LMFDB - Embedded newform 4704.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(1,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, 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("4704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.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:	$37.5616291108$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{9} -5.65685 q^{11} +5.65685 q^{13} +5.65685 q^{17} -4.00000 q^{19} -5.65685 q^{23} -5.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -8.00000 q^{31} -5.65685 q^{33} +2.00000 q^{37} +5.65685 q^{39} -5.65685 q^{41} -8.00000 q^{47} +5.65685 q^{51} -2.00000 q^{53} -4.00000 q^{57} -4.00000 q^{59} +5.65685 q^{61} +11.3137 q^{67} -5.65685 q^{69} +5.65685 q^{71} -11.3137 q^{73} -5.00000 q^{75} +11.3137 q^{79} +1.00000 q^{81} +12.0000 q^{83} -6.00000 q^{87} +5.65685 q^{89} -8.00000 q^{93} -5.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} - 8 q^{19} - 10 q^{25} + 2 q^{27} - 12 q^{29} - 16 q^{31} + 4 q^{37} - 16 q^{47} - 4 q^{53} - 8 q^{57} - 8 q^{59} - 10 q^{75} + 2 q^{81} + 24 q^{83} - 12 q^{87} - 16 q^{93}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 - 8 * q^19 - 10 * q^25 + 2 * q^27 - 12 * q^29 - 16 * q^31 + 4 * q^37 - 16 * q^47 - 4 * q^53 - 8 * q^57 - 8 * q^59 - 10 * q^75 + 2 * q^81 + 24 * q^83 - 12 * q^87 - 16 * q^93

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
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.65685	−1.70561	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	−5.65685	−1.70561	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.65685	1.37199	0.685994	−	0.727607i	$-0.259367\pi$
$17$	5.65685	1.37199	0.685994	+	0.727607i	$0.259367\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	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
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−5.65685	−0.984732
$34$	0	0
$35$	0	0
$36$	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
$39$	5.65685	0.905822
$40$	0	0
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	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
$49$	0	0
$50$	0	0
$51$	5.65685	0.792118
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	5.65685	0.724286	0.362143	−	0.932123i	$-0.382045\pi$
$61$	5.65685	0.724286	0.362143	+	0.932123i	$0.382045\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	−5.65685	−0.681005
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	−11.3137	−1.32417	−0.662085	−	0.749429i	$-0.730328\pi$
$73$	−11.3137	−1.32417	−0.662085	+	0.749429i	$0.730328\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	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
$87$	−6.00000	−0.643268
$88$	0	0
$89$	5.65685	0.599625	0.299813	−	0.953998i	$-0.403076\pi$
$89$	5.65685	0.599625	0.299813	+	0.953998i	$0.403076\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	−5.65685	−0.568535
$100$	0	0
$101$	−11.3137	−1.12576	−0.562878	−	0.826540i	$-0.690306\pi$
$101$	−11.3137	−1.12576	−0.562878	+	0.826540i	$0.690306\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.65685	−0.546869	−0.273434	−	0.961891i	$-0.588160\pi$
$107$	−5.65685	−0.546869	−0.273434	+	0.961891i	$0.588160\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.65685	0.522976
$118$	0	0
$119$	0	0
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	−5.65685	−0.510061
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.3137	−1.00393	−0.501965	−	0.864888i	$-0.667389\pi$
$127$	−11.3137	−1.00393	−0.501965	+	0.864888i	$0.667389\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−32.0000	−2.67597
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−22.6274	−1.84139	−0.920697	−	0.390279i	$-0.872378\pi$
$151$	−22.6274	−1.84139	−0.920697	+	0.390279i	$0.872378\pi$
$152$	0	0
$153$	5.65685	0.457330
