LMFDB - Embedded newform 9072.2.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$9072 = 2^{4} \cdot 3^{4} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9072.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:	$72.4402847137$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - 3x - 1$ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.879385 q^{5} -1.00000 q^{7} -3.87939 q^{11} -5.45336 q^{13} +1.65270 q^{17} -2.41147 q^{19} -3.16250 q^{23} -4.22668 q^{25} -6.04963 q^{29} +4.55438 q^{31} +0.879385 q^{35} -4.55438 q^{37} -1.18479 q^{41} -0.184793 q^{43} +1.02229 q^{47} +1.00000 q^{49} +7.29086 q^{53} +3.41147 q^{55} -6.66044 q^{59} -2.59627 q^{61} +4.79561 q^{65} +2.95811 q^{67} +3.68004 q^{71} -12.7811 q^{73} +3.87939 q^{77} +5.95811 q^{79} +0.218941 q^{83} -1.45336 q^{85} +11.0273 q^{89} +5.45336 q^{91} +2.12061 q^{95} +12.5030 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 3 q^{5} - 3 q^{7} - 6 q^{11} - 3 q^{13} + 6 q^{17} + 3 q^{19} - 12 q^{23} - 6 q^{25} + 9 q^{29} + 3 q^{31} - 3 q^{35} - 3 q^{37} + 3 q^{43} - 3 q^{47} + 3 q^{49} + 6 q^{53} + 3 q^{59} + 6 q^{61} + 15 q^{65}+ \cdots - 3 q^{97}+O(q^{100})$ 3 * q + 3 * q^5 - 3 * q^7 - 6 * q^11 - 3 * q^13 + 6 * q^17 + 3 * q^19 - 12 * q^23 - 6 * q^25 + 9 * q^29 + 3 * q^31 - 3 * q^35 - 3 * q^37 + 3 * q^43 - 3 * q^47 + 3 * q^49 + 6 * q^53 + 3 * q^59 + 6 * q^61 + 15 * q^65 + 12 * q^67 - 9 * q^71 - 21 * q^73 + 6 * q^77 + 21 * q^79 + 18 * q^83 + 9 * q^85 + 12 * q^89 + 3 * q^91 + 12 * q^95 - 3 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.879385	−0.393273	−0.196637	−	0.980476i	$-0.563002\pi$
$5$	−0.879385	−0.393273	−0.196637	+	0.980476i	$0.563002\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.87939	−1.16968	−0.584839	−	0.811149i	$-0.698842\pi$
$11$	−3.87939	−1.16968	−0.584839	+	0.811149i	$0.698842\pi$
$12$	0	0
$13$	−5.45336	−1.51249	−0.756245	−	0.654288i	$-0.772968\pi$
$13$	−5.45336	−1.51249	−0.756245	+	0.654288i	$0.772968\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.65270	0.400840	0.200420	−	0.979710i	$-0.435769\pi$
$17$	1.65270	0.400840	0.200420	+	0.979710i	$0.435769\pi$
$18$	0	0
$19$	−2.41147	−0.553230	−0.276615	−	0.960981i	$-0.589213\pi$
$19$	−2.41147	−0.553230	−0.276615	+	0.960981i	$0.589213\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.16250	−0.659428	−0.329714	−	0.944081i	$-0.606952\pi$
$23$	−3.16250	−0.659428	−0.329714	+	0.944081i	$0.606952\pi$
$24$	0	0
$25$	−4.22668	−0.845336
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.04963	−1.12339	−0.561694	−	0.827345i	$-0.689850\pi$
$29$	−6.04963	−1.12339	−0.561694	+	0.827345i	$0.689850\pi$
$30$	0	0
$31$	4.55438	0.817990	0.408995	−	0.912537i	$-0.365879\pi$
$31$	4.55438	0.817990	0.408995	+	0.912537i	$0.365879\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.879385	0.148643
$36$	0	0
$37$	−4.55438	−0.748735	−0.374368	−	0.927280i	$-0.622140\pi$
$37$	−4.55438	−0.748735	−0.374368	+	0.927280i	$0.622140\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.18479	−0.185034	−0.0925168	−	0.995711i	$-0.529491\pi$
$41$	−1.18479	−0.185034	−0.0925168	+	0.995711i	$0.529491\pi$
$42$	0	0
$43$	−0.184793	−0.0281806	−0.0140903	−	0.999901i	$-0.504485\pi$
$43$	−0.184793	−0.0281806	−0.0140903	+	0.999901i	$0.504485\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.02229	0.149116	0.0745581	−	0.997217i	$-0.476245\pi$
$47$	1.02229	0.149116	0.0745581	+	0.997217i	$0.476245\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.29086	1.00148	0.500738	−	0.865599i	$-0.333062\pi$
$53$	7.29086	1.00148	0.500738	+	0.865599i	$0.333062\pi$
$54$	0	0
$55$	3.41147	0.460003
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.66044	−0.867116	−0.433558	−	0.901126i	$-0.642742\pi$
$59$	−6.66044	−0.867116	−0.433558	+	0.901126i	$0.642742\pi$
$60$	0	0
$61$	−2.59627	−0.332418	−0.166209	−	0.986091i	$-0.553153\pi$
$61$	−2.59627	−0.332418	−0.166209	+	0.986091i	$0.553153\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.79561	0.594822
$66$	0	0
$67$	2.95811	0.361391	0.180695	−	0.983539i	$-0.442165\pi$
$67$	2.95811	0.361391	0.180695	+	0.983539i	$0.442165\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.68004	0.436741	0.218370	−	0.975866i	$-0.429926\pi$
$71$	3.68004	0.436741	0.218370	+	0.975866i	$0.429926\pi$
$72$	0	0
$73$	−12.7811	−1.49591	−0.747955	−	0.663750i	$-0.768964\pi$
$73$	−12.7811	−1.49591	−0.747955	+	0.663750i	$0.768964\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.87939	0.442097
$78$	0	0
$79$	5.95811	0.670340	0.335170	−	0.942158i	$-0.391206\pi$
$79$	5.95811	0.670340	0.335170	+	0.942158i	$0.391206\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.218941	0.0240319	0.0120159	−	0.999928i	$-0.496175\pi$
