LMFDB - Embedded newform 60.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,4,Mod(1,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("60.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	60.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:	$3.54011460034$
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	$\chi$	$=$	60.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +5.00000 q^{5} +32.0000 q^{7} +9.00000 q^{9} +36.0000 q^{11} -10.0000 q^{13} -15.0000 q^{15} -78.0000 q^{17} +140.000 q^{19} -96.0000 q^{21} -192.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +6.00000 q^{29} -16.0000 q^{31} -108.000 q^{33} +160.000 q^{35} -34.0000 q^{37} +30.0000 q^{39} -390.000 q^{41} -52.0000 q^{43} +45.0000 q^{45} +408.000 q^{47} +681.000 q^{49} +234.000 q^{51} -114.000 q^{53} +180.000 q^{55} -420.000 q^{57} +516.000 q^{59} -58.0000 q^{61} +288.000 q^{63} -50.0000 q^{65} -892.000 q^{67} +576.000 q^{69} -120.000 q^{71} -646.000 q^{73} -75.0000 q^{75} +1152.00 q^{77} -1168.00 q^{79} +81.0000 q^{81} -732.000 q^{83} -390.000 q^{85} -18.0000 q^{87} -1590.00 q^{89} -320.000 q^{91} +48.0000 q^{93} +700.000 q^{95} +194.000 q^{97} +324.000 q^{99} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	32.0000	1.72784	0.863919	−	0.503631i	$-0.168003\pi$
$7$	32.0000	1.72784	0.863919	+	0.503631i	$0.168003\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	36.0000	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$11$	36.0000	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$12$	0	0
$13$	−10.0000	−0.213346	−0.106673	−	0.994294i	$-0.534020\pi$
$13$	−10.0000	−0.213346	−0.106673	+	0.994294i	$0.534020\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−78.0000	−1.11281	−0.556405	−	0.830911i	$-0.687820\pi$
$17$	−78.0000	−1.11281	−0.556405	+	0.830911i	$0.687820\pi$
$18$	0	0
$19$	140.000	1.69043	0.845216	−	0.534425i	$-0.179472\pi$
$19$	140.000	1.69043	0.845216	+	0.534425i	$0.179472\pi$
$20$	0	0
$21$	−96.0000	−0.997567
$22$	0	0
$23$	−192.000	−1.74064	−0.870321	−	0.492485i	$-0.836089\pi$
$23$	−192.000	−1.74064	−0.870321	+	0.492485i	$0.836089\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	6.00000	0.0384197	0.0192099	−	0.999815i	$-0.493885\pi$
$29$	6.00000	0.0384197	0.0192099	+	0.999815i	$0.493885\pi$
$30$	0	0
$31$	−16.0000	−0.0926995	−0.0463498	−	0.998925i	$-0.514759\pi$
$31$	−16.0000	−0.0926995	−0.0463498	+	0.998925i	$0.514759\pi$
$32$	0	0
$33$	−108.000	−0.569709
$34$	0	0
$35$	160.000	0.772712
$36$	0	0
$37$	−34.0000	−0.151069	−0.0755347	−	0.997143i	$-0.524066\pi$
$37$	−34.0000	−0.151069	−0.0755347	+	0.997143i	$0.524066\pi$
$38$	0	0
$39$	30.0000	0.123176
$40$	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
$43$	−52.0000	−0.184417	−0.0922084	−	0.995740i	$-0.529393\pi$
$43$	−52.0000	−0.184417	−0.0922084	+	0.995740i	$0.529393\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	408.000	1.26623	0.633116	−	0.774057i	$-0.281776\pi$
$47$	408.000	1.26623	0.633116	+	0.774057i	$0.281776\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	234.000	0.642481
$52$	0	0
$53$	−114.000	−0.295455	−0.147727	−	0.989028i	$-0.547196\pi$
$53$	−114.000	−0.295455	−0.147727	+	0.989028i	$0.547196\pi$
$54$	0	0
$55$	180.000	0.441294
$56$	0	0
$57$	−420.000	−0.975971
$58$	0	0
$59$	516.000	1.13860	0.569301	−	0.822129i	$-0.307214\pi$
$59$	516.000	1.13860	0.569301	+	0.822129i	$0.307214\pi$
$60$	0	0
$61$	−58.0000	−0.121740	−0.0608700	−	0.998146i	$-0.519388\pi$
$61$	−58.0000	−0.121740	−0.0608700	+	0.998146i	$0.519388\pi$
$62$	0	0
$63$	288.000	0.575946
$64$	0	0
$65$	−50.0000	−0.0954113
$66$	0	0
$67$	−892.000	−1.62649	−0.813247	−	0.581918i	$-0.802302\pi$
$67$	−892.000	−1.62649	−0.813247	+	0.581918i	$0.802302\pi$
$68$	0	0
$69$	576.000	1.00496
$70$	0	0
$71$	−120.000	−0.200583	−0.100291	−	0.994958i	$-0.531978\pi$
$71$	−120.000	−0.200583	−0.100291	+	0.994958i	$0.531978\pi$
$72$	0	0
$73$	−646.000	−1.03573	−0.517867	−	0.855461i	$-0.673274\pi$
$73$	−646.000	−1.03573	−0.517867	+	0.855461i	$0.673274\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	1152.00	1.70497
$78$	0	0
$79$	−1168.00	−1.66342	−0.831711	−	0.555209i	$-0.812638\pi$
$79$	−1168.00	−1.66342	−0.831711	+	0.555209i	$0.812638\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−732.000	−0.968041	−0.484021	−	0.875057i	$-0.660824\pi$
$83$	−732.000	−0.968041	−0.484021	+	0.875057i	$0.660824\pi$
$84$	0	0
$85$	−390.000	−0.497664
$86$	0	0
$87$	−18.0000	−0.0221816
$88$	0	0
