LMFDB - Embedded newform 120.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,4,Mod(1,120)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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)

$f(q)$

$=$

$q+3.00000 q^{3} +5.00000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +5.00000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +20.0000 q^{11} +22.0000 q^{13} +15.0000 q^{15} -14.0000 q^{17} +76.0000 q^{19} +24.0000 q^{21} +56.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -154.000 q^{29} +160.000 q^{31} +60.0000 q^{33} +40.0000 q^{35} -162.000 q^{37} +66.0000 q^{39} -390.000 q^{41} +388.000 q^{43} +45.0000 q^{45} -544.000 q^{47} -279.000 q^{49} -42.0000 q^{51} -210.000 q^{53} +100.000 q^{55} +228.000 q^{57} -380.000 q^{59} -794.000 q^{61} +72.0000 q^{63} +110.000 q^{65} -148.000 q^{67} +168.000 q^{69} -840.000 q^{71} +858.000 q^{73} +75.0000 q^{75} +160.000 q^{77} +144.000 q^{79} +81.0000 q^{81} +316.000 q^{83} -70.0000 q^{85} -462.000 q^{87} +1098.00 q^{89} +176.000 q^{91} +480.000 q^{93} +380.000 q^{95} +994.000 q^{97} +180.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	20.0000	0.548202	0.274101	−	0.961701i	$-0.411620\pi$
$11$	20.0000	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$12$	0	0
$13$	22.0000	0.469362	0.234681	−	0.972072i	$-0.424595\pi$
$13$	22.0000	0.469362	0.234681	+	0.972072i	$0.424595\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−14.0000	−0.199735	−0.0998676	−	0.995001i	$-0.531842\pi$
$17$	−14.0000	−0.199735	−0.0998676	+	0.995001i	$0.531842\pi$
$18$	0	0
$19$	76.0000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	76.0000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	24.0000	0.249392
$22$	0	0
$23$	56.0000	0.507687	0.253844	−	0.967245i	$-0.418305\pi$
$23$	56.0000	0.507687	0.253844	+	0.967245i	$0.418305\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−154.000	−0.986106	−0.493053	−	0.869999i	$-0.664119\pi$
$29$	−154.000	−0.986106	−0.493053	+	0.869999i	$0.664119\pi$
$30$	0	0
$31$	160.000	0.926995	0.463498	−	0.886098i	$-0.346594\pi$
$31$	160.000	0.926995	0.463498	+	0.886098i	$0.346594\pi$
$32$	0	0
$33$	60.0000	0.316505
$34$	0	0
$35$	40.0000	0.193178
$36$	0	0
$37$	−162.000	−0.719801	−0.359900	−	0.932991i	$-0.617189\pi$
$37$	−162.000	−0.719801	−0.359900	+	0.932991i	$0.617189\pi$
$38$	0	0
$39$	66.0000	0.270986
$40$	0	0
$41$	−390.000	−1.48556	−0.742778	−	0.669538i	$-0.766492\pi$
$41$	−390.000	−1.48556	−0.742778	+	0.669538i	$0.766492\pi$
$42$	0	0
$43$	388.000	1.37603	0.688017	−	0.725695i	$-0.258482\pi$
$43$	388.000	1.37603	0.688017	+	0.725695i	$0.258482\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−544.000	−1.68831	−0.844155	−	0.536099i	$-0.819897\pi$
$47$	−544.000	−1.68831	−0.844155	+	0.536099i	$0.819897\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	−42.0000	−0.115317
$52$	0	0
$53$	−210.000	−0.544259	−0.272129	−	0.962261i	$-0.587728\pi$
$53$	−210.000	−0.544259	−0.272129	+	0.962261i	$0.587728\pi$
$54$	0	0
$55$	100.000	0.245164
$56$	0	0
$57$	228.000	0.529813
$58$	0	0
$59$	−380.000	−0.838505	−0.419252	−	0.907870i	$-0.637708\pi$
$59$	−380.000	−0.838505	−0.419252	+	0.907870i	$0.637708\pi$
$60$	0	0
$61$	−794.000	−1.66658	−0.833289	−	0.552837i	$-0.813545\pi$
$61$	−794.000	−1.66658	−0.833289	+	0.552837i	$0.813545\pi$
$62$	0	0
$63$	72.0000	0.143986
$64$	0	0
$65$	110.000	0.209905
$66$	0	0
$67$	−148.000	−0.269867	−0.134933	−	0.990855i	$-0.543082\pi$
$67$	−148.000	−0.269867	−0.134933	+	0.990855i	$0.543082\pi$
$68$	0	0
$69$	168.000	0.293113
$70$	0	0
$71$	−840.000	−1.40408	−0.702040	−	0.712138i	$-0.747727\pi$
$71$	−840.000	−1.40408	−0.702040	+	0.712138i	$0.747727\pi$
$72$	0	0
$73$	858.000	1.37563	0.687817	−	0.725884i	$-0.258569\pi$
$73$	858.000	1.37563	0.687817	+	0.725884i	$0.258569\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	160.000	0.236801
$78$	0	0
$79$	144.000	0.205079	0.102540	−	0.994729i	$-0.467303\pi$
$79$	144.000	0.205079	0.102540	+	0.994729i	$0.467303\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	316.000	0.417898	0.208949	−	0.977927i	$-0.432996\pi$
$83$	316.000	0.417898	0.208949	+	0.977927i	$0.432996\pi$
$84$	0	0
$85$	−70.0000	−0.0893243
$86$	0	0
$87$	−462.000	−0.569329
$88$	0	0
$89$	1098.00	1.30773	0.653864	−	0.756612i	$-0.273147\pi$
$89$	1098.00	1.30773	0.653864	+	0.756612i	$0.273147\pi$
