LMFDB - Embedded newform 1728.4.c.j.1727.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(1727,1728)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1728.1727");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$101.955300490$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + \cdots + 9496 $ x^12 + 3x^10 - 12x^9 + 73x^8 - 12x^7 + 589x^6 + 84x^5 + 2452x^4 + 852x^3 + 6854x^2 - 888x + 9496
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{12} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1727.5
Root			$-0.453986 - 2.07664i$ of defining polynomial
Character	$\chi$	$=$	1728.1727
Dual form			1728.4.c.j.1727.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.49508i q^{5} -26.1852i q^{7} +O(q^{10})$	$q-1.49508i q^{5} -26.1852i q^{7} +56.3941 q^{11} +41.3170 q^{13} +51.0410i q^{17} +79.0640i q^{19} -27.3688 q^{23} +122.765 q^{25} +134.567i q^{29} -187.192i q^{31} -39.1491 q^{35} +196.585 q^{37} +298.015i q^{41} +465.576i q^{43} +373.845 q^{47} -342.667 q^{49} +620.093i q^{53} -84.3140i q^{55} -321.152 q^{59} -674.699 q^{61} -61.7724i q^{65} -576.075i q^{67} -223.813 q^{71} +70.1371 q^{73} -1476.69i q^{77} +1052.32i q^{79} +1219.05 q^{83} +76.3107 q^{85} -1340.64i q^{89} -1081.89i q^{91} +118.207 q^{95} -576.059 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 72 q^{13} - 384 q^{25} + 240 q^{37} + 288 q^{49} - 144 q^{61} + 156 q^{73} + 168 q^{85} + 516 q^{97}+O(q^{100}) $ 12 * q + 72 * q^13 - 384 * q^25 + 240 * q^37 + 288 * q^49 - 144 * q^61 + 156 * q^73 + 168 * q^85 + 516 * q^97

Display 100 coefficients 10 coefficients

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
$5$	− 1.49508i	− 0.133724i	−0.997762	−	0.0668622i	$-0.978701\pi$
$5$	− 1.49508i	− 0.133724i	0.997762	−	0.0668622i	$-0.0212988\pi$
$6$	0	0
$7$	− 26.1852i	− 1.41387i	−0.707279	−	0.706935i	$-0.750077\pi$
$7$	− 26.1852i	− 1.41387i	0.707279	−	0.706935i	$-0.249923\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	56.3941	1.54577	0.772885	−	0.634546i	$-0.218813\pi$
$11$	56.3941	1.54577	0.772885	+	0.634546i	$0.218813\pi$
$12$	0	0
$13$	41.3170	0.881482	0.440741	−	0.897634i	$-0.354716\pi$
$13$	41.3170	0.881482	0.440741	+	0.897634i	$0.354716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	51.0410i	0.728192i	0.931361	+	0.364096i	$0.118622\pi$
$17$	51.0410i	0.728192i	−0.931361	+	0.364096i	$0.881378\pi$
$18$	0	0
$19$	79.0640i	0.954659i	0.878724	+	0.477330i	$0.158395\pi$
$19$	79.0640i	0.954659i	−0.878724	+	0.477330i	$0.841605\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−27.3688	−0.248121	−0.124061	−	0.992275i	$-0.539592\pi$
$23$	−27.3688	−0.248121	−0.124061	+	0.992275i	$0.539592\pi$
$24$	0	0
$25$	122.765	0.982118
$26$	0	0
$27$	0	0
$28$	0	0
$29$	134.567i	0.861674i	0.902430	+	0.430837i	$0.141782\pi$
$29$	134.567i	0.861674i	−0.902430	+	0.430837i	$0.858218\pi$
$30$	0	0
$31$	− 187.192i	− 1.08454i	−0.840204	−	0.542270i	$-0.817565\pi$
$31$	− 187.192i	− 1.08454i	0.840204	−	0.542270i	$-0.182435\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−39.1491	−0.189069
$36$	0	0
$37$	196.585	0.873469	0.436734	−	0.899590i	$-0.356135\pi$
$37$	196.585	0.873469	0.436734	+	0.899590i	$0.356135\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	298.015i	1.13517i	0.823313	+	0.567587i	$0.192123\pi$
$41$	298.015i	1.13517i	−0.823313	+	0.567587i	$0.807877\pi$
$42$	0	0
$43$	465.576i	1.65115i	0.564289	+	0.825577i	$0.309150\pi$
$43$	465.576i	1.65115i	−0.564289	+	0.825577i	$0.690850\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	373.845	1.16023	0.580116	−	0.814534i	$-0.303007\pi$
$47$	373.845	1.16023	0.580116	+	0.814534i	$0.303007\pi$
$48$	0	0
$49$	−342.667	−0.999028
$50$	0	0
$51$	0	0
$52$	0	0
$53$	620.093i	1.60710i	0.595237	+	0.803550i	$0.297058\pi$
$53$	620.093i	1.60710i	−0.595237	+	0.803550i	$0.702942\pi$
$54$	0	0
$55$	− 84.3140i	− 0.206707i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−321.152	−0.708652	−0.354326	−	0.935122i	$-0.615290\pi$
$59$	−321.152	−0.708652	−0.354326	+	0.935122i	$0.615290\pi$
$60$	0	0
$61$	−674.699	−1.41617	−0.708085	−	0.706127i	$-0.750441\pi$
$61$	−674.699	−1.41617	−0.708085	+	0.706127i	$0.750441\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 61.7724i	− 0.117876i
$66$	0	0
$67$	− 576.075i	− 1.05043i	−0.850970	−	0.525215i	$-0.823985\pi$
$67$	− 576.075i	− 1.05043i	0.850970	−	0.525215i	$-0.176015\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−223.813	−0.374110	−0.187055	−	0.982349i	$-0.559894\pi$
$71$	−223.813	−0.374110	−0.187055	+	0.982349i	$0.559894\pi$
$72$	0	0
$73$	70.1371	0.112451	0.0562255	−	0.998418i	$-0.482093\pi$
