LMFDB - Embedded newform 1568.3.g.c.687.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(687,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, 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("1568.687");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.g (of order $2$, degree $1$, 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:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			687.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1568.687
Dual form			1568.3.g.c.687.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +5.19615i q^{5} -8.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +5.19615i q^{5} -8.00000 q^{9} -17.0000 q^{11} +13.8564i q^{13} -5.19615i q^{15} -25.0000 q^{17} +7.00000 q^{19} +5.19615i q^{23} -2.00000 q^{25} +17.0000 q^{27} -13.8564i q^{29} -32.9090i q^{31} +17.0000 q^{33} -8.66025i q^{37} -13.8564i q^{39} +26.0000 q^{41} -14.0000 q^{43} -41.5692i q^{45} -50.2295i q^{47} +25.0000 q^{51} +91.7987i q^{53} -88.3346i q^{55} -7.00000 q^{57} +55.0000 q^{59} -22.5167i q^{61} -72.0000 q^{65} -17.0000 q^{67} -5.19615i q^{69} +119.000 q^{73} +2.00000 q^{75} +74.4782i q^{79} +55.0000 q^{81} -110.000 q^{83} -129.904i q^{85} +13.8564i q^{87} +71.0000 q^{89} +32.9090i q^{93} +36.3731i q^{95} -22.0000 q^{97} +136.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 16 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 16 * q^9	$ 2 q - 2 q^{3} - 16 q^{9} - 34 q^{11} - 50 q^{17} + 14 q^{19} - 4 q^{25} + 34 q^{27} + 34 q^{33} + 52 q^{41} - 28 q^{43} + 50 q^{51} - 14 q^{57} + 110 q^{59} - 144 q^{65} - 34 q^{67} + 238 q^{73} + 4 q^{75} + 110 q^{81} - 220 q^{83} + 142 q^{89} - 44 q^{97} + 272 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 16 * q^9 - 34 * q^11 - 50 * q^17 + 14 * q^19 - 4 * q^25 + 34 * q^27 + 34 * q^33 + 52 * q^41 - 28 * q^43 + 50 * q^51 - 14 * q^57 + 110 * q^59 - 144 * q^65 - 34 * q^67 + 238 * q^73 + 4 * q^75 + 110 * q^81 - 220 * q^83 + 142 * q^89 - 44 * q^97 + 272 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−1.00000	−0.333333	−0.166667	−	0.986013i	$-0.553300\pi$
$3$	−1.00000	−0.333333	−0.166667	+	0.986013i	$0.553300\pi$
$4$	0	0
$5$	5.19615i	1.03923i	0.854400	+	0.519615i	$0.173925\pi$
$5$	5.19615i	1.03923i	−0.854400	+	0.519615i	$0.826075\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−8.00000	−0.888889
$10$	0	0
$11$	−17.0000	−1.54545	−0.772727	−	0.634738i	$-0.781108\pi$
$11$	−17.0000	−1.54545	−0.772727	+	0.634738i	$0.781108\pi$
$12$	0	0
$13$	13.8564i	1.06588i	0.846154	+	0.532939i	$0.178912\pi$
$13$	13.8564i	1.06588i	−0.846154	+	0.532939i	$0.821088\pi$
$14$	0	0
$15$	− 5.19615i	− 0.346410i
$16$	0	0
$17$	−25.0000	−1.47059	−0.735294	−	0.677748i	$-0.762956\pi$
$17$	−25.0000	−1.47059	−0.735294	+	0.677748i	$0.762956\pi$
$18$	0	0
$19$	7.00000	0.368421	0.184211	−	0.982887i	$-0.441027\pi$
$19$	7.00000	0.368421	0.184211	+	0.982887i	$0.441027\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.19615i	0.225920i	0.993600	+	0.112960i	$0.0360331\pi$
$23$	5.19615i	0.225920i	−0.993600	+	0.112960i	$0.963967\pi$
$24$	0	0
$25$	−2.00000	−0.0800000
$26$	0	0
$27$	17.0000	0.629630
$28$	0	0
$29$	− 13.8564i	− 0.477807i	−0.971043	−	0.238904i	$-0.923212\pi$
$29$	− 13.8564i	− 0.477807i	0.971043	−	0.238904i	$-0.0767880\pi$
$30$	0	0
$31$	− 32.9090i	− 1.06158i	−0.847504	−	0.530790i	$-0.821895\pi$
$31$	− 32.9090i	− 1.06158i	0.847504	−	0.530790i	$-0.178105\pi$
$32$	0	0
$33$	17.0000	0.515152
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.66025i	− 0.234061i	−0.993128	−	0.117030i	$-0.962662\pi$
$37$	− 8.66025i	− 0.234061i	0.993128	−	0.117030i	$-0.0373375\pi$
$38$	0	0
$39$	− 13.8564i	− 0.355292i
$40$	0	0
$41$	26.0000	0.634146	0.317073	−	0.948401i	$-0.397300\pi$
$41$	26.0000	0.634146	0.317073	+	0.948401i	$0.397300\pi$
$42$	0	0
$43$	−14.0000	−0.325581	−0.162791	−	0.986661i	$-0.552050\pi$
$43$	−14.0000	−0.325581	−0.162791	+	0.986661i	$0.552050\pi$
$44$	0	0
$45$	− 41.5692i	− 0.923760i
$46$	0	0
$47$	− 50.2295i	− 1.06871i	−0.845259	−	0.534356i	$-0.820554\pi$
$47$	− 50.2295i	− 1.06871i	0.845259	−	0.534356i	$-0.179446\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	25.0000	0.490196
$52$	0	0
$53$	91.7987i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$53$	91.7987i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$54$	0	0
$55$	− 88.3346i	− 1.60608i
$56$	0	0
$57$	−7.00000	−0.122807
$58$	0	0
$59$	55.0000	0.932203	0.466102	−	0.884731i	$-0.345658\pi$
$59$	55.0000	0.932203	0.466102	+	0.884731i	$0.345658\pi$
$60$	0	0
$61$	− 22.5167i	− 0.369126i	−0.982821	−	0.184563i	$-0.940913\pi$
$61$	− 22.5167i	− 0.369126i	0.982821	−	0.184563i	$-0.0590869\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−72.0000	−1.10769
$66$	0	0
$67$	−17.0000	−0.253731	−0.126866	−	0.991920i	$-0.540492\pi$
$67$	−17.0000	−0.253731	−0.126866	+	0.991920i	$0.540492\pi$
$68$	0	0
$69$	− 5.19615i	− 0.0753066i
