LMFDB - Embedded newform 5472.2.g.b.2737.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5472,2,Mod(2737,5472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

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

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5472.2737"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 4x^12 + 4x^11 - 10x^10 + 24x^9 - 40x^8 + 48x^7 - 40x^6 + 32x^5 + 64x^4 - 128x^3 + 192x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2737.5
Root			$-0.466170 - 1.33517i$ of defining polynomial
Character	$\chi$	$=$	5472.2737
Dual form			5472.2.g.b.2737.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.10882i q^{5} +2.73436 q^{7} -4.66474i q^{11} -4.47791i q^{13} +6.85237 q^{17} -1.00000i q^{19} -1.20416 q^{23} +0.552894 q^{25} -9.57484i q^{29} +5.35842 q^{31} -5.76625i q^{35} -1.09693i q^{37} -7.33797 q^{41} +7.64408i q^{43} +7.56486 q^{47} +0.476702 q^{49} +3.11949i q^{53} -9.83708 q^{55} +10.2442i q^{59} -0.722061i q^{61} -9.44309 q^{65} -6.13698i q^{67} +4.62247 q^{71} -6.19270 q^{73} -12.7551i q^{77} +3.26723 q^{79} +8.97934i q^{83} -14.4504i q^{85} -0.620707 q^{89} -12.2442i q^{91} -2.10882 q^{95} -1.67284 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{7} + 8 q^{17} - 24 q^{25} - 16 q^{31} - 16 q^{41} + 24 q^{47} + 24 q^{49} - 16 q^{55} - 16 q^{65} + 48 q^{71} + 48 q^{79} + 16 q^{89} + 16 q^{95} + 32 q^{97}+O(q^{100}) $ 16 * q + 8 * q^7 + 8 * q^17 - 24 * q^25 - 16 * q^31 - 16 * q^41 + 24 * q^47 + 24 * q^49 - 16 * q^55 - 16 * q^65 + 48 * q^71 + 48 * q^79 + 16 * q^89 + 16 * q^95 + 32 * 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
$5$	− 2.10882i	− 0.943091i	−0.881842	−	0.471546i	$-0.843696\pi$
$5$	− 2.10882i	− 0.943091i	0.881842	−	0.471546i	$-0.156304\pi$
$6$	0	0
$7$	2.73436	1.03349	0.516745	−	0.856140i	$-0.327144\pi$
$7$	2.73436	1.03349	0.516745	+	0.856140i	$0.327144\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.66474i	− 1.40647i	−0.710957	−	0.703236i	$-0.751738\pi$
$11$	− 4.66474i	− 1.40647i	0.710957	−	0.703236i	$-0.248262\pi$
$12$	0	0
$13$	− 4.47791i	− 1.24195i	−0.783831	−	0.620974i	$-0.786737\pi$
$13$	− 4.47791i	− 1.24195i	0.783831	−	0.620974i	$-0.213263\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.85237	1.66194	0.830972	−	0.556314i	$-0.187785\pi$
$17$	6.85237	1.66194	0.830972	+	0.556314i	$0.187785\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.20416	−0.251085	−0.125543	−	0.992088i	$-0.540067\pi$
$23$	−1.20416	−0.251085	−0.125543	+	0.992088i	$0.540067\pi$
$24$	0	0
$25$	0.552894	0.110579
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 9.57484i	− 1.77800i	−0.457904	−	0.889002i	$-0.651400\pi$
$29$	− 9.57484i	− 1.77800i	0.457904	−	0.889002i	$-0.348600\pi$
$30$	0	0
$31$	5.35842	0.962400	0.481200	−	0.876611i	$-0.340201\pi$
$31$	5.35842	0.962400	0.481200	+	0.876611i	$0.340201\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 5.76625i	− 0.974675i
$36$	0	0
$37$	− 1.09693i	− 0.180335i	−0.995927	−	0.0901674i	$-0.971260\pi$
$37$	− 1.09693i	− 0.180335i	0.995927	−	0.0901674i	$-0.0287402\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.33797	−1.14600	−0.573000	−	0.819556i	$-0.694220\pi$
$41$	−7.33797	−1.14600	−0.573000	+	0.819556i	$0.694220\pi$
$42$	0	0
$43$	7.64408i	1.16571i	0.812576	+	0.582856i	$0.198065\pi$
$43$	7.64408i	1.16571i	−0.812576	+	0.582856i	$0.801935\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.56486	1.10345	0.551724	−	0.834027i	$-0.313970\pi$
$47$	7.56486	1.10345	0.551724	+	0.834027i	$0.313970\pi$
$48$	0	0
$49$	0.476702	0.0681003
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.11949i	0.428495i	0.976779	+	0.214248i	$0.0687300\pi$
$53$	3.11949i	0.428495i	−0.976779	+	0.214248i	$0.931270\pi$
$54$	0	0
$55$	−9.83708	−1.32643
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.2442i	1.33368i	0.745201	+	0.666840i	$0.232354\pi$
$59$	10.2442i	1.33368i	−0.745201	+	0.666840i	$0.767646\pi$
$60$	0	0
$61$	− 0.722061i	− 0.0924504i	−0.998931	−	0.0462252i	$-0.985281\pi$
$61$	− 0.722061i	− 0.0924504i	0.998931	−	0.0462252i	$-0.0147192\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−9.44309	−1.17127
$66$	0	0
$67$	− 6.13698i	− 0.749752i	−0.927075	−	0.374876i	$-0.877685\pi$
$67$	− 6.13698i	− 0.749752i	0.927075	−	0.374876i	$-0.122315\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.62247	0.548587	0.274293	−	0.961646i	$-0.411556\pi$
$71$	4.62247	0.548587	0.274293	+	0.961646i	$0.411556\pi$
$72$	0	0
$73$	−6.19270	−0.724801	−0.362400	−	0.932022i	$-0.618043\pi$
$73$	−6.19270	−0.724801	−0.362400	+	0.932022i	$0.618043\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 12.7551i	− 1.45357i
$78$	0	0
