LMFDB - Embedded newform 120.4.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [120,4,Mod(1,120)] 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("120.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 120 = 2^{3} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	120.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,-5,0,20] 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:	$7.08022920069$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	120.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -5.00000 q^{5} +20.0000 q^{7} +9.00000 q^{9} +16.0000 q^{11} +58.0000 q^{13} -15.0000 q^{15} +38.0000 q^{17} +4.00000 q^{19} +60.0000 q^{21} -80.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +82.0000 q^{29} -8.00000 q^{31} +48.0000 q^{33} -100.000 q^{35} +426.000 q^{37} +174.000 q^{39} -246.000 q^{41} -524.000 q^{43} -45.0000 q^{45} -464.000 q^{47} +57.0000 q^{49} +114.000 q^{51} -702.000 q^{53} -80.0000 q^{55} +12.0000 q^{57} -592.000 q^{59} +574.000 q^{61} +180.000 q^{63} -290.000 q^{65} -172.000 q^{67} -240.000 q^{69} +768.000 q^{71} -558.000 q^{73} +75.0000 q^{75} +320.000 q^{77} +408.000 q^{79} +81.0000 q^{81} +164.000 q^{83} -190.000 q^{85} +246.000 q^{87} -510.000 q^{89} +1160.00 q^{91} -24.0000 q^{93} -20.0000 q^{95} +514.000 q^{97} +144.000 q^{99} +O(q^{100})$

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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	20.0000	1.07990	0.539949	−	0.841698i	$-0.318443\pi$
$7$	20.0000	1.07990	0.539949	+	0.841698i	$0.318443\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	16.0000	0.438562	0.219281	−	0.975662i	$-0.429629\pi$
$11$	16.0000	0.438562	0.219281	+	0.975662i	$0.429629\pi$
$12$	0	0
$13$	58.0000	1.23741	0.618704	−	0.785624i	$-0.287658\pi$
$13$	58.0000	1.23741	0.618704	+	0.785624i	$0.287658\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	38.0000	0.542138	0.271069	−	0.962560i	$-0.412623\pi$
$17$	38.0000	0.542138	0.271069	+	0.962560i	$0.412623\pi$
$18$	0	0
$19$	4.00000	0.0482980	0.0241490	−	0.999708i	$-0.492312\pi$
$19$	4.00000	0.0482980	0.0241490	+	0.999708i	$0.492312\pi$
$20$	0	0
$21$	60.0000	0.623480
$22$	0	0
$23$	−80.0000	−0.725268	−0.362634	−	0.931932i	$-0.618122\pi$
$23$	−80.0000	−0.725268	−0.362634	+	0.931932i	$0.618122\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	82.0000	0.525070	0.262535	−	0.964923i	$-0.415442\pi$
$29$	82.0000	0.525070	0.262535	+	0.964923i	$0.415442\pi$
$30$	0	0
$31$	−8.00000	−0.0463498	−0.0231749	−	0.999731i	$-0.507377\pi$
$31$	−8.00000	−0.0463498	−0.0231749	+	0.999731i	$0.507377\pi$
$32$	0	0
$33$	48.0000	0.253204
$34$	0	0
$35$	−100.000	−0.482945
$36$	0	0
$37$	426.000	1.89281	0.946405	−	0.322982i	$-0.104685\pi$
$37$	426.000	1.89281	0.946405	+	0.322982i	$0.104685\pi$
$38$	0	0
$39$	174.000	0.714418
$40$	0	0
$41$	−246.000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−246.000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−524.000	−1.85835	−0.929177	−	0.369634i	$-0.879483\pi$
$43$	−524.000	−1.85835	−0.929177	+	0.369634i	$0.879483\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−464.000	−1.44003	−0.720014	−	0.693959i	$-0.755865\pi$
$47$	−464.000	−1.44003	−0.720014	+	0.693959i	$0.755865\pi$
$48$	0	0
$49$	57.0000	0.166181
$50$	0	0
$51$	114.000	0.313004
$52$	0	0
$53$	−702.000	−1.81938	−0.909690	−	0.415288i	$-0.863681\pi$
$53$	−702.000	−1.81938	−0.909690	+	0.415288i	$0.863681\pi$
$54$	0	0
$55$	−80.0000	−0.196131
$56$	0	0
$57$	12.0000	0.0278849
$58$	0	0
$59$	−592.000	−1.30630	−0.653151	−	0.757228i	$-0.726553\pi$
$59$	−592.000	−1.30630	−0.653151	+	0.757228i	$0.726553\pi$
$60$	0	0
$61$	574.000	1.20481	0.602403	−	0.798192i	$-0.294210\pi$
$61$	574.000	1.20481	0.602403	+	0.798192i	$0.294210\pi$
$62$	0	0
$63$	180.000	0.359966
$64$	0	0
$65$	−290.000	−0.553386
$66$	0	0
$67$	−172.000	−0.313629	−0.156815	−	0.987628i	$-0.550122\pi$
$67$	−172.000	−0.313629	−0.156815	+	0.987628i	$0.550122\pi$
$68$	0	0
$69$	−240.000	−0.418733
$70$	0	0
$71$	768.000	1.28373	0.641865	−	0.766818i	$-0.278161\pi$
$71$	768.000	1.28373	0.641865	+	0.766818i	$0.278161\pi$
$72$	0	0
$73$	−558.000	−0.894643	−0.447322	−	0.894373i	$-0.647622\pi$
$73$	−558.000	−0.894643	−0.447322	+	0.894373i	$0.647622\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	320.000	0.473602
$78$	0	0
$79$	408.000	0.581058	0.290529	−	0.956866i	$-0.406169\pi$
$79$	408.000	0.581058	0.290529	+	0.956866i	$0.406169\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	164.000	0.216884	0.108442	−	0.994103i	$-0.465414\pi$