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	119.000	1.63014	0.815068	−	0.579365i	$-0.196699\pi$
$73$	119.000	1.63014	0.815068	+	0.579365i	$0.196699\pi$
$74$	0	0
$75$	2.00000	0.0266667
$76$	0	0
$77$	0	0
$78$	0	0
$79$	74.4782i	0.942762i	0.881930	+	0.471381i	$0.156244\pi$
$79$	74.4782i	0.942762i	−0.881930	+	0.471381i	$0.843756\pi$
$80$	0	0
$81$	55.0000	0.679012
$82$	0	0
$83$	−110.000	−1.32530	−0.662651	−	0.748929i	$-0.730569\pi$
$83$	−110.000	−1.32530	−0.662651	+	0.748929i	$0.730569\pi$
$84$	0	0
$85$	− 129.904i	− 1.52828i
$86$	0	0
$87$	13.8564i	0.159269i
$88$	0	0
$89$	71.0000	0.797753	0.398876	−	0.917005i	$-0.369400\pi$
$89$	71.0000	0.797753	0.398876	+	0.917005i	$0.369400\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	32.9090i	0.353860i
$94$	0	0
$95$	36.3731i	0.382874i
$96$	0	0
$97$	−22.0000	−0.226804	−0.113402	−	0.993549i	$-0.536175\pi$
$97$	−22.0000	−0.226804	−0.113402	+	0.993549i	$0.536175\pi$
$98$	0	0
$99$	136.000	1.37374
$100$	0	0
$101$	77.9423i	0.771706i	0.922560	+	0.385853i	$0.126093\pi$
$101$	77.9423i	0.771706i	−0.922560	+	0.385853i	$0.873907\pi$
$102$	0	0
$103$	− 161.081i	− 1.56389i	−0.623347	−	0.781945i	$-0.714228\pi$
$103$	− 161.081i	− 1.56389i	0.623347	−	0.781945i	$-0.285772\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−65.0000	−0.607477	−0.303738	−	0.952755i	$-0.598235\pi$
$107$	−65.0000	−0.607477	−0.303738	+	0.952755i	$0.598235\pi$
$108$	0	0
$109$	8.66025i	0.0794519i	0.999211	+	0.0397259i	$0.0126485\pi$
$109$	8.66025i	0.0794519i	−0.999211	+	0.0397259i	$0.987352\pi$
$110$	0	0
$111$	8.66025i	0.0780203i
$112$	0	0
$113$	122.000	1.07965	0.539823	−	0.841779i	$-0.318491\pi$
$113$	122.000	1.07965	0.539823	+	0.841779i	$0.318491\pi$
$114$	0	0
$115$	−27.0000	−0.234783
$116$	0	0
$117$	− 110.851i	− 0.947447i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	168.000	1.38843
$122$	0	0
$123$	−26.0000	−0.211382
$124$	0	0
$125$	119.512i	0.956092i
$126$	0	0
$127$	− 166.277i	− 1.30927i	−0.755947	−	0.654633i	$-0.772823\pi$
$127$	− 166.277i	− 1.30927i	0.755947	−	0.654633i	$-0.227177\pi$
$128$	0	0
$129$	14.0000	0.108527
$130$	0	0
$131$	−17.0000	−0.129771	−0.0648855	−	0.997893i	$-0.520668\pi$
$131$	−17.0000	−0.129771	−0.0648855	+	0.997893i	$0.520668\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	88.3346i	0.654330i
$136$	0	0
$137$	−145.000	−1.05839	−0.529197	−	0.848499i	$-0.677507\pi$
$137$	−145.000	−1.05839	−0.529197	+	0.848499i	$0.677507\pi$
$138$	0	0
$139$	82.0000	0.589928	0.294964	−	0.955508i	$-0.404692\pi$
$139$	82.0000	0.589928	0.294964	+	0.955508i	$0.404692\pi$
$140$	0	0
$141$	50.2295i	0.356237i
$142$	0	0
$143$	− 235.559i	− 1.64727i
$144$	0	0
$145$	72.0000	0.496552
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.19615i	0.0348735i	0.999848	+	0.0174368i	$0.00555057\pi$
$149$	5.19615i	0.0348735i	−0.999848	+	0.0174368i	$0.994449\pi$
$150$	0	0
$151$	36.3731i	0.240881i	0.992721	+	0.120441i	$0.0384307\pi$
$151$	36.3731i	0.240881i	−0.992721	+	0.120441i	$0.961569\pi$
$152$	0	0
$153$	200.000	1.30719
$154$	0	0
$155$	171.000	1.10323
$156$	0	0
$157$	− 310.037i	− 1.97476i	−0.158373	−	0.987379i	$-0.550625\pi$
$157$	− 310.037i	− 1.97476i	0.158373	−	0.987379i	$-0.449375\pi$
$158$	0	0
$159$	− 91.7987i	− 0.577350i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−17.0000	−0.104294	−0.0521472	−	0.998639i	$-0.516607\pi$
$163$	−17.0000	−0.104294	−0.0521472	+	0.998639i	$0.516607\pi$
$164$	0	0
$165$	88.3346i	0.535361i
$166$	0	0
$167$	− 13.8564i	− 0.0829725i	−0.999139	−	0.0414862i	$-0.986791\pi$
$167$	− 13.8564i	− 0.0829725i	0.999139	−	0.0414862i	$-0.0132093\pi$
$168$	0	0
$169$	−23.0000	−0.136095
$170$	0	0
$171$	−56.0000	−0.327485
$172$	0	0
$173$	− 105.655i	− 0.610723i	−0.952236	−	0.305362i	$-0.901223\pi$
$173$	− 105.655i	− 0.610723i	0.952236	−	0.305362i	$-0.0987773\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−55.0000	−0.310734
$178$	0	0
$179$	−89.0000	−0.497207	−0.248603	−	0.968605i	$-0.579972\pi$
$179$	−89.0000	−0.497207	−0.248603	+	0.968605i	$0.579972\pi$
$180$	0	0
$181$	− 249.415i	− 1.37799i	−0.724768	−	0.688993i	$-0.758053\pi$
$181$	− 249.415i	− 1.37799i	0.724768	−	0.688993i	$-0.241947\pi$
$182$	0	0
$183$	22.5167i	0.123042i
$184$	0	0
$185$	45.0000	0.243243
$186$	0	0
$187$	425.000	2.27273
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 216.506i	− 1.13354i	−0.823876	−	0.566771i	$-0.808193\pi$
$191$	− 216.506i	− 1.13354i	0.823876	−	0.566771i	$-0.191807\pi$
$192$	0	0
$193$	−73.0000	−0.378238	−0.189119	−	0.981954i	$-0.560563\pi$
$193$	−73.0000	−0.378238	−0.189119	+	0.981954i	$0.560563\pi$
$194$	0	0
$195$	72.0000	0.369231
$196$	0	0
$197$	− 207.846i	− 1.05506i	−0.849538	−	0.527528i	$-0.823119\pi$
$197$	− 207.846i	− 1.05506i	0.849538	−	0.527528i	$-0.176881\pi$
$198$	0	0
$199$	64.0859i	0.322040i	0.986951	+	0.161020i	$0.0514783\pi$
$199$	64.0859i	0.322040i	−0.986951	+	0.161020i	$0.948522\pi$