$79$	3.26723	0.367592	0.183796	−	0.982964i	$-0.441161\pi$
$79$	3.26723	0.367592	0.183796	+	0.982964i	$0.441161\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.97934i	0.985611i	0.870140	+	0.492805i	$0.164029\pi$
$83$	8.97934i	0.985611i	−0.870140	+	0.492805i	$0.835971\pi$
$84$	0	0
$85$	− 14.4504i	− 1.56736i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.620707	−0.0657948	−0.0328974	−	0.999459i	$-0.510473\pi$
$89$	−0.620707	−0.0657948	−0.0328974	+	0.999459i	$0.510473\pi$
$90$	0	0
$91$	− 12.2442i	− 1.28354i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.10882	−0.216360
$96$	0	0
$97$	−1.67284	−0.169851	−0.0849257	−	0.996387i	$-0.527065\pi$
$97$	−1.67284	−0.169851	−0.0849257	+	0.996387i	$0.527065\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.30571i	0.726946i	0.931605	+	0.363473i	$0.118409\pi$
$101$	7.30571i	0.726946i	−0.931605	+	0.363473i	$0.881591\pi$
$102$	0	0
$103$	8.77332	0.864461	0.432231	−	0.901763i	$-0.357727\pi$
$103$	8.77332	0.864461	0.432231	+	0.901763i	$0.357727\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.91355i	0.958379i	0.877711	+	0.479190i	$0.159069\pi$
$107$	9.91355i	0.958379i	−0.877711	+	0.479190i	$0.840931\pi$
$108$	0	0
$109$	− 2.84589i	− 0.272587i	−0.990669	−	0.136293i	$-0.956481\pi$
$109$	− 2.84589i	− 0.272587i	0.990669	−	0.136293i	$-0.0435189\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.58487	0.149092	0.0745462	−	0.997218i	$-0.476249\pi$
$113$	1.58487	0.149092	0.0745462	+	0.997218i	$0.476249\pi$
$114$	0	0
$115$	2.53936i	0.236797i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	18.7368	1.71760
$120$	0	0
$121$	−10.7598	−0.978163
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.7100i	− 1.04738i
$126$	0	0
$127$	−14.2933	−1.26833	−0.634163	−	0.773199i	$-0.718655\pi$
$127$	−14.2933	−1.26833	−0.634163	+	0.773199i	$0.718655\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.609006i	0.0532091i	0.999646	+	0.0266046i	$0.00846949\pi$
$131$	0.609006i	0.0532091i	−0.999646	+	0.0266046i	$0.991531\pi$
$132$	0	0
$133$	− 2.73436i	− 0.237099i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.1589	−1.55142	−0.775710	−	0.631089i	$-0.782608\pi$
$137$	−18.1589	−1.55142	−0.775710	+	0.631089i	$0.782608\pi$
$138$	0	0
$139$	1.93421i	0.164058i	0.996630	+	0.0820289i	$0.0261400\pi$
$139$	1.93421i	0.164058i	−0.996630	+	0.0820289i	$0.973860\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−20.8883	−1.74677
$144$	0	0
$145$	−20.1916	−1.67682
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.5775i	1.27616i	0.769970	+	0.638080i	$0.220271\pi$
$149$	15.5775i	1.27616i	−0.769970	+	0.638080i	$0.779729\pi$
$150$	0	0
$151$	19.9746	1.62551	0.812756	−	0.582605i	$-0.197966\pi$
$151$	19.9746	1.62551	0.812756	+	0.582605i	$0.197966\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 11.2999i	− 0.907631i
$156$	0	0
$157$	− 20.8872i	− 1.66698i	−0.552536	−	0.833489i	$-0.686340\pi$
$157$	− 20.8872i	− 1.66698i	0.552536	−	0.833489i	$-0.313660\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.29261	−0.259494
$162$	0	0
$163$	2.54835i	0.199602i	0.995007	+	0.0998010i	$0.0318206\pi$
$163$	2.54835i	0.199602i	−0.995007	+	0.0998010i	$0.968179\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.6644	−1.44429	−0.722146	−	0.691740i	$-0.756844\pi$
$167$	−18.6644	−1.44429	−0.722146	+	0.691740i	$0.756844\pi$
$168$	0	0
$169$	−7.05167	−0.542436
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.42370i	0.640442i	0.947343	+	0.320221i	$0.103757\pi$
$173$	8.42370i	0.640442i	−0.947343	+	0.320221i	$0.896243\pi$
$174$	0	0
$175$	1.51181	0.114282
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.6809i	1.24679i	0.781908	+	0.623394i	$0.214247\pi$
$179$	16.6809i	1.24679i	−0.781908	+	0.623394i	$0.785753\pi$
$180$	0	0
$181$	− 4.65888i	− 0.346292i	−0.984896	−	0.173146i	$-0.944607\pi$
$181$	− 4.65888i	− 0.346292i	0.984896	−	0.173146i	$-0.0553932\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.31323	−0.170072
$186$	0	0
$187$	− 31.9645i	− 2.33748i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.0609	1.45155	0.725777	−	0.687930i	$-0.241480\pi$
$191$	20.0609	1.45155	0.725777	+	0.687930i	$0.241480\pi$
$192$	0	0
$193$	1.85117	0.133250	0.0666251	−	0.997778i	$-0.478777\pi$
$193$	1.85117	0.133250	0.0666251	+	0.997778i	$0.478777\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.224883i	0.0160222i	0.999968	+	0.00801112i	$0.00255005\pi$
$197$	0.224883i	0.0160222i	−0.999968	+	0.00801112i	$0.997450\pi$
$198$	0	0
$199$	1.77685	0.125957	0.0629787	−	0.998015i	$-0.479940\pi$
$199$	1.77685	0.125957	0.0629787	+	0.998015i	$0.479940\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 26.1810i	− 1.83755i