$73$	70.1371	0.112451	0.0562255	+	0.998418i	$0.482093\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1476.69i	− 2.18552i
$78$	0	0
$79$	1052.32i	1.49868i	0.662187	+	0.749338i	$0.269628\pi$
$79$	1052.32i	1.49868i	−0.662187	+	0.749338i	$0.730372\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1219.05	1.61214	0.806070	−	0.591820i	$-0.201591\pi$
$83$	1219.05	1.61214	0.806070	+	0.591820i	$0.201591\pi$
$84$	0	0
$85$	76.3107	0.0973771
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1340.64i	− 1.59672i	−0.602183	−	0.798358i	$-0.705702\pi$
$89$	− 1340.64i	− 1.59672i	0.602183	−	0.798358i	$-0.294298\pi$
$90$	0	0
$91$	− 1081.89i	− 1.24630i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	118.207	0.127661
$96$	0	0
$97$	−576.059	−0.602989	−0.301494	−	0.953468i	$-0.597485\pi$
$97$	−576.059	−0.602989	−0.301494	+	0.953468i	$0.597485\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	116.079i	0.114359i	0.998364	+	0.0571797i	$0.0182108\pi$
$101$	116.079i	0.114359i	−0.998364	+	0.0571797i	$0.981789\pi$
$102$	0	0
$103$	165.074i	0.157915i	0.996878	+	0.0789573i	$0.0251591\pi$
$103$	165.074i	0.157915i	−0.996878	+	0.0789573i	$0.974841\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	936.718	0.846318	0.423159	−	0.906056i	$-0.360921\pi$
$107$	936.718	0.846318	0.423159	+	0.906056i	$0.360921\pi$
$108$	0	0
$109$	−346.957	−0.304885	−0.152443	−	0.988312i	$-0.548714\pi$
$109$	−346.957	−0.304885	−0.152443	+	0.988312i	$0.548714\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1462.22i	1.21729i	0.793443	+	0.608645i	$0.208287\pi$
$113$	1462.22i	1.21729i	−0.793443	+	0.608645i	$0.791713\pi$
$114$	0	0
$115$	40.9187i	0.0331799i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1336.52	1.02957
$120$	0	0
$121$	1849.30	1.38941
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 370.429i	− 0.265058i
$126$	0	0
$127$	− 265.004i	− 0.185160i	−0.995705	−	0.0925800i	$-0.970489\pi$
$127$	− 265.004i	− 0.185160i	0.995705	−	0.0925800i	$-0.0295114\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1151.86	0.768231	0.384115	−	0.923285i	$-0.374507\pi$
$131$	1151.86	0.768231	0.384115	+	0.923285i	$0.374507\pi$
$132$	0	0
$133$	2070.31	1.34976
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2348.67i	− 1.46467i	−0.680943	−	0.732337i	$-0.738430\pi$
$137$	− 2348.67i	− 1.46467i	0.680943	−	0.732337i	$-0.261570\pi$
$138$	0	0
$139$	− 215.240i	− 0.131341i	−0.997841	−	0.0656706i	$-0.979081\pi$
$139$	− 215.240i	− 0.131341i	0.997841	−	0.0656706i	$-0.0209187\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2330.04	1.36257
$144$	0	0
$145$	201.190	0.115227
$146$	0	0
$147$	0	0
$148$	0	0
$149$	219.954i	0.120935i	0.998170	+	0.0604674i	$0.0192591\pi$
$149$	219.954i	0.120935i	−0.998170	+	0.0604674i	$0.980741\pi$
$150$	0	0
$151$	1148.02i	0.618703i	0.950948	+	0.309351i	$0.100112\pi$
$151$	1148.02i	0.618703i	−0.950948	+	0.309351i	$0.899888\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−279.869	−0.145030
$156$	0	0
$157$	276.320	0.140463	0.0702316	−	0.997531i	$-0.477626\pi$
$157$	276.320	0.140463	0.0702316	+	0.997531i	$0.477626\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	716.659i	0.350811i
$162$	0	0
$163$	1419.15i	0.681941i	0.940074	+	0.340971i	$0.110756\pi$
$163$	1419.15i	0.681941i	−0.940074	+	0.340971i	$0.889244\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3508.69	1.62581	0.812906	−	0.582394i	$-0.197884\pi$
$167$	3508.69	1.62581	0.812906	+	0.582394i	$0.197884\pi$
$168$	0	0
$169$	−489.908	−0.222990
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 330.965i	− 0.145450i	−0.997352	−	0.0727248i	$-0.976831\pi$
$173$	− 330.965i	− 0.145450i	0.997352	−	0.0727248i	$-0.0231695\pi$
$174$	0	0
$175$	− 3214.62i	− 1.38859i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1136.75	0.474662	0.237331	−	0.971429i	$-0.423727\pi$
$179$	1136.75	0.474662	0.237331	+	0.971429i	$0.423727\pi$
$180$	0	0
$181$	−2056.92	−0.844692	−0.422346	−	0.906435i	$-0.638793\pi$
$181$	−2056.92	−0.844692	−0.422346	+	0.906435i	$0.638793\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 293.911i	− 0.116804i
$186$	0	0
$187$	2878.42i	1.12562i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1983.23	−0.751316	−0.375658	−	0.926758i	$-0.622583\pi$
$191$	−1983.23	−0.751316	−0.375658	+	0.926758i	$0.622583\pi$
$192$	0	0
$193$	−189.908	−0.0708283	−0.0354141	−	0.999373i	$-0.511275\pi$
$193$	−189.908	−0.0708283	−0.0354141	+	0.999373i	$0.511275\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2160.42i	0.781337i	0.920531	+	0.390669i	$0.127756\pi$
$197$	2160.42i	0.781337i	−0.920531	+	0.390669i	$0.872244\pi$
$198$	0	0
$199$	− 2656.23i	− 0.946205i	−0.881007	−	0.473103i	$-0.843134\pi$
