LMFDB - Embedded newform 1728.3.b.b.1567.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(1567,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("1728.1567");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.b (of order $2$, degree $1$, 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:	$4$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1728.1567
Dual form			1728.3.b.b.1567.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410i q^{5} -1.00000i q^{7} +O(q^{10})$	$q+3.46410i q^{5} -1.00000i q^{7} -17.3205 q^{11} +1.73205i q^{13} -6.00000 q^{17} +1.73205 q^{19} -30.0000i q^{23} +13.0000 q^{25} -20.7846i q^{29} -14.0000i q^{31} +3.46410 q^{35} +19.0526i q^{37} +48.0000 q^{41} +20.7846 q^{43} +66.0000i q^{47} +48.0000 q^{49} -48.4974i q^{53} -60.0000i q^{55} +31.1769 q^{59} +43.3013i q^{61} -6.00000 q^{65} +60.6218 q^{67} -48.0000i q^{71} +49.0000 q^{73} +17.3205i q^{77} -83.0000i q^{79} -13.8564 q^{83} -20.7846i q^{85} -66.0000 q^{89} +1.73205 q^{91} +6.00000i q^{95} +107.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{17} + 52 q^{25} + 192 q^{41} + 192 q^{49} - 24 q^{65} + 196 q^{73} - 264 q^{89} + 428 q^{97}+O(q^{100}) $ 4 * q - 24 * q^17 + 52 * q^25 + 192 * q^41 + 192 * q^49 - 24 * q^65 + 196 * q^73 - 264 * q^89 + 428 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$-1$	$1$

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
$5$	3.46410i	0.692820i	0.938083	+	0.346410i	$0.112599\pi$
$5$	3.46410i	0.692820i	−0.938083	+	0.346410i	$0.887401\pi$
$6$	0	0
$7$	− 1.00000i	− 0.142857i	−0.997446	−	0.0714286i	$-0.977244\pi$
$7$	− 1.00000i	− 0.142857i	0.997446	−	0.0714286i	$-0.0227558\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−17.3205	−1.57459	−0.787296	−	0.616575i	$-0.788520\pi$
$11$	−17.3205	−1.57459	−0.787296	+	0.616575i	$0.788520\pi$
$12$	0	0
$13$	1.73205i	0.133235i	0.997779	+	0.0666173i	$0.0212207\pi$
$13$	1.73205i	0.133235i	−0.997779	+	0.0666173i	$0.978779\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.00000	−0.352941	−0.176471	−	0.984306i	$-0.556468\pi$
$17$	−6.00000	−0.352941	−0.176471	+	0.984306i	$0.556468\pi$
$18$	0	0
$19$	1.73205	0.0911606	0.0455803	−	0.998961i	$-0.485486\pi$
$19$	1.73205	0.0911606	0.0455803	+	0.998961i	$0.485486\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 30.0000i	− 1.30435i	−0.758069	−	0.652174i	$-0.773857\pi$
$23$	− 30.0000i	− 1.30435i	0.758069	−	0.652174i	$-0.226143\pi$
$24$	0	0
$25$	13.0000	0.520000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 20.7846i	− 0.716711i	−0.933585	−	0.358355i	$-0.883338\pi$
$29$	− 20.7846i	− 0.716711i	0.933585	−	0.358355i	$-0.116662\pi$
$30$	0	0
$31$	− 14.0000i	− 0.451613i	−0.974172	−	0.225806i	$-0.927498\pi$
$31$	− 14.0000i	− 0.451613i	0.974172	−	0.225806i	$-0.0725017\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.46410	0.0989743
$36$	0	0
$37$	19.0526i	0.514934i	0.966287	+	0.257467i	$0.0828879\pi$
$37$	19.0526i	0.514934i	−0.966287	+	0.257467i	$0.917112\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	48.0000	1.17073	0.585366	−	0.810769i	$-0.300951\pi$
$41$	48.0000	1.17073	0.585366	+	0.810769i	$0.300951\pi$
$42$	0	0
$43$	20.7846	0.483363	0.241682	−	0.970356i	$-0.422301\pi$
$43$	20.7846	0.483363	0.241682	+	0.970356i	$0.422301\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	66.0000i	1.40426i	0.712051	+	0.702128i	$0.247766\pi$
$47$	66.0000i	1.40426i	−0.712051	+	0.702128i	$0.752234\pi$
$48$	0	0
$49$	48.0000	0.979592
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 48.4974i	− 0.915046i	−0.889198	−	0.457523i	$-0.848737\pi$
$53$	− 48.4974i	− 0.915046i	0.889198	−	0.457523i	$-0.151263\pi$
$54$	0	0
$55$	− 60.0000i	− 1.09091i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	31.1769	0.528422	0.264211	−	0.964465i	$-0.414888\pi$
$59$	31.1769	0.528422	0.264211	+	0.964465i	$0.414888\pi$
$60$	0	0
$61$	43.3013i	0.709857i	0.934893	+	0.354928i	$0.115495\pi$
$61$	43.3013i	0.709857i	−0.934893	+	0.354928i	$0.884505\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.00000	−0.0923077
$66$	0	0
$67$	60.6218	0.904803	0.452401	−	0.891814i	$-0.350567\pi$
$67$	60.6218	0.904803	0.452401	+	0.891814i	$0.350567\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 48.0000i	− 0.676056i	−0.941136	−	0.338028i	$-0.890240\pi$
$71$	− 48.0000i	− 0.676056i	0.941136	−	0.338028i	$-0.109760\pi$
$72$	0	0
$73$	49.0000	0.671233	0.335616	−	0.941999i	$-0.391055\pi$
$73$	49.0000	0.671233	0.335616	+	0.941999i	$0.391055\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17.3205i	0.224942i
$78$	0	0
$79$	− 83.0000i	− 1.05063i	−0.850907	−	0.525316i	$-0.823947\pi$
