LMFDB - Embedded newform 2736.2.d.a.2015.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2015,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, 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("2736.2015");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.d (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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 12x^{10} + 47x^{8} - 44x^{6} + 81x^{4} + 848x^{2} + 1600 $ x^12 - 12x^10 + 47x^8 - 44x^6 + 81x^4 + 848*x^2 + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2015.1
Root			$-1.49887 - 1.07270i$ of defining polynomial
Character	$\chi$	$=$	2736.2015
Dual form			2736.2.d.a.2015.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.55961i q^{5} -3.63675i q^{7} +O(q^{10})$	$q-3.55961i q^{5} -3.63675i q^{7} -4.12147 q^{11} -1.20541 q^{13} -1.58353i q^{17} -1.00000i q^{19} -3.68078 q^{23} -7.67079 q^{25} -0.561860i q^{29} +5.20541i q^{31} -12.9454 q^{35} +9.27349 q^{37} -3.72892i q^{41} -9.46538i q^{43} -6.33989 q^{47} -6.22593 q^{49} +11.9719i q^{53} +14.6708i q^{55} +10.2863 q^{59} +2.60270 q^{61} +4.29078i q^{65} -2.06808i q^{67} -7.36156 q^{71} -0.363253 q^{73} +14.9887i q^{77} +11.6573i q^{79} -3.68078 q^{83} -5.63675 q^{85} -6.79970i q^{89} +4.38377i q^{91} -3.55961 q^{95} +8.86267 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 4 q^{25} + 8 q^{37} - 52 q^{49} + 24 q^{61} - 56 q^{73} - 16 q^{85} + 32 q^{97}+O(q^{100}) $ 12 * q + 4 * q^25 + 8 * q^37 - 52 * q^49 + 24 * q^61 - 56 * q^73 - 16 * q^85 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$5$	− 3.55961i	− 1.59190i	−0.605360	−	0.795952i	$-0.706971\pi$
$5$	− 3.55961i	− 1.59190i	0.605360	−	0.795952i	$-0.293029\pi$
$6$	0	0
$7$	− 3.63675i	− 1.37456i	−0.726392	−	0.687281i	$-0.758804\pi$
$7$	− 3.63675i	− 1.37456i	0.726392	−	0.687281i	$-0.241196\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.12147	−1.24267	−0.621334	−	0.783546i	$-0.713409\pi$
$11$	−4.12147	−1.24267	−0.621334	+	0.783546i	$0.713409\pi$
$12$	0	0
$13$	−1.20541	−0.334321	−0.167160	−	0.985930i	$-0.553460\pi$
$13$	−1.20541	−0.334321	−0.167160	+	0.985930i	$0.553460\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.58353i	− 0.384063i	−0.981389	−	0.192031i	$-0.938492\pi$
$17$	− 1.58353i	− 0.384063i	0.981389	−	0.192031i	$-0.0615075\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.68078	−0.767496	−0.383748	−	0.923438i	$-0.625367\pi$
$23$	−3.68078	−0.767496	−0.383748	+	0.923438i	$0.625367\pi$
$24$	0	0
$25$	−7.67079	−1.53416
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.561860i	− 0.104335i	−0.998638	−	0.0521674i	$-0.983387\pi$
$29$	− 0.561860i	− 0.104335i	0.998638	−	0.0521674i	$-0.0166129\pi$
$30$	0	0
$31$	5.20541i	0.934919i	0.884014	+	0.467460i	$0.154831\pi$
$31$	5.20541i	0.934919i	−0.884014	+	0.467460i	$0.845169\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−12.9454	−2.18817
$36$	0	0
$37$	9.27349	1.52455	0.762276	−	0.647252i	$-0.224082\pi$
$37$	9.27349	1.52455	0.762276	+	0.647252i	$0.224082\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.72892i	− 0.582360i	−0.956668	−	0.291180i	$-0.905952\pi$
$41$	− 3.72892i	− 0.582360i	0.956668	−	0.291180i	$-0.0940478\pi$
$42$	0	0
$43$	− 9.46538i	− 1.44346i	−0.692176	−	0.721728i	$-0.743348\pi$
$43$	− 9.46538i	− 1.44346i	0.692176	−	0.721728i	$-0.256652\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.33989	−0.924768	−0.462384	−	0.886680i	$-0.653006\pi$
$47$	−6.33989	−0.924768	−0.462384	+	0.886680i	$0.653006\pi$
$48$	0	0
$49$	−6.22593	−0.889418
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.9719i	1.64446i	0.569155	+	0.822230i	$0.307270\pi$
$53$	11.9719i	1.64446i	−0.569155	+	0.822230i	$0.692730\pi$
$54$	0	0
$55$	14.6708i	1.97821i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.2863	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$59$	10.2863	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$60$	0	0
$61$	2.60270	0.333242	0.166621	−	0.986021i	$-0.446714\pi$
$61$	2.60270	0.333242	0.166621	+	0.986021i	$0.446714\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.29078i	0.532206i
$66$	0	0
$67$	− 2.06808i	− 0.252657i	−0.991988	−	0.126328i	$-0.959681\pi$
$67$	− 2.06808i	− 0.252657i	0.991988	−	0.126328i	$-0.0403193\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.36156	−0.873657	−0.436828	−	0.899545i	$-0.643898\pi$
$71$	−7.36156	−0.873657	−0.436828	+	0.899545i	$0.643898\pi$
$72$	0	0
$73$	−0.363253	−0.0425156	−0.0212578	−	0.999774i	$-0.506767\pi$
$73$	−0.363253	−0.0425156	−0.0212578	+	0.999774i	$0.506767\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.9887i	1.70812i
