LMFDB - Embedded newform 648.2.i.a.217.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,2,Mod(217,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("648.217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$5.17430605098$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	648.217
Dual form			648.2.i.a.433.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 3.46410i) q^{5} +(1.50000 - 2.59808i) q^{7} +O(q^{10})$	$q+(-2.00000 - 3.46410i) q^{5} +(1.50000 - 2.59808i) q^{7} +(-2.00000 + 3.46410i) q^{11} +(-0.500000 - 0.866025i) q^{13} -4.00000 q^{17} -1.00000 q^{19} +(-2.00000 - 3.46410i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(2.00000 + 3.46410i) q^{31} -12.0000 q^{35} -9.00000 q^{37} +(4.00000 - 6.92820i) q^{43} +(6.00000 - 10.3923i) q^{47} +(-1.00000 - 1.73205i) q^{49} -8.00000 q^{53} +16.0000 q^{55} +(-2.00000 - 3.46410i) q^{59} +(2.50000 - 4.33013i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(-5.50000 - 9.52628i) q^{67} +8.00000 q^{71} +1.00000 q^{73} +(6.00000 + 10.3923i) q^{77} +(2.50000 - 4.33013i) q^{79} +(-4.00000 + 6.92820i) q^{83} +(8.00000 + 13.8564i) q^{85} +12.0000 q^{89} -3.00000 q^{91} +(2.00000 + 3.46410i) q^{95} +(-2.50000 + 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 3 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 + 3 * q^7	$ 2 q - 4 q^{5} + 3 q^{7} - 4 q^{11} - q^{13} - 8 q^{17} - 2 q^{19} - 4 q^{23} - 11 q^{25} + 4 q^{31} - 24 q^{35} - 18 q^{37} + 8 q^{43} + 12 q^{47} - 2 q^{49} - 16 q^{53} + 32 q^{55} - 4 q^{59} + 5 q^{61} - 4 q^{65} - 11 q^{67} + 16 q^{71} + 2 q^{73} + 12 q^{77} + 5 q^{79} - 8 q^{83} + 16 q^{85} + 24 q^{89} - 6 q^{91} + 4 q^{95} - 5 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + 3 * q^7 - 4 * q^11 - q^13 - 8 * q^17 - 2 * q^19 - 4 * q^23 - 11 * q^25 + 4 * q^31 - 24 * q^35 - 18 * q^37 + 8 * q^43 + 12 * q^47 - 2 * q^49 - 16 * q^53 + 32 * q^55 - 4 * q^59 + 5 * q^61 - 4 * q^65 - 11 * q^67 + 16 * q^71 + 2 * q^73 + 12 * q^77 + 5 * q^79 - 8 * q^83 + 16 * q^85 + 24 * q^89 - 6 * q^91 + 4 * q^95 - 5 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.0599153	−	0.998203i	$-0.519083\pi$
$6$		0			0
$7$	1.50000	−	2.59808i	0.566947	−	0.981981i	−0.429919	−	0.902867i	$-0.641458\pi$
$7$	1.50000	−	2.59808i	0.566947	−	0.981981i	0.996866	−	0.0791130i	$-0.0252088\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	0.389338	+	0.921095i	$0.372704\pi$
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	−0.992361	+	0.123371i	$0.960630\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	0.788320	−	0.615265i	$-0.210951\pi$
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	−0.926995	+	0.375073i	$0.877618\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.00000			−0.970143			−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000			−0.970143			−0.485071	+	0.874475i	$0.661206\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−12.0000			−2.02837
$36$		0			0
$37$	−9.00000			−1.47959			−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000			−1.47959			−0.739795	+	0.672832i	$0.765078\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	4.00000	−	6.92820i	0.609994	−	1.05654i	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	4.00000	−	6.92820i	0.609994	−	1.05654i	0.991241	−	0.132068i	$-0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.00000	−	10.3923i	0.875190	−	1.51587i	0.0186297	−	0.999826i	$-0.494070\pi$
$47$	6.00000	−	10.3923i	0.875190	−	1.51587i	0.856560	−	0.516047i	$-0.172597\pi$
$48$		0			0
$49$	−1.00000	−	1.73205i	−0.142857	−	0.247436i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.00000			−1.09888			−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000			−1.09888			−0.549442	+	0.835532i	$0.685160\pi$
$54$		0			0
$55$	16.0000			2.15744
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	0.705965	−	0.708247i	$-0.250514\pi$
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	−0.966342	+	0.257260i	$0.917180\pi$
$60$		0			0
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	−0.660415	−	0.750901i	$-0.729619\pi$
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	0.980507	+	0.196485i	$0.0629528\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000	+	3.46410i	−0.248069	+	0.429669i
$66$		0			0
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	0.305424	−	0.952217i	$-0.401202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$	1.00000			0.117041			0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000			0.117041			0.0585206	+	0.998286i	$0.481362\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.00000	+	10.3923i	0.683763	+	1.18431i
$78$		0			0
$79$	2.50000	−	4.33013i	0.281272	−	0.487177i	−0.690426	−	0.723403i	$-0.742577\pi$
$79$	2.50000	−	4.33013i	0.281272	−	0.487177i	0.971698	+	0.236225i	$0.0759104\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.00000	+	6.92820i	−0.439057	+	0.760469i	−0.997617	−	0.0689950i	$-0.978021\pi$
$83$	−4.00000	+	6.92820i	−0.439057	+	0.760469i	0.558560	+	0.829464i	$0.311354\pi$
$84$		0			0
$85$	8.00000	+	13.8564i	0.867722	+	1.50294i
$86$		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
$91$	−3.00000			−0.314485
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00000	+	3.46410i	0.205196	+	0.355409i
