LMFDB - Embedded newform 1296.2.i.a.433.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,2,Mod(433,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("1296.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1296.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:	$10.3486121020$
Analytic rank:	$1$
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			433.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1296.433
Dual form			1296.2.i.a.865.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 3.46410i) q^{5} +(-1.50000 - 2.59808i) q^{7} +O(q^{10})$	$q+(-2.00000 + 3.46410i) q^{5} +(-1.50000 - 2.59808i) q^{7} +(2.00000 + 3.46410i) q^{11} +(-0.500000 + 0.866025i) q^{13} -4.00000 q^{17} +1.00000 q^{19} +(2.00000 - 3.46410i) q^{23} +(-5.50000 - 9.52628i) q^{25} +(-2.00000 + 3.46410i) q^{31} +12.0000 q^{35} -9.00000 q^{37} +(-4.00000 - 6.92820i) q^{43} +(-6.00000 - 10.3923i) q^{47} +(-1.00000 + 1.73205i) q^{49} -8.00000 q^{53} -16.0000 q^{55} +(2.00000 - 3.46410i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(5.50000 - 9.52628i) q^{67} -8.00000 q^{71} +1.00000 q^{73} +(6.00000 - 10.3923i) q^{77} +(-2.50000 - 4.33013i) q^{79} +(4.00000 + 6.92820i) q^{83} +(8.00000 - 13.8564i) q^{85} +12.0000 q^{89} +3.00000 q^{91} +(-2.00000 + 3.46410i) q^{95} +(-2.50000 - 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.0599153	+	0.998203i	$0.519083\pi$
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.834512	+	0.550990i	$0.814250\pi$
$6$		0			0
$7$	−1.50000	−	2.59808i	−0.566947	−	0.981981i	−0.996866	−	0.0791130i	$-0.974791\pi$
$7$	−1.50000	−	2.59808i	−0.566947	−	0.981981i	0.429919	−	0.902867i	$-0.358542\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	0.992361	+	0.123371i	$0.0393705\pi$
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	−0.389338	+	0.921095i	$0.627296\pi$
$12$		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
$17$	−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	$-0.463407\pi$
$19$	1.00000			0.229416			0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		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	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$		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	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	0.381246	−	0.924473i	$-0.375495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.856560	−	0.516047i	$-0.827403\pi$
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.0186297	−	0.999826i	$-0.505930\pi$
$48$		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	$-0.749486\pi$
$59$	2.00000	−	3.46410i	0.260378	−	0.450988i	0.966342	+	0.257260i	$0.0828195\pi$
$60$		0			0
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	0.980507	−	0.196485i	$-0.0629528\pi$
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	−0.660415	+	0.750901i	$0.729619\pi$
$62$		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	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	5.50000	−	9.52628i	0.671932	−	1.16382i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		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	$-0.257423\pi$
$79$	−2.50000	−	4.33013i	−0.281272	−	0.487177i	−0.971698	+	0.236225i	$0.924090\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.00000	+	6.92820i	0.439057	+	0.760469i	0.997617	−	0.0689950i	$-0.0219793\pi$
$83$	4.00000	+	6.92820i	0.439057	+	0.760469i	−0.558560	+	0.829464i	$0.688646\pi$
$84$		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	0.710742	−	0.703452i	$-0.248359\pi$
$97$	−2.50000	−	4.33013i	−0.253837	−	0.439658i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	0.500000	−	0.866025i	0.0492665	−	0.0853320i	−0.840341	−	0.542059i	$-0.817645\pi$