$79$	− 83.0000i	− 1.05063i	0.850907	−	0.525316i	$-0.176053\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.8564	−0.166945	−0.0834723	−	0.996510i	$-0.526601\pi$
$83$	−13.8564	−0.166945	−0.0834723	+	0.996510i	$0.526601\pi$
$84$	0	0
$85$	− 20.7846i	− 0.244525i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−66.0000	−0.741573	−0.370787	−	0.928718i	$-0.620912\pi$
$89$	−66.0000	−0.741573	−0.370787	+	0.928718i	$0.620912\pi$
$90$	0	0
$91$	1.73205	0.0190335
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.00000i	0.0631579i
$96$	0	0
$97$	107.000	1.10309	0.551546	−	0.834144i	$-0.314038\pi$
$97$	107.000	1.10309	0.551546	+	0.834144i	$0.314038\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	159.349i	1.57771i	0.614580	+	0.788855i	$0.289326\pi$
$101$	159.349i	1.57771i	−0.614580	+	0.788855i	$0.710674\pi$
$102$	0	0
$103$	− 95.0000i	− 0.922330i	−0.887314	−	0.461165i	$-0.847432\pi$
$103$	− 95.0000i	− 0.922330i	0.887314	−	0.461165i	$-0.152568\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.3205	0.161874	0.0809370	−	0.996719i	$-0.474209\pi$
$107$	17.3205	0.161874	0.0809370	+	0.996719i	$0.474209\pi$
$108$	0	0
$109$	187.061i	1.71616i	0.513516	+	0.858080i	$0.328343\pi$
$109$	187.061i	1.71616i	−0.513516	+	0.858080i	$0.671657\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−150.000	−1.32743	−0.663717	−	0.747984i	$-0.731022\pi$
$113$	−150.000	−1.32743	−0.663717	+	0.747984i	$0.731022\pi$
$114$	0	0
$115$	103.923	0.903679
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.00000i	0.0504202i
$120$	0	0
$121$	179.000	1.47934
$122$	0	0
$123$	0	0
$124$	0	0
$125$	131.636i	1.05309i
$126$	0	0
$127$	− 190.000i	− 1.49606i	−0.663663	−	0.748031i	$-0.730999\pi$
$127$	− 190.000i	− 1.49606i	0.663663	−	0.748031i	$-0.269001\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−48.4974	−0.370209	−0.185105	−	0.982719i	$-0.559262\pi$
$131$	−48.4974	−0.370209	−0.185105	+	0.982719i	$0.559262\pi$
$132$	0	0
$133$	− 1.73205i	− 0.0130229i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	222.000	1.62044	0.810219	−	0.586127i	$-0.199348\pi$
$137$	222.000	1.62044	0.810219	+	0.586127i	$0.199348\pi$
$138$	0	0
$139$	164.545	1.18378	0.591888	−	0.806020i	$-0.298383\pi$
$139$	164.545	1.18378	0.591888	+	0.806020i	$0.298383\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 30.0000i	− 0.209790i
$144$	0	0
$145$	72.0000	0.496552
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 96.9948i	− 0.650972i	−0.945547	−	0.325486i	$-0.894472\pi$
$149$	− 96.9948i	− 0.650972i	0.945547	−	0.325486i	$-0.105528\pi$
$150$	0	0
$151$	− 109.000i	− 0.721854i	−0.932594	−	0.360927i	$-0.882460\pi$
$151$	− 109.000i	− 0.721854i	0.932594	−	0.360927i	$-0.117540\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	48.4974	0.312887
$156$	0	0
$157$	62.3538i	0.397158i	0.980085	+	0.198579i	$0.0636327\pi$
$157$	62.3538i	0.397158i	−0.980085	+	0.198579i	$0.936367\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−30.0000	−0.186335
$162$	0	0
$163$	−84.8705	−0.520678	−0.260339	−	0.965517i	$-0.583834\pi$
$163$	−84.8705	−0.520678	−0.260339	+	0.965517i	$0.583834\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	234.000i	1.40120i	0.713555	+	0.700599i	$0.247084\pi$
$167$	234.000i	1.40120i	−0.713555	+	0.700599i	$0.752916\pi$
$168$	0	0
$169$	166.000	0.982249
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 180.133i	− 1.04123i	−0.853791	−	0.520616i	$-0.825702\pi$
$173$	− 180.133i	− 1.04123i	0.853791	−	0.520616i	$-0.174298\pi$
$174$	0	0
$175$	− 13.0000i	− 0.0742857i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	193.990	1.08374	0.541871	−	0.840462i	$-0.317716\pi$
$179$	193.990	1.08374	0.541871	+	0.840462i	$0.317716\pi$
$180$	0	0
$181$	164.545i	0.909087i	0.890724	+	0.454544i	$0.150198\pi$
$181$	164.545i	0.909087i	−0.890724	+	0.454544i	$0.849802\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−66.0000	−0.356757
$186$	0	0
$187$	103.923	0.555738
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 198.000i	− 1.03665i	−0.855184	−	0.518325i	$-0.826556\pi$
$191$	− 198.000i	− 1.03665i	0.855184	−	0.518325i	$-0.173444\pi$
$192$	0	0
$193$	227.000	1.17617	0.588083	−	0.808801i	$-0.299883\pi$
$193$	227.000	1.17617	0.588083	+	0.808801i	$0.299883\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 232.095i	− 1.17815i	−0.808080	−	0.589073i	$-0.799493\pi$
$197$	− 232.095i	− 1.17815i	0.808080	−	0.589073i	$-0.200507\pi$
$198$	0	0
$199$	− 239.000i	− 1.20101i	−0.799623	−	0.600503i	$-0.794967\pi$
$199$	− 239.000i	− 1.20101i	0.799623	−	0.600503i	$-0.205033\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−20.7846	−0.102387
