$	−1032.00	−0.410131
$186$	0	0
$187$	12.0000	0.00469266
$188$	0	0
$189$	108.000	0.0415653
$190$	0	0
$191$	3472.00	1.31531	0.657657	−	0.753317i	$-0.271547\pi$
$191$	3472.00	1.31531	0.657657	+	0.753317i	$0.271547\pi$
$192$	0	0
$193$	−310.000	−0.115618	−0.0578090	−	0.998328i	$-0.518411\pi$
$193$	−310.000	−0.115618	−0.0578090	+	0.998328i	$0.518411\pi$
$194$	0	0
$195$	156.000	0.0572892
$196$	0	0
$197$	1020.00	0.368893	0.184447	−	0.982843i	$-0.440951\pi$
$197$	1020.00	0.368893	0.184447	+	0.982843i	$0.440951\pi$
$198$	0	0
$199$	3256.00	1.15986	0.579929	−	0.814667i	$-0.303080\pi$
$199$	3256.00	1.15986	0.579929	+	0.814667i	$0.303080\pi$
$200$	0	0
$201$	1344.00	0.471634
$202$	0	0
$203$	56.0000	0.0193617
$204$	0	0
$205$	336.000	0.114474
$206$	0	0
$207$	180.000	0.0604390
$208$	0	0
$209$	−72.0000	−0.0238294
$210$	0	0
$211$	4564.00	1.48909	0.744547	−	0.667570i	$-0.232666\pi$
$211$	4564.00	1.48909	0.744547	+	0.667570i	$0.232666\pi$
$212$	0	0
$213$	2778.00	0.893640
$214$	0	0
$215$	752.000	0.238539
$216$	0	0
$217$	−608.000	−0.190202
$218$	0	0
$219$	−762.000	−0.235120
$220$	0	0
$221$	78.0000	0.0237414
$222$	0	0
$223$	72.0000	0.0216210	0.0108105	−	0.999942i	$-0.496559\pi$
$223$	72.0000	0.0216210	0.0108105	+	0.999942i	$0.496559\pi$
$224$	0	0
$225$	−981.000	−0.290667
$226$	0	0
$227$	−2694.00	−0.787696	−0.393848	−	0.919176i	$-0.628856\pi$
$227$	−2694.00	−0.787696	−0.393848	+	0.919176i	$0.628856\pi$
$228$	0	0
$229$	5922.00	1.70889	0.854447	−	0.519538i	$-0.173896\pi$
$229$	5922.00	1.70889	0.854447	+	0.519538i	$0.173896\pi$
$230$	0	0
$231$	−24.0000	−0.00683586
$232$	0	0
$233$	−5122.00	−1.44014	−0.720072	−	0.693900i	$-0.755891\pi$
$233$	−5122.00	−1.44014	−0.720072	+	0.693900i	$0.755891\pi$
$234$	0	0
$235$	−1016.00	−0.282028
$236$	0	0
$237$	3984.00	1.09194
$238$	0	0
$239$	−5022.00	−1.35919	−0.679595	−	0.733588i	$-0.737844\pi$
$239$	−5022.00	−1.35919	−0.679595	+	0.733588i	$0.737844\pi$
$240$	0	0
$241$	−1218.00	−0.325553	−0.162777	−	0.986663i	$-0.552045\pi$
$241$	−1218.00	−0.325553	−0.162777	+	0.986663i	$0.552045\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−1308.00	−0.341082
$246$	0	0
$247$	−468.000	−0.120559
$248$	0	0
$249$	558.000	0.142015
$250$	0	0
$251$	2112.00	0.531109	0.265554	−	0.964096i	$-0.414445\pi$
$251$	2112.00	0.531109	0.265554	+	0.964096i	$0.414445\pi$
$252$	0	0
$253$	−40.0000	−0.00993984
$254$	0	0
$255$	72.0000	0.0176816
$256$	0	0
$257$	2814.00	0.683006	0.341503	−	0.939881i	$-0.389064\pi$
$257$	2814.00	0.683006	0.341503	+	0.939881i	$0.389064\pi$
$258$	0	0
$259$	1032.00	0.247588
$260$	0	0
$261$	−126.000	−0.0298820
$262$	0	0
$263$	4044.00	0.948151	0.474076	−	0.880484i	$-0.342782\pi$
$263$	4044.00	0.948151	0.474076	+	0.880484i	$0.342782\pi$
$264$	0	0
$265$	1464.00	0.339369
$266$	0	0
$267$	1008.00	0.231043
$268$	0	0
$269$	−1470.00	−0.333188	−0.166594	−	0.986026i	$-0.553277\pi$
$269$	−1470.00	−0.333188	−0.166594	+	0.986026i	$0.553277\pi$
$270$	0	0
$271$	1844.00	0.413340	0.206670	−	0.978411i	$-0.433737\pi$
$271$	1844.00	0.413340	0.206670	+	0.978411i	$0.433737\pi$
$272$	0	0
$273$	−156.000	−0.0345844
$274$	0	0
$275$	218.000	0.0478033
$276$	0	0
$277$	5766.00	1.25071	0.625353	−	0.780342i	$-0.284955\pi$
$277$	5766.00	1.25071	0.625353	+	0.780342i	$0.284955\pi$
$278$	0	0
$279$	1368.00	0.293548
$280$	0	0
$281$	−7468.00	−1.58542	−0.792711	−	0.609598i	$-0.791331\pi$
$281$	−7468.00	−1.58542	−0.792711	+	0.609598i	$0.791331\pi$
