LMFDB - Embedded newform 5148.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, 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("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.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:	$41.1069869606$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.585786 q^{5} -2.82843 q^{7} +O(q^{10})$	$q+0.585786 q^{5} -2.82843 q^{7} +1.00000 q^{11} -1.00000 q^{13} -2.24264 q^{17} -0.828427 q^{23} -4.65685 q^{25} +3.41421 q^{29} -2.58579 q^{31} -1.65685 q^{35} -4.82843 q^{37} +7.65685 q^{41} +8.24264 q^{43} +8.00000 q^{47} +1.00000 q^{49} +13.3137 q^{53} +0.585786 q^{55} +2.82843 q^{59} -8.82843 q^{61} -0.585786 q^{65} -4.24264 q^{67} +5.17157 q^{71} +5.65685 q^{73} -2.82843 q^{77} +7.07107 q^{79} +6.34315 q^{83} -1.31371 q^{85} -7.89949 q^{89} +2.82843 q^{91} -7.17157 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5}+O(q^{10}) $ 2 * q + 4 * q^5	$ 2 q + 4 q^{5} + 2 q^{11} - 2 q^{13} + 4 q^{17} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 8 q^{31} + 8 q^{35} - 4 q^{37} + 4 q^{41} + 8 q^{43} + 16 q^{47} + 2 q^{49} + 4 q^{53} + 4 q^{55} - 12 q^{61} - 4 q^{65} + 16 q^{71} + 24 q^{83} + 20 q^{85} + 4 q^{89} - 20 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 + 2 * q^11 - 2 * q^13 + 4 * q^17 + 4 * q^23 + 2 * q^25 + 4 * q^29 - 8 * q^31 + 8 * q^35 - 4 * q^37 + 4 * q^41 + 8 * q^43 + 16 * q^47 + 2 * q^49 + 4 * q^53 + 4 * q^55 - 12 * q^61 - 4 * q^65 + 16 * q^71 + 24 * q^83 + 20 * q^85 + 4 * q^89 - 20 * 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
$5$	0.585786	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$5$	0.585786	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.24264	−0.543920	−0.271960	−	0.962309i	$-0.587672\pi$
$17$	−2.24264	−0.543920	−0.271960	+	0.962309i	$0.587672\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.828427	−0.172739	−0.0863695	−	0.996263i	$-0.527527\pi$
$23$	−0.828427	−0.172739	−0.0863695	+	0.996263i	$0.527527\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.41421	0.634004	0.317002	−	0.948425i	$-0.397324\pi$
$29$	3.41421	0.634004	0.317002	+	0.948425i	$0.397324\pi$
$30$	0	0
$31$	−2.58579	−0.464421	−0.232210	−	0.972666i	$-0.574596\pi$
$31$	−2.58579	−0.464421	−0.232210	+	0.972666i	$0.574596\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.65685	−0.280059
$36$	0	0
$37$	−4.82843	−0.793789	−0.396894	−	0.917864i	$-0.629912\pi$
$37$	−4.82843	−0.793789	−0.396894	+	0.917864i	$0.629912\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	0	0
$43$	8.24264	1.25699	0.628495	−	0.777813i	$-0.283671\pi$
$43$	8.24264	1.25699	0.628495	+	0.777813i	$0.283671\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.3137	1.82878	0.914389	−	0.404836i	$-0.132671\pi$
$53$	13.3137	1.82878	0.914389	+	0.404836i	$0.132671\pi$
$54$	0	0
$55$	0.585786	0.0789874
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	−8.82843	−1.13036	−0.565182	−	0.824966i	$-0.691194\pi$
$61$	−8.82843	−1.13036	−0.565182	+	0.824966i	$0.691194\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.585786	−0.0726579
$66$	0	0
$67$	−4.24264	−0.518321	−0.259161	−	0.965834i	$-0.583446\pi$
$67$	−4.24264	−0.518321	−0.259161	+	0.965834i	$0.583446\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.17157	0.613753	0.306876	−	0.951749i	$-0.400716\pi$
$71$	5.17157	0.613753	0.306876	+	0.951749i	$0.400716\pi$
$72$	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
$77$	−2.82843	−0.322329
$78$	0	0
$79$	7.07107	0.795557	0.397779	−	0.917481i	$-0.369781\pi$
$79$	7.07107	0.795557	0.397779	+	0.917481i	$0.369781\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.34315	0.696251	0.348125	−	0.937448i	$-0.386818\pi$
$83$	6.34315	0.696251	0.348125	+	0.937448i	$0.386818\pi$
$84$	0	0
$85$	−1.31371	−0.142492
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.89949	−0.837345	−0.418672	−	0.908137i	$-0.637504\pi$
$89$	−7.89949	−0.837345	−0.418672	+	0.908137i	$0.637504\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.17157	−0.728163	−0.364081	−	0.931367i	$-0.618617\pi$
