LMFDB - Embedded newform 2592.2.r.b.433.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2592,2,Mod(433,2592)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2592.433"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$2592 = 2^{5} \cdot 3^{4}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2592.r (of order $6$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-6,0,2,0,0,0,-6,0,12,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{2} + 1$ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 648)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			433.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2592.433
Dual form			2592.2.r.b.2161.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.23205 - 1.86603i) q^{5} +(-0.366025 - 0.633975i) q^{7} +(-4.09808 + 2.36603i) q^{11} +(2.13397 + 1.23205i) q^{13} +3.73205 q^{17} -3.26795i q^{19} +(-4.36603 + 7.56218i) q^{23} +(4.46410 + 7.73205i) q^{25} +(-4.50000 + 2.59808i) q^{29} +(1.00000 - 1.73205i) q^{31} +2.73205i q^{35} -5.00000i q^{37} +(2.26795 - 3.92820i) q^{41} +(-8.36603 + 4.83013i) q^{43} +(-1.73205 - 3.00000i) q^{47} +(3.23205 - 5.59808i) q^{49} -0.928203i q^{53} +17.6603 q^{55} +(7.26795 + 4.19615i) q^{59} +(7.79423 - 4.50000i) q^{61} +(-4.59808 - 7.96410i) q^{65} +(4.09808 + 2.36603i) q^{67} +5.66025 q^{71} +9.00000 q^{73} +(3.00000 + 1.73205i) q^{77} +(2.63397 + 4.56218i) q^{79} +(12.1244 - 7.00000i) q^{83} +(-12.0622 - 6.96410i) q^{85} +11.7321 q^{89} -1.80385i q^{91} +(-6.09808 + 10.5622i) q^{95} +(4.00000 + 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 6 q^{5} + 2 q^{7} - 6 q^{11} + 12 q^{13} + 8 q^{17} - 14 q^{23} + 4 q^{25} - 18 q^{29} + 4 q^{31} + 16 q^{41} - 30 q^{43} + 6 q^{49} + 36 q^{55} + 36 q^{59} - 8 q^{65} + 6 q^{67} - 12 q^{71} + 36 q^{73}+ \cdots + 16 q^{97}+O(q^{100})$ 4 * q - 6 * q^5 + 2 * q^7 - 6 * q^11 + 12 * q^13 + 8 * q^17 - 14 * q^23 + 4 * q^25 - 18 * q^29 + 4 * q^31 + 16 * q^41 - 30 * q^43 + 6 * q^49 + 36 * q^55 + 36 * q^59 - 8 * q^65 + 6 * q^67 - 12 * q^71 + 36 * q^73 + 12 * q^77 + 14 * q^79 - 24 * q^85 + 40 * q^89 - 14 * q^95 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$ .

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	−3.23205	−	1.86603i	−1.44542	−	0.834512i	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−3.23205	−	1.86603i	−1.44542	−	0.834512i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$	−0.366025	−	0.633975i	−0.138345	−	0.239620i	0.788526	−	0.615002i	$-0.210845\pi$
$7$	−0.366025	−	0.633975i	−0.138345	−	0.239620i	−0.926870	+	0.375382i	$0.877511\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.09808	+	2.36603i	−1.23562	+	0.713384i	−0.968195	−	0.250196i	$-0.919505\pi$
$11$	−4.09808	+	2.36603i	−1.23562	+	0.713384i	−0.267421	+	0.963580i	$0.586172\pi$
$12$		0			0
$13$	2.13397	+	1.23205i	0.591858	+	0.341709i	0.765832	−	0.643041i	$-0.222327\pi$
$13$	2.13397	+	1.23205i	0.591858	+	0.341709i	−0.173974	+	0.984750i	$0.555661\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.73205			0.905155			0.452578	−	0.891725i	$-0.350505\pi$
$17$	3.73205			0.905155			0.452578	+	0.891725i	$0.350505\pi$
$18$		0			0
$19$		−	3.26795i		−	0.749719i	−0.927082	−	0.374859i	$-0.877691\pi$
$19$		−	3.26795i		−	0.749719i	0.927082	−	0.374859i	$-0.122309\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.36603	+	7.56218i	−0.910379	+	1.57682i	−0.0968500	+	0.995299i	$0.530877\pi$
$23$	−4.36603	+	7.56218i	−0.910379	+	1.57682i	−0.813529	+	0.581524i	$0.802457\pi$
$24$		0			0
$25$	4.46410	+	7.73205i	0.892820	+	1.54641i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.50000	+	2.59808i	−0.835629	+	0.482451i	−0.855776	−	0.517346i	$-0.826920\pi$
$29$	−4.50000	+	2.59808i	−0.835629	+	0.482451i	0.0201471	+	0.999797i	$0.493587\pi$
$30$		0			0
$31$	1.00000	−	1.73205i	0.179605	−	0.311086i	−0.762140	−	0.647412i	$-0.775851\pi$
$31$	1.00000	−	1.73205i	0.179605	−	0.311086i	0.941745	+	0.336327i	$0.109185\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			2.73205i			0.461801i
$36$		0			0
$37$		−	5.00000i		−	0.821995i	−0.911636	−	0.410997i	$-0.865181\pi$
$37$		−	5.00000i		−	0.821995i	0.911636	−	0.410997i	$-0.134819\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.26795	−	3.92820i	0.354194	−	0.613482i	−0.632786	−	0.774327i	$-0.718089\pi$
$41$	2.26795	−	3.92820i	0.354194	−	0.613482i	0.986980	+	0.160845i	$0.0514219\pi$
$42$		0			0
$43$	−8.36603	+	4.83013i	−1.27581	+	0.736587i	−0.976075	−	0.217436i	$-0.930231\pi$
$43$	−8.36603	+	4.83013i	−1.27581	+	0.736587i	−0.299732	+	0.954023i	$0.596897\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.73205	−	3.00000i	−0.252646	−	0.437595i	0.711608	−	0.702577i	$-0.247967\pi$
$47$	−1.73205	−	3.00000i	−0.252646	−	0.437595i	−0.964253	+	0.264982i	$0.914634\pi$
$48$		0			0
$49$	3.23205	−	5.59808i	0.461722	−	0.799725i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	0.928203i		−	0.127499i	−0.997966	−	0.0637493i	$-0.979694\pi$
$53$		−	0.928203i		−	0.127499i	0.997966	−	0.0637493i	$-0.0203058\pi$
$54$		0			0
$55$	17.6603			2.38131
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.26795	+	4.19615i	0.946206	+	0.546293i	0.891900	−	0.452232i	$-0.149372\pi$
$59$	7.26795	+	4.19615i	0.946206	+	0.546293i	0.0543060	+	0.998524i	$0.482705\pi$
$60$		0			0
$61$	7.79423	−	4.50000i	0.997949	−	0.576166i	0.0903080	−	0.995914i	$-0.471215\pi$
$61$	7.79423	−	4.50000i	0.997949	−	0.576166i	0.907641	+	0.419748i	$0.137882\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.59808	−	7.96410i	−0.570321	−	0.987825i
$66$		0			0
$67$	4.09808	+	2.36603i	0.500660	+	0.289056i	0.728986	−	0.684529i	$-0.239992\pi$
$67$	4.09808	+	2.36603i	0.500660	+	0.289056i	−0.228326	+	0.973585i	$0.573325\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.66025			0.671749			0.335874	−	0.941907i	$-0.390968\pi$
