LMFDB - Embedded newform 1728.4.f.h.863.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,4,Mod(863,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.863"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-384,0,304,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1248,0,-720] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)

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

Self dual:	no
Analytic conductor:	$101.955300490$
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} - 29x^{14} + 601x^{12} - 5608x^{10} + 37420x^{8} - 128832x^{6} + 318736x^{4} - 389376x^{2} + 331776 $ x^16 - 29x^14 + 601x^12 - 5608x^10 + 37420x^8 - 128832x^6 + 318736x^4 - 389376*x^2 + 331776
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			863.10
Root			$3.59605 - 2.07618i$ of defining polynomial
Character	$\chi$	$=$	1728.863
Dual form			1728.4.f.h.863.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+8.20969 q^{5} +10.6450i q^{7} -61.3898i q^{11} +46.7024i q^{13} -23.7154i q^{17} +145.390 q^{19} +57.4601 q^{23} -57.6010 q^{25} -48.8645 q^{29} +247.178i q^{31} +87.3922i q^{35} -71.1248i q^{37} -394.685i q^{41} +158.068 q^{43} -478.509 q^{47} +229.684 q^{49} +273.440 q^{53} -503.991i q^{55} +355.565i q^{59} -810.872i q^{61} +383.413i q^{65} -240.607 q^{67} +985.700 q^{71} +416.165 q^{73} +653.495 q^{77} +659.086i q^{79} -493.999i q^{83} -194.696i q^{85} -302.839i q^{89} -497.148 q^{91} +1193.61 q^{95} +537.681 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 384 q^{23} + 304 q^{25} + 1248 q^{47} - 720 q^{49} + 5088 q^{71} - 128 q^{73} + 11712 q^{95} + 4592 q^{97}+O(q^{100}) $ 16 * q - 384 * q^23 + 304 * q^25 + 1248 * q^47 - 720 * q^49 + 5088 * q^71 - 128 * q^73 + 11712 * q^95 + 4592 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	8.20969	0.734297	0.367149	−	0.930162i	$-0.380334\pi$
$5$	8.20969	0.734297	0.367149	+	0.930162i	$0.380334\pi$
$6$	0	0
$7$	10.6450i	0.574777i	0.957814	+	0.287388i	$0.0927870\pi$
$7$	10.6450i	0.574777i	−0.957814	+	0.287388i	$0.907213\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 61.3898i	− 1.68270i	−0.540490	−	0.841351i	$-0.681761\pi$
$11$	− 61.3898i	− 1.68270i	0.540490	−	0.841351i	$-0.318239\pi$
$12$	0	0
$13$	46.7024i	0.996379i	0.867068	+	0.498190i	$0.166002\pi$
$13$	46.7024i	0.996379i	−0.867068	+	0.498190i	$0.833998\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 23.7154i	− 0.338343i	−0.985587	−	0.169172i	$-0.945891\pi$
$17$	− 23.7154i	− 0.338343i	0.985587	−	0.169172i	$-0.0541092\pi$
$18$	0	0
$19$	145.390	1.75552	0.877758	−	0.479103i	$-0.159038\pi$
$19$	145.390	1.75552	0.877758	+	0.479103i	$0.159038\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	57.4601	0.520925	0.260462	−	0.965484i	$-0.416125\pi$
$23$	57.4601	0.520925	0.260462	+	0.965484i	$0.416125\pi$
$24$	0	0
$25$	−57.6010	−0.460808
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−48.8645	−0.312893	−0.156447	−	0.987686i	$-0.550004\pi$
$29$	−48.8645	−0.312893	−0.156447	+	0.987686i	$0.550004\pi$
$30$	0	0
$31$	247.178i	1.43208i	0.698058	+	0.716041i	$0.254048\pi$
$31$	247.178i	1.43208i	−0.698058	+	0.716041i	$0.745952\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	87.3922i	0.422057i
$36$	0	0
$37$	− 71.1248i	− 0.316023i	−0.987437	−	0.158011i	$-0.949492\pi$
$37$	− 71.1248i	− 0.316023i	0.987437	−	0.158011i	$-0.0505083\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 394.685i	− 1.50340i	−0.659504	−	0.751701i	$-0.729234\pi$
$41$	− 394.685i	− 1.50340i	0.659504	−	0.751701i	$-0.270766\pi$
$42$	0	0
$43$	158.068	0.560583	0.280292	−	0.959915i	$-0.409569\pi$
$43$	158.068	0.560583	0.280292	+	0.959915i	$0.409569\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−478.509	−1.48506	−0.742528	−	0.669815i	$-0.766373\pi$
$47$	−478.509	−1.48506	−0.742528	+	0.669815i	$0.766373\pi$
$48$	0	0
$49$	229.684	0.669632
$50$	0	0
$51$	0	0
$52$	0	0
$53$	273.440	0.708677	0.354339	−	0.935117i	$-0.384706\pi$
$53$	273.440	0.708677	0.354339	+	0.935117i	$0.384706\pi$
$54$	0	0
$55$	− 503.991i	− 1.23560i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	355.565i	0.784587i	0.919840	+	0.392293i	$0.128318\pi$
$59$	355.565i	0.784587i	−0.919840	+	0.392293i	$0.871682\pi$
$60$	0	0
$61$	− 810.872i	− 1.70199i	−0.525171	−	0.850996i	$-0.675999\pi$
$61$	− 810.872i	− 1.70199i	0.525171	−	0.850996i	$-0.324001\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	383.413i	0.731638i
$66$	0	0
$67$	−240.607	−0.438729	−0.219364	−	0.975643i	$-0.570398\pi$
$67$	−240.607	−0.438729	−0.219364	+	0.975643i	$0.570398\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	985.700	1.64762	0.823811	−	0.566865i	$-0.191844\pi$
