LMFDB - Embedded newform 693.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("693.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$693 = 3^{2} \cdot 7 \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	693.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:	$5.53363286007$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	693.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{11} -4.00000 q^{13} +4.00000 q^{16} -2.00000 q^{17} -6.00000 q^{19} -2.00000 q^{20} +5.00000 q^{23} -4.00000 q^{25} +2.00000 q^{28} -10.0000 q^{29} +1.00000 q^{31} -1.00000 q^{35} -5.00000 q^{37} +2.00000 q^{41} -8.00000 q^{43} -2.00000 q^{44} -8.00000 q^{47} +1.00000 q^{49} +8.00000 q^{52} +6.00000 q^{53} +1.00000 q^{55} -3.00000 q^{59} -2.00000 q^{61} -8.00000 q^{64} -4.00000 q^{65} -3.00000 q^{67} +4.00000 q^{68} -1.00000 q^{71} +10.0000 q^{73} +12.0000 q^{76} -1.00000 q^{77} +6.00000 q^{79} +4.00000 q^{80} -12.0000 q^{83} -2.00000 q^{85} +15.0000 q^{89} +4.00000 q^{91} -10.0000 q^{92} -6.00000 q^{95} -5.00000 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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	2.00000	0.377964
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	8.00000	1.10940
$53$	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
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	0	0
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	0	0
$76$	12.0000	1.37649
$77$	−1.00000	−0.113961
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	−10.0000	−1.04257
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	0	0
$99$	0	0
$100$	8.00000	0.800000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	−4.00000	−0.377964
$113$	19.0000	1.78737	0.893685	−	0.448695i	$-0.148111\pi$
$113$	19.0000	1.78737	0.893685	+	0.448695i	$0.148111\pi$
$114$	0	0
$115$	5.00000	0.466252
$116$	20.0000	1.85695
$117$	0	0
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	−2.00000	−0.179605
$125$	−9.00000	−0.804984
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	−10.0000	−0.830455
$146$	0	0
$147$	0	0
$148$	10.0000	0.821995
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.00000	−0.394055
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	16.0000	1.21999
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	4.00000	0.301511
$177$	0	0
$178$	0	0
$179$	−1.00000	−0.0747435	−0.0373718	−	0.999301i	$-0.511899\pi$
$179$	−1.00000	−0.0747435	−0.0373718	+	0.999301i	$0.511899\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.00000	−0.367607
$186$	0	0
$187$	−2.00000	−0.146254
$188$	16.0000	1.16692
$189$	0	0
$190$	0	0
$191$	−5.00000	−0.361787	−0.180894	−	0.983503i	$-0.557899\pi$
$191$	−5.00000	−0.361787	−0.180894	+	0.983503i	$0.557899\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	−2.00000	−0.142857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.0000	0.701862
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	−16.0000	−1.10940
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−1.00000	−0.0678844
$218$	0	0
$219$	0	0
$220$	−2.00000	−0.134840
$221$	8.00000	0.538138
$222$	0	0
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	0	0
$243$	0	0
$244$	4.00000	0.256074
$245$	1.00000	0.0638877
$246$	0	0
$247$	24.0000	1.52708
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	0	0
$253$	5.00000	0.314347
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	5.00000	0.310685
$260$	8.00000	0.496139
$261$	0	0
$262$	0	0
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	6.00000	0.366508
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−8.00000	−0.485071
$273$	0	0
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	24.0000	1.44202	0.721010	−	0.692925i	$-0.243678\pi$