$89$	−1590.00	−1.89370	−0.946852	−	0.321669i	$-0.895756\pi$
$89$	−1590.00	−1.89370	−0.946852	+	0.321669i	$0.895756\pi$
$90$	0	0
$91$	−320.000	−0.368628
$92$	0	0
$93$	48.0000	0.0535201
$94$	0	0
$95$	700.000	0.755984
$96$	0	0
$97$	194.000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	194.000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	324.000	0.328921
$100$	0	0
$101$	798.000	0.786178	0.393089	−	0.919500i	$-0.371406\pi$
$101$	798.000	0.786178	0.393089	+	0.919500i	$0.371406\pi$
$102$	0	0
$103$	272.000	0.260203	0.130102	−	0.991501i	$-0.458470\pi$
$103$	272.000	0.260203	0.130102	+	0.991501i	$0.458470\pi$
$104$	0	0
$105$	−480.000	−0.446126
$106$	0	0
$107$	156.000	0.140945	0.0704724	−	0.997514i	$-0.477549\pi$
$107$	156.000	0.140945	0.0704724	+	0.997514i	$0.477549\pi$
$108$	0	0
$109$	1622.00	1.42532	0.712658	−	0.701512i	$-0.247491\pi$
$109$	1622.00	1.42532	0.712658	+	0.701512i	$0.247491\pi$
$110$	0	0
$111$	102.000	0.0872199
$112$	0	0
$113$	1074.00	0.894101	0.447051	−	0.894509i	$-0.352474\pi$
$113$	1074.00	0.894101	0.447051	+	0.894509i	$0.352474\pi$
$114$	0	0
$115$	−960.000	−0.778439
$116$	0	0
$117$	−90.0000	−0.0711154
$118$	0	0
$119$	−2496.00	−1.92276
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	1170.00	0.857686
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1528.00	−1.06762	−0.533811	−	0.845604i	$-0.679241\pi$
$127$	−1528.00	−1.06762	−0.533811	+	0.845604i	$0.679241\pi$
$128$	0	0
$129$	156.000	0.106473
$130$	0	0
$131$	2412.00	1.60868	0.804341	−	0.594168i	$-0.202518\pi$
$131$	2412.00	1.60868	0.804341	+	0.594168i	$0.202518\pi$
$132$	0	0
$133$	4480.00	2.92079
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	2106.00	1.31334	0.656671	−	0.754178i	$-0.271964\pi$
$137$	2106.00	1.31334	0.656671	+	0.754178i	$0.271964\pi$
$138$	0	0
$139$	−556.000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−556.000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−1224.00	−0.731060
$142$	0	0
$143$	−360.000	−0.210522
$144$	0	0
$145$	30.0000	0.0171818
$146$	0	0
$147$	−2043.00	−1.14628
$148$	0	0
$149$	−2418.00	−1.32946	−0.664732	−	0.747081i	$-0.731454\pi$
$149$	−2418.00	−1.32946	−0.664732	+	0.747081i	$0.731454\pi$
$150$	0	0
$151$	2840.00	1.53057	0.765285	−	0.643692i	$-0.222598\pi$
$151$	2840.00	1.53057	0.765285	+	0.643692i	$0.222598\pi$
$152$	0	0
$153$	−702.000	−0.370937
$154$	0	0
$155$	−80.0000	−0.0414565
$156$	0	0
$157$	2054.00	1.04412	0.522061	−	0.852908i	$-0.325163\pi$
$157$	2054.00	1.04412	0.522061	+	0.852908i	$0.325163\pi$
$158$	0	0
$159$	342.000	0.170581
$160$	0	0
$161$	−6144.00	−3.00755
$162$	0	0
$163$	−460.000	−0.221043	−0.110521	−	0.993874i	$-0.535252\pi$
$163$	−460.000	−0.221043	−0.110521	+	0.993874i	$0.535252\pi$
$164$	0	0
$165$	−540.000	−0.254781
$166$	0	0
$167$	2016.00	0.934148	0.467074	−	0.884218i	$-0.345308\pi$
$167$	2016.00	0.934148	0.467074	+	0.884218i	$0.345308\pi$
$168$	0	0
$169$	−2097.00	−0.954483
$170$	0	0
$171$	1260.00	0.563477
$172$	0	0
$173$	−618.000	−0.271593	−0.135797	−	0.990737i	$-0.543359\pi$
$173$	−618.000	−0.271593	−0.135797	+	0.990737i	$0.543359\pi$
$174$	0	0
$175$	800.000	0.345568
$176$	0	0
$177$	−1548.00	−0.657372
$178$	0	0
$179$	−2964.00	−1.23765	−0.618826	−	0.785528i	$-0.712391\pi$
$179$	−2964.00	−1.23765	−0.618826	+	0.785528i	$0.712391\pi$
$180$	0	0
$181$	−370.000	−0.151944	−0.0759721	−	0.997110i	$-0.524206\pi$
$181$	−370.000	−0.151944	−0.0759721	+	0.997110i	$0.524206\pi$
$182$	0	0
$183$	174.000	0.0702866
$184$	0	0
$185$	−170.000	−0.0675603
$186$	0	0
$187$	−2808.00	−1.09808
$188$	0	0
$189$	−864.000	−0.332522
$190$	0	0
$191$	1104.00	0.418234	0.209117	−	0.977891i	$-0.432941\pi$
$191$	1104.00	0.418234	0.209117	+	0.977891i	$0.432941\pi$
$192$	0	0
$193$	−2398.00	−0.894362	−0.447181	−	0.894444i	$-0.647572\pi$
$193$	−2398.00	−0.894362	−0.447181	+	0.894444i	$0.647572\pi$
$194$	0	0
$195$	150.000	0.0550858
$196$	0	0
$197$	1278.00	0.462202	0.231101	−	0.972930i	$-0.425767\pi$
$197$	1278.00	0.462202	0.231101	+	0.972930i	$0.425767\pi$
$198$	0	0
$199$	4472.00	1.59302	0.796512	−	0.604623i	$-0.206676\pi$
$199$	4472.00	1.59302	0.796512	+	0.604623i	$0.206676\pi$
$200$	0	0
$201$	2676.00	0.939057
$202$	0	0
$203$	192.000	0.0663830
$204$	0	0
$205$	−1950.00	−0.664361
$206$	0	0
$207$	−1728.00	−0.580214
$208$	0	0
$209$	5040.00	1.66806
$210$	0	0
$211$	1340.00	0.437201	0.218600	−	0.975814i	$-0.429851\pi$
$211$	1340.00	0.437201	0.218600	+	0.975814i	$0.429851\pi$
$212$	0	0
$213$	360.000	0.115807
$214$	0	0
$215$	−260.000	−0.0824737
$216$	0	0
