LMFDB - Embedded newform 1152.2.k.a.865.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(289,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1152.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.k (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.19876631285$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			865.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1152.865
Dual form			1152.2.k.a.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.00000i) q^{5} +2.00000i q^{7} +O(q^{10})$	$q+(-1.00000 - 1.00000i) q^{5} +2.00000i q^{7} +(-1.00000 - 1.00000i) q^{11} +(1.00000 - 1.00000i) q^{13} +2.00000 q^{17} +(3.00000 - 3.00000i) q^{19} +6.00000i q^{23} -3.00000i q^{25} +(3.00000 - 3.00000i) q^{29} +8.00000 q^{31} +(2.00000 - 2.00000i) q^{35} +(-3.00000 - 3.00000i) q^{37} +(5.00000 + 5.00000i) q^{43} +8.00000 q^{47} +3.00000 q^{49} +(-5.00000 - 5.00000i) q^{53} +2.00000i q^{55} +(3.00000 + 3.00000i) q^{59} +(9.00000 - 9.00000i) q^{61} -2.00000 q^{65} +(-5.00000 + 5.00000i) q^{67} -10.0000i q^{71} -4.00000i q^{73} +(2.00000 - 2.00000i) q^{77} +(1.00000 - 1.00000i) q^{83} +(-2.00000 - 2.00000i) q^{85} -4.00000i q^{89} +(2.00000 + 2.00000i) q^{91} -6.00000 q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} - 2 q^{11} + 2 q^{13} + 4 q^{17} + 6 q^{19} + 6 q^{29} + 16 q^{31} + 4 q^{35} - 6 q^{37} + 10 q^{43} + 16 q^{47} + 6 q^{49} - 10 q^{53} + 6 q^{59} + 18 q^{61} - 4 q^{65} - 10 q^{67} + 4 q^{77} + 2 q^{83} - 4 q^{85} + 4 q^{91} - 12 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^11 + 2 * q^13 + 4 * q^17 + 6 * q^19 + 6 * q^29 + 16 * q^31 + 4 * q^35 - 6 * q^37 + 10 * q^43 + 16 * q^47 + 6 * q^49 - 10 * q^53 + 6 * q^59 + 18 * q^61 - 4 * q^65 - 10 * q^67 + 4 * q^77 + 2 * q^83 - 4 * q^85 + 4 * q^91 - 12 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$5$	−1.00000	−	1.00000i	−0.447214	−	0.447214i	0.447214	−	0.894427i	$-0.352416\pi$
$5$	−1.00000	−	1.00000i	−0.447214	−	0.447214i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	−	1.00000i	−0.301511	−	0.301511i	0.540094	−	0.841605i	$-0.318389\pi$
$11$	−1.00000	−	1.00000i	−0.301511	−	0.301511i	−0.841605	+	0.540094i	$0.818389\pi$
$12$		0			0
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$	3.00000	−	3.00000i	0.688247	−	0.688247i	−0.273597	−	0.961844i	$-0.588214\pi$
$19$	3.00000	−	3.00000i	0.688247	−	0.688247i	0.961844	+	0.273597i	$0.0882135\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$		−	3.00000i		−	0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.00000	−	3.00000i	0.557086	−	0.557086i	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	3.00000	−	3.00000i	0.557086	−	0.557086i	0.928477	+	0.371391i	$0.121119\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000	−	2.00000i	0.338062	−	0.338062i
$36$		0			0
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	0.416115	−	0.909312i	$-0.363391\pi$
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	−0.909312	+	0.416115i	$0.863391\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	5.00000	+	5.00000i	0.762493	+	0.762493i	0.976772	−	0.214280i	$-0.0687403\pi$
$43$	5.00000	+	5.00000i	0.762493	+	0.762493i	−0.214280	+	0.976772i	$0.568740\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.00000	−	5.00000i	−0.686803	−	0.686803i	0.274721	−	0.961524i	$-0.411414\pi$
$53$	−5.00000	−	5.00000i	−0.686803	−	0.686803i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$			2.00000i			0.269680i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.00000	+	3.00000i	0.390567	+	0.390567i	0.874889	−	0.484323i	$-0.160934\pi$
$59$	3.00000	+	3.00000i	0.390567	+	0.390567i	−0.484323	+	0.874889i	$0.660934\pi$
$60$		0			0
$61$	9.00000	−	9.00000i	1.15233	−	1.15233i	0.166248	−	0.986084i	$-0.446835\pi$
$61$	9.00000	−	9.00000i	1.15233	−	1.15233i	0.986084	−	0.166248i	$-0.0531652\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000			−0.248069
$66$		0			0
