LMFDB - Embedded newform 1296.5.g.c.1135.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,5,Mod(1135,1296)] 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("1296.1135"); 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([1, 0, 0])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1296.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$133.967472157$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-19}, \sqrt{-39})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 25 $ x^4 + 29*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1135.2
Root			$-5.30195i$ of defining polynomial
Character	$\chi$	$=$	1296.1135
Dual form			1296.5.g.c.1135.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-39.2213 q^{5} +95.4351i q^{7} -61.4853i q^{11} -119.664 q^{13} +147.885 q^{17} +566.196i q^{19} +735.945i q^{23} +913.312 q^{25} +1070.53 q^{29} +967.179i q^{31} -3743.09i q^{35} +705.648 q^{37} +2045.95 q^{41} +1915.12i q^{43} +1963.77i q^{47} -6706.85 q^{49} +3497.90 q^{53} +2411.53i q^{55} +5645.38i q^{59} +4317.55 q^{61} +4693.38 q^{65} +2397.93i q^{67} +4667.25i q^{71} -2687.64 q^{73} +5867.85 q^{77} +425.861i q^{79} -6994.30i q^{83} -5800.25 q^{85} -7129.82 q^{89} -11420.1i q^{91} -22207.0i q^{95} +2277.85 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 48 q^{5} - 152 q^{13} + 156 q^{17} + 1040 q^{25} + 1560 q^{29} + 536 q^{37} + 2304 q^{41} - 9188 q^{49} + 2232 q^{53} + 3224 q^{61} + 10716 q^{65} - 7484 q^{73} + 5832 q^{77} - 13728 q^{85} - 20244 q^{89}+ \cdots - 8528 q^{97}+O(q^{100}) $ 4 * q - 48 * q^5 - 152 * q^13 + 156 * q^17 + 1040 * q^25 + 1560 * q^29 + 536 * q^37 + 2304 * q^41 - 9188 * q^49 + 2232 * q^53 + 3224 * q^61 + 10716 * q^65 - 7484 * q^73 + 5832 * q^77 - 13728 * q^85 - 20244 * q^89 - 8528 * 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
$5$	−39.2213	−1.56885	−0.784426	−	0.620222i	$-0.787042\pi$
$5$	−39.2213	−1.56885	−0.784426	+	0.620222i	$0.787042\pi$
$6$	0	0
$7$	95.4351i	1.94765i	0.227289	+	0.973827i	$0.427014\pi$
$7$	95.4351i	1.94765i	−0.227289	+	0.973827i	$0.572986\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 61.4853i	− 0.508143i	−0.967185	−	0.254071i	$-0.918230\pi$
$11$	− 61.4853i	− 0.508143i	0.967185	−	0.254071i	$-0.0817698\pi$
$12$	0	0
$13$	−119.664	−0.708071	−0.354035	−	0.935232i	$-0.615191\pi$
$13$	−119.664	−0.708071	−0.354035	+	0.935232i	$0.615191\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	147.885	0.511714	0.255857	−	0.966715i	$-0.417642\pi$
$17$	147.885	0.511714	0.255857	+	0.966715i	$0.417642\pi$
$18$	0	0
$19$	566.196i	1.56841i	0.620502	+	0.784205i	$0.286929\pi$
$19$	566.196i	1.56841i	−0.620502	+	0.784205i	$0.713071\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	735.945i	1.39120i	0.718429	+	0.695600i	$0.244861\pi$
$23$	735.945i	1.39120i	−0.718429	+	0.695600i	$0.755139\pi$
$24$	0	0
$25$	913.312	1.46130
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1070.53	1.27293	0.636464	−	0.771306i	$-0.280396\pi$
$29$	1070.53	1.27293	0.636464	+	0.771306i	$0.280396\pi$
$30$	0	0
$31$	967.179i	1.00643i	0.864161	+	0.503215i	$0.167850\pi$
$31$	967.179i	1.00643i	−0.864161	+	0.503215i	$0.832150\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3743.09i	− 3.05558i
$36$	0	0
$37$	705.648	0.515447	0.257724	−	0.966219i	$-0.417028\pi$
$37$	705.648	0.515447	0.257724	+	0.966219i	$0.417028\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2045.95	1.21710	0.608552	−	0.793514i	$-0.291751\pi$
$41$	2045.95	1.21710	0.608552	+	0.793514i	$0.291751\pi$
$42$	0	0
$43$	1915.12i	1.03576i	0.855454	+	0.517879i	$0.173278\pi$
$43$	1915.12i	1.03576i	−0.855454	+	0.517879i	$0.826722\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1963.77i	0.888987i	0.895782	+	0.444494i	$0.146616\pi$
$47$	1963.77i	0.888987i	−0.895782	+	0.444494i	$0.853384\pi$
$48$	0	0
$49$	−6706.85	−2.79336
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3497.90	1.24525	0.622624	−	0.782521i	$-0.286067\pi$
$53$	3497.90	1.24525	0.622624	+	0.782521i	$0.286067\pi$
$54$	0	0
$55$	2411.53i	0.797201i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5645.38i	1.62177i	0.585207	+	0.810884i	$0.301013\pi$
$59$	5645.38i	1.62177i	−0.585207	+	0.810884i	$0.698987\pi$
$60$	0	0
$61$	4317.55	1.16032	0.580160	−	0.814503i	$-0.302990\pi$
$61$	4317.55	1.16032	0.580160	+	0.814503i	$0.302990\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4693.38	1.11086
$66$	0	0
$67$	2397.93i	0.534178i	0.963672	+	0.267089i	$0.0860618\pi$
$67$	2397.93i	0.534178i	−0.963672	+	0.267089i	$0.913938\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4667.25i	0.925857i	0.886396	+	0.462929i	$0.153201\pi$