$83$	164.000	0.216884	0.108442	+	0.994103i	$0.465414\pi$
$84$	0	0
$85$	−190.000	−0.242452
$86$	0	0
$87$	246.000	0.303149
$88$	0	0
$89$	−510.000	−0.607415	−0.303707	−	0.952765i	$-0.598224\pi$
$89$	−510.000	−0.607415	−0.303707	+	0.952765i	$0.598224\pi$
$90$	0	0
$91$	1160.00	1.33628
$92$	0	0
$93$	−24.0000	−0.0267600
$94$	0	0
$95$	−20.0000	−0.0215995
$96$	0	0
$97$	514.000	0.538029	0.269014	−	0.963136i	$-0.413302\pi$
$97$	514.000	0.538029	0.269014	+	0.963136i	$0.413302\pi$
$98$	0	0
$99$	144.000	0.146187
$100$	0	0
$101$	666.000	0.656133	0.328067	−	0.944655i	$-0.393603\pi$
$101$	666.000	0.656133	0.328067	+	0.944655i	$0.393603\pi$
$102$	0	0
$103$	−1100.00	−1.05229	−0.526147	−	0.850394i	$-0.676364\pi$
$103$	−1100.00	−1.05229	−0.526147	+	0.850394i	$0.676364\pi$
$104$	0	0
$105$	−300.000	−0.278829
$106$	0	0
$107$	1212.00	1.09503	0.547516	−	0.836795i	$-0.315573\pi$
$107$	1212.00	1.09503	0.547516	+	0.836795i	$0.315573\pi$
$108$	0	0
$109$	2078.00	1.82602	0.913011	−	0.407936i	$-0.133751\pi$
$109$	2078.00	1.82602	0.913011	+	0.407936i	$0.133751\pi$
$110$	0	0
$111$	1278.00	1.09281
$112$	0	0
$113$	−1458.00	−1.21378	−0.606890	−	0.794786i	$-0.707583\pi$
$113$	−1458.00	−1.21378	−0.606890	+	0.794786i	$0.707583\pi$
$114$	0	0
$115$	400.000	0.324349
$116$	0	0
$117$	522.000	0.412469
$118$	0	0
$119$	760.000	0.585455
$120$	0	0
$121$	−1075.00	−0.807663
$122$	0	0
$123$	−738.000	−0.541002
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2436.00	−1.70205	−0.851024	−	0.525127i	$-0.824018\pi$
$127$	−2436.00	−1.70205	−0.851024	+	0.525127i	$0.824018\pi$
$128$	0	0
$129$	−1572.00	−1.07292
$130$	0	0
$131$	2544.00	1.69672	0.848360	−	0.529420i	$-0.177590\pi$
$131$	2544.00	1.69672	0.848360	+	0.529420i	$0.177590\pi$
$132$	0	0
$133$	80.0000	0.0521570
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	694.000	0.432791	0.216396	−	0.976306i	$-0.430570\pi$
$137$	694.000	0.432791	0.216396	+	0.976306i	$0.430570\pi$
$138$	0	0
$139$	516.000	0.314867	0.157434	−	0.987530i	$-0.449678\pi$
$139$	516.000	0.314867	0.157434	+	0.987530i	$0.449678\pi$
$140$	0	0
$141$	−1392.00	−0.831401
$142$	0	0
$143$	928.000	0.542680
$144$	0	0
$145$	−410.000	−0.234818
$146$	0	0
$147$	171.000	0.0959445
$148$	0	0
$149$	770.000	0.423361	0.211681	−	0.977339i	$-0.432106\pi$
$149$	770.000	0.423361	0.211681	+	0.977339i	$0.432106\pi$
$150$	0	0
$151$	−424.000	−0.228507	−0.114254	−	0.993452i	$-0.536448\pi$
$151$	−424.000	−0.228507	−0.114254	+	0.993452i	$0.536448\pi$
$152$	0	0
$153$	342.000	0.180713
$154$	0	0
$155$	40.0000	0.0207282
$156$	0	0
$157$	922.000	0.468685	0.234343	−	0.972154i	$-0.424706\pi$
$157$	922.000	0.468685	0.234343	+	0.972154i	$0.424706\pi$
$158$	0	0
$159$	−2106.00	−1.05042
$160$	0	0
$161$	−1600.00	−0.783215
$162$	0	0
$163$	−3788.00	−1.82024	−0.910120	−	0.414345i	$-0.864011\pi$
$163$	−3788.00	−1.82024	−0.910120	+	0.414345i	$0.864011\pi$
$164$	0	0
$165$	−240.000	−0.113236
$166$	0	0
$167$	−48.0000	−0.0222416	−0.0111208	−	0.999938i	$-0.503540\pi$
$167$	−48.0000	−0.0222416	−0.0111208	+	0.999938i	$0.503540\pi$
$168$	0	0
$169$	1167.00	0.531179
$170$	0	0
$171$	36.0000	0.0160993
$172$	0	0
$173$	3242.00	1.42477	0.712384	−	0.701790i	$-0.247616\pi$
$173$	3242.00	1.42477	0.712384	+	0.701790i	$0.247616\pi$
$174$	0	0
$175$	500.000	0.215980
$176$	0	0
$177$	−1776.00	−0.754194
$178$	0	0
$179$	−2728.00	−1.13911	−0.569554	−	0.821954i	$-0.692884\pi$
$179$	−2728.00	−1.13911	−0.569554	+	0.821954i	$0.692884\pi$
$180$	0	0
$181$	−4090.00	−1.67960	−0.839799	−	0.542897i	$-0.817327\pi$
$181$	−4090.00	−1.67960	−0.839799	+	0.542897i	$0.817327\pi$
$182$	0	0
$183$	1722.00	0.695595
$184$	0	0
$185$	−2130.00	−0.846490
$186$	0	0
$187$	608.000	0.237761
$188$	0	0
$189$	540.000	0.207827
$190$	0	0
$191$	−1480.00	−0.560676	−0.280338	−	0.959901i	$-0.590446\pi$
$191$	−1480.00	−0.560676	−0.280338	+	0.959901i	$0.590446\pi$
$192$	0	0
$193$	−1622.00	−0.604944	−0.302472	−	0.953158i	$-0.597812\pi$
$193$	−1622.00	−0.604944	−0.302472	+	0.953158i	$0.597812\pi$
$194$	0	0
$195$	−870.000	−0.319497
$196$	0	0
$197$	2530.00	0.915000	0.457500	−	0.889210i	$-0.348745\pi$
$197$	2530.00	0.915000	0.457500	+	0.889210i	$0.348745\pi$
$198$	0	0
$199$	−2440.00	−0.869181	−0.434590	−	0.900628i	$-0.643107\pi$
$199$	−2440.00	−0.869181	−0.434590	+	0.900628i	$0.643107\pi$
$200$	0	0
$201$	−516.000	−0.181074
$202$	0	0
$203$	1640.00	0.567022
$204$	0	0
$205$	1230.00	0.419058
$206$	0	0