$90$	0	0
$91$	176.000	0.202745
$92$	0	0
$93$	480.000	0.535201
$94$	0	0
$95$	380.000	0.410391
$96$	0	0
$97$	994.000	1.04047	0.520234	−	0.854024i	$-0.325845\pi$
$97$	994.000	1.04047	0.520234	+	0.854024i	$0.325845\pi$
$98$	0	0
$99$	180.000	0.182734
$100$	0	0
$101$	−834.000	−0.821645	−0.410822	−	0.911715i	$-0.634758\pi$
$101$	−834.000	−0.821645	−0.410822	+	0.911715i	$0.634758\pi$
$102$	0	0
$103$	1672.00	1.59949	0.799743	−	0.600343i	$-0.204969\pi$
$103$	1672.00	1.59949	0.799743	+	0.600343i	$0.204969\pi$
$104$	0	0
$105$	120.000	0.111531
$106$	0	0
$107$	−732.000	−0.661356	−0.330678	−	0.943744i	$-0.607277\pi$
$107$	−732.000	−0.661356	−0.330678	+	0.943744i	$0.607277\pi$
$108$	0	0
$109$	−970.000	−0.852378	−0.426189	−	0.904634i	$-0.640144\pi$
$109$	−970.000	−0.852378	−0.426189	+	0.904634i	$0.640144\pi$
$110$	0	0
$111$	−486.000	−0.415577
$112$	0	0
$113$	1938.00	1.61338	0.806689	−	0.590976i	$-0.201257\pi$
$113$	1938.00	1.61338	0.806689	+	0.590976i	$0.201257\pi$
$114$	0	0
$115$	280.000	0.227045
$116$	0	0
$117$	198.000	0.156454
$118$	0	0
$119$	−112.000	−0.0862775
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	−1170.00	−0.857686
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−528.000	−0.368917	−0.184458	−	0.982840i	$-0.559053\pi$
$127$	−528.000	−0.368917	−0.184458	+	0.982840i	$0.559053\pi$
$128$	0	0
$129$	1164.00	0.794453
$130$	0	0
$131$	636.000	0.424180	0.212090	−	0.977250i	$-0.431973\pi$
$131$	636.000	0.424180	0.212090	+	0.977250i	$0.431973\pi$
$132$	0	0
$133$	608.000	0.396393
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	1754.00	1.09383	0.546914	−	0.837189i	$-0.315803\pi$
$137$	1754.00	1.09383	0.546914	+	0.837189i	$0.315803\pi$
$138$	0	0
$139$	−2508.00	−1.53040	−0.765201	−	0.643792i	$-0.777360\pi$
$139$	−2508.00	−1.53040	−0.765201	+	0.643792i	$0.777360\pi$
$140$	0	0
$141$	−1632.00	−0.974746
$142$	0	0
$143$	440.000	0.257305
$144$	0	0
$145$	−770.000	−0.441000
$146$	0	0
$147$	−837.000	−0.469623
$148$	0	0
$149$	1486.00	0.817033	0.408516	−	0.912751i	$-0.366046\pi$
$149$	1486.00	0.817033	0.408516	+	0.912751i	$0.366046\pi$
$150$	0	0
$151$	2120.00	1.14254	0.571269	−	0.820763i	$-0.306451\pi$
$151$	2120.00	1.14254	0.571269	+	0.820763i	$0.306451\pi$
$152$	0	0
$153$	−126.000	−0.0665784
$154$	0	0
$155$	800.000	0.414565
$156$	0	0
$157$	−1850.00	−0.940421	−0.470210	−	0.882554i	$-0.655822\pi$
$157$	−1850.00	−0.940421	−0.470210	+	0.882554i	$0.655822\pi$
$158$	0	0
$159$	−630.000	−0.314228
$160$	0	0
$161$	448.000	0.219300
$162$	0	0
$163$	−1172.00	−0.563179	−0.281589	−	0.959535i	$-0.590862\pi$
$163$	−1172.00	−0.563179	−0.281589	+	0.959535i	$0.590862\pi$
$164$	0	0
$165$	300.000	0.141545
$166$	0	0
$167$	−1656.00	−0.767336	−0.383668	−	0.923471i	$-0.625339\pi$
$167$	−1656.00	−0.767336	−0.383668	+	0.923471i	$0.625339\pi$
$168$	0	0
$169$	−1713.00	−0.779700
$170$	0	0
$171$	684.000	0.305888
$172$	0	0
$173$	−2666.00	−1.17163	−0.585816	−	0.810444i	$-0.699226\pi$
$173$	−2666.00	−1.17163	−0.585816	+	0.810444i	$0.699226\pi$
$174$	0	0
$175$	200.000	0.0863919
$176$	0	0
$177$	−1140.00	−0.484111
$178$	0	0
$179$	1132.00	0.472680	0.236340	−	0.971670i	$-0.424052\pi$
$179$	1132.00	0.472680	0.236340	+	0.971670i	$0.424052\pi$
$180$	0	0
$181$	−2866.00	−1.17695	−0.588475	−	0.808515i	$-0.700272\pi$
$181$	−2866.00	−1.17695	−0.588475	+	0.808515i	$0.700272\pi$
$182$	0	0
$183$	−2382.00	−0.962199
$184$	0	0
$185$	−810.000	−0.321905
$186$	0	0
$187$	−280.000	−0.109495
$188$	0	0
$189$	216.000	0.0831306
$190$	0	0
$191$	1888.00	0.715240	0.357620	−	0.933867i	$-0.383588\pi$
$191$	1888.00	0.715240	0.357620	+	0.933867i	$0.383588\pi$
$192$	0	0
$193$	1282.00	0.478137	0.239068	−	0.971003i	$-0.423158\pi$
$193$	1282.00	0.478137	0.239068	+	0.971003i	$0.423158\pi$
$194$	0	0
$195$	330.000	0.121189
$196$	0	0
$197$	350.000	0.126581	0.0632905	−	0.997995i	$-0.479841\pi$
$197$	350.000	0.126581	0.0632905	+	0.997995i	$0.479841\pi$
$198$	0	0
$199$	−3400.00	−1.21115	−0.605577	−	0.795787i	$-0.707058\pi$
$199$	−3400.00	−1.21115	−0.605577	+	0.795787i	$0.707058\pi$
$200$	0	0
$201$	−444.000	−0.155808
$202$	0	0
$203$	−1232.00	−0.425958
$204$	0	0
$205$	−1950.00	−0.664361
$206$	0	0
$207$	504.000	0.169229
$208$	0	0
$209$	1520.00	0.503065
$210$	0	0
$211$	4652.00	1.51781	0.758903	−	0.651204i	$-0.225736\pi$
$211$	4652.00	1.51781	0.758903	+	0.651204i	$0.225736\pi$