$200$	0	0
$201$	17.0000	0.0845771
$202$	0	0
$203$	0	0
$204$	0	0
$205$	135.100i	0.659024i
$206$	0	0
$207$	− 41.5692i	− 0.200817i
$208$	0	0
$209$	−119.000	−0.569378
$210$	0	0
$211$	−302.000	−1.43128	−0.715640	−	0.698470i	$-0.753865\pi$
$211$	−302.000	−1.43128	−0.715640	+	0.698470i	$0.753865\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 72.7461i	− 0.338354i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−119.000	−0.543379
$220$	0	0
$221$	− 346.410i	− 1.56747i
$222$	0	0
$223$	138.564i	0.621364i	0.950514	+	0.310682i	$0.100557\pi$
$223$	138.564i	0.621364i	−0.950514	+	0.310682i	$0.899443\pi$
$224$	0	0
$225$	16.0000	0.0711111
$226$	0	0
$227$	55.0000	0.242291	0.121145	−	0.992635i	$-0.461343\pi$
$227$	55.0000	0.242291	0.121145	+	0.992635i	$0.461343\pi$
$228$	0	0
$229$	− 327.358i	− 1.42951i	−0.699375	−	0.714755i	$-0.746538\pi$
$229$	− 327.358i	− 1.42951i	0.699375	−	0.714755i	$-0.253462\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−385.000	−1.65236	−0.826180	−	0.563406i	$-0.809491\pi$
$233$	−385.000	−1.65236	−0.826180	+	0.563406i	$0.809491\pi$
$234$	0	0
$235$	261.000	1.11064
$236$	0	0
$237$	− 74.4782i	− 0.314254i
$238$	0	0
$239$	− 429.549i	− 1.79727i	−0.438693	−	0.898637i	$-0.644558\pi$
$239$	− 429.549i	− 1.79727i	0.438693	−	0.898637i	$-0.355442\pi$
$240$	0	0
$241$	−145.000	−0.601660	−0.300830	−	0.953678i	$-0.597264\pi$
$241$	−145.000	−0.601660	−0.300830	+	0.953678i	$0.597264\pi$
$242$	0	0
$243$	−208.000	−0.855967
$244$	0	0
$245$	0	0
$246$	0	0
$247$	96.9948i	0.392692i
$248$	0	0
$249$	110.000	0.441767
$250$	0	0
$251$	58.0000	0.231076	0.115538	−	0.993303i	$-0.463141\pi$
$251$	58.0000	0.231076	0.115538	+	0.993303i	$0.463141\pi$
$252$	0	0
$253$	− 88.3346i	− 0.349149i
$254$	0	0
$255$	129.904i	0.509427i
$256$	0	0
$257$	119.000	0.463035	0.231518	−	0.972831i	$-0.425631\pi$
$257$	119.000	0.463035	0.231518	+	0.972831i	$0.425631\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	110.851i	0.424717i
$262$	0	0
$263$	327.358i	1.24471i	0.782737	+	0.622353i	$0.213823\pi$
$263$	327.358i	1.24471i	−0.782737	+	0.622353i	$0.786177\pi$
$264$	0	0
$265$	−477.000	−1.80000
$266$	0	0
$267$	−71.0000	−0.265918
$268$	0	0
$269$	133.368i	0.495791i	0.968787	+	0.247896i	$0.0797390\pi$
$269$	133.368i	0.495791i	−0.968787	+	0.247896i	$0.920261\pi$
$270$	0	0
$271$	434.745i	1.60422i	0.597174	+	0.802112i	$0.296290\pi$
$271$	434.745i	1.60422i	−0.597174	+	0.802112i	$0.703710\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	34.0000	0.123636
$276$	0	0
$277$	202.650i	0.731588i	0.930696	+	0.365794i	$0.119203\pi$
$277$	202.650i	0.731588i	−0.930696	+	0.365794i	$0.880797\pi$
$278$	0	0
$279$	263.272i	0.943626i
$280$	0	0
$281$	74.0000	0.263345	0.131673	−	0.991293i	$-0.457965\pi$
$281$	74.0000	0.263345	0.131673	+	0.991293i	$0.457965\pi$
$282$	0	0
$283$	463.000	1.63604	0.818021	−	0.575188i	$-0.195071\pi$
$283$	463.000	1.63604	0.818021	+	0.575188i	$0.195071\pi$
$284$	0	0
$285$	− 36.3731i	− 0.127625i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	336.000	1.16263
$290$	0	0
$291$	22.0000	0.0756014
$292$	0	0
$293$	− 110.851i	− 0.378332i	−0.981945	−	0.189166i	$-0.939422\pi$
$293$	− 110.851i	− 0.378332i	0.981945	−	0.189166i	$-0.0605784\pi$
$294$	0	0
$295$	285.788i	0.968774i
$296$	0	0
$297$	−289.000	−0.973064
$298$	0	0
$299$	−72.0000	−0.240803
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 77.9423i	− 0.257235i
$304$	0	0
$305$	117.000	0.383607
$306$	0	0
$307$	274.000	0.892508	0.446254	−	0.894906i	$-0.352758\pi$
$307$	274.000	0.892508	0.446254	+	0.894906i	$0.352758\pi$
$308$	0	0
$309$	161.081i	0.521297i
$310$	0	0
$311$	50.2295i	0.161510i	0.996734	+	0.0807548i	$0.0257331\pi$
$311$	50.2295i	0.161510i	−0.996734	+	0.0807548i	$0.974267\pi$
$312$	0	0
$313$	−409.000	−1.30671	−0.653355	−	0.757052i	$-0.726639\pi$
$313$	−409.000	−1.30671	−0.653355	+	0.757052i	$0.726639\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 188.794i	− 0.595563i	−0.954634	−	0.297782i	$-0.903753\pi$
$317$	− 188.794i	− 0.595563i	0.954634	−	0.297782i	$-0.0962467\pi$
$318$	0	0
$319$	235.559i	0.738429i
$320$	0	0
$321$	65.0000	0.202492
$322$	0	0
$323$	−175.000	−0.541796
$324$	0	0
$325$	− 27.7128i	− 0.0852702i
$326$	0	0
$327$	− 8.66025i	− 0.0264840i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	295.000	0.891239	0.445619	−	0.895223i	$-0.352983\pi$
$331$	295.000	0.891239	0.445619	+	0.895223i	$0.352983\pi$
$332$	0	0
$333$	69.2820i	0.208054i
$334$	0	0
$335$	− 88.3346i	− 0.263685i
$336$	0	0
$337$	26.0000	0.0771513	0.0385757	−	0.999256i	$-0.487718\pi$
$337$	26.0000	0.0771513	0.0385757	+	0.999256i	$0.487718\pi$
$338$	0	0
$339$	−122.000	−0.359882
$340$	0	0
$341$	559.452i	1.64062i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	27.0000	0.0782609
$346$	0	0
$347$	−377.000	−1.08646	−0.543228	−	0.839585i	$-0.682798\pi$
$347$	−377.000	−1.08646	−0.543228	+	0.839585i	$0.682798\pi$