$83$	0.218941	0.0240319	0.0120159	+	0.999928i	$0.496175\pi$
$84$	0	0
$85$	−1.45336	−0.157639
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.0273	1.16890	0.584448	−	0.811431i	$-0.301311\pi$
$89$	11.0273	1.16890	0.584448	+	0.811431i	$0.301311\pi$
$90$	0	0
$91$	5.45336	0.571668
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.12061	0.217570
$96$	0	0
$97$	12.5030	1.26949	0.634743	−	0.772723i	$-0.281106\pi$
$97$	12.5030	1.26949	0.634743	+	0.772723i	$0.281106\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.71688	−0.966866	−0.483433	−	0.875381i	$-0.660610\pi$
$101$	−9.71688	−0.966866	−0.483433	+	0.875381i	$0.660610\pi$
$102$	0	0
$103$	−6.59627	−0.649949	−0.324975	−	0.945723i	$-0.605356\pi$
$103$	−6.59627	−0.649949	−0.324975	+	0.945723i	$0.605356\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.38919	−0.230971	−0.115486	−	0.993309i	$-0.536842\pi$
$107$	−2.38919	−0.230971	−0.115486	+	0.993309i	$0.536842\pi$
$108$	0	0
$109$	3.95811	0.379118	0.189559	−	0.981869i	$-0.439294\pi$
$109$	3.95811	0.379118	0.189559	+	0.981869i	$0.439294\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.4534	1.54780	0.773901	−	0.633306i	$-0.218303\pi$
$113$	16.4534	1.54780	0.773901	+	0.633306i	$0.218303\pi$
$114$	0	0
$115$	2.78106	0.259335
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.65270	−0.151503
$120$	0	0
$121$	4.04963	0.368148
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.11381	0.725721
$126$	0	0
$127$	−17.6536	−1.56651	−0.783253	−	0.621702i	$-0.786441\pi$
$127$	−17.6536	−1.56651	−0.783253	+	0.621702i	$0.786441\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	19.1976	1.67730	0.838650	−	0.544670i	$-0.183345\pi$
$131$	19.1976	1.67730	0.838650	+	0.544670i	$0.183345\pi$
$132$	0	0
$133$	2.41147	0.209101
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.1557	1.55115	0.775573	−	0.631258i	$-0.217461\pi$
$137$	18.1557	1.55115	0.775573	+	0.631258i	$0.217461\pi$
$138$	0	0
$139$	−22.0574	−1.87088	−0.935441	−	0.353483i	$-0.884997\pi$
$139$	−22.0574	−1.87088	−0.935441	+	0.353483i	$0.884997\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	21.1557	1.76913
$144$	0	0
$145$	5.31996	0.441798
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.1557	−1.24160	−0.620802	−	0.783968i	$-0.713193\pi$
$149$	−15.1557	−1.24160	−0.620802	+	0.783968i	$0.713193\pi$
$150$	0	0
$151$	18.9564	1.54265	0.771323	−	0.636444i	$-0.219595\pi$
$151$	18.9564	1.54265	0.771323	+	0.636444i	$0.219595\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.00505	−0.321694
$156$	0	0
$157$	−18.0574	−1.44114	−0.720568	−	0.693385i	$-0.756119\pi$
$157$	−18.0574	−1.44114	−0.720568	+	0.693385i	$0.756119\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.16250	0.249240
$162$	0	0
$163$	−0.958111	−0.0750450	−0.0375225	−	0.999296i	$-0.511947\pi$
$163$	−0.958111	−0.0750450	−0.0375225	+	0.999296i	$0.511947\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.8384	−1.53514	−0.767572	−	0.640963i	$-0.778535\pi$
$167$	−19.8384	−1.53514	−0.767572	+	0.640963i	$0.778535\pi$
$168$	0	0
$169$	16.7392	1.28763
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.6827	1.72454	0.862268	−	0.506452i	$-0.169043\pi$
$173$	22.6827	1.72454	0.862268	+	0.506452i	$0.169043\pi$
$174$	0	0
$175$	4.22668	0.319507
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.34730	0.549163	0.274581	−	0.961564i	$-0.411461\pi$
$179$	7.34730	0.549163	0.274581	+	0.961564i	$0.411461\pi$
$180$	0	0
$181$	−3.44562	−0.256111	−0.128056	−	0.991767i	$-0.540874\pi$
$181$	−3.44562	−0.256111	−0.128056	+	0.991767i	$0.540874\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.00505	0.294457
$186$	0	0
$187$	−6.41147	−0.468853
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.65776	−0.409381	−0.204690	−	0.978827i	$-0.565619\pi$
$191$	−5.65776	−0.409381	−0.204690	+	0.978827i	$0.565619\pi$
$192$	0	0
$193$	9.59627	0.690754	0.345377	−	0.938464i	$-0.387751\pi$
$193$	9.59627	0.690754	0.345377	+	0.938464i	$0.387751\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.31996	0.592772	0.296386	−	0.955068i	$-0.404218\pi$
$197$	8.31996	0.592772	0.296386	+	0.955068i	$0.404218\pi$
$198$	0	0
$199$	−6.59627	−0.467597	−0.233798	−	0.972285i	$-0.575116\pi$
$199$	−6.59627	−0.467597	−0.233798	+	0.972285i	$0.575116\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.04963	0.424601
$204$	0	0
$205$	1.04189	0.0727687
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.35504	0.647101
$210$	0	0
$211$	3.36959	0.231972	0.115986	−	0.993251i	$-0.462997\pi$
$211$	3.36959	0.231972	0.115986	+	0.993251i	$0.462997\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.162504	0.0110827
