LMFDB - Embedded newform 272.2.o.c.225.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [272,2,Mod(81,272)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("272.81"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(272, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	272.o (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$2.17193093498$
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 34)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			225.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	272.225
Dual form			272.2.o.c.81.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{5} +3.00000i q^{9} +(4.00000 + 4.00000i) q^{11} +4.00000 q^{13} +(-1.00000 - 4.00000i) q^{17} +4.00000i q^{19} +(-4.00000 - 4.00000i) q^{23} +3.00000i q^{25} +(-3.00000 + 3.00000i) q^{29} +(4.00000 - 4.00000i) q^{31} +(3.00000 - 3.00000i) q^{37} +(1.00000 + 1.00000i) q^{41} -4.00000i q^{43} +(-3.00000 - 3.00000i) q^{45} -8.00000 q^{47} -7.00000i q^{49} +4.00000i q^{53} -8.00000 q^{55} +4.00000i q^{59} +(-9.00000 - 9.00000i) q^{61} +(-4.00000 + 4.00000i) q^{65} +12.0000 q^{67} +(4.00000 - 4.00000i) q^{71} +(5.00000 - 5.00000i) q^{73} +(-8.00000 - 8.00000i) q^{79} -9.00000 q^{81} -4.00000i q^{83} +(5.00000 + 3.00000i) q^{85} +(-4.00000 - 4.00000i) q^{95} +(3.00000 - 3.00000i) q^{97} +(-12.0000 + 12.0000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 8 q^{11} + 8 q^{13} - 2 q^{17} - 8 q^{23} - 6 q^{29} + 8 q^{31} + 6 q^{37} + 2 q^{41} - 6 q^{45} - 16 q^{47} - 16 q^{55} - 18 q^{61} - 8 q^{65} + 24 q^{67} + 8 q^{71} + 10 q^{73} - 16 q^{79}+ \cdots - 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 + 8 * q^11 + 8 * q^13 - 2 * q^17 - 8 * q^23 - 6 * q^29 + 8 * q^31 + 6 * q^37 + 2 * q^41 - 6 * q^45 - 16 * q^47 - 16 * q^55 - 18 * q^61 - 8 * q^65 + 24 * q^67 + 8 * q^71 + 10 * q^73 - 16 * q^79 - 18 * q^81 + 10 * q^85 - 8 * q^95 + 6 * q^97 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/272\mathbb{Z}\right)^\times$.

$n$	$69$	$239$	$241$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		0			0
$5$	−1.00000	+	1.00000i	−0.447214	+	0.447214i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−1.00000	+	1.00000i	−0.447214	+	0.447214i	0.447214	+	0.894427i	$0.352416\pi$
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$	4.00000	+	4.00000i	1.20605	+	1.20605i	0.972297	+	0.233748i	$0.0750991\pi$
$11$	4.00000	+	4.00000i	1.20605	+	1.20605i	0.233748	+	0.972297i	$0.424901\pi$
$12$		0			0
$13$	4.00000			1.10940			0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000			1.10940			0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	−	4.00000i	−0.242536	−	0.970143i
$17$	−1.00000	−	4.00000i	−0.242536	−	0.970143i
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000	−	4.00000i	−0.834058	−	0.834058i	0.154011	−	0.988069i	$-0.450781\pi$
$23$	−4.00000	−	4.00000i	−0.834058	−	0.834058i	−0.988069	+	0.154011i	$0.950781\pi$
$24$		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	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−3.00000	+	3.00000i	−0.557086	+	0.557086i	0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$	4.00000	−	4.00000i	0.718421	−	0.718421i	−0.249861	−	0.968282i	$-0.580385\pi$
$31$	4.00000	−	4.00000i	0.718421	−	0.718421i	0.968282	+	0.249861i	$0.0803848\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	−0.416115	−	0.909312i	$-0.636609\pi$
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	0.909312	+	0.416115i	$0.136609\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.00000	+	1.00000i	0.156174	+	0.156174i	0.780869	−	0.624695i	$-0.214777\pi$
$41$	1.00000	+	1.00000i	0.156174	+	0.156174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$	−3.00000	−	3.00000i	−0.447214	−	0.447214i
$46$		0			0
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4.00000i			0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$			4.00000i			0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$		0			0
$58$		0			0
$59$			4.00000i			0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$			4.00000i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$		0			0
$61$	−9.00000	−	9.00000i	−1.15233	−	1.15233i	−0.986084	−	0.166248i	$-0.946835\pi$
$61$	−9.00000	−	9.00000i	−1.15233	−	1.15233i	−0.166248	−	0.986084i	$-0.553165\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.00000	+	4.00000i	−0.496139	+	0.496139i
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.00000	−	4.00000i	0.474713	−	0.474713i	−0.428723	−	0.903436i	$-0.641036\pi$
$71$	4.00000	−	4.00000i	0.474713	−	0.474713i	0.903436	+	0.428723i	$0.141036\pi$
$72$		0			0
$73$	5.00000	−	5.00000i	0.585206	−	0.585206i	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	5.00000	−	5.00000i	0.585206	−	0.585206i	0.936329	+	0.351123i	$0.114200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.00000	−	8.00000i	−0.900070	−	0.900070i	0.0953714	−	0.995442i	$-0.469596\pi$
$79$	−8.00000	−	8.00000i	−0.900070	−	0.900070i	−0.995442	+	0.0953714i	$0.969596\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		0			0
