LMFDB - Embedded newform 120.4.a.b.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:	$1$
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)

$f(q)$

$=$

$q-3.00000 q^{3} -5.00000 q^{5} +20.0000 q^{7} +9.00000 q^{9} -56.0000 q^{11} -86.0000 q^{13} +15.0000 q^{15} -106.000 q^{17} +4.00000 q^{19} -60.0000 q^{21} +136.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -206.000 q^{29} -152.000 q^{31} +168.000 q^{33} -100.000 q^{35} +282.000 q^{37} +258.000 q^{39} -246.000 q^{41} +412.000 q^{43} -45.0000 q^{45} +40.0000 q^{47} +57.0000 q^{49} +318.000 q^{51} -126.000 q^{53} +280.000 q^{55} -12.0000 q^{57} +56.0000 q^{59} -2.00000 q^{61} +180.000 q^{63} +430.000 q^{65} -388.000 q^{67} -408.000 q^{69} -672.000 q^{71} +1170.00 q^{73} -75.0000 q^{75} -1120.00 q^{77} +408.000 q^{79} +81.0000 q^{81} +668.000 q^{83} +530.000 q^{85} +618.000 q^{87} +66.0000 q^{89} -1720.00 q^{91} +456.000 q^{93} -20.0000 q^{95} -926.000 q^{97} -504.000 q^{99} +O(q^{100})$

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
$11$	−56.0000	−1.53497	−0.767483	−	0.641069i	$-0.778491\pi$
$11$	−56.0000	−1.53497	−0.767483	+	0.641069i	$0.778491\pi$
$12$	0	0
$13$	−86.0000	−1.83478	−0.917389	−	0.397992i	$-0.869707\pi$
$13$	−86.0000	−1.83478	−0.917389	+	0.397992i	$0.869707\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−106.000	−1.51228	−0.756140	−	0.654409i	$-0.772917\pi$
$17$	−106.000	−1.51228	−0.756140	+	0.654409i	$0.772917\pi$
$18$	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
$23$	136.000	1.23295	0.616477	−	0.787373i	$-0.288559\pi$
$23$	136.000	1.23295	0.616477	+	0.787373i	$0.288559\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−206.000	−1.31908	−0.659539	−	0.751671i	$-0.729248\pi$
$29$	−206.000	−1.31908	−0.659539	+	0.751671i	$0.729248\pi$
$30$	0	0
$31$	−152.000	−0.880645	−0.440323	−	0.897840i	$-0.645136\pi$
$31$	−152.000	−0.880645	−0.440323	+	0.897840i	$0.645136\pi$
$32$	0	0
$33$	168.000	0.886214
$34$	0	0
$35$	−100.000	−0.482945
$36$	0	0
$37$	282.000	1.25299	0.626493	−	0.779427i	$-0.284490\pi$
$37$	282.000	1.25299	0.626493	+	0.779427i	$0.284490\pi$
$38$	0	0
$39$	258.000	1.05931
$40$	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
$43$	412.000	1.46115	0.730575	−	0.682833i	$-0.239252\pi$
$43$	412.000	1.46115	0.730575	+	0.682833i	$0.239252\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	40.0000	0.124140	0.0620702	−	0.998072i	$-0.480230\pi$
$47$	40.0000	0.124140	0.0620702	+	0.998072i	$0.480230\pi$
$48$	0	0
$49$	57.0000	0.166181
$50$	0	0
$51$	318.000	0.873116
$52$	0	0
$53$	−126.000	−0.326555	−0.163278	−	0.986580i	$-0.552207\pi$
$53$	−126.000	−0.326555	−0.163278	+	0.986580i	$0.552207\pi$
$54$	0	0
$55$	280.000	0.686458
$56$	0	0
$57$	−12.0000	−0.0278849
$58$	0	0
$59$	56.0000	0.123569	0.0617846	−	0.998090i	$-0.480321\pi$
$59$	56.0000	0.123569	0.0617846	+	0.998090i	$0.480321\pi$
$60$	0	0
$61$	−2.00000	−0.00419793	−0.00209897	−	0.999998i	$-0.500668\pi$
$61$	−2.00000	−0.00419793	−0.00209897	+	0.999998i	$0.500668\pi$
$62$	0	0
$63$	180.000	0.359966
$64$	0	0
$65$	430.000	0.820537
$66$	0	0
$67$	−388.000	−0.707489	−0.353744	−	0.935342i	$-0.615092\pi$
$67$	−388.000	−0.707489	−0.353744	+	0.935342i	$0.615092\pi$
$68$	0	0
$69$	−408.000	−0.711847
$70$	0	0
$71$	−672.000	−1.12326	−0.561632	−	0.827387i	$-0.689826\pi$
$71$	−672.000	−1.12326	−0.561632	+	0.827387i	$0.689826\pi$
$72$	0	0
$73$	1170.00	1.87586	0.937932	−	0.346818i	$-0.112738\pi$
$73$	1170.00	1.87586	0.937932	+	0.346818i	$0.112738\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−1120.00	−1.65761
$78$	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
$83$	668.000	0.883404	0.441702	−	0.897162i	$-0.354375\pi$
$83$	668.000	0.883404	0.441702	+	0.897162i	$0.354375\pi$
$84$	0	0
$85$	530.000	0.676313
$86$	0	0
$87$	618.000	0.761570
$88$	0	0
$89$	66.0000	0.0786066	0.0393033	−	0.999227i	$-0.487486\pi$
$89$	66.0000	0.0786066	0.0393033	+	0.999227i	$0.487486\pi$
$90$	0	0
$91$	−1720.00	−1.98137
$92$	0	0
$93$	456.000	0.508441
$94$	0	0
$95$	−20.0000	−0.0215995
$96$	0	0
$97$	−926.000	−0.969289	−0.484645	−	0.874711i	$-0.661051\pi$
$97$	−926.000	−0.969289	−0.484645	+	0.874711i	$0.661051\pi$
$98$	0	0
$99$	−504.000	−0.511656
$100$	0	0
$101$	−198.000	−0.195067	−0.0975333	−	0.995232i	$-0.531095\pi$