$216$	0	0
$217$	−4.55438	−0.309171
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.01279	−0.606266
$222$	0	0
$223$	6.27631	0.420293	0.210146	−	0.977670i	$-0.432606\pi$
$223$	6.27631	0.420293	0.210146	+	0.977670i	$0.432606\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.16250	0.409020	0.204510	−	0.978865i	$-0.434440\pi$
$227$	6.16250	0.409020	0.204510	+	0.978865i	$0.434440\pi$
$228$	0	0
$229$	23.3851	1.54533	0.772664	−	0.634815i	$-0.218924\pi$
$229$	23.3851	1.54533	0.772664	+	0.634815i	$0.218924\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.52528	−0.558510	−0.279255	−	0.960217i	$-0.590087\pi$
$233$	−8.52528	−0.558510	−0.279255	+	0.960217i	$0.590087\pi$
$234$	0	0
$235$	−0.898986	−0.0586434
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.5621	−0.941945	−0.470973	−	0.882148i	$-0.656097\pi$
$239$	−14.5621	−0.941945	−0.470973	+	0.882148i	$0.656097\pi$
$240$	0	0
$241$	−5.40373	−0.348085	−0.174043	−	0.984738i	$-0.555683\pi$
$241$	−5.40373	−0.348085	−0.174043	+	0.984738i	$0.555683\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.879385	−0.0561819
$246$	0	0
$247$	13.1506	0.836755
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0669	0.761654	0.380827	−	0.924646i	$-0.375639\pi$
$251$	12.0669	0.761654	0.380827	+	0.924646i	$0.375639\pi$
$252$	0	0
$253$	12.2686	0.771318
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.5662	0.659104	0.329552	−	0.944137i	$-0.393102\pi$
$257$	10.5662	0.659104	0.329552	+	0.944137i	$0.393102\pi$
$258$	0	0
$259$	4.55438	0.282995
$260$	0	0
$261$	0	0
$262$	0	0
$263$	28.3533	1.74834	0.874169	−	0.485622i	$-0.161407\pi$
$263$	28.3533	1.74834	0.874169	+	0.485622i	$0.161407\pi$
$264$	0	0
$265$	−6.41147	−0.393854
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.48339	0.456271	0.228135	−	0.973629i	$-0.426737\pi$
$269$	7.48339	0.456271	0.228135	+	0.973629i	$0.426737\pi$
$270$	0	0
$271$	−13.6382	−0.828459	−0.414229	−	0.910172i	$-0.635949\pi$
$271$	−13.6382	−0.828459	−0.414229	+	0.910172i	$0.635949\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	16.3969	0.988772
$276$	0	0
$277$	−6.15064	−0.369556	−0.184778	−	0.982780i	$-0.559157\pi$
$277$	−6.15064	−0.369556	−0.184778	+	0.982780i	$0.559157\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.31221	0.197590	0.0987951	−	0.995108i	$-0.468501\pi$
$281$	3.31221	0.197590	0.0987951	+	0.995108i	$0.468501\pi$
$282$	0	0
$283$	−29.0232	−1.72525	−0.862626	−	0.505843i	$-0.831182\pi$
$283$	−29.0232	−1.72525	−0.862626	+	0.505843i	$0.831182\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.18479	0.0699361
$288$	0	0
$289$	−14.2686	−0.839328
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.41921	−0.491856	−0.245928	−	0.969288i	$-0.579093\pi$
$293$	−8.41921	−0.491856	−0.245928	+	0.969288i	$0.579093\pi$
$294$	0	0
$295$	5.85710	0.341013
$296$	0	0
$297$	0	0
$298$	0	0
$299$	17.2463	0.997378
$300$	0	0
$301$	0.184793	0.0106513
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.28312	0.130731
$306$	0	0
$307$	12.6878	0.724130	0.362065	−	0.932153i	$-0.382072\pi$
$307$	12.6878	0.724130	0.362065	+	0.932153i	$0.382072\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.4902	0.935073	0.467537	−	0.883974i	$-0.345142\pi$
$311$	16.4902	0.935073	0.467537	+	0.883974i	$0.345142\pi$
$312$	0	0
$313$	28.5185	1.61196	0.805980	−	0.591943i	$-0.201639\pi$
$313$	28.5185	1.61196	0.805980	+	0.591943i	$0.201639\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.8949	−1.45440	−0.727200	−	0.686425i	$-0.759179\pi$
$317$	−25.8949	−1.45440	−0.727200	+	0.686425i	$0.759179\pi$
$318$	0	0
$319$	23.4688	1.31400
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.98545	−0.221756
$324$	0	0
$325$	23.0496	1.27856
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.02229	−0.0563606
$330$	0	0
$331$	−8.21894	−0.451754	−0.225877	−	0.974156i	$-0.572525\pi$
$331$	−8.21894	−0.451754	−0.225877	+	0.974156i	$0.572525\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.60132	−0.142125
$336$	0	0
$337$	4.57129	0.249014	0.124507	−	0.992219i	$-0.460265\pi$
$337$	4.57129	0.249014	0.124507	+	0.992219i	$0.460265\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−17.6682	−0.956786
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.4662	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$347$	−22.4662	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$348$	0	0
$349$	26.0993	1.39706	0.698531	−	0.715580i	$-0.253838\pi$
$349$	26.0993	1.39706	0.698531	+	0.715580i	$0.253838\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.355037	−0.0188967	−0.00944836	−	0.999955i	$-0.503008\pi$