$277$	24.0000	1.44202	0.721010	+	0.692925i	$0.243678\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	−20.0000	−1.17041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−3.00000	−0.174667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−20.0000	−1.15663
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	0	0
$304$	−24.0000	−1.37649
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−23.0000	−1.30004	−0.650018	−	0.759918i	$-0.725239\pi$
$313$	−23.0000	−1.30004	−0.650018	+	0.759918i	$0.725239\pi$
$314$	0	0
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−9.00000	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$318$	0	0
$319$	−10.0000	−0.559893
$320$	−8.00000	−0.447214
$321$	0	0
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	16.0000	0.887520
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	24.0000	1.31717
$333$	0	0
$334$	0	0
$335$	−3.00000	−0.163908
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	0	0
$340$	4.00000	0.216930
$341$	1.00000	0.0541530
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	0	0
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	0	0
$355$	−1.00000	−0.0530745
$356$	−30.0000	−1.59000
$357$	0	0
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	−8.00000	−0.419314
$365$	10.0000	0.523424
$366$	0	0
$367$	−11.0000	−0.574195	−0.287098	−	0.957901i	$-0.592690\pi$
$367$	−11.0000	−0.574195	−0.287098	+	0.957901i	$0.592690\pi$
$368$	20.0000	1.04257
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	40.0000	2.06010
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	12.0000	0.615587
$381$	0	0
$382$	0	0
$383$	−17.0000	−0.868659	−0.434330	−	0.900754i	$-0.643015\pi$
$383$	−17.0000	−0.868659	−0.434330	+	0.900754i	$0.643015\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	0	0
$388$	10.0000	0.507673
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	−10.0000	−0.505722
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.00000	0.301893
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	0	0
$399$	0	0
$400$	−16.0000	−0.800000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	−24.0000	−1.19404
$405$	0	0
$406$	0	0
$407$	−5.00000	−0.247841
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	24.0000	1.18240
$413$	3.00000	0.147620
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.00000	0.388057
$426$	0	0
$427$	2.00000	0.0967868
$428$	−20.0000	−0.966736
$429$	0	0
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	0	0
$436$	−8.00000	−0.383131
$437$	−30.0000	−1.43509
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	0	0
$445$	15.0000	0.711068
$446$	0	0
$447$	0	0
$448$	8.00000	0.377964
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	−38.0000	−1.78737
$453$	0	0
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	0	0
$460$	−10.0000	−0.466252
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	−40.0000	−1.85695
$465$	0	0
$466$	0	0
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	0	0
$469$	3.00000	0.138527
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	24.0000	1.10120
$476$	−4.00000	−0.183340
$477$	0	0
$478$	0	0
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−5.00000	−0.227038
$486$	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
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	0	0
$496$	4.00000	0.179605
$497$	1.00000	0.0448561
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	18.0000	0.804984
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	0	0
$508$	−4.00000	−0.177471
$509$	31.0000	1.37405	0.687025	−	0.726633i	$-0.258916\pi$
$509$	31.0000	1.37405	0.687025	+	0.726633i	$0.258916\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	−8.00000	−0.351840
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.00000	−0.306676	−0.153338	−	0.988174i	$-0.549002\pi$
$521$	−7.00000	−0.306676	−0.153338	+	0.988174i	$0.549002\pi$
$522$	0	0
$523$	32.0000	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$524$	36.0000	1.57267