$101$	−198.000	−0.195067	−0.0975333	+	0.995232i	$0.531095\pi$
$102$	0	0
$103$	−1532.00	−1.46556	−0.732779	−	0.680467i	$-0.761777\pi$
$103$	−1532.00	−1.46556	−0.732779	+	0.680467i	$0.761777\pi$
$104$	0	0
$105$	300.000	0.278829
$106$	0	0
$107$	−444.000	−0.401150	−0.200575	−	0.979678i	$-0.564281\pi$
$107$	−444.000	−0.401150	−0.200575	+	0.979678i	$0.564281\pi$
$108$	0	0
$109$	62.0000	0.0544819	0.0272409	−	0.999629i	$-0.491328\pi$
$109$	62.0000	0.0544819	0.0272409	+	0.999629i	$0.491328\pi$
$110$	0	0
$111$	−846.000	−0.723412
$112$	0	0
$113$	414.000	0.344653	0.172327	−	0.985040i	$-0.444872\pi$
$113$	414.000	0.344653	0.172327	+	0.985040i	$0.444872\pi$
$114$	0	0
$115$	−680.000	−0.551394
$116$	0	0
$117$	−774.000	−0.611593
$118$	0	0
$119$	−2120.00	−1.63311
$120$	0	0
$121$	1805.00	1.35612
$122$	0	0
$123$	738.000	0.541002
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−996.000	−0.695911	−0.347956	−	0.937511i	$-0.613124\pi$
$127$	−996.000	−0.695911	−0.347956	+	0.937511i	$0.613124\pi$
$128$	0	0
$129$	−1236.00	−0.843595
$130$	0	0
$131$	−264.000	−0.176075	−0.0880374	−	0.996117i	$-0.528059\pi$
$131$	−264.000	−0.176075	−0.0880374	+	0.996117i	$0.528059\pi$
$132$	0	0
$133$	80.0000	0.0521570
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	2278.00	1.42060	0.710302	−	0.703897i	$-0.248558\pi$
$137$	2278.00	1.42060	0.710302	+	0.703897i	$0.248558\pi$
$138$	0	0
$139$	1812.00	1.10570	0.552848	−	0.833282i	$-0.313541\pi$
$139$	1812.00	1.10570	0.552848	+	0.833282i	$0.313541\pi$
$140$	0	0
$141$	−120.000	−0.0716725
$142$	0	0
$143$	4816.00	2.81632
$144$	0	0
$145$	1030.00	0.589909
$146$	0	0
$147$	−171.000	−0.0959445
$148$	0	0
$149$	−1534.00	−0.843424	−0.421712	−	0.906730i	$-0.638571\pi$
$149$	−1534.00	−0.843424	−0.421712	+	0.906730i	$0.638571\pi$
$150$	0	0
$151$	−3016.00	−1.62542	−0.812711	−	0.582668i	$-0.802009\pi$
$151$	−3016.00	−1.62542	−0.812711	+	0.582668i	$0.802009\pi$
$152$	0	0
$153$	−954.000	−0.504094
$154$	0	0
$155$	760.000	0.393837
$156$	0	0
$157$	−1814.00	−0.922121	−0.461060	−	0.887369i	$-0.652531\pi$
$157$	−1814.00	−0.922121	−0.461060	+	0.887369i	$0.652531\pi$
$158$	0	0
$159$	378.000	0.188537
$160$	0	0
$161$	2720.00	1.33147
$162$	0	0
$163$	−1844.00	−0.886093	−0.443047	−	0.896499i	$-0.646102\pi$
$163$	−1844.00	−0.886093	−0.443047	+	0.896499i	$0.646102\pi$
$164$	0	0
$165$	−840.000	−0.396327
$166$	0	0
$167$	3768.00	1.74597	0.872984	−	0.487749i	$-0.162182\pi$
$167$	3768.00	1.74597	0.872984	+	0.487749i	$0.162182\pi$
$168$	0	0
$169$	5199.00	2.36641
$170$	0	0
$171$	36.0000	0.0160993
$172$	0	0
$173$	938.000	0.412224	0.206112	−	0.978528i	$-0.433919\pi$
$173$	938.000	0.412224	0.206112	+	0.978528i	$0.433919\pi$
$174$	0	0
$175$	500.000	0.215980
$176$	0	0
$177$	−168.000	−0.0713427
$178$	0	0
$179$	3968.00	1.65688	0.828442	−	0.560075i	$-0.189228\pi$
$179$	3968.00	1.65688	0.828442	+	0.560075i	$0.189228\pi$
$180$	0	0
$181$	−3514.00	−1.44306	−0.721529	−	0.692384i	$-0.756560\pi$
$181$	−3514.00	−1.44306	−0.721529	+	0.692384i	$0.756560\pi$
$182$	0	0
$183$	6.00000	0.00242368
$184$	0	0
$185$	−1410.00	−0.560353
$186$	0	0
$187$	5936.00	2.32130
$188$	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
$193$	−2774.00	−1.03460	−0.517298	−	0.855806i	$-0.673062\pi$
$193$	−2774.00	−1.03460	−0.517298	+	0.855806i	$0.673062\pi$
$194$	0	0
$195$	−1290.00	−0.473738
$196$	0	0
$197$	−3806.00	−1.37648	−0.688239	−	0.725484i	$-0.741616\pi$
$197$	−3806.00	−1.37648	−0.688239	+	0.725484i	$0.741616\pi$
$198$	0	0
$199$	−856.000	−0.304926	−0.152463	−	0.988309i	$-0.548720\pi$
$199$	−856.000	−0.304926	−0.152463	+	0.988309i	$0.548720\pi$
$200$	0	0
$201$	1164.00	0.408469
$202$	0	0
$203$	−4120.00	−1.42447
$204$	0	0
$205$	1230.00	0.419058
$206$	0	0
$207$	1224.00	0.410985
$208$	0	0
$209$	−224.000	−0.0741359
$210$	0	0
$211$	3020.00	0.985334	0.492667	−	0.870218i	$-0.336022\pi$
$211$	3020.00	0.985334	0.492667	+	0.870218i	$0.336022\pi$
$212$	0	0
$213$	2016.00	0.648517
$214$	0	0
$215$	−2060.00	−0.653446
$216$	0	0
$217$	−3040.00	−0.951008
$218$	0	0
$219$	−3510.00	−1.08303
$220$	0	0
$221$	9116.00	2.77470
$222$	0	0
$223$	−1684.00	−0.505690	−0.252845	−	0.967507i	$-0.581366\pi$
$223$	−1684.00	−0.505690	−0.252845	+	0.967507i	$0.581366\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	2004.00	0.585948	0.292974	−	0.956120i	$-0.405355\pi$
