LMFDB - Embedded newform 6048.2.h.e.2591.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(2591,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("6048.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.h (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.e.2591.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{5} -1.00000i q^{7} +O(q^{10})$	$q-2.82843i q^{5} -1.00000i q^{7} -6.29253 q^{11} +5.44949 q^{13} -0.317837i q^{17} +4.00000i q^{19} -0.317837 q^{23} -3.00000 q^{25} +5.19615i q^{29} +8.79796i q^{31} -2.82843 q^{35} +0.898979 q^{37} +3.46410i q^{41} -5.44949i q^{43} -12.5851 q^{47} -1.00000 q^{49} +8.02458i q^{53} +17.7980i q^{55} -5.83183 q^{59} +12.8990 q^{61} -15.4135i q^{65} +3.44949i q^{67} +2.51059 q^{71} -4.00000 q^{73} +6.29253i q^{77} -10.0000i q^{79} +3.46410 q^{83} -0.898979 q^{85} +10.7101i q^{89} -5.44949i q^{91} +11.3137 q^{95} -15.7980 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 24 q^{13} - 24 q^{25} - 32 q^{37} - 8 q^{49} + 64 q^{61} - 32 q^{73} + 32 q^{85} - 48 q^{97}+O(q^{100}) $ 8 * q + 24 * q^13 - 24 * q^25 - 32 * q^37 - 8 * q^49 + 64 * q^61 - 32 * q^73 + 32 * q^85 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 2.82843i	− 1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	− 2.82843i	− 1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.29253	−1.89727	−0.948634	−	0.316374i	$-0.897534\pi$
$11$	−6.29253	−1.89727	−0.948634	+	0.316374i	$0.897534\pi$
$12$	0	0
$13$	5.44949	1.51142	0.755708	−	0.654908i	$-0.227293\pi$
$13$	5.44949	1.51142	0.755708	+	0.654908i	$0.227293\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.317837i	− 0.0770869i	−0.999257	−	0.0385434i	$-0.987728\pi$
$17$	− 0.317837i	− 0.0770869i	0.999257	−	0.0385434i	$-0.0122718\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.317837	−0.0662736	−0.0331368	−	0.999451i	$-0.510550\pi$
$23$	−0.317837	−0.0662736	−0.0331368	+	0.999451i	$0.510550\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.19615i	0.964901i	0.875923	+	0.482451i	$0.160253\pi$
$29$	5.19615i	0.964901i	−0.875923	+	0.482451i	$0.839747\pi$
$30$	0	0
$31$	8.79796i	1.58016i	0.613004	+	0.790080i	$0.289961\pi$
$31$	8.79796i	1.58016i	−0.613004	+	0.790080i	$0.710039\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	0.898979	0.147791	0.0738957	−	0.997266i	$-0.476457\pi$
$37$	0.898979	0.147791	0.0738957	+	0.997266i	$0.476457\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.46410i	0.541002i	0.962720	+	0.270501i	$0.0871893\pi$
$41$	3.46410i	0.541002i	−0.962720	+	0.270501i	$0.912811\pi$
$42$	0	0
$43$	− 5.44949i	− 0.831039i	−0.909584	−	0.415520i	$-0.863600\pi$
$43$	− 5.44949i	− 0.831039i	0.909584	−	0.415520i	$-0.136400\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.5851	−1.83572	−0.917860	−	0.396905i	$-0.870084\pi$
$47$	−12.5851	−1.83572	−0.917860	+	0.396905i	$0.870084\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.02458i	1.10226i	0.834419	+	0.551130i	$0.185803\pi$
$53$	8.02458i	1.10226i	−0.834419	+	0.551130i	$0.814197\pi$
$54$	0	0
$55$	17.7980i	2.39988i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.83183	−0.759239	−0.379620	−	0.925143i	$-0.623945\pi$
$59$	−5.83183	−0.759239	−0.379620	+	0.925143i	$0.623945\pi$
$60$	0	0
$61$	12.8990	1.65155	0.825773	−	0.564003i	$-0.190739\pi$
$61$	12.8990	1.65155	0.825773	+	0.564003i	$0.190739\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 15.4135i	− 1.91181i
$66$	0	0
$67$	3.44949i	0.421422i	0.977548	+	0.210711i	$0.0675779\pi$
$67$	3.44949i	0.421422i	−0.977548	+	0.210711i	$0.932422\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.51059	0.297952	0.148976	−	0.988841i	$-0.452402\pi$
$71$	2.51059	0.297952	0.148976	+	0.988841i	$0.452402\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.29253i	0.717100i
$78$	0	0
$79$	− 10.0000i	− 1.12509i	−0.826767	−	0.562544i	$-0.809823\pi$
$79$	− 10.0000i	− 1.12509i	0.826767	−	0.562544i	$-0.190177\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.46410	0.380235	0.190117	−	0.981761i	$-0.439113\pi$
$83$	3.46410	0.380235	0.190117	+	0.981761i	$0.439113\pi$
$84$	0	0
$85$	−0.898979	−0.0975080
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.7101i	1.13527i	0.823279	+	0.567636i	$0.192142\pi$
$89$	10.7101i	1.13527i	−0.823279	+	0.567636i	$0.807858\pi$
$90$	0	0
$91$	− 5.44949i	− 0.571262i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	11.3137	1.16076