$103$	0.500000	−	0.866025i	0.0492665	−	0.0853320i	0.889607	+	0.456727i	$0.150978\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		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	0.432670	+	0.901553i	$0.357572\pi$
$113$	−6.00000	+	10.3923i	−0.564433	+	0.977626i	−0.997102	+	0.0760733i	$0.975762\pi$
$114$		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	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.00000	+	13.8564i	−0.698963	+	1.21064i	0.269863	+	0.962899i	$0.413022\pi$
$131$	−8.00000	+	13.8564i	−0.698963	+	1.21064i	−0.968826	+	0.247741i	$0.920312\pi$
$132$		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	0.999893	+	0.0146279i	$0.00465636\pi$
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	−0.487278	+	0.873247i	$0.662010\pi$
$138$		0			0
$139$	4.50000	−	7.79423i	0.381685	−	0.661098i	−0.609618	−	0.792695i	$-0.708677\pi$
$139$	4.50000	−	7.79423i	0.381685	−	0.661098i	0.991303	+	0.131597i	$0.0420106\pi$
$140$		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	−0.982054	−	0.188602i	$-0.939604\pi$
$149$	−4.00000	+	6.92820i	−0.327693	+	0.567581i	0.654361	+	0.756182i	$0.272938\pi$
$150$		0			0
$151$	−0.500000	−	0.866025i	−0.0406894	−	0.0704761i	0.844963	−	0.534824i	$-0.179622\pi$
$151$	−0.500000	−	0.866025i	−0.0406894	−	0.0704761i	−0.885653	+	0.464348i	$0.846289\pi$
$152$		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	−0.823359	−	0.567521i	$-0.807902\pi$
$157$	1.00000	−	1.73205i	0.0798087	−	0.138233i	0.903167	+	0.429289i	$0.141236\pi$
$158$		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	$-0.300134\pi$
$163$	15.0000			1.17489			0.587445	+	0.809264i	$0.300134\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000	+	10.3923i	−0.464294	+	0.804181i	−0.999169	−	0.0407502i	$-0.987025\pi$
$167$	−6.00000	+	10.3923i	−0.464294	+	0.804181i	0.534875	+	0.844931i	$0.320359\pi$
$168$		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	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		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	$-0.854202\pi$
$179$	−24.0000			−1.79384			−0.896922	+	0.442189i	$0.854202\pi$
$180$		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	$-0.0237173\pi$
$191$	6.00000	+	10.3923i	0.434145	+	0.751961i	−0.563081	+	0.826402i	$0.690384\pi$
$192$		0			0
$193$	−11.5000	+	19.9186i	−0.827788	+	1.43377i	0.0719816	+	0.997406i	$0.477068\pi$
$193$	−11.5000	+	19.9186i	−0.827788	+	1.43377i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		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	$-0.846597\pi$
$199$	−25.0000			−1.77220			−0.886102	+	0.463491i	$0.846597\pi$
$200$		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	$-0.685655\pi$
$211$	6.50000	−	11.2583i	0.447478	−	0.775055i	0.998221	+	0.0596196i	$0.0189888\pi$
$212$		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	$-0.123907\pi$
$223$	2.00000	+	3.46410i	0.133930	+	0.231973i	−0.791258	+	0.611482i	$0.790574\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0000	−	20.7846i	−0.796468	−	1.37952i	−0.921903	−	0.387421i	$-0.873366\pi$
$227$	−12.0000	−	20.7846i	−0.796468	−	1.37952i	0.125435	−	0.992102i	$-0.459967\pi$
$228$		0			0
$229$	−5.00000	+	8.66025i	−0.330409	+	0.572286i	−0.982592	−	0.185776i	$-0.940520\pi$
$229$	−5.00000	+	8.66025i	−0.330409	+	0.572286i	0.652183	+	0.758062i	$0.273853\pi$
$230$		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	$-0.750027\pi$
$239$	4.00000	−	6.92820i	0.258738	−	0.448148i	0.965904	+	0.258900i	$0.0833599\pi$
$240$		0			0