$348$	0	0
$349$	− 96.9948i	− 0.277922i	−0.990298	−	0.138961i	$-0.955624\pi$
$349$	− 96.9948i	− 0.277922i	0.990298	−	0.138961i	$-0.0443763\pi$
$350$	0	0
$351$	235.559i	0.671108i
$352$	0	0
$353$	503.000	1.42493	0.712465	−	0.701708i	$-0.247579\pi$
$353$	503.000	1.42493	0.712465	+	0.701708i	$0.247579\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	185.329i	0.516238i	0.966113	+	0.258119i	$0.0831027\pi$
$359$	185.329i	0.516238i	−0.966113	+	0.258119i	$0.916897\pi$
$360$	0	0
$361$	−312.000	−0.864266
$362$	0	0
$363$	−168.000	−0.462810
$364$	0	0
$365$	618.342i	1.69409i
$366$	0	0
$367$	− 296.181i	− 0.807032i	−0.914973	−	0.403516i	$-0.867788\pi$
$367$	− 296.181i	− 0.807032i	0.914973	−	0.403516i	$-0.132212\pi$
$368$	0	0
$369$	−208.000	−0.563686
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 119.512i	− 0.320406i	−0.987084	−	0.160203i	$-0.948785\pi$
$373$	− 119.512i	− 0.320406i	0.987084	−	0.160203i	$-0.0512149\pi$
$374$	0	0
$375$	− 119.512i	− 0.318697i
$376$	0	0
$377$	192.000	0.509284
$378$	0	0
$379$	634.000	1.67282	0.836412	−	0.548102i	$-0.184649\pi$
$379$	634.000	1.67282	0.836412	+	0.548102i	$0.184649\pi$
$380$	0	0
$381$	166.277i	0.436422i
$382$	0	0
$383$	− 244.219i	− 0.637648i	−0.947814	−	0.318824i	$-0.896712\pi$
$383$	− 244.219i	− 0.637648i	0.947814	−	0.318824i	$-0.103288\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	112.000	0.289406
$388$	0	0
$389$	− 587.165i	− 1.50942i	−0.656057	−	0.754711i	$-0.727777\pi$
$389$	− 587.165i	− 1.50942i	0.656057	−	0.754711i	$-0.272223\pi$
$390$	0	0
$391$	− 129.904i	− 0.332235i
$392$	0	0
$393$	17.0000	0.0432570
$394$	0	0
$395$	−387.000	−0.979747
$396$	0	0
$397$	240.755i	0.606436i	0.952921	+	0.303218i	$0.0980610\pi$
$397$	240.755i	0.606436i	−0.952921	+	0.303218i	$0.901939\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	119.000	0.296758	0.148379	−	0.988931i	$-0.452594\pi$
$401$	119.000	0.296758	0.148379	+	0.988931i	$0.452594\pi$
$402$	0	0
$403$	456.000	1.13151
$404$	0	0
$405$	285.788i	0.705650i
$406$	0	0
$407$	147.224i	0.361731i
$408$	0	0
$409$	−145.000	−0.354523	−0.177262	−	0.984164i	$-0.556724\pi$
$409$	−145.000	−0.354523	−0.177262	+	0.984164i	$0.556724\pi$
$410$	0	0
$411$	145.000	0.352798
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 571.577i	− 1.37729i
$416$	0	0
$417$	−82.0000	−0.196643
$418$	0	0
$419$	−302.000	−0.720764	−0.360382	−	0.932805i	$-0.617354\pi$
$419$	−302.000	−0.720764	−0.360382	+	0.932805i	$0.617354\pi$
$420$	0	0
$421$	401.836i	0.954479i	0.878773	+	0.477240i	$0.158363\pi$
$421$	401.836i	0.954479i	−0.878773	+	0.477240i	$0.841637\pi$
$422$	0	0
$423$	401.836i	0.949966i
$424$	0	0
$425$	50.0000	0.117647
$426$	0	0
$427$	0	0
$428$	0	0
$429$	235.559i	0.549088i
$430$	0	0
$431$	− 808.868i	− 1.87672i	−0.345655	−	0.938362i	$-0.612343\pi$
$431$	− 808.868i	− 1.87672i	0.345655	−	0.938362i	$-0.387657\pi$
$432$	0	0
$433$	410.000	0.946882	0.473441	−	0.880825i	$-0.343012\pi$
$433$	410.000	0.946882	0.473441	+	0.880825i	$0.343012\pi$
$434$	0	0
$435$	−72.0000	−0.165517
$436$	0	0
$437$	36.3731i	0.0832336i
$438$	0	0
$439$	490.170i	1.11656i	0.829652	+	0.558281i	$0.188539\pi$
$439$	490.170i	1.11656i	−0.829652	+	0.558281i	$0.811461\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−401.000	−0.905192	−0.452596	−	0.891716i	$-0.649502\pi$
$443$	−401.000	−0.905192	−0.452596	+	0.891716i	$0.649502\pi$
$444$	0	0
$445$	368.927i	0.829049i
$446$	0	0
$447$	− 5.19615i	− 0.0116245i
$448$	0	0
$449$	−310.000	−0.690423	−0.345212	−	0.938525i	$-0.612193\pi$
$449$	−310.000	−0.690423	−0.345212	+	0.938525i	$0.612193\pi$
$450$	0	0
$451$	−442.000	−0.980044
$452$	0	0
$453$	− 36.3731i	− 0.0802937i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	167.000	0.365427	0.182713	−	0.983166i	$-0.441512\pi$
$457$	167.000	0.365427	0.182713	+	0.983166i	$0.441512\pi$
$458$	0	0
$459$	−425.000	−0.925926
$460$	0	0
$461$	− 13.8564i	− 0.0300573i	−0.999887	−	0.0150286i	$-0.995216\pi$
$461$	− 13.8564i	− 0.0300573i	0.999887	−	0.0150286i	$-0.00478394\pi$
$462$	0	0
$463$	609.682i	1.31681i	0.752665	+	0.658404i	$0.228768\pi$
$463$	609.682i	1.31681i	−0.752665	+	0.658404i	$0.771232\pi$
$464$	0	0
$465$	−171.000	−0.367742
$466$	0	0
$467$	−785.000	−1.68094	−0.840471	−	0.541856i	$-0.817722\pi$
$467$	−785.000	−1.68094	−0.840471	+	0.541856i	$0.817722\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	310.037i	0.658253i
$472$	0	0
$473$	238.000	0.503171
$474$	0	0
$475$	−14.0000	−0.0294737
$476$	0	0
$477$	− 734.390i	− 1.53960i
$478$	0	0
$479$	618.342i	1.29090i	0.763802	+	0.645451i	$0.223331\pi$
$479$	618.342i	1.29090i	−0.763802	+	0.645451i	$0.776669\pi$
$480$	0	0
$481$	120.000	0.249480
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 114.315i	− 0.235702i
$486$	0	0
$487$	− 393.176i	− 0.807342i	−0.914904	−	0.403671i	$-0.867734\pi$
$487$	− 393.176i	− 0.807342i	0.914904	−	0.403671i	$-0.132266\pi$
$488$	0	0
$489$	17.0000	0.0347648
