LMFDB - Embedded newform 1728.3.q.d.1601.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(449,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1728.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$1728 = 2^{6} \cdot 3^{3}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	1728.q (of order $6$ , 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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{2} + 4$ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$3^{2}$
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1601.1
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1728.1601
Dual form			1728.3.q.d.449.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 2.59808i) q^{5} +(-3.17423 + 5.49794i) q^{7} +(8.17423 + 4.71940i) q^{11} +(9.84847 + 17.0580i) q^{13} +1.90702i q^{17} -4.69694 q^{19} +(-8.17423 + 4.71940i) q^{23} +(1.00000 - 1.73205i) q^{25} +(-2.84847 - 1.64456i) q^{29} +(20.5227 + 35.5464i) q^{31} -32.9876i q^{35} -17.3031 q^{37} +(53.5454 - 30.9145i) q^{41} +(0.477296 - 0.826701i) q^{43} +(12.2196 + 7.05501i) q^{47} +(4.34847 + 7.53177i) q^{49} -9.53512i q^{53} -49.0454 q^{55} +(79.2650 - 45.7637i) q^{59} +(-37.5454 + 65.0306i) q^{61} +(-88.6362 - 51.1741i) q^{65} +(15.4773 + 26.8075i) q^{67} -85.9026i q^{71} -96.0908 q^{73} +(-51.8939 + 29.9609i) q^{77} +(-14.8712 + 25.7576i) q^{79} +(-76.1288 - 43.9530i) q^{83} +(-4.95459 - 8.58161i) q^{85} +41.3766i q^{89} -125.045 q^{91} +(21.1362 - 12.2030i) q^{95} +(-47.9393 + 83.0333i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 18 q^{5} + 2 q^{7} + 18 q^{11} + 10 q^{13} + 40 q^{19} - 18 q^{23} + 4 q^{25} + 18 q^{29} + 38 q^{31} - 128 q^{37} + 126 q^{41} + 46 q^{43} - 54 q^{47} - 12 q^{49} - 108 q^{55} + 126 q^{59} - 62 q^{61}+ \cdots + 14 q^{97}+O(q^{100})$ 4 * q - 18 * q^5 + 2 * q^7 + 18 * q^11 + 10 * q^13 + 40 * q^19 - 18 * q^23 + 4 * q^25 + 18 * q^29 + 38 * q^31 - 128 * q^37 + 126 * q^41 + 46 * q^43 - 54 * q^47 - 12 * q^49 - 108 * q^55 + 126 * q^59 - 62 * q^61 - 90 * q^65 + 106 * q^67 - 208 * q^73 - 90 * q^77 + 14 * q^79 - 378 * q^83 - 108 * q^85 - 412 * q^91 - 180 * q^95 + 14 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$ .

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	−4.50000	+	2.59808i	−0.900000	+	0.519615i	−0.877200	−	0.480125i	$-0.840591\pi$
$5$	−4.50000	+	2.59808i	−0.900000	+	0.519615i	−0.0227998	+	0.999740i	$0.507258\pi$
$6$		0			0
$7$	−3.17423	+	5.49794i	−0.453462	+	0.785419i	−0.998598	−	0.0529281i	$-0.983145\pi$
$7$	−3.17423	+	5.49794i	−0.453462	+	0.785419i	0.545136	+	0.838347i	$0.316478\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.17423	+	4.71940i	0.743112	+	0.429036i	0.823200	−	0.567752i	$-0.192187\pi$
$11$	8.17423	+	4.71940i	0.743112	+	0.429036i	−0.0800876	+	0.996788i	$0.525520\pi$
$12$		0			0
$13$	9.84847	+	17.0580i	0.757575	+	1.31216i	0.944084	+	0.329704i	$0.106949\pi$
$13$	9.84847	+	17.0580i	0.757575	+	1.31216i	−0.186510	+	0.982453i	$0.559718\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.90702i			0.112178i	0.998426	+	0.0560889i	$0.0178630\pi$
$17$			1.90702i			0.112178i	−0.998426	+	0.0560889i	$0.982137\pi$
$18$		0			0
$19$	−4.69694			−0.247207			−0.123604	−	0.992332i	$-0.539445\pi$
$19$	−4.69694			−0.247207			−0.123604	+	0.992332i	$0.539445\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.17423	+	4.71940i	−0.355402	+	0.205191i	−0.667062	−	0.745002i	$-0.732448\pi$
$23$	−8.17423	+	4.71940i	−0.355402	+	0.205191i	0.311660	+	0.950194i	$0.399115\pi$
$24$		0			0
$25$	1.00000	−	1.73205i	0.0400000	−	0.0692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.84847	−	1.64456i	−0.0982231	−	0.0567091i	0.450084	−	0.892986i	$-0.351394\pi$
$29$	−2.84847	−	1.64456i	−0.0982231	−	0.0567091i	−0.548307	+	0.836277i	$0.684727\pi$
$30$		0			0
$31$	20.5227	+	35.5464i	0.662023	+	1.14666i	0.980083	+	0.198587i	$0.0636351\pi$
$31$	20.5227	+	35.5464i	0.662023	+	1.14666i	−0.318061	+	0.948070i	$0.603032\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	32.9876i		−	0.942503i
$36$		0			0
$37$	−17.3031			−0.467650			−0.233825	−	0.972279i	$-0.575124\pi$
$37$	−17.3031			−0.467650			−0.233825	+	0.972279i	$0.575124\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	53.5454	−	30.9145i	1.30599	−	0.754011i	0.324562	−	0.945864i	$-0.394783\pi$
$41$	53.5454	−	30.9145i	1.30599	−	0.754011i	0.981424	+	0.191853i	$0.0614498\pi$
$42$		0			0
$43$	0.477296	−	0.826701i	0.0110999	−	0.0192256i	−0.860422	−	0.509582i	$-0.829800\pi$
$43$	0.477296	−	0.826701i	0.0110999	−	0.0192256i	0.871522	+	0.490356i	$0.163133\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	12.2196	+	7.05501i	0.259992	+	0.150107i	0.624331	−	0.781160i	$-0.285372\pi$
$47$	12.2196	+	7.05501i	0.259992	+	0.150107i	−0.364339	+	0.931267i	$0.618705\pi$
$48$		0			0
$49$	4.34847	+	7.53177i	0.0887443	+	0.153710i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.53512i		−	0.179908i	−0.995946	−	0.0899539i	$-0.971328\pi$
$53$		−	9.53512i		−	0.179908i	0.995946	−	0.0899539i	$-0.0286720\pi$
$54$		0			0
$55$	−49.0454			−0.891735
$56$		0			0
$57$		0			0
$58$		0			0
$59$	79.2650	−	45.7637i	1.34348	−	0.775656i	0.356160	−	0.934425i	$-0.384086\pi$
$59$	79.2650	−	45.7637i	1.34348	−	0.775656i	0.987316	+	0.158769i	$0.0507526\pi$
$60$		0			0
$61$	−37.5454	+	65.0306i	−0.615498	+	1.06607i	0.374798	+	0.927106i	$0.377712\pi$
$61$	−37.5454	+	65.0306i	−0.615498	+	1.06607i	−0.990297	+	0.138968i	$0.955621\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−88.6362	−	51.1741i	−1.36363	−	0.787295i
$66$		0			0
$67$	15.4773	+	26.8075i	0.231004	+	0.400111i	0.958104	−	0.286421i	$-0.0924655\pi$
$67$	15.4773	+	26.8075i	0.231004	+	0.400111i	−0.727100	+	0.686532i	$0.759132\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	85.9026i		−	1.20990i	−0.796265	−	0.604948i	$-0.793194\pi$
$71$		−	85.9026i		−	1.20990i	0.796265	−	0.604948i	$-0.206806\pi$
$72$		0			0