$96$	0	0
$97$	−15.7980	−1.60404	−0.802020	−	0.597297i	$-0.796241\pi$
$97$	−15.7980	−1.60404	−0.802020	+	0.597297i	$0.796241\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.65685i	0.562878i	0.959579	+	0.281439i	$0.0908117\pi$
$101$	5.65685i	0.562878i	−0.959579	+	0.281439i	$0.909188\pi$
$102$	0	0
$103$	3.89898i	0.384178i	0.981378	+	0.192089i	$0.0615262\pi$
$103$	3.89898i	0.384178i	−0.981378	+	0.192089i	$0.938474\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.2419	1.76351	0.881756	−	0.471706i	$-0.156362\pi$
$107$	18.2419	1.76351	0.881756	+	0.471706i	$0.156362\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.48528i	0.798228i	0.916901	+	0.399114i	$0.130682\pi$
$113$	8.48528i	0.798228i	−0.916901	+	0.399114i	$0.869318\pi$
$114$	0	0
$115$	0.898979i	0.0838303i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.317837	−0.0291361
$120$	0	0
$121$	28.5959	2.59963
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 5.65685i	− 0.505964i
$126$	0	0
$127$	10.8990i	0.967128i	0.875309	+	0.483564i	$0.160658\pi$
$127$	10.8990i	0.967128i	−0.875309	+	0.483564i	$0.839342\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.36773	0.206869	0.103435	−	0.994636i	$-0.467017\pi$
$131$	2.36773	0.206869	0.103435	+	0.994636i	$0.467017\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.5133i	1.66713i	0.552421	+	0.833565i	$0.313704\pi$
$137$	19.5133i	1.66713i	−0.552421	+	0.833565i	$0.686296\pi$
$138$	0	0
$139$	− 15.7980i	− 1.33997i	−0.742377	−	0.669983i	$-0.766302\pi$
$139$	− 15.7980i	− 1.33997i	0.742377	−	0.669983i	$-0.233698\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−34.2911	−2.86756
$144$	0	0
$145$	14.6969	1.22051
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8.02458i	− 0.657399i	−0.944435	−	0.328700i	$-0.893390\pi$
$149$	− 8.02458i	− 0.657399i	0.944435	−	0.328700i	$-0.106610\pi$
$150$	0	0
$151$	5.79796i	0.471831i	0.971774	+	0.235916i	$0.0758089\pi$
$151$	5.79796i	0.471831i	−0.971774	+	0.235916i	$0.924191\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	24.8844	1.99876
$156$	0	0
$157$	16.1464	1.28863	0.644313	−	0.764762i	$-0.277144\pi$
$157$	16.1464	1.28863	0.644313	+	0.764762i	$0.277144\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.317837i	0.0250491i
$162$	0	0
$163$	2.55051i	0.199771i	0.994999	+	0.0998857i	$0.0318477\pi$
$163$	2.55051i	0.199771i	−0.994999	+	0.0998857i	$0.968152\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0492	1.24192	0.620961	−	0.783842i	$-0.286743\pi$
$167$	16.0492	1.24192	0.620961	+	0.783842i	$0.286743\pi$
$168$	0	0
$169$	16.6969	1.28438
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.921404i	0.0700530i	0.999386	+	0.0350265i	$0.0111516\pi$
$173$	0.921404i	0.0700530i	−0.999386	+	0.0350265i	$0.988848\pi$
$174$	0	0
$175$	3.00000i	0.226779i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.0280	0.824270	0.412135	−	0.911123i	$-0.364783\pi$
$179$	11.0280	0.824270	0.412135	+	0.911123i	$0.364783\pi$
$180$	0	0
$181$	−9.65153	−0.717393	−0.358696	−	0.933454i	$-0.616779\pi$
$181$	−9.65153	−0.717393	−0.358696	+	0.933454i	$0.616779\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.54270i	− 0.186943i
$186$	0	0
$187$	2.00000i	0.146254i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.0280	−0.797957	−0.398978	−	0.916960i	$-0.630635\pi$
$191$	−11.0280	−0.797957	−0.398978	+	0.916960i	$0.630635\pi$
$192$	0	0
$193$	−3.89898	−0.280655	−0.140327	−	0.990105i	$-0.544815\pi$
$193$	−3.89898	−0.280655	−0.140327	+	0.990105i	$0.544815\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 16.9706i	− 1.20910i	−0.796566	−	0.604551i	$-0.793352\pi$
$197$	− 16.9706i	− 1.20910i	0.796566	−	0.604551i	$-0.206648\pi$
$198$	0	0
$199$	20.7980i	1.47433i	0.675714	+	0.737164i	$0.263836\pi$
$199$	20.7980i	1.47433i	−0.675714	+	0.737164i	$0.736164\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.19615	0.364698
$204$	0	0
$205$	9.79796	0.684319
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 25.1701i	− 1.74105i
$210$	0	0
$211$	14.1464i	0.973880i	0.873435	+	0.486940i	$0.161887\pi$
$211$	14.1464i	0.973880i	−0.873435	+	0.486940i	$0.838113\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−15.4135	−1.05119
$216$	0	0
$217$	8.79796	0.597244
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.73205i	− 0.116510i
$222$	0	0
$223$	9.79796i	0.656120i	0.944657	+	0.328060i	$0.106395\pi$
