LMFDB - Embedded newform 12.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [12,4,Mod(1,12)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$12 = 2^{2} \cdot 3$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	12.a (trivial)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.708022920069$
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	$\chi$	$=$	12.1

$q$ -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} -18.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +36.0000 q^{11} -10.0000 q^{13} -54.0000 q^{15} +18.0000 q^{17} -100.000 q^{19} +24.0000 q^{21} +72.0000 q^{23} +199.000 q^{25} +27.0000 q^{27} -234.000 q^{29} -16.0000 q^{31} +108.000 q^{33} -144.000 q^{35} -226.000 q^{37} -30.0000 q^{39} +90.0000 q^{41} +452.000 q^{43} -162.000 q^{45} +432.000 q^{47} -279.000 q^{49} +54.0000 q^{51} +414.000 q^{53} -648.000 q^{55} -300.000 q^{57} -684.000 q^{59} +422.000 q^{61} +72.0000 q^{63} +180.000 q^{65} +332.000 q^{67} +216.000 q^{69} -360.000 q^{71} +26.0000 q^{73} +597.000 q^{75} +288.000 q^{77} +512.000 q^{79} +81.0000 q^{81} -1188.00 q^{83} -324.000 q^{85} -702.000 q^{87} -630.000 q^{89} -80.0000 q^{91} -48.0000 q^{93} +1800.00 q^{95} -1054.00 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
$5$	−18.0000	−1.60997	−0.804984	−	0.593296i	$-0.797826\pi$
$5$	−18.0000	−1.60997	−0.804984	+	0.593296i	$0.797826\pi$
$6$	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
$11$	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
$15$	−54.0000	−0.929516
$16$	0	0
$17$	18.0000	0.256802	0.128401	−	0.991722i	$-0.459015\pi$
$17$	18.0000	0.256802	0.128401	+	0.991722i	$0.459015\pi$
$18$	0	0
$19$	−100.000	−1.20745	−0.603726	−	0.797192i	$-0.706318\pi$
$19$	−100.000	−1.20745	−0.603726	+	0.797192i	$0.706318\pi$
$20$	0	0
$21$	24.0000	0.249392
$22$	0	0
$23$	72.0000	0.652741	0.326370	−	0.945242i	$-0.394174\pi$
$23$	72.0000	0.652741	0.326370	+	0.945242i	$0.394174\pi$
$24$	0	0
$25$	199.000	1.59200
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−234.000	−1.49837	−0.749185	−	0.662361i	$-0.769554\pi$
$29$	−234.000	−1.49837	−0.749185	+	0.662361i	$0.769554\pi$
$30$	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
$35$	−144.000	−0.695441
$36$	0	0
$37$	−226.000	−1.00417	−0.502083	−	0.864819i	$-0.667433\pi$
$37$	−226.000	−1.00417	−0.502083	+	0.864819i	$0.667433\pi$
$38$	0	0
$39$	−30.0000	−0.123176
$40$	0	0
$41$	90.0000	0.342820	0.171410	−	0.985200i	$-0.445168\pi$
$41$	90.0000	0.342820	0.171410	+	0.985200i	$0.445168\pi$
$42$	0	0
$43$	452.000	1.60301	0.801504	−	0.597989i	$-0.204033\pi$
$43$	452.000	1.60301	0.801504	+	0.597989i	$0.204033\pi$
$44$	0	0
$45$	−162.000	−0.536656
$46$	0	0
$47$	432.000	1.34072	0.670358	−	0.742038i	$-0.266140\pi$
$47$	432.000	1.34072	0.670358	+	0.742038i	$0.266140\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	54.0000	0.148265
$52$	0	0
$53$	414.000	1.07297	0.536484	−	0.843911i	$-0.319752\pi$
$53$	414.000	1.07297	0.536484	+	0.843911i	$0.319752\pi$
$54$	0	0
$55$	−648.000	−1.58866
$56$	0	0
$57$	−300.000	−0.697122
$58$	0	0
$59$	−684.000	−1.50931	−0.754654	−	0.656123i	$-0.772195\pi$
$59$	−684.000	−1.50931	−0.754654	+	0.656123i	$0.772195\pi$
$60$	0	0
$61$	422.000	0.885763	0.442882	−	0.896580i	$-0.353956\pi$
$61$	422.000	0.885763	0.442882	+	0.896580i	$0.353956\pi$
$62$	0	0
$63$	72.0000	0.143986
$64$	0	0
$65$	180.000	0.343481
$66$	0	0
$67$	332.000	0.605377	0.302688	−	0.953090i	$-0.402116\pi$
$67$	332.000	0.605377	0.302688	+	0.953090i	$0.402116\pi$
$68$	0	0
$69$	216.000	0.376860
$70$	0	0
$71$	−360.000	−0.601748	−0.300874	−	0.953664i	$-0.597278\pi$
$71$	−360.000	−0.601748	−0.300874	+	0.953664i	$0.597278\pi$
$72$	0	0
$73$	26.0000	0.0416859	0.0208429	−	0.999783i	$-0.493365\pi$
$73$	26.0000	0.0416859	0.0208429	+	0.999783i	$0.493365\pi$
$74$	0	0
$75$	597.000	0.919142
$76$	0	0
$77$	288.000	0.426242
$78$	0	0
$79$	512.000	0.729171	0.364585	−	0.931170i	$-0.381211\pi$
$79$	512.000	0.729171	0.364585	+	0.931170i	$0.381211\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1188.00	−1.57108	−0.785542	−	0.618809i	$-0.787616\pi$
$83$	−1188.00	−1.57108	−0.785542	+	0.618809i	$0.787616\pi$
$84$	0	0
$85$	−324.000	−0.413444
$86$	0	0
$87$	−702.000	−0.865084
$88$	0	0
$89$	−630.000	−0.750336	−0.375168	−	0.926957i	$-0.622415\pi$
$89$	−630.000	−0.750336	−0.375168	+	0.926957i	$0.622415\pi$
$90$	0	0
$91$	−80.0000	−0.0921569
$92$	0	0
$93$	−48.0000	−0.0535201
$94$	0	0
$95$	1800.00	1.94396
$96$	0	0
$97$	−1054.00	−1.10327	−0.551637	−	0.834085i	$-0.685996\pi$
$97$	−1054.00	−1.10327	−0.551637	+	0.834085i	$0.685996\pi$