$204$	0	0
$205$	15.4744i	1.08078i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.66474	−0.322667
$210$	0	0
$211$	17.0194i	1.17166i	0.810433	+	0.585831i	$0.199232\pi$
$211$	17.0194i	1.17166i	−0.810433	+	0.585831i	$0.800768\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.1200	1.09937
$216$	0	0
$217$	14.6518	0.994630
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 30.6843i	− 2.06405i
$222$	0	0
$223$	−13.6634	−0.914971	−0.457486	−	0.889217i	$-0.651250\pi$
$223$	−13.6634	−0.914971	−0.457486	+	0.889217i	$0.651250\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.46152i	0.362494i	0.983438	+	0.181247i	$0.0580133\pi$
$227$	5.46152i	0.362494i	−0.983438	+	0.181247i	$0.941987\pi$
$228$	0	0
$229$	− 3.22791i	− 0.213306i	−0.994296	−	0.106653i	$-0.965987\pi$
$229$	− 3.22791i	− 0.213306i	0.994296	−	0.106653i	$-0.0340135\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.8344	1.16837	0.584185	−	0.811621i	$-0.301414\pi$
$233$	17.8344	1.16837	0.584185	+	0.811621i	$0.301414\pi$
$234$	0	0
$235$	− 15.9529i	− 1.04065i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.82556	0.118086	0.0590428	−	0.998255i	$-0.481195\pi$
$239$	1.82556	0.118086	0.0590428	+	0.998255i	$0.481195\pi$
$240$	0	0
$241$	6.47608	0.417161	0.208581	−	0.978005i	$-0.433116\pi$
$241$	6.47608	0.417161	0.208581	+	0.978005i	$0.433116\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.00528i	− 0.0642248i
$246$	0	0
$247$	−4.47791	−0.284923
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3.10899i	− 0.196238i	−0.995175	−	0.0981189i	$-0.968717\pi$
$251$	− 3.10899i	− 0.196238i	0.995175	−	0.0981189i	$-0.0312825\pi$
$252$	0	0
$253$	5.61711i	0.353145i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.75606	−0.359053	−0.179526	−	0.983753i	$-0.557457\pi$
$257$	−5.75606	−0.359053	−0.179526	+	0.983753i	$0.557457\pi$
$258$	0	0
$259$	− 2.99941i	− 0.186374i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.58595	−0.282782	−0.141391	−	0.989954i	$-0.545157\pi$
$263$	−4.58595	−0.282782	−0.141391	+	0.989954i	$0.545157\pi$
$264$	0	0
$265$	6.57844	0.404110
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 11.3913i	− 0.694542i	−0.937765	−	0.347271i	$-0.887108\pi$
$269$	− 11.3913i	− 0.694542i	0.937765	−	0.347271i	$-0.112892\pi$
$270$	0	0
$271$	−25.1252	−1.52625	−0.763124	−	0.646252i	$-0.776336\pi$
$271$	−25.1252	−1.52625	−0.763124	+	0.646252i	$0.776336\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.57910i	− 0.155526i
$276$	0	0
$277$	− 18.2998i	− 1.09953i	−0.835321	−	0.549763i	$-0.814718\pi$
$277$	− 18.2998i	− 1.09953i	0.835321	−	0.549763i	$-0.185282\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.0708	1.07802	0.539008	−	0.842301i	$-0.318799\pi$
$281$	18.0708	1.07802	0.539008	+	0.842301i	$0.318799\pi$
$282$	0	0
$283$	15.3972i	0.915269i	0.889140	+	0.457634i	$0.151303\pi$
$283$	15.3972i	0.915269i	−0.889140	+	0.457634i	$0.848697\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.0646	−1.18438
$288$	0	0
$289$	29.9550	1.76206
$290$	0	0
$291$	0	0
$292$	0	0
$293$	11.4407i	0.668374i	0.942507	+	0.334187i	$0.108462\pi$
$293$	11.4407i	0.668374i	−0.942507	+	0.334187i	$0.891538\pi$
$294$	0	0
$295$	21.6031	1.25778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.39214i	0.311835i
$300$	0	0
$301$	20.9016i	1.20475i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.52269	−0.0871892
$306$	0	0
$307$	− 21.3453i	− 1.21824i	−0.793077	−	0.609122i	$-0.791522\pi$
$307$	− 21.3453i	− 1.21824i	0.793077	−	0.609122i	$-0.208478\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.7295	−1.23216	−0.616082	−	0.787682i	$-0.711281\pi$
$311$	−21.7295	−1.23216	−0.616082	+	0.787682i	$0.711281\pi$
$312$	0	0
$313$	34.1009	1.92750	0.963748	−	0.266814i	$-0.0859710\pi$
$313$	34.1009	1.92750	0.963748	+	0.266814i	$0.0859710\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 22.2281i	− 1.24845i	−0.781244	−	0.624226i	$-0.785414\pi$
$317$	− 22.2281i	− 1.24845i	0.781244	−	0.624226i	$-0.214586\pi$
$318$	0	0
$319$	−44.6641	−2.50071
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6.85237i	− 0.381276i
$324$	0	0
$325$	− 2.47581i	− 0.137333i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	20.6850	1.14040
$330$	0	0
$331$	24.1969i	1.32998i	0.746852	+	0.664991i	$0.231564\pi$
$331$	24.1969i	1.32998i	−0.746852	+	0.664991i	$0.768436\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−12.9418	−0.707084
$336$	0	0
$337$	−34.2735	−1.86700	−0.933498	−	0.358583i	$-0.883260\pi$
$337$	−34.2735	−1.86700	−0.933498	+	0.358583i	$0.883260\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 24.9956i	− 1.35359i
$342$	0	0
$343$	−17.8370	−0.963108
