LMFDB - Embedded newform 1296.5.e.c.161.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,5,Mod(161,1296)] 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("1296.161"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1296.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,24,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$133.967472157$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.39400128.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 $ x^6 - x^5 + 11x^4 + 14x^3 + 98x^2 + 20x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{9} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$-1.28901 + 2.23263i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.5.e.c.161.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-16.0226i q^{5} -72.4837 q^{7} -96.1576i q^{11} +153.906 q^{13} +72.7905i q^{17} +190.660 q^{19} -14.4552i q^{23} +368.276 q^{25} +716.262i q^{29} +302.568 q^{31} +1161.38i q^{35} +826.277 q^{37} -556.119i q^{41} -892.680 q^{43} +3955.43i q^{47} +2852.89 q^{49} -1966.96i q^{53} -1540.69 q^{55} +5414.94i q^{59} -1712.42 q^{61} -2465.97i q^{65} +4634.49 q^{67} -6697.12i q^{71} -4823.86 q^{73} +6969.86i q^{77} +5728.79 q^{79} -2832.23i q^{83} +1166.29 q^{85} -14277.7i q^{89} -11155.7 q^{91} -3054.87i q^{95} +7165.31 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 24 q^{7} + 12 q^{13} + 258 q^{19} + 546 q^{25} + 2580 q^{31} + 12 q^{37} - 570 q^{43} + 3726 q^{49} - 2016 q^{55} - 7260 q^{61} - 10110 q^{67} - 14622 q^{73} + 9528 q^{79} - 24732 q^{85} - 34836 q^{91}+ \cdots + 57918 q^{97}+O(q^{100}) $ 6 * q + 24 * q^7 + 12 * q^13 + 258 * q^19 + 546 * q^25 + 2580 * q^31 + 12 * q^37 - 570 * q^43 + 3726 * q^49 - 2016 * q^55 - 7260 * q^61 - 10110 * q^67 - 14622 * q^73 + 9528 * q^79 - 24732 * q^85 - 34836 * q^91 + 57918 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	− 16.0226i	− 0.640904i	−0.947265	−	0.320452i	$-0.896165\pi$
$5$	− 16.0226i	− 0.640904i	0.947265	−	0.320452i	$-0.103835\pi$
$6$	0	0
$7$	−72.4837	−1.47926	−0.739630	−	0.673014i	$-0.764999\pi$
$7$	−72.4837	−1.47926	−0.739630	+	0.673014i	$0.764999\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 96.1576i	− 0.794691i	−0.917669	−	0.397345i	$-0.869932\pi$
$11$	− 96.1576i	− 0.794691i	0.917669	−	0.397345i	$-0.130068\pi$
$12$	0	0
$13$	153.906	0.910686	0.455343	−	0.890316i	$-0.349517\pi$
$13$	153.906	0.910686	0.455343	+	0.890316i	$0.349517\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	72.7905i	0.251870i	0.992038	+	0.125935i	$0.0401931\pi$
$17$	72.7905i	0.251870i	−0.992038	+	0.125935i	$0.959807\pi$
$18$	0	0
$19$	190.660	0.528145	0.264072	−	0.964503i	$-0.414934\pi$
$19$	190.660	0.528145	0.264072	+	0.964503i	$0.414934\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 14.4552i	− 0.0273256i	−0.999907	−	0.0136628i	$-0.995651\pi$
$23$	− 14.4552i	− 0.0273256i	0.999907	−	0.0136628i	$-0.00434914\pi$
$24$	0	0
$25$	368.276	0.589242
$26$	0	0
$27$	0	0
$28$	0	0
$29$	716.262i	0.851679i	0.904799	+	0.425839i	$0.140021\pi$
$29$	716.262i	0.851679i	−0.904799	+	0.425839i	$0.859979\pi$
$30$	0	0
$31$	302.568	0.314847	0.157423	−	0.987531i	$-0.449681\pi$
$31$	302.568	0.314847	0.157423	+	0.987531i	$0.449681\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1161.38i	0.948063i
$36$	0	0
$37$	826.277	0.603562	0.301781	−	0.953377i	$-0.402419\pi$
$37$	826.277	0.603562	0.301781	+	0.953377i	$0.402419\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 556.119i	− 0.330826i	−0.986224	−	0.165413i	$-0.947104\pi$
$41$	− 556.119i	− 0.330826i	0.986224	−	0.165413i	$-0.0528958\pi$
$42$	0	0
$43$	−892.680	−0.482791	−0.241395	−	0.970427i	$-0.577605\pi$
$43$	−892.680	−0.482791	−0.241395	+	0.970427i	$0.577605\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3955.43i	1.79060i	0.445465	+	0.895299i	$0.353038\pi$
$47$	3955.43i	1.79060i	−0.445465	+	0.895299i	$0.646962\pi$
$48$	0	0
$49$	2852.89	1.18821
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1966.96i	− 0.700234i	−0.936706	−	0.350117i	$-0.886142\pi$
$53$	− 1966.96i	− 0.700234i	0.936706	−	0.350117i	$-0.113858\pi$
$54$	0	0
$55$	−1540.69	−0.509320
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5414.94i	1.55557i	0.628530	+	0.777785i	$0.283657\pi$
$59$	5414.94i	1.55557i	−0.628530	+	0.777785i	$0.716343\pi$
$60$	0	0
$61$	−1712.42	−0.460204	−0.230102	−	0.973166i	$-0.573906\pi$
$61$	−1712.42	−0.460204	−0.230102	+	0.973166i	$0.573906\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2465.97i	− 0.583663i
$66$	0	0
$67$	4634.49	1.03241	0.516205	−	0.856465i	$-0.327344\pi$
$67$	4634.49	1.03241	0.516205	+	0.856465i	$0.327344\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 6697.12i	− 1.32853i	−0.747497	−	0.664265i	$-0.768744\pi$
