LMFDB - Embedded newform 6048.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(1,6048)]

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([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("6048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.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:	$48.2935231425$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 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	$\chi$	$=$	6048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.82843 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-3.82843 q^{5} +1.00000 q^{7} -0.828427 q^{11} -2.82843 q^{13} +5.82843 q^{17} +1.17157 q^{19} -4.00000 q^{23} +9.65685 q^{25} -2.82843 q^{29} -1.17157 q^{31} -3.82843 q^{35} -8.65685 q^{37} -1.82843 q^{41} +2.17157 q^{43} -2.65685 q^{47} +1.00000 q^{49} -2.00000 q^{53} +3.17157 q^{55} -10.6569 q^{59} +8.82843 q^{61} +10.8284 q^{65} +4.00000 q^{67} -9.65685 q^{71} +5.65685 q^{73} -0.828427 q^{77} +3.82843 q^{79} -16.6569 q^{83} -22.3137 q^{85} -6.00000 q^{89} -2.82843 q^{91} -4.48528 q^{95} +10.1421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7	$ 2 q - 2 q^{5} + 2 q^{7} + 4 q^{11} + 6 q^{17} + 8 q^{19} - 8 q^{23} + 8 q^{25} - 8 q^{31} - 2 q^{35} - 6 q^{37} + 2 q^{41} + 10 q^{43} + 6 q^{47} + 2 q^{49} - 4 q^{53} + 12 q^{55} - 10 q^{59} + 12 q^{61} + 16 q^{65} + 8 q^{67} - 8 q^{71} + 4 q^{77} + 2 q^{79} - 22 q^{83} - 22 q^{85} - 12 q^{89} + 8 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 + 4 * q^11 + 6 * q^17 + 8 * q^19 - 8 * q^23 + 8 * q^25 - 8 * q^31 - 2 * q^35 - 6 * q^37 + 2 * q^41 + 10 * q^43 + 6 * q^47 + 2 * q^49 - 4 * q^53 + 12 * q^55 - 10 * q^59 + 12 * q^61 + 16 * q^65 + 8 * q^67 - 8 * q^71 + 4 * q^77 + 2 * q^79 - 22 * q^83 - 22 * q^85 - 12 * q^89 + 8 * q^95 - 8 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.82843	−1.71212	−0.856062	−	0.516873i	$-0.827096\pi$
$5$	−3.82843	−1.71212	−0.856062	+	0.516873i	$0.827096\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.82843	1.41360	0.706801	−	0.707413i	$-0.250138\pi$
$17$	5.82843	1.41360	0.706801	+	0.707413i	$0.250138\pi$
$18$	0	0
$19$	1.17157	0.268777	0.134389	−	0.990929i	$-0.457093\pi$
$19$	1.17157	0.268777	0.134389	+	0.990929i	$0.457093\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	9.65685	1.93137
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.82843	−0.525226	−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843	−0.525226	−0.262613	+	0.964901i	$0.584584\pi$
$30$	0	0
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.82843	−0.647122
$36$	0	0
$37$	−8.65685	−1.42318	−0.711589	−	0.702596i	$-0.752024\pi$
$37$	−8.65685	−1.42318	−0.711589	+	0.702596i	$0.752024\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.82843	−0.285552	−0.142776	−	0.989755i	$-0.545603\pi$
$41$	−1.82843	−0.285552	−0.142776	+	0.989755i	$0.545603\pi$
$42$	0	0
$43$	2.17157	0.331162	0.165581	−	0.986196i	$-0.447050\pi$
$43$	2.17157	0.331162	0.165581	+	0.986196i	$0.447050\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.65685	−0.387542	−0.193771	−	0.981047i	$-0.562072\pi$
$47$	−2.65685	−0.387542	−0.193771	+	0.981047i	$0.562072\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	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
$55$	3.17157	0.427655
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.6569	−1.38740	−0.693702	−	0.720262i	$-0.744022\pi$
$59$	−10.6569	−1.38740	−0.693702	+	0.720262i	$0.744022\pi$
$60$	0	0
$61$	8.82843	1.13036	0.565182	−	0.824966i	$-0.308806\pi$
$61$	8.82843	1.13036	0.565182	+	0.824966i	$0.308806\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.8284	1.34310
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.65685	−1.14606	−0.573029	−	0.819535i	$-0.694232\pi$
$71$	−9.65685	−1.14606	−0.573029	+	0.819535i	$0.694232\pi$
$72$	0	0
$73$	5.65685	0.662085	0.331042	−	0.943616i	$-0.392600\pi$
$73$	5.65685	0.662085	0.331042	+	0.943616i	$0.392600\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.828427	−0.0944080
$78$	0	0
$79$	3.82843	0.430732	0.215366	−	0.976533i	$-0.430906\pi$
$79$	3.82843	0.430732	0.215366	+	0.976533i	$0.430906\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.6569	−1.82833	−0.914164	−	0.405345i	$-0.867151\pi$