$207$	−720.000	−0.241756
$208$	0	0
$209$	64.0000	0.0211817
$210$	0	0
$211$	−148.000	−0.0482879	−0.0241439	−	0.999708i	$-0.507686\pi$
$211$	−148.000	−0.0482879	−0.0241439	+	0.999708i	$0.507686\pi$
$212$	0	0
$213$	2304.00	0.741162
$214$	0	0
$215$	2620.00	0.831081
$216$	0	0
$217$	−160.000	−0.0500530
$218$	0	0
$219$	−1674.00	−0.516523
$220$	0	0
$221$	2204.00	0.670847
$222$	0	0
$223$	−676.000	−0.202997	−0.101498	−	0.994836i	$-0.532364\pi$
$223$	−676.000	−0.202997	−0.101498	+	0.994836i	$0.532364\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−6276.00	−1.83503	−0.917517	−	0.397696i	$-0.869810\pi$
$227$	−6276.00	−1.83503	−0.917517	+	0.397696i	$0.869810\pi$
$228$	0	0
$229$	6190.00	1.78623	0.893115	−	0.449828i	$-0.148515\pi$
$229$	6190.00	1.78623	0.893115	+	0.449828i	$0.148515\pi$
$230$	0	0
$231$	960.000	0.273434
$232$	0	0
$233$	5406.00	1.52000	0.759998	−	0.649926i	$-0.225200\pi$
$233$	5406.00	1.52000	0.759998	+	0.649926i	$0.225200\pi$
$234$	0	0
$235$	2320.00	0.644000
$236$	0	0
$237$	1224.00	0.335474
$238$	0	0
$239$	−600.000	−0.162388	−0.0811941	−	0.996698i	$-0.525873\pi$
$239$	−600.000	−0.162388	−0.0811941	+	0.996698i	$0.525873\pi$
$240$	0	0
$241$	−1054.00	−0.281718	−0.140859	−	0.990030i	$-0.544986\pi$
$241$	−1054.00	−0.281718	−0.140859	+	0.990030i	$0.544986\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−285.000	−0.0743183
$246$	0	0
$247$	232.000	0.0597644
$248$	0	0
$249$	492.000	0.125218
$250$	0	0
$251$	−2232.00	−0.561285	−0.280643	−	0.959812i	$-0.590548\pi$
$251$	−2232.00	−0.561285	−0.280643	+	0.959812i	$0.590548\pi$
$252$	0	0
$253$	−1280.00	−0.318075
$254$	0	0
$255$	−570.000	−0.139980
$256$	0	0
$257$	3630.00	0.881063	0.440531	−	0.897737i	$-0.354790\pi$
$257$	3630.00	0.881063	0.440531	+	0.897737i	$0.354790\pi$
$258$	0	0
$259$	8520.00	2.04404
$260$	0	0
$261$	738.000	0.175023
$262$	0	0
$263$	6960.00	1.63183	0.815916	−	0.578170i	$-0.196233\pi$
$263$	6960.00	1.63183	0.815916	+	0.578170i	$0.196233\pi$
$264$	0	0
$265$	3510.00	0.813651
$266$	0	0
$267$	−1530.00	−0.350691
$268$	0	0
$269$	−2062.00	−0.467369	−0.233685	−	0.972312i	$-0.575078\pi$
$269$	−2062.00	−0.467369	−0.233685	+	0.972312i	$0.575078\pi$
$270$	0	0
$271$	−2544.00	−0.570247	−0.285124	−	0.958491i	$-0.592035\pi$
$271$	−2544.00	−0.570247	−0.285124	+	0.958491i	$0.592035\pi$
$272$	0	0
$273$	3480.00	0.771499
$274$	0	0
$275$	400.000	0.0877124
$276$	0	0
$277$	−694.000	−0.150536	−0.0752679	−	0.997163i	$-0.523981\pi$
$277$	−694.000	−0.150536	−0.0752679	+	0.997163i	$0.523981\pi$
$278$	0	0
$279$	−72.0000	−0.0154499
$280$	0	0
$281$	−1982.00	−0.420769	−0.210385	−	0.977619i	$-0.567472\pi$
$281$	−1982.00	−0.420769	−0.210385	+	0.977619i	$0.567472\pi$
$282$	0	0
$283$	5228.00	1.09814	0.549068	−	0.835778i	$-0.314983\pi$
$283$	5228.00	1.09814	0.549068	+	0.835778i	$0.314983\pi$
$284$	0	0
$285$	−60.0000	−0.0124705
$286$	0	0
$287$	−4920.00	−1.01191
$288$	0	0
$289$	−3469.00	−0.706086
$290$	0	0
$291$	1542.00	0.310631
$292$	0	0
$293$	−7454.00	−1.48624	−0.743118	−	0.669160i	$-0.766654\pi$
$293$	−7454.00	−1.48624	−0.743118	+	0.669160i	$0.766654\pi$
$294$	0	0
$295$	2960.00	0.584196
$296$	0	0
$297$	432.000	0.0844013
$298$	0	0
$299$	−4640.00	−0.897452
$300$	0	0
$301$	−10480.0	−2.00683
$302$	0	0
$303$	1998.00	0.378819
$304$	0	0
$305$	−2870.00	−0.538806
$306$	0	0
$307$	−1316.00	−0.244652	−0.122326	−	0.992490i	$-0.539035\pi$
$307$	−1316.00	−0.244652	−0.122326	+	0.992490i	$0.539035\pi$
$308$	0	0
$309$	−3300.00	−0.607542
$310$	0	0
$311$	−832.000	−0.151699	−0.0758495	−	0.997119i	$-0.524167\pi$
$311$	−832.000	−0.151699	−0.0758495	+	0.997119i	$0.524167\pi$
$312$	0	0
$313$	6770.00	1.22257	0.611283	−	0.791412i	$-0.290654\pi$
$313$	6770.00	1.22257	0.611283	+	0.791412i	$0.290654\pi$
$314$	0	0
$315$	−900.000	−0.160982
$316$	0	0
$317$	−6582.00	−1.16619	−0.583095	−	0.812404i	$-0.698158\pi$
$317$	−6582.00	−1.16619	−0.583095	+	0.812404i	$0.698158\pi$
$318$	0	0
$319$	1312.00	0.230276
$320$	0	0
$321$	3636.00	0.632217
$322$	0	0
$323$	152.000	0.0261842
$324$	0	0
$325$	1450.00	0.247482
$326$	0	0
$327$	6234.00	1.05425
$328$	0	0
$329$	−9280.00	−1.55508
$330$	0	0
$331$	11292.0	1.87512	0.937560	−	0.347825i	$-0.113080\pi$
$331$	11292.0	1.87512	0.937560	+	0.347825i	$0.113080\pi$
$332$	0	0
$333$	3834.00	0.630937
$334$	0	0
$335$	860.000	0.140259
$336$	0	0
$337$	−8006.00	−1.29411	−0.647054	−	0.762444i	$-0.723999\pi$
$337$	−8006.00	−1.29411	−0.647054	+	0.762444i	$0.723999\pi$
$338$	0	0
$339$	−4374.00	−0.700776