$98$	0	0
$99$	324.000	0.328921
$100$	0	0
$101$	558.000	0.549733	0.274867	−	0.961482i	$-0.411366\pi$
$101$	558.000	0.549733	0.274867	+	0.961482i	$0.411366\pi$
$102$	0	0
$103$	8.00000	0.00765304	0.00382652	−	0.999993i	$-0.498782\pi$
$103$	8.00000	0.00765304	0.00382652	+	0.999993i	$0.498782\pi$
$104$	0	0
$105$	−432.000	−0.401513
$106$	0	0
$107$	1764.00	1.59376	0.796880	−	0.604138i	$-0.206482\pi$
$107$	1764.00	1.59376	0.796880	+	0.604138i	$0.206482\pi$
$108$	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
$111$	−678.000	−0.579756
$112$	0	0
$113$	−1134.00	−0.944051	−0.472025	−	0.881585i	$-0.656477\pi$
$113$	−1134.00	−0.944051	−0.472025	+	0.881585i	$0.656477\pi$
$114$	0	0
$115$	−1296.00	−1.05089
$116$	0	0
$117$	−90.0000	−0.0711154
$118$	0	0
$119$	144.000	0.110928
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	270.000	0.197927
$124$	0	0
$125$	−1332.00	−0.953102
$126$	0	0
$127$	−592.000	−0.413634	−0.206817	−	0.978380i	$-0.566310\pi$
$127$	−592.000	−0.413634	−0.206817	+	0.978380i	$0.566310\pi$
$128$	0	0
$129$	1356.00	0.925497
$130$	0	0
$131$	−1908.00	−1.27254	−0.636270	−	0.771466i	$-0.719524\pi$
$131$	−1908.00	−1.27254	−0.636270	+	0.771466i	$0.719524\pi$
$132$	0	0
$133$	−800.000	−0.521570
$134$	0	0
$135$	−486.000	−0.309839
$136$	0	0
$137$	954.000	0.594932	0.297466	−	0.954732i	$-0.403858\pi$
$137$	954.000	0.594932	0.297466	+	0.954732i	$0.403858\pi$
$138$	0	0
$139$	2564.00	1.56457	0.782286	−	0.622919i	$-0.214053\pi$
$139$	2564.00	1.56457	0.782286	+	0.622919i	$0.214053\pi$
$140$	0	0
$141$	1296.00	0.774063
$142$	0	0
$143$	−360.000	−0.210522
$144$	0	0
$145$	4212.00	2.41233
$146$	0	0
$147$	−837.000	−0.469623
$148$	0	0
$149$	−738.000	−0.405767	−0.202884	−	0.979203i	$-0.565031\pi$
$149$	−738.000	−0.405767	−0.202884	+	0.979203i	$0.565031\pi$
$150$	0	0
$151$	−2440.00	−1.31500	−0.657498	−	0.753456i	$-0.728385\pi$
$151$	−2440.00	−1.31500	−0.657498	+	0.753456i	$0.728385\pi$
$152$	0	0
$153$	162.000	0.0856008
$154$	0	0
$155$	288.000	0.149243
$156$	0	0
$157$	−2554.00	−1.29829	−0.649145	−	0.760665i	$-0.724873\pi$
$157$	−2554.00	−1.29829	−0.649145	+	0.760665i	$0.724873\pi$
$158$	0	0
$159$	1242.00	0.619478
$160$	0	0
$161$	576.000	0.281958
$162$	0	0
$163$	−820.000	−0.394033	−0.197016	−	0.980400i	$-0.563125\pi$
$163$	−820.000	−0.394033	−0.197016	+	0.980400i	$0.563125\pi$
$164$	0	0
$165$	−1944.00	−0.917213
$166$	0	0
$167$	1944.00	0.900786	0.450393	−	0.892830i	$-0.351284\pi$
$167$	1944.00	0.900786	0.450393	+	0.892830i	$0.351284\pi$
$168$	0	0
$169$	−2097.00	−0.954483
$170$	0	0
$171$	−900.000	−0.402484
$172$	0	0
$173$	−1242.00	−0.545824	−0.272912	−	0.962039i	$-0.587987\pi$
$173$	−1242.00	−0.545824	−0.272912	+	0.962039i	$0.587987\pi$
$174$	0	0
$175$	1592.00	0.687679
$176$	0	0
$177$	−2052.00	−0.871400
$178$	0	0
$179$	1116.00	0.465999	0.232999	−	0.972477i	$-0.425146\pi$
$179$	1116.00	0.465999	0.232999	+	0.972477i	$0.425146\pi$
$180$	0	0
$181$	1070.00	0.439406	0.219703	−	0.975567i	$-0.429491\pi$
$181$	1070.00	0.439406	0.219703	+	0.975567i	$0.429491\pi$
$182$	0	0
$183$	1266.00	0.511396
$184$	0	0
$185$	4068.00	1.61668
$186$	0	0
$187$	648.000	0.253403
$188$	0	0
$189$	216.000	0.0831306
$190$	0	0
$191$	−576.000	−0.218209	−0.109104	−	0.994030i	$-0.534798\pi$
$191$	−576.000	−0.218209	−0.109104	+	0.994030i	$0.534798\pi$
$192$	0	0
$193$	−1342.00	−0.500514	−0.250257	−	0.968179i	$-0.580515\pi$
$193$	−1342.00	−0.500514	−0.250257	+	0.968179i	$0.580515\pi$
$194$	0	0
$195$	540.000	0.198309
$196$	0	0
$197$	1422.00	0.514281	0.257140	−	0.966374i	$-0.417220\pi$
$197$	1422.00	0.514281	0.257140	+	0.966374i	$0.417220\pi$
$198$	0	0
$199$	872.000	0.310625	0.155313	−	0.987865i	$-0.450361\pi$
$199$	872.000	0.310625	0.155313	+	0.987865i	$0.450361\pi$
$200$	0	0
$201$	996.000	0.349515
$202$	0	0
$203$	−1872.00	−0.647235
$204$	0	0
$205$	−1620.00	−0.551930
$206$	0	0
$207$	648.000	0.217580
$208$	0	0
$209$	−3600.00	−1.19147
$210$	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
$213$	−1080.00	−0.347420
$214$	0	0
$215$	−8136.00	−2.58079
$216$	0	0
$217$	−128.000	−0.0400424
$218$	0	0
$219$	78.0000	0.0240674
$220$	0	0
$221$	−180.000	−0.0547878
$222$	0	0
$223$	4880.00	1.46542	0.732711	−	0.680540i	$-0.238255\pi$
$223$	4880.00	1.46542	0.732711	+	0.680540i	$0.238255\pi$
$224$	0	0
$225$	1791.00	0.530667
$226$	0	0
$227$	2700.00	0.789451	0.394725	−	0.918799i	$-0.370840\pi$
$227$	2700.00	0.789451	0.394725	+	0.918799i	$0.370840\pi$
$228$	0	0
$229$	254.000	0.0732960	0.0366480	−	0.999328i	$-0.488332\pi$
$229$	254.000	0.0732960	0.0366480	+	0.999328i	$0.488332\pi$
$230$	0	0