$71$	4667.25i	0.925857i	−0.886396	+	0.462929i	$0.846799\pi$
$72$	0	0
$73$	−2687.64	−0.504342	−0.252171	−	0.967683i	$-0.581145\pi$
$73$	−2687.64	−0.504342	−0.252171	+	0.967683i	$0.581145\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5867.85	0.989687
$78$	0	0
$79$	425.861i	0.0682360i	0.999418	+	0.0341180i	$0.0108622\pi$
$79$	425.861i	0.0682360i	−0.999418	+	0.0341180i	$0.989138\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6994.30i	− 1.01528i	−0.861568	−	0.507642i	$-0.830517\pi$
$83$	− 6994.30i	− 1.01528i	0.861568	−	0.507642i	$-0.169483\pi$
$84$	0	0
$85$	−5800.25	−0.802803
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7129.82	−0.900116	−0.450058	−	0.892999i	$-0.648597\pi$
$89$	−7129.82	−0.900116	−0.450058	+	0.892999i	$0.648597\pi$
$90$	0	0
$91$	− 11420.1i	− 1.37908i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 22207.0i	− 2.46060i
$96$	0	0
$97$	2277.85	0.242093	0.121047	−	0.992647i	$-0.461375\pi$
$97$	2277.85	0.242093	0.121047	+	0.992647i	$0.461375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10470.0	1.02637	0.513183	−	0.858279i	$-0.328466\pi$
$101$	10470.0	1.02637	0.513183	+	0.858279i	$0.328466\pi$
$102$	0	0
$103$	− 10725.7i	− 1.01100i	−0.862828	−	0.505498i	$-0.831309\pi$
$103$	− 10725.7i	− 1.01100i	0.862828	−	0.505498i	$-0.168691\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14838.1i	1.29602i	0.761632	+	0.648009i	$0.224398\pi$
$107$	14838.1i	1.29602i	−0.761632	+	0.648009i	$0.775602\pi$
$108$	0	0
$109$	3108.14	0.261606	0.130803	−	0.991408i	$-0.458244\pi$
$109$	3108.14	0.261606	0.130803	+	0.991408i	$0.458244\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−22826.3	−1.78764	−0.893818	−	0.448431i	$-0.851983\pi$
$113$	−22826.3	−1.78764	−0.893818	+	0.448431i	$0.851983\pi$
$114$	0	0
$115$	− 28864.7i	− 2.18259i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14113.4i	0.996642i
$120$	0	0
$121$	10860.6	0.741791
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11308.0	−0.723709
$126$	0	0
$127$	818.872i	0.0507702i	0.999678	+	0.0253851i	$0.00808119\pi$
$127$	818.872i	0.0507702i	−0.999678	+	0.0253851i	$0.991919\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1149.44i	0.0669796i	0.999439	+	0.0334898i	$0.0106621\pi$
$131$	1149.44i	0.0669796i	−0.999439	+	0.0334898i	$0.989338\pi$
$132$	0	0
$133$	−54035.0	−3.05472
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2463.88	0.131274	0.0656370	−	0.997844i	$-0.479092\pi$
$137$	2463.88	0.131274	0.0656370	+	0.997844i	$0.479092\pi$
$138$	0	0
$139$	− 29355.5i	− 1.51936i	−0.650298	−	0.759679i	$-0.725356\pi$
$139$	− 29355.5i	− 1.51936i	0.650298	−	0.759679i	$-0.274644\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7357.57i	0.359801i
$144$	0	0
$145$	−41987.7	−1.99704
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−102.860	−0.00463314	−0.00231657	−	0.999997i	$-0.500737\pi$
$149$	−102.860	−0.00463314	−0.00231657	+	0.999997i	$0.500737\pi$
$150$	0	0
$151$	41850.3i	1.83546i	0.397205	+	0.917730i	$0.369980\pi$
$151$	41850.3i	1.83546i	−0.397205	+	0.917730i	$0.630020\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 37934.0i	− 1.57894i
$156$	0	0
$157$	25172.7	1.02125	0.510623	−	0.859804i	$-0.329415\pi$
$157$	25172.7	1.02125	0.510623	+	0.859804i	$0.329415\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−70235.0	−2.70958
$162$	0	0
$163$	4059.59i	0.152794i	0.997077	+	0.0763971i	$0.0243417\pi$
$163$	4059.59i	0.152794i	−0.997077	+	0.0763971i	$0.975658\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3210.38i	0.115113i	0.998342	+	0.0575565i	$0.0183309\pi$
$167$	3210.38i	0.115113i	−0.998342	+	0.0575565i	$0.981669\pi$
$168$	0	0
$169$	−14241.5	−0.498636
$170$	0	0
$171$	0	0
$172$	0	0
$173$	23891.0	0.798256	0.399128	−	0.916895i	$-0.369313\pi$
$173$	23891.0	0.798256	0.399128	+	0.916895i	$0.369313\pi$
$174$	0	0
$175$	87162.0i	2.84610i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 42054.9i	− 1.31254i	−0.754528	−	0.656268i	$-0.772134\pi$
$179$	− 42054.9i	− 1.31254i	0.754528	−	0.656268i	$-0.227866\pi$
$180$	0	0
$181$	2337.41	0.0713474	0.0356737	−	0.999363i	$-0.488642\pi$
$181$	2337.41	0.0713474	0.0356737	+	0.999363i	$0.488642\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−27676.4	−0.808661
$186$	0	0
$187$	− 9092.77i	− 0.260024i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	51678.6i	1.41659i	0.705917	+	0.708294i	$0.250535\pi$
$191$	51678.6i	1.41659i	−0.705917	+	0.708294i	$0.749465\pi$
$192$	0	0
$193$	−39478.2	−1.05985	−0.529923	−	0.848046i	$-0.677779\pi$
$193$	−39478.2	−1.05985	−0.529923	+	0.848046i	$0.677779\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−51268.2	−1.32104	−0.660519	−	0.750809i	$-0.729664\pi$
