LMFDB - Embedded newform 1728.4.d.i.865.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(865,1728)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("1728.865");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.d (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:	$101.955300490$
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} + 170x^{12} + 7609x^{8} + 59868x^{4} + 104976 $ x^16 + 170x^12 + 7609x^8 + 59868*x^4 + 104976
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			865.8
Root			$-0.890014 + 0.890014i$ of defining polynomial
Character	$\chi$	$=$	1728.865
Dual form			1728.4.d.i.865.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.31976i q^{5} +2.47210 q^{7} +O(q^{10})$	$q-1.31976i q^{5} +2.47210 q^{7} +35.9931i q^{11} -65.5540i q^{13} +87.1175 q^{17} +9.13947i q^{19} -49.3929 q^{23} +123.258 q^{25} +122.355i q^{29} -138.042 q^{31} -3.26258i q^{35} +90.2859i q^{37} -93.1282 q^{41} -191.655i q^{43} +275.910 q^{47} -336.889 q^{49} -648.331i q^{53} +47.5023 q^{55} -173.633i q^{59} +297.865i q^{61} -86.5156 q^{65} +658.458i q^{67} +558.113 q^{71} +275.325 q^{73} +88.9784i q^{77} +625.878 q^{79} +119.135i q^{83} -114.974i q^{85} +551.721 q^{89} -162.056i q^{91} +12.0619 q^{95} +74.6796 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 240 q^{17} - 304 q^{25} + 1008 q^{41} + 1616 q^{49} + 2736 q^{65} + 128 q^{73} + 5856 q^{89} - 2576 q^{97}+O(q^{100}) $ 16 * q - 240 * q^17 - 304 * q^25 + 1008 * q^41 + 1616 * q^49 + 2736 * q^65 + 128 * q^73 + 5856 * q^89 - 2576 * 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
$5$	− 1.31976i	− 0.118043i	−0.998257	−	0.0590215i	$-0.981202\pi$
$5$	− 1.31976i	− 0.118043i	0.998257	−	0.0590215i	$-0.0187980\pi$
$6$	0	0
$7$	2.47210	0.133481	0.0667404	−	0.997770i	$-0.478740\pi$
$7$	2.47210	0.133481	0.0667404	+	0.997770i	$0.478740\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	35.9931i	0.986575i	0.869866	+	0.493288i	$0.164205\pi$
$11$	35.9931i	0.986575i	−0.869866	+	0.493288i	$0.835795\pi$
$12$	0	0
$13$	− 65.5540i	− 1.39857i	−0.714843	−	0.699285i	$-0.753502\pi$
$13$	− 65.5540i	− 1.39857i	0.714843	−	0.699285i	$-0.246498\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	87.1175	1.24289	0.621444	−	0.783459i	$-0.286546\pi$
$17$	87.1175	1.24289	0.621444	+	0.783459i	$0.286546\pi$
$18$	0	0
$19$	9.13947i	0.110355i	0.998477	+	0.0551773i	$0.0175724\pi$
$19$	9.13947i	0.110355i	−0.998477	+	0.0551773i	$0.982428\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−49.3929	−0.447789	−0.223894	−	0.974613i	$-0.571877\pi$
$23$	−49.3929	−0.447789	−0.223894	+	0.974613i	$0.571877\pi$
$24$	0	0
$25$	123.258	0.986066
$26$	0	0
$27$	0	0
$28$	0	0
$29$	122.355i	0.783471i	0.920078	+	0.391736i	$0.128125\pi$
$29$	122.355i	0.783471i	−0.920078	+	0.391736i	$0.871875\pi$
$30$	0	0
$31$	−138.042	−0.799775	−0.399887	−	0.916564i	$-0.630951\pi$
$31$	−138.042	−0.799775	−0.399887	+	0.916564i	$0.630951\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.26258i	− 0.0157565i
$36$	0	0
$37$	90.2859i	0.401160i	0.979677	+	0.200580i	$0.0642826\pi$
$37$	90.2859i	0.401160i	−0.979677	+	0.200580i	$0.935717\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−93.1282	−0.354736	−0.177368	−	0.984145i	$-0.556758\pi$
$41$	−93.1282	−0.354736	−0.177368	+	0.984145i	$0.556758\pi$
$42$	0	0
$43$	− 191.655i	− 0.679701i	−0.940480	−	0.339850i	$-0.889624\pi$
$43$	− 191.655i	− 0.679701i	0.940480	−	0.339850i	$-0.110376\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	275.910	0.856289	0.428145	−	0.903710i	$-0.359167\pi$
$47$	275.910	0.856289	0.428145	+	0.903710i	$0.359167\pi$
$48$	0	0
$49$	−336.889	−0.982183
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 648.331i	− 1.68029i	−0.542366	−	0.840143i	$-0.682471\pi$
$53$	− 648.331i	− 1.68029i	0.542366	−	0.840143i	$-0.317529\pi$
$54$	0	0
$55$	47.5023	0.116458
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 173.633i	− 0.383137i	−0.981479	−	0.191569i	$-0.938643\pi$
$59$	− 173.633i	− 0.383137i	0.981479	−	0.191569i	$-0.0613575\pi$
$60$	0	0
$61$	297.865i	0.625208i	0.949884	+	0.312604i	$0.101201\pi$
$61$	297.865i	0.625208i	−0.949884	+	0.312604i	$0.898799\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−86.5156	−0.165091
$66$	0	0
$67$	658.458i	1.20065i	0.799757	+	0.600324i	$0.204962\pi$
$67$	658.458i	1.20065i	−0.799757	+	0.600324i	$0.795038\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	558.113	0.932899	0.466450	−	0.884548i	$-0.345533\pi$
$71$	558.113	0.932899	0.466450	+	0.884548i	$0.345533\pi$
$72$	0	0
$73$	275.325	0.441429	0.220715	−	0.975338i	$-0.429161\pi$
$73$	275.325	0.441429	0.220715	+	0.975338i	$0.429161\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	88.9784i	0.131689i