$97$	−7.17157	−0.728163	−0.364081	+	0.931367i	$0.618617\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.72792	−0.669453	−0.334727	−	0.942315i	$-0.608644\pi$
$101$	−6.72792	−0.669453	−0.334727	+	0.942315i	$0.608644\pi$
$102$	0	0
$103$	2.82843	0.278693	0.139347	−	0.990244i	$-0.455500\pi$
$103$	2.82843	0.278693	0.139347	+	0.990244i	$0.455500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.82843	−0.660129	−0.330064	−	0.943958i	$-0.607070\pi$
$107$	−6.82843	−0.660129	−0.330064	+	0.943958i	$0.607070\pi$
$108$	0	0
$109$	7.31371	0.700526	0.350263	−	0.936651i	$-0.386092\pi$
$109$	7.31371	0.700526	0.350263	+	0.936651i	$0.386092\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.9706	1.40831	0.704156	−	0.710045i	$-0.251326\pi$
$113$	14.9706	1.40831	0.704156	+	0.710045i	$0.251326\pi$
$114$	0	0
$115$	−0.485281	−0.0452527
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.34315	0.581475
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	7.07107	0.627456	0.313728	−	0.949513i	$-0.398422\pi$
$127$	7.07107	0.627456	0.313728	+	0.949513i	$0.398422\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.07107	0.0915075	0.0457537	−	0.998953i	$-0.485431\pi$
$137$	1.07107	0.0915075	0.0457537	+	0.998953i	$0.485431\pi$
$138$	0	0
$139$	−0.242641	−0.0205805	−0.0102903	−	0.999947i	$-0.503276\pi$
$139$	−0.242641	−0.0205805	−0.0102903	+	0.999947i	$0.503276\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.4853	1.18668	0.593340	−	0.804952i	$-0.297809\pi$
$149$	14.4853	1.18668	0.593340	+	0.804952i	$0.297809\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.51472	−0.121665
$156$	0	0
$157$	17.6569	1.40917	0.704585	−	0.709619i	$-0.251133\pi$
$157$	17.6569	1.40917	0.704585	+	0.709619i	$0.251133\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.34315	0.184666
$162$	0	0
$163$	12.7279	0.996928	0.498464	−	0.866910i	$-0.333898\pi$
$163$	12.7279	0.996928	0.498464	+	0.866910i	$0.333898\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.100505	−0.00764126	−0.00382063	−	0.999993i	$-0.501216\pi$
$173$	−0.100505	−0.00764126	−0.00382063	+	0.999993i	$0.501216\pi$
$174$	0	0
$175$	13.1716	0.995677
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.828427	0.0619196	0.0309598	−	0.999521i	$-0.490144\pi$
$179$	0.828427	0.0619196	0.0309598	+	0.999521i	$0.490144\pi$
$180$	0	0
$181$	−8.97056	−0.666777	−0.333388	−	0.942790i	$-0.608192\pi$
$181$	−8.97056	−0.666777	−0.333388	+	0.942790i	$0.608192\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.82843	−0.207950
$186$	0	0
$187$	−2.24264	−0.163998
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.6569	0.988175	0.494088	−	0.869412i	$-0.335502\pi$
$191$	13.6569	0.988175	0.494088	+	0.869412i	$0.335502\pi$
$192$	0	0
$193$	5.65685	0.407189	0.203595	−	0.979055i	$-0.434738\pi$
$193$	5.65685	0.407189	0.203595	+	0.979055i	$0.434738\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.1421	−1.15008	−0.575040	−	0.818125i	$-0.695013\pi$
$197$	−16.1421	−1.15008	−0.575040	+	0.818125i	$0.695013\pi$
$198$	0	0
$199$	12.4853	0.885058	0.442529	−	0.896754i	$-0.354081\pi$
$199$	12.4853	0.885058	0.442529	+	0.896754i	$0.354081\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.65685	−0.677778
$204$	0	0
$205$	4.48528	0.313266
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	13.4142	0.923473	0.461736	−	0.887017i	$-0.347227\pi$
$211$	13.4142	0.923473	0.461736	+	0.887017i	$0.347227\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.82843	0.329296
$216$	0	0
$217$	7.31371	0.496487
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.24264	0.150856
$222$	0	0
$223$	−2.10051	−0.140660	−0.0703301	−	0.997524i	$-0.522405\pi$
$223$	−2.10051	−0.140660	−0.0703301	+	0.997524i	$0.522405\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.1421	1.07139	0.535696	−	0.844411i	$-0.320049\pi$
$227$	16.1421	1.07139	0.535696	+	0.844411i	$0.320049\pi$
$228$	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
$233$	−11.4142	−0.747770	−0.373885	−	0.927475i	$-0.621975\pi$
$233$	−11.4142	−0.747770	−0.373885	+	0.927475i	$0.621975\pi$
$234$	0	0