$231$	864.000	0.246091
$232$	0	0
$233$	4410.00	1.23995	0.619976	−	0.784621i	$-0.287142\pi$
$233$	4410.00	1.23995	0.619976	+	0.784621i	$0.287142\pi$
$234$	0	0
$235$	−7776.00	−2.15851
$236$	0	0
$237$	1536.00	0.420987
$238$	0	0
$239$	−3888.00	−1.05228	−0.526138	−	0.850399i	$-0.676360\pi$
$239$	−3888.00	−1.05228	−0.526138	+	0.850399i	$0.676360\pi$
$240$	0	0
$241$	5138.00	1.37331	0.686655	−	0.726984i	$-0.259078\pi$
$241$	5138.00	1.37331	0.686655	+	0.726984i	$0.259078\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	5022.00	1.30957
$246$	0	0
$247$	1000.00	0.257605
$248$	0	0
$249$	−3564.00	−0.907066
$250$	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
$253$	2592.00	0.644101
$254$	0	0
$255$	−972.000	−0.238702
$256$	0	0
$257$	−5886.00	−1.42863	−0.714316	−	0.699823i	$-0.753262\pi$
$257$	−5886.00	−1.42863	−0.714316	+	0.699823i	$0.753262\pi$
$258$	0	0
$259$	−1808.00	−0.433759
$260$	0	0
$261$	−2106.00	−0.499456
$262$	0	0
$263$	2232.00	0.523312	0.261656	−	0.965161i	$-0.415731\pi$
$263$	2232.00	0.523312	0.261656	+	0.965161i	$0.415731\pi$
$264$	0	0
$265$	−7452.00	−1.72744
$266$	0	0
$267$	−1890.00	−0.433206
$268$	0	0
$269$	−666.000	−0.150954	−0.0754772	−	0.997148i	$-0.524048\pi$
$269$	−666.000	−0.150954	−0.0754772	+	0.997148i	$0.524048\pi$
$270$	0	0
$271$	−5536.00	−1.24092	−0.620458	−	0.784240i	$-0.713053\pi$
$271$	−5536.00	−1.24092	−0.620458	+	0.784240i	$0.713053\pi$
$272$	0	0
$273$	−240.000	−0.0532068
$274$	0	0
$275$	7164.00	1.57093
$276$	0	0
$277$	2126.00	0.461151	0.230576	−	0.973054i	$-0.425939\pi$
$277$	2126.00	0.461151	0.230576	+	0.973054i	$0.425939\pi$
$278$	0	0
$279$	−144.000	−0.0308998
$280$	0	0
$281$	−2934.00	−0.622875	−0.311437	−	0.950267i	$-0.600810\pi$
$281$	−2934.00	−0.622875	−0.311437	+	0.950267i	$0.600810\pi$
$282$	0	0
$283$	2036.00	0.427659	0.213830	−	0.976871i	$-0.431406\pi$
$283$	2036.00	0.427659	0.213830	+	0.976871i	$0.431406\pi$
$284$	0	0
$285$	5400.00	1.12235
$286$	0	0
$287$	720.000	0.148085
$288$	0	0
$289$	−4589.00	−0.934053
$290$	0	0
$291$	−3162.00	−0.636975
$292$	0	0
$293$	2286.00	0.455800	0.227900	−	0.973684i	$-0.426814\pi$
$293$	2286.00	0.455800	0.227900	+	0.973684i	$0.426814\pi$
$294$	0	0
$295$	12312.0	2.42994
$296$	0	0
$297$	972.000	0.189903
$298$	0	0
$299$	−720.000	−0.139260
$300$	0	0
$301$	3616.00	0.692434
$302$	0	0
$303$	1674.00	0.317389
$304$	0	0
$305$	−7596.00	−1.42605
$306$	0	0
$307$	1244.00	0.231267	0.115633	−	0.993292i	$-0.463110\pi$
$307$	1244.00	0.231267	0.115633	+	0.993292i	$0.463110\pi$
$308$	0	0
$309$	24.0000	0.00441849
$310$	0	0
$311$	1224.00	0.223173	0.111586	−	0.993755i	$-0.464407\pi$
$311$	1224.00	0.223173	0.111586	+	0.993755i	$0.464407\pi$
$312$	0	0
$313$	1898.00	0.342752	0.171376	−	0.985206i	$-0.445179\pi$
$313$	1898.00	0.342752	0.171376	+	0.985206i	$0.445179\pi$
$314$	0	0
$315$	−1296.00	−0.231814
$316$	0	0
$317$	−9162.00	−1.62331	−0.811655	−	0.584137i	$-0.801433\pi$
$317$	−9162.00	−1.62331	−0.811655	+	0.584137i	$0.801433\pi$
$318$	0	0
$319$	−8424.00	−1.47854
$320$	0	0
$321$	5292.00	0.920158
$322$	0	0
$323$	−1800.00	−0.310076
$324$	0	0
$325$	−1990.00	−0.339647
$326$	0	0
$327$	4866.00	0.822906
$328$	0	0
$329$	3456.00	0.579135
$330$	0	0
$331$	−4348.00	−0.722017	−0.361009	−	0.932562i	$-0.617568\pi$
$331$	−4348.00	−0.722017	−0.361009	+	0.932562i	$0.617568\pi$
$332$	0	0
$333$	−2034.00	−0.334722
$334$	0	0
$335$	−5976.00	−0.974638
$336$	0	0
$337$	7154.00	1.15639	0.578195	−	0.815899i	$-0.303757\pi$
$337$	7154.00	1.15639	0.578195	+	0.815899i	$0.303757\pi$
$338$	0	0
$339$	−3402.00	−0.545048
$340$	0	0
$341$	−576.000	−0.0914726
$342$	0	0
$343$	−4976.00	−0.783320
$344$	0	0
$345$	−3888.00	−0.606733
$346$	0	0
$347$	−1836.00	−0.284039	−0.142020	−	0.989864i	$-0.545360\pi$
$347$	−1836.00	−0.284039	−0.142020	+	0.989864i	$0.545360\pi$
$348$	0	0
$349$	5894.00	0.904007	0.452004	−	0.892016i	$-0.350709\pi$
$349$	5894.00	0.904007	0.452004	+	0.892016i	$0.350709\pi$
$350$	0	0
$351$	−270.000	−0.0410585
$352$	0	0
$353$	11106.0	1.67454	0.837270	−	0.546789i	$-0.184150\pi$
$353$	11106.0	1.67454	0.837270	+	0.546789i	$0.184150\pi$
$354$	0	0
$355$	6480.00	0.968796
$356$	0	0
$357$	432.000	0.0640444
$358$	0	0
$359$	13176.0	1.93705	0.968527	−	0.248907i	$-0.0800713\pi$
$359$	13176.0	1.93705	0.968527	+	0.248907i	$0.0800713\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	0	0
$363$	−105.000	−0.0151820
$364$	0	0
$365$	−468.000	−0.0671130
$366$	0	0
$367$	−6112.00	−0.869329	−0.434665	−	0.900592i	$-0.643133\pi$
$367$	−6112.00	−0.869329	−0.434665	+	0.900592i	$0.643133\pi$
$368$	0	0
$369$	810.000	0.114273
$370$	0	0