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.9233i	− 1.44532i	−0.691206	−	0.722658i	$-0.742920\pi$
$347$	− 26.9233i	− 1.44532i	0.691206	−	0.722658i	$-0.257080\pi$
$348$	0	0
$349$	− 8.65769i	− 0.463436i	−0.972783	−	0.231718i	$-0.925565\pi$
$349$	− 8.65769i	− 0.463436i	0.972783	−	0.231718i	$-0.0744346\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.11265	0.485018	0.242509	−	0.970149i	$-0.422030\pi$
$353$	9.11265	0.485018	0.242509	+	0.970149i	$0.422030\pi$
$354$	0	0
$355$	− 9.74795i	− 0.517367i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.29978	0.226934	0.113467	−	0.993542i	$-0.463804\pi$
$359$	4.29978	0.226934	0.113467	+	0.993542i	$0.463804\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.0593i	0.683553i
$366$	0	0
$367$	21.2873	1.11119	0.555594	−	0.831454i	$-0.312491\pi$
$367$	21.2873	1.11119	0.555594	+	0.831454i	$0.312491\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.52980i	0.442845i
$372$	0	0
$373$	11.1076i	0.575129i	0.957761	+	0.287565i	$0.0928456\pi$
$373$	11.1076i	0.575129i	−0.957761	+	0.287565i	$0.907154\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−42.8753	−2.20819
$378$	0	0
$379$	− 20.6908i	− 1.06281i	−0.847117	−	0.531407i	$-0.821664\pi$
$379$	− 20.6908i	− 1.06281i	0.847117	−	0.531407i	$-0.178336\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12.1869	−0.622720	−0.311360	−	0.950292i	$-0.600785\pi$
$383$	−12.1869	−0.622720	−0.311360	+	0.950292i	$0.600785\pi$
$384$	0	0
$385$	−26.8981	−1.37085
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.1769i	1.32722i	0.748078	+	0.663611i	$0.230977\pi$
$389$	26.1769i	1.32722i	−0.748078	+	0.663611i	$0.769023\pi$
$390$	0	0
$391$	−8.25137	−0.417290
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 6.88999i	− 0.346673i
$396$	0	0
$397$	14.8451i	0.745055i	0.928021	+	0.372528i	$0.121509\pi$
$397$	14.8451i	0.745055i	−0.928021	+	0.372528i	$0.878491\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−34.4111	−1.71841	−0.859205	−	0.511632i	$-0.829041\pi$
$401$	−34.4111	−1.71841	−0.859205	+	0.511632i	$0.829041\pi$
$402$	0	0
$403$	− 23.9945i	− 1.19525i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.11691	−0.253636
$408$	0	0
$409$	1.43283	0.0708487	0.0354243	−	0.999372i	$-0.488722\pi$
$409$	1.43283	0.0708487	0.0354243	+	0.999372i	$0.488722\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	28.0113i	1.37834i
$414$	0	0
$415$	18.9358	0.929521
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 33.6886i	− 1.64579i	−0.568190	−	0.822897i	$-0.692356\pi$
$419$	− 33.6886i	− 1.64579i	0.568190	−	0.822897i	$-0.307644\pi$
$420$	0	0
$421$	− 0.237430i	− 0.0115716i	−0.999983	−	0.00578582i	$-0.998158\pi$
$421$	− 0.237430i	− 0.0115716i	0.999983	−	0.00578582i	$-0.00184169\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.78863	0.183776
$426$	0	0
$427$	− 1.97437i	− 0.0955465i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.0743	0.629766	0.314883	−	0.949130i	$-0.398035\pi$
$431$	13.0743	0.629766	0.314883	+	0.949130i	$0.398035\pi$
$432$	0	0
$433$	−9.67718	−0.465056	−0.232528	−	0.972590i	$-0.574700\pi$
$433$	−9.67718	−0.465056	−0.232528	+	0.972590i	$0.574700\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.20416i	0.0576030i
$438$	0	0
$439$	−17.7983	−0.849468	−0.424734	−	0.905318i	$-0.639632\pi$
$439$	−17.7983	−0.849468	−0.424734	+	0.905318i	$0.639632\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.30468i	0.157010i	0.996914	+	0.0785050i	$0.0250147\pi$
$443$	3.30468i	0.157010i	−0.996914	+	0.0785050i	$0.974985\pi$
$444$	0	0
$445$	1.30896i	0.0620505i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	29.6543	1.39947	0.699735	−	0.714402i	$-0.253301\pi$
$449$	29.6543	1.39947	0.699735	+	0.714402i	$0.253301\pi$
$450$	0	0
$451$	34.2297i	1.61182i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−25.8208	−1.21050
$456$	0	0
$457$	−20.2310	−0.946368	−0.473184	−	0.880964i	$-0.656895\pi$
$457$	−20.2310	−0.946368	−0.473184	+	0.880964i	$0.656895\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 41.2429i	− 1.92087i	−0.278502	−	0.960436i	$-0.589838\pi$
$461$	− 41.2429i	− 1.92087i	0.278502	−	0.960436i	$-0.410162\pi$
$462$	0	0
$463$	16.5232	0.767896	0.383948	−	0.923355i	$-0.374564\pi$
$463$	16.5232	0.767896	0.383948	+	0.923355i	$0.374564\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.9967i	1.15671i	0.815786	+	0.578355i	$0.196305\pi$
$467$	24.9967i	1.15671i	−0.815786	+	0.578355i	$0.803695\pi$
$468$	0	0
$469$	− 16.7807i	− 0.774860i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	35.6576	1.63954
$474$	0	0
$475$	− 0.552894i	− 0.0253685i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.0298	0.458273	0.229137	−	0.973394i	$-0.426410\pi$