$96$		0			0
$97$	−2.50000	+	4.33013i	−0.253837	+	0.439658i	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−2.50000	+	4.33013i	−0.253837	+	0.439658i	0.710742	+	0.703452i	$0.248359\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−0.500000	−	0.866025i	−0.0492665	−	0.0853320i	0.840341	−	0.542059i	$-0.182355\pi$
$103$	−0.500000	−	0.866025i	−0.0492665	−	0.0853320i	−0.889607	+	0.456727i	$0.849022\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	−14.0000			−1.34096			−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000			−1.34096			−0.670478	+	0.741929i	$0.733911\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000	−	10.3923i	−0.564433	−	0.977626i	−0.997102	−	0.0760733i	$-0.975762\pi$
$113$	−6.00000	−	10.3923i	−0.564433	−	0.977626i	0.432670	−	0.901553i	$-0.357572\pi$
$114$		0			0
$115$	−8.00000	+	13.8564i	−0.746004	+	1.29212i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.00000	+	10.3923i	−0.550019	+	0.952661i
$120$		0			0
$121$	−2.50000	−	4.33013i	−0.227273	−	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	4.00000			0.354943			0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000			0.354943			0.177471	+	0.984126i	$0.443208\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.00000	+	13.8564i	0.698963	+	1.21064i	0.968826	+	0.247741i	$0.0796882\pi$
$131$	8.00000	+	13.8564i	0.698963	+	1.21064i	−0.269863	+	0.962899i	$0.586978\pi$
$132$		0			0
$133$	−1.50000	+	2.59808i	−0.130066	+	0.225282i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	−4.50000	−	7.79423i	−0.381685	−	0.661098i	0.609618	−	0.792695i	$-0.291323\pi$
$139$	−4.50000	−	7.79423i	−0.381685	−	0.661098i	−0.991303	+	0.131597i	$0.957989\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.00000			0.334497
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.00000	−	6.92820i	−0.327693	−	0.567581i	0.654361	−	0.756182i	$-0.272938\pi$
$149$	−4.00000	−	6.92820i	−0.327693	−	0.567581i	−0.982054	+	0.188602i	$0.939604\pi$
$150$		0			0
$151$	0.500000	−	0.866025i	0.0406894	−	0.0704761i	−0.844963	−	0.534824i	$-0.820378\pi$
$151$	0.500000	−	0.866025i	0.0406894	−	0.0704761i	0.885653	+	0.464348i	$0.153711\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.00000	−	13.8564i	0.642575	−	1.11297i
$156$		0			0
$157$	1.00000	+	1.73205i	0.0798087	+	0.138233i	0.903167	−	0.429289i	$-0.141236\pi$
$157$	1.00000	+	1.73205i	0.0798087	+	0.138233i	−0.823359	+	0.567521i	$0.807902\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$	−15.0000			−1.17489			−0.587445	−	0.809264i	$-0.699866\pi$
$163$	−15.0000			−1.17489			−0.587445	+	0.809264i	$0.699866\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	0.999169	−	0.0407502i	$-0.0129748\pi$
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	−0.534875	+	0.844931i	$0.679641\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	16.5000	+	28.5788i	1.24728	+	2.16036i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$		0			0
$181$	21.0000			1.56092			0.780459	−	0.625207i	$-0.214986\pi$
$181$	21.0000			1.56092			0.780459	+	0.625207i	$0.214986\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	18.0000	+	31.1769i	1.32339	+	2.29217i
$186$		0			0
$187$	8.00000	−	13.8564i	0.585018	−	1.01328i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	−0.997225	−	0.0744412i	$-0.976283\pi$
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	0.563081	+	0.826402i	$0.309616\pi$
$192$		0			0
$193$	−11.5000	−	19.9186i	−0.827788	−	1.43377i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−11.5000	−	19.9186i	−0.827788	−	1.43377i	0.0719816	−	0.997406i	$-0.477068\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	25.0000			1.77220			0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000			1.77220			0.886102	+	0.463491i	$0.153403\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		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
$211$	−6.50000	−	11.2583i	−0.447478	−	0.775055i	0.550743	−	0.834675i	$-0.314345\pi$
$211$	−6.50000	−	11.2583i	−0.447478	−	0.775055i	−0.998221	+	0.0596196i	$0.981011\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−32.0000			−2.18238
$216$		0			0
$217$	12.0000			0.814613
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.00000	+	3.46410i	0.134535	+	0.233021i
$222$		0			0
$223$	−2.00000	+	3.46410i	−0.133930	+	0.231973i	−0.925188	−	0.379509i	$-0.876093\pi$
$223$	−2.00000	+	3.46410i	−0.133930	+	0.231973i	0.791258	+	0.611482i	$0.209426\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.0000	−	20.7846i	0.796468	−	1.37952i	−0.125435	−	0.992102i	$-0.540033\pi$
$227$	12.0000	−	20.7846i	0.796468	−	1.37952i	0.921903	−	0.387421i	$-0.126634\pi$
$228$		0			0
$229$	−5.00000	−	8.66025i	−0.330409	−	0.572286i	0.652183	−	0.758062i	$-0.273853\pi$
$229$	−5.00000	−	8.66025i	−0.330409	−	0.572286i	−0.982592	+	0.185776i	$0.940520\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	16.0000			1.04819			0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000			1.04819			0.524097	+	0.851658i	$0.324403\pi$
$234$		0			0
$235$	−48.0000			−3.13117
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.00000	−	6.92820i	−0.258738	−	0.448148i	0.707166	−	0.707048i	$-0.249973\pi$
$239$	−4.00000	−	6.92820i	−0.258738	−	0.448148i	−0.965904	+	0.258900i	$0.916640\pi$