$199$	− 2656.23i	− 0.946205i	0.881007	−	0.473103i	$-0.156866\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3523.68	1.21829
$204$	0	0
$205$	445.558	0.151801
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4458.75i	1.47568i
$210$	0	0
$211$	1001.20i	0.326662i	0.986571	+	0.163331i	$0.0522239\pi$
$211$	1001.20i	0.326662i	−0.986571	+	0.163331i	$0.947776\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	696.075	0.220800
$216$	0	0
$217$	−4901.68	−1.53340
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2108.86i	0.641888i
$222$	0	0
$223$	− 3193.62i	− 0.959015i	−0.877538	−	0.479507i	$-0.840815\pi$
$223$	− 3193.62i	− 0.959015i	0.877538	−	0.479507i	$-0.159185\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1714.72	0.501366	0.250683	−	0.968069i	$-0.419345\pi$
$227$	1714.72	0.501366	0.250683	+	0.968069i	$0.419345\pi$
$228$	0	0
$229$	407.497	0.117590	0.0587951	−	0.998270i	$-0.481274\pi$
$229$	407.497	0.117590	0.0587951	+	0.998270i	$0.481274\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3210.54i	0.902702i	0.892346	+	0.451351i	$0.149058\pi$
$233$	3210.54i	0.902702i	−0.892346	+	0.451351i	$0.850942\pi$
$234$	0	0
$235$	− 558.930i	− 0.155151i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1561.19	0.422532	0.211266	−	0.977429i	$-0.432241\pi$
$239$	1561.19	0.422532	0.211266	+	0.977429i	$0.432241\pi$
$240$	0	0
$241$	1460.89	0.390475	0.195238	−	0.980756i	$-0.437452\pi$
$241$	1460.89	0.390475	0.195238	+	0.980756i	$0.437452\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	512.316i	0.133594i
$246$	0	0
$247$	3266.69i	0.841515i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6868.44	1.72722	0.863609	−	0.504162i	$-0.168198\pi$
$251$	6868.44	1.72722	0.863609	+	0.504162i	$0.168198\pi$
$252$	0	0
$253$	−1543.44	−0.383539
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4450.84i	− 1.08029i	−0.841570	−	0.540147i	$-0.818368\pi$
$257$	− 4450.84i	− 1.08029i	0.841570	−	0.540147i	$-0.181632\pi$
$258$	0	0
$259$	− 5147.62i	− 1.23497i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6525.31	1.52991	0.764957	−	0.644081i	$-0.222760\pi$
$263$	6525.31	1.52991	0.764957	+	0.644081i	$0.222760\pi$
$264$	0	0
$265$	927.091	0.214909
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3222.22i	− 0.730343i	−0.930940	−	0.365171i	$-0.881010\pi$
$269$	− 3222.22i	− 0.730343i	0.930940	−	0.365171i	$-0.118990\pi$
$270$	0	0
$271$	7368.93i	1.65177i	0.563837	+	0.825886i	$0.309325\pi$
$271$	7368.93i	1.65177i	−0.563837	+	0.825886i	$0.690675\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6923.21	1.51813
$276$	0	0
$277$	−9078.61	−1.96924	−0.984622	−	0.174699i	$-0.944105\pi$
$277$	−9078.61	−1.96924	−0.984622	+	0.174699i	$0.944105\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2613.34i	0.554801i	0.960754	+	0.277400i	$0.0894729\pi$
$281$	2613.34i	0.554801i	−0.960754	+	0.277400i	$0.910527\pi$
$282$	0	0
$283$	− 927.970i	− 0.194919i	−0.995239	−	0.0974596i	$-0.968928\pi$
$283$	− 927.970i	− 0.194919i	0.995239	−	0.0974596i	$-0.0310717\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7803.60	1.60499
$288$	0	0
$289$	2307.81	0.469736
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5351.73i	1.06707i	0.845778	+	0.533535i	$0.179137\pi$
$293$	5351.73i	1.06707i	−0.845778	+	0.533535i	$0.820863\pi$
$294$	0	0
$295$	480.150i	0.0947641i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1130.80	−0.218714
$300$	0	0
$301$	12191.2	2.33452
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1008.73i	0.189377i
$306$	0	0
$307$	1892.38i	0.351804i	0.984408	+	0.175902i	$0.0562842\pi$
$307$	1892.38i	0.351804i	−0.984408	+	0.175902i	$0.943716\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5645.22	1.02930	0.514648	−	0.857402i	$-0.327923\pi$
$311$	5645.22	1.02930	0.514648	+	0.857402i	$0.327923\pi$
$312$	0	0
$313$	−818.001	−0.147719	−0.0738597	−	0.997269i	$-0.523532\pi$
$313$	−818.001	−0.147719	−0.0738597	+	0.997269i	$0.523532\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1946.72i	− 0.344917i	−0.985017	−	0.172458i	$-0.944829\pi$
$317$	− 1946.72i	− 0.344917i	0.985017	−	0.172458i	$-0.0551710\pi$
$318$	0	0
$319$	7588.81i	1.33195i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4035.51	−0.695176
$324$	0	0
$325$	5072.27	0.865719
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 9789.22i	− 1.64042i
$330$	0	0
$331$	− 3404.83i	− 0.565396i	−0.959209	−	0.282698i	$-0.908771\pi$
$331$	− 3404.83i	− 0.565396i	0.959209	−	0.282698i	$-0.0912295\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−861.281	−0.140468
$336$	0	0
$337$	7072.15	1.14316	0.571579	−	0.820547i	$-0.306331\pi$