$353$	−0.355037	−0.0188967	−0.00944836	+	0.999955i	$0.503008\pi$
$354$	0	0
$355$	−3.23618	−0.171758
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.45605	−0.287959	−0.143980	−	0.989581i	$-0.545990\pi$
$359$	−5.45605	−0.287959	−0.143980	+	0.989581i	$0.545990\pi$
$360$	0	0
$361$	−13.1848	−0.693936
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.2395	0.588301
$366$	0	0
$367$	−10.9240	−0.570226	−0.285113	−	0.958494i	$-0.592031\pi$
$367$	−10.9240	−0.570226	−0.285113	+	0.958494i	$0.592031\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7.29086	−0.378523
$372$	0	0
$373$	1.73143	0.0896500	0.0448250	−	0.998995i	$-0.485727\pi$
$373$	1.73143	0.0896500	0.0448250	+	0.998995i	$0.485727\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.9908	1.69911
$378$	0	0
$379$	12.1334	0.623251	0.311626	−	0.950205i	$-0.399127\pi$
$379$	12.1334	0.623251	0.311626	+	0.950205i	$0.399127\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.71183	0.445154	0.222577	−	0.974915i	$-0.428553\pi$
$383$	8.71183	0.445154	0.222577	+	0.974915i	$0.428553\pi$
$384$	0	0
$385$	−3.41147	−0.173865
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.64321	0.184718	0.0923590	−	0.995726i	$-0.470559\pi$
$389$	3.64321	0.184718	0.0923590	+	0.995726i	$0.470559\pi$
$390$	0	0
$391$	−5.22668	−0.264325
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.23947	−0.263627
$396$	0	0
$397$	−15.4456	−0.775194	−0.387597	−	0.921829i	$-0.626695\pi$
$397$	−15.4456	−0.775194	−0.387597	+	0.921829i	$0.626695\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.4219	0.919946	0.459973	−	0.887933i	$-0.347859\pi$
$401$	18.4219	0.919946	0.459973	+	0.887933i	$0.347859\pi$
$402$	0	0
$403$	−24.8367	−1.23720
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17.6682	0.875779
$408$	0	0
$409$	−28.6364	−1.41598	−0.707989	−	0.706223i	$-0.750398\pi$
$409$	−28.6364	−1.41598	−0.707989	+	0.706223i	$0.750398\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.66044	0.327739
$414$	0	0
$415$	−0.192533	−0.00945109
$416$	0	0
$417$	0	0
$418$	0	0
$419$	34.6955	1.69499	0.847494	−	0.530805i	$-0.178111\pi$
$419$	34.6955	1.69499	0.847494	+	0.530805i	$0.178111\pi$
$420$	0	0
$421$	−27.4020	−1.33549	−0.667745	−	0.744390i	$-0.732740\pi$
$421$	−27.4020	−1.33549	−0.667745	+	0.744390i	$0.732740\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.98545	−0.338844
$426$	0	0
$427$	2.59627	0.125642
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.5921	−1.28090	−0.640449	−	0.768000i	$-0.721252\pi$
$431$	−26.5921	−1.28090	−0.640449	+	0.768000i	$0.721252\pi$
$432$	0	0
$433$	37.1830	1.78690	0.893451	−	0.449160i	$-0.148277\pi$
$433$	37.1830	1.78690	0.893451	+	0.449160i	$0.148277\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.62630	0.364815
$438$	0	0
$439$	−25.0746	−1.19675	−0.598373	−	0.801218i	$-0.704186\pi$
$439$	−25.0746	−1.19675	−0.598373	+	0.801218i	$0.704186\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.04458	−0.0971408	−0.0485704	−	0.998820i	$-0.515467\pi$
$443$	−2.04458	−0.0971408	−0.0485704	+	0.998820i	$0.515467\pi$
$444$	0	0
$445$	−9.69728	−0.459695
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.2344	−0.482992	−0.241496	−	0.970402i	$-0.577638\pi$
$449$	−10.2344	−0.482992	−0.241496	+	0.970402i	$0.577638\pi$
$450$	0	0
$451$	4.59627	0.216430
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.79561	−0.224822
$456$	0	0
$457$	−42.5945	−1.99249	−0.996244	−	0.0865948i	$-0.972401\pi$
$457$	−42.5945	−1.99249	−0.996244	+	0.0865948i	$0.972401\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.504748	0.0235084	0.0117542	−	0.999931i	$-0.496258\pi$
$461$	0.504748	0.0235084	0.0117542	+	0.999931i	$0.496258\pi$
$462$	0	0
$463$	−2.68004	−0.124552	−0.0622761	−	0.998059i	$-0.519836\pi$
$463$	−2.68004	−0.124552	−0.0622761	+	0.998059i	$0.519836\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.4165	1.45378	0.726892	−	0.686752i	$-0.240964\pi$
$467$	31.4165	1.45378	0.726892	+	0.686752i	$0.240964\pi$
$468$	0	0
$469$	−2.95811	−0.136593
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.716881	0.0329622
$474$	0	0
$475$	10.1925	0.467666
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16.4406	0.751189	0.375594	−	0.926784i	$-0.377439\pi$
$479$	16.4406	0.751189	0.375594	+	0.926784i	$0.377439\pi$
$480$	0	0
$481$	24.8367	1.13245
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.9949	−0.499255
$486$	0	0
$487$	2.97535	0.134826	0.0674129	−	0.997725i	$-0.478526\pi$
$487$	2.97535	0.134826	0.0674129	+	0.997725i	$0.478526\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	26.4861	1.19530	0.597650	−	0.801757i	$-0.296101\pi$