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.9706	1.35440	0.677199	−	0.735800i	$-0.263194\pi$
$157$	16.9706	1.35440	0.677199	+	0.735800i	$0.263194\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.3137	−0.886158	−0.443079	−	0.896483i	$-0.646114\pi$
$163$	−11.3137	−0.886158	−0.443079	+	0.896483i	$0.646114\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−22.6274	−1.72033	−0.860165	−	0.510015i	$-0.829640\pi$
$173$	−22.6274	−1.72033	−0.860165	+	0.510015i	$0.829640\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	5.65685	0.422813	0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685	0.422813	0.211407	+	0.977398i	$0.432196\pi$
$180$	0	0
$181$	−16.9706	−1.26141	−0.630706	−	0.776022i	$-0.717235\pi$
$181$	−16.9706	−1.26141	−0.630706	+	0.776022i	$0.717235\pi$
$182$	0	0
$183$	5.65685	0.418167
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−32.0000	−2.34007
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.65685	0.409316	0.204658	−	0.978834i	$-0.434392\pi$
$191$	5.65685	0.409316	0.204658	+	0.978834i	$0.434392\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−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
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	11.3137	0.798007
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5.65685	−0.393179
$208$	0	0
$209$	22.6274	1.56517
$210$	0	0
$211$	22.6274	1.55774	0.778868	−	0.627188i	$-0.215794\pi$
$211$	22.6274	1.55774	0.778868	+	0.627188i	$0.215794\pi$
$212$	0	0
$213$	5.65685	0.387601
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−11.3137	−0.764510
$220$	0	0
$221$	32.0000	2.15255
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−16.9706	−1.12145	−0.560723	−	0.828003i	$-0.689477\pi$
$229$	−16.9706	−1.12145	−0.560723	+	0.828003i	$0.689477\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	11.3137	0.734904
$238$	0	0
$239$	−5.65685	−0.365911	−0.182956	−	0.983121i	$-0.558567\pi$
$239$	−5.65685	−0.365911	−0.182956	+	0.983121i	$0.558567\pi$
$240$	0	0
$241$	22.6274	1.45756	0.728780	−	0.684748i	$-0.240088\pi$
$241$	22.6274	1.45756	0.728780	+	0.684748i	$0.240088\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−22.6274	−1.43975
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	32.0000	2.01182
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.65685	0.352865	0.176432	−	0.984313i	$-0.443544\pi$
$257$	5.65685	0.352865	0.176432	+	0.984313i	$0.443544\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	16.9706	1.04645	0.523225	−	0.852195i	$-0.324729\pi$
$263$	16.9706	1.04645	0.523225	+	0.852195i	$0.324729\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.65685	0.346194
$268$	0	0
$269$	11.3137	0.689809	0.344904	−	0.938638i	$-0.387911\pi$
$269$	11.3137	0.689809	0.344904	+	0.938638i	$0.387911\pi$
$270$	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
$275$	28.2843	1.70561
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	11.3137	0.660954	0.330477	−	0.943814i	$-0.392790\pi$
$293$	11.3137	0.660954	0.330477	+	0.943814i	$0.392790\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.65685	−0.328244
$298$	0	0
$299$	−32.0000	−1.85061
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−11.3137	−0.649956
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	0	0
$313$	−11.3137	−0.639489	−0.319744	−	0.947504i	$-0.603597\pi$
$313$	−11.3137	−0.639489	−0.319744	+	0.947504i	$0.603597\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	33.9411	1.90034
$320$	0	0
$321$	−5.65685	−0.315735
$322$	0	0
$323$	−22.6274	−1.25902
$324$	0	0
$325$	−28.2843	−1.56893
$326$	0	0
$327$	−6.00000	−0.331801
$328$	0	0
$329$	0	0
$330$	0	0
$331$	22.6274	1.24372	0.621858	−	0.783130i	$-0.286378\pi$
$331$	22.6274	1.24372	0.621858	+	0.783130i	$0.286378\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	0	0
