LMFDB - Embedded newform 3240.2.q.o.1081.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3240,2,Mod(1081,3240)] 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("3240.1081"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3240, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,1,0,-2,0,0,0,1,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$25.8715302549$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3240.1081
Dual form			3240.2.q.o.2161.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(0.500000 + 0.866025i) q^{11} +(-0.500000 + 0.866025i) q^{13} +1.00000 q^{17} +4.00000 q^{19} +(0.500000 - 0.866025i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-2.50000 - 4.33013i) q^{29} +(-0.500000 + 0.866025i) q^{31} -2.00000 q^{35} +6.00000 q^{37} +(-3.50000 - 6.06218i) q^{43} +(3.50000 + 6.06218i) q^{47} +(1.50000 - 2.59808i) q^{49} +12.0000 q^{53} +1.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(-5.00000 - 8.66025i) q^{61} +(0.500000 + 0.866025i) q^{65} +(2.00000 - 3.46410i) q^{67} -12.0000 q^{71} +6.00000 q^{73} +(1.00000 - 1.73205i) q^{77} +(-7.50000 - 12.9904i) q^{79} +(1.00000 + 1.73205i) q^{83} +(0.500000 - 0.866025i) q^{85} +12.0000 q^{89} +2.00000 q^{91} +(2.00000 - 3.46410i) q^{95} +(-5.00000 - 8.66025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} - 2 q^{7} + q^{11} - q^{13} + 2 q^{17} + 8 q^{19} + q^{23} - q^{25} - 5 q^{29} - q^{31} - 4 q^{35} + 12 q^{37} - 7 q^{43} + 7 q^{47} + 3 q^{49} + 24 q^{53} + 2 q^{55} - 4 q^{59} - 10 q^{61}+ \cdots - 10 q^{97}+O(q^{100}) $ 2 * q + q^5 - 2 * q^7 + q^11 - q^13 + 2 * q^17 + 8 * q^19 + q^23 - q^25 - 5 * q^29 - q^31 - 4 * q^35 + 12 * q^37 - 7 * q^43 + 7 * q^47 + 3 * q^49 + 24 * q^53 + 2 * q^55 - 4 * q^59 - 10 * q^61 + q^65 + 4 * q^67 - 24 * q^71 + 12 * q^73 + 2 * q^77 - 15 * q^79 + 2 * q^83 + q^85 + 24 * q^89 + 4 * q^91 + 4 * q^95 - 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	0.612801	−	0.790237i	$-0.290043\pi$
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	−0.990766	+	0.135583i	$0.956709\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	0.931505	−	0.363727i	$-0.118496\pi$
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	−0.780750	+	0.624844i	$0.785163\pi$
$12$		0			0
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	−0.926995	−	0.375073i	$-0.877618\pi$
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	0.788320	+	0.615265i	$0.210951\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000			0.242536			0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000			0.242536			0.121268	+	0.992620i	$0.461304\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.500000	−	0.866025i	0.104257	−	0.180579i	−0.809177	−	0.587565i	$-0.800087\pi$
$23$	0.500000	−	0.866025i	0.104257	−	0.180579i	0.913434	+	0.406986i	$0.133420\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.50000	−	4.33013i	−0.464238	−	0.804084i	0.534928	−	0.844897i	$-0.320339\pi$
$29$	−2.50000	−	4.33013i	−0.464238	−	0.804084i	−0.999167	+	0.0408130i	$0.987005\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	−0.907428	−	0.420208i	$-0.861957\pi$
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	0.817625	+	0.575751i	$0.195290\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00000			−0.338062
$36$		0			0
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	−3.50000	−	6.06218i	−0.533745	−	0.924473i	−0.999223	−	0.0394140i	$-0.987451\pi$
$43$	−3.50000	−	6.06218i	−0.533745	−	0.924473i	0.465478	−	0.885059i	$-0.345882\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.50000	+	6.06218i	0.510527	+	0.884260i	0.999926	+	0.0121990i	$0.00388317\pi$
$47$	3.50000	+	6.06218i	0.510527	+	0.884260i	−0.489398	+	0.872060i	$0.662783\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.0000			1.64833			0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000			1.64833			0.824163	+	0.566352i	$0.191646\pi$
$54$		0			0
$55$	1.00000			0.134840
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	−0.966342	−	0.257260i	$-0.917180\pi$
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	0.705965	+	0.708247i	$0.250514\pi$
$60$		0			0
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	−0.985391	−	0.170305i	$-0.945525\pi$
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	0.345207	−	0.938527i	$-0.387809\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.500000	+	0.866025i	0.0620174	+	0.107417i
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.00000	−	1.73205i	0.113961	−	0.197386i
$78$		0			0
$79$	−7.50000	−	12.9904i	−0.843816	−	1.46153i	−0.886646	−	0.462450i	$-0.846971\pi$
$79$	−7.50000	−	12.9904i	−0.843816	−	1.46153i	0.0428296	−	0.999082i	$-0.486363\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.00000	+	1.73205i	0.109764	+	0.190117i	0.915675	−	0.401920i	$-0.131657\pi$
$83$	1.00000	+	1.73205i	0.109764	+	0.190117i	−0.805910	+	0.592037i	$0.798324\pi$
$84$		0			0
$85$	0.500000	−	0.866025i	0.0542326	−	0.0939336i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00000	−	3.46410i	0.205196	−	0.355409i
$96$		0			0
$97$	−5.00000	−	8.66025i	−0.507673	−	0.879316i	−0.999961	−	0.00888289i	$-0.997172\pi$
$97$	−5.00000	−	8.66025i	−0.507673	−	0.879316i	0.492287	−	0.870433i	$-0.336161\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	−0.949595	−	0.313478i	$-0.898506\pi$