$235$	4.68629	0.305700
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.14214	0.526671	0.263335	−	0.964704i	$-0.415177\pi$
$239$	8.14214	0.526671	0.263335	+	0.964704i	$0.415177\pi$
$240$	0	0
$241$	−11.6569	−0.750884	−0.375442	−	0.926846i	$-0.622509\pi$
$241$	−11.6569	−0.750884	−0.375442	+	0.926846i	$0.622509\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.585786	0.0374245
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.3431	−0.905331	−0.452666	−	0.891680i	$-0.649527\pi$
$251$	−14.3431	−0.905331	−0.452666	+	0.891680i	$0.649527\pi$
$252$	0	0
$253$	−0.828427	−0.0520828
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.82843	−0.301189	−0.150595	−	0.988596i	$-0.548119\pi$
$257$	−4.82843	−0.301189	−0.150595	+	0.988596i	$0.548119\pi$
$258$	0	0
$259$	13.6569	0.848596
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.48528	0.523225	0.261612	−	0.965173i	$-0.415746\pi$
$263$	8.48528	0.523225	0.261612	+	0.965173i	$0.415746\pi$
$264$	0	0
$265$	7.79899	0.479088
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.17157	0.193374	0.0966871	−	0.995315i	$-0.469175\pi$
$269$	3.17157	0.193374	0.0966871	+	0.995315i	$0.469175\pi$
$270$	0	0
$271$	−9.65685	−0.586612	−0.293306	−	0.956019i	$-0.594755\pi$
$271$	−9.65685	−0.586612	−0.293306	+	0.956019i	$0.594755\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.65685	−0.280819
$276$	0	0
$277$	−18.9706	−1.13983	−0.569915	−	0.821703i	$-0.693024\pi$
$277$	−18.9706	−1.13983	−0.569915	+	0.821703i	$0.693024\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.3431	0.736330	0.368165	−	0.929760i	$-0.379986\pi$
$281$	12.3431	0.736330	0.368165	+	0.929760i	$0.379986\pi$
$282$	0	0
$283$	12.9289	0.768545	0.384273	−	0.923220i	$-0.374452\pi$
$283$	12.9289	0.768545	0.384273	+	0.923220i	$0.374452\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−21.6569	−1.27836
$288$	0	0
$289$	−11.9706	−0.704151
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.4853	1.07992	0.539961	−	0.841690i	$-0.318439\pi$
$293$	18.4853	1.07992	0.539961	+	0.841690i	$0.318439\pi$
$294$	0	0
$295$	1.65685	0.0964658
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.828427	0.0479092
$300$	0	0
$301$	−23.3137	−1.34378
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.17157	−0.296123
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.1716	1.08712	0.543560	−	0.839370i	$-0.317076\pi$
$311$	19.1716	1.08712	0.543560	+	0.839370i	$0.317076\pi$
$312$	0	0
$313$	22.6274	1.27898	0.639489	−	0.768801i	$-0.279146\pi$
$313$	22.6274	1.27898	0.639489	+	0.768801i	$0.279146\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.0711	1.18347	0.591735	−	0.806133i	$-0.298443\pi$
$317$	21.0711	1.18347	0.591735	+	0.806133i	$0.298443\pi$
$318$	0	0
$319$	3.41421	0.191159
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.65685	0.258316
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−22.6274	−1.24749
$330$	0	0
$331$	0.443651	0.0243853	0.0121926	−	0.999926i	$-0.496119\pi$
$331$	0.443651	0.0243853	0.0121926	+	0.999926i	$0.496119\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.48528	−0.135785
$336$	0	0
$337$	26.4853	1.44275	0.721373	−	0.692547i	$-0.243512\pi$
$337$	26.4853	1.44275	0.721373	+	0.692547i	$0.243512\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.58579	−0.140028
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.3431	0.769980	0.384990	−	0.922921i	$-0.374205\pi$
$347$	14.3431	0.769980	0.384990	+	0.922921i	$0.374205\pi$
$348$	0	0
$349$	31.9411	1.70977	0.854885	−	0.518818i	$-0.173628\pi$
$349$	31.9411	1.70977	0.854885	+	0.518818i	$0.173628\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.7279	0.570990	0.285495	−	0.958380i	$-0.407842\pi$
$353$	10.7279	0.570990	0.285495	+	0.958380i	$0.407842\pi$
$354$	0	0
$355$	3.02944	0.160786
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.1421	−0.640837	−0.320419	−	0.947276i	$-0.603824\pi$
$359$	−12.1421	−0.640837	−0.320419	+	0.947276i	$0.603824\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.31371	0.173447
$366$	0	0
$367$	31.3137	1.63456	0.817281	−	0.576239i	$-0.195480\pi$