$491$	26.4861	1.19530	0.597650	+	0.801757i	$0.296101\pi$
$492$	0	0
$493$	−9.99825	−0.450298
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.68004	−0.165073
$498$	0	0
$499$	13.4439	0.601830	0.300915	−	0.953651i	$-0.402708\pi$
$499$	13.4439	0.601830	0.300915	+	0.953651i	$0.402708\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.6631	1.01050	0.505250	−	0.862973i	$-0.331400\pi$
$503$	22.6631	1.01050	0.505250	+	0.862973i	$0.331400\pi$
$504$	0	0
$505$	8.54488	0.380242
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.54757	0.423189	0.211594	−	0.977358i	$-0.432134\pi$
$509$	9.54757	0.423189	0.211594	+	0.977358i	$0.432134\pi$
$510$	0	0
$511$	12.7811	0.565401
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.80066	0.255608
$516$	0	0
$517$	−3.96585	−0.174418
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.11287	−0.136377	−0.0681887	−	0.997672i	$-0.521722\pi$
$521$	−3.11287	−0.136377	−0.0681887	+	0.997672i	$0.521722\pi$
$522$	0	0
$523$	16.1489	0.706142	0.353071	−	0.935597i	$-0.385137\pi$
$523$	16.1489	0.706142	0.353071	+	0.935597i	$0.385137\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.52704	0.327883
$528$	0	0
$529$	−12.9986	−0.565155
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.46110	0.279861
$534$	0	0
$535$	2.10101	0.0908348
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.87939	−0.167097
$540$	0	0
$541$	−5.01548	−0.215632	−0.107816	−	0.994171i	$-0.534386\pi$
$541$	−5.01548	−0.215632	−0.107816	+	0.994171i	$0.534386\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.48070	−0.149097
$546$	0	0
$547$	−16.4780	−0.704549	−0.352275	−	0.935897i	$-0.614592\pi$
$547$	−16.4780	−0.704549	−0.352275	+	0.935897i	$0.614592\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14.5885	0.621492
$552$	0	0
$553$	−5.95811	−0.253365
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−34.5631	−1.46448	−0.732242	−	0.681045i	$-0.761526\pi$
$557$	−34.5631	−1.46448	−0.732242	+	0.681045i	$0.761526\pi$
$558$	0	0
$559$	1.00774	0.0426229
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.2104	1.56823	0.784115	−	0.620615i	$-0.213117\pi$
$563$	37.2104	1.56823	0.784115	+	0.620615i	$0.213117\pi$
$564$	0	0
$565$	−14.4688	−0.608709
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0.404667	0.0169645	0.00848226	−	0.999964i	$-0.497300\pi$
$569$	0.404667	0.0169645	0.00848226	+	0.999964i	$0.497300\pi$
$570$	0	0
$571$	37.7793	1.58101	0.790507	−	0.612453i	$-0.209817\pi$
$571$	37.7793	1.58101	0.790507	+	0.612453i	$0.209817\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13.3669	0.557438
$576$	0	0
$577$	−2.21120	−0.0920535	−0.0460267	−	0.998940i	$-0.514656\pi$
$577$	−2.21120	−0.0920535	−0.0460267	+	0.998940i	$0.514656\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.218941	−0.00908320
$582$	0	0
$583$	−28.2841	−1.17141
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.2098	−0.999244	−0.499622	−	0.866243i	$-0.666528\pi$
$587$	−24.2098	−0.999244	−0.499622	+	0.866243i	$0.666528\pi$
$588$	0	0
$589$	−10.9828	−0.452537
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.2385	0.502577	0.251288	−	0.967912i	$-0.419146\pi$
$593$	12.2385	0.502577	0.251288	+	0.967912i	$0.419146\pi$
$594$	0	0
$595$	1.45336	0.0595821
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−39.6168	−1.61870	−0.809349	−	0.587328i	$-0.800180\pi$
$599$	−39.6168	−1.61870	−0.809349	+	0.587328i	$0.800180\pi$
$600$	0	0
$601$	−30.0077	−1.22404	−0.612021	−	0.790842i	$-0.709643\pi$
$601$	−30.0077	−1.22404	−0.612021	+	0.790842i	$0.709643\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.56118	−0.144783
$606$	0	0
$607$	19.4843	0.790844	0.395422	−	0.918499i	$-0.370598\pi$
$607$	19.4843	0.790844	0.395422	+	0.918499i	$0.370598\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.57491	−0.225537
$612$	0	0
$613$	−18.5276	−0.748325	−0.374162	−	0.927363i	$-0.622070\pi$
$613$	−18.5276	−0.748325	−0.374162	+	0.927363i	$0.622070\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.8402	1.12080	0.560402	−	0.828221i	$-0.310647\pi$
$617$	27.8402	1.12080	0.560402	+	0.828221i	$0.310647\pi$
$618$	0	0
$619$	44.9813	1.80795	0.903976	−	0.427583i	$-0.140635\pi$
$619$	44.9813	1.80795	0.903976	+	0.427583i	$0.140635\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11.0273	−0.441801
$624$	0	0
$625$	13.9982	0.559930
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.52704	−0.300123
$630$	0	0
$631$	−9.43613	−0.375646	−0.187823	−	0.982203i	$-0.560143\pi$
$631$	−9.43613	−0.375646	−0.187823	+	0.982203i	$0.560143\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15.5243	0.616065
$636$	0	0
$637$	−5.45336	−0.216070