$71$	985.700	1.64762	0.823811	+	0.566865i	$0.191844\pi$
$72$	0	0
$73$	416.165	0.667239	0.333619	−	0.942708i	$-0.391730\pi$
$73$	416.165	0.667239	0.333619	+	0.942708i	$0.391730\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	653.495	0.967177
$78$	0	0
$79$	659.086i	0.938645i	0.883027	+	0.469322i	$0.155502\pi$
$79$	659.086i	0.938645i	−0.883027	+	0.469322i	$0.844498\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 493.999i	− 0.653294i	−0.945146	−	0.326647i	$-0.894081\pi$
$83$	− 493.999i	− 0.653294i	0.945146	−	0.326647i	$-0.105919\pi$
$84$	0	0
$85$	− 194.696i	− 0.248445i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 302.839i	− 0.360683i	−0.983604	−	0.180342i	$-0.942280\pi$
$89$	− 302.839i	− 0.360683i	0.983604	−	0.180342i	$-0.0577203\pi$
$90$	0	0
$91$	−497.148	−0.572695
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1193.61	1.28907
$96$	0	0
$97$	537.681	0.562817	0.281409	−	0.959588i	$-0.409198\pi$
$97$	537.681	0.562817	0.281409	+	0.959588i	$0.409198\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−36.4333	−0.0358936	−0.0179468	−	0.999839i	$-0.505713\pi$
$101$	−36.4333	−0.0358936	−0.0179468	+	0.999839i	$0.505713\pi$
$102$	0	0
$103$	− 852.950i	− 0.815958i	−0.912991	−	0.407979i	$-0.866234\pi$
$103$	− 852.950i	− 0.815958i	0.912991	−	0.407979i	$-0.133766\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1305.74i	1.17973i	0.807503	+	0.589864i	$0.200819\pi$
$107$	1305.74i	1.17973i	−0.807503	+	0.589864i	$0.799181\pi$
$108$	0	0
$109$	753.700i	0.662306i	0.943577	+	0.331153i	$0.107438\pi$
$109$	753.700i	0.662306i	−0.943577	+	0.331153i	$0.892562\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1431.45i	− 1.19168i	−0.803103	−	0.595840i	$-0.796819\pi$
$113$	− 1431.45i	− 1.19168i	0.803103	−	0.595840i	$-0.203181\pi$
$114$	0	0
$115$	471.730	0.382513
$116$	0	0
$117$	0	0
$118$	0	0
$119$	252.451	0.194472
$120$	0	0
$121$	−2437.70	−1.83148
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1499.10	−1.07267
$126$	0	0
$127$	1187.29i	0.829563i	0.909921	+	0.414782i	$0.136142\pi$
$127$	1187.29i	0.829563i	−0.909921	+	0.414782i	$0.863858\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	318.570i	0.212470i	0.994341	+	0.106235i	$0.0338797\pi$
$131$	318.570i	0.212470i	−0.994341	+	0.106235i	$0.966120\pi$
$132$	0	0
$133$	1547.68i	1.00903i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2531.92i	− 1.57895i	−0.613781	−	0.789476i	$-0.710352\pi$
$137$	− 2531.92i	− 1.57895i	0.613781	−	0.789476i	$-0.289648\pi$
$138$	0	0
$139$	1223.07	0.746329	0.373165	−	0.927765i	$-0.378273\pi$
$139$	1223.07	0.746329	0.373165	+	0.927765i	$0.378273\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2867.05	1.67661
$144$	0	0
$145$	−401.162	−0.229756
$146$	0	0
$147$	0	0
$148$	0	0
$149$	775.698	0.426494	0.213247	−	0.976998i	$-0.431596\pi$
$149$	775.698	0.426494	0.213247	+	0.976998i	$0.431596\pi$
$150$	0	0
$151$	− 787.215i	− 0.424256i	−0.977242	−	0.212128i	$-0.931961\pi$
$151$	− 787.215i	− 0.424256i	0.977242	−	0.212128i	$-0.0680394\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2029.26i	1.05157i
$156$	0	0
$157$	3280.61i	1.66765i	0.552028	+	0.833826i	$0.313854\pi$
$157$	3280.61i	1.66765i	−0.552028	+	0.833826i	$0.686146\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	611.664i	0.299415i
$162$	0	0
$163$	−1146.44	−0.550894	−0.275447	−	0.961316i	$-0.588826\pi$
$163$	−1146.44	−0.550894	−0.275447	+	0.961316i	$0.588826\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	179.881	0.0833510	0.0416755	−	0.999131i	$-0.486730\pi$
$167$	179.881	0.0833510	0.0416755	+	0.999131i	$0.486730\pi$
$168$	0	0
$169$	15.8818	0.00722885
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4130.86	1.81539	0.907697	−	0.419626i	$-0.137839\pi$
$173$	4130.86	1.81539	0.907697	+	0.419626i	$0.137839\pi$
$174$	0	0
$175$	− 613.163i	− 0.264862i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1232.13i	0.514489i	0.966346	+	0.257244i	$0.0828145\pi$
$179$	1232.13i	0.514489i	−0.966346	+	0.257244i	$0.917185\pi$
$180$	0	0
$181$	− 1739.78i	− 0.714457i	−0.934017	−	0.357229i	$-0.883722\pi$
$181$	− 1739.78i	− 0.714457i	0.934017	−	0.357229i	$-0.116278\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 583.912i	− 0.232055i
$186$	0	0
$187$	−1455.89	−0.569331
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4019.49	1.52272	0.761362	−	0.648328i	$-0.224531\pi$
$191$	4019.49	1.52272	0.761362	+	0.648328i	$0.224531\pi$
$192$	0	0
$193$	2091.11	0.779904	0.389952	−	0.920835i	$-0.372492\pi$
$193$	2091.11	0.779904	0.389952	+	0.920835i	$0.372492\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5104.49	1.84609	0.923046	−	0.384689i	$-0.125691\pi$
$197$	5104.49	1.84609	0.923046	+	0.384689i	$0.125691\pi$