$525$	0	0
$526$	0	0
$527$	−2.00000	−0.0871214
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	0	0
$532$	−12.0000	−0.520266
$533$	−8.00000	−0.346518
$534$	0	0
$535$	10.0000	0.432338
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	−24.0000	−1.02617	−0.513083	−	0.858339i	$-0.671497\pi$
$547$	−24.0000	−1.02617	−0.513083	+	0.858339i	$0.671497\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	0	0
$551$	60.0000	2.55609
$552$	0	0
$553$	−6.00000	−0.255146
$554$	0	0
$555$	0	0
$556$	20.0000	0.848189
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	−4.00000	−0.169031
$561$	0	0
$562$	0	0
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	0	0
$565$	19.0000	0.799336
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	8.00000	0.334497
$573$	0	0
$574$	0	0
$575$	−20.0000	−0.834058
$576$	0	0
$577$	−25.0000	−1.04076	−0.520382	−	0.853934i	$-0.674210\pi$
$577$	−25.0000	−1.04076	−0.520382	+	0.853934i	$0.674210\pi$
$578$	0	0
$579$	0	0
$580$	20.0000	0.830455
$581$	12.0000	0.497844
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	0	0
$591$	0	0
$592$	−20.0000	−0.821995
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	−44.0000	−1.80231
$597$	0	0
$598$	0	0
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	0	0
$604$	−12.0000	−0.488273
$605$	1.00000	0.0406558
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	32.0000	1.29458
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	−2.00000	−0.0803219
$621$	0	0
$622$	0	0
$623$	−15.0000	−0.600962
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	−14.0000	−0.558661
$629$	10.0000	0.398726
$630$	0	0
$631$	27.0000	1.07485	0.537427	−	0.843311i	$-0.319397\pi$
$631$	27.0000	1.07485	0.537427	+	0.843311i	$0.319397\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.00000	0.0793676
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	0	0
$643$	−29.0000	−1.14365	−0.571824	−	0.820376i	$-0.693764\pi$
$643$	−29.0000	−1.14365	−0.571824	+	0.820376i	$0.693764\pi$
$644$	10.0000	0.394055
$645$	0	0
$646$	0	0
$647$	21.0000	0.825595	0.412798	−	0.910823i	$-0.364552\pi$
$647$	21.0000	0.825595	0.412798	+	0.910823i	$0.364552\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	17.0000	0.665261	0.332631	−	0.943057i	$-0.392064\pi$
$653$	17.0000	0.665261	0.332631	+	0.943057i	$0.392064\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	8.00000	0.312348
$657$	0	0
$658$	0	0
$659$	2.00000	0.0779089	0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000	0.0779089	0.0389545	+	0.999241i	$0.487597\pi$
$660$	0	0
$661$	35.0000	1.36134	0.680671	−	0.732589i	$-0.261688\pi$
$661$	35.0000	1.36134	0.680671	+	0.732589i	$0.261688\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.00000	0.232670
$666$	0	0
$667$	−50.0000	−1.93601
$668$	−4.00000	−0.154765
$669$	0	0
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	4.00000	0.154189	0.0770943	−	0.997024i	$-0.475436\pi$
$673$	4.00000	0.154189	0.0770943	+	0.997024i	$0.475436\pi$
$674$	0	0
$675$	0	0
$676$	−6.00000	−0.230769
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	0	0
$679$	5.00000	0.191882
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	3.00000	0.114624
$686$	0	0
$687$	0	0
$688$	−32.0000	−1.21999
$689$	−24.0000	−0.914327
$690$	0	0
$691$	15.0000	0.570627	0.285313	−	0.958434i	$-0.407902\pi$
$691$	15.0000	0.570627	0.285313	+	0.958434i	$0.407902\pi$
$692$	32.0000	1.21646
$693$	0	0
$694$	0	0
$695$	−10.0000	−0.379322
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	0	0
$700$	−8.00000	−0.302372
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	30.0000	1.13147
$704$	−8.00000	−0.301511
$705$	0	0
$706$	0	0
$707$	−12.0000	−0.451306
$708$	0	0
$709$	39.0000	1.46468	0.732338	−	0.680941i	$-0.238429\pi$