$479$	10.0298	0.458273	0.229137	+	0.973394i	$0.426410\pi$
$480$	0	0
$481$	−4.91197	−0.223966
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.52772i	0.160185i
$486$	0	0
$487$	−41.6248	−1.88620	−0.943101	−	0.332508i	$-0.892105\pi$
$487$	−41.6248	−1.88620	−0.943101	+	0.332508i	$0.892105\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 34.6939i	− 1.56572i	−0.622200	−	0.782858i	$-0.713761\pi$
$491$	− 34.6939i	− 1.56572i	0.622200	−	0.782858i	$-0.286239\pi$
$492$	0	0
$493$	− 65.6104i	− 2.95494i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.6395	0.566959
$498$	0	0
$499$	− 14.1270i	− 0.632413i	−0.948690	−	0.316207i	$-0.897591\pi$
$499$	− 14.1270i	− 0.632413i	0.948690	−	0.316207i	$-0.102409\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.01157	0.134279	0.0671395	−	0.997744i	$-0.478613\pi$
$503$	3.01157	0.134279	0.0671395	+	0.997744i	$0.478613\pi$
$504$	0	0
$505$	15.4064	0.685576
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 2.38122i	− 0.105546i	−0.998607	−	0.0527730i	$-0.983194\pi$
$509$	− 2.38122i	− 0.105546i	0.998607	−	0.0527730i	$-0.0168060\pi$
$510$	0	0
$511$	−16.9330	−0.749074
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 18.5013i	− 0.815266i
$516$	0	0
$517$	− 35.2881i	− 1.55197i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17.9133	−0.784798	−0.392399	−	0.919795i	$-0.628355\pi$
$521$	−17.9133	−0.784798	−0.392399	+	0.919795i	$0.628355\pi$
$522$	0	0
$523$	− 18.9602i	− 0.829071i	−0.910033	−	0.414536i	$-0.863944\pi$
$523$	− 18.9602i	− 0.829071i	0.910033	−	0.414536i	$-0.136056\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36.7178	1.59945
$528$	0	0
$529$	−21.5500	−0.936956
$530$	0	0
$531$	0	0
$532$	0	0
$533$	32.8588i	1.42327i
$534$	0	0
$535$	20.9059	0.903839
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.22369i	− 0.0957811i
$540$	0	0
$541$	34.2027i	1.47049i	0.677802	+	0.735245i	$0.262933\pi$
$541$	34.2027i	1.47049i	−0.677802	+	0.735245i	$0.737067\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.00145	−0.257074
$546$	0	0
$547$	− 0.645113i	− 0.0275830i	−0.999905	−	0.0137915i	$-0.995610\pi$
$547$	− 0.645113i	− 0.0275830i	0.999905	−	0.0137915i	$-0.00439011\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.57484	−0.407902
$552$	0	0
$553$	8.93377	0.379903
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.2596i	0.434714i	0.976092	+	0.217357i	$0.0697436\pi$
$557$	10.2596i	0.434714i	−0.976092	+	0.217357i	$0.930256\pi$
$558$	0	0
$559$	34.2295	1.44775
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.9293i	0.544906i	0.962169	+	0.272453i	$0.0878350\pi$
$563$	12.9293i	0.544906i	−0.962169	+	0.272453i	$0.912165\pi$
$564$	0	0
$565$	− 3.34221i	− 0.140608i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.32637	−0.265215	−0.132608	−	0.991169i	$-0.542335\pi$
$569$	−6.32637	−0.265215	−0.132608	+	0.991169i	$0.542335\pi$
$570$	0	0
$571$	− 31.1997i	− 1.30566i	−0.757502	−	0.652832i	$-0.773581\pi$
$571$	− 31.1997i	− 1.30566i	0.757502	−	0.652832i	$-0.226419\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.665774	−0.0277647
$576$	0	0
$577$	5.48996	0.228550	0.114275	−	0.993449i	$-0.463545\pi$
$577$	5.48996	0.228550	0.114275	+	0.993449i	$0.463545\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	24.5527i	1.01862i
$582$	0	0
$583$	14.5516	0.602666
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.4576i	0.720552i	0.932846	+	0.360276i	$0.117317\pi$
$587$	17.4576i	0.720552i	−0.932846	+	0.360276i	$0.882683\pi$
$588$	0	0
$589$	− 5.35842i	− 0.220790i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.7657	0.688486	0.344243	−	0.938881i	$-0.388136\pi$
$593$	16.7657	0.688486	0.344243	+	0.938881i	$0.388136\pi$
$594$	0	0
$595$	− 39.5125i	− 1.61985i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.357982	−0.0146267	−0.00731337	−	0.999973i	$-0.502328\pi$
$599$	−0.357982	−0.0146267	−0.00731337	+	0.999973i	$0.502328\pi$
$600$	0	0
$601$	22.8949	0.933903	0.466952	−	0.884283i	$-0.345352\pi$
$601$	22.8949	0.933903	0.466952	+	0.884283i	$0.345352\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	22.6904i	0.922497i
$606$	0	0
$607$	−1.33917	−0.0543551	−0.0271775	−	0.999631i	$-0.508652\pi$
$607$	−1.33917	−0.0543551	−0.0271775	+	0.999631i	$0.508652\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 33.8747i	− 1.37043i
$612$	0	0
$613$	29.8667i	1.20631i	0.797625	+	0.603153i	$0.206089\pi$
$613$	29.8667i	1.20631i	−0.797625	+	0.603153i	$0.793911\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.4935	1.46917	0.734587	−	0.678514i	$-0.237376\pi$
$617$	36.4935	1.46917	0.734587	+	0.678514i	$0.237376\pi$
$618$	0	0
$619$	− 8.88669i	− 0.357186i	−0.983923	−	0.178593i	$-0.942845\pi$
