LMFDB - Embedded newform 1728.4.d.i.865.11

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			865.11
Root			$-2.23276 - 2.23276i$ of defining polynomial
Character	$\chi$	$=$	1728.865
Dual form			1728.4.d.i.865.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.1512i q^{5} -33.8599 q^{7} +O(q^{10})$	$q+10.1512i q^{5} -33.8599 q^{7} -13.5846i q^{11} -45.9227i q^{13} -131.190 q^{17} +7.89999i q^{19} -28.2430 q^{23} +21.9537 q^{25} +238.102i q^{29} -67.2350 q^{31} -343.718i q^{35} +80.2173i q^{37} -299.356 q^{41} -378.069i q^{43} -367.377 q^{47} +803.492 q^{49} +28.5042i q^{53} +137.899 q^{55} -597.359i q^{59} -142.614i q^{61} +466.169 q^{65} -656.298i q^{67} +982.700 q^{71} -900.999 q^{73} +459.971i q^{77} -381.751 q^{79} +650.106i q^{83} -1331.73i q^{85} +1233.60 q^{89} +1554.94i q^{91} -80.1942 q^{95} -25.6927 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	10.1512i	0.907949i	0.891015	+	0.453974i	$0.149994\pi$
$5$	10.1512i	0.907949i	−0.891015	+	0.453974i	$0.850006\pi$
$6$	0	0
$7$	−33.8599	−1.82826	−0.914131	−	0.405419i	$-0.867126\pi$
$7$	−33.8599	−1.82826	−0.914131	+	0.405419i	$0.867126\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 13.5846i	− 0.372354i	−0.982516	−	0.186177i	$-0.940390\pi$
$11$	− 13.5846i	− 0.372354i	0.982516	−	0.186177i	$-0.0596098\pi$
$12$	0	0
$13$	− 45.9227i	− 0.979743i	−0.871795	−	0.489871i	$-0.837044\pi$
$13$	− 45.9227i	− 0.979743i	0.871795	−	0.489871i	$-0.162956\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−131.190	−1.87166	−0.935828	−	0.352456i	$-0.885347\pi$
$17$	−131.190	−1.87166	−0.935828	+	0.352456i	$0.885347\pi$
$18$	0	0
$19$	7.89999i	0.0953886i	0.998862	+	0.0476943i	$0.0151873\pi$
$19$	7.89999i	0.0953886i	−0.998862	+	0.0476943i	$0.984813\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−28.2430	−0.256047	−0.128023	−	0.991771i	$-0.540863\pi$
$23$	−28.2430	−0.256047	−0.128023	+	0.991771i	$0.540863\pi$
$24$	0	0
$25$	21.9537	0.175629
$26$	0	0
$27$	0	0
$28$	0	0
$29$	238.102i	1.52463i	0.647204	+	0.762317i	$0.275938\pi$
$29$	238.102i	1.52463i	−0.647204	+	0.762317i	$0.724062\pi$
$30$	0	0
$31$	−67.2350	−0.389541	−0.194770	−	0.980849i	$-0.562396\pi$
$31$	−67.2350	−0.389541	−0.194770	+	0.980849i	$0.562396\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 343.718i	− 1.65997i
$36$	0	0
$37$	80.2173i	0.356423i	0.983992	+	0.178211i	$0.0570311\pi$
$37$	80.2173i	0.356423i	−0.983992	+	0.178211i	$0.942969\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−299.356	−1.14028	−0.570140	−	0.821547i	$-0.693111\pi$
$41$	−299.356	−1.14028	−0.570140	+	0.821547i	$0.693111\pi$
$42$	0	0
$43$	− 378.069i	− 1.34081i	−0.741994	−	0.670407i	$-0.766120\pi$
$43$	− 378.069i	− 1.34081i	0.741994	−	0.670407i	$-0.233880\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−367.377	−1.14016	−0.570079	−	0.821590i	$-0.693088\pi$
$47$	−367.377	−1.14016	−0.570079	+	0.821590i	$0.693088\pi$
$48$	0	0
$49$	803.492	2.34254
$50$	0	0
$51$	0	0
$52$	0	0
$53$	28.5042i	0.0738746i	0.999318	+	0.0369373i	$0.0117602\pi$
$53$	28.5042i	0.0738746i	−0.999318	+	0.0369373i	$0.988240\pi$
$54$	0	0
$55$	137.899	0.338079
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 597.359i	− 1.31813i	−0.752087	−	0.659063i	$-0.770953\pi$
$59$	− 597.359i	− 1.31813i	0.752087	−	0.659063i	$-0.229047\pi$
$60$	0	0
$61$	− 142.614i	− 0.299342i	−0.988736	−	0.149671i	$-0.952179\pi$
$61$	− 142.614i	− 0.299342i	0.988736	−	0.149671i	$-0.0478214\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	466.169	0.889556
$66$	0	0
$67$	− 656.298i	− 1.19671i	−0.801231	−	0.598355i	$-0.795821\pi$
$67$	− 656.298i	− 1.19671i	0.801231	−	0.598355i	$-0.204179\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	982.700	1.64261	0.821303	−	0.570493i	$-0.193248\pi$
$71$	982.700	1.64261	0.821303	+	0.570493i	$0.193248\pi$
$72$	0	0
$73$	−900.999	−1.44457	−0.722287	−	0.691593i	$-0.756909\pi$
$73$	−900.999	−1.44457	−0.722287	+	0.691593i	$0.756909\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	459.971i	0.680761i
$78$	0	0
$79$	−381.751	−0.543675	−0.271837	−	0.962343i	$-0.587631\pi$
$79$	−381.751	−0.543675	−0.271837	+	0.962343i	$0.587631\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	650.106i	0.859740i	0.902891	+	0.429870i	$0.141441\pi$
$83$	650.106i	0.859740i	−0.902891	+	0.429870i	$0.858559\pi$
$84$	0	0
$85$	− 1331.73i	− 1.69937i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1233.60	1.46923	0.734615	−	0.678484i	$-0.237363\pi$
$89$	1233.60	1.46923	0.734615	+	0.678484i	$0.237363\pi$
$90$	0	0