$367$	31.3137	1.63456	0.817281	+	0.576239i	$0.195480\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−37.6569	−1.95505
$372$	0	0
$373$	−31.4558	−1.62872	−0.814361	−	0.580359i	$-0.802912\pi$
$373$	−31.4558	−1.62872	−0.814361	+	0.580359i	$0.802912\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.41421	−0.175841
$378$	0	0
$379$	−29.6985	−1.52551	−0.762754	−	0.646688i	$-0.776153\pi$
$379$	−29.6985	−1.52551	−0.762754	+	0.646688i	$0.776153\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.7990	−1.01168	−0.505841	−	0.862627i	$-0.668818\pi$
$383$	−19.7990	−1.01168	−0.505841	+	0.862627i	$0.668818\pi$
$384$	0	0
$385$	−1.65685	−0.0844411
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−21.7990	−1.10525	−0.552626	−	0.833429i	$-0.686374\pi$
$389$	−21.7990	−1.10525	−0.552626	+	0.833429i	$0.686374\pi$
$390$	0	0
$391$	1.85786	0.0939562
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.14214	0.208413
$396$	0	0
$397$	1.79899	0.0902887	0.0451444	−	0.998980i	$-0.485625\pi$
$397$	1.79899	0.0902887	0.0451444	+	0.998980i	$0.485625\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.2132	−1.15921	−0.579606	−	0.814897i	$-0.696794\pi$
$401$	−23.2132	−1.15921	−0.579606	+	0.814897i	$0.696794\pi$
$402$	0	0
$403$	2.58579	0.128807
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.82843	−0.239336
$408$	0	0
$409$	13.6569	0.675288	0.337644	−	0.941274i	$-0.390370\pi$
$409$	13.6569	0.675288	0.337644	+	0.941274i	$0.390370\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	3.71573	0.182398
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.1716	−1.13201	−0.566003	−	0.824403i	$-0.691511\pi$
$419$	−23.1716	−1.13201	−0.566003	+	0.824403i	$0.691511\pi$
$420$	0	0
$421$	−14.6863	−0.715766	−0.357883	−	0.933766i	$-0.616501\pi$
$421$	−14.6863	−0.715766	−0.357883	+	0.933766i	$0.616501\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10.4437	0.506591
$426$	0	0
$427$	24.9706	1.20841
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.34315	0.112865	0.0564327	−	0.998406i	$-0.482027\pi$
$431$	2.34315	0.112865	0.0564327	+	0.998406i	$0.482027\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	6.58579	0.314322	0.157161	−	0.987573i	$-0.449766\pi$
$439$	6.58579	0.314322	0.157161	+	0.987573i	$0.449766\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.686292	0.0326067	0.0163033	−	0.999867i	$-0.494810\pi$
$443$	0.686292	0.0326067	0.0163033	+	0.999867i	$0.494810\pi$
$444$	0	0
$445$	−4.62742	−0.219361
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−11.4142	−0.538670	−0.269335	−	0.963047i	$-0.586804\pi$
$449$	−11.4142	−0.538670	−0.269335	+	0.963047i	$0.586804\pi$
$450$	0	0
$451$	7.65685	0.360547
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.65685	0.0776745
$456$	0	0
$457$	2.68629	0.125659	0.0628297	−	0.998024i	$-0.479988\pi$
$457$	2.68629	0.125659	0.0628297	+	0.998024i	$0.479988\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.9706	1.06985	0.534923	−	0.844901i	$-0.320341\pi$
$461$	22.9706	1.06985	0.534923	+	0.844901i	$0.320341\pi$
$462$	0	0
$463$	27.0711	1.25810	0.629050	−	0.777365i	$-0.283444\pi$
$463$	27.0711	1.25810	0.629050	+	0.777365i	$0.283444\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−27.4558	−1.27050	−0.635252	−	0.772305i	$-0.719104\pi$
$467$	−27.4558	−1.27050	−0.635252	+	0.772305i	$0.719104\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.24264	0.378997
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.14214	0.189259	0.0946295	−	0.995513i	$-0.469833\pi$
$479$	4.14214	0.189259	0.0946295	+	0.995513i	$0.469833\pi$
$480$	0	0
$481$	4.82843	0.220157
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.20101	−0.190758
$486$	0	0
$487$	19.5563	0.886183	0.443091	−	0.896476i	$-0.353882\pi$
$487$	19.5563	0.886183	0.443091	+	0.896476i	$0.353882\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.9706	0.585353	0.292677	−	0.956211i	$-0.405454\pi$
$491$	12.9706	0.585353	0.292677	+	0.956211i	$0.405454\pi$
$492$	0	0
$493$	−7.65685	−0.344847
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.6274	−0.656129
