LMFDB - Embedded newform 5472.2.d.g.2015.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5472,2,Mod(2015,5472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5472.2015"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 5x^{12} + 4x^{10} - 20x^{8} + 16x^{6} + 80x^{4} - 256x^{2} + 256 $ x^16 - 4x^14 + 5x^12 + 4x^10 - 20x^8 + 16x^6 + 80x^4 - 256*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2015.13
Root			$0.310269 + 1.37976i$ of defining polynomial
Character	$\chi$	$=$	5472.2015
Dual form			5472.2.d.g.2015.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.62563i q^{5} -0.712386i q^{7} -3.62563 q^{11} -3.02497 q^{13} +5.59147i q^{17} +1.00000i q^{19} -2.37259 q^{23} -8.14521 q^{25} +1.44953i q^{29} -1.02497i q^{31} +2.58285 q^{35} -3.02497 q^{37} +2.01493i q^{41} -7.78986i q^{43} +6.97037 q^{47} +6.49251 q^{49} -2.40790i q^{53} -13.1452i q^{55} -5.80173 q^{59} -11.8398 q^{61} -10.9674i q^{65} -5.42477i q^{67} +1.87005 q^{71} +5.49251 q^{73} +2.58285i q^{77} -7.07492i q^{79} +5.20102 q^{83} -20.2726 q^{85} -2.68355i q^{89} +2.15495i q^{91} -3.62563 q^{95} +12.9850 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{13} - 48 q^{25} - 16 q^{37} + 8 q^{49} - 80 q^{61} - 8 q^{73} - 152 q^{85} + 16 q^{97}+O(q^{100}) $ 16 * q - 16 * q^13 - 48 * q^25 - 16 * q^37 + 8 * q^49 - 80 * q^61 - 8 * q^73 - 152 * q^85 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	3.62563i	1.62143i	0.585440	+	0.810716i	$0.300922\pi$
$5$	3.62563i	1.62143i	−0.585440	+	0.810716i	$0.699078\pi$
$6$	0	0
$7$	− 0.712386i	− 0.269257i	−0.990896	−	0.134628i	$-0.957016\pi$
$7$	− 0.712386i	− 0.269257i	0.990896	−	0.134628i	$-0.0429840\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.62563	−1.09317	−0.546585	−	0.837404i	$-0.684072\pi$
$11$	−3.62563	−1.09317	−0.546585	+	0.837404i	$0.684072\pi$
$12$	0	0
$13$	−3.02497	−0.838976	−0.419488	−	0.907761i	$-0.637790\pi$
$13$	−3.02497	−0.838976	−0.419488	+	0.907761i	$0.637790\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.59147i	1.35613i	0.735001	+	0.678066i	$0.237182\pi$
$17$	5.59147i	1.35613i	−0.735001	+	0.678066i	$0.762818\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.37259	−0.494719	−0.247359	−	0.968924i	$-0.579563\pi$
$23$	−2.37259	−0.494719	−0.247359	+	0.968924i	$0.579563\pi$
$24$	0	0
$25$	−8.14521	−1.62904
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.44953i	0.269171i	0.990902	+	0.134585i	$0.0429703\pi$
$29$	1.44953i	0.269171i	−0.990902	+	0.134585i	$0.957030\pi$
$30$	0	0
$31$	− 1.02497i	− 0.184090i	−0.995755	−	0.0920452i	$-0.970660\pi$
$31$	− 1.02497i	− 0.184090i	0.995755	−	0.0920452i	$-0.0293404\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.58285	0.436581
$36$	0	0
$37$	−3.02497	−0.497302	−0.248651	−	0.968593i	$-0.579987\pi$
$37$	−3.02497	−0.497302	−0.248651	+	0.968593i	$0.579987\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.01493i	0.314680i	0.987545	+	0.157340i	$0.0502918\pi$
$41$	2.01493i	0.314680i	−0.987545	+	0.157340i	$0.949708\pi$
$42$	0	0
$43$	− 7.78986i	− 1.18794i	−0.804486	−	0.593971i	$-0.797559\pi$
$43$	− 7.78986i	− 1.18794i	0.804486	−	0.593971i	$-0.202441\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.97037	1.01673	0.508367	−	0.861141i	$-0.330249\pi$
$47$	6.97037	1.01673	0.508367	+	0.861141i	$0.330249\pi$
$48$	0	0
$49$	6.49251	0.927501
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.40790i	− 0.330751i	−0.986231	−	0.165376i	$-0.947116\pi$
$53$	− 2.40790i	− 0.330751i	0.986231	−	0.165376i	$-0.0528836\pi$
$54$	0	0
$55$	− 13.1452i	− 1.77250i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.80173	−0.755322	−0.377661	−	0.925944i	$-0.623271\pi$
$59$	−5.80173	−0.755322	−0.377661	+	0.925944i	$0.623271\pi$
$60$	0	0
$61$	−11.8398	−1.51593	−0.757966	−	0.652294i	$-0.773807\pi$
$61$	−11.8398	−1.51593	−0.757966	+	0.652294i	$0.773807\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 10.9674i	− 1.36034i
$66$	0	0
$67$	− 5.42477i	− 0.662741i	−0.943501	−	0.331371i	$-0.892489\pi$
$67$	− 5.42477i	− 0.662741i	0.943501	−	0.331371i	$-0.107511\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.87005	0.221934	0.110967	−	0.993824i	$-0.464605\pi$
$71$	1.87005	0.221934	0.110967	+	0.993824i	$0.464605\pi$
$72$	0	0
$73$	5.49251	0.642849	0.321425	−	0.946935i	$-0.395838\pi$
$73$	5.49251	0.642849	0.321425	+	0.946935i	$0.395838\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.58285i	0.294343i
$78$	0	0
$79$	− 7.07492i	− 0.795990i	−0.917388	−	0.397995i	$-0.869706\pi$
$79$	− 7.07492i	− 0.795990i	0.917388	−	0.397995i	$-0.130294\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.20102	0.570886	0.285443	−	0.958396i	$-0.407859\pi$