$91$	1554.94i	1.79123i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−80.1942	−0.0866079
$96$	0	0
$97$	−25.6927	−0.0268938	−0.0134469	−	0.999910i	$-0.504280\pi$
$97$	−25.6927	−0.0268938	−0.0134469	+	0.999910i	$0.504280\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 556.136i	− 0.547897i	−0.961744	−	0.273949i	$-0.911670\pi$
$101$	− 556.136i	− 0.547897i	0.961744	−	0.273949i	$-0.0883299\pi$
$102$	0	0
$103$	−855.126	−0.818039	−0.409020	−	0.912526i	$-0.634129\pi$
$103$	−855.126	−0.818039	−0.409020	+	0.912526i	$0.634129\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1423.76i	1.28636i	0.765716	+	0.643179i	$0.222385\pi$
$107$	1423.76i	1.28636i	−0.765716	+	0.643179i	$0.777615\pi$
$108$	0	0
$109$	1772.20i	1.55730i	0.627460	+	0.778649i	$0.284095\pi$
$109$	1772.20i	1.55730i	−0.627460	+	0.778649i	$0.715905\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	348.197	0.289873	0.144936	−	0.989441i	$-0.453702\pi$
$113$	348.197	0.289873	0.144936	+	0.989441i	$0.453702\pi$
$114$	0	0
$115$	− 286.700i	− 0.232477i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4442.07	3.42188
$120$	0	0
$121$	1146.46	0.861352
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1491.75i	1.06741i
$126$	0	0
$127$	1541.87	1.07731	0.538657	−	0.842525i	$-0.318932\pi$
$127$	1541.87	1.07731	0.538657	+	0.842525i	$0.318932\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 190.595i	− 0.127117i	−0.997978	−	0.0635585i	$-0.979755\pi$
$131$	− 190.595i	− 0.127117i	0.997978	−	0.0635585i	$-0.0202449\pi$
$132$	0	0
$133$	− 267.493i	− 0.174395i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2328.41	1.45204	0.726020	−	0.687674i	$-0.241368\pi$
$137$	2328.41	1.45204	0.726020	+	0.687674i	$0.241368\pi$
$138$	0	0
$139$	− 1310.45i	− 0.799645i	−0.916593	−	0.399822i	$-0.869072\pi$
$139$	− 1310.45i	− 0.799645i	0.916593	−	0.399822i	$-0.130928\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−623.839	−0.364812
$144$	0	0
$145$	−2417.01	−1.38429
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2479.21i	1.36312i	0.731763	+	0.681559i	$0.238698\pi$
$149$	2479.21i	1.36312i	−0.731763	+	0.681559i	$0.761302\pi$
$150$	0	0
$151$	2277.83	1.22760	0.613799	−	0.789462i	$-0.289640\pi$
$151$	2277.83	1.22760	0.613799	+	0.789462i	$0.289640\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 682.514i	− 0.353683i
$156$	0	0
$157$	2918.51i	1.48358i	0.670631	+	0.741791i	$0.266023\pi$
$157$	2918.51i	1.48358i	−0.670631	+	0.741791i	$0.733977\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	956.306	0.468121
$162$	0	0
$163$	3451.21i	1.65840i	0.558949	+	0.829202i	$0.311205\pi$
$163$	3451.21i	1.65840i	−0.558949	+	0.829202i	$0.688795\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2215.29	1.02649	0.513246	−	0.858241i	$-0.328443\pi$
$167$	2215.29	1.02649	0.513246	+	0.858241i	$0.328443\pi$
$168$	0	0
$169$	88.1083	0.0401039
$170$	0	0
$171$	0	0
$172$	0	0
$173$	379.191i	0.166644i	0.996523	+	0.0833218i	$0.0265529\pi$
$173$	379.191i	0.166644i	−0.996523	+	0.0833218i	$0.973447\pi$
$174$	0	0
$175$	−743.349	−0.321097
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4053.71i	− 1.69267i	−0.532648	−	0.846337i	$-0.678803\pi$
$179$	− 4053.71i	− 1.69267i	0.532648	−	0.846337i	$-0.321197\pi$
$180$	0	0
$181$	660.485i	0.271235i	0.990761	+	0.135617i	$0.0433018\pi$
$181$	660.485i	0.271235i	−0.990761	+	0.135617i	$0.956698\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−814.299	−0.323613
$186$	0	0
$187$	1782.15i	0.696920i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−583.007	−0.220863	−0.110432	−	0.993884i	$-0.535223\pi$
$191$	−583.007	−0.220863	−0.110432	+	0.993884i	$0.535223\pi$
$192$	0	0
$193$	−2720.37	−1.01459	−0.507296	−	0.861772i	$-0.669355\pi$
$193$	−2720.37	−1.01459	−0.507296	+	0.861772i	$0.669355\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3698.56i	1.33762i	0.743432	+	0.668812i	$0.233197\pi$
$197$	3698.56i	1.33762i	−0.743432	+	0.668812i	$0.766803\pi$
$198$	0	0
$199$	−2153.47	−0.767112	−0.383556	−	0.923518i	$-0.625301\pi$
$199$	−2153.47	−0.767112	−0.383556	+	0.923518i	$0.625301\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 8062.10i	− 2.78743i
$204$	0	0
$205$	− 3038.81i	− 1.03532i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	107.318	0.0355183
$210$	0	0
$211$	− 469.208i	− 0.153088i	−0.997066	−	0.0765442i	$-0.975611\pi$
$211$	− 469.208i	− 0.153088i	0.997066	−	0.0765442i	$-0.0243886\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3837.84	1.21739
$216$	0	0
$217$	2276.57	0.712182
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6024.58i	1.83374i
$222$	0	0