$204$	0	0
$205$	166.277i	0.811107i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−30.0000	−0.143541
$210$	0	0
$211$	292.717	1.38728	0.693641	−	0.720321i	$-0.256005\pi$
$211$	292.717	1.38728	0.693641	+	0.720321i	$0.256005\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	72.0000i	0.334884i
$216$	0	0
$217$	−14.0000	−0.0645161
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 10.3923i	− 0.0470240i
$222$	0	0
$223$	− 130.000i	− 0.582960i	−0.956577	−	0.291480i	$-0.905852\pi$
$223$	− 130.000i	− 0.582960i	0.956577	−	0.291480i	$-0.0941476\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−41.5692	−0.183124	−0.0915622	−	0.995799i	$-0.529186\pi$
$227$	−41.5692	−0.183124	−0.0915622	+	0.995799i	$0.529186\pi$
$228$	0	0
$229$	− 332.554i	− 1.45220i	−0.687589	−	0.726100i	$-0.741331\pi$
$229$	− 332.554i	− 1.45220i	0.687589	−	0.726100i	$-0.258669\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	276.000	1.18455	0.592275	−	0.805736i	$-0.298230\pi$
$233$	276.000	1.18455	0.592275	+	0.805736i	$0.298230\pi$
$234$	0	0
$235$	−228.631	−0.972897
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 204.000i	− 0.853556i	−0.904356	−	0.426778i	$-0.859648\pi$
$239$	− 204.000i	− 0.853556i	0.904356	−	0.426778i	$-0.140352\pi$
$240$	0	0
$241$	133.000	0.551867	0.275934	−	0.961177i	$-0.411013\pi$
$241$	133.000	0.551867	0.275934	+	0.961177i	$0.411013\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	166.277i	0.678681i
$246$	0	0
$247$	3.00000i	0.0121457i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−457.261	−1.82176	−0.910879	−	0.412673i	$-0.864595\pi$
$251$	−457.261	−1.82176	−0.910879	+	0.412673i	$0.864595\pi$
$252$	0	0
$253$	519.615i	2.05382i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	300.000	1.16732	0.583658	−	0.812000i	$-0.301621\pi$
$257$	300.000	1.16732	0.583658	+	0.812000i	$0.301621\pi$
$258$	0	0
$259$	19.0526	0.0735620
$260$	0	0
$261$	0	0
$262$	0	0
$263$	84.0000i	0.319392i	0.987166	+	0.159696i	$0.0510513\pi$
$263$	84.0000i	0.319392i	−0.987166	+	0.159696i	$0.948949\pi$
$264$	0	0
$265$	168.000	0.633962
$266$	0	0
$267$	0	0
$268$	0	0
$269$	93.5307i	0.347698i	0.984772	+	0.173849i	$0.0556205\pi$
$269$	93.5307i	0.347698i	−0.984772	+	0.173849i	$0.944380\pi$
$270$	0	0
$271$	− 265.000i	− 0.977860i	−0.872323	−	0.488930i	$-0.837387\pi$
$271$	− 265.000i	− 0.977860i	0.872323	−	0.488930i	$-0.162613\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−225.167	−0.818788
$276$	0	0
$277$	− 311.769i	− 1.12552i	−0.826620	−	0.562760i	$-0.809739\pi$
$277$	− 311.769i	− 1.12552i	0.826620	−	0.562760i	$-0.190261\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−540.000	−1.92171	−0.960854	−	0.277055i	$-0.910642\pi$
$281$	−540.000	−1.92171	−0.960854	+	0.277055i	$0.910642\pi$
$282$	0	0
$283$	353.338	1.24855	0.624273	−	0.781206i	$-0.285395\pi$
$283$	353.338	1.24855	0.624273	+	0.781206i	$0.285395\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 48.0000i	− 0.167247i
$288$	0	0
$289$	−253.000	−0.875433
$290$	0	0
$291$	0	0
$292$	0	0
$293$	155.885i	0.532029i	0.963969	+	0.266015i	$0.0857069\pi$
$293$	155.885i	0.532029i	−0.963969	+	0.266015i	$0.914293\pi$
$294$	0	0
$295$	108.000i	0.366102i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	51.9615	0.173784
$300$	0	0
$301$	− 20.7846i	− 0.0690519i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−150.000	−0.491803
$306$	0	0
$307$	−353.338	−1.15094	−0.575470	−	0.817823i	$-0.695181\pi$
$307$	−353.338	−1.15094	−0.575470	+	0.817823i	$0.695181\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 462.000i	− 1.48553i	−0.669552	−	0.742765i	$-0.733514\pi$
$311$	− 462.000i	− 1.48553i	0.669552	−	0.742765i	$-0.266486\pi$
$312$	0	0
$313$	−35.0000	−0.111821	−0.0559105	−	0.998436i	$-0.517806\pi$
$313$	−35.0000	−0.111821	−0.0559105	+	0.998436i	$0.517806\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 263.272i	− 0.830510i	−0.909705	−	0.415255i	$-0.863692\pi$
$317$	− 263.272i	− 0.830510i	0.909705	−	0.415255i	$-0.136308\pi$
$318$	0	0
$319$	360.000i	1.12853i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−10.3923	−0.0321743
$324$	0	0
$325$	22.5167i	0.0692820i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	66.0000	0.200608
$330$	0	0
$331$	39.8372	0.120354	0.0601770	−	0.998188i	$-0.480833\pi$
$331$	39.8372	0.120354	0.0601770	+	0.998188i	$0.480833\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	210.000i	0.626866i
$336$	0	0
$337$	59.0000	0.175074	0.0875371	−	0.996161i	$-0.472100\pi$
$337$	59.0000	0.175074	0.0875371	+	0.996161i	$0.472100\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	242.487i	0.711106i
$342$	0	0