$371$	3312.00	0.463478
$372$	0	0
$373$	−13618.0	−1.89038	−0.945192	−	0.326515i	$-0.894126\pi$
$373$	−13618.0	−1.89038	−0.945192	+	0.326515i	$0.894126\pi$
$374$	0	0
$375$	−3996.00	−0.550273
$376$	0	0
$377$	2340.00	0.319671
$378$	0	0
$379$	692.000	0.0937880	0.0468940	−	0.998900i	$-0.485068\pi$
$379$	692.000	0.0937880	0.0468940	+	0.998900i	$0.485068\pi$
$380$	0	0
$381$	−1776.00	−0.238812
$382$	0	0
$383$	−8064.00	−1.07585	−0.537926	−	0.842992i	$-0.680792\pi$
$383$	−8064.00	−1.07585	−0.537926	+	0.842992i	$0.680792\pi$
$384$	0	0
$385$	−5184.00	−0.686237
$386$	0	0
$387$	4068.00	0.534336
$388$	0	0
$389$	12654.0	1.64931	0.824657	−	0.565633i	$-0.191368\pi$
$389$	12654.0	1.64931	0.824657	+	0.565633i	$0.191368\pi$
$390$	0	0
$391$	1296.00	0.167625
$392$	0	0
$393$	−5724.00	−0.734701
$394$	0	0
$395$	−9216.00	−1.17394
$396$	0	0
$397$	−106.000	−0.0134005	−0.00670024	−	0.999978i	$-0.502133\pi$
$397$	−106.000	−0.0134005	−0.00670024	+	0.999978i	$0.502133\pi$
$398$	0	0
$399$	−2400.00	−0.301129
$400$	0	0
$401$	−4014.00	−0.499874	−0.249937	−	0.968262i	$-0.580410\pi$
$401$	−4014.00	−0.499874	−0.249937	+	0.968262i	$0.580410\pi$
$402$	0	0
$403$	160.000	0.0197771
$404$	0	0
$405$	−1458.00	−0.178885
$406$	0	0
$407$	−8136.00	−0.990876
$408$	0	0
$409$	3914.00	0.473190	0.236595	−	0.971608i	$-0.423968\pi$
$409$	3914.00	0.473190	0.236595	+	0.971608i	$0.423968\pi$
$410$	0	0
$411$	2862.00	0.343484
$412$	0	0
$413$	−5472.00	−0.651960
$414$	0	0
$415$	21384.0	2.52940
$416$	0	0
$417$	7692.00	0.903307
$418$	0	0
$419$	4428.00	0.516282	0.258141	−	0.966107i	$-0.416890\pi$
$419$	4428.00	0.516282	0.258141	+	0.966107i	$0.416890\pi$
$420$	0	0
$421$	−15490.0	−1.79320	−0.896599	−	0.442843i	$-0.853970\pi$
$421$	−15490.0	−1.79320	−0.896599	+	0.442843i	$0.853970\pi$
$422$	0	0
$423$	3888.00	0.446906
$424$	0	0
$425$	3582.00	0.408829
$426$	0	0
$427$	3376.00	0.382614
$428$	0	0
$429$	−1080.00	−0.121545
$430$	0	0
$431$	6768.00	0.756388	0.378194	−	0.925726i	$-0.376545\pi$
$431$	6768.00	0.756388	0.378194	+	0.925726i	$0.376545\pi$
$432$	0	0
$433$	1298.00	0.144060	0.0720299	−	0.997402i	$-0.477052\pi$
$433$	1298.00	0.144060	0.0720299	+	0.997402i	$0.477052\pi$
$434$	0	0
$435$	12636.0	1.39276
$436$	0	0
$437$	−7200.00	−0.788153
$438$	0	0
$439$	−2248.00	−0.244399	−0.122200	−	0.992506i	$-0.538995\pi$
$439$	−2248.00	−0.244399	−0.122200	+	0.992506i	$0.538995\pi$
$440$	0	0
$441$	−2511.00	−0.271137
$442$	0	0
$443$	−9612.00	−1.03088	−0.515440	−	0.856926i	$-0.672372\pi$
$443$	−9612.00	−1.03088	−0.515440	+	0.856926i	$0.672372\pi$
$444$	0	0
$445$	11340.0	1.20802
$446$	0	0
$447$	−2214.00	−0.234270
$448$	0	0
$449$	162.000	0.0170273	0.00851364	−	0.999964i	$-0.497290\pi$
$449$	162.000	0.0170273	0.00851364	+	0.999964i	$0.497290\pi$
$450$	0	0
$451$	3240.00	0.338283
$452$	0	0
$453$	−7320.00	−0.759213
$454$	0	0
$455$	1440.00	0.148370
$456$	0	0
$457$	1370.00	0.140232	0.0701159	−	0.997539i	$-0.477663\pi$
$457$	1370.00	0.140232	0.0701159	+	0.997539i	$0.477663\pi$
$458$	0	0
$459$	486.000	0.0494217
$460$	0	0
$461$	−15354.0	−1.55121	−0.775604	−	0.631220i	$-0.782555\pi$
$461$	−15354.0	−1.55121	−0.775604	+	0.631220i	$0.782555\pi$
$462$	0	0
$463$	−13024.0	−1.30729	−0.653646	−	0.756800i	$-0.726762\pi$
$463$	−13024.0	−1.30729	−0.653646	+	0.756800i	$0.726762\pi$
$464$	0	0
$465$	864.000	0.0861657
$466$	0	0
$467$	−14436.0	−1.43045	−0.715223	−	0.698896i	$-0.753675\pi$
$467$	−14436.0	−1.43045	−0.715223	+	0.698896i	$0.753675\pi$
$468$	0	0
$469$	2656.00	0.261498
$470$	0	0
$471$	−7662.00	−0.749568
$472$	0	0
$473$	16272.0	1.58179
$474$	0	0
$475$	−19900.0	−1.92226
$476$	0	0
$477$	3726.00	0.357656
$478$	0	0
$479$	12096.0	1.15382	0.576911	−	0.816807i	$-0.304258\pi$
$479$	12096.0	1.15382	0.576911	+	0.816807i	$0.304258\pi$
$480$	0	0
$481$	2260.00	0.214235
$482$	0	0
$483$	1728.00	0.162788
$484$	0	0
$485$	18972.0	1.77624
$486$	0	0
$487$	6056.00	0.563498	0.281749	−	0.959488i	$-0.409085\pi$
$487$	6056.00	0.563498	0.281749	+	0.959488i	$0.409085\pi$
$488$	0	0
$489$	−2460.00	−0.227495
$490$	0	0
$491$	7524.00	0.691555	0.345777	−	0.938317i	$-0.387615\pi$
$491$	7524.00	0.691555	0.345777	+	0.938317i	$0.387615\pi$
$492$	0	0
$493$	−4212.00	−0.384785
$494$	0	0
$495$	−5832.00	−0.529553
$496$	0	0
$497$	−2880.00	−0.259931
$498$	0	0
$499$	5276.00	0.473319	0.236660	−	0.971593i	$-0.423947\pi$
$499$	5276.00	0.473319	0.236660	+	0.971593i	$0.423947\pi$
$500$	0	0
$501$	5832.00	0.520069
$502$	0	0
$503$	4968.00	0.440382	0.220191	−	0.975457i	$-0.429332\pi$
$503$	4968.00	0.440382	0.220191	+	0.975457i	$0.429332\pi$
$504$	0	0
$505$	−10044.0	−0.885054