$223$	201.276	0.0604416	0.0302208	−	0.999543i	$-0.490379\pi$
$223$	201.276	0.0604416	0.0302208	+	0.999543i	$0.490379\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2071.91i	− 0.605804i	−0.953022	−	0.302902i	$-0.902045\pi$
$227$	− 2071.91i	− 0.605804i	0.953022	−	0.302902i	$-0.0979555\pi$
$228$	0	0
$229$	− 4791.50i	− 1.38267i	−0.722534	−	0.691335i	$-0.757023\pi$
$229$	− 4791.50i	− 1.38267i	0.722534	−	0.691335i	$-0.242977\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−420.018	−0.118096	−0.0590478	−	0.998255i	$-0.518806\pi$
$233$	−420.018	−0.118096	−0.0590478	+	0.998255i	$0.518806\pi$
$234$	0	0
$235$	− 3729.31i	− 1.03521i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	213.131	0.0576832	0.0288416	−	0.999584i	$-0.490818\pi$
$239$	213.131	0.0576832	0.0288416	+	0.999584i	$0.490818\pi$
$240$	0	0
$241$	−1184.91	−0.316707	−0.158354	−	0.987382i	$-0.550619\pi$
$241$	−1184.91	−0.316707	−0.158354	+	0.987382i	$0.550619\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	8156.38i	2.12691i
$246$	0	0
$247$	362.789	0.0934563
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 435.785i	− 0.109588i	−0.998498	−	0.0547938i	$-0.982550\pi$
$251$	− 435.785i	− 0.109588i	0.998498	−	0.0547938i	$-0.0174501\pi$
$252$	0	0
$253$	383.669i	0.0953402i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2774.42	−0.673398	−0.336699	−	0.941612i	$-0.609311\pi$
$257$	−2774.42	−0.673398	−0.336699	+	0.941612i	$0.609311\pi$
$258$	0	0
$259$	− 2716.15i	− 0.651634i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2037.84	0.477789	0.238895	−	0.971046i	$-0.423215\pi$
$263$	2037.84	0.477789	0.238895	+	0.971046i	$0.423215\pi$
$264$	0	0
$265$	−289.351	−0.0670744
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2124.72i	− 0.481586i	−0.970576	−	0.240793i	$-0.922592\pi$
$269$	− 2124.72i	− 0.481586i	0.970576	−	0.240793i	$-0.0774076\pi$
$270$	0	0
$271$	−3347.92	−0.750449	−0.375225	−	0.926934i	$-0.622434\pi$
$271$	−3347.92	−0.750449	−0.375225	+	0.926934i	$0.622434\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 298.231i	− 0.0653964i
$276$	0	0
$277$	− 400.152i	− 0.0867971i	−0.999058	−	0.0433986i	$-0.986181\pi$
$277$	− 400.152i	− 0.0867971i	0.999058	−	0.0433986i	$-0.0138185\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6250.69	1.32699	0.663496	−	0.748179i	$-0.269072\pi$
$281$	6250.69	1.32699	0.663496	+	0.748179i	$0.269072\pi$
$282$	0	0
$283$	5019.04i	1.05424i	0.849790	+	0.527122i	$0.176729\pi$
$283$	5019.04i	1.05424i	−0.849790	+	0.527122i	$0.823271\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10136.2	2.08473
$288$	0	0
$289$	12297.7	2.50310
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3328.32i	0.663626i	0.943345	+	0.331813i	$0.107660\pi$
$293$	3328.32i	0.663626i	−0.943345	+	0.331813i	$0.892340\pi$
$294$	0	0
$295$	6063.89	1.19679
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1297.00i	0.250860i
$300$	0	0
$301$	12801.4i	2.45136i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1447.70	0.271787
$306$	0	0
$307$	− 7458.88i	− 1.38665i	−0.720626	−	0.693324i	$-0.756146\pi$
$307$	− 7458.88i	− 1.38665i	0.720626	−	0.693324i	$-0.243854\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7654.03	1.39556	0.697781	−	0.716311i	$-0.254171\pi$
$311$	7654.03	1.39556	0.697781	+	0.716311i	$0.254171\pi$
$312$	0	0
$313$	−4667.07	−0.842805	−0.421403	−	0.906874i	$-0.638462\pi$
$313$	−4667.07	−0.842805	−0.421403	+	0.906874i	$0.638462\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6817.04i	1.20783i	0.797048	+	0.603916i	$0.206394\pi$
$317$	6817.04i	1.20783i	−0.797048	+	0.603916i	$0.793606\pi$
$318$	0	0
$319$	3234.51	0.567704
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1036.40i	− 0.178535i
$324$	0	0
$325$	− 1008.17i	− 0.172072i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12439.3	2.08451
$330$	0	0
$331$	− 9883.48i	− 1.64122i	−0.571486	−	0.820612i	$-0.693633\pi$
$331$	− 9883.48i	− 1.64122i	0.571486	−	0.820612i	$-0.306367\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6662.20	1.08655
$336$	0	0
$337$	2441.10	0.394585	0.197292	−	0.980345i	$-0.436785\pi$
$337$	2441.10	0.394585	0.197292	+	0.980345i	$0.436785\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	913.357i	0.145047i
$342$	0	0
$343$	−15592.2	−2.45452
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8251.95i	1.27662i	0.769778	+	0.638311i	$0.220367\pi$
$347$	8251.95i	1.27662i	−0.769778	+	0.638311i	$0.779633\pi$
$348$	0	0
$349$	− 10034.7i	− 1.53910i	−0.638587	−	0.769550i	$-0.720481\pi$