$498$	0	0
$499$	−9.89949	−0.443162	−0.221581	−	0.975142i	$-0.571122\pi$
$499$	−9.89949	−0.443162	−0.221581	+	0.975142i	$0.571122\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.51472	0.335065	0.167532	−	0.985867i	$-0.446420\pi$
$503$	7.51472	0.335065	0.167532	+	0.985867i	$0.446420\pi$
$504$	0	0
$505$	−3.94113	−0.175378
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.5563	0.955468	0.477734	−	0.878504i	$-0.341458\pi$
$509$	21.5563	0.955468	0.477734	+	0.878504i	$0.341458\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.65685	0.0730097
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.9706	0.831115	0.415558	−	0.909567i	$-0.363586\pi$
$521$	18.9706	0.831115	0.415558	+	0.909567i	$0.363586\pi$
$522$	0	0
$523$	−21.8995	−0.957598	−0.478799	−	0.877925i	$-0.658928\pi$
$523$	−21.8995	−0.957598	−0.478799	+	0.877925i	$0.658928\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.79899	0.252608
$528$	0	0
$529$	−22.3137	−0.970161
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.65685	−0.331655
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.28427	0.183518
$546$	0	0
$547$	45.8995	1.96252	0.981260	−	0.192687i	$-0.0617201\pi$
$547$	45.8995	1.96252	0.981260	+	0.192687i	$0.0617201\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−20.0000	−0.850487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.51472	−0.403152	−0.201576	−	0.979473i	$-0.564606\pi$
$557$	−9.51472	−0.403152	−0.201576	+	0.979473i	$0.564606\pi$
$558$	0	0
$559$	−8.24264	−0.348627
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.65685	−0.406988	−0.203494	−	0.979076i	$-0.565230\pi$
$563$	−9.65685	−0.406988	−0.203494	+	0.979076i	$0.565230\pi$
$564$	0	0
$565$	8.76955	0.368938
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.5563	0.736000	0.368000	−	0.929826i	$-0.380043\pi$
$569$	17.5563	0.736000	0.368000	+	0.929826i	$0.380043\pi$
$570$	0	0
$571$	20.0416	0.838716	0.419358	−	0.907821i	$-0.362255\pi$
$571$	20.0416	0.838716	0.419358	+	0.907821i	$0.362255\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.85786	0.160884
$576$	0	0
$577$	−6.97056	−0.290188	−0.145094	−	0.989418i	$-0.546349\pi$
$577$	−6.97056	−0.290188	−0.145094	+	0.989418i	$0.546349\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17.9411	−0.744323
$582$	0	0
$583$	13.3137	0.551397
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.79899	−0.321899	−0.160949	−	0.986963i	$-0.551456\pi$
$587$	−7.79899	−0.321899	−0.160949	+	0.986963i	$0.551456\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.8284	1.01958	0.509791	−	0.860298i	$-0.329723\pi$
$593$	24.8284	1.01958	0.509791	+	0.860298i	$0.329723\pi$
$594$	0	0
$595$	3.71573	0.152330
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.34315	0.0957383	0.0478692	−	0.998854i	$-0.484757\pi$
$599$	2.34315	0.0957383	0.0478692	+	0.998854i	$0.484757\pi$
$600$	0	0
$601$	−47.2548	−1.92756	−0.963782	−	0.266690i	$-0.914070\pi$
$601$	−47.2548	−1.92756	−0.963782	+	0.266690i	$0.914070\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.585786	0.0238156
$606$	0	0
$607$	−29.2132	−1.18573	−0.592864	−	0.805303i	$-0.702003\pi$
$607$	−29.2132	−1.18573	−0.592864	+	0.805303i	$0.702003\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	29.9411	1.20931	0.604655	−	0.796487i	$-0.293311\pi$
$613$	29.9411	1.20931	0.604655	+	0.796487i	$0.293311\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−41.0711	−1.65346	−0.826729	−	0.562600i	$-0.809801\pi$
$617$	−41.0711	−1.65346	−0.826729	+	0.562600i	$0.809801\pi$
$618$	0	0
$619$	6.58579	0.264705	0.132353	−	0.991203i	$-0.457747\pi$
$619$	6.58579	0.264705	0.132353	+	0.991203i	$0.457747\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	22.3431	0.895159
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.8284	0.431758
$630$	0	0
$631$	−28.9289	−1.15164	−0.575821	−	0.817576i	$-0.695318\pi$
$631$	−28.9289	−1.15164	−0.575821	+	0.817576i	$0.695318\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.14214	0.164376
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	41.5980	1.64302	0.821511	−	0.570193i	$-0.193132\pi$