$73$	−96.0908			−1.31631			−0.658156	−	0.752881i	$-0.728663\pi$
$73$	−96.0908			−1.31631			−0.658156	+	0.752881i	$0.728663\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−51.8939	+	29.9609i	−0.673946	+	0.389103i
$78$		0			0
$79$	−14.8712	+	25.7576i	−0.188243	+	0.326046i	−0.944664	−	0.328038i	$-0.893612\pi$
$79$	−14.8712	+	25.7576i	−0.188243	+	0.326046i	0.756422	+	0.654084i	$0.226946\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−76.1288	−	43.9530i	−0.917215	−	0.529554i	−0.0344693	−	0.999406i	$-0.510974\pi$
$83$	−76.1288	−	43.9530i	−0.917215	−	0.529554i	−0.882745	+	0.469852i	$0.844307\pi$
$84$		0			0
$85$	−4.95459	−	8.58161i	−0.0582893	−	0.100960i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			41.3766i			0.464905i	0.972608	+	0.232453i	$0.0746751\pi$
$89$			41.3766i			0.464905i	−0.972608	+	0.232453i	$0.925325\pi$
$90$		0			0
$91$	−125.045			−1.37413
$92$		0			0
$93$		0			0
$94$		0			0
$95$	21.1362	−	12.2030i	0.222487	−	0.128453i
$96$		0			0
$97$	−47.9393	+	83.0333i	−0.494219	+	0.856013i	−0.999978	−	0.00666202i	$-0.997879\pi$
$97$	−47.9393	+	83.0333i	−0.494219	+	0.856013i	0.505758	+	0.862675i	$0.331213\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−136.772	−	78.9656i	−1.35418	−	0.781838i	−0.365350	−	0.930870i	$-0.619051\pi$
$101$	−136.772	−	78.9656i	−1.35418	−	0.781838i	−0.988832	+	0.149032i	$0.952384\pi$
$102$		0			0
$103$	−14.5681	−	25.2327i	−0.141438	−	0.244978i	0.786600	−	0.617462i	$-0.211839\pi$
$103$	−14.5681	−	25.2327i	−0.141438	−	0.244978i	−0.928038	+	0.372485i	$0.878506\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			171.805i			1.60566i	0.596210	+	0.802829i	$0.296673\pi$
$107$			171.805i			1.60566i	−0.596210	+	0.802829i	$0.703327\pi$
$108$		0			0
$109$	−116.272			−1.06672			−0.533360	−	0.845888i	$-0.679071\pi$
$109$	−116.272			−1.06672			−0.533360	+	0.845888i	$0.679071\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−175.166	+	101.132i	−1.55014	+	0.894976i	−0.552015	+	0.833834i	$0.686141\pi$
$113$	−175.166	+	101.132i	−1.55014	+	0.894976i	−0.998129	+	0.0611424i	$0.980526\pi$
$114$		0			0
$115$	24.5227	−	42.4746i	0.213241	−	0.369344i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−10.4847	−	6.05334i	−0.0881067	−	0.0508684i
$120$		0			0
$121$	−15.9546	−	27.6342i	−0.131856	−	0.228382i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	119.512i		−	0.956092i
$126$		0			0
$127$	10.0908			0.0794552			0.0397276	−	0.999211i	$-0.487351\pi$
$127$	10.0908			0.0794552			0.0397276	+	0.999211i	$0.487351\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.29567	−	2.48010i	0.0327913	−	0.0189321i	−0.483515	−	0.875336i	$-0.660640\pi$
$131$	4.29567	−	2.48010i	0.0327913	−	0.0189321i	0.516306	+	0.856404i	$0.327307\pi$
$132$		0			0
$133$	14.9092	−	25.8235i	0.112099	−	0.194161i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−203.242	−	117.342i	−1.48352	−	0.856511i	−0.483696	−	0.875236i	$-0.660706\pi$
$137$	−203.242	−	117.342i	−1.48352	−	0.856511i	−0.999825	+	0.0187249i	$0.994039\pi$
$138$		0			0
$139$	53.2650	+	92.2578i	0.383202	+	0.663725i	0.991518	−	0.129970i	$-0.0414881\pi$
$139$	53.2650	+	92.2578i	0.383202	+	0.663725i	−0.608316	+	0.793695i	$0.708155\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			185.915i			1.30011i
$144$		0			0
$145$	17.0908			0.117868
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−91.0301	+	52.5563i	−0.610940	+	0.352727i	−0.773333	−	0.634000i	$-0.781412\pi$
$149$	−91.0301	+	52.5563i	−0.610940	+	0.352727i	0.162393	+	0.986726i	$0.448079\pi$
$150$		0			0
$151$	142.614	−	247.014i	0.944460	−	1.63585i	0.187632	−	0.982239i	$-0.439919\pi$
$151$	142.614	−	247.014i	0.944460	−	1.63585i	0.756828	−	0.653614i	$-0.226748\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−184.704	−	106.639i	−1.19164	−	0.687994i
$156$		0			0
$157$	−98.5908	−	170.764i	−0.627967	−	1.08767i	−0.987959	−	0.154715i	$-0.950554\pi$
$157$	−98.5908	−	170.764i	−0.627967	−	1.08767i	0.359992	−	0.932955i	$-0.382779\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	59.9219i		−	0.372186i
$162$		0			0
$163$	249.060			1.52798			0.763988	−	0.645230i	$-0.223238\pi$
$163$	249.060			1.52798			0.763988	+	0.645230i	$0.223238\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	41.9472	−	24.2182i	0.251181	−	0.145019i	−0.369124	−	0.929380i	$-0.620342\pi$
$167$	41.9472	−	24.2182i	0.251181	−	0.145019i	0.620305	+	0.784361i	$0.287009\pi$
$168$		0			0
$169$	−109.485	+	189.633i	−0.647838	+	1.12209i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	86.9847	+	50.2206i	0.502802	+	0.290293i	0.729870	−	0.683586i	$-0.239581\pi$
$173$	86.9847	+	50.2206i	0.502802	+	0.290293i	−0.227068	+	0.973879i	$0.572914\pi$
$174$		0			0
$175$	6.34847	+	10.9959i	0.0362770	+	0.0628336i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			285.071i			1.59257i	0.604919	+	0.796287i	$0.293206\pi$
$179$			285.071i			1.59257i	−0.604919	+	0.796287i	$0.706794\pi$
$180$		0			0
$181$	−37.1214			−0.205091			−0.102545	−	0.994728i	$-0.532699\pi$
$181$	−37.1214			−0.205091			−0.102545	+	0.994728i	$0.532699\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	77.8638	−	44.9547i	0.420885	−	0.242998i
$186$		0			0
$187$	−9.00000	+	15.5885i	−0.0481283	+	0.0833607i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	15.5227	+	8.96204i	0.0812707	+	0.0469217i	0.540085	−	0.841611i	$-0.318392\pi$
$191$	15.5227	+	8.96204i	0.0812707	+	0.0469217i	−0.458814	+	0.888532i	$0.651726\pi$
$192$		0			0
$193$	47.7270	+	82.6657i	0.247290	+	0.428319i	0.962773	−	0.270311i	$-0.0871265\pi$
$193$	47.7270	+	82.6657i	0.247290	+	0.428319i	−0.715483	+	0.698630i	$0.753793\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	160.363i		−	0.814026i	−0.913422	−	0.407013i	$-0.866570\pi$
$197$		−	160.363i		−	0.814026i	0.913422	−	0.407013i	$-0.133430\pi$
$198$		0			0
$199$	6.51531			0.0327402			0.0163701	−	0.999866i	$-0.494789\pi$