$223$	9.79796i	0.656120i	−0.944657	+	0.328060i	$0.893605\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.460702	0.0305779	0.0152889	−	0.999883i	$-0.495133\pi$
$227$	0.460702	0.0305779	0.0152889	+	0.999883i	$0.495133\pi$
$228$	0	0
$229$	−3.10102	−0.204921	−0.102461	−	0.994737i	$-0.532672\pi$
$229$	−3.10102	−0.204921	−0.102461	+	0.994737i	$0.532672\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 27.0771i	− 1.77388i	−0.461882	−	0.886941i	$-0.652826\pi$
$233$	− 27.0771i	− 1.77388i	0.461882	−	0.886941i	$-0.347174\pi$
$234$	0	0
$235$	35.5959i	2.32202i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.82843	0.182956	0.0914779	−	0.995807i	$-0.470841\pi$
$239$	2.82843	0.182956	0.0914779	+	0.995807i	$0.470841\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.82843i	0.180702i
$246$	0	0
$247$	21.7980i	1.38697i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.73545	0.298899	0.149449	−	0.988769i	$-0.452250\pi$
$251$	4.73545	0.298899	0.149449	+	0.988769i	$0.452250\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.0915i	1.62754i	0.581184	+	0.813772i	$0.302590\pi$
$257$	26.0915i	1.62754i	−0.581184	+	0.813772i	$0.697410\pi$
$258$	0	0
$259$	− 0.898979i	− 0.0558599i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.6527	−1.02685	−0.513426	−	0.858134i	$-0.671624\pi$
$263$	−16.6527	−1.02685	−0.513426	+	0.858134i	$0.671624\pi$
$264$	0	0
$265$	22.6969	1.39426
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 14.4921i	− 0.883598i	−0.897114	−	0.441799i	$-0.854340\pi$
$269$	− 14.4921i	− 0.883598i	0.897114	−	0.441799i	$-0.145660\pi$
$270$	0	0
$271$	− 29.8990i	− 1.81623i	−0.418717	−	0.908117i	$-0.637520\pi$
$271$	− 29.8990i	− 1.81623i	0.418717	−	0.908117i	$-0.362480\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.8776	1.13836
$276$	0	0
$277$	30.8990	1.85654	0.928270	−	0.371907i	$-0.121296\pi$
$277$	30.8990	1.85654	0.928270	+	0.371907i	$0.121296\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 6.57826i	− 0.392426i	−0.980561	−	0.196213i	$-0.937136\pi$
$281$	− 6.57826i	− 0.392426i	0.980561	−	0.196213i	$-0.0628644\pi$
$282$	0	0
$283$	5.79796i	0.344653i	0.985040	+	0.172326i	$0.0551284\pi$
$283$	5.79796i	0.344653i	−0.985040	+	0.172326i	$0.944872\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.46410	0.204479
$288$	0	0
$289$	16.8990	0.994058
$290$	0	0
$291$	0	0
$292$	0	0
$293$	13.2207i	0.772363i	0.922423	+	0.386182i	$0.126206\pi$
$293$	13.2207i	0.772363i	−0.922423	+	0.386182i	$0.873794\pi$
$294$	0	0
$295$	16.4949i	0.960370i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.73205	−0.100167
$300$	0	0
$301$	−5.44949	−0.314103
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 36.4838i	− 2.08906i
$306$	0	0
$307$	26.6969i	1.52367i	0.647768	+	0.761837i	$0.275702\pi$
$307$	26.6969i	1.52367i	−0.647768	+	0.761837i	$0.724298\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.4203	−1.21463	−0.607316	−	0.794460i	$-0.707754\pi$
$311$	−21.4203	−1.21463	−0.607316	+	0.794460i	$0.707754\pi$
$312$	0	0
$313$	15.5959	0.881533	0.440767	−	0.897622i	$-0.354707\pi$
$313$	15.5959	0.881533	0.440767	+	0.897622i	$0.354707\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.5851i	− 0.706847i	−0.935463	−	0.353424i	$-0.885017\pi$
$317$	− 12.5851i	− 0.706847i	0.935463	−	0.353424i	$-0.114983\pi$
$318$	0	0
$319$	− 32.6969i	− 1.83068i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.27135	0.0707397
$324$	0	0
$325$	−16.3485	−0.906850
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.5851i	0.693837i
$330$	0	0
$331$	− 5.65153i	− 0.310636i	−0.987865	−	0.155318i	$-0.950360\pi$
$331$	− 5.65153i	− 0.310636i	0.987865	−	0.155318i	$-0.0496403\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.75663	0.533062
$336$	0	0
$337$	−19.8990	−1.08397	−0.541983	−	0.840389i	$-0.682326\pi$
$337$	−19.8990	−1.08397	−0.541983	+	0.840389i	$0.682326\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 55.3614i	− 2.99799i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.1489	1.08165	0.540826	−	0.841135i	$-0.318112\pi$
$347$	20.1489	1.08165	0.540826	+	0.841135i	$0.318112\pi$
$348$	0	0
$349$	28.8434	1.54395	0.771975	−	0.635653i	$-0.219269\pi$
$349$	28.8434	1.54395	0.771975	+	0.635653i	$0.219269\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 32.7019i	− 1.74055i	−0.492570	−	0.870273i	$-0.663942\pi$
$353$	− 32.7019i	− 1.74055i	0.492570	−	0.870273i	$-0.336058\pi$
$354$	0	0