$212$	0	0
$213$	−2520.00	−0.810646
$214$	0	0
$215$	1940.00	0.615381
$216$	0	0
$217$	1280.00	0.400424
$218$	0	0
$219$	2574.00	0.794223
$220$	0	0
$221$	−308.000	−0.0937481
$222$	0	0
$223$	4016.00	1.20597	0.602985	−	0.797753i	$-0.293978\pi$
$223$	4016.00	1.20597	0.602985	+	0.797753i	$0.293978\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	2316.00	0.677173	0.338587	−	0.940935i	$-0.390051\pi$
$227$	2316.00	0.677173	0.338587	+	0.940935i	$0.390051\pi$
$228$	0	0
$229$	94.0000	0.0271253	0.0135627	−	0.999908i	$-0.495683\pi$
$229$	94.0000	0.0271253	0.0135627	+	0.999908i	$0.495683\pi$
$230$	0	0
$231$	480.000	0.136717
$232$	0	0
$233$	−4230.00	−1.18934	−0.594671	−	0.803969i	$-0.702718\pi$
$233$	−4230.00	−1.18934	−0.594671	+	0.803969i	$0.702718\pi$
$234$	0	0
$235$	−2720.00	−0.755035
$236$	0	0
$237$	432.000	0.118403
$238$	0	0
$239$	2064.00	0.558615	0.279308	−	0.960202i	$-0.409895\pi$
$239$	2064.00	0.558615	0.279308	+	0.960202i	$0.409895\pi$
$240$	0	0
$241$	4562.00	1.21935	0.609677	−	0.792650i	$-0.291299\pi$
$241$	4562.00	1.21935	0.609677	+	0.792650i	$0.291299\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1395.00	−0.363768
$246$	0	0
$247$	1672.00	0.430716
$248$	0	0
$249$	948.000	0.241273
$250$	0	0
$251$	2532.00	0.636727	0.318363	−	0.947969i	$-0.396867\pi$
$251$	2532.00	0.636727	0.318363	+	0.947969i	$0.396867\pi$
$252$	0	0
$253$	1120.00	0.278315
$254$	0	0
$255$	−210.000	−0.0515714
$256$	0	0
$257$	3522.00	0.854850	0.427425	−	0.904051i	$-0.359421\pi$
$257$	3522.00	0.854850	0.427425	+	0.904051i	$0.359421\pi$
$258$	0	0
$259$	−1296.00	−0.310925
$260$	0	0
$261$	−1386.00	−0.328702
$262$	0	0
$263$	−2232.00	−0.523312	−0.261656	−	0.965161i	$-0.584269\pi$
$263$	−2232.00	−0.523312	−0.261656	+	0.965161i	$0.584269\pi$
$264$	0	0
$265$	−1050.00	−0.243400
$266$	0	0
$267$	3294.00	0.755017
$268$	0	0
$269$	2806.00	0.636003	0.318002	−	0.948090i	$-0.396988\pi$
$269$	2806.00	0.636003	0.318002	+	0.948090i	$0.396988\pi$
$270$	0	0
$271$	4848.00	1.08670	0.543349	−	0.839507i	$-0.317156\pi$
$271$	4848.00	1.08670	0.543349	+	0.839507i	$0.317156\pi$
$272$	0	0
$273$	528.000	0.117055
$274$	0	0
$275$	500.000	0.109640
$276$	0	0
$277$	7790.00	1.68973	0.844866	−	0.534978i	$-0.179680\pi$
$277$	7790.00	1.68973	0.844866	+	0.534978i	$0.179680\pi$
$278$	0	0
$279$	1440.00	0.308998
$280$	0	0
$281$	−118.000	−0.0250509	−0.0125254	−	0.999922i	$-0.503987\pi$
$281$	−118.000	−0.0250509	−0.0125254	+	0.999922i	$0.503987\pi$
$282$	0	0
$283$	−6508.00	−1.36700	−0.683499	−	0.729951i	$-0.739543\pi$
$283$	−6508.00	−1.36700	−0.683499	+	0.729951i	$0.739543\pi$
$284$	0	0
$285$	1140.00	0.236940
$286$	0	0
$287$	−3120.00	−0.641700
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	2982.00	0.600715
$292$	0	0
$293$	−8770.00	−1.74863	−0.874315	−	0.485358i	$-0.838689\pi$
$293$	−8770.00	−1.74863	−0.874315	+	0.485358i	$0.838689\pi$
$294$	0	0
$295$	−1900.00	−0.374991
$296$	0	0
$297$	540.000	0.105502
$298$	0	0
$299$	1232.00	0.238289
$300$	0	0
$301$	3104.00	0.594391
$302$	0	0
$303$	−2502.00	−0.474377
$304$	0	0
$305$	−3970.00	−0.745317
$306$	0	0
$307$	−4292.00	−0.797907	−0.398953	−	0.916971i	$-0.630626\pi$
$307$	−4292.00	−0.797907	−0.398953	+	0.916971i	$0.630626\pi$
$308$	0	0
$309$	5016.00	0.923464
$310$	0	0
$311$	−9464.00	−1.72558	−0.862788	−	0.505566i	$-0.831284\pi$
$311$	−9464.00	−1.72558	−0.862788	+	0.505566i	$0.831284\pi$
$312$	0	0
$313$	9578.00	1.72965	0.864825	−	0.502073i	$-0.167429\pi$
$313$	9578.00	1.72965	0.864825	+	0.502073i	$0.167429\pi$
$314$	0	0
$315$	360.000	0.0643927
$316$	0	0
$317$	−186.000	−0.0329552	−0.0164776	−	0.999864i	$-0.505245\pi$
$317$	−186.000	−0.0329552	−0.0164776	+	0.999864i	$0.505245\pi$
$318$	0	0
$319$	−3080.00	−0.540586
$320$	0	0
$321$	−2196.00	−0.381834
$322$	0	0
$323$	−1064.00	−0.183290
$324$	0	0
$325$	550.000	0.0938723
$326$	0	0
$327$	−2910.00	−0.492120
$328$	0	0
$329$	−4352.00	−0.729281
$330$	0	0
$331$	−492.000	−0.0817002	−0.0408501	−	0.999165i	$-0.513007\pi$
$331$	−492.000	−0.0817002	−0.0408501	+	0.999165i	$0.513007\pi$
$332$	0	0
$333$	−1458.00	−0.239934
$334$	0	0
$335$	−740.000	−0.120688
$336$	0	0
$337$	2290.00	0.370161	0.185080	−	0.982723i	$-0.440745\pi$
$337$	2290.00	0.370161	0.185080	+	0.982723i	$0.440745\pi$
$338$	0	0
$339$	5814.00	0.931484
$340$	0	0
$341$	3200.00	0.508181
$342$	0	0
$343$	−4976.00	−0.783320
$344$	0	0
