LMFDB - Embedded newform 1728.3.b.j.1567.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(1567,1728)]

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, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1728.1567");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.b (of order $2$, degree $1$, 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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.116304318664704.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64 $ x^12 - 2x^11 + x^10 + 6x^9 - 9x^8 - 2x^7 + 18x^6 - 4x^5 - 36x^4 + 48x^3 + 16x^2 - 64x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.6
Root			$0.828615 + 1.14604i$ of defining polynomial
Character	$\chi$	$=$	1728.1567
Dual form			1728.3.b.j.1567.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64040i q^{5} +8.30833i q^{7} +O(q^{10})$	$q-2.64040i q^{5} +8.30833i q^{7} -18.7629 q^{11} +22.3117i q^{13} -23.4983 q^{17} -9.93333 q^{19} -32.3517i q^{23} +18.0283 q^{25} +8.45525i q^{29} -46.9532i q^{31} +21.9373 q^{35} -28.3219i q^{37} +77.7915 q^{41} +58.4797 q^{43} -54.2933i q^{47} -20.0283 q^{49} +5.81484i q^{53} +49.5416i q^{55} -47.5795 q^{59} -27.7128i q^{61} +58.9118 q^{65} +50.6336 q^{67} +7.34824i q^{71} -29.4383 q^{73} -155.888i q^{77} -43.2050i q^{79} -10.7858 q^{83} +62.0450i q^{85} +136.703 q^{89} -185.372 q^{91} +26.2280i q^{95} -54.6465 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 48 q^{17} - 72 q^{25} + 336 q^{41} + 48 q^{49} + 912 q^{65} + 60 q^{73} + 1248 q^{89} - 204 q^{97}+O(q^{100}) $ 12 * q + 48 * q^17 - 72 * q^25 + 336 * q^41 + 48 * q^49 + 912 * q^65 + 60 * q^73 + 1248 * q^89 - 204 * 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
$5$	− 2.64040i	− 0.528081i	−0.964512	−	0.264040i	$-0.914945\pi$
$5$	− 2.64040i	− 0.528081i	0.964512	−	0.264040i	$-0.0850552\pi$
$6$	0	0
$7$	8.30833i	1.18690i	0.804870	+	0.593452i	$0.202235\pi$
$7$	8.30833i	1.18690i	−0.804870	+	0.593452i	$0.797765\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−18.7629	−1.70572	−0.852859	−	0.522141i	$-0.825133\pi$
$11$	−18.7629	−1.70572	−0.852859	+	0.522141i	$0.825133\pi$
$12$	0	0
$13$	22.3117i	1.71628i	0.513415	+	0.858141i	$0.328380\pi$
$13$	22.3117i	1.71628i	−0.513415	+	0.858141i	$0.671620\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−23.4983	−1.38225	−0.691126	−	0.722734i	$-0.742885\pi$
$17$	−23.4983	−1.38225	−0.691126	+	0.722734i	$0.742885\pi$
$18$	0	0
$19$	−9.93333	−0.522807	−0.261403	−	0.965230i	$-0.584185\pi$
$19$	−9.93333	−0.522807	−0.261403	+	0.965230i	$0.584185\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 32.3517i	− 1.40659i	−0.710896	−	0.703297i	$-0.751710\pi$
$23$	− 32.3517i	− 1.40659i	0.710896	−	0.703297i	$-0.248290\pi$
$24$	0	0
$25$	18.0283	0.721131
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.45525i	0.291560i	0.989317	+	0.145780i	$0.0465692\pi$
$29$	8.45525i	0.291560i	−0.989317	+	0.145780i	$0.953431\pi$
$30$	0	0
$31$	− 46.9532i	− 1.51462i	−0.653055	−	0.757310i	$-0.726513\pi$
$31$	− 46.9532i	− 1.51462i	0.653055	−	0.757310i	$-0.273487\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	21.9373	0.626781
$36$	0	0
$37$	− 28.3219i	− 0.765457i	−0.923861	−	0.382728i	$-0.874984\pi$
$37$	− 28.3219i	− 0.765457i	0.923861	−	0.382728i	$-0.125016\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	77.7915	1.89735	0.948677	−	0.316246i	$-0.102422\pi$
$41$	77.7915	1.89735	0.948677	+	0.316246i	$0.102422\pi$
$42$	0	0
$43$	58.4797	1.35999	0.679997	−	0.733215i	$-0.261981\pi$
$43$	58.4797	1.35999	0.679997	+	0.733215i	$0.261981\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 54.2933i	− 1.15518i	−0.816329	−	0.577588i	$-0.803994\pi$
$47$	− 54.2933i	− 1.15518i	0.816329	−	0.577588i	$-0.196006\pi$
$48$	0	0
$49$	−20.0283	−0.408740
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.81484i	0.109714i	0.998494	+	0.0548570i	$0.0174703\pi$
$53$	5.81484i	0.109714i	−0.998494	+	0.0548570i	$0.982530\pi$
$54$	0	0
$55$	49.5416i	0.900757i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−47.5795	−0.806432	−0.403216	−	0.915105i	$-0.632108\pi$
$59$	−47.5795	−0.806432	−0.403216	+	0.915105i	$0.632108\pi$
$60$	0	0
$61$	− 27.7128i	− 0.454308i	−0.973859	−	0.227154i	$-0.927058\pi$
$61$	− 27.7128i	− 0.454308i	0.973859	−	0.227154i	$-0.0729421\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	58.9118	0.906335
$66$	0	0
$67$	50.6336	0.755725	0.377862	−	0.925862i	$-0.376659\pi$
$67$	50.6336	0.755725	0.377862	+	0.925862i	$0.376659\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.34824i	0.103496i	0.998660	+	0.0517482i	$0.0164793\pi$
$71$	7.34824i	0.103496i	−0.998660	+	0.0517482i	$0.983521\pi$