$506$	0	0
$507$	−6291.00	−0.551071
$508$	0	0
$509$	10998.0	0.957717	0.478858	−	0.877892i	$-0.341051\pi$
$509$	10998.0	0.957717	0.478858	+	0.877892i	$0.341051\pi$
$510$	0	0
$511$	208.000	0.0180066
$512$	0	0
$513$	−2700.00	−0.232374
$514$	0	0
$515$	−144.000	−0.0123212
$516$	0	0
$517$	15552.0	1.32297
$518$	0	0
$519$	−3726.00	−0.315131
$520$	0	0
$521$	−8838.00	−0.743186	−0.371593	−	0.928396i	$-0.621188\pi$
$521$	−8838.00	−0.743186	−0.371593	+	0.928396i	$0.621188\pi$
$522$	0	0
$523$	22436.0	1.87583	0.937914	−	0.346869i	$-0.112755\pi$
$523$	22436.0	1.87583	0.937914	+	0.346869i	$0.112755\pi$
$524$	0	0
$525$	4776.00	0.397032
$526$	0	0
$527$	−288.000	−0.0238055
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	−6156.00	−0.503103
$532$	0	0
$533$	−900.000	−0.0731395
$534$	0	0
$535$	−31752.0	−2.56590
$536$	0	0
$537$	3348.00	0.269044
$538$	0	0
$539$	−10044.0	−0.802645
$540$	0	0
$541$	−4762.00	−0.378437	−0.189218	−	0.981935i	$-0.560595\pi$
$541$	−4762.00	−0.378437	−0.189218	+	0.981935i	$0.560595\pi$
$542$	0	0
$543$	3210.00	0.253691
$544$	0	0
$545$	−29196.0	−2.29471
$546$	0	0
$547$	−6004.00	−0.469310	−0.234655	−	0.972079i	$-0.575396\pi$
$547$	−6004.00	−0.469310	−0.234655	+	0.972079i	$0.575396\pi$
$548$	0	0
$549$	3798.00	0.295254
$550$	0	0
$551$	23400.0	1.80921
$552$	0	0
$553$	4096.00	0.314972
$554$	0	0
$555$	12204.0	0.933389
$556$	0	0
$557$	−5274.00	−0.401197	−0.200598	−	0.979674i	$-0.564289\pi$
$557$	−5274.00	−0.401197	−0.200598	+	0.979674i	$0.564289\pi$
$558$	0	0
$559$	−4520.00	−0.341996
$560$	0	0
$561$	1944.00	0.146303
$562$	0	0
$563$	−12420.0	−0.929735	−0.464867	−	0.885380i	$-0.653898\pi$
$563$	−12420.0	−0.929735	−0.464867	+	0.885380i	$0.653898\pi$
$564$	0	0
$565$	20412.0	1.51989
$566$	0	0
$567$	648.000	0.0479955
$568$	0	0
$569$	−21366.0	−1.57418	−0.787091	−	0.616837i	$-0.788414\pi$
$569$	−21366.0	−1.57418	−0.787091	+	0.616837i	$0.788414\pi$
$570$	0	0
$571$	21140.0	1.54935	0.774677	−	0.632357i	$-0.217912\pi$
$571$	21140.0	1.54935	0.774677	+	0.632357i	$0.217912\pi$
$572$	0	0
$573$	−1728.00	−0.125983
$574$	0	0
$575$	14328.0	1.03916
$576$	0	0
$577$	3266.00	0.235642	0.117821	−	0.993035i	$-0.462409\pi$
$577$	3266.00	0.235642	0.117821	+	0.993035i	$0.462409\pi$
$578$	0	0
$579$	−4026.00	−0.288972
$580$	0	0
$581$	−9504.00	−0.678644
$582$	0	0
$583$	14904.0	1.05877
$584$	0	0
$585$	1620.00	0.114494
$586$	0	0
$587$	17028.0	1.19731	0.598655	−	0.801007i	$-0.295702\pi$
$587$	17028.0	1.19731	0.598655	+	0.801007i	$0.295702\pi$
$588$	0	0
$589$	1600.00	0.111930
$590$	0	0
$591$	4266.00	0.296920
$592$	0	0
$593$	9522.00	0.659396	0.329698	−	0.944086i	$-0.393053\pi$
$593$	9522.00	0.659396	0.329698	+	0.944086i	$0.393053\pi$
$594$	0	0
$595$	−2592.00	−0.178591
$596$	0	0
$597$	2616.00	0.179340
$598$	0	0
$599$	−10296.0	−0.702309	−0.351155	−	0.936318i	$-0.614211\pi$
$599$	−10296.0	−0.702309	−0.351155	+	0.936318i	$0.614211\pi$
$600$	0	0
$601$	−3382.00	−0.229542	−0.114771	−	0.993392i	$-0.536613\pi$
$601$	−3382.00	−0.229542	−0.114771	+	0.993392i	$0.536613\pi$
$602$	0	0
$603$	2988.00	0.201792
$604$	0	0
$605$	630.000	0.0423358
$606$	0	0
$607$	−20656.0	−1.38122	−0.690611	−	0.723227i	$-0.742658\pi$
$607$	−20656.0	−1.38122	−0.690611	+	0.723227i	$0.742658\pi$
$608$	0	0
$609$	−5616.00	−0.373681
$610$	0	0
$611$	−4320.00	−0.286037
$612$	0	0
$613$	−22114.0	−1.45706	−0.728529	−	0.685015i	$-0.759795\pi$
$613$	−22114.0	−1.45706	−0.728529	+	0.685015i	$0.759795\pi$
$614$	0	0
$615$	−4860.00	−0.318657
$616$	0	0
$617$	19962.0	1.30250	0.651248	−	0.758865i	$-0.274246\pi$
$617$	19962.0	1.30250	0.651248	+	0.758865i	$0.274246\pi$
$618$	0	0
$619$	−604.000	−0.0392194	−0.0196097	−	0.999808i	$-0.506242\pi$
$619$	−604.000	−0.0392194	−0.0196097	+	0.999808i	$0.506242\pi$
$620$	0	0
$621$	1944.00	0.125620
$622$	0	0
$623$	−5040.00	−0.324115
$624$	0	0
$625$	−899.000	−0.0575360
$626$	0	0
$627$	−10800.0	−0.687895
$628$	0	0
$629$	−4068.00	−0.257872
$630$	0	0
$631$	152.000	0.00958958	0.00479479	−	0.999989i	$-0.498474\pi$
$631$	152.000	0.00958958	0.00479479	+	0.999989i	$0.498474\pi$
$632$	0	0
$633$	4020.00	0.252418
$634$	0	0
$635$	10656.0	0.665938
$636$	0	0
$637$	2790.00	0.173538
$638$	0	0
$639$	−3240.00	−0.200583
$640$	0	0
$641$	4194.00	0.258429	0.129215	−	0.991617i	$-0.458754\pi$
$641$	4194.00	0.258429	0.129215	+	0.991617i	$0.458754\pi$
$642$	0	0
$643$	−7252.00	−0.444776	−0.222388	−	0.974958i	$-0.571385\pi$
$643$	−7252.00	−0.444776	−0.222388	+	0.974958i	$0.571385\pi$
$644$	0	0
$645$	−24408.0	−1.49002
$646$	0	0
$647$	−6696.00	−0.406873	−0.203437	−	0.979088i	$-0.565211\pi$