$345$	840.000	0.131084
$346$	0	0
$347$	−6092.00	−0.942466	−0.471233	−	0.882009i	$-0.656191\pi$
$347$	−6092.00	−0.942466	−0.471233	+	0.882009i	$0.656191\pi$
$348$	0	0
$349$	5766.00	0.884375	0.442188	−	0.896923i	$-0.354203\pi$
$349$	5766.00	0.884375	0.442188	+	0.896923i	$0.354203\pi$
$350$	0	0
$351$	594.000	0.0903287
$352$	0	0
$353$	−9374.00	−1.41339	−0.706696	−	0.707517i	$-0.749815\pi$
$353$	−9374.00	−1.41339	−0.706696	+	0.707517i	$0.749815\pi$
$354$	0	0
$355$	−4200.00	−0.627924
$356$	0	0
$357$	−336.000	−0.0498123
$358$	0	0
$359$	−3528.00	−0.518665	−0.259332	−	0.965788i	$-0.583503\pi$
$359$	−3528.00	−0.518665	−0.259332	+	0.965788i	$0.583503\pi$
$360$	0	0
$361$	−1083.00	−0.157895
$362$	0	0
$363$	−2793.00	−0.403842
$364$	0	0
$365$	4290.00	0.615202
$366$	0	0
$367$	7616.00	1.08325	0.541624	−	0.840621i	$-0.317810\pi$
$367$	7616.00	1.08325	0.541624	+	0.840621i	$0.317810\pi$
$368$	0	0
$369$	−3510.00	−0.495185
$370$	0	0
$371$	−1680.00	−0.235098
$372$	0	0
$373$	3406.00	0.472804	0.236402	−	0.971655i	$-0.424032\pi$
$373$	3406.00	0.472804	0.236402	+	0.971655i	$0.424032\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−3388.00	−0.462841
$378$	0	0
$379$	−12284.0	−1.66487	−0.832436	−	0.554121i	$-0.813055\pi$
$379$	−12284.0	−1.66487	−0.832436	+	0.554121i	$0.813055\pi$
$380$	0	0
$381$	−1584.00	−0.212994
$382$	0	0
$383$	5424.00	0.723638	0.361819	−	0.932248i	$-0.382156\pi$
$383$	5424.00	0.723638	0.361819	+	0.932248i	$0.382156\pi$
$384$	0	0
$385$	800.000	0.105901
$386$	0	0
$387$	3492.00	0.458678
$388$	0	0
$389$	3486.00	0.454363	0.227182	−	0.973852i	$-0.427049\pi$
$389$	3486.00	0.454363	0.227182	+	0.973852i	$0.427049\pi$
$390$	0	0
$391$	−784.000	−0.101403
$392$	0	0
$393$	1908.00	0.244900
$394$	0	0
$395$	720.000	0.0917143
$396$	0	0
$397$	−3626.00	−0.458397	−0.229199	−	0.973380i	$-0.573611\pi$
$397$	−3626.00	−0.458397	−0.229199	+	0.973380i	$0.573611\pi$
$398$	0	0
$399$	1824.00	0.228858
$400$	0	0
$401$	5874.00	0.731505	0.365753	−	0.930712i	$-0.380812\pi$
$401$	5874.00	0.731505	0.365753	+	0.930712i	$0.380812\pi$
$402$	0	0
$403$	3520.00	0.435096
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−3240.00	−0.394597
$408$	0	0
$409$	−12662.0	−1.53080	−0.765398	−	0.643557i	$-0.777458\pi$
$409$	−12662.0	−1.53080	−0.765398	+	0.643557i	$0.777458\pi$
$410$	0	0
$411$	5262.00	0.631521
$412$	0	0
$413$	−3040.00	−0.362200
$414$	0	0
$415$	1580.00	0.186890
$416$	0	0
$417$	−7524.00	−0.883578
$418$	0	0
$419$	6396.00	0.745740	0.372870	−	0.927884i	$-0.378374\pi$
$419$	6396.00	0.745740	0.372870	+	0.927884i	$0.378374\pi$
$420$	0	0
$421$	8286.00	0.959228	0.479614	−	0.877480i	$-0.340777\pi$
$421$	8286.00	0.959228	0.479614	+	0.877480i	$0.340777\pi$
$422$	0	0
$423$	−4896.00	−0.562770
$424$	0	0
$425$	−350.000	−0.0399470
$426$	0	0
$427$	−6352.00	−0.719894
$428$	0	0
$429$	1320.00	0.148555
$430$	0	0
$431$	−4112.00	−0.459555	−0.229777	−	0.973243i	$-0.573800\pi$
$431$	−4112.00	−0.459555	−0.229777	+	0.973243i	$0.573800\pi$
$432$	0	0
$433$	5330.00	0.591555	0.295778	−	0.955257i	$-0.404421\pi$
$433$	5330.00	0.591555	0.295778	+	0.955257i	$0.404421\pi$
$434$	0	0
$435$	−2310.00	−0.254612
$436$	0	0
$437$	4256.00	0.465886
$438$	0	0
$439$	11272.0	1.22547	0.612737	−	0.790287i	$-0.290068\pi$
$439$	11272.0	1.22547	0.612737	+	0.790287i	$0.290068\pi$
$440$	0	0
$441$	−2511.00	−0.271137
$442$	0	0
$443$	14196.0	1.52251	0.761255	−	0.648452i	$-0.224583\pi$
$443$	14196.0	1.52251	0.761255	+	0.648452i	$0.224583\pi$
$444$	0	0
$445$	5490.00	0.584834
$446$	0	0
$447$	4458.00	0.471714
$448$	0	0
$449$	−5886.00	−0.618658	−0.309329	−	0.950955i	$-0.600104\pi$
$449$	−5886.00	−0.618658	−0.309329	+	0.950955i	$0.600104\pi$
$450$	0	0
$451$	−7800.00	−0.814385
$452$	0	0
$453$	6360.00	0.659644
$454$	0	0
$455$	880.000	0.0906704
$456$	0	0
$457$	−7526.00	−0.770353	−0.385177	−	0.922843i	$-0.625859\pi$
$457$	−7526.00	−0.770353	−0.385177	+	0.922843i	$0.625859\pi$
$458$	0	0
$459$	−378.000	−0.0384391
$460$	0	0
$461$	8502.00	0.858954	0.429477	−	0.903078i	$-0.358698\pi$
$461$	8502.00	0.858954	0.429477	+	0.903078i	$0.358698\pi$
$462$	0	0
$463$	12672.0	1.27196	0.635980	−	0.771705i	$-0.280596\pi$
$463$	12672.0	1.27196	0.635980	+	0.771705i	$0.280596\pi$
$464$	0	0
$465$	2400.00	0.239349
$466$	0	0
$467$	16540.0	1.63893	0.819465	−	0.573130i	$-0.194271\pi$
$467$	16540.0	1.63893	0.819465	+	0.573130i	$0.194271\pi$