$227$	2004.00	0.585948	0.292974	+	0.956120i	$0.405355\pi$
$228$	0	0
$229$	−5042.00	−1.45496	−0.727478	−	0.686131i	$-0.759307\pi$
$229$	−5042.00	−1.45496	−0.727478	+	0.686131i	$0.759307\pi$
$230$	0	0
$231$	3360.00	0.957021
$232$	0	0
$233$	−3090.00	−0.868810	−0.434405	−	0.900718i	$-0.643041\pi$
$233$	−3090.00	−0.868810	−0.434405	+	0.900718i	$0.643041\pi$
$234$	0	0
$235$	−200.000	−0.0555173
$236$	0	0
$237$	−1224.00	−0.335474
$238$	0	0
$239$	2136.00	0.578102	0.289051	−	0.957314i	$-0.406660\pi$
$239$	2136.00	0.578102	0.289051	+	0.957314i	$0.406660\pi$
$240$	0	0
$241$	98.0000	0.0261939	0.0130970	−	0.999914i	$-0.495831\pi$
$241$	98.0000	0.0261939	0.0130970	+	0.999914i	$0.495831\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−285.000	−0.0743183
$246$	0	0
$247$	−344.000	−0.0886162
$248$	0	0
$249$	−2004.00	−0.510033
$250$	0	0
$251$	−5040.00	−1.26742	−0.633709	−	0.773571i	$-0.718468\pi$
$251$	−5040.00	−1.26742	−0.633709	+	0.773571i	$0.718468\pi$
$252$	0	0
$253$	−7616.00	−1.89254
$254$	0	0
$255$	−1590.00	−0.390469
$256$	0	0
$257$	−1986.00	−0.482036	−0.241018	−	0.970521i	$-0.577481\pi$
$257$	−1986.00	−0.482036	−0.241018	+	0.970521i	$0.577481\pi$
$258$	0	0
$259$	5640.00	1.35310
$260$	0	0
$261$	−1854.00	−0.439692
$262$	0	0
$263$	1416.00	0.331994	0.165997	−	0.986126i	$-0.446916\pi$
$263$	1416.00	0.331994	0.165997	+	0.986126i	$0.446916\pi$
$264$	0	0
$265$	630.000	0.146040
$266$	0	0
$267$	−198.000	−0.0453835
$268$	0	0
$269$	−6670.00	−1.51181	−0.755905	−	0.654681i	$-0.772803\pi$
$269$	−6670.00	−1.51181	−0.755905	+	0.654681i	$0.772803\pi$
$270$	0	0
$271$	48.0000	0.0107594	0.00537969	−	0.999986i	$-0.498288\pi$
$271$	48.0000	0.0107594	0.00537969	+	0.999986i	$0.498288\pi$
$272$	0	0
$273$	5160.00	1.14395
$274$	0	0
$275$	−1400.00	−0.306993
$276$	0	0
$277$	6938.00	1.50492	0.752462	−	0.658636i	$-0.228866\pi$
$277$	6938.00	1.50492	0.752462	+	0.658636i	$0.228866\pi$
$278$	0	0
$279$	−1368.00	−0.293548
$280$	0	0
$281$	−1694.00	−0.359628	−0.179814	−	0.983701i	$-0.557550\pi$
$281$	−1694.00	−0.359628	−0.179814	+	0.983701i	$0.557550\pi$
$282$	0	0
$283$	−6364.00	−1.33675	−0.668376	−	0.743824i	$-0.733010\pi$
$283$	−6364.00	−1.33675	−0.668376	+	0.743824i	$0.733010\pi$
$284$	0	0
$285$	60.0000	0.0124705
$286$	0	0
$287$	−4920.00	−1.01191
$288$	0	0
$289$	6323.00	1.28699
$290$	0	0
$291$	2778.00	0.559619
$292$	0	0
$293$	−3134.00	−0.624881	−0.312441	−	0.949937i	$-0.601147\pi$
$293$	−3134.00	−0.624881	−0.312441	+	0.949937i	$0.601147\pi$
$294$	0	0
$295$	−280.000	−0.0552618
$296$	0	0
$297$	1512.00	0.295405
$298$	0	0
$299$	−11696.0	−2.26220
$300$	0	0
$301$	8240.00	1.57789
$302$	0	0
$303$	594.000	0.112622
$304$	0	0
$305$	10.0000	0.00187737
$306$	0	0
$307$	−236.000	−0.0438737	−0.0219369	−	0.999759i	$-0.506983\pi$
$307$	−236.000	−0.0438737	−0.0219369	+	0.999759i	$0.506983\pi$
$308$	0	0
$309$	4596.00	0.846140
$310$	0	0
$311$	3776.00	0.688480	0.344240	−	0.938882i	$-0.388137\pi$
$311$	3776.00	0.688480	0.344240	+	0.938882i	$0.388137\pi$
$312$	0	0
$313$	−7918.00	−1.42988	−0.714939	−	0.699187i	$-0.753546\pi$
$313$	−7918.00	−1.42988	−0.714939	+	0.699187i	$0.753546\pi$
$314$	0	0
$315$	−900.000	−0.160982
$316$	0	0
$317$	4362.00	0.772853	0.386426	−	0.922320i	$-0.373709\pi$
$317$	4362.00	0.772853	0.386426	+	0.922320i	$0.373709\pi$
$318$	0	0
$319$	11536.0	2.02474
$320$	0	0
$321$	1332.00	0.231604
$322$	0	0
$323$	−424.000	−0.0730402
$324$	0	0
$325$	−2150.00	−0.366956
$326$	0	0
$327$	−186.000	−0.0314551
$328$	0	0
$329$	800.000	0.134059
$330$	0	0
$331$	7980.00	1.32514	0.662569	−	0.749001i	$-0.269466\pi$
$331$	7980.00	1.32514	0.662569	+	0.749001i	$0.269466\pi$
$332$	0	0
$333$	2538.00	0.417662
$334$	0	0
$335$	1940.00	0.316399
$336$	0	0
$337$	−8294.00	−1.34066	−0.670331	−	0.742062i	$-0.733848\pi$
$337$	−8294.00	−1.34066	−0.670331	+	0.742062i	$0.733848\pi$
$338$	0	0
$339$	−1242.00	−0.198986
$340$	0	0
$341$	8512.00	1.35176
$342$	0	0
$343$	−5720.00	−0.900440
$344$	0	0
$345$	2040.00	0.318348
$346$	0	0
$347$	−964.000	−0.149136	−0.0745681	−	0.997216i	$-0.523758\pi$
$347$	−964.000	−0.149136	−0.0745681	+	0.997216i	$0.523758\pi$
$348$	0	0
$349$	8670.00	1.32978	0.664892	−	0.746940i	$-0.268478\pi$
$349$	8670.00	1.32978	0.664892	+	0.746940i	$0.268478\pi$