$71$	− 6697.12i	− 1.32853i	0.747497	−	0.664265i	$-0.231256\pi$
$72$	0	0
$73$	−4823.86	−0.905208	−0.452604	−	0.891712i	$-0.649505\pi$
$73$	−4823.86	−0.905208	−0.452604	+	0.891712i	$0.649505\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6969.86i	1.17555i
$78$	0	0
$79$	5728.79	0.917929	0.458964	−	0.888455i	$-0.348221\pi$
$79$	5728.79	0.917929	0.458964	+	0.888455i	$0.348221\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2832.23i	− 0.411124i	−0.978644	−	0.205562i	$-0.934098\pi$
$83$	− 2832.23i	− 0.411124i	0.978644	−	0.205562i	$-0.0659022\pi$
$84$	0	0
$85$	1166.29	0.161425
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 14277.7i	− 1.80251i	−0.433290	−	0.901255i	$-0.642647\pi$
$89$	− 14277.7i	− 1.80251i	0.433290	−	0.901255i	$-0.357353\pi$
$90$	0	0
$91$	−11155.7	−1.34714
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 3054.87i	− 0.338490i
$96$	0	0
$97$	7165.31	0.761538	0.380769	−	0.924670i	$-0.375659\pi$
$97$	7165.31	0.761538	0.380769	+	0.924670i	$0.375659\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1959.20i	− 0.192060i	−0.995378	−	0.0960298i	$-0.969386\pi$
$101$	− 1959.20i	− 0.192060i	0.995378	−	0.0960298i	$-0.0306144\pi$
$102$	0	0
$103$	5155.48	0.485953	0.242976	−	0.970032i	$-0.421876\pi$
$103$	5155.48	0.485953	0.242976	+	0.970032i	$0.421876\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 9117.08i	− 0.796321i	−0.917316	−	0.398161i	$-0.869649\pi$
$107$	− 9117.08i	− 0.796321i	0.917316	−	0.398161i	$-0.130351\pi$
$108$	0	0
$109$	−16161.1	−1.36024	−0.680122	−	0.733099i	$-0.738073\pi$
$109$	−16161.1	−1.36024	−0.680122	+	0.733099i	$0.738073\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 21099.2i	− 1.65238i	−0.563393	−	0.826189i	$-0.690504\pi$
$113$	− 21099.2i	− 1.65238i	0.563393	−	0.826189i	$-0.309496\pi$
$114$	0	0
$115$	−231.611	−0.0175131
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 5276.13i	− 0.372582i
$120$	0	0
$121$	5394.72	0.368467
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 15914.9i	− 1.01855i
$126$	0	0
$127$	20660.9	1.28098	0.640489	−	0.767968i	$-0.278732\pi$
$127$	20660.9	1.28098	0.640489	+	0.767968i	$0.278732\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3201.45i	0.186554i	0.995640	+	0.0932768i	$0.0297342\pi$
$131$	3201.45i	0.186554i	−0.995640	+	0.0932768i	$0.970266\pi$
$132$	0	0
$133$	−13819.8	−0.781263
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3871.92i	− 0.206293i	−0.994666	−	0.103147i	$-0.967109\pi$
$137$	− 3871.92i	− 0.206293i	0.994666	−	0.103147i	$-0.0328911\pi$
$138$	0	0
$139$	11679.2	0.604484	0.302242	−	0.953231i	$-0.402265\pi$
$139$	11679.2	0.604484	0.302242	+	0.953231i	$0.402265\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 14799.2i	− 0.723714i
$144$	0	0
$145$	11476.4	0.545844
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15091.8i	0.679778i	0.940466	+	0.339889i	$0.110390\pi$
$149$	15091.8i	0.679778i	−0.940466	+	0.339889i	$0.889610\pi$
$150$	0	0
$151$	−30255.3	−1.32693	−0.663465	−	0.748207i	$-0.730915\pi$
$151$	−30255.3	−1.32693	−0.663465	+	0.748207i	$0.730915\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 4847.92i	− 0.201787i
$156$	0	0
$157$	20622.7	0.836656	0.418328	−	0.908296i	$-0.362616\pi$
$157$	20622.7	0.836656	0.418328	+	0.908296i	$0.362616\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1047.77i	0.0404216i
$162$	0	0
$163$	−39790.7	−1.49764	−0.748818	−	0.662776i	$-0.769378\pi$
$163$	−39790.7	−1.49764	−0.748818	+	0.662776i	$0.769378\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 27450.7i	− 0.984283i	−0.870515	−	0.492141i	$-0.836214\pi$
$167$	− 27450.7i	− 0.984283i	0.870515	−	0.492141i	$-0.163786\pi$
$168$	0	0
$169$	−4873.95	−0.170651
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10079.7i	0.336787i	0.985720	+	0.168393i	$0.0538578\pi$
$173$	10079.7i	0.336787i	−0.985720	+	0.168393i	$0.946142\pi$
$174$	0	0
$175$	−26694.0	−0.871642
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1215.45i	− 0.0379343i	−0.999820	−	0.0189672i	$-0.993962\pi$
$179$	− 1215.45i	− 0.0379343i	0.999820	−	0.0189672i	$-0.00603779\pi$
$180$	0	0
$181$	28359.9	0.865661	0.432831	−	0.901475i	$-0.357515\pi$
$181$	28359.9	0.865661	0.432831	+	0.901475i	$0.357515\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 13239.1i	− 0.386826i
$186$	0	0
$187$	6999.36	0.200159
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9751.84i	0.267313i	0.991028	+	0.133656i	$0.0426719\pi$
$191$	9751.84i	0.267313i	−0.991028	+	0.133656i	$0.957328\pi$
$192$	0	0
$193$	−53402.8	−1.43367	−0.716836	−	0.697242i	$-0.754410\pi$
$193$	−53402.8	−1.43367	−0.716836	+	0.697242i	$0.754410\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	68537.7i	1.76603i	0.469349	+	0.883013i	$0.344489\pi$