$340$	0	0
$341$	−128.000	−0.0203272
$342$	0	0
$343$	−5720.00	−0.900440
$344$	0	0
$345$	1200.00	0.187263
$346$	0	0
$347$	−316.000	−0.0488869	−0.0244435	−	0.999701i	$-0.507781\pi$
$347$	−316.000	−0.0488869	−0.0244435	+	0.999701i	$0.507781\pi$
$348$	0	0
$349$	4926.00	0.755538	0.377769	−	0.925900i	$-0.376691\pi$
$349$	4926.00	0.755538	0.377769	+	0.925900i	$0.376691\pi$
$350$	0	0
$351$	1566.00	0.238139
$352$	0	0
$353$	2438.00	0.367597	0.183798	−	0.982964i	$-0.441161\pi$
$353$	2438.00	0.367597	0.183798	+	0.982964i	$0.441161\pi$
$354$	0	0
$355$	−3840.00	−0.574102
$356$	0	0
$357$	2280.00	0.338012
$358$	0	0
$359$	−3336.00	−0.490438	−0.245219	−	0.969468i	$-0.578860\pi$
$359$	−3336.00	−0.490438	−0.245219	+	0.969468i	$0.578860\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	−3225.00	−0.466305
$364$	0	0
$365$	2790.00	0.400097
$366$	0	0
$367$	44.0000	0.00625826	0.00312913	−	0.999995i	$-0.499004\pi$
$367$	44.0000	0.00625826	0.00312913	+	0.999995i	$0.499004\pi$
$368$	0	0
$369$	−2214.00	−0.312348
$370$	0	0
$371$	−14040.0	−1.96475
$372$	0	0
$373$	−11966.0	−1.66106	−0.830531	−	0.556973i	$-0.811963\pi$
$373$	−11966.0	−1.66106	−0.830531	+	0.556973i	$0.811963\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	4756.00	0.649725
$378$	0	0
$379$	12676.0	1.71800	0.859001	−	0.511975i	$-0.171086\pi$
$379$	12676.0	1.71800	0.859001	+	0.511975i	$0.171086\pi$
$380$	0	0
$381$	−7308.00	−0.982678
$382$	0	0
$383$	6672.00	0.890139	0.445070	−	0.895496i	$-0.353179\pi$
$383$	6672.00	0.890139	0.445070	+	0.895496i	$0.353179\pi$
$384$	0	0
$385$	−1600.00	−0.211801
$386$	0	0
$387$	−4716.00	−0.619452
$388$	0	0
$389$	354.000	0.0461401	0.0230701	−	0.999734i	$-0.492656\pi$
$389$	354.000	0.0461401	0.0230701	+	0.999734i	$0.492656\pi$
$390$	0	0
$391$	−3040.00	−0.393195
$392$	0	0
$393$	7632.00	0.979602
$394$	0	0
$395$	−2040.00	−0.259857
$396$	0	0
$397$	−5054.00	−0.638924	−0.319462	−	0.947599i	$-0.603502\pi$
$397$	−5054.00	−0.638924	−0.319462	+	0.947599i	$0.603502\pi$
$398$	0	0
$399$	240.000	0.0301129
$400$	0	0
$401$	10266.0	1.27845	0.639226	−	0.769019i	$-0.279255\pi$
$401$	10266.0	1.27845	0.639226	+	0.769019i	$0.279255\pi$
$402$	0	0
$403$	−464.000	−0.0573536
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	6816.00	0.830114
$408$	0	0
$409$	−1526.00	−0.184489	−0.0922443	−	0.995736i	$-0.529404\pi$
$409$	−1526.00	−0.184489	−0.0922443	+	0.995736i	$0.529404\pi$
$410$	0	0
$411$	2082.00	0.249872
$412$	0	0
$413$	−11840.0	−1.41067
$414$	0	0
$415$	−820.000	−0.0969933
$416$	0	0
$417$	1548.00	0.181789
$418$	0	0
$419$	2064.00	0.240652	0.120326	−	0.992734i	$-0.461606\pi$
$419$	2064.00	0.240652	0.120326	+	0.992734i	$0.461606\pi$
$420$	0	0
$421$	4590.00	0.531361	0.265680	−	0.964061i	$-0.414403\pi$
$421$	4590.00	0.531361	0.265680	+	0.964061i	$0.414403\pi$
$422$	0	0
$423$	−4176.00	−0.480010
$424$	0	0
$425$	950.000	0.108428
$426$	0	0
$427$	11480.0	1.30107
$428$	0	0
$429$	2784.00	0.313317
$430$	0	0
$431$	−5536.00	−0.618700	−0.309350	−	0.950948i	$-0.600111\pi$
$431$	−5536.00	−0.618700	−0.309350	+	0.950948i	$0.600111\pi$
$432$	0	0
$433$	1850.00	0.205324	0.102662	−	0.994716i	$-0.467264\pi$
$433$	1850.00	0.205324	0.102662	+	0.994716i	$0.467264\pi$
$434$	0	0
$435$	−1230.00	−0.135572
$436$	0	0
$437$	−320.000	−0.0350290
$438$	0	0
$439$	11704.0	1.27244	0.636220	−	0.771507i	$-0.280497\pi$
$439$	11704.0	1.27244	0.636220	+	0.771507i	$0.280497\pi$
$440$	0	0
$441$	513.000	0.0553936
$442$	0	0
$443$	6948.00	0.745168	0.372584	−	0.927998i	$-0.378472\pi$
$443$	6948.00	0.745168	0.372584	+	0.927998i	$0.378472\pi$
$444$	0	0
$445$	2550.00	0.271644
$446$	0	0
$447$	2310.00	0.244428
$448$	0	0
$449$	12090.0	1.27074	0.635370	−	0.772208i	$-0.280848\pi$
$449$	12090.0	1.27074	0.635370	+	0.772208i	$0.280848\pi$
$450$	0	0
$451$	−3936.00	−0.410951
$452$	0	0
$453$	−1272.00	−0.131929
$454$	0	0
$455$	−5800.00	−0.597600
$456$	0	0
$457$	11626.0	1.19002	0.595012	−	0.803717i	$-0.297147\pi$
$457$	11626.0	1.19002	0.595012	+	0.803717i	$0.297147\pi$
$458$	0	0
$459$	1026.00	0.104335
$460$	0	0
$461$	16314.0	1.64820	0.824098	−	0.566447i	$-0.191682\pi$
$461$	16314.0	1.64820	0.824098	+	0.566447i	$0.191682\pi$
$462$	0	0
$463$	−15756.0	−1.58152	−0.790760	−	0.612127i	$-0.790314\pi$
$463$	−15756.0	−1.58152	−0.790760	+	0.612127i	$0.790314\pi$
$464$	0	0
$465$	120.000	0.0119675
$466$	0	0
$467$	5684.00	0.563221	0.281610	−	0.959529i	$-0.409131\pi$
$467$	5684.00	0.563221	0.281610	+	0.959529i	$0.409131\pi$
$468$	0	0