$350$	0	0
$351$	2322.00	0.353103
$352$	0	0
$353$	−2314.00	−0.348900	−0.174450	−	0.984666i	$-0.555815\pi$
$353$	−2314.00	−0.348900	−0.174450	+	0.984666i	$0.555815\pi$
$354$	0	0
$355$	3360.00	0.502339
$356$	0	0
$357$	6360.00	0.942876
$358$	0	0
$359$	−1896.00	−0.278738	−0.139369	−	0.990240i	$-0.544507\pi$
$359$	−1896.00	−0.278738	−0.139369	+	0.990240i	$0.544507\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	−5415.00	−0.782958
$364$	0	0
$365$	−5850.00	−0.838912
$366$	0	0
$367$	1484.00	0.211074	0.105537	−	0.994415i	$-0.466344\pi$
$367$	1484.00	0.211074	0.105537	+	0.994415i	$0.466344\pi$
$368$	0	0
$369$	−2214.00	−0.312348
$370$	0	0
$371$	−2520.00	−0.352647
$372$	0	0
$373$	12370.0	1.71714	0.858571	−	0.512694i	$-0.171352\pi$
$373$	12370.0	1.71714	0.858571	+	0.512694i	$0.171352\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	17716.0	2.42021
$378$	0	0
$379$	5620.00	0.761689	0.380844	−	0.924639i	$-0.375633\pi$
$379$	5620.00	0.761689	0.380844	+	0.924639i	$0.375633\pi$
$380$	0	0
$381$	2988.00	0.401784
$382$	0	0
$383$	5880.00	0.784475	0.392238	−	0.919864i	$-0.371701\pi$
$383$	5880.00	0.784475	0.392238	+	0.919864i	$0.371701\pi$
$384$	0	0
$385$	5600.00	0.741305
$386$	0	0
$387$	3708.00	0.487050
$388$	0	0
$389$	2082.00	0.271367	0.135683	−	0.990752i	$-0.456677\pi$
$389$	2082.00	0.271367	0.135683	+	0.990752i	$0.456677\pi$
$390$	0	0
$391$	−14416.0	−1.86457
$392$	0	0
$393$	792.000	0.101657
$394$	0	0
$395$	−2040.00	−0.259857
$396$	0	0
$397$	−1742.00	−0.220223	−0.110111	−	0.993919i	$-0.535121\pi$
$397$	−1742.00	−0.220223	−0.110111	+	0.993919i	$0.535121\pi$
$398$	0	0
$399$	−240.000	−0.0301129
$400$	0	0
$401$	−3270.00	−0.407222	−0.203611	−	0.979052i	$-0.565268\pi$
$401$	−3270.00	−0.407222	−0.203611	+	0.979052i	$0.565268\pi$
$402$	0	0
$403$	13072.0	1.61579
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−15792.0	−1.92329
$408$	0	0
$409$	−6134.00	−0.741581	−0.370791	−	0.928716i	$-0.620913\pi$
$409$	−6134.00	−0.741581	−0.370791	+	0.928716i	$0.620913\pi$
$410$	0	0
$411$	−6834.00	−0.820186
$412$	0	0
$413$	1120.00	0.133442
$414$	0	0
$415$	−3340.00	−0.395070
$416$	0	0
$417$	−5436.00	−0.638374
$418$	0	0
$419$	−10392.0	−1.21165	−0.605826	−	0.795597i	$-0.707157\pi$
$419$	−10392.0	−1.21165	−0.605826	+	0.795597i	$0.707157\pi$
$420$	0	0
$421$	−12690.0	−1.46906	−0.734528	−	0.678578i	$-0.762596\pi$
$421$	−12690.0	−1.46906	−0.734528	+	0.678578i	$0.762596\pi$
$422$	0	0
$423$	360.000	0.0413801
$424$	0	0
$425$	−2650.00	−0.302456
$426$	0	0
$427$	−40.0000	−0.00453334
$428$	0	0
$429$	−14448.0	−1.62600
$430$	0	0
$431$	−7408.00	−0.827914	−0.413957	−	0.910297i	$-0.635854\pi$
$431$	−7408.00	−0.827914	−0.413957	+	0.910297i	$0.635854\pi$
$432$	0	0
$433$	−5062.00	−0.561811	−0.280906	−	0.959735i	$-0.590635\pi$
$433$	−5062.00	−0.561811	−0.280906	+	0.959735i	$0.590635\pi$
$434$	0	0
$435$	−3090.00	−0.340584
$436$	0	0
$437$	544.000	0.0595493
$438$	0	0
$439$	−7160.00	−0.778424	−0.389212	−	0.921148i	$-0.627253\pi$
$439$	−7160.00	−0.778424	−0.389212	+	0.921148i	$0.627253\pi$
$440$	0	0
$441$	513.000	0.0553936
$442$	0	0
$443$	17100.0	1.83396	0.916981	−	0.398930i	$-0.130618\pi$
$443$	17100.0	1.83396	0.916981	+	0.398930i	$0.130618\pi$
$444$	0	0
$445$	−330.000	−0.0351539
$446$	0	0
$447$	4602.00	0.486951
$448$	0	0
$449$	8634.00	0.907491	0.453746	−	0.891131i	$-0.350087\pi$
$449$	8634.00	0.907491	0.453746	+	0.891131i	$0.350087\pi$
$450$	0	0
$451$	13776.0	1.43833
$452$	0	0
$453$	9048.00	0.938437
$454$	0	0
$455$	8600.00	0.886097
$456$	0	0
$457$	2986.00	0.305644	0.152822	−	0.988254i	$-0.451164\pi$
$457$	2986.00	0.305644	0.152822	+	0.988254i	$0.451164\pi$
$458$	0	0
$459$	2862.00	0.291039
$460$	0	0
$461$	−2406.00	−0.243077	−0.121539	−	0.992587i	$-0.538783\pi$
$461$	−2406.00	−0.243077	−0.121539	+	0.992587i	$0.538783\pi$
$462$	0	0
$463$	−14316.0	−1.43698	−0.718489	−	0.695538i	$-0.755166\pi$
$463$	−14316.0	−1.43698	−0.718489	+	0.695538i	$0.755166\pi$
$464$	0	0
$465$	−2280.00	−0.227382
$466$	0	0
$467$	−292.000	−0.0289339	−0.0144670	−	0.999895i	$-0.504605\pi$
$467$	−292.000	−0.0289339	−0.0144670	+	0.999895i	$0.504605\pi$
$468$	0	0
$469$	−7760.00	−0.764016
$470$	0	0
$471$	5442.00	0.532387
$472$	0	0
$473$	−23072.0	−2.24282
$474$	0	0
$475$	100.000	0.00965961