$490$	0	0
$491$	−422.000	−0.859470	−0.429735	−	0.902955i	$-0.641393\pi$
$491$	−422.000	−0.859470	−0.429735	+	0.902955i	$0.641393\pi$
$492$	0	0
$493$	346.410i	0.702658i
$494$	0	0
$495$	706.677i	1.42763i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−65.0000	−0.130261	−0.0651303	−	0.997877i	$-0.520746\pi$
$499$	−65.0000	−0.130261	−0.0651303	+	0.997877i	$0.520746\pi$
$500$	0	0
$501$	13.8564i	0.0276575i
$502$	0	0
$503$	− 249.415i	− 0.495855i	−0.968779	−	0.247928i	$-0.920250\pi$
$503$	− 249.415i	− 0.495855i	0.968779	−	0.247928i	$-0.0797496\pi$
$504$	0	0
$505$	−405.000	−0.801980
$506$	0	0
$507$	23.0000	0.0453649
$508$	0	0
$509$	545.596i	1.07190i	0.844250	+	0.535949i	$0.180046\pi$
$509$	545.596i	1.07190i	−0.844250	+	0.535949i	$0.819954\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	119.000	0.231969
$514$	0	0
$515$	837.000	1.62524
$516$	0	0
$517$	853.901i	1.65165i
$518$	0	0
$519$	105.655i	0.203574i
$520$	0	0
$521$	−25.0000	−0.0479846	−0.0239923	−	0.999712i	$-0.507638\pi$
$521$	−25.0000	−0.0479846	−0.0239923	+	0.999712i	$0.507638\pi$
$522$	0	0
$523$	−593.000	−1.13384	−0.566922	−	0.823772i	$-0.691866\pi$
$523$	−593.000	−1.13384	−0.566922	+	0.823772i	$0.691866\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	822.724i	1.56115i
$528$	0	0
$529$	502.000	0.948960
$530$	0	0
$531$	−440.000	−0.828625
$532$	0	0
$533$	360.267i	0.675922i
$534$	0	0
$535$	− 337.750i	− 0.631308i
$536$	0	0
$537$	89.0000	0.165736
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 756.906i	− 1.39909i	−0.714590	−	0.699544i	$-0.753387\pi$
$541$	− 756.906i	− 1.39909i	0.714590	−	0.699544i	$-0.246613\pi$
$542$	0	0
$543$	249.415i	0.459328i
$544$	0	0
$545$	−45.0000	−0.0825688
$546$	0	0
$547$	−662.000	−1.21024	−0.605119	−	0.796135i	$-0.706874\pi$
$547$	−662.000	−1.21024	−0.605119	+	0.796135i	$0.706874\pi$
$548$	0	0
$549$	180.133i	0.328112i
$550$	0	0
$551$	− 96.9948i	− 0.176034i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−45.0000	−0.0810811
$556$	0	0
$557$	590.629i	1.06038i	0.847880	+	0.530188i	$0.177879\pi$
$557$	590.629i	1.06038i	−0.847880	+	0.530188i	$0.822121\pi$
$558$	0	0
$559$	− 193.990i	− 0.347030i
$560$	0	0
$561$	−425.000	−0.757576
$562$	0	0
$563$	−737.000	−1.30906	−0.654529	−	0.756037i	$-0.727133\pi$
$563$	−737.000	−1.30906	−0.654529	+	0.756037i	$0.727133\pi$
$564$	0	0
$565$	633.931i	1.12200i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−121.000	−0.212654	−0.106327	−	0.994331i	$-0.533909\pi$
$569$	−121.000	−0.212654	−0.106327	+	0.994331i	$0.533909\pi$
$570$	0	0
$571$	−737.000	−1.29072	−0.645359	−	0.763879i	$-0.723292\pi$
$571$	−737.000	−1.29072	−0.645359	+	0.763879i	$0.723292\pi$
$572$	0	0
$573$	216.506i	0.377847i
$574$	0	0
$575$	− 10.3923i	− 0.0180736i
$576$	0	0
$577$	47.0000	0.0814558	0.0407279	−	0.999170i	$-0.487032\pi$
$577$	47.0000	0.0814558	0.0407279	+	0.999170i	$0.487032\pi$
$578$	0	0
$579$	73.0000	0.126079
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 1560.58i	− 2.67681i
$584$	0	0
$585$	576.000	0.984615
$586$	0	0
$587$	−446.000	−0.759796	−0.379898	−	0.925028i	$-0.624041\pi$
$587$	−446.000	−0.759796	−0.379898	+	0.925028i	$0.624041\pi$
$588$	0	0
$589$	− 230.363i	− 0.391108i
$590$	0	0
$591$	207.846i	0.351685i
$592$	0	0
$593$	215.000	0.362563	0.181282	−	0.983431i	$-0.441975\pi$
$593$	215.000	0.362563	0.181282	+	0.983431i	$0.441975\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 64.0859i	− 0.107347i
$598$	0	0
$599$	− 282.324i	− 0.471326i	−0.971835	−	0.235663i	$-0.924274\pi$
$599$	− 282.324i	− 0.471326i	0.971835	−	0.235663i	$-0.0757262\pi$
$600$	0	0
$601$	266.000	0.442596	0.221298	−	0.975206i	$-0.428971\pi$
$601$	266.000	0.442596	0.221298	+	0.975206i	$0.428971\pi$
$602$	0	0
$603$	136.000	0.225539
$604$	0	0
$605$	872.954i	1.44290i
$606$	0	0
$607$	− 659.911i	− 1.08717i	−0.839355	−	0.543584i	$-0.817067\pi$
$607$	− 659.911i	− 1.08717i	0.839355	−	0.543584i	$-0.182933\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	696.000	1.13912
$612$	0	0
$613$	− 698.016i	− 1.13869i	−0.822099	−	0.569345i	$-0.807197\pi$
$613$	− 698.016i	− 1.13869i	0.822099	−	0.569345i	$-0.192803\pi$
$614$	0	0
$615$	− 135.100i	− 0.219675i
$616$	0	0
$617$	−118.000	−0.191248	−0.0956240	−	0.995418i	$-0.530485\pi$
$617$	−118.000	−0.191248	−0.0956240	+	0.995418i	$0.530485\pi$
$618$	0	0
$619$	919.000	1.48465	0.742326	−	0.670039i	$-0.233722\pi$
$619$	919.000	1.48465	0.742326	+	0.670039i	$0.233722\pi$
$620$	0	0
$621$	88.3346i	0.142246i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−671.000	−1.07360
$626$	0	0
$627$	119.000	0.189793
$628$	0	0
$629$	216.506i	0.344207i
$630$	0	0
$631$	− 166.277i	− 0.263513i	−0.991282	−	0.131757i	$-0.957938\pi$
$631$	− 166.277i	− 0.263513i	0.991282	−	0.131757i	$-0.0420617\pi$
$632$	0	0
$633$	302.000	0.477093
$634$	0	0
$635$	864.000	1.36063
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.00000	−0.00156006	−0.000780031	−	1.00000i	$-0.500248\pi$