$282$	0	0
$283$	−1228.00	−0.257940	−0.128970	−	0.991648i	$-0.541167\pi$
$283$	−1228.00	−0.257940	−0.128970	+	0.991648i	$0.541167\pi$
$284$	0	0
$285$	−432.000	−0.0897876
$286$	0	0
$287$	−336.000	−0.0691061
$288$	0	0
$289$	−4877.00	−0.992673
$290$	0	0
$291$	−1842.00	−0.371065
$292$	0	0
$293$	6608.00	1.31755	0.658777	−	0.752338i	$-0.271074\pi$
$293$	6608.00	1.31755	0.658777	+	0.752338i	$0.271074\pi$
$294$	0	0
$295$	−2200.00	−0.434200
$296$	0	0
$297$	54.0000	0.0105502
$298$	0	0
$299$	−260.000	−0.0502883
$300$	0	0
$301$	−752.000	−0.144002
$302$	0	0
$303$	4818.00	0.913488
$304$	0	0
$305$	−56.0000	−0.0105133
$306$	0	0
$307$	−7664.00	−1.42478	−0.712390	−	0.701784i	$-0.752387\pi$
$307$	−7664.00	−1.42478	−0.712390	+	0.701784i	$0.752387\pi$
$308$	0	0
$309$	624.000	0.114881
$310$	0	0
$311$	2340.00	0.426653	0.213327	−	0.976981i	$-0.431570\pi$
$311$	2340.00	0.426653	0.213327	+	0.976981i	$0.431570\pi$
$312$	0	0
$313$	6710.00	1.21173	0.605865	−	0.795567i	$-0.292827\pi$
$313$	6710.00	1.21173	0.605865	+	0.795567i	$0.292827\pi$
$314$	0	0
$315$	−144.000	−0.0257571
$316$	0	0
$317$	4164.00	0.737771	0.368886	−	0.929475i	$-0.379739\pi$
$317$	4164.00	0.737771	0.368886	+	0.929475i	$0.379739\pi$
$318$	0	0
$319$	28.0000	0.00491442
$320$	0	0
$321$	−744.000	−0.129365
$322$	0	0
$323$	−216.000	−0.0372092
$324$	0	0
$325$	1417.00	0.241849
$326$	0	0
$327$	1626.00	0.274979
$328$	0	0
$329$	1016.00	0.170255
$330$	0	0
$331$	10072.0	1.67253	0.836265	−	0.548326i	$-0.184735\pi$
$331$	10072.0	1.67253	0.836265	+	0.548326i	$0.184735\pi$
$332$	0	0
$333$	−2322.00	−0.382117
$334$	0	0
$335$	−1792.00	−0.292261
$336$	0	0
$337$	2990.00	0.483311	0.241655	−	0.970362i	$-0.422310\pi$
$337$	2990.00	0.483311	0.241655	+	0.970362i	$0.422310\pi$
$338$	0	0
$339$	6126.00	0.981471
$340$	0	0
$341$	−304.000	−0.0482772
$342$	0	0
$343$	2680.00	0.421885
$344$	0	0
$345$	−240.000	−0.0374527
$346$	0	0
$347$	−6564.00	−1.01549	−0.507743	−	0.861508i	$-0.669520\pi$
$347$	−6564.00	−1.01549	−0.507743	+	0.861508i	$0.669520\pi$
$348$	0	0
$349$	−674.000	−0.103376	−0.0516882	−	0.998663i	$-0.516460\pi$
$349$	−674.000	−0.103376	−0.0516882	+	0.998663i	$0.516460\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	−10732.0	−1.61815	−0.809075	−	0.587706i	$-0.800031\pi$
$353$	−10732.0	−1.61815	−0.809075	+	0.587706i	$0.800031\pi$
$354$	0	0
$355$	−3704.00	−0.553769
$356$	0	0
$357$	−72.0000	−0.0106741
$358$	0	0
$359$	4842.00	0.711841	0.355921	−	0.934516i	$-0.384167\pi$
$359$	4842.00	0.711841	0.355921	+	0.934516i	$0.384167\pi$
$360$	0	0
$361$	−5563.00	−0.811051
$362$	0	0
$363$	3981.00	0.575615
$364$	0	0
$365$	1016.00	0.145698
$366$	0	0
$367$	6280.00	0.893224	0.446612	−	0.894728i	$-0.352630\pi$
$367$	6280.00	0.893224	0.446612	+	0.894728i	$0.352630\pi$
$368$	0	0
$369$	756.000	0.106655
$370$	0	0
$371$	−1464.00	−0.204871
$372$	0	0
$373$	6434.00	0.893136	0.446568	−	0.894750i	$-0.352646\pi$
$373$	6434.00	0.893136	0.446568	+	0.894750i	$0.352646\pi$
$374$	0	0
$375$	2808.00	0.386679
$376$	0	0
$377$	182.000	0.0248633
$378$	0	0
$379$	9068.00	1.22900	0.614501	−	0.788916i	$-0.289357\pi$
$379$	9068.00	1.22900	0.614501	+	0.788916i	$0.289357\pi$
$380$	0	0
$381$	−1464.00	−0.196858
$382$	0	0
$383$	−3162.00	−0.421855	−0.210928	−	0.977502i	$-0.567648\pi$
$383$	−3162.00	−0.421855	−0.210928	+	0.977502i	$0.567648\pi$
$384$	0	0
$385$	32.0000	0.00423603
$386$	0	0