$85$	5.00000	+	3.00000i	0.542326	+	0.325396i
$86$		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
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.00000	−	4.00000i	−0.410391	−	0.410391i
$96$		0			0
$97$	3.00000	−	3.00000i	0.304604	−	0.304604i	−0.538208	−	0.842812i	$-0.680899\pi$
$97$	3.00000	−	3.00000i	0.304604	−	0.304604i	0.842812	+	0.538208i	$0.180899\pi$
$98$		0			0
$99$	−12.0000	+	12.0000i	−1.20605	+	1.20605i
$100$		0			0
$101$	12.0000			1.19404			0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000			1.19404			0.597022	+	0.802225i	$0.296350\pi$
$102$		0			0
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.00000	+	8.00000i	−0.773389	+	0.773389i	−0.978697	−	0.205308i	$-0.934180\pi$
$107$	−8.00000	+	8.00000i	−0.773389	+	0.773389i	0.205308	+	0.978697i	$0.434180\pi$
$108$		0			0
$109$	3.00000	+	3.00000i	0.287348	+	0.287348i	0.836031	−	0.548683i	$-0.184871\pi$
$109$	3.00000	+	3.00000i	0.287348	+	0.287348i	−0.548683	+	0.836031i	$0.684871\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.00000	−	1.00000i	−0.0940721	−	0.0940721i	0.658505	−	0.752577i	$-0.271189\pi$
$113$	−1.00000	−	1.00000i	−0.0940721	−	0.0940721i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$	8.00000			0.746004
$116$		0			0
$117$			12.0000i			1.10940i
$118$		0			0
$119$		0			0
$120$		0			0
$121$			21.0000i			1.90909i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.00000	−	8.00000i	−0.715542	−	0.715542i
$126$		0			0
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.00000	−	4.00000i	0.349482	−	0.349482i	−0.510435	−	0.859916i	$-0.670516\pi$
$131$	4.00000	−	4.00000i	0.349482	−	0.349482i	0.859916	+	0.510435i	$0.170516\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.00000			0.683486			0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000			0.683486			0.341743	+	0.939793i	$0.388983\pi$
$138$		0			0
$139$	8.00000	−	8.00000i	0.678551	−	0.678551i	−0.281121	−	0.959672i	$-0.590706\pi$
$139$	8.00000	−	8.00000i	0.678551	−	0.678551i	0.959672	+	0.281121i	$0.0907063\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	16.0000	+	16.0000i	1.33799	+	1.33799i
$144$		0			0
$145$		−	6.00000i		−	0.498273i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	12.0000	−	3.00000i	0.970143	−	0.242536i
$154$		0			0
$155$			8.00000i			0.642575i
$156$		0			0
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.00000	−	4.00000i	−0.313304	−	0.313304i	0.532884	−	0.846188i	$-0.321108\pi$
$163$	−4.00000	−	4.00000i	−0.313304	−	0.313304i	−0.846188	+	0.532884i	$0.821108\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.00000	+	8.00000i	−0.619059	+	0.619059i	−0.945290	−	0.326231i	$-0.894221\pi$
$167$	−8.00000	+	8.00000i	−0.619059	+	0.619059i	0.326231	+	0.945290i	$0.394221\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	−12.0000			−0.917663
$172$		0			0
$173$	5.00000	−	5.00000i	0.380143	−	0.380143i	−0.491011	−	0.871154i	$-0.663372\pi$
$173$	5.00000	−	5.00000i	0.380143	−	0.380143i	0.871154	+	0.491011i	$0.163372\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			4.00000i			0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$			4.00000i			0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$		0			0
$181$	11.0000	+	11.0000i	0.817624	+	0.817624i	0.985763	−	0.168140i	$-0.0537759\pi$
$181$	11.0000	+	11.0000i	0.817624	+	0.817624i	−0.168140	+	0.985763i	$0.553776\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			6.00000i			0.441129i
$186$		0			0
$187$	12.0000	−	20.0000i	0.877527	−	1.46254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$	−11.0000	−	11.0000i	−0.791797	−	0.791797i	0.189989	−	0.981786i	$-0.439155\pi$
$193$	−11.0000	−	11.0000i	−0.791797	−	0.791797i	−0.981786	+	0.189989i	$0.939155\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.00000	+	5.00000i	0.356235	+	0.356235i	0.862423	−	0.506188i	$-0.168946\pi$
$197$	5.00000	+	5.00000i	0.356235	+	0.356235i	−0.506188	+	0.862423i	$0.668946\pi$
$198$		0			0
$199$	−12.0000	+	12.0000i	−0.850657	+	0.850657i	−0.990214	−	0.139557i	$-0.955432\pi$
$199$	−12.0000	+	12.0000i	−0.850657	+	0.850657i	0.139557	+	0.990214i	$0.455432\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2.00000			−0.139686
$206$		0			0
$207$	12.0000	−	12.0000i	0.834058	−	0.834058i
$208$		0			0
$209$	−16.0000	+	16.0000i	−1.10674	+	1.10674i
$210$		0			0
$211$	−16.0000	−	16.0000i	−1.10149	−	1.10149i	−0.994232	−	0.107254i	$-0.965794\pi$
$211$	−16.0000	−	16.0000i	−1.10149	−	1.10149i	−0.107254	−	0.994232i	$-0.534206\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000	+	4.00000i	0.272798	+	0.272798i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.00000	−	16.0000i	−0.269069	−	1.07628i
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$		0			0
$225$	−9.00000			−0.600000
$226$		0			0
$227$	−20.0000	−	20.0000i	−1.32745	−	1.32745i	−0.907591	−	0.419856i	$-0.862081\pi$
$227$	−20.0000	−	20.0000i	−1.32745	−	1.32745i	−0.419856	−	0.907591i	$-0.637919\pi$
$228$		0			0