$72$	0	0
$73$	−29.4383	−0.403265	−0.201632	−	0.979461i	$-0.564625\pi$
$73$	−29.4383	−0.403265	−0.201632	+	0.979461i	$0.564625\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 155.888i	− 2.02452i
$78$	0	0
$79$	− 43.2050i	− 0.546899i	−0.961886	−	0.273450i	$-0.911835\pi$
$79$	− 43.2050i	− 0.546899i	0.961886	−	0.273450i	$-0.0881647\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.7858	−0.129950	−0.0649748	−	0.997887i	$-0.520697\pi$
$83$	−10.7858	−0.129950	−0.0649748	+	0.997887i	$0.520697\pi$
$84$	0	0
$85$	62.0450i	0.729941i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	136.703	1.53599	0.767996	−	0.640454i	$-0.221254\pi$
$89$	136.703	1.53599	0.767996	+	0.640454i	$0.221254\pi$
$90$	0	0
$91$	−185.372	−2.03706
$92$	0	0
$93$	0	0
$94$	0	0
$95$	26.2280i	0.276084i
$96$	0	0
$97$	−54.6465	−0.563366	−0.281683	−	0.959508i	$-0.590893\pi$
$97$	−54.6465	−0.563366	−0.281683	+	0.959508i	$0.590893\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 123.165i	− 1.21946i	−0.792611	−	0.609728i	$-0.791279\pi$
$101$	− 123.165i	− 1.21946i	0.792611	−	0.609728i	$-0.208721\pi$
$102$	0	0
$103$	82.0848i	0.796940i	0.917181	+	0.398470i	$0.130459\pi$
$103$	82.0848i	0.796940i	−0.917181	+	0.398470i	$0.869541\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−180.466	−1.68660	−0.843299	−	0.537445i	$-0.819390\pi$
$107$	−180.466	−1.68660	−0.843299	+	0.537445i	$0.819390\pi$
$108$	0	0
$109$	22.2137i	0.203796i	0.994795	+	0.101898i	$0.0324915\pi$
$109$	22.2137i	0.203796i	−0.994795	+	0.101898i	$0.967509\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	139.381	1.23346	0.616732	−	0.787173i	$-0.288456\pi$
$113$	139.381	1.23346	0.616732	+	0.787173i	$0.288456\pi$
$114$	0	0
$115$	−85.4215	−0.742795
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 195.231i	− 1.64060i
$120$	0	0
$121$	231.046	1.90947
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 113.612i	− 0.908896i
$126$	0	0
$127$	80.3083i	0.632349i	0.948701	+	0.316175i	$0.102399\pi$
$127$	80.3083i	0.632349i	−0.948701	+	0.316175i	$0.897601\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	55.7583	0.425636	0.212818	−	0.977092i	$-0.431736\pi$
$131$	55.7583	0.425636	0.212818	+	0.977092i	$0.431736\pi$
$132$	0	0
$133$	− 82.5293i	− 0.620521i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.28641	−0.0312877	−0.0156438	−	0.999878i	$-0.504980\pi$
$137$	−4.28641	−0.0312877	−0.0156438	+	0.999878i	$0.504980\pi$
$138$	0	0
$139$	−7.84615	−0.0564471	−0.0282236	−	0.999602i	$-0.508985\pi$
$139$	−7.84615	−0.0564471	−0.0282236	+	0.999602i	$0.508985\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 418.631i	− 2.92749i
$144$	0	0
$145$	22.3253	0.153967
$146$	0	0
$147$	0	0
$148$	0	0
$149$	269.056i	1.80574i	0.429912	+	0.902871i	$0.358545\pi$
$149$	269.056i	1.80574i	−0.429912	+	0.902871i	$0.641455\pi$
$150$	0	0
$151$	− 179.626i	− 1.18958i	−0.803881	−	0.594789i	$-0.797235\pi$
$151$	− 179.626i	− 1.18958i	0.803881	−	0.594789i	$-0.202765\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−123.976	−0.799842
$156$	0	0
$157$	− 145.890i	− 0.929238i	−0.885511	−	0.464619i	$-0.846191\pi$
$157$	− 145.890i	− 0.929238i	0.885511	−	0.464619i	$-0.153809\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	268.788	1.66949
$162$	0	0
$163$	100.496	0.616540	0.308270	−	0.951299i	$-0.400250\pi$
$163$	100.496	0.616540	0.308270	+	0.951299i	$0.400250\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 154.410i	− 0.924611i	−0.886721	−	0.462306i	$-0.847022\pi$
$167$	− 154.410i	− 0.924611i	0.886721	−	0.462306i	$-0.152978\pi$
$168$	0	0
$169$	−328.810	−1.94562
$170$	0	0
$171$	0	0
$172$	0	0
$173$	57.6442i	0.333203i	0.986024	+	0.166602i	$0.0532794\pi$
$173$	57.6442i	0.333203i	−0.986024	+	0.166602i	$0.946721\pi$
$174$	0	0
$175$	149.785i	0.855913i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−83.2753	−0.465225	−0.232613	−	0.972569i	$-0.574727\pi$
$179$	−83.2753	−0.465225	−0.232613	+	0.972569i	$0.574727\pi$
$180$	0	0
$181$	227.811i	1.25862i	0.777153	+	0.629311i	$0.216663\pi$
$181$	227.811i	1.25862i	−0.777153	+	0.629311i	$0.783337\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−74.7813	−0.404223
$186$	0	0
$187$	440.896	2.35773
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 28.0652i	− 0.146938i	−0.997297	−	0.0734692i	$-0.976593\pi$
$191$	− 28.0652i	− 0.146938i	0.997297	−	0.0734692i	$-0.0234071\pi$
$192$	0	0
$193$	−310.951	−1.61115	−0.805573	−	0.592496i	$-0.798142\pi$
$193$	−310.951	−1.61115	−0.805573	+	0.592496i	$0.798142\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 177.071i	− 0.898838i	−0.893321	−	0.449419i	$-0.851631\pi$
$197$	− 177.071i	− 0.898838i	0.893321	−	0.449419i	$-0.148369\pi$