$387$	1692.00	0.222246
$388$	0	0
$389$	−3666.00	−0.477824	−0.238912	−	0.971041i	$-0.576791\pi$
$389$	−3666.00	−0.477824	−0.238912	+	0.971041i	$0.576791\pi$
$390$	0	0
$391$	−120.000	−0.0155209
$392$	0	0
$393$	5232.00	0.671551
$394$	0	0
$395$	−5312.00	−0.676647
$396$	0	0
$397$	11054.0	1.39744	0.698721	−	0.715394i	$-0.253753\pi$
$397$	11054.0	1.39744	0.698721	+	0.715394i	$0.253753\pi$
$398$	0	0
$399$	432.000	0.0542031
$400$	0	0
$401$	−5328.00	−0.663510	−0.331755	−	0.943366i	$-0.607641\pi$
$401$	−5328.00	−0.663510	−0.331755	+	0.943366i	$0.607641\pi$
$402$	0	0
$403$	−1976.00	−0.244247
$404$	0	0
$405$	324.000	0.0397523
$406$	0	0
$407$	516.000	0.0628432
$408$	0	0
$409$	−12074.0	−1.45971	−0.729854	−	0.683603i	$-0.760412\pi$
$409$	−12074.0	−1.45971	−0.729854	+	0.683603i	$0.760412\pi$
$410$	0	0
$411$	2484.00	0.298118
$412$	0	0
$413$	2200.00	0.262118
$414$	0	0
$415$	−744.000	−0.0880037
$416$	0	0
$417$	−1212.00	−0.142331
$418$	0	0
$419$	−13584.0	−1.58382	−0.791911	−	0.610636i	$-0.790914\pi$
$419$	−13584.0	−1.58382	−0.791911	+	0.610636i	$0.790914\pi$
$420$	0	0
$421$	−7406.00	−0.857355	−0.428677	−	0.903458i	$-0.641020\pi$
$421$	−7406.00	−0.857355	−0.428677	+	0.903458i	$0.641020\pi$
$422$	0	0
$423$	−2286.00	−0.262764
$424$	0	0
$425$	654.000	0.0746439
$426$	0	0
$427$	56.0000	0.00634667
$428$	0	0
$429$	−78.0000	−0.00877826
$430$	0	0
$431$	10134.0	1.13257	0.566285	−	0.824210i	$-0.308380\pi$
$431$	10134.0	1.13257	0.566285	+	0.824210i	$0.308380\pi$
$432$	0	0
$433$	9406.00	1.04393	0.521967	−	0.852966i	$-0.325198\pi$
$433$	9406.00	1.04393	0.521967	+	0.852966i	$0.325198\pi$
$434$	0	0
$435$	168.000	0.0185172
$436$	0	0
$437$	720.000	0.0788153
$438$	0	0
$439$	−4088.00	−0.444441	−0.222220	−	0.974996i	$-0.571330\pi$
$439$	−4088.00	−0.444441	−0.222220	+	0.974996i	$0.571330\pi$
$440$	0	0
$441$	−2943.00	−0.317784
$442$	0	0
$443$	5328.00	0.571424	0.285712	−	0.958315i	$-0.407770\pi$
$443$	5328.00	0.571424	0.285712	+	0.958315i	$0.407770\pi$
$444$	0	0
$445$	−1344.00	−0.143172
$446$	0	0
$447$	−8784.00	−0.929461
$448$	0	0
$449$	13160.0	1.38320	0.691602	−	0.722279i	$-0.256905\pi$
$449$	13160.0	1.38320	0.691602	+	0.722279i	$0.256905\pi$
$450$	0	0
$451$	−168.000	−0.0175406
$452$	0	0
$453$	5832.00	0.604881
$454$	0	0
$455$	208.000	0.0214312
$456$	0	0
$457$	−9146.00	−0.936175	−0.468087	−	0.883682i	$-0.655057\pi$
$457$	−9146.00	−0.936175	−0.468087	+	0.883682i	$0.655057\pi$
$458$	0	0
$459$	162.000	0.0164739
$460$	0	0
$461$	5580.00	0.563745	0.281873	−	0.959452i	$-0.409044\pi$
$461$	5580.00	0.563745	0.281873	+	0.959452i	$0.409044\pi$
$462$	0	0
$463$	−14788.0	−1.48436	−0.742178	−	0.670203i	$-0.766207\pi$
$463$	−14788.0	−1.48436	−0.742178	+	0.670203i	$0.766207\pi$
$464$	0	0
$465$	−1824.00	−0.181905
$466$	0	0
$467$	−12376.0	−1.22632	−0.613162	−	0.789957i	$-0.710103\pi$
$467$	−12376.0	−1.22632	−0.613162	+	0.789957i	$0.710103\pi$
$468$	0	0
$469$	1792.00	0.176433
$470$	0	0
$471$	−10770.0	−1.05362
$472$	0	0
$473$	−376.000	−0.0365507
$474$	0	0
$475$	−3924.00	−0.379043
$476$	0	0
$477$	3294.00	0.316188
$478$	0	0
$479$	−834.000	−0.0795541	−0.0397771	−	0.999209i	$-0.512665\pi$
$479$	−834.000	−0.0795541	−0.0397771	+	0.999209i	$0.512665\pi$
$480$	0	0
$481$	3354.00	0.317940
$482$	0	0
$483$	240.000	0.0226095
$484$	0	0
$485$	2456.00	0.229941
$486$	0	0
$487$	13192.0	1.22749	0.613744	−	0.789505i	$-0.289663\pi$
$487$	13192.0	1.22749	0.613744	+	0.789505i	$0.289663\pi$