$71$	5.66025			0.671749			0.335874	+	0.941907i	$0.390968\pi$
$72$		0			0
$73$	9.00000			1.05337			0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000			1.05337			0.526685	+	0.850060i	$0.323435\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.00000	+	1.73205i	0.341882	+	0.197386i
$78$		0			0
$79$	2.63397	+	4.56218i	0.296345	+	0.513285i	0.975297	−	0.220898i	$-0.0708988\pi$
$79$	2.63397	+	4.56218i	0.296345	+	0.513285i	−0.678952	+	0.734183i	$0.737565\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.1244	−	7.00000i	1.33082	−	0.768350i	0.345395	−	0.938457i	$-0.387745\pi$
$83$	12.1244	−	7.00000i	1.33082	−	0.768350i	0.985426	+	0.170107i	$0.0544115\pi$
$84$		0			0
$85$	−12.0622	−	6.96410i	−1.30833	−	0.755363i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.7321			1.24359			0.621797	−	0.783178i	$-0.286403\pi$
$89$	11.7321			1.24359			0.621797	+	0.783178i	$0.286403\pi$
$90$		0			0
$91$		−	1.80385i		−	0.189095i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.09808	+	10.5622i	−0.625649	+	1.08366i
$96$		0			0
$97$	4.00000	+	6.92820i	0.406138	+	0.703452i	0.994453	−	0.105180i	$-0.0335417\pi$
$97$	4.00000	+	6.92820i	0.406138	+	0.703452i	−0.588315	+	0.808632i	$0.700208\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.92820	−	4.00000i	0.689382	−	0.398015i	−0.113998	−	0.993481i	$-0.536366\pi$
$101$	6.92820	−	4.00000i	0.689382	−	0.398015i	0.803380	+	0.595466i	$0.203033\pi$
$102$		0			0
$103$	−1.73205	+	3.00000i	−0.170664	+	0.295599i	−0.938652	−	0.344865i	$-0.887925\pi$
$103$	−1.73205	+	3.00000i	−0.170664	+	0.295599i	0.767988	+	0.640464i	$0.221258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	10.3923i		−	1.00466i	−0.864675	−	0.502331i	$-0.832476\pi$
$107$		−	10.3923i		−	1.00466i	0.864675	−	0.502331i	$-0.167524\pi$
$108$		0			0
$109$			12.8564i			1.23142i	0.787973	+	0.615710i	$0.211131\pi$
$109$			12.8564i			1.23142i	−0.787973	+	0.615710i	$0.788869\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.86603	+	6.69615i	−0.363685	+	0.629921i	−0.988564	−	0.150800i	$-0.951815\pi$
$113$	−3.86603	+	6.69615i	−0.363685	+	0.629921i	0.624879	+	0.780721i	$0.285148\pi$
$114$		0			0
$115$	28.2224	−	16.2942i	2.63176	−	1.51944i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.36603	−	2.36603i	−0.125223	−	0.216893i
$120$		0			0
$121$	5.69615	−	9.86603i	0.517832	−	0.896911i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	14.6603i		−	1.31125i
$126$		0			0
$127$	13.6603			1.21215			0.606076	−	0.795407i	$-0.292743\pi$
$127$	13.6603			1.21215			0.606076	+	0.795407i	$0.292743\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.169873	+	0.0980762i	0.0148419	+	0.00856896i	0.507403	−	0.861709i	$-0.330606\pi$
$131$	0.169873	+	0.0980762i	0.0148419	+	0.00856896i	−0.492561	+	0.870278i	$0.663939\pi$
$132$		0			0
$133$	−2.07180	+	1.19615i	−0.179648	+	0.103720i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.59808	−	14.8923i	−0.734583	−	1.27234i	−0.954906	−	0.296908i	$-0.904044\pi$
$137$	−8.59808	−	14.8923i	−0.734583	−	1.27234i	0.220323	−	0.975427i	$-0.429289\pi$
$138$		0			0
$139$	−16.8564	−	9.73205i	−1.42974	−	0.825462i	−0.432642	−	0.901566i	$-0.642419\pi$
$139$	−16.8564	−	9.73205i	−1.42974	−	0.825462i	−0.997100	+	0.0761041i	$0.975752\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.6603			−0.975079
$144$		0			0
$145$	19.3923			1.61044
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.50000	+	2.59808i	0.368654	+	0.212843i	0.672870	−	0.739760i	$-0.265061\pi$
$149$	4.50000	+	2.59808i	0.368654	+	0.212843i	−0.304216	+	0.952603i	$0.598394\pi$
$150$		0			0
$151$	6.92820	+	12.0000i	0.563809	+	0.976546i	0.997159	+	0.0753205i	$0.0239980\pi$
$151$	6.92820	+	12.0000i	0.563809	+	0.976546i	−0.433350	+	0.901226i	$0.642669\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.46410	+	3.73205i	−0.519209	+	0.299766i
$156$		0			0
$157$	2.13397	+	1.23205i	0.170310	+	0.0983284i	0.582732	−	0.812664i	$-0.301984\pi$
$157$	2.13397	+	1.23205i	0.170310	+	0.0983284i	−0.412422	+	0.910993i	$0.635317\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.39230			0.503784
$162$		0			0
$163$		−	6.53590i		−	0.511931i	−0.966686	−	0.255966i	$-0.917607\pi$
$163$		−	6.53590i		−	0.511931i	0.966686	−	0.255966i	$-0.0823934\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.36603	−	12.7583i	0.570000	−	0.987269i	−0.426565	−	0.904457i	$-0.640276\pi$
$167$	7.36603	−	12.7583i	0.570000	−	0.987269i	0.996565	−	0.0828123i	$-0.0263902\pi$
$168$		0			0
$169$	−3.46410	−	6.00000i	−0.266469	−	0.461538i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.50000	−	0.866025i	0.114043	−	0.0658427i	−0.441894	−	0.897067i	$-0.645693\pi$
$173$	1.50000	−	0.866025i	0.114043	−	0.0658427i	0.555936	+	0.831225i	$0.312360\pi$
$174$		0			0
$175$	3.26795	−	5.66025i	0.247034	−	0.427875i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			14.0000i			1.04641i	0.852207	+	0.523205i	$0.175264\pi$
$179$			14.0000i			1.04641i	−0.852207	+	0.523205i	$0.824736\pi$
$180$		0			0
$181$		−	6.53590i		−	0.485810i	−0.970050	−	0.242905i	$-0.921900\pi$
$181$		−	6.53590i		−	0.485810i	0.970050	−	0.242905i	$-0.0781002\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.33013	+	16.1603i	−0.685965	+	1.18813i
$186$		0			0
$187$	−15.2942	+	8.83013i	−1.11842	+	0.645723i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.29423	−	14.3660i	−0.600149	−	1.03949i	−0.992798	−	0.119801i	$-0.961774\pi$
$191$	−8.29423	−	14.3660i	−0.600149	−	1.03949i	0.392649	−	0.919689i	$-0.371559\pi$
$192$		0			0
$193$	−5.42820	+	9.40192i	−0.390731	+	0.676765i	−0.992546	−	0.121870i	$-0.961111\pi$
$193$	−5.42820	+	9.40192i	−0.390731	+	0.676765i	0.601815	+	0.798635i	$0.294444\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.12436i			0.293848i	0.989148	+	0.146924i	$0.0469373\pi$