$241$	7.50000	+	12.9904i	0.483117	+	0.836784i	0.999812	−	0.0193858i	$-0.00617107\pi$
$241$	7.50000	+	12.9904i	0.483117	+	0.836784i	−0.516695	+	0.856170i	$0.672838\pi$
$242$		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	$-0.331509\pi$
$251$	16.0000			1.00991			0.504956	+	0.863145i	$0.331509\pi$
$252$		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	−0.963391	−	0.268101i	$-0.913604\pi$
$257$	−4.00000	+	6.92820i	−0.249513	+	0.432169i	0.713878	+	0.700270i	$0.246937\pi$
$258$		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	$-0.245997\pi$
$263$	−4.00000	−	6.92820i	−0.246651	−	0.427211i	−0.962594	+	0.270947i	$0.912663\pi$
$264$		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	$-0.225531\pi$
$271$	25.0000			1.51864			0.759321	+	0.650716i	$0.225531\pi$
$272$		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	0.675810	−	0.737075i	$-0.263794\pi$
$277$	−5.00000	−	8.66025i	−0.300421	−	0.520344i	−0.976231	+	0.216731i	$0.930460\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.0000	−	27.7128i	−0.954480	−	1.65321i	−0.735554	−	0.677466i	$-0.763078\pi$
$281$	−16.0000	−	27.7128i	−0.954480	−	1.65321i	−0.218926	−	0.975741i	$-0.570255\pi$
$282$		0			0
$283$	−8.00000	+	13.8564i	−0.475551	+	0.823678i	−0.999608	−	0.0280052i	$-0.991084\pi$
$283$	−8.00000	+	13.8564i	−0.475551	+	0.823678i	0.524057	+	0.851683i	$0.324418\pi$
$284$		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	−0.986341	−	0.164714i	$-0.947330\pi$
$293$	−6.00000	+	10.3923i	−0.350524	+	0.607125i	0.635818	+	0.771839i	$0.280663\pi$
$294$		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	$-0.722829\pi$
$311$	6.00000	−	10.3923i	0.340229	−	0.589294i	0.984475	+	0.175525i	$0.0561621\pi$
$312$		0			0
$313$	−1.50000	−	2.59808i	−0.0847850	−	0.146852i	0.820515	−	0.571626i	$-0.193687\pi$
$313$	−1.50000	−	2.59808i	−0.0847850	−	0.146852i	−0.905300	+	0.424774i	$0.860354\pi$
$314$		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
$319$		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	$-0.0119287\pi$
$331$	8.50000	+	14.7224i	0.467202	+	0.809218i	−0.532096	+	0.846684i	$0.678595\pi$
$332$		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	−0.822274	−	0.569091i	$-0.807295\pi$
$337$	1.50000	−	2.59808i	0.0817102	−	0.141526i	0.903985	+	0.427565i	$0.140628\pi$
$338$		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	0.340293	+	0.940319i	$0.389474\pi$
$347$	−12.0000	+	20.7846i	−0.644194	+	1.11578i	−0.984487	+	0.175457i	$0.943860\pi$
$348$		0			0
$349$	16.5000	+	28.5788i	0.883225	+	1.52979i	0.847735	+	0.530420i	$0.177966\pi$
$349$	16.5000	+	28.5788i	0.883225	+	1.52979i	0.0354898	+	0.999370i	$0.488701\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.00000	−	6.92820i	−0.212899	−	0.368751i	0.739722	−	0.672913i	$-0.234957\pi$
$353$	−4.00000	−	6.92820i	−0.212899	−	0.368751i	−0.952620	+	0.304162i	$0.901624\pi$
$354$		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	$-0.602564\pi$
$359$	−12.0000			−0.633336			−0.316668	+	0.948536i	$0.602564\pi$
$360$		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	−0.992776	−	0.119982i	$-0.961716\pi$
$367$	−11.5000	−	19.9186i	−0.600295	−	1.03974i	0.392481	−	0.919760i	$-0.371617\pi$
$368$		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	−0.852791	−	0.522253i	$-0.825092\pi$
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	0.878680	+	0.477412i	$0.158425\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−11.0000			−0.565032			−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000			−0.565032			−0.282516	+	0.959263i	$0.591169\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		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	0.810859	−	0.585241i	$-0.199000\pi$