$217$	−512.000	−0.160170
$218$	0	0
$219$	1938.00	0.597981
$220$	0	0
$221$	780.000	0.237414
$222$	0	0
$223$	2360.00	0.708687	0.354344	−	0.935115i	$-0.384704\pi$
$223$	2360.00	0.708687	0.354344	+	0.935115i	$0.384704\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	1380.00	0.403497	0.201748	−	0.979437i	$-0.435338\pi$
$227$	1380.00	0.403497	0.201748	+	0.979437i	$0.435338\pi$
$228$	0	0
$229$	1694.00	0.488833	0.244416	−	0.969670i	$-0.421404\pi$
$229$	1694.00	0.488833	0.244416	+	0.969670i	$0.421404\pi$
$230$	0	0
$231$	−3456.00	−0.984364
$232$	0	0
$233$	−5190.00	−1.45926	−0.729631	−	0.683841i	$-0.760308\pi$
$233$	−5190.00	−1.45926	−0.729631	+	0.683841i	$0.760308\pi$
$234$	0	0
$235$	2040.00	0.566276
$236$	0	0
$237$	3504.00	0.960377
$238$	0	0
$239$	2352.00	0.636562	0.318281	−	0.947996i	$-0.396895\pi$
$239$	2352.00	0.636562	0.318281	+	0.947996i	$0.396895\pi$
$240$	0	0
$241$	−3502.00	−0.936032	−0.468016	−	0.883720i	$-0.655031\pi$
$241$	−3502.00	−0.936032	−0.468016	+	0.883720i	$0.655031\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	3405.00	0.887908
$246$	0	0
$247$	−1400.00	−0.360647
$248$	0	0
$249$	2196.00	0.558899
$250$	0	0
$251$	4788.00	1.20405	0.602024	−	0.798478i	$-0.294361\pi$
$251$	4788.00	1.20405	0.602024	+	0.798478i	$0.294361\pi$
$252$	0	0
$253$	−6912.00	−1.71760
$254$	0	0
$255$	1170.00	0.287326
$256$	0	0
$257$	1506.00	0.365532	0.182766	−	0.983156i	$-0.441495\pi$
$257$	1506.00	0.365532	0.182766	+	0.983156i	$0.441495\pi$
$258$	0	0
$259$	−1088.00	−0.261023
$260$	0	0
$261$	54.0000	0.0128066
$262$	0	0
$263$	−432.000	−0.101286	−0.0506431	−	0.998717i	$-0.516127\pi$
$263$	−432.000	−0.101286	−0.0506431	+	0.998717i	$0.516127\pi$
$264$	0	0
$265$	−570.000	−0.132131
$266$	0	0
$267$	4770.00	1.09333
$268$	0	0
$269$	54.0000	0.0122395	0.00611977	−	0.999981i	$-0.498052\pi$
$269$	54.0000	0.0122395	0.00611977	+	0.999981i	$0.498052\pi$
$270$	0	0
$271$	−6496.00	−1.45610	−0.728051	−	0.685522i	$-0.759574\pi$
$271$	−6496.00	−1.45610	−0.728051	+	0.685522i	$0.759574\pi$
$272$	0	0
$273$	960.000	0.212827
$274$	0	0
$275$	900.000	0.197353
$276$	0	0
$277$	−466.000	−0.101080	−0.0505401	−	0.998722i	$-0.516094\pi$
$277$	−466.000	−0.101080	−0.0505401	+	0.998722i	$0.516094\pi$
$278$	0	0
$279$	−144.000	−0.0308998
$280$	0	0
$281$	−4854.00	−1.03048	−0.515241	−	0.857045i	$-0.672298\pi$
$281$	−4854.00	−1.03048	−0.515241	+	0.857045i	$0.672298\pi$
$282$	0	0
$283$	−4516.00	−0.948581	−0.474290	−	0.880368i	$-0.657295\pi$
$283$	−4516.00	−0.948581	−0.474290	+	0.880368i	$0.657295\pi$
$284$	0	0
$285$	−2100.00	−0.436468
$286$	0	0
$287$	−12480.0	−2.56680
$288$	0	0
$289$	1171.00	0.238347
$290$	0	0
$291$	−582.000	−0.117242
$292$	0	0
$293$	8574.00	1.70955	0.854775	−	0.518998i	$-0.173695\pi$
$293$	8574.00	1.70955	0.854775	+	0.518998i	$0.173695\pi$
$294$	0	0
$295$	2580.00	0.509198
$296$	0	0
$297$	−972.000	−0.189903
$298$	0	0
$299$	1920.00	0.371359
$300$	0	0
$301$	−1664.00	−0.318642
$302$	0	0
$303$	−2394.00	−0.453900
$304$	0	0
$305$	−290.000	−0.0544438
$306$	0	0
$307$	3476.00	0.646208	0.323104	−	0.946363i	$-0.395274\pi$
$307$	3476.00	0.646208	0.323104	+	0.946363i	$0.395274\pi$
$308$	0	0
$309$	−816.000	−0.150229
$310$	0	0
$311$	2424.00	0.441969	0.220985	−	0.975277i	$-0.429073\pi$
$311$	2424.00	0.441969	0.220985	+	0.975277i	$0.429073\pi$
$312$	0	0
$313$	−1558.00	−0.281353	−0.140676	−	0.990056i	$-0.544928\pi$
$313$	−1558.00	−0.281353	−0.140676	+	0.990056i	$0.544928\pi$
$314$	0	0
$315$	1440.00	0.257571
$316$	0	0
$317$	−8538.00	−1.51275	−0.756375	−	0.654138i	$-0.773032\pi$
$317$	−8538.00	−1.51275	−0.756375	+	0.654138i	$0.773032\pi$
$318$	0	0
$319$	216.000	0.0379112
$320$	0	0
$321$	−468.000	−0.0813745
$322$	0	0
$323$	−10920.0	−1.88113
$324$	0	0
$325$	−250.000	−0.0426692
$326$	0	0
$327$	−4866.00	−0.822906
$328$	0	0
$329$	13056.0	2.18784
$330$	0	0
$331$	−988.000	−0.164065	−0.0820323	−	0.996630i	$-0.526141\pi$
$331$	−988.000	−0.164065	−0.0820323	+	0.996630i	$0.526141\pi$
$332$	0	0
$333$	−306.000	−0.0503564
$334$	0	0
$335$	−4460.00	−0.727391
$336$	0	0
$337$	2546.00	0.411541	0.205771	−	0.978600i	$-0.434030\pi$
$337$	2546.00	0.411541	0.205771	+	0.978600i	$0.434030\pi$
$338$	0	0
$339$	−3222.00	−0.516209
$340$	0	0
$341$	−576.000	−0.0914726
$342$	0	0
$343$	10816.0	1.70265
$344$	0	0
$345$	2880.00	0.449432
$346$	0	0
$347$	8556.00	1.32366	0.661830	−	0.749654i	$-0.269780\pi$
$347$	8556.00	1.32366	0.661830	+	0.749654i	$0.269780\pi$
$348$	0	0
$349$	−3706.00	−0.568417	−0.284209	−	0.958762i	$-0.591731\pi$