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	45.2548	2.45069
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.65685	0.303676	0.151838	−	0.988405i	$-0.451481\pi$
$347$	5.65685	0.303676	0.151838	+	0.988405i	$0.451481\pi$
$348$	0	0
$349$	−5.65685	−0.302804	−0.151402	−	0.988472i	$-0.548379\pi$
$349$	−5.65685	−0.302804	−0.151402	+	0.988472i	$0.548379\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	−28.2843	−1.50542	−0.752710	−	0.658352i	$-0.771254\pi$
$353$	−28.2843	−1.50542	−0.752710	+	0.658352i	$0.771254\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	21.0000	1.10221
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	0	0
$369$	−5.65685	−0.294484
$370$	0	0
$371$	0	0
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−33.9411	−1.74806
$378$	0	0
$379$	−22.6274	−1.16229	−0.581146	−	0.813799i	$-0.697396\pi$
$379$	−22.6274	−1.16229	−0.581146	+	0.813799i	$0.697396\pi$
$380$	0	0
$381$	−11.3137	−0.579619
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−32.0000	−1.61831
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.9706	0.851728	0.425864	−	0.904787i	$-0.359970\pi$
$397$	16.9706	0.851728	0.425864	+	0.904787i	$0.359970\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	−45.2548	−2.25430
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.3137	−0.560800
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−28.2843	−1.37199
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−32.0000	−1.54497
$430$	0	0
$431$	−16.9706	−0.817443	−0.408722	−	0.912659i	$-0.634025\pi$
$431$	−16.9706	−0.817443	−0.408722	+	0.912659i	$0.634025\pi$
$432$	0	0
$433$	−11.3137	−0.543702	−0.271851	−	0.962339i	$-0.587636\pi$
$433$	−11.3137	−0.543702	−0.271851	+	0.962339i	$0.587636\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	22.6274	1.08242
$438$	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
$441$	0	0
$442$	0	0
$443$	−16.9706	−0.806296	−0.403148	−	0.915135i	$-0.632084\pi$
$443$	−16.9706	−0.806296	−0.403148	+	0.915135i	$0.632084\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	32.0000	1.50682
$452$	0	0
$453$	−22.6274	−1.06313
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	0	0
$459$	5.65685	0.264039
$460$	0	0
$461$	−22.6274	−1.05386	−0.526932	−	0.849907i	$-0.676658\pi$
$461$	−22.6274	−1.05386	−0.526932	+	0.849907i	$0.676658\pi$
$462$	0	0
$463$	33.9411	1.57738	0.788689	−	0.614792i	$-0.210760\pi$
$463$	33.9411	1.57738	0.788689	+	0.614792i	$0.210760\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	16.9706	0.781962
$472$	0	0
$473$	0	0
$474$	0	0
$475$	20.0000	0.917663
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	11.3137	0.515861
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	−11.3137	−0.511624
$490$	0	0
$491$	28.2843	1.27645	0.638226	−	0.769849i	$-0.279669\pi$
$491$	28.2843	1.27645	0.638226	+	0.769849i	$0.279669\pi$
$492$	0	0
$493$	−33.9411	−1.52863
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−22.6274	−1.01294	−0.506471	−	0.862257i	$-0.669050\pi$
$499$	−22.6274	−1.01294	−0.506471	+	0.862257i	$0.669050\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	19.0000	0.843820
$508$	0	0
$509$	11.3137	0.501471	0.250736	−	0.968056i	$-0.419328\pi$
$509$	11.3137	0.501471	0.250736	+	0.968056i	$0.419328\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	45.2548	1.99031
$518$	0	0
$519$	−22.6274	−0.993233
$520$	0	0
$521$	28.2843	1.23916	0.619578	−	0.784935i	$-0.287304\pi$
$521$	28.2843	1.23916	0.619578	+	0.784935i	$0.287304\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−45.2548	−1.97133
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−32.0000	−1.38607
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.65685	0.244111
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	−16.9706	−0.728277