$647$	−6696.00	−0.406873	−0.203437	+	0.979088i	$0.565211\pi$
$648$	0	0
$649$	−24624.0	−1.48933
$650$	0	0
$651$	−384.000	−0.0231185
$652$	0	0
$653$	28422.0	1.70328	0.851638	−	0.524131i	$-0.175610\pi$
$653$	28422.0	1.70328	0.851638	+	0.524131i	$0.175610\pi$
$654$	0	0
$655$	34344.0	2.04875
$656$	0	0
$657$	234.000	0.0138953
$658$	0	0
$659$	−19908.0	−1.17679	−0.588396	−	0.808573i	$-0.700240\pi$
$659$	−19908.0	−1.17679	−0.588396	+	0.808573i	$0.700240\pi$
$660$	0	0
$661$	14318.0	0.842520	0.421260	−	0.906940i	$-0.361588\pi$
$661$	14318.0	0.842520	0.421260	+	0.906940i	$0.361588\pi$
$662$	0	0
$663$	−540.000	−0.0316318
$664$	0	0
$665$	14400.0	0.839711
$666$	0	0
$667$	−16848.0	−0.978047
$668$	0	0
$669$	14640.0	0.846061
$670$	0	0
$671$	15192.0	0.874040
$672$	0	0
$673$	30050.0	1.72116	0.860581	−	0.509313i	$-0.170101\pi$
$673$	30050.0	1.72116	0.860581	+	0.509313i	$0.170101\pi$
$674$	0	0
$675$	5373.00	0.306381
$676$	0	0
$677$	22158.0	1.25790	0.628952	−	0.777444i	$-0.283484\pi$
$677$	22158.0	1.25790	0.628952	+	0.777444i	$0.283484\pi$
$678$	0	0
$679$	−8432.00	−0.476569
$680$	0	0
$681$	8100.00	0.455790
$682$	0	0
$683$	−3132.00	−0.175465	−0.0877325	−	0.996144i	$-0.527962\pi$
$683$	−3132.00	−0.175465	−0.0877325	+	0.996144i	$0.527962\pi$
$684$	0	0
$685$	−17172.0	−0.957822
$686$	0	0
$687$	762.000	0.0423175
$688$	0	0
$689$	−4140.00	−0.228914
$690$	0	0
$691$	−20932.0	−1.15237	−0.576187	−	0.817318i	$-0.695460\pi$
$691$	−20932.0	−1.15237	−0.576187	+	0.817318i	$0.695460\pi$
$692$	0	0
$693$	2592.00	0.142081
$694$	0	0
$695$	−46152.0	−2.51891
$696$	0	0
$697$	1620.00	0.0880371
$698$	0	0
$699$	13230.0	0.715886
$700$	0	0
$701$	−21834.0	−1.17640	−0.588202	−	0.808714i	$-0.700164\pi$
$701$	−21834.0	−1.17640	−0.588202	+	0.808714i	$0.700164\pi$
$702$	0	0
$703$	22600.0	1.21248
$704$	0	0
$705$	−23328.0	−1.24622
$706$	0	0
$707$	4464.00	0.237463
$708$	0	0
$709$	12446.0	0.659266	0.329633	−	0.944109i	$-0.393075\pi$
$709$	12446.0	0.659266	0.329633	+	0.944109i	$0.393075\pi$
$710$	0	0
$711$	4608.00	0.243057
$712$	0	0
$713$	−1152.00	−0.0605088
$714$	0	0
$715$	6480.00	0.338935
$716$	0	0
$717$	−11664.0	−0.607531
$718$	0	0
$719$	−12528.0	−0.649813	−0.324907	−	0.945746i	$-0.605333\pi$
$719$	−12528.0	−0.649813	−0.324907	+	0.945746i	$0.605333\pi$
$720$	0	0
$721$	64.0000	0.00330580
$722$	0	0
$723$	15414.0	0.792881
$724$	0	0
$725$	−46566.0	−2.38540
$726$	0	0
$727$	11576.0	0.590550	0.295275	−	0.955412i	$-0.404589\pi$
$727$	11576.0	0.590550	0.295275	+	0.955412i	$0.404589\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	8136.00	0.411656
$732$	0	0
$733$	−29338.0	−1.47834	−0.739170	−	0.673519i	$-0.764782\pi$
$733$	−29338.0	−1.47834	−0.739170	+	0.673519i	$0.764782\pi$
$734$	0	0
$735$	15066.0	0.756079
$736$	0	0
$737$	11952.0	0.597364
$738$	0	0
$739$	2540.00	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$739$	2540.00	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$740$	0	0
$741$	3000.00	0.148728
$742$	0	0
$743$	−18792.0	−0.927876	−0.463938	−	0.885868i	$-0.653564\pi$
$743$	−18792.0	−0.927876	−0.463938	+	0.885868i	$0.653564\pi$
$744$	0	0
$745$	13284.0	0.653273
$746$	0	0
$747$	−10692.0	−0.523695
$748$	0	0
$749$	14112.0	0.688440
$750$	0	0
$751$	4832.00	0.234783	0.117392	−	0.993086i	$-0.462547\pi$
$751$	4832.00	0.234783	0.117392	+	0.993086i	$0.462547\pi$
$752$	0	0
$753$	14364.0	0.695157
$754$	0	0
$755$	43920.0	2.11710
$756$	0	0
$757$	−20818.0	−0.999529	−0.499764	−	0.866161i	$-0.666580\pi$
$757$	−20818.0	−0.999529	−0.499764	+	0.866161i	$0.666580\pi$
$758$	0	0
$759$	7776.00	0.371872
$760$	0	0
$761$	12042.0	0.573617	0.286808	−	0.957988i	$-0.407406\pi$
$761$	12042.0	0.573617	0.286808	+	0.957988i	$0.407406\pi$
$762$	0	0
$763$	12976.0	0.615679
$764$	0	0
$765$	−2916.00	−0.137815
$766$	0	0
$767$	6840.00	0.322005
$768$	0	0
$769$	13058.0	0.612332	0.306166	−	0.951978i	$-0.400954\pi$
$769$	13058.0	0.612332	0.306166	+	0.951978i	$0.400954\pi$
$770$	0	0
$771$	−17658.0	−0.824821
$772$	0	0
$773$	−11826.0	−0.550261	−0.275130	−	0.961407i	$-0.588721\pi$
$773$	−11826.0	−0.550261	−0.275130	+	0.961407i	$0.588721\pi$
$774$	0	0
$775$	−3184.00	−0.147578
$776$	0	0
$777$	−5424.00	−0.250431
$778$	0	0
$779$	−9000.00	−0.413939
$780$	0	0
$781$	−12960.0	−0.593784
$782$	0	0
$783$	−6318.00	−0.288361
$784$	0	0
$785$	45972.0	2.09021
$786$	0	0
$787$	11996.0	0.543343	0.271672	−	0.962390i	$-0.412424\pi$
$787$	11996.0	0.543343	0.271672	+	0.962390i	$0.412424\pi$
$788$	0	0
$789$	6696.00	0.302134
$790$	0	0
$791$	−9072.00	−0.407792
$792$	0	0
$793$	−4220.00	−0.188974
$794$	0	0
$795$	−22356.0	−0.997340
$796$	0	0