$349$	−3706.00	−0.568417	−0.284209	+	0.958762i	$0.591731\pi$
$350$	0	0
$351$	270.000	0.0410585
$352$	0	0
$353$	11394.0	1.71796	0.858982	−	0.512005i	$-0.171097\pi$
$353$	11394.0	1.71796	0.858982	+	0.512005i	$0.171097\pi$
$354$	0	0
$355$	−600.000	−0.0897034
$356$	0	0
$357$	7488.00	1.11010
$358$	0	0
$359$	−264.000	−0.0388117	−0.0194058	−	0.999812i	$-0.506177\pi$
$359$	−264.000	−0.0388117	−0.0194058	+	0.999812i	$0.506177\pi$
$360$	0	0
$361$	12741.0	1.85756
$362$	0	0
$363$	105.000	0.0151820
$364$	0	0
$365$	−3230.00	−0.463194
$366$	0	0
$367$	10232.0	1.45533	0.727665	−	0.685933i	$-0.240606\pi$
$367$	10232.0	1.45533	0.727665	+	0.685933i	$0.240606\pi$
$368$	0	0
$369$	−3510.00	−0.495185
$370$	0	0
$371$	−3648.00	−0.510498
$372$	0	0
$373$	−562.000	−0.0780141	−0.0390070	−	0.999239i	$-0.512419\pi$
$373$	−562.000	−0.0780141	−0.0390070	+	0.999239i	$0.512419\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−60.0000	−0.00819670
$378$	0	0
$379$	−7228.00	−0.979624	−0.489812	−	0.871828i	$-0.662935\pi$
$379$	−7228.00	−0.979624	−0.489812	+	0.871828i	$0.662935\pi$
$380$	0	0
$381$	4584.00	0.616392
$382$	0	0
$383$	−5736.00	−0.765263	−0.382632	−	0.923901i	$-0.624982\pi$
$383$	−5736.00	−0.765263	−0.382632	+	0.923901i	$0.624982\pi$
$384$	0	0
$385$	5760.00	0.762485
$386$	0	0
$387$	−468.000	−0.0614723
$388$	0	0
$389$	−9186.00	−1.19730	−0.598649	−	0.801012i	$-0.704295\pi$
$389$	−9186.00	−1.19730	−0.598649	+	0.801012i	$0.704295\pi$
$390$	0	0
$391$	14976.0	1.93700
$392$	0	0
$393$	−7236.00	−0.928773
$394$	0	0
$395$	−5840.00	−0.743905
$396$	0	0
$397$	−394.000	−0.0498093	−0.0249047	−	0.999690i	$-0.507928\pi$
$397$	−394.000	−0.0498093	−0.0249047	+	0.999690i	$0.507928\pi$
$398$	0	0
$399$	−13440.0	−1.68632
$400$	0	0
$401$	−1614.00	−0.200996	−0.100498	−	0.994937i	$-0.532044\pi$
$401$	−1614.00	−0.200996	−0.100498	+	0.994937i	$0.532044\pi$
$402$	0	0
$403$	160.000	0.0197771
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−1224.00	−0.149070
$408$	0	0
$409$	1034.00	0.125007	0.0625037	−	0.998045i	$-0.480091\pi$
$409$	1034.00	0.125007	0.0625037	+	0.998045i	$0.480091\pi$
$410$	0	0
$411$	−6318.00	−0.758258
$412$	0	0
$413$	16512.0	1.96732
$414$	0	0
$415$	−3660.00	−0.432921
$416$	0	0
$417$	1668.00	0.195881
$418$	0	0
$419$	3708.00	0.432333	0.216167	−	0.976356i	$-0.430645\pi$
$419$	3708.00	0.432333	0.216167	+	0.976356i	$0.430645\pi$
$420$	0	0
$421$	−4930.00	−0.570721	−0.285360	−	0.958420i	$-0.592113\pi$
$421$	−4930.00	−0.570721	−0.285360	+	0.958420i	$0.592113\pi$
$422$	0	0
$423$	3672.00	0.422077
$424$	0	0
$425$	−1950.00	−0.222562
$426$	0	0
$427$	−1856.00	−0.210347
$428$	0	0
$429$	1080.00	0.121545
$430$	0	0
$431$	−2592.00	−0.289680	−0.144840	−	0.989455i	$-0.546267\pi$
$431$	−2592.00	−0.289680	−0.144840	+	0.989455i	$0.546267\pi$
$432$	0	0
$433$	2162.00	0.239952	0.119976	−	0.992777i	$-0.461718\pi$
$433$	2162.00	0.239952	0.119976	+	0.992777i	$0.461718\pi$
$434$	0	0
$435$	−90.0000	−0.00991993
$436$	0	0
$437$	−26880.0	−2.94244
$438$	0	0
$439$	1352.00	0.146987	0.0734937	−	0.997296i	$-0.476585\pi$
$439$	1352.00	0.146987	0.0734937	+	0.997296i	$0.476585\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	0	0
$443$	5532.00	0.593303	0.296652	−	0.954986i	$-0.404130\pi$
$443$	5532.00	0.593303	0.296652	+	0.954986i	$0.404130\pi$
$444$	0	0
$445$	−7950.00	−0.846890
$446$	0	0
$447$	7254.00	0.767567
$448$	0	0
$449$	−3198.00	−0.336131	−0.168066	−	0.985776i	$-0.553752\pi$
$449$	−3198.00	−0.336131	−0.168066	+	0.985776i	$0.553752\pi$
$450$	0	0
$451$	−14040.0	−1.46589
$452$	0	0
$453$	−8520.00	−0.883674
$454$	0	0
$455$	−1600.00	−0.164855
$456$	0	0
$457$	−1510.00	−0.154562	−0.0772810	−	0.997009i	$-0.524624\pi$
$457$	−1510.00	−0.154562	−0.0772810	+	0.997009i	$0.524624\pi$
$458$	0	0
$459$	2106.00	0.214160
$460$	0	0
$461$	16086.0	1.62516	0.812581	−	0.582848i	$-0.198062\pi$
$461$	16086.0	1.62516	0.812581	+	0.582848i	$0.198062\pi$
$462$	0	0
$463$	5384.00	0.540423	0.270211	−	0.962801i	$-0.412906\pi$
$463$	5384.00	0.540423	0.270211	+	0.962801i	$0.412906\pi$
$464$	0	0
$465$	240.000	0.0239349
$466$	0	0
$467$	−2604.00	−0.258027	−0.129014	−	0.991643i	$-0.541181\pi$
$467$	−2604.00	−0.258027	−0.129014	+	0.991643i	$0.541181\pi$
$468$	0	0
$469$	−28544.0	−2.81032
$470$	0	0
$471$	−6162.00	−0.602824
$472$	0	0
$473$	−1872.00	−0.181976
$474$	0	0
$475$	3500.00	0.338086
$476$	0	0
$477$	−1026.00	−0.0984849
$478$	0	0
$479$	11136.0	1.06225	0.531124	−	0.847294i	$-0.321770\pi$
$479$	11136.0	1.06225	0.531124	+	0.847294i	$0.321770\pi$
