LMFDB - Embedded newform 1728.2.d.b.865.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,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, 2, names="a")

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

chi := DirichletCharacter("1728.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			865.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1728.865
Dual form			1728.2.d.b.865.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{5} -3.00000 q^{7} +O(q^{10})$	$q+3.00000i q^{5} -3.00000 q^{7} -3.00000i q^{11} -6.00000i q^{13} +6.00000 q^{17} +2.00000i q^{19} -6.00000 q^{23} -4.00000 q^{25} -6.00000i q^{29} -3.00000 q^{31} -9.00000i q^{35} +6.00000i q^{37} +6.00000 q^{41} -8.00000i q^{43} +12.0000 q^{47} +2.00000 q^{49} -3.00000i q^{53} +9.00000 q^{55} -12.0000i q^{59} +18.0000 q^{65} +4.00000i q^{67} +6.00000 q^{71} +11.0000 q^{73} +9.00000i q^{77} +12.0000 q^{79} -9.00000i q^{83} +18.0000i q^{85} +18.0000i q^{91} -6.00000 q^{95} -17.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{7}+O(q^{10}) $ 2 * q - 6 * q^7	$ 2 q - 6 q^{7} + 12 q^{17} - 12 q^{23} - 8 q^{25} - 6 q^{31} + 12 q^{41} + 24 q^{47} + 4 q^{49} + 18 q^{55} + 36 q^{65} + 12 q^{71} + 22 q^{73} + 24 q^{79} - 12 q^{95} - 34 q^{97}+O(q^{100}) $ 2 * q - 6 * q^7 + 12 * q^17 - 12 * q^23 - 8 * q^25 - 6 * q^31 + 12 * q^41 + 24 * q^47 + 4 * q^49 + 18 * q^55 + 36 * q^65 + 12 * q^71 + 22 * q^73 + 24 * q^79 - 12 * q^95 - 34 * 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
$5$	3.00000i	1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$	3.00000i	1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.00000i	− 0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$	− 3.00000i	− 0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 9.00000i	− 1.52128i
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 3.00000i	− 0.412082i	−0.978543	−	0.206041i	$-0.933942\pi$
$53$	− 3.00000i	− 0.412082i	0.978543	−	0.206041i	$-0.0660580\pi$
$54$	0	0
$55$	9.00000	1.21356
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	18.0000	2.23263
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.00000i	1.02565i
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	0	0
$85$	18.0000i	1.95237i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	18.0000i	1.88691i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 3.00000i	− 0.298511i	−0.988799	−	0.149256i	$-0.952312\pi$
$101$	− 3.00000i	− 0.298511i	0.988799	−	0.149256i	$-0.0476877\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.00000i	− 0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$	− 3.00000i	− 0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$	0	0
$109$	− 6.00000i	− 0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	− 6.00000i	− 0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	− 18.0000i	− 1.67851i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−18.0000	−1.65006
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.00000i	0.268328i
$126$	0	0
$127$	−15.0000	−1.33103	−0.665517	−	0.746382i	$-0.731789\pi$
$127$	−15.0000	−1.33103	−0.665517	+	0.746382i	$0.731789\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 9.00000i	− 0.786334i	−0.919467	−	0.393167i	$-0.871379\pi$
$131$	− 9.00000i	− 0.786334i	0.919467	−	0.393167i	$-0.128621\pi$
$132$	0	0
$133$	− 6.00000i	− 0.520266i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	− 4.00000i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	− 4.00000i	− 0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−18.0000	−1.50524
$144$	0	0
$145$	18.0000	1.49482
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 9.00000i	− 0.737309i	−0.929567	−	0.368654i	$-0.879819\pi$
$149$	− 9.00000i	− 0.737309i	0.929567	−	0.368654i	$-0.120181\pi$
$150$	0	0
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 9.00000i	− 0.722897i
$156$	0	0