$355$	− 7.10102i	− 0.376883i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.61037	−0.348882	−0.174441	−	0.984668i	$-0.555812\pi$
$359$	−6.61037	−0.348882	−0.174441	+	0.984668i	$0.555812\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.3137i	0.592187i
$366$	0	0
$367$	34.7980i	1.81644i	0.418494	+	0.908219i	$0.362558\pi$
$367$	34.7980i	1.81644i	−0.418494	+	0.908219i	$0.637442\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.02458	0.416615
$372$	0	0
$373$	−32.2929	−1.67206	−0.836030	−	0.548683i	$-0.815129\pi$
$373$	−32.2929	−1.67206	−0.836030	+	0.548683i	$0.815129\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	28.3164i	1.45837i
$378$	0	0
$379$	− 10.0000i	− 0.513665i	−0.966456	−	0.256833i	$-0.917321\pi$
$379$	− 10.0000i	− 0.513665i	0.966456	−	0.256833i	$-0.0826790\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	17.8920	0.914237	0.457118	−	0.889406i	$-0.348882\pi$
$383$	17.8920	0.914237	0.457118	+	0.889406i	$0.348882\pi$
$384$	0	0
$385$	17.7980	0.907068
$386$	0	0
$387$	0	0
$388$	0	0
$389$	36.4838i	1.84980i	0.380206	+	0.924902i	$0.375853\pi$
$389$	36.4838i	1.84980i	−0.380206	+	0.924902i	$0.624147\pi$
$390$	0	0
$391$	0.101021i	0.00510883i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−28.2843	−1.42314
$396$	0	0
$397$	−15.1010	−0.757898	−0.378949	−	0.925417i	$-0.623715\pi$
$397$	−15.1010	−0.757898	−0.378949	+	0.925417i	$0.623715\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 11.6637i	− 0.582455i	−0.956654	−	0.291228i	$-0.905936\pi$
$401$	− 11.6637i	− 0.582455i	0.956654	−	0.291228i	$-0.0940637\pi$
$402$	0	0
$403$	47.9444i	2.38828i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.65685	−0.280400
$408$	0	0
$409$	−28.6969	−1.41897	−0.709486	−	0.704719i	$-0.751073\pi$
$409$	−28.6969	−1.41897	−0.709486	+	0.704719i	$0.751073\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.83183i	0.286965i
$414$	0	0
$415$	− 9.79796i	− 0.480963i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	30.6520	1.49745	0.748724	−	0.662882i	$-0.230667\pi$
$419$	30.6520	1.49745	0.748724	+	0.662882i	$0.230667\pi$
$420$	0	0
$421$	−31.5959	−1.53989	−0.769945	−	0.638110i	$-0.779717\pi$
$421$	−31.5959	−1.53989	−0.769945	+	0.638110i	$0.779717\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.953512i	0.0462521i
$426$	0	0
$427$	− 12.8990i	− 0.624225i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.7272	−1.28740	−0.643702	−	0.765276i	$-0.722602\pi$
$431$	−26.7272	−1.28740	−0.643702	+	0.765276i	$0.722602\pi$
$432$	0	0
$433$	−19.5959	−0.941720	−0.470860	−	0.882208i	$-0.656056\pi$
$433$	−19.5959	−0.941720	−0.470860	+	0.882208i	$0.656056\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.27135i	− 0.0608169i
$438$	0	0
$439$	− 11.4949i	− 0.548622i	−0.961641	−	0.274311i	$-0.911550\pi$
$439$	− 11.4949i	− 0.548622i	0.961641	−	0.274311i	$-0.0884497\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.285729	−0.0135754	−0.00678770	−	0.999977i	$-0.502161\pi$
$443$	−0.285729	−0.0135754	−0.00678770	+	0.999977i	$0.502161\pi$
$444$	0	0
$445$	30.2929	1.43602
$446$	0	0
$447$	0	0
$448$	0	0
$449$	29.2699i	1.38133i	0.723174	+	0.690666i	$0.242682\pi$
$449$	29.2699i	1.38133i	−0.723174	+	0.690666i	$0.757318\pi$
$450$	0	0
$451$	− 21.7980i	− 1.02643i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15.4135	−0.722595
$456$	0	0
$457$	−10.3031	−0.481957	−0.240978	−	0.970530i	$-0.577468\pi$
$457$	−10.3031	−0.481957	−0.240978	+	0.970530i	$0.577468\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.92820i	0.322679i	0.986899	+	0.161339i	$0.0515813\pi$
$461$	6.92820i	0.322679i	−0.986899	+	0.161339i	$0.948419\pi$
$462$	0	0
$463$	31.5959i	1.46839i	0.678940	+	0.734193i	$0.262439\pi$
$463$	31.5959i	1.46839i	−0.678940	+	0.734193i	$0.737561\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.84961	−0.363236	−0.181618	−	0.983369i	$-0.558134\pi$
$467$	−7.84961	−0.363236	−0.181618	+	0.983369i	$0.558134\pi$
$468$	0	0
$469$	3.44949	0.159283
$470$	0	0
$471$	0	0
$472$	0	0
$473$	34.2911i	1.57671i
$474$	0	0
$475$	− 12.0000i	− 0.550598i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.94258	0.271524	0.135762	−	0.990742i	$-0.456652\pi$
$479$	5.94258	0.271524	0.135762	+	0.990742i	$0.456652\pi$
$480$	0	0
$481$	4.89898	0.223374
$482$	0	0
$483$	0	0
$484$	0	0
$485$	44.6834i	2.02897i
$486$	0	0
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	24.5344	1.10722	0.553612	−	0.832775i	$-0.313249\pi$