$197$	−51268.2	−1.32104	−0.660519	+	0.750809i	$0.729664\pi$
$198$	0	0
$199$	56884.0i	1.43643i	0.695822	+	0.718214i	$0.255040\pi$
$199$	56884.0i	1.43643i	−0.695822	+	0.718214i	$0.744960\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	102166.i	2.47922i
$204$	0	0
$205$	−80244.9	−1.90946
$206$	0	0
$207$	0	0
$208$	0	0
$209$	34812.7	0.796977
$210$	0	0
$211$	− 5262.54i	− 0.118203i	−0.998252	−	0.0591017i	$-0.981176\pi$
$211$	− 5262.54i	− 0.118203i	0.998252	−	0.0591017i	$-0.0188236\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 75113.4i	− 1.62495i
$216$	0	0
$217$	−92302.8	−1.96018
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−17696.5	−0.362329
$222$	0	0
$223$	22951.7i	0.461535i	0.973009	+	0.230767i	$0.0741236\pi$
$223$	22951.7i	0.461535i	−0.973009	+	0.230767i	$0.925876\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19133.2i	0.371309i	0.982615	+	0.185655i	$0.0594406\pi$
$227$	19133.2i	0.371309i	−0.982615	+	0.185655i	$0.940559\pi$
$228$	0	0
$229$	−68010.9	−1.29690	−0.648452	−	0.761256i	$-0.724583\pi$
$229$	−68010.9	−1.29690	−0.648452	+	0.761256i	$0.724583\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5496.60	0.101247	0.0506235	−	0.998718i	$-0.483879\pi$
$233$	5496.60	0.101247	0.0506235	+	0.998718i	$0.483879\pi$
$234$	0	0
$235$	− 77021.7i	− 1.39469i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 41030.8i	− 0.718315i	−0.933277	−	0.359157i	$-0.883064\pi$
$239$	− 41030.8i	− 0.718315i	0.933277	−	0.359157i	$-0.116936\pi$
$240$	0	0
$241$	71605.5	1.23286	0.616428	−	0.787411i	$-0.288579\pi$
$241$	71605.5	1.23286	0.616428	+	0.787411i	$0.288579\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	263052.	4.38237
$246$	0	0
$247$	− 67753.3i	− 1.11055i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6291.66i	0.0998660i	0.998753	+	0.0499330i	$0.0159008\pi$
$251$	6291.66i	0.0998660i	−0.998753	+	0.0499330i	$0.984099\pi$
$252$	0	0
$253$	45249.8	0.706929
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10475.8	0.158607	0.0793033	−	0.996851i	$-0.474730\pi$
$257$	10475.8	0.158607	0.0793033	+	0.996851i	$0.474730\pi$
$258$	0	0
$259$	67343.5i	1.00391i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 25865.1i	− 0.373941i	−0.982365	−	0.186971i	$-0.940133\pi$
$263$	− 25865.1i	− 0.373941i	0.982365	−	0.186971i	$-0.0598669\pi$
$264$	0	0
$265$	−137192.	−1.95361
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−55591.1	−0.768246	−0.384123	−	0.923282i	$-0.625496\pi$
$269$	−55591.1	−0.768246	−0.384123	+	0.923282i	$0.625496\pi$
$270$	0	0
$271$	49383.5i	0.672424i	0.941786	+	0.336212i	$0.109146\pi$
$271$	49383.5i	0.672424i	−0.941786	+	0.336212i	$0.890854\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 56155.2i	− 0.742548i
$276$	0	0
$277$	−27278.6	−0.355519	−0.177759	−	0.984074i	$-0.556885\pi$
$277$	−27278.6	−0.355519	−0.177759	+	0.984074i	$0.556885\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	53783.9	0.681145	0.340573	−	0.940218i	$-0.389379\pi$
$281$	53783.9	0.681145	0.340573	+	0.940218i	$0.389379\pi$
$282$	0	0
$283$	45742.8i	0.571149i	0.958356	+	0.285575i	$0.0921844\pi$
$283$	45742.8i	0.571149i	−0.958356	+	0.285575i	$0.907816\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	195255.i	2.37050i
$288$	0	0
$289$	−61650.9	−0.738149
$290$	0	0
$291$	0	0
$292$	0	0
$293$	148321.	1.72770	0.863849	−	0.503750i	$-0.168047\pi$
$293$	148321.	1.72770	0.863849	+	0.503750i	$0.168047\pi$
$294$	0	0
$295$	− 221419.i	− 2.54432i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 88066.1i	− 0.985068i
$300$	0	0
$301$	−182769.	−2.01730
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−169340.	−1.82037
$306$	0	0
$307$	76869.7i	0.815602i	0.913071	+	0.407801i	$0.133704\pi$
$307$	76869.7i	0.815602i	−0.913071	+	0.407801i	$0.866296\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 120301.i	− 1.24379i	−0.783100	−	0.621896i	$-0.786363\pi$
$311$	− 120301.i	− 1.24379i	0.783100	−	0.621896i	$-0.213637\pi$
$312$	0	0
$313$	21203.1	0.216427	0.108213	−	0.994128i	$-0.465487\pi$
$313$	21203.1	0.216427	0.108213	+	0.994128i	$0.465487\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	136811.	1.36146	0.680728	−	0.732536i	$-0.261663\pi$
$317$	136811.	1.36146	0.680728	+	0.732536i	$0.261663\pi$
$318$	0	0
$319$	− 65822.0i	− 0.646830i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	83732.1i	0.802577i
$324$	0	0
$325$	−109290.	−1.03470
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−187413.	−1.73144
$330$	0	0
$331$	− 15705.6i	− 0.143350i	−0.997428	−	0.0716751i	$-0.977166\pi$
$331$	− 15705.6i	− 0.143350i	0.997428	−	0.0716751i	$-0.0228345\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 94049.8i	− 0.838047i