$197$	68537.7i	1.76603i	−0.469349	+	0.883013i	$0.655511\pi$
$198$	0	0
$199$	8237.42	0.208010	0.104005	−	0.994577i	$-0.466834\pi$
$199$	8237.42	0.208010	0.104005	+	0.994577i	$0.466834\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 51917.3i	− 1.25985i
$204$	0	0
$205$	−8910.47	−0.212028
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 18333.4i	− 0.419712i
$210$	0	0
$211$	−40787.4	−0.916138	−0.458069	−	0.888917i	$-0.651459\pi$
$211$	−40787.4	−0.916138	−0.458069	+	0.888917i	$0.651459\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14303.1i	0.309423i
$216$	0	0
$217$	−21931.2	−0.465740
$218$	0	0
$219$	0	0
$220$	0	0
$221$	11202.9i	0.229375i
$222$	0	0
$223$	59919.6	1.20492	0.602461	−	0.798148i	$-0.294187\pi$
$223$	59919.6	1.20492	0.602461	+	0.798148i	$0.294187\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 98810.4i	− 1.91757i	−0.284134	−	0.958785i	$-0.591706\pi$
$227$	− 98810.4i	− 1.91757i	0.284134	−	0.958785i	$-0.408294\pi$
$228$	0	0
$229$	−5317.94	−0.101408	−0.0507040	−	0.998714i	$-0.516147\pi$
$229$	−5317.94	−0.101408	−0.0507040	+	0.998714i	$0.516147\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 18467.8i	− 0.340176i	−0.985429	−	0.170088i	$-0.945595\pi$
$233$	− 18467.8i	− 0.340176i	0.985429	−	0.170088i	$-0.0544052\pi$
$234$	0	0
$235$	63376.3	1.14760
$236$	0	0
$237$	0	0
$238$	0	0
$239$	25846.2i	0.452482i	0.974071	+	0.226241i	$0.0726437\pi$
$239$	25846.2i	0.452482i	−0.974071	+	0.226241i	$0.927356\pi$
$240$	0	0
$241$	83072.3	1.43028	0.715142	−	0.698979i	$-0.246362\pi$
$241$	83072.3	1.43028	0.715142	+	0.698979i	$0.246362\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 45710.7i	− 0.761527i
$246$	0	0
$247$	29343.8	0.480974
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 49051.6i	− 0.778585i	−0.921114	−	0.389292i	$-0.872720\pi$
$251$	− 49051.6i	− 0.778585i	0.921114	−	0.389292i	$-0.127280\pi$
$252$	0	0
$253$	−1389.98	−0.0217154
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 97105.7i	− 1.47021i	−0.677955	−	0.735104i	$-0.737133\pi$
$257$	− 97105.7i	− 1.47021i	0.677955	−	0.735104i	$-0.262867\pi$
$258$	0	0
$259$	−59891.6	−0.892825
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 41478.0i	− 0.599662i	−0.953992	−	0.299831i	$-0.903070\pi$
$263$	− 41478.0i	− 0.599662i	0.953992	−	0.299831i	$-0.0969303\pi$
$264$	0	0
$265$	−31515.8	−0.448783
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 115737.i	− 1.59944i	−0.600370	−	0.799722i	$-0.704980\pi$
$269$	− 115737.i	− 1.59944i	0.600370	−	0.799722i	$-0.295020\pi$
$270$	0	0
$271$	5077.71	0.0691400	0.0345700	−	0.999402i	$-0.488994\pi$
$271$	5077.71	0.0691400	0.0345700	+	0.999402i	$0.488994\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 35412.5i	− 0.468265i
$276$	0	0
$277$	−42385.8	−0.552409	−0.276205	−	0.961099i	$-0.589077\pi$
$277$	−42385.8	−0.552409	−0.276205	+	0.961099i	$0.589077\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 41456.6i	− 0.525027i	−0.964928	−	0.262513i	$-0.915449\pi$
$281$	− 41456.6i	− 0.525027i	0.964928	−	0.262513i	$-0.0845514\pi$
$282$	0	0
$283$	−38388.0	−0.479316	−0.239658	−	0.970857i	$-0.577035\pi$
$283$	−38388.0	−0.479316	−0.239658	+	0.970857i	$0.577035\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	40309.6i	0.489378i
$288$	0	0
$289$	78222.5	0.936561
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12978.2i	0.151175i	0.997139	+	0.0755876i	$0.0240833\pi$
$293$	12978.2i	0.151175i	−0.997139	+	0.0755876i	$0.975917\pi$
$294$	0	0
$295$	86761.4	0.996971
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 2224.75i	− 0.0248850i
$300$	0	0
$301$	64704.8	0.714173
$302$	0	0
$303$	0	0
$304$	0	0
$305$	27437.4i	0.294947i
$306$	0	0
$307$	126105.	1.33799	0.668997	−	0.743265i	$-0.266724\pi$
$307$	126105.	1.33799	0.668997	+	0.743265i	$0.266724\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 99803.5i	− 1.03187i	−0.856628	−	0.515935i	$-0.827445\pi$
$311$	− 99803.5i	− 1.03187i	0.856628	−	0.515935i	$-0.172555\pi$
$312$	0	0
$313$	−2225.35	−0.0227148	−0.0113574	−	0.999936i	$-0.503615\pi$
$313$	−2225.35	−0.0227148	−0.0113574	+	0.999936i	$0.503615\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 168069.i	− 1.67251i	−0.548342	−	0.836254i	$-0.684741\pi$
$317$	− 168069.i	− 1.67251i	0.548342	−	0.836254i	$-0.315259\pi$
$318$	0	0
$319$	68874.0	0.676821
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13878.3i	0.133024i
$324$	0	0
$325$	56679.9	0.536615
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 286704.i	− 2.64876i
$330$	0	0
$331$	−6758.86	−0.0616904	−0.0308452	−	0.999524i	$-0.509820\pi$
$331$	−6758.86	−0.0616904	−0.0308452	+	0.999524i	$0.509820\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 74256.6i	− 0.661676i