$641$	−1.00000	−0.00156006	−0.000780031		1.00000i	$0.500248\pi$
$642$	0	0
$643$	514.000	0.799378	0.399689	−	0.916651i	$-0.369118\pi$
$643$	514.000	0.799378	0.399689	+	0.916651i	$0.369118\pi$
$644$	0	0
$645$	72.7461i	0.112785i
$646$	0	0
$647$	− 60.6218i	− 0.0936967i	−0.998902	−	0.0468484i	$-0.985082\pi$
$647$	− 60.6218i	− 0.0936967i	0.998902	−	0.0468484i	$-0.0149178\pi$
$648$	0	0
$649$	−935.000	−1.44068
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 327.358i	− 0.501313i	−0.968076	−	0.250657i	$-0.919353\pi$
$653$	− 327.358i	− 0.501313i	0.968076	−	0.250657i	$-0.0806465\pi$
$654$	0	0
$655$	− 88.3346i	− 0.134862i
$656$	0	0
$657$	−952.000	−1.44901
$658$	0	0
$659$	−542.000	−0.822458	−0.411229	−	0.911532i	$-0.634900\pi$
$659$	−542.000	−0.822458	−0.411229	+	0.911532i	$0.634900\pi$
$660$	0	0
$661$	− 1182.99i	− 1.78970i	−0.446368	−	0.894849i	$-0.647283\pi$
$661$	− 1182.99i	− 1.78970i	0.446368	−	0.894849i	$-0.352717\pi$
$662$	0	0
$663$	346.410i	0.522489i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	72.0000	0.107946
$668$	0	0
$669$	− 138.564i	− 0.207121i
$670$	0	0
$671$	382.783i	0.570467i
$672$	0	0
$673$	218.000	0.323923	0.161961	−	0.986797i	$-0.448218\pi$
$673$	218.000	0.323923	0.161961	+	0.986797i	$0.448218\pi$
$674$	0	0
$675$	−34.0000	−0.0503704
$676$	0	0
$677$	642.591i	0.949174i	0.880209	+	0.474587i	$0.157403\pi$
$677$	642.591i	0.949174i	−0.880209	+	0.474587i	$0.842597\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−55.0000	−0.0807636
$682$	0	0
$683$	367.000	0.537335	0.268668	−	0.963233i	$-0.413417\pi$
$683$	367.000	0.537335	0.268668	+	0.963233i	$0.413417\pi$
$684$	0	0
$685$	− 753.442i	− 1.09992i
$686$	0	0
$687$	327.358i	0.476503i
$688$	0	0
$689$	−1272.00	−1.84615
$690$	0	0
$691$	−497.000	−0.719247	−0.359624	−	0.933097i	$-0.617095\pi$
$691$	−497.000	−0.719247	−0.359624	+	0.933097i	$0.617095\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	426.084i	0.613071i
$696$	0	0
$697$	−650.000	−0.932568
$698$	0	0
$699$	385.000	0.550787
$700$	0	0
$701$	332.554i	0.474399i	0.971461	+	0.237200i	$0.0762295\pi$
$701$	332.554i	0.474399i	−0.971461	+	0.237200i	$0.923770\pi$
$702$	0	0
$703$	− 60.6218i	− 0.0862330i
$704$	0	0
$705$	−261.000	−0.370213
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 396.640i	− 0.559435i	−0.960082	−	0.279718i	$-0.909759\pi$
$709$	− 396.640i	− 0.559435i	0.960082	−	0.279718i	$-0.0902409\pi$
$710$	0	0
$711$	− 595.825i	− 0.838011i
$712$	0	0
$713$	171.000	0.239832
$714$	0	0
$715$	1224.00	1.71189
$716$	0	0
$717$	429.549i	0.599091i
$718$	0	0
$719$	− 64.0859i	− 0.0891320i	−0.999006	−	0.0445660i	$-0.985810\pi$
$719$	− 64.0859i	− 0.0891320i	0.999006	−	0.0445660i	$-0.0141905\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	145.000	0.200553
$724$	0	0
$725$	27.7128i	0.0382246i
$726$	0	0
$727$	55.4256i	0.0762388i	0.999273	+	0.0381194i	$0.0121367\pi$
$727$	55.4256i	0.0762388i	−0.999273	+	0.0381194i	$0.987863\pi$
$728$	0	0
$729$	−287.000	−0.393690
$730$	0	0
$731$	350.000	0.478796
$732$	0	0
$733$	− 826.188i	− 1.12713i	−0.826071	−	0.563566i	$-0.809429\pi$
$733$	− 826.188i	− 1.12713i	0.826071	−	0.563566i	$-0.190571\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	289.000	0.392130
$738$	0	0
$739$	−713.000	−0.964817	−0.482409	−	0.875946i	$-0.660238\pi$
$739$	−713.000	−0.964817	−0.482409	+	0.875946i	$0.660238\pi$
$740$	0	0
$741$	− 96.9948i	− 0.130897i
$742$	0	0
$743$	− 637.395i	− 0.857866i	−0.903336	−	0.428933i	$-0.858890\pi$
$743$	− 637.395i	− 0.857866i	0.903336	−	0.428933i	$-0.141110\pi$
$744$	0	0
$745$	−27.0000	−0.0362416
$746$	0	0
$747$	880.000	1.17805
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1169.13i	1.55677i	0.627787	+	0.778385i	$0.283961\pi$
$751$	1169.13i	1.55677i	−0.627787	+	0.778385i	$0.716039\pi$
$752$	0	0
$753$	−58.0000	−0.0770252
$754$	0	0
$755$	−189.000	−0.250331
$756$	0	0
$757$	1039.23i	1.37283i	0.727211	+	0.686414i	$0.240816\pi$
$757$	1039.23i	1.37283i	−0.727211	+	0.686414i	$0.759184\pi$
$758$	0	0
$759$	88.3346i	0.116383i
$760$	0	0
$761$	863.000	1.13403	0.567017	−	0.823706i	$-0.308097\pi$
$761$	863.000	1.13403	0.567017	+	0.823706i	$0.308097\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1039.23i	1.35847i
$766$	0	0
$767$	762.102i	0.993615i
$768$	0	0
$769$	410.000	0.533160	0.266580	−	0.963813i	$-0.414106\pi$
$769$	410.000	0.533160	0.266580	+	0.963813i	$0.414106\pi$
$770$	0	0
$771$	−119.000	−0.154345
$772$	0	0
$773$	798.475i	1.03296i	0.856300	+	0.516478i	$0.172757\pi$
$773$	798.475i	1.03296i	−0.856300	+	0.516478i	$0.827243\pi$
$774$	0	0
$775$	65.8179i	0.0849264i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	182.000	0.233633
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 235.559i	− 0.300842i
$784$	0	0
$785$	1611.00	2.05223
$786$	0	0
$787$	31.0000	0.0393901	0.0196950	−	0.999806i	$-0.493730\pi$