$468$	0	0
$469$	−1184.00	−0.116572
$470$	0	0
$471$	−5550.00	−0.542952
$472$	0	0
$473$	7760.00	0.754345
$474$	0	0
$475$	1900.00	0.183533
$476$	0	0
$477$	−1890.00	−0.181420
$478$	0	0
$479$	8864.00	0.845525	0.422763	−	0.906241i	$-0.361060\pi$
$479$	8864.00	0.845525	0.422763	+	0.906241i	$0.361060\pi$
$480$	0	0
$481$	−3564.00	−0.337847
$482$	0	0
$483$	1344.00	0.126613
$484$	0	0
$485$	4970.00	0.465311
$486$	0	0
$487$	3688.00	0.343161	0.171580	−	0.985170i	$-0.445113\pi$
$487$	3688.00	0.343161	0.171580	+	0.985170i	$0.445113\pi$
$488$	0	0
$489$	−3516.00	−0.325151
$490$	0	0
$491$	−16140.0	−1.48348	−0.741739	−	0.670688i	$-0.765999\pi$
$491$	−16140.0	−1.48348	−0.741739	+	0.670688i	$0.765999\pi$
$492$	0	0
$493$	2156.00	0.196960
$494$	0	0
$495$	900.000	0.0817212
$496$	0	0
$497$	−6720.00	−0.606505
$498$	0	0
$499$	1580.00	0.141745	0.0708723	−	0.997485i	$-0.477422\pi$
$499$	1580.00	0.141745	0.0708723	+	0.997485i	$0.477422\pi$
$500$	0	0
$501$	−4968.00	−0.443022
$502$	0	0
$503$	15000.0	1.32966	0.664828	−	0.746996i	$-0.268505\pi$
$503$	15000.0	1.32966	0.664828	+	0.746996i	$0.268505\pi$
$504$	0	0
$505$	−4170.00	−0.367451
$506$	0	0
$507$	−5139.00	−0.450160
$508$	0	0
$509$	20486.0	1.78394	0.891971	−	0.452094i	$-0.149323\pi$
$509$	20486.0	1.78394	0.891971	+	0.452094i	$0.149323\pi$
$510$	0	0
$511$	6864.00	0.594218
$512$	0	0
$513$	2052.00	0.176604
$514$	0	0
$515$	8360.00	0.715312
$516$	0	0
$517$	−10880.0	−0.925535
$518$	0	0
$519$	−7998.00	−0.676442
$520$	0	0
$521$	7706.00	0.647996	0.323998	−	0.946058i	$-0.394973\pi$
$521$	7706.00	0.647996	0.323998	+	0.946058i	$0.394973\pi$
$522$	0	0
$523$	−3932.00	−0.328746	−0.164373	−	0.986398i	$-0.552560\pi$
$523$	−3932.00	−0.328746	−0.164373	+	0.986398i	$0.552560\pi$
$524$	0	0
$525$	600.000	0.0498784
$526$	0	0
$527$	−2240.00	−0.185154
$528$	0	0
$529$	−9031.00	−0.742254
$530$	0	0
$531$	−3420.00	−0.279502
$532$	0	0
$533$	−8580.00	−0.697263
$534$	0	0
$535$	−3660.00	−0.295767
$536$	0	0
$537$	3396.00	0.272902
$538$	0	0
$539$	−5580.00	−0.445914
$540$	0	0
$541$	−23930.0	−1.90172	−0.950860	−	0.309620i	$-0.899798\pi$
$541$	−23930.0	−1.90172	−0.950860	+	0.309620i	$0.899798\pi$
$542$	0	0
$543$	−8598.00	−0.679513
$544$	0	0
$545$	−4850.00	−0.381195
$546$	0	0
$547$	11468.0	0.896410	0.448205	−	0.893931i	$-0.352063\pi$
$547$	11468.0	0.896410	0.448205	+	0.893931i	$0.352063\pi$
$548$	0	0
$549$	−7146.00	−0.555526
$550$	0	0
$551$	−11704.0	−0.904913
$552$	0	0
$553$	1152.00	0.0885859
$554$	0	0
$555$	−2430.00	−0.185852
$556$	0	0
$557$	−11498.0	−0.874660	−0.437330	−	0.899301i	$-0.644076\pi$
$557$	−11498.0	−0.874660	−0.437330	+	0.899301i	$0.644076\pi$
$558$	0	0
$559$	8536.00	0.645857
$560$	0	0
$561$	−840.000	−0.0632172
$562$	0	0
$563$	16988.0	1.27169	0.635843	−	0.771819i	$-0.280653\pi$
$563$	16988.0	1.27169	0.635843	+	0.771819i	$0.280653\pi$
$564$	0	0
$565$	9690.00	0.721525
$566$	0	0
$567$	648.000	0.0479955
$568$	0	0
$569$	−17366.0	−1.27947	−0.639737	−	0.768594i	$-0.720957\pi$
$569$	−17366.0	−1.27947	−0.639737	+	0.768594i	$0.720957\pi$
$570$	0	0
$571$	−24860.0	−1.82199	−0.910997	−	0.412413i	$-0.864686\pi$
$571$	−24860.0	−1.82199	−0.910997	+	0.412413i	$0.864686\pi$
$572$	0	0
$573$	5664.00	0.412944
$574$	0	0
$575$	1400.00	0.101537
$576$	0	0
$577$	−26302.0	−1.89769	−0.948845	−	0.315744i	$-0.897746\pi$
$577$	−26302.0	−1.89769	−0.948845	+	0.315744i	$0.897746\pi$
$578$	0	0
$579$	3846.00	0.276052
$580$	0	0
$581$	2528.00	0.180515
$582$	0	0
$583$	−4200.00	−0.298364
$584$	0	0
$585$	990.000	0.0699683
$586$	0	0
$587$	7812.00	0.549294	0.274647	−	0.961545i	$-0.411439\pi$
$587$	7812.00	0.549294	0.274647	+	0.961545i	$0.411439\pi$
$588$	0	0
$589$	12160.0	0.850669
$590$	0	0
$591$	1050.00	0.0730816
$592$	0	0
$593$	7986.00	0.553028	0.276514	−	0.961010i	$-0.410821\pi$
$593$	7986.00	0.553028	0.276514	+	0.961010i	$0.410821\pi$
$594$	0	0
$595$	−560.000	−0.0385845
$596$	0	0
$597$	−10200.0	−0.699260
$598$	0	0
$599$	−21048.0	−1.43572	−0.717861	−	0.696186i	$-0.754879\pi$
$599$	−21048.0	−1.43572	−0.717861	+	0.696186i	$0.754879\pi$
$600$	0	0
$601$	1738.00	0.117961	0.0589804	−	0.998259i	$-0.481215\pi$
$601$	1738.00	0.117961	0.0589804	+	0.998259i	$0.481215\pi$
$602$	0	0
$603$	−1332.00	−0.0899556
$604$	0	0
$605$	−4655.00	−0.312814
$606$	0	0
$607$	18576.0	1.24214	0.621068	−	0.783757i	$-0.286699\pi$