$476$	0	0
$477$	−1134.00	−0.108852
$478$	0	0
$479$	14056.0	1.34078	0.670391	−	0.742008i	$-0.266126\pi$
$479$	14056.0	1.34078	0.670391	+	0.742008i	$0.266126\pi$
$480$	0	0
$481$	−24252.0	−2.29895
$482$	0	0
$483$	−8160.00	−0.768722
$484$	0	0
$485$	4630.00	0.433479
$486$	0	0
$487$	−11204.0	−1.04251	−0.521254	−	0.853401i	$-0.674536\pi$
$487$	−11204.0	−1.04251	−0.521254	+	0.853401i	$0.674536\pi$
$488$	0	0
$489$	5532.00	0.511586
$490$	0	0
$491$	4608.00	0.423536	0.211768	−	0.977320i	$-0.432078\pi$
$491$	4608.00	0.423536	0.211768	+	0.977320i	$0.432078\pi$
$492$	0	0
$493$	21836.0	1.99482
$494$	0	0
$495$	2520.00	0.228819
$496$	0	0
$497$	−13440.0	−1.21301
$498$	0	0
$499$	2468.00	0.221409	0.110704	−	0.993853i	$-0.464689\pi$
$499$	2468.00	0.221409	0.110704	+	0.993853i	$0.464689\pi$
$500$	0	0
$501$	−11304.0	−1.00803
$502$	0	0
$503$	12192.0	1.08074	0.540372	−	0.841426i	$-0.318283\pi$
$503$	12192.0	1.08074	0.540372	+	0.841426i	$0.318283\pi$
$504$	0	0
$505$	990.000	0.0872365
$506$	0	0
$507$	−15597.0	−1.36625
$508$	0	0
$509$	1714.00	0.149257	0.0746284	−	0.997211i	$-0.476223\pi$
$509$	1714.00	0.149257	0.0746284	+	0.997211i	$0.476223\pi$
$510$	0	0
$511$	23400.0	2.02574
$512$	0	0
$513$	−108.000	−0.00929496
$514$	0	0
$515$	7660.00	0.655417
$516$	0	0
$517$	−2240.00	−0.190551
$518$	0	0
$519$	−2814.00	−0.237998
$520$	0	0
$521$	−18014.0	−1.51479	−0.757397	−	0.652955i	$-0.773529\pi$
$521$	−18014.0	−1.51479	−0.757397	+	0.652955i	$0.773529\pi$
$522$	0	0
$523$	−16748.0	−1.40027	−0.700133	−	0.714013i	$-0.746876\pi$
$523$	−16748.0	−1.40027	−0.700133	+	0.714013i	$0.746876\pi$
$524$	0	0
$525$	−1500.00	−0.124696
$526$	0	0
$527$	16112.0	1.33178
$528$	0	0
$529$	6329.00	0.520178
$530$	0	0
$531$	504.000	0.0411897
$532$	0	0
$533$	21156.0	1.71926
$534$	0	0
$535$	2220.00	0.179400
$536$	0	0
$537$	−11904.0	−0.956602
$538$	0	0
$539$	−3192.00	−0.255082
$540$	0	0
$541$	−14018.0	−1.11401	−0.557006	−	0.830508i	$-0.688050\pi$
$541$	−14018.0	−1.11401	−0.557006	+	0.830508i	$0.688050\pi$
$542$	0	0
$543$	10542.0	0.833150
$544$	0	0
$545$	−310.000	−0.0243650
$546$	0	0
$547$	−412.000	−0.0322045	−0.0161022	−	0.999870i	$-0.505126\pi$
$547$	−412.000	−0.0322045	−0.0161022	+	0.999870i	$0.505126\pi$
$548$	0	0
$549$	−18.0000	−0.00139931
$550$	0	0
$551$	−824.000	−0.0637089
$552$	0	0
$553$	8160.00	0.627484
$554$	0	0
$555$	4230.00	0.323520
$556$	0	0
$557$	18218.0	1.38586	0.692928	−	0.721007i	$-0.256321\pi$
$557$	18218.0	1.38586	0.692928	+	0.721007i	$0.256321\pi$
$558$	0	0
$559$	−35432.0	−2.68088
$560$	0	0
$561$	−17808.0	−1.34020
$562$	0	0
$563$	23524.0	1.76096	0.880478	−	0.474087i	$-0.157222\pi$
$563$	23524.0	1.76096	0.880478	+	0.474087i	$0.157222\pi$
$564$	0	0
$565$	−2070.00	−0.154134
$566$	0	0
$567$	1620.00	0.119989
$568$	0	0
$569$	23330.0	1.71888	0.859442	−	0.511234i	$-0.170811\pi$
$569$	23330.0	1.71888	0.859442	+	0.511234i	$0.170811\pi$
$570$	0	0
$571$	−13124.0	−0.961860	−0.480930	−	0.876759i	$-0.659701\pi$
$571$	−13124.0	−0.961860	−0.480930	+	0.876759i	$0.659701\pi$
$572$	0	0
$573$	4440.00	0.323706
$574$	0	0
$575$	3400.00	0.246591
$576$	0	0
$577$	11714.0	0.845165	0.422582	−	0.906324i	$-0.361124\pi$
$577$	11714.0	0.845165	0.422582	+	0.906324i	$0.361124\pi$
$578$	0	0
$579$	8322.00	0.597324
$580$	0	0
$581$	13360.0	0.953987
$582$	0	0
$583$	7056.00	0.501252
$584$	0	0
$585$	3870.00	0.273512
$586$	0	0
$587$	−17628.0	−1.23950	−0.619749	−	0.784800i	$-0.712766\pi$
$587$	−17628.0	−1.23950	−0.619749	+	0.784800i	$0.712766\pi$
$588$	0	0
$589$	−608.000	−0.0425335
$590$	0	0
$591$	11418.0	0.794710
$592$	0	0
$593$	−2802.00	−0.194038	−0.0970188	−	0.995283i	$-0.530931\pi$
$593$	−2802.00	−0.194038	−0.0970188	+	0.995283i	$0.530931\pi$
$594$	0	0
$595$	10600.0	0.730349
$596$	0	0
$597$	2568.00	0.176049
$598$	0	0
$599$	−2664.00	−0.181716	−0.0908582	−	0.995864i	$-0.528961\pi$
$599$	−2664.00	−0.181716	−0.0908582	+	0.995864i	$0.528961\pi$
$600$	0	0
$601$	23962.0	1.62634	0.813170	−	0.582026i	$-0.197740\pi$
$601$	23962.0	1.62634	0.813170	+	0.582026i	$0.197740\pi$
$602$	0	0
$603$	−3492.00	−0.235830
$604$	0	0
$605$	−9025.00	−0.606477
$606$	0	0
$607$	11940.0	0.798401	0.399201	−	0.916864i	$-0.369288\pi$
$607$	11940.0	0.798401	0.399201	+	0.916864i	$0.369288\pi$
$608$	0	0
$609$	12360.0	0.822418
$610$	0	0