$349$	− 10034.7i	− 1.53910i	0.638587	−	0.769550i	$-0.279519\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	815.263	0.122924	0.0614618	−	0.998109i	$-0.480424\pi$
$353$	815.263	0.122924	0.0614618	+	0.998109i	$0.480424\pi$
$354$	0	0
$355$	9975.55i	1.49140i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7902.99	−1.16185	−0.580925	−	0.813957i	$-0.697309\pi$
$359$	−7902.99	−1.16185	−0.580925	+	0.813957i	$0.697309\pi$
$360$	0	0
$361$	6796.59	0.990901
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 9146.20i	− 1.31160i
$366$	0	0
$367$	328.415	0.0467115	0.0233557	−	0.999727i	$-0.492565\pi$
$367$	328.415	0.0467115	0.0233557	+	0.999727i	$0.492565\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 965.149i	− 0.135062i
$372$	0	0
$373$	− 11569.9i	− 1.60608i	−0.595924	−	0.803041i	$-0.703214\pi$
$373$	− 11569.9i	− 1.60608i	0.595924	−	0.803041i	$-0.296786\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10934.3	1.49375
$378$	0	0
$379$	26.9740i	0.00365584i	0.999998	+	0.00182792i	$0.000581845\pi$
$379$	26.9740i	0.00365584i	−0.999998	+	0.00182792i	$0.999418\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10627.4	1.41785	0.708924	−	0.705285i	$-0.249181\pi$
$383$	10627.4	1.41785	0.708924	+	0.705285i	$0.249181\pi$
$384$	0	0
$385$	−4669.25	−0.618096
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 8604.92i	− 1.12156i	−0.827965	−	0.560780i	$-0.810501\pi$
$389$	− 8604.92i	− 1.12156i	0.827965	−	0.560780i	$-0.189499\pi$
$390$	0	0
$391$	3705.19	0.479232
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 3875.22i	− 0.493629i
$396$	0	0
$397$	− 1354.63i	− 0.171251i	−0.996327	−	0.0856256i	$-0.972711\pi$
$397$	− 1354.63i	− 0.171251i	0.996327	−	0.0856256i	$-0.0272889\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10242.5	1.27552	0.637762	−	0.770234i	$-0.279861\pi$
$401$	10242.5	1.27552	0.637762	+	0.770234i	$0.279861\pi$
$402$	0	0
$403$	3087.61i	0.381650i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1089.72	0.132716
$408$	0	0
$409$	−11355.2	−1.37281	−0.686406	−	0.727219i	$-0.740813\pi$
$409$	−11355.2	−1.37281	−0.686406	+	0.727219i	$0.740813\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	20226.5i	2.40988i
$414$	0	0
$415$	−6599.34	−0.780600
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1194.81i	0.139309i	0.997571	+	0.0696543i	$0.0221896\pi$
$419$	1194.81i	0.139309i	−0.997571	+	0.0696543i	$0.977810\pi$
$420$	0	0
$421$	6624.50i	0.766884i	0.923565	+	0.383442i	$0.125261\pi$
$421$	6624.50i	0.766884i	−0.923565	+	0.383442i	$0.874739\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2880.10	−0.328718
$426$	0	0
$427$	4828.89i	0.547275i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12152.0	1.35810	0.679051	−	0.734091i	$-0.262391\pi$
$431$	12152.0	1.35810	0.679051	+	0.734091i	$0.262391\pi$
$432$	0	0
$433$	−56.5109	−0.00627192	−0.00313596	−	0.999995i	$-0.500998\pi$
$433$	−56.5109	−0.00627192	−0.00313596	+	0.999995i	$0.500998\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 223.120i	− 0.0244239i
$438$	0	0
$439$	10326.6	1.12269	0.561347	−	0.827581i	$-0.310283\pi$
$439$	10326.6	1.12269	0.561347	+	0.827581i	$0.310283\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 16354.5i	− 1.75401i	−0.480484	−	0.877004i	$-0.659539\pi$
$443$	− 16354.5i	− 1.75401i	0.480484	−	0.877004i	$-0.340461\pi$
$444$	0	0
$445$	12522.5i	1.33399i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9058.38	−0.952096	−0.476048	−	0.879419i	$-0.657931\pi$
$449$	−9058.38	−0.952096	−0.476048	+	0.879419i	$0.657931\pi$
$450$	0	0
$451$	4066.62i	0.424589i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15784.4	−1.62634
$456$	0	0
$457$	−9478.41	−0.970200	−0.485100	−	0.874459i	$-0.661217\pi$
$457$	−9478.41	−0.970200	−0.485100	+	0.874459i	$0.661217\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 4920.18i	− 0.497084i	−0.968621	−	0.248542i	$-0.920049\pi$
$461$	− 4920.18i	− 0.497084i	0.968621	−	0.248542i	$-0.0799514\pi$
$462$	0	0
$463$	−1311.62	−0.131654	−0.0658271	−	0.997831i	$-0.520969\pi$
$463$	−1311.62	−0.131654	−0.0658271	+	0.997831i	$0.520969\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5211.07i	0.516359i	0.966097	+	0.258179i	$0.0831225\pi$
$467$	5211.07i	0.516359i	−0.966097	+	0.258179i	$0.916877\pi$
$468$	0	0
$469$	22222.2i	2.18790i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5135.90	−0.499258
$474$	0	0
$475$	173.434i	0.0167530i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4097.70	−0.390874	−0.195437	−	0.980716i	$-0.562613\pi$
$479$	−4097.70	−0.390874	−0.195437	+	0.980716i	$0.562613\pi$
$480$	0	0
$481$	3683.79	0.349202