$343$	− 97.0000i	− 0.282799i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	630.466	1.81691	0.908453	−	0.417987i	$-0.137264\pi$
$347$	630.466	1.81691	0.908453	+	0.417987i	$0.137264\pi$
$348$	0	0
$349$	185.329i	0.531030i	0.964107	+	0.265515i	$0.0855420\pi$
$349$	185.329i	0.531030i	−0.964107	+	0.265515i	$0.914458\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.0000	−0.0339943	−0.0169972	−	0.999856i	$-0.505411\pi$
$353$	−12.0000	−0.0339943	−0.0169972	+	0.999856i	$0.505411\pi$
$354$	0	0
$355$	166.277	0.468386
$356$	0	0
$357$	0	0
$358$	0	0
$359$	318.000i	0.885794i	0.896573	+	0.442897i	$0.146049\pi$
$359$	318.000i	0.885794i	−0.896573	+	0.442897i	$0.853951\pi$
$360$	0	0
$361$	−358.000	−0.991690
$362$	0	0
$363$	0	0
$364$	0	0
$365$	169.741i	0.465044i
$366$	0	0
$367$	109.000i	0.297003i	0.988912	+	0.148501i	$0.0474449\pi$
$367$	109.000i	0.297003i	−0.988912	+	0.148501i	$0.952555\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−48.4974	−0.130721
$372$	0	0
$373$	− 729.193i	− 1.95494i	−0.211070	−	0.977471i	$-0.567695\pi$
$373$	− 729.193i	− 1.95494i	0.211070	−	0.977471i	$-0.432305\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	36.0000	0.0954907
$378$	0	0
$379$	−438.209	−1.15622	−0.578112	−	0.815957i	$-0.696210\pi$
$379$	−438.209	−1.15622	−0.578112	+	0.815957i	$0.696210\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	516.000i	1.34726i	0.739069	+	0.673629i	$0.235266\pi$
$383$	516.000i	1.34726i	−0.739069	+	0.673629i	$0.764734\pi$
$384$	0	0
$385$	−60.0000	−0.155844
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 543.864i	− 1.39811i	−0.715069	−	0.699054i	$-0.753605\pi$
$389$	− 543.864i	− 1.39811i	0.715069	−	0.699054i	$-0.246395\pi$
$390$	0	0
$391$	180.000i	0.460358i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	287.520	0.727900
$396$	0	0
$397$	685.892i	1.72769i	0.503759	+	0.863844i	$0.331950\pi$
$397$	685.892i	1.72769i	−0.503759	+	0.863844i	$0.668050\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	252.000	0.628429	0.314214	−	0.949352i	$-0.398259\pi$
$401$	252.000	0.628429	0.314214	+	0.949352i	$0.398259\pi$
$402$	0	0
$403$	24.2487	0.0601705
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 330.000i	− 0.810811i
$408$	0	0
$409$	−11.0000	−0.0268949	−0.0134474	−	0.999910i	$-0.504281\pi$
$409$	−11.0000	−0.0268949	−0.0134474	+	0.999910i	$0.504281\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 31.1769i	− 0.0754889i
$414$	0	0
$415$	− 48.0000i	− 0.115663i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	135.100	0.322434	0.161217	−	0.986919i	$-0.448458\pi$
$419$	135.100	0.322434	0.161217	+	0.986919i	$0.448458\pi$
$420$	0	0
$421$	− 601.022i	− 1.42760i	−0.700347	−	0.713802i	$-0.746971\pi$
$421$	− 601.022i	− 1.42760i	0.700347	−	0.713802i	$-0.253029\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−78.0000	−0.183529
$426$	0	0
$427$	43.3013	0.101408
$428$	0	0
$429$	0	0
$430$	0	0
$431$	486.000i	1.12761i	0.825908	+	0.563805i	$0.190663\pi$
$431$	486.000i	1.12761i	−0.825908	+	0.563805i	$0.809337\pi$
$432$	0	0
$433$	−494.000	−1.14088	−0.570439	−	0.821340i	$-0.693227\pi$
$433$	−494.000	−1.14088	−0.570439	+	0.821340i	$0.693227\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 51.9615i	− 0.118905i
$438$	0	0
$439$	610.000i	1.38952i	0.719241	+	0.694761i	$0.244490\pi$
$439$	610.000i	1.38952i	−0.719241	+	0.694761i	$0.755510\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	145.492	0.328425	0.164212	−	0.986425i	$-0.447492\pi$
$443$	145.492	0.328425	0.164212	+	0.986425i	$0.447492\pi$
$444$	0	0
$445$	− 228.631i	− 0.513777i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	366.000	0.815145	0.407572	−	0.913173i	$-0.366375\pi$
$449$	366.000	0.815145	0.407572	+	0.913173i	$0.366375\pi$
$450$	0	0
$451$	−831.384	−1.84342
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.00000i	0.0131868i
$456$	0	0
$457$	−398.000	−0.870897	−0.435449	−	0.900214i	$-0.643410\pi$
$457$	−398.000	−0.870897	−0.435449	+	0.900214i	$0.643410\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	710.141i	1.54044i	0.637781	+	0.770218i	$0.279853\pi$
$461$	710.141i	1.54044i	−0.637781	+	0.770218i	$0.720147\pi$
$462$	0	0
$463$	875.000i	1.88985i	0.327289	+	0.944924i	$0.393865\pi$
$463$	875.000i	1.88985i	−0.327289	+	0.944924i	$0.606135\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	349.874	0.749195	0.374598	−	0.927187i	$-0.377781\pi$
$467$	349.874	0.749195	0.374598	+	0.927187i	$0.377781\pi$
$468$	0	0
$469$	− 60.6218i	− 0.129258i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−360.000	−0.761099
$474$	0	0
$475$	22.5167	0.0474035