$78$	0	0
$79$	11.6573i	1.31154i	0.754959	+	0.655772i	$0.227657\pi$
$79$	11.6573i	1.31154i	−0.754959	+	0.655772i	$0.772343\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.68078	−0.404018	−0.202009	−	0.979384i	$-0.564747\pi$
$83$	−3.68078	−0.404018	−0.202009	+	0.979384i	$0.564747\pi$
$84$	0	0
$85$	−5.63675	−0.611391
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 6.79970i	− 0.720767i	−0.932804	−	0.360383i	$-0.882646\pi$
$89$	− 6.79970i	− 0.720767i	0.932804	−	0.360383i	$-0.117354\pi$
$90$	0	0
$91$	4.38377i	0.459544i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.55961	−0.365208
$96$	0	0
$97$	8.86267	0.899868	0.449934	−	0.893062i	$-0.351447\pi$
$97$	8.86267	0.899868	0.449934	+	0.893062i	$0.351447\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 15.6526i	− 1.55750i	−0.627337	−	0.778748i	$-0.715855\pi$
$101$	− 15.6526i	− 1.55750i	0.627337	−	0.778748i	$-0.284145\pi$
$102$	0	0
$103$	17.6843i	1.74249i	0.490851	+	0.871244i	$0.336686\pi$
$103$	17.6843i	1.74249i	−0.490851	+	0.871244i	$0.663314\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.24293	−0.796874	−0.398437	−	0.917196i	$-0.630447\pi$
$107$	−8.24293	−0.796874	−0.398437	+	0.917196i	$0.630447\pi$
$108$	0	0
$109$	14.5470	1.39335	0.696674	−	0.717388i	$-0.254662\pi$
$109$	14.5470	1.39335	0.696674	+	0.717388i	$0.254662\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.97127i	0.373586i	0.982399	+	0.186793i	$0.0598094\pi$
$113$	3.97127i	0.373586i	−0.982399	+	0.186793i	$0.940191\pi$
$114$	0	0
$115$	13.1021i	1.22178i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.75890	−0.527918
$120$	0	0
$121$	5.98648	0.544225
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.50695i	0.850328i
$126$	0	0
$127$	16.8627i	1.49632i	0.663518	+	0.748160i	$0.269063\pi$
$127$	16.8627i	1.49632i	−0.663518	+	0.748160i	$0.730937\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.38323	−0.732446	−0.366223	−	0.930527i	$-0.619349\pi$
$131$	−8.38323	−0.732446	−0.366223	+	0.930527i	$0.619349\pi$
$132$	0	0
$133$	−3.63675	−0.315346
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.61255i	− 0.137769i	−0.997625	−	0.0688846i	$-0.978056\pi$
$137$	− 1.61255i	− 0.137769i	0.997625	−	0.0688846i	$-0.0219440\pi$
$138$	0	0
$139$	3.63675i	0.308465i	0.988035	+	0.154232i	$0.0492905\pi$
$139$	3.63675i	0.308465i	−0.988035	+	0.154232i	$0.950710\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.96805	0.415450
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 0.702162i	− 0.0575234i	−0.999586	−	0.0287617i	$-0.990844\pi$
$149$	− 0.702162i	− 0.0575234i	0.999586	−	0.0287617i	$-0.00915639\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	18.5292	1.48830
$156$	0	0
$157$	14.8627	1.18617	0.593085	−	0.805140i	$-0.297910\pi$
$157$	14.8627	1.18617	0.593085	+	0.805140i	$0.297910\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	13.3861i	1.05497i
$162$	0	0
$163$	− 13.7253i	− 1.07505i	−0.843247	−	0.537526i	$-0.819359\pi$
$163$	− 13.7253i	− 1.07505i	0.843247	−	0.537526i	$-0.180641\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.4539	1.66015	0.830077	−	0.557649i	$-0.188296\pi$
$167$	21.4539	1.66015	0.830077	+	0.557649i	$0.188296\pi$
$168$	0	0
$169$	−11.5470	−0.888230
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 21.0381i	− 1.59950i	−0.600334	−	0.799749i	$-0.704966\pi$
$173$	− 21.0381i	− 1.59950i	0.600334	−	0.799749i	$-0.295034\pi$
$174$	0	0
$175$	27.8967i	2.10879i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.5292	−1.38494	−0.692469	−	0.721448i	$-0.743477\pi$
$179$	−18.5292	−1.38494	−0.692469	+	0.721448i	$0.743477\pi$
$180$	0	0
$181$	−18.9308	−1.40711	−0.703556	−	0.710640i	$-0.748406\pi$
$181$	−18.9308	−1.40711	−0.703556	+	0.710640i	$0.748406\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 33.0100i	− 2.42694i
$186$	0	0
$187$	6.52647i	0.477263i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.9124	−1.94732	−0.973658	−	0.228014i	$-0.926777\pi$
$191$	−26.9124	−1.94732	−0.973658	+	0.228014i	$0.926777\pi$
$192$	0	0
$193$	−12.4519	−0.896304	−0.448152	−	0.893957i	$-0.647918\pi$
$193$	−12.4519	−0.896304	−0.448152	+	0.893957i	$0.647918\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 9.99578i	− 0.712170i	−0.934454	−	0.356085i	$-0.884111\pi$
$197$	− 9.99578i	− 0.712170i	0.934454	−	0.356085i	$-0.115889\pi$
$198$	0	0
$199$	− 24.0124i	− 1.70219i	−0.525011	−	0.851096i	$-0.675939\pi$
$199$	− 24.0124i	− 1.70219i	0.525011	−	0.851096i	$-0.324061\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.04334	−0.143415