$83$	5.20102	0.570886	0.285443	+	0.958396i	$0.407859\pi$
$84$	0	0
$85$	−20.2726	−2.19887
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 2.68355i	− 0.284455i	−0.989834	−	0.142228i	$-0.954573\pi$
$89$	− 2.68355i	− 0.284455i	0.989834	−	0.142228i	$-0.0454265\pi$
$90$	0	0
$91$	2.15495i	0.225900i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.62563	−0.371982
$96$	0	0
$97$	12.9850	1.31843	0.659214	−	0.751955i	$-0.270889\pi$
$97$	12.9850	1.31843	0.659214	+	0.751955i	$0.270889\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 4.80804i	− 0.478418i	−0.970968	−	0.239209i	$-0.923112\pi$
$101$	− 4.80804i	− 0.478418i	0.970968	−	0.239209i	$-0.0768881\pi$
$102$	0	0
$103$	4.53527i	0.446873i	0.974718	+	0.223437i	$0.0717276\pi$
$103$	4.53527i	0.446873i	−0.974718	+	0.223437i	$0.928272\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.3220	−1.86792	−0.933962	−	0.357372i	$-0.883673\pi$
$107$	−19.3220	−1.86792	−0.933962	+	0.357372i	$0.883673\pi$
$108$	0	0
$109$	−8.44974	−0.809339	−0.404669	−	0.914463i	$-0.632613\pi$
$109$	−8.44974	−0.809339	−0.404669	+	0.914463i	$0.632613\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.91675i	0.180313i	0.995928	+	0.0901564i	$0.0287367\pi$
$113$	1.91675i	0.180313i	−0.995928	+	0.0901564i	$0.971263\pi$
$114$	0	0
$115$	− 8.60213i	− 0.802153i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.98329	0.365147
$120$	0	0
$121$	2.14521	0.195019
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.4034i	− 1.01995i
$126$	0	0
$127$	− 2.90560i	− 0.257831i	−0.991656	−	0.128915i	$-0.958850\pi$
$127$	− 2.90560i	− 0.257831i	0.991656	−	0.128915i	$-0.0411495\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	19.8785	1.73679	0.868396	−	0.495872i	$-0.165151\pi$
$131$	19.8785	1.73679	0.868396	+	0.495872i	$0.165151\pi$
$132$	0	0
$133$	0.712386	0.0617717
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 7.83296i	− 0.669215i	−0.942358	−	0.334607i	$-0.891396\pi$
$137$	− 7.83296i	− 0.669215i	0.942358	−	0.334607i	$-0.108604\pi$
$138$	0	0
$139$	12.5275i	1.06257i	0.847195	+	0.531283i	$0.178290\pi$
$139$	12.5275i	1.06257i	−0.847195	+	0.531283i	$0.821710\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.9674	0.917143
$144$	0	0
$145$	−5.25546	−0.436442
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.56247i	0.373772i	0.982382	+	0.186886i	$0.0598395\pi$
$149$	4.56247i	0.373772i	−0.982382	+	0.186886i	$0.940161\pi$
$150$	0	0
$151$	− 14.5852i	− 1.18693i	−0.804861	−	0.593464i	$-0.797760\pi$
$151$	− 14.5852i	− 1.18693i	0.804861	−	0.593464i	$-0.202240\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.71617	0.298490
$156$	0	0
$157$	−19.6102	−1.56506	−0.782532	−	0.622611i	$-0.786072\pi$
$157$	−19.6102	−1.56506	−0.782532	+	0.622611i	$0.786072\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.69020i	0.133206i
$162$	0	0
$163$	− 7.51030i	− 0.588252i	−0.955767	−	0.294126i	$-0.904972\pi$
$163$	− 7.51030i	− 0.588252i	0.955767	−	0.294126i	$-0.0950285\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.0707	−0.934058	−0.467029	−	0.884242i	$-0.654676\pi$
$167$	−12.0707	−0.934058	−0.467029	+	0.884242i	$0.654676\pi$
$168$	0	0
$169$	−3.84954	−0.296119
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.60934i	− 0.274413i	−0.990542	−	0.137207i	$-0.956188\pi$
$173$	− 3.60934i	− 0.274413i	0.990542	−	0.137207i	$-0.0438124\pi$
$174$	0	0
$175$	5.80253i	0.438630i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	23.8719	1.78427	0.892136	−	0.451767i	$-0.149206\pi$
$179$	23.8719	1.78427	0.892136	+	0.451767i	$0.149206\pi$
$180$	0	0
$181$	−3.09440	−0.230005	−0.115002	−	0.993365i	$-0.536688\pi$
$181$	−3.09440	−0.230005	−0.115002	+	0.993365i	$0.536688\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 10.9674i	− 0.806342i
$186$	0	0
$187$	− 20.2726i	− 1.48248i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.1533	−1.31353	−0.656764	−	0.754096i	$-0.728075\pi$
$191$	−18.1533	−1.31353	−0.656764	+	0.754096i	$0.728075\pi$
$192$	0	0
$193$	11.4552	0.824566	0.412283	−	0.911056i	$-0.364732\pi$
$193$	11.4552	0.824566	0.412283	+	0.911056i	$0.364732\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.8543i	− 1.62830i	−0.580653	−	0.814151i	$-0.697203\pi$
$197$	− 22.8543i	− 1.62830i	0.580653	−	0.814151i	$-0.302797\pi$
$198$	0	0
$199$	14.8748i	1.05444i	0.849728	+	0.527222i	$0.176766\pi$
$199$	14.8748i	1.05444i	−0.849728	+	0.527222i	$0.823234\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.03262	0.0724760
$204$	0	0
$205$	−7.30540	−0.510232