$480$	0	0
$481$	340.000	0.0322301
$482$	0	0
$483$	18432.0	1.73641
$484$	0	0
$485$	970.000	0.0908153
$486$	0	0
$487$	14624.0	1.36073	0.680366	−	0.732872i	$-0.261821\pi$
$487$	14624.0	1.36073	0.680366	+	0.732872i	$0.261821\pi$
$488$	0	0
$489$	1380.00	0.127619
$490$	0	0
$491$	11844.0	1.08862	0.544310	−	0.838884i	$-0.316792\pi$
$491$	11844.0	1.08862	0.544310	+	0.838884i	$0.316792\pi$
$492$	0	0
$493$	−468.000	−0.0427539
$494$	0	0
$495$	1620.00	0.147098
$496$	0	0
$497$	−3840.00	−0.346575
$498$	0	0
$499$	−11284.0	−1.01231	−0.506154	−	0.862443i	$-0.668933\pi$
$499$	−11284.0	−1.01231	−0.506154	+	0.862443i	$0.668933\pi$
$500$	0	0
$501$	−6048.00	−0.539331
$502$	0	0
$503$	4032.00	0.357412	0.178706	−	0.983903i	$-0.442809\pi$
$503$	4032.00	0.357412	0.178706	+	0.983903i	$0.442809\pi$
$504$	0	0
$505$	3990.00	0.351589
$506$	0	0
$507$	6291.00	0.551071
$508$	0	0
$509$	−17562.0	−1.52932	−0.764658	−	0.644436i	$-0.777092\pi$
$509$	−17562.0	−1.52932	−0.764658	+	0.644436i	$0.777092\pi$
$510$	0	0
$511$	−20672.0	−1.78958
$512$	0	0
$513$	−3780.00	−0.325324
$514$	0	0
$515$	1360.00	0.116367
$516$	0	0
$517$	14688.0	1.24947
$518$	0	0
$519$	1854.00	0.156805
$520$	0	0
$521$	3162.00	0.265892	0.132946	−	0.991123i	$-0.457556\pi$
$521$	3162.00	0.265892	0.132946	+	0.991123i	$0.457556\pi$
$522$	0	0
$523$	6764.00	0.565524	0.282762	−	0.959190i	$-0.408749\pi$
$523$	6764.00	0.565524	0.282762	+	0.959190i	$0.408749\pi$
$524$	0	0
$525$	−2400.00	−0.199513
$526$	0	0
$527$	1248.00	0.103157
$528$	0	0
$529$	24697.0	2.02983
$530$	0	0
$531$	4644.00	0.379534
$532$	0	0
$533$	3900.00	0.316938
$534$	0	0
$535$	780.000	0.0630324
$536$	0	0
$537$	8892.00	0.714559
$538$	0	0
$539$	24516.0	1.95914
$540$	0	0
$541$	17798.0	1.41441	0.707205	−	0.707009i	$-0.249956\pi$
$541$	17798.0	1.41441	0.707205	+	0.707009i	$0.249956\pi$
$542$	0	0
$543$	1110.00	0.0877250
$544$	0	0
$545$	8110.00	0.637421
$546$	0	0
$547$	−19996.0	−1.56301	−0.781506	−	0.623898i	$-0.785548\pi$
$547$	−19996.0	−1.56301	−0.781506	+	0.623898i	$0.785548\pi$
$548$	0	0
$549$	−522.000	−0.0405800
$550$	0	0
$551$	840.000	0.0649459
$552$	0	0
$553$	−37376.0	−2.87412
$554$	0	0
$555$	510.000	0.0390059
$556$	0	0
$557$	11094.0	0.843928	0.421964	−	0.906613i	$-0.361341\pi$
$557$	11094.0	0.843928	0.421964	+	0.906613i	$0.361341\pi$
$558$	0	0
$559$	520.000	0.0393446
$560$	0	0
$561$	8424.00	0.633978
$562$	0	0
$563$	900.000	0.0673721	0.0336860	−	0.999432i	$-0.489275\pi$
$563$	900.000	0.0673721	0.0336860	+	0.999432i	$0.489275\pi$
$564$	0	0
$565$	5370.00	0.399854
$566$	0	0
$567$	2592.00	0.191982
$568$	0	0
$569$	7914.00	0.583079	0.291540	−	0.956559i	$-0.405832\pi$
$569$	7914.00	0.583079	0.291540	+	0.956559i	$0.405832\pi$
$570$	0	0
$571$	−2380.00	−0.174431	−0.0872153	−	0.996189i	$-0.527797\pi$
$571$	−2380.00	−0.174431	−0.0872153	+	0.996189i	$0.527797\pi$
$572$	0	0
$573$	−3312.00	−0.241467
$574$	0	0
$575$	−4800.00	−0.348128
$576$	0	0
$577$	−25726.0	−1.85613	−0.928065	−	0.372417i	$-0.878529\pi$
$577$	−25726.0	−1.85613	−0.928065	+	0.372417i	$0.878529\pi$
$578$	0	0
$579$	7194.00	0.516360
$580$	0	0
$581$	−23424.0	−1.67262
$582$	0	0
$583$	−4104.00	−0.291544
$584$	0	0
$585$	−450.000	−0.0318038
$586$	0	0
$587$	3612.00	0.253975	0.126987	−	0.991904i	$-0.459469\pi$
$587$	3612.00	0.253975	0.126987	+	0.991904i	$0.459469\pi$
$588$	0	0
$589$	−2240.00	−0.156702
$590$	0	0
$591$	−3834.00	−0.266852
$592$	0	0
$593$	2898.00	0.200686	0.100343	−	0.994953i	$-0.468006\pi$
$593$	2898.00	0.200686	0.100343	+	0.994953i	$0.468006\pi$
$594$	0	0
$595$	−12480.0	−0.859883
$596$	0	0
$597$	−13416.0	−0.919733
$598$	0	0
$599$	2664.00	0.181716	0.0908582	−	0.995864i	$-0.471039\pi$
$599$	2664.00	0.181716	0.0908582	+	0.995864i	$0.471039\pi$
$600$	0	0
$601$	−502.000	−0.0340716	−0.0170358	−	0.999855i	$-0.505423\pi$
$601$	−502.000	−0.0340716	−0.0170358	+	0.999855i	$0.505423\pi$
$602$	0	0
$603$	−8028.00	−0.542165
$604$	0	0
$605$	−175.000	−0.0117599
$606$	0	0
$607$	7976.00	0.533337	0.266669	−	0.963788i	$-0.414077\pi$
$607$	7976.00	0.533337	0.266669	+	0.963788i	$0.414077\pi$
$608$	0	0
$609$	−576.000	−0.0383263
$610$	0	0
$611$	−4080.00	−0.270146
$612$	0	0
$613$	20414.0	1.34505	0.672523	−	0.740076i	$-0.265210\pi$
$613$	20414.0	1.34505	0.672523	+	0.740076i	$0.265210\pi$
$614$	0	0
$615$	5850.00	0.383569
$616$	0	0
$617$	−6342.00	−0.413808	−0.206904	−	0.978361i	$-0.566339\pi$
$617$	−6342.00	−0.413808	−0.206904	+	0.978361i	$0.566339\pi$
$618$	0	0
$619$	22676.0	1.47242	0.736208	−	0.676755i	$-0.236615\pi$