$619$	− 8.88669i	− 0.357186i	0.983923	−	0.178593i	$-0.0571546\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.69723	−0.0679982
$624$	0	0
$625$	−21.9298	−0.877194
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 7.51659i	− 0.299706i
$630$	0	0
$631$	28.1724	1.12153	0.560763	−	0.827977i	$-0.310508\pi$
$631$	28.1724	1.12153	0.560763	+	0.827977i	$0.310508\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	30.1420i	1.19615i
$636$	0	0
$637$	− 2.13463i	− 0.0845771i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.58435	−0.141573	−0.0707867	−	0.997491i	$-0.522551\pi$
$641$	−3.58435	−0.141573	−0.0707867	+	0.997491i	$0.522551\pi$
$642$	0	0
$643$	19.8837i	0.784137i	0.919936	+	0.392069i	$0.128240\pi$
$643$	19.8837i	0.784137i	−0.919936	+	0.392069i	$0.871760\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	35.7712	1.40631	0.703156	−	0.711036i	$-0.251774\pi$
$647$	35.7712	1.40631	0.703156	+	0.711036i	$0.251774\pi$
$648$	0	0
$649$	47.7865	1.87578
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 8.57378i	− 0.335518i	−0.985828	−	0.167759i	$-0.946347\pi$
$653$	− 8.57378i	− 0.335518i	0.985828	−	0.167759i	$-0.0536531\pi$
$654$	0	0
$655$	1.28428	0.0501810
$656$	0	0
$657$	0	0
$658$	0	0
$659$	22.9993i	0.895927i	0.894052	+	0.447964i	$0.147851\pi$
$659$	22.9993i	0.895927i	−0.894052	+	0.447964i	$0.852149\pi$
$660$	0	0
$661$	28.4720i	1.10743i	0.832706	+	0.553716i	$0.186791\pi$
$661$	28.4720i	1.10743i	−0.832706	+	0.553716i	$0.813209\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.76625	−0.223606
$666$	0	0
$667$	11.5297i	0.446431i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.36822	−0.130029
$672$	0	0
$673$	6.63685	0.255832	0.127916	−	0.991785i	$-0.459171\pi$
$673$	6.63685	0.255832	0.127916	+	0.991785i	$0.459171\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.0823i	1.11772i	0.829261	+	0.558862i	$0.188762\pi$
$677$	29.0823i	1.11772i	−0.829261	+	0.558862i	$0.811238\pi$
$678$	0	0
$679$	−4.57415	−0.175540
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 4.63939i	− 0.177521i	−0.996053	−	0.0887607i	$-0.971709\pi$
$683$	− 4.63939i	− 0.177521i	0.996053	−	0.0887607i	$-0.0282906\pi$
$684$	0	0
$685$	38.2938i	1.46313i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13.9688	0.532169
$690$	0	0
$691$	5.11245i	0.194487i	0.995261	+	0.0972434i	$0.0310025\pi$
$691$	5.11245i	0.194487i	−0.995261	+	0.0972434i	$0.968997\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.07890	0.154721
$696$	0	0
$697$	−50.2825	−1.90459
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.27938i	0.199399i	0.995018	+	0.0996997i	$0.0317882\pi$
$701$	5.27938i	0.199399i	−0.995018	+	0.0996997i	$0.968212\pi$
$702$	0	0
$703$	−1.09693	−0.0413716
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.9764i	0.751290i
$708$	0	0
$709$	28.1441i	1.05697i	0.848942	+	0.528486i	$0.177240\pi$
$709$	28.1441i	1.05697i	−0.848942	+	0.528486i	$0.822760\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.45241	−0.241645
$714$	0	0
$715$	44.0495i	1.64736i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.51160	0.354723	0.177361	−	0.984146i	$-0.443244\pi$
$719$	9.51160	0.354723	0.177361	+	0.984146i	$0.443244\pi$
$720$	0	0
$721$	23.9894	0.893412
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 5.29387i	− 0.196609i
$726$	0	0
$727$	−7.06196	−0.261914	−0.130957	−	0.991388i	$-0.541805\pi$
$727$	−7.06196	−0.261914	−0.130957	+	0.991388i	$0.541805\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	52.3801i	1.93735i
$732$	0	0
$733$	− 7.62443i	− 0.281615i	−0.990037	−	0.140807i	$-0.955030\pi$
$733$	− 7.62443i	− 0.281615i	0.990037	−	0.140807i	$-0.0449698\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−28.6274	−1.05450
$738$	0	0
$739$	45.0131i	1.65583i	0.560851	+	0.827917i	$0.310474\pi$
$739$	45.0131i	1.65583i	−0.560851	+	0.827917i	$0.689526\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.3137	0.561805	0.280903	−	0.959736i	$-0.409366\pi$
$743$	15.3137	0.561805	0.280903	+	0.959736i	$0.409366\pi$
$744$	0	0
$745$	32.8501	1.20354
$746$	0	0
$747$	0	0
$748$	0	0
$749$	27.1072i	0.990475i
$750$	0	0
$751$	−4.09580	−0.149458	−0.0747290	−	0.997204i	$-0.523809\pi$
$751$	−4.09580	−0.149458	−0.0747290	+	0.997204i	$0.523809\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 42.1228i	− 1.53301i
$756$	0	0
$757$	9.51720i	0.345909i	0.984930	+	0.172954i	$0.0553313\pi$
$757$	9.51720i	0.345909i	−0.984930	+	0.172954i	$0.944669\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.4841	1.39505	0.697523	−	0.716562i	$-0.254285\pi$
$761$	38.4841	1.39505	0.697523	+	0.716562i	$0.254285\pi$
$762$	0	0
$763$	− 7.78167i	− 0.281715i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	45.8726	1.65636