$336$	0	0
$337$	−222093.	−1.95557	−0.977787	−	0.209600i	$-0.932784\pi$
$337$	−222093.	−1.95557	−0.977787	+	0.209600i	$0.932784\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 29094.2i	− 0.250206i
$342$	0	0
$343$	−32754.4	−0.278408
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 162548.i	− 1.34996i	−0.737835	−	0.674981i	$-0.764152\pi$
$347$	− 162548.i	− 1.34996i	0.737835	−	0.674981i	$-0.235848\pi$
$348$	0	0
$349$	−72696.9	−0.596850	−0.298425	−	0.954433i	$-0.596461\pi$
$349$	−72696.9	−0.596850	−0.298425	+	0.954433i	$0.596461\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	128629.i	1.03226i	0.856509	+	0.516131i	$0.172628\pi$
$353$	128629.i	1.03226i	−0.856509	+	0.516131i	$0.827372\pi$
$354$	0	0
$355$	−107305.	−0.851461
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 95302.8i	− 0.739463i	−0.929139	−	0.369732i	$-0.879450\pi$
$359$	− 95302.8i	− 0.739463i	0.929139	−	0.369732i	$-0.120550\pi$
$360$	0	0
$361$	−93969.7	−0.721063
$362$	0	0
$363$	0	0
$364$	0	0
$365$	77290.7i	0.580152i
$366$	0	0
$367$	−104146.	−0.773233	−0.386616	−	0.922241i	$-0.626356\pi$
$367$	−104146.	−0.773233	−0.386616	+	0.922241i	$0.626356\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	142572.i	1.03583i
$372$	0	0
$373$	245831.	1.76693	0.883463	−	0.468501i	$-0.155206\pi$
$373$	245831.	1.76693	0.883463	+	0.468501i	$0.155206\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	110237.i	0.775612i
$378$	0	0
$379$	−200830.	−1.39814	−0.699070	−	0.715053i	$-0.746403\pi$
$379$	−200830.	−1.39814	−0.699070	+	0.715053i	$0.746403\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	135338.i	0.922615i	0.887240	+	0.461308i	$0.152620\pi$
$383$	135338.i	0.922615i	−0.887240	+	0.461308i	$0.847380\pi$
$384$	0	0
$385$	111675.	0.753417
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 202177.i	− 1.33608i	−0.744126	−	0.668039i	$-0.767134\pi$
$389$	− 202177.i	− 1.33608i	0.744126	−	0.668039i	$-0.232866\pi$
$390$	0	0
$391$	1052.20	0.00688251
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 91790.2i	− 0.588304i
$396$	0	0
$397$	102384.	0.649608	0.324804	−	0.945781i	$-0.394702\pi$
$397$	102384.	0.649608	0.324804	+	0.945781i	$0.394702\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	90097.2i	0.560302i	0.959956	+	0.280151i	$0.0903846\pi$
$401$	90097.2i	0.560302i	−0.959956	+	0.280151i	$0.909615\pi$
$402$	0	0
$403$	46567.0	0.286727
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 79452.8i	− 0.479645i
$408$	0	0
$409$	149722.	0.895033	0.447517	−	0.894276i	$-0.352309\pi$
$409$	149722.	0.895033	0.447517	+	0.894276i	$0.352309\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 392495.i	− 2.30109i
$414$	0	0
$415$	−45379.7	−0.263491
$416$	0	0
$417$	0	0
$418$	0	0
$419$	126351.i	0.719697i	0.933011	+	0.359848i	$0.117172\pi$
$419$	126351.i	0.719697i	−0.933011	+	0.359848i	$0.882828\pi$
$420$	0	0
$421$	−215289.	−1.21467	−0.607335	−	0.794446i	$-0.707762\pi$
$421$	−215289.	−1.21467	−0.607335	+	0.794446i	$0.707762\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	26807.0i	0.148413i
$426$	0	0
$427$	124123.	0.680762
$428$	0	0
$429$	0	0
$430$	0	0
$431$	121972.i	0.656607i	0.944572	+	0.328304i	$0.106477\pi$
$431$	121972.i	0.656607i	−0.944572	+	0.328304i	$0.893523\pi$
$432$	0	0
$433$	−152419.	−0.812951	−0.406475	−	0.913662i	$-0.633242\pi$
$433$	−152419.	−0.812951	−0.406475	+	0.913662i	$0.633242\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2756.04i	− 0.0144319i
$438$	0	0
$439$	−231128.	−1.19929	−0.599643	−	0.800267i	$-0.704691\pi$
$439$	−231128.	−1.19929	−0.599643	+	0.800267i	$0.704691\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20185.7i	0.102858i	0.998677	+	0.0514289i	$0.0163775\pi$
$443$	20185.7i	0.102858i	−0.998677	+	0.0514289i	$0.983622\pi$
$444$	0	0
$445$	−228766.	−1.15524
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 92962.8i	− 0.461123i	−0.973058	−	0.230561i	$-0.925944\pi$
$449$	− 92962.8i	− 0.461123i	0.973058	−	0.230561i	$-0.0740562\pi$
$450$	0	0
$451$	−53475.0	−0.262905
$452$	0	0
$453$	0	0
$454$	0	0
$455$	178743.i	0.863388i
$456$	0	0
$457$	220887.	1.05764	0.528819	−	0.848735i	$-0.322635\pi$
$457$	220887.	1.05764	0.528819	+	0.848735i	$0.322635\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	106195.i	0.499690i	0.968286	+	0.249845i	$0.0803797\pi$
$461$	106195.i	0.499690i	−0.968286	+	0.249845i	$0.919620\pi$
$462$	0	0
$463$	68179.9	0.318049	0.159025	−	0.987275i	$-0.449165\pi$
$463$	68179.9	0.318049	0.159025	+	0.987275i	$0.449165\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 224350.i	− 1.02871i	−0.857578	−	0.514353i	$-0.828032\pi$
$467$	− 224350.i	− 1.02871i	0.857578	−	0.514353i	$-0.171968\pi$