$619$	22676.0	1.47242	0.736208	+	0.676755i	$0.236615\pi$
$620$	0	0
$621$	5184.00	0.334987
$622$	0	0
$623$	−50880.0	−3.27201
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−15120.0	−0.963054
$628$	0	0
$629$	2652.00	0.168112
$630$	0	0
$631$	−7048.00	−0.444654	−0.222327	−	0.974972i	$-0.571365\pi$
$631$	−7048.00	−0.444654	−0.222327	+	0.974972i	$0.571365\pi$
$632$	0	0
$633$	−4020.00	−0.252418
$634$	0	0
$635$	−7640.00	−0.477455
$636$	0	0
$637$	−6810.00	−0.423582
$638$	0	0
$639$	−1080.00	−0.0668609
$640$	0	0
$641$	−20286.0	−1.25000	−0.624999	−	0.780625i	$-0.714901\pi$
$641$	−20286.0	−1.25000	−0.624999	+	0.780625i	$0.714901\pi$
$642$	0	0
$643$	−16108.0	−0.987928	−0.493964	−	0.869482i	$-0.664452\pi$
$643$	−16108.0	−0.987928	−0.493964	+	0.869482i	$0.664452\pi$
$644$	0	0
$645$	780.000	0.0476162
$646$	0	0
$647$	27456.0	1.66833	0.834163	−	0.551518i	$-0.185951\pi$
$647$	27456.0	1.66833	0.834163	+	0.551518i	$0.185951\pi$
$648$	0	0
$649$	18576.0	1.12353
$650$	0	0
$651$	1536.00	0.0924740
$652$	0	0
$653$	−12522.0	−0.750419	−0.375210	−	0.926940i	$-0.622429\pi$
$653$	−12522.0	−0.750419	−0.375210	+	0.926940i	$0.622429\pi$
$654$	0	0
$655$	12060.0	0.719425
$656$	0	0
$657$	−5814.00	−0.345245
$658$	0	0
$659$	−16308.0	−0.963990	−0.481995	−	0.876174i	$-0.660088\pi$
$659$	−16308.0	−0.963990	−0.481995	+	0.876174i	$0.660088\pi$
$660$	0	0
$661$	32078.0	1.88758	0.943789	−	0.330547i	$-0.107233\pi$
$661$	32078.0	1.88758	0.943789	+	0.330547i	$0.107233\pi$
$662$	0	0
$663$	−2340.00	−0.137071
$664$	0	0
$665$	22400.0	1.30622
$666$	0	0
$667$	−1152.00	−0.0668750
$668$	0	0
$669$	−7080.00	−0.409161
$670$	0	0
$671$	−2088.00	−0.120129
$672$	0	0
$673$	4610.00	0.264045	0.132023	−	0.991247i	$-0.457853\pi$
$673$	4610.00	0.264045	0.132023	+	0.991247i	$0.457853\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	10782.0	0.612091	0.306046	−	0.952017i	$-0.400994\pi$
$677$	10782.0	0.612091	0.306046	+	0.952017i	$0.400994\pi$
$678$	0	0
$679$	6208.00	0.350871
$680$	0	0
$681$	−4140.00	−0.232959
$682$	0	0
$683$	2892.00	0.162019	0.0810097	−	0.996713i	$-0.474186\pi$
$683$	2892.00	0.162019	0.0810097	+	0.996713i	$0.474186\pi$
$684$	0	0
$685$	10530.0	0.587344
$686$	0	0
$687$	−5082.00	−0.282228
$688$	0	0
$689$	1140.00	0.0630342
$690$	0	0
$691$	−29572.0	−1.62803	−0.814017	−	0.580841i	$-0.802724\pi$
$691$	−29572.0	−1.62803	−0.814017	+	0.580841i	$0.802724\pi$
$692$	0	0
$693$	10368.0	0.568323
$694$	0	0
$695$	−2780.00	−0.151729
$696$	0	0
$697$	30420.0	1.65314
$698$	0	0
$699$	15570.0	0.842506
$700$	0	0
$701$	5766.00	0.310669	0.155334	−	0.987862i	$-0.450354\pi$
$701$	5766.00	0.310669	0.155334	+	0.987862i	$0.450354\pi$
$702$	0	0
$703$	−4760.00	−0.255372
$704$	0	0
$705$	−6120.00	−0.326940
$706$	0	0
$707$	25536.0	1.35839
$708$	0	0
$709$	3326.00	0.176178	0.0880892	−	0.996113i	$-0.471924\pi$
$709$	3326.00	0.176178	0.0880892	+	0.996113i	$0.471924\pi$
$710$	0	0
$711$	−10512.0	−0.554474
$712$	0	0
$713$	3072.00	0.161357
$714$	0	0
$715$	−1800.00	−0.0941485
$716$	0	0
$717$	−7056.00	−0.367519
$718$	0	0
$719$	−7728.00	−0.400843	−0.200421	−	0.979710i	$-0.564231\pi$
$719$	−7728.00	−0.400843	−0.200421	+	0.979710i	$0.564231\pi$
$720$	0	0
$721$	8704.00	0.449589
$722$	0	0
$723$	10506.0	0.540418
$724$	0	0
$725$	150.000	0.00768395
$726$	0	0
$727$	−21616.0	−1.10274	−0.551371	−	0.834260i	$-0.685895\pi$
$727$	−21616.0	−1.10274	−0.551371	+	0.834260i	$0.685895\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	4056.00	0.205221
$732$	0	0
$733$	10118.0	0.509846	0.254923	−	0.966961i	$-0.417950\pi$
$733$	10118.0	0.509846	0.254923	+	0.966961i	$0.417950\pi$
$734$	0	0
$735$	−10215.0	−0.512634
$736$	0	0
$737$	−32112.0	−1.60497
$738$	0	0
$739$	10460.0	0.520673	0.260336	−	0.965518i	$-0.416167\pi$
$739$	10460.0	0.520673	0.260336	+	0.965518i	$0.416167\pi$
$740$	0	0
$741$	4200.00	0.208220
$742$	0	0
$743$	17232.0	0.850849	0.425424	−	0.904994i	$-0.360125\pi$
$743$	17232.0	0.850849	0.425424	+	0.904994i	$0.360125\pi$
$744$	0	0
$745$	−12090.0	−0.594555
$746$	0	0
$747$	−6588.00	−0.322680
$748$	0	0
$749$	4992.00	0.243530
$750$	0	0
$751$	26912.0	1.30763	0.653817	−	0.756653i	$-0.273167\pi$
$751$	26912.0	1.30763	0.653817	+	0.756653i	$0.273167\pi$
$752$	0	0
$753$	−14364.0	−0.695157
$754$	0	0
$755$	14200.0	0.684491
$756$	0	0
$757$	13838.0	0.664400	0.332200	−	0.943209i	$-0.392209\pi$
$757$	13838.0	0.664400	0.332200	+	0.943209i	$0.392209\pi$
$758$	0	0
$759$	20736.0	0.991659
$760$	0	0
$761$	−17238.0	−0.821126	−0.410563	−	0.911832i	$-0.634668\pi$