$157$	− 12.0000i	− 0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	− 12.0000i	− 0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.0000	1.41860
$162$	0	0
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.0000i	1.59660i	0.602260	+	0.798300i	$0.294267\pi$
$173$	21.0000i	1.59660i	−0.602260	+	0.798300i	$0.705733\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.00000i	0.672692i	0.941739	+	0.336346i	$0.109191\pi$
$179$	9.00000i	0.672692i	−0.941739	+	0.336346i	$0.890809\pi$
$180$	0	0
$181$	− 24.0000i	− 1.78391i	−0.452128	−	0.891953i	$-0.649335\pi$
$181$	− 24.0000i	− 1.78391i	0.452128	−	0.891953i	$-0.350665\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−18.0000	−1.32339
$186$	0	0
$187$	− 18.0000i	− 1.31629i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.00000i	0.641223i	0.947211	+	0.320612i	$0.103888\pi$
$197$	9.00000i	0.641223i	−0.947211	+	0.320612i	$0.896112\pi$
$198$	0	0
$199$	−21.0000	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$199$	−21.0000	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.0000i	1.26335i
$204$	0	0
$205$	18.0000i	1.25717i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	22.0000i	1.51454i	0.653101	+	0.757271i	$0.273468\pi$
$211$	22.0000i	1.51454i	−0.653101	+	0.757271i	$0.726532\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	24.0000	1.63679
$216$	0	0
$217$	9.00000	0.610960
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 36.0000i	− 2.42162i
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	− 12.0000i	− 0.792982i	−0.918039	−	0.396491i	$-0.870228\pi$
$229$	− 12.0000i	− 0.792982i	0.918039	−	0.396491i	$-0.129772\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	36.0000i	2.34838i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.00000i	0.383326i
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 12.0000i	− 0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$	− 12.0000i	− 0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$	0	0
$253$	18.0000i	1.13165i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	− 18.0000i	− 1.11847i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 18.0000i	− 1.09748i	−0.835993	−	0.548740i	$-0.815108\pi$
$269$	− 18.0000i	− 1.09748i	0.835993	−	0.548740i	$-0.184892\pi$
$270$	0	0
$271$	27.0000	1.64013	0.820067	−	0.572268i	$-0.193936\pi$
$271$	27.0000	1.64013	0.820067	+	0.572268i	$0.193936\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.0000i	0.723627i
$276$	0	0
$277$	− 12.0000i	− 0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$	− 12.0000i	− 0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	− 22.0000i	− 1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	− 22.0000i	− 1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−18.0000	−1.06251
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	36.0000	2.09600
$296$	0	0
$297$	0	0
$298$	0	0
$299$	36.0000i	2.08193i
$300$	0	0
$301$	24.0000i	1.38334i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	−17.0000	−0.960897	−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000	−0.960897	−0.480448	+	0.877023i	$0.659526\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.00000i	− 0.505490i	−0.967533	−	0.252745i	$-0.918667\pi$
$317$	− 9.00000i	− 0.505490i	0.967533	−	0.252745i	$-0.0813334\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.0000i	0.667698i
$324$	0	0
$325$	24.0000i	1.33128i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−36.0000	−1.98474
$330$	0	0
$331$	− 28.0000i	− 1.53902i	−0.638635	−	0.769510i	$-0.720501\pi$
$331$	− 28.0000i	− 1.53902i	0.638635	−	0.769510i	$-0.279499\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.00000i	0.487377i
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.0000i	0.805242i	0.915367	+	0.402621i	$0.131901\pi$
$347$	15.0000i	0.805242i	−0.915367	+	0.402621i	$0.868099\pi$
$348$	0	0
$349$	6.00000i	0.321173i	0.987022	+	0.160586i	$0.0513385\pi$