$491$	24.5344	1.10722	0.553612	+	0.832775i	$0.313249\pi$
$492$	0	0
$493$	1.65153	0.0743812
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.51059i	− 0.112615i
$498$	0	0
$499$	35.7980i	1.60254i	0.598305	+	0.801268i	$0.295841\pi$
$499$	35.7980i	1.60254i	−0.598305	+	0.801268i	$0.704159\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.37113	−0.239487	−0.119743	−	0.992805i	$-0.538207\pi$
$503$	−5.37113	−0.239487	−0.119743	+	0.992805i	$0.538207\pi$
$504$	0	0
$505$	16.0000	0.711991
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.285729i	0.0126647i	0.999980	+	0.00633236i	$0.00201567\pi$
$509$	0.285729i	0.0126647i	−0.999980	+	0.00633236i	$0.997984\pi$
$510$	0	0
$511$	4.00000i	0.176950i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.0280	0.485951
$516$	0	0
$517$	79.1918	3.48285
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 11.6315i	− 0.509587i	−0.966995	−	0.254794i	$-0.917992\pi$
$521$	− 11.6315i	− 0.509587i	0.966995	−	0.254794i	$-0.0820075\pi$
$522$	0	0
$523$	19.7980i	0.865704i	0.901465	+	0.432852i	$0.142493\pi$
$523$	19.7980i	0.865704i	−0.901465	+	0.432852i	$0.857507\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.79632	0.121810
$528$	0	0
$529$	−22.8990	−0.995608
$530$	0	0
$531$	0	0
$532$	0	0
$533$	18.8776i	0.817679i
$534$	0	0
$535$	− 51.5959i	− 2.23069i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.29253	0.271038
$540$	0	0
$541$	40.4949	1.74101	0.870506	−	0.492158i	$-0.163792\pi$
$541$	40.4949	1.74101	0.870506	+	0.492158i	$0.163792\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.6274i	0.969252i
$546$	0	0
$547$	15.3939i	0.658195i	0.944296	+	0.329097i	$0.106744\pi$
$547$	15.3939i	0.658195i	−0.944296	+	0.329097i	$0.893256\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−20.7846	−0.885454
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 0.746431i	− 0.0316273i	−0.999875	−	0.0158136i	$-0.994966\pi$
$557$	− 0.746431i	− 0.0316273i	0.999875	−	0.0158136i	$-0.00503385\pi$
$558$	0	0
$559$	− 29.6969i	− 1.25605i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.9951	1.05342	0.526710	−	0.850045i	$-0.323425\pi$
$563$	24.9951	1.05342	0.526710	+	0.850045i	$0.323425\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 2.19275i	− 0.0919250i	−0.998943	−	0.0459625i	$-0.985365\pi$
$569$	− 2.19275i	− 0.0919250i	0.998943	−	0.0459625i	$-0.0146355\pi$
$570$	0	0
$571$	11.2474i	0.470691i	0.971912	+	0.235346i	$0.0756222\pi$
$571$	11.2474i	0.470691i	−0.971912	+	0.235346i	$0.924378\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.953512	0.0397642
$576$	0	0
$577$	5.30306	0.220769	0.110385	−	0.993889i	$-0.464792\pi$
$577$	5.30306	0.220769	0.110385	+	0.993889i	$0.464792\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 3.46410i	− 0.143715i
$582$	0	0
$583$	− 50.4949i	− 2.09128i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.2385	−0.628961	−0.314480	−	0.949264i	$-0.601830\pi$
$587$	−15.2385	−0.628961	−0.314480	+	0.949264i	$0.601830\pi$
$588$	0	0
$589$	−35.1918	−1.45005
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26.0915i	1.07145i	0.844392	+	0.535725i	$0.179962\pi$
$593$	26.0915i	1.07145i	−0.844392	+	0.535725i	$0.820038\pi$
$594$	0	0
$595$	0.898979i	0.0368546i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.6513	1.82440	0.912201	−	0.409744i	$-0.134382\pi$
$599$	44.6513	1.82440	0.912201	+	0.409744i	$0.134382\pi$
$600$	0	0
$601$	−16.8990	−0.689324	−0.344662	−	0.938727i	$-0.612006\pi$
$601$	−16.8990	−0.689324	−0.344662	+	0.938727i	$0.612006\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 80.8815i	− 3.28830i
$606$	0	0
$607$	19.0000i	0.771186i	0.922669	+	0.385593i	$0.126003\pi$
$607$	19.0000i	0.771186i	−0.922669	+	0.385593i	$0.873997\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−68.5821	−2.77454
$612$	0	0
$613$	−20.0000	−0.807792	−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000	−0.807792	−0.403896	+	0.914805i	$0.632344\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.2699i	1.17836i	0.808001	+	0.589181i	$0.200549\pi$
$617$	29.2699i	1.17836i	−0.808001	+	0.589181i	$0.799451\pi$
$618$	0	0
$619$	29.5959i	1.18956i	0.803888	+	0.594780i	$0.202761\pi$
$619$	29.5959i	1.18956i	−0.803888	+	0.594780i	$0.797239\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.7101	0.429093
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 0.285729i	− 0.0113928i