$67$	−5.00000	+	5.00000i	−0.610847	+	0.610847i	−0.943167	−	0.332320i	$-0.892169\pi$
$67$	−5.00000	+	5.00000i	−0.610847	+	0.610847i	0.332320	+	0.943167i	$0.392169\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.0000i		−	1.18678i	−0.804914	−	0.593391i	$-0.797789\pi$
$71$		−	10.0000i		−	1.18678i	0.804914	−	0.593391i	$-0.202211\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.00000	−	2.00000i	0.227921	−	0.227921i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.00000	−	1.00000i	0.109764	−	0.109764i	−0.650092	−	0.759856i	$-0.725269\pi$
$83$	1.00000	−	1.00000i	0.109764	−	0.109764i	0.759856	+	0.650092i	$0.225269\pi$
$84$		0			0
$85$	−2.00000	−	2.00000i	−0.216930	−	0.216930i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.00000i		−	0.423999i	−0.977270	−	0.212000i	$-0.932002\pi$
$89$		−	4.00000i		−	0.423999i	0.977270	−	0.212000i	$-0.0679975\pi$
$90$		0			0
$91$	2.00000	+	2.00000i	0.209657	+	0.209657i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.00000			−0.615587
$96$		0			0
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.0000	+	11.0000i	1.09454	+	1.09454i	0.995037	+	0.0995037i	$0.0317255\pi$
$101$	11.0000	+	11.0000i	1.09454	+	1.09454i	0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.00000	+	7.00000i	0.676716	+	0.676716i	0.959256	−	0.282540i	$-0.0911770\pi$
$107$	7.00000	+	7.00000i	0.676716	+	0.676716i	−0.282540	+	0.959256i	$0.591177\pi$
$108$		0			0
$109$	−3.00000	+	3.00000i	−0.287348	+	0.287348i	−0.836031	−	0.548683i	$-0.815129\pi$
$109$	−3.00000	+	3.00000i	−0.287348	+	0.287348i	0.548683	+	0.836031i	$0.315129\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	6.00000	−	6.00000i	0.559503	−	0.559503i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			4.00000i			0.366679i
$120$		0			0
$121$		−	9.00000i		−	0.818182i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.00000	+	8.00000i	−0.715542	+	0.715542i
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11.0000	+	11.0000i	−0.961074	+	0.961074i	−0.999270	−	0.0381958i	$-0.987839\pi$
$131$	−11.0000	+	11.0000i	−0.961074	+	0.961074i	0.0381958	+	0.999270i	$0.487839\pi$
$132$		0			0
$133$	6.00000	+	6.00000i	0.520266	+	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	8.00000i		−	0.683486i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$		−	8.00000i		−	0.683486i	0.939793	−	0.341743i	$-0.111017\pi$
$138$		0			0
$139$	−3.00000	−	3.00000i	−0.254457	−	0.254457i	0.568338	−	0.822795i	$-0.307586\pi$
$139$	−3.00000	−	3.00000i	−0.254457	−	0.254457i	−0.822795	+	0.568338i	$0.807586\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.00000			−0.167248
$144$		0			0
$145$	−6.00000			−0.498273
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.00000	+	7.00000i	0.573462	+	0.573462i	0.933094	−	0.359632i	$-0.117098\pi$
$149$	7.00000	+	7.00000i	0.573462	+	0.573462i	−0.359632	+	0.933094i	$0.617098\pi$
$150$		0			0
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.00000	−	8.00000i	−0.642575	−	0.642575i
$156$		0			0
$157$	−15.0000	+	15.0000i	−1.19713	+	1.19713i	−0.222108	+	0.975022i	$0.571294\pi$
$157$	−15.0000	+	15.0000i	−1.19713	+	1.19713i	−0.975022	+	0.222108i	$0.928706\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$	−1.00000	+	1.00000i	−0.0783260	+	0.0783260i	−0.745184	−	0.666858i	$-0.767639\pi$
$163$	−1.00000	+	1.00000i	−0.0783260	+	0.0783260i	0.666858	+	0.745184i	$0.267639\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2.00000i		−	0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$		−	2.00000i		−	0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$		0			0
$169$			11.0000i			0.846154i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.00000	+	1.00000i	−0.0760286	+	0.0760286i	−0.744099	−	0.668070i	$-0.767121\pi$
$173$	−1.00000	+	1.00000i	−0.0760286	+	0.0760286i	0.668070	+	0.744099i	$0.267121\pi$
$174$		0			0
$175$	6.00000			0.453557
$176$		0			0
$177$		0			0
$178$		0			0