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 3.62563i	− 0.250790i
$210$	0	0
$211$	− 27.7457i	− 1.91009i	−0.296462	−	0.955045i	$-0.595807\pi$
$211$	− 27.7457i	− 1.91009i	0.296462	−	0.955045i	$-0.404193\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	28.2432	1.92617
$216$	0	0
$217$	−0.730176	−0.0495676
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 16.9141i	− 1.13776i
$222$	0	0
$223$	5.85504i	0.392082i	0.980596	+	0.196041i	$0.0628086\pi$
$223$	5.85504i	0.392082i	−0.980596	+	0.196041i	$0.937191\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.91675	−0.127219	−0.0636096	−	0.997975i	$-0.520261\pi$
$227$	−1.91675	−0.127219	−0.0636096	+	0.997975i	$0.520261\pi$
$228$	0	0
$229$	−2.17968	−0.144037	−0.0720186	−	0.997403i	$-0.522944\pi$
$229$	−2.17968	−0.144037	−0.0720186	+	0.997403i	$0.522944\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 5.72223i	− 0.374876i	−0.982276	−	0.187438i	$-0.939982\pi$
$233$	− 5.72223i	− 0.374876i	0.982276	−	0.187438i	$-0.0600184\pi$
$234$	0	0
$235$	25.2720i	1.64856i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.6748	1.01392	0.506959	−	0.861970i	$-0.330770\pi$
$239$	15.6748	1.01392	0.506959	+	0.861970i	$0.330770\pi$
$240$	0	0
$241$	−13.3359	−0.859039	−0.429519	−	0.903058i	$-0.641317\pi$
$241$	−13.3359	−0.859039	−0.429519	+	0.903058i	$0.641317\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	23.5394i	1.50388i
$246$	0	0
$247$	− 3.02497i	− 0.192474i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	13.9245	0.878904	0.439452	−	0.898266i	$-0.355173\pi$
$251$	13.9245	0.878904	0.439452	+	0.898266i	$0.355173\pi$
$252$	0	0
$253$	8.60213	0.540812
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.2246i	0.637791i	0.947790	+	0.318896i	$0.103312\pi$
$257$	10.2246i	0.637791i	−0.947790	+	0.318896i	$0.896688\pi$
$258$	0	0
$259$	2.15495i	0.133902i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.5365	0.834697	0.417348	−	0.908747i	$-0.362959\pi$
$263$	13.5365	0.834697	0.417348	+	0.908747i	$0.362959\pi$
$264$	0	0
$265$	8.73018	0.536290
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 15.9557i	− 0.972835i	−0.873727	−	0.486417i	$-0.838303\pi$
$269$	− 15.9557i	− 0.972835i	0.873727	−	0.486417i	$-0.161697\pi$
$270$	0	0
$271$	29.3254i	1.78139i	0.454602	+	0.890695i	$0.349782\pi$
$271$	29.3254i	1.78139i	−0.454602	+	0.890695i	$0.650218\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	29.5315	1.78082
$276$	0	0
$277$	6.55050	0.393581	0.196791	−	0.980446i	$-0.436948\pi$
$277$	6.55050	0.393581	0.196791	+	0.980446i	$0.436948\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 27.4106i	− 1.63518i	−0.575799	−	0.817591i	$-0.695309\pi$
$281$	− 27.4106i	− 1.63518i	0.575799	−	0.817591i	$-0.304691\pi$
$282$	0	0
$283$	10.2496i	0.609275i	0.952468	+	0.304637i	$0.0985353\pi$
$283$	10.2496i	0.609275i	−0.952468	+	0.304637i	$0.901465\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.43541	0.0847296
$288$	0	0
$289$	−14.2646	−0.839093
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.1421i	0.826192i	0.910687	+	0.413096i	$0.135553\pi$
$293$	14.1421i	0.826192i	−0.910687	+	0.413096i	$0.864447\pi$
$294$	0	0
$295$	− 21.0350i	− 1.22470i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.17701	0.415057
$300$	0	0
$301$	−5.54939	−0.319861
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 42.9268i	− 2.45798i
$306$	0	0
$307$	1.28592i	0.0733915i	0.999326	+	0.0366957i	$0.0116832\pi$
$307$	1.28592i	0.0733915i	−0.999326	+	0.0366957i	$0.988317\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.34334	−0.246288	−0.123144	−	0.992389i	$-0.539298\pi$
$311$	−4.34334	−0.246288	−0.123144	+	0.992389i	$0.539298\pi$
$312$	0	0
$313$	−0.0998878	−0.00564599	−0.00282300	−	0.999996i	$-0.500899\pi$
$313$	−0.0998878	−0.00564599	−0.00282300	+	0.999996i	$0.500899\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.4759i	1.65553i	0.561074	+	0.827766i	$0.310388\pi$
$317$	29.4759i	1.65553i	−0.561074	+	0.827766i	$0.689612\pi$
$318$	0	0
$319$	− 5.25546i	− 0.294249i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.59147	−0.311118
$324$	0	0
$325$	24.6390	1.36673
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 4.96560i	− 0.273762i
$330$	0	0
$331$	− 8.62517i	− 0.474082i	−0.971500	−	0.237041i	$-0.923822\pi$
$331$	− 8.62517i	− 0.474082i	0.971500	−	0.237041i	$-0.0761776\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	19.6682	1.07459
$336$	0	0
$337$	2.06942	0.112729	0.0563644	−	0.998410i	$-0.482049\pi$
$337$	2.06942	0.112729	0.0563644	+	0.998410i	$0.482049\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.71617i	0.201242i