$78$	0	0
$79$	625.878	0.891351	0.445676	−	0.895195i	$-0.352964\pi$
$79$	625.878	0.891351	0.445676	+	0.895195i	$0.352964\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	119.135i	0.157551i	0.996892	+	0.0787755i	$0.0251010\pi$
$83$	119.135i	0.157551i	−0.996892	+	0.0787755i	$0.974899\pi$
$84$	0	0
$85$	− 114.974i	− 0.146714i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	551.721	0.657104	0.328552	−	0.944486i	$-0.393439\pi$
$89$	551.721	0.657104	0.328552	+	0.944486i	$0.393439\pi$
$90$	0	0
$91$	− 162.056i	− 0.186682i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	12.0619	0.0130266
$96$	0	0
$97$	74.6796	0.0781708	0.0390854	−	0.999236i	$-0.487556\pi$
$97$	74.6796	0.0781708	0.0390854	+	0.999236i	$0.487556\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 855.709i	− 0.843032i	−0.906821	−	0.421516i	$-0.861498\pi$
$101$	− 855.709i	− 0.843032i	0.906821	−	0.421516i	$-0.138502\pi$
$102$	0	0
$103$	183.007	0.175070	0.0875349	−	0.996161i	$-0.472101\pi$
$103$	183.007	0.175070	0.0875349	+	0.996161i	$0.472101\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1395.85i	1.26114i	0.776133	+	0.630569i	$0.217179\pi$
$107$	1395.85i	1.26114i	−0.776133	+	0.630569i	$0.782821\pi$
$108$	0	0
$109$	− 590.120i	− 0.518562i	−0.965802	−	0.259281i	$-0.916514\pi$
$109$	− 590.120i	− 0.518562i	0.965802	−	0.259281i	$-0.0834856\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	606.251	0.504702	0.252351	−	0.967636i	$-0.418796\pi$
$113$	606.251	0.504702	0.252351	+	0.967636i	$0.418796\pi$
$114$	0	0
$115$	65.1868i	0.0528583i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	215.363	0.165902
$120$	0	0
$121$	35.4971	0.0266695
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 327.641i	− 0.234441i
$126$	0	0
$127$	−1258.67	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$127$	−1258.67	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1933.54i	1.28958i	0.764361	+	0.644788i	$0.223054\pi$
$131$	1933.54i	1.28958i	−0.764361	+	0.644788i	$0.776946\pi$
$132$	0	0
$133$	22.5937i	0.0147302i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2156.67	1.34494	0.672469	−	0.740125i	$-0.265234\pi$
$137$	2156.67	1.34494	0.672469	+	0.740125i	$0.265234\pi$
$138$	0	0
$139$	− 2153.75i	− 1.31424i	−0.753787	−	0.657119i	$-0.771775\pi$
$139$	− 2153.75i	− 1.31424i	0.753787	−	0.657119i	$-0.228225\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2359.49	1.37979
$144$	0	0
$145$	161.479	0.0924832
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2268.88i	− 1.24748i	−0.781633	−	0.623738i	$-0.785613\pi$
$149$	− 2268.88i	− 1.24748i	0.781633	−	0.623738i	$-0.214387\pi$
$150$	0	0
$151$	−1519.19	−0.818743	−0.409372	−	0.912368i	$-0.634252\pi$
$151$	−1519.19	−0.818743	−0.409372	+	0.912368i	$0.634252\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	182.182i	0.0944078i
$156$	0	0
$157$	66.6451i	0.0338781i	0.999857	+	0.0169390i	$0.00539212\pi$
$157$	66.6451i	0.0338781i	−0.999857	+	0.0169390i	$0.994608\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−122.104	−0.0597712
$162$	0	0
$163$	220.564i	0.105987i	0.998595	+	0.0529937i	$0.0168763\pi$
$163$	220.564i	0.105987i	−0.998595	+	0.0529937i	$0.983124\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3322.97	1.53976	0.769878	−	0.638191i	$-0.220317\pi$
$167$	3322.97	1.53976	0.769878	+	0.638191i	$0.220317\pi$
$168$	0	0
$169$	−2100.33	−0.955998
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3687.70i	− 1.62064i	−0.585988	−	0.810320i	$-0.699293\pi$
$173$	− 3687.70i	− 1.62064i	0.585988	−	0.810320i	$-0.300707\pi$
$174$	0	0
$175$	304.706	0.131621
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1344.27i	− 0.561315i	−0.959808	−	0.280658i	$-0.909447\pi$
$179$	− 1344.27i	− 0.561315i	0.959808	−	0.280658i	$-0.0905525\pi$
$180$	0	0
$181$	− 168.428i	− 0.0691665i	−0.999402	−	0.0345832i	$-0.988990\pi$
$181$	− 168.428i	− 0.0691665i	0.999402	−	0.0345832i	$-0.0110104\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	119.156	0.0473541
$186$	0	0
$187$	3135.63i	1.22620i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2866.99	1.08611	0.543057	−	0.839696i	$-0.317267\pi$
$191$	2866.99	1.08611	0.543057	+	0.839696i	$0.317267\pi$
$192$	0	0
$193$	3021.80	1.12701	0.563507	−	0.826111i	$-0.309452\pi$
$193$	3021.80	1.12701	0.563507	+	0.826111i	$0.309452\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 782.353i	− 0.282946i	−0.989942	−	0.141473i	$-0.954816\pi$
$197$	− 782.353i	− 0.282946i	0.989942	−	0.141473i	$-0.0451838\pi$
$198$	0	0
$199$	1925.31	0.685836	0.342918	−	0.939365i	$-0.388585\pi$