$468$	0	0
$469$	−335925.	−1.52720
$470$	0	0
$471$	0	0
$472$	0	0
$473$	85838.0i	0.383669i
$474$	0	0
$475$	70215.6	0.311205
$476$	0	0
$477$	0	0
$478$	0	0
$479$	109169.i	0.475802i	0.971289	+	0.237901i	$0.0764594\pi$
$479$	109169.i	0.475802i	−0.971289	+	0.237901i	$0.923541\pi$
$480$	0	0
$481$	127169.	0.549656
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 114807.i	− 0.488073i
$486$	0	0
$487$	−106309.	−0.448240	−0.224120	−	0.974562i	$-0.571951\pi$
$487$	−106309.	−0.448240	−0.224120	+	0.974562i	$0.571951\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 74446.6i	− 0.308803i	−0.988008	−	0.154402i	$-0.950655\pi$
$491$	− 74446.6i	− 0.308803i	0.988008	−	0.154402i	$-0.0493450\pi$
$492$	0	0
$493$	−52137.1	−0.214513
$494$	0	0
$495$	0	0
$496$	0	0
$497$	485432.i	1.96524i
$498$	0	0
$499$	111207.	0.446612	0.223306	−	0.974748i	$-0.428315\pi$
$499$	111207.	0.446612	0.223306	+	0.974748i	$0.428315\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 115897.i	− 0.458077i	−0.973417	−	0.229038i	$-0.926442\pi$
$503$	− 115897.i	− 0.458077i	0.973417	−	0.229038i	$-0.0735581\pi$
$504$	0	0
$505$	−31391.5	−0.123092
$506$	0	0
$507$	0	0
$508$	0	0
$509$	196644.i	0.759007i	0.925190	+	0.379503i	$0.123905\pi$
$509$	196644.i	0.759007i	−0.925190	+	0.379503i	$0.876095\pi$
$510$	0	0
$511$	349651.	1.33904
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 82604.1i	− 0.311449i
$516$	0	0
$517$	380345.	1.42297
$518$	0	0
$519$	0	0
$520$	0	0
$521$	299306.i	1.10266i	0.834289	+	0.551328i	$0.185879\pi$
$521$	299306.i	1.10266i	−0.834289	+	0.551328i	$0.814121\pi$
$522$	0	0
$523$	162892.	0.595520	0.297760	−	0.954641i	$-0.403760\pi$
$523$	162892.	0.595520	0.297760	+	0.954641i	$0.403760\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22024.1i	0.0793006i
$528$	0	0
$529$	279632.	0.999253
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 85590.0i	− 0.301279i
$534$	0	0
$535$	−146079.	−0.510365
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 274327.i	− 0.944257i
$540$	0	0
$541$	−62918.7	−0.214974	−0.107487	−	0.994207i	$-0.534280\pi$
$541$	−62918.7	−0.214974	−0.107487	+	0.994207i	$0.534280\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	258942.i	0.871786i
$546$	0	0
$547$	14807.1	0.0494876	0.0247438	−	0.999694i	$-0.492123\pi$
$547$	14807.1	0.0494876	0.0247438	+	0.999694i	$0.492123\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	136563.i	0.449810i
$552$	0	0
$553$	−415244.	−1.35785
$554$	0	0
$555$	0	0
$556$	0	0
$557$	254392.i	0.819962i	0.912094	+	0.409981i	$0.134465\pi$
$557$	254392.i	0.819962i	−0.912094	+	0.409981i	$0.865535\pi$
$558$	0	0
$559$	−137389.	−0.439671
$560$	0	0
$561$	0	0
$562$	0	0
$563$	93307.5i	0.294374i	0.989109	+	0.147187i	$0.0470220\pi$
$563$	93307.5i	0.294374i	−0.989109	+	0.147187i	$0.952978\pi$
$564$	0	0
$565$	−338064.	−1.05902
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 158024.i	− 0.488089i	−0.969764	−	0.244045i	$-0.921526\pi$
$569$	− 158024.i	− 0.488089i	0.969764	−	0.244045i	$-0.0784744\pi$
$570$	0	0
$571$	438375.	1.34454	0.672270	−	0.740306i	$-0.265319\pi$
$571$	438375.	1.34454	0.672270	+	0.740306i	$0.265319\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 5323.52i	− 0.0161014i
$576$	0	0
$577$	214770.	0.645094	0.322547	−	0.946554i	$-0.395461\pi$
$577$	214770.	0.645094	0.322547	+	0.946554i	$0.395461\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	205291.i	0.608158i
$582$	0	0
$583$	−189138.	−0.556469
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 656412.i	− 1.90502i	−0.304502	−	0.952512i	$-0.598490\pi$
$587$	− 656412.i	− 1.90502i	0.304502	−	0.952512i	$-0.401510\pi$
$588$	0	0
$589$	57687.6	0.166285
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 118369.i	− 0.336612i	−0.985735	−	0.168306i	$-0.946170\pi$
$593$	− 118369.i	− 0.336612i	0.985735	−	0.168306i	$-0.0538296\pi$
$594$	0	0
$595$	−84537.3	−0.238789
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 354139.i	− 0.987006i	−0.869744	−	0.493503i	$-0.835716\pi$
$599$	− 354139.i	− 0.987006i	0.869744	−	0.493503i	$-0.164284\pi$
$600$	0	0
$601$	−595639.	−1.64905	−0.824526	−	0.565824i	$-0.808558\pi$
$601$	−595639.	−1.64905	−0.824526	+	0.565824i	$0.808558\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 86437.5i	− 0.236152i
$606$	0	0
$607$	−9721.62	−0.0263853	−0.0131926	−	0.999913i	$-0.504199\pi$
$607$	−9721.62	−0.0263853	−0.0131926	+	0.999913i	$0.504199\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	608765.i	1.63067i
$612$	0	0
$613$	194450.	0.517473	0.258736	−	0.965948i	$-0.416694\pi$
$613$	194450.	0.517473	0.258736	+	0.965948i	$0.416694\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 487639.i	− 1.28094i	−0.767984	−	0.640469i	$-0.778740\pi$