$179$	17.0000	−	17.0000i	1.27064	−	1.27064i	0.324887	−	0.945753i	$-0.394674\pi$
$179$	17.0000	−	17.0000i	1.27064	−	1.27064i	0.945753	−	0.324887i	$-0.105326\pi$
$180$		0			0
$181$	9.00000	+	9.00000i	0.668965	+	0.668965i	0.957476	−	0.288512i	$-0.0931604\pi$
$181$	9.00000	+	9.00000i	0.668965	+	0.668965i	−0.288512	+	0.957476i	$0.593160\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			6.00000i			0.441129i
$186$		0			0
$187$	−2.00000	−	2.00000i	−0.146254	−	0.146254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−17.0000	−	17.0000i	−1.21120	−	1.21120i	−0.970632	−	0.240567i	$-0.922666\pi$
$197$	−17.0000	−	17.0000i	−1.21120	−	1.21120i	−0.240567	−	0.970632i	$-0.577334\pi$
$198$		0			0
$199$		−	14.0000i		−	0.992434i	−0.868199	−	0.496217i	$-0.834722\pi$
$199$		−	14.0000i		−	0.992434i	0.868199	−	0.496217i	$-0.165278\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.00000	+	6.00000i	0.421117	+	0.421117i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.00000			−0.415029
$210$		0			0
$211$	−9.00000	+	9.00000i	−0.619586	+	0.619586i	−0.945425	−	0.325840i	$-0.894353\pi$
$211$	−9.00000	+	9.00000i	−0.619586	+	0.619586i	0.325840	+	0.945425i	$0.394353\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	10.0000i		−	0.681994i
$216$		0			0
$217$			16.0000i			1.08615i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.00000	−	2.00000i	0.134535	−	0.134535i
$222$		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
$227$	−15.0000	+	15.0000i	−0.995585	+	0.995585i	−0.999990	−	0.00440533i	$-0.998598\pi$
$227$	−15.0000	+	15.0000i	−0.995585	+	0.995585i	0.00440533	+	0.999990i	$0.498598\pi$
$228$		0			0
$229$	−7.00000	−	7.00000i	−0.462573	−	0.462573i	0.436925	−	0.899498i	$-0.356068\pi$
$229$	−7.00000	−	7.00000i	−0.462573	−	0.462573i	−0.899498	+	0.436925i	$0.856068\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			4.00000i			0.262049i	0.991379	+	0.131024i	$0.0418266\pi$
$233$			4.00000i			0.262049i	−0.991379	+	0.131024i	$0.958173\pi$
$234$		0			0
$235$	−8.00000	−	8.00000i	−0.521862	−	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.00000	−	3.00000i	−0.191663	−	0.191663i
$246$		0			0
$247$		−	6.00000i		−	0.381771i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.0000	−	21.0000i	−1.32551	−	1.32551i	−0.909243	−	0.416265i	$-0.863339\pi$
$251$	−21.0000	−	21.0000i	−1.32551	−	1.32551i	−0.416265	−	0.909243i	$-0.636661\pi$
$252$		0			0
$253$	6.00000	−	6.00000i	0.377217	−	0.377217i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	22.0000			1.37232			0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000			1.37232			0.686161	+	0.727450i	$0.259294\pi$
$258$		0			0
$259$	6.00000	−	6.00000i	0.372822	−	0.372822i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			6.00000i			0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$			6.00000i			0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$		0			0
$265$			10.0000i			0.614295i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.00000	−	3.00000i	0.182913	−	0.182913i	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	3.00000	−	3.00000i	0.182913	−	0.182913i	0.792624	+	0.609711i	$0.208714\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.00000	+	3.00000i	−0.180907	+	0.180907i
$276$		0			0
$277$	−3.00000	−	3.00000i	−0.180253	−	0.180253i	0.611213	−	0.791466i	$-0.290682\pi$
$277$	−3.00000	−	3.00000i	−0.180253	−	0.180253i	−0.791466	+	0.611213i	$0.790682\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			20.0000i			1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$281$			20.0000i			1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$282$		0			0
$283$	−15.0000	−	15.0000i	−0.891657	−	0.891657i	0.103022	−	0.994679i	$-0.467149\pi$
$283$	−15.0000	−	15.0000i	−0.891657	−	0.891657i	−0.994679	+	0.103022i	$0.967149\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.0000	+	15.0000i	0.876309	+	0.876309i	0.993151	−	0.116841i	$-0.0372769\pi$