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	0.203317	−	0.979113i	$-0.434828\pi$
$102$		0			0
$103$	7.00000	−	12.1244i	0.689730	−	1.19465i	−0.282194	−	0.959357i	$-0.591062\pi$
$103$	7.00000	−	12.1244i	0.689730	−	1.19465i	0.971925	−	0.235291i	$-0.0756043\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−18.0000			−1.74013			−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000			−1.74013			−0.870063	+	0.492941i	$0.835922\pi$
$108$		0			0
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.50000	−	2.59808i	0.141108	−	0.244406i	−0.786806	−	0.617200i	$-0.788267\pi$
$113$	1.50000	−	2.59808i	0.141108	−	0.244406i	0.927914	+	0.372794i	$0.121600\pi$
$114$		0			0
$115$	−0.500000	−	0.866025i	−0.0466252	−	0.0807573i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.00000	−	1.73205i	−0.0916698	−	0.158777i
$120$		0			0
$121$	5.00000	−	8.66025i	0.454545	−	0.787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−6.00000			−0.532414			−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000			−0.532414			−0.266207	+	0.963916i	$0.585770\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.50000	+	7.79423i	−0.393167	+	0.680985i	−0.992865	−	0.119241i	$-0.961954\pi$
$131$	−4.50000	+	7.79423i	−0.393167	+	0.680985i	0.599699	+	0.800226i	$0.295287\pi$
$132$		0			0
$133$	−4.00000	−	6.92820i	−0.346844	−	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.0000	+	19.0526i	0.939793	+	1.62777i	0.765855	+	0.643013i	$0.222316\pi$
$137$	11.0000	+	19.0526i	0.939793	+	1.62777i	0.173939	+	0.984757i	$0.444351\pi$
$138$		0			0
$139$	3.00000	−	5.19615i	0.254457	−	0.440732i	−0.710291	−	0.703908i	$-0.751437\pi$
$139$	3.00000	−	5.19615i	0.254457	−	0.440732i	0.964748	+	0.263176i	$0.0847700\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.00000			−0.0836242
$144$		0			0
$145$	−5.00000			−0.415227
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.50000	+	11.2583i	−0.532501	+	0.922318i	0.466779	+	0.884374i	$0.345414\pi$
$149$	−6.50000	+	11.2583i	−0.532501	+	0.922318i	−0.999280	+	0.0379444i	$0.987919\pi$
$150$		0			0
$151$	0.500000	+	0.866025i	0.0406894	+	0.0704761i	0.885653	−	0.464348i	$-0.153711\pi$
$151$	0.500000	+	0.866025i	0.0406894	+	0.0704761i	−0.844963	+	0.534824i	$0.820378\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.500000	+	0.866025i	0.0401610	+	0.0695608i
$156$		0			0
$157$	8.50000	−	14.7224i	0.678374	−	1.17498i	−0.297097	−	0.954847i	$-0.596018\pi$
$157$	8.50000	−	14.7224i	0.678374	−	1.17498i	0.975470	−	0.220131i	$-0.0706483\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00000			−0.157622
$162$		0			0
$163$	−5.00000			−0.391630			−0.195815	−	0.980641i	$-0.562735\pi$
$163$	−5.00000			−0.391630			−0.195815	+	0.980641i	$0.562735\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.00000	−	8.66025i	−0.380143	−	0.658427i	0.610939	−	0.791677i	$-0.290792\pi$
$173$	−5.00000	−	8.66025i	−0.380143	−	0.658427i	−0.991082	+	0.133250i	$0.957459\pi$
$174$		0			0
$175$	−1.00000	+	1.73205i	−0.0755929	+	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.00000			0.298974			0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000			0.298974			0.149487	+	0.988764i	$0.452238\pi$
$180$		0			0
$181$	−4.00000			−0.297318			−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000			−0.297318			−0.148659	+	0.988889i	$0.547496\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.00000	−	5.19615i	0.220564	−	0.382029i
$186$		0			0
$187$	0.500000	+	0.866025i	0.0365636	+	0.0633300i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.0000	−	19.0526i	−0.795932	−	1.37859i	−0.922246	−	0.386604i	$-0.873648\pi$
$191$	−11.0000	−	19.0526i	−0.795932	−	1.37859i	0.126314	−	0.991990i	$-0.459685\pi$
$192$		0			0
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.00000			0.569976			0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000			0.569976			0.284988	+	0.958531i	$0.408010\pi$
$198$		0			0
$199$	5.00000			0.354441			0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000			0.354441			0.177220	+	0.984171i	$0.443289\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.00000	+	8.66025i	−0.350931	+	0.607831i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.00000	+	3.46410i	0.138343	+	0.239617i
$210$		0			0
$211$	11.0000	−	19.0526i	0.757271	−	1.31163i	−0.186966	−	0.982366i	$-0.559865\pi$
$211$	11.0000	−	19.0526i	0.757271	−	1.31163i	0.944237	−	0.329266i	$-0.106801\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−7.00000			−0.477396
$216$		0			0
$217$	2.00000			0.135769
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.500000	+	0.866025i	−0.0336336	+	0.0582552i
$222$		0			0
$223$	8.00000	+	13.8564i	0.535720	+	0.927894i	0.999128	+	0.0417488i	$0.0132929\pi$
$223$	8.00000	+	13.8564i	0.535720	+	0.927894i	−0.463409	+	0.886145i	$0.653374\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.0000	+	20.7846i	0.796468	+	1.37952i	0.921903	+	0.387421i	$0.126634\pi$
$227$	12.0000	+	20.7846i	0.796468	+	1.37952i	−0.125435	+	0.992102i	$0.540033\pi$
$228$		0			0
$229$	10.0000	−	17.3205i	0.660819	−	1.14457i	−0.319582	−	0.947559i	$-0.603543\pi$
$229$	10.0000	−	17.3205i	0.660819	−	1.14457i	0.980401	−	0.197013i	$-0.0631241\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$	7.00000			0.456630