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 260.811i	− 0.0244182i
$486$	0	0
$487$	10183.4	0.947545	0.473773	−	0.880647i	$-0.342892\pi$
$487$	10183.4	0.947545	0.473773	+	0.880647i	$0.342892\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10083.3i	0.926792i	0.886151	+	0.463396i	$0.153369\pi$
$491$	10083.3i	0.926792i	−0.886151	+	0.463396i	$0.846631\pi$
$492$	0	0
$493$	− 31236.5i	− 2.85359i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−33274.1	−3.00311
$498$	0	0
$499$	− 8267.61i	− 0.741702i	−0.928692	−	0.370851i	$-0.879066\pi$
$499$	− 8267.61i	− 0.741702i	0.928692	−	0.370851i	$-0.120934\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21173.2	−1.87687	−0.938435	−	0.345455i	$-0.887725\pi$
$503$	−21173.2	−1.87687	−0.938435	+	0.345455i	$0.887725\pi$
$504$	0	0
$505$	5645.44	0.497463
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15391.3i	1.34029i	0.742231	+	0.670145i	$0.233768\pi$
$509$	15391.3i	1.34029i	−0.742231	+	0.670145i	$0.766232\pi$
$510$	0	0
$511$	30507.7	2.64106
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 8680.53i	− 0.742738i
$516$	0	0
$517$	4990.66i	0.424543i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4749.37	0.399374	0.199687	−	0.979860i	$-0.436007\pi$
$521$	4749.37	0.399374	0.199687	+	0.979860i	$0.436007\pi$
$522$	0	0
$523$	15205.4i	1.27129i	0.771982	+	0.635644i	$0.219265\pi$
$523$	15205.4i	1.27129i	−0.771982	+	0.635644i	$0.780735\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8820.53	0.729086
$528$	0	0
$529$	−11369.3	−0.934440
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13747.2i	1.11718i
$534$	0	0
$535$	−14452.9	−1.16795
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 10915.1i	− 0.872255i
$540$	0	0
$541$	11241.2i	0.893339i	0.894699	+	0.446669i	$0.147390\pi$
$541$	11241.2i	0.893339i	−0.894699	+	0.446669i	$0.852610\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−17989.9	−1.41395
$546$	0	0
$547$	− 18724.2i	− 1.46360i	−0.681521	−	0.731798i	$-0.738681\pi$
$547$	− 18724.2i	− 1.46360i	0.681521	−	0.731798i	$-0.261319\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1881.00	−0.145433
$552$	0	0
$553$	12926.0	0.993979
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 23022.4i	− 1.75133i	−0.482921	−	0.875664i	$-0.660424\pi$
$557$	− 23022.4i	− 1.75133i	0.482921	−	0.875664i	$-0.339576\pi$
$558$	0	0
$559$	−17361.9	−1.31365
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 14064.1i	− 1.05281i	−0.850234	−	0.526404i	$-0.823540\pi$
$563$	− 14064.1i	− 1.05281i	0.850234	−	0.526404i	$-0.176460\pi$
$564$	0	0
$565$	3534.61i	0.263190i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10365.8	0.763720	0.381860	−	0.924220i	$-0.375284\pi$
$569$	10365.8	0.763720	0.381860	+	0.924220i	$0.375284\pi$
$570$	0	0
$571$	− 5061.81i	− 0.370981i	−0.982646	−	0.185491i	$-0.940613\pi$
$571$	− 5061.81i	− 0.370981i	0.982646	−	0.185491i	$-0.0593874\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−620.039	−0.0449694
$576$	0	0
$577$	−16242.2	−1.17187	−0.585937	−	0.810357i	$-0.699273\pi$
$577$	−16242.2	−1.17187	−0.585937	+	0.810357i	$0.699273\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 22012.5i	− 1.57183i
$582$	0	0
$583$	387.217	0.0275075
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10850.8i	0.762964i	0.924376	+	0.381482i	$0.124586\pi$
$587$	10850.8i	0.762964i	−0.924376	+	0.381482i	$0.875414\pi$
$588$	0	0
$589$	− 531.156i	− 0.0371577i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26700.0	−1.84897	−0.924484	−	0.381221i	$-0.875504\pi$
$593$	−26700.0	−1.84897	−0.924484	+	0.381221i	$0.875504\pi$
$594$	0	0
$595$	45092.2i	3.10689i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9801.85	−0.668602	−0.334301	−	0.942466i	$-0.608500\pi$
$599$	−9801.85	−0.668602	−0.334301	+	0.942466i	$0.608500\pi$
$600$	0	0
$601$	12947.5	0.878767	0.439384	−	0.898300i	$-0.355197\pi$
$601$	12947.5	0.878767	0.439384	+	0.898300i	$0.355197\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11637.9i	0.782063i
$606$	0	0
$607$	−10742.0	−0.718294	−0.359147	−	0.933281i	$-0.616932\pi$
$607$	−10742.0	−0.718294	−0.359147	+	0.933281i	$0.616932\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16870.9i	1.11706i
$612$	0	0
$613$	16243.2i	1.07024i	0.844777	+	0.535118i	$0.179733\pi$
$613$	16243.2i	1.07024i	−0.844777	+	0.535118i	$0.820267\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16310.9	1.06427	0.532134	−	0.846660i	$-0.321390\pi$
$617$	16310.9	1.06427	0.532134	+	0.846660i	$0.321390\pi$
$618$	0	0
$619$	14103.9i	0.915808i	0.889002	+	0.457904i	$0.151400\pi$