$488$	0	0
$489$	−6852.00	−0.633657
$490$	0	0
$491$	−16568.0	−1.52282	−0.761409	−	0.648272i	$-0.775492\pi$
$491$	−16568.0	−1.52282	−0.761409	+	0.648272i	$0.775492\pi$
$492$	0	0
$493$	84.0000	0.00767377
$494$	0	0
$495$	−72.0000	−0.00653770
$496$	0	0
$497$	3704.00	0.334300
$498$	0	0
$499$	10136.0	0.909318	0.454659	−	0.890666i	$-0.349761\pi$
$499$	10136.0	0.909318	0.454659	+	0.890666i	$0.349761\pi$
$500$	0	0
$501$	9522.00	0.849125
$502$	0	0
$503$	−10412.0	−0.922959	−0.461479	−	0.887151i	$-0.652681\pi$
$503$	−10412.0	−0.922959	−0.461479	+	0.887151i	$0.652681\pi$
$504$	0	0
$505$	−6424.00	−0.566068
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−4180.00	−0.363999	−0.181999	−	0.983299i	$-0.558257\pi$
$509$	−4180.00	−0.363999	−0.181999	+	0.983299i	$0.558257\pi$
$510$	0	0
$511$	−1016.00	−0.0879554
$512$	0	0
$513$	−972.000	−0.0836547
$514$	0	0
$515$	−832.000	−0.0711889
$516$	0	0
$517$	508.000	0.0432143
$518$	0	0
$519$	4074.00	0.344564
$520$	0	0
$521$	−14610.0	−1.22855	−0.614276	−	0.789091i	$-0.710552\pi$
$521$	−14610.0	−1.22855	−0.614276	+	0.789091i	$0.710552\pi$
$522$	0	0
$523$	2172.00	0.181596	0.0907982	−	0.995869i	$-0.471058\pi$
$523$	2172.00	0.181596	0.0907982	+	0.995869i	$0.471058\pi$
$524$	0	0
$525$	−1308.00	−0.108735
$526$	0	0
$527$	−912.000	−0.0753840
$528$	0	0
$529$	−11767.0	−0.967124
$530$	0	0
$531$	−4950.00	−0.404542
$532$	0	0
$533$	−1092.00	−0.0887425
$534$	0	0
$535$	992.000	0.0801643
$536$	0	0
$537$	2124.00	0.170684
$538$	0	0
$539$	654.000	0.0522630
$540$	0	0
$541$	−11758.0	−0.934410	−0.467205	−	0.884149i	$-0.654739\pi$
$541$	−11758.0	−0.934410	−0.467205	+	0.884149i	$0.654739\pi$
$542$	0	0
$543$	1638.00	0.129454
$544$	0	0
$545$	−2168.00	−0.170398
$546$	0	0
$547$	−340.000	−0.0265765	−0.0132883	−	0.999912i	$-0.504230\pi$
$547$	−340.000	−0.0265765	−0.0132883	+	0.999912i	$0.504230\pi$
$548$	0	0
$549$	−126.000	−0.00979517
$550$	0	0
$551$	−504.000	−0.0389676
$552$	0	0
$553$	5312.00	0.408480
$554$	0	0
$555$	3096.00	0.236789
$556$	0	0
$557$	−3768.00	−0.286634	−0.143317	−	0.989677i	$-0.545777\pi$
$557$	−3768.00	−0.286634	−0.143317	+	0.989677i	$0.545777\pi$
$558$	0	0
$559$	−2444.00	−0.184920
$560$	0	0
$561$	−36.0000	−0.00270931
$562$	0	0
$563$	10172.0	0.761454	0.380727	−	0.924687i	$-0.375674\pi$
$563$	10172.0	0.761454	0.380727	+	0.924687i	$0.375674\pi$
$564$	0	0
$565$	−8168.00	−0.608195
$566$	0	0
$567$	−324.000	−0.0239977
$568$	0	0
$569$	−5506.00	−0.405665	−0.202833	−	0.979213i	$-0.565015\pi$
$569$	−5506.00	−0.405665	−0.202833	+	0.979213i	$0.565015\pi$
$570$	0	0
$571$	−2340.00	−0.171499	−0.0857495	−	0.996317i	$-0.527328\pi$
$571$	−2340.00	−0.171499	−0.0857495	+	0.996317i	$0.527328\pi$
$572$	0	0
$573$	−10416.0	−0.759397
$574$	0	0
$575$	−2180.00	−0.158108
$576$	0	0
$577$	−20094.0	−1.44978	−0.724891	−	0.688864i	$-0.758110\pi$
$577$	−20094.0	−1.44978	−0.724891	+	0.688864i	$0.758110\pi$
$578$	0	0
$579$	930.000	0.0667521
$580$	0	0
$581$	744.000	0.0531262
$582$	0	0
$583$	−732.000	−0.0520006
$584$	0	0
$585$	−468.000	−0.0330759
$586$	0	0
$587$	7118.00	0.500496	0.250248	−	0.968182i	$-0.419488\pi$
$587$	7118.00	0.500496	0.250248	+	0.968182i	$0.419488\pi$
$588$	0	0
$589$	5472.00	0.382801
$590$	0	0
$591$	−3060.00	−0.212981
$592$	0	0
$593$	−10328.0	−0.715211	−0.357606	−	0.933873i	$-0.616407\pi$
$593$	−10328.0	−0.715211	−0.357606	+	0.933873i	$0.616407\pi$
$594$	0	0
$595$	96.0000	0.00661448