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.3901	1.47682	0.738410	−	0.674352i	$-0.235577\pi$
$641$	37.3901	1.47682	0.738410	+	0.674352i	$0.235577\pi$
$642$	0	0
$643$	−1.61175	−0.0635611	−0.0317806	−	0.999495i	$-0.510118\pi$
$643$	−1.61175	−0.0635611	−0.0317806	+	0.999495i	$0.510118\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.1762	1.61880	0.809402	−	0.587255i	$-0.199791\pi$
$647$	41.1762	1.61880	0.809402	+	0.587255i	$0.199791\pi$
$648$	0	0
$649$	25.8384	1.01425
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.05199	0.119434	0.0597169	−	0.998215i	$-0.480980\pi$
$653$	3.05199	0.119434	0.0597169	+	0.998215i	$0.480980\pi$
$654$	0	0
$655$	−16.8821	−0.659637
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−41.6350	−1.62187	−0.810934	−	0.585138i	$-0.801041\pi$
$659$	−41.6350	−1.62187	−0.810934	+	0.585138i	$0.801041\pi$
$660$	0	0
$661$	20.3010	0.789616	0.394808	−	0.918764i	$-0.370811\pi$
$661$	20.3010	0.789616	0.394808	+	0.918764i	$0.370811\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.12061	−0.0822339
$666$	0	0
$667$	19.1320	0.740793
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.0719	0.388822
$672$	0	0
$673$	−0.830689	−0.0320207	−0.0160104	−	0.999872i	$-0.505096\pi$
$673$	−0.830689	−0.0320207	−0.0160104	+	0.999872i	$0.505096\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.8672	0.417660	0.208830	−	0.977952i	$-0.433034\pi$
$677$	10.8672	0.417660	0.208830	+	0.977952i	$0.433034\pi$
$678$	0	0
$679$	−12.5030	−0.479821
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−32.6946	−1.25102	−0.625512	−	0.780215i	$-0.715110\pi$
$683$	−32.6946	−1.25102	−0.625512	+	0.780215i	$0.715110\pi$
$684$	0	0
$685$	−15.9659	−0.610024
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−39.7597	−1.51472
$690$	0	0
$691$	−14.9982	−0.570560	−0.285280	−	0.958444i	$-0.592087\pi$
$691$	−14.9982	−0.570560	−0.285280	+	0.958444i	$0.592087\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	19.3969	0.735767
$696$	0	0
$697$	−1.95811	−0.0741687
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.4688	0.999714	0.499857	−	0.866108i	$-0.333386\pi$
$701$	26.4688	0.999714	0.499857	+	0.866108i	$0.333386\pi$
$702$	0	0
$703$	10.9828	0.414223
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.71688	0.365441
$708$	0	0
$709$	15.3601	0.576860	0.288430	−	0.957501i	$-0.406867\pi$
$709$	15.3601	0.576860	0.288430	+	0.957501i	$0.406867\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14.4032	−0.539405
$714$	0	0
$715$	−18.6040	−0.695750
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−26.7306	−0.996883	−0.498442	−	0.866923i	$-0.666094\pi$
$719$	−26.7306	−0.996883	−0.498442	+	0.866923i	$0.666094\pi$
$720$	0	0
$721$	6.59627	0.245658
$722$	0	0
$723$	0	0
$724$	0	0
$725$	25.5699	0.949641
$726$	0	0
$727$	45.6441	1.69285	0.846424	−	0.532510i	$-0.178751\pi$
$727$	45.6441	1.69285	0.846424	+	0.532510i	$0.178751\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.305407	−0.0112959
$732$	0	0
$733$	5.97502	0.220693	0.110346	−	0.993893i	$-0.464804\pi$
$733$	5.97502	0.220693	0.110346	+	0.993893i	$0.464804\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.4757	−0.422711
$738$	0	0
$739$	35.5963	1.30943	0.654715	−	0.755876i	$-0.272789\pi$
$739$	35.5963	1.30943	0.654715	+	0.755876i	$0.272789\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.3087	1.07523	0.537616	−	0.843190i	$-0.319325\pi$
$743$	29.3087	1.07523	0.537616	+	0.843190i	$0.319325\pi$
$744$	0	0
$745$	13.3277	0.488289
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.38919	0.0872989
$750$	0	0
$751$	17.3337	0.632515	0.316258	−	0.948673i	$-0.397574\pi$
$751$	17.3337	0.632515	0.316258	+	0.948673i	$0.397574\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−16.6699	−0.606681
$756$	0	0
$757$	−2.77156	−0.100734	−0.0503671	−	0.998731i	$-0.516039\pi$
$757$	−2.77156	−0.100734	−0.0503671	+	0.998731i	$0.516039\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.50744	0.272144	0.136072	−	0.990699i	$-0.456552\pi$
$761$	7.50744	0.272144	0.136072	+	0.990699i	$0.456552\pi$
$762$	0	0
$763$	−3.95811	−0.143293
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.3218	1.31150
$768$	0	0
$769$	2.04364	0.0736957	0.0368478	−	0.999321i	$-0.488268\pi$
$769$	2.04364	0.0736957	0.0368478	+	0.999321i	$0.488268\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−24.9418	−0.897094	−0.448547	−	0.893759i	$-0.648058\pi$
$773$	−24.9418	−0.897094	−0.448547	+	0.893759i	$0.648058\pi$
$774$	0	0
$775$	−19.2499	−0.691477
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.85710	0.102366
$780$	0	0
$781$	−14.2763	−0.510847
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.8794	0.566760