$229$			6.00000i			0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$			6.00000i			0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.00000	+	5.00000i	−0.327561	+	0.327561i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−5.00000	+	5.00000i	−0.327561	+	0.327561i	0.524097	+	0.851658i	$0.324403\pi$
$234$		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
$241$	21.0000	−	21.0000i	1.35273	−	1.35273i	0.470134	−	0.882595i	$-0.344206\pi$
$241$	21.0000	−	21.0000i	1.35273	−	1.35273i	0.882595	−	0.470134i	$-0.155794\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.00000	+	7.00000i	0.447214	+	0.447214i
$246$		0			0
$247$			16.0000i			1.01806i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$		−	32.0000i		−	2.01182i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−9.00000	−	9.00000i	−0.557086	−	0.557086i
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	−4.00000	−	4.00000i	−0.245718	−	0.245718i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.0000	+	13.0000i	−0.792624	+	0.792624i	−0.981920	−	0.189296i	$-0.939379\pi$
$269$	−13.0000	+	13.0000i	−0.792624	+	0.792624i	0.189296	+	0.981920i	$0.439379\pi$
$270$		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
$275$	−12.0000	+	12.0000i	−0.723627	+	0.723627i
$276$		0			0
$277$	−7.00000	+	7.00000i	−0.420589	+	0.420589i	−0.885407	−	0.464817i	$-0.846120\pi$
$277$	−7.00000	+	7.00000i	−0.420589	+	0.420589i	0.464817	+	0.885407i	$0.346120\pi$
$278$		0			0
$279$	12.0000	+	12.0000i	0.718421	+	0.718421i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−4.00000	−	4.00000i	−0.237775	−	0.237775i	0.578153	−	0.815928i	$-0.303774\pi$
$283$	−4.00000	−	4.00000i	−0.237775	−	0.237775i	−0.815928	+	0.578153i	$0.803774\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−15.0000	+	8.00000i	−0.882353	+	0.470588i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$	−4.00000	−	4.00000i	−0.232889	−	0.232889i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.0000	−	16.0000i	−0.925304	−	0.925304i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	18.0000			1.03068
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.0000	−	24.0000i	1.36092	−	1.36092i	0.488162	−	0.872753i	$-0.337667\pi$
$311$	24.0000	−	24.0000i	1.36092	−	1.36092i	0.872753	−	0.488162i	$-0.162333\pi$
$312$		0			0
$313$	9.00000	+	9.00000i	0.508710	+	0.508710i	0.914130	−	0.405420i	$-0.132875\pi$
$313$	9.00000	+	9.00000i	0.508710	+	0.508710i	−0.405420	+	0.914130i	$0.632875\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.0000	−	15.0000i	−0.842484	−	0.842484i	0.146697	−	0.989181i	$-0.453136\pi$
$317$	−15.0000	−	15.0000i	−0.842484	−	0.842484i	−0.989181	+	0.146697i	$0.953136\pi$
$318$		0			0
$319$	−24.0000			−1.34374
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.0000	−	4.00000i	0.890264	−	0.222566i
$324$		0			0
$325$			12.0000i			0.665640i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			20.0000i			1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$			20.0000i			1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$		0			0
$333$	9.00000	+	9.00000i	0.493197	+	0.493197i
$334$		0			0
$335$	−12.0000	+	12.0000i	−0.655630	+	0.655630i
$336$		0			0
$337$	−7.00000	+	7.00000i	−0.381314	+	0.381314i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−7.00000	+	7.00000i	−0.381314	+	0.381314i	0.490261	+	0.871576i	$0.336901\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	32.0000			1.73290
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	4.00000i		−	0.214115i	−0.994253	−	0.107058i	$-0.965857\pi$
$349$		−	4.00000i		−	0.214115i	0.994253	−	0.107058i	$-0.0341429\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.0000			0.745145			0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000			0.745145			0.372572	+	0.928003i	$0.378476\pi$
$354$		0			0
$355$			8.00000i			0.424596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			24.0000i			1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$			24.0000i			1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$			10.0000i			0.523424i
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$	−3.00000	+	3.00000i	−0.156174	+	0.156174i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	4.00000			0.207112			0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000			0.207112			0.103556	+	0.994624i	$0.466978\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.0000	+	12.0000i	−0.618031	+	0.618031i
$378$		0			0
$379$	−12.0000	+	12.0000i	−0.616399	+	0.616399i	−0.944606	−	0.328207i	$-0.893556\pi$
$379$	−12.0000	+	12.0000i	−0.616399	+	0.616399i	0.328207	+	0.944606i	$0.393556\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	24.0000i		−	1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$		−	24.0000i		−	1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	12.0000			0.609994
$388$		0			0
$389$			6.00000i			0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$			6.00000i			0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$		0			0