$611$	−3440.00	−0.227770
$612$	0	0
$613$	16794.0	1.10653	0.553265	−	0.833005i	$-0.313382\pi$
$613$	16794.0	1.10653	0.553265	+	0.833005i	$0.313382\pi$
$614$	0	0
$615$	−3690.00	−0.241943
$616$	0	0
$617$	−20706.0	−1.35104	−0.675520	−	0.737341i	$-0.736081\pi$
$617$	−20706.0	−1.35104	−0.675520	+	0.737341i	$0.736081\pi$
$618$	0	0
$619$	10724.0	0.696339	0.348170	−	0.937432i	$-0.386803\pi$
$619$	10724.0	0.696339	0.348170	+	0.937432i	$0.386803\pi$
$620$	0	0
$621$	−3672.00	−0.237282
$622$	0	0
$623$	1320.00	0.0848871
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	672.000	0.0428024
$628$	0	0
$629$	−29892.0	−1.89487
$630$	0	0
$631$	−5744.00	−0.362385	−0.181193	−	0.983448i	$-0.557996\pi$
$631$	−5744.00	−0.362385	−0.181193	+	0.983448i	$0.557996\pi$
$632$	0	0
$633$	−9060.00	−0.568883
$634$	0	0
$635$	4980.00	0.311221
$636$	0	0
$637$	−4902.00	−0.304905
$638$	0	0
$639$	−6048.00	−0.374421
$640$	0	0
$641$	27906.0	1.71953	0.859767	−	0.510687i	$-0.170609\pi$
$641$	27906.0	1.71953	0.859767	+	0.510687i	$0.170609\pi$
$642$	0	0
$643$	20556.0	1.26073	0.630365	−	0.776299i	$-0.282905\pi$
$643$	20556.0	1.26073	0.630365	+	0.776299i	$0.282905\pi$
$644$	0	0
$645$	6180.00	0.377267
$646$	0	0
$647$	−10224.0	−0.621247	−0.310624	−	0.950533i	$-0.600538\pi$
$647$	−10224.0	−0.621247	−0.310624	+	0.950533i	$0.600538\pi$
$648$	0	0
$649$	−3136.00	−0.189675
$650$	0	0
$651$	9120.00	0.549064
$652$	0	0
$653$	−12982.0	−0.777986	−0.388993	−	0.921241i	$-0.627177\pi$
$653$	−12982.0	−0.777986	−0.388993	+	0.921241i	$0.627177\pi$
$654$	0	0
$655$	1320.00	0.0787430
$656$	0	0
$657$	10530.0	0.625288
$658$	0	0
$659$	−1512.00	−0.0893766	−0.0446883	−	0.999001i	$-0.514229\pi$
$659$	−1512.00	−0.0893766	−0.0446883	+	0.999001i	$0.514229\pi$
$660$	0	0
$661$	16710.0	0.983273	0.491637	−	0.870800i	$-0.336399\pi$
$661$	16710.0	0.983273	0.491637	+	0.870800i	$0.336399\pi$
$662$	0	0
$663$	−27348.0	−1.60197
$664$	0	0
$665$	−400.000	−0.0233253
$666$	0	0
$667$	−28016.0	−1.62636
$668$	0	0
$669$	5052.00	0.291961
$670$	0	0
$671$	112.000	0.00644368
$672$	0	0
$673$	7962.00	0.456036	0.228018	−	0.973657i	$-0.426775\pi$
$673$	7962.00	0.456036	0.228018	+	0.973657i	$0.426775\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	12226.0	0.694067	0.347033	−	0.937853i	$-0.387189\pi$
$677$	12226.0	0.694067	0.347033	+	0.937853i	$0.387189\pi$
$678$	0	0
$679$	−18520.0	−1.04673
$680$	0	0
$681$	−6012.00	−0.338297
$682$	0	0
$683$	−8748.00	−0.490092	−0.245046	−	0.969511i	$-0.578803\pi$
$683$	−8748.00	−0.490092	−0.245046	+	0.969511i	$0.578803\pi$
$684$	0	0
$685$	−11390.0	−0.635313
$686$	0	0
$687$	15126.0	0.840019
$688$	0	0
$689$	10836.0	0.599156
$690$	0	0
$691$	−7324.00	−0.403210	−0.201605	−	0.979467i	$-0.564616\pi$
$691$	−7324.00	−0.403210	−0.201605	+	0.979467i	$0.564616\pi$
$692$	0	0
$693$	−10080.0	−0.552536
$694$	0	0
$695$	−9060.00	−0.494483
$696$	0	0
$697$	26076.0	1.41707
$698$	0	0
$699$	9270.00	0.501607
$700$	0	0
$701$	−21934.0	−1.18179	−0.590896	−	0.806748i	$-0.701226\pi$
$701$	−21934.0	−1.18179	−0.590896	+	0.806748i	$0.701226\pi$
$702$	0	0
$703$	1128.00	0.0605168
$704$	0	0
$705$	600.000	0.0320529
$706$	0	0
$707$	−3960.00	−0.210652
$708$	0	0
$709$	−10690.0	−0.566250	−0.283125	−	0.959083i	$-0.591371\pi$
$709$	−10690.0	−0.566250	−0.283125	+	0.959083i	$0.591371\pi$
$710$	0	0
$711$	3672.00	0.193686
$712$	0	0
$713$	−20672.0	−1.08580
$714$	0	0
$715$	−24080.0	−1.25950
$716$	0	0
$717$	−6408.00	−0.333767
$718$	0	0
$719$	13792.0	0.715375	0.357688	−	0.933841i	$-0.383565\pi$
$719$	13792.0	0.715375	0.357688	+	0.933841i	$0.383565\pi$
$720$	0	0
$721$	−30640.0	−1.58265
$722$	0	0
$723$	−294.000	−0.0151231
$724$	0	0
$725$	−5150.00	−0.263815
$726$	0	0
$727$	−24004.0	−1.22457	−0.612283	−	0.790639i	$-0.709749\pi$
$727$	−24004.0	−1.22457	−0.612283	+	0.790639i	$0.709749\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−43672.0	−2.20967
$732$	0	0
$733$	8562.00	0.431439	0.215719	−	0.976455i	$-0.430790\pi$
$733$	8562.00	0.431439	0.215719	+	0.976455i	$0.430790\pi$
$734$	0	0
$735$	855.000	0.0429077
$736$	0	0
$737$	21728.0	1.08597
$738$	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
$741$	1032.00	0.0511626
$742$	0	0
$743$	−22224.0	−1.09733	−0.548667	−	0.836041i	$-0.684865\pi$
$743$	−22224.0	−1.09733	−0.548667	+	0.836041i	$0.684865\pi$