$197$			4.12436i			0.293848i	−0.989148	+	0.146924i	$0.953063\pi$
$198$		0			0
$199$	−2.39230			−0.169586			−0.0847930	−	0.996399i	$-0.527023\pi$
$199$	−2.39230			−0.169586			−0.0847930	+	0.996399i	$0.527023\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.29423	+	1.90192i	0.231210	+	0.133489i
$204$		0			0
$205$	−14.6603	+	8.46410i	−1.02392	+	0.591158i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.73205	+	13.3923i	0.534837	+	0.926365i
$210$		0			0
$211$	22.5622	+	13.0263i	1.55324	+	0.896766i	0.997875	+	0.0651625i	$0.0207566\pi$
$211$	22.5622	+	13.0263i	1.55324	+	0.896766i	0.555370	+	0.831604i	$0.312577\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	36.0526			2.45876
$216$		0			0
$217$	−1.46410			−0.0993897
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.96410	+	4.59808i	0.535723	+	0.309300i
$222$		0			0
$223$	11.5622	+	20.0263i	0.774261	+	1.34106i	0.935209	+	0.354096i	$0.115211\pi$
$223$	11.5622	+	20.0263i	0.774261	+	1.34106i	−0.160948	+	0.986963i	$0.551455\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.83013	+	3.36603i	−0.386959	+	0.223411i	−0.680842	−	0.732431i	$-0.738386\pi$
$227$	−5.83013	+	3.36603i	−0.386959	+	0.223411i	0.293883	+	0.955842i	$0.405053\pi$
$228$		0			0
$229$	9.86603	+	5.69615i	0.651965	+	0.376412i	0.789209	−	0.614125i	$-0.210491\pi$
$229$	9.86603	+	5.69615i	0.651965	+	0.376412i	−0.137243	+	0.990537i	$0.543824\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.73205			−0.244495			−0.122247	−	0.992500i	$-0.539010\pi$
$233$	−3.73205			−0.244495			−0.122247	+	0.992500i	$0.539010\pi$
$234$		0			0
$235$			12.9282i			0.843343i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.92820	+	6.80385i	−0.254094	+	0.440104i	−0.964649	−	0.263538i	$-0.915111\pi$
$239$	−3.92820	+	6.80385i	−0.254094	+	0.440104i	0.710555	+	0.703642i	$0.248444\pi$
$240$		0			0
$241$	2.23205	+	3.86603i	0.143779	+	0.249033i	0.928917	−	0.370289i	$-0.120741\pi$
$241$	2.23205	+	3.86603i	0.143779	+	0.249033i	−0.785138	+	0.619321i	$0.787408\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−20.8923	+	12.0622i	−1.33476	+	0.770624i
$246$		0			0
$247$	4.02628	−	6.97372i	0.256186	−	0.443727i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	7.26795i		−	0.458749i	−0.973338	−	0.229374i	$-0.926332\pi$
$251$		−	7.26795i		−	0.458749i	0.973338	−	0.229374i	$-0.0736680\pi$
$252$		0			0
$253$		−	41.3205i		−	2.59780i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.66987	+	2.89230i	−0.104164	+	0.180417i	−0.913396	−	0.407072i	$-0.866550\pi$
$257$	−1.66987	+	2.89230i	−0.104164	+	0.180417i	0.809232	+	0.587489i	$0.199883\pi$
$258$		0			0
$259$	−3.16987	+	1.83013i	−0.196966	+	0.113719i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	8.12436	+	14.0718i	0.500969	+	0.867704i	0.999999	+	0.00111953i	$0.000356359\pi$
$263$	8.12436	+	14.0718i	0.500969	+	0.867704i	−0.499030	+	0.866585i	$0.666310\pi$
$264$		0			0
$265$	−1.73205	+	3.00000i	−0.106399	+	0.184289i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			25.7321i			1.56891i	0.620185	+	0.784455i	$0.287057\pi$
$269$			25.7321i			1.56891i	−0.620185	+	0.784455i	$0.712943\pi$
$270$		0			0
$271$	−10.7321			−0.651926			−0.325963	−	0.945383i	$-0.605688\pi$
$271$	−10.7321			−0.651926			−0.325963	+	0.945383i	$0.605688\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−36.5885	−	21.1244i	−2.20637	−	1.27385i
$276$		0			0
$277$	8.53590	−	4.92820i	0.512872	−	0.296107i	−0.221141	−	0.975242i	$-0.570978\pi$
$277$	8.53590	−	4.92820i	0.512872	−	0.296107i	0.734014	+	0.679135i	$0.237645\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.86603	+	11.8923i	0.409593	+	0.709435i	0.994844	−	0.101417i	$-0.0323375\pi$
$281$	6.86603	+	11.8923i	0.409593	+	0.709435i	−0.585251	+	0.810852i	$0.699004\pi$
$282$		0			0
$283$	−5.53590	−	3.19615i	−0.329075	−	0.189992i	0.326355	−	0.945247i	$-0.394179\pi$
$283$	−5.53590	−	3.19615i	−0.329075	−	0.189992i	−0.655430	+	0.755256i	$0.727513\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.32051			−0.196003
$288$		0			0
$289$	−3.07180			−0.180694
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.89230	−	1.66987i	−0.168970	−	0.0975550i	0.413130	−	0.910672i	$-0.364436\pi$
$293$	−2.89230	−	1.66987i	−0.168970	−	0.0975550i	−0.582100	+	0.813117i	$0.697769\pi$
$294$		0			0
$295$	−15.6603	−	27.1244i	−0.911775	−	1.57924i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−18.6340	+	10.7583i	−1.07763	+	0.622170i
$300$		0			0
$301$	6.12436	+	3.53590i	0.353002	+	0.203806i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−33.5885			−1.92327
$306$		0			0
$307$		−	14.7846i		−	0.843802i	−0.906642	−	0.421901i	$-0.861363\pi$
$307$		−	14.7846i		−	0.843802i	0.906642	−	0.421901i	$-0.138637\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.66025	−	16.7321i	0.547783	−	0.948788i	−0.450643	−	0.892704i	$-0.648805\pi$
$311$	9.66025	−	16.7321i	0.547783	−	0.948788i	0.998426	−	0.0560835i	$-0.0178613\pi$
$312$		0			0
$313$	−1.23205	−	2.13397i	−0.0696396	−	0.120619i	0.829103	−	0.559096i	$-0.188852\pi$
$313$	−1.23205	−	2.13397i	−0.0696396	−	0.120619i	−0.898743	+	0.438476i	$0.855518\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.03590	−	0.598076i	0.0581818	−	0.0335913i	−0.470627	−	0.882332i	$-0.655972\pi$
$317$	1.03590	−	0.598076i	0.0581818	−	0.0335913i	0.528809	+	0.848741i	$0.322639\pi$
$318$		0			0
$319$	12.2942	−	21.2942i	0.688345	−	1.19225i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	12.1962i		−	0.678612i
$324$		0			0
$325$			22.0000i			1.22034i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.26795	+	2.19615i	−0.0699043	+	0.121078i
$330$		0			0
$331$	−11.3660	+	6.56218i	−0.624733	+	0.360690i	−0.778710	−	0.627385i	$-0.784125\pi$
$331$	−11.3660	+	6.56218i	−0.624733	+	0.360690i	0.153976	+	0.988075i	$0.450792\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.83013	−	15.2942i	−0.482441	−	0.835613i