$617$	− 487639.i	− 1.28094i	0.767984	−	0.640469i	$-0.221260\pi$
$618$	0	0
$619$	−17256.5	−0.0450371	−0.0225185	−	0.999746i	$-0.507168\pi$
$619$	−17256.5	−0.0450371	−0.0225185	+	0.999746i	$0.507168\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.03490e6i	2.66638i
$624$	0	0
$625$	−24825.0	−0.0635520
$626$	0	0
$627$	0	0
$628$	0	0
$629$	60145.1i	0.152019i
$630$	0	0
$631$	429836.	1.07955	0.539777	−	0.841808i	$-0.318509\pi$
$631$	429836.	1.07955	0.539777	+	0.841808i	$0.318509\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 331041.i	− 0.820984i
$636$	0	0
$637$	439076.	1.08208
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 32308.0i	− 0.0786310i	−0.999227	−	0.0393155i	$-0.987482\pi$
$641$	− 32308.0i	− 0.0786310i	0.999227	−	0.0393155i	$-0.0125177\pi$
$642$	0	0
$643$	−277048.	−0.670091	−0.335045	−	0.942202i	$-0.608752\pi$
$643$	−277048.	−0.670091	−0.335045	+	0.942202i	$0.608752\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	125452.i	0.299689i	0.988710	+	0.149844i	$0.0478773\pi$
$647$	125452.i	0.299689i	−0.988710	+	0.149844i	$0.952123\pi$
$648$	0	0
$649$	520687.	1.23620
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 283982.i	− 0.665985i	−0.942930	−	0.332992i	$-0.891942\pi$
$653$	− 283982.i	− 0.665985i	0.942930	−	0.332992i	$-0.108058\pi$
$654$	0	0
$655$	51295.5	0.119563
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 359272.i	− 0.827280i	−0.910440	−	0.413640i	$-0.864257\pi$
$659$	− 359272.i	− 0.827280i	0.910440	−	0.413640i	$-0.135743\pi$
$660$	0	0
$661$	197804.	0.452723	0.226362	−	0.974043i	$-0.427317\pi$
$661$	197804.	0.452723	0.226362	+	0.974043i	$0.427317\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	221429.i	0.500715i
$666$	0	0
$667$	10353.7	0.0232726
$668$	0	0
$669$	0	0
$670$	0	0
$671$	164662.i	0.365720i
$672$	0	0
$673$	238072.	0.525628	0.262814	−	0.964847i	$-0.415349\pi$
$673$	238072.	0.525628	0.262814	+	0.964847i	$0.415349\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	250150.i	0.545787i	0.962044	+	0.272894i	$0.0879807\pi$
$677$	250150.i	0.545787i	−0.962044	+	0.272894i	$0.912019\pi$
$678$	0	0
$679$	−519368.	−1.12651
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 267358.i	− 0.573129i	−0.958061	−	0.286564i	$-0.907487\pi$
$683$	− 267358.i	− 0.573129i	0.958061	−	0.286564i	$-0.0925132\pi$
$684$	0	0
$685$	−62038.2	−0.132214
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 302727.i	− 0.637694i
$690$	0	0
$691$	−336686.	−0.705129	−0.352565	−	0.935787i	$-0.614690\pi$
$691$	−336686.	−0.705129	−0.352565	+	0.935787i	$0.614690\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 187132.i	− 0.387417i
$696$	0	0
$697$	40480.2	0.0833253
$698$	0	0
$699$	0	0
$700$	0	0
$701$	635795.i	1.29384i	0.762557	+	0.646921i	$0.223944\pi$
$701$	635795.i	1.29384i	−0.762557	+	0.646921i	$0.776056\pi$
$702$	0	0
$703$	157538.	0.318768
$704$	0	0
$705$	0	0
$706$	0	0
$707$	142010.i	0.284106i
$708$	0	0
$709$	431026.	0.857455	0.428728	−	0.903434i	$-0.358962\pi$
$709$	431026.	0.857455	0.428728	+	0.903434i	$0.358962\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 4373.69i	− 0.00860337i
$714$	0	0
$715$	−237122.	−0.463831
$716$	0	0
$717$	0	0
$718$	0	0
$719$	798370.i	1.54435i	0.635409	+	0.772176i	$0.280832\pi$
$719$	798370.i	1.54435i	−0.635409	+	0.772176i	$0.719168\pi$
$720$	0	0
$721$	−373688.	−0.718850
$722$	0	0
$723$	0	0
$724$	0	0
$725$	263782.i	0.501845i
$726$	0	0
$727$	−317113.	−0.599992	−0.299996	−	0.953940i	$-0.596985\pi$
$727$	−317113.	−0.599992	−0.299996	+	0.953940i	$0.596985\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 64978.7i	− 0.121601i
$732$	0	0
$733$	−977843.	−1.81996	−0.909979	−	0.414654i	$-0.863902\pi$
$733$	−977843.	−1.81996	−0.909979	+	0.414654i	$0.863902\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 445641.i	− 0.820446i
$738$	0	0
$739$	392565.	0.718825	0.359412	−	0.933179i	$-0.382977\pi$
$739$	392565.	0.718825	0.359412	+	0.933179i	$0.382977\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.03860e6i	1.88135i	0.339314	+	0.940673i	$0.389805\pi$
$743$	1.03860e6i	1.88135i	−0.339314	+	0.940673i	$0.610195\pi$
$744$	0	0
$745$	241809.	0.435673
$746$	0	0
$747$	0	0
$748$	0	0
$749$	660840.i	1.17797i
$750$	0	0
$751$	−319565.	−0.566604	−0.283302	−	0.959031i	$-0.591430\pi$
$751$	−319565.	−0.566604	−0.283302	+	0.959031i	$0.591430\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	484769.i	0.850435i
$756$	0	0
$757$	500321.	0.873085	0.436543	−	0.899684i	$-0.356203\pi$
$757$	500321.	0.873085	0.436543	+	0.899684i	$0.356203\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	904875.i	1.56250i	0.624220	+	0.781249i	$0.285417\pi$
$761$	904875.i	1.56250i	−0.624220	+	0.781249i	$0.714583\pi$