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−33.9411	−1.45122	−0.725609	−	0.688107i	$-0.758442\pi$
$547$	−33.9411	−1.45122	−0.725609	+	0.688107i	$0.758442\pi$
$548$	0	0
$549$	5.65685	0.241429
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.0000	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$557$	−10.0000	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−32.0000	−1.35104
$562$	0	0
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	33.9411	1.42039	0.710196	−	0.704004i	$-0.248606\pi$
$571$	33.9411	1.42039	0.710196	+	0.704004i	$0.248606\pi$
$572$	0	0
$573$	5.65685	0.236318
$574$	0	0
$575$	28.2843	1.17954
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	0	0
$581$	0	0
$582$	0	0
$583$	11.3137	0.468566
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	−5.65685	−0.232299	−0.116150	−	0.993232i	$-0.537055\pi$
$593$	−5.65685	−0.232299	−0.116150	+	0.993232i	$0.537055\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−24.0000	−0.982255
$598$	0	0
$599$	−16.9706	−0.693398	−0.346699	−	0.937976i	$-0.612698\pi$
$599$	−16.9706	−0.693398	−0.346699	+	0.937976i	$0.612698\pi$
$600$	0	0
$601$	45.2548	1.84598	0.922992	−	0.384820i	$-0.125737\pi$
$601$	45.2548	1.84598	0.922992	+	0.384820i	$0.125737\pi$
$602$	0	0
$603$	11.3137	0.460730
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−45.2548	−1.83081
$612$	0	0
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	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
$621$	−5.65685	−0.227002
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	22.6274	0.903652
$628$	0	0
$629$	11.3137	0.451107
$630$	0	0
$631$	11.3137	0.450392	0.225196	−	0.974314i	$-0.427698\pi$
$631$	11.3137	0.450392	0.225196	+	0.974314i	$0.427698\pi$
$632$	0	0
$633$	22.6274	0.899359
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.65685	0.223782
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	22.6274	0.888204
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−11.3137	−0.441390
$658$	0	0
$659$	39.5980	1.54252	0.771259	−	0.636521i	$-0.219627\pi$
$659$	39.5980	1.54252	0.771259	+	0.636521i	$0.219627\pi$
$660$	0	0
$661$	50.9117	1.98024	0.990118	−	0.140240i	$-0.0447873\pi$
$661$	50.9117	1.98024	0.990118	+	0.140240i	$0.0447873\pi$
$662$	0	0
$663$	32.0000	1.24278
$664$	0	0
$665$	0	0
$666$	0	0
$667$	33.9411	1.31421
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−32.0000	−1.23535
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−45.2548	−1.73928	−0.869642	−	0.493682i	$-0.835651\pi$
$677$	−45.2548	−1.73928	−0.869642	+	0.493682i	$0.835651\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	5.65685	0.216454	0.108227	−	0.994126i	$-0.465483\pi$
$683$	5.65685	0.216454	0.108227	+	0.994126i	$0.465483\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−16.9706	−0.647467
$688$	0	0
$689$	−11.3137	−0.431018
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−32.0000	−1.21209
$698$	0	0
$699$	−26.0000	−0.983410
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	11.3137	0.424297
$712$	0	0
$713$	45.2548	1.69481
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.65685	−0.211259
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	22.6274	0.841523
$724$	0	0
$725$	30.0000	1.11417
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−5.65685	−0.208941	−0.104470	−	0.994528i	$-0.533315\pi$
$733$	−5.65685	−0.208941	−0.104470	+	0.994528i	$0.533315\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−64.0000	−2.35747
$738$	0	0
$739$	−11.3137	−0.416181	−0.208091	−	0.978110i	$-0.566725\pi$
$739$	−11.3137	−0.416181	−0.208091	+	0.978110i	$0.566725\pi$
$740$	0	0
$741$	−22.6274	−0.831239
$742$	0	0
$743$	−39.5980	−1.45271	−0.726354	−	0.687320i	$-0.758787\pi$
$743$	−39.5980	−1.45271	−0.726354	+	0.687320i	$0.758787\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	0	0