$469$	−3440.00	−0.338688
$470$	0	0
$471$	2766.00	0.270596
$472$	0	0
$473$	−8384.00	−0.815004
$474$	0	0
$475$	100.000	0.00965961
$476$	0	0
$477$	−6318.00	−0.606460
$478$	0	0
$479$	−3368.00	−0.321269	−0.160634	−	0.987014i	$-0.551354\pi$
$479$	−3368.00	−0.321269	−0.160634	+	0.987014i	$0.551354\pi$
$480$	0	0
$481$	24708.0	2.34218
$482$	0	0
$483$	−4800.00	−0.452190
$484$	0	0
$485$	−2570.00	−0.240614
$486$	0	0
$487$	−5588.00	−0.519952	−0.259976	−	0.965615i	$-0.583715\pi$
$487$	−5588.00	−0.519952	−0.259976	+	0.965615i	$0.583715\pi$
$488$	0	0
$489$	−11364.0	−1.05092
$490$	0	0
$491$	10584.0	0.972809	0.486405	−	0.873734i	$-0.338308\pi$
$491$	10584.0	0.972809	0.486405	+	0.873734i	$0.338308\pi$
$492$	0	0
$493$	3116.00	0.284660
$494$	0	0
$495$	−720.000	−0.0653770
$496$	0	0
$497$	15360.0	1.38630
$498$	0	0
$499$	−12220.0	−1.09628	−0.548139	−	0.836388i	$-0.684663\pi$
$499$	−12220.0	−1.09628	−0.548139	+	0.836388i	$0.684663\pi$
$500$	0	0
$501$	−144.000	−0.0128412
$502$	0	0
$503$	16152.0	1.43177	0.715887	−	0.698216i	$-0.246023\pi$
$503$	16152.0	1.43177	0.715887	+	0.698216i	$0.246023\pi$
$504$	0	0
$505$	−3330.00	−0.293432
$506$	0	0
$507$	3501.00	0.306676
$508$	0	0
$509$	10642.0	0.926716	0.463358	−	0.886171i	$-0.346644\pi$
$509$	10642.0	0.926716	0.463358	+	0.886171i	$0.346644\pi$
$510$	0	0
$511$	−11160.0	−0.966124
$512$	0	0
$513$	108.000	0.00929496
$514$	0	0
$515$	5500.00	0.470600
$516$	0	0
$517$	−7424.00	−0.631542
$518$	0	0
$519$	9726.00	0.822590
$520$	0	0
$521$	22882.0	1.92414	0.962072	−	0.272797i	$-0.0879487\pi$
$521$	22882.0	1.92414	0.962072	+	0.272797i	$0.0879487\pi$
$522$	0	0
$523$	−10052.0	−0.840427	−0.420213	−	0.907425i	$-0.638045\pi$
$523$	−10052.0	−0.840427	−0.420213	+	0.907425i	$0.638045\pi$
$524$	0	0
$525$	1500.00	0.124696
$526$	0	0
$527$	−304.000	−0.0251280
$528$	0	0
$529$	−5767.00	−0.473987
$530$	0	0
$531$	−5328.00	−0.435434
$532$	0	0
$533$	−14268.0	−1.15950
$534$	0	0
$535$	−6060.00	−0.489713
$536$	0	0
$537$	−8184.00	−0.657664
$538$	0	0
$539$	912.000	0.0728806
$540$	0	0
$541$	−6530.00	−0.518940	−0.259470	−	0.965751i	$-0.583548\pi$
$541$	−6530.00	−0.518940	−0.259470	+	0.965751i	$0.583548\pi$
$542$	0	0
$543$	−12270.0	−0.969717
$544$	0	0
$545$	−10390.0	−0.816621
$546$	0	0
$547$	16652.0	1.30162	0.650812	−	0.759239i	$-0.274429\pi$
$547$	16652.0	1.30162	0.650812	+	0.759239i	$0.274429\pi$
$548$	0	0
$549$	5166.00	0.401602
$550$	0	0
$551$	328.000	0.0253598
$552$	0	0
$553$	8160.00	0.627484
$554$	0	0
$555$	−6390.00	−0.488721
$556$	0	0
$557$	−12886.0	−0.980247	−0.490123	−	0.871653i	$-0.663048\pi$
$557$	−12886.0	−0.980247	−0.490123	+	0.871653i	$0.663048\pi$
$558$	0	0
$559$	−30392.0	−2.29954
$560$	0	0
$561$	1824.00	0.137272
$562$	0	0
$563$	−11108.0	−0.831521	−0.415761	−	0.909474i	$-0.636485\pi$
$563$	−11108.0	−0.831521	−0.415761	+	0.909474i	$0.636485\pi$
$564$	0	0
$565$	7290.00	0.542819
$566$	0	0
$567$	1620.00	0.119989
$568$	0	0
$569$	−9214.00	−0.678859	−0.339430	−	0.940631i	$-0.610234\pi$
$569$	−9214.00	−0.678859	−0.339430	+	0.940631i	$0.610234\pi$
$570$	0	0
$571$	−4052.00	−0.296972	−0.148486	−	0.988915i	$-0.547440\pi$
$571$	−4052.00	−0.296972	−0.148486	+	0.988915i	$0.547440\pi$
$572$	0	0
$573$	−4440.00	−0.323706
$574$	0	0
$575$	−2000.00	−0.145054
$576$	0	0
$577$	−8446.00	−0.609379	−0.304689	−	0.952452i	$-0.598553\pi$
$577$	−8446.00	−0.609379	−0.304689	+	0.952452i	$0.598553\pi$
$578$	0	0
$579$	−4866.00	−0.349264
$580$	0	0
$581$	3280.00	0.234212
$582$	0	0
$583$	−11232.0	−0.797911
$584$	0	0
$585$	−2610.00	−0.184462
$586$	0	0
$587$	2172.00	0.152722	0.0763612	−	0.997080i	$-0.475670\pi$
$587$	2172.00	0.152722	0.0763612	+	0.997080i	$0.475670\pi$
$588$	0	0
$589$	−32.0000	−0.00223860
$590$	0	0
$591$	7590.00	0.528276
$592$	0	0
$593$	−1218.00	−0.0843461	−0.0421731	−	0.999110i	$-0.513428\pi$
$593$	−1218.00	−0.0843461	−0.0421731	+	0.999110i	$0.513428\pi$
$594$	0	0
$595$	−3800.00	−0.261823
$596$	0	0
$597$	−7320.00	−0.501822
$598$	0	0
$599$	21240.0	1.44882	0.724410	−	0.689370i	$-0.242112\pi$
$599$	21240.0	1.44882	0.724410	+	0.689370i	$0.242112\pi$
$600$	0	0
$601$	17626.0	1.19631	0.598153	−	0.801382i	$-0.295902\pi$
$601$	17626.0	1.19631	0.598153	+	0.801382i	$0.295902\pi$
$602$	0	0
$603$	−1548.00	−0.104543
$604$	0	0
$605$	5375.00	0.361198
$606$	0	0
$607$	2580.00	0.172519	0.0862594	−	0.996273i	$-0.472509\pi$
$607$	2580.00	0.172519	0.0862594	+	0.996273i	$0.472509\pi$
$608$	0	0
$609$	4920.00	0.327370
$610$	0	0
$611$	−26912.0	−1.78190
$612$	0	0