$391$	−12.0000	+	20.0000i	−0.606866	+	1.01144i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.0000			0.805047
$396$		0			0
$397$	15.0000	+	15.0000i	0.752828	+	0.752828i	0.975006	−	0.222178i	$-0.0713165\pi$
$397$	15.0000	+	15.0000i	0.752828	+	0.752828i	−0.222178	+	0.975006i	$0.571317\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−19.0000	−	19.0000i	−0.948815	−	0.948815i	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	−19.0000	−	19.0000i	−0.948815	−	0.948815i	−0.998752	+	0.0499376i	$0.984098\pi$
$402$		0			0
$403$	16.0000	−	16.0000i	0.797017	−	0.797017i
$404$		0			0
$405$	9.00000	−	9.00000i	0.447214	−	0.447214i
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$	−10.0000			−0.494468			−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	4.00000	+	4.00000i	0.196352	+	0.196352i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.0000	+	12.0000i	0.586238	+	0.586238i	0.936611	−	0.350372i	$-0.113945\pi$
$419$	12.0000	+	12.0000i	0.586238	+	0.586238i	−0.350372	+	0.936611i	$0.613945\pi$
$420$		0			0
$421$	−28.0000			−1.36464			−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000			−1.36464			−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$		−	24.0000i		−	1.16692i
$424$		0			0
$425$	12.0000	−	3.00000i	0.582086	−	0.145521i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	24.0000	+	24.0000i	1.15604	+	1.15604i	0.985320	+	0.170720i	$0.0546093\pi$
$431$	24.0000	+	24.0000i	1.15604	+	1.15604i	0.170720	+	0.985320i	$0.445391\pi$
$432$		0			0
$433$			14.0000i			0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$			14.0000i			0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.0000	−	16.0000i	0.765384	−	0.765384i
$438$		0			0
$439$	−12.0000	+	12.0000i	−0.572729	+	0.572729i	−0.932890	−	0.360161i	$-0.882722\pi$
$439$	−12.0000	+	12.0000i	−0.572729	+	0.572729i	0.360161	+	0.932890i	$0.382722\pi$
$440$		0			0
$441$	21.0000			1.00000
$442$		0			0
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.00000	+	3.00000i	0.141579	+	0.141579i	0.774344	−	0.632765i	$-0.218080\pi$
$449$	3.00000	+	3.00000i	0.141579	+	0.141579i	−0.632765	+	0.774344i	$0.718080\pi$
$450$		0			0
$451$			8.00000i			0.376705i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			22.0000i			1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$			22.0000i			1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			20.0000i			0.931493i	0.884918	+	0.465746i	$0.154214\pi$
$461$			20.0000i			0.931493i	−0.884918	+	0.465746i	$0.845786\pi$
$462$		0			0
$463$	−24.0000			−1.11537			−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000			−1.11537			−0.557687	+	0.830051i	$0.688311\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			28.0000i			1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$			28.0000i			1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	16.0000	−	16.0000i	0.735681	−	0.735681i
$474$		0			0
$475$	−12.0000			−0.550598
$476$		0			0
$477$	−12.0000			−0.549442
$478$		0			0
$479$	8.00000	−	8.00000i	0.365529	−	0.365529i	−0.500314	−	0.865844i	$-0.666782\pi$
$479$	8.00000	−	8.00000i	0.365529	−	0.365529i	0.865844	+	0.500314i	$0.166782\pi$
$480$		0			0
$481$	12.0000	−	12.0000i	0.547153	−	0.547153i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			6.00000i			0.272446i
$486$		0			0
$487$	−20.0000	−	20.0000i	−0.906287	−	0.906287i	0.0896838	−	0.995970i	$-0.471414\pi$
$487$	−20.0000	−	20.0000i	−0.906287	−	0.906287i	−0.995970	+	0.0896838i	$0.971414\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			20.0000i			0.902587i	0.892375	+	0.451294i	$0.149037\pi$
$491$			20.0000i			0.902587i	−0.892375	+	0.451294i	$0.850963\pi$
$492$		0			0
$493$	15.0000	+	9.00000i	0.675566	+	0.405340i
$494$		0			0
$495$		−	24.0000i		−	1.07872i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−8.00000	−	8.00000i	−0.358129	−	0.358129i	0.504994	−	0.863123i	$-0.331495\pi$
$499$	−8.00000	−	8.00000i	−0.358129	−	0.358129i	−0.863123	+	0.504994i	$0.831495\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.00000	−	4.00000i	−0.178351	−	0.178351i	0.612286	−	0.790637i	$-0.290250\pi$
$503$	−4.00000	−	4.00000i	−0.178351	−	0.178351i	−0.790637	+	0.612286i	$0.790250\pi$
$504$		0			0
$505$	−12.0000	+	12.0000i	−0.533993	+	0.533993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.0000	+	16.0000i	−0.705044	+	0.705044i
$516$		0			0
$517$	−32.0000	−	32.0000i	−1.40736	−	1.40736i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.00000	+	1.00000i	0.0438108	+	0.0438108i	0.728673	−	0.684862i	$-0.240138\pi$
$521$	1.00000	+	1.00000i	0.0438108	+	0.0438108i	−0.684862	+	0.728673i	$0.740138\pi$
$522$		0			0
$523$	36.0000			1.57417			0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000			1.57417			0.787085	+	0.616844i	$0.211589\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−20.0000	−	12.0000i	−0.871214	−	0.522728i
$528$		0			0
$529$			9.00000i			0.391304i
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$	4.00000	+	4.00000i	0.173259	+	0.173259i
$534$		0			0