$641$	41.5980	1.64302	0.821511	+	0.570193i	$0.193132\pi$
$642$	0	0
$643$	23.5563	0.928972	0.464486	−	0.885581i	$-0.346239\pi$
$643$	23.5563	0.928972	0.464486	+	0.885581i	$0.346239\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.3431	−0.721143	−0.360572	−	0.932731i	$-0.617418\pi$
$647$	−18.3431	−0.721143	−0.360572	+	0.932731i	$0.617418\pi$
$648$	0	0
$649$	2.82843	0.111025
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23.1716	0.906774	0.453387	−	0.891314i	$-0.350216\pi$
$653$	23.1716	0.906774	0.453387	+	0.891314i	$0.350216\pi$
$654$	0	0
$655$	−4.68629	−0.183109
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−31.1127	−1.21198	−0.605989	−	0.795473i	$-0.707223\pi$
$659$	−31.1127	−1.21198	−0.605989	+	0.795473i	$0.707223\pi$
$660$	0	0
$661$	3.17157	0.123360	0.0616799	−	0.998096i	$-0.480354\pi$
$661$	3.17157	0.123360	0.0616799	+	0.998096i	$0.480354\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.82843	−0.109517
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.82843	−0.340818
$672$	0	0
$673$	−0.343146	−0.0132273	−0.00661365	−	0.999978i	$-0.502105\pi$
$673$	−0.343146	−0.0132273	−0.00661365	+	0.999978i	$0.502105\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.2426	0.854854	0.427427	−	0.904050i	$-0.359420\pi$
$677$	22.2426	0.854854	0.427427	+	0.904050i	$0.359420\pi$
$678$	0	0
$679$	20.2843	0.778439
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.17157	−0.0448290	−0.0224145	−	0.999749i	$-0.507135\pi$
$683$	−1.17157	−0.0448290	−0.0224145	+	0.999749i	$0.507135\pi$
$684$	0	0
$685$	0.627417	0.0239724
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−13.3137	−0.507212
$690$	0	0
$691$	−39.5563	−1.50479	−0.752397	−	0.658710i	$-0.771103\pi$
$691$	−39.5563	−1.50479	−0.752397	+	0.658710i	$0.771103\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.142136	−0.00539151
$696$	0	0
$697$	−17.1716	−0.650420
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−21.7574	−0.821764	−0.410882	−	0.911689i	$-0.634779\pi$
$701$	−21.7574	−0.821764	−0.410882	+	0.911689i	$0.634779\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.0294	0.715676
$708$	0	0
$709$	−3.37258	−0.126660	−0.0633300	−	0.997993i	$-0.520172\pi$
$709$	−3.37258	−0.126660	−0.0633300	+	0.997993i	$0.520172\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.14214	0.0802236
$714$	0	0
$715$	−0.585786	−0.0219072
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.6863	0.473119	0.236559	−	0.971617i	$-0.423980\pi$
$719$	12.6863	0.473119	0.236559	+	0.971617i	$0.423980\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−15.8995	−0.590492
$726$	0	0
$727$	−22.8284	−0.846659	−0.423330	−	0.905976i	$-0.639139\pi$
$727$	−22.8284	−0.846659	−0.423330	+	0.905976i	$0.639139\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−18.4853	−0.683703
$732$	0	0
$733$	−30.6274	−1.13125	−0.565625	−	0.824663i	$-0.691365\pi$
$733$	−30.6274	−1.13125	−0.565625	+	0.824663i	$0.691365\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.24264	−0.156280
$738$	0	0
$739$	14.8284	0.545473	0.272736	−	0.962089i	$-0.412071\pi$
$739$	14.8284	0.545473	0.272736	+	0.962089i	$0.412071\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.9411	1.53867	0.769335	−	0.638845i	$-0.220588\pi$
$743$	41.9411	1.53867	0.769335	+	0.638845i	$0.220588\pi$
$744$	0	0
$745$	8.48528	0.310877
$746$	0	0
$747$	0	0
$748$	0	0
$749$	19.3137	0.705708
$750$	0	0
$751$	3.31371	0.120919	0.0604595	−	0.998171i	$-0.480743\pi$
$751$	3.31371	0.120919	0.0604595	+	0.998171i	$0.480743\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.02944	0.255827
$756$	0	0
$757$	32.2843	1.17339	0.586696	−	0.809807i	$-0.300428\pi$
$757$	32.2843	1.17339	0.586696	+	0.809807i	$0.300428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	24.8284	0.900030	0.450015	−	0.893021i	$-0.351419\pi$
$761$	24.8284	0.900030	0.450015	+	0.893021i	$0.351419\pi$
$762$	0	0
$763$	−20.6863	−0.748894
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.82843	−0.102129
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.41421	0.266671	0.133335	−	0.991071i	$-0.457431\pi$