$336$		0			0
$337$	−2.00000	+	3.46410i	−0.108947	+	0.188702i	−0.915344	−	0.402673i	$-0.868081\pi$
$337$	−2.00000	+	3.46410i	−0.108947	+	0.188702i	0.806397	+	0.591375i	$0.201415\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			9.46410i			0.512510i
$342$		0			0
$343$	−9.85641			−0.532196
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.83013	+	3.36603i	0.312978	+	0.180698i	0.648258	−	0.761421i	$-0.275498\pi$
$347$	5.83013	+	3.36603i	0.312978	+	0.180698i	−0.335281	+	0.942118i	$0.608831\pi$
$348$		0			0
$349$	10.2679	−	5.92820i	0.549631	−	0.317329i	−0.199342	−	0.979930i	$-0.563881\pi$
$349$	10.2679	−	5.92820i	0.549631	−	0.317329i	0.748973	+	0.662600i	$0.230547\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.46410	−	6.00000i	−0.184376	−	0.319348i	0.758990	−	0.651102i	$-0.225693\pi$
$353$	−3.46410	−	6.00000i	−0.184376	−	0.319348i	−0.943366	+	0.331754i	$0.892360\pi$
$354$		0			0
$355$	−18.2942	−	10.5622i	−0.970957	−	0.560582i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.12436			0.0593412			0.0296706	−	0.999560i	$-0.490554\pi$
$359$	1.12436			0.0593412			0.0296706	+	0.999560i	$0.490554\pi$
$360$		0			0
$361$	8.32051			0.437921
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−29.0885	−	16.7942i	−1.52256	−	0.879050i
$366$		0			0
$367$	1.26795	+	2.19615i	0.0661864	+	0.114638i	0.897220	−	0.441585i	$-0.145583\pi$
$367$	1.26795	+	2.19615i	0.0661864	+	0.114638i	−0.831033	+	0.556223i	$0.812250\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.588457	+	0.339746i	−0.0305512	+	0.0176387i
$372$		0			0
$373$	19.8564	+	11.4641i	1.02813	+	0.593589i	0.916447	−	0.400156i	$-0.131044\pi$
$373$	19.8564	+	11.4641i	1.02813	+	0.593589i	0.111679	+	0.993744i	$0.464377\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.8038			−0.659432
$378$		0			0
$379$			10.0000i			0.513665i	0.966456	+	0.256833i	$0.0826790\pi$
$379$			10.0000i			0.513665i	−0.966456	+	0.256833i	$0.917321\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.0981	−	17.4904i	0.515988	−	0.893717i	−0.483840	−	0.875156i	$-0.660758\pi$
$383$	10.0981	−	17.4904i	0.515988	−	0.893717i	0.999828	−	0.0185603i	$-0.00590828\pi$
$384$		0			0
$385$	−6.46410	−	11.1962i	−0.329441	−	0.570609i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.6603	−	8.46410i	0.743304	−	0.429147i	−0.0799651	−	0.996798i	$-0.525481\pi$
$389$	14.6603	−	8.46410i	0.743304	−	0.429147i	0.823270	+	0.567651i	$0.192148\pi$
$390$		0			0
$391$	−16.2942	+	28.2224i	−0.824035	+	1.42727i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	19.6603i		−	0.989215i
$396$		0			0
$397$			32.4641i			1.62933i	0.579934	+	0.814663i	$0.303078\pi$
$397$			32.4641i			1.62933i	−0.579934	+	0.814663i	$0.696922\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.79423	+	8.30385i	−0.239412	+	0.414674i	−0.960546	−	0.278122i	$-0.910288\pi$
$401$	−4.79423	+	8.30385i	−0.239412	+	0.414674i	0.721133	+	0.692796i	$0.243621\pi$
$402$		0			0
$403$	4.26795	−	2.46410i	0.212602	−	0.122746i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	11.8301	+	20.4904i	0.586398	+	1.01567i
$408$		0			0
$409$	−14.3564	+	24.8660i	−0.709879	+	1.22955i	0.255023	+	0.966935i	$0.417917\pi$
$409$	−14.3564	+	24.8660i	−0.709879	+	1.22955i	−0.964902	+	0.262611i	$0.915416\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	6.14359i		−	0.302306i
$414$		0			0
$415$	−52.2487			−2.56479
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.5885	+	10.7321i	0.908106	+	0.524295i	0.879821	−	0.475305i	$-0.157662\pi$
$419$	18.5885	+	10.7321i	0.908106	+	0.524295i	0.0282844	+	0.999600i	$0.490996\pi$
$420$		0			0
$421$	−11.2583	+	6.50000i	−0.548697	+	0.316791i	−0.748596	−	0.663026i	$-0.769272\pi$
$421$	−11.2583	+	6.50000i	−0.548697	+	0.316791i	0.199899	+	0.979817i	$0.435939\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	16.6603	+	28.8564i	0.808141	+	1.39974i
$426$		0			0
$427$	−5.70577	−	3.29423i	−0.276122	−	0.159419i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−21.7128			−1.04587			−0.522935	−	0.852373i	$-0.675163\pi$
$431$	−21.7128			−1.04587			−0.522935	+	0.852373i	$0.675163\pi$
$432$		0			0
$433$	−21.7846			−1.04690			−0.523451	−	0.852056i	$-0.675356\pi$
$433$	−21.7846			−1.04690			−0.523451	+	0.852056i	$0.675356\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	24.7128	+	14.2679i	1.18217	+	0.682529i
$438$		0			0
$439$	−15.0000	−	25.9808i	−0.715911	−	1.23999i	−0.962607	−	0.270901i	$-0.912678\pi$
$439$	−15.0000	−	25.9808i	−0.715911	−	1.23999i	0.246696	−	0.969093i	$-0.420655\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	25.3923	−	14.6603i	1.20642	−	0.696530i	0.244448	−	0.969662i	$-0.421393\pi$
$443$	25.3923	−	14.6603i	1.20642	−	0.696530i	0.961976	+	0.273133i	$0.0880598\pi$
$444$		0			0
$445$	−37.9186	−	21.8923i	−1.79751	−	1.03779i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	24.9282			1.17643			0.588217	−	0.808703i	$-0.299830\pi$
$449$	24.9282			1.17643			0.588217	+	0.808703i	$0.299830\pi$
$450$		0			0
$451$			21.4641i			1.01071i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.36603	+	5.83013i	−0.157802	+	0.273321i
$456$		0			0
$457$	3.16025	+	5.47372i	0.147830	+	0.256050i	0.930425	−	0.366481i	$-0.119438\pi$
$457$	3.16025	+	5.47372i	0.147830	+	0.256050i	−0.782595	+	0.622531i	$0.786104\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	35.7846	−	20.6603i	1.66666	−	0.962244i	0.697233	−	0.716844i	$-0.254414\pi$
$461$	35.7846	−	20.6603i	1.66666	−	0.962244i	0.969422	−	0.245400i	$-0.0789193\pi$
$462$		0			0
$463$	6.73205	−	11.6603i	0.312865	−	0.541898i	−0.666116	−	0.745848i	$-0.732045\pi$
$463$	6.73205	−	11.6603i	0.312865	−	0.541898i	0.978981	+	0.203950i	$0.0653779\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			24.7846i			1.14689i	0.819242	+	0.573447i	$0.194394\pi$