$337$	7072.15	1.14316	0.571579	+	0.820547i	$0.306331\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 10556.6i	− 1.67645i
$342$	0	0
$343$	− 8.73194i	− 0.00137458i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5783.02	−0.894665	−0.447332	−	0.894368i	$-0.647626\pi$
$347$	−5783.02	−0.894665	−0.447332	+	0.894368i	$0.647626\pi$
$348$	0	0
$349$	−1748.60	−0.268196	−0.134098	−	0.990968i	$-0.542814\pi$
$349$	−1748.60	−0.268196	−0.134098	+	0.990968i	$0.542814\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 9552.31i	− 1.44028i	−0.693830	−	0.720139i	$-0.744078\pi$
$353$	− 9552.31i	− 1.44028i	0.693830	−	0.720139i	$-0.255922\pi$
$354$	0	0
$355$	334.620i	0.0500276i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	921.392	0.135457	0.0677287	−	0.997704i	$-0.478425\pi$
$359$	921.392	0.135457	0.0677287	+	0.997704i	$0.478425\pi$
$360$	0	0
$361$	607.881	0.0886254
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 104.861i	− 0.0150375i
$366$	0	0
$367$	− 8490.66i	− 1.20765i	−0.797115	−	0.603827i	$-0.793642\pi$
$367$	− 8490.66i	− 1.20765i	0.797115	−	0.603827i	$-0.206358\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	16237.3	2.27223
$372$	0	0
$373$	−2824.92	−0.392142	−0.196071	−	0.980590i	$-0.562818\pi$
$373$	−2824.92	−0.392142	−0.196071	+	0.980590i	$0.562818\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5559.92i	0.759550i
$378$	0	0
$379$	− 5322.35i	− 0.721348i	−0.932692	−	0.360674i	$-0.882547\pi$
$379$	− 5322.35i	− 0.721348i	0.932692	−	0.360674i	$-0.117453\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8320.49	−1.11007	−0.555035	−	0.831827i	$-0.687295\pi$
$383$	−8320.49	−1.11007	−0.555035	+	0.831827i	$0.687295\pi$
$384$	0	0
$385$	−2207.78	−0.292257
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 8358.52i	− 1.08944i	−0.838617	−	0.544722i	$-0.816635\pi$
$389$	− 8358.52i	− 1.08944i	0.838617	−	0.544722i	$-0.183365\pi$
$390$	0	0
$391$	− 1396.93i	− 0.180680i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1573.31	0.200410
$396$	0	0
$397$	−7283.17	−0.920735	−0.460367	−	0.887728i	$-0.652282\pi$
$397$	−7283.17	−0.920735	−0.460367	+	0.887728i	$0.652282\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1549.41i	0.192952i	0.995335	+	0.0964759i	$0.0307571\pi$
$401$	1549.41i	0.192952i	−0.995335	+	0.0964759i	$0.969243\pi$
$402$	0	0
$403$	− 7734.22i	− 0.956003i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11086.2	1.35018
$408$	0	0
$409$	3291.67	0.397952	0.198976	−	0.980004i	$-0.436238\pi$
$409$	3291.67	0.397952	0.198976	+	0.980004i	$0.436238\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8409.45i	1.00194i
$414$	0	0
$415$	− 1822.58i	− 0.215583i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5299.78	0.617927	0.308964	−	0.951074i	$-0.400018\pi$
$419$	5299.78	0.617927	0.308964	+	0.951074i	$0.400018\pi$
$420$	0	0
$421$	10681.4	1.23653	0.618265	−	0.785970i	$-0.287836\pi$
$421$	10681.4	1.23653	0.618265	+	0.785970i	$0.287836\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6266.04i	0.715170i
$426$	0	0
$427$	17667.2i	2.00228i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9866.13	−1.10263	−0.551316	−	0.834296i	$-0.685874\pi$
$431$	−9866.13	−1.10263	−0.551316	+	0.834296i	$0.685874\pi$
$432$	0	0
$433$	−12560.2	−1.39400	−0.697002	−	0.717069i	$-0.745483\pi$
$433$	−12560.2	−1.39400	−0.697002	+	0.717069i	$0.745483\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2163.89i	− 0.236871i
$438$	0	0
$439$	− 929.228i	− 0.101024i	−0.998723	−	0.0505121i	$-0.983915\pi$
$439$	− 929.228i	− 0.101024i	0.998723	−	0.0505121i	$-0.0160854\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4373.85	−0.469092	−0.234546	−	0.972105i	$-0.575360\pi$
$443$	−4373.85	−0.469092	−0.234546	+	0.972105i	$0.575360\pi$
$444$	0	0
$445$	−2004.37	−0.213520
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 14149.5i	− 1.48721i	−0.668618	−	0.743606i	$-0.733114\pi$
$449$	− 14149.5i	− 1.48721i	0.668618	−	0.743606i	$-0.266886\pi$
$450$	0	0
$451$	16806.3i	1.75472i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1617.52	−0.166661
$456$	0	0
$457$	−12582.4	−1.28792	−0.643960	−	0.765059i	$-0.722710\pi$
$457$	−12582.4	−1.28792	−0.643960	+	0.765059i	$0.722710\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10602.2i	1.07114i	0.844492	+	0.535568i	$0.179902\pi$
$461$	10602.2i	1.07114i	−0.844492	+	0.535568i	$0.820098\pi$
$462$	0	0
$463$	− 5080.88i	− 0.509997i	−0.966941	−	0.254999i	$-0.917925\pi$
$463$	− 5080.88i	− 0.509997i	0.966941	−	0.254999i	$-0.0820750\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6903.92	0.684102	0.342051	−	0.939681i	$-0.388879\pi$
$467$	6903.92	0.684102	0.342051	+	0.939681i	$0.388879\pi$
$468$	0	0