$293$	15.0000	+	15.0000i	0.876309	+	0.876309i	−0.116841	+	0.993151i	$0.537277\pi$
$294$		0			0
$295$		−	6.00000i		−	0.349334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000	+	6.00000i	0.346989	+	0.346989i
$300$		0			0
$301$	−10.0000	+	10.0000i	−0.576390	+	0.576390i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−18.0000			−1.03068
$306$		0			0
$307$	−5.00000	+	5.00000i	−0.285365	+	0.285365i	−0.835244	−	0.549879i	$-0.814674\pi$
$307$	−5.00000	+	5.00000i	−0.285365	+	0.285365i	0.549879	+	0.835244i	$0.314674\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			30.0000i			1.70114i	0.525859	+	0.850572i	$0.323744\pi$
$311$			30.0000i			1.70114i	−0.525859	+	0.850572i	$0.676256\pi$
$312$		0			0
$313$			16.0000i			0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$			16.0000i			0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.00000	+	5.00000i	−0.280828	+	0.280828i	−0.833439	−	0.552611i	$-0.813631\pi$
$317$	−5.00000	+	5.00000i	−0.280828	+	0.280828i	0.552611	+	0.833439i	$0.313631\pi$
$318$		0			0
$319$	−6.00000			−0.335936
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.00000	−	6.00000i	0.333849	−	0.333849i
$324$		0			0
$325$	−3.00000	−	3.00000i	−0.166410	−	0.166410i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			16.0000i			0.882109i
$330$		0			0
$331$	1.00000	+	1.00000i	0.0549650	+	0.0549650i	0.734055	−	0.679090i	$-0.237625\pi$
$331$	1.00000	+	1.00000i	0.0549650	+	0.0549650i	−0.679090	+	0.734055i	$0.737625\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10.0000			0.546358
$336$		0			0
$337$	18.0000			0.980522			0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000			0.980522			0.490261	+	0.871576i	$0.336901\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.00000	−	8.00000i	−0.433224	−	0.433224i
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.0000	−	13.0000i	−0.697877	−	0.697877i	0.266076	−	0.963952i	$-0.414273\pi$
$347$	−13.0000	−	13.0000i	−0.697877	−	0.697877i	−0.963952	+	0.266076i	$0.914273\pi$
$348$		0			0
$349$	−3.00000	+	3.00000i	−0.160586	+	0.160586i	−0.782826	−	0.622240i	$-0.786223\pi$
$349$	−3.00000	+	3.00000i	−0.160586	+	0.160586i	0.622240	+	0.782826i	$0.286223\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$	−10.0000	+	10.0000i	−0.530745	+	0.530745i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	26.0000i		−	1.37223i	−0.727494	−	0.686114i	$-0.759315\pi$
$359$		−	26.0000i		−	1.37223i	0.727494	−	0.686114i	$-0.240685\pi$
$360$		0			0
$361$			1.00000i			0.0526316i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.00000	+	4.00000i	−0.209370	+	0.209370i
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.0000	−	10.0000i	0.519174	−	0.519174i
$372$		0			0
$373$	5.00000	+	5.00000i	0.258890	+	0.258890i	0.824603	−	0.565712i	$-0.191399\pi$
$373$	5.00000	+	5.00000i	0.258890	+	0.258890i	−0.565712	+	0.824603i	$0.691399\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	6.00000i		−	0.309016i
$378$		0			0
$379$	−3.00000	−	3.00000i	−0.154100	−	0.154100i	0.625847	−	0.779946i	$-0.284754\pi$
$379$	−3.00000	−	3.00000i	−0.154100	−	0.154100i	−0.779946	+	0.625847i	$0.784754\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.0000			−0.817562			−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000			−0.817562			−0.408781	+	0.912633i	$0.634046\pi$
$384$		0			0
$385$	−4.00000			−0.203859
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.0000	−	13.0000i	−0.659126	−	0.659126i	0.296047	−	0.955173i	$-0.404331\pi$
$389$	−13.0000	−	13.0000i	−0.659126	−	0.659126i	−0.955173	+	0.296047i	$0.904331\pi$
$390$		0			0
$391$			12.0000i			0.606866i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	5.00000	−	5.00000i	0.250943	−	0.250943i	−0.570414	−	0.821357i	$-0.693217\pi$
$397$	5.00000	−	5.00000i	0.250943	−	0.250943i	0.821357	+	0.570414i	$0.193217\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$	8.00000	−	8.00000i	0.398508	−	0.398508i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6.00000i			0.297409i