$773$	7.41421	0.266671	0.133335	+	0.991071i	$0.457431\pi$
$774$	0	0
$775$	12.0416	0.432548
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	5.17157	0.185053
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.3431	0.369163
$786$	0	0
$787$	23.7990	0.848342	0.424171	−	0.905582i	$-0.360565\pi$
$787$	23.7990	0.848342	0.424171	+	0.905582i	$0.360565\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−42.3431	−1.50555
$792$	0	0
$793$	8.82843	0.313507
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22.4853	−0.796470	−0.398235	−	0.917284i	$-0.630377\pi$
$797$	−22.4853	−0.796470	−0.398235	+	0.917284i	$0.630377\pi$
$798$	0	0
$799$	−17.9411	−0.634711
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.65685	0.199626
$804$	0	0
$805$	1.37258	0.0483772
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.3848	−0.576058	−0.288029	−	0.957622i	$-0.593000\pi$
$809$	−16.3848	−0.576058	−0.288029	+	0.957622i	$0.593000\pi$
$810$	0	0
$811$	−40.2843	−1.41457	−0.707286	−	0.706927i	$-0.750081\pi$
$811$	−40.2843	−1.41457	−0.707286	+	0.706927i	$0.750081\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.45584	0.261167
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.62742	−0.161498	−0.0807490	−	0.996734i	$-0.525731\pi$
$821$	−4.62742	−0.161498	−0.0807490	+	0.996734i	$0.525731\pi$
$822$	0	0
$823$	−49.4558	−1.72392	−0.861961	−	0.506974i	$-0.830764\pi$
$823$	−49.4558	−1.72392	−0.861961	+	0.506974i	$0.830764\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.7696	1.06996	0.534981	−	0.844864i	$-0.320319\pi$
$827$	30.7696	1.06996	0.534981	+	0.844864i	$0.320319\pi$
$828$	0	0
$829$	−18.6863	−0.649002	−0.324501	−	0.945885i	$-0.605196\pi$
$829$	−18.6863	−0.649002	−0.324501	+	0.945885i	$0.605196\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.24264	−0.0777029
$834$	0	0
$835$	6.62742	0.229351
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.3431	1.04756	0.523781	−	0.851853i	$-0.324521\pi$
$839$	30.3431	1.04756	0.523781	+	0.851853i	$0.324521\pi$
$840$	0	0
$841$	−17.3431	−0.598040
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.585786	0.0201517
$846$	0	0
$847$	−2.82843	−0.0971859
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	5.31371	0.181938	0.0909690	−	0.995854i	$-0.471004\pi$
$853$	5.31371	0.181938	0.0909690	+	0.995854i	$0.471004\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.9289	−0.373325	−0.186663	−	0.982424i	$-0.559767\pi$
$857$	−10.9289	−0.373325	−0.186663	+	0.982424i	$0.559767\pi$
$858$	0	0
$859$	−17.1716	−0.585887	−0.292943	−	0.956130i	$-0.594635\pi$
$859$	−17.1716	−0.585887	−0.292943	+	0.956130i	$0.594635\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	52.7696	1.79630	0.898148	−	0.439693i	$-0.144913\pi$
$863$	52.7696	1.79630	0.898148	+	0.439693i	$0.144913\pi$
$864$	0	0
$865$	−0.0588745	−0.00200179
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.07107	0.239870
$870$	0	0
$871$	4.24264	0.143756
$872$	0	0
$873$	0	0
$874$	0	0
$875$	16.0000	0.540899
$876$	0	0
$877$	16.3431	0.551869	0.275934	−	0.961176i	$-0.411013\pi$
$877$	16.3431	0.551869	0.275934	+	0.961176i	$0.411013\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−46.9706	−1.58248	−0.791239	−	0.611507i	$-0.790564\pi$
$881$	−46.9706	−1.58248	−0.791239	+	0.611507i	$0.790564\pi$
$882$	0	0
$883$	−29.1716	−0.981702	−0.490851	−	0.871244i	$-0.663314\pi$
$883$	−29.1716	−0.981702	−0.490851	+	0.871244i	$0.663314\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.48528	0.284908	0.142454	−	0.989801i	$-0.454501\pi$
$887$	8.48528	0.284908	0.142454	+	0.989801i	$0.454501\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0.485281	0.0162212
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.82843	−0.294445
$900$	0	0
$901$	−29.8579	−0.994710
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.25483	−0.174677
$906$	0	0
$907$	−11.3137	−0.375666	−0.187833	−	0.982201i	$-0.560146\pi$
$907$	−11.3137	−0.375666	−0.187833	+	0.982201i	$0.560146\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.2843	0.406996	0.203498	−	0.979075i	$-0.434769\pi$