$797$	6966.00	0.309596	0.154798	−	0.987946i	$-0.450527\pi$
$797$	6966.00	0.309596	0.154798	+	0.987946i	$0.450527\pi$
$798$	0	0
$799$	7776.00	0.344299
$800$	0	0
$801$	−5670.00	−0.250112
$802$	0	0
$803$	936.000	0.0411342
$804$	0	0
$805$	−10368.0	−0.453943
$806$	0	0
$807$	−1998.00	−0.0871536
$808$	0	0
$809$	−40806.0	−1.77338	−0.886689	−	0.462367i	$-0.847000\pi$
$809$	−40806.0	−1.77338	−0.886689	+	0.462367i	$0.847000\pi$
$810$	0	0
$811$	−17980.0	−0.778500	−0.389250	−	0.921132i	$-0.627266\pi$
$811$	−17980.0	−0.778500	−0.389250	+	0.921132i	$0.627266\pi$
$812$	0	0
$813$	−16608.0	−0.716443
$814$	0	0
$815$	14760.0	0.634381
$816$	0	0
$817$	−45200.0	−1.93555
$818$	0	0
$819$	−720.000	−0.0307190
$820$	0	0
$821$	−12834.0	−0.545566	−0.272783	−	0.962076i	$-0.587944\pi$
$821$	−12834.0	−0.545566	−0.272783	+	0.962076i	$0.587944\pi$
$822$	0	0
$823$	−37864.0	−1.60371	−0.801857	−	0.597516i	$-0.796154\pi$
$823$	−37864.0	−1.60371	−0.801857	+	0.597516i	$0.796154\pi$
$824$	0	0
$825$	21492.0	0.906976
$826$	0	0
$827$	42516.0	1.78770	0.893849	−	0.448368i	$-0.147995\pi$
$827$	42516.0	1.78770	0.893849	+	0.448368i	$0.147995\pi$
$828$	0	0
$829$	45638.0	1.91203	0.956015	−	0.293317i	$-0.0947591\pi$
$829$	45638.0	1.91203	0.956015	+	0.293317i	$0.0947591\pi$
$830$	0	0
$831$	6378.00	0.266246
$832$	0	0
$833$	−5022.00	−0.208886
$834$	0	0
$835$	−34992.0	−1.45024
$836$	0	0
$837$	−432.000	−0.0178400
$838$	0	0
$839$	17496.0	0.719939	0.359970	−	0.932964i	$-0.382787\pi$
$839$	17496.0	0.719939	0.359970	+	0.932964i	$0.382787\pi$
$840$	0	0
$841$	30367.0	1.24511
$842$	0	0
$843$	−8802.00	−0.359617
$844$	0	0
$845$	37746.0	1.53669
$846$	0	0
$847$	−280.000	−0.0113588
$848$	0	0
$849$	6108.00	0.246909
$850$	0	0
$851$	−16272.0	−0.655461
$852$	0	0
$853$	32174.0	1.29146	0.645731	−	0.763565i	$-0.276553\pi$
$853$	32174.0	1.29146	0.645731	+	0.763565i	$0.276553\pi$
$854$	0	0
$855$	16200.0	0.647986
$856$	0	0
$857$	−38934.0	−1.55188	−0.775939	−	0.630807i	$-0.782724\pi$
$857$	−38934.0	−1.55188	−0.775939	+	0.630807i	$0.782724\pi$
$858$	0	0
$859$	29780.0	1.18286	0.591432	−	0.806355i	$-0.298563\pi$
$859$	29780.0	1.18286	0.591432	+	0.806355i	$0.298563\pi$
$860$	0	0
$861$	2160.00	0.0854966
$862$	0	0
$863$	−48096.0	−1.89711	−0.948556	−	0.316611i	$-0.897455\pi$
$863$	−48096.0	−1.89711	−0.948556	+	0.316611i	$0.897455\pi$
$864$	0	0
$865$	22356.0	0.878759
$866$	0	0
$867$	−13767.0	−0.539275
$868$	0	0
$869$	18432.0	0.719520
$870$	0	0
$871$	−3320.00	−0.129155
$872$	0	0
$873$	−9486.00	−0.367758
$874$	0	0
$875$	−10656.0	−0.411701
$876$	0	0
$877$	21302.0	0.820202	0.410101	−	0.912040i	$-0.365493\pi$
$877$	21302.0	0.820202	0.410101	+	0.912040i	$0.365493\pi$
$878$	0	0
$879$	6858.00	0.263157
$880$	0	0
$881$	−7470.00	−0.285665	−0.142832	−	0.989747i	$-0.545621\pi$
$881$	−7470.00	−0.285665	−0.142832	+	0.989747i	$0.545621\pi$
$882$	0	0
$883$	764.000	0.0291174	0.0145587	−	0.999894i	$-0.495366\pi$
$883$	764.000	0.0291174	0.0145587	+	0.999894i	$0.495366\pi$
$884$	0	0
$885$	36936.0	1.40293
$886$	0	0
$887$	32328.0	1.22375	0.611876	−	0.790954i	$-0.290415\pi$
$887$	32328.0	1.22375	0.611876	+	0.790954i	$0.290415\pi$
$888$	0	0
$889$	−4736.00	−0.178673
$890$	0	0
$891$	2916.00	0.109640
$892$	0	0
$893$	−43200.0	−1.61885
$894$	0	0
$895$	−20088.0	−0.750243
$896$	0	0
$897$	−2160.00	−0.0804017
$898$	0	0
$899$	3744.00	0.138898
$900$	0	0
$901$	7452.00	0.275541
$902$	0	0
$903$	10848.0	0.399777
$904$	0	0
$905$	−19260.0	−0.707430
$906$	0	0
$907$	−36316.0	−1.32950	−0.664748	−	0.747068i	$-0.731461\pi$
$907$	−36316.0	−1.32950	−0.664748	+	0.747068i	$0.731461\pi$
$908$	0	0
$909$	5022.00	0.183244
$910$	0	0
$911$	−13392.0	−0.487044	−0.243522	−	0.969895i	$-0.578303\pi$
$911$	−13392.0	−0.487044	−0.243522	+	0.969895i	$0.578303\pi$
$912$	0	0
$913$	−42768.0	−1.55029
$914$	0	0
$915$	−22788.0	−0.823331
$916$	0	0
$917$	−15264.0	−0.549686
$918$	0	0
$919$	38072.0	1.36657	0.683286	−	0.730151i	$-0.260550\pi$
$919$	38072.0	1.36657	0.683286	+	0.730151i	$0.260550\pi$
$920$	0	0
$921$	3732.00	0.133522
$922$	0	0
$923$	3600.00	0.128381
$924$	0	0
$925$	−44974.0	−1.59863
$926$	0	0
$927$	72.0000	0.00255101
$928$	0	0
$929$	−12798.0	−0.451979	−0.225990	−	0.974130i	$-0.572562\pi$
$929$	−12798.0	−0.451979	−0.225990	+	0.974130i	$0.572562\pi$
$930$	0	0
$931$	27900.0	0.982154
$932$	0	0
$933$	3672.00	0.128849
$934$	0	0
$935$	−11664.0	−0.407972
$936$	0	0
$937$	34874.0	1.21588	0.607942	−	0.793981i	$-0.291995\pi$
$937$	34874.0	1.21588	0.607942	+	0.793981i	$0.291995\pi$
$938$	0	0
$939$	5694.00	0.197888
$940$	0	0
$941$	17190.0	0.595513	0.297757	−	0.954642i	$-0.403762\pi$