$535$		−	16.0000i		−	0.691740i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	28.0000	−	28.0000i	1.20605	−	1.20605i
$540$		0			0
$541$	−9.00000	+	9.00000i	−0.386940	+	0.386940i	−0.873595	−	0.486654i	$-0.838217\pi$
$541$	−9.00000	+	9.00000i	−0.386940	+	0.386940i	0.486654	+	0.873595i	$0.338217\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.00000			−0.257012
$546$		0			0
$547$	12.0000	−	12.0000i	0.513083	−	0.513083i	−0.402387	−	0.915470i	$-0.631819\pi$
$547$	12.0000	−	12.0000i	0.513083	−	0.513083i	0.915470	+	0.402387i	$0.131819\pi$
$548$		0			0
$549$	27.0000	−	27.0000i	1.15233	−	1.15233i
$550$		0			0
$551$	−12.0000	−	12.0000i	−0.511217	−	0.511217i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	28.0000			1.18640			0.593199	−	0.805056i	$-0.297865\pi$
$557$	28.0000			1.18640			0.593199	+	0.805056i	$0.297865\pi$
$558$		0			0
$559$		−	16.0000i		−	0.676728i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	4.00000i		−	0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$		−	4.00000i		−	0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$		0			0
$565$	2.00000			0.0841406
$566$		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
$571$	24.0000	+	24.0000i	1.00437	+	1.00437i	0.999990	+	0.00437833i	$0.00139367\pi$
$571$	24.0000	+	24.0000i	1.00437	+	1.00437i	0.00437833	+	0.999990i	$0.498606\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	12.0000	−	12.0000i	0.500435	−	0.500435i
$576$		0			0
$577$	−32.0000			−1.33218			−0.666089	−	0.745873i	$-0.732033\pi$
$577$	−32.0000			−1.33218			−0.666089	+	0.745873i	$0.732033\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−16.0000	+	16.0000i	−0.662652	+	0.662652i
$584$		0			0
$585$	−12.0000	−	12.0000i	−0.496139	−	0.496139i
$586$		0			0
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$	16.0000	+	16.0000i	0.659269	+	0.659269i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	16.0000i		−	0.657041i	−0.944497	−	0.328521i	$-0.893450\pi$
$593$		−	16.0000i		−	0.657041i	0.944497	−	0.328521i	$-0.106550\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−40.0000			−1.63436			−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000			−1.63436			−0.817178	+	0.576386i	$0.804463\pi$
$600$		0			0
$601$	−19.0000	−	19.0000i	−0.775026	−	0.775026i	0.203954	−	0.978980i	$-0.434621\pi$
$601$	−19.0000	−	19.0000i	−0.775026	−	0.775026i	−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$			36.0000i			1.46603i
$604$		0			0
$605$	−21.0000	−	21.0000i	−0.853771	−	0.853771i
$606$		0			0
$607$	−28.0000	+	28.0000i	−1.13648	+	1.13648i	−0.147409	+	0.989076i	$0.547093\pi$
$607$	−28.0000	+	28.0000i	−1.13648	+	1.13648i	−0.989076	+	0.147409i	$0.952907\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−32.0000			−1.29458
$612$		0			0
$613$	−26.0000			−1.05013			−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000			−1.05013			−0.525065	+	0.851062i	$0.675959\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.00000	−	3.00000i	0.120775	−	0.120775i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	3.00000	−	3.00000i	0.120775	−	0.120775i	0.764911	+	0.644136i	$0.222783\pi$
$618$		0			0
$619$	−8.00000	−	8.00000i	−0.321547	−	0.321547i	0.527813	−	0.849360i	$-0.323012\pi$
$619$	−8.00000	−	8.00000i	−0.321547	−	0.321547i	−0.849360	+	0.527813i	$0.823012\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.0000	−	9.00000i	−0.598089	−	0.358854i
$630$		0			0
$631$		−	40.0000i		−	1.59237i	−0.605050	−	0.796187i	$-0.706847\pi$
$631$		−	40.0000i		−	1.59237i	0.605050	−	0.796187i	$-0.293153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.00000	−	8.00000i	−0.317470	−	0.317470i
$636$		0			0
$637$		−	28.0000i		−	1.10940i
$638$		0			0
$639$	12.0000	+	12.0000i	0.474713	+	0.474713i
$640$		0			0
$641$	11.0000	−	11.0000i	0.434474	−	0.434474i	−0.455673	−	0.890147i	$-0.650601\pi$
$641$	11.0000	−	11.0000i	0.434474	−	0.434474i	0.890147	+	0.455673i	$0.150601\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		0			0
$649$	−16.0000	+	16.0000i	−0.628055	+	0.628055i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	29.0000	+	29.0000i	1.13486	+	1.13486i	0.989358	+	0.145499i	$0.0464789\pi$
$653$	29.0000	+	29.0000i	1.13486	+	1.13486i	0.145499	+	0.989358i	$0.453521\pi$
$654$		0			0
$655$			8.00000i			0.312586i
$656$		0			0
$657$	15.0000	+	15.0000i	0.585206	+	0.585206i
$658$		0			0
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$			20.0000i			0.777910i	0.921257	+	0.388955i	$0.127164\pi$
$661$			20.0000i			0.777910i	−0.921257	+	0.388955i	$0.872836\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	24.0000			0.929284
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	72.0000i		−	2.77953i
$672$		0			0
$673$	19.0000	+	19.0000i	0.732396	+	0.732396i	0.971094	−	0.238698i	$-0.0767205\pi$
$673$	19.0000	+	19.0000i	0.732396	+	0.732396i	−0.238698	+	0.971094i	$0.576721\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13.0000	−	13.0000i	0.499631	−	0.499631i	−0.411692	−	0.911323i	$-0.635062\pi$