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.00000	+	15.5885i	−0.582162	+	1.00833i	0.413061	+	0.910703i	$0.364460\pi$
$239$	−9.00000	+	15.5885i	−0.582162	+	1.00833i	−0.995223	+	0.0976302i	$0.968874\pi$
$240$		0			0
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	0.774202	−	0.632938i	$-0.218151\pi$
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	−0.935242	+	0.354010i	$0.884818\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.50000	−	2.59808i	−0.0958315	−	0.165985i
$246$		0			0
$247$	−2.00000	+	3.46410i	−0.127257	+	0.220416i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.0000			−1.32551			−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000			−1.32551			−0.662754	+	0.748837i	$0.730613\pi$
$252$		0			0
$253$	1.00000			0.0628695
$254$		0			0
$255$		0			0
$256$		0			0
$257$	13.5000	−	23.3827i	0.842107	−	1.45857i	−0.0460033	−	0.998941i	$-0.514648\pi$
$257$	13.5000	−	23.3827i	0.842107	−	1.45857i	0.888110	−	0.459631i	$-0.152018\pi$
$258$		0			0
$259$	−6.00000	−	10.3923i	−0.372822	−	0.645746i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	0.619586	−	0.784929i	$-0.287301\pi$
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	−0.989561	+	0.144112i	$0.953967\pi$
$264$		0			0
$265$	6.00000	−	10.3923i	0.368577	−	0.638394i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	31.0000			1.89010			0.945052	−	0.326921i	$-0.106011\pi$
$269$	31.0000			1.89010			0.945052	+	0.326921i	$0.106011\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.500000	−	0.866025i	0.0301511	−	0.0522233i
$276$		0			0
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.00000	−	1.73205i	−0.0596550	−	0.103325i	0.834656	−	0.550772i	$-0.185667\pi$
$281$	−1.00000	−	1.73205i	−0.0596550	−	0.103325i	−0.894311	+	0.447447i	$0.852333\pi$
$282$		0			0
$283$	8.00000	−	13.8564i	0.475551	−	0.823678i	−0.524057	−	0.851683i	$-0.675582\pi$
$283$	8.00000	−	13.8564i	0.475551	−	0.823678i	0.999608	+	0.0280052i	$0.00891551\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.0000			−0.941176
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.00000	+	1.73205i	−0.0584206	+	0.101187i	−0.893757	−	0.448552i	$-0.851940\pi$
$293$	−1.00000	+	1.73205i	−0.0584206	+	0.101187i	0.835336	+	0.549740i	$0.185273\pi$
$294$		0			0
$295$	2.00000	+	3.46410i	0.116445	+	0.201688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.500000	+	0.866025i	0.0289157	+	0.0500835i
$300$		0			0
$301$	−7.00000	+	12.1244i	−0.403473	+	0.698836i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.0000			−0.572598
$306$		0			0
$307$	−5.00000			−0.285365			−0.142683	−	0.989769i	$-0.545573\pi$
$307$	−5.00000			−0.285365			−0.142683	+	0.989769i	$0.545573\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.00000	−	6.92820i	0.226819	−	0.392862i	−0.730044	−	0.683400i	$-0.760501\pi$
$311$	4.00000	−	6.92820i	0.226819	−	0.392862i	0.956864	+	0.290537i	$0.0938340\pi$
$312$		0			0
$313$	−4.00000	−	6.92820i	−0.226093	−	0.391605i	0.730554	−	0.682855i	$-0.239262\pi$
$313$	−4.00000	−	6.92820i	−0.226093	−	0.391605i	−0.956647	+	0.291250i	$0.905929\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.0000	+	20.7846i	0.673987	+	1.16738i	0.976764	+	0.214318i	$0.0687530\pi$
$317$	12.0000	+	20.7846i	0.673987	+	1.16738i	−0.302777	+	0.953062i	$0.597914\pi$
$318$		0			0
$319$	2.50000	−	4.33013i	0.139973	−	0.242441i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.00000			0.222566
$324$		0			0
$325$	1.00000			0.0554700
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.00000	−	12.1244i	0.385922	−	0.668437i
$330$		0			0
$331$	9.00000	+	15.5885i	0.494685	+	0.856819i	0.999981	−	0.00612670i	$-0.00195020\pi$
$331$	9.00000	+	15.5885i	0.494685	+	0.856819i	−0.505296	+	0.862946i	$0.668617\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.00000	−	3.46410i	−0.109272	−	0.189264i
$336$		0			0
$337$	4.00000	−	6.92820i	0.217894	−	0.377403i	−0.736270	−	0.676688i	$-0.763415\pi$
$337$	4.00000	−	6.92820i	0.217894	−	0.377403i	0.954164	+	0.299285i	$0.0967480\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.00000			−0.0541530
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	−0.996826	−	0.0796169i	$-0.974630\pi$
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	0.567363	+	0.823468i	$0.307964\pi$
$348$		0			0
$349$	9.00000	+	15.5885i	0.481759	+	0.834431i	0.999781	−	0.0209364i	$-0.00666475\pi$
$349$	9.00000	+	15.5885i	0.481759	+	0.834431i	−0.518022	+	0.855367i	$0.673331\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.50000	+	14.7224i	0.452409	+	0.783596i	0.998535	−	0.0541072i	$-0.0172313\pi$
$353$	8.50000	+	14.7224i	0.452409	+	0.783596i	−0.546126	+	0.837703i	$0.683898\pi$
$354$		0			0
$355$	−6.00000	+	10.3923i	−0.318447	+	0.551566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.00000	−	5.19615i	0.157027	−	0.271979i
$366$		0			0
$367$	−1.00000	−	1.73205i	−0.0521996	−	0.0904123i	0.838745	−	0.544524i	$-0.183290\pi$
$367$	−1.00000	−	1.73205i	−0.0521996	−	0.0904123i	−0.890945	+	0.454112i	$0.849957\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.0000	−	20.7846i	−0.623009	−	1.07908i
$372$		0			0
$373$	−9.50000	+	16.4545i	−0.491891	+	0.851981i	−0.999956	−	0.00933789i	$-0.997028\pi$