$240$		0			0
$241$	7.50000	−	12.9904i	0.483117	−	0.836784i	−0.516695	−	0.856170i	$-0.672838\pi$
$241$	7.50000	−	12.9904i	0.483117	−	0.836784i	0.999812	+	0.0193858i	$0.00617107\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.00000	+	6.92820i	−0.255551	+	0.442627i
$246$		0			0
$247$	0.500000	+	0.866025i	0.0318142	+	0.0551039i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−16.0000			−1.00991			−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000			−1.00991			−0.504956	+	0.863145i	$0.668491\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.00000	−	6.92820i	−0.249513	−	0.432169i	0.713878	−	0.700270i	$-0.246937\pi$
$257$	−4.00000	−	6.92820i	−0.249513	−	0.432169i	−0.963391	+	0.268101i	$0.913604\pi$
$258$		0			0
$259$	−13.5000	+	23.3827i	−0.838849	+	1.45293i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.00000	−	6.92820i	0.246651	−	0.427211i	−0.715944	−	0.698158i	$-0.754003\pi$
$263$	4.00000	−	6.92820i	0.246651	−	0.427211i	0.962594	+	0.270947i	$0.0873367\pi$
$264$		0			0
$265$	16.0000	+	27.7128i	0.982872	+	1.70238i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.00000			−0.243884			−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000			−0.243884			−0.121942	+	0.992537i	$0.538912\pi$
$270$		0			0
$271$	−25.0000			−1.51864			−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000			−1.51864			−0.759321	+	0.650716i	$0.774469\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−22.0000	−	38.1051i	−1.32665	−	2.29783i
$276$		0			0
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	−0.976231	−	0.216731i	$-0.930460\pi$
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	0.675810	+	0.737075i	$0.263794\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.0000	+	27.7128i	−0.954480	+	1.65321i	−0.218926	+	0.975741i	$0.570255\pi$
$281$	−16.0000	+	27.7128i	−0.954480	+	1.65321i	−0.735554	+	0.677466i	$0.763078\pi$
$282$		0			0
$283$	8.00000	+	13.8564i	0.475551	+	0.823678i	0.999608	−	0.0280052i	$-0.00891551\pi$
$283$	8.00000	+	13.8564i	0.475551	+	0.823678i	−0.524057	+	0.851683i	$0.675582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.00000	−	10.3923i	−0.350524	−	0.607125i	0.635818	−	0.771839i	$-0.280663\pi$
$293$	−6.00000	−	10.3923i	−0.350524	−	0.607125i	−0.986341	+	0.164714i	$0.947330\pi$
$294$		0			0
$295$	−8.00000	+	13.8564i	−0.465778	+	0.806751i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.00000	+	3.46410i	−0.115663	+	0.200334i
$300$		0			0
$301$	−12.0000	−	20.7846i	−0.691669	−	1.19800i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−20.0000			−1.14520
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.00000	−	10.3923i	−0.340229	−	0.589294i	0.644246	−	0.764818i	$-0.277171\pi$
$311$	−6.00000	−	10.3923i	−0.340229	−	0.589294i	−0.984475	+	0.175525i	$0.943838\pi$
$312$		0			0
$313$	−1.50000	+	2.59808i	−0.0847850	+	0.146852i	−0.905300	−	0.424774i	$-0.860354\pi$
$313$	−1.50000	+	2.59808i	−0.0847850	+	0.146852i	0.820515	+	0.571626i	$0.193687\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.0000	−	20.7846i	0.673987	−	1.16738i	−0.302777	−	0.953062i	$-0.597914\pi$
$317$	12.0000	−	20.7846i	0.673987	−	1.16738i	0.976764	−	0.214318i	$-0.0687530\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.00000			0.222566
$324$		0			0
$325$	11.0000			0.610170
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−18.0000	−	31.1769i	−0.992372	−	1.71884i
$330$		0			0
$331$	−8.50000	+	14.7224i	−0.467202	+	0.809218i	−0.999298	−	0.0374662i	$-0.988071\pi$
$331$	−8.50000	+	14.7224i	−0.467202	+	0.809218i	0.532096	+	0.846684i	$0.321405\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−22.0000	+	38.1051i	−1.20199	+	2.08190i
$336$		0			0
$337$	1.50000	+	2.59808i	0.0817102	+	0.141526i	0.903985	−	0.427565i	$-0.140628\pi$
$337$	1.50000	+	2.59808i	0.0817102	+	0.141526i	−0.822274	+	0.569091i	$0.807295\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−16.0000			−0.866449
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	0.984487	+	0.175457i	$0.0561403\pi$
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	−0.340293	+	0.940319i	$0.610526\pi$
$348$		0			0
$349$	16.5000	−	28.5788i	0.883225	−	1.52979i	0.0354898	−	0.999370i	$-0.488701\pi$
$349$	16.5000	−	28.5788i	0.883225	−	1.52979i	0.847735	−	0.530420i	$-0.177966\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.00000	+	6.92820i	−0.212899	+	0.368751i	−0.952620	−	0.304162i	$-0.901624\pi$
$353$	−4.00000	+	6.92820i	−0.212899	+	0.368751i	0.739722	+	0.672913i	$0.234957\pi$
$354$		0			0
$355$	−16.0000	−	27.7128i	−0.849192	−	1.47084i
$356$		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
$361$	−18.0000			−0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.00000	−	3.46410i	−0.104685	−	0.181319i
$366$		0			0
$367$	11.5000	−	19.9186i	0.600295	−	1.03974i	−0.392481	−	0.919760i	$-0.628383\pi$
$367$	11.5000	−	19.9186i	0.600295	−	1.03974i	0.992776	−	0.119982i	$-0.0382835\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.0000	+	20.7846i	−0.623009	+	1.07908i
$372$		0			0
$373$	0.500000	+	0.866025i	0.0258890	+	0.0448411i	0.878680	−	0.477412i	$-0.158425\pi$