$204$	0	0
$205$	−13.2735	−0.927061
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.12147i	0.285088i
$210$	0	0
$211$	− 16.6151i	− 1.14383i	−0.820313	−	0.571914i	$-0.806201\pi$
$211$	− 16.6151i	− 1.14383i	0.820313	−	0.571914i	$-0.193799\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−33.6930	−2.29784
$216$	0	0
$217$	18.9308	1.28510
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.90880i	0.128400i
$222$	0	0
$223$	11.6573i	0.780628i	0.920682	+	0.390314i	$0.127634\pi$
$223$	11.6573i	0.780628i	−0.920682	+	0.390314i	$0.872366\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.8908	−1.71843	−0.859215	−	0.511614i	$-0.829048\pi$
$227$	−25.8908	−1.71843	−0.859215	+	0.511614i	$0.829048\pi$
$228$	0	0
$229$	−24.3961	−1.61214	−0.806071	−	0.591819i	$-0.798410\pi$
$229$	−24.3961	−1.61214	−0.806071	+	0.591819i	$0.798410\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.07332i	0.266852i	0.991059	+	0.133426i	$0.0425979\pi$
$233$	4.07332i	0.266852i	−0.991059	+	0.133426i	$0.957402\pi$
$234$	0	0
$235$	22.5675i	1.47214i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.3079	0.731450	0.365725	−	0.930723i	$-0.380821\pi$
$239$	11.3079	0.731450	0.365725	+	0.930723i	$0.380821\pi$
$240$	0	0
$241$	−20.4789	−1.31916	−0.659581	−	0.751633i	$-0.729266\pi$
$241$	−20.4789	−1.31916	−0.659581	+	0.751633i	$0.729266\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	22.1618i	1.41587i
$246$	0	0
$247$	1.20541i	0.0766984i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.43968	0.595828	0.297914	−	0.954593i	$-0.403709\pi$
$251$	9.43968	0.595828	0.297914	+	0.954593i	$0.403709\pi$
$252$	0	0
$253$	15.1702	0.953743
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.35834i	0.521379i	0.965423	+	0.260690i	$0.0839499\pi$
$257$	8.35834i	0.521379i	−0.965423	+	0.260690i	$0.916050\pi$
$258$	0	0
$259$	− 33.7253i	− 2.09559i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.6940	−1.52270	−0.761349	−	0.648342i	$-0.775463\pi$
$263$	−24.6940	−1.52270	−0.761349	+	0.648342i	$0.775463\pi$
$264$	0	0
$265$	42.6151	2.61782
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.2097i	1.11026i	0.831762	+	0.555132i	$0.187332\pi$
$269$	18.2097i	1.11026i	−0.831762	+	0.555132i	$0.812668\pi$
$270$	0	0
$271$	− 8.51994i	− 0.517549i	−0.965938	−	0.258775i	$-0.916681\pi$
$271$	− 8.51994i	− 0.517549i	0.965938	−	0.258775i	$-0.0833187\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	31.6149	1.90645
$276$	0	0
$277$	−25.6015	−1.53825	−0.769124	−	0.639100i	$-0.779307\pi$
$277$	−25.6015	−1.53825	−0.769124	+	0.639100i	$0.779307\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 8.01971i	− 0.478416i	−0.970968	−	0.239208i	$-0.923112\pi$
$281$	− 8.01971i	− 0.478416i	0.970968	−	0.239208i	$-0.0768877\pi$
$282$	0	0
$283$	− 0.945441i	− 0.0562006i	−0.999605	−	0.0281003i	$-0.991054\pi$
$283$	− 0.945441i	− 0.0562006i	0.999605	−	0.0281003i	$-0.00894579\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−13.5611	−0.800489
$288$	0	0
$289$	14.4924	0.852496
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 15.1389i	− 0.884425i	−0.896910	−	0.442212i	$-0.854194\pi$
$293$	− 15.1389i	− 0.884425i	0.896910	−	0.442212i	$-0.145806\pi$
$294$	0	0
$295$	− 36.6151i	− 2.13181i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.43685	0.256590
$300$	0	0
$301$	−34.4232	−1.98412
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 9.26460i	− 0.530490i
$306$	0	0
$307$	− 14.5470i	− 0.830240i	−0.909767	−	0.415120i	$-0.863740\pi$
$307$	− 14.5470i	− 0.830240i	0.909767	−	0.415120i	$-0.136260\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.3079	0.641215	0.320607	−	0.947212i	$-0.396113\pi$
$311$	11.3079	0.641215	0.320607	+	0.947212i	$0.396113\pi$
$312$	0	0
$313$	−10.8627	−0.613995	−0.306997	−	0.951710i	$-0.599324\pi$
$313$	−10.8627	−0.613995	−0.306997	+	0.951710i	$0.599324\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.83165i	− 0.102876i	−0.998676	−	0.0514378i	$-0.983620\pi$
$317$	− 1.83165i	− 0.102876i	0.998676	−	0.0514378i	$-0.0163804\pi$
$318$	0	0
$319$	2.31569i	0.129654i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.58353	−0.0881100
$324$	0	0
$325$	9.24644	0.512900
$326$	0	0
$327$	0	0
$328$	0	0
$329$	23.0566i	1.27115i
$330$	0	0
$331$	− 32.4789i	− 1.78520i	−0.450848	−	0.892601i	$-0.648878\pi$
$331$	− 32.4789i	− 1.78520i	0.450848	−	0.892601i	$-0.351122\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.36156	−0.402205
$336$	0	0
$337$	18.5880	1.01255	0.506277	−	0.862371i	$-0.331021\pi$