$744$	0	0
$745$	7670.00	0.377191
$746$	0	0
$747$	6012.00	0.294468
$748$	0	0
$749$	−8880.00	−0.433202
$750$	0	0
$751$	11544.0	0.560914	0.280457	−	0.959867i	$-0.409514\pi$
$751$	11544.0	0.560914	0.280457	+	0.959867i	$0.409514\pi$
$752$	0	0
$753$	15120.0	0.731744
$754$	0	0
$755$	15080.0	0.726910
$756$	0	0
$757$	−3814.00	−0.183120	−0.0915602	−	0.995800i	$-0.529185\pi$
$757$	−3814.00	−0.183120	−0.0915602	+	0.995800i	$0.529185\pi$
$758$	0	0
$759$	22848.0	1.09266
$760$	0	0
$761$	−25662.0	−1.22240	−0.611200	−	0.791476i	$-0.709313\pi$
$761$	−25662.0	−1.22240	−0.611200	+	0.791476i	$0.709313\pi$
$762$	0	0
$763$	1240.00	0.0588349
$764$	0	0
$765$	4770.00	0.225438
$766$	0	0
$767$	−4816.00	−0.226722
$768$	0	0
$769$	30658.0	1.43765	0.718827	−	0.695189i	$-0.244679\pi$
$769$	30658.0	1.43765	0.718827	+	0.695189i	$0.244679\pi$
$770$	0	0
$771$	5958.00	0.278304
$772$	0	0
$773$	−30894.0	−1.43749	−0.718745	−	0.695274i	$-0.755283\pi$
$773$	−30894.0	−1.43749	−0.718745	+	0.695274i	$0.755283\pi$
$774$	0	0
$775$	−3800.00	−0.176129
$776$	0	0
$777$	−16920.0	−0.781212
$778$	0	0
$779$	−984.000	−0.0452573
$780$	0	0
$781$	37632.0	1.72417
$782$	0	0
$783$	5562.00	0.253857
$784$	0	0
$785$	9070.00	0.412385
$786$	0	0
$787$	21596.0	0.978163	0.489081	−	0.872238i	$-0.337332\pi$
$787$	21596.0	0.978163	0.489081	+	0.872238i	$0.337332\pi$
$788$	0	0
$789$	−4248.00	−0.191677
$790$	0	0
$791$	8280.00	0.372191
$792$	0	0
$793$	172.000	0.00770227
$794$	0	0
$795$	−1890.00	−0.0843162
$796$	0	0
$797$	−8646.00	−0.384262	−0.192131	−	0.981369i	$-0.561540\pi$
$797$	−8646.00	−0.384262	−0.192131	+	0.981369i	$0.561540\pi$
$798$	0	0
$799$	−4240.00	−0.187735
$800$	0	0
$801$	594.000	0.0262022
$802$	0	0
$803$	−65520.0	−2.87939
$804$	0	0
$805$	−13600.0	−0.595450
$806$	0	0
$807$	20010.0	0.872844
$808$	0	0
$809$	24954.0	1.08447	0.542235	−	0.840227i	$-0.317578\pi$
$809$	24954.0	1.08447	0.542235	+	0.840227i	$0.317578\pi$
$810$	0	0
$811$	40004.0	1.73210	0.866048	−	0.499960i	$-0.166652\pi$
$811$	40004.0	1.73210	0.866048	+	0.499960i	$0.166652\pi$
$812$	0	0
$813$	−144.000	−0.00621193
$814$	0	0
$815$	9220.00	0.396273
$816$	0	0
$817$	1648.00	0.0705707
$818$	0	0
$819$	−15480.0	−0.660458
$820$	0	0
$821$	16570.0	0.704381	0.352191	−	0.935928i	$-0.385437\pi$
$821$	16570.0	0.704381	0.352191	+	0.935928i	$0.385437\pi$
$822$	0	0
$823$	−4388.00	−0.185852	−0.0929259	−	0.995673i	$-0.529622\pi$
$823$	−4388.00	−0.185852	−0.0929259	+	0.995673i	$0.529622\pi$
$824$	0	0
$825$	4200.00	0.177243
$826$	0	0
$827$	14364.0	0.603972	0.301986	−	0.953312i	$-0.402350\pi$
$827$	14364.0	0.603972	0.301986	+	0.953312i	$0.402350\pi$
$828$	0	0
$829$	−21170.0	−0.886929	−0.443465	−	0.896292i	$-0.646251\pi$
$829$	−21170.0	−0.886929	−0.443465	+	0.896292i	$0.646251\pi$
$830$	0	0
$831$	−20814.0	−0.868868
$832$	0	0
$833$	−6042.00	−0.251312
$834$	0	0
$835$	−18840.0	−0.780820
$836$	0	0
$837$	4104.00	0.169480
$838$	0	0
$839$	10664.0	0.438811	0.219405	−	0.975634i	$-0.429588\pi$
$839$	10664.0	0.438811	0.219405	+	0.975634i	$0.429588\pi$
$840$	0	0
$841$	18047.0	0.739965
$842$	0	0
$843$	5082.00	0.207632
$844$	0	0
$845$	−25995.0	−1.05829
$846$	0	0
$847$	36100.0	1.46448
$848$	0	0
$849$	19092.0	0.771774
$850$	0	0
$851$	38352.0	1.54488
$852$	0	0
$853$	−3190.00	−0.128046	−0.0640232	−	0.997948i	$-0.520393\pi$
$853$	−3190.00	−0.128046	−0.0640232	+	0.997948i	$0.520393\pi$
$854$	0	0
$855$	−180.000	−0.00719985
$856$	0	0
$857$	20814.0	0.829630	0.414815	−	0.909906i	$-0.363846\pi$
$857$	20814.0	0.829630	0.414815	+	0.909906i	$0.363846\pi$
$858$	0	0
$859$	−18988.0	−0.754205	−0.377103	−	0.926172i	$-0.623080\pi$
$859$	−18988.0	−0.754205	−0.377103	+	0.926172i	$0.623080\pi$
$860$	0	0
$861$	14760.0	0.584227
$862$	0	0
$863$	11664.0	0.460078	0.230039	−	0.973181i	$-0.426115\pi$
$863$	11664.0	0.460078	0.230039	+	0.973181i	$0.426115\pi$
$864$	0	0
$865$	−4690.00	−0.184352
$866$	0	0
$867$	−18969.0	−0.743046
$868$	0	0
$869$	−22848.0	−0.891905
$870$	0	0
$871$	33368.0	1.29808
$872$	0	0
$873$	−8334.00	−0.323096
$874$	0	0
$875$	−2500.00	−0.0965891
$876$	0	0
$877$	−8246.00	−0.317500	−0.158750	−	0.987319i	$-0.550746\pi$
$877$	−8246.00	−0.317500	−0.158750	+	0.987319i	$0.550746\pi$
$878$	0	0
$879$	9402.00	0.360775
$880$	0	0
$881$	22890.0	0.875350	0.437675	−	0.899133i	$-0.355802\pi$