$336$	0	0
$337$	−148152.	−1.30451	−0.652257	−	0.757998i	$-0.726178\pi$
$337$	−148152.	−1.30451	−0.652257	+	0.757998i	$0.726178\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	59467.3	0.511410
$342$	0	0
$343$	− 410929.i	− 3.49284i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 128291.i	− 1.06546i	−0.846284	−	0.532732i	$-0.821165\pi$
$347$	− 128291.i	− 1.06546i	0.846284	−	0.532732i	$-0.178835\pi$
$348$	0	0
$349$	−119096.	−0.977792	−0.488896	−	0.872342i	$-0.662600\pi$
$349$	−119096.	−0.977792	−0.488896	+	0.872342i	$0.662600\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	207280.	1.66344	0.831722	−	0.555193i	$-0.187355\pi$
$353$	207280.	1.66344	0.831722	+	0.555193i	$0.187355\pi$
$354$	0	0
$355$	− 183056.i	− 1.45253i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 67284.6i	− 0.522067i	−0.965330	−	0.261034i	$-0.915937\pi$
$359$	− 67284.6i	− 0.522067i	0.965330	−	0.261034i	$-0.0840634\pi$
$360$	0	0
$361$	−190257.	−1.45991
$362$	0	0
$363$	0	0
$364$	0	0
$365$	105413.	0.791239
$366$	0	0
$367$	171363.i	1.27228i	0.771572	+	0.636142i	$0.219471\pi$
$367$	171363.i	1.27228i	−0.771572	+	0.636142i	$0.780529\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	333823.i	2.42531i
$372$	0	0
$373$	132742.	0.954094	0.477047	−	0.878878i	$-0.341707\pi$
$373$	132742.	0.954094	0.477047	+	0.878878i	$0.341707\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−128104.	−0.901323
$378$	0	0
$379$	195728.i	1.36262i	0.731996	+	0.681309i	$0.238589\pi$
$379$	195728.i	1.36262i	−0.731996	+	0.681309i	$0.761411\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	277896.i	1.89446i	0.320556	+	0.947230i	$0.396130\pi$
$383$	277896.i	1.89446i	−0.320556	+	0.947230i	$0.603870\pi$
$384$	0	0
$385$	−230145.	−1.55267
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−130440.	−0.862009	−0.431005	−	0.902350i	$-0.641841\pi$
$389$	−130440.	−0.862009	−0.431005	+	0.902350i	$0.641841\pi$
$390$	0	0
$391$	108835.i	0.711896i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 16702.8i	− 0.107052i
$396$	0	0
$397$	62335.7	0.395509	0.197754	−	0.980252i	$-0.436635\pi$
$397$	62335.7	0.395509	0.197754	+	0.980252i	$0.436635\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−111606.	−0.694065	−0.347033	−	0.937853i	$-0.612811\pi$
$401$	−111606.	−0.694065	−0.347033	+	0.937853i	$0.612811\pi$
$402$	0	0
$403$	− 115736.i	− 0.712624i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 43386.9i	− 0.261921i
$408$	0	0
$409$	168750.	1.00878	0.504390	−	0.863476i	$-0.331717\pi$
$409$	168750.	1.00878	0.504390	+	0.863476i	$0.331717\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−538767.	−3.15864
$414$	0	0
$415$	274325.i	1.59283i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 24779.9i	− 0.141147i	−0.997507	−	0.0705736i	$-0.977517\pi$
$419$	− 24779.9i	− 0.141147i	0.997507	−	0.0705736i	$-0.0224830\pi$
$420$	0	0
$421$	−85242.0	−0.480938	−0.240469	−	0.970657i	$-0.577301\pi$
$421$	−85242.0	−0.480938	−0.240469	+	0.970657i	$0.577301\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	135065.	0.747766
$426$	0	0
$427$	412046.i	2.25990i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 240656.i	− 1.29551i	−0.761847	−	0.647757i	$-0.775707\pi$
$431$	− 240656.i	− 1.29551i	0.761847	−	0.647757i	$-0.224293\pi$
$432$	0	0
$433$	155361.	0.828640	0.414320	−	0.910131i	$-0.364019\pi$
$433$	155361.	0.828640	0.414320	+	0.910131i	$0.364019\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−416689.	−2.18197
$438$	0	0
$439$	70986.9i	0.368340i	0.982894	+	0.184170i	$0.0589598\pi$
$439$	70986.9i	0.368340i	−0.982894	+	0.184170i	$0.941040\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 101898.i	− 0.519226i	−0.965713	−	0.259613i	$-0.916405\pi$
$443$	− 101898.i	− 0.519226i	0.965713	−	0.259613i	$-0.0835950\pi$
$444$	0	0
$445$	279641.	1.41215
$446$	0	0
$447$	0	0
$448$	0	0
$449$	218354.	1.08310	0.541551	−	0.840668i	$-0.317837\pi$
$449$	218354.	1.08310	0.541551	+	0.840668i	$0.317837\pi$
$450$	0	0
$451$	− 125796.i	− 0.618462i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	447913.i	2.16357i
$456$	0	0
$457$	−310389.	−1.48619	−0.743095	−	0.669186i	$-0.766643\pi$
$457$	−310389.	−1.48619	−0.743095	+	0.669186i	$0.766643\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	88024.7	0.414193	0.207097	−	0.978321i	$-0.433599\pi$
$461$	88024.7	0.414193	0.207097	+	0.978321i	$0.433599\pi$
$462$	0	0
$463$	− 87008.4i	− 0.405881i	−0.979191	−	0.202941i	$-0.934950\pi$
$463$	− 87008.4i	− 0.405881i	0.979191	−	0.202941i	$-0.0650499\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 169998.i	− 0.779488i	−0.920923	−	0.389744i	$-0.872564\pi$