$469$	−15084.7	−1.48517
$470$	0	0
$471$	0	0
$472$	0	0
$473$	26255.8i	2.55231i
$474$	0	0
$475$	9706.27i	0.937588i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5114.56	0.487871	0.243936	−	0.969791i	$-0.421561\pi$
$479$	5114.56	0.487871	0.243936	+	0.969791i	$0.421561\pi$
$480$	0	0
$481$	8122.29	0.769947
$482$	0	0
$483$	0	0
$484$	0	0
$485$	861.257i	0.0806344i
$486$	0	0
$487$	− 15594.2i	− 1.45101i	−0.688218	−	0.725504i	$-0.741607\pi$
$487$	− 15594.2i	− 1.45101i	0.688218	−	0.725504i	$-0.258393\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10521.0	−0.967023	−0.483511	−	0.875338i	$-0.660639\pi$
$491$	−10521.0	−0.967023	−0.483511	+	0.875338i	$0.660639\pi$
$492$	0	0
$493$	−6868.46	−0.627464
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5860.61i	0.528942i
$498$	0	0
$499$	− 8984.43i	− 0.806009i	−0.915198	−	0.403004i	$-0.867966\pi$
$499$	− 8984.43i	− 0.806009i	0.915198	−	0.403004i	$-0.132034\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4299.12	0.381090	0.190545	−	0.981678i	$-0.438974\pi$
$503$	4299.12	0.381090	0.190545	+	0.981678i	$0.438974\pi$
$504$	0	0
$505$	173.548	0.0152926
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2984.83i	0.259922i	0.991519	+	0.129961i	$0.0414853\pi$
$509$	2984.83i	0.259922i	−0.991519	+	0.129961i	$0.958515\pi$
$510$	0	0
$511$	− 1836.56i	− 0.158991i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	246.799	0.0211170
$516$	0	0
$517$	21082.7	1.79345
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 12255.7i	− 1.03058i	−0.857016	−	0.515290i	$-0.827684\pi$
$521$	− 12255.7i	− 1.03058i	0.857016	−	0.515290i	$-0.172316\pi$
$522$	0	0
$523$	− 15148.5i	− 1.26654i	−0.773932	−	0.633269i	$-0.781713\pi$
$523$	− 15148.5i	− 1.26654i	0.773932	−	0.633269i	$-0.218287\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9554.49	0.789754
$528$	0	0
$529$	−11417.9	−0.938436
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12313.1i	1.00064i
$534$	0	0
$535$	− 1400.47i	− 0.113173i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−19324.4	−1.54427
$540$	0	0
$541$	−5723.84	−0.454875	−0.227437	−	0.973793i	$-0.573035\pi$
$541$	−5723.84	−0.454875	−0.227437	+	0.973793i	$0.573035\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	518.730i	0.0407706i
$546$	0	0
$547$	8367.43i	0.654051i	0.945016	+	0.327025i	$0.106046\pi$
$547$	8367.43i	0.654051i	−0.945016	+	0.327025i	$0.893954\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10639.4	−0.822605
$552$	0	0
$553$	27555.3	2.11893
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15824.2i	1.20375i	0.798589	+	0.601877i	$0.205580\pi$
$557$	15824.2i	1.20375i	−0.798589	+	0.601877i	$0.794420\pi$
$558$	0	0
$559$	19236.2i	1.45546i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2781.72	−0.208233	−0.104117	−	0.994565i	$-0.533202\pi$
$563$	−2781.72	−0.208233	−0.104117	+	0.994565i	$0.533202\pi$
$564$	0	0
$565$	2186.14	0.162781
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 4687.63i	− 0.345370i	−0.984977	−	0.172685i	$-0.944756\pi$
$569$	− 4687.63i	− 0.345370i	0.984977	−	0.172685i	$-0.0552443\pi$
$570$	0	0
$571$	− 15169.4i	− 1.11177i	−0.831259	−	0.555885i	$-0.812379\pi$
$571$	− 15169.4i	− 1.11177i	0.831259	−	0.555885i	$-0.187621\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3359.92	−0.243684
$576$	0	0
$577$	−14452.3	−1.04273	−0.521367	−	0.853332i	$-0.674578\pi$
$577$	−14452.3	−1.04273	−0.521367	+	0.853332i	$0.674578\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 31921.0i	− 2.27936i
$582$	0	0
$583$	34969.6i	2.48421i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7405.55	−0.520715	−0.260358	−	0.965512i	$-0.583840\pi$
$587$	−7405.55	−0.520715	−0.260358	+	0.965512i	$0.583840\pi$
$588$	0	0
$589$	14800.2	1.03537
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19888.1i	1.37725i	0.725120	+	0.688623i	$0.241784\pi$
$593$	19888.1i	1.37725i	−0.725120	+	0.688623i	$0.758216\pi$
$594$	0	0
$595$	− 1998.21i	− 0.137679i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2335.49	0.159308	0.0796539	−	0.996823i	$-0.474618\pi$
$599$	2335.49	0.159308	0.0796539	+	0.996823i	$0.474618\pi$
$600$	0	0
$601$	24547.9	1.66611	0.833054	−	0.553192i	$-0.186590\pi$
$601$	24547.9	1.66611	0.833054	+	0.553192i	$0.186590\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2764.86i	− 0.185798i
$606$	0	0
$607$	− 18328.9i	− 1.22561i	−0.790233	−	0.612806i	$-0.790041\pi$
$607$	− 18328.9i	− 1.22561i	0.790233	−	0.612806i	$-0.209959\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	15446.1	1.02272
$612$	0	0
$613$	−6450.01	−0.424981	−0.212491	−	0.977163i	$-0.568157\pi$