$349$	6.00000i	0.321173i	−0.987022	+	0.160586i	$0.948662\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	18.0000i	0.955341i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	33.0000i	1.72730i
$366$	0	0
$367$	9.00000	0.469796	0.234898	−	0.972020i	$-0.424524\pi$
$367$	9.00000	0.469796	0.234898	+	0.972020i	$0.424524\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.00000i	0.467257i
$372$	0	0
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−36.0000	−1.85409
$378$	0	0
$379$	16.0000i	0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$	16.0000i	0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	−27.0000	−1.37605
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.0000i	0.760530i	0.924878	+	0.380265i	$0.124167\pi$
$389$	15.0000i	0.760530i	−0.924878	+	0.380265i	$0.875833\pi$
$390$	0	0
$391$	−36.0000	−1.82060
$392$	0	0
$393$	0	0
$394$	0	0
$395$	36.0000i	1.81136i
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	18.0000i	0.896644i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	18.0000	0.892227
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	36.0000i	1.77144i
$414$	0	0
$415$	27.0000	1.32538
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	− 12.0000i	− 0.584844i	−0.956289	−	0.292422i	$-0.905539\pi$
$421$	− 12.0000i	− 0.584844i	0.956289	−	0.292422i	$-0.0944612\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−24.0000	−1.16417
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 12.0000i	− 0.574038i
$438$	0	0
$439$	−9.00000	−0.429547	−0.214773	−	0.976664i	$-0.568901\pi$
$439$	−9.00000	−0.429547	−0.214773	+	0.976664i	$0.568901\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	− 18.0000i	− 0.847587i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−54.0000	−2.53156
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 27.0000i	− 1.25752i	−0.777601	−	0.628758i	$-0.783564\pi$
$461$	− 27.0000i	− 1.25752i	0.777601	−	0.628758i	$-0.216436\pi$
$462$	0	0
$463$	3.00000	0.139422	0.0697109	−	0.997567i	$-0.477792\pi$
$463$	3.00000	0.139422	0.0697109	+	0.997567i	$0.477792\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	39.0000i	1.80470i	0.430999	+	0.902352i	$0.358161\pi$
$467$	39.0000i	1.80470i	−0.430999	+	0.902352i	$0.641839\pi$
$468$	0	0
$469$	− 12.0000i	− 0.554109i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	− 8.00000i	− 0.367065i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 51.0000i	− 2.31579i
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.00000i	0.135388i	0.997706	+	0.0676941i	$0.0215642\pi$
$491$	3.00000i	0.135388i	−0.997706	+	0.0676941i	$0.978436\pi$
$492$	0	0
$493$	− 36.0000i	− 1.62136i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.0000	−0.807410
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	9.00000	0.400495
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.0000i	0.664863i	0.943127	+	0.332432i	$0.107869\pi$
$509$	15.0000i	0.664863i	−0.943127	+	0.332432i	$0.892131\pi$
$510$	0	0
$511$	−33.0000	−1.45983
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 36.0000i	− 1.58328i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	34.0000i	1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$	34.0000i	1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.0000	−0.784092
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 36.0000i	− 1.55933i
$534$	0	0
$535$	9.00000	0.389104
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 6.00000i	− 0.258438i
$540$	0	0
$541$	− 12.0000i	− 0.515920i	−0.966156	−	0.257960i	$-0.916950\pi$
$541$	− 12.0000i	− 0.515920i	0.966156	−	0.257960i	$-0.0830503\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.0000	0.771035
$546$	0	0
$547$	− 10.0000i	− 0.427569i	−0.976881	−	0.213785i	$-0.931421\pi$
$547$	− 10.0000i	− 0.427569i	0.976881	−	0.213785i	$-0.0685791\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	−36.0000	−1.53088
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 27.0000i	− 1.14403i	−0.820244	−	0.572013i	$-0.806163\pi$