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 228.000i	− 0.475992i	−0.971266	−	0.237996i	$-0.923510\pi$
$479$	− 228.000i	− 0.475992i	0.971266	−	0.237996i	$-0.0764905\pi$
$480$	0	0
$481$	−33.0000	−0.0686071
$482$	0	0
$483$	0	0
$484$	0	0
$485$	370.659i	0.764245i
$486$	0	0
$487$	251.000i	0.515400i	0.966225	+	0.257700i	$0.0829647\pi$
$487$	251.000i	0.515400i	−0.966225	+	0.257700i	$0.917035\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−703.213	−1.43220	−0.716102	−	0.697995i	$-0.754076\pi$
$491$	−703.213	−1.43220	−0.716102	+	0.697995i	$0.754076\pi$
$492$	0	0
$493$	124.708i	0.252957i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−48.0000	−0.0965795
$498$	0	0
$499$	374.123	0.749745	0.374873	−	0.927076i	$-0.377686\pi$
$499$	374.123	0.749745	0.374873	+	0.927076i	$0.377686\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 762.000i	− 1.51491i	−0.652887	−	0.757455i	$-0.726442\pi$
$503$	− 762.000i	− 1.51491i	0.652887	−	0.757455i	$-0.273558\pi$
$504$	0	0
$505$	−552.000	−1.09307
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 204.382i	− 0.401536i	−0.979639	−	0.200768i	$-0.935656\pi$
$509$	− 204.382i	− 0.401536i	0.979639	−	0.200768i	$-0.0643438\pi$
$510$	0	0
$511$	− 49.0000i	− 0.0958904i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	329.090	0.639009
$516$	0	0
$517$	− 1143.15i	− 2.21113i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−726.000	−1.39347	−0.696737	−	0.717327i	$-0.745366\pi$
$521$	−726.000	−1.39347	−0.696737	+	0.717327i	$0.745366\pi$
$522$	0	0
$523$	−833.116	−1.59296	−0.796478	−	0.604667i	$-0.793306\pi$
$523$	−833.116	−1.59296	−0.796478	+	0.604667i	$0.793306\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	84.0000i	0.159393i
$528$	0	0
$529$	−371.000	−0.701323
$530$	0	0
$531$	0	0
$532$	0	0
$533$	83.1384i	0.155982i
$534$	0	0
$535$	60.0000i	0.112150i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−831.384	−1.54246
$540$	0	0
$541$	646.055i	1.19419i	0.802172	+	0.597093i	$0.203678\pi$
$541$	646.055i	1.19419i	−0.802172	+	0.597093i	$0.796322\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−648.000	−1.18899
$546$	0	0
$547$	−313.501	−0.573128	−0.286564	−	0.958061i	$-0.592513\pi$
$547$	−313.501	−0.573128	−0.286564	+	0.958061i	$0.592513\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 36.0000i	− 0.0653358i
$552$	0	0
$553$	−83.0000	−0.150090
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 682.428i	− 1.22518i	−0.790399	−	0.612592i	$-0.790127\pi$
$557$	− 682.428i	− 1.22518i	0.790399	−	0.612592i	$-0.209873\pi$
$558$	0	0
$559$	36.0000i	0.0644007i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−651.251	−1.15675	−0.578376	−	0.815770i	$-0.696313\pi$
$563$	−651.251	−1.15675	−0.578376	+	0.815770i	$0.696313\pi$
$564$	0	0
$565$	− 519.615i	− 0.919673i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	846.000	1.48682	0.743409	−	0.668837i	$-0.233207\pi$
$569$	846.000	1.48682	0.743409	+	0.668837i	$0.233207\pi$
$570$	0	0
$571$	375.855	0.658240	0.329120	−	0.944288i	$-0.393248\pi$
$571$	375.855	0.658240	0.329120	+	0.944288i	$0.393248\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 390.000i	− 0.678261i
$576$	0	0
$577$	−791.000	−1.37088	−0.685442	−	0.728127i	$-0.740391\pi$
$577$	−791.000	−1.37088	−0.685442	+	0.728127i	$0.740391\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.8564i	0.0238492i
$582$	0	0
$583$	840.000i	1.44082i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−273.664	−0.466208	−0.233104	−	0.972452i	$-0.574888\pi$
$587$	−273.664	−0.466208	−0.233104	+	0.972452i	$0.574888\pi$
$588$	0	0
$589$	− 24.2487i	− 0.0411693i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	768.000	1.29511	0.647555	−	0.762019i	$-0.275792\pi$
$593$	768.000	1.29511	0.647555	+	0.762019i	$0.275792\pi$
$594$	0	0
$595$	−20.7846	−0.0349321
$596$	0	0
$597$	0	0
$598$	0	0
$599$	564.000i	0.941569i	0.882248	+	0.470785i	$0.156029\pi$
$599$	564.000i	0.941569i	−0.882248	+	0.470785i	$0.843971\pi$
$600$	0	0
$601$	−178.000	−0.296173	−0.148087	−	0.988974i	$-0.547311\pi$
$601$	−178.000	−0.296173	−0.148087	+	0.988974i	$0.547311\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	620.074i	1.02492i
$606$	0	0
$607$	1103.00i	1.81713i	0.417740	+	0.908567i	$0.362822\pi$
$607$	1103.00i	1.81713i	−0.417740	+	0.908567i	$0.637178\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−114.315	−0.187096
$612$	0	0
$613$	− 226.899i	− 0.370145i	−0.982725	−	0.185072i	$-0.940748\pi$
$613$	− 226.899i	− 0.370145i	0.982725	−	0.185072i	$-0.0592519\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−162.000	−0.262561	−0.131280	−	0.991345i	$-0.541909\pi$