$408$		0			0
$409$		−	16.0000i		−	0.791149i	−0.918434	−	0.395575i	$-0.870545\pi$
$409$		−	16.0000i		−	0.791149i	0.918434	−	0.395575i	$-0.129455\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.00000	+	6.00000i	−0.295241	+	0.295241i
$414$		0			0
$415$	−2.00000			−0.0981761
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.00000	+	3.00000i	−0.146560	+	0.146560i	−0.776579	−	0.630020i	$-0.783047\pi$
$419$	−3.00000	+	3.00000i	−0.146560	+	0.146560i	0.630020	+	0.776579i	$0.283047\pi$
$420$		0			0
$421$	9.00000	+	9.00000i	0.438633	+	0.438633i	0.891552	−	0.452919i	$-0.149617\pi$
$421$	9.00000	+	9.00000i	0.438633	+	0.438633i	−0.452919	+	0.891552i	$0.649617\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	6.00000i		−	0.291043i
$426$		0			0
$427$	18.0000	+	18.0000i	0.871081	+	0.871081i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	32.0000			1.54139			0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000			1.54139			0.770693	+	0.637207i	$0.219910\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18.0000	+	18.0000i	0.861057	+	0.861057i
$438$		0			0
$439$		−	14.0000i		−	0.668184i	−0.942541	−	0.334092i	$-0.891570\pi$
$439$		−	14.0000i		−	0.668184i	0.942541	−	0.334092i	$-0.108430\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.0000	+	15.0000i	0.712672	+	0.712672i	0.967093	−	0.254422i	$-0.0818852\pi$
$443$	15.0000	+	15.0000i	0.712672	+	0.712672i	−0.254422	+	0.967093i	$0.581885\pi$
$444$		0			0
$445$	−4.00000	+	4.00000i	−0.189618	+	0.189618i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	4.00000i		−	0.187523i
$456$		0			0
$457$		−	32.0000i		−	1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$		−	32.0000i		−	1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11.0000	−	11.0000i	0.512321	−	0.512321i	−0.402916	−	0.915237i	$-0.632003\pi$
$461$	11.0000	−	11.0000i	0.512321	−	0.512321i	0.915237	+	0.402916i	$0.132003\pi$
$462$		0			0
$463$	16.0000			0.743583			0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000			0.743583			0.371792	+	0.928316i	$0.378744\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.00000	−	5.00000i	0.231372	−	0.231372i	−0.581893	−	0.813265i	$-0.697688\pi$
$467$	5.00000	−	5.00000i	0.231372	−	0.231372i	0.813265	+	0.581893i	$0.197688\pi$
$468$		0			0
$469$	−10.0000	−	10.0000i	−0.461757	−	0.461757i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	10.0000i		−	0.459800i
$474$		0			0
$475$	−9.00000	−	9.00000i	−0.412948	−	0.412948i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−40.0000			−1.82765			−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000			−1.82765			−0.913823	+	0.406112i	$0.866884\pi$
$480$		0			0
$481$	−6.00000			−0.273576
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.00000	+	2.00000i	0.0908153	+	0.0908153i
$486$		0			0
$487$			2.00000i			0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$			2.00000i			0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	19.0000	+	19.0000i	0.857458	+	0.857458i	0.991038	−	0.133580i	$-0.0426473\pi$
$491$	19.0000	+	19.0000i	0.857458	+	0.857458i	−0.133580	+	0.991038i	$0.542647\pi$
$492$		0			0
$493$	6.00000	−	6.00000i	0.270226	−	0.270226i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	20.0000			0.897123
$498$		0			0
$499$	23.0000	−	23.0000i	1.02962	−	1.02962i	0.0300737	−	0.999548i	$-0.490426\pi$
$499$	23.0000	−	23.0000i	1.02962	−	1.02962i	0.999548	−	0.0300737i	$-0.00957421\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			6.00000i			0.267527i	0.991013	+	0.133763i	$0.0427062\pi$
$503$			6.00000i			0.267527i	−0.991013	+	0.133763i	$0.957294\pi$
$504$		0			0
$505$		−	22.0000i		−	0.978987i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	23.0000	−	23.0000i	1.01946	−	1.01946i	0.0196502	−	0.999807i	$-0.493745\pi$