$199$	1925.31	0.685836	0.342918	+	0.939365i	$0.388585\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	302.472i	0.104578i
$204$	0	0
$205$	122.907i	0.0418741i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−328.958	−0.108873
$210$	0	0
$211$	− 1361.40i	− 0.444183i	−0.975026	−	0.222092i	$-0.928712\pi$
$211$	− 1361.40i	− 0.444183i	0.975026	−	0.222092i	$-0.0712884\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−252.939	−0.0802339
$216$	0	0
$217$	−341.252	−0.106755
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 5710.90i	− 1.73827i
$222$	0	0
$223$	5314.09	1.59577	0.797887	−	0.602806i	$-0.205951\pi$
$223$	5314.09	1.59577	0.797887	+	0.602806i	$0.205951\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3437.08i	1.00496i	0.864587	+	0.502482i	$0.167580\pi$
$227$	3437.08i	1.00496i	−0.864587	+	0.502482i	$0.832420\pi$
$228$	0	0
$229$	880.344i	0.254038i	0.991900	+	0.127019i	$0.0405409\pi$
$229$	880.344i	0.254038i	−0.991900	+	0.127019i	$0.959459\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2858.44	0.803701	0.401850	−	0.915705i	$-0.368367\pi$
$233$	2858.44	0.803701	0.401850	+	0.915705i	$0.368367\pi$
$234$	0	0
$235$	− 364.135i	− 0.101079i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5350.87	1.44820	0.724098	−	0.689697i	$-0.242256\pi$
$239$	5350.87	1.44820	0.724098	+	0.689697i	$0.242256\pi$
$240$	0	0
$241$	−2790.27	−0.745798	−0.372899	−	0.927872i	$-0.621636\pi$
$241$	−2790.27	−0.745798	−0.372899	+	0.927872i	$0.621636\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	444.612i	0.115940i
$246$	0	0
$247$	599.129	0.154339
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3752.93i	0.943757i	0.881664	+	0.471878i	$0.156424\pi$
$251$	3752.93i	0.943757i	−0.881664	+	0.471878i	$0.843576\pi$
$252$	0	0
$253$	− 1777.80i	− 0.441777i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6299.43	1.52898	0.764490	−	0.644636i	$-0.222991\pi$
$257$	6299.43	1.52898	0.764490	+	0.644636i	$0.222991\pi$
$258$	0	0
$259$	223.196i	0.0535471i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7614.33	1.78525	0.892623	−	0.450805i	$-0.148863\pi$
$263$	7614.33	1.78525	0.892623	+	0.450805i	$0.148863\pi$
$264$	0	0
$265$	−855.642	−0.198346
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 6081.62i	− 1.37845i	−0.724547	−	0.689225i	$-0.757951\pi$
$269$	− 6081.62i	− 1.37845i	0.724547	−	0.689225i	$-0.242049\pi$
$270$	0	0
$271$	−7989.97	−1.79098	−0.895491	−	0.445080i	$-0.853175\pi$
$271$	−7989.97	−1.79098	−0.895491	+	0.445080i	$0.853175\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4436.45i	0.972828i
$276$	0	0
$277$	6087.60i	1.32046i	0.751061	+	0.660232i	$0.229542\pi$
$277$	6087.60i	1.32046i	−0.751061	+	0.660232i	$0.770458\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6106.28	1.29633	0.648167	−	0.761498i	$-0.275536\pi$
$281$	6106.28	1.29633	0.648167	+	0.761498i	$0.275536\pi$
$282$	0	0
$283$	5424.68i	1.13945i	0.821836	+	0.569724i	$0.192950\pi$
$283$	5424.68i	1.13945i	−0.821836	+	0.569724i	$0.807050\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−230.222	−0.0473504
$288$	0	0
$289$	2676.45	0.544770
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1513.45i	− 0.301763i	−0.988552	−	0.150882i	$-0.951789\pi$
$293$	− 1513.45i	− 0.301763i	0.988552	−	0.150882i	$-0.0482113\pi$
$294$	0	0
$295$	−229.154	−0.0452267
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3237.90i	0.626264i
$300$	0	0
$301$	− 473.790i	− 0.0907269i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	393.110	0.0738014
$306$	0	0
$307$	− 6543.54i	− 1.21648i	−0.793753	−	0.608241i	$-0.791876\pi$
$307$	− 6543.54i	− 1.21648i	0.793753	−	0.608241i	$-0.208124\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1284.46	−0.234196	−0.117098	−	0.993120i	$-0.537359\pi$
$311$	−1284.46	−0.234196	−0.117098	+	0.993120i	$0.537359\pi$
$312$	0	0
$313$	4013.86	0.724847	0.362423	−	0.932014i	$-0.381949\pi$
$313$	4013.86	0.724847	0.362423	+	0.932014i	$0.381949\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2203.61i	− 0.390433i	−0.980760	−	0.195216i	$-0.937459\pi$
$317$	− 2203.61i	− 0.390433i	0.980760	−	0.195216i	$-0.0625409\pi$
$318$	0	0
$319$	−4403.92	−0.772953
$320$	0	0
$321$	0	0
$322$	0	0
$323$	796.208i	0.137159i
$324$	0	0
$325$	− 8080.07i	− 1.37908i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	682.076	0.114298
$330$	0	0
$331$	− 2467.37i	− 0.409725i	−0.978791	−	0.204862i	$-0.934325\pi$
$331$	− 2467.37i	− 0.409725i	0.978791	−	0.204862i	$-0.0656747\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	869.007	0.141728
$336$	0	0
$337$	−7160.82	−1.15749	−0.578746	−	0.815508i	$-0.696458\pi$
$337$	−7160.82	−1.15749	−0.578746	+	0.815508i	$0.696458\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 4968.55i	− 0.789038i