$617$	−162.000	−0.262561	−0.131280	+	0.991345i	$0.541909\pi$
$618$	0	0
$619$	−351.606	−0.568023	−0.284012	−	0.958821i	$-0.591665\pi$
$619$	−351.606	−0.568023	−0.284012	+	0.958821i	$0.591665\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	66.0000i	0.105939i
$624$	0	0
$625$	−131.000	−0.209600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 114.315i	− 0.181741i
$630$	0	0
$631$	− 85.0000i	− 0.134707i	−0.997729	−	0.0673534i	$-0.978545\pi$
$631$	− 85.0000i	− 0.134707i	0.997729	−	0.0673534i	$-0.0214555\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	658.179	1.03650
$636$	0	0
$637$	83.1384i	0.130516i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	720.000	1.12324	0.561622	−	0.827394i	$-0.310177\pi$
$641$	720.000	1.12324	0.561622	+	0.827394i	$0.310177\pi$
$642$	0	0
$643$	−394.908	−0.614164	−0.307082	−	0.951683i	$-0.599353\pi$
$643$	−394.908	−0.614164	−0.307082	+	0.951683i	$0.599353\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	312.000i	0.482226i	0.970497	+	0.241113i	$0.0775124\pi$
$647$	312.000i	0.482226i	−0.970497	+	0.241113i	$0.922488\pi$
$648$	0	0
$649$	−540.000	−0.832049
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 720.533i	− 1.10342i	−0.834036	−	0.551710i	$-0.813976\pi$
$653$	− 720.533i	− 1.10342i	0.834036	−	0.551710i	$-0.186024\pi$
$654$	0	0
$655$	− 168.000i	− 0.256489i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−200.918	−0.304883	−0.152441	−	0.988312i	$-0.548714\pi$
$659$	−200.918	−0.304883	−0.152441	+	0.988312i	$0.548714\pi$
$660$	0	0
$661$	− 933.575i	− 1.41237i	−0.708028	−	0.706184i	$-0.750415\pi$
$661$	− 933.575i	− 1.41237i	0.708028	−	0.706184i	$-0.249585\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.00000	0.00902256
$666$	0	0
$667$	−623.538	−0.934840
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 750.000i	− 1.11773i
$672$	0	0
$673$	−323.000	−0.479941	−0.239970	−	0.970780i	$-0.577138\pi$
$673$	−323.000	−0.479941	−0.239970	+	0.970780i	$0.577138\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 737.854i	− 1.08989i	−0.838473	−	0.544944i	$-0.816551\pi$
$677$	− 737.854i	− 1.08989i	0.838473	−	0.544944i	$-0.183449\pi$
$678$	0	0
$679$	− 107.000i	− 0.157585i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1060.02	1.55200	0.775999	−	0.630734i	$-0.217246\pi$
$683$	1060.02	1.55200	0.775999	+	0.630734i	$0.217246\pi$
$684$	0	0
$685$	769.031i	1.12267i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	84.0000	0.121916
$690$	0	0
$691$	561.184	0.812134	0.406067	−	0.913843i	$-0.366900\pi$
$691$	561.184	0.812134	0.406067	+	0.913843i	$0.366900\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	570.000i	0.820144i
$696$	0	0
$697$	−288.000	−0.413199
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 557.720i	− 0.795607i	−0.917471	−	0.397803i	$-0.869773\pi$
$701$	− 557.720i	− 0.795607i	0.917471	−	0.397803i	$-0.130227\pi$
$702$	0	0
$703$	33.0000i	0.0469417i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	159.349	0.225387
$708$	0	0
$709$	− 912.791i	− 1.28743i	−0.765264	−	0.643717i	$-0.777391\pi$
$709$	− 912.791i	− 1.28743i	0.765264	−	0.643717i	$-0.222609\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−420.000	−0.589060
$714$	0	0
$715$	103.923	0.145347
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 432.000i	− 0.600834i	−0.953808	−	0.300417i	$-0.902874\pi$
$719$	− 432.000i	− 0.600834i	0.953808	−	0.300417i	$-0.0971259\pi$
$720$	0	0
$721$	−95.0000	−0.131761
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 270.200i	− 0.372690i
$726$	0	0
$727$	758.000i	1.04264i	0.853361	+	0.521320i	$0.174560\pi$
$727$	758.000i	1.04264i	−0.853361	+	0.521320i	$0.825440\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−124.708	−0.170599
$732$	0	0
$733$	270.200i	0.368622i	0.982868	+	0.184311i	$0.0590054\pi$
$733$	270.200i	0.368622i	−0.982868	+	0.184311i	$0.940995\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1050.00	−1.42469
$738$	0	0
$739$	−852.169	−1.15314	−0.576569	−	0.817048i	$-0.695609\pi$
$739$	−852.169	−1.15314	−0.576569	+	0.817048i	$0.695609\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	150.000i	0.201884i	0.994892	+	0.100942i	$0.0321857\pi$
$743$	150.000i	0.201884i	−0.994892	+	0.100942i	$0.967814\pi$
$744$	0	0
$745$	336.000	0.451007
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 17.3205i	− 0.0231248i
$750$	0	0
$751$	457.000i	0.608522i	0.952589	+	0.304261i	$0.0984095\pi$
$751$	457.000i	0.608522i	−0.952589	+	0.304261i	$0.901591\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	377.587	0.500115
$756$	0	0
$757$	− 621.806i	− 0.821409i	−0.911769	−	0.410704i	$-0.865283\pi$
