LMFDB - Embedded newform 7728.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{13} +4.00000 q^{17} +3.00000 q^{19} +1.00000 q^{21} -1.00000 q^{23} -5.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +2.00000 q^{31} +1.00000 q^{33} -2.00000 q^{37} -2.00000 q^{39} +1.00000 q^{41} +8.00000 q^{43} +5.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} +3.00000 q^{53} -3.00000 q^{57} -5.00000 q^{59} +13.0000 q^{61} -1.00000 q^{63} +1.00000 q^{69} -16.0000 q^{73} +5.00000 q^{75} +1.00000 q^{77} +2.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +6.00000 q^{87} +6.00000 q^{89} -2.00000 q^{91} -2.00000 q^{93} +10.0000 q^{97} -1.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.00000	0.729325	0.364662	−	0.931140i	$-0.381184\pi$
$47$	5.00000	0.729325	0.364662	+	0.931140i	$0.381184\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	5.00000	0.577350
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	0	0
$103$	−11.0000	−1.08386	−0.541931	−	0.840423i	$-0.682307\pi$
$103$	−11.0000	−1.08386	−0.541931	+	0.840423i	$0.682307\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−1.00000	−0.0901670
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.0000	1.62328	0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000	1.62328	0.811640	+	0.584158i	$0.198575\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−5.00000	−0.421076
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	17.0000	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$149$	17.0000	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$150$	0	0
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823	0.0386912	−	0.999251i	$-0.487681\pi$
$167$	1.00000	0.0773823	0.0386912	+	0.999251i	$0.487681\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	0	0
$177$	5.00000	0.375823
$178$	0	0
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−13.0000	−0.960988
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	13.0000	0.935760	0.467880	−	0.883792i	$-0.345018\pi$
$193$	13.0000	0.935760	0.467880	+	0.883792i	$0.345018\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−3.00000	−0.207514
$210$	0	0
$211$	19.0000	1.30801	0.654007	−	0.756489i	$-0.273087\pi$
$211$	19.0000	1.30801	0.654007	+	0.756489i	$0.273087\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	16.0000	1.08118
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.00000	−0.129914
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	15.0000	0.966235	0.483117	−	0.875556i	$-0.339504\pi$
$241$	15.0000	0.966235	0.483117	+	0.875556i	$0.339504\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	14.0000	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	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
$259$	2.00000	0.124274
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	19.0000	1.17159	0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000	1.17159	0.585795	+	0.810459i	$0.300782\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	22.0000	1.34136	0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000	1.34136	0.670682	+	0.741745i	$0.266002\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	5.00000	0.301511
$276$	0	0
$277$	31.0000	1.86261	0.931305	−	0.364241i	$-0.118672\pi$
$277$	31.0000	1.86261	0.931305	+	0.364241i	$0.118672\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−9.00000	−0.517036
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	11.0000	0.625768
$310$	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
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	−10.0000	−0.554700
$326$	0	0
$327$	4.00000	0.221201
$328$	0	0
$329$	−5.00000	−0.275659
$330$	0	0
$331$	−5.00000	−0.274825	−0.137412	−	0.990514i	$-0.543879\pi$
$331$	−5.00000	−0.274825	−0.137412	+	0.990514i	$0.543879\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	14.0000	0.760376
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	−22.0000	−1.17094	−0.585471	−	0.810693i	$-0.699090\pi$
$353$	−22.0000	−1.17094	−0.585471	+	0.810693i	$0.699090\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−29.0000	−1.51379	−0.756894	−	0.653538i	$-0.773284\pi$
$367$	−29.0000	−1.51379	−0.756894	+	0.653538i	$0.773284\pi$
$368$	0	0
$369$	1.00000	0.0520579
$370$	0	0
$371$	−3.00000	−0.155752
$372$	0	0
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	0	0
$383$	30.0000	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$383$	30.0000	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.00000	0.406663
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	15.0000	0.756650
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	0	0
$399$	3.00000	0.150188
$400$	0	0
$401$	25.0000	1.24844	0.624220	−	0.781248i	$-0.285417\pi$
$401$	25.0000	1.24844	0.624220	+	0.781248i	$0.285417\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	−19.0000	−0.937201
$412$	0	0
$413$	5.00000	0.246034
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	0	0
$423$	5.00000	0.243108
$424$	0	0
$425$	−20.0000	−0.970143
$426$	0	0
$427$	−13.0000	−0.629114
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	23.0000	1.10787	0.553936	−	0.832560i	$-0.313125\pi$
$431$	23.0000	1.10787	0.553936	+	0.832560i	$0.313125\pi$
$432$	0	0
$433$	−37.0000	−1.77811	−0.889053	−	0.457804i	$-0.848636\pi$
$433$	−37.0000	−1.77811	−0.889053	+	0.457804i	$0.848636\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.00000	−0.143509
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−17.0000	−0.804072