$342$	0	0
$343$	−1680.75	−0.264583
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1843.40i	0.285184i	0.989782	+	0.142592i	$0.0455436\pi$
$347$	1843.40i	0.285184i	−0.989782	+	0.142592i	$0.954456\pi$
$348$	0	0
$349$	5209.52i	0.799023i	0.916728	+	0.399512i	$0.130820\pi$
$349$	5209.52i	0.799023i	−0.916728	+	0.399512i	$0.869180\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8650.90	−1.30437	−0.652183	−	0.758062i	$-0.726146\pi$
$353$	−8650.90	−1.30437	−0.652183	+	0.758062i	$0.726146\pi$
$354$	0	0
$355$	− 736.576i	− 0.110122i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12565.4	1.84728	0.923641	−	0.383260i	$-0.125199\pi$
$359$	12565.4	1.84728	0.923641	+	0.383260i	$0.125199\pi$
$360$	0	0
$361$	6775.47	0.987822
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 363.363i	− 0.0521076i
$366$	0	0
$367$	10796.9	1.53567	0.767837	−	0.640646i	$-0.221333\pi$
$367$	10796.9	1.53567	0.767837	+	0.640646i	$0.221333\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1602.74i	− 0.224286i
$372$	0	0
$373$	− 8397.99i	− 1.16577i	−0.812555	−	0.582884i	$-0.801924\pi$
$373$	− 8397.99i	− 1.16577i	0.812555	−	0.582884i	$-0.198076\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8020.83	1.09574
$378$	0	0
$379$	242.519i	0.0328690i	0.999865	+	0.0164345i	$0.00523150\pi$
$379$	242.519i	0.0328690i	−0.999865	+	0.0164345i	$0.994768\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3601.64	0.480510	0.240255	−	0.970710i	$-0.422769\pi$
$383$	3601.64	0.480510	0.240255	+	0.970710i	$0.422769\pi$
$384$	0	0
$385$	117.430	0.0155449
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 13291.8i	− 1.73245i	−0.499656	−	0.866224i	$-0.666540\pi$
$389$	− 13291.8i	− 1.73245i	0.499656	−	0.866224i	$-0.333460\pi$
$390$	0	0
$391$	−4302.99	−0.556551
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 826.008i	− 0.105218i
$396$	0	0
$397$	− 3255.27i	− 0.411530i	−0.978601	−	0.205765i	$-0.934032\pi$
$397$	− 3255.27i	− 0.411530i	0.978601	−	0.205765i	$-0.0659682\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8209.13	−1.02231	−0.511153	−	0.859490i	$-0.670781\pi$
$401$	−8209.13	−1.02231	−0.511153	+	0.859490i	$0.670781\pi$
$402$	0	0
$403$	9049.19i	1.11854i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3249.67	−0.395774
$408$	0	0
$409$	4554.98	0.550683	0.275342	−	0.961346i	$-0.411209\pi$
$409$	4554.98	0.550683	0.275342	+	0.961346i	$0.411209\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 429.238i	− 0.0511414i
$414$	0	0
$415$	157.229	0.0185978
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 7550.04i	− 0.880295i	−0.897925	−	0.440148i	$-0.854926\pi$
$419$	− 7550.04i	− 0.880295i	0.897925	−	0.440148i	$-0.145074\pi$
$420$	0	0
$421$	11044.7i	1.27859i	0.768961	+	0.639296i	$0.220774\pi$
$421$	11044.7i	1.27859i	−0.768961	+	0.639296i	$0.779226\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10737.9	1.22557
$426$	0	0
$427$	736.351i	0.0834532i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9156.47	1.02332	0.511661	−	0.859187i	$-0.329030\pi$
$431$	9156.47	1.02332	0.511661	+	0.859187i	$0.329030\pi$
$432$	0	0
$433$	−2250.45	−0.249768	−0.124884	−	0.992171i	$-0.539856\pi$
$433$	−2250.45	−0.249768	−0.124884	+	0.992171i	$0.539856\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 451.425i	− 0.0494156i
$438$	0	0
$439$	11848.2	1.28812	0.644061	−	0.764974i	$-0.277248\pi$
$439$	11848.2	1.28812	0.644061	+	0.764974i	$0.277248\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 1274.32i	− 0.136670i	−0.997662	−	0.0683348i	$-0.978231\pi$
$443$	− 1274.32i	− 0.136670i	0.997662	−	0.0683348i	$-0.0217686\pi$
$444$	0	0
$445$	− 728.139i	− 0.0775665i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8201.95	−0.862080	−0.431040	−	0.902333i	$-0.641853\pi$
$449$	−8201.95	−0.862080	−0.431040	+	0.902333i	$0.641853\pi$
$450$	0	0
$451$	− 3351.97i	− 0.349974i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−213.875	−0.0220365
$456$	0	0
$457$	−16627.3	−1.70195	−0.850976	−	0.525205i	$-0.823989\pi$
$457$	−16627.3	−1.70195	−0.850976	+	0.525205i	$0.823989\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8128.27i	0.821196i	0.911816	+	0.410598i	$0.134680\pi$
$461$	8128.27i	0.821196i	−0.911816	+	0.410598i	$0.865320\pi$
$462$	0	0
$463$	1415.26	0.142057	0.0710286	−	0.997474i	$-0.477372\pi$
$463$	1415.26	0.142057	0.0710286	+	0.997474i	$0.477372\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17479.2i	1.73200i	0.500048	+	0.865998i	$0.333316\pi$
$467$	17479.2i	1.73200i	−0.500048	+	0.865998i	$0.666684\pi$
$468$	0	0
$469$	1627.77i	0.160263i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6898.26	0.670576