$83$	−16.6569	−1.82833	−0.914164	+	0.405345i	$0.867151\pi$
$84$	0	0
$85$	−22.3137	−2.42026
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.48528	−0.460180
$96$	0	0
$97$	10.1421	1.02978	0.514889	−	0.857257i	$-0.327833\pi$
$97$	10.1421	1.02978	0.514889	+	0.857257i	$0.327833\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.9706	1.48963	0.744813	−	0.667273i	$-0.232539\pi$
$101$	14.9706	1.48963	0.744813	+	0.667273i	$0.232539\pi$
$102$	0	0
$103$	13.6569	1.34565	0.672825	−	0.739802i	$-0.265081\pi$
$103$	13.6569	1.34565	0.672825	+	0.739802i	$0.265081\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.65685	0.546869	0.273434	−	0.961891i	$-0.411840\pi$
$107$	5.65685	0.546869	0.273434	+	0.961891i	$0.411840\pi$
$108$	0	0
$109$	0.656854	0.0629152	0.0314576	−	0.999505i	$-0.489985\pi$
$109$	0.656854	0.0629152	0.0314576	+	0.999505i	$0.489985\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.8284	−1.20680	−0.603398	−	0.797440i	$-0.706187\pi$
$113$	−12.8284	−1.20680	−0.603398	+	0.797440i	$0.706187\pi$
$114$	0	0
$115$	15.3137	1.42801
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.82843	0.534291
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−17.8284	−1.59462
$126$	0	0
$127$	17.8284	1.58202	0.791009	−	0.611805i	$-0.209556\pi$
$127$	17.8284	1.58202	0.791009	+	0.611805i	$0.209556\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	1.17157	0.101588
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.17157	0.783580	0.391790	−	0.920055i	$-0.371856\pi$
$137$	9.17157	0.783580	0.391790	+	0.920055i	$0.371856\pi$
$138$	0	0
$139$	16.8284	1.42737	0.713684	−	0.700468i	$-0.247025\pi$
$139$	16.8284	1.42737	0.713684	+	0.700468i	$0.247025\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.34315	0.195944
$144$	0	0
$145$	10.8284	0.899252
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.14214	0.667030	0.333515	−	0.942745i	$-0.391765\pi$
$149$	8.14214	0.667030	0.333515	+	0.942745i	$0.391765\pi$
$150$	0	0
$151$	12.1716	0.990509	0.495254	−	0.868748i	$-0.335075\pi$
$151$	12.1716	0.990509	0.495254	+	0.868748i	$0.335075\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.48528	0.360266
$156$	0	0
$157$	−2.82843	−0.225733	−0.112867	−	0.993610i	$-0.536003\pi$
$157$	−2.82843	−0.225733	−0.112867	+	0.993610i	$0.536003\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	15.4853	1.21290	0.606450	−	0.795121i	$-0.292593\pi$
$163$	15.4853	1.21290	0.606450	+	0.795121i	$0.292593\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.9706	1.23584	0.617920	−	0.786241i	$-0.287976\pi$
$167$	15.9706	1.23584	0.617920	+	0.786241i	$0.287976\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	9.65685	0.729990
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.1421	1.80447	0.902234	−	0.431247i	$-0.141926\pi$
$179$	24.1421	1.80447	0.902234	+	0.431247i	$0.141926\pi$
$180$	0	0
$181$	8.34315	0.620141	0.310071	−	0.950714i	$-0.399647\pi$
$181$	8.34315	0.620141	0.310071	+	0.950714i	$0.399647\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	33.1421	2.43666
$186$	0	0
$187$	−4.82843	−0.353090
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.6274	1.49255	0.746274	−	0.665639i	$-0.231841\pi$
$191$	20.6274	1.49255	0.746274	+	0.665639i	$0.231841\pi$
$192$	0	0
$193$	−12.3137	−0.886360	−0.443180	−	0.896433i	$-0.646150\pi$
$193$	−12.3137	−0.886360	−0.443180	+	0.896433i	$0.646150\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−27.7990	−1.98060	−0.990298	−	0.138960i	$-0.955624\pi$
$197$	−27.7990	−1.98060	−0.990298	+	0.138960i	$0.955624\pi$
$198$	0	0
$199$	−14.9706	−1.06124	−0.530618	−	0.847611i	$-0.678040\pi$
$199$	−14.9706	−1.06124	−0.530618	+	0.847611i	$0.678040\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.82843	−0.198517
$204$	0	0
$205$	7.00000	0.488901
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.970563	−0.0671352
$210$	0	0
$211$	−16.9706	−1.16830	−0.584151	−	0.811645i	$-0.698572\pi$
$211$	−16.9706	−1.16830	−0.584151	+	0.811645i	$0.698572\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.31371	−0.566990
$216$	0	0
$217$	−1.17157	−0.0795315
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−16.4853	−1.10892
$222$	0	0