$509$	23.0000	−	23.0000i	1.01946	−	1.01946i	0.999807	−	0.0196502i	$-0.00625524\pi$
$510$		0			0
$511$	8.00000			0.353899
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.00000	+	6.00000i	−0.264392	+	0.264392i
$516$		0			0
$517$	−8.00000	−	8.00000i	−0.351840	−	0.351840i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	40.0000i		−	1.75243i	−0.481919	−	0.876216i	$-0.660060\pi$
$521$		−	40.0000i		−	1.75243i	0.481919	−	0.876216i	$-0.339940\pi$
$522$		0			0
$523$	25.0000	+	25.0000i	1.09317	+	1.09317i	0.995188	+	0.0979859i	$0.0312400\pi$
$523$	25.0000	+	25.0000i	1.09317	+	1.09317i	0.0979859	+	0.995188i	$0.468760\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16.0000			0.696971
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	14.0000i		−	0.605273i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.00000	−	3.00000i	−0.129219	−	0.129219i
$540$		0			0
$541$	9.00000	−	9.00000i	0.386940	−	0.386940i	−0.486654	−	0.873595i	$-0.661783\pi$
$541$	9.00000	−	9.00000i	0.386940	−	0.386940i	0.873595	+	0.486654i	$0.161783\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.00000			0.257012
$546$		0			0
$547$	−5.00000	+	5.00000i	−0.213785	+	0.213785i	−0.805873	−	0.592088i	$-0.798304\pi$
$547$	−5.00000	+	5.00000i	−0.213785	+	0.213785i	0.592088	+	0.805873i	$0.298304\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	18.0000i		−	0.766826i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.0000	+	25.0000i	−1.05928	+	1.05928i	−0.0611558	+	0.998128i	$0.519479\pi$
$557$	−25.0000	+	25.0000i	−1.05928	+	1.05928i	−0.998128	+	0.0611558i	$0.980521\pi$
$558$		0			0
$559$	10.0000			0.422955
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.0000	+	19.0000i	−0.800755	+	0.800755i	−0.983213	−	0.182459i	$-0.941594\pi$
$563$	−19.0000	+	19.0000i	−0.800755	+	0.800755i	0.182459	+	0.983213i	$0.441594\pi$
$564$		0			0
$565$	−6.00000	−	6.00000i	−0.252422	−	0.252422i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	24.0000i		−	1.00613i	−0.864248	−	0.503066i	$-0.832205\pi$
$569$		−	24.0000i		−	1.00613i	0.864248	−	0.503066i	$-0.167795\pi$
$570$		0			0
$571$	1.00000	+	1.00000i	0.0418487	+	0.0418487i	0.727721	−	0.685873i	$-0.240579\pi$
$571$	1.00000	+	1.00000i	0.0418487	+	0.0418487i	−0.685873	+	0.727721i	$0.740579\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	18.0000			0.750652
$576$		0			0
$577$	18.0000			0.749350			0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000			0.749350			0.374675	+	0.927156i	$0.377754\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.00000	+	2.00000i	0.0829740	+	0.0829740i
$582$		0			0
$583$			10.0000i			0.414158i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.00000	+	7.00000i	0.288921	+	0.288921i	0.836653	−	0.547733i	$-0.184509\pi$
$587$	7.00000	+	7.00000i	0.288921	+	0.288921i	−0.547733	+	0.836653i	$0.684509\pi$
$588$		0			0
$589$	24.0000	−	24.0000i	0.988903	−	0.988903i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−34.0000			−1.39621			−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000			−1.39621			−0.698106	+	0.715994i	$0.745974\pi$
$594$		0			0
$595$	4.00000	−	4.00000i	0.163984	−	0.163984i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			14.0000i			0.572024i	0.958226	+	0.286012i	$0.0923298\pi$
$599$			14.0000i			0.572024i	−0.958226	+	0.286012i	$0.907670\pi$
$600$		0			0
$601$			20.0000i			0.815817i	0.913023	+	0.407909i	$0.133742\pi$
$601$			20.0000i			0.815817i	−0.913023	+	0.407909i	$0.866258\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.00000	+	9.00000i	−0.365902	+	0.365902i
$606$		0			0
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.00000	−	8.00000i	0.323645	−	0.323645i
$612$		0			0
$613$	25.0000	+	25.0000i	1.00974	+	1.00974i	0.999952	+	0.00978840i	$0.00311579\pi$
$613$	25.0000	+	25.0000i	1.00974	+	1.00974i	0.00978840	+	0.999952i	$0.496884\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			12.0000i			0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$			12.0000i			0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$		0			0
