LMFDB - Embedded newform 4140.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4140.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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4140.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+1.00000 q^{5} -4.00000 q^{7} +6.00000 q^{11} -1.00000 q^{13} +2.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} -9.00000 q^{29} +5.00000 q^{31} -4.00000 q^{35} +2.00000 q^{37} +9.00000 q^{41} -4.00000 q^{43} +3.00000 q^{47} +9.00000 q^{49} +6.00000 q^{53} +6.00000 q^{55} +2.00000 q^{61} -1.00000 q^{65} -10.0000 q^{67} +3.00000 q^{71} -7.00000 q^{73} -24.0000 q^{77} -10.0000 q^{79} +12.0000 q^{83} +4.00000 q^{91} +2.00000 q^{95} +8.00000 q^{97} +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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−24.0000	−2.73505
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	23.0000	1.95083	0.975417	−	0.220366i	$-0.0707252\pi$
$139$	23.0000	1.95083	0.975417	+	0.220366i	$0.0707252\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.00000	−0.501745
$144$	0	0
$145$	−9.00000	−0.747409
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.00000	0.401610
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.0000	−1.49619	−0.748094	−	0.663593i	$-0.769031\pi$
$197$	−21.0000	−1.49619	−0.748094	+	0.663593i	$0.769031\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	36.0000	2.52670
$204$	0	0
$205$	9.00000	0.628587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	−20.0000	−1.35769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.00000	0.574989
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.00000	0.361814
$276$	0	0
$277$	−31.0000	−1.86261	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−31.0000	−1.86261	−0.931305	+	0.364241i	$0.881328\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−36.0000	−2.12501
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	16.0000	0.922225
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	0	0
$313$	32.0000	1.80875	0.904373	−	0.426742i	$-0.140339\pi$
$313$	32.0000	1.80875	0.904373	+	0.426742i	$0.140339\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−54.0000	−3.02342
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	35.0000	1.92377	0.961887	−	0.273447i	$-0.0881639\pi$
$331$	35.0000	1.92377	0.961887	+	0.273447i	$0.0881639\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.0000	−0.546358
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	30.0000	1.62459
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	36.0000	1.93258	0.966291	−	0.257454i	$-0.0828835\pi$
$347$	36.0000	1.93258	0.966291	+	0.257454i	$0.0828835\pi$
$348$	0	0
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	3.00000	0.159223
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.00000	−0.366397
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.00000	0.463524
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.0000	−0.503155
$396$	0	0
$397$	23.0000	1.15434	0.577168	−	0.816625i	$-0.304158\pi$
$397$	23.0000	1.15434	0.577168	+	0.816625i	$0.304158\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−5.00000	−0.249068
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8.00000	−0.387147
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	11.0000	0.525001	0.262501	−	0.964932i	$-0.415453\pi$
$439$	11.0000	0.525001	0.262501	+	0.964932i	$0.415453\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.00000	−0.427603	−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000	−0.427603	−0.213801	+	0.976877i	$0.568585\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	54.0000	2.54276
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	40.0000	1.84703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.00000	0.363261
$486$	0	0
$487$	−13.0000	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$487$	−13.0000	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	−19.0000	−0.850557	−0.425278	−	0.905063i	$-0.639824\pi$
$499$	−19.0000	−0.850557	−0.425278	+	0.905063i	$0.639824\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.00000	−0.176261
$516$	0	0
$517$	18.0000	0.791639
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.00000	−0.389833
$534$	0	0
$535$	18.0000	0.778208
$536$	0	0
$537$	0	0
$538$	0	0
$539$	54.0000	2.32594
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	40.0000	1.70097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−48.0000	−1.99138
$582$	0	0
$583$	36.0000	1.49097
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.0000	−1.36206	−0.681028	−	0.732257i	$-0.738467\pi$
$587$	−33.0000	−1.36206	−0.681028	+	0.732257i	$0.738467\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	0	0
$591$	0	0
$592$	0	0
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	25.0000	1.01639
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.0000	0.436522
$636$	0	0
$637$	−9.00000	−0.356593
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.0000	−0.825595	−0.412798	−	0.910823i	$-0.635448\pi$
$647$	−21.0000	−0.825595	−0.412798	+	0.910823i	$0.635448\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	0	0
$655$	15.0000	0.586098
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	9.00000	0.348481
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	−32.0000	−1.22805