$467$			24.7846i			1.14689i	−0.819242	+	0.573447i	$0.805606\pi$
$468$		0			0
$469$		−	3.46410i		−	0.159957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	22.8564	−	39.5885i	1.05094	−	1.82028i
$474$		0			0
$475$	25.2679	−	14.5885i	1.15937	−	0.669364i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.830127	−	1.43782i	−0.0379295	−	0.0656958i	0.846437	−	0.532488i	$-0.178743\pi$
$479$	−0.830127	−	1.43782i	−0.0379295	−	0.0656958i	−0.884367	+	0.466792i	$0.845410\pi$
$480$		0			0
$481$	6.16025	−	10.6699i	0.280883	−	0.486504i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	29.8564i		−	1.35571i
$486$		0			0
$487$	25.7128			1.16516			0.582579	−	0.812774i	$-0.302044\pi$
$487$	25.7128			1.16516			0.582579	+	0.812774i	$0.302044\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11.9545	+	6.90192i	0.539498	+	0.311479i	0.744876	−	0.667203i	$-0.232509\pi$
$491$	11.9545	+	6.90192i	0.539498	+	0.311479i	−0.205377	+	0.978683i	$0.565842\pi$
$492$		0			0
$493$	−16.7942	+	9.69615i	−0.756374	+	0.436693i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.07180	−	3.58846i	−0.0929328	−	0.160964i
$498$		0			0
$499$	7.09808	+	4.09808i	0.317754	+	0.183455i	0.650391	−	0.759600i	$-0.274605\pi$
$499$	7.09808	+	4.09808i	0.317754	+	0.183455i	−0.332637	+	0.943055i	$0.607938\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.679492			−0.0302970			−0.0151485	−	0.999885i	$-0.504822\pi$
$503$	−0.679492			−0.0302970			−0.0151485	+	0.999885i	$0.504822\pi$
$504$		0			0
$505$	−29.8564			−1.32859
$506$		0			0
$507$		0			0
$508$		0			0
$509$	25.3923	+	14.6603i	1.12549	+	0.649804i	0.942798	−	0.333365i	$-0.108184\pi$
$509$	25.3923	+	14.6603i	1.12549	+	0.649804i	0.182696	+	0.983169i	$0.441517\pi$
$510$		0			0
$511$	−3.29423	−	5.70577i	−0.145728	−	0.252408i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	11.1962	−	6.46410i	0.493361	−	0.284842i
$516$		0			0
$517$	14.1962	+	8.19615i	0.624346	+	0.360466i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−26.7846			−1.17346			−0.586728	−	0.809784i	$-0.699584\pi$
$521$	−26.7846			−1.17346			−0.586728	+	0.809784i	$0.699584\pi$
$522$		0			0
$523$			13.2679i			0.580167i	0.957001	+	0.290083i	$0.0936831\pi$
$523$			13.2679i			0.580167i	−0.957001	+	0.290083i	$0.906317\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.73205	−	6.46410i	0.162571	−	0.281581i
$528$		0			0
$529$	−26.6244	−	46.1147i	−1.15758	−	2.00499i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.67949	−	5.58846i	0.419265	−	0.242063i
$534$		0			0
$535$	−19.3923	+	33.5885i	−0.838402	+	1.45216i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			30.5885i			1.31754i
$540$		0			0
$541$		−	24.1769i		−	1.03945i	−0.854335	−	0.519723i	$-0.826035\pi$
$541$		−	24.1769i		−	1.03945i	0.854335	−	0.519723i	$-0.173965\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	23.9904	−	41.5526i	1.02764	−	1.77992i
$546$		0			0
$547$	−22.5167	+	13.0000i	−0.962743	+	0.555840i	−0.897016	−	0.441998i	$-0.854270\pi$
$547$	−22.5167	+	13.0000i	−0.962743	+	0.555840i	−0.0657267	+	0.997838i	$0.520937\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.49038	+	14.7058i	0.361702	+	0.626487i
$552$		0			0
$553$	1.92820	−	3.33975i	0.0819955	−	0.142020i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			13.0526i			0.553055i	0.961006	+	0.276527i	$0.0891836\pi$
$557$			13.0526i			0.553055i	−0.961006	+	0.276527i	$0.910816\pi$
$558$		0			0
$559$	−23.8038			−1.00680
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.0000	+	12.1244i	0.885044	+	0.510981i	0.872318	−	0.488938i	$-0.162616\pi$
$563$	21.0000	+	12.1244i	0.885044	+	0.510981i	0.0127261	+	0.999919i	$0.495949\pi$
$564$		0			0
$565$	24.9904	−	14.4282i	1.05135	−	0.606999i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16.7942	−	29.0885i	−0.704051	−	1.21945i	−0.967033	−	0.254651i	$-0.918039\pi$
$569$	−16.7942	−	29.0885i	−0.704051	−	1.21945i	0.262982	−	0.964801i	$-0.415294\pi$
$570$		0			0
$571$	22.7321	+	13.1244i	0.951307	+	0.549237i	0.893487	−	0.449090i	$-0.148252\pi$
$571$	22.7321	+	13.1244i	0.951307	+	0.549237i	0.0578201	+	0.998327i	$0.481585\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−77.9615			−3.25122
$576$		0			0
$577$	−45.2487			−1.88373			−0.941864	−	0.335994i	$-0.890928\pi$
$577$	−45.2487			−1.88373			−0.941864	+	0.335994i	$0.890928\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8.87564	−	5.12436i	−0.368224	−	0.212594i
$582$		0			0
$583$	2.19615	+	3.80385i	0.0909553	+	0.157539i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	36.6340	−	21.1506i	1.51205	−	0.872980i	0.512145	−	0.858899i	$-0.328851\pi$
$587$	36.6340	−	21.1506i	1.51205	−	0.872980i	0.999901	−	0.0140812i	$-0.00448233\pi$
$588$		0			0
$589$	−5.66025	−	3.26795i	−0.233227	−	0.134654i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.4115			0.591811			0.295906	−	0.955217i	$-0.404379\pi$
$593$	14.4115			0.591811			0.295906	+	0.955217i	$0.404379\pi$
$594$		0			0
$595$			10.1962i			0.418001i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.12436	−	7.14359i	0.168517	−	0.291879i	−0.769382	−	0.638789i	$-0.779436\pi$
$599$	4.12436	−	7.14359i	0.168517	−	0.291879i	0.937899	+	0.346910i	$0.112769\pi$
$600$		0			0
$601$	−8.62436	−	14.9378i	−0.351795	−	0.609326i	0.634769	−	0.772702i	$-0.281095\pi$
$601$	−8.62436	−	14.9378i	−0.351795	−	0.609326i	−0.986564	+	0.163375i	$0.947762\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−36.8205	+	21.2583i	−1.49697	+	0.864274i
$606$		0			0
$607$	−12.9545	+	22.4378i	−0.525806	+	0.910723i	0.473742	+	0.880664i	$0.342903\pi$
$607$	−12.9545	+	22.4378i	−0.525806	+	0.910723i	−0.999548	+	0.0300594i	$0.990430\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	8.53590i		−	0.345325i
$612$		0			0
$613$			24.3923i			0.985196i	0.870257	+	0.492598i	$0.163953\pi$