$223$	−10.1421	−0.679168	−0.339584	−	0.940576i	$-0.610286\pi$
$223$	−10.1421	−0.679168	−0.339584	+	0.940576i	$0.610286\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.34315	0.421009	0.210505	−	0.977593i	$-0.432489\pi$
$227$	6.34315	0.421009	0.210505	+	0.977593i	$0.432489\pi$
$228$	0	0
$229$	−21.3137	−1.40845	−0.704225	−	0.709977i	$-0.748705\pi$
$229$	−21.3137	−1.40845	−0.704225	+	0.709977i	$0.748705\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.1421	1.18853	0.594265	−	0.804269i	$-0.297443\pi$
$233$	18.1421	1.18853	0.594265	+	0.804269i	$0.297443\pi$
$234$	0	0
$235$	10.1716	0.663520
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.1421	1.04415	0.522074	−	0.852900i	$-0.325158\pi$
$239$	16.1421	1.04415	0.522074	+	0.852900i	$0.325158\pi$
$240$	0	0
$241$	10.1421	0.653312	0.326656	−	0.945143i	$-0.394078\pi$
$241$	10.1421	0.653312	0.326656	+	0.945143i	$0.394078\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.82843	−0.244589
$246$	0	0
$247$	−3.31371	−0.210846
$248$	0	0
$249$	0	0
$250$	0	0
$251$	29.9706	1.89173	0.945863	−	0.324567i	$-0.105219\pi$
$251$	29.9706	1.89173	0.945863	+	0.324567i	$0.105219\pi$
$252$	0	0
$253$	3.31371	0.208331
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	−8.65685	−0.537911
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.17157	−0.318893	−0.159446	−	0.987207i	$-0.550971\pi$
$263$	−5.17157	−0.318893	−0.159446	+	0.987207i	$0.550971\pi$
$264$	0	0
$265$	7.65685	0.470357
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.1421	0.801290	0.400645	−	0.916233i	$-0.368786\pi$
$269$	13.1421	0.801290	0.400645	+	0.916233i	$0.368786\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.00000	−0.482418
$276$	0	0
$277$	−13.3431	−0.801712	−0.400856	−	0.916141i	$-0.631287\pi$
$277$	−13.3431	−0.801712	−0.400856	+	0.916141i	$0.631287\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−0.970563	−0.0578989	−0.0289495	−	0.999581i	$-0.509216\pi$
$281$	−0.970563	−0.0578989	−0.0289495	+	0.999581i	$0.509216\pi$
$282$	0	0
$283$	28.8284	1.71367	0.856836	−	0.515589i	$-0.172427\pi$
$283$	28.8284	1.71367	0.856836	+	0.515589i	$0.172427\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.82843	−0.107929
$288$	0	0
$289$	16.9706	0.998268
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−17.8284	−1.04155	−0.520774	−	0.853695i	$-0.674357\pi$
$293$	−17.8284	−1.04155	−0.520774	+	0.853695i	$0.674357\pi$
$294$	0	0
$295$	40.7990	2.37541
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.3137	0.654289
$300$	0	0
$301$	2.17157	0.125167
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−33.7990	−1.93532
$306$	0	0
$307$	32.2843	1.84256	0.921280	−	0.388899i	$-0.127145\pi$
$307$	32.2843	1.84256	0.921280	+	0.388899i	$0.127145\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.9706	1.35925	0.679623	−	0.733562i	$-0.262143\pi$
$311$	23.9706	1.35925	0.679623	+	0.733562i	$0.262143\pi$
$312$	0	0
$313$	−20.6274	−1.16593	−0.582965	−	0.812497i	$-0.698108\pi$
$313$	−20.6274	−1.16593	−0.582965	+	0.812497i	$0.698108\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.3431	1.03025	0.515127	−	0.857114i	$-0.327745\pi$
$317$	18.3431	1.03025	0.515127	+	0.857114i	$0.327745\pi$
$318$	0	0
$319$	2.34315	0.131191
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.82843	0.379944
$324$	0	0
$325$	−27.3137	−1.51509
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.65685	−0.146477
$330$	0	0
$331$	−10.7990	−0.593566	−0.296783	−	0.954945i	$-0.595914\pi$
$331$	−10.7990	−0.593566	−0.296783	+	0.954945i	$0.595914\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15.3137	−0.836677
$336$	0	0
$337$	2.31371	0.126036	0.0630179	−	0.998012i	$-0.479927\pi$
$337$	2.31371	0.126036	0.0630179	+	0.998012i	$0.479927\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.970563	0.0525589
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.1421	−1.18865	−0.594326	−	0.804224i	$-0.702581\pi$
$347$	−22.1421	−1.18865	−0.594326	+	0.804224i	$0.702581\pi$
$348$	0	0
$349$	36.2843	1.94225	0.971126	−	0.238566i	$-0.0766774\pi$
$349$	36.2843	1.94225	0.971126	+	0.238566i	$0.0766774\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	33.1421	1.76398	0.881989	−	0.471270i	$-0.156204\pi$
$353$	33.1421	1.76398	0.881989	+	0.471270i	$0.156204\pi$