$787$	31.0000	0.0393901	0.0196950	+	0.999806i	$0.493730\pi$
$788$	0	0
$789$	− 327.358i	− 0.414902i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	312.000	0.393443
$794$	0	0
$795$	477.000	0.600000
$796$	0	0
$797$	595.825i	0.747585i	0.927512	+	0.373793i	$0.121943\pi$
$797$	595.825i	0.747585i	−0.927512	+	0.373793i	$0.878057\pi$
$798$	0	0
$799$	1255.74i	1.57164i
$800$	0	0
$801$	−568.000	−0.709114
$802$	0	0
$803$	−2023.00	−2.51930
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 133.368i	− 0.165264i
$808$	0	0
$809$	−313.000	−0.386897	−0.193449	−	0.981110i	$-0.561967\pi$
$809$	−313.000	−0.386897	−0.193449	+	0.981110i	$0.561967\pi$
$810$	0	0
$811$	1138.00	1.40321	0.701603	−	0.712568i	$-0.252468\pi$
$811$	1138.00	1.40321	0.701603	+	0.712568i	$0.252468\pi$
$812$	0	0
$813$	− 434.745i	− 0.534741i
$814$	0	0
$815$	− 88.3346i	− 0.108386i
$816$	0	0
$817$	−98.0000	−0.119951
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1224.56i	1.49155i	0.666200	+	0.745773i	$0.267920\pi$
$821$	1224.56i	1.49155i	−0.666200	+	0.745773i	$0.732080\pi$
$822$	0	0
$823$	− 116.047i	− 0.141005i	−0.997512	−	0.0705027i	$-0.977540\pi$
$823$	− 116.047i	− 0.141005i	0.997512	−	0.0705027i	$-0.0224603\pi$
$824$	0	0
$825$	−34.0000	−0.0412121
$826$	0	0
$827$	754.000	0.911729	0.455865	−	0.890049i	$-0.349330\pi$
$827$	754.000	0.911729	0.455865	+	0.890049i	$0.349330\pi$
$828$	0	0
$829$	− 905.863i	− 1.09272i	−0.837551	−	0.546359i	$-0.816014\pi$
$829$	− 905.863i	− 1.09272i	0.837551	−	0.546359i	$-0.183986\pi$
$830$	0	0
$831$	− 202.650i	− 0.243863i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	72.0000	0.0862275
$836$	0	0
$837$	− 559.452i	− 0.668402i
$838$	0	0
$839$	1053.09i	1.25517i	0.778548	+	0.627585i	$0.215956\pi$
$839$	1053.09i	1.25517i	−0.778548	+	0.627585i	$0.784044\pi$
$840$	0	0
$841$	649.000	0.771700
$842$	0	0
$843$	−74.0000	−0.0877817
$844$	0	0
$845$	− 119.512i	− 0.141434i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−463.000	−0.545347
$850$	0	0
$851$	45.0000	0.0528790
$852$	0	0
$853$	− 845.241i	− 0.990904i	−0.868635	−	0.495452i	$-0.835003\pi$
$853$	− 845.241i	− 0.990904i	0.868635	−	0.495452i	$-0.164997\pi$
$854$	0	0
$855$	− 290.985i	− 0.340333i
$856$	0	0
$857$	887.000	1.03501	0.517503	−	0.855681i	$-0.326862\pi$
$857$	887.000	1.03501	0.517503	+	0.855681i	$0.326862\pi$
$858$	0	0
$859$	1663.00	1.93597	0.967986	−	0.251004i	$-0.0807608\pi$
$859$	1663.00	1.93597	0.967986	+	0.251004i	$0.0807608\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 562.917i	− 0.652279i	−0.945322	−	0.326139i	$-0.894252\pi$
$863$	− 562.917i	− 0.652279i	0.945322	−	0.326139i	$-0.105748\pi$
$864$	0	0
$865$	549.000	0.634682
$866$	0	0
$867$	−336.000	−0.387543
$868$	0	0
$869$	− 1266.13i	− 1.45700i
$870$	0	0
$871$	− 235.559i	− 0.270447i
$872$	0	0
$873$	176.000	0.201604
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 119.512i	− 0.136273i	−0.997676	−	0.0681365i	$-0.978295\pi$
$877$	− 119.512i	− 0.136273i	0.997676	−	0.0681365i	$-0.0217054\pi$
$878$	0	0
$879$	110.851i	0.126111i
$880$	0	0
$881$	−574.000	−0.651532	−0.325766	−	0.945450i	$-0.605622\pi$
$881$	−574.000	−0.651532	−0.325766	+	0.945450i	$0.605622\pi$
$882$	0	0
$883$	−1166.00	−1.32050	−0.660249	−	0.751047i	$-0.729549\pi$
$883$	−1166.00	−1.32050	−0.660249	+	0.751047i	$0.729549\pi$
$884$	0	0
$885$	− 285.788i	− 0.322925i
$886$	0	0
$887$	545.596i	0.615103i	0.951532	+	0.307551i	$0.0995096\pi$
$887$	545.596i	0.615103i	−0.951532	+	0.307551i	$0.900490\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−935.000	−1.04938
$892$	0	0
$893$	− 351.606i	− 0.393736i
$894$	0	0
$895$	− 462.458i	− 0.516712i
$896$	0	0
$897$	72.0000	0.0802676
$898$	0	0
$899$	−456.000	−0.507230
$900$	0	0
$901$	− 2294.97i	− 2.54713i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1296.00	1.43204
$906$	0	0
$907$	−521.000	−0.574421	−0.287211	−	0.957867i	$-0.592728\pi$
$907$	−521.000	−0.574421	−0.287211	+	0.957867i	$0.592728\pi$
$908$	0	0
$909$	− 623.538i	− 0.685961i
$910$	0	0
$911$	− 1191.65i	− 1.30807i	−0.756465	−	0.654035i	$-0.773075\pi$
$911$	− 1191.65i	− 1.30807i	0.756465	−	0.654035i	$-0.226925\pi$
$912$	0	0
$913$	1870.00	2.04819
$914$	0	0
$915$	−117.000	−0.127869
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 1394.30i	− 1.51719i	−0.651560	−	0.758597i	$-0.725885\pi$
$919$	− 1394.30i	− 1.51719i	0.651560	−	0.758597i	$-0.274115\pi$
$920$	0	0
$921$	−274.000	−0.297503
$922$	0	0
$923$	0	0
$924$	0	0
$925$	17.3205i	0.0187249i
$926$	0	0
$927$	1288.65i	1.39012i
$928$	0	0
$929$	−961.000	−1.03445	−0.517223	−	0.855851i	$-0.673034\pi$
$929$	−961.000	−1.03445	−0.517223	+	0.855851i	$0.673034\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 50.2295i	− 0.0538365i
$934$	0	0
$935$	2208.36i	2.36189i
$936$	0	0
$937$	−142.000	−0.151547	−0.0757737	−	0.997125i	$-0.524143\pi$
$937$	−142.000	−0.151547	−0.0757737	+	0.997125i	$0.524143\pi$
$938$	0	0
$939$	409.000	0.435570
$940$	0	0