$199$	6.51531			0.0327402			0.0163701	+	0.999866i	$0.494789\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	18.0834	−	10.4405i	0.0890809	−	0.0514309i
$204$		0			0
$205$	−160.636	+	278.230i	−0.783591	+	1.35722i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−38.3939	−	22.1667i	−0.183703	−	0.106061i
$210$		0			0
$211$	−77.2196	−	133.748i	−0.365970	−	0.633878i	0.622961	−	0.782253i	$-0.285929\pi$
$211$	−77.2196	−	133.748i	−0.365970	−	0.633878i	−0.988931	+	0.148374i	$0.952596\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			4.96021i			0.0230707i
$216$		0			0
$217$	−260.576			−1.20081
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−32.5301	+	18.7813i	−0.147195	+	0.0849831i
$222$		0			0
$223$	−46.3865	+	80.3437i	−0.208011	+	0.360286i	−0.951088	−	0.308920i	$-0.900032\pi$
$223$	−46.3865	+	80.3437i	−0.208011	+	0.360286i	0.743077	+	0.669206i	$0.233366\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	147.053	+	84.9010i	0.647810	+	0.374013i	0.787617	−	0.616166i	$-0.211315\pi$
$227$	147.053	+	84.9010i	0.647810	+	0.374013i	−0.139807	+	0.990179i	$0.544648\pi$
$228$		0			0
$229$	203.772	+	352.944i	0.889836	+	1.54124i	0.840068	+	0.542480i	$0.182515\pi$
$229$	203.772	+	352.944i	0.889836	+	1.54124i	0.0497675	+	0.998761i	$0.484152\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			15.2562i			0.0654772i	0.999464	+	0.0327386i	$0.0104229\pi$
$233$			15.2562i			0.0654772i	−0.999464	+	0.0327386i	$0.989577\pi$
$234$		0			0
$235$	−73.3179			−0.311991
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−48.9620	+	28.2682i	−0.204862	+	0.118277i	−0.598921	−	0.800808i	$-0.704404\pi$
$239$	−48.9620	+	28.2682i	−0.204862	+	0.118277i	0.394059	+	0.919085i	$0.371070\pi$
$240$		0			0
$241$	−42.1061	+	72.9299i	−0.174714	+	0.302614i	−0.940062	−	0.341003i	$-0.889233\pi$
$241$	−42.1061	+	72.9299i	−0.174714	+	0.302614i	0.765348	+	0.643617i	$0.222567\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−39.1362	−	22.5953i	−0.159740	−	0.0922258i
$246$		0			0
$247$	−46.2577	−	80.1206i	−0.187278	−	0.324375i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	218.903i		−	0.872123i	−0.899917	−	0.436062i	$-0.856373\pi$
$251$		−	218.903i		−	0.872123i	0.899917	−	0.436062i	$-0.143627\pi$
$252$		0			0
$253$	−89.0908			−0.352138
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.1061	−	6.41212i	0.0432145	−	0.0249499i	−0.478237	−	0.878231i	$-0.658724\pi$
$257$	11.1061	−	6.41212i	0.0432145	−	0.0249499i	0.521452	+	0.853281i	$0.325391\pi$
$258$		0			0
$259$	54.9240	−	95.1311i	0.212062	−	0.367302i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−291.386	−	168.232i	−1.10793	−	0.639666i	−0.169640	−	0.985506i	$-0.554261\pi$
$263$	−291.386	−	168.232i	−1.10793	−	0.639666i	−0.938293	+	0.345840i	$0.887594\pi$
$264$		0			0
$265$	24.7730	+	42.9080i	0.0934829	+	0.161917i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	60.4468i		−	0.224709i	−0.993668	−	0.112355i	$-0.964161\pi$
$269$		−	60.4468i		−	0.224709i	0.993668	−	0.112355i	$-0.0358393\pi$
$270$		0			0
$271$	274.636			1.01342			0.506708	−	0.862118i	$-0.330862\pi$
$271$	274.636			1.01342			0.506708	+	0.862118i	$0.330862\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	16.3485	−	9.43879i	0.0594490	−	0.0343229i
$276$		0			0
$277$	−24.5000	+	42.4352i	−0.0884477	+	0.153196i	−0.906855	−	0.421442i	$-0.861524\pi$
$277$	−24.5000	+	42.4352i	−0.0884477	+	0.153196i	0.818407	+	0.574638i	$0.194857\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	297.121	+	171.543i	1.05737	+	0.610473i	0.924704	−	0.380688i	$-0.124313\pi$
$281$	297.121	+	171.543i	1.05737	+	0.610473i	0.132666	+	0.991161i	$0.457646\pi$
$282$		0			0
$283$	−171.704	−	297.401i	−0.606729	−	1.05089i	−0.991776	−	0.127988i	$-0.959148\pi$
$283$	−171.704	−	297.401i	−0.606729	−	1.05089i	0.385047	−	0.922897i	$-0.374185\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			392.519i			1.36766i
$288$		0			0
$289$	285.363			0.987416
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−248.076	+	143.226i	−0.846674	+	0.488828i	−0.859527	−	0.511090i	$-0.829242\pi$
$293$	−248.076	+	143.226i	−0.846674	+	0.488828i	0.0128532	+	0.999917i	$0.495909\pi$
$294$		0			0
$295$	−237.795	+	411.873i	−0.806085	+	1.39618i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−161.007	−	92.9577i	−0.538486	−	0.310895i
$300$		0			0
$301$	3.03010	+	5.24829i	0.0100668	+	0.0174362i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	390.183i		−	1.27929i
$306$		0			0
$307$	−154.091			−0.501924			−0.250962	−	0.967997i	$-0.580747\pi$
$307$	−154.091			−0.501924			−0.250962	+	0.967997i	$0.580747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	62.3411	−	35.9926i	0.200454	−	0.115732i	−0.396413	−	0.918072i	$-0.629745\pi$
$311$	62.3411	−	35.9926i	0.200454	−	0.115732i	0.596867	+	0.802340i	$0.296412\pi$
$312$		0			0
$313$	183.803	−	318.356i	0.587230	−	1.01711i	−0.407363	−	0.913266i	$-0.633552\pi$
$313$	183.803	−	318.356i	0.587230	−	1.01711i	0.994593	−	0.103846i	$-0.0331150\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−93.1821	−	53.7987i	−0.293950	−	0.169712i	0.345772	−	0.938319i	$-0.387617\pi$
$317$	−93.1821	−	53.7987i	−0.293950	−	0.169712i	−0.639722	+	0.768607i	$0.720950\pi$
$318$		0			0
$319$	−15.5227	−	26.8861i	−0.0486605	−	0.0842825i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	8.95717i		−	0.0277312i
$324$		0			0
$325$	39.3939			0.121212
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−77.5760	+	44.7885i	−0.235793	+	0.136135i
$330$		0			0
$331$	8.59873	−	14.8934i	0.0259780	−	0.0449953i	−0.852744	−	0.522329i	$-0.825063\pi$
$331$	8.59873	−	14.8934i	0.0259780	−	0.0449953i	0.878722	+	0.477334i	$0.158397\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−139.296	−	80.4224i	−0.415808	−	0.240067i
$336$		0			0
$337$	−182.197	−	315.574i	−0.540644	−	0.936422i	−0.998867	−	0.0475854i	$-0.984847\pi$