$768$	0	0
$769$	15.0210	0.541670	0.270835	−	0.962626i	$-0.412700\pi$
$769$	15.0210	0.541670	0.270835	+	0.962626i	$0.412700\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 38.4471i	− 1.38285i	−0.722451	−	0.691423i	$-0.756984\pi$
$773$	− 38.4471i	− 1.38285i	0.722451	−	0.691423i	$-0.243016\pi$
$774$	0	0
$775$	2.96263	0.106421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.33797i	0.262910i
$780$	0	0
$781$	− 21.5626i	− 0.771572i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−44.0472	−1.57211
$786$	0	0
$787$	− 30.8457i	− 1.09953i	−0.835319	−	0.549765i	$-0.814717\pi$
$787$	− 30.8457i	− 1.09953i	0.835319	−	0.549765i	$-0.185283\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.33361	0.154085
$792$	0	0
$793$	−3.23332	−0.114819
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 34.1285i	− 1.20889i	−0.796646	−	0.604446i	$-0.793394\pi$
$797$	− 34.1285i	− 1.20889i	0.796646	−	0.604446i	$-0.206606\pi$
$798$	0	0
$799$	51.8372	1.83387
$800$	0	0
$801$	0	0
$802$	0	0
$803$	28.8873i	1.01941i
$804$	0	0
$805$	6.94351i	0.244727i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	25.9170	0.911192	0.455596	−	0.890187i	$-0.349426\pi$
$809$	25.9170	0.911192	0.455596	+	0.890187i	$0.349426\pi$
$810$	0	0
$811$	33.7874i	1.18644i	0.805041	+	0.593219i	$0.202143\pi$
$811$	33.7874i	1.18644i	−0.805041	+	0.593219i	$0.797857\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.37400	0.188243
$816$	0	0
$817$	7.64408	0.267432
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 27.6475i	− 0.964903i	−0.875923	−	0.482451i	$-0.839746\pi$
$821$	− 27.6475i	− 0.964903i	0.875923	−	0.482451i	$-0.160254\pi$
$822$	0	0
$823$	12.4983	0.435665	0.217832	−	0.975986i	$-0.430101\pi$
$823$	12.4983	0.435665	0.217832	+	0.975986i	$0.430101\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 9.03084i	− 0.314033i	−0.987596	−	0.157017i	$-0.949812\pi$
$827$	− 9.03084i	− 0.314033i	0.987596	−	0.157017i	$-0.0501876\pi$
$828$	0	0
$829$	− 55.7684i	− 1.93692i	−0.249175	−	0.968458i	$-0.580159\pi$
$829$	− 55.7684i	− 1.93692i	0.249175	−	0.968458i	$-0.419841\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.26654	0.113179
$834$	0	0
$835$	39.3597i	1.36210i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.3128	−0.494134	−0.247067	−	0.968998i	$-0.579467\pi$
$839$	−14.3128	−0.494134	−0.247067	+	0.968998i	$0.579467\pi$
$840$	0	0
$841$	−62.6776	−2.16130
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.8707i	0.511567i
$846$	0	0
$847$	−29.4211	−1.01092
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.32089i	0.0452794i
$852$	0	0
$853$	38.0325i	1.30221i	0.758988	+	0.651105i	$0.225694\pi$
$853$	38.0325i	1.30221i	−0.758988	+	0.651105i	$0.774306\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.1869	0.792048	0.396024	−	0.918240i	$-0.370390\pi$
$857$	23.1869	0.792048	0.396024	+	0.918240i	$0.370390\pi$
$858$	0	0
$859$	46.5870i	1.58953i	0.606919	+	0.794764i	$0.292405\pi$
$859$	46.5870i	1.58953i	−0.606919	+	0.794764i	$0.707595\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−54.3149	−1.84890	−0.924451	−	0.381301i	$-0.875476\pi$
$863$	−54.3149	−1.84890	−0.924451	+	0.381301i	$0.875476\pi$
$864$	0	0
$865$	17.7640	0.603995
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 15.2408i	− 0.517008i
$870$	0	0
$871$	−27.4808	−0.931153
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 32.0194i	− 1.08245i
$876$	0	0
$877$	− 1.80324i	− 0.0608910i	−0.999536	−	0.0304455i	$-0.990307\pi$
$877$	− 1.80324i	− 0.0608910i	0.999536	−	0.0304455i	$-0.00969260\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.6372	−0.425759	−0.212880	−	0.977078i	$-0.568284\pi$
$881$	−12.6372	−0.425759	−0.212880	+	0.977078i	$0.568284\pi$
$882$	0	0
$883$	39.9276i	1.34367i	0.740700	+	0.671835i	$0.234494\pi$
$883$	39.9276i	1.34367i	−0.740700	+	0.671835i	$0.765506\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.3642	−0.482304	−0.241152	−	0.970487i	$-0.577525\pi$
$887$	−14.3642	−0.482304	−0.241152	+	0.970487i	$0.577525\pi$
$888$	0	0
$889$	−39.0830	−1.31080
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 7.56486i	− 0.253148i
$894$	0	0
$895$	35.1769	1.17584
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 51.3060i	− 1.71115i
$900$	0	0
$901$	21.3759i	0.712135i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.82473	−0.326585
$906$	0	0
$907$	− 3.32787i	− 0.110500i	−0.998473	−	0.0552501i	$-0.982404\pi$
$907$	− 3.32787i	− 0.110500i	0.998473	−	0.0552501i	$-0.0175956\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−35.8252	−1.18694	−0.593471	−	0.804855i	$-0.702243\pi$
$911$	−35.8252	−1.18694	−0.593471	+	0.804855i	$0.702243\pi$