$354$	0	0
$355$	36.9706	1.96219
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.6569	−1.45967	−0.729836	−	0.683622i	$-0.760404\pi$
$359$	−27.6569	−1.45967	−0.729836	+	0.683622i	$0.760404\pi$
$360$	0	0
$361$	−17.6274	−0.927759
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−21.6569	−1.13357
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	26.3137	1.36247	0.681236	−	0.732064i	$-0.261443\pi$
$373$	26.3137	1.36247	0.681236	+	0.732064i	$0.261443\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	−0.514719	−0.0264393	−0.0132197	−	0.999913i	$-0.504208\pi$
$379$	−0.514719	−0.0264393	−0.0132197	+	0.999913i	$0.504208\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	0	0
$385$	3.17157	0.161638
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−37.7990	−1.91648	−0.958242	−	0.285957i	$-0.907689\pi$
$389$	−37.7990	−1.91648	−0.958242	+	0.285957i	$0.907689\pi$
$390$	0	0
$391$	−23.3137	−1.17902
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14.6569	−0.737466
$396$	0	0
$397$	25.6569	1.28768	0.643840	−	0.765160i	$-0.277340\pi$
$397$	25.6569	1.28768	0.643840	+	0.765160i	$0.277340\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.68629	−0.134147	−0.0670735	−	0.997748i	$-0.521366\pi$
$401$	−2.68629	−0.134147	−0.0670735	+	0.997748i	$0.521366\pi$
$402$	0	0
$403$	3.31371	0.165068
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.17157	0.355482
$408$	0	0
$409$	−36.9706	−1.82808	−0.914038	−	0.405628i	$-0.867053\pi$
$409$	−36.9706	−1.82808	−0.914038	+	0.405628i	$0.867053\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.6569	−0.524390
$414$	0	0
$415$	63.7696	3.13032
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.34315	0.358736	0.179368	−	0.983782i	$-0.442595\pi$
$419$	7.34315	0.358736	0.179368	+	0.983782i	$0.442595\pi$
$420$	0	0
$421$	−18.9706	−0.924569	−0.462284	−	0.886732i	$-0.652970\pi$
$421$	−18.9706	−0.924569	−0.462284	+	0.886732i	$0.652970\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	56.2843	2.73019
$426$	0	0
$427$	8.82843	0.427238
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.97056	0.432097	0.216048	−	0.976383i	$-0.430683\pi$
$431$	8.97056	0.432097	0.216048	+	0.976383i	$0.430683\pi$
$432$	0	0
$433$	17.5147	0.841704	0.420852	−	0.907129i	$-0.361731\pi$
$433$	17.5147	0.841704	0.420852	+	0.907129i	$0.361731\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.68629	−0.224176
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.1716	1.38598	0.692992	−	0.720946i	$-0.256292\pi$
$443$	29.1716	1.38598	0.692992	+	0.720946i	$0.256292\pi$
$444$	0	0
$445$	22.9706	1.08891
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.82843	−0.322253	−0.161127	−	0.986934i	$-0.551513\pi$
$449$	−6.82843	−0.322253	−0.161127	+	0.986934i	$0.551513\pi$
$450$	0	0
$451$	1.51472	0.0713253
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.8284	0.507644
$456$	0	0
$457$	18.6863	0.874108	0.437054	−	0.899435i	$-0.356022\pi$
$457$	18.6863	0.874108	0.437054	+	0.899435i	$0.356022\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.4853	0.814371	0.407185	−	0.913346i	$-0.366510\pi$
$461$	17.4853	0.814371	0.407185	+	0.913346i	$0.366510\pi$
$462$	0	0
$463$	−23.8284	−1.10740	−0.553700	−	0.832716i	$-0.686785\pi$
$463$	−23.8284	−1.10740	−0.553700	+	0.832716i	$0.686785\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.2843	0.753546	0.376773	−	0.926306i	$-0.377034\pi$
$467$	16.2843	0.753546	0.376773	+	0.926306i	$0.377034\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.79899	−0.0827176
$474$	0	0
$475$	11.3137	0.519109
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.00000	0.0456912	0.0228456	−	0.999739i	$-0.492727\pi$
$479$	1.00000	0.0456912	0.0228456	+	0.999739i	$0.492727\pi$
$480$	0	0
$481$	24.4853	1.11643
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−38.8284	−1.76311
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.85786	0.0838442	0.0419221	−	0.999121i	$-0.486652\pi$
$491$	1.85786	0.0838442	0.0419221	+	0.999121i	$0.486652\pi$
$492$	0	0
$493$	−16.4853	−0.742460
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.65685	−0.433169
$498$	0	0
$499$	−29.7696	−1.33267	−0.666334	−	0.745653i	$-0.732138\pi$