$198$	0	0
$199$	− 493.677i	− 0.175859i	−0.996127	−	0.0879293i	$-0.971975\pi$
$199$	− 493.677i	− 0.175859i	0.996127	−	0.0879293i	$-0.0280249\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 520.163i	− 0.179844i
$204$	0	0
$205$	− 3240.24i	− 1.10394i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 8925.48i	− 2.95401i
$210$	0	0
$211$	−1638.64	−0.534640	−0.267320	−	0.963608i	$-0.586138\pi$
$211$	−1638.64	−0.534640	−0.267320	+	0.963608i	$0.586138\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1297.69	0.411635
$216$	0	0
$217$	−2631.22	−0.823127
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1107.57	0.337118
$222$	0	0
$223$	− 3074.33i	− 0.923196i	−0.887089	−	0.461598i	$-0.847276\pi$
$223$	− 3074.33i	− 0.923196i	0.887089	−	0.461598i	$-0.152724\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1613.66i	0.471817i	0.971775	+	0.235908i	$0.0758065\pi$
$227$	1613.66i	0.471817i	−0.971775	+	0.235908i	$0.924194\pi$
$228$	0	0
$229$	− 4.35984i	− 0.00125811i	−1.00000		0.000629054i	$-0.999800\pi$
$229$	− 4.35984i	− 0.00125811i	1.00000		0.000629054i	$-0.000200234\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2075.46i	− 0.583553i	−0.956486	−	0.291777i	$-0.905754\pi$
$233$	− 2075.46i	− 0.583553i	0.956486	−	0.291777i	$-0.0942464\pi$
$234$	0	0
$235$	−3928.41	−1.09047
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1066.01	−0.288514	−0.144257	−	0.989540i	$-0.546079\pi$
$239$	−1066.01	−0.288514	−0.144257	+	0.989540i	$0.546079\pi$
$240$	0	0
$241$	3372.15	0.901324	0.450662	−	0.892695i	$-0.351188\pi$
$241$	3372.15	0.901324	0.450662	+	0.892695i	$0.351188\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1885.63	0.491709
$246$	0	0
$247$	6790.08i	1.74916i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3234.55i	0.813399i	0.913562	+	0.406699i	$0.133320\pi$
$251$	3234.55i	0.813399i	−0.913562	+	0.406699i	$0.866680\pi$
$252$	0	0
$253$	− 3527.46i	− 0.876561i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4146.07i	− 1.00632i	−0.864192	−	0.503162i	$-0.832170\pi$
$257$	− 4146.07i	− 1.00632i	0.864192	−	0.503162i	$-0.167830\pi$
$258$	0	0
$259$	757.124	0.181642
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7374.93	1.72912	0.864558	−	0.502532i	$-0.167598\pi$
$263$	7374.93	1.72912	0.864558	+	0.502532i	$0.167598\pi$
$264$	0	0
$265$	2244.86	0.520380
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7310.05	−1.65688	−0.828441	−	0.560076i	$-0.810772\pi$
$269$	−7310.05	−1.65688	−0.828441	+	0.560076i	$0.810772\pi$
$270$	0	0
$271$	− 6232.53i	− 1.39705i	−0.715588	−	0.698523i	$-0.753841\pi$
$271$	− 6232.53i	− 1.39705i	0.715588	−	0.698523i	$-0.246159\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3536.11i	0.775402i
$276$	0	0
$277$	232.295i	0.0503871i	0.999683	+	0.0251936i	$0.00802021\pi$
$277$	232.295i	0.0503871i	−0.999683	+	0.0251936i	$0.991980\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 2063.02i	− 0.437969i	−0.975728	−	0.218985i	$-0.929726\pi$
$281$	− 2063.02i	− 0.437969i	0.975728	−	0.218985i	$-0.0702744\pi$
$282$	0	0
$283$	9043.90	1.89966	0.949830	−	0.312765i	$-0.101255\pi$
$283$	9043.90	1.89966	0.949830	+	0.312765i	$0.101255\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4201.43	0.864120
$288$	0	0
$289$	4350.58	0.885524
$290$	0	0
$291$	0	0
$292$	0	0
$293$	608.707	0.121369	0.0606844	−	0.998157i	$-0.480672\pi$
$293$	608.707	0.121369	0.0606844	+	0.998157i	$0.480672\pi$
$294$	0	0
$295$	2919.08i	0.576120i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2683.53i	0.519038i
$300$	0	0
$301$	1682.63i	0.322210i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 6657.01i	− 1.24977i
$306$	0	0
$307$	6916.53	1.28582	0.642911	−	0.765941i	$-0.277727\pi$
$307$	6916.53	1.28582	0.642911	+	0.765941i	$0.277727\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8623.54	1.57234	0.786168	−	0.618013i	$-0.212062\pi$
$311$	8623.54	1.57234	0.786168	+	0.618013i	$0.212062\pi$
$312$	0	0
$313$	−9045.12	−1.63342	−0.816710	−	0.577048i	$-0.804205\pi$
$313$	−9045.12	−1.63342	−0.816710	+	0.577048i	$0.804205\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3568.68	0.632293	0.316147	−	0.948710i	$-0.397611\pi$
$317$	3568.68	0.632293	0.316147	+	0.948710i	$0.397611\pi$
$318$	0	0
$319$	2999.78i	0.526506i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3447.99i	− 0.593968i
$324$	0	0
$325$	− 2690.11i	− 0.459139i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 5093.73i	− 0.853576i
$330$	0	0
$331$	−3121.11	−0.518284	−0.259142	−	0.965839i	$-0.583440\pi$
$331$	−3121.11	−0.518284	−0.259142	+	0.965839i	$0.583440\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1975.31	−0.322157