$941$	17190.0	0.595513	0.297757	+	0.954642i	$0.403762\pi$
$942$	0	0
$943$	6480.00	0.223773
$944$	0	0
$945$	−3888.00	−0.133838
$946$	0	0
$947$	40284.0	1.38232	0.691158	−	0.722703i	$-0.257101\pi$
$947$	40284.0	1.38232	0.691158	+	0.722703i	$0.257101\pi$
$948$	0	0
$949$	−260.000	−0.00889353
$950$	0	0
$951$	−27486.0	−0.937218
$952$	0	0
$953$	15498.0	0.526789	0.263394	−	0.964688i	$-0.415158\pi$
$953$	15498.0	0.526789	0.263394	+	0.964688i	$0.415158\pi$
$954$	0	0
$955$	10368.0	0.351310
$956$	0	0
$957$	−25272.0	−0.853634
$958$	0	0
$959$	7632.00	0.256987
$960$	0	0
$961$	−29535.0	−0.991407
$962$	0	0
$963$	15876.0	0.531253
$964$	0	0
$965$	24156.0	0.805813
$966$	0	0
$967$	37160.0	1.23577	0.617883	−	0.786270i	$-0.287991\pi$
$967$	37160.0	1.23577	0.617883	+	0.786270i	$0.287991\pi$
$968$	0	0
$969$	−5400.00	−0.179023
$970$	0	0
$971$	18468.0	0.610367	0.305183	−	0.952294i	$-0.401282\pi$
$971$	18468.0	0.610367	0.305183	+	0.952294i	$0.401282\pi$
$972$	0	0
$973$	20512.0	0.675832
$974$	0	0
$975$	−5970.00	−0.196095
$976$	0	0
$977$	10386.0	0.340100	0.170050	−	0.985435i	$-0.445607\pi$
$977$	10386.0	0.340100	0.170050	+	0.985435i	$0.445607\pi$
$978$	0	0
$979$	−22680.0	−0.740404
$980$	0	0
$981$	14598.0	0.475105
$982$	0	0
$983$	44136.0	1.43206	0.716032	−	0.698067i	$-0.245956\pi$
$983$	44136.0	1.43206	0.716032	+	0.698067i	$0.245956\pi$
$984$	0	0
$985$	−25596.0	−0.827976
$986$	0	0
$987$	10368.0	0.334364
$988$	0	0
$989$	32544.0	1.04635
$990$	0	0
$991$	−28432.0	−0.911375	−0.455687	−	0.890140i	$-0.650606\pi$
$991$	−28432.0	−0.911375	−0.455687	+	0.890140i	$0.650606\pi$
$992$	0	0
$993$	−13044.0	−0.416857
$994$	0	0
$995$	−15696.0	−0.500097
$996$	0	0
$997$	−39778.0	−1.26357	−0.631786	−	0.775143i	$-0.717678\pi$
$997$	−39778.0	−1.26357	−0.631786	+	0.775143i	$0.717678\pi$
$998$	0	0
$999$	−6102.00	−0.193252

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.4.a.a.1.1	✓	1
3.2	odd	2		36.4.a.a.1.1		1
4.3	odd	2		48.4.a.a.1.1		1
5.2	odd	4		300.4.d.e.49.1		2
5.3	odd	4		300.4.d.e.49.2		2
5.4	even	2		300.4.a.b.1.1		1
7.2	even	3		588.4.i.d.361.1		2
7.3	odd	6		588.4.i.e.373.1		2
7.4	even	3		588.4.i.d.373.1		2
7.5	odd	6		588.4.i.e.361.1		2
7.6	odd	2		588.4.a.c.1.1		1
8.3	odd	2		192.4.a.l.1.1		1
8.5	even	2		192.4.a.f.1.1		1
9.2	odd	6		324.4.e.a.109.1		2
9.4	even	3		324.4.e.h.217.1		2
9.5	odd	6		324.4.e.a.217.1		2
9.7	even	3		324.4.e.h.109.1		2
11.10	odd	2		1452.4.a.d.1.1		1
12.11	even	2		144.4.a.g.1.1		1
13.5	odd	4		2028.4.b.c.337.2		2
13.8	odd	4		2028.4.b.c.337.1		2
13.12	even	2		2028.4.a.c.1.1		1
15.2	even	4		900.4.d.c.649.2		2
15.8	even	4		900.4.d.c.649.1		2
15.14	odd	2		900.4.a.g.1.1		1
16.3	odd	4		768.4.d.j.385.1		2
16.5	even	4		768.4.d.g.385.1		2
16.11	odd	4		768.4.d.j.385.2		2
16.13	even	4		768.4.d.g.385.2		2
20.3	even	4		1200.4.f.d.49.1		2
20.7	even	4		1200.4.f.d.49.2		2
20.19	odd	2		1200.4.a.be.1.1		1
21.2	odd	6		1764.4.k.b.361.1		2
21.5	even	6		1764.4.k.o.361.1		2
21.11	odd	6		1764.4.k.b.1549.1		2
21.17	even	6		1764.4.k.o.1549.1		2
21.20	even	2		1764.4.a.b.1.1		1
24.5	odd	2		576.4.a.b.1.1		1
24.11	even	2		576.4.a.a.1.1		1
28.27	even	2		2352.4.a.bk.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.4.a.a.1.1	✓	1	1.1	even	1	trivial
36.4.a.a.1.1		1	3.2	odd	2
48.4.a.a.1.1		1	4.3	odd	2
144.4.a.g.1.1		1	12.11	even	2
192.4.a.f.1.1		1	8.5	even	2
192.4.a.l.1.1		1	8.3	odd	2
300.4.a.b.1.1		1	5.4	even	2
300.4.d.e.49.1		2	5.2	odd	4
300.4.d.e.49.2		2	5.3	odd	4
324.4.e.a.109.1		2	9.2	odd	6
324.4.e.a.217.1		2	9.5	odd	6
324.4.e.h.109.1		2	9.7	even	3
324.4.e.h.217.1		2	9.4	even	3
576.4.a.a.1.1		1	24.11	even	2
576.4.a.b.1.1		1	24.5	odd	2
588.4.a.c.1.1		1	7.6	odd	2
588.4.i.d.361.1		2	7.2	even	3
588.4.i.d.373.1		2	7.4	even	3
588.4.i.e.361.1		2	7.5	odd	6
588.4.i.e.373.1		2	7.3	odd	6
768.4.d.g.385.1		2	16.5	even	4
768.4.d.g.385.2		2	16.13	even	4
768.4.d.j.385.1		2	16.3	odd	4
768.4.d.j.385.2		2	16.11	odd	4
900.4.a.g.1.1		1	15.14	odd	2
900.4.d.c.649.1		2	15.8	even	4
900.4.d.c.649.2		2	15.2	even	4
1200.4.a.be.1.1		1	20.19	odd	2
1200.4.f.d.49.1		2	20.3	even	4
1200.4.f.d.49.2		2	20.7	even	4
1452.4.a.d.1.1		1	11.10	odd	2
1764.4.a.b.1.1		1	21.20	even	2
1764.4.k.b.361.1		2	21.2	odd	6
1764.4.k.b.1549.1		2	21.11	odd	6
1764.4.k.o.361.1		2	21.5	even	6
1764.4.k.o.1549.1		2	21.17	even	6
2028.4.a.c.1.1		1	13.12	even	2
2028.4.b.c.337.1		2	13.8	odd	4
2028.4.b.c.337.2		2	13.5	odd	4
2352.4.a.bk.1.1		1	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	12.4.a.a.1.1

Level	$12$
Weight	$4$
Character	12.1
Self dual	yes
Analytic conductor	$0.708$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$