$337$	−182.197	−	315.574i	−0.540644	−	0.936422i	0.458223	−	0.888837i	$-0.348486\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			387.419i			1.13613i
$342$		0			0
$343$	−366.287			−1.06789
$344$		0			0
$345$		0			0
$346$		0			0
$347$	505.234	−	291.697i	1.45601	−	0.840626i	0.457196	−	0.889366i	$-0.348854\pi$
$347$	505.234	−	291.697i	1.45601	−	0.840626i	0.998811	+	0.0487402i	$0.0155206\pi$
$348$		0			0
$349$	156.379	−	270.856i	0.448076	−	0.776091i	−0.550185	−	0.835043i	$-0.685443\pi$
$349$	156.379	−	270.856i	0.448076	−	0.776091i	0.998261	+	0.0589524i	$0.0187760\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	32.5760	+	18.8078i	0.0922834	+	0.0532798i	0.545431	−	0.838155i	$-0.316366\pi$
$353$	32.5760	+	18.8078i	0.0922834	+	0.0532798i	−0.453148	+	0.891435i	$0.649699\pi$
$354$		0			0
$355$	223.182	+	386.562i	0.628681	+	1.08891i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	294.028i		−	0.819019i	−0.912306	−	0.409510i	$-0.865700\pi$
$359$		−	294.028i		−	0.819019i	0.912306	−	0.409510i	$-0.134300\pi$
$360$		0			0
$361$	−338.939			−0.938889
$362$		0			0
$363$		0			0
$364$		0			0
$365$	432.409	−	249.651i	1.18468	−	0.683976i
$366$		0			0
$367$	16.6135	−	28.7755i	0.0452684	−	0.0784072i	−0.842503	−	0.538691i	$-0.818919\pi$
$367$	16.6135	−	28.7755i	0.0452684	−	0.0784072i	0.887772	+	0.460284i	$0.152252\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	52.4235	+	30.2667i	0.141303	+	0.0815814i
$372$		0			0
$373$	−112.515	−	194.881i	−0.301648	−	0.522470i	0.674861	−	0.737945i	$-0.264203\pi$
$373$	−112.515	−	194.881i	−0.301648	−	0.522470i	−0.976509	+	0.215475i	$0.930870\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	64.7858i		−	0.171846i
$378$		0			0
$379$	166.334			0.438875			0.219438	−	0.975627i	$-0.429578\pi$
$379$	166.334			0.438875			0.219438	+	0.975627i	$0.429578\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	638.249	−	368.493i	1.66645	−	0.962124i	0.696917	−	0.717152i	$-0.254555\pi$
$383$	638.249	−	368.493i	1.66645	−	0.962124i	0.969530	−	0.244972i	$-0.0787787\pi$
$384$		0			0
$385$	155.682	−	269.648i	0.404368	−	0.700386i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−146.682	−	84.6867i	−0.377074	−	0.217704i	0.299471	−	0.954106i	$-0.403190\pi$
$389$	−146.682	−	84.6867i	−0.377074	−	0.217704i	−0.676544	+	0.736402i	$0.736523\pi$
$390$		0			0
$391$	−9.00000	−	15.5885i	−0.0230179	−	0.0398682i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	154.546i		−	0.391255i
$396$		0			0
$397$	256.272			0.645523			0.322761	−	0.946480i	$-0.395389\pi$
$397$	256.272			0.645523			0.322761	+	0.946480i	$0.395389\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−226.364	+	130.691i	−0.564498	+	0.325913i	−0.754949	−	0.655784i	$-0.772338\pi$
$401$	−226.364	+	130.691i	−0.564498	+	0.325913i	0.190451	+	0.981697i	$0.439005\pi$
$402$		0			0
$403$	−404.234	+	700.155i	−1.00306	+	1.73736i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−141.439	−	81.6600i	−0.347517	−	0.200639i
$408$		0			0
$409$	221.894	+	384.331i	0.542528	+	0.939686i	0.998758	+	0.0498240i	$0.0158660\pi$
$409$	221.894	+	384.331i	0.542528	+	0.939686i	−0.456230	+	0.889862i	$0.650801\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			581.059i			1.40692i
$414$		0			0
$415$	456.773			1.10066
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.32525	−	5.38394i	0.0222560	−	0.0128495i	−0.488831	−	0.872379i	$-0.662576\pi$
$419$	9.32525	−	5.38394i	0.0222560	−	0.0128495i	0.511087	+	0.859529i	$0.329243\pi$
$420$		0			0
$421$	127.152	−	220.233i	0.302023	−	0.523119i	−0.674571	−	0.738210i	$-0.735672\pi$
$421$	127.152	−	220.233i	0.302023	−	0.523119i	0.976594	+	0.215091i	$0.0690048\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.30306	+	1.90702i	0.00777191	+	0.00448711i
$426$		0			0
$427$	−238.356	−	412.844i	−0.558210	−	0.966849i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			698.663i			1.62103i	0.585719	+	0.810514i	$0.300812\pi$
$431$			698.663i			1.62103i	−0.585719	+	0.810514i	$0.699188\pi$
$432$		0			0
$433$	211.728			0.488978			0.244489	−	0.969652i	$-0.421380\pi$
$433$	211.728			0.488978			0.244489	+	0.969652i	$0.421380\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	38.3939	−	22.1667i	0.0878578	−	0.0507247i
$438$		0			0
$439$	−139.931	+	242.368i	−0.318750	+	0.552092i	−0.980228	−	0.197874i	$-0.936596\pi$
$439$	−139.931	+	242.368i	−0.318750	+	0.552092i	0.661477	+	0.749965i	$0.269930\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−477.400	−	275.627i	−1.07765	−	0.622183i	−0.147391	−	0.989078i	$-0.547087\pi$
$443$	−477.400	−	275.627i	−1.07765	−	0.622183i	−0.930262	+	0.366895i	$0.880421\pi$
$444$		0			0
$445$	−107.499	−	186.195i	−0.241572	−	0.418415i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			542.865i			1.20905i	0.796585	+	0.604527i	$0.206638\pi$
$449$			542.865i			1.20905i	−0.796585	+	0.604527i	$0.793362\pi$
$450$		0			0
$451$	583.590			1.29399
$452$		0			0
$453$		0			0
$454$		0			0
$455$	562.704	−	324.877i	1.23671	−	0.714016i
$456$		0			0
$457$	−46.1821	+	79.9898i	−0.101055	+	0.175032i	−0.912120	−	0.409924i	$-0.865555\pi$
$457$	−46.1821	+	79.9898i	−0.101055	+	0.175032i	0.811065	+	0.584957i	$0.198888\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−199.030	−	114.910i	−0.431736	−	0.249263i	0.268350	−	0.963321i	$-0.413522\pi$
$461$	−199.030	−	114.910i	−0.431736	−	0.249263i	−0.700086	+	0.714059i	$0.746855\pi$
$462$		0			0
$463$	255.401	+	442.368i	0.551623	+	0.955438i	0.998158	+	0.0606723i	$0.0193245\pi$
$463$	255.401	+	442.368i	0.551623	+	0.955438i	−0.446535	+	0.894766i	$0.647342\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			833.657i			1.78513i	0.450915	+	0.892567i	$0.351098\pi$
$467$			833.657i			1.78513i	−0.450915	+	0.892567i	$0.648902\pi$
$468$		0			0
$469$	−196.514			−0.419007
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.80306	−	4.50510i	0.0164970	−	0.00952452i
$474$		0			0