$342$	0	0
$343$	− 9.61187i	− 0.518992i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.9319	−0.747902	−0.373951	−	0.927448i	$-0.621997\pi$
$347$	−13.9319	−0.747902	−0.373951	+	0.927448i	$0.621997\pi$
$348$	0	0
$349$	−6.30260	−0.337371	−0.168685	−	0.985670i	$-0.553952\pi$
$349$	−6.30260	−0.337371	−0.168685	+	0.985670i	$0.553952\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 34.1594i	− 1.81812i	−0.416665	−	0.909060i	$-0.636801\pi$
$353$	− 34.1594i	− 1.81812i	0.416665	−	0.909060i	$-0.363199\pi$
$354$	0	0
$355$	6.78012i	0.359851i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.0288	1.58486	0.792430	−	0.609963i	$-0.208816\pi$
$359$	30.0288	1.58486	0.792430	+	0.609963i	$0.208816\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	19.9138i	1.04234i
$366$	0	0
$367$	− 17.1543i	− 0.895448i	−0.894172	−	0.447724i	$-0.852235\pi$
$367$	− 17.1543i	− 0.895448i	0.894172	−	0.447724i	$-0.147765\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.71536	−0.0890569
$372$	0	0
$373$	−31.9905	−1.65641	−0.828203	−	0.560429i	$-0.810636\pi$
$373$	−31.9905	−1.65641	−0.828203	+	0.560429i	$0.810636\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.38479i	− 0.225828i
$378$	0	0
$379$	− 28.4792i	− 1.46288i	−0.681906	−	0.731439i	$-0.738849\pi$
$379$	− 28.4792i	− 1.46288i	0.681906	−	0.731439i	$-0.261151\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−34.6655	−1.77133	−0.885663	−	0.464329i	$-0.846296\pi$
$383$	−34.6655	−1.77133	−0.885663	+	0.464329i	$0.846296\pi$
$384$	0	0
$385$	−9.36446	−0.477257
$386$	0	0
$387$	0	0
$388$	0	0
$389$	33.1470i	1.68062i	0.542106	+	0.840310i	$0.317627\pi$
$389$	33.1470i	1.68062i	−0.542106	+	0.840310i	$0.682373\pi$
$390$	0	0
$391$	− 13.2663i	− 0.670904i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	25.6510	1.29064
$396$	0	0
$397$	12.4115	0.622914	0.311457	−	0.950260i	$-0.399183\pi$
$397$	12.4115	0.622914	0.311457	+	0.950260i	$0.399183\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 25.2269i	− 1.25977i	−0.776688	−	0.629886i	$-0.783102\pi$
$401$	− 25.2269i	− 1.25977i	0.776688	−	0.629886i	$-0.216898\pi$
$402$	0	0
$403$	3.10051i	0.154447i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.9674	0.543636
$408$	0	0
$409$	−38.3247	−1.89504	−0.947518	−	0.319701i	$-0.896417\pi$
$409$	−38.3247	−1.89504	−0.947518	+	0.319701i	$0.896417\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.13308i	0.203375i
$414$	0	0
$415$	18.8570i	0.925652i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.16025	0.447507	0.223754	−	0.974646i	$-0.428169\pi$
$419$	9.16025	0.447507	0.223754	+	0.974646i	$0.428169\pi$
$420$	0	0
$421$	−21.3154	−1.03885	−0.519424	−	0.854517i	$-0.673854\pi$
$421$	−21.3154	−1.03885	−0.519424	+	0.854517i	$0.673854\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 45.5437i	− 2.20919i
$426$	0	0
$427$	8.43451i	0.408175i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−40.5379	−1.95264	−0.976321	−	0.216328i	$-0.930592\pi$
$431$	−40.5379	−1.95264	−0.976321	+	0.216328i	$0.930592\pi$
$432$	0	0
$433$	34.0894	1.63823	0.819116	−	0.573628i	$-0.194465\pi$
$433$	34.0894	1.63823	0.819116	+	0.573628i	$0.194465\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.37259i	− 0.113496i
$438$	0	0
$439$	− 33.4652i	− 1.59721i	−0.601857	−	0.798604i	$-0.705572\pi$
$439$	− 33.4652i	− 1.59721i	0.601857	−	0.798604i	$-0.294428\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−28.0139	−1.33098	−0.665490	−	0.746407i	$-0.731777\pi$
$443$	−28.0139	−1.33098	−0.665490	+	0.746407i	$0.731777\pi$
$444$	0	0
$445$	9.72955	0.461225
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.925796i	0.0436910i	0.999761	+	0.0218455i	$0.00695419\pi$
$449$	0.925796i	0.0436910i	−0.999761	+	0.0218455i	$0.993046\pi$
$450$	0	0
$451$	− 7.30540i	− 0.343998i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.81305	−0.366281
$456$	0	0
$457$	−20.6164	−0.964393	−0.482197	−	0.876063i	$-0.660161\pi$
$457$	−20.6164	−0.964393	−0.482197	+	0.876063i	$0.660161\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.1482i	0.798673i	0.916804	+	0.399337i	$0.130760\pi$
$461$	17.1482i	0.798673i	−0.916804	+	0.399337i	$0.869240\pi$
$462$	0	0
$463$	− 17.1652i	− 0.797733i	−0.917009	−	0.398867i	$-0.869404\pi$
$463$	− 17.1652i	− 0.797733i	0.917009	−	0.398867i	$-0.130596\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.0226	0.602614	0.301307	−	0.953527i	$-0.402577\pi$
$467$	13.0226	0.602614	0.301307	+	0.953527i	$0.402577\pi$
$468$	0	0
$469$	−3.86453	−0.178448
$470$	0	0
$471$	0	0
$472$	0	0
$473$	28.2432i	1.29862i