$474$	0	0
$475$	1126.52i	0.108817i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6370.47	−0.607670	−0.303835	−	0.952725i	$-0.598267\pi$
$479$	−6370.47	−0.607670	−0.303835	+	0.952725i	$0.598267\pi$
$480$	0	0
$481$	5918.60	0.561050
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 98.5592i	− 0.00922751i
$486$	0	0
$487$	−13658.3	−1.27088	−0.635440	−	0.772150i	$-0.719181\pi$
$487$	−13658.3	−1.27088	−0.635440	+	0.772150i	$0.719181\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3242.02i	0.297984i	0.988838	+	0.148992i	$0.0476029\pi$
$491$	3242.02i	0.297984i	−0.988838	+	0.148992i	$0.952397\pi$
$492$	0	0
$493$	10659.2i	0.973767i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1379.71	0.124524
$498$	0	0
$499$	388.342i	0.0348388i	0.999848	+	0.0174194i	$0.00554506\pi$
$499$	388.342i	0.0348388i	−0.999848	+	0.0174194i	$0.994455\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12450.6	−1.10367	−0.551834	−	0.833954i	$-0.686072\pi$
$503$	−12450.6	−1.10367	−0.551834	+	0.833954i	$0.686072\pi$
$504$	0	0
$505$	−1129.33	−0.0995140
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12529.1i	1.09104i	0.838097	+	0.545521i	$0.183668\pi$
$509$	12529.1i	1.09104i	−0.838097	+	0.545521i	$0.816332\pi$
$510$	0	0
$511$	680.630	0.0589223
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 241.525i	− 0.0206658i
$516$	0	0
$517$	9930.85i	0.844794i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5185.14	0.436017	0.218009	−	0.975947i	$-0.430044\pi$
$521$	5185.14	0.436017	0.218009	+	0.975947i	$0.430044\pi$
$522$	0	0
$523$	− 10344.4i	− 0.864877i	−0.901663	−	0.432438i	$-0.857653\pi$
$523$	− 10344.4i	− 0.864877i	0.901663	−	0.432438i	$-0.142347\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12025.8	−0.994030
$528$	0	0
$529$	−9727.34	−0.799485
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6104.93i	0.496123i
$534$	0	0
$535$	1842.19	0.148869
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 12125.7i	− 0.968997i
$540$	0	0
$541$	− 21105.5i	− 1.67726i	−0.544705	−	0.838628i	$-0.683358\pi$
$541$	− 21105.5i	− 1.67726i	0.544705	−	0.838628i	$-0.316642\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−778.817	−0.0612126
$546$	0	0
$547$	12491.4i	0.976409i	0.872729	+	0.488204i	$0.162348\pi$
$547$	12491.4i	0.976409i	−0.872729	+	0.488204i	$0.837652\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1118.26	−0.0864597
$552$	0	0
$553$	1547.23	0.118978
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8317.90i	0.632748i	0.948635	+	0.316374i	$0.102465\pi$
$557$	8317.90i	0.632748i	−0.948635	+	0.316374i	$0.897535\pi$
$558$	0	0
$559$	−12563.8	−0.950609
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 8298.80i	− 0.621231i	−0.950536	−	0.310615i	$-0.899465\pi$
$563$	− 8298.80i	− 0.621231i	0.950536	−	0.310615i	$-0.100535\pi$
$564$	0	0
$565$	− 800.106i	− 0.0595765i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6133.13	−0.451870	−0.225935	−	0.974142i	$-0.572544\pi$
$569$	−6133.13	−0.451870	−0.225935	+	0.974142i	$0.572544\pi$
$570$	0	0
$571$	− 26180.9i	− 1.91880i	−0.282043	−	0.959402i	$-0.591012\pi$
$571$	− 26180.9i	− 1.91880i	0.282043	−	0.959402i	$-0.408988\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6088.09	−0.441549
$576$	0	0
$577$	22659.7	1.63489	0.817447	−	0.576004i	$-0.195389\pi$
$577$	22659.7	1.63489	0.817447	+	0.576004i	$0.195389\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	294.513i	0.0210300i
$582$	0	0
$583$	23335.4	1.65773
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4163.23i	0.292734i	0.989230	+	0.146367i	$0.0467580\pi$
$587$	4163.23i	0.292734i	−0.989230	+	0.146367i	$0.953242\pi$
$588$	0	0
$589$	− 1261.63i	− 0.0882589i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11485.2	−0.795350	−0.397675	−	0.917526i	$-0.630183\pi$
$593$	−11485.2	−0.795350	−0.397675	+	0.917526i	$0.630183\pi$
$594$	0	0
$595$	− 284.227i	− 0.0195835i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10138.6	0.691571	0.345785	−	0.938314i	$-0.387613\pi$
$599$	10138.6	0.691571	0.345785	+	0.938314i	$0.387613\pi$
$600$	0	0
$601$	−5983.08	−0.406082	−0.203041	−	0.979170i	$-0.565082\pi$
$601$	−5983.08	−0.406082	−0.203041	+	0.979170i	$0.565082\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 46.8477i	− 0.00314815i
$606$	0	0
$607$	2579.69	0.172498	0.0862492	−	0.996274i	$-0.472512\pi$
$607$	2579.69	0.172498	0.0862492	+	0.996274i	$0.472512\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 18087.0i	− 1.19758i
$612$	0	0
$613$	− 28274.9i	− 1.86299i	−0.363759	−	0.931493i	$-0.618507\pi$
$613$	− 28274.9i	− 1.86299i	0.363759	−	0.931493i	$-0.381493\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8007.60	0.522486	0.261243	−	0.965273i	$-0.415868\pi$