$373$	−9.50000	+	16.4545i	−0.491891	+	0.851981i	0.508065	+	0.861319i	$0.330361\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.00000			0.257513
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.50000	+	12.9904i	−0.383232	+	0.663777i	−0.991522	−	0.129937i	$-0.958522\pi$
$383$	−7.50000	+	12.9904i	−0.383232	+	0.663777i	0.608290	+	0.793715i	$0.291856\pi$
$384$		0			0
$385$	−1.00000	−	1.73205i	−0.0509647	−	0.0882735i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.50000	−	7.79423i	−0.228159	−	0.395183i	0.729103	−	0.684403i	$-0.239937\pi$
$389$	−4.50000	−	7.79423i	−0.228159	−	0.395183i	−0.957263	+	0.289220i	$0.906604\pi$
$390$		0			0
$391$	0.500000	−	0.866025i	0.0252861	−	0.0437968i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−15.0000			−0.754732
$396$		0			0
$397$	−11.0000			−0.552074			−0.276037	−	0.961147i	$-0.589021\pi$
$397$	−11.0000			−0.552074			−0.276037	+	0.961147i	$0.589021\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.0000	−	31.1769i	0.898877	−	1.55690i	0.0699455	−	0.997551i	$-0.477717\pi$
$401$	18.0000	−	31.1769i	0.898877	−	1.55690i	0.828932	−	0.559350i	$-0.188949\pi$
$402$		0			0
$403$	−0.500000	−	0.866025i	−0.0249068	−	0.0431398i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.00000	+	5.19615i	0.148704	+	0.257564i
$408$		0			0
$409$	−15.5000	+	26.8468i	−0.766426	+	1.32749i	0.173064	+	0.984911i	$0.444633\pi$
$409$	−15.5000	+	26.8468i	−0.766426	+	1.32749i	−0.939490	+	0.342578i	$0.888700\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.00000			0.393654
$414$		0			0
$415$	2.00000			0.0981761
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16.5000	+	28.5788i	−0.806078	+	1.39617i	0.109483	+	0.993989i	$0.465080\pi$
$419$	−16.5000	+	28.5788i	−0.806078	+	1.39617i	−0.915561	+	0.402179i	$0.868253\pi$
$420$		0			0
$421$	−6.00000	−	10.3923i	−0.292422	−	0.506490i	0.681960	−	0.731390i	$-0.261128\pi$
$421$	−6.00000	−	10.3923i	−0.292422	−	0.506490i	−0.974382	+	0.224900i	$0.927795\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.500000	−	0.866025i	−0.0242536	−	0.0420084i
$426$		0			0
$427$	−10.0000	+	17.3205i	−0.483934	+	0.838198i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.00000			−0.385346			−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000			−0.385346			−0.192673	+	0.981263i	$0.561716\pi$
$432$		0			0
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.00000	−	3.46410i	0.0956730	−	0.165710i
$438$		0			0
$439$	−8.00000	−	13.8564i	−0.381819	−	0.661330i	0.609503	−	0.792784i	$-0.291369\pi$
$439$	−8.00000	−	13.8564i	−0.381819	−	0.661330i	−0.991322	+	0.131453i	$0.958036\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	0.972626	−	0.232377i	$-0.0746503\pi$
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	−0.687557	+	0.726130i	$0.741317\pi$
$444$		0			0
$445$	6.00000	−	10.3923i	0.284427	−	0.492642i
$446$		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
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.00000	−	1.73205i	0.0468807	−	0.0811998i
$456$		0			0
$457$	−14.0000	−	24.2487i	−0.654892	−	1.13431i	−0.981921	−	0.189292i	$-0.939381\pi$
$457$	−14.0000	−	24.2487i	−0.654892	−	1.13431i	0.327028	−	0.945015i	$-0.393953\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.00000	−	15.5885i	−0.419172	−	0.726027i	0.576685	−	0.816967i	$-0.304346\pi$
$461$	−9.00000	−	15.5885i	−0.419172	−	0.726027i	−0.995856	+	0.0909401i	$0.971013\pi$
$462$		0			0
$463$	−3.00000	+	5.19615i	−0.139422	+	0.241486i	−0.927278	−	0.374374i	$-0.877858\pi$
$463$	−3.00000	+	5.19615i	−0.139422	+	0.241486i	0.787856	+	0.615859i	$0.211191\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	28.0000			1.29569			0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000			1.29569			0.647843	+	0.761774i	$0.275671\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.50000	−	6.06218i	0.160930	−	0.278739i
$474$		0			0
$475$	−2.00000	−	3.46410i	−0.0917663	−	0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.0000	−	17.3205i	−0.456912	−	0.791394i	0.541884	−	0.840453i	$-0.317711\pi$
$479$	−10.0000	−	17.3205i	−0.456912	−	0.791394i	−0.998796	+	0.0490589i	$0.984378\pi$
$480$		0			0
$481$	−3.00000	+	5.19615i	−0.136788	+	0.236924i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.0000			−0.454077
$486$		0			0
$487$	4.00000			0.181257			0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000			0.181257			0.0906287	+	0.995885i	$0.471112\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.00000	+	10.3923i	−0.270776	+	0.468998i	−0.969061	−	0.246822i	$-0.920614\pi$
$491$	−6.00000	+	10.3923i	−0.270776	+	0.468998i	0.698285	+	0.715820i	$0.253947\pi$
$492$		0			0
$493$	−2.50000	−	4.33013i	−0.112594	−	0.195019i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.0000	+	20.7846i	0.538274	+	0.932317i
$498$		0			0
$499$	−10.0000	+	17.3205i	−0.447661	+	0.775372i	−0.998233	−	0.0594153i	$-0.981076\pi$
$499$	−10.0000	+	17.3205i	−0.447661	+	0.775372i	0.550572	+	0.834788i	$0.314410\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−9.00000			−0.401290			−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000			−0.401290			−0.200645	+	0.979664i	$0.564304\pi$
$504$		0			0
$505$	−15.0000			−0.667491
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.50000	−	6.06218i	0.155135	−	0.268701i	−0.777973	−	0.628297i	$-0.783752\pi$