$607$	18576.0	1.24214	0.621068	+	0.783757i	$0.286699\pi$
$608$	0	0
$609$	−3696.00	−0.245927
$610$	0	0
$611$	−11968.0	−0.792428
$612$	0	0
$613$	−13602.0	−0.896215	−0.448107	−	0.893980i	$-0.647902\pi$
$613$	−13602.0	−0.896215	−0.448107	+	0.893980i	$0.647902\pi$
$614$	0	0
$615$	−5850.00	−0.383569
$616$	0	0
$617$	19578.0	1.27744	0.638720	−	0.769439i	$-0.279464\pi$
$617$	19578.0	1.27744	0.638720	+	0.769439i	$0.279464\pi$
$618$	0	0
$619$	12308.0	0.799193	0.399596	−	0.916691i	$-0.369150\pi$
$619$	12308.0	0.799193	0.399596	+	0.916691i	$0.369150\pi$
$620$	0	0
$621$	1512.00	0.0977045
$622$	0	0
$623$	8784.00	0.564885
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4560.00	0.290445
$628$	0	0
$629$	2268.00	0.143770
$630$	0	0
$631$	−8600.00	−0.542568	−0.271284	−	0.962499i	$-0.587448\pi$
$631$	−8600.00	−0.542568	−0.271284	+	0.962499i	$0.587448\pi$
$632$	0	0
$633$	13956.0	0.876305
$634$	0	0
$635$	−2640.00	−0.164985
$636$	0	0
$637$	−6138.00	−0.381784
$638$	0	0
$639$	−7560.00	−0.468027
$640$	0	0
$641$	6978.00	0.429976	0.214988	−	0.976617i	$-0.431029\pi$
$641$	6978.00	0.429976	0.214988	+	0.976617i	$0.431029\pi$
$642$	0	0
$643$	−7668.00	−0.470290	−0.235145	−	0.971960i	$-0.575556\pi$
$643$	−7668.00	−0.470290	−0.235145	+	0.971960i	$0.575556\pi$
$644$	0	0
$645$	5820.00	0.355290
$646$	0	0
$647$	−15384.0	−0.934787	−0.467394	−	0.884049i	$-0.654807\pi$
$647$	−15384.0	−0.934787	−0.467394	+	0.884049i	$0.654807\pi$
$648$	0	0
$649$	−7600.00	−0.459670
$650$	0	0
$651$	3840.00	0.231185
$652$	0	0
$653$	−2186.00	−0.131003	−0.0655014	−	0.997852i	$-0.520865\pi$
$653$	−2186.00	−0.131003	−0.0655014	+	0.997852i	$0.520865\pi$
$654$	0	0
$655$	3180.00	0.189699
$656$	0	0
$657$	7722.00	0.458545
$658$	0	0
$659$	−1524.00	−0.0900859	−0.0450430	−	0.998985i	$-0.514342\pi$
$659$	−1524.00	−0.0900859	−0.0450430	+	0.998985i	$0.514342\pi$
$660$	0	0
$661$	−4242.00	−0.249614	−0.124807	−	0.992181i	$-0.539831\pi$
$661$	−4242.00	−0.249614	−0.124807	+	0.992181i	$0.539831\pi$
$662$	0	0
$663$	−924.000	−0.0541255
$664$	0	0
$665$	3040.00	0.177272
$666$	0	0
$667$	−8624.00	−0.500634
$668$	0	0
$669$	12048.0	0.696267
$670$	0	0
$671$	−15880.0	−0.913622
$672$	0	0
$673$	24354.0	1.39491	0.697457	−	0.716626i	$-0.254315\pi$
$673$	24354.0	1.39491	0.697457	+	0.716626i	$0.254315\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−322.000	−0.0182799	−0.00913993	−	0.999958i	$-0.502909\pi$
$677$	−322.000	−0.0182799	−0.00913993	+	0.999958i	$0.502909\pi$
$678$	0	0
$679$	7952.00	0.449440
$680$	0	0
$681$	6948.00	0.390966
$682$	0	0
$683$	−7932.00	−0.444377	−0.222189	−	0.975004i	$-0.571320\pi$
$683$	−7932.00	−0.444377	−0.222189	+	0.975004i	$0.571320\pi$
$684$	0	0
$685$	8770.00	0.489174
$686$	0	0
$687$	282.000	0.0156608
$688$	0	0
$689$	−4620.00	−0.255454
$690$	0	0
$691$	20684.0	1.13872	0.569361	−	0.822088i	$-0.307191\pi$
$691$	20684.0	1.13872	0.569361	+	0.822088i	$0.307191\pi$
$692$	0	0
$693$	1440.00	0.0789337
$694$	0	0
$695$	−12540.0	−0.684416
$696$	0	0
$697$	5460.00	0.296718
$698$	0	0
$699$	−12690.0	−0.686666
$700$	0	0
$701$	25222.0	1.35895	0.679473	−	0.733700i	$-0.262208\pi$
$701$	25222.0	1.35895	0.679473	+	0.733700i	$0.262208\pi$
$702$	0	0
$703$	−12312.0	−0.660535
$704$	0	0
$705$	−8160.00	−0.435920
$706$	0	0
$707$	−6672.00	−0.354917
$708$	0	0
$709$	23678.0	1.25423	0.627113	−	0.778928i	$-0.284236\pi$
$709$	23678.0	1.25423	0.627113	+	0.778928i	$0.284236\pi$
$710$	0	0
$711$	1296.00	0.0683598
$712$	0	0
$713$	8960.00	0.470624
$714$	0	0
$715$	2200.00	0.115070
$716$	0	0
$717$	6192.00	0.322517
$718$	0	0
$719$	8432.00	0.437358	0.218679	−	0.975797i	$-0.429825\pi$
$719$	8432.00	0.437358	0.218679	+	0.975797i	$0.429825\pi$
$720$	0	0
$721$	13376.0	0.690913
$722$	0	0
$723$	13686.0	0.703994
$724$	0	0
$725$	−3850.00	−0.197221
$726$	0	0
$727$	8312.00	0.424037	0.212019	−	0.977266i	$-0.431996\pi$
$727$	8312.00	0.424037	0.212019	+	0.977266i	$0.431996\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−5432.00	−0.274842
$732$	0	0
$733$	−26298.0	−1.32516	−0.662578	−	0.748993i	$-0.730538\pi$
$733$	−26298.0	−1.32516	−0.662578	+	0.748993i	$0.730538\pi$
$734$	0	0
$735$	−4185.00	−0.210022
$736$	0	0
$737$	−2960.00	−0.147942
$738$	0	0
$739$	16956.0	0.844028	0.422014	−	0.906589i	$-0.361323\pi$
$739$	16956.0	0.844028	0.422014	+	0.906589i	$0.361323\pi$
$740$	0	0
$741$	5016.00	0.248674
$742$	0	0