$881$	22890.0	0.875350	0.437675	+	0.899133i	$0.355802\pi$
$882$	0	0
$883$	33548.0	1.27857	0.639287	−	0.768969i	$-0.279230\pi$
$883$	33548.0	1.27857	0.639287	+	0.768969i	$0.279230\pi$
$884$	0	0
$885$	840.000	0.0319054
$886$	0	0
$887$	−32264.0	−1.22133	−0.610665	−	0.791889i	$-0.709098\pi$
$887$	−32264.0	−1.22133	−0.610665	+	0.791889i	$0.709098\pi$
$888$	0	0
$889$	−19920.0	−0.751513
$890$	0	0
$891$	−4536.00	−0.170552
$892$	0	0
$893$	160.000	0.00599574
$894$	0	0
$895$	−19840.0	−0.740981
$896$	0	0
$897$	35088.0	1.30608
$898$	0	0
$899$	31312.0	1.16164
$900$	0	0
$901$	13356.0	0.493843
$902$	0	0
$903$	−24720.0	−0.910997
$904$	0	0
$905$	17570.0	0.645355
$906$	0	0
$907$	51228.0	1.87541	0.937706	−	0.347431i	$-0.112946\pi$
$907$	51228.0	1.87541	0.937706	+	0.347431i	$0.112946\pi$
$908$	0	0
$909$	−1782.00	−0.0650222
$910$	0	0
$911$	−2144.00	−0.0779735	−0.0389868	−	0.999240i	$-0.512413\pi$
$911$	−2144.00	−0.0779735	−0.0389868	+	0.999240i	$0.512413\pi$
$912$	0	0
$913$	−37408.0	−1.35600
$914$	0	0
$915$	−30.0000	−0.00108390
$916$	0	0
$917$	−5280.00	−0.190143
$918$	0	0
$919$	33584.0	1.20548	0.602739	−	0.797939i	$-0.294076\pi$
$919$	33584.0	1.20548	0.602739	+	0.797939i	$0.294076\pi$
$920$	0	0
$921$	708.000	0.0253305
$922$	0	0
$923$	57792.0	2.06094
$924$	0	0
$925$	7050.00	0.250597
$926$	0	0
$927$	−13788.0	−0.488519
$928$	0	0
$929$	−3590.00	−0.126786	−0.0633929	−	0.997989i	$-0.520192\pi$
$929$	−3590.00	−0.126786	−0.0633929	+	0.997989i	$0.520192\pi$
$930$	0	0
$931$	228.000	0.00802621
$932$	0	0
$933$	−11328.0	−0.397494
$934$	0	0
$935$	−29680.0	−1.03812
$936$	0	0
$937$	−21686.0	−0.756084	−0.378042	−	0.925788i	$-0.623403\pi$
$937$	−21686.0	−0.756084	−0.378042	+	0.925788i	$0.623403\pi$
$938$	0	0
$939$	23754.0	0.825540
$940$	0	0
$941$	−5174.00	−0.179243	−0.0896215	−	0.995976i	$-0.528566\pi$
$941$	−5174.00	−0.179243	−0.0896215	+	0.995976i	$0.528566\pi$
$942$	0	0
$943$	−33456.0	−1.15533
$944$	0	0
$945$	2700.00	0.0929429
$946$	0	0
$947$	35524.0	1.21898	0.609490	−	0.792793i	$-0.291374\pi$
$947$	35524.0	1.21898	0.609490	+	0.792793i	$0.291374\pi$
$948$	0	0
$949$	−100620.	−3.44179
$950$	0	0
$951$	−13086.0	−0.446207
$952$	0	0
$953$	−16122.0	−0.547999	−0.273999	−	0.961730i	$-0.588347\pi$
$953$	−16122.0	−0.547999	−0.273999	+	0.961730i	$0.588347\pi$
$954$	0	0
$955$	7400.00	0.250742
$956$	0	0
$957$	−34608.0	−1.16898
$958$	0	0
$959$	45560.0	1.53411
$960$	0	0
$961$	−6687.00	−0.224464
$962$	0	0
$963$	−3996.00	−0.133717
$964$	0	0
$965$	13870.0	0.462685
$966$	0	0
$967$	19188.0	0.638102	0.319051	−	0.947738i	$-0.396636\pi$
$967$	19188.0	0.638102	0.319051	+	0.947738i	$0.396636\pi$
$968$	0	0
$969$	1272.00	0.0421698
$970$	0	0
$971$	−38464.0	−1.27123	−0.635617	−	0.772004i	$-0.719254\pi$
$971$	−38464.0	−1.27123	−0.635617	+	0.772004i	$0.719254\pi$
$972$	0	0
$973$	36240.0	1.19404
$974$	0	0
$975$	6450.00	0.211862
$976$	0	0
$977$	−43930.0	−1.43853	−0.719266	−	0.694735i	$-0.755522\pi$
$977$	−43930.0	−1.43853	−0.719266	+	0.694735i	$0.755522\pi$
$978$	0	0
$979$	−3696.00	−0.120659
$980$	0	0
$981$	558.000	0.0181606
$982$	0	0
$983$	17328.0	0.562235	0.281118	−	0.959673i	$-0.409295\pi$
$983$	17328.0	0.562235	0.281118	+	0.959673i	$0.409295\pi$
$984$	0	0
$985$	19030.0	0.615580
$986$	0	0
$987$	−2400.00	−0.0773990
$988$	0	0
$989$	56032.0	1.80153
$990$	0	0
$991$	−18160.0	−0.582110	−0.291055	−	0.956706i	$-0.594006\pi$
$991$	−18160.0	−0.582110	−0.291055	+	0.956706i	$0.594006\pi$
$992$	0	0
$993$	−23940.0	−0.765068
$994$	0	0
$995$	4280.00	0.136367
$996$	0	0
$997$	−9102.00	−0.289131	−0.144565	−	0.989495i	$-0.546178\pi$
$997$	−9102.00	−0.289131	−0.144565	+	0.989495i	$0.546178\pi$
$998$	0	0
$999$	−7614.00	−0.241137

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.b.1.1	✓	1	1.1	even	1	trivial
240.4.a.g.1.1		1	4.3	odd	2
360.4.a.n.1.1		1	3.2	odd	2
600.4.a.i.1.1		1	5.4	even	2
600.4.f.a.49.1		2	5.3	odd	4
600.4.f.a.49.2		2	5.2	odd	4
720.4.a.q.1.1		1	12.11	even	2
960.4.a.k.1.1		1	8.3	odd	2
960.4.a.bj.1.1		1	8.5	even	2
1200.4.a.p.1.1		1	20.19	odd	2
1200.4.f.t.49.1		2	20.7	even	4
1200.4.f.t.49.2		2	20.3	even	4
1800.4.a.f.1.1		1	15.14	odd	2
1800.4.f.v.649.1		2	15.8	even	4
1800.4.f.v.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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