$509$	3.50000	−	6.06218i	0.155135	−	0.268701i	0.933108	+	0.359596i	$0.117085\pi$
$510$		0			0
$511$	−6.00000	−	10.3923i	−0.265424	−	0.459728i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.00000	−	12.1244i	−0.308457	−	0.534263i
$516$		0			0
$517$	−3.50000	+	6.06218i	−0.153930	+	0.266614i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−28.0000			−1.22670			−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000			−1.22670			−0.613351	+	0.789810i	$0.710179\pi$
$522$		0			0
$523$	1.00000			0.0437269			0.0218635	−	0.999761i	$-0.493040\pi$
$523$	1.00000			0.0437269			0.0218635	+	0.999761i	$0.493040\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.500000	+	0.866025i	−0.0217803	+	0.0377247i
$528$		0			0
$529$	11.0000	+	19.0526i	0.478261	+	0.828372i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−9.00000	+	15.5885i	−0.389104	+	0.673948i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.00000			0.129219
$540$		0			0
$541$	34.0000			1.46177			0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000			1.46177			0.730887	+	0.682498i	$0.239107\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.00000	+	3.46410i	−0.0856706	+	0.148386i
$546$		0			0
$547$	18.5000	+	32.0429i	0.791003	+	1.37006i	0.925347	+	0.379122i	$0.123774\pi$
$547$	18.5000	+	32.0429i	0.791003	+	1.37006i	−0.134344	+	0.990935i	$0.542893\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.0000	−	17.3205i	−0.426014	−	0.737878i
$552$		0			0
$553$	−15.0000	+	25.9808i	−0.637865	+	1.10481i
$554$		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
$559$	7.00000			0.296068
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.0000	+	32.9090i	−0.800755	+	1.38695i	0.118366	+	0.992970i	$0.462235\pi$
$563$	−19.0000	+	32.9090i	−0.800755	+	1.38695i	−0.919120	+	0.393977i	$0.871099\pi$
$564$		0			0
$565$	−1.50000	−	2.59808i	−0.0631055	−	0.109302i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	16.0000	+	27.7128i	0.670755	+	1.16178i	0.977690	+	0.210051i	$0.0673631\pi$
$569$	16.0000	+	27.7128i	0.670755	+	1.16178i	−0.306935	+	0.951730i	$0.599304\pi$
$570$		0			0
$571$	−11.0000	+	19.0526i	−0.460336	+	0.797325i	−0.998978	−	0.0452101i	$-0.985604\pi$
$571$	−11.0000	+	19.0526i	−0.460336	+	0.797325i	0.538642	+	0.842535i	$0.318938\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.00000			−0.0417029
$576$		0			0
$577$	32.0000			1.33218			0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000			1.33218			0.666089	+	0.745873i	$0.267967\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.00000	−	3.46410i	0.0829740	−	0.143715i
$582$		0			0
$583$	6.00000	+	10.3923i	0.248495	+	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21.0000	+	36.3731i	0.866763	+	1.50128i	0.865286	+	0.501278i	$0.167137\pi$
$587$	21.0000	+	36.3731i	0.866763	+	1.50128i	0.00147660	+	0.999999i	$0.499530\pi$
$588$		0			0
$589$	−2.00000	+	3.46410i	−0.0824086	+	0.142736i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−15.0000			−0.615976			−0.307988	−	0.951390i	$-0.599656\pi$
$593$	−15.0000			−0.615976			−0.307988	+	0.951390i	$0.599656\pi$
$594$		0			0
$595$	−2.00000			−0.0819920
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.0000	−	25.9808i	0.612883	−	1.06155i	−0.377869	−	0.925859i	$-0.623343\pi$
$599$	15.0000	−	25.9808i	0.612883	−	1.06155i	0.990752	−	0.135686i	$-0.0433238\pi$
$600$		0			0
$601$	17.5000	+	30.3109i	0.713840	+	1.23641i	0.963405	+	0.268049i	$0.0863789\pi$
$601$	17.5000	+	30.3109i	0.713840	+	1.23641i	−0.249565	+	0.968358i	$0.580288\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.00000	−	8.66025i	−0.203279	−	0.352089i
$606$		0			0
$607$	−15.0000	+	25.9808i	−0.608831	+	1.05453i	0.382602	+	0.923913i	$0.375028\pi$
$607$	−15.0000	+	25.9808i	−0.608831	+	1.05453i	−0.991433	+	0.130613i	$0.958305\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.00000			−0.283190
$612$		0			0
$613$	−15.0000			−0.605844			−0.302922	−	0.953015i	$-0.597962\pi$
$613$	−15.0000			−0.605844			−0.302922	+	0.953015i	$0.597962\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.50000	−	7.79423i	0.181163	−	0.313784i	−0.761114	−	0.648618i	$-0.775347\pi$
$617$	4.50000	−	7.79423i	0.181163	−	0.313784i	0.942277	+	0.334835i	$0.108680\pi$
$618$		0			0
$619$	−19.0000	−	32.9090i	−0.763674	−	1.32272i	−0.940945	−	0.338561i	$-0.890060\pi$
$619$	−19.0000	−	32.9090i	−0.763674	−	1.32272i	0.177270	−	0.984162i	$-0.443273\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.0000	−	20.7846i	−0.480770	−	0.832718i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.00000			0.239236
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.00000	+	5.19615i	−0.119051	+	0.206203i
$636$		0			0
$637$	1.50000	+	2.59808i	0.0594322	+	0.102940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	−0.993899	−	0.110291i	$-0.964822\pi$
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	0.401435	−	0.915888i	$-0.368512\pi$
$642$		0			0
$643$	4.50000	−	7.79423i	0.177463	−	0.307374i	−0.763548	−	0.645751i	$-0.776544\pi$
$643$	4.50000	−	7.79423i	0.177463	−	0.307374i	0.941011	+	0.338377i	$0.109878\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		0			0
$649$	−4.00000			−0.157014
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.00000	+	3.46410i	−0.0782660	+	0.135561i	−0.902502	−	0.430686i	$-0.858272\pi$