$619$	17.0000	+	17.0000i	0.683288	+	0.683288i	0.960740	−	0.277452i	$-0.0894899\pi$
$619$	17.0000	+	17.0000i	0.683288	+	0.683288i	−0.277452	+	0.960740i	$0.589490\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.00000			0.320513
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.00000	−	6.00000i	−0.239236	−	0.239236i
$630$		0			0
$631$			10.0000i			0.398094i	0.979990	+	0.199047i	$0.0637846\pi$
$631$			10.0000i			0.398094i	−0.979990	+	0.199047i	$0.936215\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.00000	+	8.00000i	0.317470	+	0.317470i
$636$		0			0
$637$	3.00000	−	3.00000i	0.118864	−	0.118864i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$		0			0
$643$	−21.0000	+	21.0000i	−0.828159	+	0.828159i	−0.987262	−	0.159103i	$-0.949140\pi$
$643$	−21.0000	+	21.0000i	−0.828159	+	0.828159i	0.159103	+	0.987262i	$0.449140\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	42.0000i		−	1.65119i	−0.564263	−	0.825595i	$-0.690840\pi$
$647$		−	42.0000i		−	1.65119i	0.564263	−	0.825595i	$-0.309160\pi$
$648$		0			0
$649$		−	6.00000i		−	0.235521i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	19.0000	−	19.0000i	0.743527	−	0.743527i	−0.229728	−	0.973255i	$-0.573784\pi$
$653$	19.0000	−	19.0000i	0.743527	−	0.743527i	0.973255	+	0.229728i	$0.0737835\pi$
$654$		0			0
$655$	22.0000			0.859611
$656$		0			0
$657$		0			0
$658$		0			0
$659$	17.0000	−	17.0000i	0.662226	−	0.662226i	−0.293678	−	0.955904i	$-0.594879\pi$
$659$	17.0000	−	17.0000i	0.662226	−	0.662226i	0.955904	+	0.293678i	$0.0948794\pi$
$660$		0			0
$661$	9.00000	+	9.00000i	0.350059	+	0.350059i	0.860132	−	0.510072i	$-0.170381\pi$
$661$	9.00000	+	9.00000i	0.350059	+	0.350059i	−0.510072	+	0.860132i	$0.670381\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	12.0000i		−	0.465340i
$666$		0			0
$667$	18.0000	+	18.0000i	0.696963	+	0.696963i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−18.0000			−0.694882
$672$		0			0
$673$	14.0000			0.539660			0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000			0.539660			0.269830	+	0.962908i	$0.413032\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.00000	+	3.00000i	0.115299	+	0.115299i	0.762402	−	0.647103i	$-0.224020\pi$
$677$	3.00000	+	3.00000i	0.115299	+	0.115299i	−0.647103	+	0.762402i	$0.724020\pi$
$678$		0			0
$679$		−	4.00000i		−	0.153506i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−5.00000	−	5.00000i	−0.191320	−	0.191320i	0.604946	−	0.796266i	$-0.293195\pi$
$683$	−5.00000	−	5.00000i	−0.191320	−	0.191320i	−0.796266	+	0.604946i	$0.793195\pi$
$684$		0			0
$685$	−8.00000	+	8.00000i	−0.305664	+	0.305664i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−10.0000			−0.380970
$690$		0			0
$691$	−9.00000	+	9.00000i	−0.342376	+	0.342376i	−0.857260	−	0.514884i	$-0.827835\pi$
$691$	−9.00000	+	9.00000i	−0.342376	+	0.342376i	0.514884	+	0.857260i	$0.327835\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			6.00000i			0.227593i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	31.0000	−	31.0000i	1.17085	−	1.17085i	0.188847	−	0.982006i	$-0.439525\pi$
$701$	31.0000	−	31.0000i	1.17085	−	1.17085i	0.982006	−	0.188847i	$-0.0604752\pi$
$702$		0			0
$703$	−18.0000			−0.678883
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.0000	+	22.0000i	−0.827395	+	0.827395i
$708$		0			0
$709$	−27.0000	−	27.0000i	−1.01401	−	1.01401i	−0.999901	−	0.0141058i	$-0.995510\pi$
$709$	−27.0000	−	27.0000i	−1.01401	−	1.01401i	−0.0141058	−	0.999901i	$-0.504490\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			48.0000i			1.79761i
$714$		0			0
$715$	2.00000	+	2.00000i	0.0747958	+	0.0747958i
$716$		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
$721$	12.0000			0.446903
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.00000	−	9.00000i	−0.334252	−	0.334252i
$726$		0			0