$617$	8007.60	0.522486	0.261243	+	0.965273i	$0.415868\pi$
$618$	0	0
$619$	27366.1i	1.77696i	0.458915	+	0.888480i	$0.348238\pi$
$619$	27366.1i	1.77696i	−0.458915	+	0.888480i	$0.651762\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1363.91	0.0877107
$624$	0	0
$625$	14974.9	0.958392
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7865.48i	0.498596i
$630$	0	0
$631$	−27939.7	−1.76270	−0.881349	−	0.472466i	$-0.843364\pi$
$631$	−27939.7	−1.76270	−0.881349	+	0.472466i	$0.843364\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1661.14i	0.103812i
$636$	0	0
$637$	22084.4i	1.37365i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−28620.9	−1.76358	−0.881792	−	0.471639i	$-0.843663\pi$
$641$	−28620.9	−1.76358	−0.881792	+	0.471639i	$0.843663\pi$
$642$	0	0
$643$	4650.43i	0.285218i	0.989779	+	0.142609i	$0.0455491\pi$
$643$	4650.43i	0.285218i	−0.989779	+	0.142609i	$0.954451\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22838.1	−1.38772	−0.693862	−	0.720108i	$-0.744092\pi$
$647$	−22838.1	−1.38772	−0.693862	+	0.720108i	$0.744092\pi$
$648$	0	0
$649$	6249.59	0.377994
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 8551.87i	− 0.512497i	−0.966611	−	0.256248i	$-0.917513\pi$
$653$	− 8551.87i	− 0.512497i	0.966611	−	0.256248i	$-0.0824865\pi$
$654$	0	0
$655$	2551.81	0.152225
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 10145.1i	− 0.599689i	−0.953988	−	0.299845i	$-0.903065\pi$
$659$	− 10145.1i	− 0.599689i	0.953988	−	0.299845i	$-0.0969348\pi$
$660$	0	0
$661$	− 17372.6i	− 1.02226i	−0.859503	−	0.511131i	$-0.829227\pi$
$661$	− 17372.6i	− 1.02226i	0.859503	−	0.511131i	$-0.170773\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	29.8182	0.00173880
$666$	0	0
$667$	− 6043.45i	− 0.350829i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10721.1	−0.616815
$672$	0	0
$673$	−19271.9	−1.10383	−0.551914	−	0.833901i	$-0.686102\pi$
$673$	−19271.9	−1.10383	−0.551914	+	0.833901i	$0.686102\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30042.8i	1.70552i	0.522300	+	0.852762i	$0.325074\pi$
$677$	30042.8i	1.70552i	−0.522300	+	0.852762i	$0.674926\pi$
$678$	0	0
$679$	184.615	0.0104343
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 17811.9i	− 0.997880i	−0.866636	−	0.498940i	$-0.833723\pi$
$683$	− 17811.9i	− 0.997880i	0.866636	−	0.498940i	$-0.166277\pi$
$684$	0	0
$685$	− 2846.28i	− 0.158760i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−42500.7	−2.35000
$690$	0	0
$691$	2461.63i	0.135521i	0.997702	+	0.0677604i	$0.0215854\pi$
$691$	2461.63i	0.135521i	−0.997702	+	0.0677604i	$0.978415\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2842.44	−0.155137
$696$	0	0
$697$	−8113.10	−0.440897
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13292.5i	0.716190i	0.933685	+	0.358095i	$0.116574\pi$
$701$	13292.5i	0.716190i	−0.933685	+	0.358095i	$0.883426\pi$
$702$	0	0
$703$	−825.165	−0.0442699
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2115.40i	− 0.112529i
$708$	0	0
$709$	27124.4i	1.43678i	0.695640	+	0.718390i	$0.255121\pi$
$709$	27124.4i	1.43678i	−0.695640	+	0.718390i	$0.744879\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6818.28	0.358130
$714$	0	0
$715$	− 3113.96i	− 0.162875i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−30827.3	−1.59898	−0.799489	−	0.600680i	$-0.794897\pi$
$719$	−30827.3	−1.59898	−0.799489	+	0.600680i	$0.794897\pi$
$720$	0	0
$721$	452.411	0.0233684
$722$	0	0
$723$	0	0
$724$	0	0
$725$	15081.2i	0.772554i
$726$	0	0
$727$	29322.3	1.49588	0.747939	−	0.663767i	$-0.231043\pi$
$727$	29322.3	1.49588	0.747939	+	0.663767i	$0.231043\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 16696.5i	− 0.844791i
$732$	0	0
$733$	5700.95i	0.287271i	0.989631	+	0.143635i	$0.0458792\pi$
$733$	5700.95i	0.287271i	−0.989631	+	0.143635i	$0.954121\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−23699.9	−1.18453
$738$	0	0
$739$	− 26824.2i	− 1.33524i	−0.744502	−	0.667621i	$-0.767313\pi$
$739$	− 26824.2i	− 1.33524i	0.744502	−	0.667621i	$-0.232687\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−25329.2	−1.25066	−0.625328	−	0.780362i	$-0.715035\pi$
$743$	−25329.2	−1.25066	−0.625328	+	0.780362i	$0.715035\pi$
$744$	0	0
$745$	−2994.38	−0.147256
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3450.68i	0.168338i
$750$	0	0
$751$	−9862.86	−0.479229	−0.239614	−	0.970868i	$-0.577021\pi$
$751$	−9862.86	−0.479229	−0.239614	+	0.970868i	$0.577021\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2004.97i	0.0966469i
$756$	0	0
$757$	12425.8i	0.596595i	0.954473	+	0.298298i	$0.0964188\pi$