$336$	0	0
$337$	−9558.00	−1.54498	−0.772489	−	0.635028i	$-0.780988\pi$
$337$	−9558.00	−1.54498	−0.772489	+	0.635028i	$0.780988\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	15174.2	2.40977
$342$	0	0
$343$	6096.22i	0.959665i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1275.37i	0.197306i	0.995122	+	0.0986531i	$0.0314534\pi$
$347$	1275.37i	0.197306i	−0.995122	+	0.0986531i	$0.968547\pi$
$348$	0	0
$349$	4453.58i	0.683080i	0.939867	+	0.341540i	$0.110948\pi$
$349$	4453.58i	0.683080i	−0.939867	+	0.341540i	$0.889052\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8349.61i	1.25894i	0.777026	+	0.629469i	$0.216727\pi$
$353$	8349.61i	1.25894i	−0.777026	+	0.629469i	$0.783273\pi$
$354$	0	0
$355$	8092.29	1.20984
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7012.99	1.03101	0.515503	−	0.856888i	$-0.327605\pi$
$359$	7012.99	1.03101	0.515503	+	0.856888i	$0.327605\pi$
$360$	0	0
$361$	14279.3	2.08184
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3416.59	0.489951
$366$	0	0
$367$	1863.72i	0.265083i	0.991177	+	0.132541i	$0.0423138\pi$
$367$	1863.72i	0.265083i	−0.991177	+	0.132541i	$0.957686\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2910.77i	0.407331i
$372$	0	0
$373$	7551.94i	1.04832i	0.851619	+	0.524162i	$0.175621\pi$
$373$	7551.94i	1.04832i	−0.851619	+	0.524162i	$0.824379\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2282.09i	− 0.311760i
$378$	0	0
$379$	−1940.37	−0.262982	−0.131491	−	0.991317i	$-0.541976\pi$
$379$	−1940.37	−0.262982	−0.131491	+	0.991317i	$0.541976\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10872.4	1.45053	0.725267	−	0.688468i	$-0.241716\pi$
$383$	10872.4	1.45053	0.725267	+	0.688468i	$0.241716\pi$
$384$	0	0
$385$	5364.99	0.710195
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7412.83	−0.966184	−0.483092	−	0.875570i	$-0.660486\pi$
$389$	−7412.83	−0.966184	−0.483092	+	0.875570i	$0.660486\pi$
$390$	0	0
$391$	− 1362.69i	− 0.176251i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5410.89i	0.689244i
$396$	0	0
$397$	− 3060.22i	− 0.386872i	−0.981113	−	0.193436i	$-0.938037\pi$
$397$	− 3060.22i	− 0.386872i	0.981113	−	0.193436i	$-0.0619632\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2599.05i	0.323667i	0.986818	+	0.161833i	$0.0517407\pi$
$401$	2599.05i	0.323667i	−0.986818	+	0.161833i	$0.948259\pi$
$402$	0	0
$403$	−11543.8	−1.42690
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4366.33	−0.531772
$408$	0	0
$409$	−10830.9	−1.30942	−0.654712	−	0.755879i	$-0.727210\pi$
$409$	−10830.9	−1.30942	−0.654712	+	0.755879i	$0.727210\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3784.99	−0.450962
$414$	0	0
$415$	− 4055.58i	− 0.479712i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15554.1i	1.81353i	0.421641	+	0.906763i	$0.361454\pi$
$419$	15554.1i	1.81353i	−0.421641	+	0.906763i	$0.638546\pi$
$420$	0	0
$421$	− 10236.7i	− 1.18505i	−0.805551	−	0.592527i	$-0.798130\pi$
$421$	− 10236.7i	− 1.18505i	0.805551	−	0.592527i	$-0.201870\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1366.03i	0.155911i
$426$	0	0
$427$	8631.75	0.978266
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9116.85	−1.01889	−0.509447	−	0.860502i	$-0.670150\pi$
$431$	−9116.85	−1.01889	−0.509447	+	0.860502i	$0.670150\pi$
$432$	0	0
$433$	−2826.42	−0.313693	−0.156846	−	0.987623i	$-0.550133\pi$
$433$	−2826.42	−0.313693	−0.156846	+	0.987623i	$0.550133\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8354.15	0.914492
$438$	0	0
$439$	− 11154.6i	− 1.21271i	−0.795192	−	0.606357i	$-0.792630\pi$
$439$	− 11154.6i	− 1.21271i	0.795192	−	0.606357i	$-0.207370\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6265.18i	− 0.671936i	−0.941873	−	0.335968i	$-0.890937\pi$
$443$	− 6265.18i	− 0.671936i	0.941873	−	0.335968i	$-0.109063\pi$
$444$	0	0
$445$	− 2486.21i	− 0.264849i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 12554.1i	− 1.31952i	−0.751475	−	0.659761i	$-0.770657\pi$
$449$	− 12554.1i	− 1.31952i	0.751475	−	0.659761i	$-0.229343\pi$
$450$	0	0
$451$	−24229.6	−2.52978
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4081.43	−0.420528
$456$	0	0
$457$	9759.09	0.998930	0.499465	−	0.866334i	$-0.333530\pi$
$457$	9759.09	0.998930	0.499465	+	0.866334i	$0.333530\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13116.4	−1.32515	−0.662575	−	0.748996i	$-0.730536\pi$
$461$	−13116.4	−1.32515	−0.662575	+	0.748996i	$0.730536\pi$
$462$	0	0
$463$	3376.95i	0.338964i	0.985533	+	0.169482i	$0.0542094\pi$
$463$	3376.95i	0.338964i	−0.985533	+	0.169482i	$0.945791\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3513.24i	0.348122i	0.984735	+	0.174061i	$0.0556891\pi$