$727$			2.00000i			0.0741759i	0.999312	+	0.0370879i	$0.0118082\pi$
$727$			2.00000i			0.0741759i	−0.999312	+	0.0370879i	$0.988192\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.0000	+	10.0000i	0.369863	+	0.369863i
$732$		0			0
$733$	21.0000	−	21.0000i	0.775653	−	0.775653i	−0.203436	−	0.979088i	$-0.565211\pi$
$733$	21.0000	−	21.0000i	0.775653	−	0.775653i	0.979088	+	0.203436i	$0.0652108\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.0000			0.368355
$738$		0			0
$739$	23.0000	−	23.0000i	0.846069	−	0.846069i	−0.143571	−	0.989640i	$-0.545859\pi$
$739$	23.0000	−	23.0000i	0.846069	−	0.846069i	0.989640	+	0.143571i	$0.0458586\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			46.0000i			1.68758i	0.536676	+	0.843788i	$0.319680\pi$
$743$			46.0000i			1.68758i	−0.536676	+	0.843788i	$0.680320\pi$
$744$		0			0
$745$		−	14.0000i		−	0.512920i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−14.0000	+	14.0000i	−0.511549	+	0.511549i
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10.0000	−	10.0000i	0.363937	−	0.363937i
$756$		0			0
$757$	−23.0000	−	23.0000i	−0.835949	−	0.835949i	0.152374	−	0.988323i	$-0.451308\pi$
$757$	−23.0000	−	23.0000i	−0.835949	−	0.835949i	−0.988323	+	0.152374i	$0.951308\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−6.00000	−	6.00000i	−0.217215	−	0.217215i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.00000			0.216647
$768$		0			0
$769$	−50.0000			−1.80305			−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000			−1.80305			−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.00000	−	5.00000i	−0.179838	−	0.179838i	0.611448	−	0.791285i	$-0.290588\pi$
$773$	−5.00000	−	5.00000i	−0.179838	−	0.179838i	−0.791285	+	0.611448i	$0.790588\pi$
$774$		0			0
$775$		−	24.0000i		−	0.862105i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−10.0000	+	10.0000i	−0.357828	+	0.357828i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	30.0000			1.07075
$786$		0			0
$787$	15.0000	−	15.0000i	0.534692	−	0.534692i	−0.387273	−	0.921965i	$-0.626583\pi$
$787$	15.0000	−	15.0000i	0.534692	−	0.534692i	0.921965	+	0.387273i	$0.126583\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			12.0000i			0.426671i
$792$		0			0
$793$		−	18.0000i		−	0.639199i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−25.0000	+	25.0000i	−0.885545	+	0.885545i	−0.994091	−	0.108546i	$-0.965381\pi$
$797$	−25.0000	+	25.0000i	−0.885545	+	0.885545i	0.108546	+	0.994091i	$0.465381\pi$
$798$		0

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{5} +2.00000i q^{7} +O(q^{10})\)	\(q+(-1.00000 - 1.00000i) q^{5} +2.00000i q^{7} +(-1.00000 - 1.00000i) q^{11} +(1.00000 - 1.00000i) q^{13} +2.00000 q^{17} +(3.00000 - 3.00000i) q^{19} +6.00000i q^{23} -3.00000i q^{25} +(3.00000 - 3.00000i) q^{29} +8.00000 q^{31} +(2.00000 - 2.00000i) q^{35} +(-3.00000 - 3.00000i) q^{37} +(5.00000 + 5.00000i) q^{43} +8.00000 q^{47} +3.00000 q^{49} +(-5.00000 - 5.00000i) q^{53} +2.00000i q^{55} +(3.00000 + 3.00000i) q^{59} +(9.00000 - 9.00000i) q^{61} -2.00000 q^{65} +(-5.00000 + 5.00000i) q^{67} -10.0000i q^{71} -4.00000i q^{73} +(2.00000 - 2.00000i) q^{77} +(1.00000 - 1.00000i) q^{83} +(-2.00000 - 2.00000i) q^{85} -4.00000i q^{89} +(2.00000 + 2.00000i) q^{91} -6.00000 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} - 2 q^{11} + 2 q^{13} + 4 q^{17} + 6 q^{19} + 6 q^{29} + 16 q^{31} + 4 q^{35} - 6 q^{37} + 10 q^{43} + 16 q^{47} + 6 q^{49} - 10 q^{53} + 6 q^{59} + 18 q^{61} - 4 q^{65} - 10 q^{67} + 4 q^{77} + 2 q^{83} - 4 q^{85} + 4 q^{91} - 12 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^11 + 2 * q^13 + 4 * q^17 + 6 * q^19 + 6 * q^29 + 16 * q^31 + 4 * q^35 - 6 * q^37 + 10 * q^43 + 16 * q^47 + 6 * q^49 - 10 * q^53 + 6 * q^59 + 18 * q^61 - 4 * q^65 - 10 * q^67 + 4 * q^77 + 2 * q^83 - 4 * q^85 + 4 * q^91 - 12 * q^95 - 4 * q^97

Introduction

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