$653$	−2.00000	+	3.46410i	−0.0782660	+	0.135561i	0.824236	+	0.566247i	$0.191605\pi$
$654$		0			0
$655$	4.50000	+	7.79423i	0.175830	+	0.304546i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.0000	+	20.7846i	0.467454	+	0.809653i	0.999309	−	0.0371821i	$-0.0118382\pi$
$659$	12.0000	+	20.7846i	0.467454	+	0.809653i	−0.531855	+	0.846836i	$0.678505\pi$
$660$		0			0
$661$	−21.0000	+	36.3731i	−0.816805	+	1.41475i	0.0912190	+	0.995831i	$0.470924\pi$
$661$	−21.0000	+	36.3731i	−0.816805	+	1.41475i	−0.908024	+	0.418917i	$0.862410\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.00000			−0.310227
$666$		0			0
$667$	−5.00000			−0.193601
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.00000	−	8.66025i	0.193023	−	0.334325i
$672$		0			0
$673$	3.00000	+	5.19615i	0.115642	+	0.200297i	0.918036	−	0.396497i	$-0.129774\pi$
$673$	3.00000	+	5.19615i	0.115642	+	0.200297i	−0.802395	+	0.596794i	$0.796441\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.00000	+	3.46410i	0.0768662	+	0.133136i	0.901896	−	0.431953i	$-0.142175\pi$
$677$	2.00000	+	3.46410i	0.0768662	+	0.133136i	−0.825030	+	0.565089i	$0.808842\pi$
$678$		0			0
$679$	−10.0000	+	17.3205i	−0.383765	+	0.664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.0000			−0.918334			−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000			−0.918334			−0.459167	+	0.888350i	$0.651852\pi$
$684$		0			0
$685$	22.0000			0.840577
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.00000	+	10.3923i	−0.228582	+	0.395915i
$690$		0			0
$691$	5.00000	+	8.66025i	0.190209	+	0.329452i	0.945319	−	0.326146i	$-0.105750\pi$
$691$	5.00000	+	8.66025i	0.190209	+	0.329452i	−0.755110	+	0.655598i	$0.772417\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.00000	−	5.19615i	−0.113796	−	0.197101i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−11.0000			−0.415464			−0.207732	−	0.978186i	$-0.566608\pi$
$701$	−11.0000			−0.415464			−0.207732	+	0.978186i	$0.566608\pi$
$702$		0			0
$703$	24.0000			0.905177
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−15.0000	+	25.9808i	−0.564133	+	0.977107i
$708$		0			0
$709$	26.0000	+	45.0333i	0.976450	+	1.69126i	0.675063	+	0.737760i	$0.264116\pi$
$709$	26.0000	+	45.0333i	0.976450	+	1.69126i	0.301388	+	0.953502i	$0.402550\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.500000	+	0.866025i	0.0187251	+	0.0324329i
$714$		0			0
$715$	−0.500000	+	0.866025i	−0.0186989	+	0.0323875i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.00000			0.0745874			0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000			0.0745874			0.0372937	+	0.999304i	$0.488126\pi$
$720$		0			0
$721$	−28.0000			−1.04277
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.50000	+	4.33013i	−0.0928477	+	0.160817i
$726$		0			0
$727$	14.0000	+	24.2487i	0.519231	+	0.899335i	0.999750	+	0.0223506i	$0.00711500\pi$
$727$	14.0000	+	24.2487i	0.519231	+	0.899335i	−0.480519	+	0.876984i	$0.659552\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.50000	−	6.06218i	−0.129452	−	0.224218i
$732$		0			0
$733$	−21.0000	+	36.3731i	−0.775653	+	1.34347i	0.158774	+	0.987315i	$0.449246\pi$
$733$	−21.0000	+	36.3731i	−0.775653	+	1.34347i	−0.934427	+	0.356155i	$0.884088\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.00000			0.147342
$738$		0			0
$739$	12.0000			0.441427			0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000			0.441427			0.220714	+	0.975339i	$0.429161\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	25.5000	−	44.1673i	0.935504	−	1.62034i	0.161772	−	0.986828i	$-0.448279\pi$
$743$	25.5000	−	44.1673i	0.935504	−	1.62034i	0.773732	−	0.633513i	$-0.218388\pi$
$744$		0			0
$745$	6.50000	+	11.2583i	0.238142	+	0.412473i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.0000	+	31.1769i	0.657706	+	1.13918i
$750$		0			0
$751$	−1.50000	+	2.59808i	−0.0547358	+	0.0948051i	−0.892095	−	0.451848i	$-0.850765\pi$
$751$	−1.50000	+	2.59808i	−0.0547358	+	0.0948051i	0.837359	+	0.546653i	$0.184098\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.00000			0.0363937
$756$		0			0
$757$	−1.00000			−0.0363456			−0.0181728	−	0.999835i	$-0.505785\pi$
$757$	−1.00000			−0.0363456			−0.0181728	+	0.999835i	$0.505785\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.00000	+	5.19615i	−0.108750	+	0.188360i	−0.915264	−	0.402854i	$-0.868018\pi$
$761$	−3.00000	+	5.19615i	−0.108750	+	0.188360i	0.806514	+	0.591215i	$0.201351\pi$
$762$		0			0
$763$	4.00000	+	6.92820i	0.144810	+	0.250818i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.00000	−	3.46410i	−0.0722158	−	0.125081i
$768$		0			0
$769$	−3.50000	+	6.06218i	−0.126213	+	0.218608i	−0.922207	−	0.386698i	$-0.873616\pi$
$769$	−3.50000	+	6.06218i	−0.126213	+	0.218608i	0.795993	+	0.605305i	$0.206949\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$		0			0
$775$	1.00000			0.0359211
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−6.00000	−	10.3923i	−0.214697	−	0.371866i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.50000	−	14.7224i	−0.303378	−	0.525466i
$786$		0			0
$787$	−23.5000	+	40.7032i	−0.837685	+	1.45091i	0.0541413	+	0.998533i	$0.482758\pi$
$787$	−23.5000	+	40.7032i	−0.837685	+	1.45091i	−0.891826	+	0.452379i	$0.850575\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.00000			−0.213335
$792$		0			0
$793$	10.0000			0.355110