$596$	0	0
$597$	−9768.00	−0.669644
$598$	0	0
$599$	19732.0	1.34596	0.672978	−	0.739662i	$-0.265015\pi$
$599$	19732.0	1.34596	0.672978	+	0.739662i	$0.265015\pi$
$600$	0	0
$601$	−12026.0	−0.816224	−0.408112	−	0.912932i	$-0.633813\pi$
$601$	−12026.0	−0.816224	−0.408112	+	0.912932i	$0.633813\pi$
$602$	0	0
$603$	−4032.00	−0.272298
$604$	0	0
$605$	−5308.00	−0.356696
$606$	0	0
$607$	−17016.0	−1.13782	−0.568911	−	0.822399i	$-0.692635\pi$
$607$	−17016.0	−1.13782	−0.568911	+	0.822399i	$0.692635\pi$
$608$	0	0
$609$	−168.000	−0.0111785
$610$	0	0
$611$	3302.00	0.218633
$612$	0	0
$613$	11654.0	0.767864	0.383932	−	0.923361i	$-0.374570\pi$
$613$	11654.0	0.767864	0.383932	+	0.923361i	$0.374570\pi$
$614$	0	0
$615$	−1008.00	−0.0660918
$616$	0	0
$617$	11612.0	0.757669	0.378834	−	0.925465i	$-0.376325\pi$
$617$	11612.0	0.757669	0.378834	+	0.925465i	$0.376325\pi$
$618$	0	0
$619$	−4024.00	−0.261290	−0.130645	−	0.991429i	$-0.541705\pi$
$619$	−4024.00	−0.261290	−0.130645	+	0.991429i	$0.541705\pi$
$620$	0	0
$621$	−540.000	−0.0348945
$622$	0	0
$623$	1344.00	0.0864305
$624$	0	0
$625$	9881.00	0.632384
$626$	0	0
$627$	216.000	0.0137579
$628$	0	0
$629$	1548.00	0.0981285
$630$	0	0
$631$	1088.00	0.0686412	0.0343206	−	0.999411i	$-0.489073\pi$
$631$	1088.00	0.0686412	0.0343206	+	0.999411i	$0.489073\pi$
$632$	0	0
$633$	−13692.0	−0.859729
$634$	0	0
$635$	1952.00	0.121989
$636$	0	0
$637$	4251.00	0.264412
$638$	0	0
$639$	−8334.00	−0.515944
$640$	0	0
$641$	−7078.00	−0.436138	−0.218069	−	0.975933i	$-0.569976\pi$
$641$	−7078.00	−0.436138	−0.218069	+	0.975933i	$0.569976\pi$
$642$	0	0
$643$	−8336.00	−0.511259	−0.255630	−	0.966775i	$-0.582283\pi$
$643$	−8336.00	−0.511259	−0.255630	+	0.966775i	$0.582283\pi$
$644$	0	0
$645$	−2256.00	−0.137721
$646$	0	0
$647$	−32.0000	−0.00194444	−0.000972218	−	1.00000i	$-0.500309\pi$
$647$	−32.0000	−0.00194444	−0.000972218		1.00000i	$0.500309\pi$
$648$	0	0
$649$	1100.00	0.0665312
$650$	0	0
$651$	1824.00	0.109813
$652$	0	0
$653$	−15822.0	−0.948182	−0.474091	−	0.880476i	$-0.657223\pi$
$653$	−15822.0	−0.948182	−0.474091	+	0.880476i	$0.657223\pi$
$654$	0	0
$655$	−6976.00	−0.416145
$656$	0	0
$657$	2286.00	0.135746
$658$	0	0
$659$	−21540.0	−1.27326	−0.636631	−	0.771169i	$-0.719672\pi$
$659$	−21540.0	−1.27326	−0.636631	+	0.771169i	$0.719672\pi$
$660$	0	0
$661$	8270.00	0.486635	0.243317	−	0.969947i	$-0.421764\pi$
$661$	8270.00	0.486635	0.243317	+	0.969947i	$0.421764\pi$
$662$	0	0
$663$	−234.000	−0.0137071
$664$	0	0
$665$	−576.000	−0.0335885
$666$	0	0
$667$	−280.000	−0.0162543
$668$	0	0
$669$	−216.000	−0.0124829
$670$	0	0
$671$	28.0000	0.00161092
$672$	0	0
$673$	8482.00	0.485820	0.242910	−	0.970049i	$-0.421898\pi$
$673$	8482.00	0.485820	0.242910	+	0.970049i	$0.421898\pi$
$674$	0	0
$675$	2943.00	0.167816
$676$	0	0
$677$	2550.00	0.144763	0.0723814	−	0.997377i	$-0.476940\pi$
$677$	2550.00	0.144763	0.0723814	+	0.997377i	$0.476940\pi$
$678$	0	0
$679$	−2456.00	−0.138811
$680$	0	0
$681$	8082.00	0.454777
$682$	0	0
$683$	31534.0	1.76664	0.883320	−	0.468771i	$-0.155303\pi$
$683$	31534.0	1.76664	0.883320	+	0.468771i	$0.155303\pi$
$684$	0	0
$685$	−3312.00	−0.184737
$686$	0	0
$687$	−17766.0	−0.986631
$688$	0	0
$689$	−4758.00	−0.263085
$690$	0	0
$691$	−33832.0	−1.86256	−0.931281	−	0.364302i	$-0.881307\pi$
$691$	−33832.0	−1.86256	−0.931281	+	0.364302i	$0.881307\pi$
$692$	0	0
$693$	72.0000	0.00394669
$694$	0	0