$709$	39.0000	1.46468	0.732338	+	0.680941i	$0.238429\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.00000	0.187251
$714$	0	0
$715$	−4.00000	−0.149592
$716$	2.00000	0.0747435
$717$	0	0
$718$	0	0
$719$	11.0000	0.410231	0.205115	−	0.978738i	$-0.434243\pi$
$719$	11.0000	0.410231	0.205115	+	0.978738i	$0.434243\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	0	0
$724$	−10.0000	−0.371647
$725$	40.0000	1.48556
$726$	0	0
$727$	−19.0000	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.00000	−0.110506
$738$	0	0
$739$	−18.0000	−0.662141	−0.331070	−	0.943606i	$-0.607410\pi$
$739$	−18.0000	−0.662141	−0.331070	+	0.943606i	$0.607410\pi$
$740$	10.0000	0.367607
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	22.0000	0.806018
$746$	0	0
$747$	0	0
$748$	4.00000	0.146254
$749$	−10.0000	−0.365392
$750$	0	0
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	−32.0000	−1.16692
$753$	0	0
$754$	0	0
$755$	6.00000	0.218362
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	10.0000	0.361787
$765$	0	0
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	0	0
$772$	−28.0000	−1.00774
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−1.00000	−0.0357828
$782$	0	0
$783$	0	0
$784$	4.00000	0.142857
$785$	7.00000	0.249841
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	36.0000	1.28245
$789$	0	0
$790$	0	0
$791$	−19.0000	−0.675562
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	0	0
$796$	16.0000	0.567105
$797$	−23.0000	−0.814702	−0.407351	−	0.913272i	$-0.633547\pi$
$797$	−23.0000	−0.814702	−0.407351	+	0.913272i	$0.633547\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.0000	0.352892
$804$	0	0
$805$	−5.00000	−0.176227
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	−20.0000	−0.701862
$813$	0	0
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−25.0000	−0.871445	−0.435723	−	0.900081i	$-0.643507\pi$
$823$	−25.0000	−0.871445	−0.435723	+	0.900081i	$0.643507\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	0	0
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	0	0
$831$	0	0
$832$	32.0000	1.10940
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	2.00000	0.0692129
$836$	12.0000	0.415029
$837$	0	0
$838$	0	0
$839$	−45.0000	−1.55357	−0.776786	−	0.629764i	$-0.783151\pi$
$839$	−45.0000	−1.55357	−0.776786	+	0.629764i	$0.783151\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	0	0
$844$	4.00000	0.137686
$845$	3.00000	0.103203
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	24.0000	0.824163
$849$	0	0
$850$	0	0
$851$	−25.0000	−0.856989
$852$	0	0
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	0	0
$859$	55.0000	1.87658	0.938288	−	0.345855i	$-0.112411\pi$
$859$	55.0000	1.87658	0.938288	+	0.345855i	$0.112411\pi$
$860$	16.0000	0.545595
$861$	0	0
$862$	0	0
$863$	−52.0000	−1.77010	−0.885050	−	0.465495i	$-0.845876\pi$
$863$	−52.0000	−1.77010	−0.885050	+	0.465495i	$0.845876\pi$
$864$	0	0
$865$	−16.0000	−0.544016
$866$	0	0
$867$	0	0
$868$	2.00000	0.0678844
$869$	6.00000	0.203536
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	0	0
$880$	4.00000	0.134840
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	−16.0000	−0.538138
$885$	0	0
$886$	0	0
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	0	0
$892$	−2.00000	−0.0669650
$893$	48.0000	1.60626
$894$	0	0
$895$	−1.00000	−0.0334263
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.0000	−0.333519
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.00000	0.166206
$906$	0	0
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	8.00000	0.265489
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	0	0
$916$	14.0000	0.462573
$917$	18.0000	0.594412
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.00000	0.131662
$924$	0	0
$925$	20.0000	0.657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	12.0000	0.393073
$933$	0	0
$934$	0	0