$757$	12425.8i	0.596595i	−0.954473	+	0.298298i	$0.903581\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38708.2	1.84385	0.921926	−	0.387365i	$-0.126615\pi$
$761$	38708.2	1.84385	0.921926	+	0.387365i	$0.126615\pi$
$762$	0	0
$763$	− 1458.83i	− 0.0692180i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11382.3	−0.535844
$768$	0	0
$769$	−17734.4	−0.831626	−0.415813	−	0.909450i	$-0.636503\pi$
$769$	−17734.4	−0.831626	−0.415813	+	0.909450i	$0.636503\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32273.9i	1.50170i	0.660475	+	0.750848i	$0.270355\pi$
$773$	32273.9i	1.50170i	−0.660475	+	0.750848i	$0.729645\pi$
$774$	0	0
$775$	−17014.8	−0.788631
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 851.143i	− 0.0391468i
$780$	0	0
$781$	20088.2i	0.920375i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	87.9556	0.00399907
$786$	0	0
$787$	24819.5i	1.12417i	0.827080	+	0.562084i	$0.190000\pi$
$787$	24819.5i	1.12417i	−0.827080	+	0.562084i	$0.810000\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1498.71	0.0673679
$792$	0	0
$793$	19526.2	0.874397
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15582.6i	0.692553i	0.938132	+	0.346277i	$0.112554\pi$
$797$	15582.6i	0.692553i	−0.938132	+	0.346277i	$0.887446\pi$
$798$	0	0
$799$	24036.6	1.06427
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9909.80i	0.435503i
$804$	0	0
$805$	161.148i	0.00705556i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	13815.9	0.600422	0.300211	−	0.953873i	$-0.402943\pi$
$809$	13815.9	0.600422	0.300211	+	0.953873i	$0.402943\pi$
$810$	0	0
$811$	26934.2i	1.16620i	0.812400	+	0.583100i	$0.198161\pi$
$811$	26934.2i	1.16620i	−0.812400	+	0.583100i	$0.801839\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	291.092	0.0125111
$816$	0	0
$817$	1751.63	0.0750081
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8300.96i	0.352869i	0.984312	+	0.176434i	$0.0564564\pi$
$821$	8300.96i	0.352869i	−0.984312	+	0.176434i	$0.943544\pi$
$822$	0	0
$823$	−27975.6	−1.18489	−0.592447	−	0.805609i	$-0.701838\pi$
$823$	−27975.6	−1.18489	−0.592447	+	0.805609i	$0.701838\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11664.8i	0.490476i	0.969463	+	0.245238i	$0.0788662\pi$
$827$	11664.8i	0.490476i	−0.969463	+	0.245238i	$0.921134\pi$
$828$	0	0
$829$	− 22133.6i	− 0.927298i	−0.886019	−	0.463649i	$-0.846540\pi$
$829$	− 22133.6i	− 0.927298i	0.886019	−	0.463649i	$-0.153460\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−29348.9	−1.22074
$834$	0	0
$835$	− 4385.53i	− 0.181757i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−44278.9	−1.82202	−0.911012	−	0.412380i	$-0.864698\pi$
$839$	−44278.9	−1.82202	−0.911012	+	0.412380i	$0.864698\pi$
$840$	0	0
$841$	9418.38	0.386173
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2771.93i	0.112849i
$846$	0	0
$847$	87.7524	0.00355987
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 4459.48i	− 0.179635i
$852$	0	0
$853$	32745.5i	1.31440i	0.753715	+	0.657202i	$0.228260\pi$
$853$	32745.5i	1.31440i	−0.753715	+	0.657202i	$0.771740\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11799.2	0.470305	0.235153	−	0.971958i	$-0.424441\pi$
$857$	11799.2	0.470305	0.235153	+	0.971958i	$0.424441\pi$
$858$	0	0
$859$	6975.49i	0.277067i	0.990358	+	0.138534i	$0.0442389\pi$
$859$	6975.49i	0.277067i	−0.990358	+	0.138534i	$0.955761\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8615.25	0.339822	0.169911	−	0.985459i	$-0.445652\pi$
$863$	8615.25	0.339822	0.169911	+	0.985459i	$0.445652\pi$
$864$	0	0
$865$	−4866.88	−0.191305
$866$	0	0
$867$	0	0
$868$	0	0
$869$	22527.3i	0.879385i
$870$	0	0
$871$	43164.6	1.67919
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 809.961i	− 0.0312934i
$876$	0	0
$877$	− 32197.0i	− 1.23970i	−0.784720	−	0.619850i	$-0.787193\pi$
$877$	− 32197.0i	− 1.23970i	0.784720	−	0.619850i	$-0.212807\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30615.2	−1.17077	−0.585387	−	0.810754i	$-0.699057\pi$
$881$	−30615.2	−1.17077	−0.585387	+	0.810754i	$0.699057\pi$
$882$	0	0
$883$	− 37640.3i	− 1.43454i	−0.696796	−	0.717269i	$-0.745392\pi$
$883$	− 37640.3i	− 1.43454i	0.696796	−	0.717269i	$-0.254608\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21465.3	−0.812553	−0.406276	−	0.913750i	$-0.633173\pi$
$887$	−21465.3	−0.812553	−0.406276	+	0.913750i	$0.633173\pi$
$888$	0	0
$889$	−3111.55	−0.117388
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2521.67i	0.0944955i
$894$	0	0
$895$	−1774.11	−0.0662593
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 16890.0i	− 0.626600i
$900$	0	0
$901$	− 56481.0i	− 2.08841i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−222.284	−0.00816462
$906$	0	0