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.0000	+	31.1769i	−0.637593	+	1.10434i	0.348367	+	0.937358i	$0.386736\pi$
$797$	−18.0000	+	31.1769i	−0.637593	+	1.10434i	−0.985959	+	0.166985i	$0.946597\pi$
$798$		0			0
$799$	3.50000	+	6.06218i	0.123821	+	0.214464i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.00000	+	5.19615i	0.105868	+	0.183368i
$804$		0			0
$805$	−1.00000	+	1.73205i	−0.0352454	+	0.0610468i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−10.0000			−0.351147			−0.175574	−	0.984466i	$-0.556178\pi$
$811$	−10.0000			−0.351147			−0.175574	+	0.984466i	$0.556178\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.50000	+	4.33013i	−0.0875712	+	0.151678i
$816$		0			0
$817$	−14.0000	−	24.2487i	−0.489798	−	0.848355i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19.0000	−	32.9090i	−0.663105	−	1.14853i	−0.979795	−	0.200002i	$-0.935905\pi$
$821$	−19.0000	−	32.9090i	−0.663105	−	1.14853i	0.316691	−	0.948529i	$-0.397428\pi$
$822$		0			0
$823$	10.0000	−	17.3205i	0.348578	−	0.603755i	−0.637419	−	0.770517i	$-0.719998\pi$
$823$	10.0000	−	17.3205i	0.348578	−	0.603755i	0.985997	+	0.166762i	$0.0533313\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	30.0000			1.04320			0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000			1.04320			0.521601	+	0.853189i	$0.325335\pi$
$828$		0			0
$829$	−20.0000			−0.694629			−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000			−0.694629			−0.347314	+	0.937749i	$0.612906\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.50000	−	2.59808i	0.0519719	−	0.0900180i
$834$		0			0
$835$	−6.00000	−	10.3923i	−0.207639	−	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	13.0000	+	22.5167i	0.448810	+	0.777361i	0.998309	−	0.0581329i	$-0.0185147\pi$
$839$	13.0000	+	22.5167i	0.448810	+	0.777361i	−0.549499	+	0.835494i	$0.685181\pi$
$840$		0			0
$841$	2.00000	−	3.46410i	0.0689655	−	0.119452i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.0000			0.412813
$846$		0			0
$847$	−20.0000			−0.687208
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.00000	−	5.19615i	0.102839	−	0.178122i
$852$		0			0
$853$	15.5000	+	26.8468i	0.530710	+	0.919216i	0.999358	+	0.0358315i	$0.0114080\pi$
$853$	15.5000	+	26.8468i	0.530710	+	0.919216i	−0.468648	+	0.883385i	$0.655259\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.00000	+	5.19615i	0.102478	+	0.177497i	0.912705	−	0.408619i	$-0.133990\pi$
$857$	3.00000	+	5.19615i	0.102478	+	0.177497i	−0.810227	+	0.586116i	$0.800656\pi$
$858$		0			0
$859$	11.0000	−	19.0526i	0.375315	−	0.650065i	−0.615059	−	0.788481i	$-0.710868\pi$
$859$	11.0000	−	19.0526i	0.375315	−	0.650065i	0.990374	+	0.138416i	$0.0442012\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	21.0000			0.714848			0.357424	−	0.933942i	$-0.383655\pi$
$863$	21.0000			0.714848			0.357424	+	0.933942i	$0.383655\pi$
$864$		0			0
$865$	−10.0000			−0.340010
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.50000	−	12.9904i	0.254420	−	0.440668i
$870$		0			0
$871$	2.00000	+	3.46410i	0.0677674	+	0.117377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.00000	+	1.73205i	0.0338062	+	0.0585540i
$876$		0			0
$877$	26.5000	−	45.8993i	0.894841	−	1.54991i	0.0608407	−	0.998147i	$-0.480622\pi$
$877$	26.5000	−	45.8993i	0.894841	−	1.54991i	0.834001	−	0.551763i	$-0.186045\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	−52.0000			−1.74994			−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000			−1.74994			−0.874970	+	0.484178i	$0.839119\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.50000	−	16.4545i	0.318979	−	0.552487i	−0.661296	−	0.750125i	$-0.729993\pi$
$887$	9.50000	−	16.4545i	0.318979	−	0.552487i	0.980275	+	0.197637i	$0.0633268\pi$
$888$		0			0
$889$	6.00000	+	10.3923i	0.201234	+	0.348547i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	14.0000	+	24.2487i	0.468492	+	0.811452i
$894$		0			0
$895$	2.00000	−	3.46410i	0.0668526	−	0.115792i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.00000			0.166759
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.00000	+	3.46410i	−0.0664822	+	0.115151i
$906$		0			0
$907$	22.5000	+	38.9711i	0.747100	+	1.29402i	0.949207	+	0.314652i	$0.101888\pi$
$907$	22.5000	+	38.9711i	0.747100	+	1.29402i	−0.202107	+	0.979363i	$0.564779\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−28.0000	−	48.4974i	−0.927681	−	1.60679i	−0.787191	−	0.616709i	$-0.788465\pi$
$911$	−28.0000	−	48.4974i	−0.927681	−	1.60679i	−0.140490	−	0.990082i	$-0.544868\pi$
$912$		0			0
$913$	−1.00000	+	1.73205i	−0.0330952	+	0.0573225i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.0000			0.594412
$918$		0			0
$919$	35.0000			1.15454			0.577272	−	0.816552i	$-0.304117\pi$
$919$	35.0000			1.15454			0.577272	+	0.816552i	$0.304117\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.00000	−	10.3923i	0.197492	−	0.342067i
$924$		0			0
$925$	−3.00000	−	5.19615i	−0.0986394	−	0.170848i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.0000	−	17.3205i	−0.328089	−	0.568267i	0.654043	−	0.756457i	$-0.273071\pi$
$929$	−10.0000	−	17.3205i	−0.328089	−	0.568267i	−0.982133	+	0.188190i	$0.939738\pi$
$930$		0			0
$931$	6.00000	−	10.3923i	0.196642	−	0.340594i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.00000			0.0327035
$936$		0			0