$757$	− 621.806i	− 0.821409i	0.911769	−	0.410704i	$-0.134717\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1458.00	1.91590	0.957950	−	0.286935i	$-0.0926364\pi$
$761$	1458.00	1.91590	0.957950	+	0.286935i	$0.0926364\pi$
$762$	0	0
$763$	187.061	0.245166
$764$	0	0
$765$	0	0
$766$	0	0
$767$	54.0000i	0.0704042i
$768$	0	0
$769$	47.0000	0.0611183	0.0305592	−	0.999533i	$-0.490271\pi$
$769$	47.0000	0.0611183	0.0305592	+	0.999533i	$0.490271\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	706.677i	0.914200i	0.889415	+	0.457100i	$0.151112\pi$
$773$	706.677i	0.914200i	−0.889415	+	0.457100i	$0.848888\pi$
$774$	0	0
$775$	− 182.000i	− 0.234839i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	83.1384	0.106725
$780$	0	0
$781$	831.384i	1.06451i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−216.000	−0.275159
$786$	0	0
$787$	708.409	0.900138	0.450069	−	0.892994i	$-0.351399\pi$
$787$	708.409	0.900138	0.450069	+	0.892994i	$0.351399\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	150.000i	0.189633i
$792$	0	0
$793$	−75.0000	−0.0945776
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1129.30i	1.41693i	0.705743	+	0.708467i	$0.250613\pi$
$797$	1129.30i	1.41693i	−0.705743	+	0.708467i	$0.749387\pi$
$798$	0	0
$799$	− 396.000i	− 0.495620i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−848.705	−1.05692
$804$	0	0
$805$	− 103.923i	− 0.129097i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−288.000	−0.355995	−0.177998	−	0.984031i	$-0.556962\pi$
$809$	−288.000	−0.355995	−0.177998	+	0.984031i	$0.556962\pi$
$810$	0	0
$811$	187.061	0.230655	0.115328	−	0.993328i	$-0.463208\pi$
$811$	187.061	0.230655	0.115328	+	0.993328i	$0.463208\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 294.000i	− 0.360736i
$816$	0	0
$817$	36.0000	0.0440636
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 114.315i	− 0.139239i	−0.997574	−	0.0696196i	$-0.977821\pi$
$821$	− 114.315i	− 0.139239i	0.997574	−	0.0696196i	$-0.0221785\pi$
$822$	0	0
$823$	− 601.000i	− 0.730255i	−0.930957	−	0.365128i	$-0.881025\pi$
$823$	− 601.000i	− 0.730255i	0.930957	−	0.365128i	$-0.118975\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−820.992	−0.992735	−0.496368	−	0.868112i	$-0.665333\pi$
$827$	−820.992	−0.992735	−0.496368	+	0.868112i	$0.665333\pi$
$828$	0	0
$829$	642.591i	0.775140i	0.921840	+	0.387570i	$0.126685\pi$
$829$	642.591i	0.775140i	−0.921840	+	0.387570i	$0.873315\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−288.000	−0.345738
$834$	0	0
$835$	−810.600	−0.970778
$836$	0	0
$837$	0	0
$838$	0	0
$839$	972.000i	1.15852i	0.815142	+	0.579261i	$0.196659\pi$
$839$	972.000i	1.15852i	−0.815142	+	0.579261i	$0.803341\pi$
$840$	0	0
$841$	409.000	0.486326
$842$	0	0
$843$	0	0
$844$	0	0
$845$	575.041i	0.680522i
$846$	0	0
$847$	− 179.000i	− 0.211334i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	571.577	0.671653
$852$	0	0
$853$	− 334.286i	− 0.391894i	−0.980614	−	0.195947i	$-0.937222\pi$
$853$	− 334.286i	− 0.391894i	0.980614	−	0.195947i	$-0.0627781\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.0000	−0.0280047	−0.0140023	−	0.999902i	$-0.504457\pi$
$857$	−24.0000	−0.0280047	−0.0140023	+	0.999902i	$0.504457\pi$
$858$	0	0
$859$	105.655	0.122998	0.0614989	−	0.998107i	$-0.480412\pi$
$859$	105.655	0.122998	0.0614989	+	0.998107i	$0.480412\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 150.000i	− 0.173812i	−0.996217	−	0.0869061i	$-0.972302\pi$
$863$	− 150.000i	− 0.173812i	0.996217	−	0.0869061i	$-0.0276980\pi$
$864$	0	0
$865$	624.000	0.721387
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1437.60i	1.65432i
$870$	0	0
$871$	105.000i	0.120551i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	131.636	0.150441
$876$	0	0
$877$	39.8372i	0.0454244i	0.999742	+	0.0227122i	$0.00723013\pi$
$877$	39.8372i	0.0454244i	−0.999742	+	0.0227122i	$0.992770\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1014.00	−1.15096	−0.575482	−	0.817814i	$-0.695186\pi$
$881$	−1014.00	−1.15096	−0.575482	+	0.817814i	$0.695186\pi$
$882$	0	0
$883$	476.314	0.539427	0.269713	−	0.962941i	$-0.413071\pi$
$883$	476.314	0.539427	0.269713	+	0.962941i	$0.413071\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 768.000i	− 0.865840i	−0.901432	−	0.432920i	$-0.857483\pi$
$887$	− 768.000i	− 0.865840i	0.901432	−	0.432920i	$-0.142517\pi$
$888$	0	0
$889$	−190.000	−0.213723
$890$	0	0
$891$	0	0
$892$	0	0
$893$	114.315i	0.128013i
$894$	0	0
$895$	672.000i	0.750838i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−290.985	−0.323676
$900$	0	0
$901$	290.985i	0.322957i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−570.000	−0.629834
$906$	0	0