$467$	3513.24i	0.348122i	−0.984735	+	0.174061i	$0.944311\pi$
$468$	0	0
$469$	− 2561.26i	− 0.252171i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 9703.73i	− 0.943294i
$474$	0	0
$475$	−8374.62	−0.808956
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5710.95	0.544760	0.272380	−	0.962190i	$-0.412189\pi$
$479$	5710.95	0.544760	0.272380	+	0.962190i	$0.412189\pi$
$480$	0	0
$481$	3321.70	0.314878
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4414.20	0.413275
$486$	0	0
$487$	− 18464.0i	− 1.71804i	−0.511943	−	0.859019i	$-0.671074\pi$
$487$	− 18464.0i	− 1.71804i	0.511943	−	0.859019i	$-0.328926\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 20448.8i	− 1.87951i	−0.341843	−	0.939757i	$-0.611051\pi$
$491$	− 20448.8i	− 1.87951i	0.341843	−	0.939757i	$-0.388949\pi$
$492$	0	0
$493$	1158.84i	0.105865i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10492.8i	0.947014i
$498$	0	0
$499$	10287.5	0.922910	0.461455	−	0.887164i	$-0.347328\pi$
$499$	10287.5	0.922910	0.461455	+	0.887164i	$0.347328\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9960.84	−0.882966	−0.441483	−	0.897270i	$-0.645547\pi$
$503$	−9960.84	−0.882966	−0.441483	+	0.897270i	$0.645547\pi$
$504$	0	0
$505$	−299.106	−0.0263565
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13898.1	−1.21026	−0.605129	−	0.796128i	$-0.706878\pi$
$509$	−13898.1	−1.21026	−0.605129	+	0.796128i	$0.706878\pi$
$510$	0	0
$511$	4430.08i	0.383513i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 7002.46i	− 0.599156i
$516$	0	0
$517$	29375.5i	2.49891i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 8999.30i	− 0.756750i	−0.925652	−	0.378375i	$-0.876483\pi$
$521$	− 8999.30i	− 0.756750i	0.925652	−	0.378375i	$-0.123517\pi$
$522$	0	0
$523$	−10212.0	−0.853801	−0.426901	−	0.904299i	$-0.640395\pi$
$523$	−10212.0	−0.853801	−0.426901	+	0.904299i	$0.640395\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5861.94	0.484536
$528$	0	0
$529$	−8865.33	−0.728638
$530$	0	0
$531$	0	0
$532$	0	0
$533$	18432.8	1.49796
$534$	0	0
$535$	10719.7i	0.866271i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 14100.2i	− 1.12679i
$540$	0	0
$541$	− 13697.9i	− 1.08857i	−0.838900	−	0.544286i	$-0.816801\pi$
$541$	− 13697.9i	− 1.08857i	0.838900	−	0.544286i	$-0.183199\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6187.64i	0.486329i
$546$	0	0
$547$	11731.3	0.916991	0.458496	−	0.888697i	$-0.348388\pi$
$547$	11731.3	0.916991	0.458496	+	0.888697i	$0.348388\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7104.42	−0.549289
$552$	0	0
$553$	−7015.97	−0.539511
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−286.608	−0.0218025	−0.0109012	−	0.999941i	$-0.503470\pi$
$557$	−286.608	−0.0218025	−0.0109012	+	0.999941i	$0.503470\pi$
$558$	0	0
$559$	7382.14i	0.558553i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2157.26i	− 0.161488i	−0.996735	−	0.0807440i	$-0.974270\pi$
$563$	− 2157.26i	− 0.161488i	0.996735	−	0.0807440i	$-0.0257296\pi$
$564$	0	0
$565$	− 11751.8i	− 0.875047i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7387.12i	0.544260i	0.962260	+	0.272130i	$0.0877281\pi$
$569$	7387.12i	0.544260i	−0.962260	+	0.272130i	$0.912272\pi$
$570$	0	0
$571$	−4047.04	−0.296608	−0.148304	−	0.988942i	$-0.547381\pi$
$571$	−4047.04	−0.296608	−0.148304	+	0.988942i	$0.547381\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3309.76	−0.240046
$576$	0	0
$577$	−19124.2	−1.37981	−0.689904	−	0.723901i	$-0.742347\pi$
$577$	−19124.2	−1.37981	−0.689904	+	0.723901i	$0.742347\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5258.62	0.375498
$582$	0	0
$583$	− 16786.4i	− 1.19249i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15564.6i	1.09441i	0.836998	+	0.547205i	$0.184308\pi$
$587$	15564.6i	1.09441i	−0.836998	+	0.547205i	$0.815692\pi$
$588$	0	0
$589$	35937.3i	2.51404i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	27560.4i	1.90855i	0.298925	+	0.954276i	$0.403372\pi$
$593$	27560.4i	1.90855i	−0.298925	+	0.954276i	$0.596628\pi$
$594$	0	0
$595$	2072.55	0.142800
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14408.7	−0.982843	−0.491422	−	0.870922i	$-0.663523\pi$
$599$	−14408.7	−0.982843	−0.491422	+	0.870922i	$0.663523\pi$
$600$	0	0
$601$	11543.6	0.783482	0.391741	−	0.920076i	$-0.371873\pi$
$601$	11543.6	0.783482	0.391741	+	0.920076i	$0.371873\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−20012.8	−1.34485
$606$	0	0
$607$	23862.3i	1.59562i	0.602911	+	0.797809i	$0.294007\pi$
$607$	23862.3i	1.59562i	−0.602911	+	0.797809i	$0.705993\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 22347.5i	− 1.47968i
$612$	0	0
$613$	− 14046.2i	− 0.925481i	−0.886494	−	0.462741i	$-0.846866\pi$