$475$	−4.69694	+	8.13534i	−0.00988829	+	0.0171270i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−569.144	−	328.595i	−1.18819	−	0.686003i	−0.230296	−	0.973121i	$-0.573969\pi$
$479$	−569.144	−	328.595i	−1.18819	−	0.686003i	−0.957895	+	0.287118i	$0.907303\pi$
$480$		0			0
$481$	−170.409	−	295.156i	−0.354280	−	0.613631i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	498.200i		−	1.02722i
$486$		0			0
$487$	−351.666			−0.722107			−0.361054	−	0.932545i	$-0.617583\pi$
$487$	−351.666			−0.722107			−0.361054	+	0.932545i	$0.617583\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	212.539	−	122.709i	0.432869	−	0.249917i	−0.267699	−	0.963503i	$-0.586263\pi$
$491$	212.539	−	122.709i	0.432869	−	0.249917i	0.700568	+	0.713586i	$0.252930\pi$
$492$		0			0
$493$	3.13622	−	5.43210i	0.00636151	−	0.0110185i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	472.287	+	272.675i	0.950276	+	0.548642i
$498$		0			0
$499$	315.113	+	545.792i	0.631489	+	1.09377i	0.987247	+	0.159193i	$0.0508892\pi$
$499$	315.113	+	545.792i	0.631489	+	1.09377i	−0.355758	+	0.934578i	$0.615777\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			286.891i			0.570360i	0.958474	+	0.285180i	$0.0920534\pi$
$503$			286.891i			0.570360i	−0.958474	+	0.285180i	$0.907947\pi$
$504$		0			0
$505$	820.635			1.62502
$506$		0			0
$507$		0			0
$508$		0			0
$509$	755.454	−	436.161i	1.48419	−	0.856898i	0.484353	−	0.874873i	$-0.339055\pi$
$509$	755.454	−	436.161i	1.48419	−	0.856898i	0.999838	+	0.0179741i	$0.00572163\pi$
$510$		0			0
$511$	305.015	−	528.301i	0.596898	−	1.03386i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	131.113	+	75.6981i	0.254588	+	0.146987i
$516$		0			0
$517$	66.5908	+	115.339i	0.128802	+	0.223092i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			206.132i			0.395646i	0.980238	+	0.197823i	$0.0633872\pi$
$521$			206.132i			0.395646i	−0.980238	+	0.197823i	$0.936613\pi$
$522$		0			0
$523$	−884.817			−1.69181			−0.845906	−	0.533333i	$-0.820939\pi$
$523$	−884.817			−1.69181			−0.845906	+	0.533333i	$0.820939\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−67.7878	+	39.1373i	−0.128630	+	0.0742643i
$528$		0			0
$529$	−219.955	+	380.973i	−0.415793	+	0.720175i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1054.68	+	608.920i	1.97876	+	1.14244i
$534$		0			0
$535$	−446.363	−	773.124i	−0.834324	−	1.44509i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			82.0886i			0.152298i
$540$		0			0
$541$	509.151			0.941129			0.470565	−	0.882365i	$-0.344050\pi$
$541$	509.151			0.941129			0.470565	+	0.882365i	$0.344050\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	523.226	−	302.085i	0.960048	−	0.554284i
$546$		0			0
$547$	274.022	−	474.620i	0.500955	−	0.867679i	−0.499045	−	0.866576i	$-0.666316\pi$
$547$	274.022	−	474.620i	0.500955	−	0.867679i	0.999999	−	0.00110267i	$-0.000350992\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	13.3791	+	7.72442i	0.0242815	+	0.0140189i
$552$		0			0
$553$	−94.4092	−	163.522i	−0.170722	−	0.295699i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			406.542i			0.729879i	0.931031	+	0.364939i	$0.118910\pi$
$557$			406.542i			0.729879i	−0.931031	+	0.364939i	$0.881090\pi$
$558$		0			0
$559$	18.8025			0.0336360
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−525.220	+	303.236i	−0.932895	+	0.538607i	−0.887726	−	0.460372i	$-0.847716\pi$
$563$	−525.220	+	303.236i	−0.932895	+	0.538607i	−0.0451687	+	0.998979i	$0.514383\pi$
$564$		0			0
$565$	525.499	−	910.191i	0.930087	−	1.61096i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	224.954	+	129.877i	0.395350	+	0.228255i	0.684476	−	0.729036i	$-0.260031\pi$
$569$	224.954	+	129.877i	0.395350	+	0.228255i	−0.289126	+	0.957291i	$0.593365\pi$
$570$		0			0
$571$	−43.9166	−	76.0657i	−0.0769117	−	0.133215i	0.825004	−	0.565126i	$-0.191173\pi$
$571$	−43.9166	−	76.0657i	−0.0769117	−	0.133215i	−0.901916	+	0.431911i	$0.857839\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			18.8776i			0.0328306i
$576$		0			0
$577$	−132.091			−0.228927			−0.114463	−	0.993427i	$-0.536515\pi$
$577$	−132.091			−0.228927			−0.114463	+	0.993427i	$0.536515\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	483.302	−	279.034i	0.831844	−	0.480266i
$582$		0			0
$583$	45.0000	−	77.9423i	0.0771870	−	0.133692i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−491.614	−	283.833i	−0.837502	−	0.483532i	0.0189125	−	0.999821i	$-0.493980\pi$
$587$	−491.614	−	283.833i	−0.837502	−	0.483532i	−0.856414	+	0.516289i	$0.827313\pi$
$588$		0			0
$589$	−96.3939	−	166.959i	−0.163657	−	0.283462i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	77.0321i		−	0.129902i	−0.997888	−	0.0649512i	$-0.979311\pi$
$593$		−	77.0321i		−	0.129902i	0.997888	−	0.0649512i	$-0.0206892\pi$
$594$		0			0
$595$	62.9082			0.105728
$596$		0			0
$597$		0			0
$598$		0			0
$599$	764.917	−	441.625i	1.27699	−	0.737270i	0.300696	−	0.953720i	$-0.402781\pi$
$599$	764.917	−	441.625i	1.27699	−	0.737270i	0.976294	+	0.216450i	$0.0694479\pi$
$600$		0			0
$601$	397.545	−	688.569i	0.661473	−	1.14571i	−0.318755	−	0.947837i	$-0.603265\pi$
$601$	397.545	−	688.569i	0.661473	−	1.14571i	0.980229	−	0.197868i	$-0.0634018\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	143.591	+	82.9025i	0.237341	+	0.137029i
$606$		0			0
$607$	148.372	+	256.987i	0.244434	+	0.423373i	0.961972	−	0.273147i	$-0.0880644\pi$
$607$	148.372	+	256.987i	0.244434	+	0.423373i	−0.717538	+	0.696519i	$0.754731\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			277.924i			0.454868i
$612$		0			0
$613$	517.181			0.843688			0.421844	−	0.906668i	$-0.361383\pi$
$613$	517.181			0.843688			0.421844	+	0.906668i	$0.361383\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	229.909	−	132.738i	0.372623	−	0.215134i	−0.301981	−	0.953314i	$-0.597648\pi$
$617$	229.909	−	132.738i	0.372623	−	0.215134i	0.674604	+	0.738180i	$0.264314\pi$
$618$		0			0
$619$	−98.5227	+	170.646i	−0.159164	+	0.275681i	−0.934568	−	0.355786i	$-0.884213\pi$