$613$	−14166.0	−0.933376	−0.466688	−	0.884422i	$-0.654553\pi$
$613$	−14166.0	−0.933376	−0.466688	+	0.884422i	$0.654553\pi$
$614$	0	0
$615$	3690.00	0.241943
$616$	0	0
$617$	−21426.0	−1.39802	−0.699010	−	0.715112i	$-0.746376\pi$
$617$	−21426.0	−1.39802	−0.699010	+	0.715112i	$0.746376\pi$
$618$	0	0
$619$	3668.00	0.238173	0.119087	−	0.992884i	$-0.462003\pi$
$619$	3668.00	0.238173	0.119087	+	0.992884i	$0.462003\pi$
$620$	0	0
$621$	−2160.00	−0.139578
$622$	0	0
$623$	−10200.0	−0.655946
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	192.000	0.0122293
$628$	0	0
$629$	16188.0	1.02617
$630$	0	0
$631$	20032.0	1.26381	0.631903	−	0.775048i	$-0.282274\pi$
$631$	20032.0	1.26381	0.631903	+	0.775048i	$0.282274\pi$
$632$	0	0
$633$	−444.000	−0.0278790
$634$	0	0
$635$	12180.0	0.761179
$636$	0	0
$637$	3306.00	0.205633
$638$	0	0
$639$	6912.00	0.427910
$640$	0	0
$641$	7458.00	0.459553	0.229776	−	0.973243i	$-0.426201\pi$
$641$	7458.00	0.459553	0.229776	+	0.973243i	$0.426201\pi$
$642$	0	0
$643$	7092.00	0.434963	0.217481	−	0.976064i	$-0.430216\pi$
$643$	7092.00	0.434963	0.217481	+	0.976064i	$0.430216\pi$
$644$	0	0
$645$	7860.00	0.479825
$646$	0	0
$647$	3384.00	0.205624	0.102812	−	0.994701i	$-0.467216\pi$
$647$	3384.00	0.205624	0.102812	+	0.994701i	$0.467216\pi$
$648$	0	0
$649$	−9472.00	−0.572894
$650$	0	0
$651$	−480.000	−0.0288981
$652$	0	0
$653$	−29398.0	−1.76177	−0.880883	−	0.473335i	$-0.843050\pi$
$653$	−29398.0	−1.76177	−0.880883	+	0.473335i	$0.843050\pi$
$654$	0	0
$655$	−12720.0	−0.758796
$656$	0	0
$657$	−5022.00	−0.298214
$658$	0	0
$659$	−6624.00	−0.391554	−0.195777	−	0.980648i	$-0.562723\pi$
$659$	−6624.00	−0.391554	−0.195777	+	0.980648i	$0.562723\pi$
$660$	0	0
$661$	8646.00	0.508760	0.254380	−	0.967104i	$-0.418129\pi$
$661$	8646.00	0.508760	0.254380	+	0.967104i	$0.418129\pi$
$662$	0	0
$663$	6612.00	0.387313
$664$	0	0
$665$	−400.000	−0.0233253
$666$	0	0
$667$	−6560.00	−0.380816
$668$	0	0
$669$	−2028.00	−0.117200
$670$	0	0
$671$	9184.00	0.528382
$672$	0	0
$673$	28698.0	1.64372	0.821862	−	0.569686i	$-0.192935\pi$
$673$	28698.0	1.64372	0.821862	+	0.569686i	$0.192935\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	19426.0	1.10281	0.551405	−	0.834238i	$-0.314092\pi$
$677$	19426.0	1.10281	0.551405	+	0.834238i	$0.314092\pi$
$678$	0	0
$679$	10280.0	0.581016
$680$	0	0
$681$	−18828.0	−1.05946
$682$	0	0
$683$	8604.00	0.482025	0.241012	−	0.970522i	$-0.422521\pi$
$683$	8604.00	0.482025	0.241012	+	0.970522i	$0.422521\pi$
$684$	0	0
$685$	−3470.00	−0.193550
$686$	0	0
$687$	18570.0	1.03128
$688$	0	0
$689$	−40716.0	−2.25132
$690$	0	0
$691$	12980.0	0.714591	0.357296	−	0.933991i	$-0.383699\pi$
$691$	12980.0	0.714591	0.357296	+	0.933991i	$0.383699\pi$
$692$	0	0
$693$	2880.00	0.157867
$694$	0	0
$695$	−2580.00	−0.140813
$696$	0	0
$697$	−9348.00	−0.508007
$698$	0	0
$699$	16218.0	0.877570
$700$	0	0
$701$	−19630.0	−1.05765	−0.528827	−	0.848730i	$-0.677368\pi$
$701$	−19630.0	−1.05765	−0.528827	+	0.848730i	$0.677368\pi$
$702$	0	0
$703$	1704.00	0.0914190
$704$	0	0
$705$	6960.00	0.371814
$706$	0	0
$707$	13320.0	0.708558
$708$	0	0
$709$	8030.00	0.425350	0.212675	−	0.977123i	$-0.431782\pi$
$709$	8030.00	0.425350	0.212675	+	0.977123i	$0.431782\pi$
$710$	0	0
$711$	3672.00	0.193686
$712$	0	0
$713$	640.000	0.0336160
$714$	0	0
$715$	−4640.00	−0.242694
$716$	0	0
$717$	−1800.00	−0.0937549
$718$	0	0
$719$	22720.0	1.17846	0.589230	−	0.807965i	$-0.299431\pi$
$719$	22720.0	1.17846	0.589230	+	0.807965i	$0.299431\pi$
$720$	0	0
$721$	−22000.0	−1.13637
$722$	0	0
$723$	−3162.00	−0.162650
$724$	0	0
$725$	2050.00	0.105014
$726$	0	0
$727$	27116.0	1.38332	0.691662	−	0.722221i	$-0.256879\pi$
$727$	27116.0	1.38332	0.691662	+	0.722221i	$0.256879\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−19912.0	−1.00749
$732$	0	0
$733$	30882.0	1.55614	0.778071	−	0.628176i	$-0.216198\pi$
$733$	30882.0	1.55614	0.778071	+	0.628176i	$0.216198\pi$
$734$	0	0
$735$	−855.000	−0.0429077
$736$	0	0
$737$	−2752.00	−0.137546
$738$	0	0
$739$	−13836.0	−0.688722	−0.344361	−	0.938837i	$-0.611904\pi$
$739$	−13836.0	−0.688722	−0.344361	+	0.938837i	$0.611904\pi$
$740$	0	0
$741$	696.000	0.0345050
$742$	0	0
$743$	32712.0	1.61519	0.807595	−	0.589737i	$-0.200769\pi$
$743$	32712.0	1.61519	0.807595	+	0.589737i	$0.200769\pi$
$744$	0	0
$745$	−3850.00	−0.189333
$746$	0	0
$747$	1476.00	0.0722945
$748$	0	0
$749$	24240.0	1.18252
$750$	0	0
$751$	−8472.00	−0.411648	−0.205824	−	0.978589i	$-0.565987\pi$