$907$	−209.578	−0.231067	−0.115534	−	0.993304i	$-0.536858\pi$
$907$	−209.578	−0.231067	−0.115534	+	0.993304i	$0.536858\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	600.000i	0.658617i	0.944222	+	0.329308i	$0.106816\pi$
$911$	600.000i	0.658617i	−0.944222	+	0.329308i	$0.893184\pi$
$912$	0	0
$913$	240.000	0.262870
$914$	0	0
$915$	0	0
$916$	0	0
$917$	48.4974i	0.0528870i
$918$	0	0
$919$	422.000i	0.459195i	0.973286	+	0.229597i	$0.0737409\pi$
$919$	422.000i	0.459195i	−0.973286	+	0.229597i	$0.926259\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	83.1384	0.0900741
$924$	0	0
$925$	247.683i	0.267766i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1116.00	−1.20129	−0.600646	−	0.799515i	$-0.705090\pi$
$929$	−1116.00	−1.20129	−0.600646	+	0.799515i	$0.705090\pi$
$930$	0	0
$931$	83.1384	0.0893001
$932$	0	0
$933$	0	0
$934$	0	0
$935$	360.000i	0.385027i
$936$	0	0
$937$	−275.000	−0.293490	−0.146745	−	0.989174i	$-0.546880\pi$
$937$	−275.000	−0.293490	−0.146745	+	0.989174i	$0.546880\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 329.090i	− 0.349723i	−0.984593	−	0.174862i	$-0.944052\pi$
$941$	− 329.090i	− 0.349723i	0.984593	−	0.174862i	$-0.0559478\pi$
$942$	0	0
$943$	− 1440.00i	− 1.52704i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	162.813	0.171925	0.0859624	−	0.996298i	$-0.472604\pi$
$947$	162.813	0.171925	0.0859624	+	0.996298i	$0.472604\pi$
$948$	0	0
$949$	84.8705i	0.0894315i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1506.00	1.58027	0.790136	−	0.612931i	$-0.210010\pi$
$953$	1506.00	1.58027	0.790136	+	0.612931i	$0.210010\pi$
$954$	0	0
$955$	685.892	0.718212
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 222.000i	− 0.231491i
$960$	0	0
$961$	765.000	0.796046
$962$	0	0
$963$	0	0
$964$	0	0
$965$	786.351i	0.814872i
$966$	0	0
$967$	− 1307.00i	− 1.35160i	−0.737084	−	0.675801i	$-0.763798\pi$
$967$	− 1307.00i	− 1.35160i	0.737084	−	0.675801i	$-0.236202\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	627.002	0.645729	0.322864	−	0.946445i	$-0.395354\pi$
$971$	627.002	0.645729	0.322864	+	0.946445i	$0.395354\pi$
$972$	0	0
$973$	− 164.545i	− 0.169111i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	840.000	0.859775	0.429887	−	0.902883i	$-0.358553\pi$
$977$	840.000	0.859775	0.429887	+	0.902883i	$0.358553\pi$
$978$	0	0
$979$	1143.15	1.16767
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1134.00i	− 1.15361i	−0.816881	−	0.576806i	$-0.804299\pi$
$983$	− 1134.00i	− 1.15361i	0.816881	−	0.576806i	$-0.195701\pi$
$984$	0	0
$985$	804.000	0.816244
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 623.538i	− 0.630473i
$990$	0	0
$991$	− 1355.00i	− 1.36731i	−0.729807	−	0.683653i	$-0.760390\pi$
$991$	− 1355.00i	− 1.36731i	0.729807	−	0.683653i	$-0.239610\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	827.920	0.832081
$996$	0	0
$997$	394.908i	0.396096i	0.980192	+	0.198048i	$0.0634602\pi$
$997$	394.908i	0.396096i	−0.980192	+	0.198048i	$0.936540\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.b.b.1567.3	yes	4
3.2	odd	2		1728.3.b.e.1567.1	yes	4
4.3	odd	2	inner	1728.3.b.b.1567.4	yes	4
8.3	odd	2	inner	1728.3.b.b.1567.2	yes	4
8.5	even	2	inner	1728.3.b.b.1567.1	✓	4
12.11	even	2		1728.3.b.e.1567.2	yes	4
24.5	odd	2		1728.3.b.e.1567.3	yes	4
24.11	even	2		1728.3.b.e.1567.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.b.b.1567.1	✓	4	8.5	even	2	inner
1728.3.b.b.1567.2	yes	4	8.3	odd	2	inner
1728.3.b.b.1567.3	yes	4	1.1	even	1	trivial
1728.3.b.b.1567.4	yes	4	4.3	odd	2	inner
1728.3.b.e.1567.1	yes	4	3.2	odd	2
1728.3.b.e.1567.2	yes	4	12.11	even	2
1728.3.b.e.1567.3	yes	4	24.5	odd	2
1728.3.b.e.1567.4	yes	4	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.46410i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+3.46410i q^{5} -1.00000i q^{7} -17.3205 q^{11} +1.73205i q^{13} -6.00000 q^{17} +1.73205 q^{19} -30.0000i q^{23} +13.0000 q^{25} -20.7846i q^{29} -14.0000i q^{31} +3.46410 q^{35} +19.0526i q^{37} +48.0000 q^{41} +20.7846 q^{43} +66.0000i q^{47} +48.0000 q^{49} -48.4974i q^{53} -60.0000i q^{55} +31.1769 q^{59} +43.3013i q^{61} -6.00000 q^{65} +60.6218 q^{67} -48.0000i q^{71} +49.0000 q^{73} +17.3205i q^{77} -83.0000i q^{79} -13.8564 q^{83} -20.7846i q^{85} -66.0000 q^{89} +1.73205 q^{91} +6.00000i q^{95} +107.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{17} + 52 q^{25} + 192 q^{41} + 192 q^{49} - 24 q^{65} + 196 q^{73} - 264 q^{89} + 428 q^{97}+O(q^{100}) \) 4 * q - 24 * q^17 + 52 * q^25 + 192 * q^41 + 192 * q^49 - 24 * q^65 + 196 * q^73 - 264 * q^89 + 428 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more