$619$	−98.5227	+	170.646i	−0.159164	+	0.275681i	0.775403	+	0.631466i	$0.217547\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−227.486	−	131.339i	−0.365146	−	0.210817i
$624$		0			0
$625$	335.500	+	581.103i	0.536800	+	0.929765i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	32.9973i		−	0.0524600i
$630$		0			0
$631$	−160.879			−0.254958			−0.127479	−	0.991841i	$-0.540689\pi$
$631$	−160.879			−0.254958			−0.127479	+	0.991841i	$0.540689\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−45.4087	+	26.2167i	−0.0715097	+	0.0412862i
$636$		0			0
$637$	−85.6515	+	148.353i	−0.134461	+	0.232893i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	267.894	+	154.669i	0.417931	+	0.241293i	0.694192	−	0.719790i	$-0.255762\pi$
$641$	267.894	+	154.669i	0.417931	+	0.241293i	−0.276261	+	0.961083i	$0.589095\pi$
$642$		0			0
$643$	197.296	+	341.726i	0.306836	+	0.531456i	0.977668	−	0.210153i	$-0.0673963\pi$
$643$	197.296	+	341.726i	0.306836	+	0.531456i	−0.670832	+	0.741609i	$0.734063\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			418.736i			0.647196i	0.946195	+	0.323598i	$0.104892\pi$
$647$			418.736i			0.647196i	−0.946195	+	0.323598i	$0.895108\pi$
$648$		0			0
$649$	863.908			1.33114
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−459.621	+	265.363i	−0.703861	+	0.406375i	−0.808784	−	0.588106i	$-0.799874\pi$
$653$	−459.621	+	265.363i	−0.703861	+	0.406375i	0.104923	+	0.994480i	$0.466540\pi$
$654$		0			0
$655$	−12.8870	+	22.3209i	−0.0196748	+	0.0340778i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	310.204	+	179.096i	0.470719	+	0.271770i	0.716541	−	0.697545i	$-0.245724\pi$
$659$	310.204	+	179.096i	0.470719	+	0.271770i	−0.245822	+	0.969315i	$0.579058\pi$
$660$		0			0
$661$	−111.136	−	192.493i	−0.168133	−	0.291214i	0.769631	−	0.638489i	$-0.220440\pi$
$661$	−111.136	−	192.493i	−0.168133	−	0.291214i	−0.937763	+	0.347275i	$0.887107\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			154.941i			0.232994i
$666$		0			0
$667$	31.0454			0.0465448
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−613.810	+	354.383i	−0.914769	+	0.528142i
$672$		0			0
$673$	144.606	−	250.464i	0.214867	−	0.372161i	−0.738364	−	0.674402i	$-0.764401\pi$
$673$	144.606	−	250.464i	0.214867	−	0.372161i	0.953231	+	0.302241i	$0.0977348\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	402.227	+	232.226i	0.594131	+	0.343022i	0.766729	−	0.641971i	$-0.221883\pi$
$677$	402.227	+	232.226i	0.594131	+	0.343022i	−0.172598	+	0.984992i	$0.555216\pi$
$678$		0			0
$679$	−304.341	−	527.134i	−0.448220	−	0.776339i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1126.36i		−	1.64913i	−0.565767	−	0.824565i	$-0.691420\pi$
$683$		−	1126.36i		−	1.64913i	0.565767	−	0.824565i	$-0.308580\pi$
$684$		0			0
$685$	1219.45			1.78022
$686$		0			0
$687$		0			0
$688$		0			0
$689$	162.650	−	93.9063i	0.236067	−	0.136294i
$690$		0			0
$691$	518.841	−	898.658i	0.750855	−	1.30052i	−0.196554	−	0.980493i	$-0.562975\pi$
$691$	518.841	−	898.658i	0.750855	−	1.30052i	0.947409	−	0.320025i	$-0.103691\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−479.385	−	276.773i	−0.689763	−	0.398235i
$696$		0			0
$697$	58.9546	+	102.112i	0.0845833	+	0.146503i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			778.180i			1.11010i	0.831817	+	0.555050i	$0.187301\pi$
$701$			778.180i			1.11010i	−0.831817	+	0.555050i	$0.812699\pi$
$702$		0			0
$703$	81.2714			0.115607
$704$		0			0
$705$		0			0
$706$		0			0
$707$	868.296	−	501.311i	1.22814	−	0.709068i
$708$		0			0
$709$	−586.014	+	1015.01i	−0.826536	+	1.43160i	0.0742031	+	0.997243i	$0.476359\pi$
$709$	−586.014	+	1015.01i	−0.826536	+	1.43160i	−0.900739	+	0.434360i	$0.856975\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−335.515	−	193.710i	−0.470568	−	0.271682i
$714$		0			0
$715$	−483.022	−	836.619i	−0.675556	−	1.17010i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			515.416i			0.716851i	0.933558	+	0.358426i	$0.116686\pi$
$719$			515.416i			0.716851i	−0.933558	+	0.358426i	$0.883314\pi$
$720$		0			0
$721$	184.970			0.256547
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.69694	+	3.28913i	−0.00785785	+	0.00453673i
$726$		0			0
$727$	420.704	−	728.681i	0.578685	−	1.00231i	−0.416945	−	0.908932i	$-0.636899\pi$
$727$	420.704	−	728.681i	0.578685	−	1.00231i	0.995630	−	0.0933809i	$-0.0297674\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.57654	+	0.910215i	0.00215669	+	0.00124516i
$732$		0			0
$733$	303.181	+	525.125i	0.413617	+	0.716405i	0.995282	−	0.0970229i	$-0.0309320\pi$
$733$	303.181	+	525.125i	0.413617	+	0.716405i	−0.581665	+	0.813428i	$0.697599\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			292.174i			0.396437i
$738$		0			0
$739$	389.362			0.526877			0.263439	−	0.964676i	$-0.415143\pi$
$739$	389.362			0.526877			0.263439	+	0.964676i	$0.415143\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−904.779	+	522.375i	−1.21774	+	0.703061i	−0.964434	−	0.264325i	$-0.914851\pi$
$743$	−904.779	+	522.375i	−1.21774	+	0.703061i	−0.253304	+	0.967387i	$0.581517\pi$
$744$		0			0
$745$	273.090	−	473.006i	0.366564	−	0.634908i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−944.574	−	545.350i	−1.26111	−	0.728105i
$750$		0			0
$751$	645.916	+	1118.76i	0.860074	+	1.48969i	0.871857	+	0.489761i	$0.162916\pi$
$751$	645.916	+	1118.76i	0.860074	+	1.48969i	−0.0117826	+	0.999931i	$0.503751\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1482.08i			1.96302i
$756$		0			0
$757$	−1042.36			−1.37697			−0.688483	−	0.725252i	$-0.741723\pi$
$757$	−1042.36			−1.37697			−0.688483	+	0.725252i	$0.741723\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	281.607	−	162.586i	0.370048	−	0.213647i	−0.303431	−	0.952853i	$-0.598132\pi$
$761$	281.607	−	162.586i	0.370048	−	0.213647i	0.673479	+	0.739206i	$0.264799\pi$
$762$		0			0