$198$	0	0
$199$	− 183.708i	− 0.923156i	−0.887100	−	0.461578i	$-0.847283\pi$
$199$	− 183.708i	− 0.923156i	0.887100	−	0.461578i	$-0.152717\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−70.2489	−0.346054
$204$	0	0
$205$	− 205.401i	− 1.00196i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	186.378	0.891761
$210$	0	0
$211$	−198.513	−0.940821	−0.470411	−	0.882448i	$-0.655894\pi$
$211$	−198.513	−0.940821	−0.470411	+	0.882448i	$0.655894\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 154.410i	− 0.718186i
$216$	0	0
$217$	390.103	1.79771
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 524.286i	− 2.37233i
$222$	0	0
$223$	− 2.68500i	− 0.0120403i	−0.999982	−	0.00602017i	$-0.998084\pi$
$223$	− 2.68500i	− 0.0120403i	0.999982	−	0.00602017i	$-0.00191629\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−337.120	−1.48511	−0.742555	−	0.669786i	$-0.766386\pi$
$227$	−337.120	−1.48511	−0.742555	+	0.669786i	$0.766386\pi$
$228$	0	0
$229$	− 46.9618i	− 0.205073i	−0.994729	−	0.102537i	$-0.967304\pi$
$229$	− 46.9618i	− 0.205073i	0.994729	−	0.102537i	$-0.0326959\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−86.6965	−0.372088	−0.186044	−	0.982541i	$-0.559567\pi$
$233$	−86.6965	−0.372088	−0.186044	+	0.982541i	$0.559567\pi$
$234$	0	0
$235$	−143.356	−0.610026
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 149.408i	− 0.625138i	−0.949895	−	0.312569i	$-0.898811\pi$
$239$	− 149.408i	− 0.625138i	0.949895	−	0.312569i	$-0.101189\pi$
$240$	0	0
$241$	338.559	1.40481	0.702405	−	0.711778i	$-0.252110\pi$
$241$	338.559	1.40481	0.702405	+	0.711778i	$0.252110\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	52.8827i	0.215848i
$246$	0	0
$247$	− 221.629i	− 0.897284i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	117.462	0.467976	0.233988	−	0.972239i	$-0.424822\pi$
$251$	117.462	0.467976	0.233988	+	0.972239i	$0.424822\pi$
$252$	0	0
$253$	607.011i	2.39925i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	79.7319	0.310241	0.155120	−	0.987896i	$-0.450423\pi$
$257$	79.7319	0.310241	0.155120	+	0.987896i	$0.450423\pi$
$258$	0	0
$259$	235.308	0.908524
$260$	0	0
$261$	0	0
$262$	0	0
$263$	70.1112i	0.266582i	0.991077	+	0.133291i	$0.0425546\pi$
$263$	70.1112i	0.266582i	−0.991077	+	0.133291i	$0.957445\pi$
$264$	0	0
$265$	15.3535	0.0579379
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 128.891i	− 0.479147i	−0.970878	−	0.239574i	$-0.922992\pi$
$269$	− 128.891i	− 0.479147i	0.970878	−	0.239574i	$-0.0770077\pi$
$270$	0	0
$271$	334.953i	1.23599i	0.786182	+	0.617995i	$0.212055\pi$
$271$	334.953i	1.23599i	−0.786182	+	0.617995i	$0.787945\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−338.262	−1.23005
$276$	0	0
$277$	− 490.465i	− 1.77063i	−0.464991	−	0.885315i	$-0.653942\pi$
$277$	− 490.465i	− 1.77063i	0.464991	−	0.885315i	$-0.346058\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	496.474	1.76681	0.883406	−	0.468608i	$-0.155244\pi$
$281$	496.474	1.76681	0.883406	+	0.468608i	$0.155244\pi$
$282$	0	0
$283$	−323.319	−1.14247	−0.571235	−	0.820787i	$-0.693535\pi$
$283$	−323.319	−1.14247	−0.571235	+	0.820787i	$0.693535\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	646.317i	2.25198i
$288$	0	0
$289$	263.170	0.910621
$290$	0	0
$291$	0	0
$292$	0	0
$293$	238.720i	0.814742i	0.913263	+	0.407371i	$0.133554\pi$
$293$	238.720i	0.814742i	−0.913263	+	0.407371i	$0.866446\pi$
$294$	0	0
$295$	125.629i	0.425861i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	721.819	2.41411
$300$	0	0
$301$	485.868i	1.61418i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−73.1730	−0.239912
$306$	0	0
$307$	−196.695	−0.640699	−0.320349	−	0.947299i	$-0.603800\pi$
$307$	−196.695	−0.640699	−0.320349	+	0.947299i	$0.603800\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 20.0012i	− 0.0643126i	−0.999483	−	0.0321563i	$-0.989763\pi$
$311$	− 20.0012i	− 0.0643126i	0.999483	−	0.0321563i	$-0.0102374\pi$
$312$	0	0
$313$	−341.608	−1.09140	−0.545700	−	0.837981i	$-0.683736\pi$
$313$	−341.608	−1.09140	−0.545700	+	0.837981i	$0.683736\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 481.401i	− 1.51861i	−0.650732	−	0.759307i	$-0.725538\pi$
$317$	− 481.401i	− 1.51861i	0.650732	−	0.759307i	$-0.274462\pi$
$318$	0	0
$319$	− 158.645i	− 0.497319i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	233.416	0.722651
$324$	0	0
$325$	402.240i	1.23766i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	451.086	1.37108
$330$	0	0
$331$	91.4065	0.276152	0.138076	−	0.990422i	$-0.455908\pi$
$331$	91.4065	0.276152	0.138076	+	0.990422i	$0.455908\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 133.693i	− 0.399084i