$448$	0	0
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	−1.00000	−0.0470882
$452$	0	0
$453$	13.0000	0.610793
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	11.0000	0.511213	0.255607	−	0.966781i	$-0.417725\pi$
$463$	11.0000	0.511213	0.255607	+	0.966781i	$0.417725\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.00000	−0.0460776
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−15.0000	−0.688247
$476$	0	0
$477$	3.00000	0.137361
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	0	0
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	0	0
$489$	−11.0000	−0.497437
$490$	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
$493$	−24.0000	−1.08091
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	33.0000	1.46270	0.731350	−	0.682003i	$-0.238891\pi$
$509$	33.0000	1.46270	0.731350	+	0.682003i	$0.238891\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	0	0
$513$	−3.00000	−0.132453
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.00000	−0.219900
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	0	0
$523$	19.0000	0.830812	0.415406	−	0.909636i	$-0.363640\pi$
$523$	19.0000	0.830812	0.415406	+	0.909636i	$0.363640\pi$
$524$	0	0
$525$	−5.00000	−0.218218
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.00000	−0.216982
$532$	0	0
$533$	2.00000	0.0866296
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.0000	0.690451
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	0	0
$549$	13.0000	0.554826
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	−2.00000	−0.0850487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.0000	0.423714	0.211857	−	0.977301i	$-0.432049\pi$
$557$	10.0000	0.423714	0.211857	+	0.977301i	$0.432049\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	4.00000	0.168880
$562$	0	0
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−27.0000	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$569$	−27.0000	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	3.00000	0.125327
$574$	0	0
$575$	5.00000	0.208514
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	0	0
$579$	−13.0000	−0.540262
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	−3.00000	−0.124247
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.0000	−1.44460	−0.722302	−	0.691577i	$-0.756916\pi$
$587$	−35.0000	−1.44460	−0.722302	+	0.691577i	$0.756916\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.0000	−0.450200
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	24.0000	0.978980	0.489490	−	0.872009i	$-0.337183\pi$
$601$	24.0000	0.978980	0.489490	+	0.872009i	$0.337183\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	10.0000	0.404557
$612$	0	0
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	46.0000	1.85189	0.925945	−	0.377658i	$-0.123271\pi$
$617$	46.0000	1.85189	0.925945	+	0.377658i	$0.123271\pi$
$618$	0	0
$619$	12.0000	0.482321	0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000	0.482321	0.241160	+	0.970485i	$0.422472\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	3.00000	0.119808
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	50.0000	1.99047	0.995234	−	0.0975126i	$-0.0310886\pi$
$631$	50.0000	1.99047	0.995234	+	0.0975126i	$0.0310886\pi$
$632$	0	0
$633$	−19.0000	−0.755182
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	45.0000	1.77739	0.888697	−	0.458496i	$-0.151612\pi$
$641$	45.0000	1.77739	0.888697	+	0.458496i	$0.151612\pi$
$642$	0	0
$643$	9.00000	0.354925	0.177463	−	0.984128i	$-0.443211\pi$
$643$	9.00000	0.354925	0.177463	+	0.984128i	$0.443211\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	0	0
$649$	5.00000	0.196267
$650$	0	0
$651$	2.00000	0.0783862
$652$	0	0
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−16.0000	−0.624219
$658$	0	0
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	13.0000	0.505641	0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000	0.505641	0.252821	+	0.967513i	$0.418642\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.00000	0.232321
$668$	0	0
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	−13.0000	−0.501859
$672$	0	0
$673$	−31.0000	−1.19496	−0.597481	−	0.801883i	$-0.703832\pi$
$673$	−31.0000	−1.19496	−0.597481	+	0.801883i	$0.703832\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	4.00000	0.153732	0.0768662	−	0.997041i	$-0.475509\pi$
$677$	4.00000	0.153732	0.0768662	+	0.997041i	$0.475509\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	−2.00000	−0.0766402
$682$	0	0
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−13.0000	−0.495981
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	−24.0000	−0.907763
$700$	0	0
$701$	−45.0000	−1.69963	−0.849813	−	0.527084i	$-0.823285\pi$
$701$	−45.0000	−1.69963	−0.849813	+	0.527084i	$0.823285\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.00000	−0.338480
$708$	0	0
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	0	0
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4.00000	−0.149383
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	11.0000	0.409661
$722$	0	0
$723$	−15.0000	−0.557856
$724$	0	0
$725$	30.0000	1.11417
$726$	0	0
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	32.0000	1.18356
$732$	0	0
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	3.00000	0.110059	0.0550297	−	0.998485i	$-0.482475\pi$
$743$	3.00000	0.110059	0.0550297	+	0.998485i	$0.482475\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−6.00000	−0.219529