$373$	0.500000	+	0.866025i	0.0258890	+	0.0448411i	−0.852791	+	0.522253i	$0.825092\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	11.0000			0.565032			0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000			0.565032			0.282516	+	0.959263i	$0.408831\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$	24.0000	−	41.5692i	1.22315	−	2.11856i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.00000	+	3.46410i	−0.101404	+	0.175637i	−0.912263	−	0.409604i	$-0.865667\pi$
$389$	−2.00000	+	3.46410i	−0.101404	+	0.175637i	0.810859	+	0.585241i	$0.199000\pi$
$390$		0			0
$391$	8.00000	+	13.8564i	0.404577	+	0.700749i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−20.0000			−1.00631
$396$		0			0
$397$	14.0000			0.702640			0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000			0.702640			0.351320	+	0.936255i	$0.385733\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	−0.992932	−	0.118686i	$-0.962132\pi$
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	0.393680	−	0.919247i	$-0.371202\pi$
$402$		0			0
$403$	2.00000	−	3.46410i	0.0996271	−	0.172559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.0000	−	31.1769i	0.892227	−	1.54538i
$408$		0			0
$409$	19.5000	+	33.7750i	0.964213	+	1.67007i	0.711715	+	0.702468i	$0.247919\pi$
$409$	19.5000	+	33.7750i	0.964213	+	1.67007i	0.252498	+	0.967597i	$0.418748\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12.0000			−0.590481
$414$		0			0
$415$	32.0000			1.57082
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	0.974546	−	0.224189i	$-0.0719734\pi$
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	−0.681426	+	0.731887i	$0.738640\pi$
$420$		0			0
$421$	−8.50000	+	14.7224i	−0.414265	+	0.717527i	−0.995351	−	0.0963145i	$-0.969295\pi$
$421$	−8.50000	+	14.7224i	−0.414265	+	0.717527i	0.581086	+	0.813842i	$0.302628\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.0000	−	38.1051i	1.06716	−	1.84837i
$426$		0			0
$427$	−7.50000	−	12.9904i	−0.362950	−	0.628649i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−28.0000			−1.34871			−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000			−1.34871			−0.674356	+	0.738406i	$0.735579\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.00000	+	3.46410i	0.0956730	+	0.165710i
$438$		0			0
$439$	−18.0000	+	31.1769i	−0.859093	+	1.48799i	0.0137020	+	0.999906i	$0.495638\pi$
$439$	−18.0000	+	31.1769i	−0.859093	+	1.48799i	−0.872795	+	0.488087i	$0.837695\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.00000	+	6.92820i	−0.190046	+	0.329169i	−0.945265	−	0.326303i	$-0.894197\pi$
$443$	−4.00000	+	6.92820i	−0.190046	+	0.329169i	0.755219	+	0.655472i	$0.227530\pi$
$444$		0			0
$445$	−24.0000	−	41.5692i	−1.13771	−	1.97057i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	4.00000			0.188772			0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000			0.188772			0.0943858	+	0.995536i	$0.469911\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.00000	+	10.3923i	0.281284	+	0.487199i
$456$		0			0
$457$	−19.0000	+	32.9090i	−0.888783	+	1.53942i	−0.0474665	+	0.998873i	$0.515115\pi$
$457$	−19.0000	+	32.9090i	−0.888783	+	1.53942i	−0.841316	+	0.540544i	$0.818219\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.0000	+	24.2487i	−0.652045	+	1.12938i	0.330581	+	0.943778i	$0.392755\pi$
$461$	−14.0000	+	24.2487i	−0.652045	+	1.12938i	−0.982626	+	0.185597i	$0.940578\pi$
$462$		0			0
$463$	9.50000	+	16.4545i	0.441502	+	0.764705i	0.997801	−	0.0662777i	$-0.0211123\pi$
$463$	9.50000	+	16.4545i	0.441502	+	0.764705i	−0.556299	+	0.830982i	$0.687779\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	−33.0000			−1.52380
$470$		0			0
$471$		0			0
$472$		0			0
$473$	16.0000	+	27.7128i	0.735681	+	1.27424i
$474$		0			0
$475$	5.50000	−	9.52628i	0.252357	−	0.437096i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	20.0000	−	34.6410i	0.913823	−	1.58279i	0.105208	−	0.994450i	$-0.466449\pi$
$479$	20.0000	−	34.6410i	0.913823	−	1.58279i	0.808615	−	0.588338i	$-0.200218\pi$
$480$		0			0
$481$	4.50000	+	7.79423i	0.205182	+	0.355386i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	20.0000			0.908153
$486$		0			0
$487$	−11.0000			−0.498458			−0.249229	−	0.968445i	$-0.580177\pi$
$487$	−11.0000			−0.498458			−0.249229	+	0.968445i	$0.580177\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.00000	−	10.3923i	−0.270776	−	0.468998i	0.698285	−	0.715820i	$-0.253947\pi$
$491$	−6.00000	−	10.3923i	−0.270776	−	0.468998i	−0.969061	+	0.246822i	$0.920614\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.0000	−	20.7846i	0.538274	−	0.932317i
$498$		0			0
$499$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$499$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	36.0000			1.60516			0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000			1.60516			0.802580	+	0.596544i	$0.203460\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.00000	+	10.3923i	0.265945	+	0.460631i	0.967811	−	0.251679i	$-0.0809826\pi$
$509$	6.00000	+	10.3923i	0.265945	+	0.460631i	−0.701866	+	0.712309i	$0.747649\pi$
$510$		0			0
$511$	1.50000	−	2.59808i	0.0663561	−	0.114932i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.00000	+	3.46410i	−0.0881305	+	0.152647i
$516$		0			0
$517$	24.0000	+	41.5692i	1.05552	+	1.82821i
$518$		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