$763$	369.076	−	639.258i	0.483717	−	0.837822i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1561.28	+	901.405i	2.03557	+	1.17523i
$768$		0			0
$769$	−171.348	−	296.783i	−0.222819	−	0.385934i	0.732844	−	0.680397i	$-0.238193\pi$
$769$	−171.348	−	296.783i	−0.222819	−	0.385934i	−0.955663	+	0.294463i	$0.904859\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	532.579i		−	0.688977i	−0.938791	−	0.344488i	$-0.888052\pi$
$773$		−	532.579i		−	0.688977i	0.938791	−	0.344488i	$-0.111948\pi$
$774$		0			0
$775$	82.0908			0.105924
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−251.499	+	145.203i	−0.322849	+	0.186397i
$780$		0			0
$781$	405.409	−	702.188i	0.519089	−	0.899089i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	887.317	+	512.293i	1.13034	+	0.652602i
$786$		0			0
$787$	−51.9768	−	90.0264i	−0.0660442	−	0.114392i	0.831113	−	0.556104i	$-0.187704\pi$
$787$	−51.9768	−	90.0264i	−0.0660442	−	0.114392i	−0.897157	+	0.441712i	$0.854371\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1284.07i		−	1.62335i
$792$		0			0
$793$	−1479.06			−1.86514
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−956.331	+	552.138i	−1.19991	+	0.692770i	−0.960536	−	0.278156i	$-0.910277\pi$
$797$	−956.331	+	552.138i	−1.19991	+	0.692770i	−0.239378	+	0.970927i	$0.576943\pi$
$798$		0			0
$799$	−13.4541	+	23.3031i	−0.0168386	+	0.0291654i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−785.469	−	453.491i	−0.978168	−	0.564746i
$804$		0			0
$805$	155.682	+	269.648i	0.193393	+	0.334967i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			256.465i			0.317015i	0.987358	+	0.158508i	$0.0506683\pi$
$809$			256.465i			0.317015i	−0.987358	+	0.158508i	$0.949332\pi$
$810$		0			0
$811$	−735.362			−0.906735			−0.453368	−	0.891324i	$-0.649778\pi$
$811$	−735.362			−0.906735			−0.453368	+	0.891324i	$0.649778\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1120.77	+	647.077i	−1.37518	+	0.793960i
$816$		0			0
$817$	−2.24183	+	3.88296i	−0.00274398	+	0.00475271i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1078.45	+	622.645i	1.31358	+	0.758398i	0.982688	−	0.185269i	$-0.0593157\pi$
$821$	1078.45	+	622.645i	1.31358	+	0.758398i	0.330896	+	0.943667i	$0.392649\pi$
$822$		0			0
$823$	771.129	+	1335.63i	0.936973	+	1.62288i	0.771077	+	0.636742i	$0.219718\pi$
$823$	771.129	+	1335.63i	0.936973	+	1.62288i	0.165896	+	0.986143i	$0.446948\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			955.707i			1.15563i	0.816167	+	0.577815i	$0.196095\pi$
$827$			955.707i			1.15563i	−0.816167	+	0.577815i	$0.803905\pi$
$828$		0			0
$829$	−1082.88			−1.30625			−0.653123	−	0.757252i	$-0.726542\pi$
$829$	−1082.88			−1.30625			−0.653123	+	0.757252i	$0.726542\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−14.3633	+	8.29263i	−0.0172428	+	0.00995514i
$834$		0			0
$835$	−125.842	+	217.964i	−0.150708	+	0.261035i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−903.778	−	521.797i	−1.07721	−	0.621927i	−0.147067	−	0.989127i	$-0.546983\pi$
$839$	−903.778	−	521.797i	−1.07721	−	0.621927i	−0.930142	+	0.367200i	$0.880317\pi$
$840$		0			0
$841$	−415.091	−	718.958i	−0.493568	−	0.854885i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1137.80i		−	1.34651i
$846$		0			0
$847$	202.574			0.239167
$848$		0			0
$849$		0			0
$850$		0			0
$851$	141.439	−	81.6600i	0.166204	−	0.0959577i
$852$		0			0
$853$	236.909	−	410.338i	0.277736	−	0.481053i	−0.693086	−	0.720855i	$-0.743749\pi$
$853$	236.909	−	410.338i	0.277736	−	0.481053i	0.970822	+	0.239802i	$0.0770827\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−793.939	−	458.381i	−0.926417	−	0.534867i	−0.0407403	−	0.999170i	$-0.512972\pi$
$857$	−793.939	−	458.381i	−0.926417	−	0.534867i	−0.885677	+	0.464303i	$0.846305\pi$
$858$		0			0
$859$	478.901	+	829.480i	0.557510	+	0.965635i	0.997704	+	0.0677322i	$0.0215764\pi$
$859$	478.901	+	829.480i	0.557510	+	0.965635i	−0.440194	+	0.897903i	$0.645090\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			524.200i			0.607416i	0.952765	+	0.303708i	$0.0982247\pi$
$863$			524.200i			0.607416i	−0.952765	+	0.303708i	$0.901775\pi$
$864$		0			0
$865$	−521.908			−0.603362
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−243.121	+	140.366i	−0.279771	+	0.161526i
$870$		0			0
$871$	−304.855	+	528.025i	−0.350006	+	0.606228i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	657.067	+	379.358i	0.750933	+	0.433551i
$876$		0			0
$877$	503.878	+	872.742i	0.574547	+	0.995145i	0.996091	+	0.0883370i	$0.0281552\pi$
$877$	503.878	+	872.742i	0.574547	+	0.995145i	−0.421543	+	0.906808i	$0.638511\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1536.71i		−	1.74428i	−0.489254	−	0.872141i	$-0.662731\pi$
$881$		−	1536.71i		−	1.74428i	0.489254	−	0.872141i	$-0.337269\pi$
$882$		0			0
$883$	294.213			0.333197			0.166599	−	0.986025i	$-0.446722\pi$
$883$	294.213			0.333197			0.166599	+	0.986025i	$0.446722\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	497.794	−	287.402i	0.561211	−	0.324015i	−0.192420	−	0.981313i	$-0.561634\pi$
$887$	497.794	−	287.402i	0.561211	−	0.324015i	0.753631	+	0.657297i	$0.228300\pi$
$888$		0			0
$889$	−32.0306	+	55.4787i	−0.0360299	+	0.0624057i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−57.3949	−	33.1370i	−0.0642720	−	0.0371075i
$894$		0			0
$895$	−740.636	−	1282.82i	−0.827526	−	1.43332i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	135.004i		−	0.150171i
$900$		0			0
$901$	18.1837			0.0201817
$902$		0			0
$903$		0			0
$904$		0			0
$905$	167.046	−	96.4443i	0.184582	−	0.106568i
$906$		0			0
$907$	−255.037	+	441.737i	−0.281187	+	0.487031i	−0.971677	−	0.236311i	$-0.924062\pi$
$907$	−255.037	+	441.737i	−0.281187	+	0.487031i	0.690490	+	0.723342i	$0.257395\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	803.127	+	463.685i	0.881588	+	0.508985i	0.871182	−	0.490961i	$-0.163354\pi$
$911$	803.127	+	463.685i	0.881588	+	0.508985i	0.0104064	+	0.999946i	$0.496687\pi$
$912$		0			0