$761$	−17238.0	−0.821126	−0.410563	+	0.911832i	$0.634668\pi$
$762$	0	0
$763$	51904.0	2.46271
$764$	0	0
$765$	−3510.00	−0.165888
$766$	0	0
$767$	−5160.00	−0.242916
$768$	0	0
$769$	21698.0	1.01749	0.508745	−	0.860917i	$-0.330110\pi$
$769$	21698.0	1.01749	0.508745	+	0.860917i	$0.330110\pi$
$770$	0	0
$771$	−4518.00	−0.211040
$772$	0	0
$773$	18366.0	0.854565	0.427283	−	0.904118i	$-0.359471\pi$
$773$	18366.0	0.854565	0.427283	+	0.904118i	$0.359471\pi$
$774$	0	0
$775$	−400.000	−0.0185399
$776$	0	0
$777$	3264.00	0.150702
$778$	0	0
$779$	−54600.0	−2.51123
$780$	0	0
$781$	−4320.00	−0.197928
$782$	0	0
$783$	−162.000	−0.00739388
$784$	0	0
$785$	10270.0	0.466945
$786$	0	0
$787$	−30316.0	−1.37312	−0.686562	−	0.727071i	$-0.740881\pi$
$787$	−30316.0	−1.37312	−0.686562	+	0.727071i	$0.740881\pi$
$788$	0	0
$789$	1296.00	0.0584776
$790$	0	0
$791$	34368.0	1.54486
$792$	0	0
$793$	580.000	0.0259728
$794$	0	0
$795$	1710.00	0.0762861
$796$	0	0
$797$	7494.00	0.333063	0.166531	−	0.986036i	$-0.446743\pi$
$797$	7494.00	0.333063	0.166531	+	0.986036i	$0.446743\pi$
$798$	0	0
$799$	−31824.0	−1.40908
$800$	0	0
$801$	−14310.0	−0.631235
$802$	0	0
$803$	−23256.0	−1.02203
$804$	0	0
$805$	−30720.0	−1.34502
$806$	0	0
$807$	−162.000	−0.00706651
$808$	0	0
$809$	−11526.0	−0.500906	−0.250453	−	0.968129i	$-0.580579\pi$
$809$	−11526.0	−0.500906	−0.250453	+	0.968129i	$0.580579\pi$
$810$	0	0
$811$	−33820.0	−1.46434	−0.732171	−	0.681121i	$-0.761493\pi$
$811$	−33820.0	−1.46434	−0.732171	+	0.681121i	$0.761493\pi$
$812$	0	0
$813$	19488.0	0.840681
$814$	0	0
$815$	−2300.00	−0.0988534
$816$	0	0
$817$	−7280.00	−0.311744
$818$	0	0
$819$	−2880.00	−0.122876
$820$	0	0
$821$	19566.0	0.831739	0.415870	−	0.909424i	$-0.363477\pi$
$821$	19566.0	0.831739	0.415870	+	0.909424i	$0.363477\pi$
$822$	0	0
$823$	−40096.0	−1.69825	−0.849124	−	0.528193i	$-0.822870\pi$
$823$	−40096.0	−1.69825	−0.849124	+	0.528193i	$0.822870\pi$
$824$	0	0
$825$	−2700.00	−0.113942
$826$	0	0
$827$	31884.0	1.34065	0.670324	−	0.742069i	$-0.266155\pi$
$827$	31884.0	1.34065	0.670324	+	0.742069i	$0.266155\pi$
$828$	0	0
$829$	−24442.0	−1.02401	−0.512006	−	0.858982i	$-0.671097\pi$
$829$	−24442.0	−1.02401	−0.512006	+	0.858982i	$0.671097\pi$
$830$	0	0
$831$	1398.00	0.0583587
$832$	0	0
$833$	−53118.0	−2.20940
$834$	0	0
$835$	10080.0	0.417764
$836$	0	0
$837$	432.000	0.0178400
$838$	0	0
$839$	−31944.0	−1.31446	−0.657228	−	0.753691i	$-0.728271\pi$
$839$	−31944.0	−1.31446	−0.657228	+	0.753691i	$0.728271\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	14562.0	0.594949
$844$	0	0
$845$	−10485.0	−0.426858
$846$	0	0
$847$	−1120.00	−0.0454352
$848$	0	0
$849$	13548.0	0.547663
$850$	0	0
$851$	6528.00	0.262958
$852$	0	0
$853$	17486.0	0.701887	0.350943	−	0.936397i	$-0.385861\pi$
$853$	17486.0	0.701887	0.350943	+	0.936397i	$0.385861\pi$
$854$	0	0
$855$	6300.00	0.251995
$856$	0	0
$857$	43434.0	1.73125	0.865623	−	0.500697i	$-0.166923\pi$
$857$	43434.0	1.73125	0.865623	+	0.500697i	$0.166923\pi$
$858$	0	0
$859$	10820.0	0.429771	0.214886	−	0.976639i	$-0.431062\pi$
$859$	10820.0	0.429771	0.214886	+	0.976639i	$0.431062\pi$
$860$	0	0
$861$	37440.0	1.48194
$862$	0	0
$863$	29976.0	1.18238	0.591191	−	0.806532i	$-0.298658\pi$
$863$	29976.0	1.18238	0.591191	+	0.806532i	$0.298658\pi$
$864$	0	0
$865$	−3090.00	−0.121460
$866$	0	0
$867$	−3513.00	−0.137610
$868$	0	0
$869$	−42048.0	−1.64140
$870$	0	0
$871$	8920.00	0.347007
$872$	0	0
$873$	1746.00	0.0676897
$874$	0	0
$875$	4000.00	0.154542
$876$	0	0
$877$	−40522.0	−1.56024	−0.780120	−	0.625630i	$-0.784842\pi$
$877$	−40522.0	−1.56024	−0.780120	+	0.625630i	$0.784842\pi$
$878$	0	0
$879$	−25722.0	−0.987010
$880$	0	0
$881$	15570.0	0.595422	0.297711	−	0.954656i	$-0.403777\pi$
$881$	15570.0	0.595422	0.297711	+	0.954656i	$0.403777\pi$
$882$	0	0
$883$	−1084.00	−0.0413131	−0.0206566	−	0.999787i	$-0.506576\pi$
$883$	−1084.00	−0.0413131	−0.0206566	+	0.999787i	$0.506576\pi$
$884$	0	0
$885$	−7740.00	−0.293986
$886$	0	0
$887$	−8208.00	−0.310708	−0.155354	−	0.987859i	$-0.549652\pi$
$887$	−8208.00	−0.310708	−0.155354	+	0.987859i	$0.549652\pi$
$888$	0	0
$889$	−48896.0	−1.84468
$890$	0	0
$891$	2916.00	0.109640
$892$	0	0
$893$	57120.0	2.14048
$894$	0	0
$895$	−14820.0	−0.553495
$896$	0	0
$897$	−5760.00	−0.214404
$898$	0	0
$899$	−96.0000	−0.00356149
$900$	0	0
$901$	8892.00	0.328785
$902$	0	0
$903$	4992.00	0.183968
$904$	0	0
$905$	−1850.00	−0.0679515
$906$	0	0
$907$	34076.0	1.24749	0.623746	−	0.781627i	$-0.285610\pi$