$337$	18.5880	1.01255	0.506277	+	0.862371i	$0.331021\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 21.4539i	− 1.16179i
$342$	0	0
$343$	− 2.81511i	− 0.152002i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.38323	0.450036	0.225018	−	0.974355i	$-0.427756\pi$
$347$	8.38323	0.450036	0.225018	+	0.974355i	$0.427756\pi$
$348$	0	0
$349$	35.4572	1.89798	0.948991	−	0.315303i	$-0.102106\pi$
$349$	35.4572	1.89798	0.948991	+	0.315303i	$0.102106\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	31.6920i	1.68680i	0.537288	+	0.843399i	$0.319449\pi$
$353$	31.6920i	1.68680i	−0.537288	+	0.843399i	$0.680551\pi$
$354$	0	0
$355$	26.2043i	1.39078i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.1140	−0.797688	−0.398844	−	0.917019i	$-0.630588\pi$
$359$	−15.1140	−0.797688	−0.398844	+	0.917019i	$0.630588\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.29304i	0.0676807i
$366$	0	0
$367$	− 4.38377i	− 0.228831i	−0.993433	−	0.114415i	$-0.963500\pi$
$367$	− 4.38377i	− 0.228831i	0.993433	−	0.114415i	$-0.0364995\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	43.5386	2.26041
$372$	0	0
$373$	−0.794590	−0.0411423	−0.0205712	−	0.999788i	$-0.506548\pi$
$373$	−0.794590	−0.0411423	−0.0205712	+	0.999788i	$0.506548\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.677272i	0.0348813i
$378$	0	0
$379$	21.8205i	1.12084i	0.828208	+	0.560421i	$0.189361\pi$
$379$	21.8205i	1.12084i	−0.828208	+	0.560421i	$0.810639\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−16.7665	−0.856726	−0.428363	−	0.903607i	$-0.640910\pi$
$383$	−16.7665	−0.856726	−0.428363	+	0.903607i	$0.640910\pi$
$384$	0	0
$385$	53.3539	2.71917
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.9686i	1.21526i	0.794222	+	0.607628i	$0.207879\pi$
$389$	23.9686i	1.21526i	−0.794222	+	0.607628i	$0.792121\pi$
$390$	0	0
$391$	5.82863i	0.294767i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	41.4953	2.08785
$396$	0	0
$397$	36.1837	1.81601	0.908005	−	0.418960i	$-0.137605\pi$
$397$	36.1837	1.81601	0.908005	+	0.418960i	$0.137605\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 20.7958i	− 1.03849i	−0.854625	−	0.519246i	$-0.826213\pi$
$401$	− 20.7958i	− 1.03849i	0.854625	−	0.519246i	$-0.173787\pi$
$402$	0	0
$403$	− 6.27465i	− 0.312563i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−38.2204	−1.89451
$408$	0	0
$409$	−15.6162	−0.772173	−0.386086	−	0.922463i	$-0.626173\pi$
$409$	−15.6162	−0.772173	−0.386086	+	0.922463i	$0.626173\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 37.4086i	− 1.84076i
$414$	0	0
$415$	13.1021i	0.643158i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	17.2419	0.842323	0.421162	−	0.906986i	$-0.361622\pi$
$419$	17.2419	0.842323	0.421162	+	0.906986i	$0.361622\pi$
$420$	0	0
$421$	26.9308	1.31252	0.656262	−	0.754533i	$-0.272136\pi$
$421$	26.9308	1.31252	0.656262	+	0.754533i	$0.272136\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.1469i	0.589213i
$426$	0	0
$427$	− 9.46538i	− 0.458062i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.1676	0.537926	0.268963	−	0.963150i	$-0.413319\pi$
$431$	11.1676	0.537926	0.268963	+	0.963150i	$0.413319\pi$
$432$	0	0
$433$	−34.7512	−1.67004	−0.835019	−	0.550221i	$-0.814543\pi$
$433$	−34.7512	−1.67004	−0.835019	+	0.550221i	$0.814543\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.68078i	0.176076i
$438$	0	0
$439$	− 29.3416i	− 1.40040i	−0.713948	−	0.700199i	$-0.753095\pi$
$439$	− 29.3416i	− 1.40040i	0.713948	−	0.700199i	$-0.246905\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	37.5489	1.78400	0.892000	−	0.452035i	$-0.149302\pi$
$443$	37.5489	1.78400	0.892000	+	0.452035i	$0.149302\pi$
$444$	0	0
$445$	−24.2043	−1.14739
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.5483i	0.875350i	0.899133	+	0.437675i	$0.144198\pi$
$449$	18.5483i	0.875350i	−0.899133	+	0.437675i	$0.855802\pi$
$450$	0	0
$451$	15.3686i	0.723680i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	15.6045	0.731550
$456$	0	0
$457$	−28.4994	−1.33315	−0.666573	−	0.745439i	$-0.732240\pi$
$457$	−28.4994	−1.33315	−0.666573	+	0.745439i	$0.732240\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 16.7033i	− 0.777951i	−0.921248	−	0.388976i	$-0.872829\pi$
$461$	− 16.7033i	− 0.777951i	0.921248	−	0.388976i	$-0.127171\pi$
$462$	0	0
$463$	− 13.6015i	− 0.632117i	−0.948740	−	0.316059i	$-0.897640\pi$
$463$	− 13.6015i	− 0.632117i	0.948740	−	0.316059i	$-0.102360\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.5264	−0.625926	−0.312963	−	0.949765i	$-0.601322\pi$
$467$	−13.5264	−0.625926	−0.312963	+	0.949765i	$0.601322\pi$