$786$	0	0
$787$	−7.10700	−0.253337	−0.126669	−	0.991945i	$-0.540428\pi$
$787$	−7.10700	−0.253337	−0.126669	+	0.991945i	$0.540428\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16.4534	−0.585014
$792$	0	0
$793$	14.1584	0.502779
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.6628	1.19240	0.596199	−	0.802837i	$-0.296677\pi$
$797$	33.6628	1.19240	0.596199	+	0.802837i	$0.296677\pi$
$798$	0	0
$799$	1.68954	0.0597716
$800$	0	0
$801$	0	0
$802$	0	0
$803$	49.5827	1.74973
$804$	0	0
$805$	−2.78106	−0.0980195
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−12.8161	−0.450592	−0.225296	−	0.974290i	$-0.572335\pi$
$809$	−12.8161	−0.450592	−0.225296	+	0.974290i	$0.572335\pi$
$810$	0	0
$811$	26.1239	0.917335	0.458667	−	0.888608i	$-0.348327\pi$
$811$	26.1239	0.917335	0.458667	+	0.888608i	$0.348327\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.842549	0.0295132
$816$	0	0
$817$	0.445622	0.0155904
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27.6641	0.965483	0.482741	−	0.875763i	$-0.339641\pi$
$821$	27.6641	0.965483	0.482741	+	0.875763i	$0.339641\pi$
$822$	0	0
$823$	−27.8324	−0.970178	−0.485089	−	0.874465i	$-0.661213\pi$
$823$	−27.8324	−0.970178	−0.485089	+	0.874465i	$0.661213\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.65507	−0.161873	−0.0809363	−	0.996719i	$-0.525791\pi$
$827$	−4.65507	−0.161873	−0.0809363	+	0.996719i	$0.525791\pi$
$828$	0	0
$829$	−9.97359	−0.346397	−0.173199	−	0.984887i	$-0.555410\pi$
$829$	−9.97359	−0.346397	−0.173199	+	0.984887i	$0.555410\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.65270	0.0572628
$834$	0	0
$835$	17.4456	0.603731
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.72967	0.232334	0.116167	−	0.993230i	$-0.462939\pi$
$839$	6.72967	0.232334	0.116167	+	0.993230i	$0.462939\pi$
$840$	0	0
$841$	7.59802	0.262001
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−14.7202	−0.506390
$846$	0	0
$847$	−4.04963	−0.139147
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.4032	0.493737
$852$	0	0
$853$	5.79055	0.198265	0.0991324	−	0.995074i	$-0.468393\pi$
$853$	5.79055	0.198265	0.0991324	+	0.995074i	$0.468393\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.9077	−1.19242	−0.596211	−	0.802827i	$-0.703328\pi$
$857$	−34.9077	−1.19242	−0.596211	+	0.802827i	$0.703328\pi$
$858$	0	0
$859$	12.6149	0.430416	0.215208	−	0.976568i	$-0.430957\pi$
$859$	12.6149	0.430416	0.215208	+	0.976568i	$0.430957\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.2053	0.823959	0.411979	−	0.911193i	$-0.364838\pi$
$863$	24.2053	0.823959	0.411979	+	0.911193i	$0.364838\pi$
$864$	0	0
$865$	−19.9469	−0.678214
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−23.1138	−0.784082
$870$	0	0
$871$	−16.1317	−0.546600
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.11381	−0.274297
$876$	0	0
$877$	−1.12567	−0.0380111	−0.0190055	−	0.999819i	$-0.506050\pi$
$877$	−1.12567	−0.0380111	−0.0190055	+	0.999819i	$0.506050\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.38331	−0.147678	−0.0738388	−	0.997270i	$-0.523525\pi$
$881$	−4.38331	−0.147678	−0.0738388	+	0.997270i	$0.523525\pi$
$882$	0	0
$883$	6.88949	0.231850	0.115925	−	0.993258i	$-0.463017\pi$
$883$	6.88949	0.231850	0.115925	+	0.993258i	$0.463017\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39.0752	−1.31202	−0.656009	−	0.754753i	$-0.727757\pi$
$887$	−39.0752	−1.31202	−0.656009	+	0.754753i	$0.727757\pi$
$888$	0	0
$889$	17.6536	0.592084
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.46522	−0.0824955
$894$	0	0
$895$	−6.46110	−0.215971
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−27.5523	−0.918921
$900$	0	0
$901$	12.0496	0.401431
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.03003	0.100722
$906$	0	0
$907$	−42.4938	−1.41098	−0.705492	−	0.708718i	$-0.749274\pi$
$907$	−42.4938	−1.41098	−0.705492	+	0.708718i	$0.749274\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	15.4935	0.513322	0.256661	−	0.966501i	$-0.417378\pi$
$911$	15.4935	0.513322	0.256661	+	0.966501i	$0.417378\pi$
$912$	0	0
$913$	−0.849356	−0.0281096
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19.1976	−0.633960
$918$	0	0
$919$	−6.52940	−0.215385	−0.107693	−	0.994184i	$-0.534346\pi$
$919$	−6.52940	−0.215385	−0.107693	+	0.994184i	$0.534346\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−20.0686	−0.660567
$924$	0	0
$925$	19.2499	0.632933
$926$	0	0
$927$	0	0
$928$	0	0
$929$	58.2772	1.91201	0.956007	−	0.293343i	$-0.0947679\pi$
$929$	58.2772	1.91201	0.956007	+	0.293343i	$0.0947679\pi$
$930$	0	0
$931$	−2.41147	−0.0790329
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.63816	0.184387