$499$	−29.7696	−1.33267	−0.666334	+	0.745653i	$0.732138\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−35.6274	−1.58855	−0.794274	−	0.607560i	$-0.792149\pi$
$503$	−35.6274	−1.58855	−0.794274	+	0.607560i	$0.792149\pi$
$504$	0	0
$505$	−57.3137	−2.55043
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.4853	0.509076	0.254538	−	0.967063i	$-0.418077\pi$
$509$	11.4853	0.509076	0.254538	+	0.967063i	$0.418077\pi$
$510$	0	0
$511$	5.65685	0.250244
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−52.2843	−2.30392
$516$	0	0
$517$	2.20101	0.0968003
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.4558	0.545700	0.272850	−	0.962057i	$-0.412034\pi$
$521$	12.4558	0.545700	0.272850	+	0.962057i	$0.412034\pi$
$522$	0	0
$523$	−44.4264	−1.94263	−0.971316	−	0.237794i	$-0.923576\pi$
$523$	−44.4264	−1.94263	−0.971316	+	0.237794i	$0.923576\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.82843	−0.297451
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.17157	0.224006
$534$	0	0
$535$	−21.6569	−0.936307
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.828427	−0.0356829
$540$	0	0
$541$	29.2843	1.25903	0.629515	−	0.776989i	$-0.283254\pi$
$541$	29.2843	1.25903	0.629515	+	0.776989i	$0.283254\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.51472	−0.107719
$546$	0	0
$547$	−11.4853	−0.491075	−0.245538	−	0.969387i	$-0.578964\pi$
$547$	−11.4853	−0.491075	−0.245538	+	0.969387i	$0.578964\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.31371	−0.141169
$552$	0	0
$553$	3.82843	0.162801
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.6863	−0.537535	−0.268768	−	0.963205i	$-0.586616\pi$
$557$	−12.6863	−0.537535	−0.268768	+	0.963205i	$0.586616\pi$
$558$	0	0
$559$	−6.14214	−0.259785
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16.6863	−0.703243	−0.351622	−	0.936142i	$-0.614370\pi$
$563$	−16.6863	−0.703243	−0.351622	+	0.936142i	$0.614370\pi$
$564$	0	0
$565$	49.1127	2.06619
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.6569	1.57866	0.789329	−	0.613971i	$-0.210429\pi$
$569$	37.6569	1.57866	0.789329	+	0.613971i	$0.210429\pi$
$570$	0	0
$571$	41.7696	1.74800	0.874001	−	0.485925i	$-0.161517\pi$
$571$	41.7696	1.74800	0.874001	+	0.485925i	$0.161517\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−38.6274	−1.61087
$576$	0	0
$577$	−22.9706	−0.956277	−0.478139	−	0.878284i	$-0.658688\pi$
$577$	−22.9706	−0.956277	−0.478139	+	0.878284i	$0.658688\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−16.6569	−0.691043
$582$	0	0
$583$	1.65685	0.0686199
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.9706	1.19574	0.597872	−	0.801592i	$-0.296013\pi$
$587$	28.9706	1.19574	0.597872	+	0.801592i	$0.296013\pi$
$588$	0	0
$589$	−1.37258	−0.0565563
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5.48528	−0.225254	−0.112627	−	0.993637i	$-0.535926\pi$
$593$	−5.48528	−0.225254	−0.112627	+	0.993637i	$0.535926\pi$
$594$	0	0
$595$	−22.3137	−0.914773
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.8284	1.34133	0.670667	−	0.741759i	$-0.266008\pi$
$599$	32.8284	1.34133	0.670667	+	0.741759i	$0.266008\pi$
$600$	0	0
$601$	−43.1127	−1.75860	−0.879302	−	0.476265i	$-0.841990\pi$
$601$	−43.1127	−1.75860	−0.879302	+	0.476265i	$0.841990\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	39.4853	1.60530
$606$	0	0
$607$	24.1421	0.979899	0.489950	−	0.871751i	$-0.337015\pi$
$607$	24.1421	0.979899	0.489950	+	0.871751i	$0.337015\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.51472	0.304013
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.6863	0.510731	0.255365	−	0.966845i	$-0.417804\pi$
$617$	12.6863	0.510731	0.255365	+	0.966845i	$0.417804\pi$
$618$	0	0
$619$	−6.82843	−0.274458	−0.137229	−	0.990539i	$-0.543820\pi$
$619$	−6.82843	−0.274458	−0.137229	+	0.990539i	$0.543820\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−50.4558	−2.01181
$630$	0	0
$631$	34.1127	1.35801	0.679003	−	0.734136i	$-0.262413\pi$
$631$	34.1127	1.35801	0.679003	+	0.734136i	$0.262413\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−68.2548	−2.70861
$636$	0	0
$637$	−2.82843	−0.112066
$638$	0	0
$639$	0	0
$640$	0	0
$641$	39.4558	1.55841	0.779206	−	0.626768i	$-0.215623\pi$