$762$	0	0
$763$	1.17141e6	2.01215
$764$	0	0
$765$	0	0
$766$	0	0
$767$	833392.i	1.41664i
$768$	0	0
$769$	143377.	0.242453	0.121227	−	0.992625i	$-0.461317\pi$
$769$	143377.	0.242453	0.121227	+	0.992625i	$0.461317\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	710063.i	1.18833i	0.804342	+	0.594166i	$0.202518\pi$
$773$	710063.i	1.18833i	−0.804342	+	0.594166i	$0.797482\pi$
$774$	0	0
$775$	111428.	0.185521
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 106030.i	− 0.174724i
$780$	0	0
$781$	−643979.	−1.05577
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 330430.i	− 0.536216i
$786$	0	0
$787$	601711.	0.971490	0.485745	−	0.874101i	$-0.338548\pi$
$787$	601711.	0.971490	0.485745	+	0.874101i	$0.338548\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.52935e6i	2.44429i
$792$	0	0
$793$	−263552.	−0.419102
$794$	0	0
$795$	0	0
$796$	0	0
$797$	429811.i	0.676645i	0.941030	+	0.338322i	$0.109859\pi$
$797$	429811.i	0.676645i	−0.941030	+	0.338322i	$0.890141\pi$
$798$	0	0
$799$	−287918.	−0.450999
$800$	0	0
$801$	0	0
$802$	0	0
$803$	463850.i	0.719361i
$804$	0	0
$805$	16788.0	0.0259064
$806$	0	0
$807$	0	0
$808$	0	0
$809$	629639.i	0.962042i	0.876709	+	0.481021i	$0.159734\pi$
$809$	629639.i	0.962042i	−0.876709	+	0.481021i	$0.840266\pi$
$810$	0	0
$811$	−577760.	−0.878428	−0.439214	−	0.898383i	$-0.644743\pi$
$811$	−577760.	−0.878428	−0.439214	+	0.898383i	$0.644743\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	637550.i	0.959841i
$816$	0	0
$817$	−170199.	−0.254984
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44719.6i	0.0663455i	0.999450	+	0.0331727i	$0.0105612\pi$
$821$	44719.6i	0.0663455i	−0.999450	+	0.0331727i	$0.989439\pi$
$822$	0	0
$823$	292578.	0.431958	0.215979	−	0.976398i	$-0.430706\pi$
$823$	292578.	0.431958	0.215979	+	0.976398i	$0.430706\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 806491.i	− 1.17920i	−0.807694	−	0.589601i	$-0.799285\pi$
$827$	− 806491.i	− 1.17920i	0.807694	−	0.589601i	$-0.200715\pi$
$828$	0	0
$829$	204904.	0.298154	0.149077	−	0.988826i	$-0.452370\pi$
$829$	204904.	0.298154	0.149077	+	0.988826i	$0.452370\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	207663.i	0.299274i
$834$	0	0
$835$	−439831.	−0.630831
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 771105.i	− 1.09544i	−0.836661	−	0.547721i	$-0.815495\pi$
$839$	− 771105.i	− 1.09544i	0.836661	−	0.547721i	$-0.184505\pi$
$840$	0	0
$841$	194250.	0.274643
$842$	0	0
$843$	0	0
$844$	0	0
$845$	78093.4i	0.109371i
$846$	0	0
$847$	−391030.	−0.545058
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 11944.0i	− 0.0164927i
$852$	0	0
$853$	1.08237e6	1.48757	0.743784	−	0.668420i	$-0.233029\pi$
$853$	1.08237e6	1.48757	0.743784	+	0.668420i	$0.233029\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 533345.i	− 0.726183i	−0.931753	−	0.363092i	$-0.881721\pi$
$857$	− 533345.i	− 0.726183i	0.931753	−	0.363092i	$-0.118279\pi$
$858$	0	0
$859$	−1.13105e6	−1.53283	−0.766417	−	0.642343i	$-0.777962\pi$
$859$	−1.13105e6	−1.53283	−0.766417	+	0.642343i	$0.777962\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	110025.i	0.147730i	0.997268	+	0.0738649i	$0.0235334\pi$
$863$	110025.i	0.147730i	−0.997268	+	0.0738649i	$0.976467\pi$
$864$	0	0
$865$	161503.	0.215848
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 550867.i	− 0.729470i
$870$	0	0
$871$	713275.	0.940202
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.15357e6i	1.50670i
$876$	0	0
$877$	−408753.	−0.531450	−0.265725	−	0.964049i	$-0.585611\pi$
$877$	−408753.	−0.531450	−0.265725	+	0.964049i	$0.585611\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	235809.i	0.303815i	0.988395	+	0.151907i	$0.0485416\pi$
$881$	235809.i	0.303815i	−0.988395	+	0.151907i	$0.951458\pi$
$882$	0	0
$883$	273719.	0.351061	0.175531	−	0.984474i	$-0.443836\pi$
$883$	273719.	0.351061	0.175531	+	0.984474i	$0.443836\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	371896.i	0.472687i	0.971670	+	0.236344i	$0.0759491\pi$
$887$	371896.i	0.472687i	−0.971670	+	0.236344i	$0.924051\pi$
$888$	0	0
$889$	−1.49758e6	−1.89490
$890$	0	0
$891$	0	0
$892$	0	0
$893$	754144.i	0.945695i
$894$	0	0
$895$	−19474.7	−0.0243122
$896$	0	0
$897$	0	0
$898$	0	0
$899$	216718.i	0.268148i
$900$	0	0
$901$	143176.	0.176368
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 454400.i	− 0.554806i
$906$	0	0
$907$	143931.	0.174960	0.0874801	−	0.996166i	$-0.472119\pi$
$907$	143931.	0.174960	0.0874801	+	0.996166i	$0.472119\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	891661.i	1.07439i	0.843457	+	0.537196i	$0.180517\pi$
$911$	891661.i	1.07439i	−0.843457	+	0.537196i	$0.819483\pi$
$912$	0	0
$913$	−272340.	−0.326716