$389$	−2.00000	−	3.46410i	−0.101404	−	0.175637i	−0.912263	+	0.409604i	$0.865667\pi$
$390$		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	0.393680	+	0.919247i	$0.371202\pi$
$401$	−12.0000	+	20.7846i	−0.599251	+	1.03793i	−0.992932	+	0.118686i	$0.962132\pi$
$402$		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	0.252498	−	0.967597i	$-0.418748\pi$
$409$	19.5000	−	33.7750i	0.964213	−	1.67007i	0.711715	−	0.702468i	$-0.247919\pi$
$410$		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	$-0.928027\pi$
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	0.681426	+	0.731887i	$0.261360\pi$
$420$		0			0
$421$	−8.50000	−	14.7224i	−0.414265	−	0.717527i	0.581086	−	0.813842i	$-0.302628\pi$
$421$	−8.50000	−	14.7224i	−0.414265	−	0.717527i	−0.995351	+	0.0963145i	$0.969295\pi$
$422$		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	$-0.264421\pi$
$431$	28.0000			1.34871			0.674356	+	0.738406i	$0.264421\pi$
$432$		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	0.872795	+	0.488087i	$0.162305\pi$
$439$	18.0000	+	31.1769i	0.859093	+	1.48799i	−0.0137020	+	0.999906i	$0.504362\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.00000	+	6.92820i	0.190046	+	0.329169i	0.945265	−	0.326303i	$-0.105803\pi$
$443$	4.00000	+	6.92820i	0.190046	+	0.329169i	−0.755219	+	0.655472i	$0.772470\pi$
$444$		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	−0.841316	−	0.540544i	$-0.818219\pi$
$457$	−19.0000	−	32.9090i	−0.888783	−	1.53942i	−0.0474665	−	0.998873i	$-0.515115\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.0000	−	24.2487i	−0.652045	−	1.12938i	−0.982626	−	0.185597i	$-0.940578\pi$
$461$	−14.0000	−	24.2487i	−0.652045	−	1.12938i	0.330581	−	0.943778i	$-0.392755\pi$
$462$		0			0
$463$	−9.50000	+	16.4545i	−0.441502	+	0.764705i	−0.997801	−	0.0662777i	$-0.978888\pi$
$463$	−9.50000	+	16.4545i	−0.441502	+	0.764705i	0.556299	+	0.830982i	$0.312221\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		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	−0.808615	−	0.588338i	$-0.799782\pi$
$479$	−20.0000	−	34.6410i	−0.913823	−	1.58279i	−0.105208	−	0.994450i	$-0.533551\pi$
$480$		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	$-0.419823\pi$
$487$	11.0000			0.498458			0.249229	+	0.968445i	$0.419823\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	−0.698285	−	0.715820i	$-0.746053\pi$
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	0.969061	+	0.246822i	$0.0793863\pi$
$492$		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	$-0.833333\pi$
$499$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−36.0000			−1.60516			−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000			−1.60516			−0.802580	+	0.596544i	$0.796540\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.00000	−	10.3923i	0.265945	−	0.460631i	−0.701866	−	0.712309i	$-0.747649\pi$
$509$	6.00000	−	10.3923i	0.265945	−	0.460631i	0.967811	+	0.251679i	$0.0809826\pi$
$510$		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	$-0.281395\pi$
$523$	29.0000			1.26808			0.634041	+	0.773300i	$0.281395\pi$
$524$		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	$-0.214481\pi$
$547$	−3.50000	−	6.06218i	−0.149649	−	0.259200i	−0.931098	+	0.364770i	$0.881148\pi$
$548$		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	0.302348	+	0.953198i	$0.402230\pi$
$563$	−16.0000	+	27.7128i	−0.674320	+	1.16796i	−0.976667	+	0.214758i	$0.931104\pi$
$564$		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	−0.994634	−	0.103454i	$-0.967011\pi$