$907$	8317.31i	0.304489i	0.988343	+	0.152245i	$0.0486502\pi$
$907$	8317.31i	0.304489i	−0.988343	+	0.152245i	$0.951350\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−28334.9	−1.03049	−0.515246	−	0.857043i	$-0.672299\pi$
$911$	−28334.9	−1.03049	−0.515246	+	0.857043i	$0.672299\pi$
$912$	0	0
$913$	−4288.03	−0.155436
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4779.91i	0.172134i
$918$	0	0
$919$	−18433.3	−0.661652	−0.330826	−	0.943692i	$-0.607327\pi$
$919$	−18433.3	−0.661652	−0.330826	+	0.943692i	$0.607327\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 36586.6i	− 1.30473i
$924$	0	0
$925$	11128.5i	0.395570i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−20036.0	−0.707600	−0.353800	−	0.935321i	$-0.615111\pi$
$929$	−20036.0	−0.707600	−0.353800	+	0.935321i	$0.615111\pi$
$930$	0	0
$931$	− 3078.99i	− 0.108388i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4138.28	0.144745
$936$	0	0
$937$	24844.2	0.866196	0.433098	−	0.901347i	$-0.357420\pi$
$937$	24844.2	0.866196	0.433098	+	0.901347i	$0.357420\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12898.8i	0.446853i	0.974721	+	0.223426i	$0.0717243\pi$
$941$	12898.8i	0.446853i	−0.974721	+	0.223426i	$0.928276\pi$
$942$	0	0
$943$	4599.88	0.158847
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 45388.5i	− 1.55747i	−0.627350	−	0.778737i	$-0.715861\pi$
$947$	− 45388.5i	− 1.55747i	0.627350	−	0.778737i	$-0.284139\pi$
$948$	0	0
$949$	− 18048.7i	− 0.617370i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	13191.2	0.448378	0.224189	−	0.974546i	$-0.428027\pi$
$953$	13191.2	0.448378	0.224189	+	0.974546i	$0.428027\pi$
$954$	0	0
$955$	− 3783.74i	− 0.128208i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5331.49	0.179523
$960$	0	0
$961$	−10735.5	−0.360360
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 3988.05i	− 0.133036i
$966$	0	0
$967$	−3495.31	−0.116237	−0.0581186	−	0.998310i	$-0.518510\pi$
$967$	−3495.31	−0.116237	−0.0581186	+	0.998310i	$0.518510\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	21176.9i	0.699896i	0.936769	+	0.349948i	$0.113801\pi$
$971$	21176.9i	0.699896i	−0.936769	+	0.349948i	$0.886199\pi$
$972$	0	0
$973$	− 5324.29i	− 0.175425i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−58221.0	−1.90650	−0.953252	−	0.302178i	$-0.902286\pi$
$977$	−58221.0	−1.90650	−0.953252	+	0.302178i	$0.902286\pi$
$978$	0	0
$979$	19858.1i	0.648283i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32317.8	1.04860	0.524302	−	0.851533i	$-0.324326\pi$
$983$	32317.8	1.04860	0.524302	+	0.851533i	$0.324326\pi$
$984$	0	0
$985$	−1032.52	−0.0333998
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9466.41i	0.304362i
$990$	0	0
$991$	−43887.8	−1.40680	−0.703401	−	0.710793i	$-0.748336\pi$
$991$	−43887.8	−1.40680	−0.703401	+	0.710793i	$0.748336\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2540.94i	− 0.0809581i
$996$	0	0
$997$	19646.1i	0.624071i	0.950070	+	0.312036i	$0.101011\pi$
$997$	19646.1i	0.624071i	−0.950070	+	0.312036i	$0.898989\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.d.i.865.8	yes	16
3.2	odd	2		1728.4.d.k.865.10	yes	16
4.3	odd	2	inner	1728.4.d.i.865.7	✓	16
8.3	odd	2	inner	1728.4.d.i.865.9	yes	16
8.5	even	2	inner	1728.4.d.i.865.10	yes	16
12.11	even	2		1728.4.d.k.865.9	yes	16
24.5	odd	2		1728.4.d.k.865.8	yes	16
24.11	even	2		1728.4.d.k.865.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.d.i.865.7	✓	16	4.3	odd	2	inner
1728.4.d.i.865.8	yes	16	1.1	even	1	trivial
1728.4.d.i.865.9	yes	16	8.3	odd	2	inner
1728.4.d.i.865.10	yes	16	8.5	even	2	inner
1728.4.d.k.865.7	yes	16	24.11	even	2
1728.4.d.k.865.8	yes	16	24.5	odd	2
1728.4.d.k.865.9	yes	16	12.11	even	2
1728.4.d.k.865.10	yes	16	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.31976i q^{5} +2.47210 q^{7} +O(q^{10})\)	\(q-1.31976i q^{5} +2.47210 q^{7} +35.9931i q^{11} -65.5540i q^{13} +87.1175 q^{17} +9.13947i q^{19} -49.3929 q^{23} +123.258 q^{25} +122.355i q^{29} -138.042 q^{31} -3.26258i q^{35} +90.2859i q^{37} -93.1282 q^{41} -191.655i q^{43} +275.910 q^{47} -336.889 q^{49} -648.331i q^{53} +47.5023 q^{55} -173.633i q^{59} +297.865i q^{61} -86.5156 q^{65} +658.458i q^{67} +558.113 q^{71} +275.325 q^{73} +88.9784i q^{77} +625.878 q^{79} +119.135i q^{83} -114.974i q^{85} +551.721 q^{89} -162.056i q^{91} +12.0619 q^{95} +74.6796 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 240 q^{17} - 304 q^{25} + 1008 q^{41} + 1616 q^{49} + 2736 q^{65} + 128 q^{73} + 5856 q^{89} - 2576 q^{97}+O(q^{100}) \) 16 * q - 240 * q^17 - 304 * q^25 + 1008 * q^41 + 1616 * q^49 + 2736 * q^65 + 128 * q^73 + 5856 * q^89 - 2576 * q^97

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