$467$	− 169998.i	− 0.779488i	0.920923	−	0.389744i	$-0.127436\pi$
$468$	0	0
$469$	−228846.	−1.04039
$470$	0	0
$471$	0	0
$472$	0	0
$473$	117751.	0.526313
$474$	0	0
$475$	517113.i	2.29192i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 392018.i	− 1.70858i	−0.519799	−	0.854288i	$-0.673993\pi$
$479$	− 392018.i	− 1.70858i	0.519799	−	0.854288i	$-0.326007\pi$
$480$	0	0
$481$	−84440.6	−0.364973
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−89340.4	−0.379808
$486$	0	0
$487$	191123.i	0.805851i	0.915233	+	0.402925i	$0.132007\pi$
$487$	191123.i	0.805851i	−0.915233	+	0.402925i	$0.867993\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 264539.i	− 1.09730i	−0.836051	−	0.548652i	$-0.815141\pi$
$491$	− 264539.i	− 1.09730i	0.836051	−	0.548652i	$-0.184859\pi$
$492$	0	0
$493$	158316.	0.651375
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−445419.	−1.80325
$498$	0	0
$499$	201153.i	0.807840i	0.914794	+	0.403920i	$0.132353\pi$
$499$	201153.i	0.807840i	−0.914794	+	0.403920i	$0.867647\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 377433.i	− 1.49178i	−0.666071	−	0.745889i	$-0.732025\pi$
$503$	− 377433.i	− 1.49178i	0.666071	−	0.745889i	$-0.267975\pi$
$504$	0	0
$505$	−410645.	−1.61022
$506$	0	0
$507$	0	0
$508$	0	0
$509$	84475.5	0.326058	0.163029	−	0.986621i	$-0.447874\pi$
$509$	84475.5	0.326058	0.163029	+	0.986621i	$0.447874\pi$
$510$	0	0
$511$	− 256495.i	− 0.982284i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	420674.i	1.58610i
$516$	0	0
$517$	120743.	0.451732
$518$	0	0
$519$	0	0
$520$	0	0
$521$	115569.	0.425760	0.212880	−	0.977078i	$-0.431716\pi$
$521$	115569.	0.425760	0.212880	+	0.977078i	$0.431716\pi$
$522$	0	0
$523$	197602.i	0.722416i	0.932485	+	0.361208i	$0.117636\pi$
$523$	197602.i	0.722416i	−0.932485	+	0.361208i	$0.882364\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	143032.i	0.515004i
$528$	0	0
$529$	−261774.	−0.935439
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−244827.	−0.861795
$534$	0	0
$535$	− 581970.i	− 2.03326i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	412373.i	1.41943i
$540$	0	0
$541$	402304.	1.37455	0.687274	−	0.726398i	$-0.258807\pi$
$541$	402304.	1.37455	0.687274	+	0.726398i	$0.258807\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−121905.	−0.410421
$546$	0	0
$547$	− 353602.i	− 1.18179i	−0.806748	−	0.590895i	$-0.798775\pi$
$547$	− 353602.i	− 1.18179i	0.806748	−	0.590895i	$-0.201225\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	606132.i	1.99647i
$552$	0	0
$553$	−40642.1	−0.132900
$554$	0	0
$555$	0	0
$556$	0	0
$557$	85669.4	0.276131	0.138065	−	0.990423i	$-0.455912\pi$
$557$	85669.4	0.276131	0.138065	+	0.990423i	$0.455912\pi$
$558$	0	0
$559$	− 229170.i	− 0.733390i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 268439.i	− 0.846893i	−0.905921	−	0.423446i	$-0.860820\pi$
$563$	− 268439.i	− 0.846893i	0.905921	−	0.423446i	$-0.139180\pi$
$564$	0	0
$565$	895278.	2.80454
$566$	0	0
$567$	0	0
$568$	0	0
$569$	460185.	1.42137	0.710687	−	0.703509i	$-0.248384\pi$
$569$	460185.	1.42137	0.710687	+	0.703509i	$0.248384\pi$
$570$	0	0
$571$	− 312881.i	− 0.959636i	−0.877368	−	0.479818i	$-0.840703\pi$
$571$	− 312881.i	− 0.959636i	0.877368	−	0.479818i	$-0.159297\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	672147.i	2.03296i
$576$	0	0
$577$	629966.	1.89219	0.946097	−	0.323884i	$-0.104989\pi$
$577$	629966.	1.89219	0.946097	+	0.323884i	$0.104989\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	667501.	1.97742
$582$	0	0
$583$	− 215070.i	− 0.632764i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 575551.i	− 1.67035i	−0.549984	−	0.835175i	$-0.685366\pi$
$587$	− 575551.i	− 1.67035i	0.549984	−	0.835175i	$-0.314634\pi$
$588$	0	0
$589$	−547613.	−1.57850
$590$	0	0
$591$	0	0
$592$	0	0
$593$	501917.	1.42732	0.713662	−	0.700490i	$-0.247035\pi$
$593$	501917.	1.42732	0.713662	+	0.700490i	$0.247035\pi$
$594$	0	0
$595$	− 553548.i	− 1.56358i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 606236.i	− 1.68962i	−0.535070	−	0.844808i	$-0.679715\pi$
$599$	− 606236.i	− 1.68962i	0.535070	−	0.844808i	$-0.320285\pi$
$600$	0	0
$601$	−25098.1	−0.0694850	−0.0347425	−	0.999396i	$-0.511061\pi$
$601$	−25098.1	−0.0694850	−0.0347425	+	0.999396i	$0.511061\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−425965.	−1.16376
$606$	0	0
$607$	484991.i	1.31631i	0.752884	+	0.658153i	$0.228662\pi$
$607$	484991.i	1.31631i	−0.752884	+	0.658153i	$0.771338\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 234993.i	− 0.629466i
$612$	0	0
$613$	−95820.3	−0.254998	−0.127499	−	0.991839i	$-0.540695\pi$