$569$	−14.0000	−	24.2487i	−0.586911	−	1.01656i	0.407724	−	0.913105i	$-0.366323\pi$
$570$		0			0
$571$	−16.5000	+	28.5788i	−0.690504	+	1.19599i	0.281170	+	0.959658i	$0.409278\pi$
$571$	−16.5000	+	28.5788i	−0.690504	+	1.19599i	−0.971673	+	0.236329i	$0.924056\pi$
$572$		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	0.995726	+	0.0923513i	$0.0294383\pi$
$587$	14.0000	+	24.2487i	0.577842	+	1.00085i	−0.417885	+	0.908500i	$0.637228\pi$
$588$		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	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−5.00000	−	8.66025i	−0.203954	−	0.353259i	0.745845	−	0.666120i	$-0.232046\pi$
$601$	−5.00000	−	8.66025i	−0.203954	−	0.353259i	−0.949799	+	0.312861i	$0.898713\pi$
$602$		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	$-0.800978\pi$
$607$	2.50000	−	4.33013i	0.101472	−	0.175754i	0.912291	+	0.409542i	$0.134311\pi$
$608$		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	0.234464	+	0.972125i	$0.424666\pi$
$617$	−18.0000	+	31.1769i	−0.724653	+	1.25514i	−0.959117	+	0.283011i	$0.908667\pi$
$618$		0			0
$619$	−8.50000	−	14.7224i	−0.341644	−	0.591744i	0.643094	−	0.765787i	$-0.277650\pi$
$619$	−8.50000	−	14.7224i	−0.341644	−	0.591744i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		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	$-0.544496\pi$
$631$	−7.00000			−0.278666			−0.139333	+	0.990246i	$0.544496\pi$
$632$		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	−0.925995	−	0.377535i	$-0.876772\pi$
$641$	−20.0000	−	34.6410i	−0.789953	−	1.36824i	0.136043	−	0.990703i	$-0.456562\pi$
$642$		0			0
$643$	−12.0000	+	20.7846i	−0.473234	+	0.819665i	−0.999531	−	0.0306359i	$-0.990247\pi$
$643$	−12.0000	+	20.7846i	−0.473234	+	0.819665i	0.526297	+	0.850301i	$0.323580\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−32.0000			−1.25805			−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000			−1.25805			−0.629025	+	0.777385i	$0.716546\pi$
$648$		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	−0.999396	−	0.0347583i	$-0.988934\pi$
$653$	−12.0000	+	20.7846i	−0.469596	+	0.813365i	0.529799	+	0.848123i	$0.322267\pi$
$654$		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	$-0.321495\pi$
$659$	−12.0000	−	20.7846i	−0.467454	−	0.809653i	−0.999309	+	0.0371821i	$0.988162\pi$
$660$		0			0
$661$	6.50000	−	11.2583i	0.252821	−	0.437898i	−0.711481	−	0.702706i	$-0.751975\pi$
$661$	6.50000	−	11.2583i	0.252821	−	0.437898i	0.964301	+	0.264807i	$0.0853084\pi$
$662$		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	0.622770	−	0.782405i	$-0.286007\pi$
$673$	−9.50000	−	16.4545i	−0.366198	−	0.634274i	−0.988968	+	0.148132i	$0.952674\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	22.0000	+	38.1051i	0.845529	+	1.46450i	0.885161	+	0.465284i	$0.154048\pi$
$677$	22.0000	+	38.1051i	0.845529	+	1.46450i	−0.0396326	+	0.999214i	$0.512619\pi$
$678$		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	$-0.599028\pi$
$683$	−16.0000			−0.612223			−0.306111	+	0.951996i	$0.599028\pi$
$684$		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	−0.942420	−	0.334431i	$-0.891456\pi$
$691$	−20.0000	−	34.6410i	−0.760836	−	1.31781i	0.181584	−	0.983375i	$-0.441877\pi$
$692$		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	0.924234	−	0.381827i	$-0.124705\pi$
$709$	3.50000	+	6.06218i	0.131445	+	0.227670i	−0.792789	+	0.609497i	$0.791372\pi$
$710$		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	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		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	$-0.0952359\pi$
$727$	6.00000	+	10.3923i	0.222528	+	0.385429i	−0.733047	+	0.680178i	$0.761903\pi$