$523$	−29.0000			−1.26808			−0.634041	−	0.773300i	$-0.718605\pi$
$523$	−29.0000			−1.26808			−0.634041	+	0.773300i	$0.718605\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.00000	−	13.8564i	−0.348485	−	0.603595i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−24.0000	−	41.5692i	−1.03761	−	1.79719i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	8.00000			0.344584
$540$		0			0
$541$	9.00000			0.386940			0.193470	−	0.981106i	$-0.438026\pi$
$541$	9.00000			0.386940			0.193470	+	0.981106i	$0.438026\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	28.0000	+	48.4974i	1.19939	+	2.07740i
$546$		0			0
$547$	3.50000	−	6.06218i	0.149649	−	0.259200i	−0.781449	−	0.623970i	$-0.785519\pi$
$547$	3.50000	−	6.06218i	0.149649	−	0.259200i	0.931098	+	0.364770i	$0.118852\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−7.50000	−	12.9904i	−0.318932	−	0.552407i
$554$		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
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.0000	+	27.7128i	0.674320	+	1.16796i	0.976667	+	0.214758i	$0.0688963\pi$
$563$	16.0000	+	27.7128i	0.674320	+	1.16796i	−0.302348	+	0.953198i	$0.597770\pi$
$564$		0			0
$565$	−24.0000	+	41.5692i	−1.00969	+	1.74883i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14.0000	+	24.2487i	−0.586911	+	1.01656i	0.407724	+	0.913105i	$0.366323\pi$
$569$	−14.0000	+	24.2487i	−0.586911	+	1.01656i	−0.994634	+	0.103454i	$0.967011\pi$
$570$		0			0
$571$	16.5000	+	28.5788i	0.690504	+	1.19599i	0.971673	+	0.236329i	$0.0759443\pi$
$571$	16.5000	+	28.5788i	0.690504	+	1.19599i	−0.281170	+	0.959658i	$0.590722\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	44.0000			1.83493
$576$		0			0
$577$	−13.0000			−0.541197			−0.270599	−	0.962692i	$-0.587222\pi$
$577$	−13.0000			−0.541197			−0.270599	+	0.962692i	$0.587222\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.0000	+	20.7846i	0.497844	+	0.862291i
$582$		0			0
$583$	16.0000	−	27.7128i	0.662652	−	1.14775i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.0000	+	24.2487i	−0.577842	+	1.00085i	0.417885	+	0.908500i	$0.362772\pi$
$587$	−14.0000	+	24.2487i	−0.577842	+	1.00085i	−0.995726	+	0.0923513i	$0.970562\pi$
$588$		0			0
$589$	−2.00000	−	3.46410i	−0.0824086	−	0.142736i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	40.0000			1.64260			0.821302	−	0.570494i	$-0.193248\pi$
$593$	40.0000			1.64260			0.821302	+	0.570494i	$0.193248\pi$
$594$		0			0
$595$	48.0000			1.96781
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−5.00000	+	8.66025i	−0.203954	+	0.353259i	−0.949799	−	0.312861i	$-0.898713\pi$
$601$	−5.00000	+	8.66025i	−0.203954	+	0.353259i	0.745845	+	0.666120i	$0.232046\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.0000	+	17.3205i	−0.406558	+	0.704179i
$606$		0			0
$607$	−2.50000	−	4.33013i	−0.101472	−	0.175754i	0.810819	−	0.585296i	$-0.199022\pi$
$607$	−2.50000	−	4.33013i	−0.101472	−	0.175754i	−0.912291	+	0.409542i	$0.865689\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000			−0.485468
$612$		0			0
$613$	35.0000			1.41364			0.706818	−	0.707395i	$-0.250130\pi$
$613$	35.0000			1.41364			0.706818	+	0.707395i	$0.250130\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.0000	−	31.1769i	−0.724653	−	1.25514i	−0.959117	−	0.283011i	$-0.908667\pi$
$617$	−18.0000	−	31.1769i	−0.724653	−	1.25514i	0.234464	−	0.972125i	$-0.424666\pi$
$618$		0			0
$619$	8.50000	−	14.7224i	0.341644	−	0.591744i	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	8.50000	−	14.7224i	0.341644	−	0.591744i	0.984738	+	0.174042i	$0.0556830\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.0000	−	31.1769i	0.721155	−	1.24908i
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	36.0000			1.43541
$630$		0			0
$631$	7.00000			0.278666			0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000			0.278666			0.139333	+	0.990246i	$0.455504\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.00000	−	13.8564i	−0.317470	−	0.549875i
$636$		0			0
$637$	−1.00000	+	1.73205i	−0.0396214	+	0.0686264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20.0000	+	34.6410i	−0.789953	+	1.36824i	0.136043	+	0.990703i	$0.456562\pi$
$641$	−20.0000	+	34.6410i	−0.789953	+	1.36824i	−0.925995	+	0.377535i	$0.876772\pi$
$642$		0			0
$643$	12.0000	+	20.7846i	0.473234	+	0.819665i	0.999531	−	0.0306359i	$-0.00975325\pi$
$643$	12.0000	+	20.7846i	0.473234	+	0.819665i	−0.526297	+	0.850301i	$0.676420\pi$
$644$		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			0.628055
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.0000	−	20.7846i	−0.469596	−	0.813365i	0.529799	−	0.848123i	$-0.322267\pi$
$653$	−12.0000	−	20.7846i	−0.469596	−	0.813365i	−0.999396	+	0.0347583i	$0.988934\pi$
$654$		0			0
$655$	32.0000	−	55.4256i	1.25034	−	2.16566i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.0000	−	20.7846i	0.467454	−	0.809653i	−0.531855	−	0.846836i	$-0.678505\pi$
$659$	12.0000	−	20.7846i	0.467454	−	0.809653i	0.999309	+	0.0371821i	$0.0118382\pi$
$660$		0			0
$661$	6.50000	+	11.2583i	0.252821	+	0.437898i	0.964301	−	0.264807i	$-0.0853084\pi$
$661$	6.50000	+	11.2583i	0.252821	+	0.437898i	−0.711481	+	0.702706i	$0.751975\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.0000			0.465340