$474$	0	0
$475$	− 8.14521i	− 0.373728i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.21902	0.0556983	0.0278491	−	0.999612i	$-0.491134\pi$
$479$	1.21902	0.0556983	0.0278491	+	0.999612i	$0.491134\pi$
$480$	0	0
$481$	9.15046	0.417225
$482$	0	0
$483$	0	0
$484$	0	0
$485$	47.0789i	2.13774i
$486$	0	0
$487$	− 12.8400i	− 0.581838i	−0.956748	−	0.290919i	$-0.906039\pi$
$487$	− 12.8400i	− 0.581838i	0.956748	−	0.290919i	$-0.0939610\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−36.8932	−1.66497	−0.832484	−	0.554049i	$-0.813082\pi$
$491$	−36.8932	−1.66497	−0.832484	+	0.554049i	$0.813082\pi$
$492$	0	0
$493$	−8.10500	−0.365031
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.33220i	− 0.0597573i
$498$	0	0
$499$	− 19.7279i	− 0.883140i	−0.897227	−	0.441570i	$-0.854422\pi$
$499$	− 19.7279i	− 0.883140i	0.897227	−	0.441570i	$-0.145578\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7.66905	−0.341946	−0.170973	−	0.985276i	$-0.554691\pi$
$503$	−7.66905	−0.341946	−0.170973	+	0.985276i	$0.554691\pi$
$504$	0	0
$505$	17.4322	0.775723
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3.58541i	− 0.158920i	−0.996838	−	0.0794602i	$-0.974680\pi$
$509$	− 3.58541i	− 0.158920i	0.996838	−	0.0794602i	$-0.0253197\pi$
$510$	0	0
$511$	− 3.91279i	− 0.173091i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.4432	−0.724574
$516$	0	0
$517$	−25.2720	−1.11146
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.03580i	0.352055i	0.984385	+	0.176027i	$0.0563247\pi$
$521$	8.03580i	0.352055i	−0.984385	+	0.176027i	$0.943675\pi$
$522$	0	0
$523$	38.9000i	1.70098i	0.525994	+	0.850489i	$0.323694\pi$
$523$	38.9000i	1.70098i	−0.525994	+	0.850489i	$0.676306\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.73110	0.249651
$528$	0	0
$529$	−17.3708	−0.755253
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 6.09511i	− 0.264009i
$534$	0	0
$535$	− 70.0543i	− 3.02871i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−23.5394	−1.01392
$540$	0	0
$541$	−18.9477	−0.814627	−0.407314	−	0.913288i	$-0.633534\pi$
$541$	−18.9477	−0.814627	−0.407314	+	0.913288i	$0.633534\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 30.6357i	− 1.31229i
$546$	0	0
$547$	− 27.1543i	− 1.16104i	−0.814248	−	0.580518i	$-0.802850\pi$
$547$	− 27.1543i	− 1.16104i	0.814248	−	0.580518i	$-0.197150\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.44953	−0.0617520
$552$	0	0
$553$	−5.04007	−0.214326
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.2083i	0.898624i	0.893375	+	0.449312i	$0.148331\pi$
$557$	21.2083i	0.898624i	−0.893375	+	0.449312i	$0.851669\pi$
$558$	0	0
$559$	23.5641i	0.996656i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15.9521	−0.672299	−0.336150	−	0.941809i	$-0.609125\pi$
$563$	−15.9521	−0.672299	−0.336150	+	0.941809i	$0.609125\pi$
$564$	0	0
$565$	−6.94943	−0.292365
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 32.2576i	− 1.35231i	−0.736759	−	0.676155i	$-0.763645\pi$
$569$	− 32.2576i	− 1.35231i	0.736759	−	0.676155i	$-0.236355\pi$
$570$	0	0
$571$	− 35.1243i	− 1.46991i	−0.678117	−	0.734954i	$-0.737204\pi$
$571$	− 35.1243i	− 1.46991i	0.678117	−	0.734954i	$-0.262796\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	19.3252	0.805918
$576$	0	0
$577$	−24.3956	−1.01560	−0.507800	−	0.861475i	$-0.669541\pi$
$577$	−24.3956	−1.01560	−0.507800	+	0.861475i	$0.669541\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 3.70513i	− 0.153715i
$582$	0	0
$583$	8.73018i	0.361567i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.49483	0.102972	0.0514862	−	0.998674i	$-0.483604\pi$
$587$	2.49483	0.102972	0.0514862	+	0.998674i	$0.483604\pi$
$588$	0	0
$589$	1.02497	0.0422332
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.7502i	0.811045i	0.914085	+	0.405523i	$0.132910\pi$
$593$	19.7502i	0.811045i	−0.914085	+	0.405523i	$0.867090\pi$
$594$	0	0
$595$	14.4419i	0.592062i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17.6392	−0.720717	−0.360359	−	0.932814i	$-0.617346\pi$
$599$	−17.6392	−0.720717	−0.360359	+	0.932814i	$0.617346\pi$
$600$	0	0
$601$	32.6308	1.33104	0.665519	−	0.746381i	$-0.268210\pi$
$601$	32.6308	1.33104	0.665519	+	0.746381i	$0.268210\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.77773i	0.316210i
$606$	0	0
$607$	22.3109i	0.905571i	0.891619	+	0.452786i	$0.149570\pi$
$607$	22.3109i	0.905571i	−0.891619	+	0.452786i	$0.850430\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.0852	−0.853015
$612$	0	0
$613$	33.4305	1.35025	0.675123	−	0.737705i	$-0.264091\pi$