$613$	−6450.01	−0.424981	−0.212491	+	0.977163i	$0.568157\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6724.96i	− 0.438795i	−0.975636	−	0.219398i	$-0.929591\pi$
$617$	− 6724.96i	− 0.438795i	0.975636	−	0.219398i	$-0.0704092\pi$
$618$	0	0
$619$	− 3136.55i	− 0.203665i	−0.994802	−	0.101832i	$-0.967529\pi$
$619$	− 3136.55i	− 0.203665i	0.994802	−	0.101832i	$-0.0324705\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−35105.0	−2.25755
$624$	0	0
$625$	14791.8	0.946673
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10033.9i	0.636053i
$630$	0	0
$631$	− 6061.57i	− 0.382420i	−0.981549	−	0.191210i	$-0.938759\pi$
$631$	− 6061.57i	− 0.382420i	0.981549	−	0.191210i	$-0.0612412\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−396.204	−0.0247604
$636$	0	0
$637$	−14157.9	−0.880625
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 18603.3i	− 1.14631i	−0.819446	−	0.573157i	$-0.805719\pi$
$641$	− 18603.3i	− 1.14631i	0.819446	−	0.573157i	$-0.194281\pi$
$642$	0	0
$643$	− 21602.4i	− 1.32491i	−0.749103	−	0.662453i	$-0.769515\pi$
$643$	− 21602.4i	− 1.32491i	0.749103	−	0.662453i	$-0.230485\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−672.754	−0.0408790	−0.0204395	−	0.999791i	$-0.506507\pi$
$647$	−672.754	−0.0408790	−0.0204395	+	0.999791i	$0.506507\pi$
$648$	0	0
$649$	−18111.1	−1.09541
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 3322.88i	− 0.199134i	−0.995031	−	0.0995668i	$-0.968254\pi$
$653$	− 3322.88i	− 0.199134i	0.995031	−	0.0995668i	$-0.0317457\pi$
$654$	0	0
$655$	− 1722.12i	− 0.102731i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−25093.7	−1.48332	−0.741662	−	0.670773i	$-0.765962\pi$
$659$	−25093.7	−1.48332	−0.741662	+	0.670773i	$0.765962\pi$
$660$	0	0
$661$	28966.3	1.70447	0.852237	−	0.523157i	$-0.175246\pi$
$661$	28966.3	1.70447	0.852237	+	0.523157i	$0.175246\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3095.29i	− 0.180496i
$666$	0	0
$667$	− 3682.95i	− 0.213800i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−38049.1	−2.18907
$672$	0	0
$673$	−4710.36	−0.269793	−0.134897	−	0.990860i	$-0.543070\pi$
$673$	−4710.36	−0.269793	−0.134897	+	0.990860i	$0.543070\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4784.53i	0.271617i	0.990735	+	0.135808i	$0.0433631\pi$
$677$	4784.53i	0.271617i	−0.990735	+	0.135808i	$0.956637\pi$
$678$	0	0
$679$	15084.2i	0.852548i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18019.8	−1.00953	−0.504764	−	0.863258i	$-0.668420\pi$
$683$	−18019.8	−1.00953	−0.504764	+	0.863258i	$0.668420\pi$
$684$	0	0
$685$	−3511.46	−0.195863
$686$	0	0
$687$	0	0
$688$	0	0
$689$	25620.4i	1.41663i
$690$	0	0
$691$	16956.5i	0.933511i	0.884386	+	0.466756i	$0.154577\pi$
$691$	16956.5i	0.933511i	−0.884386	+	0.466756i	$0.845423\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−321.802	−0.0175635
$696$	0	0
$697$	−15211.0	−0.826625
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 30620.5i	− 1.64982i	−0.565266	−	0.824909i	$-0.691226\pi$
$701$	− 30620.5i	− 1.64982i	0.565266	−	0.824909i	$-0.308774\pi$
$702$	0	0
$703$	15542.8i	0.833865i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3039.56	0.161689
$708$	0	0
$709$	−24192.8	−1.28149	−0.640747	−	0.767752i	$-0.721375\pi$
$709$	−24192.8	−1.28149	−0.640747	+	0.767752i	$0.721375\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5123.23i	0.269098i
$714$	0	0
$715$	− 3483.60i	− 0.182209i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−37146.6	−1.92675	−0.963377	−	0.268151i	$-0.913588\pi$
$719$	−37146.6	−1.92675	−0.963377	+	0.268151i	$0.913588\pi$
$720$	0	0
$721$	4322.49	0.223271
$722$	0	0
$723$	0	0
$724$	0	0
$725$	16520.1i	0.846265i
$726$	0	0
$727$	− 14614.3i	− 0.745551i	−0.927922	−	0.372775i	$-0.878406\pi$
$727$	− 14614.3i	− 0.745551i	0.927922	−	0.372775i	$-0.121594\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−23763.5	−1.20236
$732$	0	0
$733$	8044.73	0.405374	0.202687	−	0.979244i	$-0.435033\pi$
$733$	8044.73	0.405374	0.202687	+	0.979244i	$0.435033\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 32487.3i	− 1.62372i
$738$	0	0
$739$	− 7025.01i	− 0.349688i	−0.984596	−	0.174844i	$-0.944058\pi$
$739$	− 7025.01i	− 0.349688i	0.984596	−	0.174844i	$-0.0559420\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27063.4	1.33629	0.668143	−	0.744033i	$-0.267089\pi$
$743$	27063.4	1.33629	0.668143	+	0.744033i	$0.267089\pi$
$744$	0	0
$745$	328.849	0.0161719
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 24528.2i	− 1.19658i
$750$	0	0
$751$	11434.2i	0.555579i	0.960642	+	0.277789i	$0.0896017\pi$
$751$	11434.2i	0.555579i	−0.960642	+	0.277789i	$0.910398\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1716.38	0.0827357
$756$	0	0