$336$	0	0
$337$	−194.220	−0.576320	−0.288160	−	0.957582i	$-0.593044\pi$
$337$	−194.220	−0.576320	−0.288160	+	0.957582i	$0.593044\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	880.979i	2.58352i
$342$	0	0
$343$	240.707i	0.701768i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	68.5829	0.197645	0.0988226	−	0.995105i	$-0.468492\pi$
$347$	68.5829	0.197645	0.0988226	+	0.995105i	$0.468492\pi$
$348$	0	0
$349$	− 536.933i	− 1.53849i	−0.638955	−	0.769244i	$-0.720633\pi$
$349$	− 536.933i	− 1.53849i	0.638955	−	0.769244i	$-0.279367\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−486.832	−1.37913	−0.689563	−	0.724226i	$-0.742197\pi$
$353$	−486.832	−1.37913	−0.689563	+	0.724226i	$0.742197\pi$
$354$	0	0
$355$	19.4023	0.0546544
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.1271i	0.0867049i	0.999060	+	0.0433525i	$0.0138038\pi$
$359$	31.1271i	0.0867049i	−0.999060	+	0.0433525i	$0.986196\pi$
$360$	0	0
$361$	−262.329	−0.726673
$362$	0	0
$363$	0	0
$364$	0	0
$365$	77.7291i	0.212956i
$366$	0	0
$367$	− 656.463i	− 1.78873i	−0.447340	−	0.894364i	$-0.647629\pi$
$367$	− 656.463i	− 1.78873i	0.447340	−	0.894364i	$-0.352371\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−48.3116	−0.130220
$372$	0	0
$373$	− 285.162i	− 0.764508i	−0.924057	−	0.382254i	$-0.875148\pi$
$373$	− 285.162i	− 0.764508i	0.924057	−	0.382254i	$-0.124852\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−188.651	−0.500399
$378$	0	0
$379$	−96.1985	−0.253822	−0.126911	−	0.991914i	$-0.540506\pi$
$379$	−96.1985	−0.253822	−0.126911	+	0.991914i	$0.540506\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 542.017i	− 1.41519i	−0.706619	−	0.707594i	$-0.749780\pi$
$383$	− 542.017i	− 1.41519i	0.706619	−	0.707594i	$-0.250220\pi$
$384$	0	0
$385$	−411.608	−1.06911
$386$	0	0
$387$	0	0
$388$	0	0
$389$	251.966i	0.647729i	0.946104	+	0.323864i	$0.104982\pi$
$389$	251.966i	0.647729i	−0.946104	+	0.323864i	$0.895018\pi$
$390$	0	0
$391$	760.209i	1.94427i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−114.079	−0.288807
$396$	0	0
$397$	− 534.951i	− 1.34748i	−0.738966	−	0.673742i	$-0.764686\pi$
$397$	− 534.951i	− 1.34748i	0.738966	−	0.673742i	$-0.235314\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−586.919	−1.46364	−0.731819	−	0.681499i	$-0.761328\pi$
$401$	−586.919	−1.46364	−0.731819	+	0.681499i	$0.761328\pi$
$402$	0	0
$403$	1047.60	2.59951
$404$	0	0
$405$	0	0
$406$	0	0
$407$	531.401i	1.30565i
$408$	0	0
$409$	364.507	0.891214	0.445607	−	0.895229i	$-0.352988\pi$
$409$	364.507	0.891214	0.445607	+	0.895229i	$0.352988\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 395.306i	− 0.957157i
$414$	0	0
$415$	28.4789i	0.0686239i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	181.773	0.433827	0.216913	−	0.976191i	$-0.430401\pi$
$419$	181.773	0.433827	0.216913	+	0.976191i	$0.430401\pi$
$420$	0	0
$421$	− 168.535i	− 0.400320i	−0.979763	−	0.200160i	$-0.935854\pi$
$421$	− 168.535i	− 0.400320i	0.979763	−	0.200160i	$-0.0641462\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−423.633	−0.996785
$426$	0	0
$427$	230.247	0.539220
$428$	0	0
$429$	0	0
$430$	0	0
$431$	373.929i	0.867585i	0.901013	+	0.433792i	$0.142825\pi$
$431$	373.929i	0.867585i	−0.901013	+	0.433792i	$0.857175\pi$
$432$	0	0
$433$	483.636	1.11694	0.558471	−	0.829524i	$-0.311388\pi$
$433$	483.636	1.11694	0.558471	+	0.829524i	$0.311388\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	321.360i	0.735377i
$438$	0	0
$439$	− 449.117i	− 1.02305i	−0.859270	−	0.511523i	$-0.829081\pi$
$439$	− 449.117i	− 1.02305i	0.859270	−	0.511523i	$-0.170919\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−361.663	−0.816396	−0.408198	−	0.912893i	$-0.633843\pi$
$443$	−361.663	−0.816396	−0.408198	+	0.912893i	$0.633843\pi$
$444$	0	0
$445$	− 360.952i	− 0.811128i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−589.191	−1.31223	−0.656115	−	0.754661i	$-0.727801\pi$
$449$	−589.191	−1.31223	−0.656115	+	0.754661i	$0.727801\pi$
$450$	0	0
$451$	−1459.59	−3.23635
$452$	0	0
$453$	0	0
$454$	0	0
$455$	489.458i	1.07573i
$456$	0	0
$457$	−452.703	−0.990597	−0.495299	−	0.868723i	$-0.664941\pi$
$457$	−452.703	−0.990597	−0.495299	+	0.868723i	$0.664941\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 290.060i	− 0.629197i	−0.949225	−	0.314598i	$-0.898130\pi$
$461$	− 290.060i	− 0.629197i	0.949225	−	0.314598i	$-0.101870\pi$
$462$	0	0
$463$	− 389.267i	− 0.840749i	−0.907351	−	0.420374i	$-0.861899\pi$
$463$	− 389.267i	− 0.840749i	0.907351	−	0.420374i	$-0.138101\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−175.827	−0.376504	−0.188252	−	0.982121i	$-0.560282\pi$
$467$	−175.827	−0.376504	−0.188252	+	0.982121i	$0.560282\pi$