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.0000	+	17.3205i	0.386046	+	0.668651i
$672$		0			0
$673$	−9.50000	+	16.4545i	−0.366198	+	0.634274i	−0.988968	−	0.148132i	$-0.952674\pi$
$673$	−9.50000	+	16.4545i	−0.366198	+	0.634274i	0.622770	+	0.782405i	$0.286007\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	22.0000	−	38.1051i	0.845529	−	1.46450i	−0.0396326	−	0.999214i	$-0.512619\pi$
$677$	22.0000	−	38.1051i	0.845529	−	1.46450i	0.885161	−	0.465284i	$-0.154048\pi$
$678$		0			0
$679$	7.50000	+	12.9904i	0.287824	+	0.498525i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	16.0000			0.612223			0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000			0.612223			0.306111	+	0.951996i	$0.400972\pi$
$684$		0			0
$685$	−48.0000			−1.83399
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.00000	+	6.92820i	0.152388	+	0.263944i
$690$		0			0
$691$	20.0000	−	34.6410i	0.760836	−	1.31781i	−0.181584	−	0.983375i	$-0.558123\pi$
$691$	20.0000	−	34.6410i	0.760836	−	1.31781i	0.942420	−	0.334431i	$-0.108544\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−18.0000	+	31.1769i	−0.682779	+	1.18261i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−36.0000			−1.35970			−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000			−1.35970			−0.679851	+	0.733351i	$0.737955\pi$
$702$		0			0
$703$	9.00000			0.339441
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	3.50000	−	6.06218i	0.131445	−	0.227670i	−0.792789	−	0.609497i	$-0.791372\pi$
$709$	3.50000	−	6.06218i	0.131445	−	0.227670i	0.924234	+	0.381827i	$0.124705\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8.00000	−	13.8564i	0.299602	−	0.518927i
$714$		0			0
$715$	−8.00000	−	13.8564i	−0.299183	−	0.518200i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.00000			−0.298350			−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000			−0.298350			−0.149175	+	0.988811i	$0.547662\pi$
$720$		0			0
$721$	−3.00000			−0.111726
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−6.00000	+	10.3923i	−0.222528	+	0.385429i	−0.955575	−	0.294749i	$-0.904764\pi$
$727$	−6.00000	+	10.3923i	−0.222528	+	0.385429i	0.733047	+	0.680178i	$0.238097\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−16.0000	+	27.7128i	−0.591781	+	1.02500i
$732$		0			0
$733$	−1.00000	−	1.73205i	−0.0369358	−	0.0639748i	0.846967	−	0.531646i	$-0.178426\pi$
$733$	−1.00000	−	1.73205i	−0.0369358	−	0.0639748i	−0.883902	+	0.467671i	$0.845093\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	44.0000			1.62076
$738$		0			0
$739$	32.0000			1.17714			0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000			1.17714			0.588570	+	0.808447i	$0.299691\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.00000	−	3.46410i	−0.0733729	−	0.127086i	0.827005	−	0.562195i	$-0.190043\pi$
$743$	−2.00000	−	3.46410i	−0.0733729	−	0.127086i	−0.900378	+	0.435110i	$0.856710\pi$
$744$		0			0
$745$	−16.0000	+	27.7128i	−0.586195	+	1.01532i
$746$		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	0.837359	−	0.546653i	$-0.184098\pi$
$751$	−1.50000	−	2.59808i	−0.0547358	−	0.0948051i	−0.892095	+	0.451848i	$0.850765\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4.00000			−0.145575
$756$		0			0
$757$	−31.0000			−1.12671			−0.563357	−	0.826214i	$-0.690490\pi$
$757$	−31.0000			−1.12671			−0.563357	+	0.826214i	$0.690490\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.00000	+	3.46410i	0.0724999	+	0.125574i	0.899996	−	0.435897i	$-0.143569\pi$
$761$	2.00000	+	3.46410i	0.0724999	+	0.125574i	−0.827496	+	0.561471i	$0.810236\pi$
$762$		0			0
$763$	−21.0000	+	36.3731i	−0.760251	+	1.31679i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.00000	+	3.46410i	−0.0722158	+	0.125081i
$768$		0			0
$769$	−23.5000	−	40.7032i	−0.847432	−	1.46779i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−23.5000	−	40.7032i	−0.847432	−	1.46779i	0.0360609	−	0.999350i	$-0.488519\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−24.0000			−0.863220			−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000			−0.863220			−0.431610	+	0.902060i	$0.642054\pi$
$774$		0			0
$775$	−44.0000			−1.58053
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−16.0000	+	27.7128i	−0.572525	+	0.991642i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.00000	−	6.92820i	0.142766	−	0.247278i
$786$		0			0
$787$	−3.50000	−	6.06218i	−0.124762	−	0.216093i	0.796878	−	0.604140i	$-0.206483\pi$
$787$	−3.50000	−	6.06218i	−0.124762	−	0.216093i	−0.921640	+	0.388047i	$0.873150\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−36.0000			−1.28001
$792$		0			0
$793$	−5.00000			−0.177555
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8.00000	−	13.8564i	−0.283375	−	0.490819i	0.688839	−	0.724914i	$-0.258121\pi$
$797$	−8.00000	−	13.8564i	−0.283375	−	0.490819i	−0.972214	+	0.234095i	$0.924787\pi$
$798$		0			0
$799$	−24.0000	+	41.5692i	−0.849059	+	1.47061i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.00000	+	3.46410i	−0.0705785	+	0.122245i
$804$		0			0
$805$	24.0000	+	41.5692i	0.845889	+	1.46512i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−40.0000			−1.40633			−0.703163	−	0.711029i	$-0.748229\pi$
$809$	−40.0000			−1.40633			−0.703163	+	0.711029i	$0.748229\pi$
$810$		0			0