$941$	− 1224.56i	− 1.30134i	−0.759361	−	0.650669i	$-0.774488\pi$
$941$	− 1224.56i	− 1.30134i	0.759361	−	0.650669i	$-0.225512\pi$
$942$	0	0
$943$	135.100i	0.143266i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	175.000	0.184794	0.0923970	−	0.995722i	$-0.470547\pi$
$947$	175.000	0.184794	0.0923970	+	0.995722i	$0.470547\pi$
$948$	0	0
$949$	1648.91i	1.73753i
$950$	0	0
$951$	188.794i	0.198521i
$952$	0	0
$953$	−454.000	−0.476390	−0.238195	−	0.971217i	$-0.576556\pi$
$953$	−454.000	−0.476390	−0.238195	+	0.971217i	$0.576556\pi$
$954$	0	0
$955$	1125.00	1.17801
$956$	0	0
$957$	− 235.559i	− 0.246143i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−122.000	−0.126951
$962$	0	0
$963$	520.000	0.539979
$964$	0	0
$965$	− 379.319i	− 0.393077i
$966$	0	0
$967$	− 720.533i	− 0.745122i	−0.928008	−	0.372561i	$-0.878480\pi$
$967$	− 720.533i	− 0.745122i	0.928008	−	0.372561i	$-0.121520\pi$
$968$	0	0
$969$	175.000	0.180599
$970$	0	0
$971$	1639.00	1.68795	0.843975	−	0.536382i	$-0.180209\pi$
$971$	1639.00	1.68795	0.843975	+	0.536382i	$0.180209\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	27.7128i	0.0284234i
$976$	0	0
$977$	−793.000	−0.811668	−0.405834	−	0.913947i	$-0.633019\pi$
$977$	−793.000	−0.811668	−0.405834	+	0.913947i	$0.633019\pi$
$978$	0	0
$979$	−1207.00	−1.23289
$980$	0	0
$981$	− 69.2820i	− 0.0706239i
$982$	0	0
$983$	− 1543.26i	− 1.56995i	−0.619530	−	0.784973i	$-0.712677\pi$
$983$	− 1543.26i	− 1.56995i	0.619530	−	0.784973i	$-0.287323\pi$
$984$	0	0
$985$	1080.00	1.09645
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 72.7461i	− 0.0735552i
$990$	0	0
$991$	895.470i	0.903603i	0.892119	+	0.451801i	$0.149218\pi$
$991$	895.470i	0.903603i	−0.892119	+	0.451801i	$0.850782\pi$
$992$	0	0
$993$	−295.000	−0.297080
$994$	0	0
$995$	−333.000	−0.334673
$996$	0	0
$997$	− 795.011i	− 0.797404i	−0.917081	−	0.398702i	$-0.869461\pi$
$997$	− 795.011i	− 0.797404i	0.917081	−	0.398702i	$-0.130539\pi$
$998$	0	0
$999$	− 147.224i	− 0.147372i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.g.c.687.2		2
4.3	odd	2		392.3.g.e.99.2		2
7.2	even	3		224.3.o.b.207.1		2
7.4	even	3		224.3.o.a.79.1		2
7.6	odd	2		1568.3.g.f.687.1		2
8.3	odd	2	inner	1568.3.g.c.687.1		2
8.5	even	2		392.3.g.e.99.1		2
28.3	even	6		392.3.k.a.275.1		2
28.11	odd	6		56.3.k.a.51.1	yes	2
28.19	even	6		392.3.k.c.67.1		2
28.23	odd	6		56.3.k.b.11.1	yes	2
28.27	even	2		392.3.g.d.99.2		2
56.5	odd	6		392.3.k.a.67.1		2
56.11	odd	6		224.3.o.b.79.1		2
56.13	odd	2		392.3.g.d.99.1		2
56.27	even	2		1568.3.g.f.687.2		2
56.37	even	6		56.3.k.a.11.1	✓	2
56.45	odd	6		392.3.k.c.275.1		2
56.51	odd	6		224.3.o.a.207.1		2
56.53	even	6		56.3.k.b.51.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.k.a.11.1	✓	2	56.37	even	6
56.3.k.a.51.1	yes	2	28.11	odd	6
56.3.k.b.11.1	yes	2	28.23	odd	6
56.3.k.b.51.1	yes	2	56.53	even	6
224.3.o.a.79.1		2	7.4	even	3
224.3.o.a.207.1		2	56.51	odd	6
224.3.o.b.79.1		2	56.11	odd	6
224.3.o.b.207.1		2	7.2	even	3
392.3.g.d.99.1		2	56.13	odd	2
392.3.g.d.99.2		2	28.27	even	2
392.3.g.e.99.1		2	8.5	even	2
392.3.g.e.99.2		2	4.3	odd	2
392.3.k.a.67.1		2	56.5	odd	6
392.3.k.a.275.1		2	28.3	even	6
392.3.k.c.67.1		2	28.19	even	6
392.3.k.c.275.1		2	56.45	odd	6
1568.3.g.c.687.1		2	8.3	odd	2	inner
1568.3.g.c.687.2		2	1.1	even	1	trivial
1568.3.g.f.687.1		2	7.6	odd	2
1568.3.g.f.687.2		2	56.27	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +5.19615i q^{5} -8.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +5.19615i q^{5} -8.00000 q^{9} -17.0000 q^{11} +13.8564i q^{13} -5.19615i q^{15} -25.0000 q^{17} +7.00000 q^{19} +5.19615i q^{23} -2.00000 q^{25} +17.0000 q^{27} -13.8564i q^{29} -32.9090i q^{31} +17.0000 q^{33} -8.66025i q^{37} -13.8564i q^{39} +26.0000 q^{41} -14.0000 q^{43} -41.5692i q^{45} -50.2295i q^{47} +25.0000 q^{51} +91.7987i q^{53} -88.3346i q^{55} -7.00000 q^{57} +55.0000 q^{59} -22.5167i q^{61} -72.0000 q^{65} -17.0000 q^{67} -5.19615i q^{69} +119.000 q^{73} +2.00000 q^{75} +74.4782i q^{79} +55.0000 q^{81} -110.000 q^{83} -129.904i q^{85} +13.8564i q^{87} +71.0000 q^{89} +32.9090i q^{93} +36.3731i q^{95} -22.0000 q^{97} +136.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 16 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 16 * q^9	\( 2 q - 2 q^{3} - 16 q^{9} - 34 q^{11} - 50 q^{17} + 14 q^{19} - 4 q^{25} + 34 q^{27} + 34 q^{33} + 52 q^{41} - 28 q^{43} + 50 q^{51} - 14 q^{57} + 110 q^{59} - 144 q^{65} - 34 q^{67} + 238 q^{73} + 4 q^{75} + 110 q^{81} - 220 q^{83} + 142 q^{89} - 44 q^{97} + 272 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 16 * q^9 - 34 * q^11 - 50 * q^17 + 14 * q^19 - 4 * q^25 + 34 * q^27 + 34 * q^33 + 52 * q^41 - 28 * q^43 + 50 * q^51 - 14 * q^57 + 110 * q^59 - 144 * q^65 - 34 * q^67 + 238 * q^73 + 4 * q^75 + 110 * q^81 - 220 * q^83 + 142 * q^89 - 44 * q^97 + 272 * q^99

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