$641$	39.4558	1.55841	0.779206	+	0.626768i	$0.215623\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.3431	0.406631	0.203316	−	0.979113i	$-0.434828\pi$
$647$	10.3431	0.406631	0.203316	+	0.979113i	$0.434828\pi$
$648$	0	0
$649$	8.82843	0.346546
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.4853	0.410321	0.205160	−	0.978728i	$-0.434228\pi$
$653$	10.4853	0.410321	0.205160	+	0.978728i	$0.434228\pi$
$654$	0	0
$655$	−15.3137	−0.598356
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23.6569	−0.921540	−0.460770	−	0.887520i	$-0.652427\pi$
$659$	−23.6569	−0.921540	−0.460770	+	0.887520i	$0.652427\pi$
$660$	0	0
$661$	45.1127	1.75468	0.877340	−	0.479869i	$-0.159316\pi$
$661$	45.1127	1.75468	0.877340	+	0.479869i	$0.159316\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.48528	−0.173932
$666$	0	0
$667$	11.3137	0.438069
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.31371	−0.282343
$672$	0	0
$673$	−24.3431	−0.938359	−0.469180	−	0.883103i	$-0.655450\pi$
$673$	−24.3431	−0.938359	−0.469180	+	0.883103i	$0.655450\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.68629	0.103243	0.0516213	−	0.998667i	$-0.483561\pi$
$677$	2.68629	0.103243	0.0516213	+	0.998667i	$0.483561\pi$
$678$	0	0
$679$	10.1421	0.389219
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−26.8284	−1.02656	−0.513281	−	0.858221i	$-0.671570\pi$
$683$	−26.8284	−1.02656	−0.513281	+	0.858221i	$0.671570\pi$
$684$	0	0
$685$	−35.1127	−1.34159
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.65685	0.215509
$690$	0	0
$691$	−15.1716	−0.577154	−0.288577	−	0.957457i	$-0.593182\pi$
$691$	−15.1716	−0.577154	−0.288577	+	0.957457i	$0.593182\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−64.4264	−2.44383
$696$	0	0
$697$	−10.6569	−0.403657
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29.5147	−1.11476	−0.557378	−	0.830259i	$-0.688192\pi$
$701$	−29.5147	−1.11476	−0.557378	+	0.830259i	$0.688192\pi$
$702$	0	0
$703$	−10.1421	−0.382518
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.9706	0.563026
$708$	0	0
$709$	−25.2843	−0.949571	−0.474785	−	0.880102i	$-0.657474\pi$
$709$	−25.2843	−0.949571	−0.474785	+	0.880102i	$0.657474\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.68629	0.175503
$714$	0	0
$715$	−8.97056	−0.335480
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.6569	0.472021	0.236011	−	0.971751i	$-0.424160\pi$
$719$	12.6569	0.472021	0.236011	+	0.971751i	$0.424160\pi$
$720$	0	0
$721$	13.6569	0.508608
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−27.3137	−1.01441
$726$	0	0
$727$	−18.1421	−0.672855	−0.336427	−	0.941709i	$-0.609219\pi$
$727$	−18.1421	−0.672855	−0.336427	+	0.941709i	$0.609219\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12.6569	0.468131
$732$	0	0
$733$	25.3137	0.934983	0.467492	−	0.883998i	$-0.345158\pi$
$733$	25.3137	0.934983	0.467492	+	0.883998i	$0.345158\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.31371	−0.122062
$738$	0	0
$739$	−29.9411	−1.10140	−0.550701	−	0.834703i	$-0.685640\pi$
$739$	−29.9411	−1.10140	−0.550701	+	0.834703i	$0.685640\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.82843	−0.250511	−0.125255	−	0.992125i	$-0.539975\pi$
$743$	−6.82843	−0.250511	−0.125255	+	0.992125i	$0.539975\pi$
$744$	0	0
$745$	−31.1716	−1.14204
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.65685	0.206697
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−46.5980	−1.69587
$756$	0	0
$757$	9.62742	0.349914	0.174957	−	0.984576i	$-0.444021\pi$
$757$	9.62742	0.349914	0.174957	+	0.984576i	$0.444021\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.82843	0.283780	0.141890	−	0.989882i	$-0.454682\pi$
$761$	7.82843	0.283780	0.141890	+	0.989882i	$0.454682\pi$
$762$	0	0
$763$	0.656854	0.0237797
$764$	0	0
$765$	0	0
$766$	0	0
$767$	30.1421	1.08837
$768$	0	0
$769$	15.6569	0.564601	0.282300	−	0.959326i	$-0.408903\pi$
$769$	15.6569	0.564601	0.282300	+	0.959326i	$0.408903\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−32.1716	−1.15713	−0.578566	−	0.815636i	$-0.696387\pi$
$773$	−32.1716	−1.15713	−0.578566	+	0.815636i	$0.696387\pi$
$774$	0	0
$775$	−11.3137	−0.406400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.14214	−0.0767500
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.8284	0.386483