$630$	0	0
$631$	36.4949i	1.45284i	0.687252	+	0.726419i	$0.258817\pi$
$631$	36.4949i	1.45284i	−0.687252	+	0.726419i	$0.741183\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	30.8270	1.22333
$636$	0	0
$637$	−5.44949	−0.215917
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.1489i	0.795835i	0.917421	+	0.397918i	$0.130267\pi$
$641$	20.1489i	0.795835i	−0.917421	+	0.397918i	$0.869733\pi$
$642$	0	0
$643$	− 48.4949i	− 1.91245i	−0.292629	−	0.956226i	$-0.594530\pi$
$643$	− 48.4949i	− 1.91245i	0.292629	−	0.956226i	$-0.405470\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.94258	0.233627	0.116814	−	0.993154i	$-0.462732\pi$
$647$	5.94258	0.233627	0.116814	+	0.993154i	$0.462732\pi$
$648$	0	0
$649$	36.6969	1.44048
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25.3451i	0.991830i	0.868371	+	0.495915i	$0.165167\pi$
$653$	25.3451i	0.991830i	−0.868371	+	0.495915i	$0.834833\pi$
$654$	0	0
$655$	− 6.69694i	− 0.261671i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20.7846	−0.809653	−0.404827	−	0.914393i	$-0.632668\pi$
$659$	−20.7846	−0.809653	−0.404827	+	0.914393i	$0.632668\pi$
$660$	0	0
$661$	12.8990	0.501712	0.250856	−	0.968024i	$-0.419288\pi$
$661$	12.8990	0.501712	0.250856	+	0.968024i	$0.419288\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 11.3137i	− 0.438727i
$666$	0	0
$667$	− 1.65153i	− 0.0639475i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−81.1672	−3.13342
$672$	0	0
$673$	−28.5959	−1.10229	−0.551146	−	0.834409i	$-0.685809\pi$
$673$	−28.5959	−1.10229	−0.551146	+	0.834409i	$0.685809\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.3485i	1.08952i	0.838592	+	0.544760i	$0.183379\pi$
$677$	28.3485i	1.08952i	−0.838592	+	0.544760i	$0.816621\pi$
$678$	0	0
$679$	15.7980i	0.606270i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	38.0409	1.45559	0.727797	−	0.685792i	$-0.240544\pi$
$683$	38.0409	1.45559	0.727797	+	0.685792i	$0.240544\pi$
$684$	0	0
$685$	55.1918	2.10877
$686$	0	0
$687$	0	0
$688$	0	0
$689$	43.7299i	1.66598i
$690$	0	0
$691$	28.2020i	1.07286i	0.843946	+	0.536428i	$0.180227\pi$
$691$	28.2020i	1.07286i	−0.843946	+	0.536428i	$0.819773\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−44.6834	−1.69494
$696$	0	0
$697$	1.10102	0.0417041
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 5.65685i	− 0.213656i	−0.994277	−	0.106828i	$-0.965931\pi$
$701$	− 5.65685i	− 0.213656i	0.994277	−	0.106828i	$-0.0340695\pi$
$702$	0	0
$703$	3.59592i	0.135623i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.65685	0.212748
$708$	0	0
$709$	−41.5959	−1.56217	−0.781084	−	0.624426i	$-0.785333\pi$
$709$	−41.5959	−1.56217	−0.781084	+	0.624426i	$0.785333\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.79632i	− 0.104723i
$714$	0	0
$715$	96.9898i	3.62721i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.3349	−0.609189	−0.304594	−	0.952482i	$-0.598521\pi$
$719$	−16.3349	−0.609189	−0.304594	+	0.952482i	$0.598521\pi$
$720$	0	0
$721$	3.89898	0.145206
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 15.5885i	− 0.578941i
$726$	0	0
$727$	− 9.49490i	− 0.352146i	−0.984377	−	0.176073i	$-0.943660\pi$
$727$	− 9.49490i	− 0.352146i	0.984377	−	0.176073i	$-0.0563395\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.73205	−0.0640622
$732$	0	0
$733$	0.348469	0.0128710	0.00643550	−	0.999979i	$-0.497952\pi$
$733$	0.348469	0.0128710	0.00643550	+	0.999979i	$0.497952\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 21.7060i	− 0.799551i
$738$	0	0
$739$	− 43.7980i	− 1.61113i	−0.592505	−	0.805567i	$-0.701861\pi$
$739$	− 43.7980i	− 1.61113i	0.592505	−	0.805567i	$-0.298139\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.3376	−1.22304	−0.611518	−	0.791230i	$-0.709441\pi$
$743$	−33.3376	−1.22304	−0.611518	+	0.791230i	$0.709441\pi$
$744$	0	0
$745$	−22.6969	−0.831551
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 18.2419i	− 0.666545i
$750$	0	0
$751$	44.8990i	1.63839i	0.573517	+	0.819194i	$0.305579\pi$
$751$	44.8990i	1.63839i	−0.573517	+	0.819194i	$0.694421\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.3991	0.596825
$756$	0	0
$757$	−33.1918	−1.20638	−0.603189	−	0.797598i	$-0.706103\pi$
$757$	−33.1918	−1.20638	−0.603189	+	0.797598i	$0.706103\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 48.1154i	− 1.74418i	−0.489345	−	0.872090i	$-0.662764\pi$