$695$	1616.00	0.0881991
$696$	0	0
$697$	−504.000	−0.0273893
$698$	0	0
$699$	15366.0	0.831467
$700$	0	0
$701$	19422.0	1.04645	0.523223	−	0.852196i	$-0.324729\pi$
$701$	19422.0	1.04645	0.523223	+	0.852196i	$0.324729\pi$
$702$	0	0
$703$	−9288.00	−0.498298
$704$	0	0
$705$	3048.00	0.162829
$706$	0	0
$707$	6424.00	0.341725
$708$	0	0
$709$	−1894.00	−0.100325	−0.0501627	−	0.998741i	$-0.515974\pi$
$709$	−1894.00	−0.100325	−0.0501627	+	0.998741i	$0.515974\pi$
$710$	0	0
$711$	−11952.0	−0.630429
$712$	0	0
$713$	3040.00	0.159676
$714$	0	0
$715$	104.000	0.00543969
$716$	0	0
$717$	15066.0	0.784728
$718$	0	0
$719$	20156.0	1.04547	0.522734	−	0.852496i	$-0.324912\pi$
$719$	20156.0	1.04547	0.522734	+	0.852496i	$0.324912\pi$
$720$	0	0
$721$	832.000	0.0429754
$722$	0	0
$723$	3654.00	0.187958
$724$	0	0
$725$	1526.00	0.0781713
$726$	0	0
$727$	−11128.0	−0.567696	−0.283848	−	0.958869i	$-0.591611\pi$
$727$	−11128.0	−0.567696	−0.283848	+	0.958869i	$0.591611\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1128.00	−0.0570733
$732$	0	0
$733$	16202.0	0.816418	0.408209	−	0.912888i	$-0.366153\pi$
$733$	16202.0	0.816418	0.408209	+	0.912888i	$0.366153\pi$
$734$	0	0
$735$	3924.00	0.196924
$736$	0	0
$737$	896.000	0.0447823
$738$	0	0
$739$	5328.00	0.265215	0.132607	−	0.991169i	$-0.457665\pi$
$739$	5328.00	0.265215	0.132607	+	0.991169i	$0.457665\pi$
$740$	0	0
$741$	1404.00	0.0696049
$742$	0	0
$743$	20482.0	1.01132	0.505661	−	0.862732i	$-0.331249\pi$
$743$	20482.0	1.01132	0.505661	+	0.862732i	$0.331249\pi$
$744$	0	0
$745$	11712.0	0.575966
$746$	0	0
$747$	−1674.00	−0.0819926
$748$	0	0
$749$	−992.000	−0.0483937
$750$	0	0
$751$	−8040.00	−0.390657	−0.195329	−	0.980738i	$-0.562577\pi$
$751$	−8040.00	−0.390657	−0.195329	+	0.980738i	$0.562577\pi$
$752$	0	0
$753$	−6336.00	−0.306636
$754$	0	0
$755$	−7776.00	−0.374831
$756$	0	0
$757$	−15822.0	−0.759657	−0.379829	−	0.925057i	$-0.624017\pi$
$757$	−15822.0	−0.759657	−0.379829	+	0.925057i	$0.624017\pi$
$758$	0	0
$759$	120.000	0.00573877
$760$	0	0
$761$	−1452.00	−0.0691655	−0.0345828	−	0.999402i	$-0.511010\pi$
$761$	−1452.00	−0.0691655	−0.0345828	+	0.999402i	$0.511010\pi$
$762$	0	0
$763$	2168.00	0.102866
$764$	0	0
$765$	−216.000	−0.0102085
$766$	0	0
$767$	7150.00	0.336599
$768$	0	0
$769$	32298.0	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	32298.0	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	0	0
$771$	−8442.00	−0.394334
$772$	0	0
$773$	18736.0	0.871781	0.435891	−	0.900000i	$-0.356433\pi$
$773$	18736.0	0.871781	0.435891	+	0.900000i	$0.356433\pi$
$774$	0	0
$775$	−16568.0	−0.767923
$776$	0	0
$777$	−3096.00	−0.142945
$778$	0	0
$779$	3024.00	0.139083
$780$	0	0
$781$	1852.00	0.0848525
$782$	0	0
$783$	378.000	0.0172524
$784$	0	0
$785$	14360.0	0.652905
$786$	0	0
$787$	40816.0	1.84871	0.924354	−	0.381536i	$-0.124605\pi$
$787$	40816.0	1.84871	0.924354	+	0.381536i	$0.124605\pi$
$788$	0	0
$789$	−12132.0	−0.547415
$790$	0	0
$791$	8168.00	0.367156
$792$	0	0
$793$	182.000	0.00815008
$794$	0	0
$795$	−4392.00	−0.195935
$796$	0	0
$797$	4518.00	0.200798	0.100399	−	0.994947i	$-0.467988\pi$
$797$	4518.00	0.200798	0.100399	+	0.994947i	$0.467988\pi$
$798$	0	0
$799$	1524.00	0.0674784
$800$	0	0
$801$	−3024.00	−0.133393
$802$	0	0
$803$	−508.000	−0.0223249
$804$	0	0
$805$	−320.000	−0.0140106
$806$	0	0
$807$	4410.00	0.192366
$808$	0	0
$809$	−5058.00	−0.219814	−0.109907	−	0.993942i	$-0.535055\pi$