$619$	14103.9i	0.915808i	−0.889002	+	0.457904i	$0.848600\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−41769.6	−2.68614
$624$	0	0
$625$	−12398.8	−0.793525
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 10523.7i	− 0.667101i
$630$	0	0
$631$	−12131.1	−0.765341	−0.382670	−	0.923885i	$-0.624996\pi$
$631$	−12131.1	−0.765341	−0.382670	+	0.923885i	$0.624996\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15651.8i	0.978146i
$636$	0	0
$637$	− 36898.5i	− 2.29509i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18764.5	1.15625	0.578124	−	0.815949i	$-0.303785\pi$
$641$	18764.5	1.15625	0.578124	+	0.815949i	$0.303785\pi$
$642$	0	0
$643$	− 30709.2i	− 1.88344i	−0.336393	−	0.941722i	$-0.609207\pi$
$643$	− 30709.2i	− 1.88344i	0.336393	−	0.941722i	$-0.390793\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−164.430	−0.00999133	−0.00499567	−	0.999988i	$-0.501590\pi$
$647$	−164.430	−0.00999133	−0.00499567	+	0.999988i	$0.501590\pi$
$648$	0	0
$649$	−8114.85	−0.490810
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 31130.3i	− 1.86558i	−0.360425	−	0.932788i	$-0.617368\pi$
$653$	− 31130.3i	− 1.86558i	0.360425	−	0.932788i	$-0.382632\pi$
$654$	0	0
$655$	1934.76	0.115416
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9482.21i	0.560507i	0.959926	+	0.280254i	$0.0904186\pi$
$659$	9482.21i	0.560507i	−0.959926	+	0.280254i	$0.909581\pi$
$660$	0	0
$661$	− 22841.2i	− 1.34405i	−0.740527	−	0.672026i	$-0.765424\pi$
$661$	− 22841.2i	− 1.34405i	0.740527	−	0.672026i	$-0.234576\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2715.37	0.158342
$666$	0	0
$667$	− 6724.71i	− 0.390378i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1937.35	−0.111461
$672$	0	0
$673$	−2738.50	−0.156852	−0.0784261	−	0.996920i	$-0.524989\pi$
$673$	−2738.50	−0.156852	−0.0784261	+	0.996920i	$0.524989\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 19290.9i	− 1.09514i	−0.836760	−	0.547570i	$-0.815553\pi$
$677$	− 19290.9i	− 1.09514i	0.836760	−	0.547570i	$-0.184447\pi$
$678$	0	0
$679$	869.951	0.0491688
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 11873.0i	− 0.665167i	−0.943074	−	0.332584i	$-0.892080\pi$
$683$	− 11873.0i	− 0.665167i	0.943074	−	0.332584i	$-0.107920\pi$
$684$	0	0
$685$	23636.1i	1.31838i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1308.99	0.0723781
$690$	0	0
$691$	− 27541.3i	− 1.51624i	−0.652117	−	0.758118i	$-0.726119\pi$
$691$	− 27541.3i	− 1.51624i	0.652117	−	0.758118i	$-0.273881\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13302.6	0.726036
$696$	0	0
$697$	39272.4	2.13421
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 21855.2i	− 1.17754i	−0.808299	−	0.588772i	$-0.799612\pi$
$701$	− 21855.2i	− 1.17754i	0.808299	−	0.588772i	$-0.200388\pi$
$702$	0	0
$703$	−633.716	−0.0339986
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18830.7i	1.00170i
$708$	0	0
$709$	− 19143.2i	− 1.01402i	−0.861941	−	0.507008i	$-0.830752\pi$
$709$	− 19143.2i	− 1.01402i	0.861941	−	0.507008i	$-0.169248\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1898.92	0.0997407
$714$	0	0
$715$	− 6332.70i	− 0.331230i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2602.05	−0.134965	−0.0674827	−	0.997720i	$-0.521497\pi$
$719$	−2602.05	−0.134965	−0.0674827	+	0.997720i	$0.521497\pi$
$720$	0	0
$721$	28954.5	1.49559
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5227.21i	0.267771i
$726$	0	0
$727$	27517.5	1.40381	0.701903	−	0.712273i	$-0.252334\pi$
$727$	27517.5	1.40381	0.701903	+	0.712273i	$0.252334\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	49598.7i	2.50954i
$732$	0	0
$733$	7060.72i	0.355789i	0.984050	+	0.177895i	$0.0569286\pi$
$733$	7060.72i	0.355789i	−0.984050	+	0.177895i	$0.943071\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8915.52	−0.445600
$738$	0	0
$739$	7853.91i	0.390948i	0.980709	+	0.195474i	$0.0626246\pi$
$739$	7853.91i	0.390948i	−0.980709	+	0.195474i	$0.937375\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2484.32	−0.122666	−0.0613329	−	0.998117i	$-0.519535\pi$
$743$	−2484.32	−0.122666	−0.0613329	+	0.998117i	$0.519535\pi$
$744$	0	0
$745$	−25166.9	−1.23764
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 48208.4i	− 2.35180i
$750$	0	0
$751$	−27913.9	−1.35632	−0.678158	−	0.734916i	$-0.737221\pi$
$751$	−27913.9	−1.35632	−0.678158	+	0.734916i	$0.737221\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	23122.7i	1.11460i
$756$	0	0
$757$	36850.4i	1.76929i	0.466269	+	0.884643i	$0.345598\pi$