$468$	0	0
$469$	−7.52110	−0.347292
$470$	0	0
$471$	0	0
$472$	0	0
$473$	39.0112i	1.79374i
$474$	0	0
$475$	7.67079i	0.351960i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13.9671	0.638171	0.319085	−	0.947726i	$-0.396624\pi$
$479$	13.9671	0.638171	0.319085	+	0.947726i	$0.396624\pi$
$480$	0	0
$481$	−11.1784	−0.509689
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 31.5476i	− 1.43250i
$486$	0	0
$487$	35.7934i	1.62196i	0.585077	+	0.810978i	$0.301064\pi$
$487$	35.7934i	1.62196i	−0.585077	+	0.810978i	$0.698936\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.4549	−0.562083	−0.281041	−	0.959696i	$-0.590680\pi$
$491$	−12.4549	−0.562083	−0.281041	+	0.959696i	$0.590680\pi$
$492$	0	0
$493$	−0.889723	−0.0400711
$494$	0	0
$495$	0	0
$496$	0	0
$497$	26.7721i	1.20089i
$498$	0	0
$499$	− 24.4584i	− 1.09491i	−0.836836	−	0.547454i	$-0.815597\pi$
$499$	− 24.4584i	− 1.09491i	0.836836	−	0.547454i	$-0.184403\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21.3286	0.950996	0.475498	−	0.879717i	$-0.342268\pi$
$503$	21.3286	0.950996	0.475498	+	0.879717i	$0.342268\pi$
$504$	0	0
$505$	−55.7172	−2.47938
$506$	0	0
$507$	0	0
$508$	0	0
$509$	41.4761i	1.83840i	0.393794	+	0.919199i	$0.371162\pi$
$509$	41.4761i	1.83840i	−0.393794	+	0.919199i	$0.628838\pi$
$510$	0	0
$511$	1.32106i	0.0584403i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	62.9492	2.77387
$516$	0	0
$517$	26.1296	1.14918
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 16.8436i	− 0.737933i	−0.929443	−	0.368966i	$-0.879712\pi$
$521$	− 16.8436i	− 0.737933i	0.929443	−	0.368966i	$-0.120288\pi$
$522$	0	0
$523$	− 8.09513i	− 0.353975i	−0.984213	−	0.176988i	$-0.943365\pi$
$523$	− 8.09513i	− 0.353975i	0.984213	−	0.176988i	$-0.0566353\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.24293	0.359068
$528$	0	0
$529$	−9.45185	−0.410950
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.49488i	0.194695i
$534$	0	0
$535$	29.3416i	1.26855i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	25.6599	1.10525
$540$	0	0
$541$	−45.6344	−1.96198	−0.980989	−	0.194065i	$-0.937833\pi$
$541$	−45.6344	−1.96198	−0.980989	+	0.194065i	$0.937833\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 51.7815i	− 2.21808i
$546$	0	0
$547$	− 14.7946i	− 0.632571i	−0.948664	−	0.316285i	$-0.897564\pi$
$547$	− 14.7946i	− 0.632571i	0.948664	−	0.316285i	$-0.102436\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.561860	−0.0239360
$552$	0	0
$553$	42.3945	1.80280
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 34.4682i	− 1.46046i	−0.683199	−	0.730232i	$-0.739412\pi$
$557$	− 34.4682i	− 1.46046i	0.683199	−	0.730232i	$-0.260588\pi$
$558$	0	0
$559$	11.4097i	0.482577i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.43685	0.186991	0.0934955	−	0.995620i	$-0.470196\pi$
$563$	4.43685	0.186991	0.0934955	+	0.995620i	$0.470196\pi$
$564$	0	0
$565$	14.1362	0.594713
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 32.6407i	− 1.36837i	−0.729309	−	0.684184i	$-0.760158\pi$
$569$	− 32.6407i	− 1.36837i	0.729309	−	0.684184i	$-0.239842\pi$
$570$	0	0
$571$	8.51994i	0.356548i	0.983981	+	0.178274i	$0.0570514\pi$
$571$	8.51994i	0.356548i	−0.983981	+	0.178274i	$0.942949\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	28.2345	1.17746
$576$	0	0
$577$	14.6027	0.607919	0.303959	−	0.952685i	$-0.401691\pi$
$577$	14.6027	0.607919	0.303959	+	0.952685i	$0.401691\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.3861i	0.555348i
$582$	0	0
$583$	− 49.3416i	− 2.04352i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.2552	1.57896	0.789480	−	0.613777i	$-0.210350\pi$
$587$	38.2552	1.57896	0.789480	+	0.613777i	$0.210350\pi$
$588$	0	0
$589$	5.20541	0.214485
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.9185i	1.26967i	0.772648	+	0.634835i	$0.218932\pi$
$593$	30.9185i	1.26967i	−0.772648	+	0.634835i	$0.781068\pi$
$594$	0	0
$595$	20.4994i	0.840394i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.36156	0.300785	0.150393	−	0.988626i	$-0.451946\pi$
$599$	7.36156	0.300785	0.150393	+	0.988626i	$0.451946\pi$
$600$	0	0
$601$	−13.8886	−0.566526	−0.283263	−	0.959042i	$-0.591417\pi$
$601$	−13.8886	−0.566526	−0.283263	+	0.959042i	$0.591417\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 21.3095i	− 0.866354i
$606$	0	0
$607$	1.06924i	0.0433992i	0.999765	+	0.0216996i	$0.00690774\pi$
$607$	1.06924i	0.0433992i	−0.999765	+	0.0216996i	$0.993092\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.64217	0.309169
$612$	0	0
$613$	−6.35510	−0.256680	−0.128340	−	0.991730i	$-0.540965\pi$