$912$	0	0
$913$	41.8863	1.38623
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.66524i	0.0549910i
$918$	0	0
$919$	48.3186	1.59388	0.796942	−	0.604055i	$-0.206449\pi$
$919$	48.3186	1.59388	0.796942	+	0.604055i	$0.206449\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 20.6990i	− 0.681316i
$924$	0	0
$925$	− 0.606487i	− 0.0199412i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	23.2127	0.761585	0.380792	−	0.924661i	$-0.375651\pi$
$929$	23.2127	0.761585	0.380792	+	0.924661i	$0.375651\pi$
$930$	0	0
$931$	− 0.476702i	− 0.0156233i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−67.4073	−2.20445
$936$	0	0
$937$	7.10467	0.232099	0.116050	−	0.993243i	$-0.462977\pi$
$937$	7.10467	0.232099	0.116050	+	0.993243i	$0.462977\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	55.5352i	1.81040i	0.424989	+	0.905199i	$0.360278\pi$
$941$	55.5352i	1.81040i	−0.424989	+	0.905199i	$0.639722\pi$
$942$	0	0
$943$	8.83612	0.287744
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.5988i	0.506894i	0.967349	+	0.253447i	$0.0815643\pi$
$947$	15.5988i	0.506894i	−0.967349	+	0.253447i	$0.918436\pi$
$948$	0	0
$949$	27.7304i	0.900165i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21.1243	−0.684282	−0.342141	−	0.939649i	$-0.611152\pi$
$953$	−21.1243	−0.684282	−0.342141	+	0.939649i	$0.611152\pi$
$954$	0	0
$955$	− 42.3047i	− 1.36895i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−49.6529	−1.60338
$960$	0	0
$961$	−2.28738	−0.0737863
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 3.90378i	− 0.125667i
$966$	0	0
$967$	−46.1207	−1.48314	−0.741570	−	0.670875i	$-0.765919\pi$
$967$	−46.1207	−1.48314	−0.741570	+	0.670875i	$0.765919\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 22.3253i	− 0.716454i	−0.933635	−	0.358227i	$-0.883381\pi$
$971$	− 22.3253i	− 0.716454i	0.933635	−	0.358227i	$-0.116619\pi$
$972$	0	0
$973$	5.28882i	0.169552i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.2650	−0.456379	−0.228190	−	0.973617i	$-0.573281\pi$
$977$	−14.2650	−0.456379	−0.228190	+	0.973617i	$0.573281\pi$
$978$	0	0
$979$	2.89544i	0.0925385i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	12.7242	0.405838	0.202919	−	0.979196i	$-0.434957\pi$
$983$	12.7242	0.405838	0.202919	+	0.979196i	$0.434957\pi$
$984$	0	0
$985$	0.474237	0.0151104
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 9.20472i	− 0.292693i
$990$	0	0
$991$	10.8169	0.343611	0.171806	−	0.985131i	$-0.445040\pi$
$991$	10.8169	0.343611	0.171806	+	0.985131i	$0.445040\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 3.74705i	− 0.118789i
$996$	0	0
$997$	− 5.39134i	− 0.170746i	−0.996349	−	0.0853728i	$-0.972792\pi$
$997$	− 5.39134i	− 0.170746i	0.996349	−	0.0853728i	$-0.0272081\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.g.b.2737.5		16
3.2	odd	2		608.2.c.b.305.9		16
4.3	odd	2		1368.2.g.b.685.1		16
8.3	odd	2		1368.2.g.b.685.2		16
8.5	even	2	inner	5472.2.g.b.2737.12		16
12.11	even	2		152.2.c.b.77.16	yes	16
24.5	odd	2		608.2.c.b.305.8		16
24.11	even	2		152.2.c.b.77.15	✓	16
48.5	odd	4		4864.2.a.bp.1.5		8
48.11	even	4		4864.2.a.bq.1.4		8
48.29	odd	4		4864.2.a.bn.1.4		8
48.35	even	4		4864.2.a.bo.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.15	✓	16	24.11	even	2
152.2.c.b.77.16	yes	16	12.11	even	2
608.2.c.b.305.8		16	24.5	odd	2
608.2.c.b.305.9		16	3.2	odd	2
1368.2.g.b.685.1		16	4.3	odd	2
1368.2.g.b.685.2		16	8.3	odd	2
4864.2.a.bn.1.4		8	48.29	odd	4
4864.2.a.bo.1.5		8	48.35	even	4
4864.2.a.bp.1.5		8	48.5	odd	4
4864.2.a.bq.1.4		8	48.11	even	4
5472.2.g.b.2737.5		16	1.1	even	1	trivial
5472.2.g.b.2737.12		16	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.10882i q^{5} +2.73436 q^{7} -4.66474i q^{11} -4.47791i q^{13} +6.85237 q^{17} -1.00000i q^{19} -1.20416 q^{23} +0.552894 q^{25} -9.57484i q^{29} +5.35842 q^{31} -5.76625i q^{35} -1.09693i q^{37} -7.33797 q^{41} +7.64408i q^{43} +7.56486 q^{47} +0.476702 q^{49} +3.11949i q^{53} -9.83708 q^{55} +10.2442i q^{59} -0.722061i q^{61} -9.44309 q^{65} -6.13698i q^{67} +4.62247 q^{71} -6.19270 q^{73} -12.7551i q^{77} +3.26723 q^{79} +8.97934i q^{83} -14.4504i q^{85} -0.620707 q^{89} -12.2442i q^{91} -2.10882 q^{95} -1.67284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7} + 8 q^{17} - 24 q^{25} - 16 q^{31} - 16 q^{41} + 24 q^{47} + 24 q^{49} - 16 q^{55} - 16 q^{65} + 48 q^{71} + 48 q^{79} + 16 q^{89} + 16 q^{95} + 32 q^{97}+O(q^{100}) \) 16 * q + 8 * q^7 + 8 * q^17 - 24 * q^25 - 16 * q^31 - 16 * q^41 + 24 * q^47 + 24 * q^49 - 16 * q^55 - 16 * q^65 + 48 * q^71 + 48 * q^79 + 16 * q^89 + 16 * q^95 + 32 * q^97

Introduction

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