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.00000	−0.114792	−0.0573959	−	0.998351i	$-0.518280\pi$
$683$	−3.00000	−0.114792	−0.0573959	+	0.998351i	$0.518280\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.0000	0.872440
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−24.0000	−0.902613
$708$	0	0
$709$	32.0000	1.20179	0.600893	−	0.799330i	$-0.294812\pi$
$709$	32.0000	1.20179	0.600893	+	0.799330i	$0.294812\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.00000	−0.187251
$714$	0	0
$715$	−6.00000	−0.224387
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.00000	−0.334252
$726$	0	0
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−40.0000	−1.47743	−0.738717	−	0.674016i	$-0.764568\pi$
$733$	−40.0000	−1.47743	−0.738717	+	0.674016i	$0.764568\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−60.0000	−2.21013
$738$	0	0
$739$	−25.0000	−0.919640	−0.459820	−	0.888012i	$-0.652086\pi$
$739$	−25.0000	−0.919640	−0.459820	+	0.888012i	$0.652086\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−72.0000	−2.63082
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.00000	0.181969
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	−80.0000	−2.89619
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	5.00000	0.179605
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.0000	0.644917
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.00000	−0.142766
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	48.0000	1.70668
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−42.0000	−1.48215
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−19.0000	−0.667180	−0.333590	−	0.942718i	$-0.608260\pi$
$811$	−19.0000	−0.667180	−0.333590	+	0.942718i	$0.608260\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.00000	−0.0350285
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	24.0000	0.830554
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	−100.000	−3.43604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	0	0
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.0000	−1.33221	−0.666107	−	0.745856i	$-0.732041\pi$
$857$	−39.0000	−1.33221	−0.666107	+	0.745856i	$0.732041\pi$
$858$	0	0
$859$	5.00000	0.170598	0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000	0.170598	0.0852989	+	0.996355i	$0.472815\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.0000	1.53182	0.765909	−	0.642949i	$-0.222289\pi$
$863$	45.0000	1.53182	0.765909	+	0.642949i	$0.222289\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−60.0000	−2.03536
$870$	0	0
$871$	10.0000	0.338837
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.00000	−0.135225
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	0	0
$889$	−44.0000	−1.47571
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.00000	0.200782
$894$	0	0
$895$	−15.0000	−0.501395
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−45.0000	−1.50083
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.00000	0.0664822
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−60.0000	−1.98137
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.00000	−0.0987462
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−39.0000	−1.27955	−0.639774	−	0.768563i	$-0.720972\pi$
$929$	−39.0000	−1.27955	−0.639774	+	0.768563i	$0.720972\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	0	0
$943$	−9.00000	−0.293080
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33.0000	−1.07236	−0.536178	−	0.844105i	$-0.680132\pi$
$947$	−33.0000	−1.07236	−0.536178	+	0.844105i	$0.680132\pi$
$948$	0	0
$949$	7.00000	0.227230
$950$	0	0
$951$	0	0
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	18.0000	0.582466
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−48.0000	−1.55000
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19.0000	−0.611632
$966$	0	0
$967$	23.0000	0.739630	0.369815	−	0.929105i	$-0.379421\pi$
$967$	23.0000	0.739630	0.369815	+	0.929105i	$0.379421\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	0	0
$973$	−92.0000	−2.94938
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−54.0000	−1.72233	−0.861166	−	0.508323i	$-0.830265\pi$
$983$	−54.0000	−1.72233	−0.861166	+	0.508323i	$0.830265\pi$
$984$	0	0
$985$	−21.0000	−0.669116
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.0000	0.443830
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	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	4140.2.a.f.1.1		1
3.2	odd	2		460.2.a.c.1.1	✓	1
12.11	even	2		1840.2.a.c.1.1		1
15.2	even	4		2300.2.c.d.1749.1		2
15.8	even	4		2300.2.c.d.1749.2		2
15.14	odd	2		2300.2.a.d.1.1		1
24.5	odd	2		7360.2.a.i.1.1		1
24.11	even	2		7360.2.a.v.1.1		1
60.59	even	2		9200.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.a.c.1.1	✓	1	3.2	odd	2
1840.2.a.c.1.1		1	12.11	even	2
2300.2.a.d.1.1		1	15.14	odd	2
2300.2.c.d.1749.1		2	15.2	even	4
2300.2.c.d.1749.2		2	15.8	even	4
4140.2.a.f.1.1		1	1.1	even	1	trivial
7360.2.a.i.1.1		1	24.5	odd	2
7360.2.a.v.1.1		1	24.11	even	2
9200.2.a.y.1.1		1	60.59	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 $\F_{q}$

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	4140.2.a.f.1.1

Level	$4140$
Weight	$2$
Character	4140.1
Self dual	yes
Analytic conductor	$33.058$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$