$750$	0	0
$751$	22.6274	0.825686	0.412843	−	0.910802i	$-0.364536\pi$
$751$	22.6274	0.825686	0.412843	+	0.910802i	$0.364536\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	0	0
$756$	0	0
$757$	50.0000	1.81728	0.908640	−	0.417579i	$-0.137121\pi$
$757$	50.0000	1.81728	0.908640	+	0.417579i	$0.137121\pi$
$758$	0	0
$759$	32.0000	1.16153
$760$	0	0
$761$	−5.65685	−0.205061	−0.102530	−	0.994730i	$-0.532694\pi$
$761$	−5.65685	−0.205061	−0.102530	+	0.994730i	$0.532694\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22.6274	−0.817029
$768$	0	0
$769$	−33.9411	−1.22395	−0.611974	−	0.790878i	$-0.709624\pi$
$769$	−33.9411	−1.22395	−0.611974	+	0.790878i	$0.709624\pi$
$770$	0	0
$771$	5.65685	0.203727
$772$	0	0
$773$	−11.3137	−0.406926	−0.203463	−	0.979083i	$-0.565220\pi$
$773$	−11.3137	−0.406926	−0.203463	+	0.979083i	$0.565220\pi$
$774$	0	0
$775$	40.0000	1.43684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	22.6274	0.810711
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	0	0
$786$	0	0
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	0	0
$789$	16.9706	0.604168
$790$	0	0
$791$	0	0
$792$	0	0
$793$	32.0000	1.13635
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22.6274	−0.801504	−0.400752	−	0.916187i	$-0.631251\pi$
$797$	−22.6274	−0.801504	−0.400752	+	0.916187i	$0.631251\pi$
$798$	0	0
$799$	−45.2548	−1.60100
$800$	0	0
$801$	5.65685	0.199875
$802$	0	0
$803$	64.0000	2.25851
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11.3137	0.398261
$808$	0	0
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	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
$823$	33.9411	1.18311	0.591557	−	0.806263i	$-0.298514\pi$
$823$	33.9411	1.18311	0.591557	+	0.806263i	$0.298514\pi$
$824$	0	0
$825$	28.2843	0.984732
$826$	0	0
$827$	−28.2843	−0.983540	−0.491770	−	0.870725i	$-0.663650\pi$
$827$	−28.2843	−0.983540	−0.491770	+	0.870725i	$0.663650\pi$
$828$	0	0
$829$	16.9706	0.589412	0.294706	−	0.955588i	$-0.404778\pi$
$829$	16.9706	0.589412	0.294706	+	0.955588i	$0.404778\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−10.0000	−0.344418
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−11.3137	−0.387829
$852$	0	0
$853$	−50.9117	−1.74318	−0.871592	−	0.490233i	$-0.836912\pi$
$853$	−50.9117	−1.74318	−0.871592	+	0.490233i	$0.836912\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28.2843	−0.966172	−0.483086	−	0.875573i	$-0.660484\pi$
$857$	−28.2843	−0.966172	−0.483086	+	0.875573i	$0.660484\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.9706	0.577685	0.288842	−	0.957377i	$-0.406730\pi$
$863$	16.9706	0.577685	0.288842	+	0.957377i	$0.406730\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15.0000	0.509427
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	64.0000	2.16856
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	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
$879$	11.3137	0.381602
$880$	0	0
$881$	−5.65685	−0.190584	−0.0952921	−	0.995449i	$-0.530379\pi$
$881$	−5.65685	−0.190584	−0.0952921	+	0.995449i	$0.530379\pi$
$882$	0	0
$883$	22.6274	0.761473	0.380737	−	0.924684i	$-0.375670\pi$
$883$	22.6274	0.761473	0.380737	+	0.924684i	$0.375670\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40.0000	1.34307	0.671534	−	0.740973i	$-0.265636\pi$
$887$	40.0000	1.34307	0.671534	+	0.740973i	$0.265636\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5.65685	−0.189512
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−32.0000	−1.06845
$898$	0	0
$899$	48.0000	1.60089
$900$	0	0
$901$	−11.3137	−0.376914
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	−11.3137	−0.375252
$910$	0	0
$911$	−16.9706	−0.562260	−0.281130	−	0.959670i	$-0.590709\pi$
$911$	−16.9706	−0.562260	−0.281130	+	0.959670i	$0.590709\pi$
$912$	0	0
$913$	−67.8823	−2.24657