$613$	−95820.3	−0.254998	−0.127499	+	0.991839i	$0.540695\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	495158.	1.30069	0.650345	−	0.759639i	$-0.274624\pi$
$617$	495158.	1.30069	0.650345	+	0.759639i	$0.274624\pi$
$618$	0	0
$619$	164955.i	0.430512i	0.976558	+	0.215256i	$0.0690585\pi$
$619$	164955.i	0.430512i	−0.976558	+	0.215256i	$0.930941\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 680435.i	− 1.75312i
$624$	0	0
$625$	−127307.	−0.325905
$626$	0	0
$627$	0	0
$628$	0	0
$629$	104355.	0.263762
$630$	0	0
$631$	− 658369.i	− 1.65352i	−0.562552	−	0.826762i	$-0.690180\pi$
$631$	− 658369.i	− 1.65352i	0.562552	−	0.826762i	$-0.309820\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 32117.2i	− 0.0796509i
$636$	0	0
$637$	802568.	1.97790
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−257045.	−0.625594	−0.312797	−	0.949820i	$-0.601266\pi$
$641$	−257045.	−0.625594	−0.312797	+	0.949820i	$0.601266\pi$
$642$	0	0
$643$	− 188005.i	− 0.454723i	−0.973810	−	0.227362i	$-0.926990\pi$
$643$	− 188005.i	− 0.454723i	0.973810	−	0.227362i	$-0.0730099\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	737262.i	1.76122i	0.473844	+	0.880609i	$0.342866\pi$
$647$	737262.i	1.76122i	−0.473844	+	0.880609i	$0.657134\pi$
$648$	0	0
$649$	347108.	0.824090
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−414420.	−0.971884	−0.485942	−	0.873991i	$-0.661523\pi$
$653$	−414420.	−0.971884	−0.485942	+	0.873991i	$0.661523\pi$
$654$	0	0
$655$	− 45082.4i	− 0.105081i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 215117.i	− 0.495341i	−0.968844	−	0.247671i	$-0.920335\pi$
$659$	− 215117.i	− 0.495341i	0.968844	−	0.247671i	$-0.0796651\pi$
$660$	0	0
$661$	313934.	0.718515	0.359258	−	0.933238i	$-0.383030\pi$
$661$	313934.	0.718515	0.359258	+	0.933238i	$0.383030\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.11932e6	4.79241
$666$	0	0
$667$	787853.i	1.77090i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 265466.i	− 0.589608i
$672$	0	0
$673$	372599.	0.822643	0.411322	−	0.911490i	$-0.365067\pi$
$673$	372599.	0.822643	0.411322	+	0.911490i	$0.365067\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	287283.	0.626805	0.313402	−	0.949620i	$-0.398531\pi$
$677$	287283.	0.626805	0.313402	+	0.949620i	$0.398531\pi$
$678$	0	0
$679$	217387.i	0.471514i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	303751.i	0.651143i	0.945517	+	0.325572i	$0.105557\pi$
$683$	303751.i	0.651143i	−0.945517	+	0.325572i	$0.894443\pi$
$684$	0	0
$685$	−96636.7	−0.205950
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−418573.	−0.881724
$690$	0	0
$691$	− 605388.i	− 1.26788i	−0.773383	−	0.633940i	$-0.781437\pi$
$691$	− 605388.i	− 1.26788i	0.773383	−	0.633940i	$-0.218563\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.15136e6i	2.38365i
$696$	0	0
$697$	302566.	0.622809
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−607994.	−1.23727	−0.618633	−	0.785680i	$-0.712313\pi$
$701$	−607994.	−1.23727	−0.618633	+	0.785680i	$0.712313\pi$
$702$	0	0
$703$	399535.i	0.808433i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	999201.i	1.99900i
$708$	0	0
$709$	565849.	1.12566	0.562831	−	0.826572i	$-0.309712\pi$
$709$	565849.	1.12566	0.562831	+	0.826572i	$0.309712\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−711791.	−1.40015
$714$	0	0
$715$	− 288574.i	− 0.564475i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 24589.7i	− 0.0475660i	−0.999717	−	0.0237830i	$-0.992429\pi$
$719$	− 24589.7i	− 0.0475660i	0.999717	−	0.0237830i	$-0.00757107\pi$
$720$	0	0
$721$	1.02360e6	1.96907
$722$	0	0
$723$	0	0
$724$	0	0
$725$	977730.	1.86013
$726$	0	0
$727$	879787.i	1.66459i	0.554329	+	0.832297i	$0.312975\pi$
$727$	879787.i	1.66459i	−0.554329	+	0.832297i	$0.687025\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	283217.i	0.530011i
$732$	0	0
$733$	−104477.	−0.194452	−0.0972260	−	0.995262i	$-0.530997\pi$
$733$	−104477.	−0.194452	−0.0972260	+	0.995262i	$0.530997\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	147437.	0.271439
$738$	0	0
$739$	− 588536.i	− 1.07767i	−0.842412	−	0.538833i	$-0.818865\pi$
$739$	− 588536.i	− 1.07767i	0.842412	−	0.538833i	$-0.181135\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 505323.i	− 0.915360i	−0.889117	−	0.457680i	$-0.848681\pi$
$743$	− 505323.i	− 0.915360i	0.889117	−	0.457680i	$-0.151319\pi$
$744$	0	0
$745$	4034.32	0.00726871
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.41608e6	−2.52420
$750$	0	0
$751$	− 74574.4i	− 0.132224i	−0.997812	−	0.0661119i	$-0.978941\pi$
$751$	− 74574.4i	− 0.132224i	0.997812	−	0.0661119i	$-0.0210594\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1.64142e6i	− 2.87957i
$756$	0	0
$757$	55418.8	0.0967086	0.0483543	−	0.998830i	$-0.484602\pi$