$728$		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	−0.883902	−	0.467671i	$-0.845093\pi$
$733$	−1.00000	+	1.73205i	−0.0369358	+	0.0639748i	0.846967	+	0.531646i	$0.178426\pi$
$734$		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	$-0.700309\pi$
$739$	−32.0000			−1.17714			−0.588570	+	0.808447i	$0.700309\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.00000	−	3.46410i	0.0733729	−	0.127086i	−0.827005	−	0.562195i	$-0.809957\pi$
$743$	2.00000	−	3.46410i	0.0733729	−	0.127086i	0.900378	+	0.435110i	$0.143290\pi$
$744$		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	$-0.815902\pi$
$751$	1.50000	−	2.59808i	0.0547358	−	0.0948051i	0.892095	+	0.451848i	$0.149235\pi$
$752$		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	−0.827496	−	0.561471i	$-0.810236\pi$
$761$	2.00000	−	3.46410i	0.0724999	−	0.125574i	0.899996	+	0.435897i	$0.143569\pi$
$762$		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	0.0360609	+	0.999350i	$0.488519\pi$
$769$	−23.5000	+	40.7032i	−0.847432	+	1.46779i	−0.883493	+	0.468445i	$0.844814\pi$
$770$		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	$-0.793517\pi$
$787$	3.50000	−	6.06218i	0.124762	−	0.216093i	0.921640	+	0.388047i	$0.126850\pi$
$788$		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	−0.972214	−	0.234095i	$-0.924787\pi$
$797$	−8.00000	+	13.8564i	−0.283375	+	0.490819i	0.688839	+	0.724914i	$0.258121\pi$
$798$		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	$-0.747841\pi$
$811$	−40.0000			−1.40459			−0.702295	+	0.711886i	$0.747841\pi$
$812$		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	0.951526	−	0.307568i	$-0.0995151\pi$
$821$	6.00000	+	10.3923i	0.209401	+	0.362694i	−0.742125	+	0.670262i	$0.766182\pi$
$822$		0			0
$823$	−12.5000	+	21.6506i	−0.435723	+	0.754694i	−0.997354	−	0.0726937i	$-0.976840\pi$
$823$	−12.5000	+	21.6506i	−0.435723	+	0.754694i	0.561632	+	0.827387i	$0.310174\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20.0000			−0.695468			−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000			−0.695468			−0.347734	+	0.937593i	$0.613049\pi$
$828$		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	$-0.0306980\pi$
$839$	12.0000	+	20.7846i	0.414286	+	0.717564i	−0.581067	+	0.813856i	$0.697365\pi$
$840$		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	0.874458	−	0.485101i	$-0.161217\pi$
$853$	0.500000	+	0.866025i	0.0171197	+	0.0296521i	−0.857339	+	0.514753i	$0.827884\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12.0000	−	20.7846i	−0.409912	−	0.709989i	0.584967	−	0.811057i	$-0.301107\pi$
$857$	−12.0000	−	20.7846i	−0.409912	−	0.709989i	−0.994880	+	0.101068i	$0.967774\pi$
$858$		0			0
$859$	−13.5000	+	23.3827i	−0.460614	+	0.797807i	−0.998992	−	0.0448968i	$-0.985704\pi$
$859$	−13.5000	+	23.3827i	−0.460614	+	0.797807i	0.538378	+	0.842704i	$0.319037\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	44.0000			1.49778			0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000			1.49778			0.748889	+	0.662696i	$0.230588\pi$
$864$		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	−0.440566	−	0.897720i	$-0.645222\pi$
$877$	16.5000	−	28.5788i	0.557165	−	0.965039i	0.997732	−	0.0673186i	$-0.0214444\pi$
$878$		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	$-0.462421\pi$
$883$	7.00000			0.235569			0.117784	+	0.993039i	$0.462421\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	28.0000	−	48.4974i	0.940148	−	1.62838i	0.174962	−	0.984575i	$-0.444020\pi$
$887$	28.0000	−	48.4974i	0.940148	−	1.62838i	0.765186	−	0.643809i	$-0.222647\pi$
$888$		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	$-0.193120\pi$