$757$	−22087.2	−1.06046	−0.530232	−	0.847853i	$-0.677895\pi$
$757$	−22087.2	−1.06046	−0.530232	+	0.847853i	$0.677895\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 35524.0i	− 1.69218i	−0.533043	−	0.846088i	$-0.678952\pi$
$761$	− 35524.0i	− 1.69218i	0.533043	−	0.846088i	$-0.321048\pi$
$762$	0	0
$763$	9085.16i	0.431068i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13269.0	−0.624664
$768$	0	0
$769$	21213.0	0.994745	0.497372	−	0.867537i	$-0.334298\pi$
$769$	21213.0	0.994745	0.497372	+	0.867537i	$0.334298\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14370.7i	0.668663i	0.942456	+	0.334332i	$0.108511\pi$
$773$	14370.7i	0.668663i	−0.942456	+	0.334332i	$0.891489\pi$
$774$	0	0
$775$	− 22980.6i	− 1.06515i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−23562.3	−1.08371
$780$	0	0
$781$	−12621.8	−0.578287
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 413.122i	− 0.0187834i
$786$	0	0
$787$	41642.9i	1.88616i	0.332564	+	0.943081i	$0.392086\pi$
$787$	41642.9i	1.88616i	−0.332564	+	0.943081i	$0.607914\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	38288.5	1.72109
$792$	0	0
$793$	−27876.5	−1.24833
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 23527.1i	− 1.04564i	−0.852445	−	0.522818i	$-0.824881\pi$
$797$	− 23527.1i	− 1.04564i	0.852445	−	0.522818i	$-0.175119\pi$
$798$	0	0
$799$	19081.4i	0.844872i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3955.32	0.173823
$804$	0	0
$805$	1071.47	0.0469120
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7508.22i	0.326298i	0.986601	+	0.163149i	$0.0521651\pi$
$809$	7508.22i	0.326298i	−0.986601	+	0.163149i	$0.947835\pi$
$810$	0	0
$811$	15834.0i	0.685581i	0.939412	+	0.342791i	$0.111372\pi$
$811$	15834.0i	0.685581i	−0.939412	+	0.342791i	$0.888628\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2121.75	0.0911923
$816$	0	0
$817$	−36810.3	−1.57629
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 13279.1i	− 0.564486i	−0.959343	−	0.282243i	$-0.908922\pi$
$821$	− 13279.1i	− 0.564486i	0.959343	−	0.282243i	$-0.0910784\pi$
$822$	0	0
$823$	− 27997.4i	− 1.18582i	−0.805270	−	0.592909i	$-0.797979\pi$
$823$	− 27997.4i	− 1.18582i	0.805270	−	0.592909i	$-0.202021\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35177.8	−1.47914	−0.739571	−	0.673078i	$-0.764972\pi$
$827$	−35177.8	−1.47914	−0.739571	+	0.673078i	$0.764972\pi$
$828$	0	0
$829$	13390.3	0.560996	0.280498	−	0.959855i	$-0.409500\pi$
$829$	13390.3	0.560996	0.280498	+	0.959855i	$0.409500\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 17490.1i	− 0.727484i
$834$	0	0
$835$	− 5245.79i	− 0.217411i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8466.38	−0.348381	−0.174191	−	0.984712i	$-0.555731\pi$
$839$	−8466.38	−0.348381	−0.174191	+	0.984712i	$0.555731\pi$
$840$	0	0
$841$	6280.62	0.257518
$842$	0	0
$843$	0	0
$844$	0	0
$845$	732.454i	0.0298192i
$846$	0	0
$847$	− 48424.4i	− 1.96444i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5380.29	−0.216726
$852$	0	0
$853$	16448.2	0.660229	0.330114	−	0.943941i	$-0.392913\pi$
$853$	16448.2	0.660229	0.330114	+	0.943941i	$0.392913\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13593.7i	0.541832i	0.962603	+	0.270916i	$0.0873266\pi$
$857$	13593.7i	0.541832i	−0.962603	+	0.270916i	$0.912673\pi$
$858$	0	0
$859$	15107.9i	0.600086i	0.953926	+	0.300043i	$0.0970011\pi$
$859$	15107.9i	0.600086i	−0.953926	+	0.300043i	$0.902999\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−43717.5	−1.72440	−0.862202	−	0.506565i	$-0.830915\pi$
$863$	−43717.5	−1.72440	−0.862202	+	0.506565i	$0.830915\pi$
$864$	0	0
$865$	−494.820	−0.0194502
$866$	0	0
$867$	0	0
$868$	0	0
$869$	59344.8i	2.31661i
$870$	0	0
$871$	− 23801.7i	− 0.925935i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9699.78	−0.374757
$876$	0	0
$877$	23868.3	0.919012	0.459506	−	0.888175i	$-0.348026\pi$
$877$	23868.3	0.919012	0.459506	+	0.888175i	$0.348026\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	48039.2i	1.83710i	0.395310	+	0.918548i	$0.370637\pi$
$881$	48039.2i	1.83710i	−0.395310	+	0.918548i	$0.629363\pi$
$882$	0	0
$883$	− 10091.3i	− 0.384595i	−0.981337	−	0.192298i	$-0.938406\pi$
$883$	− 10091.3i	− 0.384595i	0.981337	−	0.192298i	$-0.0615939\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37951.2	−1.43661	−0.718307	−	0.695726i	$-0.755083\pi$
$887$	−37951.2	−1.43661	−0.718307	+	0.695726i	$0.755083\pi$
$888$	0	0
$889$	−6939.20	−0.261792
$890$	0	0
$891$	0	0
$892$	0	0
$893$	29557.7i	1.10763i
$894$	0	0
$895$	− 1699.53i	− 0.0634739i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	25190.0	0.934520
$900$	0	0
$901$	−31650.2	−1.17028