$757$	36850.4i	1.76929i	−0.466269	+	0.884643i	$0.654402\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	24465.1	1.16539	0.582694	−	0.812691i	$-0.301998\pi$
$761$	24465.1	1.16539	0.582694	+	0.812691i	$0.301998\pi$
$762$	0	0
$763$	− 60006.3i	− 2.84715i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−27432.3	−1.29143
$768$	0	0
$769$	38131.1	1.78809	0.894047	−	0.447974i	$-0.147854\pi$
$769$	38131.1	1.78809	0.894047	+	0.447974i	$0.147854\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 13060.0i	− 0.607680i	−0.952723	−	0.303840i	$-0.901731\pi$
$773$	− 13060.0i	− 0.607680i	0.952723	−	0.303840i	$-0.0982688\pi$
$774$	0	0
$775$	−1476.06	−0.0684148
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2364.91i	− 0.108770i
$780$	0	0
$781$	− 13349.5i	− 0.611631i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−29626.3	−1.34702
$786$	0	0
$787$	18883.6i	0.855310i	0.903942	+	0.427655i	$0.140660\pi$
$787$	18883.6i	0.855310i	−0.903942	+	0.427655i	$0.859340\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11789.9	−0.529963
$792$	0	0
$793$	−6549.22	−0.293278
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6999.63i	0.311091i	0.987829	+	0.155546i	$0.0497136\pi$
$797$	6999.63i	0.311091i	−0.987829	+	0.155546i	$0.950286\pi$
$798$	0	0
$799$	48196.1	2.13399
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12239.7i	0.537894i
$804$	0	0
$805$	9707.62i	0.425030i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24272.5	1.05485	0.527426	−	0.849601i	$-0.323157\pi$
$809$	24272.5	1.05485	0.527426	+	0.849601i	$0.323157\pi$
$810$	0	0
$811$	22038.3i	0.954217i	0.878845	+	0.477108i	$0.158315\pi$
$811$	22038.3i	0.954217i	−0.878845	+	0.477108i	$0.841685\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−35033.9	−1.50575
$816$	0	0
$817$	2986.74	0.127898
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 12819.5i	− 0.544950i	−0.962163	−	0.272475i	$-0.912158\pi$
$821$	− 12819.5i	− 0.544950i	0.962163	−	0.272475i	$-0.0878422\pi$
$822$	0	0
$823$	−3627.33	−0.153634	−0.0768170	−	0.997045i	$-0.524476\pi$
$823$	−3627.33	−0.153634	−0.0768170	+	0.997045i	$0.524476\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40155.1i	1.68843i	0.536007	+	0.844213i	$0.319932\pi$
$827$	40155.1i	1.68843i	−0.536007	+	0.844213i	$0.680068\pi$
$828$	0	0
$829$	− 15348.4i	− 0.643030i	−0.946904	−	0.321515i	$-0.895808\pi$
$829$	− 15348.4i	− 0.643030i	0.946904	−	0.321515i	$-0.104192\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−105410.	−4.38443
$834$	0	0
$835$	22487.8i	0.932002i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	45000.1	1.85170	0.925850	−	0.377891i	$-0.123351\pi$
$839$	45000.1	1.85170	0.925850	+	0.377891i	$0.123351\pi$
$840$	0	0
$841$	−32303.4	−1.32451
$842$	0	0
$843$	0	0
$844$	0	0
$845$	894.403i	0.0364123i
$846$	0	0
$847$	−38819.0	−1.57478
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 2265.58i	− 0.0912609i
$852$	0	0
$853$	− 22419.3i	− 0.899910i	−0.893051	−	0.449955i	$-0.851440\pi$
$853$	− 22419.3i	− 0.899910i	0.893051	−	0.449955i	$-0.148560\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22129.7	0.882073	0.441037	−	0.897489i	$-0.354611\pi$
$857$	22129.7	0.882073	0.441037	+	0.897489i	$0.354611\pi$
$858$	0	0
$859$	16643.6i	0.661086i	0.943791	+	0.330543i	$0.107232\pi$
$859$	16643.6i	0.661086i	−0.943791	+	0.330543i	$0.892768\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−43612.7	−1.72027	−0.860135	−	0.510066i	$-0.829621\pi$
$863$	−43612.7	−1.72027	−0.860135	+	0.510066i	$0.829621\pi$
$864$	0	0
$865$	−3849.23	−0.151304
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5185.91i	0.202440i
$870$	0	0
$871$	−30139.0	−1.17247
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 50510.6i	− 1.95151i
$876$	0	0
$877$	− 24856.8i	− 0.957075i	−0.878067	−	0.478537i	$-0.841167\pi$
$877$	− 24856.8i	− 0.957075i	0.878067	−	0.478537i	$-0.158833\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9284.63	−0.355059	−0.177530	−	0.984115i	$-0.556811\pi$
$881$	−9284.63	−0.355059	−0.177530	+	0.984115i	$0.556811\pi$
$882$	0	0
$883$	− 10002.7i	− 0.381220i	−0.981666	−	0.190610i	$-0.938953\pi$
$883$	− 10002.7i	− 0.381220i	0.981666	−	0.190610i	$-0.0610465\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17532.9	−0.663695	−0.331848	−	0.943333i	$-0.607672\pi$
$887$	−17532.9	−0.663695	−0.331848	+	0.943333i	$0.607672\pi$
$888$	0	0
$889$	−52207.6	−1.96961
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2902.28i	− 0.108758i
$894$	0	0
$895$	41149.9	1.53686
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 16008.8i	− 0.593907i
$900$	0	0