$907$	−2.50000	−	4.33013i	−0.0830111	−	0.143780i	−0.904542	+	0.426385i	$0.859787\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	8.00000	+	13.8564i	0.265052	+	0.459083i	0.967577	−	0.252575i	$-0.0812776\pi$
$911$	8.00000	+	13.8564i	0.265052	+	0.459083i	−0.702525	+	0.711659i	$0.747944\pi$
$912$		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	$-0.392995\pi$
$919$	20.0000			0.659739			0.329870	+	0.944027i	$0.392995\pi$
$920$		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	0.981597	+	0.190965i	$0.0611616\pi$
$929$	20.0000	+	34.6410i	0.656179	+	1.13653i	−0.325418	+	0.945570i	$0.605505\pi$
$930$		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	−0.947095	−	0.320953i	$-0.895997\pi$
$941$	−6.00000	+	10.3923i	−0.195594	+	0.338779i	0.751501	+	0.659732i	$0.229330\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.0000	−	31.1769i	−0.584921	−	1.01311i	−0.994885	−	0.101012i	$-0.967792\pi$
$947$	−18.0000	−	31.1769i	−0.584921	−	1.01311i	0.409964	−	0.912102i	$-0.365541\pi$
$948$		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	0.200396	−	0.979715i	$-0.435777\pi$
$967$	29.5000	−	51.0955i	0.948656	−	1.64312i	0.748260	−	0.663405i	$-0.230890\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		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	−0.607704	−	0.794164i	$-0.707909\pi$
$977$	12.0000	−	20.7846i	0.383914	−	0.664959i	0.991618	+	0.129205i	$0.0412426\pi$
$978$		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	$-0.105374\pi$
$983$	6.00000	+	10.3923i	0.191370	+	0.331463i	−0.754334	+	0.656490i	$0.772040\pi$
$984$		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	$-0.629975\pi$
$991$	−25.0000			−0.794151			−0.397076	+	0.917786i	$0.629975\pi$
$992$		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	0.995077	−	0.0991016i	$-0.0315969\pi$
$997$	13.0000	+	22.5167i	0.411714	+	0.713110i	−0.583363	+	0.812211i	$0.698264\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.2.i.a.433.1		2
3.2	odd	2		1296.2.i.q.433.1		2
4.3	odd	2		648.2.i.a.433.1		2
9.2	odd	6		1296.2.i.q.865.1		2
9.4	even	3		432.2.a.h.1.1		1
9.5	odd	6		432.2.a.a.1.1		1
9.7	even	3	inner	1296.2.i.a.865.1		2
12.11	even	2		648.2.i.h.433.1		2
36.7	odd	6		648.2.i.a.217.1		2
36.11	even	6		648.2.i.h.217.1		2
36.23	even	6		216.2.a.a.1.1	✓	1
36.31	odd	6		216.2.a.d.1.1	yes	1
72.5	odd	6		1728.2.a.bb.1.1		1
72.13	even	6		1728.2.a.b.1.1		1
72.59	even	6		1728.2.a.ba.1.1		1
72.67	odd	6		1728.2.a.a.1.1		1
180.23	odd	12		5400.2.f.e.649.2		2
180.59	even	6		5400.2.a.bn.1.1		1
180.67	even	12		5400.2.f.v.649.1		2
180.103	even	12		5400.2.f.v.649.2		2
180.139	odd	6		5400.2.a.bp.1.1		1
180.167	odd	12		5400.2.f.e.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.a.a.1.1	✓	1	36.23	even	6
216.2.a.d.1.1	yes	1	36.31	odd	6
432.2.a.a.1.1		1	9.5	odd	6
432.2.a.h.1.1		1	9.4	even	3
648.2.i.a.217.1		2	36.7	odd	6
648.2.i.a.433.1		2	4.3	odd	2
648.2.i.h.217.1		2	36.11	even	6
648.2.i.h.433.1		2	12.11	even	2
1296.2.i.a.433.1		2	1.1	even	1	trivial
1296.2.i.a.865.1		2	9.7	even	3	inner
1296.2.i.q.433.1		2	3.2	odd	2
1296.2.i.q.865.1		2	9.2	odd	6
1728.2.a.a.1.1		1	72.67	odd	6
1728.2.a.b.1.1		1	72.13	even	6
1728.2.a.ba.1.1		1	72.59	even	6
1728.2.a.bb.1.1		1	72.5	odd	6
5400.2.a.bn.1.1		1	180.59	even	6
5400.2.a.bp.1.1		1	180.139	odd	6
5400.2.f.e.649.1		2	180.167	odd	12
5400.2.f.e.649.2		2	180.23	odd	12
5400.2.f.v.649.1		2	180.67	even	12
5400.2.f.v.649.2		2	180.103	even	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.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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