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	36.0000	1.17607	0.588034	−	0.808836i	$-0.299902\pi$
$937$	36.0000	1.17607	0.588034	+	0.808836i	$0.299902\pi$
$938$	0	0
$939$	0	0
$940$	16.0000	0.521862
$941$	−58.0000	−1.89075	−0.945373	−	0.325991i	$-0.894302\pi$
$941$	−58.0000	−1.89075	−0.945373	+	0.325991i	$0.894302\pi$
$942$	0	0
$943$	10.0000	0.325645
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	−5.00000	−0.162478	−0.0812391	−	0.996695i	$-0.525888\pi$
$947$	−5.00000	−0.162478	−0.0812391	+	0.996695i	$0.525888\pi$
$948$	0	0
$949$	−40.0000	−1.29845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	0	0
$955$	−5.00000	−0.161796
$956$	8.00000	0.258738
$957$	0	0
$958$	0	0
$959$	−3.00000	−0.0968751
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	0	0
$964$	24.0000	0.772988
$965$	14.0000	0.450676
$966$	0	0
$967$	−34.0000	−1.09337	−0.546683	−	0.837340i	$-0.684110\pi$
$967$	−34.0000	−1.09337	−0.546683	+	0.837340i	$0.684110\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.0000	−0.930654	−0.465327	−	0.885139i	$-0.654063\pi$
$971$	−29.0000	−0.930654	−0.465327	+	0.885139i	$0.654063\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	0	0
$975$	0	0
$976$	−8.00000	−0.256074
$977$	31.0000	0.991778	0.495889	−	0.868386i	$-0.334842\pi$
$977$	31.0000	0.991778	0.495889	+	0.868386i	$0.334842\pi$
$978$	0	0
$979$	15.0000	0.479402
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	0	0
$983$	27.0000	0.861166	0.430583	−	0.902551i	$-0.358308\pi$
$983$	27.0000	0.861166	0.430583	+	0.902551i	$0.358308\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	−48.0000	−1.52708
$989$	−40.0000	−1.27193
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	−12.0000	−0.380044	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	−12.0000	−0.380044	−0.190022	+	0.981780i	$0.560856\pi$
$998$	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	693.2.a.c.1.1		1
3.2	odd	2		77.2.a.a.1.1	✓	1
7.6	odd	2		4851.2.a.j.1.1		1
11.10	odd	2		7623.2.a.j.1.1		1
12.11	even	2		1232.2.a.l.1.1		1
15.2	even	4		1925.2.b.e.1849.2		2
15.8	even	4		1925.2.b.e.1849.1		2
15.14	odd	2		1925.2.a.h.1.1		1
21.2	odd	6		539.2.e.f.67.1		2
21.5	even	6		539.2.e.c.67.1		2
21.11	odd	6		539.2.e.f.177.1		2
21.17	even	6		539.2.e.c.177.1		2
21.20	even	2		539.2.a.c.1.1		1
24.5	odd	2		4928.2.a.bj.1.1		1
24.11	even	2		4928.2.a.a.1.1		1
33.2	even	10		847.2.f.h.323.1		4
33.5	odd	10		847.2.f.i.729.1		4
33.8	even	10		847.2.f.h.372.1		4
33.14	odd	10		847.2.f.i.372.1		4
33.17	even	10		847.2.f.h.729.1		4
33.20	odd	10		847.2.f.i.323.1		4
33.26	odd	10		847.2.f.i.148.1		4
33.29	even	10		847.2.f.h.148.1		4
33.32	even	2		847.2.a.b.1.1		1
84.83	odd	2		8624.2.a.a.1.1		1
231.230	odd	2		5929.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.a.a.1.1	✓	1	3.2	odd	2
539.2.a.c.1.1		1	21.20	even	2
539.2.e.c.67.1		2	21.5	even	6
539.2.e.c.177.1		2	21.17	even	6
539.2.e.f.67.1		2	21.2	odd	6
539.2.e.f.177.1		2	21.11	odd	6
693.2.a.c.1.1		1	1.1	even	1	trivial
847.2.a.b.1.1		1	33.32	even	2
847.2.f.h.148.1		4	33.29	even	10
847.2.f.h.323.1		4	33.2	even	10
847.2.f.h.372.1		4	33.8	even	10
847.2.f.h.729.1		4	33.17	even	10
847.2.f.i.148.1		4	33.26	odd	10
847.2.f.i.323.1		4	33.20	odd	10
847.2.f.i.372.1		4	33.14	odd	10
847.2.f.i.729.1		4	33.5	odd	10
1232.2.a.l.1.1		1	12.11	even	2
1925.2.a.h.1.1		1	15.14	odd	2
1925.2.b.e.1849.1		2	15.8	even	4
1925.2.b.e.1849.2		2	15.2	even	4
4851.2.a.j.1.1		1	7.6	odd	2
4928.2.a.a.1.1		1	24.11	even	2
4928.2.a.bj.1.1		1	24.5	odd	2
5929.2.a.f.1.1		1	231.230	odd	2
7623.2.a.j.1.1		1	11.10	odd	2
8624.2.a.a.1.1		1	84.83	odd	2

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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	693.2.a.c.1.1

Level	$693$
Weight	$2$
Character	693.1
Self dual	yes
Analytic conductor	$5.534$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$