$468$	0	0
$469$	420.680i	0.896972i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1097.25	−2.31976
$474$	0	0
$475$	−179.081	−0.377012
$476$	0	0
$477$	0	0
$478$	0	0
$479$	321.473i	0.671134i	0.942016	+	0.335567i	$0.108928\pi$
$479$	321.473i	0.671134i	−0.942016	+	0.335567i	$0.891072\pi$
$480$	0	0
$481$	631.909	1.31374
$482$	0	0
$483$	0	0
$484$	0	0
$485$	144.289i	0.297503i
$486$	0	0
$487$	244.014i	0.501055i	0.968109	+	0.250528i	$0.0806041\pi$
$487$	244.014i	0.501055i	−0.968109	+	0.250528i	$0.919396\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	239.563	0.487909	0.243955	−	0.969787i	$-0.421555\pi$
$491$	239.563	0.487909	0.243955	+	0.969787i	$0.421555\pi$
$492$	0	0
$493$	− 198.684i	− 0.403010i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−61.0516	−0.122840
$498$	0	0
$499$	134.164	0.268865	0.134433	−	0.990923i	$-0.457079\pi$
$499$	134.164	0.268865	0.134433	+	0.990923i	$0.457079\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 214.317i	− 0.426078i	−0.977044	−	0.213039i	$-0.931664\pi$
$503$	− 214.317i	− 0.426078i	0.977044	−	0.213039i	$-0.0683362\pi$
$504$	0	0
$505$	−325.206	−0.643971
$506$	0	0
$507$	0	0
$508$	0	0
$509$	453.973i	0.891892i	0.895060	+	0.445946i	$0.147133\pi$
$509$	453.973i	0.891892i	−0.895060	+	0.445946i	$0.852867\pi$
$510$	0	0
$511$	− 244.583i	− 0.478637i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	216.737	0.420849
$516$	0	0
$517$	1018.70i	1.97040i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	97.5418	0.187220	0.0936102	−	0.995609i	$-0.470159\pi$
$521$	97.5418	0.187220	0.0936102	+	0.995609i	$0.470159\pi$
$522$	0	0
$523$	−476.762	−0.911590	−0.455795	−	0.890085i	$-0.650645\pi$
$523$	−476.762	−0.911590	−0.455795	+	0.890085i	$0.650645\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1103.32i	2.09359i
$528$	0	0
$529$	−517.630	−0.978507
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1735.66i	3.25639i
$534$	0	0
$535$	476.503i	0.890660i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	375.788	0.697195
$540$	0	0
$541$	987.446i	1.82522i	0.408826	+	0.912612i	$0.365938\pi$
$541$	987.446i	1.82522i	−0.408826	+	0.912612i	$0.634062\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	58.6532	0.107621
$546$	0	0
$547$	396.486	0.724837	0.362418	−	0.932016i	$-0.381951\pi$
$547$	396.486	0.724837	0.362418	+	0.932016i	$0.381951\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 83.9888i	− 0.152430i
$552$	0	0
$553$	358.961	0.649117
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 488.922i	− 0.877778i	−0.898541	−	0.438889i	$-0.855372\pi$
$557$	− 488.922i	− 0.877778i	0.898541	−	0.438889i	$-0.144628\pi$
$558$	0	0
$559$	1304.78i	2.33413i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	417.788	0.742075	0.371037	−	0.928618i	$-0.379002\pi$
$563$	417.788	0.742075	0.371037	+	0.928618i	$0.379002\pi$
$564$	0	0
$565$	− 368.023i	− 0.651369i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−493.590	−0.867469	−0.433735	−	0.901041i	$-0.642804\pi$
$569$	−493.590	−0.867469	−0.433735	+	0.901041i	$0.642804\pi$
$570$	0	0
$571$	687.730	1.20443	0.602215	−	0.798334i	$-0.294285\pi$
$571$	687.730	1.20443	0.602215	+	0.798334i	$0.294285\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 583.244i	− 1.01434i
$576$	0	0
$577$	206.939	0.358647	0.179324	−	0.983790i	$-0.442609\pi$
$577$	206.939	0.358647	0.179324	+	0.983790i	$0.442609\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 89.6120i	− 0.154238i
$582$	0	0
$583$	− 109.103i	− 0.187141i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	879.623	1.49851	0.749253	−	0.662284i	$-0.230413\pi$
$587$	879.623	1.49851	0.749253	+	0.662284i	$0.230413\pi$
$588$	0	0
$589$	466.402i	0.791854i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	444.195	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$593$	444.195	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$594$	0	0
$595$	−515.490	−0.866369
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1025.55i	1.71211i	0.516888	+	0.856053i	$0.327090\pi$
$599$	1025.55i	1.71211i	−0.516888	+	0.856053i	$0.672910\pi$
$600$	0	0
$601$	−18.5617	−0.0308846	−0.0154423	−	0.999881i	$-0.504916\pi$
$601$	−18.5617	−0.0308846	−0.0154423	+	0.999881i	$0.504916\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 610.055i	− 1.00836i
$606$	0	0
$607$	− 648.184i	− 1.06785i	−0.845533	−	0.533924i	$-0.820717\pi$
$607$	− 648.184i	− 1.06785i	0.845533	−	0.533924i	$-0.179283\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1211.37	1.98261
$612$	0	0
$613$	− 616.049i	− 1.00497i	−0.864585	−	0.502487i	$-0.832418\pi$