$677$	13.0000	−	13.0000i	0.499631	−	0.499631i	0.911323	+	0.411692i	$0.135062\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−20.0000	+	20.0000i	−0.765279	+	0.765279i	−0.977271	−	0.211993i	$-0.932005\pi$
$683$	−20.0000	+	20.0000i	−0.765279	+	0.765279i	0.211993	+	0.977271i	$0.432005\pi$
$684$		0			0
$685$	−8.00000	+	8.00000i	−0.305664	+	0.305664i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			16.0000i			0.609551i
$690$		0			0
$691$	24.0000	+	24.0000i	0.913003	+	0.913003i	0.996507	−	0.0835044i	$-0.0266113\pi$
$691$	24.0000	+	24.0000i	0.913003	+	0.913003i	−0.0835044	+	0.996507i	$0.526611\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			16.0000i			0.606915i
$696$		0			0
$697$	3.00000	−	5.00000i	0.113633	−	0.189389i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.0000			0.453234			0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000			0.453234			0.226617	+	0.973984i	$0.427233\pi$
$702$		0			0
$703$	12.0000	+	12.0000i	0.452589	+	0.452589i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−23.0000	+	23.0000i	−0.863783	+	0.863783i	−0.991775	−	0.127992i	$-0.959147\pi$
$709$	−23.0000	+	23.0000i	−0.863783	+	0.863783i	0.127992	+	0.991775i	$0.459147\pi$
$710$		0			0
$711$	24.0000	−	24.0000i	0.900070	−	0.900070i
$712$		0			0
$713$	−32.0000			−1.19841
$714$		0			0
$715$	−32.0000			−1.19673
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.00000	−	8.00000i	0.298350	−	0.298350i	−0.542018	−	0.840367i	$-0.682339\pi$
$719$	8.00000	−	8.00000i	0.298350	−	0.298350i	0.840367	+	0.542018i	$0.182339\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.00000	−	9.00000i	−0.334252	−	0.334252i
$726$		0			0
$727$	−48.0000			−1.78022			−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000			−1.78022			−0.890111	+	0.455744i	$0.849373\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	−16.0000	+	4.00000i	−0.591781	+	0.147945i
$732$		0			0
$733$		−	36.0000i		−	1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$		−	36.0000i		−	1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	48.0000	+	48.0000i	1.76810	+	1.76810i
$738$		0			0
$739$			4.00000i			0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$			4.00000i			0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$	10.0000	−	10.0000i	0.366372	−	0.366372i
$746$		0			0
$747$	12.0000			0.439057
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−16.0000	+	16.0000i	−0.583848	+	0.583848i	−0.935959	−	0.352110i	$-0.885464\pi$
$751$	−16.0000	+	16.0000i	−0.583848	+	0.583848i	0.352110	+	0.935959i	$0.385464\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			22.0000i			0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$			22.0000i			0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8.00000			−0.290000			−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000			−0.290000			−0.145000	+	0.989432i	$0.546318\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−9.00000	+	15.0000i	−0.325396	+	0.542326i
$766$		0			0
$767$			16.0000i			0.577727i
$768$		0			0
$769$	40.0000			1.44244			0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000			1.44244			0.721218	+	0.692708i	$0.243582\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$		0			0
$775$	12.0000	+	12.0000i	0.431053	+	0.431053i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.00000	+	4.00000i	−0.143315	+	0.143315i
$780$		0			0
$781$	32.0000			1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.00000	−	2.00000i	0.0713831	−	0.0713831i
$786$		0			0
$787$	12.0000	−	12.0000i	0.427754	−	0.427754i	−0.460109	−	0.887863i	$-0.652190\pi$
$787$	12.0000	−	12.0000i	0.427754	−	0.427754i	0.887863	+	0.460109i	$0.152190\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−36.0000	−	36.0000i	−1.27840	−	1.27840i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			12.0000i			0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$			12.0000i			0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$		0			0
$799$	8.00000	+	32.0000i	0.283020	+	1.13208i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	40.0000			1.41157
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	23.0000	+	23.0000i	0.808637	+	0.808637i	0.984428	−	0.175791i	$-0.0562482\pi$
$809$	23.0000	+	23.0000i	0.808637	+	0.808637i	−0.175791	+	0.984428i	$0.556248\pi$
$810$		0			0
$811$	4.00000	−	4.00000i	0.140459	−	0.140459i	−0.633381	−	0.773840i	$-0.718333\pi$
$811$	4.00000	−	4.00000i	0.140459	−	0.140459i	0.773840	+	0.633381i	$0.218333\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.00000			0.280228
$816$		0			0
$817$	16.0000			0.559769
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19.0000	+	19.0000i	−0.663105	+	0.663105i	−0.956111	−	0.293006i	$-0.905344\pi$
$821$	−19.0000	+	19.0000i	−0.663105	+	0.663105i	0.293006	+	0.956111i	$0.405344\pi$
$822$		0			0
$823$	36.0000	+	36.0000i	1.25488	+	1.25488i	0.953506	+	0.301376i	$0.0974458\pi$
$823$	36.0000	+	36.0000i	1.25488	+	1.25488i	0.301376	+	0.953506i	$0.402554\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.0000	+	20.0000i	0.695468	+	0.695468i	0.963430	−	0.267961i	$-0.0863500\pi$