$761$	− 48.1154i	− 1.74418i	0.489345	−	0.872090i	$-0.337236\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31.7805	−1.14753
$768$	0	0
$769$	−4.89898	−0.176662	−0.0883309	−	0.996091i	$-0.528153\pi$
$769$	−4.89898	−0.176662	−0.0883309	+	0.996091i	$0.528153\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.4626i	1.13163i	0.824531	+	0.565816i	$0.191439\pi$
$773$	31.4626i	1.13163i	−0.824531	+	0.565816i	$0.808561\pi$
$774$	0	0
$775$	− 26.3939i	− 0.948096i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.8564	−0.496457
$780$	0	0
$781$	−15.7980	−0.565295
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 45.6690i	− 1.63000i
$786$	0	0
$787$	− 21.5959i	− 0.769811i	−0.922956	−	0.384906i	$-0.874234\pi$
$787$	− 21.5959i	− 0.769811i	0.922956	−	0.384906i	$-0.125766\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.48528	0.301702
$792$	0	0
$793$	70.2929	2.49617
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.4558i	0.901692i	0.892602	+	0.450846i	$0.148878\pi$
$797$	25.4558i	0.901692i	−0.892602	+	0.450846i	$0.851122\pi$
$798$	0	0
$799$	4.00000i	0.141510i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	25.1701	0.888234
$804$	0	0
$805$	0.898979	0.0316849
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 27.6486i	− 0.972073i	−0.873939	−	0.486036i	$-0.838442\pi$
$809$	− 27.6486i	− 0.972073i	0.873939	−	0.486036i	$-0.161558\pi$
$810$	0	0
$811$	− 24.4949i	− 0.860132i	−0.902797	−	0.430066i	$-0.858490\pi$
$811$	− 24.4949i	− 0.860132i	0.902797	−	0.430066i	$-0.141510\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.21393	0.252693
$816$	0	0
$817$	21.7980	0.762614
$818$	0	0
$819$	0	0
$820$	0	0
$821$	37.0087i	1.29161i	0.763501	+	0.645807i	$0.223479\pi$
$821$	37.0087i	1.29161i	−0.763501	+	0.645807i	$0.776521\pi$
$822$	0	0
$823$	− 21.1010i	− 0.735535i	−0.929918	−	0.367768i	$-0.880122\pi$
$823$	− 21.1010i	− 0.735535i	0.929918	−	0.367768i	$-0.119878\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.7552	1.31288	0.656438	−	0.754380i	$-0.272062\pi$
$827$	37.7552	1.31288	0.656438	+	0.754380i	$0.272062\pi$
$828$	0	0
$829$	−46.6969	−1.62185	−0.810926	−	0.585149i	$-0.801036\pi$
$829$	−46.6969	−1.62185	−0.810926	+	0.585149i	$0.801036\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.317837i	0.0110124i
$834$	0	0
$835$	− 45.3939i	− 1.57092i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.46410	−0.119594	−0.0597970	−	0.998211i	$-0.519045\pi$
$839$	−3.46410	−0.119594	−0.0597970	+	0.998211i	$0.519045\pi$
$840$	0	0
$841$	2.00000	0.0689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 47.2261i	− 1.62463i
$846$	0	0
$847$	− 28.5959i	− 0.982567i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−0.285729	−0.00979467
$852$	0	0
$853$	−31.6515	−1.08373	−0.541864	−	0.840466i	$-0.682281\pi$
$853$	−31.6515	−1.08373	−0.541864	+	0.840466i	$0.682281\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.7101i	0.365851i	0.983127	+	0.182926i	$0.0585568\pi$
$857$	10.7101i	0.365851i	−0.983127	+	0.182926i	$0.941443\pi$
$858$	0	0
$859$	− 42.2929i	− 1.44301i	−0.692407	−	0.721507i	$-0.743450\pi$
$859$	− 42.2929i	− 1.44301i	0.692407	−	0.721507i	$-0.256550\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.7949	−1.04827	−0.524135	−	0.851635i	$-0.675611\pi$
$863$	−30.7949	−1.04827	−0.524135	+	0.851635i	$0.675611\pi$
$864$	0	0
$865$	2.60612	0.0886108
$866$	0	0
$867$	0	0
$868$	0	0
$869$	62.9253i	2.13459i
$870$	0	0
$871$	18.7980i	0.636945i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.65685	−0.191237
$876$	0	0
$877$	29.1010	0.982672	0.491336	−	0.870970i	$-0.336509\pi$
$877$	29.1010	0.982672	0.491336	+	0.870970i	$0.336509\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 41.5371i	− 1.39942i	−0.714427	−	0.699710i	$-0.753312\pi$
$881$	− 41.5371i	− 1.39942i	0.714427	−	0.699710i	$-0.246688\pi$
$882$	0	0
$883$	− 1.65153i	− 0.0555784i	−0.999614	−	0.0277892i	$-0.991153\pi$
$883$	− 1.65153i	− 0.0555784i	0.999614	−	0.0277892i	$-0.00884672\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.3991	−0.550628	−0.275314	−	0.961354i	$-0.588782\pi$
$887$	−16.3991	−0.550628	−0.275314	+	0.961354i	$0.588782\pi$
$888$	0	0
$889$	10.8990	0.365540
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 50.3402i	− 1.68457i
$894$	0	0
$895$	− 31.1918i	− 1.04263i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−45.7155	−1.52470
$900$	0	0
$901$	2.55051	0.0849698
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27.2987i	0.907438i