$613$	−6.35510	−0.256680	−0.128340	+	0.991730i	$0.540965\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 11.8025i	− 0.475152i	−0.971369	−	0.237576i	$-0.923647\pi$
$617$	− 11.8025i	− 0.475152i	0.971369	−	0.237576i	$-0.0763529\pi$
$618$	0	0
$619$	− 15.8638i	− 0.637621i	−0.947818	−	0.318811i	$-0.896717\pi$
$619$	− 15.8638i	− 0.637621i	0.947818	−	0.318811i	$-0.103283\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24.7288	−0.990738
$624$	0	0
$625$	−4.51295	−0.180518
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 14.6849i	− 0.585524i
$630$	0	0
$631$	− 37.3621i	− 1.48736i	−0.668535	−	0.743681i	$-0.733078\pi$
$631$	− 37.3621i	− 1.48736i	0.668535	−	0.743681i	$-0.266922\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	60.0245	2.38200
$636$	0	0
$637$	7.50479	0.297351
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 24.3513i	− 0.961817i	−0.876771	−	0.480908i	$-0.840307\pi$
$641$	− 24.3513i	− 0.961817i	0.876771	−	0.480908i	$-0.159693\pi$
$642$	0	0
$643$	15.4924i	0.610962i	0.952198	+	0.305481i	$0.0988172\pi$
$643$	15.4924i	0.610962i	−0.952198	+	0.305481i	$0.901183\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.5320	−0.925140	−0.462570	−	0.886583i	$-0.653073\pi$
$647$	−23.5320	−0.925140	−0.462570	+	0.886583i	$0.653073\pi$
$648$	0	0
$649$	−42.3945	−1.66413
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.4772i	0.879602i	0.898095	+	0.439801i	$0.144951\pi$
$653$	22.4772i	0.879602i	−0.898095	+	0.439801i	$0.855049\pi$
$654$	0	0
$655$	29.8410i	1.16598i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.3786	−0.949656	−0.474828	−	0.880078i	$-0.657490\pi$
$659$	−24.3786	−0.949656	−0.474828	+	0.880078i	$0.657490\pi$
$660$	0	0
$661$	−8.93076	−0.347366	−0.173683	−	0.984802i	$-0.555567\pi$
$661$	−8.93076	−0.347366	−0.173683	+	0.984802i	$0.555567\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.9454i	0.502000i
$666$	0	0
$667$	2.06808i	0.0800765i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.7270	−0.414110
$672$	0	0
$673$	23.3416	0.899752	0.449876	−	0.893091i	$-0.351468\pi$
$673$	23.3416	0.899752	0.449876	+	0.893091i	$0.351468\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.6670i	0.678996i	0.940607	+	0.339498i	$0.110257\pi$
$677$	17.6670i	0.678996i	−0.940607	+	0.339498i	$0.889743\pi$
$678$	0	0
$679$	− 32.2313i	− 1.23692i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.5272	1.39767	0.698837	−	0.715281i	$-0.253701\pi$
$683$	36.5272	1.39767	0.698837	+	0.715281i	$0.253701\pi$
$684$	0	0
$685$	−5.74003	−0.219315
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 14.4310i	− 0.549777i
$690$	0	0
$691$	− 34.6708i	− 1.31894i	−0.751731	−	0.659469i	$-0.770781\pi$
$691$	− 34.6708i	− 1.31894i	0.751731	−	0.659469i	$-0.229219\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.9454	0.491046
$696$	0	0
$697$	−5.90487	−0.223663
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.7738i	1.01123i	0.862759	+	0.505616i	$0.168735\pi$
$701$	26.7738i	1.01123i	−0.862759	+	0.505616i	$0.831265\pi$
$702$	0	0
$703$	− 9.27349i	− 0.349756i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−56.9247	−2.14087
$708$	0	0
$709$	−17.5481	−0.659034	−0.329517	−	0.944150i	$-0.606886\pi$
$709$	−17.5481	−0.659034	−0.329517	+	0.944150i	$0.606886\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 19.1600i	− 0.717547i
$714$	0	0
$715$	− 17.6843i	− 0.661356i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	23.5320	0.877597	0.438798	−	0.898586i	$-0.355404\pi$
$719$	23.5320	0.877597	0.438798	+	0.898586i	$0.355404\pi$
$720$	0	0
$721$	64.3134	2.39516
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.30991i	0.160066i
$726$	0	0
$727$	− 15.4924i	− 0.574582i	−0.957843	−	0.287291i	$-0.907245\pi$
$727$	− 15.4924i	− 0.574582i	0.957843	−	0.287291i	$-0.0927547\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−14.9887	−0.554378
$732$	0	0
$733$	−3.27349	−0.120909	−0.0604546	−	0.998171i	$-0.519255\pi$
$733$	−3.27349	−0.120909	−0.0604546	+	0.998171i	$0.519255\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.52353i	0.313968i
$738$	0	0
$739$	− 0.499421i	− 0.0183715i	−0.999958	−	0.00918574i	$-0.997076\pi$
$739$	− 0.499421i	− 0.0183715i	0.999958	−	0.00918574i	$-0.00292395\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.6364	0.390213	0.195107	−	0.980782i	$-0.437495\pi$
$743$	10.6364	0.390213	0.195107	+	0.980782i	$0.437495\pi$
$744$	0	0
$745$	−2.49942	−0.0915717
$746$	0	0
$747$	0	0
$748$	0	0
$749$	29.9775i	1.09535i
$750$	0	0
$751$	20.8216i	0.759792i	0.925029	+	0.379896i	$0.124040\pi$