$613$			24.3923i			0.985196i	−0.870257	+	0.492598i	$0.836047\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.0622	+	22.6244i	−0.525863	+	0.910822i	0.473683	+	0.880696i	$0.342924\pi$
$617$	−13.0622	+	22.6244i	−0.525863	+	0.910822i	−0.999546	+	0.0301266i	$0.990409\pi$
$618$		0			0
$619$	5.41154	−	3.12436i	0.217508	−	0.125578i	−0.387288	−	0.921959i	$-0.626588\pi$
$619$	5.41154	−	3.12436i	0.217508	−	0.125578i	0.604796	+	0.796380i	$0.293255\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.29423	−	7.43782i	−0.172045	−	0.297990i
$624$		0			0
$625$	−5.03590	+	8.72243i	−0.201436	+	0.348897i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	18.6603i		−	0.744033i
$630$		0			0
$631$	−0.784610			−0.0312348			−0.0156174	−	0.999878i	$-0.504971\pi$
$631$	−0.784610			−0.0312348			−0.0156174	+	0.999878i	$0.504971\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−44.1506	−	25.4904i	−1.75206	−	1.01155i
$636$		0			0
$637$	13.7942	−	7.96410i	0.546547	−	0.315549i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	11.5263	+	19.9641i	0.455261	+	0.788535i	0.998703	−	0.0509118i	$-0.0162127\pi$
$641$	11.5263	+	19.9641i	0.455261	+	0.788535i	−0.543442	+	0.839446i	$0.682879\pi$
$642$		0			0
$643$	−5.53590	−	3.19615i	−0.218315	−	0.126044i	0.386855	−	0.922141i	$-0.373561\pi$
$643$	−5.53590	−	3.19615i	−0.218315	−	0.126044i	−0.605170	+	0.796097i	$0.706895\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	49.7654			1.95648			0.978239	−	0.207480i	$-0.0665261\pi$
$647$	49.7654			1.95648			0.978239	+	0.207480i	$0.0665261\pi$
$648$		0			0
$649$	−39.7128			−1.55886
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−30.8038	−	17.7846i	−1.20545	−	0.695966i	−0.243686	−	0.969854i	$-0.578357\pi$
$653$	−30.8038	−	17.7846i	−1.20545	−	0.695966i	−0.961761	+	0.273889i	$0.911690\pi$
$654$		0			0
$655$	−0.366025	−	0.633975i	−0.0143018	−	0.0247714i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.70577	+	5.02628i	−0.339129	+	0.195796i	−0.659887	−	0.751365i	$-0.729396\pi$
$659$	−8.70577	+	5.02628i	−0.339129	+	0.195796i	0.320758	+	0.947161i	$0.396062\pi$
$660$		0			0
$661$	−27.8660	−	16.0885i	−1.08386	−	0.625768i	−0.151927	−	0.988392i	$-0.548548\pi$
$661$	−27.8660	−	16.0885i	−1.08386	−	0.625768i	−0.931936	+	0.362623i	$0.881881\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.92820			0.346221
$666$		0			0
$667$		−	45.3731i		−	1.75685i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−21.2942	+	36.8827i	−0.822055	+	1.42384i
$672$		0			0
$673$	18.6244	+	32.2583i	0.717916	+	1.24347i	0.961824	+	0.273669i	$0.0882372\pi$
$673$	18.6244	+	32.2583i	0.717916	+	1.24347i	−0.243908	+	0.969798i	$0.578429\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−25.3923	+	14.6603i	−0.975906	+	0.563439i	−0.901031	−	0.433754i	$-0.857189\pi$
$677$	−25.3923	+	14.6603i	−0.975906	+	0.563439i	−0.0748741	+	0.997193i	$0.523855\pi$
$678$		0			0
$679$	2.92820	−	5.07180i	0.112374	−	0.194638i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			34.1051i			1.30500i	0.757790	+	0.652498i	$0.226279\pi$
$683$			34.1051i			1.30500i	−0.757790	+	0.652498i	$0.773721\pi$
$684$		0			0
$685$			64.1769i			2.45207i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.14359	−	1.98076i	0.0435674	−	0.0754610i
$690$		0			0
$691$	−11.0263	+	6.36603i	−0.419459	+	0.242175i	−0.694846	−	0.719159i	$-0.744527\pi$
$691$	−11.0263	+	6.36603i	−0.419459	+	0.242175i	0.275387	+	0.961334i	$0.411194\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	36.3205	+	62.9090i	1.37772	+	2.38627i
$696$		0			0
$697$	8.46410	−	14.6603i	0.320601	−	0.555297i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	10.1244i		−	0.382392i	−0.981552	−	0.191196i	$-0.938763\pi$
$701$		−	10.1244i		−	0.382392i	0.981552	−	0.191196i	$-0.0612365\pi$
$702$		0			0
$703$	−16.3397			−0.616265
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.07180	−	2.92820i	−0.190745	−	0.110126i
$708$		0			0
$709$	−41.0429	+	23.6962i	−1.54140	+	0.889928i	−0.542649	+	0.839959i	$0.682579\pi$
$709$	−41.0429	+	23.6962i	−1.54140	+	0.889928i	−0.998751	+	0.0499682i	$0.984088\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8.73205	+	15.1244i	0.327018	+	0.566412i
$714$		0			0
$715$	37.6865	+	21.7583i	1.40940	+	0.813715i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−48.0526			−1.79206			−0.896029	−	0.443995i	$-0.853561\pi$
$719$	−48.0526			−1.79206			−0.896029	+	0.443995i	$0.853561\pi$
$720$		0			0
$721$	2.53590			0.0944418
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−40.1769	−	23.1962i	−1.49213	−	0.861483i
$726$		0			0
$727$	−24.7583	−	42.8827i	−0.918236	−	1.59043i	−0.802094	−	0.597198i	$-0.796281\pi$
$727$	−24.7583	−	42.8827i	−0.918236	−	1.59043i	−0.116142	−	0.993233i	$-0.537053\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−31.2224	+	18.0263i	−1.15480	+	0.666726i
$732$		0			0
$733$	25.5167	+	14.7321i	0.942479	+	0.544141i	0.890737	−	0.454520i	$-0.150189\pi$
$733$	25.5167	+	14.7321i	0.942479	+	0.544141i	0.0517427	+	0.998660i	$0.483522\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−22.3923			−0.824831
$738$		0			0
$739$			9.60770i			0.353425i	0.984263	+	0.176712i	$0.0565462\pi$
$739$			9.60770i			0.353425i	−0.984263	+	0.176712i	$0.943454\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.3923	−	26.6603i	0.564689	−	0.978070i	−0.432390	−	0.901687i	$-0.642329\pi$
$743$	15.3923	−	26.6603i	0.564689	−	0.978070i	0.997079	−	0.0763830i	$-0.0243372\pi$
$744$		0			0
$745$	−9.69615	−	16.7942i	−0.355240	−	0.615293i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.58846	+	3.80385i	−0.240737	+	0.138990i
$750$		0			0
$751$	−8.75833	+	15.1699i	−0.319596	+	0.553557i	−0.980404	−	0.196999i	$-0.936881\pi$
$751$	−8.75833	+	15.1699i	−0.319596	+	0.553557i	0.660808	+	0.750555i	$0.270214\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	51.7128i		−	1.88202i