$937$	−14.0000			−0.457360			−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000			−0.457360			−0.228680	+	0.973502i	$0.573441\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.50000	+	14.7224i	−0.277092	+	0.479938i	−0.970661	−	0.240453i	$-0.922704\pi$
$941$	−8.50000	+	14.7224i	−0.277092	+	0.479938i	0.693569	+	0.720390i	$0.256037\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.00000	−	12.1244i	−0.227469	−	0.393989i	0.729588	−	0.683887i	$-0.239712\pi$
$947$	−7.00000	−	12.1244i	−0.227469	−	0.393989i	−0.957057	+	0.289898i	$0.906379\pi$
$948$		0			0
$949$	−3.00000	+	5.19615i	−0.0973841	+	0.168674i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−21.0000			−0.680257			−0.340128	−	0.940379i	$-0.610471\pi$
$953$	−21.0000			−0.680257			−0.340128	+	0.940379i	$0.610471\pi$
$954$		0			0
$955$	−22.0000			−0.711903
$956$		0			0
$957$		0			0
$958$		0			0
$959$	22.0000	−	38.1051i	0.710417	−	1.23048i
$960$		0			0
$961$	15.0000	+	25.9808i	0.483871	+	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.00000	−	1.73205i	−0.0321911	−	0.0557567i
$966$		0			0
$967$	3.00000	−	5.19615i	0.0964735	−	0.167097i	−0.813749	−	0.581216i	$-0.802577\pi$
$967$	3.00000	−	5.19615i	0.0964735	−	0.167097i	0.910223	+	0.414119i	$0.135910\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−53.0000			−1.70085			−0.850425	−	0.526096i	$-0.823655\pi$
$971$	−53.0000			−1.70085			−0.850425	+	0.526096i	$0.823655\pi$
$972$		0			0
$973$	−12.0000			−0.384702
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−30.5000	+	52.8275i	−0.975781	+	1.69010i	−0.298450	+	0.954425i	$0.596469\pi$
$977$	−30.5000	+	52.8275i	−0.975781	+	1.69010i	−0.677332	+	0.735678i	$0.736864\pi$
$978$		0			0
$979$	6.00000	+	10.3923i	0.191761	+	0.332140i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.5000	+	28.5788i	0.526268	+	0.911523i	0.999532	+	0.0306024i	$0.00974257\pi$
$983$	16.5000	+	28.5788i	0.526268	+	0.911523i	−0.473263	+	0.880921i	$0.656924\pi$
$984$		0			0
$985$	4.00000	−	6.92820i	0.127451	−	0.220751i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7.00000			−0.222587
$990$		0			0
$991$	−55.0000			−1.74713			−0.873566	−	0.486705i	$-0.838199\pi$
$991$	−55.0000			−1.74713			−0.873566	+	0.486705i	$0.838199\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.50000	−	4.33013i	0.0792553	−	0.137274i
$996$		0			0
$997$	−19.5000	−	33.7750i	−0.617571	−	1.06966i	−0.989928	−	0.141575i	$-0.954783\pi$
$997$	−19.5000	−	33.7750i	−0.617571	−	1.06966i	0.372356	−	0.928090i	$-0.378550\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.q.o.1081.1		2
3.2	odd	2		3240.2.q.c.1081.1		2
9.2	odd	6		3240.2.q.c.2161.1		2
9.4	even	3		1080.2.a.f.1.1	✓	1
9.5	odd	6		1080.2.a.k.1.1	yes	1
9.7	even	3	inner	3240.2.q.o.2161.1		2
36.23	even	6		2160.2.a.n.1.1		1
36.31	odd	6		2160.2.a.d.1.1		1
45.4	even	6		5400.2.a.m.1.1		1
45.13	odd	12		5400.2.f.m.649.1		2
45.14	odd	6		5400.2.a.n.1.1		1
45.22	odd	12		5400.2.f.m.649.2		2
45.23	even	12		5400.2.f.p.649.1		2
45.32	even	12		5400.2.f.p.649.2		2
72.5	odd	6		8640.2.a.v.1.1		1
72.13	even	6		8640.2.a.cb.1.1		1
72.59	even	6		8640.2.a.i.1.1		1
72.67	odd	6		8640.2.a.bk.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.a.f.1.1	✓	1	9.4	even	3
1080.2.a.k.1.1	yes	1	9.5	odd	6
2160.2.a.d.1.1		1	36.31	odd	6
2160.2.a.n.1.1		1	36.23	even	6
3240.2.q.c.1081.1		2	3.2	odd	2
3240.2.q.c.2161.1		2	9.2	odd	6
3240.2.q.o.1081.1		2	1.1	even	1	trivial
3240.2.q.o.2161.1		2	9.7	even	3	inner
5400.2.a.m.1.1		1	45.4	even	6
5400.2.a.n.1.1		1	45.14	odd	6
5400.2.f.m.649.1		2	45.13	odd	12
5400.2.f.m.649.2		2	45.22	odd	12
5400.2.f.p.649.1		2	45.23	even	12
5400.2.f.p.649.2		2	45.32	even	12
8640.2.a.i.1.1		1	72.59	even	6
8640.2.a.v.1.1		1	72.5	odd	6
8640.2.a.bk.1.1		1	72.67	odd	6
8640.2.a.cb.1.1		1	72.13	even	6

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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(0.500000 + 0.866025i) q^{11} +(-0.500000 + 0.866025i) q^{13} +1.00000 q^{17} +4.00000 q^{19} +(0.500000 - 0.866025i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-2.50000 - 4.33013i) q^{29} +(-0.500000 + 0.866025i) q^{31} -2.00000 q^{35} +6.00000 q^{37} +(-3.50000 - 6.06218i) q^{43} +(3.50000 + 6.06218i) q^{47} +(1.50000 - 2.59808i) q^{49} +12.0000 q^{53} +1.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(-5.00000 - 8.66025i) q^{61} +(0.500000 + 0.866025i) q^{65} +(2.00000 - 3.46410i) q^{67} -12.0000 q^{71} +6.00000 q^{73} +(1.00000 - 1.73205i) q^{77} +(-7.50000 - 12.9904i) q^{79} +(1.00000 + 1.73205i) q^{83} +(0.500000 - 0.866025i) q^{85} +12.0000 q^{89} +2.00000 q^{91} +(2.00000 - 3.46410i) q^{95} +(-5.00000 - 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} - 2 q^{7} + q^{11} - q^{13} + 2 q^{17} + 8 q^{19} + q^{23} - q^{25} - 5 q^{29} - q^{31} - 4 q^{35} + 12 q^{37} - 7 q^{43} + 7 q^{47} + 3 q^{49} + 24 q^{53} + 2 q^{55} - 4 q^{59} - 10 q^{61}+ \cdots - 10 q^{97}+O(q^{100}) \) 2 * q + q^5 - 2 * q^7 + q^11 - q^13 + 2 * q^17 + 8 * q^19 + q^23 - q^25 - 5 * q^29 - q^31 - 4 * q^35 + 12 * q^37 - 7 * q^43 + 7 * q^47 + 3 * q^49 + 24 * q^53 + 2 * q^55 - 4 * q^59 - 10 * q^61 + q^65 + 4 * q^67 - 24 * q^71 + 12 * q^73 + 2 * q^77 - 15 * q^79 + 2 * q^83 + q^85 + 24 * q^89 + 4 * q^91 + 4 * q^95 - 10 * q^97

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