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3075.26i	0.112956i
$906$	0	0
$907$	23697.4i	0.867540i	0.901024	+	0.433770i	$0.142817\pi$
$907$	23697.4i	0.867540i	−0.901024	+	0.433770i	$0.857183\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	11833.0	0.430347	0.215174	−	0.976576i	$-0.430968\pi$
$911$	11833.0	0.430347	0.215174	+	0.976576i	$0.430968\pi$
$912$	0	0
$913$	68747.0	2.49200
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 30161.6i	− 1.08618i
$918$	0	0
$919$	51576.2i	1.85130i	0.378386	+	0.925648i	$0.376479\pi$
$919$	51576.2i	1.85130i	−0.378386	+	0.925648i	$0.623521\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9247.29	−0.329771
$924$	0	0
$925$	24133.7	0.857849
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46577.3i	1.64494i	0.568807	+	0.822471i	$0.307405\pi$
$929$	46577.3i	1.64494i	−0.568807	+	0.822471i	$0.692595\pi$
$930$	0	0
$931$	− 27092.6i	− 0.953731i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4303.47	0.150523
$936$	0	0
$937$	11409.2	0.397783	0.198891	−	0.980022i	$-0.436266\pi$
$937$	11409.2	0.397783	0.198891	+	0.980022i	$0.436266\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32444.9i	1.12399i	0.827141	+	0.561994i	$0.189966\pi$
$941$	32444.9i	1.12399i	−0.827141	+	0.561994i	$0.810034\pi$
$942$	0	0
$943$	− 8156.32i	− 0.281661i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22327.6	0.766154	0.383077	−	0.923716i	$-0.374864\pi$
$947$	22327.6	0.766154	0.383077	+	0.923716i	$0.374864\pi$
$948$	0	0
$949$	2897.85	0.0991236
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15314.3i	0.520546i	0.965535	+	0.260273i	$0.0838126\pi$
$953$	15314.3i	0.520546i	−0.965535	+	0.260273i	$0.916187\pi$
$954$	0	0
$955$	2965.09i	0.100469i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−61500.4	−2.07086
$960$	0	0
$961$	−5250.01	−0.176228
$962$	0	0
$963$	0	0
$964$	0	0
$965$	283.928i	0.00947147i
$966$	0	0
$967$	16913.6i	0.562465i	0.959640	+	0.281232i	$0.0907431\pi$
$967$	16913.6i	0.562465i	−0.959640	+	0.281232i	$0.909257\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−56451.5	−1.86572	−0.932861	−	0.360237i	$-0.882696\pi$
$971$	−56451.5	−1.86572	−0.932861	+	0.360237i	$0.882696\pi$
$972$	0	0
$973$	−5636.12	−0.185699
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 7397.50i	− 0.242238i	−0.992638	−	0.121119i	$-0.961352\pi$
$977$	− 7397.50i	− 0.242238i	0.992638	−	0.121119i	$-0.0386483\pi$
$978$	0	0
$979$	− 75604.3i	− 2.46816i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18261.7	0.592532	0.296266	−	0.955105i	$-0.404258\pi$
$983$	18261.7	0.592532	0.296266	+	0.955105i	$0.404258\pi$
$984$	0	0
$985$	3230.01	0.104484
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 12742.3i	− 0.409687i
$990$	0	0
$991$	7020.98i	0.225054i	0.993649	+	0.112527i	$0.0358945\pi$
$991$	7020.98i	0.225054i	−0.993649	+	0.112527i	$0.964105\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3971.28	−0.126531
$996$	0	0
$997$	−6983.34	−0.221830	−0.110915	−	0.993830i	$-0.535378\pi$
$997$	−6983.34	−0.221830	−0.110915	+	0.993830i	$0.535378\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.c.j.1727.5		12
3.2	odd	2	inner	1728.4.c.j.1727.7		12
4.3	odd	2	inner	1728.4.c.j.1727.6		12
8.3	odd	2		108.4.b.b.107.5	✓	12
8.5	even	2		108.4.b.b.107.7	yes	12
12.11	even	2	inner	1728.4.c.j.1727.8		12
24.5	odd	2		108.4.b.b.107.6	yes	12
24.11	even	2		108.4.b.b.107.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.b.b.107.5	✓	12	8.3	odd	2
108.4.b.b.107.6	yes	12	24.5	odd	2
108.4.b.b.107.7	yes	12	8.5	even	2
108.4.b.b.107.8	yes	12	24.11	even	2
1728.4.c.j.1727.5		12	1.1	even	1	trivial
1728.4.c.j.1727.6		12	4.3	odd	2	inner
1728.4.c.j.1727.7		12	3.2	odd	2	inner
1728.4.c.j.1727.8		12	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.49508i q^{5} -26.1852i q^{7} +O(q^{10})\)	\(q-1.49508i q^{5} -26.1852i q^{7} +56.3941 q^{11} +41.3170 q^{13} +51.0410i q^{17} +79.0640i q^{19} -27.3688 q^{23} +122.765 q^{25} +134.567i q^{29} -187.192i q^{31} -39.1491 q^{35} +196.585 q^{37} +298.015i q^{41} +465.576i q^{43} +373.845 q^{47} -342.667 q^{49} +620.093i q^{53} -84.3140i q^{55} -321.152 q^{59} -674.699 q^{61} -61.7724i q^{65} -576.075i q^{67} -223.813 q^{71} +70.1371 q^{73} -1476.69i q^{77} +1052.32i q^{79} +1219.05 q^{83} +76.3107 q^{85} -1340.64i q^{89} -1081.89i q^{91} +118.207 q^{95} -576.059 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 72 q^{13} - 384 q^{25} + 240 q^{37} + 288 q^{49} - 144 q^{61} + 156 q^{73} + 168 q^{85} + 516 q^{97}+O(q^{100}) \) 12 * q + 72 * q^13 - 384 * q^25 + 240 * q^37 + 288 * q^49 - 144 * q^61 + 156 * q^73 + 168 * q^85 + 516 * q^97

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