$613$	− 14046.2i	− 0.925481i	0.886494	−	0.462741i	$-0.153134\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	28656.3i	1.86979i	0.354926	+	0.934895i	$0.384506\pi$
$617$	28656.3i	1.86979i	−0.354926	+	0.934895i	$0.615494\pi$
$618$	0	0
$619$	15085.3	0.979532	0.489766	−	0.871854i	$-0.337082\pi$
$619$	15085.3	0.979532	0.489766	+	0.871854i	$0.337082\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3223.72	0.207312
$624$	0	0
$625$	−5107.00	−0.326848
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1686.76	−0.106924
$630$	0	0
$631$	609.214i	0.0384349i	0.999815	+	0.0192175i	$0.00611748\pi$
$631$	609.214i	0.0384349i	−0.999815	+	0.0192175i	$0.993883\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9747.25i	0.609146i
$636$	0	0
$637$	10726.8i	0.667207i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 25186.8i	− 1.55198i	−0.630744	−	0.775991i	$-0.717250\pi$
$641$	− 25186.8i	− 1.55198i	0.630744	−	0.775991i	$-0.282750\pi$
$642$	0	0
$643$	30058.0	1.84350	0.921750	−	0.387783i	$-0.126759\pi$
$643$	30058.0	1.84350	0.921750	+	0.387783i	$0.126759\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21515.3	−1.30735	−0.653673	−	0.756777i	$-0.726773\pi$
$647$	−21515.3	−1.30735	−0.653673	+	0.756777i	$0.726773\pi$
$648$	0	0
$649$	21828.1	1.32023
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25835.0	−1.54824	−0.774120	−	0.633039i	$-0.781807\pi$
$653$	−25835.0	−1.54824	−0.774120	+	0.633039i	$0.781807\pi$
$654$	0	0
$655$	2615.36i	0.156016i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 246.463i	− 0.0145688i	−0.999973	−	0.00728440i	$-0.997681\pi$
$659$	− 246.463i	− 0.0145688i	0.999973	−	0.00728440i	$-0.00231872\pi$
$660$	0	0
$661$	10481.9i	0.616791i	0.951258	+	0.308396i	$0.0997921\pi$
$661$	10481.9i	0.616791i	−0.951258	+	0.308396i	$0.900208\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12706.0i	0.740928i
$666$	0	0
$667$	−2807.76	−0.162994
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−49779.3	−2.86395
$672$	0	0
$673$	8394.98	0.480836	0.240418	−	0.970669i	$-0.422715\pi$
$673$	8394.98	0.480836	0.240418	+	0.970669i	$0.422715\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18608.4	1.05639	0.528197	−	0.849122i	$-0.322868\pi$
$677$	18608.4	1.05639	0.528197	+	0.849122i	$0.322868\pi$
$678$	0	0
$679$	5723.62i	0.323494i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15757.8i	0.882803i	0.897310	+	0.441402i	$0.145519\pi$
$683$	15757.8i	0.882803i	−0.897310	+	0.441402i	$0.854481\pi$
$684$	0	0
$685$	− 20786.3i	− 1.15942i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12770.3i	0.706111i
$690$	0	0
$691$	−24645.1	−1.35679	−0.678396	−	0.734696i	$-0.737325\pi$
$691$	−24645.1	−1.35679	−0.678396	+	0.734696i	$0.737325\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10041.1	0.548027
$696$	0	0
$697$	−9360.13	−0.508666
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−23093.0	−1.24424	−0.622118	−	0.782923i	$-0.713728\pi$
$701$	−23093.0	−1.24424	−0.622118	+	0.782923i	$0.713728\pi$
$702$	0	0
$703$	− 10340.9i	− 0.554783i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 387.833i	− 0.0206308i
$708$	0	0
$709$	− 24428.3i	− 1.29397i	−0.762503	−	0.646985i	$-0.776030\pi$
$709$	− 24428.3i	− 1.29397i	0.762503	−	0.646985i	$-0.223970\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	14202.9i	0.746007i
$714$	0	0
$715$	23537.6	1.23113
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4996.04	−0.259139	−0.129569	−	0.991570i	$-0.541360\pi$
$719$	−4996.04	−0.259139	−0.129569	+	0.991570i	$0.541360\pi$
$720$	0	0
$721$	9079.67	0.468994
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2814.64	0.144184
$726$	0	0
$727$	11510.2i	0.587195i	0.955929	+	0.293598i	$0.0948526\pi$
$727$	11510.2i	0.587195i	−0.955929	+	0.293598i	$0.905147\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 3748.64i	− 0.189670i
$732$	0	0
$733$	− 2628.53i	− 0.132452i	−0.997805	−	0.0662259i	$-0.978904\pi$
$733$	− 2628.53i	− 0.132452i	0.997805	−	0.0662259i	$-0.0210958\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14770.8i	0.738249i
$738$	0	0
$739$	−30793.5	−1.53282	−0.766412	−	0.642350i	$-0.777960\pi$
$739$	−30793.5	−1.53282	−0.766412	+	0.642350i	$0.777960\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15961.2	0.788103	0.394052	−	0.919088i	$-0.371073\pi$
$743$	15961.2	0.788103	0.394052	+	0.919088i	$0.371073\pi$
$744$	0	0
$745$	6368.24	0.313174
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13899.6	−0.678080
$750$	0	0
$751$	− 26899.6i	− 1.30703i	−0.756913	−	0.653516i	$-0.773293\pi$
$751$	− 26899.6i	− 1.30703i	0.756913	−	0.653516i	$-0.226707\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 6462.79i	− 0.311530i
$756$	0	0