$557$	− 27.0000i	− 1.14403i	0.820244	−	0.572013i	$-0.193837\pi$
$558$	0	0
$559$	−48.0000	−2.03018
$560$	0	0
$561$	0	0
$562$	0	0
$563$	21.0000i	0.885044i	0.896758	+	0.442522i	$0.145916\pi$
$563$	21.0000i	0.885044i	−0.896758	+	0.442522i	$0.854084\pi$
$564$	0	0
$565$	54.0000i	2.27180i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	− 22.0000i	− 0.920671i	−0.887745	−	0.460336i	$-0.847729\pi$
$571$	− 22.0000i	− 0.920671i	0.887745	−	0.460336i	$-0.152271\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24.0000	1.00087
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	27.0000i	1.12015i
$582$	0	0
$583$	−9.00000	−0.372742
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.0000i	1.36206i	0.732257	+	0.681028i	$0.238467\pi$
$587$	33.0000i	1.36206i	−0.732257	+	0.681028i	$0.761533\pi$
$588$	0	0
$589$	− 6.00000i	− 0.247226i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	− 54.0000i	− 2.21378i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.00000i	0.243935i
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 72.0000i	− 2.91281i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	26.0000i	1.04503i	0.852631	+	0.522514i	$0.175006\pi$
$619$	26.0000i	1.04503i	−0.852631	+	0.522514i	$0.824994\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	36.0000i	1.43541i
$630$	0	0
$631$	21.0000	0.835997	0.417998	−	0.908448i	$-0.362732\pi$
$631$	21.0000	0.835997	0.417998	+	0.908448i	$0.362732\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 45.0000i	− 1.78577i
$636$	0	0
$637$	− 12.0000i	− 0.475457i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	32.0000i	1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$	32.0000i	1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.00000i	0.352197i	0.984373	+	0.176099i	$0.0563478\pi$
$653$	9.00000i	0.352197i	−0.984373	+	0.176099i	$0.943652\pi$
$654$	0	0
$655$	27.0000	1.05498
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9.00000i	0.350590i	0.984516	+	0.175295i	$0.0560880\pi$
$659$	9.00000i	0.350590i	−0.984516	+	0.175295i	$0.943912\pi$
$660$	0	0
$661$	18.0000i	0.700119i	0.936727	+	0.350059i	$0.113839\pi$
$661$	18.0000i	0.700119i	−0.936727	+	0.350059i	$0.886161\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	18.0000	0.698010
$666$	0	0
$667$	36.0000i	1.39393i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	37.0000	1.42625	0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000	1.42625	0.713123	+	0.701039i	$0.247280\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	51.0000	1.95720
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	− 36.0000i	− 1.37549i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−18.0000	−0.685745
$690$	0	0
$691$	8.00000i	0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	8.00000i	0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27.0000i	1.01978i	0.860241	+	0.509888i	$0.170313\pi$
$701$	27.0000i	1.01978i	−0.860241	+	0.509888i	$0.829687\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.00000i	0.338480i
$708$	0	0
$709$	48.0000i	1.80268i	0.433114	+	0.901339i	$0.357415\pi$
$709$	48.0000i	1.80268i	−0.433114	+	0.901339i	$0.642585\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	18.0000	0.674105
$714$	0	0
$715$	− 54.0000i	− 2.01949i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	24.0000i	0.891338i
$726$	0	0
$727$	−27.0000	−1.00137	−0.500687	−	0.865628i	$-0.666919\pi$
$727$	−27.0000	−1.00137	−0.500687	+	0.865628i	$0.666919\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 48.0000i	− 1.77534i
$732$	0	0
$733$	30.0000i	1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$	30.0000i	1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	− 38.0000i	− 1.39785i	−0.715194	−	0.698926i	$-0.753662\pi$
$739$	− 38.0000i	− 1.39785i	0.715194	−	0.698926i	$-0.246338\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	27.0000	0.989203