$757$	55418.8	0.0967086	0.0483543	+	0.998830i	$0.484602\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−78533.4	−0.135608	−0.0678040	−	0.997699i	$-0.521599\pi$
$761$	−78533.4	−0.135608	−0.0678040	+	0.997699i	$0.521599\pi$
$762$	0	0
$763$	296626.i	0.509518i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 675548.i	− 1.14833i
$768$	0	0
$769$	−55527.1	−0.0938972	−0.0469486	−	0.998897i	$-0.514950\pi$
$769$	−55527.1	−0.0938972	−0.0469486	+	0.998897i	$0.514950\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−290784.	−0.486645	−0.243322	−	0.969945i	$-0.578237\pi$
$773$	−290784.	−0.486645	−0.243322	+	0.969945i	$0.578237\pi$
$774$	0	0
$775$	883336.i	1.47069i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.15841e6i	1.90892i
$780$	0	0
$781$	286967.	0.470468
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−987307.	−1.60219
$786$	0	0
$787$	286886.i	0.463191i	0.972812	+	0.231596i	$0.0743946\pi$
$787$	286886.i	0.463191i	−0.972812	+	0.231596i	$0.925605\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 2.17843e6i	− 3.48170i
$792$	0	0
$793$	−516655.	−0.821588
$794$	0	0
$795$	0	0
$796$	0	0
$797$	284903.	0.448518	0.224259	−	0.974530i	$-0.428004\pi$
$797$	284903.	0.448518	0.224259	+	0.974530i	$0.428004\pi$
$798$	0	0
$799$	290413.i	0.454907i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	165250.i	0.256278i
$804$	0	0
$805$	2.75471e6	4.25093
$806$	0	0
$807$	0	0
$808$	0	0
$809$	824554.	1.25986	0.629929	−	0.776652i	$-0.283084\pi$
$809$	824554.	1.25986	0.629929	+	0.776652i	$0.283084\pi$
$810$	0	0
$811$	− 377547.i	− 0.574022i	−0.957927	−	0.287011i	$-0.907338\pi$
$811$	− 377547.i	− 0.574022i	0.957927	−	0.287011i	$-0.0926617\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 159222.i	− 0.239711i
$816$	0	0
$817$	−1.08433e6	−1.62449
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−444605.	−0.659611	−0.329805	−	0.944049i	$-0.606983\pi$
$821$	−444605.	−0.659611	−0.329805	+	0.944049i	$0.606983\pi$
$822$	0	0
$823$	227777.i	0.336287i	0.985763	+	0.168143i	$0.0537772\pi$
$823$	227777.i	0.336287i	−0.985763	+	0.168143i	$0.946223\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.03418e6i	1.51212i	0.654504	+	0.756059i	$0.272878\pi$
$827$	1.03418e6i	1.51212i	−0.654504	+	0.756059i	$0.727122\pi$
$828$	0	0
$829$	1.25037e6	1.81940	0.909702	−	0.415261i	$-0.136310\pi$
$829$	1.25037e6	1.81940	0.909702	+	0.415261i	$0.136310\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−991845.	−1.42940
$834$	0	0
$835$	− 125915.i	− 0.180595i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	291698.i	0.414390i	0.978300	+	0.207195i	$0.0664335\pi$
$839$	291698.i	0.414390i	−0.978300	+	0.207195i	$0.933567\pi$
$840$	0	0
$841$	438760.	0.620347
$842$	0	0
$843$	0	0
$844$	0	0
$845$	558572.	0.782286
$846$	0	0
$847$	1.03648e6i	1.44475i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	519318.i	0.717091i
$852$	0	0
$853$	−495456.	−0.680937	−0.340469	−	0.940256i	$-0.610586\pi$
$853$	−495456.	−0.680937	−0.340469	+	0.940256i	$0.610586\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−748306.	−1.01887	−0.509434	−	0.860510i	$-0.670145\pi$
$857$	−748306.	−1.01887	−0.509434	+	0.860510i	$0.670145\pi$
$858$	0	0
$859$	1.01314e6i	1.37305i	0.727108	+	0.686523i	$0.240864\pi$
$859$	1.01314e6i	1.37305i	−0.727108	+	0.686523i	$0.759136\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	517950.i	0.695450i	0.937597	+	0.347725i	$0.113046\pi$
$863$	517950.i	0.695450i	−0.937597	+	0.347725i	$0.886954\pi$
$864$	0	0
$865$	−937036.	−1.25235
$866$	0	0
$867$	0	0
$868$	0	0
$869$	26184.2	0.0346737
$870$	0	0
$871$	− 286945.i	− 0.378236i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.07918e6i	− 1.40954i
$876$	0	0
$877$	712741.	0.926686	0.463343	−	0.886179i	$-0.346650\pi$
$877$	712741.	0.926686	0.463343	+	0.886179i	$0.346650\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	154178.	0.198641	0.0993207	−	0.995055i	$-0.468333\pi$
$881$	154178.	0.198641	0.0993207	+	0.995055i	$0.468333\pi$
$882$	0	0
$883$	671899.i	0.861753i	0.902411	+	0.430876i	$0.141796\pi$
$883$	671899.i	0.861753i	−0.902411	+	0.430876i	$0.858204\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 166555.i	− 0.211695i	−0.994382	−	0.105848i	$-0.966244\pi$
$887$	− 166555.i	− 0.211695i	0.994382	−	0.105848i	$-0.0337555\pi$
$888$	0	0
$889$	−78149.1	−0.0988827
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.11188e6	−1.39430
$894$	0	0
$895$	1.64945e6i	2.05917i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.03540e6i	1.28111i
$900$	0	0
$901$	517288.	0.637211
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−91676.4	−0.111934
$906$	0	0