$751$	−8472.00	−0.411648	−0.205824	+	0.978589i	$0.565987\pi$
$752$	0	0
$753$	−6696.00	−0.324058
$754$	0	0
$755$	2120.00	0.102192
$756$	0	0
$757$	9866.00	0.473693	0.236847	−	0.971547i	$-0.423886\pi$
$757$	9866.00	0.473693	0.236847	+	0.971547i	$0.423886\pi$
$758$	0	0
$759$	−3840.00	−0.183641
$760$	0	0
$761$	−3774.00	−0.179773	−0.0898866	−	0.995952i	$-0.528650\pi$
$761$	−3774.00	−0.179773	−0.0898866	+	0.995952i	$0.528650\pi$
$762$	0	0
$763$	41560.0	1.97192
$764$	0	0
$765$	−1710.00	−0.0808172
$766$	0	0
$767$	−34336.0	−1.61643
$768$	0	0
$769$	−28670.0	−1.34443	−0.672215	−	0.740356i	$-0.734657\pi$
$769$	−28670.0	−1.34443	−0.672215	+	0.740356i	$0.734657\pi$
$770$	0	0
$771$	10890.0	0.508682
$772$	0	0
$773$	−3246.00	−0.151036	−0.0755178	−	0.997144i	$-0.524061\pi$
$773$	−3246.00	−0.151036	−0.0755178	+	0.997144i	$0.524061\pi$
$774$	0	0
$775$	−200.000	−0.00926995
$776$	0	0
$777$	25560.0	1.18013
$778$	0	0
$779$	−984.000	−0.0452573
$780$	0	0
$781$	12288.0	0.562995
$782$	0	0
$783$	2214.00	0.101050
$784$	0	0
$785$	−4610.00	−0.209602
$786$	0	0
$787$	−19372.0	−0.877430	−0.438715	−	0.898626i	$-0.644566\pi$
$787$	−19372.0	−0.877430	−0.438715	+	0.898626i	$0.644566\pi$
$788$	0	0
$789$	20880.0	0.942139
$790$	0	0
$791$	−29160.0	−1.31076
$792$	0	0
$793$	33292.0	1.49084
$794$	0	0
$795$	10530.0	0.469762
$796$	0	0
$797$	−11814.0	−0.525061	−0.262530	−	0.964924i	$-0.584557\pi$
$797$	−11814.0	−0.525061	−0.262530	+	0.964924i	$0.584557\pi$
$798$	0	0
$799$	−17632.0	−0.780695
$800$	0	0
$801$	−4590.00	−0.202472
$802$	0	0
$803$	−8928.00	−0.392357
$804$	0	0
$805$	8000.00	0.350265
$806$	0	0
$807$	−6186.00	−0.269836
$808$	0	0
$809$	−30054.0	−1.30611	−0.653055	−	0.757311i	$-0.726513\pi$
$809$	−30054.0	−1.30611	−0.653055	+	0.757311i	$0.726513\pi$
$810$	0	0
$811$	2852.00	0.123486	0.0617431	−	0.998092i	$-0.480334\pi$
$811$	2852.00	0.123486	0.0617431	+	0.998092i	$0.480334\pi$
$812$	0	0
$813$	−7632.00	−0.329232
$814$	0	0
$815$	18940.0	0.814036
$816$	0	0
$817$	−2096.00	−0.0897549
$818$	0	0
$819$	10440.0	0.445425
$820$	0	0
$821$	2170.00	0.0922455	0.0461227	−	0.998936i	$-0.485313\pi$
$821$	2170.00	0.0922455	0.0461227	+	0.998936i	$0.485313\pi$
$822$	0	0
$823$	19804.0	0.838790	0.419395	−	0.907804i	$-0.362242\pi$
$823$	19804.0	0.838790	0.419395	+	0.907804i	$0.362242\pi$
$824$	0	0
$825$	1200.00	0.0506408
$826$	0	0
$827$	5508.00	0.231598	0.115799	−	0.993273i	$-0.463057\pi$
$827$	5508.00	0.231598	0.115799	+	0.993273i	$0.463057\pi$
$828$	0	0
$829$	33262.0	1.39353	0.696765	−	0.717299i	$-0.254622\pi$
$829$	33262.0	1.39353	0.696765	+	0.717299i	$0.254622\pi$
$830$	0	0
$831$	−2082.00	−0.0869119
$832$	0	0
$833$	2166.00	0.0900930
$834$	0	0
$835$	240.000	0.00994676
$836$	0	0
$837$	−216.000	−0.00892001
$838$	0	0
$839$	−4600.00	−0.189284	−0.0946422	−	0.995511i	$-0.530171\pi$
$839$	−4600.00	−0.189284	−0.0946422	+	0.995511i	$0.530171\pi$
$840$	0	0
$841$	−17665.0	−0.724302
$842$	0	0
$843$	−5946.00	−0.242931
$844$	0	0
$845$	−5835.00	−0.237550
$846$	0	0
$847$	−21500.0	−0.872195
$848$	0	0
$849$	15684.0	0.634009
$850$	0	0
$851$	−34080.0	−1.37279
$852$	0	0
$853$	−4198.00	−0.168507	−0.0842537	−	0.996444i	$-0.526851\pi$
$853$	−4198.00	−0.168507	−0.0842537	+	0.996444i	$0.526851\pi$
$854$	0	0
$855$	−180.000	−0.00719985
$856$	0	0
$857$	−5826.00	−0.232220	−0.116110	−	0.993236i	$-0.537042\pi$
$857$	−5826.00	−0.232220	−0.116110	+	0.993236i	$0.537042\pi$
$858$	0	0
$859$	−3004.00	−0.119319	−0.0596596	−	0.998219i	$-0.519002\pi$
$859$	−3004.00	−0.119319	−0.0596596	+	0.998219i	$0.519002\pi$
$860$	0	0
$861$	−14760.0	−0.584227
$862$	0	0
$863$	−36936.0	−1.45691	−0.728457	−	0.685092i	$-0.759762\pi$
$863$	−36936.0	−1.45691	−0.728457	+	0.685092i	$0.759762\pi$
$864$	0	0
$865$	−16210.0	−0.637175
$866$	0	0
$867$	−10407.0	−0.407659
$868$	0	0
$869$	6528.00	0.254830
$870$	0	0
$871$	−9976.00	−0.388087
$872$	0	0
$873$	4626.00	0.179343
$874$	0	0
$875$	−2500.00	−0.0965891
$876$	0	0
$877$	5434.00	0.209228	0.104614	−	0.994513i	$-0.466639\pi$
$877$	5434.00	0.209228	0.104614	+	0.994513i	$0.466639\pi$
$878$	0	0
$879$	−22362.0	−0.858079
$880$	0	0
$881$	−4758.00	−0.181954	−0.0909768	−	0.995853i	$-0.528999\pi$
$881$	−4758.00	−0.181954	−0.0909768	+	0.995853i	$0.528999\pi$
$882$	0	0
$883$	15476.0	0.589818	0.294909	−	0.955525i	$-0.404711\pi$
$883$	15476.0	0.589818	0.294909	+	0.955525i	$0.404711\pi$
$884$	0	0
$885$	8880.00	0.337286
$886$	0	0
$887$	−27440.0	−1.03872	−0.519360	−	0.854555i	$-0.673830\pi$
$887$	−27440.0	−1.03872	−0.519360	+	0.854555i	$0.673830\pi$