$786$	0	0
$787$	41.9411	1.49504	0.747520	−	0.664239i	$-0.231244\pi$
$787$	41.9411	1.49504	0.747520	+	0.664239i	$0.231244\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.8284	−0.456126
$792$	0	0
$793$	−24.9706	−0.886731
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.31371	−0.329908	−0.164954	−	0.986301i	$-0.552748\pi$
$797$	−9.31371	−0.329908	−0.164954	+	0.986301i	$0.552748\pi$
$798$	0	0
$799$	−15.4853	−0.547830
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.68629	−0.165376
$804$	0	0
$805$	15.3137	0.539737
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−35.9411	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$809$	−35.9411	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$810$	0	0
$811$	30.7696	1.08047	0.540233	−	0.841516i	$-0.318336\pi$
$811$	30.7696	1.08047	0.540233	+	0.841516i	$0.318336\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−59.2843	−2.07664
$816$	0	0
$817$	2.54416	0.0890087
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.3431	−0.360978	−0.180489	−	0.983577i	$-0.557768\pi$
$821$	−10.3431	−0.360978	−0.180489	+	0.983577i	$0.557768\pi$
$822$	0	0
$823$	−31.8284	−1.10947	−0.554735	−	0.832027i	$-0.687180\pi$
$823$	−31.8284	−1.10947	−0.554735	+	0.832027i	$0.687180\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.82843	0.306994	0.153497	−	0.988149i	$-0.450946\pi$
$827$	8.82843	0.306994	0.153497	+	0.988149i	$0.450946\pi$
$828$	0	0
$829$	14.8284	0.515013	0.257506	−	0.966277i	$-0.417099\pi$
$829$	14.8284	0.515013	0.257506	+	0.966277i	$0.417099\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.82843	0.201943
$834$	0	0
$835$	−61.1421	−2.11591
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.68629	−0.0582172	−0.0291086	−	0.999576i	$-0.509267\pi$
$839$	−1.68629	−0.0582172	−0.0291086	+	0.999576i	$0.509267\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19.1421	0.658509
$846$	0	0
$847$	−10.3137	−0.354383
$848$	0	0
$849$	0	0
$850$	0	0
$851$	34.6274	1.18701
$852$	0	0
$853$	−18.9706	−0.649540	−0.324770	−	0.945793i	$-0.605287\pi$
$853$	−18.9706	−0.649540	−0.324770	+	0.945793i	$0.605287\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.8579	0.507535	0.253767	−	0.967265i	$-0.418330\pi$
$857$	14.8579	0.507535	0.253767	+	0.967265i	$0.418330\pi$
$858$	0	0
$859$	−0.343146	−0.0117080	−0.00585399	−	0.999983i	$-0.501863\pi$
$859$	−0.343146	−0.0117080	−0.00585399	+	0.999983i	$0.501863\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.3137	1.40633	0.703167	−	0.711024i	$-0.251768\pi$
$863$	41.3137	1.40633	0.703167	+	0.711024i	$0.251768\pi$
$864$	0	0
$865$	7.65685	0.260341
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.17157	−0.107588
$870$	0	0
$871$	−11.3137	−0.383350
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−17.8284	−0.602711
$876$	0	0
$877$	1.97056	0.0665412	0.0332706	−	0.999446i	$-0.489408\pi$
$877$	1.97056	0.0665412	0.0332706	+	0.999446i	$0.489408\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.97056	0.234844	0.117422	−	0.993082i	$-0.462537\pi$
$881$	6.97056	0.234844	0.117422	+	0.993082i	$0.462537\pi$
$882$	0	0
$883$	−8.79899	−0.296110	−0.148055	−	0.988979i	$-0.547301\pi$
$883$	−8.79899	−0.296110	−0.148055	+	0.988979i	$0.547301\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.5980	0.893073	0.446536	−	0.894765i	$-0.352657\pi$
$887$	26.5980	0.893073	0.446536	+	0.894765i	$0.352657\pi$
$888$	0	0
$889$	17.8284	0.597946
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.11270	−0.104162
$894$	0	0
$895$	−92.4264	−3.08947
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.31371	0.110518
$900$	0	0
$901$	−11.6569	−0.388346
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31.9411	−1.06176
$906$	0	0
$907$	43.7696	1.45334	0.726672	−	0.686984i	$-0.241066\pi$
$907$	43.7696	1.45334	0.726672	+	0.686984i	$0.241066\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−13.5147	−0.447763	−0.223881	−	0.974616i	$-0.571873\pi$
$911$	−13.5147	−0.447763	−0.223881	+	0.974616i	$0.571873\pi$
$912$	0	0
$913$	13.7990	0.456680
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.00000	0.132092
$918$	0	0
$919$	51.0833	1.68508	0.842541	−	0.538633i	$-0.181059\pi$