$613$	− 616.049i	− 1.00497i	0.864585	−	0.502487i	$-0.167582\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−321.805	−0.521564	−0.260782	−	0.965398i	$-0.583980\pi$
$617$	−321.805	−0.521564	−0.260782	+	0.965398i	$0.583980\pi$
$618$	0	0
$619$	769.275	1.24277	0.621386	−	0.783505i	$-0.286570\pi$
$619$	769.275	1.24277	0.621386	+	0.783505i	$0.286570\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1135.78i	1.82307i
$624$	0	0
$625$	150.725	0.241160
$626$	0	0
$627$	0	0
$628$	0	0
$629$	665.516i	1.05805i
$630$	0	0
$631$	− 766.730i	− 1.21510i	−0.794280	−	0.607551i	$-0.792152\pi$
$631$	− 766.730i	− 1.21510i	0.794280	−	0.607551i	$-0.207848\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	212.046	0.333931
$636$	0	0
$637$	− 446.864i	− 0.701513i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	436.825	0.681474	0.340737	−	0.940159i	$-0.389323\pi$
$641$	436.825	0.681474	0.340737	+	0.940159i	$0.389323\pi$
$642$	0	0
$643$	−276.284	−0.429680	−0.214840	−	0.976649i	$-0.568923\pi$
$643$	−276.284	−0.429680	−0.214840	+	0.976649i	$0.568923\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	795.622i	1.22971i	0.788640	+	0.614855i	$0.210785\pi$
$647$	795.622i	1.22971i	−0.788640	+	0.614855i	$0.789215\pi$
$648$	0	0
$649$	892.729	1.37555
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 305.664i	− 0.468091i	−0.972226	−	0.234046i	$-0.924804\pi$
$653$	− 305.664i	− 0.468091i	0.972226	−	0.234046i	$-0.0751965\pi$
$654$	0	0
$655$	− 147.224i	− 0.224770i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	153.889	0.233519	0.116760	−	0.993160i	$-0.462749\pi$
$659$	153.889	0.233519	0.116760	+	0.993160i	$0.462749\pi$
$660$	0	0
$661$	− 38.0847i	− 0.0576167i	−0.999585	−	0.0288084i	$-0.990829\pi$
$661$	− 38.0847i	− 0.0576167i	0.999585	−	0.0288084i	$-0.00917126\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−217.911	−0.327685
$666$	0	0
$667$	273.541	0.410107
$668$	0	0
$669$	0	0
$670$	0	0
$671$	519.973i	0.774922i
$672$	0	0
$673$	392.195	0.582757	0.291378	−	0.956608i	$-0.405886\pi$
$673$	392.195	0.582757	0.291378	+	0.956608i	$0.405886\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	917.262i	1.35489i	0.735573	+	0.677446i	$0.236913\pi$
$677$	917.262i	1.35489i	−0.735573	+	0.677446i	$0.763087\pi$
$678$	0	0
$679$	− 454.021i	− 0.668661i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1280.10	−1.87424	−0.937119	−	0.349010i	$-0.886518\pi$
$683$	−1280.10	−1.87424	−0.937119	+	0.349010i	$0.886518\pi$
$684$	0	0
$685$	11.3179i	0.0165224i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−129.739	−0.188300
$690$	0	0
$691$	1147.79	1.66106	0.830531	−	0.556972i	$-0.188037\pi$
$691$	1147.79	1.66106	0.830531	+	0.556972i	$0.188037\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.7170i	0.0298086i
$696$	0	0
$697$	−1827.97	−2.62262
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 450.531i	− 0.642697i	−0.946961	−	0.321349i	$-0.895864\pi$
$701$	− 450.531i	− 0.642697i	0.946961	−	0.321349i	$-0.104136\pi$
$702$	0	0
$703$	281.331i	0.400186i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1023.30	1.44738
$708$	0	0
$709$	− 1071.57i	− 1.51139i	−0.654926	−	0.755693i	$-0.727300\pi$
$709$	− 1071.57i	− 1.51139i	0.654926	−	0.755693i	$-0.272700\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1519.02	−2.13046
$714$	0	0
$715$	−1105.36	−1.54595
$716$	0	0
$717$	0	0
$718$	0	0
$719$	569.979i	0.792739i	0.918091	+	0.396370i	$0.129730\pi$
$719$	569.979i	0.792739i	−0.918091	+	0.396370i	$0.870270\pi$
$720$	0	0
$721$	−681.987	−0.945891
$722$	0	0
$723$	0	0
$724$	0	0
$725$	152.433i	0.210253i
$726$	0	0
$727$	− 699.311i	− 0.961914i	−0.876744	−	0.480957i	$-0.840289\pi$
$727$	− 699.311i	− 0.961914i	0.876744	−	0.480957i	$-0.159711\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1374.17	−1.87985
$732$	0	0
$733$	− 175.235i	− 0.239065i	−0.992830	−	0.119533i	$-0.961860\pi$
$733$	− 175.235i	− 0.239065i	0.992830	−	0.119533i	$-0.0381396\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−950.032	−1.28905
$738$	0	0
$739$	975.643	1.32022	0.660110	−	0.751169i	$-0.270510\pi$
$739$	975.643	1.32022	0.660110	+	0.751169i	$0.270510\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 880.230i	− 1.18470i	−0.805682	−	0.592349i	$-0.798201\pi$
$743$	− 880.230i	− 1.18470i	0.805682	−	0.592349i	$-0.201799\pi$
$744$	0	0
$745$	710.415	0.953578
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1499.37i	− 2.00183i
$750$	0	0
$751$	− 48.7538i	− 0.0649186i	−0.999473	−	0.0324593i	$-0.989666\pi$
$751$	− 48.7538i	− 0.0649186i	0.999473	−	0.0324593i	$-0.0103339\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−474.286	−0.628194
$756$	0	0
$757$	1194.56i	1.57802i	0.614382	+	0.789009i	$0.289406\pi$