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 232053.i	− 0.275961i
$918$	0	0
$919$	582156.	0.689300	0.344650	−	0.938731i	$-0.387998\pi$
$919$	582156.	0.689300	0.344650	+	0.938731i	$0.387998\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.03073e6i	− 1.20987i
$924$	0	0
$925$	304298.	0.355644
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 771371.i	− 0.893782i	−0.894588	−	0.446891i	$-0.852531\pi$
$929$	− 771371.i	− 0.893782i	0.894588	−	0.446891i	$-0.147469\pi$
$930$	0	0
$931$	543932.	0.627546
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 112148.i	− 0.128283i
$936$	0	0
$937$	−1.46097e6	−1.66403	−0.832015	−	0.554753i	$-0.812813\pi$
$937$	−1.46097e6	−1.66403	−0.832015	+	0.554753i	$0.812813\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 169346.i	− 0.191248i	−0.995418	−	0.0956240i	$-0.969515\pi$
$941$	− 169346.i	− 0.191248i	0.995418	−	0.0956240i	$-0.0304846\pi$
$942$	0	0
$943$	−8038.83	−0.00904002
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 593888.i	− 0.662224i	−0.943591	−	0.331112i	$-0.892576\pi$
$947$	− 593888.i	− 0.662224i	0.943591	−	0.331112i	$-0.107424\pi$
$948$	0	0
$949$	−742420.	−0.824361
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.25477e6i	− 1.38159i	−0.723050	−	0.690796i	$-0.757260\pi$
$953$	− 1.25477e6i	− 1.38159i	0.723050	−	0.690796i	$-0.242740\pi$
$954$	0	0
$955$	156250.	0.171322
$956$	0	0
$957$	0	0
$958$	0	0
$959$	280651.i	0.305161i
$960$	0	0
$961$	−831974.	−0.900872
$962$	0	0
$963$	0	0
$964$	0	0
$965$	855652.i	0.918846i
$966$	0	0
$967$	1.05786e6	1.13129	0.565645	−	0.824649i	$-0.308627\pi$
$967$	1.05786e6	1.13129	0.565645	+	0.824649i	$0.308627\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 775745.i	− 0.822774i	−0.911461	−	0.411387i	$-0.865044\pi$
$971$	− 775745.i	− 0.822774i	0.911461	−	0.411387i	$-0.134956\pi$
$972$	0	0
$973$	−846555.	−0.894189
$974$	0	0
$975$	0	0
$976$	0	0
$977$	496516.i	0.520168i	0.965586	+	0.260084i	$0.0837503\pi$
$977$	496516.i	0.520168i	−0.965586	+	0.260084i	$0.916250\pi$
$978$	0	0
$979$	−1.37291e6	−1.43244
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 300964.i	− 0.311464i	−0.987799	−	0.155732i	$-0.950226\pi$
$983$	− 300964.i	− 0.311464i	0.987799	−	0.155732i	$-0.0497736\pi$
$984$	0	0
$985$	1.09815e6	1.13185
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12903.9i	0.0131925i
$990$	0	0
$991$	−1.70094e6	−1.73197	−0.865987	−	0.500067i	$-0.833309\pi$
$991$	−1.70094e6	−1.73197	−0.865987	+	0.500067i	$0.833309\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 131985.i	− 0.133315i
$996$	0	0
$997$	−625837.	−0.629609	−0.314805	−	0.949156i	$-0.601939\pi$
$997$	−625837.	−0.629609	−0.314805	+	0.949156i	$0.601939\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.5.e.c.161.2		6
3.2	odd	2	inner	1296.5.e.c.161.5		6
4.3	odd	2		81.5.b.a.80.5		6
9.2	odd	6		432.5.q.a.17.1		6
9.4	even	3		432.5.q.a.305.1		6
9.5	odd	6		144.5.q.a.65.3		6
9.7	even	3		144.5.q.a.113.3		6
12.11	even	2		81.5.b.a.80.2		6
36.7	odd	6		9.5.d.a.5.3	yes	6
36.11	even	6		27.5.d.a.17.1		6
36.23	even	6		9.5.d.a.2.3	✓	6
36.31	odd	6		27.5.d.a.8.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.5.d.a.2.3	✓	6	36.23	even	6
9.5.d.a.5.3	yes	6	36.7	odd	6
27.5.d.a.8.1		6	36.31	odd	6
27.5.d.a.17.1		6	36.11	even	6
81.5.b.a.80.2		6	12.11	even	2
81.5.b.a.80.5		6	4.3	odd	2
144.5.q.a.65.3		6	9.5	odd	6
144.5.q.a.113.3		6	9.7	even	3
432.5.q.a.17.1		6	9.2	odd	6
432.5.q.a.305.1		6	9.4	even	3
1296.5.e.c.161.2		6	1.1	even	1	trivial
1296.5.e.c.161.5		6	3.2	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-16.0226i q^{5} -72.4837 q^{7} -96.1576i q^{11} +153.906 q^{13} +72.7905i q^{17} +190.660 q^{19} -14.4552i q^{23} +368.276 q^{25} +716.262i q^{29} +302.568 q^{31} +1161.38i q^{35} +826.277 q^{37} -556.119i q^{41} -892.680 q^{43} +3955.43i q^{47} +2852.89 q^{49} -1966.96i q^{53} -1540.69 q^{55} +5414.94i q^{59} -1712.42 q^{61} -2465.97i q^{65} +4634.49 q^{67} -6697.12i q^{71} -4823.86 q^{73} +6969.86i q^{77} +5728.79 q^{79} -2832.23i q^{83} +1166.29 q^{85} -14277.7i q^{89} -11155.7 q^{91} -3054.87i q^{95} +7165.31 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 24 q^{7} + 12 q^{13} + 258 q^{19} + 546 q^{25} + 2580 q^{31} + 12 q^{37} - 570 q^{43} + 3726 q^{49} - 2016 q^{55} - 7260 q^{61} - 10110 q^{67} - 14622 q^{73} + 9528 q^{79} - 24732 q^{85} - 34836 q^{91}+ \cdots + 57918 q^{97}+O(q^{100}) \) 6 * q + 24 * q^7 + 12 * q^13 + 258 * q^19 + 546 * q^25 + 2580 * q^31 + 12 * q^37 - 570 * q^43 + 3726 * q^49 - 2016 * q^55 - 7260 * q^61 - 10110 * q^67 - 14622 * q^73 + 9528 * q^79 - 24732 * q^85 - 34836 * q^91 + 57918 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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