$751$	20.8216i	0.759792i	−0.925029	+	0.379896i	$0.875960\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8.12380	−0.295265	−0.147632	−	0.989042i	$-0.547165\pi$
$757$	−8.12380	−0.295265	−0.147632	+	0.989042i	$0.547165\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 17.9151i	− 0.649421i	−0.945813	−	0.324710i	$-0.894733\pi$
$761$	− 17.9151i	− 0.649421i	0.945813	−	0.324710i	$-0.105267\pi$
$762$	0	0
$763$	− 52.9037i	− 1.91524i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.3992	−0.447708
$768$	0	0
$769$	10.3010	0.371464	0.185732	−	0.982600i	$-0.440534\pi$
$769$	10.3010	0.371464	0.185732	+	0.982600i	$0.440534\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 13.7844i	− 0.495789i	−0.968787	−	0.247895i	$-0.920261\pi$
$773$	− 13.7844i	− 0.495789i	0.968787	−	0.247895i	$-0.0797387\pi$
$774$	0	0
$775$	− 39.9296i	− 1.43431i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.72892	−0.133603
$780$	0	0
$781$	30.3404	1.08567
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 52.9052i	− 1.88827i
$786$	0	0
$787$	42.0681i	1.49957i	0.661684	+	0.749783i	$0.269842\pi$
$787$	42.0681i	1.49957i	−0.661684	+	0.749783i	$0.730158\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.4425	0.513517
$792$	0	0
$793$	−3.13733	−0.111410
$794$	0	0
$795$	0	0
$796$	0	0
$797$	48.1106i	1.70417i	0.523407	+	0.852083i	$0.324661\pi$
$797$	48.1106i	1.70417i	−0.523407	+	0.852083i	$0.675339\pi$
$798$	0	0
$799$	10.0394i	0.355169i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.49714	0.0528328
$804$	0	0
$805$	47.6491	1.67941
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 8.64471i	− 0.303932i	−0.988386	−	0.151966i	$-0.951440\pi$
$809$	− 8.64471i	− 0.303932i	0.988386	−	0.151966i	$-0.0485604\pi$
$810$	0	0
$811$	− 30.5880i	− 1.07409i	−0.843553	−	0.537045i	$-0.819540\pi$
$811$	− 30.5880i	− 1.07409i	0.843553	−	0.537045i	$-0.180460\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−48.8568	−1.71138
$816$	0	0
$817$	−9.46538	−0.331152
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 10.3784i	− 0.362210i	−0.983464	−	0.181105i	$-0.942033\pi$
$821$	− 10.3784i	− 0.362210i	0.983464	−	0.181105i	$-0.0579674\pi$
$822$	0	0
$823$	− 3.63675i	− 0.126769i	−0.997989	−	0.0633845i	$-0.979811\pi$
$823$	− 3.63675i	− 0.126769i	0.997989	−	0.0633845i	$-0.0201895\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.52353	−0.296392	−0.148196	−	0.988958i	$-0.547347\pi$
$827$	−8.52353	−0.296392	−0.148196	+	0.988958i	$0.547347\pi$
$828$	0	0
$829$	22.0681	0.766456	0.383228	−	0.923654i	$-0.374812\pi$
$829$	22.0681	0.766456	0.383228	+	0.923654i	$0.374812\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	9.85895i	0.341592i
$834$	0	0
$835$	− 76.3675i	− 2.64281i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	54.1750	1.87033	0.935165	−	0.354212i	$-0.115251\pi$
$839$	54.1750	1.87033	0.935165	+	0.354212i	$0.115251\pi$
$840$	0	0
$841$	28.6843	0.989114
$842$	0	0
$843$	0	0
$844$	0	0
$845$	41.1027i	1.41398i
$846$	0	0
$847$	− 21.7713i	− 0.748071i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−34.1337	−1.17009
$852$	0	0
$853$	25.2325	0.863943	0.431971	−	0.901887i	$-0.357818\pi$
$853$	25.2325	0.863943	0.431971	+	0.901887i	$0.357818\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.9852i	0.785159i	0.919718	+	0.392579i	$0.128417\pi$
$857$	22.9852i	0.785159i	−0.919718	+	0.392579i	$0.871583\pi$
$858$	0	0
$859$	− 6.12964i	− 0.209140i	−0.994518	−	0.104570i	$-0.966653\pi$
$859$	− 6.12964i	− 0.209140i	0.994518	−	0.104570i	$-0.0333467\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.4859	−0.561185	−0.280593	−	0.959827i	$-0.590531\pi$
$863$	−16.4859	−0.561185	−0.280593	+	0.959827i	$0.590531\pi$
$864$	0	0
$865$	−74.8874	−2.54625
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 48.0450i	− 1.62982i
$870$	0	0
$871$	2.49289i	0.0844683i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	34.5744	1.16883
$876$	0	0
$877$	−12.9718	−0.438026	−0.219013	−	0.975722i	$-0.570284\pi$
$877$	−12.9718	−0.438026	−0.219013	+	0.975722i	$0.570284\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.11960i	0.0377201i	0.999822	+	0.0188601i	$0.00600370\pi$
$881$	1.11960i	0.0377201i	−0.999822	+	0.0188601i	$0.993996\pi$
$882$	0	0
$883$	− 25.4572i	− 0.856704i	−0.903612	−	0.428352i	$-0.859094\pi$
$883$	− 25.4572i	− 0.856704i	0.903612	−	0.428352i	$-0.140906\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.0470	−1.00888	−0.504440	−	0.863447i	$-0.668301\pi$
$887$	−30.0470	−1.00888	−0.504440	+	0.863447i	$0.668301\pi$
$888$	0	0
$889$	61.3253	2.05678