$613$	33.4305	1.35025	0.675123	+	0.737705i	$0.264091\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.5157i	1.47007i	0.678032	+	0.735033i	$0.262833\pi$
$617$	36.5157i	1.47007i	−0.678032	+	0.735033i	$0.737167\pi$
$618$	0	0
$619$	− 8.30990i	− 0.334003i	−0.985957	−	0.167001i	$-0.946592\pi$
$619$	− 8.30990i	− 0.334003i	0.985957	−	0.167001i	$-0.0534085\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.91172	−0.0765915
$624$	0	0
$625$	0.618370	0.0247348
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 16.9141i	− 0.674407i
$630$	0	0
$631$	22.8978i	0.911547i	0.890096	+	0.455773i	$0.150637\pi$
$631$	22.8978i	0.911547i	−0.890096	+	0.455773i	$0.849363\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.5346	0.418055
$636$	0	0
$637$	−19.6396	−0.778151
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.0028i	1.46152i	0.682633	+	0.730761i	$0.260835\pi$
$641$	37.0028i	1.46152i	−0.682633	+	0.730761i	$0.739165\pi$
$642$	0	0
$643$	21.3196i	0.840764i	0.907347	+	0.420382i	$0.138104\pi$
$643$	21.3196i	0.840764i	−0.907347	+	0.420382i	$0.861896\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.7225	−0.736058	−0.368029	−	0.929814i	$-0.619967\pi$
$647$	−18.7225	−0.736058	−0.368029	+	0.929814i	$0.619967\pi$
$648$	0	0
$649$	21.0350	0.825694
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.2472i	1.37933i	0.724128	+	0.689665i	$0.242242\pi$
$653$	35.2472i	1.37933i	−0.724128	+	0.689665i	$0.757758\pi$
$654$	0	0
$655$	72.0721i	2.81609i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.70944	0.339272	0.169636	−	0.985507i	$-0.445741\pi$
$659$	8.70944	0.339272	0.169636	+	0.985507i	$0.445741\pi$
$660$	0	0
$661$	28.8951	1.12389	0.561945	−	0.827175i	$-0.310053\pi$
$661$	28.8951	1.12389	0.561945	+	0.827175i	$0.310053\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.58285i	0.100159i
$666$	0	0
$667$	− 3.43914i	− 0.133164i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	42.9268	1.65717
$672$	0	0
$673$	18.5557	0.715271	0.357636	−	0.933861i	$-0.383583\pi$
$673$	18.5557	0.715271	0.357636	+	0.933861i	$0.383583\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.8964i	1.14901i	0.818500	+	0.574506i	$0.194806\pi$
$677$	29.8964i	1.14901i	−0.818500	+	0.574506i	$0.805194\pi$
$678$	0	0
$679$	− 9.25034i	− 0.354996i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−44.0313	−1.68481	−0.842406	−	0.538844i	$-0.818861\pi$
$683$	−44.0313	−1.68481	−0.842406	+	0.538844i	$0.818861\pi$
$684$	0	0
$685$	28.3994	1.08509
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.28384i	0.277492i
$690$	0	0
$691$	47.0102i	1.78835i	0.447714	+	0.894177i	$0.352238\pi$
$691$	47.0102i	1.78835i	−0.447714	+	0.894177i	$0.647762\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−45.4200	−1.72288
$696$	0	0
$697$	−11.2664	−0.426747
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.9199i	0.827902i	0.910299	+	0.413951i	$0.135852\pi$
$701$	21.9199i	0.827902i	−0.910299	+	0.413951i	$0.864148\pi$
$702$	0	0
$703$	− 3.02497i	− 0.114089i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.42518	−0.128817
$708$	0	0
$709$	18.2709	0.686179	0.343090	−	0.939303i	$-0.388527\pi$
$709$	18.2709	0.686179	0.343090	+	0.939303i	$0.388527\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.43184i	0.0910730i
$714$	0	0
$715$	39.7639i	1.48708i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	13.9067	0.518633	0.259316	−	0.965792i	$-0.416503\pi$
$719$	13.9067	0.518633	0.259316	+	0.965792i	$0.416503\pi$
$720$	0	0
$721$	3.23086	0.120324
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 11.8067i	− 0.438490i
$726$	0	0
$727$	− 21.8754i	− 0.811313i	−0.914026	−	0.405657i	$-0.867043\pi$
$727$	− 21.8754i	− 0.811313i	0.914026	−	0.405657i	$-0.132957\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	43.5568	1.61101
$732$	0	0
$733$	−11.3093	−0.417718	−0.208859	−	0.977946i	$-0.566975\pi$
$733$	−11.3093	−0.417718	−0.208859	+	0.977946i	$0.566975\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.6682i	0.724489i
$738$	0	0
$739$	34.0877i	1.25394i	0.779045	+	0.626968i	$0.215704\pi$
$739$	34.0877i	1.25394i	−0.779045	+	0.626968i	$0.784296\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19.6958	−0.722568	−0.361284	−	0.932456i	$-0.617662\pi$
$743$	−19.6958	−0.722568	−0.361284	+	0.932456i	$0.617662\pi$
$744$	0	0
$745$	−16.5418	−0.606046
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13.7647i	0.502951i
$750$	0	0
$751$	− 20.2149i	− 0.737651i	−0.929499	−	0.368826i	$-0.879760\pi$
$751$	− 20.2149i	− 0.737651i	0.929499	−	0.368826i	$-0.120240\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	52.8806	1.92452
$756$	0	0