$888$	0	0
$889$	−48720.0	−1.83804
$890$	0	0
$891$	1296.00	0.0487291
$892$	0	0
$893$	−1856.00	−0.0695506
$894$	0	0
$895$	13640.0	0.509424
$896$	0	0
$897$	−13920.0	−0.518144
$898$	0	0
$899$	−656.000	−0.0243368
$900$	0	0
$901$	−26676.0	−0.986356
$902$	0	0
$903$	−31440.0	−1.15865
$904$	0	0
$905$	20450.0	0.751139
$906$	0	0
$907$	−48924.0	−1.79106	−0.895532	−	0.444997i	$-0.853205\pi$
$907$	−48924.0	−1.79106	−0.895532	+	0.444997i	$0.853205\pi$
$908$	0	0
$909$	5994.00	0.218711
$910$	0	0
$911$	−3440.00	−0.125107	−0.0625534	−	0.998042i	$-0.519924\pi$
$911$	−3440.00	−0.125107	−0.0625534	+	0.998042i	$0.519924\pi$
$912$	0	0
$913$	2624.00	0.0951169
$914$	0	0
$915$	−8610.00	−0.311080
$916$	0	0
$917$	50880.0	1.83229
$918$	0	0
$919$	−27184.0	−0.975753	−0.487877	−	0.872913i	$-0.662228\pi$
$919$	−27184.0	−0.975753	−0.487877	+	0.872913i	$0.662228\pi$
$920$	0	0
$921$	−3948.00	−0.141250
$922$	0	0
$923$	44544.0	1.58850
$924$	0	0
$925$	10650.0	0.378562
$926$	0	0
$927$	−9900.00	−0.350764
$928$	0	0
$929$	42490.0	1.50059	0.750297	−	0.661101i	$-0.229911\pi$
$929$	42490.0	1.50059	0.750297	+	0.661101i	$0.229911\pi$
$930$	0	0
$931$	228.000	0.00802621
$932$	0	0
$933$	−2496.00	−0.0875835
$934$	0	0
$935$	−3040.00	−0.106330
$936$	0	0
$937$	37354.0	1.30235	0.651175	−	0.758928i	$-0.274276\pi$
$937$	37354.0	1.30235	0.651175	+	0.758928i	$0.274276\pi$
$938$	0	0
$939$	20310.0	0.705849
$940$	0	0
$941$	−24470.0	−0.847714	−0.423857	−	0.905729i	$-0.639324\pi$
$941$	−24470.0	−0.847714	−0.423857	+	0.905729i	$0.639324\pi$
$942$	0	0
$943$	19680.0	0.679607
$944$	0	0
$945$	−2700.00	−0.0929429
$946$	0	0
$947$	−34100.0	−1.17012	−0.585059	−	0.810991i	$-0.698929\pi$
$947$	−34100.0	−1.17012	−0.585059	+	0.810991i	$0.698929\pi$
$948$	0	0
$949$	−32364.0	−1.10704
$950$	0	0
$951$	−19746.0	−0.673300
$952$	0	0
$953$	1878.00	0.0638346	0.0319173	−	0.999491i	$-0.489839\pi$
$953$	1878.00	0.0638346	0.0319173	+	0.999491i	$0.489839\pi$
$954$	0	0
$955$	7400.00	0.250742
$956$	0	0
$957$	3936.00	0.132950
$958$	0	0
$959$	13880.0	0.467371
$960$	0	0
$961$	−29727.0	−0.997852
$962$	0	0
$963$	10908.0	0.365011
$964$	0	0
$965$	8110.00	0.270539
$966$	0	0
$967$	38484.0	1.27980	0.639898	−	0.768460i	$-0.278977\pi$
$967$	38484.0	1.27980	0.639898	+	0.768460i	$0.278977\pi$
$968$	0	0
$969$	456.000	0.0151175
$970$	0	0
$971$	45272.0	1.49624	0.748119	−	0.663564i	$-0.230957\pi$
$971$	45272.0	1.49624	0.748119	+	0.663564i	$0.230957\pi$
$972$	0	0
$973$	10320.0	0.340025
$974$	0	0
$975$	4350.00	0.142884
$976$	0	0
$977$	−25354.0	−0.830242	−0.415121	−	0.909766i	$-0.636261\pi$
$977$	−25354.0	−0.830242	−0.415121	+	0.909766i	$0.636261\pi$
$978$	0	0
$979$	−8160.00	−0.266389
$980$	0	0
$981$	18702.0	0.608674
$982$	0	0
$983$	−18744.0	−0.608180	−0.304090	−	0.952643i	$-0.598352\pi$
$983$	−18744.0	−0.608180	−0.304090	+	0.952643i	$0.598352\pi$
$984$	0	0
$985$	−12650.0	−0.409201
$986$	0	0
$987$	−27840.0	−0.897829
$988$	0	0
$989$	41920.0	1.34780
$990$	0	0
$991$	59600.0	1.91045	0.955225	−	0.295880i	$-0.0956127\pi$
$991$	59600.0	1.91045	0.955225	+	0.295880i	$0.0956127\pi$
$992$	0	0
$993$	33876.0	1.08260
$994$	0	0
$995$	12200.0	0.388710
$996$	0	0
$997$	−17886.0	−0.568160	−0.284080	−	0.958801i	$-0.591688\pi$
$997$	−17886.0	−0.568160	−0.284080	+	0.958801i	$0.591688\pi$
$998$	0	0
$999$	11502.0	0.364271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.4.a.e.1.1	✓	1
3.2	odd	2		360.4.a.m.1.1		1
4.3	odd	2		240.4.a.a.1.1		1
5.2	odd	4		600.4.f.f.49.1		2
5.3	odd	4		600.4.f.f.49.2		2
5.4	even	2		600.4.a.a.1.1		1
8.3	odd	2		960.4.a.bd.1.1		1
8.5	even	2		960.4.a.q.1.1		1
12.11	even	2		720.4.a.s.1.1		1
15.2	even	4		1800.4.f.k.649.2		2
15.8	even	4		1800.4.f.k.649.1		2
15.14	odd	2		1800.4.a.e.1.1		1
20.3	even	4		1200.4.f.h.49.1		2
20.7	even	4		1200.4.f.h.49.2		2
20.19	odd	2		1200.4.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.e.1.1	✓	1	1.1	even	1	trivial
240.4.a.a.1.1		1	4.3	odd	2
360.4.a.m.1.1		1	3.2	odd	2
600.4.a.a.1.1		1	5.4	even	2
600.4.f.f.49.1		2	5.2	odd	4
600.4.f.f.49.2		2	5.3	odd	4
720.4.a.s.1.1		1	12.11	even	2
960.4.a.q.1.1		1	8.5	even	2
960.4.a.bd.1.1		1	8.3	odd	2
1200.4.a.bj.1.1		1	20.19	odd	2
1200.4.f.h.49.1		2	20.3	even	4
1200.4.f.h.49.2		2	20.7	even	4
1800.4.a.e.1.1		1	15.14	odd	2
1800.4.f.k.649.1		2	15.8	even	4
1800.4.f.k.649.2		2	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	120.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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