$907$	− 405449.i	− 0.492858i	−0.969161	−	0.246429i	$-0.920743\pi$
$907$	− 405449.i	− 0.492858i	0.969161	−	0.246429i	$-0.0792572\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.02635e6i	1.23669i	0.785909	+	0.618343i	$0.212196\pi$
$911$	1.02635e6i	1.23669i	−0.785909	+	0.618343i	$0.787804\pi$
$912$	0	0
$913$	−430046.	−0.515910
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−109697.	−0.130453
$918$	0	0
$919$	− 750897.i	− 0.889098i	−0.895755	−	0.444549i	$-0.853364\pi$
$919$	− 750897.i	− 0.889098i	0.895755	−	0.444549i	$-0.146636\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 558501.i	− 0.655572i
$924$	0	0
$925$	644476.	0.753223
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−803129.	−0.930580	−0.465290	−	0.885158i	$-0.654050\pi$
$929$	−803129.	−0.930580	−0.465290	+	0.885158i	$0.654050\pi$
$930$	0	0
$931$	− 3.79739e6i	− 4.38113i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	356630.i	0.407939i
$936$	0	0
$937$	−412932.	−0.470326	−0.235163	−	0.971956i	$-0.575562\pi$
$937$	−412932.	−0.470326	−0.235163	+	0.971956i	$0.575562\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−305243.	−0.344720	−0.172360	−	0.985034i	$-0.555139\pi$
$941$	−305243.	−0.344720	−0.172360	+	0.985034i	$0.555139\pi$
$942$	0	0
$943$	1.50571e6i	1.69324i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 713232.i	− 0.795300i	−0.917537	−	0.397650i	$-0.869826\pi$
$947$	− 713232.i	− 0.795300i	0.917537	−	0.397650i	$-0.130174\pi$
$948$	0	0
$949$	321614.	0.357110
$950$	0	0
$951$	0	0
$952$	0	0
$953$	69136.5	0.0761241	0.0380620	−	0.999275i	$-0.487882\pi$
$953$	69136.5	0.0761241	0.0380620	+	0.999275i	$0.487882\pi$
$954$	0	0
$955$	− 2.02690e6i	− 2.22242i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	235141.i	0.255677i
$960$	0	0
$961$	−11914.8	−0.0129014
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.54839e6	1.66274
$966$	0	0
$967$	− 385873.i	− 0.412659i	−0.978483	−	0.206329i	$-0.933848\pi$
$967$	− 385873.i	− 0.412659i	0.978483	−	0.206329i	$-0.0661518\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 345117.i	− 0.366039i	−0.983109	−	0.183020i	$-0.941413\pi$
$971$	− 345117.i	− 0.366039i	0.983109	−	0.183020i	$-0.0585872\pi$
$972$	0	0
$973$	2.80155e6	2.95918
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.14503e6	−1.19957	−0.599786	−	0.800160i	$-0.704748\pi$
$977$	−1.14503e6	−1.19957	−0.599786	+	0.800160i	$0.704748\pi$
$978$	0	0
$979$	438379.i	0.457388i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	951377.i	0.984568i	0.870435	+	0.492284i	$0.163838\pi$
$983$	951377.i	0.984568i	−0.870435	+	0.492284i	$0.836162\pi$
$984$	0	0
$985$	2.01080e6	2.07251
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.40942e6	−1.44095
$990$	0	0
$991$	− 1.15391e6i	− 1.17497i	−0.809236	−	0.587484i	$-0.800119\pi$
$991$	− 1.15391e6i	− 1.17497i	0.809236	−	0.587484i	$-0.199881\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2.23106e6i	− 2.25354i
$996$	0	0
$997$	1.36515e6	1.37338	0.686688	−	0.726953i	$-0.259064\pi$
$997$	1.36515e6	1.37338	0.686688	+	0.726953i	$0.259064\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.5.g.c.1135.2	yes	4
3.2	odd	2		1296.5.g.e.1135.4	yes	4
4.3	odd	2	inner	1296.5.g.c.1135.1	✓	4
12.11	even	2		1296.5.g.e.1135.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1296.5.g.c.1135.1	✓	4	4.3	odd	2	inner
1296.5.g.c.1135.2	yes	4	1.1	even	1	trivial
1296.5.g.e.1135.3	yes	4	12.11	even	2
1296.5.g.e.1135.4	yes	4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1296.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-39.2213 q^{5} +95.4351i q^{7} -61.4853i q^{11} -119.664 q^{13} +147.885 q^{17} +566.196i q^{19} +735.945i q^{23} +913.312 q^{25} +1070.53 q^{29} +967.179i q^{31} -3743.09i q^{35} +705.648 q^{37} +2045.95 q^{41} +1915.12i q^{43} +1963.77i q^{47} -6706.85 q^{49} +3497.90 q^{53} +2411.53i q^{55} +5645.38i q^{59} +4317.55 q^{61} +4693.38 q^{65} +2397.93i q^{67} +4667.25i q^{71} -2687.64 q^{73} +5867.85 q^{77} +425.861i q^{79} -6994.30i q^{83} -5800.25 q^{85} -7129.82 q^{89} -11420.1i q^{91} -22207.0i q^{95} +2277.85 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 48 q^{5} - 152 q^{13} + 156 q^{17} + 1040 q^{25} + 1560 q^{29} + 536 q^{37} + 2304 q^{41} - 9188 q^{49} + 2232 q^{53} + 3224 q^{61} + 10716 q^{65} - 7484 q^{73} + 5832 q^{77} - 13728 q^{85} - 20244 q^{89}+ \cdots - 8528 q^{97}+O(q^{100}) \) 4 * q - 48 * q^5 - 152 * q^13 + 156 * q^17 + 1040 * q^25 + 1560 * q^29 + 536 * q^37 + 2304 * q^41 - 9188 * q^49 + 2232 * q^53 + 3224 * q^61 + 10716 * q^65 - 7484 * q^73 + 5832 * q^77 - 13728 * q^85 - 20244 * q^89 - 8528 * q^97

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