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−22.6274	−0.746410	−0.373205	−	0.927749i	$-0.621741\pi$
$919$	−22.6274	−0.746410	−0.373205	+	0.927749i	$0.621741\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	32.0000	1.05329
$924$	0	0
$925$	−10.0000	−0.328798
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	39.5980	1.29917	0.649584	−	0.760290i	$-0.274943\pi$
$929$	39.5980	1.29917	0.649584	+	0.760290i	$0.274943\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−16.0000	−0.523816
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.6274	−0.739205	−0.369603	−	0.929190i	$-0.620506\pi$
$937$	−22.6274	−0.739205	−0.369603	+	0.929190i	$0.620506\pi$
$938$	0	0
$939$	−11.3137	−0.369209
$940$	0	0
$941$	−45.2548	−1.47527	−0.737633	−	0.675202i	$-0.764056\pi$
$941$	−45.2548	−1.47527	−0.737633	+	0.675202i	$0.764056\pi$
$942$	0	0
$943$	32.0000	1.04206
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.9706	0.551469	0.275735	−	0.961234i	$-0.411079\pi$
$947$	16.9706	0.551469	0.275735	+	0.961234i	$0.411079\pi$
$948$	0	0
$949$	−64.0000	−2.07753
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	33.9411	1.09716
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−5.65685	−0.182290
$964$	0	0
$965$	0	0
$966$	0	0
$967$	33.9411	1.09147	0.545737	−	0.837957i	$-0.316250\pi$
$967$	33.9411	1.09147	0.545737	+	0.837957i	$0.316250\pi$
$968$	0	0
$969$	−22.6274	−0.726897
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−28.2843	−0.905822
$976$	0	0
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	0	0
$979$	−32.0000	−1.02272
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	8.00000	0.255160	0.127580	−	0.991828i	$-0.459279\pi$
$983$	8.00000	0.255160	0.127580	+	0.991828i	$0.459279\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−22.6274	−0.718784	−0.359392	−	0.933187i	$-0.617016\pi$
$991$	−22.6274	−0.718784	−0.359392	+	0.933187i	$0.617016\pi$
$992$	0	0
$993$	22.6274	0.718059
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.65685	0.179154	0.0895772	−	0.995980i	$-0.471448\pi$
$997$	5.65685	0.179154	0.0895772	+	0.995980i	$0.471448\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.a.bp.1.1	yes	2
4.3	odd	2		4704.2.a.bk.1.2	yes	2
7.6	odd	2		4704.2.a.bk.1.1	✓	2
8.3	odd	2		9408.2.a.dz.1.1		2
8.5	even	2		9408.2.a.dm.1.2		2
28.27	even	2	inner	4704.2.a.bp.1.2	yes	2
56.13	odd	2		9408.2.a.dz.1.2		2
56.27	even	2		9408.2.a.dm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4704.2.a.bk.1.1	✓	2	7.6	odd	2
4704.2.a.bk.1.2	yes	2	4.3	odd	2
4704.2.a.bp.1.1	yes	2	1.1	even	1	trivial
4704.2.a.bp.1.2	yes	2	28.27	even	2	inner
9408.2.a.dm.1.1		2	56.27	even	2
9408.2.a.dm.1.2		2	8.5	even	2
9408.2.a.dz.1.1		2	8.3	odd	2
9408.2.a.dz.1.2		2	56.13	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{9} -5.65685 q^{11} +5.65685 q^{13} +5.65685 q^{17} -4.00000 q^{19} -5.65685 q^{23} -5.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -8.00000 q^{31} -5.65685 q^{33} +2.00000 q^{37} +5.65685 q^{39} -5.65685 q^{41} -8.00000 q^{47} +5.65685 q^{51} -2.00000 q^{53} -4.00000 q^{57} -4.00000 q^{59} +5.65685 q^{61} +11.3137 q^{67} -5.65685 q^{69} +5.65685 q^{71} -11.3137 q^{73} -5.00000 q^{75} +11.3137 q^{79} +1.00000 q^{81} +12.0000 q^{83} -6.00000 q^{87} +5.65685 q^{89} -8.00000 q^{93} -5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} - 8 q^{19} - 10 q^{25} + 2 q^{27} - 12 q^{29} - 16 q^{31} + 4 q^{37} - 16 q^{47} - 4 q^{53} - 8 q^{57} - 8 q^{59} - 10 q^{75} + 2 q^{81} + 24 q^{83} - 12 q^{87} - 16 q^{93}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 - 8 * q^19 - 10 * q^25 + 2 * q^27 - 12 * q^29 - 16 * q^31 + 4 * q^37 - 16 * q^47 - 4 * q^53 - 8 * q^57 - 8 * q^59 - 10 * q^75 + 2 * q^81 + 24 * q^83 - 12 * q^87 - 16 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more