$757$	− 8061.29i	− 0.387045i	−0.981096	−	0.193522i	$-0.938009\pi$
$757$	− 8061.29i	− 0.387045i	0.981096	−	0.193522i	$-0.0619912\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25377.0i	1.20883i	0.796671	+	0.604413i	$0.206592\pi$
$761$	25377.0i	1.20883i	−0.796671	+	0.604413i	$0.793408\pi$
$762$	0	0
$763$	−8023.14	−0.380678
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16605.8	−0.781746
$768$	0	0
$769$	−33750.4	−1.58267	−0.791334	−	0.611385i	$-0.790613\pi$
$769$	−33750.4	−1.58267	−0.791334	+	0.611385i	$0.790613\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−21502.9	−1.00052	−0.500262	−	0.865874i	$-0.666763\pi$
$773$	−21502.9	−1.00052	−0.500262	+	0.865874i	$0.666763\pi$
$774$	0	0
$775$	− 14237.7i	− 0.659915i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 57383.4i	− 2.63925i
$780$	0	0
$781$	− 60511.9i	− 2.77245i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	26932.8i	1.22455i
$786$	0	0
$787$	25947.2	1.17525	0.587623	−	0.809135i	$-0.300064\pi$
$787$	25947.2	1.17525	0.587623	+	0.809135i	$0.300064\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15237.8	0.684950
$792$	0	0
$793$	37869.7	1.69583
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31007.1	−1.37808	−0.689040	−	0.724724i	$-0.741967\pi$
$797$	−31007.1	−1.37808	−0.689040	+	0.724724i	$0.741967\pi$
$798$	0	0
$799$	11348.0i	0.502459i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 25548.3i	− 1.12276i
$804$	0	0
$805$	5021.57i	0.219860i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	37947.6i	1.64916i	0.565749	+	0.824578i	$0.308587\pi$
$809$	37947.6i	1.64916i	−0.565749	+	0.824578i	$0.691413\pi$
$810$	0	0
$811$	−24889.4	−1.07766	−0.538832	−	0.842413i	$-0.681134\pi$
$811$	−24889.4	−1.07766	−0.538832	+	0.842413i	$0.681134\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9411.88	−0.404520
$816$	0	0
$817$	22981.5	0.984113
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24317.3	1.03371	0.516856	−	0.856072i	$-0.327102\pi$
$821$	24317.3	1.03371	0.516856	+	0.856072i	$0.327102\pi$
$822$	0	0
$823$	39460.8i	1.67134i	0.549229	+	0.835672i	$0.314921\pi$
$823$	39460.8i	1.67134i	−0.549229	+	0.835672i	$0.685079\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3828.93i	− 0.160997i	−0.996755	−	0.0804987i	$-0.974349\pi$
$827$	− 3828.93i	− 0.160997i	0.996755	−	0.0804987i	$-0.0256513\pi$
$828$	0	0
$829$	3707.97i	0.155348i	0.996979	+	0.0776738i	$0.0247493\pi$
$829$	3707.97i	0.155348i	−0.996979	+	0.0776738i	$0.975251\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5447.05i	− 0.226566i
$834$	0	0
$835$	1476.77	0.0612044
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−27919.4	−1.14885	−0.574426	−	0.818557i	$-0.694775\pi$
$839$	−27919.4	−1.14885	−0.574426	+	0.818557i	$0.694775\pi$
$840$	0	0
$841$	−22001.3	−0.902098
$842$	0	0
$843$	0	0
$844$	0	0
$845$	130.384	0.00530812
$846$	0	0
$847$	− 25949.4i	− 1.05269i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 4086.84i	− 0.164624i
$852$	0	0
$853$	32727.5i	1.31368i	0.754030	+	0.656840i	$0.228107\pi$
$853$	32727.5i	1.31368i	−0.754030	+	0.656840i	$0.771893\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 19168.1i	− 0.764024i	−0.924157	−	0.382012i	$-0.875231\pi$
$857$	− 19168.1i	− 0.764024i	0.924157	−	0.382012i	$-0.124769\pi$
$858$	0	0
$859$	17878.7	0.710144	0.355072	−	0.934839i	$-0.384456\pi$
$859$	17878.7	0.710144	0.355072	+	0.934839i	$0.384456\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22113.8	−0.872261	−0.436130	−	0.899883i	$-0.643651\pi$
$863$	−22113.8	−0.872261	−0.436130	+	0.899883i	$0.643651\pi$
$864$	0	0
$865$	33913.0	1.33304
$866$	0	0
$867$	0	0
$868$	0	0
$869$	40461.1	1.57946
$870$	0	0
$871$	− 11236.9i	− 0.437140i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 15957.9i	− 0.616544i
$876$	0	0
$877$	40530.5i	1.56057i	0.625426	+	0.780284i	$0.284925\pi$
$877$	40530.5i	1.56057i	−0.625426	+	0.780284i	$0.715075\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 8719.62i	− 0.333452i	−0.986003	−	0.166726i	$-0.946680\pi$
$881$	− 8719.62i	− 0.333452i	0.986003	−	0.166726i	$-0.0533196\pi$
$882$	0	0
$883$	−6891.50	−0.262647	−0.131324	−	0.991340i	$-0.541923\pi$
$883$	−6891.50	−0.262647	−0.131324	+	0.991340i	$0.541923\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29891.9	−1.13153	−0.565767	−	0.824565i	$-0.691420\pi$
$887$	−29891.9	−1.13153	−0.565767	+	0.824565i	$0.691420\pi$
$888$	0	0
$889$	−12638.7	−0.476814
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−69570.5	−2.60704
$894$	0	0
$895$	10115.4i	0.377787i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 12078.2i	− 0.448089i
$900$	0	0
$901$	− 6484.75i	− 0.239776i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 14283.0i	− 0.524624i