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.00000i	0.328853i
$750$	0	0
$751$	−3.00000	−0.109472	−0.0547358	−	0.998501i	$-0.517432\pi$
$751$	−3.00000	−0.109472	−0.0547358	+	0.998501i	$0.517432\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 27.0000i	− 0.982631i
$756$	0	0
$757$	36.0000i	1.30844i	0.756303	+	0.654221i	$0.227003\pi$
$757$	36.0000i	1.30844i	−0.756303	+	0.654221i	$0.772997\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	0	0
$763$	18.0000i	0.651644i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−72.0000	−2.59977
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	12.0000	0.431053
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.0000i	0.429945i
$780$	0	0
$781$	− 18.0000i	− 0.644091i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	36.0000	1.28490
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−54.0000	−1.92002
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 51.0000i	− 1.80651i	−0.429101	−	0.903256i	$-0.641170\pi$
$797$	− 51.0000i	− 1.80651i	0.429101	−	0.903256i	$-0.358830\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 33.0000i	− 1.16454i
$804$	0	0
$805$	54.0000i	1.90325i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	− 38.0000i	− 1.33436i	−0.744896	−	0.667180i	$-0.767501\pi$
$811$	− 38.0000i	− 1.33436i	0.744896	−	0.667180i	$-0.232499\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.00000	−0.210171
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.0000i	1.04701i	0.852023	+	0.523504i	$0.175375\pi$
$821$	30.0000i	1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$	0	0
$823$	−21.0000	−0.732014	−0.366007	−	0.930612i	$-0.619275\pi$
$823$	−21.0000	−0.732014	−0.366007	+	0.930612i	$0.619275\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	0	0
$829$	− 12.0000i	− 0.416777i	−0.978046	−	0.208389i	$-0.933178\pi$
$829$	− 12.0000i	− 0.416777i	0.978046	−	0.208389i	$-0.0668219\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	12.0000	0.415775
$834$	0	0
$835$	72.0000i	2.49166i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 69.0000i	− 2.37367i
$846$	0	0
$847$	−6.00000	−0.206162
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 36.0000i	− 1.23406i
$852$	0	0
$853$	− 36.0000i	− 1.23262i	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	− 36.0000i	− 1.23262i	0.787505	−	0.616308i	$-0.211372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	4.00000i	0.136478i	0.997669	+	0.0682391i	$0.0217381\pi$
$859$	4.00000i	0.136478i	−0.997669	+	0.0682391i	$0.978262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−63.0000	−2.14206
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 36.0000i	− 1.22122i
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 9.00000i	− 0.304256i
$876$	0	0
$877$	24.0000i	0.810422i	0.914223	+	0.405211i	$0.132802\pi$
$877$	24.0000i	0.810422i	−0.914223	+	0.405211i	$0.867198\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	− 34.0000i	− 1.14419i	−0.820187	−	0.572096i	$-0.806131\pi$
$883$	− 34.0000i	− 1.14419i	0.820187	−	0.572096i	$-0.193869\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	45.0000	1.50925
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.0000i	0.803129i
$894$	0	0
$895$	−27.0000	−0.902510
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18.0000i	0.600334i
$900$	0	0
$901$	− 18.0000i	− 0.599667i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	72.0000	2.39336
$906$	0	0
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	−27.0000	−0.893570
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.0000i	0.891619i
$918$	0	0
$919$	−9.00000	−0.296883	−0.148441	−	0.988921i	$-0.547426\pi$
$919$	−9.00000	−0.296883	−0.148441	+	0.988921i	$0.547426\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 36.0000i	− 1.18495i
$924$	0	0
$925$	− 24.0000i	− 0.789115i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	4.00000i	0.131095i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	54.0000	1.76599