$757$	1194.56i	1.57802i	−0.614382	+	0.789009i	$0.710594\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1032.85	1.35722	0.678612	−	0.734497i	$-0.262582\pi$
$761$	1032.85	1.35722	0.678612	+	0.734497i	$0.262582\pi$
$762$	0	0
$763$	−184.559	−0.241886
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1061.58i	− 1.38406i
$768$	0	0
$769$	−200.830	−0.261157	−0.130579	−	0.991438i	$-0.541684\pi$
$769$	−200.830	−0.261157	−0.130579	+	0.991438i	$0.541684\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	949.438i	1.22825i	0.789208	+	0.614126i	$0.210491\pi$
$773$	949.438i	1.22825i	−0.789208	+	0.614126i	$0.789509\pi$
$774$	0	0
$775$	− 846.486i	− 1.09224i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−772.729	−0.991950
$780$	0	0
$781$	− 137.874i	− 0.176536i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−385.210	−0.490713
$786$	0	0
$787$	−646.331	−0.821259	−0.410630	−	0.911802i	$-0.634691\pi$
$787$	−646.331	−0.821259	−0.410630	+	0.911802i	$0.634691\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1158.03i	1.46400i
$792$	0	0
$793$	618.319	0.779721
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 37.5148i	− 0.0470700i	−0.999723	−	0.0235350i	$-0.992508\pi$
$797$	− 37.5148i	− 0.0470700i	0.999723	−	0.0235350i	$-0.00749211\pi$
$798$	0	0
$799$	1275.80i	1.59674i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	552.348	0.687856
$804$	0	0
$805$	− 709.709i	− 0.881626i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−572.790	−0.708023	−0.354011	−	0.935241i	$-0.615183\pi$
$809$	−572.790	−0.708023	−0.354011	+	0.935241i	$0.615183\pi$
$810$	0	0
$811$	−486.350	−0.599692	−0.299846	−	0.953988i	$-0.596935\pi$
$811$	−486.350	−0.599692	−0.299846	+	0.953988i	$0.596935\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 265.350i	− 0.325583i
$816$	0	0
$817$	−580.898	−0.711014
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 311.021i	− 0.378832i	−0.981897	−	0.189416i	$-0.939341\pi$
$821$	− 311.021i	− 0.378832i	0.981897	−	0.189416i	$-0.0606595\pi$
$822$	0	0
$823$	824.124i	1.00137i	0.865631	+	0.500683i	$0.166918\pi$
$823$	824.124i	1.00137i	−0.865631	+	0.500683i	$0.833082\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	918.441	1.11057	0.555285	−	0.831660i	$-0.312609\pi$
$827$	918.441	1.11057	0.555285	+	0.831660i	$0.312609\pi$
$828$	0	0
$829$	937.997i	1.13148i	0.824584	+	0.565740i	$0.191409\pi$
$829$	937.997i	1.13148i	−0.824584	+	0.565740i	$0.808591\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	470.630	0.564982
$834$	0	0
$835$	−407.705	−0.488269
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 612.844i	− 0.730446i	−0.930920	−	0.365223i	$-0.880993\pi$
$839$	− 612.844i	− 0.730446i	0.930920	−	0.365223i	$-0.119007\pi$
$840$	0	0
$841$	769.509	0.914993
$842$	0	0
$843$	0	0
$844$	0	0
$845$	868.191i	1.02744i
$846$	0	0
$847$	1919.61i	2.26636i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−916.261	−1.07669
$852$	0	0
$853$	491.372i	0.576051i	0.957623	+	0.288026i	$0.0929989\pi$
$853$	491.372i	0.576051i	−0.957623	+	0.288026i	$0.907001\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	642.820	0.750082	0.375041	−	0.927008i	$-0.377629\pi$
$857$	642.820	0.750082	0.375041	+	0.927008i	$0.377629\pi$
$858$	0	0
$859$	−443.260	−0.516018	−0.258009	−	0.966142i	$-0.583066\pi$
$859$	−443.260	−0.516018	−0.258009	+	0.966142i	$0.583066\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 565.494i	− 0.655265i	−0.944805	−	0.327632i	$-0.893749\pi$
$863$	− 565.494i	− 0.655265i	0.944805	−	0.327632i	$-0.106251\pi$
$864$	0	0
$865$	152.204	0.175958
$866$	0	0
$867$	0	0
$868$	0	0
$869$	810.652i	0.932856i
$870$	0	0
$871$	1129.72i	1.29704i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	943.925	1.07877
$876$	0	0
$877$	330.215i	0.376528i	0.982118	+	0.188264i	$0.0602861\pi$
$877$	330.215i	0.376528i	−0.982118	+	0.188264i	$0.939714\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−358.037	−0.406398	−0.203199	−	0.979137i	$-0.565134\pi$
$881$	−358.037	−0.406398	−0.203199	+	0.979137i	$0.565134\pi$
$882$	0	0
$883$	−266.351	−0.301643	−0.150822	−	0.988561i	$-0.548192\pi$
$883$	−266.351	−0.301643	−0.150822	+	0.988561i	$0.548192\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1375.40i	− 1.55062i	−0.631581	−	0.775310i	$-0.717594\pi$
$887$	− 1375.40i	− 1.55062i	0.631581	−	0.775310i	$-0.282406\pi$
$888$	0	0
$889$	−667.228	−0.750537
$890$	0	0
$891$	0	0
$892$	0	0
$893$	539.313i	0.603934i
$894$	0	0
$895$	219.880i	0.245676i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	397.001	0.441603
$900$	0	0
$901$	− 136.639i	− 0.151652i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	601.512	0.664654
$906$	0	0