$906$	0	0
$907$	−23084.5	−0.845103	−0.422551	−	0.906339i	$-0.638865\pi$
$907$	−23084.5	−0.845103	−0.422551	+	0.906339i	$0.638865\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44048.4	−1.60196	−0.800981	−	0.598690i	$-0.795688\pi$
$911$	−44048.4	−1.60196	−0.800981	+	0.598690i	$0.795688\pi$
$912$	0	0
$913$	−30326.5	−1.09930
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3391.19	−0.122123
$918$	0	0
$919$	23633.3i	0.848304i	0.905591	+	0.424152i	$0.139428\pi$
$919$	23633.3i	0.848304i	−0.905591	+	0.424152i	$0.860572\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	46034.6i	1.64166i
$924$	0	0
$925$	4096.86i	0.145626i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	25750.4i	0.909410i	0.890642	+	0.454705i	$0.150255\pi$
$929$	25750.4i	0.909410i	−0.890642	+	0.454705i	$0.849745\pi$
$930$	0	0
$931$	33393.8	1.17555
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11952.4	−0.418058
$936$	0	0
$937$	−49356.4	−1.72081	−0.860407	−	0.509607i	$-0.829791\pi$
$937$	−49356.4	−1.72081	−0.860407	+	0.509607i	$0.829791\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−23129.7	−0.801282	−0.400641	−	0.916235i	$-0.631213\pi$
$941$	−23129.7	−0.801282	−0.400641	+	0.916235i	$0.631213\pi$
$942$	0	0
$943$	− 22678.7i	− 0.783159i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 5076.56i	− 0.174199i	−0.996200	−	0.0870993i	$-0.972240\pi$
$947$	− 5076.56i	− 0.174199i	0.996200	−	0.0870993i	$-0.0277597\pi$
$948$	0	0
$949$	19435.9i	0.664823i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22339.4i	0.759334i	0.925123	+	0.379667i	$0.123961\pi$
$953$	22339.4i	0.759334i	−0.925123	+	0.379667i	$0.876039\pi$
$954$	0	0
$955$	32998.8	1.11813
$956$	0	0
$957$	0	0
$958$	0	0
$959$	26952.3	0.907545
$960$	0	0
$961$	−31306.1	−1.05086
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17167.4	0.572681
$966$	0	0
$967$	35904.7i	1.19402i	0.802233	+	0.597011i	$0.203645\pi$
$967$	35904.7i	1.19402i	−0.802233	+	0.597011i	$0.796355\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37659.4i	1.24464i	0.782762	+	0.622321i	$0.213810\pi$
$971$	37659.4i	1.24464i	−0.782762	+	0.622321i	$0.786190\pi$
$972$	0	0
$973$	13019.6i	0.428972i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 19792.1i	− 0.648111i	−0.946038	−	0.324055i	$-0.894954\pi$
$977$	− 19792.1i	− 0.648111i	0.946038	−	0.324055i	$-0.105046\pi$
$978$	0	0
$979$	−18591.2	−0.606922
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−45179.0	−1.46591	−0.732953	−	0.680279i	$-0.761859\pi$
$983$	−45179.0	−1.46591	−0.732953	+	0.680279i	$0.761859\pi$
$984$	0	0
$985$	41906.3	1.35558
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9082.58	0.292022
$990$	0	0
$991$	− 48553.2i	− 1.55635i	−0.628046	−	0.778176i	$-0.716145\pi$
$991$	− 48553.2i	− 1.55635i	0.628046	−	0.778176i	$-0.283855\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 4052.94i	− 0.129132i
$996$	0	0
$997$	36769.8i	1.16801i	0.811749	+	0.584007i	$0.198516\pi$
$997$	36769.8i	1.16801i	−0.811749	+	0.584007i	$0.801484\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.f.h.863.10	yes	16
3.2	odd	2		1728.4.f.j.863.8	yes	16
4.3	odd	2		1728.4.f.j.863.9	yes	16
8.3	odd	2		1728.4.f.j.863.7	yes	16
8.5	even	2	inner	1728.4.f.h.863.8	yes	16
12.11	even	2	inner	1728.4.f.h.863.7	✓	16
24.5	odd	2		1728.4.f.j.863.10	yes	16
24.11	even	2	inner	1728.4.f.h.863.9	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.f.h.863.7	✓	16	12.11	even	2	inner
1728.4.f.h.863.8	yes	16	8.5	even	2	inner
1728.4.f.h.863.9	yes	16	24.11	even	2	inner
1728.4.f.h.863.10	yes	16	1.1	even	1	trivial
1728.4.f.j.863.7	yes	16	8.3	odd	2
1728.4.f.j.863.8	yes	16	3.2	odd	2
1728.4.f.j.863.9	yes	16	4.3	odd	2
1728.4.f.j.863.10	yes	16	24.5	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.20969 q^{5} +10.6450i q^{7} -61.3898i q^{11} +46.7024i q^{13} -23.7154i q^{17} +145.390 q^{19} +57.4601 q^{23} -57.6010 q^{25} -48.8645 q^{29} +247.178i q^{31} +87.3922i q^{35} -71.1248i q^{37} -394.685i q^{41} +158.068 q^{43} -478.509 q^{47} +229.684 q^{49} +273.440 q^{53} -503.991i q^{55} +355.565i q^{59} -810.872i q^{61} +383.413i q^{65} -240.607 q^{67} +985.700 q^{71} +416.165 q^{73} +653.495 q^{77} +659.086i q^{79} -493.999i q^{83} -194.696i q^{85} -302.839i q^{89} -497.148 q^{91} +1193.61 q^{95} +537.681 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 384 q^{23} + 304 q^{25} + 1248 q^{47} - 720 q^{49} + 5088 q^{71} - 128 q^{73} + 11712 q^{95} + 4592 q^{97}+O(q^{100}) \) 16 * q - 384 * q^23 + 304 * q^25 + 1248 * q^47 - 720 * q^49 + 5088 * q^71 - 128 * q^73 + 11712 * q^95 + 4592 * q^97

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