$827$	20.0000	+	20.0000i	0.695468	+	0.695468i	−0.267961	+	0.963430i	$0.586350\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−28.0000	+	7.00000i	−0.970143	+	0.242536i
$834$		0			0
$835$		−	16.0000i		−	0.553703i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	32.0000	+	32.0000i	1.10476	+	1.10476i	0.993828	+	0.110935i	$0.0353845\pi$
$839$	32.0000	+	32.0000i	1.10476	+	1.10476i	0.110935	+	0.993828i	$0.464615\pi$
$840$		0			0
$841$			11.0000i			0.379310i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.00000	+	3.00000i	−0.103203	+	0.103203i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.0000			−0.822709
$852$		0			0
$853$	−25.0000	+	25.0000i	−0.855984	+	0.855984i	−0.990862	−	0.134878i	$-0.956936\pi$
$853$	−25.0000	+	25.0000i	−0.855984	+	0.855984i	0.134878	+	0.990862i	$0.456936\pi$
$854$		0			0
$855$	12.0000	−	12.0000i	0.410391	−	0.410391i
$856$		0			0
$857$	−15.0000	−	15.0000i	−0.512390	−	0.512390i	0.402868	−	0.915258i	$-0.368013\pi$
$857$	−15.0000	−	15.0000i	−0.512390	−	0.512390i	−0.915258	+	0.402868i	$0.868013\pi$
$858$		0			0
$859$		−	36.0000i		−	1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$		−	36.0000i		−	1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$			10.0000i			0.340010i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	64.0000i		−	2.17105i
$870$		0			0
$871$	48.0000			1.62642
$872$		0			0
$873$	9.00000	+	9.00000i	0.304604	+	0.304604i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	35.0000	+	35.0000i	1.18187	+	1.18187i	0.979260	+	0.202606i	$0.0649409\pi$
$877$	35.0000	+	35.0000i	1.18187	+	1.18187i	0.202606	+	0.979260i	$0.435059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−39.0000	+	39.0000i	−1.31394	+	1.31394i	−0.395460	+	0.918483i	$0.629415\pi$
$881$	−39.0000	+	39.0000i	−1.31394	+	1.31394i	−0.918483	+	0.395460i	$0.870585\pi$
$882$		0			0
$883$	−44.0000			−1.48072			−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000			−1.48072			−0.740359	+	0.672212i	$0.765344\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.0000	+	28.0000i	−0.940148	+	0.940148i	−0.998307	−	0.0581593i	$-0.981477\pi$
$887$	−28.0000	+	28.0000i	−0.940148	+	0.940148i	0.0581593	+	0.998307i	$0.481477\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−36.0000	−	36.0000i	−1.20605	−	1.20605i
$892$		0			0
$893$		−	32.0000i		−	1.07084i
$894$		0			0
$895$	−4.00000	−	4.00000i	−0.133705	−	0.133705i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			24.0000i			0.800445i
$900$		0			0
$901$	16.0000	−	4.00000i	0.533037	−	0.133259i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−22.0000			−0.731305
$906$		0			0
$907$	−20.0000	−	20.0000i	−0.664089	−	0.664089i	0.292252	−	0.956341i	$-0.405595\pi$
$907$	−20.0000	−	20.0000i	−0.664089	−	0.664089i	−0.956341	+	0.292252i	$0.905595\pi$
$908$		0			0
$909$			36.0000i			1.19404i
$910$		0			0
$911$	4.00000	+	4.00000i	0.132526	+	0.132526i	0.770258	−	0.637732i	$-0.220127\pi$
$911$	4.00000	+	4.00000i	0.132526	+	0.132526i	−0.637732	+	0.770258i	$0.720127\pi$
$912$		0			0
$913$	16.0000	−	16.0000i	0.529523	−	0.529523i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	16.0000	−	16.0000i	0.526646	−	0.526646i
$924$		0			0
$925$	9.00000	+	9.00000i	0.295918	+	0.295918i
$926$		0			0
$927$			48.0000i			1.57653i
$928$		0			0
$929$	13.0000	+	13.0000i	0.426516	+	0.426516i	0.887440	−	0.460924i	$-0.152482\pi$
$929$	13.0000	+	13.0000i	0.426516	+	0.426516i	−0.460924	+	0.887440i	$0.652482\pi$
$930$		0			0
$931$	28.0000			0.917663
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8.00000	+	32.0000i	0.261628	+	1.04651i
$936$		0			0
$937$		−	48.0000i		−	1.56809i	−0.620703	−	0.784046i	$-0.713153\pi$
$937$		−	48.0000i		−	1.56809i	0.620703	−	0.784046i	$-0.286847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.00000	+	1.00000i	0.0325991	+	0.0325991i	0.723218	−	0.690619i	$-0.242662\pi$
$941$	1.00000	+	1.00000i	0.0325991	+	0.0325991i	−0.690619	+	0.723218i	$0.742662\pi$
$942$		0			0
$943$		−	8.00000i		−	0.260516i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.0000	−	12.0000i	0.389948	−	0.389948i	−0.484721	−	0.874669i	$-0.661079\pi$
$947$	12.0000	−	12.0000i	0.389948	−	0.389948i	0.874669	+	0.484721i	$0.161079\pi$
$948$		0			0
$949$	20.0000	−	20.0000i	0.649227	−	0.649227i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	24.0000			0.777436			0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000			0.777436			0.388718	+	0.921357i	$0.372918\pi$
$954$		0			0
$955$	−8.00000	+	8.00000i	−0.258874	+	0.258874i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	1.00000i		−	0.0322581i
$962$		0			0
$963$	−24.0000	−	24.0000i	−0.773389	−	0.773389i
$964$		0			0
$965$	22.0000			0.708205
$966$		0			0
$967$		−	32.0000i		−	1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$		−	32.0000i		−	1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	20.0000i		−	0.641831i	−0.947108	−	0.320915i	$-0.896010\pi$