$743$	−17880.0	−0.882845	−0.441422	−	0.897299i	$-0.645526\pi$
$743$	−17880.0	−0.882845	−0.441422	+	0.897299i	$0.645526\pi$
$744$	0	0
$745$	7430.00	0.365388
$746$	0	0
$747$	2844.00	0.139299
$748$	0	0
$749$	−5856.00	−0.285679
$750$	0	0
$751$	22032.0	1.07052	0.535259	−	0.844688i	$-0.320214\pi$
$751$	22032.0	1.07052	0.535259	+	0.844688i	$0.320214\pi$
$752$	0	0
$753$	7596.00	0.367614
$754$	0	0
$755$	10600.0	0.510958
$756$	0	0
$757$	11534.0	0.553779	0.276889	−	0.960902i	$-0.410696\pi$
$757$	11534.0	0.553779	0.276889	+	0.960902i	$0.410696\pi$
$758$	0	0
$759$	3360.00	0.160685
$760$	0	0
$761$	38250.0	1.82203	0.911013	−	0.412378i	$-0.135302\pi$
$761$	38250.0	1.82203	0.911013	+	0.412378i	$0.135302\pi$
$762$	0	0
$763$	−7760.00	−0.368192
$764$	0	0
$765$	−630.000	−0.0297748
$766$	0	0
$767$	−8360.00	−0.393562
$768$	0	0
$769$	19330.0	0.906447	0.453223	−	0.891397i	$-0.350274\pi$
$769$	19330.0	0.906447	0.453223	+	0.891397i	$0.350274\pi$
$770$	0	0
$771$	10566.0	0.493548
$772$	0	0
$773$	−40674.0	−1.89255	−0.946276	−	0.323361i	$-0.895187\pi$
$773$	−40674.0	−1.89255	−0.946276	+	0.323361i	$0.895187\pi$
$774$	0	0
$775$	4000.00	0.185399
$776$	0	0
$777$	−3888.00	−0.179513
$778$	0	0
$779$	−29640.0	−1.36324
$780$	0	0
$781$	−16800.0	−0.769720
$782$	0	0
$783$	−4158.00	−0.189776
$784$	0	0
$785$	−9250.00	−0.420569
$786$	0	0
$787$	7004.00	0.317237	0.158619	−	0.987340i	$-0.449296\pi$
$787$	7004.00	0.317237	0.158619	+	0.987340i	$0.449296\pi$
$788$	0	0
$789$	−6696.00	−0.302134
$790$	0	0
$791$	15504.0	0.696914
$792$	0	0
$793$	−17468.0	−0.782228
$794$	0	0
$795$	−3150.00	−0.140527
$796$	0	0
$797$	12198.0	0.542127	0.271064	−	0.962561i	$-0.412625\pi$
$797$	12198.0	0.542127	0.271064	+	0.962561i	$0.412625\pi$
$798$	0	0
$799$	7616.00	0.337215
$800$	0	0
$801$	9882.00	0.435909
$802$	0	0
$803$	17160.0	0.754126
$804$	0	0
$805$	2240.00	0.0980741
$806$	0	0
$807$	8418.00	0.367197
$808$	0	0
$809$	−25734.0	−1.11837	−0.559184	−	0.829044i	$-0.688885\pi$
$809$	−25734.0	−1.11837	−0.559184	+	0.829044i	$0.688885\pi$
$810$	0	0
$811$	15668.0	0.678394	0.339197	−	0.940715i	$-0.389845\pi$
$811$	15668.0	0.678394	0.339197	+	0.940715i	$0.389845\pi$
$812$	0	0
$813$	14544.0	0.627405
$814$	0	0
$815$	−5860.00	−0.251861
$816$	0	0
$817$	29488.0	1.26274
$818$	0	0
$819$	1584.00	0.0675817
$820$	0	0
$821$	−34450.0	−1.46445	−0.732225	−	0.681063i	$-0.761518\pi$
$821$	−34450.0	−1.46445	−0.732225	+	0.681063i	$0.761518\pi$
$822$	0	0
$823$	−38792.0	−1.64302	−0.821509	−	0.570195i	$-0.806868\pi$
$823$	−38792.0	−1.64302	−0.821509	+	0.570195i	$0.806868\pi$
$824$	0	0
$825$	1500.00	0.0633010
$826$	0	0
$827$	−20460.0	−0.860295	−0.430147	−	0.902759i	$-0.641538\pi$
$827$	−20460.0	−0.860295	−0.430147	+	0.902759i	$0.641538\pi$
$828$	0	0
$829$	5542.00	0.232185	0.116093	−	0.993238i	$-0.462963\pi$
$829$	5542.00	0.232185	0.116093	+	0.993238i	$0.462963\pi$
$830$	0	0
$831$	23370.0	0.975567
$832$	0	0
$833$	3906.00	0.162467
$834$	0	0
$835$	−8280.00	−0.343163
$836$	0	0
$837$	4320.00	0.178400
$838$	0	0
$839$	25240.0	1.03860	0.519298	−	0.854593i	$-0.326194\pi$
$839$	25240.0	1.03860	0.519298	+	0.854593i	$0.326194\pi$
$840$	0	0
$841$	−673.000	−0.0275944
$842$	0	0
$843$	−354.000	−0.0144631
$844$	0	0
$845$	−8565.00	−0.348692
$846$	0	0
$847$	−7448.00	−0.302144
$848$	0	0
$849$	−19524.0	−0.789237
$850$	0	0
$851$	−9072.00	−0.365434
$852$	0	0
$853$	−37330.0	−1.49842	−0.749212	−	0.662331i	$-0.769567\pi$
$853$	−37330.0	−1.49842	−0.749212	+	0.662331i	$0.769567\pi$
$854$	0	0
$855$	3420.00	0.136797
$856$	0	0
$857$	−3894.00	−0.155212	−0.0776059	−	0.996984i	$-0.524728\pi$
$857$	−3894.00	−0.155212	−0.0776059	+	0.996984i	$0.524728\pi$
$858$	0	0
$859$	20324.0	0.807271	0.403636	−	0.914920i	$-0.367746\pi$
$859$	20324.0	0.807271	0.403636	+	0.914920i	$0.367746\pi$
$860$	0	0
$861$	−9360.00	−0.370485
$862$	0	0
$863$	6288.00	0.248026	0.124013	−	0.992281i	$-0.460424\pi$
$863$	6288.00	0.248026	0.124013	+	0.992281i	$0.460424\pi$
$864$	0	0
$865$	−13330.0	−0.523969
$866$	0	0
$867$	−14151.0	−0.554317
$868$	0	0
$869$	2880.00	0.112425
$870$	0	0
$871$	−3256.00	−0.126665
$872$	0	0
$873$	8946.00	0.346823
$874$	0	0
$875$	1000.00	0.0386356
$876$	0	0
$877$	−24650.0	−0.949112	−0.474556	−	0.880225i	$-0.657391\pi$
$877$	−24650.0	−0.949112	−0.474556	+	0.880225i	$0.657391\pi$
$878$	0	0
$879$	−26310.0	−1.00957
$880$	0	0