$913$	−414.863	−	718.564i	−0.454396	−	0.787036i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			31.4897i			0.0343399i
$918$		0			0
$919$	−1240.63			−1.34998			−0.674991	−	0.737826i	$-0.735853\pi$
$919$	−1240.63			−1.34998			−0.674991	+	0.737826i	$0.735853\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1465.33	−	846.010i	1.58757	−	0.916587i
$924$		0			0
$925$	−17.3031	+	29.9698i	−0.0187060	+	0.0323998i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−293.576	−	169.496i	−0.316013	−	0.182450i	0.333601	−	0.942714i	$-0.391736\pi$
$929$	−293.576	−	169.496i	−0.316013	−	0.182450i	−0.649614	+	0.760264i	$0.725069\pi$
$930$		0			0
$931$	−20.4245	−	35.3763i	−0.0219382	−	0.0379981i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	93.5307i		−	0.100033i
$936$		0			0
$937$	1322.21			1.41111			0.705556	−	0.708655i	$-0.250698\pi$
$937$	1322.21			1.41111			0.705556	+	0.708655i	$0.250698\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−310.984	+	179.547i	−0.330482	+	0.190804i	−0.656055	−	0.754713i	$-0.727776\pi$
$941$	−310.984	+	179.547i	−0.330482	+	0.190804i	0.325573	+	0.945517i	$0.394443\pi$
$942$		0			0
$943$	−291.795	+	505.404i	−0.309433	+	0.535953i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−671.855	−	387.896i	−0.709457	−	0.409605i	0.101403	−	0.994845i	$-0.467667\pi$
$947$	−671.855	−	387.896i	−0.709457	−	0.409605i	−0.810860	+	0.585240i	$0.801000\pi$
$948$		0			0
$949$	−946.347	−	1639.12i	−0.997205	−	1.72721i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			465.082i			0.488019i	0.969773	+	0.244010i	$0.0784628\pi$
$953$			465.082i			0.488019i	−0.969773	+	0.244010i	$0.921537\pi$
$954$		0			0
$955$	−93.1362			−0.0975248
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1290.28	−	744.942i	1.34544	−	0.776791i
$960$		0			0
$961$	−361.863	+	626.765i	−0.376548	+	0.652200i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−429.543	−	247.997i	−0.445123	−	0.256992i
$966$		0			0
$967$	−612.113	−	1060.21i	−0.633002	−	1.09639i	−0.986935	−	0.161121i	$-0.948489\pi$
$967$	−612.113	−	1060.21i	−0.633002	−	1.09639i	0.353933	−	0.935271i	$-0.384844\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	658.702i		−	0.678375i	−0.940719	−	0.339188i	$-0.889848\pi$
$971$		−	658.702i		−	0.678375i	0.940719	−	0.339188i	$-0.110152\pi$
$972$		0			0
$973$	−676.303			−0.695070
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1314.92	+	759.170i	−1.34588	+	0.777042i	−0.987663	−	0.156597i	$-0.949948\pi$
$977$	−1314.92	+	759.170i	−1.34588	+	0.777042i	−0.358214	+	0.933639i	$0.616614\pi$
$978$		0			0
$979$	−195.272	+	338.222i	−0.199461	+	0.345477i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−716.930	−	413.920i	−0.729329	−	0.421078i	0.0888477	−	0.996045i	$-0.471682\pi$
$983$	−716.930	−	413.920i	−0.729329	−	0.421078i	−0.818177	+	0.574967i	$0.805015\pi$
$984$		0			0
$985$	416.636	+	721.634i	0.422980	+	0.732624i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			9.01020i			0.00911041i
$990$		0			0
$991$	429.546			0.433447			0.216723	−	0.976233i	$-0.430463\pi$
$991$	429.546			0.433447			0.216723	+	0.976233i	$0.430463\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−29.3189	+	16.9273i	−0.0294662	+	0.0170123i
$996$		0			0
$997$	−347.499	+	601.886i	−0.348545	+	0.603697i	−0.985991	−	0.166798i	$-0.946657\pi$
$997$	−347.499	+	601.886i	−0.348545	+	0.603697i	0.637447	+	0.770495i	$0.279991\pi$
$998$		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	1728.3.q.d.1601.1		4
3.2	odd	2		576.3.q.f.65.1		4
4.3	odd	2		1728.3.q.c.1601.2		4
8.3	odd	2		432.3.q.d.305.2		4
8.5	even	2		54.3.d.a.35.2		4
9.4	even	3		576.3.q.f.257.1		4
9.5	odd	6	inner	1728.3.q.d.449.1		4
12.11	even	2		576.3.q.e.65.2		4
24.5	odd	2		18.3.d.a.11.1	yes	4
24.11	even	2		144.3.q.c.65.1		4
36.23	even	6		1728.3.q.c.449.2		4
36.31	odd	6		576.3.q.e.257.2		4
40.13	odd	4		1350.3.k.a.899.3		8
40.29	even	2		1350.3.i.b.251.1		4
40.37	odd	4		1350.3.k.a.899.2		8
72.5	odd	6		54.3.d.a.17.2		4
72.11	even	6		1296.3.e.g.161.1		4
72.13	even	6		18.3.d.a.5.1	✓	4
72.29	odd	6		162.3.b.a.161.3		4
72.43	odd	6		1296.3.e.g.161.3		4
72.59	even	6		432.3.q.d.17.2		4
72.61	even	6		162.3.b.a.161.2		4
72.67	odd	6		144.3.q.c.113.1		4
120.29	odd	2		450.3.i.b.101.2		4
120.53	even	4		450.3.k.a.299.2		8
120.77	even	4		450.3.k.a.299.3		8
360.13	odd	12		450.3.k.a.149.3		8
360.77	even	12		1350.3.k.a.449.3		8
360.149	odd	6		1350.3.i.b.1151.1		4
360.157	odd	12		450.3.k.a.149.2		8
360.229	even	6		450.3.i.b.401.2		4
360.293	even	12		1350.3.k.a.449.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.d.a.5.1	✓	4	72.13	even	6
18.3.d.a.11.1	yes	4	24.5	odd	2
54.3.d.a.17.2		4	72.5	odd	6
54.3.d.a.35.2		4	8.5	even	2
144.3.q.c.65.1		4	24.11	even	2
144.3.q.c.113.1		4	72.67	odd	6
162.3.b.a.161.2		4	72.61	even	6
162.3.b.a.161.3		4	72.29	odd	6
432.3.q.d.17.2		4	72.59	even	6
432.3.q.d.305.2		4	8.3	odd	2
450.3.i.b.101.2		4	120.29	odd	2
450.3.i.b.401.2		4	360.229	even	6
450.3.k.a.149.2		8	360.157	odd	12
450.3.k.a.149.3		8	360.13	odd	12
450.3.k.a.299.2		8	120.53	even	4
450.3.k.a.299.3		8	120.77	even	4
576.3.q.e.65.2		4	12.11	even	2
576.3.q.e.257.2		4	36.31	odd	6
576.3.q.f.65.1		4	3.2	odd	2
576.3.q.f.257.1		4	9.4	even	3
1296.3.e.g.161.1		4	72.11	even	6
1296.3.e.g.161.3		4	72.43	odd	6
1350.3.i.b.251.1		4	40.29	even	2
1350.3.i.b.1151.1		4	360.149	odd	6
1350.3.k.a.449.2		8	360.293	even	12
1350.3.k.a.449.3		8	360.77	even	12
1350.3.k.a.899.2		8	40.37	odd	4
1350.3.k.a.899.3		8	40.13	odd	4
1728.3.q.c.449.2		4	36.23	even	6
1728.3.q.c.1601.2		4	4.3	odd	2
1728.3.q.d.449.1		4	9.5	odd	6	inner
1728.3.q.d.1601.1		4	1.1	even	1	trivial

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

Downloads

Learn more

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	1728.3.q.d.1601.1

Level	$1728$
Weight	$3$
Character	1728.1601
Analytic conductor	$47.085$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$