$907$	−106.524	−0.117446	−0.0587230	−	0.998274i	$-0.518703\pi$
$907$	−106.524	−0.117446	−0.0587230	+	0.998274i	$0.518703\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 842.877i	− 0.925222i	−0.886561	−	0.462611i	$-0.846913\pi$
$911$	− 842.877i	− 0.925222i	0.886561	−	0.462611i	$-0.153087\pi$
$912$	0	0
$913$	202.373	0.221657
$914$	0	0
$915$	0	0
$916$	0	0
$917$	463.258i	0.505189i
$918$	0	0
$919$	− 1388.07i	− 1.51041i	−0.655489	−	0.755205i	$-0.727537\pi$
$919$	− 1388.07i	− 1.51041i	0.655489	−	0.755205i	$-0.272463\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−163.951	−0.177629
$924$	0	0
$925$	− 510.595i	− 0.551995i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−557.068	−0.599643	−0.299821	−	0.953995i	$-0.596927\pi$
$929$	−557.068	−0.599643	−0.299821	+	0.953995i	$0.596927\pi$
$930$	0	0
$931$	198.947	0.213692
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1164.14i	− 1.24507i
$936$	0	0
$937$	876.585	0.935523	0.467761	−	0.883855i	$-0.345061\pi$
$937$	876.585	0.935523	0.467761	+	0.883855i	$0.345061\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1290.14i	− 1.37103i	−0.728058	−	0.685515i	$-0.759577\pi$
$941$	− 1290.14i	− 1.37103i	0.728058	−	0.685515i	$-0.240423\pi$
$942$	0	0
$943$	− 2516.69i	− 2.66881i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−850.851	−0.898470	−0.449235	−	0.893414i	$-0.648303\pi$
$947$	−850.851	−0.898470	−0.449235	+	0.893414i	$0.648303\pi$
$948$	0	0
$949$	− 656.818i	− 0.692116i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	297.237	0.311896	0.155948	−	0.987765i	$-0.450157\pi$
$953$	297.237	0.311896	0.155948	+	0.987765i	$0.450157\pi$
$954$	0	0
$955$	−74.1036	−0.0775954
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 35.6129i	− 0.0371355i
$960$	0	0
$961$	−1243.61	−1.29408
$962$	0	0
$963$	0	0
$964$	0	0
$965$	821.037i	0.850815i
$966$	0	0
$967$	943.377i	0.975570i	0.872964	+	0.487785i	$0.162195\pi$
$967$	943.377i	0.975570i	−0.872964	+	0.487785i	$0.837805\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	709.360	0.730546	0.365273	−	0.930900i	$-0.380976\pi$
$971$	709.360	0.730546	0.365273	+	0.930900i	$0.380976\pi$
$972$	0	0
$973$	− 65.1884i	− 0.0669973i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−295.503	−0.302459	−0.151230	−	0.988499i	$-0.548323\pi$
$977$	−295.503	−0.302459	−0.151230	+	0.988499i	$0.548323\pi$
$978$	0	0
$979$	−2564.95	−2.61997
$980$	0	0
$981$	0	0
$982$	0	0
$983$	559.879i	0.569561i	0.958593	+	0.284781i	$0.0919208\pi$
$983$	559.879i	0.569561i	−0.958593	+	0.284781i	$0.908079\pi$
$984$	0	0
$985$	−467.539	−0.474659
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1891.92i	− 1.91296i
$990$	0	0
$991$	− 1327.50i	− 1.33955i	−0.742563	−	0.669776i	$-0.766390\pi$
$991$	− 1327.50i	− 1.33955i	0.742563	−	0.669776i	$-0.233610\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−485.064	−0.487501
$996$	0	0
$997$	878.699i	0.881343i	0.897668	+	0.440672i	$0.145260\pi$
$997$	878.699i	0.881343i	−0.897668	+	0.440672i	$0.854740\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.b.j.1567.6	yes	12
3.2	odd	2		1728.3.b.i.1567.8	yes	12
4.3	odd	2	inner	1728.3.b.j.1567.5	yes	12
8.3	odd	2	inner	1728.3.b.j.1567.7	yes	12
8.5	even	2	inner	1728.3.b.j.1567.8	yes	12
12.11	even	2		1728.3.b.i.1567.7	yes	12
24.5	odd	2		1728.3.b.i.1567.6	yes	12
24.11	even	2		1728.3.b.i.1567.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.b.i.1567.5	✓	12	24.11	even	2
1728.3.b.i.1567.6	yes	12	24.5	odd	2
1728.3.b.i.1567.7	yes	12	12.11	even	2
1728.3.b.i.1567.8	yes	12	3.2	odd	2
1728.3.b.j.1567.5	yes	12	4.3	odd	2	inner
1728.3.b.j.1567.6	yes	12	1.1	even	1	trivial
1728.3.b.j.1567.7	yes	12	8.3	odd	2	inner
1728.3.b.j.1567.8	yes	12	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.64040i q^{5} +8.30833i q^{7} +O(q^{10})\)	\(q-2.64040i q^{5} +8.30833i q^{7} -18.7629 q^{11} +22.3117i q^{13} -23.4983 q^{17} -9.93333 q^{19} -32.3517i q^{23} +18.0283 q^{25} +8.45525i q^{29} -46.9532i q^{31} +21.9373 q^{35} -28.3219i q^{37} +77.7915 q^{41} +58.4797 q^{43} -54.2933i q^{47} -20.0283 q^{49} +5.81484i q^{53} +49.5416i q^{55} -47.5795 q^{59} -27.7128i q^{61} +58.9118 q^{65} +50.6336 q^{67} +7.34824i q^{71} -29.4383 q^{73} -155.888i q^{77} -43.2050i q^{79} -10.7858 q^{83} +62.0450i q^{85} +136.703 q^{89} -185.372 q^{91} +26.2280i q^{95} -54.6465 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 48 q^{17} - 72 q^{25} + 336 q^{41} + 48 q^{49} + 912 q^{65} + 60 q^{73} + 1248 q^{89} - 204 q^{97}+O(q^{100}) \) 12 * q + 48 * q^17 - 72 * q^25 + 336 * q^41 + 48 * q^49 + 912 * q^65 + 60 * q^73 + 1248 * q^89 - 204 * q^97

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