$756$		0			0
$757$		−	18.2487i		−	0.663261i	−0.943409	−	0.331630i	$-0.892401\pi$
$757$		−	18.2487i		−	0.663261i	0.943409	−	0.331630i	$-0.107599\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	17.9904	−	31.1603i	0.652151	−	1.12956i	−0.330449	−	0.943824i	$-0.607200\pi$
$761$	17.9904	−	31.1603i	0.652151	−	1.12956i	0.982600	−	0.185735i	$-0.0594664\pi$
$762$		0			0
$763$	8.15064	−	4.70577i	0.295073	−	0.170360i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.3397	+	17.9090i	0.373347	+	0.646655i
$768$		0			0
$769$	19.6962	−	34.1147i	0.710261	−	1.23021i	−0.254497	−	0.967073i	$-0.581910\pi$
$769$	19.6962	−	34.1147i	0.710261	−	1.23021i	0.964759	−	0.263135i	$-0.0847566\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			40.6603i			1.46245i	0.682138	+	0.731224i	$0.261051\pi$
$773$			40.6603i			1.46245i	−0.682138	+	0.731224i	$0.738949\pi$
$774$		0			0
$775$	17.8564			0.641421
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.8372	−	7.41154i	−0.459939	−	0.265546i
$780$		0			0
$781$	−23.1962	+	13.3923i	−0.830024	+	0.479214i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.59808	−	7.96410i	−0.164112	−	0.284251i
$786$		0			0
$787$	−12.5096	−	7.22243i	−0.445920	−	0.257452i	0.260186	−	0.965559i	$-0.416216\pi$
$787$	−12.5096	−	7.22243i	−0.445920	−	0.257452i	−0.706105	+	0.708107i	$0.749550\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.66025			0.201255
$792$		0			0
$793$	22.1769			0.787525
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.6244	+	6.13397i	0.376334	+	0.217277i	0.676222	−	0.736698i	$-0.263616\pi$
$797$	10.6244	+	6.13397i	0.376334	+	0.217277i	−0.299888	+	0.953974i	$0.596949\pi$
$798$		0			0
$799$	−6.46410	−	11.1962i	−0.228683	−	0.396091i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−36.8827	+	21.2942i	−1.30156	+	0.751457i
$804$		0			0
$805$	−20.6603	−	11.9282i	−0.728178	−	0.420414i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	49.0526			1.72460			0.862298	−	0.506401i	$-0.169024\pi$
$809$	49.0526			1.72460			0.862298	+	0.506401i	$0.169024\pi$
$810$		0			0
$811$		−	48.7846i		−	1.71306i	−0.516098	−	0.856530i	$-0.672616\pi$
$811$		−	48.7846i		−	1.71306i	0.516098	−	0.856530i	$-0.327384\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−12.1962	+	21.1244i	−0.427213	+	0.739954i
$816$		0			0
$817$	15.7846	+	27.3397i	0.552234	+	0.956497i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.37564	+	0.794229i	−0.0480103	+	0.0277188i	−0.523813	−	0.851833i	$-0.675491\pi$
$821$	−1.37564	+	0.794229i	−0.0480103	+	0.0277188i	0.475803	+	0.879552i	$0.342158\pi$
$822$		0			0
$823$	−8.80385	+	15.2487i	−0.306883	+	0.531537i	−0.977679	−	0.210105i	$-0.932619\pi$
$823$	−8.80385	+	15.2487i	−0.306883	+	0.531537i	0.670796	+	0.741642i	$0.265953\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12.3397i			0.429095i	0.976714	+	0.214548i	$0.0688277\pi$
$827$			12.3397i			0.429095i	−0.976714	+	0.214548i	$0.931172\pi$
$828$		0			0
$829$		−	53.1769i		−	1.84691i	−0.383706	−	0.923455i	$-0.625352\pi$
$829$		−	53.1769i		−	1.84691i	0.383706	−	0.923455i	$-0.374648\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	12.0622	−	20.8923i	0.417930	−	0.723875i
$834$		0			0
$835$	−47.6147	+	27.4904i	−1.64778	+	0.951344i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−20.9019	−	36.2032i	−0.721615	−	1.24987i	−0.960352	−	0.278789i	$-0.910067\pi$
$839$	−20.9019	−	36.2032i	−0.721615	−	1.24987i	0.238738	−	0.971084i	$-0.423266\pi$
$840$		0			0
$841$	−1.00000	+	1.73205i	−0.0344828	+	0.0597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			25.8564i			0.889487i
$846$		0			0
$847$	−8.33975			−0.286557
$848$		0			0
$849$		0			0
$850$		0			0
$851$	37.8109	+	21.8301i	1.29614	+	0.748327i
$852$		0			0
$853$	4.39230	−	2.53590i	0.150390	−	0.0868275i	−0.422917	−	0.906168i	$-0.638994\pi$
$853$	4.39230	−	2.53590i	0.150390	−	0.0868275i	0.573307	+	0.819341i	$0.305660\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.0621778	+	0.107695i	0.00212395	+	0.00367880i	0.867085	−	0.498159i	$-0.165991\pi$
$857$	0.0621778	+	0.107695i	0.00212395	+	0.00367880i	−0.864961	+	0.501838i	$0.832657\pi$
$858$		0			0
$859$	−1.73205	−	1.00000i	−0.0590968	−	0.0341196i	0.470160	−	0.882581i	$-0.344196\pi$
$859$	−1.73205	−	1.00000i	−0.0590968	−	0.0341196i	−0.529257	+	0.848461i	$0.677529\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−25.5167			−0.868597			−0.434299	−	0.900769i	$-0.643004\pi$
$863$	−25.5167			−0.868597			−0.434299	+	0.900769i	$0.643004\pi$
$864$		0			0
$865$	−6.46410			−0.219786
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−21.5885	−	12.4641i	−0.732338	−	0.422816i
$870$		0			0
$871$	5.83013	+	10.0981i	0.197546	+	0.342160i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9.29423	+	5.36603i	−0.314202	+	0.181405i
$876$		0			0
$877$	49.4545	+	28.5526i	1.66996	+	0.964151i	0.967656	+	0.252274i	$0.0811783\pi$
$877$	49.4545	+	28.5526i	1.66996	+	0.964151i	0.702303	+	0.711878i	$0.252155\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27.8564			−0.938506			−0.469253	−	0.883064i	$-0.655477\pi$
$881$	−27.8564			−0.938506			−0.469253	+	0.883064i	$0.655477\pi$
$882$		0			0
$883$			1.32051i			0.0444386i	0.999753	+	0.0222193i	$0.00707321\pi$
$883$			1.32051i			0.0444386i	−0.999753	+	0.0222193i	$0.992927\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.09808	+	1.90192i	−0.0368698	+	0.0638604i	−0.883871	−	0.467730i	$-0.845072\pi$
$887$	−1.09808	+	1.90192i	−0.0368698	+	0.0638604i	0.847002	+	0.531590i	$0.178405\pi$
$888$		0			0
$889$	−5.00000	−	8.66025i	−0.167695	−	0.290456i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.80385	+	5.66025i	−0.328073	+	0.189413i
$894$		0			0
$895$	26.1244	−	45.2487i	0.873241	−	1.51250i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			10.3923i			0.346603i