$811$	40.0000			1.40459			0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000			1.40459			0.702295	+	0.711886i	$0.252159\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	30.0000	+	51.9615i	1.05085	+	1.82013i
$816$		0			0
$817$	−4.00000	+	6.92820i	−0.139942	+	0.242387i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.00000	−	10.3923i	0.209401	−	0.362694i	−0.742125	−	0.670262i	$-0.766182\pi$
$821$	6.00000	−	10.3923i	0.209401	−	0.362694i	0.951526	+	0.307568i	$0.0995151\pi$
$822$		0			0
$823$	12.5000	+	21.6506i	0.435723	+	0.754694i	0.997354	−	0.0726937i	$-0.0231595\pi$
$823$	12.5000	+	21.6506i	0.435723	+	0.754694i	−0.561632	+	0.827387i	$0.689826\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.0000			0.695468			0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000			0.695468			0.347734	+	0.937593i	$0.386951\pi$
$828$		0			0
$829$	−35.0000			−1.21560			−0.607800	−	0.794090i	$-0.707948\pi$
$829$	−35.0000			−1.21560			−0.607800	+	0.794090i	$0.707948\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.00000	+	6.92820i	0.138592	+	0.240048i
$834$		0			0
$835$	24.0000	−	41.5692i	0.830554	−	1.43856i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.0000	+	20.7846i	−0.414286	+	0.717564i	−0.995353	−	0.0962912i	$-0.969302\pi$
$839$	−12.0000	+	20.7846i	−0.414286	+	0.717564i	0.581067	+	0.813856i	$0.302635\pi$
$840$		0			0
$841$	14.5000	+	25.1147i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−48.0000			−1.65125
$846$		0			0
$847$	−15.0000			−0.515406
$848$		0			0
$849$		0			0
$850$		0			0
$851$	18.0000	+	31.1769i	0.617032	+	1.06873i
$852$		0			0
$853$	0.500000	−	0.866025i	0.0171197	−	0.0296521i	−0.857339	−	0.514753i	$-0.827884\pi$
$853$	0.500000	−	0.866025i	0.0171197	−	0.0296521i	0.874458	+	0.485101i	$0.161217\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12.0000	+	20.7846i	−0.409912	+	0.709989i	−0.994880	−	0.101068i	$-0.967774\pi$
$857$	−12.0000	+	20.7846i	−0.409912	+	0.709989i	0.584967	+	0.811057i	$0.301107\pi$
$858$		0			0
$859$	13.5000	+	23.3827i	0.460614	+	0.797807i	0.998992	−	0.0448968i	$-0.0142959\pi$
$859$	13.5000	+	23.3827i	0.460614	+	0.797807i	−0.538378	+	0.842704i	$0.680963\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−44.0000			−1.49778			−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000			−1.49778			−0.748889	+	0.662696i	$0.769412\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.0000	+	17.3205i	0.339227	+	0.587558i
$870$		0			0
$871$	−5.50000	+	9.52628i	−0.186360	+	0.322786i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	36.0000	−	62.3538i	1.21702	−	2.10794i
$876$		0			0
$877$	16.5000	+	28.5788i	0.557165	+	0.965039i	0.997732	+	0.0673186i	$0.0214444\pi$
$877$	16.5000	+	28.5788i	0.557165	+	0.965039i	−0.440566	+	0.897720i	$0.645222\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−28.0000			−0.943344			−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000			−0.943344			−0.471672	+	0.881774i	$0.656349\pi$
$882$		0			0
$883$	−7.00000			−0.235569			−0.117784	−	0.993039i	$-0.537579\pi$
$883$	−7.00000			−0.235569			−0.117784	+	0.993039i	$0.537579\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.0000	−	48.4974i	−0.940148	−	1.62838i	−0.765186	−	0.643809i	$-0.777353\pi$
$887$	−28.0000	−	48.4974i	−0.940148	−	1.62838i	−0.174962	−	0.984575i	$-0.555980\pi$
$888$		0			0
$889$	6.00000	−	10.3923i	0.201234	−	0.348547i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.00000	+	10.3923i	−0.200782	+	0.347765i
$894$		0			0
$895$	−48.0000	−	83.1384i	−1.60446	−	2.77901i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	32.0000			1.06607
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−42.0000	−	72.7461i	−1.39613	−	2.41816i
$906$		0			0
$907$	2.50000	−	4.33013i	0.0830111	−	0.143780i	−0.821531	−	0.570164i	$-0.806880\pi$
$907$	2.50000	−	4.33013i	0.0830111	−	0.143780i	0.904542	+	0.426385i	$0.140213\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8.00000	+	13.8564i	−0.265052	+	0.459083i	−0.967577	−	0.252575i	$-0.918722\pi$
$911$	−8.00000	+	13.8564i	−0.265052	+	0.459083i	0.702525	+	0.711659i	$0.252056\pi$
$912$		0			0
$913$	−16.0000	−	27.7128i	−0.529523	−	0.917160i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	48.0000			1.58510
$918$		0			0
$919$	−20.0000			−0.659739			−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000			−0.659739			−0.329870	+	0.944027i	$0.607005\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.00000	−	6.92820i	−0.131662	−	0.228045i
$924$		0			0
$925$	49.5000	−	85.7365i	1.62755	−	2.81900i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20.0000	−	34.6410i	0.656179	−	1.13653i	−0.325418	−	0.945570i	$-0.605505\pi$
$929$	20.0000	−	34.6410i	0.656179	−	1.13653i	0.981597	−	0.190965i	$-0.0611616\pi$
$930$		0			0
$931$	1.00000	+	1.73205i	0.0327737	+	0.0567657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−64.0000			−2.09302
$936$		0			0
$937$	51.0000			1.66610			0.833049	−	0.553200i	$-0.186593\pi$
$937$	51.0000			1.66610			0.833049	+	0.553200i	$0.186593\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.00000	−	10.3923i	−0.195594	−	0.338779i	0.751501	−	0.659732i	$-0.229330\pi$
$941$	−6.00000	−	10.3923i	−0.195594	−	0.338779i	−0.947095	+	0.320953i	$0.895997\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.0000	−	31.1769i	0.584921	−	1.01311i	−0.409964	−	0.912102i	$-0.634459\pi$