$757$	−36.0308	−1.30956	−0.654781	−	0.755818i	$-0.727239\pi$
$757$	−36.0308	−1.30956	−0.654781	+	0.755818i	$0.727239\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.7030i	0.931733i	0.884855	+	0.465866i	$0.154257\pi$
$761$	25.7030i	0.931733i	−0.884855	+	0.465866i	$0.845743\pi$
$762$	0	0
$763$	6.01948i	0.217920i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17.5501	0.633697
$768$	0	0
$769$	−37.5745	−1.35497	−0.677485	−	0.735537i	$-0.736930\pi$
$769$	−37.5745	−1.35497	−0.677485	+	0.735537i	$0.736930\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 42.1410i	− 1.51570i	−0.652426	−	0.757852i	$-0.726249\pi$
$773$	− 42.1410i	− 1.51570i	0.652426	−	0.757852i	$-0.273751\pi$
$774$	0	0
$775$	8.34861i	0.299891i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.01493	−0.0721924
$780$	0	0
$781$	−6.78012	−0.242612
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 71.0993i	− 2.53764i
$786$	0	0
$787$	− 43.7352i	− 1.55899i	−0.626408	−	0.779495i	$-0.715476\pi$
$787$	− 43.7352i	− 1.55899i	0.626408	−	0.779495i	$-0.284524\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.36547	0.0485504
$792$	0	0
$793$	35.8151	1.27183
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 29.0518i	− 1.02907i	−0.857471	−	0.514533i	$-0.827965\pi$
$797$	− 29.0518i	− 1.02907i	0.857471	−	0.514533i	$-0.172035\pi$
$798$	0	0
$799$	38.9746i	1.37882i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−19.9138	−0.702743
$804$	0	0
$805$	−6.12804	−0.215985
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 17.3434i	− 0.609763i	−0.952390	−	0.304881i	$-0.901383\pi$
$809$	− 17.3434i	− 0.609763i	0.952390	−	0.304881i	$-0.0986169\pi$
$810$	0	0
$811$	− 8.37082i	− 0.293939i	−0.989141	−	0.146970i	$-0.953048\pi$
$811$	− 8.37082i	− 0.293939i	0.989141	−	0.146970i	$-0.0469520\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	27.2296	0.953810
$816$	0	0
$817$	7.78986	0.272533
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 8.62252i	− 0.300928i	−0.988615	−	0.150464i	$-0.951923\pi$
$821$	− 8.62252i	− 0.300928i	0.988615	−	0.150464i	$-0.0480768\pi$
$822$	0	0
$823$	− 24.7324i	− 0.862116i	−0.902324	−	0.431058i	$-0.858140\pi$
$823$	− 24.7324i	− 0.862116i	0.902324	−	0.431058i	$-0.141860\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.8072	0.549669	0.274835	−	0.961491i	$-0.411377\pi$
$827$	15.8072	0.549669	0.274835	+	0.961491i	$0.411377\pi$
$828$	0	0
$829$	5.14607	0.178731	0.0893653	−	0.995999i	$-0.471516\pi$
$829$	5.14607	0.178731	0.0893653	+	0.995999i	$0.471516\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	36.3027i	1.25781i
$834$	0	0
$835$	− 43.7639i	− 1.51451i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.5782	−0.503295	−0.251647	−	0.967819i	$-0.580972\pi$
$839$	−14.5782	−0.503295	−0.251647	+	0.967819i	$0.580972\pi$
$840$	0	0
$841$	26.8989	0.927547
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 13.9570i	− 0.480137i
$846$	0	0
$847$	− 1.52822i	− 0.0525101i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.17701	0.246025
$852$	0	0
$853$	−47.0905	−1.61235	−0.806174	−	0.591678i	$-0.798466\pi$
$853$	−47.0905	−1.61235	−0.806174	+	0.591678i	$0.798466\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 12.4133i	− 0.424032i	−0.977266	−	0.212016i	$-0.931997\pi$
$857$	− 12.4133i	− 0.424032i	0.977266	−	0.212016i	$-0.0680029\pi$
$858$	0	0
$859$	3.49251i	0.119163i	0.998223	+	0.0595814i	$0.0189766\pi$
$859$	3.49251i	0.119163i	−0.998223	+	0.0595814i	$0.981023\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.4899	−0.561323	−0.280662	−	0.959807i	$-0.590554\pi$
$863$	−16.4899	−0.561323	−0.280662	+	0.959807i	$0.590554\pi$
$864$	0	0
$865$	13.0861	0.444942
$866$	0	0
$867$	0	0
$868$	0	0
$869$	25.6510i	0.870152i
$870$	0	0
$871$	16.4098i	0.556024i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.12360	−0.274628
$876$	0	0
$877$	30.6247	1.03412	0.517061	−	0.855949i	$-0.327026\pi$
$877$	30.6247	1.03412	0.517061	+	0.855949i	$0.327026\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 26.2241i	− 0.883512i	−0.897135	−	0.441756i	$-0.854356\pi$
$881$	− 26.2241i	− 0.883512i	0.897135	−	0.441756i	$-0.145644\pi$
$882$	0	0
$883$	19.0471i	0.640987i	0.947251	+	0.320494i	$0.103849\pi$
$883$	19.0471i	0.640987i	−0.947251	+	0.320494i	$0.896151\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−54.6058	−1.83348	−0.916741	−	0.399481i	$-0.869190\pi$
$887$	−54.6058	−1.83348	−0.916741	+	0.399481i	$0.869190\pi$
$888$	0	0
$889$	−2.06991	−0.0694226
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.97037i	0.233255i
$894$	0	0
$895$	86.5509i	2.89308i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.48573	0.0495518