$900$		0			0
$901$		−	3.46410i		−	0.115406i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.1962	+	21.1244i	−0.405414	+	0.702197i
$906$		0			0
$907$	28.0526	−	16.1962i	0.931470	−	0.537784i	0.0441938	−	0.999023i	$-0.485928\pi$
$907$	28.0526	−	16.1962i	0.931470	−	0.537784i	0.887276	+	0.461239i	$0.152595\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8.36603	−	14.4904i	−0.277179	−	0.480088i	0.693504	−	0.720453i	$-0.256066\pi$
$911$	−8.36603	−	14.4904i	−0.277179	−	0.480088i	−0.970683	+	0.240365i	$0.922733\pi$
$912$		0			0
$913$	−33.1244	+	57.3731i	−1.09626	+	1.89877i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	0.143594i		−	0.00474188i
$918$		0			0
$919$	−16.9808			−0.560144			−0.280072	−	0.959979i	$-0.590358\pi$
$919$	−16.9808			−0.560144			−0.280072	+	0.959979i	$0.590358\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.0788	+	6.97372i	0.397580	+	0.229543i
$924$		0			0
$925$	38.6603	−	22.3205i	1.27114	−	0.733894i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.258330	−	0.447441i	−0.00847554	−	0.0146801i	0.861757	−	0.507322i	$-0.169365\pi$
$929$	−0.258330	−	0.447441i	−0.00847554	−	0.0146801i	−0.870232	+	0.492642i	$0.836031\pi$
$930$		0			0
$931$	−18.2942	−	10.5622i	−0.599569	−	0.346161i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	65.9090			2.15545
$936$		0			0
$937$	−7.24871			−0.236805			−0.118403	−	0.992966i	$-0.537777\pi$
$937$	−7.24871			−0.236805			−0.118403	+	0.992966i	$0.537777\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15.9115	+	9.18653i	0.518701	+	0.299472i	0.736403	−	0.676543i	$-0.236523\pi$
$941$	15.9115	+	9.18653i	0.518701	+	0.299472i	−0.217702	+	0.976015i	$0.569856\pi$
$942$		0			0
$943$	19.8038	+	34.3013i	0.644902	+	1.11700i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.4904	−	10.0981i	0.568361	−	0.328143i	−0.188133	−	0.982143i	$-0.560244\pi$
$947$	17.4904	−	10.0981i	0.568361	−	0.328143i	0.756494	+	0.654000i	$0.226910\pi$
$948$		0			0
$949$	19.2058	+	11.0885i	0.623446	+	0.359947i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.6603			−0.863610			−0.431805	−	0.901967i	$-0.642123\pi$
$953$	−26.6603			−0.863610			−0.431805	+	0.901967i	$0.642123\pi$
$954$		0			0
$955$			61.9090i			2.00333i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.29423	+	10.9019i	−0.203251	+	0.352041i
$960$		0			0
$961$	13.5000	+	23.3827i	0.435484	+	0.754280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	35.0885	−	20.2583i	1.12954	−	0.652139i
$966$		0			0
$967$	5.09808	−	8.83013i	0.163943	−	0.283958i	−0.772336	−	0.635214i	$-0.780912\pi$
$967$	5.09808	−	8.83013i	0.163943	−	0.283958i	0.936279	+	0.351256i	$0.114245\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			51.1244i			1.64066i	0.571891	+	0.820329i	$0.306210\pi$
$971$			51.1244i			1.64066i	−0.571891	+	0.820329i	$0.693790\pi$
$972$		0			0
$973$			14.2487i			0.456793i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11.3923	+	19.7321i	−0.364472	+	0.631284i	−0.988691	−	0.149965i	$-0.952084\pi$
$977$	−11.3923	+	19.7321i	−0.364472	+	0.631284i	0.624219	+	0.781249i	$0.285417\pi$
$978$		0			0
$979$	−48.0788	+	27.7583i	−1.53661	+	0.887160i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	0.732051	+	1.26795i	0.0233488	+	0.0404413i	0.877464	−	0.479643i	$-0.159234\pi$
$983$	0.732051	+	1.26795i	0.0233488	+	0.0404413i	−0.854115	+	0.520084i	$0.825901\pi$
$984$		0			0
$985$	7.69615	−	13.3301i	0.245220	−	0.424733i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	84.3538i		−	2.68230i
$990$		0			0
$991$	−2.58846			−0.0822251			−0.0411125	−	0.999155i	$-0.513090\pi$
$991$	−2.58846			−0.0822251			−0.0411125	+	0.999155i	$0.513090\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.73205	+	4.46410i	0.245122	+	0.141522i
$996$		0			0
$997$	24.1865	−	13.9641i	0.765995	−	0.442248i	−0.0654489	−	0.997856i	$-0.520848\pi$
$997$	24.1865	−	13.9641i	0.765995	−	0.442248i	0.831444	+	0.555608i	$0.187515\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.r.b.433.1		4
3.2	odd	2		2592.2.r.k.433.2		4
4.3	odd	2		648.2.n.b.109.2		4
8.3	odd	2		648.2.n.m.109.2		4
8.5	even	2		2592.2.r.j.433.2		4
9.2	odd	6		2592.2.r.a.2161.1		4
9.4	even	3		2592.2.d.h.1297.4		4
9.5	odd	6		2592.2.d.g.1297.1		4
9.7	even	3		2592.2.r.j.2161.2		4
12.11	even	2		648.2.n.l.109.1		4
24.5	odd	2		2592.2.r.a.433.1		4
24.11	even	2		648.2.n.a.109.1		4
36.7	odd	6		648.2.n.m.541.1		4
36.11	even	6		648.2.n.a.541.2		4
36.23	even	6		648.2.d.i.325.4	yes	4
36.31	odd	6		648.2.d.e.325.1	✓	4
72.5	odd	6		2592.2.d.g.1297.4		4
72.11	even	6		648.2.n.l.541.1		4
72.13	even	6		2592.2.d.h.1297.1		4
72.29	odd	6		2592.2.r.k.2161.2		4
72.43	odd	6		648.2.n.b.541.2		4
72.59	even	6		648.2.d.i.325.3	yes	4
72.61	even	6	inner	2592.2.r.b.2161.1		4
72.67	odd	6		648.2.d.e.325.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.d.e.325.1	✓	4	36.31	odd	6
648.2.d.e.325.2	yes	4	72.67	odd	6
648.2.d.i.325.3	yes	4	72.59	even	6
648.2.d.i.325.4	yes	4	36.23	even	6
648.2.n.a.109.1		4	24.11	even	2
648.2.n.a.541.2		4	36.11	even	6
648.2.n.b.109.2		4	4.3	odd	2
648.2.n.b.541.2		4	72.43	odd	6
648.2.n.l.109.1		4	12.11	even	2
648.2.n.l.541.1		4	72.11	even	6
648.2.n.m.109.2		4	8.3	odd	2
648.2.n.m.541.1		4	36.7	odd	6
2592.2.d.g.1297.1		4	9.5	odd	6
2592.2.d.g.1297.4		4	72.5	odd	6
2592.2.d.h.1297.1		4	72.13	even	6
2592.2.d.h.1297.4		4	9.4	even	3
2592.2.r.a.433.1		4	24.5	odd	2
2592.2.r.a.2161.1		4	9.2	odd	6
2592.2.r.b.433.1		4	1.1	even	1	trivial
2592.2.r.b.2161.1		4	72.61	even	6	inner
2592.2.r.j.433.2		4	8.5	even	2
2592.2.r.j.2161.2		4	9.7	even	3
2592.2.r.k.433.2		4	3.2	odd	2
2592.2.r.k.2161.2		4	72.29	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2592.2.r.b.433.1

Level	$2592$
Weight	$2$
Character	2592.433
Analytic conductor	$20.697$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$