$911$	12.2843	0.406996	0.203498	+	0.979075i	$0.434769\pi$
$912$	0	0
$913$	6.34315	0.209927
$914$	0	0
$915$	0	0
$916$	0	0
$917$	22.6274	0.747223
$918$	0	0
$919$	−38.5858	−1.27283	−0.636414	−	0.771348i	$-0.719583\pi$
$919$	−38.5858	−1.27283	−0.636414	+	0.771348i	$0.719583\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.17157	−0.170224
$924$	0	0
$925$	22.4853	0.739311
$926$	0	0
$927$	0	0
$928$	0	0
$929$	33.0711	1.08503	0.542513	−	0.840047i	$-0.317473\pi$
$929$	33.0711	1.08503	0.542513	+	0.840047i	$0.317473\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.31371	−0.0429629
$936$	0	0
$937$	−34.2843	−1.12002	−0.560009	−	0.828486i	$-0.689202\pi$
$937$	−34.2843	−1.12002	−0.560009	+	0.828486i	$0.689202\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	−6.34315	−0.206561
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34.6274	−1.12524	−0.562620	−	0.826716i	$-0.690207\pi$
$947$	−34.6274	−1.12524	−0.562620	+	0.826716i	$0.690207\pi$
$948$	0	0
$949$	−5.65685	−0.183629
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−32.5858	−1.05556	−0.527779	−	0.849382i	$-0.676975\pi$
$953$	−32.5858	−1.05556	−0.527779	+	0.849382i	$0.676975\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.02944	−0.0978256
$960$	0	0
$961$	−24.3137	−0.784313
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.31371	0.106672
$966$	0	0
$967$	−38.4264	−1.23571	−0.617855	−	0.786292i	$-0.711998\pi$
$967$	−38.4264	−1.23571	−0.617855	+	0.786292i	$0.711998\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.8284	1.18188	0.590940	−	0.806715i	$-0.298757\pi$
$971$	36.8284	1.18188	0.590940	+	0.806715i	$0.298757\pi$
$972$	0	0
$973$	0.686292	0.0220015
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.5563	0.561677	0.280839	−	0.959755i	$-0.409387\pi$
$977$	17.5563	0.561677	0.280839	+	0.959755i	$0.409387\pi$
$978$	0	0
$979$	−7.89949	−0.252469
$980$	0	0
$981$	0	0
$982$	0	0
$983$	51.5980	1.64572	0.822860	−	0.568244i	$-0.192377\pi$
$983$	51.5980	1.64572	0.822860	+	0.568244i	$0.192377\pi$
$984$	0	0
$985$	−9.45584	−0.301288
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.82843	−0.217131
$990$	0	0
$991$	25.4558	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$991$	25.4558	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.31371	0.231860
$996$	0	0
$997$	−32.1421	−1.01795	−0.508976	−	0.860781i	$-0.669976\pi$
$997$	−32.1421	−1.01795	−0.508976	+	0.860781i	$0.669976\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.j.1.1		2
3.2	odd	2		1716.2.a.e.1.2	✓	2
12.11	even	2		6864.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.e.1.2	✓	2	3.2	odd	2
5148.2.a.j.1.1		2	1.1	even	1	trivial
6864.2.a.bb.1.2		2	12.11	even	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:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q+0.585786 q^{5} -2.82843 q^{7} +O(q^{10})\)	\(q+0.585786 q^{5} -2.82843 q^{7} +1.00000 q^{11} -1.00000 q^{13} -2.24264 q^{17} -0.828427 q^{23} -4.65685 q^{25} +3.41421 q^{29} -2.58579 q^{31} -1.65685 q^{35} -4.82843 q^{37} +7.65685 q^{41} +8.24264 q^{43} +8.00000 q^{47} +1.00000 q^{49} +13.3137 q^{53} +0.585786 q^{55} +2.82843 q^{59} -8.82843 q^{61} -0.585786 q^{65} -4.24264 q^{67} +5.17157 q^{71} +5.65685 q^{73} -2.82843 q^{77} +7.07107 q^{79} +6.34315 q^{83} -1.31371 q^{85} -7.89949 q^{89} +2.82843 q^{91} -7.17157 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5}+O(q^{10}) \) 2 * q + 4 * q^5	\( 2 q + 4 q^{5} + 2 q^{11} - 2 q^{13} + 4 q^{17} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 8 q^{31} + 8 q^{35} - 4 q^{37} + 4 q^{41} + 8 q^{43} + 16 q^{47} + 2 q^{49} + 4 q^{53} + 4 q^{55} - 12 q^{61} - 4 q^{65} + 16 q^{71} + 24 q^{83} + 20 q^{85} + 4 q^{89} - 20 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 + 2 * q^11 - 2 * q^13 + 4 * q^17 + 4 * q^23 + 2 * q^25 + 4 * q^29 - 8 * q^31 + 8 * q^35 - 4 * q^37 + 4 * q^41 + 8 * q^43 + 16 * q^47 + 2 * q^49 + 4 * q^53 + 4 * q^55 - 12 * q^61 - 4 * q^65 + 16 * q^71 + 24 * q^83 + 20 * q^85 + 4 * q^89 - 20 * q^97

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