$919$	51.0833	1.68508	0.842541	+	0.538633i	$0.181059\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	27.3137	0.899042
$924$	0	0
$925$	−83.5980	−2.74868
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.1716	0.333718	0.166859	−	0.985981i	$-0.446637\pi$
$929$	10.1716	0.333718	0.166859	+	0.985981i	$0.446637\pi$
$930$	0	0
$931$	1.17157	0.0383968
$932$	0	0
$933$	0	0
$934$	0	0
$935$	18.4853	0.604533
$936$	0	0
$937$	38.8284	1.26847	0.634235	−	0.773141i	$-0.281315\pi$
$937$	38.8284	1.26847	0.634235	+	0.773141i	$0.281315\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.20101	0.104350	0.0521750	−	0.998638i	$-0.483385\pi$
$941$	3.20101	0.104350	0.0521750	+	0.998638i	$0.483385\pi$
$942$	0	0
$943$	7.31371	0.238167
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.94113	−0.323043	−0.161522	−	0.986869i	$-0.551640\pi$
$947$	−9.94113	−0.323043	−0.161522	+	0.986869i	$0.551640\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	0	0
$951$	0	0
$952$	0	0
$953$	19.3137	0.625632	0.312816	−	0.949814i	$-0.398728\pi$
$953$	19.3137	0.625632	0.312816	+	0.949814i	$0.398728\pi$
$954$	0	0
$955$	−78.9706	−2.55543
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.17157	0.296166
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	0	0
$964$	0	0
$965$	47.1421	1.51756
$966$	0	0
$967$	5.65685	0.181912	0.0909561	−	0.995855i	$-0.471008\pi$
$967$	5.65685	0.181912	0.0909561	+	0.995855i	$0.471008\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−0.656854	−0.0210795	−0.0105397	−	0.999944i	$-0.503355\pi$
$971$	−0.656854	−0.0210795	−0.0105397	+	0.999944i	$0.503355\pi$
$972$	0	0
$973$	16.8284	0.539495
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.02944	−0.288877	−0.144439	−	0.989514i	$-0.546138\pi$
$977$	−9.02944	−0.288877	−0.144439	+	0.989514i	$0.546138\pi$
$978$	0	0
$979$	4.97056	0.158860
$980$	0	0
$981$	0	0
$982$	0	0
$983$	59.2843	1.89087	0.945437	−	0.325804i	$-0.105635\pi$
$983$	59.2843	1.89087	0.945437	+	0.325804i	$0.105635\pi$
$984$	0	0
$985$	106.426	3.39103
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.68629	−0.276208
$990$	0	0
$991$	−57.4264	−1.82421	−0.912105	−	0.409957i	$-0.865544\pi$
$991$	−57.4264	−1.82421	−0.912105	+	0.409957i	$0.865544\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	57.3137	1.81697
$996$	0	0
$997$	−46.1421	−1.46134	−0.730668	−	0.682733i	$-0.760791\pi$
$997$	−46.1421	−1.46134	−0.730668	+	0.682733i	$0.760791\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.bb.1.1	yes	2
3.2	odd	2		6048.2.a.bi.1.2	yes	2
4.3	odd	2		6048.2.a.y.1.1	✓	2
12.11	even	2		6048.2.a.bh.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.y.1.1	✓	2	4.3	odd	2
6048.2.a.bb.1.1	yes	2	1.1	even	1	trivial
6048.2.a.bh.1.2	yes	2	12.11	even	2
6048.2.a.bi.1.2	yes	2	3.2	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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.a (trivial)

\(f(q)\)	\(=\)	\(q-3.82843 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-3.82843 q^{5} +1.00000 q^{7} -0.828427 q^{11} -2.82843 q^{13} +5.82843 q^{17} +1.17157 q^{19} -4.00000 q^{23} +9.65685 q^{25} -2.82843 q^{29} -1.17157 q^{31} -3.82843 q^{35} -8.65685 q^{37} -1.82843 q^{41} +2.17157 q^{43} -2.65685 q^{47} +1.00000 q^{49} -2.00000 q^{53} +3.17157 q^{55} -10.6569 q^{59} +8.82843 q^{61} +10.8284 q^{65} +4.00000 q^{67} -9.65685 q^{71} +5.65685 q^{73} -0.828427 q^{77} +3.82843 q^{79} -16.6569 q^{83} -22.3137 q^{85} -6.00000 q^{89} -2.82843 q^{91} -4.48528 q^{95} +10.1421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7	\( 2 q - 2 q^{5} + 2 q^{7} + 4 q^{11} + 6 q^{17} + 8 q^{19} - 8 q^{23} + 8 q^{25} - 8 q^{31} - 2 q^{35} - 6 q^{37} + 2 q^{41} + 10 q^{43} + 6 q^{47} + 2 q^{49} - 4 q^{53} + 12 q^{55} - 10 q^{59} + 12 q^{61} + 16 q^{65} + 8 q^{67} - 8 q^{71} + 4 q^{77} + 2 q^{79} - 22 q^{83} - 22 q^{85} - 12 q^{89} + 8 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 + 4 * q^11 + 6 * q^17 + 8 * q^19 - 8 * q^23 + 8 * q^25 - 8 * q^31 - 2 * q^35 - 6 * q^37 + 2 * q^41 + 10 * q^43 + 6 * q^47 + 2 * q^49 - 4 * q^53 + 12 * q^55 - 10 * q^59 + 12 * q^61 + 16 * q^65 + 8 * q^67 - 8 * q^71 + 4 * q^77 + 2 * q^79 - 22 * q^83 - 22 * q^85 - 12 * q^89 + 8 * q^95 - 8 * q^97

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