$936$	0	0
$937$	32.4175	1.05903	0.529516	−	0.848300i	$-0.322374\pi$
$937$	32.4175	1.05903	0.529516	+	0.848300i	$0.322374\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−27.3226	−0.890693	−0.445346	−	0.895358i	$-0.646919\pi$
$941$	−27.3226	−0.890693	−0.445346	+	0.895358i	$0.646919\pi$
$942$	0	0
$943$	3.74691	0.122016
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.2131	1.24176	0.620879	−	0.783906i	$-0.286776\pi$
$947$	38.2131	1.24176	0.620879	+	0.783906i	$0.286776\pi$
$948$	0	0
$949$	69.6998	2.26255
$950$	0	0
$951$	0	0
$952$	0	0
$953$	58.9377	1.90918	0.954590	−	0.297924i	$-0.0962943\pi$
$953$	58.9377	1.90918	0.954590	+	0.297924i	$0.0962943\pi$
$954$	0	0
$955$	4.97535	0.160998
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−18.1557	−0.586278
$960$	0	0
$961$	−10.2576	−0.330892
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8.43882	−0.271655
$966$	0	0
$967$	−24.7187	−0.794901	−0.397451	−	0.917624i	$-0.630105\pi$
$967$	−24.7187	−0.794901	−0.397451	+	0.917624i	$0.630105\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8.17623	−0.262388	−0.131194	−	0.991357i	$-0.541881\pi$
$971$	−8.17623	−0.262388	−0.131194	+	0.991357i	$0.541881\pi$
$972$	0	0
$973$	22.0574	0.707127
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.8485	−0.507040	−0.253520	−	0.967330i	$-0.581588\pi$
$977$	−15.8485	−0.507040	−0.253520	+	0.967330i	$0.581588\pi$
$978$	0	0
$979$	−42.7793	−1.36723
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−53.3063	−1.70021	−0.850104	−	0.526615i	$-0.823461\pi$
$983$	−53.3063	−1.70021	−0.850104	+	0.526615i	$0.823461\pi$
$984$	0	0
$985$	−7.31645	−0.233121
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.584407	0.0185831
$990$	0	0
$991$	−40.2094	−1.27730	−0.638648	−	0.769499i	$-0.720506\pi$
$991$	−40.2094	−1.27730	−0.638648	+	0.769499i	$0.720506\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.80066	0.183893
$996$	0	0
$997$	28.7202	0.909577	0.454789	−	0.890599i	$-0.349715\pi$
$997$	28.7202	0.909577	0.454789	+	0.890599i	$0.349715\pi$
$998$	0	0
$999$	0	0

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	9072.2.a.ca.1.1		3
3.2	odd	2		9072.2.a.bs.1.3		3
4.3	odd	2		567.2.a.h.1.3		3
9.2	odd	6		3024.2.r.k.1009.1		6
9.4	even	3		1008.2.r.h.673.3		6
9.5	odd	6		3024.2.r.k.2017.1		6
9.7	even	3		1008.2.r.h.337.3		6
12.11	even	2		567.2.a.c.1.1		3
28.27	even	2		3969.2.a.q.1.3		3
36.7	odd	6		63.2.f.a.22.1	✓	6
36.11	even	6		189.2.f.b.64.3		6
36.23	even	6		189.2.f.b.127.3		6
36.31	odd	6		63.2.f.a.43.1	yes	6
84.83	odd	2		3969.2.a.l.1.1		3
252.11	even	6		1323.2.h.c.226.1		6
252.23	even	6		1323.2.h.c.802.1		6
252.31	even	6		441.2.g.b.79.1		6
252.47	odd	6		1323.2.g.e.361.3		6
252.59	odd	6		1323.2.g.e.667.3		6
252.67	odd	6		441.2.g.c.79.1		6
252.79	odd	6		441.2.g.c.67.1		6
252.83	odd	6		1323.2.f.d.442.3		6
252.95	even	6		1323.2.g.d.667.3		6
252.103	even	6		441.2.h.e.214.3		6
252.115	even	6		441.2.h.e.373.3		6
252.131	odd	6		1323.2.h.b.802.1		6
252.139	even	6		441.2.f.c.295.1		6
252.151	odd	6		441.2.h.d.373.3		6
252.167	odd	6		1323.2.f.d.883.3		6
252.187	even	6		441.2.g.b.67.1		6
252.191	even	6		1323.2.g.d.361.3		6
252.223	even	6		441.2.f.c.148.1		6
252.227	odd	6		1323.2.h.b.226.1		6
252.247	odd	6		441.2.h.d.214.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.f.a.22.1	✓	6	36.7	odd	6
63.2.f.a.43.1	yes	6	36.31	odd	6
189.2.f.b.64.3		6	36.11	even	6
189.2.f.b.127.3		6	36.23	even	6
441.2.f.c.148.1		6	252.223	even	6
441.2.f.c.295.1		6	252.139	even	6
441.2.g.b.67.1		6	252.187	even	6
441.2.g.b.79.1		6	252.31	even	6
441.2.g.c.67.1		6	252.79	odd	6
441.2.g.c.79.1		6	252.67	odd	6
441.2.h.d.214.3		6	252.247	odd	6
441.2.h.d.373.3		6	252.151	odd	6
441.2.h.e.214.3		6	252.103	even	6
441.2.h.e.373.3		6	252.115	even	6
567.2.a.c.1.1		3	12.11	even	2
567.2.a.h.1.3		3	4.3	odd	2
1008.2.r.h.337.3		6	9.7	even	3
1008.2.r.h.673.3		6	9.4	even	3
1323.2.f.d.442.3		6	252.83	odd	6
1323.2.f.d.883.3		6	252.167	odd	6
1323.2.g.d.361.3		6	252.191	even	6
1323.2.g.d.667.3		6	252.95	even	6
1323.2.g.e.361.3		6	252.47	odd	6
1323.2.g.e.667.3		6	252.59	odd	6
1323.2.h.b.226.1		6	252.227	odd	6
1323.2.h.b.802.1		6	252.131	odd	6
1323.2.h.c.226.1		6	252.11	even	6
1323.2.h.c.802.1		6	252.23	even	6
3024.2.r.k.1009.1		6	9.2	odd	6
3024.2.r.k.2017.1		6	9.5	odd	6
3969.2.a.l.1.1		3	84.83	odd	2
3969.2.a.q.1.3		3	28.27	even	2
9072.2.a.bs.1.3		3	3.2	odd	2
9072.2.a.ca.1.1		3	1.1	even	1	trivial

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	9072.2.a.ca.1.1

Level	$9072$
Weight	$2$
Character	9072.1
Self dual	yes
Analytic conductor	$72.440$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$