$906$	0	0
$907$	− 41.1918i	− 1.36775i	−0.729598	−	0.683876i	$-0.760293\pi$
$907$	− 41.1918i	− 1.36775i	0.729598	−	0.683876i	$-0.239707\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19.2275	−0.637037	−0.318518	−	0.947917i	$-0.603185\pi$
$911$	−19.2275	−0.637037	−0.318518	+	0.947917i	$0.603185\pi$
$912$	0	0
$913$	−21.7980	−0.721407
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.36773i	− 0.0781892i
$918$	0	0
$919$	− 21.5959i	− 0.712384i	−0.934413	−	0.356192i	$-0.884075\pi$
$919$	− 21.5959i	− 0.712384i	0.934413	−	0.356192i	$-0.115925\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13.6814	0.450330
$924$	0	0
$925$	−2.69694	−0.0886748
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 56.9185i	− 1.86744i	−0.358011	−	0.933718i	$-0.616545\pi$
$929$	− 56.9185i	− 1.86744i	0.358011	−	0.933718i	$-0.383455\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.65685	0.184999
$936$	0	0
$937$	17.3031	0.565266	0.282633	−	0.959228i	$-0.408792\pi$
$937$	17.3031	0.565266	0.282633	+	0.959228i	$0.408792\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 21.4203i	− 0.698281i	−0.937070	−	0.349141i	$-0.886474\pi$
$941$	− 21.4203i	− 0.698281i	0.937070	−	0.349141i	$-0.113526\pi$
$942$	0	0
$943$	− 1.10102i	− 0.0358542i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.38551	−0.142510	−0.0712549	−	0.997458i	$-0.522700\pi$
$947$	−4.38551	−0.142510	−0.0712549	+	0.997458i	$0.522700\pi$
$948$	0	0
$949$	−21.7980	−0.707592
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 13.5065i	− 0.437517i	−0.975779	−	0.218759i	$-0.929799\pi$
$953$	− 13.5065i	− 0.437517i	0.975779	−	0.218759i	$-0.0702007\pi$
$954$	0	0
$955$	31.1918i	1.00934i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	19.5133	0.630116
$960$	0	0
$961$	−46.4041	−1.49691
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.0280i	0.355003i
$966$	0	0
$967$	− 19.1010i	− 0.614247i	−0.951670	−	0.307124i	$-0.900633\pi$
$967$	− 19.1010i	− 0.614247i	0.951670	−	0.307124i	$-0.0993665\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.38891	0.237121	0.118561	−	0.992947i	$-0.462172\pi$
$971$	7.38891	0.237121	0.118561	+	0.992947i	$0.462172\pi$
$972$	0	0
$973$	−15.7980	−0.506459
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 15.1278i	− 0.483980i	−0.970279	−	0.241990i	$-0.922200\pi$
$977$	− 15.1278i	− 0.483980i	0.970279	−	0.241990i	$-0.0778001\pi$
$978$	0	0
$979$	− 67.3939i	− 2.15392i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.09978	0.130763	0.0653813	−	0.997860i	$-0.479174\pi$
$983$	4.09978	0.130763	0.0653813	+	0.997860i	$0.479174\pi$
$984$	0	0
$985$	−48.0000	−1.52941
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.73205i	0.0550760i
$990$	0	0
$991$	46.4949i	1.47696i	0.674276	+	0.738480i	$0.264456\pi$
$991$	46.4949i	1.47696i	−0.674276	+	0.738480i	$0.735544\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	58.8255	1.86489
$996$	0	0
$997$	31.4495	0.996015	0.498008	−	0.867173i	$-0.334065\pi$
$997$	31.4495	0.996015	0.498008	+	0.867173i	$0.334065\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.e.2591.1	✓	8
3.2	odd	2	inner	6048.2.h.e.2591.6	yes	8
4.3	odd	2	inner	6048.2.h.e.2591.4	yes	8
12.11	even	2	inner	6048.2.h.e.2591.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.e.2591.1	✓	8	1.1	even	1	trivial
6048.2.h.e.2591.4	yes	8	4.3	odd	2	inner
6048.2.h.e.2591.6	yes	8	3.2	odd	2	inner
6048.2.h.e.2591.7	yes	8	12.11	even	2	inner

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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-2.82843i q^{5} -1.00000i q^{7} -6.29253 q^{11} +5.44949 q^{13} -0.317837i q^{17} +4.00000i q^{19} -0.317837 q^{23} -3.00000 q^{25} +5.19615i q^{29} +8.79796i q^{31} -2.82843 q^{35} +0.898979 q^{37} +3.46410i q^{41} -5.44949i q^{43} -12.5851 q^{47} -1.00000 q^{49} +8.02458i q^{53} +17.7980i q^{55} -5.83183 q^{59} +12.8990 q^{61} -15.4135i q^{65} +3.44949i q^{67} +2.51059 q^{71} -4.00000 q^{73} +6.29253i q^{77} -10.0000i q^{79} +3.46410 q^{83} -0.898979 q^{85} +10.7101i q^{89} -5.44949i q^{91} +11.3137 q^{95} -15.7980 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 24 q^{13} - 24 q^{25} - 32 q^{37} - 8 q^{49} + 64 q^{61} - 32 q^{73} + 32 q^{85} - 48 q^{97}+O(q^{100}) \) 8 * q + 24 * q^13 - 24 * q^25 - 32 * q^37 - 8 * q^49 + 64 * q^61 - 32 * q^73 + 32 * q^85 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more