$907$	34076.0	1.24749	0.623746	+	0.781627i	$0.285610\pi$
$908$	0	0
$909$	7182.00	0.262059
$910$	0	0
$911$	−15072.0	−0.548142	−0.274071	−	0.961709i	$-0.588370\pi$
$911$	−15072.0	−0.548142	−0.274071	+	0.961709i	$0.588370\pi$
$912$	0	0
$913$	−26352.0	−0.955229
$914$	0	0
$915$	870.000	0.0314331
$916$	0	0
$917$	77184.0	2.77954
$918$	0	0
$919$	24392.0	0.875536	0.437768	−	0.899088i	$-0.355769\pi$
$919$	24392.0	0.875536	0.437768	+	0.899088i	$0.355769\pi$
$920$	0	0
$921$	−10428.0	−0.373088
$922$	0	0
$923$	1200.00	0.0427936
$924$	0	0
$925$	−850.000	−0.0302139
$926$	0	0
$927$	2448.00	0.0867345
$928$	0	0
$929$	13602.0	0.480374	0.240187	−	0.970727i	$-0.422791\pi$
$929$	13602.0	0.480374	0.240187	+	0.970727i	$0.422791\pi$
$930$	0	0
$931$	95340.0	3.35622
$932$	0	0
$933$	−7272.00	−0.255171
$934$	0	0
$935$	−14040.0	−0.491077
$936$	0	0
$937$	−47974.0	−1.67262	−0.836309	−	0.548259i	$-0.815291\pi$
$937$	−47974.0	−1.67262	−0.836309	+	0.548259i	$0.815291\pi$
$938$	0	0
$939$	4674.00	0.162439
$940$	0	0
$941$	−48330.0	−1.67430	−0.837148	−	0.546976i	$-0.815779\pi$
$941$	−48330.0	−1.67430	−0.837148	+	0.546976i	$0.815779\pi$
$942$	0	0
$943$	74880.0	2.58582
$944$	0	0
$945$	−4320.00	−0.148709
$946$	0	0
$947$	−9324.00	−0.319946	−0.159973	−	0.987121i	$-0.551141\pi$
$947$	−9324.00	−0.319946	−0.159973	+	0.987121i	$0.551141\pi$
$948$	0	0
$949$	6460.00	0.220970
$950$	0	0
$951$	25614.0	0.873387
$952$	0	0
$953$	−14838.0	−0.504355	−0.252177	−	0.967681i	$-0.581147\pi$
$953$	−14838.0	−0.504355	−0.252177	+	0.967681i	$0.581147\pi$
$954$	0	0
$955$	5520.00	0.187040
$956$	0	0
$957$	−648.000	−0.0218881
$958$	0	0
$959$	67392.0	2.26924
$960$	0	0
$961$	−29535.0	−0.991407
$962$	0	0
$963$	1404.00	0.0469816
$964$	0	0
$965$	−11990.0	−0.399971
$966$	0	0
$967$	11360.0	0.377780	0.188890	−	0.981998i	$-0.439511\pi$
$967$	11360.0	0.377780	0.188890	+	0.981998i	$0.439511\pi$
$968$	0	0
$969$	32760.0	1.08607
$970$	0	0
$971$	−6972.00	−0.230424	−0.115212	−	0.993341i	$-0.536755\pi$
$971$	−6972.00	−0.230424	−0.115212	+	0.993341i	$0.536755\pi$
$972$	0	0
$973$	−17792.0	−0.586213
$974$	0	0
$975$	750.000	0.0246351
$976$	0	0
$977$	−41166.0	−1.34802	−0.674011	−	0.738722i	$-0.735430\pi$
$977$	−41166.0	−1.34802	−0.674011	+	0.738722i	$0.735430\pi$
$978$	0	0
$979$	−57240.0	−1.86864
$980$	0	0
$981$	14598.0	0.475105
$982$	0	0
$983$	22464.0	0.728881	0.364441	−	0.931227i	$-0.381260\pi$
$983$	22464.0	0.728881	0.364441	+	0.931227i	$0.381260\pi$
$984$	0	0
$985$	6390.00	0.206703
$986$	0	0
$987$	−39168.0	−1.26315
$988$	0	0
$989$	9984.00	0.321004
$990$	0	0
$991$	−10192.0	−0.326700	−0.163350	−	0.986568i	$-0.552230\pi$
$991$	−10192.0	−0.326700	−0.163350	+	0.986568i	$0.552230\pi$
$992$	0	0
$993$	2964.00	0.0947228
$994$	0	0
$995$	22360.0	0.712422
$996$	0	0
$997$	−322.000	−0.0102285	−0.00511426	−	0.999987i	$-0.501628\pi$
$997$	−322.000	−0.0102285	−0.00511426	+	0.999987i	$0.501628\pi$
$998$	0	0
$999$	918.000	0.0290733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.a.b.1.1	✓	1
3.2	odd	2		180.4.a.c.1.1		1
4.3	odd	2		240.4.a.j.1.1		1
5.2	odd	4		300.4.d.d.49.2		2
5.3	odd	4		300.4.d.d.49.1		2
5.4	even	2		300.4.a.e.1.1		1
8.3	odd	2		960.4.a.a.1.1		1
8.5	even	2		960.4.a.bb.1.1		1
9.2	odd	6		1620.4.i.g.1081.1		2
9.4	even	3		1620.4.i.a.541.1		2
9.5	odd	6		1620.4.i.g.541.1		2
9.7	even	3		1620.4.i.a.1081.1		2
12.11	even	2		720.4.a.c.1.1		1
15.2	even	4		900.4.d.b.649.2		2
15.8	even	4		900.4.d.b.649.1		2
15.14	odd	2		900.4.a.b.1.1		1
20.3	even	4		1200.4.f.e.49.2		2
20.7	even	4		1200.4.f.e.49.1		2
20.19	odd	2		1200.4.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.a.b.1.1	✓	1	1.1	even	1	trivial
180.4.a.c.1.1		1	3.2	odd	2
240.4.a.j.1.1		1	4.3	odd	2
300.4.a.e.1.1		1	5.4	even	2
300.4.d.d.49.1		2	5.3	odd	4
300.4.d.d.49.2		2	5.2	odd	4
720.4.a.c.1.1		1	12.11	even	2
900.4.a.b.1.1		1	15.14	odd	2
900.4.d.b.649.1		2	15.8	even	4
900.4.d.b.649.2		2	15.2	even	4
960.4.a.a.1.1		1	8.3	odd	2
960.4.a.bb.1.1		1	8.5	even	2
1200.4.a.s.1.1		1	20.19	odd	2
1200.4.f.e.49.1		2	20.7	even	4
1200.4.f.e.49.2		2	20.3	even	4
1620.4.i.a.541.1		2	9.4	even	3
1620.4.i.a.1081.1		2	9.7	even	3
1620.4.i.g.541.1		2	9.5	odd	6
1620.4.i.g.1081.1		2	9.2	odd	6

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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	60.a (trivial)

Introduction

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Representations

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Database

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