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	−14.0000	−0.510188
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	0	0
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	39.0000	1.41375	0.706874	−	0.707339i	$-0.250105\pi$
$761$	39.0000	1.41375	0.706874	+	0.707339i	$0.250105\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.0000	−0.361079
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	0	0
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	0	0
$779$	3.00000	0.107486
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−39.0000	−1.39020	−0.695100	−	0.718913i	$-0.744640\pi$
$787$	−39.0000	−1.39020	−0.695100	+	0.718913i	$0.744640\pi$
$788$	0	0
$789$	−19.0000	−0.676418
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	26.0000	0.923287
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	0	0
$799$	20.0000	0.707549
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	16.0000	0.564628
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−22.0000	−0.774437
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	6.00000	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	0	0
$816$	0	0
$817$	24.0000	0.839654
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	0	0
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	0	0
$825$	−5.00000	−0.174078
$826$	0	0
$827$	15.0000	0.521601	0.260801	−	0.965393i	$-0.416014\pi$
$827$	15.0000	0.521601	0.260801	+	0.965393i	$0.416014\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	−31.0000	−1.07538
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−50.0000	−1.72619	−0.863096	−	0.505040i	$-0.831478\pi$
$839$	−50.0000	−1.72619	−0.863096	+	0.505040i	$0.831478\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19.0000	−0.649028	−0.324514	−	0.945881i	$-0.605201\pi$
$857$	−19.0000	−0.649028	−0.324514	+	0.945881i	$0.605201\pi$
$858$	0	0
$859$	−56.0000	−1.91070	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−56.0000	−1.91070	−0.955348	+	0.295484i	$0.904519\pi$
$860$	0	0
$861$	1.00000	0.0340799
$862$	0	0
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−2.00000	−0.0678454
$870$	0	0
$871$	0	0
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	0	0
$876$	0	0
$877$	55.0000	1.85722	0.928609	−	0.371060i	$-0.121005\pi$
$877$	55.0000	1.85722	0.928609	+	0.371060i	$0.121005\pi$
$878$	0	0
$879$	16.0000	0.539667
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	−48.0000	−1.61533	−0.807664	−	0.589643i	$-0.799269\pi$
$883$	−48.0000	−1.61533	−0.807664	+	0.589643i	$0.799269\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.0000	0.537227	0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000	0.537227	0.268614	+	0.963248i	$0.413434\pi$
$888$	0	0
$889$	−7.00000	−0.234772
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	15.0000	0.501956
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.00000	0.0667781
$898$	0	0
$899$	−12.0000	−0.400222
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	0	0
$906$	0	0
$907$	48.0000	1.59381	0.796907	−	0.604102i	$-0.206468\pi$
$907$	48.0000	1.59381	0.796907	+	0.604102i	$0.206468\pi$
$908$	0	0
$909$	9.00000	0.298511
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.0000	0.495344
$918$	0	0
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	0	0
$923$	0	0
$924$	0	0
$925$	10.0000	0.328798
$926$	0	0
$927$	−11.0000	−0.361287
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	0	0
$933$	21.0000	0.687509
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.0000	0.620703	0.310351	−	0.950622i	$-0.399553\pi$
$937$	19.0000	0.620703	0.310351	+	0.950622i	$0.399553\pi$
$938$	0	0
$939$	−1.00000	−0.0326338
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	0	0
$943$	−1.00000	−0.0325645
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	0	0
$949$	−32.0000	−1.03876
$950$	0	0
$951$	2.00000	0.0648544
$952$	0	0
$953$	−53.0000	−1.71684	−0.858419	−	0.512949i	$-0.828553\pi$
$953$	−53.0000	−1.71684	−0.858419	+	0.512949i	$0.828553\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6.00000	−0.193952
$958$	0	0
$959$	−19.0000	−0.613542
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−4.00000	−0.128898
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	0	0
$971$	22.0000	0.706014	0.353007	−	0.935621i	$-0.385159\pi$
$971$	22.0000	0.706014	0.353007	+	0.935621i	$0.385159\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	10.0000	0.320256
$976$	0	0
$977$	−9.00000	−0.287936	−0.143968	−	0.989582i	$-0.545986\pi$
$977$	−9.00000	−0.287936	−0.143968	+	0.989582i	$0.545986\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	−10.0000	−0.318950	−0.159475	−	0.987202i	$-0.550980\pi$
$983$	−10.0000	−0.318950	−0.159475	+	0.987202i	$0.550980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.00000	0.159152
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	0	0
$993$	5.00000	0.158670
$994$	0	0
$995$	0	0
$996$	0	0
$997$	24.0000	0.760088	0.380044	−	0.924968i	$-0.375909\pi$
$997$	24.0000	0.760088	0.380044	+	0.924968i	$0.375909\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.e.1.1		1
4.3	odd	2		483.2.a.a.1.1	✓	1
12.11	even	2		1449.2.a.c.1.1		1
28.27	even	2		3381.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.a.1.1	✓	1	4.3	odd	2
1449.2.a.c.1.1		1	12.11	even	2
3381.2.a.m.1.1		1	28.27	even	2
7728.2.a.e.1.1		1	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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