$936$	0	0
$937$	−11.0000	−0.359354	−0.179677	−	0.983726i	$-0.557505\pi$
$937$	−11.0000	−0.359354	−0.179677	+	0.983726i	$0.557505\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 15.0000i	− 0.488986i	−0.969651	−	0.244493i	$-0.921378\pi$
$941$	− 15.0000i	− 0.488986i	0.969651	−	0.244493i	$-0.0786215\pi$
$942$	0	0
$943$	−36.0000	−1.17232
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 15.0000i	− 0.487435i	−0.969846	−	0.243717i	$-0.921633\pi$
$947$	− 15.0000i	− 0.487435i	0.969846	−	0.243717i	$-0.0783669\pi$
$948$	0	0
$949$	− 66.0000i	− 2.14245i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	0	0
$955$	18.0000i	0.582466i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15.0000i	0.482867i
$966$	0	0
$967$	−27.0000	−0.868261	−0.434131	−	0.900850i	$-0.642944\pi$
$967$	−27.0000	−0.868261	−0.434131	+	0.900850i	$0.642944\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.00000i	0.288824i	0.989518	+	0.144412i	$0.0461290\pi$
$971$	9.00000i	0.288824i	−0.989518	+	0.144412i	$0.953871\pi$
$972$	0	0
$973$	12.0000i	0.384702i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	−27.0000	−0.860292
$986$	0	0
$987$	0	0
$988$	0	0
$989$	48.0000i	1.52631i
$990$	0	0
$991$	27.0000	0.857683	0.428842	−	0.903380i	$-0.358922\pi$
$991$	27.0000	0.857683	0.428842	+	0.903380i	$0.358922\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 63.0000i	− 1.99723i
$996$	0	0
$997$	− 18.0000i	− 0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$	− 18.0000i	− 0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.d.b.865.2	yes	2
3.2	odd	2		1728.2.d.a.865.1	✓	2
4.3	odd	2		1728.2.d.d.865.2	yes	2
8.3	odd	2		1728.2.d.d.865.1	yes	2
8.5	even	2	inner	1728.2.d.b.865.1	yes	2
12.11	even	2		1728.2.d.c.865.1	yes	2
16.3	odd	4		6912.2.a.r.1.1		1
16.5	even	4		6912.2.a.g.1.1		1
16.11	odd	4		6912.2.a.a.1.1		1
16.13	even	4		6912.2.a.x.1.1		1
24.5	odd	2		1728.2.d.a.865.2	yes	2
24.11	even	2		1728.2.d.c.865.2	yes	2
48.5	odd	4		6912.2.a.w.1.1		1
48.11	even	4		6912.2.a.q.1.1		1
48.29	odd	4		6912.2.a.h.1.1		1
48.35	even	4		6912.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.2.d.a.865.1	✓	2	3.2	odd	2
1728.2.d.a.865.2	yes	2	24.5	odd	2
1728.2.d.b.865.1	yes	2	8.5	even	2	inner
1728.2.d.b.865.2	yes	2	1.1	even	1	trivial
1728.2.d.c.865.1	yes	2	12.11	even	2
1728.2.d.c.865.2	yes	2	24.11	even	2
1728.2.d.d.865.1	yes	2	8.3	odd	2
1728.2.d.d.865.2	yes	2	4.3	odd	2
6912.2.a.a.1.1		1	16.11	odd	4
6912.2.a.b.1.1		1	48.35	even	4
6912.2.a.g.1.1		1	16.5	even	4
6912.2.a.h.1.1		1	48.29	odd	4
6912.2.a.q.1.1		1	48.11	even	4
6912.2.a.r.1.1		1	16.3	odd	4
6912.2.a.w.1.1		1	48.5	odd	4
6912.2.a.x.1.1		1	16.13	even	4

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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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{5} -3.00000 q^{7} +O(q^{10})\)	\(q+3.00000i q^{5} -3.00000 q^{7} -3.00000i q^{11} -6.00000i q^{13} +6.00000 q^{17} +2.00000i q^{19} -6.00000 q^{23} -4.00000 q^{25} -6.00000i q^{29} -3.00000 q^{31} -9.00000i q^{35} +6.00000i q^{37} +6.00000 q^{41} -8.00000i q^{43} +12.0000 q^{47} +2.00000 q^{49} -3.00000i q^{53} +9.00000 q^{55} -12.0000i q^{59} +18.0000 q^{65} +4.00000i q^{67} +6.00000 q^{71} +11.0000 q^{73} +9.00000i q^{77} +12.0000 q^{79} -9.00000i q^{83} +18.0000i q^{85} +18.0000i q^{91} -6.00000 q^{95} -17.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{7}+O(q^{10}) \) 2 * q - 6 * q^7	\( 2 q - 6 q^{7} + 12 q^{17} - 12 q^{23} - 8 q^{25} - 6 q^{31} + 12 q^{41} + 24 q^{47} + 4 q^{49} + 18 q^{55} + 36 q^{65} + 12 q^{71} + 22 q^{73} + 24 q^{79} - 12 q^{95} - 34 q^{97}+O(q^{100}) \) 2 * q - 6 * q^7 + 12 * q^17 - 12 * q^23 - 8 * q^25 - 6 * q^31 + 12 * q^41 + 24 * q^47 + 4 * q^49 + 18 * q^55 + 36 * q^65 + 12 * q^71 + 22 * q^73 + 24 * q^79 - 12 * q^95 - 34 * q^97

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