$971$		−	20.0000i		−	0.641831i	0.947108	−	0.320915i	$-0.103990\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	8.00000i		−	0.255943i	−0.991778	−	0.127971i	$-0.959153\pi$
$977$		−	8.00000i		−	0.255943i	0.991778	−	0.127971i	$-0.0408466\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−9.00000	+	9.00000i	−0.287348	+	0.287348i
$982$		0			0
$983$	20.0000	−	20.0000i	0.637901	−	0.637901i	−0.312136	−	0.950037i	$-0.601045\pi$
$983$	20.0000	−	20.0000i	0.637901	−	0.637901i	0.950037	+	0.312136i	$0.101045\pi$
$984$		0			0
$985$	−10.0000			−0.318626
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.0000	+	16.0000i	−0.508770	+	0.508770i
$990$		0			0
$991$	24.0000	−	24.0000i	0.762385	−	0.762385i	−0.214368	−	0.976753i	$-0.568769\pi$
$991$	24.0000	−	24.0000i	0.762385	−	0.762385i	0.976753	+	0.214368i	$0.0687691\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	24.0000i		−	0.760851i
$996$		0			0
$997$	−35.0000	−	35.0000i	−1.10846	−	1.10846i	−0.993353	−	0.115108i	$-0.963279\pi$
$997$	−35.0000	−	35.0000i	−1.10846	−	1.10846i	−0.115108	−	0.993353i	$-0.536721\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.2.o.c.225.1		2
3.2	odd	2		2448.2.be.j.1585.1		2
4.3	odd	2		34.2.c.b.21.1	yes	2
8.3	odd	2		1088.2.o.k.769.1		2
8.5	even	2		1088.2.o.i.769.1		2
12.11	even	2		306.2.g.d.55.1		2
17.8	even	8		4624.2.a.l.1.1		2
17.9	even	8		4624.2.a.l.1.2		2
17.13	even	4	inner	272.2.o.c.81.1		2
20.3	even	4		850.2.g.a.599.1		2
20.7	even	4		850.2.g.d.599.1		2
20.19	odd	2		850.2.h.c.701.1		2
51.47	odd	4		2448.2.be.j.1441.1		2
68.3	even	16		578.2.d.f.155.1		8
68.7	even	16		578.2.d.f.179.2		8
68.11	even	16		578.2.d.f.399.1		8
68.15	odd	8		578.2.b.c.577.2		2
68.19	odd	8		578.2.b.c.577.1		2
68.23	even	16		578.2.d.f.399.2		8
68.27	even	16		578.2.d.f.179.1		8
68.31	even	16		578.2.d.f.155.2		8
68.39	even	16		578.2.d.f.423.2		8
68.43	odd	8		578.2.a.b.1.2		2
68.47	odd	4		34.2.c.b.13.1	✓	2
68.55	odd	4		578.2.c.b.251.1		2
68.59	odd	8		578.2.a.b.1.1		2
68.63	even	16		578.2.d.f.423.1		8
68.67	odd	2		578.2.c.b.327.1		2
136.13	even	4		1088.2.o.i.897.1		2
136.115	odd	4		1088.2.o.k.897.1		2
204.47	even	4		306.2.g.d.217.1		2
204.59	even	8		5202.2.a.bb.1.2		2
204.179	even	8		5202.2.a.bb.1.1		2
340.47	even	4		850.2.g.a.149.1		2
340.183	even	4		850.2.g.d.149.1		2
340.319	odd	4		850.2.h.c.251.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
34.2.c.b.13.1	✓	2	68.47	odd	4
34.2.c.b.21.1	yes	2	4.3	odd	2
272.2.o.c.81.1		2	17.13	even	4	inner
272.2.o.c.225.1		2	1.1	even	1	trivial
306.2.g.d.55.1		2	12.11	even	2
306.2.g.d.217.1		2	204.47	even	4
578.2.a.b.1.1		2	68.59	odd	8
578.2.a.b.1.2		2	68.43	odd	8
578.2.b.c.577.1		2	68.19	odd	8
578.2.b.c.577.2		2	68.15	odd	8
578.2.c.b.251.1		2	68.55	odd	4
578.2.c.b.327.1		2	68.67	odd	2
578.2.d.f.155.1		8	68.3	even	16
578.2.d.f.155.2		8	68.31	even	16
578.2.d.f.179.1		8	68.27	even	16
578.2.d.f.179.2		8	68.7	even	16
578.2.d.f.399.1		8	68.11	even	16
578.2.d.f.399.2		8	68.23	even	16
578.2.d.f.423.1		8	68.63	even	16
578.2.d.f.423.2		8	68.39	even	16
850.2.g.a.149.1		2	340.47	even	4
850.2.g.a.599.1		2	20.3	even	4
850.2.g.d.149.1		2	340.183	even	4
850.2.g.d.599.1		2	20.7	even	4
850.2.h.c.251.1		2	340.319	odd	4
850.2.h.c.701.1		2	20.19	odd	2
1088.2.o.i.769.1		2	8.5	even	2
1088.2.o.i.897.1		2	136.13	even	4
1088.2.o.k.769.1		2	8.3	odd	2
1088.2.o.k.897.1		2	136.115	odd	4
2448.2.be.j.1441.1		2	51.47	odd	4
2448.2.be.j.1585.1		2	3.2	odd	2
4624.2.a.l.1.1		2	17.8	even	8
4624.2.a.l.1.2		2	17.9	even	8
5202.2.a.bb.1.1		2	204.179	even	8
5202.2.a.bb.1.2		2	204.59	even	8

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	272.o (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{5} +3.00000i q^{9} +(4.00000 + 4.00000i) q^{11} +4.00000 q^{13} +(-1.00000 - 4.00000i) q^{17} +4.00000i q^{19} +(-4.00000 - 4.00000i) q^{23} +3.00000i q^{25} +(-3.00000 + 3.00000i) q^{29} +(4.00000 - 4.00000i) q^{31} +(3.00000 - 3.00000i) q^{37} +(1.00000 + 1.00000i) q^{41} -4.00000i q^{43} +(-3.00000 - 3.00000i) q^{45} -8.00000 q^{47} -7.00000i q^{49} +4.00000i q^{53} -8.00000 q^{55} +4.00000i q^{59} +(-9.00000 - 9.00000i) q^{61} +(-4.00000 + 4.00000i) q^{65} +12.0000 q^{67} +(4.00000 - 4.00000i) q^{71} +(5.00000 - 5.00000i) q^{73} +(-8.00000 - 8.00000i) q^{79} -9.00000 q^{81} -4.00000i q^{83} +(5.00000 + 3.00000i) q^{85} +(-4.00000 - 4.00000i) q^{95} +(3.00000 - 3.00000i) q^{97} +(-12.0000 + 12.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 8 q^{11} + 8 q^{13} - 2 q^{17} - 8 q^{23} - 6 q^{29} + 8 q^{31} + 6 q^{37} + 2 q^{41} - 6 q^{45} - 16 q^{47} - 16 q^{55} - 18 q^{61} - 8 q^{65} + 24 q^{67} + 8 q^{71} + 10 q^{73} - 16 q^{79}+ \cdots - 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + 8 * q^11 + 8 * q^13 - 2 * q^17 - 8 * q^23 - 6 * q^29 + 8 * q^31 + 6 * q^37 + 2 * q^41 - 6 * q^45 - 16 * q^47 - 16 * q^55 - 18 * q^61 - 8 * q^65 + 24 * q^67 + 8 * q^71 + 10 * q^73 - 16 * q^79 - 18 * q^81 + 10 * q^85 - 8 * q^95 + 6 * q^97 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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