$947$	18.0000	−	31.1769i	0.584921	−	1.01311i	0.994885	−	0.101012i	$-0.0322080\pi$
$948$		0			0
$949$	−0.500000	−	0.866025i	−0.0162307	−	0.0281124i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−36.0000			−1.16615			−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000			−1.16615			−0.583077	+	0.812417i	$0.698151\pi$
$954$		0			0
$955$	48.0000			1.55324
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.0000	−	31.1769i	−0.581250	−	1.00676i
$960$		0			0
$961$	7.50000	−	12.9904i	0.241935	−	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−46.0000	+	79.6743i	−1.48079	+	2.56481i
$966$		0			0
$967$	−29.5000	−	51.0955i	−0.948656	−	1.64312i	−0.748260	−	0.663405i	$-0.769110\pi$
$967$	−29.5000	−	51.0955i	−0.948656	−	1.64312i	−0.200396	−	0.979715i	$-0.564223\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$		0			0
$973$	−27.0000			−0.865580
$974$		0			0
$975$		0			0
$976$		0			0
$977$	12.0000	+	20.7846i	0.383914	+	0.664959i	0.991618	−	0.129205i	$-0.0412426\pi$
$977$	12.0000	+	20.7846i	0.383914	+	0.664959i	−0.607704	+	0.794164i	$0.707909\pi$
$978$		0			0
$979$	−24.0000	+	41.5692i	−0.767043	+	1.32856i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	−0.945705	−	0.325027i	$-0.894626\pi$
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	0.754334	+	0.656490i	$0.227960\pi$
$984$		0			0
$985$	24.0000	+	41.5692i	0.764704	+	1.32451i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32.0000			−1.01754
$990$		0			0
$991$	25.0000			0.794151			0.397076	−	0.917786i	$-0.370025\pi$
$991$	25.0000			0.794151			0.397076	+	0.917786i	$0.370025\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−50.0000	−	86.6025i	−1.58511	−	2.74549i
$996$		0			0
$997$	13.0000	−	22.5167i	0.411714	−	0.713110i	−0.583363	−	0.812211i	$-0.698264\pi$
$997$	13.0000	−	22.5167i	0.411714	−	0.713110i	0.995077	+	0.0991016i	$0.0315969\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.i.a.217.1		2
3.2	odd	2		648.2.i.h.217.1		2
4.3	odd	2		1296.2.i.a.865.1		2
9.2	odd	6		216.2.a.a.1.1	✓	1
9.4	even	3	inner	648.2.i.a.433.1		2
9.5	odd	6		648.2.i.h.433.1		2
9.7	even	3		216.2.a.d.1.1	yes	1
12.11	even	2		1296.2.i.q.865.1		2
36.7	odd	6		432.2.a.h.1.1		1
36.11	even	6		432.2.a.a.1.1		1
36.23	even	6		1296.2.i.q.433.1		2
36.31	odd	6		1296.2.i.a.433.1		2
45.2	even	12		5400.2.f.e.649.1		2
45.7	odd	12		5400.2.f.v.649.1		2
45.29	odd	6		5400.2.a.bn.1.1		1
45.34	even	6		5400.2.a.bp.1.1		1
45.38	even	12		5400.2.f.e.649.2		2
45.43	odd	12		5400.2.f.v.649.2		2
72.11	even	6		1728.2.a.bb.1.1		1
72.29	odd	6		1728.2.a.ba.1.1		1
72.43	odd	6		1728.2.a.b.1.1		1
72.61	even	6		1728.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.a.a.1.1	✓	1	9.2	odd	6
216.2.a.d.1.1	yes	1	9.7	even	3
432.2.a.a.1.1		1	36.11	even	6
432.2.a.h.1.1		1	36.7	odd	6
648.2.i.a.217.1		2	1.1	even	1	trivial
648.2.i.a.433.1		2	9.4	even	3	inner
648.2.i.h.217.1		2	3.2	odd	2
648.2.i.h.433.1		2	9.5	odd	6
1296.2.i.a.433.1		2	36.31	odd	6
1296.2.i.a.865.1		2	4.3	odd	2
1296.2.i.q.433.1		2	36.23	even	6
1296.2.i.q.865.1		2	12.11	even	2
1728.2.a.a.1.1		1	72.61	even	6
1728.2.a.b.1.1		1	72.43	odd	6
1728.2.a.ba.1.1		1	72.29	odd	6
1728.2.a.bb.1.1		1	72.11	even	6
5400.2.a.bn.1.1		1	45.29	odd	6
5400.2.a.bp.1.1		1	45.34	even	6
5400.2.f.e.649.1		2	45.2	even	12
5400.2.f.e.649.2		2	45.38	even	12
5400.2.f.v.649.1		2	45.7	odd	12
5400.2.f.v.649.2		2	45.43	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{5} +(1.50000 - 2.59808i) q^{7} +O(q^{10})\)	\(q+(-2.00000 - 3.46410i) q^{5} +(1.50000 - 2.59808i) q^{7} +(-2.00000 + 3.46410i) q^{11} +(-0.500000 - 0.866025i) q^{13} -4.00000 q^{17} -1.00000 q^{19} +(-2.00000 - 3.46410i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(2.00000 + 3.46410i) q^{31} -12.0000 q^{35} -9.00000 q^{37} +(4.00000 - 6.92820i) q^{43} +(6.00000 - 10.3923i) q^{47} +(-1.00000 - 1.73205i) q^{49} -8.00000 q^{53} +16.0000 q^{55} +(-2.00000 - 3.46410i) q^{59} +(2.50000 - 4.33013i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(-5.50000 - 9.52628i) q^{67} +8.00000 q^{71} +1.00000 q^{73} +(6.00000 + 10.3923i) q^{77} +(2.50000 - 4.33013i) q^{79} +(-4.00000 + 6.92820i) q^{83} +(8.00000 + 13.8564i) q^{85} +12.0000 q^{89} -3.00000 q^{91} +(2.00000 + 3.46410i) q^{95} +(-2.50000 + 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 3 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 + 3 * q^7	\( 2 q - 4 q^{5} + 3 q^{7} - 4 q^{11} - q^{13} - 8 q^{17} - 2 q^{19} - 4 q^{23} - 11 q^{25} + 4 q^{31} - 24 q^{35} - 18 q^{37} + 8 q^{43} + 12 q^{47} - 2 q^{49} - 16 q^{53} + 32 q^{55} - 4 q^{59} + 5 q^{61} - 4 q^{65} - 11 q^{67} + 16 q^{71} + 2 q^{73} + 12 q^{77} + 5 q^{79} - 8 q^{83} + 16 q^{85} + 24 q^{89} - 6 q^{91} + 4 q^{95} - 5 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + 3 * q^7 - 4 * q^11 - q^13 - 8 * q^17 - 2 * q^19 - 4 * q^23 - 11 * q^25 + 4 * q^31 - 24 * q^35 - 18 * q^37 + 8 * q^43 + 12 * q^47 - 2 * q^49 - 16 * q^53 + 32 * q^55 - 4 * q^59 + 5 * q^61 - 4 * q^65 - 11 * q^67 + 16 * q^71 + 2 * q^73 + 12 * q^77 + 5 * q^79 - 8 * q^83 + 16 * q^85 + 24 * q^89 - 6 * q^91 + 4 * q^95 - 5 * q^97

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