$809$	−5058.00	−0.219814	−0.109907	+	0.993942i	$0.535055\pi$
$810$	0	0
$811$	22564.0	0.976978	0.488489	−	0.872570i	$-0.337548\pi$
$811$	22564.0	0.976978	0.488489	+	0.872570i	$0.337548\pi$
$812$	0	0
$813$	−5532.00	−0.238642
$814$	0	0
$815$	9136.00	0.392663
$816$	0	0
$817$	6768.00	0.289819
$818$	0	0
$819$	468.000	0.0199673
$820$	0	0
$821$	32584.0	1.38513	0.692564	−	0.721357i	$-0.256481\pi$
$821$	32584.0	1.38513	0.692564	+	0.721357i	$0.256481\pi$
$822$	0	0
$823$	9288.00	0.393389	0.196695	−	0.980465i	$-0.436979\pi$
$823$	9288.00	0.393389	0.196695	+	0.980465i	$0.436979\pi$
$824$	0	0
$825$	−654.000	−0.0275992
$826$	0	0
$827$	−20586.0	−0.865593	−0.432796	−	0.901492i	$-0.642473\pi$
$827$	−20586.0	−0.865593	−0.432796	+	0.901492i	$0.642473\pi$
$828$	0	0
$829$	−46118.0	−1.93214	−0.966070	−	0.258280i	$-0.916844\pi$
$829$	−46118.0	−1.93214	−0.966070	+	0.258280i	$0.916844\pi$
$830$	0	0
$831$	−17298.0	−0.722095
$832$	0	0
$833$	1962.00	0.0816078
$834$	0	0
$835$	−12696.0	−0.526183
$836$	0	0
$837$	−4104.00	−0.169480
$838$	0	0
$839$	39230.0	1.61427	0.807133	−	0.590369i	$-0.201018\pi$
$839$	39230.0	1.61427	0.807133	+	0.590369i	$0.201018\pi$
$840$	0	0
$841$	−24193.0	−0.991964
$842$	0	0
$843$	22404.0	0.915344
$844$	0	0
$845$	676.000	0.0275208
$846$	0	0
$847$	5308.00	0.215331
$848$	0	0
$849$	3684.00	0.148922
$850$	0	0
$851$	−5160.00	−0.207853
$852$	0	0
$853$	−18674.0	−0.749573	−0.374786	−	0.927111i	$-0.622284\pi$
$853$	−18674.0	−0.749573	−0.374786	+	0.927111i	$0.622284\pi$
$854$	0	0
$855$	1296.00	0.0518389
$856$	0	0
$857$	41678.0	1.66125	0.830626	−	0.556830i	$-0.187983\pi$
$857$	41678.0	1.66125	0.830626	+	0.556830i	$0.187983\pi$
$858$	0	0
$859$	14740.0	0.585474	0.292737	−	0.956193i	$-0.405434\pi$
$859$	14740.0	0.585474	0.292737	+	0.956193i	$0.405434\pi$
$860$	0	0
$861$	1008.00	0.0398984
$862$	0	0
$863$	24982.0	0.985396	0.492698	−	0.870200i	$-0.336011\pi$
$863$	24982.0	0.985396	0.492698	+	0.870200i	$0.336011\pi$
$864$	0	0
$865$	−5432.00	−0.213519
$866$	0	0
$867$	14631.0	0.573120
$868$	0	0
$869$	2656.00	0.103681
$870$	0	0
$871$	5824.00	0.226566
$872$	0	0
$873$	5526.00	0.214235
$874$	0	0
$875$	3744.00	0.144652
$876$	0	0
$877$	1134.00	0.0436630	0.0218315	−	0.999762i	$-0.493050\pi$
$877$	1134.00	0.0436630	0.0218315	+	0.999762i	$0.493050\pi$
$878$	0	0
$879$	−19824.0	−0.760690
$880$	0	0
$881$	34950.0	1.33654	0.668272	−	0.743917i	$-0.267034\pi$
$881$	34950.0	1.33654	0.668272	+	0.743917i	$0.267034\pi$
$882$	0	0
$883$	3068.00	0.116927	0.0584634	−	0.998290i	$-0.481380\pi$
$883$	3068.00	0.116927	0.0584634	+	0.998290i	$0.481380\pi$
$884$	0	0
$885$	6600.00	0.250685
$886$	0	0
$887$	14080.0	0.532988	0.266494	−	0.963837i	$-0.414135\pi$
$887$	14080.0	0.532988	0.266494	+	0.963837i	$0.414135\pi$
$888$	0	0
$889$	−1952.00	−0.0736423
$890$	0	0
$891$	−162.000	−0.00609114
$892$	0	0
$893$	−9144.00	−0.342657
$894$	0	0
$895$	−2832.00	−0.105769
$896$	0	0
$897$	780.000	0.0290339
$898$	0	0
$899$	−2128.00	−0.0789464
$900$	0	0
$901$	−2196.00	−0.0811980
$902$	0	0
$903$	2256.00	0.0831395
$904$	0	0
$905$	−2184.00	−0.0802195
$906$	0	0
$907$	24876.0	0.910688	0.455344	−	0.890316i	$-0.349516\pi$
$907$	24876.0	0.910688	0.455344	+	0.890316i	$0.349516\pi$
$908$	0	0
$909$	−14454.0	−0.527403
$910$	0	0
$911$	−51456.0	−1.87136	−0.935682	−	0.352843i	$-0.885215\pi$
$911$	−51456.0	−1.87136	−0.935682	+	0.352843i	$0.885215\pi$
$912$	0	0
$913$	372.000	0.0134846