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.33989i	0.212156i
$894$	0	0
$895$	65.9566i	2.20469i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.92471	0.0975446
$900$	0	0
$901$	18.9578	0.631576
$902$	0	0
$903$	0	0
$904$	0	0
$905$	67.3860i	2.23999i
$906$	0	0
$907$	− 54.7242i	− 1.81709i	−0.417790	−	0.908543i	$-0.637195\pi$
$907$	− 54.7242i	− 1.81709i	0.417790	−	0.908543i	$-0.362805\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39.4519	−1.30710	−0.653550	−	0.756883i	$-0.726721\pi$
$911$	−39.4519	−1.30710	−0.653550	+	0.756883i	$0.726721\pi$
$912$	0	0
$913$	15.1702	0.502061
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.4877i	1.00679i
$918$	0	0
$919$	− 9.58918i	− 0.316318i	−0.987414	−	0.158159i	$-0.949444\pi$
$919$	− 9.58918i	− 0.316318i	0.987414	−	0.158159i	$-0.0505558\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.87370	0.292081
$924$	0	0
$925$	−71.1350	−2.33890
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.1169i	0.692824i	0.938083	+	0.346412i	$0.112600\pi$
$929$	21.1169i	0.692824i	−0.938083	+	0.346412i	$0.887400\pi$
$930$	0	0
$931$	6.22593i	0.204047i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	23.2317	0.759756
$936$	0	0
$937$	−40.1837	−1.31275	−0.656373	−	0.754437i	$-0.727910\pi$
$937$	−40.1837	−1.31275	−0.656373	+	0.754437i	$0.727910\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	9.78244i	0.318899i	0.987206	+	0.159449i	$0.0509718\pi$
$941$	9.78244i	0.318899i	−0.987206	+	0.159449i	$0.949028\pi$
$942$	0	0
$943$	13.7253i	0.446959i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.8918	0.548909	0.274454	−	0.961600i	$-0.411503\pi$
$947$	16.8918	0.548909	0.274454	+	0.961600i	$0.411503\pi$
$948$	0	0
$949$	0.437869	0.0142138
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 16.3009i	− 0.528038i	−0.964518	−	0.264019i	$-0.914952\pi$
$953$	− 16.3009i	− 0.528038i	0.964518	−	0.264019i	$-0.0850481\pi$
$954$	0	0
$955$	95.7976i	3.09994i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.86443	−0.189372
$960$	0	0
$961$	3.90371	0.125926
$962$	0	0
$963$	0	0
$964$	0	0
$965$	44.3237i	1.42683i
$966$	0	0
$967$	11.2325i	0.361212i	0.983556	+	0.180606i	$0.0578058\pi$
$967$	11.2325i	0.361212i	−0.983556	+	0.180606i	$0.942194\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−25.6102	−0.821869	−0.410935	−	0.911665i	$-0.634798\pi$
$971$	−25.6102	−0.821869	−0.410935	+	0.911665i	$0.634798\pi$
$972$	0	0
$973$	13.2259	0.424004
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24.6401i	− 0.788307i	−0.919045	−	0.394153i	$-0.871038\pi$
$977$	− 24.6401i	− 0.788307i	0.919045	−	0.394153i	$-0.128962\pi$
$978$	0	0
$979$	28.0247i	0.895674i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.92471	0.0932838	0.0466419	−	0.998912i	$-0.485148\pi$
$983$	2.92471	0.0932838	0.0466419	+	0.998912i	$0.485148\pi$
$984$	0	0
$985$	−35.5810	−1.13371
$986$	0	0
$987$	0	0
$988$	0	0
$989$	34.8400i	1.10785i
$990$	0	0
$991$	2.06808i	0.0656948i	0.999460	+	0.0328474i	$0.0104575\pi$
$991$	2.06808i	0.0656948i	−0.999460	+	0.0328474i	$0.989542\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−85.4745	−2.70973
$996$	0	0
$997$	19.8081	0.627329	0.313665	−	0.949534i	$-0.398443\pi$
$997$	19.8081	0.627329	0.313665	+	0.949534i	$0.398443\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.d.a.2015.1	✓	12
3.2	odd	2	inner	2736.2.d.a.2015.11	yes	12
4.3	odd	2	inner	2736.2.d.a.2015.2	yes	12
12.11	even	2	inner	2736.2.d.a.2015.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.d.a.2015.1	✓	12	1.1	even	1	trivial
2736.2.d.a.2015.2	yes	12	4.3	odd	2	inner
2736.2.d.a.2015.11	yes	12	3.2	odd	2	inner
2736.2.d.a.2015.12	yes	12	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.55961i q^{5} -3.63675i q^{7} +O(q^{10})\)	\(q-3.55961i q^{5} -3.63675i q^{7} -4.12147 q^{11} -1.20541 q^{13} -1.58353i q^{17} -1.00000i q^{19} -3.68078 q^{23} -7.67079 q^{25} -0.561860i q^{29} +5.20541i q^{31} -12.9454 q^{35} +9.27349 q^{37} -3.72892i q^{41} -9.46538i q^{43} -6.33989 q^{47} -6.22593 q^{49} +11.9719i q^{53} +14.6708i q^{55} +10.2863 q^{59} +2.60270 q^{61} +4.29078i q^{65} -2.06808i q^{67} -7.36156 q^{71} -0.363253 q^{73} +14.9887i q^{77} +11.6573i q^{79} -3.68078 q^{83} -5.63675 q^{85} -6.79970i q^{89} +4.38377i q^{91} -3.55961 q^{95} +8.86267 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 4 q^{25} + 8 q^{37} - 52 q^{49} + 24 q^{61} - 56 q^{73} - 16 q^{85} + 32 q^{97}+O(q^{100}) \) 12 * q + 4 * q^25 + 8 * q^37 - 52 * q^49 + 24 * q^61 - 56 * q^73 - 16 * q^85 + 32 * q^97

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