$881$	9426.00	0.360465	0.180233	−	0.983624i	$-0.442315\pi$
$881$	9426.00	0.360465	0.180233	+	0.983624i	$0.442315\pi$
$882$	0	0
$883$	−9316.00	−0.355049	−0.177525	−	0.984116i	$-0.556809\pi$
$883$	−9316.00	−0.355049	−0.177525	+	0.984116i	$0.556809\pi$
$884$	0	0
$885$	−5700.00	−0.216501
$886$	0	0
$887$	6968.00	0.263768	0.131884	−	0.991265i	$-0.457897\pi$
$887$	6968.00	0.263768	0.131884	+	0.991265i	$0.457897\pi$
$888$	0	0
$889$	−4224.00	−0.159357
$890$	0	0
$891$	1620.00	0.0609114
$892$	0	0
$893$	−41344.0	−1.54930
$894$	0	0
$895$	5660.00	0.211389
$896$	0	0
$897$	3696.00	0.137576
$898$	0	0
$899$	−24640.0	−0.914116
$900$	0	0
$901$	2940.00	0.108708
$902$	0	0
$903$	9312.00	0.343172
$904$	0	0
$905$	−14330.0	−0.526348
$906$	0	0
$907$	−35964.0	−1.31661	−0.658305	−	0.752751i	$-0.728726\pi$
$907$	−35964.0	−1.31661	−0.658305	+	0.752751i	$0.728726\pi$
$908$	0	0
$909$	−7506.00	−0.273882
$910$	0	0
$911$	47888.0	1.74160	0.870801	−	0.491635i	$-0.163601\pi$
$911$	47888.0	1.74160	0.870801	+	0.491635i	$0.163601\pi$
$912$	0	0
$913$	6320.00	0.229093
$914$	0	0
$915$	−11910.0	−0.430309
$916$	0	0
$917$	5088.00	0.183229
$918$	0	0
$919$	−12760.0	−0.458013	−0.229006	−	0.973425i	$-0.573548\pi$
$919$	−12760.0	−0.458013	−0.229006	+	0.973425i	$0.573548\pi$
$920$	0	0
$921$	−12876.0	−0.460672
$922$	0	0
$923$	−18480.0	−0.659021
$924$	0	0
$925$	−4050.00	−0.143960
$926$	0	0
$927$	15048.0	0.533162
$928$	0	0
$929$	−25054.0	−0.884817	−0.442409	−	0.896814i	$-0.645876\pi$
$929$	−25054.0	−0.884817	−0.442409	+	0.896814i	$0.645876\pi$
$930$	0	0
$931$	−21204.0	−0.746437
$932$	0	0
$933$	−28392.0	−0.996262
$934$	0	0
$935$	−1400.00	−0.0489678
$936$	0	0
$937$	28282.0	0.986054	0.493027	−	0.870014i	$-0.335890\pi$
$937$	28282.0	0.986054	0.493027	+	0.870014i	$0.335890\pi$
$938$	0	0
$939$	28734.0	0.998614
$940$	0	0
$941$	−30634.0	−1.06125	−0.530627	−	0.847605i	$-0.678043\pi$
$941$	−30634.0	−1.06125	−0.530627	+	0.847605i	$0.678043\pi$
$942$	0	0
$943$	−21840.0	−0.754198
$944$	0	0
$945$	1080.00	0.0371771
$946$	0	0
$947$	48572.0	1.66671	0.833357	−	0.552735i	$-0.186416\pi$
$947$	48572.0	1.66671	0.833357	+	0.552735i	$0.186416\pi$
$948$	0	0
$949$	18876.0	0.645670
$950$	0	0
$951$	−558.000	−0.0190267
$952$	0	0
$953$	12906.0	0.438685	0.219342	−	0.975648i	$-0.429609\pi$
$953$	12906.0	0.438685	0.219342	+	0.975648i	$0.429609\pi$
$954$	0	0
$955$	9440.00	0.319865
$956$	0	0
$957$	−9240.00	−0.312107
$958$	0	0
$959$	14032.0	0.472489
$960$	0	0
$961$	−4191.00	−0.140680
$962$	0	0
$963$	−6588.00	−0.220452
$964$	0	0
$965$	6410.00	0.213829
$966$	0	0
$967$	−5880.00	−0.195541	−0.0977705	−	0.995209i	$-0.531171\pi$
$967$	−5880.00	−0.195541	−0.0977705	+	0.995209i	$0.531171\pi$
$968$	0	0
$969$	−3192.00	−0.105822
$970$	0	0
$971$	55444.0	1.83242	0.916211	−	0.400695i	$-0.131231\pi$
$971$	55444.0	1.83242	0.916211	+	0.400695i	$0.131231\pi$
$972$	0	0
$973$	−20064.0	−0.661071
$974$	0	0
$975$	1650.00	0.0541972
$976$	0	0
$977$	32050.0	1.04951	0.524755	−	0.851254i	$-0.324157\pi$
$977$	32050.0	1.04951	0.524755	+	0.851254i	$0.324157\pi$
$978$	0	0
$979$	21960.0	0.716900
$980$	0	0
$981$	−8730.00	−0.284126
$982$	0	0
$983$	29880.0	0.969506	0.484753	−	0.874651i	$-0.338910\pi$
$983$	29880.0	0.969506	0.484753	+	0.874651i	$0.338910\pi$
$984$	0	0
$985$	1750.00	0.0566088
$986$	0	0
$987$	−13056.0	−0.421051
$988$	0	0
$989$	21728.0	0.698595
$990$	0	0
$991$	5216.00	0.167196	0.0835982	−	0.996500i	$-0.473359\pi$
$991$	5216.00	0.167196	0.0835982	+	0.996500i	$0.473359\pi$
$992$	0	0
$993$	−1476.00	−0.0471696
$994$	0	0
$995$	−17000.0	−0.541644
$996$	0	0
$997$	6750.00	0.214418	0.107209	−	0.994237i	$-0.465809\pi$
$997$	6750.00	0.214418	0.107209	+	0.994237i	$0.465809\pi$
$998$	0	0
$999$	−4374.00	−0.138526

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.f.1.1	✓	1	1.1	even	1	trivial
240.4.a.d.1.1		1	4.3	odd	2
360.4.a.e.1.1		1	3.2	odd	2
600.4.a.d.1.1		1	5.4	even	2
600.4.f.g.49.1		2	5.2	odd	4
600.4.f.g.49.2		2	5.3	odd	4
720.4.a.f.1.1		1	12.11	even	2
960.4.a.g.1.1		1	8.5	even	2
960.4.a.v.1.1		1	8.3	odd	2
1200.4.a.bf.1.1		1	20.19	odd	2
1200.4.f.g.49.1		2	20.3	even	4
1200.4.f.g.49.2		2	20.7	even	4
1800.4.a.k.1.1		1	15.14	odd	2
1800.4.f.h.649.1		2	15.8	even	4
1800.4.f.h.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 $\F_{q}$

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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