$914$	0	0
$915$	168.000	0.00606985
$916$	0	0
$917$	6976.00	0.251219
$918$	0	0
$919$	31032.0	1.11388	0.556938	−	0.830554i	$-0.311976\pi$
$919$	31032.0	1.11388	0.556938	+	0.830554i	$0.311976\pi$
$920$	0	0
$921$	22992.0	0.822597
$922$	0	0
$923$	12038.0	0.429291
$924$	0	0
$925$	28122.0	0.999617
$926$	0	0
$927$	−1872.00	−0.0663264
$928$	0	0
$929$	50820.0	1.79478	0.897390	−	0.441239i	$-0.145461\pi$
$929$	50820.0	1.79478	0.897390	+	0.441239i	$0.145461\pi$
$930$	0	0
$931$	−11772.0	−0.414406
$932$	0	0
$933$	−7020.00	−0.246328
$934$	0	0
$935$	48.0000	0.00167890
$936$	0	0
$937$	5982.00	0.208563	0.104281	−	0.994548i	$-0.466746\pi$
$937$	5982.00	0.208563	0.104281	+	0.994548i	$0.466746\pi$
$938$	0	0
$939$	−20130.0	−0.699593
$940$	0	0
$941$	−20224.0	−0.700620	−0.350310	−	0.936634i	$-0.613924\pi$
$941$	−20224.0	−0.700620	−0.350310	+	0.936634i	$0.613924\pi$
$942$	0	0
$943$	1680.00	0.0580152
$944$	0	0
$945$	432.000	0.0148709
$946$	0	0
$947$	−8478.00	−0.290917	−0.145458	−	0.989364i	$-0.546466\pi$
$947$	−8478.00	−0.290917	−0.145458	+	0.989364i	$0.546466\pi$
$948$	0	0
$949$	−3302.00	−0.112948
$950$	0	0
$951$	−12492.0	−0.425953
$952$	0	0
$953$	40918.0	1.39083	0.695417	−	0.718607i	$-0.255220\pi$
$953$	40918.0	1.39083	0.695417	+	0.718607i	$0.255220\pi$
$954$	0	0
$955$	13888.0	0.470581
$956$	0	0
$957$	−84.0000	−0.00283734
$958$	0	0
$959$	3312.00	0.111522
$960$	0	0
$961$	−6687.00	−0.224464
$962$	0	0
$963$	2232.00	0.0746887
$964$	0	0
$965$	−1240.00	−0.0413648
$966$	0	0
$967$	4624.00	0.153772	0.0768862	−	0.997040i	$-0.475502\pi$
$967$	4624.00	0.153772	0.0768862	+	0.997040i	$0.475502\pi$
$968$	0	0
$969$	648.000	0.0214827
$970$	0	0
$971$	−15300.0	−0.505665	−0.252832	−	0.967510i	$-0.581362\pi$
$971$	−15300.0	−0.505665	−0.252832	+	0.967510i	$0.581362\pi$
$972$	0	0
$973$	−1616.00	−0.0532442
$974$	0	0
$975$	−4251.00	−0.139632
$976$	0	0
$977$	19584.0	0.641298	0.320649	−	0.947198i	$-0.396099\pi$
$977$	19584.0	0.641298	0.320649	+	0.947198i	$0.396099\pi$
$978$	0	0
$979$	672.000	0.0219379
$980$	0	0
$981$	−4878.00	−0.158759
$982$	0	0
$983$	17582.0	0.570477	0.285238	−	0.958457i	$-0.407927\pi$
$983$	17582.0	0.570477	0.285238	+	0.958457i	$0.407927\pi$
$984$	0	0
$985$	4080.00	0.131979
$986$	0	0
$987$	−3048.00	−0.0982968
$988$	0	0
$989$	3760.00	0.120891
$990$	0	0
$991$	−47904.0	−1.53554	−0.767770	−	0.640725i	$-0.778634\pi$
$991$	−47904.0	−1.53554	−0.767770	+	0.640725i	$0.778634\pi$
$992$	0	0
$993$	−30216.0	−0.965635
$994$	0	0
$995$	13024.0	0.414963
$996$	0	0
$997$	−44578.0	−1.41605	−0.708024	−	0.706189i	$-0.750413\pi$
$997$	−44578.0	−1.41605	−0.708024	+	0.706189i	$0.750413\pi$
$998$	0	0
$999$	6966.00	0.220615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.c.1.1		1
3.2	odd	2		1872.4.a.f.1.1		1
4.3	odd	2		78.4.a.f.1.1	✓	1
8.3	odd	2		2496.4.a.c.1.1		1
8.5	even	2		2496.4.a.l.1.1		1
12.11	even	2		234.4.a.c.1.1		1
20.19	odd	2		1950.4.a.a.1.1		1
52.31	even	4		1014.4.b.g.337.1		2
52.47	even	4		1014.4.b.g.337.2		2
52.51	odd	2		1014.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.f.1.1	✓	1	4.3	odd	2
234.4.a.c.1.1		1	12.11	even	2
624.4.a.c.1.1		1	1.1	even	1	trivial
1014.4.a.e.1.1		1	52.51	odd	2
1014.4.b.g.337.1		2	52.31	even	4
1014.4.b.g.337.2		2	52.47	even	4
1872.4.a.f.1.1		1	3.2	odd	2
1950.4.a.a.1.1		1	20.19	odd	2
2496.4.a.c.1.1		1	8.3	odd	2
2496.4.a.l.1.1		1	8.5	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more