$900$	0	0
$901$	13.4637	0.448542
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 11.2191i	− 0.372937i
$906$	0	0
$907$	25.8955i	0.859845i	0.902866	+	0.429923i	$0.141459\pi$
$907$	25.8955i	0.859845i	−0.902866	+	0.429923i	$0.858541\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−24.4982	−0.811660	−0.405830	−	0.913949i	$-0.633018\pi$
$911$	−24.4982	−0.811660	−0.405830	+	0.913949i	$0.633018\pi$
$912$	0	0
$913$	−18.8570	−0.624075
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 14.1612i	− 0.467643i
$918$	0	0
$919$	16.4487i	0.542594i	0.962496	+	0.271297i	$0.0874526\pi$
$919$	16.4487i	0.542594i	−0.962496	+	0.271297i	$0.912547\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.65685	−0.186198
$924$	0	0
$925$	24.6390	0.810126
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.72877i	0.155146i	0.996987	+	0.0775729i	$0.0247170\pi$
$929$	4.72877i	0.155146i	−0.996987	+	0.0775729i	$0.975283\pi$
$930$	0	0
$931$	6.49251i	0.212783i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	73.5011	2.40374
$936$	0	0
$937$	24.6519	0.805344	0.402672	−	0.915344i	$-0.368082\pi$
$937$	24.6519	0.805344	0.402672	+	0.915344i	$0.368082\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	49.5230i	1.61440i	0.590276	+	0.807201i	$0.299019\pi$
$941$	49.5230i	1.61440i	−0.590276	+	0.807201i	$0.700981\pi$
$942$	0	0
$943$	− 4.78061i	− 0.155678i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.5926	1.44906	0.724532	−	0.689241i	$-0.242056\pi$
$947$	44.5926	1.44906	0.724532	+	0.689241i	$0.242056\pi$
$948$	0	0
$949$	−16.6147	−0.539335
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 40.8327i	− 1.32270i	−0.750077	−	0.661350i	$-0.769984\pi$
$953$	− 40.8327i	− 1.32270i	0.750077	−	0.661350i	$-0.230016\pi$
$954$	0	0
$955$	− 65.8173i	− 2.12980i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.58009	−0.180191
$960$	0	0
$961$	29.9494	0.966111
$962$	0	0
$963$	0	0
$964$	0	0
$965$	41.5325i	1.33698i
$966$	0	0
$967$	13.0705i	0.420320i	0.977667	+	0.210160i	$0.0673985\pi$
$967$	13.0705i	0.420320i	−0.977667	+	0.210160i	$0.932601\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42.7868	−1.37309	−0.686547	−	0.727086i	$-0.740874\pi$
$971$	−42.7868	−1.37309	−0.686547	+	0.727086i	$0.740874\pi$
$972$	0	0
$973$	8.92439	0.286103
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.3789i	1.29184i	0.763407	+	0.645918i	$0.223525\pi$
$977$	40.3789i	1.29184i	−0.763407	+	0.645918i	$0.776475\pi$
$978$	0	0
$979$	9.72955i	0.310958i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15.5352	0.495494	0.247747	−	0.968825i	$-0.420310\pi$
$983$	15.5352	0.495494	0.247747	+	0.968825i	$0.420310\pi$
$984$	0	0
$985$	82.8613	2.64018
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.4821i	0.587698i
$990$	0	0
$991$	− 33.5241i	− 1.06493i	−0.846452	−	0.532465i	$-0.821266\pi$
$991$	− 33.5241i	− 1.06493i	0.846452	−	0.532465i	$-0.178734\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−53.9304	−1.70971
$996$	0	0
$997$	−6.03909	−0.191260	−0.0956300	−	0.995417i	$-0.530487\pi$
$997$	−6.03909	−0.191260	−0.0956300	+	0.995417i	$0.530487\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.d.g.2015.13	yes	16
3.2	odd	2	inner	5472.2.d.g.2015.3	✓	16
4.3	odd	2	inner	5472.2.d.g.2015.14	yes	16
12.11	even	2	inner	5472.2.d.g.2015.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5472.2.d.g.2015.3	✓	16	3.2	odd	2	inner
5472.2.d.g.2015.4	yes	16	12.11	even	2	inner
5472.2.d.g.2015.13	yes	16	1.1	even	1	trivial
5472.2.d.g.2015.14	yes	16	4.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.62563i q^{5} -0.712386i q^{7} -3.62563 q^{11} -3.02497 q^{13} +5.59147i q^{17} +1.00000i q^{19} -2.37259 q^{23} -8.14521 q^{25} +1.44953i q^{29} -1.02497i q^{31} +2.58285 q^{35} -3.02497 q^{37} +2.01493i q^{41} -7.78986i q^{43} +6.97037 q^{47} +6.49251 q^{49} -2.40790i q^{53} -13.1452i q^{55} -5.80173 q^{59} -11.8398 q^{61} -10.9674i q^{65} -5.42477i q^{67} +1.87005 q^{71} +5.49251 q^{73} +2.58285i q^{77} -7.07492i q^{79} +5.20102 q^{83} -20.2726 q^{85} -2.68355i q^{89} +2.15495i q^{91} -3.62563 q^{95} +12.9850 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{13} - 48 q^{25} - 16 q^{37} + 8 q^{49} - 80 q^{61} - 8 q^{73} - 152 q^{85} + 16 q^{97}+O(q^{100}) \) 16 * q - 16 * q^13 - 48 * q^25 - 16 * q^37 + 8 * q^49 - 80 * q^61 - 8 * q^73 - 152 * q^85 + 16 * q^97

Introduction

L-functions

Modular forms

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Representations

Groups

Database

Properties

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