$901$	− 3739.46i	− 0.138268i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6704.70	−0.246267
$906$	0	0
$907$	− 6351.57i	− 0.232525i	−0.993219	−	0.116263i	$-0.962909\pi$
$907$	− 6351.57i	− 0.232525i	0.993219	−	0.116263i	$-0.0370914\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4753.51	−0.172877	−0.0864384	−	0.996257i	$-0.527549\pi$
$911$	−4753.51	−0.172877	−0.0864384	+	0.996257i	$0.527549\pi$
$912$	0	0
$913$	8831.41	0.320128
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6453.51i	0.232403i
$918$	0	0
$919$	−41240.4	−1.48030	−0.740150	−	0.672442i	$-0.765245\pi$
$919$	−41240.4	−1.48030	−0.740150	+	0.672442i	$0.765245\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 45128.2i	− 1.60933i
$924$	0	0
$925$	1761.06i	0.0625983i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−33423.7	−1.18040	−0.590202	−	0.807255i	$-0.700952\pi$
$929$	−33423.7	−1.18040	−0.590202	+	0.807255i	$0.700952\pi$
$930$	0	0
$931$	6347.58i	0.223452i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18090.9	−0.632767
$936$	0	0
$937$	52959.8	1.84645	0.923223	−	0.384264i	$-0.125545\pi$
$937$	52959.8	1.84645	0.923223	+	0.384264i	$0.125545\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 27256.8i	− 0.944258i	−0.881529	−	0.472129i	$-0.843486\pi$
$941$	− 27256.8i	− 0.944258i	0.881529	−	0.472129i	$-0.156514\pi$
$942$	0	0
$943$	8454.71	0.291965
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3537.26i	0.121379i	0.998157	+	0.0606893i	$0.0193299\pi$
$947$	3537.26i	0.121379i	−0.998157	+	0.0606893i	$0.980670\pi$
$948$	0	0
$949$	41376.3i	1.41531i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−23661.6	−0.804275	−0.402138	−	0.915579i	$-0.631733\pi$
$953$	−23661.6	−0.804275	−0.402138	+	0.915579i	$0.631733\pi$
$954$	0	0
$955$	− 5918.20i	− 0.200532i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−78839.7	−2.65471
$960$	0	0
$961$	−25270.5	−0.848258
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 27614.9i	− 0.921198i
$966$	0	0
$967$	16800.1	0.558691	0.279345	−	0.960191i	$-0.409883\pi$
$967$	16800.1	0.558691	0.279345	+	0.960191i	$0.409883\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28036.7i	0.926611i	0.886199	+	0.463306i	$0.153337\pi$
$971$	28036.7i	0.926611i	−0.886199	+	0.463306i	$0.846663\pi$
$972$	0	0
$973$	44371.6i	1.46196i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8961.37	−0.293449	−0.146724	−	0.989177i	$-0.546873\pi$
$977$	−8961.37	−0.293449	−0.146724	+	0.989177i	$0.546873\pi$
$978$	0	0
$979$	− 16757.9i	− 0.547074i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−33372.7	−1.08283	−0.541415	−	0.840755i	$-0.682111\pi$
$983$	−33372.7	−1.08283	−0.541415	+	0.840755i	$0.682111\pi$
$984$	0	0
$985$	−37544.8	−1.21449
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10677.8i	0.343311i
$990$	0	0
$991$	−39288.7	−1.25938	−0.629691	−	0.776846i	$-0.716818\pi$
$991$	−39288.7	−1.25938	−0.629691	+	0.776846i	$0.716818\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 21860.2i	− 0.696499i
$996$	0	0
$997$	− 26951.7i	− 0.856137i	−0.903746	−	0.428068i	$-0.859194\pi$
$997$	− 26951.7i	− 0.856137i	0.903746	−	0.428068i	$-0.140806\pi$
$998$	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.865.11	yes	16
3.2	odd	2		1728.4.d.k.865.5	yes	16
4.3	odd	2	inner	1728.4.d.i.865.12	yes	16
8.3	odd	2	inner	1728.4.d.i.865.6	yes	16
8.5	even	2	inner	1728.4.d.i.865.5	✓	16
12.11	even	2		1728.4.d.k.865.6	yes	16
24.5	odd	2		1728.4.d.k.865.11	yes	16
24.11	even	2		1728.4.d.k.865.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.d.i.865.5	✓	16	8.5	even	2	inner
1728.4.d.i.865.6	yes	16	8.3	odd	2	inner
1728.4.d.i.865.11	yes	16	1.1	even	1	trivial
1728.4.d.i.865.12	yes	16	4.3	odd	2	inner
1728.4.d.k.865.5	yes	16	3.2	odd	2
1728.4.d.k.865.6	yes	16	12.11	even	2
1728.4.d.k.865.11	yes	16	24.5	odd	2
1728.4.d.k.865.12	yes	16	24.11	even	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)

\(f(q)\)	\(=\)	\(q+10.1512i q^{5} -33.8599 q^{7} +O(q^{10})\)	\(q+10.1512i q^{5} -33.8599 q^{7} -13.5846i q^{11} -45.9227i q^{13} -131.190 q^{17} +7.89999i q^{19} -28.2430 q^{23} +21.9537 q^{25} +238.102i q^{29} -67.2350 q^{31} -343.718i q^{35} +80.2173i q^{37} -299.356 q^{41} -378.069i q^{43} -367.377 q^{47} +803.492 q^{49} +28.5042i q^{53} +137.899 q^{55} -597.359i q^{59} -142.614i q^{61} +466.169 q^{65} -656.298i q^{67} +982.700 q^{71} -900.999 q^{73} +459.971i q^{77} -381.751 q^{79} +650.106i q^{83} -1331.73i q^{85} +1233.60 q^{89} +1554.94i q^{91} -80.1942 q^{95} -25.6927 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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